99.99 % 3 purity) was subsequently deposited by electron-beam evaporation onto the PMMA thin film at room temperature with an oblique incidence angle of 80◦, a deposition rate of 0.1 nm/s, and a target thickness of about 12 nm. The deposited Au film was plasma treated in O2 at a power of 200W until the entire PMMA layer was removed. In the case of Ag and Cu films, in order to avoid fast oxidation, the plasma treatment was performed in N2 at 200W until the PMMA layer was completely removed (as previously demonstrated, this ensures that the oxidation is kept to a very low level2). The bi-layer structure was obtained by repeating the preparation using the first porous layer as substrate for the deposition of the second porous layer on top. Spectroscopic measurements Raman spectroscopy measurements were performed using a HORIBA LabRAM HR Evolution Raman spectrometer (HORIBA Jobin Yvon, Kyoto, Japan) with a 50X long-focal-length objective (NA = 0.75) , a 100μm-hole aperture, and a 600 gr/mm grating. SERS spectra of Rhodamine 6G (R6G) were acquired under different conditions to characterize system performance: 100 nM R6G with laser power attenuated to 10% at 532 nm excitation; 1 μM R6G with 5% laser power at 633 nm; and 0.1 mM R6G with full laser power (no attenuation) at 785 nm. For each substrate, a Raman map was collected over a 20 × 20 μm2 area with a step size of 5 μm, resulting in 25 individual spectra. The final spectrum for each substrate was generated by averaging these 25 spectra after pre-processing. Spectra of the protein ADAMTS3 were obtained by averaging five randomly collected spectra, using a 5% ND filter and 60s acquisition time for both 532nm and 633nm excitations. For 785nm excitation, a 100% ND filter (no attenuation) and 120s acquisition time were used. Data Analysis The raw spectra were baseline-corrected and smoothed using HORIBA LabSpec6 software. Subsequently, all data averaging, normalization, spectral deconvolution, peak fitting, and graphical plotting were performed using OriginPro 2025b. Numerical modelling 4 To explore the plasmonic properties of porous NPMs, we performed a numerical investigation of the electromagnetic response using a finite element method (FEM) commercial software, employing COMSOL Multiphysics. In particular, following a procedure reported in detail in our recently reported works1,2 nanometric pores and irregularities of NPMs have been numerically reproduced from scanning electron microscopy (SEM) images of the experimental samples. However, continuum modeling, such as FEM, might fail in describing the response properties of bimetallic nanostructured materials3-5. Therefore, to validate the FEM procedure, the optical response of NPMs has also been calculated using a fully atomistic approach, named frequency-dependent fluctuating charges and dipoles (ωFQFμ)5-7. This method provides a reliable description of noble metal nanoparticles, comparable to ab initio5 but at significantly lower computational cost, and can thus handle systems composed of thousands of atoms811. Remarkably, it can also be applied to bimetallic systems and, in the specific case of Au/Ag nanoparticles, yields results in almost perfect agreement with experiment12. For cross-validation of the COMSOL results, we have adopted an idealized picture in which the same porous structure is generated and then considered either as Ag or Au, which are characterized by the same lattice constant (4.08 Å). Such structures were created by introducing randomly generated nanometric pores, thereby generating realistic surface roughness and porosity. In particular, we modeled a 7 × 7 × 4 (nm3) structure, composed of 6584 atoms, a size unfeasible for ab initio atomistic models (see SI for a graphical representation). The same nanoporous geometry was then used to simulate Ag/Ag, Au/Au, and mixed Au/Ag bilayers, allowing for a direct comparison of their optical response and a detailed analysis of the different compositional effects. All ωFQFμ calculations are performed using the Ag/Au parameters reported in Ref. 12, with the open-source plasmonX software6. Dielectric Permittivity and Ellipsometry. Spectroscopic ellipsometry was employed to determine the dielectric permittivity of nanoporous Ag/Ag, Ag/Au, and Ag/Cu bilayers. The measurements provided both the real and imaginary components of the dielectric function over the visible range. The experimental data were fitted using a model composed of Lorentzian and Gaussian oscillators to capture the main features of each alloy's optical response. The resulting dielectric permittivities accurately reproduce the experimental ellipsometric parameters Ψ and 5 Δ. Complete fitting details and oscillator parameters are reported in the Supporting Information (Table S1). Introduction Over the last decades, research on nanoporous metals (NPMs) has grown steadily, driven by their broad range of applications in sensing13-15, catalysys16,17, photonics18,19 and biomedicine20. The main characteristic of NPMs is their high surface-to-volume ratio that, combined with unique chemical reactivity and mass transport properties, makes this family of nanostructured materials highly interesting for advanced research. NPMs can be prepared following different strategies such as template-assisted physical vapor deposition, electrochemical processes, and chemical dealloying18,21. The latter is probably the most used method to prepare NPMs through selective dissolution of the more chemically active component from precursor alloys. Dealloying methods have been applied to prepare NPMs as gold13,18,22, copper23,24, silver25,26, and aluminum27,28. A specific aspect of dealloying is the presence of residual less noble metal at the end of the process. This effect is difficult to avoid and leads to resulting nanoporous structures with varying alloy compositions29,30. Interestingly, this residual \"contamination\" in the final porous film, long considered as a limitation in the preparation of pure porous structures, has recently been shown to enhance plasmonic properties of nanoporous gold containing residual amount of the less noble metal due to modulation of its permittivity by the alloying components.31-33 The combination of two or more metals in alloys is a strategy to provide a way to tune the electronic state through charge transfer between the metals, making it possible to alter the chemical interaction between molecules and the surface of the alloy. In particular, the adsorption behavior of molecules having either high electronegativity (e.g., oxygen and halogens), strong polarity (e.g., carbonyl and nitro groups), partial charge (e.g., carboxylate and hydride-like hydrogen), or high electron density (e.g., alkenes and arenes) is affected by the alloy composition. For instance, palladium-silver alloy promoted the adsorption of negatively charged molecules (carboxylate intermediate) due to the high electronegativity of silver. The alteration of the electronic state by alloying on the molecular detection capability has been recently studied by La et al.33 In particular, a SERS enhancement of up to 6 orders of magnitude for nanoporous gold containing silver was observed and attributed to the narrowing of the electronic structure and its 6 alignment close to the Fermi level. These results suggest that the electronic state modulation of the metal structure can affect molecule-surface interactions. Here, we decided to explore bi-metallic nanoporous systems from a different point of view. Instead of using chemical dealloying, we used a recently reported dry-synthesis method to prepare highly pure layers of plasmonic metals such as Au, Ag, and Cu1-3. The peculiarity of this approach lies in the possibility to produce stacked nanoporous layers, either composed of the same metal or of different metals deposited sequentially on top of one another, as demonstrated previously in refs. 2,3 At the best of our knowledge, this is the first example of perfectly stacked nanoporous metal layers, and it represents a valuable platform to investigate how the use of two different metals can impact the performance in terms of SERS enhancements at different wavelengths across the visible spectral range. Experimental results and numerical simulations will be presented to shed light on the behavior of these novel plasmonic nanoporous platforms. Results and discussion Figure 1 illustrates the planar morphologies of the samples used for these experiments. While the first row (Fig. 1(a-c)) reports the single NPM films, respectively Au, Ag, and Cu, the other micrographs show the obtained morphologies for the bi-metallic layered films. As can be seen, the single metal films present a nanoporous structure with fully connected ligaments of diameters of a few tens of nm. In the case of bi-metallic layered films, the morphologies vary significantly among the samples. In all the bimetallic samples (Fig. 1(d-i)), it is possible to observe the two layers of metals stacked one on the other, but not mixed, with a gap between the 2 metallic nanostructures that can only be due to a potential small oxide layer surrounding the Ag and Cu nanoporous structures. 7 Figure 1: (a-e) SEM micrographs of the investigated samples; porous layers of (a) Au; (b) Ag; (c) Cu; (d) Au/Ag; (e) Ag/Au; (f) Au/Cu; (g) Cu/Au; (h) Ag/Cu; (i) Cu/Ag. Following the same approach used in our recent works3,4, we performed numerical studies of the electromagnetic response of the porous bi-layer system. This enables us to obtain an overview of the electromagnetic (EM) field distributions at the wavelengths of excitation used in the experiments. The nanometric structural features of the layers were reproduced directly from the SEM images of the experimental samples in the finite element method (FEM)-based model, developed using COMSOL Multiphysics. Specifically, the top-view SEM micrographs of the Au, Ag, and Cu films used in the experiments were imported into the FEM solver and extruded along the thickness direction of the layer to build the 3D numerical geometry. As previously shown3, this method enabled us to numerically 8 account for the actual discontinuities and profile of the materials down to the nanoscale. Figures 2, 3, 4, and 5 report an estimation of the EM field enhancement distribution along the surfaces of the two (the top and bottom) porous layers in the different cases, taking into consideration three wavelengths of excitation (550, 630, and 780 nm). Figure 2: (a,b) EM field enhancement distributions for (a) homometallic nanoporous bilayer structures composed of Cu (top), Au (middle), and Ag (bottom), calculated with FEM at excitation wavelengths of 550 nm, 630 nm, and 780 nm (from left to right); (b) homometallic nanoporous bilayers composed of Au (top) and Ag (bottom), obtained using the ωFQFμ approach at the same excitation wavelengths. All 9 maps are normalized to the same color scale to highlight spatial regions of strongest field enhancement localization. (c,d) Maximum field enhancement values extracted from FEM (c) and ωFQFμ (d) simulations for each structure. Figure 2 shows the EM field distributions at wavelengths of 550, 630, and 780 nm for bi-layer nanoporous structures made of a single metal: Cu (a, top), Au (a, middle), and Ag (a, bottom). For the specific case of Au and Ag, the EM distributions are also computed using the ωFQFμ method, for comparison (Fig. 2b). In all cases, the EM enhancement profiles were evaluated on the surface of the bilayer structure. The color scale indicates a normalized intensity of the EM field, calculated as the ratio of the electric field in the presence of nanoporous metals to the incident field. All field maps are normalized to the same values, pointing out the regions where the EM field is highly localized. Nevertheless, at the FEM level (Fig. 2a), each homometallic sample is characterized by its own geometry and plasmonic properties; therefore, the diagram of the maximum intensities of the EM field is shown for each case (Fig. 2c). The strongest enhancement across all studied wavelengths is observed for Ag at 630 nm, whereas Cu shows the weakest response, with almost negligible activity at 550 nm. For Cu and Au, the localized enhanced fields are mainly observed in the pore gaps, whereas in nanoporous Ag, there is a large number of \"hot spots\" strongly affecting the distribution of the field. Similar results are also obtained at the ωFQFμ level (see Fig. 2b) for Ag and Au NPMs. In this case, the electric field shows a different enhancement distribution depending on the metal and the laser frequency, reflecting a different induced density on the surface. In particular, as for FEM results, the response of Ag NPM is significantly larger than that of Au NPM, and also more hot-spots are observed. Nevertheless, the trends of the maximum electric field enhancements do not correlate exactly with those computed by using FEM (see Fig. 2c-d): in fact, in both Ag and Au NPMs, the maximum enhanced field is reported at 780 nm. This can, however, be due to the diverse plasmonic response provided by the two models, and by the small model structure considered in ωFQFμ calculations. Understanding the optical response of homometallic nanoporous structures at different excitation wavelengths is crucial for a deeper comprehension of the processes occurring across the layers in bimetallic configurations and for disentangling the individual contribution of each metal. Accordingly, new plasmonic effects emerge in Au/Ag, Ag/Au, Au/Cu, Cu/Au, Ag/Cu, and Cu/Ag NPM systems as a result 10 of interlayer coupling and interactions between the constituent metals. Figure 3 depicts the EM field enhancement distributions for Au/Ag and Ag/Au NPMs (a-b) and field enhancement maxima (c-d) at different wavelengths computed by using FEM (a-c) and at the ωFQFμ level (b-d). At both levels, in the Au/Ag bilayer, the silver layer (Figure 3a), which is placed on top of Au displays significantly reduced EM field intensity at all studied wavelengths relative to the homometallic Ag layer (Figure 2a-b). Nevertheless, the Ag layer exhibits stronger plasmonic activity, despite the less intense localization of hot spots caused by the electrodynamical interaction between two porous layers consisting of two different plasmonic materials. Interestingly, at the FEM level, illumination of the Au/Ag NPM at 780 nm produces a field enhancement comparable to that at 630 nm (Fig. 3c, top), which can likely be associated with the interaction between plasmonic modes of silver and gold. This is also confirmed by the field maps computed at the ωFQFμ level and by the maximum field enhancements, which are similar at all considered frequencies (Fig. 3d, top). The same interaction induces a stronger enhancement for Au layer (Figure 3c-d) at 550 nm than in the homometallic layer (Figure 2c-d) and rises with increasing wavelength, a trend that is perfectly reproduced qualitatively by both methods. 11 Figure 3: (a-b) The enhanced EM field distribution of Au and Ag layers for Au/Ag (top) and Ag/Au (bottom) NPMs calculated at the FEM (a) and ωFQFμ (b) levels of theory. (c-d) FEM c) and ωFQFμ (d) field enhancement maxima for each structure on top and bottom sides. In contrast, when the Au layer is on top of Ag (Figure 3a-b, bottom), the plasmonic response remains weaker at 550 and 780 nm, but increases significantly at 630 nm because of interlayer coupling. This remarkable result, reproduced by both numerical methods (see Fig. 3c-d, top), indicates that the stacking sequence strongly affects the resonance conditions. As a result, at 630 nm, silver has a weaker plasmonic response than in the homometallic Ag case. Between two systems, Au/Ag and Ag/Au configurations at 550 nm, the enhancement is better in the first configuration when the Ag layer is on top, consistent with the superior visible range plasmonic activity of Ag and its more efficient near-field localization. The wavelength-dependent trends of the enhanced field maxima predicted by the two numerical methods are qualitatively identical, validating the FEM-based approach for the subsequent analyses. 12 Although Au and Cu do not show the strong plasmonic response of Ag (Figure 2), their coupling still produces noteworthy effects. Figure 4 compares the EM field enhancement distributions and maxima for Au/Cu and Cu/Au bilayer configurations. In the case of Cu as upper layer (Figure 4 (a-c)), low plasmonic enhancement of Cu is compensated by coupling with underlying Au, especially at longer wavelengths (780 nm). When Au is on top of Cu (Figure 4 (g-i)), instead, the enhancement is mainly determined by the Au layer itself, while Cu acts as a substrate preventing additional EM field enhancement, possibly because of significant losses in the Cu layer. Interestingly, the field EM enhancement intensities of Cu are mostly the same in combination with Ag (Figure 5). Figure 4: The enhanced EM field distribution of Au and Cu layers for(a-f) Au/Cu and (g-l) Cu/Au NPMs calculated as a ratio of the electric field in the presence of nanoporous metals to the incident field for different wavelengths. 13 In bimetallic structures Ag/Cu (Fig. 5 (a-f)) and Cu/Ag (Fig. 5(g-l)), the EM field enhancement is dominated by silver, while copper mainly shows negligible activity, as can be seen from a comparison between the EM field enhancement maxima of these configurations and those of the homometallic structure composed of Ag (Fig. 2a, bottom). For Cu located on top of Ag, the lower Ag layer still supports a strong EM field localization, though slightly reduced due to copper-induced losses. In contrast, when Ag is deposited on top of Cu, the EM field enhancement is intensively concentrated in the upper Ag layer, but the overall response, as mentioned earlier, is reduced compared to Ag alone due to the lossy copper layer. These results confirm that the stacking order determines whether the Cu layer suppresses the Ag layer's resonances or favors the interlayer EM interactions. These conclusions are also supported by ellipsometric investigations that allowed us to retrieve the dielectric permittivities of the bi-metallic combinations (see Supporting Information). In particular, homometallic bi-layer Ag/Ag exhibits a pronounced metallic response with a negative real permittivity throughout the visible range, confirming its validity as a plasmonic material and efficient field confinement. Ag/Au shows broader dispersion and larger optical losses, reflecting its intrinsically damped plasmonic character. Conversely, Ag/Cu displays a largely suppressed plasmonic response, with the real part of ε only weakly negative and significant absorption across the visible spectrum. 14 Figure 5: The enhanced EM field distribution of Ag and Cu layers for(a-f) Ag/Cu and (g-l) Cu/Ag NPMs calculated as a ratio of the electric field in the presence of nanoporous metals to the incident field for different wavelengths. The numerical results provide a framework for understanding and analyzing data during SERS experiments on the investigated bi-metallic NPMs. As predicted by our calculations, the efficiency of EM field enhancement and distribution in bi-layer systems is determined not only by the type of metal, but also by the order of the layers and the coupling efficiency between them. In homometallic structures, the enhancement is defined by plasmonic properties of each metal; moreover, the Fermi level remains the same. At the excitation energies explored here, in Au-Ag systems, silver dominates in terms of 15 enhancement intensity, whereas the enhancement of gold is explained by the coupling of plasmonic modes. In the case of Ag-Cu, the interlayer coupling is weak, and the enhancement is determined mainly by silver. For Au-Cu, gold dominates in intensity. Additionally, the observed enhancement is still higher than in homometallic structures but remains significantly lower compared to configurations containing Ag. Starting from the numerical prediction of the EM field distributions in our samples, the experimental performance in terms of SERS enhancement of the NPM films was examined under conditions as close as possible to the simulated ones. In particular, three excitation wavelengths very close to those analyzed in the simulations, i.e., 532 nm, 633 nm, and 785 nm, were employed experimentally, and Rhodamine 6G (R6G) was used as the molecular probe. The average SERS spectra of R6G, measured across the series of NPMs configurations with distinct bi-layer stacking sequences (e.g., Au/Cu, Ag/Au), are compared in Fig. 6 (a-c) to evaluate their corresponding SERS enhancement performances. Fig. 6 (d-f) presents the intensities of five characteristic R6G Raman bands at 620, 1280, 1360, 1509, and 1649 cm-1 for the top three NPMs configurations, which were selected based on their highest overall SERS intensities under identical measurement conditions, as marked by asterisks in Figures 6 (a-c). To amid potential peak shifts, each intensity value was determined by taking the maximum within a ±10 cm-1 window around the nominal wavenumber. The SERS spectra and corresponding peak intensities of R6G in Fig. 6 vary markedly across the various substrates and excitation wavelengths. Specifically, under 532 nm excitation, the average spectra of 100 nM R6G show that samples containing Ag as the bottom layer, combined with either Ag or Cu layers on top, yield the highest signal enhancement. The Au NPMs rank next in terms of enhancement efficacy, whereas other configurations of homometallic NPMs demonstrate remarkably inferior performances. Notably, the homometallic Cu structure yields the weakest R6G signal. As illustrated by the histogram in Fig. 6(d), the Ag/Cu NPM configuration delivers the most intense peaks for nearly all extracted characteristic bands of R6G, except for that at 1360 cm-1. Under 633 nm excitation, with 1 μM R6G, it is evident that the Ag NPM significantly outperforms the Au nanoporous structure. In contrast, the Cu/Ag and other configurations show weaker enhancements, with the homometallic Cu structure yielding the lowest overall enhancement. These observations are further supported by the extracted intensities of the five 16 characteristic Raman bands of R6G in Fig. 6(e), which clearly show that the homometallic Ag NPM produces signals approximately 2.5 times stronger than those obtained from the homometallic Au substrate. These results are consistent with the simulation data presented in Fig. 2. Figure 6. SERS spectra of R6G on various substrates under excitation wavelengths of (a) 532nm, (b) 633nm and (c) 785nm; (d-f): Comparison of the five characteristic R6G band intensities for the top three NPMs configurations identified in Figures 6 (a-c), respectively. Under 785 nm excitation with 100 μM R6G, a different trend emerges: the Ag/Au structure (with Ag beneath Au) exhibits the strongest R6G signal enhancement, despite its negligible performance at shorter wavelengths. This result highlights the significant influence of the excitation wavelength on the SERS efficiency across different materials. Conversely, under 785 nm laser excitation, the enhancement from the homometallic Ag structure is even weaker than that of the Cu/Au structure, which appears inconsistent with the simulation results in Fig. 2. We attribute this discrepancy to the potential oxidation of the exposed Ag nanostructured top layer, as there was a significant time gap (more than one month) 17 between substrate fabrication and measurement. In contrast, the Ag/Au structure, protected by the chemically inert Au top layer, likely maintained its plasmonic properties. To verify this hypothesis, a supplementary experiment was conducted using NPM substrates within two weeks from fabrication (see Supplementary Information), showing that the homometallic Ag structure demonstrates significantly stronger enhancement than Ag/Au, which is consistent with the simulations in Fig. 2. To further assess SERS performance, based on the above discussion, we evaluated the signal uniformity, limit of detection (LOD), and concentration dependence of Ag/Cu, homometallic Ag (bilayer Ag/Ag), and Ag/Au NPMs under 532 nm, 633 nm, and 785 nm excitation, respectively. Results are summarized in Figure 7. Fig. 7 (a-c) reports the averaged SERS spectra of R6G at varying concentrations measured under consistent experimental conditions. The lowest detectable concentrations of R6G were determined to be as low as 10 nM, 0.1 μM, and 10 μM for 532, 633, and 785 nm excitation wavelengths, respectively. Concentration calibration curves are also shown in Fig. 7(d-f) left panels, derived from the five characteristic SERS bands at 620, 1280, 1360, 1509, and 1649 cm-1 in the SERS spectra shown in Fig. 7 (a-c). We found that the intensities of five characteristic SERS bands of R6G exhibit an essentially linear relationship with the logarithm of the corresponding R6G concentrations, which is consistent with the results in the literature34. Fig. 7 (d-f) Right panels displayed the intensity distributions of these five SERS bands across multiple randomly selected spectra on different nanoporous substrates. For each R6G characteristic peak, the relative standard deviation (RSD) of the intensities across the spectra was calculated. Subsequently, the average RSD (mean RSD) for the five peaks was determined, yielding values of 9.98%, 7.76%, and 15.53% for Ag/Cu, Ag, and Ag/Au substrates, respectively. These mean 18 RSD values demonstrate excellent reproducibility and uniformity of the SERS response across the different nanoporous substrates. Figure 7: (a-c) Concentration-dependent SERS spectra of R6G on Ag/Cu, Ag/Ag and Ag/Au NPMs under 532, 633 and 785nm, respectively. (d-f) Left panels: Calibration curves for the five characteristic bands (620, 1280, 1360, 1509 and 1649 cm-1) derived from the corresponding SERS spectra shown in (a-c); Right panels: Intensity distribution of these bands from randomly selected spectra for 1μM, 10 μM and 1 mM R6G on the respective substrates. The long-term stability of the fabricated porous substrates was evaluated by measuring the SERS spectra of R6G on films stored for several months (Fig. S2, Supporting Information). Furthermore, the mechanical durability was assessed by testing the SERS enhancement after ultrasonic washing. The substrates maintained significant SERS activity even after 40 minutes of ultrasonication, demonstrating robust performance under harsh conditions (Fig. S3, Supporting Information). This combination of long19 term and mechanical stability makes these substrates highly suitable for practical applications, such as photocatalytic or biochemical testing. Figure 8: (a-c) Normalized SERS spectra of ADAMTS3 at various concentrations on Ag/Cu, Ag/Ag, and Ag/Au NPMs under 532, 633, and 785nm excitation, respectively. (d-f) Fit-integrated areas under the corresponding characteristics bands marked with dot lines in (a-c). The translation of SERS substrates from conceptual fabrication to practical application is a critical step in biosensing, particularly for the early screening and diagnosis of cancers and related biomarkers3537. The ability to detect low-concentration biomarkers with high specificity and sensitivity provides a crucial, time-sensitive advantage for early diagnosis and subsequent therapeutic intervention. In a recent work, we reported a DNA origami-based SERS platform for the sensitive detection of ADAMTS3 protein, a promising biomarker associated with hepatocellular carcinoma (HCC)38. Here, we build upon our previous findings by deploying the novel bi-layer metal nanoporous films as a superior SERS 20 substrate to re-investigate the detection of the same ADAMTS3 protein. As discussed above, these substrates offer large, uniform active area and long-term stability, therefore they can be interesting for applied SERS biosensing. Figure 8 presents the analytical performance of the bi-layer NPM SERS substrates for the detection of ADAMTS3. Fig. 8 (a-c) displays the normalized mean SERS spectra of ADAMTS3 protein at various concentrations, acquired using 532, 633, and 785 nm laser excitation wavelengths, respectively. Each set of spectra corresponds to the optimally designed bi-layer configuration (e.g. Ag/Cu, homometallic Ag/Ag, and Ag/Au) for the respective laser wavelength, as detailed in the previous sections. The fitted areas under the characteristic SERS bands of ADAMTS3 protein (marked by dotted lines in a-c) at 1098, 1258, and 1328 cm-1 were plotted in Fig. 8(d-f) as a function of logarithmic concentration with error bars. The peak fitting was performed using a Gaussian function, consistent with our previous methodology39. As shown in the figure, the characteristic SERS peaks of the ADAMTS3 protein remain clearly distinguishable at a concentration of 10-9 M using the three optimal enhancement substrates with the magnification of Ag/Cu, Ag/Ag, and Ag/Au under 532, 633, and 785 nm excitation, respectively. Notably, with 633 nm excitation, these characteristic peaks are still detectable even at a lower concentration of 10-10 M, representing an enhancement factor comparable to that achieved in our recent work using an optimized Au-dimer/DNA origami as a SERS platform38. Furthermore, the fitted areas of the characteristic peaks at 1098, 1258, and 1328 cm-1 were plotted against the logarithmic concentration. Under 633 nm excitation, the data points exhibit smaller error bars compared to those at 532 and 785 nm, and follow an approximately linear relationship. This indicates that the peak areas increase nearly linearly with increasing protein concentration, which is consistent with our previous findings38. In contrast, this linear relationship is not always observed under 532 nm and 785 nm excitation. The fitted areas at 1098 and 1258cm-1 exhibit an approximate linear trend ranging from 10-7 to 10-9 M, but with a notable deviation occurring at 10-6 M at 785nm excitation, and all these fitted areas are associated with relatively large error bars. Under 532nm laser excitation, the relationship between the fitted area and logarithmic concentration deviates entirely from linearity for all three characteristic peaks, accompanied by significantly larger error bars compared to other wavelengths. This pronounced nonlinearity and data scatter can be attributed to the wavelength-specific electronic interaction between 21 the protein and the corresponding bi-metallic substrate. As highlighted in the introduction section, the adsorption behavior of molecules with polar or partially charged groups-such as those present in the ADAMTS3 protein-is highly sensitive to the electronic state of the alloy surface. We hypothesize that under 532 nm excitation, the specific bi-layer configuration (e.g., Ag/Cu) exhibits an electronic structure that leads to non-uniform or saturable adsorption of protein molecules across the concentration series. This results in inconsistent SERS responses and larger variability in measured intensities, thereby disrupting the linear correlation and increasing observational uncertainties. Conclusions In summary, we conducted a comprehensive investigation of novel plasmonic platforms composed of stacked layers of noble plasmonic metals (NPMs). By preparing multiple configurations (Au/Ag, Ag/Au, Au/Cu, Cu/Au, Ag/Cu, and Cu/Ag), we were able to systematically explore how the choice of metals and their stacking order influence metal-metal interactions. Extensive numerical simulations, carried out using both finite element methods (FEM, COMSOL) and fully atomistic models, provided valuable insights into the optical properties of these nanostructures under excitation at different energies within the visible range. These models proved effective in describing the behavior of porous systems, highlighting their potential not only for enhanced spectroscopy but also for a variety of additional applications. To experimentally validate the platforms, we performed SERS measurements on both a standard probe molecule (R6G) and a biologically relevant target (the ADAMTS3 biomarker). These experiments demonstrated the strong performance of the bilayer systems as SERS substrates and offered initial evidence of how bimetallic interactions may facilitate interlayer interactions at the metal-metal interface. Altogether, this work provides a promising first step toward the study of complex multimaterials plasmonic platforms for SERS, combining the advantages of reproducibility, simplicity, and low fabrication cost. 22 Author contribution YZ performed the Raman experiments and analyzed the data, AS and TG performed the numerical simulations, AS, AA, LB, NM and IM supported the numerical simulations and results discussion, CW and HJ supported the experimental measurements, AD, GL, SW and RK supported the samples fabrication, ZJ, ZZ, and DG supervised the work. YZ and AS equally contributed to the present work. All authors contributed to the writing of the manuscript. Supporting Information Supporting Information is available free of charge at: ........ Acknowledgment The authors thank the National Natural Science Foundation of China (No.22202167), the National Key Research and Development Project of China (No. 2023YFF0613603), and the HORIZON-MSCADN2022: DYNAMO, grant Agreement 101072818. NM and BJ acknowledge the 'Excellence by Choice' Programme at Umeà University funded by Kempestielserna (grant no. JCK-2130.3). The authors thank the Clean Room Facility of IIT. References (1) Kwon, H.; Barad, H.-N.; Silva Olaya, A. R.; Alarcón-Correa, M.; Hahn, K.; Richter, G.; Wittstock, G.; Fischer, P. Dry Synthesis of Pure and Ultrathin Nanoporous Metallic Films. ACS Appl. Mater. Interfaces 2023, 15 (4), 5620-5627. https://doi.org/10.1021/acsami.2c19584. (2) Tapani, T.; Caligiuri, V.; Zou, Y.; Griesi, A.; Ivanov, Y. P.; Cuscunà, M.; Balestra, G.; Lin, H.; Sapunova, A.; Franceschini, P.; Tognazzi, A.; De Angelis, C.; Divitini, G.; Carzino, R.; Kwon, H.; Fischer, P.; Krahne, R.; Maccaferri, N.; Garoli, D. Disordered Plasmonic System with Dense Copper Nano-Island Morphology. Nanophotonics 2025, 14 (12), 2151-2160. https://doi.org/10.1515/nanoph-2024-0743. (3) Caligiuri, V.; Kwon, H.; Griesi, A.; Ivanov, Y. 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Nanoscale Adv. 2025, 7 (17), 5212-5220. https://doi.org/10.1039/D5NA00486A. 1. Additional SERS Experiments 26 - R6G test under 785nm excitation with new set of substrates Fig. S1. SERS spectra of R6G at different concentrations obtained from NPMs with varying configuration (Ag/Cu, Ag/Au and Ag/Ag) under 785nm excitation. The SERS performance of different bi-layer nanoporous substrates (Ag/Ag, Ag/Au, Ag/Cu) was re-evaluated using R6G under 785 nm excitation, as shown in Fig. S1. A direct comparison reveals that the Ag/Ag substrate possesses the highest enhancement capability, which is different from the previous results in Figure 6(c). It successfully detected R6G at a concentration of 0.1 μM, whereas under identical conditions, the Ag/Au and Ag/Cu substrates only yielded measurable signals at a higher concentration of 1 μM. 27 This result clearly indicates that the Ag/Ag bi-layer structure offers superior SERS sensitivity over the other two configurations at 785nm excitation. - Long-term stability of the substrates After initial characterization, we re-evaluated the SERS performance of 100nM R6G under 532 nm excitation on various bi-layer nanoporous substrates following a two-month storage period under ambient conditions. The four substrates exhibiting the highest enhancement were selected for further analysis, as shown in Fig. S2. Both the bi-layer Ag and bi-layer Au substrates maintained the strongest SERS enhancement after storage. In contrast, the Au/Cu and Cu/Au configurations also demonstrated considerable enhancement following the two-month period. Notably, the Ag/Cu substrate, which initially ranked third in enhancement intensity in Figure 6(a) and Figure 6(d), showed a significant reduction in SERS performance after storage. We attribute these observations to the protective role of gold against oxidation, which helps preserve the nano-structural integrity of the underlying porous film. In comparison, silver and copper are more susceptible to oxidation over time, leading to diminished SERS activity in Ag/Cu bilayers after prolonged exposure to air. 28 Fig. S2. Normalized SERS spectra of 100nM R6G acquired from the selected bilayer nanoporous substrates, before (short dot line) and after (solid line) two months storage period. Spectra were collected under 532nm excitation with all measurements using a 10% laser filter except for the poststorage Au/Cu and Cu/Au samples, for which a 100% laser power was employed. All spectra are normalized to [0,1] range for comparative purposes. 29 Fig. S3 Evolution of SERS performance after storage. Mean SERS spectra of 0.1mM R6G on bi-layer nanoporous substrates with different configurations under 633nm excitation (5% laser filter), acquired after a two-month storage period. 30 Fig. S4 Evolution of SERS performance after storage under 785nm laser excitation. a) Mean SERS spectra of 0.1mM R6G obtained initially (dotted line) and after storage (solid line); b) Corresponding normalized post-storage spectra. Having established the variation in initial SERS enhancement across substrates in Figure 6(b), we proceeded to evaluate its subsequent evolution of the same substrates following a two-month storage period under 633nm excitation. As shown in Fig. S3, Ag/Au bilayer emerges as the most effective substrate, exhibiting the highest SERS intensity, followed by Au/Ag configuration. In contrast, the initially superior configurations (Ag/Ag, Au/Au, and Cu/Ag, marked by asterisks) displayed a considerable decline after storage. This marked contrast underscores the significant role of structural stability and oxidation resistance in maintaining SERS activity over time. The results obtained under 785 nm excitation differed from those under 532 nm and 633 nm, as shown in Fig. S4. For comparison, we evaluated the top four enhanced SERS configurations-Ag/Au, Cu/Au, Au/Ag, and Au/Cu-before and after a two-month storage period. As illustrated in Fig. S4(a), the SERS intensity of R6G decreased in all configurations between the initial and post-storage measurements. However, configurations with Ag/Au and Cu/Au consistently exhibited stronger enhancement than the others, both initially and after storage. In Fig. S4(b), after normalization of the poststorage spectra, it can be observed that Ag/Au and Cu/Au configurations maintained superior enhancement performance before and after storage. In contrast, the double-layer silver porous structure, which initially ranked third in enhancement (as shown in the main text Figure 6(c)), exhibited weaker signals than both Au/Ag and Au/Cu after two months of storage. 31 - Influence of washing to the substrates Fig. S5 Evaluation of the washing effect on substrate stability. Mean SERS spectra of the background signal and 10-6 M ADAMTS3 were acquired under 633 nm excitation after 40 minutes of sequential ultrasonication in ethanol and ultrapure water. The results for both Ag/Au and Ag/Ag porous substrates are shown, with error bars representing the standard deviation across multiple measurements. 2. Additional data calculations 32 Figure S6. Ag/Ag (a), Ag/Au (b), Au/Ag (c), Au/Au (d) NPM structures exploited in ωFQFμ calculations. 33 Figure S7. The enhanced EM field distribution of Au and Ag bottom layers for Au/Ag (top) and Ag/Au (bottom) NPMs calculated at the FEM (a) and ωFQFμ (b) levels of theory. 3. Ellipsometric characterization Figure S8: Real (black) and imaginary (red) parts of the dielectric permittivity for Ag/Ag (a), Ag/Au (d), and Ag/Cu (g) nanoporous bilayers, obtained by fitting ellipsometric data using a combination of Lorentzian and Gaussian oscillators. Corresponding experimental (dots) and fitted (solid lines) ellipsometric angles Ψ and Δ are shown in panels (b-c), (e-f), and (h-i), respectively. The oscillator parameters are reported in Table S1. The dielectric functions of the three nanoporous bilayers, Ag/Ag, Ag/Au, and Ag/Cu, shown in Figure S8.a,d,g, respectively, were modeled using a set of Lorentzian and 34 Gaussian oscillators to capture the key features of their optical response. For Ag/Ag (Figure S8a), the response is dominated by a single Lorentzian centered at 1.297 eV, accounting for the low-energy plasmonic resonance typical of nanostructured silver. A Gaussian oscillator with negative amplitude was included at 2.84 eV to compensate for the overestimated ε'' background induced by the Lorentzian tail, ensuring better agreement with the experimental data in the near-UV region. In the case of Ag/Au (Figure S8d), a broader Lorentzian (En0 = 1.20 eV, Br = 1.44 eV) and two Gaussians were required to account for the increased damping and the presence of interband transitions characteristic of gold. The higher complexity of this system reflects the hybridized electronic structure of the alloy. Finally, the Ag/Cu (Figure S8g) sample shows a markedly suppressed optical response, with three weak oscillators spread across the visible range. The low oscillator strengths and absence of sharp features are consistent with stronger damping and reduced plasmonic activity due to the presence of copper. In all cases, the fitted dielectric functions reproduce the experimental ellipsometric parameters Ψ and Δ, shown in Figures S8b,c, S8e,f, and S8h,i with high fidelity across the measured spectral range. 35 Table S1: Parameters of the Lorentzian and Gaussian oscillators used to model the dielectric functions of the Ag/Ag, Ag/Au, and Ag/Cu bilayers. En0: central energy (eV), Br: broadening (eV), A0: amplitude (dimensionless). Sample Oscillator Type A0 En0 (eV) Br (eV) Ag/Ag Lorentzian 10.01 1.297 0.200 Gaussian -1.40 2.840 1.023 Ag/Au Lorentzian 13.17 1.200 1.440 Gaussian 1.07 2.880 0.644 Gaussian 2.98 4.580 1.745 Ag/Cu Lorentzian 0.115 3.210 0.960 Gaussian 0.021 1.970 0.440"}}},{"rowIdx":281,"cells":{"id":{"kind":"string","value":"2510.14700"},"raw":{"kind":"string","value":"LLM Agents for Automated Web Vulnerability Reproduction:\nAre We There Yet?\nBIN LIU, Harbin Institute of Technology, Shenzhen, China\nYANJIE ZHAO, Huazhong University of Science and Technology, China\nGUOAI XU, Harbin Institute of Technology, Shenzhen, China\nHAOYU WANG, Huazhong University of Science and Technology, China\nLarge language model (LLM) agents have demonstrated remarkable capabilities in software engineering and\ncybersecurity tasks, including code generation, vulnerability discovery and automated testing. One critical\nbut underexplored application is automated web vulnerability reproduction, which transforms vulnerability\nreports into working exploits. Although recent advances suggest promising potential, significant challenges\nremain in applying LLM agents to real-world web vulnerability reproduction scenarios.\nIn this paper, we present the first comprehensive evaluation of state-of-the-art LLM agents for automated\nweb vulnerability reproduction. We first systematically assess 20 agents from software engineering, cyberse-\ncurity, and general domains across 16 dimensions, including technical capabilities, environment adaptability,\nand user experience factors, on 3 representative web vulnerabilities. Based on the results, we select three top-\nperforming agents (OpenHands, SWE-agent, and CAI) for in-depth evaluation on our constructed benchmark\ndataset of 80 real-world CVEs spanning 7 vulnerability types and 6 web technologies. Our results reveal that\nwhile LLM agents achieve reasonable success on simple library-based vulnerabilities, they consistently fail\non complex service-based vulnerabilities requiring multi-component environments. Furthermore, complex\nenvironment configurations and authentication barriers create a critical gap where agents can execute exploit\ncode but fail to trigger actual vulnerabilities. We observe that agents demonstrate high sensitivity to input\nguidance, with performance degrading by over 33.3% under incomplete authentication information. Our\nfindings highlight the significant gap between current LLM agent capabilities and the demands of reliable\nautomated vulnerability reproduction, emphasizing the need for advances in environmental adaptation and\nautonomous problem-solving capabilities.\nCCS Concepts: • Software and its engineering;\nAdditional Key Words and Phrases: Web Vulnerability, LLM Agents, Vulnerability Management\nACM Reference Format:\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang. 2025. LLM Agents for Automated Web Vulnerability\nReproduction: Are We There Yet?. 1, 1 (October 2025), 22 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn\n1\nIntroduction\nRecently, Large language model (LLM) agents have achieved remarkable success across software\nengineering and cybersecurity domains. In software engineering, these agents have demonstrated\nstrong capabilities in code generation [15, 19, 31], bug fixing [36, 83, 90], issue resolution [37, 86]\nAuthors’ Contact Information: Bin Liu, Harbin Institute of Technology, Shenzhen, China, liubin1999@stu.hit.edu.cn; Yanjie\nZhao, Huazhong University of Science and Technology, China, yanjie_zhao@hust.edu.cn; Guoai Xu, Harbin Institute of\nTechnology, Shenzhen, China, xga@hit.edu.cn; Haoyu Wang, Huazhong University of Science and Technology, China,\nhaoyuwang@hust.edu.cn.\nPermission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee\nprovided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the\nfull citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored.\nAbstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires\nprior specific permission and/or a fee. Request permissions from permissions@acm.org.\n© 2025 Copyright held by the owner/author(s). Publication rights licensed to ACM.\nACM XXXX-XXXX/2025/10-ART\nhttps://doi.org/10.1145/nnnnnnn.nnnnnnn\n, Vol. 1, No. 1, Article . Publication date: October 2025.\narXiv:2510.14700v1 [cs.SE] 16 Oct 2025\n\n2\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nand code review [24, 71]. In cybersecurity, they have shown impressive performance in vulnerability\ndetection [30, 41, 68], fuzzing [18, 47, 74], exploit generation [17, 80], malware analysis [11, 49]\nand penetration testing [28, 40]. These successes demonstrate their ability to understand complex\ntechnical contexts, reason about software behavior, and automate sophisticated workflows.\nWeb applications have become the backbone of modern digital infrastructure, representing\nthe dominant form of deployed applications in enterprise environments [69]. This widespread\nadoption has made them prime targets for cyberattacks, with web vulnerabilities consistently\ndominating security incident reports [72]. The OWASP Top 10 identifies critical issues such as\nSQL injection, Cross-Site Scripting (XSS), and Cross-Site Request Forgery (CSRF) as the majority\nof reported vulnerabilities [55], while the National Vulnerability Database confirms that web\napplication vulnerabilities represent the fastest-growing category of security threats [52].\nHowever, effectively managing these web vulnerabilities relies heavily on successful reproduction,\nwhich involves transforming vulnerability reports into working Proof-of-Concept (PoC) exploits\nthat validate threats and enable effective patching. This process presents unique challenges that are\ndifferent from other security tasks where LLM agents have succeeded. Manual reproduction requires\nexpertise in environment configuration, dependency resolution, authentication mechanisms, and\nexploit crafting, often taking days or weeks per vulnerability [20, 61]. Automated reproduction\nfaces additional complexities: software version incompatibilities, incomplete PoC code, complex\ndeployment, and network setups. These factors lead to low success rates in existing automated\nreproduction methods, highlighting the substantial technical barriers.\nWhile LLM agents have remarkable capabilities in related security and software engineering\ntasks, their effectiveness for comprehensive web vulnerability reproduction remains underexplored.\nExisting cybersecurity benchmarks for LLM agents suffer from significant limitations that fail\nto capture real-world complexity. Many evaluations rely on simplified Capture-the-Flag (CTF)\nscenarios [10, 64, 73, 85, 88] that focus on individual challenges rather than end-to-end reproduction\nprocesses. Additionally, current benchmarks often evaluate agents on small, isolated code snippets\nor fragments, limiting their scope to specific programming languages such as C/C++ [7, 38, 50, 79],\nJava [42] or JavaScript [14], rather than the diverse components of production web applications.\nHowever, end-to-end vulnerability reproduction requires agents to handle complex multi-step\nworkflows involving environment setup, dependency management, and real-world deployment\nuncertainties. These capabilities extend far beyond current simplified evaluation scenarios, creating\na significant gap in agent ability for practical vulnerability reproduction applications.\nTo address this gap, we present the first comprehensive empirical study examining LLM agents’\nability for automated web vulnerability reproduction. We systematically evaluate 20 representative\nagents across 16 core dimensions on 3 representative CVEs, then conduct in-depth assessment\nof the top 3 agents using our constructed benchmark dataset of 80 real-world CVEs. We assess\ncurrent agent capabilities and identify fundamental limitations that prevent reliable automated\nvulnerability reproduction. In summary, our main contributions include:\n• We construct a comprehensive benchmark dataset comprising 80 real-world CVEs covering 7\nvulnerability types and 6 web technologies with complete reproduction environments.\n• We conduct the first systematic evaluation of 20 state-of-the-art LLM agents for automated\nweb vulnerability reproduction across 16 core dimensions on 3 representative CVEs, establish-\ning a rigorous evaluation framework. From this evaluation, we select the three top-performing\nagents (OpenHands, SWE-agent, and CAI) for in-depth analysis.\n• We provide critical empirical insights revealing the significant gap between current LLM\nagent capabilities and practical requirements, identifying key failure modes and documenting\nover 33.3% performance degradation under incomplete authentication information.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n3\n2\nBackground\n2.1\nLLM Agents\nLLMs exhibit impressive performance across multiple domains, including natural language compre-\nhension, code synthesis, and reasoning processes. Based on these foundational abilities, researchers\nhave developed LLM-based agent systems [39, 65, 75] that leverage LLMs as core reasoning engines\nfor autonomous decision-making, while utilizing various tools and operations to interact with\nexternal environments in order to accomplish sophisticated tasks.\n2.1.1\nCore Mechanisms. LLM agents combine LLM reasoning with action execution capabilities\nthrough several key mechanisms. ReAct [87] introduces iterative reasoning-action cycles, where\nmodels generate reasoning traces before executing actions, with results integrated into context\nfor subsequent cycles. Toolformer [63] demonstrates autonomous tool usage, enhancing factual\naccuracy and computational abilities.\nFor complex tasks, simple reactive loops prove insufficient, requiring explicit planning capabili-\nties. Early systems like BabyAGI [51] and Auto-GPT [26] introduced dynamic task lists where agents\nexecute, reorderand generate tasks continuously. Advanced frameworks such as AgentVerse [16]\nand ChatDev [62] adopt multi-agent architectures where controller agents decompose goals into\nsubtasks distributed to task-specific agents, enabling coordination across different domains. Re-\nflexion [66] addresses failure handling through reflection mechanisms that analyze past mistakes\nand store self-reflections in memory for improving future planning decisions. This self-correction\ncapability enables iterative problem-solving and strategy refinement over time.\n2.1.2\nRepresentative Agents and Frameworks. LLM agents have emerged across diverse application\ndomains. Software engineering agents assist with code generation, debugging, and repository-level\nanalysis. Cybersecurity agents focus on threat detection, vulnerability analysis, and automated\npenetration testing. General-purpose agents aim for broad applicability, emphasizing flexible tool\nuse and adaptive reasoning across various tasks. Additionally, agent orchestration frameworks like\nLangChain [13] and AutoGen [81] provide infrastructure for building and coordinating multi-agent\nsystems, enabling developers to construct domain-specific agents with standardized tooling and\ncommunication protocols. A summary of representative agents and frameworks is shown in Table 1.\nTable 1. Representative LLM agents and orchestration frameworks across different domains.\nDomain\nAgent\nYear\nDomain\nAgent\nYear\nSoftware\nEngineering\nAGENTLESS [82]\n2024\nCybersecurity\nPentestGPT [17]\n2023\nOpenHands [78]\n2024\nCAI [48]\n2024\nSWE-agent [84]\n2024\nPentestAgent [59]\n2024\nAutoCodeRover [89]\n2024\nNebula [53]\n2024\nAider [23]\n2023\nHackingBuddyGPT [27]\n2023\nRepoAgent [45]\n2024\nPentAGI [57]\n2024\nChatDev [62]\n2023\nEnIGMA [2]\n2024\nMetaGPT [29]\n2024\nAI-OPS [4]\n2024\nGeneral-Purpose\nAgents\nAutoGPT [26]\n2023\nOrchestration\nFrameworks\nLangChain [13]\n2022\nBabyAGI [51]\n2023\nAutoGen [81]\n2023\nAgentVerse [16]\n2023\nCrewAI [32]\n2023\nAIlice [5]\n2024\nLangGraph [33]\n2024\n2.2\nWeb Vulnerability Reproduction\nWeb vulnerability reproduction involves constructing operational PoC exploits from natural lan-\nguage vulnerability reports to validate security flaws and guide remediation efforts.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n4\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nThe common approach relies on manual analysis by security researchers using tools like Burp\nSuite [60] or ZAP [56] to intercept, analyze and craft HTTP/S requests. While effective for complex\nlogic-based flaws, this method suffers from inefficiency and high labor costs that limit scalability.\nSemi-automated systems target common vulnerability classes like XSS or SQL Injection by using\nNLP to extract key entities from reports and inserting them into predefined attack templates [9].\nAdvanced works like Wang et al. [77] extract contextual information for targeted web fuzzing.\nCommercial tools like Nessus [70] and Acunetix [34] represent this approach. While faster for simple\ncases, these methods lack robustness for vulnerabilities requiring complex, multi-step interactions.\nProgram analysis based automated exploit generation (AEG) [8] achieved significant success\nin binary applications through symbolic execution. Frameworks like Mayhem [12] and angr [67]\nexemplify this approach. However, their application to web vulnerabilities is limited due to the\nimmense state space of web applications and inability to reason about high-level logic flaws.\nLLM-based methods represent the newest paradigm, framing reproduction as an agentic problem\nwhere LLMs act as reasoning engines. LLMs have been explored for exploiting vulnerabilities in\nweb applications based on known vulnerability descriptions with one-day setting [21, 22] and\ninvestigating agent teams with hierarchical planning for zero-day scenarios [91]. Recent work [80]\nshowed AI systems can generate working exploits for published CVEs in 10-15 minutes, while\nsystems like PentestGPT [17] and PentestAgent [40] demonstrate autonomous penetration testing\ncapabilities. This methodology combines high-level contextual reasoning with machine execution\nspeed and scalability. All the above methods are shown in Table 2.\nTable 2. Web vulnerability reproduction methods and tools, including manual, semi-automated, program\nanalysis-based, and LLM-based methods.\nTool\nCategory\nYear\nTool\nCategory\nYear\nBurp Suite [60]\nManual\n2003\nAEG [8]\nProgram analysis\nbased\n2011\nOWASP ZAP [56]\n2010\nMayhem [12]\n2012\nNessus [70]\nSemi-automated\n2024\nangr [67]\n2016\nAcunetix [34]\n2024\nAutoExploit [46]\n2018\nWebsite Vulnerability Scanner [58]\n2024\nPentestGPT [17]\nLLM-based\n2023\nWang et al. [77]\n2023\nPentestAgent [40]\n2024\nAydin et al. [9]\n2014\n2.3\nMotivation\nAutomated web vulnerability reproduction remains a critical bottleneck in security operations.\nManual analysis lacks scalability, semi-automated tools are brittle with poor comprehension, and\nformal methods like AEG cannot handle web application complexity. This validation gap prevents\nsecurity teams from efficiently confirming and prioritizing threats.\nLLM-based agents offer a promising solution through semantic comprehension, logical reasoning,\nand tool invocation capabilities that address existing limitations. Recent advances have produced\ngeneral-purpose agents (as shown in Table 1) claiming broad problem-solving capabilities, and\nspecialized vulnerability analysis agents (as illustrated in Table 2) targeting security workflows\nspecifically. However, despite theoretical promise and agent proliferation, the practical effectiveness\nof LLM agents in web vulnerability reproduction lacks empirical validation. This creates a critical\ngap between claimed potential and demonstrated capability in this demanding security domain.\nWe conduct the first systematic empirical study evaluating LLM agents’ ability to repro-\nduce web vulnerabilities from natural language descriptions. Our comprehensive framework\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n5\nassesses success rates, reasoning processes, and failure modes, establishing empirical foundations\nfor understanding current capabilities and limitations in automated vulnerability validation.\n3\nEmpirical Study Design\nWe design a comprehensive empirical study to systematically evaluate LLM agents for web vulner-\nability reproduction. Our study follows the workflow illustrated in Figure 1.\nOur study evaluates agents according to the following four criteria:\nC1: Effectiveness. We compare the effectiveness of different agents on web vulnerability\nreproduction by measuring success rates across individual stages (environment setup, vulnerability\nlocalization, and PoC generation with verification) and full end-to-end reproduction.\nC2: Compatibility. We evaluate the adaptability of different agents across various vulnerability\ntypes and web technologies.\nC3: Efficiency. We compare the efficiency of different agents by measuring execution time, USD\ncost, and token consumption, which are critical factors for the practical deployment of LLM agents.\nC4: Robustness. We evaluate the robustness of different agents by analyzing their performance\nunder varying conditions, including model capabilities and input quality.\n§ 3.1 Data Preparation\nVulnerability Database & Code Repository\n§ 3.2 Agent Selection\n§ 3.3 Task Specification\n80 Web Vulnerability-PoC Datasets\n7 Vulnerability Types\nAPI Fetch\nVulnerability Report\nPre-patch Codebase\nManual Construction\n/source\ndocker\n-compose.yml\nDockerfile\nPoC exploits\n& Oracle\nMetadata\nScripts\nGPT-4.1\nClaude-Sonnet-4\nGemini-2.5-pro\nFoundation LLMs\nIntegration\n§ 4 & § 5 Experimental \nSetup & Results\nTarget Web Application Container\nLibrary Application\nWeb Application\nOther\nConfigurations\nDatabase\nInput\nEnv setup task\nLocalization task\nGeneration & Attack\nPoC gen task\nReproduction task\nOutput Example\n{\n\"stage\": \"e2e\",\n\"success\": ,\n\"files\": [{\"path\": \"./.bench/e2e/Dockerfile\",…},\n...\n{\"path\": “./.bench/e2e/steps.md”,…}],\n\"script_files\": {\n\"./.bench/e2e/run.sh\": \"commands\"},\n\"run_commands\": [\n\"bash ./.bench/e2e/run.sh\"],\n}\nC1: Effectiveness\nC2: Compatibility\nC3: Efficiency\nC4: Robustness\nEvaluation Criteria\nAgents\n3 agents selected \nfrom 20 candidates \nfor further analysis\nFig. 1. Systematic workflow for evaluating LLM agents in web vulnerability reproduction tasks.\n3.1\nData Preparation\nTo effectively evaluate the performance of LLM agents in vulnerability reproduction, we have\ncurated a benchmark dataset comprising real-world web application vulnerabilities. Our dataset con-\nstruction follows a structured multi-round filtering strategy to ensure reliability and reproducibility\nacross diverse security scenarios in the real world.\nData Collection Process. We employ a systematic multi-round filtering strategy for data\ncollection. Initially, we search the OSV database’s Git ecosystem for service-based web vulnerability\nkeywords, including SSRF, CSRF, SQL Injection, Prototype Pollution, Path Traversal, Remote Code\nExecution, and Cross-Site Scripting, limiting our scope to vulnerabilities disclosed after 2024. This\ninitial search yields 744 candidate entries. To ensure manageable experimental environments,\nwe further filter out projects with repository sizes exceeding 30MB, resulting in 224 remaining\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n6\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\ncandidates. Finally, we apply three stringent criteria for final selection: (1) only vulnerabilities\naffecting web-based applications are included, (2) vulnerabilities must be available in accessible Git\nrepositories where the specific vulnerable commit can be checked out with complete source code\naccess, and (3) vulnerability reports must be disclosed after 2024 to reflect current threat landscapes.\nThrough this systematic filtering process, we construct a high-quality dataset comprising 80\nvulnerabilities.\nVulnerability Type Selection. Our curation process focuses on 7 critical web vulnerability\ntypes strategically selected to comprehensively evaluate agent capabilities across different vulnera-\nbility paradigms. We include six mainstream service-based vulnerability types: Cross-Site Request\nForgery (CSRF), Path Traversal, Remote Code Execution (RCE), SQL Injection (SQLI), Server-Side\nRequest Forgery (SSRF) and Cross-Site Scripting (XSS). These represent the most prevalent web\napplication security flaws that occur at the service level. Additionally, to provide a more holistic\nassessment of agent performance, we incorporate Prototype Pollution as a representative library-\nbased vulnerability type that manifests through dependencies and third-party packages rather than\ndirect application code. This approach ensures our evaluation captures both traditional web service\nvulnerabilities and modern JavaScript ecosystem threats.\nEnvironment Construction. For each selected vulnerability, we execute a comprehensive data\npreparation pipeline. After fetching the vulnerability reports via APIs, we identify the corresponding\npre-patch codebase from the linked Git repositories and check out the exact vulnerable commit.\nWe utilize Git’s diff capabilities to identify the precise code changes between vulnerable and\npatched versions, allowing us to pinpoint the exact vulnerable code sections, including specific files,\nfunctions and line numbers. We then construct fully containerized environments using Docker,\nmanually developing Dockerfile and docker-compose.yml configurations for each vulnerability to\nensure reproducible deployment across different systems. Each environment is thoroughly tested\nto verify successful vulnerability reproduction before exploit development.\nPoC Development. For PoC exploit construction, we develop PoC exploits based on existing\nvulnerability reports. When existing PoCs are available in the reports, we adapt and refine them for\nour containers; otherwise, we manually create new exploits from scratch. Our PoC development\nfollows vulnerability-specific methodologies: CSRF vulnerabilities are implemented using HTML-\nbased attack vectors, Prototype Pollution exploits are crafted in JavaScript to demonstrate object\nmanipulation, while other vulnerability categories utilize Python-based HTTP request construction\nfor systematic exploitation. Each PoC is accompanied by automated verification oracles that confirm\nsuccessful exploitation, enabling reliable automated assessment.\nOur construction process generates dataset entries containing vulnerable source code directories,\nDocker configuration files, environment setup scripts, executable PoC exploit code with verification\noracles, and structured metadata including CVE identifier, vulnerability description, affected file\npaths, vulnerable function names, specific line numbers, corresponding GitHub repository URLs,\nvulnerable commit hashes and temporal information. Our final dataset comprises 80 vulnerabilities\nspanning 6 popular web technologies including Python, PHP, Java, JavaScript, TypeScript, and Go,\ncovering 7 vulnerability types.\n3.2\nAgent Selection\nRecall that in Table 1, we listed 20 representative agents. These agents share common charac-\nteristics: they are open-source projects available on GitHub with over 1,000 stars for software\nengineering/general-purpose agents and over 100 stars for cybersecurity agents, released or up-\ndated after 2023, and demonstrating active maintenance. All agents demonstrate basic operational\ncapabilities for automated task execution.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n7\nInitial Capability Assessment. We first conduct a preliminary capability assessment of these\n20 representative agents using 16 core dimensions and 3 representative CVEs for testing. The 16\nevaluation dimensions include source code access, tool integration, script execution, file operations,\nweb requests, security domain focus, custom structured output, custom task support, runtime\nstability, error handling, session management, efficiency metrics, log persistence, latest LLM support,\nuser interface, and installation convenience. We select 3 representative CVEs for testing: CVE-\n2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and CVE-2024-41112 (RCE), which represent\ndifferent vulnerability types and complexity levels.\nAgent Selection Results. We conduct a comprehensive evaluation of these 20 agents spanning\nmultiple architectures and application domains, with detailed results presented in Table 3 (§ 5.1).\nThrough this evaluation round, we identify the 3 best-performing agents. Subsequently, we conduct\nan in-depth empirical evaluation of these 3 selected agents using the complete 80-vulnerability\ndataset described in § 3.1. Therefore, our further empirical evaluation focuses on these three\npromising agents that demonstrate adequate capabilities and performance.\nOutputs (all saved under ./.bench/e2e/):\n- Intermediate artifacts\n- steps.md (the reproduction steps and \nkey commands).\n- result.json (consolidated end-to-end \nresult).\nUser Prompt\nYou are participating in web vulnerability reproduction in a safe container.\nRules:\n- Never invent placeholder hostnames. Derive service_url from docker-compose \nport mapping (localhost) or explicit run.sh output.\n- You can edit/create files under the workspace and execute shell commands.\n- Do NOT modify any source code files; only write under ${stage_dir}.\n- Do NOT access files outside the current working directory.\nCategory: RCE\nVulnerability description: MailDev 2 through 2.1.0 allows Remote Code \nExecution via a crafted Content-ID header for an e-mail attachment, leading to \nlib/mailserver.js writing arbitrary code into the routes.js file.\nCodebase Directory: ${codebase_dir}\nStage: End-to-End Reproduction.\nSteps: Starting from the CVE folder and the codebase, autonomously:\n1) Build/run the environment from scratch based on c\n2) Localize the vulnerable code region (files/functions/lines)\n3) Generate a PoC and an in-code verification oracle under ./.bench/e2e/ to \nassess exploit success.\n4) Execute the PoC, collect evidence/logs/state, and decide success/failure.\n5) Produce a concise step-by-step reproduction document \n(./.bench/e2e/steps.md).\nOutputs (all saved under ./.bench/e2e/):\n- Intermediate artifacts\n- steps.md (the reproduction steps and key commands).\n- result.json (consolidated end-to-end result).\nResponse.png\nAgent\n{\n\"stage\" : \"e2e\",\n\"success\": ,\n\"files\": [\n{\"path\": \"./.bench/e2e/Dockerfile\", \"status\": \n\"created|updated|copied\"},\n],\n\"run_commands\": [\n\"bash ./.bench/e2e/run.sh\"\n],\n\"artifacts\": {\n\"service_url\": \"http://localhost:\",\n\"exec_success\": ,\n\"signals\": {\"status\": 200, \"log_regex\": \"...\"},\n\"observed\": {\"status\": 200, \"log\": \"...\"}\n},\n\"steps\": [\n{\"name\": \"env_setup\", \"success\": },\n{\"name\": \"localization\", \"success\": \n},\n{\"name\": \"poc_and_verify\", \"success\": \n, \"notes\": \"...\"}\n],\n…\n}\nJSON Response\nVulnerability Information\nReproduction Requirements\nOutput Schema\nFig. 2. Prompt template and example agent response.\n3.3\nTask Specification\nWe design four evaluation tasks of increasing complexity to systematically assess agent capabilities\nacross the vulnerability reproduction pipeline. The first three tasks are independent, single-stage\nevaluations where each task receives expert-curated ground truth from prerequisite stages as\ninput, ensuring that only the current stage’s capabilities are measured. The final task represents a\ncomprehensive end-to-end evaluation requiring agents to autonomously navigate the complete\nvulnerability reproduction workflow.\nEnvironment Setup. This foundational task evaluates an agent’s ability to establish a functional\ntesting environment from source code alone. Given only the vulnerable application’s codebase,\nagents must generate the necessary containerization files (Dockerfile, docker-compose.yml) and\ndeployment scripts (run.sh) to create a running instance container of the application.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n8\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nVulnerability Localization. With access to both the running environment and source code,\nagents must perform code analysis to pinpoint the vulnerability’s location. Given the CVE descrip-\ntion, agents analyze the codebase to produce a ranked list of suspicious files, functions, and specific\nline numbers that correspond to the security flaw.\nProof-of-Concept (PoC) Exploits Generation. This task tests an agent’s exploit development\ncapabilities. Provided with the vulnerability’s exact location (file and line number), agents must\ncraft a working Proof-of-Concept exploits that successfully triggers the exploit and includes a\nverification oracle to programmatically confirm successful exploitation.\nEnd-to-End Reproduction. As the most challenging evaluation, this task provides agents only\nwith the source code and original CVE description. Agents must autonomously perform all previous\nstages—from environment setup through PoC execution—and produce a complete, documented\nworkflow of their reproduction process.\nTo ensure fair evaluation across different agents, we establish a standardized input-output\ninterface that provides consistent task specifications and response formats. Each agent receives\nidentical prompts and must produce outputs in the same structured format, enabling objective\ncomparison of capabilities across all web vulnerability reproduction tasks.\nAs illustrated in Figure 2, the prompt contains structured vulnerability information including\nCVE category, detailed description, codebase directory, and reproduction steps. The prompt also\nspecifies operational constraints such as workspace restrictions and file access limitations. Agents\nmust respond with structured JSON outputs that include stage indicators, success flags, file paths,\nexecution commands, and artifact metadata. This standardized format enables automated assessment\nof task completion. It also allows verification that agents produce the required code artifacts\nincluding Dockerfiles, execution scripts, PoC exploits, and verification oracles.\n4\nExperimental Setup\n4.1\nStudied Dataset and Agents\nAs introduced in § 3.1, we evaluate agent performance on a curated benchmark of 80 real-world\nvulnerabilities across diverse categories and complexity levels. Following the setup described in § 3.2,\nwe assess the performance of 20 popular LLM agents across software engineering, cybersecurity, and\ngeneral-purpose, each integrated with three foundation LLMs: GPT-4.1 [54], Claude-Sonnet-4 [6],\nand Gemini-2.5-Pro [25].\n4.2\nMetrics\nTo comprehensively evaluate agent performance across the four criteria (C1-C4), we measure\nsuccess rates at different granularities and additional efficiency metrics:\n4.2.1\nSuccess Rate Metrics. We measure the percentage of successful completions across different\nstages of the vulnerability reproduction pipeline, with all results verified through manual inspection:\n• Environment Setup Success Rate: The percentage of vulnerabilities for which an agent\nsuccessfully generates and deploys a functional vulnerable environment, verified through\nmanual testing of application accessibility and configuration correctness.\n• Vulnerability Localization Accuracy (Top-K): The percentage of cases where the ground-\ntruth vulnerability location appears in the agent’s top-K predictions. We evaluate this metric\nat three granularity levels:\n– File-level Top-K: Ground-truth vulnerable file present in top-K file predictions.\n– Function-level Top-K: Ground-truth vulnerable function present in top-K function predic-\ntions.\n– Line-level Top-K: Ground-truth vulnerable line present in top-K line predictions.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n9\n• PoC Generation Success Rate: We decompose this into two complementary sub-metrics:\n– Execution Success Rate: The percentage of generated PoCs that execute without runtime\nerrors, verified through both automated testing and manual execution.\n– Vulnerability Trigger Success Rate: The percentage of executable PoCs that successfully\ntrigger the vulnerability, confirmed through manual verification of exploitation effects.\n• End-to-End Success Rate: The percentage of vulnerabilities for which an agent successfully\ncompletes the entire reproduction workflow from environment setup to verified exploitation,\nwith manual confirmation of successful vulnerability triggering.\n4.2.2\nEfficiency Metrics. To assess operational costs and practical deployment considerations, we\nalso measure:\n• Execution Time: Total wall-clock time required for task completion.\n• Token Consumption: Total API tokens consumed during execution.\n• Financial Cost: Estimated USD cost based on publicly available API pricing.\n4.3\nImplementation Details\nAll experiments are conducted on a standardized Linux machine with Ubuntu 20.04 LTS, equipped\nwith an Intel Xeon Platinum 8358P CPU (2.60GHz, 128 cores) and 2TB memory. We use Docker En-\ngine 26.1.0 to containerize environments, ensuring isolation and reproducibility. Agent frameworks\nare executed using Python 3.11 with dependencies installed from their official repositories.\nTo mitigate inherent randomness in LLM outputs, each experiment undergoes three independent\nexecutions, with final metrics calculated as averages across all. A strict 60-minute timeout is enforced\nper execution, where any run exceeding this limit is automatically terminated and classified as a\nfailure. All generated artifacts, execution logs, and API interactions are systematically captured and\nstored for comprehensive post-hoc analysis. To maintain experimental feasibility, we impose budget\nconstraints of $2 per individual stage and $5 for complete end-to-end reproduction workflows.\nTo ensure fair comparison across all agents, browser-based interactions are strictly prohibited,\nrestricting all agents to command-line interfaces only.\n5\nExperimental Results\n5.1\nC1: Effectiveness\nMotivation. Manual reproduction of security vulnerabilities represents a well-documented\nbottleneck in software development, consuming substantial expert time and delaying remediation\nefforts. Previous approaches relied heavily on manual processes where security researchers must\ninterpret vulnerability reports and develop proof-of-concept code. However, recent advances in LLM\nagents present opportunities to automate these complex tasks. Therefore, this criterion evaluates\nthe effectiveness of current agents in performing core vulnerability reproduction tasks.\nApproach. As described in § 4, we evaluate agent effectiveness across the four-stage vulnerability\nreproduction pipeline using the success rate metrics defined in § 4.2. Each agent-model combination\nis assessed on environment setup, vulnerability localization (measured by file-level, function-level\nand line-level Top-3 accuracy), PoC generation with execution verification, vulnerability trigger\nverification, and end-to-end reproduction success, which is defined in § 3.3.\nResults. Table 3 shows the capability assessment results of 20 agents across 16 dimensions\nusing three representative CVEs: CVE-2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and\nCVE-2024-41112 (RCE). The analysis reveals severe capability gaps across the agent landscape.\nSoftware engineering agents show poor performance with only 2 out of 8 agents meeting basic\nrequirements. Most fail in critical areas like tool integaration and session management, with agents\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n10\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nTable 3. Comparison of 20 agents for web vulnerability reproduction with CVE-2024-7065, CVE-2024-7458\nand CVE-2024-41112.\nAgent\nSCA TI\nSE\nFO\nWR\nSDF\nCSO CTS RS\nEH SM EM\nLP\nLLS\nUI\nEI\nSoftware Engineering Agents\nAGENTLESS\n✓\n✗\n✗\n✗\n✗\n✗\n✗\n✗\nG\n#\n✗\n✗\n✗\n✗\nG\n#\nC\n✓\nOpenHands\n✓\n✓\n✓\n✓\n✓\n✗\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\nC/W\n✓\nSWE-agent\n✓\nG\n#\n✓\n✓\n✓\n✗\nG\n#\n✓\n✓\n✓\n✓\n✓\n✓\n✓\nC\n✓\nAutoCodeRover\n✓\n✗\nG\n#\n✓\n✗\n✗\n✗\nG\n#\n✓\n✗\n✗\n✗\n✓\nG\n#\nC\n✓\nAider\nG\n#\n✓\nG\n#\n✓\n✗\n✗\nG\n#\n✓\n✓\n✓\n✗\n✓\n✓\n✓\nC\n✓\nRepoAgent\n✓\n✗\nG\n#\nG\n#\n✗\n✗\n✗\nG\n#\n✓\n✗\n✗\n✗\n✗\nG\n#\nC\n✓\nChatDev\n✗\n✗\n✗\nG\n#\n✗\n✗\nG\n#\nG\n#\nG\n#\n✗\n✗\n✗\n✗\n✗\nC/W\nG\n#\nMetaGPT\nG\n#\n✓\nG\n#\nG\n#\n✗\n✗\nG\n#\nG\n#\n✓\n✓\n✗\n✗\n✓\nG\n#\nC\n✓\nCybersecurity Agents\nPentestGPT\nG\n#\n✓\n✓\n✓\n✓\n✓\nG\n#\n✗\n✓\n✓\n✓\n✗\n✓\n✓\nC\n✓\nCAI\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\n✓\nC\n✓\nPentestAgent\n✗\n✓\n✓\n✓\n✓\n✓\n✗\n✗\n✗\n✗\n✗\n✗\n✗\nG\n#\nC\n✓\nNebula\n✓\n✓\n✓\nG\n#\n✗\n✓\n✗\n✗\n✗\n✗\n✗\n✗\n✗\n✗\nC\nG\n#\nHackingBuddyGPT\n✗\n✓\n✓\n✓\n✓\n✓\nG\n#\nG\n#\n✓\n✓\n✓\n✗\n✓\n✓\nC\nG\n#\nPentAGI\n✗\nG\n#\n✓\n✓\n✓\n✓\n✗\n✗\n✓\n✓\n✓\n✗\n✓\n✓\nC\n✗\nEnIGMA\n✗\n✓\n✓\n✓\n✓\n✓\n✗\n✗\n✓\n✓\n✓\n✓\n✓\n✗\nC\nG\n#\nAI-OPS\n✗\n✗\n✗\n✗\nG\n#\nG\n#\n✗\n✗\n✓\n✗\n✓\n✗\n✗\n✗\nC\nG\n#\nGeneral-purpose Agents\nAutoGPT\n✓\nG\n#\nG\n#\n✓\n✓\n✗\nG\n#\nG\n#\n✗\n✗\n✓\n✗\n✓\n✓\nC/W\n✗\nBabyAGI\n✓\nG\n#\nG\n#\nG\n#\n✗\n✗\n✗\n✗\nG\n#\nG\n#\n✓\n✗\n✗\nG\n#\nC\n✓\nAgentVerse\n✓\n✗\n✗\n✗\n✗\n✗\nG\n#\nG\n#\nG\n#\n✗\n✓\n✗\n✗\n✗\nC/W\nG\n#\nAIlice\n✓\n✓\n✓\nG\n#\n✗\n✗\n✓\n✓\nG\n#\n✗\n✓\n✗\n✓\n✗\nC/W\n✗\nNote: Symbols represent Good (✓), Average (G\n#), and Poor (✗) qualitative assessment.\nAbbreviations:\nSCA: Source Code Access\nRS:\nRuntime Stability\nTI:\nTool Integration\nEH:\nError Handling\nSE:\nScript Execution\nSM:\nSession Management\nFO:\nFile Operation\nEM:\nEfficiency Metrics\nWR:\nWeb Request\nLP:\nLog Persistence\nSDF:\nSecurity Domain Focus\nLLS: Latest LLM Support\nCSO: Custom Structured Output\nUI:\nUser Interface (C: CLI, C/W: CLI/WEB)\nCTS:\nCustom Task Support\nEI:\nEase of Install\nlike AGENTLESS [82] and ChatDev [62] lacking over 10 essential capabilities. Cybersecurity agents\ndemonstrate domain knowledge but face limitations in technical implementation capabilities, most\nlack essential technical features such as structured output and custom task support. Except for CAI\nwhich achieves full capability, all other cybersecurity agents fail to support custom task types and\nstructured output formats, limiting their adaptability to diverse web vulnerability reproduction\nscenarios. General-purpose agents offer broad applicability and flexible task handling but struggle\nwith specialized technical requirements. For example, AutoGPT [26] demonstrates reasonable\nperformance in 8-9 dimensions but lacks security domain focus and has installation challenges.\nOnly three agents demonstrate comprehensive capabilities: SWE-agent [84], OpenHands [78],\nand CAI [48]. These agents form the foundation of our further empirical evaluation:\n- SWE-agent employs a carefully designed Agent-Computer Interface (ACI) that enables au-\ntonomous interaction with software repositories through specialized commands for code navigation,\nediting, and testing. The framework demonstrates exceptional performance in repository-level vul-\nnerability discovery through its integrated file viewer, search capabilities, and context management\nsystem that maintains concise yet informative interaction histories.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n11\n- OpenHands provides a comprehensive platform for AI software development agents, featuring\nan event-stream architecture that facilitates complex multi-turn interactions. The framework\nexcels in code generation, debugging, and repository manipulation through its sandboxed Docker\nenvironment and extensive tool integration, including IPython execution, web browsing capabilities,\nand multi-agent delegation patterns.\n- CAI represents a specialized cybersecurity AI framework designed for autonomous penetration\ntesting and vulnerability assessment. Built around six fundamental pillars (Agents, Tools, Handoffs,\nPatterns, Turns, and Human-In-The-Loop), CAI demonstrates superior performance in offensive se-\ncurity operations, achieving human-competitive results across diverse Capture-The-Flag challenges\nand real-world bug bounty scenarios.\nTo ensure experimental consistency and fair comparison across all subsequent evaluations, we\nconfigure each framework with identical LLM foundations: GPT-4.1[54], Claude-Sonnet-\n4[6] and Gemini-2.5-Pro[25] for the following experiments.\nTable 4 presents the comprehensive effectiveness evaluation results across these three selected\nagent-model combinations. The results reveal significant challenges in automated vulnerability\nreproduction, with substantial performance variations across different pipeline stages and agent-\nmodel configurations.\nTable 4. Effectiveness results: success rates (%) across vulnerability reproduction pipeline.\nAgent\nModel\nEnv Setup\nVulnerability Localization\nPoC Generation\nEnd-to-End\nSetup\nFile Level\nFunc Level\nLine Level\nExecution\nTrigger\nSuccess@1\nSuccess@3\nOpenHands\nClaude-Sonnet-4\n28.8\n46.3\n36.3\n23.8\n70.0\n21.3\n10.0\n22.5\nGemini-2.5-Pro\n26.3\n40.0\n27.5\n17.5\n52.5\n10.0\n6.3\n13.8\nGPT-4.1\n32.5\n53.8\n43.8\n31.3\n56.3\n16.3\n13.8\n20.0\nSWE-agent\nClaude-Sonnet-4\n22.5\n50.0\n40.0\n27.5\n58.8\n11.3\n8.8\n15.0\nGemini-2.5-Pro\n20.0\n37.5\n30.0\n21.3\n56.3\n8.8\n5.0\n11.3\nGPT-4.1\n25.0\n46.3\n36.3\n25.0\n66.3\n13.8\n11.3\n18.8\nCAI\nClaude-Sonnet-4\n35.0\n56.3\n46.3\n32.5\n56.3\n15.0\n7.5\n12.5\nGemini-2.5-Pro\n38.8\n50.0\n38.8\n26.3\n33.8\n17.5\n3.8\n10.0\nGPT-4.1\n36.3\n58.8\n48.8\n35.0\n25.0\n20.0\n10.0\n16.3\nExamining the overall end-to-end performance, OpenHands demonstrates the strongest capabili-\nties, particularly when paired with Claude-Sonnet-4, achieving the highest Success@3 rate of 22.5%.\nOpenHands maintains consistent superiority across different models, with Success@3 rates of\n22.5% (Claude-Sonnet-4), 20.0% (GPT-4.1), and 13.8% (Gemini-2.5-Pro). In comparison, SWE-agent\nachieves maximum Success@3 rates of 18.8% (GPT-4.1), while CAI peaks at 16.3% (GPT-4.1). The\nSuccess@1 results show similar patterns, with OpenHands + GPT-4.1 leading at 13.8%, followed\nby OpenHands + Claude-Sonnet-4 at 10.0%. Notably, even the best-performing combinations fail\nto achieve successful vulnerability reproduction in more than three-quarters of cases, indicating\nsubstantial room for improvement in current agent capabilities.\nEnvironment setup performance varies significantly across agents, with CAI demonstrating the\nmost consistent results, achieving rates above 35% across all models and peaking at 38.8% with\nGemini-2.5-Pro. OpenHands shows moderate performance ranging from 26.3% to 32.5%, while\nSWE-agent exhibits the lowest and most variable setup success rates, spanning 20.0% to 25.0%.\nVulnerability localization reveals a clear performance hierarchy and degradation pattern. CAI\npaired with GPT-4.1 leads in localization precision across all granularities: 58.8% file-level, 48.8%\nfunction-level, and 35.0% line-level accuracy. However, the degradation from file-level to line-level\nrepresents a consistent 35–40% relative decrease across all combinations. GPT-4.1 consistently\noutperforms other models in localization tasks, while Gemini-2.5-Pro shows the weakest localization\ncapabilities across all agents.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n12\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nThe PoC generation stage reveals OpenHands’ key strength and a critical bottleneck. OpenHands\n+ Claude-Sonnet-4 achieves the highest PoC execution rate at 70.0%, significantly outperforming\nother combinations. However, a substantial execution-to-trigger gap exists across all agents: while\nexecution rates range from 25.0% to 70.0%, trigger rates remain uniformly low at 8.8% to 21.3%.\nRemarkably, OpenHands + Claude-Sonnet-4 also leads in trigger success at 21.3%, demonstrating\nsuperior PoC quality despite the overall low trigger rates.\nModel-specific patterns emerge clearly. GPT-4.1 excels in localization precision but shows moder-\nate end-to-end performance. Claude-Sonnet-4 demonstrates exceptional synergy with OpenHands,\nparticularly in PoC generation and end-to-end orchestration. Gemini-2.5-Pro consistently under-\nperforms across most metrics, especially in PoC-related tasks.\nThe results highlight OpenHands’ superior workflow orchestration capabilities. While CAI\nachieves the highest individual stage performance in environment setup and localization, Open-\nHands translates moderate individual performance into the strongest end-to-end success rates,\nsuggesting more effective inter-stage coordination and error recovery mechanisms.\nWe further analyze the underlying causes of these performance differences by examining the\ntool utilization patterns across agents. Table 5 reveals that while all agents share basic execution\ncapabilities, OpenHands possesses unique workflow control tools that fundamentally enhance its\nvulnerability reproduction effectiveness. The think tool enables structured reasoning processes\nthat strengthen agent memory and decision-making throughout the multi-stage pipeline, while the\nfinish tool provides systematic task summarization and completion mechanisms. These capabilities\nexplain OpenHands’ superior ability to maintain context across pipeline stages and achieve higher\nend-to-end success rates despite comparable basic tool availability.\nUnexpectedly, CAI’s extensive domain-specific security tool suite remains largely unutilized\nin our task specifications, revealing a critical gap between tool availability and practical applica-\ntion. While CAI provides numerous specialized cybersecurity tools (such as exploitation, privilege\nscalation [48]) theoretically suitable for vulnerability reproduction, the agent fails to effectively\nleverage these capabilities within the structured task framework, resulting in lower end-to-end\nperformance despite superior individual stage metrics in environment setup and localization. Addi-\ntionally, OpenHands demonstrates superior compliance with output formatting requirements by\ncorrectly generating artifacts and JSON outputs as specified in task prompts, indicating more robust\ninstruction following capabilities. This analysis reveals that tool ecosystem effectiveness of agent\ndepends not merely on capability breadth but on workflow integration, reasoning enhancement,\nand systematic task completion mechanisms that enable consistent execution across complex\nmulti-stage security tasks.\nTable 5. The comparison of tool utilization across agent frameworks during task execution.\nTool Type\nTool Capability\nCAI\nOpenHands\nSWE-agent\nBasic Execution\nexecute_bash\n✓\n✓\n✓\nstr_replace_editor\n✓\n✓\n✓\nFile Operations\n✓\n✓\n✓\nWorkflow Control\nthink (Reasoning Enhancement)\n×\n✓\n×\nfinish (Task Summary)\n×\n✓\n×\nDomain-Specific\nSecurity/Vulnerability Tools\n9\n×\n×\nUsed in Tasks\n0\nN/A\nN/A\nTask Execution\nClear Task Termination\n✓\n✓\n✓\nStructured Reasoning\n×\n✓\n×\nCorrect Artifacts Output\n×\n✓\n×\nProper JSON Format\n✓\n✓\n×\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n13\nFindings for C1: Current LLM agents achieve limited web vulnerability reproduction effective-\nness with end-to-end success rates below 25%. OpenHands demonstrates superior performance\nthrough structured reasoning and systematic completion mechanisms, while tool availability\nalone does not ensure effectiveness. The primary bottleneck remains the exploitation gap\nwhere agents can execute PoCs but struggle to trigger actual vulnerabilities.\n5.2\nC2: Compatibility\nMotivation. Real-world vulnerability reproduction environments exhibit significant diversity\nacross vulnerability types and web technologies. Different vulnerability categories require distinct\nunderstanding patterns, exploitation techniques, and verification approaches. Similarly, vulnerabili-\nties manifest differently across web technologies due to language-specific features, frameworks,\nand coding patterns. A practical vulnerability reproduction agent must demonstrate consistent\nperformance across this diversity rather than excelling only in specific domains. Understanding\ncompatibility helps assess whether current agents can serve as general tools or require specialized\nadaptations.\nApproach. As shown in § 3.1, our dataset covers diverse vulnerability types and web technolo-\ngies. We conduct categorical evaluation across these dimensions to assess agent compatibility. For\nvulnerability type compatibility, we analyze reproduction success rates across the seven vulnerabil-\nity categories in our dataset. For programming language compatibility, we evaluate performance\nacross the six primary languages. To ensure fairness and reproducibility, we use the top-performing\ncombination from § 5.1, specifically OpenHands with Claude-Sonnet-4.\nResults. Table 6 presents stage-wise success rates across different vulnerability types. The results\nreveal substantial performance variations, with end-to-end success rates (Success@3) ranging from\n0% to 63%. Library-based Prototype Pollution achieves the highest reproduction success rate at\n63%. Among service-based web vulnerabilities, CSRF performs best with 60%, followed by Path\nTraversal at 36%. Other service-based vulnerabilities demonstrate considerably lower success rates:\nXSS at 16%, SSRF at 15%, RCE at 8%, and SQL Injection with the lowest at 0%.\nThe low success rates for XSS and SSRF reflect agents’ difficulties with complex multi-component\nenvironments requiring database systems, frontend-backend configurations, and middleware ser-\nvices. Agents struggle to extract configurations from large repositories, perform data imports, and\nhandle intricate deployment procedures. SQL Injection faces similar environment setup challenges,\ncompounded by database-dependent architectures. Beyond environment setup, a critical bottleneck\nemerges in vulnerability triggering due to API authentication requirements, as agents without\nbrowser interaction capabilities cannot perform automated login procedures like human users.\nIn contrast, CSRF achieves perfect file localization due to clear vulnerability descriptions enabling\nprecise interface targeting. Prototype Pollution maintains consistent performance across stages,\nbenefiting from simpler library-based repositories requiring only dependency installation rather\nthan complex infrastructure.\nWhile PoC execution rates remain high across vulnerability types, the dramatic drop from\nexecution to trigger phases reveals that agents can run exploit code but struggle with the precise\nenvironmental conditions and interaction sequences needed for actual vulnerability manifestation.\nFigure 3 demonstrates end-to-end reproduction success rates across web technologies. PHP\nachieves the highest success rate at 38.9%, while TypeScript shows the lowest at 11.1%. PHP’s\nsuperior performance stems from its prevalence in web security research, providing agents with\nabundant training data and well-documented exploit patterns, combined with its simpler syntax and\ndirect web-oriented nature. JavaScript demonstrates strong performance through extensive client-\nside vulnerability examples and straightforward DOM manipulation patterns. TypeScript’s poor\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n14\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nTable 6. Vulnerability reproduction success rates by vulnerability types.\nVulnerability Type\nEnvironment\nCode Localization\nPoC Generation\nEnd-to-End\nSetup\nFile\nFunction\nLine\nExecution\nTrigger\nSuccess@1\nSuccess@3\nPROTOTYPE_POLLUTION\n75%\n88%\n75%\n63%\n88%\n50%\n38%\n63%\nCSRF\n80%\n100%\n80%\n60%\n80%\n60%\n40%\n60%\nPATH_TRAVERSAL\n45%\n82%\n73%\n45%\n82%\n27%\n18%\n36%\nXSS\n21%\n53%\n37%\n21%\n79%\n11%\n5%\n16%\nSSRF\n15%\n31%\n23%\n8%\n77%\n8%\n0%\n15%\nRCE\n8%\n17%\n8%\n8%\n67%\n8%\n0%\n8%\nSQLI\n8%\n0%\n0%\n0%\n25%\n0%\n0%\n0%\nTOTAL\n29%\n46%\n36%\n24%\n70%\n18%\n10%\n23%\nPHP\nJavaScript\nJava\nPython\nTypeScript\nOthers\n0\n5\n10\n15\n20\n25\n30\n35\n40\nEnd-to-End Success Rate (%)\n38.9%\n23.5%\n18.2%\n22.2%\n11.1%\n14.3%\nAverage: 22.5%\nFig. 3. Comparison of vulnerability reproduction success rates between different web technologies.\nperformance shows that its additional complexity and stricter syntax requirements hinder agents\ndespite its similarity to JavaScript. The results indicate that agent performance varies significantly\nacross both vulnerability types and web technologies, with success rates influenced by environment\ncomplexity and the specific requirements of each vulnerability category.\nFindings for C2: Agent performance exhibits substantial variation across vulnerability types\nand web technologies. Service-based vulnerabilities face significant challenges from complex\nenvironment setup, authentication requirements, and multi-component dependencies, while\nlibrary-based vulnerabilities achieve higher success through simpler deployment requirements.\nLanguage compatibility depends on security research prevalence and syntactic complexity,\nwith web-oriented languages showing better agent adaptation.\n5.3\nC3: Efficiency\nMotivation. While effectiveness measures reproduction success rates, efficiency evaluates the\ncomputational and economic costs of achieving successful reproductions. Understanding resource\nrequirements is crucial for practical deployment, as organizations must balance reproduction\ncapabilities with operational costs. We analyze efficiency across successful reproduction attempts\nto assess real-world viability of different agent-model combinations.\nApproach. We adopt the efficiency evaluation methodology from § 4.2, measuring token con-\nsumption, execution time, and monetary cost across the vulnerability reproduction pipeline. To\ndo so, we focus exclusively on the successful reproduction cases achieved by our three founda-\ntional agents as identified in § 5.1. To ensure a fair comparison, all tests are conducted using the\nClaude-Sonnet-4 model, and measurements are averaged across successful attempts for each agent.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n15\nResults. Table 7 presents efficiency metrics for all three agents calculated from successful\nreproduction attempts. Environment setup demonstrates the highest resource requirements across\nall agents, while vulnerability localization proves most economical. End-to-end reproduction costs\nrange from $1.68 to $2.19 with execution times between 25.7-34.0 minutes. OpenHands consistently\nrequires the highest resources ($2.19 total), followed by SWE-agent ($1.90), while CAI demonstrates\nthe most efficient utilization ($1.68).\nTable 7. Efficiency analysis of successful vulnerability reproduction cases.\nAgent\nEnv\nLoc\nPoC\nE2E\nT\nTi\nC\nT\nTi\nC\nT\nTi\nC\nT\nTi\nC\nOpenHands\n78.3\n18.2\n1.17\n15.2\n4.3\n0.23\n52.7\n11.5\n0.79\n146.2\n34.0\n2.19\nSWE-agent\n65.4\n15.8\n0.98\n12.8\n3.7\n0.19\n48.9\n10.2\n0.73\n127.1\n29.7\n1.90\nCAI\n56.8\n13.1\n0.85\n11.1\n3.2\n0.17\n44.2\n9.4\n0.66\n112.1\n25.7\n1.68\nEnv: Environment Setup, Loc: Vulnerability Localization, PoC: PoC Generation, E2E: End-to-End Reproduction.\nT: Tokens (K), Ti: Time (min), C: Cost ($). Averages from successful cases only.\nCosts based on Claude Sonnet pricing ($3/1M input, $15/1M output tokens).\nThe cost variations stem from architectural differences between agents. OpenHands incorpo-\nrates extensive internal reasoning including explicit \"think\" and \"finish\" operations that generate\nsubstantial token overhead. SWE-agent employs numerous system prompts to enforce tool usage\ncompliance, creating intermediate communication costs. CAI, despite having the most comprehen-\nsive tool suite, exhibits minimal tool invocation in vulnerability reproduction tasks, resulting in\nstreamlined execution. These design choices create a trade-off between computational efficiency and\nreproduction capability, with OpenHands prioritizing thorough reasoning, SWE-agent emphasizing\ntool compliance, and CAI optimizing for direct execution.\nFindings for C3: Web vulnerability reproduction requires substantial computational resources,\nwith end-to-end costs ranging from $1.68 to $2.19 per successful case. Environment setup\nconsumes the most resources across all agents, while agent architectural differences create\nsignificant efficiency variations in the reproduction pipeline.\n5.4\nC4: Robustness Analysis\nMotivation. Robustness measures how well agents perform under challenging conditions. In\nvulnerability reproduction, agents face obstacles like limited processing capabilities, browser secu-\nrity restrictions, or incomplete information. These challenges are inherent to real-world security\nresearch environments where perfect conditions rarely exist. A robust agent should maintain rea-\nsonable performance across different scenarios, adapting to unexpected failures and environmental\nconstraints. Understanding robustness helps us assess agent reliability for practical deployment\nand identify which architectures handle reproduction challenges most effectively.\nApproach. We design two experiments to test the robustness of our three selected agents\nacross key challenge areas: First, we compare agent performance using Claude-Sonnet-4 with and\nwithout enhanced reasoning across different vulnerability reproduction stages in 10 vulnerabilities.\nSecond, we test agent performance under different authentication scenarios using 12 SQL Injection\nvulnerabilities, comparing three cases: manually providing login tokens, prompting agents to login\nthemselves, and providing no authentication help.\nResults. Figure 4 shows mixed results when comparing models with and without enhanced rea-\nsoning. Enhanced reasoning helps in early stages, with OpenHands improving from 60% to 70% in\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n16\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\nEnvironment Setup and from 40% to 70% in File Localization, while improvements decrease in later\nstages with PoC Execution showing smaller gaps and Trigger Success showing no improvement\nacross all agents. SWE-Agent shows the most consistent benefit from enhanced reasoning across\nlocalization stages, while OpenHands demonstrates strong baseline performance even without\nreasoning due to its built-in \"think\" operations, and CAI shows more variable results with minimal\nreasoning benefits. The limited reasoning impact reflects both agent design differences and task\ncomplexity variations, as reasoning enhancements prove most effective for analytical tasks like code\nlocalization but provide little benefit for execution-oriented activities like vulnerability triggering.\nWithin our 10-vulnerability sample, agents demonstrate robustness across both reasoning config-\nurations, suggesting that architectural design choices may be more influential than underlying\nmodel reasoning capabilities for practical vulnerability reproduction tasks.\nEnvironment\nFile Localization\nFunction Localization\nLine Localization\nPoC Execution\nTrigger Success\n0\n10\n20\n30\n40\n50\n60\n70\n80\nSuccess Rate (%)\nOpenhands (w/ reasonning)\nOpenhands (w/o reasonning)\nSWE-Agent (w/ reasonning)\nSWE-Agent (w/o reasonning)\nCAI (w/ reasonning)\nCAI (w/o reasonning)\nFig. 4. Performance comparison between agents using Claude-Sonnet-4 with and without reasoning across\ndifferent vulnerability reproduction stages.\nTable 8 shows authentication context impacts vulnerability reproduction success. Manual token\nprovision works best, with Openhands + Claude-Sonnet-4 achieving 67% success. Requiring agents\nto handle login autonomously results in an average performance degradation of 33.3% compared to\nmanual token provision, with success rates dropping to 25-50% across agent-model combinations.\nRemoving authentication guidance causes near-complete failure, with only three configurations\nmaintaining 8% success rates. This reveals that current agents struggle with autonomous authentica-\ntion discovery and rely heavily on explicit guidance. The consistent degradation across architectures\nand models indicates that authentication handling is a widespread limitation.\nFindings for C4: Current agents show robustness to reasoning changes but high sensitivity to\nprompt variations. While reasoning versus non-reasoning modes produce similar performance,\nincomplete authentication results in severe performance drops, highlighting agents’ limited\ncapacity for problem-solving without explicit guidance.\n6\nDiscussion\nOur empirical evaluation reveals that current open-source agents face substantial limitations\nin vulnerability reproduction beyond specific ecosystems. While agents demonstrate reasonable\nsuccess in library-based JavaScript environments and relatively straightforward CSRF scenarios,\nthey struggle significantly with complex web vulnerabilities that require intricate environment\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n17\nTable 8. Comparison of trigger rate in authentication robustness analysis.\nAgent\nModel\nManual Token\nAgent Login\nNo Auth Info\nOpenhands\nClaude-Sonnet-4\n67%\n50%\n8%\nGemini-2.5-Pro\n58%\n42%\n8%\nGPT-4.1\n50%\n33%\n0%\nSWE-Agent\nClaude-Sonnet-4\n58%\n33%\n8%\nGemini-2.5-Pro\n50%\n25%\n0%\nGPT-4.1\n42%\n17%\n0%\nCAI\nClaude-Sonnet-4\n50%\n25%\n0%\nGemini-2.5-Pro\n42%\n17%\n0%\nGPT-4.1\n33%\n8%\n0%\nsetup and multi-step exploitation chains. The stark performance gaps—ranging from 0% for SQL\ninjection to 63% for prototype pollution—indicate that pure code-based approaches are insufficient\nfor comprehensive vulnerability reproduction. Traditional injection vulnerabilities, despite their\nwell-documented nature, consistently fail due to environment configuration challenges and the\ncomplex interplay between application deployment, database setup, and exploitation context. This\nsuggests that current agent architectures fundamentally lack the systems-level understanding\nnecessary for realistic security testing scenarios.\nThe integration of Model Context Protocol (MCP)[1] with browser automation tools like Play-\nwright or Chrome presents a breakthrough for vulnerability reproduction. By enabling agents\nto control browser instances, capture network traffic, and monitor application state changes in\nMCP-based approaches can bridge the gap between static code analysis and dynamic exploitation.\nThis browser-centric methodology allows agents to observe runtime behavior of web applications,\nintercept HTTP requests and responses, and validate success through DOM manipulation and state\ninspection. The ability to interact with web interfaces while monitoring network activity represents\na shift from traditional code-only approaches, unlocking automated reproduction capabilities for\ncomplex web vulnerabilities that have resisted agent-based solutions.\n7\nThreats to Validity\nInternal Validity. The primary internal threat stems from potential data leakage in LLM training\ndatasets. Many CVEs in our benchmark were disclosed before the training cutoff dates of evaluated\nmodels, creating a risk that agents may leverage memorized exploit patterns rather than demon-\nstrating genuine reasoning capabilities. Although we deliberately avoided referencing specific\nPoCs in our prompts, models may still access historical vulnerability data through their pre-trained\nknowledge, potentially inflating performance metrics. Additionally, our evaluation employs binary\nsuccess metrics that only capture whether vulnerability reproduction succeeds or fails, providing a\nsingle-dimensional assessment that may miss important aspects of the reproduction process.\nExternal Validity. Our findings may have limited generalizability due to several factors. First,\nour CVE selection, while diverse across vulnerability types and web technologies, represents a\ncurated subset that may not fully reflect the complexity distribution of real-world security scenarios.\nSecond, the containerized evaluation environment, despite efforts toward realism, may still differ\nfrom production deployments in ways that affect agent performance. Finally, the rapid evolution of\nLLM capabilities means our findings represent a temporal snapshot that may not predict future\nagent performance across different models or architectures.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\n18\nBin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang\n8\nRelated Work\n8.1\nSoftware Engineering Benchmarks for LLM Agents\nSoftware engineering represents a critical application domain for evaluating LLM agents, with\nbenchmarks spanning different complexity levels and task types. SWE-bench [37] evaluates agents’\nabilities to solve real-world GitHub issues by generating patches for bug fixes, representing one\nof the most challenging repository-level tasks that require understanding complex codebases and\ncoordinating changes across multiple files. Traditional coding benchmarks like HumanEval fo-\ncus on isolated function generation from natural language descriptions, while MBPP evaluates\nbasic Python programming tasks. Additional benchmarks including BigCodeBench [92], Live-\nCodeBench [35], and EvalPlus [43] extend evaluation across multiple languages and enhanced test\ncoverage. However, these SE benchmarks exhibit significant limitations: many focus on narrow\ntask scopes rather than diverse professional tasks, traditional benchmarks often involve only local\nedits to individual functions rather than repository-wide reasoning, and they frequently ignore\nautonomous environment setup and end-to-end task solving capabilities essential for practical soft-\nware engineering. Recent advances like AgentBench [44] provide more comprehensive evaluation\nacross diverse environments, highlighting the evolution toward more realistic and challenging\nevaluation frameworks for software engineering agents.\n8.2\nCybersecurity Benchmarks for LLM Agents\nVarious specialized benchmarks have been developed to evaluate LLM agents in cybersecurity.\nInterCode-CTF [85] manually evaluated LLMs on Capture-the-Flag challenges requiring reverse\nengineering and exploitation skills, while NYU CTF Bench [64] expanded this with an automated\nframework for diverse offensive security tasks across cryptography, forensics and binary exploita-\ntion. The CyberSecEval series [10, 73] focuses on evaluating cybersecurity risks including prompt\ninjection, code interpreter abuse and automated social engineering. ARVO [50] provides repro-\nducible memory vulnerabilities with triggering inputs and developer patches. SEC-bench [38]\nevaluates PoC generation and vulnerability patching tasks using data from OSS-FUZZ project,\nSecVulEval [3] benchmarks statement-level vulnerability detection in C/C++ code. CyberGym [79]\nassesses vulnerability reproduction across large software projects. Most existing benchmarks exhibit\nlimitations in capturing real-world cybersecurity complexity, often focusing on isolated tasks or\nsynthetic environments. The most closely related work is CVE-bench [76], which remains limited\nin both scope and scale, capturing only a narrow slice of agent performance in web vulnerability\nreproduction scenarios.\n9\nConclusion\nOur evaluation of 20 LLM agents across 16 core dimensions on 3 representative CVEs identified three\ntop performers (OpenHands, SWE-agent, and CAI) for testing on 80 real-world CVEs spanning 6\nweb technologies and 7 vulnerability types. Results show agents handle simple library-based vulner-\nabilities well but consistently fail on complex service-based scenarios requiring multi-component\nenvironments and authentication. The gap between executing exploit code and triggering actual vul-\nnerabilities reveals fundamental limitations in environmental adaptation. These findings highlight\nthe need for significant advances before automated vulnerability reproduction becomes practical.\nCurrent agents lack the robustness required for real-world deployment, showing high sensitivity\nto input guidance and poor performance under incomplete information.\nData Availability\nThe artifact of this paper can be accessed via https://figshare.com/s/7e55eaaaca0b0146ee62.\n, Vol. 1, No. 1, Article . Publication date: October 2025.\n\nLLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?\n19\nReferences\n[1] 2025. Model Context Protocol Specification. Technical Report. 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BIN LIU, Harbin (LLM) agents have demonstrated remarkable capabilities in software engineering and cybersecurity tasks, including code generation, vulnerability discovery and automated testing. One critical but underexplored application is automated web vulnerability reproduction, which transforms vulnerability reports into working exploits. Although recent advances suggest promising potential, significant challenges remain in applying LLM agents to real-world web vulnerability reproduction scenarios. In this paper, we present the first comprehensive evaluation of state-of-the-art LLM agents for automated web vulnerability reproduction. We first systematically assess 20 agents from software engineering, cybersecurity, and general domains across 16 dimensions, including technical capabilities, environment adaptability, and user experience factors, on 3 representative web vulnerabilities. Based on the results, we select three topperforming agents (OpenHands, SWE-agent, and CAI) for in-depth evaluation on our constructed benchmark dataset of 80 real-world CVEs spanning 7 vulnerability types and 6 web technologies. Our results reveal that while LLM agents achieve reasonable success on simple library-based vulnerabilities, they consistently fail on complex service-based vulnerabilities requiring multi-component environments. Furthermore, complex environment configurations and authentication barriers create a critical gap where agents can execute exploit code but fail to trigger actual vulnerabilities. We observe that agents demonstrate high sensitivity to input guidance, with performance degrading by over 33.3% under incomplete authentication information. Our findings highlight the significant gap between current LLM agent capabilities and the demands of reliable automated vulnerability reproduction, emphasizing the need for advances in environmental adaptation and autonomous problem-solving capabilities. CCS Concepts: • Software and its engineering; Additional Key Words and Phrases: Web Vulnerability, LLM Agents, Vulnerability Management ACM Reference Format: Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang. 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?. 1, 1 (October 2025), 22 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn 1 Introduction Recently, Large language model (LLM) agents have achieved remarkable success across software engineering and cybersecurity domains. In software engineering, these agents have demonstrated strong capabilities in code generation [15, 19, 31], bug fixing [36, 83, 90], issue resolution [37, 86] Authors' Contact Information: Bin Liu, Harbin ; Yanjie Zhao, Huazhong ; Guoai Xu, Harbin ; Haoyu Wang, Huazhong . Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from . Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM XXXX-XXXX/2025/10-ART https://doi.org/10.1145/nnnnnnn.nnnnnnn , Vol. 1, No. 1, Article . Publication date: October 2025. 16 Oct 2025 2 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang and code review [24, 71]. In cybersecurity, they have shown impressive performance in vulnerability detection [30, 41, 68], fuzzing [18, 47, 74], exploit generation [17, 80], malware analysis [11, 49] and penetration testing [28, 40]. These successes demonstrate their ability to understand complex technical contexts, reason about software behavior, and automate sophisticated workflows. Web applications have become the backbone of modern digital infrastructure, representing the dominant form of deployed applications in enterprise environments [69]. This widespread adoption has made them prime targets for cyberattacks, with web vulnerabilities consistently dominating security incident reports [72]. The OWASP Top 10 identifies critical issues such as SQL injection, Cross-Site Scripting (XSS), and Cross-Site Request Forgery (CSRF) as the majority of reported vulnerabilities [55], while the National Vulnerability Database confirms that web application vulnerabilities represent the fastest-growing category of security threats [52]. However, effectively managing these web vulnerabilities relies heavily on successful reproduction, which involves transforming vulnerability reports into working Proof-of-Concept (PoC) exploits that validate threats and enable effective patching. This process presents unique challenges that are different from other security tasks where LLM agents have succeeded. Manual reproduction requires expertise in environment configuration, dependency resolution, authentication mechanisms, and exploit crafting, often taking days or weeks per vulnerability [20, 61]. Automated reproduction faces additional complexities: software version incompatibilities, incomplete PoC code, complex deployment, and network setups. These factors lead to low success rates in existing automated reproduction methods, highlighting the substantial technical barriers. While LLM agents have remarkable capabilities in related security and software engineering tasks, their effectiveness for comprehensive web vulnerability reproduction remains underexplored. Existing cybersecurity benchmarks for LLM agents suffer from significant limitations that fail to capture real-world complexity. Many evaluations rely on simplified Capture-the-Flag (CTF) scenarios [10, 64, 73, 85, 88] that focus on individual challenges rather than end-to-end reproduction processes. Additionally, current benchmarks often evaluate agents on small, isolated code snippets or fragments, limiting their scope to specific programming languages such as C/C++ [7, 38, 50, 79], Java [42] or JavaScript [14], rather than the diverse components of production web applications. However, end-to-end vulnerability reproduction requires agents to handle complex multi-step workflows involving environment setup, dependency management, and real-world deployment uncertainties. These capabilities extend far beyond current simplified evaluation scenarios, creating a significant gap in agent ability for practical vulnerability reproduction applications. To address this gap, we present the first comprehensive empirical study examining LLM agents' ability for automated web vulnerability reproduction. We systematically evaluate 20 representative agents across 16 core dimensions on 3 representative CVEs, then conduct in-depth assessment of the top 3 agents using our constructed benchmark dataset of 80 real-world CVEs. We assess current agent capabilities and identify fundamental limitations that prevent reliable automated vulnerability reproduction. In summary, our main contributions include: • We construct a comprehensive benchmark dataset comprising 80 real-world CVEs covering 7 vulnerability types and 6 web technologies with complete reproduction environments. • We conduct the first systematic evaluation of 20 state-of-the-art LLM agents for automated web vulnerability reproduction across 16 core dimensions on 3 representative CVEs, establishing a rigorous evaluation framework. From this evaluation, we select the three top-performing agents (OpenHands, SWE-agent, and CAI) for in-depth analysis. • We provide critical empirical insights revealing the significant gap between current LLM agent capabilities and practical requirements, identifying key failure modes and documenting over 33.3% performance degradation under incomplete authentication information. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 3 2 Background 2.1 LLM Agents LLMs exhibit impressive performance across multiple domains, including natural language comprehension, code synthesis, and reasoning processes. Based on these foundational abilities, researchers have developed LLM-based agent systems [39, 65, 75] that leverage LLMs as core reasoning engines for autonomous decision-making, while utilizing various tools and operations to interact with external environments in order to accomplish sophisticated tasks. 2.1.1 Core Mechanisms. LLM agents combine LLM reasoning with action execution capabilities through several key mechanisms. ReAct [87] introduces iterative reasoning-action cycles, where models generate reasoning traces before executing actions, with results integrated into context for subsequent cycles. Toolformer [63] demonstrates autonomous tool usage, enhancing factual accuracy and computational abilities. For complex tasks, simple reactive loops prove insufficient, requiring explicit planning capabilities. Early systems like BabyAGI [51] and Auto-GPT [26] introduced dynamic task lists where agents execute, reorderand generate tasks continuously. Advanced frameworks such as AgentVerse [16] and ChatDev [62] adopt multi-agent architectures where controller agents decompose goals into subtasks distributed to task-specific agents, enabling coordination across different domains. Reflexion [66] addresses failure handling through reflection mechanisms that analyze past mistakes and store self-reflections in memory for improving future planning decisions. This self-correction capability enables iterative problem-solving and strategy refinement over time. 2.1.2 Representative Agents and Frameworks. LLM agents have emerged across diverse application domains. Software engineering agents assist with code generation, debugging, and repository-level analysis. Cybersecurity agents focus on threat detection, vulnerability analysis, and automated penetration testing. General-purpose agents aim for broad applicability, emphasizing flexible tool use and adaptive reasoning across various tasks. Additionally, agent orchestration frameworks like LangChain [13] and AutoGen [81] provide infrastructure for building and coordinating multi-agent systems, enabling developers to construct domain-specific agents with standardized tooling and communication protocols. A summary of representative agents and frameworks is shown in Table 1. Table 1. Representative LLM agents and orchestration frameworks across different domains. Domain Agent Year Domain Agent Year Software Engineering AGENTLESS [82] 2024 Cybersecurity PentestGPT [17] 2023 OpenHands [78] 2024 CAI [48] 2024 SWE-agent [84] 2024 PentestAgent [59] 2024 AutoCodeRover [89] 2024 Nebula [53] 2024 Aider [23] 2023 HackingBuddyGPT [27] 2023 RepoAgent [45] 2024 PentAGI [57] 2024 ChatDev [62] 2023 EnIGMA [2] 2024 MetaGPT [29] 2024 AI-OPS [4] 2024 General-Purpose Agents AutoGPT [26] 2023 Orchestration Frameworks LangChain [13] 2022 BabyAGI [51] 2023 AutoGen [81] 2023 AgentVerse [16] 2023 CrewAI [32] 2023 AIlice [5] 2024 LangGraph [33] 2024 2.2 Web Vulnerability Reproduction Web vulnerability reproduction involves constructing operational PoC exploits from natural language vulnerability reports to validate security flaws and guide remediation efforts. , Vol. 1, No. 1, Article . Publication date: October 2025. 4 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang The common approach relies on manual analysis by security researchers using tools like Burp Suite [60] or ZAP [56] to intercept, analyze and craft HTTP/S requests. While effective for complex logic-based flaws, this method suffers from inefficiency and high labor costs that limit scalability. Semi-automated systems target common vulnerability classes like XSS or SQL Injection by using NLP to extract key entities from reports and inserting them into predefined attack templates [9]. Advanced works like Wang et al. [77] extract contextual information for targeted web fuzzing. Commercial tools like Nessus [70] and Acunetix [34] represent this approach. While faster for simple cases, these methods lack robustness for vulnerabilities requiring complex, multi-step interactions. Program analysis based automated exploit generation (AEG) [8] achieved significant success in binary applications through symbolic execution. Frameworks like Mayhem [12] and angr [67] exemplify this approach. However, their application to web vulnerabilities is limited due to the immense state space of web applications and inability to reason about high-level logic flaws. LLM-based methods represent the newest paradigm, framing reproduction as an agentic problem where LLMs act as reasoning engines. LLMs have been explored for exploiting vulnerabilities in web applications based on known vulnerability descriptions with one-day setting [21, 22] and investigating agent teams with hierarchical planning for zero-day scenarios [91]. Recent work [80] showed AI systems can generate working exploits for published CVEs in 10-15 minutes, while systems like PentestGPT [17] and PentestAgent [40] demonstrate autonomous penetration testing capabilities. This methodology combines high-level contextual reasoning with machine execution speed and scalability. All the above methods are shown in Table 2. Table 2. Web vulnerability reproduction methods and tools, including manual, semi-automated, program analysis-based, and LLM-based methods. Tool Category Year Tool Category Year Burp Suite [60] Manual 2003 AEG [8] Program analysis based 2011 OWASP ZAP [56] 2010 Mayhem [12] 2012 Nessus [70] Semi-automated 2024 angr [67] 2016 Acunetix [34] 2024 AutoExploit [46] 2018 Website Vulnerability Scanner [58] 2024 PentestGPT [17] LLM-based 2023 Wang et al. [77] 2023 PentestAgent [40] 2024 Aydin et al. [9] 2014 2.3 Motivation Automated web vulnerability reproduction remains a critical bottleneck in security operations. Manual analysis lacks scalability, semi-automated tools are brittle with poor comprehension, and formal methods like AEG cannot handle web application complexity. This validation gap prevents security teams from efficiently confirming and prioritizing threats. LLM-based agents offer a promising solution through semantic comprehension, logical reasoning, and tool invocation capabilities that address existing limitations. Recent advances have produced general-purpose agents (as shown in Table 1) claiming broad problem-solving capabilities, and specialized vulnerability analysis agents (as illustrated in Table 2) targeting security workflows specifically. However, despite theoretical promise and agent proliferation, the practical effectiveness of LLM agents in web vulnerability reproduction lacks empirical validation. This creates a critical gap between claimed potential and demonstrated capability in this demanding security domain. We conduct the first systematic empirical study evaluating LLM agents' ability to reproduce web vulnerabilities from natural language descriptions. Our comprehensive framework , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 5 assesses success rates, reasoning processes, and failure modes, establishing empirical foundations for understanding current capabilities and limitations in automated vulnerability validation. 3 Empirical Study Design We design a comprehensive empirical study to systematically evaluate LLM agents for web vulnerability reproduction. Our study follows the workflow illustrated in Figure 1. Our study evaluates agents according to the following four criteria: C1: Effectiveness. We compare the effectiveness of different agents on web vulnerability reproduction by measuring success rates across individual stages (environment setup, vulnerability localization, and PoC generation with verification) and full end-to-end reproduction. C2: Compatibility. We evaluate the adaptability of different agents across various vulnerability types and web technologies. C3: Efficiency. We compare the efficiency of different agents by measuring execution time, USD cost, and token consumption, which are critical factors for the practical deployment of LLM agents. C4: Robustness. We evaluate the robustness of different agents by analyzing their performance under varying conditions, including model capabilities and input quality. § 3.1 Data Preparation Vulnerability Database & Code Repository § 3.2 Agent Selection § 3.3 Task Specification 80 Web Vulnerability-PoC Datasets 7 Vulnerability Types API Fetch Vulnerability Report Pre-patch Codebase Manual Construction /source docker -compose.yml Dockerfile PoC exploits & Oracle Metadata Scripts GPT-4.1 Claude-Sonnet-4 Gemini-2.5-pro Foundation LLMs Integration § 4 & § 5 Experimental Setup & Results Target Web Application Container Library Application Web Application Other Configurations Database Input Env setup task Localization task Generation & Attack PoC gen task Reproduction task Output Example { \"stage\": \"e2e\", \"success\": , \"files\": [{\"path\": \"./.bench/e2e/Dockerfile\",...}, ... {\"path\": \"./.bench/e2e/steps.md\",...}], \"script_files\": { \"./.bench/e2e/run.sh\": \"commands\"}, \"run_commands\": [ \"bash ./.bench/e2e/run.sh\"], } C1: Effectiveness C2: Compatibility C3: Efficiency C4: Robustness Evaluation Criteria Agents 3 agents selected from 20 candidates for further analysis Fig. 1. Systematic workflow for evaluating LLM agents in web vulnerability reproduction tasks. 3.1 Data Preparation To effectively evaluate the performance of LLM agents in vulnerability reproduction, we have curated a benchmark dataset comprising real-world web application vulnerabilities. Our dataset construction follows a structured multi-round filtering strategy to ensure reliability and reproducibility across diverse security scenarios in the real world. Data Collection Process. We employ a systematic multi-round filtering strategy for data collection. Initially, we search the OSV database's Git ecosystem for service-based web vulnerability keywords, including SSRF, CSRF, SQL Injection, Prototype Pollution, Path Traversal, Remote Code Execution, and Cross-Site Scripting, limiting our scope to vulnerabilities disclosed after 2024. This initial search yields 744 candidate entries. To ensure manageable experimental environments, we further filter out projects with repository sizes exceeding 30MB, resulting in 224 remaining , Vol. 1, No. 1, Article . Publication date: October 2025. 6 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang candidates. Finally, we apply three stringent criteria for final selection: (1) only vulnerabilities affecting web-based applications are included, (2) vulnerabilities must be available in accessible Git repositories where the specific vulnerable commit can be checked out with complete source code access, and (3) vulnerability reports must be disclosed after 2024 to reflect current threat landscapes. Through this systematic filtering process, we construct a high-quality dataset comprising 80 vulnerabilities. Vulnerability Type Selection. Our curation process focuses on 7 critical web vulnerability types strategically selected to comprehensively evaluate agent capabilities across different vulnerability paradigms. We include six mainstream service-based vulnerability types: Cross-Site Request Forgery (CSRF), Path Traversal, Remote Code Execution (RCE), SQL Injection (SQLI), Server-Side Request Forgery (SSRF) and Cross-Site Scripting (XSS). These represent the most prevalent web application security flaws that occur at the service level. Additionally, to provide a more holistic assessment of agent performance, we incorporate Prototype Pollution as a representative librarybased vulnerability type that manifests through dependencies and third-party packages rather than direct application code. This approach ensures our evaluation captures both traditional web service vulnerabilities and modern JavaScript ecosystem threats. Environment Construction. For each selected vulnerability, we execute a comprehensive data preparation pipeline. After fetching the vulnerability reports via APIs, we identify the corresponding pre-patch codebase from the linked Git repositories and check out the exact vulnerable commit. We utilize Git's diff capabilities to identify the precise code changes between vulnerable and patched versions, allowing us to pinpoint the exact vulnerable code sections, including specific files, functions and line numbers. We then construct fully containerized environments using Docker, manually developing Dockerfile and docker-compose.yml configurations for each vulnerability to ensure reproducible deployment across different systems. Each environment is thoroughly tested to verify successful vulnerability reproduction before exploit development. PoC Development. For PoC exploit construction, we develop PoC exploits based on existing vulnerability reports. When existing PoCs are available in the reports, we adapt and refine them for our containers; otherwise, we manually create new exploits from scratch. Our PoC development follows vulnerability-specific methodologies: CSRF vulnerabilities are implemented using HTMLbased attack vectors, Prototype Pollution exploits are crafted in JavaScript to demonstrate object manipulation, while other vulnerability categories utilize Python-based HTTP request construction for systematic exploitation. Each PoC is accompanied by automated verification oracles that confirm successful exploitation, enabling reliable automated assessment. Our construction process generates dataset entries containing vulnerable source code directories, Docker configuration files, environment setup scripts, executable PoC exploit code with verification oracles, and structured metadata including CVE identifier, vulnerability description, affected file paths, vulnerable function names, specific line numbers, corresponding GitHub repository URLs, vulnerable commit hashes and temporal information. Our final dataset comprises 80 vulnerabilities spanning 6 popular web technologies including Python, PHP, Java, JavaScript, TypeScript, and Go, covering 7 vulnerability types. 3.2 Agent Selection Recall that in Table 1, we listed 20 representative agents. These agents share common characteristics: they are open-source projects available on GitHub with over 1,000 stars for software engineering/general-purpose agents and over 100 stars for cybersecurity agents, released or updated after 2023, and demonstrating active maintenance. All agents demonstrate basic operational capabilities for automated task execution. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 7 Initial Capability Assessment. We first conduct a preliminary capability assessment of these 20 representative agents using 16 core dimensions and 3 representative CVEs for testing. The 16 evaluation dimensions include source code access, tool integration, script execution, file operations, web requests, security domain focus, custom structured output, custom task support, runtime stability, error handling, session management, efficiency metrics, log persistence, latest LLM support, user interface, and installation convenience. We select 3 representative CVEs for testing: CVE2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and CVE-2024-41112 (RCE), which represent different vulnerability types and complexity levels. Agent Selection Results. We conduct a comprehensive evaluation of these 20 agents spanning multiple architectures and application domains, with detailed results presented in Table 3 (§ 5.1). Through this evaluation round, we identify the 3 best-performing agents. Subsequently, we conduct an in-depth empirical evaluation of these 3 selected agents using the complete 80-vulnerability dataset described in § 3.1. Therefore, our further empirical evaluation focuses on these three promising agents that demonstrate adequate capabilities and performance. Outputs (all saved under ./.bench/e2e/): - Intermediate artifacts - steps.md (the reproduction steps and key commands). - result.json (consolidated end-to-end result). User Prompt You are participating in web vulnerability reproduction in a safe container. Rules: - Never invent placeholder hostnames. Derive service_url from docker-compose port mapping (localhost) or explicit run.sh output. - You can edit/create files under the workspace and execute shell commands. - Do NOT modify any source code files; only write under {codebase_dir} Stage: End-to-End Reproduction. Steps: Starting from the CVE folder and the codebase, autonomously: 1) Build/run the environment from scratch based on c 2) Localize the vulnerable code region (files/functions/lines) 3) Generate a PoC and an in-code verification oracle under ./.bench/e2e/ to assess exploit success. 4) Execute the PoC, collect evidence/logs/state, and decide success/failure. 5) Produce a concise step-by-step reproduction document (./.bench/e2e/steps.md). Outputs (all saved under ./.bench/e2e/): - Intermediate artifacts - steps.md (the reproduction steps and key commands). - result.json (consolidated end-to-end result). Response.png Agent { \"stage\" : \"e2e\", \"success\": , \"files\": [ {\"path\": \"./.bench/e2e/Dockerfile\", \"status\": \"created|updated|copied\"}, ], \"run_commands\": [ \"bash ./.bench/e2e/run.sh\" ], \"artifacts\": { \"service_url\": \"http://localhost: \", \"exec_success\": , \"signals\": {\"status\": 200, \"log_regex\": \"...\"}, \"observed\": {\"status\": 200, \"log\": \"...\"} }, \"steps\": [ {\"name\": \"env_setup\", \"success\": }, {\"name\": \"localization\", \"success\": }, {\"name\": \"poc_and_verify\", \"success\": , \"notes\": \"...\"} ], ... } JSON Response Vulnerability Information Reproduction Requirements Output Schema Fig. 2. Prompt template and example agent response. 3.3 Task Specification We design four evaluation tasks of increasing complexity to systematically assess agent capabilities across the vulnerability reproduction pipeline. The first three tasks are independent, single-stage evaluations where each task receives expert-curated ground truth from prerequisite stages as input, ensuring that only the current stage's capabilities are measured. The final task represents a comprehensive end-to-end evaluation requiring agents to autonomously navigate the complete vulnerability reproduction workflow. Environment Setup. This foundational task evaluates an agent's ability to establish a functional testing environment from source code alone. Given only the vulnerable application's codebase, agents must generate the necessary containerization files (Dockerfile, docker-compose.yml) and deployment scripts (run.sh) to create a running instance container of the application. , Vol. 1, No. 1, Article . Publication date: October 2025. 8 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Vulnerability Localization. With access to both the running environment and source code, agents must perform code analysis to pinpoint the vulnerability's location. Given the CVE description, agents analyze the codebase to produce a ranked list of suspicious files, functions, and specific line numbers that correspond to the security flaw. Proof-of-Concept (PoC) Exploits Generation. This task tests an agent's exploit development capabilities. Provided with the vulnerability's exact location (file and line number), agents must craft a working Proof-of-Concept exploits that successfully triggers the exploit and includes a verification oracle to programmatically confirm successful exploitation. End-to-End Reproduction. As the most challenging evaluation, this task provides agents only with the source code and original CVE description. Agents must autonomously perform all previous stages-from environment setup through PoC execution-and produce a complete, documented workflow of their reproduction process. To ensure fair evaluation across different agents, we establish a standardized input-output interface that provides consistent task specifications and response formats. Each agent receives identical prompts and must produce outputs in the same structured format, enabling objective comparison of capabilities across all web vulnerability reproduction tasks. As illustrated in Figure 2, the prompt contains structured vulnerability information including CVE category, detailed description, codebase directory, and reproduction steps. The prompt also specifies operational constraints such as workspace restrictions and file access limitations. Agents must respond with structured JSON outputs that include stage indicators, success flags, file paths, execution commands, and artifact metadata. This standardized format enables automated assessment of task completion. It also allows verification that agents produce the required code artifacts including Dockerfiles, execution scripts, PoC exploits, and verification oracles. 4 Experimental Setup 4.1 Studied Dataset and Agents As introduced in § 3.1, we evaluate agent performance on a curated benchmark of 80 real-world vulnerabilities across diverse categories and complexity levels. Following the setup described in § 3.2, we assess the performance of 20 popular LLM agents across software engineering, cybersecurity, and general-purpose, each integrated with three foundation LLMs: GPT-4.1 [54], Claude-Sonnet-4 [6], and Gemini-2.5-Pro [25]. 4.2 Metrics To comprehensively evaluate agent performance across the four criteria (C1-C4), we measure success rates at different granularities and additional efficiency metrics: 4.2.1 Success Rate Metrics. We measure the percentage of successful completions across different stages of the vulnerability reproduction pipeline, with all results verified through manual inspection: • Environment Setup Success Rate: The percentage of vulnerabilities for which an agent successfully generates and deploys a functional vulnerable environment, verified through manual testing of application accessibility and configuration correctness. • Vulnerability Localization Accuracy (Top-K): The percentage of cases where the groundtruth vulnerability location appears in the agent's top-K predictions. We evaluate this metric at three granularity levels: - File-level Top-K: Ground-truth vulnerable file present in top-K file predictions. - Function-level Top-K: Ground-truth vulnerable function present in top-K function predictions. - Line-level Top-K: Ground-truth vulnerable line present in top-K line predictions. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 9 • PoC Generation Success Rate: We decompose this into two complementary sub-metrics: - Execution Success Rate: The percentage of generated PoCs that execute without runtime errors, verified through both automated testing and manual execution. - Vulnerability Trigger Success Rate: The percentage of executable PoCs that successfully trigger the vulnerability, confirmed through manual verification of exploitation effects. • End-to-End Success Rate: The percentage of vulnerabilities for which an agent successfully completes the entire reproduction workflow from environment setup to verified exploitation, with manual confirmation of successful vulnerability triggering. 4.2.2 Efficiency Metrics. To assess operational costs and practical deployment considerations, we also measure: • Execution Time: Total wall-clock time required for task completion. • Token Consumption: Total API tokens consumed during execution. • Financial Cost: Estimated USD cost based on publicly available API pricing. 4.3 Implementation Details All experiments are conducted on a standardized Linux machine with Ubuntu 20.04 LTS, equipped with an Intel Xeon Platinum 8358P CPU (2.60GHz, 128 cores) and 2TB memory. We use Docker Engine 26.1.0 to containerize environments, ensuring isolation and reproducibility. Agent frameworks are executed using Python 3.11 with dependencies installed from their official repositories. To mitigate inherent randomness in LLM outputs, each experiment undergoes three independent executions, with final metrics calculated as averages across all. A strict 60-minute timeout is enforced per execution, where any run exceeding this limit is automatically terminated and classified as a failure. All generated artifacts, execution logs, and API interactions are systematically captured and stored for comprehensive post-hoc analysis. To maintain experimental feasibility, we impose budget constraints of 5 for complete end-to-end reproduction workflows. To ensure fair comparison across all agents, browser-based interactions are strictly prohibited, restricting all agents to command-line interfaces only. 5 Experimental Results 5.1 C1: Effectiveness Motivation. Manual reproduction of security vulnerabilities represents a well-documented bottleneck in software development, consuming substantial expert time and delaying remediation efforts. Previous approaches relied heavily on manual processes where security researchers must interpret vulnerability reports and develop proof-of-concept code. However, recent advances in LLM agents present opportunities to automate these complex tasks. Therefore, this criterion evaluates the effectiveness of current agents in performing core vulnerability reproduction tasks. Approach. As described in § 4, we evaluate agent effectiveness across the four-stage vulnerability reproduction pipeline using the success rate metrics defined in § 4.2. Each agent-model combination is assessed on environment setup, vulnerability localization (measured by file-level, function-level and line-level Top-3 accuracy), PoC generation with execution verification, vulnerability trigger verification, and end-to-end reproduction success, which is defined in § 3.3. Results. Table 3 shows the capability assessment results of 20 agents across 16 dimensions using three representative CVEs: CVE-2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and CVE-2024-41112 (RCE). The analysis reveals severe capability gaps across the agent landscape. Software engineering agents show poor performance with only 2 out of 8 agents meeting basic requirements. Most fail in critical areas like tool integaration and session management, with agents , Vol. 1, No. 1, Article . Publication date: October 2025. 10 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Table 3. Comparison of 20 agents for web vulnerability reproduction with CVE-2024-7065, CVE-2024-7458 and CVE-2024-41112. Agent SCA TI SE FO WR SDF CSO CTS RS EH SM EM LP LLS UI EI Software Engineering Agents AGENTLESS ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ G # ✗ ✗ ✗ ✗ G # C ✓ OpenHands ✓ ✓ ✓ ✓ ✓ ✗ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ C/W ✓ SWE-agent ✓ G # ✓ ✓ ✓ ✗ G # ✓ ✓ ✓ ✓ ✓ ✓ ✓ C ✓ AutoCodeRover ✓ ✗ G # ✓ ✗ ✗ ✗ G # ✓ ✗ ✗ ✗ ✓ G # C ✓ Aider G # ✓ G # ✓ ✗ ✗ G # ✓ ✓ ✓ ✗ ✓ ✓ ✓ C ✓ RepoAgent ✓ ✗ G # G # ✗ ✗ ✗ G # ✓ ✗ ✗ ✗ ✗ G # C ✓ ChatDev ✗ ✗ ✗ G # ✗ ✗ G # G # G # ✗ ✗ ✗ ✗ ✗ C/W G # MetaGPT G # ✓ G # G # ✗ ✗ G # G # ✓ ✓ ✗ ✗ ✓ G # C ✓ Cybersecurity Agents PentestGPT G # ✓ ✓ ✓ ✓ ✓ G # ✗ ✓ ✓ ✓ ✗ ✓ ✓ C ✓ CAI ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ C ✓ PentestAgent ✗ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ G # C ✓ Nebula ✓ ✓ ✓ G # ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ C G # HackingBuddyGPT ✗ ✓ ✓ ✓ ✓ ✓ G # G # ✓ ✓ ✓ ✗ ✓ ✓ C G # PentAGI ✗ G # ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✓ ✗ ✓ ✓ C ✗ EnIGMA ✗ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✓ ✓ ✓ ✗ C G # AI-OPS ✗ ✗ ✗ ✗ G # G # ✗ ✗ ✓ ✗ ✓ ✗ ✗ ✗ C G # General-purpose Agents AutoGPT ✓ G # G # ✓ ✓ ✗ G # G # ✗ ✗ ✓ ✗ ✓ ✓ C/W ✗ BabyAGI ✓ G # G # G # ✗ ✗ ✗ ✗ G # G # ✓ ✗ ✗ G # C ✓ AgentVerse ✓ ✗ ✗ ✗ ✗ ✗ G # G # G # ✗ ✓ ✗ ✗ ✗ C/W G # AIlice ✓ ✓ ✓ G # ✗ ✗ ✓ ✓ G # ✗ ✓ ✗ ✓ ✗ C/W ✗ Note: Symbols represent Good (✓), Average (G #), and Poor (✗) qualitative assessment. Abbreviations: SCA: Source Code Access RS: Runtime Stability TI: Tool Integration EH: Error Handling SE: Script Execution SM: Session Management FO: File Operation EM: Efficiency Metrics WR: Web Request LP: Log Persistence SDF: Security Domain Focus LLS: Latest LLM Support CSO: Custom Structured Output UI: User Interface (C: CLI, C/W: CLI/WEB) CTS: Custom Task Support EI: Ease of Install like AGENTLESS [82] and ChatDev [62] lacking over 10 essential capabilities. Cybersecurity agents demonstrate domain knowledge but face limitations in technical implementation capabilities, most lack essential technical features such as structured output and custom task support. Except for CAI which achieves full capability, all other cybersecurity agents fail to support custom task types and structured output formats, limiting their adaptability to diverse web vulnerability reproduction scenarios. General-purpose agents offer broad applicability and flexible task handling but struggle with specialized technical requirements. For example, AutoGPT [26] demonstrates reasonable performance in 8-9 dimensions but lacks security domain focus and has installation challenges. Only three agents demonstrate comprehensive capabilities: SWE-agent [84], OpenHands [78], and CAI [48]. These agents form the foundation of our further empirical evaluation: - SWE-agent employs a carefully designed Agent-Computer Interface (ACI) that enables autonomous interaction with software repositories through specialized commands for code navigation, editing, and testing. The framework demonstrates exceptional performance in repository-level vulnerability discovery through its integrated file viewer, search capabilities, and context management system that maintains concise yet informative interaction histories. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 11 - OpenHands provides a comprehensive platform for AI software development agents, featuring an event-stream architecture that facilitates complex multi-turn interactions. The framework excels in code generation, debugging, and repository manipulation through its sandboxed Docker environment and extensive tool integration, including IPython execution, web browsing capabilities, and multi-agent delegation patterns. - CAI represents a specialized cybersecurity AI framework designed for autonomous penetration testing and vulnerability assessment. Built around six fundamental pillars (Agents, Tools, Handoffs, Patterns, Turns, and Human-In-The-Loop), CAI demonstrates superior performance in offensive security operations, achieving human-competitive results across diverse Capture-The-Flag challenges and real-world bug bounty scenarios. To ensure experimental consistency and fair comparison across all subsequent evaluations, we configure each framework with identical LLM foundations: GPT-4.1[54], Claude-Sonnet4[6] and Gemini-2.5-Pro[25] for the following experiments. Table 4 presents the comprehensive effectiveness evaluation results across these three selected agent-model combinations. The results reveal significant challenges in automated vulnerability reproduction, with substantial performance variations across different pipeline stages and agentmodel configurations. Table 4. Effectiveness results: success rates (%) across vulnerability reproduction pipeline. Agent Model Env Setup Vulnerability Localization PoC Generation End-to-End Setup File Level Func Level Line Level Execution Trigger Success@1 Success@3 OpenHands Claude-Sonnet-4 28.8 46.3 36.3 23.8 70.0 21.3 10.0 22.5 Gemini-2.5-Pro 26.3 40.0 27.5 17.5 52.5 10.0 6.3 13.8 GPT-4.1 32.5 53.8 43.8 31.3 56.3 16.3 13.8 20.0 SWE-agent Claude-Sonnet-4 22.5 50.0 40.0 27.5 58.8 11.3 8.8 15.0 Gemini-2.5-Pro 20.0 37.5 30.0 21.3 56.3 8.8 5.0 11.3 GPT-4.1 25.0 46.3 36.3 25.0 66.3 13.8 11.3 18.8 CAI Claude-Sonnet-4 35.0 56.3 46.3 32.5 56.3 15.0 7.5 12.5 Gemini-2.5-Pro 38.8 50.0 38.8 26.3 33.8 17.5 3.8 10.0 GPT-4.1 36.3 58.8 48.8 35.0 25.0 20.0 10.0 16.3 Examining the overall end-to-end performance, OpenHands demonstrates the strongest capabilities, particularly when paired with Claude-Sonnet-4, achieving the highest Success@3 rate of 22.5%. OpenHands maintains consistent superiority across different models, with Success@3 rates of 22.5% (Claude-Sonnet-4), 20.0% (GPT-4.1), and 13.8% (Gemini-2.5-Pro). In comparison, SWE-agent achieves maximum Success@3 rates of 18.8% (GPT-4.1), while CAI peaks at 16.3% (GPT-4.1). The Success@1 results show similar patterns, with OpenHands + GPT-4.1 leading at 13.8%, followed by OpenHands + Claude-Sonnet-4 at 10.0%. Notably, even the best-performing combinations fail to achieve successful vulnerability reproduction in more than three-quarters of cases, indicating substantial room for improvement in current agent capabilities. Environment setup performance varies significantly across agents, with CAI demonstrating the most consistent results, achieving rates above 35% across all models and peaking at 38.8% with Gemini-2.5-Pro. OpenHands shows moderate performance ranging from 26.3% to 32.5%, while SWE-agent exhibits the lowest and most variable setup success rates, spanning 20.0% to 25.0%. Vulnerability localization reveals a clear performance hierarchy and degradation pattern. CAI paired with GPT-4.1 leads in localization precision across all granularities: 58.8% file-level, 48.8% function-level, and 35.0% line-level accuracy. However, the degradation from file-level to line-level represents a consistent 35-40% relative decrease across all combinations. GPT-4.1 consistently outperforms other models in localization tasks, while Gemini-2.5-Pro shows the weakest localization capabilities across all agents. , Vol. 1, No. 1, Article . Publication date: October 2025. 12 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang The PoC generation stage reveals OpenHands' key strength and a critical bottleneck. OpenHands + Claude-Sonnet-4 achieves the highest PoC execution rate at 70.0%, significantly outperforming other combinations. However, a substantial execution-to-trigger gap exists across all agents: while execution rates range from 25.0% to 70.0%, trigger rates remain uniformly low at 8.8% to 21.3%. Remarkably, OpenHands + Claude-Sonnet-4 also leads in trigger success at 21.3%, demonstrating superior PoC quality despite the overall low trigger rates. Model-specific patterns emerge clearly. GPT-4.1 excels in localization precision but shows moderate end-to-end performance. Claude-Sonnet-4 demonstrates exceptional synergy with OpenHands, particularly in PoC generation and end-to-end orchestration. Gemini-2.5-Pro consistently underperforms across most metrics, especially in PoC-related tasks. The results highlight OpenHands' superior workflow orchestration capabilities. While CAI achieves the highest individual stage performance in environment setup and localization, OpenHands translates moderate individual performance into the strongest end-to-end success rates, suggesting more effective inter-stage coordination and error recovery mechanisms. We further analyze the underlying causes of these performance differences by examining the tool utilization patterns across agents. Table 5 reveals that while all agents share basic execution capabilities, OpenHands possesses unique workflow control tools that fundamentally enhance its vulnerability reproduction effectiveness. The think tool enables structured reasoning processes that strengthen agent memory and decision-making throughout the multi-stage pipeline, while the finish tool provides systematic task summarization and completion mechanisms. These capabilities explain OpenHands' superior ability to maintain context across pipeline stages and achieve higher end-to-end success rates despite comparable basic tool availability. Unexpectedly, CAI's extensive domain-specific security tool suite remains largely unutilized in our task specifications, revealing a critical gap between tool availability and practical application. While CAI provides numerous specialized cybersecurity tools (such as exploitation, privilege scalation [48]) theoretically suitable for vulnerability reproduction, the agent fails to effectively leverage these capabilities within the structured task framework, resulting in lower end-to-end performance despite superior individual stage metrics in environment setup and localization. Additionally, OpenHands demonstrates superior compliance with output formatting requirements by correctly generating artifacts and JSON outputs as specified in task prompts, indicating more robust instruction following capabilities. This analysis reveals that tool ecosystem effectiveness of agent depends not merely on capability breadth but on workflow integration, reasoning enhancement, and systematic task completion mechanisms that enable consistent execution across complex multi-stage security tasks. Table 5. The comparison of tool utilization across agent frameworks during task execution. Tool Type Tool Capability CAI OpenHands SWE-agent Basic Execution execute_bash ✓ ✓ ✓ str_replace_editor ✓ ✓ ✓ File Operations ✓ ✓ ✓ Workflow Control think (Reasoning Enhancement) × ✓ × finish (Task Summary) × ✓ × Domain-Specific Security/Vulnerability Tools 9 × × Used in Tasks 0 N/A N/A Task Execution Clear Task Termination ✓ ✓ ✓ Structured Reasoning × ✓ × Correct Artifacts Output × ✓ × Proper JSON Format ✓ ✓ × , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 13 Findings for C1: Current LLM agents achieve limited web vulnerability reproduction effectiveness with end-to-end success rates below 25%. OpenHands demonstrates superior performance through structured reasoning and systematic completion mechanisms, while tool availability alone does not ensure effectiveness. The primary bottleneck remains the exploitation gap where agents can execute PoCs but struggle to trigger actual vulnerabilities. 5.2 C2: Compatibility Motivation. Real-world vulnerability reproduction environments exhibit significant diversity across vulnerability types and web technologies. Different vulnerability categories require distinct understanding patterns, exploitation techniques, and verification approaches. Similarly, vulnerabilities manifest differently across web technologies due to language-specific features, frameworks, and coding patterns. A practical vulnerability reproduction agent must demonstrate consistent performance across this diversity rather than excelling only in specific domains. Understanding compatibility helps assess whether current agents can serve as general tools or require specialized adaptations. Approach. As shown in § 3.1, our dataset covers diverse vulnerability types and web technologies. We conduct categorical evaluation across these dimensions to assess agent compatibility. For vulnerability type compatibility, we analyze reproduction success rates across the seven vulnerability categories in our dataset. For programming language compatibility, we evaluate performance across the six primary languages. To ensure fairness and reproducibility, we use the top-performing combination from § 5.1, specifically OpenHands with Claude-Sonnet-4. Results. Table 6 presents stage-wise success rates across different vulnerability types. The results reveal substantial performance variations, with end-to-end success rates (Success@3) ranging from 0% to 63%. Library-based Prototype Pollution achieves the highest reproduction success rate at 63%. Among service-based web vulnerabilities, CSRF performs best with 60%, followed by Path Traversal at 36%. Other service-based vulnerabilities demonstrate considerably lower success rates: XSS at 16%, SSRF at 15%, RCE at 8%, and SQL Injection with the lowest at 0%. The low success rates for XSS and SSRF reflect agents' difficulties with complex multi-component environments requiring database systems, frontend-backend configurations, and middleware services. Agents struggle to extract configurations from large repositories, perform data imports, and handle intricate deployment procedures. SQL Injection faces similar environment setup challenges, compounded by database-dependent architectures. Beyond environment setup, a critical bottleneck emerges in vulnerability triggering due to API authentication requirements, as agents without browser interaction capabilities cannot perform automated login procedures like human users. In contrast, CSRF achieves perfect file localization due to clear vulnerability descriptions enabling precise interface targeting. Prototype Pollution maintains consistent performance across stages, benefiting from simpler library-based repositories requiring only dependency installation rather than complex infrastructure. While PoC execution rates remain high across vulnerability types, the dramatic drop from execution to trigger phases reveals that agents can run exploit code but struggle with the precise environmental conditions and interaction sequences needed for actual vulnerability manifestation. Figure 3 demonstrates end-to-end reproduction success rates across web technologies. PHP achieves the highest success rate at 38.9%, while TypeScript shows the lowest at 11.1%. PHP's superior performance stems from its prevalence in web security research, providing agents with abundant training data and well-documented exploit patterns, combined with its simpler syntax and direct web-oriented nature. JavaScript demonstrates strong performance through extensive clientside vulnerability examples and straightforward DOM manipulation patterns. TypeScript's poor , Vol. 1, No. 1, Article . Publication date: October 2025. 14 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Table 6. Vulnerability reproduction success rates by vulnerability types. Vulnerability Type Environment Code Localization PoC Generation End-to-End Setup File Function Line Execution Trigger Success@1 Success@3 PROTOTYPE_POLLUTION 75% 88% 75% 63% 88% 50% 38% 63% CSRF 80% 100% 80% 60% 80% 60% 40% 60% PATH_TRAVERSAL 45% 82% 73% 45% 82% 27% 18% 36% XSS 21% 53% 37% 21% 79% 11% 5% 16% SSRF 15% 31% 23% 8% 77% 8% 0% 15% RCE 8% 17% 8% 8% 67% 8% 0% 8% SQLI 8% 0% 0% 0% 25% 0% 0% 0% TOTAL 29% 46% 36% 24% 70% 18% 10% 23% PHP JavaScript Java Python TypeScript Others 0 5 10 15 20 25 30 35 40 End-to-End Success Rate (%) 38.9% 23.5% 18.2% 22.2% 11.1% 14.3% Average: 22.5% Fig. 3. Comparison of vulnerability reproduction success rates between different web technologies. performance shows that its additional complexity and stricter syntax requirements hinder agents despite its similarity to JavaScript. The results indicate that agent performance varies significantly across both vulnerability types and web technologies, with success rates influenced by environment complexity and the specific requirements of each vulnerability category. Findings for C2: Agent performance exhibits substantial variation across vulnerability types and web technologies. Service-based vulnerabilities face significant challenges from complex environment setup, authentication requirements, and multi-component dependencies, while library-based vulnerabilities achieve higher success through simpler deployment requirements. Language compatibility depends on security research prevalence and syntactic complexity, with web-oriented languages showing better agent adaptation. 5.3 C3: Efficiency Motivation. While effectiveness measures reproduction success rates, efficiency evaluates the computational and economic costs of achieving successful reproductions. Understanding resource requirements is crucial for practical deployment, as organizations must balance reproduction capabilities with operational costs. We analyze efficiency across successful reproduction attempts to assess real-world viability of different agent-model combinations. Approach. We adopt the efficiency evaluation methodology from § 4.2, measuring token consumption, execution time, and monetary cost across the vulnerability reproduction pipeline. To do so, we focus exclusively on the successful reproduction cases achieved by our three foundational agents as identified in § 5.1. To ensure a fair comparison, all tests are conducted using the Claude-Sonnet-4 model, and measurements are averaged across successful attempts for each agent. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 15 Results. Table 7 presents efficiency metrics for all three agents calculated from successful reproduction attempts. Environment setup demonstrates the highest resource requirements across all agents, while vulnerability localization proves most economical. End-to-end reproduction costs range from 2.19 with execution times between 25.7-34.0 minutes. OpenHands consistently requires the highest resources ( 1.90), while CAI demonstrates the most efficient utilization ( ). Averages from successful cases only. Costs based on Claude Sonnet pricing ( 15/1M output tokens). The cost variations stem from architectural differences between agents. OpenHands incorporates extensive internal reasoning including explicit \"think\" and \"finish\" operations that generate substantial token overhead. SWE-agent employs numerous system prompts to enforce tool usage compliance, creating intermediate communication costs. CAI, despite having the most comprehensive tool suite, exhibits minimal tool invocation in vulnerability reproduction tasks, resulting in streamlined execution. These design choices create a trade-off between computational efficiency and reproduction capability, with OpenHands prioritizing thorough reasoning, SWE-agent emphasizing tool compliance, and CAI optimizing for direct execution. Findings for C3: Web vulnerability reproduction requires substantial computational resources, with end-to-end costs ranging from 2.19 per successful case. Environment setup consumes the most resources across all agents, while agent architectural differences create significant efficiency variations in the reproduction pipeline. 5.4 C4: Robustness Analysis Motivation. Robustness measures how well agents perform under challenging conditions. In vulnerability reproduction, agents face obstacles like limited processing capabilities, browser security restrictions, or incomplete information. These challenges are inherent to real-world security research environments where perfect conditions rarely exist. A robust agent should maintain reasonable performance across different scenarios, adapting to unexpected failures and environmental constraints. Understanding robustness helps us assess agent reliability for practical deployment and identify which architectures handle reproduction challenges most effectively. Approach. We design two experiments to test the robustness of our three selected agents across key challenge areas: First, we compare agent performance using Claude-Sonnet-4 with and without enhanced reasoning across different vulnerability reproduction stages in 10 vulnerabilities. Second, we test agent performance under different authentication scenarios using 12 SQL Injection vulnerabilities, comparing three cases: manually providing login tokens, prompting agents to login themselves, and providing no authentication help. Results. Figure 4 shows mixed results when comparing models with and without enhanced reasoning. Enhanced reasoning helps in early stages, with OpenHands improving from 60% to 70% in , Vol. 1, No. 1, Article . Publication date: October 2025. 16 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Environment Setup and from 40% to 70% in File Localization, while improvements decrease in later stages with PoC Execution showing smaller gaps and Trigger Success showing no improvement across all agents. SWE-Agent shows the most consistent benefit from enhanced reasoning across localization stages, while OpenHands demonstrates strong baseline performance even without reasoning due to its built-in \"think\" operations, and CAI shows more variable results with minimal reasoning benefits. The limited reasoning impact reflects both agent design differences and task complexity variations, as reasoning enhancements prove most effective for analytical tasks like code localization but provide little benefit for execution-oriented activities like vulnerability triggering. Within our 10-vulnerability sample, agents demonstrate robustness across both reasoning configurations, suggesting that architectural design choices may be more influential than underlying model reasoning capabilities for practical vulnerability reproduction tasks. Environment File Localization Function Localization Line Localization PoC Execution Trigger Success 0 10 20 30 40 50 60 70 80 Success Rate (%) Openhands (w/ reasonning) Openhands (w/o reasonning) SWE-Agent (w/ reasonning) SWE-Agent (w/o reasonning) CAI (w/ reasonning) CAI (w/o reasonning) Fig. 4. Performance comparison between agents using Claude-Sonnet-4 with and without reasoning across different vulnerability reproduction stages. Table 8 shows authentication context impacts vulnerability reproduction success. Manual token provision works best, with Openhands + Claude-Sonnet-4 achieving 67% success. Requiring agents to handle login autonomously results in an average performance degradation of 33.3% compared to manual token provision, with success rates dropping to 25-50% across agent-model combinations. Removing authentication guidance causes near-complete failure, with only three configurations maintaining 8% success rates. This reveals that current agents struggle with autonomous authentication discovery and rely heavily on explicit guidance. The consistent degradation across architectures and models indicates that authentication handling is a widespread limitation. Findings for C4: Current agents show robustness to reasoning changes but high sensitivity to prompt variations. While reasoning versus non-reasoning modes produce similar performance, incomplete authentication results in severe performance drops, highlighting agents' limited capacity for problem-solving without explicit guidance. 6 Discussion Our empirical evaluation reveals that current open-source agents face substantial limitations in vulnerability reproduction beyond specific ecosystems. While agents demonstrate reasonable success in library-based JavaScript environments and relatively straightforward CSRF scenarios, they struggle significantly with complex web vulnerabilities that require intricate environment , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 17 Table 8. Comparison of trigger rate in authentication robustness analysis. Agent Model Manual Token Agent Login No Auth Info Openhands Claude-Sonnet-4 67% 50% 8% Gemini-2.5-Pro 58% 42% 8% GPT-4.1 50% 33% 0% SWE-Agent Claude-Sonnet-4 58% 33% 8% Gemini-2.5-Pro 50% 25% 0% GPT-4.1 42% 17% 0% CAI Claude-Sonnet-4 50% 25% 0% Gemini-2.5-Pro 42% 17% 0% GPT-4.1 33% 8% 0% setup and multi-step exploitation chains. The stark performance gaps-ranging from 0% for SQL injection to 63% for prototype pollution-indicate that pure code-based approaches are insufficient for comprehensive vulnerability reproduction. Traditional injection vulnerabilities, despite their well-documented nature, consistently fail due to environment configuration challenges and the complex interplay between application deployment, database setup, and exploitation context. This suggests that current agent architectures fundamentally lack the systems-level understanding necessary for realistic security testing scenarios. The integration of Model Context Protocol (MCP)[1] with browser automation tools like Playwright or Chrome presents a breakthrough for vulnerability reproduction. By enabling agents to control browser instances, capture network traffic, and monitor application state changes in MCP-based approaches can bridge the gap between static code analysis and dynamic exploitation. This browser-centric methodology allows agents to observe runtime behavior of web applications, intercept HTTP requests and responses, and validate success through DOM manipulation and state inspection. The ability to interact with web interfaces while monitoring network activity represents a shift from traditional code-only approaches, unlocking automated reproduction capabilities for complex web vulnerabilities that have resisted agent-based solutions. 7 Threats to Validity Internal Validity. The primary internal threat stems from potential data leakage in LLM training datasets. Many CVEs in our benchmark were disclosed before the training cutoff dates of evaluated models, creating a risk that agents may leverage memorized exploit patterns rather than demonstrating genuine reasoning capabilities. Although we deliberately avoided referencing specific PoCs in our prompts, models may still access historical vulnerability data through their pre-trained knowledge, potentially inflating performance metrics. Additionally, our evaluation employs binary success metrics that only capture whether vulnerability reproduction succeeds or fails, providing a single-dimensional assessment that may miss important aspects of the reproduction process. External Validity. Our findings may have limited generalizability due to several factors. First, our CVE selection, while diverse across vulnerability types and web technologies, represents a curated subset that may not fully reflect the complexity distribution of real-world security scenarios. Second, the containerized evaluation environment, despite efforts toward realism, may still differ from production deployments in ways that affect agent performance. Finally, the rapid evolution of LLM capabilities means our findings represent a temporal snapshot that may not predict future agent performance across different models or architectures. , Vol. 1, No. 1, Article . Publication date: October 2025. 18 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang 8 Related Work 8.1 Software Engineering Benchmarks for LLM Agents Software engineering represents a critical application domain for evaluating LLM agents, with benchmarks spanning different complexity levels and task types. SWE-bench [37] evaluates agents' abilities to solve real-world GitHub issues by generating patches for bug fixes, representing one of the most challenging repository-level tasks that require understanding complex codebases and coordinating changes across multiple files. Traditional coding benchmarks like HumanEval focus on isolated function generation from natural language descriptions, while MBPP evaluates basic Python programming tasks. Additional benchmarks including BigCodeBench [92], LiveCodeBench [35], and EvalPlus [43] extend evaluation across multiple languages and enhanced test coverage. However, these SE benchmarks exhibit significant limitations: many focus on narrow task scopes rather than diverse professional tasks, traditional benchmarks often involve only local edits to individual functions rather than repository-wide reasoning, and they frequently ignore autonomous environment setup and end-to-end task solving capabilities essential for practical software engineering. Recent advances like AgentBench [44] provide more comprehensive evaluation across diverse environments, highlighting the evolution toward more realistic and challenging evaluation frameworks for software engineering agents. 8.2 Cybersecurity Benchmarks for LLM Agents Various specialized benchmarks have been developed to evaluate LLM agents in cybersecurity. InterCode-CTF [85] manually evaluated LLMs on Capture-the-Flag challenges requiring reverse engineering and exploitation skills, while NYU CTF Bench [64] expanded this with an automated framework for diverse offensive security tasks across cryptography, forensics and binary exploitation. The CyberSecEval series [10, 73] focuses on evaluating cybersecurity risks including prompt injection, code interpreter abuse and automated social engineering. ARVO [50] provides reproducible memory vulnerabilities with triggering inputs and developer patches. SEC-bench [38] evaluates PoC generation and vulnerability patching tasks using data from OSS-FUZZ project, SecVulEval [3] benchmarks statement-level vulnerability detection in C/C++ code. CyberGym [79] assesses vulnerability reproduction across large software projects. Most existing benchmarks exhibit limitations in capturing real-world cybersecurity complexity, often focusing on isolated tasks or synthetic environments. The most closely related work is CVE-bench [76], which remains limited in both scope and scale, capturing only a narrow slice of agent performance in web vulnerability reproduction scenarios. 9 Conclusion Our evaluation of 20 LLM agents across 16 core dimensions on 3 representative CVEs identified three top performers (OpenHands, SWE-agent, and CAI) for testing on 80 real-world CVEs spanning 6 web technologies and 7 vulnerability types. Results show agents handle simple library-based vulnerabilities well but consistently fail on complex service-based scenarios requiring multi-component environments and authentication. The gap between executing exploit code and triggering actual vulnerabilities reveals fundamental limitations in environmental adaptation. These findings highlight the need for significant advances before automated vulnerability reproduction becomes practical. Current agents lack the robustness required for real-world deployment, showing high sensitivity to input guidance and poor performance under incomplete information. Data Availability The artifact of this paper can be accessed via https://figshare.com/s/7e55eaaaca0b0146ee62. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 19 References [1] 2025. Model Context Protocol Specification. Technical Report. 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Publication date: October 2025."}}},{"rowIdx":282,"cells":{"id":{"kind":"string","value":"2510.14699"},"raw":{"kind":"string","value":"Efficient Verification of Metric Temporal\nProperties with Past in Pointwise Semantics\nS Akshay #\nDepartment of CSE, Indian Institute of Technology Bombay, Mumbai, India\nPrerak Contractor #\nDepartment of CSE, Indian Institute of Technology Bombay, Mumbai, India\nPaul Gastin #\nUniversité Paris-Saclay, ENS Paris-Saclay, CNRS, LMF, 91190, Gif-sur-Yvette, France\nCNRS, ReLaX, IRL 2000, Siruseri, India\nR Govind #\nUppsala University, Sweden\nB Srivathsan #\nChennai Mathematical Institute, India\nCNRS, ReLaX, IRL 2000, Siruseri, India\nAbstract\nModel checking for real-timed systems is a rich and diverse topic. Among the different logics\nconsidered, Metric Interval Temporal Logic (MITL) is a powerful and commonly used logic, which\ncan succinctly encode many interesting timed properties especially when past and future modalities\nare used together. In this work, we develop a new approach for MITL model checking in the pointwise\nsemantics, where our focus is on integrating past and maximizing determinism in the translated\nautomata.\nTowards this goal, we define synchronous networks of timed automata with shared variables\nand show that the past fragment of MITL can be translated in linear time to synchronous networks\nof deterministic timed automata. Moreover determinism can be preserved even when the logic is\nextended with future modalities at the top-level of the formula. We further extend this approach to\nthe full MITL with past, translating it into networks of generalized timed automata (GTA) with\nfuture clocks (which extend timed automata and event clock automata). We present an SCC-based\nliveness algorithm to analyse GTA. We implement our translation in a prototype tool which handles\nboth finite and infinite timed words and supports past modalities. Our experimental evaluation\ndemonstrates that our approach significantly outperforms the state-of-the-art in MITL satisfiability\nchecking in pointwise semantics on a benchmark suite of 72 formulas. Finally, we implement an\nend-to-end model checking algorithm for pointwise semantics and demonstrate its effectiveness on\ntwo well-known benchmarks.\n2012 ACM Subject Classification Theory of computation →Timed and hybrid models; Theory of\ncomputation →Quantitative automata; Theory of computation →Logic and verification\nKeywords and phrases Real-time systems, Timed automata, Liveness, Event-clock automata, Clocks,\nTimers, Verification, Zones, Simulations\n1\nIntroduction\nLinear Temporal Logic (LTL) is a standard specification formalism. A central ingredient\nin the model-checking approach is a translation from LTL to automata.\nThe study of\nalgorithmic techniques to optimize LTL-to-automata translations spans almost four decades\n– see [57] for a recent survey of this topic. Many of the theoretical techniques have also\nbeen incorporated in practical model-checking tools [36, 22, 28, 56, 37]. For model-checking\nsystems with timing constraints, the de-facto automaton model would be timed automata\n(TA) [4, 3]. State-of-the-art timed automata tools [41, 34, 56, 37] implement analysis methods\narXiv:2510.14699v1 [cs.FL] 16 Oct 2025\n\n2\nEfficient Verification of MTL with Past in Pointwise Semantics\nfor models represented as (networks of) timed automata (see Section 7 of [25] for the practical\napplications of these tools). Properties checked on timed models are typically reachability,\nliveness, fragments of timed CTL [41] or (untimed) LTL. On the other hand, there is a very\nlarge body of foundational work on timed logics [53, 35, 54]. The two main timed logics are\nMetric Temporal Logic [40] (MTL) and Timed Propositional Temporal Logic [8] (TPTL). In\norder to use these logics for real-time model-checking, we need an efficient translation from\ntimed logics to timed automata. The foundations for MTL-to-TA (resp. TPTL-to-TA) have\nbeen laid out in [5] (resp. [8]).\nBroadly, there are two ways to interpret MTL formulae: over continuous (dense/super-\ndense) timed signals [5, 45] or pointwise timed words [6, 58, 20]. In continuous (state-based)\nsemantics, the formula is interpreted over signals, which are functions that associate the\nvaluation of the propositions to all timestamps. Over the pointwise semantics, the formula\nis interpreted over sequences of events (timed words), each consisting of a timestamp and a\nset of propositions. Thus, in the continuous semantics, the system is under observation at\nall times, while in the pointwise semantics, it is observed only when it executes a discrete\ntransition. However, as noted by Ferrére et al. [30], in the continuous semantics, arbitrary\nMTL formulae can be simplified to the [0, ∞] fragment, while as pointed out in [2], this\nsimplification is not possible in the pointwise semantics. In this work, we only consider\npointwise semantics as our focus is on models represented in tools like UPPAAL, PAT,\nTChecker whose behaviours are given as timed words, as in the classical timed automata\nmodel of Alur and Dill [4].\nIn the pointwise semantics, it is known that MTL model checking is undecidable over\ninfinite words, and over finite words it is decidable but the complexity is Ackermanian [7, 50,\n48, 49]. Thus, the literature often considers the more efficient MITL fragment, where Until\nand Since operators must use non-punctual intervals (either [0, 0] or non-singleton intervals).\nThe classical approach for model checking for MITL involves translating MITL formula into\ntimed automata (TA) and then using state-of-the-art TA tools such as UPPAAL [41, 12]\nor TChecker [34]. The best known tool that accomplishes this translation in the pointwise\nsemantics, called MightyL [20], goes via alternating 1-clock automata and compiles to\ntimed automata. Recently, an extension of timed automata called generalized timed automata\n(GTA) was introduced in [1]. GTA are equipped with richer features than TA, and enable\na different translation from MITL to GTA, which can then be analyzed using a GTA\nanalysis tool from [1]. The theoretical foundations for such an MITL-to-GTA translation\nwhich goes via transducers was presented in [2], where it was argued that the MITL-to-\nGTA translation has clear theoretical benefits over the MITL to 1-clock Alternating Timed\nAutomata translation [18, 19] on which MightyL is based on. However, this latter approach\nwas not implemented. Moreover, neither of these approaches considered Past modalities, and\nwere restricted only to Future.\nOn the other hand, from the classical work of Maler et al [44], in the signal semantics, it\nis known that having only Past modalities leads to efficient algorithms and in particular it\ngives a translation to deterministic (timed) automata albeit over signals. Our first question\nis whether the same holds under the pointwise semantics, and whether determinism can be\nextended beyond just the past fragment. Next, if we cannot guarantee determinism, can we\nreduce non-deterministic branching? Indeed, even if it does not change the expressiveness,\nseveral works have highlighted the usefulness and succinctness of combining Past and Future\noperators, both in the untimed setting [43, 46, 10] and in the timed setting [39, 30].\nIn this work, we address these questions both for past and future in the pointwise\nsemantics and to build the foundations for a new efficient model checking algorithm for\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n3\nMITL with past that exploits determinism as much as possible. We show that for the past\nfragment of MITL, we can obtain a linear-time translation to networks of deterministic timed\nautomata. Further, even when we have to handle future operators, where non-determinism\ncannot be avoided, we propose techniques to reduce non-deterministic branching in the\nresulting automata, using various ideas including sharing components, automata, clocks and\npredictions. We start by defining a formal semantics for a model of synchronous network\nof timed automata with shared variables, where each component of the network owns some\nclocks which only it can reset but can be tested by all components. Moreover it allows for\ninstantaneous programs on transitions which have sequences of tests and updates, like in [2].\nOur main contributions are the following:\n1. We identify a fragment of metric temporal logic with past and future for which we provide\na linear time translation to networks of deterministic timed automata. In this fragment,\nthat we call detMITL+p, the outermost temporal modalities are future (next, until) and\ncan even have punctual intervals, while inner temporal modalities should be past: Since\nwith non-punctual interval or Yesterday. This fragment already subsumes many safety\nproperties, extending e.g., the G(pLTL) fragment defined in [9].\n2. We present a new linear-time algorithm to translate detMITL+p formula to networks\nof deterministic timed automata. This algorithm is in two steps. First we show how\nthe outermost temporal modalities can be handled with respect to initial satisfiability.\nNext we show how the past temporal modalities are handled for general satisfiability.\nWe obtain a synchronous network of deterministic automata with shared variables that\ncaptures the semantics. We start with Yesterday allowing an arbitrary interval constraint.\nThen, we deal with Since operators with one-sided intervals of the form [0, a] or [a, ∞).\nFor these simpler cases, we construct networks where each automaton has at most 1\nclock and 2 states, but critically uses the sharing feature across components. Finally,\nthe automaton for Since with a general two-sided interval [a,b] is the most complicated\npart of the construction and results in a network whose one component uses O(a/(b−a))\nclocks. Our construction works for both finite and infinite words.\n3. Next, we extend this algorithm to handle future modalities. To achieve this we move\nfrom networks of timed automata to networks of generalized timed automata (GTA) [1],\nwhich have future clocks in addition to usual (history) clocks. We obtain a translation\nfrom general MITL+p (with Past and allowing punctual intervals at the outermost level)\nto networks of GTA with shared variable. While non-determinism cannot be avoided\nhere, we show that we can reduce non-deterministic branching in two ways: first by using\nshared variables to share predictions, and second, by using non-Zenoness assumption and\nthe semantics of future clocks to obtain a 2-state automaton for Until (which results in\nan exponential improvement in running time for the model checking algorithm). Again\nour model checking algorithm works for infinite words and is easily adaptable for finite\ntimed words.\n4. Since our model checking approach ultimately reduces to checking reachability and liveness\nfor GTAs, we propose an improved liveness-checking algorithm that extends Couvreur’s\nSCC algorithm [26] (which is the state-of-the-art for timed automata) to the setting of\nGTAs, marking the first adaptation of Couvreur’s procedure to automata with event\nclocks or diagonal constraints.\n5. Finally, we implement our algorithms in a prototype tool, called Tempora, built on top of\nthe open-source timed automata library Tchecker. Tempora supports satisfiability and\nmodel checking for detMITL+p and general MITL+p, for both finite and infinite words, in\nthe pointwise semantics. We compare against MightyL which translates MITL formula\n\n4\nEfficient Verification of MTL with Past in Pointwise Semantics\nto timed automata (and uses UPPAAL to check satisfiability with finite words and\nOPAAL for infinite words). Our comparison on a suite of 72 benchmarks (22 new and 50\ntaken from the literature) shows that our pipeline (Tempora+Tchecker) significantly\noutperforms MightyL pipeline; it is faster in all benchmarks for finite words and all but\n2 for infinite words, and most often explores a smaller state space (in zones). We also\nimplement the theoretical translation to GTA presented in [2] and demonstrate an order\nof magnitude improvement. Finally, we also implement an end-to-end model checking\nalgorithm for MITL+p over the pointwise semantics and evaluate our model-checking\npipeline on two models: Fischer’s mutual exclusion algorithm and the Dining Philosophers\nproblem.\nRelated works. Model checking for real-time systems has been extensively studied, with\nMetric Interval Temporal Logic (MITL) emerging as a widely used specification language\ndue to its expressiveness for capturing timed properties. A central challenge lies in efficiently\ntranslating MITL formulas into automata-based representations, enabling algorithmic veri-\nfication. For a comprehensive overview of the topic, we refer the reader to the survey by\nBouyer [14].\nOver continuous/signal semantics, we already discussed several works that focus on\nobtaining networks of timed automata for various fragments of MITL with and without\npast including [30, 44, 45]. Most of these employ construction of compositional testers or\ntransducers for timed modalities, but do not focus on sharing e.g., of networks and subformula.\nA different purely logic based approach was considered in [47], where the timed automata\nand MITL formula, still with signal semantics, are translated to an intermediate logical\nlanguage, which is then encoded into an SMT solver. This work also included synchronization\nprimitives and was implemented in a Java tool.\nComing back to pointwise semantics, in [5] that introduced MITL, the authors propose\nthe first such translation, which is known to be notoriously complicated, spanning several\npages, and acknowledged to be difficult in the follow-up works by different authors [45]. The\ncurrent state-of-the-art for MITL model checking is the procedure proposed by Brihaye et al.\nin [20], where they translate MITL formulae to a network of timed automata, which can be\nplugged in a standard timed automata tool implementing non-emptiness. Their procedure\nand the associated tool, called MightyL [20] is based on a series of works that study a\ntranslation of MITL to timed automata [18, 19], via alternating timed automata [49]. [17]\nstudies a zone-based algorithm directly for 1-clock alternating timed automata, and focusses\non a fragment of MTL.\nSeveral tools support MITL model checking under continuous semantics, including [38,\n23, 24, 52, 42, 47]. Under pointwise semantics, to the best of our knowledge, MightyL [20]\nis the only tool that can handle the full future fragment of MITL (even if it only solves\nsatisfiability and does not handle past operators). The other works (to our knowledge) that\noffer tool support are for restricted fragments such as MITL(0.∞) [21], or MITL over untimed\nwords [59].\nStructure of the paper. In Section 2, we formalize the notion of a synchronous network\nof timed automata with shared variables, whose transitions are instantaneous programs, and\ndefine the model checking problem in terms of such networks. In Section 3 we describe the\ndetMITL+p logic followed by the translation to synchronous networks of deterministic timed\nautomata in Section 4. In Section 5, we explain how to integrate future modalities to get\na translation from MITL+p to generalized timed automata with future clocks. Section 6\ndescribes our adaptation of Couvreur’s SCC algorithm to the setting of GTAs. Section 7\ncontains our experimental evaluation and comparisons and we end with a brief conclusion in\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n5\nSection 8.\nTo improve the flow and readability of the paper, we have moved some proofs and\ntechnical details to the Appendix. Moreover, some benchmark details and extended results\nare presented in the Appendix.\n2\nSynchronous network of timed automata with shared variables\nWe define the model of network of timed systems that we use in this paper. The presentation\nis different from previous works [11, 16, 33, 2], and is geared towards providing a clean and\nconcise explanation of our logic-to-automata translation. Before giving the formal syntax,\nwe highlight the main aspects in our definition.\nEach component in the network is an automaton with a finite set of locations, real\nvalued variables (called clocks) and finitely valued variables (mainly boolean variables and\nthe state variable ranging over the finite set of locations). The semantics of the network is\nsynchronous: either time elapses in all components of the network (all clocks are increased\nby the same amount), or all components execute simultaneously an instantaneous transition.\nThe components of the system communicate via shared variables (clocks, states, boolean\nvariables, . . . ). The paradigm is concurrent-read owner-write (CROW). Hence, each variable\nis owned by a unique component of the system. A component may read all variables of the\nsystem, but it may only modify its owned variables.\nA transition of a component A is of the form (ℓ, prog, ℓ′) where ℓ, ℓ′ are locations of A\nand prog is an instantaneous program, which is a sequence of tests and updates. For instance,\nif x, y are clocks owned by A and out is a boolean variable owned by A, then a purely\nlocal program could be A.x ∈(3, 5]; A.y := 2; A.out := 0 where A.x ∈(3, 5] is a test which\nsucceeds if the current value of clock x is within the interval, and A.y := 2 and A.out := 0\nare updates. Tests may also use variables owned by other components. For instance, if\ncomponent B owns clock z, then a program of A could be of the form\nA.x ∈(3, 5] ∧B.out• = 1 ∧B.z > 2 ∧B.st• = ℓ1; A.y := 2; A.out := 0\nwhere B.st means the state of B. Since we use a synchronous semantics, all components of the\nnetwork simultaneously take a transition and execute the associated programs. By default,\na variable such as A.x or B.z used in a test refers to the value of the variable before the\nsynchronous transition is taken. It is sometimes convenient to refer to the value a variable\nwill have after the synchronous transition is taken. We do so by writing B.out• or B.st•.\nWe give now the formal syntax and semantics for a network of timed systems. A concrete\nexample follows in Figure 1 and Example 3.\n▶Definition 1 (Timed automaton with shared variables). A timed automaton with shared\nvariables is a tuple A = (Q, X, V, ∆, init) where A is the name, Q is a finite set of locations\nover which ranges the state variable A.st, X is a finite set of clock variables owned by A, V\nis a finite set of boolean variables owned by A, ∆is the finite set of transitions, and init is\nan initial condition for the variables owned by A.\nA transition is a triple (ℓ, prog, ℓ′) where ℓ, ℓ′ ∈Q are locations and prog is an instantan-\neous program. A program is a sequence of tests and updates. An atomic update for a boolean\nvariable v ∈V owned by A has the form A.v := c (or simply v := c when A is clear from the\ncontext) with c ∈{0, 1}. An atomic update for a clock variable x ∈X owned by A has the\nform A.x := c (or simply x := c) with c ∈N = N ∪{+∞}. We assume that each variable is\nupdated at most once in a program. An atomic test for a clock variable x ∈X owned by A\nhas the form A.x ∈I (or simply x ∈I) where I is an integer bounded interval whose end\n\n6\nEfficient Verification of MTL with Past in Pointwise Semantics\nℓ0\nA1\nx = +∞\nq•; x := 0\n¬q•\nℓ0\nℓ1\nA2\ny = +∞\n¬q•\nq•; y := 0\n¬p• ∧q•; y := 0\np•\n¬p• ∧¬q•\nℓ0\nℓ1\nℓ2\nA3\nz = 0\nz ≤100 ∧¬r•\nz ≤100 ∧r• ∧A1.x < 2 ∧A2.st• = ℓ1 ∧A2.y• ≥5\nz > 100\nz ≤100 ∧r• ∧(A1.x ≥2 ∨A2.st• = ℓ0 ∨A2.y• < 5)\nℓ0\nMp\np := 0\np := 1\nFigure 1 Synchronous Network of timed automata with shared variables. The model M is not\ndrawn and owns the boolean variables Prop = {p, q, r}. To lighten the figures above, we simply write\np•, q• and r• instead of M.p•, M.q• and M.r•. The automata A1, A2, A3 own respectively the\nclock variables x, y and z. The names and initial conditions are depicted with an incoming arrow to\na location. For instance, the initial condition of A2 is st = ℓ0 ∧y = +∞. Notice that A3 refers to\nthe post values of the state A2.st• and clock A2.y• of A2.\npoints come from N (e.g., [0, 3) or [2, +∞) or (2, 5], etc). A may also test a clock variable\nowned by another component with N.x ∈I or N.x• ∈I where N is a name and I is an\ninteger bounded interval. Similarly, an atomic test for a boolean variable b owned by A has\nthe form b = 0 or b = 1 (also written b and ¬b) and if the variable is not owned by A we\nuse N.b or N.b•. We write N.st = ℓ, N.st ̸= ℓ, N.st• = ℓand N.st• ̸= ℓfor testing whether\ncomponent N is in location ℓor not. Finally, a test in a program is a boolean combination\nof atomic tests.\nThe initialisation init of A is a conjunction of atomic tests, one for each variable of A,\nincluding the state variable. For instance, init could be st = ℓ0 ∧y = +∞∧out = 0.\n▶Definition 2 (Synchronous network of timed automata). A network is a tuple A =\n(A1, . . . , An) where each Ai = (Qi, Xi, Vi, ∆i, initi) is a timed automaton with shared vari-\nables. The network is closed if the references to external variables in programs are all of\nthe form Ai.st or Ai.st•, Ai.x or Ai.x• with x ∈Xi, Ai.v or Ai.v• with v ∈Vi, for some\n1 ≤i ≤n. The network is partially ordered if there is a partial order ⪯on its components\nsuch that whenever component A refers to a variable owned by component B, we have B ⪯A.\nIn this paper, we will only use partially ordered networks.\nWe describe now the semantics of a network. A configuration C of A is a valuation of\nall variables of the network: for all 1 ≤i ≤n, we have C(Ai.st) ∈Qi, C(Ai.x) ∈R+ =\nR+∪{+∞} for x ∈Xi, and C(Ai.v) ∈{0, 1} for v ∈Vi. A configuration is initial if it satisfies\nall initial conditions of its components. For instance, if init2 is st = ℓ0 ∧y = +∞∧out = 0,\nthen it is satisfied at C if C(A2.st) = ℓ0, C(A2.y) = +∞and C(A2.out) = 0.\nThe network is synchronous, i.e., all components move simultaneously, and has two types\nof transitions: time elapse and discrete. A time elapse by some non-negative real number\nδ ≥0 is written C\nδ−→C′ where C′ coincides with C on all variables other than clocks, and\nthe value of each clock is advanced by the same quantity δ: C′(Ai.x) = C(Ai.x) + δ for all\n1 ≤i ≤n and x ∈Xi (with +∞+ δ = +∞).\nA discrete transition occurs in the network when each component executes simultaneously\na transition. Let t = (t1, . . . , tn) with ti = (ℓi, progi, ℓ′\ni) ∈∆i for all 1 ≤i ≤n. Let C, C′ be\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n7\ntwo configurations. There is a discrete transition C\nt−→C′ if the following holds:\nC(Ai.st) = ℓi and C′(Ai.st) = ℓ′\ni for all 1 ≤i ≤n,\nfor all i and x ∈Xi ∪Vi, we have C′(Ai.x) = c if progi contains an update Ai.x := c and\nC′(Ai.x) = C(Ai.x) otherwise (recall that each variable is updated at most once),\nall tests occurring in the programs evaluate to true:\nfor all i and clock x ∈Xi and integer bounded interval I, an atomic test Ai.x ∈I (resp.\nAi.x• ∈I) evaluates to true if C(Ai.x) ∈I (resp. C′(Ai.x) ∈I),\nfor all i and boolean variables v ∈Vi and c ∈{0, 1}, an atomic test Ai.v = c (resp.\nAi.v• = c) evaluates to true if C(Ai.v) = c (resp. C′(Ai.v) = c),\nfor all i, an atomic test Ai.st = ℓ(resp. Ai.st• = ℓ) evaluates to true if C(Ai.st) = ℓ(resp.\nC′(Ai.st) = ℓ).\n▶Example 3. We consider the specification given by the sentence\nΦ = G≤100\n\u0000r −→(YP<2 q ∧p S≥5 q)\n\u0001\n.\nWe assume that the model M owns the boolean variables Prop = {p, q, r}, also called atomic\npropositions. The sentence Φ says that during the first 100 units of time, whenever an event\nsatisfies r (i.e., M sets r to true, or r was true and is not updated by M) then some event\nless than 2 time units in the strict past satisfies q (written as YP<2 q), and some other event\nat least 5 time units in the past satisfies q and since then all events satisfy p (written as\np S≥5 q).\nWe give in Figure 1 a network of timed automata which implements the specification\nΦ. Component A1 records with clock x the time elapsed since the latest q event occurred.\nClock x is initialized to +∞and remains so until the first q event is seen. Then it is reset to\n0 whenever a q event occurs.\nComponent A2 is in location ℓ1 iff p since q holds (there is a q event in the past such\nthat since then all events satisfy p). When A2 is in location ℓ1, clock y records the time\nelapsed since the earliest witness of p S q, i.e., the earliest q event such that since then all\nevents satisfy p.\nFinally, A3 is in location ℓ2 if the sequence of events seen so far violates Φ: within the\nfirst 100 time units, there is an r event such that the closest q event is more than 2 time\nunits in the strict past (A1.x ≥2) or p S q does not hold (A2.st• = ℓ0) or the earliest witness\nof p S q is less than 5 time units in the past. Component A3 is in location ℓ1 if Φ is definitely\nsatisfied, more than 100 time units has elapsed without violating the property. Finally, A3\nstays in location ℓ0 if the sequence of events seen so far satisfies Φ and at most 100 time\nunits has elapsed.\n◀\n2.1\nThe model checking problem\nIn this paper, we are interested in the model checking problem: given a model M owning a\nset Prop of atomic propositions, and a specification Φ on Prop, do all behaviours of M satisfy\nΦ. Behaviours of a model are formalized as timed words over sets of atomic propositions.\nThe alphabet Σ that we will consider is 2Prop, the set of subsets of Prop. A timed word over\nΣ, denoted by w ∈(Σ × R)∞, is an infinite sequence of pairs of letter and timestamp of the\nform (a0, τ0)(a1, τ1)(a2, τ2) . . . , where each ai is a subset of Prop and τi is the timestamp of\nthe i-th letter; timestamps are assumed to be non-decreasing, i.e., 0 ≤τ0 ≤τ1 ≤τ2 · · · .\n\n8\nEfficient Verification of MTL with Past in Pointwise Semantics\nA run ρ of the model M is an alternating sequence of delay and discrete transitions,\nstarting from an initial configuration. Formally, a run is an infinite sequence\nρ := C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · ·\nwhere C0 is an initial configuration, each δi ≥0 is a time delay, and each ti is a tuple\nof transitions (one from each component of the model) as defined above, such that each\nstep respects the semantics of time elapse and discrete transitions. The timed word wρ =\n(a0, τ0)(a1, τ1) · · · associated to the above run ρ is given as follows: for all i ≥0\nτi = δ0 + δ1 + · · · + δi (the timestamp of occurrence of ti) and\nai is the set of atomic propositions p such that Ci+1(p) = 1 (the propositions that are\ntrue after executing ti).\nThe timed word wρ is a behaviour of the model M.\nThe model checking problem asks whether all (infinite, non-zeno) behaviours generated\nby all possible runs of the model M satisfy a given specification Φ, written as M |= Φ.\nIn this work, we are interested in specifications given by fragments of MTL, which we will\ndefine in the later sections. To solve the model checking problem, we will construct from the\nspecification Φ a network of timed automata A = (A1, . . . , An). Then, we will consider the\nclosed network (M, A) and check whether some (bad) cycle may be iterated infinitely often.\nFor instance, to check whether a given model M satisfies the specification Φ from\nExample 3, we consider the closed network N = (M, A1, A2, A3) where the automata\nA1, A2, A3 are given in Figure 1. Then, M ̸|= Φ iff a configuration C of N where A3 is in\nlocation ℓ2 is visited infinitely often.\nWe may also check whether a specification is satisfiable by considering a universal\nmodel allowing all sequences of events. This is achieved by considering, for each atomic\nproposition p ∈Prop, an automaton Mp as shown in Figure 1. Then, the model is the\nnetwork M = (Mp)p∈Prop.\n3\nThe detMITL+p fragment\nWe present a fragment of MTL for which we can construct a network (A1, A2, . . . , An) such\nthat each Ai is deterministic. For the formula in Example 3, the automata A1, A2 and A3\nshown in Figure 1 are deterministic. When (A1, . . . , An) are deterministic, the product with\nthe model (M, A1, . . . , An) does not further “multiply” the runs of M: for every run σ of\nM, there is at most one run ρ of (M, A1, . . . , An) whose projection leads to σ. On the other\nhand, if some Ai is non-deterministic, the run σ can branch out into multiple runs in the\nproduct, leading to an explosion in the state-space (of the zone graph that is computed).\nThe set of MTL formulae over the atomic propositions Prop is defined as\nφ := p | φ ∧φ | ¬φ | YI φ | XI φ | φ UI φ | φ SI φ\nwhere p ∈Prop, and I is an interval with end-points from N = N ∪{∞}. Note that, our\ndefinition of MTL is with both past and future modalities. The pointwise semantics of MTL\nformulae is defined inductively as follows. A timed word w = (a0, τ0)(a1, τ1)(a2, τ2) · · · is\nsaid to satisfy the MTL formula φ at position i ≥0, denoted as (w, i) |= φ if (omitting the\nclassical boolean connectives)\n(w, i) |= p if p ∈ai\n(w, i) |= YI φ if i > 0, (w, i −1) |= φ and τi −τi−1 ∈I.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n9\n(w, i) |= XI φ if (w, i + 1) |= φ and τi+1 −τi ∈I.\n(w, i) |= φ1 UI φ2 if there exists j ≥i s.t. (w, j) |= φ2, τj −τi ∈I and (w, k) |= φ1 for all\ni ≤k < j.\n(w, i) |= φ1 SI φ2 if there exists 0 ≤j ≤i s.t. (w, j)|=φ2, τi −τj ∈I and (w, k)|=φ1 for\nall j < k ≤i.\nA word w initially satisfies φ, written as w |= φ if (w, 0) |= φ. Finally, a model M satisfies\nMTL formula φ, denoted as M |= φ if wρ |= φ for every run ρ of M. We rewrite the\nderived operators G, F, P, H in terms of U or S, in the standard manner: FI φ = true UI φ,\nGI φ = ¬ FI ¬φ, PI φ = true SI φ and HI φ = ¬ PI ¬φ.\n▶Definition 4 (detMITL+p). The fragment detMITL+p consists of formulas where the\noutermost temporal operators are future operators XI, UI and may use arbitrary intervals\nI, while all inner temporal operators are restricted to past operators. Inner Since operators\nmust use non-punctual intervals (either [0, 0] or non-singleton intervals). Formally, formulas\nare built as follows:\n(Sentences)\nΦ := p | Φ ∧Φ | ¬Φ | XI φ | φ UI φ\n(Point-formulas)\nφ := p | φ ∧φ | ¬φ | YI φ | φ SI′ φ\nwhere I, I′ are integer-bounded intervals, and I′ is either [0, 0] or a non-singleton interval.\nSentences are evaluated at the first position (w, 0) of a word w, whereas point-formulas\ncan be checked anywhere in the word. The formula in Example 3 is a sentence in the\ndetMITL+p fragment: the derived GI operator induces an outer UI formula, and all the inner\noperators are past. The formula YP<2 q can be rewritten as P(0,2) q ∨Y[0,0] P[0,0] q.\n4\nFrom detMITL+p to network of deterministic timed automata\nFor a detMITL+p sentence Φ we build a deterministic network NΦ and specify a set BadΦ\nof bad configurations. Our objective is to achieve a construction that entails the following\ntheorem for detMITL+p sentences.\n▶Theorem 5. Let Φ be a detMITL+p sentence and M a timed model. Then: M ̸|= Φ iff\nthere exists an infinite run of the closed network (M, NΦ) which eventually remains in bad\nconfigurations.\nGiven a sentence Φ, we construct the following deterministic automata:\nAisat\np\nAisat\nXI\nAisat\nUI\nfor every atomic proposition p ∈Prop that is not under the scope of a temporal operator,\nand every subformula XI and UI appearing in the sentence. The automaton Aisat\np\ntakes the\natomic proposition p as input. The automaton Aisat\nXI\ntakes as input the boolean variable b1\ncorresponding to the argument of XI. Automaton Aisat\nUI\ntakes two boolean variables b1, b2 as\ninputs corresponding to the first and second arguments of UI. Notice that these operators\nappear only at the outermost level of Φ and will be evaluated at the first position in a word.\nThe automata for these outermost operations will be called initial satisfiability automata,\nand are superscripted with an isat. Apart from these, we build an automaton Ainit which\nresets a clock xinit at the first action. All the isat automata make use of this single clock xinit\nin their tests. The isat automata do not own any other clocks. Finally, for every temporal\npoint-formula φ appearing as a subformula of Φ, we build a deterministic automaton Aφ.\nFor these point-formula automata we also allow for sharing of clocks, which we explain later.\n\n10\nEfficient Verification of MTL with Past in Pointwise Semantics\n1\n2\nAinit\nxinit := 0\n1\n2\n3\n4\nAisat\nXI\ntrue\nb•\n1 ∧x•\ninit ∈I\n¬b•\n1 ∨x•\ninit /∈I\nFigure 3 Left: Clock xinit is reset on the first event and measures time elapsed for all the initial\nsatisfiability automata (Aisat\nXI\nand Aisat\nUI ).\nRight: Automaton Aisat\nXI\nfor initial satisfiability for XI b1. The boolean variable b1 is not owned by\nAisat\nXI\nand stands for the argument of XI. In Aisat\nXI , we simply write xinit instead of Ainit.xinit. The\nautomaton is deterministic and complete. If a run reaches state 3 (resp. 4) then XI b1 was initially\ntrue (resp. false).\nBad configurations.\nThe collection of all the isat automata and the point-formula automata for Φ form the\nnetwork NΦ. States of each of the isat automata will be marked either true or false. A\nconfiguration C of NΦ is bad if Φ evaluates to false when we replace every outermost p, and\noperators XI, UI of Φ by the truth value of the state the corresponding isat automata are in.\nFor example if Φ = p ∧XI q, a configuration C1 where Aisat\np\n= true and Aisat\nXI q = true is not\nbad, whereas C2 with Aisat\np\n= false will be bad. The set of bad configurations of Φ will be\ncalled BadΦ. This notion naturally extends to the product (M, NΦ): a configuration of the\nproduct is bad if its projection to NΦ evaluates to false.\n1\n2\n3\nAisat\np\ntrue\np•\n¬p•\nFigure 2 Automaton Aisat\np\n.\nWe describe the initial satisfiability automata in\nSection 4.1 and show some local correctness properties.\nThe automata for past operators are presented in\nSections 4.2, 4.2.2 and 4.2.3. Finally, in Section 4.4,\nwe present the proof of Theorem 5 which shows the\nglobal correctness of the construction, by making use\nof the intermediate local correctness lemmas shown\nin the previous sections.\n4.1\nInitial satisfiability of sentences\nFigure 2 depicts the automaton Aisat\np\nwhich takes as input the atomic proposition p. The\nproperty that it satisfies is stated in the following lemma, whose proof is straightforward.\n▶Lemma 6. For every timed word w = (a0, τ0)(a1, τ1) · · · over Prop with p ∈Prop, there is\na unique run C0\nδ0,t0\n−−−→C1\nδ1,t1\n−−−→C2 · · · of Aisat\np\nover w. Moreover, w |= p iff for all i ≥1,\nconfiguration Ci satisfies Aisat\np\n.st = 2.\nFigure 3 describes Ainit and Aisat\nXI . The invariant satisfied by Aisat\nXI\nis explained in the\ncaption of Figure 3 and formalized below.\n▶Lemma 7. Let Prop′ = Prop ⊎{b1} be an augmented set of atomic propositions. For\nevery word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0\nδ0,t0\n−−−→C1\nδ1,t1\n−−−→C2 · · ·\nof (Ainit, Aisat\nXI ) over w. Moreover, w |= XI b1 iff for all i ≥2, configuration Ci satisfies\nAisat\nXI .st = 3.\nFinally, Figure 4 gives the deterministic automaton for the initial satisfiability of UI.\nCorrectness of the automaton is formalized below.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n11\n1\n2\n3\nAisat\nUI\ntrue\n(x•\ninit ∈I ∧b•\n1 ∧¬b•\n2)\n∨(x•\ninit < I ∧b•\n1)\nx•\ninit ∈I ∧b•\n2\nx•\ninit > I ∨\n(x•\ninit < I ∧¬b•\n1) ∨\n(x•\ninit ∈I ∧¬b•\n1 ∧¬b•\n2)\nFigure 4 Automaton Aisat\nUI\nfor the initial satisfiability for b1 UI b2 (with I ̸= ∅). The boolean\nvariables b1, b2 are not owned by Aisat\nUI\nand stand for the left and right arguments of UI. The\nautomaton is deterministic and complete. We write xinit < I (resp. xinit > I) to state that xinit is\nsmaller (resp. larger) than all values in I. For instance, if I = [b, c) with b < c then xinit < I means\nxinit < b and xinit > I means xinit ≥c. If a run reaches state 2 (resp. 3) then we know for sure that\nb1 UI b2 was initially true (resp. false). State 1 means we still do not know whether b1 UI b2 was\ninitially true. If the run stays forever in state 1, b1 UI b2 was initially false.\nℓ0\nℓ1\nAY\n¬b•\n1\nb•\n1\nb•\n1\n¬b•\n1\n1\nxlast = +∞\nAlast\nxlast := 0\nFigure 5 Left: Automaton AY for the untimed Yesterday operator Y b1. The boolean variable b1\nis not owned by AY and stands for the argument of Y.\nRight: Sharer automaton Alast with clock xlast used by all Yesterday operators.\n▶Lemma 8. Let Prop′ = Prop ⊎{b1, b2} be an augmented set of propositions. For every\nword w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0\nδ0,t0\n−−−→C1\nδ1,t1\n−−−→C2 · · · of\n(Ainit, Aisat\nUI ) over w. Moreover, w |= b1 UI b2 iff there exists an i ≥1 s.t. for all k ≥i,\nconfiguration Ck satisfies Aisat\nUI .st = 2,\nProof. Suppose w |= b1 UI b2. Then, (1) there exists an i ≥0 such that b2 ∈ai, (2) for all\n0 ≤j < i, we have b1 ∈aj, and (3) τi −τ0 ∈I.\nWe assume that i ≥0 is minimal with the above property. Hence, in the run of w,\ntransitions tj for 0 ≤j < i are self-loops on state 1. Indeed, b•\n1 is true at Cj+1 and either\nτj −τ0 < I or τj −τ0 ∈I and b•\n2 is false at Cj+1 due to the minimality of i.\ntransition ti goes to state 2 since b•\n2 is true at Ci+1 and τi −τ0 ∈I.\nHence configuration Ci+1 satisfies Aisat\nUI .st = 2.\nConversely, suppose the automaton moves to state 2 for the first time after reading the\nprefix (a0, τ0)(a1, τ1) · · · (ai, τi). Since Aisat\nUI\nstarts at state 1, the run loops around 1 on all\nevents a0 upto ai−1 and then a transition is made to state 2. In the loop b1 is true, and\nin the transition that moves to 2, we have b2 to be true. The transition to 2 also checks\nfor xinit ∈I. Since xinit is reset at a0, we get τi −τ0 ∈I. Hence, the run shows the three\nproperties listed above and w |= b1 UI b2.\n◀\n4.2\nPast operators\n4.2.1\nYesterday operator.\nThe automaton for the Yesterday operator Y, depicted in Figure 5 (left), has two states ℓ0 and\nℓ1. The states remember whether the argument b1 of Y was true or not at the previous letter:\nall transitions reading a b1 enter ℓ1, and the ones that read ¬b1 go to ℓ0. To incorporate the\ntiming constraint, we use a single state automaton Alast, depicted in Figure 5 (right), which\n\n12\nEfficient Verification of MTL with Past in Pointwise Semantics\nresets a clock xlast in every step. Now, for an arbitrary interval constraint I, the current\nposition satisfies YI b1 if the previous letter had a b1 and the value of clock xlast (which was\nreset at the previous position) when the current position is read falls in the interval I.\n▶Lemma 9. Let Prop′ = Prop ⊎{b1}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′,\nwe have a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (Alast, AY). Moreover,\nfor all i ≥0, (w, i) |= Y b1 iff Ci(AY.st) = C′\ni(AY.st) = ℓ1,\nfor all i ≥1, τi −τi−1 ∈I iff C′\ni(Alast.xlast) ∈I.\nLet outYI = (AY.st = ℓ1 ∧Alast.xlast ∈I). Then, for all i ≥0, we may check if (w, i) |= YI b1\nwith the test outYI during transition ti from C′\ni to Ci+1.\n4.2.2\nSince operator with a one-sided interval.\nThe formula b1 SI b2 essentially asks that there exists a position j in the past, such that b2 is\ntrue at that position, and at all positions between j and the current position, b1 is true, and\nthe time elapsed between the position j and the current position lies in the interval I. Note\nthat there could be multiple candidates for the position j. Let us first consider the untimed\nversion of the Since operator given in Figure 6 (left). The transition from ℓ0 to ℓ1 on b2 is\ntaken with the first b2 that could be a witness. When the automaton remains in ℓ1, each\ntime a b2 is seen, the automaton encounters a fresh witness.\nWhen we move to the timed setting, some of these witnesses may satisfy the timing\nconstraint while others may not, and so this information must be tracked carefully. We will\nfirst consider the simpler case where the intervals are one-sided, i.e., of the form [0, c), [0, c],\n[b, +∞) or (b, +∞).\nWhen the interval is one-sided, it is sufficient to track the two extreme b2 witnesses: the\nearliest witness and the last possible witness at each point. The idea is simply that if the\nearliest witness does not satisfy a lower-bound constraint, then no later witness could satisfy\nit. Analogously, if the last witness does not satisfy an upper-bound, no earlier witness could\nhave satisfied it.\nIn the automaton Alast\nS\nof Figure 6 (middle), clock x maintains the time since the latest\nb2 (which may serve as a witness for b1 S b2 when AS is in state ℓ1). For an upper-bound\ninterval SI where I = [0, c) or I = [0, c], we check the value of x, i.e., the time since the last\nb2. The output is 1 if the target state of AS is ℓ1 and x ∈I.\n▶Lemma 10. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′\nthere is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (AS, Alast\nS ). Moreover, for all\ni ≥0:\nℓ0\nℓ1\nAS\n¬b•\n2\nb•\n2\nb•\n1 ∨b•\n2\n¬b•\n1 ∧¬b•\n2\nℓ0\nAlast\nS\nx = +∞\nb•\n2; x := 0\n¬b•\n2\nℓ0\nAfirst\nS\ny = +∞\nb•\n2 ∧(¬b•\n1 ∨AS.st = ℓ0);\ny := 0\n¬b•\n2 ∨(b•\n1 ∧AS.st = ℓ1)\nFigure 6 Left: Automaton for the untimed Since operator b1 S b2.\nMiddle: Automaton recording with clock x the time since the last (most recent) occurrence of b2.\nRight: Automaton recording with clock y time since the first (earliest) b2 which may serve as a\nwitness of b1 SI b2.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n13\n1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,\n2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Alast\nS .x) = τi −τj where j ≤i, b2 ∈aj and b2 /∈ak\nfor all j < k ≤i.\n3. Let I be an upper-bound interval and outSI = (AS.st• = ℓ1 ∧Alast\nS .x• ∈I). Then, for all\ni ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′\ni\nto Ci+1.\nIn the automaton Afirst\nS\nof Figure 6 (right), clock y maintains the time since the earliest\nb2 which may serve as a witness for b1 S b2: clock y is reset when reading b2 while AS is in\nstate ℓ0 (the previous position did not satisfy b1 S b2) or when reading ¬b1 ∧b2 while AS is\nin state ℓ1. For a lower-bound interval SI where I = [b, +∞) or I = (b, +∞) with b ∈N, we\ncheck the value of y, i.e., the time since the earliest q. The output is 1 if the target state of\nAS is ℓ1 and y ∈I.\n▶Lemma 11. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′\nthere is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (AS, Afirst\nS\n). Moreover, for all\ni ≥0:\n1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,\n2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Afirst\nS\n.y) = τi −τj where j is the earliest witness at\ni of b1 S b2: j ≤i, b2 ∈aj and b1 ∈ak for all j < k ≤i, and for all 0 ≤j′ < j either\nb2 /∈aj′ or b1 /∈ak′ for some j′ < k′ ≤i.\n3. Let I be a lower-bound interval and outSI = (AS.st• = ℓ1 ∧Afirst\nS\n.y• ∈I). Then, for all\ni ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′\ni\nto Ci+1.\n4.2.3\nSince with a two-sided interval.\nThe general case with a nonempty non-singleton interval I is much harder. We may have\na position satisfying b1 SI b2 but neither the earliest nor the last witness of b2 satisfy both\nthe upper and lower constraints of I. In such a case, there could still be another witness in\nbetween which does satisfy the constraint. We need to change our perspective and look at\neach b2 event as a potential witness for b1 SI b2. However, we cannot simply start a new clock\nat each b2 event, as we only have finitely many clocks at our disposal. We discuss below our\nsolution for this general case.\nLet I be a non-singleton interval with lower bound b and upper bound c. We assume\n0 /∈I and c ̸= +∞, otherwise it is a simple upper-bound or lower-bound constraint which\nwas already solved in Section 4.2.2. Since both b, c are integers and I is non-singleton, we\nhave c −b ≥1.\nt\nt + (c −b)\nt′\nt + b\nt + c\nt′ + b\nt′ + c\nFigure 7 Illustrating the idea for SI\nThe idea of our approach is illustrated in\nFigure 7. Assume there is a b2 at t, represen-\nted by the red dot. Assuming I = [b, c], this\nb2 is a witness for the events happening in\nthe interval [t + b, t + c] (marked by the red\nline), assuming all the intermediate points\ncontain b1. Now, consider the last b2-event in the interval [t, t + (c −b)]. This is marked\nby the green dot in the figure, at time t′. This b2 event acts as a witness for the events in\n[t′ + b, t′ + c], marked by the green line in the figure. For any point to the right of t′ + c,\na valid b2 witness should necessarily be to the right of t + (c −b). This observation leads\nto the following construction: when the first b2 is seen (the red dot), as long as there is a\n\n14\nEfficient Verification of MTL with Past in Pointwise Semantics\ncontinuous sequence of b1’s following it, break the timeline into blocks of c −b and store the\nearliest and latest b2 within these blocks. When the values become sufficiently large, the\nearliest block becomes irrelevant, allowing to reuse clocks. Coupled with c −b ≥1, we are\nable to implement this idea with a bounded number of clocks.\nLet I be a non-singleton interval with lower and upper bounds b, c ∈N and 0 /∈I. Since\nI is non-singleton, we have c −b ≥1. Let k = 2 +\nj\nb\nc −b\nk\n.1 The automaton Agen\nSI\nfor\nb1 SI b2 has a single location ℓ0 and uses clocks x1, y1, . . . , xk, yk. The initial condition at\nℓ0 is x1 = y1 = · · · = xk = yk = +∞(all clocks are initially inactive). Automaton Agen\nSI is\nsynchronized with AS (untimed since) from Figure 6 (left). Its transitions are defined in\nAlgorithm 1 with ◁= ≤if b ∈I and ◁= < otherwise.\nAlgorithm 1 Transitions of automaton Agen\nSI .\n1: if AS.st• = ℓ0 then\n▷b1 S b2 does not hold\n2:\nx1, y1, . . . , xk, yk := +∞\n3: else if ¬b•\n1 ∨AS.st = ℓ0 then\n▷b1 S b2 holds (AS.st• = ℓ1), new earliest witness\n4:\nx1, y1 := 0; x2, y2, . . . , xk, yk := +∞\n5: else\n▷b1 S b2 holds (AS.st• = ℓ1) and there are past witnesses (b•\n1 ∧AS.st = ℓ1)\n6:\nif b ◁xn ̸= +∞∧(n = k ∨¬(b ◁xn+1 ̸= +∞)) then\n▷Clock shift with 1 < n ≤k\n7:\n[x1, y1, . . . , xk, yk] := [xn, yn, . . . , xk, yk, +∞, . . . , +∞]\n8:\nend if\n9:\nif b•\n2 then\n▷new witness (not earliest)\n10:\nif xj < c −b then yj := 0\n▷1 ≤j ≤k\n11:\nif c −b ≤xj ̸= +∞∧xj+1 = +∞then xj+1, yj+1 := 0\n▷1 ≤j < k\n12:\nelse\n▷not a new witness (¬b•\n2)\n13:\nSkip\n14:\nend if\n15: end if\nNotice that the automaton is deterministic and complete. Consequently, every word will\nhave exactly one run in the automaton. We give some intuitions before the formal statement\nin Lemma 12.\nThe test of Line 1 is satisfied when the current position does not satisfy b1 S b2. In this\ncase, the new clocks are not needed and we set them to +∞, meaning inactive.\nWhen the test of Line 3 succeeds, the current position satisfies b1 S b2 and there are no\nother witness in the past. So we set x1, y1 to 0 and the other clocks are inactive.\nIf we have b ◁xn for some active clock with n > 1 then xn satisfies the lower bound con-\nstraint of I and it may serve as a witness instead of the earlier clocks x1, y1, . . . , xn−1, yn−1\n(notice that xn < yn−1 ≤xn−1 < · · · < y1 ≤x1 and if one of these clocks satisfies the\nupper bound constraint of I then so does xn). In this case, clocks x1, y1, . . . , xn−1, yn−1\nare not needed anymore and we reduce the set of active clocks with the clock shift in\nLines (6–8).\nWhen the test of Line 9 succeeds, the existing witnesses stay and the current position is a\nnew witness. Let x1, y1, . . . , xj, yj be the active clocks.\n1 If the interval I is not of the form (b, c) (i.e., open on both sides) then we can construct an automaton\nAgen\nSI with k = 1 +\nl\nb\nc −b\nm\nby changing xj < c −b on Line 12 and c −b ≤xj on Line 13 to xj ≤c −b\nand c −b < xj respectively.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n15\nIf xj < c −b (Line 10) then we simply reset yj to 0 since the current position is now\nthe last witness in the block of length c −b which started when xj was reset. Otherwise\nxj ≥c −b (Line 11) and the current position starts a new block by setting xj+1, yj+1 to\n0 (new pair of active clocks). When this occurs, we have j < k thanks to the clock shift\nof Lines (6–8).\nThe last possibility (Lines 12,13), the current position satisfies b1 S b2 but is not a new\nwitness since b2 doest not hold. In that case, we do not update the clocks since the set of\nwitnesses does not change.\n▶Lemma 12. Let I be a non-singleton interval with lower and upper bounds b, c ∈N and\n0 /∈I. Let ◁= ≤if b ∈I and ◁= < otherwise. Let Prop′ = Prop ⊎{b1, b2}. For every word\nw = (a0, τ0)(a1, τ1) · · · over Prop′ there is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · ·\nof (AS, Agen\nSI ).\nMoreover, let outSI = (AS.st• = ℓ1 ∧(Agen\nSI .x•\n1 ∈I ∨Agen\nSI .y•\n1 ∈I)). Then, for all i ≥0,\nwe may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′\ni to Ci+1.\nThe proof of Lemma 12 will be given after the following technical lemma which establishes\nall the necessary invariants of the automaton Agen\nSI .\n▶Lemma 13. Let I be a non-singleton interval with lower and upper bounds b, c ∈N and\n0 /∈I. Let ◁= ≤if b ∈I and ◁= < otherwise. Let Prop′ = Prop ⊎{b1, b2}. For every word\nw = (a0, τ0)(a1, τ1) · · · over Prop′ there is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · ·\nof (AS, Agen\nSI ). Moreover, for all i ≥0 we have:\n1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1.\n2. When Ci+1(AS.st) = ℓ0 then configuration Ci+1 satisfies x1 = y1 = · · · = xk = yk = +∞.\n3. When Ci+1(AS.st) = ℓ1 then for some 1 ≤j ≤k, configuration Ci+1 satisfies the following:\nyj ≤xj < yj−1 ≤xj−1 < · · · < y1 ≤x1 < +∞, xj+1 = yj+1 = · · · = xk = yk = +∞,\nxj′ −yj′ < c −b for all 1 ≤j′ ≤j, and xj′ −xj′+1 ≥c −b for all 1 ≤j′ < j,\nthere are indices i1 ≤i′\n1 < i2 ≤i′\n2 < · · · < ij ≤i′\nj ≤i such that\nb1 ∈ai′ for all i′ ∈(i1, i],\nb2 ∈ai′ for all i′ ∈{i1, i′\n1, . . . , ij, i′\nj},\nb2 /∈ai′ for all i′ ∈(i′\n1, i2) ∪· · · ∪(i′\nj−1, ij) ∪(i′\nj, i],\nx1 = τi −τi1, y1 = τi −τi′\n1, . . . , xj = τi −τij, yj = τi −τi′\nj,\nb ◁x1 or (w, i1) ̸|= b1 ∧Y(b1 S b2). Moreover, if j > 1 then b ̸ ◁x2.\nProof. 1. was already stated in Lemma 10. 2. follows directly from Lines (1–2) of Algorithm 1.\n3. The proof is by induction on i. The base case is i = 0. The initial state of AS is ℓ0,\ni.e., C0(AS.st) = ℓ0. Now, C1(AS.st) = ℓ1 iff b2 ∈a0. In this case, Line 4 of Algorithm 1 is\ntaken and we easily check that item (3) of Lemma 12 holds with j = 1 and i1 = i′\n1 = 0.\nConsider now i > 0 and assume that item (3) hold for i −1. If Ci(AS.st) = ℓ0 then we\nargue exactly as in the base case above. The interesting case is Ci(AS.st) = ℓ1. By induction,\nthere is 1 ≤j ≤k and indices i1 ≤i′\n1 < i2 ≤i′\n2 < · · · < ij ≤i′\nj ≤i −1 satisfying all the\nconditions of item (3) for i −1.\nIf b1 /∈ai then Line 4 of Algorithm 1 is taken. Notice that b2 ∈ai since the selfloop on ℓ1\nof AS is taken. We easily check that item (3) of Lemma 12 holds with j = 1 and i1 = i′\n1 = i.\nIt remains to deal with the case Ci+1(AS.st) = Ci(AS.st) = ℓ1 and b1 ∈ai.\nAssume first that no clock shift occurs: we have b ̸ ◁C′\ni(xj′) for all 1 < j′ ≤j.\n\n16\nEfficient Verification of MTL with Past in Pointwise Semantics\nIf b2 ∈ai then Lines (10–11) of Algorithm 1 are executed.\nRecall that C′\ni satisfies xj′ −xj′+1 ≥c −b for all 1 ≤j′ < j. Hence, if C′\ni satisfies\nxj′ < c −b it must be with j′ = j. In this case, Line 10 resets clock yj to 0. We see easily\nthat Ci+1 satisfies Item 3 with the same j and the sequence of indices i1 ≤i′\n1 < i2 ≤i′\n2 <\n· · · < ij ≤i′′\nj = i. Note that the last index in the sequence has been updated: i′′\nj = i. Since\nxj < c −b when yj is reset, Ci+1 satisfies xj −yj < c −b.\nAssume now that C′\ni satisfies xj ≥c −b. Assume towards a contradiction that j = k.\nThen, C′\ni satisfies x2 −xk ≥(k −2)(c −b). Since we had no clock shift, we also have x2 ≤b.\nTogether with c −b ≤xk we get (k −1)(c −b) ≤x2 ≤b. This is a contradiction with\nk = 2 +\nj\nb\nc −b\nk\n. Hence, j < k and Line 11 will reset clocks xj+1, yj+1 to 0. We easily check\nthat Item 3 is satisfied with j + 1 and the sequence of indices i1 ≤i′\n1 < i2 ≤i′\n2 < · · · <\nij ≤i′\nj ≤ij+1 = i′\nj+1 = i. In particular, Cj+1 satisfies xj −xj+1 = xj ≥c −b. And also, if\nj + 1 = 2, we have b ̸ ◁0 = x2 at Ci+1 since 0 /∈I.\nIf b2 /∈ai then Line 13 of Algorithm 1 is taken and we easily check that item (3) holds for\ni with the same j and indices i1 ≤i′\n1 < i2 ≤i′\n2 < · · · < ij ≤i′\nj. In particular, since clocks\nare not updated, for all 1 ≤j′ ≤j, we get Ci+1(xj′) = Ci(xj′) + τi −τi−1 = τi −τij′ and\nsimilarly for yj′.\nFinally, assume that a clock shift occurs: the test of Line 6 evaluates to true with some\n1 < n ≤j. Let j′′ = j −n + 1. Then, 1 ≤j′′ < j and after the clock shift we have xj′′ ̸= +∞\nand xj′′+1 = +∞. We can easily check that, after the clock shift, Item 3 holds with j′′ and\nthe sequence of indices in ≤i′\nn < in+1 ≤i′\nn+1 < · · · < ij ≤i′\nj ≤i. In particular, the value of\nx1 after the clock shift is the value of xn at C′\ni, and b is smaller than (◁) this value due to\nthe condition of the clock shift. Similarly, if j′′ > 1 then b is smaller than (◁) the value of x2\nafter the clock shift which is the value of xn+1 at C′\ni. Therefore, the last bullet of Item 3\nholds after the clock shift, hence also at Ci+1.\nNow, we can prove as above, in the case of no clock shift, that after executing Lines (9–14)\nof Algorithm 1, Item 3 holds at Ci+1. This concludes the inductive proof of Item (3).\n◀\nProof of Lemma 12. Fix i ≥0. By Lemma 13, position i satisfies the untimed since b1 S b2\niff Ci+1(AS.st) = ℓ1 (corresponding to AS.st• = ℓ1 in outSI). We assume that we are in\nthis case. The only question is whether there is a witnessing b2 within the interval I. We\nclaim that this can be determined by looking solely at the values of clocks x1 and y1 at\nconfiguration Ci+1 (corresponding to the test Agen\nSI .x•\n1 ∈I ∨Agen\nSI .y•\n1 ∈I in outSI).\nWe can view this as follows. Suppose there is in the past a b2 event witnessing b1 SI b2:\nb2 ∈ai′′ for some i′′ ≤i and b1 ∈ai′ for all i′′ < i′ ≤i and τi −τi′′ ∈I. Let 1 ≤j ≤k and\ni1, i′\n1, . . . , ij, i′\nj be the indices given by Lemma 13 (3).\nIf i′′ < i1 then (w, i1) |= b1 ∧Y(b1 S b2). Therefore in this case we have b ◁x1 at Ci+1.\nNow, b ◁τi −τi1 ≤τi −τi′′ ∈I. We deduce that i1 is also a witness of (w, i) |= b1 SI b2.\nIf i′\n1 < i′′ then we must have j ≥2. Moreover, i2 ≤i′′ since there are no b2 in (i′\n1, i2).\nUsing b ̸ ◁x2 at Ci+1, we deduce that b ̸ ◁τi −τi2 and then b ̸ ◁τi −τi′′, a contradiction with\nτi −τi′′ ∈I.\nFinally, if i1 < i′′ < i′\n1 then using x1 −y1 < c −b at Ci+1 we get τi′\n1 −τi1 < c −b. Using\nτi1 ≤τi′′ ≤τi′\n1 and τi −τi′′ ∈I, we deduce that either i1 or i′\n1 (or both) is also a witness of\n(w, i) |= b1 SI b2.\n◀\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n17\n4.3\nA full example\nWe explain the construction of a network NΦ associated with a sentence Φ on an ex-\nample. We use the template automata defined in the previous sections. We instantiate the\nformal arguments b1, b2 of these template automata with actual test formulas. For instance,\nAY[b•\n1/(¬p• ∧q•)] denotes a copy of the automaton AY from Figure 5 (left) in which we\nsubstitute the test ¬p• ∧q• for the formal argument b•\n1.\nWe consider the sentence Φ = (Φ1 ∧Φ2) →Φ3 where Φ1 = r and\nΦ2 = FI6(p ∧φ4) = true UI6 (p ∧φ4)\nΦ3 = FI7(φ3 ∧φ5) = true UI7 (φ3 ∧φ5)\nφ4 = (p ∨q) SI4 φ3\nφ5 = (φ1 ∨φ2) SI5 (q ∨φ3)\nφ1 = YI1(¬p ∧q)\nφ2 = YI2(¬p ∧q)\nφ3 = YI3 r\nwith I4 = [0, c4) an upper bound constraint and I5 = [c5, +∞) a lower bound constraint.\nThe network NΦ does not contain automata for the boolean connectives. Instead, we\nuse the power of tests in automata, which allow arbitrary boolean combinations of atomic\ntests.\nA temporal subformula is either an atomic proposition or a formula where the\noutermost connective is a temporal modality (Yesterday, Since, neXt or Until). The temporal\nsubformulas of the sentence Φ are the atomic propositions p, q, r, the point formulas φ1, . . . , φ5\nand the sentences Φ2, Φ3.\nThe network NΦ does not need automata for the atomic propositions since they are\nowned by the model M. For each temporal subformula, we add to the network the automata\nrequired to determine the truth value of the subformula. These automata are described\nin the previous sections. For instance, the temporal subformula φ4 requires a copy of the\nautomata AS and Alast\nS\nfrom Figure 6 (with suitable substitutions of the formal arguments).\nLemma 10 provides the test expression which allows to determine the truth value of φ4.\nFor the sentence Φ = (Φ1 ∧Φ2) →Φ3, we construct the network\nNΦ = (B0, B1, B2, B3, B4, B5, B6, B7, A1, A2, A3)\nwhere\nB1 := Alast\nB2 := AY[b•\n1/(¬p• ∧q•)]\nout1 = (B2.st = ℓ1 ∧B1.xlast ∈I1)\nout2 = (B2.st = ℓ1 ∧B1.xlast ∈I2)\nB3 := AY[b•\n1/r•]\nout3 = (B3.st = ℓ1 ∧B1.xlast ∈I3)\nB4 := AS[b•\n1/(p• ∨q•), b•\n2/out3]\nB5 := Alast\nS [b•\n2/out3]\nB6 := AS[b•\n1/(out1 ∨out2), b•\n2/(q• ∨out3)]\nB7 := Afirst\nS\n[b•\n1/(out1 ∨out2), b•\n2/(q• ∨out3)]\nout4 = (B4.st• = ℓ1 ∧B5.x• ∈I4)\nout5 = (B6.st• = ℓ1 ∧B7.y• ∈I5)\nB0 := Ainit\nA1 := Aisat\nr\nA2 := Aisat\nUI6 [b•\n1/true, b•\n2/(p• ∧out4)]\nA3 := Aisat\nUI7 [b•\n1/true, b•\n2/(out3 ∧out5)]\nIn the automata above, we use ghost variables (outi)1≤i≤5 to make the definitions easier to\nread. When we expand these definitions, we get for instance\nB4 = AS[b•\n1/(p• ∨q•), b•\n2/(B3.st = ℓ1 ∧B1.xlast ∈I3)]\nA2 := Aisat\nUI6 [b•\n1/true, b•\n2/(p• ∧(B4.st• = ℓ1 ∧B5.x• ∈I4))]\nNotice that the automaton B2 is shared: it is used in out1, out2. Similarly, B0 and B1 are\nshared.\n\n18\nEfficient Verification of MTL with Past in Pointwise Semantics\nIt remains to define the set BadΦ of bad configurations for the sentence Φ. A configuration\nC is bad if Φ evaluates to false at C. Hence, BadΦ is the set of configurations C which satisfy\nA1.st = 2 ∧A2.st = 2 ∧¬(A3.st = 2) (i.e., Φ1 ∧Φ2 ∧¬Φ3).\n4.4\nProof of Theorem 5\nWe give now the general construction for an arbitrary sentence Φ. Let φ1, . . . , φn be the\ntemporal point subformulas of Φ other than the atomic propositions. We assume that if φj\nis a subformula of φi then j ≤i. Let Φ1, . . . , Φm be the temporal subsentences of Φ. We\ndefine the network\nNΦ = (B0, B1, B2, . . . , B2n−1, B2n, A1, . . . , Am)\ntogether with the ghost variables out1, . . . , outn. For each 1 ≤i ≤n, we define B2i−1, B2i\nand outi as follows:\nIf φi = YIi φ′\ni then we let B2i = Alast. The automaton B2i−1 is a copy of the untimed\nAY with a suitable substitution of its formal argument b1 that we define now. The\nformula φ′\ni is a boolean combination of temporal subformulas. Let φ′′\ni be the same boolean\ncombination in which an outermost temporal subformula φj is replaced with outj and an\noutermost atomic proposition p is replaced with p•. For instance, if φ′\ni = (q ∧φ2) ∨φ4\nthen φ′′\ni = (q• ∧out2) ∨out4. Then, B2i−1 = AY[b•\n1/φ′′\ni ]. Finally, outi = (B2i−1.st =\nℓ1 ∧B2i.xlast ∈Ii).\nIf φi = φ′\ni SIi ψ′\ni and I is an upper-bound interval. Then we let B2i−1 = AS[b•\n1/φ′′\ni , b•\n2/ψ′′\ni ],\nB2i = Alast\nS [b•\n2/ψ′′\ni ] and outi = (B2i−1.st• = ℓ1 ∧B2i.x• ∈Ii).\nIf φi = φ′\ni SIi ψ′\ni and I is a lower-bound interval. Then we let B2i−1 = AS[b•\n1/φ′′\ni , b•\n2/ψ′′\ni ],\nB2i = Afirst\nS\n[b•\n1/φ′′\ni , b•\n2/ψ′′\ni ] and outi = (B2i−1.st• = ℓ1 ∧B2i.y• ∈Ii).\nIf φi = φ′\ni SIi ψ′\ni and I is a non-singleton interval which is neither an upper-bound nor a\nlower-bound. Then we let B2i−1 = AS[b•\n1/φ′′\ni , b•\n2/ψ′′\ni ], B2i = Agen\nSIi [b•\n1/φ′′\ni , b•\n2/ψ′′\ni , AS.st/B2i−1.st]\nand outi = (B2i−1.st• = ℓ1 ∧(B2i.x•\n1 ∈Ii ∨B2i.y•\n1 ∈Ii)).\nNext, we let B0 = Ainit and for 1 ≤i ≤m we define\nIf Φi = p is an atomic proposition, then we let Ai = Aisat\np\n.\nIf Φi = XJi φ′\ni then we let Ai = Aisat\nXJi [b•\n1/φ′′\ni ].\nIf Φi = φ′\ni UJi ψ′\ni then we let Ai = Aisat\nUJi [b•\n1/φ′′\ni , b•\n2/ψ′′\ni ].\nFinally, as explained in Section 4, the set BadΦ of bad configurations consists of all configur-\nations C such that Φ evaluates to false when we replace each temporal subsentence Φi of Φ\nby Ai.st = ℓi where ℓi is the state labelled true in Ai (ℓi = 3 if Φi is a next subformula, and\nℓi = 2 otherwise).\nNow that we have constructed the deterministic network NΦ, we prove its correctness.\n▶Lemma 14. Let Φ be a sentence in detMITL+p. The network NΦ constructed above is\ndeterministic and complete. Hence, for every word w = (a0, τ0)(a1, τ1) · · · over Prop there is\na unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of NΦ. Moreover,\n1. for all temporal point subformula φj of Φ (with 1 ≤j ≤n) and for all position i ≥0 in\nw, we have (w, i) |= φj iff evaluating the test outj during transition ti from C′\ni to Ci+1\nyields true.\n2. w ̸|= Φ iff there is k ≥0 such that Ci ∈BadΦ for all i ≥k.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n19\nProof. All automata used in the network NΦ are deterministic and complete. Hence, so is\nthe network and each timed word has indeed a unique run.\n1. The proof is by induction on j. It follows easily from the correctness lemmas in the\nprevious sections: Lemma 9 for yesterday with an arbitrary interval, Lemma 10 for since\nwith an upper-bounded interval, Lemma 11 for since with a lower-bounded interval and\nLemma 12 for since with a non-singleton interval which is neither upper-bounded nor\nlower-bounded.\n2. Using (1) and the lemmas for initial satisfiability (Lemma 6, Lemma 7 and Lemma 8), we\ndeduce that for each temporal subsentences Φj of Φ (with 1 ≤j ≤m), we have w |= Φj iff\nthe automaton Aj eventually stabilises in a state labelled true, i.e., state 2 for an atomic\nproposition or for an until modality, and state 3 for next. Notice that each automaton\nAj must eventually stabilise in some state since each cycle in these automata are actually\nself-loops. Now, using the definition of BadΦ, we deduce that w ̸|= Φ iff the run for w\neventually stays in bad configurations.\n◀\n▶Remark 15. The number of automata in the network NΦ is O(|Φ|). The automata B2i\nhave a single state and own clocks, the automata B2i−1 have two states and do not own\nclocks, and the automata Ai have 3 or 4 states and do not own clocks. The number of clocks\nowned by B2i is 1 unless it comes from some SI where I is a two-sided interval with lower\nand upper bounds b, c ∈N such that c−b ≥1. In that case, the number of clocks is 2k where\nk = 2 +\nj\nb\nc −b\nk\n≤2 + b. Hence, the total number of clocks in the network is O(Φ) if Φ does\nnot use since with two-sided intervals, or if k = O(1) for all since with two-sided intervals, or\nif the constants are given in unary (actually, only the lower constants of two-sided intervals).\n▶Remark 16. We reduce further the size of the network NΦ by removing duplicates. For\ninstance, if Φ has several occurrences of Y, possibly with different arguments and intervals,\nthe automaton Alast occurs as many times in the network constructed above. In practice,\nwe use a single occurrence of Alast adapting accordingly the atomic tests xlast ∈Ii in the\ncorresponding ghost variables outi. We also remove duplicate occurrences of AY automata\nand of since automata (AS, Alast\nS , Afirst\nS\n).\n5\nIntegrating the Future\nOur goal in this section is to include future modalities in the logic and obtain an efficient\nmodel checking procedure for a general fragment of MTL with past and future. To do so,\nwe consider the logic detMITL+p from Definition 4 and allow Point formulas to also have\nXI and UI′ operators. This results in a restriction of MTL in which, except in the topmost\nlevel, the intervals in Since and Until are restricted to be [0, 0] or non-singleton. We call this\nfragment MITL+p. This fragment subsumes the well-known MITL fragment and even goes\nbeyond extending it with past.\nWe recall that in the pointwise semantics, MTL model checking is undecidable over infinite\nwords, and over finite words it is decidable but the complexity is Ackermanian [7, 50, 48, 49].\nHence, the literature often considers the MITL fragment in the pointwise semantics, with\nmultiple known approaches for building efficient model checking algorithms. Of these we\nhighlight two recent ones, both of which only work with future modalities. The first from [20],\nworks by translating MITL into a network of timed automata (via 1-clock alternating\nautomata) and was implemented in a state-of-the-art tool MightyL. The second from [2],\nconverts MITL into generalized timed transducers, which use future clocks defined in [1].\nHowever, this approach has not yet been implemented.\n\n20\nEfficient Verification of MTL with Past in Pointwise Semantics\nIn the rest of this section, we combine ideas from [2] and from [20], with our ideas for\ninitial satisfiability and past modalities presented in the previous sections, to develop a new\nconstruction from formulas in MITL+p directly to network of generalized timed automata\nwith shared variables, without the explicit use of transducers. As we are dealing with future\nmodalities, non-determinism cannot be avoided, but we show two algorithmic improvements,\nboth designed to reduce non-deterministic branching. Our experimental results later showcase\nimproved performance (compared to either of the two approaches above) in number/type\nof benchmarks solved and time taken, thus highlighting the significant impact of these\nimprovements.\n5.1\nIncluding Future clocks in the Model\nFirst, we lift the definitions to include future clocks. Thus the set of clocks X is now partitioned\ninto history clocks XH, which are just clocks as we saw till now and XF , future clocks, that\ntake non-positive values. A future clock takes values in the interval R−= (−∞, 0] ∪{−∞},\nwhere value −∞signifies that this clock is undefined or inactive. Thus, a generalized timed\nautomaton with shared variables (GTA) is a tuple A = (Q, X, V, ∆, init), where X = XH ·∪XF\nis the set of clocks owned by A.\nNow, an atomic update for a future clock variable y ∈XF owned by A is non-deterministic\n(unlike to normal/history clocks) and has the form A.y :∈I (or simply y :∈I), for some\ninteger bounded interval I with end-points in Z−= {0, −1, −2, . . . , −∞}. Further, an atomic\ntest for y ∈XF owned by A has the form A.y ∈I (or simply y ∈I) where I ⊆R−is an\ninteger bounded interval. For simplicity, in examples we often write −y ∈I with endpoints\nin N instead. As before A may also test a future clock variable owned by another component\nwith N.y ∈I or N.y• ∈I where N is a name. As before a network of GTA with shared\nvariables is just a tuple A = (A1, . . . , An) where each Ai is a GTA with shared variables.\nThe semantics of a network of GTA is the same as for network of TA as before with two\nkey differences: First, in the definition of a configuration C for any 1 ≤i ≤n, we have in\naddition C(Ai.y) ∈R−for y ∈XF\ni . Second, a time elapse transition C\nδ−→C′ can be taken\nonly if (i) C′ coincides with C on all variables other than clocks, (ii) the value of each clock\n(regardless of whether it is future or history) is advanced by the same quantity δ and (iii)\nC(Ai.y) + δ ≤0 for all 1 ≤i ≤n and y ∈XF\ni (with −∞+ δ = −∞). The third condition is\nnew and means that an elapse can occur only if it does not force a future clock to take a\npositive value (same requirement as in [1, 2] for future clocks).\nFinally, we recall that our TA (and GTA) are always strongly non-Zeno, i.e., every\naccepting run is non-Zeno (a timed run is Zeno if P\ni≥0 δi is bounded). As remarked in\nLemma 5 of [2], this assumption (not a severe restriction, since every GTA can be converted\neasily to a strongly non-Zeno one) has interesting consequence for future clocks: if they are\nnot ultimately −∞, they should be released infinitely often. If not, there is a last point\nwhere a future clock is released to a finite value, and the entire suffix of the run lies within\nthis finite time, contradicting non-Zenoness.\n5.2\nModel checking the future\nFor the model checking procedure, the idea at the high level is as before. Given a formula in\nMITL+p, we translate it to a network of GTA with shared variables and then build a closed\nnetwork with a model and check existence of a bad cycle of configurations. As explained\nin [2], the definition of a bad cycle now also has to take into account future clocks. The goal\nof this section is to prove the following theorem.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n21\nℓ0\nℓ1\nAX\n¬b•\n1\nb•\n1\nb•\n1\n¬b•\n1\n1\nAnext\nxnext = 0; release(xnext)\nFigure 8 Left: Automaton AX for the untimed Next operator X b1. The boolean variable b1 is\nnot owned by AX and stands for the argument of X.\nRight: Automaton Anext with clock xnext used (shared) by all Next operators.\n▶Theorem 17. Let Φ be a MITL+p sentence and M a timed model with history and future\nclocks. Then, we can construct a network of GTA with shared variables NΦ of size linear\nin size of Φ (assuming unary encoding of constants in the formula)2 such that M ̸|= Φ iff\nthere exists a reachable cycle of the closed network (M, NΦ), which eventually remains in\nbad configurations and where every future clock x of the network is either released during the\ncycle or is undefined −∞.\n5.2.1\nNext with sharing.\nAs for Yesterday, for any occurrence of the Next operator XI, we have an automaton depicted\nin Figure 8 (left), which has two states ℓ0 and ℓ1 and captures the untimed semantics of\nNext. The states guess whether the argument b1 of X will be true or not at the next letter\nwithout worrying about the timing constraints. Then, we use a single state automaton Anext,\ndepicted in Figure 8 (right), which releases clock xnext whenever it reaches 0. Thus, it always\npredicts the time to the next event. It will be shared across all occurrences of Next in the\nentire network, to check all the timing constraints. Now, for an arbitrary interval constraint\nI, the current position satisfies XI b1 if the next letter has a b1 and the value of clock xnext\n(released here) falls in the interval I. Formally,\n▶Lemma 18. Let Prop′ = Prop ⊎{b1}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′,\nwe have a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (AX, Anext). Moreover, for all\ni ≥0,\n1. (w, i) |= X b1 iff Ci+1(AX.st) = ℓ1,\n2. τi+1 −τi ∈I iff Ci+1(Anext.xnext) ∈I.\nLet outXI = (AX.st• = ℓ1 ∧−Anext.x•\nnext ∈I).\nThen, for all i ≥0, we may check if\n(w, i) |= XI b1 with the test outXI during transition ti from C′\ni to Ci+1.\n5.2.2\nUntil with sharing.\nThe formula b1 UI b2 asks that there exists a position j in the future (there could be many\nof them), such that b2 is true at that position, and b1 is true at all positions between the\ncurrent position and j, and the time elapsed between the current position and position j\nlies in the interval I. As for Since, we first consider the simpler case where intervals are\none-sided, i.e., of the form [0, c), [0, c], [b, +∞) or (b, +∞). Again it is sufficient to track the\ntwo extreme b2 witnesses: the earliest witness and the last possible witness at each point. As\n2 In fact, we can also do the same with binary encoding of constants with one caveat: the lower bounds\non intervals used in Since and Until operators should be bounded by a constant.\n\n22\nEfficient Verification of MTL with Past in Pointwise Semantics\nℓ0\nAfirst\nU\nb•\n2;\nx = 0; release(x)\n¬b•\n2\nℓ0\nℓ1\nAU\nAfirst\nU .x ̸= −∞\n¬b•\n2\n¬b•\n1 ∧¬b•\n2\nb•\n2\nb•\n1 ∨b•\n2\nℓ0\nAlast\nU\nb•\n2 ∧(¬b•\n1 ∨AU.st• = ℓ0);\ny = 0; release(y)\n¬b•\n2 ∨(b•\n1 ∧AU.st = ℓ1)\nFigure 9 Left: Afirst\nU\nguessing time to the earliest b2 with future clock x. release(x) stands for the\nnon-deterministic update x :∈R−, allowing x to be set to any non-positive value, including −∞.\nMiddle: AU for untimed until but which uses future clock x in the invariant of state ℓ1.\nRight: Alast\nU\nguessing with future clock y time to the latest witness of b1 UI b2.\nshown in Figure 9, we introduce three automata: (i) Afirst\nU\nwhich owns and uses future clock\nx to track the time to the first b2, (ii) Alast\nU\nwhich owns and uses future clock y and tracks\nthe time to the latest witness of b1 U b2 and (iii) a GTA AU which captures the semantics of\nthe untimed Until operator, using clock x in the invariant of state ℓ1.\nAs we are dealing with future, all three automata are non-deterministic. However, the\nonly non-determinism in Afirst\nU\nand Alast\nU\nlies in the non-deterministic updates (release). The\nzone automata3 built from them are actually deterministic since the release operation results\nin a unique zone.\nUnlike in the previous section, we cannot hope to get a completely\ndeterministic network since with future modalities we must have non-determinism (or pay an\nexponential price to remove non-determinism). But we can try to limit the non-deterministic\nbranching for better performance in practice. Towards this, our networks of GTA with shared\nvariables for each future modality allow sharing of clocks and predictions (compared e.g.,\nto using combined transducers as done in [2]). The advantage of using different automata\nis that we can now share the prediction of the earliest witness without insisting to share\nthe latest witness at a point. This results in increased sharing and thus further decrease in\nnon-deterministic branching. Our experimental results in the next section validate these\ntheoretical insights.\n▶Remark 19. The most interesting automaton is AU in Figure 9 (middle), which uses clock x\nfrom Afirst\nU , but does not own any clock. Notice that it is a two-state automaton and does not\nrequire any Büchi acceptance condition. In fact, in the untimed setting (i.e., in the absence\nof clocks), Until requires some Büchi acceptance condition and 3 states if the acceptance\ncondition is state-based. Lifting this to the timed setting gives a 3-state transducer with a\nBüchi acceptance condition in [2]. However, we show in this paper that, surprisingly, 2 states\nare sufficient and no acceptance condition is required, by exploiting clocks and non-zenoness.\nThe main idea is that in state ℓ1, we employ an invariant4 implying that b2 will eventually\noccur. Under the non-zenoness assumption, stating that clock x has a finite (not −∞) value,\nforces clock x to be eventually released. Hence, Afirst\nU\nmust eventually take the loop labeled\nb2. We highlight that going from 3 states to 2 states represents an exponential improvement\nin non-deterministic branching and runtime as we go from 3O(|Φ|) to 2O(|Φ|) in the number of\nzones that we will have to explore during reachability checking. Also, reducing the number\nof Büchi acceptance conditions makes the liveness algorithm more efficient. Further, we also\n3 Zone automata are finite abstractions of (generalized) timed automata, used for efficient algorithms\nchecking reachability [27, 1]\n4 Note that, though we did not introduce state invariant in the model, it can easily be simulated with\nguards on all incoming transitions –for succinctness we chose to present it this way.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n23\nnote that if we remove the invariant, and use of the clock, we obtain the automaton for\nuntimed Weak-Until.\nThe following lemma shows correctness of the above construction:\n▶Lemma 20. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′\nthere is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (Afirst\nU , AU, Alast\nU ). Moreover, for\nall i ≥0:\n1. (w, i) |= b1 U b2 iff Ci(AU.st) = C′\ni(AU.st) = ℓ1,\n2. If I is an upper-bound interval, we set outUI = (AU.st = ℓ1 ∧−Afirst\nU .x ∈I). Then, for all\ni ≥0, we may check if (w, i) |= b1 UI b2 with the test outUI during transition ti from C′\ni\nto Ci+1.\n3. If I is a lower-bound interval, we set outUI = (AU.st = ℓ1 ∧−Alast\nU .y ∈I). Then, for all\ni ≥0, we may check if (w, i) |= b1 UI b2 with the test outUI during transition ti from C′\ni\nto Ci+1.\nFor the most difficult case of the General Until operator, we note that it follows as a dual\nof the Since case from Section 4.2.3. To avoid repetitiveness, we omit a formal description of\nthis. Finally, combining all the above lemmas, we may prove Theorem 17 following the same\nlines of the proof of Theorem 5 in Section 4.4.\n▶Remark 21. Note that Theorem 5 and Theorem 17 can be extended to the case of finite\nwords, where we want to reach a bad configuration instead of eventually remaining in bad\nconfigurations. The only modification required concerns the future modalities operator. We\ncan use the same automata defined above, but the definition of a bad configuration C for\nfinite words asks, in addition to the existing conditions, that (1) for each occurrence of AX\nthe final state is ℓ0 (C(AX.st) = ℓ0) and (2) for each occurrence of AU the final state is ℓ0\n(C(AU.st) = ℓ0).\n6\nImproved simulation check and liveness for GTA\nIn this section, we describe our technical contributions that have gone into the development\nof the LiveRGTA tool. We start with an overview of these contributions.\nA liveness algorithm for timed automata or GTA computes a symbolic abstraction called\nthe zone graph. Zone graphs are known to be infinite and termination mechanisms for zone\ncomputations have seen a long line of work (see [15] for a survey). A node in the zone graph is\nof the form (q, Z) where q is a control state of the network of automata and Z is a constraint\nsystem on clocks that represents the set of configurations abstracted by the current node. For\nreachability, termination is guaranteed by using finite simulation relations, which are binary\nrelations on nodes (q, Z) ⪯(q, Z′). For liveness, to ensure soundness, simulation needs to be\nreplaced with a mutual simulation: (q, Z) ⪯(q, Z′) and (q, Z′) ⪯(q, Z). The procedure for\nchecking (q, Z) ⪯(q, Z′) is the most used operation in the zone graph computation and it is\nextremely crucial to have efficient algorithms for the same. When the automaton contains no\ndiagonal constraints (guards of the form x −y ◁c), the check can be done in O(|X|2) where\nX is the total number of clocks. When diagonal constraints are present, the check is known\nto be NP-hard, and it has a running time of O(|X|2 · 2d) where d is the number of diagonals\npresent. Essentially, the check calls the diagonal-free module at most 2d times.\nIn the LiveRGTA tool, we have implemented an improved simulation check for GTAs with\nfuture clocks, compared to the simulation check of [1]. This results in coarser simulations and\nsmaller parts of the zone graph being explored. There are cases where the gain is exponential,\n\n24\nEfficient Verification of MTL with Past in Pointwise Semantics\nas for instance with GTAs of the form of Figure 10. We also use a new direct algorithm for\nthe mutual simulation check, that bypasses the need to call the basic simulation check twice.\nThis reduces the computation time by half.\nℓ0\nℓ1\nℓ2\n· · ·\nℓn−1\nℓn\nz = −1\nx1 = −3\nx1 = −1\nx2 = −3\nx2 = −1\nxn = −3\nxn = −1\nFigure 10 A GTA that shows the exponential improvement produced smaller G-sets.\nFinally, another important contribution is that we lift and adapt Couvreur’s algorithm\n[26] for Büchi liveness to the setting of generalized timed automata.\nWe present below our procedure for the liveness problem of safe GTAs. Decidability of\nliveness for safe GTAs was recently shown in [2, Theorem 11]. A GTA is live, i.e., it admits\nan infinite accepting non-Zeno run, iff in its zone graph there is a reachable accepting cycle,\ni.e., a reachable zone (q, Z) and a path (q, Z)\nt1\n−→(q1, Z1)\nt2\n−→· · · (qk, Zk)\ntk+1\n−−−→(q, Z′) such\nthat (q, Z) ⪯(q, Z′) and (q, Z′) ⪯(q, Z), each Büchi condition is witnessed along this cycle,\nand each future clock x ∈XF is either released along the cycle or could take value −∞\n(there is a valuation vx ∈Z with vx(x) = −∞).\nThere are several ways to implement this liveness check.\nOur choice is to extend\nCouvreur’s SCC algorithm [26, 55, 31] (the state-of-the-art approach for checking liveness of\ntimed automata) to the setting of GTAs. Our procedure can be thought of as an adaptation\nof the idea for checking liveness of GTA proposed in [2] to the well-studied framework of\nCouvreur’s SCC algorithm. On top of the usual liveness algorithm, we need to perform extra\nchecks to determine if future clocks are released in SCCs.\nThe idea for checking liveness in [2] ultimately boils down to identifying cycles in zone\ngraphs that satisfy certain special properties. Couvreur’s SCC algorithm provides an optimal\nmethod for detecting cycles. Algorithm 2, given in Appendix B, is an SCC-based liveness\nalgorithm to check if a GTA A has an accepting cycle with respect to some generalized Büchi\nacceptance condition. It leverages the various optimizations of Couvreur’s SCC algorithm [26]\nthat makes liveness checking more efficient for timed automata. Notice that, to the best\nof our knowledge, Couvreur’s algorithm has not been previously extended to event-clock\nautomata or timed automata with diagonal constraints. Hence, Algorithm 2 is the first\nextension of Couvreur’s liveness procedure for these classes.\n7\nImplementation and Experimental evaluation\nIn this section, we present an experimental evaluation of our prototype implementation tool\nTempora for satisfiability and model checking for MITL+p formulas. The tool contains two\nmodules:\nMITL2GTA that implements our optimized translation from detMITL+p to networks\nof deterministic timed automata (from Section 4) and the optimized translation from\nMITL+p to networks of GTA (from Section 5), using the improvements (sharing and\ntwo-state until) for future modalities.\nLiveRGTA where we have implemented the new SCC-based liveness algorithm (discussed\nin Section 6) and an improved reachability algorithm for GTA. This is an enhanced version\nof GTA from [1], built on top of the open-source timed automata library Tchecker [34].\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n25\nBy combining these two modules, we obtain a new pipeline Tempora(MITL2GTA+LiveRGTA)\nfor satisfiability and model checking for MITL+p formulas (with both past and future modal-\nities) over timed words under pointwise semantics.\nWe organize our evaluation into three parts as follows:\nFirst, we evaluate the performance of the detMITL+p to deterministic timed automata\ntranslation on a set of benchmarks, which include both standalone past modalities and\ncombinations with future modalities. To the best of our knowledge, in the pointwise\nsemantics, there are no other tools that tackle this fragment.\nNext, we compare our MITL+p to GTA translation against two competing approaches in\nthe pointwise semantics, namely:\n1. MightyL [20]: a state-of-the-art tool that translates MITL to OCATA and then to\nnetworks of timed automata. The resulting automata are analyzed using UPPAAL [41]\n(for finite words) and OPAAL [37] (for infinite words).\n2. Baseline GTA Translation an implementation of the (theoretical) translation from\nMITL to GTA proposed in [2].\nFinally, to demonstrate the full scope of our implementation, we present our full model-\nchecking pipeline on two classical examples, namely Fischer and Dining Philosophers with\nMITL+p formulae capturing some standard properties.\nExperimental setup\nOur tool Tempora takes as input a MITL+p formula and applies the appropriate translation\ndescribed in Sections 4 and 5, producing a network of GTA (which is deterministic, when\nthe formula is in detMITL+p). Then, checking satisfiability of a formula over finite timed\nwords in the pointwise semantics reduces to checking emptiness of the corresponding network\nof GTA. We also adapted our translation to work with finite timed words as explained in\nRemark 21. We use an improved version of GTA reachability algorithm from [1], which is\nbuilt on the open source tool Tchecker [34]. Over infinite words, checking satisfiability\nreduces to checking Büchi acceptance in GTA. For this, we implement the liveness algorithm\nfrom [2], again on top of TChecker. Finally, we implemented a prototype model-checking\nalgorithm for MITL+p formulae w.r.t. a network of GTA with the pointwise semantics.\nAll experiments were run on an Ubuntu 18.04 machine with an Intel i7 3.40GHz processor\nand 32 GB RAM. In this section, we describe the benchmarks and present an analysis of the\nresults. Details of our implementation, including the various engineering optimisations that\nwe employ, are discussed in Appendix C.\nBenchmarks. We conduct our experiments on a suite of 72 benchmarks, which we have\ngrouped into three groups: (1) benchmarks designed to demonstrate past operators (alongside\nfuture modalities), (2) benchmarks from MightyL [20] that are themselves based on examples\nfrom [32, 51], and (3) benchmarks from Acacia synthesis tool’s benchmark suite [13].\nGroup 1. This group consists of 22 MITL formulae designed to highlight various strengths of\nour translation. These include formulae featuring conjunctions of temporal operators, the\nsimultaneous use of past and future modalities (for which, to the best of our knowledge, no\nexisting tool supports), and other representative constructs. We rewrite the derived operators\nG, F, R in terms of U, in the standard way. We use the following shorthand for common\npatterns: ρ(n) = X(n) ∧G(p1 ∨p2), where X(n) = X[2,3]\n\u0000(p1\n=⇒X(n −1)) ∧(p2\n=⇒\n¬ X(n −1))\n\u0001\nwith base case of X(1) = p1, η(n) = G(Vn\ni=1(pi =⇒(ai S[0,2] bi))) ∧G(Wn\ni=1 pi),\nReq-grant = G[2,∞)\n\u0000(R =⇒F[4,5] G)\n\u0001\n, and Spot-ex = G[0,10] F[1,2](a ⇐⇒X b).\nGroup 2. This group contains all 33 benchmarks from [20], accompanying the tool MightyL.\nThese are based on temporal patterns from [32, 51], and include families of parameterized\n\n26\nEfficient Verification of MTL with Past in Pointwise Semantics\nformulae: F(k, I) = Vk\ni=1 FI pi, G(k, I) = Vk\ni=1 GI pi, U(k, I) = (· · · (p1 UI p2) · · · ) UI pk,\nand R(k, I) = (· · · (p1 RI p2) · · · ) RI pk, µ(k, I) = Vk\ni=1 F[3i−1,3i] ti ∧G ¬p and θ(k, I) =\n¬((Vk\ni=1 G F pi) =⇒G(q =⇒FI r)).\nGroup 3.\nThis group includes 17 benchmarks drawn from the Acacia synthesis tool’s\nbenchmark suite [13], which are also featured in the SyntComp competition suite [29], with\nadded timing intervals. These include formulae such as\nAcacia1 =\n\u0000F[0,2] G p\n\u0001\n⇐⇒(G F q) ,\nAcacia2 =\n\u0000G\n\u0000F[0,2]\n\u0000p =⇒X[1,2] X[1,2] X[1,2] q\n\u0001\u0001\u0001\n⇐⇒(G F r) ,\nAcacia3 =\n\u0000((p U[0,2] q) U[0,2] r) ∧((q U[0,2] r) U[0,2] p) ∧((r U[0,2] p) U[0,2] q)\n\u0001\n⇐⇒G F s,\nAcacia4 =\n\u0000G(p =⇒F[3,4] q) ∧G(¬p =⇒F[3,4] ¬q)\n\u0001\n⇐⇒G F r.\nThe complete list of all 72 benchmark formulae are given in Tables A and B of Appendix C.\nOur results\nWe present the results of our experiments in Tables 1 to 3, which show the number of\nnodes stored and the time taken by each tool to check satisfiability of the formulae over\nfinite and infinite words, respectively. We also present a log-scale plot of the time taken\nby MightyL and Tempora in Figure 11, which shows the performance of our tool on the\nvarious benchmarks. We discuss our experimental results in detail below. In all the tables, we\nhighlight in green the rows corresponding to formulae belonging to the detMITL+p fragment.\nSatisfiability for the past fragment of detMITL+p. The first set of experiments, presented\nin Table 1, evaluates the performance of our tool Tempora on detMITL+p formulae involving\npast operators. We find that Tempora successfully handles all benchmarks in this category.\nWe note that MightyL could not be run on these, as it does not support past modalities\n(listed as part of future work in [20]). The last but two formulae are specifically designed to\nstress-test the handling of multiple nested past operators. Tempora processes each of them\nefficiently, storing only a small number of nodes and completing each run in under 30 seconds.\nThe final formula in this set is more complex, combining several past operators with a\nconjunction of future modalities. Here too, Tempora performs efficiently, requiring only 186\nnodes and less than 2 ms to check satisfiability. These results demonstrate the effectiveness of\nour deterministic translation, which avoids the exponential blowup typically associated with\nnon-deterministic approaches and enables scalable analysis even for complicated formulae. To\nsee this, consider, for instance, the formula F(pU[1,2]q∧pU[2,3]q∧pU[3,4]q∧pU[4,5]q), obtained\nby replacing the past operator S with the future operator U in the 5th entry Tempora\nsolves this by generating 28,030 nodes in 784 ms. Another example is the nested formula\nF(p U[1,2] (p U[1,2] (p U[1,2] q))) which results in 299,130 nodes and takes 9 seconds, compared\nto the past version F(p S[1,2] (p S[1,2] (p S[1,2] q))) which generates only 1,303 nodes in 29.9\nmilliseconds.\nSatisfiability for full MITL+p and comparisons. For the second set of experiments, we\nevaluate the performance of Tempora on the satisfiability problem for MITL+p formulae\nover both finite and infinite timed words. We compare our tool against MightyL and\nthe baseline implementation of the translation proposed in [2]. For finite timed words, we\ncompare against MightyL using UPPAAL’s reachability analysis; for infinite timed words,\nwe use MightyL with OPAAL’s liveness checking algorithm.\nFigure 11(left) compares the performance of our tool and the MightyL pipeline on all\n73 benchmarks for finite timed words, and Figure 11(right) presents the same comparison\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n27\nFormula\nTempora\nStored nodes\nTime in ms.\nF[0,20](Y[2,3] p1 ∨Y[4,5] p2) ∨Y[6,7] p3)\n287\n1.5\nG[0,20](Y[2,3] p1 ∨Y[4,5] p2) ∨Y[6,7] p3)\n262\n3.1\nF(p S[1,2] (p S[1,2] (p S[1,2] q)))\n1,303\n29.9\nF(p S[1,∞) (p S[1,∞) (p S[1,∞) q)))\n66\n0.5\nF(p S[1,2] q ∧p S[2,3] q ∧p S[3,4] q ∧p S[4,5] q)\n1,571\n50.9\nF(p S[1,∞) q ∧p S[2,∞) q ∧p S[3,∞) q ∧p S[4,∞) q)\n68\n0.8\nF(p S[1,2] q ∧p S[2,3] q ∧p S[3,4] q ∧p S[4,5] q ∧p S[5,6] q)\n27,737\n1,762.1\nF(p S[1,2] q ∧p S[2,3] q ∧p S[3,4] q ∧p S[4,5] q ∧p S[5,6] q ∧p S[6,7] q)\n167,077\n21,490.5\nG(¬p1 ∨(a1 S[0,2] b1)) ∧(¬p2 ∨(a2 S[0,2] b2))∧\n(¬p3 ∨(a3 S[0,2] b3)) ∧(¬p4 ∨(a4 S[0,2] b4)) ∧G(p1 ∨p2 ∨p3 ∨p4)\n186\n1.67\nTable 1 Experimental results obtained by running our prototype implementation Tempora for\nsatisfiability of detMITL+p formulae over finite timed words. Timeout is set to 300 seconds.\nfor infinite words. The time reported is the time after the translation (to TA in case of\nMightyL and GTA for Tempora), that is, the time for zone graph exploration to find a\nreachable node (for finite timed words) or a live cycle (for infinite timed words). This is\nbecause, the time for the translation itself is rather small in most cases (of the order of\n3-4ms), except for the cases shown in Table 4 in Appendix C. For OPAAL, we report the\nsum of time taken by OPAAL to generate C++ code, the g++ compiler to compile the\ngenerated code and LTSmin to check liveness. We note that OPAAL is a multi-core tool\nand uses 8 cores on our machine. On 10 of the benchmarks, OPAAL gave a segmentation\nfault (which we mark as timeout) and UPPAAL reported an overflow error on the output\ngenerated by MightyL for 2 benchmarks.\nTable 2 presents selected results reporting the number of nodes stored and time taken by\neach tool to check satisfiability over infinite words, while the full set of results appears in\nAppendix C\nAs can be seen from Figure 11, in the finite timed words case, Tempora is always faster\nthan MightyL + UPPAAL and for benchmarks where both tools take more than 1s, we are\nsignificantly faster, as the plot is in log-scale. Further, UPPAAL times out in several cases,\nwhile we never do so on the benchmarks tested. Thus, we scale better on larger and more\ncomplex formulae. For the infinite case, except for 2 formulae we are always better. Note that\nthe plot is only on the 63 benchmarks on which all three tools work, and therefore, we exclude\nthe formulae containing past operators. Our performance is significantly better in terms\nof time-taken as can be seen from the figure. Further, out of the 72 benchmarks, OPAAL\ntimed out or seg-faulted on 18 benchmarks whereas we timeout on only 4 benchmarks (of\nFigure 11 Log-scale plot MightyL vs Tempora comparison for time taken to check satisfiability\non finite (left) and infinite (right) timed words, after the translation.\nGroup 1 refers to our\nbenchmarks, while Group 2 are MightyL benchmarks [20] are Group 3 benchmarks derived from\nAcacia [13]. Timeout set to 300s.\n\n28\nEfficient Verification of MTL with Past in Pointwise Semantics\nFormula\nBaseline\nMightyL+ OPAAL\nTempora\nStored\nTime in\nStored\nTime in\nStored\nTime in\nnodes\nms.\nnodes\nms.\nnodes\nms.\nF(5, [2, ∞))\n-\nTO\n3069\n373 + 820 + 85\n200\n2.38\nF(5, [0, 2])\n952\n5.99\n2693\n214 + 702 + 82\n68\n0.49\nU(5, [2, ∞))\n567,192\n5,404\n9,383\n382 + 960 + 114\n4,075\n32.70\nU(5, [1, 2])\n-\nTO\n-\nTO\n465,185\n8,750\nµ(2)\n953,492\n24,977\n964\n68,462 + 20,584+ 103\n158\n0.90\nµ(3)\n-\nTO\n-\nTO\n315\n3.34\nθ(3)\n-\nTO\n42,804\n836 + 1,057 + 225\n330,044\n2977\nG(5, [0, 2])\n4,793\n34.6\n5,913\n257 + 760+ 101\n224\n1.45\nG(5, [1, 2])\n-\nTO\n-\nTO\n225\n2.45\nR(5, [2, ∞))\n29,685\n278.3\n1,059\n231 + 726 + 77\n3,119\n27.75\nR(5, [1, 2])\n-\nTO\n-\nTO\n1,093,187\n23,474\np U[11,12] q\n1,484\n53\n-\nTO\n220\n1.37\nReq-grant\n-\nTO\n-\nTO\n444\n4.00\nSpot-ex\n117,319\n1,141\n7,921\n74,585 + 20,184 + 238\n504\n3.50\nη(4)\n-\nTO\n-\nNA\n249\n1.49\nρ(4)\n-\nTO\n-\nTO\n1933\n15.93\nAcacia1\n-\nTO\n4,943\n258 + 755 + 96\n390\n2.54\nAcacia2\n-\nTO\n742,427\n426 + 949 + 1,683\n231\n1.94\nAcacia3\n-\nTO\n7,271,276\n611 + 1,143 + 21,349\n918\n7.41\nAcacia4\n-\nTO\n-\nTO\n4,159\n55.71\nTable 2 Selected set of experimental results obtained by running our prototype implementation\nTempora, MightyL and, the baseline implementation of the translation proposed in [2] for\nsatisfiability over infinite traces. Timeout is set to 300 seconds. The complete table is available\nin Table 8 in Appendix C.\nwhich OPAAL also times out/segfaults on 3). Finally, Table 2 (and the extended tables\nin Appendix C) show that Tempora is also significantly better than the Baseline (in fact\nthe MightyL pipeline also outperforms Baseline significantly on most benchmarks) on all\nbenchmarks.\nIn summary, both for finite words and infinite words over the pointwise semantics,\nTempora significantly outperforms MightyL and the baseline for checking satisfiability,\nboth in terms of time taken and size of stored nodes.\nModel checking. Finally, going beyond satisfiability, we evaluate our model-checking\npipeline on a set of MITL+p formulas and two models: Fischer’s mutual exclusion algorithm\nand the Dining Philosophers problem.\nThe model-checking pipeline is implemented in\nTempora and uses Tchecker for the zone graph exploration. To the best of our knowledge,\nthere is no existing tool among UPPAAL, MightyL, and OPAAL that can perform complete\nmodel checking for pointwise semantics with a model and a MITL+p formula. As a result,\nwe are unable to provide a comparison for our model-checking pipeline. Nevertheless, as a\nproof of concept, we evaluate our pipeline on two models (Fischer and Dining Philosophers)\nagainst a set of formulas, with results in Table 3.\nFor the Fischer’s mutual exclusion algorithm, we consider a model with 3 to 7 processes and\ncheck various properties. The first property checks whether the first process can eventually\nenter the waiting state after making a request, while the second property checks whether\nit can enter the critical section. We also check the classical safety property for mutual\nexclusion protocols that ensures that no two processes can be in the critical section at the\nsame time. Our experimental results show that for Fischer with 6 processes, the property\nG(req1 →F[0,5] wait1) can be checked to hold in 4.6 ms, while for Fischer with 7 processes, it\ntakes 5.3 ms. We also check the property that no two processes can be in the critical section\nat the same time, which is reported to hold true for Fischer with 5 processes in 15.51 seconds.\nFor the Dining Philosophers problem, we consider models with 2 and 8 philosophers. We\ncheck properties such as whether at least one philosopher is eating at some point in time, and\nwhether both philosophers are guaranteed to eat within a certain window of time. For the\nmodel with 8 philosophers, we find that the property F[0,20](W8\ni=1 eatingi) does not hold true\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n29\nSl.\nModels\nFormula\nTempora+ Tchecker\nNo.\nStored nodes\nTime (ms)\n1\nFischer (3)\nG(¬req1 ∨F[0,20] wait1)\n32,817\n221.3\n2\nFischer (4)\nG(¬req1 ∨F[0,20] wait1)\n449,108\n5,723\n3\nFischer (6)\nG(¬req1 ∨F[0,5] wait1)\n362\n4.6\n4\nFischer (7)\nG(¬req1 ∨F[0,5] wait1)\n391\n5.3\n5\nFischer (3)\nG(¬req1 ∨F[0,20] cs)\n33,918\n209.69\n6\nFischer (4)\nG(¬req1 ∨F[0,20] cs)\n352,774\n3,749.6\n7\nFischer (5)\nG(¬((cs1 ∧(cs2 ∨cs3 ∨cs4 ∨cs5))∨\n(cs2 ∧(cs1 ∨cs3 ∨cs4 ∨cs5)) ∨(cs3 ∧(cs1 ∨cs2 ∨cs4 ∨cs5))∨\n(cs4 ∧(cs1 ∨cs2 ∨cs3 ∨cs5)) ∨(cs5 ∧(cs1 ∨cs2 ∨cs3 ∨cs4))))\n527,829\n15,514\n8\nDin-Phil(8, 10, 3, 1)\nF[0,20](eating1 ∨eating2 ∨eating3 ∨eating4\n∨eating5 ∨eating6 ∨eating7 ∨eating8)\n371\n5.8\n9\nDin-Phil(2, 10, 3, 1)\nG[10,∞)(F[0,30](eating1) ∧F[0,30](eating2))\n4,904\n31.35\nTable 3 Experimental results obtained by running our prototype implementation Tempora, and\nthe GTA liveness algorithm implementation in Tchecker for checking liveness. The numbers in the\nbrackets represent parameterization, for Fischer’s it is #-processes and for Dining Philosophers, it\nrepresents the number of philosophers, timeout before releasing fork, time to eat and delay time to\nrelease fork.\nwithin 0.01 seconds, indicating that no philosopher is guaranteed to eat within 20 time units.\nFor the model with 2 philosophers, we check whether both philosophers are guaranteed to\neat within a certain time window, which does not hold true within 0.03 seconds. The results\nindicate that our model-checking pipeline can efficiently handle the verification of MITL+p\nproperties on these models, demonstrating its potential for practical applications in verifying\ntimed systems.\nAdditional results, including ablation studies, are presented in Section C.4 of Appendix C.\nSummary of Experimental Evaluation.\nIn summary, our experiments demonstrate the practicality and effectiveness of Tempora\nin various settings. Tempora handles all standard MITL benchmarks used in prior works,\nand crucially extends support to the richer MITL+p fragment - including past operators and\nextending timed modalities to allow punctual constraints at the top level. This is beyond\nthe capabilities of state-of-the-art tools for pointwise MITL logics such as MightyL, which\ndo not support past operators or the punctual timed operators. Further, on benchmarks\nwhere MightyL does work, Tempora outperforms it significantly both in the size of\nzone graph computed and time taken to check satisfiability. Finally, Tempora can be\neffectively used for model checking full MITL+p specifications, including both safety and\nliveness properties on standard benchmarks such as Fischer’s mutual exclusion protocol and\nthe Dining Philosophers.\n8\nConclusion\nIn this paper, we developed a new synchronization-based model checking technique that\nallows translating formulas in MITL with past modalities (and punctuality at the outermost\nlevel) to networks of small (generalized) timed automata with the general theme of making\nit “more deterministic”. While we get deterministic automata for a large fragment, we show\nimprovements even in the general case, especially a surprising 2-state until automaton that\ntakes advantage of future clocks. All the technical developments have resulted in the tool\nTempora, which outperforms the current state-of-the-art tools. 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Then, we know that p ∈a0, i.e., the atomic\nproposition p holds at the first position of the word. By construction, Aisat\np\n(Figure 2), the\nautomaton starts in state 1 and checks whether p holds at the first position. If p ∈a0, then\nAisat\np\ntakes the transition to state 2, and from then on, continues to remain in state 2 on all\nsubsequent inputs. Hence, configuration Ci satisfies Aisat\np\n.st = 2 for i ≥1.\nIn the other direction, suppose that for all i ≥1, the configuration Ci satisfies Aisat\np\n.st = 2.\nSince the only way for the automaton to reach state 2 from the initial state 1 is via a transition\nthat checks p ∈a0, it must be the case that p ∈a0. Therefore, w |= p.\n◀\n▶Lemma 7. Let Prop′ = Prop ⊎{b1} be an augmented set of atomic propositions. For\nevery word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0\nδ0,t0\n−−−→C1\nδ1,t1\n−−−→C2 · · ·\nof (Ainit, Aisat\nXI ) over w. Moreover, w |= XI b1 iff for all i ≥2, configuration Ci satisfies\nAisat\nXI .st = 3.\nProof. Suppose that w |= XI b1. Then, this means that the timed word w satisfies (1)\nb1 ∈a1, and (2) δ1 = τ1 −τ0 ∈I. Consider the run of the automaton (Ainit, Aisat\nXI ) over w.\nThe automaton starts in state 1 at configuration C0. From state 1, the automaton moves to\nstate 2 after reading the first letter (a0, τ0), resetting clock xinit. Then, from state 2, since b1\nholds at the next letter a1 and the clock value of xinit is δ1 = τ1 −τ0 ∈I, the automaton\nmoves to state 3. Since state 3 is a sink-state with a self-loop, Aisat\nXI\nremains in state 3 for\nall i ≥2. Therefore, for all i ≥2, the configuration Ci satisfies Aisat\nXI .st = 3.\nConversely, suppose that for all i ≥2, the configuration Ci satisfies Aisat\nXI .st = 3. By\nconstruction of Aisat\nXI , state 3 can only be reached from state 2 if (a1, τ1) satisfies the following:\nb1 ∈a1 and xinit ∈I. Further, since xinit was reset on reading (a0, τ0), this implies that\nτ1 −τ0 ∈I. Therefore, w |= XI b1.\n◀\n▶Lemma 9. Let Prop′ = Prop ⊎{b1}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′,\nwe have a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (Alast, AY). Moreover,\nfor all i ≥0, (w, i) |= Y b1 iff Ci(AY.st) = C′\ni(AY.st) = ℓ1,\nfor all i ≥1, τi −τi−1 ∈I iff C′\ni(Alast.xlast) ∈I.\nLet outYI = (AY.st = ℓ1 ∧Alast.xlast ∈I). Then, for all i ≥0, we may check if (w, i) |= YI b1\nwith the test outYI during transition ti from C′\ni to Ci+1.\nProof. Since the network is deterministic and complete, there is a unique run of the network\n(Alast, AY) on the given input word w.\nNote that the network (Alast, AY) maintains two components:\nAlast stores the time elapsed since the previous position using the clock xlast.\nAY allows to check whether the formula Y b1 holds by storing the value of b1 at the\nprevious position.\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n35\nBy construction, the state of AY is updated as follows. If b1 was true at ai−1, then after\nprocessing (ai−1, τi−1), AY enters state ℓ1 and remains there at both Ci and C′\ni. Similarly, if\nb1 was not true at ai−1, then after processing (ai−1, τi−1), AY enters state ℓ0 and remains\nthere at both Ci and C′\ni.\nSimilarly, Alast resets xlast upon reading each letter of the input word. Hence, at C′\ni, we\nhave Alast.xlast = δi = τi −τi−1. Therefore, for all i ≥1, τi −τi−1 ∈I iff C′\ni(Alast.xlast) ∈I.\nThe formula YI b1 is true at i iff\nb1 held at position i −1, which translates to AY being in state ℓ1 at configuration C′\ni,\nτi −τi−1 ∈I (i.e., Alast.xlast ∈I at C′\ni).\nTherefore, the test AY.st = ℓ1 ∧Alast.xlast ∈I precisely checks (w, i) |= YI b1 during the\ntransition ti from C′\ni to Ci+1.\n◀\n▶Lemma 10. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′\nthere is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (AS, Alast\nS ). Moreover, for all\ni ≥0:\n1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,\n2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Alast\nS .x) = τi −τj where j ≤i, b2 ∈aj and b2 /∈ak\nfor all j < k ≤i.\n3. Let I be an upper-bound interval and outSI = (AS.st• = ℓ1 ∧Alast\nS .x• ∈I). Then, for all\ni ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′\ni\nto Ci+1.\nProof.\n1. From the semantics of the untimed S operator, (w, i) |= b1 S b2 if there exists 0 ≤j ≤i\nsuch that b2 ∈aj, and b1 ∈ak for all j < k ≤i.\nFrom Figure 6, it can be seen that AS moves to state ℓ1 upon seeing b2, and stays in ℓ1 as\nlong as b1 ∨b2 holds. AS returns to ℓ0 if it sees a position for which b1 ∨b2 does not hold.\nThus, if Ci+1(AS.st) = ℓ1, there exists some j ≤i where b2 ∈aj and if we take the largest\nsuch j, at all positions in (j, i], b1 holds, which implies that (w, i) |= b1 S b2.\nConversely, suppose that (w, i) |= b1 S b2. This means, that there was some position j ≤i\nsuch that (w, j) |= b2, and at all positions between j and i, b1 holds. Then, from the\nconstruction, AS must be in state ℓ1 at Ci+1.\n2. Alast\nS .x is reset whenever b2 is observed, and is not updated until the next occurrence of\nb2. Thus, Alast\nS\nstores the total time elapsed since the most recent occurrence of b2. We\nhave already seen from the previous item that if Ci+1(AS.st) = ℓ1, then there exists some\nj ≤i where b2 ∈aj. Therefore, when Ci+1(AS.st) = ℓ1, the last observed b2 was at some\nposition j ≤i with no b2 between j and i, then Ci+1(Alast\nS .x) = τi −τj.\n3. Recall that b1 SI b2 holds at (w, i) if and only if the following two conditions are satisfied\nthe untimed condition for b1 S b2 is true, which from (i) we know is the case if the\nautomaton AS is in state ℓ1 at configuration Ci+1, and\nthe timed constraint is satisfied, i.e., τi −τj ∈I, where j is the most recent position\nwhere b2 was true. This holds because I is an upper-bounded interval. From (2), we\nknow that when Ci+1(AS.st) = ℓ1, we have Ci+1(Alast\nS .x) = τi −τj.\nFrom the above, it is easy to see that the test outSI = (AS.st• = ℓ1 ∧Alast\nS .x• ∈I) is\nsatisfied during transition ti from C′\ni to Ci+1 exactly when (w, i) |= b1 SI b2.\n◀\n\n36\nEfficient Verification of MTL with Past in Pointwise Semantics\n▶Lemma 11. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′\nthere is a unique run C0\nδ0\n−→C′\n0\nt0\n−→C1\nδ1\n−→C′\n1\nt1\n−→C2 · · · of (AS, Afirst\nS\n). Moreover, for all\ni ≥0:\n1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,\n2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Afirst\nS\n.y) = τi −τj where j is the earliest witness at\ni of b1 S b2: j ≤i, b2 ∈aj and b1 ∈ak for all j < k ≤i, and for all 0 ≤j′ < j either\nb2 /∈aj′ or b1 /∈ak′ for some j′ < k′ ≤i.\n3. Let I be a lower-bound interval and outSI = (AS.st• = ℓ1 ∧Afirst\nS\n.y• ∈I). Then, for all\ni ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′\ni\nto Ci+1.\nProof.\n1. This is already shown in the proof of Lemma 10 given above.\n2. First, notice that if Ci+1(AS.st) = ℓ1, the automaton AS moved from ℓ0 to ℓ1 in the\npast, i.e., with some ti′ with i′ ≤i. During this transition ti′, the automaton Afirst\nS\ntakes\nthe loop which resets y. Hence, y was reset at least once in the past of i. Let j ≤i\nbe the position where y was last reset by Afirst\nS\n. We have b2 ∈aj and either b1 /∈aj or\nCj(AS.st) = C′\nj(AS.st) = ℓ0.\nNow we focus on the two parts of the statement that we need to show.\nWe already saw that b2 ∈aj. Next, we show that b1 ∈ak for all j < k ≤i. Assume\nthat b1 /∈ak for some j < k ≤i. If b2 ∈ak then Afirst\nS\ntakes the loop which resets y\nduring transition tk, a contradiction with the maximality of j. Hence, b2 /∈ak and\ntransition tk takes AS to state ℓ0. So we must have some k < i′ ≤i such that ti′ takes\nAS from ℓ0 to ℓ1. As explained above, Afirst\nS\nmust take the loop which resets y during\ntransition ti′, again a contradiction with the maximality of j. Therefore, b1 ∈ak for\nall j < k ≤i.\nFor the second part, suppose towards a contradiction that there was a position\n0 ≤j′ < j such that b2 ∈aj′ and b1 ∈ak′ for all j′ < k′ ≤i. Since b2 ∈aj′ and\nb1 ∈ak′ for all j′ < k′ ≤i, automaton AS must be in state ℓ1 from Cj′+1 to Ci+1.\nSince j′ < j ≤i, automaton AS is in state ℓ1 at configurations Cj and C′\nj. Further,\nwe know that y was reset by Afirst\nS\nduring transition tj. Since C′\nj(AS.st) = ℓ1, this is\nonly possible if b1 ̸∈aj, a contradiction. Therefore, for all 0 ≤j′ < j either b2 /∈aj′ or\nb1 /∈ak′ for some j′ < k′ ≤i.\n3. The proof proceeds as in the proof of Lemma 10, as we have shown the untimed and\ntimed aspects separately above. For the sake of completeness, we give the arguments\nbelow: b1 SI b2 holds at (w, i) if and only if the following two conditions are satisfied\nthe untimed aspect follows from the fact that automaton AS is in state ℓ1 at configur-\nation Ci+1 (shown in the first part), and\nthe timed constraint is satisfied, i.e., τi −τj ∈I, where j is the earliest position where\nb2 is seen such that this b2 can act as a witness for b1 S b2: This holds because I is\na lower-bounded interval. From (ii), we know that when Ci+1(AS.st) = ℓ1, we have\nCi+1(Afirst\nS\n.y) = τi −τj.\nFrom the above, it is easy to see that the test outSI = (AS.st• = ℓ1 ∧Afirst\nS\n.y• ∈I) is\nsatisfied during transition ti from C′\ni to Ci+1 exactly when (w, i) |= b1 SI b2.\n◀\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n37\nB\nAppendix for liveness\nFor the purpose of the algorithm, we consider the finite graph G defined as the zone graph\nof A quotiented by the mutual simulation relation. The finiteness and soundness of this\nabstraction for liveness was shown in [2].\nAlgorithm 2 SCC decomposition based algorithm for GTA liveness with generalized Büchi\nacceptance condition.\n1: Global variables\n2: count: Integer initialized to 0\n3: Roots: Stack storing tuples (η, labels, releasedclocks) where η is a potential root\nnode, labels is the set of labels seen in SCC rooted at η, and releasedclocks is the\nset of clocks released in SCC rooted at η\n4: CurrentReleasedClocks: Stack storing the set of future clocks that have been\nreleased in the current SCC\n5: Active: Stack storing the nodes which are active in the SCC exploration of the current\nroot node\n6: Todo: Stack storing pairs (η, succ) where η is a node and succ is a list of successor\nnode and transition.\n7: function GTASCC(G, L)\n▷Outputs if there is a live component in G which visits all\nlabels from L\n8:\nfor Each initial node η in G do Dfs(η, L) end for\n9: end function\n10: function Dfs(η, L)\n▷Perform DFS starting from η\n11:\nPush(η, ∅)\n12:\nwhile Todo is not empty do\n13:\n(ν, succ) ←Todo.top()\n14:\nif succ is empty then\n15:\nif (ν, _, _) = Roots.top() then\n▷SCC rooted at u is explored\n16:\nClose(s)\n17:\nend if\n18:\nTodo.pop()\n19:\nelse\n20:\n(t, χ) ←succ.FirstAndRemove()\n▷Explore along transition ν\nt−→χ\n21:\nif χ.dfsnum = 0 then\n▷New node\n22:\nPush(χ, released(t))\n▷released(t) is the set of future clocks released in t\n23:\nelse if χ.current then\n▷Cycle detected\n24:\nCurrentReleasedClocks.push(released(t))\n25:\nMergeSCC(χ, L)\n26:\nend if\n27:\nend if\n28:\nend while\n29: end function\n\n38\nEfficient Verification of MTL with Past in Pointwise Semantics\n30: function Push(η, Xrel) ▷η is a new node, Xrel is the set of clocks released on transition\nto η\n31:\ncount ←count + 1; η.dfsnum ←count; η.current ←true\n32:\nActive.push(η)\n33:\nTodo.push((η, next(η)))\n▷next(η) returns the successors of η\n34:\nCurrentReleasedClocks.push(Xrel)\n35:\nRoots.push((η, labels(η), ∅))\n36: end function\n37: function Close(η)\n▷Closes the SCC of η\n38:\nRoots.Pop()\n39:\nCurrentReleasedClocks.Pop()\n40:\nµ ←null\n41:\nrepeat\n42:\nµ ←Active.top()\n43:\nµ.current = false\n44:\nActive.pop()\n45:\nuntil µ ̸= η\n46: end function\n47: function MergeSCC(v, L)\n48:\nA ←∅, R ←∅\n49:\n(µ, _, _) ←Roots.top()\n50:\nrepeat\n51:\nc ←CurrentReleasedClocks.Top()\n52:\nCurrentReleasedClocks.Pop()\n53:\nR ←R ∪c\n54:\n(µ, l, c) ←Roots.Top()\n55:\nRoots.Pop()\n56:\nA ←A ∪l, R ←R ∪c\n57:\nuntil µ.dfsnum > η.dfsnum\n58:\nif L ⊆A and XF = R ∪InfClocks(µ) = XF then\n▷InfClocks(µ) is the set of\nfuture clocks which can take value −∞in zone of u\n59:\noutput Accepting SCC detected\n60:\nend if\n61:\nRoots.Push((µ, A, R))\n62: end function\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n39\nC\nAppendix for Implementation and Experimental evaluation\nC.1\nImplementation details.\nOur tool, Tempora, takes as input a MITL+p formula ψ and generates a synchronous network\nof generalized timed automata with shared variables based on the translation described in\nSections 4 and 5. The execution of this network is simulated using an array of shared variables.\nEach generalized timed automaton writes its output to a dedicated shared variable, which\ncan then be accessed by its parent automaton. Synchronization is enforced by sequentially\nexecuting each automaton in bottom-to-top order. A shared control variable tracks the\nactive automaton during execution, and a transition in an automaton is executed when its\nID matches the value of the control variable. Upon taking the transition, the control variable\nis updated to the ID of the next automaton. A controller is used to reset the execution\nafter every automaton takes a transition and alternates between time elapse and automaton\nexecution. A master automaton evaluates the top-level boolean subformula, ensuring the\nformula is satisfied at the start and, for finite timed words, that all automata have performed\na transition in the last round.\nTo avoid constructing automata for boolean operators, we use symbolic labels to represent\ntemporal operators and atomic propositions, generating boolean expressions with these labels.\nFor model checking, we construct the network of automata for ¬ψ, where ψ is the input\nspecification. The input model and the network of automata generated by our translation\nare executed alternately, with transitions of the model updating atomic propositions in the\nshared variable array, which the network of timed automata reads.\nC.2\nComparison of translation times\nIn Table 4, we discuss the time for the translation for a few instances. We remark that the\ntime for the translation is rather small in most cases.\nSl.\nFormula\nTime taken by\nNo.\nMightyL (ms)\nTempora (ms)\n1\n(((((p0 U[1,2] q1) U[1,2] q2) U[1,2] q3) U[1,2] q4) U[1,2] q5)\n50\n5\n2\nG[0,10] F[1,2]((a ∧(X b)) ∨(¬a∧!(X b)))\n15\n3\n3\nG[2,∞)((¬request ∨F[4,5] grant))\n42\n4\n4\nF[1,2] p1 ∧F[1,2] p2 ∧F[1,2] p3 ∧F[1,2] p4 ∧F[1,2] p5\n49\n3\nTable 4 Comparison of time taken by Tempora, and MightyL for translation of MITL formulae\nto generalized timed automata (timed automata respectively).\nC.3\nEngineering optimizations.\nWe have implemented several optimizations to improve performance, of which we discuss one\nsuch key optimization here. First, in addition to the locations of each automaton, the global\nstate in our translation also stores the output of each automaton in the array of shared\nvariables. Each shared variable doubles the global state space of the network. However, at\nany point in the sequential execution, the subsequent behaviour of the network depends only\non the outputs of the previous automaton which are the direct dependencies of the remaining\nautomata in the round. Therefore, once the last parent of an automaton reads a shared\nvariable, its value can be discarded and the variable reused to store the output of a different\nautomaton. We use static liveness analysis to determine when shared variables become dead\n\n40\nEfficient Verification of MTL with Past in Pointwise Semantics\nand reuse them, reducing the number of auxiliary variables. This optimization minimizes\nthe state size and the overall state space of the zone graph.\nC.4\nAblation Study\nTo better understand the contribution of individual improvements in Tempora, we perform\nablation studies by selectively disabling them and measuring their impact on performance.\nFor the ablation study, we consider (a) one of our theoretical contributions, namely sharing\nof predictions, (b) specialized automata for the F operator and (c) finally, an engineering\noptimization, namely memory reuse discussed above.\nFor each feature, we compare the performance of Tempora with the feature, Tempora\nwith the feature turned off, and the state-of-the-art tool MightyL. Our results demonstrate\nthat each of these improvements plays a critical role in enabling scalability, often yielding\nexponential improvements.\nSharing Predictions. In the absence of prediction sharing, each until subformula in the\nnetwork of automata independently guesses the time of satisfaction. As a result, multiple non-\ndeterministic predictions must be resolved during reachability analysis, significantly increasing\nthe complexity of the state space. With prediction sharing enabled, a single dedicated\nautomaton generates the prediction, and the remaining components deterministically consume\nthis information.\nThis reduces the overall non-determinism in the system and leads to a much smaller\nproduct automaton. Empirical results confirm this effect: benchmark families involving\nmultiple until operators show exponential gains in verification time when prediction sharing\nis enabled.\nWe depict the results in Figure 12 (left) for ϕ(n) = Vn\ni=1 p U[0,i] q.\nFigure 12 Comparison of performance of Tempora with and without features. The left figure\nshows the impact of sharing predictions. The middle figure shows the impact of specialized automata\nfor F operator. The right figure shows the impact of memory reuse. The x-axis represents the\nparameter in the formula, while the y-axis shows the time taken in milliseconds.\nSpecialized Automaton for F While the eventually operator FI p can be encoded as\ntrue UI p, this transformation can be an overkill for untimed or simple timing intervals like\n[0, ∞). Tempora implements specialized automata for such cases that are smaller and\nstructurally simpler than general-purpose until-based encodings.\nFor instance, the automaton for F[0,a] p simply waits up to a time units for p to become\ntrue, without introducing intermediate obligations or control states.\nThese specialized\nconstructions avoid unnecessary non-determinism and reduce clock usage. Our benchmarks\nshow that formulas involving eventualities with small intervals benefit substantially from\nthis, often reducing the number of locations and transitions in the resulting automaton\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n41\nby an order of magnitude. The results are shown in Figure 12 (middle) for the formula\nϕ(n) = VN\ni=1 F[0,i] p.\nReusing Memory The memory optimization addresses inefficiencies in how truth values of\natomic propositions are stored during automaton execution. Consider the formula ϕ(n) =\nVn\ni=1 X pi, where each X pi depends only on the value of pi at the previous timestamp. In\na naive implementation, each pi is assigned a unique index in the state array, even though\ntheir lifetimes do not overlap.\nTempora incorporates a live-range analysis to reuse memory indices once a variable\nbecomes dead — i.e., no longer accessed in the remaining evaluation of the round. In the\nabove example, all pi can safely share a single index. This drastically reduces the size of the\nmemory array and thus the number of reachable states. Our experiments demonstrate that\nthis reuse yields exponential improvements in model-checking time for formulas with long\nchains of independent next operators, as depicted in Figure 12 (right).\nC.5\nBenchmarks and Extended Experimental Results\nWe next briefly describe the additional benchmarks we use to evaluate our performance.\nOur benchmarks. We constructed several MITL+p formulae to show the benefits of our\nimprovements. To show the benefit of sharing clocks, we use a disjunction of multiple X\noperators: G[0,2](Wn\ni=1 X[2i,2i+1] pi), where each Next automaton uses a single shared clock.\nA conjunction of multiple U operators with the same left and right children: G(Vn\ni=1(pi →\na U[0,i] b)) ∧G(Wn\ni=1 pi) shows the benefit of using sharer automata, where each Until\nautomaton uses a common sharer automaton, as well as using a smaller 2 state automaton\nfor Until. The power of using deterministic automata for top level operators can be seen\nfor a formula with conjunction of multiple U operators having general intervals at the top\nlevel: Vn\ni=1(p U[2i,2i+1] q). We show the benefit of reusing shared variables for formulae where\natomic propositions have short live ranges: Wn\ni=1 G[2,∞)(pi). In all these formulae, we try\nto ensure that they cannot be easily rewritten to a simpler formula, and are not trivially\nsatisfiable/unsatisfiable.\nAcacia Benchmarks. Each LTL formula in the original Acacia benchmark is of the form\nψ ⇐⇒G F acc, where ψ is an LTL specification. We modify these examples by adding\nintervals to temporal operators of ψ. For several of these specifications, we add intervals to\nthe operators specifying occurrences of atomic propositions, which adds stricter constraints\nfor the atomic propositions to occur within an interval. For some of the benchmarks, we also\nadd general intervals to nested U operators to add complexity to the formula.\nThe complete experimental results are presented in Table A (for finite words) and Table\nB for infinite words.\n\n42\nEfficient Verification of MTL with Past in Pointwise Semantics\nSl. No\nFormula\nBaseline\nMightyL+UPPAAL\nTEMPORA\nNodes Stored\nTime (ms)\nNodes Stored\nTime (ms)\nNodes Stored\nTime (ms)\n1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3)\n32341\n179.083\n11239\n66.00\n2744\n25.4233\n2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5)\n1927857\n24478.2\n243106\n1,348.00\n24076\n209.238\n3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3)\n70615\n480.662\n3440\n17.00\n329\n1.57934\n4\nG[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 || p2 || \np3)\n35079\n230.365\n3440\n20.00\n345\n2.02975\n5\n(X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || \natomic) && (!p2) || !(atomic))))))&& G(p1 || p2)\n25210\n165.833\n885\n5.00\n437\n2.63777\n6\n(X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) \n&& (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X\n[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !\n(atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) \n|| !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, \n3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) \n&& (!p2) || !(atomic))))))))))))&& G(p1 || p2)\n-\nTO\n3517869\n43,896.00\n3509\n21.3181\n7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q\n5441\n225,448.00\n1990\n12.3429\n8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5\n2130\n14.2971\n3712\n25.00\n160\n0.791932\n9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5\n80500\n619.015\n3872\n23.00\n20\n0.133869\n10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5)\n118296\n984.777\n16182\n129.00\n10777\n84.286\n11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) ))\n48083\n416.384\n479764\n215,172.00\n3367\n33.0925\n12 G[2, Inf) ( (!request || F[4, 5] grant))\n643\n4.67705\n60\n307.00\n71\n0.511877\n13 p U[11, 12] q\n144\n2.10697\n57\n1,408.00\n38\n0.305787\n14\nG((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 || p3 || \np4)\n-\nN/A\n-\nN/A\n186\n1.67275\n15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q)))\n-\nN/A\n-\nN/A\n1303\n29.9574\n16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q)\n-\nN/A\n-\nN/A\n1571\n50.9912\n17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q)\n-\nN/A\n-\nN/A\n27737\n1762.13\n18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q)\n-\nN/A\n-\nN/A\n167077\n21490.5\n19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)\n-\nN/A\n-\nN/A\n262\n3.18676\n20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)\n-\nN/A\n-\nN/A\n286\n1.53985\n21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf) q && p S[4, Inf) q)\n-\nN/A\n-\nN/A\n68\n0.798897\n22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q)))\n-\nN/A\n-\nN/A\n66\n0.49014\nMightyL benchmarks\n23 F p1 && F p2 && F p3 && F p4 && F p5\n9292\n44.3188\n3263\n15.00\n157\n0.530741\n24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5\n1241\n9.18455\n3263\n16.00\n160\n0.900405\n25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5\n74516\n556.001\n6592\n44.00\n177\n0.806017\n26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3\n47766\n524.722\n455\n1,362.00\n59\n0.34313\n27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5\n-\nTO\n6623\n31,749.00\n191\n1.73696\n28 (((((p0 U q1) U q2) U q3) U q4) U q5)\n25972\n101.764\n8133\n50.00\n4063\n21.7138\n29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5)\n104149\n549.412\n8133\n40.00\n6303\n33.4324\n30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5)\n48557\n319.548\n18204\n136.00\n12496\n82.9332\n31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3)\n16160\n185.278\n1140\n3,278.00\n3238\n35.6263\n32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4)\n162606\n3074.69\n5006\n19,078.00\n27758\n434.539\n33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5)\n1629834\n62186.7\n22228\n107,447.00\n235536\n6455.47\n34 G(!p) && F[0, 3] t1\n110\n0.790953\n55\n2.00\n21\n0.118925\n35 G(!p) && F[0, 3] t1 && F[3, 6] t2\n2396\n12.5292\n627\n254.00\n174\n1.31732\n36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3\n48222\n364.599\n2450\n6,690.00\n419\n2.98024\n37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4\n-\nTO\n48383\n274,012.00\n2079\n11.2824\n38 !(!(G F p1) || (G(!q || (F[100, 1000] r))))\n6916\n38.5607\n463\n3.00\n271\n1.66032\n39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r))))\n65149\n418.318\n3423\n32.00\n1005\n5.93152\n40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r))))\n561208\n5684.74\n26047\n280.00\n3397\n18.338\n41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r))))\n4651945\n126995\n202623\n2,359.00\n10953\n59.6345\n42 F[0, 30] (!p || G[0, 20] p)\n31\n0.32008\n31\n2.00\n17\n0.118925\n43 G[0, 30](!p) || F[0, 20] p\n37\n0.303894\n32\n2.00\n12\n0.152296\n44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5)\n179753\n1216.62\n8128\n44.00\n6318\n37.332\n45 (((((p0 R q1) R q2) R q3) R q4) R q5)\n47841\n250.798\n8128\n33.00\n4079\n19.3469\n46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3)\n7571\n69.5354\n560\n15.00\n546\n5.74616\n47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5)\n375155\n6503.65\n9792\n393.00\n12042\n230.663\n48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4)\n54947\n678.074\n2368\n74.00\n2628\n38.1337\n49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5)\n78161\n587.855\n9792\n55.00\n5892\n39.5866\n50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5\n2735492\n78412.1\n3232\n132.00\n20\n0.252719\n51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5\n70930\n578.802\n3232\n18.00\n20\n0.136226\n52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5\n1815\n14.362\n3134\n19.00\n160\n0.809136\n53 G p1 && G p2 && G p3 && G p4 && G p5\n12897\n75.8965\n3134\n14.00\n158\n1.06356\n54 !((F[0, 30] p) || !(F[0, 20] p))\n60\n0.568014\n69\n2.00\n49\n0.272393\n55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p))\n1272\n10.5949\n495\n221.00\n224\n0.991115\nAcacia benchmarks\n56 ((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F(a)))\n3047327\n70749.6\n54777\n82,277.00\n530\n3.40564\n57\n(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F a))) || (!\n(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) && !((G (F a))))\n-\nTO\n3193831\n41,472.00\n3455\n19.531\n58\n((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q && (X \nF[2, Inf) r))))) && !(G(F(a))))\n-\nTO\n214577\n96,763.00\n1059\n7.08385\n59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a)))\n466847\n5451.49\n13712\n21,448.00\n225\n1.61587\n60\n((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r U[0, 3] \ns)))) && !(G(F a)))\n-\nTO\n-\nError\n564\n5.20393\n61\n((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G(Fa))) || \n(!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && !(G(Fa)))\n-\nTO\n-\nError\n3276\n19.6802\n62\n((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, \n3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F\n[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && !(G(F a)))\n-\nTO\n-\nTO\n366144\n3712.65\n63\n(((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && (G(F\n(a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) \n&& !(G(F(a))))\n-\nTO\n-\nTO\n362442\n8380.15\n64\n(((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, 2]q)) || \n(G(F[1, 2]r))) && !(G(F(a))))\n-\nTO\n-\nTO\n7362\n86.5996\n65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a)))\n51613\n392.448\n6641\n72.00\n61\n0.507483\n66\n(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, \n2] (!r) )) && ! (G( F (a))))\n-\nTO\n-\nTO\n3153\n40.0667\n67\n(( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((!(G(2, Inf) \np) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a))))\n885283\n14219.3\n99857\n945.00\n87\n0.747122\n68\n(((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (!((G[0, 2] \nr) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a))))\n-\nTO\n-\nTO\n5053\n39.7407\n69\n(((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && \n!(G(F(a))))\n6802985\n269414\n51641\n19,852.00\n444\n2.52895\n70\n(((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2](X[1, 2]\n(X[1, 2] q))))))) && !(G(F a)))\n-\nTO\n430184\n3,928.00\n1061\n4.97043\n71\n(((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) && (G(p \n|| (F[3, 4] (!q))))) && !(G(F a)))\n-\nTO\n-\nTO\n1917\n23.3503\n72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a)))\n13546\n82.0316\n3511\n20.00\n96\n0.430384\nTable A: Experimental results obtained by running TEMPORA, MightyL + UPPAAL and, the baseline implementation of the translation for satisfiability for finite traces. (Timeout is set to 300s)\n\nS. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan\n43\nSl. No\nFormula\nBaseline\nMightyL + OPAAL\nTEMPORA\nNodes Stored\nTime (ms)\nNodes Stored \n(Reporting state \nspace given by \nopaal) -> Non \ndeterministic, since \nmulti threaded. \nNumbers change \nacross different \nexperiments\nTime (ms). Note: opaal is multicore, and hence \nuses 8 cores of local machine\nTotal time (ms)\nNodes Stored\nTime (ms)\nopaal\ng++ \nltsmin -> the \nactual \nexploration\n1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3)\n2292\n14.1087\n1161\n172.79\n661.47\n76.03\n910.28\n212\n1.08509\n2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5)\n29455\n193.183\n1861\n260.97\n710.76\n80.51\n1,052.25\n552\n3.17967\n3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3)\n1310856\n16076.9\n1162\n227.10\n700.43\n76.16\n1,003.69\n458\n2.42018\n4\nG[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 \n|| p2 || p3)\n1587059\n21828.1\n1450\n263.38\n718.80\n77.70\n1,059.88\n1044\n8.24895\n5\n(X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!\np1) || atomic) && (!p2) || !(atomic))))))&& G(p1 || p2)\n44857\n350.62\n1893\n203.98\n663.58\n82.98\n950.54\n168\n0.987026\n6\n(X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !\n(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !\n((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!\np1) || atomic) && (!p2) || !(atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || \n(X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && \n(!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !\n(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))))))))&& G(p1 || \np2)\n-\nTO\n-\n-\n-\nSegfault\n0.00\n1933\n15.9299\n7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q\n-\nTO\n-\n-\n-\nTO\n0.00\n456\n4.17251\n8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5\n2346\n16.1747\n969\n301.04\n736.75\n75.30\n1,113.09\n224\n1.30602\n9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5\n-\nTO\n1143\n272.19\n721.28\n78.98\n1,072.45\n178\n1.58184\n10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5)\n-\nTO\n7003\n404.46\n851.76\n106.54\n1,362.75\n1087\n9.45395\n11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) ))\n117319\n1140.97\n7921\n74,584.66\n20,184.02\n238.35\n95,007.03\n504\n3.50233\n12 G[2, Inf) ( (!request || F[4, 5] grant))\n-\nTO\n-\n-\n-\nTO\n0.00\n444\n4.00833\n13 p U[11, 12] q\n1360\n21.8606\n-\n-\n-\nTO\n0.00\n220\n1.3685\n14\nG((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 \n|| p3 || p4)\n-\nN/A\n-\n-\n-\nN/A\n798\n2.19275\n15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q)))\n-\nN/A\n-\n-\n-\nN/A\n14440\n1721.87\n16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q)\n-\nN/A\n-\n-\n-\nN/A\n1410\n51.2159\n17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q)\n-\nN/A\n-\n-\n-\nN/A\n2310\n132.646\n18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q)\n-\nN/A\n-\n-\n-\nN/A\n3927\n418.381\n19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)\n-\nN/A\n-\n-\n-\nN/A\n91\n0.509345\n20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)\n-\nN/A\n-\n-\n-\nN/A\n267\n1.44564\n21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf9 q && p S[4, Inf) q)\n-\nN/A\n-\n-\n-\nN/A\n114\n0.645279\n22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q)))\n-\nN/A\n-\n-\n-\nN/A\n108\n0.73709\nMightyL benchmarks\n23 F p1 && F p2 && F p3 && F p4 && F p5\n822\n5.11449\n2165\n190.21\n698.33\n79.48\n968.01\n64\n0.407546\n24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5\n952\n5.99807\n2693\n214.01\n701.74\n82.35\n998.09\n68\n0.485472\n25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5\n-\nTO\n3069\n372.57\n820.01\n85.33\n1,277.91\n200\n2.38852\n26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3\n-\nTO\n-\n-\n-\nSegfault\n0.00\n102\n0.528738\n27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5\n-\nTO\n-\n-\n-\nTO\n0.00\n158\n1.05082\n28 (((((p0 U q1) U q2) U q3) U q4) U q5)\n1828\n10.7493\n1560\n196.70\n704.35\n78.12\n979.17\n18689\n91.4055\n29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5)\n47166\n298.922\n1477\n219.19\n699.27\n76.57\n995.03\n2668\n19.0006\n30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5)\n567192\n5404.53\n9383\n382.47\n960.14\n113.60\n1,456.21\n4075\n32.6984\n31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3)\n-\nTO\n-\n-\n-\nSegfault\n0.00\n3254\n30.8566\n32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4)\n-\nTO\n-\n-\n-\nTO\n0.00\n36124\n451.716\n33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5)\n-\nTO\n-\n-\n-\nTO\n0.00\n465185\n8750.47\n34 G(!p) && F[0, 3] t1\n310\n1.95935\n184\n117.86\n612.55\n72.49\n802.89\n32\n0.235228\n35 G(!p) && F[0, 3] t1 && F[3, 6] t2\n953492\n24977.1\n964\n68,462.12\n20,584.03\n102.84\n89,149.00\n158\n0.901839\n36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3\n-\nTO\n-\n-\n-\nSegfault\n0.00\n315\n3.3407\n37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4\n-\nTO\n-\n-\n-\nTO\n0.00\n522\n4.21687\n38 !(!(G F p1) || (G(!q || (F[100, 1000] r))))\n-\nTO\n1935\n692.31\n954.65\n79.29\n1,726.25\n2823\n18.9429\n39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r))))\n-\nTO\n4212\n749.15\n1,006.85\n89.55\n1,845.55\n28570\n196.333\n40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r))))\n-\nTO\n42804\n836.05\n1,056.50\n225.40\n2,117.95\n330044\n2976.99\n41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r))))\n-\nTO\n149029\n906.87\n1,133.70\n473.50\n2,514.08\n3819965\n88366.7\n42 F[0, 30] (!p || G[0, 20] p)\n9432\n52.7811\n43\n131.32\n615.22\n69.64\n816.18\n704\n5.34067\n43 G[0, 30](!p) || F[0, 20] p\n61\n0.756158\n37\n126.61\n617.54\n75.80\n819.95\n27\n0.295317\n44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5)\n145353\n1141.14\n2267\n260.88\n746.28\n86.45\n1,093.60\n113881\n905.989\n45 (((((p0 R q1) R q2) R q3) R q4) R q5)\n22354\n140.505\n1521\n185.82\n690.21\n79.60\n955.64\n11957\n61.4713\n46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3)\n1086882\n15332.4\n453\n1,588.79\n1,496.63\n78.06\n3,163.48\n5985\n55.9789\n47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5)\n-\nTO\n-\n-\n-\nSegfault\n0.00\n1093187\n23474.1\n48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4)\n-\nTO\n-\n-\n-\nSegfault\n0.00\n79258\n1018.67\n49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5)\n29685\n278.315\n1059\n230.60\n725.70\n77.18\n1,033.48\n3119\n27.748\n50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5\n-\nTO\n-\n-\n-\nSegfault\n0.00\n225\n2.45304\n51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5\n-\nTO\n1549\n226.81\n724.91\n82.29\n1,034.01\n178\n1.07516\n52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5\n4793\n34.5633\n5913\n257.33\n760.33\n100.74\n1,118.41\n224\n1.45227\n53 G p1 && G p2 && G p3 && G p4 && G p5\n195653\n1451.52\n8413\n186.71\n694.95\n111.65\n993.31\n64\n0.61791\n54 !((F[0, 30] p) || !(F[0, 20] p))\n7069\n37.444\n100\n126.35\n622.49\n72.87\n821.71\n629\n10.9368\n55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p))\n3072735\n56119.3\n624\n62,506.68\n20,677.93\n112.71\n83,297.33\n1669\n19.924\nAcacia benchmarks\n56\n((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F\n(a)))\n-\nTO\n76875\n150,058.47\n42,824.04\n2,635.25\n195,517.76\n4561\n37.7644\n57\n(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F \na))) || (!(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) \n&& !((G (F a))))\n-\nTO\n7271276\n611.46\n1,143.21\n21,348.99\n23,103.66\n918\n7.40638\n58\n((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q \n&& (X F[2, Inf) r))))) && !(G(F(a))))\n-\nTO\n523958\n67,992.90\n22,837.11\n10,320.40\n101,150.41\n4614\n32.6728\n59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a)))\n-\nTO\n22945\n116,791.31\n39,078.93\n840.78\n156,711.03\n1617\n12.6589\n60\n((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r \nU[0, 3] s)))) && !(G(F a)))\n-\nTO\n-\n-\n-\nSegfault\n0.00\n820\n6.57777\n61\n((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G\n(Fa))) || (!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]\ns)))))))))) && !(G(Fa)))\n-\nTO\n6733\n183,618.00\n14,086.56\n323.78\n198,028.34\n4658\n32.0218\n62\n((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v \n&& (X[2, 3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, \nInf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && \n!(G(F a)))\n-\nTO\n-\n-\n-\nSegfault\n0.00\n54366\n545.194\n63\n(((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) \n&& (G(F(a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && \n(G(F[2, 4] u))) && !(G(F(a))))\n-\nTO\n-\n-\n-\nTO\n0.00\n-\nTO\n64\n(((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, \n2]q)) || (G(F[1, 2]r))) && !(G(F(a))))\n-\nTO\n-\n-\n-\nTO\n0.00\n-\nTO\n65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a)))\n-\nTO\n11391\n515.86\n1,265.08\n159.79\n1,940.73\n246\n1.536\n66\n(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) \n) U[1, 2] (!r) )) && ! (G( F (a))))\n-\nTO\n-\n-\n-\nSegfault\n0.00\n-\nTO\n67\n(( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((!\n(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a))))\n-\nTO\n250875\n516.21\n950.14\n750.60\n2,216.94\n3484\n25.3826\n68\n(((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (!\n((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a))))\n-\nTO\n-\n-\n-\nSegfault\n0.00\n2598\n25.7481\n69\n(((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && \n(G r)) && !(G(F(a))))\n-\nTO\n35945\n63,521.58\n21,527.32\n788.87\n85,837.77\n785\n5.94832\n70\n(((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2]\n(X[1, 2](X[1, 2] q))))))) && !(G(F a)))\n-\nTO\n742427\n426.25\n948.75\n1,682.79\n3,057.79\n231\n1.94063\n71\n(((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) \n&& (G(p || (F[3, 4] (!q))))) && !(G(F a)))\n-\nTO\n-\n-\n-\nTO\n0.00\n4159\n55.7087\n72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a)))\n1802521\n22455.7\n4943\n257.75\n755.07\n95.59\n1,108.41\n390\n2.54234\nTable B: Experimental results obtained by running TEMPORA , MightyL + OPPAAL and, the baseline implementation of the translation for satisfiability for infinite traces. (Timeout is set to 300s)\n"},"cleaned":{"kind":"string","value":"Efficient Verification of Metric Temporal Properties with Past in Pointwise Semantics S Akshay # # # Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMF, 91190, Gif-sur-Yvette, France CNRS, ReLaX, IRL 2000, Siruseri, India R Govind # Uppsala University, Sweden B Srivathsan # Chennai Mathematical Institute, India CNRS, ReLaX, IRL 2000, Siruseri, India Abstract Model checking for real-timed systems is a rich and diverse topic. Among the different logics considered, Metric Interval Temporal Logic (MITL) is a powerful and commonly used logic, which can succinctly encode many interesting timed properties especially when past and future modalities are used together. In this work, we develop a new approach for MITL model checking in the pointwise semantics, where our focus is on integrating past and maximizing determinism in the translated automata. Towards this goal, we define synchronous networks of timed automata with shared variables and show that the past fragment of MITL can be translated in linear time to synchronous networks of deterministic timed automata. Moreover determinism can be preserved even when the logic is extended with future modalities at the top-level of the formula. We further extend this approach to the full MITL with past, translating it into networks of generalized timed automata (GTA) with future clocks (which extend timed automata and event clock automata). We present an SCC-based liveness algorithm to analyse GTA. We implement our translation in a prototype tool which handles both finite and infinite timed words and supports past modalities. Our experimental evaluation demonstrates that our approach significantly outperforms the state-of-the-art in MITL satisfiability checking in pointwise semantics on a benchmark suite of 72 formulas. Finally, we implement an end-to-end model checking algorithm for pointwise semantics and demonstrate its effectiveness on two well-known benchmarks. 2012 ACM Subject Classification Theory of computation →Timed and hybrid models; Theory of computation →Quantitative automata; Theory of computation →Logic and verification Keywords and phrases Real-time systems, Timed automata, Liveness, Event-clock automata, Clocks, Timers, Verification, Zones, Simulations 1 Introduction Linear Temporal Logic (LTL) is a standard specification formalism. A central ingredient in the model-checking approach is a translation from LTL to automata. The study of algorithmic techniques to optimize LTL-to-automata translations spans almost four decades - see [57] for a recent survey of this topic. Many of the theoretical techniques have also been incorporated in practical model-checking tools [36, 22, 28, 56, 37]. For model-checking systems with timing constraints, the de-facto automaton model would be timed automata (TA) [4, 3]. State-of-the-art timed automata tools [41, 34, 56, 37] implement analysis methods 16 Oct 2025 2 Efficient Verification of MTL with Past in Pointwise Semantics for models represented as (networks of) timed automata (see Section 7 of [25] for the practical applications of these tools). Properties checked on timed models are typically reachability, liveness, fragments of timed CTL [41] or (untimed) LTL. On the other hand, there is a very large body of foundational work on timed logics [53, 35, 54]. The two main timed logics are Metric Temporal Logic [40] (MTL) and Timed Propositional Temporal Logic [8] (TPTL). In order to use these logics for real-time model-checking, we need an efficient translation from timed logics to timed automata. The foundations for MTL-to-TA (resp. TPTL-to-TA) have been laid out in [5] (resp. [8]). Broadly, there are two ways to interpret MTL formulae: over continuous (dense/superdense) timed signals [5, 45] or pointwise timed words [6, 58, 20]. In continuous (state-based) semantics, the formula is interpreted over signals, which are functions that associate the valuation of the propositions to all timestamps. Over the pointwise semantics, the formula is interpreted over sequences of events (timed words), each consisting of a timestamp and a set of propositions. Thus, in the continuous semantics, the system is under observation at all times, while in the pointwise semantics, it is observed only when it executes a discrete transition. However, as noted by Ferrére et al. [30], in the continuous semantics, arbitrary MTL formulae can be simplified to the [0, ∞] fragment, while as pointed out in [2], this simplification is not possible in the pointwise semantics. In this work, we only consider pointwise semantics as our focus is on models represented in tools like UPPAAL, PAT, TChecker whose behaviours are given as timed words, as in the classical timed automata model of Alur and Dill [4]. In the pointwise semantics, it is known that MTL model checking is undecidable over infinite words, and over finite words it is decidable but the complexity is Ackermanian [7, 50, 48, 49]. Thus, the literature often considers the more efficient MITL fragment, where Until and Since operators must use non-punctual intervals (either [0, 0] or non-singleton intervals). The classical approach for model checking for MITL involves translating MITL formula into timed automata (TA) and then using state-of-the-art TA tools such as UPPAAL [41, 12] or TChecker [34]. The best known tool that accomplishes this translation in the pointwise semantics, called MightyL [20], goes via alternating 1-clock automata and compiles to timed automata. Recently, an extension of timed automata called generalized timed automata (GTA) was introduced in [1]. GTA are equipped with richer features than TA, and enable a different translation from MITL to GTA, which can then be analyzed using a GTA analysis tool from [1]. The theoretical foundations for such an MITL-to-GTA translation which goes via transducers was presented in [2], where it was argued that the MITL-toGTA translation has clear theoretical benefits over the MITL to 1-clock Alternating Timed Automata translation [18, 19] on which MightyL is based on. However, this latter approach was not implemented. Moreover, neither of these approaches considered Past modalities, and were restricted only to Future. On the other hand, from the classical work of Maler et al [44], in the signal semantics, it is known that having only Past modalities leads to efficient algorithms and in particular it gives a translation to deterministic (timed) automata albeit over signals. Our first question is whether the same holds under the pointwise semantics, and whether determinism can be extended beyond just the past fragment. Next, if we cannot guarantee determinism, can we reduce non-deterministic branching? Indeed, even if it does not change the expressiveness, several works have highlighted the usefulness and succinctness of combining Past and Future operators, both in the untimed setting [43, 46, 10] and in the timed setting [39, 30]. In this work, we address these questions both for past and future in the pointwise semantics and to build the foundations for a new efficient model checking algorithm for S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 3 MITL with past that exploits determinism as much as possible. We show that for the past fragment of MITL, we can obtain a linear-time translation to networks of deterministic timed automata. Further, even when we have to handle future operators, where non-determinism cannot be avoided, we propose techniques to reduce non-deterministic branching in the resulting automata, using various ideas including sharing components, automata, clocks and predictions. We start by defining a formal semantics for a model of synchronous network of timed automata with shared variables, where each component of the network owns some clocks which only it can reset but can be tested by all components. Moreover it allows for instantaneous programs on transitions which have sequences of tests and updates, like in [2]. Our main contributions are the following: 1. We identify a fragment of metric temporal logic with past and future for which we provide a linear time translation to networks of deterministic timed automata. In this fragment, that we call detMITL+p, the outermost temporal modalities are future (next, until) and can even have punctual intervals, while inner temporal modalities should be past: Since with non-punctual interval or Yesterday. This fragment already subsumes many safety properties, extending e.g., the G(pLTL) fragment defined in [9]. 2. We present a new linear-time algorithm to translate detMITL+p formula to networks of deterministic timed automata. This algorithm is in two steps. First we show how the outermost temporal modalities can be handled with respect to initial satisfiability. Next we show how the past temporal modalities are handled for general satisfiability. We obtain a synchronous network of deterministic automata with shared variables that captures the semantics. We start with Yesterday allowing an arbitrary interval constraint. Then, we deal with Since operators with one-sided intervals of the form [0, a] or [a, ∞). For these simpler cases, we construct networks where each automaton has at most 1 clock and 2 states, but critically uses the sharing feature across components. Finally, the automaton for Since with a general two-sided interval [a,b] is the most complicated part of the construction and results in a network whose one component uses O(a/(b-a)) clocks. Our construction works for both finite and infinite words. 3. Next, we extend this algorithm to handle future modalities. To achieve this we move from networks of timed automata to networks of generalized timed automata (GTA) [1], which have future clocks in addition to usual (history) clocks. We obtain a translation from general MITL+p (with Past and allowing punctual intervals at the outermost level) to networks of GTA with shared variable. While non-determinism cannot be avoided here, we show that we can reduce non-deterministic branching in two ways: first by using shared variables to share predictions, and second, by using non-Zenoness assumption and the semantics of future clocks to obtain a 2-state automaton for Until (which results in an exponential improvement in running time for the model checking algorithm). Again our model checking algorithm works for infinite words and is easily adaptable for finite timed words. 4. Since our model checking approach ultimately reduces to checking reachability and liveness for GTAs, we propose an improved liveness-checking algorithm that extends Couvreur's SCC algorithm [26] (which is the state-of-the-art for timed automata) to the setting of GTAs, marking the first adaptation of Couvreur's procedure to automata with event clocks or diagonal constraints. 5. Finally, we implement our algorithms in a prototype tool, called Tempora, built on top of the open-source timed automata library Tchecker. Tempora supports satisfiability and model checking for detMITL+p and general MITL+p, for both finite and infinite words, in the pointwise semantics. We compare against MightyL which translates MITL formula 4 Efficient Verification of MTL with Past in Pointwise Semantics to timed automata (and uses UPPAAL to check satisfiability with finite words and OPAAL for infinite words). Our comparison on a suite of 72 benchmarks (22 new and 50 taken from the literature) shows that our pipeline (Tempora+Tchecker) significantly outperforms MightyL pipeline; it is faster in all benchmarks for finite words and all but 2 for infinite words, and most often explores a smaller state space (in zones). We also implement the theoretical translation to GTA presented in [2] and demonstrate an order of magnitude improvement. Finally, we also implement an end-to-end model checking algorithm for MITL+p over the pointwise semantics and evaluate our model-checking pipeline on two models: Fischer's mutual exclusion algorithm and the Dining Philosophers problem. Related works. Model checking for real-time systems has been extensively studied, with Metric Interval Temporal Logic (MITL) emerging as a widely used specification language due to its expressiveness for capturing timed properties. A central challenge lies in efficiently translating MITL formulas into automata-based representations, enabling algorithmic verification. For a comprehensive overview of the topic, we refer the reader to the survey by Bouyer [14]. Over continuous/signal semantics, we already discussed several works that focus on obtaining networks of timed automata for various fragments of MITL with and without past including [30, 44, 45]. Most of these employ construction of compositional testers or transducers for timed modalities, but do not focus on sharing e.g., of networks and subformula. A different purely logic based approach was considered in [47], where the timed automata and MITL formula, still with signal semantics, are translated to an intermediate logical language, which is then encoded into an SMT solver. This work also included synchronization primitives and was implemented in a Java tool. Coming back to pointwise semantics, in [5] that introduced MITL, the authors propose the first such translation, which is known to be notoriously complicated, spanning several pages, and acknowledged to be difficult in the follow-up works by different authors [45]. The current state-of-the-art for MITL model checking is the procedure proposed by Brihaye et al. in [20], where they translate MITL formulae to a network of timed automata, which can be plugged in a standard timed automata tool implementing non-emptiness. Their procedure and the associated tool, called MightyL [20] is based on a series of works that study a translation of MITL to timed automata [18, 19], via alternating timed automata [49]. [17] studies a zone-based algorithm directly for 1-clock alternating timed automata, and focusses on a fragment of MTL. Several tools support MITL model checking under continuous semantics, including [38, 23, 24, 52, 42, 47]. Under pointwise semantics, to the best of our knowledge, MightyL [20] is the only tool that can handle the full future fragment of MITL (even if it only solves satisfiability and does not handle past operators). The other works (to our knowledge) that offer tool support are for restricted fragments such as MITL(0.∞) [21], or MITL over untimed words [59]. Structure of the paper. In Section 2, we formalize the notion of a synchronous network of timed automata with shared variables, whose transitions are instantaneous programs, and define the model checking problem in terms of such networks. In Section 3 we describe the detMITL+p logic followed by the translation to synchronous networks of deterministic timed automata in Section 4. In Section 5, we explain how to integrate future modalities to get a translation from MITL+p to generalized timed automata with future clocks. Section 6 describes our adaptation of Couvreur's SCC algorithm to the setting of GTAs. Section 7 contains our experimental evaluation and comparisons and we end with a brief conclusion in S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 5 Section 8. To improve the flow and readability of the paper, we have moved some proofs and technical details to the Appendix. Moreover, some benchmark details and extended results are presented in the Appendix. 2 Synchronous network of timed automata with shared variables We define the model of network of timed systems that we use in this paper. The presentation is different from previous works [11, 16, 33, 2], and is geared towards providing a clean and concise explanation of our logic-to-automata translation. Before giving the formal syntax, we highlight the main aspects in our definition. Each component in the network is an automaton with a finite set of locations, real valued variables (called clocks) and finitely valued variables (mainly boolean variables and the state variable ranging over the finite set of locations). The semantics of the network is synchronous: either time elapses in all components of the network (all clocks are increased by the same amount), or all components execute simultaneously an instantaneous transition. The components of the system communicate via shared variables (clocks, states, boolean variables, . . . ). The paradigm is concurrent-read owner-write (CROW). Hence, each variable is owned by a unique component of the system. A component may read all variables of the system, but it may only modify its owned variables. A transition of a component A is of the form (l, prog, l′) where l, l′ are locations of A and prog is an instantaneous program, which is a sequence of tests and updates. For instance, if x, y are clocks owned by A and out is a boolean variable owned by A, then a purely local program could be A.x ∈(3, 5]; A.y := 2; A.out := 0 where A.x ∈(3, 5] is a test which succeeds if the current value of clock x is within the interval, and A.y := 2 and A.out := 0 are updates. Tests may also use variables owned by other components. For instance, if component B owns clock z, then a program of A could be of the form A.x ∈(3, 5] ∧B.out• = 1 ∧B.z > 2 ∧B.st• = l1; A.y := 2; A.out := 0 where B.st means the state of B. Since we use a synchronous semantics, all components of the network simultaneously take a transition and execute the associated programs. By default, a variable such as A.x or B.z used in a test refers to the value of the variable before the synchronous transition is taken. It is sometimes convenient to refer to the value a variable will have after the synchronous transition is taken. We do so by writing B.out• or B.st•. We give now the formal syntax and semantics for a network of timed systems. A concrete example follows in Figure 1 and Example 3. ▶Definition 1 (Timed automaton with shared variables). A timed automaton with shared variables is a tuple A = (Q, X, V, ∆, init) where A is the name, Q is a finite set of locations over which ranges the state variable A.st, X is a finite set of clock variables owned by A, V is a finite set of boolean variables owned by A, ∆is the finite set of transitions, and init is an initial condition for the variables owned by A. A transition is a triple (l, prog, l′) where l, l′ ∈Q are locations and prog is an instantaneous program. A program is a sequence of tests and updates. An atomic update for a boolean variable v ∈V owned by A has the form A.v := c (or simply v := c when A is clear from the context) with c ∈{0, 1}. An atomic update for a clock variable x ∈X owned by A has the form A.x := c (or simply x := c) with c ∈N = N ∪{+∞}. We assume that each variable is updated at most once in a program. An atomic test for a clock variable x ∈X owned by A has the form A.x ∈I (or simply x ∈I) where I is an integer bounded interval whose end 6 Efficient Verification of MTL with Past in Pointwise Semantics l0 A1 x = +∞ q•; x := 0 ¬q• l0 l1 A2 y = +∞ ¬q• q•; y := 0 ¬p• ∧q•; y := 0 p• ¬p• ∧¬q• l0 l1 l2 A3 z = 0 z ≤100 ∧¬r• z ≤100 ∧r• ∧A1.x 100 z ≤100 ∧r• ∧(A1.x ≥2 ∨A2.st• = l0 ∨A2.y• 0, (w, i -1) |= φ and τi -τi-1 ∈I. S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 9 (w, i) |= XI φ if (w, i + 1) |= φ and τi+1 -τi ∈I. (w, i) |= φ1 UI φ2 if there exists j ≥i s.t. (w, j) |= φ2, τj -τi ∈I and (w, k) |= φ1 for all i ≤k I ∨ (x• init I) to state that xinit is smaller (resp. larger) than all values in I. For instance, if I = [b, c) with b I means xinit ≥c. If a run reaches state 2 (resp. 3) then we know for sure that b1 UI b2 was initially true (resp. false). State 1 means we still do not know whether b1 UI b2 was initially true. If the run stays forever in state 1, b1 UI b2 was initially false. l0 l1 AY ¬b• 1 b• 1 b• 1 ¬b• 1 1 xlast = +∞ Alast xlast := 0 Figure 5 Left: Automaton AY for the untimed Yesterday operator Y b1. The boolean variable b1 is not owned by AY and stands for the argument of Y. Right: Sharer automaton Alast with clock xlast used by all Yesterday operators. ▶Lemma 8. Let Prop′ = Prop ⊎{b1, b2} be an augmented set of propositions. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0 δ0,t0 ---→C1 δ1,t1 ---→C2 · · · of (Ainit, Aisat UI ) over w. Moreover, w |= b1 UI b2 iff there exists an i ≥1 s.t. for all k ≥i, configuration Ck satisfies Aisat UI .st = 2, Proof. Suppose w |= b1 UI b2. Then, (1) there exists an i ≥0 such that b2 ∈ai, (2) for all 0 ≤j 1 then xn satisfies the lower bound constraint of I and it may serve as a witness instead of the earlier clocks x1, y1, . . . , xn-1, yn-1 (notice that xn 1 then b ̸ ◁x2. Proof. 1. was already stated in Lemma 10. 2. follows directly from Lines (1-2) of Algorithm 1. 3. The proof is by induction on i. The base case is i = 0. The initial state of AS is l0, i.e., C0(AS.st) = l0. Now, C1(AS.st) = l1 iff b2 ∈a0. In this case, Line 4 of Algorithm 1 is taken and we easily check that item (3) of Lemma 12 holds with j = 1 and i1 = i′ 1 = 0. Consider now i > 0 and assume that item (3) hold for i -1. If Ci(AS.st) = l0 then we argue exactly as in the base case above. The interesting case is Ci(AS.st) = l1. By induction, there is 1 ≤j ≤k and indices i1 ≤i′ 1 1 then b is smaller than (◁) the value of x2 after the clock shift which is the value of xn+1 at C′ i. Therefore, the last bullet of Item 3 holds after the clock shift, hence also at Ci+1. Now, we can prove as above, in the case of no clock shift, that after executing Lines (9-14) of Algorithm 1, Item 3 holds at Ci+1. This concludes the inductive proof of Item (3). ◀ Proof of Lemma 12. Fix i ≥0. By Lemma 13, position i satisfies the untimed since b1 S b2 iff Ci+1(AS.st) = l1 (corresponding to AS.st• = l1 in outSI). We assume that we are in this case. The only question is whether there is a witnessing b2 within the interval I. We claim that this can be determined by looking solely at the values of clocks x1 and y1 at configuration Ci+1 (corresponding to the test Agen SI .x• 1 ∈I ∨Agen SI .y• 1 ∈I in outSI). We can view this as follows. Suppose there is in the past a b2 event witnessing b1 SI b2: b2 ∈ai′′ for some i′′ ≤i and b1 ∈ai′ for all i′′ η.dfsnum 58: if L ⊆A and XF = R ∪InfClocks(μ) = XF then ▷InfClocks(μ) is the set of future clocks which can take value -∞in zone of u 59: output Accepting SCC detected 60: end if 61: Roots.Push((μ, A, R)) 62: end function S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 39 C Appendix for Implementation and Experimental evaluation C.1 Implementation details. Our tool, Tempora, takes as input a MITL+p formula ψ and generates a synchronous network of generalized timed automata with shared variables based on the translation described in Sections 4 and 5. The execution of this network is simulated using an array of shared variables. Each generalized timed automaton writes its output to a dedicated shared variable, which can then be accessed by its parent automaton. Synchronization is enforced by sequentially executing each automaton in bottom-to-top order. A shared control variable tracks the active automaton during execution, and a transition in an automaton is executed when its ID matches the value of the control variable. Upon taking the transition, the control variable is updated to the ID of the next automaton. A controller is used to reset the execution after every automaton takes a transition and alternates between time elapse and automaton execution. A master automaton evaluates the top-level boolean subformula, ensuring the formula is satisfied at the start and, for finite timed words, that all automata have performed a transition in the last round. To avoid constructing automata for boolean operators, we use symbolic labels to represent temporal operators and atomic propositions, generating boolean expressions with these labels. For model checking, we construct the network of automata for ¬ψ, where ψ is the input specification. The input model and the network of automata generated by our translation are executed alternately, with transitions of the model updating atomic propositions in the shared variable array, which the network of timed automata reads. C.2 Comparison of translation times In Table 4, we discuss the time for the translation for a few instances. We remark that the time for the translation is rather small in most cases. Sl. Formula Time taken by No. MightyL (ms) Tempora (ms) 1 (((((p0 U[1,2] q1) U[1,2] q2) U[1,2] q3) U[1,2] q4) U[1,2] q5) 50 5 2 G[0,10] F[1,2]((a ∧(X b)) ∨(¬a∧!(X b))) 15 3 3 G[2,∞)((¬request ∨F[4,5] grant)) 42 4 4 F[1,2] p1 ∧F[1,2] p2 ∧F[1,2] p3 ∧F[1,2] p4 ∧F[1,2] p5 49 3 Table 4 Comparison of time taken by Tempora, and MightyL for translation of MITL formulae to generalized timed automata (timed automata respectively). C.3 Engineering optimizations. We have implemented several optimizations to improve performance, of which we discuss one such key optimization here. First, in addition to the locations of each automaton, the global state in our translation also stores the output of each automaton in the array of shared variables. Each shared variable doubles the global state space of the network. However, at any point in the sequential execution, the subsequent behaviour of the network depends only on the outputs of the previous automaton which are the direct dependencies of the remaining automata in the round. Therefore, once the last parent of an automaton reads a shared variable, its value can be discarded and the variable reused to store the output of a different automaton. We use static liveness analysis to determine when shared variables become dead 40 Efficient Verification of MTL with Past in Pointwise Semantics and reuse them, reducing the number of auxiliary variables. This optimization minimizes the state size and the overall state space of the zone graph. C.4 Ablation Study To better understand the contribution of individual improvements in Tempora, we perform ablation studies by selectively disabling them and measuring their impact on performance. For the ablation study, we consider (a) one of our theoretical contributions, namely sharing of predictions, (b) specialized automata for the F operator and (c) finally, an engineering optimization, namely memory reuse discussed above. For each feature, we compare the performance of Tempora with the feature, Tempora with the feature turned off, and the state-of-the-art tool MightyL. Our results demonstrate that each of these improvements plays a critical role in enabling scalability, often yielding exponential improvements. Sharing Predictions. In the absence of prediction sharing, each until subformula in the network of automata independently guesses the time of satisfaction. As a result, multiple nondeterministic predictions must be resolved during reachability analysis, significantly increasing the complexity of the state space. With prediction sharing enabled, a single dedicated automaton generates the prediction, and the remaining components deterministically consume this information. This reduces the overall non-determinism in the system and leads to a much smaller product automaton. Empirical results confirm this effect: benchmark families involving multiple until operators show exponential gains in verification time when prediction sharing is enabled. We depict the results in Figure 12 (left) for φ(n) = Vn i=1 p U[0,i] q. Figure 12 Comparison of performance of Tempora with and without features. The left figure shows the impact of sharing predictions. The middle figure shows the impact of specialized automata for F operator. The right figure shows the impact of memory reuse. The x-axis represents the parameter in the formula, while the y-axis shows the time taken in milliseconds. Specialized Automaton for F While the eventually operator FI p can be encoded as true UI p, this transformation can be an overkill for untimed or simple timing intervals like [0, ∞). Tempora implements specialized automata for such cases that are smaller and structurally simpler than general-purpose until-based encodings. For instance, the automaton for F[0,a] p simply waits up to a time units for p to become true, without introducing intermediate obligations or control states. These specialized constructions avoid unnecessary non-determinism and reduce clock usage. Our benchmarks show that formulas involving eventualities with small intervals benefit substantially from this, often reducing the number of locations and transitions in the resulting automaton S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 41 by an order of magnitude. The results are shown in Figure 12 (middle) for the formula φ(n) = VN i=1 F[0,i] p. Reusing Memory The memory optimization addresses inefficiencies in how truth values of atomic propositions are stored during automaton execution. Consider the formula φ(n) = Vn i=1 X pi, where each X pi depends only on the value of pi at the previous timestamp. In a naive implementation, each pi is assigned a unique index in the state array, even though their lifetimes do not overlap. Tempora incorporates a live-range analysis to reuse memory indices once a variable becomes dead - i.e., no longer accessed in the remaining evaluation of the round. In the above example, all pi can safely share a single index. This drastically reduces the size of the memory array and thus the number of reachable states. Our experiments demonstrate that this reuse yields exponential improvements in model-checking time for formulas with long chains of independent next operators, as depicted in Figure 12 (right). C.5 Benchmarks and Extended Experimental Results We next briefly describe the additional benchmarks we use to evaluate our performance. Our benchmarks. We constructed several MITL+p formulae to show the benefits of our improvements. To show the benefit of sharing clocks, we use a disjunction of multiple X operators: G[0,2](Wn i=1 X[2i,2i+1] pi), where each Next automaton uses a single shared clock. A conjunction of multiple U operators with the same left and right children: G(Vn i=1(pi → a U[0,i] b)) ∧G(Wn i=1 pi) shows the benefit of using sharer automata, where each Until automaton uses a common sharer automaton, as well as using a smaller 2 state automaton for Until. The power of using deterministic automata for top level operators can be seen for a formula with conjunction of multiple U operators having general intervals at the top level: Vn i=1(p U[2i,2i+1] q). We show the benefit of reusing shared variables for formulae where atomic propositions have short live ranges: Wn i=1 G[2,∞)(pi). In all these formulae, we try to ensure that they cannot be easily rewritten to a simpler formula, and are not trivially satisfiable/unsatisfiable. Acacia Benchmarks. Each LTL formula in the original Acacia benchmark is of the form ψ ⇐⇒G F acc, where ψ is an LTL specification. We modify these examples by adding intervals to temporal operators of ψ. For several of these specifications, we add intervals to the operators specifying occurrences of atomic propositions, which adds stricter constraints for the atomic propositions to occur within an interval. For some of the benchmarks, we also add general intervals to nested U operators to add complexity to the formula. The complete experimental results are presented in Table A (for finite words) and Table B for infinite words. 42 Efficient Verification of MTL with Past in Pointwise Semantics Sl. No Formula Baseline MightyL+UPPAAL TEMPORA Nodes Stored Time (ms) Nodes Stored Time (ms) Nodes Stored Time (ms) 1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3) 32341 179.083 11239 66.00 2744 25.4233 2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5) 1927857 24478.2 243106 1,348.00 24076 209.238 3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3) 70615 480.662 3440 17.00 329 1.57934 4 G[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 || p2 || p3) 35079 230.365 3440 20.00 345 2.02975 5 (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))&& G(p1 || p2) 25210 165.833 885 5.00 437 2.63777 6 (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X [2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || ! (atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))))))))&& G(p1 || p2) - TO 3517869 43,896.00 3509 21.3181 7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q 5441 225,448.00 1990 12.3429 8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5 2130 14.2971 3712 25.00 160 0.791932 9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5 80500 619.015 3872 23.00 20 0.133869 10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5) 118296 984.777 16182 129.00 10777 84.286 11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) )) 48083 416.384 479764 215,172.00 3367 33.0925 12 G[2, Inf) ( (!request || F[4, 5] grant)) 643 4.67705 60 307.00 71 0.511877 13 p U[11, 12] q 144 2.10697 57 1,408.00 38 0.305787 14 G((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 || p3 || p4) - N/A - N/A 186 1.67275 15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q))) - N/A - N/A 1303 29.9574 16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q) - N/A - N/A 1571 50.9912 17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q) - N/A - N/A 27737 1762.13 18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q) - N/A - N/A 167077 21490.5 19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - N/A 262 3.18676 20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - N/A 286 1.53985 21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf) q && p S[4, Inf) q) - N/A - N/A 68 0.798897 22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q))) - N/A - N/A 66 0.49014 MightyL benchmarks 23 F p1 && F p2 && F p3 && F p4 && F p5 9292 44.3188 3263 15.00 157 0.530741 24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5 1241 9.18455 3263 16.00 160 0.900405 25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5 74516 556.001 6592 44.00 177 0.806017 26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 47766 524.722 455 1,362.00 59 0.34313 27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5 - TO 6623 31,749.00 191 1.73696 28 (((((p0 U q1) U q2) U q3) U q4) U q5) 25972 101.764 8133 50.00 4063 21.7138 29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5) 104149 549.412 8133 40.00 6303 33.4324 30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5) 48557 319.548 18204 136.00 12496 82.9332 31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) 16160 185.278 1140 3,278.00 3238 35.6263 32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) 162606 3074.69 5006 19,078.00 27758 434.539 33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5) 1629834 62186.7 22228 107,447.00 235536 6455.47 34 G(!p) && F[0, 3] t1 110 0.790953 55 2.00 21 0.118925 35 G(!p) && F[0, 3] t1 && F[3, 6] t2 2396 12.5292 627 254.00 174 1.31732 36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 48222 364.599 2450 6,690.00 419 2.98024 37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4 - TO 48383 274,012.00 2079 11.2824 38 !(!(G F p1) || (G(!q || (F[100, 1000] r)))) 6916 38.5607 463 3.00 271 1.66032 39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r)))) 65149 418.318 3423 32.00 1005 5.93152 40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r)))) 561208 5684.74 26047 280.00 3397 18.338 41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r)))) 4651945 126995 202623 2,359.00 10953 59.6345 42 F[0, 30] (!p || G[0, 20] p) 31 0.32008 31 2.00 17 0.118925 43 G[0, 30](!p) || F[0, 20] p 37 0.303894 32 2.00 12 0.152296 44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5) 179753 1216.62 8128 44.00 6318 37.332 45 (((((p0 R q1) R q2) R q3) R q4) R q5) 47841 250.798 8128 33.00 4079 19.3469 46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) 7571 69.5354 560 15.00 546 5.74616 47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5) 375155 6503.65 9792 393.00 12042 230.663 48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) 54947 678.074 2368 74.00 2628 38.1337 49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5) 78161 587.855 9792 55.00 5892 39.5866 50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5 2735492 78412.1 3232 132.00 20 0.252719 51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5 70930 578.802 3232 18.00 20 0.136226 52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5 1815 14.362 3134 19.00 160 0.809136 53 G p1 && G p2 && G p3 && G p4 && G p5 12897 75.8965 3134 14.00 158 1.06356 54 !((F[0, 30] p) || !(F[0, 20] p)) 60 0.568014 69 2.00 49 0.272393 55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p)) 1272 10.5949 495 221.00 224 0.991115 Acacia benchmarks 56 ((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F(a))) 3047327 70749.6 54777 82,277.00 530 3.40564 57 (((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F a))) || (! (((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) && !((G (F a)))) - TO 3193831 41,472.00 3455 19.531 58 ((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && !(G(F(a)))) - TO 214577 96,763.00 1059 7.08385 59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a))) 466847 5451.49 13712 21,448.00 225 1.61587 60 ((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r U[0, 3] s)))) && !(G(F a))) - TO - Error 564 5.20393 61 ((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G(Fa))) || (!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && !(G(Fa))) - TO - Error 3276 19.6802 62 ((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F [2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && !(G(F a))) - TO - TO 366144 3712.65 63 (((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && (G(F (a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && !(G(F(a)))) - TO - TO 362442 8380.15 64 (((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && !(G(F(a)))) - TO - TO 7362 86.5996 65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a))) 51613 392.448 6641 72.00 61 0.507483 66 (G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) )) && ! (G( F (a)))) - TO - TO 3153 40.0667 67 (( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a)))) 885283 14219.3 99857 945.00 87 0.747122 68 (((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (!((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a)))) - TO - TO 5053 39.7407 69 (((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && !(G(F(a)))) 6802985 269414 51641 19,852.00 444 2.52895 70 (((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2](X[1, 2] (X[1, 2] q))))))) && !(G(F a))) - TO 430184 3,928.00 1061 4.97043 71 (((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && !(G(F a))) - TO - TO 1917 23.3503 72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a))) 13546 82.0316 3511 20.00 96 0.430384 Table A: Experimental results obtained by running TEMPORA, MightyL + UPPAAL and, the baseline implementation of the translation for satisfiability for finite traces. (Timeout is set to 300s) S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 43 Sl. No Formula Baseline MightyL + OPAAL TEMPORA Nodes Stored Time (ms) Nodes Stored (Reporting state space given by opaal) -> Non deterministic, since multi threaded. Numbers change across different experiments Time (ms). Note: opaal is multicore, and hence uses 8 cores of local machine Total time (ms) Nodes Stored Time (ms) opaal g++ ltsmin -> the actual exploration 1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3) 2292 14.1087 1161 172.79 661.47 76.03 910.28 212 1.08509 2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5) 29455 193.183 1861 260.97 710.76 80.51 1,052.25 552 3.17967 3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3) 1310856 16076.9 1162 227.10 700.43 76.16 1,003.69 458 2.42018 4 G[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 || p2 || p3) 1587059 21828.1 1450 263.38 718.80 77.70 1,059.88 1044 8.24895 5 (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((! p1) || atomic) && (!p2) || !(atomic))))))&& G(p1 || p2) 44857 350.62 1893 203.98 663.58 82.98 950.54 168 0.987026 6 (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || ! (atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || ! ((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((! p1) || atomic) && (!p2) || !(atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || ! (atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))))))))&& G(p1 || p2) - TO - - - Segfault 0.00 1933 15.9299 7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q - TO - - - TO 0.00 456 4.17251 8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5 2346 16.1747 969 301.04 736.75 75.30 1,113.09 224 1.30602 9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5 - TO 1143 272.19 721.28 78.98 1,072.45 178 1.58184 10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5) - TO 7003 404.46 851.76 106.54 1,362.75 1087 9.45395 11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) )) 117319 1140.97 7921 74,584.66 20,184.02 238.35 95,007.03 504 3.50233 12 G[2, Inf) ( (!request || F[4, 5] grant)) - TO - - - TO 0.00 444 4.00833 13 p U[11, 12] q 1360 21.8606 - - - TO 0.00 220 1.3685 14 G((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 || p3 || p4) - N/A - - - N/A 798 2.19275 15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q))) - N/A - - - N/A 14440 1721.87 16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q) - N/A - - - N/A 1410 51.2159 17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q) - N/A - - - N/A 2310 132.646 18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q) - N/A - - - N/A 3927 418.381 19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - - - N/A 91 0.509345 20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - - - N/A 267 1.44564 21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf9 q && p S[4, Inf) q) - N/A - - - N/A 114 0.645279 22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q))) - N/A - - - N/A 108 0.73709 MightyL benchmarks 23 F p1 && F p2 && F p3 && F p4 && F p5 822 5.11449 2165 190.21 698.33 79.48 968.01 64 0.407546 24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5 952 5.99807 2693 214.01 701.74 82.35 998.09 68 0.485472 25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5 - TO 3069 372.57 820.01 85.33 1,277.91 200 2.38852 26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 - TO - - - Segfault 0.00 102 0.528738 27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5 - TO - - - TO 0.00 158 1.05082 28 (((((p0 U q1) U q2) U q3) U q4) U q5) 1828 10.7493 1560 196.70 704.35 78.12 979.17 18689 91.4055 29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5) 47166 298.922 1477 219.19 699.27 76.57 995.03 2668 19.0006 30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5) 567192 5404.53 9383 382.47 960.14 113.60 1,456.21 4075 32.6984 31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) - TO - - - Segfault 0.00 3254 30.8566 32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) - TO - - - TO 0.00 36124 451.716 33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5) - TO - - - TO 0.00 465185 8750.47 34 G(!p) && F[0, 3] t1 310 1.95935 184 117.86 612.55 72.49 802.89 32 0.235228 35 G(!p) && F[0, 3] t1 && F[3, 6] t2 953492 24977.1 964 68,462.12 20,584.03 102.84 89,149.00 158 0.901839 36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 - TO - - - Segfault 0.00 315 3.3407 37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4 - TO - - - TO 0.00 522 4.21687 38 !(!(G F p1) || (G(!q || (F[100, 1000] r)))) - TO 1935 692.31 954.65 79.29 1,726.25 2823 18.9429 39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r)))) - TO 4212 749.15 1,006.85 89.55 1,845.55 28570 196.333 40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r)))) - TO 42804 836.05 1,056.50 225.40 2,117.95 330044 2976.99 41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r)))) - TO 149029 906.87 1,133.70 473.50 2,514.08 3819965 88366.7 42 F[0, 30] (!p || G[0, 20] p) 9432 52.7811 43 131.32 615.22 69.64 816.18 704 5.34067 43 G[0, 30](!p) || F[0, 20] p 61 0.756158 37 126.61 617.54 75.80 819.95 27 0.295317 44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5) 145353 1141.14 2267 260.88 746.28 86.45 1,093.60 113881 905.989 45 (((((p0 R q1) R q2) R q3) R q4) R q5) 22354 140.505 1521 185.82 690.21 79.60 955.64 11957 61.4713 46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) 1086882 15332.4 453 1,588.79 1,496.63 78.06 3,163.48 5985 55.9789 47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5) - TO - - - Segfault 0.00 1093187 23474.1 48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) - TO - - - Segfault 0.00 79258 1018.67 49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5) 29685 278.315 1059 230.60 725.70 77.18 1,033.48 3119 27.748 50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5 - TO - - - Segfault 0.00 225 2.45304 51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5 - TO 1549 226.81 724.91 82.29 1,034.01 178 1.07516 52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5 4793 34.5633 5913 257.33 760.33 100.74 1,118.41 224 1.45227 53 G p1 && G p2 && G p3 && G p4 && G p5 195653 1451.52 8413 186.71 694.95 111.65 993.31 64 0.61791 54 !((F[0, 30] p) || !(F[0, 20] p)) 7069 37.444 100 126.35 622.49 72.87 821.71 629 10.9368 55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p)) 3072735 56119.3 624 62,506.68 20,677.93 112.71 83,297.33 1669 19.924 Acacia benchmarks 56 ((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F (a))) - TO 76875 150,058.47 42,824.04 2,635.25 195,517.76 4561 37.7644 57 (((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F a))) || (!(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) && !((G (F a)))) - TO 7271276 611.46 1,143.21 21,348.99 23,103.66 918 7.40638 58 ((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && !(G(F(a)))) - TO 523958 67,992.90 22,837.11 10,320.40 101,150.41 4614 32.6728 59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a))) - TO 22945 116,791.31 39,078.93 840.78 156,711.03 1617 12.6589 60 ((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r U[0, 3] s)))) && !(G(F a))) - TO - - - Segfault 0.00 820 6.57777 61 ((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G (Fa))) || (!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2] s)))))))))) && !(G(Fa))) - TO 6733 183,618.00 14,086.56 323.78 198,028.34 4658 32.0218 62 ((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && !(G(F a))) - TO - - - Segfault 0.00 54366 545.194 63 (((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && (G(F(a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && !(G(F(a)))) - TO - - - TO 0.00 - TO 64 (((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && !(G(F(a)))) - TO - - - TO 0.00 - TO 65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a))) - TO 11391 515.86 1,265.08 159.79 1,940.73 246 1.536 66 (G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) )) && ! (G( F (a)))) - TO - - - Segfault 0.00 - TO 67 (( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((! (G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a)))) - TO 250875 516.21 950.14 750.60 2,216.94 3484 25.3826 68 (((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (! ((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a)))) - TO - - - Segfault 0.00 2598 25.7481 69 (((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && !(G(F(a)))) - TO 35945 63,521.58 21,527.32 788.87 85,837.77 785 5.94832 70 (((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2] (X[1, 2](X[1, 2] q))))))) && !(G(F a))) - TO 742427 426.25 948.75 1,682.79 3,057.79 231 1.94063 71 (((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && !(G(F a))) - TO - - - TO 0.00 4159 55.7087 72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a))) 1802521 22455.7 4943 257.75 755.07 95.59 1,108.41 390 2.54234 Table B: Experimental results obtained by running TEMPORA , MightyL + OPPAAL and, the baseline implementation of the translation for satisfiability for infinite traces. (Timeout is set to 300s)"}}},{"rowIdx":283,"cells":{"id":{"kind":"string","value":"2510.14697"},"raw":{"kind":"string","value":"Preprint\nPURIFYING TASK VECTORS IN KNOWLEDGE-AWARE\nSUBSPACE FOR MODEL MERGING\nBang An1\nYibo Yang1∗\nPhilip Torr2\nBernard Ghanem1\n1King Abdullah University of Science and Technology (KAUST)\n2University of Oxford\n{bang.an, yibo.yang, bernard.ghanem}@kaust.edu.sa\nphilip.torr@eng.ox.ac.uk\nABSTRACT\nModel merging aims to integrate task-specific abilities from individually fine-\ntuned models into a single model without extra training. In recent model merg-\ning methods, task vector has become a fundamental building block, as it can en-\ncapsulate the residual information from finetuning. However, the merged model\noften suffers from notable performance degradation due to the conflicts caused\nby task-irrelevant redundancy in task vectors. Existing efforts in overcoming re-\ndundancy by randomly dropping elements in the parameter space involves ran-\ndomness and lacks knowledge awareness. To address these challenges, in this\nstudy, we propose Purifying TAsk VEctors (PAVE) in knowledge-aware sub-\nspace. Concretely, we sample some training examples from each task, and feed\nthem into their corresponding fine-tuned models to acquire the covariance ma-\ntrices before linear layers. We then perform a context-oriented singular value\ndecomposition, which accentuates the weight components most relevant to the\ntarget knowledge. As a result, we can split fine-tuned model weights into task-\nrelevant and redundant components in the knowledge-aware subspace, and pu-\nrify the task vector by pruning the redundant components. To induce fair prun-\ning efforts across models, we further introduce a spectral rank allocation strat-\negy by optimizing a normalized activated pruning error. The task vector purifi-\ncation by our method as a plug-and-play scheme is applicable across various\ntask vector-based merging methods to improve their performance.\nIn experi-\nments, we demonstrate the effectiveness of PAVE across a diverse set of merg-\ning methods, tasks, and model architectures. Remarkably, when integrated on\nthe state-of-the-art merging method EMR-Merging with RoBERTa, our method\nyields a significant performance improvement of 4.1% (from 80.18% w/o PAVE\nto 84.28%) on the GLUE benchmark, approaching to the average performance\nof 8 individual models, 85.55%. Our code and models are released at: https:\n//github.com/ABgit111/purified_task_vector.\n1\nINTRODUCTION\nTransformer-based models, including large language models (LLMs) (Devlin et al., 2019; Liu et al.,\n2019; He et al., 2020; Radford et al., 2019; Touvron et al., 2023; Achiam et al., 2023) and vi-\nsion models (Dosovitskiy et al., 2020; Radford et al., 2021) have found widespread applications\nacross various domains. Supervised fine-tuning (Radford et al., 2018; Hu et al., 2022) is a common\ntechnique to enhance downstream task performance. Despite the effectiveness of multi-task learn-\ning (MTL) in acquiring versatile abilities across multiple domains (Caruana, 1997; Zhang & Yang,\n2018; Vandenhende et al., 2021), its applicability will be limited when only individually fine-tuned\nmodels are accessible due to data privacy concerns or computational constraints. To address this\nchallenge, model merging has been proposed, aiming to construct a single model by combining\nparameters from domain-specific models, without relying on any training or fine-tuning efforts.\nEarly model merging methods focus on performing weighted (Matena & Raffel, 2022) or regularized\n(Jin et al., 2022) averaging in the parameter space of fine-tuned models, yet the conflicts inherent in\n∗Corresponding author\n1\narXiv:2510.14697v1 [cs.AI] 16 Oct 2025\n\nPreprint\nKnowledge-aware subspace\n∆𝑊= 𝑊!\" −𝑊#\nTask vector\nNaive\nDARE\nPAVE (ours)\nFormulation\nAddress redundancy\nKnowledge awareness\n∆𝑊$%&'=\n1\n1 −𝑝Drop 𝑊!\" −𝑊#\n∆𝑊(%)'= SVD* 𝑊!\"𝐶𝐶+, −𝑊#\n𝑊!\"\n𝑊#\nFine-tune\nSVD$ 𝑊!\"𝐶𝐶%&\nRedundancy w.r.t. the \ntarget knowledge\n❌\n✅(in parameter space)\n✅(in subspace)\n❌\n❌\n✅\nTarget knowledge as \ncharacterized by 𝐶\n➕\nSVD $:( 𝑊!\"𝐶𝐶%&\nNaive task vector\nPurified task vector\nFigure 1: An illustration of our method and a comparison between PAVE and prior task vectors.\nDARE addresses redundancy by randomly dropping elements in the parameter space without knowl-\nedge awareness. PAVE refines task vectors by identifying the target knowledge-relevant components\nwithin knowledge-aware subspace and removing the redundant components.\nnaive parameter averaging often lead to performance degradation of merged models. Subsequently,\nTask Arithmetic (Ilharco et al., 2022) has achieved great success by introducing task vector, defined\nas the difference between the fine-tuned model weights for a certain task and the base model. Since\ntask vector can capture the update direction of each task, it allows for efficient composition of\nknowledge from multiple fine-tuned models while preserving their distinctive features, and thus has\nbecome a fundamental building block in the follow-up studies (Ilharco et al., 2022; Cheng et al.,\n2025; Du et al., 2024; Huang et al., 2024). However, the task vector constructed by subtracting the\nbase model weights from the fine-tuned weights contains substantial redundancy, which means that\nnot all parameter shifts during fine-tuning contribute to task-specific abilities (Yu et al., 2024). The\nredundant or noisy components may interfere with the valuable components from other task vectors\nduring merging. As a result, performing model merging based on such task vectors still encounters\nconflicts and yields suboptimal performance (Yadav et al., 2023).\nTo address the redundancy in task vectors, DARE (Yu et al., 2024) proposes to randomly mask\ntask vectors by Dropout (Srivastava et al., 2014) to induce sparsity and reduce conflict. Despite the\nimproved performance, we point out that: (1) random masks lack knowledge awareness and may\nfail to accurately locate the parameter elements most relevant to a specific task; (2) task-relevant and\nredundant elements are not necessarily separable in the parameter space, and actually task-specific\nability lies in a low-rank subspace of parameters (Hu et al., 2022), which means that masking in\nthe parameter space may not effectively disentangle abilities and resolve conflicts. Therefore, how\nto purify task vectors to make them precisely align with task-specific abilities is still a fundamental\nchallenge in the model merging problem.\nTo this end, in this paper, we propose purifying task vectors in knowledge-aware subspace (PAVE).\nInstead of addressing redundant and noisy elements in the original parameter space, we aim to\ndecompose fine-tuned LLM parameters into subspace such that the decomposed components are\nordered by their contribution to a specific task. Naively performing singular value decomposition\n(SVD) on the fine-tuned weights is inadequate because it only captures the spectral structure of the\nweight matrix itself and the resulting subspace is not knowledge-aware. In contrast, we adopt the\ncontext-oriented decomposition that is initially proposed in a recent study for low-rank adaptation\n(Yang et al., 2024b). Specifically, given multiple fine-tuned models on different tasks, we randomly\nsample a small number of training samples for each task, and feed them into the corresponding\nfine-tuned model. After obtaining the activation of each layer X, we calculate the covariance matrix\nC = XXT and then perform the context-oriented decomposition by applying SVD to the product of\nthe fine-tuned weight matrix WFT and the covariance matrix C from its input, i.e., SVD(WFTC). By\ndoing so, the decomposed subspace is spanned by singular vectors that are ordered by their contri-\nbutions to the task associated with C. It has been observed that the first several components account\nfor the majority of the learned task-relevant knowledge. Therefore, we filter out the redundant and\n2\n\nPreprint\nnoisy directions by retaining only the largest r singular values and their singular vectors. We then\npost-multiply by C−1 to reconstruct the fine-tuned model weights, which yields our purified task\nvector formulated as ∆WPAVE = SVDr(WFTC)C−1 −WB. The illustration of our method and its\ncomparison with existing task vector constructions are shown in Figure 1.\nTo induce fair pruning efforts across models, we further introduce a spectral rank allocation strategy\nthat allocates the number of preserved ranks based on their spectral distribution. It aims to minimize\nthe summation of a normalized activated pruning error from each individual model. This objective\ncan be efficiently optimized using a greedy algorithm that iteratively excludes components with the\nsmallest normalized singular value. It is noteworthy that PAVE as a plug-and-play method can be\nintegrated on any task vector-based merging methods to improve their performance. We evaluate\nthe performance of PAVE in combination with several popular merging strategies, including Task\nArithmetic (Ilharco et al., 2022), Ties-Merging (Yadav et al., 2023), and EMR-Merging (Huang\net al., 2024). Experiments are conducted on the GLUE benchmark with two widely used language\nmodels, RoBERTa (Liu et al., 2019) and DeBERTa (He et al., 2020), and also on generation tasks,\nMath and Coding, with LLaMA-2-7B (Touvron et al., 2023). Additionally, we apply our approach\nto ViT-B/16, ViT-B/32, and ViT-L/14 (Dosovitskiy et al., 2020) on 8 vision tasks.\nThe contributions of this study are summarized as follows:\n• We propose PAVE, a plug-and-play method that can improve the performance of modern\nmodel merging methods by purifying task vectors and resolving task-irrelevant redundan-\ncies in knowledge-aware subspace.\n• We design a spectral rank allocation strategy to further refine the redundancy pruning pro-\ncess and improve the performance.\n• We demonstrate the effectiveness of PAVE with a wide range of merging methods, tasks,\nand model architectures. Notably, PAVE improves the performance of EMR-Merging on\nGLUE with RoBERTa from 80.18% to 84.28%, which is close to the average result of 8\nindividual models 85.55%.\n2\nRELATED WORK\nModel Merging. Model merging has emerged as an active area due to its great potential in inte-\ngrating abilities from multiple specialized models into a single model (Wortsman et al., 2022; Yadav\net al., 2023; Yang et al., 2023; Xiao et al., 2023). Different approaches have been developed in-\ncluding optimization-driven methods (Matena & Raffel, 2022; Jin et al., 2022; Wang et al., 2024a),\ntest-time adaptation methods (Yang et al., 2023; 2024a), and task vector-based methods (Ilharco\net al., 2022; Yadav et al., 2023; Wei et al., 2025). In particular, task vector has become an influen-\ntial tool, as it encodes the essential information obtained during fine-tuning and can be effectively\ninjected to a merged model using appropriate techniques. However, during the merging process,\nevidences have shown that there is noise and redundancy in the plain task vectors, which can cause\nconflicts and performance degradation. To address this issue, Ties-Merging (Yadav et al., 2023)\nidentifies redundancy in model parameters and proposes a trimming and sign-selection scheme to\nresolve conflicts. EMR-Merging (Huang et al., 2024) employs a strategy of electing, masking, and\nrescaling to integrate lightweight task-specific components, aiming to minimize the divergence be-\ntween the merged model and each individual model. DARE (Yu et al., 2024), as a plug-and-play\nmethod, resolves conflict issues by introducing random Dropout (Srivastava et al., 2014) to the task\nvectors. Nevertheless, former methods mainly focus on reducing redundancy by removing elements\nfrom the task vectors, which can be considered as sparsity-based approaches. In contrast, our method\npurifies task vectors in a knowledge-aware subspace by performing context-oriented decomposition\nand pruning the task-irrelevant weight components.\nLow-Rank Decomposition for LLMs. Low-rank decomposition based on SVD has been widely\nadopted in LLMs across multiple applications. For example, in low-rank adaptation (Hu et al.,\n2022; Liu et al., 2024; Zhang et al., 2023; Yang et al., 2024b; 2025), SVD is often leveraged to\ncreate lightweight low-rank adapters for improving fine-tuning performance (Meng et al., 2024)\nor preserving task-specific abilities (Yang et al., 2024b). In model compression, various methods\nhave been proposed to prune model parameters based on the importance provided by singular values\n(Yuan et al., 2023; Wang et al., 2024b; Hsu et al., 2022; Li et al., 2025). Similar techniques have also\n3\n\nPreprint\nbeen developed for quantization (Li et al., 2024b;a). However, when it comes to model merging,\nredundancy purification still relies mostly on sparsity-based approaches—removing or rescaling task\nvectors in the parameter space. While decomposition approach is adopted in Twin Merging (Lu et al.,\n2024), TSV-Merge (Gargiulo et al., 2025), and KnOTS (Stoica et al., 2024) as a pre-processing step,\nthe implementations in these works still lack knowledge awareness. As a comparison, our proposed\nmethod performs task vector purification in knowledge-aware subspace to extract and highlight the\nparts within a task vector that truly matter for its corresponding task, leading to smoother and more\neffective merging.\n3\nMETHODOLOGY\n3.1\nPRELIMINARIES\nGiven K different models {M{W l\n1}L\nl=1, M{W l\n2}L\nl=1, · · · , M{W l\nK}L\nl=1} that are fine-tuned on K\ndifferent tasks {T1, T2, · · · , TK} from the same base model M{W l\nB}L\nl=1, where L represents the\nnumber of layers, model merging aims to produce a single model that simultaneously achieves good\nperformance on these tasks without extra training or fine-tuning. The merged model M{W l\nM}L\nl=1\ncan be given by:\nM{W l\nM}L\nl=1 = ϕ(M{W l\n1}L\nl=1, · · · , M{W l\nK}L\nl=1, M{W l\nB}L\nl=1),\n(1)\nwhere ϕ refers to a merging method.\nMost model merging methods focus on combining the parameters of fine-tuned models. Task vector\ncaptures the task-specific changes introduced during fine-tuning and has become a common and es-\nsential tool in modern model merging methods. It is typically defined as the element-wise difference\nbetween a fine-tuned model and the base model,\n∆W = WFT −WB,\n(2)\nwhere WFT denotes the fine-tuned model weights. Based on task vector, different merging methods\nhave been proposed to integrate these task-specific parameter variations into a merged model. For\nexample, in Task Arithmetic (Ilharco et al., 2022), the merged model weights can be computed as:\nWM = WB + λ\nK\nX\ni=1\n∆Wi,\n(3)\nwhere λ is a hyperparameter.\nHowever, recent studies have shown that naively calculating the difference between WFT and WB\ncannot precisely represent the shifting direction of a certain task in the parameter space, as the\nweight change caused by fine-tuning contains redundant components, e.g., task-irrelevant abilities\nor noisy elements (Yadav et al., 2023; Yu et al., 2024). As a result, there still exist conflicts in the\nsubsequent merging procedure using these task vectors. DARE (Yu et al., 2024) proposes to sparsify\ntask vectors by randomly dropping elements and rescaling. Their task vector can be formulated as:\n∆WDARE =\n1\n1 −pDrop(WFT −WB),\n(4)\nwhere Drop denotes the Dropout operation with rate p, and\n1\n1−p is a rescaling coefficient. Despite its\neffectiveness in reducing conflict, several limitations remain as follows. First, the Dropout operation\ninvolves randomness and lacks knowledge awareness, and thus it cannot locate the valuable compo-\nnents in a targeted manner. Second, task-specific abilities actually lie in the subspace of pre-trained\nparameters (Raghu et al., 2017; Li et al., 2018), suggesting that binary masking in the parameter\nspace may not truly separate redundancy or irrelevant components from the required ability. This\nanalysis motivates us to resort to subspace to purify task vectors.\n3.2\nPURIFYING TASK VECTORS IN KNOWLEDGE-AWARE SUBSPACE\nDirectly performing singular value decomposition (SVD) on fine-tuned model weights constructs\na subspace, but such subspace is not necessarily aligned with task-specific knowledge, as it only\n4\n\nPreprint\n×𝐶!\n\"!\n×𝐶#\n\"!\n×𝐶$\n\"!\nSVD\n𝑊!𝐶!\n𝑊#𝐶#\n𝑊$𝐶$\n𝑊!\n𝑊#\n𝑊$\n𝑈!\nRank Allocation\nSVD\nSVD\n𝑈#\n𝑈$\nΣ!\nΣ#\nΣ$\n𝑉!\n%\n𝑉#\n%\n𝑉$\n%\n𝑈![: 𝑟!]\n𝑈#[: 𝑟#]\n𝑈$[: 𝑟$]\nΣ![: 𝑟!]\nΣ#[: 𝑟#]\nΣ$[: 𝑟$]\n𝑉!\n%[: 𝑟!]\n𝑉#\n%[: 𝑟#]\n𝑉$\n%[: 𝑟$]\n𝐶!\n𝐶#\n𝐶$\nSVD&! 𝑊!𝐶! 𝐶!\n\"!\nSVD&\" 𝑊#𝐶# 𝐶#\n\"!\nSVD&# 𝑊$𝐶$ 𝐶$\n\"!\n⋮\n⋮\n⋮\n⋮\n⋮\nTruncate\nMerging \nMethod\n𝑊'\nTask 2\nTask 𝐾\nTask 1\nsample\nsample\nsample\nFigure 2: An overall illustration of our proposed method, PAVE, which serves as a plug-and-play\nmethod for task-vector based merging methods. It begins by identifying task-aware components\nthrough a decomposition approach, and then eliminates noise and less effective components via\nrank-based pruning. A rank allocation strategy is designed to optimize the preserved rank ratios\nacross individual models. Overall, rather than employing the plain task vector Wk −WB, PAVE\nperforms model merging based on purified task vectors SVDrk(WkCk)C−1\nk\n−WB, 1 ≤k ≤K.\nreflects the spectral distribution of the parameter matrix itself and pre-trained LLM weights handle\na broad range of tasks. To this end, we leverage the context-oriented singular value decomposition\n(CO-SVD) (Yang et al., 2024b) to decompose fine-tuned weights into a knowledge-aware subspace.\nConcretely, for each task Tk, 1 ≤k ≤K, we sample some training examples from this task and feed\nthem into the fine-tuned model Mk. After acquiring the input activation of each layer X (layer index\nis omitted for simplicity), we calculate the covariance matrix as C = XXT . Projecting the weight\nmatrix of a linear layer onto the covariance matrix of its input activation can accentuate the task-\nspecific components related to the input. Therefore, we apply SVD on the product of each weight\nmatrix and its corresponding covariance matrix, i.e., SVD(WFTC). The decomposed components\nare ordered by the singular values that reflect their contributions to the task as characterized by C.\nSuch subspace enjoys a low-rank structure, which means that the the knowledge-aware abilities can\nbe condensed into the first several leading singular values and singular vectors, as will be shown\nin Section 4.1. Hence, we filter out the task-irrelevant components by keeping only the first r\ncomponents, and then reconstruct the weight matrix through right multiplication with C−1, i.e., the\ninverse of covariance matrix C. Consequently, the purified weight matrix can be formulated as:\nW †\nFT = SVDr(WFTC)C−1 =\nr\nX\ni=1\nσiuivT\ni C−1,\n(5)\nwhere σi denotes the i-th singular value, ui and vi are the left and right singular vectors, respectively.\nWe use the purified fine-tuned weight matrix W †\nFT to construct our task vector as:\n∆WPAVE = W †\nFT −WB = SVDr(WFTC)C−1 −WB,\n(6)\nwhere WB is the base model weights before fine-tuning, and ∆WPAVE represents our purified task\nvector, which can be used for any task vector-based merging methods, such as Ties-Merging (Yadav\net al., 2023), Task Arithmetic (Ilharco et al., 2022), and EMR-Merging (Huang et al., 2024).\nThe effectiveness of CO-SVD in identifying task-specific directions of weight matrices has been\nverified by Yang et al. (2024b) for model adaptation. In addition, C = XXT is a symmetric pos-\nitive semi-definite matrix, which enables efficient inverse operation with numerical stability. When\nC is not invertible, we repeatedly add positive elements on its diagonal until it is invertible. In exper-\niments, we show that our method can be integrated on various model merging methods to improve\nthe performance of the merged models.\n3.3\nSPECTRAL RANK ALLOCATION\n5\n\nPreprint\nAlgorithm 1 Spectral rank allocation strategy\nRequire: Individual models {M{W l\n1}L\nl=1, M{W l\n2}L\nl=1, · · · , M{W l\nK}L\nl=1}, data sampling from target tasks\n{X1, X2, · · · , XK}, a preserved rank ratio ρ, a stopping ratio γ.\n1: Feed Xi into M{W l\ni }L\nl=1. Calculate covariance matrices for all layers and for all models, {Cl\ni}K,L\ni,l , and\nthe rank of each layer {Rl}L\nl=1.\n2: for l = 1 to L do\n3:\nApply SVD to W l\ni Cl\ni and compute {σl\ni,j}K,Rl\ni,j\n, sort them in descending order.\n4:\nNormalize {σl\ni,j}K,Rl\ni,j\nby the largest singular value, {sl\ni,j}K,Rl\ni,j\n=\n\u001a\nσl\ni,j\nσl\ni,max\n\u001bK,Rl\ni,j\n5:\nInitialize {rl\ni}K,L\ni,l\n= {Rl}K,L\ni,l\n6:\nInitialize index set I = {1, 2, · · · , K}\n7:\nwhile\nPk\ni=1 rl\ni\nKRl\n> ρ do\n8:\nPick up the smallest sl\ni,rl\ni across i ∈I, suppose it is sl\nt,rl\ni\n9:\nrl\nt = rl\nt −1\n10:\nif rl\nt ≤γRl then\n11:\nremove t from I\n12:\nend if\n13:\nend while\n14: end for\n15: Output: the rank ratios for each individual model, {ρi}K\ni =\nn PL\nl=1 rl\ni\nPL\nl=1 Rl\noK\ni .\nWe have introduced the construction of our purified task vector in Section 3.2. However, when\nmerging multiple fine-tuned models, assigning the rank r to each model is highly non-trivial. To\noptimize rank allocation under a specified preserved rank ratio ρ, we propose a layer-wise optimiza-\ntion framework that determines the preserved rank of linear layer l by minimizing a summation of\nnormalized activated pruning error from different models. This strategy enables an adaptive preser-\nvation of task-specific knowledge during the merging process.\nWe begin by introduce the activated pruning error with rank rl\ni as:\n∥SVDrl\ni(W l\ni Cl\ni) −W l\ni Cl\ni∥2\nF ,\n(7)\nwhere i refers to the i-th model. In practice, we further normalize it with σmax(W l\ni Cl\ni) to ensure\ncomparability across different models with various spectral magnitudes. This normalization bounds\nthe scaled spectrum within a range from the inverse of its condition number to 1, thus enabling a fair\nassignment of rank across models. The normalized activated pruning error is defined as:\n∥SVDrl\ni(W l\ni Cl\ni) −W l\ni Cl\ni∥2\nF\n(σmax(W l\ni Cl\ni))2\n=\nrl\ni\nX\nj=1\n(σj(W l\ni Cl\ni))2\n(σmax(W l\ni Cl\ni))2 =\nrl\ni\nX\nj=1\n \nσl\ni,j\nσl\ni,max\n!2\n.\n(8)\nBuilding on this formulation, we define the overall objective for the l-th layer as:\nmin\nPK\ni=1 rl\ni\nKRl\n<ρ\nK\nX\ni=1\n∥SVDrl\ni(W l\ni Cl\ni) −W l\ni Cl\ni∥2\nF\n(σmax(W l\ni Cl\ni))2\n,\n(9)\nwhere Rl refers to full rank. Minimizing the objective Eq. (9) can be effectively approached using a\ngreedy algorithm, and the whole procedure has been summarized in Algorithm 1. In this procedure,\nfor each layer l, we iteratively remove the component associated with the smallest normalized sin-\ngular value, defined as sl\nt,rl\ni =\nσl\ni,j\nσl\ni,max , thereby achieving optimality of Eq. (9) through the greedy\nprocess. To prevent excessive rank pruning that causes performance degradation, we introduce a\nthreshold γ which serves as a stopping ratio. This rank strategy enables spectrum-based rank allo-\ncation for model merging while respecting the global rank constraint.\n6\n\nPreprint\n96\n192\n384\n512\nPruned ranks\n0.5\n0.6\n0.7\n0.8\n0.9\nPerformance\nSST-2\nSVD\nCO-SVD random\nASVD with SST-2\nCO-SVD with SST-2\nCO-SVD with COLA\n96\n192\n384\n512\nPruned ranks\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\nPerformance\nMRPC\nSVD\nCO-SVD random\nASVD with MRPC\nCO-SVD with MRPC\nCO-SVD with COLA\n96\n192\n384\n512\nPruned ranks\n0.5\n0.6\n0.7\n0.8\n0.9\nPerformance\nQQP\nSVD\nCO-SVD random\nASVD with QQP\nCO-SVD with QQP\nCO-SVD with COLA\n96\n192\n384\n512\nPruned ranks\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\nPerformance\nMNLI\nSVD\nCO-SVD random\nASVD with MNLI\nCO-SVD with MNLI\nCO-SVD with COLA\n96\n192\n384\n512\nPruned ranks\n0.5\n0.6\n0.7\n0.8\n0.9\nPerformance\nQNLI\nSVD\nCO-SVD random\nASVD with QNLI\nCO-SVD with QNLI\nCO-SVD with COLA\n96\n192\n384\n512\nPruned ranks\n0.45\n0.50\n0.55\n0.60\n0.65\n0.70\n0.75\nPerformance\nRTE\nSVD\nCO-SVD random\nASVD with RTE\nCO-SVD with RTE\nCO-SVD with COLA\nFigure 3: Model rank pruning with different decomposition methods. The results show that the\ncontext-oriented decomposition (CO-SVD) is more effective in aligning the subspace with the target\nknowledge. The pruned ranks, 96, 192, 384 and 512, correspond to removing 1/8, 1/4, 1/2 and 2/3\nof the full rank, respectively.\n4\nEXPERIMENTS\nIn this section, we provide the results on the GLUE benchmarks using RoBERTa-base (Liu et al.,\n2019) and DeBERTa-base (He et al., 2020), as well as the generation benchmarks with LLaMA2-7B-\nhf (Touvron et al., 2023) on mathematical solving and code generation tasks. We evaluate PAVE with\nvarious merging methods, including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav\net al., 2023), and EMR-Merging (Huang et al., 2024). For further detailed experimental settings and\nhyperparameters, please refer to Appendix A.\n4.1\nEFFECT OF DECOMPOSITION IN KNOWLEDGE-AWARE SUBSPACES\nWe demonstrate the effectiveness of decomposition in knowledge-aware subspaces through a se-\nries of numerical experiments. As concluded in Figure 3, we prune individual RoBERTa models\nfine-tuned on six GLUE tasks using various decomposition methods. The decomposition methods\nmentioned in Figure 3 are summarized as follows: SVD (Meng et al., 2024) applies singular value\ndecomposition directly to the weights. ASVD (Yuan et al., 2023) introduces a diagonal scaling ma-\ntrix, where each diagonal entry corresponds to the average absolute value of the associated feature.\nCO-SVD (Yang et al., 2024b) leverages covariance matrices. In the CO-SVD random variant, the\ncovariance matrix is replaced by a sampled random square matrix from [-1,1], serving as a task-\nagnostic baseline. CO-SVD with CoLA derives the covariance matrix using samples from CoLA,\nhighlighting the impact of selecting samples from an appropriate task.\nFrom Figure 3, it is evident that, for a given rank, decomposition methods that incorporate task\ninformation (CO-SVD, ASVD) outperform those that do not (SVD, CO-SVD random). In par-\nticular, CO-SVD demonstrates superior performance compared to ASVD, as CO-SVD preserves\nthe context information more effectively through task-specific activations. This pattern holds con-\nsistently across all datasets evaluated, reinforcing the importance of using the correct task data to\nguide decomposition. Notably, performance advantages of CO-SVD become even more prominent\nat pruning larger ranks, where other methods exhibit sharp degradation, indicating CO-SVD’s ef-\nfectiveness in condensing knowledge awareness. It is noteworthy that the input must originate from\nthe corresponding task to construct knowledge-aware subspace. When using CoLA samples to gen-\nerate the covariance matrix, performance drops significantly across all tasks. These results suggest\nknowledge-awareness is crucial for successful rank preservation, and emphasize the importance of\nintroducing knowledge-aware subspaces for purifying task vectors.\n7\n\nPreprint\nTable 1: Results of merging fine-tuned RoBERTa models on eight datasets from GLUE benchmark.\nThe number of examples used to generate context matrix is 4096.\nMethods\nCoLA\nSST-2\nMRPC\nSTSB\nQQP\nMNLI\nQNLI\nRTE\nAverage\nIndividual\n60.18\n94.04\n89.22\n90.63\n91.41\n87.20\n92.71\n79.06\n85.55\nEMR-Merging\n39.96\n93.35\n86.27\n82.77\n89.72\n85.45\n89.57\n74.37\n80.18\nEMR-Merging W/ DARE\n40.58\n93.23\n86.52\n82.86\n89.93\n85.98\n89.68\n75.09\n80.48+0.30\nEMR-Merging W/ PAVE\n60.42\n93.69\n87.75\n89.73\n89.13\n85.27\n92.09\n76.17\n84.28+4.10\nTask Arithmetic\n18.78\n85.89\n79.90\n74.03\n83.78\n59.08\n69.67\n62.09\n66.65\nTask Arithmetic W/ DARE\n18.26\n85.67\n80.39\n73.79\n83.77\n59.20\n69.58\n61.01\n66.45−0.20\nTask Arithmetic W/ PAVE\n18.96\n85.67\n80.64\n74.14\n83.77\n58.97\n69.91\n61.73\n66.72+0.07\nTies-Merging\n20.48\n84.40\n81.13\n58.19\n85.70\n64.65\n74.81\n42.96\n64.04\nTies-Merging W/ DARE\n17.56\n83.83\n80.39\n63.60\n85.79\n64.82\n0.7324\n44.77\n64.25+0.21\nTies-Merging W/ PAVE\n25.45\n82.45\n79.41\n64.03\n85.43\n69.93\n77.19\n51.62\n66.93+2.89\nTable 2: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark.\nThe number of examples used to generate context matrix is 4096.\nMethods\nCoLA\nSST-2\nMRPC\nSTSB\nQQP\nMNLI\nQNLI\nRTE\nAverage\nIndividual\n59.10\n95.06\n89.21\n91.31\n91.50\n88.52\n93.30\n69.31\n84.66\nEMR-Merging\n47.05\n94.72\n71.32\n87.57\n88.36\n87.38\n92.29\n64.26\n79.11\nEMR-Merging W/ DARE\n47.76\n94.61\n72.06\n87.30\n88.77\n87.67\n92.28\n66.43\n79.61+0.50\nEMR-Merging W/ PAVE\n61.32\n94.50\n84.31\n89.54\n90.34\n87.50\n93.01\n60.65\n82.64+3.53\nTask Arithmetic\n3.03\n75.00\n68.38\n52.72\n63.44\n43.46\n51.16\n50.90\n51.01\nTask Arithmetic W/ DARE\n2.84\n68.81\n68.38\n54.24\n64.38\n42.23\n55.15\n49.46\n50.68−0.33\nTask Arithmetic W/ PAVE\n2.18\n75.46\n68.38\n55.90\n64.00\n43.48\n61.25\n51.99\n52.83+1.82\nTies-Merging\n5.36\n66.28\n68.63\n19.09\n63.68\n40.44\n55.39\n48.74\n45.95\nTies-Merging W/ DARE\n6.03\n56.42\n68.87\n0.61\n66.28\n42.11\n54.00\n51.26\n43.19−2.76\nTies-Merging W/ PAVE\n1.39\n68.35\n68.63\n36.76\n63.12\n44.01\n59.36\n50.90\n49.06+3.11\n4.2\nMODEL MERGING FOR GLUE TASKS\nTo evaluate the effectivness of PAVE, we first present results on GLUE tasks using RoBERTa as\nthe backbone model. The numerical results are summarized in Table 1. We observe that PAVE\nyields a notable average performance gain of +4.10% when applied to EMR-Merging, reducing\nthe gap to just 1.27% from the individual models’ average performance. For specific tasks such\nas CoLA, PAVE significantly improves the score from 39.96% to 60.42%, even surpassing the in-\ndividual model’s score of 60.18%. These improvements are not limited to EMR-Merging. PAVE\nalso shows improvements with Ties-Merging and Task Arithmetic strategies. For instance, when\nintegrating on Ties-Merging, PAVE achieves an average gain of +2.89%, outperforming both the\nbaseline and the DARE-enhanced variant.\nSimilar to the experimental setting for RoBERTa, we evaluate our method on DeBERTa and present\nthe results in Table 2. PAVE cooperates effectively with EMR-Merging, achieving a substantial\naverage performance gain of +3.53%, narrowing the average gap to 2.02% compared to the aver-\naged performance (84.66%) of individual models. Notably, on CoLA, PAVE boosts performance\nto a new high of 61.32%, even surpassing the individual model’s score of 59.10%, also achieving\nan improvement of 14.27% over the EMR-Merging baseline. Beyond EMR-Merging, PAVE also\nimproves results in Task Arithmetic and Ties-Merging settings. In Task Arithmetic, it raises the\naverage performance by +1.82%, while in Ties-Merging, it achieves a considerable gain of +3.11%\nover the original baseline.\n4.3\nMODEL MERGING FOR GENERATIVE TASKS\nFor generative tasks, we conduct experiments merging the math model WizardMath-7B-V1.0 (Luo\net al., 2023) and the code model LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). As shown\n8\n\nPreprint\nTable 3: Results of merging fine-tuned LLaMA-2-7B models on Math and Code tasks. The number\nof examples used to generate the context matrix is 512.\nMethods\nMathematical Solving\nCode Generating\nGSM8K\nMATH\nHuman Eval\nMBPP\nIndividual Math\n54.9\n10.7\n-\n-\nIndividual Code\n-\n-\n39.6\n30.1\nAverage Merging\n32.2\n5.0\n29.2\n29.0\nAverage Merging w/ DARE\n31.9−0.3\n5.0+0.0\n29.8+0.6\n28.6−0.4\nAverage Merging w/ PAVE\n32.4+0.2\n5.0+0.0\n31.1+1.9\n29.0+0.0\nTask Arithmetic\n34.3\n4.1\n33.5\n27.6\nTask Arithmetic w/ DARE\n32.3−2.0\n4.4+0.3\n32.3−1.2\n28.6+1.0\nTask Arithmetic w/ PAVE\n34.5+0.2\n4.2+0.1\n35.4+1.9\n28.0+0.4\nTable 4: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark.\nThe number of examples used to generate context matrix is 4096.\nMethods\nCoLA\nSST-2\nMRPC\nSTSB\nQQP\nMNLI\nQNLI\nRTE\nAverage\nIndividual\n59.10\n95.06\n89.21\n91.31\n91.50\n88.52\n93.30\n69.31\n84.66\nEMR-Merging\n47.05\n94.72\n71.32\n87.57\n88.36\n87.38\n92.29\n64.26\n79.11\nEMR-Merging W/ PAVE plain\n59.88\n94.61\n82.60\n88.83\n90.40\n87.35\n92.44\n58.48\n81.82\nEMR-Merging W/ PAVE\n61.32\n94.50\n84.31\n89.54\n90.34\n87.50\n93.01\n60.65\n82.64\nTask Arithmetic\n3.03\n75.00\n68.38\n52.72\n63.44\n43.46\n51.16\n50.90\n51.01\nTask Arithmetic W/ PAVE plain\n2.56\n74.73\n68.38\n54.83\n64.03\n42.84\n60.19\n51.16\n52.34\nTask Arithmetic W/ PAVE\n2.18\n75.46\n68.38\n55.90\n64.00\n43.48\n61.25\n51.99\n52.83\nTies-Merging\n5.36\n66.28\n68.63\n19.09\n63.68\n40.44\n55.39\n48.74\n45.95\nTies-Merging W/ PAVE plain\n1.04\n70.07\n68.87\n31.33\n63.16\n43.89\n57.79\n50.18\n48.29\nTies-Merging W/ PAVE\n1.39\n68.35\n68.63\n36.76\n63.12\n44.01\n59.36\n50.90\n49.06\nin Table 3, PAVE demonstrates a marked improvement of +1.9% on Human Eval compared to Aver-\nage Merging and Task Arithmetic. Notably, PAVE achieves the highest Human Eval score (35.4%)\namong all merging strategies, indicating its effectiveness of our purified task vectors in code gener-\nation tasks. In the mathematical solving tasks, PAVE consistently maintains or modestly improves\nperformance across both GSM8K (+0.2%) and MATH (+0.1%) compared to the Task Arithmetic\nbaseline. It is worth noting that, in contrast to the randomness in DARE, PAVE is more stable and\npractical, as none of the merged models show any degradation on the individual tasks. This sta-\nbility, combined with consistent improvements across multiple tasks, highlights the robustness and\npractical advantage of PAVE.\n4.4\nABLATION STUDIES\nWe evaluate our rank allocation strategy Algorithm 1 in Table 4, using the same DeBERTa ex-\nperiment settings on models and merging methods in Section 4.2. The plain method uses a fixed\npreserved rank ratio across all layers and models. Compared to this baseline, our adaptive strategy\nconsistently improves performance across different merging settings. In EMR-Merging, the rank\nstrategy improves the average performance from 81.82% to 82.64% (+0.82%). In other merging\nmethods like Task Arithmetic and Ties-Merging, the gains are +0.49% (from 52.34% to 52.83%)\nand +0.77% (from 48.29% to 49.06%), respectively. These results suggest that uniform pruning\nrepresents a naive and suboptimal approach, while our rank allocation strategy enables better per-\nformance under the same pruning budget. Additionally, we include further experiments, presenting\nnumerical results when PAVE is integrated with alternative decomposition methods, as well as an\nanalysis of sample size and runtime performance. Please refer to Appendix B for the results.\n9\n\nPreprint\n5\nCONCLUSION\nIn this paper, we propose the purified task vector (PAVE) as a plug-in method for task vector-based\nmerging approaches. PAVE employs context-oriented decomposition to identify and prune redun-\ndancies of the task vectors in knowledge-aware subspace, thereby mitigating conflicts arising from\nredundancy. To further optimize the rank allocation across different fine-tuned models and improve\nthe merging performance, we introduce a spectral rank allocation strategy that optimizes the sum-\nmation of normalized activated pruning errors. In experiment, we demonstrate the effectiveness\nof PAVE with a wide range of merging methods, tasks and model architectures. Furthermore, we\nexplore the integration of alternative decomposition techniques within the PAVE framework and\nconfirm its compatibility with existing model merging approaches.\nREFERENCES\nJosh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Ale-\nman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. 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For evaluation, CoLA is\nassessed using the Matthews correlation coefficient, STS-B is measured by the average of Pearson\nand Spearman correlation coefficients, and the remaining tasks are evaluated based on accuracy. The\nRoBERTa checkpoints used for model merging are publicly available via the links in (Huang et al.,\n2024). The DeBERTa checkpoints are obtained by training DeBERTa-base individually on each\nGLUE task for 5 epochs using the AdamW (Loshchilov & Hutter, 2017) optimizer with a learning\nrate of 2 × 10−5, a warm-up phase of 100 steps, and a weight decay of 10−2.\nFollowing the setting in Huang et al. (2024), we report the numerical experiments of PAVE with\nvarious merging methods including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav\net al., 2023) and EMR-Merging (Huang et al., 2024). We also evaluate these methods with plug-in\nmethod DARE (Yu et al., 2024). Following the standard setting of each method, the hyperparameter\nλ in Task Arithmetic and Ties-Merging is selected from {0.1, 0.3, 0.5, 0.7 0.9} based on the best\nperformance. Specifically, for the RoBERTa experiments, we set λ = 0.3 for Task Arithmetic and\nλ = 0.9 for Ties-Merging; for BERT experiments, we set λ = 0.1 and λ = 0.5, respectively. The\ndropout ratio for DARE is fixed to 0.1 to avoid instability. The preserved rank ratio ρ of PAVE is\nchosen from { 7\n8, 15\n16, 31\n32, 63\n64}, and the stopping ratio is set to γ = ρ −1−ρ\n2 .\nTo attain optimal performance on GLUE benchmarks, PAVE progressively progressively assigns full\nrank to ∆WPAVE according to the performance of the plain merging method. This design reflects the\nobservation that certain tasks are more sensitive and cannot tolerate aggressive pruning. At each\nstep, we apply full rank to the the task whose performance is furthest from that of its individual\nmodel. This process continues until all task vectors are full rank. As a result, we obtain K merged\nmodels and select the one with the highest average performance. Since model merging is not com-\nputationally expensive and requires no training, the computational cost of such procedure remains\nacceptable.\nA.2\nMODEL MERGING ON GENERATION BENCHMARKS\nFollowing the setup of DARE presented in Yu et al. (2024), we evaluate PAVE on mathematical\nsolving and code generation tasks, as described in Section 4.3. For mathematical solving, we con-\nsider GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021), while for code generation,\nwe use HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021). We adopt the LLaMA-\n2-7B (Touvron et al., 2023) model as the base model. For mathematical solving, we utilize the\nfine-tuned model WizardMath-7B-V1.0 (Luo et al., 2023), while for code generation, we employ\nLLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). Additionally, code samples are drawn\nfrom the CodeFeedback dataset (Zheng et al., 2024) to support covariance matrix generation. The\nTask Arithmetic hyperparameter λ is selected from the set {0.5, 1}, and the dropout ratio for DARE\nis fixed at 0.2. The pruning rank for PAVE is chosen from {4, 6, 8, 16}.\nB\nADDITIONAL ANALYSIS AND RESULTS\nB.1\nTASK VECTOR WITH VARIOUS DECOMPOSITION METHODS\nIn this section, we present numerical experiments evaluating the performance of various decompo-\nsition methods applied to our decomposition-based task vector. The setting of models and merging\nmethods follow the DeBERTa settings in Section 4.2. The evaluated decomposition methods in-\nclude: The basic SVD method (Meng et al., 2024), directly approximates the fine-tuned weights\nvia a plain SVD decomposition. ASVD (Yuan et al., 2023), incorporates an input-dependent di-\nagonal scaling matrix derived from the average absolute values of the previous layer’s outputs.\nSVD-LLM (Wang et al., 2024b), applies Cholesky decomposition to the previous layer’s output.\n15\n\nPreprint\nTable 5: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark.\nThe number of examples used to generate context matrix is 4096.\nMethods\nCoLA\nSST-2\nMRPC\nSTSB\nQQP\nMNLI\nQNLI\nRTE\nAverage\nIndividual\n60.18\n94.04\n89.22\n90.63\n91.41\n87.20\n92.71\n79.06\n85.55\nEMR-Merging\n47.05\n94.72\n71.32\n87.57\n88.36\n87.38\n92.29\n64.26\n79.11\nEMR-Merging W/ SVD\n62.20\n93.58\n83.82\n89.40\n89.45\n76.72\n92.53\n58.48\n80.77\nEMR-Merging W/ ASVD\n62.33\n95.07\n84.56\n89.73\n89.98\n87.44\n92.86\n61.01\n82.87\nEMR-Merging W/ SVD-LLM\n61.82\n94.72\n84.07\n89.55\n90.10\n87.44\n92.90\n61.37\n82.74\nEMR-Merging W/ PAVE\n61.32\n94.50\n84.31\n89.54\n90.34\n87.50\n93.01\n60.65\n82.64\nTask Arithmetic\n3.03\n75.00\n68.38\n52.72\n63.44\n43.46\n51.16\n50.90\n51.01\nTask Arithmetic W/ SVD\n2.96\n70.30\n68.38\n37.90\n64.30\n42.88\n67.01\n53.07\n50.85\nTask Arithmetic W/ ASVD\n2.53\n76.03\n66.38\n55.60\n64.08\n43.23\n61.27\n51.62\n52.59\nTask Arithmetic W/ SVD-LLM\n2.74\n74.89\n68.38\n55.48\n64.02\n43.54\n61.45\n51.62\n52.76\nTask Arithmetic W/ PAVE\n2.18\n75.46\n68.38\n55.90\n64.00\n43.48\n61.25\n51.99\n52.83\nTies-Merging\n5.36\n66.28\n68.63\n19.09\n63.68\n40.44\n55.39\n48.74\n45.95\nTies-Merging W/ SVD\n-3.29\n53.90\n68.38\n6.15\n62.98\n35.34\n54.22\n52.71\n41.29\nTies-Merging W/ ASVD\n-1.74\n66.86\n68.63\n17.19\n62.84\n40.33\n53.10\n46.57\n44.22\nTies-Merging W/ SVD-LLM\n1.13\n67.89\n68.63\n23.33\n63.09\n43.02\n57.84\n48.74\n46.70\nTies-Merging W/ PAVE\n1.39\n68.35\n68.63\n36.76\n63.12\n44.01\n59.36\n50.90\n49.06\n64\n128\n256\n512\n1024\n2048\n4096\nSamples\n88.7\n88.8\n88.9\n89.0\n89.1\n89.2\nPerformance\nEMR-Merging W/ PAVE\nStd\nEMR-Merging\n64\n128\n256\n512\n1024\n2048\n4096\nSamples\n2.0\n4.0\n6.0\n8.0\n10.0\n12.0\n14.0\nTime (min)\nPAVE Runtime\nFigure 4: Performance and runtime analysis of PAVE across different sample sizes. Left: Mean per-\nformance (solid line) and standard deviation (shaded region) of EMR-Merging with PAVE, showing\nconsistent improvements as more samples are used. Right: PAVE runtime grows with number of\nsamples.\nCO-SVD (Yang et al., 2024b), as the default decomposition method embedded in PAVE, utilizes the\ncovariance matrix of the output of the previous layer as the activation matrix. Among the evaluated\nmethods, CO-SVD demonstrates the most consistent performance on Task Arithmetic and Ties-\nMerging, while also exhibiting performance comparable to EMR-Merging. These results highlight\nthe strong potential of our decomposition-based purified task vector framework, which is designed\nto integrate seamlessly with a variety of decomposition methods.\nB.2\nANALYSIS OF SAMPLE SIZE AND TIME EFFICIENCY\nWe next analyze the impact of sample size on the stability, efficiency, and reproducibility of PAVE.\nAs illustrated in Figure 4, increasing the sample size consistently reduces the variance and im-\nproves the overall performance of EMR-Merging with PAVE, highlighting its stability and robust-\nness. When the sample size is very small, due to the regularization, the method degenerates to plain\nSVD, which explains the lower performance observed in this regime. Importantly, we observe that\n16\n\nPreprint\nTable 6: Results of merging fine-tuned ViT models on 8 image classification benchmarks. The\nnumber of samples used to generate context matrix is 4096.\nMethods\nSUN397\nCars\nRESISC45\nEuroSAT\nSVHN\nGTSRB\nMNIST\nDTD\nAverage\nIndividual ViT-B/32\n75.3\n77.7\n96.1\n99.7\n97.5\n98.7\n99.7\n79.4\n90.5\nEMR-Merging\n75.2\n72.8\n93.5\n99.5\n96.9\n98.1\n99.6\n74.4\n88.7\nEMR-Merging W/PAVE\n76.3\n74.3\n93.8\n99.3\n96.7\n97.6\n99.6\n76.1\n89.2+0.5\nIndividual ViT-B/16\n79.0\n87.0\n96.4\n99.1\n97.7\n99.0\n99.7\n82.3\n92.5\nEMR-Merging\n78.5\n82.6\n95.4\n99.1\n97.6\n98.8\n99.6\n78.3\n91.2\nEMR-Merging W/PAVE\n80.2\n82.8\n96.1\n99.2\n97.6\n98.7\n99.6\n78.9\n91.7+0.5\nIndividual ViT-L/14\n82.3\n92.4\n97.4\n100\n98.1\n99.2\n99.7\n84.1\n94.2\nEMR-Merging\n83.1\n90.7\n96.7\n99.7\n97.9\n99.1\n99.6\n82.7\n93.7\nEMR-Merging W/PAVE\n83.6\n91.7\n97.2\n99.8\n99.7\n98.9\n99.7\n82.2\n93.9+0.2\neven a moderate sample size (256–512) is sufficient to yield substantial performance gains, offering\na practical trade-off between accuracy and computational cost. This finding suggests that PAVE\ncan be effectively deployed in practice without incurring excessive runtime overhead, while still\npreserving most of the performance benefits of using larger sample sizes. Throughout our numer-\nical experiments, we increased the number of samples up to 4096 ensure a stable and consistent\ndemonstration of our method’s performance. Employing larger sample sizes also helps ensure the\nreproducibility of our results. All runtime measurements were conducted on a workstation equipped\nwith a single NVIDIA A100 GPU and 8 CPUs, with 512 GB of RAM.\nB.3\nMODEL MERGING FOR VIT MODELS\nSettings. Following the setup of EMR-Merging presented in Huang et al. (2024), we employ ViT-\nB/32, ViT-B/16 and ViT-L/14 (Radford et al., 2021) as the base model to evaluate the performance\nof PAVE across 8 image classification tasks: SUN397 (Xiao et al., 2010), Cars Krause et al. (2013),\nRESISC45 (Cheng et al., 2017), EuroSAT (Helber et al., 2019), SVHN (Netzer et al., 2011), GT-\nSRB (Stallkamp et al., 2011), MNIST (LeCun, 1998), and DTD (Cimpoi et al., 2014). The preserved\nrank ratio ρ of PAVE is chosen from { 7\n8, 15\n16, 31\n32, 63\n64}, and the stopping ratio is set to γ = ρ −1−ρ\n2 .\nThe checkpoints are publicly accessible via the link provided in Ilharco et al. (2022).\nResults. As shown in Table 6, while the plain EMR-Merging method already demonstrates strong\nperformance, PAVE still consistently enhances the performance across all ViT variants. For ViT-\nB/32, PAVE improves the average accuracy by +0.5% (from 88.7% to 89.2). On the more capable\nViT-B/16 model, PAVE again contributes an improvement of +0.5%, increasing the average from\n91.2% to 91.7%.\n17\n"},"cleaned":{"kind":"string","value":"Preprint PURIFYING TASK VECTORS IN KNOWLEDGE-AWARE SUBSPACE FOR MODEL MERGING Bang An1 Yibo Yang1∗ Philip Torr2 Bernard Ghanem1 1King Abdullah (KAUST) 2 {bang.an, yibo.yang, ABSTRACT Model merging aims to integrate task-specific abilities from individually finetuned models into a single model without extra training. In recent model merging methods, task vector has become a fundamental building block, as it can encapsulate the residual information from finetuning. However, the merged model often suffers from notable performance degradation due to the conflicts caused by task-irrelevant redundancy in task vectors. Existing efforts in overcoming redundancy by randomly dropping elements in the parameter space involves randomness and lacks knowledge awareness. To address these challenges, in this study, we propose Purifying TAsk VEctors (PAVE) in knowledge-aware subspace. Concretely, we sample some training examples from each task, and feed them into their corresponding fine-tuned models to acquire the covariance matrices before linear layers. We then perform a context-oriented singular value decomposition, which accentuates the weight components most relevant to the target knowledge. As a result, we can split fine-tuned model weights into taskrelevant and redundant components in the knowledge-aware subspace, and purify the task vector by pruning the redundant components. To induce fair pruning efforts across models, we further introduce a spectral rank allocation strategy by optimizing a normalized activated pruning error. The task vector purification by our method as a plug-and-play scheme is applicable across various task vector-based merging methods to improve their performance. In experiments, we demonstrate the effectiveness of PAVE across a diverse set of merging methods, tasks, and model architectures. Remarkably, when integrated on the state-of-the-art merging method EMR-Merging with RoBERTa, our method yields a significant performance improvement of 4.1% (from 80.18% w/o PAVE to 84.28%) on the GLUE benchmark, approaching to the average performance of 8 individual models, 85.55%. Our code and models are released at: https: //github.com/ABgit111/purified_task_vector. 1 INTRODUCTION Transformer-based models, including large language models (LLMs) (Devlin et al., 2019; Liu et al., 2019; He et al., 2020; Radford et al., 2019; Touvron et al., 2023; Achiam et al., 2023) and vision models (Dosovitskiy et al., 2020; Radford et al., 2021) have found widespread applications across various domains. Supervised fine-tuning (Radford et al., 2018; Hu et al., 2022) is a common technique to enhance downstream task performance. Despite the effectiveness of multi-task learning (MTL) in acquiring versatile abilities across multiple domains (Caruana, 1997; Zhang & Yang, 2018; Vandenhende et al., 2021), its applicability will be limited when only individually fine-tuned models are accessible due to data privacy concerns or computational constraints. To address this challenge, model merging has been proposed, aiming to construct a single model by combining parameters from domain-specific models, without relying on any training or fine-tuning efforts. Early model merging methods focus on performing weighted (Matena & Raffel, 2022) or regularized (Jin et al., 2022) averaging in the parameter space of fine-tuned models, yet the conflicts inherent in ∗Corresponding author 1 16 Oct 2025 Preprint Knowledge-aware subspace ∆W= W!\" -W# Task vector Naive DARE PAVE (ours) Formulation Address redundancy Knowledge awareness ∆W W!\"CC%& Redundancy w.r.t. the target knowledge ❌ ✅(in parameter space) ✅(in subspace) ❌ ❌ ✅ Target knowledge as characterized by C ➕ SVD \"! SVD W!C! W#C# W W! W# W Σ! Σ# Σ % U![: r!] U#[: r#] U ] Σ![: r!] Σ#[: r#] Σ ] V! %[: r!] V# %[: r#] V ] C! C# C C \"! ⋮ ⋮ ⋮ ⋮ ⋮ Truncate Merging Method W' Task 2 Task K Task 1 sample sample sample Figure 2: An overall illustration of our proposed method, PAVE, which serves as a plug-and-play method for task-vector based merging methods. It begins by identifying task-aware components through a decomposition approach, and then eliminates noise and less effective components via rank-based pruning. A rank allocation strategy is designed to optimize the preserved rank ratios across individual models. Overall, rather than employing the plain task vector Wk -WB, PAVE performs model merging based on purified task vectors SVDrk(WkCk)C-1 k -WB, 1 ≤k ≤K. reflects the spectral distribution of the parameter matrix itself and pre-trained LLM weights handle a broad range of tasks. To this end, we leverage the context-oriented singular value decomposition (CO-SVD) (Yang et al., 2024b) to decompose fine-tuned weights into a knowledge-aware subspace. Concretely, for each task Tk, 1 ≤k ≤K, we sample some training examples from this task and feed them into the fine-tuned model Mk. After acquiring the input activation of each layer X (layer index is omitted for simplicity), we calculate the covariance matrix as C = XXT . Projecting the weight matrix of a linear layer onto the covariance matrix of its input activation can accentuate the taskspecific components related to the input. Therefore, we apply SVD on the product of each weight matrix and its corresponding covariance matrix, i.e., SVD(WFTC). The decomposed components are ordered by the singular values that reflect their contributions to the task as characterized by C. Such subspace enjoys a low-rank structure, which means that the the knowledge-aware abilities can be condensed into the first several leading singular values and singular vectors, as will be shown in Section 4.1. Hence, we filter out the task-irrelevant components by keeping only the first r components, and then reconstruct the weight matrix through right multiplication with C-1, i.e., the inverse of covariance matrix C. Consequently, the purified weight matrix can be formulated as: W † FT = SVDr(WFTC)C-1 = r X i=1 σiuivT i C-1, (5) where σi denotes the i-th singular value, ui and vi are the left and right singular vectors, respectively. We use the purified fine-tuned weight matrix W † FT to construct our task vector as: ∆WPAVE = W † FT -WB = SVDr(WFTC)C-1 -WB, (6) where WB is the base model weights before fine-tuning, and ∆WPAVE represents our purified task vector, which can be used for any task vector-based merging methods, such as Ties-Merging (Yadav et al., 2023), Task Arithmetic (Ilharco et al., 2022), and EMR-Merging (Huang et al., 2024). The effectiveness of CO-SVD in identifying task-specific directions of weight matrices has been verified by Yang et al. (2024b) for model adaptation. In addition, C = XXT is a symmetric positive semi-definite matrix, which enables efficient inverse operation with numerical stability. When C is not invertible, we repeatedly add positive elements on its diagonal until it is invertible. In experiments, we show that our method can be integrated on various model merging methods to improve the performance of the merged models. 3.3 SPECTRAL RANK ALLOCATION 5 Preprint Algorithm 1 Spectral rank allocation strategy Require: Individual models {M{W l 1}L l=1, M{W l 2}L l=1, · · · , M{W l K}L l=1}, data sampling from target tasks {X1, X2, · · · , XK}, a preserved rank ratio ρ, a stopping ratio γ. 1: Feed Xi into M{W l i }L l=1. Calculate covariance matrices for all layers and for all models, {Cl i}K,L i,l , and the rank of each layer {Rl}L l=1. 2: for l = 1 to L do 3: Apply SVD to W l i Cl i and compute {σl i,j}K,Rl i,j , sort them in descending order. 4: Normalize {σl i,j}K,Rl i,j by the largest singular value, {sl i,j}K,Rl i,j = σl i,j σl i,max K,Rl i,j 5: Initialize {rl i}K,L i,l = {Rl}K,L i,l 6: Initialize index set I = {1, 2, · · · , K} 7: while Pk i=1 rl i KRl > ρ do 8: Pick up the smallest sl i,rl i across i ∈I, suppose it is sl t,rl i 9: rl t = rl t -1 10: if rl t ≤γRl then 11: remove t from I 12: end if 13: end while 14: end for 15: Output: the rank ratios for each individual model, {ρi}K i = n PL l=1 rl i PL l=1 Rl oK i . We have introduced the construction of our purified task vector in Section 3.2. However, when merging multiple fine-tuned models, assigning the rank r to each model is highly non-trivial. To optimize rank allocation under a specified preserved rank ratio ρ, we propose a layer-wise optimization framework that determines the preserved rank of linear layer l by minimizing a summation of normalized activated pruning error from different models. This strategy enables an adaptive preservation of task-specific knowledge during the merging process. We begin by introduce the activated pruning error with rank rl i as: ∥SVDrl i(W l i Cl i) -W l i Cl i∥2 F , (7) where i refers to the i-th model. In practice, we further normalize it with σmax(W l i Cl i) to ensure comparability across different models with various spectral magnitudes. This normalization bounds the scaled spectrum within a range from the inverse of its condition number to 1, thus enabling a fair assignment of rank across models. The normalized activated pruning error is defined as: ∥SVDrl i(W l i Cl i) -W l i Cl i∥2 F (σmax(W l i Cl i))2 = rl i X j=1 (σj(W l i Cl i))2 (σmax(W l i Cl i))2 = rl i X j=1 σl i,j σl i,max !2 . (8) Building on this formulation, we define the overall objective for the l-th layer as: min PK i=1 rl i KRl <ρ K X i=1 ∥SVDrl i(W l i Cl i) -W l i Cl i∥2 F (σmax(W l i Cl i))2 , (9) where Rl refers to full rank. Minimizing the objective Eq. (9) can be effectively approached using a greedy algorithm, and the whole procedure has been summarized in Algorithm 1. In this procedure, for each layer l, we iteratively remove the component associated with the smallest normalized singular value, defined as sl t,rl i = σl i,j σl i,max , thereby achieving optimality of Eq. (9) through the greedy process. To prevent excessive rank pruning that causes performance degradation, we introduce a threshold γ which serves as a stopping ratio. This rank strategy enables spectrum-based rank allocation for model merging while respecting the global rank constraint. 6 Preprint 96 192 384 512 Pruned ranks 0.5 0.6 0.7 0.8 0.9 Performance SST-2 SVD CO-SVD random ASVD with SST-2 CO-SVD with SST-2 CO-SVD with COLA 96 192 384 512 Pruned ranks 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Performance MRPC SVD CO-SVD random ASVD with MRPC CO-SVD with MRPC CO-SVD with COLA 96 192 384 512 Pruned ranks 0.5 0.6 0.7 0.8 0.9 Performance QQP SVD CO-SVD random ASVD with QQP CO-SVD with QQP CO-SVD with COLA 96 192 384 512 Pruned ranks 0.3 0.4 0.5 0.6 0.7 0.8 Performance MNLI SVD CO-SVD random ASVD with MNLI CO-SVD with MNLI CO-SVD with COLA 96 192 384 512 Pruned ranks 0.5 0.6 0.7 0.8 0.9 Performance QNLI SVD CO-SVD random ASVD with QNLI CO-SVD with QNLI CO-SVD with COLA 96 192 384 512 Pruned ranks 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Performance RTE SVD CO-SVD random ASVD with RTE CO-SVD with RTE CO-SVD with COLA Figure 3: Model rank pruning with different decomposition methods. The results show that the context-oriented decomposition (CO-SVD) is more effective in aligning the subspace with the target knowledge. The pruned ranks, 96, 192, 384 and 512, correspond to removing 1/8, 1/4, 1/2 and 2/3 of the full rank, respectively. 4 EXPERIMENTS In this section, we provide the results on the GLUE benchmarks using RoBERTa-base (Liu et al., 2019) and DeBERTa-base (He et al., 2020), as well as the generation benchmarks with LLaMA2-7Bhf (Touvron et al., 2023) on mathematical solving and code generation tasks. We evaluate PAVE with various merging methods, including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav et al., 2023), and EMR-Merging (Huang et al., 2024). For further detailed experimental settings and hyperparameters, please refer to Appendix A. 4.1 EFFECT OF DECOMPOSITION IN KNOWLEDGE-AWARE SUBSPACES We demonstrate the effectiveness of decomposition in knowledge-aware subspaces through a series of numerical experiments. As concluded in Figure 3, we prune individual RoBERTa models fine-tuned on six GLUE tasks using various decomposition methods. The decomposition methods mentioned in Figure 3 are summarized as follows: SVD (Meng et al., 2024) applies singular value decomposition directly to the weights. ASVD (Yuan et al., 2023) introduces a diagonal scaling matrix, where each diagonal entry corresponds to the average absolute value of the associated feature. CO-SVD (Yang et al., 2024b) leverages covariance matrices. In the CO-SVD random variant, the covariance matrix is replaced by a sampled random square matrix from [-1,1], serving as a taskagnostic baseline. CO-SVD with CoLA derives the covariance matrix using samples from CoLA, highlighting the impact of selecting samples from an appropriate task. From Figure 3, it is evident that, for a given rank, decomposition methods that incorporate task information (CO-SVD, ASVD) outperform those that do not (SVD, CO-SVD random). In particular, CO-SVD demonstrates superior performance compared to ASVD, as CO-SVD preserves the context information more effectively through task-specific activations. This pattern holds consistently across all datasets evaluated, reinforcing the importance of using the correct task data to guide decomposition. Notably, performance advantages of CO-SVD become even more prominent at pruning larger ranks, where other methods exhibit sharp degradation, indicating CO-SVD's effectiveness in condensing knowledge awareness. It is noteworthy that the input must originate from the corresponding task to construct knowledge-aware subspace. When using CoLA samples to generate the covariance matrix, performance drops significantly across all tasks. These results suggest knowledge-awareness is crucial for successful rank preservation, and emphasize the importance of introducing knowledge-aware subspaces for purifying task vectors. 7 Preprint Table 1: Results of merging fine-tuned RoBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 60.18 94.04 89.22 90.63 91.41 87.20 92.71 79.06 85.55 EMR-Merging 39.96 93.35 86.27 82.77 89.72 85.45 89.57 74.37 80.18 EMR-Merging W/ DARE 40.58 93.23 86.52 82.86 89.93 85.98 89.68 75.09 80.48+0.30 EMR-Merging W/ PAVE 60.42 93.69 87.75 89.73 89.13 85.27 92.09 76.17 84.28+4.10 Task Arithmetic 18.78 85.89 79.90 74.03 83.78 59.08 69.67 62.09 66.65 Task Arithmetic W/ DARE 18.26 85.67 80.39 73.79 83.77 59.20 69.58 61.01 66.45-0.20 Task Arithmetic W/ PAVE 18.96 85.67 80.64 74.14 83.77 58.97 69.91 61.73 66.72+0.07 Ties-Merging 20.48 84.40 81.13 58.19 85.70 64.65 74.81 42.96 64.04 Ties-Merging W/ DARE 17.56 83.83 80.39 63.60 85.79 64.82 0.7324 44.77 64.25+0.21 Ties-Merging W/ PAVE 25.45 82.45 79.41 64.03 85.43 69.93 77.19 51.62 66.93+2.89 Table 2: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 59.10 95.06 89.21 91.31 91.50 88.52 93.30 69.31 84.66 EMR-Merging 47.05 94.72 71.32 87.57 88.36 87.38 92.29 64.26 79.11 EMR-Merging W/ DARE 47.76 94.61 72.06 87.30 88.77 87.67 92.28 66.43 79.61+0.50 EMR-Merging W/ PAVE 61.32 94.50 84.31 89.54 90.34 87.50 93.01 60.65 82.64+3.53 Task Arithmetic 3.03 75.00 68.38 52.72 63.44 43.46 51.16 50.90 51.01 Task Arithmetic W/ DARE 2.84 68.81 68.38 54.24 64.38 42.23 55.15 49.46 50.68-0.33 Task Arithmetic W/ PAVE 2.18 75.46 68.38 55.90 64.00 43.48 61.25 51.99 52.83+1.82 Ties-Merging 5.36 66.28 68.63 19.09 63.68 40.44 55.39 48.74 45.95 Ties-Merging W/ DARE 6.03 56.42 68.87 0.61 66.28 42.11 54.00 51.26 43.19-2.76 Ties-Merging W/ PAVE 1.39 68.35 68.63 36.76 63.12 44.01 59.36 50.90 49.06+3.11 4.2 MODEL MERGING FOR GLUE TASKS To evaluate the effectivness of PAVE, we first present results on GLUE tasks using RoBERTa as the backbone model. The numerical results are summarized in Table 1. We observe that PAVE yields a notable average performance gain of +4.10% when applied to EMR-Merging, reducing the gap to just 1.27% from the individual models' average performance. For specific tasks such as CoLA, PAVE significantly improves the score from 39.96% to 60.42%, even surpassing the individual model's score of 60.18%. These improvements are not limited to EMR-Merging. PAVE also shows improvements with Ties-Merging and Task Arithmetic strategies. For instance, when integrating on Ties-Merging, PAVE achieves an average gain of +2.89%, outperforming both the baseline and the DARE-enhanced variant. Similar to the experimental setting for RoBERTa, we evaluate our method on DeBERTa and present the results in Table 2. PAVE cooperates effectively with EMR-Merging, achieving a substantial average performance gain of +3.53%, narrowing the average gap to 2.02% compared to the averaged performance (84.66%) of individual models. Notably, on CoLA, PAVE boosts performance to a new high of 61.32%, even surpassing the individual model's score of 59.10%, also achieving an improvement of 14.27% over the EMR-Merging baseline. Beyond EMR-Merging, PAVE also improves results in Task Arithmetic and Ties-Merging settings. In Task Arithmetic, it raises the average performance by +1.82%, while in Ties-Merging, it achieves a considerable gain of +3.11% over the original baseline. 4.3 MODEL MERGING FOR GENERATIVE TASKS For generative tasks, we conduct experiments merging the math model WizardMath-7B-V1.0 (Luo et al., 2023) and the code model LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). As shown 8 Preprint Table 3: Results of merging fine-tuned LLaMA-2-7B models on Math and Code tasks. The number of examples used to generate the context matrix is 512. Methods Mathematical Solving Code Generating GSM8K MATH Human Eval MBPP Individual Math 54.9 10.7 - - Individual Code - - 39.6 30.1 Average Merging 32.2 5.0 29.2 29.0 Average Merging w/ DARE 31.9-0.3 5.0+0.0 29.8+0.6 28.6-0.4 Average Merging w/ PAVE 32.4+0.2 5.0+0.0 31.1+1.9 29.0+0.0 Task Arithmetic 34.3 4.1 33.5 27.6 Task Arithmetic w/ DARE 32.3-2.0 4.4+0.3 32.3-1.2 28.6+1.0 Task Arithmetic w/ PAVE 34.5+0.2 4.2+0.1 35.4+1.9 28.0+0.4 Table 4: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 59.10 95.06 89.21 91.31 91.50 88.52 93.30 69.31 84.66 EMR-Merging 47.05 94.72 71.32 87.57 88.36 87.38 92.29 64.26 79.11 EMR-Merging W/ PAVE plain 59.88 94.61 82.60 88.83 90.40 87.35 92.44 58.48 81.82 EMR-Merging W/ PAVE 61.32 94.50 84.31 89.54 90.34 87.50 93.01 60.65 82.64 Task Arithmetic 3.03 75.00 68.38 52.72 63.44 43.46 51.16 50.90 51.01 Task Arithmetic W/ PAVE plain 2.56 74.73 68.38 54.83 64.03 42.84 60.19 51.16 52.34 Task Arithmetic W/ PAVE 2.18 75.46 68.38 55.90 64.00 43.48 61.25 51.99 52.83 Ties-Merging 5.36 66.28 68.63 19.09 63.68 40.44 55.39 48.74 45.95 Ties-Merging W/ PAVE plain 1.04 70.07 68.87 31.33 63.16 43.89 57.79 50.18 48.29 Ties-Merging W/ PAVE 1.39 68.35 68.63 36.76 63.12 44.01 59.36 50.90 49.06 in Table 3, PAVE demonstrates a marked improvement of +1.9% on Human Eval compared to Average Merging and Task Arithmetic. Notably, PAVE achieves the highest Human Eval score (35.4%) among all merging strategies, indicating its effectiveness of our purified task vectors in code generation tasks. In the mathematical solving tasks, PAVE consistently maintains or modestly improves performance across both GSM8K (+0.2%) and MATH (+0.1%) compared to the Task Arithmetic baseline. It is worth noting that, in contrast to the randomness in DARE, PAVE is more stable and practical, as none of the merged models show any degradation on the individual tasks. This stability, combined with consistent improvements across multiple tasks, highlights the robustness and practical advantage of PAVE. 4.4 ABLATION STUDIES We evaluate our rank allocation strategy Algorithm 1 in Table 4, using the same DeBERTa experiment settings on models and merging methods in Section 4.2. The plain method uses a fixed preserved rank ratio across all layers and models. Compared to this baseline, our adaptive strategy consistently improves performance across different merging settings. In EMR-Merging, the rank strategy improves the average performance from 81.82% to 82.64% (+0.82%). In other merging methods like Task Arithmetic and Ties-Merging, the gains are +0.49% (from 52.34% to 52.83%) and +0.77% (from 48.29% to 49.06%), respectively. These results suggest that uniform pruning represents a naive and suboptimal approach, while our rank allocation strategy enables better performance under the same pruning budget. Additionally, we include further experiments, presenting numerical results when PAVE is integrated with alternative decomposition methods, as well as an analysis of sample size and runtime performance. Please refer to Appendix B for the results. 9 Preprint 5 CONCLUSION In this paper, we propose the purified task vector (PAVE) as a plug-in method for task vector-based merging approaches. 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The performance of each method is evaluated by eight tasks in the GLUE benchmark tasks, including CoLA (Warstadt et al., 2019), SST-2 (Socher et al., 2013), MRPC (Dolan & Brockett, 2005), STS-B (Cer et al., 2017), QQP (Iyer et al., 2017), MNLI (Williams et al., 2017), QNLI (Rajpurkar et al., 2016), RTE (Giampiccolo et al., 2007). For evaluation, CoLA is assessed using the Matthews correlation coefficient, STS-B is measured by the average of Pearson and Spearman correlation coefficients, and the remaining tasks are evaluated based on accuracy. The RoBERTa checkpoints used for model merging are publicly available via the links in (Huang et al., 2024). The DeBERTa checkpoints are obtained by training DeBERTa-base individually on each GLUE task for 5 epochs using the AdamW (Loshchilov & Hutter, 2017) optimizer with a learning rate of 2 × 10-5, a warm-up phase of 100 steps, and a weight decay of 10-2. Following the setting in Huang et al. (2024), we report the numerical experiments of PAVE with various merging methods including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav et al., 2023) and EMR-Merging (Huang et al., 2024). We also evaluate these methods with plug-in method DARE (Yu et al., 2024). Following the standard setting of each method, the hyperparameter λ in Task Arithmetic and Ties-Merging is selected from {0.1, 0.3, 0.5, 0.7 0.9} based on the best performance. Specifically, for the RoBERTa experiments, we set λ = 0.3 for Task Arithmetic and λ = 0.9 for Ties-Merging; for BERT experiments, we set λ = 0.1 and λ = 0.5, respectively. The dropout ratio for DARE is fixed to 0.1 to avoid instability. The preserved rank ratio ρ of PAVE is chosen from { 7 8, 15 16, 31 32, 63 64}, and the stopping ratio is set to γ = ρ -1-ρ 2 . To attain optimal performance on GLUE benchmarks, PAVE progressively progressively assigns full rank to ∆WPAVE according to the performance of the plain merging method. This design reflects the observation that certain tasks are more sensitive and cannot tolerate aggressive pruning. At each step, we apply full rank to the the task whose performance is furthest from that of its individual model. This process continues until all task vectors are full rank. As a result, we obtain K merged models and select the one with the highest average performance. Since model merging is not computationally expensive and requires no training, the computational cost of such procedure remains acceptable. A.2 MODEL MERGING ON GENERATION BENCHMARKS Following the setup of DARE presented in Yu et al. (2024), we evaluate PAVE on mathematical solving and code generation tasks, as described in Section 4.3. For mathematical solving, we consider GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021), while for code generation, we use HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021). We adopt the LLaMA2-7B (Touvron et al., 2023) model as the base model. For mathematical solving, we utilize the fine-tuned model WizardMath-7B-V1.0 (Luo et al., 2023), while for code generation, we employ LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). Additionally, code samples are drawn from the CodeFeedback dataset (Zheng et al., 2024) to support covariance matrix generation. The Task Arithmetic hyperparameter λ is selected from the set {0.5, 1}, and the dropout ratio for DARE is fixed at 0.2. The pruning rank for PAVE is chosen from {4, 6, 8, 16}. B ADDITIONAL ANALYSIS AND RESULTS B.1 TASK VECTOR WITH VARIOUS DECOMPOSITION METHODS In this section, we present numerical experiments evaluating the performance of various decomposition methods applied to our decomposition-based task vector. The setting of models and merging methods follow the DeBERTa settings in Section 4.2. The evaluated decomposition methods include: The basic SVD method (Meng et al., 2024), directly approximates the fine-tuned weights via a plain SVD decomposition. ASVD (Yuan et al., 2023), incorporates an input-dependent diagonal scaling matrix derived from the average absolute values of the previous layer's outputs. SVD-LLM (Wang et al., 2024b), applies Cholesky decomposition to the previous layer's output. 15 Preprint Table 5: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 60.18 94.04 89.22 90.63 91.41 87.20 92.71 79.06 85.55 EMR-Merging 47.05 94.72 71.32 87.57 88.36 87.38 92.29 64.26 79.11 EMR-Merging W/ SVD 62.20 93.58 83.82 89.40 89.45 76.72 92.53 58.48 80.77 EMR-Merging W/ ASVD 62.33 95.07 84.56 89.73 89.98 87.44 92.86 61.01 82.87 EMR-Merging W/ SVD-LLM 61.82 94.72 84.07 89.55 90.10 87.44 92.90 61.37 82.74 EMR-Merging W/ PAVE 61.32 94.50 84.31 89.54 90.34 87.50 93.01 60.65 82.64 Task Arithmetic 3.03 75.00 68.38 52.72 63.44 43.46 51.16 50.90 51.01 Task Arithmetic W/ SVD 2.96 70.30 68.38 37.90 64.30 42.88 67.01 53.07 50.85 Task Arithmetic W/ ASVD 2.53 76.03 66.38 55.60 64.08 43.23 61.27 51.62 52.59 Task Arithmetic W/ SVD-LLM 2.74 74.89 68.38 55.48 64.02 43.54 61.45 51.62 52.76 Task Arithmetic W/ PAVE 2.18 75.46 68.38 55.90 64.00 43.48 61.25 51.99 52.83 Ties-Merging 5.36 66.28 68.63 19.09 63.68 40.44 55.39 48.74 45.95 Ties-Merging W/ SVD -3.29 53.90 68.38 6.15 62.98 35.34 54.22 52.71 41.29 Ties-Merging W/ ASVD -1.74 66.86 68.63 17.19 62.84 40.33 53.10 46.57 44.22 Ties-Merging W/ SVD-LLM 1.13 67.89 68.63 23.33 63.09 43.02 57.84 48.74 46.70 Ties-Merging W/ PAVE 1.39 68.35 68.63 36.76 63.12 44.01 59.36 50.90 49.06 64 128 256 512 1024 2048 4096 Samples 88.7 88.8 88.9 89.0 89.1 89.2 Performance EMR-Merging W/ PAVE Std EMR-Merging 64 128 256 512 1024 2048 4096 Samples 2.0 4.0 6.0 8.0 10.0 12.0 14.0 Time (min) PAVE Runtime Figure 4: Performance and runtime analysis of PAVE across different sample sizes. Left: Mean performance (solid line) and standard deviation (shaded region) of EMR-Merging with PAVE, showing consistent improvements as more samples are used. Right: PAVE runtime grows with number of samples. CO-SVD (Yang et al., 2024b), as the default decomposition method embedded in PAVE, utilizes the covariance matrix of the output of the previous layer as the activation matrix. Among the evaluated methods, CO-SVD demonstrates the most consistent performance on Task Arithmetic and TiesMerging, while also exhibiting performance comparable to EMR-Merging. These results highlight the strong potential of our decomposition-based purified task vector framework, which is designed to integrate seamlessly with a variety of decomposition methods. B.2 ANALYSIS OF SAMPLE SIZE AND TIME EFFICIENCY We next analyze the impact of sample size on the stability, efficiency, and reproducibility of PAVE. As illustrated in Figure 4, increasing the sample size consistently reduces the variance and improves the overall performance of EMR-Merging with PAVE, highlighting its stability and robustness. When the sample size is very small, due to the regularization, the method degenerates to plain SVD, which explains the lower performance observed in this regime. Importantly, we observe that 16 Preprint Table 6: Results of merging fine-tuned ViT models on 8 image classification benchmarks. The number of samples used to generate context matrix is 4096. Methods SUN397 Cars RESISC45 EuroSAT SVHN GTSRB MNIST DTD Average Individual ViT-B/32 75.3 77.7 96.1 99.7 97.5 98.7 99.7 79.4 90.5 EMR-Merging 75.2 72.8 93.5 99.5 96.9 98.1 99.6 74.4 88.7 EMR-Merging W/PAVE 76.3 74.3 93.8 99.3 96.7 97.6 99.6 76.1 89.2+0.5 Individual ViT-B/16 79.0 87.0 96.4 99.1 97.7 99.0 99.7 82.3 92.5 EMR-Merging 78.5 82.6 95.4 99.1 97.6 98.8 99.6 78.3 91.2 EMR-Merging W/PAVE 80.2 82.8 96.1 99.2 97.6 98.7 99.6 78.9 91.7+0.5 Individual ViT-L/14 82.3 92.4 97.4 100 98.1 99.2 99.7 84.1 94.2 EMR-Merging 83.1 90.7 96.7 99.7 97.9 99.1 99.6 82.7 93.7 EMR-Merging W/PAVE 83.6 91.7 97.2 99.8 99.7 98.9 99.7 82.2 93.9+0.2 even a moderate sample size (256-512) is sufficient to yield substantial performance gains, offering a practical trade-off between accuracy and computational cost. This finding suggests that PAVE can be effectively deployed in practice without incurring excessive runtime overhead, while still preserving most of the performance benefits of using larger sample sizes. Throughout our numerical experiments, we increased the number of samples up to 4096 ensure a stable and consistent demonstration of our method's performance. Employing larger sample sizes also helps ensure the reproducibility of our results. All runtime measurements were conducted on a workstation equipped with a single NVIDIA A100 GPU and 8 CPUs, with 512 GB of RAM. B.3 MODEL MERGING FOR VIT MODELS Settings. Following the setup of EMR-Merging presented in Huang et al. (2024), we employ ViTB/32, ViT-B/16 and ViT-L/14 (Radford et al., 2021) as the base model to evaluate the performance of PAVE across 8 image classification tasks: SUN397 (Xiao et al., 2010), Cars Krause et al. (2013), RESISC45 (Cheng et al., 2017), EuroSAT (Helber et al., 2019), SVHN (Netzer et al., 2011), GTSRB (Stallkamp et al., 2011), MNIST (LeCun, 1998), and DTD (Cimpoi et al., 2014). The preserved rank ratio ρ of PAVE is chosen from { 7 8, 15 16, 31 32, 63 64}, and the stopping ratio is set to γ = ρ -1-ρ 2 . The checkpoints are publicly accessible via the link provided in Ilharco et al. (2022). Results. As shown in Table 6, while the plain EMR-Merging method already demonstrates strong performance, PAVE still consistently enhances the performance across all ViT variants. For ViTB/32, PAVE improves the average accuracy by +0.5% (from 88.7% to 89.2). On the more capable ViT-B/16 model, PAVE again contributes an improvement of +0.5%, increasing the average from 91.2% to 91.7%. 17"}}},{"rowIdx":284,"cells":{"id":{"kind":"string","value":"2510.14698"},"raw":{"kind":"string","value":"FedPPA: Progressive Parameter Alignment for\nPersonalized Federated Learning\nMaulidi Adi Prasetia∗, Muhamad Risqi U. Saputra†, and Guntur Dharma Putra∗‡\n∗Universitas Gadjah Mada, Indonesia †Monash University, Indonesia\nmaulidiadiprasetia2000@mail.ugm.ac.id, risqi.saputra@monash.edu, gdputra@ugm.ac.id\nAbstract—Federated Learning (FL) is designed as a decen-\ntralized, privacy-preserving machine learning paradigm that\nenables multiple clients to collaboratively train a model without\nsharing their data. In real-world scenarios, however, clients\noften have heterogeneous computational resources and hold non-\nindependent and identically distributed data (non-IID), which\nposes significant challenges during training. Personalized Fed-\nerated Learning (PFL) has emerged to address these issues by\ncustomizing models for each client based on their unique data dis-\ntribution. Despite its potential, existing PFL approaches typically\noverlook the coexistence of model and data heterogeneity arising\nfrom clients with diverse computational capabilities. To overcome\nthis limitation, we propose a novel method, called Progressive\nParameter Alignment (FedPPA), which progressively aligns the\nweights of common layers across clients with the global model’s\nweights. Our approach not only mitigates inconsistencies between\nglobal and local models during client updates, but also pre-\nserves client’s local knowledge, thereby enhancing personalization\nrobustness in non-IID settings. To further enhance the global\nmodel performance while retaining strong personalization, we\nalso integrate entropy-based weighted averaging into the FedPPA\nframework. Experiments on three image classification datasets,\nincluding MNIST, FMNIST, and CIFAR-10, demonstrate that\nFedPPA consistently outperforms existing FL algorithms, achiev-\ning superior performance in personalized adaptation.\nIndex Terms—Distributed training, personalized Federated\nlearning, inconsistent knowledge, parameter alignment, and\nweighted entropy-based average.\nI. INTRODUCTION\nFederated Learning (FL) is a distributed machine learning\nframework in which multiple clients collaboratively train a\nshared model without directly sharing their data, thereby\nenhancing overall model performance while preserving data\nprivacy [1]. FL works through iterative training rounds be-\ntween participating clients and a central aggregation server. In\neach round, clients train models locally using their private data\nand transmit the resulting model to the server. Subsequently,\nthe server aggregates these models to generate a global model,\nwhich is then redistributed to the clients for further local train-\ning in the next round [2], [3]. However, in real-world scenarios,\nparticipating clients often have incomplete or sparse data, i.e.,\nnon-independent and identically distributed data (non-IID),\nwhich would degrade the overall model performance.\nPersonalized Federated Learning (PFL) addresses these data\nheterogeneity issues in the clients by grouping the models into\na shared base layer and a personalized head layer [4]. The base\n‡Corresponding author.\nlayer, shared by the server and all clients, encodes the common\ndeep learning layer of the entire clients, while the head layer\nis locally customized and is fine-tuned to capture the unique\ncharacteristics of the client’s data. These personalization layers\nenable each client’s model to more accurately align with\nlocal data while still gaining advantages from the collective\nknowledge shared across the networks. While PFL yields\nstrong performance, these schemes typically demand all clients\nto have identical architecture. However, in practice, clients\noften possess heterogeneous computational resources, making\nit impractical to enforce a uniform model architecture across\nall clients, leading to each client having distinct architectures.\nConsequently, it becomes challenging to implement this per-\nsonalization strategy that demands consistent model structures.\nRecent studies [5], [6] proposed PFL methods to address the\nissue of heterogeneous model architectures between clients.\nFlexiFed identifies common components within clients’ lo-\ncal models and facilitates joint training on these layers to\neffectively fuse knowledge while maintaining architectural\nflexibility [5]. In addition, FedADP introduces a comprehen-\nsive model strategy to manage a variety of heterogeneous\nnetwork architectures and ensures that devices with limited\ncomputational capacity can still significantly contribute to the\nglobal model’s training [6]. However, these approaches often\noverlook the potential parameter misalignment between the\nserver (global) model and the individual clients’ model during\nlocal updates, potentially leading to inconsistencies between\nexisting client’s parameters and the updated parameters from\nthe global model. To address this knowledge inconsistency,\nFedAS proposes a solution to align the global model’s param-\neter with those of each client during local training, mitigating\ninconsistencies and improving overall model performance [7].\nHowever, this solution heavily relies on the assumption that\nevery client uses the same model architecture, which is\nquite unlikely given the significant variations in computational\npower commonly found in real-world scenarios.\nIn this paper, we propose FedPPA: a PFL approach to\nsimultaneously address heterogeneity in client models and\nnon-IID nature of client data, while also resolving the knowl-\nedge inconsistency between the server and the client models\nduring local updates. FedPPA progressively aligns the weights\nof common layers across clients with the global model’s\nweights, mitigating inconsistencies between global and local\nmodels during client updates. While our main target is to\narXiv:2510.14698v1 [cs.LG] 16 Oct 2025\n\nimprove personalization effectiveness and address the knowl-\nedge discrepancy problem, our approach also preserves client’s\nlocal knowledge, thereby enhancing personalization robustness\nin non-IID settings. In addition, we propose FedPPA+, in\nwhich we incorporate an entropy-based weighted averaging to\nFedPPA, retaining strong personalization with a better trade-\noff to global model performance. The entropy values are\ncalculated from each client’s variability in label distribution,\nwhere more labels exist in client’s dataset equal to a higher\ncontribution weight.\nIn summary, we propose the following contributions:\n• We develop a PFL approach, termed FedPPA, which\nprogressively aligns client’s common layers with the\nglobal model parameters during local updates, which is\nsuitable for FL with heterogeneous model architectures\nand non-IID data distribution.\n• We incorporate an entropy-based weighted averaging in\nthe global aggregation mechanism to better adapt to non-\nIID data distribution.\n• We perform a comprehensive experiment and analysis on\npublicly available data sets for image classification (i.e.,\nMNIST, F-MNIST, and CIFAR-10) using heterogeneous\nmodel architectures (i.e., VGG-11, VGG-13, VGG-16,\nand VGG-19).\nThe remainder of this paper is structured as follows. We\npresent the related work in Section II and discuss our proposed\nsolution in Section III. The implementation of our solution\nwith the corresponding evaluation is presented in Section IV,\nwhile the conclusion of the paper is presented in Section V.\nII. RELATED WORK\nFL was introduced as a modern distributed AI training\nparadigm designed to preserve user privacy by keeping data\nlocalized on user devices [8]. Instead of sharing raw data\nwith a central server, FL enables collaborative model training\nacross multiple clients, ensuring high levels of data privacy and\nsecurity. While FL has successfully introduced collaborative\ntraining across clients, achieving high model performance\nusing this approach typically relies on the assumption that\nthe datasets used for training are independent and identically\ndistributed (IID). In real-world applications, IID assumption\nis rarely met, resulting in non-IID data distributions across\nclients, which can significantly degrade the performance of\nFL model. Previous studies have shown that non-IID data\ndistribution can reduce FL model performance for up to 29%\n[9].\nIn order to address the performance degradation of standard\nFL model due to the non-IID data distribution among clients,\nPFL was introduced by Arivazhagan et al. [4] by incorporating\npersonalization layers. These personalization layers allow each\nclient’s model to better fit local data while still benefiting from\nshared knowledge across the networks. Since then, various\nPFL methods have been developed to address the diverse\nchallenges in distributed and personalized model training. For\ninstance, Hierarchical PFL (HPFL) [10]–[14] was developed\nto enable hierarchical weight aggregation among groups of\nclients through edge servers, which act as an intermediate\naggregation points. In this approach, clients with similar\nresources or data characteristics are clustered together, and\nthey only allow to communicate with the server via the edge\nservers, avoiding direct communication of each client with the\ncloud server. This approach not only reduces communication\ncosts and latency but also lessens the computational load on\nthe central server, especially when training involves a large\nnumber of clients. However, clustering similar clients might\nlead to knowledge inconsistency across clusters due to limited\ninformation sharing, resulting in knowledge isolation. Addi-\ntionally, the HPFL approach is best suited for edge computing\narchitectures and may be challenging to be implemented in\nmore general or less structured environments.\nConventional PFL personalizes each client’s model by di-\nviding it into two parts: a shared base layer and a personalized\nhead layer. While this approach has demonstrated good per-\nformance, it relies on the assumption that all clients use the\nsame model architecture. In practice, however, clients often\nhave varying computational resources, making it difficult to\nenforce a uniform model structure. To address this limitation,\nrecent researches, such as [5], [6], [15]–[18], have begun\nexploring PFL methods that support heterogeneous model\narchitectures across clients. One notable example is FlexiFed,\nproposed by [5]. FlexiFed allows each client to use different\nmodel architectures during PFL training by introducing three\nmodel aggregation strategies. One such strategy, called Max-\nCommon, clusters a subset of model architectures based on\ntheir common layers and aggregates weights only from these\ncommon layers. This approach facilitates effective federated\ntraining across clients with heterogeneous models, ensuring\nboth flexibility and consistent performance. Nevertheless, this\napproach might overlook the possibility of parameter mis-\nalignment between the global model and each client’s local\nmodel during local updates, which can lead to inconsistencies\nin shared knowledge. Previous work, such as FedAS [7],\nhas attempted to resolve this problem by aligning the global\nmodel’s weights with those of the client during local training.\nHowever, this solution is limited to scenarios where clients\nshare a uniform model architecture. In contrast, our work\naims to tackle the challenges of PFL in the presence of both\nheterogeneous client models and non-IID data, while also\nresolving the knowledge inconsistency between the server and\nthe client models during local updates.\nIII. METHODOLOGY\nFigure 1 shows an overview of our proposed framework that\nhighlights our two key contributions (depicted as yellow box),\nnamely 1) Progressive Parameter Alignment (PPA) and 2)\nentropy-based weighted averaging. Compared to the standard\nPFL methods, which straightforwardly override the client’s\nbase model with the global (server) model’s weights during\nclient update, our method progressively aligns common (and\nclustered) layers in the clients with the global model, address-\ning the knowledge-inconsistency problem between global and\nclients’ models more effectively. By re-aligning the aggregated\n\nNon-IID Data \nAmong \nClients\nHeterogeneous client’s \nmodel architecture\nClient A\nClient B\nClient C\nClient D\nEntropy-based \nWeighted \nAveraging\nCommon Layers 1\nCommon Layers 2\nCloud Server\nProgressive \nParameter \nAlignment\nGlobal \nAggregation\nClient Local \nUpdate\n(ē: 0.35)\n(ē: 0.2)\n(ē: 0.25)\n(ē: 0.2)\n𝓂i\nt\nℒ\nAlignment for \ncommon layers 1\nAlignment for \ncommon layers 2\nAlignment for \ncommon layers 2\nFig. 1: Overview of the proposed framework, where common layers are extracted from each client’s heterogeneous models.\nEach client is assigned to an entropy-based contribution weight during global aggregation. The updated model is then returned\nto the clients, after which parameter alignment is progressively carried out locally based on each client’s distinct dataset.\nknowledge from the central server with each client’s local\ndata before overwriting the client’s existing local knowledge,\nour method ensures that clients benefit from the latest server-\naggregated information without sacrificing the personalized\nknowledge manifested in the local models. Note that this\nalignment occurs iteratively from the first common layer until\nthe end, which matches and synchronizes the most similar\nlayers as much as possible.\nTo better handle the non-IID data distribution, we employ an\nentropy-based weighting to determine the contribution weight\nof each client based on the variability of its local data. Incorpo-\nrating these weights yields more robust performance in highly\nskewed non-IID environments compared to the conventional\nPFL. The entropy-based weights ensure that clients with richer,\nmore diverse datasets play a significant role in the knowledge\naggregation process. In summary, the steps performed in each\ntraining round are described as follows:\n• Calculation of client’s weight: In this step, we calcu-\nlate client’s weight contribution via the entropy-based\nmethod. Intuitively, we put higher weights to the clients\nwith more diverse datasets.\n• Local training: Each client trains their model locally\nusing their personalized dataset. Note that the client’s\nlocal dataset is divided into training and evaluation/testing\ndataset.\n• Clustering common layers: All layers from each client\nare compared to identify a list of common layers. Both\nthe architecture and the order of layers must match, e.g.,\nclients with the same layer types but arranged differently\nare not considered to share common layers. Only clients\ncontaining the identified common layers would contribute\nto the aggregation process in the server.\n• Global aggregation: In the server, the client’s model\nparameters is aggregated based on the client’s layer\nsimilarity. Note that the aggregation would leverage an\nentropy-based method, applied to the corresponding com-\nmon layers.\n• Progressive parameter alignment: Before updating each\nclient’s common layer parameters, parameter alignment\nprocess is performed. Using the list of common layers\nidentified in the previous step, the previous client’s model\nfeatures are extracted for each common layer. These\noutput common features is used to align the client’s model\nparameters by minimizing its differences with the global\nmodel features.\n• Model distribution: In the final step, the aligned param-\neters of the common layers are distributed back to each\nclient.\nA. Progressive Parameter Alignment — FedPPA\nConventional PFL methods typically split the client models\ninto the base and the head (personalization) layers and over-\nwrite their parameters directly during local updates. However,\nthis approach could lead to inconsistent knowledge between\nlocal and global models, as the client already is trained\nwith their personalized dataset. We introduce a progressive\nparameter alignment approach, namely FedPPA, which aligns\nthe aggregated global knowledge with client’s personalized\nknowledge progressively for each common layer. In practice,\n\nFedPPA works by minimizing the differences between the\nglobal features and the features of the client model before\nperforming local updates [7]. The goal is to allow each client to\nshare common knowledge for the same task without sacrificing\npersonalized capabilities tailored to their unique dataset.\nWe summarize the notations used in this paper in Table I\nand describe the complete workflow of FedPPA in Algorithm\n1. FedPPA begins by extracting the common layers of each\nclient using the following function\nTABLE I: Notations used in this paper and their definitions\nNotation\nDefinition\nNotation\nDefinition\nM\nNumber of clients\nCl\nNumber of contribu-\ntors for layer l\nt\nNumber of rounds\nLi\nTotal layers of client\ni\nmt\ni\nClient’s model i at\nround t\nH(l)\ni\nOutput features of\nlayer l in client i\nmt+1\ni\nUpdated\nclient’s\nmodel\nDi\nDataset of client i\nl\nLayer of model m\nei\nEntropy\nvalue\nof\nclient i\nlt+1\ni\n, Cl = ExtractMaxCommonLayers(mt\ni, mt\ni+1),\n(1)\nwhere lt+1\ni\nis a set of common layers extracted on the\nclient i and Cl represents the number of clients contributing\nto lt+1\ni\n. We adopt a layer-level extraction method from [5]\ninto our ExtractMaxCommonLayers function. Subsequently,\neach model parameter from the extracted common layers is\naveraged using the following formula\nmt+1\ni\n=\n\n\n\n1\nCl\nX\nli∈lt+1\ni\nli\n\n\n\nLi\n,\n(2)\nwhere mt+1\ni\nis the updated parameter from client i and Li\nis the number of layers in the model. This computation is\nperformed iteratively for each layer in the client’s local model.\nIf li is selected as the common layers, it is incorporated into\nlt+1\ni\n, which means that the parameter (i.e., the \"knowledge\")\nfrom li is added to the aggregated parameters. Before aligning\nthe aggregated client’s model parameters in each common\nlayer with the global, we need to extract the current client’s\nlocal features using the following formula\nH(l)\ni\n= {h(l)(x; mt\ni)|x ∈Di} ∈Rdl×Ni,\n(3)\nwhere H(l)\ni\nis the feature extracted from layer l in the local\nmodel mt\ni using the personalized data set of the client Di.\nThis equation is also used to extract the features from the\nupdated and aggregated client model mt+1\ni\nto ensure that\nthe dimensions of the data are equal. Finally, each client’s\ncommon layers are aligned with mt\ni using the following\nequations\nmt+1\ni\n←mt+1\ni\n−∇L(f(mt+1\ni\n, Di)H(l)\ni )\n(4)\nand\nL(f(mt+1\ni\n, Di), H(l)\ni ) = ∥f(mt+1\ni\n, Di) −H(l)\ni ∥2\n2,\n(5)\nwhere L is the L2 loss function used to align the features\nbetween mt+1\ni\nand mt\ni. The goal of this alignment is to\nmake mt+1\ni\ncloser to mt\ni by minimizing the differences of\nthe features using Equation (5).\nB. Entropy-Based Client Weighting\nConventional FL methods typically assume equal contribu-\ntion from all clients during global aggregation. However, under\nnon-IID data distributions and heterogeneous model architec-\ntures, client contributions should be weighted differently rather\nthan treated uniformly. To this end, we proposed to leverage\nEntropy-based weighting to treat each client differently based\non their dataset variability. Entropy, which quantifies the\nuncertainty or randomness of information, provides a natural\nmeasure for weighting each client’s influence.\nThe Shannon’s entropy [19] method is employed to quantify\nhow random and uncertain a piece of information is. In the\ncontext of a non-IID data distribution, a higher entropy value\nindicates that a client’s overall data are highly uncertain and\ndiverse. To ensure fair comparison across clients, we normalize\neach client’s entropy-based weight so that the total sums\nto one. Based on this procedure, we propose the following\nmethod for computing client contribution weights using the\nShannon entropy formula\nei =\nH(D)i\nPM\ni=1 H(D)i\n,\n(6)\nwhere H(D)i is the Shannon Entropy of each client i, which\ncan be calculated from [20]–[22]. As can be seen, each client’s\nentropy value is divided with the sum of all client’s entropy\nvalues to obtain the normalized entropy for each client, which\nis denoted as ei.\nFigure 2 shows the distribution of the contributions based\non weighted entropy from each client, applied to different non-\nIID data with different Dirichlet values of α (see Section 4.2\nAlgorithm 1 FedPPA Algorithm\n1: t = 1\n▷Round counter\n2: W = ∅\n▷Store aligned models\n3:\n4: for i = 1,..,M do:\n5:\nlt+1\ni\n, Cl = ExtractMaxCommonLayers(mt\ni, mt\ni+1)\ncf. (1)\n6:\nCalculate mt+1\ni\ncf. (2)\n▷Aggregate model\n7:\nUpdate mt+1\ni\n, cf. (4)\n▷Parameter alignment\n8:\nAdd aligned model mt+1\ni\nto W\n9: end for\n10:\n11: return W\n\nClient 1\n11%\nClient 2\n14%\nClient 3\n12%\nClient 4\n14%\nClient 5\n14%\nClient 6\n12%\nClient 7\n14%\nClient 8\n10%\n(a) α = 0.5\nClient 1\n11%\nClient 2\n6%\nClient 3\n17%\nClient 4\n11%\nClient 5\n13%\nClient 6\n13%\nClient 7\n17%\nClient 8\n11%\n(b) α = 0.1\nClient 1\n15%\nClient 2\n10%\nClient 3\n5%\nClient 4\n15%\nClient 5\n10%\nClient 6\n15%\nClient 7\n10%\nClient 8\n20%\n(c) α = 0.01\nFig. 2: Distribution of weighted contributions from each client, calculated using Shanon Entropy, applied to different non-IID\ndata with different α values.\nfor more information on how the non-IID data are generated).\nIt can be seen that each client exhibits a distinct contribution\nweight that reflects the variability of its class labels. When a\nlarge value of α is used, the data distribution approaches IID,\nresulting in relatively uniform client contribution. On the other\nhand, when α decreases and the data become more non-IID,\nthe variation in contribution weights across clients becomes\nmore pronounced. This demonstrates that the entropy-based\nmethod effectively captures the degree of uncertainty and\nrandomness in the client’s data.\nC. FedPPA with Entropy-Guided Weighted Averaging\nWe propose FedPPA+, an extension of FedPPA that incorpo-\nrates the client’s entropy value as the weighting factor during\nthe global knowledge aggregation. Each client would have\ndifferent aggregation weights based on their data distribution,\nas mentioned in the previous section. In general, FedPPA+\nfollows the same steps as FedPPA, which can be found in\nAlgorithm 2, except that the global aggregation approach is\nnow leveraging the entropy-based weighted averaging. The\nglobal aggregation process of FedPPA+ is formulated as\nfollows:\nmt+1\ni\n=\n\n\nei ×\n\n1\nCl\nX\nli∈lt+1\ni\nli\n\n\n\n\n\nLi\n(7)\nwhere ei is the entropy value of client i. The only difference\nwith FedPPA is that the averaged knowledge is not directly\nused as the new global model. Instead, it must first be weighted\nby ei.\nIV. EXPERIMENTAL EVALUATION\nA. Dataset, Baselines, and Metrics\nWe performed our experiments using publicly available\ndataset for image classification: 1) MNIST [24], a widely used\ndataset of 70K grayscale handwritten digit images (0–9); 2)\nFashion-MNIST (F-MNIST) [25], a dataset of 70K grayscale\nimages representing 10 categories of clothing items, serving\nas a more challenging version of MNIST; and 3) CIFAR-\n10 [26], which contains 60K 32×32 color images across 10\nobject categories and is considered as the most challenging\nAlgorithm 2 FedPPA+ Algorithm\n1: t = 1\n▷Round counter\n2: W = ∅\n▷Store aligned models\n3:\n4: Calculate e for all clients using cf. (6)\n5: for i = 1,..,M do:\n6:\nlt+1\ni\n, Cl = ExtractMaxCommonLayers(mt\ni, mt\ni+1)\ncf. (1)\n7:\nCalculate mt+1\ni\nwith ei cf. 7\n▷Aggregate model\n8:\nUpdate mt+1\ni\ncf. 4\n▷Parameter Alignment\n9:\nAdd aligned model mt+1\ni\nto W\n10: end for\n11:\n12: return W\ndataset compared to MNIST and F-MNIST. As baselines, we\ncompared our approach with Max-Common [5] and FedAvg\n[23], which represent the FL and PFL approaches, respectively.\nFinally, the classification accuracy was used as the primary\nevaluation metric to assess and compare the performance of\neach method. For a comprehensive evaluation, we measured\nboth personalization performance (the average accuracy across\nindividual clients) and global performance (the accuracy of the\nglobal model evaluated on the full dataset on the cloud server).\nB. Experimental Setup\nWe conducted our experiments under conditions where\nclients had both non-IID data distributions and heterogeneous\nmodel architectures. Table II summarizes the experimental\nsetup which shows the number of clients, the number of com-\nTABLE II: Experimental Scenario Setup\nScenario\nDirichlet α\nDataset\nClients\nRounds\n1\n0.5\nMNIST\n8\n51\nF-MNIST\nCIFAR-10\n2\n0.1\nMNIST\nF-MNIST\nCIFAR-10\n3\n0.01\nMNIST\nF-MNIST\nCIFAR-10\n\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(a) MNIST α = 0.5\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(b) F-MNIST α = 0.5\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(c) CIFAR-10 α = 0.5\nFig. 3: Model accuracy and convergence of our methods in Scenario 1 (α = 0.5) with two baseline models [5], [23]. While\nall methods can achieve good results, FedPPA and FedPPA+ surpass the baseline performances after 10 training iterations.\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(a) MNIST α = 0.1\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(b) F-MNIST α = 0.1\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(c) CIFAR-10 α = 0.1\nFig. 4: Model accuracy and convergence in Scenario 2 (α = 0.1). FedPPA and FedPPA+ exhibit stable performance relative\nto the baselines. As smaller α means more non-IIDness, our methods demonstrated superior performance in personalization.\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(a) MNIST α = 0.01\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(b) F-MNIST α = 0.01\n0\n10\n20\n30\n40\n50\nRounds\n0\n20\n40\n60\n80\n100\nAccuracy (%)\nMax Common\nFedAvg\nFedPPA\nFedPPA+\n(c) CIFAR-10 α = 0.01\nFig. 5: Model accuracy and convergence in Scenario 3 (α = 0.01). As α further decreases, the disparity between our methods\nand baseline models widens, highlighting the advantages of our proposed methods in handling personalization.\nmunication rounds, the dataset, and the α value. To simulate\na non-IID data distribution for each dataset, we leveraged the\nDirichlet distribution-based method [7], in which the param-\neter α is used to characterize the distribution, influencing the\nshape and concentration of probabilities between categories.\nFigure 6 depicts the label distribution across clients with\nα ∈{0.5, 0.1, 0.01}, where it represents label distributions\nfrom almost IID to highly non-IID. Note that each scenario is\nrun for 10 epochs in each client’s local training.\nTo simulate heterogeneous client model architectures, we\nutilized the VGG model for image classification with different\ndepth, namely VGG-11, VGG-13, VGG-16, and VGG-19.\nSince we used a total of 8 clients and only 4 different\nmodel architectures were available, we assigned every 2 clients\nwith the same model architecture. As one of the baselines,\nsince FedAVG aggregates weights by averaging across client\nmodels, this implies that weight aggregation can only occur\namong clients sharing the same architecture. On the other\nhand, similar to our approach, Max-Common was executed\nas is, since the model has the ability to handle architecture\nheterogeneity.\nWe implemented our proposed framework using PyTorch\nlibrary, where some of our implementations were inspired\nby the work of [7] and [5]. The experiment was conducted\nusing an Nvidia RTX3060 GPU with 12GB of memory, a\nCUDA Version 12.3, Python v3.10.12, and PyTorch v2.2.0.\nThe configuration includes 8 clients, each of whom locally\ntrains their model for 10 epochs using the same hyperparam-\n\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nProportion\n1\n2\n3\n4\n5\n6\n7\n8\nClient ID\n(a) α = 0.5\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nProportion\n1\n2\n3\n4\n5\n6\n7\n8\nClient ID\n(b) α = 0.1\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nProportion\n1\n2\n3\n4\n5\n6\n7\n8\nClient ID\n(c) α = 0.01\nFig. 6: Distribution of labels among clients in the MNIST dataset using the Dirichlet Distribution approach.\neters. Stochastic Gradient Descent (SGD) was applied as an\noptimizer function with a learning rate of 0.01, a momentum\nof 0.9, and a learning rate decay of 5e−4. The entire PFL\ntraining process was executed for 51 rounds in total.\nC. Experimental Results\nTable III shows the personalization performance of our ap-\nproach, in terms of the average prediction accuracy, compared\nto the baselines in Scenario 1 (α = 0.5), 2 (α = 0.1), and\n3 (α = 0.01), respectively. As we can see, in all datasets,\nour approaches, both FedPPA and FedPPA+, show superior\npersonalization performance, except for MNIST with α = 0.5\nalthough the differences with other approaches are relatively\nsmall (less than 0.5%). As expected, Max-Common, repre-\nsenting a PFL approach, achieves overall better performance\ncompared to FedAVG, which serves as a baseline for standard\nFL methods.\nTo gain a clearer understanding of each model’s perfor-\nmance over time during training, Figure 3, 4, and 5 illustrate\nthe personalization performance of our approach across all\nscenarios, from the first until the last round. From these\nfigures, our approaches generally converge faster than the\nbaselines, achieving peak accuracy earlier in most scenarios.\nNotably, the performance gap between our model and the\nbaselines widens as the value of α decreases (highly non-IID),\nreaching nearly 25% differences in the most extreme case (See\nCIFAR-10 with α = 0.01 in Table III). On the other hand, our\nmodel show only a slight performance improvement over the\nbaselines at higher value of α, where the data distribution is\nTABLE III: Accuracy of Comparison in Personalized Evalua-\ntion\nDirichlet α\nDataset\nFedAvg\n[23]\nMax-\nCommon\n[5]\nFedPPA\n(Ours)\nFedPPA+\n(Ours)\n0.5\nMNIST\n99.75\n99.62\n99.37\n99.37\nF-MNIST\n92.37\n92.62\n92.62\n93.37\nCIFAR-10\n72.62\n74.25\n77.87\n76.37\n0.1\nMNIST\n99.75\n99.62\n99.37\n99.37\nF-MNIST\n96.37\n96.24\n97.00\n97.25\nCIFAR-10\n78.25\n80.75\n91.62\n90.87\n0.01\nMNIST\n93.25\n89.00\n100.0\n100.0\nF-MNIST\n94.12\n95.62\n100.0\n100.0\nCIFAR-10\n63.37\n73.37\n98.87\n98.62\n1 Green color indicates that our methods outperform the baselines.\n2 Red color indicates that our method performance is below the baselines.\ncloser to IID, as seen in Figure 3. This indicates that our model\nshows a strong robustness in personalization performance\nin an extreme non-IID scenario compared to the baselines.\nFinally, all compared models demonstrated relatively lower\nperformance on the CIFAR-10 dataset, which is expected given\nthat CIFAR-10 is considered more challenging than MNIST\nand F-MNIST.\nThe performance of our methods improved significantly as\nthe level of non-IIDness increased in the dataset. As shown\nin Scenario 3 (α = 0.01) of Table III, our approaches reach\na prediction accuracy of up to 100% for both MNIST and\nF-MNIST. In such extreme non-IID settings, some clients\npossess only one or two labels, as illustrated in Fig. 6(c).\nUnder these circumstances, our highly personalized approach\ncan achieve perfect predictions, given a minimal number of\nclasses.\nTo highlight the performance differences between FedPPA\nand FedPPA+, Table IV presents the global performance\nevaluation between both methods across all tested datasets\nwith α = 0.5. As shown, FedPPA+ outperforms FedPPA on\ntwo of the datasets, indicating its enhanced capability in han-\ndling non-IID data distributions. Although FedPPA+ achieves\nslightly lower accuracy on the MNIST dataset, the difference\nis minimal (less than 2%). Nevertheless, further research is\nrequired to better balance personalized adaptation and global\nmodel performance, aiming to improve both personalization\nand global accuracy.\nLimitations. We acknowledge two primary limitations in\nthe present study. First, since FedPPA and FedPPA+ were\nevaluated only on VGG model variations, these evaluations\nmay not capture unforeseen heterogeneity in other scenarios,\nwhich may yield different results, for instance, when the\nproposed methods are tested on transformer-based models.\nSecondly, we conducted our evaluations on only eight FL\nTABLE IV: Global Performance of FedPPA and FedPPA+\nwith α = 0.5\nDataset\nFedPPA\nFedPPA+\nAccuracy\nRound\nAccuracy\nRound\nMNIST\n79.25\n50\n77.62\n36\nF-MNIST\n62.0\n50\n63.0\n51\nCIFAR-10\n38.87\n49\n40.62\n51\n1 Green color indicates that FedPPA+ performs better than FedPPA.\n2 Red colored indicates that FedPPA+ performance is below FedPPA.\n\nnodes, which may require further exploration in scenarios\ninvolving hundreds of participating nodes to confirm the\nconvergence of the model performance. Further research may\nbe directed towards validating these methods across more\ndiverse model architectures and at a larger scale.\nV. CONCLUSION\nIn this paper, we introduced a novel PFL method, termed\nFedPPA, designed to operate effectively in the presence of\nboth heterogeneous client models and non-IID data, while also\naddressing the knowledge inconsistency between server and\nclient models during local updates. FedPPA works by progres-\nsively aligning the weights of common layers between clients\nand the global model, thereby improving personalization per-\nformance under non-IID conditions. We further extended our\napproach with FedPPA+, which enhances model robustness by\nincorporating an entropy-based weighting mechanism during\nglobal model aggregation. Experimental results in three image\nclassification datasets, namely MNIST, F-MNIST, and CIFAR-\n10, demonstrated that our methods consistently outperform\nbaseline models in terms of personalization performance. 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Saputra†, and Guntur Dharma Putra∗‡ ∗Universitas Gadjah Mada, Indonesia †Monash University, Indonesia , , Abstract-Federated Learning (FL) is designed as a decentralized, privacy-preserving machine learning paradigm that enables multiple clients to collaboratively train a model without sharing their data. In real-world scenarios, however, clients often have heterogeneous computational resources and hold nonindependent and identically distributed data (non-IID), which poses significant challenges during training. Personalized Federated Learning (PFL) has emerged to address these issues by customizing models for each client based on their unique data distribution. Despite its potential, existing PFL approaches typically overlook the coexistence of model and data heterogeneity arising from clients with diverse computational capabilities. To overcome this limitation, we propose a novel method, called Progressive Parameter Alignment (FedPPA), which progressively aligns the weights of common layers across clients with the global model's weights. Our approach not only mitigates inconsistencies between global and local models during client updates, but also preserves client's local knowledge, thereby enhancing personalization robustness in non-IID settings. To further enhance the global model performance while retaining strong personalization, we also integrate entropy-based weighted averaging into the FedPPA framework. Experiments on three image classification datasets, including MNIST, FMNIST, and CIFAR-10, demonstrate that FedPPA consistently outperforms existing FL algorithms, achieving superior performance in personalized adaptation. Index Terms-Distributed training, personalized Federated learning, inconsistent knowledge, parameter alignment, and weighted entropy-based average. I. INTRODUCTION Federated Learning (FL) is a distributed machine learning framework in which multiple clients collaboratively train a shared model without directly sharing their data, thereby enhancing overall model performance while preserving data privacy [1]. FL works through iterative training rounds between participating clients and a central aggregation server. In each round, clients train models locally using their private data and transmit the resulting model to the server. Subsequently, the server aggregates these models to generate a global model, which is then redistributed to the clients for further local training in the next round [2], [3]. However, in real-world scenarios, participating clients often have incomplete or sparse data, i.e., non-independent and identically distributed data (non-IID), which would degrade the overall model performance. Personalized Federated Learning (PFL) addresses these data heterogeneity issues in the clients by grouping the models into a shared base layer and a personalized head layer [4]. The base ‡Corresponding author. layer, shared by the server and all clients, encodes the common deep learning layer of the entire clients, while the head layer is locally customized and is fine-tuned to capture the unique characteristics of the client's data. These personalization layers enable each client's model to more accurately align with local data while still gaining advantages from the collective knowledge shared across the networks. While PFL yields strong performance, these schemes typically demand all clients to have identical architecture. However, in practice, clients often possess heterogeneous computational resources, making it impractical to enforce a uniform model architecture across all clients, leading to each client having distinct architectures. Consequently, it becomes challenging to implement this personalization strategy that demands consistent model structures. Recent studies [5], [6] proposed PFL methods to address the issue of heterogeneous model architectures between clients. FlexiFed identifies common components within clients' local models and facilitates joint training on these layers to effectively fuse knowledge while maintaining architectural flexibility [5]. In addition, FedADP introduces a comprehensive model strategy to manage a variety of heterogeneous network architectures and ensures that devices with limited computational capacity can still significantly contribute to the global model's training [6]. However, these approaches often overlook the potential parameter misalignment between the server (global) model and the individual clients' model during local updates, potentially leading to inconsistencies between existing client's parameters and the updated parameters from the global model. To address this knowledge inconsistency, FedAS proposes a solution to align the global model's parameter with those of each client during local training, mitigating inconsistencies and improving overall model performance [7]. However, this solution heavily relies on the assumption that every client uses the same model architecture, which is quite unlikely given the significant variations in computational power commonly found in real-world scenarios. In this paper, we propose FedPPA: a PFL approach to simultaneously address heterogeneity in client models and non-IID nature of client data, while also resolving the knowledge inconsistency between the server and the client models during local updates. FedPPA progressively aligns the weights of common layers across clients with the global model's weights, mitigating inconsistencies between global and local models during client updates. While our main target is to 16 Oct 2025 improve personalization effectiveness and address the knowledge discrepancy problem, our approach also preserves client's local knowledge, thereby enhancing personalization robustness in non-IID settings. In addition, we propose FedPPA+, in which we incorporate an entropy-based weighted averaging to FedPPA, retaining strong personalization with a better tradeoff to global model performance. The entropy values are calculated from each client's variability in label distribution, where more labels exist in client's dataset equal to a higher contribution weight. In summary, we propose the following contributions: • We develop a PFL approach, termed FedPPA, which progressively aligns client's common layers with the global model parameters during local updates, which is suitable for FL with heterogeneous model architectures and non-IID data distribution. • We incorporate an entropy-based weighted averaging in the global aggregation mechanism to better adapt to nonIID data distribution. • We perform a comprehensive experiment and analysis on publicly available data sets for image classification (i.e., MNIST, F-MNIST, and CIFAR-10) using heterogeneous model architectures (i.e., VGG-11, VGG-13, VGG-16, and VGG-19). The remainder of this paper is structured as follows. We present the related work in Section II and discuss our proposed solution in Section III. The implementation of our solution with the corresponding evaluation is presented in Section IV, while the conclusion of the paper is presented in Section V. II. RELATED WORK FL was introduced as a modern distributed AI training paradigm designed to preserve user privacy by keeping data localized on user devices [8]. Instead of sharing raw data with a central server, FL enables collaborative model training across multiple clients, ensuring high levels of data privacy and security. While FL has successfully introduced collaborative training across clients, achieving high model performance using this approach typically relies on the assumption that the datasets used for training are independent and identically distributed (IID). In real-world applications, IID assumption is rarely met, resulting in non-IID data distributions across clients, which can significantly degrade the performance of FL model. Previous studies have shown that non-IID data distribution can reduce FL model performance for up to 29% [9]. In order to address the performance degradation of standard FL model due to the non-IID data distribution among clients, PFL was introduced by Arivazhagan et al. [4] by incorporating personalization layers. These personalization layers allow each client's model to better fit local data while still benefiting from shared knowledge across the networks. Since then, various PFL methods have been developed to address the diverse challenges in distributed and personalized model training. For instance, Hierarchical PFL (HPFL) [10]-[14] was developed to enable hierarchical weight aggregation among groups of clients through edge servers, which act as an intermediate aggregation points. In this approach, clients with similar resources or data characteristics are clustered together, and they only allow to communicate with the server via the edge servers, avoiding direct communication of each client with the cloud server. This approach not only reduces communication costs and latency but also lessens the computational load on the central server, especially when training involves a large number of clients. However, clustering similar clients might lead to knowledge inconsistency across clusters due to limited information sharing, resulting in knowledge isolation. Additionally, the HPFL approach is best suited for edge computing architectures and may be challenging to be implemented in more general or less structured environments. Conventional PFL personalizes each client's model by dividing it into two parts: a shared base layer and a personalized head layer. While this approach has demonstrated good performance, it relies on the assumption that all clients use the same model architecture. In practice, however, clients often have varying computational resources, making it difficult to enforce a uniform model structure. To address this limitation, recent researches, such as [5], [6], [15]-[18], have begun exploring PFL methods that support heterogeneous model architectures across clients. One notable example is FlexiFed, proposed by [5]. FlexiFed allows each client to use different model architectures during PFL training by introducing three model aggregation strategies. One such strategy, called MaxCommon, clusters a subset of model architectures based on their common layers and aggregates weights only from these common layers. This approach facilitates effective federated training across clients with heterogeneous models, ensuring both flexibility and consistent performance. Nevertheless, this approach might overlook the possibility of parameter misalignment between the global model and each client's local model during local updates, which can lead to inconsistencies in shared knowledge. Previous work, such as FedAS [7], has attempted to resolve this problem by aligning the global model's weights with those of the client during local training. However, this solution is limited to scenarios where clients share a uniform model architecture. In contrast, our work aims to tackle the challenges of PFL in the presence of both heterogeneous client models and non-IID data, while also resolving the knowledge inconsistency between the server and the client models during local updates. III. METHODOLOGY Figure 1 shows an overview of our proposed framework that highlights our two key contributions (depicted as yellow box), namely 1) Progressive Parameter Alignment (PPA) and 2) entropy-based weighted averaging. Compared to the standard PFL methods, which straightforwardly override the client's base model with the global (server) model's weights during client update, our method progressively aligns common (and clustered) layers in the clients with the global model, addressing the knowledge-inconsistency problem between global and clients' models more effectively. By re-aligning the aggregated Non-IID Data Among Clients Heterogeneous client's model architecture Client A Client B Client C Client D Entropy-based Weighted Averaging Common Layers 1 Common Layers 2 Cloud Server Progressive Parameter Alignment Global Aggregation Client Local Update (ē: 0.35) (ē: 0.2) (ē: 0.25) (ē: 0.2) mi t L Alignment for common layers 1 Alignment for common layers 2 Alignment for common layers 2 Fig. 1: Overview of the proposed framework, where common layers are extracted from each client's heterogeneous models. Each client is assigned to an entropy-based contribution weight during global aggregation. The updated model is then returned to the clients, after which parameter alignment is progressively carried out locally based on each client's distinct dataset. knowledge from the central server with each client's local data before overwriting the client's existing local knowledge, our method ensures that clients benefit from the latest serveraggregated information without sacrificing the personalized knowledge manifested in the local models. Note that this alignment occurs iteratively from the first common layer until the end, which matches and synchronizes the most similar layers as much as possible. To better handle the non-IID data distribution, we employ an entropy-based weighting to determine the contribution weight of each client based on the variability of its local data. Incorporating these weights yields more robust performance in highly skewed non-IID environments compared to the conventional PFL. The entropy-based weights ensure that clients with richer, more diverse datasets play a significant role in the knowledge aggregation process. In summary, the steps performed in each training round are described as follows: • Calculation of client's weight: In this step, we calculate client's weight contribution via the entropy-based method. Intuitively, we put higher weights to the clients with more diverse datasets. • Local training: Each client trains their model locally using their personalized dataset. Note that the client's local dataset is divided into training and evaluation/testing dataset. • Clustering common layers: All layers from each client are compared to identify a list of common layers. Both the architecture and the order of layers must match, e.g., clients with the same layer types but arranged differently are not considered to share common layers. Only clients containing the identified common layers would contribute to the aggregation process in the server. • Global aggregation: In the server, the client's model parameters is aggregated based on the client's layer similarity. Note that the aggregation would leverage an entropy-based method, applied to the corresponding common layers. • Progressive parameter alignment: Before updating each client's common layer parameters, parameter alignment process is performed. Using the list of common layers identified in the previous step, the previous client's model features are extracted for each common layer. These output common features is used to align the client's model parameters by minimizing its differences with the global model features. • Model distribution: In the final step, the aligned parameters of the common layers are distributed back to each client. A. Progressive Parameter Alignment - FedPPA Conventional PFL methods typically split the client models into the base and the head (personalization) layers and overwrite their parameters directly during local updates. However, this approach could lead to inconsistent knowledge between local and global models, as the client already is trained with their personalized dataset. We introduce a progressive parameter alignment approach, namely FedPPA, which aligns the aggregated global knowledge with client's personalized knowledge progressively for each common layer. In practice, FedPPA works by minimizing the differences between the global features and the features of the client model before performing local updates [7]. The goal is to allow each client to share common knowledge for the same task without sacrificing personalized capabilities tailored to their unique dataset. We summarize the notations used in this paper in Table I and describe the complete workflow of FedPPA in Algorithm 1. FedPPA begins by extracting the common layers of each client using the following function TABLE I: Notations used in this paper and their definitions Notation Definition Notation Definition M Number of clients Cl Number of contributors for layer l t Number of rounds Li Total layers of client i mt i Client's model i at round t H(l) i Output features of layer l in client i mt+1 i Updated client's model Di Dataset of client i l Layer of model m ei Entropy value of client i lt+1 i , Cl = ExtractMaxCommonLayers(mt i, mt i+1), (1) where lt+1 i is a set of common layers extracted on the client i and Cl represents the number of clients contributing to lt+1 i . We adopt a layer-level extraction method from [5] into our ExtractMaxCommonLayers function. Subsequently, each model parameter from the extracted common layers is averaged using the following formula mt+1 i = 1 Cl X li∈lt+1 i li Li , (2) where mt+1 i is the updated parameter from client i and Li is the number of layers in the model. This computation is performed iteratively for each layer in the client's local model. If li is selected as the common layers, it is incorporated into lt+1 i , which means that the parameter (i.e., the \"knowledge\") from li is added to the aggregated parameters. Before aligning the aggregated client's model parameters in each common layer with the global, we need to extract the current client's local features using the following formula H(l) i = {h(l)(x; mt i)|x ∈Di} ∈Rdl×Ni, (3) where H(l) i is the feature extracted from layer l in the local model mt i using the personalized data set of the client Di. This equation is also used to extract the features from the updated and aggregated client model mt+1 i to ensure that the dimensions of the data are equal. Finally, each client's common layers are aligned with mt i using the following equations mt+1 i ←mt+1 i -∇L(f(mt+1 i , Di)H(l) i ) (4) and L(f(mt+1 i , Di), H(l) i ) = ∥f(mt+1 i , Di) -H(l) i ∥2 2, (5) where L is the L2 loss function used to align the features between mt+1 i and mt i. The goal of this alignment is to make mt+1 i closer to mt i by minimizing the differences of the features using Equation (5). B. Entropy-Based Client Weighting Conventional FL methods typically assume equal contribution from all clients during global aggregation. However, under non-IID data distributions and heterogeneous model architectures, client contributions should be weighted differently rather than treated uniformly. To this end, we proposed to leverage Entropy-based weighting to treat each client differently based on their dataset variability. Entropy, which quantifies the uncertainty or randomness of information, provides a natural measure for weighting each client's influence. The Shannon's entropy [19] method is employed to quantify how random and uncertain a piece of information is. In the context of a non-IID data distribution, a higher entropy value indicates that a client's overall data are highly uncertain and diverse. To ensure fair comparison across clients, we normalize each client's entropy-based weight so that the total sums to one. Based on this procedure, we propose the following method for computing client contribution weights using the Shannon entropy formula ei = H(D)i PM i=1 H(D)i , (6) where H(D)i is the Shannon Entropy of each client i, which can be calculated from [20]-[22]. As can be seen, each client's entropy value is divided with the sum of all client's entropy values to obtain the normalized entropy for each client, which is denoted as ei. Figure 2 shows the distribution of the contributions based on weighted entropy from each client, applied to different nonIID data with different Dirichlet values of α (see Section 4.2 Algorithm 1 FedPPA Algorithm 1: t = 1 ▷Round counter 2: W = ∅ ▷Store aligned models 3: 4: for i = 1,..,M do: 5: lt+1 i , Cl = ExtractMaxCommonLayers(mt i, mt i+1) cf. (1) 6: Calculate mt+1 i cf. (2) ▷Aggregate model 7: Update mt+1 i , cf. (4) ▷Parameter alignment 8: Add aligned model mt+1 i to W 9: end for 10: 11: return W Client 1 11% Client 2 14% Client 3 12% Client 4 14% Client 5 14% Client 6 12% Client 7 14% Client 8 10% (a) α = 0.5 Client 1 11% Client 2 6% Client 3 17% Client 4 11% Client 5 13% Client 6 13% Client 7 17% Client 8 11% (b) α = 0.1 Client 1 15% Client 2 10% Client 3 5% Client 4 15% Client 5 10% Client 6 15% Client 7 10% Client 8 20% (c) α = 0.01 Fig. 2: Distribution of weighted contributions from each client, calculated using Shanon Entropy, applied to different non-IID data with different α values. for more information on how the non-IID data are generated). It can be seen that each client exhibits a distinct contribution weight that reflects the variability of its class labels. When a large value of α is used, the data distribution approaches IID, resulting in relatively uniform client contribution. On the other hand, when α decreases and the data become more non-IID, the variation in contribution weights across clients becomes more pronounced. This demonstrates that the entropy-based method effectively captures the degree of uncertainty and randomness in the client's data. C. FedPPA with Entropy-Guided Weighted Averaging We propose FedPPA+, an extension of FedPPA that incorporates the client's entropy value as the weighting factor during the global knowledge aggregation. Each client would have different aggregation weights based on their data distribution, as mentioned in the previous section. In general, FedPPA+ follows the same steps as FedPPA, which can be found in Algorithm 2, except that the global aggregation approach is now leveraging the entropy-based weighted averaging. The global aggregation process of FedPPA+ is formulated as follows: mt+1 i = ei × 1 Cl X li∈lt+1 i li Li (7) where ei is the entropy value of client i. The only difference with FedPPA is that the averaged knowledge is not directly used as the new global model. Instead, it must first be weighted by ei. IV. EXPERIMENTAL EVALUATION A. Dataset, Baselines, and Metrics We performed our experiments using publicly available dataset for image classification: 1) MNIST [24], a widely used dataset of 70K grayscale handwritten digit images (0-9); 2) Fashion-MNIST (F-MNIST) [25], a dataset of 70K grayscale images representing 10 categories of clothing items, serving as a more challenging version of MNIST; and 3) CIFAR10 [26], which contains 60K 32×32 color images across 10 object categories and is considered as the most challenging Algorithm 2 FedPPA+ Algorithm 1: t = 1 ▷Round counter 2: W = ∅ ▷Store aligned models 3: 4: Calculate e for all clients using cf. (6) 5: for i = 1,..,M do: 6: lt+1 i , Cl = ExtractMaxCommonLayers(mt i, mt i+1) cf. (1) 7: Calculate mt+1 i with ei cf. 7 ▷Aggregate model 8: Update mt+1 i cf. 4 ▷Parameter Alignment 9: Add aligned model mt+1 i to W 10: end for 11: 12: return W dataset compared to MNIST and F-MNIST. As baselines, we compared our approach with Max-Common [5] and FedAvg [23], which represent the FL and PFL approaches, respectively. Finally, the classification accuracy was used as the primary evaluation metric to assess and compare the performance of each method. For a comprehensive evaluation, we measured both personalization performance (the average accuracy across individual clients) and global performance (the accuracy of the global model evaluated on the full dataset on the cloud server). B. Experimental Setup We conducted our experiments under conditions where clients had both non-IID data distributions and heterogeneous model architectures. Table II summarizes the experimental setup which shows the number of clients, the number of comTABLE II: Experimental Scenario Setup Scenario Dirichlet α Dataset Clients Rounds 1 0.5 MNIST 8 51 F-MNIST CIFAR-10 2 0.1 MNIST F-MNIST CIFAR-10 3 0.01 MNIST F-MNIST CIFAR-10 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (a) MNIST α = 0.5 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (b) F-MNIST α = 0.5 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (c) CIFAR-10 α = 0.5 Fig. 3: Model accuracy and convergence of our methods in Scenario 1 (α = 0.5) with two baseline models [5], [23]. While all methods can achieve good results, FedPPA and FedPPA+ surpass the baseline performances after 10 training iterations. 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (a) MNIST α = 0.1 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (b) F-MNIST α = 0.1 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (c) CIFAR-10 α = 0.1 Fig. 4: Model accuracy and convergence in Scenario 2 (α = 0.1). FedPPA and FedPPA+ exhibit stable performance relative to the baselines. As smaller α means more non-IIDness, our methods demonstrated superior performance in personalization. 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (a) MNIST α = 0.01 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (b) F-MNIST α = 0.01 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (c) CIFAR-10 α = 0.01 Fig. 5: Model accuracy and convergence in Scenario 3 (α = 0.01). As α further decreases, the disparity between our methods and baseline models widens, highlighting the advantages of our proposed methods in handling personalization. munication rounds, the dataset, and the α value. To simulate a non-IID data distribution for each dataset, we leveraged the Dirichlet distribution-based method [7], in which the parameter α is used to characterize the distribution, influencing the shape and concentration of probabilities between categories. Figure 6 depicts the label distribution across clients with α ∈{0.5, 0.1, 0.01}, where it represents label distributions from almost IID to highly non-IID. Note that each scenario is run for 10 epochs in each client's local training. To simulate heterogeneous client model architectures, we utilized the VGG model for image classification with different depth, namely VGG-11, VGG-13, VGG-16, and VGG-19. Since we used a total of 8 clients and only 4 different model architectures were available, we assigned every 2 clients with the same model architecture. As one of the baselines, since FedAVG aggregates weights by averaging across client models, this implies that weight aggregation can only occur among clients sharing the same architecture. On the other hand, similar to our approach, Max-Common was executed as is, since the model has the ability to handle architecture heterogeneity. We implemented our proposed framework using PyTorch library, where some of our implementations were inspired by the work of [7] and [5]. The experiment was conducted using an Nvidia RTX3060 GPU with 12GB of memory, a CUDA Version 12.3, Python v3.10.12, and PyTorch v2.2.0. The configuration includes 8 clients, each of whom locally trains their model for 10 epochs using the same hyperparam0.0 0.2 0.4 0.6 0.8 1.0 Proportion 1 2 3 4 5 6 7 8 Client ID (a) α = 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Proportion 1 2 3 4 5 6 7 8 Client ID (b) α = 0.1 0.0 0.2 0.4 0.6 0.8 1.0 Proportion 1 2 3 4 5 6 7 8 Client ID (c) α = 0.01 Fig. 6: Distribution of labels among clients in the MNIST dataset using the Dirichlet Distribution approach. eters. Stochastic Gradient Descent (SGD) was applied as an optimizer function with a learning rate of 0.01, a momentum of 0.9, and a learning rate decay of 5e-4. The entire PFL training process was executed for 51 rounds in total. C. Experimental Results Table III shows the personalization performance of our approach, in terms of the average prediction accuracy, compared to the baselines in Scenario 1 (α = 0.5), 2 (α = 0.1), and 3 (α = 0.01), respectively. As we can see, in all datasets, our approaches, both FedPPA and FedPPA+, show superior personalization performance, except for MNIST with α = 0.5 although the differences with other approaches are relatively small (less than 0.5%). As expected, Max-Common, representing a PFL approach, achieves overall better performance compared to FedAVG, which serves as a baseline for standard FL methods. To gain a clearer understanding of each model's performance over time during training, Figure 3, 4, and 5 illustrate the personalization performance of our approach across all scenarios, from the first until the last round. From these figures, our approaches generally converge faster than the baselines, achieving peak accuracy earlier in most scenarios. Notably, the performance gap between our model and the baselines widens as the value of α decreases (highly non-IID), reaching nearly 25% differences in the most extreme case (See CIFAR-10 with α = 0.01 in Table III). On the other hand, our model show only a slight performance improvement over the baselines at higher value of α, where the data distribution is TABLE III: Accuracy of Comparison in Personalized Evaluation Dirichlet α Dataset FedAvg [23] MaxCommon [5] FedPPA (Ours) FedPPA+ (Ours) 0.5 MNIST 99.75 99.62 99.37 99.37 F-MNIST 92.37 92.62 92.62 93.37 CIFAR-10 72.62 74.25 77.87 76.37 0.1 MNIST 99.75 99.62 99.37 99.37 F-MNIST 96.37 96.24 97.00 97.25 CIFAR-10 78.25 80.75 91.62 90.87 0.01 MNIST 93.25 89.00 100.0 100.0 F-MNIST 94.12 95.62 100.0 100.0 CIFAR-10 63.37 73.37 98.87 98.62 1 Green color indicates that our methods outperform the baselines. 2 Red color indicates that our method performance is below the baselines. closer to IID, as seen in Figure 3. This indicates that our model shows a strong robustness in personalization performance in an extreme non-IID scenario compared to the baselines. Finally, all compared models demonstrated relatively lower performance on the CIFAR-10 dataset, which is expected given that CIFAR-10 is considered more challenging than MNIST and F-MNIST. The performance of our methods improved significantly as the level of non-IIDness increased in the dataset. As shown in Scenario 3 (α = 0.01) of Table III, our approaches reach a prediction accuracy of up to 100% for both MNIST and F-MNIST. In such extreme non-IID settings, some clients possess only one or two labels, as illustrated in Fig. 6(c). Under these circumstances, our highly personalized approach can achieve perfect predictions, given a minimal number of classes. To highlight the performance differences between FedPPA and FedPPA+, Table IV presents the global performance evaluation between both methods across all tested datasets with α = 0.5. As shown, FedPPA+ outperforms FedPPA on two of the datasets, indicating its enhanced capability in handling non-IID data distributions. Although FedPPA+ achieves slightly lower accuracy on the MNIST dataset, the difference is minimal (less than 2%). Nevertheless, further research is required to better balance personalized adaptation and global model performance, aiming to improve both personalization and global accuracy. Limitations. We acknowledge two primary limitations in the present study. First, since FedPPA and FedPPA+ were evaluated only on VGG model variations, these evaluations may not capture unforeseen heterogeneity in other scenarios, which may yield different results, for instance, when the proposed methods are tested on transformer-based models. Secondly, we conducted our evaluations on only eight FL TABLE IV: Global Performance of FedPPA and FedPPA+ with α = 0.5 Dataset FedPPA FedPPA+ Accuracy Round Accuracy Round MNIST 79.25 50 77.62 36 F-MNIST 62.0 50 63.0 51 CIFAR-10 38.87 49 40.62 51 1 Green color indicates that FedPPA+ performs better than FedPPA. 2 Red colored indicates that FedPPA+ performance is below FedPPA. nodes, which may require further exploration in scenarios involving hundreds of participating nodes to confirm the convergence of the model performance. Further research may be directed towards validating these methods across more diverse model architectures and at a larger scale. V. CONCLUSION In this paper, we introduced a novel PFL method, termed FedPPA, designed to operate effectively in the presence of both heterogeneous client models and non-IID data, while also addressing the knowledge inconsistency between server and client models during local updates. FedPPA works by progressively aligning the weights of common layers between clients and the global model, thereby improving personalization performance under non-IID conditions. We further extended our approach with FedPPA+, which enhances model robustness by incorporating an entropy-based weighting mechanism during global model aggregation. Experimental results in three image classification datasets, namely MNIST, F-MNIST, and CIFAR10, demonstrated that our methods consistently outperform baseline models in terms of personalization performance. As FedPPA and FedPPA+ primarily aim to improve personalization effectiveness and address the issue of knowledge discrepancies, future work may be directed toward balancing personalization and global performance. ACKNOWLEDGMENT We express our gratitude to the Indonesia Endowment Fund for Education (LPDP) under the Ministry of Finance of the Republic of Indonesia for providing the scholarship and supporting this research. REFERENCES [1] M. N. P. 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Kaggle."}}},{"rowIdx":285,"cells":{"id":{"kind":"string","value":"2510.14702"},"raw":{"kind":"string","value":"Cognitive-Aligned Spatio-Temporal Large Language Models For\nNext Point-of-Interest Prediction\nPenglong Zhai∗\nAMAP, Alibaba Group\nBeijing, China\nzhaipenglong.zpl@alibaba-inc.com\nJie Li∗\nAMAP, Alibaba Group\nBeijing, China\nlj313796@alibaba-inc.com\nFanyi Di∗\nAMAP, Alibaba Group\nBeijing, China\ndifanyi.dfy@alibaba-inc.com\nYue Liu\nAMAP, Alibaba Group\nBeijing, China\nly355576@alibaba-inc.com\nYifang Yuan\nAMAP, Alibaba Group\nBeijing, China\nyuanyifang.yyf@alibaba-inc.com\nJie Huang\nAMAP, Alibaba Group\nBeijing, China\njielu.hj@alibaba-inc.com\nPeng Wu\nAMAP, Alibaba Group\nBeijing, China\nzhouxiao.wp@alibaba-inc.com\nSicong Wang\nAMAP, Alibaba Group\nBeijing, China\nwsc488938@alibaba-inc.com\nMingyang Yin\nAMAP, Alibaba Group\nBeijing, China\nhengyang.ymy@alibaba-inc.com\nTingting Hu\nAMAP, Alibaba Group\nBeijing, China\njingting.htt@alibaba-inc.com\nYao Xu\nAMAP, Alibaba Group\nBeijing, China\nxuenuo.xy@alibaba-inc.com\nXin Li†\nAMAP, Alibaba Group\nBeijing, China\nbeilai.bl@alibaba-inc.com\nAbstract\nThe next point-of-interest (POI) recommendation task aims to pre-\ndict the users’ immediate next destinations based on their prefer-\nences and historical check-ins, holding significant value in location-\nbased services. Recently, large language models (LLMs) have shown\ngreat potential in recommender systems, which treat the next POI\nprediction in a generative manner. However, these LLMs, pretrained\nprimarily on vast corpora of unstructured text, lack the native un-\nderstanding of structured geographical entities and sequential mo-\nbility patterns required for next POI prediction tasks. Moreover, in\nindustrial-scale POI prediction applications, incorporating world\nknowledge and alignment of human cognition, such as seasons,\nweather conditions, holidays, and users’ profiles (such as habits,\noccupation, and preferences), can enhance the user experience\nwhile improving recommendation performance. To address these\nissues, we propose CoAST (Cognitive-Aligned Spatial-Temporal\nLLMs), a framework employing natural language as an interface,\nallowing for the incorporation of world knowledge, spatio-temporal\ntrajectory patterns, profiles, and situational information. Specifi-\ncally, CoAST mainly comprises of 2 stages: (1) Recommendation\n∗Contributed equally to this research.\n†Corresponding Author.\nPermission to make digital or hard copies of all or part of this work for personal or\nclassroom use is granted without fee provided that copies are not made or distributed\nfor profit or commercial advantage and that copies bear this notice and the full citation\non the first page. Copyrights for components of this work owned by others than the\nauthor(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or\nrepublish, to post on servers or to redistribute to lists, requires prior specific permission\nand/or a fee. Request permissions from permissions@acm.org.\nConference acronym ’XX, Woodstock, NY\n© 2018 Copyright held by the owner/author(s). Publication rights licensed to ACM.\nACM ISBN 978-1-4503-XXXX-X/2018/06\nhttps://doi.org/XXXXXXX.XXXXXXX\nKnowledge Acquisition through continued pretraining on the en-\nriched spatial-temporal trajectory data of the desensitized users;\n(2) Cognitive Alignment to align cognitive judgments with human\npreferences using enriched training data through Supervised Fine-\nTuning (SFT) and a subsequent Reinforcement Learning (RL) phase.\nExtensive offline experiments on various real-world datasets and\nonline experiments deployed in \"Guess Where You Go\" of AMAP1\nApp homepage demonstrate the effectiveness of CoAST.\nCCS Concepts\n• Information systems →Location based services.\nKeywords\nGenerative Recommendation, Next POI Prediction, Cognitive Rec-\nommendation, Large Language Models\nACM Reference Format:\nPenglong Zhai, Jie Li, Fanyi Di, Yue Liu, Yifang Yuan, Jie Huang, Peng\nWu, Sicong Wang, Mingyang Yin, Tingting Hu, Yao Xu, and Xin Li. 2018.\nCognitive-Aligned Spatio-Temporal Large Language Models For Next Point-\nof-Interest Prediction. In Proceedings of Make sure to enter the correct con-\nference title from your rights confirmation email (Conference acronym ’XX).\nACM, New York, NY, USA, 12 pages. https://doi.org/XXXXXXX.XXXXXXX\n1\nIntroduction\nNext Point-of-Interest (POI) prediction, as a variant of POI recom-\nmendation task, aiming to predict users’ future locations based\non their historical sequential check-ins and other contextual in-\nformation, is a fundamental task in Location-Based Services (LBS)\n[9, 10, 19, 24, 25, 32, 36, 38, 40, 43]. For large-scale navigation plat-\nforms such as AMAP, which processes billions of daily requests\n1https://amap.com/\narXiv:2510.14702v1 [cs.AI] 16 Oct 2025\n\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nZhai et al.\nfrom hundreds of millions of active users in China, making accu-\nrate next POI prediction a critical capability for efficient destination\nplanning and personalized recommendation. Thus, next POI predic-\ntion has been a popular research direction, capitalizing on develop-\nments in mobile and localization techniques, as they provide rich\nlocation-based information. Much of the existing research in next\nPOI recommendation has predominantly focused on sequential rec-\nommendation and graph neural networks, tackling the challenge of\ndata sparsity, particularly for cold-start users and those with sparse\nhistorical trajectories [8, 29, 33, 45, 48].\nHowever, these approaches often fail to encapsulate the rich\ncontextual information embedded within Location-Based Social\nNetwork (LBSN) data, such as the world knowledge and contextual\ninformation, which are critical for improving the user experience\nin industrial applications. Recently, Large Language Models (LLMs)\nhave demonstrated remarkable capabilities across various domains\n[1, 27, 31, 41], offering new opportunities for this direction. How-\never, their direct application to POI prediction task remains several\nkey challenges. First, POIs are typically indexed by discrete iden-\ntifiers outside the LLM vocabulary, impeding the model’s ability\nto capture semantic relationships [2, 47, 50]. Second, the intricate\ninteractions between the billions of users and POIs, coupled with\nhighly personalized and context-dependent behavioral patterns,\nintroduce complex collaborative signals that are difficult for LLMs\nto encode and reason effectively [23, 25].\nAnother critical challenge, stemming from the real-world deploy-\nment, is that recommended POIs must be cognitively plausible to\nusers. Recommendations that violate human intuition or common-\nsense reasoning, which we term human cognitive preference in this\nwork, even if statistically likely, can appear nonsensical or irrele-\nvant, severely degrading the user experience and eroding trust in\nthe system [25]. In industrial applications, these logical lapses are\ncritical failures, as they directly risk frustrating users and reducing\nretention.\nBased on these observations, we formulate the task of human\ncognitive preference alignment for individual POI prediction as: to\ninvestigate issues of personality and individual differences (who),\nenvironmental and social impacts (what), spatio-temporal features\nand relationships (when and where), and corresponding actions\n(how).\nTo address the aforementioned challenges and build a POI rec-\nommendation system that aligns with human cognitive preferences,\nwe introduce CoAST (Cognitive-Aligned Spatio-Temporal LLM), a\nnovel framework that endows a base LLM with the ability to gen-\nerate cognitively plausible POI recommendations. CoAST begins\nby adapting the LLM to the mobility domain through continued\npretraining on an enriched corpus of spatio-temporal tokenized POI\nsequences, allowing it to master complex user behavior patterns.\nFurthermore, we explicitly align the model with human common-\nsense. This is achieved through a multi-stage fine-tuning process,\nutilizing a curated dataset of cognitively-aware samples for both\nSupervised Fine-Tuning (SFT) and a subsequent Reinforcement\nLearning (RL) phase.\nOur main contributions as as follows:\n• To our knowledge, this is the first generic LLM-based cognitive-\naligned framework for the field of next POI prediction. The\nproposed framework enhances the comprehension of human\ncognitive preference in recommendation.\n• We propose CoAST, a novel end-to-end LLM-based frame-\nwork for large-scale next POI prediction that effectively\nbridges the semantic gap between world and recommen-\ndation knowledge, which demonstrates significant improve-\nment on human cognitive alignment and applicable to prac-\ntical applicability of LLMs in LBS applications.\n• Through a series of inference optimization strategy, CoAST\nis deployed in AMAP’s online production environment, a\nleading real-world LBS platform. Compared traditional cas-\ncade ranking system, long term online A/B test achieves a\n4.23% and 4.59% increase in P-CTR and U-CTR, which is a\nsubstantial improvement.\nThe remainder of this paper is organized as follows: Section\n2 reviews related literature, Section 3 details problem definition\nand the detailed categories of cognitive recommendation, Section\n4 elaborates our methodology, Section 5 reports empirical results,\nand Section 6 concludes the paper.\n2\nRelated Work\n2.1\nNext POI Prediction\nNext POI recommendation extends beyond conventional item rec-\nommendation by incorporating inherent geographical and tempo-\nral characteristics. Early methods relied on user check-in histo-\nries using Markov chains, or RNNs to capture sequential behav-\niors. Spatial-temporal modeling has been extensively studied with\nworks like ST-GRAT [26] employing graph attention networks for\nspatial-temporal dependencies, STAN [25] introducing hierarchical\nattention mechanisms, and Sun et al. [30] focusing on periodic pat-\nterns in human mobility through time-aware embeddings. Recent\nLLM-based approaches have focused on addressing the semantic\ngap in POI recommendation. For example, GNPR-SID [32] proposed\nsemantic POI IDs using residual quantized variational autoencoders,\nachieving improvements by mapping POIs into discrete semantic\nspaces. LLM4POI [19] leveraged pretrained LLMs to preserve het-\nerogeneous data in original format, effectively utilizing contextual\ninformation.\n2.2\nLLMs for Recommendation\nThe integration of Large Language Models (LLMs) into recommen-\ndation systems has emerged as a transformative research direction.\nP5 [12] pioneered the \"Pretrain, Personalized Prompt, and Predict\"\nparadigm, establishing the foundation for treating recommenda-\ntion as a language processing task by converting all recommen-\ndation data into natural language sequences. Building upon this,\n[2] addressed the knowledge gap between LLMs’ capabilities and\nrecommendation-specific requirements by incorporating collabo-\nrative filtering operations like Masked Item Modeling (MIM) and\nBayesian Personalized Ranking (BPR) through natural language sim-\nulation. More recently, GenRank [17] advanced generative ranking\nsystems for large-scale deployment. OneRec [5, 53] proposed a rev-\nolutionary end-to-end framework that replaces traditional cascaded\narchitectures, achieving significant improvements in large-scale\nonline deployments.\n\nCognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\n3\nPreliminaries\n3.1\nProblem Formulation\nGiven a collection of POIs I = {𝑝1, 𝑝2, ..., 𝑝𝑁} and a group of users\nU = {𝑢1,𝑢2, ...,𝑢𝑀}. Each POI can be characterized by a tuple\n𝑝𝑖= (𝑙𝑜𝑛,𝑙𝑎𝑡,𝑐), where 𝑐represents side information of the POI\nincluding the descriptive texts and pictures, 𝑙𝑜𝑛and 𝑙𝑎𝑡denotes\nthe geographical coordinates (longitude and latitude). Each user 𝑢\nhas the check-ins history ℎ𝑢= {(𝑝1,𝑡1), . . . , (𝑝𝑚,𝑡𝑚)} of length 𝑚,\nwhere 𝑝𝑖,𝑖= 1, . . . ,𝑚reflects the 𝑖-th POI recently interacted by\nthe user and 𝑡𝑖refers to the check-in timestamps. The objective of\nnext POI prediction task is to predict the next POI 𝑝that the user 𝑢\nwill visit at time 𝑡given the recent check-in record ℎ𝑢, the user’s\nrequest location 𝑙and the contextual situation 𝑠.\n3.2\nCognitive-Aligned POI Prediction\nCompared to traditional accuracy-oriented recommender systems\nthat rely on rapidly capturing users’ immediate interests, next POI\nprediction is a complex issue in simulating individual behaviors and\ncontextual information. Empirically, we category cognitive-aligned\nrecommendations into the following four scenarios. Recommended\nPOIs that do not align with these categories may result in a dimin-\nished user experience or decreased user engagement. To summa-\nrize, our approach formulates the next POI recommendation task\nas a decision-making problem. The model synthesizes a diverse set\nof contextual factors—including temporal (when), spatial (where),\nuser-specific (who), and other situational inputs—to determine the\noptimal user action (what to do and how).\n3.2.1\nTemporal Consistency. User behavior often exhibits certain\nregularities. For example, office workers typically commute to work-\nplaces in the morning and return home at noon. Recommending\nirrelevant POIs during these commuting periods may lead to poor\nuser experiences. We define these observed regularity patterns in\nuser mobility as temporal consistency. Formally, at the given time\n𝑡, the temporal consistency score (denoted as TCS) of the predicted\nPOI ˆ𝑝can be formulated as,\nTCS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) =\nÍ\n𝑖=𝑚I(𝑡𝑖= 𝑡) · sim(𝑝𝑖, ˆ𝑝∗)\nÍ𝑚\n𝑖=1 I(𝑡𝑖= 𝑡)\n(1)\nwhere sim(·) denotes the function measuring the similarity of the\ninputs pair.\n3.2.2\nSpatial Clustering. Check-ins of human are empirically demon-\nstrated to occur within a certain range [20, 21]; that is, the distance\nof a user’s next visit POI is typically distributed as a negative expo-\nnential relative to the user’s usual activity space. Recommending\na POI that is far from the user’s position (or their active area) is\ninappropriate. We define the spatial clustering phenomenon of user\nmobility as spatial consistency. Formally, given the location 𝑝𝑢of\nthe user 𝑢, the spatial consistency score of the predicted POI ˆ𝑝is\nformulated as,\nSCS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) = 𝑒−𝑑(𝑝𝑢, ˆ𝑝)/𝑑𝑢\n(2)\nwhere 𝑒denotes the exponential function, 𝑑(·) is the distance func-\ntion, and 𝑑𝑢represents the user’s historical average movement\ndistance.\n3.2.3\nProfile Alignment. Behaviors of human often correlate with\ntheir profile tags. For instance, suggesting family-oriented venues or\nrestaurants to university students represents a cognitive mismatch\nin recommendation logic. We define the user profile alignment\nscore to reflect the alignment between recommended item ˆ𝑝and\nuser characteristics:\nPAS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) = LLM𝑢(𝑢, ˆ𝑝)\n(3)\nwhere a LLM model is utilized to evaluate the reasonableness of\nthe recommended POI ˆ𝑝given the profile of user 𝑢and the score\n∈{0, 1}.\n3.2.4\nSituational Awareness. Situations such as weather, traffic,\nand the conditions of the POIs could have a significant impact\non human’s traveling decisions. For example, recommending an\noutdoor tennis court on a rainy day is considered inappropriate. We\ndefine the recommendation system’s Awareness to these contextual\nsituations as situational awareness. Similar to definition of profile\nalignment, we also utilize a LLM to score this metric,\nSAS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) = LLM𝑠(𝑠, ˆ𝑝)\n(4)\nwhere the score ∈{0, 1}.\nThe definitions articulated above serve a dual purpose in our\nmethodology. They directly guide the creation of a training set of\ncognitively plausible examples for the model’s alignment phase.\nSimultaneously, these definitions are translated into quantitative\nmetrics, providing a principled framework for evaluating the cog-\nnitive capabilities of our proposed model.\n4\nMethodologies\nThe overall framework is presented in Figure 1. The framework\nconsists of two main modules: (1) Spatio-temporal Knowledge Ac-\nquisition module enables the LLMs to incorporate the linguistic and\ncollaborative semantics and (2) Cognitive Alignment to adjusting\nthe ST-LLM obtained from step (1) to generate results that better\nalign with human cognition. We finally detail our inference opti-\nmization strategy in online deployment since this is a bottleneck of\nlarge-scale application of generative models.\n4.1\nSpatio-temporal Knowledge Acquisition\nConsidering the huge amount of the POI corpus in large industrial\nLBS applications, we firstly convert POI data into Semantic IDs\n(SIDs) to reduce the total number of tokens. Building on these SIDs,\nwe propose to enable the LLM to incorporate the recommendation\nknowledge by continued pretraining on an enriched corpus.\n4.1.1\nSemantic ID generation. A well crafted SID representation\ncan boost the LLM’s recommendation efficiency by capturing the\nsemantic and spatial relationships among POIs. There has already\nbeen extensive research in this area [22, 32, 34, 49, 51, 54], so we\ndo not elaborate on it here. In brief, we adopt the RQ-VAE, which\nadopts the multi-modal information of the POIs to learning the\ncodebooks. Specifically, we incorporate the spatial location infor-\nmation and descriptive titles/pictures of the POIs to generate the\nidentifiers with residual quantization process. As a result, we get a\nset of codebooks\n𝐶𝑙= {𝒆𝑙\n𝑘|𝑘= 1, ..., 𝐾},𝑙∈[1, 𝐿]\n\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nZhai et al.\nFigure 1: Offline Training and Online Service Procedure of proposed method CoAST. Offline Training: (1) POIs are tokenized as\nSemantic IDs (SIDs); (2) a base LLM is continually pretrained on the collected corpus, yielding ST-LLM; (3) multi-level user\nprofiles are distilled from the global tracking database with the annotations of LLM; (4) cognitive-aligned data are produced\nthrough LLM annotation and human screening; (6) ST-LLM is further fine-tuned and aligned to finish this procedure. Online\nServing: CoAST is stored in a model engine and invoked through a data-service engine that converts real-time context—user\nprofile, recent check-ins, current time, location, and weather—into an instruction prompt. The model predicts the next POI in\nSID form, which is mapped back to a readable location and returned to the mobile client.\nwhere 𝐿represents the number of codebooks and 𝐾denotes the\nsize of each codebook. Therefore, a POI can be tokenized as an\nordered sequence of tokens, such as \" \", in\nwhich the number of the above tokens represents the indices of the\nresponding codebook.\n4.1.2\nContinued Pretraining on Spatio-temporal Corpus. After ac-\nquiring SIDs of the POIs, a simple method is to incorporate these\nSIDs into the vocabulary of large language models (LLMs). This\nallows the LLMs to perform recommendation tasks in a generative\nmanner, ultimately moving away from using item indices. Never-\ntheless, there are two remaining concerns: (1) these newly added\nindices are out-of-vocabulary tokens for LLMs, incorporating these\nlinguistic and collaborative semantics becomes necessary. (2) Rec-\nommendation tasks present unique technical challenges that distin-\nguish them from conventional NLP tasks, particularly in modeling\nspatio-temporal user behaviors and handling data sparsity.\nTo achieve this incorporation, we implemented continued pre-\ntraining [14] on the recommendation corpus, aiming to adjust LLMs\nto improve understanding of linguistic and collaborative semantics.\nOur approach to recommendation domain-adaptive pretraining is\nstraightforward—we continue pretraining a base model (like Qwen3\n[41]) on a large corpus of unlabeled spatio-temporal recommenda-\ntion domain-specific text. Specifically, the corpus are designed and\ncollected mainly focusing on 2 objectives: clear index-language-\nspace aligning (finding the POI with its description, location and\nits category, or outputting the description and location given the\nPOI) and sequential POI prediction (predicting the next POI given\nthe historical POI check-ins).\nWhile understanding the spatial relationships between POIs is\nfundamental to LBS recommendation, directly instilling this knowl-\nedge into LLMs is difficult. To address this, our approach tokenizes\neach POI’s Geohash and integrates these tokens into our pretraining\ndata. This process enables the model to learn a unified embedding\nspace where the POI’s identity (SID), its semantic meaning, and its\nlocation (via Geohash) are jointly represented and aligned. Table 1\nillustrates the utilized corpus in this stage.\nTable 1: Recommendation corpus collected for continuous\npretraining based on a base model. The words in orange\ndenote the newly added tokens in the base LLM.\nSID-Location-Description Alignment Corpus\nThe Temple of Heaven Park is the world’s largest surviving\nancient architectural complex dedicated to the worship of\nheaven. Originally constructed during the Ming Dynasty, it\nserved as the site where emperors of the Ming and Qing dy-\nnasties held ceremonies to “sacrifice to heaven” and “pray for\na good harvest.” Located in of Beijing, and the ID is\n.\nSpatio-Temporal Behavior Sequence Recommendation\nCorpus\nHere is user uid_256’s historical POI check-ins. At 7 AM\nJune 10, the user ride to company ,\nat 21 PM June 11, the user navigated to the hotel\n, at 15 PM June 20, the user searched\nan office ,..., at 19 PM July 20, the user\nwalked to home .\nMasked prediction has recently been applied to adapt pretrained\nmodels to specific downstream tasks by continuing to pretrain\nmodels on in-domain unlabelled data [6, 15, 37]. Inspired by these\nworks, we continue our pretraining with masking some original\ntokens of the base LLM. However, a crucial consideration in our\npretraining phase is the token composition of the training corpus,\n\nCognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nwhich presents a delicate trade-off between newly introduced POI\ntokens (SIDs and Geohash tokens) and the original tokens within\nthe pretraining corpus.\nThrough the comprehensive continued pretraining on the corpus\nof spatio-temporal recommendation domain, the model is adapted\nto learn information centered around generic behavioral patterns.\nHowever, its outputs may exhibit cognitive mis-alignment with\nhuman reasoning principles. This necessitates a specialized cog-\nnitive calibration process to bridge the divergent representations\nbetween machine-learned patterns and anthropic decision-making\nframeworks.\n4.2\nCognitive Preference Alignment\n4.2.1\nUser Profile Generation. Understanding user historical POI\ncheck-ins is crucial for modeling the trajectory patterns of users.\nHowever, it is challenging to effectively utilize extended user be-\nhavior sequences, which often extends to lengths of 10,000 in large-\nscale recommendation systems [11, 16]. Yet, LLMs face difficulties\nin processing these long sequences, and this hinders their ability\nto grasp the patterns. To counteract this, we have developed multi-\nlevel profile of the users by condensing extensive user behavior\ninto refined profiles that are more amenable to LLMs. The utilized\nprofiles can be mainly divided into three main categories: (1) Static\nProfile, (2) Long-term Preference Profile, and (3) Periodic Demand\nProfile.\nStatic profile, such as POIs of the users’ home and company,\nare critical for next POI prediction. However, the labeled data is\npretty limited. To expand the coverage of static profiles, we first\nemploy LLM-based summarization of user behavioral sequences to\ngenerate preliminary annotations. These labeled patterns are then\nutilized to train models (like a MOE-Transformer network [7]) that\nsubsequently perform inference across the entire user population,\nproducing comprehensive static profiles that maintain semantic\nconsistency with human cognitive patterns.\nLong-term preference profile refers to stable and long-term in-\ntentions derived from long-term behaviors, which focuses more\non stability, accuracy, and long-term relevance, requiring a more\nrigorous reasoning and calibration process [39, 46].\nPeriodic demand profile identifies recurring consumption pat-\nterns that exhibit temporal periodicity correlated with interval cy-\ncles (e.g., weekly/monthly), holiday events, and workday rhythms.\nWe employs a LLM to summarize the extended behavioral sequences\nthrough scheduled interval sampling, implementing an automated\nperiodicity and long-term preference detection mechanism.\n4.2.2\nCognitive-aligned Data Generation. Within the vast amount\nof POI check-in data left by users, a significant portion is considered\nnoise (such as accidental clicks), or does not align with the human\ncognition process. Therefore, we intervene in data collection by\ncombining the world knowledge of LLMs and human annotations.\nFollowing similar process of [39], we collect the data from tradi-\ntional CTR-oriented recommender systems that align with human\ncognition preference defined in Section 3.\nIn particular, LLMs is utilized to access the users’ historical check-\nins, profile information and other contextual information (like time\nand weather) to conduct chain-of-thought (CoT) reasoning [35],\nthereby evaluating the potential candidate items aligned with hu-\nman cognition for individual users. Subsequently, human annota-\ntors review the reasoning process and outcomes generated by the\nLLMs to refine and improve the accuracy of serendipity labels. This\nannotated dataset—which contains user history, profiles, candidate\nitems, contextual conditions, and cognition annotations—is then\nutilized to further fine-tune the model after continued pretraining\nphase that can identify the examples aligned with human cogni-\ntion for the subsequent training. This approach enables scalable\ndiscovery of cognitive-aligned items while reducing the need for\nextensive manual annotation.\n4.2.3\nSupervised Fine-tuning. After obtaining cognitive-aligned\ndata, with each record consisting of user𝑢, their multi-level profiles,\nhistorical POI check-ins ℎ𝑢, the current situation of the user (time,\nlocation, weather and so on) and the selected cognitive-aligned POI\n𝑝𝑖, we begin the process of fine-tuning by aligning the inputs. Ini-\ntially, all of the inputs are transformed into a text-based supervised\nfine-tuning (SFT) dataset, where each record is a pair (𝑥,𝑦), with\n𝑦denoting the output SID of the target POI 𝑝𝑖, and 𝑥as the input\nprompt. For example,\nInstruction:\nBased on the user’s profile: {user-profile}, and his historical\nPOI check-ins: {check-ins}. Now the time is {time}, he is in\n{user-location}, and the weather is {weather}. Please recom-\nmend the next possible POI that the user may expect?\nResponse:\n\nSubsequently, LLM is fine-tuned under supervision to enhance\nits ability to produce the desired responses.\n4.2.4\nCognitive Preference Alignment. Following the standard par-\nadigm for human preference alignment, we continue to leverage\nReinforcement Learning (RL) to further refine the model’s outputs\nand explicitly align its recommendations with human cognitive\npreferences.\nAfter obtaining datasets comprising human-annotated prefer-\nences over pairs of model-generated responses, we train the model\nwith the preference alignment algorithm. We choose Direct Prefer-\nence Optimization (DPO) [28] in this phase, since it has been widely\nemployed to align the model’s outputs more closely with human\npreference.\nL = −E(𝑥,𝑦𝑤,𝑦𝑙)∼𝐷\n\u0014\nlog𝜎\n\u0012\n𝛽𝜋𝜃(𝑦𝑤| 𝑥)\n𝜋ref(𝑦𝑤| 𝑥) −𝛽𝜋𝜃(𝑦𝑙| 𝑥)\n𝜋ref(𝑦𝑙| 𝑥)\n\u0013\u0015\n(5)\nwhere 𝐷denote the preference dataset, 𝑥denotes the input prompt,\nand 𝑦𝑤and 𝑦𝑙represent the preferred and dis-preferred responses\nannotated by human and LLM. 𝜋𝜃is the trainable LLM for pref-\nerences alignment, while 𝜋ref is a fixed reference policy. 𝜏is the\nhyper-parameter of the regularization term balancing learning hu-\nman preferences and staying close to the reference policy, thereby\npreventing overfitting.\nIn summary, after the Supervised Fine-Tuning (SFT) phase, our\nmodel adheres to the required output format (a list of 𝐿valid SIDs)\nand already possesses a baseline level of cognitive awareness from\nthe training data. The subsequent DPO phase then significantly\n\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nZhai et al.\nenhances this cognitive reasoning while preserving the model’s\nrecommendation performance.\n4.3\nOnline Deployment\nA primary challenge for the large-scale application of generative\nmodels is their inference latency, particularly in recommendation\nscenarios with high Queries Per Second (QPS). To enable the large-\nscale online deployment of our generative model, inspired by recent\nworks [13, 52], we implemented a dual-pronged optimization strat-\negy targeting both system architecture and algorithmic efficiency,\nachieving significant breakthroughs in stability and performance.\n4.3.1\nPrefill-Decoding Decoupling. We employ Prefill-Decoding\n(PD) Decoupling [52] to separate the inference process into two\ndistinct stages: a computationally-intensive Prefill stage for prompt\nprocessing and a memory-bound Decoding stage for iterative token\ngeneration. By applying dedicated optimization strategies to each\nstage, this approach mitigates resource contention and latency\njitter common in unified pipelines. The resulting improvements in\nresource scheduling and concurrency led to a 10-fold increase in\nthroughput, drastically reducing the per-inference cost on the same\nhardware.\n4.3.2\nMulti-Token Prediction. Furthermore, we leverage Multi-Token\nPrediction (MTP) to overcome the sequential dependencies of tra-\nditional auto-regressive generation [13]. Specifically, we adopt\nMTP Eagle, an optimized variant of the Eagle speculative decoding\nmethod. Its key innovation lies in a single, reusable MTP module\nthat shares the same KV cache for predicting multiple draft tokens,\nthus eliminating the redundant computational overhead of vanilla\nMTP. Its chained decoding strategy, which only requires the last\npredicted token and its hidden state to proceed, significantly sim-\nplifies implementation by obviating the need to store historical\nstates. MTP Eagle is fully compatible with existing checkpoints\n(e.g., DeepSeek-V3) and substantially reduces both memory foot-\nprint and computational complexity, enhancing the engineering\nfeasibility and resource efficiency of our deployment.\nCollectively, these optimizations create a highly efficient and\nstable inference framework. This is crucial for deploying large-scale\ngenerative models in production, ensuring both a responsive user\nexperience and the economic viability of the system in a real-world\nadvertising setting.\n5\nExperiments\n5.1\nExperimental Settings\n5.1.1\nScenario. Our method is performed on AMAP, a prominent\nnavigation and mapping platform hosting billions of users and POIs.\nWe focus on the \"Guess Where You Go\" card presented on the\nhomepage of the AMAP App, aiming to forecast the next subse-\nquent POI using a user’s previous check-ins and profile information.\nThe objective of this scenario is to predict the next POI the users\nare most likely to visit. By displaying the POIs on the homepage,\nwe aim to reduce the users’ search effort, thereby enhancing user\nexperience and long-term retention. Critically, our goal transcends\nmere prediction accuracy; we strive to build a model that captures\nthe underlying dynamics of user mobility patterns and intents. This\nallows the system to develop a deeper \"cognitive\" understanding,\nultimately facilitating more intelligent and context-aware decision-\nmaking for the user.\nThe current deployed framework, which serves as our primary\nbaseline, employs a traditional cascaded ranking system for recom-\nmendation. As illustrated in Figure 2, the cascaded ranking system\nincludes three stages: (1) Retrieval stage to recall a coarse-grained\nPOI corpus based on the user’s preference, historical check-ins, and\nlocation [18, 44]; (2) Ranking stage to obtain top 10 POIs the users\nmost likely visit with a ranking model based on behavior sequence\ntransformer [3]; and (3) Cognitive Re-ranking stage with distance,\nquality, and exposure filters to yield the most cognitively plausible\ndestination.\nFigure 2: Cascade ranking system of \"Guess Where You Go\"\ndeployed on AMAP, which includes three stages from the left\nto the right: Retrieval, Ranking, and Cognitive Re-ranking.\nWe evaluate our approach on three real-world datasets: Foursquare-\nNYC [42], Foursquare-TKY [42], and Gowalla-CA [4] and a large-\nscale proprietary dataset by sampling desensitized users from AMAP\n(named Beijing). The details of the utilized dataset is illustrated in\nAppendix A.\n5.1.3\nImplementation Details. The quantization module consists\nof 3 codebook layers and for the generative recommendation mod-\nule, we employ the Qwen [27, 41] as the base model. We employ a\nconstant learning rate schedule with a learning rate of 1e-5, com-\nbined with a warm-up phase of 20 steps. The model is trained on\n200× NVIDIA H20 GPUs, with a batch size of 16 per GPU, gradient\naccumulation steps set to 8, and a sequence length of 4096 tokens.\nEach input consists of the most recent 50 check-in records of the\nuser.\n5.1.4\nEvaluation Metric. Following previous works, we employ the\naccuracy metric for the top-1 recommendation accuracy (Acc@1)\nand the four cognitive metrics (TCS, SCS, PAS and SAS) defined\nin 3.2 as our metrics to evaluate the effect of the cognitive recom-\nmendation. Detailed information of the metrics is illustrated in\nAppendix C.\n5.1.5\nBaseline Methods. The details of the baseline methods in this\npart is illustrated in Appendix B.\n5.1.6\nOur Models. In our experiments we consider three versions\nof our model: (i) CoAST-0.5B: the model trained with Qwen2.5-0.5B\n[27] as our base LLM. (ii) CoAST-1.5B: with Qwen2.5-1.5B as our\nbase LLM. (iii) CoAST-3B: A variation with Qwen2.5-3B as our base\nLLM. (iv) CoAST-7B: A second variation on Qwen2.5-7B. Below,\nunless stated otherwise, when we say “our model,” we refer to the\nCoAST-0.5B.\n\nCognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nTable 2: Comparison of different models on three datasets:\nNYC, TKY, and CA. We present the model inputs including\nthe POI and User representation approach and whether visit\ntimestamps are utilized. RID, SID, UID and UP represent\nRandom one-hot ID, Semantic ID, User one-hot ID and User\nProfiles, respectively. The results for the baseline methods\nare borrowed from [19, 32].\nModel\nInputs\nAcc@1\nPOI\nUser\nTime\nNYC\nTKY\nCA\nPRME\nRID\nUID\n×\n0.1159\n0.1052\n0.0521\nPLSPL\nRID\nUID\n×\n0.1917\n0.1889\n0.1072\nSTAN\nRID\nUID\n×\n0.2231\n0.1963\n0.1104\nGETNext\nRID\nUID\n×\n0.2435\n0.1829\n0.1357\nSTHGCN\nRID\nUID\n×\n0.2734\n0.2950\n0.1730\nTPG\nRID\nUID\n✓\n0.2555\n0.1420\n0.1749\nROTAN\nRID\nUID\n✓\n0.3106\n0.2458\n0.2199\nLLM4POI\nRID\nUID\n✓\n0.3372\n0.3035\n0.2065\nGNPR-SID\nSID\nUID\n✓\n0.3618\n0.3062\n0.2403\nCoAST-0.5B\nSID\nUP\n✓\n0.3705\n0.3125\n0.2555\nCoAST-1.5B\nSID\nUP\n✓\n0.3727\n0.3170\n0.2585\nCoAST-3B\nSID\nUP\n✓\n0.3872\n0.3202\n0.2611\nCoAST-7B\nSID\nUP\n✓\n0.4027\n0.3310\n0.2721\nTable 3: Comparison results of different versions of CoAST\non Beijing dataset with the Cascade Ranking baseline system.\n’a-’ denotes average.\nBaseline\nCoAST-0.5B\nCoAST-1.5B\nCoAST-3B\nCoAST-7B\nAcc@1\n0.2675\n0.2820\n0.2862\n0.2923\n0.3071\na-TCS\n0.5561\n0.6071\n0.6021\n0.6632\n0.7543\na-SCS\n0.8022\n0.8212\n0.8231\n0.8452\n0.8838\na-PAS\n0.6232\n0.6285\n0.6240\n0.6531\n0.6970\na-SAS\n0.2472\n0.2840\n0.2988\n0.3240\n0.3725\n5.2\nOffline Experiments\n5.2.1\nOverall Performance. Our method substantially outperforms\nall baseline methods across all of the evaluated datasets. Table 2\nillustrate the comparison results on NYC, TKY, and CA datasets.\nSpecifically, compared to the state-of-the-art GSNR-SID which uti-\nlizes LLaMA-7B as its base model, we observe a substantial im-\nprovement on all of the versions of CoAST. Notably, CoAST-7B\ndemonstrated the most significant improvements, achieving gains\nof 11.3%, 8.1%, and 13.2% over the SOTA, respectively.\nTable 3 details the performance comparison between our pro-\nposed model and the cascade ranking system on the Beijing dataset.\nIn addition to the standard Acc@1 metric, we evaluate the models\non four cognitive-aware recommendation metrics (where ’a-’ de-\nnotes average). Consistent with our findings on other datasets, our\nmodel significantly outperforms the traditional cascade ranking\nsystem on Acc@1. More critically, it demonstrates substantial gains\non the cognitive metrics, achieving remarkable uplifts of 35.6% in\ntemporal consistence and 50.68% in situational awareness perfor-\nmance. These results highlight a key limitation of conventional\ncollaborative filtering-based systems: their struggle to capture per-\nsonalized, periodic user needs and inability to effectively model\nsituational or contextual information. This validates the superiority\nof our generative approach, which excels in understanding these\ncomplex, dynamic aspects of user behavior.\n5.2.2\nAblation Study. To evaluate the impact of individual modules,\nwe developed various versions of CoAST. Initially, certain essential\nmodules were removed: \"w/o UP\" eliminates user profiling from\nthe prompt. \"w/o S-ST\" eliminates situational and spatio-temporal\ndescriptions from the prompt. \"w/o SLD\" exclude the SID-Location-\nDescription alignment corpus in the continued pretraining stage.\n\"w/o CPT\" exclude the Continued Pretraining stage in the align-\nment stage, only applying SFT and DPO. As illustrated in Table 4,\nremoving any module leads to noticeable performance degradation,\nemphasizing the importance of each component. Significantly, the\nlack of SFT results in the most considerable decrease in all met-\nrics, demonstrating the essential role of preference alignment in\nrecommendation efficiency.\nTable 4: Ablation study of CoAST (with the online deploy-\nment version CoAST-3B).\nAcc@1\na-TCS\na-SCS\na-PAS\na-SAS\nw/o UP\n0.2720\n0.6435\n0.8211\n0.5920\n0.2923\nw/o S-ST\n0.2708\n0.5905\n0.7938\n0.6496\n0.2822\nw/o SLD\n0.2825\n0.6078\n0.8240\n0.6501\n0.3020\nw/o CPT\n0.2632\n0.5875\n0.7820\n0.6020\n0.2875\nCoAST\n0.2923\n0.6632\n0.8452\n0.6531\n0.3040\n5.2.3\nCase Study. To provide a qualitative and intuitive demon-\nstration of our model’s advantages, we conduct a case study that\ncontrasts the recommendation lists generated by our model with\nthose from a baseline model for a representative user. Detailed\ninformation of the case is illustrated in Appendix D.\n5.2.4\nFurther Analysis. In this section, we analyze the impact on\nthe performance of sequence length (input tokens) and number of\nutilized tokens in continued pretraining stage. Figure 3 presents the\nresults. Our investigation into the impact of input sequence length\nshows that while longer context is beneficial, its utility eventually\nsaturates. The model’s performance gains are most pronounced\nup to a context window of 2048 tokens (∼30 user historical check-\nins) and largely stabilize beyond 4096 tokens (∼50 check-ins). This\nsuggests that a moderately-sized history is sufficient to encapsulate\nthe user’s core preferences. Further extending the context window\nprovides limited new signal, as this long-term information is already\nimplicitly encoded in the user’s learned representation.\nThe volume of tokens involved in pretraining phase also proved\nto be a key determinant of performance. We observed a nearly linear\nimprovement in model effectiveness as the training corpus size\nincreased. Constrained by our training budget, the final model was\ntrained to convergence on a dataset of 10B tokens. The consistent,\n\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nZhai et al.\nnon-saturating growth suggests that performance is largely data-\nbound, with significant potential for improvement as more data\nbecomes available.\nFigure 3: Impact on the performance of sequence length\n(input tokens) and number of tokens involved in contin-\nued pretraining stage. a-CAS denotes the average Cognition\nAlignment score computed by averaging the four cognitive\nscores.\n5.3\nOnline Experiments\n5.3.1\nEfficiency Analysis. To evaluate the online inference effi-\nciency, we compare different variants of our model with the efficiency-\nSOTA cascade ranking system. Before deploying our model, we con-\nduct a load test under the settings with a 50 QPS on 2× NVIDIA H20\nGPUs. Figure 4 presents the inference p99 latency results of different\nmethods. As shown, the Cascade Ranking system, a widely adopted\nparadigm in industrial recommender systems, demonstrates ex-\nceptional efficiency with its latency consistently around 20ms. As\na comparison, while CoAST-7B achieved the best offline perfor-\nmance among all our variants, its high computational cost posed\na significant challenge for online serving. Even after applying the\nacceleration strategies described in Section 4, its latency remained\nabove 120ms, rendering large-scale deployment infeasible.\nTo strike a practical balance between efficacy and efficiency, we\nidentified CoAST-0.5B as the optimal candidate. It delivers compet-\nitive performance while maintaining a manageable inference cost\n(∼30ms). Consequently, we deployed CoAST-0.5B to production\nand conducted a online A/B test against the state-of-the-art (SOTA)\nCascade Ranking baseline.\n5.3.2\nOnline A/B Test. To assess the online effect of CoAST, we\nimplemented a short-term and long-term A/B testing on \"Guess\nFigure 4: Comparison of Model Inference p99 Latency (ms)\nunder a 50 QPS Load Test on 2× NVIDIA H20 GPUs.\nWhere You Go\" card of AMAP’s homepage. We examined CoAST-\n0.5B, trained based on a Qwen2.5-0.5B model, compared to the\ntraditional cascade ranking baseline as shown in Table 5.\nTable 5: Online improvement over traditional Cascade Rank-\ning baseline of short and long term A/B test. % denotes rela-\ntive improvement. short and long represents short-term and\nlong-term A/B online test, respectively.\nP-CTR U-CTR SC-Rate RE-Rate AC-Rate NF-Rate\nshort +5.06% +4.65%\n+0.69%\n+0.31%\n+0.77%\n-5.11%\nlong\n+4.23% +4.59%\n+1.01%\n+1.65%\n+5.99%\n-5.85%\nIt is worth mentioning that this baseline already includes user\nprofiles, trending topics and seasonal features for improving cogni-\ntion in the cognitive re-ranking stage. For short-term (one week)\nonline test, CoAST demonstrate significant improvements in met-\nrics like P-CTR (PV CTR) and U-CTR (UV CTR). This indicates that\nCoAST has increased the share of cognitive-aligned recommen-\ndations that effectively attracted user attention in terms of clicks.\nAdditionally, it slightly enhances the user engagement metrics,\nsuch as SC-Rate (Scroll Rate), RE-Rate (User Retention Rate), and\nAC-Rate (User Active Rate), implying heightened user involvement.\nNotably, the effects on general utility metrics such as CTR and NF-\nRate (Users’ Negative Feedback Rate) are more modest, while the\nimprovements on these engagement-related metrics often become\napparent only over the long term.\nTo investigate the long-term effects of the proposed method\nin large-scale RSs, we maintain a small traffic for our baselines\nand compare it with CoAST for over one month. Compared to the\nbaseline and the performance on the short-term test, our method\nhas achieved a significant enhancement in the user engagement\nmetrics, with a 5.85% average decrease in NF-Rate (Users’ Negative\nFeedback Rate). This demonstrates that CoAST could help to miti-\ngate counter-intuitive recommendations and effectively improve\nuser engagement while also boosting CTR, thereby enhancing the\noverall user experience.\n6\nConclusion\nIn this paper, we proposed a LLM-based next POI prediction ap-\nproach, named CoAST. In order to adapt LLMs to sequential recom-\nmendation tasks, we focused on a comprehensive pretraining task\nto adapt a base LLM to align language and collaborative semantics\nfor recommendation. This task include two categories of corpus:\nsequential POI prediction and explicit index-language alignment.\nBased on the learned item indices, our approach employed these\nalignment tasks to effectively adapt LLMs for sequential recom-\nmendation. Furthermore, we give complete definition and metrics\non cognitive-aligned next POI prediction task. For constructing\na cognitive-aligned POI recommendation system in industrial ap-\nplication, we design a multi-stage fine-tuning process, utilizing\na curated dataset of cognitively-aware samples for both Super-\nvised Fine-Tuning (SFT) and a subsequent Reinforcement Learning\n(RL) phase. This procedure explicitly align the model with human\ncommon-sense. Extensive offline experiments on three datasets\ndemonstrated the effectiveness of our approach and Online A/B\n\nCognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\ntest achieves substantial improvements compared to traditional\ncascade ranking recommendation system.\nAcknowledgments\nWe would like to acknowledge the discussions of Jiawei Xue, Ning\nWang, and Fangfang Chen throughout this work. 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OneRec\nTechnical Report. arXiv:2506.13695 [cs.IR] https://arxiv.org/abs/2506.13695\n[54] Jieming Zhu, Mengqun Jin, Qijiong Liu, Zexuan Qiu, Zhenhua Dong, and Xiu Li.\n2024. CoST: Contrastive Quantization based Semantic Tokenization for Genera-\ntive Recommendation. In Proceedings of the 18th ACM Conference on Recommender\nSystems (Bari, Italy) (RecSys ’24). Association for Computing Machinery, New\nYork, NY, USA, 969–974. doi:10.1145/3640457.3688178\n\nCognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nFigure 5: Case comparison of recommendation results of CoAST and the traditional basline RSs. The top row represents the\nuser’s real-time check-ins with different action types. The middle and bottom row represent the recommendation results of\ntraditional basline RSs and CoAST, respectively.\nA\nDataset\nTo validate our model’s performance in a real-world application,\nwe constructed a large-scale proprietary dataset by sampling de-\nsensitized users from AMAP (named Beijing). On this dataset, we\nconducted both extensive offline evaluations and an online deploy-\nment. Furthermore, to ensure a comprehensive and fair comparison\nagainst existing baselines, we also evaluate our approach on three\nreal-world datasets: Foursquare-NYC [42], Foursquare-TKY [42],\nand Gowalla-CA [4].\nThe pre-processing follows the settings in [32]. we pre-process\neach dataset by removing POIs with fewer than 10 interactions\nand users with fewer than 10 check-ins. The data is then sorted by\ntime, with 80% used for training, 10% for validation, and 10% for\ntesting. Users and POIs that do not appear in the training set are\nremoved from the test set to ensure consistency between training\nand evaluation. Subsequently, the check-in records are grouped by\nuser and ordered chronologically. For each user, the last visited POI\nis held out as the ground truth during evaluation, while the preced-\ning visits are used as the input sequence. Note that in the training\nset, the user’s visit records are also treated as historical sequences\nand concatenated with the test set to the specified length before\nbeing fed into the model. Detailed dataset statistics are provided in\nTable 6.\nTable 6: Statistics of the preprocessed datasets. Avg.len de-\nnotes the average length of item sequences.\nDataset\n#Users\n#Items\n#Inter.\nAvg.len\nSparsity\nNYC\n1,083\n5,135\n104,074\n136\n98.1013%\nTKY\n2,293\n7,873\n361,430\n195\n97.9798%\nCA\n6,592\n14,027\n349,375\n53\n99.9996%\nBeijing\n10,000k\n1,520k\n38,0000k\n38\n99.9999%\nB\nBaseline\nFollowing the settings of [19, 32], we compare our proposed method\nwith various next POI recommendation baselines, which can be\nspecified as the following four categories: (1) Traditional meth-\nods: PRME [9], a ranking-based metric embedding method; and\nPLSPL [38], a personalized sequential recommendation model. (2)\nTransformer-based methods: STAN [25], a spatio-temporal atten-\ntion network; and GETNext [43], a Transformer-based next POI\nrecommendation model. (3) GCN-based method: STHGCN [40],\na spatio-temporal hierarchical graph convolutional network. (4)\nTime-aware methods: TPG [24], a timestamp-guided model; and\nROTAN [10], a time-aware POI recommendation framework. (5)\nLLM-based method: LLM4POI [19], a method based on LLMs; and\nGNPR-SID [32], a state-of-the-art based on Semantics codes and\ngenerative recommendation framework, which is pretty closely\nrelated to our work.\nC\nEvaluation Metric\nFollowing previous works, we employ the accuracy metric for the\ntop-1 recommendation, denoted as Acc@1, at the specified time.\nAcc@1 = 1\n𝑛\n𝑛\n∑︁\n𝑖=1\nI(𝑦𝑖= ˆ𝑦𝑖)\nwhere 𝑛is the number of test samples, 𝑦𝑖and ˆ𝑦𝑖represent the\nground-truth label and the predict label of the 𝑖-th sample, respec-\ntively. I is the indicator function, which equals 1 when the condition\nis true and 0 otherwise. Otherwise, we take the four cognitive cat-\negories (TCS, SCS, PAS and SAS) defined in 3.2 as our metrics to\nevaluate the effect of the cognitive recommendation.\nD\nCase Study\nFigure 5 illustrates the case comparison of recommendation results\nof CoAST and the traditional basline RSs. In the traditional cas-\ncade ranking RSs, the model collapses to the user’s dominant daily\nrhythm: go to company at morning and back to home at afternoon.\nHowever, during other times of the day, conventional CTR-oriented\nmodels tend to be dominated by users’ most recent interactions.\nThis short-term focus often leads to recommendations that fail\nto capture the user’s broader, long-term interests, resulting in a\ncounter-intuitive user experience.\n\nConference acronym ’XX, June 03–05, 2018, Woodstock, NY\nZhai et al.\nIn contrast, our proposed generative model achieves a better\nalignment of the user’s temporal context with their holistic prefer-\nence profile, thereby fostering a more well-rounded and sustainable\nrecommendation ecosystem.\n"},"cleaned":{"kind":"string","value":"Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Penglong Zhai∗ AMAP, Alibaba Group Beijing, China Jie Li∗ AMAP, Alibaba Group Beijing, China Fanyi Di∗ AMAP, Alibaba Group Beijing, China Yue Liu AMAP, Alibaba Group Beijing, China Yifang Yuan AMAP, Alibaba Group Beijing, China Jie Huang AMAP, Alibaba Group Beijing, China Peng Wu AMAP, Alibaba Group Beijing, China Sicong Wang AMAP, Alibaba Group Beijing, China Mingyang Yin AMAP, Alibaba Group Beijing, China Tingting Hu AMAP, Alibaba Group Beijing, China Yao Xu AMAP, Alibaba Group Beijing, China Xin Li† AMAP, Alibaba Group Beijing, China Abstract The next point-of-interest (POI) recommendation task aims to predict the users' immediate next destinations based on their preferences and historical check-ins, holding significant value in locationbased services. Recently, large language models (LLMs) have shown great potential in recommender systems, which treat the next POI prediction in a generative manner. However, these LLMs, pretrained primarily on vast corpora of unstructured text, lack the native understanding of structured geographical entities and sequential mobility patterns required for next POI prediction tasks. Moreover, in industrial-scale POI prediction applications, incorporating world knowledge and alignment of human cognition, such as seasons, weather conditions, holidays, and users' profiles (such as habits, occupation, and preferences), can enhance the user experience while improving recommendation performance. To address these issues, we propose CoAST (Cognitive-Aligned Spatial-Temporal LLMs), a framework employing natural language as an interface, allowing for the incorporation of world knowledge, spatio-temporal trajectory patterns, profiles, and situational information. Specifically, CoAST mainly comprises of 2 stages: (1) Recommendation ∗Contributed equally to this research. †Corresponding Author. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from . Conference acronym 'XX, Woodstock, NY Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-1-4503-XXXX-X/2018/06 https://doi.org/XXXXXXX.XXXXXXX Knowledge Acquisition through continued pretraining on the enriched spatial-temporal trajectory data of the desensitized users; (2) Cognitive Alignment to align cognitive judgments with human preferences using enriched training data through Supervised FineTuning (SFT) and a subsequent Reinforcement Learning (RL) phase. Extensive offline experiments on various real-world datasets and online experiments deployed in \"Guess Where You Go\" of AMAP1 App homepage demonstrate the effectiveness of CoAST. CCS Concepts • Information systems →Location based services. Keywords Generative Recommendation, Next POI Prediction, Cognitive Recommendation, Large Language Models ACM Reference Format: Penglong Zhai, Jie Li, Fanyi Di, Yue Liu, Yifang Yuan, Jie Huang, Peng Wu, Sicong Wang, Mingyang Yin, Tingting Hu, Yao Xu, and Xin Li. 2018. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Pointof-Interest Prediction. In Proceedings of Make sure to enter the correct conference title from your rights confirmation email (Conference acronym 'XX). ACM, New York, NY, USA, 12 pages. https://doi.org/XXXXXXX.XXXXXXX 1 Introduction Next Point-of-Interest (POI) prediction, as a variant of POI recommendation task, aiming to predict users' future locations based on their historical sequential check-ins and other contextual information, is a fundamental task in Location-Based Services (LBS) [9, 10, 19, 24, 25, 32, 36, 38, 40, 43]. For large-scale navigation platforms such as AMAP, which processes billions of daily requests 1https://amap.com/ 16 Oct 2025 Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. from hundreds of millions of active users in China, making accurate next POI prediction a critical capability for efficient destination planning and personalized recommendation. Thus, next POI prediction has been a popular research direction, capitalizing on developments in mobile and localization techniques, as they provide rich location-based information. Much of the existing research in next POI recommendation has predominantly focused on sequential recommendation and graph neural networks, tackling the challenge of data sparsity, particularly for cold-start users and those with sparse historical trajectories [8, 29, 33, 45, 48]. However, these approaches often fail to encapsulate the rich contextual information embedded within Location-Based Social Network (LBSN) data, such as the world knowledge and contextual information, which are critical for improving the user experience in industrial applications. Recently, Large Language Models (LLMs) have demonstrated remarkable capabilities across various domains [1, 27, 31, 41], offering new opportunities for this direction. However, their direct application to POI prediction task remains several key challenges. First, POIs are typically indexed by discrete identifiers outside the LLM vocabulary, impeding the model's ability to capture semantic relationships [2, 47, 50]. Second, the intricate interactions between the billions of users and POIs, coupled with highly personalized and context-dependent behavioral patterns, introduce complex collaborative signals that are difficult for LLMs to encode and reason effectively [23, 25]. Another critical challenge, stemming from the real-world deployment, is that recommended POIs must be cognitively plausible to users. Recommendations that violate human intuition or commonsense reasoning, which we term human cognitive preference in this work, even if statistically likely, can appear nonsensical or irrelevant, severely degrading the user experience and eroding trust in the system [25]. In industrial applications, these logical lapses are critical failures, as they directly risk frustrating users and reducing retention. Based on these observations, we formulate the task of human cognitive preference alignment for individual POI prediction as: to investigate issues of personality and individual differences (who), environmental and social impacts (what), spatio-temporal features and relationships (when and where), and corresponding actions (how). To address the aforementioned challenges and build a POI recommendation system that aligns with human cognitive preferences, we introduce CoAST (Cognitive-Aligned Spatio-Temporal LLM), a novel framework that endows a base LLM with the ability to generate cognitively plausible POI recommendations. CoAST begins by adapting the LLM to the mobility domain through continued pretraining on an enriched corpus of spatio-temporal tokenized POI sequences, allowing it to master complex user behavior patterns. Furthermore, we explicitly align the model with human commonsense. This is achieved through a multi-stage fine-tuning process, utilizing a curated dataset of cognitively-aware samples for both Supervised Fine-Tuning (SFT) and a subsequent Reinforcement Learning (RL) phase. Our main contributions as as follows: • To our knowledge, this is the first generic LLM-based cognitivealigned framework for the field of next POI prediction. The proposed framework enhances the comprehension of human cognitive preference in recommendation. • We propose CoAST, a novel end-to-end LLM-based framework for large-scale next POI prediction that effectively bridges the semantic gap between world and recommendation knowledge, which demonstrates significant improvement on human cognitive alignment and applicable to practical applicability of LLMs in LBS applications. • Through a series of inference optimization strategy, CoAST is deployed in AMAP's online production environment, a leading real-world LBS platform. Compared traditional cascade ranking system, long term online A/B test achieves a 4.23% and 4.59% increase in P-CTR and U-CTR, which is a substantial improvement. The remainder of this paper is organized as follows: Section 2 reviews related literature, Section 3 details problem definition and the detailed categories of cognitive recommendation, Section 4 elaborates our methodology, Section 5 reports empirical results, and Section 6 concludes the paper. 2 Related Work 2.1 Next POI Prediction Next POI recommendation extends beyond conventional item recommendation by incorporating inherent geographical and temporal characteristics. Early methods relied on user check-in histories using Markov chains, or RNNs to capture sequential behaviors. Spatial-temporal modeling has been extensively studied with works like ST-GRAT [26] employing graph attention networks for spatial-temporal dependencies, STAN [25] introducing hierarchical attention mechanisms, and Sun et al. [30] focusing on periodic patterns in human mobility through time-aware embeddings. Recent LLM-based approaches have focused on addressing the semantic gap in POI recommendation. For example, GNPR-SID [32] proposed semantic POI IDs using residual quantized variational autoencoders, achieving improvements by mapping POIs into discrete semantic spaces. LLM4POI [19] leveraged pretrained LLMs to preserve heterogeneous data in original format, effectively utilizing contextual information. 2.2 LLMs for Recommendation The integration of Large Language Models (LLMs) into recommendation systems has emerged as a transformative research direction. P5 [12] pioneered the \"Pretrain, Personalized Prompt, and Predict\" paradigm, establishing the foundation for treating recommendation as a language processing task by converting all recommendation data into natural language sequences. Building upon this, [2] addressed the knowledge gap between LLMs' capabilities and recommendation-specific requirements by incorporating collaborative filtering operations like Masked Item Modeling (MIM) and Bayesian Personalized Ranking (BPR) through natural language simulation. More recently, GenRank [17] advanced generative ranking systems for large-scale deployment. OneRec [5, 53] proposed a revolutionary end-to-end framework that replaces traditional cascaded architectures, achieving significant improvements in large-scale online deployments. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY 3 Preliminaries 3.1 Problem Formulation Given a collection of POIs I = {p1, p2, ..., pN} and a group of users U = {u1,u2, ...,uM}. Each POI can be characterized by a tuple pi= (lon,lat,c), where crepresents side information of the POI including the descriptive texts and pictures, lonand latdenotes the geographical coordinates (longitude and latitude). Each user u has the check-ins history hu= {(p1,t1), . . . , (pm,tm)} of length m, where pi,i= 1, . . . ,mreflects the i-th POI recently interacted by the user and tirefers to the check-in timestamps. The objective of next POI prediction task is to predict the next POI pthat the user u will visit at time tgiven the recent check-in record hu, the user's request location land the contextual situation s. 3.2 Cognitive-Aligned POI Prediction Compared to traditional accuracy-oriented recommender systems that rely on rapidly capturing users' immediate interests, next POI prediction is a complex issue in simulating individual behaviors and contextual information. Empirically, we category cognitive-aligned recommendations into the following four scenarios. Recommended POIs that do not align with these categories may result in a diminished user experience or decreased user engagement. To summarize, our approach formulates the next POI recommendation task as a decision-making problem. The model synthesizes a diverse set of contextual factors-including temporal (when), spatial (where), user-specific (who), and other situational inputs-to determine the optimal user action (what to do and how). 3.2.1 Temporal Consistency. User behavior often exhibits certain regularities. For example, office workers typically commute to workplaces in the morning and return home at noon. Recommending irrelevant POIs during these commuting periods may lead to poor user experiences. We define these observed regularity patterns in user mobility as temporal consistency. Formally, at the given time t, the temporal consistency score (denoted as TCS) of the predicted POI ˆpcan be formulated as, TCS(u,l,s,t, ˆp) = Í i=mI(ti= t) · sim(pi, ˆp∗) Ím i=1 I(ti= t) (1) where sim(·) denotes the function measuring the similarity of the inputs pair. 3.2.2 Spatial Clustering. Check-ins of human are empirically demonstrated to occur within a certain range [20, 21]; that is, the distance of a user's next visit POI is typically distributed as a negative exponential relative to the user's usual activity space. Recommending a POI that is far from the user's position (or their active area) is inappropriate. We define the spatial clustering phenomenon of user mobility as spatial consistency. Formally, given the location puof the user u, the spatial consistency score of the predicted POI ˆpis formulated as, SCS(u,l,s,t, ˆp) = e-d(pu, ˆp)/du (2) where edenotes the exponential function, d(·) is the distance function, and durepresents the user's historical average movement distance. 3.2.3 Profile Alignment. Behaviors of human often correlate with their profile tags. For instance, suggesting family-oriented venues or restaurants to university students represents a cognitive mismatch in recommendation logic. We define the user profile alignment score to reflect the alignment between recommended item ˆpand user characteristics: PAS(u,l,s,t, ˆp) = LLMu(u, ˆp) (3) where a LLM model is utilized to evaluate the reasonableness of the recommended POI ˆpgiven the profile of user uand the score ∈{0, 1}. 3.2.4 Situational Awareness. Situations such as weather, traffic, and the conditions of the POIs could have a significant impact on human's traveling decisions. For example, recommending an outdoor tennis court on a rainy day is considered inappropriate. We define the recommendation system's Awareness to these contextual situations as situational awareness. Similar to definition of profile alignment, we also utilize a LLM to score this metric, SAS(u,l,s,t, ˆp) = LLMs(s, ˆp) (4) where the score ∈{0, 1}. The definitions articulated above serve a dual purpose in our methodology. They directly guide the creation of a training set of cognitively plausible examples for the model's alignment phase. Simultaneously, these definitions are translated into quantitative metrics, providing a principled framework for evaluating the cognitive capabilities of our proposed model. 4 Methodologies The overall framework is presented in Figure 1. The framework consists of two main modules: (1) Spatio-temporal Knowledge Acquisition module enables the LLMs to incorporate the linguistic and collaborative semantics and (2) Cognitive Alignment to adjusting the ST-LLM obtained from step (1) to generate results that better align with human cognition. We finally detail our inference optimization strategy in online deployment since this is a bottleneck of large-scale application of generative models. 4.1 Spatio-temporal Knowledge Acquisition Considering the huge amount of the POI corpus in large industrial LBS applications, we firstly convert POI data into Semantic IDs (SIDs) to reduce the total number of tokens. Building on these SIDs, we propose to enable the LLM to incorporate the recommendation knowledge by continued pretraining on an enriched corpus. 4.1.1 Semantic ID generation. A well crafted SID representation can boost the LLM's recommendation efficiency by capturing the semantic and spatial relationships among POIs. There has already been extensive research in this area [22, 32, 34, 49, 51, 54], so we do not elaborate on it here. In brief, we adopt the RQ-VAE, which adopts the multi-modal information of the POIs to learning the codebooks. Specifically, we incorporate the spatial location information and descriptive titles/pictures of the POIs to generate the identifiers with residual quantization process. As a result, we get a set of codebooks Cl= {el k|k= 1, ..., K},l∈[1, L] Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. Figure 1: Offline Training and Online Service Procedure of proposed method CoAST. Offline Training: (1) POIs are tokenized as Semantic IDs (SIDs); (2) a base LLM is continually pretrained on the collected corpus, yielding ST-LLM; (3) multi-level user profiles are distilled from the global tracking database with the annotations of LLM; (4) cognitive-aligned data are produced through LLM annotation and human screening; (6) ST-LLM is further fine-tuned and aligned to finish this procedure. Online Serving: CoAST is stored in a model engine and invoked through a data-service engine that converts real-time context-user profile, recent check-ins, current time, location, and weather-into an instruction prompt. The model predicts the next POI in SID form, which is mapped back to a readable location and returned to the mobile client. where Lrepresents the number of codebooks and Kdenotes the size of each codebook. Therefore, a POI can be tokenized as an ordered sequence of tokens, such as \" \", in which the number of the above tokens represents the indices of the responding codebook. 4.1.2 Continued Pretraining on Spatio-temporal Corpus. After acquiring SIDs of the POIs, a simple method is to incorporate these SIDs into the vocabulary of large language models (LLMs). This allows the LLMs to perform recommendation tasks in a generative manner, ultimately moving away from using item indices. Nevertheless, there are two remaining concerns: (1) these newly added indices are out-of-vocabulary tokens for LLMs, incorporating these linguistic and collaborative semantics becomes necessary. (2) Recommendation tasks present unique technical challenges that distinguish them from conventional NLP tasks, particularly in modeling spatio-temporal user behaviors and handling data sparsity. To achieve this incorporation, we implemented continued pretraining [14] on the recommendation corpus, aiming to adjust LLMs to improve understanding of linguistic and collaborative semantics. Our approach to recommendation domain-adaptive pretraining is straightforward-we continue pretraining a base model (like Qwen3 [41]) on a large corpus of unlabeled spatio-temporal recommendation domain-specific text. Specifically, the corpus are designed and collected mainly focusing on 2 objectives: clear index-languagespace aligning (finding the POI with its description, location and its category, or outputting the description and location given the POI) and sequential POI prediction (predicting the next POI given the historical POI check-ins). While understanding the spatial relationships between POIs is fundamental to LBS recommendation, directly instilling this knowledge into LLMs is difficult. To address this, our approach tokenizes each POI's Geohash and integrates these tokens into our pretraining data. This process enables the model to learn a unified embedding space where the POI's identity (SID), its semantic meaning, and its location (via Geohash) are jointly represented and aligned. Table 1 illustrates the utilized corpus in this stage. Table 1: Recommendation corpus collected for continuous pretraining based on a base model. The words in orange denote the newly added tokens in the base LLM. SID-Location-Description Alignment Corpus The Temple of Heaven Park is the world's largest surviving ancient architectural complex dedicated to the worship of heaven. Originally constructed during the Ming Dynasty, it served as the site where emperors of the Ming and Qing dynasties held ceremonies to \"sacrifice to heaven\" and \"pray for a good harvest.\" Located in of Beijing, and the ID is . Spatio-Temporal Behavior Sequence Recommendation Corpus Here is user uid_256's historical POI check-ins. At 7 AM June 10, the user ride to company , at 21 PM June 11, the user navigated to the hotel , at 15 PM June 20, the user searched an office ,..., at 19 PM July 20, the user walked to home . Masked prediction has recently been applied to adapt pretrained models to specific downstream tasks by continuing to pretrain models on in-domain unlabelled data [6, 15, 37]. Inspired by these works, we continue our pretraining with masking some original tokens of the base LLM. However, a crucial consideration in our pretraining phase is the token composition of the training corpus, Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY which presents a delicate trade-off between newly introduced POI tokens (SIDs and Geohash tokens) and the original tokens within the pretraining corpus. Through the comprehensive continued pretraining on the corpus of spatio-temporal recommendation domain, the model is adapted to learn information centered around generic behavioral patterns. However, its outputs may exhibit cognitive mis-alignment with human reasoning principles. This necessitates a specialized cognitive calibration process to bridge the divergent representations between machine-learned patterns and anthropic decision-making frameworks. 4.2 Cognitive Preference Alignment 4.2.1 User Profile Generation. Understanding user historical POI check-ins is crucial for modeling the trajectory patterns of users. However, it is challenging to effectively utilize extended user behavior sequences, which often extends to lengths of 10,000 in largescale recommendation systems [11, 16]. Yet, LLMs face difficulties in processing these long sequences, and this hinders their ability to grasp the patterns. To counteract this, we have developed multilevel profile of the users by condensing extensive user behavior into refined profiles that are more amenable to LLMs. The utilized profiles can be mainly divided into three main categories: (1) Static Profile, (2) Long-term Preference Profile, and (3) Periodic Demand Profile. Static profile, such as POIs of the users' home and company, are critical for next POI prediction. However, the labeled data is pretty limited. To expand the coverage of static profiles, we first employ LLM-based summarization of user behavioral sequences to generate preliminary annotations. These labeled patterns are then utilized to train models (like a MOE-Transformer network [7]) that subsequently perform inference across the entire user population, producing comprehensive static profiles that maintain semantic consistency with human cognitive patterns. Long-term preference profile refers to stable and long-term intentions derived from long-term behaviors, which focuses more on stability, accuracy, and long-term relevance, requiring a more rigorous reasoning and calibration process [39, 46]. Periodic demand profile identifies recurring consumption patterns that exhibit temporal periodicity correlated with interval cycles (e.g., weekly/monthly), holiday events, and workday rhythms. We employs a LLM to summarize the extended behavioral sequences through scheduled interval sampling, implementing an automated periodicity and long-term preference detection mechanism. 4.2.2 Cognitive-aligned Data Generation. Within the vast amount of POI check-in data left by users, a significant portion is considered noise (such as accidental clicks), or does not align with the human cognition process. Therefore, we intervene in data collection by combining the world knowledge of LLMs and human annotations. Following similar process of [39], we collect the data from traditional CTR-oriented recommender systems that align with human cognition preference defined in Section 3. In particular, LLMs is utilized to access the users' historical checkins, profile information and other contextual information (like time and weather) to conduct chain-of-thought (CoT) reasoning [35], thereby evaluating the potential candidate items aligned with human cognition for individual users. Subsequently, human annotators review the reasoning process and outcomes generated by the LLMs to refine and improve the accuracy of serendipity labels. This annotated dataset-which contains user history, profiles, candidate items, contextual conditions, and cognition annotations-is then utilized to further fine-tune the model after continued pretraining phase that can identify the examples aligned with human cognition for the subsequent training. This approach enables scalable discovery of cognitive-aligned items while reducing the need for extensive manual annotation. 4.2.3 Supervised Fine-tuning. After obtaining cognitive-aligned data, with each record consisting of useru, their multi-level profiles, historical POI check-ins hu, the current situation of the user (time, location, weather and so on) and the selected cognitive-aligned POI pi, we begin the process of fine-tuning by aligning the inputs. Initially, all of the inputs are transformed into a text-based supervised fine-tuning (SFT) dataset, where each record is a pair (x,y), with ydenoting the output SID of the target POI pi, and xas the input prompt. For example, Instruction: Based on the user's profile: {user-profile}, and his historical POI check-ins: {check-ins}. Now the time is {time}, he is in {user-location}, and the weather is {weather}. Please recommend the next possible POI that the user may expect? Response: Subsequently, LLM is fine-tuned under supervision to enhance its ability to produce the desired responses. 4.2.4 Cognitive Preference Alignment. Following the standard paradigm for human preference alignment, we continue to leverage Reinforcement Learning (RL) to further refine the model's outputs and explicitly align its recommendations with human cognitive preferences. After obtaining datasets comprising human-annotated preferences over pairs of model-generated responses, we train the model with the preference alignment algorithm. We choose Direct Preference Optimization (DPO) [28] in this phase, since it has been widely employed to align the model's outputs more closely with human preference. L = -E(x,yw,yl)∼D logσ βπθ(yw| x) πref(yw| x) -βπθ(yl| x) πref(yl| x) (5) where Ddenote the preference dataset, xdenotes the input prompt, and ywand ylrepresent the preferred and dis-preferred responses annotated by human and LLM. πθis the trainable LLM for preferences alignment, while πref is a fixed reference policy. τis the hyper-parameter of the regularization term balancing learning human preferences and staying close to the reference policy, thereby preventing overfitting. In summary, after the Supervised Fine-Tuning (SFT) phase, our model adheres to the required output format (a list of Lvalid SIDs) and already possesses a baseline level of cognitive awareness from the training data. The subsequent DPO phase then significantly Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. enhances this cognitive reasoning while preserving the model's recommendation performance. 4.3 Online Deployment A primary challenge for the large-scale application of generative models is their inference latency, particularly in recommendation scenarios with high Queries Per Second (QPS). To enable the largescale online deployment of our generative model, inspired by recent works [13, 52], we implemented a dual-pronged optimization strategy targeting both system architecture and algorithmic efficiency, achieving significant breakthroughs in stability and performance. 4.3.1 Prefill-Decoding Decoupling. We employ Prefill-Decoding (PD) Decoupling [52] to separate the inference process into two distinct stages: a computationally-intensive Prefill stage for prompt processing and a memory-bound Decoding stage for iterative token generation. By applying dedicated optimization strategies to each stage, this approach mitigates resource contention and latency jitter common in unified pipelines. The resulting improvements in resource scheduling and concurrency led to a 10-fold increase in throughput, drastically reducing the per-inference cost on the same hardware. 4.3.2 Multi-Token Prediction. Furthermore, we leverage Multi-Token Prediction (MTP) to overcome the sequential dependencies of traditional auto-regressive generation [13]. Specifically, we adopt MTP Eagle, an optimized variant of the Eagle speculative decoding method. Its key innovation lies in a single, reusable MTP module that shares the same KV cache for predicting multiple draft tokens, thus eliminating the redundant computational overhead of vanilla MTP. Its chained decoding strategy, which only requires the last predicted token and its hidden state to proceed, significantly simplifies implementation by obviating the need to store historical states. MTP Eagle is fully compatible with existing checkpoints (e.g., DeepSeek-V3) and substantially reduces both memory footprint and computational complexity, enhancing the engineering feasibility and resource efficiency of our deployment. Collectively, these optimizations create a highly efficient and stable inference framework. This is crucial for deploying large-scale generative models in production, ensuring both a responsive user experience and the economic viability of the system in a real-world advertising setting. 5 Experiments 5.1 Experimental Settings 5.1.1 Scenario. Our method is performed on AMAP, a prominent navigation and mapping platform hosting billions of users and POIs. We focus on the \"Guess Where You Go\" card presented on the homepage of the AMAP App, aiming to forecast the next subsequent POI using a user's previous check-ins and profile information. The objective of this scenario is to predict the next POI the users are most likely to visit. By displaying the POIs on the homepage, we aim to reduce the users' search effort, thereby enhancing user experience and long-term retention. Critically, our goal transcends mere prediction accuracy; we strive to build a model that captures the underlying dynamics of user mobility patterns and intents. This allows the system to develop a deeper \"cognitive\" understanding, ultimately facilitating more intelligent and context-aware decisionmaking for the user. The current deployed framework, which serves as our primary baseline, employs a traditional cascaded ranking system for recommendation. As illustrated in Figure 2, the cascaded ranking system includes three stages: (1) Retrieval stage to recall a coarse-grained POI corpus based on the user's preference, historical check-ins, and location [18, 44]; (2) Ranking stage to obtain top 10 POIs the users most likely visit with a ranking model based on behavior sequence transformer [3]; and (3) Cognitive Re-ranking stage with distance, quality, and exposure filters to yield the most cognitively plausible destination. Figure 2: Cascade ranking system of \"Guess Where You Go\" deployed on AMAP, which includes three stages from the left to the right: Retrieval, Ranking, and Cognitive Re-ranking. We evaluate our approach on three real-world datasets: FoursquareNYC [42], Foursquare-TKY [42], and Gowalla-CA [4] and a largescale proprietary dataset by sampling desensitized users from AMAP (named Beijing). The details of the utilized dataset is illustrated in Appendix A. 5.1.3 Implementation Details. The quantization module consists of 3 codebook layers and for the generative recommendation module, we employ the Qwen [27, 41] as the base model. We employ a constant learning rate schedule with a learning rate of 1e-5, combined with a warm-up phase of 20 steps. The model is trained on 200× NVIDIA H20 GPUs, with a batch size of 16 per GPU, gradient accumulation steps set to 8, and a sequence length of 4096 tokens. Each input consists of the most recent 50 check-in records of the user. 5.1.4 Evaluation Metric. Following previous works, we employ the accuracy metric for the top-1 recommendation accuracy (Acc@1) and the four cognitive metrics (TCS, SCS, PAS and SAS) defined in 3.2 as our metrics to evaluate the effect of the cognitive recommendation. Detailed information of the metrics is illustrated in Appendix C. 5.1.5 Baseline Methods. The details of the baseline methods in this part is illustrated in Appendix B. 5.1.6 Our Models. In our experiments we consider three versions of our model: (i) CoAST-0.5B: the model trained with Qwen2.5-0.5B [27] as our base LLM. (ii) CoAST-1.5B: with Qwen2.5-1.5B as our base LLM. (iii) CoAST-3B: A variation with Qwen2.5-3B as our base LLM. (iv) CoAST-7B: A second variation on Qwen2.5-7B. Below, unless stated otherwise, when we say \"our model,\" we refer to the CoAST-0.5B. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Table 2: Comparison of different models on three datasets: NYC, TKY, and CA. We present the model inputs including the POI and User representation approach and whether visit timestamps are utilized. RID, SID, UID and UP represent Random one-hot ID, Semantic ID, User one-hot ID and User Profiles, respectively. The results for the baseline methods are borrowed from [19, 32]. Model Inputs Acc@1 POI User Time NYC TKY CA PRME RID UID × 0.1159 0.1052 0.0521 PLSPL RID UID × 0.1917 0.1889 0.1072 STAN RID UID × 0.2231 0.1963 0.1104 GETNext RID UID × 0.2435 0.1829 0.1357 STHGCN RID UID × 0.2734 0.2950 0.1730 TPG RID UID ✓ 0.2555 0.1420 0.1749 ROTAN RID UID ✓ 0.3106 0.2458 0.2199 LLM4POI RID UID ✓ 0.3372 0.3035 0.2065 GNPR-SID SID UID ✓ 0.3618 0.3062 0.2403 CoAST-0.5B SID UP ✓ 0.3705 0.3125 0.2555 CoAST-1.5B SID UP ✓ 0.3727 0.3170 0.2585 CoAST-3B SID UP ✓ 0.3872 0.3202 0.2611 CoAST-7B SID UP ✓ 0.4027 0.3310 0.2721 Table 3: Comparison results of different versions of CoAST on Beijing dataset with the Cascade Ranking baseline system. 'a-' denotes average. Baseline CoAST-0.5B CoAST-1.5B CoAST-3B CoAST-7B Acc@1 0.2675 0.2820 0.2862 0.2923 0.3071 a-TCS 0.5561 0.6071 0.6021 0.6632 0.7543 a-SCS 0.8022 0.8212 0.8231 0.8452 0.8838 a-PAS 0.6232 0.6285 0.6240 0.6531 0.6970 a-SAS 0.2472 0.2840 0.2988 0.3240 0.3725 5.2 Offline Experiments 5.2.1 Overall Performance. Our method substantially outperforms all baseline methods across all of the evaluated datasets. Table 2 illustrate the comparison results on NYC, TKY, and CA datasets. Specifically, compared to the state-of-the-art GSNR-SID which utilizes LLaMA-7B as its base model, we observe a substantial improvement on all of the versions of CoAST. Notably, CoAST-7B demonstrated the most significant improvements, achieving gains of 11.3%, 8.1%, and 13.2% over the SOTA, respectively. Table 3 details the performance comparison between our proposed model and the cascade ranking system on the Beijing dataset. In addition to the standard Acc@1 metric, we evaluate the models on four cognitive-aware recommendation metrics (where 'a-' denotes average). Consistent with our findings on other datasets, our model significantly outperforms the traditional cascade ranking system on Acc@1. More critically, it demonstrates substantial gains on the cognitive metrics, achieving remarkable uplifts of 35.6% in temporal consistence and 50.68% in situational awareness performance. These results highlight a key limitation of conventional collaborative filtering-based systems: their struggle to capture personalized, periodic user needs and inability to effectively model situational or contextual information. This validates the superiority of our generative approach, which excels in understanding these complex, dynamic aspects of user behavior. 5.2.2 Ablation Study. To evaluate the impact of individual modules, we developed various versions of CoAST. Initially, certain essential modules were removed: \"w/o UP\" eliminates user profiling from the prompt. \"w/o S-ST\" eliminates situational and spatio-temporal descriptions from the prompt. \"w/o SLD\" exclude the SID-LocationDescription alignment corpus in the continued pretraining stage. \"w/o CPT\" exclude the Continued Pretraining stage in the alignment stage, only applying SFT and DPO. As illustrated in Table 4, removing any module leads to noticeable performance degradation, emphasizing the importance of each component. Significantly, the lack of SFT results in the most considerable decrease in all metrics, demonstrating the essential role of preference alignment in recommendation efficiency. Table 4: Ablation study of CoAST (with the online deployment version CoAST-3B). Acc@1 a-TCS a-SCS a-PAS a-SAS w/o UP 0.2720 0.6435 0.8211 0.5920 0.2923 w/o S-ST 0.2708 0.5905 0.7938 0.6496 0.2822 w/o SLD 0.2825 0.6078 0.8240 0.6501 0.3020 w/o CPT 0.2632 0.5875 0.7820 0.6020 0.2875 CoAST 0.2923 0.6632 0.8452 0.6531 0.3040 5.2.3 Case Study. To provide a qualitative and intuitive demonstration of our model's advantages, we conduct a case study that contrasts the recommendation lists generated by our model with those from a baseline model for a representative user. Detailed information of the case is illustrated in Appendix D. 5.2.4 Further Analysis. In this section, we analyze the impact on the performance of sequence length (input tokens) and number of utilized tokens in continued pretraining stage. Figure 3 presents the results. Our investigation into the impact of input sequence length shows that while longer context is beneficial, its utility eventually saturates. The model's performance gains are most pronounced up to a context window of 2048 tokens (∼30 user historical checkins) and largely stabilize beyond 4096 tokens (∼50 check-ins). This suggests that a moderately-sized history is sufficient to encapsulate the user's core preferences. Further extending the context window provides limited new signal, as this long-term information is already implicitly encoded in the user's learned representation. The volume of tokens involved in pretraining phase also proved to be a key determinant of performance. We observed a nearly linear improvement in model effectiveness as the training corpus size increased. Constrained by our training budget, the final model was trained to convergence on a dataset of 10B tokens. The consistent, Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. non-saturating growth suggests that performance is largely databound, with significant potential for improvement as more data becomes available. Figure 3: Impact on the performance of sequence length (input tokens) and number of tokens involved in continued pretraining stage. a-CAS denotes the average Cognition Alignment score computed by averaging the four cognitive scores. 5.3 Online Experiments 5.3.1 Efficiency Analysis. To evaluate the online inference efficiency, we compare different variants of our model with the efficiencySOTA cascade ranking system. Before deploying our model, we conduct a load test under the settings with a 50 QPS on 2× NVIDIA H20 GPUs. Figure 4 presents the inference p99 latency results of different methods. As shown, the Cascade Ranking system, a widely adopted paradigm in industrial recommender systems, demonstrates exceptional efficiency with its latency consistently around 20ms. As a comparison, while CoAST-7B achieved the best offline performance among all our variants, its high computational cost posed a significant challenge for online serving. Even after applying the acceleration strategies described in Section 4, its latency remained above 120ms, rendering large-scale deployment infeasible. To strike a practical balance between efficacy and efficiency, we identified CoAST-0.5B as the optimal candidate. It delivers competitive performance while maintaining a manageable inference cost (∼30ms). Consequently, we deployed CoAST-0.5B to production and conducted a online A/B test against the state-of-the-art (SOTA) Cascade Ranking baseline. 5.3.2 Online A/B Test. To assess the online effect of CoAST, we implemented a short-term and long-term A/B testing on \"Guess Figure 4: Comparison of Model Inference p99 Latency (ms) under a 50 QPS Load Test on 2× NVIDIA H20 GPUs. Where You Go\" card of AMAP's homepage. We examined CoAST0.5B, trained based on a Qwen2.5-0.5B model, compared to the traditional cascade ranking baseline as shown in Table 5. Table 5: Online improvement over traditional Cascade Ranking baseline of short and long term A/B test. % denotes relative improvement. short and long represents short-term and long-term A/B online test, respectively. P-CTR U-CTR SC-Rate RE-Rate AC-Rate NF-Rate short +5.06% +4.65% +0.69% +0.31% +0.77% -5.11% long +4.23% +4.59% +1.01% +1.65% +5.99% -5.85% It is worth mentioning that this baseline already includes user profiles, trending topics and seasonal features for improving cognition in the cognitive re-ranking stage. For short-term (one week) online test, CoAST demonstrate significant improvements in metrics like P-CTR (PV CTR) and U-CTR (UV CTR). This indicates that CoAST has increased the share of cognitive-aligned recommendations that effectively attracted user attention in terms of clicks. Additionally, it slightly enhances the user engagement metrics, such as SC-Rate (Scroll Rate), RE-Rate (User Retention Rate), and AC-Rate (User Active Rate), implying heightened user involvement. Notably, the effects on general utility metrics such as CTR and NFRate (Users' Negative Feedback Rate) are more modest, while the improvements on these engagement-related metrics often become apparent only over the long term. To investigate the long-term effects of the proposed method in large-scale RSs, we maintain a small traffic for our baselines and compare it with CoAST for over one month. Compared to the baseline and the performance on the short-term test, our method has achieved a significant enhancement in the user engagement metrics, with a 5.85% average decrease in NF-Rate (Users' Negative Feedback Rate). This demonstrates that CoAST could help to mitigate counter-intuitive recommendations and effectively improve user engagement while also boosting CTR, thereby enhancing the overall user experience. 6 Conclusion In this paper, we proposed a LLM-based next POI prediction approach, named CoAST. In order to adapt LLMs to sequential recommendation tasks, we focused on a comprehensive pretraining task to adapt a base LLM to align language and collaborative semantics for recommendation. This task include two categories of corpus: sequential POI prediction and explicit index-language alignment. Based on the learned item indices, our approach employed these alignment tasks to effectively adapt LLMs for sequential recommendation. Furthermore, we give complete definition and metrics on cognitive-aligned next POI prediction task. For constructing a cognitive-aligned POI recommendation system in industrial application, we design a multi-stage fine-tuning process, utilizing a curated dataset of cognitively-aware samples for both Supervised Fine-Tuning (SFT) and a subsequent Reinforcement Learning (RL) phase. This procedure explicitly align the model with human common-sense. Extensive offline experiments on three datasets demonstrated the effectiveness of our approach and Online A/B Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY test achieves substantial improvements compared to traditional cascade ranking recommendation system. Acknowledgments We would like to acknowledge the discussions of Jiawei Xue, Ning Wang, and Fangfang Chen throughout this work. 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CoST: Contrastive Quantization based Semantic Tokenization for Generative Recommendation. In Proceedings of the 18th ACM Conference on Recommender Systems (Bari, Italy) (RecSys '24). Association for Computing Machinery, New York, NY, USA, 969-974. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Figure 5: Case comparison of recommendation results of CoAST and the traditional basline RSs. The top row represents the user's real-time check-ins with different action types. The middle and bottom row represent the recommendation results of traditional basline RSs and CoAST, respectively. A Dataset To validate our model's performance in a real-world application, we constructed a large-scale proprietary dataset by sampling desensitized users from AMAP (named Beijing). On this dataset, we conducted both extensive offline evaluations and an online deployment. Furthermore, to ensure a comprehensive and fair comparison against existing baselines, we also evaluate our approach on three real-world datasets: Foursquare-NYC [42], Foursquare-TKY [42], and Gowalla-CA [4]. The pre-processing follows the settings in [32]. we pre-process each dataset by removing POIs with fewer than 10 interactions and users with fewer than 10 check-ins. The data is then sorted by time, with 80% used for training, 10% for validation, and 10% for testing. Users and POIs that do not appear in the training set are removed from the test set to ensure consistency between training and evaluation. Subsequently, the check-in records are grouped by user and ordered chronologically. For each user, the last visited POI is held out as the ground truth during evaluation, while the preceding visits are used as the input sequence. Note that in the training set, the user's visit records are also treated as historical sequences and concatenated with the test set to the specified length before being fed into the model. Detailed dataset statistics are provided in Table 6. Table 6: Statistics of the preprocessed datasets. Avg.len denotes the average length of item sequences. Dataset #Users #Items #Inter. Avg.len Sparsity NYC 1,083 5,135 104,074 136 98.1013% TKY 2,293 7,873 361,430 195 97.9798% CA 6,592 14,027 349,375 53 99.9996% Beijing 10,000k 1,520k 38,0000k 38 99.9999% B Baseline Following the settings of [19, 32], we compare our proposed method with various next POI recommendation baselines, which can be specified as the following four categories: (1) Traditional methods: PRME [9], a ranking-based metric embedding method; and PLSPL [38], a personalized sequential recommendation model. (2) Transformer-based methods: STAN [25], a spatio-temporal attention network; and GETNext [43], a Transformer-based next POI recommendation model. (3) GCN-based method: STHGCN [40], a spatio-temporal hierarchical graph convolutional network. (4) Time-aware methods: TPG [24], a timestamp-guided model; and ROTAN [10], a time-aware POI recommendation framework. (5) LLM-based method: LLM4POI [19], a method based on LLMs; and GNPR-SID [32], a state-of-the-art based on Semantics codes and generative recommendation framework, which is pretty closely related to our work. C Evaluation Metric Following previous works, we employ the accuracy metric for the top-1 recommendation, denoted as Acc@1, at the specified time. Acc@1 = 1 n n ∑︁ i=1 I(yi= ˆyi) where nis the number of test samples, yiand ˆyirepresent the ground-truth label and the predict label of the i-th sample, respectively. I is the indicator function, which equals 1 when the condition is true and 0 otherwise. Otherwise, we take the four cognitive categories (TCS, SCS, PAS and SAS) defined in 3.2 as our metrics to evaluate the effect of the cognitive recommendation. D Case Study Figure 5 illustrates the case comparison of recommendation results of CoAST and the traditional basline RSs. In the traditional cascade ranking RSs, the model collapses to the user's dominant daily rhythm: go to company at morning and back to home at afternoon. However, during other times of the day, conventional CTR-oriented models tend to be dominated by users' most recent interactions. This short-term focus often leads to recommendations that fail to capture the user's broader, long-term interests, resulting in a counter-intuitive user experience. Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. In contrast, our proposed generative model achieves a better alignment of the user's temporal context with their holistic preference profile, thereby fostering a more well-rounded and sustainable recommendation ecosystem."}}},{"rowIdx":286,"cells":{"id":{"kind":"string","value":"2510.14694"},"raw":{"kind":"string","value":"Response to Discussions of “Causal and Counterfactual Views of\nMissing Data Models”\nRazieh Nabi,1 Rohit Bhattacharya,2 Ilya Shpitser,3 James M. Robins4\n1Department of Biostatistics and Bioinformatics, Emory University, Atlanta, GA, U.S.A.\n2Department of Computer Science, Williams College, Williamstown, MA, U.S.A.\n3Department of Computer Science, Johns Hopkins University, Baltimore, MD, U.S.A.\n4Departments of Biostatistics and Epidemiology, Harvard University, Boston, MA, U.S.A.\n1\nIntroduction\nWe are grateful to the discussants – Levis and Kennedy [2025], Luo and Geng [2025], Wang\nand van der Laan [2025], and Yang and Kim [2025] – for their thoughtful comments on our\npaper [Nabi et al., 2025]. Below we summarize our main contributions before responding to each\ndiscussion in turn.\nGraphical models have emerged as an important tool for clarifying identifying assumptions made\nin both causal inference [Pearl and Robins, 1995, Pearl, 2000] and missing data [Robins and\nGill, 1997, Bhattacharya et al., 2019, Nabi et al., 2020, Malinsky et al., 2021, Mohan and Pearl,\n2021]. Our paper [Nabi et al., 2025] shows how recent techniques motivated by causal graphical\nmodeling may be fruitfully applied to obtain identification in missing data models. These recent\ntechniques allow us to obtain novel identification results that would not be possible to obtain\nin standard causal inference problems, except by imposing implausible additional assumptions,\nsuch as rank preservation.\nSpecifically, we formalize how identifiability of the target (i.e., complete) data law can be viewed\nas identification of a joint distribution over counterfactuals L(1), the variables that would have\nbeen observed if all missingness indicators R were set to one (that is, if no data were missing).\nThis reframing yields a counterfactual analogue of the g-formula. When assumptions encoded in\nthe Markov properties of a missing data DAG (m-DAG) allow us to express the counterfactual\ng-formula in terms of the factuals, we obtain (nonparametric) identification. That is, noting\nthat the target law p(l(1)) satisfies p(l(1)) = p(l, R = 1)/p(R = 1 | l(1)), it follows that when the\nmissingness selection model p(R = 1 | L(1)) is identified from the observed law, so is p(l(1)).\nIn our paper we used the word “nonparametric” in two different ways. The joint distribution\nof (L, L(1), R) is Markov to a given m-DAG if it factorizes as the product of the conditional\ndensities of each variable given its parents. In the graphical causal modeling literature, the\nmodel is said to be “nonparametric” just when these conditional densities are left unrestricted\n(with the exception that L is a deterministic function of its parents). The term “nonparametric\nidentification” refers to identification in such a model. In contrast, in the missing data literature\nin statistics, a model is said to be nonparametric if it places no restrictions on either the observed\ndata law or the target data law p(l(1)). It is said to be nonparametric just identified (NPI) if\nthe target law is identified from the observed data law. The permutation model [Robins, 1997]\nwe discuss in our paper is known to be NPI. In contrast, all the other identified missing data\nmodels in our paper place testable restrictions on the joint distribution of the observed data. In\n1\narXiv:2510.14694v1 [stat.ME] 16 Oct 2025\n\nfact, we conjecture that the permutation model is the only m-DAG model that is NPI. In the\nfollowing, we use nonparametric in the graphical causal modeling sense.\nA key message of our paper is that features specific to missing data models (in particular the\npartial observability of the counterfactuals through the proxies L and the structural restriction\nthat R and L do not cause L(1)) can deliver parameter identification in settings where analogous\nparameters in hidden variable causal DAGs are not identified. We catalog several such non-\nparametric identification techniques in m-DAGs, and we clarify similarities and differences with\nstandard causal identification, including when additional assumptions like rank preservation\nmight be needed for causal analogues.\nEach discussion extends or challenges our framework in important ways. Levis and Kennedy\n[2025] highlight identification strategies complementary to ours, based on the existence of\ninstrumental and shadow variables, discuss semiparametric estimation theory for functionals\narising from such strategies, as well as discuss connections between m-DAGs and Single World\nIntervention Graphs (SWIGs). Luo and Geng [2025] analyze self-censoring MNAR mechanisms\nwith binary variables, deriving identifiability results that leverage auxiliary variables. Wang and\nvan der Laan [2025] emphasize, as we do, viewing missingness as interventions, and connect our\nperspective to censoring in survival models. Yang and Kim [2025] and Wang and van der Laan\n[2025] both examine challenges of applying m-DAGs in applications, particularly assumption\nvalidation, scalability, and sensitivity analysis.\n2\nResponse to the discussions\n2.1\nOn the discussion by Levis and Kennedy\nLevis and Kennedy [2025] draw attention to additional causal identification tools and highlight\nimplications for estimation. In particular, they emphasize the relevance of instrumental variables,\nshadow variables, and SWIGs as complementary devices for reasoning about identification in\nmissing data problems. They also consider how identification results derived from m-DAGs can\nbe translated into practical estimation procedures with desirable asymptotic properties. Their\ndiscussion situates our contribution within a broader pipeline of causal inference methods and\npoints toward promising directions for future methodological development.\nWe appreciate Levis and Kennedy’s thoughtful remarks. On their first point, we agree that\ninstrumental variables [Sun et al., 2018], shadow variables [Miao et al., 2024], and related\ntools [Li et al., 2023] are useful complements. Our emphasis in the paper, however, was on\na nonparametric identification framework: aside from the Markov restrictions encoded via\nindependence assumptions in m-DAGs, we make no further distributional assumptions. In\ncontrast, the use of IVs or shadow variables begins in settings where nonparametric identification\nfails, and proceeds by imposing additional restrictions (such as functional form constraints e.g.,\nhomogeneity assumptions when using of IVs), relevance assumptions which result in generic\nidentification, or restrictions on the support of variables to restore identification. These extra\ningredients move the analysis outside the purely nonparametric domain that was our focus.\nOn estimation, we appreciate the worked example they provide. While our goal in this paper was\nprimarily conceptual, highlighting the philosophical parallels and differences between missing data\nand causal identification, their discussion underscores the importance of connecting identification\nresults to practical estimation and efficiency theory. We agree this is important future work.\nWe also agree with Levis and Kennedy that causal inference problems in the presence of\nconfounding often occur together with censoring, and that obtaining identification when both\ncomplications are present is challenging. We also appreciate their worked example illustrating\nthese complications.\n2\n\nA\nY\nA\na\nY (a)\nA\n1\nY (1) + A\n0\nY (0)\nR\nL\nR\nr = 1\nL(1)\nA\nX\nY\nA\na\nX\nY (a)\nA\n1\nX\nY (1) +\nA\n0\nX\nY (0)\nR\nX\nL\nR\nr = 1\nX\nL(1)\n(a)\n(b)\n(c)\n(d)\nFigure 1: (a) Ignorable treatment model with SWIGs; (b) Its MCAR analogue, with a single\nrelevant SWIG; (c) Conditionally ignorable treatment model with SWIGs; (d) Its MAR analogue,\nagain with only one relevant SWIG.\nWe would like to offer two notes of caution regarding generalizing the construction presented by\nLevis and Kennedy, first on the appropriate generalization of the SWIG splitting construction\nfrom causal diagrams to m-DAGs, and second on the general utility of using SWIGs for obtaining\nidentification in missing data settings.\nOn the translation of m-DAGs to SWIGs\nFor simple MAR models where the m-DAG’s structure resembles that of the standard condition-\nally ignorable model in causal inference, the translation of an m-DAG to an equivalent SWIG\nis fairly natural. There are, however, important distinctions even in these simple settings. In\ncausal inference, SWIGs provide a template indexed by treatment interventions (two templates\nif the treatment is binary) [Richardson and Robins, 2013]; in the missing data analogue, only the\ntemplate for R = 1 is meaningful, since the counterfactual L(0) is not defined. This difference is\nhighlighted in Figures 1(a, b) for a causal model satisfying ignorability and an analogous missing\ndata model that is MCAR. Similarly, Figures 1(c, d) highlight the same for a conditionally\nignorable causal model and a MAR missing data model.\nCaution must be exercised when SWIGs are constructed from m-DAGs representing MNAR\nmechanisms. For instance, in a missing data model with self-censoring, a natural idea for what\nthe SWIG should look like would be the graph shown in Figure 2(a). However, if we try to\ncollapse the random and fixed nodes in this SWIG to reconstruct the observed data graph, we\nobtain the graph shown in Figure 2(b), which erroneously implies a cycle in the underlying data\ngenerating process.\nThe problem arises from conflating the full data variable L(1) in m-DAGs, and the observed\ndata variable L relabeled to be counterfactual in the SWIG construction. An appropriate view\nof m-DAGs that avoids these types of difficulties is to view full data variables such as L(1)\nas unobserved confounders U, with additional structure imposed by missing data consistency.\nIn this view, the SWIG corresponding to the self-censoring model is better represented by\nFigure 2(c), where the variable L(1) is viewed as an unobserved confounder U with special\nstructure, which influences both the observability indicator R, and the observed proxy variable\nL. The m-DAG corresponding to this view of the self-censoring model is shown in Figure 2(d).\nSWIGs do not make identification arguments for MNAR problems clearer\nThe primary motivation for SWIGs in the causal inference context is to provide a graphical\nrepresentation for independences that arise in identification arguments. The function of m-DAGs\nis to provide precisely the same graphical representation in missing data problems, rendering\nSWIGs unnecessary.\n3\n\nR\nr = 1\nL(1)\n(a)\nR\nL\n(b)\nR\nr = 1\nU ≡L(1)\nL(1)\n(c)\nR\nU ≡L(1)\nL\n(d)\nFigure 2: (a) A possible SWIG representation of a self-censoring missing data model; (b)\nStitching SWIGs into an observed-data model produces a cycle; (c) A SWIG that draws a\ndistinction between the full data variable L(1), viewed as an unobserved confounder U with extra\nrestrictions, and the observed variable L under an intervention where R is set to 1; (d) The full\ndata graph equating the full data variable L(1) with an unobserved confounder U with special\nstructure.\nU1 ≡L(1)\n1\nU2 ≡L(1)\n2\nR1\nR2\nL1\nL2\n(a)\nU1 ≡L(1)\n1\nU2 ≡L(1)\n2\nR1\nR2\nL(1)\n1\nL(1)\n2\nr1 = 1\nr2 = 1\n(b)\nFigure 3: (a) The block-parallel MNAR model, labeling missing variables as unmeasured for the\npurposes of constructing a SWIG; (b) The corresponding SWIG obtained from (a).\nTake, for example, the independences that can be read from the m-DAG in Figure 3(a) and the\ncorresponding SWIG in Figure 3(b). From either graph, we are able to extract independences\nof the form Rk ⊥⊥L(1)\nk , R−k | L(1)\n−k, for k ∈{1, 2} that define the binary block-parallel MNAR\nmodel. That is, the m-DAG itself is just as expressive as the SWIG for reading off independences\nimportant for identification in missing data models. In fact, we now argue that attempting\nidentification using standard causal inference arguments from the SWIG may be problematic\nfor missing data problems due to the single counterfactual nature of missing data, which as we\ndiscussed in Section 6 of our paper is more akin to the rank preservation assumption in causal\ninference.\nTo obtain the identification of the target law p(l(1)\n1 , l(1)\n2 ) of the block-parallel MNAR model, we\nshowed in Section 5.2 of our paper that a parallel use of the g-formula was required, which never\narose in standard causal inference settings. If just the marginal p(l(1)\n2 ) is desired, this is obtained\nby marginalization. Now suppose instead, we attempted to perform identification arguments\nfor p(l(1)\n2 ) using the SWIG, in a manner similar to the derivation of the adjustment formula for\ncausal inference. This would proceed as follows. First notice from the SWIG in Figure 3(b) that\nL(1)\n2\n⊥⊥R2 | U1, R1 – this is analogous to conditional ignorability except U1 is observed only\nwhen R1 = 1. Thus we have,\np(l(1)\n2 ) =\nX\nr1,u1\np(r1, u1, l(1)\n2 )\n=\nX\nr1,u1\np(r1, u1) p(l(1)\n2\n| r1, u1)\n=\nX\nr1,u1\np(r1, u1) p(l(1)\n2\n| r1, u1, r2 = 1)\n=\nX\nr1,u1\np(r1, u1) p(l2 | r1, u1, r2 = 1).\nDespite the final expression above appearing to be a function that is devoid of counterfactuals,\nthe appearance of the variable U1 in the expression prevents identification, as U1 is observed\n4\n\nonly when R1 = 1. Since R1 cannot be set to the value 1 in the formula above, we are unable to\nestablish identification via the arguments presented. In fact, no sequential strategy that we are\naware of based on the SWIG in Figure 3(b) would help establish identification. That is, the\nparallel fixing arguments presented in Section 5.2 of our paper are required here.\nIn short, while applying the SWIG construction to m-DAGs offers useful insights in causal\nproblems with simple types of missingness, such as MCAR or MAR, we caution that such\nconstructions must be done with care, as Levis and Kennedy have done.\nTo illustrate, Figure 4(a) extends the example of Levis and Kennedy to include missingness in\nboth the outcome Y and treatment A. The goal here is to identify the average causal effect of A\non Y in the absence of censoring.\nJust as in the previous example shown in Figures 3(a, b), an argument based solely on SWIGs\ndoes not yield identification. Instead, identification strategies that leverage restrictions encoded\nin both m-DAGs and SWIGs are necessary.\nIn particular, we first obtain the target law\np(X, A(1), Y (1)) by applying inverse probability weighting with the product of the propensity\nscores for R1 and R2, resulting in the graph in Figure 4(b). We then apply the g-formula on\nA(1) to adjust for confounding by X, taking advantage of the restrictions in the SWIG shown in\nFigure 4(c). This yields p(Y (1,a)), the distribution of the outcome Y , had it not been censored\nand had treatment been set to value a, from which the average causal effect of interest may be\nobtained.\nFinally, we note that there is currently no complete graphical identification theory for causal\nparameters associated with arbitrary m-DAGs that encode both MNAR and (possible) con-\nfounding by unmeasured common causes U. We expect this theory to employ both SWIG and\nm-DAG constructions, as in the worked example of Levis and Kennedy, and our examples above.\n2.2\nOn the discussion by Luo and Geng\nLuo and Geng [2025] extend our framework by focusing on self-censoring MNAR mechanisms,\nwhere the missingness of a variable depends directly on its unobserved value. They study\nsuch models with binary outcomes and establish identifiability results under explicit, testable\nconditions. Their discussion highlights how auxiliary variables, either baseline or follow-up\nmeasurements, can be leveraged to restore identification. In doing so, they illustrate that\nself-censoring structures, which were not emphasized in our paper, can still yield identification\nin important cases. They also point to future directions for extending these ideas beyond the\nbinary case to more general settings.\nWe thank Luo and Geng for their examples of self-censoring mechanisms. In our paper, we noted\nself-censoring as a form of MNAR but did not explore it further, since m-DAG models with\nself centering are never nonparametrically identified [Mohan et al., 2013]. Since our emphasis\nwas on nonparametric identification, we did not employ any restrictions not implied by the\nm-DAG factorization, including (i) relevant restrictions needed by instrumental variable or\nproxy methods, or (ii) constraints implied by state space restrictions. As Luo and Geng [2025]\nillustrated, such assumptions can sometimes yield identifiability, and their results highlight\nconcrete testable conditions under which this occurs.\nWe fully agree that self-censoring is of practical importance, with income surveys, sensitive\nhealth questions, and other social science settings providing common examples. Their discussion\nusefully demonstrates how auxiliary information, through baseline or follow-up variables, can be\nleveraged to address such scenarios.\nMore broadly, we believe that in MNAR submodels that identify the target law, restrictions\nimplied by the m-DAG always yield testable implications for the observed data law, with the\n5\n\nA(1)\nX\nY (1)\nRA\nRY\nA\nY\n(a)\nA(1) = A\nX\nY (1) = Y\nrA = 1\nrY = 1\nA\nY\n(b)\nA(1) = A\na\nX\nY (1,a) = Y (a)\nrA = 1\nrY = 1\nA\nY\n(c)\nFigure 4: (a) Extension of Levis and Kennedy’s example to have two missing variables. (b)\nGraph obtained after fixing RA and RY in parallel. (c) SWIG obtained by splitting the treatment\nvariable A after already having fixed RA and RY .\nexception of the permutation model [Robins, 1997]. Examples of such tests have been developed\nin prior work on graphical models [Mohan and Pearl, 2014, Nabi and Bhattacharya, 2023, Guo\net al., 2023, Chen et al., 2023].\nOverall, we view their discussion as complementary to our focus: while our framework aimed to\ncatalog nonparametric identification results, their work illustrates how additional structure can\nboth sharpen identifiability and connect to practical applications.\n2.3\nOn the discussion by Wang and van der Laan\nWang and van der Laan [2025], like us, approach missingness through the lens of interventions.\nThey emphasize the close relationship between missing data models and multivariate censoring\nmodels in survival analysis. They carefully prove that the permutation missingness model\nis MNAR rather than MAR and flesh out our example of why the permutation model is\nsubstantively plausible in some real world applications. They also discuss substantive settings\nwhere the block sequential model is plausible.\nThey are less certain that the other identifiable MNAR models discussed in our paper are\nsubstantively plausible. In our view, it is difficult to establish plausibility of any missing data\nmodel in practice, whether it is formulated graphically or not. One advantage of the sort of\ntheory we present is a single formulation for a large class of identifiable models (including existing\nmodels in the literature such as those described in [Zhou et al., 2010, Robins, 1997]). Since no\nsingle model may be plausible in applications, an alternative strategy is to perform the analysis\nof interest under a wide class of identifiable missingness models, as a form of sensitivity analysis.\nWe make one final observation regarding the plausibility of (or the lack there of) the identifiable\nMNAR models considered in the paper. Identification in every case we discuss relies on an\nassumption that one or more counterfactuals L(1)\nj\nsuffice to control confounding of the “effect”\nof Rk on Lk where j ̸= k. But why should we privilege L(1)\nj\n(or a set of such variables) as a\nset sufficient for adjustment? It seems much more plausible that there exist other unmeasured\ncommon causes of Rk and Lk (equivalently L(1)\nk ) that would also need to be adjusted for to\neliminate confounding. Are we assuming the L(1)\nj\nsuffice to control confounding because we\nbelieve it to be so or rather because we want to achieve identification for identification’s sake? In\nfact, similar plausibility concerns arise in non-graphical identifiable models, such as non-monotone\nMAR. In our view, plausible models of missingness feature a description of data generation\nthat follows a temporal order (often representable as a DAG). Unfortunately, to be realistic,\neven models of this type would feature unobserved confounding of the sort that would prevent\nidentifiability.\n6\n\n2.4\nOn the discussion by Yang and Kim\nYang and Kim [2025] emphasize the practical challenges of applying the m-DAG framework.\nThey point out that while m-DAGs provide a principled way to encode assumptions and extend\nidentification theory, their utility in practice depends critically on correct graph specification,\nwhich may be difficult to achieve in applied settings. They raise concerns about scalability in\nhigh-dimensional problems, where the number of variables and missingness patterns grows quickly,\nand about the feasibility of validating the conditional independence assumptions encoded in an\nm-DAG. They further stress the importance of developing systematic approaches to sensitivity\nanalysis, noting that small misspecifications of the graph can have consequences for identification\nand downstream tasks.\nWe thank Yang and Kim for raising important questions about the practical utility of m-DAGs,\nand graphical modeling more broadly. Our goal in this paper was primarily conceptual: to\nprovide a general framework that connects causal and missing data perspectives and to catalog\nidentification results that arise from this connection. Their emphasis on practical feasibility\nis appreciated, and complementary to our focus. That said, continuing work in the graphical\nmodeling community over the last two decades has rendered much of the critiques of the graphical\nmodeling approach to causal inference and missing data problems out of date.\nIdentification of causal and missing data parameters must, by necessity, rely on equality re-\nstrictions in the full data distribution. Graphical models aim to encode these restrictions in a\nway that allows the construction of a plausible data generating mechanism consistent with a\ntemporal order of events. Models that obtain identifiability without a corresponding graphical\nrepresentation, for instance the non-monotone missing at random (MAR) model, often lack such\na plausible mechanism. Indeed, existing efforts that argue for the plausibility of models such as\nnon-monotone MAR embed them into a graphical model or a mixture of such models [Robins\nand Gill, 1997].\nOver the last decade, an explosion of easy to use open-source packages that take advantage of\ngraphical models have been developed for all tasks in causal inference and missing data, including\nestablishing identification, constructing estimators and applying them to data, conducting\nsensitivity analyses, establishing bounds on non-identified parameters, and model selection. A\nnon-exhaustive list of these packages includes: Ananke [Lee et al., 2023], autobounds [Duarte\net al., 2024], dosearch [Tikka et al., 2021], DAGitty [Textor et al., 2016], software developed\nas part of the Tetrad project (equipped with Python and R interfaces) [Ramsey and Andrews,\n2023], as well the pcalg package [Kalisch et al., 2012], and extensions to deal with missing values\n[Andrews et al., 2024].\nFurther, sensitivity analyses results can be fruitfully applied to graphical models. For example,\nnongraphical results in Robins et al. [1999] can easily be represented in graphical structures that\nencode MAR, conditional or sequential ignorability, and other types of Markov restrictions. For\ninstance, the permutation model in Robins [1997] is shown to be a graphical model in our paper.\nIn addition, graphical models allow a particularly powerful form of nonparametric sensitivity\nanalysis, where consistent inferences are made in union models defined over a potentially large\nclass of graphs that the analyst is uncertain about; see for example, [VanderWeele and Shpitser,\n2011, Yang et al., 2024, Shpitser and VanderWeele, 2010, Shpitser et al., 2010, Chang et al.,\n2024, Wang et al., 2025]. For instance, the result in [VanderWeele and Shpitser, 2011] states\nthat identification of the causal effect by covariate adjustment may be formulated without\nprecise knowledge of the graph, but only via the set of common causes of the treatment and\noutcome, provided an adjustment set exists. The method developed by Yang et al. [2024] provide\nsimilar robustness guarantees with more complex types of identifying functionals, including the\nfront-door functional, and the ratio functional arising in instrumental variable analysis.\n7\n\nWe share the authors’ view that algorithmic and computational advances will be needed to bring\ngraphical identification methods to bear on high-dimensional problems with complex missingness\npatterns. We see this as an exciting frontier for future work, and one where continued integration\nof causal inference tools with missing data methodology will be especially fruitful. We believe\nsmoothing and sparsity methods, as well as sum-product algorithm type methods developed in\nthe graphical modeling literature may all be relevant for these advances [Lee et al., 2021].\n3\nConclusion\nWe thank the discussants for their thoughtful contributions. 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Robins4 1 .S.A. 2 .S.A. 3 .S.A. 4Departments of Biostatistics and Epidemiology, Harvard University, Boston, MA, U.S.A. 1 Introduction We are grateful to the discussants - Levis and Kennedy [2025], Luo and Geng [2025], Wang and van der Laan [2025], and Yang and Kim [2025] - for their thoughtful comments on our paper [Nabi et al., 2025]. Below we summarize our main contributions before responding to each discussion in turn. Graphical models have emerged as an important tool for clarifying identifying assumptions made in both causal inference [Pearl and Robins, 1995, Pearl, 2000] and missing data [Robins and Gill, 1997, Bhattacharya et al., 2019, Nabi et al., 2020, Malinsky et al., 2021, Mohan and Pearl, 2021]. Our paper [Nabi et al., 2025] shows how recent techniques motivated by causal graphical modeling may be fruitfully applied to obtain identification in missing data models. These recent techniques allow us to obtain novel identification results that would not be possible to obtain in standard causal inference problems, except by imposing implausible additional assumptions, such as rank preservation. Specifically, we formalize how identifiability of the target (i.e., complete) data law can be viewed as identification of a joint distribution over counterfactuals L(1), the variables that would have been observed if all missingness indicators R were set to one (that is, if no data were missing). This reframing yields a counterfactual analogue of the g-formula. When assumptions encoded in the Markov properties of a missing data DAG (m-DAG) allow us to express the counterfactual g-formula in terms of the factuals, we obtain (nonparametric) identification. That is, noting that the target law p(l(1)) satisfies p(l(1)) = p(l, R = 1)/p(R = 1 | l(1)), it follows that when the missingness selection model p(R = 1 | L(1)) is identified from the observed law, so is p(l(1)). In our paper we used the word \"nonparametric\" in two different ways. The joint distribution of (L, L(1), R) is Markov to a given m-DAG if it factorizes as the product of the conditional densities of each variable given its parents. In the graphical causal modeling literature, the model is said to be \"nonparametric\" just when these conditional densities are left unrestricted (with the exception that L is a deterministic function of its parents). The term \"nonparametric identification\" refers to identification in such a model. In contrast, in the missing data literature in statistics, a model is said to be nonparametric if it places no restrictions on either the observed data law or the target data law p(l(1)). It is said to be nonparametric just identified (NPI) if the target law is identified from the observed data law. The permutation model [Robins, 1997] we discuss in our paper is known to be NPI. In contrast, all the other identified missing data models in our paper place testable restrictions on the joint distribution of the observed data. In 1 16 Oct 2025 fact, we conjecture that the permutation model is the only m-DAG model that is NPI. In the following, we use nonparametric in the graphical causal modeling sense. A key message of our paper is that features specific to missing data models (in particular the partial observability of the counterfactuals through the proxies L and the structural restriction that R and L do not cause L(1)) can deliver parameter identification in settings where analogous parameters in hidden variable causal DAGs are not identified. We catalog several such nonparametric identification techniques in m-DAGs, and we clarify similarities and differences with standard causal identification, including when additional assumptions like rank preservation might be needed for causal analogues. Each discussion extends or challenges our framework in important ways. Levis and Kennedy [2025] highlight identification strategies complementary to ours, based on the existence of instrumental and shadow variables, discuss semiparametric estimation theory for functionals arising from such strategies, as well as discuss connections between m-DAGs and Single World Intervention Graphs (SWIGs). Luo and Geng [2025] analyze self-censoring MNAR mechanisms with binary variables, deriving identifiability results that leverage auxiliary variables. Wang and van der Laan [2025] emphasize, as we do, viewing missingness as interventions, and connect our perspective to censoring in survival models. Yang and Kim [2025] and Wang and van der Laan [2025] both examine challenges of applying m-DAGs in applications, particularly assumption validation, scalability, and sensitivity analysis. 2 Response to the discussions 2.1 On the discussion by Levis and Kennedy Levis and Kennedy [2025] draw attention to additional causal identification tools and highlight implications for estimation. In particular, they emphasize the relevance of instrumental variables, shadow variables, and SWIGs as complementary devices for reasoning about identification in missing data problems. They also consider how identification results derived from m-DAGs can be translated into practical estimation procedures with desirable asymptotic properties. Their discussion situates our contribution within a broader pipeline of causal inference methods and points toward promising directions for future methodological development. We appreciate Levis and Kennedy's thoughtful remarks. On their first point, we agree that instrumental variables [Sun et al., 2018], shadow variables [Miao et al., 2024], and related tools [Li et al., 2023] are useful complements. Our emphasis in the paper, however, was on a nonparametric identification framework: aside from the Markov restrictions encoded via independence assumptions in m-DAGs, we make no further distributional assumptions. In contrast, the use of IVs or shadow variables begins in settings where nonparametric identification fails, and proceeds by imposing additional restrictions (such as functional form constraints e.g., homogeneity assumptions when using of IVs), relevance assumptions which result in generic identification, or restrictions on the support of variables to restore identification. These extra ingredients move the analysis outside the purely nonparametric domain that was our focus. On estimation, we appreciate the worked example they provide. While our goal in this paper was primarily conceptual, highlighting the philosophical parallels and differences between missing data and causal identification, their discussion underscores the importance of connecting identification results to practical estimation and efficiency theory. We agree this is important future work. We also agree with Levis and Kennedy that causal inference problems in the presence of confounding often occur together with censoring, and that obtaining identification when both complications are present is challenging. We also appreciate their worked example illustrating these complications. 2 A Y A a Y (a) A 1 Y (1) + A 0 Y (0) R L R r = 1 L(1) A X Y A a X Y (a) A 1 X Y (1) + A 0 X Y (0) R X L R r = 1 X L(1) (a) (b) (c) (d) Figure 1: (a) Ignorable treatment model with SWIGs; (b) Its MCAR analogue, with a single relevant SWIG; (c) Conditionally ignorable treatment model with SWIGs; (d) Its MAR analogue, again with only one relevant SWIG. We would like to offer two notes of caution regarding generalizing the construction presented by Levis and Kennedy, first on the appropriate generalization of the SWIG splitting construction from causal diagrams to m-DAGs, and second on the general utility of using SWIGs for obtaining identification in missing data settings. On the translation of m-DAGs to SWIGs For simple MAR models where the m-DAG's structure resembles that of the standard conditionally ignorable model in causal inference, the translation of an m-DAG to an equivalent SWIG is fairly natural. There are, however, important distinctions even in these simple settings. In causal inference, SWIGs provide a template indexed by treatment interventions (two templates if the treatment is binary) [Richardson and Robins, 2013]; in the missing data analogue, only the template for R = 1 is meaningful, since the counterfactual L(0) is not defined. This difference is highlighted in Figures 1(a, b) for a causal model satisfying ignorability and an analogous missing data model that is MCAR. Similarly, Figures 1(c, d) highlight the same for a conditionally ignorable causal model and a MAR missing data model. Caution must be exercised when SWIGs are constructed from m-DAGs representing MNAR mechanisms. For instance, in a missing data model with self-censoring, a natural idea for what the SWIG should look like would be the graph shown in Figure 2(a). However, if we try to collapse the random and fixed nodes in this SWIG to reconstruct the observed data graph, we obtain the graph shown in Figure 2(b), which erroneously implies a cycle in the underlying data generating process. The problem arises from conflating the full data variable L(1) in m-DAGs, and the observed data variable L relabeled to be counterfactual in the SWIG construction. An appropriate view of m-DAGs that avoids these types of difficulties is to view full data variables such as L(1) as unobserved confounders U, with additional structure imposed by missing data consistency. In this view, the SWIG corresponding to the self-censoring model is better represented by Figure 2(c), where the variable L(1) is viewed as an unobserved confounder U with special structure, which influences both the observability indicator R, and the observed proxy variable L. The m-DAG corresponding to this view of the self-censoring model is shown in Figure 2(d). SWIGs do not make identification arguments for MNAR problems clearer The primary motivation for SWIGs in the causal inference context is to provide a graphical representation for independences that arise in identification arguments. The function of m-DAGs is to provide precisely the same graphical representation in missing data problems, rendering SWIGs unnecessary. 3 R r = 1 L(1) (a) R L (b) R r = 1 U ≡L(1) L(1) (c) R U ≡L(1) L (d) Figure 2: (a) A possible SWIG representation of a self-censoring missing data model; (b) Stitching SWIGs into an observed-data model produces a cycle; (c) A SWIG that draws a distinction between the full data variable L(1), viewed as an unobserved confounder U with extra restrictions, and the observed variable L under an intervention where R is set to 1; (d) The full data graph equating the full data variable L(1) with an unobserved confounder U with special structure. U1 ≡L(1) 1 U2 ≡L(1) 2 R1 R2 L1 L2 (a) U1 ≡L(1) 1 U2 ≡L(1) 2 R1 R2 L(1) 1 L(1) 2 r1 = 1 r2 = 1 (b) Figure 3: (a) The block-parallel MNAR model, labeling missing variables as unmeasured for the purposes of constructing a SWIG; (b) The corresponding SWIG obtained from (a). Take, for example, the independences that can be read from the m-DAG in Figure 3(a) and the corresponding SWIG in Figure 3(b). From either graph, we are able to extract independences of the form Rk ⊥⊥L(1) k , R-k | L(1) -k, for k ∈{1, 2} that define the binary block-parallel MNAR model. That is, the m-DAG itself is just as expressive as the SWIG for reading off independences important for identification in missing data models. In fact, we now argue that attempting identification using standard causal inference arguments from the SWIG may be problematic for missing data problems due to the single counterfactual nature of missing data, which as we discussed in Section 6 of our paper is more akin to the rank preservation assumption in causal inference. To obtain the identification of the target law p(l(1) 1 , l(1) 2 ) of the block-parallel MNAR model, we showed in Section 5.2 of our paper that a parallel use of the g-formula was required, which never arose in standard causal inference settings. If just the marginal p(l(1) 2 ) is desired, this is obtained by marginalization. Now suppose instead, we attempted to perform identification arguments for p(l(1) 2 ) using the SWIG, in a manner similar to the derivation of the adjustment formula for causal inference. This would proceed as follows. First notice from the SWIG in Figure 3(b) that L(1) 2 ⊥⊥R2 | U1, R1 - this is analogous to conditional ignorability except U1 is observed only when R1 = 1. Thus we have, p(l(1) 2 ) = X r1,u1 p(r1, u1, l(1) 2 ) = X r1,u1 p(r1, u1) p(l(1) 2 | r1, u1) = X r1,u1 p(r1, u1) p(l(1) 2 | r1, u1, r2 = 1) = X r1,u1 p(r1, u1) p(l2 | r1, u1, r2 = 1). Despite the final expression above appearing to be a function that is devoid of counterfactuals, the appearance of the variable U1 in the expression prevents identification, as U1 is observed 4 only when R1 = 1. Since R1 cannot be set to the value 1 in the formula above, we are unable to establish identification via the arguments presented. In fact, no sequential strategy that we are aware of based on the SWIG in Figure 3(b) would help establish identification. That is, the parallel fixing arguments presented in Section 5.2 of our paper are required here. In short, while applying the SWIG construction to m-DAGs offers useful insights in causal problems with simple types of missingness, such as MCAR or MAR, we caution that such constructions must be done with care, as Levis and Kennedy have done. To illustrate, Figure 4(a) extends the example of Levis and Kennedy to include missingness in both the outcome Y and treatment A. The goal here is to identify the average causal effect of A on Y in the absence of censoring. Just as in the previous example shown in Figures 3(a, b), an argument based solely on SWIGs does not yield identification. Instead, identification strategies that leverage restrictions encoded in both m-DAGs and SWIGs are necessary. In particular, we first obtain the target law p(X, A(1), Y (1)) by applying inverse probability weighting with the product of the propensity scores for R1 and R2, resulting in the graph in Figure 4(b). We then apply the g-formula on A(1) to adjust for confounding by X, taking advantage of the restrictions in the SWIG shown in Figure 4(c). This yields p(Y (1,a)), the distribution of the outcome Y , had it not been censored and had treatment been set to value a, from which the average causal effect of interest may be obtained. Finally, we note that there is currently no complete graphical identification theory for causal parameters associated with arbitrary m-DAGs that encode both MNAR and (possible) confounding by unmeasured common causes U. We expect this theory to employ both SWIG and m-DAG constructions, as in the worked example of Levis and Kennedy, and our examples above. 2.2 On the discussion by Luo and Geng Luo and Geng [2025] extend our framework by focusing on self-censoring MNAR mechanisms, where the missingness of a variable depends directly on its unobserved value. They study such models with binary outcomes and establish identifiability results under explicit, testable conditions. Their discussion highlights how auxiliary variables, either baseline or follow-up measurements, can be leveraged to restore identification. In doing so, they illustrate that self-censoring structures, which were not emphasized in our paper, can still yield identification in important cases. They also point to future directions for extending these ideas beyond the binary case to more general settings. We thank Luo and Geng for their examples of self-censoring mechanisms. In our paper, we noted self-censoring as a form of MNAR but did not explore it further, since m-DAG models with self centering are never nonparametrically identified [Mohan et al., 2013]. Since our emphasis was on nonparametric identification, we did not employ any restrictions not implied by the m-DAG factorization, including (i) relevant restrictions needed by instrumental variable or proxy methods, or (ii) constraints implied by state space restrictions. As Luo and Geng [2025] illustrated, such assumptions can sometimes yield identifiability, and their results highlight concrete testable conditions under which this occurs. We fully agree that self-censoring is of practical importance, with income surveys, sensitive health questions, and other social science settings providing common examples. Their discussion usefully demonstrates how auxiliary information, through baseline or follow-up variables, can be leveraged to address such scenarios. More broadly, we believe that in MNAR submodels that identify the target law, restrictions implied by the m-DAG always yield testable implications for the observed data law, with the 5 A(1) X Y (1) RA RY A Y (a) A(1) = A X Y (1) = Y rA = 1 rY = 1 A Y (b) A(1) = A a X Y (1,a) = Y (a) rA = 1 rY = 1 A Y (c) Figure 4: (a) Extension of Levis and Kennedy's example to have two missing variables. (b) Graph obtained after fixing RA and RY in parallel. (c) SWIG obtained by splitting the treatment variable A after already having fixed RA and RY . exception of the permutation model [Robins, 1997]. Examples of such tests have been developed in prior work on graphical models [Mohan and Pearl, 2014, Nabi and Bhattacharya, 2023, Guo et al., 2023, Chen et al., 2023]. Overall, we view their discussion as complementary to our focus: while our framework aimed to catalog nonparametric identification results, their work illustrates how additional structure can both sharpen identifiability and connect to practical applications. 2.3 On the discussion by Wang and van der Laan Wang and van der Laan [2025], like us, approach missingness through the lens of interventions. They emphasize the close relationship between missing data models and multivariate censoring models in survival analysis. They carefully prove that the permutation missingness model is MNAR rather than MAR and flesh out our example of why the permutation model is substantively plausible in some real world applications. They also discuss substantive settings where the block sequential model is plausible. They are less certain that the other identifiable MNAR models discussed in our paper are substantively plausible. In our view, it is difficult to establish plausibility of any missing data model in practice, whether it is formulated graphically or not. One advantage of the sort of theory we present is a single formulation for a large class of identifiable models (including existing models in the literature such as those described in [Zhou et al., 2010, Robins, 1997]). Since no single model may be plausible in applications, an alternative strategy is to perform the analysis of interest under a wide class of identifiable missingness models, as a form of sensitivity analysis. We make one final observation regarding the plausibility of (or the lack there of) the identifiable MNAR models considered in the paper. Identification in every case we discuss relies on an assumption that one or more counterfactuals L(1) j suffice to control confounding of the \"effect\" of Rk on Lk where j ̸= k. But why should we privilege L(1) j (or a set of such variables) as a set sufficient for adjustment? It seems much more plausible that there exist other unmeasured common causes of Rk and Lk (equivalently L(1) k ) that would also need to be adjusted for to eliminate confounding. Are we assuming the L(1) j suffice to control confounding because we believe it to be so or rather because we want to achieve identification for identification's sake? In fact, similar plausibility concerns arise in non-graphical identifiable models, such as non-monotone MAR. In our view, plausible models of missingness feature a description of data generation that follows a temporal order (often representable as a DAG). Unfortunately, to be realistic, even models of this type would feature unobserved confounding of the sort that would prevent identifiability. 6 2.4 On the discussion by Yang and Kim Yang and Kim [2025] emphasize the practical challenges of applying the m-DAG framework. They point out that while m-DAGs provide a principled way to encode assumptions and extend identification theory, their utility in practice depends critically on correct graph specification, which may be difficult to achieve in applied settings. They raise concerns about scalability in high-dimensional problems, where the number of variables and missingness patterns grows quickly, and about the feasibility of validating the conditional independence assumptions encoded in an m-DAG. They further stress the importance of developing systematic approaches to sensitivity analysis, noting that small misspecifications of the graph can have consequences for identification and downstream tasks. We thank Yang and Kim for raising important questions about the practical utility of m-DAGs, and graphical modeling more broadly. Our goal in this paper was primarily conceptual: to provide a general framework that connects causal and missing data perspectives and to catalog identification results that arise from this connection. Their emphasis on practical feasibility is appreciated, and complementary to our focus. That said, continuing work in the graphical modeling community over the last two decades has rendered much of the critiques of the graphical modeling approach to causal inference and missing data problems out of date. Identification of causal and missing data parameters must, by necessity, rely on equality restrictions in the full data distribution. Graphical models aim to encode these restrictions in a way that allows the construction of a plausible data generating mechanism consistent with a temporal order of events. Models that obtain identifiability without a corresponding graphical representation, for instance the non-monotone missing at random (MAR) model, often lack such a plausible mechanism. Indeed, existing efforts that argue for the plausibility of models such as non-monotone MAR embed them into a graphical model or a mixture of such models [Robins and Gill, 1997]. Over the last decade, an explosion of easy to use open-source packages that take advantage of graphical models have been developed for all tasks in causal inference and missing data, including establishing identification, constructing estimators and applying them to data, conducting sensitivity analyses, establishing bounds on non-identified parameters, and model selection. A non-exhaustive list of these packages includes: Ananke [Lee et al., 2023], autobounds [Duarte et al., 2024], dosearch [Tikka et al., 2021], DAGitty [Textor et al., 2016], software developed as part of the Tetrad project (equipped with Python and R interfaces) [Ramsey and Andrews, 2023], as well the pcalg package [Kalisch et al., 2012], and extensions to deal with missing values [Andrews et al., 2024]. Further, sensitivity analyses results can be fruitfully applied to graphical models. For example, nongraphical results in Robins et al. [1999] can easily be represented in graphical structures that encode MAR, conditional or sequential ignorability, and other types of Markov restrictions. For instance, the permutation model in Robins [1997] is shown to be a graphical model in our paper. In addition, graphical models allow a particularly powerful form of nonparametric sensitivity analysis, where consistent inferences are made in union models defined over a potentially large class of graphs that the analyst is uncertain about; see for example, [VanderWeele and Shpitser, 2011, Yang et al., 2024, Shpitser and VanderWeele, 2010, Shpitser et al., 2010, Chang et al., 2024, Wang et al., 2025]. For instance, the result in [VanderWeele and Shpitser, 2011] states that identification of the causal effect by covariate adjustment may be formulated without precise knowledge of the graph, but only via the set of common causes of the treatment and outcome, provided an adjustment set exists. The method developed by Yang et al. [2024] provide similar robustness guarantees with more complex types of identifying functionals, including the front-door functional, and the ratio functional arising in instrumental variable analysis. 7 We share the authors' view that algorithmic and computational advances will be needed to bring graphical identification methods to bear on high-dimensional problems with complex missingness patterns. We see this as an exciting frontier for future work, and one where continued integration of causal inference tools with missing data methodology will be especially fruitful. We believe smoothing and sparsity methods, as well as sum-product algorithm type methods developed in the graphical modeling literature may all be relevant for these advances [Lee et al., 2021]. 3 Conclusion We thank the discussants for their thoughtful contributions. Together, their contributions point toward a roadmap: combine graphical identification, auxiliary information, intervention-based censoring perspectives, efficiency-oriented estimation, and structured sensitivity analysis into a practical toolkit for MNAR problems. References Ryan M. Andrews, Christine W. Bang, Vanessa Didelez, Janine Witte, and Ronja Foraita. Software application profile: tpc and micd-R packages for causal discovery with incomplete cohort data. International Journal of Epidemiology, 53(5):dyae113, 2024. Rohit Bhattacharya, Razieh Nabi, Ilya Shpitser, and James Robins. Identification in missing data models represented by directed acyclic graphs. 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Statistical Science, 25(4):517-532, 2010. 10"}}},{"rowIdx":287,"cells":{"id":{"kind":"string","value":"2510.14693"},"raw":{"kind":"string","value":"FibRace: a large-scale benchmark of client-side\nproving on mobile devices\nSimon Malatrait\nsimon@kakarot.org\nKKRT Labs\nAlex Sirac\nalex@hyli.org\nHyli\nOctober 9, 2025\nAbstract\nFibRace, jointly developed by KKRT Labs and Hyli, was the first large-scale\nexperiment to test client-side proof generation on smartphones using Cairo M.\nPresented as a mobile game in which players proved Fibonacci numbers and\nclimbed a leaderboard, FibRace served a dual purpose: to engage the public\nand to provide empirical benchmarking. Over a three-week campaign (September\n11-30, 2025), 6,047 players across 99 countries generated 2,195,488 proofs on\n1,420 unique device models. The results show that most modern smartphones can\ncomplete a proof in under 5 seconds, confirming that mobile devices are now\ncapable of producing zero-knowledge proofs reliably, without the need\nfor remote provers or specialized hardware. Performance was correlated primarily\nwith RAM capacity and SoC (System on Chip) performance: devices with at least\n3 GB of RAM proved stably, when Apple’s A19 Pro and M-series chips achieved\nthe fastest proving times. Hyli’s blockchain natively verified every proof onchain\nwithout congestion. FibRace provides the most comprehensive dataset to date on\nmobile proving performance, establishing a practical baseline for future research in\nlightweight provers, proof-powered infrastructure, and privacy-preserving mobile\napplications.\nKeywords zero-knowledge proofs · client-side proving · mobile cryptography ·\nblockchain · Hyli · KKRT Labs\n\nContents\n1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1\n2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n2.2 The FibRace gamified campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2\n2.3 Proof generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.4 Data collection and dataset structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.5 Benchmark parameters and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n2.6 Ethical and experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.7 Data availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3 Benchmark results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3.1 Global participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n3.2 Proving performance distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n3.3 Hardware correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n3.4 Reliability and crashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3.5 Summary interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n4 Discussion and implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n4.1 Viability of client-side proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n4.2 Implications for developers and ecosystem builders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n4.3 Broader impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n4.4 Implications for Hyli and KKRT Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n5 Limitations of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n5.1 Technical limitations . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .. .. .. .. . . . . . .. .. .. .. . . . . . . . . . .. . . . 10\n5.2 Game design limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\nBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\nTables\nTable 1 Collected datasets during FibRace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\nTable 2 FibRace benchmarking assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\nFigures\nFigure 1 FibRace downloads by country. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\nFigure 2 FibRace Cairo M proof generation duration distribution (central 99%). . . . . . . . . . . . 5\nFigure 3 Key performance indicators per RAM capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\nFigure 4 Proof volume per device manufacturer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\nFigure 5 Best 50 SoCs for Cairo M proof generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n\n1 Introduction\nZero-knowledge proofs have become a cornerstone of blockchain scalability and privacy. In recent\nyears, most progress in proving technology has focused on server-side proving, powering large-\nscale systems such as Layer 2 roll-ups and blockchain settlement layers. These developments,\nspurred by launches of high-throughput proving networks in 2024 and 2025, have dramatically\nreduced latency and cost for infrastructure-level proofs.\nYet this focus leaves a gap in a critical area: client-side proving.\nServer-based models assume proofs are generated in centralized environments or distributed\nnetworks, but not on user devices. This architecture works well for throughput, not for privacy.\nSome use cases, especially those involving personal identity and private data, can only be\nrealized if proofs are generated locally. Confidentiality is no longer guaranteed when sensitive\ndata leaves the user’s device for processing elsewhere.\nClient-side proving minimizes friction for both developers and users. Developers no longer need\nto maintain or pay for remote-proving infrastructure: computation is delegated to end-user\ndevices. Users, in turn, gain stronger privacy guarantees: proofs are generated and owned locally,\nwithout ever exposing private data to external servers or custodial intermediaries.\nRecent incidents highlight this need vividly: as we write this report in October 2025, millions of\nDiscord users’ government IDs were leaked following a third-party verification breach, reigniting\ndebates about how identity management should be handled online (Chia 2025). Similarly,\nsystems like Google’s zkWallet with Ligero and other emerging proof-of-age or proof-of-identity\napplications show that for many privacy-preserving use cases, the proof must be generated by\nthe user, not by a remote prover (Kuhn 2025).\nDespite the progress in ZK proving systems, client-side proving remains a technical challenge.\nFor instance, a 2025 benchmark across several proving schemes showed that in-browser proving\nfor p256 ECDSA signatures was still largely impractical, with only Noir achieving limited\nusability (Vlad 2025).\nTo achieve privacy and scalability at a global level, proof generation must become viable across\nthe full spectrum of consumer devices, including low-end smartphones. KKRT Labs’s Cairo\nM, a ZK-VM optimized for mobile hardware, was designed to address this challenge (KKRT\nLabs 2025).\nDeveloping for client-side proving inevitably raises a central question: Can we generate\nproofs directly on consumer devices, at scale, and under real-world conditions?\nIn September 2025, FibRace was created as a joint initiative by KKRT Labs, developers\nof the Cairo M prover, and Hyli, the proof-powered blockchain for the next generation of\napplications. By disguising a client-side proving benchmark as a mobile game, it allowed us to\ntest thousands of device models worldwide and begin answering this question experimentally.\n1\n\n2 Methodology\n2.1 Overview\nThe FibRace campaign was designed as a public mobile game and a controlled benchmark\nof client-side proving using Cairo M.\nPlayers interacted with a game interface, computing Fibonacci numbers and climbing\nleaderboards (Sirac 2025). Each interaction generated a data point for the study. This dual\nframing allowed the experiment to reach thousands of devices under natural, real-world usage\nconditions.\n2.2 The FibRace gamified campaign\nFibRace was a mobile game designed to turn a serious experiment in zero-knowledge\nbenchmarking into an accessible and playful experience. Its core challenge was simple: prove\nFibonacci numbers as quickly as possible and climb the leaderboard.\nEach proof was generated locally on the player’s smartphone using Cairo M, KKRT Labs’s\nprover optimized for mobile hardware (Walter 2025). Once a proof was produced, it was\ntransmitted to Hyli’s proof-powered blockchain, which natively verifies Cairo M proofs and\nsettles them onchain. Every successful proof advanced the player in the game and added a real\ndata point to the benchmark.\nThe campaign ran from September 11 to 30, 2025, and reached a diverse global audience. Players\nfrom 99 countries participated across Android and iOS devices, 5,156 of them generating over\n2 million proofs on 1,420 unique device models.\n2.2.1 Gameplay\nParticipants could:\n1. Generate a proof of the computation of the 𝑛-th term of the Fibonacci sequence, for a\nrandomly chosen 𝑛.\n2. Mint a collector card that displayed their number and proving time, designed for easy\nsharing on social media.\n3. Compete on leaderboards tracking three metrics (fastest proof, highest proving\nfrequency, and largest collection) to encourage repeated participation.\nNo financial rewards, airdrops, or tradeable assets were offered. This ensured that gameplay\nreflected genuine engagement and natural device diversity rather than speculative incentives.\n2.2.2 Benchmark connection\nEvery player session generated measurable data collected automatically through the app’s\nleaderboard and device-logging modules. This structure enabled KKRT Labs and Hyli to gather\nan unprecedented dataset on real-world proving performance.\nBy merging gaming dynamics with experimental rigor, FibRace shows that large-scale\nbenchmarking of zero-knowledge provers can occur in open, decentralized conditions, engaging\nusers not as test subjects, but as participants in a game.\n2\n\n2.3 Proof generation\nEach game session consisted of the following sequence:\n1. Sampling: The app pseudo-randomly sampled a 𝑛 in [1..100, 000] for which to compute the\n𝑛-th term of the Fibonacci sequence 𝐹𝑛 on the M31 prime field.\n2. Proof generation: The player’s smartphone executed the Cairo M program locally to prove\nthe correct computation of 𝐹𝑛, with 96 security bits. The higher the number 𝑛, the more\nresource-intensive the proving.\n3. Verification and settlement: The proof was sent to Hyli, where it was natively verified and\nsettled on the Hyli testnet blockchain.\n4. Data logging: Metadata, including device model, OS version, RAM, SoC information, proof\nduration, and frequency, was recorded client-side and uploaded once verification succeeded.\nAll computation and proof generation occurred entirely on-device. No remote provers, cloud\nrelays, or trusted execution environments were used.\n2.4 Data collection and dataset structure\nFour datasets were collected during the campaign, described in Table 1. The initial analysis\nwas performed in a Jupyter Notebook, as detailed in (Malatrait 2025):\nDataset\nDescription\nmintedItems\nOne record per successfully generated and verified proof (2,195,488 rows)\nplayers\nPlayer profiles, including historical best proof and mint counters\ndevices\nHardware and OS metadata for each registered device (1,420 unique models)\ncrashLogs\nReports of failed or incomplete proof attempts, including device identifiers\nTable 1: Collected datasets during FibRace.\nEach entry is timestamped.\n2.5 Benchmark parameters and assumptions\nThe campaign defined a set of baseline assumptions to guide analysis and interpretation,\nsummarized in Table 2.\nA proof is deemed valid if successfully generated, sent to Hyli, verified by Hyli, and settled\nonchain on Hyli. The sampling window was the duration of the FibRace campaign, September 11\nto 30, 2025. The program used for the benchmark would be a fixed Cairo M program computing\nthe randomly chosen 𝑛-th term of the Fibonacci sequence.\n3\n\nParameter\nDescription\nAssumed / Target Value\nMinimum usable device RAM\nWe \nconsider \nthat \nclient-side\nproving is viable if successful\non enough devices with a RAM\nhigher than this.\n≥ 3 GB\nAcceptable crash rate\nWe considered Cairo M was\nsuccessful on device models under\na certain failure threshold.\n< 1% failure rate\nTarget proof duration\nWe considered Cairo M offered\na smooth user experience if the\nmedian proving time was under\nthis threshold.\n≤ 5 s per proof\nTable 2: FibRace benchmarking assumptions.\nKKRT Labs used these thresholds to evaluate Cairo M’s performance. Hyli used them to\ncharacterize onchain verification throughput.\n2.6 Ethical and experimental considerations\n• Voluntary participation: Players opted in by installing the application. No rewards,\nairdrops, or financial incentives were offered, and this was made clear in the launch posts as\nwell as in the application’s interface.\n• Data minimization: Only technical and performance metrics were collected. No personal\nidentifiers were stored. FibRace did not enable account creation or data portability.\n• Cross-platform coverage: Android and iOS were supported, ensuring SoC diversity: Apple\nSilicon, Qualcomm Snapdragon, MediaTek, Huawei Kirin, and Samsung Exynos.\n2.7 Data availability\nThe performance report and raw datasets are publicly available at https://github.com/kkrt-\nlabs/cairo-m/tree/main/docs/fibrace.\n3 Benchmark results\n3.1 Global participation\nAcross the 19-day campaign, 6,047 player names were registered. While some players\nregistered multiple player names (due to multi-device use or reinstalling the application), we\nwill refer to them as « players » in the remainder of this report. Of these players, 5,156 generated\nat least one valid proof.\nThe players used 1,420 unique device models. Device models are provided by the\nmanufacturers themselves and read from the OS API. According to the app stores’ data, the\ngame was installed by players in 99 countries. The Figure 1 is an heatmap of the number of\nFibRace downloads per country.\nIn total, 2,195,488 proofs were generated on FibRace during the campaign.\n4\n\nFigure 1: FibRace downloads by country.\n3.2 Proving performance distribution\nThe duration of proof generation followed a right-skewed distribution, shown in Figure 2, with\nmost devices completing a proof within 3 to 10 seconds. The average frequency (Cairo M\ncycles per second) was 30 kHz, but the median was closer to 10 kHz, reflecting the heavy tail\nof slower devices.\nHigh-performance clusters (≤ 3 s proofs) were led by the most recent iPhone and iPad models\nrunning Apple A19 Pro, M1–M4 chips, followed by older ones and high-end Android phones.\nMid-range Android devices populated the 5–10 s band. Older devices (with an average proving\ntime of more than 10s) showed increased crash incidence and throttling effects.\nFigure 2: FibRace Cairo M proof generation duration distribution (central 99%).\n5\n\n3.3 Hardware correlations\n3.3.1 RAM Capacity\nFigure 3 displays key performance indicators, grouped by RAM buckets.\nAt least 3 GB of available memory is required for successful proving on Cairo M.\nDevices with 3–4 GB RAM were the most prone to crashes due to Cairo M’s memory footprint,\nwhile stability scaled steadily with RAM until plateauing beyond 6 GB.\nFigure 3: Key performance indicators per RAM capacity.\n3.3.2 Device and manufacturer trends\nMost proofs were generated on Xiaomi, Apple, and Samsung devices, as per Figure 4, reflecting\nglobal market share rather than platform bias (Kuzmin 2025). Within each brand, newer chip\ngenerations demonstrated predictable improvements in both duration and frequency.\nFigure 4: Proof volume per device manufacturer.\n6\n\nFigure 5 ranks the 50 best SoC, performance wise. It has been sorted based on proof duration\nas well as frequency, with best score as well as best average.\nApple SoCs dominated the upper performance tier, with the latest iPhone chips and M-series\non iPad. iPads with the Apple M4 SoC and iPhones with the A19 Pro achieved the fastest\nproving times, approaching the limits of what Cairo M can deliver on mobile hardware.\nFollowing the most recent Apple SoCs, high-end SoCs from Qualcomm, MediaTek, Kirin, and\nExynos achieved great performance, with a proving time under 3 seconds.\nFigure 5: Best 50 SoCs for Cairo M proof generation.\n3.4 Reliability and crashes\n3.4.1 Reliability per device\nStable devices maintained success rates above 95%, confirming that Cairo M can operate\nreliably on modern smartphones without external proving support.\n7\n\nThe overall crash rate during the FibRace campaign is 0.72%, representing the number of\nrecorded crash events (15,762 occurrences) over the total number of proof successfully generated.\nThe majority of these occurred on devices with limited RAM (3-4 GB) as shown in Figure 3,\nwhich are often over five years old.\nCrash rates correlated strongly with memory pressure rather than CPU limitations, suggesting\nthat future optimizations should prioritize reducing the RAM footprint.\n3.4.2 Temporal trends\nHourly aggregates showed distinct activity peaks corresponding to regional evening hours,\ndemonstrating genuine player engagement rather than automated testing.\nProof throughput and crash incidence followed inversely proportional patterns: as participation\nincreased, minor rises in crash frequency indicated thermal throttling or sustained high-load\nconditions on some devices.\nWe also encountered a crash error on higher-RAM devices when minting many times in a row\n(usually more than a hundred). While unable to identify the root cause of these crashes with\ncertainty, we assume that, although the spawned Rust thread ought to be killed at the end\nof each generated proof, it may not have been properly done, which could lead to OOM-type\ncrashes. The short delay incurred when verifying and settling the proof onchain between two\nproof generation might be the cause. If this delay is too short, the thread may not properly\nclose. Due to the short campaign duration and to the benchmark’s focus on one-time proving,\nwe did not investigate the issue further.\n3.5 Summary interpretation\n• Performance: Modern smartphones can generate Cairo M proofs in under 5 seconds on\naverage.\n• Accessibility: Proofs succeeded on 1,400+ device models, proving wide feasibility.\n• Reliability: Devices with ≥ 3 GB RAM show consistent stability; below that threshold,\nproving fails systematically.\n• Hardware Insight: CPU frequency correlates strongly with proof frequency, confirming\nCairo M’s effective scaling on mobile hardware.\n4 Discussion and implications\n4.1 Viability of client-side proving\nThe FibRace experiment demonstrates that client-side proving using Cairo M is technically\nviable and globally accessible.\nOver 2 million proofs were generated, totaling 333 billions Cairo M cycles on more than 1,400\ndistinct device models, confirming that local proof generation can operate reliably across diverse\nhardware and network environments.\nThe results validate that the majority of modern smartphones can generate zero-knowledge\nproofs in under 5 seconds, which is well within the user-acceptable latency for casual or\ntransactional applications. Devices with less than 3 GB of RAM consistently failed to complete\na proof, delineating a clear lower bound for current implementations.\n8\n\nFrom a research perspective, FibRace provides the first large-scale, open dataset that quantifies\nthe distribution of proof generation times across consumer hardware. This enables empirical\ncalibration of future prover optimizations and helps refine benchmarks for memory footprint,\ncycle counts, and acceptable crash rates.\n4.2 Implications for developers and ecosystem builders\nFor developers, these results are a strong signal that end-user devices can handle proving\nworkloads previously reserved for servers or specialized hardware.\nClient-side proving simplifies architecture, reduces cost, and improves privacy, aligning with the\nlong-term goal of self-sovereign computation.\nIn practical terms, this means applications can now:\n• Deploy zero-knowledge solutions without relying on remote provers.\n• Execute verifiable interactions offline, syncing proofs asynchronously to Hyli.\n• Build new interaction models (e.g., games, attestations, on-device identity proofs) that no\nlonger depend on cloud resources.\nFor ecosystem builders, FibRace establishes a methodology for gamified benchmarking,\ncollecting real-world performance data in a playful and opt-in format. This approach can be\nreplicated for other proving schemes or for measuring device-specific bottlenecks, such as GPU\nacceleration or energy efficiency.\n4.3 Broader impact\nFibRace bridges the gap between cryptographic research and user experience. Re-framing\nbenchmarking as a public game showcased how ZK technology can reach mainstream audiences\nwithout compromising scientific rigor.\nFuture iterations of this model could benchmark additional proving schemes, introduce\ncontrolled parameters (e.g., energy consumption, thermal throttling), and work upon different\ncircuits.\n4.4 Implications for Hyli and KKRT Labs\nFor Hyli, the experiment validated the scalability of its onchain proof verification layer. Over\ntwo million verifications were processed without congestion, even during peaks with 260,000\nproofs per hour, demonstrating that Hyli’s implementation of the Autobahn consensus can\nabsorb high-frequency proof submissions from a large user base (Giridharan et al. 2024).\nFor KKRT Labs, FibRace served as a proof-of-concept for Cairo M, showing that its\nmobile optimization can sustain multi-million-proof throughput across everyday devices. The\nbenchmark offers valuable feedback for future improvements.\nTogether, Hyli and KKRT Labs demonstrated that a proof-powered L1 and a mobile-optimized\nprover can function seamlessly, paving the way for scalable, privacy-preserving applications that\ncan be deployed to consumer hardware.\n9\n\n5 Limitations of the experiment\nWhile FibRace demonstrates the viability of client-side proving on mobile devices, several\nlimitations must be acknowledged to correctly interpret the dataset and results.\n5.1 Technical limitations\n5.1.1 Benchmark circuit scope\nThe experiment focused exclusively on proving computations of terms of the Fibonacci sequence,\nan arithmetic circuit with low memory usage.\nWhile representative for testing Cairo M baseline performance, and while we benched a wide\nrange of witnesses, resulting in traces of 8 to 800,000 Cairo M cycles, it does not capture all of the\nCairo M capabilities that can be found in real-world proving workloads, such as SHA2 hashes.\n5.1.2 Memory footprint\nCairo M’s memory consumption constrained participation on low-end devices. Proofs could not\nbe generated on devices with less than 3 GB of RAM, and devices with 3–4 GB of RAM showed\nthe highest crash rate, respectively 4.2 and 4.4 crashes per generated proof, due to OOM errors\non the largest witnesses (𝑛 near the 100,000 upper-bound).\nIt showcases that devices with limited available memory can generate proofs for most workloads\nof the benchmark, but not for the most expensive ones, due to memory constraints.\n5.1.3 Energy and thermal effects\nFibRace did not record power draw or thermal throttling. Anecdotal player feedback suggested\nnoticeable heat and battery drain during extended play sessions, but these factors were not\nquantified.\nOther performance factors were not measured, including battery state, energy saving mode\nactivation, and CPU temperature.\nTherefore we do not have a quantitative performance degradation ratio on our assumption that\nthese factors lead to degraded performance.\nFuture benchmarks should include energy-use metrics to more effectively evaluate sustainability,\nuser comfort and performance impact.\n5.2 Game design limitations\n5.2.1 Gameplay constraints\nBecause FibRace was designed as a game, participants could interrupt or close the app at any\npoint, including by force-quitting the app while generating or sending a proof. Such interruptions\nare indistinguishable from technical crashes in the dataset, introducing ambiguity in interpreting\nsome failure events.\n5.2.2 No user accounts\nThe game operated without persistent user accounts to reduce friction and prevent airdrop\nspeculation. While this ensured privacy and simplicity, it made longitudinal tracking of player\nbehavior and device switching impossible. This did not affect the benchmark, but made it less\nenjoyable for some players.\n10\n\n5.3 Summary\nThese limitations do not diminish the core finding that modern smartphones can locally perform\nzero-knowledge proving. They do, however, highlight key areas for future experiments:\n• measuring more complex circuits,\n• measuring circuits with real-life applications,\n• quantifying energy and heat effect on performance.\nTogether, these refinements would strengthen the empirical foundation laid by FibRace and\nsupport the next generation of client-side proving research.\n6 Conclusion\nFibRace demonstrates that client-side proving is no longer theoretical: it is operational,\nmeasurable, and already viable across a wide range of consumer devices.\nOver the course of three weeks, more than 5,000 participants generated 2.2 million Cairo M\nproofs on 1,420 unique device models, confirming that proof generation can occur reliably on\nmodern smartphones without the need for centralized infrastructure or trusted hardware.\nThe results validate a key assumption behind both KKRT Labs and Hyli: that distributed\nproving and native onchain verification can coexist efficiently. Cairo M made on-device proof\ngeneration accessible even to mid-range devices, while Hyli’s proof-powered Layer 1 absorbed\nmillions of verifications without network congestion.\nBeyond its immediate findings, FibRace established a new model for open benchmarking\nthrough gamified participation. By blending experimentation and gameplay, it produced one of\nthe most extensive public datasets on client-side proving to date, an approach that future ZK\necosystems can replicate for other circuits, provers, or device classes.\nThe experiment’s limitations define clear next steps. Expanding to real-world use cases, and\nintegrating thermal and battery instrumentation.\nUltimately, FibRace offers a simple but powerful insight: zero-knowledge proofs can now\nrun on users’ own devices.\nThis shift opens a new frontier for privacy-preserving applications and proof-native blockchains,\nwhere the path to scalable cryptography becomes user-driven.\n11\n\nBibliography\n[1] O. Chia, “ID photos of 70,000 users may have been leaked, Discord says,” BBC News,\nOct. 2025.\n[2] D. Kuhn, “Google Wallet integrates ZK-proofs, a tech incubated by crypto industry,” The\nBlock, May 2025.\n[3] Vlad, “Benchmarking in-browser p256 ECDSA proving systems,” Hyli, Mar. 2025.\n[4] KKRT Labs, “Kakarot presents Cairo M: The Frontier zkVM Built for Your Phone.” Oct.\n08, 2025.\n[5] A. Sirac, “Launching FibRace: Prove to win!,” Hyli, Sep. 2025.\n[6] C. Walter, “Cairo M Design Document.” Sep. 29, 2025.\n[7] S. Malatrait, “FibRace performance report.” Oct. 01, 2025.\n[8] A. Kuzmin, “Baseline Benchmarking Hardware.” Mar. 31, 2025.\n[9] N. Giridharan, F. Suri-Payer, I. Abraham, L. Alvisi, and N. Crooks, “Autobahn: Seamless\nhigh speed BFT,” in Proceedings of the ACM SIGOPS 30th Symposium on Operating\nSystems Principles, New York, NY, USA, Nov. 2024, pp. 1–23.\n12\n"},"cleaned":{"kind":"string","value":"FibRace: a large-scale benchmark of client-side proving on mobile devices Simon Malatrait KKRT Labs Alex Sirac Hyli October 9, 2025 Abstract FibRace, jointly developed by KKRT Labs and Hyli, was the first large-scale experiment to test client-side proof generation on smartphones using Cairo M. Presented as a mobile game in which players proved Fibonacci numbers and climbed a leaderboard, FibRace served a dual purpose: to engage the public and to provide empirical benchmarking. Over a three-week campaign (September 11-30, 2025), 6,047 players across 99 countries generated 2,195,488 proofs on 1,420 unique device models. The results show that most modern smartphones can complete a proof in under 5 seconds, confirming that mobile devices are now capable of producing zero-knowledge proofs reliably, without the need for remote provers or specialized hardware. Performance was correlated primarily with RAM capacity and SoC (System on Chip) performance: devices with at least 3 GB of RAM proved stably, when Apple's A19 Pro and M-series chips achieved the fastest proving times. Hyli's blockchain natively verified every proof onchain without congestion. FibRace provides the most comprehensive dataset to date on mobile proving performance, establishing a practical baseline for future research in lightweight provers, proof-powered infrastructure, and privacy-preserving mobile applications. Keywords zero-knowledge proofs · client-side proving · mobile cryptography · blockchain · Hyli · KKRT Labs Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The FibRace gamified campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Proof generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Data collection and dataset structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5 Benchmark parameters and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.6 Ethical and experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.7 Data availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Benchmark results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1 Global participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Proving performance distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Hardware correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Reliability and crashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.5 Summary interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Discussion and implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1 Viability of client-side proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Implications for developers and ecosystem builders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Broader impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4 Implications for Hyli and KKRT Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Limitations of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.1 Technical limitations . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .. .. .. .. . . . . . .. .. .. .. . . . . . . . . . .. . . . 10 5.2 Game design limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Tables Table 1 Collected datasets during FibRace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Table 2 FibRace benchmarking assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Figures Figure 1 FibRace downloads by country. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 2 FibRace Cairo M proof generation duration distribution (central 99%). . . . . . . . . . . . 5 Figure 3 Key performance indicators per RAM capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 4 Proof volume per device manufacturer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 5 Best 50 SoCs for Cairo M proof generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Introduction Zero-knowledge proofs have become a cornerstone of blockchain scalability and privacy. In recent years, most progress in proving technology has focused on server-side proving, powering largescale systems such as Layer 2 roll-ups and blockchain settlement layers. These developments, spurred by launches of high-throughput proving networks in 2024 and 2025, have dramatically reduced latency and cost for infrastructure-level proofs. Yet this focus leaves a gap in a critical area: client-side proving. Server-based models assume proofs are generated in centralized environments or distributed networks, but not on user devices. This architecture works well for throughput, not for privacy. Some use cases, especially those involving personal identity and private data, can only be realized if proofs are generated locally. Confidentiality is no longer guaranteed when sensitive data leaves the user's device for processing elsewhere. Client-side proving minimizes friction for both developers and users. Developers no longer need to maintain or pay for remote-proving infrastructure: computation is delegated to end-user devices. Users, in turn, gain stronger privacy guarantees: proofs are generated and owned locally, without ever exposing private data to external servers or custodial intermediaries. Recent incidents highlight this need vividly: as we write this report in October 2025, millions of Discord users' government IDs were leaked following a third-party verification breach, reigniting debates about how identity management should be handled online (Chia 2025). Similarly, systems like Google's zkWallet with Ligero and other emerging proof-of-age or proof-of-identity applications show that for many privacy-preserving use cases, the proof must be generated by the user, not by a remote prover (Kuhn 2025). Despite the progress in ZK proving systems, client-side proving remains a technical challenge. For instance, a 2025 benchmark across several proving schemes showed that in-browser proving for p256 ECDSA signatures was still largely impractical, with only Noir achieving limited usability (Vlad 2025). To achieve privacy and scalability at a global level, proof generation must become viable across the full spectrum of consumer devices, including low-end smartphones. KKRT Labs's Cairo M, a ZK-VM optimized for mobile hardware, was designed to address this challenge (KKRT Labs 2025). Developing for client-side proving inevitably raises a central question: Can we generate proofs directly on consumer devices, at scale, and under real-world conditions? In September 2025, FibRace was created as a joint initiative by KKRT Labs, developers of the Cairo M prover, and Hyli, the proof-powered blockchain for the next generation of applications. By disguising a client-side proving benchmark as a mobile game, it allowed us to test thousands of device models worldwide and begin answering this question experimentally. 1 2 Methodology 2.1 Overview The FibRace campaign was designed as a public mobile game and a controlled benchmark of client-side proving using Cairo M. Players interacted with a game interface, computing Fibonacci numbers and climbing leaderboards (Sirac 2025). Each interaction generated a data point for the study. This dual framing allowed the experiment to reach thousands of devices under natural, real-world usage conditions. 2.2 The FibRace gamified campaign FibRace was a mobile game designed to turn a serious experiment in zero-knowledge benchmarking into an accessible and playful experience. Its core challenge was simple: prove Fibonacci numbers as quickly as possible and climb the leaderboard. Each proof was generated locally on the player's smartphone using Cairo M, KKRT Labs's prover optimized for mobile hardware (Walter 2025). Once a proof was produced, it was transmitted to Hyli's proof-powered blockchain, which natively verifies Cairo M proofs and settles them onchain. Every successful proof advanced the player in the game and added a real data point to the benchmark. The campaign ran from September 11 to 30, 2025, and reached a diverse global audience. Players from 99 countries participated across Android and iOS devices, 5,156 of them generating over 2 million proofs on 1,420 unique device models. 2.2.1 Gameplay Participants could: 1. Generate a proof of the computation of the n-th term of the Fibonacci sequence, for a randomly chosen n. 2. Mint a collector card that displayed their number and proving time, designed for easy sharing on social media. 3. Compete on leaderboards tracking three metrics (fastest proof, highest proving frequency, and largest collection) to encourage repeated participation. No financial rewards, airdrops, or tradeable assets were offered. This ensured that gameplay reflected genuine engagement and natural device diversity rather than speculative incentives. 2.2.2 Benchmark connection Every player session generated measurable data collected automatically through the app's leaderboard and device-logging modules. This structure enabled KKRT Labs and Hyli to gather an unprecedented dataset on real-world proving performance. By merging gaming dynamics with experimental rigor, FibRace shows that large-scale benchmarking of zero-knowledge provers can occur in open, decentralized conditions, engaging users not as test subjects, but as participants in a game. 2 2.3 Proof generation Each game session consisted of the following sequence: 1. Sampling: The app pseudo-randomly sampled a n in [1..100, 000] for which to compute the n-th term of the Fibonacci sequence Fn on the M31 prime field. 2. Proof generation: The player's smartphone executed the Cairo M program locally to prove the correct computation of Fn, with 96 security bits. The higher the number n, the more resource-intensive the proving. 3. Verification and settlement: The proof was sent to Hyli, where it was natively verified and settled on the Hyli testnet blockchain. 4. Data logging: Metadata, including device model, OS version, RAM, SoC information, proof duration, and frequency, was recorded client-side and uploaded once verification succeeded. All computation and proof generation occurred entirely on-device. No remote provers, cloud relays, or trusted execution environments were used. 2.4 Data collection and dataset structure Four datasets were collected during the campaign, described in Table 1. The initial analysis was performed in a Jupyter Notebook, as detailed in (Malatrait 2025): Dataset Description mintedItems One record per successfully generated and verified proof (2,195,488 rows) players Player profiles, including historical best proof and mint counters devices Hardware and OS metadata for each registered device (1,420 unique models) crashLogs Reports of failed or incomplete proof attempts, including device identifiers Table 1: Collected datasets during FibRace. Each entry is timestamped. 2.5 Benchmark parameters and assumptions The campaign defined a set of baseline assumptions to guide analysis and interpretation, summarized in Table 2. A proof is deemed valid if successfully generated, sent to Hyli, verified by Hyli, and settled onchain on Hyli. The sampling window was the duration of the FibRace campaign, September 11 to 30, 2025. The program used for the benchmark would be a fixed Cairo M program computing the randomly chosen n-th term of the Fibonacci sequence. 3 Parameter Description Assumed / Target Value Minimum usable device RAM We consider that client-side proving is viable if successful on enough devices with a RAM higher than this. ≥ 3 GB Acceptable crash rate We considered Cairo M was successful on device models under a certain failure threshold. < 1% failure rate Target proof duration We considered Cairo M offered a smooth user experience if the median proving time was under this threshold. ≤ 5 s per proof Table 2: FibRace benchmarking assumptions. KKRT Labs used these thresholds to evaluate Cairo M's performance. Hyli used them to characterize onchain verification throughput. 2.6 Ethical and experimental considerations • Voluntary participation: Players opted in by installing the application. No rewards, airdrops, or financial incentives were offered, and this was made clear in the launch posts as well as in the application's interface. • Data minimization: Only technical and performance metrics were collected. No personal identifiers were stored. FibRace did not enable account creation or data portability. • Cross-platform coverage: Android and iOS were supported, ensuring SoC diversity: Apple Silicon, Qualcomm Snapdragon, MediaTek, Huawei Kirin, and Samsung Exynos. 2.7 Data availability The performance report and raw datasets are publicly available at https://github.com/kkrtlabs/cairo-m/tree/main/docs/fibrace. 3 Benchmark results 3.1 Global participation Across the 19-day campaign, 6,047 player names were registered. While some players registered multiple player names (due to multi-device use or reinstalling the application), we will refer to them as « players » in the remainder of this report. Of these players, 5,156 generated at least one valid proof. The players used 1,420 unique device models. Device models are provided by the manufacturers themselves and read from the OS API. According to the app stores' data, the game was installed by players in 99 countries. The Figure 1 is an heatmap of the number of FibRace downloads per country. In total, 2,195,488 proofs were generated on FibRace during the campaign. 4 Figure 1: FibRace downloads by country. 3.2 Proving performance distribution The duration of proof generation followed a right-skewed distribution, shown in Figure 2, with most devices completing a proof within 3 to 10 seconds. The average frequency (Cairo M cycles per second) was 30 kHz, but the median was closer to 10 kHz, reflecting the heavy tail of slower devices. High-performance clusters (≤ 3 s proofs) were led by the most recent iPhone and iPad models running Apple A19 Pro, M1-M4 chips, followed by older ones and high-end Android phones. Mid-range Android devices populated the 5-10 s band. Older devices (with an average proving time of more than 10s) showed increased crash incidence and throttling effects. Figure 2: FibRace Cairo M proof generation duration distribution (central 99%). 5 3.3 Hardware correlations 3.3.1 RAM Capacity Figure 3 displays key performance indicators, grouped by RAM buckets. At least 3 GB of available memory is required for successful proving on Cairo M. Devices with 3-4 GB RAM were the most prone to crashes due to Cairo M's memory footprint, while stability scaled steadily with RAM until plateauing beyond 6 GB. Figure 3: Key performance indicators per RAM capacity. 3.3.2 Device and manufacturer trends Most proofs were generated on Xiaomi, Apple, and Samsung devices, as per Figure 4, reflecting global market share rather than platform bias (Kuzmin 2025). Within each brand, newer chip generations demonstrated predictable improvements in both duration and frequency. Figure 4: Proof volume per device manufacturer. 6 Figure 5 ranks the 50 best SoC, performance wise. It has been sorted based on proof duration as well as frequency, with best score as well as best average. Apple SoCs dominated the upper performance tier, with the latest iPhone chips and M-series on iPad. iPads with the Apple M4 SoC and iPhones with the A19 Pro achieved the fastest proving times, approaching the limits of what Cairo M can deliver on mobile hardware. Following the most recent Apple SoCs, high-end SoCs from Qualcomm, MediaTek, Kirin, and Exynos achieved great performance, with a proving time under 3 seconds. Figure 5: Best 50 SoCs for Cairo M proof generation. 3.4 Reliability and crashes 3.4.1 Reliability per device Stable devices maintained success rates above 95%, confirming that Cairo M can operate reliably on modern smartphones without external proving support. 7 The overall crash rate during the FibRace campaign is 0.72%, representing the number of recorded crash events (15,762 occurrences) over the total number of proof successfully generated. The majority of these occurred on devices with limited RAM (3-4 GB) as shown in Figure 3, which are often over five years old. Crash rates correlated strongly with memory pressure rather than CPU limitations, suggesting that future optimizations should prioritize reducing the RAM footprint. 3.4.2 Temporal trends Hourly aggregates showed distinct activity peaks corresponding to regional evening hours, demonstrating genuine player engagement rather than automated testing. Proof throughput and crash incidence followed inversely proportional patterns: as participation increased, minor rises in crash frequency indicated thermal throttling or sustained high-load conditions on some devices. We also encountered a crash error on higher-RAM devices when minting many times in a row (usually more than a hundred). While unable to identify the root cause of these crashes with certainty, we assume that, although the spawned Rust thread ought to be killed at the end of each generated proof, it may not have been properly done, which could lead to OOM-type crashes. The short delay incurred when verifying and settling the proof onchain between two proof generation might be the cause. If this delay is too short, the thread may not properly close. Due to the short campaign duration and to the benchmark's focus on one-time proving, we did not investigate the issue further. 3.5 Summary interpretation • Performance: Modern smartphones can generate Cairo M proofs in under 5 seconds on average. • Accessibility: Proofs succeeded on 1,400+ device models, proving wide feasibility. • Reliability: Devices with ≥ 3 GB RAM show consistent stability; below that threshold, proving fails systematically. • Hardware Insight: CPU frequency correlates strongly with proof frequency, confirming Cairo M's effective scaling on mobile hardware. 4 Discussion and implications 4.1 Viability of client-side proving The FibRace experiment demonstrates that client-side proving using Cairo M is technically viable and globally accessible. Over 2 million proofs were generated, totaling 333 billions Cairo M cycles on more than 1,400 distinct device models, confirming that local proof generation can operate reliably across diverse hardware and network environments. The results validate that the majority of modern smartphones can generate zero-knowledge proofs in under 5 seconds, which is well within the user-acceptable latency for casual or transactional applications. Devices with less than 3 GB of RAM consistently failed to complete a proof, delineating a clear lower bound for current implementations. 8 From a research perspective, FibRace provides the first large-scale, open dataset that quantifies the distribution of proof generation times across consumer hardware. This enables empirical calibration of future prover optimizations and helps refine benchmarks for memory footprint, cycle counts, and acceptable crash rates. 4.2 Implications for developers and ecosystem builders For developers, these results are a strong signal that end-user devices can handle proving workloads previously reserved for servers or specialized hardware. Client-side proving simplifies architecture, reduces cost, and improves privacy, aligning with the long-term goal of self-sovereign computation. In practical terms, this means applications can now: • Deploy zero-knowledge solutions without relying on remote provers. • Execute verifiable interactions offline, syncing proofs asynchronously to Hyli. • Build new interaction models (e.g., games, attestations, on-device identity proofs) that no longer depend on cloud resources. For ecosystem builders, FibRace establishes a methodology for gamified benchmarking, collecting real-world performance data in a playful and opt-in format. This approach can be replicated for other proving schemes or for measuring device-specific bottlenecks, such as GPU acceleration or energy efficiency. 4.3 Broader impact FibRace bridges the gap between cryptographic research and user experience. Re-framing benchmarking as a public game showcased how ZK technology can reach mainstream audiences without compromising scientific rigor. Future iterations of this model could benchmark additional proving schemes, introduce controlled parameters (e.g., energy consumption, thermal throttling), and work upon different circuits. 4.4 Implications for Hyli and KKRT Labs For Hyli, the experiment validated the scalability of its onchain proof verification layer. Over two million verifications were processed without congestion, even during peaks with 260,000 proofs per hour, demonstrating that Hyli's implementation of the Autobahn consensus can absorb high-frequency proof submissions from a large user base (Giridharan et al. 2024). For KKRT Labs, FibRace served as a proof-of-concept for Cairo M, showing that its mobile optimization can sustain multi-million-proof throughput across everyday devices. The benchmark offers valuable feedback for future improvements. Together, Hyli and KKRT Labs demonstrated that a proof-powered L1 and a mobile-optimized prover can function seamlessly, paving the way for scalable, privacy-preserving applications that can be deployed to consumer hardware. 9 5 Limitations of the experiment While FibRace demonstrates the viability of client-side proving on mobile devices, several limitations must be acknowledged to correctly interpret the dataset and results. 5.1 Technical limitations 5.1.1 Benchmark circuit scope The experiment focused exclusively on proving computations of terms of the Fibonacci sequence, an arithmetic circuit with low memory usage. While representative for testing Cairo M baseline performance, and while we benched a wide range of witnesses, resulting in traces of 8 to 800,000 Cairo M cycles, it does not capture all of the Cairo M capabilities that can be found in real-world proving workloads, such as SHA2 hashes. 5.1.2 Memory footprint Cairo M's memory consumption constrained participation on low-end devices. Proofs could not be generated on devices with less than 3 GB of RAM, and devices with 3-4 GB of RAM showed the highest crash rate, respectively 4.2 and 4.4 crashes per generated proof, due to OOM errors on the largest witnesses (n near the 100,000 upper-bound). It showcases that devices with limited available memory can generate proofs for most workloads of the benchmark, but not for the most expensive ones, due to memory constraints. 5.1.3 Energy and thermal effects FibRace did not record power draw or thermal throttling. Anecdotal player feedback suggested noticeable heat and battery drain during extended play sessions, but these factors were not quantified. Other performance factors were not measured, including battery state, energy saving mode activation, and CPU temperature. Therefore we do not have a quantitative performance degradation ratio on our assumption that these factors lead to degraded performance. Future benchmarks should include energy-use metrics to more effectively evaluate sustainability, user comfort and performance impact. 5.2 Game design limitations 5.2.1 Gameplay constraints Because FibRace was designed as a game, participants could interrupt or close the app at any point, including by force-quitting the app while generating or sending a proof. Such interruptions are indistinguishable from technical crashes in the dataset, introducing ambiguity in interpreting some failure events. 5.2.2 No user accounts The game operated without persistent user accounts to reduce friction and prevent airdrop speculation. While this ensured privacy and simplicity, it made longitudinal tracking of player behavior and device switching impossible. This did not affect the benchmark, but made it less enjoyable for some players. 10 5.3 Summary These limitations do not diminish the core finding that modern smartphones can locally perform zero-knowledge proving. They do, however, highlight key areas for future experiments: • measuring more complex circuits, • measuring circuits with real-life applications, • quantifying energy and heat effect on performance. Together, these refinements would strengthen the empirical foundation laid by FibRace and support the next generation of client-side proving research. 6 Conclusion FibRace demonstrates that client-side proving is no longer theoretical: it is operational, measurable, and already viable across a wide range of consumer devices. Over the course of three weeks, more than 5,000 participants generated 2.2 million Cairo M proofs on 1,420 unique device models, confirming that proof generation can occur reliably on modern smartphones without the need for centralized infrastructure or trusted hardware. The results validate a key assumption behind both KKRT Labs and Hyli: that distributed proving and native onchain verification can coexist efficiently. Cairo M made on-device proof generation accessible even to mid-range devices, while Hyli's proof-powered Layer 1 absorbed millions of verifications without network congestion. Beyond its immediate findings, FibRace established a new model for open benchmarking through gamified participation. By blending experimentation and gameplay, it produced one of the most extensive public datasets on client-side proving to date, an approach that future ZK ecosystems can replicate for other circuits, provers, or device classes. The experiment's limitations define clear next steps. Expanding to real-world use cases, and integrating thermal and battery instrumentation. Ultimately, FibRace offers a simple but powerful insight: zero-knowledge proofs can now run on users' own devices. This shift opens a new frontier for privacy-preserving applications and proof-native blockchains, where the path to scalable cryptography becomes user-driven. 11 Bibliography [1] O. Chia, \"ID photos of 70,000 users may have been leaked, Discord says,\" BBC News, Oct. 2025. [2] D. Kuhn, \"Google Wallet integrates ZK-proofs, a tech incubated by crypto industry,\" The Block, May 2025. [3] Vlad, \"Benchmarking in-browser p256 ECDSA proving systems,\" Hyli, Mar. 2025. [4] KKRT Labs, \"Kakarot presents Cairo M: The Frontier zkVM Built for Your Phone.\" Oct. 08, 2025. [5] A. Sirac, \"Launching FibRace: Prove to win!,\" Hyli, Sep. 2025. [6] C. Walter, \"Cairo M Design Document.\" Sep. 29, 2025. [7] S. Malatrait, \"FibRace performance report.\" Oct. 01, 2025. [8] A. Kuzmin, \"Baseline Benchmarking Hardware.\" Mar. 31, 2025. [9] N. Giridharan, F. Suri-Payer, I. Abraham, L. Alvisi, and N. Crooks, \"Autobahn: Seamless high speed BFT,\" in Proceedings of the ACM SIGOPS 30th Symposium on Operating Systems Principles, New York, NY, USA, Nov. 2024, pp. 1-23. 12"}}},{"rowIdx":288,"cells":{"id":{"kind":"string","value":"2510.14696"},"raw":{"kind":"string","value":"High-Resolution PTDF-Based Planning of Storage\nand Transmission Under High Renewables\nKevin Wu\nPascal Van Hentenryck\nH. Milton Stewart School of Industrial and Systems Engineering\nGeorgia Institute of Technology, Atlanta, GA, United States\n{kwu381, pvh}@gatech.edu\nRabab Haider\nCivil and Environmental Engineering\nUniversity of Michigan, Ann Arbor, MI, USA\nrababh@umich.edu\nAbstract—Transmission Expansion Planning (TEP) optimizes\npower grid upgrades and investments to ensure reliable, effi-\ncient, and cost-effective electricity delivery while addressing grid\nconstraints. To support growing demand and renewable energy\nintegration, energy storage is emerging as a pivotal asset that\nprovides temporal flexibility and alleviates congestion. This paper\ndevelops a multiperiod, two-stage PTDF formulation that co-\noptimizes transmission upgrades and storage siting/sizing. To\nensure scalability, a trust-region, multicut Benders scheme warm-\nstarted from per-representative-day optima is proposed. Applied\nto a 2,000-bus synthetic Texas system under high-renewable\nprojections, the method attains final optimality gaps below 1%\nand yields a plan with storage at about 180 nodes (32% of peak\nrenewable capacity). These results demonstrate that the proposed\nPTDF-based methodology efficiently handles large distributed\nstorage fleets, demonstrating scalability at high spatial resolution.\nIndex Terms—transmission expansion planning, storage plan-\nning, co-optimization, PTDF, renewables.\nNOMENCLATURE\na) Sets:\nN\nSet of buses; N = {1, ..., N}; indexed by i\nE\nSet of branches; E = {1, ..., E}; indexed by ij\nS\nSet of scenarios; S = {1, ..., S}; indexed by s\nT\nSet of hours; T = {1, ..., T}; indexed by t\nb) Parameters:\npd\ni,s,t\nLoad at node i in scenario s at hour t\n¯\npg\ni,s,t\nmin output of generator i in scenario s at hour t\n¯pg\ni,s,t\nmax output of generator i in scenario s at hour t\nccap\nij\nLine capacity upgrade cost of edge ij\ncstor\ni\nStorage energy rating cost at node i\ncG\ni (·)\nProduction cost function of generator i\nΩS(·)\nProbability mass function for set of scenarios S\nλ\nLoad shedding penalty cost\n∆c\nij\nCapacity upgrade increment of branch ij\n¯pf\nij\nThermal limit of branch ij\nThis work was partially funded by NSF award 2112533.\nc) Variables:\nγij\nCapacity upgrade level of branch ij\nσi\nStorage energy rating at node i\nsoci,s,t\nStorage SoC at node i in scenario s at hour t\nchi,s,t\nStorage charge at node i in scenario s at hour t\ndisi,s,t\nStorage discharge at node i in scenario s at hour t\npg\ni,s,t\nOutput of generator i in scenario s at hour t\npf\nij,s,t\nPower flow on branch ij in scenario s at hour t\nξi,s,t\nLoad shed at node i in scenario s at hour t\nI. INTRODUCTION\nThe U.S. electric grid is rapidly evolving with rising elec-\ntricity demand and renewable energy penetration, requiring\nsubstantial transmission infrastructure investments to maintain\nreliable and affordable service [1]. This presents a critical\nchallenge for system operators who must solve Transmission\nExpansion Planning (TEP) problems to identify cost-effective\nstrategies for grid upgrades to ensure capability of delivering\npower from generators to load centers. TEP is modeled as\na mixed-integer program (MIP) that is difficult to solve for\nlarge-scale systems due to discrete investment decisions. They\nare made more computationally challenging by including\ninvestment decisions for generation [2], FACTS devices [3],\nand energy storage [4]. Given the critical role of storage in\nproviding flexibility under high renewable penetration, this\nwork focuses on TEP with storage.\nThe TEP+Storage literature typically adopts at least one\nof three simplifications: (i) reduced spatial granularity; (ii)\nrelaxation of DC power flow to a transport model; or (iii)\nsettings whose economics yield sparse storage deployments.\nThese choices reduce problem dimensionality and computa-\ntional burden. This work address these limits: a PTDF-based,\nhigh-resolution model under high-renewable scenarios that\nendogenously deploys a large distributed storage fleet, creat-\ning a more demanding computational setting. A trust-region-\nstabilized Benders scheme addresses this regime, attaining\nfinal optimality gaps below 1% on a 2,000-bus system with\nstorage at about 180 nodes (32% of peak renewable capacity).\nA. Related Work\nHigh-fidelity TEP+Storage formulations are often limited to\nsmall systems due to computational complexity. For example,\narXiv:2510.14696v1 [eess.SY] 16 Oct 2025\n\n[5] uses a linearized AC model on a 24-bus system, and\n[6] solves a security-constrained DC model on a 53-bus\nsystem; these preserve grid physics but do not readily scale\nto thousands of buses. To reach larger systems, many studies\nadopt transport relaxations; e.g., [7] scales to the 8,870-bus\nCATS using progressive hedging. Storage deployment is also\nfrequently sparse: [8] studies merchant storage with a tri-level\nDC model and, under their economic assumptions and 240-\nbus case, report profitable deployment at ≤7 sites (3% of\npeak renewables); while [9] solves a DC model with binary\nstorage siting/sizing and line undergrounding on CATS using\nBenders under wildfire risk and report deployments at ≤9 sites\ntotaling 2.6 GWh (≈11% of peak renewables). Collectively,\nthese works show large-scale DCOPF feasibility, but it has\ntypically involved reduced flow-model fidelity and/or sparse\nstorage arising from the assumed economics or risk setting,\nwhich can alter congestion patterns and investment signals\nwhile easing computation burden.\nB. Contributions\nTo address these limitations, this paper proposes a com-\nbined a PTDF-based DC formulation with a stabilized (trust-\nregion) Benders approach. The methodology is demonstrated\nin a high-renewables setting on a 2,000-bus network, thereby\naddressing simultaneous gaps in spatial resolution, network-\nfidelity, and storage scalability. The contributions are:\n• A multiperiod, two-stage PTDF model that co-optimizes\nline upgrades and storage siting/sizing with hourly dis-\npatch and storage dynamics, preserving DC physics while\nremaining tractable at scale\n• A trust-region, multicut Benders scheme, warm-started\nfrom per-representative day optima, that mitigates dual\ndegeneracy and oscillatory master-iterates, strengthens\ncuts, and improves convergence predictability\n• A multiyear evaluation on a 2,000-bus synthetic Texas\nsystem under high-renewable scenarios that endogenously\nyields a large distributed storage fleet (≥179 devices, 32%\nof peak renewables) while attaining final optimality gaps\n≤1%, demonstrating quality of solution.\nThe paper proceeds as follows. §II formulates the two-stage\nTEP+Storage model; §III details the Benders methodology;\n§IV describes the Texas case study; §V reports optimality\ngaps, runtime, and investment decisions; and §VI concludes.\nII. FORMULATION\nPTDF-based DCOPF is well established for TEP [10],\nyet large-scale storage integration within PTDF frameworks\nremains comparatively unexplored. This section presents the\nPTDF-based TEP+Storage formulation. The formulation as-\nsumes that at most one generator is connected to each bus,\nwithout loss of generality.\nA. The Two-Stage TEP+Storage PTDF Formulation\nThe TEP+Storage model is a two-stage program. The\ninvestment stage selects transmission upgrades and storage\nsiting/sizing to minimize capital expenditures (CapEx). The\nrecourse stage fixes those investments and minimizes opera-\ntional costs (OpEx) over the investment period H, the plan-\nning horizon during which CapEx decisions apply, thereby\ncapturing the coupling between long-term planning and short-\nterm operations.\n1) Investment decisions and variables: This study prior-\nitizes capacity upgrades of existing transmission lines over\nnew line additions, as new construction often faces economic,\nregulatory, and social barriers [11]. Upgrades are represented\nby the discrete decision variable γij for each branch ij ∈E,\nwith allowable levels of enhancement ∆c\nij (MW per line).\nStorage is modeled as short-duration lithium-ion batteries\ndue to their technological maturity and widespread deploy-\nment [12]. Sizing decisions are captured by σi, denoting the\ninstalled energy capacity (MWh) at node i. Each unit has a\n4-hour duration, giving a power rating of 1\n4σi. Hourly storage\noperations are modeled by chi,t, disi,t, and soci,t, representing\ncharge, discharge, and state of charge (SoC), respectively.\nOther key variables include nodal generation dispatch pg\nand line power flows pf (auxiliary via PTDF).\n2) Objective: The TEP+Storage model minimizes the sum\nof CapEx and OpEx. The CapEx is a linear function of first-\nstage decisions (γ, σ) covering line and storage investments:\nCapEx = f(γ, σ) := P\nij∈E ccap\nij γij + P\ni∈N cstor\ni\nσi\n(1)\nGiven investments (γ, σ), the operating cost OpExs(γ, σ) of\nscenario-s includes daily generation expenses, storage charge\nand discharge costs, and penalties for load shedding using a\nper-unit penalty λ for load shed ξi,s,t. It is computed over\nthe investment period via scaling factor α (e.g., α = 365\nfor one year). The aggregate operating cost OpEx(γ, σ) is\na probability-weighted sum across scenarios s ∈S.\nOpExs(γ, σ) := GenEx + StorEx + Penalties\n(2)\n= α P\ni∈N ,t∈T [cG\ni (pg\ni,s,t) + cS\ni (chi,s,t + disi,s,t) + λ ξi,s,t],\nOpEx(γ, σ) := P\ns∈S ΩS(s) OpExs(γ, σ)\n(3)\n3) Constraints: The DC power flow model uses the Power\nTransfer Distribution Factor (PTDF) matrix Φ, which maps\nnet nodal injections to line flows. For scenario s and hour t,\nlet injs,t := [pg\ni,s,t +disi,s,t −pd\ni,s,t −chi,s,t +ξi,s,t]N\ni=1 denote\nthe vector of net injections. The operational constraints are:\n[pf\nij,s,t]E\nij=1 = Φ × injs,t\n(4)\n1⊤injs,t = 0\n(5)\n−¯pf\nij −γij∆c\nij ≤pf\nij,s,t ≤¯pf\nij + γij∆c\nij\n(6)\n¯\npg\ni,s,t ≤pg\ni,s,t ≤¯pg\ni,s,t\n(7)\nsoci,s,t = soci,s,t−1 + chi,s,tη −disi,s,t/η\n(8)\nsoci,s,1 = chi,s,1η −disi,s,1/η\n(9)\nsoci,s,T = 0\n(10)\n0 ≤soci,s,t ≤σi\n(11)\n0 ≤chi,s,t ≤σi/4\n(12)\n0 ≤disi,s,t ≤σi/4\n(13)\n\nModel 1 Two-Stage TEP+Storage Formulation\nmin\n(1) + (3)\ns.t.\n(4) −(5),\n∀s ∈S, t ∈T\n(6),\n∀ij ∈E, s ∈S, t ∈T\n(7),\n∀i ∈N, s ∈S, t ∈T\n(8)\n∀i ∈N, s ∈S, t ∈{2, ..., T}\n(9) −(10),\n∀i ∈N, s ∈S\n(11) −(13),\n∀i ∈N, s ∈S, t ∈T\nwhere (4) computes the vector of line flows; (5) imposes\npower balance; (6) enforces thermal flow limits; (7) bounds\ngenerator outputs, with renewable generators modeled with\n¯\npg = 0 and scenario-dependent ¯pg (curtailment allowed), and\nconventional units dispatchable within their limits; (8)–(13)\ngovern storage, including SoC with efficiency losses, a zero-\nSoC boundary at the start and end of each representative day,\nand charge/discharge limits. Model 1 outlines the full two-\nstage TEP+Storage formulation.\nRemark. To maintain recourse feasibility, load shed slack\nvariables ξi,s,t are included and penalized in OpEx by a large\ncoefficient λ (see §II-A2). Any plan with ξ > 0 is rejected;\nλ is increased (doubled) and the model is re-solved until zero\nload shed is achieved. This approach preserves feasibility and\nis guaranteed to terminate when a no-shed solution exists,\nconsistent with industry practice where TEP must serve future\nload by system design.\nB. Scalable PTDF Reformulation\nThe PTDF model is scaled to high resolution systems\nusing a sparse representation of the PTDF matrix and lazy\nconstraint generation. A PTDF cutoff is a common sparsifi-\ncation approach, where entries of Φ with magnitudes below\na threshold kp (typically 0.01–0.05) are set to zero, thereby\nsparsifying the matrix and improving computational efficiency\nby removing negligible effects [10]. For very large systems,\nhowever, even the sparse PTDF can remain expensive. To\naddress this, the PTDF flow (4) and thermal-limit constraints\n(6) are enforced lazily: the model is first solved without these\nconstraints, then any violated line-flow inequalities are itera-\ntively added to model until no violations remain. Specifically,\nif branch ij in scenario s at time t violates its upper bound,\nthe following inequality constraint is added to the model:\npf\nij,s,t = P\nk∈N PTDFij,kinjs,t\nk\n≤¯pf\nij + γij∆c\nij,\nwith the inequality reversed for lower bound violations. Lazy\nconstraints preserve model equivalence while omitting non-\nbinding limits, yielding smaller and more tractable formula-\ntions. The set of lazy constraints is further reduced by adding\nonly the kv most severe violations per iteration, where hyper-\nparameter kv can be tuned: smaller values produce a sparser\nmodel at the cost of more iterations to convergence. Both\ntechniques (cutoff and lazy constraints) are used throughout\nthe remainder of the paper.\nFig. 1. Benders Schematic Flow Diagram.\nModel 2 Benders Master Problem (MP)\nzMP\nk\n= min\n(1) + P\ns∈S ΩS(s)θs + ρ\ns.t.\nρ ≤0\n(14)\nS\ns∈S Ck\ns\nIII. METHODOLOGY\nThe two-stage TEP+Storage model grows linearly with\nthe number of second-stage scenarios; combined with the\nnetwork’s high spatial resolution, even a modest number\nof scenarios yields very large instances. To address this, a\nBenders decomposition (BD) is employed. The baseline BD\nformulation (§III-A) decomposes the problem into a master\nproblem (MP) and subproblems (SPs) linked by investment\nand operational decisions. Two methodological enhancements\nare introduced to improve convergence and stability: a warm-\nstart module (§III-B) that initializes the MP using relaxed in-\nvestment solutions, and a trust-region Benders module (§III-C)\nthat stabilizes the iterative process using a proximal constraint\nand restricted candidate set. §III-D presents a valid global\nlower bound used to assess optimality gaps.\nA. Multicut Benders for TEP+Storage\nThe BD algorithm alternates between the MP, which selects\nfirst-stage investments to minimize CapEx, and scenario SPs,\nwhich–given (γ, σ)–optimize continuous operational variables\nand minimize OpEx. Each SP is a linear program, and SPs\nare solved in parallel. From each SP, feasibility and optimality\ncuts are generated and added to the MP. Under the multicut\nscheme, the MP retains all accumulated cuts per scenario. Let\nC k\ns denote the set of cuts for scenario s generated through\niteration k −1 (with C 1\ns = ∅); at iteration k the MP imposes\nC k\ns for all s ∈S. Figure 1 summarizes the algorithm.\n1) Benders MP: The MP co-optimizes line and storage\ninvestments to minimize CapEx. To capture operational and\nfeasibility constraints, the MP includes proxy variables θs\nan underestimator of OpExs for each scenario s, and ρ\nan underestimator of scenario-wide load-shed. The MP at\niteration k is presented in Model 2, where (14) prevents load\nshedding in the final solution. After solving the MP, the new\ninvestment point (γk, σk) is broadcast to the SPs, which are\nsolved in parallel to generate the next set of cuts in C k+1\ns\n.\n2) Benders SP: The scenario-s SP optimizes operations\nunder the MP decisions (γk, σk), fixed in the SP with dual\ndual multipliers πs,k\nγ\nand πs,k\nσ\nin (15)-(16). Each SP runs\ntwo phases: Phase I enforces load feasibility and Phase II\n\nModel 3 Benders Subproblem (SP)\nmin\nOpExs(γk, σk)\ns.t.\n(4) −(5),\n∀t ∈T\n(6),\n∀ij ∈E, t ∈T\n(7),\n∀i ∈N, t ∈T\n(8),\n∀i ∈N, t ∈{2, ..., T}\n(9) −(10),\n∀i ∈N\n(11) −(13),\n∀i ∈N, t ∈T\nγij = γk\nij\n(πs,k\nγ ),\n∀ij ∈E\n(15)\nσi = σk\ni\n(πs,k\nσ ),\n∀i ∈N\n(16)\noptimizes dispatch. In Phase I, the SP minimizes load shed,\nP\ni∈N,t∈T ξi,s,t. If the optimum exhibits load shed, the SP\nterminates and a feasibility cut is generated for shed under-\nestimator ρ. If no shed occurs, the SP proceeds to Phase II\nto minimize OpExs (Model 3) and generate an optimality cut\nfor θs. Both feasibility and optimality cuts take an affine form,\nwhere ψ ∈{I, II} indicates the Phase, ζs = ρ in Phase I and\nζs = θs in Phase II:\nζs ≥βk\ns + (πs,k\nγ )⊤γ + (πs,k\nσ )⊤σ\nβk\ns := vk,ψ\ns\n−(πs,k\nγ )⊤γk −(πs,k\nσ )⊤σk\nvk,I\ns\n= P\ni,t ξi,s,t,\nvk,II\ns\n= OpExs(γk, σk)\nThis two-phase design prioritizes feasibility (no shed) while\nmaintaining complete recourse via ξ, avoiding classical\ninfeasibility-ray cuts which can be numerically fragile [13].\n3) Convergence of Benders: At iteration k, the MP opti-\nmum zMP\nk\nis a valid lower bound LBk for Model 1, since the\nvariables {θs} underestimate the true scenario costs {OpExs},\ngiving LBk := zMP\nk\n. If all SPs are feasible (Phase I yields\nP\ni∈N,t∈T ξi,s,t = 0 for every s ∈S), a feasible two-stage\nobjective at (γk, σk) is ˆzk = f(γk, σk) + OpEx(γk, σk), and\nis used to maintain a non-increasing upper bound UBk :=\nmin{UBk−1, ˆzk}. The algorithm terminates when the relative\ngap falls below a threshold ϵ: (UBk −LBk)/UBk ≤ϵ.\nRemark. Lazy constraints integrate naturally with BD: per\nscenario flow/limit constraints identified in one iteration are\nretained for subsequent iterations, accelerating SP solves.\nB. Warm-Start from Relaxed Two-Stage Model (WS Module)\nIn the baseline BD formulation, early iterations are\nfeasibility-cut dominated: with no initial cuts, the MP selects\na no-expansion plan, causing heavy load shedding in the SPs\nuntil a feasible investment set is learned. The WS accelerates\nthis by solving, in parallel for each s ∈S, the continuous\nrelaxation of Model 1 to obtain (γs, σs). A conservative, non-\nanticipative design is then formed by component-wise aggre-\ngation of the per-scenario optimal investments, and rounded\nup to the admissible investment granularity:\n¯γ :=\n\u0006\nmaxs∈S γs\u0007\n,\n¯σ :=\n\u0006\nmaxs∈S σs\u0007\n.\nFig. 2. Workflow of the proposed methodology.\nThe Benders MP is initialized at (¯γ, ¯σ), which reduces early\ninfeasibility cuts and accelerates convergence.\nC. Trust-Region-Stabilized Benders (TR-Benders Module)\nOnce feasibility is achieved, the BD formulation exhibits\nstrong dual degeneracy–many line-storage configurations with\nnearly identical operating costs–yielding flat recourse value\nfunctions, oscillatory MP iterates, and weak cuts that slow\nconvergence. The TR-Benders addresses this issue by aug-\nmenting the BD framework with a proximal trust-region (TR)\nconstraint on first-stage decisions and a restricted candidate\nset. This stabilization mechanism limits drastic investment\nchanges between iterations, improving numerical stability and\nensuring more consistent convergence behavior.\nThe MP is stabilized around an anchor point (˘γ, ˘σ) defined\nas the incumbent investment that attains the best known upper\nbound; it is initialized at (¯γ, ¯σ) and updated only upon a strict\nimprovement in the upper bound. Inspired by stabilized BD\n(e.g., [14]), the method adapts to discrete first-stage variables\nwith an ℓ1 trust-region (linear, sparsity-promoting):\nP\nij∈E\n\f\fγij −˘γij\n\f\f ≤rγ\nk\n(17)\nP\ni∈N\n\f\fσi −˘σi\n\f\f ≤rσ\n(18)\nAbsolute values are linearized in the standard way for MIPs.\nA fixed storage radius rσ = 2 (permitting limited re-siting) is\nused, while the transmission radius rγ\nk is dynamic: (i) resetting\nto 1 whenever the anchor point updates, and (ii) increasing\nby 1 whenever the MP revisits the same solution, to limit\nlocal oscillations. To further reduce degeneracy, investment\ndecisions are restricted to a candidate set derived from the\nper-scenario relaxations:\nEc := { ij ∈E | ∃s ∈S : γs\nij > 0}\nNc := { i ∈N | ∃s ∈S : σs\ni > 0}\nFinally, a level-set constraint prunes regions already known to\nbe suboptimal:\nf(γ, σ) + P\ns∈S ΩS(s) θs ≤UBk\n(19)\nTogether, the TR, candidate sets, and level-set constraint damp\noscillations and weaken the effects of dual degeneracy. Figure\n2 summarizes the full workflow of the proposed methodology.\n\nD. Global Lower Bound and Optimality Assessment\nA limitation of the TR stabilization is that the master’s value\nno longer certifies a global lower bound: with the TR active,\nLBk is a local lower bound over the current region around\nthe anchor point. However, a valid global lower bound can be\nconstructed as below.\n1) First-Stage Lower Bound: With load shed disallowed by\n(14), the minimum feasible investment decision is constrained\nby the first-stage investments of the most challenging repre-\nsentative day (typically characterized by high load and/or low\nrenewable generation). This is formalized in Theorem 1.\nTheorem 1. For each s ∈S, define the day-s feasible set and\nthe all-scenarios feasible set:\nΣs\nfeas := {(γ, σ) : recourse is feasible for scenario s}\nΣfeas := T\ns∈S Σs\nfeas\nGiven the feasible sets, define the corresponding minimum\nfirst stage costs for feasible day-s and all-scenarios:\ncfeas\ns\n:=\nmin\n(γ,σ)∈Σs\nfeas\nf(γ, σ),\ncfeas :=\nmin\n(γ,σ)∈Σfeas f(γ, σ),\nand first-stage optimal investments for scenario s:\n(γs\nfeas, σs\nfeas) ∈arg min(γ,σ)∈Σs\nfeas f(γ, σ)\nLet ˆs ∈arg maxs∈S cfeas\ns\n. Then\nmaxs∈S cfeas\ns\n≤f(γ, σ)\n∀(γ, σ) ∈Σfeas\nIn particular, f(γˆs\nfeas, σˆs\nfeas) = maxs cfeas\ns\nis a valid lower\nbound on any year-feasible first-stage plan’s cost.\nProof. Since Σfeas ⊆Σs\nfeas for every s,\ncfeas =\nmin\n(γ,σ)∈Σfeas f(γ, σ) ≥\nmin\n(γ,σ)∈Σs\nfeas\nf(γ, σ) = cfeas\ns\n∀s,\nyielding maxs cfeas\ns\n≤cfeas. Choosing ˆs ∈arg maxs cs gives\nthe stated bound with (γˆs\nfeas, σˆs\nfeas).\n2) Second-Stage Lower Bound: The per-scenario opera-\ntional cost is monotone nonincreasing in investment capac-\nity, as per Theorem 2. Let (γmax, σmax) denote the com-\nponentwise upper bounds on line and storage capacities\n(not restricted to the candidate set) of Model 1. Then a\nvalid lower bound on the aggregated second-stage cost is\nP\ns∈S ΩS(s) qs(γmax, σmax).\nTheorem 2. Let qs(γ, σ) be the optimal value of Model 3.\nIf γ′ ⪰γ and σ′ ⪰σ (componentwise), then qs(γ′, σ′) ≤\nqs(γ, σ). In particular, qs(·) is monotone nonincreasing in γ\nand σ.\nProof. For fixed s ∈S, increasing γ relaxes the thermal limit\nconstraints (6), and increasing σ relaxes the storage constraints\n(11)-(13). Hence the feasible region under (γ′, σ′) contains\nthat under (γ, σ). Because the SP objective is minimized and\ndepends on γ, σ only through these constraints, enlarging the\nfeasible set cannot increase the optimal value, yielding the\nclaim.\nFig. 3. Generator sites of the system. Black circles indicate nonrenewable,\nblue circles indicate wind, and red circles indicate solar generators.\n3) Combined Bound: Define z∗as the optimal final so-\nlution’s objective, with optimal line and storage investments\n(γ∗, σ∗) ∈Σfeas.\nz∗:= f(γ∗, σ∗) + P\ns∈S ΩS(s) qs(γ∗, σ∗)\nSince Theorem 1 and 2 show, respectively, that\nf(γˆs\nfeas, σˆs\nfeas) ≤f(γ∗, σ∗),\nqs(γmax, σmax) ≤qs(γ∗, σ∗)\n∀s ∈S,\na valid combined lower bound can be constructed:\nLB∗:= f(γˆs\nfeas, σˆs\nfeas) + P\ns∈S ΩS(s) qs(γmax, σmax) ≤z∗\n4) Optimality Gap: Each cfeas\ns\nis computed under the integer\nrelaxation of the first-stage investments (as in the WS). Being\na relaxation, this provides a valid lower bound for the year-\nwide plan, which must satisfy discretized first-stage decisions.\nThe reported optimality gaps are therefore conservative upper\nbounds: (UBk −LB∗)/UBk.\nIV. CASE STUDY\nThe proposed methodology is tested on a large-scale power\nsystem case study: the ACTIVSg2000 test case, a 2000-bus\nmodel of the U.S. Texas grid. The TEP+Storage model is\napplied in a rolling-horizon manner. The formulation itself has\nyear-agnostic investment variables, but is solved sequentially\nat 5-year decision points Y = {2030, 2035, 2040, 2045} with\ninvestment periods of length α = 1825 days. Investments\n(and their capital costs) are cumulative: capacity installed\nin an earlier year remains available in subsequent years,\nensuring monotone, non-decreasing expansion across planning\nperiods. This application reflects practical staging, but it is not\nguaranteed optimal relative to an omniscient, all-years model;\nquantifying any cross-period optimality gap is left for future\nwork.\na) Network Data: The synthetic Texas system is built\nfrom public information and statistical analysis of real power\nsystems [15]. The system has 2000 buses, 3206 lines, 423\nnon-renewable generators, and 175 renewable generators.\nFigure 3 shows the location of solar, wind, and nonrenew-\nable generators, overlaid on the grid topology. The future\ngeneration mix from 2030 to 2045 follows EIA outlook\n\nTABLE I\nGENERATION BASE CAPACITY (GW) AND GENERATION/LOAD SCALING\nType\nCapacity\n2022\n2030\n2035\n2040\n2045\nCoal\n14.5\n1.00\n0.82\n0.82\n0.82\n0.82\nNatural Gas\n75.1\n1.00\n0.79\n0.73\n0.71\n0.72\nNuclear\n7.1\n1.00\n0.98\n0.90\n0.80\n0.80\nSolar\n2.5\n1.00\n4.51\n6.00\n6.87\n8.04\nWind\n15.5\n1.00\n2.02\n2.23\n2.26\n2.32\nLoad\n-\n1.00\n1.13\n1.21\n1.31\n1.41\nFig. 4. Representative days by average demand vs. wind in 2022. Red circles\nindicate selected representative days. Transparent blue circles indicate non-\nselected days.\nprojections [16]. All generation resources are uniformly scaled\nto reflect projected growth or planned decommissioning. Load\ngrowth is modeled with a year-to-year increase of 1.5% and\nis scaled uniformly across all buses. Table I shows the total\ngenerator capacities (GW) of the network and summarizes the\nscaling for each generation type and load relative to the base\nyear, 2022.\nb) Scenario selection: Scenarios are 24-h representative\ndays sampled from a year of load/wind/solar data. This re-\nduces the computational time required for evaluating a full\nyear of operations, while still capturing intertemporal storage\ndynamics. In this case study, kr = 18 days are selected by\na max-min diversity criterion on normalized daily profiles–\nmaximizing the minimum pairwise Euclidean distance–and\nassigned equal weights. Figure 4 shows the chosen days\n(red circles) in average wind-load space–the main drivers of\ninvestment (solar capacity is comparatively smaller). Days that\nrequire storage operation to avoid load shed are labeled; these\ngenerally coincide with higher demand. August 11 (08–11)\nis the most challenging day, exhibiting the highest average\ndemand.\nc) Investments parameters: Table II lists capital and\noperating costs for line upgrades and storage. Line upgrades\nare discretized to capacity increases of 30, 60, and 90%,\nreflecting the enhancements achievable through typical line\nreconductoring [11]. Storage units are 250 MWh each, limited\nto 12 per node (3 GWh total), with charge and discharge\nefficiencies of η = 95%. Line upgrade costs are adapted from\n[17]. Storage costs are adapted from [18].\nd) Computational Setup and Hyperparameters:\nThe\nTEP+Storage model is implemented in Julia with JuMP [19]\nand solved with Gurobi [20] on Georgia Tech’s PACE Phoenix\nTABLE II\nCOST CONFIGURATION\nCosts\nCapacity Upgrades\n$1243/MW-km\nStorage Cost\n$1,000,000/MWh\nCharge/Discharge Cost\n$68.50/MWh\ncluster [21] (24-core Intel Xeon, Linux); each run used 24\nthreads and 288 GB RAM. TR-Benders employed a 0.1% MP\noptimality-gap tolerance, with wall-clock limits of 24 hours\nfor 2030/2035/2040 and 72 hours for 2045 to accommodate\nlonger per-iteration solves. The per-iteration added-violations\nquota was set to kv = 32 and the PTDF cutoff to kp = 0.005,\nreducing nonzeros in Φ by 85.5%.\nV. RESULTS\nWith the proposed methodology, the year-wise TEP+Storage\nplanning problem is both tractable and achieves optimality\nguarantees of ≤1%. Results are organized as follows: op-\ntimality gaps (§V-A), computational performance (§V-B), and\ninvestment outcomes (§V-C).\nA. Optimality Gaps\nAll optimality gaps are computed relative to the lower\nbound defined in §III-D, which bounds the best achievable\nobjective for each 5-year decision given previously committed\ninvestments. Final gap values are reported in the rightmost\ncolumn of Table III. Across all investment periods, the WS\ninitializes runs near optimality, yielding initial gaps below\n10%. The subsequent TR-Benders reduces these gaps to\nbelow 1% in every period, validating the approach and the\ninvestment solutions presented below.\nB. Computational Results\nUnder\nthe\nscalable\nPTDF\nreformulation\n(§II-B),\nthe\nTEP+Storage model is tractable at scale. A tunable PTDF\ncutoff kp provides planners with a direct accuracy-scalability\ntrade-off (less straightforward in phase-angle DC formula-\ntions). The following sections describe the computational\nresults from the WS and TR-Benders modules, respectively.\n1) WS Module: Figure 5 summarizes computation times\nfor the relaxed two-stage model in the WS, reported per\nrepresentative day and 5-year investment period. Runtime is\ndominated by the iterative discovery of violated thermal-limit\nconstraints; consequently, more congested days take longer to\nsolve. Average time-to-solve falls from 2030 to 2040 because\ninvestments made in earlier 5-year periods remain in place\nin subsequent periods, relieving congestion and simplifying\ndecisions for most representative days. Outliers persist on\ndays that combine heavy storage utilization with elevated\ncongestion (e.g., 08–11 with its high average demand).\n2) TR-Benders Module: The TR-Benders demonstrates im-\nproved tractability by decoupling first-stage investment de-\ncisions from per-scenario operations. Table IV reports time\nlimits, iteration counts, and average seconds per iteration\nfor each 5-year investment period. Iteration time increases\n\nTABLE III\nINVESTMENT DECISIONS AND COSTS FOR 2030-2045 FOR MOST CHALLENGING REP. DAY, WS, AND TR-BENDERS PLANS.\nMode\n# Lines\n# Storage units /\nTotal capacity (GWh)\nCapEx\nOpEx\nOpt. Gap\nLines ($B)\nStorage ($B)\nGen ($B)\nStorOpEx ($B)\n2030\n08-11\n183\n0 / 0\n0.14\n0\n46.49\n0\n-\nWarm-start\n291\n0 / 0\n0.37\n0\n28.80\n0\n1.3%\nTR-Benders\n238\n0 / 0\n0.24\n0\n28.86\n0\n1.0%\n2035\n08-11\n334\n33 / 28.91\n0.36\n28.92\n50.91\n7.27\n-\nWarm-start\n381\n33 / 32.75\n0.53\n32.77\n30.39\n0.40\n6.3%\nTR-Benders\n352\n32 / 29.25\n0.40\n29.27\n30.42\n0.41\n0.8%\n2040\n08-11\n425\n118 / 98.07\n0.47\n98.12\n56.60\n24.71\n-\nWarm-start\n462\n118 / 111.75\n0.66\n111.81\n33.35\n1.64\n9.4%\nTR-Benders\n441\n108 / 98.50\n0.53\n98.55\n33.38\n1.65\n0.4%\n2045\n08-11\n476\n184 / 167.30\n0.58\n167.40\n62.6\n42.01\n-\nWarmstart\n508\n184 / 183.75\n0.72\n183.84\n36.63\n3.66\n7.4%\nTR-Benders\n499\n179 / 167.75\n0.66\n167.84\n36.64\n3.66\n0.3%\nFig. 5.\nWS computation times by representative day for each 5-year\ninvestment period.\nTABLE IV\nTR-BENDERS COMPUTATIONAL PERFORMANCE FOR EACH 5-YEAR\nINVESTMENT PERIOD.\n2030\n2035\n2040\n2045\nTime Limit (Hrs.)\n24\n24\n24\n72\nIterations\n1964\n1198\n332\n102\nAvg. Seconds/Iter\n44\n72\n260\n2541\nover the planning horizon because progress each round is\npaced by the most challenging SP/representative day: the\nalgorithm advances only after all cuts are generated. The\nslowest iterations coincide with intensive storage operation\nand pronounced congestion, which prolong SP solves due\nto storage time-coupling and ongoing discovery of violated\nthermal-limit constraints. Further computational improvement\nmay be attained by pre-screening thermal violations.\nC. Investment Results\nThe TR-Benders produces a feasible, near-optimal year-\nwide plan for 2030–2045 with zero load shed across all\nrepresentative days. Figure 6 maps the geographic distribution\nof investments. Transmission upgrades concentrate in West\nTexas, strengthening transfer from wind/solar regions, while\nstorage appears beginning in 2035 and grows considerably\nwith increasing load. Figure 7 reports storage energy through-\nputs for each planning year, defined as the total GWh charged\nand discharged during the representative day. Only a small\nfraction of days require storage operation to eliminate load\nshedding; 08–11 exhibits the largest throughput and aligns\nwith its computational difficulty (Fig. 5). Consequently, mul-\ntiyear investment choices are largely driven by this hardest\nday.\nTable III summarizes the selected upgrades, storage sit-\ning/sizing, and costs; its rightmost column displays the fi-\nnal optimality gaps referenced earlier. For completeness, the\nsingle-day (08-11) WS results are also listed by period.\nThe WS intentionally over-invests. After the TR-Benders, the\nnear-optimal plan reduces CapEx relative to the WS, while\nincurring only negligible increases in OpEx, yielding a lower\ntotal cost. The 08–11 single-day results show comparatively\nhigher OpEx than both the WS and TR-Benders plans because\nthey optimize solely for the highest-demand day rather than the\nequally weighted set of representative days. Because storage\ndominates investment costs, most of the residual gap is driven\nby reconciling storage capacity rather than line upgrades.\nThe storage capacity of TR-Benders approaches–but remains\nabove–the requirement implied by the hardest day, consistent\nwith Theorem 1 ensuring zero load-shed feasibility.\nBy 2045, the optimized plan yields 167.75 GWh of storage\ndistributed across 179 nodes, which is equivalent to 32%\nof the scaled peak renewable capacity. This underscores the\nrobustness of the proposed formulation and its tractability at\nhigh spatial and storage resolution.\nVI. CONCLUSION\nThis paper presents a PTDF-based TEP+Storage for joint\ntransmission and storage expansion and applies it to a 2,000-\nbus synthetic Texas system. To ensure computational feasi-\nbility, it introduces a trust-region, multicut Benders scheme\nwarm-started from per-representative-day optima. The method\nattains final optimality gaps ≤1% and remains tractable for a\nlarge distributed storage fleet deployed at ≥179 nodes (≈32%\nof peak renewable capacity). These results demonstrate the\nproposed PTDF-based methodology effectively handles high\nspatial resolution and large distributed storage deployments in\njoint TEP+Storage planning.\nTable IV shows that computational challenges persist, par-\nticularly under high storage penetration and network conges-\n\n(a) 2030 TEP+Storage\n(b) 2035 TEP+Storage\n(c) 2040 TEP+Storage\n(d) 2045 TEP+Storage\nFig. 6.\nUpgrades from 2030 to 2045 in 5-year intervals. Red lines denote capacity upgrade locations, with their thickness corresponding to one of three\npossible upgrade levels. Circles indicate storage locations, with their size and color representing the number of installed units (250MWh each).\nFig. 7.\nStorage energy throughput (GWh) for representative days with\nnonzero utilization across the 2030–2045 planning horizon. Values reflect the\nrepresentative day under the investment plan in effect for that year.\ntion, and will intensify as temporal, operational, and spatial\nfidelity increase. Priorities include accelerating SP solves–\ne.g., pre-screening and dynamic pruning of thermal/flow\nconstraints–and exploring unified Benders variants that com-\nbine feasibility and optimality cuts. Another direction is to\nintegrate generation expansion, which may relieve conges-\ntion and further improve lazy-constraint performance. 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This paper develops a multiperiod, two-stage PTDF formulation that cooptimizes transmission upgrades and storage siting/sizing. To ensure scalability, a trust-region, multicut Benders scheme warmstarted from per-representative-day optima is proposed. Applied to a 2,000-bus synthetic Texas system under high-renewable projections, the method attains final optimality gaps below 1% and yields a plan with storage at about 180 nodes (32% of peak renewable capacity). These results demonstrate that the proposed PTDF-based methodology efficiently handles large distributed storage fleets, demonstrating scalability at high spatial resolution. Index Terms-transmission expansion planning, storage planning, co-optimization, PTDF, renewables. NOMENCLATURE a) Sets: N Set of buses; N = {1, ..., N}; indexed by i E Set of branches; E = {1, ..., E}; indexed by ij S Set of scenarios; S = {1, ..., S}; indexed by s T Set of hours; T = {1, ..., T}; indexed by t b) Parameters: pd i,s,t Load at node i in scenario s at hour t ̄ pg i,s,t min output of generator i in scenario s at hour t ̄pg i,s,t max output of generator i in scenario s at hour t ccap ij Line capacity upgrade cost of edge ij cstor i Storage energy rating cost at node i cG i (·) Production cost function of generator i ΩS(·) Probability mass function for set of scenarios S λ Load shedding penalty cost ∆c ij Capacity upgrade increment of branch ij ̄pf ij Thermal limit of branch ij This work was partially funded by NSF award 2112533. c) Variables: γij Capacity upgrade level of branch ij σi Storage energy rating at node i soci,s,t Storage SoC at node i in scenario s at hour t chi,s,t Storage charge at node i in scenario s at hour t disi,s,t Storage discharge at node i in scenario s at hour t pg i,s,t Output of generator i in scenario s at hour t pf ij,s,t Power flow on branch ij in scenario s at hour t ξi,s,t Load shed at node i in scenario s at hour t I. INTRODUCTION The U.S. electric grid is rapidly evolving with rising electricity demand and renewable energy penetration, requiring substantial transmission infrastructure investments to maintain reliable and affordable service [1]. This presents a critical challenge for system operators who must solve Transmission Expansion Planning (TEP) problems to identify cost-effective strategies for grid upgrades to ensure capability of delivering power from generators to load centers. TEP is modeled as a mixed-integer program (MIP) that is difficult to solve for large-scale systems due to discrete investment decisions. They are made more computationally challenging by including investment decisions for generation [2], FACTS devices [3], and energy storage [4]. Given the critical role of storage in providing flexibility under high renewable penetration, this work focuses on TEP with storage. The TEP+Storage literature typically adopts at least one of three simplifications: (i) reduced spatial granularity; (ii) relaxation of DC power flow to a transport model; or (iii) settings whose economics yield sparse storage deployments. These choices reduce problem dimensionality and computational burden. This work address these limits: a PTDF-based, high-resolution model under high-renewable scenarios that endogenously deploys a large distributed storage fleet, creating a more demanding computational setting. A trust-regionstabilized Benders scheme addresses this regime, attaining final optimality gaps below 1% on a 2,000-bus system with storage at about 180 nodes (32% of peak renewable capacity). A. Related Work High-fidelity TEP+Storage formulations are often limited to small systems due to computational complexity. For example, 16 Oct 2025 [5] uses a linearized AC model on a 24-bus system, and [6] solves a security-constrained DC model on a 53-bus system; these preserve grid physics but do not readily scale to thousands of buses. To reach larger systems, many studies adopt transport relaxations; e.g., [7] scales to the 8,870-bus CATS using progressive hedging. Storage deployment is also frequently sparse: [8] studies merchant storage with a tri-level DC model and, under their economic assumptions and 240bus case, report profitable deployment at ≤7 sites (3% of peak renewables); while [9] solves a DC model with binary storage siting/sizing and line undergrounding on CATS using Benders under wildfire risk and report deployments at ≤9 sites totaling 2.6 GWh (≈11% of peak renewables). Collectively, these works show large-scale DCOPF feasibility, but it has typically involved reduced flow-model fidelity and/or sparse storage arising from the assumed economics or risk setting, which can alter congestion patterns and investment signals while easing computation burden. B. Contributions To address these limitations, this paper proposes a combined a PTDF-based DC formulation with a stabilized (trustregion) Benders approach. The methodology is demonstrated in a high-renewables setting on a 2,000-bus network, thereby addressing simultaneous gaps in spatial resolution, networkfidelity, and storage scalability. The contributions are: • A multiperiod, two-stage PTDF model that co-optimizes line upgrades and storage siting/sizing with hourly dispatch and storage dynamics, preserving DC physics while remaining tractable at scale • A trust-region, multicut Benders scheme, warm-started from per-representative day optima, that mitigates dual degeneracy and oscillatory master-iterates, strengthens cuts, and improves convergence predictability • A multiyear evaluation on a 2,000-bus synthetic Texas system under high-renewable scenarios that endogenously yields a large distributed storage fleet (≥179 devices, 32% of peak renewables) while attaining final optimality gaps ≤1%, demonstrating quality of solution. The paper proceeds as follows. §II formulates the two-stage TEP+Storage model; §III details the Benders methodology; §IV describes the Texas case study; §V reports optimality gaps, runtime, and investment decisions; and §VI concludes. II. FORMULATION PTDF-based DCOPF is well established for TEP [10], yet large-scale storage integration within PTDF frameworks remains comparatively unexplored. This section presents the PTDF-based TEP+Storage formulation. The formulation assumes that at most one generator is connected to each bus, without loss of generality. A. The Two-Stage TEP+Storage PTDF Formulation The TEP+Storage model is a two-stage program. The investment stage selects transmission upgrades and storage siting/sizing to minimize capital expenditures (CapEx). The recourse stage fixes those investments and minimizes operational costs (OpEx) over the investment period H, the planning horizon during which CapEx decisions apply, thereby capturing the coupling between long-term planning and shortterm operations. 1) Investment decisions and variables: This study prioritizes capacity upgrades of existing transmission lines over new line additions, as new construction often faces economic, regulatory, and social barriers [11]. Upgrades are represented by the discrete decision variable γij for each branch ij ∈E, with allowable levels of enhancement ∆c ij (MW per line). Storage is modeled as short-duration lithium-ion batteries due to their technological maturity and widespread deployment [12]. Sizing decisions are captured by σi, denoting the installed energy capacity (MWh) at node i. Each unit has a 4-hour duration, giving a power rating of 1 4σi. Hourly storage operations are modeled by chi,t, disi,t, and soci,t, representing charge, discharge, and state of charge (SoC), respectively. Other key variables include nodal generation dispatch pg and line power flows pf (auxiliary via PTDF). 2) Objective: The TEP+Storage model minimizes the sum of CapEx and OpEx. The CapEx is a linear function of firststage decisions (γ, σ) covering line and storage investments: CapEx = f(γ, σ) := P ij∈E ccap ij γij + P i∈N cstor i σi (1) Given investments (γ, σ), the operating cost OpExs(γ, σ) of scenario-s includes daily generation expenses, storage charge and discharge costs, and penalties for load shedding using a per-unit penalty λ for load shed ξi,s,t. It is computed over the investment period via scaling factor α (e.g., α = 365 for one year). The aggregate operating cost OpEx(γ, σ) is a probability-weighted sum across scenarios s ∈S. OpExs(γ, σ) := GenEx + StorEx + Penalties (2) = α P i∈N ,t∈T [cG i (pg i,s,t) + cS i (chi,s,t + disi,s,t) + λ ξi,s,t], OpEx(γ, σ) := P s∈S ΩS(s) OpExs(γ, σ) (3) 3) Constraints: The DC power flow model uses the Power Transfer Distribution Factor (PTDF) matrix Φ, which maps net nodal injections to line flows. For scenario s and hour t, let injs,t := [pg i,s,t +disi,s,t -pd i,s,t -chi,s,t +ξi,s,t]N i=1 denote the vector of net injections. The operational constraints are: [pf ij,s,t]E ij=1 = Φ × injs,t (4) 1⊤injs,t = 0 (5) - ̄pf ij -γij∆c ij ≤pf ij,s,t ≤ ̄pf ij + γij∆c ij (6) ̄ pg i,s,t ≤pg i,s,t ≤ ̄pg i,s,t (7) soci,s,t = soci,s,t-1 + chi,s,tη -disi,s,t/η (8) soci,s,1 = chi,s,1η -disi,s,1/η (9) soci,s,T = 0 (10) 0 ≤soci,s,t ≤σi (11) 0 ≤chi,s,t ≤σi/4 (12) 0 ≤disi,s,t ≤σi/4 (13) Model 1 Two-Stage TEP+Storage Formulation min (1) + (3) s.t. (4) -(5), ∀s ∈S, t ∈T (6), ∀ij ∈E, s ∈S, t ∈T (7), ∀i ∈N, s ∈S, t ∈T (8) ∀i ∈N, s ∈S, t ∈{2, ..., T} (9) -(10), ∀i ∈N, s ∈S (11) -(13), ∀i ∈N, s ∈S, t ∈T where (4) computes the vector of line flows; (5) imposes power balance; (6) enforces thermal flow limits; (7) bounds generator outputs, with renewable generators modeled with ̄ pg = 0 and scenario-dependent ̄pg (curtailment allowed), and conventional units dispatchable within their limits; (8)-(13) govern storage, including SoC with efficiency losses, a zeroSoC boundary at the start and end of each representative day, and charge/discharge limits. Model 1 outlines the full twostage TEP+Storage formulation. Remark. To maintain recourse feasibility, load shed slack variables ξi,s,t are included and penalized in OpEx by a large coefficient λ (see §II-A2). Any plan with ξ > 0 is rejected; λ is increased (doubled) and the model is re-solved until zero load shed is achieved. This approach preserves feasibility and is guaranteed to terminate when a no-shed solution exists, consistent with industry practice where TEP must serve future load by system design. B. Scalable PTDF Reformulation The PTDF model is scaled to high resolution systems using a sparse representation of the PTDF matrix and lazy constraint generation. A PTDF cutoff is a common sparsification approach, where entries of Φ with magnitudes below a threshold kp (typically 0.01-0.05) are set to zero, thereby sparsifying the matrix and improving computational efficiency by removing negligible effects [10]. For very large systems, however, even the sparse PTDF can remain expensive. To address this, the PTDF flow (4) and thermal-limit constraints (6) are enforced lazily: the model is first solved without these constraints, then any violated line-flow inequalities are iteratively added to model until no violations remain. Specifically, if branch ij in scenario s at time t violates its upper bound, the following inequality constraint is added to the model: pf ij,s,t = P k∈N PTDFij,kinjs,t k ≤ ̄pf ij + γij∆c ij, with the inequality reversed for lower bound violations. Lazy constraints preserve model equivalence while omitting nonbinding limits, yielding smaller and more tractable formulations. The set of lazy constraints is further reduced by adding only the kv most severe violations per iteration, where hyperparameter kv can be tuned: smaller values produce a sparser model at the cost of more iterations to convergence. Both techniques (cutoff and lazy constraints) are used throughout the remainder of the paper. Fig. 1. Benders Schematic Flow Diagram. Model 2 Benders Master Problem (MP) zMP k = min (1) + P s∈S ΩS(s)θs + ρ s.t. ρ ≤0 (14) S s∈S Ck s III. METHODOLOGY The two-stage TEP+Storage model grows linearly with the number of second-stage scenarios; combined with the network's high spatial resolution, even a modest number of scenarios yields very large instances. To address this, a Benders decomposition (BD) is employed. The baseline BD formulation (§III-A) decomposes the problem into a master problem (MP) and subproblems (SPs) linked by investment and operational decisions. Two methodological enhancements are introduced to improve convergence and stability: a warmstart module (§III-B) that initializes the MP using relaxed investment solutions, and a trust-region Benders module (§III-C) that stabilizes the iterative process using a proximal constraint and restricted candidate set. §III-D presents a valid global lower bound used to assess optimality gaps. A. Multicut Benders for TEP+Storage The BD algorithm alternates between the MP, which selects first-stage investments to minimize CapEx, and scenario SPs, which-given (γ, σ)-optimize continuous operational variables and minimize OpEx. Each SP is a linear program, and SPs are solved in parallel. From each SP, feasibility and optimality cuts are generated and added to the MP. Under the multicut scheme, the MP retains all accumulated cuts per scenario. Let C k s denote the set of cuts for scenario s generated through iteration k -1 (with C 1 s = ∅); at iteration k the MP imposes C k s for all s ∈S. Figure 1 summarizes the algorithm. 1) Benders MP: The MP co-optimizes line and storage investments to minimize CapEx. To capture operational and feasibility constraints, the MP includes proxy variables θs an underestimator of OpExs for each scenario s, and ρ an underestimator of scenario-wide load-shed. The MP at iteration k is presented in Model 2, where (14) prevents load shedding in the final solution. After solving the MP, the new investment point (γk, σk) is broadcast to the SPs, which are solved in parallel to generate the next set of cuts in C k+1 s . 2) Benders SP: The scenario-s SP optimizes operations under the MP decisions (γk, σk), fixed in the SP with dual dual multipliers πs,k γ and πs,k σ in (15)-(16). Each SP runs two phases: Phase I enforces load feasibility and Phase II Model 3 Benders Subproblem (SP) min OpExs(γk, σk) s.t. (4) -(5), ∀t ∈T (6), ∀ij ∈E, t ∈T (7), ∀i ∈N, t ∈T (8), ∀i ∈N, t ∈{2, ..., T} (9) -(10), ∀i ∈N (11) -(13), ∀i ∈N, t ∈T γij = γk ij (πs,k γ ), ∀ij ∈E (15) σi = σk i (πs,k σ ), ∀i ∈N (16) optimizes dispatch. In Phase I, the SP minimizes load shed, P i∈N,t∈T ξi,s,t. If the optimum exhibits load shed, the SP terminates and a feasibility cut is generated for shed underestimator ρ. If no shed occurs, the SP proceeds to Phase II to minimize OpExs (Model 3) and generate an optimality cut for θs. Both feasibility and optimality cuts take an affine form, where ψ ∈{I, II} indicates the Phase, ζs = ρ in Phase I and ζs = θs in Phase II: ζs ≥βk s + (πs,k γ )⊤γ + (πs,k σ )⊤σ βk s := vk,ψ s -(πs,k γ )⊤γk -(πs,k σ )⊤σk vk,I s = P i,t ξi,s,t, vk,II s = OpExs(γk, σk) This two-phase design prioritizes feasibility (no shed) while maintaining complete recourse via ξ, avoiding classical infeasibility-ray cuts which can be numerically fragile [13]. 3) Convergence of Benders: At iteration k, the MP optimum zMP k is a valid lower bound LBk for Model 1, since the variables {θs} underestimate the true scenario costs {OpExs}, giving LBk := zMP k . If all SPs are feasible (Phase I yields P i∈N,t∈T ξi,s,t = 0 for every s ∈S), a feasible two-stage objective at (γk, σk) is ˆzk = f(γk, σk) + OpEx(γk, σk), and is used to maintain a non-increasing upper bound UBk := min{UBk-1, ˆzk}. The algorithm terminates when the relative gap falls below a threshold ε: (UBk -LBk)/UBk ≤ε. Remark. Lazy constraints integrate naturally with BD: per scenario flow/limit constraints identified in one iteration are retained for subsequent iterations, accelerating SP solves. B. Warm-Start from Relaxed Two-Stage Model (WS Module) In the baseline BD formulation, early iterations are feasibility-cut dominated: with no initial cuts, the MP selects a no-expansion plan, causing heavy load shedding in the SPs until a feasible investment set is learned. The WS accelerates this by solving, in parallel for each s ∈S, the continuous relaxation of Model 1 to obtain (γs, σs). A conservative, nonanticipative design is then formed by component-wise aggregation of the per-scenario optimal investments, and rounded up to the admissible investment granularity: ̄γ := maxs∈S γs , ̄σ := maxs∈S σs . Fig. 2. Workflow of the proposed methodology. The Benders MP is initialized at ( ̄γ, ̄σ), which reduces early infeasibility cuts and accelerates convergence. C. Trust-Region-Stabilized Benders (TR-Benders Module) Once feasibility is achieved, the BD formulation exhibits strong dual degeneracy-many line-storage configurations with nearly identical operating costs-yielding flat recourse value functions, oscillatory MP iterates, and weak cuts that slow convergence. The TR-Benders addresses this issue by augmenting the BD framework with a proximal trust-region (TR) constraint on first-stage decisions and a restricted candidate set. This stabilization mechanism limits drastic investment changes between iterations, improving numerical stability and ensuring more consistent convergence behavior. The MP is stabilized around an anchor point ( ̆γ, ̆σ) defined as the incumbent investment that attains the best known upper bound; it is initialized at ( ̄γ, ̄σ) and updated only upon a strict improvement in the upper bound. Inspired by stabilized BD (e.g., [14]), the method adapts to discrete first-stage variables with an l1 trust-region (linear, sparsity-promoting): P ij∈E γij - ̆γij ≤rγ k (17) P i∈N σi - ̆σi ≤rσ (18) Absolute values are linearized in the standard way for MIPs. A fixed storage radius rσ = 2 (permitting limited re-siting) is used, while the transmission radius rγ k is dynamic: (i) resetting to 1 whenever the anchor point updates, and (ii) increasing by 1 whenever the MP revisits the same solution, to limit local oscillations. To further reduce degeneracy, investment decisions are restricted to a candidate set derived from the per-scenario relaxations: Ec := { ij ∈E | ∃s ∈S : γs ij > 0} Nc := { i ∈N | ∃s ∈S : σs i > 0} Finally, a level-set constraint prunes regions already known to be suboptimal: f(γ, σ) + P s∈S ΩS(s) θs ≤UBk (19) Together, the TR, candidate sets, and level-set constraint damp oscillations and weaken the effects of dual degeneracy. Figure 2 summarizes the full workflow of the proposed methodology. D. Global Lower Bound and Optimality Assessment A limitation of the TR stabilization is that the master's value no longer certifies a global lower bound: with the TR active, LBk is a local lower bound over the current region around the anchor point. However, a valid global lower bound can be constructed as below. 1) First-Stage Lower Bound: With load shed disallowed by (14), the minimum feasible investment decision is constrained by the first-stage investments of the most challenging representative day (typically characterized by high load and/or low renewable generation). This is formalized in Theorem 1. Theorem 1. For each s ∈S, define the day-s feasible set and the all-scenarios feasible set: Σs feas := {(γ, σ) : recourse is feasible for scenario s} Σfeas := T s∈S Σs feas Given the feasible sets, define the corresponding minimum first stage costs for feasible day-s and all-scenarios: cfeas s := min (γ,σ)∈Σs feas f(γ, σ), cfeas := min (γ,σ)∈Σfeas f(γ, σ), and first-stage optimal investments for scenario s: (γs feas, σs feas) ∈arg min(γ,σ)∈Σs feas f(γ, σ) Let ˆs ∈arg maxs∈S cfeas s . Then maxs∈S cfeas s ≤f(γ, σ) ∀(γ, σ) ∈Σfeas In particular, f(γˆs feas, σˆs feas) = maxs cfeas s is a valid lower bound on any year-feasible first-stage plan's cost. Proof. Since Σfeas ⊆Σs feas for every s, cfeas = min (γ,σ)∈Σfeas f(γ, σ) ≥ min (γ,σ)∈Σs feas f(γ, σ) = cfeas s ∀s, yielding maxs cfeas s ≤cfeas. Choosing ˆs ∈arg maxs cs gives the stated bound with (γˆs feas, σˆs feas). 2) Second-Stage Lower Bound: The per-scenario operational cost is monotone nonincreasing in investment capacity, as per Theorem 2. Let (γmax, σmax) denote the componentwise upper bounds on line and storage capacities (not restricted to the candidate set) of Model 1. Then a valid lower bound on the aggregated second-stage cost is P s∈S ΩS(s) qs(γmax, σmax). Theorem 2. Let qs(γ, σ) be the optimal value of Model 3. If γ′ ⪰γ and σ′ ⪰σ (componentwise), then qs(γ′, σ′) ≤ qs(γ, σ). In particular, qs(·) is monotone nonincreasing in γ and σ. Proof. For fixed s ∈S, increasing γ relaxes the thermal limit constraints (6), and increasing σ relaxes the storage constraints (11)-(13). Hence the feasible region under (γ′, σ′) contains that under (γ, σ). Because the SP objective is minimized and depends on γ, σ only through these constraints, enlarging the feasible set cannot increase the optimal value, yielding the claim. Fig. 3. Generator sites of the system. Black circles indicate nonrenewable, blue circles indicate wind, and red circles indicate solar generators. 3) Combined Bound: Define z∗as the optimal final solution's objective, with optimal line and storage investments (γ∗, σ∗) ∈Σfeas. z∗:= f(γ∗, σ∗) + P s∈S ΩS(s) qs(γ∗, σ∗) Since Theorem 1 and 2 show, respectively, that f(γˆs feas, σˆs feas) ≤f(γ∗, σ∗), qs(γmax, σmax) ≤qs(γ∗, σ∗) ∀s ∈S, a valid combined lower bound can be constructed: LB∗:= f(γˆs feas, σˆs feas) + P s∈S ΩS(s) qs(γmax, σmax) ≤z∗ 4) Optimality Gap: Each cfeas s is computed under the integer relaxation of the first-stage investments (as in the WS). Being a relaxation, this provides a valid lower bound for the yearwide plan, which must satisfy discretized first-stage decisions. The reported optimality gaps are therefore conservative upper bounds: (UBk -LB∗)/UBk. IV. CASE STUDY The proposed methodology is tested on a large-scale power system case study: the ACTIVSg2000 test case, a 2000-bus model of the U.S. Texas grid. The TEP+Storage model is applied in a rolling-horizon manner. The formulation itself has year-agnostic investment variables, but is solved sequentially at 5-year decision points Y = {2030, 2035, 2040, 2045} with investment periods of length α = 1825 days. Investments (and their capital costs) are cumulative: capacity installed in an earlier year remains available in subsequent years, ensuring monotone, non-decreasing expansion across planning periods. This application reflects practical staging, but it is not guaranteed optimal relative to an omniscient, all-years model; quantifying any cross-period optimality gap is left for future work. a) Network Data: The synthetic Texas system is built from public information and statistical analysis of real power systems [15]. The system has 2000 buses, 3206 lines, 423 non-renewable generators, and 175 renewable generators. Figure 3 shows the location of solar, wind, and nonrenewable generators, overlaid on the grid topology. The future generation mix from 2030 to 2045 follows EIA outlook TABLE I GENERATION BASE CAPACITY (GW) AND GENERATION/LOAD SCALING Type Capacity 2022 2030 2035 2040 2045 Coal 14.5 1.00 0.82 0.82 0.82 0.82 Natural Gas 75.1 1.00 0.79 0.73 0.71 0.72 Nuclear 7.1 1.00 0.98 0.90 0.80 0.80 Solar 2.5 1.00 4.51 6.00 6.87 8.04 Wind 15.5 1.00 2.02 2.23 2.26 2.32 Load - 1.00 1.13 1.21 1.31 1.41 Fig. 4. Representative days by average demand vs. wind in 2022. Red circles indicate selected representative days. Transparent blue circles indicate nonselected days. projections [16]. All generation resources are uniformly scaled to reflect projected growth or planned decommissioning. Load growth is modeled with a year-to-year increase of 1.5% and is scaled uniformly across all buses. Table I shows the total generator capacities (GW) of the network and summarizes the scaling for each generation type and load relative to the base year, 2022. b) Scenario selection: Scenarios are 24-h representative days sampled from a year of load/wind/solar data. This reduces the computational time required for evaluating a full year of operations, while still capturing intertemporal storage dynamics. In this case study, kr = 18 days are selected by a max-min diversity criterion on normalized daily profilesmaximizing the minimum pairwise Euclidean distance-and assigned equal weights. Figure 4 shows the chosen days (red circles) in average wind-load space-the main drivers of investment (solar capacity is comparatively smaller). Days that require storage operation to avoid load shed are labeled; these generally coincide with higher demand. August 11 (08-11) is the most challenging day, exhibiting the highest average demand. c) Investments parameters: Table II lists capital and operating costs for line upgrades and storage. Line upgrades are discretized to capacity increases of 30, 60, and 90%, reflecting the enhancements achievable through typical line reconductoring [11]. Storage units are 250 MWh each, limited to 12 per node (3 GWh total), with charge and discharge efficiencies of η = 95%. Line upgrade costs are adapted from [17]. Storage costs are adapted from [18]. d) Computational Setup and Hyperparameters: The TEP+Storage model is implemented in Julia with JuMP [19] and solved with Gurobi [20] on Georgia Tech's PACE Phoenix TABLE II COST CONFIGURATION Costs Capacity Upgrades 1,000,000/MWh Charge/Discharge Cost B) Storage ( B) StorOpEx ($B) 2030 08-11 183 0 / 0 0.14 0 46.49 0 - Warm-start 291 0 / 0 0.37 0 28.80 0 1.3% TR-Benders 238 0 / 0 0.24 0 28.86 0 1.0% 2035 08-11 334 33 / 28.91 0.36 28.92 50.91 7.27 - Warm-start 381 33 / 32.75 0.53 32.77 30.39 0.40 6.3% TR-Benders 352 32 / 29.25 0.40 29.27 30.42 0.41 0.8% 2040 08-11 425 118 / 98.07 0.47 98.12 56.60 24.71 - Warm-start 462 118 / 111.75 0.66 111.81 33.35 1.64 9.4% TR-Benders 441 108 / 98.50 0.53 98.55 33.38 1.65 0.4% 2045 08-11 476 184 / 167.30 0.58 167.40 62.6 42.01 - Warmstart 508 184 / 183.75 0.72 183.84 36.63 3.66 7.4% TR-Benders 499 179 / 167.75 0.66 167.84 36.64 3.66 0.3% Fig. 5. WS computation times by representative day for each 5-year investment period. TABLE IV TR-BENDERS COMPUTATIONAL PERFORMANCE FOR EACH 5-YEAR INVESTMENT PERIOD. 2030 2035 2040 2045 Time Limit (Hrs.) 24 24 24 72 Iterations 1964 1198 332 102 Avg. Seconds/Iter 44 72 260 2541 over the planning horizon because progress each round is paced by the most challenging SP/representative day: the algorithm advances only after all cuts are generated. The slowest iterations coincide with intensive storage operation and pronounced congestion, which prolong SP solves due to storage time-coupling and ongoing discovery of violated thermal-limit constraints. Further computational improvement may be attained by pre-screening thermal violations. C. Investment Results The TR-Benders produces a feasible, near-optimal yearwide plan for 2030-2045 with zero load shed across all representative days. Figure 6 maps the geographic distribution of investments. Transmission upgrades concentrate in West Texas, strengthening transfer from wind/solar regions, while storage appears beginning in 2035 and grows considerably with increasing load. Figure 7 reports storage energy throughputs for each planning year, defined as the total GWh charged and discharged during the representative day. Only a small fraction of days require storage operation to eliminate load shedding; 08-11 exhibits the largest throughput and aligns with its computational difficulty (Fig. 5). Consequently, multiyear investment choices are largely driven by this hardest day. Table III summarizes the selected upgrades, storage siting/sizing, and costs; its rightmost column displays the final optimality gaps referenced earlier. For completeness, the single-day (08-11) WS results are also listed by period. The WS intentionally over-invests. After the TR-Benders, the near-optimal plan reduces CapEx relative to the WS, while incurring only negligible increases in OpEx, yielding a lower total cost. The 08-11 single-day results show comparatively higher OpEx than both the WS and TR-Benders plans because they optimize solely for the highest-demand day rather than the equally weighted set of representative days. Because storage dominates investment costs, most of the residual gap is driven by reconciling storage capacity rather than line upgrades. The storage capacity of TR-Benders approaches-but remains above-the requirement implied by the hardest day, consistent with Theorem 1 ensuring zero load-shed feasibility. By 2045, the optimized plan yields 167.75 GWh of storage distributed across 179 nodes, which is equivalent to 32% of the scaled peak renewable capacity. This underscores the robustness of the proposed formulation and its tractability at high spatial and storage resolution. VI. CONCLUSION This paper presents a PTDF-based TEP+Storage for joint transmission and storage expansion and applies it to a 2,000bus synthetic Texas system. To ensure computational feasibility, it introduces a trust-region, multicut Benders scheme warm-started from per-representative-day optima. The method attains final optimality gaps ≤1% and remains tractable for a large distributed storage fleet deployed at ≥179 nodes (≈32% of peak renewable capacity). These results demonstrate the proposed PTDF-based methodology effectively handles high spatial resolution and large distributed storage deployments in joint TEP+Storage planning. Table IV shows that computational challenges persist, particularly under high storage penetration and network conges- (a) 2030 TEP+Storage (b) 2035 TEP+Storage (c) 2040 TEP+Storage (d) 2045 TEP+Storage Fig. 6. Upgrades from 2030 to 2045 in 5-year intervals. Red lines denote capacity upgrade locations, with their thickness corresponding to one of three possible upgrade levels. Circles indicate storage locations, with their size and color representing the number of installed units (250MWh each). Fig. 7. Storage energy throughput (GWh) for representative days with nonzero utilization across the 2030-2045 planning horizon. Values reflect the representative day under the investment plan in effect for that year. tion, and will intensify as temporal, operational, and spatial fidelity increase. Priorities include accelerating SP solvese.g., pre-screening and dynamic pruning of thermal/flow constraints-and exploring unified Benders variants that combine feasibility and optimality cuts. Another direction is to integrate generation expansion, which may relieve congestion and further improve lazy-constraint performance. 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Akimov1, †\n1Experimentelle Physik 2, Technische Universit¨at Dortmund, 44227 Dortmund, Germany\n2Spin Optics Laboratory, St. Petersburg State University, Peterhof, 198504 St. Petersburg, Russia\n3ITMO University, 199034 St.\nPetersburg, Russia\n4St. Petersburg State Institute of Technology, 190013 St. Petersburg , Russia\n5Department of Physics, Jackson State University, Jackson, 39217 Mississippi, USA\n6Condensed Matter Theory, Department of Physics,\nTechnische Universit¨at Dortmund, 44227 Dortmund, Germany\n7Research Center FEMS, Technische Universit¨at Dortmund, 44227 Dortmund, Germany\n(Dated: October 17, 2025)\nExciton–phonon interactions govern the energy level spectrum and thus the optical response in\nsemiconductors.\nIn this respect, lead-halide perovskite nanocrystals represent a unique system,\nfor which the interaction with optical phonons is particularly strong, giving rise to a ladder of\nmultiple exciton states which can be optically excited with femtosecond pulses. We establish a new\nregime of coherent exciton-polaron dynamics with exceptionally long coherence times (T2 ≈300 ps)\nin an ensemble of CsPbI3 nanocrystals embedded in a glass matrix.\nUsing transient two-pulse\nphoton echo at 2 K temperature, we observe quantum beats between the exciton-polaron states.\nWithin a four-level model, we directly quantify the exciton–phonon coupling strength through the\nHuang–Rhys factors of 0.05 ÷ 0.1 and 0.02 ÷ 0.04 for low-energy optical phonons with energies of\n3.2 and 5.1 meV, respectively. The pronounced size dependence of both coupling strengths and\nphonon lifetimes offers a path to tune the optical transitions between polaron states and to tailor\nthe coherent optical dynamics in perovskite semiconductors for solid-state quantum technologies.\nI.\nINTRODUCTION\nLead halide perovskite semiconductors attract close at-\ntention due to their intriguing optical and electrical prop-\nerties with huge potential for photovoltaic and light emit-\nting applications [1, 2]. These materials provide a unique\nplayground for polaron physics in the solid state gov-\nerned by the interaction of electronic excitations with the\nphonons of the crystal lattice [3]. In particular, several\nstudies indicate that low-energy optical phonons in halide\nperovskites have an important influence on the conduc-\ntivity, while acoustic phonons have a minor effect [4, 5].\nThis is in contrast to conventional semiconductors such\nas GaAs, where the interaction of electrons with acous-\ntic phonons is dominant at cryogenic temperatures [6, 7].\nFurthermore, the carrier mobility is determined not by a\nfew specific optical phonons, but by a large multitude of\nmodes due to the complex phonon spectrum [8].\nPerovskite nanocrystals (NC) belong to the class of\nquantum emitters with discrete energy level spectrum,\nlarge spectral tunability, and high quantum yield, which\nmakes them attractive as single photon sources [9, 10].\nAt cryogenic temperatures, the elementary optical exci-\ntation in NCs is an exciton (electron-hole pair) which\npossesses long optical and spin coherence times up to\nhundreds of ps, comparable to its radiative lifetime [11–\n13].\nLong-lived coherent excitons are often considered\nas qubits and are appealing for possible applications\n∗correspondence address: artur.trifonov@tu-dortmund.de\n† correspondence address: ilja.akimov@tu-dortmund.de\nin quantum communication [14].\nHowever, the strong\nexciton-phonon interaction leads to formation of exciton-\npolarons which modifies the energy level spectrum. In\nparticular, the photoluminescence spectra of single NCs\nshow pronounced optical phonon replicas which confirm\nthe substantial exciton-phonon coupling with Huang-\nRhys factors in the range from 0.05 to 0.4 in lead halide\nperovskite NCs [15–19].\nRecently, electron-phonon in-\nteraction dominated by optical phonons with a similar\nHuang-Rhys factor of about 0.4 was demonstrated in\nCsPbBr3 NCs [20] and other inorganic lead halide per-\novskite NCs [21] using Raman spectroscopy. Compared\nto InGaAs quantum dots [22, 23], these values are sig-\nnificantly larger and comparable to those reported in\nII-VI nanostructures [24]. A distinctive feature of per-\novskite nanocrystals is the existence of low-energy optical\nphonons with a strong coupling to phonons. This combi-\nnation substantially modifies the exciton level spectrum\nin the vicinity of the ground state [25] and represents an\nattractive testbed for investigation of the quantum dy-\nnamics under resonant femtosecond excitation markedly\ndifferent from previously studied systems.\nTime-resolved studies can provide direct access to the\nexciton dynamics. It is well established that scattering\non incoherent phonons with a thermal distribution causes\nloss of exciton coherence [26, 27].\nYet, less is known\nabout the quantum dynamics of exciton-polarons, i.e.,\nthe hybrid excitations of an exciton and a phonon. Re-\ncently, transient four-wave mixing and two-dimensional\nFourier spectroscopy revealed the importance of coherent\nexciton-phonon coupling for the ground and excited (ex-\nciton and biexciton) states in lead halide perovskites [28–\narXiv:2510.14695v1 [cond-mat.mes-hall] 16 Oct 2025\n\n2\n31]. However, most of these studies were limited to the\ncase of short optical coherence of excitons (≤100 fs) and\ntherefore reported on the coherent evolution of either the\nexciton or the ground electronic state modulated by the\nfrequencies of phonons with much slower decoherence.\nIn other words, the lack of optical coherence restricted\nthe observations to the phonon dynamics, while the gen-\nuine coherent evolution of the coupled exciton–phonon\nsystem has remained inaccessible. This scenario was also\ndocumented for bulk semiconductors, nanostructures and\norganic molecules [32–36]. However, the opposite quan-\ntum dynamics limit, where pronounced quantum beats of\nexciton-polarons are expected, has not yet been uncov-\nered. Furthermore, another general question arises, con-\nnected to the limitations of coherent control for resonant\nexcitation with femtosecond pulses, since time-resolved\nstudies reported quite short coherence times up to 20 ps\nin lead halide perovskites [28, 37], in contrast with pho-\ntoluminescence data of single NCs [11–13].\nIn this work we establish a new regime of coherent dy-\nnamics governed by exciton-polarons in NCs with an ex-\nceptionally long coherence time of the zero-phonon exci-\nton (T2,X ≈300 ps) on the one hand and a discrete spec-\ntrum of spectrally narrow optical phonon modes on the\nother hand. The regime manifests in long-lived exciton-\npolaron quantum beats, which are detected by tran-\nsient two-pulse photon echoes in an ensemble of CsPbI3\nnanocrystals embedded in a glass matrix at low temper-\nature, T = 2 K. The oscillatory signal is dictated by\noptical phonons with low frequencies corresponding to\n3.2 and 5.1 meV energy, and can be described by the\nsolution of the Lindblad equations for a set of four-level\nsystems that include a phonon ladder in the ground and\nexcited exciton-polaron states. Different sizes, ranging\nfrom 10 to 12 nm, were selectively addressed within the\nsame ensemble of nanocrystals by varying the laser pho-\nton energy. From the relative amplitude of oscillations\nwe directly evaluate the strength of the exciton-phonon\ncoupling with Huang–Rhys factors of 0.06 and 0.02 for\noptical phonons with 3.2 meV and 5.1 meV energy, re-\nspectively.\nCorrespondingly, the electron–phonon cou-\npling strength doubles reaching values of 0.12 and 0.04\nas the NC size is reduced from 12 to 10 nm. The theo-\nretical analysis of the size dependence shows that both\nthe deformation potential and the Fr¨ohlich mechanism\nof electron-phonon interaction qualitatively agree with\nthe experimental data in the regime of weak exciton con-\nfinement.\nOur study demonstrates that the dynamics\nof a pure quantum state following resonant optical ex-\ncitation requires consideration of polaron effects, whose\nstrength and coherence can be controlled by composition\nand size of the NCs. This opens up new possibilities for\non-demand coherent control and coherent phonon gener-\nation in lead-halide perovskite NCs.\nII.\nPHOTON ECHO AND LONG COHERENCE\nOF ZERO-PHONON EXCITON\nThe ensemble of CsPbI3 NCs is synthesized in a flu-\norophosphate glass matrix by rapid cooling of a glass\nmelt enriched with the materials needed for the per-\novskite crystallization. The details of synthesis are given\nin Ref. [38]. The NC size is about 9 ÷ 13 nm in diameter\n(see Fig. S6 in Ref. 39). The low-temperature photolumi-\nnescence (PL) spectrum is dominated by a 60 meV broad\nband centered around 1.75 eV, as shown in Fig. 1(a). The\nbroadening originates from fluctuations of the NC size,\nwhich we estimate to be in the order of 20%. A single NC\nis schematically illustrated in the inset of Fig. 1(b). Note\nthat the cubic crystal structure is shown for illustrative\npurposes as the actual crystallographic phase depends on\nthe growth conditions and the size of the nanocrystals\n[40–43].\nWe perform transient four-wave mixing (FWM) exper-\niments in transmission geometry as shown schematically\nin Fig. 1(b). The NCs are resonantly excited with a se-\nquence of two 100 fs laser-pulses with photon energies\nhν <1.76 eV in the low energy flank of the PL band.\nThe sample temperature is kept at T = 2 K. The electric\nfield amplitude of the FWM signal EFWM(t) is resolved in\ntime using heterodyne detection where the signal field is\ntemporally overlapped with a strong reference pulse with\namplitude Eref(t−τref) (see Methods and Refs. 44–47 for\ndetails). Here, time t = 0 corresponds to excitation with\nthe first pulse, while τref is the delay of the reference\npulse with respect to the first pulse in the excitation se-\nquence. The resulting FWM signal is shown in Fig. 1(c),\nwhich gives a two-dimensional plot of EFWM as function\nof the delay time between the first and second pulse τ12\n(vertical axis) and the reference delay time τref (horizon-\ntal axis). The FWM signal demonstrates the expected\npeak centered at time τref = 2τ12, corresponding to emis-\nsion of a photon echo (PE) from the NC ensemble [48].\nThis behavior arises from the inhomogeneous broaden-\ning of the optical transitions resulting from fluctuations\nof the NC size as demonstrated for similar halide per-\novskite NCs [28, 37]. Interestingly, during the first tens\nof ps the PE amplitude shows pronounced high frequency\noscillations with increasing τ12 which we will discuss be-\nlow.\nOn a longer time scale, the amplitude of the photon\necho decays exponentially with increasing delay time τ12\nas shown in Fig. 1(d).\nUsing exponential decay fits,\nshown with the dashed lines, we obtain T2 = 230÷330 ps\nfor photon energies in the range 1.724 ÷ 1.75 eV, show-\ning a weak decrease with increasing hν.\nThe popula-\ntion relaxation time T1, measured in three-pulse experi-\nments, is found to be T1 = 800 ÷ 850 ps, and is compa-\nrable to the exciton lifetime measured by time-resolved\nPL on the same sample.\nTherefore, we conclude that\nthe signal at τref > 50 ps is due to the zero-phonon ex-\nciton transition with long-lived optical coherence time\nT2 and exciton recombination lifetime T1.\nIf the co-\n\n3\n1.65\n1.7\n1.75\n1.8\n1.85\n0\n0.5\n1\nEnergy (eV)\nPL intensity (arb. units)\nτ12\nτref\nLead\nIodine\nCaesium\n1\n2\n3\n4\n5\n6\n7\n−0.5\n0\n0.5\n0.5\n1\n1.5\n2\nτ12 (ps)\nτref - 2τ12 (ps)\n E *\nFWM E ref (arb. units)\n(a)\n(b)\n(c)\n0\n50\n100\n1\n234\n0.1\nτref (ps)\n E *\nFWM E ref (arb. units)\n1.75 eV\n1.74 eV\n1.724 eV\nT2 = 276 ps\n(d)\n10\n12\n8\n9\n15\nNC diameter (nm)\nT2 = 330 ps\nT2 = 230 ps\n1\n2\nref\nPE\nk2\nk1\n2k2 - k1\nT = 2K\nFigure 1. Photon echoes in CsPbI3 nanocrystals. (a) Typical photoluminescence (PL) spectrum for excitation with photon\nenergy of 2.33 eV. Vertical dashed line indicates the highest photon energy used in the four-wave mixing FWM experiment.\nThe scale on top corresponds to the NC diameter. (b) Schematic representation of the experimental geometry, where the\nlaser pulses hit the sample with wave vectors k1, k2.\nThe photon echo is detected along the direction 2k2 −k1 using a\nheterodyning technique by overlapping it with a reference pulse. Inset shows schematically a CsPbI3 perovskite nanocrystal.\n(c) Two-dimensional plot of the FWM electric field amplitude E∗\nFWM as function of delay between the two pulses τ12 and the\nreference time τref. The photon echo (PE) signal forms at the time τref = 2τ12. The amplitude of the PE shows oscillations\nduring the initial evolution when τ12 is scanned. The signal is recorded in linearly co-polarized configuration. Photon energy\nhν = 1.736eV. (d) Decay of two-pulse photon echo amplitudes for excitation with different photon energies. Black dashed lines\nare fits with exponential functions, from which the labeld exciton coherence times T2 are extracted. Temperature T = 2 K.\nherent dynamics would be governed only by the pop-\nulation decay we would expect to the T2 = 2T1 rela-\ntion to hold, which typically holds for excitons in self-\nassembled InGaAs quantum dots at T = 2 K [49]. Here,\nwe observe a different situation, where elastic scattering\nprocesses (”pure dephasing”) with a decay constant of\nTp = [1/T2 −1/(2T1)]−1 = 330 ps mainly govern the ex-\nciton coherence. Nevertheless, to the best of our knowl-\nedge, the homogeneous linewidth of the zero-phonon ex-\nciton of Γ2 = 2ℏ/T2 = 4.8 µeV demonstrated here has a\nrecord low value for perovskite nanocrystals [11–13].\nIII.\nCOHERENT EXCITON-PHONON\nDYNAMICS\nAs follows from Fig. 1(c) for short delay times τ12 ≲\n10 ps, oscillations of the PE signal with multiple frequen-\ncies are observed. For deeper insight into the origin of\nthe oscillations, we analyze polarization-resolved PE sig-\nnals. These results are summarized in Fig. 2. We employ\ntwo configurations, where the first and second pulses are\nlinearly co- or cross- polarized, while the detection polar-\nization in all cases coincides with that of the first pulse\n(see Methods). The PE amplitudes as function of τ12 in\nco-(∥) and cross-(×) polarized configurations are shown\nin Fig. 2(a). The oscillations can be grouped into two\ncategories: slow oscillations with a period of 1–10 ps and\nfast oscillations with a period of less than 1 ps. It is evi-\ndent that the fast oscillations in the co-polarized config-\nuration are in phase and nearly identical in both curves.\nWe will show below that these oscillations correspond to\nquantum beats between exciton-polaron states.\nBy contrast, the slow oscillations possess opposite\nphases.\nThe co-polarized A∥-signal starts maximum\nvalue, while the A×-signal starts from zero.\nSubse-\nquently, the minimum of the slow oscillations in ∥-\nconfiguration around τ12 = 5 ps corresponds to a maxi-\nmum in the ×-configuration. This antiphase behavior is\nrelated to quantum beats between the orthogonally po-\nlarized bright exciton states [37]. In perovskite nanocrys-\ntals, the bright exciton fine structure comprises three lin-\nearly polarized states [15, 50], split by the energies δ1 and\nδ2 as shown in Fig. 3(a). This energy level scheme is sim-\nilar to the fine structure splitting in self-assembled quan-\ntum dots [51, 52] and in II-VI colloidal NCs [53, 54]. In\norder to isolate the oscillations due to the fine structure\nsplitting in the data, we model the expected polariza-\ntion dependence of the PE signal, accounting also for the\nrandom orientation of the nanocrystals (for details see\nSec. S1 of the Supplementary Information). We obtain\nthe following equations for the amplitude of the phonon\n\n4\n0\n10\n20\n30\n40\n0\n0.2\n0.4\n0.6\n0.8\n1\nτ12 (ps)\nPE ampl. (arb. units)\nco-linear (||)\ncross-linear (x)\n0\n10\n20\n30\n40\n0\n0.5\n1\n1.5\nτ12 (ps)\nAmplitude (arb. units)\nρ\nΣ\nFit\n0\n2\n4\n6\n8\n0\n2\n4\n6\n8\n10\nEnergy (meV)\nIntensity (arb. units)\n0\n2\n4\n6\n8\n0\n0.5\n1\n1.5\n2\nEnergy (meV)\nFFT (arb. units)\nρ\nΣ\nFit\n(a)\n(b)\n(c)\n(d)\nRaman\n(x2)\nFigure 2. Quantum beats of exciton-polarons. (a) Initial range of the PE dynamics in the co-linear (∥) and cross-linear\n(×) polarization configurations shown with blue and red lines, respectively. Photon energy hν = 1.746 eV. The amplitude of\nthe × signal is multiplied by two for clarity. (b) Dynamics of ρ (black) and Σ (green) as defined by Eqs. 2 and 3, respectively.\nThe dashed red curve is a fit using the polaron model with Eq. 5 with the following parameters for two phonon modes:\nℏΩ1 = 3.2 meV, SHR1 = 0.097, τph1 = 5.1 ps; ℏΩ2 = 5.1 meV, SHR2 = 0.032, τph2 = 10 ps. (c) Fast Fourier transform (FFT)\namplitude spectra of ρ (black), Σ (green), and the fit curves in panel (b). Vertical dashed lines mark the peaks corresponding to\noptical phonons. (d) Raman spectrum measured at photon energy hν = 1.734 eV. Vertical dashed lines indicate the positions\nof peaks corresponding to the optically active optical phonon modes that couple most strongly to the exciton.\necho signal in two polarization configurations:\nA∥=\n\"\n18\n15 +\nX\ni\n4\n15 cos (δiτ12) e\n−2τ12\nt∗\ni\n#\nΨ0(τ12) ,\n(1a)\nA× =\n\"\n6\n15 −\nX\ni\n2\n15 cos (δiτ12) e\n−2τ12\nt∗\ni\n#\nΨ0(τ12) ,\n(1b)\nwhere the index i = 1, 2, 3 corresponds to the beats be-\ntween the three optical transitions (δ3 = δ1 + δ2), and t∗\ni\nis the dephasing time of the beats caused by the disper-\nsion of the splitting energies ℏδi in the ensemble. The\ncommon factor Ψ0(τ12) is the same for both polariza-\ntion configurations and will be discussed below, see also\nSupplementary Information S1. Following Eqs. (1) it is\npossible to distinguish between quantum beats related to\nthe exciton fine structure and other polarization insen-\nsitive contributions to the PE signal by introducing the\npolarization contrast\nρ = A∥−3A×\nA∥+ 2A×\n,\n(2)\nand the polarization sum\nΣ = A∥+ 2A×\n2\n= Ψ0(τ12) .\n(3)\nIt follows that the temporal evolution of ρ exhibits oscil-\nlations at frequencies corresponding to the energy split-\ntings δi of the exciton fine structure and is independent\nof Ψ0(τ12). In contrast, the polarization sum Σ is given\nby the intrinsic coherent dynamics Ψ0(τ12) only.\nFigure 2(b) shows the dynamics of these quantities,\ncalculated from the data in Fig. 2(a).\nIn full accord\nwith our expectations we obtain that the high-frequency\noscillations are absent in the dynamics of ρ, while the\nlow-frequency oscillations are still present. On the other\nhand, the low-frequency oscillations vanish in the Σ tran-\nsient in contrast to the high-frequency components. This\nbecomes even more clear from the fast Fourier transform\n(FFT) spectra of ρ and Σ shown in Fig. 2(c). From the\nfit of the ρ transient we evaluate δ1 = 0.25 meV and\nδ2 = 0.55 meV, which are in agreement with the exci-\nton fine structure splitting evaluated by two-dimensional\nFourier spectroscopy on CsPbI3 NCs of similar size with\na diameter of about 9 nm [37]. A detailed discussion of\nthe fine exciton structure combining pump-probe and PE\nstudies will be published elsewhere. In what follows, we\nfocus on the dynamics of Ψ0(τ12), which is independent\nof the spin level structure of the exciton.\nThe FFT spectrum of Σ, shown by the green line in\nFigure 2(c), exhibits several spectrally narrow features.\nThese features are consistent with the peaks in the Ra-\n\n5\nman spectrum in Figure 2(d). The latter originates from\nlight scattering on optically active phonons.\nThe fea-\ntures at energies below 0.5 meV are attributed to con-\nfined acoustic phonons [55]. Below we focus on the two\nmost prominent features marked by the vertical dashed\nlines with energies of ℏΩ1 = 3.2 meV and ℏΩ2 = 5.1meV,\ncorresponding to the energies of optically active optical\nphonons in the vicinity of the Γ point. Indeed optical\nphonon modes with energies close to 3 and 5 meV in\nCsPbI3 were observed in Ref. [21] and calculated [8]. Op-\ntical phonons with similar energies were also detected by\nRaman spectroscopy in other lead halide perovskites such\nas FAPbI3 [56, 57], CsPbBr3 [19], and CsPbCl3 [17, 58].\nThus, we conclude that the high frequency THz oscilla-\ntions in the two-pulse coherent optical response are due to\nthe coherent evolution of the coupled, hybridized exciton-\nphonon system.\nIV.\nEXCITON-POLARON QUANTUM BEATS\nIn order to describe the quantum dynamics of the cou-\npled exciton-phonon system we consider the structure of\nthe optically excited energy levels in a single NC, which\nrequires the involvement of polaron states. To that end,\nwe introduce the four-level scheme shown in Fig. 3(b)\nwhich comprises the ground state |0⟩, the phonon mode\n|0′⟩, the exciton polaron state |X⟩(“exciton”) and the\nfirst excited exciton polaron state |X′⟩(see details in the\nSupplementary Information S2 S2.A). We neglect the ex-\nciton fine structure splitting to concentrate on the deriva-\ntion of Ψ0(τ12) in Eq. (1).\nInitially, we consider the\nexciton coupling to a single phonon mode in a NC. At\nthe next stage, to compare with the experimental data,\nwe perform summation over different phonon modes Ωi\nwhere we neglect the interaction between them\nΨ0(τ12) =\nX\ni\nΨ0(τ12; Ωi) .\n(4)\nThe optical transitions between all energy levels are\nallowed using the same polarization.\nIn the polaron\nmodel [59], the transition probability amplitude is pro-\nportional to the product of the dipole matrix element and\nthe overlap of the wavefunctions associated with vibronic\nmodes shifted due to polaron formation (see Fig. 3(c) and\nEq. S16), which depends on the strength of the exciton-\nphonon interaction given by the Huang-Rhys factor SHR\n(Condon principle). The pulse duration is assumed to be\nshort as compared with all other characteristic times in\nthe system, i.e. we can assume the δ-pulse limit. This is\njustified because the laser pulse duration is τd ≈100 fs\n(see Methods) and the fit to experimental data gives re-\nlaxation times in the order of 10 ps. The spectral width of\nthe laser pulse ∼h/τd is about 15 meV, which is larger\nthan the phonon energies ℏΩi.\nTherefore, we assume\nthat all four transitions between states |0⟩, |0′⟩and |X⟩,\n|X′⟩are covered by the spectral width of the laser. Note\nthat a not too large Huang-Rhys factor means that states\ninvolving multiple phonons (i.e. 2ℏΩ1, etc.) may be ne-\nglected. However, the spectral width of the laser pulse is\nlarge enough to cover them as well. We neglect the possi-\nble difference of phonon energy in the ground and polaron\nstate as we do not find corresponding frequency changes\nin our experimental data and in previous reports in other\nperovskites [56, 57]. Similar assumptions were made in\nthe model developed for desscribing 2D spectroscopy in\nRef. 60.\nIn the polaron model, we trace the evolution of the\ndensity matrix components which contribute to the PE.\nThe result arises from the coherent off-diagonal terms\nand all levels contribute to the result in first order in the\nHuang-Rhys factor SHR, which quantifies the exciton-\nphonon interaction. Solution of the Lindblad equation\nin Supplementary Information S2 gives the following ex-\npression for the amplitude of the phonon echo signal from\na NC with single phonon mode Ωi\nΨ0(τ12; Ωi) = e−(1+SHR)γ0τ12\n\"\n1 + SHRe−(γph−SHRγ0)τ12\n+SHR cos (Ωiτ12)e−\nγph−2SHRγ0\n2\nτ12h\n2+e−SHRγ0τ12+e−γphτ12i\n+ SHR cos (2Ωiτ12)e−(γph−SHRγ0)τ12\n#\n.\n(5)\nHere T2,0 = 1/γ0 is the coherence time associated with\nthe zero-phonon optical transition, and τph = 1/γph is\nthe lifetime of the optical phonon.\nIn the simplified\nmodel we do not include other sources of decoherence.\nThe analytical expression fully supports the experimen-\ntal observations for Σ = Ψ0(τ12) shown in Fig. 2(b). The\nfirst term in Eq. 5 corresponds to the long-lived zero-\nphonon coherence which decays exponentially with the\ntime T2 = T2,0/(1 + SHR) ∼300 ps, as evaluated in\nsection\nII. The last two terms correspond to high fre-\nquency oscillations at the single and double frequency of\nthe phonon mode Ωi, respectively. They appear due to\nquantum interference of excitations of different exciton-\npolaron states, i.e.\ndue to quantum beats of exciton-\npolarons.\nThese oscillations are superimposed on the\nlong-lived signal and decay with a shorter time τph. The\nrelative amplitude of the oscillatory signal and the long-\nlived plateau allows one to measure the Huang-Rhys fac-\ntor SHR.\nUsing Eqs. 4 and 5 for two independent phonon modes\nwith ℏΩ1 = 3.2 meV and ℏΩ2 = 5.1 meV we obtain\nexcellent agreement with the experimental data.\nThe\nphonon frequencies are taken from the peak positions in\nthe Fourier spectrum of Fig. 2(c). The Huang-Rhys fac-\ntors and phonon lifetimes are evaluated from the best fit\nto the experimentally measured transients, see Fig. 2(b).\nWe emphasize that in contrast to previous reports [32–\n36], our system demonstrates long-lived coherent dynam-\nics, which is attributed to an exceptionally long zero-\nphonon exciton coherence (∼300 ps) and a relatively\nlong phonon lifetime (∼10 ps). Here, we stress that the\n\n6\nlow-temperature regime with T = 2 K is essential, as the\ncoherence of the zero-phonon excitons rapidly vanishes\nwith increasing temperature. Furthermore, the optical\nphonons each are represented by well-defined frequen-\ncies due to their flat dispersion around the Γ-point, as\nconfirmed by the Raman spectrum in Fig. 2(d), mak-\ning CsPbI3 NCs embedded in a glass matrix particularly\ninteresting for investigating coupled exciton-phonon dy-\nnamics. The peak widths in the Raman spectrum can be\nrecalculated into phonon lifetimes using the following re-\nlation τph = 2ℏ/∆ER [61], where ∆ER is the full width\nat half maximum in energy units. This yields phonon\nlifetimes of approximately 7 ps and 11 ps for the 3.2 and\n5.1 meV modes, respectively. These values are in excel-\nlent agreement with the decay times of the PE oscillatory\nsignals.\nIt should be noted that the Raman peaks in\nFig. 2(d) appear narrower than those in the FFT spectra\nin Fig. 2(c), which originates from the difference between\nthe power spectral and the amplitude spectral represen-\ntations, respectively.\nThe proposed polaron model (i) provides an intuitive\npicture of the microscopic processes involved, (ii) has\na simple analytical solution, and (iii) accounts explic-\nitly for the coherence decay through simple equations.\nEquation (5) gives a result that closely matches that in\nRefs. [32, 62] and is in effect equivalent within the accu-\nracy limits of the models. Note that a direct comparison\nbetween the results obtained by the two approaches is\nnot straightforward: Eq. (5) is obtained from the exact\nsolution of the Lindblad equations for a four-level sys-\ntem, while Ref. 62 considers the quasi-classical evolution\nof a complex system averaged over the phonon subsys-\ntem.\nFor a detailed discussion and comparison of the\ntwo approaches see e.g. Ref. 63.\nV.\nNC SIZE DEPENDENCE\nAbove, we discussed the PE dependence measured for\nthe fixed laser pulse photon energy of hν = 1.734 eV\nand explained how the phonon parameters can be eval-\nuated from the PE signal. However, the sample under\nstudy contains NCs with different sizes which can be se-\nlectively excited by tuning the laser photon energy. We\nmeasured PE transients at photon energies in the range\nof 1.72÷1.76 eV. The photon energy can be recalculated\ninto a NC diameter using the parameters from Ref. 39,\nwhere the same sample was studied (sample #3). Fig-\nures 4(a) and\n4(b) show the dependence of the evalu-\nated Huang-Rhys factors SHR and phonon lifetimes τph\nas function of the NC diameter a, for phonon modes with\nenergies ℏΩ1 = 3.2 meV and ℏΩ2 = 5.1 meV. We note\nthat the frequencies of these modes do not depend on a\nwithin the accuracy of the experiment. The Huang–Rhys\nfactors for both phonon modes exhibit a clear size de-\npendence, increasing with decreasing NC size.\nCorre-\nspondingly, the phonon lifetimes decrease with decreas-\ning a.\nAn increase in the electron–phonon coupling\n|0>\n|0'>\n|X'>\n|X>\n(b)\nγph\nγph\nγ0\nSHRγ0\n(1-SHR)2γ0\n|Z>\n|Y>\n|X>\n|0>\nγ0\nγ0\nγ0\n(a)\nδ2\nδ1\nħΩ\nħΩ\nConfiguration coordinate Q\nEnergy\nEX\nSHRħΩ\nħΩ\n|0>\n|0'>\n|X'>\n|X>\n(c)\nFigure 3. Level schemes. (a) Energy levels of the bright ex-\nciton fine structure. (b) Energy level structure of the excitons\ninteracting with optical phonons. We assume this level struc-\nture for each phonon mode, independent of the exciton po-\nlarization. The probability of the diagonal transitions is pro-\nportional to the Huang-Rhys factor SHR. γph is the phonon\ndecay rate and γ0 is the exciton decay rate, mostly determined\nby its oscillator strength. (c) Sketch of adiabatic potential of\nan exciton interacting with a particular phonon mode. The\nsolid parabolas represent the potential for phonons for the\nground state of the crystal (no exciton, bottom) and the ex-\ncited state (with one exciton, top). The solid parabolas are\nseparated by the energy difference EX corresponding to the\nenergy of zero-phonon exciton transition. ℏΩdenotes the en-\nergy of the optical phonon.\nstrength with decreasing NC size was previously reported\nfor CdSe nanocrystals [64], PbS quantum dots [65], and\nperovskite nanocrystals, both for optical and acoustic\nphonons [16, 19]. However, to the best of our knowledge,\ndirect measurements of the associated phonon relaxation\ntimes have not been reported.\nIt is also important to\nnote that, in contrast to previous studies, our PE exper-\niments selectively probe the coherent dynamics of each\nindividual phonon mode.\nIn Ref. 19, the dominant mechanism of electron-\nphonon coupling is assigned to the optical deformation\npotential [66]. In this case, the size dependence of the\ninteraction can be estimated from the phonon normal-\nization condition [67] which leads to SHR ∼a−3.\nFor\ncompleteness, let us discuss the size dependence of the\ninteraction between charge carriers and optical phonons\n\n7\nfor the Fr¨holich mechanism following Takagahara [68].\nIn the original work, the interaction of charge carriers\nwith phonons is rewritten as potential for an electron\nin the field induced by the phonon mode polarization\nP(r). This potential is found from the Poisson equation\n∇2φ(r) = 4 π∇· P(r) which leads to\nφ(re) =\nZ\ndr ∇· P(r)\n|r −re| = −\nZ\ndr P(r) · ∇\n1\n|r −re| . (6)\nFrom Eq. (6) the strength of electron-phonon interaction\n∆e ∼a−1 which leads to Se\nHR = ∆2\ne/2 ∼a−2. This esti-\nmate is valid also for excitons in the strong confinement\nregime.\nFor weakly confined excitons, the change of exciton\nenergy ∆is proportional to the difference of the electro-\nstatic potential for electron and hole φ(re) −φ(rh). It is\nfound to be the sum of Eq. (6) for electron and hole:\n∆∼φ(re)−φ(rh) =\nZ\ndr P(r)·∇\n\u0012\n1\n|r −rh| −\n1\n|r −re|\n\u0013\n.\n(7)\nIn the weak confinement regime when the exciton Bohr\nradius aB is small compared with the NC diameter aB ≪\na,\n1\n|r −rh| ≈\n1\n|r −re| + (re −rh) · ∇\n1\n|r −re| .\n(8)\nSubstituting Eq. (8) into Eq. (7) we obtain\nφ(re) −φ(rh) ≈\nZ\ndr P(r) · ∇\n\u0014\n(re −rh) · ∇\n1\n|r −re|\n\u0015\n.\nAs a result, in the weak confinement regime, the size\ndependence of this expression is estimated as\n∆∼φ(re) −φ(rh) ∝a3 · a−3/2 · aB\na3 ∝a−3/2.\nHere, the first size factor comes from the integral, the\nsecond one from the normalization of the phonon mode,\nand the last one from the gradient of the expression in\nthe square brackets. As a result the Huang-Rhys factor\nscales as SHR = ∆2/2 ∼a−3.\nThus, in the weak confinement regime both mecha-\nnisms, the deformation potential and the Fr¨ohlich inter-\naction, lead to similar dependences of the Huang-Rhys\nfactor on the NC size SHR ∼a−3. In the strong con-\nfinement regime, the Fr¨ohlich interaction is expected to\nhave a weaker size dependence, while the deformation\npotential mechanism should show the same scaling. The\ndashed lines in Fig. 4 represent fits of the experimental\ndata by the empirical relation SHR ∝a−3. The experi-\nmental data in Fig. 4(a) even shows a steeper dependence\nof the Huang–Rhys factor on NC size, with pronounced\ndeviation from the a−3 dependence, particularly for small\nNCs. This observation suggests that further analysis is\nrequired for smaller NCs to identify the mechanisms of\nelectron–phonon interaction in perovskite NCs.\n0\n0.02\n0.04\n0.06\n0.08\n0.1\n0.12\nSHR\nħΩ1 = 3.2 meV\nħΩ2 = 5.1 meV\n10\n10.5\n11\n11.5\n12\n0\n5\n10\n15\n20\nNC diameter (nm)\nτph (ps)\n(a)\n(b)\nFigure 4.\nNC size dependence.\nDependence of (a) the\nHuang-Rhys factor, SHR, and (b) the phonon lifetime, τph,\nfor the two phonon modes with energies ℏΩ1 = 3.2 meV\n(blue dots) and ℏΩ2 = 5.1 meV (red dots) on NC diameter a.\nDashed lines in (a) are fits proportional to a−3. Dashed lines\nin (b) are guides to the eye.\nVI.\nDISCUSSION\nTo summarize, we have revealed a new regime of co-\nherent exciton-phonon dynamics following resonant opti-\ncal excitation with femtosecond pulses in an ensemble of\nCsPbI3 nanocrystals embedded in a glass matrix. Quan-\ntum beats between exciton-polaron states manifest them-\nselves in two-pulse photon echo experiments as oscilla-\ntions with high frequencies corresponding to two optical\nphonon modes at 3.2 and 5.1 meV. A polarization re-\nsolved analysis of the photon echo signal has allowed us\nto unambiguously isolate the quantum beat signatures of\nexciton-polarons from slower beats arising from exciton\nfine-structure splittings. This unique regime of exciton-\npolaron quantum beats is established at the low temper-\nature of 2 K, where an exceptionally long zero-phonon\nexciton coherence time is achieved (T2 ≈300 ps). The\npresence of spectrally narrow optical phonon modes with\nlifetimes of about 10 ps, combined with a relatively strong\nexciton-phonon coupling (Huang-Rhys factors up to 0.1),\nleads to a pronounced modulation of the photon echo sig-\nnal at the phonon frequencies. Furthermore, the strong\nsize dependence of the Huang–Rhys factor makes it pos-\nsible to tune the strength of the optical transitions be-\ntween different polaron states, offering a route to control\nthe quantum dynamics for targeted applications.\nWe emphasize that the observed optical response\nmarkedly differs from previous studies in, e.g., III–V self-\nassembled quantum dots, where the dynamics are mainly\ngoverned by exciton–acoustic phonon coupling [26, 69,\n70].\nThere, the continuous phonon spectrum leads to\na partial loss of coherence within the first few picosec-\n\n8\nonds after pulsed excitation.\nIn our case no such be-\nhavior is observed in the temporal evolution of Ψ0(τ12)\n[see Fig. 2(b)]. Instead, the coherent dynamics is dom-\ninated by exciton–polaron states, i.e., interactions with\ndiscrete optical phonon modes, which can be consistently\ndescribed within a four-level model for each phonon fre-\nquency. The desired quantum evolution can be achieved\nby tuning both the exciton–phonon coupling strength\nand, most importantly, the phonon lifetime, which limits\nthe evolution of the pure quantum state following res-\nonant excitation with a short optical pulse.\nNotably,\nthe phonon lifetime exhibits a pronounced dependence on\nnanocrystal size. For the optical phonon mode with an\nenergy of 5.1 meV, we observe an almost twofold increase\nof τph as the size increases from 10 to 12 nm. In layered\nperovskites, phonon lifetimes on the order of ∼10 ps were\nreported [71], with values extended up to 75 ps [72], ap-\nproaching the record-high ∼200 ps observed in bulk ZnO\n[61].\nTherefore, the strong size dependence of phonon\nlifetimes, along with reports of extended phonon coher-\nence in layered perovskites, gives a promising outlook for\nlong-lived and tunable coherent control of the quantum\ndynamics in perovskite nanocrystals.\nVII.\nMETHODS\nSamples. The studied CsPbI3 nanocrystals embed-\nded in fluorophosphate Ba(PO3)2-AlF3 glass were syn-\nthesized by rapid cooling of a glass melt enriched with\nthe components needed for the perovskite crystallization.\nDetails of the method are given in Refs. [38, 73]. The\ndata on NC sizes evaluated from scanning transmission\nelectron microscopy can be found in the supporting in-\nformation section S5 in Ref. [39]. Samples of fluorophos-\nphate (FP) glass with the composition 35P2O5–35BaO–\n5AlF3–10Ga2O3–10PbF2–5Cs2O (mol.\n%) doped with\nBaI2 were synthesized using the melt-quench technique.\nThe glass synthesis was performed in a closed glassy car-\nbon crucible at the temperature of T = 1050◦C. Techno-\nlogical code of the studied sample is EK8. More infor-\nmation on its optical and spin-dependent properties can\nbe found in Refs. 39 and 74.\nThe sample is placed in the variable temperature insert\nof a bath cryostat and kept in superfluid helium at the\ntemperature of 2 K. The low energy exciton optical tran-\nsitions in nanocrystals are located in the spectral range of\n1.7–1.8 eV as follows from the photoluminescence spec-\ntrum, see Fig. 1(a).\nPhoton echo. Two-pulse photon echoes were mea-\nsured using a transient four-wave mixing technique with\nheterodyne detection [44–47].\nWe used a Ti:Sapphire\nlaser as source of optical pulses with duration of ∼100 fs\nat the repetition rate of 75.75 MHz. All optical experi-\nments were carried out at photon energies below 1.76 eV.\nThe sample is excited resonantly by a sequence of two\npulses separated in time by the delay τ12, giving rise to\nthe formation of a four-wave mixing signal with electric\nfield amplitude EFWM(t). The pulses are focused into a\nspot of about 200 µm diameter using a 0.5-m parabolic\naluminum mirror. The incidence angles of the pulses are\nclose to normal and equal to approximately 1/50 rad and\n2/50 rad, corresponding to the in-plane wavevectors k1\nand k2. An additional reference pulse is used for detec-\ntion and its delay with respect to the first pulse is given\nby the time τref. The delay times τ12 and τref are con-\ntrolled using mechanical delay lines. The four-wave mix-\ning signal is measured in transmission geometry in the\ndirection ≈3/50 rad, which corresponds to the phase-\nmatching direction 2k2 −k1. Optical heterodyne detec-\ntion is used to resolve the enhanced signal in time. By\nmixing with a relatively strong reference pulse and scan-\nning τref, the temporal profile of the photon echo pulse\nis measured.\nPolarization sensitive measurements are performed\nby\nselecting\nproper\npolarization\nconfigurations\n(H-\nhorizontal or V-vertical) for each of the beams using\nGlan-Thompson prism in conjunction with half-wave\nplates.\nIn the detection path we also select the linear\npolarization by setting the polarization of the reference\nbeam. We adopt a three-letter notation for the PE polar-\nization configuration. The first two symbols correspond\nto the polarizations of the pulses exciting the sample,\nand the third symbol corresponds to the detection po-\nlarization. For example, in the HHH configuration, the\npolarizations of both excitation pulses and the detection\nare horizontal (H). In the HVH configuration, the sec-\nond excitation pulse is vertically (V) polarized, while the\nfirst pulse and detection are horizontally polarized. HHH\nis accordingly termed co-linearly polarized (∥) and HVH\ncross-linearly polarized (×) in Fig. 2. We note that rota-\ntion of the sample does not lead to changes in the signal,\ni.e. the orientation of the sample is not important.\nRaman spectroscopy.\nThe Raman scattering sig-\nnal was excited by a single-frequency Ti:Sapphire laser.\nThe laser beam with the power of about 1 mW was\nfocused on the sample to a spot with a diameter of\nabout 300 µm. The scattered light was analyzed by a\nJobin-Yvon U1000 double monochromator with 1-meter\nfocal length, allowing the high resolution of 0.2 cm−1\n(0.024 meV). The Raman signal was detected by a cooled\nGaAs photomultiplier and conventional photon-counting\nelectronics. The Raman spectra were measured for co-\npolarized linear polarizations of excitation and detection.\nVIII.\nACKNOWLEDGEMENTS\nThe authors are thankful to I. A. Yugova and N.\nE. Kopteva for fruitful discussions.\nWe acknowledge\nthe financial support by the Deutsche Forschungsge-\nmeinschaft:\nA.V.T., M.A.H, S.G., M.O.N and I.A.A\n(project AK40/13-1, no. 506623857) and D.R.Y. (project\nYA65/28-1, no.\n527080192).\nThe work of S.V.G. was\nsupported by the NSF through DMR-2100248. A.V.T,\nE.V.K. and M.S.K. acknowledge the Saint-Petersburg\n\n9\nState University (Grant No. 125022803069-4).\n[1] M. A. Green, A. Ho-Baillie, and H. J. Snaith, The emer-\ngence of perovskite solar cells, Nature Photonics 8, 506\n(2014).\n[2] B. R. Sutherland and E. H. Sargent, Perovskite photonic\nsources, Nature Photonics 10, 295 (2016).\n[3] Y. Yamada and Y. 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For the photon echo calculations the following procedure\nis used: the dynamics is calculated in the eigenbasis of NC {|0⟩, |x′⟩, |y′⟩, |z′⟩} then the equations for the density\nmatrix components are rotated to the laboratory frame {|0⟩, |x⟩, |y⟩, |z⟩}. The 1st pulse is assumed to be polarized\nalong z axis, the 2nd pulse at time τ is either polarized along z or x axis for HHH and HVH schemes respectively.\nThe polarization along z is detected at the photon echo time 2τ. The orientation of a nanocrystal in the laboratory\nreference frame is described by three Euler angles α, β, and γ. The rotation matrix relating coordinates in xyz and\nx′y′z′ bases is [75]:\nR(α, β, γ) =\n\n\ncαcβcγ −sαsγ −cαcβsγ −sαcγ cαsβ\nsαcβcγ + cαsγ −sαcβsγ + cαcγ sαsβ\n−sβcγ\nsβsγ\ncβ\n\n,\n(S1)\nhere and below in this section we use sξ ≡sin ξ, cξ ≡cos ξ (ξ = α, β, γ) to shorten the notation.\nIn the eigenbasis of NC, the dynamics is given by (we skip common factor e−γt):\nρ0x′ = e−t/T2eiω1tρ0\n0x′ ,\nρ0y′ = e−t/T2eiω2tρ0\n0y′ ,\nρ0z′ = e−t/T2eiω3tρ0\n0z′ ;\n(S2)\nρx′0 = e−t/T2e−iω1tρ0\nx′0 ,\nρy′0 = e−t/T2e−iω2tρ0\ny′0 ,\nρz′0 = e−t/T2e−iω3tρ0\nz′0 .\n(S3)\nHere ℏωi (i = 1, 2, 3) are the three exciton energies for excitons linerly polarized along x′, y′, z′ respectively. We\nassume that the diplole matrix element (and, consequently, radiative decay time) as well as the decoherence time T2\nis the same for all three exciton states. After 1st pulse the components of density matrix are\nρ1+\n0z = 1 ,\nρ1+\n0y = ρ1+\n0x = 0 .\n(S4)\nAfter 2nd pulse we have two options: for HHH scheme\nρ2+\nz0 = ρ2−\n0z ,\nρ2+\ny0 = ρ2+\nx0 = 0 .\n(S5)\nand for HVH scheme\nρ2+\nx0 = ρ2−\n0x ,\nρ2+\ny0 = ρ2+\nz0 = 0 .\n(S6)\nRotating the dynamics equations (S2) into laboratory coordinate frame using (S1) gives rather lengthy equations for\nthe amplitude of the signal:\nA∥(α, β, γ) ∝ρz0 =\n\u0002\nc4\nαs4\nβ + s4\nαs4\nβ + c4\nβ\n+ 2s2\nαc2\nαs4\nβ cos(ω1 −ω2)τ + 2c2\nαs2\nβc2\nβ cos(ω1 −ω3)τ + 2s2\nαs2\nβc2\nβ cos(ω2 −ω3)τ\n\u0003\ne−2τ/T2\n(S7)\nA×(α, β, γ) ∝ρz0 =\n\u0002\nc2\nαs2\nβ(cαcβcγ −sαsγ)2 + s2\nαs2\nβ(sαcβcγ + cαsγ)2 + s2\nβc2\nβc2\nγ\n+ 2sαcαs2\nβ(cαcβcγ −sαsγ)(sαcβcγ + cαsγ) cos(ω1 −ω2)τ −2cαs2\nβcβcγ(cαcβcγ −sαsγ) cos(ω1 −ω3)τ\n−2sαs2\nβcβcγ(sαcβcγ + sαsγ) cos(ω2 −ω3)τ\n\u0003\ne−2τ/T2\n(S8)\nThe result should be averaged over directions using:\n⟨A⟩ang ≡\n1\n8π2\nZ 2π\n0\ndα\nZ π\n0\nsin βdβ\nZ 2π\n0\ndγ A(α, β, γ)\n(S9)\n\nS2\nAssuming the constant energies of three exciton components, but the arbitrary direction of the NCs, the amplitude\nof the echo is proporitional to\n\nA∥\n\u000b\nang ∝\n\n6\n5 + 4\n15\nX\ni>j\ncos(ωi −ωj)τ\n\ne−2τ\nT2 ,\n(S10)\n⟨A×⟩ang ∝\n\n2\n5 −2\n15\nX\ni>j\ncos(ωi −ωj)τ\n\ne−2τ\nT2 .\n(S11)\nNow we also take into account the possible deviation of exciton energies. The absolute value of the exciton energy is\nnot important for the phonon echo, only the splittings. E.g., the amplitude averaged over distribution of splittings\nbetween 1st and 2nd levels ω12 = ω1 −ω2 is given by\n⟨A⟩ω12 =\nZ\nW(ω12)A(ω12, ω13, ω23 = ω12 −ω13)dω12\n(S12)\nAssuming the distribution function of the splittings to be Lorentzian,\nW(ω) = t∗\ni\nπ\n1\n(ω −δi)2 +\n√\n2/(t∗\ni )2 ,\n(S13)\nand that all three splittings are not correlated, the PE signal averaged over both the angles and the splittings\ndistribution is given by\n\nA∥\n\u000b\n∝\n\n18\n15 + 4\n15\nX\ni=1,2,3\ncos(δiτ)e\n−2τ\nt∗\ni\n\ne−2τ\nT2 ,\n(S14)\n⟨A×⟩∝\n\n6\n15 −2\n15\nX\ni=1,2,3\ncos(δiτ)e\n−2τ\nt∗\ni\n\ne−2τ\nT2 ,\n(S15)\nwhere the parameters t∗\ni work as an effective signal dephasing times and δi are the average splittings. An interesting\nresult is that even though the PE signal is stable with respect to the level disorder, the disorder of its fine structure\ngives the contribution to the additional decay, not of the full signal, but of its oscillating components.\nS2.\nMODEL OF PHOTON ECHO OF EXCITON-POLARON STATES.\nIn this section we outline the derivation of PE signal in the polaron model. We assume the exciton-polaron with\nrelatively small Huang-Rhys factor which allows to perform calculations in four (six with account on light polarization)\nlevel scheme.\nS2.A.\nScheme of the levels and dynamics of the system\nWe consider the four-level system shown in Fig. 3b. There are: ground level |0⟩, statwe with one phonon |0′⟩,\npolaron state |X⟩, and excited state of polaron |X′⟩. The energies of corresponding states are 0, ℏΩ, ωX, ωX + ℏΩ.\nIn the calculations below we assume that spectral width of the laser pulse covers at least |X⟩and |X′⟩levels, this is\nin contrast to standard approximations for three-level atoms [76].\nDue to polaron formation, allowed transitions are not only X →0 (with probability γ0) and X′ →0′ (with\nprobability γ1), but also X →0′ and X′ →0 (in this model, probability is the same γ′). In polaron model [59], the\ntransition probability is proportional to product of dipole matrix element |d|2 and the overlap of oscillator functions\nshifted due to exciton formation which leads to extra factor\nγ0 ∝|d|2e−2SHR\n(S16a)\nγ′ ∝|d|2SHRe−2SHR\n(S16b)\nγ1 ∝|d|2(1 −SHR)2e−2SHR\n(S16c)\n\nS3\nwhere d is the dipole matrix elements of the optical transition, and we defined Huang-Rhys factor as SHR = ϵpol/(ℏΩ)\n(ϵpol is the polaron formation energy). Note that in this approximation polaron formation energy should be small\nwhich means that γ′ ≪γ0, γ1.\nIn addition, we assume that the phonon may be emitted which leads to transitions 0′ →0 and X′ →X with the\nsame probability γph.\nTo trace the polarization of the signal, we need to consider two polarization states of exciton, from now on the\nenumeration 0 and 0′ are ground state and state with phonon, 1(2) and 1′(2′) are right(left) circularly polarized states\nof exciton(polaron) and first oscillator level of exciton(polaron), making the model effectively six-level. Still, we call\nthe model four-level as here we neglect the splittings between levels (as the role of the fine structure is excluded, see\nSec. S1) and we need it only to demonstrate that the phonon structure does not change the polarization structure of\nthe signal. Below, we extensively omit states 2 and 2′ when equations are same.\nThe Lindblad equation, with all levels considered is:\ndρ\ndt = −i\nℏ[H0, ρ] + Ld[ˆρ] + Γ ⊙ˆρ .\n(S17)\nFor convenience and compactness of the notation, with the use of ⊙for the Hadamard product (element-wise product\nor Schur product).\nThe Hamiltonian is\nH0 =\n\n\n\n\n\n\n\n0\n0\n0\n0\n0\n0\n0 ℏΩ\n0\n0\n0\n0\n0\n0\nℏωX\n0\n0\n0\n0\n0\n0\nℏωX\n0\n0\n0\n0\n0\n0\nℏωX + ℏΩ\n0\n0\n0\n0\n0\n0\nℏωX + ℏΩ\n\n\n\n\n\n\n\n(S18)\nand the “inflow”diagonal part of superoperator is\nLd[ˆρ] =\n\n\n\n\n\n\n\nγphρ0′0′ + γ0ρX + γ′ρX′\n0\n0\n0\n0 0\n0\nγ1ρX′ + γ′ρX\n0\n0\n0 0\n0\n0\nγphρ1′1′\n0\n0 0\n0\n0\n0\nγphρ2′2′ 0 0\n0\n0\n0\n0\n0 0\n0\n0\n0\n0\n0 0\n\n\n\n\n\n\n\n(S19)\nwhere for convenience we introduced ρX = ρ11 + ρ22 and ρX′ = ρ1′1′ + ρ2′2′, and the matrix in the “outflow” part is\nΓ = −1\n2\n\n\n\n\n\n\n\n0\nγph\nγ′\n0\nγ′\n0\nγph + γ′\n1\nγph + γ′\n1\nγph\n2γph\nγph + γ′\n0\nγph + γ′\n0\n2γph + γ′\n1\n2γph + γ′\n1\nγ′\n0\nγph + γ′\n0\n2γ′\n0\n2γ′\n0\nγph + γ′\n1 + γ′\n0 γph + γ′\n0 + γ′\n1\nγ′\n0\nγph + γ′\n0\n2γ′\n0\n2γ′\n0\nγph + γ′\n1 + γ′\n0 γph + γ′\n0 + γ′\n1\nγph + γ′\n1 2γph + γ′\n1 γph + γ′\n0 + γ′\n1 γph + γ′\n0 + γ′\n1\n2γph + 2γ′\n1\n2γph + 2γ′\n1\nγph + γ′\n1 2γph + γ′\n1 γph + γ′\n0 + γ′\n1 γph + γ′\n0 + γ′\n1\n2γph + 2γ′\n1\n2γph + 2γ′\n1\n\n\n\n\n\n\n\n(S20)\nwhere for convenience we defined\nγ′\n0 = γ0 + γ′ ,\nγ′\n1 = γ1 + γ′ .\n(S21)\nS2.B.\nEffect of the pulse on density matrix\nNow we consider light-mediated transitions in the system shown in Fig. 3b. This is again generalization of the\ntwo-level system, check e.g. chapters 2, 6 and 10 of [76]. The pulse is approximated by the plane wave with the\nsmooth enveope E0(z, t) and electric field given by\nE(r, t) = E0(k · r/k, t)ei(k·r−ωt) + c.c.\n(S22)\nAssuming the pulse duration is short compared with all coherence and relaxation times, the evolution of density\nmatrix (DM) during the pulse may be found from the Schr¨odinger equation\ndρ\ndt = −i\nℏ[H0 + V, ρ] ,\n(S23)\n\nS4\nwhere, in rotating wave approximation,\nV = ℏ\n\n\n\n\n\n\n\n\n0\n0\n˜f ∗\n+\n˜f ∗\n−\ns · ˜f ∗\n+ s · ˜f ∗\n+\n0\n0\ns · ˜f ∗\n+ s · ˜f ∗\n−\n˜ff ∗\n+\n˜f ∗\n+\n˜f+\ns · ˜f+\n0\n0\n0\n0\n˜f−\ns · ˜f−\n0\n0\n0\n0\ns · ˜f+\n˜f+\n0\n0\n0\n0\ns · ˜f−\n˜f−\n0\n0\n0\n0\n\n\n\n\n\n\n\n\n,\n(S24)\n˜f± = dE±\n0 e−iωt ≡f±e−iωt ,\n(S25)\nwhere d is the dipole matrix element for the transitions 1, 2 →0, s = √SHR, E±\n0 are circular components of the\nenvelope (S22).\nSolution of Eq. (S23) is surprisingly complex in general case, for two-level system it may be found in Ref. 76, but\nwe are interested in four-level system and the case of small pulse area when the full solution is surplus. When the\npulse area θ± (equal to f±tp for square pulse of duration tp) is small compared with Rabi frequency, the first order\ncorrection to the density matrix may be obtained as\nρ+ = −i\nℏ\n\u0002\nH0 + V, ρ−\u0003\ntp ≡ˆ\nAρ−,\n(S26)\nwhere ρ−is the density matrix before pulse and ρ+ is the density matrix after pulse. Full solution of Eq. (S26) is too\nlengthy for our needs, below we will only give the results relevant for the calculations.\nFor the PE we will need the action of the light pulse in the second order over pulse area. It may be shown that up\nto second order, it may be calculated as\nρ+ = ˆ\nA ˆ\nAρ−.\n(S27)\nBelow we also give only part of this result relevant for photon echo calculations.\nS2.C.\nPhoton echo\nThe signal in two circular polarizations is proportional [76] to the corresponding components of polarization vector\nwhich is given by the following components of the density matrix:\nP+ ∼ρ10 + ρ1′0′ + sρ1′0 + sρ10′ ,\n(S28a)\nP−∼ρ20 + ρ2′0′ + sρ2′0 + sρ20′ .\n(S28b)\nNote the extra factor s in Eq. (S28) which originates from the ratio of diplole matrix elements for phonon-assisted\ntransitions to the phonon-less transitions. Thus, we are interested in the following components of the density matrix\nat times after second pulse: ρ10, ρ20, ρ1′0, ρ2′0, ρ10′, ρ20′, ρ1′0′, ρ2′0′.\na.\nDensity matrix after 1st pulse\nThe components of DM after 1st pulse are given by (S26) which leads to:\nρ1+\n01 = iθ∗\n1+ ,\nρ1+\n01′ = isθ∗\n1+ ,\nρ1+\n02 = iθ∗\n1−,\nρ1+\n02′ = isθ∗\n1−.\n(S29)\nb.\nRelaxation of DM components after 1st impulse\nOne needs to trace the evolution of the following DM compo-\nnents: ρ01, ρ01′, ρ02, ρ02′. Below we give results only for right polarization, dynamics of other components is the same.\nWriting the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components:\nρ01 (t ∈(0, τ12)) = ρ1+\n01 eiωXte−\nγ′\n0\n2 t ,\n(S30a)\nρ01′ (t ∈(0, τ12)) = ρ1+\n01′ei(ωX+Ω)te−\nγ′\n1+γph\n2\nt ,\n(S30b)\nThis leads to the following components of the DM at the time of 2nd pulse t = τ12\nρ2−\n01 = ρ1+\n01 εXΓ0 ,\nρ2−\n01′ = ρ1+\n01′εXεΓ1Γp ,\n(S31)\nwhere for convenience we defined\nεX = eiωXτ12 , ε = eiΩτ12 , Γ0 = e−\nγ′\n0\n2 τ12 ,\nΓ1 = e−\nγ′\n1\n2 τ12 ,\nΓp = e−\nγph\n2 τ12 .\n(S32)\nNote that Γ0 and Γ1 are almost equal and defined by the exciton coherence time (under our assumptions half of its\nlifetime). The difference between them is Γ1/Γ0 = e−γ1−γ0\n2\nτ12 = e\nSHRγ0\n2\nτ12 which is much smaller than all typical\ntimes in the system since SHR is small.\n\nS5\nc.\nDM after 2nd pulse\nFrom (S27) we may compute the components relevant for the PE signal after the 2nd\npulse:\nρ2+\n10 =\n\u0002\nθ2\n2+(ρ2−\n01 + sρ2−\n01′) + θ2+θ2−(ρ2−\n02 + sρ2−\n02′)\n\u0003\n(S33a)\nρ2+\n10′ =\n\u0002\nθ2\n2+(sρ2−\n01 + ρ2−\n01′) + θ2+θ2−(sρ2−\n02 + ρ2−\n02′)\n\u0003\n(S33b)\nρ2+\n1′0 = s\n\u0002\nθ2\n2+(ρ2−\n01 + sρ2−\n01′) + θ2+θ2−(ρ2−\n02 + sρ2−\n02′)\n\u0003\n(S33c)\nρ2+\n1′0′ = s\n\u0002\nθ2\n2+(ρ2−\n01′ + sρ2−\n01 ) + θ2+θ2−(ρ2−\n02′ + sρ2−\n02 )\n\u0003\n(S33d)\nand\nρ2+\n20 =\n\u0002\nθ2+θ2−(ρ2−\n01 + sρ2−\n01′) + θ2\n2−(ρ2−\n02 + sρ2−\n02′)\n\u0003\n(S34a)\nρ2+\n2′0 =\n\u0002\nθ2+θ2−(sρ2−\n01 + ρ2−\n01′) + θ2\n2−(sρ2−\n02 + ρ2−\n02′)\n\u0003\n(S34b)\nρ2+\n20′ = s\n\u0002\nθ2+θ2−(ρ2−\n01 + sρ2−\n01′) + θ2\n2−(ρ2−\n02 + sρ2−\n02′)\n\u0003\n(S34c)\nρ2+\n2′0′ = s\n\u0002\nθ2+θ2−(ρ2−\n01′ + sρ2−\n01 ) + θ2\n2−(ρ2−\n02′ + sρ2−\n02 )\n\u0003\n(S34d)\nd.\nRelaxation of DM components after 2nd impulse\nWe we need to know the evolution of the DM components\nwhich enter (S28). Below we give only components entering P+. Dynamics of components entering P−is the same.\nWriting the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components:\nρ10 (t > t2) = ρ2+\n10 e−iωX(t−t2)e−\nγ′\n0\n2 (t−t2) ,\n(S35a)\nρ1′0 (t > t2) = ρ2+\n1′0e−i(ωX+Ω)(t−t2)e−\nγ′\n1+γph\n2\n(t−t2) ,\n(S35b)\nρ10′ (t > t2) = ρ2+\n10′e−i(ωX−Ω)(t−t2)e−\nγ′\n0+γph\n2\n(t−t2) ,\n(S35c)\nρ1′0′ (t > t2) = ρ2+\n1′0′e−iωX(t−t2)e−\nγ′\n1+2γph\n2\n(t−t2) ,\n(S35d)\nWhich gives the DM components at the time of echo signal t −t2 = τ12:\nρP E\n10 = ρ2+\n10 ε∗\nXΓ0 ,\nρP E\n1′0 = ρ2+\n1′0ε∗\nXε∗Γ1Γp ,\n(S36a)\nρP E\n10′ = ρ2+\n10′ε∗\nXεΓ0Γp ,\nρP E\n1′0′ = ρ2+\n1′0′ε∗\nXΓ1Γ2\np .\n(S36b)\nThe evolution of components corresponding to second circular polarization is the same.\ne.\nContribution of different paths to PE\nTo get the final result, we substitude (S29) into (S31) and then the result\ninto (S33) and then from (S36) we get the DM at the time of PE. The amplitude of the signal in two polarizations is\nthen obtained from Eqs. (S28):\nP+ ∼ρPE\n10 + ρPE\n1′0′ + sρPE\n1′0 + sρPE\n10′ ,\n(S37a)\nP−∼ρPE\n20 + ρPE\n2′0′ + sρPE\n2′0 + sρPE\n20′ .\n(S37b)\nCollecting all terms in the result we have for contributions to P+ from ρ10, ρ1′0′, ρ10′, ρ1′0 originating from the DM\ncomponents after 1st pulse given below in the beginning of each line:\nρ01,02 →ρPE\n10\nΓ2\n0A+ ,\n(S38a)\nρ01′,02′ →ρPE\n10\ns2εΓ0Γ1ΓpA+ ,\n(S38b)\nρ01,02 →ρPE\n10′\ns2εΓ2\n0ΓpA+ ,\n(S38c)\nρ01′,02′ →ρPE\n10′\ns2ε2Γ0Γ1Γ2\npA+ ,\n(S38d)\nρ01,02 →ρPE\n1′0\ns2ε∗Γ0Γ1ΓpA+ ,\n(S38e)\nρ01′,02′ →ρPE\n1′0\ns4Γ2\n1Γ2\npA+ ,\n(S38f)\nρ01,02 →ρPE\n1′0′\ns2Γ0Γ1Γ2\npA+ ,\n(S38g)\nρ01′,02′ →ρPE\n1′0′\ns2εΓ2\n1Γ3\npA+ ,\n(S38h)\nwhere\nA+ = θ2+\n\u0000θ2+θ∗\n1+ + θ2−θ∗\n1−\n\u0001\n.\n(S39a)\n\nS6\nNote that the term (S38f) may be dropped as it exceeds precision of equations.\nThe contribution to P−from ρ20, ρ2′0′, ρ20′, ρ2′0 is the same as (S38) but instead of A+ the polarization dependence\nis given by\nA−= θ2−\n\u0000θ2−θ∗\n1−+ θ2+θ∗\n1+\n\u0001\n.\n(S39b)\nIt is convenient to write both as\nA± = θ2±A0 ,\nA0 = θ2−θ∗\n1−+ θ2+θ∗\n1+ ,\n(S40)\nto see that the polarization dependence of the PE signal coincides with second impulse polarization while its amplitude\ndepends on respective polarization of first and second pulses.\nAs a result, the amplitude of the non-oscillating signal is\nPP E,0\n±\n∼\n\u0002\nΓ2\n0 + SHRΓ0Γ1Γ2\np\n\u0003\nA± .\n(S41)\nThe amplitude of the signal oscillating at Ωis\nPP E,Ω\n±\n∼SHRΓp\n\u0002\nΓ0Γ1 (ε + ε∗) + ε\n\u0000Γ2\n0 + Γ2\n1Γ2\np\n\u0001\u0003\nA± .\n(S42)\nand of the signal oscillating at 2Ωis\nPP E,2Ω\n±\n∼SHRΓ0Γ1Γ2\npε2A± .\n(S43)\nf.\nResult\nLet us first consider the general scheme of the polarization dependence of PE signal. Following [47],\nwe consider linearly polarized excitation pulses with second pulse rotated to angle ϕ and linearly polarized detection\nwith polarized rotated to ϕd. Then\nθ1+ = θ1−= 1/\n√\n2 , θ2+ = eiϕ/\n√\n2 , θ2−= e−iϕ/\n√\n2 .\n(S44)\nWhich gives A0 = cos ϕ and\nA± = e±iϕ\n√\n2 cos ϕ .\n(S45)\nAmplitude of linearly-polarized heterodyne detection of main PE signal is proportional to\nP 0\nd =\n\f\f\f\f\nei(ϕ−ϕd) + e−i(ϕd−ϕ)\n2\ncos ϕ\n\f\f\f\f = |cos (ϕ −ϕd) cos ϕ|\n(S46)\nWe have two configurations: HHH and HVH. First is ϕ = ϕd = 0 and second is ϕd = 0, ϕ = π/2. This gives\nP 0,HHH\nd\n= 1 ,\nP 0,HV H\nd\n= 0.\n(S47)\nAs long as polarization dependence on the excitation/detection angles (S46) is real (which is the case for HHH\nand HVH configurations), the absolute value of product is the product of absolute values and signal is zero for HVH\nscheme and for HHH scheme (we omit index HHH below) is:\nPP E\nd\n= Γ2\n0\n\f\f\f\nh\n1 + SHReΓΓ2\np\ni\n+ SHRΓp\nh\neΓ (ε + ε∗) + ε(1 + eΓ2Γ2\np)\ni\n+ SHReΓΓ2\npε2\f\f\f\n(S48)\nwhere we introduced\neΓ = Γ1/Γ0 = eSHRγ0τ12 .\n(S49)\nNote that eΓ is 1 as compared with both, Γ0 and Γp.\nTo write explicitly the result for the PE amplitude, we use the fact that for t ≪1 and arbitrary complex ϵ:\n|1 + tϵ| = 1 + tRe(ϵ) + t2\n2 (Im(ϵ))2 + O\n\u0000t3\u0001\n.\n(S50)\nWe keep second order terms to highlight the fact that even for single frequency of the complex amplitude, the absolute\nvalue in (S48) will result in second order in SHR signal at double frequency.\n\nS7\nUsing (S50) we may write the final result (keeping only terms to the first order in SHR)\nPP E\nd\n= Γ2\n0\nn\n1 + SHRΓp\nh\neΓΓp +\n\u0010\n1 + 2eΓ + eΓ2Γ2\np\n\u0011\ncos(Ωτ12) + eΓΓp cos(2Ωτ12)\nio\n(S51)\nOr, explicitly,\nPP E\nd\n= e−γ′\n0τ12\n\"\n1 + SHRe−\nγph\n2 τ12\n \ne−\nγph−2SHRγ0\n2\nτ12\n+\n\u0010\n1 + 2eSHRγ0τ12 + e−(γph−2SHRγ0)τ12\u0011\ncos(Ωτ12) + e−\nγph−2SHRγ0\n2\nτ12 cos(2Ωτ12)\n!#\n,\n(S52)\nand, finally, with the help of Eqs.(S16,S21):\nPP E\nd\n= e−(1+SHR)γ0τ12\n\"\n1 + SHRe−(γph−SHRγ0)τ12\n+ SHR cos(Ωτ12)e−\nγph−2SHRγ0\n2\nτ12 \u0000e−SHRγ0τ12 + 2 + e−γphτ12\u0001\n+ SHR cos(2Ωτ12)e−(γph−SHRγ0)τ12\n#\n≡Ψ0(τ12; Ω) .\n(S53)\n"},"cleaned":{"kind":"string","value":"Quantum beats of exciton-polarons in CsPbI3 perovskite nanocrystals A. V. Trifonov,1, 2, ∗M. O. Nestoklon,1 M.-A. Hollberg,1 S. Grisard,1 D. Kudlacik,1 E. V. Kolobkova,3, 4 M. S. Kuznetsova,2 S. V. Goupalov,5 J. M. Kaspari,6 D. E. Reiter,6 D. R. Yakovlev,1 M. Bayer,1, 7 and I. A. Akimov1, † 1Experimentelle Physik 2, Technische Universit ̈at Dortmund, 44227 Dortmund, Germany 2Spin Optics Laboratory, St. Petersburg State University, Peterhof, 198504 St. Petersburg, Russia 3ITMO University, 199034 St. Petersburg, Russia 4St. Petersburg State 190013 St. Petersburg , Russia 5 39217 Mississippi, USA 6Condensed Matter Theory, ̈at Dortmund, 44227 Dortmund, Germany 7Research Center FEMS, Technische Universit ̈at Dortmund, 44227 Dortmund, Germany (Dated: October 17, 2025) Exciton-phonon interactions govern the energy level spectrum and thus the optical response in semiconductors. In this respect, lead-halide perovskite nanocrystals represent a unique system, for which the interaction with optical phonons is particularly strong, giving rise to a ladder of multiple exciton states which can be optically excited with femtosecond pulses. We establish a new regime of coherent exciton-polaron dynamics with exceptionally long coherence times (T2 ≈300 ps) in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix. Using transient two-pulse photon echo at 2 K temperature, we observe quantum beats between the exciton-polaron states. Within a four-level model, we directly quantify the exciton-phonon coupling strength through the Huang-Rhys factors of 0.05 ÷ 0.1 and 0.02 ÷ 0.04 for low-energy optical phonons with energies of 3.2 and 5.1 meV, respectively. The pronounced size dependence of both coupling strengths and phonon lifetimes offers a path to tune the optical transitions between polaron states and to tailor the coherent optical dynamics in perovskite semiconductors for solid-state quantum technologies. I. INTRODUCTION Lead halide perovskite semiconductors attract close attention due to their intriguing optical and electrical properties with huge potential for photovoltaic and light emitting applications [1, 2]. These materials provide a unique playground for polaron physics in the solid state governed by the interaction of electronic excitations with the phonons of the crystal lattice [3]. In particular, several studies indicate that low-energy optical phonons in halide perovskites have an important influence on the conductivity, while acoustic phonons have a minor effect [4, 5]. This is in contrast to conventional semiconductors such as GaAs, where the interaction of electrons with acoustic phonons is dominant at cryogenic temperatures [6, 7]. Furthermore, the carrier mobility is determined not by a few specific optical phonons, but by a large multitude of modes due to the complex phonon spectrum [8]. Perovskite nanocrystals (NC) belong to the class of quantum emitters with discrete energy level spectrum, large spectral tunability, and high quantum yield, which makes them attractive as single photon sources [9, 10]. At cryogenic temperatures, the elementary optical excitation in NCs is an exciton (electron-hole pair) which possesses long optical and spin coherence times up to hundreds of ps, comparable to its radiative lifetime [1113]. Long-lived coherent excitons are often considered as qubits and are appealing for possible applications ∗correspondence address: † correspondence address: in quantum communication [14]. However, the strong exciton-phonon interaction leads to formation of excitonpolarons which modifies the energy level spectrum. In particular, the photoluminescence spectra of single NCs show pronounced optical phonon replicas which confirm the substantial exciton-phonon coupling with HuangRhys factors in the range from 0.05 to 0.4 in lead halide perovskite NCs [15-19]. Recently, electron-phonon interaction dominated by optical phonons with a similar Huang-Rhys factor of about 0.4 was demonstrated in CsPbBr3 NCs [20] and other inorganic lead halide perovskite NCs [21] using Raman spectroscopy. Compared to InGaAs quantum dots [22, 23], these values are significantly larger and comparable to those reported in II-VI nanostructures [24]. A distinctive feature of perovskite nanocrystals is the existence of low-energy optical phonons with a strong coupling to phonons. This combination substantially modifies the exciton level spectrum in the vicinity of the ground state [25] and represents an attractive testbed for investigation of the quantum dynamics under resonant femtosecond excitation markedly different from previously studied systems. Time-resolved studies can provide direct access to the exciton dynamics. It is well established that scattering on incoherent phonons with a thermal distribution causes loss of exciton coherence [26, 27]. Yet, less is known about the quantum dynamics of exciton-polarons, i.e., the hybrid excitations of an exciton and a phonon. Recently, transient four-wave mixing and two-dimensional Fourier spectroscopy revealed the importance of coherent exciton-phonon coupling for the ground and excited (exciton and biexciton) states in lead halide perovskites [2816 Oct 2025 2 31]. However, most of these studies were limited to the case of short optical coherence of excitons (≤100 fs) and therefore reported on the coherent evolution of either the exciton or the ground electronic state modulated by the frequencies of phonons with much slower decoherence. In other words, the lack of optical coherence restricted the observations to the phonon dynamics, while the genuine coherent evolution of the coupled exciton-phonon system has remained inaccessible. This scenario was also documented for bulk semiconductors, nanostructures and organic molecules [32-36]. However, the opposite quantum dynamics limit, where pronounced quantum beats of exciton-polarons are expected, has not yet been uncovered. Furthermore, another general question arises, connected to the limitations of coherent control for resonant excitation with femtosecond pulses, since time-resolved studies reported quite short coherence times up to 20 ps in lead halide perovskites [28, 37], in contrast with photoluminescence data of single NCs [11-13]. In this work we establish a new regime of coherent dynamics governed by exciton-polarons in NCs with an exceptionally long coherence time of the zero-phonon exciton (T2,X ≈300 ps) on the one hand and a discrete spectrum of spectrally narrow optical phonon modes on the other hand. The regime manifests in long-lived excitonpolaron quantum beats, which are detected by transient two-pulse photon echoes in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix at low temperature, T = 2 K. The oscillatory signal is dictated by optical phonons with low frequencies corresponding to 3.2 and 5.1 meV energy, and can be described by the solution of the Lindblad equations for a set of four-level systems that include a phonon ladder in the ground and excited exciton-polaron states. Different sizes, ranging from 10 to 12 nm, were selectively addressed within the same ensemble of nanocrystals by varying the laser photon energy. From the relative amplitude of oscillations we directly evaluate the strength of the exciton-phonon coupling with Huang-Rhys factors of 0.06 and 0.02 for optical phonons with 3.2 meV and 5.1 meV energy, respectively. Correspondingly, the electron-phonon coupling strength doubles reaching values of 0.12 and 0.04 as the NC size is reduced from 12 to 10 nm. The theoretical analysis of the size dependence shows that both the deformation potential and the Fr ̈ohlich mechanism of electron-phonon interaction qualitatively agree with the experimental data in the regime of weak exciton confinement. Our study demonstrates that the dynamics of a pure quantum state following resonant optical excitation requires consideration of polaron effects, whose strength and coherence can be controlled by composition and size of the NCs. This opens up new possibilities for on-demand coherent control and coherent phonon generation in lead-halide perovskite NCs. II. PHOTON ECHO AND LONG COHERENCE OF ZERO-PHONON EXCITON The ensemble of CsPbI3 NCs is synthesized in a fluorophosphate glass matrix by rapid cooling of a glass melt enriched with the materials needed for the perovskite crystallization. The details of synthesis are given in Ref. [38]. The NC size is about 9 ÷ 13 nm in diameter (see Fig. S6 in Ref. 39). The low-temperature photoluminescence (PL) spectrum is dominated by a 60 meV broad band centered around 1.75 eV, as shown in Fig. 1(a). The broadening originates from fluctuations of the NC size, which we estimate to be in the order of 20%. A single NC is schematically illustrated in the inset of Fig. 1(b). Note that the cubic crystal structure is shown for illustrative purposes as the actual crystallographic phase depends on the growth conditions and the size of the nanocrystals [40-43]. We perform transient four-wave mixing (FWM) experiments in transmission geometry as shown schematically in Fig. 1(b). The NCs are resonantly excited with a sequence of two 100 fs laser-pulses with photon energies hν 50 ps is due to the zero-phonon exciton transition with long-lived optical coherence time T2 and exciton recombination lifetime T1. If the co3 1.65 1.7 1.75 1.8 1.85 0 0.5 1 Energy (eV) PL intensity (arb. units) τ12 τref Lead Iodine Caesium 1 2 3 4 5 6 7 -0.5 0 0.5 0.5 1 1.5 2 τ12 (ps) τref - 2τ12 (ps) E * FWM E ref (arb. units) (a) (b) (c) 0 50 100 1 234 0.1 τref (ps) E * FWM E ref (arb. units) 1.75 eV 1.74 eV 1.724 eV T2 = 276 ps (d) 10 12 8 9 15 NC diameter (nm) T2 = 330 ps T2 = 230 ps 1 2 ref PE k2 k1 2k2 - k1 T = 2K Figure 1. Photon echoes in CsPbI3 nanocrystals. (a) Typical photoluminescence (PL) spectrum for excitation with photon energy of 2.33 eV. Vertical dashed line indicates the highest photon energy used in the four-wave mixing FWM experiment. The scale on top corresponds to the NC diameter. (b) Schematic representation of the experimental geometry, where the laser pulses hit the sample with wave vectors k1, k2. The photon echo is detected along the direction 2k2 -k1 using a heterodyning technique by overlapping it with a reference pulse. Inset shows schematically a CsPbI3 perovskite nanocrystal. (c) Two-dimensional plot of the FWM electric field amplitude E∗ FWM as function of delay between the two pulses τ12 and the reference time τref. The photon echo (PE) signal forms at the time τref = 2τ12. The amplitude of the PE shows oscillations during the initial evolution when τ12 is scanned. The signal is recorded in linearly co-polarized configuration. Photon energy hν = 1.736eV. (d) Decay of two-pulse photon echo amplitudes for excitation with different photon energies. Black dashed lines are fits with exponential functions, from which the labeld exciton coherence times T2 are extracted. Temperature T = 2 K. herent dynamics would be governed only by the population decay we would expect to the T2 = 2T1 relation to hold, which typically holds for excitons in selfassembled InGaAs quantum dots at T = 2 K [49]. Here, we observe a different situation, where elastic scattering processes (\"pure dephasing\") with a decay constant of Tp = [1/T2 -1/(2T1)]-1 = 330 ps mainly govern the exciton coherence. Nevertheless, to the best of our knowledge, the homogeneous linewidth of the zero-phonon exciton of Γ2 = 2ħ/T2 = 4.8 μeV demonstrated here has a record low value for perovskite nanocrystals [11-13]. III. COHERENT EXCITON-PHONON DYNAMICS As follows from Fig. 1(c) for short delay times τ12 ≲ 10 ps, oscillations of the PE signal with multiple frequencies are observed. For deeper insight into the origin of the oscillations, we analyze polarization-resolved PE signals. These results are summarized in Fig. 2. We employ two configurations, where the first and second pulses are linearly co- or cross- polarized, while the detection polarization in all cases coincides with that of the first pulse (see Methods). The PE amplitudes as function of τ12 in co-(∥) and cross-(×) polarized configurations are shown in Fig. 2(a). The oscillations can be grouped into two categories: slow oscillations with a period of 1-10 ps and fast oscillations with a period of less than 1 ps. It is evident that the fast oscillations in the co-polarized configuration are in phase and nearly identical in both curves. We will show below that these oscillations correspond to quantum beats between exciton-polaron states. By contrast, the slow oscillations possess opposite phases. The co-polarized A∥-signal starts maximum value, while the A×-signal starts from zero. Subsequently, the minimum of the slow oscillations in ∥- configuration around τ12 = 5 ps corresponds to a maximum in the ×-configuration. This antiphase behavior is related to quantum beats between the orthogonally polarized bright exciton states [37]. In perovskite nanocrystals, the bright exciton fine structure comprises three linearly polarized states [15, 50], split by the energies δ1 and δ2 as shown in Fig. 3(a). This energy level scheme is similar to the fine structure splitting in self-assembled quantum dots [51, 52] and in II-VI colloidal NCs [53, 54]. In order to isolate the oscillations due to the fine structure splitting in the data, we model the expected polarization dependence of the PE signal, accounting also for the random orientation of the nanocrystals (for details see Sec. S1 of the Supplementary Information). We obtain the following equations for the amplitude of the phonon 4 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 τ12 (ps) PE ampl. (arb. units) co-linear (||) cross-linear (x) 0 10 20 30 40 0 0.5 1 1.5 τ12 (ps) Amplitude (arb. units) ρ Σ Fit 0 2 4 6 8 0 2 4 6 8 10 Energy (meV) Intensity (arb. units) 0 2 4 6 8 0 0.5 1 1.5 2 Energy (meV) FFT (arb. units) ρ Σ Fit (a) (b) (c) (d) Raman (x2) Figure 2. Quantum beats of exciton-polarons. (a) Initial range of the PE dynamics in the co-linear (∥) and cross-linear (×) polarization configurations shown with blue and red lines, respectively. Photon energy hν = 1.746 eV. The amplitude of the × signal is multiplied by two for clarity. (b) Dynamics of ρ (black) and Σ (green) as defined by Eqs. 2 and 3, respectively. The dashed red curve is a fit using the polaron model with Eq. 5 with the following parameters for two phonon modes: ħΩ1 = 3.2 meV, SHR1 = 0.097, τph1 = 5.1 ps; ħΩ2 = 5.1 meV, SHR2 = 0.032, τph2 = 10 ps. (c) Fast Fourier transform (FFT) amplitude spectra of ρ (black), Σ (green), and the fit curves in panel (b). Vertical dashed lines mark the peaks corresponding to optical phonons. (d) Raman spectrum measured at photon energy hν = 1.734 eV. Vertical dashed lines indicate the positions of peaks corresponding to the optically active optical phonon modes that couple most strongly to the exciton. echo signal in two polarization configurations: A∥= \" 18 15 + X i 4 15 cos (δiτ12) e -2τ12 t∗ i # Ψ0(τ12) , (1a) A× = \" 6 15 - X i 2 15 cos (δiτ12) e -2τ12 t∗ i # Ψ0(τ12) , (1b) where the index i = 1, 2, 3 corresponds to the beats between the three optical transitions (δ3 = δ1 + δ2), and t∗ i is the dephasing time of the beats caused by the dispersion of the splitting energies ħδi in the ensemble. The common factor Ψ0(τ12) is the same for both polarization configurations and will be discussed below, see also Supplementary Information S1. Following Eqs. (1) it is possible to distinguish between quantum beats related to the exciton fine structure and other polarization insensitive contributions to the PE signal by introducing the polarization contrast ρ = A∥-3A× A∥+ 2A× , (2) and the polarization sum Σ = A∥+ 2A× 2 = Ψ0(τ12) . (3) It follows that the temporal evolution of ρ exhibits oscillations at frequencies corresponding to the energy splittings δi of the exciton fine structure and is independent of Ψ0(τ12). In contrast, the polarization sum Σ is given by the intrinsic coherent dynamics Ψ0(τ12) only. Figure 2(b) shows the dynamics of these quantities, calculated from the data in Fig. 2(a). In full accord with our expectations we obtain that the high-frequency oscillations are absent in the dynamics of ρ, while the low-frequency oscillations are still present. On the other hand, the low-frequency oscillations vanish in the Σ transient in contrast to the high-frequency components. This becomes even more clear from the fast Fourier transform (FFT) spectra of ρ and Σ shown in Fig. 2(c). From the fit of the ρ transient we evaluate δ1 = 0.25 meV and δ2 = 0.55 meV, which are in agreement with the exciton fine structure splitting evaluated by two-dimensional Fourier spectroscopy on CsPbI3 NCs of similar size with a diameter of about 9 nm [37]. A detailed discussion of the fine exciton structure combining pump-probe and PE studies will be published elsewhere. In what follows, we focus on the dynamics of Ψ0(τ12), which is independent of the spin level structure of the exciton. The FFT spectrum of Σ, shown by the green line in Figure 2(c), exhibits several spectrally narrow features. These features are consistent with the peaks in the Ra5 man spectrum in Figure 2(d). The latter originates from light scattering on optically active phonons. The features at energies below 0.5 meV are attributed to confined acoustic phonons [55]. Below we focus on the two most prominent features marked by the vertical dashed lines with energies of ħΩ1 = 3.2 meV and ħΩ2 = 5.1meV, corresponding to the energies of optically active optical phonons in the vicinity of the Γ point. Indeed optical phonon modes with energies close to 3 and 5 meV in CsPbI3 were observed in Ref. [21] and calculated [8]. Optical phonons with similar energies were also detected by Raman spectroscopy in other lead halide perovskites such as FAPbI3 [56, 57], CsPbBr3 [19], and CsPbCl3 [17, 58]. Thus, we conclude that the high frequency THz oscillations in the two-pulse coherent optical response are due to the coherent evolution of the coupled, hybridized excitonphonon system. IV. EXCITON-POLARON QUANTUM BEATS In order to describe the quantum dynamics of the coupled exciton-phonon system we consider the structure of the optically excited energy levels in a single NC, which requires the involvement of polaron states. To that end, we introduce the four-level scheme shown in Fig. 3(b) which comprises the ground state |0⟩, the phonon mode |0′⟩, the exciton polaron state |X⟩(\"exciton\") and the first excited exciton polaron state |X′⟩(see details in the Supplementary Information S2 S2.A). We neglect the exciton fine structure splitting to concentrate on the derivation of Ψ0(τ12) in Eq. (1). Initially, we consider the exciton coupling to a single phonon mode in a NC. At the next stage, to compare with the experimental data, we perform summation over different phonon modes Ωi where we neglect the interaction between them Ψ0(τ12) = X i Ψ0(τ12; Ωi) . (4) The optical transitions between all energy levels are allowed using the same polarization. In the polaron model [59], the transition probability amplitude is proportional to the product of the dipole matrix element and the overlap of the wavefunctions associated with vibronic modes shifted due to polaron formation (see Fig. 3(c) and Eq. S16), which depends on the strength of the excitonphonon interaction given by the Huang-Rhys factor SHR (Condon principle). The pulse duration is assumed to be short as compared with all other characteristic times in the system, i.e. we can assume the δ-pulse limit. This is justified because the laser pulse duration is τd ≈100 fs (see Methods) and the fit to experimental data gives relaxation times in the order of 10 ps. The spectral width of the laser pulse ∼h/τd is about 15 meV, which is larger than the phonon energies ħΩi. Therefore, we assume that all four transitions between states |0⟩, |0′⟩and |X⟩, |X′⟩are covered by the spectral width of the laser. Note that a not too large Huang-Rhys factor means that states involving multiple phonons (i.e. 2ħΩ1, etc.) may be neglected. However, the spectral width of the laser pulse is large enough to cover them as well. We neglect the possible difference of phonon energy in the ground and polaron state as we do not find corresponding frequency changes in our experimental data and in previous reports in other perovskites [56, 57]. Similar assumptions were made in the model developed for desscribing 2D spectroscopy in Ref. 60. In the polaron model, we trace the evolution of the density matrix components which contribute to the PE. The result arises from the coherent off-diagonal terms and all levels contribute to the result in first order in the Huang-Rhys factor SHR, which quantifies the excitonphonon interaction. Solution of the Lindblad equation in Supplementary Information S2 gives the following expression for the amplitude of the phonon echo signal from a NC with single phonon mode Ωi Ψ0(τ12; Ωi) = e-(1+SHR)γ0τ12 \" 1 + SHRe-(γph-SHRγ0)τ12 +SHR cos (Ωiτ12)eγph-2SHRγ0 2 τ12h 2+e-SHRγ0τ12+e-γphτ12i + SHR cos (2Ωiτ12)e-(γph-SHRγ0)τ12 # . (5) Here T2,0 = 1/γ0 is the coherence time associated with the zero-phonon optical transition, and τph = 1/γph is the lifetime of the optical phonon. In the simplified model we do not include other sources of decoherence. The analytical expression fully supports the experimental observations for Σ = Ψ0(τ12) shown in Fig. 2(b). The first term in Eq. 5 corresponds to the long-lived zerophonon coherence which decays exponentially with the time T2 = T2,0/(1 + SHR) ∼300 ps, as evaluated in section II. The last two terms correspond to high frequency oscillations at the single and double frequency of the phonon mode Ωi, respectively. They appear due to quantum interference of excitations of different excitonpolaron states, i.e. due to quantum beats of excitonpolarons. These oscillations are superimposed on the long-lived signal and decay with a shorter time τph. The relative amplitude of the oscillatory signal and the longlived plateau allows one to measure the Huang-Rhys factor SHR. Using Eqs. 4 and 5 for two independent phonon modes with ħΩ1 = 3.2 meV and ħΩ2 = 5.1 meV we obtain excellent agreement with the experimental data. The phonon frequencies are taken from the peak positions in the Fourier spectrum of Fig. 2(c). The Huang-Rhys factors and phonon lifetimes are evaluated from the best fit to the experimentally measured transients, see Fig. 2(b). We emphasize that in contrast to previous reports [3236], our system demonstrates long-lived coherent dynamics, which is attributed to an exceptionally long zerophonon exciton coherence (∼300 ps) and a relatively long phonon lifetime (∼10 ps). Here, we stress that the 6 low-temperature regime with T = 2 K is essential, as the coherence of the zero-phonon excitons rapidly vanishes with increasing temperature. Furthermore, the optical phonons each are represented by well-defined frequencies due to their flat dispersion around the Γ-point, as confirmed by the Raman spectrum in Fig. 2(d), making CsPbI3 NCs embedded in a glass matrix particularly interesting for investigating coupled exciton-phonon dynamics. The peak widths in the Raman spectrum can be recalculated into phonon lifetimes using the following relation τph = 2ħ/∆ER [61], where ∆ER is the full width at half maximum in energy units. This yields phonon lifetimes of approximately 7 ps and 11 ps for the 3.2 and 5.1 meV modes, respectively. These values are in excellent agreement with the decay times of the PE oscillatory signals. It should be noted that the Raman peaks in Fig. 2(d) appear narrower than those in the FFT spectra in Fig. 2(c), which originates from the difference between the power spectral and the amplitude spectral representations, respectively. The proposed polaron model (i) provides an intuitive picture of the microscopic processes involved, (ii) has a simple analytical solution, and (iii) accounts explicitly for the coherence decay through simple equations. Equation (5) gives a result that closely matches that in Refs. [32, 62] and is in effect equivalent within the accuracy limits of the models. Note that a direct comparison between the results obtained by the two approaches is not straightforward: Eq. (5) is obtained from the exact solution of the Lindblad equations for a four-level system, while Ref. 62 considers the quasi-classical evolution of a complex system averaged over the phonon subsystem. For a detailed discussion and comparison of the two approaches see e.g. Ref. 63. V. NC SIZE DEPENDENCE Above, we discussed the PE dependence measured for the fixed laser pulse photon energy of hν = 1.734 eV and explained how the phonon parameters can be evaluated from the PE signal. However, the sample under study contains NCs with different sizes which can be selectively excited by tuning the laser photon energy. We measured PE transients at photon energies in the range of 1.72÷1.76 eV. The photon energy can be recalculated into a NC diameter using the parameters from Ref. 39, where the same sample was studied (sample #3). Figures 4(a) and 4(b) show the dependence of the evaluated Huang-Rhys factors SHR and phonon lifetimes τph as function of the NC diameter a, for phonon modes with energies ħΩ1 = 3.2 meV and ħΩ2 = 5.1 meV. We note that the frequencies of these modes do not depend on a within the accuracy of the experiment. The Huang-Rhys factors for both phonon modes exhibit a clear size dependence, increasing with decreasing NC size. Correspondingly, the phonon lifetimes decrease with decreasing a. An increase in the electron-phonon coupling |0> |0'> |X'> |X> (b) γph γph γ0 SHRγ0 (1-SHR)2γ0 |Z> |Y> |X> |0> γ0 γ0 γ0 (a) δ2 δ1 ħΩ ħΩ Configuration coordinate Q Energy EX SHRħΩ ħΩ |0> |0'> |X'> |X> (c) Figure 3. Level schemes. (a) Energy levels of the bright exciton fine structure. (b) Energy level structure of the excitons interacting with optical phonons. We assume this level structure for each phonon mode, independent of the exciton polarization. The probability of the diagonal transitions is proportional to the Huang-Rhys factor SHR. γph is the phonon decay rate and γ0 is the exciton decay rate, mostly determined by its oscillator strength. (c) Sketch of adiabatic potential of an exciton interacting with a particular phonon mode. The solid parabolas represent the potential for phonons for the ground state of the crystal (no exciton, bottom) and the excited state (with one exciton, top). The solid parabolas are separated by the energy difference EX corresponding to the energy of zero-phonon exciton transition. ħΩdenotes the energy of the optical phonon. strength with decreasing NC size was previously reported for CdSe nanocrystals [64], PbS quantum dots [65], and perovskite nanocrystals, both for optical and acoustic phonons [16, 19]. However, to the best of our knowledge, direct measurements of the associated phonon relaxation times have not been reported. It is also important to note that, in contrast to previous studies, our PE experiments selectively probe the coherent dynamics of each individual phonon mode. In Ref. 19, the dominant mechanism of electronphonon coupling is assigned to the optical deformation potential [66]. In this case, the size dependence of the interaction can be estimated from the phonon normalization condition [67] which leads to SHR ∼a-3. For completeness, let us discuss the size dependence of the interaction between charge carriers and optical phonons 7 for the Fr ̈holich mechanism following Takagahara [68]. In the original work, the interaction of charge carriers with phonons is rewritten as potential for an electron in the field induced by the phonon mode polarization P(r). This potential is found from the Poisson equation ∇2φ(r) = 4 π∇· P(r) which leads to φ(re) = Z dr ∇· P(r) |r -re| = - Z dr P(r) · ∇ 1 |r -re| . (6) From Eq. (6) the strength of electron-phonon interaction ∆e ∼a-1 which leads to Se HR = ∆2 e/2 ∼a-2. This estimate is valid also for excitons in the strong confinement regime. For weakly confined excitons, the change of exciton energy ∆is proportional to the difference of the electrostatic potential for electron and hole φ(re) -φ(rh). It is found to be the sum of Eq. (6) for electron and hole: ∆∼φ(re)-φ(rh) = Z dr P(r)·∇ 1 |r -rh| - 1 |r -re| . (7) In the weak confinement regime when the exciton Bohr radius aB is small compared with the NC diameter aB ≪ a, 1 |r -rh| ≈ 1 |r -re| + (re -rh) · ∇ 1 |r -re| . (8) Substituting Eq. (8) into Eq. (7) we obtain φ(re) -φ(rh) ≈ Z dr P(r) · ∇ (re -rh) · ∇ 1 |r -re| . As a result, in the weak confinement regime, the size dependence of this expression is estimated as ∆∼φ(re) -φ(rh) ∝a3 · a-3/2 · aB a3 ∝a-3/2. Here, the first size factor comes from the integral, the second one from the normalization of the phonon mode, and the last one from the gradient of the expression in the square brackets. As a result the Huang-Rhys factor scales as SHR = ∆2/2 ∼a-3. Thus, in the weak confinement regime both mechanisms, the deformation potential and the Fr ̈ohlich interaction, lead to similar dependences of the Huang-Rhys factor on the NC size SHR ∼a-3. In the strong confinement regime, the Fr ̈ohlich interaction is expected to have a weaker size dependence, while the deformation potential mechanism should show the same scaling. The dashed lines in Fig. 4 represent fits of the experimental data by the empirical relation SHR ∝a-3. The experimental data in Fig. 4(a) even shows a steeper dependence of the Huang-Rhys factor on NC size, with pronounced deviation from the a-3 dependence, particularly for small NCs. This observation suggests that further analysis is required for smaller NCs to identify the mechanisms of electron-phonon interaction in perovskite NCs. 0 0.02 0.04 0.06 0.08 0.1 0.12 SHR ħΩ1 = 3.2 meV ħΩ2 = 5.1 meV 10 10.5 11 11.5 12 0 5 10 15 20 NC diameter (nm) τph (ps) (a) (b) Figure 4. NC size dependence. Dependence of (a) the Huang-Rhys factor, SHR, and (b) the phonon lifetime, τph, for the two phonon modes with energies ħΩ1 = 3.2 meV (blue dots) and ħΩ2 = 5.1 meV (red dots) on NC diameter a. Dashed lines in (a) are fits proportional to a-3. Dashed lines in (b) are guides to the eye. VI. DISCUSSION To summarize, we have revealed a new regime of coherent exciton-phonon dynamics following resonant optical excitation with femtosecond pulses in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix. Quantum beats between exciton-polaron states manifest themselves in two-pulse photon echo experiments as oscillations with high frequencies corresponding to two optical phonon modes at 3.2 and 5.1 meV. A polarization resolved analysis of the photon echo signal has allowed us to unambiguously isolate the quantum beat signatures of exciton-polarons from slower beats arising from exciton fine-structure splittings. This unique regime of excitonpolaron quantum beats is established at the low temperature of 2 K, where an exceptionally long zero-phonon exciton coherence time is achieved (T2 ≈300 ps). The presence of spectrally narrow optical phonon modes with lifetimes of about 10 ps, combined with a relatively strong exciton-phonon coupling (Huang-Rhys factors up to 0.1), leads to a pronounced modulation of the photon echo signal at the phonon frequencies. Furthermore, the strong size dependence of the Huang-Rhys factor makes it possible to tune the strength of the optical transitions between different polaron states, offering a route to control the quantum dynamics for targeted applications. We emphasize that the observed optical response markedly differs from previous studies in, e.g., III-V selfassembled quantum dots, where the dynamics are mainly governed by exciton-acoustic phonon coupling [26, 69, 70]. There, the continuous phonon spectrum leads to a partial loss of coherence within the first few picosec8 onds after pulsed excitation. In our case no such behavior is observed in the temporal evolution of Ψ0(τ12) [see Fig. 2(b)]. Instead, the coherent dynamics is dominated by exciton-polaron states, i.e., interactions with discrete optical phonon modes, which can be consistently described within a four-level model for each phonon frequency. The desired quantum evolution can be achieved by tuning both the exciton-phonon coupling strength and, most importantly, the phonon lifetime, which limits the evolution of the pure quantum state following resonant excitation with a short optical pulse. Notably, the phonon lifetime exhibits a pronounced dependence on nanocrystal size. For the optical phonon mode with an energy of 5.1 meV, we observe an almost twofold increase of τph as the size increases from 10 to 12 nm. In layered perovskites, phonon lifetimes on the order of ∼10 ps were reported [71], with values extended up to 75 ps [72], approaching the record-high ∼200 ps observed in bulk ZnO [61]. Therefore, the strong size dependence of phonon lifetimes, along with reports of extended phonon coherence in layered perovskites, gives a promising outlook for long-lived and tunable coherent control of the quantum dynamics in perovskite nanocrystals. VII. METHODS Samples. The studied CsPbI3 nanocrystals embedded in fluorophosphate Ba(PO3)2-AlF3 glass were synthesized by rapid cooling of a glass melt enriched with the components needed for the perovskite crystallization. Details of the method are given in Refs. [38, 73]. The data on NC sizes evaluated from scanning transmission electron microscopy can be found in the supporting information section S5 in Ref. [39]. Samples of fluorophosphate (FP) glass with the composition 35P2O5-35BaO5AlF3-10Ga2O3-10PbF2-5Cs2O (mol. %) doped with BaI2 were synthesized using the melt-quench technique. The glass synthesis was performed in a closed glassy carbon crucible at the temperature of T = 1050◦C. Technological code of the studied sample is EK8. More information on its optical and spin-dependent properties can be found in Refs. 39 and 74. The sample is placed in the variable temperature insert of a bath cryostat and kept in superfluid helium at the temperature of 2 K. The low energy exciton optical transitions in nanocrystals are located in the spectral range of 1.7-1.8 eV as follows from the photoluminescence spectrum, see Fig. 1(a). Photon echo. Two-pulse photon echoes were measured using a transient four-wave mixing technique with heterodyne detection [44-47]. We used a Ti:Sapphire laser as source of optical pulses with duration of ∼100 fs at the repetition rate of 75.75 MHz. All optical experiments were carried out at photon energies below 1.76 eV. The sample is excited resonantly by a sequence of two pulses separated in time by the delay τ12, giving rise to the formation of a four-wave mixing signal with electric field amplitude EFWM(t). The pulses are focused into a spot of about 200 μm diameter using a 0.5-m parabolic aluminum mirror. The incidence angles of the pulses are close to normal and equal to approximately 1/50 rad and 2/50 rad, corresponding to the in-plane wavevectors k1 and k2. An additional reference pulse is used for detection and its delay with respect to the first pulse is given by the time τref. The delay times τ12 and τref are controlled using mechanical delay lines. The four-wave mixing signal is measured in transmission geometry in the direction ≈3/50 rad, which corresponds to the phasematching direction 2k2 -k1. Optical heterodyne detection is used to resolve the enhanced signal in time. By mixing with a relatively strong reference pulse and scanning τref, the temporal profile of the photon echo pulse is measured. Polarization sensitive measurements are performed by selecting proper polarization configurations (Hhorizontal or V-vertical) for each of the beams using Glan-Thompson prism in conjunction with half-wave plates. In the detection path we also select the linear polarization by setting the polarization of the reference beam. We adopt a three-letter notation for the PE polarization configuration. The first two symbols correspond to the polarizations of the pulses exciting the sample, and the third symbol corresponds to the detection polarization. For example, in the HHH configuration, the polarizations of both excitation pulses and the detection are horizontal (H). In the HVH configuration, the second excitation pulse is vertically (V) polarized, while the first pulse and detection are horizontally polarized. HHH is accordingly termed co-linearly polarized (∥) and HVH cross-linearly polarized (×) in Fig. 2. We note that rotation of the sample does not lead to changes in the signal, i.e. the orientation of the sample is not important. Raman spectroscopy. The Raman scattering signal was excited by a single-frequency Ti:Sapphire laser. The laser beam with the power of about 1 mW was focused on the sample to a spot with a diameter of about 300 μm. The scattered light was analyzed by a Jobin-Yvon U1000 double monochromator with 1-meter focal length, allowing the high resolution of 0.2 cm-1 (0.024 meV). The Raman signal was detected by a cooled GaAs photomultiplier and conventional photon-counting electronics. The Raman spectra were measured for copolarized linear polarizations of excitation and detection. VIII. ACKNOWLEDGEMENTS The authors are thankful to I. A. Yugova and N. E. Kopteva for fruitful discussions. We acknowledge the financial support by the Deutsche Forschungsgemeinschaft: A.V.T., M.A.H, S.G., M.O.N and I.A.A (project AK40/13-1, no. 506623857) and D.R.Y. 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Khersonskii, Quantum theory of angular momentum (World Scientific, 1988). [76] P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics (Princeton University Press, 2011). S1 Supplementary Information: Quantum beats of exciton-polarons in CsPbI3 perovskite nanocrystals S1. FINE STRUCTURE OF EXCITON To take into account the fine structure of the exciton, we assume that the three-fold bright exciton state is split into mutually orthogonal linearly polarized components. For the photon echo calculations the following procedure is used: the dynamics is calculated in the eigenbasis of NC {|0⟩, |x′⟩, |y′⟩, |z′⟩} then the equations for the density matrix components are rotated to the laboratory frame {|0⟩, |x⟩, |y⟩, |z⟩}. The 1st pulse is assumed to be polarized along z axis, the 2nd pulse at time τ is either polarized along z or x axis for HHH and HVH schemes respectively. The polarization along z is detected at the photon echo time 2τ. The orientation of a nanocrystal in the laboratory reference frame is described by three Euler angles α, β, and γ. The rotation matrix relating coordinates in xyz and x′y′z′ bases is [75]: R(α, β, γ) = cαcβcγ -sαsγ -cαcβsγ -sαcγ cαsβ sαcβcγ + cαsγ -sαcβsγ + cαcγ sαsβ -sβcγ sβsγ cβ , (S1) here and below in this section we use sξ ≡sin ξ, cξ ≡cos ξ (ξ = α, β, γ) to shorten the notation. In the eigenbasis of NC, the dynamics is given by (we skip common factor e-γt): ρ0x′ = e-t/T2eiω1tρ0 0x′ , ρ0y′ = e-t/T2eiω2tρ0 0y′ , ρ0z′ = e-t/T2eiω3tρ0 0z′ ; (S2) ρx′0 = e-t/T2e-iω1tρ0 x′0 , ρy′0 = e-t/T2e-iω2tρ0 y′0 , ρz′0 = e-t/T2e-iω3tρ0 z′0 . (S3) Here ħωi (i = 1, 2, 3) are the three exciton energies for excitons linerly polarized along x′, y′, z′ respectively. We assume that the diplole matrix element (and, consequently, radiative decay time) as well as the decoherence time T2 is the same for all three exciton states. After 1st pulse the components of density matrix are ρ1+ 0z = 1 , ρ1+ 0y = ρ1+ 0x = 0 . (S4) After 2nd pulse we have two options: for HHH scheme ρ2+ z0 = ρ20z , ρ2+ y0 = ρ2+ x0 = 0 . (S5) and for HVH scheme ρ2+ x0 = ρ20x , ρ2+ y0 = ρ2+ z0 = 0 . (S6) Rotating the dynamics equations (S2) into laboratory coordinate frame using (S1) gives rather lengthy equations for the amplitude of the signal: A∥(α, β, γ) ∝ρz0 = c4 αs4 β + s4 αs4 β + c4 β + 2s2 αc2 αs4 β cos(ω1 -ω2)τ + 2c2 αs2 βc2 β cos(ω1 -ω3)τ + 2s2 αs2 βc2 β cos(ω2 -ω3)τ e-2τ/T2 (S7) A×(α, β, γ) ∝ρz0 = c2 αs2 β(cαcβcγ -sαsγ)2 + s2 αs2 β(sαcβcγ + cαsγ)2 + s2 βc2 βc2 γ + 2sαcαs2 β(cαcβcγ -sαsγ)(sαcβcγ + cαsγ) cos(ω1 -ω2)τ -2cαs2 βcβcγ(cαcβcγ -sαsγ) cos(ω1 -ω3)τ -2sαs2 βcβcγ(sαcβcγ + sαsγ) cos(ω2 -ω3)τ e-2τ/T2 (S8) The result should be averaged over directions using: ⟨A⟩ang ≡ 1 8π2 Z 2π 0 dα Z π 0 sin βdβ Z 2π 0 dγ A(α, β, γ) (S9) S2 Assuming the constant energies of three exciton components, but the arbitrary direction of the NCs, the amplitude of the echo is proporitional to A∥ ang ∝ 6 5 + 4 15 X i>j cos(ωi -ωj)τ e-2τ T2 , (S10) ⟨A×⟩ang ∝ 2 5 -2 15 X i>j cos(ωi -ωj)τ e-2τ T2 . (S11) Now we also take into account the possible deviation of exciton energies. The absolute value of the exciton energy is not important for the phonon echo, only the splittings. E.g., the amplitude averaged over distribution of splittings between 1st and 2nd levels ω12 = ω1 -ω2 is given by ⟨A⟩ω12 = Z W(ω12)A(ω12, ω13, ω23 = ω12 -ω13)dω12 (S12) Assuming the distribution function of the splittings to be Lorentzian, W(ω) = t∗ i π 1 (ω -δi)2 + √ 2/(t∗ i )2 , (S13) and that all three splittings are not correlated, the PE signal averaged over both the angles and the splittings distribution is given by A∥ ∝ 18 15 + 4 15 X i=1,2,3 cos(δiτ)e -2τ t∗ i e-2τ T2 , (S14) ⟨A×⟩∝ 6 15 -2 15 X i=1,2,3 cos(δiτ)e -2τ t∗ i e-2τ T2 , (S15) where the parameters t∗ i work as an effective signal dephasing times and δi are the average splittings. An interesting result is that even though the PE signal is stable with respect to the level disorder, the disorder of its fine structure gives the contribution to the additional decay, not of the full signal, but of its oscillating components. S2. MODEL OF PHOTON ECHO OF EXCITON-POLARON STATES. In this section we outline the derivation of PE signal in the polaron model. We assume the exciton-polaron with relatively small Huang-Rhys factor which allows to perform calculations in four (six with account on light polarization) level scheme. S2.A. Scheme of the levels and dynamics of the system We consider the four-level system shown in Fig. 3b. There are: ground level |0⟩, statwe with one phonon |0′⟩, polaron state |X⟩, and excited state of polaron |X′⟩. The energies of corresponding states are 0, ħΩ, ωX, ωX + ħΩ. In the calculations below we assume that spectral width of the laser pulse covers at least |X⟩and |X′⟩levels, this is in contrast to standard approximations for three-level atoms [76]. Due to polaron formation, allowed transitions are not only X →0 (with probability γ0) and X′ →0′ (with probability γ1), but also X →0′ and X′ →0 (in this model, probability is the same γ′). In polaron model [59], the transition probability is proportional to product of dipole matrix element |d|2 and the overlap of oscillator functions shifted due to exciton formation which leads to extra factor γ0 ∝|d|2e-2SHR (S16a) γ′ ∝|d|2SHRe-2SHR (S16b) γ1 ∝|d|2(1 -SHR)2e-2SHR (S16c) S3 where d is the dipole matrix elements of the optical transition, and we defined Huang-Rhys factor as SHR = εpol/(ħΩ) (εpol is the polaron formation energy). Note that in this approximation polaron formation energy should be small which means that γ′ ≪γ0, γ1. In addition, we assume that the phonon may be emitted which leads to transitions 0′ →0 and X′ →X with the same probability γph. To trace the polarization of the signal, we need to consider two polarization states of exciton, from now on the enumeration 0 and 0′ are ground state and state with phonon, 1(2) and 1′(2′) are right(left) circularly polarized states of exciton(polaron) and first oscillator level of exciton(polaron), making the model effectively six-level. Still, we call the model four-level as here we neglect the splittings between levels (as the role of the fine structure is excluded, see Sec. S1) and we need it only to demonstrate that the phonon structure does not change the polarization structure of the signal. Below, we extensively omit states 2 and 2′ when equations are same. The Lindblad equation, with all levels considered is: dρ dt = -i ħ[H0, ρ] + Ld[ˆρ] + Γ ⊙ˆρ . (S17) For convenience and compactness of the notation, with the use of ⊙for the Hadamard product (element-wise product or Schur product). The Hamiltonian is H0 = 0 0 0 0 0 0 0 ħΩ 0 0 0 0 0 0 ħωX 0 0 0 0 0 0 ħωX 0 0 0 0 0 0 ħωX + ħΩ 0 0 0 0 0 0 ħωX + ħΩ (S18) and the \"inflow\"diagonal part of superoperator is Ld[ˆρ] = γphρ0′0′ + γ0ρX + γ′ρX′ 0 0 0 0 0 0 γ1ρX′ + γ′ρX 0 0 0 0 0 0 γphρ1′1′ 0 0 0 0 0 0 γphρ2′2′ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (S19) where for convenience we introduced ρX = ρ11 + ρ22 and ρX′ = ρ1′1′ + ρ2′2′, and the matrix in the \"outflow\" part is Γ = -1 2 0 γph γ′ 0 γ′ 0 γph + γ′ 1 γph + γ′ 1 γph 2γph γph + γ′ 0 γph + γ′ 0 2γph + γ′ 1 2γph + γ′ 1 γ′ 0 γph + γ′ 0 2γ′ 0 2γ′ 0 γph + γ′ 1 + γ′ 0 γph + γ′ 0 + γ′ 1 γ′ 0 γph + γ′ 0 2γ′ 0 2γ′ 0 γph + γ′ 1 + γ′ 0 γph + γ′ 0 + γ′ 1 γph + γ′ 1 2γph + γ′ 1 γph + γ′ 0 + γ′ 1 γph + γ′ 0 + γ′ 1 2γph + 2γ′ 1 2γph + 2γ′ 1 γph + γ′ 1 2γph + γ′ 1 γph + γ′ 0 + γ′ 1 γph + γ′ 0 + γ′ 1 2γph + 2γ′ 1 2γph + 2γ′ 1 (S20) where for convenience we defined γ′ 0 = γ0 + γ′ , γ′ 1 = γ1 + γ′ . (S21) S2.B. Effect of the pulse on density matrix Now we consider light-mediated transitions in the system shown in Fig. 3b. This is again generalization of the two-level system, check e.g. chapters 2, 6 and 10 of [76]. The pulse is approximated by the plane wave with the smooth enveope E0(z, t) and electric field given by E(r, t) = E0(k · r/k, t)ei(k·r-ωt) + c.c. (S22) Assuming the pulse duration is short compared with all coherence and relaxation times, the evolution of density matrix (DM) during the pulse may be found from the Schr ̈odinger equation dρ dt = -i ħ[H0 + V, ρ] , (S23) S4 where, in rotating wave approximation, V = ħ 0 0 ̃f ∗ + ̃f ∗ - s · ̃f ∗ + s · ̃f ∗ + 0 0 s · ̃f ∗ + s · ̃f ∗ - ̃ff ∗ + ̃f ∗ + ̃f+ s · ̃f+ 0 0 0 0 ̃fs · ̃f0 0 0 0 s · ̃f+ ̃f+ 0 0 0 0 s · ̃f- ̃f0 0 0 0 , (S24) ̃f± = dE± 0 e-iωt ≡f±e-iωt , (S25) where d is the dipole matrix element for the transitions 1, 2 →0, s = √SHR, E± 0 are circular components of the envelope (S22). Solution of Eq. (S23) is surprisingly complex in general case, for two-level system it may be found in Ref. 76, but we are interested in four-level system and the case of small pulse area when the full solution is surplus. When the pulse area θ± (equal to f±tp for square pulse of duration tp) is small compared with Rabi frequency, the first order correction to the density matrix may be obtained as ρ+ = -i ħ H0 + V, ρ- tp ≡ˆ Aρ-, (S26) where ρ-is the density matrix before pulse and ρ+ is the density matrix after pulse. Full solution of Eq. (S26) is too lengthy for our needs, below we will only give the results relevant for the calculations. For the PE we will need the action of the light pulse in the second order over pulse area. It may be shown that up to second order, it may be calculated as ρ+ = ˆ A ˆ Aρ-. (S27) Below we also give only part of this result relevant for photon echo calculations. S2.C. Photon echo The signal in two circular polarizations is proportional [76] to the corresponding components of polarization vector which is given by the following components of the density matrix: P+ ∼ρ10 + ρ1′0′ + sρ1′0 + sρ10′ , (S28a) P-∼ρ20 + ρ2′0′ + sρ2′0 + sρ20′ . (S28b) Note the extra factor s in Eq. (S28) which originates from the ratio of diplole matrix elements for phonon-assisted transitions to the phonon-less transitions. Thus, we are interested in the following components of the density matrix at times after second pulse: ρ10, ρ20, ρ1′0, ρ2′0, ρ10′, ρ20′, ρ1′0′, ρ2′0′. a. Density matrix after 1st pulse The components of DM after 1st pulse are given by (S26) which leads to: ρ1+ 01 = iθ∗ 1+ , ρ1+ 01′ = isθ∗ 1+ , ρ1+ 02 = iθ∗ 1-, ρ1+ 02′ = isθ∗ 1-. (S29) b. Relaxation of DM components after 1st impulse One needs to trace the evolution of the following DM components: ρ01, ρ01′, ρ02, ρ02′. Below we give results only for right polarization, dynamics of other components is the same. Writing the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components: ρ01 (t ∈(0, τ12)) = ρ1+ 01 eiωXteγ′ 0 2 t , (S30a) ρ01′ (t ∈(0, τ12)) = ρ1+ 01′ei(ωX+Ω)teγ′ 1+γph 2 t , (S30b) This leads to the following components of the DM at the time of 2nd pulse t = τ12 ρ201 = ρ1+ 01 εXΓ0 , ρ201′ = ρ1+ 01′εXεΓ1Γp , (S31) where for convenience we defined εX = eiωXτ12 , ε = eiΩτ12 , Γ0 = eγ′ 0 2 τ12 , Γ1 = eγ′ 1 2 τ12 , Γp = eγph 2 τ12 . (S32) Note that Γ0 and Γ1 are almost equal and defined by the exciton coherence time (under our assumptions half of its lifetime). The difference between them is Γ1/Γ0 = e-γ1-γ0 2 τ12 = e SHRγ0 2 τ12 which is much smaller than all typical times in the system since SHR is small. S5 c. DM after 2nd pulse From (S27) we may compute the components relevant for the PE signal after the 2nd pulse: ρ2+ 10 = θ2 2+(ρ201 + sρ201′) + θ2+θ2-(ρ202 + sρ202′) (S33a) ρ2+ 10′ = θ2 2+(sρ201 + ρ201′) + θ2+θ2-(sρ202 + ρ202′) (S33b) ρ2+ 1′0 = s θ2 2+(ρ201 + sρ201′) + θ2+θ2-(ρ202 + sρ202′) (S33c) ρ2+ 1′0′ = s θ2 2+(ρ201′ + sρ201 ) + θ2+θ2-(ρ202′ + sρ202 ) (S33d) and ρ2+ 20 = θ2+θ2-(ρ201 + sρ201′) + θ2 2-(ρ202 + sρ202′) (S34a) ρ2+ 2′0 = θ2+θ2-(sρ201 + ρ201′) + θ2 2-(sρ202 + ρ202′) (S34b) ρ2+ 20′ = s θ2+θ2-(ρ201 + sρ201′) + θ2 2-(ρ202 + sρ202′) (S34c) ρ2+ 2′0′ = s θ2+θ2-(ρ201′ + sρ201 ) + θ2 2-(ρ202′ + sρ202 ) (S34d) d. Relaxation of DM components after 2nd impulse We we need to know the evolution of the DM components which enter (S28). Below we give only components entering P+. Dynamics of components entering P-is the same. Writing the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components: ρ10 (t > t2) = ρ2+ 10 e-iωX(t-t2)eγ′ 0 2 (t-t2) , (S35a) ρ1′0 (t > t2) = ρ2+ 1′0e-i(ωX+Ω)(t-t2)eγ′ 1+γph 2 (t-t2) , (S35b) ρ10′ (t > t2) = ρ2+ 10′e-i(ωX-Ω)(t-t2)eγ′ 0+γph 2 (t-t2) , (S35c) ρ1′0′ (t > t2) = ρ2+ 1′0′e-iωX(t-t2)eγ′ 1+2γph 2 (t-t2) , (S35d) Which gives the DM components at the time of echo signal t -t2 = τ12: ρP E 10 = ρ2+ 10 ε∗ XΓ0 , ρP E 1′0 = ρ2+ 1′0ε∗ Xε∗Γ1Γp , (S36a) ρP E 10′ = ρ2+ 10′ε∗ XεΓ0Γp , ρP E 1′0′ = ρ2+ 1′0′ε∗ XΓ1Γ2 p . (S36b) The evolution of components corresponding to second circular polarization is the same. e. Contribution of different paths to PE To get the final result, we substitude (S29) into (S31) and then the result into (S33) and then from (S36) we get the DM at the time of PE. The amplitude of the signal in two polarizations is then obtained from Eqs. (S28): P+ ∼ρPE 10 + ρPE 1′0′ + sρPE 1′0 + sρPE 10′ , (S37a) P-∼ρPE 20 + ρPE 2′0′ + sρPE 2′0 + sρPE 20′ . (S37b) Collecting all terms in the result we have for contributions to P+ from ρ10, ρ1′0′, ρ10′, ρ1′0 originating from the DM components after 1st pulse given below in the beginning of each line: ρ01,02 →ρPE 10 Γ2 0A+ , (S38a) ρ01′,02′ →ρPE 10 s2εΓ0Γ1ΓpA+ , (S38b) ρ01,02 →ρPE 10′ s2εΓ2 0ΓpA+ , (S38c) ρ01′,02′ →ρPE 10′ s2ε2Γ0Γ1Γ2 pA+ , (S38d) ρ01,02 →ρPE 1′0 s2ε∗Γ0Γ1ΓpA+ , (S38e) ρ01′,02′ →ρPE 1′0 s4Γ2 1Γ2 pA+ , (S38f) ρ01,02 →ρPE 1′0′ s2Γ0Γ1Γ2 pA+ , (S38g) ρ01′,02′ →ρPE 1′0′ s2εΓ2 1Γ3 pA+ , (S38h) where A+ = θ2+ θ2+θ∗ 1+ + θ2-θ∗ 1- . (S39a) S6 Note that the term (S38f) may be dropped as it exceeds precision of equations. The contribution to P-from ρ20, ρ2′0′, ρ20′, ρ2′0 is the same as (S38) but instead of A+ the polarization dependence is given by A-= θ2- θ2-θ∗ 1-+ θ2+θ∗ 1+ . (S39b) It is convenient to write both as A± = θ2±A0 , A0 = θ2-θ∗ 1-+ θ2+θ∗ 1+ , (S40) to see that the polarization dependence of the PE signal coincides with second impulse polarization while its amplitude depends on respective polarization of first and second pulses. As a result, the amplitude of the non-oscillating signal is PP E,0 ± ∼ Γ2 0 + SHRΓ0Γ1Γ2 p A± . (S41) The amplitude of the signal oscillating at Ωis PP E,Ω ± ∼SHRΓp Γ0Γ1 (ε + ε∗) + ε Γ2 0 + Γ2 1Γ2 p A± . (S42) and of the signal oscillating at 2Ωis PP E,2Ω ± ∼SHRΓ0Γ1Γ2 pε2A± . (S43) f. Result Let us first consider the general scheme of the polarization dependence of PE signal. Following [47], we consider linearly polarized excitation pulses with second pulse rotated to angle φ and linearly polarized detection with polarized rotated to φd. Then θ1+ = θ1-= 1/ √ 2 , θ2+ = eiφ/ √ 2 , θ2-= e-iφ/ √ 2 . (S44) Which gives A0 = cos φ and A± = e±iφ √ 2 cos φ . (S45) Amplitude of linearly-polarized heterodyne detection of main PE signal is proportional to P 0 d = ei(φ-φd) + e-i(φd-φ) 2 cos φ = |cos (φ -φd) cos φ| (S46) We have two configurations: HHH and HVH. First is φ = φd = 0 and second is φd = 0, φ = π/2. This gives P 0,HHH d = 1 , P 0,HV H d = 0. (S47) As long as polarization dependence on the excitation/detection angles (S46) is real (which is the case for HHH and HVH configurations), the absolute value of product is the product of absolute values and signal is zero for HVH scheme and for HHH scheme (we omit index HHH below) is: PP E d = Γ2 0 h 1 + SHReΓΓ2 p i + SHRΓp h eΓ (ε + ε∗) + ε(1 + eΓ2Γ2 p) i + SHReΓΓ2 pε2 (S48) where we introduced eΓ = Γ1/Γ0 = eSHRγ0τ12 . (S49) Note that eΓ is 1 as compared with both, Γ0 and Γp. To write explicitly the result for the PE amplitude, we use the fact that for t ≪1 and arbitrary complex ε: |1 + tε| = 1 + tRe(ε) + t2 2 (Im(ε))2 + O t3 . (S50) We keep second order terms to highlight the fact that even for single frequency of the complex amplitude, the absolute value in (S48) will result in second order in SHR signal at double frequency. S7 Using (S50) we may write the final result (keeping only terms to the first order in SHR) PP E d = Γ2 0 n 1 + SHRΓp h eΓΓp + 1 + 2eΓ + eΓ2Γ2 p cos(Ωτ12) + eΓΓp cos(2Ωτ12) io (S51) Or, explicitly, PP E d = e-γ′ 0τ12 \" 1 + SHReγph 2 τ12 eγph-2SHRγ0 2 τ12 + 1 + 2eSHRγ0τ12 + e-(γph-2SHRγ0)τ12 cos(Ωτ12) + eγph-2SHRγ0 2 τ12 cos(2Ωτ12) !# , (S52) and, finally, with the help of Eqs.(S16,S21): PP E d = e-(1+SHR)γ0τ12 \" 1 + SHRe-(γph-SHRγ0)τ12 + SHR cos(Ωτ12)eγph-2SHRγ0 2 τ12 e-SHRγ0τ12 + 2 + e-γphτ12 + SHR cos(2Ωτ12)e-(γph-SHRγ0)τ12 # ≡Ψ0(τ12; Ω) . (S53)"}}},{"rowIdx":290,"cells":{"id":{"kind":"string","value":"2510.14691"},"raw":{"kind":"string","value":"If You Hold Me Without Hurting Me: Pathways to Designing\nGame Audio for Healthy Escapism and Player Well-being∗\nCAIO NUNES, Federal University of Ceará, Brazil\nBOSCO BORGES, Federal University of Ceará, Brazil\nGEORGIA CRUZ, Federal University of Ceará, Brazil\nTICIANNE DARIN, Federal University of Ceará, Brazil\nEscapism in games can support recovery or lead to harmful avoidance. Self-regulation, understood as combining\nautonomy with positive outcomes, is key to this distinction. We argue that audio, often overlooked, plays a\ncentral role in regulation. It can modulate arousal, mark transitions, and provide closure, yet its contribution\nto well-being remains underexplored. This paper identifies methodological and accessibility gaps that limit\nrecognition of audio’s potential and outlines ways to address them. We aim to encourage researchers and\ndevelopers to integrate audio more deliberately into the design and study of healthier escapist play.\nCCS Concepts: • Human-centered computing →HCI design and evaluation methods; • Applied\ncomputing →Sound and music computing.\nAdditional Key Words and Phrases: Escapism, Emotions, Self-Regulation, Game Audio\n1\nINTRODUCTION\nEscapism has long been recognized as one of the main motivations for gameplay, appearing across\nmodels of player motivation and media use [4, 7, 19]. It is often described as the tendency to\ndetach from everyday concerns or stressors, yet its consequences are ambivalent. On the one\nhand, escapism can offer distraction, stress relief, and opportunities for affective regulation [2]. On\nthe other hand, it can reinforce avoidance and interfere with everyday functioning. Recent work\nemphasizes this tension by distinguishing between healthy escapism, when play supports recovery\nand well-being, and subversive escapism, when it undermines them [9]. This perspective shifts the\ndebate from whether escapism is good or bad to a more pressing question: \"Which factors determine\nwhen escapist play sustains well-being and when it becomes harmful?\"\nOne of the most decisive factors in this distinction is self-regulation. In Self-Determination\nTheory (SDT), self-regulation refers to the extent to which behavior is experienced as autonomous,\nreflecting choice and volition rather than external pressure or compulsion [17]. Applied to games,\nthis perspective captures whether players can autonomously decide when to begin, how long\nto continue, and when to disengage. Research consistently shows that difficulties in regulation\nare linked to excessive or compulsive gaming [6]. At the same time, successful self-regulation\nallows games to function as bounded recesses that facilitate recovery, coping, and positive mood\nmanagement [9, 13]. Framing escapism through this lens highlights autonomy as a key condition\nfor distinguishing between healthy and subversive play. However, a player could autonomously\nchoose to play for a duration that negatively impacts their life responsibilities. In this case, the\nbehavior is self-determined, but the outcome is detrimental. Hence, a healthy escapism also requires\npositive outcomes, where psychological relief (e.g., reported mood improvement) is supported by\nphysiological recovery (e.g., stress marker reduction), independent of the feeling of volition during\nplay. Consequently, we draw attention to how the design of game elements can either scaffold a\nplayer’s ability to self-regulate or undermine it.\nWe argue that audio is a key game element that can be intentionally designed to struc-\nture the escapist experience itself, facilitating autonomous self-regulation and positive\n∗Presented and discussed at the CHI PLAY 2025 Workshop Exploring Future Directions for Healthy Escapism and Self-Regulation\nin Games, Pittsburgh, USA, October 13, 2025.\n1\narXiv:2510.14691v1 [cs.HC] 16 Oct 2025\n\nNunes et al.\noutcomes. While often perceived as secondary to visual or mechanical elements, audio possesses\nunique qualities that make it a potent regulatory channel. Unlike visual cues, which require focused\nattention, sound is pervasive and processed with a lower cognitive load, capable of influencing\narousal and emotional states subconsciously, creating a direct tension between systemic regulation\nand player autonomy that design must address. The well-documented link between music and\nphysiological responses, from emotion regulation and stress recovery to everyday relaxation [5, 8].\nThis makes audio a prime channel for influencing the very recovery and emotion regulation that\nare central to healthy escapism [9]. Besides, audio’s ability to structure time and rhythm transitions\noffers a subtle yet material means of scaffolding player awareness without disrupting immersion.\nYet, despite its unique potential, the role of audio in player well-being remains poorly understood.\nMuch of the game-audio literature concentrates on immersion, presence, arousal, and affect, seldom\nlinking these outcomes to well-being or self-regulatory processes [10]. This gap is especially evident\nin discussions of escapism and self-regulation, which often emphasize constructs such as absorption,\naffective modulation, stress relief, and distraction [3, 18]. While these emphases are not identical,\nthey share a concern with how players become deeply involved and how their emotional states are\nshaped during play.\nThis lack of synthesis leaves a critical gap, preventing designers and researchers from systemat-\nically using audio as a tool for player well-being. This position paper moves to fill that gap. We\ncontend that audio is not a peripheral feature but a central, designable mechanism for scaffolding\nhealthy escapism in games. To substantiate this claim, the remainder of this paper will first identify\nmethodological and accessibility gaps previously identified in game audio research that perpetuate\nthis problem. We then propose three concrete directions for future work focused on using audio to\nsupport self-regulation and positive emotional outcomes.\n2\nHOW RESEARCH GAPS OBSCURE AUDIO’S ROLE IN HEALTHY ESCAPISM\nIn a previous study, we reviewed how audio and player experience are evaluated [10]. The results\nshow a fragmented evidence base where game audio is routinely linked to immersion and affect, but\nits deeper impact is unclear. A key problem is that self-report measures often fail to converge with\nphysiological data; for example, players may report feeling calm while their arousal markers remain\nelevated [10]. These inconsistencies make it challenging to draw conclusions about outcomes\nsuch as coping, stress recovery, or emotion regulation. This methodological fragmentation is\ncompounded by other persistent issues, including a lack of personalization and the tendency for\naudio to be overshadowed in high-sensory contexts [11, 14–16]. Together, these findings reveal that\naudio is seen as an immersive aesthetic but not perceived as a potential lever for self-regulatory\noutcomes.\nThe real-world impact of this research gap is starkly visible in games designed for sensitive\naudiences. Our literature review (n=49) on games for autistic players [1], for instance, shows\nthat audio is often absent from game design descriptions and from user evaluation, despite its\npotential to support engagement or, if mishandled, trigger overload in audiences with auditory\nhypersensitivity. Across the surveyed studies, only 14% (n=7) explicitly evaluated audio; when\nthey did, its uses ranged from background music to movement sonification, yet the impacts on\nthe target audience were seldom reported in depth [12]. Such an absence of systematic evaluation\nand reporting reveals a double gap: a lack of design transparency and potentially a neglect of\naccessibility. For that audience, without customization options or sensitivity to diverse auditory\nneeds, audio risks becoming a barrier instead of a resource. These findings highlight how sound\ndesign, if overlooked, can miss therapeutic opportunities and expose players to overstimulation or\ndiscomfort. This case starkly illustrates how neglecting audio can undermine a player’s ability to\nself-regulate, turning a potentially healthy escape into a subversive, dysregulating experience.\n2\n\nIf You Hold Me Without Hurting Me\nWhile these examples are not exhaustive, the gaps identified in the literature prevent us from\nmeasuring whether audio supports well-being or causes distress. Methodological blind spots mean\nthat we lack ways to evaluate if audio is contributing to the positive outcome of healthy escapism,\nwhether audio supports coping and recovery or simply intensifies immersion. Accessibility gaps\ncan imply that sound becomes overwhelming or exclusionary, rather than restorative. In both cases,\nescapism risks sliding toward avoidance and overuse – not because audio cannot support healthier\noutcomes, but because research and design fall short in providing the tools to differentiate them.\nThe absence of variation, closure, and inclusive controls leaves players to self-manage through\nworkarounds such as muting or replacing game soundtracks [16], undermining the potential of\naudio to contribute directly to constructive escapist experiences.\nUltimately, players are left without the necessary scaffolds to exercise their autonomy due to a\nlack of key tools, such as methods for measuring player responses to audio cues or designs that\nutilize audio to create natural stopping points. For instance, if safe exit points are signaled by a\nshift to calmer music in a hub area, it creates a clear closing motif after a mission, or even a brief\n\"breathing window\"\" of silence after a moment of high tension, all of which provide a sense of\nclosure that could help players disengage with a sense of accomplishment. Without such intentional\ndesign, poorly tested or repetitive audio can instead reinforce compulsive immersion or force\navoidance, both of which undermine self-regulation. These blind spots leave players without the\nnecessary scaffolds to disengage autonomously, showing how overlooked audio can shift escapism\nfrom recovery to overuse.\n3\nTOWARDS SELF-REGULATION AND HEALTHY ESCAPISM WITH AUDIO\nThe gaps highlighted reveal a significant methodological deficiency: despite audio’s clear influence,\nwe remain unable to reliably determine its impact on recovery versus its potential to undermine\nregulation. Strategies are absent that repurpose sound as a design utility for player empowerment\nand healthy boundaries. To resolve this issue, we propose three core directions focusing on ad-\ndressing methodological pitfalls, enhancing reporting transparency, and solidifying collaborative\nties between research and industry.\nFirst, adopt multi-modal evaluation: self-reports, behavioral traces, and psychophysiological\nmarkers each provide partial insights, but their combination offers a fuller picture of audio’s\nregulatory role. Logs of muting, playlist swaps, or sudden volume drops can reveal when players\nregulate their own auditory load. Coupled with physiological markers and short probes, such as “Do\nyou feel calmer?” or “Are you ready to stop?” one can test whether sound genuinely supports coping\nor merely shifts physiology without providing felt relief. These behaviors can act as practical,\nscalable proxies for laboratory measures. Convergences and divergences are equally informative,\nallowing researchers to identify when design succeeds and giving developers concrete indicators\nof overload or recovery.\nSecond, report design intentions transparently. Findings remain anecdotal when studies only state\nthat sound was present. Reporting must clarify how audio was crafted and with what regulatory\npurpose to subconsciously guide players toward recovery, or to provide explicit cues for conscious\nself-regulation: a low-tempo loop added after combat to support recovery, a brief silence window\nfollowing repeated failures to encourage pause, or a closing motif marking the transition to a\nsafe hub area. By grounding technical decisions in regulatory intent, researchers can accumulate\nevidence across contexts and developers can align design choices with player well-being, especially\nin sensitive cases where unexamined audio risks overstimulation [1].\nThird, foster a shared agenda between research and industry. Rather than prescribing isolated\ntechniques, advancing this field requires collaboration and teamwork. Initiatives such as joint\n3\n\nNunes et al.\nworkshops, repositories of customizable regulatory audio patterns and adaptive sound systems, or\nco-produced guidelines, similar to existing accessibility standards, can foster cumulative knowledge.\nTogether, these commitments elevate audio from a mere backdrop to an active mediator of\nescapist play. They reposition sound as a critical lever for discerning the balance between recovery\nand over-engagement, underscoring that without systematic evidence, designers still lack the means\nto use it intentionally for player well-being. To drive this agenda, we invite the community to\nconsider:\n(1) Which combinations of self-report, behavioral, and physiological measures best capture\naudio’s regulatory role?\n(2) How might soundscapes, finality cues in UI sounds, and music be designed to make transi-\ntions between immersion and disengagement more intentional?\n(3) What forms of player control over audio foster genuine autonomy in shaping escapist\nexperiences?\n(4) Which standardized or recurring sound patterns are being used unreflectively, and how\nmight their effects on well-being be better understood?\nAUTHOR BIOGRAPHIES\nCaio Nunes is a researcher with a Master’s degree in Computer Science from Federal University of\nCeará, Brazil. His work focuses on Human–Computer Interaction (HCI) and Game User Research,\nexploring audio, musical visualizations, motivation, and well-being in emotional communication.\nBosco Borges is a Master’s student at Federal University of Ceará, Brazil. His work focuses on\nHCI and Game User Research, examining how sound and interaction design can support player\nexperience, promote healthier engagement, and cater to diverse audiences.\nGeorgia Cruz is an Adjunct Professor at Federal University of Ceará, Brazil. Her work focuses on\nUX, Audiovisual, and Communicability, exploring user experience, gender in games, and critical\nplay, with an emphasis on how interactive media can foster reflection and influence well-being.\nTicianne Darin is an Associate Professor of HCI at the Federal University of Ceará. Her game\nresearch examines the foundational components of player experience, from motivation and engage-\nment to the pursuit of eudaimonia through reflexive games. She is currently focused on developing\nethical alternatives to manipulative design to foster digital well-being. Her leadership in the Brazil-\nian research community includes chairing the nation’s primary HCI symposia and founding the\nWIPlay workshop series, a dedicated venue to unite the country’s HCI and games researchers. As a\nUNESCO consultant, she helps shape national policy on digital rights, focusing on implementing\n\"Child and Adolescent Rights by Design\" frameworks for games and online services.\nREFERENCES\n[1] Bosco Borges and Ticianne Darin. 2025. Sounds in Game Design and Evaluation for Autistic People: A Literature\nReview. In International Workshop on Interaction and Player Research in Game Development. Springer, 29–48.\n[2] Lorelle Bowditch, Janine Chapman, and Anjum Naweed. 2018. Do coping strategies moderate the relationship between\nescapism and negative gaming outcomes in World of Warcraft (MMORPG) players? Computers in Human Behavior 86\n(2018), 69–76.\n[3] Gordon Calleja. 2010. Digital games and escapism. Games and Culture 5, 4 (2010), 335–353.\n[4] Scott Caplan, Dmitri Williams, and Nick Yee. 2009. Problematic Internet use and psychosocial well-being among MMO\nplayers. Computers in human behavior 25, 6 (2009), 1312–1319.\n[5] Hiroharu Kamioka, Kiichiro Tsutani, Minoru Yamada, Hyuntae Park, Hiroyasu Okuizumi, Koki Tsuruoka, Takuya\nHonda, Shinpei Okada, Sang-Jun Park, Jun Kitayuguchi, et al. 2014. Effectiveness of music therapy: a summary of\nsystematic reviews based on randomized controlled trials of music interventions. Patient preference and adherence\n(2014), 727–754.\n4\n\nIf You Hold Me Without Hurting Me\n[6] Daniel Kardefelt-Winther. 2014. A conceptual and methodological critique of internet addiction research: Towards a\nmodel of compensatory internet use. Computers in human behavior 31 (2014), 351–354.\n[7] Christoph Klimmt, Dorothée Hefner, and Peter Vorderer. 2009. The video game experience as “true” identification: A\ntheory of enjoyable alterations of players’ self-perception. Communication theory 19, 4 (2009), 351–373.\n[8] Stefan Koelsch. 2015. Music-evoked emotions: principles, brain correlates, and implications for therapy. Annals of the\nNew York Academy of Sciences 1337, 1 (2015), 193–201.\n[9] Mehmet Kosa and Ahmet Uysal. 2020. Four pillars of healthy escapism in games: Emotion regulation, mood manage-\nment, coping, and recovery. In Game user experience and player-centered design. Springer, 63–76.\n[10] Caio Nunes and Ticianne Darin. 2024. Echoes of player experience: A literature review on audio assessment and\nplayer experience in games. Proceedings of the ACM on Human-Computer Interaction 8, CHI PLAY (2024), 1–27.\n[11] Anthony Prechtl, Robin Laney, Alistair Willis, and Robert Samuels. 2014. Methodological approaches to the evaluation\nof game music systems. In Proceedings of the 9th audio mostly: a conference on interaction with sound. 1–8.\n[12] Grazia Ragone, Judith Good, and Katherine Howland. 2020. OSMoSIS: Interactive sound generation system for children\nwith autism. In Proceedings of the 2020 ACM Interaction Design and Children Conference: Extended Abstracts. 151–156.\n[13] Leonard Reinecke. 2009. Games and recovery: The use of video and computer games to recuperate from stress and\nstrain. Journal of media psychology 21, 3 (2009), 126–142.\n[14] Giovanni Ribeiro, Katja Rogers, Maximilian Altmeyer, Thomas Terkildsen, and Lennart E Nacke. 2020. Game atmo-\nsphere: effects of audiovisual thematic cohesion on player experience and psychophysiology. In Proceedings of the\nannual symposium on computer-human interaction in play. 107–119.\n[15] Katja Rogers, Giovanni Ribeiro, Rina R Wehbe, Michael Weber, and Lennart E Nacke. 2018. Vanishing importance:\nstudying immersive effects of game audio perception on player experiences in virtual reality. In Proceedings of the 2018\nCHI conference on human factors in computing systems. 1–13.\n[16] Katja Rogers and Michael Weber. 2019. Audio habits and motivations in video game players. In Proceedings of the 14th\nInternational Audio Mostly Conference: A Journey in Sound. 45–52.\n[17] Richard M Ryan and Edward L Deci. 2017. Self-determination theory: Basic psychological needs in motivation, development,\nand wellness. Guilford Publications.\n[18] Frode Stenseng, Jonas Falch-Madsen, and Beate Wold Hygen. 2021. Are there two types of escapism? Exploring a\ndualistic model of escapism in digital gaming and online streaming. Psychology of Popular Media 10, 3 (2021), 319.\n[19] Nick Yee. 2006. Motivations for play in online games. CyberPsychology & behavior 9, 6 (2006), 772–775.\nReceived 20 February 2025; revised 12 March 2025; accepted 5 June 2025\n5\n"},"cleaned":{"kind":"string","value":"If You Hold Me Without Hurting Me: Pathways to Designing Game Audio for Healthy Escapism and Player Well-being∗ CAIO NUNES, Federal á, Brazil BOSCO BORGES, Federal á, Brazil GEORGIA CRUZ, Federal á, Brazil TICIANNE DARIN, Federal á, Brazil Escapism in games can support recovery or lead to harmful avoidance. Self-regulation, understood as combining autonomy with positive outcomes, is key to this distinction. We argue that audio, often overlooked, plays a central role in regulation. It can modulate arousal, mark transitions, and provide closure, yet its contribution to well-being remains underexplored. This paper identifies methodological and accessibility gaps that limit recognition of audio's potential and outlines ways to address them. We aim to encourage researchers and developers to integrate audio more deliberately into the design and study of healthier escapist play. CCS Concepts: • Human-centered computing →HCI design and evaluation methods; • Applied computing →Sound and music computing. Additional Key Words and Phrases: Escapism, Emotions, Self-Regulation, Game Audio 1 INTRODUCTION Escapism has long been recognized as one of the main motivations for gameplay, appearing across models of player motivation and media use [4, 7, 19]. It is often described as the tendency to detach from everyday concerns or stressors, yet its consequences are ambivalent. On the one hand, escapism can offer distraction, stress relief, and opportunities for affective regulation [2]. On the other hand, it can reinforce avoidance and interfere with everyday functioning. Recent work emphasizes this tension by distinguishing between healthy escapism, when play supports recovery and well-being, and subversive escapism, when it undermines them [9]. This perspective shifts the debate from whether escapism is good or bad to a more pressing question: \"Which factors determine when escapist play sustains well-being and when it becomes harmful?\" One of the most decisive factors in this distinction is self-regulation. In Self-Determination Theory (SDT), self-regulation refers to the extent to which behavior is experienced as autonomous, reflecting choice and volition rather than external pressure or compulsion [17]. Applied to games, this perspective captures whether players can autonomously decide when to begin, how long to continue, and when to disengage. Research consistently shows that difficulties in regulation are linked to excessive or compulsive gaming [6]. At the same time, successful self-regulation allows games to function as bounded recesses that facilitate recovery, coping, and positive mood management [9, 13]. Framing escapism through this lens highlights autonomy as a key condition for distinguishing between healthy and subversive play. However, a player could autonomously choose to play for a duration that negatively impacts their life responsibilities. In this case, the behavior is self-determined, but the outcome is detrimental. Hence, a healthy escapism also requires positive outcomes, where psychological relief (e.g., reported mood improvement) is supported by physiological recovery (e.g., stress marker reduction), independent of the feeling of volition during play. Consequently, we draw attention to how the design of game elements can either scaffold a player's ability to self-regulate or undermine it. We argue that audio is a key game element that can be intentionally designed to structure the escapist experience itself, facilitating autonomous self-regulation and positive ∗Presented and discussed at the CHI PLAY 2025 Workshop Exploring Future Directions for Healthy Escapism and Self-Regulation in Games, Pittsburgh, USA, October 13, 2025. 1 16 Oct 2025 Nunes et al. outcomes. While often perceived as secondary to visual or mechanical elements, audio possesses unique qualities that make it a potent regulatory channel. Unlike visual cues, which require focused attention, sound is pervasive and processed with a lower cognitive load, capable of influencing arousal and emotional states subconsciously, creating a direct tension between systemic regulation and player autonomy that design must address. The well-documented link between music and physiological responses, from emotion regulation and stress recovery to everyday relaxation [5, 8]. This makes audio a prime channel for influencing the very recovery and emotion regulation that are central to healthy escapism [9]. Besides, audio's ability to structure time and rhythm transitions offers a subtle yet material means of scaffolding player awareness without disrupting immersion. Yet, despite its unique potential, the role of audio in player well-being remains poorly understood. Much of the game-audio literature concentrates on immersion, presence, arousal, and affect, seldom linking these outcomes to well-being or self-regulatory processes [10]. This gap is especially evident in discussions of escapism and self-regulation, which often emphasize constructs such as absorption, affective modulation, stress relief, and distraction [3, 18]. While these emphases are not identical, they share a concern with how players become deeply involved and how their emotional states are shaped during play. This lack of synthesis leaves a critical gap, preventing designers and researchers from systematically using audio as a tool for player well-being. This position paper moves to fill that gap. We contend that audio is not a peripheral feature but a central, designable mechanism for scaffolding healthy escapism in games. To substantiate this claim, the remainder of this paper will first identify methodological and accessibility gaps previously identified in game audio research that perpetuate this problem. We then propose three concrete directions for future work focused on using audio to support self-regulation and positive emotional outcomes. 2 HOW RESEARCH GAPS OBSCURE AUDIO'S ROLE IN HEALTHY ESCAPISM In a previous study, we reviewed how audio and player experience are evaluated [10]. The results show a fragmented evidence base where game audio is routinely linked to immersion and affect, but its deeper impact is unclear. A key problem is that self-report measures often fail to converge with physiological data; for example, players may report feeling calm while their arousal markers remain elevated [10]. These inconsistencies make it challenging to draw conclusions about outcomes such as coping, stress recovery, or emotion regulation. This methodological fragmentation is compounded by other persistent issues, including a lack of personalization and the tendency for audio to be overshadowed in high-sensory contexts [11, 14-16]. Together, these findings reveal that audio is seen as an immersive aesthetic but not perceived as a potential lever for self-regulatory outcomes. The real-world impact of this research gap is starkly visible in games designed for sensitive audiences. Our literature review (n=49) on games for autistic players [1], for instance, shows that audio is often absent from game design descriptions and from user evaluation, despite its potential to support engagement or, if mishandled, trigger overload in audiences with auditory hypersensitivity. Across the surveyed studies, only 14% (n=7) explicitly evaluated audio; when they did, its uses ranged from background music to movement sonification, yet the impacts on the target audience were seldom reported in depth [12]. Such an absence of systematic evaluation and reporting reveals a double gap: a lack of design transparency and potentially a neglect of accessibility. For that audience, without customization options or sensitivity to diverse auditory needs, audio risks becoming a barrier instead of a resource. These findings highlight how sound design, if overlooked, can miss therapeutic opportunities and expose players to overstimulation or discomfort. This case starkly illustrates how neglecting audio can undermine a player's ability to self-regulate, turning a potentially healthy escape into a subversive, dysregulating experience. 2 If You Hold Me Without Hurting Me While these examples are not exhaustive, the gaps identified in the literature prevent us from measuring whether audio supports well-being or causes distress. Methodological blind spots mean that we lack ways to evaluate if audio is contributing to the positive outcome of healthy escapism, whether audio supports coping and recovery or simply intensifies immersion. Accessibility gaps can imply that sound becomes overwhelming or exclusionary, rather than restorative. In both cases, escapism risks sliding toward avoidance and overuse - not because audio cannot support healthier outcomes, but because research and design fall short in providing the tools to differentiate them. The absence of variation, closure, and inclusive controls leaves players to self-manage through workarounds such as muting or replacing game soundtracks [16], undermining the potential of audio to contribute directly to constructive escapist experiences. Ultimately, players are left without the necessary scaffolds to exercise their autonomy due to a lack of key tools, such as methods for measuring player responses to audio cues or designs that utilize audio to create natural stopping points. For instance, if safe exit points are signaled by a shift to calmer music in a hub area, it creates a clear closing motif after a mission, or even a brief \"breathing window\"\" of silence after a moment of high tension, all of which provide a sense of closure that could help players disengage with a sense of accomplishment. Without such intentional design, poorly tested or repetitive audio can instead reinforce compulsive immersion or force avoidance, both of which undermine self-regulation. These blind spots leave players without the necessary scaffolds to disengage autonomously, showing how overlooked audio can shift escapism from recovery to overuse. 3 TOWARDS SELF-REGULATION AND HEALTHY ESCAPISM WITH AUDIO The gaps highlighted reveal a significant methodological deficiency: despite audio's clear influence, we remain unable to reliably determine its impact on recovery versus its potential to undermine regulation. Strategies are absent that repurpose sound as a design utility for player empowerment and healthy boundaries. To resolve this issue, we propose three core directions focusing on addressing methodological pitfalls, enhancing reporting transparency, and solidifying collaborative ties between research and industry. First, adopt multi-modal evaluation: self-reports, behavioral traces, and psychophysiological markers each provide partial insights, but their combination offers a fuller picture of audio's regulatory role. Logs of muting, playlist swaps, or sudden volume drops can reveal when players regulate their own auditory load. Coupled with physiological markers and short probes, such as \"Do you feel calmer?\" or \"Are you ready to stop?\" one can test whether sound genuinely supports coping or merely shifts physiology without providing felt relief. These behaviors can act as practical, scalable proxies for laboratory measures. Convergences and divergences are equally informative, allowing researchers to identify when design succeeds and giving developers concrete indicators of overload or recovery. Second, report design intentions transparently. Findings remain anecdotal when studies only state that sound was present. Reporting must clarify how audio was crafted and with what regulatory purpose to subconsciously guide players toward recovery, or to provide explicit cues for conscious self-regulation: a low-tempo loop added after combat to support recovery, a brief silence window following repeated failures to encourage pause, or a closing motif marking the transition to a safe hub area. By grounding technical decisions in regulatory intent, researchers can accumulate evidence across contexts and developers can align design choices with player well-being, especially in sensitive cases where unexamined audio risks overstimulation [1]. Third, foster a shared agenda between research and industry. Rather than prescribing isolated techniques, advancing this field requires collaboration and teamwork. Initiatives such as joint 3 Nunes et al. workshops, repositories of customizable regulatory audio patterns and adaptive sound systems, or co-produced guidelines, similar to existing accessibility standards, can foster cumulative knowledge. Together, these commitments elevate audio from a mere backdrop to an active mediator of escapist play. They reposition sound as a critical lever for discerning the balance between recovery and over-engagement, underscoring that without systematic evidence, designers still lack the means to use it intentionally for player well-being. To drive this agenda, we invite the community to consider: (1) Which combinations of self-report, behavioral, and physiological measures best capture audio's regulatory role? (2) How might soundscapes, finality cues in UI sounds, and music be designed to make transitions between immersion and disengagement more intentional? (3) What forms of player control over audio foster genuine autonomy in shaping escapist experiences? (4) Which standardized or recurring sound patterns are being used unreflectively, and how might their effects on well-being be better understood? AUTHOR BIOGRAPHIES Caio Nunes is a researcher with a Master's degree in Computer Science from Federal á, Brazil. His work focuses on Human-Computer Interaction (HCI) and Game User Research, exploring audio, musical visualizations, motivation, and well-being in emotional communication. Bosco Borges is a Master's student at Federal á, Brazil. His work focuses on HCI and Game User Research, examining how sound and interaction design can support player experience, promote healthier engagement, and cater to diverse audiences. Georgia Cruz is an Adjunct Professor at Federal á, Brazil. Her work focuses on UX, Audiovisual, and Communicability, exploring user experience, gender in games, and critical play, with an emphasis on how interactive media can foster reflection and influence well-being. Ticianne Darin is an Associate Professor of HCI at the Federal á. Her game research examines the foundational components of player experience, from motivation and engagement to the pursuit of eudaimonia through reflexive games. She is currently focused on developing ethical alternatives to manipulative design to foster digital well-being. Her leadership in the Brazilian research community includes chairing the nation's primary HCI symposia and founding the WIPlay workshop series, a dedicated venue to unite the country's HCI and games researchers. As a UNESCO consultant, she helps shape national policy on digital rights, focusing on implementing \"Child and Adolescent Rights by Design\" frameworks for games and online services. REFERENCES [1] Bosco Borges and Ticianne Darin. 2025. Sounds in Game Design and Evaluation for Autistic People: A Literature Review. In International Workshop on Interaction and Player Research in Game Development. Springer, 29-48. [2] Lorelle Bowditch, Janine Chapman, and Anjum Naweed. 2018. Do coping strategies moderate the relationship between escapism and negative gaming outcomes in World of Warcraft (MMORPG) players? Computers in Human Behavior 86 (2018), 69-76. [3] Gordon Calleja. 2010. Digital games and escapism. Games and Culture 5, 4 (2010), 335-353. [4] Scott Caplan, Dmitri Williams, and Nick Yee. 2009. Problematic Internet use and psychosocial well-being among MMO players. Computers in human behavior 25, 6 (2009), 1312-1319. [5] Hiroharu Kamioka, Kiichiro Tsutani, Minoru Yamada, Hyuntae Park, Hiroyasu Okuizumi, Koki Tsuruoka, Takuya Honda, Shinpei Okada, Sang-Jun Park, Jun Kitayuguchi, et al. 2014. Effectiveness of music therapy: a summary of systematic reviews based on randomized controlled trials of music interventions. Patient preference and adherence (2014), 727-754. 4 If You Hold Me Without Hurting Me [6] Daniel Kardefelt-Winther. 2014. A conceptual and methodological critique of internet addiction research: Towards a model of compensatory internet use. Computers in human behavior 31 (2014), 351-354. [7] Christoph Klimmt, Dorothée Hefner, and Peter Vorderer. 2009. The video game experience as \"true\" identification: A theory of enjoyable alterations of players' self-perception. Communication theory 19, 4 (2009), 351-373. [8] Stefan Koelsch. 2015. Music-evoked emotions: principles, brain correlates, and implications for therapy. Annals of the New York Academy of Sciences 1337, 1 (2015), 193-201. [9] Mehmet Kosa and Ahmet Uysal. 2020. Four pillars of healthy escapism in games: Emotion regulation, mood management, coping, and recovery. In Game user experience and player-centered design. Springer, 63-76. [10] Caio Nunes and Ticianne Darin. 2024. Echoes of player experience: A literature review on audio assessment and player experience in games. Proceedings of the ACM on Human-Computer Interaction 8, CHI PLAY (2024), 1-27. [11] Anthony Prechtl, Robin Laney, Alistair Willis, and Robert Samuels. 2014. Methodological approaches to the evaluation of game music systems. In Proceedings of the 9th audio mostly: a conference on interaction with sound. 1-8. [12] Grazia Ragone, Judith Good, and Katherine Howland. 2020. OSMoSIS: Interactive sound generation system for children with autism. In Proceedings of the 2020 ACM Interaction Design and Children Conference: Extended Abstracts. 151-156. [13] Leonard Reinecke. 2009. Games and recovery: The use of video and computer games to recuperate from stress and strain. Journal of media psychology 21, 3 (2009), 126-142. [14] Giovanni Ribeiro, Katja Rogers, Maximilian Altmeyer, Thomas Terkildsen, and Lennart E Nacke. 2020. Game atmosphere: effects of audiovisual thematic cohesion on player experience and psychophysiology. In Proceedings of the annual symposium on computer-human interaction in play. 107-119. [15] Katja Rogers, Giovanni Ribeiro, Rina R Wehbe, Michael Weber, and Lennart E Nacke. 2018. Vanishing importance: studying immersive effects of game audio perception on player experiences in virtual reality. In Proceedings of the 2018 CHI conference on human factors in computing systems. 1-13. [16] Katja Rogers and Michael Weber. 2019. Audio habits and motivations in video game players. In Proceedings of the 14th International Audio Mostly Conference: A Journey in Sound. 45-52. [17] Richard M Ryan and Edward L Deci. 2017. Self-determination theory: Basic psychological needs in motivation, development, and wellness. Guilford Publications. [18] Frode Stenseng, Jonas Falch-Madsen, and Beate Wold Hygen. 2021. Are there two types of escapism? Exploring a dualistic model of escapism in digital gaming and online streaming. Psychology of Popular Media 10, 3 (2021), 319. [19] Nick Yee. 2006. Motivations for play in online games. CyberPsychology & behavior 9, 6 (2006), 772-775. Received 20 February 2025; revised 12 March 2025; accepted 5 June 2025 5"}}},{"rowIdx":291,"cells":{"id":{"kind":"string","value":"2510.14690"},"raw":{"kind":"string","value":"CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\nON COMPACT HERMITIAN MANIFOLDS\nYUETONG FANG\nAbstract. Let (X, ω) be a compact Hermitian manifold of dimension n. We derive an\nL∞-estimate for bounded solutions to the complex m-th Hessian equations on X, assuming\na positive right-hand side in the Orlicz space L\nn\nm (log L)n(h◦log ◦log L)n, where the associ-\nated weight satisfies Ko lodziej’s Condition. Building upon this estimate, we then establish\nthe existence of continuous solutions to the complex Hessian equation under the prescribed\nassumptions.\nIntroduction\nLet (X, ω) be a compact complex Hermitian manifold of dimension n equipped with a\nHermitian metric ω. Fix 0 < m ≤n and let dVX be a smooth volume form. The problem\nof interest in this paper is to find a constant c > 0 and continuous (ω, m)-subharmonic\nsolutions to the following complex m-Hessian equation\n(ω + ddcu)m ∧ωn−m = ceλufdVX, λ ≥0,\n(Hessian)\nunder the assumption that the densities fn/m belong to an Orlicz space Lχ with the weight\nχ satisfying Ko lodziej’s condition, which will be explained later.\nWhen m = 1, this simplifies to the Poisson equation, while when m = n, it reduces to the\nMonge–Amp`ere equation. Yau’s solution [Yau78] of the Calabi Conjecture has motivated\nmany subsequent studies on Monge–Amp`ere equations over the past fifty years. As presented\nby B locki [B lo12], using the simplifications due to Kazdan and to Aubin–Bourguignon, Yau’s\na priori estimate can be extended to cases where the densities f belong to Lp for p > n.\nOne of the breakthrough was achieved by Ko lodziej [Ko l98], who applied pluripotential\ntheory to obtain an L∞-estimate for Monge–Amp`ere equations when f belongs to some\nOrlicz space, with the associated weight specifically satisfying Condition (K). As remarked\nby Ko lodziej [Ko l98], such an a priori estimate is ”almost sharp”.\nThis integrability condition on densities f can be extended in the context of complex\nHessian equations. We say that a convex weight χ satisfies Ko lodziej’s condition (Condition\n(K)) if it is increasing and\nχ(t)\n∼\nt→+∞t(log(t + 1))n (h ◦log ◦log(t + 3))n ,\nwhere the function h : R+ →R+ satisfying\n´ +∞h−1(t)dt < +∞. In this paper, we focus on\nthe case where fn/m ∈Lχ. We refer the reader to Section 1 for more detailed discussions\non Orlicz spaces and Condition (K).\n2020 Mathematics Subject Classification. 32W20, 32U05, 32Q15, 35A23.\nKey words and phrases. Hermitian Geometry, Monge–Amp`ere equations and complex Hessian equations,\nweak solution, a priori estimate, stability estimate.\n1\narXiv:2510.14690v1 [math.CV] 16 Oct 2025\n\n2\nYUETONG FANG\nThe study of weak solutions to complex Monge–Amp`ere equations plays an important role\nin the analysis of Ricci flows (see [TW15]). On compact K¨ahler manifolds, one can study the\nmodulus of continuity for solutions to complex Monge–Amp`ere equations. H¨older regularity\nfor such equations with densities in Lp was established in [DDG+14], and has been further\ngeneralized to the case where the densities belong to certain Orlicz spaces, in [GPTW21]\nvia a PDE approach, and in [GGZ25] using pluripotential theory.\nTo prove the existence of continuous solution to the complex Monge–Amp`ere equation,\na key ingredient is the L∞-estimate.\nVarious techniques have been developed to obtain\nL∞-estimate for Monge–Amp`ere equations on compact manifolds, including Yau’s [Yau78]\noriginal proof, which employs Moser’s iteration scheme. Later, Ko lodziej [Ko l98] developed\nL∞-estimates using properties of capacity and established an optimal integrability condition\nfor the density f (see also [EGZ08, DP10] for related generalizations).\nInspired by the\nwork of Chen–Cheng [CC21] on constant scalar curvature K¨ahler metrics, a very different\napproach was introduced by Guo–Phong–Tong [GPT23], by comparing the given equation\nto an auxiliary complex Monge–Amp`ere equation.\nThis PDE technique has since been\ngeneralized in [GP24, Qia24].\nAn efficient approach was developed by Guedj–Lu [GL23,\nGL24], using properties of envelopes, which can be adapted to the case where f ∈L logp L\nwith p > n, as noted in [GL24, Section 2]. More recently, they generalized this result to\nOrlicz spaces satisfying Condition (K) on compact K¨ahler manifolds in [GL25b], using a\nnew approach that reduces Ko lodziej’s criterion to Yau’s theorem.\nOn the other hand, complex m-Hessian equations are natural generalizations of Monge–\nAmp`ere equations. B locki [B lo05] established potential theory for m-Hessian equations in\ndomains of Cn. Later, Dinew–Ko lodziej [DK14] obtained a priori estimates and proved a\nstability result for weak solutions of the complex Hessian equation in domains of Cn and\ncompact K¨ahler manifolds.\nThe modulus of continuity of solutions to complex Hessian\nequations has been studied in [Ngu14,Cha16] with densities f ∈Lp, p > n/m, and has been\nextended recently in [˚AC25] to the cases where f ∈Ln/m logp(L), p > 2n. On compact\nHermitian manifolds, Ko lodziej and Nguyen proved the existence of weak solutions with a\nnonnegative right-hand side belonging to Lp, p > n/m, in [KN16], while the corresponding\nresult for Monge–Amp`ere equations was established in [KN15].\nThe purpose of this paper is to extend the above results to prove the continuity of bounded\nsolutions to the complex Hessian equation. We begin by establishing a subsolution lemma for\nthe Monge–Amp`ere equation, generalizing Lemma 2.1 of [GL23] to the case where f ∈Lχ,\nwith χ satisfying Condition (K). The lemma is stated as follows:\nLemma A (Subsolution Lemma). Assume χ satisfies Condition (K). Then there exist a\nuniform constant s = s(χ) > 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω),\n−1 ≤ψ ≤0, satisfying\n(ω + ddcψ)n ≥s(χ)\ng\n∥g∥χ\ndVX.\nWe obtain the Subsolution Lemma by gluing the solutions to the Dirichlet problem in\nlocal context:\n(ddcu)n =\nf\n∥f∥χ\ndVX, in Ω;\nu = 0, on ∂Ω.\n(0.1)\nCompared with the case where the densities f ∈Lp, working with Orlicz norms is more\ninvolved, as it requires controlling the Orlicz norm of the solution u associated to the convex\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n3\nconjugate of χ. To simplify the argument, we adapt an approach from the recent article by\nGuedj–Lu [GL25b] to derive a local L∞-estimate for problem 0.1.\nAs a consequence of Lemma A, we obtain a uniform estimate for the complex Hessian\nequation on a Hermitian manifold, where the densities f satisfy Ko lodziej’s optimal condi-\ntion.\nTheorem B (L∞-estimate). Let φ be a bounded (ω, m)-subharmonic function satisfying\n(ω + ddcφ)m ∧ωn−m ≤fdVX,\n(Hess)\nwhere 0 ≤fn/m ∈Lχ with the weight χ satisfying Condition (K). Then\nOscX(φ) ≤C,\nwhere C depends on ω, n, m and\n\r\r\rf\nn\nm\n\r\r\r\nχ.\nBy generalizing an idea in [GL25a], we demonstrate that the L∞-estimate for complex\nHessian equations can be reduced to the corresponding estimate for an associated Monge–\nAmp`ere equation:\n(ω + ddcφ)n ≤fdVX,\n(0.2)\nwhere f ∈Lχ, and χ satisfies Condition (K). To obtain the L∞-estimate for the Monge–\nAmp`ere equation, our main tool is the domination principle. The key novelty of our approach\nlies in constructing an appropriate subsolution. Our approach avoids potential computa-\ntional difficulties associated with Orlicz norms and presents a different method compared to\nKo lodziej’s original proof.\nOnce the L∞-estimate is obtained, we can derive the existence and regularity of bounded\nsolutions. This leads us to the second main result of the study:\nTheorem C. Let 0 ≤fn/m ∈Lχ, with the weight function χ satisfying Condition (K).\nThen there exist a constant c > 0 and a solution u ∈SHm(X, ω) ∩C0(X) satisfying the\ncomplex Hessian equation\n(ω + ddcu)m ∧ωn−m = ceλufdVX, λ ≥0.\nMoreover, all bounded solutions to the above complex Hessian equation are continuous.\nTo prove the first statement of Theorem C, an essential component is to establish a\nstability theorem (Theorem 3.1). For f ∈Lp, with p > n/m, such a stability result was\nproven in [KN16]. In the case where f ∈Lχ and the weight function χ satisfies Condition\n(K), the continuity of solutions to Monge–Amp`ere equations was studied in [KN21] by\nadapting the method in [KN19, Theorem 3.1].\nWe point out that our approach to the\nstability estimate is fundamentally different and remains of interest in the case where the\ndensities f ∈Lp, as discussed in Remark 3.2. Our main tool will be the Domination Principle\nand the subsolution (Lemma A). Having established the stability estimate (Theorem 3.1),\nwe approximate the density f by the sequence of bounded functions min(f, j). We show\nthat the corresponding solutions uj converge uniformly to an (ω, m)-subharmonic function\nu solving the equation (Hessian), thereby obtaining a continuous solution. The continuity\nof all bounded solutions follows from the Domination Principle (Proposition 1.8).\n\n4\nYUETONG FANG\nOrganization of the paper. The paper is organized as follows. Section 1 gives preliminary\nmaterial on (ω, m)-subharmonic functions, as well as key concepts of Orlicz spaces. A local\nL∞-estimate is established, and Lemma A and Theorem B are proved in Section 2.\nIn\nsection 3, we establish a stability estimate and subsequently investigate the continuity of\nbounded solutions, leading to the proof of Theorem C.\nAcknowledgment. The author thanks her supervisor, Hoang-Chinh Lu, for suggesting the\nproblem, valuable discussions during the preparation of the work, and useful comments that\nimproved the presentation. This research is part of a PhD program funded by the PhD\nscholarship CSC-202308070110, and is partially supported by the projects Centre Henri\nLebesgue ANR-11-LABX-0020-01 and the KRIS project of Fondation Charles Defforey.\n1. Preliminaries\nIn the whole paper, we let (X, ω) denote a compact Hermitian manifold of complex\ndimension n ∈N∗, equipped with a Hermitian form ω. We denote by dVX a smooth volume\nform on X. We use differential operators d = ∂+ ¯∂, and dc = i(¯∂−∂), so that ddc = 2i∂¯∂.\n1.1. (ω, m)-subharmonic functions.\n1.1.1. Definition. We recall the definition of (ω, m)-subharmonic functions following [GN18]\nand briefly review some results related to both ω-plurisubharmonic and (ω, m)-subharmonic\nfunctions that will be used later.\nFix an integer 1 ≤m ≤n. Fix Ωan open set in Cn. Given a hermitian metric ω on Ω, a\nreal (1, 1)-form α on X is called m-positive with respect to ω if at all points in X,\nαk ∧ωn−k ≥0, ∀k = 1, · · · , m.\nA C2(Ω) function u : Ω→R is called harmonic with respect to ω if ddcu ∧ωn−1 = 0 at all\npoints in Ω.\nDefinition 1.1. A function u : Ω→{−∞} ∪R is subharmonic with respect to ω if:\n(a) u is upper semicontinuous and u ∈L1\nloc(Ω);\n(b) for every relatively compact open set D ⋐Ωand every function h ∈C0(D) that is\nharmonic with respect to ω on D the following implication holds:\nu ≤h on ∂D =⇒u ≤h in D.\nDefinition 1.2. A function φ : Ω→{−∞} ∪R is quasi-subharmonic with respect to ω if\nlocally φ = u + ρ, where u is subharmonic with respect to ω and ρ is smooth.\nA function φ is α-subharmonic with respect to ω if φ is quasi-subharmonic with respect\nto ω and (α + ddcφ) ∧ωn−1 ≥0 in the sense of distributions.\nThe positive cone Γm(ω) associated with the metric ω is defined by\nn\nγ real (1, 1)-form : γk ∧ωn−k > 0, k = 1, · · · , m\no\n.\nIt follows from G˚arding’s inequality [G˚ar59] that if γ0, γ1, · · · , γm−1 ∈Γm(ω), then\nγ0 ∧γ1 ∧· · · ∧γm−1 ∧ωn−m > 0.\nOne can write eω = γ1 ∧· · · ∧γm−1 ∧ω, and it is a strictly positive (n −1, n −1)-form on Ω.\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n5\nDefinition 1.3. A function φ : Ω→[−∞, +∞) is called (α, m)-subharmonic with respect to\nω if φ is α-subharmonic with respect to eω in Ωfor all eω of the form eωn−1 = γ1∧· · ·∧γm−1∧ω,\nwhere γ1, · · · , γm−1 ∈Γm(ω).\nA function u : X →R ∪{−∞} is called (α, m)-subharmonic on X if u is (α, m)-\nsubharmonic on each local chart U of X.\nThe set of all locally integrable functions on U which are (α, m)-subharmonic with respect\nto ω in U is denoted by SHω,m(U, α). In many contexts, such as [GL25a], the forms ω and\nα may differ.\nHowever, in this paper, we focus on the set SHω,m(U, ω). To simplify the notations, we\ndenote by SHm(U, ω) the set of all (ω, m)-subharmonic functions with respect to ω in U.\nThe set of all ω-plurisubharmonic functions on U is denoted by PSH(U, ω) = SHn(U, ω).\nFor notational convenience, we denote ωu := ω + ddcu.\nRemark 1.4. By G˚arding’s inequality [G˚ar59], if u ∈C2(X), then u is (ω, m)-subharmonic\nwith respect to ω on X if and only if the associated form ωu belongs to the closure of Γm(ω).\nIt is well known that when n = m, the set PSH(X, ω) is a closed subset of Lr(X), for all\nr ≥1. For (ω, m)-subharmonic functions, the following L1-compactness result was proved\nin [KN16, lemma 3.3].\nLemma 1.5. Let u ∈SHm(X, ω) be normalized by supX u = 0. Then there exists a uniform\nconstant A > 0 depending only on X, ω such that\nˆ\nX\n|u|ωn ≤A.\n1.1.2. Domination principle. Analogous to the Monge–Amp`ere operators, the complex Hes-\nsian operators also satisfy a domination principle. The domination principle for Hessian op-\nerator associated with continuous (ω, m)-subharmonic functions is established by Guedj–Lu\nin [GL25a, Section 2]. Following the demonstration that the Hessian operators for bounded\nfunctions are well-defined, we show the domination principle associated with bounded (ω, m)-\nsh functions here.\nLemma 1.6 (Maximum principle). Assume that u and v are bounded (ω, m)-subharmonic\nfunctions. Then\n(ω + ddc(max(u, v)))m ∧ωn−m ≥1{u>v}(ω + ddcu)m ∧ωn−m + 1{v≥u}(ω + ddcv)m ∧ωn−m.\nIn particular, if u ≤v, then,\n1{v=u}(ω + ddcu)m ∧ωn−m ≤1{v=u}(ω + ddcv)m ∧ωn−m\nIn the case where u and v are continuous, the above lemma is proven in [GL25a, Lemma\n1.10]. When u is bounded, thanks to the continuity of complex Hessian operators ( [KN23]),\nwe can approximate u by a sequence of decreasing smooth (ω, m)-subharmonic functions.\nThe comparison principle for u, v ∈SHm(Ω) ∩L∞(Ω) is introduced in [KN23, Section 6].\nProposition 1.7 (Comparison principle). Let u, v be bounded m-subharmonic functions\nwith respect to ω and such that lim infz→∂Ωu(z) −v(z) ≥0. Assume\n(ddcv)m ∧ωn−m ≥(ddcu)m ∧ωn−m in Ω.\nThen it follows that u ≥v in Ω.\nProof. See [KN23, Corollary 6.2].\n□\n\n6\nYUETONG FANG\nWe will need a global version of domination principle:\nProposition 1.8 (Domination principle). Assume that u, v ∈PSH(X, ω) ∩L∞(X) satisfy\n1{u 0. If e−λvωm\nv ∧ωn−m ≥e−λuωm\nu ∧ωn−m, then u ≥v.\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n7\nProof. Fix C > 0. On the set {u < v −C} we have\nωm\nu ∧ωn−m ≤eλ(u−v)ωm\nv ∧ωn−m ≤e−λCωm\nv ∧ωn−m.\nIt follows from Proposition 1.8 that u ≥v −C. Since this holds for all C > 0, we conclude\nthat u ≥v.\n□\nCorollary 1.10. Assume u, v are bounded (ω, m)-subharmonic functions on X such that\n(ω + ddcu)m ∧ωn−m ≤c (ω + ddcv)m ∧ωn−m\nfor some positive constant c. Then c ≥1.\n1.1.3. Cegrell classes. In Section 2, we study the Monge–Amp`ere equation in a local context.\nHere, we recall some essential definitions of the Cegrell classes [Ceg98,Ceg04].\nFix a bounded hyperconvex domain Ω⊂Cn. Let T (Ω) denote the set of bounded plurisub-\nharmonic functions u in Ωsuch that limz→ζ u(z) = 0 for every ζ ∈∂Ω, and\n´\nΩ(ddcu)n < +∞.\nA function u belongs to F(Ω) iff there exists a sequence uj ∈T (Ω) decreasing to u in all\nof Ω, which satisfies supj\n´\nΩ(ddcuj)n < +∞.\nAs demonstrated by Cegrell [Ceg98,Ceg04], the complex Monge–Amp`ere operator can be\ndefined for functions belonging to the class F(Ω) even when these functions are unbounded.\nThe Dirichlet problem in F(Ω) was solved in [Ceg04]:\nTheorem 1.11. Assume that µ is a positive measure on Ω. If µ(Ω) < +∞and µ vanishes\non all pluripolar sets, then there exists a unique function u ∈F(Ω) such that (ddcu)n = µ.\nProof. See [Ceg04, Lemma 5.14].\n□\n1.2. Orlicz space. We recall some essential facts about Orlicz spaces that will be used\nthroughout, following [Dar19]. Let µ be a probability measure on X. A weight χ : R∗→\nR∗∪{+∞} is called normalized Young weight if it is convex, lower semi-continuous, non-\ntrivial satisfying the normalizing condition\nχ(0) = 0 and 1 ∈∂χ(1).\nRecall that ∂χ(l) ⊂R denotes the set of subgradients of χ at l, which means that a ∈∂χ(t)\nif and only if χ(t) + ab ≤χ(t + b) for all b ∈R. Its conjugate convex weight χ∗is the\nLegendre transform of χ:\nχ∗(h) = sup\nt∈R∗{ht −χ(t)}.\nOne can easily verify that the following propositions hold:\nProposition 1.12. Let χ be a normalized Young weight. Then\n(i) χ∗is also a normalized Young weight;\n(ii) (Fenchel–Moreau theorem). The biconjugate weight χ∗∗is also a normalized Young\nweight, and χ∗∗= χ.\nOne can define a function space with respect to the Young function. Let Lχ(µ) denote\nthe space of measurable functions defined as follows:\nLχ(µ) :=\n\u001a\nf : X →R∗∪{+∞} : ∃s > 0,\nˆ\nX\nχ(sf) < +∞\n\u001b\n.\nOne may also define the norm (also known as Luxembourg norm) on Lχ(µ).\n∥f∥χ,µ := inf\n\u001a\nr > 0 :\nˆ\nX\nχ\n\u0012f\nr\n\u0013\n≤χ(1)\n\u001b\n.\n\n8\nYUETONG FANG\nWe remark that (Lχ(µ), ∥·∥χ,µ) constitutes a normed space and is complete, hence a Banach\nspace. In the remainder of the paper, we fix µ = dVX, a smooth probability measure, and\nwrite (Lχ, ∥· ∥χ) for simplicity.\nIn [Ko l98], S.Ko lodziej developped a technique using pluripotential theory that enables\none to bound the solution to the Monge–Amp`ere equation by the Orlicz norm, ∥· ∥χ, of\nthe right-hand side, associated with a weaker integrable condition. We thus introduce the\nCondition (K).\nDefinition 1.13 (Condition (K)). A weight χ : R+ →R+ satisfies condition (K) if it is\nconvex increasing and\nχ(t)\n∼\nt→+∞tn/m (log(1 + t))n (h ◦log ◦log(1 + t))n , where\nˆ +∞dt\nh(t) < +∞.\nExample 1.14. We discuss and present a few typical examples:\n(i) When χ(t) = tp\np for p > 1, its convex conjugate is given by χ∗(t) = tq\nq , where q =\np\np−1\nis the H¨older conjugate of p. The Orlicz norm ∥· ∥χ here is the usual Lp norm.\n(ii) The weight χ defined in the Condition (K) above is a Young function hence induce\nan Orlicz space.\nIn particular, the weight χ(t) = t logp(1 + t) with p > n is a\nYoung function. One can check that as t →+∞, the conjugate weight grows like\nχ∗(t) = t1−1\np exp(t\n1\np ).\nWe will focus on the Orlicz space (Lχ, ∥· ∥χ) in section 3. It is important to note that in\nsuch an Orlicz space, the H¨older inequality (or H¨older–Young inequality) holds.\nProposition 1.15 (H¨older–Young inequality). For f ∈Lχ and g ∈Lχ∗, we have\nˆ\nX\n|fg|dVX ≤∥f∥χ∥g∥χ∗.\nProof. See [Dar19, Proposition 1.3].\n□\nWe will need the following additive Young inequality (also known as Fenchel–Young in-\nequality).\nProposition 1.16. For every a, b ∈R+, it holds that\nab ≤χ(a) + χ∗(b),\nwith equality if and only if b ∈∂χ(a).\nIn particular, let f, g : X →R+ be two measurable functions. Then\nf\n\u0000(χ∗)−1 ◦g\n\u0001\n≤χ ◦f + g.\n1.3. Envelopes. Given a Lebesgue measurable function l : X →R, the ω-plurisubharmonic\nenvelope of l is defined by\nPω(l) := sup{u ∈PSH(X, ω) : u ≤l}.\nThe (ω, m)-subharmonic envelope of l is defined by\nPω,m(l) := sup{u ∈SHm(X, ω) : u ≤l}.\nWe now summarize the basic properties of these envelopes, following [GL25a].\nProposition 1.17. If l : X →R is continuous, then\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n9\n(1) The complex Monge–Amp`ere measure (ω +ddcPω(l))n is concentrated on the contact\nset {Pω(l) = l}.\n(2) The complex Hessian measure (ω + ddcPω,m(l))m ∧ωn−m is concentrated on the\ncontact set {Pω,m(l) = l}.\n(3) If l is C1,1-smooth, then Pω(l) is C1,1-smooth and\n(ω + ddcPω(l))n = 1{Pω(l)=l}(ω + ddcl)n.\n(4) If l is C1,1-smooth, then Pω,m(l) is C1,1-smooth and\n(ω + ddcPω,m(l))m ∧ωn−m = 1{Pω,m(l)=l}(ω + ddcl)m ∧ωn−m.\nFor detailed proofs and additional discussion, we refer the reader to [Ber19,CM21,CZ19,\nGL22, GLZ19, Tos18]. As a consequence of the preceding results, we will make use of the\nfollowing statements in the sequel.\nLemma 1.18. Fix λ > 0. Let u, v ∈SHm(X, ω) ∩L∞(X).\n(1) If (ω + ddcu)m ∧ωn−m ≥eλufdVX and (ω + ddcv)m ∧ωn−m ≥eλugdVX, then\n(ω + ddc(max(u, v)))m ∧ωn−m ≥eλ max(u,v) min(f, g)dVX.\n(2) If (ω + ddcu)m ∧ωn−m ≤eλufdVX, and (ω + ddcv)m ∧ωn−m ≤eλvgdVX, then\n(ω + ddc(Pω,m(min(u, v)))m ∧ωn−m ≤eλPω,m(min(u,v)) max(f, g)dVX.\nThe proof follows the same strategy as that for the Monge–Amp`ere measures presented\nin [GL22, Lemma 1.9].\nProof. Let ψ := max(u, v). As a consequence of the maximum principle (Lemma 1.6), we\nhave\n(ω + ddcψ)m ∧ωn−m ≥1{u≥v}(ω + ddcu)m ∧ωn−m + 1{v>u}(ω + ddcv)m ∧ωn−m\n≥1{u≥v}eλufdVX + 1{v>u}eλvgdVX\n=1{u≥v}eλψfdVX + 1{v>u}eλψgdVX ≥eλψ min(f, g)dVX.\nSet φ := Pω,m(min(u, v)). We prove the second statement. It follows from the proposi-\ntion 1.17 that the complex Hessian measure (ω + ddcφ)m ∧ωn−m has support in\n{Pω,m(min(u, v)) = min(u, v)} = {Pω,m(min(u, v)) = u < v} ∪{Pω,m(min(u, v)) = v}.\nWe therefore obtain that\n(ω + ddcφ)m ∧ωn−m ≤1{φ=u 0 such that ∥f∥χ ≤C0\n´\nX χ(f)dVX for all f ∈Lχ.\n2.1. Local L∞-bound of Monge–Amp`ere operators. Let Ωbe the unit ball in Cn. We\ndefine g :=\nf\n∥f∥χ to normalize f. We now consider the following Dirichlet problem to find a\nplurisubharmonic solution u:\n(ddcu)n = gdVX, in Ω;\nu = 0, on ∂Ω.\n(LocMA)\nWe now establish a local uniform estimate for u.\nWe start by establishing a useful re-\nsult, which, for K¨ahler manifold, was shown in [DDL21, Theorem 3.3] and [GL25b, Lemma\n2.2]. We present here a local version. Its proof is simpler than that of the global case, as\nplurisubharmonicity is preserved under addition.\nLemma 2.1. Assume that f ∈Lp(Ω, dVX) for some p > 1, and that u is normalized so that\nsupΩu = 0 and u = 0 on ∂Ω. Suppose that there exists a bounded function v ∈PSH(Ω)\nnormalized so that supΩv = 0 and v = 0 on ∂Ω, such that\n(ddcu)n ≤(ddcv)n + fdVX,\nthen u ≥−C, where C = C(n, p, ∥f∥p, ∥v∥∞).\nProof. Let ρ ∈PSH(Ω) be the solution to the Monge–Amp`ere equation\n(ddcρ)n = fdVX in Ω; ρ = 0 on ∂Ω.\nThe uniform bound for ρ can be deduced from the results in [Ko l98,GL23]. Consider now\nψ = v + ρ. It is sufficient to show that ψ ≤u. Indeed, ψ satisfies\n(ddcψ)n ≥(ddcv)n + (ddcρ)n = (ddcv)n + fdVX ≥(ddcu)n.\nIt thus follows from the Comparison Principle (Proposition 1.7) that u ≥ψ on Ω.\n□\nWe are now prepared to prove a local uniform estimate.\nLemma 2.2 (Local uniform estimate for Monge–Amp`ere equations). Assume that u is a\nbounded plurisubharmonic solution to the Dirichlet problem (LocMA). Then\nOscΩu ≤C,\nwhere C = C(n, Ω).\nProof. Let v ∈F(Ω) be such that\n(ddcv)n = χ (g) dVX, in Ω; v = 0, on ∂Ω.\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n11\nThe existence of v ∈F(Ω) is ensured by Theorem 1.11. Let α > 0 be fixed (to be specified\nlater). On the set {log (g + 3) < −αv} we have\n(ddcu)n ≤(e−αv −3)dVX.\nWe are going to show that e−αv ∈L2. To see this, observe that\nˆ\nΩ\n(ddcv)n =\nˆ\nΩ\nχ (g) dVX ≤χ(1).\n(2.1)\nWe now choose α =\n1\nχ(1). It follows from [˚ACK+09, Theorem B] that\nˆ\nΩ\ne−2αv ≤C0(n, Ω).\nWe proceed to bound 1{log(g+3)≥−αv}(ddcu)n. Let Θ : R−→R−be a convex increasing\nfunction. Then Θ ◦v satisfies\nΘ ◦v ∈PSH(Ω), and (ddc(Θ ◦v))n =\n\u0000Θ′′(v)dv ∧dcv + Θ′(v)ddcv\n\u0001n ≥(Θ′ ◦v)n(ddcv)n.\nOn the set {log (g + 3) ≥−αv}, we deduce that\n(ddc(Θ ◦v))n ≥(Θ′ ◦v)n(ddcv)n ≥\n\u0012\nΘ′\n\u0012−log (g + 3)\nα\n\u0013\u0013n\nχ (g) dVX.\n(2.2)\nWe aim to construct a uniformly bounded function Θ such that, for every t > 1, the following\ninequality is satisfied:\n\u0012\nΘ′\n\u0012−t\nα\n\u0013\u0013n\ntn(h ◦log t)n ≥2.\n(2.3)\nTo see this, let x = log t and let\nb(x) = −Θ\n\u0012−ex\nα\n\u0013\n.\nThen the condition (2.3) is equivalent to b′(x)h(x) ≥21/n\nα , for all x > 0. We define b by\nb(x) :=\nˆ x\n0\n21/n\nαh(s)ds.\nSince h−1 is integrable, the function b(x) remains bounded, which in turn implies that Θ is\nuniformly bounded, as claimed. Taking t = log (g + 3) in (2.3) yields that\n\u0012\nΘ′\n\u0012−log (g + 3)\nα\n\u0013\u0013n\n(log (g + 3))n (h ◦log log (g + 3))n ≥2.\n(2.4)\nWithout loss of generality, we assume that there exist k0 > 0 such that\nχ(k) ≥1\n2k logn(k + 3)(h ◦log ◦log(k + 3))n, ∀k ≥k0.\nFrom (2.2) and (2.4), it follows that on the set {log (g + 3) ≥−αv} ∩{g ≥k0},\n(ddc(Θ ◦v))n ≥gdVX = (ddcu)n.\nOn the other hand, on the set {log (g + 3) ≥−αv} ∩{g < k0}, we deduce that\n(ddcu)n < k0dVX.\nFrom the preceding arguments, we arrive at the result that\n(ddcu)n ≤(e−αv + k0)dVX + (ddc(Θ ◦v))n .\n(2.5)\n\n12\nYUETONG FANG\nIt follows from Lemma 2.1 that OscΩu ≤C(n, Ω).\n□\n2.2. Global L∞- bounds. We now provide a proof of the global L∞bound for the solution\nto the Monge–Amp`ere equations on Hermitian manifolds, that only relies on local solutions.\nThe strategy is to construct a bounded subsolution by solving equations on each small ball,\nand then to show that the uniform boundedness of these subsolutions implies a uniform\nbound for the solution, using domination principle. This method is adapted from the work\nof Guedj–Lu, in [GL23]. We start by establishing a subsolution lemma.\nLemma 2.3 (Subsolution Lemma). Assume χ satisfies Condition (K). Then there exist a\nuniform constant s = s(χ) > 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω),\n−1 ≤ψ ≤0, with\n(ω + ddcψ)n ≥s(χ)\ng\n∥g∥χ\ndVX.\nProof. Cover X by balls, X ⊂∪N\nj=1Bj. Choose B′\nj = {ρj < 0}, where ρj : X →R are\nsmooth functions such that ω + ddcρj > 0 in a neighborhood of B′\nj, and such that Bj ⋐B\n′\nj.\nIn B′\nj, let vj ∈PSH(B′\nj) be solution to Monge–Amp`ere equation\n(ddcvj)n =\ng\n∥g∥χ\ndVX in B′\nj; vj = −1 on ∂Ω.\nThe local L∞-estimate (Lemma 2.2) implies OscB′\njvj ≤C′\nj. We define ψj := max(vj, ξjρj),\nwhere ξj > 1 for all j. Thus, ψj coincides with vj in Bj, and coincides with ξjρj in X \\ B′\nj\nand in the neighborhood of ∂B′\nj. Moreover, ψj is δ−1ω-plurisubharmonic for some uniform\nconstant δ > 0 in Bj. Taking ψ := 1\nN\nPN\nj=1 δψj, and taking 0 < s(χ) ≤δn\nNn , we infer that in\nBj,\nωn\nψ ≥δn\nNn (ω + ddcψj)n ≥s(χ)(ddcvj)ndVX = s(χ)\ng\n∥g∥χ\ndVX.\nTherefore, ψ is the subsolution that we are looking for.\n□\nTheorem 2.4 (A priori estimates for Monge–Amp`ere equations). Let χ be as above, and\nlet f ∈Lχ with f ≥0. Suppose that φ ∈PSH(X, ω) ∩L∞(X) satisfies\n(ω + ddcφ)n ≤fdVX.\nLet s := s(χ) be the constant in Lemma 2.3 and assume that A > 0 satisfies\n∥1{f>A}f∥χ ≤s(χ)\n4 .\nThen OscX(φ) ≤C, where C depends on A, and χ.\nProof. We consider the function\ng =\n1{f>A}f\n∥1{f>A}f∥χ\n.\nApplying the Subsolution Lemma (Lemma 2.3) to g yields that there exists subsolution\nψ ∈PSH(X, ω), such that\n(ω + ddcψ)n ≥sgdVX.\nThe dominated convergence theorem implies that\nlim\nt→+∞\nˆ\nX\nχ(1{f>t}f)dVX =\nˆ\nX\nlim\nt→+∞χ\n\u00001{f>t}f\n\u0001\ndVX = 0.\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n13\nWe can thus choose A sufficiently large, depending only on f and s, such that\n∥1{f>A}f∥χ ≤s\n4.\nIt follows that\n(ω + ddcψ)n ≥41{f>A}(ω + ddcφ)n.\nTherefore,\n(ω + ddcφ)n ≤1\n4(ω + ddcψ)n + AdVX.\nFix B > 0 large enough. It follows from [GL23, Lemma 2.5] that there exist ρ ∈PSH(X, ω), −1 ≤\nρ ≤0 such that\n(ω + ddcρ)n ≥4A1{φ<−B}dVX.\nNormalize φ such that supX φ = 0. Then on the set {φ < ψ+ρ\n2\n−B} ⊂{φ < −B}, we have\n(ω + ddcφ)n ≤1\n4(ω + ddcψ)n + 1\n4(ω + ddcρ)n ≤1\n2\n\u0012\nω + ddc ψ + ρ\n2\n\u0013n\n.\nWe conclude from the domination principle (Proposition 1.8) that φ ≥ψ+ρ\n2 −B ≥−1−B.\n□\n2.3. Global L∞- estimates for Hessian equations. The uniform bound for the Hessian\nequation follows the method of [GL25a, Theorem 3.1], which assumes f ∈Lp with p > n/m.\nA generalization to Orlicz spaces is outlined in Remark 3.2 therein. It has also been shown in\nthe recent article [GL25b, Theorem B] that this approach can be adapted in certain general\ncomplex geometric PDEs. We provide a more detailed presentation in the proof of Theorem\nB here.\nThe following lemma will be used to obtain a uniform estimate for bounded solutions.\nLemma 2.5. Assume f ∈L1(dVX). Let φ ∈SHm(X, ω) ∩L∞(X) be such that\n(ω + ddcφ)m ∧ωn−m = fωn.\nThen, we have (ω + ddcu)n ≤fn/mωn, where u = Pω(φ).\nProof. The proof proceeds in two steps.\nWe first treat the case where the density f is\nbounded. In the general case, we approximate a given f ∈L1 by a sequence of bounded\ndensities. Normalize the solution φ such that supX φ = 0.\nStep 1. Assume that f is bounded. Let {fj} be a sequence of smooth, uniformly bounded\nfunctions such that fj →f in L2n. It follows from [KN16, Lemma 3.20] that there exists a\ndecreasing sequence of (ω, m)-subharmonic functions {ψj} such that ψj →φ. We consider\nthe following complex Hessian equation:\n(ω + ddcφj)m ∧ωn−m = eφj−ψjfjωn.\nThe existence of smooth, (ω, m)-subharmonic solution φj follows from [Sz´e18]. Then, from\n[KN16, Theorem 3.9] we deduce that φj converges uniformly to eφ ∈SHm(X, ω) ∩C0(X),\nwhich is the solution to\n(ω + ddc eφ)m ∧ωn−m = eeφ−φfωn.\nBy the domination principle (Proposition 1.8), we obtain that eφ = φ. Taking uj = Pω(φj),\nit follows from Proposition 1.17 that uj ∈C1,1(X). The mixed Monge–Amp`ere inequality\n[Ngu16, Lemma 1.19] implies that\n(ω + ddcuj)n ≤e\nn(φj−ψj)\nm\nfn/m\nj\nωn.\n\n14\nYUETONG FANG\nSince uj converges uniformly to u, passing to the limit, we conclude that\n(ω + ddcu)n ≤fn/mωn.\nStep 2. Assume now that f ∈L1(dVX). We consider the equation given by\n(ω + ddcφj)m ∧ωn−m = eφj−φ min(f, j)ωn.\nTheorem 3.19 of [KN15] ensures the existence of a bounded solution φj, satisfying φj ≥φ.\nThe domination principle yields that φj is decreasing to a (ω, m)-subharmonic function eφ.\nIt follows from the domination principle again that eφ = φ. Taking uj := Pω(φj), then uj\ndecreases to u. Moreover, Step 1 yields\n(ω + ddcuj)n ≤en/m(φj−φ)fn/m\nj\nωn.\nApplying the continuity of Monge–Amp`ere operator along monotone sequences [BT76,BT82],\nwe obtain\n(ω + ddcu)n ≤fn/mωn,\nas required.\n□\nProof of Theorem B. Normalize φ such that supX φ = 0. Let u = Pω (φ) and let C denote\nthe contact set C := {u = φ}. We remark that u ∈PSH(X, ω) ⊂SHm(X, ω). Therefore, it\nfollows from the mixed Monge–Amp`ere inequality [Ngu16, Lemma 1.19] and proposition 1.17\nthat\n1C(ω + ddcu)m ∧ωn−m ≤1C(ω + ddcφ)m ∧ωn−m ≤fdVX\nWe remark that hX := ωn/dVX ≥δ0 is positive on X. Lemma 2.5 ensures us to take\ng := (ω + ddcu)n /dVX. Then, the mixed Monge–Amp`ere inequality yields that\nδ1−m/n\n0\ngm/n ≤h1−m/n\nX\ngm/n ≤f,\nhence g ≤C0fn/m, so that g ∈Lχ0, where χ0 = t logn(1 + t)[h ◦log ◦log(3 + t)]n satisfies\nCondition (K). It follows from Theorem 2.4 that\nOscX(u) ≤A.\nIt suffices to bound supX u from below. Set\nv−\nA(ω) := inf\n\u001aˆ\nX\n(ω + ddcφ)n : φ ∈L∞(X) ∩PSH(X, ω), −A ≤φ ≤0\n\u001b\n.\nRecall that v−\nA(ω) is strictly positive by [GL22, Proposition 3.4]. We denote by (χ∗\n0)−1 the\ninverse of χ∗\n0. Then it follows from additive H¨older–Young inequality (Proposition 1.16) that\nv−\nA(ω)(χ∗\n0)−1\n\u0012\n−sup\nX\nu\n\u0013\n≤\nˆ\nX\n(χ∗\n0)−1(−u) (ω + ddcu)n\n=\nˆ\nX\n(χ∗\n0)−1(−u)gdVX\n≤\nˆ\nX\nχ0(g)dVX +\nˆ\nX\n(−u)dVX.\nWe remark that\n´\nX(−u)dVX is uniformly bounded from above by Lemma 1.5. Thus we\nobtain a uniform lower bound of supX u, which yields together with OscXu ≤A that |u| has\na uniform bound, and since u ≤φ, the solution φ is also uniformly bounded.\n□\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n15\n3. Weak solution\n3.1. Stability estimates.\nTheorem 3.1. Let u, v ∈SHm(X, ω)∩L∞(X) be such that supX u = 0 and v ≤0. Assume\nthat u ∈SHm(X, ω) is a solution to the complex Hessian equation\nωm\nu ∧ωn−m = eλufωn,\n(3.1)\nwhere λ ≥0 and fn/m ∈Lχ with χ satisfying Condition (K). Then,\nsup\nX\n(v −u)+ ≤C∥(v −u)+fn/m∥\n1\nn+1\nχ\n,\nwhere the positive constant C depends only on n, m, ω, ∥f\nn\nm ∥χ, and supX |v|.\nProof. We consider only the case λ = 0; when λ > 0, the L∞-estimate implies that eλuf ≤f,\nallowing the problem to be reduced to the case λ = 0. We now fix ϵ > 0, to be chosen later,\nand define gϵ := 1{u u + C2δ + ϵ} ⊂{u < v −ϵ}.\nFrom the (ω, m)-subharmonicity of v and ψ, and applying [KN15, Lemma 2.3], we derive\nthat\n1{φ>u+C2δ+ϵ}ωm\nφ ∧ωn−m =1{φ>u+C2δ+ϵ} [(1 −δ)ωv + δωψ]m ∧ωn−m\n=1{φ>u+C2δ+ϵ}\nm\nX\nk=0\n\u0012m\nk\n\u0013\n[(1 −δ)ωv]k ∧(δωψ)m−k ∧ωn−m\n≥1{φ>u+C2δ+ϵ}δm(ωψ)m ∧ωn−m.\nIf ∥gϵ∥χ is not small, then the stability property follows trivially by adjusting C. Hence, we\ncan assume that ∥gϵ∥χ <\ns\n2n/m . Choosing δ = 21/m∥gϵ∥1/n\nχ\ns1/n\n, we obtain that\n1{φ>u+C2δ+ϵ}δm(ωψ)m ∧ωn−m ≥21{φ>u+C2δ+ϵ}fωn\n=21{φ>u+C2δ+ϵ}ωm\nu ∧ωn−m.\nIt follows from domination principle (Proposition 1.8) that\nφ ≤u + C2δ + ϵ.\n\n16\nYUETONG FANG\nHence,\n(v −u)+ ≤C3\n \n21/m\ns1/n\n!\n∥gϵ∥1/n\nχ\n+ ϵ\n≤C4∥gϵ∥1/n\nχ\n+ ϵ\n≤C4(ϵ)−1/n∥(v −u)+fn/m∥1/n\nχ\n+ ϵ.\nOptimizing the right hand side by choosing\nϵ =\n \n∥C4(v −u)+fn/m∥1/n\nχ\nn\n!\nn\nn+1\n,\nwe obtain\n(v −u)+ ≤C\n\u0010\n∥(v −u)+fn/m∥χ\n\u0011\n1\nn+1 .\nThe conclusion follows by taking the supremum of the left-hand side.\n□\nRemark 3.2. When χ(t) = tp/p, with p > 1, the Orlicz norm ∥· ∥χ is simply the Lebesgue\nLp-norm. Assume fn/m ∈Lp, for p > 1. Let χq(t) = tq/q for some 1 < q < p. Then\nψ ∈PSH(X, ω) is a subsolution such that\n(ω + ddcψ)n ≥s′gωn, where g =\n1{u 0 such that\n(ω + ddcuj)m ∧ωn−m = cjeλujfjdVX, λ ≥0.\nWe normalized uj such that supX uj = 0. Our first step is to bound the coefficients cj from\nabove and below. This requires an inequality involving mixed Hessian measures. It’s well\nknown that if φj ∈SHm(X, ω)∩C∞(X) and (ω +ddcφj)m ∧ωn−m = fjdVX, then G˚arding’s\ninequality [G˚ar59] holds:\n(ω + ddcφ1) ∧· · · ∧(ω + ddcφm) ∧ωn−m ≥(f1 · · · fm)1/mdVX.\nIn the case where φj’s are not smooth, we can apply an approximation argument to obtain\nthe following inequality.\nProposition 3.4. Let u ∈SHm(X, ω) be bounded functions such that\n(ω + ddcφ)m ∧ωn−m ≥fωn.\nThen we have that (ω + ddcφ) ∧ωn−1 ≥f1/mωn.\nProof. Let {gj} be a sequence of bounded, positive and smooth functions converging to f.\nLet vj be a sequence of smooth, (ω, m)-subharmonic functions converging to φ. We consider\nthe following Hessian equation:\n(ω + ddcφj)m ∧ωn−m = eφj−vjgjωn.\nThe existence of smooth solution φj follows from [Sz´e18]. Applying G˚arding’s inequality\n[G˚ar59] yields\n(ω + ddcφj) ∧ωn−1 ≥e\nφj−vj\nm\ng1/m\nj\nωn.\nMoreover, we deduce from [KN16, Theorem 3.19] that φj converges uniformly to eφ ∈\nSHm(X, ω). Passing to the limit as j →+∞, we obtain\n(ω + ddceu)m ∧ωn−m = eeφ−φfωn.\nThe domination principle (Proposition 1.8) then implies eφ = φ. Consequently, we conclude\nthat\n(ω + ddcφ) ∧ωn−1 ≥f1/mωn.\n□\nWe now turn to obtaining uniform bounds for cj.\nLemma 3.5. The sequence of constant {cj} is uniformly bounded. In particular, the Orlicz\nnorm ∥cjfj∥χ is uniformly bounded from above.\nProof. We first bound cj from above. Fix δ0 > 0 such that δ0 ≤ωn/dVX. Proposition 3.4\nimplies that for each j,\n(ω + ddcuj) ∧ωn−1 ≥c1/m\nj\nf1/m\nj\nωn ≥δ0c1/m\nj\nf1/m\nj\ndVX.\n\n18\nYUETONG FANG\nThus, by taking the integral of both sides:\nδ0c1/m\nj\nˆ\nX\nf1/m\nj\ndVX ≤\nˆ\nX\n(ω + ddcuj) ∧ωn−1.\nNote that uj ∈SHm(X, ω) ⊂SH1(X, ω), and the sequence (uj) is relatively compact in L1.\nMoreover, we have\n´\nX f1/m\nj\nωn →\n´\nX f1/mωn > 0 as j →+∞. It follows that there exist j0\nlarge enough such that for each j > j0, the following inequality holds:\nc1/m\nj\n≤\n2\nδ0\n´\nX f1/mdVX\nˆ\nX\n(ω + ddcuj) ∧ωn−1.\nIt suffices to bound the right hand side. Indeed, it follows from Stokes’ theorem that\nˆ\nX\n(ω + ddcuj) ∧ωn−1 =\nˆ\nX\nωn +\nˆ\nX\nujddc(ωn−1) ≤\nˆ\nX\nωn + B\nˆ\nX\n|uj|ωn.\nBy the L1-compactness result for (ω, 1)-subharmonic functions (Lemma 1.5), we conclude\nthat the right hand side above is bounded. Thus cj is bounded from above.\nWe next show that {cj} is bounded away from 0. Let gj = fn/m\nj\n∈Lχ. By the subsolution\nLemma (Lemma 2.3), there exists a uniform constant δ > 0, such that\n(ω + ddcvj)n ≥δjgjdVX.\nThen, it follows from mixed Monge–Amp`ere inequality [Ngu16, Lemma 1.9] that\nωm\nvj ∧ωn−m ≥δm/n\nj\ngm/n\nj\nωn ≥δm/n\nj\nfjωn.\nThe domination principle (Proposition 1.8) yields that\nδm/n\nj\n≤cj.\nWe thus obtain a uniform lower bound for cj’s, since δj’s are strictly positive, uniform\nconstants.\nThe uniqueness of cj for each fj follows directly from domination principle\n(Proposition 1.8).\n□\nUniform bound of uj. It follows from Theorem B that {uj} have uniform bound:\n−C ≤uj ≤0.\nBy extracting, we can assume that {uj} converges in L1(X) to a bounded (ω, m)-subharmonic\nfunction u, as j →+∞.\nExistence of solutions. We now show that, by construction, the limit u is a solution to\nthe Hessian equation\n(ω + ddcu)m ∧ωn−m = ceλufdVX, λ ≥0.\nWe first consider the simpler case in which λ > 0.\nTheorem 3.6. Let λ > 0. Then u ∈SHm(X, ω) ∩L∞(X) is the unique solution such that\n(ω + ddcu)m ∧ωn−m = eλufdVX.\nMoreover u is uniformly bounded.\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n19\nProof. To simplify the notation, we assume that λ = 1. Then each uj satisfies\n(ω + ddcuj)m ∧ωn−m = eujfjdVX.\nFrom the domination principle (Proposition 1.8), we deduce that uj decreases to some\nfunctions u ∈SHm(X, ω). Passing to the limit, it follows from the continuity of Hessian\noperator along decreasing sequence ( [KN23, Section 5]) that\n(ω + ddcu)m ∧ωn−m = eufdVX.\nApplying Theorem B, we obtain a uniform bound for u.\n□\nMonotonicity of the sequence uj is not guaranteed when λ = 0.\nOur approach is to\nconstruct a supersolution and a subsolution and demonstrate that they coincide, as in [GL24,\nSection 3] and [BGL25, Theorem 4.5].\nTheorem 3.7 (Existence of bounded solution). Let f, u and the constant c be as above.\nThen u ∈SHm(X, ω) ∩L∞(X) is a solution to\n(ω + ddcu)m ∧ωn−m = cfdVX.\nMoreover, u is uniformly bounded.\nProof. We define\nρj :=\n \nsup\nk≥j\nuk\n!∗\n, vj := Pω,m\n\u0012\ninf\nk≥j uk\n\u0013\n,\nwhere Pω,m (infk≥j uk) denotes the largest (ω, m)-subharmonic function lying below infk≥j uk.\nObserve that ρj is a decreasing sequence that converges to u, and vj is an increasing se-\nquence that converges to an (ω, m)-subharmonic function v. In particular, by construction,\nwe have u ≥v. We are going to show that u = v.\nSet c−\nj f−\nj := infk≥j ckfk. Then it follows from Lemma 1.18 that\n(ω + ddcρj)m ∧ωn−m ≥c−\nj f−\nj dVX.\nThen we deduce from the continuity of Hessian operator along monotone sequences (see\n[KN23, Lemma 5.1]) that\n(ω + ddcu)m ∧ωn−m ≥cfdVX.\n(3.2)\nOn the other hand, we write (ω + ddcvj)m ∧ωn−m = evj−vjcjfjdVX.\nDefine c+\nj f+\nj\n:=\nsupk≥j ckfk. It also follows from Lemma 1.18 that\n(ω + ddcvj)m ∧ωn−m ≤evj−infk≥j ukc+\nj f+\nj dVX.\nLet j →+∞. Together with (3.2) we deduce that\n(ω + ddcv)m ∧ωn−m ≤ev−ucfdVX ≤ev−u(ω + ddcu)m ∧ωn−m.\nBy the domination principle (Proposition 1.8), we conclude that v ≥u. Hence, v = u is a\nbounded solution to the complex Hessian equation:\n(ω + ddcu)m ∧ωn−m = cfdVX.\nThe uniform bound for u follows from Theorem B.\n□\n\n20\nYUETONG FANG\nProof of Theorem C. Existence of continuous solutions. We now prove that the sequence\n{uj} converges uniformly to u as j →+∞. We recall that when λ > 0, the L∞-estimate\nimplies that eλuf ≤f, allowing us to reduce to the case where λ = 0. Fix j0 large enough.\nFor each j > j0, we infer by Theorem 3.1 that\n(uj −u)+ ≤C1\n\r\r\r(uj −u)+f\nn\nm\n\r\r\r\n1\nn+1\nχ\n, (u −uj)+ ≤C1\n\r\r\r(u −uj)+f\nn\nm\nj\n\r\r\r\n1\nn+1\nχ\n.\nWe are going to show that\n\r\r\r(uj −u)+f\nn\nm\n\r\r\r\nχ →0 as j →+∞. Fixing s > 0, we proceed to\ncompute the corresponding integrability.\nˆ\nX\nχ\n\u0012(uj −u)+f\nn\nm\ns\n\u0013\ndVX\n=\nˆ\nX\n\u0012(uj −u)+f\nn\nm\ns\n\u0013\nlogn\n\u0012\n1 + (uj −u)+f\nn\nm\ns\n\u0013 \u0012\nh ◦log ◦log\n\u0012\n3 + (uj −u)+f\nn\nm\ns\n\u0013\u0013n\ndVX.\nAs u and uj’s are uniformly bounded, it follows that (uj −u)+ ≤C0 for some uniform constant C0,\nand since h is increasing, we deduce that\nˆ\nX\nχ\n\u0012(uj −u)+f\nn\nm\ns\n\u0013\ndVX ≤\nˆ\nX\n\u0012(uj −u)+f\nn\nm\ns\n\u0013\nlogn\n\u0012\n1 + C0f\nn\nm\ns\n\u0013\n·\n\u0012\nh ◦log ◦log\n\u0012\n3 + C0f\nn\nm\ns\n\u0013\u0013n\ndVX.\nBy the dominated convergence theorem, we conclude that\n\r\r\r\r\n(uj −u)+f\nn\nm\ns\n\r\r\r\r\nχ\n→0, for all 0 < s < +∞,\nand consequently\n∥(uj −u)+f\nn\nm ∥\n1\nn+1\nχ\n→0.\nApplying the same argument, we obtain that\n\r\r\r(u −uj)+f\nn\nm\nj\n\r\r\r\n1\nn+1\nχ\n≤\n\r\r(u −uj)+f\nn\nm \r\r\n1\nn+1\nχ\n→0. There-\nfore, {uj}∞\nj=1 is a Cauchy sequence in SHm(X, ω) ∩C0(X). It follows that the bounded solution u\nto the complex Hessian equation\n(ω + ddcu)m ∧ωn−m = ceλufωn, where f ∈Lχ,\nis continuous.\nContinuity of bounded solutions. Assume now that v ∈SHm(X, ω) is another bounded solution\nto the complex Hessian equation\n(ω + ddcv)m ∧ωn−m = cfdVX.\nThen v is continuous. Indeed, the uniform bound for v ensures that ef = e−vf ∈Lχ. We deduce\nfrom the previous argument that there exist eu ∈SHm(X, ω) ∩C0(X) solving\n(ω + ddceu)m ∧ωn−m = eeu efdVX.\nIt follows that\n(ω + ddceu)m ∧ωn−m = eeu−vfdVX = eeu−v(ω + ddcv)m ∧ωn−m.\nApplying the Domination Principle (Proposition 1.8) yields that v = eu. Therefore, v ∈C0(X), as\nclaimed.\n□\n\nCONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS\n21\nReferences\n[˚AC25]\nPer ˚Ahag and Rafa l Czy˙z. Continuity of solutions to complex Hessian equations via the\ndinew–Ko lodziej estimate. Ann. Fenn. Math., 50:201–214, 2025.\n[˚ACK+09]\nPer ˚Ahag, Urban Cegrell, S lawomir Ko lodziej, Hoang-Hiep Pha.m, and Ahmed Zeriahi. 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Math., 31(3):339–411, 1978.\nUniversit´e d’Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France\nEmail address: yuetong.fang@univ-angers.fr\n"},"cleaned":{"kind":"string","value":"CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS ON COMPACT HERMITIAN MANIFOLDS YUETONG FANG Abstract. Let (X, ω) be a compact Hermitian manifold of dimension n. We derive an L∞-estimate for bounded solutions to the complex m-th Hessian equations on X, assuming a positive right-hand side in the Orlicz space L n m (log L)n(h◦log ◦log L)n, where the associated weight satisfies Ko lodziej's Condition. Building upon this estimate, we then establish the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions. Introduction Let (X, ω) be a compact complex Hermitian manifold of dimension n equipped with a Hermitian metric ω. Fix 0 0 and continuous (ω, m)-subharmonic solutions to the following complex m-Hessian equation (ω + ddcu)m ∧ωn-m = ceλufdVX, λ ≥0, (Hessian) under the assumption that the densities fn/m belong to an Orlicz space Lχ with the weight χ satisfying Ko lodziej's condition, which will be explained later. When m = 1, this simplifies to the Poisson equation, while when m = n, it reduces to the Monge-Amp`ere equation. Yau's solution [Yau78] of the Calabi Conjecture has motivated many subsequent studies on Monge-Amp`ere equations over the past fifty years. As presented by B locki [B lo12], using the simplifications due to Kazdan and to Aubin-Bourguignon, Yau's a priori estimate can be extended to cases where the densities f belong to Lp for p > n. One of the breakthrough was achieved by Ko lodziej [Ko l98], who applied pluripotential theory to obtain an L∞-estimate for Monge-Amp`ere equations when f belongs to some Orlicz space, with the associated weight specifically satisfying Condition (K). As remarked by Ko lodziej [Ko l98], such an a priori estimate is \"almost sharp\". This integrability condition on densities f can be extended in the context of complex Hessian equations. We say that a convex weight χ satisfies Ko lodziej's condition (Condition (K)) if it is increasing and χ(t) ∼ t→+∞t(log(t + 1))n (h ◦log ◦log(t + 3))n , where the function h : R+ →R+ satisfying ́ +∞h-1(t)dt n, as noted in [GL24, Section 2]. More recently, they generalized this result to Orlicz spaces satisfying Condition (K) on compact K ̈ahler manifolds in [GL25b], using a new approach that reduces Ko lodziej's criterion to Yau's theorem. On the other hand, complex m-Hessian equations are natural generalizations of MongeAmp`ere equations. B locki [B lo05] established potential theory for m-Hessian equations in domains of Cn. Later, Dinew-Ko lodziej [DK14] obtained a priori estimates and proved a stability result for weak solutions of the complex Hessian equation in domains of Cn and compact K ̈ahler manifolds. The modulus of continuity of solutions to complex Hessian equations has been studied in [Ngu14,Cha16] with densities f ∈Lp, p > n/m, and has been extended recently in [ ̊AC25] to the cases where f ∈Ln/m logp(L), p > 2n. On compact Hermitian manifolds, Ko lodziej and Nguyen proved the existence of weak solutions with a nonnegative right-hand side belonging to Lp, p > n/m, in [KN16], while the corresponding result for Monge-Amp`ere equations was established in [KN15]. The purpose of this paper is to extend the above results to prove the continuity of bounded solutions to the complex Hessian equation. We begin by establishing a subsolution lemma for the Monge-Amp`ere equation, generalizing Lemma 2.1 of [GL23] to the case where f ∈Lχ, with χ satisfying Condition (K). The lemma is stated as follows: Lemma A (Subsolution Lemma). Assume χ satisfies Condition (K). Then there exist a uniform constant s = s(χ) > 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω), -1 ≤ψ ≤0, satisfying (ω + ddcψ)n ≥s(χ) g ∥g∥χ dVX. We obtain the Subsolution Lemma by gluing the solutions to the Dirichlet problem in local context: (ddcu)n = f ∥f∥χ dVX, in Ω; u = 0, on ∂Ω. (0.1) Compared with the case where the densities f ∈Lp, working with Orlicz norms is more involved, as it requires controlling the Orlicz norm of the solution u associated to the convex CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 3 conjugate of χ. To simplify the argument, we adapt an approach from the recent article by Guedj-Lu [GL25b] to derive a local L∞-estimate for problem 0.1. As a consequence of Lemma A, we obtain a uniform estimate for the complex Hessian equation on a Hermitian manifold, where the densities f satisfy Ko lodziej's optimal condition. Theorem B (L∞-estimate). Let φ be a bounded (ω, m)-subharmonic function satisfying (ω + ddcφ)m ∧ωn-m ≤fdVX, (Hess) where 0 ≤fn/m ∈Lχ with the weight χ satisfying Condition (K). Then OscX(φ) ≤C, where C depends on ω, n, m and f n m χ. By generalizing an idea in [GL25a], we demonstrate that the L∞-estimate for complex Hessian equations can be reduced to the corresponding estimate for an associated MongeAmp`ere equation: (ω + ddcφ)n ≤fdVX, (0.2) where f ∈Lχ, and χ satisfies Condition (K). To obtain the L∞-estimate for the MongeAmp`ere equation, our main tool is the domination principle. The key novelty of our approach lies in constructing an appropriate subsolution. Our approach avoids potential computational difficulties associated with Orlicz norms and presents a different method compared to Ko lodziej's original proof. Once the L∞-estimate is obtained, we can derive the existence and regularity of bounded solutions. This leads us to the second main result of the study: Theorem C. Let 0 ≤fn/m ∈Lχ, with the weight function χ satisfying Condition (K). Then there exist a constant c > 0 and a solution u ∈SHm(X, ω) ∩C0(X) satisfying the complex Hessian equation (ω + ddcu)m ∧ωn-m = ceλufdVX, λ ≥0. Moreover, all bounded solutions to the above complex Hessian equation are continuous. To prove the first statement of Theorem C, an essential component is to establish a stability theorem (Theorem 3.1). For f ∈Lp, with p > n/m, such a stability result was proven in [KN16]. In the case where f ∈Lχ and the weight function χ satisfies Condition (K), the continuity of solutions to Monge-Amp`ere equations was studied in [KN21] by adapting the method in [KN19, Theorem 3.1]. We point out that our approach to the stability estimate is fundamentally different and remains of interest in the case where the densities f ∈Lp, as discussed in Remark 3.2. Our main tool will be the Domination Principle and the subsolution (Lemma A). Having established the stability estimate (Theorem 3.1), we approximate the density f by the sequence of bounded functions min(f, j). We show that the corresponding solutions uj converge uniformly to an (ω, m)-subharmonic function u solving the equation (Hessian), thereby obtaining a continuous solution. The continuity of all bounded solutions follows from the Domination Principle (Proposition 1.8). 4 YUETONG FANG Organization of the paper. The paper is organized as follows. Section 1 gives preliminary material on (ω, m)-subharmonic functions, as well as key concepts of Orlicz spaces. A local L∞-estimate is established, and Lemma A and Theorem B are proved in Section 2. In section 3, we establish a stability estimate and subsequently investigate the continuity of bounded solutions, leading to the proof of Theorem C. Acknowledgment. The author thanks her supervisor, Hoang-Chinh Lu, for suggesting the problem, valuable discussions during the preparation of the work, and useful comments that improved the presentation. This research is part of a PhD program funded by the PhD scholarship CSC-202308070110, and is partially supported by the projects Centre Henri Lebesgue ANR-11-LABX-0020-01 and the KRIS project of Fondation Charles Defforey. 1. Preliminaries In the whole paper, we let (X, ω) denote a compact Hermitian manifold of complex dimension n ∈N∗, equipped with a Hermitian form ω. We denote by dVX a smooth volume form on X. We use differential operators d = ∂+ ̄∂, and dc = i( ̄∂-∂), so that ddc = 2i∂ ̄∂. 1.1. (ω, m)-subharmonic functions. 1.1.1. Definition. We recall the definition of (ω, m)-subharmonic functions following [GN18] and briefly review some results related to both ω-plurisubharmonic and (ω, m)-subharmonic functions that will be used later. Fix an integer 1 ≤m ≤n. Fix Ωan open set in Cn. Given a hermitian metric ω on Ω, a real (1, 1)-form α on X is called m-positive with respect to ω if at all points in X, αk ∧ωn-k ≥0, ∀k = 1, · · · , m. A C2(Ω) function u : Ω→R is called harmonic with respect to ω if ddcu ∧ωn-1 = 0 at all points in Ω. Definition 1.1. A function u : Ω→{-∞} ∪R is subharmonic with respect to ω if: (a) u is upper semicontinuous and u ∈L1 loc(Ω); (b) for every relatively compact open set D ⋐Ωand every function h ∈C0(D) that is harmonic with respect to ω on D the following implication holds: u ≤h on ∂D =⇒u ≤h in D. Definition 1.2. A function φ : Ω→{-∞} ∪R is quasi-subharmonic with respect to ω if locally φ = u + ρ, where u is subharmonic with respect to ω and ρ is smooth. A function φ is α-subharmonic with respect to ω if φ is quasi-subharmonic with respect to ω and (α + ddcφ) ∧ωn-1 ≥0 in the sense of distributions. The positive cone Γm(ω) associated with the metric ω is defined by n γ real (1, 1)-form : γk ∧ωn-k > 0, k = 1, · · · , m o . It follows from G ̊arding's inequality [G ̊ar59] that if γ0, γ1, · · · , γm-1 ∈Γm(ω), then γ0 ∧γ1 ∧· · · ∧γm-1 ∧ωn-m > 0. One can write eω = γ1 ∧· · · ∧γm-1 ∧ω, and it is a strictly positive (n -1, n -1)-form on Ω. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 5 Definition 1.3. A function φ : Ω→[-∞, +∞) is called (α, m)-subharmonic with respect to ω if φ is α-subharmonic with respect to eω in Ωfor all eω of the form eωn-1 = γ1∧· · ·∧γm-1∧ω, where γ1, · · · , γm-1 ∈Γm(ω). A function u : X →R ∪{-∞} is called (α, m)-subharmonic on X if u is (α, m)- subharmonic on each local chart U of X. The set of all locally integrable functions on U which are (α, m)-subharmonic with respect to ω in U is denoted by SHω,m(U, α). In many contexts, such as [GL25a], the forms ω and α may differ. However, in this paper, we focus on the set SHω,m(U, ω). To simplify the notations, we denote by SHm(U, ω) the set of all (ω, m)-subharmonic functions with respect to ω in U. The set of all ω-plurisubharmonic functions on U is denoted by PSH(U, ω) = SHn(U, ω). For notational convenience, we denote ωu := ω + ddcu. Remark 1.4. By G ̊arding's inequality [G ̊ar59], if u ∈C2(X), then u is (ω, m)-subharmonic with respect to ω on X if and only if the associated form ωu belongs to the closure of Γm(ω). It is well known that when n = m, the set PSH(X, ω) is a closed subset of Lr(X), for all r ≥1. For (ω, m)-subharmonic functions, the following L1-compactness result was proved in [KN16, lemma 3.3]. Lemma 1.5. Let u ∈SHm(X, ω) be normalized by supX u = 0. Then there exists a uniform constant A > 0 depending only on X, ω such that ˆ X |u|ωn ≤A. 1.1.2. Domination principle. Analogous to the Monge-Amp`ere operators, the complex Hessian operators also satisfy a domination principle. The domination principle for Hessian operator associated with continuous (ω, m)-subharmonic functions is established by Guedj-Lu in [GL25a, Section 2]. Following the demonstration that the Hessian operators for bounded functions are well-defined, we show the domination principle associated with bounded (ω, m)- sh functions here. Lemma 1.6 (Maximum principle). Assume that u and v are bounded (ω, m)-subharmonic functions. Then (ω + ddc(max(u, v)))m ∧ωn-m ≥1{u>v}(ω + ddcu)m ∧ωn-m + 1{v≥u}(ω + ddcv)m ∧ωn-m. In particular, if u ≤v, then, 1{v=u}(ω + ddcu)m ∧ωn-m ≤1{v=u}(ω + ddcv)m ∧ωn-m In the case where u and v are continuous, the above lemma is proven in [GL25a, Lemma 1.10]. When u is bounded, thanks to the continuity of complex Hessian operators ( [KN23]), we can approximate u by a sequence of decreasing smooth (ω, m)-subharmonic functions. The comparison principle for u, v ∈SHm(Ω) ∩L∞(Ω) is introduced in [KN23, Section 6]. Proposition 1.7 (Comparison principle). Let u, v be bounded m-subharmonic functions with respect to ω and such that lim infz→∂Ωu(z) -v(z) ≥0. Assume (ddcv)m ∧ωn-m ≥(ddcu)m ∧ωn-m in Ω. Then it follows that u ≥v in Ω. Proof. See [KN23, Corollary 6.2]. □ 6 YUETONG FANG We will need a global version of domination principle: Proposition 1.8 (Domination principle). Assume that u, v ∈PSH(X, ω) ∩L∞(X) satisfy 1{u 0. If e-λvωm v ∧ωn-m ≥e-λuωm u ∧ωn-m, then u ≥v. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 7 Proof. Fix C > 0. On the set {u 0, we conclude that u ≥v. □ Corollary 1.10. Assume u, v are bounded (ω, m)-subharmonic functions on X such that (ω + ddcu)m ∧ωn-m ≤c (ω + ddcv)m ∧ωn-m for some positive constant c. Then c ≥1. 1.1.3. Cegrell classes. In Section 2, we study the Monge-Amp`ere equation in a local context. Here, we recall some essential definitions of the Cegrell classes [Ceg98,Ceg04]. Fix a bounded hyperconvex domain Ω⊂Cn. Let T (Ω) denote the set of bounded plurisubharmonic functions u in Ωsuch that limz→ζ u(z) = 0 for every ζ ∈∂Ω, and ́ Ω(ddcu)n 0, ˆ X χ(sf) 0 : ˆ X χ f r ≤χ(1) . 8 YUETONG FANG We remark that (Lχ(μ), ∥·∥χ,μ) constitutes a normed space and is complete, hence a Banach space. In the remainder of the paper, we fix μ = dVX, a smooth probability measure, and write (Lχ, ∥· ∥χ) for simplicity. In [Ko l98], S.Ko lodziej developped a technique using pluripotential theory that enables one to bound the solution to the Monge-Amp`ere equation by the Orlicz norm, ∥· ∥χ, of the right-hand side, associated with a weaker integrable condition. We thus introduce the Condition (K). Definition 1.13 (Condition (K)). A weight χ : R+ →R+ satisfies condition (K) if it is convex increasing and χ(t) ∼ t→+∞tn/m (log(1 + t))n (h ◦log ◦log(1 + t))n , where ˆ +∞dt h(t) 1, its convex conjugate is given by χ∗(t) = tq q , where q = p p-1 is the H ̈older conjugate of p. The Orlicz norm ∥· ∥χ here is the usual Lp norm. (ii) The weight χ defined in the Condition (K) above is a Young function hence induce an Orlicz space. In particular, the weight χ(t) = t logp(1 + t) with p > n is a Young function. One can check that as t →+∞, the conjugate weight grows like χ∗(t) = t1-1 p exp(t 1 p ). We will focus on the Orlicz space (Lχ, ∥· ∥χ) in section 3. It is important to note that in such an Orlicz space, the H ̈older inequality (or H ̈older-Young inequality) holds. Proposition 1.15 (H ̈older-Young inequality). For f ∈Lχ and g ∈Lχ∗, we have ˆ X |fg|dVX ≤∥f∥χ∥g∥χ∗. Proof. See [Dar19, Proposition 1.3]. □ We will need the following additive Young inequality (also known as Fenchel-Young inequality). Proposition 1.16. For every a, b ∈R+, it holds that ab ≤χ(a) + χ∗(b), with equality if and only if b ∈∂χ(a). In particular, let f, g : X →R+ be two measurable functions. Then f (χ∗)-1 ◦g ≤χ ◦f + g. 1.3. Envelopes. Given a Lebesgue measurable function l : X →R, the ω-plurisubharmonic envelope of l is defined by Pω(l) := sup{u ∈PSH(X, ω) : u ≤l}. The (ω, m)-subharmonic envelope of l is defined by Pω,m(l) := sup{u ∈SHm(X, ω) : u ≤l}. We now summarize the basic properties of these envelopes, following [GL25a]. Proposition 1.17. If l : X →R is continuous, then CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 9 (1) The complex Monge-Amp`ere measure (ω +ddcPω(l))n is concentrated on the contact set {Pω(l) = l}. (2) The complex Hessian measure (ω + ddcPω,m(l))m ∧ωn-m is concentrated on the contact set {Pω,m(l) = l}. (3) If l is C1,1-smooth, then Pω(l) is C1,1-smooth and (ω + ddcPω(l))n = 1{Pω(l)=l}(ω + ddcl)n. (4) If l is C1,1-smooth, then Pω,m(l) is C1,1-smooth and (ω + ddcPω,m(l))m ∧ωn-m = 1{Pω,m(l)=l}(ω + ddcl)m ∧ωn-m. For detailed proofs and additional discussion, we refer the reader to [Ber19,CM21,CZ19, GL22, GLZ19, Tos18]. As a consequence of the preceding results, we will make use of the following statements in the sequel. Lemma 1.18. Fix λ > 0. Let u, v ∈SHm(X, ω) ∩L∞(X). (1) If (ω + ddcu)m ∧ωn-m ≥eλufdVX and (ω + ddcv)m ∧ωn-m ≥eλugdVX, then (ω + ddc(max(u, v)))m ∧ωn-m ≥eλ max(u,v) min(f, g)dVX. (2) If (ω + ddcu)m ∧ωn-m ≤eλufdVX, and (ω + ddcv)m ∧ωn-m ≤eλvgdVX, then (ω + ddc(Pω,m(min(u, v)))m ∧ωn-m ≤eλPω,m(min(u,v)) max(f, g)dVX. The proof follows the same strategy as that for the Monge-Amp`ere measures presented in [GL22, Lemma 1.9]. Proof. Let ψ := max(u, v). As a consequence of the maximum principle (Lemma 1.6), we have (ω + ddcψ)m ∧ωn-m ≥1{u≥v}(ω + ddcu)m ∧ωn-m + 1{v>u}(ω + ddcv)m ∧ωn-m ≥1{u≥v}eλufdVX + 1{v>u}eλvgdVX =1{u≥v}eλψfdVX + 1{v>u}eλψgdVX ≥eλψ min(f, g)dVX. Set φ := Pω,m(min(u, v)). We prove the second statement. It follows from the proposition 1.17 that the complex Hessian measure (ω + ddcφ)m ∧ωn-m has support in {Pω,m(min(u, v)) = min(u, v)} = {Pω,m(min(u, v)) = u 0 such that ∥f∥χ ≤C0 ́ X χ(f)dVX for all f ∈Lχ. 2.1. Local L∞-bound of Monge-Amp`ere operators. Let Ωbe the unit ball in Cn. We define g := f ∥f∥χ to normalize f. We now consider the following Dirichlet problem to find a plurisubharmonic solution u: (ddcu)n = gdVX, in Ω; u = 0, on ∂Ω. (LocMA) We now establish a local uniform estimate for u. We start by establishing a useful result, which, for K ̈ahler manifold, was shown in [DDL21, Theorem 3.3] and [GL25b, Lemma 2.2]. We present here a local version. Its proof is simpler than that of the global case, as plurisubharmonicity is preserved under addition. Lemma 2.1. Assume that f ∈Lp(Ω, dVX) for some p > 1, and that u is normalized so that supΩu = 0 and u = 0 on ∂Ω. Suppose that there exists a bounded function v ∈PSH(Ω) normalized so that supΩv = 0 and v = 0 on ∂Ω, such that (ddcu)n ≤(ddcv)n + fdVX, then u ≥-C, where C = C(n, p, ∥f∥p, ∥v∥∞). Proof. Let ρ ∈PSH(Ω) be the solution to the Monge-Amp`ere equation (ddcρ)n = fdVX in Ω; ρ = 0 on ∂Ω. The uniform bound for ρ can be deduced from the results in [Ko l98,GL23]. Consider now ψ = v + ρ. It is sufficient to show that ψ ≤u. Indeed, ψ satisfies (ddcψ)n ≥(ddcv)n + (ddcρ)n = (ddcv)n + fdVX ≥(ddcu)n. It thus follows from the Comparison Principle (Proposition 1.7) that u ≥ψ on Ω. □ We are now prepared to prove a local uniform estimate. Lemma 2.2 (Local uniform estimate for Monge-Amp`ere equations). Assume that u is a bounded plurisubharmonic solution to the Dirichlet problem (LocMA). Then OscΩu ≤C, where C = C(n, Ω). Proof. Let v ∈F(Ω) be such that (ddcv)n = χ (g) dVX, in Ω; v = 0, on ∂Ω. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 11 The existence of v ∈F(Ω) is ensured by Theorem 1.11. Let α > 0 be fixed (to be specified later). On the set {log (g + 3) 1, the following inequality is satisfied: Θ′ -t α n tn(h ◦log t)n ≥2. (2.3) To see this, let x = log t and let b(x) = -Θ -ex α . Then the condition (2.3) is equivalent to b′(x)h(x) ≥21/n α , for all x > 0. We define b by b(x) := ˆ x 0 21/n αh(s)ds. Since h-1 is integrable, the function b(x) remains bounded, which in turn implies that Θ is uniformly bounded, as claimed. Taking t = log (g + 3) in (2.3) yields that Θ′ -log (g + 3) α n (log (g + 3))n (h ◦log log (g + 3))n ≥2. (2.4) Without loss of generality, we assume that there exist k0 > 0 such that χ(k) ≥1 2k logn(k + 3)(h ◦log ◦log(k + 3))n, ∀k ≥k0. From (2.2) and (2.4), it follows that on the set {log (g + 3) ≥-αv} ∩{g ≥k0}, (ddc(Θ ◦v))n ≥gdVX = (ddcu)n. On the other hand, on the set {log (g + 3) ≥-αv} ∩{g 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω), -1 ≤ψ ≤0, with (ω + ddcψ)n ≥s(χ) g ∥g∥χ dVX. Proof. Cover X by balls, X ⊂∪N j=1Bj. Choose B′ j = {ρj 0 in a neighborhood of B′ j, and such that Bj ⋐B ′ j. In B′ j, let vj ∈PSH(B′ j) be solution to Monge-Amp`ere equation (ddcvj)n = g ∥g∥χ dVX in B′ j; vj = -1 on ∂Ω. The local L∞-estimate (Lemma 2.2) implies OscB′ jvj ≤C′ j. We define ψj := max(vj, ξjρj), where ξj > 1 for all j. Thus, ψj coincides with vj in Bj, and coincides with ξjρj in X \\ B′ j and in the neighborhood of ∂B′ j. Moreover, ψj is δ-1ω-plurisubharmonic for some uniform constant δ > 0 in Bj. Taking ψ := 1 N PN j=1 δψj, and taking 0 0 satisfies ∥1{f>A}f∥χ ≤s(χ) 4 . Then OscX(φ) ≤C, where C depends on A, and χ. Proof. We consider the function g = 1{f>A}f ∥1{f>A}f∥χ . Applying the Subsolution Lemma (Lemma 2.3) to g yields that there exists subsolution ψ ∈PSH(X, ω), such that (ω + ddcψ)n ≥sgdVX. The dominated convergence theorem implies that lim t→+∞ ˆ X χ(1{f>t}f)dVX = ˆ X lim t→+∞χ 1{f>t}f dVX = 0. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 13 We can thus choose A sufficiently large, depending only on f and s, such that ∥1{f>A}f∥χ ≤s 4. It follows that (ω + ddcψ)n ≥41{f>A}(ω + ddcφ)n. Therefore, (ω + ddcφ)n ≤1 4(ω + ddcψ)n + AdVX. Fix B > 0 large enough. It follows from [GL23, Lemma 2.5] that there exist ρ ∈PSH(X, ω), -1 ≤ ρ ≤0 such that (ω + ddcρ)n ≥4A1{φ n/m. A generalization to Orlicz spaces is outlined in Remark 3.2 therein. It has also been shown in the recent article [GL25b, Theorem B] that this approach can be adapted in certain general complex geometric PDEs. We provide a more detailed presentation in the proof of Theorem B here. The following lemma will be used to obtain a uniform estimate for bounded solutions. Lemma 2.5. Assume f ∈L1(dVX). Let φ ∈SHm(X, ω) ∩L∞(X) be such that (ω + ddcφ)m ∧ωn-m = fωn. Then, we have (ω + ddcu)n ≤fn/mωn, where u = Pω(φ). Proof. The proof proceeds in two steps. We first treat the case where the density f is bounded. In the general case, we approximate a given f ∈L1 by a sequence of bounded densities. Normalize the solution φ such that supX φ = 0. Step 1. Assume that f is bounded. Let {fj} be a sequence of smooth, uniformly bounded functions such that fj →f in L2n. It follows from [KN16, Lemma 3.20] that there exists a decreasing sequence of (ω, m)-subharmonic functions {ψj} such that ψj →φ. We consider the following complex Hessian equation: (ω + ddcφj)m ∧ωn-m = eφj-ψjfjωn. The existence of smooth, (ω, m)-subharmonic solution φj follows from [Sz ́e18]. Then, from [KN16, Theorem 3.9] we deduce that φj converges uniformly to eφ ∈SHm(X, ω) ∩C0(X), which is the solution to (ω + ddc eφ)m ∧ωn-m = eeφ-φfωn. By the domination principle (Proposition 1.8), we obtain that eφ = φ. Taking uj = Pω(φj), it follows from Proposition 1.17 that uj ∈C1,1(X). The mixed Monge-Amp`ere inequality [Ngu16, Lemma 1.19] implies that (ω + ddcuj)n ≤e n(φj-ψj) m fn/m j ωn. 14 YUETONG FANG Since uj converges uniformly to u, passing to the limit, we conclude that (ω + ddcu)n ≤fn/mωn. Step 2. Assume now that f ∈L1(dVX). We consider the equation given by (ω + ddcφj)m ∧ωn-m = eφj-φ min(f, j)ωn. Theorem 3.19 of [KN15] ensures the existence of a bounded solution φj, satisfying φj ≥φ. The domination principle yields that φj is decreasing to a (ω, m)-subharmonic function eφ. It follows from the domination principle again that eφ = φ. Taking uj := Pω(φj), then uj decreases to u. Moreover, Step 1 yields (ω + ddcuj)n ≤en/m(φj-φ)fn/m j ωn. Applying the continuity of Monge-Amp`ere operator along monotone sequences [BT76,BT82], we obtain (ω + ddcu)n ≤fn/mωn, as required. □ Proof of Theorem B. Normalize φ such that supX φ = 0. Let u = Pω (φ) and let C denote the contact set C := {u = φ}. We remark that u ∈PSH(X, ω) ⊂SHm(X, ω). Therefore, it follows from the mixed Monge-Amp`ere inequality [Ngu16, Lemma 1.19] and proposition 1.17 that 1C(ω + ddcu)m ∧ωn-m ≤1C(ω + ddcφ)m ∧ωn-m ≤fdVX We remark that hX := ωn/dVX ≥δ0 is positive on X. Lemma 2.5 ensures us to take g := (ω + ddcu)n /dVX. Then, the mixed Monge-Amp`ere inequality yields that δ1-m/n 0 gm/n ≤h1-m/n X gm/n ≤f, hence g ≤C0fn/m, so that g ∈Lχ0, where χ0 = t logn(1 + t)[h ◦log ◦log(3 + t)]n satisfies Condition (K). It follows from Theorem 2.4 that OscX(u) ≤A. It suffices to bound supX u from below. Set vA(ω) := inf ˆ X (ω + ddcφ)n : φ ∈L∞(X) ∩PSH(X, ω), -A ≤φ ≤0 . Recall that vA(ω) is strictly positive by [GL22, Proposition 3.4]. We denote by (χ∗ 0)-1 the inverse of χ∗ 0. Then it follows from additive H ̈older-Young inequality (Proposition 1.16) that vA(ω)(χ∗ 0)-1 -sup X u ≤ ˆ X (χ∗ 0)-1(-u) (ω + ddcu)n = ˆ X (χ∗ 0)-1(-u)gdVX ≤ ˆ X χ0(g)dVX + ˆ X (-u)dVX. We remark that ́ X(-u)dVX is uniformly bounded from above by Lemma 1.5. Thus we obtain a uniform lower bound of supX u, which yields together with OscXu ≤A that |u| has a uniform bound, and since u ≤φ, the solution φ is also uniformly bounded. □ CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 15 3. Weak solution 3.1. Stability estimates. Theorem 3.1. Let u, v ∈SHm(X, ω)∩L∞(X) be such that supX u = 0 and v ≤0. Assume that u ∈SHm(X, ω) is a solution to the complex Hessian equation ωm u ∧ωn-m = eλufωn, (3.1) where λ ≥0 and fn/m ∈Lχ with χ satisfying Condition (K). Then, sup X (v -u)+ ≤C∥(v -u)+fn/m∥ 1 n+1 χ , where the positive constant C depends only on n, m, ω, ∥f n m ∥χ, and supX |v|. Proof. We consider only the case λ = 0; when λ > 0, the L∞-estimate implies that eλuf ≤f, allowing the problem to be reduced to the case λ = 0. We now fix ε > 0, to be chosen later, and define gε := 1{u u + C2δ + ε} ⊂{u u+C2δ+ε}ωm φ ∧ωn-m =1{φ>u+C2δ+ε} [(1 -δ)ωv + δωψ]m ∧ωn-m =1{φ>u+C2δ+ε} m X k=0 m k [(1 -δ)ωv]k ∧(δωψ)m-k ∧ωn-m ≥1{φ>u+C2δ+ε}δm(ωψ)m ∧ωn-m. If ∥gε∥χ is not small, then the stability property follows trivially by adjusting C. Hence, we can assume that ∥gε∥χ u+C2δ+ε}δm(ωψ)m ∧ωn-m ≥21{φ>u+C2δ+ε}fωn =21{φ>u+C2δ+ε}ωm u ∧ωn-m. It follows from domination principle (Proposition 1.8) that φ ≤u + C2δ + ε. 16 YUETONG FANG Hence, (v -u)+ ≤C3 21/m s1/n ! ∥gε∥1/n χ + ε ≤C4∥gε∥1/n χ + ε ≤C4(ε)-1/n∥(v -u)+fn/m∥1/n χ + ε. Optimizing the right hand side by choosing ε = ∥C4(v -u)+fn/m∥1/n χ n ! n n+1 , we obtain (v -u)+ ≤C ∥(v -u)+fn/m∥χ 1 n+1 . The conclusion follows by taking the supremum of the left-hand side. □ Remark 3.2. When χ(t) = tp/p, with p > 1, the Orlicz norm ∥· ∥χ is simply the Lebesgue Lp-norm. Assume fn/m ∈Lp, for p > 1. Let χq(t) = tq/q for some 1 0 such that (ω + ddcuj)m ∧ωn-m = cjeλujfjdVX, λ ≥0. We normalized uj such that supX uj = 0. Our first step is to bound the coefficients cj from above and below. This requires an inequality involving mixed Hessian measures. It's well known that if φj ∈SHm(X, ω)∩C∞(X) and (ω +ddcφj)m ∧ωn-m = fjdVX, then G ̊arding's inequality [G ̊ar59] holds: (ω + ddcφ1) ∧· · · ∧(ω + ddcφm) ∧ωn-m ≥(f1 · · · fm)1/mdVX. In the case where φj's are not smooth, we can apply an approximation argument to obtain the following inequality. Proposition 3.4. Let u ∈SHm(X, ω) be bounded functions such that (ω + ddcφ)m ∧ωn-m ≥fωn. Then we have that (ω + ddcφ) ∧ωn-1 ≥f1/mωn. Proof. Let {gj} be a sequence of bounded, positive and smooth functions converging to f. Let vj be a sequence of smooth, (ω, m)-subharmonic functions converging to φ. We consider the following Hessian equation: (ω + ddcφj)m ∧ωn-m = eφj-vjgjωn. The existence of smooth solution φj follows from [Sz ́e18]. Applying G ̊arding's inequality [G ̊ar59] yields (ω + ddcφj) ∧ωn-1 ≥e φj-vj m g1/m j ωn. Moreover, we deduce from [KN16, Theorem 3.19] that φj converges uniformly to eφ ∈ SHm(X, ω). Passing to the limit as j →+∞, we obtain (ω + ddceu)m ∧ωn-m = eeφ-φfωn. The domination principle (Proposition 1.8) then implies eφ = φ. Consequently, we conclude that (ω + ddcφ) ∧ωn-1 ≥f1/mωn. □ We now turn to obtaining uniform bounds for cj. Lemma 3.5. The sequence of constant {cj} is uniformly bounded. In particular, the Orlicz norm ∥cjfj∥χ is uniformly bounded from above. Proof. We first bound cj from above. Fix δ0 > 0 such that δ0 ≤ωn/dVX. Proposition 3.4 implies that for each j, (ω + ddcuj) ∧ωn-1 ≥c1/m j f1/m j ωn ≥δ0c1/m j f1/m j dVX. 18 YUETONG FANG Thus, by taking the integral of both sides: δ0c1/m j ˆ X f1/m j dVX ≤ ˆ X (ω + ddcuj) ∧ωn-1. Note that uj ∈SHm(X, ω) ⊂SH1(X, ω), and the sequence (uj) is relatively compact in L1. Moreover, we have ́ X f1/m j ωn → ́ X f1/mωn > 0 as j →+∞. It follows that there exist j0 large enough such that for each j > j0, the following inequality holds: c1/m j ≤ 2 δ0 ́ X f1/mdVX ˆ X (ω + ddcuj) ∧ωn-1. It suffices to bound the right hand side. Indeed, it follows from Stokes' theorem that ˆ X (ω + ddcuj) ∧ωn-1 = ˆ X ωn + ˆ X ujddc(ωn-1) ≤ ˆ X ωn + B ˆ X |uj|ωn. By the L1-compactness result for (ω, 1)-subharmonic functions (Lemma 1.5), we conclude that the right hand side above is bounded. Thus cj is bounded from above. We next show that {cj} is bounded away from 0. Let gj = fn/m j ∈Lχ. By the subsolution Lemma (Lemma 2.3), there exists a uniform constant δ > 0, such that (ω + ddcvj)n ≥δjgjdVX. Then, it follows from mixed Monge-Amp`ere inequality [Ngu16, Lemma 1.9] that ωm vj ∧ωn-m ≥δm/n j gm/n j ωn ≥δm/n j fjωn. The domination principle (Proposition 1.8) yields that δm/n j ≤cj. We thus obtain a uniform lower bound for cj's, since δj's are strictly positive, uniform constants. The uniqueness of cj for each fj follows directly from domination principle (Proposition 1.8). □ Uniform bound of uj. It follows from Theorem B that {uj} have uniform bound: -C ≤uj ≤0. By extracting, we can assume that {uj} converges in L1(X) to a bounded (ω, m)-subharmonic function u, as j →+∞. Existence of solutions. We now show that, by construction, the limit u is a solution to the Hessian equation (ω + ddcu)m ∧ωn-m = ceλufdVX, λ ≥0. We first consider the simpler case in which λ > 0. Theorem 3.6. Let λ > 0. Then u ∈SHm(X, ω) ∩L∞(X) is the unique solution such that (ω + ddcu)m ∧ωn-m = eλufdVX. Moreover u is uniformly bounded. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 19 Proof. To simplify the notation, we assume that λ = 1. Then each uj satisfies (ω + ddcuj)m ∧ωn-m = eujfjdVX. From the domination principle (Proposition 1.8), we deduce that uj decreases to some functions u ∈SHm(X, ω). Passing to the limit, it follows from the continuity of Hessian operator along decreasing sequence ( [KN23, Section 5]) that (ω + ddcu)m ∧ωn-m = eufdVX. Applying Theorem B, we obtain a uniform bound for u. □ Monotonicity of the sequence uj is not guaranteed when λ = 0. Our approach is to construct a supersolution and a subsolution and demonstrate that they coincide, as in [GL24, Section 3] and [BGL25, Theorem 4.5]. Theorem 3.7 (Existence of bounded solution). Let f, u and the constant c be as above. Then u ∈SHm(X, ω) ∩L∞(X) is a solution to (ω + ddcu)m ∧ωn-m = cfdVX. Moreover, u is uniformly bounded. Proof. We define ρj := sup k≥j uk !∗ , vj := Pω,m inf k≥j uk , where Pω,m (infk≥j uk) denotes the largest (ω, m)-subharmonic function lying below infk≥j uk. Observe that ρj is a decreasing sequence that converges to u, and vj is an increasing sequence that converges to an (ω, m)-subharmonic function v. In particular, by construction, we have u ≥v. We are going to show that u = v. Set cj fj := infk≥j ckfk. Then it follows from Lemma 1.18 that (ω + ddcρj)m ∧ωn-m ≥cj fj dVX. Then we deduce from the continuity of Hessian operator along monotone sequences (see [KN23, Lemma 5.1]) that (ω + ddcu)m ∧ωn-m ≥cfdVX. (3.2) On the other hand, we write (ω + ddcvj)m ∧ωn-m = evj-vjcjfjdVX. Define c+ j f+ j := supk≥j ckfk. It also follows from Lemma 1.18 that (ω + ddcvj)m ∧ωn-m ≤evj-infk≥j ukc+ j f+ j dVX. Let j →+∞. Together with (3.2) we deduce that (ω + ddcv)m ∧ωn-m ≤ev-ucfdVX ≤ev-u(ω + ddcu)m ∧ωn-m. By the domination principle (Proposition 1.8), we conclude that v ≥u. Hence, v = u is a bounded solution to the complex Hessian equation: (ω + ddcu)m ∧ωn-m = cfdVX. The uniform bound for u follows from Theorem B. □ 20 YUETONG FANG Proof of Theorem C. Existence of continuous solutions. We now prove that the sequence {uj} converges uniformly to u as j →+∞. We recall that when λ > 0, the L∞-estimate implies that eλuf ≤f, allowing us to reduce to the case where λ = 0. Fix j0 large enough. For each j > j0, we infer by Theorem 3.1 that (uj -u)+ ≤C1 (uj -u)+f n m 1 n+1 χ , (u -uj)+ ≤C1 (u -uj)+f n m j 1 n+1 χ . We are going to show that (uj -u)+f n m χ →0 as j →+∞. Fixing s > 0, we proceed to compute the corresponding integrability. ˆ X χ (uj -u)+f n m s dVX = ˆ X (uj -u)+f n m s logn 1 + (uj -u)+f n m s h ◦log ◦log 3 + (uj -u)+f n m s n dVX. As u and uj's are uniformly bounded, it follows that (uj -u)+ ≤C0 for some uniform constant C0, and since h is increasing, we deduce that ˆ X χ (uj -u)+f n m s dVX ≤ ˆ X (uj -u)+f n m s logn 1 + C0f n m s · h ◦log ◦log 3 + C0f n m s n dVX. By the dominated convergence theorem, we conclude that (uj -u)+f n m s χ →0, for all 0 < s < +∞, and consequently ∥(uj -u)+f n m ∥ 1 n+1 χ →0. Applying the same argument, we obtain that (u -uj)+f n m j 1 n+1 χ ≤ (u -uj)+f n m 1 n+1 χ →0. Therefore, {uj}∞ j=1 is a Cauchy sequence in SHm(X, ω) ∩C0(X). It follows that the bounded solution u to the complex Hessian equation (ω + ddcu)m ∧ωn-m = ceλufωn, where f ∈Lχ, is continuous. Continuity of bounded solutions. Assume now that v ∈SHm(X, ω) is another bounded solution to the complex Hessian equation (ω + ddcv)m ∧ωn-m = cfdVX. 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China\n2Department of Physics, Chemistry and Pharmacy,\nUniversity of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark\n3Department of Physics & Research and Education Center for Natural Sciences,\nKeio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan\n4International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2),\nHiroshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8511, Japan\n(Dated: October, 2025)\nMagnetic skyrmions extended to three dimensions form string-like objects whose fundamental role\nremains largely unexplored. We show that skyrmion strings can terminate on a N´eel-type domain\nwall (DW), realizing a magnetic analogue of a Dirichlet(D)-brane soliton. While an isolated N´eel DW\ntends to rotate into a Bloch DW, the N´eel DW is stabilized when a skyrmion string ends on it. Unlike\nfield-theory D-branes, the Bloch-type DMI produces linear rather than logarithmic DW bending,\nand the strings retain finite width far from the DW, circumventing singular behavior. Furthermore,\nthe repulsive interaction between strings allows periodic multi-junction solutions, yielding a square\nlattice of alternating strings and local DW deformations. These results establish magnetic skyrmion\nstrings as fundamental strings that can end on a D-brane.\nSuperstring theory predicts the fundamental con-\nstituents not to be point particles, but strings, which\nare generally found in two varieties: Closed strings and\nopen strings [1, 2]. Excitations of the strings correspond\nto point particles at low energies. The former need no\nextra ingredients in the theory and can e.g. describe the\npropagation of gravitons in spacetime. The latter, how-\never, predict and require the existence of hypersurfaces\ncalled Dirichlet(D)-branes on which strings terminate, pi-\noneered and much developed by Polchinski [3, 4]. String\nexcitations induce a gauge theory on D-branes, described\nby the Dirac-Born-Infeld (DBI) action. The discovery of\nD-branes was a key ingredient for the 2nd revolution of\nstring theory in the ’90ies. A field theoretical analog of\nD-branes was proposed in Ref. [5] as a string ending on\na domain wall (DW), referred to as a D-brane soliton,\nin the O(3) nonlinear sigma model (NLσM) or the CP 1\nmodel with a quadratic potential, known as the easy-axis\nanisotropy. In fact, the D-brane soliton coincides with\nthe so-called BIon [6] when it is viewed from the DBI ac-\ntion on the DW, thereby justifying to be identified with\na D-brane. D-brane solitons were further generalized to\na U(1) gauge theory [7] and the CP N or Grassmannian\nmodel [8, 9]; see also Refs. [10–12].\nThis brings us to consider analogous systems of D-\nbranes in more down-to-earth systems. The purpose of\nthis Letter is to propose a D-brane-string junction in\nchiral magnets with a Dzyaloshinskii-Moriya-Interaction\n(DMI) [13, 14] in three dimensions. Chiral magnets are\nknown to admit various topological defects and solitons\n[15]. With an easy-axis potential, they admit magnetic\nDWs. These have, in particular, been a subject of intense\nstudy due to their application to magnetic memory (stor-\nage) [16, 17]. The ground state is in the ferromagnetic\nphase for small DMI while for large enough DMI it is\nin a spiral phase or a chiral soliton lattice (CSL) [18–23].\nThe other more well-studied topological solitons are mag-\nnetic skyrmions [24–30], which are a 2-dimensional vari-\nant of the original 3-dimensional skyrmions proposed by\nSkyrme himself as a simple model for nucleons [31, 32].1\nWith a certain Zeeman magnetic field, a skyrmion lattice\nis the ground state [26–30, 34]. Isolated skyrmions have\nbeen also studied [35]. They have been extensively stud-\nied in particular for applications to nanotechnology such\nas information carriers in magnetic storage devices [36].\nSee Ref. [36] for a review of magnetic skyrmions. The co-\nexistence of a DW and skyrmions in two dimensions gives\nrise to their composite states called DW-skyrmions [37–\n52] (see also Refs. [53, 54]) which are experimentally ob-\nserved [55–57].\nSimilar DW-skyrmions (bimerons) are\nproposed for an easy-plane anisotropy [58]. Such DW-\nskyrmions are expected to be useful for manupulating\nskyrmions circumventing the skyrmion Hall effect [59–\n62].\nRecently, much attention has been paid to topological\ndefects in three spatial dimensions; Hopfions [63, 64] and\nmonopoles [65–68] have been studied in chiral magnets.\nIn addition, DWs and skyrmions are extended objects\nin three dimensions, which are two dimensional hyper-\nsurfaces and strings [69], respectively. Several aspects of\nskyrmion strings have been studied thus far, such as a\n1 Baby-skyrmions were latter considered in the 1990’ies and on-\nward [33], which differ from the magnetic skyrmions in the way\nthey are stabilized: The baby-skyrmions have a Skyrme term like\n3D skyrmions and the magnetic skyrmions are stabilized by the\nDMI. An important consequence is that the DMI breaks parity,\nunlike the Skyrme term, and this implies that only skyrmions\nand not anti-skyrmions exist in the chiral magnets.\narXiv:2510.14689v1 [cond-mat.mes-hall] 16 Oct 2025\n\n2\nskyrmion string ending on a monopole (torons) [65, 66],\nKelvin modes propagating along a string [70, 71], braid-\ning skyrmion strings [72], etc. One of fundamental ques-\ntions may be how skyrmion strings differ from or are sim-\nilar to fundamental strings in string theory. We should\nask whether skyrmion strings can end on a D-brane or\nsome alternative extended structure. One hint to attack\nthis problem is a D-brane soliton in field theory [5, 7, 8].\nIn this Letter, we show that indeed skyrmion strings\ncan end on a N´eel DW, forming a stable D-brane soliton,\nand thus can be regarded as fundamental strings. While\na single N´eel DW is unstable to rotate to a Bloch DW, it\ncan be stabilized once a string terminates on it. There are\noutstanding differences between the field theory D-brane\nsoliton and the magnetic D-brane soliton. In the former,\nthe string gives rise to a logarithmic bending of the DW,\nwhile in the latter we find a linear bending of the DW\ndue to the Bloch-type DMI. We also have stable finite-\nwidth strings far from the DW, in contrast to a singular\nbehaviour in the former. We also consider the possibil-\nity of having more than one string-DW junction. Due\nto the repuslive nature of the force between strings, we\nconstruct a solution with periodic boundary conditions\nin the (x, y)-plane and find a square lattice of strings\npointing up- and downwards from the DW, that locally\nhas a linear bending around each of the strings. We thus\nestablish the notion that magnetic skyrmion strings are\nfundamental strings that can end on a D-brane.\nThe ferromagnetic chiral magnetic system is described\nby a gradient term (Heisenberg interaction), the DMI\nand a potential (on-site interaction) that we take here to\nbe the easy-axis anisotropy term\nE = 1\n2∂in·∂in+κn·∇×n+ 1\n2(1−n2\n3),\ni = 1, 2, 3, (1)\nwhere the total energy is given by E =\nR\nE d3x and\nthe magnetization vector or sigma model field, n =\n(n1, n2, n3), is subject to the usual sigma model con-\nstraint: n · n = 1.\nThe energy and length units are\nrescaled so as to have a canonical kinetic term and a po-\ntential coupling constant of unity: The only remaining\ncoupling constant of the theory (κ) is a combination of\nthe physical constants. In this Letter, we consider the\nferromagnetic phase κ ∈[0.3, κcrit] with κcrit ≈0.6 being\nthe critical coupling beyond which a CSL-type phase is\nthe ground state (for κ > κcrit) [18–22]. We also concen-\ntrate on static configurations and do not discuss dynam-\nics.\nThe topology of the theory can be classified into codi-\nmensions. First, drawing a line in the z-direction, the\nimportance of the potential comes into light: The DW\n(of codimension one) is supported by the existence of\ntwo generate and disconnected vacua\nπ0(Mvac) = π0\n\u0000{n3 = 1 ∪n3 = −1}\n\u0001\n≃Z2.\n(2)\nTaking instead a slice, say in the (x, y)-plane, the topol-\nogy of 2-dimensional skyrmions is given by the 2nd ho-\nmotopy group that supports also NLσM lumps:\nπ2(S2) ≃Z ∋Qskyrmion.\n(3)\nThis π2 corresponds to the point-compactified (x, y)-\nplane, also known as the extended complex plane, and\nis topologically identical to the Riemann sphere (S2).\nFinally, the 3-dimensional space mapped to the sigma\nmodel target space being a 2-sphere, gives rise to the\nHopf invariant π3(S2) = Z ∋QHopfion, which supports\nHopfions that are knotted string-like solutions [63, 64] –\nwe will not discuss those further here. In this Letter, we\nconstruct the string-DW junction, which is combination\nof the two former types of soliton.\nLet us briefly review the field theory D-brane soliton\nsolution, which is the string-DW junction for κ := 0,\nthat endows the DW with a logarithmic bending. The\nsolution is\nη = ρeiθ±z,\n(4)\nwhere we have used the Riemann sphere coordinate\nη = n1 + in2\n1 + n3\n,\n(5)\non the sigma model target space and cylindrical coordi-\nnates on space: x+iy = ρeiθ. To observe the logarithmic\nbeing of the DW, we can simply analyze the iso-curve\ncorresponding to the field at a fixed level set, say |η| = 1:\nlog ρ = ±z.\n(6)\nThis equation also reveals that far from the junction, on\nthe string side, the string becomes infinitesimally small\nin size.\nWe now turn to the case of κ ̸= 0 and will for concrete-\nness work in the ferromagnetic phase with κ ∈[0.3, κcrit].\nFirstly, we have stable skyrmion strings in this case with\nκ > 0 turned on.\nThis can be understood simply by\ncomparing each slice at a fixed z to the usual 2D mag-\nnetic skyrmions and see that up to a choice of potential,\nthese are identical. The DWs perpendicular to a vector\nin the (x, y)-plane are of the Bloch-type, for which DMI\ncontributes negatively to the energy, and are stable. In\nthe presence of a Bloch DW, the skyrmion string is re-\npelled [43] and does not automatically flow to the DW to\nbecome a DW-skyrmion [37–52, 55–57].2 On the other\nhand, the DW is of N´eel-type when perpendicular to the\nz-direction, which is actually unstable and tends to ro-\ntate to a Bloch DW, as the N´eel DW’s DMI energy is\n2 In the CSL phase, the Bloch DWs have negative energy and\nconstitute a CSL [18–22], and the DW-skyrmion string, if formed,\nis unstable to split into a bimeron (pair) [42].\n\n3\npositive. Nevertheless, we show the N´eel DW is stabi-\nlized in the ferromagnetic phase when skyrmion strings\nterminate on it. One essential point is that the N´eel DW\nhas an exact U(1) modulus on it while the Bloch DW\ndoes not – this facilitates that the winding of the string\ncan be connected to the DW.\nFirst, to establish that the bending of the wall will be\nlinear, we cannot simply fix the field value of the solution,\nsince we are not aware of analytic exact solutions to the\nequations of motion in this case. However, using the idea\nof fixing a curve in the equation of motion, we arrive at\nleading order at the solution of the curve for |η| = 1:\nκr + cr log r + czz = 0,\n(7)\nwhere dimensional analysis3 fixes cr/cz = 1 when κ = 0.\nThis reproduces the logarithmic bending of the DW when\nκ = 0 and predicts a linear bending of the DW when\nκ ̸= 0.\nIn order to confirm our prediction, we turn to numeri-\ncal computations of the equation of motion coming from\nvarying the energy functional (1):\neom := ∂2\ni η −2(∂iη)2¯η\n1 + η¯η\n−(1 −η¯η)η\n1 + η¯η\n= 0.\n(8)\nWe discretize a box typically of the size 511 × 511 × 767\nwhere the derivatives of the equation of motion are ap-\nproximated by a 4th order 5-point stencil finite-difference\nscheme, with lattice spacing h = 0.15. The method for\nfinding solutions utilized here is the so-called arrested-\nNewton flow, where we solve\n∂2\nτη = eom,\n(9)\nmonitoring the static energy given by Eq. (1) at every\nstep of the flow in τ (τ is a fictitious time parameter).\nIf the static energy rises, we stop the flow by setting\n∂τη = 0 and then restart the algorithm.\nIn Fig. 1, we display the numerical results for the\nstring-DW junction (trumpet) for various values of κ =\n0.35, 0.4, 0.45, 0.5, illustrating the dependence of the\nslope of the linear bending of the wall on the DMI cou-\npling κ, with the steepest slopes corresponding to the\nlargest values of κ. The figure shows the isosurface of\nthe n3 field at n3 = 0; this is the midway between the\ntwo vacua of the potential in Eq. (1). Clearly the DW\nobeys a linear relation between the z-coordinate and the\nradius from the string junction’s center when the radius\nis sufficiently large (say of the order of 10 or larger). The\nsize (thickness) of the string also depends approximately\nlinearly on κ, which can be seen from Derrick’s theorem\n3 More specifically, by reinstating units into the equations, the\nratio cz/cr = m with m2 being the coupling of the potential\nterm. Our dimensionless equations correspond to m = 1.\nFIG. 1. String-DW junctions (trumpets) for various strength\nof the DMI κ = 0.35 (blue), 0.4 (red), 0.45 (magenta), 0.5\n(yellow). The slope of the linear bending is nearly linearly\nproportional to κ.\n(also known as a virial law in other fields of physics) [73].\nIn a linear theory or an integrable theory, we would ex-\npect the constants cr and cz to truly be constants (like\nin the field theory D-brane soliton case), but due to non-\nlinearities of system we find they have some dependence\non κ. Fitting to the slope data of the solutions in Fig. 1,\nwe find\ncz ≈3.4 −5.2κ,\ncr/cz ≈1 + 12.6κ.\n(10)\nwhich is consistent with the logarithmic bending with\ncoefficient cr/cz = 1 when κ →0.\nInterestingly, the\napproximate relation between cz and κ predicts a criti-\ncal value of the coupling, which indeed exists due to the\ntransition to the CSL: The linear fit predicts κcrit ≈0.65,\nwhereas in the full nonlinear theory we find it to be closer\nto κcrit ∼0.6.\nWe find also a lower critical value for\nκ, where the string-DW junction ceases to exist near\nκcrit,junc ≳0.3, although the skyrmion string itself still\nexists at κ = 0.3.\nFig. 2 shows the detailed information of the entire 3-\nvector field, n, at the string-slice in the inlet at the top-\nright corner, at the wall-slice at the bottom of the figure\nand finally in a vertical cross section in the top of the fig-\nure. In this figure, the vector n is mapped onto real space\n\n4\nFIG. 2.\nString-DW junction with κ = 0.5.\nArrows are\nmapped from n to x and the colors are in one-to-one cor-\nrespondence with the arrows: n1 = 1 7→red, n2 = 1 7→green,\nn1 = −1 7→cyan, n2 = −1 7→magenta, n3 = 1 7→white and\nn3 = −1 7→black. The inlet and the bottom are horizontal\ncross sections, whereas the top is a vertical cross section of\nthe string-DW junction.\n(R3) using arrows, with the up vacuum (n3 = 1) marked\nby ⊙, the down vacuum (n3 = −1) marked by ⊗and the\nvectors in the plane corresponding to (n1, n2) when n3 is\nsmall. The color scheme is in one-to-one correspondence\nwith the arrows and is described in the caption of the\nfigure. There are about 15 times more lattice points in\nthe solution, than arrows depicted on the figure.\nThis string junction turns out to repel other string\njunctions, both of strings pointing upwards (skyrmion\nstrings) and strings pointing downwards (anti-skyrmion\nstrings). Packing the string junctions into a given vol-\nume, however, has solutions which can most easily be\nfound by considering periodic boundary conditions in the\n(x, y)-plane.\nSuch a configuration of multi-string-DW\njunctions is shown in Fig. 3, with the upper part of the\nspace being mostly in the up vacuum (n3 = 1) and the\nlower part mostly in the down vacuum (n3 = −1). The\nright-hand side of the figure shows the spin interpreta-\ntion of the map from n to real space, see the caption of\nFig. 2 for details of the color scheme.\nAlthough we mentioned in the introduction, that the\nchiral DMI supports only skyrmion strings and no anti-\nskyrmion string, this is indeed the case in the up-vacuum\n(upper part of Fig. 3). However, in the down-vacuum\n(lower part of Fig. 3), the situation is reversed and the\nDMI supports only anti-skyrmion strings.\nInterestingly, if we impose periodic boundary condi-\ntions also in the z-direction, we obtain a fully periodic\nconfiguration just like that of Fig. 3, but with the anti-\nDW placed equidistantly from the DW (the one visible\nin Fig. 3). This means that one could have n layers of\ncyclic DW-anti-DW configurations with skyrmion strings\nand anti-skyrmion strings stretched between them.\nReal-world experimentally realizable physical systems\nfor studying D-brane analogs etc. are hard to come by.\nIt is yet unclear if and how this could be utilized for\napplications in real-world spintronics. However, the pos-\nsibility of realizing a D-brane-string junction in a chi-\nral magnet is very interesting for fundamental physics.\nOne of the important directions to explore is how to\ncreate a D-brane soliton. If a N´eel wall moving under\nan applied current passes through a fixed impurity or\nmonopole, it will be bent and eventually would create\na D-brane soliton. Another direction entails excitations\non a D-brane soliton. Kelvin modes propagating along a\nskyrmion string [70, 71] are the so-called type-B Nambu-\nGoldstone modes [74, 75] with a fractional dispersion re-\nlation [76, 77] as well as ripple modes on a domain wall\n[77–79].\nIt will be worthwhile to study how they are\ntransformed to each other when propagating along a D-\nbrane soliton. D-brane analogs have been also proposed\nin Bose-Einstein condensates [80–83]. Condensed matter\nD-branes open up a new avenue of composite topologi-\ncal objects to simulate fundamental physics, which could\nbe realized in other systems such as liquid crystals and\noptics.\nAcknowledgments\nS. B. G. thanks the Outstanding Talent Program of\nHenan University for partial support.\nThe work of\nM. 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A 88, 013620 (2013).\n"},"cleaned":{"kind":"string","value":"Magnetic D-brane solitons: skyrmion strings ending on a N ́eel wall in chiral magnets Sven Bjarke Gudnason1, 2 and Muneto Nitta3, 4 1 475004, P. R. China 2 55, 5230 Odense M, Denmark 3 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan 4International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8511, Japan (Dated: October, 2025) Magnetic skyrmions extended to three dimensions form string-like objects whose fundamental role remains largely unexplored. We show that skyrmion strings can terminate on a N ́eel-type domain wall (DW), realizing a magnetic analogue of a Dirichlet(D)-brane soliton. While an isolated N ́eel DW tends to rotate into a Bloch DW, the N ́eel DW is stabilized when a skyrmion string ends on it. Unlike field-theory D-branes, the Bloch-type DMI produces linear rather than logarithmic DW bending, and the strings retain finite width far from the DW, circumventing singular behavior. Furthermore, the repulsive interaction between strings allows periodic multi-junction solutions, yielding a square lattice of alternating strings and local DW deformations. These results establish magnetic skyrmion strings as fundamental strings that can end on a D-brane. Superstring theory predicts the fundamental constituents not to be point particles, but strings, which are generally found in two varieties: Closed strings and open strings [1, 2]. Excitations of the strings correspond to point particles at low energies. The former need no extra ingredients in the theory and can e.g. describe the propagation of gravitons in spacetime. The latter, however, predict and require the existence of hypersurfaces called Dirichlet(D)-branes on which strings terminate, pioneered and much developed by Polchinski [3, 4]. String excitations induce a gauge theory on D-branes, described by the Dirac-Born-Infeld (DBI) action. The discovery of D-branes was a key ingredient for the 2nd revolution of string theory in the '90ies. A field theoretical analog of D-branes was proposed in Ref. [5] as a string ending on a domain wall (DW), referred to as a D-brane soliton, in the O(3) nonlinear sigma model (NLσM) or the CP 1 model with a quadratic potential, known as the easy-axis anisotropy. In fact, the D-brane soliton coincides with the so-called BIon [6] when it is viewed from the DBI action on the DW, thereby justifying to be identified with a D-brane. D-brane solitons were further generalized to a U(1) gauge theory [7] and the CP N or Grassmannian model [8, 9]; see also Refs. [10-12]. This brings us to consider analogous systems of Dbranes in more down-to-earth systems. The purpose of this Letter is to propose a D-brane-string junction in chiral magnets with a Dzyaloshinskii-Moriya-Interaction (DMI) [13, 14] in three dimensions. Chiral magnets are known to admit various topological defects and solitons [15]. With an easy-axis potential, they admit magnetic DWs. These have, in particular, been a subject of intense study due to their application to magnetic memory (storage) [16, 17]. The ground state is in the ferromagnetic phase for small DMI while for large enough DMI it is in a spiral phase or a chiral soliton lattice (CSL) [18-23]. The other more well-studied topological solitons are magnetic skyrmions [24-30], which are a 2-dimensional variant of the original 3-dimensional skyrmions proposed by Skyrme himself as a simple model for nucleons [31, 32].1 With a certain Zeeman magnetic field, a skyrmion lattice is the ground state [26-30, 34]. Isolated skyrmions have been also studied [35]. They have been extensively studied in particular for applications to nanotechnology such as information carriers in magnetic storage devices [36]. See Ref. [36] for a review of magnetic skyrmions. The coexistence of a DW and skyrmions in two dimensions gives rise to their composite states called DW-skyrmions [3752] (see also Refs. [53, 54]) which are experimentally observed [55-57]. Similar DW-skyrmions (bimerons) are proposed for an easy-plane anisotropy [58]. Such DWskyrmions are expected to be useful for manupulating skyrmions circumventing the skyrmion Hall effect [5962]. Recently, much attention has been paid to topological defects in three spatial dimensions; Hopfions [63, 64] and monopoles [65-68] have been studied in chiral magnets. In addition, DWs and skyrmions are extended objects in three dimensions, which are two dimensional hypersurfaces and strings [69], respectively. Several aspects of skyrmion strings have been studied thus far, such as a 1 Baby-skyrmions were latter considered in the 1990'ies and onward [33], which differ from the magnetic skyrmions in the way they are stabilized: The baby-skyrmions have a Skyrme term like 3D skyrmions and the magnetic skyrmions are stabilized by the DMI. An important consequence is that the DMI breaks parity, unlike the Skyrme term, and this implies that only skyrmions and not anti-skyrmions exist in the chiral magnets. 16 Oct 2025 2 skyrmion string ending on a monopole (torons) [65, 66], Kelvin modes propagating along a string [70, 71], braiding skyrmion strings [72], etc. One of fundamental questions may be how skyrmion strings differ from or are similar to fundamental strings in string theory. We should ask whether skyrmion strings can end on a D-brane or some alternative extended structure. One hint to attack this problem is a D-brane soliton in field theory [5, 7, 8]. In this Letter, we show that indeed skyrmion strings can end on a N ́eel DW, forming a stable D-brane soliton, and thus can be regarded as fundamental strings. While a single N ́eel DW is unstable to rotate to a Bloch DW, it can be stabilized once a string terminates on it. There are outstanding differences between the field theory D-brane soliton and the magnetic D-brane soliton. In the former, the string gives rise to a logarithmic bending of the DW, while in the latter we find a linear bending of the DW due to the Bloch-type DMI. We also have stable finitewidth strings far from the DW, in contrast to a singular behaviour in the former. We also consider the possibility of having more than one string-DW junction. Due to the repuslive nature of the force between strings, we construct a solution with periodic boundary conditions in the (x, y)-plane and find a square lattice of strings pointing up- and downwards from the DW, that locally has a linear bending around each of the strings. We thus establish the notion that magnetic skyrmion strings are fundamental strings that can end on a D-brane. The ferromagnetic chiral magnetic system is described by a gradient term (Heisenberg interaction), the DMI and a potential (on-site interaction) that we take here to be the easy-axis anisotropy term E = 1 2∂in·∂in+κn·∇×n+ 1 2(1-n2 3), i = 1, 2, 3, (1) where the total energy is given by E = R E d3x and the magnetization vector or sigma model field, n = (n1, n2, n3), is subject to the usual sigma model constraint: n · n = 1. The energy and length units are rescaled so as to have a canonical kinetic term and a potential coupling constant of unity: The only remaining coupling constant of the theory (κ) is a combination of the physical constants. In this Letter, we consider the ferromagnetic phase κ ∈[0.3, κcrit] with κcrit ≈0.6 being the critical coupling beyond which a CSL-type phase is the ground state (for κ > κcrit) [18-22]. We also concentrate on static configurations and do not discuss dynamics. The topology of the theory can be classified into codimensions. First, drawing a line in the z-direction, the importance of the potential comes into light: The DW (of codimension one) is supported by the existence of two generate and disconnected vacua π0(Mvac) = π0 {n3 = 1 ∪n3 = -1} ≃Z2. (2) Taking instead a slice, say in the (x, y)-plane, the topology of 2-dimensional skyrmions is given by the 2nd homotopy group that supports also NLσM lumps: π2(S2) ≃Z ∋Qskyrmion. (3) This π2 corresponds to the point-compactified (x, y)- plane, also known as the extended complex plane, and is topologically identical to the Riemann sphere (S2). Finally, the 3-dimensional space mapped to the sigma model target space being a 2-sphere, gives rise to the Hopf invariant π3(S2) = Z ∋QHopfion, which supports Hopfions that are knotted string-like solutions [63, 64] - we will not discuss those further here. In this Letter, we construct the string-DW junction, which is combination of the two former types of soliton. Let us briefly review the field theory D-brane soliton solution, which is the string-DW junction for κ := 0, that endows the DW with a logarithmic bending. The solution is η = ρeiθ±z, (4) where we have used the Riemann sphere coordinate η = n1 + in2 1 + n3 , (5) on the sigma model target space and cylindrical coordinates on space: x+iy = ρeiθ. To observe the logarithmic being of the DW, we can simply analyze the iso-curve corresponding to the field at a fixed level set, say |η| = 1: log ρ = ±z. (6) This equation also reveals that far from the junction, on the string side, the string becomes infinitesimally small in size. We now turn to the case of κ ̸= 0 and will for concreteness work in the ferromagnetic phase with κ ∈[0.3, κcrit]. Firstly, we have stable skyrmion strings in this case with κ > 0 turned on. This can be understood simply by comparing each slice at a fixed z to the usual 2D magnetic skyrmions and see that up to a choice of potential, these are identical. The DWs perpendicular to a vector in the (x, y)-plane are of the Bloch-type, for which DMI contributes negatively to the energy, and are stable. In the presence of a Bloch DW, the skyrmion string is repelled [43] and does not automatically flow to the DW to become a DW-skyrmion [37-52, 55-57].2 On the other hand, the DW is of N ́eel-type when perpendicular to the z-direction, which is actually unstable and tends to rotate to a Bloch DW, as the N ́eel DW's DMI energy is 2 In the CSL phase, the Bloch DWs have negative energy and constitute a CSL [18-22], and the DW-skyrmion string, if formed, is unstable to split into a bimeron (pair) [42]. 3 positive. Nevertheless, we show the N ́eel DW is stabilized in the ferromagnetic phase when skyrmion strings terminate on it. One essential point is that the N ́eel DW has an exact U(1) modulus on it while the Bloch DW does not - this facilitates that the winding of the string can be connected to the DW. First, to establish that the bending of the wall will be linear, we cannot simply fix the field value of the solution, since we are not aware of analytic exact solutions to the equations of motion in this case. However, using the idea of fixing a curve in the equation of motion, we arrive at leading order at the solution of the curve for |η| = 1: κr + cr log r + czz = 0, (7) where dimensional analysis3 fixes cr/cz = 1 when κ = 0. This reproduces the logarithmic bending of the DW when κ = 0 and predicts a linear bending of the DW when κ ̸= 0. In order to confirm our prediction, we turn to numerical computations of the equation of motion coming from varying the energy functional (1): eom := ∂2 i η -2(∂iη)2 ̄η 1 + η ̄η -(1 -η ̄η)η 1 + η ̄η = 0. (8) We discretize a box typically of the size 511 × 511 × 767 where the derivatives of the equation of motion are approximated by a 4th order 5-point stencil finite-difference scheme, with lattice spacing h = 0.15. The method for finding solutions utilized here is the so-called arrestedNewton flow, where we solve ∂2 τη = eom, (9) monitoring the static energy given by Eq. (1) at every step of the flow in τ (τ is a fictitious time parameter). If the static energy rises, we stop the flow by setting ∂τη = 0 and then restart the algorithm. In Fig. 1, we display the numerical results for the string-DW junction (trumpet) for various values of κ = 0.35, 0.4, 0.45, 0.5, illustrating the dependence of the slope of the linear bending of the wall on the DMI coupling κ, with the steepest slopes corresponding to the largest values of κ. The figure shows the isosurface of the n3 field at n3 = 0; this is the midway between the two vacua of the potential in Eq. (1). Clearly the DW obeys a linear relation between the z-coordinate and the radius from the string junction's center when the radius is sufficiently large (say of the order of 10 or larger). The size (thickness) of the string also depends approximately linearly on κ, which can be seen from Derrick's theorem 3 More specifically, by reinstating units into the equations, the ratio cz/cr = m with m2 being the coupling of the potential term. Our dimensionless equations correspond to m = 1. FIG. 1. String-DW junctions (trumpets) for various strength of the DMI κ = 0.35 (blue), 0.4 (red), 0.45 (magenta), 0.5 (yellow). The slope of the linear bending is nearly linearly proportional to κ. (also known as a virial law in other fields of physics) [73]. In a linear theory or an integrable theory, we would expect the constants cr and cz to truly be constants (like in the field theory D-brane soliton case), but due to nonlinearities of system we find they have some dependence on κ. Fitting to the slope data of the solutions in Fig. 1, we find cz ≈3.4 -5.2κ, cr/cz ≈1 + 12.6κ. (10) which is consistent with the logarithmic bending with coefficient cr/cz = 1 when κ →0. Interestingly, the approximate relation between cz and κ predicts a critical value of the coupling, which indeed exists due to the transition to the CSL: The linear fit predicts κcrit ≈0.65, whereas in the full nonlinear theory we find it to be closer to κcrit ∼0.6. We find also a lower critical value for κ, where the string-DW junction ceases to exist near κcrit,junc ≳0.3, although the skyrmion string itself still exists at κ = 0.3. Fig. 2 shows the detailed information of the entire 3vector field, n, at the string-slice in the inlet at the topright corner, at the wall-slice at the bottom of the figure and finally in a vertical cross section in the top of the figure. In this figure, the vector n is mapped onto real space 4 FIG. 2. String-DW junction with κ = 0.5. Arrows are mapped from n to x and the colors are in one-to-one correspondence with the arrows: n1 = 1 7→red, n2 = 1 7→green, n1 = -1 7→cyan, n2 = -1 7→magenta, n3 = 1 7→white and n3 = -1 7→black. The inlet and the bottom are horizontal cross sections, whereas the top is a vertical cross section of the string-DW junction. (R3) using arrows, with the up vacuum (n3 = 1) marked by ⊙, the down vacuum (n3 = -1) marked by ⊗and the vectors in the plane corresponding to (n1, n2) when n3 is small. The color scheme is in one-to-one correspondence with the arrows and is described in the caption of the figure. There are about 15 times more lattice points in the solution, than arrows depicted on the figure. This string junction turns out to repel other string junctions, both of strings pointing upwards (skyrmion strings) and strings pointing downwards (anti-skyrmion strings). Packing the string junctions into a given volume, however, has solutions which can most easily be found by considering periodic boundary conditions in the (x, y)-plane. Such a configuration of multi-string-DW junctions is shown in Fig. 3, with the upper part of the space being mostly in the up vacuum (n3 = 1) and the lower part mostly in the down vacuum (n3 = -1). The right-hand side of the figure shows the spin interpretation of the map from n to real space, see the caption of Fig. 2 for details of the color scheme. Although we mentioned in the introduction, that the chiral DMI supports only skyrmion strings and no antiskyrmion string, this is indeed the case in the up-vacuum (upper part of Fig. 3). However, in the down-vacuum (lower part of Fig. 3), the situation is reversed and the DMI supports only anti-skyrmion strings. Interestingly, if we impose periodic boundary conditions also in the z-direction, we obtain a fully periodic configuration just like that of Fig. 3, but with the antiDW placed equidistantly from the DW (the one visible in Fig. 3). This means that one could have n layers of cyclic DW-anti-DW configurations with skyrmion strings and anti-skyrmion strings stretched between them. Real-world experimentally realizable physical systems for studying D-brane analogs etc. are hard to come by. It is yet unclear if and how this could be utilized for applications in real-world spintronics. However, the possibility of realizing a D-brane-string junction in a chiral magnet is very interesting for fundamental physics. One of the important directions to explore is how to create a D-brane soliton. If a N ́eel wall moving under an applied current passes through a fixed impurity or monopole, it will be bent and eventually would create a D-brane soliton. Another direction entails excitations on a D-brane soliton. Kelvin modes propagating along a skyrmion string [70, 71] are the so-called type-B NambuGoldstone modes [74, 75] with a fractional dispersion relation [76, 77] as well as ripple modes on a domain wall [77-79]. It will be worthwhile to study how they are transformed to each other when propagating along a Dbrane soliton. D-brane analogs have been also proposed in Bose-Einstein condensates [80-83]. Condensed matter D-branes open up a new avenue of composite topological objects to simulate fundamental physics, which could be realized in other systems such as liquid crystals and optics. Acknowledgments S. B. G. thanks the Outstanding Talent Program of Henan University for partial support. The work of M. N. is supported in part by JSPS KAKENHI [Grants 5 FIG. 3. Multi-string-DW junctions with κ = 0.4. Periodic boundary conditions are imposed in the x- and y-directions. 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A 88, 013620 (2013)."}}},{"rowIdx":293,"cells":{"id":{"kind":"string","value":"2510.14688"},"raw":{"kind":"string","value":"Online Reliable Anomaly Detection via\nNeuromorphic Sensing and Communications\nJunya Shiraishi, Member, IEEE, Jiechen Chen, Member, IEEE, Osvaldo Simeone, Fellow, IEEE,\nand Petar Popovski, Fellow, IEEE\nAbstract—This paper proposes a low-power online anomaly\ndetection framework based on neuromorphic wireless sensor net-\nworks, encompassing possible use cases such as brain–machine\ninterfaces and remote environmental monitoring. In the con-\nsidered system, a central reader node actively queries a subset\nof neuromorphic sensor nodes (neuro-SNs) at each time frame.\nThe neuromorphic sensors are event-driven, producing spikes in\ncorrespondence to relevant changes in the monitored system. The\nqueried neuro-SNs respond to the reader with impulse radio (IR)\ntransmissions that directly encode the sensed local events. The\nreader processes these event-driven signals to determine whether\nthe monitored environment is in a normal or anomalous state,\nwhile rigorously controlling the false discovery rate (FDR) of\ndetections below a predefined threshold. The proposed approach\nemploys an online hypothesis testing method with e-values\nto maintain FDR control without requiring knowledge of the\nanomaly rate, and it dynamically optimizes the sensor querying\nstrategy by casting it as a best-arm identification problem in a\nmulti-armed bandit framework. Extensive performance evalua-\ntion demonstrates that the proposed method can reliably detect\nanomalies under stringent FDR requirements, while efficiently\nscheduling sensor communications and achieving low detection\nlatency.\nIndex Terms—anomaly detection, false discovery rate control,\nhypothesis test, multi-armed bandit, neuromorphic sensing and\ncommunication.\nI. INTRODUCTION\nNeuromorphic technologies – encompassing sensing [1],\n[2], computing [3], [4], and communication [5], [6], [7],\n[8], [9], [10] – are emerging as a promising paradigm for\nultra-low-power, low-latency sensor networks. Inspired by the\nbrain’s event-driven information processing, a neuromorphic\nsensor records information by producing spikes at times\nmarking significant changes in the monitored environment\n[11]. Thanks to the inherent capacity of neuromorphic sensors\nto respond only to relevant events, incorporating neuromorphic\nsensors in a remote monitoring system can drastically reduce\nJ. Shiraishi and J. Chen contributed equally to this work. The work of\nJ. Shiraishi was supported by European Union’s Horizon Europe research\nand innovation funding programme under Marie Skłodowska-Curie Action\n(MSCA) Postdoctoral Fellowship, “NEUTRINAI” with grant agreement\nNo. 101151067. The work of J. Chen and O. Simeone was supported by the\nEPSRC project EP/X011852/1, and O. Simeone was also supported by an\nOpen Fellowships of the EPSRC (EP/W024101/1). The work of P. Popovski\nwas supported by the Velux Foundation, Denmark, through the Villum\nInvestigator Grant WATER, nr. 37793. (Corresponding author: Jiechen Chen)\nJ. Shiraishi and P. Popovski are with the Department of Electronic\nSystems, Aalborg University, 9220 Aalborg, Denmark (e-mail: {jush,\npetarp}@es.aau.dk). J. Chen and O. Simeone are with the King’s Com-\nmunications, Learning and Information Processing Laboratory, King’s\nCollege London, WC2R 2LS London, U.K. (e-mail: {jiechen.chen, os-\nvaldo.simeone}@kcl.ac.uk)\nNeuro-SN 1\n1\n𝐾\n𝑆\n𝑞ௌ\nଵ\n1. Query (𝒩)\n2. IR transmission\nReader\nEnvironment state\nt\nt\nNeuro-SN 𝐾\n⋮\n𝑞ௌ\n\n⋮\n𝑆መ\nanomaly \ndetection\nFig. 1: In the setup under study, a reader interrogates a subset Nf\nof the K neuromorphic sensor nodes (neuro-SN) at each frame f in\norder to determine whether the behavior of the monitored system is\nnormal or anomalous. The binary variable Sf indicates the presence\nor absence of an anomaly at frame f, while ˆSf represents the\ncorresponding estimate of the reader.\nredundant transmissions, leading to substantial energy and\nbandwidth savings and supporting the deployment of large\npopulations of wireless sensors [5], [10].\nFor example, reference [10] demonstrated an asynchronous\nwireless network of salt-grain-sized sensors that mimic neu-\nral spikes, showing that hundreds of microscale nodes can\ncommunicate sporadic bursts of data to a central receiver\nusing backscattering. Such neuromorphic networks have been\nhighlighted for their potential in brain–machine interfaces,\nwhere implantable sensors would relay neural spikes in real\ntime, as well as in other domains like wearable devices and\nenvironmental sensing [12].\nIn applications such as remote surveillance and autonomous\nsystems, timely detection of abnormal events is critical. No-\ntably, neuromorphic vision sensors can detect sudden motion\nor the appearance of an intruder – e.g., an unauthorized drone\n– by producing rapid spikes, while remaining largely inactive\nunder normal conditions [1]. In environmental monitoring, dis-\ntributed sensors might pick up abrupt changes in temperature\nor smoke indicative of a forest fire [13]. In these scenarios,\nthe system must provide reliable anomaly detection despite\nlimited energy supply and communication bandwidth.\nTo this end, the detection algorithm needs to be statistically\nrobust, minimizing false alarms to avoid wasting attention\nand energy on benign events. Furthermore, the network must\nintelligently decide which sensors to query and when, so\nas to maximize the chance of detecting an anomaly while\nconserving resources. In this regard, a common reliability\ncriterion is maintaining the false discovery rate (FDR) below\narXiv:2510.14688v1 [cs.LG] 16 Oct 2025\n\na target level, ensuring that only a small fraction of reported\nanomalies are false positives [14], [15], [16]. Achieving a low\nFDR is especially important in safety-critical applications, like\nhealth monitoring or security, to maintain trust in the system’s\nalarms.\nThis paper addresses these challenges by proposing a novel\nsolution that integrates tools from statistical FDR control\n[17] and adaptive learning [18] into a neuromorphic wireless\nsensing framework.\nA. Related Work\nNeuromorphic sensing: Neuromorphic sensing aims to em-\nulate biological sensory systems by encoding information\nthrough sparse, event-driven signals rather than continuous\nstreams. A prominent category is vision sensing, where dy-\nnamic vision sensors (DVSs) detect asynchronous changes in\nbrightness at each pixel, achieving low latency and high en-\nergy efficiency. Prophesee and IniVation are among the leading\ncompanies in this area, offering state-of-the-art event-based\ncameras for machine vision and robotic applications [19], [20].\nBeyond vision, companies like SynSense are developing ultra-\nlow-power neuromorphic processors such as Speck, which en-\nable local event-driven processing for embedded intelligence\n[21].\nNeuromorphic communication: Recent work [6] has intro-\nduced end-to-end neuromorphic communication systems that\nreplaces conventional sensing and frame-based radio with\na fully spike-driven pipeline. In it, neuromorphic sensors\nproduce asynchronous event spikes which are encoded by a\nspiking neural network (SNN) and transmitted as impulse ra-\ndio (IR) pulses, then decoded by another SNN at the receiver.\nThis all-spike design was shown to significantly lower latency\nand improve efficiency compared to conventional synchronous\ndigital communication.\nBuilding on this, reference [5] extended the approach to\na multi-user, fading-channel scenario. In it, each internet of\nthings (IoT) device uses a neuromorphic event sensor, on-\ndevice SNN, and IR transmitter, communicating over a shared\nwireless channel to a receiver that also employs an SNN. A\nkey principle in these systems is event-driven data acquisition,\nwhich allows the wireless network to inherently focus on\nthe “semantics” of the data – i.e. changes that carry new\ninformation – rather than sending redundant raw readings.\nFurther energy savings can be obtained by combining IR-\nbased communication with wake-up radio triggers [7], adopt-\ning different design criteria [22], and leveraging multi-level\nspikes [9]. In particular, reference [9] shows that associating\neach spike with a small payload can improving inference\naccuracy without increasing spike count. This study also\nincludes an experimental evaluation using software-defined\nradios.\nOnline anomaly detection: Online anomaly detection fo-\ncuses on identifying abnormal patterns in streaming data\nin real time, without requiring access to the entire dataset\nbeforehand. For example, reference [14] proposed online false\ndiscovery rate control methods for anomaly detection in time\nseries to address the challenges of rare anomalies and temporal\ndependencies. Reference [15] proposed an unsupervised real-\ntime anomaly detection algorithm for streaming data, which\nis capable of continuous learning and early detection of both\nspatial and temporal anomalies without requiring labeled data\nor batch processing.\nMulti-armed\nbandit\noptimization:\nMulti-armed\nbandit\n(MAB) optimization has been extensively studied in previous\nworks. For example, reference [18] proposed the track-and-\nstop strategy for best-arm identification under fixed-confidence\nsettings. Reference [23] studied the complexity of best-arm\nidentification in stochastic MAB under both fixed-budget and\nfixed-confidence settings. To speed up the convergence of\nMAB in unseen tasks, [24] proposed for the first time the\nuse of contextual meta-learning that incorporates auxiliary\noptimization tasks knowledge.\nB. Main Contributions\nBuilding on the above context and prior art, this pa-\nper develops a protocol for online anomaly detection via\nneuromorphic sensor networks. The main contributions are\nsummarized as follows:\n• Online anomaly detection framework based on neuromor-\nphic sensors: We introduce a novel online anomaly detec-\ntion system based on neuromorphic sensor networks. As\nshown in Fig. 1, this model implements a pull-based com-\nmunication strategy, where a central reader intermittently\nqueries neuromorphic sensor nodes and receives event-\ndriven IR spike signals. The model explicitly incorporates\nthe sparse, asynchronous nature of neuromorphic data\nand the requirement to control the false discovery rate in\ndetection decisions.\n• Adaptive FDR control with e-values: We design an adap-\ntive statistical testing framework that guarantees FDR\ncontrol at any desired level in an online setting. In\ncontrast to traditional methods that rely on fixed p-\nvalue thresholds, our framework utilizes e-values [25] to\naccumulate evidence over each frame. This allows the\ndetector to flexibly adjust its confidence as new spike\ndata arrive, ensuring that the expected proportion of false\nalarms remains below the target level even when the\nanomaly occurrence rate is unknown and nonstationary.\n• Best-arm identification for sensor scheduling: We in-\ntroduce a sensor scheduling policy based on best-arm\nidentification principles from multi-armed bandit theory\n[18]. At each time frame, the reader must decide which\nsensors to query next. Our policy dynamically learns\nthe sensors that are most likely to yield an anomaly\nindication by treating each sensor as an “arm’ with an\nunknown reward. By continuously updating estimates\nand concentrating queries on the most promising sensor,\nthe policy achieves efficient allocation of limited query\nopportunities, accelerating the detection of anomalies\nunder uncertainty.\n• Extensive performance evaluation: We provide a thor-\nough performance evaluation of the proposed system\n\nthrough analysis and simulations. The results demon-\nstrate that our approach can detect anomalies quickly and\nreliably under a variety of conditions. In particular, we\nshow that the FDR is maintained below the target level\nin all tested scenarios, validating the effectiveness of the\ne-value based control. Moreover, the adaptive scheduling\npolicy is shown to significantly reduce detection delay\nand resource usage compared to non-adaptive or random\nscheduling baselines.\nThe rest of the paper is organized as follows. Section\nII describes the pull-based neuromorphic wireless commu-\nnications system, while the online anomaly detection based\non hypothesis test framework is described in Section III.\nSection IV explains the scheduling policy optimization based\non best-arm identification. Experimental setting and results\nare described in Section V. Finally, Section VI concludes the\npaper.\nII. SYSTEM MODEL\nA. Setting\nFig. 1 shows the basic setup considered in this paper.\nWe consider a pull-based neuromorphic wireless communica-\ntions scenario where the reader conducts a continuous online\nanomaly detection at each time frame based on the deployment\nof neuromorphic sensor nodes (neuro-SN) [10].\nThe K neuro-SNs observe a common environment via\nlocal neuromorphic sensors [5], and communicate with the\nreader via IR. We define the set of node indices by K =\n{1, 2, . . . , K}. Through communication with the neuro-SNs,\nthe reader continuously monitors the environment to detect\nanomalies, while ensuring that the FDR is maintained below\na pre-determined level, denoted as α. The FDR measures\nthe fraction of detected anomalies that do not correspond to\nactual anomalies. Specific applications include real-time forest\nfire detection [13] or drone activity monitoring [5] based on\nwireless sensor networks.\nAs in [14], anomalies are defined as rare events that\nsignificantly deviate from expected patterns. In our setting,\nan anomaly causes neuromorphic sensors to produce spikes,\nalso known as events, at a significantly higher rate than\nunder normal conditions. For example, neuromorphic cameras\nproduce spikes at times in which activity is recorded in the\nmonitored scene [5]. Thus, in the presence of significant\nactivity, e.g., from unauthorized drones, the spike rate is\nincreased.\nAs illustrated in Fig. 2, we adopt a discrete time model, in\nwhich time is divided into frames indexed by f = 1, 2, . . . , F.\nEach frame consists of three phases:\n1) Downlink query signal transmission: At the beginning\nof each frame, the reader first sends a query signal to a\nselected subset of devices.\n2) Sensing and uplink IR transmission: Upon detecting\nthe query signal, e.g., via a wake-up radio [7], [26],\na neuro-SN activates its neuromorphic sensor and its IR\ntransmitter. The IR transmission phase is divided into\nt\n⋯\nt\n1-st frame\n2nd frame\n𝐹-th frame\nSensing and IR transmission\nAnomaly detection\n𝐿uplink slots\nQuery transmission\ndownlink\nanomaly \ndetection\nFig. 2: Frame structure illustrating the query transmission from the\nreader (downlink), sensing and impulse radio (IR) transmission by\nthe selected neuro-SN during L uplink slots, and anomaly detection\nat the reader.\nL time slots, during which the neuro-SNs sense and\ntransmit in an online and event-driven manner.\n3) Anomaly detection at the reader: Finally, the reader\ndetermines whether the environment is in an anomalous\nstate based on the received signals.\nB. Observation Models\nFollowing [14], we model the presence or absence of an\nanomaly in a frame indexed by f as an arbitrary, unknown\nsequence of binary state Sf, which are defined as\nSf =\n(\n0,\nno anomaly,\n1,\nanomaly.\n(1)\nWe model the observations of each neuromorphic sensor\nk at the f-th frame via a Bernoulli process with spiking\nrate determined by the current state Sf. Accordingly, for\neach sensor k, we set two spiking rates corresponding to the\nabsence or presence of an anomaly, namely, qk\n0 and qk\n1. As\nillustrated in Fig. 3, we assume the inequality qk\n0\n≤qk\n1 for\nall k\n∈\n{1, 2, . . . , K}. This indicates that anomalies are\ncharacterized by larger spiking rates.\nAccordingly, each node k obtains a spike sequence sk\nf =\n[sk\nf,1, . . . , sk\nf,L] ∈{0, 1}L, where sk\nf,l = 1 indicates that node\nk records a spike at time instant l of the f-th frame, and\nsk\nf,l = 0 indicates that no spike was recorded. This means\nthat we have\nsk\nf,l\n∼\ni.i.d. Bern(qk\nf) with qk\nf =\n(\nqk\n0\nif Sf = 0\nqk\n1\nif Sf = 1.\n(2)\nThrough an initial calibration phase, the reader can ob-\ntain the statistics on the spike rates in normal conditions.\nTherefore, the reader is assumed to know the value of the\nprobability qk\n0 for all k\n∈\n{1, 2, . . . , K}. In contrast, we\nassume the reader does not know the spiking probability qk\n1\nin the presence of anomalies.\nC. Scheduling and Impulse Radio Transmission\nIn this work, we assume time division multiple access\n(TDMA)-based communication between the reader and neuro-\nSNs, which consists of downlink phase and uplink phases.\n\nSpike rate\nNeuro-SN\n1\n2\n𝐾\n⋯\n⋯\nSpike rate for the normal state\n1\n0\nSpike rate for the anomalous state\n𝑞ଵ\nଵ\n𝑞ଵ\nଶ\n𝑞\nଵ\n𝑞\nଶ\n𝑞ଵ\n\n𝑞\n\nFig. 3: Each k-th sensor is characterized by a known spiking\nprobability qk\n0 in normal conditions, and a higher, unknown spiking\nprobability qk\n1 ≥qk\n0 in the presence of an anomaly.\n0\n1\nTransmitted bit\nReceived bit\n𝜀ଵ\n1 −𝜀ଵ\n0\n1\n1 −𝜀ଵ\n𝜀ଵ\nFig. 4: Illustration of a binary asymmetric channel.\nAt each time frame f, the reader requests data from a subset\nNf ⊆{1, . . . , K} of neuro-SN by multicasting a query signal\nthat encodes the target node IDs and its schedule information.\nThe reader schedules |Nf| = Cmax nodes, and transmits this\ncontrol information during the downlink phase. In practice, the\ncardinality Cmax of the scheduled set depends on the available\nuplink capacity.\nThe nodes that receive the query signal begin sensing over\nthe L time instants within the frame. The spike sequence sk\nf is\ntransmitted by all scheduled sensors in Nf in an online and\nevent-driven manner using IR by directly mapping a spike\nfrom the sensor to an electromagnetic pulse [5].\nAs illustrated in Fig. 4, we model the channel as a binary\nasymmetric channel, where transmitted bits may be flipped at\nthe receiver, with bit 0 flipping to 1 with probability ϵ01, and\nbit 1 flipping to 0 with probability ϵ10. The IR receiver at\nthe reader estimates the spike sequence transmitted by each\nneuro-SN. After receiving the estimated spike sequence ˆsk\nf =\n[ˆsk\nf,1, . . . , ˆsk\nf,L] ∈{0, 1}L for each node k ∈Nf at time frame\nf, the reader uses this information to determine whether the\ncurrent state Sf is normal or anomalous, as described in the\nnext section.\nIII. ONLINE ANOMALY DETECTION\nIn this section, we first present the hypothesis test frame-\nwork, assuming that the reader has collected data from the\nsubset of nodes Nf at the f-th frame. Then, we describe the\nproposed anomaly detection mechanism for online anomaly\ndetection. The problem of optimizing node scheduling will be\naddressed in the next section.\nA. Hypothesis Testing Framework\nAt the end of each frame f, the reader tests whether the\ncurrent system state is normal or anomalous. Specifically, it\nconsiders the following hypothesis\nHf : Sf = 0 (normal state)\n(3)\nagainst the alternative hypothesis that there is an anomaly, i.e.,\nSf = 1. Equivalently, the reader evaluates the null hypothesis\nfor the requested sensors in Nf as follows\nHf : sk\nf ∼\ni.i.d. Bern(qk\n0), ∀k ∈Nf\n(4)\nagainst the composite alternative hypothesis\n¯Hf : sk\nf ∼\ni.i.d. Bern(qk\n1), for some qk\n1 > qk\n0, ∀k ∈Nf.\n(5)\nIn order to conduct a hypothesis test during the f-th frame,\nthe reader uses the spike sequences ˆsf = {ˆsk\nf}k∈Nf observed\nduring this frame and creates the test statistics Xf. In addition,\nthe reader selects the rejection threshold αf by considering the\ntarget metrics, such as FDR. The null hypothesis Hf is that\nthere is no anomaly, such that if this hypothesis is rejected,\nthe reader declares an anomalous state. The decision at frame\nf is denoted as ˆSf ∈{0, 1}, which can be described as\nˆSf = h(Xf, αf). Here, h(Xf, αf) is an indicator function\nrepresenting the rejection decision based on Xf and αf. In\nthe following subsection, we first describe our target metrics\nand use test statistics based on e-values. Finally, we describe\nhow to select the rejection thresholds αf considering the target\nmetrics.\nB. Decaying Memory Time-Averaged False Discovery Rate\nLet Ff|0\nrepresent the subset of the frame indices\n{1, 2, . . . , f} for which the environment is in a normal state,\ni.e., Ff|0 = {f ′ ∈{1, 2, . . . , f} : Sf ′ = 0}. At frame f,\nthe cumulative number of anomalies declared by the reader,\naveraged with a forgetting factor 0 < δ < 1 over the past\nframes, is\nˆAf =\nf\nX\nf ′=1\nδf−f ′1( ˆSf ′ = 1),\n(6)\nwhere 1(·) is the indicator function, which equals to 1 if the\nargument is true, and equals to 0 otherwise. In a similar way,\nthe weighted time-average of incorrectly detected anomalies\nis\nFf =\nf\nX\nf ′=1\nδf−f ′1( ˆSf ′ = 1)1(Sf ′ = 0).\n(7)\nWe are interested in controlling the decaying-memory time-\naveraged FDR, which is defined as the ratio [14]\nFDRf = E\n\u0014\nFf\nmax( ˆAf, 1)\n\u0015\n,\n(8)\n\nThis quantity measures the fraction of estimated anomalies\nthat were actually normal states. Thus, controlling the FDR\nin (8) ensures that the system does not cause an excessive\nfraction of false alarms. Accordingly, the design target is\nto determine a sequence of rejection thresholds {αf}∞\nf=1\nensuring that the metric FDRf is controlled below a pre-\ndetermined level α, as\nFDRf ≤α, for all frames f = 1, . . . , F.\n(9)\nC. Decaying Memory Time-Averaged True Discovery Rate\nThe bound in (9) can be trivially guaranteed by setting αf =\n0 for all frames f. In this case, no anomalies are detected,\nand thus there is no false detection of anomalies. However,\nthis approach clearly fails to detect any potential anomalies.\nTherefore, a testing strategy must be also evaluated in terms\nof its ability to detect true anomalies. To account for this, we\nconsider the true discovery rate (TDR) as the average fraction\nof truly anomalous states that are correctly detected.\nIn a similar way to the decaying-memory time-averaged\nFDR, let Af denote the cumulative number of anomalies\nthat occur in the environment at frame f, averaged with a\nforgetting factor 0 < δ < 1, which is defined as\nAf =\nf\nX\nf ′=1\nδf−f ′1(Sf ′ = 1).\n(10)\nSimilarly, the weighted time-averaged of correctly detected\nanomaly, denoted as Tf can be expressed as\nTf =\nf\nX\nf ′=1\nδf−f ′1( ˆSf ′ = 1)1(Sf ′ = 1).\n(11)\nFinally, we define decayed-memory time-averaged TDR,\nwhich is defined as the ratio, as described below:\nTDRf = E\n\u0014\nTf\nmax(Af, 1)\n\u0015\n.\n(12)\nThis quantity measures how many actual anomaly events the\nreader successfully detects. This depends on the scheduling\npolicy, which will be described in Sec. IV.\nThe metrics FDR in (8) and TDR in (12) will be used as a\nbasis for designing rejection threshold αf. Specifically, in each\nframe, the reader sets a rejection threshold αf based on the\npast rejection history so that FDR condition in (9) is always\nguaranteed while providing high TDR. Specific algorithms\nshould be designed by taking into account the well-known\npiggybacking and α death problem [14], whose details are\nspecified in Sec. III-E.\nD. Construction of E-Values\nAn e-value ef\n∈[0, ∞] is a function of the current\nobservations Sf that satisfies the inequality E[ef] ≤1 when\nthe null hypothesis Hf is true, i.e., when the state is normal\n(Sf = 0) [25]. To construct an e-value ef for each time frame\nf, the reader estimates the spike rate from node k at the f-th\nframe as\nˆqk\nf = 1\nL\nL\nX\nl=1\nˆsk\nf,l.\n(13)\nIntuitively, a larger estimated rate ˆqk\nf suggests a higher likeli-\nhood that the environment is experiencing an anomaly.\nSince the anomalous spike rates are unknown to the reader,\nan e-value can be constructed by relying on the plug-in\nmethod [25] along with the merging of e-values for the\nscheduled sensors k ∈Nf. Specifically, for each sensor\nk ∈Nf, an e-value is obtained via the likelihood ratio\nek\nf =\n ˆψk\nf|1\nψk\nf|0\n!Lˆqk\nf \n1 −ˆψk\nf|1\n1 −ψk\nf|0\n!L(1−ˆqk\nf )\n,\n(14)\nwhere\nψk\nf|0 = qk\n0(1 −ϵ10) + (1 −qk\n0)ϵ01\n(15)\nis the spiking probability that the reader receives from the\nneuro-SN k at the f-th frame when the environment is normal\nstate and ˆψk\nf|1 is the maximum likelihood of received spike\nrate at the reader from neuro-SN k at the f-th frame when\nthe environment is anomalous state. This is obtained as\nˆψk\nf|1 = ˆqk\n1(1 −ϵ10) + (1 −ˆqk\n1)ϵ01,\n(16)\nwhere ˆqk\n1 is the maximum likelihood estimate of the spike rate\nqk\n1, which is given by\nˆqk\n1 = max(qk\n0, ˆqk\nf).\n(17)\nFinally, the reader merges the e-values {ek\nf}k∈Nf to obtain\na single e-value via the arithmetic mean [25]\n¯ef(Nf) =\n1\n|Nf|\nX\nk∈Nf\nek\nf.\n(18)\nE. Design of the Rejection Thresholds\nIn order to ensure FDR condition (9), we adopt the al-\ngorithm detailed in [14]. Accordingly, at the end of each\ntime frame f, the reader dynamically controls the decision\nthreshold αf based on the past rejection history. An anomaly\nis declared if the statistic ef is larger than a threshold 1/αf,\ni.e.,\nˆSf = 1\n\u0012\nef > 1\nαf\n\u0013\n.\n(19)\nTo describe the update of the threshold αf, let ρj denote the\ntime of j-th detected anomaly, i.e.,\nρj = min\n\n\nf ≥0\n\f\f\f\f\f\nf\nX\nf ′=1\n1\n\u0012\nef ′ >\n1\nαf ′\n\u0013\n≥j\n\n\n,\n(20)\nand let {γf}∞\nf=1 denote a non-increasing sequence summing\nto 1, where γf = 0 for all f ≤0. We specifically use the\ndefault sequence [14]\nγf ∝\nlog (max(f, 2))\nf exp\n\u0010p\nlog(f)\n\u0011.\n(21)\n\nFinally, let ˜γf = max(γf, 1−δ) as in [14]. Then, the rejection\nthreshold αf is updated as\nαf = α η ˜γf + α\nX\nj≥1\nδf−ρjγf−ρj,\n(22)\nwhere η is smoothing parameter η > 0.\nProposition 1 [14]. When the decision threshold αf is up-\ndated according to (22) at each frame f, the anomaly de-\ntection rule in (19) ensures that the decaying-memory time-\naveraged FDR defined in (8) is controlled at level α for all\nframes f, thus satisfying condition (9).\nIV. SCHEDULING POLICY OPTIMIZATION\nIn order to increase the TDR while maintaining FDR\ncontrol as in (9), an ideal scheduling policy would select at\neach frame f the subset Nf of sensors k with the highest\nspiking rates qk\n1 under an anomalous state. In fact, this would\nmaximize the chance of correctly detecting an anomaly via\nthe test (19). However, the true spiking rates qk\n1, for k ∈K,\nare not known by the reader. Furthermore, anomalies occur\narbitrarily and unpredictably, making it challenging to estimate\nthe probabilities qk\n1.\nIn this section, we address the problem of optimizing the\nset Nf by tackling these challenges. To start, we study the\nproblem of identifying the best sensor\nk∗= arg max\nk∈K\nqk\n1,\n(23)\nthus concentrating on the case Cmax = |Nf| = 1, in which the\nuplink channel supports the transmission of a single neuro-\nSN. In this context, formally, the goal in this section is to\nminimize the number of frames used to identify the best neuro-\nSN k∗with a probability of error not exceeding a threshold\nδs ∈(0, 1). An extension to any uplink capacity Cmax ≥1\nwill be also provided at the end of this section.\nA. Multi-Armed Bandit Formulation\nTo facilitate the design of a scheduling policy, we make the\nworking assumption that the anomaly process is i.i.d. across\nthe frame index f, with an unknown anomalous probability\nπ1, i.e., Sf ∼Bern(π1). Furthermore, we also assume that\nthe normal spiking probabilities of all neuro-SNs are equal,\ni.e.,\nqk\n0 = q0,\n(24)\nfor all k = 1, . . . , K. Under these simplified assumptions, the\nspike rates\n1\nL\nPL\nl=1 sk\nf,l of each neuro-SN k is i.i.d. across\nframes f, with an average\nµk = (1 −π1)q0 + π1qk\n1.\n(25)\nThis implies that the problem (23) of identifying the best\nsensor becomes equivalent to the problem of finding a node\nwhose average spike rate µk is the highest. i.e.,\nk∗= arg max\nk∈K\nµk.\n(26)\nThis optimization can be formulated as the fastest best\narm identification problem, studied in [18]. As demonstrated\ntherein, a strategy known as track-and-stop reviewed next, is\noptimal.\nB. Track-and-Stop Strategy\nAt each f-th frame, the reader estimates the spike rate µk\nof neuro-SN k as\nˆµk\nf =\n1\nN k\nf\nf\nX\nf ′=1\nˆqk\nf ′1 (k ∈Nf ′) ,\n(27)\nwith\nN k\nf =\nf\nX\nf ′=1\n1 (k ∈Nf ′)\n(28)\nbeing the number of frames in which neuro-SN k was sched-\nuled and with ˆqk\nf ′ defined in (13).\nDefine as Uf = {k : N k\nf < √f −K\n2 } the subset of nodes\nthat have been scheduled fewer than √f−K\n2 times up to frame\nf. If this set is not empty, the track-and-stop strategy selects\none of the neuro-SNs that was scheduled least frequently,\ni.e., in the smallest number of frames, emphasizing explo-\nration [18]. Otherwise, the scheduling selection accounts also\nfor the current spiking rate estimates ˆµf = {ˆµ1\nf, ˆµ2\nf, . . . , ˆµK\nf },\nconsidering both exploitation and exploration. Overall, the\nscheduling decision is\nnf =\n\n\n\n\n\narg min\nk∈K\nN k\nf ,\nif |Uf| > 0,\narg min\nk∈K\n\b\nN k\nf −f · wσ(k)(ˆµf)\n\t\n,\notherwise,\n(29)\nwhere σ : {1, 2, . . . , K} →{1, 2, . . . , K} is a function that\nreturns the rank of each node k in terms of the estimated spike\nrate ˆµk in descending order, satisfying the inequalities ˆµσ(1)\nf\n≥\nˆµσ(2)\nf\n. . . ≥ˆµσ(K)\nf\n. The function wσ(k)(ˆµf), which returns\npositive value, is detailed in [18] and satisfies the following\nproperty is true: w1(µ) ≥. . . ≥wK(µ). This way, neuro-\nSNs with a larger estimated spiking rate are prioritized by the\nselection rule (29).\nC. Extension of the Scheduling for Any Scheduling Capacity\nIn this subsection, we extend the track-and-stop strategy to\nthe case where it is possible to schedule Cmax = |Nf| ≥1\nneuro-SNs. To this end, we propose that, at the beginning\nof each frame, the reader selects Cmax = |Nf| nodes for\nscheduling by recursively applying the track-and-stop method\nin (29). Specifically, the reader selects the scheduled nodes\none-by-one, obtaining the set Nf = {ni\nf}Cmax\ni=1 . In more detail,\ndefine U i\nf = {k ∈K \\ {nj\nf}i−1\nj=1 : N k\nf < √f −K\n2 } as the\nset of neuro-SNs not yet included in the set Nf that have\nbeen scheduled in the past frames fewer than √f −K\n2 times.\n\nThen, the i-th scheduling node at the f-th frame is selected\nby applying the rule (29) as\nni\nf =\n\n\n\n\n\n\n\narg min\nk∈K\\{nj\nf }j=i−1\nj=1\nN k\nf ,\nif |U i\nf| > 0,\narg min\nk∈K\\{nj\nf }j=i−1\nj=1\nN k\nf −fwσ(k)(ˆµf),\notherwise,\n(30)\nwhere function wσ(k)(·) is as defined in (29).\nV. NUMERICAL RESULTS\nIn this section, we present numerical results to validate the\nperformance of the proposed online anomaly detection system.\nA. Simulation Setting\nIn our experiments, the anomaly process Sf is i.i.d. across\nall frames f, so that at each frame f the environment is in\nthe anomalous state with probability π1 and in the normal\nstate with probability 1 −π1, independently from frame to\nframe. We simulate a setting in which, as in (24), the spike\nrate qk\n0 of a normal state is the same across all nodes, i.e.,\nqk\n0 = q0 for all k ∈K. In contrast, the spiking rate qk\n1 under\nthe anomalous state is uniformly generated within the interval\n[q0, q0+∆max] for each neuro-SN k. Table I shows our default\nparameters setting, which are used throughout the experiments\nunless stated otherwise.\nTABLE I: Default parameter setting.\nParameters\nValues\nNumber of neuro-SN K\n5\nForgetting factor δ\n0.99 [14]\nη\n0.99\nNumber of uplink slots L\n50\nTarget FDR α\n0.1 [14]\nSimulation runs\n103\nSpiking rate under the normal state q0\n0.10\nAnomaly ratio π1\n0.05\nError rates (ϵ01, ϵ10)\n(0, 0)\nUpper bound on the spiking probability ∆max\n0.5\nChannel capacity Cmax\n1\nB. On the Validity of Online Anomaly Detection\nWe first investigate the validity of the online anomaly\ndetection algorithm introduced in Sec. III. To this end, we\ncompare the performance of the proposed scheme with adap-\ntive threshold αf described in (22) with a more conventional\nfixed-threshold strategy, in which we apply the threshold\nαf = α for all f ∈[1, F]. Note that this combined approach\ndoes not satisfy the theoretical statistical guarantee in (9), and\nits performance can only be evaluated via numerical results.\nFig. 5 shows FDR and TDR versus the time frame f\nfor fixed anomaly threshold method and for the proposed\ndynamic anomaly detection method in (22), where we set\nthe number of neuro-SN K = 1. As anticipated, the fixed-\nthreshold is not able to always satisfies the FDR constraint\nin (9). This is because this approach does not take into account\nthe past rejection history for the current anomaly decision\ntasks. A fixed threshold can achieve a higher TDR than the\nproposed method, but only by violating the FDR constraint\nin (9). In contrast, the introduced methodology, dynamic\nthreshold always satisfies the FDR constraint α, validating\nProposition 1.\nFig. 6 shows the FDR and TDR as a function of the anomaly\nprobability π1, where we set K = 1 and F = 1000. We\nobserve that the fixed-threshold approach suffers from the\nhigher FDR when the anomaly probability π1 is small, while\nthe proposed dynamic anomaly detection threshold method\ncan always satisfy the FDR constraint. When the anomaly\nprobability π1 is small, with a dynamic threshold, the reader\ntakes a conservative decision by setting a relatively smaller\nthreshold αf, will a fixed-threshold method cannot adjust to\nthe anomaly probability.\nC. Random v.s. Track-and-Stop Scheduling\nWe now study the impact of the scheduling strategy. To\nthis end, Fig. 7 shows the FDR and the TDR versus the time\nframe f for the proposed anomaly detection scheme under\na random scheduling strategy that selects Cmax neuro-SNs\nuniformly at random at every frame and for the track-and-stop\nmethod presented in Sec. IV. We set the parameters described\nin Table I with the uplink channel capacity Cmax = {1, 2}\nand anomaly probability π1 = 0.1.\nAll schedulers, when coupled with the proposed anomaly\ndetection, are seen to satisfy the FDR requirements, confirm-\ning the theoretical results in Proposition 1. Moreover, the\nfigure highlights the benefits of track-and-stop, increasing the\nTDR for both uplink capacities Cmax = 1 and Cmax = 2.\nThis advantage stems from the capacity of track-and-stop to\nselect the neuro-SNs that are most likely to contribute to the\nidentification of the neuro-SNs with highest spiking rates.\nFrom Fig. 7 we also see that TDR becomes larger as the\nnumber of scheduled nodes Cmax increases, demonstrating\nthe importance of collecting information from multiple neuro-\nSNs. This is because by calculating e-value (18) based on the\nspike rate from multiple nodes, the reader can mitigate the\nnegative impact of small e-value caused by the selection of\nnodes having small spike rate in anomalous state, qk\n1.\nD. Impact of System Parameters\nIn this subsection, we focus on the performance of track-\nand-stop scheduling, evaluating the impacts of different sys-\ntem parameters.\n1) On the FDR constraint α: Fig. 8 shows the FDR and\nTDR as a function of FDR constraint α. The figure highlights\nthe trade-off between TDR and FDR: as the allocated FDR\nconstraint α becomes larger, the reader can more easily detect\nanomalies, increasing the TDR. We can also see that TDR\nbecomes larger as the forgetting factor δ becomes smaller. In\nfact, with a smaller δ, the FDR constraint puts higher weight\non the results of the latest decisions, decreasing the importance\nof past false alarms.\n2) On the sensitivity of the sensors: We now study the\ndependence of the system performance on the parameter\ndefining the quality of the observations of sensors. First, we\n\n100\n200\n300\n400\n500\n0\n0.1\n0.2\nFixed αf = α\nDynamic αf\nFDR constraint\nf\nFDR\n100\n200\n300\n400\n500\n0.0\n0.25\n0.50\n0.75\n1.0\nFixed αf = α\nDynamic αf\nf\nTDR\nFig. 5: FDR and TDR versus the time frame f for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed dynamic\nanomaly decision threshold based on (22), where we set K = 1 and π1 = 0.05.\n0.025\n0.05\n0.075\n0.10\n0\n0.1\n0.2\n0.3\nFixed αf = α\nDynamic αf\nFDR constraint\nπ1\nFDR\n0.025\n0.05\n0.075\n0.10\n0.0\n0.25\n0.50\n0.75\n1.0\nFixed αf = α\nDynamic αf\nπ1\nTDR\nFig. 6: FDR and TDR versus the anomaly probability π1 for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed\ndynamic anomaly decision threshold based on (22), where we set K = 1 and F = 1, 000.\nvary the upper bound ∆max on the spiking rate in the presence\nof anomalies. From Fig. 9, we can see that TDR is improved as\nthe upper bound ∆max becomes larger, that is, the sensitivity\nimproves. This is because a larger upper bound ∆max tends\nto increase the difference between the spike rate of neuro-SN\nk in the normal state qk\n0 and that in the anomalous state qk\n1,\nenabling the reader to detect anomalies with higher accuracy\n(see Fig. 3).\nFig. 10 shows the TDR as a function of the frame index f\nfor different numbers of uplink slots L. From Fig. 10, we can\nsee that TDR becomes larger as L increases because the reader\ncan execute anomaly detection task based on more number\nof received spikes, realizing accurate decision based on the\ne-value in (18) and decision threshold defined in (22). For\nexample, to achieve TDR larger than 0.9 under ∆max = 0.5,\nthere is a need for L = 100 uplink slots.\n3) Impact of anomaly probability π1: Fig. 11 shows TDR\nagainst f for track-and-stop, where we set the anomaly\nprobability to π1 = {0.05, 0.10, 0.15}. As π1 increases, the\nprobability that the reader obtains the higher e-value in the\nsingle frame also increases, making it more likely that the\nreader rejects the hypothesis at the current frame. As a result,\nwe can see that TDR increases as π1 becomes larger.\n4) On the channel error rate: To evaluate the impact of\nchannel error in the binary asymmetric channel described in\nSec. II-C, we vary the error probability ϵ, by setting: (i)\nϵ01 = ϵ and ϵ10 = 0; (ii) ϵ10 = ϵ and ϵ01 = 0; (iii)\nϵ01 = ϵ10 = ϵ. From Fig. 12, we see that the error probability\nϵ10 has the largest impact on TDR. In fact, as increase in\nϵ10 causes spike losses, which may be critical for anomaly\ndetection. Conversely, as the error probability ϵ01 increases, it\nbecomes more likely that the reader mistakenly detects false\nspikes, causing the reader to mistakenly detect an anomaly\nevent when the environment is in a normal state. As the reader\nmust satisfy the FDR constraint in (9), in this case the reader\nmust take a conservative decision when the actual anomaly\nevent happens, reducing the TDR. When both errors have a\nnon-zero probability, it is affected by both spike losses and\nfalsely detected spikes, causing further decreases in the TDR.\nVI. CONCLUSION\nThis paper proposed a low-power online anomaly detection\nframework for neuromorphic wireless sensor networks. We\nhave introduced an algorithm that can control the false dis-\ncovery rate by incorporating e-values test statistics calculated\nbased on the received spike sequences at each frame. We have\nalso introduced a track-and-stop methods for scheduling nodes\n\n200\n400\n600\n800\n1,000\n0\n0.1\nFDR constraint\nf\nFDR\n200\n400\n600\n800\n1,000\n0.50\n0.75\n1.0\nRandom (Cmax = 1)\ntrack-and-stop (Cmax = 1)\nRandom (Cmax = 2)\ntrack-and-stop (Cmax = 2)\nf\nTDR\nFig. 7: FDR and TDR versus the time frame f for random sampling scheme and track-and-stop.\n0.05\n0.10\n0.15\n0.20\n0\n0.1\n0.2\nFDR constraint\nα\nFDR\n0.05\n0.10\n0.15\n0.20\n0.50\n0.75\n1.0\nδ : {0.8, 0, 9, 0.99}\nα\nTDR\nFig. 8: FDR and TDR versus the FDR constraint α for track-and-stop\n(the forgetting factor δ = {0.8, 0.9, 0.99}).\n100\n200\n300\n400\n500\n0\n0.2\n0.4\n0.6\n0.8\n1\n∆max = 0.2\n∆max = 0.3\n∆max = 0.4\n∆max = 0.5\nf\nTDR\nFig. 9: TDR versus the time frame f for track-and-stop with different\nvalues of upper bound of sensor reading ∆max.\nbased on the channel constraint. Our numerical results show\nthat the proposed approach can track the nodes having the\nlargest spike efficiently with high TDR, while maintaining the\nFDR below the predetermined threshold.\nFuture work includes designing pull-based communication\nprotocol in neuromorphic communication while taking into\naccount aspects such as information freshness and value of\ninformation. Designing neuromorphic communication proto-\n100\n200\n300\n400\n500\n0\n0.2\n0.4\n0.6\n0.8\n1\nL = 25\nL = 50\nL = 75\nL = 100\nf\nTDR\nFig. 10: TDR versus the time frame f for track-and-stop with\ndifferent values of the number of uplink slots L.\n100\n200\n300\n400\n500\n0.6\n0.7\n0.8\n0.9\n1\nπ1 = 0.05\nπ1 = 0.10\nπ1 = 0.15\nf\nTDR\nFig. 11: TDR versus the time frame f for track-and-stop scheme\nwith different values of anomaly potion π1 = {0.05, 0.10, 0.15}\ncols for heterogeneous neuro-SNs with different capabilities\nin terms of sensing and communication is also an interesting\nresearch avenue.\nREFERENCES\n[1] P. Lichtsteiner, C. Posch, and T. Delbruck, “A 128 x 128 120db 30mw\nasynchronous vision sensor that responds to relative intensity change,”\n\n0\n0.02\n0.04\n0.06\n0.08\n0.4\n0.5\n0.6\n0.7\n0.8\nϵ01 = ϵ, ϵ10 = 0\nϵ01 = 0, ϵ10 = ϵ\nϵ01 = ϵ, ϵ10 = ϵ\nϵ\nTDR\nFig. 12: TDR versus error rate ϵ for track-and-stop.\nin 2006 IEEE Int. Solid State Circuits Conf. Dig. Tech. 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Conf., 2024, pp. 4836–4841.\n"},"cleaned":{"kind":"string","value":"Online Reliable Anomaly Detection via Neuromorphic Sensing and Communications Junya Shiraishi, Member, IEEE, Jiechen Chen, Member, IEEE, Osvaldo Simeone, Fellow, IEEE, and Petar Popovski, Fellow, IEEE Abstract-This paper proposes a low-power online anomaly detection framework based on neuromorphic wireless sensor networks, encompassing possible use cases such as brain-machine interfaces and remote environmental monitoring. In the considered system, a central reader node actively queries a subset of neuromorphic sensor nodes (neuro-SNs) at each time frame. The neuromorphic sensors are event-driven, producing spikes in correspondence to relevant changes in the monitored system. The queried neuro-SNs respond to the reader with impulse radio (IR) transmissions that directly encode the sensed local events. The reader processes these event-driven signals to determine whether the monitored environment is in a normal or anomalous state, while rigorously controlling the false discovery rate (FDR) of detections below a predefined threshold. The proposed approach employs an online hypothesis testing method with e-values to maintain FDR control without requiring knowledge of the anomaly rate, and it dynamically optimizes the sensor querying strategy by casting it as a best-arm identification problem in a multi-armed bandit framework. Extensive performance evaluation demonstrates that the proposed method can reliably detect anomalies under stringent FDR requirements, while efficiently scheduling sensor communications and achieving low detection latency. Index Terms-anomaly detection, false discovery rate control, hypothesis test, multi-armed bandit, neuromorphic sensing and communication. I. INTRODUCTION Neuromorphic technologies - encompassing sensing [1], [2], computing [3], [4], and communication [5], [6], [7], [8], [9], [10] - are emerging as a promising paradigm for ultra-low-power, low-latency sensor networks. Inspired by the brain's event-driven information processing, a neuromorphic sensor records information by producing spikes at times marking significant changes in the monitored environment [11]. Thanks to the inherent capacity of neuromorphic sensors to respond only to relevant events, incorporating neuromorphic sensors in a remote monitoring system can drastically reduce J. Shiraishi and J. Chen contributed equally to this work. The work of J. Shiraishi was supported by European Union's Horizon Europe research and innovation funding programme under Marie Skłodowska-Curie Action (MSCA) Postdoctoral Fellowship, \"NEUTRINAI\" with grant agreement No. 101151067. The work of J. Chen and O. Simeone was supported by the EPSRC project EP/X011852/1, and O. Simeone was also supported by an Open Fellowships of the EPSRC (EP/W024101/1). The work of P. Popovski was supported by the Velux Foundation, Denmark, through the Villum Investigator Grant WATER, nr. 37793. (Corresponding author: Jiechen Chen) J. Shiraishi and P. Popovski are with the 9220 Aalborg, Denmark (e-mail: {jush, ). J. Chen and O. Simeone are with the King's Communications, Learning and Information Processing Laboratory, King's College London, WC2R 2LS London, U.K. (e-mail: {jiechen.chen, os- ) Neuro-SN 1 1 K S qௌ ଵ 1. Query (N) 2. IR transmission Reader Environment state t t Neuro-SN K ⋮ qௌ ⋮ Sመ anomaly detection Fig. 1: In the setup under study, a reader interrogates a subset Nf of the K neuromorphic sensor nodes (neuro-SN) at each frame f in order to determine whether the behavior of the monitored system is normal or anomalous. The binary variable Sf indicates the presence or absence of an anomaly at frame f, while ˆSf represents the corresponding estimate of the reader. redundant transmissions, leading to substantial energy and bandwidth savings and supporting the deployment of large populations of wireless sensors [5], [10]. For example, reference [10] demonstrated an asynchronous wireless network of salt-grain-sized sensors that mimic neural spikes, showing that hundreds of microscale nodes can communicate sporadic bursts of data to a central receiver using backscattering. Such neuromorphic networks have been highlighted for their potential in brain-machine interfaces, where implantable sensors would relay neural spikes in real time, as well as in other domains like wearable devices and environmental sensing [12]. In applications such as remote surveillance and autonomous systems, timely detection of abnormal events is critical. Notably, neuromorphic vision sensors can detect sudden motion or the appearance of an intruder - e.g., an unauthorized drone - by producing rapid spikes, while remaining largely inactive under normal conditions [1]. In environmental monitoring, distributed sensors might pick up abrupt changes in temperature or smoke indicative of a forest fire [13]. In these scenarios, the system must provide reliable anomaly detection despite limited energy supply and communication bandwidth. To this end, the detection algorithm needs to be statistically robust, minimizing false alarms to avoid wasting attention and energy on benign events. Furthermore, the network must intelligently decide which sensors to query and when, so as to maximize the chance of detecting an anomaly while conserving resources. In this regard, a common reliability criterion is maintaining the false discovery rate (FDR) below 16 Oct 2025 a target level, ensuring that only a small fraction of reported anomalies are false positives [14], [15], [16]. Achieving a low FDR is especially important in safety-critical applications, like health monitoring or security, to maintain trust in the system's alarms. This paper addresses these challenges by proposing a novel solution that integrates tools from statistical FDR control [17] and adaptive learning [18] into a neuromorphic wireless sensing framework. A. Related Work Neuromorphic sensing: Neuromorphic sensing aims to emulate biological sensory systems by encoding information through sparse, event-driven signals rather than continuous streams. A prominent category is vision sensing, where dynamic vision sensors (DVSs) detect asynchronous changes in brightness at each pixel, achieving low latency and high energy efficiency. Prophesee and IniVation are among the leading companies in this area, offering state-of-the-art event-based cameras for machine vision and robotic applications [19], [20]. Beyond vision, companies like SynSense are developing ultralow-power neuromorphic processors such as Speck, which enable local event-driven processing for embedded intelligence [21]. Neuromorphic communication: Recent work [6] has introduced end-to-end neuromorphic communication systems that replaces conventional sensing and frame-based radio with a fully spike-driven pipeline. In it, neuromorphic sensors produce asynchronous event spikes which are encoded by a spiking neural network (SNN) and transmitted as impulse radio (IR) pulses, then decoded by another SNN at the receiver. This all-spike design was shown to significantly lower latency and improve efficiency compared to conventional synchronous digital communication. Building on this, reference [5] extended the approach to a multi-user, fading-channel scenario. In it, each internet of things (IoT) device uses a neuromorphic event sensor, ondevice SNN, and IR transmitter, communicating over a shared wireless channel to a receiver that also employs an SNN. A key principle in these systems is event-driven data acquisition, which allows the wireless network to inherently focus on the \"semantics\" of the data - i.e. changes that carry new information - rather than sending redundant raw readings. Further energy savings can be obtained by combining IRbased communication with wake-up radio triggers [7], adopting different design criteria [22], and leveraging multi-level spikes [9]. In particular, reference [9] shows that associating each spike with a small payload can improving inference accuracy without increasing spike count. This study also includes an experimental evaluation using software-defined radios. Online anomaly detection: Online anomaly detection focuses on identifying abnormal patterns in streaming data in real time, without requiring access to the entire dataset beforehand. For example, reference [14] proposed online false discovery rate control methods for anomaly detection in time series to address the challenges of rare anomalies and temporal dependencies. Reference [15] proposed an unsupervised realtime anomaly detection algorithm for streaming data, which is capable of continuous learning and early detection of both spatial and temporal anomalies without requiring labeled data or batch processing. Multi-armed bandit optimization: Multi-armed bandit (MAB) optimization has been extensively studied in previous works. For example, reference [18] proposed the track-andstop strategy for best-arm identification under fixed-confidence settings. Reference [23] studied the complexity of best-arm identification in stochastic MAB under both fixed-budget and fixed-confidence settings. To speed up the convergence of MAB in unseen tasks, [24] proposed for the first time the use of contextual meta-learning that incorporates auxiliary optimization tasks knowledge. B. Main Contributions Building on the above context and prior art, this paper develops a protocol for online anomaly detection via neuromorphic sensor networks. The main contributions are summarized as follows: • Online anomaly detection framework based on neuromorphic sensors: We introduce a novel online anomaly detection system based on neuromorphic sensor networks. As shown in Fig. 1, this model implements a pull-based communication strategy, where a central reader intermittently queries neuromorphic sensor nodes and receives eventdriven IR spike signals. The model explicitly incorporates the sparse, asynchronous nature of neuromorphic data and the requirement to control the false discovery rate in detection decisions. • Adaptive FDR control with e-values: We design an adaptive statistical testing framework that guarantees FDR control at any desired level in an online setting. In contrast to traditional methods that rely on fixed pvalue thresholds, our framework utilizes e-values [25] to accumulate evidence over each frame. This allows the detector to flexibly adjust its confidence as new spike data arrive, ensuring that the expected proportion of false alarms remains below the target level even when the anomaly occurrence rate is unknown and nonstationary. • Best-arm identification for sensor scheduling: We introduce a sensor scheduling policy based on best-arm identification principles from multi-armed bandit theory [18]. At each time frame, the reader must decide which sensors to query next. Our policy dynamically learns the sensors that are most likely to yield an anomaly indication by treating each sensor as an \"arm' with an unknown reward. By continuously updating estimates and concentrating queries on the most promising sensor, the policy achieves efficient allocation of limited query opportunities, accelerating the detection of anomalies under uncertainty. • Extensive performance evaluation: We provide a thorough performance evaluation of the proposed system through analysis and simulations. The results demonstrate that our approach can detect anomalies quickly and reliably under a variety of conditions. In particular, we show that the FDR is maintained below the target level in all tested scenarios, validating the effectiveness of the e-value based control. Moreover, the adaptive scheduling policy is shown to significantly reduce detection delay and resource usage compared to non-adaptive or random scheduling baselines. The rest of the paper is organized as follows. Section II describes the pull-based neuromorphic wireless communications system, while the online anomaly detection based on hypothesis test framework is described in Section III. Section IV explains the scheduling policy optimization based on best-arm identification. Experimental setting and results are described in Section V. Finally, Section VI concludes the paper. II. SYSTEM MODEL A. Setting Fig. 1 shows the basic setup considered in this paper. We consider a pull-based neuromorphic wireless communications scenario where the reader conducts a continuous online anomaly detection at each time frame based on the deployment of neuromorphic sensor nodes (neuro-SN) [10]. The K neuro-SNs observe a common environment via local neuromorphic sensors [5], and communicate with the reader via IR. We define the set of node indices by K = {1, 2, . . . , K}. Through communication with the neuro-SNs, the reader continuously monitors the environment to detect anomalies, while ensuring that the FDR is maintained below a pre-determined level, denoted as α. The FDR measures the fraction of detected anomalies that do not correspond to actual anomalies. Specific applications include real-time forest fire detection [13] or drone activity monitoring [5] based on wireless sensor networks. As in [14], anomalies are defined as rare events that significantly deviate from expected patterns. In our setting, an anomaly causes neuromorphic sensors to produce spikes, also known as events, at a significantly higher rate than under normal conditions. For example, neuromorphic cameras produce spikes at times in which activity is recorded in the monitored scene [5]. Thus, in the presence of significant activity, e.g., from unauthorized drones, the spike rate is increased. As illustrated in Fig. 2, we adopt a discrete time model, in which time is divided into frames indexed by f = 1, 2, . . . , F. Each frame consists of three phases: 1) Downlink query signal transmission: At the beginning of each frame, the reader first sends a query signal to a selected subset of devices. 2) Sensing and uplink IR transmission: Upon detecting the query signal, e.g., via a wake-up radio [7], [26], a neuro-SN activates its neuromorphic sensor and its IR transmitter. The IR transmission phase is divided into t ⋯ t 1-st frame 2nd frame F-th frame Sensing and IR transmission Anomaly detection Luplink slots Query transmission downlink anomaly detection Fig. 2: Frame structure illustrating the query transmission from the reader (downlink), sensing and impulse radio (IR) transmission by the selected neuro-SN during L uplink slots, and anomaly detection at the reader. L time slots, during which the neuro-SNs sense and transmit in an online and event-driven manner. 3) Anomaly detection at the reader: Finally, the reader determines whether the environment is in an anomalous state based on the received signals. B. Observation Models Following [14], we model the presence or absence of an anomaly in a frame indexed by f as an arbitrary, unknown sequence of binary state Sf, which are defined as Sf = ( 0, no anomaly, 1, anomaly. (1) We model the observations of each neuromorphic sensor k at the f-th frame via a Bernoulli process with spiking rate determined by the current state Sf. Accordingly, for each sensor k, we set two spiking rates corresponding to the absence or presence of an anomaly, namely, qk 0 and qk 1. As illustrated in Fig. 3, we assume the inequality qk 0 ≤qk 1 for all k ∈ {1, 2, . . . , K}. This indicates that anomalies are characterized by larger spiking rates. Accordingly, each node k obtains a spike sequence sk f = [sk f,1, . . . , sk f,L] ∈{0, 1}L, where sk f,l = 1 indicates that node k records a spike at time instant l of the f-th frame, and sk f,l = 0 indicates that no spike was recorded. This means that we have sk f,l ∼ i.i.d. Bern(qk f) with qk f = ( qk 0 if Sf = 0 qk 1 if Sf = 1. (2) Through an initial calibration phase, the reader can obtain the statistics on the spike rates in normal conditions. Therefore, the reader is assumed to know the value of the probability qk 0 for all k ∈ {1, 2, . . . , K}. In contrast, we assume the reader does not know the spiking probability qk 1 in the presence of anomalies. C. Scheduling and Impulse Radio Transmission In this work, we assume time division multiple access (TDMA)-based communication between the reader and neuroSNs, which consists of downlink phase and uplink phases. Spike rate Neuro-SN 1 2 K ⋯ ⋯ Spike rate for the normal state 1 0 Spike rate for the anomalous state qଵ ଵ qଵ ଶ q ଵ q ଶ qଵ q Fig. 3: Each k-th sensor is characterized by a known spiking probability qk 0 in normal conditions, and a higher, unknown spiking probability qk 1 ≥qk 0 in the presence of an anomaly. 0 1 Transmitted bit Received bit εଵ 1 -εଵ 0 1 1 -εଵ εଵ Fig. 4: Illustration of a binary asymmetric channel. At each time frame f, the reader requests data from a subset Nf ⊆{1, . . . , K} of neuro-SN by multicasting a query signal that encodes the target node IDs and its schedule information. The reader schedules |Nf| = Cmax nodes, and transmits this control information during the downlink phase. In practice, the cardinality Cmax of the scheduled set depends on the available uplink capacity. The nodes that receive the query signal begin sensing over the L time instants within the frame. The spike sequence sk f is transmitted by all scheduled sensors in Nf in an online and event-driven manner using IR by directly mapping a spike from the sensor to an electromagnetic pulse [5]. As illustrated in Fig. 4, we model the channel as a binary asymmetric channel, where transmitted bits may be flipped at the receiver, with bit 0 flipping to 1 with probability ε01, and bit 1 flipping to 0 with probability ε10. The IR receiver at the reader estimates the spike sequence transmitted by each neuro-SN. After receiving the estimated spike sequence ˆsk f = [ˆsk f,1, . . . , ˆsk f,L] ∈{0, 1}L for each node k ∈Nf at time frame f, the reader uses this information to determine whether the current state Sf is normal or anomalous, as described in the next section. III. ONLINE ANOMALY DETECTION In this section, we first present the hypothesis test framework, assuming that the reader has collected data from the subset of nodes Nf at the f-th frame. Then, we describe the proposed anomaly detection mechanism for online anomaly detection. The problem of optimizing node scheduling will be addressed in the next section. A. Hypothesis Testing Framework At the end of each frame f, the reader tests whether the current system state is normal or anomalous. Specifically, it considers the following hypothesis Hf : Sf = 0 (normal state) (3) against the alternative hypothesis that there is an anomaly, i.e., Sf = 1. Equivalently, the reader evaluates the null hypothesis for the requested sensors in Nf as follows Hf : sk f ∼ i.i.d. Bern(qk 0), ∀k ∈Nf (4) against the composite alternative hypothesis ̄Hf : sk f ∼ i.i.d. Bern(qk 1), for some qk 1 > qk 0, ∀k ∈Nf. (5) In order to conduct a hypothesis test during the f-th frame, the reader uses the spike sequences ˆsf = {ˆsk f}k∈Nf observed during this frame and creates the test statistics Xf. In addition, the reader selects the rejection threshold αf by considering the target metrics, such as FDR. The null hypothesis Hf is that there is no anomaly, such that if this hypothesis is rejected, the reader declares an anomalous state. The decision at frame f is denoted as ˆSf ∈{0, 1}, which can be described as ˆSf = h(Xf, αf). Here, h(Xf, αf) is an indicator function representing the rejection decision based on Xf and αf. In the following subsection, we first describe our target metrics and use test statistics based on e-values. Finally, we describe how to select the rejection thresholds αf considering the target metrics. B. Decaying Memory Time-Averaged False Discovery Rate Let Ff|0 represent the subset of the frame indices {1, 2, . . . , f} for which the environment is in a normal state, i.e., Ff|0 = {f ′ ∈{1, 2, . . . , f} : Sf ′ = 0}. At frame f, the cumulative number of anomalies declared by the reader, averaged with a forgetting factor 0 1 αf . (19) To describe the update of the threshold αf, let ρj denote the time of j-th detected anomaly, i.e., ρj = min f ≥0 f X f ′=1 1 ef ′ > 1 αf ′ ≥j , (20) and let {γf}∞ f=1 denote a non-increasing sequence summing to 1, where γf = 0 for all f ≤0. We specifically use the default sequence [14] γf ∝ log (max(f, 2)) f exp p log(f) . (21) Finally, let ̃γf = max(γf, 1-δ) as in [14]. Then, the rejection threshold αf is updated as αf = α η ̃γf + α X j≥1 δf-ρjγf-ρj, (22) where η is smoothing parameter η > 0. Proposition 1 [14]. When the decision threshold αf is updated according to (22) at each frame f, the anomaly detection rule in (19) ensures that the decaying-memory timeaveraged FDR defined in (8) is controlled at level α for all frames f, thus satisfying condition (9). IV. SCHEDULING POLICY OPTIMIZATION In order to increase the TDR while maintaining FDR control as in (9), an ideal scheduling policy would select at each frame f the subset Nf of sensors k with the highest spiking rates qk 1 under an anomalous state. In fact, this would maximize the chance of correctly detecting an anomaly via the test (19). However, the true spiking rates qk 1, for k ∈K, are not known by the reader. Furthermore, anomalies occur arbitrarily and unpredictably, making it challenging to estimate the probabilities qk 1. In this section, we address the problem of optimizing the set Nf by tackling these challenges. To start, we study the problem of identifying the best sensor k∗= arg max k∈K qk 1, (23) thus concentrating on the case Cmax = |Nf| = 1, in which the uplink channel supports the transmission of a single neuroSN. In this context, formally, the goal in this section is to minimize the number of frames used to identify the best neuroSN k∗with a probability of error not exceeding a threshold δs ∈(0, 1). An extension to any uplink capacity Cmax ≥1 will be also provided at the end of this section. A. Multi-Armed Bandit Formulation To facilitate the design of a scheduling policy, we make the working assumption that the anomaly process is i.i.d. across the frame index f, with an unknown anomalous probability π1, i.e., Sf ∼Bern(π1). Furthermore, we also assume that the normal spiking probabilities of all neuro-SNs are equal, i.e., qk 0 = q0, (24) for all k = 1, . . . , K. Under these simplified assumptions, the spike rates 1 L PL l=1 sk f,l of each neuro-SN k is i.i.d. across frames f, with an average μk = (1 -π1)q0 + π1qk 1. (25) This implies that the problem (23) of identifying the best sensor becomes equivalent to the problem of finding a node whose average spike rate μk is the highest. i.e., k∗= arg max k∈K μk. (26) This optimization can be formulated as the fastest best arm identification problem, studied in [18]. As demonstrated therein, a strategy known as track-and-stop reviewed next, is optimal. B. Track-and-Stop Strategy At each f-th frame, the reader estimates the spike rate μk of neuro-SN k as ˆμk f = 1 N k f f X f ′=1 ˆqk f ′1 (k ∈Nf ′) , (27) with N k f = f X f ′=1 1 (k ∈Nf ′) (28) being the number of frames in which neuro-SN k was scheduled and with ˆqk f ′ defined in (13). Define as Uf = {k : N k f 0, arg min k∈K N k f -f · wσ(k)(ˆμf) , otherwise, (29) where σ : {1, 2, . . . , K} →{1, 2, . . . , K} is a function that returns the rank of each node k in terms of the estimated spike rate ˆμk in descending order, satisfying the inequalities ˆμσ(1) f ≥ ˆμσ(2) f . . . ≥ˆμσ(K) f . The function wσ(k)(ˆμf), which returns positive value, is detailed in [18] and satisfies the following property is true: w1(μ) ≥. . . ≥wK(μ). This way, neuroSNs with a larger estimated spiking rate are prioritized by the selection rule (29). C. Extension of the Scheduling for Any Scheduling Capacity In this subsection, we extend the track-and-stop strategy to the case where it is possible to schedule Cmax = |Nf| ≥1 neuro-SNs. To this end, we propose that, at the beginning of each frame, the reader selects Cmax = |Nf| nodes for scheduling by recursively applying the track-and-stop method in (29). Specifically, the reader selects the scheduled nodes one-by-one, obtaining the set Nf = {ni f}Cmax i=1 . In more detail, define U i f = {k ∈K \\ {nj f}i-1 j=1 : N k f 0, arg min k∈K\\{nj f }j=i-1 j=1 N k f -fwσ(k)(ˆμf), otherwise, (30) where function wσ(k)(·) is as defined in (29). V. NUMERICAL RESULTS In this section, we present numerical results to validate the performance of the proposed online anomaly detection system. A. Simulation Setting In our experiments, the anomaly process Sf is i.i.d. across all frames f, so that at each frame f the environment is in the anomalous state with probability π1 and in the normal state with probability 1 -π1, independently from frame to frame. We simulate a setting in which, as in (24), the spike rate qk 0 of a normal state is the same across all nodes, i.e., qk 0 = q0 for all k ∈K. In contrast, the spiking rate qk 1 under the anomalous state is uniformly generated within the interval [q0, q0+∆max] for each neuro-SN k. Table I shows our default parameters setting, which are used throughout the experiments unless stated otherwise. TABLE I: Default parameter setting. Parameters Values Number of neuro-SN K 5 Forgetting factor δ 0.99 [14] η 0.99 Number of uplink slots L 50 Target FDR α 0.1 [14] Simulation runs 103 Spiking rate under the normal state q0 0.10 Anomaly ratio π1 0.05 Error rates (ε01, ε10) (0, 0) Upper bound on the spiking probability ∆max 0.5 Channel capacity Cmax 1 B. On the Validity of Online Anomaly Detection We first investigate the validity of the online anomaly detection algorithm introduced in Sec. III. To this end, we compare the performance of the proposed scheme with adaptive threshold αf described in (22) with a more conventional fixed-threshold strategy, in which we apply the threshold αf = α for all f ∈[1, F]. Note that this combined approach does not satisfy the theoretical statistical guarantee in (9), and its performance can only be evaluated via numerical results. Fig. 5 shows FDR and TDR versus the time frame f for fixed anomaly threshold method and for the proposed dynamic anomaly detection method in (22), where we set the number of neuro-SN K = 1. As anticipated, the fixedthreshold is not able to always satisfies the FDR constraint in (9). This is because this approach does not take into account the past rejection history for the current anomaly decision tasks. A fixed threshold can achieve a higher TDR than the proposed method, but only by violating the FDR constraint in (9). In contrast, the introduced methodology, dynamic threshold always satisfies the FDR constraint α, validating Proposition 1. Fig. 6 shows the FDR and TDR as a function of the anomaly probability π1, where we set K = 1 and F = 1000. We observe that the fixed-threshold approach suffers from the higher FDR when the anomaly probability π1 is small, while the proposed dynamic anomaly detection threshold method can always satisfy the FDR constraint. When the anomaly probability π1 is small, with a dynamic threshold, the reader takes a conservative decision by setting a relatively smaller threshold αf, will a fixed-threshold method cannot adjust to the anomaly probability. C. Random v.s. Track-and-Stop Scheduling We now study the impact of the scheduling strategy. To this end, Fig. 7 shows the FDR and the TDR versus the time frame f for the proposed anomaly detection scheme under a random scheduling strategy that selects Cmax neuro-SNs uniformly at random at every frame and for the track-and-stop method presented in Sec. IV. We set the parameters described in Table I with the uplink channel capacity Cmax = {1, 2} and anomaly probability π1 = 0.1. All schedulers, when coupled with the proposed anomaly detection, are seen to satisfy the FDR requirements, confirming the theoretical results in Proposition 1. Moreover, the figure highlights the benefits of track-and-stop, increasing the TDR for both uplink capacities Cmax = 1 and Cmax = 2. This advantage stems from the capacity of track-and-stop to select the neuro-SNs that are most likely to contribute to the identification of the neuro-SNs with highest spiking rates. From Fig. 7 we also see that TDR becomes larger as the number of scheduled nodes Cmax increases, demonstrating the importance of collecting information from multiple neuroSNs. This is because by calculating e-value (18) based on the spike rate from multiple nodes, the reader can mitigate the negative impact of small e-value caused by the selection of nodes having small spike rate in anomalous state, qk 1. D. Impact of System Parameters In this subsection, we focus on the performance of trackand-stop scheduling, evaluating the impacts of different system parameters. 1) On the FDR constraint α: Fig. 8 shows the FDR and TDR as a function of FDR constraint α. The figure highlights the trade-off between TDR and FDR: as the allocated FDR constraint α becomes larger, the reader can more easily detect anomalies, increasing the TDR. We can also see that TDR becomes larger as the forgetting factor δ becomes smaller. In fact, with a smaller δ, the FDR constraint puts higher weight on the results of the latest decisions, decreasing the importance of past false alarms. 2) On the sensitivity of the sensors: We now study the dependence of the system performance on the parameter defining the quality of the observations of sensors. First, we 100 200 300 400 500 0 0.1 0.2 Fixed αf = α Dynamic αf FDR constraint f FDR 100 200 300 400 500 0.0 0.25 0.50 0.75 1.0 Fixed αf = α Dynamic αf f TDR Fig. 5: FDR and TDR versus the time frame f for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed dynamic anomaly decision threshold based on (22), where we set K = 1 and π1 = 0.05. 0.025 0.05 0.075 0.10 0 0.1 0.2 0.3 Fixed αf = α Dynamic αf FDR constraint π1 FDR 0.025 0.05 0.075 0.10 0.0 0.25 0.50 0.75 1.0 Fixed αf = α Dynamic αf π1 TDR Fig. 6: FDR and TDR versus the anomaly probability π1 for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed dynamic anomaly decision threshold based on (22), where we set K = 1 and F = 1, 000. vary the upper bound ∆max on the spiking rate in the presence of anomalies. From Fig. 9, we can see that TDR is improved as the upper bound ∆max becomes larger, that is, the sensitivity improves. This is because a larger upper bound ∆max tends to increase the difference between the spike rate of neuro-SN k in the normal state qk 0 and that in the anomalous state qk 1, enabling the reader to detect anomalies with higher accuracy (see Fig. 3). Fig. 10 shows the TDR as a function of the frame index f for different numbers of uplink slots L. From Fig. 10, we can see that TDR becomes larger as L increases because the reader can execute anomaly detection task based on more number of received spikes, realizing accurate decision based on the e-value in (18) and decision threshold defined in (22). For example, to achieve TDR larger than 0.9 under ∆max = 0.5, there is a need for L = 100 uplink slots. 3) Impact of anomaly probability π1: Fig. 11 shows TDR against f for track-and-stop, where we set the anomaly probability to π1 = {0.05, 0.10, 0.15}. As π1 increases, the probability that the reader obtains the higher e-value in the single frame also increases, making it more likely that the reader rejects the hypothesis at the current frame. As a result, we can see that TDR increases as π1 becomes larger. 4) On the channel error rate: To evaluate the impact of channel error in the binary asymmetric channel described in Sec. II-C, we vary the error probability ε, by setting: (i) ε01 = ε and ε10 = 0; (ii) ε10 = ε and ε01 = 0; (iii) ε01 = ε10 = ε. From Fig. 12, we see that the error probability ε10 has the largest impact on TDR. In fact, as increase in ε10 causes spike losses, which may be critical for anomaly detection. Conversely, as the error probability ε01 increases, it becomes more likely that the reader mistakenly detects false spikes, causing the reader to mistakenly detect an anomaly event when the environment is in a normal state. As the reader must satisfy the FDR constraint in (9), in this case the reader must take a conservative decision when the actual anomaly event happens, reducing the TDR. When both errors have a non-zero probability, it is affected by both spike losses and falsely detected spikes, causing further decreases in the TDR. VI. CONCLUSION This paper proposed a low-power online anomaly detection framework for neuromorphic wireless sensor networks. We have introduced an algorithm that can control the false discovery rate by incorporating e-values test statistics calculated based on the received spike sequences at each frame. We have also introduced a track-and-stop methods for scheduling nodes 200 400 600 800 1,000 0 0.1 FDR constraint f FDR 200 400 600 800 1,000 0.50 0.75 1.0 Random (Cmax = 1) track-and-stop (Cmax = 1) Random (Cmax = 2) track-and-stop (Cmax = 2) f TDR Fig. 7: FDR and TDR versus the time frame f for random sampling scheme and track-and-stop. 0.05 0.10 0.15 0.20 0 0.1 0.2 FDR constraint α FDR 0.05 0.10 0.15 0.20 0.50 0.75 1.0 δ : {0.8, 0, 9, 0.99} α TDR Fig. 8: FDR and TDR versus the FDR constraint α for track-and-stop (the forgetting factor δ = {0.8, 0.9, 0.99}). 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 ∆max = 0.2 ∆max = 0.3 ∆max = 0.4 ∆max = 0.5 f TDR Fig. 9: TDR versus the time frame f for track-and-stop with different values of upper bound of sensor reading ∆max. based on the channel constraint. Our numerical results show that the proposed approach can track the nodes having the largest spike efficiently with high TDR, while maintaining the FDR below the predetermined threshold. Future work includes designing pull-based communication protocol in neuromorphic communication while taking into account aspects such as information freshness and value of information. Designing neuromorphic communication proto100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 L = 25 L = 50 L = 75 L = 100 f TDR Fig. 10: TDR versus the time frame f for track-and-stop with different values of the number of uplink slots L. 100 200 300 400 500 0.6 0.7 0.8 0.9 1 π1 = 0.05 π1 = 0.10 π1 = 0.15 f TDR Fig. 11: TDR versus the time frame f for track-and-stop scheme with different values of anomaly potion π1 = {0.05, 0.10, 0.15} cols for heterogeneous neuro-SNs with different capabilities in terms of sensing and communication is also an interesting research avenue. REFERENCES [1] P. Lichtsteiner, C. Posch, and T. 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Conf., 2024, pp. 4836-4841."}}},{"rowIdx":294,"cells":{"id":{"kind":"string","value":"2510.14687"},"raw":{"kind":"string","value":"Astronomy & Astrophysics manuscript no. Article_v1\n©ESO 2025\nOctober 17, 2025\nSearch for exocomets transits in Kepler light curves\nTen new transits identified\nP. Dumond1, 2, A. Lecavelier des Etangs1, F. Kiefer1, 3, G. Hébrard1, and V. Caillé1, 4\n1 Institut d’astrophysique de Paris, CNRS, Sorbonne Université, 98bis boulevard Arago, 75014 Paris, FRANCE\n2 CRAL, Ecole normale supérieure de Lyon, Université de Lyon, UMR, CNRS 5574, F-69364 Lyon Cedex 07, France\n3 LIRA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195\nMeudon, France\n4 Institut Pierre-Simon Laplace, Sorbonne-Université, CNRS, Paris, France,\nOctober 17, 2025\nABSTRACT\nThe Kepler mission, despite its conclusion over a decade ago, continues to offer a rich dataset for uncovering new astrophysical\nobjects and phenomena. In this study, we conducted a comprehensive search for exocometary transit signatures within the Kepler\nlight curves, using a machine learning approach based on a neural network trained on a library of theoretical exocomet transit light\ncurves. By analyzing the light curves of 201,820 stars, we identified candidate events through the neural network and subjected the\noutput to filtering and visual inspection to mitigate false positives.\nOur results are presented into three catalogs of increasing ambiguity. The first-tier catalog includes 17 high-confidence exocometary\ntransit events, comprising 7 previously reported events and 10 newly identified ones, each associated with a different host star. The\nsecond-tier catalog lists 30 lower-confidence events that remain consistent with possible exocometary transits. The third-tier catalog\nconsists of 49 more symmetric photometric events that could be either exocometary transits, exoplanet mono-transits, or false positives\ndue to eclipsing binaries mimicking transits.\nContrary to previous studies, which suggested that the cometary activity was favored by stellar youth, we find a broad age distribution\namong candidate host stars, including several red giants. This challenges the general idea of a decline in cometary activity with stellar\nage and underlines the need for further investigation into the temporal evolution of exocometary activity in planetary systems.\nKey words. Methods: data analysis - Techniques: photometric - Surveys - Comets: general - Planetary systems - Exocomets - Transit\nphotometry\n1. Introduction\nExocomets are rocky or icy minor bodies that become active\nwhen they approach their parent star on a highly eccentric or-\nbit (Strøm et al. 2020). Producing a huge tail of gas and dust\nclose to the periastron, when they transit in front of the star,\nthey become detectable using spectroscopy (for the gaseous tail)\nand photometry (for the dust tail). The first exocomets were thus\nidentified in the 1980s in the β Pic system, through the detec-\ntion of variable absorption features in the Ca ii lines of the stellar\nspectrum (Ferlet et al. 1987; Kiefer et al. 2014a). Later on, a few\nother exocometary systems have been found using spectroscopy,\nlike HD 172555 (Kiefer et al. 2014b; Grady et al. 2018), and\n49 Cet (Montgomery & Welsh 2012; Miles et al. 2016). A list of\nspectroscopically identified exocometary systems can be found\nin Strøm et al. (2020).\nFor the detection of exocomets using photometry, one needs\nto obtain photometric measurements at high accuracy, typically\nat 10−3–10−4 level over the full transit duration, that is, for one to\nthree days (Lecavelier des Etangs et al. 1999, hereafter LdE99a).\nThis capability has been reached only during the last decade with\nthe space missions like Corot, Kepler, TESS, and Cheops. Detec-\ntions have thus been obtained with Kepler toward KIC 3542116\nand KIC 11084727 (Rappaport et al. 2018), and KIC 8027456\n(Kennedy et al. 2019), with TESS toward β Pic (Zieba et al.\n2019; Pavlenko et al. 2022; Lecavelier des Etangs et al. 2022),\nand with Cheops toward HD 172555 (Kiefer et al. 2023). Here\nit must also be mentioned the detection with Kepler of a photo-\nmetric event, possibly periodic, that is interpreted by the transit\nof a string of 5 to 7 exocomets in front of KIC 8462852 (Kiefer\net al. 2017).\nAll these events have been identified thanks to the particular\nshape of the light curve of an exocomet transit, as theoretically\npredicted at the end of the 90’s in LdE99a : the characteristic\nof an exocomet transit light curve is the asymmetry caused by\nthe cometary tail passing in front of the star after the nucleus.\nThis causes a sharp decrease of the star light followed by a slow\nreturn to the normal brightness (see also Lecavelier des Etangs\n1999, hereafter LdE99b). However, exocomet transit light curves\ncan also be more symmetric when the longitude of the periastron\nof the orbit is about 90◦from the line of sight, because here the\ncometary tail is aligned with the line of sight. In this case, the\nshape of the transit light curve is similar to that of an exoplanet\ntransit (LdE99a).\nThe detection of exocometary transits based on photometry\nis complementary to the detection through spectroscopy. It al-\nlows exploring the physical characteristics of dust particles, such\nas the dust particle size distribution, the distance to the host star\nat the time of transit, and the dust production rate (Luk’yanyk\net al. 2024). Indeed, because transits give direct access to the ge-\nometrical extent and the optical thickness of the transiting dust\ncloud, coupled with models of dust production, this allows the\nArticle number, page 1 of 18\narXiv:2510.14687v1 [astro-ph.EP] 16 Oct 2025\n\nA&A proofs: manuscript no. Article_v1\nestimate of the dust production rate and the size of the comets\nnucleus. In the case of β Pic, statistical analysis of photometric\nevents allowed the estimate of the nucleus size distribution that\nis found to be similar to that of asteroids and comets in the Solar\nsystem as set by collisional equilibrium (Lecavelier des Etangs\net al. 2022).\nAlthough the characterization of exocomets paves the way\nto understanding the dynamical and chemical processes occur-\nring in young planetary systems, only a very limited number\nof exocometary systems have been identified to date. As a re-\nsult, our current understanding of the diversity, frequency, and\nevolutionary significance of exocometary activity remains in-\ncomplete. Systematic searches have been performed using spec-\ntroscopy in the Ca ii line (Montgomery & Welsh 2012; Welsh &\nMontgomery 2013, 2018; Rebollido et al. 2020; Bendahan-West\net al. 2025), resulting in a few detections (see review in Strøm\net al. 2020). Spectroscopy is efficient in detecting tiny amount\nof gas and thus allows sensitive search ; the result is that de-\ntectable spectroscopic transits seem to be more frequent than de-\ntectable photometric events: for instance in the case of β Pic, sev-\neral exocomet transits are routinely detected every day in spec-\ntroscopy (Kiefer et al. 2014a), while only 30 photometric transits\nhave been detected in 156 days of TESS observations (Lecave-\nlier des Etangs et al. 2022). However, spectroscopic searches for\nexocomets are inherently limited by their need to observe each\ntarget star individually and the need for long observation time\nwith high spectral resolution. In contrast, photometric surveys\ncan monitor several thousands of stars simultaneously, making\nthem significantly more efficient for large-scale searches for ex-\nocometary transits.\nIn that spirit, Rappaport et al. (2018) and Kennedy et al.\n(2019) have undertaken a deep search for exocometary transits\nin the Kepler light curves. To identify exocometary transits, they\nused a modified model of planetary transit or a combination of\nexponential and Gaussian tails, as also done later on by Zieba\net al. (2019), Pavlenko et al. (2022), and Lecavelier des Etangs\net al. (2022). Although this empirical approach does not carry\nphysical information about the cometary bodies and tails, these\nare simple and easy to implement to identify candidates in the\nlarge number (∼200 000) of light curves provided by Kepler.\nHere we propose to use an automated search without the use\nof analytical functions to describe the exocometary transit light\ncurves. For that we combined a search algorithm based on a neu-\nral network together with a library of photometric transits to train\nthe network. The library is an updated version of the library of\nLdE99b, which has been recalculated using faster CPUs than the\nones used 20 years ago, covering a wide range of parameters and\ntherefore a wide range of transit shapes that need to be identified.\nWe apply this automated search to the Kepler data (Borucki et al.\n2010) with the aim of producing a new catalog of exocometary\nsystems candidates. The methods are described in Sect. 2, 3 and\n4, the search in Kepler data is presented in Sect. 5 and the final\ncatalog of candidates, divided in three tiers, is given in Sect.6.\nThe results are discussed in Sect.7.\n2. Training of the neural network\n2.1. Training principle\nThe method used for the detection of exocomets in Kepler data\nis based on a neural network using supervised learning. For the\nnetwork learning, we used a library of theoretical light curves di-\nvided into three disjoint sets of labeled light curves: the training\nset, the validation set, and the test set. Each set contains the same\nnumber of light curves with and without an exocomet transit (see\nSect. 2.2). Each light curve is labeled to indicate if it includes a\ntransit or not.\nOur objective is to make the algorithm identify the curves\ncontaining an exocomet transit among all the light curves col-\nlected by the Kepler satellite and to recover the transit time when\na transit is present. Using an iterative procedure, the algorithm\noptimizes its parameters thanks to the training set. At the end of\neach iteration step called an \"epoch\", the network performance\nis tested using the validation set. This allows us to control the\noverfitting of the algorithm (e.g. it becomes specific to the train-\ning set and loses in generality) and to adjust the hyperparameters.\nThe learning phase is stopped when the network shows the best\nperformance in the validation set. The final performance of the\nalgorithm is given by its evaluation on the test set.\n2.2. Light Curve production\nThe light curves used to build the training, validation, and test\nset are the Presearch Data Conditioning (PDC) data from Kepler\nData Release 25 (Thompson et al. 2016). We used the Quarters 1\nto 17 and divided them into sub-quarters in case of interruption\nof the acquisition within a given quarter: two sub-quarters are\nseparated by at least four consecutive photometric measurements\nmissing in the light curve.\nFor each sub-quarter, a training, validation, and test set were\nconstructed to overcome their possible specificity. For each sub-\nquarter, the training set consists of 36 000 curves, while the val-\nidation and test sets contain 12 000 each. A curve consists on 10\ndays (480 photometric measurements at a cadence of one mea-\nsurement every ∼30 minutes) of acquisition of the luminosity\nflux from a star randomly chosen without repetition.\nTo ensure efficient learning and prevent the algorithm from\nfalling into obvious traps, several modifications have been made\nto the Kepler PDC data. First, all points outside the mean value\nof the light curve by more than 5-sigmas while the neighboring\npoints are not are considered to be outliers and removed. In ad-\ndition, each curve was normalized to have a zero mean and a\nstandard deviation of 1, as needed for optimal learning by the al-\ngorithm. Finally, the possible missing measurements, i.e. at most\nthree consecutive missing measurements within the same sub-\nquarter, were filled using a linear interpolation.\nFor the learning process, half of the 10-day light curves have\nbeen used with a label \"0\", indicating the absence of an exocomet\ntransit: we assume that the probability that any randomly chosen\nlight curve in the Kepler data includes an exocomet transit is ex-\ntremely low. This set of 0-labeled light curves without exocomet\ntransit is used to learn the noise patterns in the Kepler data. For\nthe other half of the 10-day light curves, we add one exocomet\ntransit by multiplying by a light curve from the theoretical tran-\nsit library (Sect. 2.3) and we label them with \"1\", indicating the\npresence of an exocomet transit.\n2.3. Simulation of exocomet transits\nIn order for the algorithm to learn to identify the shape of an ex-\nocomet transit in the light curves, we add theoretical exocomet\ntransits to half of the curves that make up the training data. These\ntheoretical exocomet transit light curves have been calculated us-\ning the method describes in LdE99a. We produced a library of\n2200 exocomet transit light curves, which is an updated version\nof the library published twenty years ago by LdE99b. Among all\navailable simulations, we selected only those with a gas produc-\nArticle number, page 2 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\ntion rate log10(P/(1kg · s−1)) between 6 and 7.5 and an impact\nparameter b < R∗, where R∗is the radius of the star. This choice\nguarantees that the used transit light curves are sufficiently deep.\nIndeed, an impact parameter greater than one stellar radius or a\ntoo low production rate leads to a noisy and very weak extinc-\ntion and would therefore have no interest for the training. Before\nadding the simulated transits to the Kepler light curves, we mul-\ntiplied the theoretical transit light curve by a random constant\nsuch that the depth of the transits is in the interval [2σ, K + 2σ]\nwhere K is the depth of the simulated transits and σ the standard\ndeviation of the Kepler light curves. We thus train our model for\na signal-to-noise ratio higher than 2 to exclude the cases where\nthe transit would be non detectable. The transit times of the the-\noretical transits added to the 10-day light curves are chosen ran-\ndomly, with a uniform distribution between the beginning and\nthe duration of the simulated transit before the end of the light\ncurve.\nFinally, in order to ensure that the training data are repre-\nsentative of the diversity of possible transits, we use a uniform\nrandom distribution of the distance of the periastron q in the set\n{0.2; 0.3,; 0.5; 0.7; 1.0} astronomical unit and of the longitude\nof the periastron ω from -157.5◦to +157.5◦with a step of 22.5◦,\nas in LdE99b.\nThe theoretical transits that have been used are totally dif-\nferent for the training, validation and the test sets. To guarantee\nthat the evaluation of the algorithm’s performance is unbiased,\nthe choice of the transits for each set has been made randomly\nfrom the whole set of theoretical transits.\n3. The neural network model\n3.1. Architecture\nThe deep learning algorithm developed in this study is mainly\nbased on the convolution layer sequence adapted for pattern\nrecognition. We relied on the architecture of existing algorithms\nfor the detection of exoplanet transits (Shallue & Vanderburg\n2018; Dattilo et al. 2019; Zucker & Giryes 2018; Chintarun-\ngruangchai & Jiang 2019). These algorithms, which all have a\nsimilar architecture, are optimized for periodic transit detection.\nThey have been modified to be efficient for the detection of sin-\ngle transits, as the orbital period of an exocomet is expected to be\nmuch longer than the Kepler mission duration. Two major mod-\nifications have been made. \"Squeeze-excited blocks\" (Hu et al.\n2018) have been added to the convolution blocks. The authors\nindeed show that such blocks can improve the performance of\na pattern matching algorithm by improving the interdependence\nrelations between the convolution layers without adding com-\nputational cost. The addition of the LSTM (Long Short-Term\nMemory) (Hochreiter & Schmidhuber 1997) and GRU (Gated\nRecurrent Unit) (Cho et al. 2014) layers also resulted in sig-\nnificant performance improvements. The algorithm was imple-\nmented in TensorFlow, an open-source machine learning frame-\nwork (Abadi et al. 2016). We optimized the neural network pa-\nrameters thanks to KerasTuner (O’Malley et al. 2019).\nFigure 1 describes our final model, which is mainly a one-\ndimensional convolutional network. The activation function of\nall the convolutional layers is the ReLu (Rectified Linear unit)\nfunction (Jarrett et al. 2009). The output is a list of two reals.\nThe first one uses a sigmoid function whose range is (0,1). It is\nthe probability returned by the algorithm that the 10 days input\nlight curve contains an exocomet transit. An output value close\nto 1 indicates high confidence that the considered light curve has\none transit, while an output value close to 0 means that there is\nFig. 1: Architecture of our best performing neural network\nmodel. The input data is a light curve of 480 timesteps, the\noutput is a list of two reals between 0 and 1: one for the\nprobability of detection of an exocometary transit and one for\nthe position of the transit in the light curve.\nConvolutional\nlayers\nare\nnoted\nConv \n, max pooling layers are noted\nMaxPool , fully con-\nnected layers are noted Dense , LSTM\nlayers are noted LSTM , GRU layers are\nnoted GRU and\nthe squeezed-excitation blocks are noted SEB.\nnone. The second output layer is activated by a linear function\nin range (0,1). It gives the position of the identified transit in the\nlight curve. If no transit is detected, the layer should return a\nvalue close to zero.\n3.2. Training\nFor each sub-quarter, we trained the model using 36 000 light\ncurves for 150 training epochs, that means that we performed\n150 complete pass through the entire training dataset to update\nthe model weight parameters. We used the Adam optimization\nalgorithm (Kingma & Ba 2015) to minimize the cross-entropy\nerror function for the classification of the light curves and the\nmean absolute error function for the position of the transit. We\nused a learning rate of 10−4 and a batch size of 5 000. The saved\nmodel in the one for which the validation loss reaches its mini-\nmum ; usually it is obtained after between 50 and 80 iterations\n(\"epochs\"). Therefore, training the model over 150 epochs is\nconsidered sufficient.\nArticle number, page 3 of 18\n\nA&A proofs: manuscript no. Article_v1\n4. Evaluation of the neural network performance\nWe evaluated the performance of our neural network with re-\nspect to several metrics in the same way as Dattilo et al. (2019)\ndid for their model. We computed all metrics over the test set\n(rather than the training or validation sets) to avoid using any\ndata that were used to optimize the parameters or hyperparame-\nters of the model.\n4.1. The metrics\nThe metrics used to evaluate the performance of a classifier are\nthe following:\n1. Accuracy: the fraction of curves correctly classified by the\nmodel.\n2. Precision (reliability): the fraction of correctly classified ex-\nocomets, i.e., the fraction of curves classified as exocomet\ncandidates (False positives + True positives NFp + NTp) that\nare indeed true positives (labeled 1):\nP =\nNTp\nNFp + NTp\n(1)\n3. Recall (completeness or True-positive rate): the fraction of\nthe curves labeled 1, which therefore includes an exocomet\ntransit (True positives + False negatives NTp + NFn), that the\nmodel classifies as exocomet candidates (NTp):\nR =\nNTp\nNTp + NFn\n(2)\n4. False-positive rate: the fraction of total labeled 0 curves,\nwhich therefore do not include any exocomet transit (False\nPositives +True Negatives NFp + NTn) that the model classi-\nfies as exocomet candidates.\nFPR =\nNFp\nNFp + NTn\n(3)\n5. AUC (area under the receiver-operator characteristic curve;\nsee Figure 2): the probability that the model, if it receives\ntwo light curves, one 1-labeled curve including an exocomet\ntransit and one 0-labeled curve without transit, would rank\nthe 1-labeled curve higher than the 0-labeled curve.\n6. Mean Absolute Error: the absolute error for all labeled 1\ncurves between the true position of the transit and the pre-\ndicted one by the model.\nThe first five metrics are related to the detection of the pres-\nence of an exocomet in the light curves, while the last metric is\nrelated to the position in time of the transit in the light curve.\nExcept for the AUC, the values of these metrics depend on the\nclassification threshold chosen for the model. This threshold is\nthe value of the probability of detection yield by the algorithm\nabove which we consider that there is a detection. In Fig. 2, we\nshow the evolution of the recall as a function of the FPR for vari-\nous thresholds, usually called the receiver-operator characteristic\n(ROC) curve. It can be seen that the algorithm differentiates well\nbetween true transits and noise, as it achieves very low FPR re-\ncall while keeping a reasonably large recall: our model reaches\nan AUC of 98.88% for the quarter Q1. We also plot the preci-\nsion as a function of the recall. This curve shows the trade-off\nbetween having no false positives (high precision) and identify-\ning all exocomet transits (high recall).\nFig. 2: The receiver-operator characteristic (ROC) curve (left\npanel) and the precision versus recall curve (right panel). The\nreceiver-operator characteristic (ROC) curve shows the recall\n(true-positive rate) of the model against the ability to recognize\nfalse positives (the false-positive rate) for different classification\nthresholds. Our model is highly successful at identifying false\npositives as shown by the high AUC value (see Table A.1). The\nplot of the fraction of exocomets that the model classified as exo-\ncomets (recall) versus the fraction of correctly classified planets\n(precision) shows the trade-off between having no false positives\n(high precision) and identifying all exocomet transits (high re-\ncall).\nConsidering the huge number of light curves in the Kepler\ndata, the goal is to drastically reduce the number of false posi-\ntive. We thus chose a classification threshold of 0.99. With this\nthreshold, our model reaches an accuracy of 89.5% for the quar-\nter Q1. The precision of the model is 99.8%, which means that\nwe reach a high true positive rate, and the recall is 79,1%, which\nmeans that about 20% of the true exocomet transits are miss-\ning in the final classification. This trade-off aims to accept some\nloss in the finding of exocomet transits of about 20% to avoid a\nlarge number of false positives. Indeed, given a large number of\nlight curves in the Kepler data and a small number of exocomet\ntransits in these data, even a tiny fraction of false positives can\nsignificantly pollute the results. The metrics values are found to\nbe similar for all quarters (see Table A.1 in Appendix A).\n4.2. Quality of the output\nThe quality of the model classification can also be visualized by\nthe histogram of the results on the test set (top panel of Fig. 3).\nThe closer the output from the neural network is to 1 (resp. 0),\nthe higher the probability that the 10-day light curve (resp. does\nnot) contains an exocomet transit. The histogram shows that the\nneural network works as expected: on the left-hand side, most\nof the light curves characterized by a low probability are mostly\nnoise (orange histogram), while on the right-hand side, the large\nmajority of the curves characterized by a probability close to 1\nincluded a simulated exocomet transit (label 1; blue histogram)\nand with very few false positives. The bottom graph of Fig. 3\nshows the mean error of the position of the transit as a function\nof the probability assigned by the algorithm when it is applied\nto the curves containing a simulated transit (label 1, with known\nposition). One can see that when the transit is not found (prob-\nability close to 0), the error on the predicted position is large,\nabout 3 days, whereas it is of the order of a few hours only when\nthe probability is above 0.99. Thus, one can be confident in the\ntime of the transit given by the algorithm for curves with a prob-\nability higher than this threshold.\nArticle number, page 4 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\nFig. 3: Histogram of the results on the test set and error on the\ntransit position found by the network.\nTop panel: Histogram of the result of the neural network applied\nto the test data. Most of the 1-labeled light curves with an exo-\ncomet transit yield a probability close to 1, while there are only a\nfew false positives (0-labeled light curves yielding a high proba-\nbility). Above the chosen threshold, there is more than two order\nof magnitude between the number of true positives and false pos-\nitive.\nBottom panel: the mean error of the position of the transit as a\nfunction of the probability of the presence of a transit assigned\nby the network to the light curves. This error is of a few hours at\nmost for the transits identified by the algorithm.\n5. Search for exocomets in Kepler data\n5.1. Application of the model to Kepler data\nTo apply the algorithm presented in the previous sections, all the\nlight curves of the Kepler data were cut into 10-day windows\nobtained in the same way as the samples used to construct the\ntraining, validation, and test sets described in Sec. 2.2. In order\nnot to lose any transits located at the edge of the windows, an\noverlap of 2 days was applied between two consecutive 10-day\ntime intervals.\nHowever, despite the level of precision achieved, the algo-\nrithm provides a large number of false positives due to the large\nnumber of 10-day windows: approximately 106 windows per\nquarter lead to about 103 positive candidate detections, among\nwhich less than a dozen real transits should be identified (see\nLdE99a and discussion in Sect. 7). To reduce the number of false\npositives, we decided to filter the candidates using constraints on\nthe shape of the light curves of the detected photometric events.\n5.2. Additional criteria to filter false positives\nAfter all the Kepler light curves have passed through the neural\nnetwork, we end up with a set of 56525 exocomet candidates.\nTo filter these candidates, we decided to add constraints on the\nshape and on the context on the transit (noise, quality flags).\n5.2.1. Filtering through the shape of the transits candidates\nThe shape of the candidate transits were determined by fitting\nthem by the model proposed by Lecavelier des Etangs et al.\n(2022) together with a second order polynomial baseline:\nfexo(t) =\n\nCt2 + Dt + E\nt < t0,\n(Ct2 + Dt + E)(1 + K(e−β(t−t0) −1))\nt1 > t > t0\n(Ct2 + Dt + E)(1 + K(e−β(t−t0) −e−β(t−t1)))\nt > t1\n(4)\nFig. 4: Distribution of the value of the parameters obtained by\nfitting the 2163 light curves in the library of the simulated ex-\nocomet transits. The parameters K, ∆T = t1 −t0, and 1/β cor-\nrespond to the depth, ingress duration, and crossing time of the\ntransit, respectively.\nThe parameters t0 and t1 represent respectively the time at which\nthe transit starts and the time at which it reaches its minimum.\nK represents the characteristic depth of the transit, while 1/β\ncorresponds to the characteristic crossing time of the transit. C, D\nand E are the coefficients of the polynomial baseline. The fit\nof the light curve by the transit model can be easily performed\nusing the position of the detected candidate in the light curve as\nprovided by the network output.\nTo accept or reject a proposed detection given by the neural\nnetwork, we fit the light curve of the corresponding photometric\nevent to obtain the values of the parameters K, β, and ∆T = t1−t0\nthat characterize the shape of the event. We then check that these\nvalues are in the domain of validity for an exocometary transit.\nThe domain of validity is obtained by considering the parame-\nters space occupied by the fits of the theoretical light curves in\nthe library of numerical simulations of exocometary transits, as\nshown in Fig. 4 and summarized in the first three lines of Table 1.\nIt is found that in the theoretical light curves K is mostly\nbetween 10−4 and 5×10−3, ∆T is between 1.7 and 17 hours (cor-\nresponding to a periastron between ∼0.02 to 2 au, Lecavelier des\nEtangs et al. 2022), and β−1 is in the domain above 0.5 hours, and\nbetween 0.3∆T −1.7 and 0.5∆T + 12. Moreover, the parameters\nC and D have to verify the condition |C|, |D| < 10−3. This ensures\nthat the background is stable enough in the neighborhoods of the\ntransit. When this is not the case, the detected transit appears to\nbe only noise.\n5.2.2. Quality flags\nWe noticed that the presence of certain quality flags indicates an\nalteration of the light curve leading to a systematic detection by\nthe network. Thus, candidates whose transit position is less than\n1 day away from a quality flag 3 (\"Spacecraft is in coarse point\")\nor 12 (\"Impulsive outlier removed before cotrending\") have been\ndiscarded from the candidate list.\n5.2.3. Presence in the list of KOIs\nTo avoid periodic transits, which constitute roughly half of the\ncandidate transits output by the algorithm, stars belonging to the\nlist of KOIs (Kepler Object of Interest) have also been removed.\nArticle number, page 5 of 18\n\nA&A proofs: manuscript no. Article_v1\nTable 1: Parameters values for filtering the candidate transits\nParameter\nLower bound\nUpper bound\nDefinition\nK\n10−4\n5 × 10−3\nTransit depth\nβ−1\n(0.5h\n0.3∆T −1.7\n0.5∆T + 12\nCharacteristic crossing time\n∆T\n1.7h\n17h\nIngress duration\n|C|, |D|\n10−3\nBaseline coefficients\n∆RMS\n36\n∆χ2\npl\n5\n∆χ2\nrp\n10\n5.2.4. Other cases of false positives\nWe have found three other circumstances that can mislead the\nalgorithm and then result in obvious false positives: (1) a signifi-\ncant background noise on a time scale close to that of a cometary\ntransit; (2) a ramp-like shape in the light curve, i.e. a very sud-\nden decrease in flux followed by a more or less rapid return\nto the normal; (3) the presence of a mono-transit of a possible\nexoplanet that results in a very symmetrical brightness dip. In\nthe following, we suggest criteria to address each of these three\ncases.\nTo characterize the presence of noise anomaly, the ∆RMS was\ncalculated. It is defined as:\n∆RMS =\nX\ni\n( fexo(ti) −¯F3)/σ\n(5)\nwhere fexo(ti) is the exocomet model fit to the flux where the\ncandidate transit is found, ¯F3 is the mean of the flux without\nthe three lowest points and σ is the standard deviation of the\nflux. Only candidate transits with ∆RMS > 36 were conserved.\nThis criterion was very useful to remove candidates that were\ndetected due to a few points significantly lower than the average.\nTo detect the presence of red-noise that mimics an exocomet\ntransit, we calculated the correlation product between the exo-\ncomet model fit to the candidate transit and the light curve over\nall the considered sub-quarters. If the maximum of the correla-\ntion max1 is reached more than one day away from the transit\ntime or if its second maximum max2 is significantly close to the\nfirst maximum ((max1−max2)/max1 < 0.15), the candidate is re-\nmoved, considering that in this case the noise in the light curve\nresembles the identified transit, which is therefore probably not\na real exocomet transit, but a background red-noise.\nThe cases of the ramp-like light curves mimicking an exo-\ncometary transit have been discarded by calculating the differ-\nence between the fit with a comet model (Eq. 4) and a fit with a\nsimple ramp model defined by:\nframp(t) =\n(Ct2 + Dt + E\nt < t0,\n(Ct2 + Dt + E)(1 −K e−β(t−t0))\nt > t0\n(6)\nWe calculate ∆χ2\nrp defined by :\n∆χ2\nrp = N\n P\ni( framp(ti) −F(ti))2\nP\ni( fexo(ti) −F(ti))2 −1\n!\n,\n(7)\nwhere N = 120 is the number of points in the fitted window.\nThe photometric events that are almost as well fitted with the\nramp model as with the exocomet model (∆χ2\nrp < 10), have been\ndiscarded from the candidate list.\nFinally, to identify the cases of a symmetrical transit that\ncould be due to the passage of an exoplanet with a long period\n(a monotransit), we calculate the difference ∆χ2\npl between the χ2\npl\nof a model of a planetary transit calculated using the equations\nof Mandel & Agol (2002) and the χ2\nexo of an exocomet transit\nmodel (Eq. 4). It can be written as:\n∆χ2\npl = N\n P\ni(fpl(ti) −F(ti))2\nP\ni(fexo(ti) −F(ti))2 −1\n!\n.\n(8)\nThe photometric events having ∆χ2\npl < 5 have a symmetrical\nshape and have therefore not been taken into account in our final\nselection of exocomets. Such transits have been taken into ac-\ncount to establish the list of symmetrical mono-transits (see Sec.\n6.3). This criterion is, however, not used as a sharp cut to distin-\nguish the exocomets transits from the symmetrical mono-transits\nbut only as a help. The final distinction has been made through\nvisual inspection.\nAfter applying these criteria, the number of candidate tran-\nsits per quarter varies between 50 and 100 per quarter, leading\nto a total of 1349 transits. It then becomes possible to proceed\nfor each of them to a visual inspection to know if the candi-\ndate is indeed or not an exocomet transit. This final inspection is\nnecessary because, although the criteria put in place make it pos-\nsible to eliminate the vast majority of false positives, there still\nremain obvious false positives. Note here that the criteria have\nbeen deliberately chosen to be conservative so as not to acciden-\ntally eliminate a true exocomet transit. The number of criteria\nwas chosen so that the final number of transits to be visually\ninspected would be no more than a few thousand. Otherwise,\nvisual inspection would be impractical. It would be more ineffi-\ncient and inaccurate to add extra criteria than to visually check\nthe current list of 1,349 transits.\nTo verify the robustness of the detected transits, we perform\nan additional test. We apply our detection procedure to the data\nafter reversing the time. If the proportion of detected exocomets\nis the same in the reversed time data as in the original data, this\nmay question the relevance of the detected transits. However,\nas this analysis is time-consuming, we performed this analysis\non half of the quarters. Of the 17 quarters, we analysed eight\nrandomly chosen quarters: Q1, Q2, Q6, Q9, Q10, Q12, Q15 and\nQ16. We found only one transit that clearly mimics an exocomet\ntransit: KIC_3129239 in Q9. Given that approximately 17 robust\ntransits have been detected in the Kepler data (see Sec. 6), we\ncan infer from this study that about 10% to 15% of these transits\ncan be false positives. Although the same number of detections\nin the time-reversed data would have disproven the exocomet\nhypothesis, the result of this test supports the idea that there is an\nastrophysical signal in the data that is consistent with exocomet\ntransits.\nArticle number, page 6 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\n6. A new catalog of transiting exocomets\nIn this section, we present the final list of exocomet transits re-\nsulting from the selection procedure. We divided the list in three\ndifferent tiers based on the likelihood of a genuine exocometary\ndetection. In the first tier, the photometric events are the ones that\nwe consider as the most likely due to the transit of an exocomet ;\nin the second tier are the events possibly due to comets, while\nthe third tier gathers the most symmetric dip in the light curve,\nfor which the transit of an exoplanet remains a possibility.\n6.1. The first-tier catalog of exocometary transits\nThe first-tier catalog of detected exocometary transits is given in\nTable 2 and the plots of the corresponding light curves are given\nin Appendix B. It is particularly noteworthy that our neural net-\nwork is able to find all the transits already identified by previous\nworks: KIC 3542116 in Quarters 1, 10 and 12 and KIC 11084727\nas found by Rappaport et al. (2018), and KIC 8027456 as found\nby Kennedy et al. (2019). This independent retrieval of the same\nexocometary transits provides confidence in our procedure.\nNonetheless, after application of all the search procedure de-\nscribed above, it appeared that two transits of KIC 3542116 iden-\ntified by Rappaport et al. (2018) in Quarters 8 were not in our\nlist of detections. These transits were eliminated by the criterion\nbased on the maximum of the correlation product that aims to\nremove the detections whose pattern occurs several times in the\nsame subquarter (Sect. 5.2.4). In fact, this criterion also has the\ndisadvantage of eliminating active cometary systems where at\nleast two comets transits occurred in the same quarter, as is the\ncase for KIC 3542166 in Quarter 8. Therefore, we made a vi-\nsual inspection of all photometric events eliminated because of\nthis criterion in order to recover possible obvious exocometary\ntransits that were wrongly rejected. We thus found the two tran-\nsits of KIC 3542116 in the Quarter 8, and in addition identify a\nnew transit in front of KIC 6263848. This last case does not cor-\nrespond to a star with frequent cometary transits, but had been\ndeleted due to an edge effect on the calculation of the correlation\nfunction. These two cases are identified by an asterisk in Table 2.\nFor all of these detection, we calculate the ∆χ2\npolynomial de-\nfined as\n∆χ2\npolynomial = N\n P\ni( fpolynomial(ti) −F(ti))2\nP\ni(fexo(ti) −F(ti))2\n−1\n!\n.\n(9)\nwhere fpolynomial is the second order polynomial that best fits\nthe detection. For all detections, we obtained large ∆χ2\npolynomial\n(larger than about 100). This highlights that the detection cannot\nbe interpreted just as noise patterns.\nThe properties of the parent stars of the identified exocomets\nas tabulated by Kepler are given in Table 3 (Brown et al. 2011).\nFor KIC 3542116 and KIC 3662483, Teff, log10 g, metallicity\nand radius are from Zhang et al. (2025). From the Teff and\nlog10 g we can obtain an estimate of the stellar type (Gray 2008),\nwhich is given in the last column of Table 3. It appears that\nmost of the stars are main-sequence stars, with the exception\nof KIC 4078638, which is likely a red giant. For these stars, the\nupper limits on the age estimate given by Zhang et al. (2025)\nare all above 109 years, with the exception of KIC 4481029 with\nan upper limit of 9 × 108 years, and KIC 8027456 with an age\nestimate of 5 × 108 years.\n6.2. The second-tier catalog of possible exocometary\nAfter visual inspection of the list of candidates produced by\nour algorithm, we came up with a list of interesting photomet-\nric events that can be due to transiting exocomets. This includes\ntransits that pass through all the criteria described in Sect. 5, but\nwhose shape or signal-to-noise ratio makes it difficult to decide\nwhether they are actually transits. The resulting list of possible\nexocometary transits in this second-tier catalog is given in Ta-\nble 4 and the plots of the corresponding light curves are given in\nAppendix C.\nThe properties of the corresponding parent stars in the sec-\nond tier catalog are given in Table 5 (Brown et al. 2011).\nHere again, estimates of the stellar types given in the last col-\numn of the table show that most of the stars are on or close\nto the main sequence, with the exception of KIC 2984102 and\nKIC 11153134, which have a log10 g below 3 and are therefore\nlikely red giants. For all stars in this second-tier catalog, the up-\nper limit of the age estimate given by Zhang et al. (2025) is above\n109 years, with the exception of KIC 2984102, KIC 5294231 and\nKIC 7183123 with age estimates of 5×108, 1×108 (with an upper\nlimit of 3 × 108) and 9 × 108 years, respectively.\n6.3. The third-tier catalog of symmetric transits\nAfter visual inspection of the list of candidates produced by our\nalgorithm, we also came up with a list of interesting photomet-\nric events that appears to be symmetrical. This is not surpris-\ning as some of the exocomets transits in the library used for\ntraining the network are indeed symmetrical (see Fig. 2 and 3\nin LdE99a). However, we cannot exclude that these are due to\nmono-transits of exoplanets or to false positives mimicking plan-\netary or cometary transits. Several scenarios might cause such\nfalse positives, including diluted or undiluted eclipsing binaries.\nTo identify these cases one needs to carry out additional anal-\nyses that are beyond the scope of the present paper (see, e.g.,\nCrossfield et al. 2016). Finally, although an exocometary tran-\nsit classification cannot be made, these detections deserve to be\nmentioned. They are given in the third-tier catalog of symmetric\ntransits listed in Table 6.\nWe compared these detected photometric events with the\nlists of single transits published by Huang et al. (2013), Wang\net al. (2015), Foreman-Mackey et al. (2016), and Herman et al.\n(2019). It appears that four events in our list were already identi-\nfied by Foreman-Mackey et al. (2016) and Herman et al. (2019).\nThe transit in front of KIC 8410697 was found in both studies.\nIt is interpreted as due to an exoplanet with a radius of 0.7 times\nthat of Jupiter. The photometric event of KIC 10321319 was\nfound by Foreman-Mackey et al. (2016) and interpreted as due\nto the transit of an exoplanet of 0.16 Jupiter radius. The photo-\nmetric event of KIC 6196417 was found by Herman et al. (2019)\nand interpreted as due to the transit of an exoplanet of about 0.7\nJupiter radius. The transit in front of KIC 10668646 at the time of\nBKJD=1449.3 was already identified by Foreman-Mackey et al.\n(2016) but the exoplanet transit scenario was rejected because of\na centroid shift in the data. However, we also identified another\nphotometric event in the same target at BKJD=196.3 with a dif-\nferent shape and a lower absorption depth. These few examples\nshow that our list of newly identified photometric events which\ncould be due to transits of exocomet with symmetrical shape of\nthe light curve deserves further investigations which are beyond\nthe scope of the present paper.\nArticle number, page 7 of 18\n\nA&A proofs: manuscript no. Article_v1\nTable 2: First-tier catalog of exocometary transits.\nStars (KIC)\nTime (BKJD)\nQuarter\nK(×10−4)\nβ−1(h)\n∆T(h)\nAD(×10−4)\n∆χ2\npl\n∆χ2\npolynomial\n†3542116\n161.5\n1\n4.74\n6.5\n9.92\n3.71\n18.32\n279.74\n†*3542116\n742.6\n8\n4.17\n1.36\n6.05\n4.12\n0.6\n348.83\n†*3542116\n792.9\n8\n8.62\n4.16\n4.77\n5.88\n15.57\n285.27\n†3542116\n992.1\n10\n10.12\n3.62\n14.68\n9.94\n28.67\n2993.05\n†3542116\n1175.8\n12\n12.98\n3.40\n9.35\n12.15\n8.58\n2670.38\n3662483\n1132.1\n12\n6.81\n6.38\n6.87\n4.49\n12.98\n204.52\n4078638\n355.5\n4\n83.21\n6.30\n3.22\n33.30\n47.89\n358.90\n4481029\n172.5\n2\n9.08\n8.57\n4.64\n3.80\n14.31\n235.74\n5206257\n925.0\n10\n10.98\n4.75\n4.47\n6.69\n5.04\n322.26\n5514200\n670.1\n7\n34.48\n12.13\n7.27\n15.53\n19.15\n155.20\n*6263848\n775.9\n8\n10.22\n3.29\n4.90\n7.92\n7.96\n151.93\n6927963\n416.8\n4\n33.60\n9.13\n2.83\n8.96\n6.54\n115.89\n7660548\n966.2\n10\n15.74\n5.95\n5.09\n9.05\n26.10\n219.47\n†8027456\n1449.0\n15\n8.0\n9.90\n14.12\n6.08\n16.78\n401.18\n8738545\n238.1\n2\n17.0\n4.92\n3.67\n8.94\n7.64\n92.53\n10484683\n309.5\n3\n29.85\n14.05\n8.01\n12.97\n87.66\n853.82\n†11084727\n1076.2\n11\n15.29\n3.92\n8.20\n13.40\n88.14\n2368.80\nNotes. The first column gives the name of the star by its KIC identifier. The second and third columns indicate the position of the transit. Columns\n4 to 6 give the fits parameters associated with the exocomet model (Eq. 4). The seventh column gives the absorption depth. The last two columns\npresent the difference of χ2 between the cometary model and, first, the planetary model (Mandel & Agol 2002) and, second, a second order\npolynomial fit.\nThe transits that were first eliminated by the criterion based on the maximum of the correlation product (see Sect. 5.2.4) are identified by an\nasterisk (*). The five transits already known are identified by the † symbol.\nTable 3: Information on the stars of the first-tier catalog.\nKepler ID\nRA\nDec\nKepler Mag\nTeff\nLog g\nMetallicity\nRadius\nStellar\n(J2000)\n(J2000)\n(K)\n(cm/s2)\n(R⊙)\ntype\n3542116\n19 22 52.939\n+38 41 41.51\n9.979\n6766\n4.208\n-0.190\n1.454\nF2 V\n3662483\n19 42 09.199\n+38 43 01.38\n10.541\n5848\n3.945\n0.107\n1.786\nG1 IV\n4078638\n19 46 16.611\n+39 06 15.23\n13.156\n4783\n2.758\n0.150\n8.331\nK3 III\n4481029\n19 43 13.210\n+39 32 32.03\n11.303\n8865\n3.968\n0.010\n2.310\nA0 IV\n5206257\n19 46 33.826\n+40 22 57.97\n12.051\n6017\n4.435\n-0.105\n1.052\nF6 V\n5514200\n18 58 08.306\n+40 46 29.32\n14.377\n5651\n4.500\n-0.243\n0.955\nG4 V\n6263848\n18 52 30.566\n+41 36 33.34\n13.172\n5850\n4.365\n-0.224\n1.137\nG2 V\n6927963\n18 56 15.946\n+42 27 52.20\n13.807\n5711\n4.441\n-0.012\n1.030\nG4 V\n7660548\n18 47 22.807\n+43 23 05.39\n13.094\n6158\n4.369\n-0.050\n1.148\nF5 V\n8027456\n19 25 15.838\n+43 51 33.55\n9.697\n8732\n3.766\n0.065\n2.974\nA0 IV\n8738545\n18 58 37.262\n+44 55 15.24\n13.227\n5091\n3.581\n-0.162\n3.101\nK1 IV\n10484683\n19 47 54.377\n+47 40 14.16\n12.588\n6326\n4.543\n-0.009\n0.934\nF5 V\n11084727\n19 28 41.191\n+48 41 15.14\n9.987\n6762\n4.067\n-0.152\n1.726\nF3 V\n7. Conclusion\nAlthough the Kepler mission ended more than 10 years ago, the\navailable data is still worth deep data mining in search of new\ndiscoveries. Here we present a new search for exocometary tran-\nsits in the Kepler light curves. We used machine learning tech-\nnique with a neural network that has been trained using a li-\nbrary of theoretical exocomets transit light curves inherited from\nthe work of LdE99b. After parsing the light curves of close to\n200 000 stars through the neural network, despite the application\nof several filters to eliminate most of the false positive, a visual\ninspection of the outcome of the network was still needed. We\nended up with three catalogs of interesting objects. The first-tier\ncatalog is composed of a total of 17 exocometary transits, includ-\ning 7 previously identified transits and 10 new transits in front of\n10 different stars. The second-tier catalog is a list of 30 photo-\nmetric events that appear of second quality and that we qualify\nas possible exocometary transits. Finally, the third-tier catalog\npresents the list of interesting photometric events that are sym-\nmetrical and may be due to transits of either an exoplanet or an\nexocomet with a periastron at 90◦from the line of sight.\nThe complete Kepler data represent a 4-year photometry sur-\nvey of more than 170 000 distinct stars. However, as noted in\nKennedy et al. (2019), not all stars were observed for the full\nduration of the mission. As a result, we can consider 150 000 as\nthe approximate number of equivalent stars that were observed\nfor the full duration of the mission. Or, in other words, our anal-\nysis covers the equivalent of about 600 000 stars-years. LdE99a\nshowed that with 30 000 stars-years survey of stars with a solar\nsystem cometary activity and a few 10−4 photometric accuracy,\nthe number of exocomets transit detections should be around 10.\nTherefore, assuming that each star harbors a cometary system\nsimilar to our own, we could expect roughly 200 detections with\nArticle number, page 8 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\nTable 4: Second-tier catalog of possible exocometary transits. Same legend as in Table 2.\nStars (KIC)\nTime (BKJD)\nQuarter\nK(×10−4)\nβ−1(h)\n∆T(h)\nAD(×10−4)\n∆χ2\npl\n∆χ2\npolynomial\n2984102\n345.7\n3\n36.42\n15.9\n10.12\n17.14\n61.90\n430.26\n4826941\n385.8\n4\n21.58\n15.22\n7.85\n8.69\n3.76\n126.17\n5195476\n1508.8\n16\n16.80\n5.09\n3.67\n8.31\n3.42\n45.92\n5294231\n411.4\n4\n4.92\n3.26\n2.13\n2.36\n31.78\n285.46\n5511746\n1105.8\n12\n13.31\n12.33\n5.06\n4.48\n2.41\n44.54\n5514277\n670.4\n7\n63.11\n13.57\n9.88\n32.64\n13.96\n160.95\n5534083\n1167.7\n12\n9.34\n10.45\n3.33\n2.55\n7.22\n43.16\n6119914\n747.2\n8\n31.41\n9.36\n6.63\n15.95\n3.54\n50.65\n7183123\n336.4\n3\n8.43\n8.36\n5.98\n4.31\n28.32\n321.42\n7201827\n409.9\n4\n43.64\n4.83\n2.83\n19.34\n5.77\n48.98\n7220968\n1325.7\n14\n18.60\n5.50\n6.23\n12.61\n2.61\n46.42\n7611858\n1275.8\n14\n6.46\n9.77\n5.19\n2.67\n7.63\n109.47\n7669613\n315.9\n3\n13.24\n13.66\n6.31\n4.89\n12.77\n121.72\n8111649\n934.0\n10\n83.04\n23.69\n10.28\n29.24\n16.36\n123.61\n8144412\n639.1\n7\n86.39\n12.00\n2.95\n18.84\n7.06\n140.49\n8374877\n769.1\n8\n7.78\n7.04\n8.46\n5.44\n13.67\n300.35\n8489495\n1351.8\n14\n16.43\n8.44\n15.04\n13.67\n10.70\n47.92\n8625941\n1505.0\n16\n5.16\n8.44\n9.80\n3.54\n8.10\n66.25\n8683719\n1340.2\n14\n19.21\n6.43\n14.25\n17.12\n7.59\n195.79\n8738569\n1462.3\n15\n13.18\n8.30\n4.30\n5.33\n8.84\n56.11\n8843356\n702.2\n7\n8.84\n10.54\n2.99\n2.18\n4.66\n46.51\n9025688\n1078.2\n11\n13.29\n7.51\n3.57\n5.03\n58.03\n58.03\n9388975\n680.3\n7\n18.41\n3.36\n2.47\n9.59\n6.76\n60.28\n9531080\n748.0\n8\n10.23\n9.31\n6.56\n5.18\n16.13\n53.73\n10329957\n1385.1\n15\n18.47\n12.90\n4.10\n5.03\n9.62\n64.04\n11124157\n704.2\n7\n15.01\n10.93\n2.00\n2.51\n11.48\n45.20\n11153134\n739.7\n8\n18.93\n3.15\n7.99\n17.44\n11.42\n345.83\n11246607\n696.4\n7\n18.06\n15.10\n8.84\n8.00\n7.04\n165.29\n11515196\n644.3\n7\n16.40\n7.98\n3.84\n6.26\n34.15\n138.30\n11701976\n931.7\n10\n23.79\n11.98\n2.70\n4.80\n13.24\n40.42\na perfect search in the Kepler data. Finding a few dozens of ex-\nocometary transits can therefore be considered as a satisfactory\nresult in agreement with what could be expected using reason-\nable estimates of the algorithm efficiency and cometary activity\nof the surveyed stars.\nUnfortunately, the Kepler stars are rather faint, and our cat-\nalogs contain stars with magnitude between 9.7 and 15.9. It is\ntherefore not possible to undertake a spectroscopic follow up\nwith the hope to confirm the exocometary nature of the photo-\nmetric events. All the more that, with only one transit per star\nin about four years, exocometary transits are rare and not pre-\ndictable in time. However, we can anticipate that the same tech-\nniques could be applied to search for exocometary transits in\ncurrent TESS data and in upcoming PLATO data. For these two\ninstruments, the surveyed star in the input catalog are signifi-\ncantly brighter that the Kepler stars. We can thus expect that the\nexocometary systems to be discovered in near future can be stud-\nied in detail, particularly through spectroscopy.\nPrevious searches (Rappaport et al. 2018; Kennedy et al.\n2019) provided a list of 3 stars that seem to be rather young.\nThis supported the anticipated conclusion that cometary activity\nis likely correlated with stellar age, with an expected decline over\ntime. However, our catalogs do not support this idea, as we found\na wide spread of age for both the first- and second-tier catalogs,\nwith possible exocometary transits in front of stars classified as\nred giants. The issue of cometary activity with age thus remains\nopen, and further studies will be needed to clarify this important\nquestion of the evolution of planetary systems.\nNote added in manuscript: After this manuscript was sub-\nmitted we became aware of a paper by Norazman et al. (2025)\nthat used Machine Learning techniques to search exocometary\ntransits in the sectors 1 to 22 of the TESS light curves. They\nfound three additional transits compared to those already known,\nwhich underlines the utility of this technique for the search of\nsuch transits.\nAcknowledgements. We thank Dr. Neda Heidari for her help in identifying al-\nready known mono-transits. We acknowledge support from the CNES (Centre\nnational d’études spatiales, France). This work has made use of the Infinity Clus-\nter hosted by Institut d’Astrophysique de Paris. We thank Dr. Guilhem Lavaux\nfor his invaluable help in making efficient use of the GPUs in this cluster. This pa-\nper includes data collected by the Kepler mission and obtained from the MAST\ndata archive at the Space Telescope Science Institute (STScI). Funding for the\nKepler mission is provided by the NASA Science Mission Directorate. STScI\nis operated by the Association of Universities for Research in Astronomy, Inc.,\nunder NASA contract NAS 5–26555. This research has made use of the NASA\nExoplanet Archive, which is operated by the California Institute of Technology,\nunder contract with the National Aeronautics and Space Administration under\nthe Exoplanet Exploration Program.\nReferences\nAbadi, M., Barham, P., Chen, J., et al. 2016, TensorFlow: A System for Large-\nScale Machine Learning\nBendahan-West, R., Kennedy, G. M., Brown, D. J. A., & Strøm, P. A. 2025,\nMNRAS, 537, 229\nBorucki, W. J., Koch, D., Basri, G., et al. 2010, Sci, 327, 977\nBrown, T. M., Latham, D. W., Everett, M. E., & Esquerdo, G. A. 2011, AJ, 142,\n112\nChintarungruangchai, P. & Jiang, I.-G. 2019, PASP, 131, 064502\nArticle number, page 9 of 18\n\nA&A proofs: manuscript no. Article_v1\nTable 5: Information on the stars of the second-tier catalog.\nKepler ID\nRA\nDec\nKepler Mag\nTeff\nLog g\nMetallicity\nRadius\nStellar\n(J2000)\n(J2000)\n(K)\n(cm/s2)\n(R⊙)\ntype\n2984102\n19 20 24.302\n+38 09 41.26\n11.271\n5064\n2.543\n-0.167\n10.715\nK1III\n4826941\n19 16 42.927\n+39 58 17.69\n13.917\n5748\n4.425\n-0.257\n1.051\nG3V\n5195476\n19 36 47.573\n+40 19 41.48\n13.925\n4848\n3.504\n-0.214\n3.406\nK2IV\n5294231\n19 46 54.941\n+40 26 00.17\n9.842\n9552\n4.082\n0.032\n2.107\nA0V\n5511746\n18 53 09.113\n+40 46 26.87\n13.216\n5928\n4.266\n-0.277\n1.290\nG1V\n5514277\n18 58 16.380\n+40 47 18.02\n15.366\n5401\n4.601\n-0.242\n0.824\nG7V\n5534083\n19 27 57.571\n+40 45 16.38\n12.602\n6416\n4.186\n-0.261\n1.458\nF4V\n6119914\n19 22 29.443\n+41 24 30.31\n15.882\n6167\n4.819\n-0.036\n0.660\nF5V\n7183123\n19 02 20.712\n+42 47 30.55\n11.207\n8005\n3.703\n-0.145\n3.020\nA0IV\n7201827\n19 29 03.487\n+42 44 03.41\n15.409\n5562\n4.815\n-0.098\n0.638\nG5V\n7220968\n19 48 59.026\n+42 44 39.52\n15.463\n5015\n4.483\n0.35\n0.921\nK1V\n7611858\n19 32 36.857\n+43 17 11.54\n11.187\n5684\n4.477\n-0.315\n0.984\nG4V\n7669613\n19 06 56.856\n+43 18 44.17\n13.044\n5754\n4.246\n-0.382\n1.313\nG3V\n8111649\n19 45 47.110\n+43 58 36.44\n15.776\n6080\n4.576\n-0.044\n0.888\nF6V\n8144412\n18 47 41.122\n+44 04 04.40\n15.213\n5721\n4.428\n0.043\n1.047\nG3V\n8374877\n19 40 27.934\n+44 19 09.62\n12.083\n6510\n4.021\n-0.505\n1.809\nF4V\n8489495\n19 18 40.234\n+44 33 44.75\n15.214\n4912\n4.328\n0.068\n1.112\nK2V\n8625941\n19 28 55.855\n+44 44 53.77\n12.783\n5689\n4.602\n-0.318\n0.844\nG4V\n8683719\n19 18 58.464\n+44 49 28.02\n14.717\n5989\n4.398\n-0.035\n1.100\nG0V\n8738569\n18 58 41.198\n+44 57 51.12\n12.771\n4907\n3.548\n0.173\n3.227\nK2IV\n8843356\n19 58 40.882\n+45 01 02.64\n12.039\n5419\n3.694\n-0.387\n2.687\nG7IV\n9025688\n19 32 42.634\n+45 23 41.57\n13.382\n6445\n4.165\n-1.598\n1.500\nF4V\n9388975\n18 56 00.802\n+45 57 05.90\n14.292\n5163\n4.493\n-0.938\n0.923\nK0V\n9531080\n19 35 07.706\n+46 09 35.64\n13.861\n6280\n4.268\n-0.346\n1.306\nF5V\n10329957\n19 10 18.223\n+47 27 04.43\n13.58\n5002\n4.567\n-0.024\n0.824\nK1V\n11124157\n18 56 39.605\n+48 45 44.21\n12.244\n4775\n3.954\n0.071\n1.869\nK3IV\n11153134\n19 53 11.854\n+48 46 20.82\n12.007\n4941\n2.719\n-0.037\n8.746\nK2III\n11246607\n19 32 21.149\n+48 58 47.78\n13.169\n6251\n4.387\n-0.253\n1.127\nF5V\n11515196\n19 42 06.725\n+49 26 57.88\n13.568\n4754\n4.15\n0.345\n1.403\nK3V\n11701976\n19 02 02.592\n+49 49 17.65\n13.403\n6210\n4.398\n-0.014\n1.111\nF5V\nCho, K., van Merriënboer, B., Gulcehre, C., et al. 2014, in Proceedings of the\n2014 Conference on Empirical Methods in Natural Language Processing\n(EMNLP), ed. 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C., et al. 2025, ApJS, 277, 6\nZieba, S., Zwintz, K., Kenworthy, M. A., & Kennedy, G. M. 2019, A&A, 625,\nL13\nZucker, S. & Giryes, R. 2018, AJ, 155, 147\nArticle number, page 10 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\nTable 6: Third-tier catalog of symmetric transits that could be either exocometary transits or exoplanet mono-transits.\nStars (KIC)\nTime (BKJD)\nQuarter\nK(×10−4)\nβ−1(h)\n∆T(h)\nAD(×10−4)\n∆χ2\npl\n∆χ2\npolynomial\n2983000\n759.1\n8\n18.38\n0.84\n16.33\n18.38\n-1.94\n483.82\n2993038\n550.9\n6\n15.22\n2.46\n4.72\n12.99\n-1.57\n1165.8\n3222471\n673.1\n7\n97.7\n5.49\n14.59\n90.83\n-70.29\n2436.25\n3346436\n270.4\n3\n8.48\n3.03\n3.05\n5.38\n3.42\n182.43\n3755854\n1492.3\n16\n3.53\n1.07\n6.11\n3.51\n1.03\n45.73\n5184479\n534.6\n5\n8.87\n1.04\n19.61\n8.87\n-11.23\n316.53\n5184479\n1203.0\n13\n12.17\n1.97\n17.91\n12.17\n-3.69\n409.71\n5305217\n673.1\n7\n33.0\n5.46\n20.87\n32.28\n-7.41\n170.59\n5456365\n270.8\n3\n35.42\n3.85\n10.37\n33.03\n-2.28\n323.73\n5905878\n1343.3\n14\n2.25\n1.46\n7.55\n2.24\n-7.35\n353.12\n5905878\n1441.3\n15\n0.89\n1.49\n8.14\n0.89\n-2.01\n60.16\n5967153\n1394.5\n15\n5.06\n4.05\n7.9\n4.34\n-22.09\n141.91\n5975275\n213.3\n2\n9.78\n2.57\n10.57\n9.62\n2.7\n179.4\nb6186417\n959.1\n10\n33.19\n1.09\n18.42\n33.19\n-2.36\n645.89\n6387193\n651.7\n7\n78.79\n2.73\n12.43\n77.96\n-48.64\n3817.28\n6515488\n1420.4\n15\n28.85\n2.98\n9.54\n27.67\n-7.02\n359.44\n6804821\n1009.1\n11\n12.97\n1.6\n22.41\n12.97\n-6.09\n317.96\n7047396\n298.8\n3\n11.35\n2.0\n5.57\n10.65\n-4.56\n305.5\n7105703\n1509.0\n16\n20.54\n2.86\n6.53\n18.45\n-4.32\n186.11\n7213651\n1239.2\n13\n21.53\n1.66\n6.78\n21.17\n-9.23\n214.24\n7465971\n342.7\n3\n11.14\n1.14\n9.75\n11.14\n-1.07\n200.89\n7983622\n471.9\n5\n65.06\n4.63\n13.76\n61.73\n-15.71\n732.08\n8005892\n608.5\n6\n96.11\n1.2\n9.78\n96.09\n-75.32\n9905.87\n8007462\n1458.6\n15\n2.44\n2.56\n5.4\n2.14\n-7.15\n93.52\n8159297\n440.7\n4\n18.1\n1.87\n4.62\n16.57\n-1.87\n132.24\n8313257\n1149.1\n12\n25.05\n2.17\n17.4\n25.05\n-8.72\n320.72\na,b8410697\n542.5\n6\n54.16\n1.0\n17.87\n54.16\n-12.07\n5694.83\n8496108\n277.7\n3\n19.41\n9.57\n3.83\n6.4\n-6.55\n62.5\n8617888\n171.8\n2\n55.2\n1.83\n5.44\n52.4\n-4.99\n149.3\n9016734\n666.0\n7\n61.61\n5.57\n13.7\n56.35\n-48.25\n1624.94\n9016734\n1230.1\n13\n27.52\n5.5\n14.31\n25.48\n-7.51\n522.79\n9413755\n1564.2\n17\n19.5\n1.63\n4.4\n18.18\n-16.75\n217.06\n9775416\n1398.1\n15\n5.04\n4.99\n10.63\n4.44\n-2.36\n141.64\n10024862\n1494.0\n16\n20.95\n1.68\n13.29\n20.94\n-2.57\n121.87\n10157075\n369.5\n4\n26.75\n1.93\n5.57\n25.27\n-2.17\n114.74\n10192453\n757.5\n8\n18.64\n5.63\n1.73\n4.94\n-29.2\n76.45\na10321319\n554.5\n6\n4.76\n4.77\n13.56\n4.48\n-29.18\n174.87\n10334763\n549.7\n6\n7.23\n0.74\n10.88\n7.23\n-3.39\n412.26\n10397849\n506.5\n5\n19.93\n2.35\n7.95\n19.25\n0.15\n289.02\n10450889\n239.6\n2\n60.03\n1.58\n2.35\n46.44\n1.45\n192.69\n10556420\n756.3\n8\n3.59\n0.64\n6.03\n3.59\n-1.17\n364.07\n10643786\n1421.2\n15\n12.62\n3.23\n5.12\n10.04\n3.16\n81.51\n10668646\n196.3\n2\n42.81\n0.93\n8.74\n42.8\n-20.57\n1376.58\na10668646\n1449.3\n15\n57.08\n1.36\n8.64\n56.98\n-38.4\n3632.55\n10909733\n898.7\n9\n3.41\n1.37\n6.17\n3.37\n4.64\n67.65\n11561379\n137.7\n1\n13.51\n2.11\n7.45\n13.12\n-1.64\n103.03\n11921843\n1234.8\n13\n3.85\n0.62\n14.83\n3.85\n-2.49\n287.33\n12066509\n632.3\n7\n38.71\n0.78\n12.41\n38.71\n-0.31\n1883.85\n12644038\n584.7\n6\n36.26\n1.49\n6.37\n35.77\n3.56\n199.23\nNotes. Four transits were already identified by Foreman-Mackey et al. (2016) (labeled with \"a\" in the first column) and Herman et al. (2019)\n(labeled with \"b\").\nArticle number, page 11 of 18\n\nA&A proofs: manuscript no. Article_v1\nAppendix A: Performance of the algorithm over all the quarters\nTable A.1: Performance of the neural network.\nQuarter\nSubquarter\nAccuracy\nRecall\nPrecision\nFPR\nAUC\nMAE\n1\n0\n0.8945\n0.7910\n0.9975\n0.0020\n0.9888\n2.3404\n2\n0\n0.8665\n0.7347\n0.9977\n0.0017\n0.9907\n2.2237\n2\n1\n0.8616\n0.7245\n0.9982\n0.0013\n0.9900\n2.0792\n2\n2\n0.8743\n0.7498\n0.9984\n0.0012\n0.9893\n2.0455\n3\n0\n0.9108\n0.8240\n0.9972\n0.0023\n0.9896\n2.0329\n3\n1\n0.9068\n0.8162\n0.9969\n0.0025\n0.9905\n2.1186\n3\n2\n0.8903\n0.7835\n0.9964\n0.0028\n0.9895\n2.1638\n4\n0\n0.8862\n0.7745\n0.9972\n0.0022\n0.9909\n2.3073\n4\n1\n0.9017\n0.8065\n0.9961\n0.0032\n0.9924\n2.4661\n4\n2\n0.8854\n0.7728\n0.9974\n0.0020\n0.9920\n1.9370\n5\n0\n0.8994\n0.8013\n0.9969\n0.0025\n0.9919\n1.9726\n5\n1\n0.8898\n0.7818\n0.9972\n0.0022\n0.9916\n2.1857\n5\n2\n0.8951\n0.7923\n0.9973\n0.0022\n0.9919\n2.5873\n6\n0\n0.8979\n0.7972\n0.9983\n0.0013\n0.9906\n2.2020\n6\n1\n0.8908\n0.7833\n0.9979\n0.0017\n0.9911\n1.9720\n6\n2\n0.8860\n0.7738\n0.9976\n0.0018\n0.9910\n2.4570\n7\n0\n0.8792\n0.7605\n0.9972\n0.0022\n0.9899\n2.1862\n7\n1\n0.8752\n0.7522\n0.9978\n0.0017\n0.9909\n1.7915\n7\n2\n0.9128\n0.8267\n0.9986\n0.0012\n0.9935\n2.1101\n8\n0\n0.9004\n0.8018\n0.9988\n0.0010\n0.9911\n2.0729\n8\n1\n0.9027\n0.8065\n0.9986\n0.0012\n0.9926\n2.1139\n9\n0\n0.9068\n0.8157\n0.9976\n0.0020\n0.9930\n2.1293\n9\n1\n0.9017\n0.8062\n0.9965\n0.0028\n0.9916\n2.3157\n9\n2\n0.8983\n0.7992\n0.9969\n0.0025\n0.9918\n2.3348\n9\n3\n0.8798\n0.7610\n0.9980\n0.0015\n0.9916\n2.2389\n10\n0\n0.8755\n0.7532\n0.9971\n0.0022\n0.9896\n2.7444\n10\n1\n0.8758\n0.7538\n0.9971\n0.0022\n0.9889\n2.0156\n10\n2\n0.8829\n0.7672\n0.9983\n0.0013\n0.9890\n2.7258\n11\n0\n0.8830\n0.7685\n0.9968\n0.0025\n0.9908\n2.0722\n11\n1\n0.8941\n0.7900\n0.9977\n0.0018\n0.9913\n1.9576\n11\n2\n0.8907\n0.7828\n0.9981\n0.0015\n0.9921\n2.4971\n12\n0\n0.8876\n0.7767\n0.9981\n0.0015\n0.9921\n2.3743\n12\n1\n0.9028\n0.8075\n0.9977\n0.0018\n0.9930\n2.3531\n12\n2\n0.9038\n0.8103\n0.9967\n0.0027\n0.9913\n2.4029\n13\n0\n0.8981\n0.8000\n0.9952\n0.0038\n0.9913\n2.8353\n13\n1\n0.8431\n0.6875\n0.9981\n0.0013\n0.9872\n2.4796\n13\n2\n0.9037\n0.8117\n0.9947\n0.0043\n0.9915\n2.5127\n14\n0\n0.8654\n0.7342\n0.9955\n0.0033\n0.9877\n2.4772\n14\n1\n0.8801\n0.7632\n0.9961\n0.0030\n0.9897\n2.4096\n14\n2\n0.8788\n0.7592\n0.9978\n0.0017\n0.9915\n3.1279\n15\n0\n0.8363\n0.6743\n0.9973\n0.0018\n0.9869\n2.5724\n15\n1\n0.8378\n0.6780\n0.9966\n0.0023\n0.9826\n3.3750\n15\n2\n0.8490\n0.7000\n0.9972\n0.0020\n0.9862\n3.1479\n16\n0\n0.8238\n0.6483\n0.9987\n0.0008\n0.9870\n2.4968\n16\n1\n0.8665\n0.7355\n0.9966\n0.0025\n0.9868\n2.1849\n16\n2\n0.8445\n0.6907\n0.9976\n0.0017\n0.9858\n2.4707\n17\n0\n0.8744\n0.7510\n0.9971\n0.0022\n0.9899\n2.3075\nArticle number, page 12 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\nAppendix B: Cometary transits light curves\nPlots of the light curves of the detected cometary transits in the first tier catalog. For each transit, the light curve is shown in the\ntop panels over a 5 days duration centered on the transit time and with an exocomet transit fit superimposed (red line). The bottom\npanels show the light curves over a larger time frame.\nArticle number, page 13 of 18\n\nA&A proofs: manuscript no. Article_v1\nArticle number, page 14 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\nArticle number, page 15 of 18\n\nA&A proofs: manuscript no. Article_v1\nAppendix C: Possible exocometary transits light curves\nPlots of the light curves of the possibly detected cometary transits in the second tier catalog. The plots cover a 5 days time frame\ncentered on the transit time and with an exocomet transit fit superimposed (red line).\nArticle number, page 16 of 18\n\nP. Dumond et al.: Search for exocomets transits in Kepler light curves\nArticle number, page 17 of 18\n\nA&A proofs: manuscript no. Article_v1\nArticle number, page 18 of 18\n"},"cleaned":{"kind":"string","value":"Astronomy & Astrophysics manuscript no. Article_v1 October 17, 2025 Search for exocomets transits in Kepler light curves Ten new transits identified P. Dumond1, 2, A. Lecavelier des Etangs1, F. Kiefer1, 3, G. Hébrard1, and V. Caillé1, 4 1 Institut d'astrophysique de Paris, CNRS, Sorbonne Université, 98bis boulevard Arago, 75014 Paris, FRANCE 2 CRAL, Ecole normale supérieure de Lyon, Université de Lyon, UMR, CNRS 5574, F-69364 Lyon Cedex 07, France 3 LIRA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France 4 Institut Pierre-Simon Laplace, Sorbonne-Université, CNRS, Paris, France, October 17, 2025 ABSTRACT The Kepler mission, despite its conclusion over a decade ago, continues to offer a rich dataset for uncovering new astrophysical objects and phenomena. In this study, we conducted a comprehensive search for exocometary transit signatures within the Kepler light curves, using a machine learning approach based on a neural network trained on a library of theoretical exocomet transit light curves. By analyzing the light curves of 201,820 stars, we identified candidate events through the neural network and subjected the output to filtering and visual inspection to mitigate false positives. Our results are presented into three catalogs of increasing ambiguity. The first-tier catalog includes 17 high-confidence exocometary transit events, comprising 7 previously reported events and 10 newly identified ones, each associated with a different host star. The second-tier catalog lists 30 lower-confidence events that remain consistent with possible exocometary transits. The third-tier catalog consists of 49 more symmetric photometric events that could be either exocometary transits, exoplanet mono-transits, or false positives due to eclipsing binaries mimicking transits. Contrary to previous studies, which suggested that the cometary activity was favored by stellar youth, we find a broad age distribution among candidate host stars, including several red giants. This challenges the general idea of a decline in cometary activity with stellar age and underlines the need for further investigation into the temporal evolution of exocometary activity in planetary systems. Key words. Methods: data analysis - Techniques: photometric - Surveys - Comets: general - Planetary systems - Exocomets - Transit photometry 1. Introduction Exocomets are rocky or icy minor bodies that become active when they approach their parent star on a highly eccentric orbit (Strøm et al. 2020). Producing a huge tail of gas and dust close to the periastron, when they transit in front of the star, they become detectable using spectroscopy (for the gaseous tail) and photometry (for the dust tail). The first exocomets were thus identified in the 1980s in the β Pic system, through the detection of variable absorption features in the Ca ii lines of the stellar spectrum (Ferlet et al. 1987; Kiefer et al. 2014a). Later on, a few other exocometary systems have been found using spectroscopy, like HD 172555 (Kiefer et al. 2014b; Grady et al. 2018), and 49 Cet (Montgomery & Welsh 2012; Miles et al. 2016). A list of spectroscopically identified exocometary systems can be found in Strøm et al. (2020). For the detection of exocomets using photometry, one needs to obtain photometric measurements at high accuracy, typically at 10-3-10-4 level over the full transit duration, that is, for one to three days (Lecavelier des Etangs et al. 1999, hereafter LdE99a). This capability has been reached only during the last decade with the space missions like Corot, Kepler, TESS, and Cheops. Detections have thus been obtained with Kepler toward KIC 3542116 and KIC 11084727 (Rappaport et al. 2018), and KIC 8027456 (Kennedy et al. 2019), with TESS toward β Pic (Zieba et al. 2019; Pavlenko et al. 2022; Lecavelier des Etangs et al. 2022), and with Cheops toward HD 172555 (Kiefer et al. 2023). Here it must also be mentioned the detection with Kepler of a photometric event, possibly periodic, that is interpreted by the transit of a string of 5 to 7 exocomets in front of KIC 8462852 (Kiefer et al. 2017). All these events have been identified thanks to the particular shape of the light curve of an exocomet transit, as theoretically predicted at the end of the 90's in LdE99a : the characteristic of an exocomet transit light curve is the asymmetry caused by the cometary tail passing in front of the star after the nucleus. This causes a sharp decrease of the star light followed by a slow return to the normal brightness (see also Lecavelier des Etangs 1999, hereafter LdE99b). However, exocomet transit light curves can also be more symmetric when the longitude of the periastron of the orbit is about 90◦from the line of sight, because here the cometary tail is aligned with the line of sight. In this case, the shape of the transit light curve is similar to that of an exoplanet transit (LdE99a). The detection of exocometary transits based on photometry is complementary to the detection through spectroscopy. It allows exploring the physical characteristics of dust particles, such as the dust particle size distribution, the distance to the host star at the time of transit, and the dust production rate (Luk'yanyk et al. 2024). Indeed, because transits give direct access to the geometrical extent and the optical thickness of the transiting dust cloud, coupled with models of dust production, this allows the Article number, page 1 of 18 16 Oct 2025 A&A proofs: manuscript no. Article_v1 estimate of the dust production rate and the size of the comets nucleus. In the case of β Pic, statistical analysis of photometric events allowed the estimate of the nucleus size distribution that is found to be similar to that of asteroids and comets in the Solar system as set by collisional equilibrium (Lecavelier des Etangs et al. 2022). Although the characterization of exocomets paves the way to understanding the dynamical and chemical processes occurring in young planetary systems, only a very limited number of exocometary systems have been identified to date. As a result, our current understanding of the diversity, frequency, and evolutionary significance of exocometary activity remains incomplete. Systematic searches have been performed using spectroscopy in the Ca ii line (Montgomery & Welsh 2012; Welsh & Montgomery 2013, 2018; Rebollido et al. 2020; Bendahan-West et al. 2025), resulting in a few detections (see review in Strøm et al. 2020). Spectroscopy is efficient in detecting tiny amount of gas and thus allows sensitive search ; the result is that detectable spectroscopic transits seem to be more frequent than detectable photometric events: for instance in the case of β Pic, several exocomet transits are routinely detected every day in spectroscopy (Kiefer et al. 2014a), while only 30 photometric transits have been detected in 156 days of TESS observations (Lecavelier des Etangs et al. 2022). However, spectroscopic searches for exocomets are inherently limited by their need to observe each target star individually and the need for long observation time with high spectral resolution. In contrast, photometric surveys can monitor several thousands of stars simultaneously, making them significantly more efficient for large-scale searches for exocometary transits. In that spirit, Rappaport et al. (2018) and Kennedy et al. (2019) have undertaken a deep search for exocometary transits in the Kepler light curves. To identify exocometary transits, they used a modified model of planetary transit or a combination of exponential and Gaussian tails, as also done later on by Zieba et al. (2019), Pavlenko et al. (2022), and Lecavelier des Etangs et al. (2022). Although this empirical approach does not carry physical information about the cometary bodies and tails, these are simple and easy to implement to identify candidates in the large number (∼200 000) of light curves provided by Kepler. Here we propose to use an automated search without the use of analytical functions to describe the exocometary transit light curves. For that we combined a search algorithm based on a neural network together with a library of photometric transits to train the network. The library is an updated version of the library of LdE99b, which has been recalculated using faster CPUs than the ones used 20 years ago, covering a wide range of parameters and therefore a wide range of transit shapes that need to be identified. We apply this automated search to the Kepler data (Borucki et al. 2010) with the aim of producing a new catalog of exocometary systems candidates. The methods are described in Sect. 2, 3 and 4, the search in Kepler data is presented in Sect. 5 and the final catalog of candidates, divided in three tiers, is given in Sect.6. The results are discussed in Sect.7. 2. Training of the neural network 2.1. Training principle The method used for the detection of exocomets in Kepler data is based on a neural network using supervised learning. For the network learning, we used a library of theoretical light curves divided into three disjoint sets of labeled light curves: the training set, the validation set, and the test set. Each set contains the same number of light curves with and without an exocomet transit (see Sect. 2.2). Each light curve is labeled to indicate if it includes a transit or not. Our objective is to make the algorithm identify the curves containing an exocomet transit among all the light curves collected by the Kepler satellite and to recover the transit time when a transit is present. Using an iterative procedure, the algorithm optimizes its parameters thanks to the training set. At the end of each iteration step called an \"epoch\", the network performance is tested using the validation set. This allows us to control the overfitting of the algorithm (e.g. it becomes specific to the training set and loses in generality) and to adjust the hyperparameters. The learning phase is stopped when the network shows the best performance in the validation set. The final performance of the algorithm is given by its evaluation on the test set. 2.2. Light Curve production The light curves used to build the training, validation, and test set are the Presearch Data Conditioning (PDC) data from Kepler Data Release 25 (Thompson et al. 2016). We used the Quarters 1 to 17 and divided them into sub-quarters in case of interruption of the acquisition within a given quarter: two sub-quarters are separated by at least four consecutive photometric measurements missing in the light curve. For each sub-quarter, a training, validation, and test set were constructed to overcome their possible specificity. For each subquarter, the training set consists of 36 000 curves, while the validation and test sets contain 12 000 each. A curve consists on 10 days (480 photometric measurements at a cadence of one measurement every ∼30 minutes) of acquisition of the luminosity flux from a star randomly chosen without repetition. To ensure efficient learning and prevent the algorithm from falling into obvious traps, several modifications have been made to the Kepler PDC data. First, all points outside the mean value of the light curve by more than 5-sigmas while the neighboring points are not are considered to be outliers and removed. In addition, each curve was normalized to have a zero mean and a standard deviation of 1, as needed for optimal learning by the algorithm. Finally, the possible missing measurements, i.e. at most three consecutive missing measurements within the same subquarter, were filled using a linear interpolation. For the learning process, half of the 10-day light curves have been used with a label \"0\", indicating the absence of an exocomet transit: we assume that the probability that any randomly chosen light curve in the Kepler data includes an exocomet transit is extremely low. This set of 0-labeled light curves without exocomet transit is used to learn the noise patterns in the Kepler data. For the other half of the 10-day light curves, we add one exocomet transit by multiplying by a light curve from the theoretical transit library (Sect. 2.3) and we label them with \"1\", indicating the presence of an exocomet transit. 2.3. Simulation of exocomet transits In order for the algorithm to learn to identify the shape of an exocomet transit in the light curves, we add theoretical exocomet transits to half of the curves that make up the training data. These theoretical exocomet transit light curves have been calculated using the method describes in LdE99a. We produced a library of 2200 exocomet transit light curves, which is an updated version of the library published twenty years ago by LdE99b. Among all available simulations, we selected only those with a gas producArticle number, page 2 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves tion rate log10(P/(1kg · s-1)) between 6 and 7.5 and an impact parameter b , max pooling layers are noted MaxPool , fully connected layers are noted Dense , LSTM layers are noted LSTM , GRU layers are noted GRU and the squeezed-excitation blocks are noted SEB. none. The second output layer is activated by a linear function in range (0,1). It gives the position of the identified transit in the light curve. If no transit is detected, the layer should return a value close to zero. 3.2. Training For each sub-quarter, we trained the model using 36 000 light curves for 150 training epochs, that means that we performed 150 complete pass through the entire training dataset to update the model weight parameters. We used the Adam optimization algorithm (Kingma & Ba 2015) to minimize the cross-entropy error function for the classification of the light curves and the mean absolute error function for the position of the transit. We used a learning rate of 10-4 and a batch size of 5 000. The saved model in the one for which the validation loss reaches its minimum ; usually it is obtained after between 50 and 80 iterations (\"epochs\"). Therefore, training the model over 150 epochs is considered sufficient. Article number, page 3 of 18 A&A proofs: manuscript no. Article_v1 4. Evaluation of the neural network performance We evaluated the performance of our neural network with respect to several metrics in the same way as Dattilo et al. (2019) did for their model. We computed all metrics over the test set (rather than the training or validation sets) to avoid using any data that were used to optimize the parameters or hyperparameters of the model. 4.1. The metrics The metrics used to evaluate the performance of a classifier are the following: 1. Accuracy: the fraction of curves correctly classified by the model. 2. Precision (reliability): the fraction of correctly classified exocomets, i.e., the fraction of curves classified as exocomet candidates (False positives + True positives NFp + NTp) that are indeed true positives (labeled 1): P = NTp NFp + NTp (1) 3. Recall (completeness or True-positive rate): the fraction of the curves labeled 1, which therefore includes an exocomet transit (True positives + False negatives NTp + NFn), that the model classifies as exocomet candidates (NTp): R = NTp NTp + NFn (2) 4. False-positive rate: the fraction of total labeled 0 curves, which therefore do not include any exocomet transit (False Positives +True Negatives NFp + NTn) that the model classifies as exocomet candidates. FPR = NFp NFp + NTn (3) 5. AUC (area under the receiver-operator characteristic curve; see Figure 2): the probability that the model, if it receives two light curves, one 1-labeled curve including an exocomet transit and one 0-labeled curve without transit, would rank the 1-labeled curve higher than the 0-labeled curve. 6. Mean Absolute Error: the absolute error for all labeled 1 curves between the true position of the transit and the predicted one by the model. The first five metrics are related to the detection of the presence of an exocomet in the light curves, while the last metric is related to the position in time of the transit in the light curve. Except for the AUC, the values of these metrics depend on the classification threshold chosen for the model. This threshold is the value of the probability of detection yield by the algorithm above which we consider that there is a detection. In Fig. 2, we show the evolution of the recall as a function of the FPR for various thresholds, usually called the receiver-operator characteristic (ROC) curve. It can be seen that the algorithm differentiates well between true transits and noise, as it achieves very low FPR recall while keeping a reasonably large recall: our model reaches an AUC of 98.88% for the quarter Q1. We also plot the precision as a function of the recall. This curve shows the trade-off between having no false positives (high precision) and identifying all exocomet transits (high recall). Fig. 2: The receiver-operator characteristic (ROC) curve (left panel) and the precision versus recall curve (right panel). The receiver-operator characteristic (ROC) curve shows the recall (true-positive rate) of the model against the ability to recognize false positives (the false-positive rate) for different classification thresholds. Our model is highly successful at identifying false positives as shown by the high AUC value (see Table A.1). The plot of the fraction of exocomets that the model classified as exocomets (recall) versus the fraction of correctly classified planets (precision) shows the trade-off between having no false positives (high precision) and identifying all exocomet transits (high recall). Considering the huge number of light curves in the Kepler data, the goal is to drastically reduce the number of false positive. We thus chose a classification threshold of 0.99. With this threshold, our model reaches an accuracy of 89.5% for the quarter Q1. The precision of the model is 99.8%, which means that we reach a high true positive rate, and the recall is 79,1%, which means that about 20% of the true exocomet transits are missing in the final classification. This trade-off aims to accept some loss in the finding of exocomet transits of about 20% to avoid a large number of false positives. Indeed, given a large number of light curves in the Kepler data and a small number of exocomet transits in these data, even a tiny fraction of false positives can significantly pollute the results. The metrics values are found to be similar for all quarters (see Table A.1 in Appendix A). 4.2. Quality of the output The quality of the model classification can also be visualized by the histogram of the results on the test set (top panel of Fig. 3). The closer the output from the neural network is to 1 (resp. 0), the higher the probability that the 10-day light curve (resp. does not) contains an exocomet transit. The histogram shows that the neural network works as expected: on the left-hand side, most of the light curves characterized by a low probability are mostly noise (orange histogram), while on the right-hand side, the large majority of the curves characterized by a probability close to 1 included a simulated exocomet transit (label 1; blue histogram) and with very few false positives. The bottom graph of Fig. 3 shows the mean error of the position of the transit as a function of the probability assigned by the algorithm when it is applied to the curves containing a simulated transit (label 1, with known position). One can see that when the transit is not found (probability close to 0), the error on the predicted position is large, about 3 days, whereas it is of the order of a few hours only when the probability is above 0.99. Thus, one can be confident in the time of the transit given by the algorithm for curves with a probability higher than this threshold. Article number, page 4 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Fig. 3: Histogram of the results on the test set and error on the transit position found by the network. Top panel: Histogram of the result of the neural network applied to the test data. Most of the 1-labeled light curves with an exocomet transit yield a probability close to 1, while there are only a few false positives (0-labeled light curves yielding a high probability). Above the chosen threshold, there is more than two order of magnitude between the number of true positives and false positive. Bottom panel: the mean error of the position of the transit as a function of the probability of the presence of a transit assigned by the network to the light curves. This error is of a few hours at most for the transits identified by the algorithm. 5. Search for exocomets in Kepler data 5.1. Application of the model to Kepler data To apply the algorithm presented in the previous sections, all the light curves of the Kepler data were cut into 10-day windows obtained in the same way as the samples used to construct the training, validation, and test sets described in Sec. 2.2. In order not to lose any transits located at the edge of the windows, an overlap of 2 days was applied between two consecutive 10-day time intervals. However, despite the level of precision achieved, the algorithm provides a large number of false positives due to the large number of 10-day windows: approximately 106 windows per quarter lead to about 103 positive candidate detections, among which less than a dozen real transits should be identified (see LdE99a and discussion in Sect. 7). To reduce the number of false positives, we decided to filter the candidates using constraints on the shape of the light curves of the detected photometric events. 5.2. Additional criteria to filter false positives After all the Kepler light curves have passed through the neural network, we end up with a set of 56525 exocomet candidates. To filter these candidates, we decided to add constraints on the shape and on the context on the transit (noise, quality flags). 5.2.1. Filtering through the shape of the transits candidates The shape of the candidate transits were determined by fitting them by the model proposed by Lecavelier des Etangs et al. (2022) together with a second order polynomial baseline: fexo(t) = Ct2 + Dt + E t t > t0 (Ct2 + Dt + E)(1 + K(e-β(t-t0) -e-β(t-t1))) t > t1 (4) Fig. 4: Distribution of the value of the parameters obtained by fitting the 2163 light curves in the library of the simulated exocomet transits. The parameters K, ∆T = t1 -t0, and 1/β correspond to the depth, ingress duration, and crossing time of the transit, respectively. The parameters t0 and t1 represent respectively the time at which the transit starts and the time at which it reaches its minimum. K represents the characteristic depth of the transit, while 1/β corresponds to the characteristic crossing time of the transit. C, D and E are the coefficients of the polynomial baseline. The fit of the light curve by the transit model can be easily performed using the position of the detected candidate in the light curve as provided by the network output. To accept or reject a proposed detection given by the neural network, we fit the light curve of the corresponding photometric event to obtain the values of the parameters K, β, and ∆T = t1-t0 that characterize the shape of the event. We then check that these values are in the domain of validity for an exocometary transit. The domain of validity is obtained by considering the parameters space occupied by the fits of the theoretical light curves in the library of numerical simulations of exocometary transits, as shown in Fig. 4 and summarized in the first three lines of Table 1. It is found that in the theoretical light curves K is mostly between 10-4 and 5×10-3, ∆T is between 1.7 and 17 hours (corresponding to a periastron between ∼0.02 to 2 au, Lecavelier des Etangs et al. 2022), and β-1 is in the domain above 0.5 hours, and between 0.3∆T -1.7 and 0.5∆T + 12. Moreover, the parameters C and D have to verify the condition |C|, |D| 36 were conserved. This criterion was very useful to remove candidates that were detected due to a few points significantly lower than the average. To detect the presence of red-noise that mimics an exocomet transit, we calculated the correlation product between the exocomet model fit to the candidate transit and the light curve over all the considered sub-quarters. If the maximum of the correlation max1 is reached more than one day away from the transit time or if its second maximum max2 is significantly close to the first maximum ((max1-max2)/max1 t0 (6) We calculate ∆χ2 rp defined by : ∆χ2 rp = N P i( framp(ti) -F(ti))2 P i( fexo(ti) -F(ti))2 -1 ! , (7) where N = 120 is the number of points in the fitted window. The photometric events that are almost as well fitted with the ramp model as with the exocomet model (∆χ2 rp < 10), have been discarded from the candidate list. Finally, to identify the cases of a symmetrical transit that could be due to the passage of an exoplanet with a long period (a monotransit), we calculate the difference ∆χ2 pl between the χ2 pl of a model of a planetary transit calculated using the equations of Mandel & Agol (2002) and the χ2 exo of an exocomet transit model (Eq. 4). It can be written as: ∆χ2 pl = N P i(fpl(ti) -F(ti))2 P i(fexo(ti) -F(ti))2 -1 ! . (8) The photometric events having ∆χ2 pl < 5 have a symmetrical shape and have therefore not been taken into account in our final selection of exocomets. Such transits have been taken into account to establish the list of symmetrical mono-transits (see Sec. 6.3). This criterion is, however, not used as a sharp cut to distinguish the exocomets transits from the symmetrical mono-transits but only as a help. The final distinction has been made through visual inspection. After applying these criteria, the number of candidate transits per quarter varies between 50 and 100 per quarter, leading to a total of 1349 transits. It then becomes possible to proceed for each of them to a visual inspection to know if the candidate is indeed or not an exocomet transit. This final inspection is necessary because, although the criteria put in place make it possible to eliminate the vast majority of false positives, there still remain obvious false positives. Note here that the criteria have been deliberately chosen to be conservative so as not to accidentally eliminate a true exocomet transit. The number of criteria was chosen so that the final number of transits to be visually inspected would be no more than a few thousand. Otherwise, visual inspection would be impractical. It would be more inefficient and inaccurate to add extra criteria than to visually check the current list of 1,349 transits. To verify the robustness of the detected transits, we perform an additional test. We apply our detection procedure to the data after reversing the time. If the proportion of detected exocomets is the same in the reversed time data as in the original data, this may question the relevance of the detected transits. However, as this analysis is time-consuming, we performed this analysis on half of the quarters. Of the 17 quarters, we analysed eight randomly chosen quarters: Q1, Q2, Q6, Q9, Q10, Q12, Q15 and Q16. We found only one transit that clearly mimics an exocomet transit: KIC_3129239 in Q9. Given that approximately 17 robust transits have been detected in the Kepler data (see Sec. 6), we can infer from this study that about 10% to 15% of these transits can be false positives. Although the same number of detections in the time-reversed data would have disproven the exocomet hypothesis, the result of this test supports the idea that there is an astrophysical signal in the data that is consistent with exocomet transits. Article number, page 6 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves 6. A new catalog of transiting exocomets In this section, we present the final list of exocomet transits resulting from the selection procedure. We divided the list in three different tiers based on the likelihood of a genuine exocometary detection. In the first tier, the photometric events are the ones that we consider as the most likely due to the transit of an exocomet ; in the second tier are the events possibly due to comets, while the third tier gathers the most symmetric dip in the light curve, for which the transit of an exoplanet remains a possibility. 6.1. The first-tier catalog of exocometary transits The first-tier catalog of detected exocometary transits is given in Table 2 and the plots of the corresponding light curves are given in Appendix B. It is particularly noteworthy that our neural network is able to find all the transits already identified by previous works: KIC 3542116 in Quarters 1, 10 and 12 and KIC 11084727 as found by Rappaport et al. (2018), and KIC 8027456 as found by Kennedy et al. (2019). This independent retrieval of the same exocometary transits provides confidence in our procedure. Nonetheless, after application of all the search procedure described above, it appeared that two transits of KIC 3542116 identified by Rappaport et al. (2018) in Quarters 8 were not in our list of detections. These transits were eliminated by the criterion based on the maximum of the correlation product that aims to remove the detections whose pattern occurs several times in the same subquarter (Sect. 5.2.4). In fact, this criterion also has the disadvantage of eliminating active cometary systems where at least two comets transits occurred in the same quarter, as is the case for KIC 3542166 in Quarter 8. Therefore, we made a visual inspection of all photometric events eliminated because of this criterion in order to recover possible obvious exocometary transits that were wrongly rejected. We thus found the two transits of KIC 3542116 in the Quarter 8, and in addition identify a new transit in front of KIC 6263848. This last case does not correspond to a star with frequent cometary transits, but had been deleted due to an edge effect on the calculation of the correlation function. These two cases are identified by an asterisk in Table 2. For all of these detection, we calculate the ∆χ2 polynomial defined as ∆χ2 polynomial = N P i( fpolynomial(ti) -F(ti))2 P i(fexo(ti) -F(ti))2 -1 ! . (9) where fpolynomial is the second order polynomial that best fits the detection. For all detections, we obtained large ∆χ2 polynomial (larger than about 100). This highlights that the detection cannot be interpreted just as noise patterns. The properties of the parent stars of the identified exocomets as tabulated by Kepler are given in Table 3 (Brown et al. 2011). For KIC 3542116 and KIC 3662483, Teff, log10 g, metallicity and radius are from Zhang et al. (2025). From the Teff and log10 g we can obtain an estimate of the stellar type (Gray 2008), which is given in the last column of Table 3. It appears that most of the stars are main-sequence stars, with the exception of KIC 4078638, which is likely a red giant. For these stars, the upper limits on the age estimate given by Zhang et al. (2025) are all above 109 years, with the exception of KIC 4481029 with an upper limit of 9 × 108 years, and KIC 8027456 with an age estimate of 5 × 108 years. 6.2. The second-tier catalog of possible exocometary After visual inspection of the list of candidates produced by our algorithm, we came up with a list of interesting photometric events that can be due to transiting exocomets. This includes transits that pass through all the criteria described in Sect. 5, but whose shape or signal-to-noise ratio makes it difficult to decide whether they are actually transits. The resulting list of possible exocometary transits in this second-tier catalog is given in Table 4 and the plots of the corresponding light curves are given in Appendix C. The properties of the corresponding parent stars in the second tier catalog are given in Table 5 (Brown et al. 2011). Here again, estimates of the stellar types given in the last column of the table show that most of the stars are on or close to the main sequence, with the exception of KIC 2984102 and KIC 11153134, which have a log10 g below 3 and are therefore likely red giants. For all stars in this second-tier catalog, the upper limit of the age estimate given by Zhang et al. (2025) is above 109 years, with the exception of KIC 2984102, KIC 5294231 and KIC 7183123 with age estimates of 5×108, 1×108 (with an upper limit of 3 × 108) and 9 × 108 years, respectively. 6.3. The third-tier catalog of symmetric transits After visual inspection of the list of candidates produced by our algorithm, we also came up with a list of interesting photometric events that appears to be symmetrical. This is not surprising as some of the exocomets transits in the library used for training the network are indeed symmetrical (see Fig. 2 and 3 in LdE99a). However, we cannot exclude that these are due to mono-transits of exoplanets or to false positives mimicking planetary or cometary transits. Several scenarios might cause such false positives, including diluted or undiluted eclipsing binaries. To identify these cases one needs to carry out additional analyses that are beyond the scope of the present paper (see, e.g., Crossfield et al. 2016). Finally, although an exocometary transit classification cannot be made, these detections deserve to be mentioned. They are given in the third-tier catalog of symmetric transits listed in Table 6. We compared these detected photometric events with the lists of single transits published by Huang et al. (2013), Wang et al. (2015), Foreman-Mackey et al. (2016), and Herman et al. (2019). It appears that four events in our list were already identified by Foreman-Mackey et al. (2016) and Herman et al. (2019). The transit in front of KIC 8410697 was found in both studies. It is interpreted as due to an exoplanet with a radius of 0.7 times that of Jupiter. The photometric event of KIC 10321319 was found by Foreman-Mackey et al. (2016) and interpreted as due to the transit of an exoplanet of 0.16 Jupiter radius. The photometric event of KIC 6196417 was found by Herman et al. (2019) and interpreted as due to the transit of an exoplanet of about 0.7 Jupiter radius. The transit in front of KIC 10668646 at the time of BKJD=1449.3 was already identified by Foreman-Mackey et al. (2016) but the exoplanet transit scenario was rejected because of a centroid shift in the data. However, we also identified another photometric event in the same target at BKJD=196.3 with a different shape and a lower absorption depth. These few examples show that our list of newly identified photometric events which could be due to transits of exocomet with symmetrical shape of the light curve deserves further investigations which are beyond the scope of the present paper. Article number, page 7 of 18 A&A proofs: manuscript no. Article_v1 Table 2: First-tier catalog of exocometary transits. Stars (KIC) Time (BKJD) Quarter K(×10-4) β-1(h) ∆T(h) AD(×10-4) ∆χ2 pl ∆χ2 polynomial †3542116 161.5 1 4.74 6.5 9.92 3.71 18.32 279.74 †*3542116 742.6 8 4.17 1.36 6.05 4.12 0.6 348.83 †*3542116 792.9 8 8.62 4.16 4.77 5.88 15.57 285.27 †3542116 992.1 10 10.12 3.62 14.68 9.94 28.67 2993.05 †3542116 1175.8 12 12.98 3.40 9.35 12.15 8.58 2670.38 3662483 1132.1 12 6.81 6.38 6.87 4.49 12.98 204.52 4078638 355.5 4 83.21 6.30 3.22 33.30 47.89 358.90 4481029 172.5 2 9.08 8.57 4.64 3.80 14.31 235.74 5206257 925.0 10 10.98 4.75 4.47 6.69 5.04 322.26 5514200 670.1 7 34.48 12.13 7.27 15.53 19.15 155.20 *6263848 775.9 8 10.22 3.29 4.90 7.92 7.96 151.93 6927963 416.8 4 33.60 9.13 2.83 8.96 6.54 115.89 7660548 966.2 10 15.74 5.95 5.09 9.05 26.10 219.47 †8027456 1449.0 15 8.0 9.90 14.12 6.08 16.78 401.18 8738545 238.1 2 17.0 4.92 3.67 8.94 7.64 92.53 10484683 309.5 3 29.85 14.05 8.01 12.97 87.66 853.82 †11084727 1076.2 11 15.29 3.92 8.20 13.40 88.14 2368.80 Notes. The first column gives the name of the star by its KIC identifier. The second and third columns indicate the position of the transit. Columns 4 to 6 give the fits parameters associated with the exocomet model (Eq. 4). The seventh column gives the absorption depth. The last two columns present the difference of χ2 between the cometary model and, first, the planetary model (Mandel & Agol 2002) and, second, a second order polynomial fit. The transits that were first eliminated by the criterion based on the maximum of the correlation product (see Sect. 5.2.4) are identified by an asterisk (*). The five transits already known are identified by the † symbol. Table 3: Information on the stars of the first-tier catalog. Kepler ID RA Dec Kepler Mag Teff Log g Metallicity Radius Stellar (J2000) (J2000) (K) (cm/s2) (R⊙) type 3542116 19 22 52.939 +38 41 41.51 9.979 6766 4.208 -0.190 1.454 F2 V 3662483 19 42 09.199 +38 43 01.38 10.541 5848 3.945 0.107 1.786 G1 IV 4078638 19 46 16.611 +39 06 15.23 13.156 4783 2.758 0.150 8.331 K3 III 4481029 19 43 13.210 +39 32 32.03 11.303 8865 3.968 0.010 2.310 A0 IV 5206257 19 46 33.826 +40 22 57.97 12.051 6017 4.435 -0.105 1.052 F6 V 5514200 18 58 08.306 +40 46 29.32 14.377 5651 4.500 -0.243 0.955 G4 V 6263848 18 52 30.566 +41 36 33.34 13.172 5850 4.365 -0.224 1.137 G2 V 6927963 18 56 15.946 +42 27 52.20 13.807 5711 4.441 -0.012 1.030 G4 V 7660548 18 47 22.807 +43 23 05.39 13.094 6158 4.369 -0.050 1.148 F5 V 8027456 19 25 15.838 +43 51 33.55 9.697 8732 3.766 0.065 2.974 A0 IV 8738545 18 58 37.262 +44 55 15.24 13.227 5091 3.581 -0.162 3.101 K1 IV 10484683 19 47 54.377 +47 40 14.16 12.588 6326 4.543 -0.009 0.934 F5 V 11084727 19 28 41.191 +48 41 15.14 9.987 6762 4.067 -0.152 1.726 F3 V 7. Conclusion Although the Kepler mission ended more than 10 years ago, the available data is still worth deep data mining in search of new discoveries. Here we present a new search for exocometary transits in the Kepler light curves. We used machine learning technique with a neural network that has been trained using a library of theoretical exocomets transit light curves inherited from the work of LdE99b. After parsing the light curves of close to 200 000 stars through the neural network, despite the application of several filters to eliminate most of the false positive, a visual inspection of the outcome of the network was still needed. We ended up with three catalogs of interesting objects. The first-tier catalog is composed of a total of 17 exocometary transits, including 7 previously identified transits and 10 new transits in front of 10 different stars. The second-tier catalog is a list of 30 photometric events that appear of second quality and that we qualify as possible exocometary transits. Finally, the third-tier catalog presents the list of interesting photometric events that are symmetrical and may be due to transits of either an exoplanet or an exocomet with a periastron at 90◦from the line of sight. The complete Kepler data represent a 4-year photometry survey of more than 170 000 distinct stars. However, as noted in Kennedy et al. (2019), not all stars were observed for the full duration of the mission. As a result, we can consider 150 000 as the approximate number of equivalent stars that were observed for the full duration of the mission. Or, in other words, our analysis covers the equivalent of about 600 000 stars-years. LdE99a showed that with 30 000 stars-years survey of stars with a solar system cometary activity and a few 10-4 photometric accuracy, the number of exocomets transit detections should be around 10. Therefore, assuming that each star harbors a cometary system similar to our own, we could expect roughly 200 detections with Article number, page 8 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Table 4: Second-tier catalog of possible exocometary transits. Same legend as in Table 2. Stars (KIC) Time (BKJD) Quarter K(×10-4) β-1(h) ∆T(h) AD(×10-4) ∆χ2 pl ∆χ2 polynomial 2984102 345.7 3 36.42 15.9 10.12 17.14 61.90 430.26 4826941 385.8 4 21.58 15.22 7.85 8.69 3.76 126.17 5195476 1508.8 16 16.80 5.09 3.67 8.31 3.42 45.92 5294231 411.4 4 4.92 3.26 2.13 2.36 31.78 285.46 5511746 1105.8 12 13.31 12.33 5.06 4.48 2.41 44.54 5514277 670.4 7 63.11 13.57 9.88 32.64 13.96 160.95 5534083 1167.7 12 9.34 10.45 3.33 2.55 7.22 43.16 6119914 747.2 8 31.41 9.36 6.63 15.95 3.54 50.65 7183123 336.4 3 8.43 8.36 5.98 4.31 28.32 321.42 7201827 409.9 4 43.64 4.83 2.83 19.34 5.77 48.98 7220968 1325.7 14 18.60 5.50 6.23 12.61 2.61 46.42 7611858 1275.8 14 6.46 9.77 5.19 2.67 7.63 109.47 7669613 315.9 3 13.24 13.66 6.31 4.89 12.77 121.72 8111649 934.0 10 83.04 23.69 10.28 29.24 16.36 123.61 8144412 639.1 7 86.39 12.00 2.95 18.84 7.06 140.49 8374877 769.1 8 7.78 7.04 8.46 5.44 13.67 300.35 8489495 1351.8 14 16.43 8.44 15.04 13.67 10.70 47.92 8625941 1505.0 16 5.16 8.44 9.80 3.54 8.10 66.25 8683719 1340.2 14 19.21 6.43 14.25 17.12 7.59 195.79 8738569 1462.3 15 13.18 8.30 4.30 5.33 8.84 56.11 8843356 702.2 7 8.84 10.54 2.99 2.18 4.66 46.51 9025688 1078.2 11 13.29 7.51 3.57 5.03 58.03 58.03 9388975 680.3 7 18.41 3.36 2.47 9.59 6.76 60.28 9531080 748.0 8 10.23 9.31 6.56 5.18 16.13 53.73 10329957 1385.1 15 18.47 12.90 4.10 5.03 9.62 64.04 11124157 704.2 7 15.01 10.93 2.00 2.51 11.48 45.20 11153134 739.7 8 18.93 3.15 7.99 17.44 11.42 345.83 11246607 696.4 7 18.06 15.10 8.84 8.00 7.04 165.29 11515196 644.3 7 16.40 7.98 3.84 6.26 34.15 138.30 11701976 931.7 10 23.79 11.98 2.70 4.80 13.24 40.42 a perfect search in the Kepler data. Finding a few dozens of exocometary transits can therefore be considered as a satisfactory result in agreement with what could be expected using reasonable estimates of the algorithm efficiency and cometary activity of the surveyed stars. Unfortunately, the Kepler stars are rather faint, and our catalogs contain stars with magnitude between 9.7 and 15.9. It is therefore not possible to undertake a spectroscopic follow up with the hope to confirm the exocometary nature of the photometric events. All the more that, with only one transit per star in about four years, exocometary transits are rare and not predictable in time. However, we can anticipate that the same techniques could be applied to search for exocometary transits in current TESS data and in upcoming PLATO data. For these two instruments, the surveyed star in the input catalog are significantly brighter that the Kepler stars. We can thus expect that the exocometary systems to be discovered in near future can be studied in detail, particularly through spectroscopy. Previous searches (Rappaport et al. 2018; Kennedy et al. 2019) provided a list of 3 stars that seem to be rather young. This supported the anticipated conclusion that cometary activity is likely correlated with stellar age, with an expected decline over time. However, our catalogs do not support this idea, as we found a wide spread of age for both the first- and second-tier catalogs, with possible exocometary transits in front of stars classified as red giants. The issue of cometary activity with age thus remains open, and further studies will be needed to clarify this important question of the evolution of planetary systems. Note added in manuscript: After this manuscript was submitted we became aware of a paper by Norazman et al. (2025) that used Machine Learning techniques to search exocometary transits in the sectors 1 to 22 of the TESS light curves. They found three additional transits compared to those already known, which underlines the utility of this technique for the search of such transits. Acknowledgements. We thank Dr. Neda Heidari for her help in identifying already known mono-transits. We acknowledge support from the CNES (Centre national d'études spatiales, France). This work has made use of the Infinity Cluster hosted by Institut d'Astrophysique de Paris. We thank Dr. Guilhem Lavaux for his invaluable help in making efficient use of the GPUs in this cluster. This paper includes data collected by the Kepler mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the Kepler mission is provided by the NASA Science Mission Directorate. 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C., et al. 2025, ApJS, 277, 6 Zieba, S., Zwintz, K., Kenworthy, M. A., & Kennedy, G. M. 2019, A&A, 625, L13 Zucker, S. & Giryes, R. 2018, AJ, 155, 147 Article number, page 10 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Table 6: Third-tier catalog of symmetric transits that could be either exocometary transits or exoplanet mono-transits. Stars (KIC) Time (BKJD) Quarter K(×10-4) β-1(h) ∆T(h) AD(×10-4) ∆χ2 pl ∆χ2 polynomial 2983000 759.1 8 18.38 0.84 16.33 18.38 -1.94 483.82 2993038 550.9 6 15.22 2.46 4.72 12.99 -1.57 1165.8 3222471 673.1 7 97.7 5.49 14.59 90.83 -70.29 2436.25 3346436 270.4 3 8.48 3.03 3.05 5.38 3.42 182.43 3755854 1492.3 16 3.53 1.07 6.11 3.51 1.03 45.73 5184479 534.6 5 8.87 1.04 19.61 8.87 -11.23 316.53 5184479 1203.0 13 12.17 1.97 17.91 12.17 -3.69 409.71 5305217 673.1 7 33.0 5.46 20.87 32.28 -7.41 170.59 5456365 270.8 3 35.42 3.85 10.37 33.03 -2.28 323.73 5905878 1343.3 14 2.25 1.46 7.55 2.24 -7.35 353.12 5905878 1441.3 15 0.89 1.49 8.14 0.89 -2.01 60.16 5967153 1394.5 15 5.06 4.05 7.9 4.34 -22.09 141.91 5975275 213.3 2 9.78 2.57 10.57 9.62 2.7 179.4 b6186417 959.1 10 33.19 1.09 18.42 33.19 -2.36 645.89 6387193 651.7 7 78.79 2.73 12.43 77.96 -48.64 3817.28 6515488 1420.4 15 28.85 2.98 9.54 27.67 -7.02 359.44 6804821 1009.1 11 12.97 1.6 22.41 12.97 -6.09 317.96 7047396 298.8 3 11.35 2.0 5.57 10.65 -4.56 305.5 7105703 1509.0 16 20.54 2.86 6.53 18.45 -4.32 186.11 7213651 1239.2 13 21.53 1.66 6.78 21.17 -9.23 214.24 7465971 342.7 3 11.14 1.14 9.75 11.14 -1.07 200.89 7983622 471.9 5 65.06 4.63 13.76 61.73 -15.71 732.08 8005892 608.5 6 96.11 1.2 9.78 96.09 -75.32 9905.87 8007462 1458.6 15 2.44 2.56 5.4 2.14 -7.15 93.52 8159297 440.7 4 18.1 1.87 4.62 16.57 -1.87 132.24 8313257 1149.1 12 25.05 2.17 17.4 25.05 -8.72 320.72 a,b8410697 542.5 6 54.16 1.0 17.87 54.16 -12.07 5694.83 8496108 277.7 3 19.41 9.57 3.83 6.4 -6.55 62.5 8617888 171.8 2 55.2 1.83 5.44 52.4 -4.99 149.3 9016734 666.0 7 61.61 5.57 13.7 56.35 -48.25 1624.94 9016734 1230.1 13 27.52 5.5 14.31 25.48 -7.51 522.79 9413755 1564.2 17 19.5 1.63 4.4 18.18 -16.75 217.06 9775416 1398.1 15 5.04 4.99 10.63 4.44 -2.36 141.64 10024862 1494.0 16 20.95 1.68 13.29 20.94 -2.57 121.87 10157075 369.5 4 26.75 1.93 5.57 25.27 -2.17 114.74 10192453 757.5 8 18.64 5.63 1.73 4.94 -29.2 76.45 a10321319 554.5 6 4.76 4.77 13.56 4.48 -29.18 174.87 10334763 549.7 6 7.23 0.74 10.88 7.23 -3.39 412.26 10397849 506.5 5 19.93 2.35 7.95 19.25 0.15 289.02 10450889 239.6 2 60.03 1.58 2.35 46.44 1.45 192.69 10556420 756.3 8 3.59 0.64 6.03 3.59 -1.17 364.07 10643786 1421.2 15 12.62 3.23 5.12 10.04 3.16 81.51 10668646 196.3 2 42.81 0.93 8.74 42.8 -20.57 1376.58 a10668646 1449.3 15 57.08 1.36 8.64 56.98 -38.4 3632.55 10909733 898.7 9 3.41 1.37 6.17 3.37 4.64 67.65 11561379 137.7 1 13.51 2.11 7.45 13.12 -1.64 103.03 11921843 1234.8 13 3.85 0.62 14.83 3.85 -2.49 287.33 12066509 632.3 7 38.71 0.78 12.41 38.71 -0.31 1883.85 12644038 584.7 6 36.26 1.49 6.37 35.77 3.56 199.23 Notes. Four transits were already identified by Foreman-Mackey et al. (2016) (labeled with \"a\" in the first column) and Herman et al. (2019) (labeled with \"b\"). Article number, page 11 of 18 A&A proofs: manuscript no. Article_v1 Appendix A: Performance of the algorithm over all the quarters Table A.1: Performance of the neural network. Quarter Subquarter Accuracy Recall Precision FPR AUC MAE 1 0 0.8945 0.7910 0.9975 0.0020 0.9888 2.3404 2 0 0.8665 0.7347 0.9977 0.0017 0.9907 2.2237 2 1 0.8616 0.7245 0.9982 0.0013 0.9900 2.0792 2 2 0.8743 0.7498 0.9984 0.0012 0.9893 2.0455 3 0 0.9108 0.8240 0.9972 0.0023 0.9896 2.0329 3 1 0.9068 0.8162 0.9969 0.0025 0.9905 2.1186 3 2 0.8903 0.7835 0.9964 0.0028 0.9895 2.1638 4 0 0.8862 0.7745 0.9972 0.0022 0.9909 2.3073 4 1 0.9017 0.8065 0.9961 0.0032 0.9924 2.4661 4 2 0.8854 0.7728 0.9974 0.0020 0.9920 1.9370 5 0 0.8994 0.8013 0.9969 0.0025 0.9919 1.9726 5 1 0.8898 0.7818 0.9972 0.0022 0.9916 2.1857 5 2 0.8951 0.7923 0.9973 0.0022 0.9919 2.5873 6 0 0.8979 0.7972 0.9983 0.0013 0.9906 2.2020 6 1 0.8908 0.7833 0.9979 0.0017 0.9911 1.9720 6 2 0.8860 0.7738 0.9976 0.0018 0.9910 2.4570 7 0 0.8792 0.7605 0.9972 0.0022 0.9899 2.1862 7 1 0.8752 0.7522 0.9978 0.0017 0.9909 1.7915 7 2 0.9128 0.8267 0.9986 0.0012 0.9935 2.1101 8 0 0.9004 0.8018 0.9988 0.0010 0.9911 2.0729 8 1 0.9027 0.8065 0.9986 0.0012 0.9926 2.1139 9 0 0.9068 0.8157 0.9976 0.0020 0.9930 2.1293 9 1 0.9017 0.8062 0.9965 0.0028 0.9916 2.3157 9 2 0.8983 0.7992 0.9969 0.0025 0.9918 2.3348 9 3 0.8798 0.7610 0.9980 0.0015 0.9916 2.2389 10 0 0.8755 0.7532 0.9971 0.0022 0.9896 2.7444 10 1 0.8758 0.7538 0.9971 0.0022 0.9889 2.0156 10 2 0.8829 0.7672 0.9983 0.0013 0.9890 2.7258 11 0 0.8830 0.7685 0.9968 0.0025 0.9908 2.0722 11 1 0.8941 0.7900 0.9977 0.0018 0.9913 1.9576 11 2 0.8907 0.7828 0.9981 0.0015 0.9921 2.4971 12 0 0.8876 0.7767 0.9981 0.0015 0.9921 2.3743 12 1 0.9028 0.8075 0.9977 0.0018 0.9930 2.3531 12 2 0.9038 0.8103 0.9967 0.0027 0.9913 2.4029 13 0 0.8981 0.8000 0.9952 0.0038 0.9913 2.8353 13 1 0.8431 0.6875 0.9981 0.0013 0.9872 2.4796 13 2 0.9037 0.8117 0.9947 0.0043 0.9915 2.5127 14 0 0.8654 0.7342 0.9955 0.0033 0.9877 2.4772 14 1 0.8801 0.7632 0.9961 0.0030 0.9897 2.4096 14 2 0.8788 0.7592 0.9978 0.0017 0.9915 3.1279 15 0 0.8363 0.6743 0.9973 0.0018 0.9869 2.5724 15 1 0.8378 0.6780 0.9966 0.0023 0.9826 3.3750 15 2 0.8490 0.7000 0.9972 0.0020 0.9862 3.1479 16 0 0.8238 0.6483 0.9987 0.0008 0.9870 2.4968 16 1 0.8665 0.7355 0.9966 0.0025 0.9868 2.1849 16 2 0.8445 0.6907 0.9976 0.0017 0.9858 2.4707 17 0 0.8744 0.7510 0.9971 0.0022 0.9899 2.3075 Article number, page 12 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Appendix B: Cometary transits light curves Plots of the light curves of the detected cometary transits in the first tier catalog. For each transit, the light curve is shown in the top panels over a 5 days duration centered on the transit time and with an exocomet transit fit superimposed (red line). The bottom panels show the light curves over a larger time frame. Article number, page 13 of 18 A&A proofs: manuscript no. Article_v1 Article number, page 14 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Article number, page 15 of 18 A&A proofs: manuscript no. Article_v1 Appendix C: Possible exocometary transits light curves Plots of the light curves of the possibly detected cometary transits in the second tier catalog. The plots cover a 5 days time frame centered on the transit time and with an exocomet transit fit superimposed (red line). Article number, page 16 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Article number, page 17 of 18 A&A proofs: manuscript no. Article_v1 Article number, page 18 of 18"}}},{"rowIdx":295,"cells":{"id":{"kind":"string","value":"2510.14684"},"raw":{"kind":"string","value":"The geometry of magnitude for finite metric spaces\nKarel Devriendt\nAbstract\nThe main result of this article is a geometric interpretation of magnitude, a real-valued\ninvariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric\nspace X such that the magnitude of X can be expressed in terms of the ‘circumradius’ of\nits embedding S.\nThe circumradius is the smallest r for which the r-thickening of S is\ncontractible. We give three applications: First, we describe the asymptotic behaviour of the\nmagnitude of tX as t →∞, in terms of the circumradius. Second, we develop a matrix\ntheory for magnitude that leads to explicit relations between the magnitude of X and the\nmagnitude of its subspaces. Third, we identify a new regime in the limiting behaviour of tX,\nand use this to show submodularity-type results for magnitude as a function on subspaces.\n1\nIntroduction\n1.1\nBackground\nMagnitude is a real-valued invariant for enriched categories (in general) and metric spaces (in\nparticular) that was introduced by Leinster in 2006, in [11].\nBefore giving a definition, we\nmention two slogans that are helpful in guiding one’s intuition on magnitude. The first slogan\nsays that\n“Magnitude is like an Euler characteristic.”\nThis statement reflects the history of magnitude [11], motivates its definition, and guides the\ndevelopment of its theory, for instance in the generalization to magnitude homology [9, 16].\nIn this paper, the connection to Euler characteristics shows itself in our study of valuative\nproperties, i.e., how magnitude behaves as a function on subspaces Y ⊆X and what kind of\ninclusion–exclusion-type relations it satisfies. A second slogan says that\n“Magnitude counts the effective number of points.”\nPerhaps more appropriate for finite spaces, this intuition explains the success of magnitude in\napplications such as quantifying (bio-)diversity [14] and in data analysis [1, 10]. The second\nslogan comes forward in this paper in our study of the asymptotic convergence of magnitude\nto the point count #X; see also Examples 1.1 and 1.4.\nAt the time of writing, an online\nbibliography [15] maintained by Leinster & Meckes counts more than 120 papers on magnitude.\n1.2\nSummary of results\nIn this article, we focus on magnitude for finite metric spaces. This setting was treated early\non in the development of magnitude, for instance by Leinster in [12, §2] and by Meckes in [18].\nIn particular, both works found that so-called positive definite metric spaces are particularly\nwell-behaved (see Definition 2.2). Our work builds forth on this analysis by drawing a strong\nconnection between magnitude of finite positive definite metric spaces and Euclidean geometry,\nvia matrix theory. We briefly describe our main result and its applications below.\n1\narXiv:2510.14684v1 [math.MG] 16 Oct 2025\n\nThe magnitude of a finite metric space (X, d) is defined based on the similarity matrix Z with\nentries zij = e−d(i,j). When Z is positive definite, we call X positive definite and define:\nThe magnitude of X = (X, d) is |X| :=\nX\ni,j∈X\n\u0000Z−1\u0001\nij.\nWe will come to a more generally applicable definition later. The main new idea in this article is\nthat there is a Euclidean embedding ϕ : X →RX−1, which we call the similarity embedding, such\nthat the magnitude of X can be expressed in terms of the circumradius R(S) of its embedding\nS = ϕ(X); the circumradius is the smallest r such that the r-thickening S\ns∈S Br(s) of S is\ncontractible. More precisely, we find:\nTheorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then\n|X| =\n1\n1 −2R(S)2 .\n(1)\nIn fact, requiring expression (1) for all subspaces Y ⊆X characterizes the embedding S (see\nTheorem 2.15). So, in a sense, the embedding ϕ naturally appears in the context of magnitude.\nExample 1.1. Let us illustrate Theorem 2.12 by a classical example in magnitude: the metric\nspace X(2) consisting of two points at distance d > 0. By inverting the positive definite similarity\nmatrix Z =\n\u0010\n1\ne−d\ne−d\n1\n\u0011\nand summing the entries of Z−1, one directly computes the magnitude\n|X(2)| = 1 + tanh(d/2).\nThe embedding of X(2) that we introduce gives two points S ⊆R at distance\n√\n1 −e−d. Since\nthe circumradius of S is half this distance, Theorem 2.12 gives the relation\n1\n1 −2R(S)2 =\n1\n1 −1−exp(−d)\n2\n=\n2\n1 + exp(−d) = 1 + tanh(d/2) = |X(2)|.\nWhat lies behind Theorem 2.12? To lift some of the mystery, let us reveal the underlying\nlinear algebra and construct the embedding ϕ of X explicitly, starting from its matrix Z:\nStep 1: Construct the centered matrix K = 1\n2(I −11T\nn )Z(I −11T\nn ), where n = #X.\nStep 2: Because Z is positive definite, K admits a square root\n√\nK.\nStep 3: Define the embedding ϕ(i) of i ∈X as the ith column of\n√\nK.\nIn other words, K is the Gram matrix of S when the origin is placed at the centroid of S. Based\non this embedding, Theorem 2.12 then follows from the observation that the circumradius of S\nis governed by very similar “equilibrium” equations to those of magnitude; see Remark 2.13.\nWe continue with a brief description of the three main applications of Theorem 2.12.\nAs a first application, we consider the asymptotic behaviour of magnitude. Much of the\ntheory of magnitude is concerned with studying the magnitude |tX| across different scales t > 0,\nbased on the metric transformation d 7→t · d. Leinster & Willerton showed in [17, Thm. 3] that\nmagnitude asymptotically counts the number of points n in the metric space X, as follows\n|tX| = n −q(tX) with q(tX) →0 as t →∞.\nWe give a description of the asymptotics of the error term q in this formula; let St := ϕ(tX).\nTheorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence\nn −|tX| = q(tX) ∼n2\n\u0012n −1\nn\n−2R(St)2\n\u0013\n.\n2\n\nGeometrically, Theorem 3.1 states that the magnitude of tX tends to the magnitude of the\ndiscrete metric space in the same way as the circumradius of St tends to the circumradius of the\nregular simplex with unit-length edges. Let us return to the example of the two-point metric\nspace to illustrate the theorem.\nExample 1.2. For the two-point metric space X(2), Theorem 3.1 correctly gives\nq(tX(2)) = 2 −|tX(2)| =\n2\n1 + exp(t · d) ∼4(1\n2 −1\n2 exp(−t · d)).\nIn other words, we find that |tX(2)| ≈2e−td asymptotically. This is illustrated in Figure 1.\nFigure 1: The magnitude |tX(2)| of the metric space with two points at distance d = 1 is well-\napproximated by 2e−t at large scales t ≫0. This is explained by Theorem 3.1.\nAs a second application, we take inspiration from Fiedler’s matrix theory for Euclidean\nsimplices in [5], and develop a matrix theory for magnitude. Generalizing the relation between\nthe pseudoinverse Gram matrix and distance matrix of a simplex (see, e.g., [2, §3.2]), we find:\nTheorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then\n \n0\n1T\n1\nZ\n!−1\n=\n \n−|X|−1\nwT /|X|\nw/|X|\n1\n2K†\n!\n.\n(3)\nIn equation (3), the vector 1 = (1, . . . , 1)T is the all-ones vector, w := (Z−1)1 is the weighting\non the metric space X and K† is the Moore–Penrose pseudoinverse of the Gram matrix K of\nthe centered point set S; more results on the matrices Z, K and K† are given in Section 4.\nExample 1.3. For the two-point metric space with distance d = −ln(δ), Theorem 4.11 reads\n\n\n\n0\n1\n1\n1\n1\nδ\n1\nδ\n1\n\n\n\n−1\n=\n1\n2\n\n\n\n−(1 + δ)\n1\n1\n1\n(1 −δ)−1\n−(1 −δ)−1\n1\n−(1 −δ)−1\n(1 −δ)−1\n\n\n.\nWhat is the use of this identity? In this article, it mainly serves as a tool to study the\nmagnitude of subspaces of X. Since any subspace Y ⊆X of a positive definite metric space is\nalso positive definite, Theorem 4.11 applies to Y as well. The subspace relation between Y and\nX translates nicely to an algebraic relation between their respective matrices, and we arrive at:\nCorollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have\n|X\\{x}| = |X|\n \n1 + 2w2\nx\n¯cx|X|\n!−1\nand\nw′\ni\n|X\\{x}| = wi\n|X| + cix\n¯cx\nwx\n|X|,\nfor any i ̸= x, and where w′ is the weighting on X\\{x}.\n3\n\nHere, we use the notation cij = −(K†)ij and ¯ci = P\nj cij. This is a corollary of Theorem\n4.16, which is an analogous statement for general subspaces. We illustrate the relation between\n|X| and |X\\{x}| in Corollary 4.18 by looking at the quantity\n2w2\nx\n¯cx|X| that controls this relation.\nExample 1.4. We consider the metric space on X = {1, 2, 3} with distances\nd(1, 2) = 2\nand\nd(1, 3) = 100\nand\nd(2, 3) = 100.\nThe points 1 and 2 are close together and equidistant from point 3. Since every three-point\nmetric space is positive definite [12, Prop. 2.4.15], its magnitude exists for all t > 0. The black\nlines in Figure 2 plot the magnitude of X: for small t we effectively see one point, for medium\nt we see two points, i.e., {1, 2} and 3, and for large t we see three points.\nFigure 2: Both panels: The magnitude |tX| (black line) of the three-point metric space in Exam-\nple 1.4; note the logarithmic scale for the horizontal axis. Left panel: The relative contribution\nwx/|tX| for points 1 and 2 (blue line) and 3 (red line). Right panel: The change contribution\n2w2\nx/(¯cx|tX|) for points 1 and 2 (blue line) and 3 (red line).\nThe left and right panel in Figure 2 compare two possible ways to measure the contribution\nof a point x to the magnitude: wx/|X| on the left, and\n2w2\nx\n¯cx|X| on the right, inspired by Corollary\n4.18. Overall, the latter appears to be more sensitive to the changing geometry across scales.\nAs a third application, we initiate the study of a new class of metric spaces; we call these\nspaces strongly positive definite and they are defined by the additional positivity conditions\nw > 0 and cij = −(K†)ij > 0, for all i ̸= j. We show that this class is closed under taking\nsubspaces, and that every metric space tX becomes strongly positive definite when t ≫0.\nAs the main application, we prove two valuative results on the magnitude of a strongly posi-\ntive definite metric space X. These results are an attempt to reproduce results for magnitude in\nthe spirit of the inclusion–exclusion property of the Euler characteristic: for (suitable) subspaces\nA, B ⊆X of a topological space X, the Euler characteristic χ satisfies\nχ(A) + χ(B) = χ(A ∪B) + χ(A ∩B).\nThis fact has inspired research into valuative properties of magnitude: Leinster showed an\ninclusion–exclusion result for magnitude of graphs in [13], Leinster and Willerton showed in [17]\nthat the magnitude of subsets of Rn satisfies inclusion–exclusion asymptotically, and Gimperlein,\nGoffeng and Louca further improved on these geometric results in [6]. Here, we consider a variant\nof inclusion–exclusion; a function f : 2X →R is called strictly submodular if1\nf(A) + f(B) > f(A ∪B) + f(A ∩B),\nfor all A, B ⊆X, and f is called increasing (resp. decreasing) if it is increasing (resp. decreasing)\nwith respect to the subset partial order. We show the following two results:\n1Later in this article we will work with a different, but equivalent definition (4).\n4\n\nTheorem 5.9. Let X be strongly positive definite. Then the function\nf : Y 7−→\n(\n−|Y |−1, if Y ̸= ∅\nα\n, if Y = ∅\nis increasing if α < −1 and strictly submodular if α < −3\n2. In particular, f is strictly submodular\non the subspaces of tX′, for any X′ and t ≫0 and α < −3\n2.\nTheorem 5.10. Let X be any metric space and t ≫0. Then the function\nf : Y 7−→\n( m−|tY |\nm2\n+ m−1\nm\nif m := #Y ̸= 0\nα\nif Y = ∅\nis increasing if α < 1\n2 and strictly submodular if α < −1\n2.\nOrganization: Section 2 starts with the definition of magnitude, positive definite metric spaces,\nand the similarity embedding ϕ; this leads to the proof of Theorem 2.12, which relates the\nmagnitude |X| and circumradius R(S). Section 3 is rather short and deals with the asymptotics\nof q(tX), in Theorem 3.1. In Section 4, we develop the matrix theory of the matrices Z, K and\nK†: we study their relations, their connections to magnitude and weightings and prove Theorem\n4.11. This is then used to derive Theorem 4.16 and its corollaries, which describe the magnitude\nand weighting of subspaces Y ⊆X in terms of the data of X. Finally, Section 5 deals with\nthe new class of strongly positive definite metric spaces. We define this class, study some of\nits properties and show that it describes the asymptotics of tX for general X. These result are\nthen used to prove the submodularity results, Theorem 5.9 and 5.10.\n2\nMagnitude and circumradius\n2.1\nDefinitions: magnitude, weighting & positive definite\nWe start with the definition of magnitude and positive definite metric spaces; recall that the\nsimilarity matrix Z = Z(X) of a finite metric space (X, d) has entries defined as zij = e−d(i,j).\nDefinition 2.1 (Weighting, magnitude). Let X be a finite metric space.\n• A vector w ∈RX that satisfies Zw = 1 is called a weighting on X.\n• The magnitude of X is the sum |X| = 1T w, for any weighting w on X.\nIf the system has no solutions, we say that the magnitude and weighting do not exist.\nBy\ndefinition, we set the magnitude of the empty metric space X = ∅to zero. We use n or #X to\ndenote the cardinality of X, and we assume that 0 < #X < ∞unless explicitly stated otherwise.\nDefinition 2.2. A metric space X is positive definite if Z(X) is positive definite.\nThe magnitude of a positive definite metric space always exists. In fact, since the similarity\nmatrix Z is invertible in this case, there is a unique weighting w = Z−11 on X, and the\nmagnitude of a positive definite metric space X has the explicit form:\n|X| =\nX\ni,j∈X\n(Z−1)ij = 1T Z−11.\nPositive definite metric spaces play an important role in the theory of magnitude. Not only\nis much of the theory of magnitude easier to develop for positive definite metric spaces, as is\nthe case in this article, but the following result due to Leinster states that any metric space\neventually becomes positive definite when scaling its metric d 7→t · d; we write tX = (X, t · d).\n5\n\nProposition 2.3 ([12, Prop. 2.4.6]). For any metric space X and t ≫0, tX is positive definite.\nThe following closure result will be useful later on.\nProposition 2.4. If X is positive definite then any subspace Y ⊆X is positive definite.\nProof. Any principal submatrix Z(Y ) of a positive definite matrix Z(X) is positive definite.\nTo conclude, we give an important class of examples of positive definite metric spaces.\nExample 2.5 (Negative type metric). A metric space (X, d) has negative type if (X,\n√\nd) admits\nan isometric Euclidean embedding. As shown by Meckes in [18], a metric space X has negative\ntype if and only if tX is positive definite for all t > 0; this is also called stably positive definite.\nThere are many natural examples of negative type metric spaces, for instance point sets in\nEuclidean, spherical or hyperbolic space. See [18, Thm. 3.6] for a list of examples.\n2.2\nSimilarity embedding of positive definite metric spaces\nIn this section, we introduce an embedding of a positive definite metric space and show its\nrelation to magnitude. This embedding makes critical use of positive definiteness.\nDefinition 2.6 (Similarity embedding). A similarity embedding of a positive definite metric\nspace X is an embedding ϕ : X →RX−1 that satisfies\n∥ϕ(i) −ϕ(j)∥2 = 1 −e−d(i,j) = 1 −zij\nfor all i, j ∈X.\nRecall the construction from the introduction, which proves existence of these embeddings\nProposition 2.7. The similarity embedding exists and is unique up to rigid transformations.\nProof. Let X be a positive definite metric space and construct the centered similarity matrix\nK := 1\n2\n\u0000I −11T\nn\n\u0001Z\n\u0000I −11T\nn\n\u0001,\nwhere n = #X is the cardinality of the metric space and I is the identity matrix.\nSince\nZ is positive definite, the matrix K is positive semidefinite with ker(K) = span(1) (see also\nProposition 4.3) and thus there exists an (n−1)×n square root\n√\nK, i.e., such that\n√\nK\nT √\nK =\nK. The embedding ϕ : X ∋i 7→(ith column of\n√\nK) ∈Rn−1 is a similarity embedding, since\n∥ϕ(i) −ϕ(j)∥2 = 1\n2(ei −ej)T √\nK\nT √\nK(ei −ej) = 1\n2(zii + zjj −2zij) = 1 −e−d(i,j),\nwhere ei is the ith basis vector. Since the embedding ϕ is defined via all pairwise distances\nbetween the points ϕ(X), it is unique up to rigid transformations, i.e., isometries of Rn−1.\nRemark 2.8 (Computation). The similarity embedding ϕ can be computed efficiently using\nstandard linear algebra methods. For instance, one can compute\n√\nK via the Cholesky decom-\nposition or an eigenvalue decomposition, for which highly optimized algorithms exist.\nProposition 2.7 associates to each positive definite metric space X a set of points S := ϕ(X)\nin RX−1. As introduced, we will show that a certain geometric invariant of this point set is\nrelated to magnitude. We first establish another important fact about the geometry of S.\nProposition 2.9. Let X be positive definite. Then S = ϕ(X) are the vertices of a simplex.\nProof. This is a consequence of positive definiteness: since Z has full rank, the matrix K has\nrank (n −1) and its square root, the (n −1) × n matrix\n√\nK, has full rank n −1. It follows that\nthe columns of\n√\nK and thus the points S are affinely independent.\n6\n\n2.3\nMagnitude and circumradius\nWe now come to the definition of circumradius. This is the geometric invariant of the point set\nS that is closely related to magnitude; we note that, by construction, the matrix (11T −Z) is\nthe matrix of squared distances between pairs of points in S, and we may thus write Z(S).\nTheorem 2.10. The following numbers are equal for the vertices S of a simplex:\n1. The radius r of the unique sphere in RS−1 that goes through S;\n2. The smallest r such that the r-thickening of S is contractible;\n3. The unique value r for which (11T −Z(S))x = 2r2 · 1 with xT 1 = 1 has a solution.\nThis common value is called the circumradius of S and is denoted by R(S).\nProof.\nFrom Euclidean geometry (or algebraic geometry), we know that there is a unique\nsphere in RS−1 that goes through the vertices S of a simplex. This is called the circumsphere\nof the simplex, and it is the smallest sphere that encloses the points S.\n(1. ⇔2.) The r-thickening of S is contractible if and only if S is contained in some radius-r\nsphere. Since the circumsphere of a simplex is the smallest sphere that contains S, its radius\nR(S) is the smallest r for which the r-thickening of S is contractible, as required.\n(1. ⇔3.) The circumsphere has center c (= the circumcenter), which satisfies the equations:\n∥c −s∥2 = r2\nfor each s ∈S,\nwhere r is the radius of the circumsphere. If we write p for the barycentric coordinate of the\ncircumcenter, i.e., the unique vector such that 1T p = 1 and\n√\nKp = c, and use the fact that for\neach i ∈X, the vector ei is the barycentric coordinate of the ith point of S, then we can rewrite\n(p −ei)T K(p −ei) = r2\nfor each i ∈X\n⇐⇒1\n2(p −ei)T Z(p −ei) = r2\nfor each i ∈X\n(since 1T (p −ei) = 0)\n⇐⇒eT\ni Zp = −r2 + 1\n2pT Zp + 1\n2\nfor each i ∈X\n⇐⇒Zp = (1\n2 −r2 + 1\n2pT Zp) · 1.\n⇐⇒Zp = (1 −2r2) · 1\n(since 1T p = 1)\n⇐⇒(11T −Z)p = 2r2 · 1.\nThus the radius r of the circumsphere satisfies the required equation (3.). Uniqueness of r in\n(3.) follows from uniqueness of the sphere that contains all vertices S of a simplex, or it can be\nderived from invertibility of Z as implied by positive definiteness.\nRemark 2.11 (Scaled embeddings). Any rescaling α · ϕ for α > 0 of the similarity embedding\nstill embeds the metric space X as the vertices of a simplex, with circumradius αR(S). Different\nchoices than α = 1 may be useful in different applications, but some of the nice theory may be\nlost, for instance, when choosing a scaling that depends on X. For instance, for α2 = |X| or\nα2 = #X, the subspace characterization in Theorem 2.15 below no longer holds. In a sense,\nX-dependent scalings are not functorial in the same way as independent scalings are.\nWe now arrive at the first main result from the introduction, and its proof.\nTheorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then\n|X| =\n1\n1 −2R(S)2 .\n(1)\n7\n\nProof. This follows immediately from the third expression for the circumradius in Theorem\n2.10. From Zx = (1−2R(S)2)1 with xT 1 = 1, we find that w = x/(1−2R(S)2) is the weighting\non X. Since x is normalized, the claimed expression for the magnitude |X| follows.\nRemark 2.13 (Equilibrium equations). With the proofs of Theorem 2.10 and Theorem 2.12\nin place, we can now appreciate where the relation between magnitude and circumradius comes\nfrom: both quantities are defined via an ‘equilibrium equation’. In the case of magnitude, the\nequilibrium equation Zw = 1 is part of the definition which Leinster arrived at by generalizing\nthe Euler characteristic. In the geometric setting, the equilibrium equation (11T −Z)p = 2r21\nencodes the fact that the circumcenter is equidistant to all vertices of a simplex.\nCorollary 2.14. Let X be a positive definite metric space. Then |X| > 1.\nProof. This follows from |X| = 1 + 2|X|R2(S) with |X| > 0 and R(S) > 0.\nWe can in fact turn Theorem 2.12 around and use it to characterize the similarity embedding.\nTheorem 2.15. Let X be a positive definite metric space. The similarity embedding ϕ is the\nunique embedding of X into RX−1, up to rigid transformation, such that\n|Y | =\n1\n1 −2R(ϕ(Y ))2 for all nonempty Y ⊆X.\nProof. We recall from Example 1.1 that the magnitude of a two-point metric space is 2/(1+e−d).\nFurthermore, the circumradius of two points at Euclidean distance d′ is d′/2.\nFor every two-point subspace Yij = {i, j} ⊆X, the requirement on the embedding ϕ gives\n2\n1 + e−d(i,j) = |Yij| =\n1\n1 −2R(ϕ(Yi,j))2 =\n2\n2 −∥ϕ(i) −ϕ(j)∥2 ⇐⇒∥ϕ(i) −ϕ(j)∥2 = 1−e−d(i,j).\nThe latter equation shows that any embedding with the prescribed conditions must be the\nsimilarity embedding, which is unique. It remains to prove existence, i.e., that the similarity\nembedding ϕ satisfies the stated relation between |Y | and R(ϕ(Y )) for subspaces Y of any\ncardinality. This follows because the restriction ϕ|Y of the similarity embedding of X onto any\nsubspace Y is the similarity embedding of Y , and R(ϕ(Y )) = R(ϕ|Y (Y )). The expression for\n|Y | in terms of R(ϕ(Y )) is then simply Theorem 2.12.\nAnother way to think about Theorem 2.12 is that it expresses functoriality of the similarity\nembedding with respect to the subspace relation. This is also reflected in the matrix identity\n(3) and its implication, e.g., Theorem 4.16.\nTo end this section, we compute the similarity embedding, magnitude and circumradius of\na three-point metric space in a numerical example.\nExample 2.16. Consider the three-point metric space X = {1, 2, 3} with metric d(1, 2) =\nln(2) ≈0.69 and d(1, 3) = d(2, 3) = ln(10) ≈4.61. Its similarity matrix, centered similarity\nmatrix and a similarity embedding are:\nZ =\n\n\n\n1\n0.5\n0.1\n0.5\n1\n0.1\n0.1\n0.1\n1\n\n\n\nK = 1\n9\n\n\n\n1.9\n−0.35\n−1.55\n−0.35\n1.9\n−1.55\n−1.55\n−1.55\n3.1\n\n\n\n√\nK =\n\u0010\nϕ(1) ϕ(2) ϕ(3)\n\u0011\n=\n √\n2/4\n−\n√\n2/4\n0\n−0.2934\n−0.2934\n0.5869\n!\n3\nMagnitude asymptotics\nAs explained in the introduction, much of the theory of magnitude is concerned with studying\nhow the magnitude |tX| behaves as a function of the scale t > 0. Our goal here is to refine the\nfollowing asymptotic formula, due to Leinster and Willerton in [17]:\n|tX| = n −q(tX) with q(tX) →0 as t →∞,\n8\n\nwhere n is the cardinality of the metric space. As a direct application of Theorem 2.12, we give\nan asymptotic expression for the remainder q(tX) in this expression; we write St := ϕ(tX).\nTheorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence\nn −|tX| = q(tX) ∼n2\n\u0012n −1\nn\n−2R(St)2\n\u0013\n.\nProof. As t →∞, tX is eventually positive definite (Proposition 2.3) and its similarity matrix\nZ(tX) converges to the identity matrix I. As a result, the simplex with vertices St converges to\nthe regular simplex with all edge lengths equal to p1 −zij = 1. From the equilibrium equation,\none computes that R(St)2 converges to n−1\n2n , the squared circumradius of the regular simplex.\nNext, based on Theorem 2.12, we can write the remainder q(tX) = n −|tX| as follows:\n|tX| =\n1\n1 −2R(St)2 =\nn\n1 + n\n\u0000n−1\nn\n−2R(St)2\u0001 ⇒q(tX) = n\n \n1 −\n1\n1 + n\n\u0000n−1\nn\n−2R(St)2\u0001\n!\nUsing the asymptotic equivalence 1 −\n1\n1+x ∼x for x →0 and the earlier result that\n\u0000 n−1\nn\n−\n2R(St)2\u0001 →0 as t →0, we then find the claimed asymptotic equivalence.\nTheorem 3.1 states that the magnitude of a metric space tX converges to the magnitude of\nthe discrete metric space (X, ∞) in the same way as the circumradius of the simplex St converges\nto the circumradius of the regular simplex. We continue with magnitude asymptotics in Section\n5, where we will further refine our understanding of the asymptotics by looking more closely at\nsome implications of the convergence of St to the regular simplex.\n4\nMatrix theory & magnitude of subspaces\nIn this section, we look more closely at the matrices Z and K that appeared in Section 2, and\nintroduce and study the matrix K†. This section is partly an intermezzo, where we develop the\nmatrix theory of magnitude for its own sake, and partly a setup for Section 5.\n4.1\nMatrix theory of Z, K and K†\nThe Moore–Penrose pseudoinverse of a matrix M is the unique matrix M† that satisfies\nMM†M = M\nand\nM†MM† = M†\nand\nMM†, M†M are symmetric.\nWe point out two consequences of this definition: (i) if M is real and symmetric, then so is M†\nand (ii) the matrices MM† and M†M are orthogonal projection matrices onto ker(M) = ker(M†)\n(see, e.g., [2, §A.3]). We recall the definition of matrices K and K†:\nDefinition 4.1. Let X be a finite metric space with similarity matrix Z. Then\n• The centered similarity matrix is K := 1\n2\n\u0000I −11T\nn\n\u0001Z\n\u0000I −11T\nn\n\u0001.\n• The pseudoinverse centered similarity matrix is K†.\nBy their construction, we immediately find the following properties of K and K†\nProposition 4.2. K and K† are real, symmetric matrices with ker(K) = ker(K†) ⊇span(1).\nProposition 4.3 (Interlacing). Let λ1 ≥λ2 ≥· · · ≥λn be the real eigenvalues of Z, and\nµ1 ≥µ2 ≥· · · ≥µn−1 the real eigenvalues of K, with one zero eigenvalue omitted2. Then\nλ1 ≥2µ1 ≥λ2 ≥2µ2 ≥· · · ≥λn−1 ≥2µn−1 ≥λn,\nwith equality throughout if Z1 = |X|−11. We have rank(K) = rank(K†) = rank(Z) −1 and if Z\nis positive definite, then K and K† are positive semidefinite with ker(K) = ker(K†) = span(1).\n2By construction we know that K has at least one eigenvalue. The eigenvalues µ1, . . . , µn−1 are all other\neigenvalues except for this one zero eigenvalue; there may be other zero eigenvalues.\n9\n\nProof. The eigenvalue inequalities are precisely the interlacing theorem for matrix projections\n[7, Thm. 2.1] applied to the matrices 2K and Z. If Z1 = |X|−1 · 1 then |X|−1 is an eigenvalue\nof Z with (normalized) eigenvector 1/√n; let’s say that it is the jth eigenvalue.\nThen Z\nhas an eigendecomposition Z = P\ni̸=j λixxT + |X|−1 11T\nn\nand thus by its construction K has\neigendecomposition K = P\ni̸=j\n1\n2λixxT , which proves that 2µi = λi for all i ̸= j.\nThe implication (Z positive definite) =⇒(K and K† positive semidefinite) follows from\nthe interlacing inequalities.\nThe rank statement holds because 2K arises from Z through a\nprojection onto span(1)⊥, where furthermore ker(Z) ̸⊇span(1) since Z is a nonnegative matrix\nwith unit-diagonal. By the rank statement, K and K† have rank n −1 when Z is positive\ndefinite, and thus ker(K) = ker(K†) = span(1). This completes the proof.\nOne example where Z1 = |X|−11 holds is when the isometry group of X acts transitively on\nits points; these are called homogeneous spaces. We continue by showing some relations between\nZ, K and K†. We start with a useful lemma:\nLemma 4.4. Let Z be invertible with nonzero magnitude. Then\nZK† = 2I −21wT\n|X| .\n(2)\nProof. We start with the definition of K in terms of Z and use KK† =\n\u0000I −11T\nn\n\u0001:\nK = 1\n2\n\u0000I −11T\nn\n\u0001Z\n\u0000I −11T\nn\n\u0001 ⇒KK† = 1\n2\n\u0000I −11T\nn\n\u0001ZK† ⇔\n\u0000I −11T\nn\n\u0001 = 1\n2\n\u0000I −11T\nn\n\u0001ZK†.\nThe third equation implies that ZK† must be of the form ZK† = 2I −21yT for some y ∈Rn.\nUsing the additional fact that wT ZK† = 0, we find that y = w/|X| as required.\nWe now show some relations between Z, K and Z−1, K†.\nProposition 4.5. The similarity and centered similarity matrices satisfy:\n2K −Z = γ1T + 1γT ,\nwhere γ := −1\nnZ1 +\n1\n2n21T Z1 · 1 = diag(K) −1\n21.\nProof. We note that Z may be singular. Starting from the definition of K, we find\n2K −Z = −1\nnZ11T −1\nn11T Z + 1\nn2 11T Z11T ,\nwhich proves the expression for 2K−Z. The second expression of γ follows from diag(Z) = 1.\nProposition 4.6. Let Z be invertible with nonzero magnitude. Then\n2Z−1 −K† = 2wwT\n|X| ,\nwhere\n\n\n\nw\n|X| = 1\n2K†γ + 1\nn1;\n|X|−1 = −1\n2γT K†γ −2\nn1T γ.\nProof. Right-multiplying equation (2) by Z−1 proves the expression for 2Z−1 −K†. We now\nprove the expression for w/|X| in terms of K† and γ.\nK† = 2Z−1 −2wwT\n|X| ⇒K†γ =\n\u00002Z−1 −2wwT\n|X|\n\u0001\u0000 −1\nnZ1 +\n1\n2n21T Z1 · 1\n\u0001\n= −2\nn1 +\n2\n|X|w ⇒1\n2K†γ = −1\nn1 + w\n|X|.\nFinally, we prove the expression for |X|−1 in terms of K† and γ.\n1\n2K†γ + 1\nn1 =\nw\n|X| ⇒1\n2γT K†γ + 1\nnγT 1 = wT γ\n|X|\n= wT\n|X|\n\u0000 −1\nnZ1 +\n1\n2n2 1T Z1 · 1\n\u0001\n= −1\n|X| +\n1\n2n2 1T Z1\n= −1\n|X| −1\nnγT 1,\n10\n\nwhich leads to the claimed expression for |X|−1 and completes the proof.\nWe focus on one further ‘object’ that turns out to play an important role in the matrix theory\nand the asymptotics of magnitude: the off-diagonal entries of matrix K†. Later, in Section 5,\nwe will show that large scales t ≫0 are marked by negativity of these entries, which in turn\nleads to additional algebraic and geometric properties. We introduce the notation\ncij := −(K†)ij = 2wiwj\n|X| −2(Z−1)ij, for i ̸= j ∈X\nand\n¯ci := (K†)ii =\nX\nj∈X\ncij.\nThe relation between ¯ci and P\nj cij follows because the rows of K† sum to zero. As a notational\nconvention, when summing over j ∈X or i, j ∈X with cij in the sum, we consider each pair\nonly once (i.e., with (i, j) = (j, i)); think of c as a function on unordered pairs of distinct points.\nWe give expressions for magnitude and weightings in terms of c.\nProposition 4.7. Let Z be invertible with nonzero magnitude. Then\nwi\n|X| = 1 −1\n2\nX\nj∈X\ncij(1 −zij)\nand\n|X|−1 = 1 −\nX\ni,j∈X\ncij(zix −zjx)2 for any x ∈X.\nProof. The expression for wi follows by taking the ith diagonal entry of equation (2). For the\nexpression of |X|−1 we start by the observation that, due to ker(K†) = span(1), we can write\nK† =\nX\ni,j∈X\ncij(ei −ej)(ei −ej)T ⇒1\n2yT K†y =\nX\ni,j∈X\ncij(yi −yj)2 for any y ∈RX.\nRight-multiplying equation (2) by Z and looking at the xth diagonal entry, we then find\n(ZK†Z)xx = 2 −2|X|−1 ⇔|X|−1 = 1 −\nX\ni,j∈X\ncij(zix −zjx)2.\nWe note that a factor of 1\n2 is incorporated in the sum because of our notational convention.\nCorollary 4.8. Let Z be invertible with nonzero magnitude. Then P\ni,j∈X cij(1 −zij) = n −1.\nProof. This follows by summing wi/|X| in Proposition 4.7, and 1T w = |X|.\nRemark 4.9 (Foster’s theorem). The direct analogue of this corollary is called Foster’s Theorem\nin other settings. One hint at the depth of this theorem is that the integer on the righthandside\nis known to count, depending on the situation, a dimension, a degree or an Euler characteristic.\nCorollary 4.10. Let Z be invertible with nonzero magnitude. Then\n|X|−1 =\n1\nn tr\n\u0010\u0000I −1\n2ZK†Z\n\u0001\u0011\n.\nProof. This follows by summing the expression for |X|−1 in Proposition 4.7 over all x ∈X.\nTo end the section, we prove a matrix identity that relates the similarity and pseudoinverse\ncentered similarity matrices of a metric space. This identity summarizes several of the expres-\nsions above, and it will be the basis to derive explicit relations between different subspaces.\nTheorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then\n \n0\n1T\n1\nZ\n!−1\n=\n \n−|X|−1\nwT /|X|\nw/|X|\n1\n2K†\n!\n.\n(3)\n11\n\nProof. We verify that multiplying the two sides of equation (3) gives the identity matrix:\n \n1Tw/|X|\n1\n21T K†\n|X|−1 · (Zw −1)\n1wT/|X| + 1\n2ZK†\n!\n=\n \n1\n0\n0\nI\n!\n.\nThe top-left equation holds by definition of magnitude. The top-right equation holds because\nker(K†) = span(1).\nThe bottom-left equation holds because Zw = 1.\nThe bottom-right\nequation holds by equation (2).\nRemark 4.12 (Cayley–Menger matrix). Theorem 4.11 mimics a matrix identity in simplex\ngeometry and algebraic graph theory; we called this the Fiedler–Bapat identity in [2, §3]. In\nthat context, the matrix\n\u0010\n0 1T\n1 D\n\u0011\nis called the Cayley–Menger matrix, where D contains the\nsquared pairwise distances between a set of points. This matrix is an important algebraic object\nin geometry; for instance, its determinant encodes affine (in-)dependence of the set of points.\nWe note the following rational expression for |X| in variables (zij) that follows from (3):\nProposition 4.13. Let X be a positive definite metric space. Then\n|X| = −det\n \n0\n1T\n1\nZ\n!\n/ det(Z).\n4.2\nMagnitude and weightings for subspaces\nWe now apply Theorem 4.11 to study the magnitude of subspaces of a metric space; we start\nby fixing some notation. We write Y ⊆X to denote the metric subspace (Y, d|Y ) ⊆(X, d), and\nwrite d for the metric d|Y ; unless stated otherwise, we assume Y nonempty. We write Y c = X\\Y\nand use the following subscript notation for submatrices:\nZ =\n \nZY Y\nZY Y c\nZY cY\nZY cY c\n!\nand\nw =\n \nwY\nwY c\n!\n.\nSince the metric on Y is simply the restriction of d, it follows that the similarity matrix Z(Y )\nis the principal submatrix ZY Y of Z(X). We recall the definition of Schur complements:\nDefinition 4.14 (Schur complement). Let M =\n\u0010 MY Y\nMY Y c\nMY cY MY cY c\n\u0011\nbe a matrix such that MY cY c\nis invertible. The Schur complement of M with respect to Y c is the matrix3 M/Y c, with rows\nand columns indexed by Y , defined by\nM/Y c := MY Y −MY Y c(MY cY c)−1MY cY .\nProposition 4.15 ([8]). Let M be a matrix such that M/Y c is well-defined. Then\n• If M is invertible, then (M−1)Y Y = (M/Y c)−1.\n• M/Y c = ((M/x1)/ . . . )/xℓwith entries Y c = {x1, . . . , xℓ} in any order.\nWe now arrive at the main result of this section.\nTheorem 4.16. Let X be a positive definite metric space. Then for any Y ⊆X we have\n|Y | = |X|\n \n1 + 2wT\nY c(K†\nY cY c)−1wYc\n|X|\n!−1\nand\nw′\n|Y | = wY\n|X| −K†\nY Y c(K†\nY cY c)−1 wY c\n|X| ,\nwhere w′ is the weighting on Y . Furthermore, K†(Y ) = K†(X)/Y c.\n3This notation is not standard; we use it to keep the focus on the underlying metric space X and its subspaces.\n12\n\nProof. Starting from Theorem 4.11 for X, combined with the expression of inverse matrix\nsubmatrices in terms of Schur complements (Proposition 4.15), we find\n \n0\n1T\n1\nZY Y\n!−1\n=\n\n\n\n−|X|−1\nwT\nY /|X|\nwT\nY c/|X|\nwY /|X|\n1\n2K†\nY Y\n1\n2K†\nY Y c\nwY c/|X|\n1\n2K†\nY cY\n1\n2K†\nY cY c\n\n\n\n.\nY c.\n=\n \n−|X|−1\nwT\nY /|X|\nwY /|X|\n1\n2K†\nY Y\n!\n−2\n \nwT\nY c/|X|\n1\n2K†\nY Y c\n!\n(K†\nY cY c)−1 \u0010\nwY c/|X|\n1\n2K†\nY cY\n\u0011\n.\nNext, Theorem 4.11 for Y states that\n \n0\n1T\n1\nZ(Y )\n!−1\n=\n \n−|Y |−1\nw′T /|Y |\nw′/|Y |\n1\n2K(Y )†\n!\n.\nThe theorem then follows by equating these two expressions, using the equality Z(Y ) = ZY Y .\nCorollary 4.17. Let X be positive definite. Then |Y | ≤|X| for all Y ⊂X, with equality if and\nonly the weighting w on X satisfies wy = 0 for all y ∈Y .\nProof. We first establish that the matrix K†\nY cY c, and thus its inverse, is positive definite:\nxT K†\nY cY cx =\n \nx\n0Y\n!T \nI −11T\nn\n!\nK†\n \nI −11T\nn\n! \nx\n0Y\n!\n> 0 for all nonzero x ∈RY c,\nwhich follows since K† is positive definite on span(1)⊥. Thus K†\nY cY c is positive definite. The\ncorollary then follows from the expression for |Y | in Theorem 4.16, and |X| > 0.\nThe special case of the subspace Y = X\\{x} for some x ∈X is particularly useful:\nCorollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have\n|X\\{x}| = |X|\n \n1 + 2w2\nx\n¯cx|X|\n!−1\nand\nw′\ni\n|X\\{x}| = wi\n|X| + cix\n¯cx\nwx\n|X|,\nfor any i ̸= x, and where w′ is the weighting on X\\{x}.\nFor the entries c of the pseudoinverse centered similarity matrix K†, we find:\nCorollary 4.19. Let X be a positive definite metric space. Then for any x ∈X we have\nc′\nij = cij + cixcjx\n¯cx\nand\n¯c′\ni = ¯ci −c2\nix\n¯cx\n,\nfor all i, j ̸= x and where c′ is defined on X\\{x}, i.e., via the entries of K(X\\{x})†.\n5\nStrongly positive definite metric spaces\nIn this section, we return to the asymptotics of |tX|. Our main contribution is to identify a new\nproperty of metric spaces, which we call “strongly positive definite”, that is satisfied by every\nmetric space tX for t ≫0. This new regime refines our understanding of the asymptotics of tX.\n13\n\n5.1\nDefinition and first properties\nWe start by defining the new class of metric spaces; recall that cij = −(K†)ij = 2wiwj\n|X| −2(Z−1)ij.\nDefinition 5.1 (Strongly positive definite). A finite metric space X is called strongly positive\ndefinite if it is positive definite with c > 0 and w > 0.\nQuestion 5.2. What is the right generalization of Definition 5.1 when X is not finite?\nStrongly positive definite metric spaces are related to other known objects and properties.\nWe summarize some of these equivalences in the two propositions below.\nProposition 5.3. The following are equivalent for a positive definite metric space X:\n1. The weighting is nonnegative w ≥0 (resp. positive w > 0);\n2. S(X) are the vertices of a simplex that contains (resp. strictly contains) its circumcenter.\nProof. In the proof of Theorem 2.12, we found that the normalized weighting w/|X| is the\nbarycentric coordinate of the circumcenter of S(X). The circumcenter lies in the simplex (resp.\nin its interior) if and only if these coordinates are nonnegative, w ≥0 (resp. positive, w > 0).\nIn the following, the terms “M-matrix, Laplacian and nonobtuse” are defined in the proof.\nProposition 5.4. The following are equivalent for a positive definite metric space X.\n1. K† has nonpositive (resp. negative) off-diagonal entries, equivalently, c ≥0 (resp. c > 0).\n2. K† is an M-matrix (resp. strict M-matrix);\n3. K† is the Laplacian matrix of a connected, positively weighted graph (resp. complete graph).\n4. S(X) are the vertices of a nonobtuse (resp. acute) simplex.\nProof. (1.⇔2.) An M-matrix is a matrix whose eigenvalues have nonnegative real part and\nwhose off-diagonal entries are non-positive. Since K† is positive semidefinite when X is positive\ndefinite, K† is indeed an M-matrix if and only if its off-diagonal entries are non-positive.\n(1.⇔3.) The Laplacian matrix of a graph G = (V, E) with edge weights ω : E →R>0 is the\nmatrix L = P\nij∈E ω(ij)(ei −ej)(ei −ej)T . We have established before that K† can be written\nin this form, with ω(ij) = cij, as a consequence of span(1) ⊆ker(K†). Thus, if c > 0 then the\nmatrix K† is a Laplacian matrix. A graph is connected if and only if ker(L) = span(1) (see,\ne.g., [5, §6]), which completes the equivalence.\n(3.⇔4.) A simplex is nonobtuse if every angle between facets is at most π/2 (i.e., nonobtuse).\nThe equivalence between Laplacian matrices of connected graphs L and nonobtuse simplices is\ndue to Fiedler [5, §6.2]; the equivalence is established via (Moore–Penrose pseudoinverse of\nLaplacian matrix)←→(centered Gram matrix of simplex).\nIf c > 0 in (1.) this corresponds to G being the complete graph in (3.) and the simplex\nhaving angles smaller than π/2 (i.e., acute) in (4.). The terminology “strict M-matrix” in (2.)\nis not standard, but we use it here for an M-matrix with negative off-diagonal entries.\nRemark 5.5 (Graphs & discrete curvature). We highlight in particular the connection with\ngraph Laplacian matrices. Many of the techniques in this article are inspired by the author’s\nprevious work in that context; see for instance [2, §2–4,§6]. Relatedly, we point out that similar\nobjects to w have recently been defined on graphs in the context of discrete curvature [3, 4, 19];\nthere, w > 0 has the natural interpretation of a positive curvature condition.\nThe following result shows that strongly positive definite metric spaces can be parametrized\n(via Z−1) by an open semi-algebraic subset of GLn(R), the space of real, invertible n×n matrices.\n14\n\nProposition 5.6. A metric space X is strongly positive definite if and only if its similarity\nmatrix Z is invertible with positive magnitude, and satisfies\n(Z−11)i(Z−11)j > (1T Z−11) · (Z−1)ij\nand\n(Z−11)i(Z−11)j > 0, for all distinct i, j ∈X.\nProof. This follows from the identities K† = 2Z−1 −2wwT /|X| and w = Z−11. Introducing\nK† and w in the inequalities above implies that c > 0 is equivalent to the first inequalities,\nwhereas the second inequalities are equivalent to w > 0. For wi · wj > 0 to imply that w > 0,\nwe use that w cannot be all negative because |X| = 1T w is positive.\nAs a setup for the next subsection, we establish two further results:\nProposition 5.7. For any metric space X and t ≫0, tX is strongly positive definite.\nProof. The result for w > 0 was shown by Leinster in [12, Prop. 2.2.6]. We use the same proof\nstrategy to show c > 0. First, let t be sufficiently large such that tX is positive definite, and\nrepresent a metric space by its invertible similarity matrix Z ∈GLn(R) ⊂Rn×n; here n = #X.\nOn the subset of invertible matrices GLn(R), the following function is continuous\nGLn(R) ∋Z 7→cij(Z) = −2 · adjij\n \n0\n1T\n1\nZ\n!\n/ det\n \n0\n1T\n1\nZ\n!\nfor any i ̸= j ∈X, where adjij denotes the (i, j)th entry of the adjugate matrix. By continuity\nand cij(I) > 0 for the identity matrix I, there is a neighbourhood U ⊂GLn(R) of the identity\nmatrix such that cij(Z) > 0 for each Z ∈U.\nSince Z(tX) →I as t →∞, we know that\nZ(tX) ∈U and thus that cij(Z(tX)) > 0 for t ≫0. This completes the proof.\nProposition 5.8. Strong positive definiteness and c > 0 are preserved under taking subspaces.\nProof. This follows from Corollaries 4.18 and 4.19, which express c′ and w′ for a subspace\nY = X \\ {x} in terms of c and w of the metric space X, or by repeated application of this\nargument following the composition property of Schur complements (Proposition 4.15).\n5.2\nMagnitude submodularity\nThis section deals with valuative properties of magnitude. We show that two magnitude-based\nset functions on the subspaces of a strongly positive definite metric space X are submodular;\nrecall that a function f on the subsets of X is called strictly submodular if it satisfies4\nf(Y ) −f(Y \\{y}) < f(Y \\{x}) −f(Y \\{x, y}), for all Y ⊆X and distinct x, y ∈Y ,\n(4)\nand f is called increasing (resp. decreasing) if it is increasing (resp. decreasing) with respect to\nthe subset partial order on 2X. We start with the inverse of magnitude as a set function:\nTheorem 5.9. Let X be strongly positive definite. Then the function\nf : Y 7−→\n(\n−|Y |−1, if Y ̸= ∅\nα\n, if Y = ∅\nis increasing if α < −1 and strictly submodular if α < −3\n2. In particular, f is strictly submodular\non the subspaces of tX′, for any X′ and t ≫0 and α < −3\n2.\n4As noted, this is a different but equivalent definition from the one given in the introduction.\n15\n\nProof. Because strong positive semidefiniteness is conserved for subspaces (Proposition 5.8), we\nmay assume without loss of generality that Y = X when checking the submodularity inequalities\n(4). For #X = 1 the statement is true since |X| = 1 and (4) holds vacuously; we distinguish\nthe two remaining cases, #X = 2 and #X > 2, and start with the latter.\n(#X > 2): We can rewrite the submodularity inequalities for f as\n−|X|−1 + |X\\{x}|−1 < −|X\\{y}|−1 + |X\\{x, y}|−1\n⇐⇒\n2w2\nx\n¯cx · |X|2 <\n2w′2\nx\n¯c′x · |X\\{y}|2\n(Corollary 4.18),\nwhere c, w are defined on X and c′, w′ on X\\{y}. For strongly positive definite X, Corollaries\n4.18 and 4.19 imply ¯c′\nx < ¯cx and w′\nx/|X\\{y}| > wx/|X|, which confirms the inequality above.\n(#X = 2): We can rewrite the submodularity inequality for f as\n−|X|−1 + |X\\{x}|−1 < −|X\\{y}|−1 −α\n⇐⇒1 −1 + exp(−d(x, y))\n2\n< −1 −α\n(Example 1.1)\n⇐⇒α < −3\n2\nThis confirms strict submodularity for #X = 2 if α < −2/3, and thus completes the proof for\nstrongly positive definite X. By Corollary 4.18, we know that f is increasing for nonempty\nsubsets and that f(Y ) ≥−1 with equality for #Y = 1; it follows that f is increasing on 2X if\nα < −1. The asymptotic result follows from Proposition 5.7.\nEquivalently, one can say that −f in Theorem 5.9 is strictly supermodular and decreasing,\nfor appropriate α. To end this section, we prove a second submodularity result. The function\nconsidered in the theorem below is inspired by the asymptotic expression in Theorem 3.1 for\nq(tX) = #X −|X| in terms of R(St)2.\nTheorem 5.10. Let X be any metric space and t ≫0. Then the function\nf : Y 7−→\n( m−|tY |\nm2\n+ m−1\nm\nif m := #Y ̸= 0\nα\nif Y = ∅\nis increasing if α < 1\n2 and strictly submodular if α < −1\n2.\nProof. We follow the strategy from the proof of Proposition 5.7 and represent a metric space\nX by its invertible similarity matrix Z ∈GLn(R). We consider three families of functions: the\nfamily FY,x,y, labeled by x ̸= y ∈Y ⊆X, the family Gx,y labeled by x ̸= y ∈X, and the family\nHY ′,Y labeled by Y ′ ⊂Y ⊆X; these are defined as\nFY,x,y = f(Y ) −f(Y \\{x}) −f(Y \\{y}) + f(Y \\{x, y})\nGx,y = f({x, y}) −f({x}) −f({y}) + f(∅)\nHY ′,Y = f(Y ′) −f(Y )\nSince |Y | = −det\n\u0010\n0\n1T\n1 ZY Y\n\u0011\n/ det(Z) by Proposition 4.13, these are continuous functions on\nGLn(R). Next, we confirm that each of these functions evaluates negatively for the discrete\nmetric space; in that case, Z = I and thus |Y | = m and f(Y ) = (m −1)/m. We find\nFY,x,y(I) = m−1\nm\n−2m−2\nm−1 + m−3\nm−2 =\n−2\nm(m−1)(m−2) < 0\nGx,y(I) = 1\n2 + α\nHY ′,Y (I) =\n( m′−1\nm′\n−m−1\nm\n= m′−m\nm·m′ < 0\nif Y ′ ̸= ∅\nα −m−1\nm\n≤α −1\n2\nif Y ′ = ∅.\n16\n\nThus F(I) < 0, G(I) < 0 if α < −1\n2 and H(I) < 0 if α < 1\n2; fix a choice of α. By continuity\nof F, G, H and negativity at the identity matrix I, there is a neighbourhood U ⊂GLn(R) of I,\nsuch that F, G, H are negative on each Z ∈U. Since Z(tX) →I as t →∞, we thus know that\nZ(tX) ∈U for t ≫0 and thus that F, G, H are negative. These are the strict submodularity\ninequalities (F, G) and the increasing inequalities (H), so this completes the proof.\nAcknowledgments\nI thank Mark Meckes for helpful discussions and encouragements. Part of this article was written\nduring a research stay at the Max Planck Institute of Molecular Cell Biology and Genetics in\nDresden. For the purpose of open access, the author has applied a CC BY public copyright li-\ncence to any author accepted manuscript arising from this submission.\nReferences\n[1] M. F. Adamer, E. D. Brouwer, L. O’Bray, and B. Rieck. The magnitude vector of images.\nJournal of Applied and Computational Topology, 8(2):251–274, 2024. 1\n[2] K. Devriendt. Graph geometry from effective resistances. PhD thesis, University of Oxford,\n2022. 3, 9, 12, 14\n[3] K. Devriendt and R. Lambiotte. Discrete curvature on graphs from the effective resistance.\nJournal of Physics: Complexity, 3(2):025008, 2022. 14\n[4] K. Devriendt, A. Ottolini, and S. Steinerberger. Graph curvature via resistance distance.\nDiscrete Applied Mathematics, 348:68–78, 2024. 14\n[5] M. Fiedler. Matrices and graphs in geometry. Encyclopedia of mathematics and its appli-\ncations ; 139. Cambridge University Press, Cambridge, 2011. 3, 14\n[6] H. Gimperlein, M. Goffeng, and N. Louca. The magnitude and spectral geometry, 2024.\narXiv:2201.11363 [math.DG]. 4\n[7] W. H. Haemers. Interlacing eigenvalues and graphs. Linear Algebra and its Applications,\n226-228:593–616, 1995. 10\n[8] E. V. Haynsworth. On the Schur complement. Basel Mathematical Notes, (20), 1968. 12\n[9] R. Hepworth and S. Willerton. Categorifying the magnitude of a graph. Homology, Homo-\ntopy and Applications, 19(2):31–60, 2017. 1\n[10] S. Huntsman.\nDiversity enhancement via magnitude.\nIn Evolutionary Multi-Criterion\nOptimization (EMO 2023), volume 13994 of Lecture Notes in Computer Science, pages\n420–432. Springer, 2023. 1\n[11] T. Leinster. The Euler characteristic of a category. Documenta Mathematica, 13:21–49,\n2008. 1\n[12] T. Leinster. The magnitude of metric spaces. Documenta Mathematica, 18:857–905, 2013.\n1, 4, 6, 15\n[13] T. Leinster. The magnitude of a graph. Mathematical Proceedings of the Cambridge Philo-\nsophical Society, 166(2):247–264, 2019. 4\n[14] T. Leinster and M. W. Meckes. Maximizing diversity in biology and beyond. Entropy,\n18(3):88, 2016. 1\n17\n\n[15] T.\nLeinster\nand\nM.\nW.\nMeckes.\nMagnitude:\na\nbibliography.\nhttps://webhomes.maths.ed.ac.uk/˜tl/magbib/, 2025. Accessed: 2025-06-23. 1\n[16] T. Leinster and M. Shulman. Magnitude homology of enriched categories and metric spaces.\nAlgebraic & Geometric Topology, 21(5):2175–2221, 2021. 1\n[17] T. Leinster and S. Willerton. On the asymptotic magnitude of subsets of Euclidean space.\nGeometriae Dedicata, 164:287–310, 2013. 2, 4, 8\n[18] M. W. Meckes. Positive definite metric spaces. Positivity : an international journal devoted\nto the theory and applications of positivity in analysis, 17(3):733–757, 2012. 1, 6\n[19] S. Steinerberger. Curvature on graphs via equilibrium measures. Journal of Graph Theory,\n103:415–436, 2022. 14\nAuthors’ address:\nKarel Devriendt, University of Oxford\ndevriendt@maths.ox.ac.uk\n18\n"},"cleaned":{"kind":"string","value":"The geometry of magnitude for finite metric spaces Karel Devriendt Abstract The main result of this article is a geometric interpretation of magnitude, a real-valued invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric space X such that the magnitude of X can be expressed in terms of the 'circumradius' of its embedding S. The circumradius is the smallest r for which the r-thickening of S is contractible. We give three applications: First, we describe the asymptotic behaviour of the magnitude of tX as t →∞, in terms of the circumradius. Second, we develop a matrix theory for magnitude that leads to explicit relations between the magnitude of X and the magnitude of its subspaces. Third, we identify a new regime in the limiting behaviour of tX, and use this to show submodularity-type results for magnitude as a function on subspaces. 1 Introduction 1.1 Background Magnitude is a real-valued invariant for enriched categories (in general) and metric spaces (in particular) that was introduced by Leinster in 2006, in [11]. Before giving a definition, we mention two slogans that are helpful in guiding one's intuition on magnitude. The first slogan says that \"Magnitude is like an Euler characteristic.\" This statement reflects the history of magnitude [11], motivates its definition, and guides the development of its theory, for instance in the generalization to magnitude homology [9, 16]. In this paper, the connection to Euler characteristics shows itself in our study of valuative properties, i.e., how magnitude behaves as a function on subspaces Y ⊆X and what kind of inclusion-exclusion-type relations it satisfies. A second slogan says that \"Magnitude counts the effective number of points.\" Perhaps more appropriate for finite spaces, this intuition explains the success of magnitude in applications such as quantifying (bio-)diversity [14] and in data analysis [1, 10]. The second slogan comes forward in this paper in our study of the asymptotic convergence of magnitude to the point count #X; see also Examples 1.1 and 1.4. At the time of writing, an online bibliography [15] maintained by Leinster & Meckes counts more than 120 papers on magnitude. 1.2 Summary of results In this article, we focus on magnitude for finite metric spaces. This setting was treated early on in the development of magnitude, for instance by Leinster in [12, §2] and by Meckes in [18]. In particular, both works found that so-called positive definite metric spaces are particularly well-behaved (see Definition 2.2). Our work builds forth on this analysis by drawing a strong connection between magnitude of finite positive definite metric spaces and Euclidean geometry, via matrix theory. We briefly describe our main result and its applications below. 1 16 Oct 2025 The magnitude of a finite metric space (X, d) is defined based on the similarity matrix Z with entries zij = e-d(i,j). When Z is positive definite, we call X positive definite and define: The magnitude of X = (X, d) is |X| := X i,j∈X Z-1 ij. We will come to a more generally applicable definition later. The main new idea in this article is that there is a Euclidean embedding φ : X →RX-1, which we call the similarity embedding, such that the magnitude of X can be expressed in terms of the circumradius R(S) of its embedding S = φ(X); the circumradius is the smallest r such that the r-thickening S s∈S Br(s) of S is contractible. More precisely, we find: Theorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then |X| = 1 1 -2R(S)2 . (1) In fact, requiring expression (1) for all subspaces Y ⊆X characterizes the embedding S (see Theorem 2.15). So, in a sense, the embedding φ naturally appears in the context of magnitude. Example 1.1. Let us illustrate Theorem 2.12 by a classical example in magnitude: the metric space X(2) consisting of two points at distance d > 0. By inverting the positive definite similarity matrix Z = 1 e-d e-d 1 and summing the entries of Z-1, one directly computes the magnitude |X(2)| = 1 + tanh(d/2). The embedding of X(2) that we introduce gives two points S ⊆R at distance √ 1 -e-d. Since the circumradius of S is half this distance, Theorem 2.12 gives the relation 1 1 -2R(S)2 = 1 1 -1-exp(-d) 2 = 2 1 + exp(-d) = 1 + tanh(d/2) = |X(2)|. What lies behind Theorem 2.12? To lift some of the mystery, let us reveal the underlying linear algebra and construct the embedding φ of X explicitly, starting from its matrix Z: Step 1: Construct the centered matrix K = 1 2(I -11T n )Z(I -11T n ), where n = #X. Step 2: Because Z is positive definite, K admits a square root √ K. Step 3: Define the embedding φ(i) of i ∈X as the ith column of √ K. In other words, K is the Gram matrix of S when the origin is placed at the centroid of S. Based on this embedding, Theorem 2.12 then follows from the observation that the circumradius of S is governed by very similar \"equilibrium\" equations to those of magnitude; see Remark 2.13. We continue with a brief description of the three main applications of Theorem 2.12. As a first application, we consider the asymptotic behaviour of magnitude. Much of the theory of magnitude is concerned with studying the magnitude |tX| across different scales t > 0, based on the metric transformation d 7→t · d. Leinster & Willerton showed in [17, Thm. 3] that magnitude asymptotically counts the number of points n in the metric space X, as follows |tX| = n -q(tX) with q(tX) →0 as t →∞. We give a description of the asymptotics of the error term q in this formula; let St := φ(tX). Theorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence n -|tX| = q(tX) ∼n2 n -1 n -2R(St)2 . 2 Geometrically, Theorem 3.1 states that the magnitude of tX tends to the magnitude of the discrete metric space in the same way as the circumradius of St tends to the circumradius of the regular simplex with unit-length edges. Let us return to the example of the two-point metric space to illustrate the theorem. Example 1.2. For the two-point metric space X(2), Theorem 3.1 correctly gives q(tX(2)) = 2 -|tX(2)| = 2 1 + exp(t · d) ∼4(1 2 -1 2 exp(-t · d)). In other words, we find that |tX(2)| ≈2e-td asymptotically. This is illustrated in Figure 1. Figure 1: The magnitude |tX(2)| of the metric space with two points at distance d = 1 is wellapproximated by 2e-t at large scales t ≫0. This is explained by Theorem 3.1. As a second application, we take inspiration from Fiedler's matrix theory for Euclidean simplices in [5], and develop a matrix theory for magnitude. Generalizing the relation between the pseudoinverse Gram matrix and distance matrix of a simplex (see, e.g., [2, §3.2]), we find: Theorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then 0 1T 1 Z !-1 = -|X|-1 wT /|X| w/|X| 1 2K† ! . (3) In equation (3), the vector 1 = (1, . . . , 1)T is the all-ones vector, w := (Z-1)1 is the weighting on the metric space X and K† is the Moore-Penrose pseudoinverse of the Gram matrix K of the centered point set S; more results on the matrices Z, K and K† are given in Section 4. Example 1.3. For the two-point metric space with distance d = -ln(δ), Theorem 4.11 reads 0 1 1 1 1 δ 1 δ 1 -1 = 1 2 -(1 + δ) 1 1 1 (1 -δ)-1 -(1 -δ)-1 1 -(1 -δ)-1 (1 -δ)-1 . What is the use of this identity? In this article, it mainly serves as a tool to study the magnitude of subspaces of X. Since any subspace Y ⊆X of a positive definite metric space is also positive definite, Theorem 4.11 applies to Y as well. The subspace relation between Y and X translates nicely to an algebraic relation between their respective matrices, and we arrive at: Corollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have |X\\{x}| = |X| 1 + 2w2 x ̄cx|X| !-1 and w′ i |X\\{x}| = wi |X| + cix ̄cx wx |X|, for any i ̸= x, and where w′ is the weighting on X\\{x}. 3 Here, we use the notation cij = -(K†)ij and ̄ci = P j cij. This is a corollary of Theorem 4.16, which is an analogous statement for general subspaces. We illustrate the relation between |X| and |X\\{x}| in Corollary 4.18 by looking at the quantity 2w2 x ̄cx|X| that controls this relation. Example 1.4. We consider the metric space on X = {1, 2, 3} with distances d(1, 2) = 2 and d(1, 3) = 100 and d(2, 3) = 100. The points 1 and 2 are close together and equidistant from point 3. Since every three-point metric space is positive definite [12, Prop. 2.4.15], its magnitude exists for all t > 0. The black lines in Figure 2 plot the magnitude of X: for small t we effectively see one point, for medium t we see two points, i.e., {1, 2} and 3, and for large t we see three points. Figure 2: Both panels: The magnitude |tX| (black line) of the three-point metric space in Example 1.4; note the logarithmic scale for the horizontal axis. Left panel: The relative contribution wx/|tX| for points 1 and 2 (blue line) and 3 (red line). Right panel: The change contribution 2w2 x/( ̄cx|tX|) for points 1 and 2 (blue line) and 3 (red line). The left and right panel in Figure 2 compare two possible ways to measure the contribution of a point x to the magnitude: wx/|X| on the left, and 2w2 x ̄cx|X| on the right, inspired by Corollary 4.18. Overall, the latter appears to be more sensitive to the changing geometry across scales. As a third application, we initiate the study of a new class of metric spaces; we call these spaces strongly positive definite and they are defined by the additional positivity conditions w > 0 and cij = -(K†)ij > 0, for all i ̸= j. We show that this class is closed under taking subspaces, and that every metric space tX becomes strongly positive definite when t ≫0. As the main application, we prove two valuative results on the magnitude of a strongly positive definite metric space X. These results are an attempt to reproduce results for magnitude in the spirit of the inclusion-exclusion property of the Euler characteristic: for (suitable) subspaces A, B ⊆X of a topological space X, the Euler characteristic χ satisfies χ(A) + χ(B) = χ(A ∪B) + χ(A ∩B). This fact has inspired research into valuative properties of magnitude: Leinster showed an inclusion-exclusion result for magnitude of graphs in [13], Leinster and Willerton showed in [17] that the magnitude of subsets of Rn satisfies inclusion-exclusion asymptotically, and Gimperlein, Goffeng and Louca further improved on these geometric results in [6]. Here, we consider a variant of inclusion-exclusion; a function f : 2X →R is called strictly submodular if1 f(A) + f(B) > f(A ∪B) + f(A ∩B), for all A, B ⊆X, and f is called increasing (resp. decreasing) if it is increasing (resp. decreasing) with respect to the subset partial order. We show the following two results: 1Later in this article we will work with a different, but equivalent definition (4). 4 Theorem 5.9. Let X be strongly positive definite. Then the function f : Y 7-→ ( -|Y |-1, if Y ̸= ∅ α , if Y = ∅ is increasing if α 0; this is also called stably positive definite. There are many natural examples of negative type metric spaces, for instance point sets in Euclidean, spherical or hyperbolic space. See [18, Thm. 3.6] for a list of examples. 2.2 Similarity embedding of positive definite metric spaces In this section, we introduce an embedding of a positive definite metric space and show its relation to magnitude. This embedding makes critical use of positive definiteness. Definition 2.6 (Similarity embedding). A similarity embedding of a positive definite metric space X is an embedding φ : X →RX-1 that satisfies ∥φ(i) -φ(j)∥2 = 1 -e-d(i,j) = 1 -zij for all i, j ∈X. Recall the construction from the introduction, which proves existence of these embeddings Proposition 2.7. The similarity embedding exists and is unique up to rigid transformations. Proof. Let X be a positive definite metric space and construct the centered similarity matrix K := 1 2 I -11T n Z I -11T n , where n = #X is the cardinality of the metric space and I is the identity matrix. Since Z is positive definite, the matrix K is positive semidefinite with ker(K) = span(1) (see also Proposition 4.3) and thus there exists an (n-1)×n square root √ K, i.e., such that √ K T √ K = K. The embedding φ : X ∋i 7→(ith column of √ K) ∈Rn-1 is a similarity embedding, since ∥φ(i) -φ(j)∥2 = 1 2(ei -ej)T √ K T √ K(ei -ej) = 1 2(zii + zjj -2zij) = 1 -e-d(i,j), where ei is the ith basis vector. Since the embedding φ is defined via all pairwise distances between the points φ(X), it is unique up to rigid transformations, i.e., isometries of Rn-1. Remark 2.8 (Computation). The similarity embedding φ can be computed efficiently using standard linear algebra methods. For instance, one can compute √ K via the Cholesky decomposition or an eigenvalue decomposition, for which highly optimized algorithms exist. Proposition 2.7 associates to each positive definite metric space X a set of points S := φ(X) in RX-1. As introduced, we will show that a certain geometric invariant of this point set is related to magnitude. We first establish another important fact about the geometry of S. Proposition 2.9. Let X be positive definite. Then S = φ(X) are the vertices of a simplex. Proof. This is a consequence of positive definiteness: since Z has full rank, the matrix K has rank (n -1) and its square root, the (n -1) × n matrix √ K, has full rank n -1. It follows that the columns of √ K and thus the points S are affinely independent. 6 2.3 Magnitude and circumradius We now come to the definition of circumradius. This is the geometric invariant of the point set S that is closely related to magnitude; we note that, by construction, the matrix (11T -Z) is the matrix of squared distances between pairs of points in S, and we may thus write Z(S). Theorem 2.10. The following numbers are equal for the vertices S of a simplex: 1. The radius r of the unique sphere in RS-1 that goes through S; 2. The smallest r such that the r-thickening of S is contractible; 3. The unique value r for which (11T -Z(S))x = 2r2 · 1 with xT 1 = 1 has a solution. This common value is called the circumradius of S and is denoted by R(S). Proof. From Euclidean geometry (or algebraic geometry), we know that there is a unique sphere in RS-1 that goes through the vertices S of a simplex. This is called the circumsphere of the simplex, and it is the smallest sphere that encloses the points S. (1. ⇔2.) The r-thickening of S is contractible if and only if S is contained in some radius-r sphere. Since the circumsphere of a simplex is the smallest sphere that contains S, its radius R(S) is the smallest r for which the r-thickening of S is contractible, as required. (1. ⇔3.) The circumsphere has center c (= the circumcenter), which satisfies the equations: ∥c -s∥2 = r2 for each s ∈S, where r is the radius of the circumsphere. If we write p for the barycentric coordinate of the circumcenter, i.e., the unique vector such that 1T p = 1 and √ Kp = c, and use the fact that for each i ∈X, the vector ei is the barycentric coordinate of the ith point of S, then we can rewrite (p -ei)T K(p -ei) = r2 for each i ∈X ⇐⇒1 2(p -ei)T Z(p -ei) = r2 for each i ∈X (since 1T (p -ei) = 0) ⇐⇒eT i Zp = -r2 + 1 2pT Zp + 1 2 for each i ∈X ⇐⇒Zp = (1 2 -r2 + 1 2pT Zp) · 1. ⇐⇒Zp = (1 -2r2) · 1 (since 1T p = 1) ⇐⇒(11T -Z)p = 2r2 · 1. Thus the radius r of the circumsphere satisfies the required equation (3.). Uniqueness of r in (3.) follows from uniqueness of the sphere that contains all vertices S of a simplex, or it can be derived from invertibility of Z as implied by positive definiteness. Remark 2.11 (Scaled embeddings). Any rescaling α · φ for α > 0 of the similarity embedding still embeds the metric space X as the vertices of a simplex, with circumradius αR(S). Different choices than α = 1 may be useful in different applications, but some of the nice theory may be lost, for instance, when choosing a scaling that depends on X. For instance, for α2 = |X| or α2 = #X, the subspace characterization in Theorem 2.15 below no longer holds. In a sense, X-dependent scalings are not functorial in the same way as independent scalings are. We now arrive at the first main result from the introduction, and its proof. Theorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then |X| = 1 1 -2R(S)2 . (1) 7 Proof. This follows immediately from the third expression for the circumradius in Theorem 2.10. From Zx = (1-2R(S)2)1 with xT 1 = 1, we find that w = x/(1-2R(S)2) is the weighting on X. Since x is normalized, the claimed expression for the magnitude |X| follows. Remark 2.13 (Equilibrium equations). With the proofs of Theorem 2.10 and Theorem 2.12 in place, we can now appreciate where the relation between magnitude and circumradius comes from: both quantities are defined via an 'equilibrium equation'. In the case of magnitude, the equilibrium equation Zw = 1 is part of the definition which Leinster arrived at by generalizing the Euler characteristic. In the geometric setting, the equilibrium equation (11T -Z)p = 2r21 encodes the fact that the circumcenter is equidistant to all vertices of a simplex. Corollary 2.14. Let X be a positive definite metric space. Then |X| > 1. Proof. This follows from |X| = 1 + 2|X|R2(S) with |X| > 0 and R(S) > 0. We can in fact turn Theorem 2.12 around and use it to characterize the similarity embedding. Theorem 2.15. Let X be a positive definite metric space. The similarity embedding φ is the unique embedding of X into RX-1, up to rigid transformation, such that |Y | = 1 1 -2R(φ(Y ))2 for all nonempty Y ⊆X. Proof. We recall from Example 1.1 that the magnitude of a two-point metric space is 2/(1+e-d). Furthermore, the circumradius of two points at Euclidean distance d′ is d′/2. For every two-point subspace Yij = {i, j} ⊆X, the requirement on the embedding φ gives 2 1 + e-d(i,j) = |Yij| = 1 1 -2R(φ(Yi,j))2 = 2 2 -∥φ(i) -φ(j)∥2 ⇐⇒∥φ(i) -φ(j)∥2 = 1-e-d(i,j). The latter equation shows that any embedding with the prescribed conditions must be the similarity embedding, which is unique. It remains to prove existence, i.e., that the similarity embedding φ satisfies the stated relation between |Y | and R(φ(Y )) for subspaces Y of any cardinality. This follows because the restriction φ|Y of the similarity embedding of X onto any subspace Y is the similarity embedding of Y , and R(φ(Y )) = R(φ|Y (Y )). The expression for |Y | in terms of R(φ(Y )) is then simply Theorem 2.12. Another way to think about Theorem 2.12 is that it expresses functoriality of the similarity embedding with respect to the subspace relation. This is also reflected in the matrix identity (3) and its implication, e.g., Theorem 4.16. To end this section, we compute the similarity embedding, magnitude and circumradius of a three-point metric space in a numerical example. Example 2.16. Consider the three-point metric space X = {1, 2, 3} with metric d(1, 2) = ln(2) ≈0.69 and d(1, 3) = d(2, 3) = ln(10) ≈4.61. Its similarity matrix, centered similarity matrix and a similarity embedding are: Z = 1 0.5 0.1 0.5 1 0.1 0.1 0.1 1 K = 1 9 1.9 -0.35 -1.55 -0.35 1.9 -1.55 -1.55 -1.55 3.1 √ K = φ(1) φ(2) φ(3) = √ 2/4 - √ 2/4 0 -0.2934 -0.2934 0.5869 ! 3 Magnitude asymptotics As explained in the introduction, much of the theory of magnitude is concerned with studying how the magnitude |tX| behaves as a function of the scale t > 0. Our goal here is to refine the following asymptotic formula, due to Leinster and Willerton in [17]: |tX| = n -q(tX) with q(tX) →0 as t →∞, 8 where n is the cardinality of the metric space. As a direct application of Theorem 2.12, we give an asymptotic expression for the remainder q(tX) in this expression; we write St := φ(tX). Theorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence n -|tX| = q(tX) ∼n2 n -1 n -2R(St)2 . Proof. As t →∞, tX is eventually positive definite (Proposition 2.3) and its similarity matrix Z(tX) converges to the identity matrix I. As a result, the simplex with vertices St converges to the regular simplex with all edge lengths equal to p1 -zij = 1. From the equilibrium equation, one computes that R(St)2 converges to n-1 2n , the squared circumradius of the regular simplex. Next, based on Theorem 2.12, we can write the remainder q(tX) = n -|tX| as follows: |tX| = 1 1 -2R(St)2 = n 1 + n n-1 n -2R(St)2 ⇒q(tX) = n 1 - 1 1 + n n-1 n -2R(St)2 ! Using the asymptotic equivalence 1 - 1 1+x ∼x for x →0 and the earlier result that n-1 n - 2R(St)2 →0 as t →0, we then find the claimed asymptotic equivalence. Theorem 3.1 states that the magnitude of a metric space tX converges to the magnitude of the discrete metric space (X, ∞) in the same way as the circumradius of the simplex St converges to the circumradius of the regular simplex. We continue with magnitude asymptotics in Section 5, where we will further refine our understanding of the asymptotics by looking more closely at some implications of the convergence of St to the regular simplex. 4 Matrix theory & magnitude of subspaces In this section, we look more closely at the matrices Z and K that appeared in Section 2, and introduce and study the matrix K†. This section is partly an intermezzo, where we develop the matrix theory of magnitude for its own sake, and partly a setup for Section 5. 4.1 Matrix theory of Z, K and K† The Moore-Penrose pseudoinverse of a matrix M is the unique matrix M† that satisfies MM†M = M and M†MM† = M† and MM†, M†M are symmetric. We point out two consequences of this definition: (i) if M is real and symmetric, then so is M† and (ii) the matrices MM† and M†M are orthogonal projection matrices onto ker(M) = ker(M†) (see, e.g., [2, §A.3]). We recall the definition of matrices K and K†: Definition 4.1. Let X be a finite metric space with similarity matrix Z. Then • The centered similarity matrix is K := 1 2 I -11T n Z I -11T n . • The pseudoinverse centered similarity matrix is K†. By their construction, we immediately find the following properties of K and K† Proposition 4.2. K and K† are real, symmetric matrices with ker(K) = ker(K†) ⊇span(1). Proposition 4.3 (Interlacing). Let λ1 ≥λ2 ≥· · · ≥λn be the real eigenvalues of Z, and μ1 ≥μ2 ≥· · · ≥μn-1 the real eigenvalues of K, with one zero eigenvalue omitted2. Then λ1 ≥2μ1 ≥λ2 ≥2μ2 ≥· · · ≥λn-1 ≥2μn-1 ≥λn, with equality throughout if Z1 = |X|-11. We have rank(K) = rank(K†) = rank(Z) -1 and if Z is positive definite, then K and K† are positive semidefinite with ker(K) = ker(K†) = span(1). 2By construction we know that K has at least one eigenvalue. The eigenvalues μ1, . . . , μn-1 are all other eigenvalues except for this one zero eigenvalue; there may be other zero eigenvalues. 9 Proof. The eigenvalue inequalities are precisely the interlacing theorem for matrix projections [7, Thm. 2.1] applied to the matrices 2K and Z. If Z1 = |X|-1 · 1 then |X|-1 is an eigenvalue of Z with (normalized) eigenvector 1/√n; let's say that it is the jth eigenvalue. Then Z has an eigendecomposition Z = P i̸=j λixxT + |X|-1 11T n and thus by its construction K has eigendecomposition K = P i̸=j 1 2λixxT , which proves that 2μi = λi for all i ̸= j. The implication (Z positive definite) =⇒(K and K† positive semidefinite) follows from the interlacing inequalities. The rank statement holds because 2K arises from Z through a projection onto span(1)⊥, where furthermore ker(Z) ̸⊇span(1) since Z is a nonnegative matrix with unit-diagonal. By the rank statement, K and K† have rank n -1 when Z is positive definite, and thus ker(K) = ker(K†) = span(1). This completes the proof. One example where Z1 = |X|-11 holds is when the isometry group of X acts transitively on its points; these are called homogeneous spaces. We continue by showing some relations between Z, K and K†. We start with a useful lemma: Lemma 4.4. Let Z be invertible with nonzero magnitude. Then ZK† = 2I -21wT |X| . (2) Proof. We start with the definition of K in terms of Z and use KK† = I -11T n : K = 1 2 I -11T n Z I -11T n ⇒KK† = 1 2 I -11T n ZK† ⇔ I -11T n = 1 2 I -11T n ZK†. The third equation implies that ZK† must be of the form ZK† = 2I -21yT for some y ∈Rn. Using the additional fact that wT ZK† = 0, we find that y = w/|X| as required. We now show some relations between Z, K and Z-1, K†. Proposition 4.5. The similarity and centered similarity matrices satisfy: 2K -Z = γ1T + 1γT , where γ := -1 nZ1 + 1 2n21T Z1 · 1 = diag(K) -1 21. Proof. We note that Z may be singular. Starting from the definition of K, we find 2K -Z = -1 nZ11T -1 n11T Z + 1 n2 11T Z11T , which proves the expression for 2K-Z. The second expression of γ follows from diag(Z) = 1. Proposition 4.6. Let Z be invertible with nonzero magnitude. Then 2Z-1 -K† = 2wwT |X| , where w |X| = 1 2K†γ + 1 n1; |X|-1 = -1 2γT K†γ -2 n1T γ. Proof. Right-multiplying equation (2) by Z-1 proves the expression for 2Z-1 -K†. We now prove the expression for w/|X| in terms of K† and γ. K† = 2Z-1 -2wwT |X| ⇒K†γ = 2Z-1 -2wwT |X| -1 nZ1 + 1 2n21T Z1 · 1 = -2 n1 + 2 |X|w ⇒1 2K†γ = -1 n1 + w |X|. Finally, we prove the expression for |X|-1 in terms of K† and γ. 1 2K†γ + 1 n1 = w |X| ⇒1 2γT K†γ + 1 nγT 1 = wT γ |X| = wT |X| -1 nZ1 + 1 2n2 1T Z1 · 1 = -1 |X| + 1 2n2 1T Z1 = -1 |X| -1 nγT 1, 10 which leads to the claimed expression for |X|-1 and completes the proof. We focus on one further 'object' that turns out to play an important role in the matrix theory and the asymptotics of magnitude: the off-diagonal entries of matrix K†. Later, in Section 5, we will show that large scales t ≫0 are marked by negativity of these entries, which in turn leads to additional algebraic and geometric properties. We introduce the notation cij := -(K†)ij = 2wiwj |X| -2(Z-1)ij, for i ̸= j ∈X and ̄ci := (K†)ii = X j∈X cij. The relation between ̄ci and P j cij follows because the rows of K† sum to zero. As a notational convention, when summing over j ∈X or i, j ∈X with cij in the sum, we consider each pair only once (i.e., with (i, j) = (j, i)); think of c as a function on unordered pairs of distinct points. We give expressions for magnitude and weightings in terms of c. Proposition 4.7. Let Z be invertible with nonzero magnitude. Then wi |X| = 1 -1 2 X j∈X cij(1 -zij) and |X|-1 = 1 - X i,j∈X cij(zix -zjx)2 for any x ∈X. Proof. The expression for wi follows by taking the ith diagonal entry of equation (2). For the expression of |X|-1 we start by the observation that, due to ker(K†) = span(1), we can write K† = X i,j∈X cij(ei -ej)(ei -ej)T ⇒1 2yT K†y = X i,j∈X cij(yi -yj)2 for any y ∈RX. Right-multiplying equation (2) by Z and looking at the xth diagonal entry, we then find (ZK†Z)xx = 2 -2|X|-1 ⇔|X|-1 = 1 - X i,j∈X cij(zix -zjx)2. We note that a factor of 1 2 is incorporated in the sum because of our notational convention. Corollary 4.8. Let Z be invertible with nonzero magnitude. Then P i,j∈X cij(1 -zij) = n -1. Proof. This follows by summing wi/|X| in Proposition 4.7, and 1T w = |X|. Remark 4.9 (Foster's theorem). The direct analogue of this corollary is called Foster's Theorem in other settings. One hint at the depth of this theorem is that the integer on the righthandside is known to count, depending on the situation, a dimension, a degree or an Euler characteristic. Corollary 4.10. Let Z be invertible with nonzero magnitude. Then |X|-1 = 1 n tr I -1 2ZK†Z . Proof. This follows by summing the expression for |X|-1 in Proposition 4.7 over all x ∈X. To end the section, we prove a matrix identity that relates the similarity and pseudoinverse centered similarity matrices of a metric space. This identity summarizes several of the expressions above, and it will be the basis to derive explicit relations between different subspaces. Theorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then 0 1T 1 Z !-1 = -|X|-1 wT /|X| w/|X| 1 2K† ! . (3) 11 Proof. We verify that multiplying the two sides of equation (3) gives the identity matrix: 1Tw/|X| 1 21T K† |X|-1 · (Zw -1) 1wT/|X| + 1 2ZK† ! = 1 0 0 I ! . The top-left equation holds by definition of magnitude. The top-right equation holds because ker(K†) = span(1). The bottom-left equation holds because Zw = 1. The bottom-right equation holds by equation (2). Remark 4.12 (Cayley-Menger matrix). Theorem 4.11 mimics a matrix identity in simplex geometry and algebraic graph theory; we called this the Fiedler-Bapat identity in [2, §3]. In that context, the matrix 0 1T 1 D is called the Cayley-Menger matrix, where D contains the squared pairwise distances between a set of points. This matrix is an important algebraic object in geometry; for instance, its determinant encodes affine (in-)dependence of the set of points. We note the following rational expression for |X| in variables (zij) that follows from (3): Proposition 4.13. Let X be a positive definite metric space. Then |X| = -det 0 1T 1 Z ! / det(Z). 4.2 Magnitude and weightings for subspaces We now apply Theorem 4.11 to study the magnitude of subspaces of a metric space; we start by fixing some notation. We write Y ⊆X to denote the metric subspace (Y, d|Y ) ⊆(X, d), and write d for the metric d|Y ; unless stated otherwise, we assume Y nonempty. We write Y c = X and use the following subscript notation for submatrices: Z = ZY Y ZY Y c ZY cY ZY cY c ! and w = wY wY c ! . Since the metric on Y is simply the restriction of d, it follows that the similarity matrix Z(Y ) is the principal submatrix ZY Y of Z(X). We recall the definition of Schur complements: Definition 4.14 (Schur complement). Let M = MY Y MY Y c MY cY MY cY c be a matrix such that MY cY c is invertible. The Schur complement of M with respect to Y c is the matrix3 M/Y c, with rows and columns indexed by Y , defined by M/Y c := MY Y -MY Y c(MY cY c)-1MY cY . Proposition 4.15 ([8]). Let M be a matrix such that M/Y c is well-defined. Then • If M is invertible, then (M-1)Y Y = (M/Y c)-1. • M/Y c = ((M/x1)/ . . . )/xlwith entries Y c = {x1, . . . , xl} in any order. We now arrive at the main result of this section. Theorem 4.16. Let X be a positive definite metric space. Then for any Y ⊆X we have |Y | = |X| 1 + 2wT Y c(K† Y cY c)-1wYc |X| !-1 and w′ |Y | = wY |X| -K† Y Y c(K† Y cY c)-1 wY c |X| , where w′ is the weighting on Y . Furthermore, K†(Y ) = K†(X)/Y c. 3This notation is not standard; we use it to keep the focus on the underlying metric space X and its subspaces. 12 Proof. Starting from Theorem 4.11 for X, combined with the expression of inverse matrix submatrices in terms of Schur complements (Proposition 4.15), we find 0 1T 1 ZY Y !-1 = -|X|-1 wT Y /|X| wT Y c/|X| wY /|X| 1 2K† Y Y 1 2K† Y Y c wY c/|X| 1 2K† Y cY 1 2K† Y cY c . Y c. = -|X|-1 wT Y /|X| wY /|X| 1 2K† Y Y ! -2 wT Y c/|X| 1 2K† Y Y c ! (K† Y cY c)-1 wY c/|X| 1 2K† Y cY . Next, Theorem 4.11 for Y states that 0 1T 1 Z(Y ) !-1 = -|Y |-1 w′T /|Y | w′/|Y | 1 2K(Y )† ! . The theorem then follows by equating these two expressions, using the equality Z(Y ) = ZY Y . Corollary 4.17. Let X be positive definite. Then |Y | ≤|X| for all Y ⊂X, with equality if and only the weighting w on X satisfies wy = 0 for all y ∈Y . Proof. We first establish that the matrix K† Y cY c, and thus its inverse, is positive definite: xT K† Y cY cx = x 0Y !T I -11T n ! K† I -11T n ! x 0Y ! > 0 for all nonzero x ∈RY c, which follows since K† is positive definite on span(1)⊥. Thus K† Y cY c is positive definite. The corollary then follows from the expression for |Y | in Theorem 4.16, and |X| > 0. The special case of the subspace Y = X\\{x} for some x ∈X is particularly useful: Corollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have |X\\{x}| = |X| 1 + 2w2 x ̄cx|X| !-1 and w′ i |X\\{x}| = wi |X| + cix ̄cx wx |X|, for any i ̸= x, and where w′ is the weighting on X\\{x}. For the entries c of the pseudoinverse centered similarity matrix K†, we find: Corollary 4.19. Let X be a positive definite metric space. Then for any x ∈X we have c′ ij = cij + cixcjx ̄cx and ̄c′ i = ̄ci -c2 ix ̄cx , for all i, j ̸= x and where c′ is defined on X\\{x}, i.e., via the entries of K(X\\{x})†. 5 Strongly positive definite metric spaces In this section, we return to the asymptotics of |tX|. Our main contribution is to identify a new property of metric spaces, which we call \"strongly positive definite\", that is satisfied by every metric space tX for t ≫0. This new regime refines our understanding of the asymptotics of tX. 13 5.1 Definition and first properties We start by defining the new class of metric spaces; recall that cij = -(K†)ij = 2wiwj |X| -2(Z-1)ij. Definition 5.1 (Strongly positive definite). A finite metric space X is called strongly positive definite if it is positive definite with c > 0 and w > 0. Question 5.2. What is the right generalization of Definition 5.1 when X is not finite? Strongly positive definite metric spaces are related to other known objects and properties. We summarize some of these equivalences in the two propositions below. Proposition 5.3. The following are equivalent for a positive definite metric space X: 1. The weighting is nonnegative w ≥0 (resp. positive w > 0); 2. S(X) are the vertices of a simplex that contains (resp. strictly contains) its circumcenter. Proof. In the proof of Theorem 2.12, we found that the normalized weighting w/|X| is the barycentric coordinate of the circumcenter of S(X). The circumcenter lies in the simplex (resp. in its interior) if and only if these coordinates are nonnegative, w ≥0 (resp. positive, w > 0). In the following, the terms \"M-matrix, Laplacian and nonobtuse\" are defined in the proof. Proposition 5.4. The following are equivalent for a positive definite metric space X. 1. K† has nonpositive (resp. negative) off-diagonal entries, equivalently, c ≥0 (resp. c > 0). 2. K† is an M-matrix (resp. strict M-matrix); 3. K† is the Laplacian matrix of a connected, positively weighted graph (resp. complete graph). 4. S(X) are the vertices of a nonobtuse (resp. acute) simplex. Proof. (1.⇔2.) An M-matrix is a matrix whose eigenvalues have nonnegative real part and whose off-diagonal entries are non-positive. Since K† is positive semidefinite when X is positive definite, K† is indeed an M-matrix if and only if its off-diagonal entries are non-positive. (1.⇔3.) The Laplacian matrix of a graph G = (V, E) with edge weights ω : E →R>0 is the matrix L = P ij∈E ω(ij)(ei -ej)(ei -ej)T . We have established before that K† can be written in this form, with ω(ij) = cij, as a consequence of span(1) ⊆ker(K†). Thus, if c > 0 then the matrix K† is a Laplacian matrix. A graph is connected if and only if ker(L) = span(1) (see, e.g., [5, §6]), which completes the equivalence. (3.⇔4.) A simplex is nonobtuse if every angle between facets is at most π/2 (i.e., nonobtuse). The equivalence between Laplacian matrices of connected graphs L and nonobtuse simplices is due to Fiedler [5, §6.2]; the equivalence is established via (Moore-Penrose pseudoinverse of Laplacian matrix)←→(centered Gram matrix of simplex). If c > 0 in (1.) this corresponds to G being the complete graph in (3.) and the simplex having angles smaller than π/2 (i.e., acute) in (4.). The terminology \"strict M-matrix\" in (2.) is not standard, but we use it here for an M-matrix with negative off-diagonal entries. Remark 5.5 (Graphs & discrete curvature). We highlight in particular the connection with graph Laplacian matrices. Many of the techniques in this article are inspired by the author's previous work in that context; see for instance [2, §2-4,§6]. Relatedly, we point out that similar objects to w have recently been defined on graphs in the context of discrete curvature [3, 4, 19]; there, w > 0 has the natural interpretation of a positive curvature condition. The following result shows that strongly positive definite metric spaces can be parametrized (via Z-1) by an open semi-algebraic subset of GLn(R), the space of real, invertible n×n matrices. 14 Proposition 5.6. A metric space X is strongly positive definite if and only if its similarity matrix Z is invertible with positive magnitude, and satisfies (Z-11)i(Z-11)j > (1T Z-11) · (Z-1)ij and (Z-11)i(Z-11)j > 0, for all distinct i, j ∈X. Proof. This follows from the identities K† = 2Z-1 -2wwT /|X| and w = Z-11. Introducing K† and w in the inequalities above implies that c > 0 is equivalent to the first inequalities, whereas the second inequalities are equivalent to w > 0. For wi · wj > 0 to imply that w > 0, we use that w cannot be all negative because |X| = 1T w is positive. As a setup for the next subsection, we establish two further results: Proposition 5.7. For any metric space X and t ≫0, tX is strongly positive definite. Proof. The result for w > 0 was shown by Leinster in [12, Prop. 2.2.6]. We use the same proof strategy to show c > 0. First, let t be sufficiently large such that tX is positive definite, and represent a metric space by its invertible similarity matrix Z ∈GLn(R) ⊂Rn×n; here n = #X. On the subset of invertible matrices GLn(R), the following function is continuous GLn(R) ∋Z 7→cij(Z) = -2 · adjij 0 1T 1 Z ! / det 0 1T 1 Z ! for any i ̸= j ∈X, where adjij denotes the (i, j)th entry of the adjugate matrix. By continuity and cij(I) > 0 for the identity matrix I, there is a neighbourhood U ⊂GLn(R) of the identity matrix such that cij(Z) > 0 for each Z ∈U. Since Z(tX) →I as t →∞, we know that Z(tX) ∈U and thus that cij(Z(tX)) > 0 for t ≫0. This completes the proof. Proposition 5.8. Strong positive definiteness and c > 0 are preserved under taking subspaces. Proof. This follows from Corollaries 4.18 and 4.19, which express c′ and w′ for a subspace Y = X \\ {x} in terms of c and w of the metric space X, or by repeated application of this argument following the composition property of Schur complements (Proposition 4.15). 5.2 Magnitude submodularity This section deals with valuative properties of magnitude. We show that two magnitude-based set functions on the subspaces of a strongly positive definite metric space X are submodular; recall that a function f on the subsets of X is called strictly submodular if it satisfies4 f(Y ) -f(Y \\{y}) 2, and start with the latter. (#X > 2): We can rewrite the submodularity inequalities for f as -|X|-1 + |X\\{x}|-1 wx/|X|, which confirms the inequality above. (#X = 2): We can rewrite the submodularity inequality for f as -|X|-1 + |X\\{x}|-1 < -|X\\{y}|-1 -α ⇐⇒1 -1 + exp(-d(x, y)) 2 < -1 -α (Example 1.1) ⇐⇒α < -3 2 This confirms strict submodularity for #X = 2 if α < -2/3, and thus completes the proof for strongly positive definite X. By Corollary 4.18, we know that f is increasing for nonempty subsets and that f(Y ) ≥-1 with equality for #Y = 1; it follows that f is increasing on 2X if α < -1. The asymptotic result follows from Proposition 5.7. Equivalently, one can say that -f in Theorem 5.9 is strictly supermodular and decreasing, for appropriate α. To end this section, we prove a second submodularity result. The function considered in the theorem below is inspired by the asymptotic expression in Theorem 3.1 for q(tX) = #X -|X| in terms of R(St)2. Theorem 5.10. Let X be any metric space and t ≫0. Then the function f : Y 7-→ ( m-|tY | m2 + m-1 m if m := #Y ̸= 0 α if Y = ∅ is increasing if α < 1 2 and strictly submodular if α < -1 2. Proof. We follow the strategy from the proof of Proposition 5.7 and represent a metric space X by its invertible similarity matrix Z ∈GLn(R). We consider three families of functions: the family FY,x,y, labeled by x ̸= y ∈Y ⊆X, the family Gx,y labeled by x ̸= y ∈X, and the family HY ′,Y labeled by Y ′ ⊂Y ⊆X; these are defined as FY,x,y = f(Y ) -f(Y \\{x}) -f(Y \\{y}) + f(Y \\{x, y}) Gx,y = f({x, y}) -f({x}) -f({y}) + f(∅) HY ′,Y = f(Y ′) -f(Y ) Since |Y | = -det 0 1T 1 ZY Y / det(Z) by Proposition 4.13, these are continuous functions on GLn(R). Next, we confirm that each of these functions evaluates negatively for the discrete metric space; in that case, Z = I and thus |Y | = m and f(Y ) = (m -1)/m. We find FY,x,y(I) = m-1 m -2m-2 m-1 + m-3 m-2 = -2 m(m-1)(m-2) < 0 Gx,y(I) = 1 2 + α HY ′,Y (I) = ( m′-1 m′ -m-1 m = m′-m m·m′ < 0 if Y ′ ̸= ∅ α -m-1 m ≤α -1 2 if Y ′ = ∅. 16 Thus F(I) < 0, G(I) < 0 if α < -1 2 and H(I) < 0 if α < 1 2; fix a choice of α. By continuity of F, G, H and negativity at the identity matrix I, there is a neighbourhood U ⊂GLn(R) of I, such that F, G, H are negative on each Z ∈U. Since Z(tX) →I as t →∞, we thus know that Z(tX) ∈U for t ≫0 and thus that F, G, H are negative. These are the strict submodularity inequalities (F, G) and the increasing inequalities (H), so this completes the proof. Acknowledgments I thank Mark Meckes for helpful discussions and encouragements. Part of this article was written during a research stay at the Max Planck . For the purpose of open access, the author has applied a CC BY public copyright licence to any author accepted manuscript arising from this submission. References [1] M. F. Adamer, E. D. Brouwer, L. O'Bray, and B. Rieck. The magnitude vector of images. Journal of Applied and Computational Topology, 8(2):251-274, 2024. 1 [2] K. Devriendt. Graph geometry from effective resistances. PhD thesis, 2022. 3, 9, 12, 14 [3] K. Devriendt and R. Lambiotte. 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Journal of Graph Theory, 103:415-436, 2022. 14 Authors' address: Karel Devriendt, 18"}}},{"rowIdx":296,"cells":{"id":{"kind":"string","value":"2510.14692"},"raw":{"kind":"string","value":"Unique Hierarchical Rotational Dynamics Induces Ultralow Lattice Thermal\nConductivity in Cyanide-bridged Framework Materials\nZhunyun Tang,1 Xiaoxia Wang,1 Jin Li,1 Chaoyu He,1 Mingxing Chen,2 Chao Tang,1, ∗and Tao Ouyang1, †\n1School of Physics and Optoelectronics & Hunan Provincial Key Laboratory of\nComputational Condensed Matter Physics and Quantum Materials Engineering,\nXiangtan University, Xiangtan 411105, Hunan, China\n2School of Physics and Electronics, Hunan Normal University, Changsha 410081, Hunan, China\n(Dated: October 17, 2025)\nThe pursuit of materials combining light constituent elements with ultralow lattice thermal con-\nductivity (κL) is crucial to advancing technologies like thermoelectrics and thermal barrier coatings,\nyet it remains a formidable challenge to date. Herein, we achieve ultralow κL in lightweight cyanide-\nbridged framework materials (CFMs) through the rational integration of properties such as the\nhierarchical vibrations exhibited in superatomic structures and rotational dynamics exhibited in\nperovskites. Unique hierarchical rotation behavior leads to multiple negative peaks in Gr¨uneisen\nparameters across a wide frequency range, thereby inducing pronounced negative thermal expan-\nsion and strong cubic anharmonicity in CFMs.\nMeanwhile, the synergistic effect between large\nfour-phonon scattering phase space (induced by phonon quasi-flat bands and wide bandgaps) and\nstrong quartic anharmonicity (associated with rotation modes) leads to giant quartic anharmonic\nscattering rates in these materials. Consequently, the κL of these CFMs decreases by one to two\norders of magnitude compared to the known perovskites or perovskite-like materials with equiva-\nlent average atomic masses. For instance, the Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 (X\n= B, Al, Ga, In) exhibit ultralow room-temperature κL values ranging from 0.35 to 0.81 W/mK.\nThis work not only establishes CFMs as a novel and rich platform for studying extreme phonon\nanharmonicity, but also provides a new paradigm for achieving ultralow thermal conductivity in\nlightweight materials via the conscious integration of hierarchical and rotational dynamics.\nI.\nINTRODUCTION\nThe pursuit of materials with ultralow lattice thermal\nconductivity (κL) is a central endeavor in condensed mat-\nter physics and materials science, driven by applications\nin thermoelectrics, thermal insulation, and barrier coat-\nings [1–4].\nConventional strategies for suppressing κL\noften rely on the incorporation of heavy constituent el-\nements or structural complexity [5–8], which typically\nconflicts with the demand for lightweight functional ma-\nterials in applications ranging from aerospace compo-\nnents to flexible electronics.\nConsequently, the explo-\nration of lightweight materials that simultaneously ex-\nhibit ultralow κL remains a formidable goal, one that\nrequires moving beyond conventional phonon scattering\nparadigms toward designs that leverage emergent lattice\ndynamics and anharmonicity.\nRecent advances highlight the role of specific bonding\ncharacteristics and dynamic structural features in driv-\ning phonon anharmonicity and collapse of thermal trans-\nport [9–16].\nSpecifically, the presence of anti-bonding\nvalence states leads to bond weakening and formation of\nsoft optical phonons, resulting in intrinsic phonon scat-\ntering rates far exceeding those in conventional semicon-\nductors [9, 10]. Alternatively, the concept of hierarchical\nvibrational architectures (inspired by superatomic crys-\ntals) has emerged as a potent design strategy [11]. Hier-\n∗tang chao@xtu.edu.cn.\n† ouyangtao@xtu.edu.cn.\narchical vibrations endow additional rotational or trans-\nlational degrees of freedom, leading to strong coupling\nbetween internal optical phonons and acoustic modes,\nfurther generating extreme phonon anharmonic scatter-\ning [12–16]. In a parallel development, it has been found\nthat low-energy optical modes associated with octahedral\nrotations in perovskites (perovskite-like materials) often\nexhibit strong anharmonicity, leading to low κL and pro-\nnounced wave-like tunneling behavior [17–21].\nAlthough these mechanisms have been individually es-\ntablished in disparate material classes, a fundamental\nquestion remains: can these distinct phonon-suppressing\nfeatures be intentionally combined within a lightweight\nmaterial system to achieve ultralow thermal conductivity\nthrough synergistic effects? Based on this design princi-\nple, we perform a systematical search in the Inorganic\nCrystal Structure Database (ICSD) and discover that\ncyanide-bridged framework materials (CFMs) present an\nexceptionally promising platform for such an integrated\napproach [22].\nTheir framework structures, compris-\ning M—C≡N—M′ linkages, the flexibility of the cyanide\nbridge allows for dynamic disorder and local distortions,\nreminiscent of perovskites, while the subunits within the\nframework suggests a pathway toward hierarchical vi-\nbration [23–25]. Despite these intriguing features, their\nthermal transport properties remain largely unexplored,\nparticularly in relation to the interplay among local ro-\ntational degrees of freedom, hierarchical vibrations, and\ncollective low-energy dynamics.\nIn this study, we fill this gap by employing first-\nprinciples calculations combined with machine learning\narXiv:2510.14692v1 [cond-mat.mtrl-sci] 16 Oct 2025\n\n2 of 10\nII\nRESULTS AND DISCUSSION\npotentials.\nTo ensure reliability, the thermal conduc-\ntivity values are cross-validated through unified phonon\ntheory and large-scale molecular dynamics simulations.\nThe results demonstrate that this family of materials\nuniquely combines the key phonon-suppressing mecha-\nnisms previously identified in separate systems: the hier-\narchical vibrational architecture characteristic of super-\natomic crystals and the rotational dynamics typical of\nperovskites. This cooperative mechanism leads to emer-\ngent phenomena, including multiple negative Gr¨uneisen\nparameter peaks across a broad frequency range, signal-\ning pronounced negative thermal expansion and strong\ncubic anharmonicity. Most notably, we identify a syn-\nergistic interplay between the large four-phonon scatter-\ning phase space (arising from phonon quasi-flat bands\nand wide bandgaps) and the strong quartic anharmonic-\nity (associated with rotation modes), leading to giant\nfour-phonon scattering rates that dominate thermal re-\nsistance. As a result, the κL of these CFMs is suppressed\nby one to two orders of magnitude compared to the\npreviously known perovskite and perovskite-like mate-\nrials with equivalent average atomic mass. Specific com-\npounds, such as Cd(CN)2, NaB(CN)4, LiIn(CN)4, and\nAgX(CN)4 (X = B, Al, Ga, In), exhibit ultralow room-\ntemperature κL values ranging from 0.35 to 0.81 W/mK,\npositioning them among the lowest known thermal con-\nductivity materials for their weight class. These findings\nprovide a promising strategy for designing lightweight\nmaterials with ultralow thermal conductivity.\nII.\nRESULTS AND DISCUSSION\nFigure 1(a) illustrates a simplified model of cubic-\nphase atomic structure, where four distinct Wyckoff po-\nsitions are represented by differently colored spheres.\nWhen positions 1 and 2 are occupied by specific tetra-\nhedral superatomic clusters ZnC4 or CdC4, and posi-\ntions 3 and 4 by ZnN4 or CdN4 clusters, these units\nconnect through —C≡N— bridges to form the classic\nZn(CN)2 or Cd(CN)2 framework materials with P−43m\nspace group, as depicted in Fig. 1(b). If positions 1 and\n2 are occupied by BC4 clusters and positions 3 and 4\nby AgN4, CuN4, or LiN4 clusters, the resulting structure\ncorresponds to AgB(CN)4, CuB(CN)4, or LiB(CN)4, re-\nspectively, all crystallizing in the P−43m space group,\nas shown in Fig. 1(c).\nIn contrast, when positions 1\nand 4 are occupied by BC4 clusters and positions 2 and\n3 by NaN4 clusters, the framework adopts the Fd−3m\nspace group, characteristic of NaB(CN)4, see Fig. 1(d).\nPrevious experimental and theoretical studies have ex-\ntensively explored the thermal expansion properties of\nthese cyanide-bridged framework materials (CFMs) [25–\n30]. The results reveal that these compounds generally\nexhibit negative thermal expansion (NTE) behavior. Fig-\nure S2 of the Supporting Information (SI) demonstrates\nthat the calculated NTE behavior exhibits markedly im-\nproved agreement with experimental data when the van\nder Waals (vdW) correction term is included in molecular\ndynamics (MD) simulations. This indicates that inter-\ncluster vdW interactions are crucial for accurately cap-\nturing the lattice dynamics and anharmonicity that drive\nthe pronounced NTE in these cyanide frameworks. No-\ntably, despite their low average atomic mass, Zn(CN)2,\nCd(CN)2, AgB(CN)4, and NaB(CN)4 hold exceptionally\npronounced NTE effects, surpassing most other known\nNTE materials [31].\nSuch remarkable behavior fore-\nshows the presence of intriguing lattice dynamics in these\nCFMs. Therefore, we select Zn(CN)2, AgB(CN)4, and\nNaB(CN)4 as representative systems and further com-\nputed their phonon spectra. As illustrated in Figs. 1(e-\ng), the presence of light elements leads to remarkably\nhigh optical phonon cutoff frequencies.\nThe hierarchi-\ncal vibrational behavior inherent to superatomic struc-\ntures gives rise to a plethora of localized phonon modes,\ninducing numerous phonon quasi-flat bands and wide\nbandgaps. Moreover, certain phonon modes exhibit pro-\nnounced temperature-dependent behavior, particularly\nthe optical branches around 2–6 THz and 10 THz, sug-\ngesting strong anharmonic effects in these specific modes.\nIn conventional perovskite or perovskite-like NTE ma-\nterials, great NTE coefficients mainly originate from\nlarge negative Gr¨uneisen parameters associated with ex-\ntremely low-frequency soft modes linked to octahedral\nrotations [21, 32, 33]. However, these traditional NTE\nmaterials typically exhibit only a limited set of rotation\nphonon modes due to constrained rotational degrees of\nfreedom. In contrast, the unique hierarchical vibrations\nconfer additional rotational freedom in CFMs, resulting\nin richer rotation phonon modes, as depicted in the insets\nof Fig. 2 (and Fig. S7 of the SI). The positions of these\nrotation modes are marked on the phonon dispersion in\nFig. S8 of the SI, with AgB(CN)4 as an example. The\ndistinctive hierarchical rotation behavior leads to multi-\nple negative peaks in the Gr¨uneisen parameter (γ) across\na broad frequency range, as shown in the middle pan-\nels of Figs. 1(e-g). This widespread negative γ distribu-\ntion significantly enhances the overall cubic anharmonic-\nity and is a key driver of the pronounced NTE observed in\nthese CFMs. Moreover, by mapping the potential energy\ncurves along the coordinates of these rotation modes, one\ncan find that the actual potential deviates significantly\nfrom the harmonic approximation (quadratic fitting), as\ndemonstrated in Figs. 2(a–h).\nIt is worth noting that\nthe fitted curves agree well with the DFT data points\nwhen the quartic term is introduced.\nThis deviation\nprovides direct evidence of strong quartic anharmonic-\nity associated with these rotation modes. Additionally,\nthe phonon quasi-flat bands and wide bandgaps prevalent\nin these materials generally expand the available phase\nspace for four-phonon scattering processes [34, 35], as\nshown in Fig. S9 of the SI. As a result, the synergistic\neffect between large four-phonon scattering phase space\nand intrinsically strong quartic anharmonicity leads to\ngiant four-phonon anharmonic scattering rates (SRs) in\nthese materials, as quantitatively shown in the right pan-\n2\n\n3 of 10\nII\nRESULTS AND DISCUSSION\nFIG. 1. (a) A simplified model of cubic crystal structure, with each sphere consisting of a tetrahedral superatom cluster.\nScheme of unit cell structure of (b) Zn(CN)2 and Cd(CN)2, (c) LiB(CN)4, CuB(CN)4, and AgB(CN)4, (d) and NaB(CN)4.\n(e-g) Calculated finite-temperature phonon dispersion, accounting for thermal expansion effects, for Zn(CN)2, AgB(CN)4, and\nNaB(CN)4, respectively. The middle and right panels of (e-g) present the mode γ and SRs (including three- and four-phonon\nprocesses) at room temperature, respectively.\nels of Figs. 1(e-g).\nWe now turn our attention to the thermal transport\nproperties of these CFMs. First, the lattice thermal con-\nductivity (κL) of Zn(CN)2, AgB(CN)4, and NaB(CN)4\nin the temperature range of 100 to 400 K is obtained\nbased on the self-consistent phonon (SCP) approach com-\nbined with the unified phonon transport theory [36, 37],\naccounting of the thermal expansion effect.\nThe re-\nsults are decomposed into particle-like (κP\nL) and wave-\nlike coherence (κC\nL) contributions, with and without four-\nphonon (4ph) scattering. All three materials exhibit low\nκL values, which is consistent with the strong anhar-\nmonicity and enhanced scattering discussed above.\nIn\nZn(CN)2, the inclusion of four-phonon processes dras-\ntically reduces κP\nL from 4.84 (3ph only; κP\n3ph) to 1.74\nW/mK (3+4ph; κP\n3+4ph) at 300 K, with a 64% sup-\npression [Fig. 3(a)]. Such suppression is even more pro-\nnounced in AgB(CN)4 and NaB(CN)4, where κP\nL is re-\nduced by 75% (from 1.35 to 0.35 W/mK) and 90% (from\n2.15 to 0.22 W/mK), respectively [Figs. 3(b-c)]. The co-\nherence contribution κC\n3+4ph remains minor in Zn(CN)2,\naccounting for only\n10% of the total κP+C\n3+4ph at 300\nK. However, in AgB(CN)4 and NaB(CN)4, κC\n3+4ph con-\ntributes significantly to the total thermal conductivity,\neven exceeding κP\n3+4ph at elevated temperatures. Con-\nsequently, κL in the two systems is increasingly domi-\nnated by wave-like phonon tunneling, leading to a weak\ntemperature dependence.\nRemarkably, NaB(CN)4 ex-\nhibits a positive temperature dependence of κL at high\ntemperatures, a behavior previously observed in other\nultralow κL materials [38–41].\nThis is mainly because\nthe stronger anharmonic scattering and smaller average\nphonon band spacing in NaB(CN)4 induce more phonon\nmodes with lifetimes below the Wigner limit [42] (see Fig.\n3\n\n4 of 10\nII\nRESULTS AND DISCUSSION\nFIG. 2. The potential well for different rotation modes. The quadratic and quartic fitting curves are represented by gray dashed\nlines and yellow solid lines, respectively. The insets label the vibration eigenvectors for different rotation modes\nS10 of the SI), resulting in glass-like thermal transport.\nHowever, we also observe that certain extreme phonon\nmodes of NaB(CN)4 enter the overdamped regime at\nhigh temperatures (Fig.\nS10 of the SI), which might\nrender the standard phonon gas model invalid [43, 44].\nTherefore, we further employ homogeneous nonequilib-\nrium molecular dynamics (HNEMD) simulations to pro-\nvide a more robust description for thermal transport\n[45]. The HNEMD results are in good agreement with\nthe unified transport theory calculations at low tempera-\ntures, but reveal a continuing decrease in κL at high tem-\nperatures, underscoring the limitations of the standard\nphonon gas model in giant anharmonic systems. Finally,\nwe extend HNEMD simulations to 15 CFMs, including\nZn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4, CuX(CN)4,\nand AgX(CN)4 (X=B, Al, Ga, In). As summarized in\nFig. 4, thermal transport is further suppressed as the\nX atom transitions from B to In.\nSeveral compounds\n[Cd(CN)2, NaB(CN)4, LiIn(CN)4, and all AgX(CN)4\nmembers] exhibit ultralow room-temperature values be-\ntween 0.35 and 0.81 W/mK. Notably, NaX(CN)4 (X=Al,\nGa, In) are excluded due to their complex thermal ex-\npansion behavior (possibly involving multiple structural\nphase transitions), as shown in Fig. S3 of the SI, which is\nbeyond the scope of this study. In contrast, AgIn(CN)4\nshows monotonic positive thermal expansion at low tem-\nperatures, but transitions to negative thermal expan-\nsion above 200 K. This appears to arise from the low-\ntemperature instability of the cubic phase, as evidenced\nby imaginary phonon modes below 200 K that vanish\nat higher temperatures [Fig. S11 of the SI], mirroring\nbehavior observed in other cubic frameworks [18, 46].\nTo elucidate the microscopic mechanism underlying\nthe gradual suppression of thermal conductivity as the\nX atom transitions from B to In in these CFMs, we an-\nalyze the evolution of chemical bonding and its impact\non lattice anharmonicity. Figure 5(a) presents the elec-\ntron localization function (ELF) for AgX(CN)4 (X=B,\nAl, Ga, In). The intense localization between B and C\natoms in AgB(CN)4 signifies strong bonding character of\nthe B–C bond. However, as B transitions to In, this local-\nization gradually diminishes, suggesting that the bond-\ning strength of X–C is progressively weakened. To fur-\nther quantify the bonding strength, the integrated crys-\ntal orbital Hamilton population (iCOHP) is calculated\n[47], a measure of bond strength, for key atomic pairs,\nas shown in inset of Fig. 5(b). The –iCOHP values for\nthe Ag–N and C–N bonds remain essentially constant\nthroughout the series. However, the –iCOHP for the X–C\nbond decreases substantially from B to In, confirming a\nsystematic chemical bond weakening. As previously es-\ntablished, the strong anharmonicity in these materials is\nintimately linked to rotation modes, which primarily in-\nvolve large-amplitude transverse vibrations of the N and\n4\n\n5 of 10\nIII\nCONCLUSION\nFIG. 3. The temperature-dependent κL of (a) Zn(CN)2, (b)\nAgB(CN)4, and (c) NaB(CN)4 is calculated based on different\nmodels are compared with molecular dynamics simulations.\nC atoms. The root-mean-square displacements (RMSD)\nof C and N atoms, calculated at 300 K, are presented in\nFig. 5(b). The RMSD values are significantly enhanced\nin AgIn(CN)4 compared to AgB(CN)4, indicating more\npronounced vibrational amplitudes and thus larger dy-\nnamic tetrahedral distortion. This enhanced flexibility\ndirectly leads to stronger lattice anharmonicity, which is\nquantitatively confirmed by the markedly increased neg-\native Gr¨uneisen parameters in AgIn(CN)4 [see Fig. 5(c)].\nConsequently, AgIn(CN)4 exhibits more pronounced neg-\native thermal expansion (Fig. S3 of the SI) and stronger\nphonon scattering rates (SRs) [Fig. 5(d)], thereby further\nsuppression of thermal transport properties.\nFinally, we plot the room-temperature thermal con-\nductivity (κL) of these CFMs as a function of average\natomic mass ( ¯\nM) in Fig. 6, alongside data for classic\nperovskites and perovskite-like materials for comparison\n[17, 19–21, 38, 48–64].\nObviously, the\n¯\nM–κL trend of\nCFMs deviates significantly from that of perovskite ma-\nterials. Despite possessing relatively low ¯\nM, the CFMs\nstill exhibit ultralow κL, roughly 1-2 orders of magnitude\nlower than those of perovskites with comparable ¯\nM.\nFIG. 4.\nThe temperature-dependent κL calculated using\nthe HNEMD method for (a) Zn(CN)2 and Cd(CN)2, (b)\nAgX(CN)4, (c) CuX(CN)4, and (d) LiX(CN)4 (X=B, Al, Ga,\nIn).\nFIG. 5. (a) The ELF, (b) RMSD of N and C atoms, (c) γ,\nand (d) phonon SRs for AgX(CN)4 (X=B, Al, Ga, In). The\ninset of (b) shows the –iCOHP for different atomic bonds.\nIII.\nCONCLUSION\nIn summary, we have unveiled the microscopic ori-\ngin of ultralow thermal conductivity (κL) in lightweight\ncyanide-bridged framework materials (CFMs), challeng-\n5\n\n6 of 10\nIV\nMETHODS\nFIG. 6. The dependence of room temperature κL on average\nmass.\nThe red spheres mark the 15 CFMs studied in this\npaper. The blue squares mark some typical perovskite and\nperovskite-like materials.\ning the conventional trade-off between low average\natomic mass and efficient thermal transport.\nOur re-\nsults reveal that the suppressed thermal transport be-\nhavior in these materials stems from exceptionally strong\nthree- and four-phonon anharmonic scattering, driven\nby abundant hierarchical rotation modes.\nRemark-\nably, four-phonon scattering rates even surpass those\nof three-phonon processes for numerous modes, which\nsignificantly suppresses thermal transport.\nSuch be-\nhavior originates from the large four-phonon scatter-\ning phase space induced by phonon quasi-flat bands\nand wide bandgaps as well as the strong quartic an-\nharmonicity accompanied by rotation modes. Moreover,\ncoherent thermal conductivity contributes significantly\nin AgB(CN)4 and NaB(CN)4, with its contribution ex-\nceeding that of the particle-like term at elevated tem-\nperatures.\nThis leads to a positive temperature de-\npendence of thermal conductivity in NaB(CN)4 at high\ntemperature regions. However, certain extreme phonon\nmodes of NaB(CN)4 enter the overdamped regime at\nhigh temperatures, which render the standard phonon\ngas model invalid. Therefore, we further employ homo-\ngeneous nonequilibrium molecular dynamics (HNEMD)\nsimulations to provide a more robust description for\nthermal transport.\nThe HNEMD simulations confirm\nthe unified transport theory results at low tempera-\ntures but reveal a continuing decrease in κL at high\ntemperatures, underscoring the limitations of the stan-\ndard phonon gas model in giant anharmonic systems.\nFinally, we extend HNEMD simulations to 15 CFMs,\nincluding Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4,\nCuX(CN)4, and AgX(CN)4 (X=B, Al, Ga, In).\nAs\nthe X atom transitions from B to In, the corresponding\nsystem exhibits weaker bonding and greater root-mean-\nsquare displacement, thereby inducing stronger anhar-\nmonic scattering and lower κL. Among them, Cd(CN)2,\nNaB(CN)4, LiIn(CN)4, and AgX(CN)4 exhibit ultralow\nroom-temperature κL values ranging from 0.35 to 0.81\nW/mK. These results shed light on the physical mecha-\nnisms of thermal transport in CFMs, and suggest a prac-\nticable strategy for developing novel lightweight materi-\nals with ultralow thermal conductivity.\nIV.\nMETHODS\nAll density-functional theory (DFT) calculations are\nimplemented using the Vienna Ab initio Simulation\nPackage (VASP) [65] with the projector augmented wave\n(PAW) [66] method. The PBE [67] exchange-correlation\nfunctional is used to relax the geometric structure. The\nkinetic energy cutoff for wave function is set to 500 eV,\nand the k-mesh of 8 × 8 × 8 is used to sample the Bril-\nlouin Zone. The DFT-D3 correction [68] is adopted to ac-\ncount for van der Waals (vdW) interactions. The ALAM-\nODE package is used to compute both the harmonic\nsecond-order interatomic force constants (IFCs) via fi-\nnite displacement method and the anharmonic higher-\norder IFCs through compressive sensing lattice dynamics\n[36, 50, 69]. High precision multi-atomic cluster expan-\nsion (MACE) machine learning potential is constructed\nto capture IFCs more efficiently [70].\nSelf-consistent\nphonon (SCP) [36, 50] calculation is performed to obtain\nthe temperature-dependent phonon dispersion and corre-\nsponding second-order IFCs. The lattice thermal conduc-\ntivity is calculated by a unified thermal transport theory\nand implemented in the modified FOURPHONON soft-\nware [37, 71–73]. The equation for κL is as follows\nκP/C\nL\n=\nℏ2\nkBT 2V N\nX\nq\nX\nj,j′\nωqj + ωqj′\n2\nνqjj′ ⊗νqjj′\n×ωqjnqj(nqj + 1) + ωqj′nqj′ (nqj′ + 1)\n4 (ωqj −ωqj′)2 + (Γqj + Γqj′)2\n(Γqj + Γqj′) ,\n(1)\nwhere the superscripts P and C represent the contribu-\ntions from populations (particle-like propagation) and\ncoherences (wave-like phonon tunneling), respectively.\nWhen j=j′, Eq. 1 is equivalent to the Peierls-Boltzmann\ntransport equation, i.e., the particle-like transport term.\nOtherwise, it corresponds to the wave-like transport\nterm.\nMoreover, the κL is also obtained using homo-\ngeneous nonequilibrium molecular dynamics (HNEMD)\nsimulations based on the trained neuroevolution poten-\ntial (NEP) and implemented in the GPUMD software\n[45, 74, 75]. To eliminate the size effects, the large su-\npercell exceeding 85 × 85 × 85 ˚A3 (containing 40960\natoms) is adopted. Corresponding supercell is also ap-\nplied to simulate the thermal expansion behavior. More\ncomputational details are provided in the Supporting In-\nformation (SI).\n6\n\n7 of 10\nIV\nMETHODS\nASSOCIATED CONTENT\nSupporting Information The Supporting Information\nis available free of charge at ***\nDetailed method description, lattice constant, root-\nmean-square error of machine learning potential models,\ntraining parameters for fitting machine learning potential\nmodels, thermal expansion, phonon spectrum, rotation\nmode eigenvectors, position markers of rotation modes\nwithin the phonon spectrum, weighted scattering phase\nspace, and phonon lifetime.\nDATA AVAILABILITY\nThe data supporting the findings of this study are\navailable within this article and its Supporting Infor-\nmation.\nThe trained NEP and MACE models are\nopenly available in the repository https://github.com/\nZhunyunTang/CFMs-ML.\nAdditional data are available\nfrom the corresponding author on reasonable request.\nCODE AVAILABILITY\nThe source code for GPUMD is available at https:\n//github.com/brucefan1983/GPUMD,\nfor ALAMODE\nis available at https://github.com/ttadano/alamode,\nfor SHENGBTE is available at https://www.shengbte.\norg,\nand\nFOURPHONON\nis\navailable\nat\nhttps:\n//github.com/FourPhonon/FourPhonon,\nfor\nPYNEP\nis\navailable\nat\nhttps://github.com/bigd4/PyNEP,\nfor\nthe\nmodified\nversion\nof\nFOURPHONON\nis\navailable\nat\nhttps://github.com/ZhunyunTang/\nFourPhonon--wigner-sampling.\nACKNOWLEDGMENTS\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No.\n52372260), the\nScience Fund for Distinguished Young Scholars of Hu-\nnan Province of China (No.\n2024JJ2048), the Hunan\nProvincial Innovation Foundation for Postgraduate (No.\nCX20240058), and the Youth Science and Technology\nTalent Project of Hunan Province (No.\n2022RC1197).\nThe calculation in this work is partly supported by\nthe high-performance computing platform of school of\nphysics and optoelectronics.\nAUTHOR CONTRIBUTIONS\nZ.T. contributed to the methodology, software, all cal-\nculations, formal analysis, visualization, and writing of\noriginal draft. 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Herein, we achieve ultralow κL in lightweight cyanidebridged framework materials (CFMs) through the rational integration of properties such as the hierarchical vibrations exhibited in superatomic structures and rotational dynamics exhibited in perovskites. Unique hierarchical rotation behavior leads to multiple negative peaks in Gr ̈uneisen parameters across a wide frequency range, thereby inducing pronounced negative thermal expansion and strong cubic anharmonicity in CFMs. Meanwhile, the synergistic effect between large four-phonon scattering phase space (induced by phonon quasi-flat bands and wide bandgaps) and strong quartic anharmonicity (associated with rotation modes) leads to giant quartic anharmonic scattering rates in these materials. Consequently, the κL of these CFMs decreases by one to two orders of magnitude compared to the known perovskites or perovskite-like materials with equivalent average atomic masses. For instance, the Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 (X = B, Al, Ga, In) exhibit ultralow room-temperature κL values ranging from 0.35 to 0.81 W/mK. This work not only establishes CFMs as a novel and rich platform for studying extreme phonon anharmonicity, but also provides a new paradigm for achieving ultralow thermal conductivity in lightweight materials via the conscious integration of hierarchical and rotational dynamics. I. INTRODUCTION The pursuit of materials with ultralow lattice thermal conductivity (κL) is a central endeavor in condensed matter physics and materials science, driven by applications in thermoelectrics, thermal insulation, and barrier coatings [1-4]. Conventional strategies for suppressing κL often rely on the incorporation of heavy constituent elements or structural complexity [5-8], which typically conflicts with the demand for lightweight functional materials in applications ranging from aerospace components to flexible electronics. Consequently, the exploration of lightweight materials that simultaneously exhibit ultralow κL remains a formidable goal, one that requires moving beyond conventional phonon scattering paradigms toward designs that leverage emergent lattice dynamics and anharmonicity. Recent advances highlight the role of specific bonding characteristics and dynamic structural features in driving phonon anharmonicity and collapse of thermal transport [9-16]. Specifically, the presence of anti-bonding valence states leads to bond weakening and formation of soft optical phonons, resulting in intrinsic phonon scattering rates far exceeding those in conventional semiconductors [9, 10]. Alternatively, the concept of hierarchical vibrational architectures (inspired by superatomic crystals) has emerged as a potent design strategy [11]. Hier- ∗tang . † . archical vibrations endow additional rotational or translational degrees of freedom, leading to strong coupling between internal optical phonons and acoustic modes, further generating extreme phonon anharmonic scattering [12-16]. In a parallel development, it has been found that low-energy optical modes associated with octahedral rotations in perovskites (perovskite-like materials) often exhibit strong anharmonicity, leading to low κL and pronounced wave-like tunneling behavior [17-21]. Although these mechanisms have been individually established in disparate material classes, a fundamental question remains: can these distinct phonon-suppressing features be intentionally combined within a lightweight material system to achieve ultralow thermal conductivity through synergistic effects? Based on this design principle, we perform a systematical search in the Inorganic Crystal Structure Database (ICSD) and discover that cyanide-bridged framework materials (CFMs) present an exceptionally promising platform for such an integrated approach [22]. Their framework structures, comprising M-C≡N-M′ linkages, the flexibility of the cyanide bridge allows for dynamic disorder and local distortions, reminiscent of perovskites, while the subunits within the framework suggests a pathway toward hierarchical vibration [23-25]. Despite these intriguing features, their thermal transport properties remain largely unexplored, particularly in relation to the interplay among local rotational degrees of freedom, hierarchical vibrations, and collective low-energy dynamics. In this study, we fill this gap by employing firstprinciples calculations combined with machine learning 16 Oct 2025 2 of 10 II RESULTS AND DISCUSSION potentials. To ensure reliability, the thermal conductivity values are cross-validated through unified phonon theory and large-scale molecular dynamics simulations. The results demonstrate that this family of materials uniquely combines the key phonon-suppressing mechanisms previously identified in separate systems: the hierarchical vibrational architecture characteristic of superatomic crystals and the rotational dynamics typical of perovskites. This cooperative mechanism leads to emergent phenomena, including multiple negative Gr ̈uneisen parameter peaks across a broad frequency range, signaling pronounced negative thermal expansion and strong cubic anharmonicity. Most notably, we identify a synergistic interplay between the large four-phonon scattering phase space (arising from phonon quasi-flat bands and wide bandgaps) and the strong quartic anharmonicity (associated with rotation modes), leading to giant four-phonon scattering rates that dominate thermal resistance. As a result, the κL of these CFMs is suppressed by one to two orders of magnitude compared to the previously known perovskite and perovskite-like materials with equivalent average atomic mass. Specific compounds, such as Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 (X = B, Al, Ga, In), exhibit ultralow roomtemperature κL values ranging from 0.35 to 0.81 W/mK, positioning them among the lowest known thermal conductivity materials for their weight class. These findings provide a promising strategy for designing lightweight materials with ultralow thermal conductivity. II. RESULTS AND DISCUSSION Figure 1(a) illustrates a simplified model of cubicphase atomic structure, where four distinct Wyckoff positions are represented by differently colored spheres. When positions 1 and 2 are occupied by specific tetrahedral superatomic clusters ZnC4 or CdC4, and positions 3 and 4 by ZnN4 or CdN4 clusters, these units connect through -C≡N- bridges to form the classic Zn(CN)2 or Cd(CN)2 framework materials with P-43m space group, as depicted in Fig. 1(b). If positions 1 and 2 are occupied by BC4 clusters and positions 3 and 4 by AgN4, CuN4, or LiN4 clusters, the resulting structure corresponds to AgB(CN)4, CuB(CN)4, or LiB(CN)4, respectively, all crystallizing in the P-43m space group, as shown in Fig. 1(c). In contrast, when positions 1 and 4 are occupied by BC4 clusters and positions 2 and 3 by NaN4 clusters, the framework adopts the Fd-3m space group, characteristic of NaB(CN)4, see Fig. 1(d). Previous experimental and theoretical studies have extensively explored the thermal expansion properties of these cyanide-bridged framework materials (CFMs) [2530]. The results reveal that these compounds generally exhibit negative thermal expansion (NTE) behavior. Figure S2 of the Supporting Information (SI) demonstrates that the calculated NTE behavior exhibits markedly improved agreement with experimental data when the van der Waals (vdW) correction term is included in molecular dynamics (MD) simulations. This indicates that intercluster vdW interactions are crucial for accurately capturing the lattice dynamics and anharmonicity that drive the pronounced NTE in these cyanide frameworks. Notably, despite their low average atomic mass, Zn(CN)2, Cd(CN)2, AgB(CN)4, and NaB(CN)4 hold exceptionally pronounced NTE effects, surpassing most other known NTE materials [31]. Such remarkable behavior foreshows the presence of intriguing lattice dynamics in these CFMs. Therefore, we select Zn(CN)2, AgB(CN)4, and NaB(CN)4 as representative systems and further computed their phonon spectra. As illustrated in Figs. 1(eg), the presence of light elements leads to remarkably high optical phonon cutoff frequencies. The hierarchical vibrational behavior inherent to superatomic structures gives rise to a plethora of localized phonon modes, inducing numerous phonon quasi-flat bands and wide bandgaps. Moreover, certain phonon modes exhibit pronounced temperature-dependent behavior, particularly the optical branches around 2-6 THz and 10 THz, suggesting strong anharmonic effects in these specific modes. In conventional perovskite or perovskite-like NTE materials, great NTE coefficients mainly originate from large negative Gr ̈uneisen parameters associated with extremely low-frequency soft modes linked to octahedral rotations [21, 32, 33]. However, these traditional NTE materials typically exhibit only a limited set of rotation phonon modes due to constrained rotational degrees of freedom. In contrast, the unique hierarchical vibrations confer additional rotational freedom in CFMs, resulting in richer rotation phonon modes, as depicted in the insets of Fig. 2 (and Fig. S7 of the SI). The positions of these rotation modes are marked on the phonon dispersion in Fig. S8 of the SI, with AgB(CN)4 as an example. The distinctive hierarchical rotation behavior leads to multiple negative peaks in the Gr ̈uneisen parameter (γ) across a broad frequency range, as shown in the middle panels of Figs. 1(e-g). This widespread negative γ distribution significantly enhances the overall cubic anharmonicity and is a key driver of the pronounced NTE observed in these CFMs. Moreover, by mapping the potential energy curves along the coordinates of these rotation modes, one can find that the actual potential deviates significantly from the harmonic approximation (quadratic fitting), as demonstrated in Figs. 2(a-h). It is worth noting that the fitted curves agree well with the DFT data points when the quartic term is introduced. This deviation provides direct evidence of strong quartic anharmonicity associated with these rotation modes. Additionally, the phonon quasi-flat bands and wide bandgaps prevalent in these materials generally expand the available phase space for four-phonon scattering processes [34, 35], as shown in Fig. S9 of the SI. As a result, the synergistic effect between large four-phonon scattering phase space and intrinsically strong quartic anharmonicity leads to giant four-phonon anharmonic scattering rates (SRs) in these materials, as quantitatively shown in the right pan2 3 of 10 II RESULTS AND DISCUSSION FIG. 1. (a) A simplified model of cubic crystal structure, with each sphere consisting of a tetrahedral superatom cluster. Scheme of unit cell structure of (b) Zn(CN)2 and Cd(CN)2, (c) LiB(CN)4, CuB(CN)4, and AgB(CN)4, (d) and NaB(CN)4. (e-g) Calculated finite-temperature phonon dispersion, accounting for thermal expansion effects, for Zn(CN)2, AgB(CN)4, and NaB(CN)4, respectively. The middle and right panels of (e-g) present the mode γ and SRs (including three- and four-phonon processes) at room temperature, respectively. els of Figs. 1(e-g). We now turn our attention to the thermal transport properties of these CFMs. First, the lattice thermal conductivity (κL) of Zn(CN)2, AgB(CN)4, and NaB(CN)4 in the temperature range of 100 to 400 K is obtained based on the self-consistent phonon (SCP) approach combined with the unified phonon transport theory [36, 37], accounting of the thermal expansion effect. The results are decomposed into particle-like (κP L) and wavelike coherence (κC L) contributions, with and without fourphonon (4ph) scattering. All three materials exhibit low κL values, which is consistent with the strong anharmonicity and enhanced scattering discussed above. In Zn(CN)2, the inclusion of four-phonon processes drastically reduces κP L from 4.84 (3ph only; κP 3ph) to 1.74 W/mK (3+4ph; κP 3+4ph) at 300 K, with a 64% suppression [Fig. 3(a)]. Such suppression is even more pronounced in AgB(CN)4 and NaB(CN)4, where κP L is reduced by 75% (from 1.35 to 0.35 W/mK) and 90% (from 2.15 to 0.22 W/mK), respectively [Figs. 3(b-c)]. The coherence contribution κC 3+4ph remains minor in Zn(CN)2, accounting for only 10% of the total κP+C 3+4ph at 300 K. However, in AgB(CN)4 and NaB(CN)4, κC 3+4ph contributes significantly to the total thermal conductivity, even exceeding κP 3+4ph at elevated temperatures. Consequently, κL in the two systems is increasingly dominated by wave-like phonon tunneling, leading to a weak temperature dependence. Remarkably, NaB(CN)4 exhibits a positive temperature dependence of κL at high temperatures, a behavior previously observed in other ultralow κL materials [38-41]. This is mainly because the stronger anharmonic scattering and smaller average phonon band spacing in NaB(CN)4 induce more phonon modes with lifetimes below the Wigner limit [42] (see Fig. 3 4 of 10 II RESULTS AND DISCUSSION FIG. 2. The potential well for different rotation modes. The quadratic and quartic fitting curves are represented by gray dashed lines and yellow solid lines, respectively. The insets label the vibration eigenvectors for different rotation modes S10 of the SI), resulting in glass-like thermal transport. However, we also observe that certain extreme phonon modes of NaB(CN)4 enter the overdamped regime at high temperatures (Fig. S10 of the SI), which might render the standard phonon gas model invalid [43, 44]. Therefore, we further employ homogeneous nonequilibrium molecular dynamics (HNEMD) simulations to provide a more robust description for thermal transport [45]. The HNEMD results are in good agreement with the unified transport theory calculations at low temperatures, but reveal a continuing decrease in κL at high temperatures, underscoring the limitations of the standard phonon gas model in giant anharmonic systems. Finally, we extend HNEMD simulations to 15 CFMs, including Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4, CuX(CN)4, and AgX(CN)4 (X=B, Al, Ga, In). As summarized in Fig. 4, thermal transport is further suppressed as the X atom transitions from B to In. Several compounds [Cd(CN)2, NaB(CN)4, LiIn(CN)4, and all AgX(CN)4 members] exhibit ultralow room-temperature values between 0.35 and 0.81 W/mK. Notably, NaX(CN)4 (X=Al, Ga, In) are excluded due to their complex thermal expansion behavior (possibly involving multiple structural phase transitions), as shown in Fig. S3 of the SI, which is beyond the scope of this study. In contrast, AgIn(CN)4 shows monotonic positive thermal expansion at low temperatures, but transitions to negative thermal expansion above 200 K. This appears to arise from the lowtemperature instability of the cubic phase, as evidenced by imaginary phonon modes below 200 K that vanish at higher temperatures [Fig. S11 of the SI], mirroring behavior observed in other cubic frameworks [18, 46]. To elucidate the microscopic mechanism underlying the gradual suppression of thermal conductivity as the X atom transitions from B to In in these CFMs, we analyze the evolution of chemical bonding and its impact on lattice anharmonicity. Figure 5(a) presents the electron localization function (ELF) for AgX(CN)4 (X=B, Al, Ga, In). The intense localization between B and C atoms in AgB(CN)4 signifies strong bonding character of the B-C bond. However, as B transitions to In, this localization gradually diminishes, suggesting that the bonding strength of X-C is progressively weakened. To further quantify the bonding strength, the integrated crystal orbital Hamilton population (iCOHP) is calculated [47], a measure of bond strength, for key atomic pairs, as shown in inset of Fig. 5(b). The -iCOHP values for the Ag-N and C-N bonds remain essentially constant throughout the series. However, the -iCOHP for the X-C bond decreases substantially from B to In, confirming a systematic chemical bond weakening. As previously established, the strong anharmonicity in these materials is intimately linked to rotation modes, which primarily involve large-amplitude transverse vibrations of the N and 4 5 of 10 III CONCLUSION FIG. 3. The temperature-dependent κL of (a) Zn(CN)2, (b) AgB(CN)4, and (c) NaB(CN)4 is calculated based on different models are compared with molecular dynamics simulations. C atoms. The root-mean-square displacements (RMSD) of C and N atoms, calculated at 300 K, are presented in Fig. 5(b). The RMSD values are significantly enhanced in AgIn(CN)4 compared to AgB(CN)4, indicating more pronounced vibrational amplitudes and thus larger dynamic tetrahedral distortion. This enhanced flexibility directly leads to stronger lattice anharmonicity, which is quantitatively confirmed by the markedly increased negative Gr ̈uneisen parameters in AgIn(CN)4 [see Fig. 5(c)]. Consequently, AgIn(CN)4 exhibits more pronounced negative thermal expansion (Fig. S3 of the SI) and stronger phonon scattering rates (SRs) [Fig. 5(d)], thereby further suppression of thermal transport properties. Finally, we plot the room-temperature thermal conductivity (κL) of these CFMs as a function of average atomic mass ( ̄ M) in Fig. 6, alongside data for classic perovskites and perovskite-like materials for comparison [17, 19-21, 38, 48-64]. Obviously, the ̄ M-κL trend of CFMs deviates significantly from that of perovskite materials. Despite possessing relatively low ̄ M, the CFMs still exhibit ultralow κL, roughly 1-2 orders of magnitude lower than those of perovskites with comparable ̄ M. FIG. 4. The temperature-dependent κL calculated using the HNEMD method for (a) Zn(CN)2 and Cd(CN)2, (b) AgX(CN)4, (c) CuX(CN)4, and (d) LiX(CN)4 (X=B, Al, Ga, In). FIG. 5. (a) The ELF, (b) RMSD of N and C atoms, (c) γ, and (d) phonon SRs for AgX(CN)4 (X=B, Al, Ga, In). The inset of (b) shows the -iCOHP for different atomic bonds. III. CONCLUSION In summary, we have unveiled the microscopic origin of ultralow thermal conductivity (κL) in lightweight cyanide-bridged framework materials (CFMs), challeng5 6 of 10 IV METHODS FIG. 6. The dependence of room temperature κL on average mass. The red spheres mark the 15 CFMs studied in this paper. The blue squares mark some typical perovskite and perovskite-like materials. ing the conventional trade-off between low average atomic mass and efficient thermal transport. Our results reveal that the suppressed thermal transport behavior in these materials stems from exceptionally strong three- and four-phonon anharmonic scattering, driven by abundant hierarchical rotation modes. Remarkably, four-phonon scattering rates even surpass those of three-phonon processes for numerous modes, which significantly suppresses thermal transport. Such behavior originates from the large four-phonon scattering phase space induced by phonon quasi-flat bands and wide bandgaps as well as the strong quartic anharmonicity accompanied by rotation modes. Moreover, coherent thermal conductivity contributes significantly in AgB(CN)4 and NaB(CN)4, with its contribution exceeding that of the particle-like term at elevated temperatures. This leads to a positive temperature dependence of thermal conductivity in NaB(CN)4 at high temperature regions. However, certain extreme phonon modes of NaB(CN)4 enter the overdamped regime at high temperatures, which render the standard phonon gas model invalid. Therefore, we further employ homogeneous nonequilibrium molecular dynamics (HNEMD) simulations to provide a more robust description for thermal transport. The HNEMD simulations confirm the unified transport theory results at low temperatures but reveal a continuing decrease in κL at high temperatures, underscoring the limitations of the standard phonon gas model in giant anharmonic systems. Finally, we extend HNEMD simulations to 15 CFMs, including Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4, CuX(CN)4, and AgX(CN)4 (X=B, Al, Ga, In). As the X atom transitions from B to In, the corresponding system exhibits weaker bonding and greater root-meansquare displacement, thereby inducing stronger anharmonic scattering and lower κL. Among them, Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 exhibit ultralow room-temperature κL values ranging from 0.35 to 0.81 W/mK. These results shed light on the physical mechanisms of thermal transport in CFMs, and suggest a practicable strategy for developing novel lightweight materials with ultralow thermal conductivity. IV. METHODS All density-functional theory (DFT) calculations are implemented using the Vienna Ab initio Simulation Package (VASP) [65] with the projector augmented wave (PAW) [66] method. The PBE [67] exchange-correlation functional is used to relax the geometric structure. The kinetic energy cutoff for wave function is set to 500 eV, and the k-mesh of 8 × 8 × 8 is used to sample the Brillouin Zone. The DFT-D3 correction [68] is adopted to account for van der Waals (vdW) interactions. The ALAMODE package is used to compute both the harmonic second-order interatomic force constants (IFCs) via finite displacement method and the anharmonic higherorder IFCs through compressive sensing lattice dynamics [36, 50, 69]. High precision multi-atomic cluster expansion (MACE) machine learning potential is constructed to capture IFCs more efficiently [70]. Self-consistent phonon (SCP) [36, 50] calculation is performed to obtain the temperature-dependent phonon dispersion and corresponding second-order IFCs. The lattice thermal conductivity is calculated by a unified thermal transport theory and implemented in the modified FOURPHONON software [37, 71-73]. The equation for κL is as follows κP/C L = ħ2 kBT 2V N X q X j,j′ ωqj + ωqj′ 2 νqjj′ ⊗νqjj′ ×ωqjnqj(nqj + 1) + ωqj′nqj′ (nqj′ + 1) 4 (ωqj -ωqj′)2 + (Γqj + Γqj′)2 (Γqj + Γqj′) , (1) where the superscripts P and C represent the contributions from populations (particle-like propagation) and coherences (wave-like phonon tunneling), respectively. When j=j′, Eq. 1 is equivalent to the Peierls-Boltzmann transport equation, i.e., the particle-like transport term. Otherwise, it corresponds to the wave-like transport term. Moreover, the κL is also obtained using homogeneous nonequilibrium molecular dynamics (HNEMD) simulations based on the trained neuroevolution potential (NEP) and implemented in the GPUMD software [45, 74, 75]. To eliminate the size effects, the large supercell exceeding 85 × 85 × 85 ̊A3 (containing 40960 atoms) is adopted. Corresponding supercell is also applied to simulate the thermal expansion behavior. More computational details are provided in the Supporting Information (SI). 6 7 of 10 IV METHODS ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge at *** Detailed method description, lattice constant, rootmean-square error of machine learning potential models, training parameters for fitting machine learning potential models, thermal expansion, phonon spectrum, rotation mode eigenvectors, position markers of rotation modes within the phonon spectrum, weighted scattering phase space, and phonon lifetime. DATA AVAILABILITY The data supporting the findings of this study are available within this article and its Supporting Information. The trained NEP and MACE models are openly available in the repository https://github.com/ ZhunyunTang/CFMs-ML. Additional data are available from the corresponding author on reasonable request. CODE AVAILABILITY The source code for GPUMD is available at https: //github.com/brucefan1983/GPUMD, for ALAMODE is available at https://github.com/ttadano/alamode, for SHENGBTE is available at https://www.shengbte. org, and FOURPHONON is available at https: //github.com/FourPhonon/FourPhonon, for PYNEP is available at https://github.com/bigd4/PyNEP, for the modified version of FOURPHONON is available at https://github.com/ZhunyunTang/ FourPhonon--wigner-sampling. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant No. 52372260), the Science Fund for Distinguished Young Scholars of Hunan Province of China (No. 2024JJ2048), the Hunan Provincial Innovation Foundation for Postgraduate (No. CX20240058), and the Youth Science and Technology Talent Project of Hunan Province (No. 2022RC1197). The calculation in this work is partly supported by the high-performance computing platform of school of physics and optoelectronics. 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Ala-Nissila, Gpumd: A package for constructing accurate machine-learned potentials and performing highly efficient atomistic simulations, The Journal of Chemical Physics 157, 114801 (2022). 10"}}},{"rowIdx":297,"cells":{"id":{"kind":"string","value":"2510.14685"},"raw":{"kind":"string","value":"Bosonic Laughlin and Moore-Read states from non-Chern flat bands\nHongyu Lu1, 2, 3 and Wang Yao1, 2, 3, ∗\n1New Cornerstone Science Lab, Department of Physics, The University of Hong Kong, Hong Kong, China\n2HK Institute of Quantum Science & Technology, The University of Hong Kong, Hong Kong, China\n3State Key Laboratory of Optical Quantum Materials, The University of Hong Kong, Hong Kong, China\n(Dated: October 17, 2025)\nThe rapid advances in the study of fractional Chern insulators (FCIs) raise a fundamental question: while\ninitially discovered in flat Chern bands motivated by their topological equivalence to Landau levels, is single-\nparticle band topology actually a prerequisite for these many-body topological orders emergent at fractional\nfillings? Here, we numerically demonstrate bosonic FCIs in two types of non-Chern flat bands in honeycomb\nlattices, using exact diagonalization and density matrix renormalization group calculations. In a gapless flat\nband with a singular band touching, we observe a Laughlin state at half filling, stabilized by onsite interactions\nfrom the hard-core limit down to arbitrarily small strength. Furthermore, we report the first example of a non-\nAbelian FCI in a non-Chern band system: a Moore-Read state at ν = 1 filling of the same singular flat band\nwith hard-core bosons. Under lattice parameters that realize a gapped trivial band (C = 0) of exact flatness,\nwe also find the Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much\nsmaller than the band gap. In this case, the FCI forms as interacting bosons spontaneously avoid the peaks\nin quantum metric and Berry curvature, preferentially occupying Brillouin zone region with relatively uniform\nquantum geometry. Our work significantly expands the landscape for (non-)Abelian FCIs and broadens the\nunderstanding of their formation beyond the Chern band paradigm.\nIntroduction.— The recent experimental discovery of frac-\ntional Chern insulators (FCI) in correlated moir´e materials\n[1–7], together with quantum simulations of their bosonic\nanalogues in artificial lattices [8, 9], have sparked widespread\ninterest in this exotic topological matter.\nThe FCI phase\nwas first conceived theoretically by translating the integer-to-\nfractional hierarchy of the quantum Hall effect in Landau lev-\nels (LLs) to lattices systems [10–15]. Initial explorations nat-\nurally focused on the scenario of isolated flat Chern band that\nresembles the LLs in topology, quantum geometry, and en-\nergy landscape [10–15]. This appears to be sufficient but not\nnecessary condition for FCI states to emerge in a Bloch band\ncontext. While having ideal quantum geometry (like that of\nLLs) is favorable for stabilizing FCI states [16–22], it is not a\nstrict requirement for their existence in a flat Chern band [23–\n25]. FCIs have also been found in models where interaction\nstrength well exceeds the band gap separating the flat Chern\nband [12, 26]. Notably, in some cases, the Chern bands them-\nselves are not present in the non-interacting limit but instead\narise from interaction-driven band renormalization [27–32].\nThe study of FCIs has also moved beyond the flat Chern\nband paradigm along different lines. Opposite to the flat band\nlimit, interaction-driven high-angular-momentum band inver-\nsion has been proposed for realizing Laughlin states at inte-\nger filling [33]. Within the flat band limit, exact diagonaliza-\ntion (ED) and density matrix renormalization group (DMRG)\ncalculations have unambiguously demonstrated FCI phases in\nfractionally filled non-Chern bands [34, 35], referring to the\nsituations where band Chern number is either undefinable or\nzero. For the former, a notable example is the singular flat\nband, with band touching to a dispersive band dictated by real-\nspace topology [36, 37]. Despite the gapless band structure\nand discontinuity of Bloch function at the singular touching,\nLaughlin-like FCI has been found in such band hosted by a\nfluxed dice lattice [34]. Even more surprisingly, FCI phases\nare also discovered in a trivial flat band of zero Chern num-\nber, which is well isolated by a band gap much larger than\nthe interaction [35]. The FCI state is shown to develop an\ninhomogeneous carrier distribution over the Brillouin zone\n(BZ), avoiding the peaks of Berry curvature and quantum met-\nric, and preferentially occupying regions of relatively uniform\nquantum geometry. Because of the exact flatness allowed in\nnon-Chern bands [38], the FCI phases are shown to persist\ndown to an arbitrarily small nearest-neighbor (NN) interaction\nin both cases [34, 35]. Non-Chern bands may greatly expand\nthe landscape for exploring FCI physics, but the findings are\nso far limited to Abelian Laughlin states, and to fermionic sys-\ntems. Non-Abelian FCIs [39–44] is of paramount importance,\nboth for fundamental interest and for fault-tolerant quantum\ncomputation [45]. And whether bosonic FCIs [46–48] can\nform in such non-Chern bands and the effect of characteris-\ntic onsite repulsion are still open questions. Addressing them\nis critical for elucidating the mechanism of FCI beyond the\nChern band paradigm and for broadening the potential exper-\nimental platforms.\nIn this work, we demonstrate bosonic FCIs in a honeycomb\nlattice that can host either type of non-Chern flat bands, us-\ning ED and iDMRG calculations. At ν = 1/2 filling of the\nsingular flat band, we identify the bosonic Laughlin state sta-\nbilized by onsite interactions from hard-core limit down to\narbitrarily small strength, with the weak-interaction stability\nenabled by the exact flatness of the band. In addition, we\nshow the existence of bosonic Moore-Read state at ν = 1 fill-\ning of the singular flat band with hard-core bosons, the first\nexample of non-Abelian FCI in non-Chern band systems. Un-\nder lattice parameters that realize a gapped trivial flat band,\nwe further demonstrate the bosonic Laughlin FCI of soft-core\nbosons in the isolated band limit where onsite interaction is\nmuch smaller than the band gap. In such a C = 0 band of\nexact flatness, we show that the interacting soft-core bosons\narXiv:2510.14685v1 [cond-mat.str-el] 16 Oct 2025\n\n2\n⃗a 1\n⃗a 2\nt1\nηt1\nt2ei 2π\n3\nηt2ei 2π\n3\nt2ei 2π\n3\nηt2ei 2π\n3\nt3\nη2t3\n(a)\nη = 0.8\n(c)\nIsolated trivial flat band\nC = 0\n-30\n0\n0.167\nΩ\nΩ\n0.167\n0\nSingular flat band\nη = 1\n(b)\nGapless\nGap: \nΔ = 0.08\nFIG. 1. Non-interacting model. (a) Two orbital model on honeycomb lattice including up to third NN hoppings with −t1 = t2 = t3 = 1.\n(b) When η = 1, the lowest band has exact flatness and a singular quadratic touching point to a dispersive band (grey) at Γ. (c) When η < 1,\nthe flat band is isolated from the dispersive band by a band gap ∆, while retaining the exact flatness, which necessarily means the band Chern\nnumber C = 0. Both (b) the singular flat band and (c) the isolated trivial flat band are color-coded by their Berry curvature distribution Ω(k).\ntend to avoid the peak of quantum metric and Berry curva-\nture, and the FCI state is formed primarily in the rest BZ re-\ngion of relatively uniform quantum geometry. This finding\nimplies the generality of the mechanism for forming bosonic\nand fermionic FCIs in such isolated C = 0 flat bands.\nModels and methods.— The two-orbital model on the honey-\ncomb lattice is:\nH =\nX\nij\ntij(b†\nibj + h.c.) + µ\nX\ni\nni +\nX\nn\nUn\nn!\nX\ni\n(b†\ni)n(bi)n.\n(1)\nThe non-interacting part including NN, second NN, and third\nNN hoppings is illustrated in Fig.1 (a). The hopping ampli-\ntude |tij| = 1 serves as the energy unit. When η = 1, this non-\ninteracting part describes a singular flat band model with the\nlower band being exactly flat [Fig.1(b)] [35]. The Bloch wave-\nfunction and quantum geometry is ill-defined at the touching\npoint at Γ, while the quantum geometry elsewhere is relatively\nuniform, and we show the Ωdistribution of the singular flat\nband in Fig.1 (b). When η < 1, the flat band is isolated\nfrom the upper band by a band gap ∆≈2(η −1)2 [Fig.1\n(c)], while retraining the exact flatness [35]. This necessarily\nmeans the band Chern number C = 0. The Ωdistribution\nof the C = 0 flat band has a negative peak at Γ, and a pos-\nitive and relatively uniform background in the rest of the BZ\n[Fig.1 (c)]. µ = (2 + η2)t fixes the non-interacting flat band\nat zero energy. In the main text, we will focus on the effect\nof onsite interactions by considering both the hard-core (with\nsome Un being infinite) and soft-core (finite Un) cases, and\nwill discuss the NN interactions in the supplementary infor-\nmation (SI) [49].\nWe label the number of two-orbital unit cells along the two\nprimitive vectors as N1 (⃗a1) and N2 (⃗a2) with a total of N1 ×\nN2 × 2 sites. The two-band ED simulations only take the\nsoft-core case such that the Hilbert space with Nb bosons is\nC(N1 × N2 × 2 + Nb −1, Nb). The iDMRG simulations\nare based on the real-space basis with the maximum onsite\noccupation as nmax and the local Hilbert-space dimension is\nd = nmax + 1. The hard-core condition here is realized by\nlimiting nmax, which means Un>nmax = ∞. To address the\n0\n2\n4\n6\n8\n10\n0.05\n0.1\n0.15\n0\n5\n10\n15\n0.08\n0.12\n0.16\n-1\n0\n1\n2\n3\n4\n0\n5\n10\nNL = 0\n1\n1\n2\n3\n5\nϕy\n0\n2π\n4π\n0\n-0.5\n-1\n2\n4\n6\nES\nES\n⟨ΔQL(ϕy)⟩\n(a)\n(b)\n(d)\n(e)\nNL\nSoft-core\nSoft-core\n(c) Hard-core\nHard-core\nFIG. 2. Laughlin FCI in the singular flat band with arbitrarily\nsmall and large onsite interactions. We consider ν = 1/2 filling\nof the singular flat band (η = 1). ED spectra of soft-core bosons\nwith a very small U2 = 0.001 interaction from N = 4 × 3 × 2 and\nN = 4 × 4 × 2 tori are shown in panels (a,b), respectively. For each\nsystem size, the spectral flow of the well gapped two-fold ground\nstates is shown in the inset. The iDMRG results of hard-core bosons\n(Un≥2 = ∞) are shown in panel (c-e). In panel (c), the momentum-\nresolved entanglement spectrum (ES) of the charge sector (NL =\n0) with the largest eigenvalue of the reduced density matrix shows\nthe characteristic counting {1, 1, 2, 3, 5, ...} of the edge conformal\nfield theory. We further show (d) the evolution of ES of different\ncharge sectors (NL = −2, −1, 0, 1) and (e) charge pumping results\nby adiabatically inserting 4π fluxes.\nsoft-core case in iDMRG, we do not need infinite local Hilbert\nspace since the maximal number of bosons in the iDMRG unit\ncell is N1 × N2 × ν, bounded by filling factor ν of the flat\nband. We keep up to D = 3000 bond dimensions in iDMRG\nsimulations with maximum truncation errors around 10−5 and\n10−6 for the Abelian and non-Abelian FCIs respectively.\nBosonic Laughlin FCI in the singular flat band.— We first\n\n3\nconsider ν = 1/2 filling of the singular flat band (η = 1) and\nstart with soft-core bosons and only very small U2 = 0.001\n(Un>2 = 0) interaction. The spectra from two-orbital ED\nsimulations of N = 4 × 3 × 2 and N = 4 × 4 × 2 tori\nare shown in Fig.2 (a,b), respectively. The two-fold ground\nstates (with their spectral flow in the insets) are well gapped\nfrom the excited states and their momenta are in agreement\nwith the generalized Pauli principle for the bosonic 1/2 FCI\non torus [14, 50, 51]. The many-body Chern number of the\nground states gives the fractional quantized Hall conductivity\nσH = 1\n2\ne2\nh , further confirming the bosonic Laughlin-like FCI.\nIn fact, due to the exact flatness of the singular flat band, ar-\nbitrarily small U2 could stabilize the FCI, which has not been\nnumerically reported in Chern-band systems. The considered\nU2 = 0.001 is already very small and the low-energy spectra\nhave well converged with decreasing U2 in the unit of En/U2\n(c.f. SI [49]).\nWe then go to the opposite limit, i.e.\nhard-core bosons\nwith Un≥2 = ∞. The iDMRG results with circumference\nN1 = 6 are shown in Fig.2(c-e). The momentum-resolved\nentanglement spectrum (ES) shows the characteristic edge-\nmode counting {1, 1, 2, 3, 5, ...}, in agreement with the edge\nconformal field theory, supporting the topological nature of\nthe ground state [Fig.2(c)] [52, 53]. After adiabatically in-\nserting 4π fluxes, the ES goes back to itself while shifted by\none charge sector and there is one boson pumped from the\nleft to the right side of the system, suggesting the FCI ground\nstate with σH =\n1\n2\ne2\nh [54]. Therefore, we have shown that\nthe Laughlin FCI could be stabilized by arbitrarily small and\nlarge onsite two-body interactions. In the SI, we further show\nthat this FCI could survive with very large NN interactions as\nwell [49].\nBosonic Moore-Read state in the singular flat band.— Fur-\nthermore, we demonstrate the existence of the MR state in the\nsingular flat band, where we consider the ν = 1 filling with\nthree-body hard-core bosons (nmax = 2), which is numeri-\ncally easier than the soft-core case for this high filling. The\nES of the identity sector ΨI ground state is shown in Fig.3(a),\nwith the characteristic edge-mode of counting {1, 2, 4, 7, ...}\nfor odd charge sectors and {1, 1, 3, 5, 10...} for even charge\nsectors. The even-odd difference can be understood from the\nroot configuration “..020202..” of ΨI depending on how it is\ncut [51, 52, 55–58]. To obtain the Ising anyon sector Ψσ, we\nconsider an iDMRG unit cell with N1 = 5 and N2 = 1 such\nthat only the root configuration “..11111..” is allowed. The\nES of the Ψσ ground state is shown in Fig.3(b), and the edge\nmode degeneracy pattern is {1, 2, 4, 8, ...} for all charge sec-\ntors, consistent with the analytical prediction of Ising anyon\nprimary field [55, 57, 58].\nThen we perform the adiabatic flux insertion. The charge\npumping results starting from the ΨI or the Ψσ ground states\nare shown in Fig.3(c). After inserting 2π (4π), there is 1 (2)\nnet charges pumped from the left to the right edge of the sys-\ntems for both cases. However, the underlying fusion rules\nof the quasiparticles are different [58]. Starting from the ΨI\n0\n2\n4\n-2\n0\n2\n0\n5\n10\n1\n1\n3\n5\n10\n1\n2\n4\n7\nNL = 0\nNL = −1\n(a) Identity sector |ΨI⟩\n(b) Ising anyon sector |Ψσ⟩\n-1\n0\n1\n2\n3\nNL = 0\n1\n2\n4\n8\n0\n5\n10\n1\n2\n4\n8\nNL = −1\nNL\nϕy\n0\n2π\n4π\n0\n2\n4\n6\n-1\n0\n1\n2\n3\nϕy\n0\n2π\n4π\n0\n-1\n-2\n⟨ΔQL(ϕy)⟩\nStarting from |ΨI⟩\n(c)\n(d)\nFIG. 3. Moore-Read state in the singular flat band. We show the\niDMRG results of ν = 1 filling of the singular flat band (η = 1)\nwith hard-core bosons (nmax = 2).\n(a) The ES of ΨI ground\nstate from the N1 = 6 cylinder, showing the edge mode counting\n{1, 2, 4, 7, ...} and {1, 1, 3, 5, 10...} for the NL = −1 and NL = 0\ncharge sectors, respectively.\n(b) The ES of the Ψσ ground state\nfrom the N1 = 5 cylinder, showing the same degeneracy pattern\n{1, 2, 4, 8, ...} in all charge sectors (NL = −1, 0 plotted for exam-\nple). (c) Charge pumping results starting from the ΨI or the Ψσ\nground state. (d) The evolution of ES after flux insertion, starting\nfrom the ΨI ground state.\nground state, the evolution of ES with flux insertion is shown\nin Fig.3(d). Although there is a net charge pumped after inser-\ntion of ϕy = 2π flux, the ES does not go back to itself and is\ndifferent from that at ϕy = 0. This is because the ground state\nadiabatically evolves into the fermion anyon sector Ψf, where\nthe ES would exhibit edge-mode counting {1, 2, 4, 7, ...} for\neven charge sectors and {1, 1, 3, 5, 10...} for odd charge sec-\ntors. The even-odd counting is shifted by one charge for the\nΨI and Ψf ground states, which is also manifested from the\n\n4\ncomparison of different ES counting in the NL = −1, 0 sec-\ntors in Fig.3(a). After inserting 4π flux, the ES goes back\nto itself with the charge sectors shifted by 2, and the ground\nstate goes back to the ΨI again. These relate to the fusion\nrules: I × f = f and f × f = I. If starting from the Ψσ,\njust as the same edge mode counting in different charge sec-\ntors (Fig.3 (b)), the ES would be the same with each integer\nnumber of 2π flux. This relates to σ × f = σ and the Ψσ\nground state always evolves into itself with flux insertion of\n2π, unlike the ΨI and Ψf ground states that evolve into each\nother. This also demonstrates that the Ising anyon does not\nrespond to the U(1) charge flux.\nBosonic Laughlin FCI in the isolated C = 0 flat band.— At\nη = 0.8, the (exactly) flat band is gapped with ∆= 0.08. Like\nthe Berry curvature, the trace of quantum metric also shows a\npeak at Γ [inset of Fig. 4(a)]. When spinless fermions form\nFCI in such C = 0 band [35], the nearest-neighbor repulsion\ntends to drive carriers away from such peak, thereby occu-\npying BZ region with relatively uniform quantum geometry.\nThis has restricted the filling factor to the low side (ν = 1/3),\notherwise the peak of quantum geometry would be less avoid-\nable. It is unclear whether such mechanism allows bosonic\nFCI to form with the characteristic on-site interaction. We ad-\ndress this open issue by considering ν = 1/2 filling of the\nisolated trivial flat band with the onsite two-body repulsion\nmuch weaker than the band gap.\nThe soft-core-boson condition is realized with onsite\nnmax = N1 × N2 × ν since each iDMRG unit cell can not\nhave more bosons. Without explicit interactions, the system\nis basically non-interacting and the ground state is topolog-\nically trivial.\nWe will consider U2 = 0.02 (Un>2 = 0),\nwhich is much smaller than the band gap and thus ensuring\nthe isolated-band limit. The results of N1 = 6 (circumfer-\nence of the infinite cylinder) and N2 = 1 (we have checked\nthat there will be no translation symmetry breaking along ⃗a2\nwith only small U2) are shown in Fig. 4. With random ini-\ntial conditions and optimization, we always obtain two nearly\ndegenerate ground states [53, 58], which actually refer to the\ntwo topological sectors of the 1/2 Laughlin state.\nTo reveal the mechanism of forming this FCI in an isolated\ntrivial band, we compute the momentum-space boson distri-\nbution n(k) = P\nα\nP\nj⟨b†\ni,αbj,α⟩e−ikrij by considering a fi-\nnite region of the infinite cylinder, where α refers to the A/B\nsublattice. The results from both topological sectors show that\nn(k) drops to 0 when approaching Γ, suggesting that this FCI\nis realized by occupying the BZ away from Γ with relatively\nuniform Ω. The n(k) from the identity sector along one k-path\nis shown in Fig.4 (a). The ES of the charge sector with largest\neigenvalue of the reduced density matrix from the identity-\nsector ground state is further shown in Fig.4 (b), with the char-\nacteristic edge mode counting {1, 1, 2, 3, 5, 7, ...}. Moreover,\nwe show the ES from the semion-sector ground state in Fig.4\n(c) with all the charge sectors having the same degeneracy pat-\ntern {1, 1, 2, 3, 5, ...}. The ES is symmetric about NL = 0.5,\nconfirming the semion anyon at each edge of the cylinder, car-\nrying the fractional charge e/2 [52, 53, 59].\n0\n2\n4\n2\n4\n6\n8\n10\n(a)\n(b)\n1\n1\n2\n3\n5\n7\nIdentity sector\nNL = 0\n0\n3\n6\n9\n0\n2\n4\n6\nSemion sector\n1\n1\n2\n3\n1\n1\n2\n3\n5\n1\n1\n2\n3\n5\n1\n1\n2\n3\n(c)\nTr(g)\n1.1\n34\nFIG. 4. Laughlin FCI in the isolated C = 0 flat band with soft-\ncore bosons. The iDMRG results at ν = 1/2 filling of the isolated\nflat band (η = 0.8 with band gap ∆= 0.08) with only U2 = 0.02.\nThe inset of panel (a) shows the trace of quantum metric of the trivial\nflat band with a peak at Γ, and the blue dashed line shows the path\nalong which we compute the momentum-space boson distribution\nn(k) in (a). n(k) is measured from the ground state in the identity\nsector for example and the bosons tend to avoid occupying around\nΓ. (b) The ES of the NL = 0 charge sector of the ground state in\nthe identity sector, showing the characteristic edge-mode counting\n{1, 1, 2, 3, 5, 7, ...}. (c) The ES of the ground state in the semion\nsector, where all charge sectors show the same degeneracy pattern\nand the ES is symmetric about NL = 0.5.\nDiscussions.— Our demonstrations of FCIs in non-Chern flat\nband systems broaden the understanding of the origins and\nmechanisms of FCI. Whereas many-body interactions are of-\nten crucial, we provide a distinct perspective centered on flat-\nband quantum geometry, pointing to FCI physics beyond band\ntopology. Due to the exact flatness of these non-Chern bands,\nFCIs could be induced by arbitrarily small onsite interactions.\nThis feature surpasses the limitations of Chern bands, which\nmust have a finite bandwidth in a finite range hopping lat-\ntice [38], thereby requiring a minimum interaction strength\nto overcome their dispersion. This enables investigation in a\nbroad range of platforms - including photonic, phononic, ex-\ncitonic and polaritonic systems [44, 60–67] - thereby signifi-\ncantly expanding the landscape for exploring bosonic FCIs.\nWe also note that in an extended Bose-Hubbard model on\nthe Kagome lattice with infinite on-site interaction and com-\npeting different-range interactions, the bosonic Laughlin state\nhas been reported [68], which is a chiral spin liquid in spin\nlanguage as the model can be mapped onto the spin-1/2 XXZ\n\n5\nmodel [69, 70]. Upon spontaneous time reversal symmetry\nbreaking, the 1/2 Laughlin state emerges at an integer filling\nof bosons per unit cell [68], which is intrinsically different\nfrom the Laughlin states in our scenarios at half flat-band fill-\nings.\nAcknowledgments — We thank Jie Wang and Nicolas Reg-\nnault for helpful discussions. We thank Zuzhang Lin, Wenqi\nYang, and Dawei Zhai for earlier collaborations. 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Panels (a) and (b) are from N = 3 × 4 × 2 and N = 4 × 4 × 2 tori, respectively.\n0\n5\n10\n0.05\n0.1\n0.15\n0\n5\n10\n0.05\n0.1\n0.15\n0\n5\n10\n0.05\n0.1\n0.15\n0\n5\n10\n0.05\n0.1\n0.15\n(a) U2 = 0.1\n(b) U2 = 0.01\n(c) U2 = 0.001\n(d) U2 = 0.0001\nFIG. S2. Laughlin FCI in the singular flat band with arbitrarily small onsite interactions. We show the spectra of the FCI at ν = 1/2 of\nthe singular flat band with different U2 from N = 3 × 4 × 2 torus. Panel (c) has been showed in Fig.2(a) of the main text.\nIn Fig. 2(a-b) of the main text, we have shown the spectral flow of the two degenerate ground states in the insets. To further\nsupport the robust many-body gap upon flux insertion, we zoom out and and add the low-energy excited states as well in Fig.\nS1.\nIn the main text, we have mentioned that due to the exact flatness, arbitrarily small onsite interactions among the soft-core\nbosons can stabilize the FCI. In the main text, the example is taken as U2 = 0.001. To further support this, we show more\nED spectra with different U2 in Fig. S2. It is clear that with the decreasing U2, the spectra almost converge, supporting that\narbitrarily small U2 could stabilize this FCI at ν = 1/2 of the singular flat band.\nRobust FCI with NN repulsion\nIn the main text, we only consider the onsite interactions. Here, in the singular flat band model, we take the hard-core boson\ncase for example to explore the NN repulsion: HV = V1\nP\n⟨ij⟩ninj. At ν = 1/2, we find that the FCI is stable even when V1\nis larger than the total width of the two bands (W = 9). We show the iDMRG results with V1 = 9 in Fig.S3. The characteristic\nedge-mode counting from the ES and the charge pumping with one boson pumped after adiabatically inserting 4π fluxes confirm\nthe robustness of the FCI with very large V1 interactions and suggest that there might be no upper limit of V1 for this FCI.\n\n8\n0\n1\n2\n3\n4\n0\n5\n10\n1\n1\n2\n3\n5\n(a)\n(b)\nFIG. S3. Robust FCI with strong NN interaction. Here, we consider the singular flat band at ν = 1/2 filling with hard-core bosons, and\nV1 = 9 is taken for these results. (a) The momentum-resolved ES. (b) The charge pumping result.\n"},"cleaned":{"kind":"string","value":"Bosonic Laughlin and Moore-Read states from non-Chern flat bands Hongyu Lu1, 2, 3 and Wang Yao1, 2, 3, ∗ 1New Cornerstone Science Lab, 2HK 3State Key Laboratory of Optical Quantum Materials, The (Dated: October 17, 2025) The rapid advances in the study of fractional Chern insulators (FCIs) raise a fundamental question: while initially discovered in flat Chern bands motivated by their topological equivalence to Landau levels, is singleparticle band topology actually a prerequisite for these many-body topological orders emergent at fractional fillings? Here, we numerically demonstrate bosonic FCIs in two types of non-Chern flat bands in honeycomb lattices, using exact diagonalization and density matrix renormalization group calculations. In a gapless flat band with a singular band touching, we observe a Laughlin state at half filling, stabilized by onsite interactions from the hard-core limit down to arbitrarily small strength. Furthermore, we report the first example of a nonAbelian FCI in a non-Chern band system: a Moore-Read state at ν = 1 filling of the same singular flat band with hard-core bosons. Under lattice parameters that realize a gapped trivial band (C = 0) of exact flatness, we also find the Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much smaller than the band gap. In this case, the FCI forms as interacting bosons spontaneously avoid the peaks in quantum metric and Berry curvature, preferentially occupying Brillouin zone region with relatively uniform quantum geometry. Our work significantly expands the landscape for (non-)Abelian FCIs and broadens the understanding of their formation beyond the Chern band paradigm. Introduction.- The recent experimental discovery of fractional Chern insulators (FCI) in correlated moir ́e materials [1-7], together with quantum simulations of their bosonic analogues in artificial lattices [8, 9], have sparked widespread interest in this exotic topological matter. The FCI phase was first conceived theoretically by translating the integer-tofractional hierarchy of the quantum Hall effect in Landau levels (LLs) to lattices systems [10-15]. Initial explorations naturally focused on the scenario of isolated flat Chern band that resembles the LLs in topology, quantum geometry, and energy landscape [10-15]. This appears to be sufficient but not necessary condition for FCI states to emerge in a Bloch band context. While having ideal quantum geometry (like that of LLs) is favorable for stabilizing FCI states [16-22], it is not a strict requirement for their existence in a flat Chern band [2325]. FCIs have also been found in models where interaction strength well exceeds the band gap separating the flat Chern band [12, 26]. Notably, in some cases, the Chern bands themselves are not present in the non-interacting limit but instead arise from interaction-driven band renormalization [27-32]. The study of FCIs has also moved beyond the flat Chern band paradigm along different lines. Opposite to the flat band limit, interaction-driven high-angular-momentum band inversion has been proposed for realizing Laughlin states at integer filling [33]. Within the flat band limit, exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations have unambiguously demonstrated FCI phases in fractionally filled non-Chern bands [34, 35], referring to the situations where band Chern number is either undefinable or zero. For the former, a notable example is the singular flat band, with band touching to a dispersive band dictated by realspace topology [36, 37]. Despite the gapless band structure and discontinuity of Bloch function at the singular touching, Laughlin-like FCI has been found in such band hosted by a fluxed dice lattice [34]. Even more surprisingly, FCI phases are also discovered in a trivial flat band of zero Chern number, which is well isolated by a band gap much larger than the interaction [35]. The FCI state is shown to develop an inhomogeneous carrier distribution over the Brillouin zone (BZ), avoiding the peaks of Berry curvature and quantum metric, and preferentially occupying regions of relatively uniform quantum geometry. Because of the exact flatness allowed in non-Chern bands [38], the FCI phases are shown to persist down to an arbitrarily small nearest-neighbor (NN) interaction in both cases [34, 35]. Non-Chern bands may greatly expand the landscape for exploring FCI physics, but the findings are so far limited to Abelian Laughlin states, and to fermionic systems. Non-Abelian FCIs [39-44] is of paramount importance, both for fundamental interest and for fault-tolerant quantum computation [45]. And whether bosonic FCIs [46-48] can form in such non-Chern bands and the effect of characteristic onsite repulsion are still open questions. Addressing them is critical for elucidating the mechanism of FCI beyond the Chern band paradigm and for broadening the potential experimental platforms. In this work, we demonstrate bosonic FCIs in a honeycomb lattice that can host either type of non-Chern flat bands, using ED and iDMRG calculations. At ν = 1/2 filling of the singular flat band, we identify the bosonic Laughlin state stabilized by onsite interactions from hard-core limit down to arbitrarily small strength, with the weak-interaction stability enabled by the exact flatness of the band. In addition, we show the existence of bosonic Moore-Read state at ν = 1 filling of the singular flat band with hard-core bosons, the first example of non-Abelian FCI in non-Chern band systems. Under lattice parameters that realize a gapped trivial flat band, we further demonstrate the bosonic Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much smaller than the band gap. In such a C = 0 band of exact flatness, we show that the interacting soft-core bosons 16 Oct 2025 2 ⃗a 1 ⃗a 2 t1 ηt1 t2ei 2π 3 ηt2ei 2π 3 t2ei 2π 3 ηt2ei 2π 3 t3 η2t3 (a) η = 0.8 (c) Isolated trivial flat band C = 0 -30 0 0.167 Ω Ω 0.167 0 Singular flat band η = 1 (b) Gapless Gap: Δ = 0.08 FIG. 1. Non-interacting model. (a) Two orbital model on honeycomb lattice including up to third NN hoppings with -t1 = t2 = t3 = 1. (b) When η = 1, the lowest band has exact flatness and a singular quadratic touching point to a dispersive band (grey) at Γ. (c) When η nmax = ∞. To address the 0 2 4 6 8 10 0.05 0.1 0.15 0 5 10 15 0.08 0.12 0.16 -1 0 1 2 3 4 0 5 10 NL = 0 1 1 2 3 5 φy 0 2π 4π 0 -0.5 -1 2 4 6 ES ES ⟨ΔQL(φy)⟩ (a) (b) (d) (e) NL Soft-core Soft-core (c) Hard-core Hard-core FIG. 2. Laughlin FCI in the singular flat band with arbitrarily small and large onsite interactions. We consider ν = 1/2 filling of the singular flat band (η = 1). ED spectra of soft-core bosons with a very small U2 = 0.001 interaction from N = 4 × 3 × 2 and N = 4 × 4 × 2 tori are shown in panels (a,b), respectively. For each system size, the spectral flow of the well gapped two-fold ground states is shown in the inset. The iDMRG results of hard-core bosons (Un≥2 = ∞) are shown in panel (c-e). In panel (c), the momentumresolved entanglement spectrum (ES) of the charge sector (NL = 0) with the largest eigenvalue of the reduced density matrix shows the characteristic counting {1, 1, 2, 3, 5, ...} of the edge conformal field theory. We further show (d) the evolution of ES of different charge sectors (NL = -2, -1, 0, 1) and (e) charge pumping results by adiabatically inserting 4π fluxes. soft-core case in iDMRG, we do not need infinite local Hilbert space since the maximal number of bosons in the iDMRG unit cell is N1 × N2 × ν, bounded by filling factor ν of the flat band. We keep up to D = 3000 bond dimensions in iDMRG simulations with maximum truncation errors around 10-5 and 10-6 for the Abelian and non-Abelian FCIs respectively. Bosonic Laughlin FCI in the singular flat band.- We first 3 consider ν = 1/2 filling of the singular flat band (η = 1) and start with soft-core bosons and only very small U2 = 0.001 (Un>2 = 0) interaction. The spectra from two-orbital ED simulations of N = 4 × 3 × 2 and N = 4 × 4 × 2 tori are shown in Fig.2 (a,b), respectively. The two-fold ground states (with their spectral flow in the insets) are well gapped from the excited states and their momenta are in agreement with the generalized Pauli principle for the bosonic 1/2 FCI on torus [14, 50, 51]. The many-body Chern number of the ground states gives the fractional quantized Hall conductivity σH = 1 2 e2 h , further confirming the bosonic Laughlin-like FCI. In fact, due to the exact flatness of the singular flat band, arbitrarily small U2 could stabilize the FCI, which has not been numerically reported in Chern-band systems. The considered U2 = 0.001 is already very small and the low-energy spectra have well converged with decreasing U2 in the unit of En/U2 (c.f. SI [49]). We then go to the opposite limit, i.e. hard-core bosons with Un≥2 = ∞. The iDMRG results with circumference N1 = 6 are shown in Fig.2(c-e). The momentum-resolved entanglement spectrum (ES) shows the characteristic edgemode counting {1, 1, 2, 3, 5, ...}, in agreement with the edge conformal field theory, supporting the topological nature of the ground state [Fig.2(c)] [52, 53]. After adiabatically inserting 4π fluxes, the ES goes back to itself while shifted by one charge sector and there is one boson pumped from the left to the right side of the system, suggesting the FCI ground state with σH = 1 2 e2 h [54]. Therefore, we have shown that the Laughlin FCI could be stabilized by arbitrarily small and large onsite two-body interactions. In the SI, we further show that this FCI could survive with very large NN interactions as well [49]. Bosonic Moore-Read state in the singular flat band.- Furthermore, we demonstrate the existence of the MR state in the singular flat band, where we consider the ν = 1 filling with three-body hard-core bosons (nmax = 2), which is numerically easier than the soft-core case for this high filling. The ES of the identity sector ΨI ground state is shown in Fig.3(a), with the characteristic edge-mode of counting {1, 2, 4, 7, ...} for odd charge sectors and {1, 1, 3, 5, 10...} for even charge sectors. The even-odd difference can be understood from the root configuration \"..020202..\" of ΨI depending on how it is cut [51, 52, 55-58]. To obtain the Ising anyon sector Ψσ, we consider an iDMRG unit cell with N1 = 5 and N2 = 1 such that only the root configuration \"..11111..\" is allowed. The ES of the Ψσ ground state is shown in Fig.3(b), and the edge mode degeneracy pattern is {1, 2, 4, 8, ...} for all charge sectors, consistent with the analytical prediction of Ising anyon primary field [55, 57, 58]. Then we perform the adiabatic flux insertion. The charge pumping results starting from the ΨI or the Ψσ ground states are shown in Fig.3(c). After inserting 2π (4π), there is 1 (2) net charges pumped from the left to the right edge of the systems for both cases. However, the underlying fusion rules of the quasiparticles are different [58]. Starting from the ΨI 0 2 4 -2 0 2 0 5 10 1 1 3 5 10 1 2 4 7 NL = 0 NL = -1 (a) Identity sector |ΨI⟩ (b) Ising anyon sector |Ψσ⟩ -1 0 1 2 3 NL = 0 1 2 4 8 0 5 10 1 2 4 8 NL = -1 NL φy 0 2π 4π 0 2 4 6 -1 0 1 2 3 φy 0 2π 4π 0 -1 -2 ⟨ΔQL(φy)⟩ Starting from |ΨI⟩ (c) (d) FIG. 3. Moore-Read state in the singular flat band. We show the iDMRG results of ν = 1 filling of the singular flat band (η = 1) with hard-core bosons (nmax = 2). (a) The ES of ΨI ground state from the N1 = 6 cylinder, showing the edge mode counting {1, 2, 4, 7, ...} and {1, 1, 3, 5, 10...} for the NL = -1 and NL = 0 charge sectors, respectively. (b) The ES of the Ψσ ground state from the N1 = 5 cylinder, showing the same degeneracy pattern {1, 2, 4, 8, ...} in all charge sectors (NL = -1, 0 plotted for example). (c) Charge pumping results starting from the ΨI or the Ψσ ground state. (d) The evolution of ES after flux insertion, starting from the ΨI ground state. ground state, the evolution of ES with flux insertion is shown in Fig.3(d). Although there is a net charge pumped after insertion of φy = 2π flux, the ES does not go back to itself and is different from that at φy = 0. This is because the ground state adiabatically evolves into the fermion anyon sector Ψf, where the ES would exhibit edge-mode counting {1, 2, 4, 7, ...} for even charge sectors and {1, 1, 3, 5, 10...} for odd charge sectors. The even-odd counting is shifted by one charge for the ΨI and Ψf ground states, which is also manifested from the 4 comparison of different ES counting in the NL = -1, 0 sectors in Fig.3(a). After inserting 4π flux, the ES goes back to itself with the charge sectors shifted by 2, and the ground state goes back to the ΨI again. These relate to the fusion rules: I × f = f and f × f = I. If starting from the Ψσ, just as the same edge mode counting in different charge sectors (Fig.3 (b)), the ES would be the same with each integer number of 2π flux. This relates to σ × f = σ and the Ψσ ground state always evolves into itself with flux insertion of 2π, unlike the ΨI and Ψf ground states that evolve into each other. This also demonstrates that the Ising anyon does not respond to the U(1) charge flux. Bosonic Laughlin FCI in the isolated C = 0 flat band.- At η = 0.8, the (exactly) flat band is gapped with ∆= 0.08. Like the Berry curvature, the trace of quantum metric also shows a peak at Γ [inset of Fig. 4(a)]. When spinless fermions form FCI in such C = 0 band [35], the nearest-neighbor repulsion tends to drive carriers away from such peak, thereby occupying BZ region with relatively uniform quantum geometry. This has restricted the filling factor to the low side (ν = 1/3), otherwise the peak of quantum geometry would be less avoidable. It is unclear whether such mechanism allows bosonic FCI to form with the characteristic on-site interaction. We address this open issue by considering ν = 1/2 filling of the isolated trivial flat band with the onsite two-body repulsion much weaker than the band gap. The soft-core-boson condition is realized with onsite nmax = N1 × N2 × ν since each iDMRG unit cell can not have more bosons. Without explicit interactions, the system is basically non-interacting and the ground state is topologically trivial. We will consider U2 = 0.02 (Un>2 = 0), which is much smaller than the band gap and thus ensuring the isolated-band limit. The results of N1 = 6 (circumference of the infinite cylinder) and N2 = 1 (we have checked that there will be no translation symmetry breaking along ⃗a2 with only small U2) are shown in Fig. 4. With random initial conditions and optimization, we always obtain two nearly degenerate ground states [53, 58], which actually refer to the two topological sectors of the 1/2 Laughlin state. To reveal the mechanism of forming this FCI in an isolated trivial band, we compute the momentum-space boson distribution n(k) = P α P j⟨b† i,αbj,α⟩e-ikrij by considering a finite region of the infinite cylinder, where α refers to the A/B sublattice. The results from both topological sectors show that n(k) drops to 0 when approaching Γ, suggesting that this FCI is realized by occupying the BZ away from Γ with relatively uniform Ω. The n(k) from the identity sector along one k-path is shown in Fig.4 (a). The ES of the charge sector with largest eigenvalue of the reduced density matrix from the identitysector ground state is further shown in Fig.4 (b), with the characteristic edge mode counting {1, 1, 2, 3, 5, 7, ...}. Moreover, we show the ES from the semion-sector ground state in Fig.4 (c) with all the charge sectors having the same degeneracy pattern {1, 1, 2, 3, 5, ...}. The ES is symmetric about NL = 0.5, confirming the semion anyon at each edge of the cylinder, carrying the fractional charge e/2 [52, 53, 59]. 0 2 4 2 4 6 8 10 (a) (b) 1 1 2 3 5 7 Identity sector NL = 0 0 3 6 9 0 2 4 6 Semion sector 1 1 2 3 1 1 2 3 5 1 1 2 3 5 1 1 2 3 (c) Tr(g) 1.1 34 FIG. 4. Laughlin FCI in the isolated C = 0 flat band with softcore bosons. The iDMRG results at ν = 1/2 filling of the isolated flat band (η = 0.8 with band gap ∆= 0.08) with only U2 = 0.02. The inset of panel (a) shows the trace of quantum metric of the trivial flat band with a peak at Γ, and the blue dashed line shows the path along which we compute the momentum-space boson distribution n(k) in (a). n(k) is measured from the ground state in the identity sector for example and the bosons tend to avoid occupying around Γ. (b) The ES of the NL = 0 charge sector of the ground state in the identity sector, showing the characteristic edge-mode counting {1, 1, 2, 3, 5, 7, ...}. (c) The ES of the ground state in the semion sector, where all charge sectors show the same degeneracy pattern and the ES is symmetric about NL = 0.5. Discussions.- Our demonstrations of FCIs in non-Chern flat band systems broaden the understanding of the origins and mechanisms of FCI. Whereas many-body interactions are often crucial, we provide a distinct perspective centered on flatband quantum geometry, pointing to FCI physics beyond band topology. Due to the exact flatness of these non-Chern bands, FCIs could be induced by arbitrarily small onsite interactions. This feature surpasses the limitations of Chern bands, which must have a finite bandwidth in a finite range hopping lattice [38], thereby requiring a minimum interaction strength to overcome their dispersion. This enables investigation in a broad range of platforms - including photonic, phononic, excitonic and polaritonic systems [44, 60-67] - thereby significantly expanding the landscape for exploring bosonic FCIs. We also note that in an extended Bose-Hubbard model on the Kagome lattice with infinite on-site interaction and competing different-range interactions, the bosonic Laughlin state has been reported [68], which is a chiral spin liquid in spin language as the model can be mapped onto the spin-1/2 XXZ 5 model [69, 70]. Upon spontaneous time reversal symmetry breaking, the 1/2 Laughlin state emerges at an integer filling of bosons per unit cell [68], which is intrinsically different from the Laughlin states in our scenarios at half flat-band fillings. Acknowledgments - We thank Jie Wang and Nicolas Regnault for helpful discussions. We thank Zuzhang Lin, Wenqi Yang, and Dawei Zhai for earlier collaborations. The work is supported by the National Natural Science Foundation of China (No. 12425406), Research Grant Council of Hong Kong (AoE/P-701/20, HKU SRFS21227S05), and New Cornerstone Science Foundation. We thank Beijing PARATERA Tech Co., Ltd. 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The parameters are the same as Fig. 2(a-b) of the main text with ν = 1/2 of the singular flat band (η = 1), and only very small U2 = 0.001 interaction. Panels (a) and (b) are from N = 3 × 4 × 2 and N = 4 × 4 × 2 tori, respectively. 0 5 10 0.05 0.1 0.15 0 5 10 0.05 0.1 0.15 0 5 10 0.05 0.1 0.15 0 5 10 0.05 0.1 0.15 (a) U2 = 0.1 (b) U2 = 0.01 (c) U2 = 0.001 (d) U2 = 0.0001 FIG. S2. Laughlin FCI in the singular flat band with arbitrarily small onsite interactions. We show the spectra of the FCI at ν = 1/2 of the singular flat band with different U2 from N = 3 × 4 × 2 torus. Panel (c) has been showed in Fig.2(a) of the main text. In Fig. 2(a-b) of the main text, we have shown the spectral flow of the two degenerate ground states in the insets. To further support the robust many-body gap upon flux insertion, we zoom out and and add the low-energy excited states as well in Fig. S1. In the main text, we have mentioned that due to the exact flatness, arbitrarily small onsite interactions among the soft-core bosons can stabilize the FCI. In the main text, the example is taken as U2 = 0.001. To further support this, we show more ED spectra with different U2 in Fig. S2. It is clear that with the decreasing U2, the spectra almost converge, supporting that arbitrarily small U2 could stabilize this FCI at ν = 1/2 of the singular flat band. Robust FCI with NN repulsion In the main text, we only consider the onsite interactions. Here, in the singular flat band model, we take the hard-core boson case for example to explore the NN repulsion: HV = V1 P ⟨ij⟩ninj. At ν = 1/2, we find that the FCI is stable even when V1 is larger than the total width of the two bands (W = 9). We show the iDMRG results with V1 = 9 in Fig.S3. The characteristic edge-mode counting from the ES and the charge pumping with one boson pumped after adiabatically inserting 4π fluxes confirm the robustness of the FCI with very large V1 interactions and suggest that there might be no upper limit of V1 for this FCI. 8 0 1 2 3 4 0 5 10 1 1 2 3 5 (a) (b) FIG. S3. Robust FCI with strong NN interaction. Here, we consider the singular flat band at ν = 1/2 filling with hard-core bosons, and V1 = 9 is taken for these results. (a) The momentum-resolved ES. (b) The charge pumping result."}}},{"rowIdx":298,"cells":{"id":{"kind":"string","value":"2510.14686"},"raw":{"kind":"string","value":"xLLM Technical Report\nTongxuan LiuB, Tao Peng, Peijun Yang, Xiaoyang Zhao, Xiusheng Lu∗, Weizhe Huang,\nZirui Liu††, Xiaoyu Chen, Zhiwei Liang, Jun Xiong, Donghe Jin, Minchao Zhang,\nJinrong Guo, Yingxu Deng, Xu Zhang, Xianzhe Dong†, Siqi Wang‡, Siyu Wu‡, Yu Wu†,\nZihan Tang∗, Yuting Zeng†, Yanshu Wang††, Jinguang Liu, Meng Kang, Menxin Li,\nYunlong Wang, Yiming Liu∗, Xiaolong Ma, Yifan Wang, Yichen Zhang∗, Jinrun Yin††,\nKeyang Zheng††, Jiawei Yin†, Jun Zhang†, Ziyue Wang†, Xiaobo Lin, Liangyu Liu†,\nLiwei Lan∗, Yang Liu†, Chunhua Peng, Han Liu, Songcheng Ren††, Xuezhu Wang‡,\nYunheng Shen∗, Yi Wang, Guyue Liu††B, Hui Chen∗B, Tong Yang††B, Hailong Yang‡B,\nJing Li†B, Guiguang Ding∗B, Ke ZhangB\nJD.com\nTHU ∗\nUSTC †\nBUAA ‡\nPKU ††\nAbstract\nWe introduce xLLM, an intelligent and efficient Large Language Model (LLM) inference frame-\nwork designed for high-performance, large-scale enterprise-grade serving, with deep optimizations\nfor diverse AI accelerators. Current mainstream inference frameworks face practical challenges. On\nthe one hand, enterprise-grade serving struggles with hybrid and dynamic workloads, strict demand\nfor high availability of services, and distributed storage management. On the other hand, infer-\nence execution is bottlenecked by underutilized AI accelerators due to new paradigms of hardwares,\nmodel architectures and inference algorithms.\nTo address these challenges, xLLM builds a novel decoupled service-engine architecture. At\nthe service layer, xLLM-Service features an intelligent scheduling module that efficiently processes\nmultimodal requests and co-locates online and offline tasks through unified elastic scheduling to\nmaximize cluster utilization.\nThis module also relies on a workload-adaptive dynamic Prefill-\nDecode (PD) disaggregation policy for instance scheduling and a novel Encode-Prefill-Decode\n(EPD) disaggregation policy designed for multimodal inputs. Furthermore, it incorporates a dis-\ntributed architecture to provide global KV Cache management for efficient AI accelerator memory\nhandling and robust fault-tolerant capabilities for high availability. At the engine layer, xLLM-\nEngine co-optimizes system and algorithm designs to fully saturate computing resources. This is\nachieved through comprehensive multi-layer execution pipeline optimizations, including overlap-\nping CPU scheduling with AI accelerator operations to minimize computational bubbles, employing\ndual-stream parallelism to overlap computation with communication, and fine-grained overlapping\nof various computational units to maximize hardware utilization. These are complemented by an\nadaptive graph mode that drastically reduces kernel launch overhead, alongside the innovative \"log-\nically contiguous, physically discrete\" xTensor memory management which resolves memory allo-\ncation conflicts. xLLM-Engine also further integrates algorithmic enhancements such as optimized\nspeculative decoding and dynamic Expert Parallel Load Balance (EPLB), collectively serving to\nsubstantially boost throughput and inference efficiency.\nExtensive evaluations demonstrate that xLLM delivers significantly superior performance and\nresource efficiency. Under identical TPOT constraints, xLLM achieves throughput up to 1.7× that\nof MindIE and 2.2× that of vLLM-Ascend with Qwen-series models, while maintaining an average\nthroughput of 1.7× that of MindIE with Deepseek-series models. We have deployed xLLM in\nproduction to support a range of core business scenarios at JD.com, covering areas including LLM,\nMultimodal Large Language Model (MLLM), and generative recommendation. These applications\nencompass the JingYan AI chatbot, marketing recommendations, product understanding, customer\nservice assistants, and more. xLLM framework is publicly available at https://github.com/\njd-opensource/xllm and https://github.com/jd-opensource/xllm-service.\nBCorresponding authors: {liutongxuan1, zhangke323}@jd.com, {guyue, yangtong}@pku.edu.cn, hai-\nlong.yang@buaa.edu.cn, lj@ustc.edu.cn, {dinggg, huichen}@tsinghua.edu.cn.\n1\narXiv:2510.14686v1 [cs.DC] 16 Oct 2025\n\nxLLM Technical Report\nContents\n1\nIntroduction\n3\n2\nSystem Overview\n5\n2.1\nxLLM-Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n5\n2.2\nxLLM-Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n6\n3\nxLLM-Service Designs\n7\n3.1\nOnline-Offline Co-location Scheduler Policy . . . . . . . . . . . . . . . . . . . . .\n7\n3.2\nDynamic PD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . .\n9\n3.3\nHybrid EPD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . .\n10\n3.4\nGlobal KV Cache Management . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n12\n3.5\nFast Fault Recovery Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . .\n13\n4\nxLLM-Engine Designs\n13\n4.1\nMulti-layer Pipeline Execution Engine . . . . . . . . . . . . . . . . . . . . . . . .\n13\n4.2\nAdaptive Graph Mode\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n15\n4.3\nEfficient Memory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n17\n4.4\nAlgorithm Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n19\n4.4.1\nOptimized Speculative Decoding . . . . . . . . . . . . . . . . . . . . . . .\n19\n4.4.2\nDynamic EP Load Balance . . . . . . . . . . . . . . . . . . . . . . . . . .\n20\n4.4.3\nHierarchical DP Load Balance . . . . . . . . . . . . . . . . . . . . . . . .\n21\n4.5\nGenerative Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n22\n4.5.1\nHost-side Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n23\n4.5.2\nDevice-side Optimization\n. . . . . . . . . . . . . . . . . . . . . . . . . .\n24\n5\nEvaluations\n24\n5.1\nMain Results\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n24\n5.1.1\nBenchmarking Performance . . . . . . . . . . . . . . . . . . . . . . . . .\n25\n5.1.2\nBusiness Serving Scenarios\n. . . . . . . . . . . . . . . . . . . . . . . . .\n26\n5.2\nAblation Study\n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n28\n6\nFuture Work\n30\n6.1\nFostering an Open and Diverse Hardware Ecosystem . . . . . . . . . . . . . . . .\n31\n6.2\nCultivating a Vibrant and Responsive Model Ecosystem . . . . . . . . . . . . . . .\n31\n6.3\nEvolving into an AI-Native Application Framework . . . . . . . . . . . . . . . . .\n32\n7\nConclusion\n32\n2\n\nxLLM Technical Report\n1\nIntroduction\nIn recent years, large language models (LLMs) with parameters ranging from billions to trillions\n(GPT [1], Claude [2], DeepSeek [3], LLaMA [4], etc.) have achieved breakthrough progress in the\nfields of natural language processing and multimodal interaction, which drives an urgent demand\nin the industry for efficient inference engines and service systems. These models are being rapidly\ndeployed in core business scenarios such as intelligent customer service [5], real-time recommen-\ndation [6], and content generation [7]. However, how to reduce the cost of model inference and\nimprove computing efficiency remains a key challenge for large-scale commercial serving.\nCurrent mainstream LLM inference frameworks (vLLM [8], SGLang [9], TensorRT-LLM [10], etc.)\nface four key challenges in enterprise-level serving scenarios: First, in an inference cluster with hy-\nbrid deployment, online inference requests exhibit significant tidal characteristics [11, 12]. Current\nscheduling systems fail to meet the service level objective (SLO) for online services while fully\nleveraging idle periods of online services to increase the throughput of offline tasks. Second, ex-\nisting Prefill-Decode (PD) disaggregation architecture [13–15] assumes static resource allocation\nfor the two phases, which cannot adapt to the dynamically changing request loads (i.e., input/output\nlengths fluctuate) in real-world applications, resulting in low AI accelerator utilization and increased\nSLO violation risks. Third, there is a lack of strategy to efficiently service multimodal requests\n(i.e., image, voice and text input [16, 17]), including parallel processing for the encode phase and\nfine-grained resource allocation accordingly. Fourth, as the scale of the inference cluster increases,\nensuring fast fault detection and service recovery for nodes or instances is critical to maintaining the\nstability of inference services.\nThe evolving computing paradigms also present significant performance challenges on existing LLM\ninference engines: First, they struggle to fully utilize the computing units of modern AI accelera-\ntors [18, 19]. Second, the All-to-All communication overhead [20] and expert parallel (EP) load\nimbalance in the Mixture of Experts (MoE) architecture [21, 22] restrict the scalability of the sys-\ntem. Third, as the model context window continues to expand, efficient KV Cache management\nbecomes critical to inference performance [23]. Forth, Due to the unpredictable nature of infer-\nence requests, conventional static scheduling and operator strategies face difficulties in effectively\nbalancing workloads across computing units in Data Parallelism (DP).\nTo address above challenges , we propose xLLM, an efficient and intelligent LLM inference frame-\nwork featuring a service-engine decoupled design. xLLM achieves efficient support for enterprise-\nlevel inference through the following innovations: at the service layer, xLLM has made ground-\nbreaking achievements in 1) unified elastic scheduling for online/offline requests, 2) workload-\nadaptive dynamic PD disaggregated architecture, 3) novel Encode-Prefill-Decode (EPD) disaggrega-\ntion for multimodal requests, and 4) a distributed cache management and fault-tolerance framework;\nat the engine layer, xLLM enhances the resource efficiency across the full-stack of “communication-\ncomputation-storage”, including 1) a multi-layer pipeline execution mechanism, 2) efficient comput-\ning and memory optimization, and 3) intelligent algorithm designs.\nSpecifically, xLLM-Service schedules online requests with preemptive execution and offline re-\nquests in a best effort manner, to maximize resource utilization while strictly ensuring SLOs for\nonline services. To address the inherent limitations of the static PD configuration, xLLM incorpo-\nrates an adaptive scheduler to dynamically adjust the proportion of PD instances for each request\nand supports fast role reversal for instances, by monitoring key metrics such as Time to First Token\n(TTFT) and Time Per Output Token (TPOT) [24]. For multimodal visual requests, xLLM auto-\nmatically selects the optimal EPD phase-disaggregation strategy based on pre-profiling, for the best\nperformance trade-off between throughput and latency. To handle failures such as hardware prob-\nlems, network faults, or software errors [25], xLLM collects node error information, evaluates KV\nrecomputation or migration costs of interrupted requests, and makes optimal global rescheduling\ndecisions. Across multiple instances, xLLM supports KV offloading and routing migration within a\nhybrid storage architecture, improving KV storage capacity and cache hit rates.\nxLLM-Engine employs a multi-layer execution pipeline that incorporates hardware-specific op-\ntimizations: (i) at the framework scheduling layer, it implements asynchronous CPU-accelerator\nscheduling to minimize computational idle time; (ii) at the model graph layer, it utilizes dual-stream\nmicro-batch parallelism to overlap computation with communication; (iii) at the operator level, it\nachieves kernel computation and memory access overlapping. For computational efficiency, xLLM-\n3\n\nxLLM Technical Report\nMulti-modal\nRequests\nxLLM Service\nFault Recovery\nGlobal Scheduler\nHybrid EPD Disagg.\nDynamic PD Disagg.\nOn/Offline Co-location\nGlobal Cache Mgr\nxLLM Engine\nAdaptive Graph\nOperator Optim.\nxTensor Memory\ndrive\ndrive\nmanage\nschedule\nSpeculative Inference\nDynamic EPLB\nPrefill\nLocal KV \nCache\nKV Cache \nTransfer\n…\nPrefill\nLocal KV \nCache\nKV Cache \nTransfer\n…\nPrefill Instance\nLocal KV \nCache\nKV Cache \nTransfer\n…\nDecode\nLocal KV \nCache\nKV Cache \nTransfer\n…\nDecode\nLocal KV \nCache\nKV Cache \nTransfer\n…\nDecode Instance\nLocal KV \nCache\nKV Cache \nTransfer\n…\nSystem-level Designs\nBatching Scheduler\nAlgorithm-driven Tech.\nco-optimize\nHierarchical DPLB\nMulti-layer Pipeline\nFigure 1: Overview of xLLM’s capabilities, with decoupled service-engine architecture.\nEngine automatically fuses small kernels into a unified computation graph and dispatches them to the\naccelerator in a single operation, significantly reducing kernel launch overhead. Regarding memory\nmanagement efficiency, xLLM-Engine introduces the xTensor memory management scheme, which\nemploys a “logically contiguous, physically discrete\"\" KV cache storage structure that resolves the\nconflict between memory contiguity requirements and dynamic allocation needs. To further enhance\nperformance, xLLM-Engine incorporates an adaptive speculative decoding mechanism, a redundant\nexpert-based load balancing algorithm for EP, and a hierarchical load balancing algorithm for DP.\nAdditionally, xLLM-Engine provides scenario-specific optimizations, such as for generative recom-\nmendation – one of JD.com’s core businesses – where it achieves 23% performance improvement\nthrough host-kernel operation overlapping.\nIn summary, we make the following contributions when developing xLLM:\nxLLM Intelligent Service Capabilities\n• We design a unified scheduling algorithm for online/offline workloads (§3.1).\n• We implement a workload-adaptive PD disaggregation policy to address scenarios with rapidly\nchanging traffic load and request input/output lengths (§3.2).\n• We propose a hybrid EPD disaggregation policy for multimodal requests, achieving intelligent\nresource allocation across different phases (§3.3).\n• We leverage multi-level KV Cache management and global KV Cache routing strategy to expand\nKV Cache capacity and improve cache hit rates (§3.4).\n• We design a multi-node fault tolerance architecture to ensure high service availability (§3.5).\nxLLM Intelligent Engine Capabilities\n• We achieve intelligent and efficient inference, through hardware-software co-design to improve\nhardware computing efficiency, including pipeline execution (§4.1), graph optimization (§4.2),\nand memory optimization (§4.3).\n• We enhance inference performance through algorithmic optimizations (§4.4), including optimized\nspeculative decoding (§4.4.1), dynamic expert parallel load balance (§4.4.2), and hierarchical data\nparallel load balance (§4.4.3).\n• We optimize online inference for generative recommendation scenarios (§4.5).\n4\n\nxLLM Technical Report\nPredictor\nProfiler\nDynamic PD \nDisaggregation\nHybrid EPD \nDisaggregation\nOn/Offline\nCo-locatiom\nPrefill Pool\nDecode pool\nEncode Pool\nGlobal Cache Management\nFault Recovery Framework\nPre-Processing \nLayer\nScheduling \nLayer\nonline\npreempt\nsupport\ndisaggregate\ndynamic PD \nallocation\nKV & Images\nResource \nLayer\nOffline Requests \nOnline Requests \nFigure 2: System workflow of xLLM-Service.\n2\nSystem Overview\nThe overall architecture of xLLM is depicted in Figure 1. Upon request arrival, xLLM-Service\nperforms intelligent scheduling to distribute each request to one of three elastic instance pools and\nmanages instance migration across these pools during runtime. xLLM-Engine then drives efficient\nrequest inference by orchestrating system- and algorithm-level optimizations.\n2.1\nxLLM-Service\nWe describe the workflow of xLLM-Service in Figure 2. The system comprises three primary lay-\ners: 1) a preprocessing layer consisting predictor and profiler, 2) a scheduling layer integrating\nthree policies (Dynamic PD Disaggregation Policy, Hybrid EPD Disaggregation Policy, and Online-\nOffline Co-location Policy), and 3) a resource layer consisting of three heterogeneous instance pools.\nSpecifically, the Dynamic PD Disaggregation Policy leverages information from the predictor to\ndynamically convert PD instances based on the workload status of the two instance types. The Hy-\nbrid EPD Disaggregation Policy utilizes the profiler to determine the optimal EPD disaggregation\nstrategy for multimodal requests. Concurrently, the Online-Offline Co-location Policy dispatches re-\nquests according to their online or offline attributes. KV and image caches are offloaded and routed\namong distributed instances, while the fault recovery framework ensures the high availability of the\nentire service.\nElastic Instance Pools.\nInstances in a cluster are partitioned into three elastic pools:\n▷Prefill Instance Pool: This pool handles the prefill phase of text requests.\n▷Decode Instance Pool: This pool handles the decode phase of text requests.\n▷Encode Instance Pool: This pool handles the encode phase of multimodal requests.\n5\n\nxLLM Technical Report\nWe design the instances in Prefill and Decode pools as stateless, i.e., instances do not require phys-\nical migration between instance pools. Instead, they achieve flexible switching between Prefill-\nDecode roles based on the type of request being processed.\nRequest Preprocessing.\nxLLM-Service implements unified resource management for both online\nand offline requests. Online requests are submitted in a preemptive and deadline-prioritized manner,\nand are co-deployed with best-effort offline tasks within the shared resource pool. During the run-\ntime of an offline request, the Unified-Scheduler dynamically scales it up/down based on the tidal\ntraffic characteristics of online requests. When an online request is submitted, it first passes through\nthe pre-processing layer, which consists of two modules:\n▷TTFT Predictor: A TTFT prediction model built for text requests. It evaluates SLO fulfillment\nby analyzing queueing delays from each prefill instance queue and request input lengths, thereby\nguiding instance allocation for the Dynamic PD Disaggregation Policy in the scheduling layer.\n▷\nEPD Profiler: A profiler for multimodal requests that uses binary search to identify optimal\ndeployment configurations: (1) EPD separation strategy, choosing from three approaches: EP-D\n(i.e., aggregated execution of Encode and Prefill phases, with Decode phase executed separately),\nED-P, or E-P-D; (2) The maximum batch size for the Encoder phase; (3) The maximum number of\ntokens for Prefill/Decode’s inputs. The Hybrid EPD Disaggregation Policy in the scheduling layer\nwill use the optimal configuration determined for phase disaggregation and task dispatching.\nIntelligent Scheduling.\nThe intelligent scheduling layer adjusts resource allocation for requests\nacross their full lifecycles. It contains three major scheduling policies designed for various scenarios:\n▷Online-Offline Co-location Policy: This policy implements a preemptive scheduler for managing\nonline and offline requests. When the load of online requests reaches the peak, they preempt some\noffline requests on PD instances. To surrender resources to offline requests, when the load on P\ninstances decreases (usually earlier than on D instances), P instances continue to process offline\nprefill requests and also migrate decode offline requests from D instances to P instances.\n▷Dynamic PD Disaggregation Policy: This adaptive scheduling policy is responsible for dynam-\nically managing P and D instance allocation. It intelligently assigns requests to suitable instances\nusing a heuristic algorithm guided by the TTFT Predictor. Furthermore, it implements a feedback\nmechanism by continuously collecting performance data from computing instances, enabling run-\ntime monitoring and adjustment of allocation decisions to maintain system efficiency.\n▷Hybrid EPD Disaggregation Policy: This multimodal policy executes the three-phase disaggre-\ngation based on strategies searched by the EPD Profiler. For the EP-D disaggregation, the fused EP\nphase executes in the P instance pool; for the ED-P disaggregation, the fused ED phase executes in\nthe D instance pool; for the E-P-D disaggregation, the three phases execute separately in the three\ninstance pools. This deployment also enables multimodal requests to benefit from the adjustment of\nDynamic PD Disaggregation Policy.\nKV-centric Storage Architecture.\nThe instance storage employs a hybrid architecture (HBM-\nDRAM-SSD) to cache KV values and image tokens. At the global level, xLLM takes the idea from\nMooncake Store [23] and extends it to domestic accelerators with specific optimizations. Routing\nand reuse of caches across instances are determined by embedded intelligent routing strategies.\nEfficient Fault-tolerant.\nThe fault recovery framework of xLLM-Service supports fault detection\nand fast recovery of instances from the three elastic pools (E, P, D). For requests on failed instances,\nthe architecture manages the migration of image caches between instances, and automatically de-\ncides the optimal KV recomputation or migration strategy for handling affected KV caches.\n2.2\nxLLM-Engine\nDuring request execution, the xLLM-Engine layer provides intelligent computing capabilities. We\nachieve various joint inference accelerations between the computing system layer and the algorithm-\ndriven layer in the following ways:\nComputing System Layer\n6\n\nxLLM Technical Report\nLatency-relaxed instance\nOnline Prefill\nLatency-strict instance\nOnline / Offline\nMixed Decode\nOffline Prefill\nOffline Decode\nResource Plane\nControl Plane\nOn-demand\nPreemption\nOffline Request \nMigration\nOffline Request \nGating \nMix decoding \nSelection\nFigure 3: Overview design of Online-Offline Co-location Scheduler Policy.\n▷Multi-layer Pipeline Execution Engine: In the framework layer, CPU tasks are scheduled asyn-\nchronously to create a pipeline with inference computation, reducing computational bubbles; in the\nmodel graph layer, a single batch is split to form a pipeline between two micro-batches, effectively\noverlapping computation and communication; in the operator kernel layer, operations are pipelined\nacross different computational units, enabling computation and memory access to overlap.\n▷\nGraph Optimization for Dynamic Inputs: Small kernels in the decoding stage are fused into a\nsingle computational graph. To handle variable sequence lengths and batch sizes, we parameterize\nthe input dimensions, and employ a multi-graph caching scheme to reduce compilation overhead.\n▷xTensor Memory Management: It uses a “logically continuous, physically discrete” KV storage\nstructure. It allocates physical memory space on-demand during token generation for each request,\nwhile asynchronously predicting and intelligently mapping the physical pages required for the next\ntoken. When a request completes, the existing physical memory will be reused to execute the next\nrequest.\nAlgorithm-Driven Layer\n▷Speculative Decoding: xLLM-Engine incorporates an optimized speculative inference algorithm\nthat generates multiple tokens at once to boost throughput [26]. It further optimizes the computing\narchitecture through ways like asynchronous CPU processing and reduced data transfer.\n▷EP Load Balance: For MoE models, xLLM implements expert weight updates based on historical\nexpert load statistics, enabling effective dynamic load balancing during inference.\n▷DP Load Balance: For data parallel (DP) deployment, xLLM achieves fine-grained load balance\nby kvcache-aware instance allocation, inter-DP requests migration, and intra-DP computing units\nallocation.\nWe will elaborate on the detailed design of xLLM-Service in §3 and outline the optimizations within\nthe xLLM-Engine in §4. A comprehensive evaluation of xLLM will be presented in §5.\n3\nxLLM-Service Designs\n3.1\nOnline-Offline Co-location Scheduler Policy\nOnline/Offline Request Characteristic.\nLLM services can be categorized into two types based\non their service modes: online and offline requests. Online requests, including those from chat-\nbots [1, 2, 27], code completion [28–30], and recommendation systems [31, 32], constitute latency-\nsensitive workloads. These services must respond immediately upon request arrival, often returning\neach generated token in real time via streaming output. Consequently, they impose strict SLO re-\nquirements on TTFT or TPOT to ensure a satisfactory user experience. In contrast, offline services,\nsuch as document analysis [33] and intelligent data annotation [34], are non-real-time workloads\nwith minimal latency constraints and thus no stringent SLO requirements.\nFurthermore, we observe that the request traffic for online services typically exhibits significant\nvolatility, including tidal variations at hourly or daily scales and sudden bursts at minute-level inter-\nvals [11, 12]. While cluster auto-scaling [35] could theoretically mitigate resource underutilization\ncaused by tidal patterns, the slow cold-start latency of instances—involving model loading and com-\n7\n\nxLLM Technical Report\nplex initialization—renders it ineffective for responding to rapid traffic spikes [36], increasing the\nrisk of service-level agreement violations.\nOnline/Offline Request Co-location Deployment.\nTo address these practical challenges, we\nadopt a hybrid deployment strategy that co-locates online and offline requests. In such a system,\noffline requests can utilize idle resources during off-peak periods of online traffic. Conversely, dur-\ning online traffic peaks, offline tasks can be preempted, as they are not bound by strict SLOs. This\napproach significantly enhances aggregate resource utilization and mitigates idleness during traffic\ntroughs. Although some recent work [37–39] also attempts to co-locate offline requests, they have\nnot explored the multi-instance scenario, especially for PD disaggregation.\nActually, the PD disaggregation architecture has demonstrated superior latency performance and is\nincreasingly becoming a mainstream design paradigm in industry [13, 23]. However, directly apply-\ning online-offline co-location to PD-disaggregated systems introduces critical PD load imbalance\nissues: (1) such systems require the load ratio between prefill and decoding stages to align with their\nrespective resource allocation ratios; otherwise, one stage may become a bottleneck, causing block-\ning or resource underutilization in the other. (2) Furthermore, as these PD load variations exhibit\nhigh volatility and burstiness similar to online traffic patterns, existing PD disaggregation techniques\nstruggle to effectively address this challenge.\nLatency-Constrained Decoupled Architecture.\nWe rethink the design of PD disaggregation ar-\nchitecture under online-offline co-location deployment. The latency advantage of PD disaggregation\nessentially stems from the separation of latency constraints: the decoding phase is highly sensitive\nto per-step latency and cannot be blocked by long-duration operations, thus necessitating decou-\npling from the prefill phase. Inspired by this insight, we propose a latency-constrained decoupled\narchitecture, as illstrated in Figure 3. This design regards cluster resources as two pools: a latency-\nrelaxed pool (corresponding to original Prefill instances) and a latency-strict pool (corresponding to\noriginal Decode instances). All tasks are then reassigned to one of the two resource pools based on\ntheir inherent latency characteristics and requirements. Within this architecture, the decoding phase\nof offline requests can be executed in either resource pool. This flexibility allows us to dynamically\nadjust the load ratio between the two pools, thereby maximizing the overall resource utilization of\nthe cluster.\nHowever, it also introduces another two challenges: (1) Complex Scheduling Space: since the de-\ncoding of offline requests can be performed on either type of instance, how to leverage this flexibility\nto design new intelligent scheduling remains unknown. (2) Strict SLO Guarantee: the execution of\noffline requests consumes resources, their prefill phase may block newly arrived online requests,\nand their decode phase on latency-strict nodes may slow down the overall response speed. Both\nscenarios can compromise the SLO satisfaction of online requests.\nSolution 1 - Performance Bottleneck Analysis.\nWe construct an LLM inference performance\nmodel based on the Roofline Model [40] and online factor learning. This model is designed to\npredict the latency, computation utilization, and memory utilization of both the prefill and decode\nphases. Since decoding operations executed on latency-strict instances typically account for a large\nproportion of workloads and are highly performance-sensitive, we set balancing computational and\nmemory resources as the optimization objective. By analyzing performance bottlenecks through\nthis model, we can select more appropriate offline requests to merge into decoding batches, thereby\nimproving resource utilization efficiency.\nSolution 2 - Efficient Preemption Mechanism.\nTo strictly ensure that the SLO of online requests\nremains unaffected, we introduce a preemption mechanism that allows online requests to preempt\noffline requests. For offline prefill tasks running on latency-relax nodes, we propose a model execu-\ntion interruption technique, which enables preemption within an acceptable latency range without\nincurring additional model maintenance overhead. For decoding tasks running on latency-strict\nnodes, we leverage the performance model to dynamically select requests for decoding batching,\nensuring that decoding latency always meets SLO constraints.\n8\n\nxLLM Technical Report\nStateless Instance\nLocal Scheduler\nrequest queue\nmigration queue\nExecution\nEngine\nSchedule Request\n& Cache Migration\nP\nD\nprefill\nunfinished\nP\nD\ndecode\nunfinished\ndecode \nfinished\nprefill \nfinished\nD\nP\nGlobal Scheduler\nInstance Scheduler\nRequest Scheduler\nInstance Monitor\nTTFT / TPOT, Req Len, Memory Util., etc.\nschedule\nschedule\nreport status\nFigure 4: Overview design of Dynamic PD Disaggregation Scheduler Policy.\n3.2\nDynamic PD Disaggregation Scheduler Policy\nInefficiency of Existing PD Disaggregation Policy under Workload Fluctuations.\nThe Prefill-\nDecode (PD) disaggregated inference architecture [13, 24, 23, 41] partitions computing instances\ninto dedicated prefill and decode instances, each handling their respective processing phases. This\ndesign mitigates interference between prefill and decode requests, achieving superior performance\ncompared to PD co-located architectures [13].\nHowever, we observe that most existing PD disaggregated systems employing static instance parti-\ntioning schemes suffer from low hardware utilization and inadequate responsiveness to traffic bursts.\nOn one hand, existing analysis-based methods [13, 24, 23, 41] typically determine the PD ratio using\nprofiling or simulator data. These approaches remain effective only when request arrival patterns and\nlength distributions remain relatively stable. Under significant workload fluctuations, pre-collected\nanalysis data often fails to accurately capture real-time request characteristics, leading to mismatches\nbetween preset load-balancing ratios and actual load requirements. On the other hand, when con-\nfronting substantial variations in input/output lengths within production workloads, existing PD dis-\naggregation architectures [13, 42] generally adopt dynamic instance type adjustment strategies [43].\nNevertheless, online switching between prefill and decode instances typically involves multiple\nsteps—including monitoring, waiting for flip conditions, and instance restarting—which introduces\nsubstantial latency overhead [15]. To address these limitations, we propose a Dynamic PD Dis-\naggregation Policy that adaptively adjusts resource allocation in response to real-time workload\ncharacteristics.\nRuntime Instance Monitor.\nPerformance indicators such as TTFT and TPOT directly reflect the\ncapability of instances to process requests in the prefill and decode phases, respectively. They serve\nas the core basis for evaluating whether an inference system can meet SLO requirements. Ideally,\nprediction-based methods could be adopted to dynamically assess these indicators. However, while\nTTFT exhibits relatively predictable characteristics (as its computation time is proportional to the\nsquare of the input sequence length [44, 41]), due to the uncertainty in the length of output tokens\nand transfer overhead, TPOT is difficult to accurately predict using traditional input/output features\nand cluster load metrics. To address this issue, we deploy additional instance monitors to collect\nreal-time performance data from each computing instance. This data includes metrics such as the\nnumber and length of prefill and decode requests, memory usage, TTFT, TPOT, and token generation\nintervals. The system can further use these real-time performance indicators to dynamically evaluate\ninstance loads and adjust scheduling strategies.\nStateless Instance and Elastic Pool.\nAs illustrated in Figure 3, Dynamic PD Disaggregation Pol-\nicy adopts a design of stateless instances and elastic instance pools to enable fast and dynamic\nswitching of instance roles. First, it treats prefill or decode phase as the request attribute rather than\ninstance attribute. Instances are designed to be stateless, allowing each instance to process both\nprefill and decode requests simultaneously. Additionally, to facilitate the management of multiple\ninstances, we further extend PD pools to four elastic instance pools (i.e., P, D, P→D, D→P) as de-\n9\n\nxLLM Technical Report\nscribed in §2.1. When flipping an instance (switching its role), we only needs to remove the instance\nfrom its original pool and move it to the new pool. This achieves zero-wait-time instance scheduling,\navoiding the overhead of instance restart or model reloading incurred in traditional systems.\nSLO-aware Instance Role Switching.\nThe instance scheduling strategy is dynamically adjusted\nstrictly based on SLO objectives: during the prefill phase, if it is predicted that existing instances\ncannot meet the TTFT requirements, the conversion of decode instances is triggered; while during\nthe decode phase, when resource shortage occurs, the average token generation interval exceeds the\nTPOT threshold, or prefill instances are idle, the conversion of prefill instances to decode instances\nwill be initiated to cope with sudden traffic surges. Specifically, when decode instances are reallo-\ncated to prefill instances, the scheduler prioritizes selecting the instance with the lightest load (i.e.,\nthe fewest tokens being processed) from the P→D pool for role conversion, and always ensures\nthat at least two decode instances are available; conversely, when prefill instances are reallocated\nto decode instances, the scheduler prioritizes scheduling from the D→P pool, which avoids local\noverload and maximizes resource utilization.\nSLO-aware Request Scheduling.\nThe request scheduling scheme adopts a two-level architecture:\n▷Global Request Scheduler: The scheduler implements a greedy strategy of prioritizing the light-\nest load, while being strictly restricted by SLO constraints. For prefill requests, the scheduler first\nevaluates the estimated queuing latency of each instance in the Prefill pool, selects the candidate in-\nstance with the smallest latency, and then invokes the TTFT prediction model for verification: if the\nestimated TTFT can still meet the SLO requirements after assigning the request to this instance, the\nrequest is allocated immediately; otherwise, it continues to find a suitable instance in the D→P pool.\nIf no instance can meet the TTFT requirements, the instance scheduling mechanism is triggered to\nallocate resources from the decode side. For decode requests, the scheduler gives priority to having\nthe original prefill instance continue processing (to avoid KV Cache transfer overhead); secondly,\nit selects the instance with the fewest running tokens in the Decode pool, and checks whether the\ntotal number of tokens in its current batch is below the memory capacity upper limit and comput-\ning throughput limit determined by pre-analysis, so as to ensure that the new request will not cause\nTPOT to exceed the standard.\n▷Local Request Scheduler: Each instance adopts a refined queue management strategy internally.\nKV Cache transfer events are placed in an independent migration queue and processed sequen-\ntially in accordance with the FCFS principle; for forward requests, an innovative scheme combining\nChunked Prefill [45] and Continuous Batching [8] is adopted: on the premise of ensuring that de-\ncode requests are prioritized to enter the running batch, the remaining computing resources are used\nto process chunked prefill requests in parallel [46].\n3.3\nHybrid EPD Disaggregation Scheduler Policy\nChallenges of Multi-modal Inference.\nThe inference process of multimodal large language mod-\nels (MLLMs) [47–52] typically consists of three phases: the image encoding phase (for extracting\nimage features), the prefill phase (for encoding images and text prompts, feeding them into the\nlanguage model to generate the first output token, and caching intermediate states), and the de-\ncode phase (for iteratively generating subsequent tokens based on the cached data). Existing main-\nstream inference engines, such as vLLM [8], Text-Generation-Inference [53], SGLang [9], and Dist-\nServe [13] are all tailored for LLMs, and thus face several challenges in handling inference tasks for\nMLLMs:\n▷Insufficient Parallelism: For example, the inference of visual models and language models can\nbe executed in parallel, thus improving the utilization of computing resources. However, most ex-\nisting inference engines [54] adopt a serial strategy, failing to exploit the inter-request parallelism\neffectively.\n▷\nCoarse-grained Scheduling: The decode phase is a memory-intensive task, which is suitable\nfor batch processing to improve throughput [55]; the prefill phase is computationally intensive,\nand it is appropriate to adopt the chunked prefill [45] together with the decode phase to balance\nlatency and throughput [56]. The computation and memory access overhead of the encode phase\nlies between the two, and it can also benefit from independent scheduling and batch processing.\nNevertheless, existing engines process encode and prefill in a combined manner, failing to perform\n10\n\nxLLM Technical Report\nTPOT & TTFT SLO Target\nED\nD\nP\n+\nEP +\nE\nP\nD\n+\n+\nEPD Profiler\nEncode Instance\nVision Model\nImage Cache\nBatch \nScheduler\nMigrate \nScheduler\nPrefill Instance\nLanguage Model\nImage Cache\nBatch \nScheduler\nMigrate \nScheduler\nKV Cache\nDecode Instance\nLanguage Model\nBatch \nScheduler\nMigrate \nScheduler\nKV Cache\nHybrid EPD Disaggregation\nasync transfer\nasync transfer\nFigure 5: Overview design of Hybrid EPD Disaggregation Policy.\nphase-specific batch processing and lacking support for chunked prefill. Coarse-grained scheduling\nmakes it difficult to finely control the execution time.\n▷Disaggregation Strategy: Existing architectures such as DistServe [13] reduce resource interfer-\nence by decoupling the prefill and decode phases. However, in multimodal inference, encoding and\nprefill jointly affect TTFT, while prefill and decode jointly determine TPOT. Under different loads,\nhow to select the optimal decoupling strategy remains a challenge [57]. For example, the perfor-\nmance of strategies such as E+P+D, EP+D, and ED+P in different tasks has not yet been evaluated,\nand in-depth analysis is required to guide system design.\nDual-stream Parallel.\nAs illustrated in Figure 5, we adopt a dual-stream parallel mode, where the\nvisual model and language model are assigned to separate execution streams: the visual stream is\ndedicated to performing computationally intensive image encoding tasks, while the language stream\nhandles the prefill and decode operations for language generation. By isolating workloads into\ndistinct streams, our system achieves concurrent execution of heterogeneous phases from different\nrequests.\nThree-Phase Disaggregation.\nTo address the differences in computation and memory access\ncharacteristics among tasks in each phase of multimodal inference, we propose a phase-aware\nscheduling strategy. Requests are subdivided into three phases within an instance: encode, prefill,\nand decode. Batch processing and scheduling optimizations are performed for each phase respec-\ntively:\n▷Optimized Batch Processing: The maximum batch size for image encoding and the token budget\nfor the language model are set according to the user’s SLO. Specifically, during system startup,\nwe use a binary search method to profile the maximum encoding batch size and the model’s token\nbudget, ensuring that the execution time of subsequent batch processing tasks in each iteration is\nless than the TPOT SLO. In each iteration, we (i) first add all running decode requests to the current\nbatch; (ii) then check for any chunked prefill tasks that have been partially computed. If such tasks\nexist, we add them to the batch; (iii) if there are no chunked prefill tasks, we check for pending\nencoding tasks and add them if any exist. This approach aims to complete requests in the prefill\nphase as quickly as possible, reducing their TTFT. New requests’ encoding phases are processed\nonly when no requests are in the prefill phase.\n▷\nPhase-aware Scheduling: To address the difficulty in selecting decoupling strategies for mul-\ntimodal inference tasks, xLLM proposes the Hybrid Encode-Prefill-Decode Disaggregation archi-\ntecture, an innovative multi-instance collaborative inference framework. In this architecture, each\ninstance only executes part of the subtasks among the three phases, while the remaining phases are\nprocessed by migrating requests to other instances, thereby avoiding resource waste and interfer-\nence. During runtime, our system automatically selects the optimal disaggregation strategy based\non historical request profiling (via EPD Profiler) to dynamically balance throughput and latency ob-\njectives. Compared with the traditional approach of binding instances to single-phase or full-phase\ntasks, this architecture significantly improves the overall system processing capability and resource\nutilization efficiency while meeting SLO requirements.\n11\n\nxLLM Technical Report\nMaster\nElection\nCache & Load\n \nMetric Sync\nService \nDiscovery\nETCD\nGlobal Multi-Level KV Cache Management\nMaster\nGlobal\nScheduler\nGlobal \nKVCache Mgr\nSelect Master\nWrite Cache &\nLoad Metrics\nWatch & Notify\nHeartBeat (Lease)\nReport Cache &\nLoad Metrics\nReq\nReq\nResp\nPrefill\nLocal KV \nCache\nKV Cache \nTransfer\n…\nPrefill\nLocal KV \nCache\nKV Cache \nTransfer\n…\nPrefill Instance\nLocal KV \nCache\nKV Cache \nTransfer\n…\nDecode\nLocal KV \nCache\nKV Cache \nTransfer\n…\nDecode\nLocal KV \nCache\nKV Cache \nTransfer\n…\nDecode Instance\nLocal KV \nCache\nKV Cache \nTransfer\n…\nSlave\nSlave\nSlave\nFigure 6: The framework of Global Multi-Level KV Cache Management.\n3.4\nGlobal KV Cache Management\nDuring the decode phase of LLMs, subsequent tokens are generated autoregressively one by one. Al-\nthough the computational cost per step is relatively low, frequent access to the historical KV Cache\nis required, making memory bandwidth the primary bottleneck. As model size expands and con-\ntext window grows, the memory consumption of the KV Cache exhibits an exponentially increasing\ntrend. For instance, a context of 128K tokens may consume over 40GB of memory, severely strain-\ning the memory resources of single-GPU devices [3].\nAlthough current mainstream optimization solutions like vLLM [8] and FasterTransformer [58] have\nmade significant progress in single-instance environments, many critical issues remain unresolved.\nIn long-context scenarios, the time required for the prefill phase increases dramatically, while the\ndecode phase suffers from intense competition for memory bandwidth. To meet stringent SLO\nrequirements (e.g., TTFT < 2s, TBT < 100ms), one instance often has to reserve excessive re-\nsources, instead of leveraging resources in other instances. This significantly restricts the overall\ncluster resource utilization. To address these challenges, we propose a global multi-level KV Cache\nmanagement system with a memory-compute integrated architecture, as illustrated in Figure 6.\nDistributed Storage and KV Transfer.\nWe adopt the Mooncake Store [23], a KV cache-centric\nstorage engine, as the underlying storage infrastructure for xLLM’s KV Cache, along with the Moon-\ncake Transfer Engine as the transmission component. Mooncake Store leverages striping and paral-\nlel I/O techniques to fully utilize the aggregated bandwidth of multiple network cards. It provides\nthree persistence strategies—Eager, Lazy, and None—to meet data durability requirements across\nvarious scenarios. Additionally, Mooncake Store offers flexible object storage capabilities, support-\ning multiple replicas and eventual consistency, which effectively mitigates hot-spot access pressure.\nAs the core transmission engine of the system, Mooncake Transfer Engine [23] automatically selects\nthe optimal transmission path based on data location and abstracts low-level complexities through a\nunified Segment and BatchTransfer interface.\nMulti-Level KV Cache Management.\nAt the global level, the system employs ETCD [59] as a\nmetadata service middleware to achieve cluster service registration, load information synchroniza-\ntion, and global cache state management. Each computing instance maintains a local multi-level\nKV Cache pool (HBM > DRAM > SSD) adhering to the strict consistency rule: “if data resides in\nHBM, it must also be present in DRAM”. Specifically, when local operations involving KV cache\n(including prefix cache [8]) loading and offloading occur, these operational events are aggregated at\nregular intervals and transmitted to the xLLM-Service via ETCD heartbeat mechanisms, enabling\nunified global monitoring and management.\n12\n\nxLLM Technical Report\nKV Cache-aware Scheduling.\nIn terms of scheduling strategy, the system implements a decision-\nmaking mechanism based on KV cache utilization, which operates through three key steps: (1)\nPrefix Matching Detection: calculating the KV cache reuse rate for each candidate node via prefix\nmatching analysis; (2) Performance Estimation: estimating the expected latency for different nodes\naccording to current load conditions and cache hit rates; (3) Optimal Node Selection: identifying\nthe node with the optimal overall performance for request processing, thereby enabling dynamic\noffloading and migration of KV cache.\n3.5\nFast Fault Recovery Architecture\nCurrent fault handling mechanisms are mainly designed for model training [60–62] and cannot be\ndirectly applied to large model inference due to its low-latency requirements. Specifically, existing\napproaches primarily adopt a checkpoint-then-recovery method [63–65] for fault handling, which\nperiodically stores model data as checkpoints in distributed storage and reloads the most recent\ncheckpoint after a failure occurs [66]. As model parameters increase, the overhead of storage and\nloading gradually grows. This is still acceptable in training since there are no strict latency require-\nments [67], but in inference, it may cause all high-priority requests on the failed instance to time\nout, resulting in severe losses.\nTo address this issue, we propose an efficient failover architecture specifically for large model infer-\nence, with targeted optimizations in two aspects: fast request migration and fast instance recovery.\nFast request migration ensures the SLO of high-priority tasks mainly through an efficient kv cache\nquick recovery strategy, while fast instance recovery achieves low-overhead recovery through effi-\ncient masking of computation and communication. Through the above optimization methods, we\nachieve fast fault recovery, greatly reducing the performance and economic losses caused by faults.\n4\nxLLM-Engine Designs\n4.1\nMulti-layer Pipeline Execution Engine\nTraditional autoregressive LLM inference relies on single-stream sequential execution. This conven-\ntional approach fails to fully exploit the parallelism capabilities of modern hardware, often leading\nto computational resource underutilization, communication stalls, and cascading blocking delays in\nheterogeneous computing environments [68, 69]. As illustrated in Figure 7, the system’s perfor-\nmance is hindered by three primary sources of inefficiency: (1) CPU-Accelerator Dependency: At\nthe framework level, a rigid dependency between the CPU and the accelerator forces the accelerator\nto remain idle during task scheduling and data processing phases, creating significant “computation\nbubbles” [68, 3]. (2) Communication Latency: In both distributed workloads and complex model\nlayers, communication delays interrupt continuous computation, preventing the full utilization of\navailable hardware resources [69, 70]. (3) Architectural Bottlenecks: On certain AI accelerators,\nthe compute-focused (tensor cores) and general-purpose (vector cores) computation units lack a\nshared high-level cache (e.g., L1 or SRAM). This architectural separation necessitates additional\ndata transfers between them, which introduces latency and leads to the underutilization of the spe-\ncialized computation units.\nTo address these challenges, we propose a three-tier asynchronous pipeline parallelism design span-\nning the framework scheduling layer, the model graph layer, and the operator layer, as shown in\nFigure 7, with the goal of maximizing hardware efficiency.\nFramework-Layer Scheduling-Execution Overlap.\nIn a typical sequential execution procedure,\nthe accelerator remains idle while the CPU performs scheduling to prepare the input data batch for\nthe next computation cycle. This serial dependency creates significant latency and underutilizes the\naccelerator.\nTo eliminate this bottleneck, we introduce an asynchronous pipelined scheduling mechanism that\ndecouples CPU scheduling from accelerator execution. The core idea is to overlap these two stages:\nwhile the accelerator is executing the forward pass for the current batch, the CPU concurrently\nassembles the batch and prepares the metadata for the next batch. This parallel workflow effectively\nhides the latency of CPU-side scheduling and data preparation. The process is as follows: (1)\nAccelerator Compute: The accelerator is busy executing the forward pass for the current iteration,\n13\n\nxLLM Technical Report\nw/o Overlap\nw/ Overlap\nCPU\nAI Accelerator \nAI accelerator bubble\nAI accelerator bubble\n… …\n… \n… \nSchedule 0\nSchedule 1\nSchedule 2\nSchedule N\nForward 0\nForward 1\nForward 2\nForward N\nSchedule 3\nSchedule 0\nForward 0\nSchedule 1\nForward 1\nSchedule N\nForward N\nFramework Layer\nCommunication\nStream\n… …\nModel Layer\nNo bubble\nAttn 0\nDispatch 0\nMoE 0\nCombine 0\nAttn 1\nDispatch 1\nMoE 1\nCombine 1\nComputation resource waste\nComputation resource waste\nAttn 0\nAttn 0\nMoE 0\nMoE 0\nAttn 1\nAttn 1\nMoE 1\nMoE 1\nFull utilization\nComputation\nStream\nDispatch 0\nDispatch 0\nCombine 0\nCombine 0\nDispatch 1\nDispatch 1\nCombine 1\nCombine 1\nComputation\nStream - 1\nOperator Layer\nVector\nMatrix\nMatrix Comp. Unit idle\nMatrix Comp. Unit idle\nAttn 1-0\nAttn 1-1\nAttn 1-2\nAttn 1-N\nMoE 1-0\nMoE 1-1\nMoE 1-2\nMoE 1-N\nFull utilization\nComputation\nStream - 2\nStream 0\nAttn 1-0\nAttn 1-1\nAttn 1-2\nAttn 1-N\nMoE 1-0\nMoE 1-1\nMoE 1-2\nMoE 1-N\nVector\nMatrix\nVector\nMatrix\nVector\nMatrix\nAI Comp. Unit\nCPU\nAI Accelerator \nBatch 0\nBatch 1\nMatrix Comp. Unit\nVector Comp. Unit\nMatrix Comp. Unit idle\nw/o Overlap\nw/ Overlap\nw/o Overlap\nw/ Overlap\nFigure 7: Multi-layer pipeline execution engine.\nwhich will eventually produce output. (2) CPU Schedule: Instead of waiting, the CPU immediately\nbegins scheduling the batch for the next iteration. To do so, it uses a set of placeholder tokens that\nstand in for the yet-to-be-computed output. This allows all CPU-side batch preparations to occur\nconcurrently with the accelerator’s computation. (3) Seamless Transition: Once the accelerator\ncompletes the forward pass and the real output is generated, a fast replacement operation swaps the\nplaceholder tokens with the real generated tokens. Since all scheduling is already complete, the\naccelerator can begin the next forward pass with minimal delay.\nBy transforming the conventional serial “prepare-then-compute\"\" workflow into a parallelized\npipeline, our method effectively masks scheduling latency behind accelerator computation time.\nThis design eliminates idle execution “bubbles\"\" in the accelerator timeline, thereby maximizing\nhardware utilization and enhancing overall system throughput.\nModel-Layer Computation-Communication Overlap.\nMaximizing the overlap between com-\nputation and communication is a critical design principle in recent LLM frameworks [3, 71]. This\nstrategy is essential for hiding the significant latency of data transfers and achieving high hard-\nware utilization. However, architectural constraints can make this challenging. For instance, on a\ntypical GPU, compute-focused units (Tensor Cores) are tightly coupled with general-purpose units\n(CUDA Cores) within a Streaming Multiprocessor (SM). Consequently, when an SM is allocated\nto a communication-related task, its powerful Tensor Cores often sit idle, leading to wasted com-\npute potential. In contrast, our target accelerator architecture allows for more flexible, independent\nallocation of its Cube (matrix) and Vector (general-purpose) units. We leverage this flexibility to\naddress the aforementioned challenges by introducing a dual-stream pipeline architecture that uses\nmicro-batch splitting to effectively overlap computation and communication.\nSpecifically, our architecture consists of a Computation Stream for compute-bound tasks (Attention,\nExpertForward, etc.) and a Communication Stream for data distribution and collection tasks (MoE\nDispatch and Combine). To enable pipelining and maximize overall pipeline throughput, we parti-\ntion a macro-batch B into n micro-batches [68] {b1, b2, · · · , bn}. The two streams asynchronously\nexecute tasks for different micro-batches. On the one hand, our scheduling policy dynamically deter-\nmines the optimal execution order of the micro-batches. On the other hand, our resource scheduling\npolicy adaptively allocates an appropriate number of AI Cores (Vector Cores and Cube Cores) to\nthe Communication Stream and Computation Stream respectively. Figure 7 illustrates the case of\nn = 2, where the Communication Stream performs the Dispatch operation for micro-batch bk while\nthe Computation Stream executes the ExpertForward pass for the preceding micro-batch bk−1.\n14\n\nxLLM Technical Report\nOperator-Layer Matrix-Vector Units Overlap.\nOn heterogeneous AI accelerators, the serial\nscheduling of matrix and vector computation units often results in significant underutilization, as\none class of units remains idle while the other is active. This inefficiency motivates a strategy of\noperator-level matrix-vector overlap to fully exploit the hardware’s parallel capabilities. However,\na naive implementation via coarse-grained parallel scheduling, without a systemic resource coordi-\nnation mechanism, is also problematic. This approach frequently leads to disordered contention for\nlimited compute units, causing resource fragmentation and access conflicts that ultimately degrade\noverall performance.\nTo address these challenges, we propose a dynamic resource allocation mechanism for computa-\ntion units (Cube and Vector Units) based on real-time load monitoring. This mechanism enables\na deep pipeline across heterogeneous compute units by dynamically and adaptively assigning the\nprecise type and quantity of resources required for each concurrent operator. By doing so, it miti-\ngates resource contention and ensures that parallel operators execute within highly overlapping time\nwindows, achieving precise computational overlap and maximizing hardware utilization.\nWe formulate the dynamic resource allocation as an optimization problem. Let C and V be the sets\nof matrix and vector operators, respectively. Let xi be the number of matrix units (Cube) allocated\nto the ith matrix operator (i ∈C), and yj be the number of vector units (Vector) allocated to the jth\nvector operator (j ∈V). For simplicity, we assume all operators begin execution simultaneously,\nwithout considering data dependencies or communication latency. Based on the known computa-\ntional workload of each operator, our mechanism seeks to find an optimal resource allocation that\nminimizes the maximum difference in execution times between any two operators. This objective,\nwhich we term the alignment loss (Lalign), effectively synchronizes the completion time of all par-\nallel kernels. The optimization problem is defined as:\nargminxi,yjLalign =\nmax\ni∈C, j∈V |Ti −Tj| ,\nTi =\nWi\nγCube · xi\n,\nTj =\nWj\nγVector · yj\n,\nX\ni∈C\nxi ≤NCube,\nX\nj∈V\nyj ≤NVector,\n(1)\nwhere T is the operator execution time, W is its computational workload, γcube and γvector are the\npeak performance per unit, and Ncube and Nvector are the total available matrix and vector units.\nThe meaning of this optimization objective is to minimize the maximum difference in execution\ntime between any pair of concurrent matrix and vector operators.\n4.2\nAdaptive Graph Mode\nThe performance of LLM inference deployment is often impeded by Host-side CPU overhead, par-\nticularly when the computation graph is composed of many fine-grained operators [72]. This bottle-\nneck manifests as both significant CPU-Accelerator synchronization latency, measured at 5 ∼50µs\nper invocation from frequent kernel launches, and suboptimal hardware utilization due to idle accel-\nerator cycles between these intermittent operator executions.\nMainstream AI accelerators such as NVIDIA GPUs (with CUDAGraph) [73] and Ascend NPUs\n(with ACLGraph) [74] employ computation graphs to try to handle above issues and enhance host-\nside scheduling performance. As illustrated in Figure 8, traditional Eager Mode relies on the CPU\nto submit a multitude of small, intensive tasks, leading to frequent launches of small kernels on the\naccelerator. In contrast, the Graph Mode enables the CPU to submit one large task, after which the\naccelerator internally executes the small kernels in a streamlined fashion. This method significantly\nreduces both launch overhead and “accelerator bubbles” (idle cycles). Specifically, the Graph Mode,\nsuch as ACLGraph [74], comprises two distinct phases: graph capture and graph execution. During\nthe graph capture phase, the entire computation flow is recorded to capture the sequence of kernel\ncalls and their dependencies, including kernel launch parameters. This recorded workflow is then\npre-compiled into a replayable directed acyclic graph (DAG) object. It is important to note that\nduring this phase, tasks are merely staged within the model’s runtime instance and are not actually\nexecuted. In the subsequent graph execution phase, the entire graph is launched via a single CPU\ncall. The accelerator then follows the pre-defined process autonomously, without real-time interven-\ntion from the CPU. This effectively consolidates many individual kernel launches into one single\ngraph launch.\n15\n\nxLLM Technical Report\nA\nLaunch A\nLaunch B\nLaunch C\nLaunch D\nLaunch E\nB\nC\nD\nE\nKernel Execution\nLaunch Overhead\nLaunch graph\nD\nKernel Execution\nLaunch Overhead\nBuild Graph\ncudaGraphExecKernelNodeSetParams\nSpeed Up\nE\nC\nB\nA\nw/o Graph\nw/ Graph\nFigure 8: Pipeline comparison between Eager Mode and Graph Mode.\nBuilding upon this foundation, we further advance the practical implementation of the ACL-\nGraph [74] and propose our Adaptive Graph Mode through addressing key challenges including\ndynamic shape adaptation, memory reuse conflicts across multiple graphs, and compatibility with\ncustom operators. This enables the pre-compilation of kernel sequences, memory copy operations,\nand synchronization points into a single computational graph, which is then dispatched to the AI ac-\ncelerator in one execution. Consequently, our approach significantly reduces kernel launch overhead\nand minimizes accelerator idle periods, maximizing overall hardware efficiency.\nDynamic Shape Adaptation via Partial Graph and Dimension Parameterization.\nACL-\nGraph [74] fixes kernel parameters and memory sizes during the graph capturing phase. However,\nin real-world scenarios, LLMs typically process inputs with variable sequence lengths and batch\nsizes, making this static capturing characteristic difficult to adapt to dynamic input requirements.\nTo address this issue, dimension parameterization can be employed, treating key dynamic dimen-\nsions—such as batch size and sequence length—as input parameters to the entire graph, thereby\nenhancing flexibility. During memory allocation and kernel configuration, these dynamic param-\neters are used to compute actual required values; for example, the memory size can be calculated\nusing the formula: alloc_size = batch_size × seq_len × hidden_size. At the graph launch stage,\nthe actual batch size and sequence length are passed as parameters to ensure the graph adapts to\nvarying input dimensions.\nAlthough parameterization can handle variations in dynamic dimensions, certain operator imple-\nmentations on different hardware may not support dynamic dimension parameterization. To over-\ncome this, we propose the Partial Graph Mode: the model is partitioned into modules with simple\ndynamic shapes (such as FFN and LayerNorm, which only have the num_tokens dimension as dy-\nnamic) and modules with complex dynamic shapes (such as Multi-Head Attention, which involves\nmultiple dynamic dimensions like batch_size, kv_seq_len, and q_seq_len). Modules with simple\ndynamic shapes are extracted and compiled for execution using Partial Graph Mode, while those\nwith complex dynamic shapes are executed in Eager mode. Our Adaptive Graph Mode dynamically\nselects the most appropriate mode above based on the current input shapes. This intelligent selection\nensures optimal performance across varying workload conditions.\nTable 1 compares the key characteristics of three dynamic shape processing mode. The traditional\nEager Mode requires no pre-compilation but incurs high scheduling overhead at runtime due to N\nkernel launches (where N is the number of operators). The Full Graph Mode, by fixing shapes,\nallows single compilation and very low launch overhead, yet its lack of dynamic adaptability limits\nits applicability. In contrast, the proposed parameterized and multi-caching Partial Graph Mode\nstrikes an optimal balance. It achieves execution performance comparable to that of a full graph\nwhile maintaining high flexibility, by trading a manageable number of pre-compilations (M ≪\nNreq, where M is the number of cached graphs and Nreq is the number of actual requests) for the\nhigh efficiency of a single graph launch.\n16\n\nxLLM Technical Report\nMethod\nEager Mode\nFull Graph Mode\nPartial Graph Mode\nCompilation Times\n0\n1\nM\nLow Launch Overhead\n×\n✓\n✓\nLow Memory Usage\n✓\n×\n✓/×\nHigh Flexibility\n✓\n×\n✓\nTable 1: Comparison of different shape handling solutions. Our Adaptive Graph Mode dynamically\nselects the most appropriate mode above based on dynamic shapes.\nEfficient and Reusable HBM Memory Pool.\nDuring the graph capture process, the specific\nvirtual addresses of input, output, and intermediate tensors are recorded. To prevent illegal memory\naccess, which can occur when the address of an input tensor changes during an actual inference\nrequest, we develop a shared HBM memory pool. First, during graph initialization, a sufficiently\nlarge, contiguous block of HBM memory is pre-allocated to serve as the graph’s memory pool.\nSubsequently, for internal address management, computation graph intercepts all tensor memory\noperations and re-describes them using offsets relative to the pool’s base address, rather than absolute\nvirtual addresses.\nBefore the graph is launched, the user-provided input tensor is copied into the HBM memory pool at\nthe offset pre-defined for the graph’s input. After graph execution is complete, the data from the out-\nput tensor is copied to the user-specified output buffer address. Furthermore, internal tensors within\nthe graph are also managed using fixed offsets within the pool, ensuring safe reuse and efficient\nmemory management. This managed memory pool guarantees that all addresses used internally by\nthe graph are known and safe, which adapts to external address changes with only the overhead of\nmemory copies before and after launch.\nIntegrating Custom Operators.\nIntegrating performance-critical custom operators, such as\nPageAttention and AllReduce, into the Adaptive Graph Mode presents a unique challenge. The\ninternal implementation of these operators often relies on CPU to perform just-in-time shape calcu-\nlations and kernel configurations based on runtime inputs. This dynamic behavior conflicts with the\npre-compiled nature of computation graph.\nTo resolve this, we modified the implementation of such operators. The first step is to identify\ncustom operators within the graph that have dynamic shape dependencies. We then refactor these\noperators to accept shape-related parameters (e.g., dimension sizes or loop counts, which can only\nbe determined at runtime) directly as kernel arguments. This approach avoids hard-coding these\nvalues on the host side during the graph capture phase. As a result of this modification, these custom\noperators can be successfully captured by our computation graph. By leveraging the graph’s param-\neterization mechanism, the required dynamic arguments for these kernels can be derived from the\ngraph’s main input parameters and passed along during execution. This method achieves seamless\nintegration of essential custom operators within the Adaptive Graph Mode framework.\n4.3\nEfficient Memory Management\nLLM inference deployment faces a critical balance between memory utilization and computational\nefficiency, particularly in autoregressive generation tasks where efficient KV Cache management\nhas become a key challenge. Traditional solutions fall into two categories: First, contiguous mem-\nory allocation approaches that statically pre-allocate memory space based on maximum sequence\nlength before inference, ensuring physical contiguity of KV Cache. While this improves compu-\ntational efficiency, it results in low memory utilization. Second, PagedAttention [75] approaches\nthat support larger batch sizes through paging mechanisms, but frequent access to block tables sac-\nrifices computational efficiency, and the increased parameters complicate operator development and\ndebugging. To address this challenge, inspired by virtual memory management in operating sys-\ntem field [76, 77], we propose the xTensor memory management scheme, which adopts a \"logically\ncontiguous, physically discrete\" KV Cache storage structure that resolves the contradiction between\n17\n\nxLLM Technical Report\nReq\nVirt_start ... Virt_end\nVirtual Address Space\nRequest R1：\nMaxSeqLen\nVirt_start ... Virt_end\nRequest R2：\nPage1:\n-PageID: P1\n-Status: Reusable\n-Owner: R1\nPhysical Page Pool\nPage2:\n-PageID: P2\n-Status: Mapped\n-Owner: R2\nScheduler\nOn-demand Mapping\nRequest R2: Write a short story about a young explorer discovering a hidden ancient civilization. \nRequest R1: I am a 25-year-old woman. Please give some recommendations for moderately priced clothes.\nSmart Scheduling:\nRequest ID Reuse\nAsync Pre-mapping\n…\n…\nInference \nRequests\nAddr\nMapping\nFigure 9: The framework of xTensor memory management.\nFeatures\nContiguous Allocation\nPaged Attention\nxTensor (Ours)\nEfficient Memory Usage\n×\n✓\n✓\nEfficient Computation\n✓\n×\n✓\nLarge Batch Support\n×\n✓\n✓\nOperator Development Complexity\n✓\n×\n✓\nTable 2: Comparison of memory management strategies.\nmemory contiguity and dynamic allocation, thereby achieving both high computational efficiency\nand high memory utilization (as shown in Table 2).\nPhysical Memory Page Management and Virtual Memory Tensor Management.\nFigure 9 il-\nlustrates the xTensor memory management framework. During service initialization, based on the\navailable memory size for KV Cache, a large number of fixed-size discrete physical memory pages\nare pre-allocated. Each physical page maintains a triple state ⟨PageID, Status, OwnerSession⟩,\nwhere PageID represents the page identifier, OwnerSession indicates the service owning the\npage, and Status ∈{Free, Allocated, Mapped, Reusable} represents states of free, allocated,\nmapped, and reusable to dynamically track physical page usage. Subsequently, each inference re-\nquest is pre-allocated with a logically contiguous virtual address space with an address range equal\nto MaxSeqLen, where this virtual address space is not actually associated with physical pages dur-\ning allocation. This decoupled design provides operators with a virtual view of KV Cache stored in\ncontiguous memory, thereby masking the discrete nature of underlying physical pages [78, 79].\nOn-demand Memory Mapping.\nDuring actual allocation, the scheduler dynamically maps phys-\nical memory pages on-demand based on the actual sequence length of requests. When a sequence\ngenerates new tokens and KV Cache needs expansion, the scheduler retrieves one or more free\npages from the pre-allocated physical page pool and maps these physical pages to the next available\ncontiguous address location in the request’s virtual address space. Since physical memory is gradu-\nally mapped as sequences grow, short sequences only require a small amount of physical memory,\nthus avoiding the memory waste of contiguous allocation strategies that still require reserving space\naccording to maximum sequence length for short sequences.\nAfter memory mapping is completed, kernel access to virtual addresses can automatically locate the\ncorresponding physical page phypageidx and offset:\nphypageidx =\n\u0016virt_addr −virt_start\npage_size\n\u0017\n,\noffset = (virt_addr −virt_start) mod page_size,\n(2)\nwhere virt_addr and virt_start represent the current virtual address and starting virtual address\nrespectively, and page_size represents the physical page size.\n18\n\nxLLM Technical Report\nRequests Schedule\nDraft model\nForward\nInput\nTarget model\nForward\nInput\nValidate\nOutput\nRequests Schedule\nDraft model\nForward\nInput\nTarget model\nForward\nInput\nValidate\nCPU thread1\nNPU\nCPU thread2\nRequests Schedule\nDraft model\nForward\nInput\nTarget model\nForward\nInput\nValidate\nDraft model\nForward\nOutput\nTarget model\nForward\nInput\nCPU\nNPU\nSchedule\n…\n…\nw/o Overlap\nw/ Overlap\nFigure 10: Comparison between original and our optimized speculative decoding.\nLow Overhead Allocation.\nIf virtual address to physical page mapping is immediately released\nupon request completion, new requests require remapping physical pages, but the Unmap oper-\nations on accelerators incur significant overhead. Therefore, under high load conditions, frequent\nMap/Unmap operations become performance bottlenecks, especially for numerous short-term con-\ncurrent requests. We adopt request ID reuse and asynchronous pre-mapping designs to reduce la-\ntency from Map/Unmap operations.\n• Physical Page Reuse. Upon request completion, physical pages are not immediately released\nbut marked as Reusable. When new requests arrive, if their required KV Cache size matches\nsome Reusable physical page set, that page set is remapped to the new request’s virtual address\nspace. This saves expensive Map and Unmap operations through fast virtual address remapping,\nparticularly suitable for scenarios with concentrated request length distributions.\n• Asynchronous Pre-mapping. During current token decoding, the next token’s required physical\npages are asynchronously predicted and mapped. Since new token writes to KV Cache occur\nafter current token decoding computation, this design can partially hide physical page mapping\noperations, significantly reducing page mapping latency.\nOperator Adaptation.\nThe decoupled design of xTensor’s physical memory pages and virtual\ncontiguous memory requires the adaptation of existing operators. On one hand, to accommodate\nKV Cache contiguity in virtual address space, operators no longer require block_table parameter\ninput. Whether for attention operators or write_kvcache operators responsible for writing KV\nCache during Prefill and Decode phases, only the starting virtual address and related offsets need to\nbe passed in, with the system automatically associating virtual addresses to corresponding physical\npages during operation. On the other hand, accelerators lack native contiguous KV FlashMLA [80]\noperators and only support PagedFlashMLA [80] operators optimized for paging. Therefore, by\nreconstructing the PagedMLAttention operator in the accelerator operator library, removing Block\nTable-related logic such as block table queries and cross-page boundary judgments, and adding\nfunctionality for direct computation based on input virtual addresses, we implement FlashMLA\noperators supporting contiguous virtual address input on accelerators.\n4.4\nAlgorithm Optimizations\n4.4.1\nOptimized Speculative Decoding\nTraditional autoregressive inference requires LLMs to generate tokens sequentially one by one, re-\nsulting in high computational latency and limited throughput. Speculative decoding technology the-\noretically breaks through this performance bottleneck through a paradigm of parallelly predicting\ncandidate token sequences and quickly verifying them [81, 82, 26]. However, in distributed de-\nployment environments, this technology faces some core challenges: First, synchronous scheduling\nbetween CPUs and accelerators in traditional multi-step inference frameworks results in insufficient\nutilization of computing resources, with accelerators often remaining idle while waiting for data\npreparation and processing. Second, existing attention mechanisms are not optimized for the char-\nacteristics of speculative inference, causing a large amount of redundant data movement. To address\nthese challenges, xLLM proposes a systematic optimization scheme based on existing frameworks.\n19\n\nxLLM Technical Report\nCompute\nLoad\ndecode\nEPLB Controller\nGlobal Expert Mapping\nPlanner\nChecker\nCompute\nLoad\ndecode\ndecode\ndecode\ndecode\ndecode\nLayer-0 Loader\nLayer-0\nWeight\nOld\nBuffer\nNew\nBuffer\nLayer-0 Rebalance\nLayer-0\nWeight\nOld\nBuffer\nNew\nBuffer\nDouble Buffer\nDouble Buffer\nLayer-1\nRebalance\n…\nDB Policy\nExpert\nRouting Map\nExpert\nLoad\nWorker-0\nReady\nAll\nReady\nExpert\nLoad\n…\nThread-0\nWorker-0\n…\nThread-1\nLayer-1\nLoader\n…\n…\nWorker-0\nReady\nAll\nReady\n…\n…\nFigure 11: Overview design of EPLB.\nAsynchronous Decoding.\nInspired by [83], xLLM implements a lightweight asynchronous frame-\nwork. For timing overlap optimization, while the accelerator executes the main model inference for\nthe current batch, the CPU processes the output decoding of the previous batch and the input prepa-\nration for the next batch in parallel—thereby minimizing the idle time of the CPU and accelerator\nas much as possible.\nMLA Optimization.\nxLLM focuses on improving the MLA [84, 3] computation process. Based\non the characteristic that speculative inference needs to process multiple tokens simultaneously on\na single sequence, we conducted in-depth optimizations on the self-attention computation in MLA.\nSpecifically, when speculatively inferring m tokens, the self-attention computation of MLA involves\nproduct operations between m+1 Q matrices and the same K matrix. By reconstructing the compu-\ntation process and optimizing the tiling strategy, we effectively reduced the data movement overhead\nof Q/K matrices. The main optimization measures include:\n▷Optimization of K matrix loading times: By adjusting the L1 cache allocation scheme to enable\nparallel loading of multiple Q matrices, and adopting a sliding window-based K matrix loading\nmethod that allows consecutive rows of the K matrix to multiply with multiple Q matrices, the\nnumber of K matrix loading operations is significantly reduced.\n▷\nQ matrix cache residency mechanism: Since there are m+1 Q matrices in speculative infer-\nence scenarios, the time to move Q matrices to L1 accounts for a larger proportion of the matrix\nmultiplication process. xLLM redesign the computing scheme to prevent softmax-V product opera-\ntions from overwriting Q matrices in the L1 cache, enabling Q matrices to remain in the cache and\nsignificantly reducing the overhead of repeated data movement, thereby effectively improving the\nutilization of the tensor core’s arithmetic logic units.\n4.4.2\nDynamic EP Load Balance\nWith the large-scale application of MoE models [85–88] in production environments, their efficient\ninference capability relies on a expert routing mechanism, which allocates input tokens to differ-\nent experts for computation [3, 18, 89]. However, in practical deployment, due to the distribution\ncharacteristics of input data and the influence of the model’s own structure, there may be significant\ndifferences in the number of tokens received by different experts [90, 91]. This imbalance leads\nto low computing resource utilization: some devices are overloaded due to processing too many\ntokens, while others remain idle, affecting the overall inference efficiency.\nTo optimize resource utilization, the industry has proposed the Expert Redundancy strategy, which\nmeans replicating hot experts and deploying replicas on multiple devices to distribute computing\nloads. Currently, DeepSeek [3] adopts two load balancing strategies: group-limited load balance\nand global load balance, which are optimized respectively for the characteristics of the prefill and\ndecode phases. Adjustments to expert redundancy (such as adding/deleting replicas) require ad-\n20\n\nxLLM Technical Report\nPreprocess\nMLA\nPreprocess\nMLA\nMLA\nMLA\nMLA\nMLA\nMLA\nMLA\nPreprocess\nMLA\nPreprocess\nMLA\nMLA\nMLA\nMLA\nMLA\nMLA\nMLA\nAggregate\ntransfer\ntransfer\n1K tokens\n3K tokens\n3K tokens\nDP-1\n(seq-1)\nDP-2\n(seq-2)\nDP-1\n(seq-1)\nDP-2\n(seq-2)\nw/o DP Load \nBalance\nw/o DP Load \nBalance\nFigure 12: Inter-DP group load migration at MLA block granularity.\nditional memory and may affect inference latency due to weight migration. How to achieve such\nadjustments efficiently and smoothly is a major challenge. We have made some adaptive improve-\nments based on DeepSeek, realizing dynamic EP load balance.\nExpert Load Statistics.\nWhen the Router distributes tokens, it records the load status of each\nexpert in real time and returns the statistical results through the model output. Each Worker asyn-\nchronously starts a thread to periodically aggregate expert loads and report them to the Controller of\nthe Master. The Controller calculates a new routing table based on the load data and then distributes\nthe updated routing table to each Worker.\nDouble Buffer Mechanism for Weight Update.\nIn general, each Worker starts an asynchronous\nthread to update the weights of single-layer experts during each decoding process. Specifically, after\na Device completes the preparation of new weights, it proactively sends a readiness notification to\nthe Controller. The Controller then verifies the weight readiness status of all Worker nodes to ensure\nglobal consistency before broadcasting an update instruction, upon which each Worker executes the\nweight update immediately. Here, we adopt a Double Buffer mechanism, meaning that the old and\nnew expert weights are stored in two separate buffers respectively. The new expert weights are\npreloaded in the spare memory space, and after the preloading is completed, the address switching\nis performed to realize the unperceived update of experts.\n4.4.3\nHierarchical DP Load Balance\nIn typical MoE model architectures like DeepSeek [3], the MoE layers use expert parallelism (EP),\nwhile the attention layers use data parallel (DP). This model requires all DP groups to synchronously\ncomplete their attention calculations in a single inference step before initiating the MoE all-to-all\ndispatch operation. This synchronization barrier means the total time for the attention phase is\ndetermined by the slowest DP group (the “straggler”). Faster DP groups cannot immediately proceed\nto the next stage after finishing their tasks; instead, they are forced into an idle state, waiting for the\nstraggler to complete [92]. This waiting directly translates into wasted computational resources and\nan increase in overall inference latency.\nIn practical scenarios, due to the dynamic and unpredictable nature of inference requests, the load\nwithin different DP groups is often difficult to balance. Particularly in the decoding phase with\nlarge-scale DP (e.g., 80 groups), the load difference between DP groups can reach tens of thousands\nof tokens, causing fast DP groups to wait for slow ones and wasting approximately 5 milliseconds of\ndelay [93]. Experimental data shows that although the difference in the number of requests processed\nbetween DP groups may be as small as four, the corresponding difference in KV Cache token count\nduring the decoding phase can be as high as 20,000. The actual computational workload is directly\nrelated to the total number of tokens the system needs to process, as this determines the memory\nfootprint of the KV Cache and the scale of matrix operations in the attention mechanism. This\nfurther establishes DP load balancing as a critical factor in determining overall system efficiency.\nCurrent mainstream frameworks like vLLM [8] and SGLang [9] use a simple static round-robin\nscheduling strategy. Once a request is assigned, its subsequent computations are fixed to a specific\n21\n\nxLLM Technical Report\nDP group, making it unable to adapt to dynamic load changes. Furthermore, at the hardware level,\nthe implementation of operators like the MLA on certain accelerators employs a “one request per\ntensor compute core” partitioning strategy. This can lead to idle compute cores within the same DP\ngroup due to differences in request lengths.\nTo address these challenges, xLLM proposes a hierarchical “defense-in-depth” strategy to tackle DP\nload imbalance issues across different time scales and granularities. The first layer is preventative,\nKV Cache-aware request scheduling; the second layer performs macroscopic correction through\ninter-DP group load balancing; and the third layer carries out microscopic correction via intra-DP\ngroup load balancing. This layered design enables the system to flexibly address the root causes of\nvarious performance degradations, thereby building a robust and efficient inference service.\nLayer 1: KV Cache-Aware Scheduling.\nxLLM’s first-layer strategy implements request load dis-\ntribution through KV Cache-aware scheduling. This mechanism moves beyond simple round-robin\nmethods. When a new request arrives, the scheduler comprehensively checks the status of all DP\ngroups, paying special attention to the remaining KV Cache capacity in each. It then preferentially\nassigns the new request to the group with the most available space. By balancing the total token load\nand its memory footprint at the system level over the long term, this strategy prevents the formation\nof severe load imbalances, thereby achieving intelligent resource allocation.\nLayer 2:\nReactive Inter-DP Group Workload Migration.\nxLLM’s second-layer strategy\nachieves reactive balancing through workload migration between DP groups. During the decoding\nprocess, the varying prompt and KV Cache lengths of selected requests cause different computa-\ntion latencies among DP groups. To counter this, xLLM’s central scheduler evaluates the current\ncomputational load of each DP group during every inference round. If a significant imbalance is\ndetected, it initiates a workload migration process, moving tasks from overloaded to under-loaded\ngroups. The scheduler also determines the migration granularity – whether to move an entire batch,\na single sequence, or a partial MLA block of a sequence.\nFigure 12 illustrates this process at the MLA block granularity. First, xLLM dispatches the input\ntokens for the migration to both the source and destination DP groups. Then, as all groups execute\nthe MLA Preprocess operation, the request’s KV Cache is transferred asynchronously, overlapping\ncommunication with computation. Next, all groups begin the attention calculation. The underloaded\ngroup prioritizes the migrated attention task, allowing it to immediately send the resulting token\nback to the source group upon completion while concurrently processing its own native attention\noperations, further overlapping overhead. Finally, all DP groups use an aggregation operator to\ngather the external computation results.\nLayer 3: Fine-Grained Kernel-Level Optimization.\nxLLM’s third-layer strategy achieves fine-\ngrained optimization through kernel-level migration within a DP group. This optimization is per-\nformed inside the matrix computation kernels of a single DP group. On one hand, during scheduling,\nit reorders requests based on their load, replacing the original compute core’s round-robin alloca-\ntion strategy. This aims to keep the total number of computation tokens assigned to each matrix\ncomputation unit as consistent as possible. On the other hand, for requests with extremely long\nsequences, it explicitly splits their computation sequence, migrating parts of the long-sequence re-\nquest’s computation to be calculated with other short-sequence computation units. Through this\nfine-grained request splitting, it directly resolves the issue of idle compute cores caused by load\nimbalance within the DP group.\n4.5\nGenerative Recommendation\nRecent advancements in generative models have spurred significant interest in generative recom-\nmendation [31, 94–98]. Beyond enhancing the retrieval and ranking stages of traditional multi-stage\npipelines [98, 94], single-stage generative recommendation frameworks directly generate a candi-\ndate item list in an auto-regressive manner [31]. These frameworks usually utilize beam search\ndecoding to directly produce diverse recommendation results, and xLLM implements extensive op-\ntimizations for these single-stage frameworks. As shown in Figure 13, in the generative recommen-\ndation scenario, xLLM is dedicated to overlapping host and device operations as much as possible to\nimprove overall execution efficiency. When a request arrives, xLLM preemptively prepares the in-\nformation required during the forward process on the CPU side. After the preparation is completed,\n22\n\nxLLM Technical Report\nPre-allocate\nScheduler\nSchedule\nBeam\nSchedule\nPre-allocate\nBeam\nPre-allocate\nSchedule\nBeam\nPre-allocate\nSchedule\nBeam\nw/o Overlap\nw/ Overlap\nPrefill\nBeam\nDecode\nBeam\nFilter Mask\nWorker\nForward\nAdd\nSampler\nLogits\nw/o Overlap\nw/ Overlap\nEngine\nDecode\nFilter Mask\nForward\nLogits\nAdd\nSampler\nFigure 13: Overview design of generative recommendation process.\nrelevant tasks are handed over to the engine for processing. The Engine executes three forward\npasses in one go. During this process, the intermediate beam search is a host operation, which by\nits nature cannot be overlapped with device operations. To optimize the processing pipeline, xLLM\ninternally employs a multi-micro-batch, multi-stream approach. During the Worker’s execution, the\nCPU operation for generating the filter mask is overlapped with the device operation for logits com-\nputation. Through an addition operation, a masked logits is generated before the sampler, thereby\nachieving data filtering and improving the accuracy and effectiveness of the recommendation results.\n4.5.1\nHost-side Optimization\nIn scenarios with large beam_width and top_k parameters, the significant increase in the number\nof candidate sequences results in substantial sorting and filtering operations on the host side. This\nshifts the computational bottleneck of the beam search algorithm from the AI accelerator to the CPU,\nultimately leading to a severe CPU-bound issue. The following outlines xLLM’s efforts to optimize\nthe beam search process in generative recommendation scenario.\nBeam Search Optimization.\nIn many fields such as natural language processing, beam search\noptimization is an efficient algorithm widely used in sequence generation tasks [99]. When process-\ning a large number of candidate sequences, one of the core operations of beam search is to select\nbeam_width sequences from numerous candidates. This process is essentially a partial sorting op-\neration, with the unique characteristic that it only needs to identify the top beam_width elements\nwithout fully sorting all candidate sequences, thereby significantly reducing computational com-\nplexity. In another key step of beam search, generating beam_width × top_k candidate sequences\nfrom the existing beam_width sequences, there exists an important property: log_probs of each se-\nquence are arranged in descending order. Leveraging this property, an optimized early termination\nmechanism can be adopted, where computations are halted prematurely if log_probs does not meet\nspecific criteria, further enhancing computational efficiency. The specific operational procedure is\nas follows:\nFirst, the existing beam_width sequences are accessed in order. For each accessed sequence, we\ncreate a small min-heap of size beam_width, which is used to dynamically maintain the set of\nlocally optimal elements currently selected. Next, the subsequent sequences are traversed. During\nthis process, if the subsequent log_probs of a sequence is smaller than the top element of the min-\nheap, it implies that, based on the current filtering criteria, the subsequent elements of this sequence\ncannot enter the current optimal set. At this point, the filtering operation for the current sequence\ncan be terminated directly, and the next existing sequence processed. Conversely, if the subsequent\nlog_probs is larger than the top element of the min-heap, the element is inserted into the min-heap,\n23\n\nxLLM Technical Report\nand the heap structure is adjusted according to the properties of the min-heap to maintain its order.\nAfter the traversal of all relevant sequences is completed, the top elements are sequentially extracted\nfrom the min-heap. These elements, in the order of extraction, form the beam_width candidate\nsequences sorted in descending order. Through this procedure, the approach ensures the selection of\nhigh-quality candidate sequences while significantly reducing unnecessary computational overhead,\nthereby improving the operational efficiency of the beam search algorithm in practical applications.\nResource Reuse and Pre-allocation.\nDuring the beam search process, the beam_width is pre-\ndetermined and fixed for each forward pass of the model, which implies that the computational\nresources required per forward propagation remains relatively constant. However, an excessively\nlarge beam_width can lead to significant wastage of both CPU computational and memory re-\nsources. To effectively mitigate this issue, xLLM incorporates a carefully designed resource reuse\nstrategy. Specifically, during the generation of new candidate sequences, the system reuses resources\npreviously occupied by older sequences without allocating new space for each candidate. This ap-\nproach avoids the overhead associated with frequently creating new data structures. Based on the\nfinal search results, the system then updates the storage areas for the old sequences with the content\nof the newly generated ones upon completion of the beam search algorithm. By doing so, xLLM\nnot only reduces memory usage but also minimizes the additional overhead on the CPU for resource\nmanagement and allocation, significantly improving the efficiency of computational resource usage.\n4.5.2\nDevice-side Optimization\nValid Item Filtering.\nIn typical single-stage generative recommendation frameworks, the valid\nitem filtering mechanism plays a crucial role. For example, OneRec [31] framework uses an ordered\ncombination of three token IDs to uniquely represent a valid product item ID. However, due to\nthe vast number of possible token ID combinations, not all of them correspond to actual and valid\nitems [100, 101]. To ensure that the results generated by the model are all valid product items, the\nsystem generates a valid mask asynchronously during the forward pass of the model. This mask is\nbased on a pre-constructed vocabulary of valid items, which is then element-wise added to the logits\noutput by the model. Through this clever design, the logits corresponding to invalid token IDs are\nadjusted, ensuring that these invalid token IDs are almost never selected in subsequent calculations.\nThis effectively filters out invalid token IDs, thereby guaranteeing the validity and accuracy of the\nfinal recommendation results.\n5\nEvaluations\nIn §5.1, we evaluate the full implementation of our xLLM framework against several base-\nline inference systems, specifically vLLM-Ascend1(v0.10.rc1) and MindIE2(v2.1.rc1), on Ascend\n910B/910C [18] instances across a range of scenarios. xLLM, MindIE, and vLLM-Ascend refer\nto the default deployments on Ascend 910B. Additionally, we denote the deployments of xLLM,\nMindIE and vLLM-Ascend on Ascend 910C as xLLM‡, MindIE‡ and vLLM-Ascend‡, respectively.\nThe testing scenarios are organized into multiple categories reflecting diverse online serving appli-\ncations, such as the JingYan AI chatbot, customer service assistants, merchant assistants, product\nunderstanding and marketing recommendation systems. Additionally, in §5.2, we conduct a detailed\nablation study to assess the individual contributions of each optimized module.\n5.1\nMain Results\nThis section benchmarks xLLM against mainstream inference frameworks to demonstrate its\nsuperior inference efficiency across diverse models, including Qwen2/3-series [102, 87] and\nDeepseek [3] models, and datasets. §5.1.1 details a fair comparison using the ShareGPT3 dataset.\nTo ensure an equitable comparison across all LLM inference frameworks, the key feature of our\nexperimental setup is that the input and output sequence lengths are fixed, while the request\n1https://github.com/vllm-project/vllm-ascend\n2https://www.hiascend.com/en/software/mindie\n3https://huggingface.co/datasets/anon8231489123/ShareGPT_Vicuna_unfiltered/blob/\nmain/ShareGPT_V3_unfiltered_cleaned_split.json\n24\n\nxLLM Technical Report\n1\n2\n4\nNumber of Accelerators\n2000\n4000\n6000\n8000\n10000\nThroughput (tokens/s)\nQwen3-0.6B\n1\n2\n4\nNumber of Accelerators\n2000\n4000\n6000\n8000\nQwen3-1.7B\n1\n2\n4\nNumber of Accelerators\n2000\n3000\n4000\n5000\n6000\nQwen3-4B\n1\n2\n4\nNumber of Accelerators\n1000\n2000\n3000\n4000\n5000\nThroughput (tokens/s)\nQwen3-8B\n1\n2\n4\nNumber of Accelerators\n1000\n2000\n3000\n4000\nQwen3-14B\n2\n4\n8\nNumber of Accelerators\n1000\n2000\n3000\nQwen3-32B\nxLLM\nxLLM\nMindIE\nMindIE\nvLLM-Ascend\nvLLM-Ascend\nFigure 14: Throughput comparison of LLM inference frameworks evaluated on Qwen3-series mod-\nels using the ShareGPT dataset, under the constraint of TPOT=50ms and input/output length=2048.\n16×910B\n8×910C\n0\n100\n200\n300\n400\nThroughout (tokens/s)\n[2048,2048] TPOT:50ms\n16×910B\n8×910C\n0\n500\n1000\n1500\n[2048,2048] TPOT:100ms\n16×910B\n8×910C\n0\n100\n200\n300\n400\n[2500,1500] TPOT:50ms\n16×910B\n8×910C\n0\n250\n500\n750\n1000\n1250\n1500\n[2500,1500] TPOT:100ms\nxLLM\nxLLM\nMindIE\nMindIE\nvLLM-Ascend\nFigure 15: Throughput comparison of LLM inference frameworks using DeepSeek-R1 on the\nShareGPT dataset constrained by given TPOT and input/output length, where [2500, 1500] means\ninput length=2500 and output length=1500. vLLM-Ascend on the 910C is excluded from the com-\nparison, as its performance with Deepseek-R1 fails to meet the required TPOT threshold.\nrate is dynamically adjusted to match the target SLO (e.g., TPOT) threshold for each frame-\nwork. We also evaluate this setup under different node configurations, including both single-node\nand multi-node setups with PD disaggregation. In §5.1.2, we evaluate xLLM on various real-world\nbusiness scenarios from JD.com, where it is currently deployed, showcasing its performance under\npractical deployment conditions.\n5.1.1\nBenchmarking Performance\nQwen3-series.\nAs shown in Figure 14, we comprehensively compares the throughput performance\nof four inference frameworks across the Qwen3-series (from 0.6B to 32B parameters) on the bench-\nmark. All tests are conducted under uniform configuration with input/output lengths set to 2048\ntokens and a TOPT constraint of 50 ms. The experimental results demonstrate that while all eval-\nuated frameworks show improved throughput with additional accelerators, xLLM and its Ascend\n910C implementation (xLLM‡) consistently deliver superior performance, confirming near-linear\nstrong scalability across various model sizes. Specifically, xLLM achieves throughput improve-\nments of up to 1.9× and 1.7× compared to vLLM-Ascend and MindIE, respectively. Similarly,\nxLLM‡ outperforms vLLM-Ascend‡ and MindIE‡ by up to 2.2× and 1.5×, respectively. Further-\n25\n\nxLLM Technical Report\nMethod\nPrompt Length\nOutput Length\nThroughput (tokens/s)\nRequest Rate (req/s)\nMindIE\n2048\n2048\n8476.44\n4.14\nxLLM\n2048\n2048\n11351.58\n5.54\nTable 3: Comparison of DeepSeek-R1 with PD disaggregation on the ShareGPT dataset constrained\nby TPOT=100ms.\n1\n2\n4\nNumber of Accelerators\n400\n600\n800\n1000\n1200\n1400\nThroughput (tokens/s)\nQwen2-7B TPOT:40ms\n1\n2\n4\nNumber of Accelerators\n0\n500\n1000\n1500\n2000\nQwen2-14B TPOT:90ms\n1\n2\n4\nNumber of Accelerators\n2000\n3000\n4000\n5000\nQwen3-8B TPOT:80ms\n2\n4\n8\nNumber of Accelerators\n1000\n2000\n3000\nQwen3-32B TPOT:80ms\nxLLM\nxLLM\nMindIE\nvLLM-Ascend\nFigure 16: Throughtput comparison of Qwen2-series and Qwen3-series models with various infer-\nence frameworks in the JingYan scenario.\nmore, xLLM‡ demonstrates steady performance improvements over the Ascend 910B-based xLLM\nin most scenarios, validating the software stack’s effective utilization of the new hardware.\nDeepSeek-R1.\nFigure 15 presents the throughput performance of various inference frameworks\nfor the DeepSeek-R1 model on the benchmark, using 16 accelerators for Ascend 910B and 8 accel-\nerators for Ascend 910C, respectively. The results demonstrate that the proposed xLLM framework\nachieves exceptional throughput on the Ascend 910 series. Quantitatively, xLLM on Ascend 910B\ndelivers an average throughput improvement of approximately 1.7× over MindIE and a significant\n12× enhancement over vLLM-Ascend. Furthermore, xLLM‡ achieves an average throughput in-\ncrease of approximately 1.4× compared to MindIE‡.\nPD Disaggregation Settings.\nTable 3 benchmarks the inference performance of the MindIE and\nxLLM frameworks on the DeepSeek-R1 model using a PD disaggregation architecture. Under iden-\ntical conditions with TPOT controlled at 100ms for 2048-length inputs/outputs, xLLM achieves\napproximately 34% higher throughput (11,351.58 vs. 8,476.44 tokens/s) and request rate (5.54 vs.\n4.14 req/s), marking a significant efficiency improvement.\n5.1.2\nBusiness Serving Scenarios\nJingYan.\nJingYan is an AI shopping assistant designed to help users discover new products, find\ninspiration, and get answers to their questions. The dataset for JingYan, consequently, consists of\nconversational logs between the model and its users, capturing these rich interactions. As illustrated\nin Figure 16, we systematically evaluate the inference throughput of four frameworks for serving\nthe Qwen2-series and Qwen3-series model in the JingYan scenario. The results demonstrate that\nboth xLLM and xLLM‡ maintain superior throughput and exhibit better scaling efficiency across\nall model sizes compared to MindIE and vLLM-Ascend. For instance, when serving the Qwen3-8B\nmodel with 4 accelerators, xLLM delivers a throughput approximately 1.6 times that of vLLM-\nAscend and significantly surpasses MindIE. The robust performance of xLLM on the 910B, and its\nenhanced results on the 910C hardware, highlight its effective adaptation to successive hardware\ngenerations. As shown in Table 4, a similar trend is observed for the DeepSeek-V3 model, where\nxLLM achieves a throughput that is over 9 times greater than vLLM-Ascend and surpasses MindIE\nby 36%.\nCustomer Service.\nThe Customer Service dataset comprises the interactive dialogues between\ncustomers and support agents. Figure 17 details the performance differences among inference frame-\n26\n\nxLLM Technical Report\nMethod\nPrompt Length\nOutput Length\nThroughput (tokens/s)\nRequest Rate (req/s)\nvLLM-Ascend\n6800\n400\n21.17\n0.11\nMindIE\n6800\n400\n144.40\n0.67\nxLLM\n6800\n400\n196.45\n0.89\nTable 4: Comparison of Deepseek-V3 model with various frameworks in the JingYan scenario con-\nstrained by TPOT=80ms.\n1\n2\n4\nNumber of Accelerators\n50\n100\n150\nThroughput (tokens/s)\nQwen3-8B\n2\n4\n8\nNumber of Accelerators\n20\n40\n60\n80\nQwen3-32B\nxLLM\nxLLM\nMindIE\nvLLM-Ascend\nFigure 17: Throughtput comparison of Qwen3-series models with various inference frameworks in\nthe customer service scenario constrained by End-to-End latency(E2E)=10s.\nworks for the Qwen-8B and Qwen-32B models in the customer service scenario. The results high-\nlight that our xLLM, particularly the xLLM‡ running on Ascend 910C, provide significantly better\nthroughput across all tested configurations. For instance, with Qwen3-32B on 8 accelerators, the\nthroughput of xLLM is 3.1 and 1.2 times greater than that of vLLM-Ascend and MindIE, respec-\ntively. It is important to note that the vLLM-Ascend framework shows a clear scaling bottleneck\nas the number of accelerators increases, whereas xLLM maintains near-linear efficiency scaling.\nThis validates the high efficiency and superiority of the xLLM framework in managing distributed\ninference for large-scale models.\nMerchant Assistant.\nTable 18 benchmarks the throughput of various inference frameworks on\nthree tasks (i.e., search terms, arrangement, intent recognition) within the merchant assistant sce-\nnario. The proposed xLLM framework achieves better or comparable performance to MindIE and\ndemonstrates a significant lead over vLLM-Ascend. Specifically, for the search terms task with four\naccelerator cards, xLLM delivers 34% higher throughput than MindIE and roughly 3.4× that of\nvLLM-Ascend.\nProduct Understanding.\nFor the product understanding scenario, the throughput comparison of\nthe Qwen2-7B model with several frameworks is shown in Table 5. The experimental results indicate\nthat xLLM outperforms MindIE and vLLM-Ascend by an average of 25% and 56%, respectively,\nacross different accelerator card counts. Moreover, the superiority of xLLM scales with the num-\nber of cards, demonstrating its effective utilization of large-scale parallel computing resources and\nthereby offering a robust solution for high-performance LLM inference.\nGenerative Recommendation.\nAs illustrated in Figure 19, the evaluation results on the Qwen-8B\nmodel demonstrate that xLLM consistently achieves lower mean end-to-end latency than MindIE\nacross various request rates and beam widths, except under very low load condition. Notably, the\nperformance advantage of xLLM becomes increasingly pronounced as the beam width (from 4 to\n128) and the request rate escalate. For instance, under the most challenging scenario with a beam\nwidth of 128 and a request rate of 8, xLLM reduces latency by approximately 23% relative to\nMindIE. This significant improvement validates that our xLLM’s host-side and device-side opti-\nmizations effectively alleviate the computational bottlenecks in generative recommendation tasks,\nmarkedly enhancing inference efficiency and scalability under heavy loads.\n27\n\nxLLM Technical Report\n1\n2\n4\nNumber of Accelerators\n100\n200\n300\nThroughput (tokens/s)\nSearch Term Qwen2.5-14B\nxLLM\nMindIE\nvLLM-Ascend\n1\n2\n4\nNumber of Accelerators\n50\n75\n100\n125\nArrangement Qwen2.5-14B\nxLLM\nMindIE\nvLLM-Ascend\n1\n2\n4\nNumber of Accelerators\n100\n150\n200\n250\n300\nIntent Recognition Qwen3-14B\nxLLM\nMindIE\nvLLM-Ascend\nFigure 18: Throughtput comparison of Qwen-series models with various inference frameworks in\nthe merchant assistant scenario constrained by End-to-End latency(E2E)=1s.\nMethod\nPrompt Length\nOutput Length\nThroughput (tokens/s)\n#accelerator=1\n#accelerator=2\n#accelerator=4\nvLLM-Ascend\n1200\n40\n795.77\n874.97\n1272.52\nMindIE\n1200\n40\n944.81\n1051.44\n1693.45\nxLLM\n1200\n40\n1001.91\n1323.90\n2425.13\nTable 5: Throughtput comparison of Qwen2-7B model with various inference frameworks in the\nproduct understanding scenario.\n5.2\nAblation Study\nImpact of MTP.\nAs shown in Figure 20, under the configuration of 1500 input length and 2500\noutput length for the DeepSeek-R1 model, enabling Multi-Token Prediction (MTP) technology sig-\nnificantly optimizes inference performance. As max concurrency increases, the MTP-enabled ver-\nsion consistently exhibits lower TPOT compared to the baseline, indicating reduced generation la-\ntency. Meanwhile, its throughput is markedly higher than the non-MTP version, with particularly\nnotable advantages beyond 32 concurrent requests. This demonstrates that MTP effectively en-\nhances computational efficiency and system throughput under high concurrency conditions.\nImpact of Dynamic PD Disaggregation Scheduler Policy.\nWe evaluate our proposed SLO-aware\nDynamic PD Disaggregation Policy against the Minimal Load and Round Robin strategies, as illus-\ntrated in Figure 21. On the Azure Code dataset characterized significant bursty traffic, our SLO-\naware policy achieves a request serving rate 1.67 times that of the Minimal Load strategy. Mean-\nwhile, relative to the Round Robin strategy, the Minimal Load strategy improves the SLO attainment\nby up to 4.3%. For the Azure Conversation dataset with stable input/output length variations, our\nSLO-aware policy results in a 1.1 times higher request serving rate compared to the Minimal Load\nstrategy. Although the Minimal Load strategy performs similarly to Round Robin, it still enhances\nthe SLO attainment by up to 2.4%. These results indicate that minimal-load scheduling is closer to\nthe optimal scheduling strategy than round-robin approach, while our adaptive instance scheduling\ndelivers the best overall performance.\nImpact of Hybrid EPD Disaggregation Scheduler Policy.\nFigure 22 demonstrates the effective-\nness of the proposed Hybrid EPD Disaggregation Policy on the TextCaps dataset, specifically in\ncontrolling TPOT and improving the SLO attainment rate. Under the configuration of 8 general-\npurpose inference instances, the removal of the hybrid EPD disaggregation method results in a drop\nin goodput from 9.5 req/s to 7.2 req/s, and a subsequent additional removal of stage-level scheduling\npolicy further decreases the goodput to 5.1 req/s. This indicates that a well-designed disaggregation\nstrategy can effectively mitigate inter-stage interference, while our stage-level scheduling policy\ncontributes to finer control over the execution time of each batch. The integration of both compo-\nnents significantly enhances the overall performance and stability of the system, particularly under\nhigh-concurrency scenarios.\nImpact of Online-Offline Co-location Scheduler Policy.\nAs shown in Figure 23, we evaluated\nthree scheduling strategies, namely the baseline P/D, online priority and our proposed Online-Offline\n28\n\nxLLM Technical Report\n1\n4\n8\nRequest Rate (req/s)\n90\n100\n110\nMean E2E (ms)\nBeam Width:4 Top-k:4\nxLLM\nMindIE\n1\n4\n8\nRequest Rate (req/s)\n100\n120\n140\nBeam Width:32 Top-k:32\nxLLM\nMindIE\n1\n4\n8\nRequest Rate (req/s)\n125\n150\n175\n200\nBeam Width:64 Top-k:64\nxLLM\nMindIE\n1\n4\n8\nRequest Rate (req/s)\n150\n200\n250\n300\n350\nBeam Width:128 Top-k:128\nxLLM\nMindIE\nFigure 19: Comparison of mean End-to-End latency(E2E) with various inference frameworks in the\ngenerative recommendation scenario. Since vLLM-Ascend does not support beam width>10, the\ncorresponding results are not plotted in the figure. Even with beam width=4 and request rate=1, its\nmean E2E far exceeds that of our framework.\n2\n8\n32\nMax Concurrency\n50\n100\n150\nTPOT (ms)\nDeepSeek-R1 [2500, 1500]\nw/ MTP\nw/o MTP\n2\n8\n32\nMax Concurrency\n100\n200\nThroughput (tokens/s)\nDeepSeek-R1 [2500, 1500]\nw/ MTP\nw/o MTP\nFigure 20: Impact of MTP on the concurrent performance of DeepSeek-R1 model.\nCo-location Scheduler Policy (denoted as xLLM-OOC), to assess the maximum achievable through-\nput for offline requests without violating the SLO of online requests. The green shaded region in the\nfigure indicates the sustainable range of offline request throughput that remains within the acceptable\nonline SLO violation threshold. When the offline query-per-second (QPS) exceeds a certain level,\nboth the baseline P/D and online priority strategies lead to a sharp increase in the online SLO viola-\ntion rate, indicating interference caused by the offline workload. In contrast, xLLM-OOC maintains\nstable SLO compliance even as offline QPS continues to rise. On our proprietary dataset, xLLM-\nOOC achieves a throughput that is three times higher than the other two methods. Furthermore, it\ndemonstrates improvements of 75% and 17% over online priority and baseline P/D, respectively, on\nthe Azure Code dataset.\nImpact of Multi-layer Pipeline Execution.\nAs evidenced by the results in Table 6, our proposed\nasynchronous scheduling mechanism delivers consistent throughput improvements across all evalu-\nated model scales. The most substantial relative gain of 17.4% is observed for the 1.5B parameter\nmodel, highlighting the method’s particular efficacy for smaller architectures where scheduling over-\nhead constitutes a larger portion of total computation time. While relative improvements moderate\nfor larger models (reaching 0.6% for 7B, 3.7% for 14B, and 6.6% for 32B), all configurations exhibit\nstatistically significant absolute gains. These results robustly validate that our method successfully\nmasks scheduling latency and eliminates computational bubbles, through leveraging placeholder\ntokens to decouple and overlap CPU scheduling with NPU execution.\nWe further assess the effectiveness of the proposed dual-stream architecture for the DeepSeek-R1\nmodel in Table 7. Experimental results from a single decoder layer show that the total communica-\ntion time increases to 12.4ms in dual-stream mode, up from 9.3ms in single-stream mode. However,\nthe computation-communication overlap mechanism successfully hides 80% of the communication\ntime, which brings the exposed communication time down to just 2.5ms, saving 6.8ms per layer. De-\nspite introducing a computational overhead of 4ms for each layer, the dual-stream strategy yields a\nnet performance gain of 172.0ms across the entire 61-layer model, clearly illustrating the capability\nof our scheduling strategy in real-world workloads.\n29\n\nxLLM Technical Report\n10\n20\n30\n40\n50\nRequest Rate (req/s)\n40\n60\n80\n100\nSLO Attainment (%)\nAzure Code\n25\n50\n75\n100\nRequest Rate (req/s)\n40\n60\n80\n100\nAzure Conversation\nOur SLO Aware\nMinimal Load\nRound Robin\nFigure 21: Performance of Dynamic PD Disaggregation Policy with different scheduling strategies.\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\nRequest Rate (req/s)\n0\n20\n40\n60\n80\n100\nSLO Attainment (%)\nOurs\nw/o hybrid EPD\nw/o hybrid EPD and sched\nFigure 22: Impact of Hybrid EPD Disaggregation Scheduler Policy.\nImpact of Adaptive Graph Mode.\nTo validate the effectiveness of the Adaptive Graph Mode opti-\nmization technique, we conduct controlled experiments on the Qwen3-1.7B and Qwen3-4B models.\nAs shown in Table 8, with both the prompt and output lengths set to 2048 tokens, enabling Adaptive\nGraph Mode result in significant performance gains for both models. The throughput of Qwen3-\n1.7B increases from 2385.58 to 3038.88 tokens/s (a 27.4% improvement), while its mean TPOT\ndecreases by 22.0%. Correspondingly, Qwen3-4B achieved an 8.5% increase in throughput and an\n8.8% reduction in latency. These results robustly demonstrate the effectiveness of Adaptive Graph\nMode as a general-purpose inference acceleration technique, with more pronounced optimization\neffects observed on the model with a smaller parameter size.\nImpact of Hierarchical DP Load Balancing.\nOur proposed hierarchical DP load balancing\nscheme is projected to increase total throughput by 5%. The kernel-level optimization yields the\nmost significant latency savings; for example, for an ultra-long 32k-token request, reordering and\nsplitting can reduce a single core’s computational load from 32k to 1.3k tokens, saving approxi-\nmately 800 microseconds. In contrast, the latency savings from inter-DP group migration are more\nmodest. Even after balancing a 20k-token difference, the total time saved over 61 Transformer lay-\ners is only about 600 microseconds. This indicates that the greatest performance bottleneck and\noptimization potential lie within the computation units themselves.\n6\nFuture Work\nAlthough proposed xLLM framework has demonstrated potential in enhancing LLM inference ef-\nficiency and reducing operational cost, achieving truly efficient and inclusive intelligent computing\ninfrastructure requires deeper collaborative innovation across Hardware, Model, and Application\nLayers. Our future research will advance xLLM from a high-performance Inference Engine toward\na comprehensive Operating System for AI that supports next-generation intelligent applications. We\noutline our planned efforts in three key directions:\n30\n\nxLLM Technical Report\n0\n3%\n5%\n10%\nOOC on Qwen2.5-7B\n0\n1\n2\n3\n0\n3%\n5%\n10%\nOnline SLOs Violation Rate\n0\n1\n2\n3\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\nOffline Request QPS (req/s)\n0\n3%\n5%\n10%\n0\n1\n2\n3\n0\n3%\n5%\n10%\nAzure Code on Qwen2.5-7B\n0\n1\n2\n3\n0\n3%\n5%\n10%\n0\n1\n2\n3\nOffline Throughput (req/s)\n0.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\nOffline Request QPS (req/s)\n0\n3%\n5%\n10%\n0\n1\n2\n3\nbase P/D - Online SLOs Violation\nbase P/D - Offline Throughput\nonline priority - Online SLOs Violation\nonline priority - Offline Throughput\nxLLM-OOC (ours) - Online SLOs Violation\nxLLM-OOC (ours) - Offline Throughput\nFigure 23: Impact of our Online-Offline Co-location Scheduler Policy.\nModel\nPrompt Length\nOutput Length\nAsync Scheduling\nThroughput (tokens/s)\nDS-Distill-Qwen-1.5B\n1000\n1000\n×\n8709.90\n1000\n1000\n✓\n10223.15\nDS-Distill-Qwen-7B\n1000\n1000\n×\n3183.68\n1000\n1000\n✓\n3201.83\nDS-Distill-Qwen-14B\n1000\n1000\n×\n1472.22\n1000\n1000\n✓\n1527.23\nDS-Distill-Qwen-32B\n1000\n1000\n×\n1415.20\n1000\n1000\n✓\n1508.76\nTable 6: Impact of asynchronous scheduling mechanism in Multi-layer Pipeline Execution.\n6.1\nFostering an Open and Diverse Hardware Ecosystem\nCurrent AI infrastructure often suffers from tight coupling with specific hardware architectures,\nintroducing computational stability risks and limiting efficient deployment in heterogeneous envi-\nronments such as edge computing. To address this, the first pillar aims to cultivate an open and\ndiverse hardware ecosystem through the following initiatives:\nUnified Hardware Abstraction Layer.\nWe will develop a high-performance, hardware-agnostic\nruntime abstraction layer that encapsulates instruction sets and memory architectures across compu-\ntational units—from domestic cloud chips to edge-side accelerators. This layer will provide unified\noperator interfaces and memory management APIs, enabling seamless migration and efficient ex-\necution across heterogeneous hardware without code modifications, thereby significantly reducing\nintegration barriers for emerging hardware.\nSoftware-Hardware Co-Design and Ecosystem Synergy.\nWe will collaborate with hardware\npartners to define more efficient and open interface standards. This co-design approach will not only\ndrive hardware innovation from the software perspective but also help establish a self-sustaining and\ncompetitive computational supply ecosystem, ultimately providing users with optimized choices\nacross cost, performance, and security considerations.\n6.2\nCultivating a Vibrant and Responsive Model Ecosystem\nThe system’s core value derives from its supported model diversity and integration efficiency. Our\nfuture efforts will focus on building an inclusive and agile model ecosystem through:\nMulti-Scenario Support.\nWe will extend the platform’s optimization and deployment capabilities\nbeyond large language models to systematically support various generative AI scenarios, including\ngenerative recommendation, text-to-image, and text-to-video generation. We will optimize execu-\n31\n\nxLLM Technical Report\nMetric\nSingle-Stream (ms)\nDual-Stream (ms)\nTotal Communication Time\n9.3\n12.4\nOverlapped Communication Ratio\n-\n80%\nExposed Communication Time\n9.3\n2.5\nTotal Computation Time\n13.0\n17.0\nReduced Time per Layer\n-\n2.8\nTotal Reduced Time (61 layers)\n-\n172.0\nTable 7: Communication and computation overheads in a single decoder layer of DeepSeek-R1\nusing Multi-layer Pipeline Execution.\nModel\nPrompt Length\nOutput Length\nAdaptive Graph Mode\nThroughput (tokens/s)\nMean TPOT (ms)\nQwen3-1.7B\n2048\n2048\n×\n2385.58\n39.27\n2048\n2048\n✓\n3038.88\n30.63\nQwen3-4B\n2048\n2048\n×\n1539.93\n55.44\n2048\n2048\n✓\n1671.39\n50.58\nTable 8: Impact of Adaptive Graph Mode.\ntion engines, request scheduling, and memory management specifically for different workloads to\nensure optimal performance across diverse applications.\n“Zero-Day” Model Integration.\nTo accommodate rapidly evolving model architectures, we will\nimplement a unified framework combining model graph representation, an extensible operator li-\nbrary, and automated compilation optimization. This will enable rapid “zero-day” integration of\nnewly released models, reducing deployment cycles from weeks to hours.\n6.3\nEvolving into an AI-Native Application Framework\nTo democratize AI adoption and accelerate value delivery, we will evolve the system from an infer-\nence engine into a full-stack AI-native application framework by:\nFramework-Native AI Middleware.\nWe will design high-level, framework-native APIs and ab-\nstractions that package complex distributed inference, multi-model orchestration, and stateful ses-\nsion management capabilities into out-of-the-box middleware services. This will enable application\ndevelopers to build sophisticated AI applications (e.g., multimodal AI agents) without managing\nunderlying infrastructure complexities.\nRapid Application Integration and Deployment.\nBuilding upon this AI-native framework, we\nwill deliver a comprehensive toolchain — including encapsulated SDKs, application templates, and\nintegrated CI/CD pipelines — to streamline development and deployment processes. Our goal is\nto enable application teams to integrate and deploy AI services within hours rather than weeks,\nsignificantly enhancing business innovation agility and fully bridging the “last mile” from model\ndevelopment to value creation.\n7\nConclusion\nWe introduce xLLM, an intelligent and efficient LLM inference framework, featuring a service-\nengine decoupled architecture.\nThe framework consists of two main components: (1) xLLM-\nService, a versatile service layer designed for efficient instance and cluster management as well\nas request scheduling. It incorporates unified elastic scheduling for co-located online and offline re-\nquests to maximize cluster utilization, a workload-adaptive PD disaggregation architecture for SLO-\naware dynamic instance scheduling, a novel Encode-Prefill-Decode (EPD) disaggregation mecha-\nnism for multimodal requests, a global KV Cache manager for efficient memory management in-\ncluding KV Cache upload and offload and a distributed fault-tolerance design to ensure high avail-\nability. (2) xLLM-Engine, a high-performance inference engine optimized for accelerating LLM\n32\n\nxLLM Technical Report\ninference across various AI accelerators. xLLM-Engine employs full-stack multi-layer execution\npipeline optimizations through asynchronizing CPU-side scheduling and AI accelerator-side model\nforwarding to minimize computational bubbles, utilizing dual-stream parallelism of micro-batches\nto overlap computation with all-to-all communication, and further overlapping various AI compu-\ntation units at the operator level. xLLM-Engine also implements an adaptive graph mode which\npre-compiles kernel sequences, memory operations, and synchronization into a single computation\ngraph to drastically reduce launch overhead and accelerator idle time. Additionally, it adopts a\n\"logically contiguous, physically discrete\" KV Cache storage strategy via proposed xTensor Mem-\nory Management, which resolves the tension between memory contiguity and dynamic allocation.\nTo further boost hardware utilization, xLLM-Engine integrates algorithmic enhancements such as\nasynchronous pipelined adaptive speculative decoding, KV cache-aware scheduling, reactive inter-\nDP group workload migration for dynamic load balancing, and dynamic MoE load balancing based\non real-time expert workload statistics and transparent weight updates. We have further extended\nxLLM to emerging generative recommendation scenarios, improving both recommendation accu-\nracy and efficiency.\nExtensive experiments demonstrate that xLLM achieves a consistent improvement compared to\nleading inference systems such as MindIE and vLLM-Ascend, evaluated across mainstream Qwen-\nseries and Deepseek-series models as well as public and real-world industrial datasets. In partic-\nular, xLLM outperforms MindIE by up to 1.7× and vLLM-Ascend by up to 2.2× in throughput.\nComprehensive ablation studies further validate the effectiveness of key components, including the\nproposed scheduling modules, multi-layer execution pipeline, optimized model tensor parallelism,\nadaptive graph mode, and others.\nBy releasing the xLLM framework as an open-source project, we intend to stimulate further in-\nnovation in developing robust, enterprise-scale inference solutions, optimizing performance on a\ndiverse range of AI accelerators, and creating tightly integrated service and engine architectures for\nnext-generation AI applications.\n33\n\nxLLM Technical Report\nReferences\n[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman,\nDiogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report.\narXiv preprint arXiv:2303.08774, 2023.\n[2] Anthropic. 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Current mainstream inference frameworks face practical challenges. On the one hand, enterprise-grade serving struggles with hybrid and dynamic workloads, strict demand for high availability of services, and distributed storage management. On the other hand, inference execution is bottlenecked by underutilized AI accelerators due to new paradigms of hardwares, model architectures and inference algorithms. To address these challenges, xLLM builds a novel decoupled service-engine architecture. At the service layer, xLLM-Service features an intelligent scheduling module that efficiently processes multimodal requests and co-locates online and offline tasks through unified elastic scheduling to maximize cluster utilization. This module also relies on a workload-adaptive dynamic PrefillDecode (PD) disaggregation policy for instance scheduling and a novel Encode-Prefill-Decode (EPD) disaggregation policy designed for multimodal inputs. Furthermore, it incorporates a distributed architecture to provide global KV Cache management for efficient AI accelerator memory handling and robust fault-tolerant capabilities for high availability. At the engine layer, xLLMEngine co-optimizes system and algorithm designs to fully saturate computing resources. This is achieved through comprehensive multi-layer execution pipeline optimizations, including overlapping CPU scheduling with AI accelerator operations to minimize computational bubbles, employing dual-stream parallelism to overlap computation with communication, and fine-grained overlapping of various computational units to maximize hardware utilization. These are complemented by an adaptive graph mode that drastically reduces kernel launch overhead, alongside the innovative \"logically contiguous, physically discrete\" xTensor memory management which resolves memory allocation conflicts. xLLM-Engine also further integrates algorithmic enhancements such as optimized speculative decoding and dynamic Expert Parallel Load Balance (EPLB), collectively serving to substantially boost throughput and inference efficiency. Extensive evaluations demonstrate that xLLM delivers significantly superior performance and resource efficiency. Under identical TPOT constraints, xLLM achieves throughput up to 1.7× that of MindIE and 2.2× that of vLLM-Ascend with Qwen-series models, while maintaining an average throughput of 1.7× that of MindIE with Deepseek-series models. We have deployed xLLM in production to support a range of core business scenarios at JD.com, covering areas including LLM, Multimodal Large Language Model (MLLM), and generative recommendation. These applications encompass the JingYan AI chatbot, marketing recommendations, product understanding, customer service assistants, and more. xLLM framework is publicly available at https://github.com/ jd-opensource/xllm and https://github.com/jd-opensource/xllm-service. BCorresponding authors: {liutongxuan1, , {guyue, , hai- , , {dinggg, . 1 16 Oct 2025 xLLM Technical Report Contents 1 Introduction 3 2 System Overview 5 2.1 xLLM-Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 xLLM-Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 xLLM-Service Designs 7 3.1 Online-Offline Co-location Scheduler Policy . . . . . . . . . . . . . . . . . . . . . 7 3.2 Dynamic PD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . . 9 3.3 Hybrid EPD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . . 10 3.4 Global KV Cache Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Fast Fault Recovery Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 xLLM-Engine Designs 13 4.1 Multi-layer Pipeline Execution Engine . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Adaptive Graph Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Efficient Memory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Algorithm Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.1 Optimized Speculative Decoding . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.2 Dynamic EP Load Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4.3 Hierarchical DP Load Balance . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5 Generative Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5.1 Host-side Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5.2 Device-side Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Evaluations 24 5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.1.1 Benchmarking Performance . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.1.2 Business Serving Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Ablation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Future Work 30 6.1 Fostering an Open and Diverse Hardware Ecosystem . . . . . . . . . . . . . . . . 31 6.2 Cultivating a Vibrant and Responsive Model Ecosystem . . . . . . . . . . . . . . . 31 6.3 Evolving into an AI-Native Application Framework . . . . . . . . . . . . . . . . . 32 7 Conclusion 32 2 xLLM Technical Report 1 Introduction In recent years, large language models (LLMs) with parameters ranging from billions to trillions (GPT [1], Claude [2], DeepSeek [3], LLaMA [4], etc.) have achieved breakthrough progress in the fields of natural language processing and multimodal interaction, which drives an urgent demand in the industry for efficient inference engines and service systems. These models are being rapidly deployed in core business scenarios such as intelligent customer service [5], real-time recommendation [6], and content generation [7]. However, how to reduce the cost of model inference and improve computing efficiency remains a key challenge for large-scale commercial serving. Current mainstream LLM inference frameworks (vLLM [8], SGLang [9], TensorRT-LLM [10], etc.) face four key challenges in enterprise-level serving scenarios: First, in an inference cluster with hybrid deployment, online inference requests exhibit significant tidal characteristics [11, 12]. Current scheduling systems fail to meet the service level objective (SLO) for online services while fully leveraging idle periods of online services to increase the throughput of offline tasks. Second, existing Prefill-Decode (PD) disaggregation architecture [13-15] assumes static resource allocation for the two phases, which cannot adapt to the dynamically changing request loads (i.e., input/output lengths fluctuate) in real-world applications, resulting in low AI accelerator utilization and increased SLO violation risks. Third, there is a lack of strategy to efficiently service multimodal requests (i.e., image, voice and text input [16, 17]), including parallel processing for the encode phase and fine-grained resource allocation accordingly. Fourth, as the scale of the inference cluster increases, ensuring fast fault detection and service recovery for nodes or instances is critical to maintaining the stability of inference services. The evolving computing paradigms also present significant performance challenges on existing LLM inference engines: First, they struggle to fully utilize the computing units of modern AI accelerators [18, 19]. Second, the All-to-All communication overhead [20] and expert parallel (EP) load imbalance in the Mixture of Experts (MoE) architecture [21, 22] restrict the scalability of the system. Third, as the model context window continues to expand, efficient KV Cache management becomes critical to inference performance [23]. Forth, Due to the unpredictable nature of inference requests, conventional static scheduling and operator strategies face difficulties in effectively balancing workloads across computing units in Data Parallelism (DP). To address above challenges , we propose xLLM, an efficient and intelligent LLM inference framework featuring a service-engine decoupled design. xLLM achieves efficient support for enterpriselevel inference through the following innovations: at the service layer, xLLM has made groundbreaking achievements in 1) unified elastic scheduling for online/offline requests, 2) workloadadaptive dynamic PD disaggregated architecture, 3) novel Encode-Prefill-Decode (EPD) disaggregation for multimodal requests, and 4) a distributed cache management and fault-tolerance framework; at the engine layer, xLLM enhances the resource efficiency across the full-stack of \"communicationcomputation-storage\", including 1) a multi-layer pipeline execution mechanism, 2) efficient computing and memory optimization, and 3) intelligent algorithm designs. Specifically, xLLM-Service schedules online requests with preemptive execution and offline requests in a best effort manner, to maximize resource utilization while strictly ensuring SLOs for online services. To address the inherent limitations of the static PD configuration, xLLM incorporates an adaptive scheduler to dynamically adjust the proportion of PD instances for each request and supports fast role reversal for instances, by monitoring key metrics such as Time to First Token (TTFT) and Time Per Output Token (TPOT) [24]. For multimodal visual requests, xLLM automatically selects the optimal EPD phase-disaggregation strategy based on pre-profiling, for the best performance trade-off between throughput and latency. To handle failures such as hardware problems, network faults, or software errors [25], xLLM collects node error information, evaluates KV recomputation or migration costs of interrupted requests, and makes optimal global rescheduling decisions. Across multiple instances, xLLM supports KV offloading and routing migration within a hybrid storage architecture, improving KV storage capacity and cache hit rates. xLLM-Engine employs a multi-layer execution pipeline that incorporates hardware-specific optimizations: (i) at the framework scheduling layer, it implements asynchronous CPU-accelerator scheduling to minimize computational idle time; (ii) at the model graph layer, it utilizes dual-stream micro-batch parallelism to overlap computation with communication; (iii) at the operator level, it achieves kernel computation and memory access overlapping. For computational efficiency, xLLM3 xLLM Technical Report Multi-modal Requests xLLM Service Fault Recovery Global Scheduler Hybrid EPD Disagg. Dynamic PD Disagg. On/Offline Co-location Global Cache Mgr xLLM Engine Adaptive Graph Operator Optim. xTensor Memory drive drive manage schedule Speculative Inference Dynamic EPLB Prefill Local KV Cache KV Cache Transfer ... Prefill Local KV Cache KV Cache Transfer ... Prefill Instance Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Instance Local KV Cache KV Cache Transfer ... System-level Designs Batching Scheduler Algorithm-driven Tech. co-optimize Hierarchical DPLB Multi-layer Pipeline Figure 1: Overview of xLLM's capabilities, with decoupled service-engine architecture. Engine automatically fuses small kernels into a unified computation graph and dispatches them to the accelerator in a single operation, significantly reducing kernel launch overhead. Regarding memory management efficiency, xLLM-Engine introduces the xTensor memory management scheme, which employs a \"logically contiguous, physically discrete\"\" KV cache storage structure that resolves the conflict between memory contiguity requirements and dynamic allocation needs. To further enhance performance, xLLM-Engine incorporates an adaptive speculative decoding mechanism, a redundant expert-based load balancing algorithm for EP, and a hierarchical load balancing algorithm for DP. Additionally, xLLM-Engine provides scenario-specific optimizations, such as for generative recommendation - one of JD.com's core businesses - where it achieves 23% performance improvement through host-kernel operation overlapping. In summary, we make the following contributions when developing xLLM: xLLM Intelligent Service Capabilities • We design a unified scheduling algorithm for online/offline workloads (§3.1). • We implement a workload-adaptive PD disaggregation policy to address scenarios with rapidly changing traffic load and request input/output lengths (§3.2). • We propose a hybrid EPD disaggregation policy for multimodal requests, achieving intelligent resource allocation across different phases (§3.3). • We leverage multi-level KV Cache management and global KV Cache routing strategy to expand KV Cache capacity and improve cache hit rates (§3.4). • We design a multi-node fault tolerance architecture to ensure high service availability (§3.5). xLLM Intelligent Engine Capabilities • We achieve intelligent and efficient inference, through hardware-software co-design to improve hardware computing efficiency, including pipeline execution (§4.1), graph optimization (§4.2), and memory optimization (§4.3). • We enhance inference performance through algorithmic optimizations (§4.4), including optimized speculative decoding (§4.4.1), dynamic expert parallel load balance (§4.4.2), and hierarchical data parallel load balance (§4.4.3). • We optimize online inference for generative recommendation scenarios (§4.5). 4 xLLM Technical Report Predictor Profiler Dynamic PD Disaggregation Hybrid EPD Disaggregation On/Offline Co-locatiom Prefill Pool Decode pool Encode Pool Global Cache Management Fault Recovery Framework Pre-Processing Layer Scheduling Layer online preempt support disaggregate dynamic PD allocation KV & Images Resource Layer Offline Requests Online Requests Figure 2: System workflow of xLLM-Service. 2 System Overview The overall architecture of xLLM is depicted in Figure 1. Upon request arrival, xLLM-Service performs intelligent scheduling to distribute each request to one of three elastic instance pools and manages instance migration across these pools during runtime. xLLM-Engine then drives efficient request inference by orchestrating system- and algorithm-level optimizations. 2.1 xLLM-Service We describe the workflow of xLLM-Service in Figure 2. The system comprises three primary layers: 1) a preprocessing layer consisting predictor and profiler, 2) a scheduling layer integrating three policies (Dynamic PD Disaggregation Policy, Hybrid EPD Disaggregation Policy, and OnlineOffline Co-location Policy), and 3) a resource layer consisting of three heterogeneous instance pools. Specifically, the Dynamic PD Disaggregation Policy leverages information from the predictor to dynamically convert PD instances based on the workload status of the two instance types. The Hybrid EPD Disaggregation Policy utilizes the profiler to determine the optimal EPD disaggregation strategy for multimodal requests. Concurrently, the Online-Offline Co-location Policy dispatches requests according to their online or offline attributes. KV and image caches are offloaded and routed among distributed instances, while the fault recovery framework ensures the high availability of the entire service. Elastic Instance Pools. Instances in a cluster are partitioned into three elastic pools: ▷Prefill Instance Pool: This pool handles the prefill phase of text requests. ▷Decode Instance Pool: This pool handles the decode phase of text requests. ▷Encode Instance Pool: This pool handles the encode phase of multimodal requests. 5 xLLM Technical Report We design the instances in Prefill and Decode pools as stateless, i.e., instances do not require physical migration between instance pools. Instead, they achieve flexible switching between PrefillDecode roles based on the type of request being processed. Request Preprocessing. xLLM-Service implements unified resource management for both online and offline requests. Online requests are submitted in a preemptive and deadline-prioritized manner, and are co-deployed with best-effort offline tasks within the shared resource pool. During the runtime of an offline request, the Unified-Scheduler dynamically scales it up/down based on the tidal traffic characteristics of online requests. When an online request is submitted, it first passes through the pre-processing layer, which consists of two modules: ▷TTFT Predictor: A TTFT prediction model built for text requests. It evaluates SLO fulfillment by analyzing queueing delays from each prefill instance queue and request input lengths, thereby guiding instance allocation for the Dynamic PD Disaggregation Policy in the scheduling layer. ▷ EPD Profiler: A profiler for multimodal requests that uses binary search to identify optimal deployment configurations: (1) EPD separation strategy, choosing from three approaches: EP-D (i.e., aggregated execution of Encode and Prefill phases, with Decode phase executed separately), ED-P, or E-P-D; (2) The maximum batch size for the Encoder phase; (3) The maximum number of tokens for Prefill/Decode's inputs. The Hybrid EPD Disaggregation Policy in the scheduling layer will use the optimal configuration determined for phase disaggregation and task dispatching. Intelligent Scheduling. The intelligent scheduling layer adjusts resource allocation for requests across their full lifecycles. It contains three major scheduling policies designed for various scenarios: ▷Online-Offline Co-location Policy: This policy implements a preemptive scheduler for managing online and offline requests. When the load of online requests reaches the peak, they preempt some offline requests on PD instances. To surrender resources to offline requests, when the load on P instances decreases (usually earlier than on D instances), P instances continue to process offline prefill requests and also migrate decode offline requests from D instances to P instances. ▷Dynamic PD Disaggregation Policy: This adaptive scheduling policy is responsible for dynamically managing P and D instance allocation. It intelligently assigns requests to suitable instances using a heuristic algorithm guided by the TTFT Predictor. Furthermore, it implements a feedback mechanism by continuously collecting performance data from computing instances, enabling runtime monitoring and adjustment of allocation decisions to maintain system efficiency. ▷Hybrid EPD Disaggregation Policy: This multimodal policy executes the three-phase disaggregation based on strategies searched by the EPD Profiler. For the EP-D disaggregation, the fused EP phase executes in the P instance pool; for the ED-P disaggregation, the fused ED phase executes in the D instance pool; for the E-P-D disaggregation, the three phases execute separately in the three instance pools. This deployment also enables multimodal requests to benefit from the adjustment of Dynamic PD Disaggregation Policy. KV-centric Storage Architecture. The instance storage employs a hybrid architecture (HBMDRAM-SSD) to cache KV values and image tokens. At the global level, xLLM takes the idea from Mooncake Store [23] and extends it to domestic accelerators with specific optimizations. Routing and reuse of caches across instances are determined by embedded intelligent routing strategies. Efficient Fault-tolerant. The fault recovery framework of xLLM-Service supports fault detection and fast recovery of instances from the three elastic pools (E, P, D). For requests on failed instances, the architecture manages the migration of image caches between instances, and automatically decides the optimal KV recomputation or migration strategy for handling affected KV caches. 2.2 xLLM-Engine During request execution, the xLLM-Engine layer provides intelligent computing capabilities. We achieve various joint inference accelerations between the computing system layer and the algorithmdriven layer in the following ways: Computing System Layer 6 xLLM Technical Report Latency-relaxed instance Online Prefill Latency-strict instance Online / Offline Mixed Decode Offline Prefill Offline Decode Resource Plane Control Plane On-demand Preemption Offline Request Migration Offline Request Gating Mix decoding Selection Figure 3: Overview design of Online-Offline Co-location Scheduler Policy. ▷Multi-layer Pipeline Execution Engine: In the framework layer, CPU tasks are scheduled asynchronously to create a pipeline with inference computation, reducing computational bubbles; in the model graph layer, a single batch is split to form a pipeline between two micro-batches, effectively overlapping computation and communication; in the operator kernel layer, operations are pipelined across different computational units, enabling computation and memory access to overlap. ▷ Graph Optimization for Dynamic Inputs: Small kernels in the decoding stage are fused into a single computational graph. To handle variable sequence lengths and batch sizes, we parameterize the input dimensions, and employ a multi-graph caching scheme to reduce compilation overhead. ▷xTensor Memory Management: It uses a \"logically continuous, physically discrete\" KV storage structure. It allocates physical memory space on-demand during token generation for each request, while asynchronously predicting and intelligently mapping the physical pages required for the next token. When a request completes, the existing physical memory will be reused to execute the next request. Algorithm-Driven Layer ▷Speculative Decoding: xLLM-Engine incorporates an optimized speculative inference algorithm that generates multiple tokens at once to boost throughput [26]. It further optimizes the computing architecture through ways like asynchronous CPU processing and reduced data transfer. ▷EP Load Balance: For MoE models, xLLM implements expert weight updates based on historical expert load statistics, enabling effective dynamic load balancing during inference. ▷DP Load Balance: For data parallel (DP) deployment, xLLM achieves fine-grained load balance by kvcache-aware instance allocation, inter-DP requests migration, and intra-DP computing units allocation. We will elaborate on the detailed design of xLLM-Service in §3 and outline the optimizations within the xLLM-Engine in §4. A comprehensive evaluation of xLLM will be presented in §5. 3 xLLM-Service Designs 3.1 Online-Offline Co-location Scheduler Policy Online/Offline Request Characteristic. LLM services can be categorized into two types based on their service modes: online and offline requests. Online requests, including those from chatbots [1, 2, 27], code completion [28-30], and recommendation systems [31, 32], constitute latencysensitive workloads. These services must respond immediately upon request arrival, often returning each generated token in real time via streaming output. Consequently, they impose strict SLO requirements on TTFT or TPOT to ensure a satisfactory user experience. In contrast, offline services, such as document analysis [33] and intelligent data annotation [34], are non-real-time workloads with minimal latency constraints and thus no stringent SLO requirements. Furthermore, we observe that the request traffic for online services typically exhibits significant volatility, including tidal variations at hourly or daily scales and sudden bursts at minute-level intervals [11, 12]. While cluster auto-scaling [35] could theoretically mitigate resource underutilization caused by tidal patterns, the slow cold-start latency of instances-involving model loading and com7 xLLM Technical Report plex initialization-renders it ineffective for responding to rapid traffic spikes [36], increasing the risk of service-level agreement violations. Online/Offline Request Co-location Deployment. To address these practical challenges, we adopt a hybrid deployment strategy that co-locates online and offline requests. In such a system, offline requests can utilize idle resources during off-peak periods of online traffic. Conversely, during online traffic peaks, offline tasks can be preempted, as they are not bound by strict SLOs. This approach significantly enhances aggregate resource utilization and mitigates idleness during traffic troughs. Although some recent work [37-39] also attempts to co-locate offline requests, they have not explored the multi-instance scenario, especially for PD disaggregation. Actually, the PD disaggregation architecture has demonstrated superior latency performance and is increasingly becoming a mainstream design paradigm in industry [13, 23]. However, directly applying online-offline co-location to PD-disaggregated systems introduces critical PD load imbalance issues: (1) such systems require the load ratio between prefill and decoding stages to align with their respective resource allocation ratios; otherwise, one stage may become a bottleneck, causing blocking or resource underutilization in the other. (2) Furthermore, as these PD load variations exhibit high volatility and burstiness similar to online traffic patterns, existing PD disaggregation techniques struggle to effectively address this challenge. Latency-Constrained Decoupled Architecture. We rethink the design of PD disaggregation architecture under online-offline co-location deployment. The latency advantage of PD disaggregation essentially stems from the separation of latency constraints: the decoding phase is highly sensitive to per-step latency and cannot be blocked by long-duration operations, thus necessitating decoupling from the prefill phase. Inspired by this insight, we propose a latency-constrained decoupled architecture, as illstrated in Figure 3. This design regards cluster resources as two pools: a latencyrelaxed pool (corresponding to original Prefill instances) and a latency-strict pool (corresponding to original Decode instances). All tasks are then reassigned to one of the two resource pools based on their inherent latency characteristics and requirements. Within this architecture, the decoding phase of offline requests can be executed in either resource pool. This flexibility allows us to dynamically adjust the load ratio between the two pools, thereby maximizing the overall resource utilization of the cluster. However, it also introduces another two challenges: (1) Complex Scheduling Space: since the decoding of offline requests can be performed on either type of instance, how to leverage this flexibility to design new intelligent scheduling remains unknown. (2) Strict SLO Guarantee: the execution of offline requests consumes resources, their prefill phase may block newly arrived online requests, and their decode phase on latency-strict nodes may slow down the overall response speed. Both scenarios can compromise the SLO satisfaction of online requests. Solution 1 - Performance Bottleneck Analysis. We construct an LLM inference performance model based on the Roofline Model [40] and online factor learning. This model is designed to predict the latency, computation utilization, and memory utilization of both the prefill and decode phases. Since decoding operations executed on latency-strict instances typically account for a large proportion of workloads and are highly performance-sensitive, we set balancing computational and memory resources as the optimization objective. By analyzing performance bottlenecks through this model, we can select more appropriate offline requests to merge into decoding batches, thereby improving resource utilization efficiency. Solution 2 - Efficient Preemption Mechanism. To strictly ensure that the SLO of online requests remains unaffected, we introduce a preemption mechanism that allows online requests to preempt offline requests. For offline prefill tasks running on latency-relax nodes, we propose a model execution interruption technique, which enables preemption within an acceptable latency range without incurring additional model maintenance overhead. For decoding tasks running on latency-strict nodes, we leverage the performance model to dynamically select requests for decoding batching, ensuring that decoding latency always meets SLO constraints. 8 xLLM Technical Report Stateless Instance Local Scheduler request queue migration queue Execution Engine Schedule Request & Cache Migration P D prefill unfinished P D decode unfinished decode finished prefill finished D P Global Scheduler Instance Scheduler Request Scheduler Instance Monitor TTFT / TPOT, Req Len, Memory Util., etc. schedule schedule report status Figure 4: Overview design of Dynamic PD Disaggregation Scheduler Policy. 3.2 Dynamic PD Disaggregation Scheduler Policy Inefficiency of Existing PD Disaggregation Policy under Workload Fluctuations. The PrefillDecode (PD) disaggregated inference architecture [13, 24, 23, 41] partitions computing instances into dedicated prefill and decode instances, each handling their respective processing phases. This design mitigates interference between prefill and decode requests, achieving superior performance compared to PD co-located architectures [13]. However, we observe that most existing PD disaggregated systems employing static instance partitioning schemes suffer from low hardware utilization and inadequate responsiveness to traffic bursts. On one hand, existing analysis-based methods [13, 24, 23, 41] typically determine the PD ratio using profiling or simulator data. These approaches remain effective only when request arrival patterns and length distributions remain relatively stable. Under significant workload fluctuations, pre-collected analysis data often fails to accurately capture real-time request characteristics, leading to mismatches between preset load-balancing ratios and actual load requirements. On the other hand, when confronting substantial variations in input/output lengths within production workloads, existing PD disaggregation architectures [13, 42] generally adopt dynamic instance type adjustment strategies [43]. Nevertheless, online switching between prefill and decode instances typically involves multiple steps-including monitoring, waiting for flip conditions, and instance restarting-which introduces substantial latency overhead [15]. To address these limitations, we propose a Dynamic PD Disaggregation Policy that adaptively adjusts resource allocation in response to real-time workload characteristics. Runtime Instance Monitor. Performance indicators such as TTFT and TPOT directly reflect the capability of instances to process requests in the prefill and decode phases, respectively. They serve as the core basis for evaluating whether an inference system can meet SLO requirements. Ideally, prediction-based methods could be adopted to dynamically assess these indicators. However, while TTFT exhibits relatively predictable characteristics (as its computation time is proportional to the square of the input sequence length [44, 41]), due to the uncertainty in the length of output tokens and transfer overhead, TPOT is difficult to accurately predict using traditional input/output features and cluster load metrics. To address this issue, we deploy additional instance monitors to collect real-time performance data from each computing instance. This data includes metrics such as the number and length of prefill and decode requests, memory usage, TTFT, TPOT, and token generation intervals. The system can further use these real-time performance indicators to dynamically evaluate instance loads and adjust scheduling strategies. Stateless Instance and Elastic Pool. As illustrated in Figure 3, Dynamic PD Disaggregation Policy adopts a design of stateless instances and elastic instance pools to enable fast and dynamic switching of instance roles. First, it treats prefill or decode phase as the request attribute rather than instance attribute. Instances are designed to be stateless, allowing each instance to process both prefill and decode requests simultaneously. Additionally, to facilitate the management of multiple instances, we further extend PD pools to four elastic instance pools (i.e., P, D, P→D, D→P) as de9 xLLM Technical Report scribed in §2.1. When flipping an instance (switching its role), we only needs to remove the instance from its original pool and move it to the new pool. This achieves zero-wait-time instance scheduling, avoiding the overhead of instance restart or model reloading incurred in traditional systems. SLO-aware Instance Role Switching. The instance scheduling strategy is dynamically adjusted strictly based on SLO objectives: during the prefill phase, if it is predicted that existing instances cannot meet the TTFT requirements, the conversion of decode instances is triggered; while during the decode phase, when resource shortage occurs, the average token generation interval exceeds the TPOT threshold, or prefill instances are idle, the conversion of prefill instances to decode instances will be initiated to cope with sudden traffic surges. Specifically, when decode instances are reallocated to prefill instances, the scheduler prioritizes selecting the instance with the lightest load (i.e., the fewest tokens being processed) from the P→D pool for role conversion, and always ensures that at least two decode instances are available; conversely, when prefill instances are reallocated to decode instances, the scheduler prioritizes scheduling from the D→P pool, which avoids local overload and maximizes resource utilization. SLO-aware Request Scheduling. The request scheduling scheme adopts a two-level architecture: ▷Global Request Scheduler: The scheduler implements a greedy strategy of prioritizing the lightest load, while being strictly restricted by SLO constraints. For prefill requests, the scheduler first evaluates the estimated queuing latency of each instance in the Prefill pool, selects the candidate instance with the smallest latency, and then invokes the TTFT prediction model for verification: if the estimated TTFT can still meet the SLO requirements after assigning the request to this instance, the request is allocated immediately; otherwise, it continues to find a suitable instance in the D→P pool. If no instance can meet the TTFT requirements, the instance scheduling mechanism is triggered to allocate resources from the decode side. For decode requests, the scheduler gives priority to having the original prefill instance continue processing (to avoid KV Cache transfer overhead); secondly, it selects the instance with the fewest running tokens in the Decode pool, and checks whether the total number of tokens in its current batch is below the memory capacity upper limit and computing throughput limit determined by pre-analysis, so as to ensure that the new request will not cause TPOT to exceed the standard. ▷Local Request Scheduler: Each instance adopts a refined queue management strategy internally. KV Cache transfer events are placed in an independent migration queue and processed sequentially in accordance with the FCFS principle; for forward requests, an innovative scheme combining Chunked Prefill [45] and Continuous Batching [8] is adopted: on the premise of ensuring that decode requests are prioritized to enter the running batch, the remaining computing resources are used to process chunked prefill requests in parallel [46]. 3.3 Hybrid EPD Disaggregation Scheduler Policy Challenges of Multi-modal Inference. The inference process of multimodal large language models (MLLMs) [47-52] typically consists of three phases: the image encoding phase (for extracting image features), the prefill phase (for encoding images and text prompts, feeding them into the language model to generate the first output token, and caching intermediate states), and the decode phase (for iteratively generating subsequent tokens based on the cached data). Existing mainstream inference engines, such as vLLM [8], Text-Generation-Inference [53], SGLang [9], and DistServe [13] are all tailored for LLMs, and thus face several challenges in handling inference tasks for MLLMs: ▷Insufficient Parallelism: For example, the inference of visual models and language models can be executed in parallel, thus improving the utilization of computing resources. However, most existing inference engines [54] adopt a serial strategy, failing to exploit the inter-request parallelism effectively. ▷ Coarse-grained Scheduling: The decode phase is a memory-intensive task, which is suitable for batch processing to improve throughput [55]; the prefill phase is computationally intensive, and it is appropriate to adopt the chunked prefill [45] together with the decode phase to balance latency and throughput [56]. The computation and memory access overhead of the encode phase lies between the two, and it can also benefit from independent scheduling and batch processing. Nevertheless, existing engines process encode and prefill in a combined manner, failing to perform 10 xLLM Technical Report TPOT & TTFT SLO Target ED D P + EP + E P D + + EPD Profiler Encode Instance Vision Model Image Cache Batch Scheduler Migrate Scheduler Prefill Instance Language Model Image Cache Batch Scheduler Migrate Scheduler KV Cache Decode Instance Language Model Batch Scheduler Migrate Scheduler KV Cache Hybrid EPD Disaggregation async transfer async transfer Figure 5: Overview design of Hybrid EPD Disaggregation Policy. phase-specific batch processing and lacking support for chunked prefill. Coarse-grained scheduling makes it difficult to finely control the execution time. ▷Disaggregation Strategy: Existing architectures such as DistServe [13] reduce resource interference by decoupling the prefill and decode phases. However, in multimodal inference, encoding and prefill jointly affect TTFT, while prefill and decode jointly determine TPOT. Under different loads, how to select the optimal decoupling strategy remains a challenge [57]. For example, the performance of strategies such as E+P+D, EP+D, and ED+P in different tasks has not yet been evaluated, and in-depth analysis is required to guide system design. Dual-stream Parallel. As illustrated in Figure 5, we adopt a dual-stream parallel mode, where the visual model and language model are assigned to separate execution streams: the visual stream is dedicated to performing computationally intensive image encoding tasks, while the language stream handles the prefill and decode operations for language generation. By isolating workloads into distinct streams, our system achieves concurrent execution of heterogeneous phases from different requests. Three-Phase Disaggregation. To address the differences in computation and memory access characteristics among tasks in each phase of multimodal inference, we propose a phase-aware scheduling strategy. Requests are subdivided into three phases within an instance: encode, prefill, and decode. Batch processing and scheduling optimizations are performed for each phase respectively: ▷Optimized Batch Processing: The maximum batch size for image encoding and the token budget for the language model are set according to the user's SLO. Specifically, during system startup, we use a binary search method to profile the maximum encoding batch size and the model's token budget, ensuring that the execution time of subsequent batch processing tasks in each iteration is less than the TPOT SLO. In each iteration, we (i) first add all running decode requests to the current batch; (ii) then check for any chunked prefill tasks that have been partially computed. If such tasks exist, we add them to the batch; (iii) if there are no chunked prefill tasks, we check for pending encoding tasks and add them if any exist. This approach aims to complete requests in the prefill phase as quickly as possible, reducing their TTFT. New requests' encoding phases are processed only when no requests are in the prefill phase. ▷ Phase-aware Scheduling: To address the difficulty in selecting decoupling strategies for multimodal inference tasks, xLLM proposes the Hybrid Encode-Prefill-Decode Disaggregation architecture, an innovative multi-instance collaborative inference framework. In this architecture, each instance only executes part of the subtasks among the three phases, while the remaining phases are processed by migrating requests to other instances, thereby avoiding resource waste and interference. During runtime, our system automatically selects the optimal disaggregation strategy based on historical request profiling (via EPD Profiler) to dynamically balance throughput and latency objectives. Compared with the traditional approach of binding instances to single-phase or full-phase tasks, this architecture significantly improves the overall system processing capability and resource utilization efficiency while meeting SLO requirements. 11 xLLM Technical Report Master Election Cache & Load Metric Sync Service Discovery ETCD Global Multi-Level KV Cache Management Master Global Scheduler Global KVCache Mgr Select Master Write Cache & Load Metrics Watch & Notify HeartBeat (Lease) Report Cache & Load Metrics Req Req Resp Prefill Local KV Cache KV Cache Transfer ... Prefill Local KV Cache KV Cache Transfer ... Prefill Instance Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Instance Local KV Cache KV Cache Transfer ... Slave Slave Slave Figure 6: The framework of Global Multi-Level KV Cache Management. 3.4 Global KV Cache Management During the decode phase of LLMs, subsequent tokens are generated autoregressively one by one. Although the computational cost per step is relatively low, frequent access to the historical KV Cache is required, making memory bandwidth the primary bottleneck. As model size expands and context window grows, the memory consumption of the KV Cache exhibits an exponentially increasing trend. For instance, a context of 128K tokens may consume over 40GB of memory, severely straining the memory resources of single-GPU devices [3]. Although current mainstream optimization solutions like vLLM [8] and FasterTransformer [58] have made significant progress in single-instance environments, many critical issues remain unresolved. In long-context scenarios, the time required for the prefill phase increases dramatically, while the decode phase suffers from intense competition for memory bandwidth. To meet stringent SLO requirements (e.g., TTFT DRAM > SSD) adhering to the strict consistency rule: \"if data resides in HBM, it must also be present in DRAM\". Specifically, when local operations involving KV cache (including prefix cache [8]) loading and offloading occur, these operational events are aggregated at regular intervals and transmitted to the xLLM-Service via ETCD heartbeat mechanisms, enabling unified global monitoring and management. 12 xLLM Technical Report KV Cache-aware Scheduling. In terms of scheduling strategy, the system implements a decisionmaking mechanism based on KV cache utilization, which operates through three key steps: (1) Prefix Matching Detection: calculating the KV cache reuse rate for each candidate node via prefix matching analysis; (2) Performance Estimation: estimating the expected latency for different nodes according to current load conditions and cache hit rates; (3) Optimal Node Selection: identifying the node with the optimal overall performance for request processing, thereby enabling dynamic offloading and migration of KV cache. 3.5 Fast Fault Recovery Architecture Current fault handling mechanisms are mainly designed for model training [60-62] and cannot be directly applied to large model inference due to its low-latency requirements. Specifically, existing approaches primarily adopt a checkpoint-then-recovery method [63-65] for fault handling, which periodically stores model data as checkpoints in distributed storage and reloads the most recent checkpoint after a failure occurs [66]. As model parameters increase, the overhead of storage and loading gradually grows. This is still acceptable in training since there are no strict latency requirements [67], but in inference, it may cause all high-priority requests on the failed instance to time out, resulting in severe losses. To address this issue, we propose an efficient failover architecture specifically for large model inference, with targeted optimizations in two aspects: fast request migration and fast instance recovery. Fast request migration ensures the SLO of high-priority tasks mainly through an efficient kv cache quick recovery strategy, while fast instance recovery achieves low-overhead recovery through efficient masking of computation and communication. Through the above optimization methods, we achieve fast fault recovery, greatly reducing the performance and economic losses caused by faults. 4 xLLM-Engine Designs 4.1 Multi-layer Pipeline Execution Engine Traditional autoregressive LLM inference relies on single-stream sequential execution. This conventional approach fails to fully exploit the parallelism capabilities of modern hardware, often leading to computational resource underutilization, communication stalls, and cascading blocking delays in heterogeneous computing environments [68, 69]. As illustrated in Figure 7, the system's performance is hindered by three primary sources of inefficiency: (1) CPU-Accelerator Dependency: At the framework level, a rigid dependency between the CPU and the accelerator forces the accelerator to remain idle during task scheduling and data processing phases, creating significant \"computation bubbles\" [68, 3]. (2) Communication Latency: In both distributed workloads and complex model layers, communication delays interrupt continuous computation, preventing the full utilization of available hardware resources [69, 70]. (3) Architectural Bottlenecks: On certain AI accelerators, the compute-focused (tensor cores) and general-purpose (vector cores) computation units lack a shared high-level cache (e.g., L1 or SRAM). This architectural separation necessitates additional data transfers between them, which introduces latency and leads to the underutilization of the specialized computation units. To address these challenges, we propose a three-tier asynchronous pipeline parallelism design spanning the framework scheduling layer, the model graph layer, and the operator layer, as shown in Figure 7, with the goal of maximizing hardware efficiency. Framework-Layer Scheduling-Execution Overlap. In a typical sequential execution procedure, the accelerator remains idle while the CPU performs scheduling to prepare the input data batch for the next computation cycle. This serial dependency creates significant latency and underutilizes the accelerator. To eliminate this bottleneck, we introduce an asynchronous pipelined scheduling mechanism that decouples CPU scheduling from accelerator execution. The core idea is to overlap these two stages: while the accelerator is executing the forward pass for the current batch, the CPU concurrently assembles the batch and prepares the metadata for the next batch. This parallel workflow effectively hides the latency of CPU-side scheduling and data preparation. The process is as follows: (1) Accelerator Compute: The accelerator is busy executing the forward pass for the current iteration, 13 xLLM Technical Report w/o Overlap w/ Overlap CPU AI Accelerator AI accelerator bubble AI accelerator bubble ... ... ... ... Schedule 0 Schedule 1 Schedule 2 Schedule N Forward 0 Forward 1 Forward 2 Forward N Schedule 3 Schedule 0 Forward 0 Schedule 1 Forward 1 Schedule N Forward N Framework Layer Communication Stream ... ... Model Layer No bubble Attn 0 Dispatch 0 MoE 0 Combine 0 Attn 1 Dispatch 1 MoE 1 Combine 1 Computation resource waste Computation resource waste Attn 0 Attn 0 MoE 0 MoE 0 Attn 1 Attn 1 MoE 1 MoE 1 Full utilization Computation Stream Dispatch 0 Dispatch 0 Combine 0 Combine 0 Dispatch 1 Dispatch 1 Combine 1 Combine 1 Computation Stream - 1 Operator Layer Vector Matrix Matrix Comp. Unit idle Matrix Comp. Unit idle Attn 1-0 Attn 1-1 Attn 1-2 Attn 1-N MoE 1-0 MoE 1-1 MoE 1-2 MoE 1-N Full utilization Computation Stream - 2 Stream 0 Attn 1-0 Attn 1-1 Attn 1-2 Attn 1-N MoE 1-0 MoE 1-1 MoE 1-2 MoE 1-N Vector Matrix Vector Matrix Vector Matrix AI Comp. Unit CPU AI Accelerator Batch 0 Batch 1 Matrix Comp. Unit Vector Comp. Unit Matrix Comp. Unit idle w/o Overlap w/ Overlap w/o Overlap w/ Overlap Figure 7: Multi-layer pipeline execution engine. which will eventually produce output. (2) CPU Schedule: Instead of waiting, the CPU immediately begins scheduling the batch for the next iteration. To do so, it uses a set of placeholder tokens that stand in for the yet-to-be-computed output. This allows all CPU-side batch preparations to occur concurrently with the accelerator's computation. (3) Seamless Transition: Once the accelerator completes the forward pass and the real output is generated, a fast replacement operation swaps the placeholder tokens with the real generated tokens. Since all scheduling is already complete, the accelerator can begin the next forward pass with minimal delay. By transforming the conventional serial \"prepare-then-compute\"\" workflow into a parallelized pipeline, our method effectively masks scheduling latency behind accelerator computation time. This design eliminates idle execution \"bubbles\"\" in the accelerator timeline, thereby maximizing hardware utilization and enhancing overall system throughput. Model-Layer Computation-Communication Overlap. Maximizing the overlap between computation and communication is a critical design principle in recent LLM frameworks [3, 71]. This strategy is essential for hiding the significant latency of data transfers and achieving high hardware utilization. However, architectural constraints can make this challenging. For instance, on a typical GPU, compute-focused units (Tensor Cores) are tightly coupled with general-purpose units (CUDA Cores) within a Streaming Multiprocessor (SM). Consequently, when an SM is allocated to a communication-related task, its powerful Tensor Cores often sit idle, leading to wasted compute potential. In contrast, our target accelerator architecture allows for more flexible, independent allocation of its Cube (matrix) and Vector (general-purpose) units. We leverage this flexibility to address the aforementioned challenges by introducing a dual-stream pipeline architecture that uses micro-batch splitting to effectively overlap computation and communication. Specifically, our architecture consists of a Computation Stream for compute-bound tasks (Attention, ExpertForward, etc.) and a Communication Stream for data distribution and collection tasks (MoE Dispatch and Combine). To enable pipelining and maximize overall pipeline throughput, we partition a macro-batch B into n micro-batches [68] {b1, b2, · · · , bn}. The two streams asynchronously execute tasks for different micro-batches. On the one hand, our scheduling policy dynamically determines the optimal execution order of the micro-batches. On the other hand, our resource scheduling policy adaptively allocates an appropriate number of AI Cores (Vector Cores and Cube Cores) to the Communication Stream and Computation Stream respectively. Figure 7 illustrates the case of n = 2, where the Communication Stream performs the Dispatch operation for micro-batch bk while the Computation Stream executes the ExpertForward pass for the preceding micro-batch bk-1. 14 xLLM Technical Report Operator-Layer Matrix-Vector Units Overlap. On heterogeneous AI accelerators, the serial scheduling of matrix and vector computation units often results in significant underutilization, as one class of units remains idle while the other is active. This inefficiency motivates a strategy of operator-level matrix-vector overlap to fully exploit the hardware's parallel capabilities. However, a naive implementation via coarse-grained parallel scheduling, without a systemic resource coordination mechanism, is also problematic. This approach frequently leads to disordered contention for limited compute units, causing resource fragmentation and access conflicts that ultimately degrade overall performance. To address these challenges, we propose a dynamic resource allocation mechanism for computation units (Cube and Vector Units) based on real-time load monitoring. This mechanism enables a deep pipeline across heterogeneous compute units by dynamically and adaptively assigning the precise type and quantity of resources required for each concurrent operator. By doing so, it mitigates resource contention and ensures that parallel operators execute within highly overlapping time windows, achieving precise computational overlap and maximizing hardware utilization. We formulate the dynamic resource allocation as an optimization problem. Let C and V be the sets of matrix and vector operators, respectively. Let xi be the number of matrix units (Cube) allocated to the ith matrix operator (i ∈C), and yj be the number of vector units (Vector) allocated to the jth vector operator (j ∈V). For simplicity, we assume all operators begin execution simultaneously, without considering data dependencies or communication latency. Based on the known computational workload of each operator, our mechanism seeks to find an optimal resource allocation that minimizes the maximum difference in execution times between any two operators. This objective, which we term the alignment loss (Lalign), effectively synchronizes the completion time of all parallel kernels. The optimization problem is defined as: argminxi,yjLalign = max i∈C, j∈V |Ti -Tj| , Ti = Wi γCube · xi , Tj = Wj γVector · yj , X i∈C xi ≤NCube, X j∈V yj ≤NVector, (1) where T is the operator execution time, W is its computational workload, γcube and γvector are the peak performance per unit, and Ncube and Nvector are the total available matrix and vector units. The meaning of this optimization objective is to minimize the maximum difference in execution time between any pair of concurrent matrix and vector operators. 4.2 Adaptive Graph Mode The performance of LLM inference deployment is often impeded by Host-side CPU overhead, particularly when the computation graph is composed of many fine-grained operators [72]. This bottleneck manifests as both significant CPU-Accelerator synchronization latency, measured at 5 ∼50μs per invocation from frequent kernel launches, and suboptimal hardware utilization due to idle accelerator cycles between these intermittent operator executions. Mainstream AI accelerators such as NVIDIA GPUs (with CUDAGraph) [73] and Ascend NPUs (with ACLGraph) [74] employ computation graphs to try to handle above issues and enhance hostside scheduling performance. As illustrated in Figure 8, traditional Eager Mode relies on the CPU to submit a multitude of small, intensive tasks, leading to frequent launches of small kernels on the accelerator. In contrast, the Graph Mode enables the CPU to submit one large task, after which the accelerator internally executes the small kernels in a streamlined fashion. This method significantly reduces both launch overhead and \"accelerator bubbles\" (idle cycles). Specifically, the Graph Mode, such as ACLGraph [74], comprises two distinct phases: graph capture and graph execution. During the graph capture phase, the entire computation flow is recorded to capture the sequence of kernel calls and their dependencies, including kernel launch parameters. This recorded workflow is then pre-compiled into a replayable directed acyclic graph (DAG) object. It is important to note that during this phase, tasks are merely staged within the model's runtime instance and are not actually executed. In the subsequent graph execution phase, the entire graph is launched via a single CPU call. The accelerator then follows the pre-defined process autonomously, without real-time intervention from the CPU. This effectively consolidates many individual kernel launches into one single graph launch. 15 xLLM Technical Report A Launch A Launch B Launch C Launch D Launch E B C D E Kernel Execution Launch Overhead Launch graph D Kernel Execution Launch Overhead Build Graph cudaGraphExecKernelNodeSetParams Speed Up E C B A w/o Graph w/ Graph Figure 8: Pipeline comparison between Eager Mode and Graph Mode. Building upon this foundation, we further advance the practical implementation of the ACLGraph [74] and propose our Adaptive Graph Mode through addressing key challenges including dynamic shape adaptation, memory reuse conflicts across multiple graphs, and compatibility with custom operators. This enables the pre-compilation of kernel sequences, memory copy operations, and synchronization points into a single computational graph, which is then dispatched to the AI accelerator in one execution. Consequently, our approach significantly reduces kernel launch overhead and minimizes accelerator idle periods, maximizing overall hardware efficiency. Dynamic Shape Adaptation via Partial Graph and Dimension Parameterization. ACLGraph [74] fixes kernel parameters and memory sizes during the graph capturing phase. However, in real-world scenarios, LLMs typically process inputs with variable sequence lengths and batch sizes, making this static capturing characteristic difficult to adapt to dynamic input requirements. To address this issue, dimension parameterization can be employed, treating key dynamic dimensions-such as batch size and sequence length-as input parameters to the entire graph, thereby enhancing flexibility. During memory allocation and kernel configuration, these dynamic parameters are used to compute actual required values; for example, the memory size can be calculated using the formula: alloc_size = batch_size × seq_len × hidden_size. At the graph launch stage, the actual batch size and sequence length are passed as parameters to ensure the graph adapts to varying input dimensions. Although parameterization can handle variations in dynamic dimensions, certain operator implementations on different hardware may not support dynamic dimension parameterization. To overcome this, we propose the Partial Graph Mode: the model is partitioned into modules with simple dynamic shapes (such as FFN and LayerNorm, which only have the num_tokens dimension as dynamic) and modules with complex dynamic shapes (such as Multi-Head Attention, which involves multiple dynamic dimensions like batch_size, kv_seq_len, and q_seq_len). Modules with simple dynamic shapes are extracted and compiled for execution using Partial Graph Mode, while those with complex dynamic shapes are executed in Eager mode. Our Adaptive Graph Mode dynamically selects the most appropriate mode above based on the current input shapes. This intelligent selection ensures optimal performance across varying workload conditions. Table 1 compares the key characteristics of three dynamic shape processing mode. The traditional Eager Mode requires no pre-compilation but incurs high scheduling overhead at runtime due to N kernel launches (where N is the number of operators). The Full Graph Mode, by fixing shapes, allows single compilation and very low launch overhead, yet its lack of dynamic adaptability limits its applicability. In contrast, the proposed parameterized and multi-caching Partial Graph Mode strikes an optimal balance. It achieves execution performance comparable to that of a full graph while maintaining high flexibility, by trading a manageable number of pre-compilations (M ≪ Nreq, where M is the number of cached graphs and Nreq is the number of actual requests) for the high efficiency of a single graph launch. 16 xLLM Technical Report Method Eager Mode Full Graph Mode Partial Graph Mode Compilation Times 0 1 M Low Launch Overhead × ✓ ✓ Low Memory Usage ✓ × ✓/× High Flexibility ✓ × ✓ Table 1: Comparison of different shape handling solutions. Our Adaptive Graph Mode dynamically selects the most appropriate mode above based on dynamic shapes. Efficient and Reusable HBM Memory Pool. During the graph capture process, the specific virtual addresses of input, output, and intermediate tensors are recorded. To prevent illegal memory access, which can occur when the address of an input tensor changes during an actual inference request, we develop a shared HBM memory pool. First, during graph initialization, a sufficiently large, contiguous block of HBM memory is pre-allocated to serve as the graph's memory pool. Subsequently, for internal address management, computation graph intercepts all tensor memory operations and re-describes them using offsets relative to the pool's base address, rather than absolute virtual addresses. Before the graph is launched, the user-provided input tensor is copied into the HBM memory pool at the offset pre-defined for the graph's input. After graph execution is complete, the data from the output tensor is copied to the user-specified output buffer address. Furthermore, internal tensors within the graph are also managed using fixed offsets within the pool, ensuring safe reuse and efficient memory management. This managed memory pool guarantees that all addresses used internally by the graph are known and safe, which adapts to external address changes with only the overhead of memory copies before and after launch. Integrating Custom Operators. Integrating performance-critical custom operators, such as PageAttention and AllReduce, into the Adaptive Graph Mode presents a unique challenge. The internal implementation of these operators often relies on CPU to perform just-in-time shape calculations and kernel configurations based on runtime inputs. This dynamic behavior conflicts with the pre-compiled nature of computation graph. To resolve this, we modified the implementation of such operators. The first step is to identify custom operators within the graph that have dynamic shape dependencies. We then refactor these operators to accept shape-related parameters (e.g., dimension sizes or loop counts, which can only be determined at runtime) directly as kernel arguments. This approach avoids hard-coding these values on the host side during the graph capture phase. As a result of this modification, these custom operators can be successfully captured by our computation graph. By leveraging the graph's parameterization mechanism, the required dynamic arguments for these kernels can be derived from the graph's main input parameters and passed along during execution. This method achieves seamless integration of essential custom operators within the Adaptive Graph Mode framework. 4.3 Efficient Memory Management LLM inference deployment faces a critical balance between memory utilization and computational efficiency, particularly in autoregressive generation tasks where efficient KV Cache management has become a key challenge. Traditional solutions fall into two categories: First, contiguous memory allocation approaches that statically pre-allocate memory space based on maximum sequence length before inference, ensuring physical contiguity of KV Cache. While this improves computational efficiency, it results in low memory utilization. Second, PagedAttention [75] approaches that support larger batch sizes through paging mechanisms, but frequent access to block tables sacrifices computational efficiency, and the increased parameters complicate operator development and debugging. To address this challenge, inspired by virtual memory management in operating system field [76, 77], we propose the xTensor memory management scheme, which adopts a \"logically contiguous, physically discrete\" KV Cache storage structure that resolves the contradiction between 17 xLLM Technical Report Req Virt_start ... Virt_end Virtual Address Space Request R1: MaxSeqLen Virt_start ... Virt_end Request R2: Page1: -PageID: P1 -Status: Reusable -Owner: R1 Physical Page Pool Page2: -PageID: P2 -Status: Mapped -Owner: R2 Scheduler On-demand Mapping Request R2: Write a short story about a young explorer discovering a hidden ancient civilization. Request R1: I am a 25-year-old woman. Please give some recommendations for moderately priced clothes. Smart Scheduling: Request ID Reuse Async Pre-mapping ... ... Inference Requests Addr Mapping Figure 9: The framework of xTensor memory management. Features Contiguous Allocation Paged Attention xTensor (Ours) Efficient Memory Usage × ✓ ✓ Efficient Computation ✓ × ✓ Large Batch Support × ✓ ✓ Operator Development Complexity ✓ × ✓ Table 2: Comparison of memory management strategies. memory contiguity and dynamic allocation, thereby achieving both high computational efficiency and high memory utilization (as shown in Table 2). Physical Memory Page Management and Virtual Memory Tensor Management. Figure 9 illustrates the xTensor memory management framework. During service initialization, based on the available memory size for KV Cache, a large number of fixed-size discrete physical memory pages are pre-allocated. Each physical page maintains a triple state ⟨PageID, Status, OwnerSession⟩, where PageID represents the page identifier, OwnerSession indicates the service owning the page, and Status ∈{Free, Allocated, Mapped, Reusable} represents states of free, allocated, mapped, and reusable to dynamically track physical page usage. Subsequently, each inference request is pre-allocated with a logically contiguous virtual address space with an address range equal to MaxSeqLen, where this virtual address space is not actually associated with physical pages during allocation. This decoupled design provides operators with a virtual view of KV Cache stored in contiguous memory, thereby masking the discrete nature of underlying physical pages [78, 79]. On-demand Memory Mapping. During actual allocation, the scheduler dynamically maps physical memory pages on-demand based on the actual sequence length of requests. When a sequence generates new tokens and KV Cache needs expansion, the scheduler retrieves one or more free pages from the pre-allocated physical page pool and maps these physical pages to the next available contiguous address location in the request's virtual address space. Since physical memory is gradually mapped as sequences grow, short sequences only require a small amount of physical memory, thus avoiding the memory waste of contiguous allocation strategies that still require reserving space according to maximum sequence length for short sequences. After memory mapping is completed, kernel access to virtual addresses can automatically locate the corresponding physical page phypageidx and offset: phypageidx = virt_addr -virt_start page_size , offset = (virt_addr -virt_start) mod page_size, (2) where virt_addr and virt_start represent the current virtual address and starting virtual address respectively, and page_size represents the physical page size. 18 xLLM Technical Report Requests Schedule Draft model Forward Input Target model Forward Input Validate Output Requests Schedule Draft model Forward Input Target model Forward Input Validate CPU thread1 NPU CPU thread2 Requests Schedule Draft model Forward Input Target model Forward Input Validate Draft model Forward Output Target model Forward Input CPU NPU Schedule ... ... w/o Overlap w/ Overlap Figure 10: Comparison between original and our optimized speculative decoding. Low Overhead Allocation. If virtual address to physical page mapping is immediately released upon request completion, new requests require remapping physical pages, but the Unmap operations on accelerators incur significant overhead. Therefore, under high load conditions, frequent Map/Unmap operations become performance bottlenecks, especially for numerous short-term concurrent requests. We adopt request ID reuse and asynchronous pre-mapping designs to reduce latency from Map/Unmap operations. • Physical Page Reuse. Upon request completion, physical pages are not immediately released but marked as Reusable. When new requests arrive, if their required KV Cache size matches some Reusable physical page set, that page set is remapped to the new request's virtual address space. This saves expensive Map and Unmap operations through fast virtual address remapping, particularly suitable for scenarios with concentrated request length distributions. • Asynchronous Pre-mapping. During current token decoding, the next token's required physical pages are asynchronously predicted and mapped. Since new token writes to KV Cache occur after current token decoding computation, this design can partially hide physical page mapping operations, significantly reducing page mapping latency. Operator Adaptation. The decoupled design of xTensor's physical memory pages and virtual contiguous memory requires the adaptation of existing operators. On one hand, to accommodate KV Cache contiguity in virtual address space, operators no longer require block_table parameter input. Whether for attention operators or write_kvcache operators responsible for writing KV Cache during Prefill and Decode phases, only the starting virtual address and related offsets need to be passed in, with the system automatically associating virtual addresses to corresponding physical pages during operation. On the other hand, accelerators lack native contiguous KV FlashMLA [80] operators and only support PagedFlashMLA [80] operators optimized for paging. Therefore, by reconstructing the PagedMLAttention operator in the accelerator operator library, removing Block Table-related logic such as block table queries and cross-page boundary judgments, and adding functionality for direct computation based on input virtual addresses, we implement FlashMLA operators supporting contiguous virtual address input on accelerators. 4.4 Algorithm Optimizations 4.4.1 Optimized Speculative Decoding Traditional autoregressive inference requires LLMs to generate tokens sequentially one by one, resulting in high computational latency and limited throughput. Speculative decoding technology theoretically breaks through this performance bottleneck through a paradigm of parallelly predicting candidate token sequences and quickly verifying them [81, 82, 26]. However, in distributed deployment environments, this technology faces some core challenges: First, synchronous scheduling between CPUs and accelerators in traditional multi-step inference frameworks results in insufficient utilization of computing resources, with accelerators often remaining idle while waiting for data preparation and processing. Second, existing attention mechanisms are not optimized for the characteristics of speculative inference, causing a large amount of redundant data movement. To address these challenges, xLLM proposes a systematic optimization scheme based on existing frameworks. 19 xLLM Technical Report Compute Load decode EPLB Controller Global Expert Mapping Planner Checker Compute Load decode decode decode decode decode Layer-0 Loader Layer-0 Weight Old Buffer New Buffer Layer-0 Rebalance Layer-0 Weight Old Buffer New Buffer Double Buffer Double Buffer Layer-1 Rebalance ... DB Policy Expert Routing Map Expert Load Worker-0 Ready All Ready Expert Load ... Thread-0 Worker-0 ... Thread-1 Layer-1 Loader ... ... Worker-0 Ready All Ready ... ... Figure 11: Overview design of EPLB. Asynchronous Decoding. Inspired by [83], xLLM implements a lightweight asynchronous framework. For timing overlap optimization, while the accelerator executes the main model inference for the current batch, the CPU processes the output decoding of the previous batch and the input preparation for the next batch in parallel-thereby minimizing the idle time of the CPU and accelerator as much as possible. MLA Optimization. xLLM focuses on improving the MLA [84, 3] computation process. Based on the characteristic that speculative inference needs to process multiple tokens simultaneously on a single sequence, we conducted in-depth optimizations on the self-attention computation in MLA. Specifically, when speculatively inferring m tokens, the self-attention computation of MLA involves product operations between m+1 Q matrices and the same K matrix. By reconstructing the computation process and optimizing the tiling strategy, we effectively reduced the data movement overhead of Q/K matrices. The main optimization measures include: ▷Optimization of K matrix loading times: By adjusting the L1 cache allocation scheme to enable parallel loading of multiple Q matrices, and adopting a sliding window-based K matrix loading method that allows consecutive rows of the K matrix to multiply with multiple Q matrices, the number of K matrix loading operations is significantly reduced. ▷ Q matrix cache residency mechanism: Since there are m+1 Q matrices in speculative inference scenarios, the time to move Q matrices to L1 accounts for a larger proportion of the matrix multiplication process. xLLM redesign the computing scheme to prevent softmax-V product operations from overwriting Q matrices in the L1 cache, enabling Q matrices to remain in the cache and significantly reducing the overhead of repeated data movement, thereby effectively improving the utilization of the tensor core's arithmetic logic units. 4.4.2 Dynamic EP Load Balance With the large-scale application of MoE models [85-88] in production environments, their efficient inference capability relies on a expert routing mechanism, which allocates input tokens to different experts for computation [3, 18, 89]. However, in practical deployment, due to the distribution characteristics of input data and the influence of the model's own structure, there may be significant differences in the number of tokens received by different experts [90, 91]. This imbalance leads to low computing resource utilization: some devices are overloaded due to processing too many tokens, while others remain idle, affecting the overall inference efficiency. To optimize resource utilization, the industry has proposed the Expert Redundancy strategy, which means replicating hot experts and deploying replicas on multiple devices to distribute computing loads. Currently, DeepSeek [3] adopts two load balancing strategies: group-limited load balance and global load balance, which are optimized respectively for the characteristics of the prefill and decode phases. Adjustments to expert redundancy (such as adding/deleting replicas) require ad20 xLLM Technical Report Preprocess MLA Preprocess MLA MLA MLA MLA MLA MLA MLA Preprocess MLA Preprocess MLA MLA MLA MLA MLA MLA MLA Aggregate transfer transfer 1K tokens 3K tokens 3K tokens DP-1 (seq-1) DP-2 (seq-2) DP-1 (seq-1) DP-2 (seq-2) w/o DP Load Balance w/o DP Load Balance Figure 12: Inter-DP group load migration at MLA block granularity. ditional memory and may affect inference latency due to weight migration. How to achieve such adjustments efficiently and smoothly is a major challenge. We have made some adaptive improvements based on DeepSeek, realizing dynamic EP load balance. Expert Load Statistics. When the Router distributes tokens, it records the load status of each expert in real time and returns the statistical results through the model output. Each Worker asynchronously starts a thread to periodically aggregate expert loads and report them to the Controller of the Master. The Controller calculates a new routing table based on the load data and then distributes the updated routing table to each Worker. Double Buffer Mechanism for Weight Update. In general, each Worker starts an asynchronous thread to update the weights of single-layer experts during each decoding process. Specifically, after a Device completes the preparation of new weights, it proactively sends a readiness notification to the Controller. The Controller then verifies the weight readiness status of all Worker nodes to ensure global consistency before broadcasting an update instruction, upon which each Worker executes the weight update immediately. Here, we adopt a Double Buffer mechanism, meaning that the old and new expert weights are stored in two separate buffers respectively. The new expert weights are preloaded in the spare memory space, and after the preloading is completed, the address switching is performed to realize the unperceived update of experts. 4.4.3 Hierarchical DP Load Balance In typical MoE model architectures like DeepSeek [3], the MoE layers use expert parallelism (EP), while the attention layers use data parallel (DP). This model requires all DP groups to synchronously complete their attention calculations in a single inference step before initiating the MoE all-to-all dispatch operation. This synchronization barrier means the total time for the attention phase is determined by the slowest DP group (the \"straggler\"). Faster DP groups cannot immediately proceed to the next stage after finishing their tasks; instead, they are forced into an idle state, waiting for the straggler to complete [92]. This waiting directly translates into wasted computational resources and an increase in overall inference latency. In practical scenarios, due to the dynamic and unpredictable nature of inference requests, the load within different DP groups is often difficult to balance. Particularly in the decoding phase with large-scale DP (e.g., 80 groups), the load difference between DP groups can reach tens of thousands of tokens, causing fast DP groups to wait for slow ones and wasting approximately 5 milliseconds of delay [93]. Experimental data shows that although the difference in the number of requests processed between DP groups may be as small as four, the corresponding difference in KV Cache token count during the decoding phase can be as high as 20,000. The actual computational workload is directly related to the total number of tokens the system needs to process, as this determines the memory footprint of the KV Cache and the scale of matrix operations in the attention mechanism. This further establishes DP load balancing as a critical factor in determining overall system efficiency. Current mainstream frameworks like vLLM [8] and SGLang [9] use a simple static round-robin scheduling strategy. Once a request is assigned, its subsequent computations are fixed to a specific 21 xLLM Technical Report DP group, making it unable to adapt to dynamic load changes. Furthermore, at the hardware level, the implementation of operators like the MLA on certain accelerators employs a \"one request per tensor compute core\" partitioning strategy. This can lead to idle compute cores within the same DP group due to differences in request lengths. To address these challenges, xLLM proposes a hierarchical \"defense-in-depth\" strategy to tackle DP load imbalance issues across different time scales and granularities. The first layer is preventative, KV Cache-aware request scheduling; the second layer performs macroscopic correction through inter-DP group load balancing; and the third layer carries out microscopic correction via intra-DP group load balancing. This layered design enables the system to flexibly address the root causes of various performance degradations, thereby building a robust and efficient inference service. Layer 1: KV Cache-Aware Scheduling. xLLM's first-layer strategy implements request load distribution through KV Cache-aware scheduling. This mechanism moves beyond simple round-robin methods. When a new request arrives, the scheduler comprehensively checks the status of all DP groups, paying special attention to the remaining KV Cache capacity in each. It then preferentially assigns the new request to the group with the most available space. By balancing the total token load and its memory footprint at the system level over the long term, this strategy prevents the formation of severe load imbalances, thereby achieving intelligent resource allocation. Layer 2: Reactive Inter-DP Group Workload Migration. xLLM's second-layer strategy achieves reactive balancing through workload migration between DP groups. During the decoding process, the varying prompt and KV Cache lengths of selected requests cause different computation latencies among DP groups. To counter this, xLLM's central scheduler evaluates the current computational load of each DP group during every inference round. If a significant imbalance is detected, it initiates a workload migration process, moving tasks from overloaded to under-loaded groups. The scheduler also determines the migration granularity - whether to move an entire batch, a single sequence, or a partial MLA block of a sequence. Figure 12 illustrates this process at the MLA block granularity. First, xLLM dispatches the input tokens for the migration to both the source and destination DP groups. Then, as all groups execute the MLA Preprocess operation, the request's KV Cache is transferred asynchronously, overlapping communication with computation. Next, all groups begin the attention calculation. The underloaded group prioritizes the migrated attention task, allowing it to immediately send the resulting token back to the source group upon completion while concurrently processing its own native attention operations, further overlapping overhead. Finally, all DP groups use an aggregation operator to gather the external computation results. Layer 3: Fine-Grained Kernel-Level Optimization. xLLM's third-layer strategy achieves finegrained optimization through kernel-level migration within a DP group. This optimization is performed inside the matrix computation kernels of a single DP group. On one hand, during scheduling, it reorders requests based on their load, replacing the original compute core's round-robin allocation strategy. This aims to keep the total number of computation tokens assigned to each matrix computation unit as consistent as possible. On the other hand, for requests with extremely long sequences, it explicitly splits their computation sequence, migrating parts of the long-sequence request's computation to be calculated with other short-sequence computation units. Through this fine-grained request splitting, it directly resolves the issue of idle compute cores caused by load imbalance within the DP group. 4.5 Generative Recommendation Recent advancements in generative models have spurred significant interest in generative recommendation [31, 94-98]. Beyond enhancing the retrieval and ranking stages of traditional multi-stage pipelines [98, 94], single-stage generative recommendation frameworks directly generate a candidate item list in an auto-regressive manner [31]. These frameworks usually utilize beam search decoding to directly produce diverse recommendation results, and xLLM implements extensive optimizations for these single-stage frameworks. As shown in Figure 13, in the generative recommendation scenario, xLLM is dedicated to overlapping host and device operations as much as possible to improve overall execution efficiency. When a request arrives, xLLM preemptively prepares the information required during the forward process on the CPU side. After the preparation is completed, 22 xLLM Technical Report Pre-allocate Scheduler Schedule Beam Schedule Pre-allocate Beam Pre-allocate Schedule Beam Pre-allocate Schedule Beam w/o Overlap w/ Overlap Prefill Beam Decode Beam Filter Mask Worker Forward Add Sampler Logits w/o Overlap w/ Overlap Engine Decode Filter Mask Forward Logits Add Sampler Figure 13: Overview design of generative recommendation process. relevant tasks are handed over to the engine for processing. The Engine executes three forward passes in one go. During this process, the intermediate beam search is a host operation, which by its nature cannot be overlapped with device operations. To optimize the processing pipeline, xLLM internally employs a multi-micro-batch, multi-stream approach. During the Worker's execution, the CPU operation for generating the filter mask is overlapped with the device operation for logits computation. Through an addition operation, a masked logits is generated before the sampler, thereby achieving data filtering and improving the accuracy and effectiveness of the recommendation results. 4.5.1 Host-side Optimization In scenarios with large beam_width and top_k parameters, the significant increase in the number of candidate sequences results in substantial sorting and filtering operations on the host side. This shifts the computational bottleneck of the beam search algorithm from the AI accelerator to the CPU, ultimately leading to a severe CPU-bound issue. The following outlines xLLM's efforts to optimize the beam search process in generative recommendation scenario. Beam Search Optimization. In many fields such as natural language processing, beam search optimization is an efficient algorithm widely used in sequence generation tasks [99]. When processing a large number of candidate sequences, one of the core operations of beam search is to select beam_width sequences from numerous candidates. This process is essentially a partial sorting operation, with the unique characteristic that it only needs to identify the top beam_width elements without fully sorting all candidate sequences, thereby significantly reducing computational complexity. In another key step of beam search, generating beam_width × top_k candidate sequences from the existing beam_width sequences, there exists an important property: log_probs of each sequence are arranged in descending order. Leveraging this property, an optimized early termination mechanism can be adopted, where computations are halted prematurely if log_probs does not meet specific criteria, further enhancing computational efficiency. The specific operational procedure is as follows: First, the existing beam_width sequences are accessed in order. For each accessed sequence, we create a small min-heap of size beam_width, which is used to dynamically maintain the set of locally optimal elements currently selected. Next, the subsequent sequences are traversed. During this process, if the subsequent log_probs of a sequence is smaller than the top element of the minheap, it implies that, based on the current filtering criteria, the subsequent elements of this sequence cannot enter the current optimal set. At this point, the filtering operation for the current sequence can be terminated directly, and the next existing sequence processed. Conversely, if the subsequent log_probs is larger than the top element of the min-heap, the element is inserted into the min-heap, 23 xLLM Technical Report and the heap structure is adjusted according to the properties of the min-heap to maintain its order. After the traversal of all relevant sequences is completed, the top elements are sequentially extracted from the min-heap. These elements, in the order of extraction, form the beam_width candidate sequences sorted in descending order. Through this procedure, the approach ensures the selection of high-quality candidate sequences while significantly reducing unnecessary computational overhead, thereby improving the operational efficiency of the beam search algorithm in practical applications. Resource Reuse and Pre-allocation. During the beam search process, the beam_width is predetermined and fixed for each forward pass of the model, which implies that the computational resources required per forward propagation remains relatively constant. However, an excessively large beam_width can lead to significant wastage of both CPU computational and memory resources. To effectively mitigate this issue, xLLM incorporates a carefully designed resource reuse strategy. Specifically, during the generation of new candidate sequences, the system reuses resources previously occupied by older sequences without allocating new space for each candidate. This approach avoids the overhead associated with frequently creating new data structures. Based on the final search results, the system then updates the storage areas for the old sequences with the content of the newly generated ones upon completion of the beam search algorithm. By doing so, xLLM not only reduces memory usage but also minimizes the additional overhead on the CPU for resource management and allocation, significantly improving the efficiency of computational resource usage. 4.5.2 Device-side Optimization Valid Item Filtering. In typical single-stage generative recommendation frameworks, the valid item filtering mechanism plays a crucial role. For example, OneRec [31] framework uses an ordered combination of three token IDs to uniquely represent a valid product item ID. However, due to the vast number of possible token ID combinations, not all of them correspond to actual and valid items [100, 101]. To ensure that the results generated by the model are all valid product items, the system generates a valid mask asynchronously during the forward pass of the model. This mask is based on a pre-constructed vocabulary of valid items, which is then element-wise added to the logits output by the model. Through this clever design, the logits corresponding to invalid token IDs are adjusted, ensuring that these invalid token IDs are almost never selected in subsequent calculations. This effectively filters out invalid token IDs, thereby guaranteeing the validity and accuracy of the final recommendation results. 5 Evaluations In §5.1, we evaluate the full implementation of our xLLM framework against several baseline inference systems, specifically vLLM-Ascend1(v0.10.rc1) and MindIE2(v2.1.rc1), on Ascend 910B/910C [18] instances across a range of scenarios. xLLM, MindIE, and vLLM-Ascend refer to the default deployments on Ascend 910B. Additionally, we denote the deployments of xLLM, MindIE and vLLM-Ascend on Ascend 910C as xLLM‡, MindIE‡ and vLLM-Ascend‡, respectively. The testing scenarios are organized into multiple categories reflecting diverse online serving applications, such as the JingYan AI chatbot, customer service assistants, merchant assistants, product understanding and marketing recommendation systems. Additionally, in §5.2, we conduct a detailed ablation study to assess the individual contributions of each optimized module. 5.1 Main Results This section benchmarks xLLM against mainstream inference frameworks to demonstrate its superior inference efficiency across diverse models, including Qwen2/3-series [102, 87] and Deepseek [3] models, and datasets. §5.1.1 details a fair comparison using the ShareGPT3 dataset. To ensure an equitable comparison across all LLM inference frameworks, the key feature of our experimental setup is that the input and output sequence lengths are fixed, while the request 1https://github.com/vllm-project/vllm-ascend 2https://www.hiascend.com/en/software/mindie 3https://huggingface.co/datasets/anon8231489123/ShareGPT_Vicuna_unfiltered/blob/ main/ShareGPT_V3_unfiltered_cleaned_split.json 24 xLLM Technical Report 1 2 4 Number of Accelerators 2000 4000 6000 8000 10000 Throughput (tokens/s) Qwen3-0.6B 1 2 4 Number of Accelerators 2000 4000 6000 8000 Qwen3-1.7B 1 2 4 Number of Accelerators 2000 3000 4000 5000 6000 Qwen3-4B 1 2 4 Number of Accelerators 1000 2000 3000 4000 5000 Throughput (tokens/s) Qwen3-8B 1 2 4 Number of Accelerators 1000 2000 3000 4000 Qwen3-14B 2 4 8 Number of Accelerators 1000 2000 3000 Qwen3-32B xLLM xLLM MindIE MindIE vLLM-Ascend vLLM-Ascend Figure 14: Throughput comparison of LLM inference frameworks evaluated on Qwen3-series models using the ShareGPT dataset, under the constraint of TPOT=50ms and input/output length=2048. 16×910B 8×910C 0 100 200 300 400 Throughout (tokens/s) [2048,2048] TPOT:50ms 16×910B 8×910C 0 500 1000 1500 [2048,2048] TPOT:100ms 16×910B 8×910C 0 100 200 300 400 [2500,1500] TPOT:50ms 16×910B 8×910C 0 250 500 750 1000 1250 1500 [2500,1500] TPOT:100ms xLLM xLLM MindIE MindIE vLLM-Ascend Figure 15: Throughput comparison of LLM inference frameworks using DeepSeek-R1 on the ShareGPT dataset constrained by given TPOT and input/output length, where [2500, 1500] means input length=2500 and output length=1500. vLLM-Ascend on the 910C is excluded from the comparison, as its performance with Deepseek-R1 fails to meet the required TPOT threshold. rate is dynamically adjusted to match the target SLO (e.g., TPOT) threshold for each framework. We also evaluate this setup under different node configurations, including both single-node and multi-node setups with PD disaggregation. In §5.1.2, we evaluate xLLM on various real-world business scenarios from JD.com, where it is currently deployed, showcasing its performance under practical deployment conditions. 5.1.1 Benchmarking Performance Qwen3-series. As shown in Figure 14, we comprehensively compares the throughput performance of four inference frameworks across the Qwen3-series (from 0.6B to 32B parameters) on the benchmark. All tests are conducted under uniform configuration with input/output lengths set to 2048 tokens and a TOPT constraint of 50 ms. The experimental results demonstrate that while all evaluated frameworks show improved throughput with additional accelerators, xLLM and its Ascend 910C implementation (xLLM‡) consistently deliver superior performance, confirming near-linear strong scalability across various model sizes. Specifically, xLLM achieves throughput improvements of up to 1.9× and 1.7× compared to vLLM-Ascend and MindIE, respectively. Similarly, xLLM‡ outperforms vLLM-Ascend‡ and MindIE‡ by up to 2.2× and 1.5×, respectively. Further25 xLLM Technical Report Method Prompt Length Output Length Throughput (tokens/s) Request Rate (req/s) MindIE 2048 2048 8476.44 4.14 xLLM 2048 2048 11351.58 5.54 Table 3: Comparison of DeepSeek-R1 with PD disaggregation on the ShareGPT dataset constrained by TPOT=100ms. 1 2 4 Number of Accelerators 400 600 800 1000 1200 1400 Throughput (tokens/s) Qwen2-7B TPOT:40ms 1 2 4 Number of Accelerators 0 500 1000 1500 2000 Qwen2-14B TPOT:90ms 1 2 4 Number of Accelerators 2000 3000 4000 5000 Qwen3-8B TPOT:80ms 2 4 8 Number of Accelerators 1000 2000 3000 Qwen3-32B TPOT:80ms xLLM xLLM MindIE vLLM-Ascend Figure 16: Throughtput comparison of Qwen2-series and Qwen3-series models with various inference frameworks in the JingYan scenario. more, xLLM‡ demonstrates steady performance improvements over the Ascend 910B-based xLLM in most scenarios, validating the software stack's effective utilization of the new hardware. DeepSeek-R1. Figure 15 presents the throughput performance of various inference frameworks for the DeepSeek-R1 model on the benchmark, using 16 accelerators for Ascend 910B and 8 accelerators for Ascend 910C, respectively. The results demonstrate that the proposed xLLM framework achieves exceptional throughput on the Ascend 910 series. Quantitatively, xLLM on Ascend 910B delivers an average throughput improvement of approximately 1.7× over MindIE and a significant 12× enhancement over vLLM-Ascend. Furthermore, xLLM‡ achieves an average throughput increase of approximately 1.4× compared to MindIE‡. PD Disaggregation Settings. Table 3 benchmarks the inference performance of the MindIE and xLLM frameworks on the DeepSeek-R1 model using a PD disaggregation architecture. Under identical conditions with TPOT controlled at 100ms for 2048-length inputs/outputs, xLLM achieves approximately 34% higher throughput (11,351.58 vs. 8,476.44 tokens/s) and request rate (5.54 vs. 4.14 req/s), marking a significant efficiency improvement. 5.1.2 Business Serving Scenarios JingYan. JingYan is an AI shopping assistant designed to help users discover new products, find inspiration, and get answers to their questions. The dataset for JingYan, consequently, consists of conversational logs between the model and its users, capturing these rich interactions. As illustrated in Figure 16, we systematically evaluate the inference throughput of four frameworks for serving the Qwen2-series and Qwen3-series model in the JingYan scenario. The results demonstrate that both xLLM and xLLM‡ maintain superior throughput and exhibit better scaling efficiency across all model sizes compared to MindIE and vLLM-Ascend. For instance, when serving the Qwen3-8B model with 4 accelerators, xLLM delivers a throughput approximately 1.6 times that of vLLMAscend and significantly surpasses MindIE. The robust performance of xLLM on the 910B, and its enhanced results on the 910C hardware, highlight its effective adaptation to successive hardware generations. As shown in Table 4, a similar trend is observed for the DeepSeek-V3 model, where xLLM achieves a throughput that is over 9 times greater than vLLM-Ascend and surpasses MindIE by 36%. Customer Service. The Customer Service dataset comprises the interactive dialogues between customers and support agents. Figure 17 details the performance differences among inference frame26 xLLM Technical Report Method Prompt Length Output Length Throughput (tokens/s) Request Rate (req/s) vLLM-Ascend 6800 400 21.17 0.11 MindIE 6800 400 144.40 0.67 xLLM 6800 400 196.45 0.89 Table 4: Comparison of Deepseek-V3 model with various frameworks in the JingYan scenario constrained by TPOT=80ms. 1 2 4 Number of Accelerators 50 100 150 Throughput (tokens/s) Qwen3-8B 2 4 8 Number of Accelerators 20 40 60 80 Qwen3-32B xLLM xLLM MindIE vLLM-Ascend Figure 17: Throughtput comparison of Qwen3-series models with various inference frameworks in the customer service scenario constrained by End-to-End latency(E2E)=10s. works for the Qwen-8B and Qwen-32B models in the customer service scenario. The results highlight that our xLLM, particularly the xLLM‡ running on Ascend 910C, provide significantly better throughput across all tested configurations. For instance, with Qwen3-32B on 8 accelerators, the throughput of xLLM is 3.1 and 1.2 times greater than that of vLLM-Ascend and MindIE, respectively. It is important to note that the vLLM-Ascend framework shows a clear scaling bottleneck as the number of accelerators increases, whereas xLLM maintains near-linear efficiency scaling. This validates the high efficiency and superiority of the xLLM framework in managing distributed inference for large-scale models. Merchant Assistant. Table 18 benchmarks the throughput of various inference frameworks on three tasks (i.e., search terms, arrangement, intent recognition) within the merchant assistant scenario. The proposed xLLM framework achieves better or comparable performance to MindIE and demonstrates a significant lead over vLLM-Ascend. Specifically, for the search terms task with four accelerator cards, xLLM delivers 34% higher throughput than MindIE and roughly 3.4× that of vLLM-Ascend. Product Understanding. For the product understanding scenario, the throughput comparison of the Qwen2-7B model with several frameworks is shown in Table 5. The experimental results indicate that xLLM outperforms MindIE and vLLM-Ascend by an average of 25% and 56%, respectively, across different accelerator card counts. Moreover, the superiority of xLLM scales with the number of cards, demonstrating its effective utilization of large-scale parallel computing resources and thereby offering a robust solution for high-performance LLM inference. Generative Recommendation. As illustrated in Figure 19, the evaluation results on the Qwen-8B model demonstrate that xLLM consistently achieves lower mean end-to-end latency than MindIE across various request rates and beam widths, except under very low load condition. Notably, the performance advantage of xLLM becomes increasingly pronounced as the beam width (from 4 to 128) and the request rate escalate. For instance, under the most challenging scenario with a beam width of 128 and a request rate of 8, xLLM reduces latency by approximately 23% relative to MindIE. This significant improvement validates that our xLLM's host-side and device-side optimizations effectively alleviate the computational bottlenecks in generative recommendation tasks, markedly enhancing inference efficiency and scalability under heavy loads. 27 xLLM Technical Report 1 2 4 Number of Accelerators 100 200 300 Throughput (tokens/s) Search Term Qwen2.5-14B xLLM MindIE vLLM-Ascend 1 2 4 Number of Accelerators 50 75 100 125 Arrangement Qwen2.5-14B xLLM MindIE vLLM-Ascend 1 2 4 Number of Accelerators 100 150 200 250 300 Intent Recognition Qwen3-14B xLLM MindIE vLLM-Ascend Figure 18: Throughtput comparison of Qwen-series models with various inference frameworks in the merchant assistant scenario constrained by End-to-End latency(E2E)=1s. Method Prompt Length Output Length Throughput (tokens/s) #accelerator=1 #accelerator=2 #accelerator=4 vLLM-Ascend 1200 40 795.77 874.97 1272.52 MindIE 1200 40 944.81 1051.44 1693.45 xLLM 1200 40 1001.91 1323.90 2425.13 Table 5: Throughtput comparison of Qwen2-7B model with various inference frameworks in the product understanding scenario. 5.2 Ablation Study Impact of MTP. As shown in Figure 20, under the configuration of 1500 input length and 2500 output length for the DeepSeek-R1 model, enabling Multi-Token Prediction (MTP) technology significantly optimizes inference performance. As max concurrency increases, the MTP-enabled version consistently exhibits lower TPOT compared to the baseline, indicating reduced generation latency. Meanwhile, its throughput is markedly higher than the non-MTP version, with particularly notable advantages beyond 32 concurrent requests. This demonstrates that MTP effectively enhances computational efficiency and system throughput under high concurrency conditions. Impact of Dynamic PD Disaggregation Scheduler Policy. We evaluate our proposed SLO-aware Dynamic PD Disaggregation Policy against the Minimal Load and Round Robin strategies, as illustrated in Figure 21. On the Azure Code dataset characterized significant bursty traffic, our SLOaware policy achieves a request serving rate 1.67 times that of the Minimal Load strategy. Meanwhile, relative to the Round Robin strategy, the Minimal Load strategy improves the SLO attainment by up to 4.3%. For the Azure Conversation dataset with stable input/output length variations, our SLO-aware policy results in a 1.1 times higher request serving rate compared to the Minimal Load strategy. Although the Minimal Load strategy performs similarly to Round Robin, it still enhances the SLO attainment by up to 2.4%. These results indicate that minimal-load scheduling is closer to the optimal scheduling strategy than round-robin approach, while our adaptive instance scheduling delivers the best overall performance. Impact of Hybrid EPD Disaggregation Scheduler Policy. Figure 22 demonstrates the effectiveness of the proposed Hybrid EPD Disaggregation Policy on the TextCaps dataset, specifically in controlling TPOT and improving the SLO attainment rate. Under the configuration of 8 generalpurpose inference instances, the removal of the hybrid EPD disaggregation method results in a drop in goodput from 9.5 req/s to 7.2 req/s, and a subsequent additional removal of stage-level scheduling policy further decreases the goodput to 5.1 req/s. This indicates that a well-designed disaggregation strategy can effectively mitigate inter-stage interference, while our stage-level scheduling policy contributes to finer control over the execution time of each batch. The integration of both components significantly enhances the overall performance and stability of the system, particularly under high-concurrency scenarios. Impact of Online-Offline Co-location Scheduler Policy. As shown in Figure 23, we evaluated three scheduling strategies, namely the baseline P/D, online priority and our proposed Online-Offline 28 xLLM Technical Report 1 4 8 Request Rate (req/s) 90 100 110 Mean E2E (ms) Beam Width:4 Top-k:4 xLLM MindIE 1 4 8 Request Rate (req/s) 100 120 140 Beam Width:32 Top-k:32 xLLM MindIE 1 4 8 Request Rate (req/s) 125 150 175 200 Beam Width:64 Top-k:64 xLLM MindIE 1 4 8 Request Rate (req/s) 150 200 250 300 350 Beam Width:128 Top-k:128 xLLM MindIE Figure 19: Comparison of mean End-to-End latency(E2E) with various inference frameworks in the generative recommendation scenario. Since vLLM-Ascend does not support beam width>10, the corresponding results are not plotted in the figure. Even with beam width=4 and request rate=1, its mean E2E far exceeds that of our framework. 2 8 32 Max Concurrency 50 100 150 TPOT (ms) DeepSeek-R1 [2500, 1500] w/ MTP w/o MTP 2 8 32 Max Concurrency 100 200 Throughput (tokens/s) DeepSeek-R1 [2500, 1500] w/ MTP w/o MTP Figure 20: Impact of MTP on the concurrent performance of DeepSeek-R1 model. Co-location Scheduler Policy (denoted as xLLM-OOC), to assess the maximum achievable throughput for offline requests without violating the SLO of online requests. The green shaded region in the figure indicates the sustainable range of offline request throughput that remains within the acceptable online SLO violation threshold. When the offline query-per-second (QPS) exceeds a certain level, both the baseline P/D and online priority strategies lead to a sharp increase in the online SLO violation rate, indicating interference caused by the offline workload. In contrast, xLLM-OOC maintains stable SLO compliance even as offline QPS continues to rise. On our proprietary dataset, xLLMOOC achieves a throughput that is three times higher than the other two methods. Furthermore, it demonstrates improvements of 75% and 17% over online priority and baseline P/D, respectively, on the Azure Code dataset. Impact of Multi-layer Pipeline Execution. As evidenced by the results in Table 6, our proposed asynchronous scheduling mechanism delivers consistent throughput improvements across all evaluated model scales. The most substantial relative gain of 17.4% is observed for the 1.5B parameter model, highlighting the method's particular efficacy for smaller architectures where scheduling overhead constitutes a larger portion of total computation time. While relative improvements moderate for larger models (reaching 0.6% for 7B, 3.7% for 14B, and 6.6% for 32B), all configurations exhibit statistically significant absolute gains. These results robustly validate that our method successfully masks scheduling latency and eliminates computational bubbles, through leveraging placeholder tokens to decouple and overlap CPU scheduling with NPU execution. We further assess the effectiveness of the proposed dual-stream architecture for the DeepSeek-R1 model in Table 7. Experimental results from a single decoder layer show that the total communication time increases to 12.4ms in dual-stream mode, up from 9.3ms in single-stream mode. However, the computation-communication overlap mechanism successfully hides 80% of the communication time, which brings the exposed communication time down to just 2.5ms, saving 6.8ms per layer. Despite introducing a computational overhead of 4ms for each layer, the dual-stream strategy yields a net performance gain of 172.0ms across the entire 61-layer model, clearly illustrating the capability of our scheduling strategy in real-world workloads. 29 xLLM Technical Report 10 20 30 40 50 Request Rate (req/s) 40 60 80 100 SLO Attainment (%) Azure Code 25 50 75 100 Request Rate (req/s) 40 60 80 100 Azure Conversation Our SLO Aware Minimal Load Round Robin Figure 21: Performance of Dynamic PD Disaggregation Policy with different scheduling strategies. 1 2 3 4 5 6 7 8 9 10 11 12 Request Rate (req/s) 0 20 40 60 80 100 SLO Attainment (%) Ours w/o hybrid EPD w/o hybrid EPD and sched Figure 22: Impact of Hybrid EPD Disaggregation Scheduler Policy. Impact of Adaptive Graph Mode. To validate the effectiveness of the Adaptive Graph Mode optimization technique, we conduct controlled experiments on the Qwen3-1.7B and Qwen3-4B models. As shown in Table 8, with both the prompt and output lengths set to 2048 tokens, enabling Adaptive Graph Mode result in significant performance gains for both models. The throughput of Qwen31.7B increases from 2385.58 to 3038.88 tokens/s (a 27.4% improvement), while its mean TPOT decreases by 22.0%. Correspondingly, Qwen3-4B achieved an 8.5% increase in throughput and an 8.8% reduction in latency. These results robustly demonstrate the effectiveness of Adaptive Graph Mode as a general-purpose inference acceleration technique, with more pronounced optimization effects observed on the model with a smaller parameter size. Impact of Hierarchical DP Load Balancing. Our proposed hierarchical DP load balancing scheme is projected to increase total throughput by 5%. The kernel-level optimization yields the most significant latency savings; for example, for an ultra-long 32k-token request, reordering and splitting can reduce a single core's computational load from 32k to 1.3k tokens, saving approximately 800 microseconds. In contrast, the latency savings from inter-DP group migration are more modest. Even after balancing a 20k-token difference, the total time saved over 61 Transformer layers is only about 600 microseconds. This indicates that the greatest performance bottleneck and optimization potential lie within the computation units themselves. 6 Future Work Although proposed xLLM framework has demonstrated potential in enhancing LLM inference efficiency and reducing operational cost, achieving truly efficient and inclusive intelligent computing infrastructure requires deeper collaborative innovation across Hardware, Model, and Application Layers. Our future research will advance xLLM from a high-performance Inference Engine toward a comprehensive Operating System for AI that supports next-generation intelligent applications. We outline our planned efforts in three key directions: 30 xLLM Technical Report 0 3% 5% 10% OOC on Qwen2.5-7B 0 1 2 3 0 3% 5% 10% Online SLOs Violation Rate 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Offline Request QPS (req/s) 0 3% 5% 10% 0 1 2 3 0 3% 5% 10% Azure Code on Qwen2.5-7B 0 1 2 3 0 3% 5% 10% 0 1 2 3 Offline Throughput (req/s) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Offline Request QPS (req/s) 0 3% 5% 10% 0 1 2 3 base P/D - Online SLOs Violation base P/D - Offline Throughput online priority - Online SLOs Violation online priority - Offline Throughput xLLM-OOC (ours) - Online SLOs Violation xLLM-OOC (ours) - Offline Throughput Figure 23: Impact of our Online-Offline Co-location Scheduler Policy. Model Prompt Length Output Length Async Scheduling Throughput (tokens/s) DS-Distill-Qwen-1.5B 1000 1000 × 8709.90 1000 1000 ✓ 10223.15 DS-Distill-Qwen-7B 1000 1000 × 3183.68 1000 1000 ✓ 3201.83 DS-Distill-Qwen-14B 1000 1000 × 1472.22 1000 1000 ✓ 1527.23 DS-Distill-Qwen-32B 1000 1000 × 1415.20 1000 1000 ✓ 1508.76 Table 6: Impact of asynchronous scheduling mechanism in Multi-layer Pipeline Execution. 6.1 Fostering an Open and Diverse Hardware Ecosystem Current AI infrastructure often suffers from tight coupling with specific hardware architectures, introducing computational stability risks and limiting efficient deployment in heterogeneous environments such as edge computing. To address this, the first pillar aims to cultivate an open and diverse hardware ecosystem through the following initiatives: Unified Hardware Abstraction Layer. We will develop a high-performance, hardware-agnostic runtime abstraction layer that encapsulates instruction sets and memory architectures across computational units-from domestic cloud chips to edge-side accelerators. This layer will provide unified operator interfaces and memory management APIs, enabling seamless migration and efficient execution across heterogeneous hardware without code modifications, thereby significantly reducing integration barriers for emerging hardware. Software-Hardware Co-Design and Ecosystem Synergy. We will collaborate with hardware partners to define more efficient and open interface standards. This co-design approach will not only drive hardware innovation from the software perspective but also help establish a self-sustaining and competitive computational supply ecosystem, ultimately providing users with optimized choices across cost, performance, and security considerations. 6.2 Cultivating a Vibrant and Responsive Model Ecosystem The system's core value derives from its supported model diversity and integration efficiency. Our future efforts will focus on building an inclusive and agile model ecosystem through: Multi-Scenario Support. We will extend the platform's optimization and deployment capabilities beyond large language models to systematically support various generative AI scenarios, including generative recommendation, text-to-image, and text-to-video generation. We will optimize execu31 xLLM Technical Report Metric Single-Stream (ms) Dual-Stream (ms) Total Communication Time 9.3 12.4 Overlapped Communication Ratio - 80% Exposed Communication Time 9.3 2.5 Total Computation Time 13.0 17.0 Reduced Time per Layer - 2.8 Total Reduced Time (61 layers) - 172.0 Table 7: Communication and computation overheads in a single decoder layer of DeepSeek-R1 using Multi-layer Pipeline Execution. Model Prompt Length Output Length Adaptive Graph Mode Throughput (tokens/s) Mean TPOT (ms) Qwen3-1.7B 2048 2048 × 2385.58 39.27 2048 2048 ✓ 3038.88 30.63 Qwen3-4B 2048 2048 × 1539.93 55.44 2048 2048 ✓ 1671.39 50.58 Table 8: Impact of Adaptive Graph Mode. tion engines, request scheduling, and memory management specifically for different workloads to ensure optimal performance across diverse applications. \"Zero-Day\" Model Integration. To accommodate rapidly evolving model architectures, we will implement a unified framework combining model graph representation, an extensible operator library, and automated compilation optimization. This will enable rapid \"zero-day\" integration of newly released models, reducing deployment cycles from weeks to hours. 6.3 Evolving into an AI-Native Application Framework To democratize AI adoption and accelerate value delivery, we will evolve the system from an inference engine into a full-stack AI-native application framework by: Framework-Native AI Middleware. We will design high-level, framework-native APIs and abstractions that package complex distributed inference, multi-model orchestration, and stateful session management capabilities into out-of-the-box middleware services. This will enable application developers to build sophisticated AI applications (e.g., multimodal AI agents) without managing underlying infrastructure complexities. Rapid Application Integration and Deployment. Building upon this AI-native framework, we will deliver a comprehensive toolchain - including encapsulated SDKs, application templates, and integrated CI/CD pipelines - to streamline development and deployment processes. Our goal is to enable application teams to integrate and deploy AI services within hours rather than weeks, significantly enhancing business innovation agility and fully bridging the \"last mile\" from model development to value creation. 7 Conclusion We introduce xLLM, an intelligent and efficient LLM inference framework, featuring a serviceengine decoupled architecture. The framework consists of two main components: (1) xLLMService, a versatile service layer designed for efficient instance and cluster management as well as request scheduling. It incorporates unified elastic scheduling for co-located online and offline requests to maximize cluster utilization, a workload-adaptive PD disaggregation architecture for SLOaware dynamic instance scheduling, a novel Encode-Prefill-Decode (EPD) disaggregation mechanism for multimodal requests, a global KV Cache manager for efficient memory management including KV Cache upload and offload and a distributed fault-tolerance design to ensure high availability. (2) xLLM-Engine, a high-performance inference engine optimized for accelerating LLM 32 xLLM Technical Report inference across various AI accelerators. xLLM-Engine employs full-stack multi-layer execution pipeline optimizations through asynchronizing CPU-side scheduling and AI accelerator-side model forwarding to minimize computational bubbles, utilizing dual-stream parallelism of micro-batches to overlap computation with all-to-all communication, and further overlapping various AI computation units at the operator level. xLLM-Engine also implements an adaptive graph mode which pre-compiles kernel sequences, memory operations, and synchronization into a single computation graph to drastically reduce launch overhead and accelerator idle time. Additionally, it adopts a \"logically contiguous, physically discrete\" KV Cache storage strategy via proposed xTensor Memory Management, which resolves the tension between memory contiguity and dynamic allocation. To further boost hardware utilization, xLLM-Engine integrates algorithmic enhancements such as asynchronous pipelined adaptive speculative decoding, KV cache-aware scheduling, reactive interDP group workload migration for dynamic load balancing, and dynamic MoE load balancing based on real-time expert workload statistics and transparent weight updates. We have further extended xLLM to emerging generative recommendation scenarios, improving both recommendation accuracy and efficiency. Extensive experiments demonstrate that xLLM achieves a consistent improvement compared to leading inference systems such as MindIE and vLLM-Ascend, evaluated across mainstream Qwenseries and Deepseek-series models as well as public and real-world industrial datasets. In particular, xLLM outperforms MindIE by up to 1.7× and vLLM-Ascend by up to 2.2× in throughput. 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Marzolo 8, 35131 Padova, Italy\ncIstituto Nazionale di Fisica Nucleare, Sezione di Padova,\nVia F. Marzolo 8, 35131 Padova, Italy\nE-mail: stefano.dinoi@kit.edu, barbaraanna.erdelyi@phd.unipd.it,\nramona.groeber@pd.infn.it\nAbstract:\nWe compute the two-loop renormalisation group equation for the effective\nHiggs to gluon coupling in Standard Model Effective Field Theory. Concretely, we present\nthe contributions generated by the operators belonging to class 3 and 7 in the Warsaw basis,\ncompleting the two-loop renormalization program of the Higgs-gluon coupling proportional\nto the top Yukawa coupling for potentially tree-level generated operators. We investigate\nthe phenomenological impact of the contributions in fits to Higgs data both in a bottom-up\napproach and a top-down approach in terms of UV models with vector-like quarks.\narXiv:2510.14680v1 [hep-ph] 16 Oct 2025\n\nContents\n1\nIntroduction\n1\n2\nNotation\n3\n3\nComputation and result\n4\n4\nPhenomenology\n8\n4.1\nFitting framework\n10\n4.2\nResults\n10\n4.2.1\nBottom-up results\n10\n4.2.2\nTop-down results\n12\n5\nConclusions\n15\nA Feynman Rules\n17\n1\nIntroduction\nThere are several reasons to believe that the Standard Model (SM) of particle physics\nrequires an extension. Despite this, no direct evidence of new physics has been observed\nso far, motivating the use of the Effective Field Theory (EFT) framework to describe\npotential beyond-the-SM (BSM) effects in the most model-independent way possible. This\napproach is valid under the assumption that the scale of new physics lies well above the\nelectroweak scale. Assuming that the Higgs boson transforms as in the SM, namely as part\nof an SU(2)L doublet, writing down all possible higher-dimensional operators consistent\nwith the symmetries of the SM leads to the so-called Standard Model EFT (SMEFT).\nConcerning collider physics, the most relevant contributions are expected to arise from\ndimension-six operators, for which a non-redundant basis was first established in Ref. [1],\nthe so-called Warsaw basis.\nDespite not being the only possible EFT extension of the SM,1 the SMEFT has become\na popular framework for global analyses [9–13], providing a self-consistent way to correlate\nvarious experimental measurements across different energy regimes.\nThe connection between the Wilson Coefficients (WCs) at different energy scales is\nencoded in the renormalisation group equations (RGEs). The complete one-loop RGEs of\ndimension-six operators have been computed in [14–16] and implemented for systematic\n1A different philosophy is embodied by the Higgs EFT [2–8] (HEFT or electroweak chiral Lagrangian),\nwhere the Goldstone bosons and the Higgs boson do not transform in the same multiplet, but the physical\nHiggs boson transforms as a singlet.\n– 1 –\n\ninclusion into phenomenological analyses in several computer codes [17–21]. Even though\nthe full set of two-loop RGEs for the dimension-six SMEFT has not been computed yet,\nmany partial results have been presented in the last years and progress towards the two-loop\nSMEFT RGEs has been reported [22–35].\nThe RGEs are thus crucial for global analyses that combine observables measured at\ndifferent energy scales [36–38], exploiting fully the variety of available data from different\nexperiments. Moreover, with the increasing precision of collider measurements, RGE effects\ncan also become sizeable in processes where the renormalisation scale varies dynamically\nover a wide range of energies. In this context, including the RGE running of WCs is essen-\ntial, as explicitly demonstrated in [39–43]. These works focus on the running proportional\nto the strong coupling constant αs, which is expected to give the dominant contributions.\nMoreover, restricting to these effects allows to solve the RGEs analytically, greatly improv-\ning the time efficiency of the computer programs employed for phenomenological studies.\nHowever, also the running effects generated by the top-quark Yukawa coupling Yt can\nproduce significant effects [27, 44]. Despite the strength of the interaction being na¨ıvely\nsmaller than in the strong case, WCs entering the RGEs via Yukawa-induced contributions\nare generally less constrained [11], potentially enhancing their impact.\nIn this work we complete the effort started in Refs. [24, 27] to determine the two-\nloop O\n\u0000g2\nsY 2\nt\n\u0001\nRGE of the Higgs-gluon coupling, which enters the dominant gluon-fusion\nproduction channel for Higgs production. Within the SMEFT framework, this coupling is\nparametrized by the operator OHG = (H†H)GA\nµνGA,µν where H denotes the Higgs doublet\nand GA\nµν the gluon field strength. We stress that this interaction generates a tree-level\ncoupling between the Higgs bosons and the gluons, which instead is loop induced in the\nSM. In weakly interacting and renormalizable UV models, the corresponding WC CHG is\nloop-generated [45, 46]. Moreover, at one-loop level, the RGE of CHG involves only WCs\nof operators that are likewise loop-generated under the same assumptions. Consequently,\nOHG does not undergo one-loop renormalisation, even when considering dimension-eight\neffects [47]. It follows that, to ensure consistency in the perturbative expansion in the loop\norder, the two-loop running effects proportional to tree-level generated operators must be\nconsidered when including RGE effects for Higgs production. Contributions belonging to\nthis category include the ones from four-quark operators [24] and Yukawa-like operators\n[27].\nIn this paper, we present the contributions from the operators with the schematic\nstructure D2H4 and D H2 ψ2. Such operators belong to, respectively, class 3 and 7 in\nthe classification of the Warsaw basis notation [1] and are the last missing contributions\nfor the complete two-loop RGE proportional to the third family Yukawa couplings Yt and\nYb involving operators that are potentially tree-level generated.\nConversely, we do not\nconsider chromomagnetic operators as they can only be generated at loop-level in weakly-\ninteracting theories [45, 46] and are expected to give a contribution at O(g2\nsYt) that would\nbe a three-loop effect.\nWe investigate the phenomenological impact of the new contributions to the RGE run-\nning by performing a fit to Higgs data both in the bottom-up and top-down approach, using\nsimplified models with heavy Vector-Like Quarks (VLQs). Such states naturally arise in a\n– 2 –\n\nvariety of frameworks proposed to address open issues of the Standard Model, such as the\nhierarchy problem, with Composite Higgs Models providing a prominent example [48–50].\nIn these models, operators belonging to class 7 are generated at tree level, which lead to a\nmodification of the couplings of quarks to massive vector, eventually in association with a\nHiggs boson. In particular, the couplings of top quarks to Z bosons are weakly constrained\nso far [37, 51], as a direct constraint comes from t¯tZ production, single top production [52]\nor Zh production [53], whereas other probes need to rely on RGE effects. Other operators\nthat couple to the third generation SU(2)L doublet can also be constrained by electroweak\nprecision data.\nConversely, class 3 operators can be constrained by electroweak preci-\nsion data and by measurements of the Higgs coupling to massive vector bosons, currently\navailable at few percent level [54, 55].\nThe paper is structured as follows. In Sec. 2 we present our notation. In Sec. 3 we\ndescribe our computation and present the result for the RGE of the Higgs-gluon coupling\nin the SMEFT. In Sec. 4 we study the phenomenological implications of the inclusion of\ntwo-loop running effects in single Higgs physics. Finally, in Sec. 5 we present our summary.\n2\nNotation\nThe SMEFT extends the SM by a tower of gauge and Lorentz invariant higher-dimensional\noperators as a series in inverse powers of a high-energy new physics scale Λ. In this work, we\nneglect lepton and baryon number violation and focus exclusively on the leading dimension-\nsix terms in the expansion:\nL = LSM +\nX\nDi=6\nCiOi.\n(2.1)\nThe operators are denoted by Oi and the associated WCs by Ci. We note that the WCs in\nthe previous formula have dimension -2 in units of energy. The operators in the previous\nexpressions are invariant under the unbroken SM group GSM = SU(3)C × SU(2)L × U(1)Y.\nWe follow the Warsaw basis of Ref. [1] for the dimension-six operators and Ref. [14]\nfor the definitions of the SM Lagrangian.\nWe consider only the third family of quarks, leading to a minimal modification of the\nnotation with respect to the original publication. In particular, we use tR, bR to denote the\nright-handed component and QL to denote the left-handed component of the third family\nof quarks. The relevant class 7 operators are defined as:\nO(1)\nHQ =\n\u0010\niH†←\n→\nDµH\n\u0011 \u0000 ¯QLγµQL\n\u0001\n,\nO(3)\nHQ =\n\u0010\niH†←\n→\nDI\nµH\n\u0011 \u0000 ¯QLγµτ IQL\n\u0001\n,\n(2.2)\nOHt =\n\u0010\ni H†←\n→\nDµH\n\u0011\n(¯tRγµtR) ,\nOHb =\n\u0010\ni H†←\n→\nDµH\n\u0011 \u0000¯bRγµbR\n\u0001\n,\n(2.3)\nOHtb =\n\u0010\ni ˜H†DµH\n\u0011\n(¯tRγµbR) ,\n(2.4)\nwhere ˜Hi = ϵijH†,j (with ϵ12 = +1) and\n\u0010\ni H†←\n→\nDµH\n\u0011\n= iH†(DµH) −i(DµH)†H,\n(2.5)\n\u0010\ni H†←\n→\nDI\nµH\n\u0011\n= iH†τ I(DµH) −i(DµH)†τ IH.\n(2.6)\n– 3 –\n\nThe operators in class 3 are defined as\nOH□=\n\u0010\nH†H\n\u0011\n□\n\u0010\nH†H\n\u0011\n,\nOHD =\n\u0010\nH†DµH\n\u0011 \u0010\nH†DµH\n\u0011∗\n.\n(2.7)\nThese two operators in the broken phase modify the kinetic term of the Higgs boson, thus\nrequiring a field redefinition to reobtain the canonical normalisation of the Higgs kinetic\nterm. In the unitary gauge, the Higgs doublet H is related to the physical Higgs field h via\nH =\n1\n√\n2\n \n0\nv + h\n\u0010\n1 + v2 CH, kin\n\u0011\n!\n,\nwhere CH, kin = CH□−1\n4CHD .\n(2.8)\nFinally, we recall that the Higgs-gluon coupling in the SMEFT is parametrized by the\noperator\nOHG =\n\u0010\nH†H\n\u0011 \u0000GA\nµνGA µν\u0001\n.\n(2.9)\nThis operator belongs to class 4 in the Warsaw basis. Other class 4 operators that will\nappear in the phenomenological study performed in this work are defined below\nOHW =\n\u0010\nH†H\n\u0011 \u0000W I\nµνW I µν\u0001\n,\nOHB =\n\u0010\nH†H\n\u0011\n(BµνBµν) ,\n(2.10)\nOHWB =\n\u0010\nH†τ IH\n\u0011 \u0000W I\nµνBµν\u0001\n.\n(2.11)\nFurthermore, the following class 5 operators will be considered, often referred to as Yukawa-\nlike operators:\nOtH =\n\u0010\nH†H\n\u0011 \u0010\n¯QL ˜HtR\n\u0011\n,\nObH =\n\u0010\nH†H\n\u0011 \u0000 ¯QLHbR\n\u0001\n.\n(2.12)\nWe remind the reader that the SM Yukawa coupling is not defined in direct analogy to the\nYukawa-like operator.\nAll the previous operators are Hermitian with the exclusion of OHtb, OtH and ObH,\nwhich enforce the addition of their complex conjugate to the Lagrangian. We use T A for\nthe SU(3)C generators and τ I for the Pauli matrices. We define the gluon field-strength\ntensor as GA\nµν = ∂µGA\nν −∂νGA\nµ −gsfABCGB\nµ GC\nν .\n3\nComputation and result\nWe describe in this section the computation of the two-loop contributions of the operators in\nEqs. (2.2)-(2.4) and Eq. (2.7) to the RGE of CHG. Practically, this requires the computation\nof the divergent parts of the two-loop corrections to a process where OHG, defined in\nEq. (2.9), enters at tree-level. We perform our computation in the unbroken phase and\nwe consider the process gg →H∗H. In dimensional regularization, the divergences are\nparametrized by poles in ϵ = 0, being D = 4 −2ϵ the number of space-time dimensions.\nWe will discuss here in detail the calculation of the terms proportional to OHt. All\nthe other operators of class 7 are obtained in analogy. At one-loop order, the contributions\nto gg →H∗H can only arise via triangle diagrams as in Fig. 1a. However, once all the\nrelevant diagrams are considered their contribution vanishes [16]. The reason is that the\n– 4 –\n\nH\nH∗\ng\ng\n(a)\nH\nH∗\ng\ng\n(b)\nH\nH∗\ng\ng\n(c)\nFigure 1: Sample of the diagrams with a single insertion of the operator OHt which vanish because\nof kinematics, see main text for details. We use a thick (thin) line to denote a left-(right-)handed\nfield, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt and\na black dot to denote a SM vertex insertion.\ng\nH\nH∗\ng\n(a)\ng\nH\nH∗\ng\n(b)\ng\nH\nH∗\ng\n(c)\nFigure 2: Sample of the two-loop order diagrams with a single insertion of the operator OHt.\nWe use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs\ndoublet, a black square to denote an insertion of OHt, a black dot to denote a SM vertex insertion\nand a white star to denote the Yukawa coupling counterterm.\ng\ng\nH\nH∗\n(a)\ng\ng\nH\nH∗\n(b)\nFigure 3: Sample of the two-loop order diagrams with a single insertion of the operators OH□and\nOHD. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the\nHiggs doublet, a white square to denote an insertion of OH□and OHD and a black dot to denote\na SM vertex insertion.\ndiagrams are purely s-channel, depending on the linear combination q1+q2 = p1+p2 where\nwe have used p1,2 (q1,2) to denote the incoming (outgoing) momenta of the gluons (Higgs\ndoublets). The scalar current gives rise to a contraction with (p1 −p2), yielding a vanishing\nresult.\nFollowing this argument, we can safely ignore all the two-loop diagrams where the\ntwo external Higgs bosons are connected to the effective vertex. This applies both to pure\nQCD (see Fig. 1b) and to mixed Yukawa-QCD (see Fig. 1c) contributions.2 We have cross-\nchecked the correctness of our reasoning by explicitly computing the O\n\u0000g4\ns\n\u0001\ndiagrams with\n2As a consequence of this statement, the class 7 pure QCD contribution to the RGE of CHG vanishes at\nall loop orders at dimension-six level.\n– 5 –\n\na QCD correction to the propagator connecting the two QCD vertices and verifying that\ntheir sum vanishes after a trivial change of variable in the momentum integration.\nOur argument allows to restrict our attention to a total of 24 diagrams containing an\ninsertion of OHt, where one of the two Higgs doublets in the EFT vertex is external and\nthe other is internal. A sample of the two families of diagrams is displayed in Figs. 2a and\n2b. Moreover, class 7 operators correct the Yukawa coupling at one-loop [14], calling for\nthe inclusion of the one-loop insertions of the one-loop Yukawa counterterms as in Fig. 2c.\nClass 3 operators modify the quadrilinear coupling, yielding 12 diagrams, of which we\npresent a sample in Fig. 3.\nWe have performed two independent computations of the class 7 operator contributions\nto check the set-up of our computation. The first one relies on FeynRules [56, 57] for the\ngeneration of the Feynman rules, where the model file is obtained by extending the SM\nmodel file in the unbroken phase provided in the additional material to MatchMakerEFT [58].\nThe resulting Feynman rules are presented in App. A. FeynArts [59] and FeynCalc [60–\n63], interfaced via FeynHelpers [64], are used to generate the diagrams and to simplify\nthe Dirac algebra. The numerical integration of the two-loop integrals is performed with\nAMFlow [65] supplemented with Blade [66].\nThe second computation relies on the same pipeline of Ref. [27]. The diagrams are\ngenerated with qgraf-3.6.10 [67] and simplified with FeynCalc. The numeric integration\nis performed with AMFlow supplemented with LiteRed2 [68, 69] and FiniteFlow [70].\nBoth computations use the Na¨ıve Dimensional Regularization (NDR) [71, 72] scheme\nfor the treatment of γ5 in a non-integer number of dimensions. This prescription is known\nto be algebraically inconsistent in presence of traces involving an odd number of γ5 in\ncombination with at least six γ matrices as in the diagrams we compute. However, we\nare interested in the contribution to the RGE of CHG, meaning that we can ignore the\ncontribution of the traces involving γ5, which would renormalize its CP-odd counterpart,\nnamely ˜CHG.\nThe Breitenlohner-Maison-t’Hooft-Veltman (BMHV) scheme [73, 74] has recently gained\npopularity as an alternative to the NDR scheme, being the only known continuation scheme\nproved to be algebraically consistent [75–79]. However, it highly increases the algebraic\ndifficulty of loop computation. We have checked that, for the present computation, no\ndifference arises between the two schemes by employing the map presented in Ref. [80].\nThis procedure is legit in this context since the leading divergence is of O (1/ϵ), as recently\nshowcased in Ref. [33].\nWe note that, in principle, all the operators considered in this work can renormalize\nredundant operators, for which we follow the notation of Ref. [58]. This renormalization\narises from subamplitudes with schematic structure ¯ψψ →H∗H, ψ = QL, tR, bR, which\naffect the operators R′\nHψ, R′′\nHψ,3 and from H →¯QLχR, χ = b, t (possibly in association\nwith one or two gluons), which impact the operators RuHDi, i = 1, 2, 3, 4. Since some of\nthe external states in these subprocesses are not on-shell, the relevant one-loop insertions\n3When ψ = QL, two possible isospin contractions are possible, generating a total of four redundant\noperators involving the left-handed quarks.\n– 6 –\n\nof redundant operators must be included in the total contribution to gg →H∗H. We have\nexplicitly checked that these contributions vanish in the case where only OHt is present.\nMoreover, we do not need to consider contributions coming from one-particle reducible\ndiagrams to OHG.\nThis conclusion stems from the fact that no redundant operator is\nreduced to OHG when the SM equations of motions are used [1].\nWe obtain\nµdCHG\ndµ\n⊃\ng2\ns\n(16π2)2\nh\n+ YbY ∗\nb (CHb −C(1)\nHQ −3C(3)\nHQ) + YtY ∗\nt (C(1)\nHQ −3C(3)\nHQ −CHt)\n+ (CHtbYbY ∗\nt + C∗\nHtbY ∗\nb Yt)\n−\n\u0012\nCH□−1\n2CHD\n\u0013\n(YbY ∗\nb + YtY ∗\nt )\n+ 3\n2 [CtHYt + C∗\ntHY ∗\nt + CbHYb + C∗\nbHY ∗\nb ]\n−4 δNDR\n\u0012\nYtY ∗\nt\n\u0012\nC(1)\nQt −1\n6C(8)\nQt\n\u0013\n+ YbY ∗\nb\n\u0012\nC(1)\nQb −1\n6C(8)\nQb\n\u0013\u0013 i\n,\n(3.1)\nδNDR =\n(\n1\n(NDR)\n0\n(BMHV).\n(3.2)\nWe use the same conventions as the one-loop results of [14–16]. The first two lines represent\nour novel result, while the contributions generated by four-quark operators and Yukawa-like\noperators (see Eq. (2.12)) have been first derived in Ref. [24] and in Ref. [27],4 respectively,\nwhich we point to for the precise definition of the operators.\nWe stress that only the\nfour-top contribution depends on the γ5 scheme, while the others do not.\nThe final amplitude cannot exhibit infrared (IR) divergences, since no corresponding\nreal-emission diagrams exist at the considered order in the coupling constants. However,\nIR divergences can still appear at intermediate stages of the calculation. As they must\nultimately cancel, one has to verify that this cancellation does not occur against ultraviolet\n(UV) divergences, as can happen, for instance, in the case of scaleless integrals.\nWe note that our result can be used to determine the analogous two-loop contributions\nfrom the operators in class 3 and 7 to OHW , OHB, OHWB (defined in Eqs. (2.10)–(2.11)),\nwhere the gluon field strength is replaced by the corresponding quantities involving elec-\ntroweak bosons. However, this would represent just a partial result, since new diagrams\narise as a consequence of the non-vanishing SU(2)L × U(1)Y charge of the Higgs doublet.\nObtaining the RGEs for such operators is hence beyond the scope of this paper. The phe-\nnomenological relevance of these operators lies in their impact in the interactions involving\nphotons or a photon and Z bosons and (at least) one Higgs boson and they lead to a shift\nin the electroweak input [81]. In the SM, such interactions are numerically dominated by\nW loops, so we expect the two-loop running contributions to be less numerically important\nthan in the gluon case such that the phenomenological study we perform in the next section\n4In the previous expression, the contribution from the operators OtH and ObH presents an extra 1/2\nwith respect to the result of Ref. [27], which will be updated.\n– 7 –\n\nremains valid omitting these sub-leading contributions. Furthermore, their contributions\nto the Higgs couplings to massive vector bosons is suppressed in the loop counting with\nrespect to the tree-level contributions from class 3 operators.\n4\nPhenomenology\nIn this section, we study the effect of the inclusion of two-loop running effects with fits to\nHiggs data at the LHC. In particular, we focus on the modifications to the signal strengths\nof a set of Higgs decay final states due to the presence of SMEFT operators.\nWe perform two different fits. In first instance, we adopt a bottom-up perspective by\nperforming the analysis in terms of the WCs to observe how their bounds are modified\nby the inclusion of the running. Secondly, we consider a top-down approach by selecting\nUV models with pairs of Vector-like Quarks (VLQs) coupled to the third-generation SM\nquarks, which generate, at tree-level, at least one of the class 7 operators considered in this\npaper.\nThe WCs that enter the observables in this section are computed at the low scale\nµR = mh = 125 GeV. Their expression in terms of the WCs at the high energy scale\nΛ = 2 TeV can be obtained with the evolution matrix formalism [20]\nCi(µR) = Uij(µR, Λ)Cj(Λ).\n(4.1)\nIn the present case, the evolution matrix Uij(µR, Λ) has been obtained with a modified\nversion of RGESolver [21] relying on a numeric solution of the RGEs. All the operators\nincluded in this section have been defined in Sec. 2.\nCross-section\nThe SM value for the Higgs boson production cross section in the dom-\ninant gluon-gluon fusion contribution is σSM = 48.68 pb at a center of mass energy of\n√s = 13 TeV. In the SMEFT, the cross-section reads\nσ =\n\u0014\n48.68 + 2.83 · 104v2CHG + 97.36v2CH,kin −112.25v2\nYt\nCtH\n\u0015\npb ,\n(4.2)\nwhere CH,kin was defined in Eq. (2.8). We have obtained this parameteric dependence on\nthe SMEFT WCs using the EFT implementation [41] in higlu [82].\nHiggs total and partial decay widths\nThe Higgs total decay width in the SMEFT\nat LO in the U(3)5 flavour-symmetric limit was computed in Ref. [83]. We adapt those\nresults, noting that quark operators of class 7 contribute only via corrections to the gauge\nboson total width except for the cases with quarks in final states from Z boson decays.\nIn this case there are some direct contributions where the Higgs boson and one Z boson\ncouples via a class 7 operator to bottom quarks, and the Z boson decays subsequently.\nThis results in\nΓh = ΓSM\nh\n\u00001 + 50.6 v2CHG −1.50 v2CHB −1.21 v2CHW + 1.21 v2CHWB + 1.83 v2CH□\n−0.43 v2CHD −48.5 v2Yb CbH + 0.00016v2 C(1)\nHq + 0.0015v2 C(3)\nHq −0.00023v2 CHb\n+2 v2\n\u0012\nBRh→γγ\n\u0014\nCH,kin + 0.53\nYt\nCtH\n\u0015\n+ BRh→gg\n\u0014\nCH,kin −2.12\nYt\nCtH\n\u0015\u0013\u0013\n,\n(4.3)\n– 8 –\n\nusing the {MW , MZ, GF } input scheme for the electroweak parameters.\nThe last line\nencodes the higher-order rescaling of the Higgs decays into photon and gluon pairs through\nCH,kin defined in Eq. (2.8), together with the additional modifications induced by shifts in\nthe top Yukawa coupling. In the UV models we consider, also the bottom Yukawa coupling\ncan be modified but we do not consider its effect in the h →gg and h →γγ decays as\nit is negligible with respect to the top Yukawa contribution. We note already from this\nformula that generically our two-loop contributions can be of the same order than the\ndirect contributions. Indeed, since the CHG has a large pre-factor, even the two-loop factor\n1/(16π2)2 log(Λ/mh) can be compensated.\nWe obtain the partial decay widths in the SMEFT rescaling the results in Ref. [83]\nto take into account the modifications of the decay into photon pairs and the top Yukawa\ncoupling, analogously to the case of the total width discussed previously. We have validated\nthese expressions to those reported in Ref. [84].5\nFor the decays into fermion pairs, introducing l ∈{µ−, τ −} , the partial decay widths\nare\nΓh→l¯l\nΓSM\nh→l¯l\n= 1 + 2v2 CH,kin,\n(4.4)\nΓh→b¯b\nΓSM\nh→b¯b\n= 1 + v2 \u00002CH,kin −2 CbH\n\u0001\n.\n(4.5)\nFor the decays into photon pairs we get\nΓh→γγ\nΓSM\nh→γγ\n= 1 + v2\n\u0012\n2 CH,kin + 0.53\nYt\nCtH −231 CHW −805 CHB + 431 CHWB\n\u0013\n.\n(4.6)\nAs for decays into pairs of W and Z pairs, we further consider their decay products in\norder to compare with experimental results. In particular, Ref. [55], whose experimental\nresults we will be using later, considers the processes h →W +W −→2l 2ν and h →ZZ →\n4l. In order to get the partial decay width for these processes, we start again from Ref. [83]\nand focus on their results for partial decay widths of the Higgs boson into four-fermion final\nstates mediated by ”only neutral”, ”only charged” and ”neutral plus charged” currents.\nIn our scenario, the first case reduces to the 4l final state, while the second and third\ncorrespond to the 2l 2ν final state. The SM partial decay width for h →ZZ →4l is then\nΓSM\n4l\n= 1.1 keV, the partial decay width for h →W +W −→2l 2ν reads ΓSM\n2l2ν = 90 keV ,\nand the corresponding SMEFT expressions read\nΓ4l\nΓSM\n4l\n=1 + v2 (2CH□+ 0.27 CHD + 0.17 CHW −1.66 CHB −0.51 CHWB\n−0.27 C(1)\nHQ −0.27 C(3)\nHQ + 0.048 CHb\n\u0011\n,\n(4.7)\nΓ2l2ν\nΓSM\n2l2ν\n=1 + v2 (2CH□−0.52 CHD −1.5 CHW + 0.003 CHB −0.023 CHWB\n−0.0046 C(1)\nHQ −0.0046 C(3)\nHQ + 0.00081 CHb\n\u0011\n.\n(4.8)\n5Ref. [84] uses the SILH basis [85], whereas Ref. [83] uses the Warsaw basis. To compare the results, we\ntranslate both sets of results into the Greens’ basis by means of Ref. [86] and compare the terms of interest.\n– 9 –\n\n4.1\nFitting framework\nAdopting the symbol Oα to identify an observable α, the χ2-function can be defined as\nχ2 =\nX\nα, β\n\u0010\nOexp\nα\n−Oth\nα\n\u0011 \u0002\nC−2\u0003\nαβ\n\u0010\nOexp\nβ\n−Oth\nβ\n\u0011\n,\n(4.9)\nwhere C−2 is the inverse of the covariance matrix and the superscript ’th’ (’exp’) means\nthat the theoretical (experimental) value of the observable Oα is being considered. The\ntheoretical value accounts for both the SM prediction and the deviations due to the insertion\nof SMEFT operators.\nFollowing Ref. [87], we construct a fit to Higgs data using the measurements of signal\nstrengths with their covariance matrix as provided in Ref. [55]. Considering the Higgs\ndecay channels h →α with α ∈{W +W −, ZZ, b¯b, τ +τ −, µ+µ−, γγ},6 for each channel it\nis given by\nµα =\nσ × BRh→α\nσSM × BRSM\nh→α\n=\nσ\nσSM\nΓSM\nh\nΓSM\nh→α\nΓh→α\nΓh\n=\nσ\nσSM\nΓSM\nh\nΓh\nΓh→α\nΓSM\nh→α\n,\n(4.10)\nin which σ(SM) and BR(SM) denote, respectively, the value in the SMEFT (SM) of the Higgs\nproduction cross-section and the branching ratio into the specific channel. We consider\nup to order O(1/Λ2) in the signal strength. Of the three multiplied ratios in the final\nexpression, the first is known from Eq. (4.2), the second from Eq. (4.3) and the third is\nselected from Eqs. (4.4)-(4.8) depending on the decay channel of interest.\nThe allowed region satisifies the condition\nχ2 < χ2\nmin + ∆χ2 (n, CL) .\n(4.11)\nIn the previous expression, n denotes the degrees of freedom (number of couplings in the\ntop-down approach, number of involved WCs in the bottom-up approach) while CL stands\nfor Confidence Level. The value χ2\nmin is the minimum of the χ2.\n4.2\nResults\nWe now present our results. We notice that the operators OHtb and OHt do not enter di-\nrectly any of the Eqs. (4.2)-(4.8), so they enter only via mixing effects with other operators.\n4.2.1\nBottom-up results\nIn this section we report our one- and two-parameter results for the bottom-up fits. We\nstress that these parameters are set to be non-zero at the high-scale Λ = 2 TeV, meaning\nthat several other operators are generated via operator mixing, governed by Eq. (4.1).\nHence, the bounds are reported for the operators at such scale.\nOne parameter fits\nWe present the individual 2σ CL interval for each WC considered\nat the energy scale Λ = 2 TeV in Fig. 4.\nWe note that the intervals for the class 3\n6For the decays into pairs of W and Z bosons, Ref. [55] considers the bosonic decay channels ZZ →4l\nand WW →lνlν. Thus, for the first two decay channels, we can use the decay widths of Eqs. (4.7) and\n(4.8) respectively.\n– 10 –\n\n−12\n−10\n−8\n−6\n−4\n−2\n0\n2\n4\n6\n8\n10\n12\nIntervals\nCHD(Λ)\nCH□(Λ)\nCtH(Λ)\nCbH(Λ)\nCHt(Λ) · 10−1\nCHb(Λ) · 10−1\nC(1)\nHQ(Λ)\nC(3)\nHQ(Λ)\nCHtb(Λ) · 10−2\n2σ 1L RGEs\n2σ 2L RGEs\nFigure 4: Allowed ranges at 2σ for the WCs obtained by performing one parameter fits. The\nyellow (orange) line is obtained by considering one-loop (two-loop) running effects.\ncoefficients are not sizeably affected by the inclusion of the two-loop effects in the RGEs.\nThese operators are already constrained by direct contributions to the couplings with the\nHiggs bosons, reducing the impact of higher order corrections. Similar conclusions apply\nto the WCs of the Yukawa-like operators CtH and CbH, which enter directly in the Higgs\nproduction cross section and in several decay channels.\nAs for the class 7 operators, CHb is not sizeably modified by the inclusion of the two-\nloop running effects, due to the bottom Yukawa suppression. Conversely, C(1)\nHQ and CHt\nshow a moderate reduction when two-loop running effects are accounted, while for C(3)\nHQ\na quite sizable reduction of the allowed range can be observed once two-loop running is\nincluded. Moreover, we note that the bounds for CHtb are looser at two-loop level, hinting\nfor some cancellations happening with respect to one-loop running effects.\nFinally, we note that the bounds we present here are much weaker than the ones\npresented in Ref. [37]. Since our focus is on the impact of the two-loop running effects in\nthe fit, we only accounted for inclusive Higgs data and not, for instance, for electroweak\nprecision data or top data as Ref. [37]. Those measurements are though sensitive to the\noperators considered here.\nTwo parameter fits\nIn Fig. 5 we allow two WCs to be non-vanishing at the energy scale\nΛ = 2 TeV. We present the results of three two-parameter fits, comparing the impact of\nincluding only one-loop or two-loop running effects. In Fig. 5a, the two class 3 operators\nare considered. The correlation between the two arises from the fact that their contribution\nappears mostly in the linear combination CH, kin = CH□−1\n4CHD. Moving on to Fig. 5b,\nwe consider the two WCs C(1)\nHQ and C(3)\nHQ which, in motivated UV extensions of the SM,\n– 11 –\n\n(a) Class 3 operators\n(b) Class 7 operators with quark doublets\n(c) Class 7 operators with quark singlets\nFigure 5: Allowed regions at 2σ obtained via a two parameter fit including one-loop (in yellow)\nand two-loop (in orange) running effects. We consider in pairs the operators belonging to class\n3 and to class 7. For class 7, we further separate between the cases in which the quarks in the\noperators are the doublets or the singlets under SU(2)L.\nappear together, as shown in Tab. 1. Finally, in Fig. 5c we compare CHt and CHb. In this\ncase, there is a significant improvement in the bounds considering that CHt does not enter\nany of the observables defined in Eqs. (4.2)–(4.8), while CHb enters but is very suppressed\ncompared to the other WCs.\n4.2.2\nTop-down results\nIn this second approach we consider UV models featuring Vector-like Quarks (VLQs) as\nthey generate at tree level both the operators in Eqs. (2.2)-(2.4) and the Yukawa-like\n– 12 –\n\nGSM\nO(1)\nHQ\nO(3)\nHQ\nOHt\nOHb\nOHtb\nU\n(3, 1) 2\n3\n✓\n✓\nQ1\n(3, 2) 1\n6\n✓\n✓\n✓\nQ7\n(3, 2) 7\n6\n✓\nT1\n(3, 3)−1\n3\n✓\n✓\nT2\n(3, 3) 2\n3\n✓\n✓\nModel\nParticle Content\n1\nU + Q1\n2\nU + Q7\n3\nQ1 + T1\n4\nQ1 + T2\n5\nQ7 + T2\nTable 1: Class 7 operators generated at the tree-level by the VLQs considered in this work, along\nwith the charges under the SM gauge group of the VLQs, with GSM = (SU(3)C, SU(2)L)U(1)Y (left).\nThe presented VLQs will be used to construct models with pairs of these particles (right).\noperators OtH and ObH [88]. The operators OHG, OHD and OH□are instead generated at\none-loop level.\nIn a similar spirit to Ref. [87], we consider models with pairs of VLQs which give rise\nto unsuppressed contributions to the operator OtH. In Tab. 1 we list the VLQs considered,\ntheir charges under the SM gauge group GSM, the class 7 operators they generate at tree-\nlevel and how the five models considered in this work are defined.\nGiven that we are interested in the RGE effects due to terms proportional to third\ngeneration Yukawa couplings, we make the simplifying assumption that the VLQs only\ncouple to the third generation SM quarks. Furthermore, we set their masses to be Λ =\n2 TeV. This value is in agreement with lower bounds on singly-produced VLQs coupled to\nthird generation quarks [89, 90] and with the bounds from VLQ pair production [90–92].\nThe interaction Lagrangians for the considered models are given for a general flavour\nstructure in Ref. [88]\n−L(int)\nM.1 = [λU]rp ¯Ur\nR ˜H†qp\nL +\n\u0002\nλu\nQ1\n\u0003\nrp ¯Qr\n1L ˜Hup\nR +\nh\nλd\nQ1\ni\nrp\n¯Qr\n1LHdp\nR+\n+ [λUQ1]rp ¯Ur ˜H†Qp\n1 + H.c. ,\n(4.12)\n−L(int)\nM.2 = [λU]rp ¯Ur\nR ˜H†qp\nL + [λQ7]rp ¯Qr\n7LHup\nR + [λUQ7]rp ¯UrH†Qp\n7 + H.c. ,\n(4.13)\n−L(int)\nM.3 =\n\u0002\nλu\nQ1\n\u0003\nrp ¯Qr\n1L ˜H up\nR +\nh\nλd\nQ1\ni\nrp\n¯Qr\n1LH dp\nR +\n[λT1]rp\n2\n¯T I r\n1RH†τ I qp\nL+\n+\n[λT1Q1]rp\n2\n¯T I r\n1 H†τ I Qp\n1 + H.c. ,\n(4.14)\n−L(int)\nM.4 =\n\u0002\nλu\nQ1\n\u0003\nrp ¯Qr\n1L ˜H up\nR +\nh\nλd\nQ1\ni\nrp\n¯Qr\n1LH dp\nR +\n[λT2]rp\n2\n¯T I r\n2R ˜H†τ I qp\nL+\n+\n[λT2Q1]rp\n2\n¯T I r\n2\n˜H†τ I Qp\n1 + H.c. ,\n(4.15)\n– 13 –\n\n(a) Model 2.\n(b) Model 3.\n(c) Model 4.\n(d) Model 5.\nFigure 6: Allowed regions at 2σ for pairs of couplings for the considered models. The orange\n(yellow) coloured areas correspond to allowed values derived accounting for two-loop (one-loop)\nrunning effects.\n−L(int)\nM.5 = [λQ7]rp ¯Qr\n7LHup\nR +\n[λT2]rp\n2\n¯T I r\n2R ˜H†τ I qp\nL +\n[λT2Q7]rp\n2\n¯T I r\n2 H†τ IQp\n7 + H.c. .\n(4.16)\nFor these equations we follow the notation of Ref. [88] and use up\nR and dp\nR for right-handed\nquark singlets and qp\nL for the left-handed quark doublets, being p, r a flavour index. For our\nphenomenological study, we only consider third generation SM quarks, namely p = r = 3.\nFor the tree-level expressions we refer to Ref. [88], whereas for the one-loop matching\nwe use SOLD [93] and Matchete [94].\nIn all models and for all operators entering Eqs. (4.2)-(4.8) we match up to the one-loop\nlevel.\n– 14 –\n\n(a) Model 1 with complete one-loop matching\n(b) Model 1 using tree-level matching\nFigure 7: Two fits performed for Model 1, considering either the complete matching up to one\nloop for all the considered operators (left) or only the tree-level matching results (right). In both\ncases, the allowed parameter space at 2σ is plotted in yellow (orange) when the one-loop (two-loop)\nrunning effects are included.\nFor each model the χ2 is a function of either three or four couplings in the UV model.\nIn Fig. 6 and Fig. 7a we present our results as two-dimensional CL regions, setting the\nremaining parameters to their best fit value.7 As can be inferred in Figs. 6–7a, the inclusion\nof two-loop running in the RGEs makes only a minor difference. While the model at tree-\nlevel matches only into operators of class 7 and the Yukawa-type operators, at one-loop level\nit matches into the class 3 operators and OHG, OHW , OHB and OHWB, which enter with\nlarge coefficients into the signal strength. This reduces the impact of the two-loop running\neffects considered here. A greater impact is observed when only the tree-level matching\nresults are considered. In order to showcase this aspect, in Fig. 7, we compare the complete\nfit performed for model 1 with a second fit in which only the tree-level generated operators\nare considered.\n5\nConclusions\nWe have analyzed the two-loop RGE of the effective Higgs-gluon coupling in the SMEFT,\nparametrized by the operator OHG. Several single and double Higgs production channels\nare directly sensitive to this coupling, that enters already at tree level whereas the SM\nsignal arises at one loop.\nAt one loop, CHG renormalizes itself and also mixes with the chromomagnetic operator.\nAdopting a loop counting in weakly coupled, renormalizable UV completions of the SM\n[45, 46], these operators are expected to arise only at the one-loop level, pushing their\n7For some models, we have explicitly checked that this approximation reproduces the behavior of the\nprofiled results.\n– 15 –\n\nRGE effects at two-loop order. Consequently, operators that can be generated at tree level\nand mix with CHG at two loop must also be consistently included under this counting.\nOur computation therefore extends the previous efforts for the computation of the two-\nloop contributions of the potentially tree-level generated four-top [24] and Yukawa-like\noperators [27], presenting for the first time the two-loop contributions of the operators\nwith schematic structure D2H4 and D H2 ψ2.\nWe addressed the phenomenological impact of the two-loop running in a fit to inclusive\nHiggs data. Indeed, we could find that bounds on the class 7 operators can shrink sizably\ndue to the two-loop running effects, if their contribution is proportional to the top Yukawa\ncoupling. In concrete models, such as the ones with VLQs presented in our phenomenolog-\nical study, this effect is though reduced, as many more operators arise when matching up\nto one-loop order. For instance, OHG is generated at one-loop level in this type of models.\nNevertheless, our results indicate that two-loop running effects can have a sizable\nimpact on global fits, once completely available. This applies in particular for operators so\nfar loosely constrained. For instance, in the case of four-top operators, it was shown that\ntwo-loop corrections to electroweak precision observables [32] can, in some cases, limit the\ncurrently allowed range [95].\nIn summary, our results complete an important piece of the two-loop RGE framework\nfor Higgs–gluon interactions, setting the stage for consistent and precise SMEFT analyses\nof Higgs data with potential phenomenological impact.\nNote added\nDuring the completion this work, we became aware of [96]. Our results are in agreement.\nAcknowledgements\nWe thank Javier Fuentes-Mart´ın for cross-checking our results with [96] and the discussions\nthat made us realise that we need to clearly state the definition of the SM Lagrangian\nthat we adopt. BE would like to thank Jonathan Ronca for several useful discussions on\nnumerical integration and Alejo N. Rossia for comments on marginalisation and profiling.\nAlso, we would like to acknowledge the Mainz Institute for Theoretical Physics (MITP)\nof the Cluster of Excellence PRISMA+ (Project ID 390831469) for its hospitality and its\npartial support during part of the completion of this work. This work received funding\nby the INFN Iniziativa Specifica APINE and by the University of Padua under the 2023\nSTARS Grants@Unipd programme (Acronym and title of the project: HiggsPairs – Precise\nTheoretical Predictions for Higgs pair production at the LHC). This work was also partially\nsupported by the Italian MUR Departments of Excellence grant 2023-2027 “Quantum\nFrontiers”. We acknowledge support by the COST Action COMETA CA22130.\nCloudVeneto is acknowledged for the use of computing and storage facilities. The Feynman\ndiagrams in this work were made using TikZ-Feynman [97].\n– 16 –\n\nA\nFeynman Rules\nIn this Appendix we present the Feynman rules associated with the effective operators\ndefined in Eqs. (2.2)-(2.4). mj (Aj) identifies an index in the fundamental (adjoint) rep-\nresentation of SU(3)C whereas ij denotes an index in the fundamental representation of\nSU(2)L. The numbers are associated to the particles in the figure on the left.\nH3,∗\nH4\ng1\ng2\n= +4i CHGδi3i4δA1A2 (pµ2\n1 pµ1\n2 −gµ1µ2p1 · p2) .\n(A.1a)\nH1\nH2, ∗\nQ4\nL\nQ3\nL\n= +i\n\u0010\nC(1)\nHqδi1i2δi3i4 + C(3)\nHqτ I\ni2i1τ I\ni4i3\n\u0011\nδm3m4\n\u0010\n/p1 −/p2\n\u0011\nPL,\n(A.1b)\nH1\nH2, ∗\nψ3\nR\nψ4\nR\n= +i CHψδi1i2δm3m4\n\u0010\n/p1 −/p2\n\u0011\nPR,\nψ = t, b ,\n(A.1c)\nH1\nH2\nd4\nR\nu3\nR\n= +i CHtbδm3m4ϵi1i2\n\u0000/p1 −/p2\n\u0001\nPR,\n(A.1d)\nH1\nH2\nH4,∗\nH3,∗\n= See Eq. (A.2).\n(A.1e)\nGiven their length, we report here the Feynman rules for the last diagrams above:\n−i\n\"\nδi1i4δi2i3\n \nCHD (p1 · p3 + p2 · p4) + CH□\n X\ni\np2\ni + 2p1 · p4 + 2p2 · p3\n!!\n+ (4 ↔3)\n#\n.\n(A.2)\nFor the sake of completeness we present the Feynman rules of the relevant SM interac-\ntions. We remind the reader that the for the Yukawa sector of the SM we adopt the same\n– 17 –\n\nnotation of Refs. 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Commun. 210 (2017)\n103 [1601.05437].\n– 23 –\n"},"cleaned":{"kind":"string","value":"KA-TP-31-2025, COMETA-2025-47 Complete two-loop Yukawa-induced running of the Higgs-gluon coupling in SMEFT Stefano Di Noi a , Barbara Anna Erdelyi b,c and Ramona Gr ̈ober b,c aInstitute for Theoretical Physics, Karlsruhe (KIT), Wolfgang-Gaede Straße 1, Karlsruhe D-76131, Germany bDipartimento di Fisica e Astronomia \"G. Galilei\", Universit`a degli Studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy cIstituto Nazionale di Fisica Nucleare, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy E-mail: , , Abstract: We compute the two-loop renormalisation group equation for the effective Higgs to gluon coupling in Standard Model Effective Field Theory. Concretely, we present the contributions generated by the operators belonging to class 3 and 7 in the Warsaw basis, completing the two-loop renormalization program of the Higgs-gluon coupling proportional to the top Yukawa coupling for potentially tree-level generated operators. We investigate the phenomenological impact of the contributions in fits to Higgs data both in a bottom-up approach and a top-down approach in terms of UV models with vector-like quarks. 16 Oct 2025 Contents 1 Introduction 1 2 Notation 3 3 Computation and result 4 4 Phenomenology 8 4.1 Fitting framework 10 4.2 Results 10 4.2.1 Bottom-up results 10 4.2.2 Top-down results 12 5 Conclusions 15 A Feynman Rules 17 1 Introduction There are several reasons to believe that the Standard Model (SM) of particle physics requires an extension. Despite this, no direct evidence of new physics has been observed so far, motivating the use of the Effective Field Theory (EFT) framework to describe potential beyond-the-SM (BSM) effects in the most model-independent way possible. This approach is valid under the assumption that the scale of new physics lies well above the electroweak scale. Assuming that the Higgs boson transforms as in the SM, namely as part of an SU(2)L doublet, writing down all possible higher-dimensional operators consistent with the symmetries of the SM leads to the so-called Standard Model EFT (SMEFT). Concerning collider physics, the most relevant contributions are expected to arise from dimension-six operators, for which a non-redundant basis was first established in Ref. [1], the so-called Warsaw basis. Despite not being the only possible EFT extension of the SM,1 the SMEFT has become a popular framework for global analyses [9-13], providing a self-consistent way to correlate various experimental measurements across different energy regimes. The connection between the Wilson Coefficients (WCs) at different energy scales is encoded in the renormalisation group equations (RGEs). The complete one-loop RGEs of dimension-six operators have been computed in [14-16] and implemented for systematic 1A different philosophy is embodied by the Higgs EFT [2-8] (HEFT or electroweak chiral Lagrangian), where the Goldstone bosons and the Higgs boson do not transform in the same multiplet, but the physical Higgs boson transforms as a singlet. - 1 - inclusion into phenomenological analyses in several computer codes [17-21]. Even though the full set of two-loop RGEs for the dimension-six SMEFT has not been computed yet, many partial results have been presented in the last years and progress towards the two-loop SMEFT RGEs has been reported [22-35]. The RGEs are thus crucial for global analyses that combine observables measured at different energy scales [36-38], exploiting fully the variety of available data from different experiments. Moreover, with the increasing precision of collider measurements, RGE effects can also become sizeable in processes where the renormalisation scale varies dynamically over a wide range of energies. In this context, including the RGE running of WCs is essential, as explicitly demonstrated in [39-43]. These works focus on the running proportional to the strong coupling constant αs, which is expected to give the dominant contributions. Moreover, restricting to these effects allows to solve the RGEs analytically, greatly improving the time efficiency of the computer programs employed for phenomenological studies. However, also the running effects generated by the top-quark Yukawa coupling Yt can produce significant effects [27, 44]. Despite the strength of the interaction being na ̈ıvely smaller than in the strong case, WCs entering the RGEs via Yukawa-induced contributions are generally less constrained [11], potentially enhancing their impact. In this work we complete the effort started in Refs. [24, 27] to determine the twoloop O g2 sY 2 t RGE of the Higgs-gluon coupling, which enters the dominant gluon-fusion production channel for Higgs production. Within the SMEFT framework, this coupling is parametrized by the operator OHG = (H†H)GA μνGA,μν where H denotes the Higgs doublet and GA μν the gluon field strength. We stress that this interaction generates a tree-level coupling between the Higgs bosons and the gluons, which instead is loop induced in the SM. In weakly interacting and renormalizable UV models, the corresponding WC CHG is loop-generated [45, 46]. Moreover, at one-loop level, the RGE of CHG involves only WCs of operators that are likewise loop-generated under the same assumptions. Consequently, OHG does not undergo one-loop renormalisation, even when considering dimension-eight effects [47]. It follows that, to ensure consistency in the perturbative expansion in the loop order, the two-loop running effects proportional to tree-level generated operators must be considered when including RGE effects for Higgs production. Contributions belonging to this category include the ones from four-quark operators [24] and Yukawa-like operators [27]. In this paper, we present the contributions from the operators with the schematic structure D2H4 and D H2 ψ2. Such operators belong to, respectively, class 3 and 7 in the classification of the Warsaw basis notation [1] and are the last missing contributions for the complete two-loop RGE proportional to the third family Yukawa couplings Yt and Yb involving operators that are potentially tree-level generated. Conversely, we do not consider chromomagnetic operators as they can only be generated at loop-level in weaklyinteracting theories [45, 46] and are expected to give a contribution at O(g2 sYt) that would be a three-loop effect. We investigate the phenomenological impact of the new contributions to the RGE running by performing a fit to Higgs data both in the bottom-up and top-down approach, using simplified models with heavy Vector-Like Quarks (VLQs). Such states naturally arise in a - 2 - variety of frameworks proposed to address open issues of the Standard Model, such as the hierarchy problem, with Composite Higgs Models providing a prominent example [48-50]. In these models, operators belonging to class 7 are generated at tree level, which lead to a modification of the couplings of quarks to massive vector, eventually in association with a Higgs boson. In particular, the couplings of top quarks to Z bosons are weakly constrained so far [37, 51], as a direct constraint comes from t ̄tZ production, single top production [52] or Zh production [53], whereas other probes need to rely on RGE effects. Other operators that couple to the third generation SU(2)L doublet can also be constrained by electroweak precision data. Conversely, class 3 operators can be constrained by electroweak precision data and by measurements of the Higgs coupling to massive vector bosons, currently available at few percent level [54, 55]. The paper is structured as follows. In Sec. 2 we present our notation. In Sec. 3 we describe our computation and present the result for the RGE of the Higgs-gluon coupling in the SMEFT. In Sec. 4 we study the phenomenological implications of the inclusion of two-loop running effects in single Higgs physics. Finally, in Sec. 5 we present our summary. 2 Notation The SMEFT extends the SM by a tower of gauge and Lorentz invariant higher-dimensional operators as a series in inverse powers of a high-energy new physics scale Λ. In this work, we neglect lepton and baryon number violation and focus exclusively on the leading dimensionsix terms in the expansion: L = LSM + X Di=6 CiOi. (2.1) The operators are denoted by Oi and the associated WCs by Ci. We note that the WCs in the previous formula have dimension -2 in units of energy. The operators in the previous expressions are invariant under the unbroken SM group GSM = SU(3)C × SU(2)L × U(1)Y. We follow the Warsaw basis of Ref. [1] for the dimension-six operators and Ref. [14] for the definitions of the SM Lagrangian. We consider only the third family of quarks, leading to a minimal modification of the notation with respect to the original publication. In particular, we use tR, bR to denote the right-handed component and QL to denote the left-handed component of the third family of quarks. The relevant class 7 operators are defined as: O(1) HQ = iH†← → DμH ̄QLγμQL , O(3) HQ = iH†← → DI μH ̄QLγμτ IQL , (2.2) OHt = i H†← → DμH ( ̄tRγμtR) , OHb = i H†← → DμH ̄bRγμbR , (2.3) OHtb = i ̃H†DμH ( ̄tRγμbR) , (2.4) where ̃Hi = εijH†,j (with ε12 = +1) and i H†← → DμH = iH†(DμH) -i(DμH)†H, (2.5) i H†← → DI μH = iH†τ I(DμH) -i(DμH)†τ IH. (2.6) - 3 - The operators in class 3 are defined as OH□= H†H □ H†H , OHD = H†DμH H†DμH ∗ . (2.7) These two operators in the broken phase modify the kinetic term of the Higgs boson, thus requiring a field redefinition to reobtain the canonical normalisation of the Higgs kinetic term. In the unitary gauge, the Higgs doublet H is related to the physical Higgs field h via H = 1 √ 2 0 v + h 1 + v2 CH, kin ! , where CH, kin = CH□-1 4CHD . (2.8) Finally, we recall that the Higgs-gluon coupling in the SMEFT is parametrized by the operator OHG = H†H GA μνGA μν . (2.9) This operator belongs to class 4 in the Warsaw basis. Other class 4 operators that will appear in the phenomenological study performed in this work are defined below OHW = H†H W I μνW I μν , OHB = H†H (BμνBμν) , (2.10) OHWB = H†τ IH W I μνBμν . (2.11) Furthermore, the following class 5 operators will be considered, often referred to as Yukawalike operators: OtH = H†H ̄QL ̃HtR , ObH = H†H ̄QLHbR . (2.12) We remind the reader that the SM Yukawa coupling is not defined in direct analogy to the Yukawa-like operator. All the previous operators are Hermitian with the exclusion of OHtb, OtH and ObH, which enforce the addition of their complex conjugate to the Lagrangian. We use T A for the SU(3)C generators and τ I for the Pauli matrices. We define the gluon field-strength tensor as GA μν = ∂μGA ν -∂νGA μ -gsfABCGB μ GC ν . 3 Computation and result We describe in this section the computation of the two-loop contributions of the operators in Eqs. (2.2)-(2.4) and Eq. (2.7) to the RGE of CHG. Practically, this requires the computation of the divergent parts of the two-loop corrections to a process where OHG, defined in Eq. (2.9), enters at tree-level. We perform our computation in the unbroken phase and we consider the process gg →H∗H. In dimensional regularization, the divergences are parametrized by poles in ε = 0, being D = 4 -2ε the number of space-time dimensions. We will discuss here in detail the calculation of the terms proportional to OHt. All the other operators of class 7 are obtained in analogy. At one-loop order, the contributions to gg →H∗H can only arise via triangle diagrams as in Fig. 1a. However, once all the relevant diagrams are considered their contribution vanishes [16]. The reason is that the - 4 - H H∗ g g (a) H H∗ g g (b) H H∗ g g (c) Figure 1: Sample of the diagrams with a single insertion of the operator OHt which vanish because of kinematics, see main text for details. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt and a black dot to denote a SM vertex insertion. g H H∗ g (a) g H H∗ g (b) g H H∗ g (c) Figure 2: Sample of the two-loop order diagrams with a single insertion of the operator OHt. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt, a black dot to denote a SM vertex insertion and a white star to denote the Yukawa coupling counterterm. g g H H∗ (a) g g H H∗ (b) Figure 3: Sample of the two-loop order diagrams with a single insertion of the operators OH□and OHD. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a white square to denote an insertion of OH□and OHD and a black dot to denote a SM vertex insertion. diagrams are purely s-channel, depending on the linear combination q1+q2 = p1+p2 where we have used p1,2 (q1,2) to denote the incoming (outgoing) momenta of the gluons (Higgs doublets). The scalar current gives rise to a contraction with (p1 -p2), yielding a vanishing result. Following this argument, we can safely ignore all the two-loop diagrams where the two external Higgs bosons are connected to the effective vertex. This applies both to pure QCD (see Fig. 1b) and to mixed Yukawa-QCD (see Fig. 1c) contributions.2 We have crosschecked the correctness of our reasoning by explicitly computing the O g4 s diagrams with 2As a consequence of this statement, the class 7 pure QCD contribution to the RGE of CHG vanishes at all loop orders at dimension-six level. - 5 - a QCD correction to the propagator connecting the two QCD vertices and verifying that their sum vanishes after a trivial change of variable in the momentum integration. Our argument allows to restrict our attention to a total of 24 diagrams containing an insertion of OHt, where one of the two Higgs doublets in the EFT vertex is external and the other is internal. A sample of the two families of diagrams is displayed in Figs. 2a and 2b. Moreover, class 7 operators correct the Yukawa coupling at one-loop [14], calling for the inclusion of the one-loop insertions of the one-loop Yukawa counterterms as in Fig. 2c. Class 3 operators modify the quadrilinear coupling, yielding 12 diagrams, of which we present a sample in Fig. 3. We have performed two independent computations of the class 7 operator contributions to check the set-up of our computation. The first one relies on FeynRules [56, 57] for the generation of the Feynman rules, where the model file is obtained by extending the SM model file in the unbroken phase provided in the additional material to MatchMakerEFT [58]. The resulting Feynman rules are presented in App. A. FeynArts [59] and FeynCalc [6063], interfaced via FeynHelpers [64], are used to generate the diagrams and to simplify the Dirac algebra. The numerical integration of the two-loop integrals is performed with AMFlow [65] supplemented with Blade [66]. The second computation relies on the same pipeline of Ref. [27]. The diagrams are generated with qgraf-3.6.10 [67] and simplified with FeynCalc. The numeric integration is performed with AMFlow supplemented with LiteRed2 [68, 69] and FiniteFlow [70]. Both computations use the Na ̈ıve Dimensional Regularization (NDR) [71, 72] scheme for the treatment of γ5 in a non-integer number of dimensions. This prescription is known to be algebraically inconsistent in presence of traces involving an odd number of γ5 in combination with at least six γ matrices as in the diagrams we compute. However, we are interested in the contribution to the RGE of CHG, meaning that we can ignore the contribution of the traces involving γ5, which would renormalize its CP-odd counterpart, namely ̃CHG. The Breitenlohner-Maison-t'Hooft-Veltman (BMHV) scheme [73, 74] has recently gained popularity as an alternative to the NDR scheme, being the only known continuation scheme proved to be algebraically consistent [75-79]. However, it highly increases the algebraic difficulty of loop computation. We have checked that, for the present computation, no difference arises between the two schemes by employing the map presented in Ref. [80]. This procedure is legit in this context since the leading divergence is of O (1/ε), as recently showcased in Ref. [33]. We note that, in principle, all the operators considered in this work can renormalize redundant operators, for which we follow the notation of Ref. [58]. This renormalization arises from subamplitudes with schematic structure ̄ψψ →H∗H, ψ = QL, tR, bR, which affect the operators R′ Hψ, R′′ Hψ,3 and from H → ̄QLχR, χ = b, t (possibly in association with one or two gluons), which impact the operators RuHDi, i = 1, 2, 3, 4. Since some of the external states in these subprocesses are not on-shell, the relevant one-loop insertions 3When ψ = QL, two possible isospin contractions are possible, generating a total of four redundant operators involving the left-handed quarks. - 6 - of redundant operators must be included in the total contribution to gg →H∗H. We have explicitly checked that these contributions vanish in the case where only OHt is present. Moreover, we do not need to consider contributions coming from one-particle reducible diagrams to OHG. This conclusion stems from the fact that no redundant operator is reduced to OHG when the SM equations of motions are used [1]. We obtain μdCHG dμ ⊃ g2 s (16π2)2 h + YbY ∗ b (CHb -C(1) HQ -3C(3) HQ) + YtY ∗ t (C(1) HQ -3C(3) HQ -CHt) + (CHtbYbY ∗ t + C∗ HtbY ∗ b Yt) - CH□-1 2CHD (YbY ∗ b + YtY ∗ t ) + 3 2 [CtHYt + C∗ tHY ∗ t + CbHYb + C∗ bHY ∗ b ] -4 δNDR YtY ∗ t C(1) Qt -1 6C(8) Qt + YbY ∗ b C(1) Qb -1 6C(8) Qb i , (3.1) δNDR = ( 1 (NDR) 0 (BMHV). (3.2) We use the same conventions as the one-loop results of [14-16]. The first two lines represent our novel result, while the contributions generated by four-quark operators and Yukawa-like operators (see Eq. (2.12)) have been first derived in Ref. [24] and in Ref. [27],4 respectively, which we point to for the precise definition of the operators. We stress that only the four-top contribution depends on the γ5 scheme, while the others do not. The final amplitude cannot exhibit infrared (IR) divergences, since no corresponding real-emission diagrams exist at the considered order in the coupling constants. However, IR divergences can still appear at intermediate stages of the calculation. As they must ultimately cancel, one has to verify that this cancellation does not occur against ultraviolet (UV) divergences, as can happen, for instance, in the case of scaleless integrals. We note that our result can be used to determine the analogous two-loop contributions from the operators in class 3 and 7 to OHW , OHB, OHWB (defined in Eqs. (2.10)-(2.11)), where the gluon field strength is replaced by the corresponding quantities involving electroweak bosons. However, this would represent just a partial result, since new diagrams arise as a consequence of the non-vanishing SU(2)L × U(1)Y charge of the Higgs doublet. Obtaining the RGEs for such operators is hence beyond the scope of this paper. The phenomenological relevance of these operators lies in their impact in the interactions involving photons or a photon and Z bosons and (at least) one Higgs boson and they lead to a shift in the electroweak input [81]. In the SM, such interactions are numerically dominated by W loops, so we expect the two-loop running contributions to be less numerically important than in the gluon case such that the phenomenological study we perform in the next section 4In the previous expression, the contribution from the operators OtH and ObH presents an extra 1/2 with respect to the result of Ref. [27], which will be updated. - 7 - remains valid omitting these sub-leading contributions. Furthermore, their contributions to the Higgs couplings to massive vector bosons is suppressed in the loop counting with respect to the tree-level contributions from class 3 operators. 4 Phenomenology In this section, we study the effect of the inclusion of two-loop running effects with fits to Higgs data at the LHC. In particular, we focus on the modifications to the signal strengths of a set of Higgs decay final states due to the presence of SMEFT operators. We perform two different fits. In first instance, we adopt a bottom-up perspective by performing the analysis in terms of the WCs to observe how their bounds are modified by the inclusion of the running. Secondly, we consider a top-down approach by selecting UV models with pairs of Vector-like Quarks (VLQs) coupled to the third-generation SM quarks, which generate, at tree-level, at least one of the class 7 operators considered in this paper. The WCs that enter the observables in this section are computed at the low scale μR = mh = 125 GeV. Their expression in terms of the WCs at the high energy scale Λ = 2 TeV can be obtained with the evolution matrix formalism [20] Ci(μR) = Uij(μR, Λ)Cj(Λ). (4.1) In the present case, the evolution matrix Uij(μR, Λ) has been obtained with a modified version of RGESolver [21] relying on a numeric solution of the RGEs. All the operators included in this section have been defined in Sec. 2. Cross-section The SM value for the Higgs boson production cross section in the dominant gluon-gluon fusion contribution is σSM = 48.68 pb at a center of mass energy of √s = 13 TeV. In the SMEFT, the cross-section reads σ = 48.68 + 2.83 · 104v2CHG + 97.36v2CH,kin -112.25v2 Yt CtH pb , (4.2) where CH,kin was defined in Eq. (2.8). We have obtained this parameteric dependence on the SMEFT WCs using the EFT implementation [41] in higlu [82]. Higgs total and partial decay widths The Higgs total decay width in the SMEFT at LO in the U(3)5 flavour-symmetric limit was computed in Ref. [83]. We adapt those results, noting that quark operators of class 7 contribute only via corrections to the gauge boson total width except for the cases with quarks in final states from Z boson decays. In this case there are some direct contributions where the Higgs boson and one Z boson couples via a class 7 operator to bottom quarks, and the Z boson decays subsequently. This results in Γh = ΓSM h 1 + 50.6 v2CHG -1.50 v2CHB -1.21 v2CHW + 1.21 v2CHWB + 1.83 v2CH□ -0.43 v2CHD -48.5 v2Yb CbH + 0.00016v2 C(1) Hq + 0.0015v2 C(3) Hq -0.00023v2 CHb +2 v2 BRh→γγ CH,kin + 0.53 Yt CtH + BRh→gg CH,kin -2.12 Yt CtH , (4.3) - 8 - using the {MW , MZ, GF } input scheme for the electroweak parameters. The last line encodes the higher-order rescaling of the Higgs decays into photon and gluon pairs through CH,kin defined in Eq. (2.8), together with the additional modifications induced by shifts in the top Yukawa coupling. In the UV models we consider, also the bottom Yukawa coupling can be modified but we do not consider its effect in the h →gg and h →γγ decays as it is negligible with respect to the top Yukawa contribution. We note already from this formula that generically our two-loop contributions can be of the same order than the direct contributions. Indeed, since the CHG has a large pre-factor, even the two-loop factor 1/(16π2)2 log(Λ/mh) can be compensated. We obtain the partial decay widths in the SMEFT rescaling the results in Ref. [83] to take into account the modifications of the decay into photon pairs and the top Yukawa coupling, analogously to the case of the total width discussed previously. We have validated these expressions to those reported in Ref. [84].5 For the decays into fermion pairs, introducing l ∈{μ-, τ -} , the partial decay widths are Γh→l ̄l ΓSM h→l ̄l = 1 + 2v2 CH,kin, (4.4) Γh→b ̄b ΓSM h→b ̄b = 1 + v2 2CH,kin -2 CbH . (4.5) For the decays into photon pairs we get Γh→γγ ΓSM h→γγ = 1 + v2 2 CH,kin + 0.53 Yt CtH -231 CHW -805 CHB + 431 CHWB . (4.6) As for decays into pairs of W and Z pairs, we further consider their decay products in order to compare with experimental results. In particular, Ref. [55], whose experimental results we will be using later, considers the processes h →W +W -→2l 2ν and h →ZZ → 4l. In order to get the partial decay width for these processes, we start again from Ref. [83] and focus on their results for partial decay widths of the Higgs boson into four-fermion final states mediated by \"only neutral\", \"only charged\" and \"neutral plus charged\" currents. In our scenario, the first case reduces to the 4l final state, while the second and third correspond to the 2l 2ν final state. The SM partial decay width for h →ZZ →4l is then ΓSM 4l = 1.1 keV, the partial decay width for h →W +W -→2l 2ν reads ΓSM 2l2ν = 90 keV , and the corresponding SMEFT expressions read Γ4l ΓSM 4l =1 + v2 (2CH□+ 0.27 CHD + 0.17 CHW -1.66 CHB -0.51 CHWB -0.27 C(1) HQ -0.27 C(3) HQ + 0.048 CHb , (4.7) Γ2l2ν ΓSM 2l2ν =1 + v2 (2CH□-0.52 CHD -1.5 CHW + 0.003 CHB -0.023 CHWB -0.0046 C(1) HQ -0.0046 C(3) HQ + 0.00081 CHb . (4.8) 5Ref. [84] uses the SILH basis [85], whereas Ref. [83] uses the Warsaw basis. To compare the results, we translate both sets of results into the Greens' basis by means of Ref. [86] and compare the terms of interest. - 9 - 4.1 Fitting framework Adopting the symbol Oα to identify an observable α, the χ2-function can be defined as χ2 = X α, β Oexp α -Oth α C-2 αβ Oexp β -Oth β , (4.9) where C-2 is the inverse of the covariance matrix and the superscript 'th' ('exp') means that the theoretical (experimental) value of the observable Oα is being considered. The theoretical value accounts for both the SM prediction and the deviations due to the insertion of SMEFT operators. Following Ref. [87], we construct a fit to Higgs data using the measurements of signal strengths with their covariance matrix as provided in Ref. [55]. Considering the Higgs decay channels h →α with α ∈{W +W -, ZZ, b ̄b, τ +τ -, μ+μ-, γγ},6 for each channel it is given by μα = σ × BRh→α σSM × BRSM h→α = σ σSM ΓSM h ΓSM h→α Γh→α Γh = σ σSM ΓSM h Γh Γh→α ΓSM h→α , (4.10) in which σ(SM) and BR(SM) denote, respectively, the value in the SMEFT (SM) of the Higgs production cross-section and the branching ratio into the specific channel. We consider up to order O(1/Λ2) in the signal strength. Of the three multiplied ratios in the final expression, the first is known from Eq. (4.2), the second from Eq. (4.3) and the third is selected from Eqs. (4.4)-(4.8) depending on the decay channel of interest. The allowed region satisifies the condition χ2 < χ2 min + ∆χ2 (n, CL) . (4.11) In the previous expression, n denotes the degrees of freedom (number of couplings in the top-down approach, number of involved WCs in the bottom-up approach) while CL stands for Confidence Level. The value χ2 min is the minimum of the χ2. 4.2 Results We now present our results. We notice that the operators OHtb and OHt do not enter directly any of the Eqs. (4.2)-(4.8), so they enter only via mixing effects with other operators. 4.2.1 Bottom-up results In this section we report our one- and two-parameter results for the bottom-up fits. We stress that these parameters are set to be non-zero at the high-scale Λ = 2 TeV, meaning that several other operators are generated via operator mixing, governed by Eq. (4.1). Hence, the bounds are reported for the operators at such scale. One parameter fits We present the individual 2σ CL interval for each WC considered at the energy scale Λ = 2 TeV in Fig. 4. We note that the intervals for the class 3 6For the decays into pairs of W and Z bosons, Ref. [55] considers the bosonic decay channels ZZ →4l and WW →lνlν. Thus, for the first two decay channels, we can use the decay widths of Eqs. (4.7) and (4.8) respectively. - 10 - -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Intervals CHD(Λ) CH□(Λ) CtH(Λ) CbH(Λ) CHt(Λ) · 10-1 CHb(Λ) · 10-1 C(1) HQ(Λ) C(3) HQ(Λ) CHtb(Λ) · 10-2 2σ 1L RGEs 2σ 2L RGEs Figure 4: Allowed ranges at 2σ for the WCs obtained by performing one parameter fits. The yellow (orange) line is obtained by considering one-loop (two-loop) running effects. coefficients are not sizeably affected by the inclusion of the two-loop effects in the RGEs. These operators are already constrained by direct contributions to the couplings with the Higgs bosons, reducing the impact of higher order corrections. Similar conclusions apply to the WCs of the Yukawa-like operators CtH and CbH, which enter directly in the Higgs production cross section and in several decay channels. As for the class 7 operators, CHb is not sizeably modified by the inclusion of the twoloop running effects, due to the bottom Yukawa suppression. Conversely, C(1) HQ and CHt show a moderate reduction when two-loop running effects are accounted, while for C(3) HQ a quite sizable reduction of the allowed range can be observed once two-loop running is included. Moreover, we note that the bounds for CHtb are looser at two-loop level, hinting for some cancellations happening with respect to one-loop running effects. Finally, we note that the bounds we present here are much weaker than the ones presented in Ref. [37]. Since our focus is on the impact of the two-loop running effects in the fit, we only accounted for inclusive Higgs data and not, for instance, for electroweak precision data or top data as Ref. [37]. Those measurements are though sensitive to the operators considered here. Two parameter fits In Fig. 5 we allow two WCs to be non-vanishing at the energy scale Λ = 2 TeV. We present the results of three two-parameter fits, comparing the impact of including only one-loop or two-loop running effects. In Fig. 5a, the two class 3 operators are considered. The correlation between the two arises from the fact that their contribution appears mostly in the linear combination CH, kin = CH□-1 4CHD. Moving on to Fig. 5b, we consider the two WCs C(1) HQ and C(3) HQ which, in motivated UV extensions of the SM, - 11 - (a) Class 3 operators (b) Class 7 operators with quark doublets (c) Class 7 operators with quark singlets Figure 5: Allowed regions at 2σ obtained via a two parameter fit including one-loop (in yellow) and two-loop (in orange) running effects. We consider in pairs the operators belonging to class 3 and to class 7. For class 7, we further separate between the cases in which the quarks in the operators are the doublets or the singlets under SU(2)L. appear together, as shown in Tab. 1. Finally, in Fig. 5c we compare CHt and CHb. In this case, there is a significant improvement in the bounds considering that CHt does not enter any of the observables defined in Eqs. (4.2)-(4.8), while CHb enters but is very suppressed compared to the other WCs. 4.2.2 Top-down results In this second approach we consider UV models featuring Vector-like Quarks (VLQs) as they generate at tree level both the operators in Eqs. (2.2)-(2.4) and the Yukawa-like - 12 - GSM O(1) HQ O(3) HQ OHt OHb OHtb U (3, 1) 2 3 ✓ ✓ Q1 (3, 2) 1 6 ✓ ✓ ✓ Q7 (3, 2) 7 6 ✓ T1 (3, 3)-1 3 ✓ ✓ T2 (3, 3) 2 3 ✓ ✓ Model Particle Content 1 U + Q1 2 U + Q7 3 Q1 + T1 4 Q1 + T2 5 Q7 + T2 Table 1: Class 7 operators generated at the tree-level by the VLQs considered in this work, along with the charges under the SM gauge group of the VLQs, with GSM = (SU(3)C, SU(2)L)U(1)Y (left). The presented VLQs will be used to construct models with pairs of these particles (right). operators OtH and ObH [88]. The operators OHG, OHD and OH□are instead generated at one-loop level. In a similar spirit to Ref. [87], we consider models with pairs of VLQs which give rise to unsuppressed contributions to the operator OtH. In Tab. 1 we list the VLQs considered, their charges under the SM gauge group GSM, the class 7 operators they generate at treelevel and how the five models considered in this work are defined. Given that we are interested in the RGE effects due to terms proportional to third generation Yukawa couplings, we make the simplifying assumption that the VLQs only couple to the third generation SM quarks. Furthermore, we set their masses to be Λ = 2 TeV. This value is in agreement with lower bounds on singly-produced VLQs coupled to third generation quarks [89, 90] and with the bounds from VLQ pair production [90-92]. The interaction Lagrangians for the considered models are given for a general flavour structure in Ref. [88] -L(int) M.1 = [λU]rp ̄Ur R ̃H†qp L + λu Q1 rp ̄Qr 1L ̃Hup R + h λd Q1 i rp ̄Qr 1LHdp R+ + [λUQ1]rp ̄Ur ̃H†Qp 1 + H.c. , (4.12) -L(int) M.2 = [λU]rp ̄Ur R ̃H†qp L + [λQ7]rp ̄Qr 7LHup R + [λUQ7]rp ̄UrH†Qp 7 + H.c. , (4.13) -L(int) M.3 = λu Q1 rp ̄Qr 1L ̃H up R + h λd Q1 i rp ̄Qr 1LH dp R + [λT1]rp 2 ̄T I r 1RH†τ I qp L+ + [λT1Q1]rp 2 ̄T I r 1 H†τ I Qp 1 + H.c. , (4.14) -L(int) M.4 = λu Q1 rp ̄Qr 1L ̃H up R + h λd Q1 i rp ̄Qr 1LH dp R + [λT2]rp 2 ̄T I r 2R ̃H†τ I qp L+ + [λT2Q1]rp 2 ̄T I r 2 ̃H†τ I Qp 1 + H.c. , (4.15) - 13 - (a) Model 2. (b) Model 3. (c) Model 4. (d) Model 5. Figure 6: Allowed regions at 2σ for pairs of couplings for the considered models. The orange (yellow) coloured areas correspond to allowed values derived accounting for two-loop (one-loop) running effects. -L(int) M.5 = [λQ7]rp ̄Qr 7LHup R + [λT2]rp 2 ̄T I r 2R ̃H†τ I qp L + [λT2Q7]rp 2 ̄T I r 2 H†τ IQp 7 + H.c. . (4.16) For these equations we follow the notation of Ref. [88] and use up R and dp R for right-handed quark singlets and qp L for the left-handed quark doublets, being p, r a flavour index. For our phenomenological study, we only consider third generation SM quarks, namely p = r = 3. For the tree-level expressions we refer to Ref. [88], whereas for the one-loop matching we use SOLD [93] and Matchete [94]. In all models and for all operators entering Eqs. (4.2)-(4.8) we match up to the one-loop level. - 14 - (a) Model 1 with complete one-loop matching (b) Model 1 using tree-level matching Figure 7: Two fits performed for Model 1, considering either the complete matching up to one loop for all the considered operators (left) or only the tree-level matching results (right). In both cases, the allowed parameter space at 2σ is plotted in yellow (orange) when the one-loop (two-loop) running effects are included. For each model the χ2 is a function of either three or four couplings in the UV model. In Fig. 6 and Fig. 7a we present our results as two-dimensional CL regions, setting the remaining parameters to their best fit value.7 As can be inferred in Figs. 6-7a, the inclusion of two-loop running in the RGEs makes only a minor difference. While the model at treelevel matches only into operators of class 7 and the Yukawa-type operators, at one-loop level it matches into the class 3 operators and OHG, OHW , OHB and OHWB, which enter with large coefficients into the signal strength. This reduces the impact of the two-loop running effects considered here. A greater impact is observed when only the tree-level matching results are considered. In order to showcase this aspect, in Fig. 7, we compare the complete fit performed for model 1 with a second fit in which only the tree-level generated operators are considered. 5 Conclusions We have analyzed the two-loop RGE of the effective Higgs-gluon coupling in the SMEFT, parametrized by the operator OHG. Several single and double Higgs production channels are directly sensitive to this coupling, that enters already at tree level whereas the SM signal arises at one loop. At one loop, CHG renormalizes itself and also mixes with the chromomagnetic operator. Adopting a loop counting in weakly coupled, renormalizable UV completions of the SM [45, 46], these operators are expected to arise only at the one-loop level, pushing their 7For some models, we have explicitly checked that this approximation reproduces the behavior of the profiled results. - 15 - RGE effects at two-loop order. Consequently, operators that can be generated at tree level and mix with CHG at two loop must also be consistently included under this counting. Our computation therefore extends the previous efforts for the computation of the twoloop contributions of the potentially tree-level generated four-top [24] and Yukawa-like operators [27], presenting for the first time the two-loop contributions of the operators with schematic structure D2H4 and D H2 ψ2. We addressed the phenomenological impact of the two-loop running in a fit to inclusive Higgs data. Indeed, we could find that bounds on the class 7 operators can shrink sizably due to the two-loop running effects, if their contribution is proportional to the top Yukawa coupling. In concrete models, such as the ones with VLQs presented in our phenomenological study, this effect is though reduced, as many more operators arise when matching up to one-loop order. For instance, OHG is generated at one-loop level in this type of models. Nevertheless, our results indicate that two-loop running effects can have a sizable impact on global fits, once completely available. This applies in particular for operators so far loosely constrained. For instance, in the case of four-top operators, it was shown that two-loop corrections to electroweak precision observables [32] can, in some cases, limit the currently allowed range [95]. In summary, our results complete an important piece of the two-loop RGE framework for Higgs-gluon interactions, setting the stage for consistent and precise SMEFT analyses of Higgs data with potential phenomenological impact. Note added During the completion this work, we became aware of [96]. Our results are in agreement. Acknowledgements We thank Javier Fuentes-Mart ́ın for cross-checking our results with [96] and the discussions that made us realise that we need to clearly state the definition of the SM Lagrangian that we adopt. BE would like to thank Jonathan Ronca for several useful discussions on numerical integration and Alejo N. Rossia for comments on marginalisation and profiling. Also, we would like to acknowledge the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 390831469) for its hospitality and its partial support during part of the completion of this work. This work received funding by the INFN Iniziativa Specifica APINE and by the 2023 STARS Grants@Unipd programme (Acronym and title of the project: HiggsPairs - Precise Theoretical Predictions for Higgs pair production at the LHC). This work was also partially supported by the Italian MUR Departments of Excellence grant 2023-2027 \"Quantum Frontiers\". We acknowledge support by the COST Action COMETA CA22130. CloudVeneto is acknowledged for the use of computing and storage facilities. The Feynman diagrams in this work were made using TikZ-Feynman [97]. - 16 - A Feynman Rules In this Appendix we present the Feynman rules associated with the effective operators defined in Eqs. (2.2)-(2.4). mj (Aj) identifies an index in the fundamental (adjoint) representation of SU(3)C whereas ij denotes an index in the fundamental representation of SU(2)L. The numbers are associated to the particles in the figure on the left. H3,∗ H4 g1 g2 = +4i CHGδi3i4δA1A2 (pμ2 1 pμ1 2 -gμ1μ2p1 · p2) . (A.1a) H1 H2, ∗ Q4 L Q3 L = +i C(1) Hqδi1i2δi3i4 + C(3) Hqτ I i2i1τ I i4i3 δm3m4 /p1 -/p2 PL, (A.1b) H1 H2, ∗ ψ3 R ψ4 R = +i CHψδi1i2δm3m4 /p1 -/p2 PR, ψ = t, b , (A.1c) H1 H2 d4 R u3 R = +i CHtbδm3m4εi1i2 /p1 -/p2 PR, (A.1d) H1 H2 H4,∗ H3,∗ = See Eq. (A.2). (A.1e) Given their length, we report here the Feynman rules for the last diagrams above: -i \" δi1i4δi2i3 CHD (p1 · p3 + p2 · p4) + CH□ X i p2 i + 2p1 · p4 + 2p2 · p3 !! + (4 ↔3) # . (A.2) For the sake of completeness we present the Feynman rules of the relevant SM interactions. We remind the reader that the for the Yukawa sector of the SM we adopt the same - 17 - notation of Refs. 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1
SkyDreamer: Interpretable End-to-End Vision-Based
Drone Racing with Model-Based Reinforcement
Learning
Aderik Verraest, Stavrow Bahnam, Robin Ferede, Guido de Croon, Christophe De Wagter
Abstract—Autonomous drone racing (ADR) systems have re-
cently achieved champion-level performance, yet remain highly
specific to drone racing. While end-to-end vision-based methods
promise broader applicability, no system to date simultane-
ously achieves full sim-to-real transfer, onboard execution, and
champion-level performance. In this work, we present Sky-
Dreamer, to the best of our knowledge, the first end-to-end
vision-based ADR policy that maps directly from pixel-level rep-
resentations to motor commands. SkyDreamer builds on informed
Dreamer, a model-based reinforcement learning approach where
the world model decodes to privileged information only available
during training. By extending this concept to end-to-end vision-
based ADR, the world model effectively functions as an implicit
state and parameter estimator, greatly improving interpretability.
SkyDreamer runs fully onboard without external aid, resolves
visual ambiguities by tracking progress using the state decoded
from the world model’s hidden state, and requires no extrinsic
camera calibration, enabling rapid deployment across different
drones without retraining. Real-world experiments show that
SkyDreamer achieves robust, high-speed flight, executing tight
maneuvers such as an inverted loop, a split-S and a ladder,
reaching speeds of up to 21 m/s and accelerations of up to 6 g.
It further demonstrates a non-trivial visual sim-to-real transfer
by operating on poor-quality segmentation masks, and exhibits
robustness to battery depletion by accurately estimating the
maximum attainable motor RPM and adjusting its flight path
in real-time. These results highlight SkyDreamer’s adaptability
to important aspects of the reality gap, bringing robustness while
still achieving extremely high-speed, agile flight.
I. INTRODUCTION
Recent advancements in autonomous drone racing (ADR)
have reached champion-level performance, even beating hu-
man world champions [1], [2]. In 2023, Kaufmann et al. [1]
became the first to defeat human champion drone racers with
autonomous drones. More recently, Bahnam et al. [2] achieved
the first champion-level victory in an externally organized
competition, relying solely on a low-cost monocular rolling
shutter camera for perception.
Despite these successes, current champion-level ADR sys-
tems [1], [2] remain modular pipelines of classical algorithms.
They are highly specialized and inflexible, relying on gate
corner detection and Perspective-n-Point (PnP) for state es-
timation. Their performance further depends on accurately
mapped track layouts, rectangular gates of known size, precise
camera calibration, and hand-tuned extended Kalman filters
(EKFs). Optimizing perception, state estimation, and control
The authors are with the Micro Air Vehicle Lab of the Faculty of Aerospace
Engineering, Delft University of Technology, 2629 HS Delft, The Netherlands
Figure 1: Real-world inverted loop track with MAVLab gates.
SkyDreamer flies agile maneuvers such as an inverted loop in
the real world without external aid (top), operates end-to-end
on poor-quality segmentation masks obtained from onboard
images (shown in red, bottom-right), and directly controls the
physical platform by outputting low-level motor commands.
In a flight area confined to 6 × 6 m, it reaches speeds of up
to 13 m/s and accelerations of up to 6 g. The racing drone is
identical to the one used in the A2RL x DCL Autonomous
Drone Championship 2025 [2] (bottom-left).
in isolation further limits overall performance. Such pipelines
cannot generalize beyond the highly structured settings they
were designed for. In contrast, humans fly with remarkable
adaptability – handling unseen tracks, different drones, and
even unstructured environments with minimal task-specific
training [3]. This gap underscores the inherent limitations in
today’s champion-level ADR systems.
To address these limitations, research has turned toward
end-to-end vision-based learning, where policies map pixel-
level representations directly to actions without explicit state
estimation [4]–[7]. Such learned representations offer greater
flexibility, potentially enhancing performance by jointly op-
timizing perception, state estimation, and control. Crucially,
arXiv:2510.14783v1 [cs.RO] 16 Oct 2025
2
Table I: Comparison of recent end-to-end vision-based ADR approaches with SkyDreamer. “Offboard” indicates that some
computation occurs offboard, while “onboard” means all computation runs onboard with no external aid during flight. “HIL”
denotes that images for the policy during real-world flight are rendered using MoCap. “Visual ambiguity” refers to situations
where visually similar observations require different flight paths, such as in the ladder inverted loop track shown in Figure 4.
The perception reward encourages the drone to look toward gates, introducing human bias. Only performance metrics from
real-world experiments are considered in this comparison.
Criterion
Geles et al. [4]
Xing et al. [5]
Romero et al. [6]
Krinner et al. [7]
SkyDreamer (ours)
Onboard
computation
Offboard
Offboard
Offboard
Offboard
Onboard
without external aid
Champion-level
No (2 g)
No (2.7 TWR) 1
No (2.7 TWR)
No
Likely (6 g)
Sim-to-real
transfer
Yes
HIL (only dynamics)
HIL (only dynamics)
HIL (only dynamics)
Yes, non-trivial
visual sim-to-real gap
Interpretability
None
None
Limited, latent →
reconstructed image
Yes, via decoding to
states
Yes, via decoding to
states and parameters
Robust to
visual ambiguity
Not demonstrated
Not demonstrated
Not demonstrated
Not demonstrated
Yes, via progress tracking
using world model
Perception
reward
Yes
Yes
No, emergent behavior
No, emergent behavior
No, emergent behavior
leveraging pixel-level inputs, such as depth maps, can general-
ize to unstructured environments while introducing a smaller
reality gap compared to raw RGB images, as demonstrated
in [8]–[11]. At the same time, end-to-end learning reduces
reliance on human-designed abstractions and reward shaping,
which can constrain system performance.
While these benefits are promising, practical implementa-
tions remain challenging. Only Geles et al. [4] manage to
bridge the visual reality gap, whereas Xing et al. [5], Romero
et al. [6] and Krinner et al. [7] still rely on hardware-in-the-
loop (HIL) simulation, where images during real-world flights
are rendered using an external motion capture (MoCap) system
rather than the real camera feed. Furthermore, as summarized
in Table I, none of these approaches perform all computa-
tion fully onboard, nor do they demonstrate champion-level
performance in the real world, as the thrust-to-weight ratio
(TWR) was deliberately software-capped. To date, no end-to-
end vision-based method simultaneously achieves full sim-to-
real transfer, onboard execution, and champion-level flight.
An additional, and often overlooked, constraint is that most
ADR approaches treat collective thrust and body rates (CTBR)
as the final control layer, relying on an inner-loop controller
such as PID or INDI at deployment to track these CTBR
actions with motor commands [1], [4]–[7]. As an alternative to
this two-stage control structure, the European Space Agency
(ESA) introduced guidance and control networks (G&CNETs),
which map state inputs directly to actuator commands [12].
For quadcopter control, this translates to generating motor
commands directly rather than relying on intermediate CTBR
actions. Doing so is required for several space applications
and simplifies modeling, as the motor response is generally
easier to capture than the complex behavior of an inner-loop
controller. Additionally, direct motor control may also enable
more optimal performance compared to approaches that rely
on inner-loop controllers.
Building on this concept, Ferede et al. [13] trained
1Xing et al. [5] did perform limited experiments in simulation demonstrat-
ing champion-level performance comparable to [1], but failed to demonstrate
such performance in the real world.
G&CNETs using reinforcement learning (RL), specifically
proximal policy optimization (PPO) [14], for ADR, demon-
strating that fully neural control policies can outperform
traditional cascaded controllers when trained under compa-
rable conditions [13], [15], [16]. The use of G&CNETs for
quadcopter control has already proven successful in ADR com-
petitions: during the A2RL x DCL Autonomous Drone Cham-
pionship in 2025, a neural end-to-end state-based controller
outperformed human world-champion pilots, underscoring its
real-world potential [2].
A promising approach to training end-to-end vision-based
ADR policies is model-based RL. Unlike model-free RL
methods, such as PPO [14] and soft actor-critic (SAC) [17],
model-based RL learns an internal model of the environment,
often referred to as a world model [18]. The core idea is to
capture environment dynamics by learning to predict future
observations – or their latent representations – based on past
observations and actions using the world model. Actor training
is thus performed entirely in latent space, where the actor
interacts with the learned world model rather than the phys-
ical or simulated environment. This is accomplished through
latent imagination, in which future latent states are predicted
from previous latent states and actions. Consequently, sample
efficiency is greatly improved: physical or simulated rollouts
are required only to train the world model, not the actor.
This results in orders-of-magnitude gains in sample efficiency
compared to model-free RL, which is particularly important
for end-to-end vision-based policies, where rendering observa-
tions at every timestep can be computationally expensive [18],
[19].
In model-based RL, the Dreamer algorithms introduced by
Hafner et al. have become among the most widely adopted
[19]–[23]. Notably, Wu et al. [23] demonstrated that a robot
could be trained to walk using only one hour of real-world
interactions – without relying on simulation – highlighting
the remarkable sample efficiency of model-based RL. More
recently, DreamerV3 has emerged as a particularly promising
variant, achieving state-of-the-art performance across more
than 150 diverse environments using a single set of hy-
3
perparameters [19]. Romero et al. [6] have already applied
DreamerV3 to end-to-end vision-based ADR, though with
limited success both in terms of performance and sim-to-real
transfer.
To further aid learning efficiency, the use of privileged
information during training has shown great promise [4], [5].
In model-based RL, this idea can be extended by allowing
the world model to decode privileged observations, such
as ground-truth states, an idea already explored for end-to-
end vision-based ADR in Krinner et al. [7]. This has been
shown to yield significant performance gains, particularly in
tasks where such states are recoverable from sequences of
observations [24], [25], as in vision-based ADR. Furthermore,
privileged reconstructions enable the world model to act as an
implicit state estimator, improving interpretability, unlike prior
approaches [4]–[6] that largely operate as black boxes.
Alternative model-based approaches such as TD-MPC2
combine latent imagination with model predictive control
(MPC) [26]. By relying on MPC rather than RL, TD-MPC2
outperforms DreamerV3 on several continuous control bench-
marks [26]. However, unlike Dreamer, TD-MPC2 lacks a
recurrent structure in latent space and is therefore unable
to model long-range temporal dependencies. Its reliance on
iterative optimization at runtime further limits its suitability
for onboard deployment. Other latent-space MPC approaches
include [27] and [28]. Notably, Zhou et al. [28] demonstrated
zero-shot planning with a pretrained billion-parameter world
model. While impressive, the scale of such models makes
deployment on resource-constrained platforms like drones
impractical.
Contributions
In this work, we present SkyDreamer, to the best of our
knowledge the first end-to-end vision-based ADR policy that
maps directly from pixels to motor commands, eliminating
the need for inner-loop controllers in end-to-end vision-based
drone racing. Beyond this main contribution, our work offers
several further advances, summarized below and in Table I:
• SkyDreamer greatly improves interpretability by decoding
privileged information, effectively turning the world model
into an implicit state and parameter estimator,
• is, to the best of our knowledge, the first end-to-end vision-
based ADR policy to reach accelerations of up to 666 g and
speeds of up to 21
21
21 m/s in the real world, on par with state-
of-the-art classical approaches [1], [2],
• demonstrates non-trivial visual sim-to-real transfer using
poor-quality segmentation masks,
• resolves visual ambiguity by explicitly tracking progress
through the state decoded from the world model’s hidden
state, laying the foundation for end-to-end vision-based
policies capable of flying arbitrary tracks,
• requires no accelerometer,
• runs fully onboard without external aid,
• estimates drone-specific parameters, such as camera
extrinsics, on the fly and operates on images standardized
via intrinsic calibration, enabling rapid deployment across
different drones without retraining.
II. METHODOLOGY
A. Problem statement
We aim to develop an end-to-end vision-based policy capa-
ble of flying a quadcopter at high speed through a known track
using only onboard computation and sensing. Specifically, the
goal is to maximize the quadcopter’s lap speed around a pre-
mapped track marked by gates, subject to the constraints of the
physical platform and the requirement to pass through all gates
in a predefined order while avoiding collisions. The policy
receives only raw sensory inputs: pixel-level representations,
body rate measurements, and motor RPM measurements.
Additionally, the drone’s camera extrinsic parameters are as-
sumed to be uncalibrated, as extrinsic calibration is both time-
consuming and must often be repeated during operation due to
changes in the camera angle from touching, crashing, or hitting
gates, making it impractical for real-world competitions. The
policy must therefore learn to handle this uncertainty. It shall
map these observations directly to motor commands, without
relying on an intermediate inner-loop controller such as PID
or INDI. During training only, the policy is granted access to
privileged information, including ground-truth states, camera
parameters, and dynamic parameters of the system.
We formalize this problem as an informed partially observ-
able Markov decision process (informed POMDP) [24], which
extends the standard POMDP [29]. The informed POMDP
was first introduced by Lambrechts et al. [24], who provide
a detailed formulation. The key distinction with a standard
POMDP is that additional, privileged information is available
during training but not at execution time. This additional
information can be exploited to improve the training process.
Formally, the informed POMDP is defined as:
eP = (S, U, I, O, T, R, eI, e
O, P, γ)
with the following components:
• state space S with states st
• action space U with actions ut
• information space I with information it only available
during training
• observation space O with observations ot
• transition distribution T(st+1 | st, ut)
• reward function rt = R(st, ut), which is designed to
reflect the task objective and is detailed further in subsec-
tion II-C.
• information distribution: eI(it | st)
• observation distribution: e
O(ot | it) available during train-
ing and execution
• initial state distribution: P(s0), which is detailed further
in Table III
• discount factor: γ ∈[0, 1)
In an Informed POMDP, the states st are never observable.
At training time, the informed POMDP does provide the policy
access to a history h+
t :
h+
t = (i0, o0, u0, r0, ..., it−1, ot−1, ut−1, rt−1, it, ot)
consisting of privileged information, observations, actions and
rewards. The observations ot are assumed to be conditionally
dependent on the states st, following the Bayesian network
4
st →it →ot. To satisfy this assumption, and following
Lambrechts et al. [24], we define the information variable as
it = [ot, o+
t ]T , a concatenation of the observations ot and
privileged observations o+
t only available during training (e.g.
position p or velocity v). At execution time, we define the
underlying execution POMDP eP as:
eP = (S, U, O, T, R, e
O, P, γ)
which provides access to a history ht:
ht = (o0, u0, ..., ot−1, ut−1, ot)
consisting of only observations and actions [6], [24].
The objective in an informed POMDP is to maximize
the expected return in the underlying execution POMDP by
learning a policy πθθθ(ut|ht), in our case parameterized by
neural network parameters θθθ. The resulting optimal policy π∗
θ
is defined by [24]:
π∗
θ = argmax
πθ
E
s0∼P (·)
ut∼πθ(·|ht)
st+1∼T (·|st,at)
ot∼e
O(·|st)
" ∞
X
t=0
γtR(st, at)
#
The optimal policy thus aims to maximize the expected return,
i.e., the cumulative sum of discounted rewards, in the execu-
tion POMDP through interaction with the informed POMDP at
training time. Although privileged information is used during
learning, the resulting optimal policy remains independent of
it at execution time.
B. Spaces
xw
yw
zw
xb
yb
zb
xc
yc
zc
xg
yg
zg
xw
yw
zw
Body
Camera
Gate
World
Figure 2: Coordinate systems. A quadcopter pitched forward
– shown as a cross with four circles – with a camera pitched
upward, is shown flying toward the gate at the bottom left.
Subscripts indicate the reference frames: w - world frame, g -
gate frame, b - body frame, and c - camera frame. Image not
to scale, the axes of the frames are not necessarily aligned.
The state space S consists of states s which are defined as:
s = [pw, pg, vw, vg,λλλ,ΩΩΩ,ωωω, ce, d,εεε, f]⊤
where
the
associated
coordinate
systems
all
follow
a
north–east–down (NED) convention illustrated in Figure 2, and
the components of s are described below:
• position:
– pw = [xw, yw, zw]⊤: position in world frame
– pg = [xg, yg, zg]⊤: position in gate frame, updated upon
passing each gate. The gate frame is a local coordinate
system attached to the next gate, which is updated each
time a gate is passed and is also shown in Figure 2
• velocity:
– vw = [vx,w, vy,w, vz,w]⊤: velocity in world frame
– vg = [vx,g, vy,g, vz,g]⊤: velocity in gate frame
• orientation λλλ = [ϕw, θw, ψw, ψg]⊤in ψ →θ →ϕ order
– ϕw, θw: roll and pitch in world frame
– ψw: yaw in world frame
– ψg: yaw in gate frame
• body rates ΩΩΩ= [p, q, r]T are the roll, pitch and yaw rate
in body frame respectively
• propeller speeds ωωω = [ω1, ω2, ω3, ω4]T are the angular
velocities of the four propellers
• camera extrinsics ce = [ϕc, θc, ψc] describes the euler
rotation from body to camera frame in ψ →θ →ϕ order
• dynamic parameters d are randomized dynamical param-
eters of the drone described in subsection II-D
• disturbances εεε = [εεεa,εεεM,εεεu]⊤
– εεεa = [εa,x, εa,y, εa,z]⊤: acceleration disturbances
– εεεM = [εM,x, εM,y, εM,z]⊤: moment disturbances
– εεεu = [εu,1, εu,1, εu,3, εu,4]: action disturbances
• flight plan vector f indicating the relative position between
gates, detailed in subsection II-H.
The
privileged
observations
are
given
by
o+
t
=
[pw, pg, vw, vg,λλλ,ΩΩΩ,ωωω, ce, d]T , where it should be noted that
ωωω and ΩΩΩrepresent the ground-truth rates, not the measured
ones. The observations are ot = [X, ˆΩΩΩ, ˆωωω]T , where X ∈
{0, 1}H×W is a binary segmentation mask of the gates, gener-
ated by the segmentation model described in subsection II-E, ˆΩΩΩ
are the body rates measured by the inertial measurement unit
(IMU), and ˆωωω are the propeller angular velocities measured
by the electronic speed controller (ESC). We do not use the
accelerometer, motivated by the observation of Bahnam et al.
[2] that it tends to saturate on the used physical platform under
high accelerations and vibrations. We define the information
variable as it = {ot, o+
t }\{X}, where the binary segmentation
mask is excluded since its information is fully captured in o+
t ,
and thus still satisfies the conditional dependence of ooot on ssst.
Moreover, including it would significantly slow down training.
Lastly, the actions are given by ut = [u1, u2, u3, u4]⊤,
which represent the normalized motor commands for each
rotor, where ui ∈[0, 1]. These commands directly specify the
desired motor power fraction, which the ESC regulates with
pulse-width modulation (PWM), without any intermediate
inner-loop controller, providing maximal control authority over
the physical platform for high-speed, agile flight.
5
C. Reward function
In RL, the reward function is a key component, hand-
designed to reflect the task objective. It encodes both the
goals and constraints of the task, assigning positive rewards for
desirable behavior and penalizing undesirable actions. Similar
to Ferede et al. [16], our reward function consists of a progress
reward, rate penalty and gate reward, and is given by:
rt = 5 · rprog −rrate + 30 · rgate
rrate =
1
2 · fc · 105 (exp (min (∥Ωt∥1, 17)) −1)
rprog = ∥pt−1,g∥2 −∥pt,g∥2
rgate =
1 −max (|yt,g|, |zt,g|)
dg
,
if gate
passed
0,
otherwise
where fc = 90 Hz is the control frequency of the policy, and
rrate is a rate penalty designed to discourage excessive angular
velocities, which helps prevent both gyroscope saturation
and excessive motion blur in the onboard camera. The term
rgate provides an additional reward upon successfully passing
through a gate, where dg is the effective size of the gate,
with the reward being maximal at the center of the gate and
decreasing linearly to zero toward its edges.
To account for the thickness of the gate and the size of the
drone, we introduce a pre-gate and post-gate, which are non-
rendered virtual gates located at a specified distance before
and after the gate, respectively. These gates provide the same
reward rgate and collectively act as collision volumes with
shaped rewards, spanning a total thickness tg.
The term rprog is a progress reward that encourages the
drone to move toward the next pre-gate, serving as a dense
shaping signal to accelerate training. Importantly, this term
does not enforce a specific trajectory: any path toward the
next gate results in the same accumulated return. For ease of
implementation, the progress reward is set to zero between the
pre- and post-gate.
Crucially, unlike Geles et al. [4] and Xing et al. [5] but
similar to Romero et al. [6] and Krinner et al. [7], there is
no explicit perception reward. We observe that the resulting
policy naturally learns to orient its camera toward the gates
to maximize gate visibility. Additionally, unlike Ferede et al.
[16], we do not include a penalty for gate collisions. In our
experiments, such penalties were found to hinder the learning
process and provided no benefit compared to the natural
truncation of the discounted return caused by an episode
termination on collision.
Finally, an episode is terminated if any of the following
conditions are met:
Gate collision:
( max(|yt,g|, |zt,g|)
dg
> 1,
if gate
passed
false,
otherwise
Ground collision:
zw > −0.5 ∧
vz,w > 1.0
∨|ϕw| > π
3 ∨|θw| > π
3
where zw = 0 denotes the ground plane. Episodes terminating
due to either condition result in a final reward of zero. The
gate collision condition also applies to pre- and post-gates. The
ground collision condition is shaped to discourage fast flight
with high pitch or roll near the ground. This is discouraged
as the drone starts from a raised podium in the real world,
and such contact forces are not simulated. Using this shaped
collision condition, the drone learns to ascend first before
accelerating forward, rather than immediately flying toward
the gate.
D. Quadcopter dynamics
Since the policy is trained in simulation, accurate modeling
of the quadcopter dynamics is crucial, particularly because
motor commands are executed directly. Our dynamics model
largely builds on the frameworks from Bahnam et al. [2] and
Ferede et al. [16]; for full details, we refer the reader to
those sources. Following the approach of Bahnam et al. [2],
we represent the rotation described by λλλ using quaternions
q
=
[qw, qx, qy, qz]T and incorporate gyroscopic effects.
Furthermore, we introduce additional disturbances εεε:
˙xw = vw
˙q = 1
2q ⊗[0 p q r]T
˙vi = ge3 + R(λλλ)F + εεεa
˙ΩΩΩ= M + εεεM
ωi = ωci −ωi
τ
where R is the rotation matrix and g is the gravitational
acceleration constant. The forces F, moments M and steady
motor response wci are given by:
F =
−kxvB
x
P4
i=1 ωi −kx2vB
x |vB
x |
−kyvB
y
P4
i=1 ωi −ky2vB
y |vB
y |
−kω (1 + kαα + khorµxx,yy) P4
i=1 ω2
i −kv2vB
z |vB
z |
with
α = tan−1 vB
z
r¯ω
µxx,yy = tan−1 vB2
x
+ vB2
y
r¯ω
with
¯ω =
4
X
i=1
ωi
M
M
M =
−kp1ω2
1 −kp2ω2
2 + kp3ω2
3 + kp4ω2
4 + Jxqr
−kq1ω2
1 + kq2ω2
2 −kq3ω2
3 + kq4ω2
4 + Jypr
−kr1ω1 + kr2ω2 + kr3ω3 −kr4ω4−
kr5 ˙ω1 + kr6 ˙ω2 + kr7 ˙ω3 −kr8 ˙ω4 + Jzpq
ωci = (ωmax −ωmin)
q
k ˜ui
2 + (1 −kl) ˜ui + ωmin
with
˜ui = min (max (ui + εu,i, 0) , 1)
where vx,b, vy,b and vz,b are the velocity components in body
frame.
The values of the parameters used for both policy training
and simulation experiments are given in Table II. These pa-
rameters together make up the previously mentioned dynamics
6
parameters d = [kw, kx, ky, . . . , ωmax, k, τ]T . The simulation
is implemented using JAX in Python, and is discretized
using fourth-order Runge–Kutta integration with a timestep
of 2.2 ms.
Table II: Quadcopter dynamical parameters used for both
policy training and simulation experiments.
Param
Value
Param
Value
kw
1.55 × 10−6
kx2
4.10 × 10−3
kx, ky
5.37 × 10−5
ky2
1.51 × 10−2
kangle
3.145
khor
7.245
kv2
0.00
Jx
−0.89
Jy
0.96
Jz
−0.34
ωmin
341.75
ωmax
3100.00
k
0.50
τ
0.03
kp1
4.99 × 10−5
kp2
3.78 × 10−5
kp3
4.82 × 10−5
kp4
3.83 × 10−5
kq1
2.05 × 10−5
kq2
2.46 × 10−5
kq3
2.02 × 10−5
kq4
2.57 × 10−5
kr1–kr4
3.38 × 10−3
kr5–kr8
3.24 × 10−4
E. Segmentation masks
As pixel level input, SkyDreamer receives a segmentation
mask X. Unlike the extrinsic parameters, we choose to cal-
ibrate the intrinsic parameters of the camera to maintain a
consistent camera matrix for SkyDreamer’s inputs, as they are
straightforward to obtain and remain constant over time. Using
these calibrated intrinsics, the RGB images are undistorted,
cropped, and mapped to a fixed-resolution image of size
H × W, such that the resulting image satisfies a nominal
pinhole camera model with intrinsic matrix:
K =
fx
0
cx
0
fy
cy
0
0
1
=
25
64W
0
0.5W
0
25
64H
0.5H
0
0
1
,
where fx and fy are the focal lengths in pixels and cx, cy
are the principal point coordinates in pixels. The factor
25
64
is empirically chosen such that an image of 64 × 64 pixels
corresponds to a focal length of 25, providing a balanced field
of view that maximizes scene coverage while avoiding black
borders in the undistorted images. By mapping all images to
a shared nominal intrinsic matrix and randomizing extrinsic
parameters during training, SkyDreamer ensures robustness
to extrinsic variations and invariance to intrinsic parameters
through calibration, enabling generalization of the same model
across different drones.
Subsequently, the fixed-resolution image is passed to a
segmentation model called GateNet that produces a binary
segmentation mask X ∈{0, 1}H×W , indicating which pixels
correspond to gates. GateNet follows a U-Net architecture,
similar to the designs used in De Wagter et al. [30] and
Bahnam et al. [2] of which the implementation details are
provided in Appendix A. The resulting masks are resized to
a resolution of 64 × 64 to speed up the training process of
SkyDreamer.
F. Visual augmentations
During training, segmentation masks are rendered in batches
using PyTorch3D. However, these masks are idealized and dif-
fer significantly from real-world masks, causing a large visual
sim-to-real gap. To bridge this gap, we employ CycleGAN
[31], a generative adversarial network (GAN) that learns to
translate images between two unpaired domains, in our case,
from synthetic segmentation masks to more realistic masks and
vice versa, without requiring paired real and rendered masks.
To further improve realism, we use a stochastic CycleGAN
(StochGAN) [32], an extension of CycleGAN that adds noise
to the inputs. This stochasticity enhances the variation and
realism of generated masks, better approximating real-world
appearance and variance. Although Almahairi et al. [32] also
proposes a more advanced AugmentedGAN, we opted for
StochGAN due to its simpler implementation and effective
performance. The GAN is trained on approximately 4000
unpaired real and rendered images, which require no manual
labeling and are therefore inexpensive to collect. Further
implementation details are provided in Appendix B.
We observed that real-life masks were often thinner, a
detail not captured well by the GAN alone. To address this,
we randomly erode the masks by 1 pixel – using average
pooling followed by thresholding – 50% of the time at a
mask resolution of 64 × 64, introducing variation in the mask
thickness. Since the erosion level typically remains stable
over short periods, we hold it fixed for an average of 100
environment steps.
Lastly, to simulate the rolling shutter effect of the onboard
camera, we account for two dominant effects: a horizontal
shear induced by the yaw rate in camera frame, and a vertical
scaling induced by the pitch rate. These arise from the line-
by-line capture of a rolling shutter camera under rotational
motion. To model these effects, we introduce a rolling shutter
parameter s ∈[0, 0.02], which also aims to capture additional
disturbances associated with rolling shutter through random-
ization.
We formalize our approximation of the rolling shutter effect
with an affine transformation given by the following matrix:
A =
1
−s rc
W
2 s rc
0
1 + s qc
−H
2 s qc
,
where rc and qc denote the yaw and pitch rates in the camera
frame, respectively. This transformation is then applied to the
rendered segmentation mask:
X′ = warp affine(X, A)
which produces a horizontally sheared and vertically scaled
segmentation mask.
G. SkyDreamer architecture
SkyDreamer’s architecture is almost entirely based on In-
formed Dreamer [24], which is largely based on DreamerV3
[19]. In this section, we provide a high level overview and
intuitive explanation of the architecture most relevant to our
work. For detailed information on the architecture, policy
7
(a) Standard DreamerV3 architecture for ADR. In standard
DreamerV3, the world model is not guided by privileged
observations o+
t .
(b) SkyDreamer at deployment. The dynamics predictor is not
used, as observations are always available. The flight plan vector
is updated based on the decoded states as indicated by the cyan
arrow.
(c) SkyDreamer during world model learning. The dynamics
predictor is trained to match the encoded inputs. Additionally, the
sequence model, encoder, decoder, reward predictor, and continue
predictor are trained.
(d) SkyDreamer during actor-critic learning. The dynamics
predictor replaces the encoder to generate imagined rollouts.
Figure 3: Comparison of standard DreamerV3 and SkyDreamer across different stages. The world model encodes segmentation
masks and measurements such as motor RPMs, body rates, and the flight plan vector into discrete representations zt. The next
hidden state ht+1 is predicted from the previous hidden state ht, action at and discrete representation zt using the sequence
model. The inputs ˆxt are reconstructed to guide the hidden representations. In SkyDreamer, the privileged observations, including
ground-truth state, camera and dynamical parameters, are decoded as o+
t . During actor-critic learning, imagined rollouts are
generated solely using the dynamics predictor without feedback from the environment. Images inspired by Hafner et al. [19].
learning and loss functions, we refer the reader to Hafner et
al. [19] and Lambrechts et al. [24].
The core component of DreamerV3 is its world model,
which encodes observations into a latent representation and
predicts future latent states based on actions, both with and
without receiving further observations. The training process
continuously alternates between three main stages: (1) data
collection, (2) world model learning, and (3) actor-critic
learning, which are illustrated in Figure 3. In the data col-
lection phase, the current policy interacts with the environ-
ment to collect rollouts, which are stored in a replay buffer,
making DreamerV3 an off-policy algorithm. During world
model learning, batches of trajectories are uniformly sampled
from the replay buffer to train the world model. Finally,
in the actor-critic phase, learning occurs entirely through
latent imagination: the world model simulates rollouts without
environment feedback, enabling sample efficient model-based
RL. All components of SkyDreamer are implemented as multi-
layer perceptrons (MLPs), unless otherwise noted.
The key element of the world model in DreamerV3 is a
recurrent state-space model (RSSM), which consists of the
following components:
8
Encoder:
zt ∼qe
θ(·|ht, ot)
Sequence model:
ht = f e
θ (ht−1, zt−1, ut−1)
Dynamics predictor:
ˆzt ∼pd
θ(·|ht)
Here, qe
θ is the encoder, which encodes the observation
ot along with the hidden state ht into a discrete stochastic
latent representation zt. This structure resembles a vector-
quantized variational autoencoder (VQ-VAE) [33]. For image
observations, it uses a convolutional neural network (CNN)
rather than an MLP. The function f e
θ is the sequence model,
implemented as a single-layer gated recurrent unit (GRU) [34],
which predicts the next hidden state ht given the previous
hidden state ht−1, previous latent state zt−1, and previous
action ut−1. Finally, pd
θ is the dynamics predictor, which
predicts an estimate of the latent representation ˆzt based solely
on the hidden state ht. This component is crucial, as it allows
the model to roll out future trajectories in latent space without
further observations from the environment, which is often
referred to as latent imagination or dreaming and is illustrated
in Figure 3d.
The RSSM also employs a decoder to guide the learning
process. Crucially, following informed Dreamer [24], Sky-
Dreamer decodes to the privileged information it rather than
only the raw observations ot:
Decoder:
it ∼pi
t(·|ht, zt)
By decoding to this privileged information, the world model
is encouraged to extract more meaningful and task-relevant
features from the inputs. This key difference from standard
DreamerV3 is visually evident when comparing Figure 3a
and Figure 3c. In addition to improving learning speed, final
performance and adaptability to drone-specific parameters, this
also results in a more interpretable world model. Whereas
decoding to images often leads to blurry reconstructions,
decoding to states such as position, velocity, and attitude
allows a direct inspection of what the world model believes
the current state to be.
Additionally, this interpretability offers a significant practi-
cal advantage during debugging: state estimation related errors
can be identified and resolved without additional real-world
flights. Instead, one can replay with an updated world model
using the same sequence of observations and actions to verify
potential improvements. This greatly accelerates the iteration
cycle when addressing vision-related sim-to-real transfer is-
sues, a common challenge in end-to-end vision-based learning
approaches.
To support actor-critic learning during latent imagination,
the world model is also trained to predict the reward and
whether a state is terminal. This is achieved using a reward
predictor pr
θ and a continue predictor pc
θ:
Reward predictor:
ˆrt ∼pr
θ(·|ht, zt)
Continue predictor:
ˆct ∼pc
θ(·|ht, zt)
where pc
θ predicts the continuation probability ˆct ∈[0, 1],
indicating whether the episode is expected to continue.
During actor-critic learning, the actor is optimized using
the Reinforce estimator [35]. Actor-critic learning is performed
entirely in latent space through imagined rollouts, as illustrated
in Figure 3d. These rollouts begin from hidden states obtained
during world model training and are propagated forward for
16 steps (0.18 s) using the dynamics predictor, without any
feedback from the environment.
The actor and critic are formalized as:
Actor:
ut ∼πθ(·|ht, zt)
Critic:
vψ(Vt|ht, zt)
where the actor is implemented as a Gaussian pol-
icy with mean µµµt = Eπθ [ut|ht, zt] and variance σσσ2
t
=
Eπθ
h
(ut −µµµt)2 | ht, zt
i
, and the critic estimates the value
of the current state, i.e. the expected discounted return Vt =
P∞
τ=0 γτrt+τ, with γ = 0.997 the discount factor.
Although the standard implementation produced performant
policies in simulation, we observed that they often converged
to near bang-bang behavior, where the motors are either fully
activated or deactivated. While such policies can be optimal in
simulation, in the real world they lead to overheating or even
burning the motors.
To mitigate this, we introduce a smoothness regularization
loss inspired by Mysore et al. [36], encouraging smoother
changes in control outputs over time. However, DreamerV3
includes an entropy loss that encourages high-variance (i.e.,
exploratory) policies to discover new strategies. Naively apply-
ing a smoothness penalty to the full policy distribution would
therefore conflict with this entropy objective.
To avoid this conflict, we apply the smoothness loss Lsmooth
only to the mean of the policy and add Lsmooth to the standard
policy loss of DreamerV3:
Lsmooth = λsmooth Et
h
||µµµt −µµµt−1||2
2
i
where λsmooth = 0.002 determines the weighting of the
regularization. This regularization is applied to the imagined
rollouts used during actor-critic learning. At inference time
and during evaluation, we use the deterministic version of the
policy (σσσt = 0) to produce even smoother control.
H. Flight Plan Logic
Another key limitation of prior end-to-end vision-based
ADR approaches is their inability to resolve visual ambigu-
ities: situations where different parts of the track produce
similar visual observations. To address this, we introduce
a flight plan logic that provides the policy with structured
information about its progress along the track, inspired by [13],
[16].
Specifically, the policy is augmented with a flight plan
vector f encoding the relative positions and yaw angles of up-
coming gates, without containing the drone’s current position,
attitude, or any related state variables. For each of the next
three gates, the input includes the difference in 3D position
9
and yaw between gate i and gate i+1, along with the absolute
position and yaw of gates i through i + 2 in the world frame:
fi =
pg
i −pg
i−1, ψψψg
i −ψψψg
i−1, . . . , ψψψg
i+2 −ψψψg
i+1,
pg
i , ψψψg
i , . . . , ψψψg
i+2
T
where pg
i denotes the position and ψψψg
i the yaw angle of gate
i in the world frame. This flight plan vector fi is provided as
input to SkyDreamer alongside the other observations ot.
To determine when a gate has been passed and update the
flight plan vector accordingly, we inspect the gate-relative
position ˆxg estimated by SkyDreamer. Once the drone has
passed through the current gate, the flight plan index i is
incremented, which is empirically defined as:
ˆxg > −0.15 m
since the nominal threshold of ˆxg > 0.0 m occasionally failed
to trigger due to small state estimation errors in SkyDreamer,
as observed in one real-world flight. During training, however,
we do not rely on SkyDreamer’s decoded states, which are
often too inaccurate in the early part of training. Instead,
the flight plan index i is incremented randomly between the
pre- and post-gate, simulating that the index does not always
increment precisely when passing the gate.
This proposed flight plan logic not only helps resolve
visual ambiguities, but also potentially enables generalization
to arbitrary tracks. Moreover, it facilitates decoding to a
global state in world frame under visual ambiguity or when
training a single model across multiple tracks. We argue that
such structured integration of track geometry is essential for
developing end-to-end vision-based ADR systems capable of
flying arbitrary unseen tracks with approximately known gate
locations.
III. EXPERIMENTS AND RESULTS
We extensively evaluate SkyDreamer on two tracks and with
two gate types, in simulation and in the real world: the ladder
inverted loop track and the inverted loop track, visualized in
Figure 4 and Figure 6. During training, additional invisible
gates are added to enforce the correct flight maneuvers. Both
tracks highlight a common challenging drone racing maneuver
at the second gate: the split-S. Within our limited 8 × 8m
testing space, we extend this split-S into a full inverted loop
to create a circular track. At the first gate, the ladder track
emphasizes precise control close to a gate, while the inverted
loop track tests high-speed flight through a gate. Finally,
we test SkyDreamer on the big track, shown in Figure 9,
demonstrating its ability to handle more complex layouts and
higher-speed flight.
Importantly, these tracks would be difficult – or even
impossible – to master for end-to-end vision-based approaches
without our proposed flight plan logic: the drone encounters
visually identical observations at different points along the
track but require taking different flight paths, which would
otherwise introduce ambiguity. By leveraging this flight plan
logic, SkyDreamer learns to resolve such ambiguity in both
simulation and the real world without external aid.
A. Experimental setup
For our experiments, we train SkyDreamer starting from
the latest JAX implementation of DreamerV32 for a total of
17 million environment steps, using the standard size12m
parameter model and default hyperparameters unless otherwise
noted. We set replay_context to 16, train_ratio to
128, slowtar to true, and use a replay buffer size of 10·106
representing 31 h of flight time.
We use environment settings and initial states as specified
in Table III. During training, 70% of initial states are sampled
from the training column, where initialization can occur in
front of any gate, and 30% from the evaluation column, where
initialization always occurs at the start gate.
Table III: Randomization, parameter and initial states. ”Train”
refers to the settings used for collecting data that is stored in
the replay buffer during training. ”Evaluation” refers to the set-
tings used during evaluation and simulation experiments. For
disturbances where a frequency is specified, 90 Hz indicates
the value changes every timestep, whereas 1 Hz indicates a
1/100 probability of change per timestep.
Parameter
Train
Evaluation
Unit
ϕc, ψc
[−5, 5]
[−5, 5]
deg
θc
[45, 55]
[45, 55]
deg
ωmin, ωmax
±20%
±20%
% rand
d \ {ωmin, ωmax}
±30%
±20%
% rand
εεεa (1Hz)
[−3, 3]
[−2, 2]
m/s2
εεεM (1Hz)
[−3, 3]
[−2, 2]
rad/s2
εεεM (90Hz)
[−125, 125]
[−100, 100]
rad/s2
εεεu (90Hz)
[−0.2, 0.2]
[−0.2, 0.2]
-
dg
0.8
1.0
m
tg
0.8
0.8
m
Initial state
xg
[−4.0, −2.0]
[−4.0, −2.0]
m
yg
[−1.0, 1.0]
[−1.0, 1.0]
m
zg
[0.0, 1.3]
[0.7, 1.3]
m
vx,w, vy,w, vz,w
0.0
0.0
m/s
p, q, r
[−0.1, 0.1]
[−0.1, 0.1]
rad/s
wi
[0.25, 0.5]
[0.25, 0.5]
ωmax
ϕw, θw, ψg
[−π/9, π/9]
[−π/9, π/9]
rad
Training is conducted in three phases on a 40GB partition
with 56 Shared Multiprocessors on an NVIDIA A100 80GB
GPU for roughly 50 hours. In the initial warm-up phase, we
follow the default DreamerV3 settings. After 8 million steps,
we increase the batch length for world model training from
64 to 256 to better capture long-term dependencies, which is
especially useful for parameter identification. After 13 million
steps, we reduce the entropy coefficient from 3·10−4 to 1·10−5
and the learning rate from 4 · 10−5 to 2 · 10−6 to stabilize
and fine-tune the policy for smoother and more fine-grained
control.
We deploy the learned policy onboard the same drone as in
Bahnam et al. [2] (also shown in Figure 1) on an NVIDIA Jet-
son Orin NX 16GB, operating entirely autonomously without
any external aid. The JAX model is first converted to PyTorch
using our custom implementation, then exported to ONNX,
2https://github.com/danijar/dreamerv3/commit/
cdf570902b1eaba193cc8ef69426cd4edde1b0bc
10
4
2
0
2
Y Position [m]
3
2
1
0
1
2
3
4
X Position [m]
Top-down View
2
1
0
1
2
Y Position [m]
-0.0
-1.0
-2.0
-3.0
-4.0
-5.0
Z Position [m]
Looping Side View
Ground Truth
SkyDreamer
Camera axis
2
4
6
8
10
12
Speed [m/s]
Figure 4: Simulated ladder inverted loop track flown by SkyDreamer. The flight begins by passing through the first gate,
followed by a tight ladder maneuver and flying over it. SkyDreamer then flies over the second gate, performs a maneuver
similar to a split-S through it, and flies back over the gate, completing the inverted loop. The lap is completed by passing
through the first gate again. In the left image, the black lines indicate ground-truth trajectory, while the colored lines show
SkyDreamer’s position and velocity estimates, with color denoting speed. The black blocks mark the gate locations with
exaggerated thickness. The black arrows indicate the camera principal axis direction, which show that SkyDreamer naturally
orients its camera toward the gates: an emergent behavior achieved without any explicit perception reward. The right image
presents a 3D render of one lap flown by SkyDreamer, where the colored line represents the ground-truth trajectory and
corresponding speed.
and finally optimized and executed with TensorRT. The full
policy, including encoder, sequence model, and actor, achieves
an average runtime of 1.3 ms in benchmark tests, while the
segmentation model averages 3 ms.
The onboard camera is a low-cost rolling shutter Arducam
IMX219 Wide Angle Camera3 with a 175◦field of view
operating at 90 Hz, which sets the control frequency of the
policy. All timing is anchored to the camera timestamps.
During training, we simulate an image delay of 33 ms and
an action delay of 11 ms to match deployment conditions.
Motor commands are sent immediately to the flight controller
with a desired execution timestamp, and sufficient buffering
is implemented to handle occasional timing variability in both
the policy and segmentation model.
Two types of gates are used in our experiments: plain square
orange gates with an inner size of 1.5m and an outer size of
2.7m shown in Figure 6, similar to those in Bahnam et al. [2],
and MAVLab gates, which resemble those in De Wagter et al.
[30], with an inner size of 1.5m and an outer size of 2.1m
shown in Figure 7.
For the orange gates, we train GateNet using only 200 man-
ually labeled real-world images, which we find sufficient to
achieve near-perfect segmentations. In contrast, the MAVLab
gates are more challenging to segment due to their dark blue
and black colors, thinner structure, the presence of logos,
and the similarity of background textures at one side of the
testing area (e.g., nearby blue machinery). For this case, we
train on 700 manually labeled real-world images combined
with 8,500 synthetically generated examples, following the
synthetic generation procedure described in Bahnam et al.
[2]. Importantly, our objective is not to optimize GateNet for
peak segmentation accuracy, as illustrated by some deliber-
3https://www.arducam.com/b0179-arducam-imx219-wide-angle-camera-
module-for-nvidia-jetson-board.html
ately retained suboptimal examples in Figure 7, but rather
to demonstrate that SkyDreamer can successfully cope with
poor-quality visual inputs and demonstrate a non-trivial visual
sim-to-real transfer.
B. Simulation
In the simulation experiments, we evaluate state and param-
eter estimation, demonstrate interpretability and an important
aspect of vision-based flight: perceptual behavior. To this
end, we investigate SkyDreamer’s performance on the ladder
inverted loop track with orange gates, shown in Figure 4. This
track is trained with a smaller tunnel size tg = 0.3m, to further
demonstrate SkyDreamer’s ability to execute tight maneuvers.
Figure 4 provides a qualitative analysis of SkyDreamer’s
flight and state estimation behavior. SkyDreamer’s position
estimate aligns closely with the ground-truth position, in-
dicating accurate state estimation. However, the estimated
states exhibit small, high-frequency jumps around the ground
truth. We hypothesize that this behavior may arise from the
discretized nature of the input representations zt. Another
possible contributing factor is that the world model may not
fully capture the underlying physics, as observed in similar
settings [37].
The maneuvers themselves are tight: SkyDreamer completes
the ladder while remaining mostly within 1 m of the gate and
executes the inverted loop in an almost perfect circle with
a radius of roughly 1.5 m, closely matching the separation
between the actual gate and the virtual gate of 2.7 m. Speeds
reach up to 13 m/s and accelerations up to 6 g within a confined
area of just 6×4 m, demonstrating both agility and high-speed
flight.
The tightness and speed of these maneuvers do not come
at the expense of perceptual behavior. The arrows in Figure 4
indicating the camera principal axis show that SkyDreamer
11
50
0
50
Error
[% of ±20%]
kx
Baseline (±1 )
SkyDreamer (±1 )
kp1
50
0
50
Error
[% of ±20%]
kw
kq1
max
2.5
0.0
2.5
Error [deg]
c
c
c
0.5
0.0
0.5
Error [m]
xw
yw
zw
50
100
150
1000
2000
Step
0.5
0.0
0.5
Error [m/s]
vx, w
50
100
150
1000
2000
Step
vy, w
50
100
150
1000
2000
Step
vz, w
Figure 5: Selected SkyDreamer parameter, extrinsic, and state estimation over time for the ladder inverted loop track with
orange gates. The left half of each plot shows the first 150 steps, highlighting the speed of convergence, while the right half
shows the remaining 1850 steps, illustrating any drift over time. The red distribution represents the baseline, computed using
the theoretical standard deviation of a uniform distribution representing the distribution of the training ranges in Table III. The
green distribution shows SkyDreamer’s estimation over time, obtained by aggregating results from 20 differently initialized
drones that completed a successful flight over 2000 timesteps each, following the evaluation parameter ranges from Table III.
Distributions such as those for kx align with the baseline distribution, indicating that SkyDreamer failed to estimate these
parameters. Other distributions are narrower, showing that SkyDreamer successfully estimated them on the fly. Parameters like
ωmax and kw converge quickly and remain stable over time, while parameters like kp1 converge more slowly and show gradual
drift over time.
consistently focuses on the gates to maximize gate visibility,
despite the absence of a perception reward. This emergent
behavior, also observed in Romero et al. [6], is thus evident
even in more complex maneuvers such as the ladder and
inverted loop.
Figure 5 shows SkyDreamer’s state and parameter esti-
mation over time. Some dynamical parameters are estimated
better than others: while drag (kx) is difficult to estimate,
performance is moderate for individual propeller responses in
roll (kp1) and pitch (kq1). Parameters such as yaw effectiveness
(e.g., kr1) are not accurately estimated.
However, SkyDreamer excels at identifying two parameters
that are crucial for thrust response: the thrust effectiveness
kw and maximum motor RPM ωmax. Correct estimation of
these parameters enables SkyDreamer to perform high-speed
maneuvers and tight cornering tailored to the specific drone
despite domain randomization, similar to Origer et al. [12].
Other parameters, such as the motor response τ, are also
estimated accurately.
Overall, parameters that have a direct and strong effect on
the drone’s inputs, like ωmax, are estimated most accurately,
whereas parameters with smaller long-term effects or those
affecting only a single motor are more challenging to estimate.
In addition to its final flight performance, the estimated
parameters converge quickly within the first 50–100 steps
(0.6 s to 1.1 s) – before the drone passes through the first
gate – enabling SkyDreamer to fly safely and at high speed
from the start. Some parameters, however, exhibit drift in
the mean of their distributions over time, indicating that
the sequence length used during world model learning (256
steps) does not necessarily generalize to the longer sequences
encountered during flight (2000 steps). It should however be
noted that the accuracy of parameter identification depends
on the training run: some runs yield more precise estimates,
while others exhibit lower accuracy and can show significant
drift over time, without resulting in a significant loss of flight
performance.
SkyDreamer also estimates the camera extrinsics over time.
Convergence is slightly slower than for some dynamical pa-
rameters, but it reaches a standard deviation of roughly 1◦over
12
4
2
0
2
Y Position [m]
3
2
1
0
1
2
3
4
X Position [m]
Top-down View
2
1
0
1
2
Y Position [m]
-0.0
-1.0
-2.0
-3.0
-4.0
-5.0
Z Position [m]
Looping Side View
MoCap
SkyDreamer
Camera axis
2
4
6
8
10
12
Speed [m/s]
Figure 6: Real-world inverted loop track flown by SkyDreamer. The flight begins by passing through the first gate. SkyDreamer
then flies over the second gate, performs a maneuver similar to a split-S through it, and flies back over the gate, completing the
inverted loop. The lap is completed by passing through the first gate. In the plots, the black lines indicate ground-truth positions
obtained using MoCap as a reference, which were not used by SkyDreamer. The colored lines show the position and velocity
estimates of SkyDreamer, with color denoting speed. The black blocks mark the gate locations with exaggerated thickness.
The black arrows indicate the camera principal axis direction, which show that SkyDreamer naturally orients its camera toward
the gates: an emergent behavior achieved without an explicit perception reward. The right image shows a composite image of
five laps flown by SkyDreamer in the real world. The overlaid trajectories of all five laps converge at the center points of the
gates, visually demonstrating that SkyDreamer consistently flies through the center of each gate.
Figure 7: Real-world inverted loop track with MAVLab gates flown by SkyDreamer. The flight begins by passing through the
first gate. SkyDreamer then flies over the second gate, performs a maneuver similar to a split-S through it, and flies back over
the gate, completing the inverted loop. The lap is completed by passing through the first gate. In the plots, the colored lines
show the position and velocity estimates of SkyDreamer, with color denoting speed. The black blocks mark the gate locations
with exaggerated thickness. Below the plots, selected suboptimal segmentations produced by GateNet are shown, with their
corresponding locations indicated by the numbers. Red pixels denote regions classified as gates by GateNet. MoCap data could
not be recorded during this flight due to technical issues, so ground-truth positions are unavailable. The right image shows a
composite image of one lap flown by SkyDreamer in the real world.
time, indicating accurate on the fly estimation of the camera’s
extrinsics. Notably, the pitch angle of the camera is estimated
most accurately, while the roll angle is slightly less accurate,
though the difference is marginal.
Finally, we evaluate the estimation of position and velocity
over time – two of the most important yet challenging state
variables for classical methods [2]. Despite randomized initial
positions, the estimates converge quickly, within at most 10
steps, to the nominal distribution, with a typical standard
deviation of only 10 −15 cm across the randomized drones,
remaining centered over time. Velocity estimates are also
accurate, with a standard deviation of approximately 0.5 m/s
in all axes, and remain centered around zero, indicating no
drift over time.
13
0
1000
Step
2000
2250
2500
2750
3000
3250
3500
[rad/s]
Charged battery
0
500
1000
1500
Step
Depleting battery
SkyDreamer
max [rad/s]
0
10
20
30
40
50
Count of RPM measurements
Figure 8: Real-world battery degradation over five laps of the
inverted loop track with orange gates. The heatmap shows
measured motor RPMs over time, where brighter regions
indicate higher occurrence at a given timestamp. The red line
denotes the maximum RPM estimated by SkyDreamer, ˆωmax.
The left-hand plot indicates a fully charged battery which is
not excessively discharged, while the right-hand plot depicts a
battery undergoing excessive discharge, resulting in reduced
RPM toward the end of the flight. However, SkyDreamer
accounts for this, correctly estimates ˆωmax, adjusts its flight
path accordingly and successfully completes all five laps with
inverted loops.
C. Real-world small tracks
In the first real-world experiments, we evaluate Sky-
Dreamer’s sim-to-real transfer, robustness, and performance
across three small tracks: the inverted loop with both orange
and MAVLab gates (Figure 6 and Figure 7), and the ladder
inverted loop track with orange gates, which uses the model
from the same training run as the simulation experiments
(Figure 4).
Table IV summarizes SkyDreamer’s performance on these
tracks. For each track, we conduct five flights of five laps
each, resulting in 25 laps per track and 75 laps in total.
SkyDreamer successfully completes all laps without crashing.
For the orange inverted loop track, one flight was performed
with a degrading battery, leading to a higher average lap time
and a larger standard deviation. Nevertheless, SkyDreamer
still successfully completed all five laps. During these flights,
SkyDreamer reached measured accelerations of up to 6 g and
estimated speeds of up to 13 m/s, demonstrating impressive
agility with flight areas typically confined to 6 × 6 m.
To demonstrate SkyDreamer’s parameter estimation in the
real world, Figure 8 illustrates its performance during flights
with and without a fully charged battery. The figures show
that the maximum RPM estimated by SkyDreamer, ˆωmax,
closely follows the actual measured RPMs, even though it was
never trained with non-constant parameters. By the final lap,
the maximum RPM drops from 3200 rad/s to 2200 rad/s, a
30% reduction, well outside the training randomization bounds
of [2480, 3720] rad/s. Despite this, SkyDreamer detects the
change, adapts its flight path, and still completes several
inverted loops. This demonstrates that online parameter es-
timation allows SkyDreamer to adapt on the fly, maintaining
both safety and agility despite a significant loss of maximum
available thrust.
Table IV: Success rates and lap times for the real-world
small tracks flown without external aid. All tracks achieved
100% success over five flights, with five laps per flight. The
± indicates the standard deviation, which is higher for the
inverted loop (orange) track due to one flight with a partially
charged battery. One flight on the ladder loop track was
excluded because the flight plan condition ˆxg > 0.0 m never
triggered, resulting in an aborted flight; the condition was
subsequently adjusted to ˆxg > −0.15 m. For the MAVLab
gates, a part of the environment resembling a gate had to be
covered, as GateNet consistently misclassified it.
Metric
Loop
(orange)
Ladder
loop
(orange)
Loop
(MAVLab)
Success rate
100%
100%
100%
Number of flights
5
5
5
Laps per flight
5
5
5
From start [s]
17.03 ± 0.76
18.87 ± 0.23
15.56 ± 0.20
From first gate [s]
16.36 ± 0.77
18.24 ± 0.22
14.88 ± 0.20
Lap 1 [s]
3.37 ± 0.08
3.78 ± 0.09
2.99 ± 0.07
Lap 2–5 [s]
3.25 ± 0.22
3.62 ± 0.06
2.97 ± 0.08
In addition to estimating parameters, Figure 6 shows that
the state estimation transfers well to the real world, where
SkyDreamer’s state estimates only show a slight offset to
the MoCap positions in the xy-plane, and an almost perfect
overlap in the yz-plane during the loop. The figure also reveals
that SkyDreamer takes less tight loops compared to the ladder
inverted loop track, which can be attributed to the larger tunnel
size (tg = 0.8 m), forcing it to take wider turns.
To visually illustrate SkyDreamer’s flight path and con-
sistency, Figure 6 shows a composite image of five laps on
the inverted loop track. Remarkably, the trajectories from all
five laps converge closely to each other and to the center
of the gates, demonstrating that SkyDreamer navigates both
consistently and safely.
While SkyDreamer generalizes well with orange gates –
whose high contrast and uniform appearance yield near-perfect
segmentation masks closely matching the non-augmented ren-
dered ones – we also evaluate it with MAVLab gates. For
these, GateNet’s performance degrades due to poor lighting,
low contrast, and distracting background elements such as
machinery that visually resemble gates. As shown in Fig-
ure 7, GateNet occasionally misclassifies background regions
as gates, produces rounded masks, and often fails to fully
align with the actual gate boundaries. However, rather than
adopting more advanced segmentation models like the Swin
Transformer V2 [38] to improve the segmentation performance
– as done in Geles et al. [4] – we intentionally retain GateNet’s
baseline performance to demonstrate that SkyDreamer can
operate in the real world with poor-quality perception.
Thanks to the use of StochGAN and additional data aug-
mentations, SkyDreamer still completes 25 consecutive laps on
the inverted loop track with MAVLab gates, as summarized in
Table IV. Despite GateNet’s poor performance, SkyDreamer
demonstrates robustness to perceptual errors and bridges a
non-trivial visual reality gap. This opens the door to using
alternative more generalizable pixel-level abstractions such as
depth maps, which could potentially enable high-speed, agile
14
Y Position [m]
8
6
4
2
0
2
4
6
8
X Position [m]
Top-down View
10
5
0
5
10
Y Position [m]
-2.0
-4.0
Z Position [m]
Side View
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Speed [m/s]
Figure 9: Real-world big track flown by SkyDreamer. In the top-down view, the flight begins in front of the top-left gate.
After the first gate, the trajectory continues with a slight left turn, two subsequent gates, a right turn, and a ladder maneuver in
which the drone passes through the lower gate, performs a full 360° left turn, and flies back over it. From there, SkyDreamer
extends the right turn, executing a steep dive into the next gate, and continues through several more gates before performing
a tight braking maneuver to initiate a sharp right turn. After the following gate, the track concludes with a split-S maneuver
over the first gate. In the plots, the colored lines show SkyDreamer’s position and velocity estimates for two real-world laps,
with color indicating speed. Black blocks mark gate locations with exaggerated thickness. The bottom-right image presents a
composite visualization of one lap from another flight flown by SkyDreamer.
flight in unstructured environments.
D. Real-world big track
Additional
real-world
experiments
demonstrate
SkyDreamer’s ability to fly the big track with MAVLab
gates illustrated in Figure 9. This track combines high-speed
flight with challenging maneuvers, including a split-S, a
sharp right turn, and a ladder. The experiments are conducted
in the real world in a larger hall and under poorer lighting
conditions,
further
highlighting
SkyDreamer’s
robustness
across environments.
We train SkyDreamer with slightly modified settings for
these experiments: we disable symlog scaling in both the
encoder and decoder [19], set the train_ratio to 64, and
train for a total of 35 million environment steps. The second
training stage begins after 23 million steps, and the third stage
after 31 million steps. Furthermore, we set tg = 0.5 and intro-
duce disturbances of ±300 rad/s to ωmax, randomly resampled
every 10 timesteps, to account for imperfect actuator response
modeling.
During these experiments, five out of six flights consisting
of two laps each were successful. The only crash occurred
with a drone whose extrinsics lay outside the training distri-
bution specified in Table III, as verified by a preflight Kalibr
calibration, likely explaining the failure. The five successful
flights were all with the same drone whose extrinsics are
within the training distribution. In three of these flights, minor
gate contact was audible once or twice, but SkyDreamer
consistently carried on successfully, highlighting its ability to
withstand and adapt to contact forces it was not explicitly
trained for. In this track, SkyDreamer reaches accelerations
comparable to those on the smaller tracks (around 6 g) while
achieving significantly higher estimated speeds of up to 21 m/s.
These results highlight SkyDreamer’s ability to combine high-
speed flight with agile and tight maneuvers.
IV. CONCLUSION
In this work, we presented SkyDreamer, to the best of
our knowledge the first end-to-end vision-based ADR policy
mapping pixel-level representations directly to motor com-
mands. By building on informed Dreamer and extending it
to end-to-end vision-based ADR, SkyDreamer provides inter-
pretability by decoding to states and drone-specific parameters,
effectively turning the world model into an implicit state and
parameter estimator. SkyDreamer runs fully onboard without
external aid, resolves visual ambiguities by tracking progress
through the world model’s decoded state, and requires no
extrinsic camera calibration, enabling rapid deployment across
different drones without retraining.
Our real-world experiments demonstrate non-trivial sim-to-
real transfer despite operating on low-quality segmentation
masks, reliable performance across diverse tracks, and robust-
ness to battery depletion. SkyDreamer successfully executes
15
tight maneuvers and reaches speeds up to 21 m/s with acceler-
ations of up to 6 g – speed and agility not previously demon-
strated by other end-to-end vision-based ADR methods in the
real world. These results highlight SkyDreamer’s adaptability
to important aspects of the reality gap, bringing robustness
while still achieving extremely high-speed, agile flight.
Despite these promising results, several limitations remain.
Parameter estimates tend to drift over time, and their quality
varies across training runs. Decoded state estimates jump
between timesteps, and SkyDreamer is still vulnerable to false
positives in segmentation masks. In addition, training remains
computationally demanding, requiring roughly 50 hours to
complete.
Future work will explore extending SkyDreamer to more
generalizable visual inputs, such as depth maps, flying un-
seen tracks, and potentially generalizing to unstructured en-
vironments or hybrid tasks by combining drone racing with
obstacle avoidance. Beyond racing, SkyDreamer represents a
promising step toward deploying end-to-end vision-based yet
interpretable policies for high-speed, agile flight in real-world
settings.
ACKNOWLEDGMENTS
We are grateful to the other members of the drone racing
team – Anton Lang, Till Blaha, Quentin Missine and Erin
Lucassen – for their valuable insights and support during
system integration and flight experiments.
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APPENDIX A
GATENET IMPLEMENTATION DETAILS
a) Network: For segmentation, we adopt a U-Net style
architecture [39], referred to as GateNet, similar to Bahnam et
al. [2]. The network consists of an encoder–decoder with skip
connections from the encoder to the corresponding decoder
layer, following the standard U-Net design [39], where channel
dimensions are scaled by a factor f. Each block consists
of double convolutional layers with batch normalization and
ReLU activation functions.
Let inc denote the initial double convolutional block.
downk denotes a max-pooling layer followed by a double
convolutional block with k output channels. Before applying
downk, the corresponding features are stored as skip connec-
tions for the decoder. upk denotes a transposed convolution
and batch normalization, followed by adding the skip connec-
tions from the corresponding encoder layer, and then a double
convolutional block with k output channels. Finally, outc-k
denotes a 1×1 convolution mapping to a single output channel
and a sigmoid activation function.
The resulting GateNet architecture is:
Encoder
Decoder
Outputs
inc-64/f
→up4-64/f
→outc4-1
↓
↑
down1-128/f
→up3-64/f
→outc3-1
↓
↑
down2-256/f
→up2-128/f
→outc2-1
↓
↑
down3-512/f
→up1-256/f
→outc1-1
↓
↗
down4-512/f
→outc0-1
The network produces five output maps {y0, y1, y2, y3, y4}
at different resolutions. These multi-scale predictions are su-
pervised with auxiliary losses to improve gradient flow and
overall segmentation accuracy.
b) Training setup: We use the same hyperparameters as
in Bahnam et al. [2]. Each output map is supervised with a
combination of Dice loss and binary cross-entropy (BCE) loss:
Li = LDice(yi, ˆyi) + 2 · LBCE(yi, ˆyi),
and we apply output-specific scaling factors to emphasize
higher-resolution predictions:
Ltotal = 4 · L0 + 2 · L1 +
4
X
i=2
Li.
All convolutional layers are initialized using Xavier uniform
initialization [40].
c) Data augmentation: We largely use the same augmen-
tations as in Bahnam et al. [2]. Since our images are captured
with a low exposure time (1 ms), we do not simulate motion
blur with KernelBlur. Instead, we apply stronger shot noise
(standard deviation of 40 for pixel values in the range 0–255
instead of 25) to mimic the increased noise associated with
low exposure times.
d) Gate-specific implementation: For MAVLab gates, we
use a resolution of 196 × 196 with f = 2. For orange gates,
we use a resolution of 384 × 384 with f = 4.
APPENDIX B
STOCHGAN IMPLEMENTATION DETAILS
a) Generator:
The generator is a CNN-based en-
coder–residual–decoder network, largely based on Zhu et
17
al. [31]. We use 6 residual blocks and images of size
64 × 64, the same resolution as those rendered and used by
SkyDreamer during training. Following a similar convention
to Zhu et al. [31], let c7s1-k denote a 7 × 7 Convolu-
tion–InstanceNorm–ReLU layer with k filters and stride 1,
using reflection padding. dk corresponds to a 3 × 3 Convolu-
tion–InstanceNorm–ReLU layer with k filters and stride 2 for
downsampling. Rk denotes a residual block consisting of two
3 × 3 Convolution–InstanceNorm–ReLU layers with k filters,
using reflection padding and a skip connection. uk represents
a 3 × 3 transposed Convolution–InstanceNorm–ReLU layer
with k filters, stride 2 for upsampling, and reflection padding.
Finally, the output layer is denoted as c7s1-k-tanh, which
follows the same convention as c7s1-k but uses a tanh
activation at the final layer.
The resulting generator architecture is:
Encoder: c7s1-32, d64, d128
Residual blocks: 6 × R128
Decoder: u64, u32, c7s1-1-tanh
To obtain a StochGAN [32], we add an additional noise
channel, uniformly sampled from [−1, 1], to the input image.
b) Discriminator: Similar to Zhu et al. [31], the discrim-
inator follows a PatchGAN design [41]. Largely following the
design and conventions in Zhu et al. [31], let Ck denote a
4 × 4 Convolution–InstanceNorm–LeakyReLU layer with k
filters and stride 2. InstanceNorm is omitted in the first Ck
layer, and LeakyReLU activations with a negative slope of 0.2
are used. After the last block, we apply zero-padding followed
by a 4×4 convolution with 1 channel (conv4-1), producing
a patch-wise realism score.
The resulting discriminator architecture is:
C32, C64, C128, C256, conv4-1
c) Training setup.: We largely follow the training proce-
dure of Zhu et al. [31], except we use the Adam optimizer with
β1 = 0.5, β2 = 0.999, and a base learning rate of 1.5 × 10−4.
|
1 SkyDreamer: Interpretable End-to-End Vision-Based Drone Racing with Model-Based Reinforcement Learning Aderik Verraest, Stavrow Bahnam, Robin Ferede, Guido de Croon, Christophe De Wagter Abstract-Autonomous drone racing (ADR) systems have recently achieved champion-level performance, yet remain highly specific to drone racing. While end-to-end vision-based methods promise broader applicability, no system to date simultaneously achieves full sim-to-real transfer, onboard execution, and champion-level performance. In this work, we present SkyDreamer, to the best of our knowledge, the first end-to-end vision-based ADR policy that maps directly from pixel-level representations to motor commands. SkyDreamer builds on informed Dreamer, a model-based reinforcement learning approach where the world model decodes to privileged information only available during training. By extending this concept to end-to-end visionbased ADR, the world model effectively functions as an implicit state and parameter estimator, greatly improving interpretability. SkyDreamer runs fully onboard without external aid, resolves visual ambiguities by tracking progress using the state decoded from the world model's hidden state, and requires no extrinsic camera calibration, enabling rapid deployment across different drones without retraining. Real-world experiments show that SkyDreamer achieves robust, high-speed flight, executing tight maneuvers such as an inverted loop, a split-S and a ladder, reaching speeds of up to 21 m/s and accelerations of up to 6 g. It further demonstrates a non-trivial visual sim-to-real transfer by operating on poor-quality segmentation masks, and exhibits robustness to battery depletion by accurately estimating the maximum attainable motor RPM and adjusting its flight path in real-time. These results highlight SkyDreamer's adaptability to important aspects of the reality gap, bringing robustness while still achieving extremely high-speed, agile flight. I. INTRODUCTION Recent advancements in autonomous drone racing (ADR) have reached champion-level performance, even beating human world champions [1], [2]. In 2023, Kaufmann et al. [1] became the first to defeat human champion drone racers with autonomous drones. More recently, Bahnam et al. [2] achieved the first champion-level victory in an externally organized competition, relying solely on a low-cost monocular rolling shutter camera for perception. Despite these successes, current champion-level ADR systems [1], [2] remain modular pipelines of classical algorithms. They are highly specialized and inflexible, relying on gate corner detection and Perspective-n-Point (PnP) for state estimation. Their performance further depends on accurately mapped track layouts, rectangular gates of known size, precise camera calibration, and hand-tuned extended Kalman filters (EKFs). Optimizing perception, state estimation, and control The authors are with the Micro Air Vehicle Lab of the Faculty of Aerospace Engineering, Delft 2629 HS Delft, The Netherlands Figure 1: Real-world inverted loop track with MAVLab gates. SkyDreamer flies agile maneuvers such as an inverted loop in the real world without external aid (top), operates end-to-end on poor-quality segmentation masks obtained from onboard images (shown in red, bottom-right), and directly controls the physical platform by outputting low-level motor commands. In a flight area confined to 6 × 6 m, it reaches speeds of up to 13 m/s and accelerations of up to 6 g. The racing drone is identical to the one used in the A2RL x DCL Autonomous Drone Championship 2025 [2] (bottom-left). in isolation further limits overall performance. Such pipelines cannot generalize beyond the highly structured settings they were designed for. In contrast, humans fly with remarkable adaptability - handling unseen tracks, different drones, and even unstructured environments with minimal task-specific training [3]. This gap underscores the inherent limitations in today's champion-level ADR systems. To address these limitations, research has turned toward end-to-end vision-based learning, where policies map pixellevel representations directly to actions without explicit state estimation [4]-[7]. Such learned representations offer greater flexibility, potentially enhancing performance by jointly optimizing perception, state estimation, and control. Crucially, 16 Oct 2025 2 Table I: Comparison of recent end-to-end vision-based ADR approaches with SkyDreamer. "Offboard" indicates that some computation occurs offboard, while "onboard" means all computation runs onboard with no external aid during flight. "HIL" denotes that images for the policy during real-world flight are rendered using MoCap. "Visual ambiguity" refers to situations where visually similar observations require different flight paths, such as in the ladder inverted loop track shown in Figure 4. The perception reward encourages the drone to look toward gates, introducing human bias. Only performance metrics from real-world experiments are considered in this comparison. Criterion Geles et al. [4] Xing et al. [5] Romero et al. [6] Krinner et al. [7] SkyDreamer (ours) Onboard computation Offboard Offboard Offboard Offboard Onboard without external aid Champion-level No (2 g) No (2.7 TWR) 1 No (2.7 TWR) No Likely (6 g) Sim-to-real transfer Yes HIL (only dynamics) HIL (only dynamics) HIL (only dynamics) Yes, non-trivial visual sim-to-real gap Interpretability None None Limited, latent → reconstructed image Yes, via decoding to states Yes, via decoding to states and parameters Robust to visual ambiguity Not demonstrated Not demonstrated Not demonstrated Not demonstrated Yes, via progress tracking using world model Perception reward Yes Yes No, emergent behavior No, emergent behavior No, emergent behavior leveraging pixel-level inputs, such as depth maps, can generalize to unstructured environments while introducing a smaller reality gap compared to raw RGB images, as demonstrated in [8]-[11]. At the same time, end-to-end learning reduces reliance on human-designed abstractions and reward shaping, which can constrain system performance. While these benefits are promising, practical implementations remain challenging. Only Geles et al. [4] manage to bridge the visual reality gap, whereas Xing et al. [5], Romero et al. [6] and Krinner et al. [7] still rely on hardware-in-theloop (HIL) simulation, where images during real-world flights are rendered using an external motion capture (MoCap) system rather than the real camera feed. Furthermore, as summarized in Table I, none of these approaches perform all computation fully onboard, nor do they demonstrate champion-level performance in the real world, as the thrust-to-weight ratio (TWR) was deliberately software-capped. To date, no end-toend vision-based method simultaneously achieves full sim-toreal transfer, onboard execution, and champion-level flight. An additional, and often overlooked, constraint is that most ADR approaches treat collective thrust and body rates (CTBR) as the final control layer, relying on an inner-loop controller such as PID or INDI at deployment to track these CTBR actions with motor commands [1], [4]-[7]. As an alternative to this two-stage control structure, the European Space Agency (ESA) introduced guidance and control networks (G&CNETs), which map state inputs directly to actuator commands [12]. For quadcopter control, this translates to generating motor commands directly rather than relying on intermediate CTBR actions. Doing so is required for several space applications and simplifies modeling, as the motor response is generally easier to capture than the complex behavior of an inner-loop controller. Additionally, direct motor control may also enable more optimal performance compared to approaches that rely on inner-loop controllers. Building on this concept, Ferede et al. [13] trained 1Xing et al. [5] did perform limited experiments in simulation demonstrating champion-level performance comparable to [1], but failed to demonstrate such performance in the real world. G&CNETs using reinforcement learning (RL), specifically proximal policy optimization (PPO) [14], for ADR, demonstrating that fully neural control policies can outperform traditional cascaded controllers when trained under comparable conditions [13], [15], [16]. The use of G&CNETs for quadcopter control has already proven successful in ADR competitions: during the A2RL x DCL Autonomous Drone Championship in 2025, a neural end-to-end state-based controller outperformed human world-champion pilots, underscoring its real-world potential [2]. A promising approach to training end-to-end vision-based ADR policies is model-based RL. Unlike model-free RL methods, such as PPO [14] and soft actor-critic (SAC) [17], model-based RL learns an internal model of the environment, often referred to as a world model [18]. The core idea is to capture environment dynamics by learning to predict future observations - or their latent representations - based on past observations and actions using the world model. Actor training is thus performed entirely in latent space, where the actor interacts with the learned world model rather than the physical or simulated environment. This is accomplished through latent imagination, in which future latent states are predicted from previous latent states and actions. Consequently, sample efficiency is greatly improved: physical or simulated rollouts are required only to train the world model, not the actor. This results in orders-of-magnitude gains in sample efficiency compared to model-free RL, which is particularly important for end-to-end vision-based policies, where rendering observations at every timestep can be computationally expensive [18], [19]. In model-based RL, the Dreamer algorithms introduced by Hafner et al. have become among the most widely adopted [19]-[23]. Notably, Wu et al. [23] demonstrated that a robot could be trained to walk using only one hour of real-world interactions - without relying on simulation - highlighting the remarkable sample efficiency of model-based RL. More recently, DreamerV3 has emerged as a particularly promising variant, achieving state-of-the-art performance across more than 150 diverse environments using a single set of hy3 perparameters [19]. Romero et al. [6] have already applied DreamerV3 to end-to-end vision-based ADR, though with limited success both in terms of performance and sim-to-real transfer. To further aid learning efficiency, the use of privileged information during training has shown great promise [4], [5]. In model-based RL, this idea can be extended by allowing the world model to decode privileged observations, such as ground-truth states, an idea already explored for end-toend vision-based ADR in Krinner et al. [7]. This has been shown to yield significant performance gains, particularly in tasks where such states are recoverable from sequences of observations [24], [25], as in vision-based ADR. Furthermore, privileged reconstructions enable the world model to act as an implicit state estimator, improving interpretability, unlike prior approaches [4]-[6] that largely operate as black boxes. Alternative model-based approaches such as TD-MPC2 combine latent imagination with model predictive control (MPC) [26]. By relying on MPC rather than RL, TD-MPC2 outperforms DreamerV3 on several continuous control benchmarks [26]. However, unlike Dreamer, TD-MPC2 lacks a recurrent structure in latent space and is therefore unable to model long-range temporal dependencies. Its reliance on iterative optimization at runtime further limits its suitability for onboard deployment. Other latent-space MPC approaches include [27] and [28]. Notably, Zhou et al. [28] demonstrated zero-shot planning with a pretrained billion-parameter world model. While impressive, the scale of such models makes deployment on resource-constrained platforms like drones impractical. Contributions In this work, we present SkyDreamer, to the best of our knowledge the first end-to-end vision-based ADR policy that maps directly from pixels to motor commands, eliminating the need for inner-loop controllers in end-to-end vision-based drone racing. Beyond this main contribution, our work offers several further advances, summarized below and in Table I: • SkyDreamer greatly improves interpretability by decoding privileged information, effectively turning the world model into an implicit state and parameter estimator, • is, to the best of our knowledge, the first end-to-end visionbased ADR policy to reach accelerations of up to 666 g and speeds of up to 21 21 21 m/s in the real world, on par with stateof-the-art classical approaches [1], [2], • demonstrates non-trivial visual sim-to-real transfer using poor-quality segmentation masks, • resolves visual ambiguity by explicitly tracking progress through the state decoded from the world model's hidden state, laying the foundation for end-to-end vision-based policies capable of flying arbitrary tracks, • requires no accelerometer, • runs fully onboard without external aid, • estimates drone-specific parameters, such as camera extrinsics, on the fly and operates on images standardized via intrinsic calibration, enabling rapid deployment across different drones without retraining. II. METHODOLOGY A. Problem statement We aim to develop an end-to-end vision-based policy capable of flying a quadcopter at high speed through a known track using only onboard computation and sensing. Specifically, the goal is to maximize the quadcopter's lap speed around a premapped track marked by gates, subject to the constraints of the physical platform and the requirement to pass through all gates in a predefined order while avoiding collisions. The policy receives only raw sensory inputs: pixel-level representations, body rate measurements, and motor RPM measurements. Additionally, the drone's camera extrinsic parameters are assumed to be uncalibrated, as extrinsic calibration is both timeconsuming and must often be repeated during operation due to changes in the camera angle from touching, crashing, or hitting gates, making it impractical for real-world competitions. The policy must therefore learn to handle this uncertainty. It shall map these observations directly to motor commands, without relying on an intermediate inner-loop controller such as PID or INDI. During training only, the policy is granted access to privileged information, including ground-truth states, camera parameters, and dynamic parameters of the system. We formalize this problem as an informed partially observable Markov decision process (informed POMDP) [24], which extends the standard POMDP [29]. The informed POMDP was first introduced by Lambrechts et al. [24], who provide a detailed formulation. The key distinction with a standard POMDP is that additional, privileged information is available during training but not at execution time. This additional information can be exploited to improve the training process. Formally, the informed POMDP is defined as: eP = (S, U, I, O, T, R, eI, e O, P, γ) with the following components: • state space S with states st • action space U with actions ut • information space I with information it only available during training • observation space O with observations ot • transition distribution T(st+1 | st, ut) • reward function rt = R(st, ut), which is designed to reflect the task objective and is detailed further in subsection II-C. • information distribution: eI(it | st) • observation distribution: e O(ot | it) available during training and execution • initial state distribution: P(s0), which is detailed further in Table III • discount factor: γ ∈[0, 1) In an Informed POMDP, the states st are never observable. At training time, the informed POMDP does provide the policy access to a history h+ t : h+ t = (i0, o0, u0, r0, ..., it-1, ot-1, ut-1, rt-1, it, ot) consisting of privileged information, observations, actions and rewards. The observations ot are assumed to be conditionally dependent on the states st, following the Bayesian network 4 st →it →ot. To satisfy this assumption, and following Lambrechts et al. [24], we define the information variable as it = [ot, o+ t ]T , a concatenation of the observations ot and privileged observations o+ t only available during training (e.g. position p or velocity v). At execution time, we define the underlying execution POMDP eP as: eP = (S, U, O, T, R, e O, P, γ) which provides access to a history ht: ht = (o0, u0, ..., ot-1, ut-1, ot) consisting of only observations and actions [6], [24]. The objective in an informed POMDP is to maximize the expected return in the underlying execution POMDP by learning a policy πθθθ(ut|ht), in our case parameterized by neural network parameters θθθ. The resulting optimal policy π∗ θ is defined by [24]: π∗ θ = argmax πθ E s0∼P (·) ut∼πθ(·|ht) st+1∼T (·|st,at) ot∼e O(·|st) " ∞ X t=0 γtR(st, at) # The optimal policy thus aims to maximize the expected return, i.e., the cumulative sum of discounted rewards, in the execution POMDP through interaction with the informed POMDP at training time. Although privileged information is used during learning, the resulting optimal policy remains independent of it at execution time. B. Spaces xw yw zw xb yb zb xc yc zc xg yg zg xw yw zw Body Camera Gate World Figure 2: Coordinate systems. A quadcopter pitched forward - shown as a cross with four circles - with a camera pitched upward, is shown flying toward the gate at the bottom left. Subscripts indicate the reference frames: w - world frame, g - gate frame, b - body frame, and c - camera frame. Image not to scale, the axes of the frames are not necessarily aligned. The state space S consists of states s which are defined as: s = [pw, pg, vw, vg,λλλ,ΩΩΩ,ωωω, ce, d,εεε, f]⊤ where the associated coordinate systems all follow a north-east-down (NED) convention illustrated in Figure 2, and the components of s are described below: • position: - pw = [xw, yw, zw]⊤: position in world frame - pg = [xg, yg, zg]⊤: position in gate frame, updated upon passing each gate. The gate frame is a local coordinate system attached to the next gate, which is updated each time a gate is passed and is also shown in Figure 2 • velocity: - vw = [vx,w, vy,w, vz,w]⊤: velocity in world frame - vg = [vx,g, vy,g, vz,g]⊤: velocity in gate frame • orientation λλλ = [φw, θw, ψw, ψg]⊤in ψ →θ →φ order - φw, θw: roll and pitch in world frame - ψw: yaw in world frame - ψg: yaw in gate frame • body rates ΩΩΩ= [p, q, r]T are the roll, pitch and yaw rate in body frame respectively • propeller speeds ωωω = [ω1, ω2, ω3, ω4]T are the angular velocities of the four propellers • camera extrinsics ce = [φc, θc, ψc] describes the euler rotation from body to camera frame in ψ →θ →φ order • dynamic parameters d are randomized dynamical parameters of the drone described in subsection II-D • disturbances εεε = [εεεa,εεεM,εεεu]⊤ - εεεa = [εa,x, εa,y, εa,z]⊤: acceleration disturbances - εεεM = [εM,x, εM,y, εM,z]⊤: moment disturbances - εεεu = [εu,1, εu,1, εu,3, εu,4]: action disturbances • flight plan vector f indicating the relative position between gates, detailed in subsection II-H. The privileged observations are given by o+ t = [pw, pg, vw, vg,λλλ,ΩΩΩ,ωωω, ce, d]T , where it should be noted that ωωω and ΩΩΩrepresent the ground-truth rates, not the measured ones. The observations are ot = [X, ˆΩΩΩ, ˆωωω]T , where X ∈ {0, 1}H×W is a binary segmentation mask of the gates, generated by the segmentation model described in subsection II-E, ˆΩΩΩ are the body rates measured by the inertial measurement unit (IMU), and ˆωωω are the propeller angular velocities measured by the electronic speed controller (ESC). We do not use the accelerometer, motivated by the observation of Bahnam et al. [2] that it tends to saturate on the used physical platform under high accelerations and vibrations. We define the information variable as it = {ot, o+ t }\{X}, where the binary segmentation mask is excluded since its information is fully captured in o+ t , and thus still satisfies the conditional dependence of ooot on ssst. Moreover, including it would significantly slow down training. Lastly, the actions are given by ut = [u1, u2, u3, u4]⊤, which represent the normalized motor commands for each rotor, where ui ∈[0, 1]. These commands directly specify the desired motor power fraction, which the ESC regulates with pulse-width modulation (PWM), without any intermediate inner-loop controller, providing maximal control authority over the physical platform for high-speed, agile flight. 5 C. Reward function In RL, the reward function is a key component, handdesigned to reflect the task objective. It encodes both the goals and constraints of the task, assigning positive rewards for desirable behavior and penalizing undesirable actions. Similar to Ferede et al. [16], our reward function consists of a progress reward, rate penalty and gate reward, and is given by: rt = 5 · rprog -rrate + 30 · rgate rrate = 1 2 · fc · 105 (exp (min (∥Ωt∥1, 17)) -1) rprog = ∥pt-1,g∥2 -∥pt,g∥2 rgate = 1 -max (|yt,g|, |zt,g|) dg , if gate passed 0, otherwise where fc = 90 Hz is the control frequency of the policy, and rrate is a rate penalty designed to discourage excessive angular velocities, which helps prevent both gyroscope saturation and excessive motion blur in the onboard camera. The term rgate provides an additional reward upon successfully passing through a gate, where dg is the effective size of the gate, with the reward being maximal at the center of the gate and decreasing linearly to zero toward its edges. To account for the thickness of the gate and the size of the drone, we introduce a pre-gate and post-gate, which are nonrendered virtual gates located at a specified distance before and after the gate, respectively. These gates provide the same reward rgate and collectively act as collision volumes with shaped rewards, spanning a total thickness tg. The term rprog is a progress reward that encourages the drone to move toward the next pre-gate, serving as a dense shaping signal to accelerate training. Importantly, this term does not enforce a specific trajectory: any path toward the next gate results in the same accumulated return. For ease of implementation, the progress reward is set to zero between the pre- and post-gate. Crucially, unlike Geles et al. [4] and Xing et al. [5] but similar to Romero et al. [6] and Krinner et al. [7], there is no explicit perception reward. We observe that the resulting policy naturally learns to orient its camera toward the gates to maximize gate visibility. Additionally, unlike Ferede et al. [16], we do not include a penalty for gate collisions. In our experiments, such penalties were found to hinder the learning process and provided no benefit compared to the natural truncation of the discounted return caused by an episode termination on collision. Finally, an episode is terminated if any of the following conditions are met: Gate collision: ( max(|yt,g|, |zt,g|) dg > 1, if gate passed false, otherwise Ground collision: zw > -0.5 ∧ vz,w > 1.0 ∨|φw| > π 3 ∨|θw| > π 3 where zw = 0 denotes the ground plane. Episodes terminating due to either condition result in a final reward of zero. The gate collision condition also applies to pre- and post-gates. The ground collision condition is shaped to discourage fast flight with high pitch or roll near the ground. This is discouraged as the drone starts from a raised podium in the real world, and such contact forces are not simulated. Using this shaped collision condition, the drone learns to ascend first before accelerating forward, rather than immediately flying toward the gate. D. Quadcopter dynamics Since the policy is trained in simulation, accurate modeling of the quadcopter dynamics is crucial, particularly because motor commands are executed directly. Our dynamics model largely builds on the frameworks from Bahnam et al. [2] and Ferede et al. [16]; for full details, we refer the reader to those sources. Following the approach of Bahnam et al. [2], we represent the rotation described by λλλ using quaternions q = [qw, qx, qy, qz]T and incorporate gyroscopic effects. Furthermore, we introduce additional disturbances εεε: ̇xw = vw ̇q = 1 2q ⊗[0 p q r]T ̇vi = ge3 + R(λλλ)F + εεεa ̇ΩΩΩ= M + εεεM ωi = ωci -ωi τ where R is the rotation matrix and g is the gravitational acceleration constant. The forces F, moments M and steady motor response wci are given by: F = -kxvB x P4 i=1 ωi -kx2vB x |vB x | -kyvB y P4 i=1 ωi -ky2vB y |vB y | -kω (1 + kαα + khorμxx,yy) P4 i=1 ω2 i -kv2vB z |vB z | with α = tan-1 vB z r ̄ω μxx,yy = tan-1 vB2 x + vB2 y r ̄ω with ̄ω = 4 X i=1 ωi M M M = -kp1ω2 1 -kp2ω2 2 + kp3ω2 3 + kp4ω2 4 + Jxqr -kq1ω2 1 + kq2ω2 2 -kq3ω2 3 + kq4ω2 4 + Jypr -kr1ω1 + kr2ω2 + kr3ω3 -kr4ω4kr5 ̇ω1 + kr6 ̇ω2 + kr7 ̇ω3 -kr8 ̇ω4 + Jzpq ωci = (ωmax -ωmin) q k ̃ui 2 + (1 -kl) ̃ui + ωmin with ̃ui = min (max (ui + εu,i, 0) , 1) where vx,b, vy,b and vz,b are the velocity components in body frame. The values of the parameters used for both policy training and simulation experiments are given in Table II. These parameters together make up the previously mentioned dynamics 6 parameters d = [kw, kx, ky, . . . , ωmax, k, τ]T . The simulation is implemented using JAX in Python, and is discretized using fourth-order Runge-Kutta integration with a timestep of 2.2 ms. Table II: Quadcopter dynamical parameters used for both policy training and simulation experiments. Param Value Param Value kw 1.55 × 10-6 kx2 4.10 × 10-3 kx, ky 5.37 × 10-5 ky2 1.51 × 10-2 kangle 3.145 khor 7.245 kv2 0.00 Jx -0.89 Jy 0.96 Jz -0.34 ωmin 341.75 ωmax 3100.00 k 0.50 τ 0.03 kp1 4.99 × 10-5 kp2 3.78 × 10-5 kp3 4.82 × 10-5 kp4 3.83 × 10-5 kq1 2.05 × 10-5 kq2 2.46 × 10-5 kq3 2.02 × 10-5 kq4 2.57 × 10-5 kr1-kr4 3.38 × 10-3 kr5-kr8 3.24 × 10-4 E. Segmentation masks As pixel level input, SkyDreamer receives a segmentation mask X. Unlike the extrinsic parameters, we choose to calibrate the intrinsic parameters of the camera to maintain a consistent camera matrix for SkyDreamer's inputs, as they are straightforward to obtain and remain constant over time. Using these calibrated intrinsics, the RGB images are undistorted, cropped, and mapped to a fixed-resolution image of size H × W, such that the resulting image satisfies a nominal pinhole camera model with intrinsic matrix: K = fx 0 cx 0 fy cy 0 0 1 = 25 64W 0 0.5W 0 25 64H 0.5H 0 0 1 , where fx and fy are the focal lengths in pixels and cx, cy are the principal point coordinates in pixels. The factor 25 64 is empirically chosen such that an image of 64 × 64 pixels corresponds to a focal length of 25, providing a balanced field of view that maximizes scene coverage while avoiding black borders in the undistorted images. By mapping all images to a shared nominal intrinsic matrix and randomizing extrinsic parameters during training, SkyDreamer ensures robustness to extrinsic variations and invariance to intrinsic parameters through calibration, enabling generalization of the same model across different drones. Subsequently, the fixed-resolution image is passed to a segmentation model called GateNet that produces a binary segmentation mask X ∈{0, 1}H×W , indicating which pixels correspond to gates. GateNet follows a U-Net architecture, similar to the designs used in De Wagter et al. [30] and Bahnam et al. [2] of which the implementation details are provided in Appendix A. The resulting masks are resized to a resolution of 64 × 64 to speed up the training process of SkyDreamer. F. Visual augmentations During training, segmentation masks are rendered in batches using PyTorch3D. However, these masks are idealized and differ significantly from real-world masks, causing a large visual sim-to-real gap. To bridge this gap, we employ CycleGAN [31], a generative adversarial network (GAN) that learns to translate images between two unpaired domains, in our case, from synthetic segmentation masks to more realistic masks and vice versa, without requiring paired real and rendered masks. To further improve realism, we use a stochastic CycleGAN (StochGAN) [32], an extension of CycleGAN that adds noise to the inputs. This stochasticity enhances the variation and realism of generated masks, better approximating real-world appearance and variance. Although Almahairi et al. [32] also proposes a more advanced AugmentedGAN, we opted for StochGAN due to its simpler implementation and effective performance. The GAN is trained on approximately 4000 unpaired real and rendered images, which require no manual labeling and are therefore inexpensive to collect. Further implementation details are provided in Appendix B. We observed that real-life masks were often thinner, a detail not captured well by the GAN alone. To address this, we randomly erode the masks by 1 pixel - using average pooling followed by thresholding - 50% of the time at a mask resolution of 64 × 64, introducing variation in the mask thickness. Since the erosion level typically remains stable over short periods, we hold it fixed for an average of 100 environment steps. Lastly, to simulate the rolling shutter effect of the onboard camera, we account for two dominant effects: a horizontal shear induced by the yaw rate in camera frame, and a vertical scaling induced by the pitch rate. These arise from the lineby-line capture of a rolling shutter camera under rotational motion. To model these effects, we introduce a rolling shutter parameter s ∈[0, 0.02], which also aims to capture additional disturbances associated with rolling shutter through randomization. We formalize our approximation of the rolling shutter effect with an affine transformation given by the following matrix: A = 1 -s rc W 2 s rc 0 1 + s qc -H 2 s qc , where rc and qc denote the yaw and pitch rates in the camera frame, respectively. This transformation is then applied to the rendered segmentation mask: X′ = warp affine(X, A) which produces a horizontally sheared and vertically scaled segmentation mask. G. SkyDreamer architecture SkyDreamer's architecture is almost entirely based on Informed Dreamer [24], which is largely based on DreamerV3 [19]. In this section, we provide a high level overview and intuitive explanation of the architecture most relevant to our work. For detailed information on the architecture, policy 7 (a) Standard DreamerV3 architecture for ADR. In standard DreamerV3, the world model is not guided by privileged observations o+ t . (b) SkyDreamer at deployment. The dynamics predictor is not used, as observations are always available. The flight plan vector is updated based on the decoded states as indicated by the cyan arrow. (c) SkyDreamer during world model learning. The dynamics predictor is trained to match the encoded inputs. Additionally, the sequence model, encoder, decoder, reward predictor, and continue predictor are trained. (d) SkyDreamer during actor-critic learning. The dynamics predictor replaces the encoder to generate imagined rollouts. Figure 3: Comparison of standard DreamerV3 and SkyDreamer across different stages. The world model encodes segmentation masks and measurements such as motor RPMs, body rates, and the flight plan vector into discrete representations zt. The next hidden state ht+1 is predicted from the previous hidden state ht, action at and discrete representation zt using the sequence model. The inputs ˆxt are reconstructed to guide the hidden representations. In SkyDreamer, the privileged observations, including ground-truth state, camera and dynamical parameters, are decoded as o+ t . During actor-critic learning, imagined rollouts are generated solely using the dynamics predictor without feedback from the environment. Images inspired by Hafner et al. [19]. learning and loss functions, we refer the reader to Hafner et al. [19] and Lambrechts et al. [24]. The core component of DreamerV3 is its world model, which encodes observations into a latent representation and predicts future latent states based on actions, both with and without receiving further observations. The training process continuously alternates between three main stages: (1) data collection, (2) world model learning, and (3) actor-critic learning, which are illustrated in Figure 3. In the data collection phase, the current policy interacts with the environment to collect rollouts, which are stored in a replay buffer, making DreamerV3 an off-policy algorithm. During world model learning, batches of trajectories are uniformly sampled from the replay buffer to train the world model. Finally, in the actor-critic phase, learning occurs entirely through latent imagination: the world model simulates rollouts without environment feedback, enabling sample efficient model-based RL. All components of SkyDreamer are implemented as multilayer perceptrons (MLPs), unless otherwise noted. The key element of the world model in DreamerV3 is a recurrent state-space model (RSSM), which consists of the following components: 8 Encoder: zt ∼qe θ(·|ht, ot) Sequence model: ht = f e θ (ht-1, zt-1, ut-1) Dynamics predictor: ˆzt ∼pd θ(·|ht) Here, qe θ is the encoder, which encodes the observation ot along with the hidden state ht into a discrete stochastic latent representation zt. This structure resembles a vectorquantized variational autoencoder (VQ-VAE) [33]. For image observations, it uses a convolutional neural network (CNN) rather than an MLP. The function f e θ is the sequence model, implemented as a single-layer gated recurrent unit (GRU) [34], which predicts the next hidden state ht given the previous hidden state ht-1, previous latent state zt-1, and previous action ut-1. Finally, pd θ is the dynamics predictor, which predicts an estimate of the latent representation ˆzt based solely on the hidden state ht. This component is crucial, as it allows the model to roll out future trajectories in latent space without further observations from the environment, which is often referred to as latent imagination or dreaming and is illustrated in Figure 3d. The RSSM also employs a decoder to guide the learning process. Crucially, following informed Dreamer [24], SkyDreamer decodes to the privileged information it rather than only the raw observations ot: Decoder: it ∼pi t(·|ht, zt) By decoding to this privileged information, the world model is encouraged to extract more meaningful and task-relevant features from the inputs. This key difference from standard DreamerV3 is visually evident when comparing Figure 3a and Figure 3c. In addition to improving learning speed, final performance and adaptability to drone-specific parameters, this also results in a more interpretable world model. Whereas decoding to images often leads to blurry reconstructions, decoding to states such as position, velocity, and attitude allows a direct inspection of what the world model believes the current state to be. Additionally, this interpretability offers a significant practical advantage during debugging: state estimation related errors can be identified and resolved without additional real-world flights. Instead, one can replay with an updated world model using the same sequence of observations and actions to verify potential improvements. This greatly accelerates the iteration cycle when addressing vision-related sim-to-real transfer issues, a common challenge in end-to-end vision-based learning approaches. To support actor-critic learning during latent imagination, the world model is also trained to predict the reward and whether a state is terminal. This is achieved using a reward predictor pr θ and a continue predictor pc θ: Reward predictor: ˆrt ∼pr θ(·|ht, zt) Continue predictor: ˆct ∼pc θ(·|ht, zt) where pc θ predicts the continuation probability ˆct ∈[0, 1], indicating whether the episode is expected to continue. During actor-critic learning, the actor is optimized using the Reinforce estimator [35]. Actor-critic learning is performed entirely in latent space through imagined rollouts, as illustrated in Figure 3d. These rollouts begin from hidden states obtained during world model training and are propagated forward for 16 steps (0.18 s) using the dynamics predictor, without any feedback from the environment. The actor and critic are formalized as: Actor: ut ∼πθ(·|ht, zt) Critic: vψ(Vt|ht, zt) where the actor is implemented as a Gaussian policy with mean μμμt = Eπθ [ut|ht, zt] and variance σσσ2 t = Eπθ h (ut -μμμt)2 | ht, zt i , and the critic estimates the value of the current state, i.e. the expected discounted return Vt = P∞ τ=0 γτrt+τ, with γ = 0.997 the discount factor. Although the standard implementation produced performant policies in simulation, we observed that they often converged to near bang-bang behavior, where the motors are either fully activated or deactivated. While such policies can be optimal in simulation, in the real world they lead to overheating or even burning the motors. To mitigate this, we introduce a smoothness regularization loss inspired by Mysore et al. [36], encouraging smoother changes in control outputs over time. However, DreamerV3 includes an entropy loss that encourages high-variance (i.e., exploratory) policies to discover new strategies. Naively applying a smoothness penalty to the full policy distribution would therefore conflict with this entropy objective. To avoid this conflict, we apply the smoothness loss Lsmooth only to the mean of the policy and add Lsmooth to the standard policy loss of DreamerV3: Lsmooth = λsmooth Et h ||μμμt -μμμt-1||2 2 i where λsmooth = 0.002 determines the weighting of the regularization. This regularization is applied to the imagined rollouts used during actor-critic learning. At inference time and during evaluation, we use the deterministic version of the policy (σσσt = 0) to produce even smoother control. H. Flight Plan Logic Another key limitation of prior end-to-end vision-based ADR approaches is their inability to resolve visual ambiguities: situations where different parts of the track produce similar visual observations. To address this, we introduce a flight plan logic that provides the policy with structured information about its progress along the track, inspired by [13], [16]. Specifically, the policy is augmented with a flight plan vector f encoding the relative positions and yaw angles of upcoming gates, without containing the drone's current position, attitude, or any related state variables. For each of the next three gates, the input includes the difference in 3D position 9 and yaw between gate i and gate i+1, along with the absolute position and yaw of gates i through i + 2 in the world frame: fi = pg i -pg i-1, ψψψg i -ψψψg i-1, . . . , ψψψg i+2 -ψψψg i+1, pg i , ψψψg i , . . . , ψψψg i+2 T where pg i denotes the position and ψψψg i the yaw angle of gate i in the world frame. This flight plan vector fi is provided as input to SkyDreamer alongside the other observations ot. To determine when a gate has been passed and update the flight plan vector accordingly, we inspect the gate-relative position ˆxg estimated by SkyDreamer. Once the drone has passed through the current gate, the flight plan index i is incremented, which is empirically defined as: ˆxg > -0.15 m since the nominal threshold of ˆxg > 0.0 m occasionally failed to trigger due to small state estimation errors in SkyDreamer, as observed in one real-world flight. During training, however, we do not rely on SkyDreamer's decoded states, which are often too inaccurate in the early part of training. Instead, the flight plan index i is incremented randomly between the pre- and post-gate, simulating that the index does not always increment precisely when passing the gate. This proposed flight plan logic not only helps resolve visual ambiguities, but also potentially enables generalization to arbitrary tracks. Moreover, it facilitates decoding to a global state in world frame under visual ambiguity or when training a single model across multiple tracks. We argue that such structured integration of track geometry is essential for developing end-to-end vision-based ADR systems capable of flying arbitrary unseen tracks with approximately known gate locations. III. EXPERIMENTS AND RESULTS We extensively evaluate SkyDreamer on two tracks and with two gate types, in simulation and in the real world: the ladder inverted loop track and the inverted loop track, visualized in Figure 4 and Figure 6. During training, additional invisible gates are added to enforce the correct flight maneuvers. Both tracks highlight a common challenging drone racing maneuver at the second gate: the split-S. Within our limited 8 × 8m testing space, we extend this split-S into a full inverted loop to create a circular track. At the first gate, the ladder track emphasizes precise control close to a gate, while the inverted loop track tests high-speed flight through a gate. Finally, we test SkyDreamer on the big track, shown in Figure 9, demonstrating its ability to handle more complex layouts and higher-speed flight. Importantly, these tracks would be difficult - or even impossible - to master for end-to-end vision-based approaches without our proposed flight plan logic: the drone encounters visually identical observations at different points along the track but require taking different flight paths, which would otherwise introduce ambiguity. By leveraging this flight plan logic, SkyDreamer learns to resolve such ambiguity in both simulation and the real world without external aid. A. Experimental setup For our experiments, we train SkyDreamer starting from the latest JAX implementation of DreamerV32 for a total of 17 million environment steps, using the standard size12m parameter model and default hyperparameters unless otherwise noted. We set replay_context to 16, train_ratio to 128, slowtar to true, and use a replay buffer size of 10·106 representing 31 h of flight time. We use environment settings and initial states as specified in Table III. During training, 70% of initial states are sampled from the training column, where initialization can occur in front of any gate, and 30% from the evaluation column, where initialization always occurs at the start gate. Table III: Randomization, parameter and initial states. "Train" refers to the settings used for collecting data that is stored in the replay buffer during training. "Evaluation" refers to the settings used during evaluation and simulation experiments. For disturbances where a frequency is specified, 90 Hz indicates the value changes every timestep, whereas 1 Hz indicates a 1/100 probability of change per timestep. Parameter Train Evaluation Unit φc, ψc [-5, 5] [-5, 5] deg θc [45, 55] [45, 55] deg ωmin, ωmax ±20% ±20% % rand d \ {ωmin, ωmax} ±30% ±20% % rand εεεa (1Hz) [-3, 3] [-2, 2] m/s2 εεεM (1Hz) [-3, 3] [-2, 2] rad/s2 εεεM (90Hz) [-125, 125] [-100, 100] rad/s2 εεεu (90Hz) [-0.2, 0.2] [-0.2, 0.2] - dg 0.8 1.0 m tg 0.8 0.8 m Initial state xg [-4.0, -2.0] [-4.0, -2.0] m yg [-1.0, 1.0] [-1.0, 1.0] m zg [0.0, 1.3] [0.7, 1.3] m vx,w, vy,w, vz,w 0.0 0.0 m/s p, q, r [-0.1, 0.1] [-0.1, 0.1] rad/s wi [0.25, 0.5] [0.25, 0.5] ωmax φw, θw, ψg [-π/9, π/9] [-π/9, π/9] rad Training is conducted in three phases on a 40GB partition with 56 Shared Multiprocessors on an NVIDIA A100 80GB GPU for roughly 50 hours. In the initial warm-up phase, we follow the default DreamerV3 settings. After 8 million steps, we increase the batch length for world model training from 64 to 256 to better capture long-term dependencies, which is especially useful for parameter identification. After 13 million steps, we reduce the entropy coefficient from 3·10-4 to 1·10-5 and the learning rate from 4 · 10-5 to 2 · 10-6 to stabilize and fine-tune the policy for smoother and more fine-grained control. We deploy the learned policy onboard the same drone as in Bahnam et al. [2] (also shown in Figure 1) on an NVIDIA Jetson Orin NX 16GB, operating entirely autonomously without any external aid. The JAX model is first converted to PyTorch using our custom implementation, then exported to ONNX, 2https://github.com/danijar/dreamerv3/commit/ cdf570902b1eaba193cc8ef69426cd4edde1b0bc 10 4 2 0 2 Y Position [m] 3 2 1 0 1 2 3 4 X Position [m] Top-down View 2 1 0 1 2 Y Position [m] -0.0 -1.0 -2.0 -3.0 -4.0 -5.0 Z Position [m] Looping Side View Ground Truth SkyDreamer Camera axis 2 4 6 8 10 12 Speed [m/s] Figure 4: Simulated ladder inverted loop track flown by SkyDreamer. The flight begins by passing through the first gate, followed by a tight ladder maneuver and flying over it. SkyDreamer then flies over the second gate, performs a maneuver similar to a split-S through it, and flies back over the gate, completing the inverted loop. The lap is completed by passing through the first gate again. In the left image, the black lines indicate ground-truth trajectory, while the colored lines show SkyDreamer's position and velocity estimates, with color denoting speed. The black blocks mark the gate locations with exaggerated thickness. The black arrows indicate the camera principal axis direction, which show that SkyDreamer naturally orients its camera toward the gates: an emergent behavior achieved without any explicit perception reward. The right image presents a 3D render of one lap flown by SkyDreamer, where the colored line represents the ground-truth trajectory and corresponding speed. and finally optimized and executed with TensorRT. The full policy, including encoder, sequence model, and actor, achieves an average runtime of 1.3 ms in benchmark tests, while the segmentation model averages 3 ms. The onboard camera is a low-cost rolling shutter Arducam IMX219 Wide Angle Camera3 with a 175◦field of view operating at 90 Hz, which sets the control frequency of the policy. All timing is anchored to the camera timestamps. During training, we simulate an image delay of 33 ms and an action delay of 11 ms to match deployment conditions. Motor commands are sent immediately to the flight controller with a desired execution timestamp, and sufficient buffering is implemented to handle occasional timing variability in both the policy and segmentation model. Two types of gates are used in our experiments: plain square orange gates with an inner size of 1.5m and an outer size of 2.7m shown in Figure 6, similar to those in Bahnam et al. [2], and MAVLab gates, which resemble those in De Wagter et al. [30], with an inner size of 1.5m and an outer size of 2.1m shown in Figure 7. For the orange gates, we train GateNet using only 200 manually labeled real-world images, which we find sufficient to achieve near-perfect segmentations. In contrast, the MAVLab gates are more challenging to segment due to their dark blue and black colors, thinner structure, the presence of logos, and the similarity of background textures at one side of the testing area (e.g., nearby blue machinery). For this case, we train on 700 manually labeled real-world images combined with 8,500 synthetically generated examples, following the synthetic generation procedure described in Bahnam et al. [2]. Importantly, our objective is not to optimize GateNet for peak segmentation accuracy, as illustrated by some deliber3https://www.arducam.com/b0179-arducam-imx219-wide-angle-cameramodule-for-nvidia-jetson-board.html ately retained suboptimal examples in Figure 7, but rather to demonstrate that SkyDreamer can successfully cope with poor-quality visual inputs and demonstrate a non-trivial visual sim-to-real transfer. B. Simulation In the simulation experiments, we evaluate state and parameter estimation, demonstrate interpretability and an important aspect of vision-based flight: perceptual behavior. To this end, we investigate SkyDreamer's performance on the ladder inverted loop track with orange gates, shown in Figure 4. This track is trained with a smaller tunnel size tg = 0.3m, to further demonstrate SkyDreamer's ability to execute tight maneuvers. Figure 4 provides a qualitative analysis of SkyDreamer's flight and state estimation behavior. SkyDreamer's position estimate aligns closely with the ground-truth position, indicating accurate state estimation. However, the estimated states exhibit small, high-frequency jumps around the ground truth. We hypothesize that this behavior may arise from the discretized nature of the input representations zt. Another possible contributing factor is that the world model may not fully capture the underlying physics, as observed in similar settings [37]. The maneuvers themselves are tight: SkyDreamer completes the ladder while remaining mostly within 1 m of the gate and executes the inverted loop in an almost perfect circle with a radius of roughly 1.5 m, closely matching the separation between the actual gate and the virtual gate of 2.7 m. Speeds reach up to 13 m/s and accelerations up to 6 g within a confined area of just 6×4 m, demonstrating both agility and high-speed flight. The tightness and speed of these maneuvers do not come at the expense of perceptual behavior. The arrows in Figure 4 indicating the camera principal axis show that SkyDreamer 11 50 0 50 Error [% of ±20%] kx Baseline (±1 ) SkyDreamer (±1 ) kp1 50 0 50 Error [% of ±20%] kw kq1 max 2.5 0.0 2.5 Error [deg] c c c 0.5 0.0 0.5 Error [m] xw yw zw 50 100 150 1000 2000 Step 0.5 0.0 0.5 Error [m/s] vx, w 50 100 150 1000 2000 Step vy, w 50 100 150 1000 2000 Step vz, w Figure 5: Selected SkyDreamer parameter, extrinsic, and state estimation over time for the ladder inverted loop track with orange gates. The left half of each plot shows the first 150 steps, highlighting the speed of convergence, while the right half shows the remaining 1850 steps, illustrating any drift over time. The red distribution represents the baseline, computed using the theoretical standard deviation of a uniform distribution representing the distribution of the training ranges in Table III. The green distribution shows SkyDreamer's estimation over time, obtained by aggregating results from 20 differently initialized drones that completed a successful flight over 2000 timesteps each, following the evaluation parameter ranges from Table III. Distributions such as those for kx align with the baseline distribution, indicating that SkyDreamer failed to estimate these parameters. Other distributions are narrower, showing that SkyDreamer successfully estimated them on the fly. Parameters like ωmax and kw converge quickly and remain stable over time, while parameters like kp1 converge more slowly and show gradual drift over time. consistently focuses on the gates to maximize gate visibility, despite the absence of a perception reward. This emergent behavior, also observed in Romero et al. [6], is thus evident even in more complex maneuvers such as the ladder and inverted loop. Figure 5 shows SkyDreamer's state and parameter estimation over time. Some dynamical parameters are estimated better than others: while drag (kx) is difficult to estimate, performance is moderate for individual propeller responses in roll (kp1) and pitch (kq1). Parameters such as yaw effectiveness (e.g., kr1) are not accurately estimated. However, SkyDreamer excels at identifying two parameters that are crucial for thrust response: the thrust effectiveness kw and maximum motor RPM ωmax. Correct estimation of these parameters enables SkyDreamer to perform high-speed maneuvers and tight cornering tailored to the specific drone despite domain randomization, similar to Origer et al. [12]. Other parameters, such as the motor response τ, are also estimated accurately. Overall, parameters that have a direct and strong effect on the drone's inputs, like ωmax, are estimated most accurately, whereas parameters with smaller long-term effects or those affecting only a single motor are more challenging to estimate. In addition to its final flight performance, the estimated parameters converge quickly within the first 50-100 steps (0.6 s to 1.1 s) - before the drone passes through the first gate - enabling SkyDreamer to fly safely and at high speed from the start. Some parameters, however, exhibit drift in the mean of their distributions over time, indicating that the sequence length used during world model learning (256 steps) does not necessarily generalize to the longer sequences encountered during flight (2000 steps). It should however be noted that the accuracy of parameter identification depends on the training run: some runs yield more precise estimates, while others exhibit lower accuracy and can show significant drift over time, without resulting in a significant loss of flight performance. SkyDreamer also estimates the camera extrinsics over time. Convergence is slightly slower than for some dynamical parameters, but it reaches a standard deviation of roughly 1◦over 12 4 2 0 2 Y Position [m] 3 2 1 0 1 2 3 4 X Position [m] Top-down View 2 1 0 1 2 Y Position [m] -0.0 -1.0 -2.0 -3.0 -4.0 -5.0 Z Position [m] Looping Side View MoCap SkyDreamer Camera axis 2 4 6 8 10 12 Speed [m/s] Figure 6: Real-world inverted loop track flown by SkyDreamer. The flight begins by passing through the first gate. SkyDreamer then flies over the second gate, performs a maneuver similar to a split-S through it, and flies back over the gate, completing the inverted loop. The lap is completed by passing through the first gate. In the plots, the black lines indicate ground-truth positions obtained using MoCap as a reference, which were not used by SkyDreamer. The colored lines show the position and velocity estimates of SkyDreamer, with color denoting speed. The black blocks mark the gate locations with exaggerated thickness. The black arrows indicate the camera principal axis direction, which show that SkyDreamer naturally orients its camera toward the gates: an emergent behavior achieved without an explicit perception reward. The right image shows a composite image of five laps flown by SkyDreamer in the real world. The overlaid trajectories of all five laps converge at the center points of the gates, visually demonstrating that SkyDreamer consistently flies through the center of each gate. Figure 7: Real-world inverted loop track with MAVLab gates flown by SkyDreamer. The flight begins by passing through the first gate. SkyDreamer then flies over the second gate, performs a maneuver similar to a split-S through it, and flies back over the gate, completing the inverted loop. The lap is completed by passing through the first gate. In the plots, the colored lines show the position and velocity estimates of SkyDreamer, with color denoting speed. The black blocks mark the gate locations with exaggerated thickness. Below the plots, selected suboptimal segmentations produced by GateNet are shown, with their corresponding locations indicated by the numbers. Red pixels denote regions classified as gates by GateNet. MoCap data could not be recorded during this flight due to technical issues, so ground-truth positions are unavailable. The right image shows a composite image of one lap flown by SkyDreamer in the real world. time, indicating accurate on the fly estimation of the camera's extrinsics. Notably, the pitch angle of the camera is estimated most accurately, while the roll angle is slightly less accurate, though the difference is marginal. Finally, we evaluate the estimation of position and velocity over time - two of the most important yet challenging state variables for classical methods [2]. Despite randomized initial positions, the estimates converge quickly, within at most 10 steps, to the nominal distribution, with a typical standard deviation of only 10 -15 cm across the randomized drones, remaining centered over time. Velocity estimates are also accurate, with a standard deviation of approximately 0.5 m/s in all axes, and remain centered around zero, indicating no drift over time. 13 0 1000 Step 2000 2250 2500 2750 3000 3250 3500 [rad/s] Charged battery 0 500 1000 1500 Step Depleting battery SkyDreamer max [rad/s] 0 10 20 30 40 50 Count of RPM measurements Figure 8: Real-world battery degradation over five laps of the inverted loop track with orange gates. The heatmap shows measured motor RPMs over time, where brighter regions indicate higher occurrence at a given timestamp. The red line denotes the maximum RPM estimated by SkyDreamer, ˆωmax. The left-hand plot indicates a fully charged battery which is not excessively discharged, while the right-hand plot depicts a battery undergoing excessive discharge, resulting in reduced RPM toward the end of the flight. However, SkyDreamer accounts for this, correctly estimates ˆωmax, adjusts its flight path accordingly and successfully completes all five laps with inverted loops. C. Real-world small tracks In the first real-world experiments, we evaluate SkyDreamer's sim-to-real transfer, robustness, and performance across three small tracks: the inverted loop with both orange and MAVLab gates (Figure 6 and Figure 7), and the ladder inverted loop track with orange gates, which uses the model from the same training run as the simulation experiments (Figure 4). Table IV summarizes SkyDreamer's performance on these tracks. For each track, we conduct five flights of five laps each, resulting in 25 laps per track and 75 laps in total. SkyDreamer successfully completes all laps without crashing. For the orange inverted loop track, one flight was performed with a degrading battery, leading to a higher average lap time and a larger standard deviation. Nevertheless, SkyDreamer still successfully completed all five laps. During these flights, SkyDreamer reached measured accelerations of up to 6 g and estimated speeds of up to 13 m/s, demonstrating impressive agility with flight areas typically confined to 6 × 6 m. To demonstrate SkyDreamer's parameter estimation in the real world, Figure 8 illustrates its performance during flights with and without a fully charged battery. The figures show that the maximum RPM estimated by SkyDreamer, ˆωmax, closely follows the actual measured RPMs, even though it was never trained with non-constant parameters. By the final lap, the maximum RPM drops from 3200 rad/s to 2200 rad/s, a 30% reduction, well outside the training randomization bounds of [2480, 3720] rad/s. Despite this, SkyDreamer detects the change, adapts its flight path, and still completes several inverted loops. This demonstrates that online parameter estimation allows SkyDreamer to adapt on the fly, maintaining both safety and agility despite a significant loss of maximum available thrust. Table IV: Success rates and lap times for the real-world small tracks flown without external aid. All tracks achieved 100% success over five flights, with five laps per flight. The ± indicates the standard deviation, which is higher for the inverted loop (orange) track due to one flight with a partially charged battery. One flight on the ladder loop track was excluded because the flight plan condition ˆxg > 0.0 m never triggered, resulting in an aborted flight; the condition was subsequently adjusted to ˆxg > -0.15 m. For the MAVLab gates, a part of the environment resembling a gate had to be covered, as GateNet consistently misclassified it. Metric Loop (orange) Ladder loop (orange) Loop (MAVLab) Success rate 100% 100% 100% Number of flights 5 5 5 Laps per flight 5 5 5 From start [s] 17.03 ± 0.76 18.87 ± 0.23 15.56 ± 0.20 From first gate [s] 16.36 ± 0.77 18.24 ± 0.22 14.88 ± 0.20 Lap 1 [s] 3.37 ± 0.08 3.78 ± 0.09 2.99 ± 0.07 Lap 2-5 [s] 3.25 ± 0.22 3.62 ± 0.06 2.97 ± 0.08 In addition to estimating parameters, Figure 6 shows that the state estimation transfers well to the real world, where SkyDreamer's state estimates only show a slight offset to the MoCap positions in the xy-plane, and an almost perfect overlap in the yz-plane during the loop. The figure also reveals that SkyDreamer takes less tight loops compared to the ladder inverted loop track, which can be attributed to the larger tunnel size (tg = 0.8 m), forcing it to take wider turns. To visually illustrate SkyDreamer's flight path and consistency, Figure 6 shows a composite image of five laps on the inverted loop track. Remarkably, the trajectories from all five laps converge closely to each other and to the center of the gates, demonstrating that SkyDreamer navigates both consistently and safely. While SkyDreamer generalizes well with orange gates - whose high contrast and uniform appearance yield near-perfect segmentation masks closely matching the non-augmented rendered ones - we also evaluate it with MAVLab gates. For these, GateNet's performance degrades due to poor lighting, low contrast, and distracting background elements such as machinery that visually resemble gates. As shown in Figure 7, GateNet occasionally misclassifies background regions as gates, produces rounded masks, and often fails to fully align with the actual gate boundaries. However, rather than adopting more advanced segmentation models like the Swin Transformer V2 [38] to improve the segmentation performance - as done in Geles et al. [4] - we intentionally retain GateNet's baseline performance to demonstrate that SkyDreamer can operate in the real world with poor-quality perception. Thanks to the use of StochGAN and additional data augmentations, SkyDreamer still completes 25 consecutive laps on the inverted loop track with MAVLab gates, as summarized in Table IV. Despite GateNet's poor performance, SkyDreamer demonstrates robustness to perceptual errors and bridges a non-trivial visual reality gap. This opens the door to using alternative more generalizable pixel-level abstractions such as depth maps, which could potentially enable high-speed, agile 14 Y Position [m] 8 6 4 2 0 2 4 6 8 X Position [m] Top-down View 10 5 0 5 10 Y Position [m] -2.0 -4.0 Z Position [m] Side View 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Speed [m/s] Figure 9: Real-world big track flown by SkyDreamer. In the top-down view, the flight begins in front of the top-left gate. After the first gate, the trajectory continues with a slight left turn, two subsequent gates, a right turn, and a ladder maneuver in which the drone passes through the lower gate, performs a full 360° left turn, and flies back over it. From there, SkyDreamer extends the right turn, executing a steep dive into the next gate, and continues through several more gates before performing a tight braking maneuver to initiate a sharp right turn. After the following gate, the track concludes with a split-S maneuver over the first gate. In the plots, the colored lines show SkyDreamer's position and velocity estimates for two real-world laps, with color indicating speed. Black blocks mark gate locations with exaggerated thickness. The bottom-right image presents a composite visualization of one lap from another flight flown by SkyDreamer. flight in unstructured environments. D. Real-world big track Additional real-world experiments demonstrate SkyDreamer's ability to fly the big track with MAVLab gates illustrated in Figure 9. This track combines high-speed flight with challenging maneuvers, including a split-S, a sharp right turn, and a ladder. The experiments are conducted in the real world in a larger hall and under poorer lighting conditions, further highlighting SkyDreamer's robustness across environments. We train SkyDreamer with slightly modified settings for these experiments: we disable symlog scaling in both the encoder and decoder [19], set the train_ratio to 64, and train for a total of 35 million environment steps. The second training stage begins after 23 million steps, and the third stage after 31 million steps. Furthermore, we set tg = 0.5 and introduce disturbances of ±300 rad/s to ωmax, randomly resampled every 10 timesteps, to account for imperfect actuator response modeling. During these experiments, five out of six flights consisting of two laps each were successful. The only crash occurred with a drone whose extrinsics lay outside the training distribution specified in Table III, as verified by a preflight Kalibr calibration, likely explaining the failure. The five successful flights were all with the same drone whose extrinsics are within the training distribution. In three of these flights, minor gate contact was audible once or twice, but SkyDreamer consistently carried on successfully, highlighting its ability to withstand and adapt to contact forces it was not explicitly trained for. In this track, SkyDreamer reaches accelerations comparable to those on the smaller tracks (around 6 g) while achieving significantly higher estimated speeds of up to 21 m/s. These results highlight SkyDreamer's ability to combine highspeed flight with agile and tight maneuvers. IV. CONCLUSION In this work, we presented SkyDreamer, to the best of our knowledge the first end-to-end vision-based ADR policy mapping pixel-level representations directly to motor commands. By building on informed Dreamer and extending it to end-to-end vision-based ADR, SkyDreamer provides interpretability by decoding to states and drone-specific parameters, effectively turning the world model into an implicit state and parameter estimator. SkyDreamer runs fully onboard without external aid, resolves visual ambiguities by tracking progress through the world model's decoded state, and requires no extrinsic camera calibration, enabling rapid deployment across different drones without retraining. Our real-world experiments demonstrate non-trivial sim-toreal transfer despite operating on low-quality segmentation masks, reliable performance across diverse tracks, and robustness to battery depletion. SkyDreamer successfully executes 15 tight maneuvers and reaches speeds up to 21 m/s with accelerations of up to 6 g - speed and agility not previously demonstrated by other end-to-end vision-based ADR methods in the real world. These results highlight SkyDreamer's adaptability to important aspects of the reality gap, bringing robustness while still achieving extremely high-speed, agile flight. Despite these promising results, several limitations remain. Parameter estimates tend to drift over time, and their quality varies across training runs. Decoded state estimates jump between timesteps, and SkyDreamer is still vulnerable to false positives in segmentation masks. In addition, training remains computationally demanding, requiring roughly 50 hours to complete. Future work will explore extending SkyDreamer to more generalizable visual inputs, such as depth maps, flying unseen tracks, and potentially generalizing to unstructured environments or hybrid tasks by combining drone racing with obstacle avoidance. Beyond racing, SkyDreamer represents a promising step toward deploying end-to-end vision-based yet interpretable policies for high-speed, agile flight in real-world settings. ACKNOWLEDGMENTS We are grateful to the other members of the drone racing team - Anton Lang, Till Blaha, Quentin Missine and Erin Lucassen - for their valuable insights and support during system integration and flight experiments. REFERENCES [1] E. Kaufmann, L. 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Available: https: //openreview.net/forum?id=Oxh5CstDJU [27] J. S. V, S. Jalagam, Y. LeCun, and V. Sobal, "Gradient-based planning with world models," 2023. [Online]. Available: https: //arxiv.org/abs/2312.17227 16 [28] G. Zhou, H. Pan, Y. LeCun, and L. Pinto, "Dino-wm: World models on pre-trained visual features enable zero-shot planning," 2025. [Online]. Available: https://arxiv.org/abs/2411.04983 [29] M. T. J. Spaan, Partially Observable Markov Decision Processes. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012, pp. 387-414. [Online]. Available: https://doi.org/10.1007/978-3-642-27645-3 12 [30] C. De Wagter, F. Paredes-Vall ́es, N. Sheth, and G. C. H. E. de Croon, "The sensing, state-estimation, and control behind the winning entry to the 2019 artificial intelligence robotic racing competition," Field Robotics, vol. 2, pp. 1263-1290, 2022. [31] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros, "Unpaired image-to-image translation using cycle-consistent adversarial networks," in Computer Vision (ICCV), 2017 IEEE International Conference on, 2017. [32] A. Almahairi, S. Rajeshwar, A. Sordoni, P. Bachman, and A. Courville, "Augmented CycleGAN: Learning many-to-many mappings from unpaired data," in Proceedings of the 35th International Conference on Machine Learning, ser. Proceedings of Machine Learning Research, J. Dy and A. Krause, Eds., vol. 80. PMLR, 7 2018, pp. 195-204. [Online]. Available: https://proceedings.mlr.press/v80/almahairi18a.html [33] A. van den Oord, O. Vinyals, and K. Kavukcuoglu, "Neural discrete representation learning." in NIPS, I. Guyon, U. von Luxburg, S. Bengio, H. M. Wallach, R. Fergus, S. V. N. Vishwanathan, and R. Garnett, Eds., 2017, pp. 6306-6315. [Online]. Available: http://dblp.uni-trier.de/db/conf/nips/nips2017.html#OordVK17 [34] J. Chung, C ̧ . G ̈ulc ̧ehre, K. Cho, and Y. Bengio, "Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling," in NIPS 2014 Workshop on Deep Learning and Representation Learning, ser. DLW 2014, Dec. 2014. [Online]. Available: http://www.dlworkshop.org/47.pdf?attredirects=0 [35] R. J. Williams, "Simple statistical gradient-following algorithms for connectionist reinforcement learning," Machine Learning, vol. 8, no. 3, pp. 229-256, 1992. [Online]. Available: https://doi.org/10.1007/ BF00992696 [36] S. Mysore, B. Mabsout, R. Mancuso, and K. Saenko, "Regularizing action policies for smooth control with reinforcement learning," in IEEE International Conference on Robotics and Automation, ICRA 2021, Xi'an, China, May 30 - June 5, 2021. IEEE, 2021, pp. 1810-1816. [Online]. Available: https://doi.org/10.1109/ICRA48506.2021.9561138 [37] K. Vafa, P. G. Chang, A. Rambachan, and S. Mullainathan, "What has a foundation model found? using inductive bias to probe for world models," 2025. [Online]. Available: https://arxiv.org/abs/2507.06952 [38] Z. Liu, H. Hu, Y. Lin, Z. Yao, Z. Xie, Y. Wei, J. Ning, Y. Cao, Z. Zhang, L. Dong, F. Wei, and B. Guo, "Swin transformer v2: Scaling up capacity and resolution," in 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022, pp. 11 999-12 009. [39] O. Ronneberger, P. Fischer, and T. Brox, "U-net: Convolutional networks for biomedical image segmentation," in Medical Image Computing and Computer-Assisted Intervention - MICCAI 2015, N. Navab, J. Hornegger, W. M. Wells, and A. F. Frangi, Eds. Cham: Springer International Publishing, 2015, pp. 234-241. [40] X. Glorot and Y. Bengio, "Understanding the difficulty of training deep feedforward neural networks," in Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, ser. Proceedings of Machine Learning Research, Y. W. Teh and M. Titterington, Eds., vol. 9. Chia Laguna Resort, Sardinia, Italy: PMLR, 5 2010, pp. 249-256. [Online]. Available: https: //proceedings.mlr.press/v9/glorot10a.html [41] P. Isola, J.-Y. Zhu, T. Zhou, and A. A. Efros, "Image-to-image translation with conditional adversarial networks," in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017, pp. 59675976. APPENDIX A GATENET IMPLEMENTATION DETAILS a) Network: For segmentation, we adopt a U-Net style architecture [39], referred to as GateNet, similar to Bahnam et al. [2]. The network consists of an encoder-decoder with skip connections from the encoder to the corresponding decoder layer, following the standard U-Net design [39], where channel dimensions are scaled by a factor f. Each block consists of double convolutional layers with batch normalization and ReLU activation functions. Let inc denote the initial double convolutional block. downk denotes a max-pooling layer followed by a double convolutional block with k output channels. Before applying downk, the corresponding features are stored as skip connections for the decoder. upk denotes a transposed convolution and batch normalization, followed by adding the skip connections from the corresponding encoder layer, and then a double convolutional block with k output channels. Finally, outc-k denotes a 1×1 convolution mapping to a single output channel and a sigmoid activation function. The resulting GateNet architecture is: Encoder Decoder Outputs inc-64/f →up4-64/f →outc4-1 ↓ ↑ down1-128/f →up3-64/f →outc3-1 ↓ ↑ down2-256/f →up2-128/f →outc2-1 ↓ ↑ down3-512/f →up1-256/f →outc1-1 ↓ ↗ down4-512/f →outc0-1 The network produces five output maps {y0, y1, y2, y3, y4} at different resolutions. These multi-scale predictions are supervised with auxiliary losses to improve gradient flow and overall segmentation accuracy. b) Training setup: We use the same hyperparameters as in Bahnam et al. [2]. Each output map is supervised with a combination of Dice loss and binary cross-entropy (BCE) loss: Li = LDice(yi, ˆyi) + 2 · LBCE(yi, ˆyi), and we apply output-specific scaling factors to emphasize higher-resolution predictions: Ltotal = 4 · L0 + 2 · L1 + 4 X i=2 Li. All convolutional layers are initialized using Xavier uniform initialization [40]. c) Data augmentation: We largely use the same augmentations as in Bahnam et al. [2]. Since our images are captured with a low exposure time (1 ms), we do not simulate motion blur with KernelBlur. Instead, we apply stronger shot noise (standard deviation of 40 for pixel values in the range 0-255 instead of 25) to mimic the increased noise associated with low exposure times. d) Gate-specific implementation: For MAVLab gates, we use a resolution of 196 × 196 with f = 2. For orange gates, we use a resolution of 384 × 384 with f = 4. APPENDIX B STOCHGAN IMPLEMENTATION DETAILS a) Generator: The generator is a CNN-based encoder-residual-decoder network, largely based on Zhu et 17 al. [31]. We use 6 residual blocks and images of size 64 × 64, the same resolution as those rendered and used by SkyDreamer during training. Following a similar convention to Zhu et al. [31], let c7s1-k denote a 7 × 7 Convolution-InstanceNorm-ReLU layer with k filters and stride 1, using reflection padding. dk corresponds to a 3 × 3 Convolution-InstanceNorm-ReLU layer with k filters and stride 2 for downsampling. Rk denotes a residual block consisting of two 3 × 3 Convolution-InstanceNorm-ReLU layers with k filters, using reflection padding and a skip connection. uk represents a 3 × 3 transposed Convolution-InstanceNorm-ReLU layer with k filters, stride 2 for upsampling, and reflection padding. Finally, the output layer is denoted as c7s1-k-tanh, which follows the same convention as c7s1-k but uses a tanh activation at the final layer. The resulting generator architecture is: Encoder: c7s1-32, d64, d128 Residual blocks: 6 × R128 Decoder: u64, u32, c7s1-1-tanh To obtain a StochGAN [32], we add an additional noise channel, uniformly sampled from [-1, 1], to the input image. b) Discriminator: Similar to Zhu et al. [31], the discriminator follows a PatchGAN design [41]. Largely following the design and conventions in Zhu et al. [31], let Ck denote a 4 × 4 Convolution-InstanceNorm-LeakyReLU layer with k filters and stride 2. InstanceNorm is omitted in the first Ck layer, and LeakyReLU activations with a negative slope of 0.2 are used. After the last block, we apply zero-padding followed by a 4×4 convolution with 1 channel (conv4-1), producing a patch-wise realism score. The resulting discriminator architecture is: C32, C64, C128, C256, conv4-1 c) Training setup.: We largely follow the training procedure of Zhu et al. [31], except we use the Adam optimizer with β1 = 0.5, β2 = 0.999, and a base learning rate of 1.5 × 10-4.
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2510.14782
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JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
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Raman-Accelerated Power Depletion of Fundamental Mode in a
Few-Mode Fiber in the Visible Spectral Range
Wasyhun A. Gemechu*, Guohao Fu, and Mario Zitelli
Abstract—We
experimentally
and
numerically
investigate
Raman-driven power depletion in the fundamental mode of
few-mode fibers (FMFs) excited by visible ultrashort pulses.
Using a tunable femtosecond laser and SMF-28 fibers operated
below the single-mode cutoff wavelength, we explore nonlinear
mode dynamics through precise coupling, holographic mode
decomposition, and spectral analysis. Experiments on 12-meter
and 50-meter fibers reveal significant energy transfer to higher-
order modes via intermodal Raman scattering. These findings
advance our understanding of nonlinear propagation in FMFs,
with important implications for high-power laser delivery and
next-generation optical communication systems.
Index Terms—Few-mode fiber (FMF), Nonlinear dynamics, In-
termodal Raman Scattering, Holographic Mode-decomposition,
Higher-Order Modes (HOMs), Energy Transfer.
I. INTRODUCTION
T
HE relentless human pursuit to master light, the most
essential medium for information and energy, has, for
decades, manifested its most refined form within the realm
of optical fibers. A delicate interplay between linear and non-
linear phenomena governs the propagation of high-intensity
light pulses through these waveguides. For a long time, single-
mode fiber (SMF) stood as the undisputed cornerstone of
optical communications and a multitude of laser applications;
its dominance was attributed to its well-defined characteristics
and exceptionally low dispersion. However, the continuous
march of technological advancement, coupled with an insa-
tiable demand for ever-greater information throughput and the
imperative for superior power handling, has compelled the
attention of optical researchers to the realms of multimode
fibers (MMFs) and few-mode fibers (FMFs) [1], [2]. These
fibers, capable of guiding multiple spatial modes, unveil a
promising trajectory toward unprecedented spectral efficiency
through mode/space division multiplexing (MDM/SDM), pro-
viding a genuine means to surpass the capacity constraints
of current optical networks [3], [4]. Beyond the demands of
telecommunications, the rich tapestry of their modal diversity
makes them uniquely advantageous for the robust transmis-
sion of high-power beams and the exploration of the fertile
landscape of complex nonlinear optical interactions [5], [6].
Manuscript received August x, 2025; revised Y Z, 2025.(Corresponding
author: Wasyhun A. Gemechu)
Wasyhun A. Gemechu is with the Department of Information Engineer-
ing, Universit`a degli Studi di Brescia, 25123 Brescia, Italy (e-mail: wasy-
hun.gemechu@unibs.it)
Guohao Fu is with the Department of Precision Instrument, Tsinghua
University, Beijing 100084, China (e-mail: fgh21@mails.tsinghua.edu.cn)
Mario Zitelli is with the Department of Information Engineering, Electron-
ics, and Telecommunications, Sapienza University of Rome, Via Eudossiana
18, 00184, Rome, Italy (e-mail: mario.zitelli@uniroma1.it)
A particularly effective strategy for investigating FMF-like
behavior involves reconfiguring conventional SMFs, coax-
ing them to operate at wavelengths below their fundamen-
tal single-mode cutoff [7]. In this specific spectral regime,
higher-order modes (HOMs), which are ordinarily confined to
evanescent decay, find themselves guided, thereby transform-
ing the SMF into a functional FMF capable of supporting
a limited number of spatial modes [8]. This change, while
brilliant, creates a slew of novel complexities. The simulta-
neous presence of numerous guided modes causes significant
intermodal dispersion, promotes subtle yet pervasive modal
coupling, and triggers a range of nonlinear effects. Self-phase
modulation (SPM), cross-phase modulation (XPM), and four-
wave mixing (FWM) are well-documented examples; however,
the phenomenon of stimulated Raman scattering (SRS) [9],
[10] is of paramount interest for our present investigation.
SRS, an inelastic scattering process fundamentally driven by
the vibrational modes of the silica matrix itself, facilitates
the transfer of optical energy from shorter to longer wave-
lengths—a phenomenon universally known as the Stokes shift.
In the intricate confines of FMFs, this energy cascade can
manifest not merely within the initially excited fundamental
mode but also across several distinct spatial modes, resulting
in complex and often counterintuitive patterns of spectral
and spatial energy redistribution [11], [12]. Such intermodal
Raman scattering is conveniently used for signal amplification
in SDM systems [13] and can, indeed, precipitate a significant
depletion of power from the fundamental mode via SRS
intermodal coupling, thereby compromising both the efficiency
and stability of FMF-based systems [14], [15].
While previous studies have rigorously documented in-
stances of beam self-cleaning phenomena—wherein the non-
linear evolution of light within a fiber appears to preferentially
favor the fundamental mode [16], [17]—our meticulous mea-
surements reveal an unexpected and opposing trend. As the
optical power introduced into solid-core FMFs increases, we
observe a strong nonlinear modal instability, characterized by
a substantial depletion of the fundamental LP01 mode. This
enigmatic behavior, we contend, originates from a subtle yet
profound mechanism: the differential Raman gain experienced
across the various spatial modes. This disparity, in turn, arises
from the unique spatial profiles of these modes and the subtle
differences in their effective refractive indices, creating a
selective amplification that ultimately undermines the stability
of the desired fundamental mode propagation.
II. EXPERIMENTAL SETUP
Figure 1 presents the experimental setup used for this
study. The laser source was a hybrid optical parametric am-
arXiv:2510.14782v1 [physics.optics] 16 Oct 2025
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
2
plifier (OPA) system (Light Conversion ORPHEUS-F), seeded
by a white-light continuum and pumped by a femtosecond
ytterbium-doped laser (Light Conversion PHAROS-SP-HP).
This system generates linearly polarized pulses of 70 fs dura-
tion at a repetition rate of 100 kHz. The central wavelength
was tunable from 650 to 900 nm and fixed at 700, 800, and
900 nm-deliberately chosen to fall below the modal cut-off
wavelength of the standard SMF-28 fiber (around 1450 nm
[18]), thereby ensuring the excitation of only a few spatial
modes. The laser beam, possessing a near-Gaussian profile
and a beam quality factor M 2 = 1.3, was precisely focused
into the fiber core using a 20× microscope objective (with a
numerical aperture of NA = 0.4). A variable neutral density
filter (VNDF), placed just before the objective, allowed fine
control of the input average power ranging from 0.1 to 5 mW,
thereby yielding peak powers close to 100 kW. The fiber itself,
a standard Alcatel SMF-28, with a core diameter of 8.8 µm
and a cladding diameter of 125µm, with a cladding refractive
index of nclad = 1.4574 at 630 nm. The relative index contrast
was ∆n = 0.003. A subtle index dip in the center of the core
is characteristic of this fiber.
Fig. 1: A schematic of the experimental setup. VNDF: Variable
neutral density filter; L1-L3: Lens; BPF: Bandpass filter;
HWP: half-wave plate; LP: Linear polarizer; FM: Flip mirror;
NF & FF camera: Near and Far field camera; SLM: Spatial
light modulator.
To investigate the beam propagation dynamics across vary-
ing interaction lengths, two distinct fiber lengths were used:
one measuring 12 meters and the other 50 meters. The cou-
pling into the fiber core required precise alignment via a three-
axis translation stage with micrometer precision. When fastid-
iously aligned on-axis, the coupling efficiency—the ratio of
transmitted to input power under linear conditions—averaged
around 75%. Furthermore, approximately ∼4% of the input
power resides within 10 nm of the 700 nm central wavelength,
a testament to the significant power carried by the spectral
components outside this narrow band. Although our primary
goal was to excite the fundamental LP01 mode, even minor
misalignment could result in contributions from HOMs.
At the fiber output end, the laser beam was collimated using
a microlens (µLens, f = 2.7 mm) and an uncoated lens (L2,
f = 400 mm). The beam was then directed towards a Hama-
matsu spatial light modulator (SLM) via a half-wave plate
(HWP) and a linear polarizer (LP). These optical elements,
combined, provide an exquisite means of finely tuning the
beam’s intensity and polarization before its interaction with the
SLM, thereby increasing modulation efficiency. A flip mirror
(FM) allowed for quick switching between two detection
pathways: one for imaging the near-field beam profile onto
a CCD camera (NF camera) and another for capturing the
Fourier transform of the beam onto a second CCD (FF camera)
positioned after a 500 mm convex lens (L3). We were able to
precisely project the output onto specified spatial modes by
adding tailored phase masks to the SLM and quantifying their
power contributions, as described in Ref. [19]. To eliminate the
unwanted specters of self-phase modulation, which emerge as
spectral broadening or temporal decoherence, a narrowband
bandpass filter (10 nm bandwidth) was strategically placed
immediately after the fiber’s exit facet, exactly centered at
the pump wavelength (700, 800, or 900 nm). This ensured
that only the desired spectral components were analyzed
downstream. The output spectrum, spanning from 600 to 1700
nm, was then recorded by an optical spectrum analyzer (OSA)
(Yokogawa AQ6370D), which allowed us to identify and
exclude parasitic signals, such as Stokes and anti-Stokes shifts.
III. EXPERIMENTAL RESULT
The guided modes supported by the optical fiber were com-
puted using the semi-vectorial finite difference method, based
on the experimentally measured refractive index profile. This
numerical approach, presented in section IV, demonstrates that
the fiber can propagate six LP modes at a wavelength of
700 nm: LP01, LP11a, LP11b, LP21a, LP21b, and LP02. To
accurately resolve the modal composition at the fiber output,
we employed a holographic mode decomposition technique, as
comprehensively delineated in Ref. [19]. The resulting power
fractions associated with each LP mode are detailed in Fig. 2.
The inset, in particular, shows a comparison between the
experimentally obtained near-field intensity profile (left panel)
and the reconstructed modal distribution generated from FF
camera data (right panel), which is perfectly consistent with
the methods given in Ref. [19]. Under conditions of low-power
linear propagation, specifically for an input power of 0.1 mW
(corresponding to a pulse energy of Ep = 1 nJ) with on-axis
coupling, the output beam predominantly maintains the spatial
profile characteristics of the fundamental mode of the fiber. As
illustrated in Fig. 2a, the LP01 mode carries most of the optical
power, with only minor contributions from HOMs. However,
as the input power increases, the modal composition undergoes
a marked evolution. At 1.6 mW (Ep = 16 nJ), the LP11a/b
modes become significantly populated, as shown in Fig. 2b.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
3
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.2
0.4
0.6
0.8
Mode Power Fraction (a.u.)
a)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
0.5
b)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.2
0.4
0.6
c)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.2
0.4
0.6
d)
Fig. 2: Experimentally measured mode power distributions for fiber length of 50 m at center wavelength of λ = 700 nm for
input average powers of a) Pin = 0.1 mW, b) Pin = 1.6 mW, c) Pin = 4.5 mW, and d) Pin = 5.0 mW.
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.2
0.4
0.6
Mode Power Fraction (a.u.)
a)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.2
0.4
0.6
d)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
0.5
Mode Power Fraction (a.u.)
b)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
0.5
e)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
0.5
Mode Power Fraction (a.u.)
c)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
0.5
f)
Fig. 3: Experimental mode power distributions at the fiber
output for two center wavelengths and varying input powers:
(a–c) At λ = 800 nm: a) Pin = 0.1 mW, b) Pin = 1.1 mW,
and c) Pin = 6.6 mW. (d–f) At λ = 900 nm: d) Pin =
0.1 mW, e) Pin
= 1.8 mW, and f) Pin
= 5.0 mW, for
50-m-long fiber.
Upon entering strongly nonlinear regime, with input powers
of 4.5 mW (Ep = 45 nJ) and 5.0 mW (Ep = 50 nJ), the
fundamental mode gets significantly depleted, and its energy
is redistributed into higher-order spatial modes, including
LP21a/b and LP02, as compellingly illustrated in Figs. 2c and
2d.
Crucially, this transformative reconfiguration of the modal
occupancy persists across distinct pump wavelengths. A strik-
ingly similar behavior is observed for input laser carrier
wavelengths of 800 nm and 900 nm, as clearly illustrated in
Fig. 3. The near-field intensity and corresponding modal de-
composition at the fiber exit facet for varying input powers are
showcased in Figs. 3(a)-(c) for 800 nm and Figs. 3(d)-(f) for
900 nm. In the linear regime, with an input power of 0.2 mW,
the beam exhibits a quintessential bell-shaped intensity profile,
a hallmark unmistakable of an output dominated by the
fundamental mode. However, as the input power exceeds
1.2 mW, the power progressively shifts toward HOMs—most
prominently LP11a/b and their coherent superposition-clearly
signaling strong nonlinear coupling. At the highest pump
levels, the output beam structure strikingly resembles that of
an orbital angular momentum (OAM) mode or vortex beam,
underscoring the pronounced, wavelength-sensitive dynamics
of nonlinear mode excitation.
Figure 4 depicts an experimentally observed spectrum with
a very broad spectrum centered at 900 nm and reaching
around 40 nm in width. The spectrum shows a notable red-
shift and an abrupt truncation on its blue flank, providing
unequivocal evidence of asymmetric spectral widening. The
lack of obvious additional sidebands strongly suggests that
this broadening arises from a complex and dynamic interplay
between SPM and SRS, rather than from phase-matched para-
metric processes. This interaction is not cumulative; instead,
it involves a competitive redistribution of energy within the
pulse envelope. Ordinarily, SPM would orchestrate a sym-
metric spectral broadening, shaping both the red and blue
wings of the spectrum equally. However, SRS intervenes
with notable efficacy, redirecting a substantial portion of the
pump pulse’s energy into red-shifted Stokes components. In
doing so, it significantly depletes the pulse’s peak intensity—a
critical factor for efficient SPM—and consequently impedes
the generation of its characteristic blue-shifted constituents.
This phenomenon is particularly pronounced in the normal
dispersion regime, where the Stokes wave lags the pump,
thereby prolonging its interaction length and increasing energy
transfer. Simultaneously, SRS induces a progressive red shift
via the Raman gain, augmenting the spectral density on the
red spectral tail. The resulting spectral shape—truncated in the
blue and expanded in the red—is the fascinating consequence
of a complicated tug-of-war in which SRS both undermines
and overwhelms SPM in a delicate energy balance.
To further investigate the spatial dynamics that accompany
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
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6
4
2
-20
Wavelength [nm]
-19
0
-18
800
-17
Intensity [a.u]
850
900
950
-16
1000
1050
-15
-14
Pin = 0.2 mW
Pin = 0.8 mW
Pin = 1.6 mW
Pin = 2.4 mW
Pin = 3.2 mW
Pin = 3.8 mW
Pin = 4.5 mW
Pin = 5.0 mW
Fig. 4: Experimental spectra obtained from a 50-m-long Alca-
tel SMF-28 as a function of the input guided power measured
at the input for a laser pulse centered at 900 nm.
this spectrum evolution, we carefully removed the band-pass
filter at the fiber’s output and recorded a series of near-field
intensity profiles corresponding to input pulses at 700 nm,
800 nm, and 900 nm. These illuminating tests unequivocally
indicated a significantly rapid depletion of the fundamental
mode at relatively low input levels, owing once again to the
presence of the Raman effect. Fig. 5a vividly illustrates how
the output beam transitions from a LP11-like mode to an
LP21-like structure with noticeable center power depletion,
a configuration distinct from the mode structure observed in
Fig. 2. This intriguing evolution is further corroborated at
longer wavelengths. Fig. 5b demonstrates that under 800 nm
excitation and low input power, the output beam exhibits
a near-Gaussian profile, indicative of the dominance of the
fundamental mode. When the input power exceeds PIn
=
1.4 mW, the beam undergoes a remarkable metamorphosis
from an LP21-like profile to a structure resembling an optical
vortex (OAM-like mode). Surprisingly, this vortex-like phase
remains stable even when subjected to further power increases
or external perturbations. A nearly similar progression occurs
at 900 nm. Figure 5c captures how a bell-shaped fundamental
mode produces a higher-order LP21-like profile with a hollow
center. Intriguingly, this transition has surprising robustness:
regardless of the initial coupling conditions—be it axial align-
ment or a deliberate lateral offset—the beam always moves
towards an OAM-like shape as the input power approaches
5 mW. This profound universality suggests an underlying
nonlinear mode-selection mechanism, intricately regulated by
the concerted interaction of SPM, SRS, and modal dispersion.
To deepen our understanding of how fiber length influ-
ences Raman-driven modal power transfer, a complementary
experiment was conducted using a 12 m-long fiber, which is
shorter than in previous experiments. This fiber was excited
with a bell-shaped beam centered at 700 nm. Fig. 6 presents
the evolution of the modal power distribution at the output,
measured for varying input power levels. At a lowest input of
PIn = 0.1 mW (Fig. 6(a)), the output remains overwhelmingly
confined to the fundamental LP01 mode. This dominance re-
Fig. 5: A fundamental mode reshaping near-field profile in
a 50 m-long fiber with input wavelengths of (a) 700 nm, (b)
800 nm, and (c) 900 nm. (d) With an offset input, the same
as (c), for different input average power PIn.
flects predominantly linear propagation, with negligible energy
transfer to HOMs. As the input increases to PIn = 2.0 mW
(Fig. 6(b)), the first signs of nonlinear behavior emerge. While
the LP01 mode still carries a substantial portion of the power,
a noticeable shift occurs: the energy is gradually diverted
into LP11a and LP11b, and to a lesser extent, LP21a, LP21b,
and LP02. This signals the beginning of intermodal Raman
interactions. Further increasing the input to PIn = 4.0 mW
(Fig. 6(c)) enhances this redistribution. The LP01 mode grad-
ually fades, replaced by more noticeable contributions from the
higher-order LP21a/b. At the highest input of PIn = 6.0 mW
(Fig. 6(d)), the modal landscape is transformed. The LP11a/b
modes now dominate the output, with significant power also
coming from the LP21a/b. The fundamental LP01 mode is
significantly reduced, indicating an effective transfer of energy
to higher-order spatial modes due to the Raman effect.
IV. NUMERICAL RESULTS
The propagation of ultrashort pulses in an Alcatel SMF-28
optical fiber operating below its cutoff wavelength, which sup-
ports M different spatial modes, can be accurately described
by the generalized multimode nonlinear Schr¨odinger equation
(GMMNLSE) [20], [21]. This model captures the complex
spatiotemporal evolution of the optical field in a few-mode
fiber (FMF). The total optical field envelope is expressed as a
superposition of the fiber’s eigenmodes:
E(x, y, z, t) =
M
X
p=1
Fp(x, y)Ap(z, t)
(1)
where Fp(x, y) denote the transverse mode profiles and
Ap(z, t) represents a slowly varying complex temporal en-
velope associated with the pth mode. For the purpose of this
study, considering M = 6 modes, the GMMNLSE for the
envelope of the pth mode, Ap(z, t), is given by:
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
5
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.2
0.4
0.6
0.8
Mode Power Fraction (a.u.)
a)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
b)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
c)
LP01
LP11a LP11b LP21a LP21b
LP02
0
0.1
0.2
0.3
0.4
d)
Fig. 6: Experimentally measured mode power distributions for fiber length of 12 m at center wavelength of λ = 700 nm for
input average powers of a) Pin = 0.1 mW, b) Pin = 2.0 mW, c) Pin = 4.0 mW, and d) Pin = 6.0 mW.
∂zAp(z, t) = i (βp
0 −β0) Ap −(βp
1 −β1) ∂Ap
∂t
+
4
X
q≥2
iq+1 βp
q
q! ∂m
t Ap
−αp
2 Ap + in2ω0
c
1 + i
ω0 ∂t
X
l,m,n
{(1 −fR)Sk
plmnAlAmA∗
n
+ fRAlSR
plmn
Z t
−∞
Am(z, t −τ)A∗
n(z, t −τ)hR(τ)dτ}
(2)
The coefficient β(p)
q
represents the qth order dispersion term
for the pth mode. The first two terms account for the phase
mismatch and the group velocity mismatch among modes.
The third term describes higher-order chromatic dispersion.
Nonlinear contributions—mediated by the Kerr coefficient
n2 = 2.7 × 10−20m2/W, central frequency ω0, and nonlin-
ear overlap integrals Sk
plmn—capture instantaneous nonlinear
effects (SPM, XPM, FWM). The Raman effect enters via
the fractional response fR ≈0.18, Raman overlap integrals
SR
plmn, and a delayed response function hR(τ) parameterized
by τ1 = 12.2 fs and τ2 = 32 fs. Modal specific attenuation is
accounted for by loss coefficients αp.
As detailed in the experimental section, the simulation
employed an Alcatel SMF-28 with a step-index core diameter
of 8.8 µm and an index contrast of ∆n = 0.003 between
its peak core and surrounding cladding. At a wavelength of
700 nm, the fiber exhibited normal chromatic dispersion, with
β2 ranging from 22.2 to 34.9 ps2/km across LP01 to LP02
modes, respectively. Furthermore, the modal delays relative to
the fundamental LP01 mode were 2.0, 2.0, -3.2, -3.2, and -
9.3 ps per meter of fiber for modes 2 through 6. Eq. 2 was
numerically solved with an integration step of 50µm, utilizing
a spatial window of 125 µm × 125 µm and a transverse grid
of 800 × 800 samples. An ultrashort pulse, with a duration
T0 = 70 fs at a full width half maximum (FWHM) and a
diameter of 2w0 = 15 µm, was launched primarily into the
fundamental mode of the FMF. The simulation explored input
pulse energies varying from 1 nJ to 30 nJ, across fiber lengths
of 1 m, 6 m, and 12 m. The initial fractional mode power
distribution is depicted in Fig. 2a.
Figures 7 and 8 reveal a fascinating set of numerical results,
clearly demonstrating the profound impact of the Raman
effect on ultrashort pulse propagation within FMFs. Consider
a fiber length of 6 m and an input pulse energy of 30 nJ.
In the absence of the Raman effect (fR = 0), the temporal
.
-LP01
........ ,LP 11a
........ LP 11 b
- - ,LP21a
- -
LP21b
- -
LP02
....... .
. ..
600
:·. .
.•••• E
.
. ..
700
800
Wavelength [nm]
a)
. .
.. ♦: •
•••
900
600
b)
700
800
900
Wavelength [nm]
Fig. 7: Raman’s transformative role in FMF dynamics. Com-
parative numerical simulations illustrate the output modal
spectral profiles after 6 m of FMF, (a) with and (b) without
the Raman nonlinearity. Input pulse with 70 fs pulsewidth and
30 nJ energy.
Fig. 8: Simulated output modal power after 6 m of FMF: (a)
with, and (b) without, the Raman nonlinearity. Input pulse with
70 fs pulsewidth and 30 nJ energy.
profiles of the modes evolve with a serene smoothness, as
evident in Fig. 8(b). Here, modal pulses undergo broadening
and distortion, a direct consequence of the interplay between
SPM and normal chromatic dispersion. This very interplay
also orchestrates a moderate blue-shift of the second modal
group (modes LP11a and LP11b) and a red-shift of the third
modal group (modes LP21a, LP21b and LP02) relative to
the fundamental mode (Fig. 7(b)). Modal dispersion cannot
temporally distinguish modal pulses due to pulse broadening;
yet, XPM induces minimal nonlinear power exchange among
modes. However, when Raman nonlinearity is introduced
(fR = 0.18), SRS emerges as the dominant effect, forcing a
net energy transfer from the fundamental mode to the second
mode group. This results in amplification of the spectra for
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
6
modes LP11a, LP11b, which concurrently leads to a depleted
spectrum at lower wavelengths for mode LP01, as illustrated
in Fig. 7(a). In the temporal domain, the instantaneous power
of the fundamental mode experiences a marked depletion
in correspondence with its trailing tail, precisely where the
modes of group 2, delayed by the modal dispersion, experience
a Raman gain (Fig. 8(a)). At sufficiently high powers and
extended interaction lengths, the onset of cascaded Raman
scattering was observed. Here, the first Stokes-shifted light acts
as a pump for subsequent Raman shifts, resulting in multiple
distinct spectral peaks. The intermodal spectrum transfer is
governed by the interaction of Raman gain and phase-matching
conditions among multiple modes at different frequencies,
producing a complex signature of spatiotemporal dynamics.
XPM, though present, plays a minor role in comparison.
Fig. 9: Simulated: (a) modal energy distribution at input and
after 6 m of FMF, when the input pulsewidth is 70 fs and 30
nJ total energy; (b) input near-field; (c) output near-field.
Figure 9(a) offers compelling evidence of the simulated
modal energy distribution of input and output when a 70-
fs pulse, with 30 nJ energy, propagated over 6 m of FMF.
In the absence of the Raman effect (not shown), the out-
put energy in each mode remains consistently aligned with
the input throughout the propagation length, exhibiting no
significant transfer of power between different spatial modes
during propagation. On the contrary, when the Raman effect is
taken into account, the modal energies undergo dramatic and
dynamic metamorphoses. The initially populated fundamental
mode (mode index 1, depicted by the red curve) experiences a
significant decrease in its energy as light propagates. Concur-
rently, the HOMs (e.g., mode group 2, represented by mode
indexes 2 and 3 in the blue curve) exhibit a noticeable increase
in their energy. This unmistakable observation is the direct
manifestation of Raman-induced inter-modal energy transfer:
energy from the depleted fundamental mode (and potentially
other pump modes) is efficiently transferred to these Stokes-
shifted HOMs. Thus, while the near field is dominated by
the fundamental mode at input or in the absence of SRS
(Fig. 9(b)), the metamorphic power of Raman nonlinearity
produce an output near field that is undeniably dominated
by modes LP11a, LP11b (Fig. 9(c)), with the contribution
of the fundamental mode rendered negligible. The efficiency
and specific target modes of this energy transfer are critically
dependent on the group velocity matching between the pump
and Stokes components in different modes, as well as the
spatial overlap integrals of the interacting modes.
In close accordance with our experimental findings, we
performed a supplementary computational analysis to investi-
gate the influence of fiber length on Raman-driven intermodal
energy transfer. Simulations were carried out for two specific
lengths—1 meter and 12 meters—at input pulse energies
of 20 nJ and 30 nJ. In the shorter fiber, energy initially
limited to the fundamental mode saw significant redistribution,
with HOMs, especially the second mode group, acquiring a
substantial share of the energy. The near-field intensity profile
matched the simulation results given in Fig. 9(c), capturing
the key aspects of modal reshaping. At 12 meters, the overall
structure in the near-field remained relatively unchanged, but
the energy transfer increased significantly in the third mode
group. These findings reveal a surprising length independence
of the nonlinear mode coupling, with Raman interactions
resulting in identical behavior of energy transfer from the
fundamental mode LP01 to the HOMs with indexes of 2
(LP11a) and 3 (LP11b), as well as a near-field profile similar
to that seen in Fig. 9(c).
Figure 10a illustrates the evolution of modal energy within
the first 6 meters of the FMF, following the launch of a 70-fs
pulse at a wavelength of 700 nm with an input energy of 30
nJ, distributed as depicted in Fig. 9. The simulation reveals a
rapid migration of energy from the fundamental LP01 mode to
the LP11 mode group. This preferential transfer is primarily
due to their significant modal overlap, which facilitates an
efficient exchange of optical power. In contrast, the higher-
order LP21 and LP02 modes within group 3 show a markedly
reduced energy coupling with the LP01 mode. This intriguing
dynamic ultimately stabilizes at a steady state, influenced by
the temporal broadening and distortion of the pulse caused by
SPM.
Turning to the frequency domain, the spectral evolution
of each mode unveils a more nuanced narrative (Fig. 10b).
While the Raman effect is universally acknowledged as a
primary driver of intensity-dependent wavelength redshift, our
findings introduce a captivating subtlety: the degree of redshift
is not uniformly dictated by intensity across all modes. The
LP01 mode, initially endowed with the lion’s share of energy,
undergoes the most significant redshift, a direct consequence
of strong intramodal Raman scattering. Yet, a curious paradox
emerges with the LP11 modes: despite siphoning off a substan-
tial portion of the LP01 mode’s energy, they exhibit the least
pronounced redshift. This seemingly counterintuitive behavior
originates from the very nature of intermodal energy transfer.
The Raman intermodal coupling between the LP01 and LP11
modes predominantly occurs at the shorter-wavelength compo-
nents of the LP01 mode. Consequently, the energy transferred
to the LP11 modes is predominantly situated within their
short-wavelength spectral regions. This spectral localization
effectively hinders the characteristic Raman redshift of the
LP11 modes, as the very process of energy acquisition acts as
a counterbalance to it. In stark contrast, mode group 3, char-
acterized by its meager overlap and weak coupling with the
LP01 mode, remains largely unaffected by this compensatory
mechanism, thus preserving its inherent Raman redshift.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025
7
(a)
(b)
Fig. 10: (a) Simulated evolution of modal energy and (b) simulated evolution of the central wavelength in the first 6 m of the
FMF, when the input pulse width is 70 fs and the total energy is 30 nJ.
V. CONCLUSION
Our combined experimental and numerical study demon-
strates that nonlinear propagation in few-mode fibers is gov-
erned by a complex interplay of SPM, SRS, and modal
dispersion, resulting in dramatic changes to both spatial and
spectral beam characteristics. As input power increases, energy
is progressively transferred from the fundamental LP01 mode
to HOMs such as LP11a/b and LP21a/2, with the emergence of
vortex-like (OAM) profiles indicating robust nonlinear mode
coupling. Spectral measurements confirm that the asymmetric
broadening, dominated by SRS, is particularly pronounced at
longer wavelengths and higher powers, with Raman-induced
redshifts effectively suppressing the blue components typically
generated by SPM. Simulations validate these findings, reveal-
ing that Raman interactions induce strong intermodal energy
transfer and significantly reshape the modal power distribution.
Notably, cascaded Raman scattering further enriches the output
spectrum, highlighting the sensitivity of pulse dynamics to
both input conditions and fiber properties. These results not
only elucidate the nonlinear mechanisms at play in multimode
fibers but also offer valuable pathways for tailoring ultrafast
light fields in advanced optical systems.
ACKNOWLEDGMENTS
We acknowledge Vincent Couderc for providing optical
fiber characterization, M. Gervaziev, D. Kharenko, and S.
Babin for providing the mode decomposition setup.
Project ECS 0000024 Rome Technopole, Funded by the
European Union – NextGenerationEU.
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JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 1 Raman-Accelerated Power Depletion of Fundamental Mode in a Few-Mode Fiber in the Visible Spectral Range Wasyhun A. Gemechu*, Guohao Fu, and Mario Zitelli Abstract-We experimentally and numerically investigate Raman-driven power depletion in the fundamental mode of few-mode fibers (FMFs) excited by visible ultrashort pulses. Using a tunable femtosecond laser and SMF-28 fibers operated below the single-mode cutoff wavelength, we explore nonlinear mode dynamics through precise coupling, holographic mode decomposition, and spectral analysis. Experiments on 12-meter and 50-meter fibers reveal significant energy transfer to higherorder modes via intermodal Raman scattering. These findings advance our understanding of nonlinear propagation in FMFs, with important implications for high-power laser delivery and next-generation optical communication systems. Index Terms-Few-mode fiber (FMF), Nonlinear dynamics, Intermodal Raman Scattering, Holographic Mode-decomposition, Higher-Order Modes (HOMs), Energy Transfer. I. INTRODUCTION T HE relentless human pursuit to master light, the most essential medium for information and energy, has, for decades, manifested its most refined form within the realm of optical fibers. A delicate interplay between linear and nonlinear phenomena governs the propagation of high-intensity light pulses through these waveguides. For a long time, singlemode fiber (SMF) stood as the undisputed cornerstone of optical communications and a multitude of laser applications; its dominance was attributed to its well-defined characteristics and exceptionally low dispersion. However, the continuous march of technological advancement, coupled with an insatiable demand for ever-greater information throughput and the imperative for superior power handling, has compelled the attention of optical researchers to the realms of multimode fibers (MMFs) and few-mode fibers (FMFs) [1], [2]. These fibers, capable of guiding multiple spatial modes, unveil a promising trajectory toward unprecedented spectral efficiency through mode/space division multiplexing (MDM/SDM), providing a genuine means to surpass the capacity constraints of current optical networks [3], [4]. Beyond the demands of telecommunications, the rich tapestry of their modal diversity makes them uniquely advantageous for the robust transmission of high-power beams and the exploration of the fertile landscape of complex nonlinear optical interactions [5], [6]. Manuscript received August x, 2025; revised Y Z, 2025.(Corresponding author: Wasyhun A. Gemechu) Wasyhun A. Gemechu is with the - ing, Universit`a degli Studi di Brescia, 25123 Brescia, Italy (e-mail: wasy- ) Guohao Fu is with the 100084, China (e-mail: ) Mario Zitelli is with the - ics, and Telecommunications, Sapienza 18, 00184, Rome, Italy (e-mail: ) A particularly effective strategy for investigating FMF-like behavior involves reconfiguring conventional SMFs, coaxing them to operate at wavelengths below their fundamental single-mode cutoff [7]. In this specific spectral regime, higher-order modes (HOMs), which are ordinarily confined to evanescent decay, find themselves guided, thereby transforming the SMF into a functional FMF capable of supporting a limited number of spatial modes [8]. This change, while brilliant, creates a slew of novel complexities. The simultaneous presence of numerous guided modes causes significant intermodal dispersion, promotes subtle yet pervasive modal coupling, and triggers a range of nonlinear effects. Self-phase modulation (SPM), cross-phase modulation (XPM), and fourwave mixing (FWM) are well-documented examples; however, the phenomenon of stimulated Raman scattering (SRS) [9], [10] is of paramount interest for our present investigation. SRS, an inelastic scattering process fundamentally driven by the vibrational modes of the silica matrix itself, facilitates the transfer of optical energy from shorter to longer wavelengths-a phenomenon universally known as the Stokes shift. In the intricate confines of FMFs, this energy cascade can manifest not merely within the initially excited fundamental mode but also across several distinct spatial modes, resulting in complex and often counterintuitive patterns of spectral and spatial energy redistribution [11], [12]. Such intermodal Raman scattering is conveniently used for signal amplification in SDM systems [13] and can, indeed, precipitate a significant depletion of power from the fundamental mode via SRS intermodal coupling, thereby compromising both the efficiency and stability of FMF-based systems [14], [15]. While previous studies have rigorously documented instances of beam self-cleaning phenomena-wherein the nonlinear evolution of light within a fiber appears to preferentially favor the fundamental mode [16], [17]-our meticulous measurements reveal an unexpected and opposing trend. As the optical power introduced into solid-core FMFs increases, we observe a strong nonlinear modal instability, characterized by a substantial depletion of the fundamental LP01 mode. This enigmatic behavior, we contend, originates from a subtle yet profound mechanism: the differential Raman gain experienced across the various spatial modes. This disparity, in turn, arises from the unique spatial profiles of these modes and the subtle differences in their effective refractive indices, creating a selective amplification that ultimately undermines the stability of the desired fundamental mode propagation. II. EXPERIMENTAL SETUP Figure 1 presents the experimental setup used for this study. The laser source was a hybrid optical parametric am16 Oct 2025 JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 2 plifier (OPA) system (Light Conversion ORPHEUS-F), seeded by a white-light continuum and pumped by a femtosecond ytterbium-doped laser (Light Conversion PHAROS-SP-HP). This system generates linearly polarized pulses of 70 fs duration at a repetition rate of 100 kHz. The central wavelength was tunable from 650 to 900 nm and fixed at 700, 800, and 900 nm-deliberately chosen to fall below the modal cut-off wavelength of the standard SMF-28 fiber (around 1450 nm [18]), thereby ensuring the excitation of only a few spatial modes. The laser beam, possessing a near-Gaussian profile and a beam quality factor M 2 = 1.3, was precisely focused into the fiber core using a 20× microscope objective (with a numerical aperture of NA = 0.4). A variable neutral density filter (VNDF), placed just before the objective, allowed fine control of the input average power ranging from 0.1 to 5 mW, thereby yielding peak powers close to 100 kW. The fiber itself, a standard Alcatel SMF-28, with a core diameter of 8.8 μm and a cladding diameter of 125μm, with a cladding refractive index of nclad = 1.4574 at 630 nm. The relative index contrast was ∆n = 0.003. A subtle index dip in the center of the core is characteristic of this fiber. Fig. 1: A schematic of the experimental setup. VNDF: Variable neutral density filter; L1-L3: Lens; BPF: Bandpass filter; HWP: half-wave plate; LP: Linear polarizer; FM: Flip mirror; NF & FF camera: Near and Far field camera; SLM: Spatial light modulator. To investigate the beam propagation dynamics across varying interaction lengths, two distinct fiber lengths were used: one measuring 12 meters and the other 50 meters. The coupling into the fiber core required precise alignment via a threeaxis translation stage with micrometer precision. When fastidiously aligned on-axis, the coupling efficiency-the ratio of transmitted to input power under linear conditions-averaged around 75%. Furthermore, approximately ∼4% of the input power resides within 10 nm of the 700 nm central wavelength, a testament to the significant power carried by the spectral components outside this narrow band. Although our primary goal was to excite the fundamental LP01 mode, even minor misalignment could result in contributions from HOMs. At the fiber output end, the laser beam was collimated using a microlens (μLens, f = 2.7 mm) and an uncoated lens (L2, f = 400 mm). The beam was then directed towards a Hamamatsu spatial light modulator (SLM) via a half-wave plate (HWP) and a linear polarizer (LP). These optical elements, combined, provide an exquisite means of finely tuning the beam's intensity and polarization before its interaction with the SLM, thereby increasing modulation efficiency. A flip mirror (FM) allowed for quick switching between two detection pathways: one for imaging the near-field beam profile onto a CCD camera (NF camera) and another for capturing the Fourier transform of the beam onto a second CCD (FF camera) positioned after a 500 mm convex lens (L3). We were able to precisely project the output onto specified spatial modes by adding tailored phase masks to the SLM and quantifying their power contributions, as described in Ref. [19]. To eliminate the unwanted specters of self-phase modulation, which emerge as spectral broadening or temporal decoherence, a narrowband bandpass filter (10 nm bandwidth) was strategically placed immediately after the fiber's exit facet, exactly centered at the pump wavelength (700, 800, or 900 nm). This ensured that only the desired spectral components were analyzed downstream. The output spectrum, spanning from 600 to 1700 nm, was then recorded by an optical spectrum analyzer (OSA) (Yokogawa AQ6370D), which allowed us to identify and exclude parasitic signals, such as Stokes and anti-Stokes shifts. III. EXPERIMENTAL RESULT The guided modes supported by the optical fiber were computed using the semi-vectorial finite difference method, based on the experimentally measured refractive index profile. This numerical approach, presented in section IV, demonstrates that the fiber can propagate six LP modes at a wavelength of 700 nm: LP01, LP11a, LP11b, LP21a, LP21b, and LP02. To accurately resolve the modal composition at the fiber output, we employed a holographic mode decomposition technique, as comprehensively delineated in Ref. [19]. The resulting power fractions associated with each LP mode are detailed in Fig. 2. The inset, in particular, shows a comparison between the experimentally obtained near-field intensity profile (left panel) and the reconstructed modal distribution generated from FF camera data (right panel), which is perfectly consistent with the methods given in Ref. [19]. Under conditions of low-power linear propagation, specifically for an input power of 0.1 mW (corresponding to a pulse energy of Ep = 1 nJ) with on-axis coupling, the output beam predominantly maintains the spatial profile characteristics of the fundamental mode of the fiber. As illustrated in Fig. 2a, the LP01 mode carries most of the optical power, with only minor contributions from HOMs. However, as the input power increases, the modal composition undergoes a marked evolution. At 1.6 mW (Ep = 16 nJ), the LP11a/b modes become significantly populated, as shown in Fig. 2b. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 3 LP01 LP11a LP11b LP21a LP21b LP02 0 0.2 0.4 0.6 0.8 Mode Power Fraction (a.u.) a) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 0.5 b) LP01 LP11a LP11b LP21a LP21b LP02 0 0.2 0.4 0.6 c) LP01 LP11a LP11b LP21a LP21b LP02 0 0.2 0.4 0.6 d) Fig. 2: Experimentally measured mode power distributions for fiber length of 50 m at center wavelength of λ = 700 nm for input average powers of a) Pin = 0.1 mW, b) Pin = 1.6 mW, c) Pin = 4.5 mW, and d) Pin = 5.0 mW. LP01 LP11a LP11b LP21a LP21b LP02 0 0.2 0.4 0.6 Mode Power Fraction (a.u.) a) LP01 LP11a LP11b LP21a LP21b LP02 0 0.2 0.4 0.6 d) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 0.5 Mode Power Fraction (a.u.) b) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 0.5 e) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 0.5 Mode Power Fraction (a.u.) c) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 0.5 f) Fig. 3: Experimental mode power distributions at the fiber output for two center wavelengths and varying input powers: (a-c) At λ = 800 nm: a) Pin = 0.1 mW, b) Pin = 1.1 mW, and c) Pin = 6.6 mW. (d-f) At λ = 900 nm: d) Pin = 0.1 mW, e) Pin = 1.8 mW, and f) Pin = 5.0 mW, for 50-m-long fiber. Upon entering strongly nonlinear regime, with input powers of 4.5 mW (Ep = 45 nJ) and 5.0 mW (Ep = 50 nJ), the fundamental mode gets significantly depleted, and its energy is redistributed into higher-order spatial modes, including LP21a/b and LP02, as compellingly illustrated in Figs. 2c and 2d. Crucially, this transformative reconfiguration of the modal occupancy persists across distinct pump wavelengths. A strikingly similar behavior is observed for input laser carrier wavelengths of 800 nm and 900 nm, as clearly illustrated in Fig. 3. The near-field intensity and corresponding modal decomposition at the fiber exit facet for varying input powers are showcased in Figs. 3(a)-(c) for 800 nm and Figs. 3(d)-(f) for 900 nm. In the linear regime, with an input power of 0.2 mW, the beam exhibits a quintessential bell-shaped intensity profile, a hallmark unmistakable of an output dominated by the fundamental mode. However, as the input power exceeds 1.2 mW, the power progressively shifts toward HOMs-most prominently LP11a/b and their coherent superposition-clearly signaling strong nonlinear coupling. At the highest pump levels, the output beam structure strikingly resembles that of an orbital angular momentum (OAM) mode or vortex beam, underscoring the pronounced, wavelength-sensitive dynamics of nonlinear mode excitation. Figure 4 depicts an experimentally observed spectrum with a very broad spectrum centered at 900 nm and reaching around 40 nm in width. The spectrum shows a notable redshift and an abrupt truncation on its blue flank, providing unequivocal evidence of asymmetric spectral widening. The lack of obvious additional sidebands strongly suggests that this broadening arises from a complex and dynamic interplay between SPM and SRS, rather than from phase-matched parametric processes. This interaction is not cumulative; instead, it involves a competitive redistribution of energy within the pulse envelope. Ordinarily, SPM would orchestrate a symmetric spectral broadening, shaping both the red and blue wings of the spectrum equally. However, SRS intervenes with notable efficacy, redirecting a substantial portion of the pump pulse's energy into red-shifted Stokes components. In doing so, it significantly depletes the pulse's peak intensity-a critical factor for efficient SPM-and consequently impedes the generation of its characteristic blue-shifted constituents. This phenomenon is particularly pronounced in the normal dispersion regime, where the Stokes wave lags the pump, thereby prolonging its interaction length and increasing energy transfer. Simultaneously, SRS induces a progressive red shift via the Raman gain, augmenting the spectral density on the red spectral tail. The resulting spectral shape-truncated in the blue and expanded in the red-is the fascinating consequence of a complicated tug-of-war in which SRS both undermines and overwhelms SPM in a delicate energy balance. To further investigate the spatial dynamics that accompany JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 4 6 4 2 -20 Wavelength [nm] -19 0 -18 800 -17 Intensity [a.u] 850 900 950 -16 1000 1050 -15 -14 Pin = 0.2 mW Pin = 0.8 mW Pin = 1.6 mW Pin = 2.4 mW Pin = 3.2 mW Pin = 3.8 mW Pin = 4.5 mW Pin = 5.0 mW Fig. 4: Experimental spectra obtained from a 50-m-long Alcatel SMF-28 as a function of the input guided power measured at the input for a laser pulse centered at 900 nm. this spectrum evolution, we carefully removed the band-pass filter at the fiber's output and recorded a series of near-field intensity profiles corresponding to input pulses at 700 nm, 800 nm, and 900 nm. These illuminating tests unequivocally indicated a significantly rapid depletion of the fundamental mode at relatively low input levels, owing once again to the presence of the Raman effect. Fig. 5a vividly illustrates how the output beam transitions from a LP11-like mode to an LP21-like structure with noticeable center power depletion, a configuration distinct from the mode structure observed in Fig. 2. This intriguing evolution is further corroborated at longer wavelengths. Fig. 5b demonstrates that under 800 nm excitation and low input power, the output beam exhibits a near-Gaussian profile, indicative of the dominance of the fundamental mode. When the input power exceeds PIn = 1.4 mW, the beam undergoes a remarkable metamorphosis from an LP21-like profile to a structure resembling an optical vortex (OAM-like mode). Surprisingly, this vortex-like phase remains stable even when subjected to further power increases or external perturbations. A nearly similar progression occurs at 900 nm. Figure 5c captures how a bell-shaped fundamental mode produces a higher-order LP21-like profile with a hollow center. Intriguingly, this transition has surprising robustness: regardless of the initial coupling conditions-be it axial alignment or a deliberate lateral offset-the beam always moves towards an OAM-like shape as the input power approaches 5 mW. This profound universality suggests an underlying nonlinear mode-selection mechanism, intricately regulated by the concerted interaction of SPM, SRS, and modal dispersion. To deepen our understanding of how fiber length influences Raman-driven modal power transfer, a complementary experiment was conducted using a 12 m-long fiber, which is shorter than in previous experiments. This fiber was excited with a bell-shaped beam centered at 700 nm. Fig. 6 presents the evolution of the modal power distribution at the output, measured for varying input power levels. At a lowest input of PIn = 0.1 mW (Fig. 6(a)), the output remains overwhelmingly confined to the fundamental LP01 mode. This dominance reFig. 5: A fundamental mode reshaping near-field profile in a 50 m-long fiber with input wavelengths of (a) 700 nm, (b) 800 nm, and (c) 900 nm. (d) With an offset input, the same as (c), for different input average power PIn. flects predominantly linear propagation, with negligible energy transfer to HOMs. As the input increases to PIn = 2.0 mW (Fig. 6(b)), the first signs of nonlinear behavior emerge. While the LP01 mode still carries a substantial portion of the power, a noticeable shift occurs: the energy is gradually diverted into LP11a and LP11b, and to a lesser extent, LP21a, LP21b, and LP02. This signals the beginning of intermodal Raman interactions. Further increasing the input to PIn = 4.0 mW (Fig. 6(c)) enhances this redistribution. The LP01 mode gradually fades, replaced by more noticeable contributions from the higher-order LP21a/b. At the highest input of PIn = 6.0 mW (Fig. 6(d)), the modal landscape is transformed. The LP11a/b modes now dominate the output, with significant power also coming from the LP21a/b. The fundamental LP01 mode is significantly reduced, indicating an effective transfer of energy to higher-order spatial modes due to the Raman effect. IV. NUMERICAL RESULTS The propagation of ultrashort pulses in an Alcatel SMF-28 optical fiber operating below its cutoff wavelength, which supports M different spatial modes, can be accurately described by the generalized multimode nonlinear Schr ̈odinger equation (GMMNLSE) [20], [21]. This model captures the complex spatiotemporal evolution of the optical field in a few-mode fiber (FMF). The total optical field envelope is expressed as a superposition of the fiber's eigenmodes: E(x, y, z, t) = M X p=1 Fp(x, y)Ap(z, t) (1) where Fp(x, y) denote the transverse mode profiles and Ap(z, t) represents a slowly varying complex temporal envelope associated with the pth mode. For the purpose of this study, considering M = 6 modes, the GMMNLSE for the envelope of the pth mode, Ap(z, t), is given by: JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 5 LP01 LP11a LP11b LP21a LP21b LP02 0 0.2 0.4 0.6 0.8 Mode Power Fraction (a.u.) a) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 b) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 c) LP01 LP11a LP11b LP21a LP21b LP02 0 0.1 0.2 0.3 0.4 d) Fig. 6: Experimentally measured mode power distributions for fiber length of 12 m at center wavelength of λ = 700 nm for input average powers of a) Pin = 0.1 mW, b) Pin = 2.0 mW, c) Pin = 4.0 mW, and d) Pin = 6.0 mW. ∂zAp(z, t) = i (βp 0 -β0) Ap -(βp 1 -β1) ∂Ap ∂t + 4 X q≥2 iq+1 βp q q! ∂m t Ap -αp 2 Ap + in2ω0 c 1 + i ω0 ∂t X l,m,n {(1 -fR)Sk plmnAlAmA∗ n + fRAlSR plmn Z t -∞ Am(z, t -τ)A∗ n(z, t -τ)hR(τ)dτ} (2) The coefficient β(p) q represents the qth order dispersion term for the pth mode. The first two terms account for the phase mismatch and the group velocity mismatch among modes. The third term describes higher-order chromatic dispersion. Nonlinear contributions-mediated by the Kerr coefficient n2 = 2.7 × 10-20m2/W, central frequency ω0, and nonlinear overlap integrals Sk plmn-capture instantaneous nonlinear effects (SPM, XPM, FWM). The Raman effect enters via the fractional response fR ≈0.18, Raman overlap integrals SR plmn, and a delayed response function hR(τ) parameterized by τ1 = 12.2 fs and τ2 = 32 fs. Modal specific attenuation is accounted for by loss coefficients αp. As detailed in the experimental section, the simulation employed an Alcatel SMF-28 with a step-index core diameter of 8.8 μm and an index contrast of ∆n = 0.003 between its peak core and surrounding cladding. At a wavelength of 700 nm, the fiber exhibited normal chromatic dispersion, with β2 ranging from 22.2 to 34.9 ps2/km across LP01 to LP02 modes, respectively. Furthermore, the modal delays relative to the fundamental LP01 mode were 2.0, 2.0, -3.2, -3.2, and - 9.3 ps per meter of fiber for modes 2 through 6. Eq. 2 was numerically solved with an integration step of 50μm, utilizing a spatial window of 125 μm × 125 μm and a transverse grid of 800 × 800 samples. An ultrashort pulse, with a duration T0 = 70 fs at a full width half maximum (FWHM) and a diameter of 2w0 = 15 μm, was launched primarily into the fundamental mode of the FMF. The simulation explored input pulse energies varying from 1 nJ to 30 nJ, across fiber lengths of 1 m, 6 m, and 12 m. The initial fractional mode power distribution is depicted in Fig. 2a. Figures 7 and 8 reveal a fascinating set of numerical results, clearly demonstrating the profound impact of the Raman effect on ultrashort pulse propagation within FMFs. Consider a fiber length of 6 m and an input pulse energy of 30 nJ. In the absence of the Raman effect (fR = 0), the temporal . -LP01 ........ ,LP 11a ........ LP 11 b - - ,LP21a - - LP21b - - LP02 ....... . . .. 600 :·. . .•••• E . . .. 700 800 Wavelength [nm] a) . . .. ♦: • ••• 900 600 b) 700 800 900 Wavelength [nm] Fig. 7: Raman's transformative role in FMF dynamics. Comparative numerical simulations illustrate the output modal spectral profiles after 6 m of FMF, (a) with and (b) without the Raman nonlinearity. Input pulse with 70 fs pulsewidth and 30 nJ energy. Fig. 8: Simulated output modal power after 6 m of FMF: (a) with, and (b) without, the Raman nonlinearity. Input pulse with 70 fs pulsewidth and 30 nJ energy. profiles of the modes evolve with a serene smoothness, as evident in Fig. 8(b). Here, modal pulses undergo broadening and distortion, a direct consequence of the interplay between SPM and normal chromatic dispersion. This very interplay also orchestrates a moderate blue-shift of the second modal group (modes LP11a and LP11b) and a red-shift of the third modal group (modes LP21a, LP21b and LP02) relative to the fundamental mode (Fig. 7(b)). Modal dispersion cannot temporally distinguish modal pulses due to pulse broadening; yet, XPM induces minimal nonlinear power exchange among modes. However, when Raman nonlinearity is introduced (fR = 0.18), SRS emerges as the dominant effect, forcing a net energy transfer from the fundamental mode to the second mode group. This results in amplification of the spectra for JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 6 modes LP11a, LP11b, which concurrently leads to a depleted spectrum at lower wavelengths for mode LP01, as illustrated in Fig. 7(a). In the temporal domain, the instantaneous power of the fundamental mode experiences a marked depletion in correspondence with its trailing tail, precisely where the modes of group 2, delayed by the modal dispersion, experience a Raman gain (Fig. 8(a)). At sufficiently high powers and extended interaction lengths, the onset of cascaded Raman scattering was observed. Here, the first Stokes-shifted light acts as a pump for subsequent Raman shifts, resulting in multiple distinct spectral peaks. The intermodal spectrum transfer is governed by the interaction of Raman gain and phase-matching conditions among multiple modes at different frequencies, producing a complex signature of spatiotemporal dynamics. XPM, though present, plays a minor role in comparison. Fig. 9: Simulated: (a) modal energy distribution at input and after 6 m of FMF, when the input pulsewidth is 70 fs and 30 nJ total energy; (b) input near-field; (c) output near-field. Figure 9(a) offers compelling evidence of the simulated modal energy distribution of input and output when a 70fs pulse, with 30 nJ energy, propagated over 6 m of FMF. In the absence of the Raman effect (not shown), the output energy in each mode remains consistently aligned with the input throughout the propagation length, exhibiting no significant transfer of power between different spatial modes during propagation. On the contrary, when the Raman effect is taken into account, the modal energies undergo dramatic and dynamic metamorphoses. The initially populated fundamental mode (mode index 1, depicted by the red curve) experiences a significant decrease in its energy as light propagates. Concurrently, the HOMs (e.g., mode group 2, represented by mode indexes 2 and 3 in the blue curve) exhibit a noticeable increase in their energy. This unmistakable observation is the direct manifestation of Raman-induced inter-modal energy transfer: energy from the depleted fundamental mode (and potentially other pump modes) is efficiently transferred to these Stokesshifted HOMs. Thus, while the near field is dominated by the fundamental mode at input or in the absence of SRS (Fig. 9(b)), the metamorphic power of Raman nonlinearity produce an output near field that is undeniably dominated by modes LP11a, LP11b (Fig. 9(c)), with the contribution of the fundamental mode rendered negligible. The efficiency and specific target modes of this energy transfer are critically dependent on the group velocity matching between the pump and Stokes components in different modes, as well as the spatial overlap integrals of the interacting modes. In close accordance with our experimental findings, we performed a supplementary computational analysis to investigate the influence of fiber length on Raman-driven intermodal energy transfer. Simulations were carried out for two specific lengths-1 meter and 12 meters-at input pulse energies of 20 nJ and 30 nJ. In the shorter fiber, energy initially limited to the fundamental mode saw significant redistribution, with HOMs, especially the second mode group, acquiring a substantial share of the energy. The near-field intensity profile matched the simulation results given in Fig. 9(c), capturing the key aspects of modal reshaping. At 12 meters, the overall structure in the near-field remained relatively unchanged, but the energy transfer increased significantly in the third mode group. These findings reveal a surprising length independence of the nonlinear mode coupling, with Raman interactions resulting in identical behavior of energy transfer from the fundamental mode LP01 to the HOMs with indexes of 2 (LP11a) and 3 (LP11b), as well as a near-field profile similar to that seen in Fig. 9(c). Figure 10a illustrates the evolution of modal energy within the first 6 meters of the FMF, following the launch of a 70-fs pulse at a wavelength of 700 nm with an input energy of 30 nJ, distributed as depicted in Fig. 9. The simulation reveals a rapid migration of energy from the fundamental LP01 mode to the LP11 mode group. This preferential transfer is primarily due to their significant modal overlap, which facilitates an efficient exchange of optical power. In contrast, the higherorder LP21 and LP02 modes within group 3 show a markedly reduced energy coupling with the LP01 mode. This intriguing dynamic ultimately stabilizes at a steady state, influenced by the temporal broadening and distortion of the pulse caused by SPM. Turning to the frequency domain, the spectral evolution of each mode unveils a more nuanced narrative (Fig. 10b). While the Raman effect is universally acknowledged as a primary driver of intensity-dependent wavelength redshift, our findings introduce a captivating subtlety: the degree of redshift is not uniformly dictated by intensity across all modes. The LP01 mode, initially endowed with the lion's share of energy, undergoes the most significant redshift, a direct consequence of strong intramodal Raman scattering. Yet, a curious paradox emerges with the LP11 modes: despite siphoning off a substantial portion of the LP01 mode's energy, they exhibit the least pronounced redshift. This seemingly counterintuitive behavior originates from the very nature of intermodal energy transfer. The Raman intermodal coupling between the LP01 and LP11 modes predominantly occurs at the shorter-wavelength components of the LP01 mode. Consequently, the energy transferred to the LP11 modes is predominantly situated within their short-wavelength spectral regions. This spectral localization effectively hinders the characteristic Raman redshift of the LP11 modes, as the very process of energy acquisition acts as a counterbalance to it. In stark contrast, mode group 3, characterized by its meager overlap and weak coupling with the LP01 mode, remains largely unaffected by this compensatory mechanism, thus preserving its inherent Raman redshift. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2025 7 (a) (b) Fig. 10: (a) Simulated evolution of modal energy and (b) simulated evolution of the central wavelength in the first 6 m of the FMF, when the input pulse width is 70 fs and the total energy is 30 nJ. V. CONCLUSION Our combined experimental and numerical study demonstrates that nonlinear propagation in few-mode fibers is governed by a complex interplay of SPM, SRS, and modal dispersion, resulting in dramatic changes to both spatial and spectral beam characteristics. As input power increases, energy is progressively transferred from the fundamental LP01 mode to HOMs such as LP11a/b and LP21a/2, with the emergence of vortex-like (OAM) profiles indicating robust nonlinear mode coupling. Spectral measurements confirm that the asymmetric broadening, dominated by SRS, is particularly pronounced at longer wavelengths and higher powers, with Raman-induced redshifts effectively suppressing the blue components typically generated by SPM. Simulations validate these findings, revealing that Raman interactions induce strong intermodal energy transfer and significantly reshape the modal power distribution. Notably, cascaded Raman scattering further enriches the output spectrum, highlighting the sensitivity of pulse dynamics to both input conditions and fiber properties. These results not only elucidate the nonlinear mechanisms at play in multimode fibers but also offer valuable pathways for tailoring ultrafast light fields in advanced optical systems. ACKNOWLEDGMENTS We acknowledge Vincent Couderc for providing optical fiber characterization, M. Gervaziev, D. Kharenko, and S. Babin for providing the mode decomposition setup. Project ECS 0000024 Rome Technopole, Funded by the European Union - NextGenerationEU. REFERENCES [1] L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nonlinear multimode photonics: nonlinear optics with many degrees of freedom, Optica, 9(7), 824-841 (2022). [2] D. J. Richardson, J. Nilsson, and W. A. Clarkson, High power fiber lasers: current status and future perspectives, J. optical society Am. B, 27(11), B63-B92 (2010). [3] N. Bai, E. Ip, Y.-K. Huang, et al., Mode-division multiplexed transmission with inline few-mode fiber amplifier, Opt. Express, 20, 2668-2680 (2012). [4] D. J. Richardson, J. M. Fini, and L. E. Nelson, Space-division multiplexing in optical fibres, Nat. photonics 7, 354-362 (2013). [5] W. H. Renninger and F. W. Wise, Optical solitons in graded-index multimode fibres, Nat. communications, 4(1), 1719 (2013). [6] L. G. Wright, D. N. Christodoulides, and F. W. Wise, Spatiotemporal mode-locking in multimode fiber lasers, Science, 358, 94-97 (2017). [7] D. Gloge, Weakly guiding fibers, Appl. Opt., 10(10), 2252-2258 (1971). [8] K.-i. Kitayama and N.-P. Diamantopoulos, Few-mode optical fibers: Original motivation and recent progress, IEEE Commun. Mag., 55(8), 163-169 (2017). [9] G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 2019), 6th ed. [10] F. M. Ferreira, C. S. Costa, S. Sygletos, and A. D. Ellis, Semi-analytical modelling of linear mode coupling in few-mode fibers, J. Light. Technol., 35(18), 4011-4022 (2017). [11] H. Pourbeyram, G. P. Agrawal, and A. Mafi, Stimulated Raman scattering cascade spanning the wavelength range of 523 to 1750 nm using a graded-index multimode optical fiber, Appl. Phys. Lett., 102(20) (2013). [12] Z. Eslami, L. Salmela, A. Filipkowski, et al., Two octave supercontinuum generation in a non-silica graded-index multimode fiber, Nat. Commun., 13(1), 2126 (2022). [13] M. Zitelli, L. Andreoli, C. Autebert, et al., Inter-modal Raman amplification in space-division multiplexed systems, Opt. Fiber Technol. 86, 103854 (2024). [14] G. Fu, D. Li, M. Gong, et al., Spatiotemporal deterioration in nonlinear ultrafast fiber amplifiers, Appl. Phys. Lett., 123(9) (2023). [15] G. Fu, W. A. Gemechu, Y. Liu, et al., Performance enhancement of nonlinear pulse amplification in large-mode-area multimode gain fibers via pump configuration optimization, Opt. & Laser Technol., 189, 113116 (2025). [16] K. Krupa, A. Tonello, B. M. Shalaby, et al., Spatial beam self-cleaning in multimode fibres, Nat. Photonics, 11(4), 237-241 (2017). [17] R. Guenard, K. Krupa, R. Dupiol, et al., Kerr self-cleaning of pulsed beam in an ytterbium doped multimode fiber, Opt. Express, 25(5), 47834792 (2017). [18] F. Mangini, M. Ferraro, M. Zitelli, et al., Experimental observation of self-imaging in SMF-28 optical fibers, Opt. Express, 29(8), 12625-12633 (2021). [19] M. Gervaziev, I. Zhdanov, D. Kharenko, et al., Mode decomposition of multimode optical fiber beams by phase-only spatial light modulator, Laser Physics Letters, 18(1), 015101 (2020). [20] F. Poletti and P. Horak, Description of ultrashort pulse propagation in multimode optical fibers, J. Opt. Soc. Am. B 25(10), 1645-1654 (2008). [21] Wright, L. G., Ziegler, Z. M., Lushnikov, et al., Multimode nonlinear fiber optics: massively parallel numerical solver, tutorial, and outlook, IEEE J. Sel. Top. Quantum Electronics, 24(3), 1-16 (2018).
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2510.14792
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COT-PL: VISUAL CHAIN-OF-THOUGHT REASONING
MEETS PSEUDO-LABELING FOR OPEN-VOCABULARY
OBJECT DETECTION
Hojun Choi1
Youngsun Lim2
Jaeyo Shin1
Hyunjung Shim1
1KAIST AI
2Boston University
{hchoi256,jaeyo shin,kateshim}@kaist.ac.kr
youngsun@bu.edu
ABSTRACT
Open-vocabulary object detection (OVD) seeks to recognize and localize object
categories beyond those seen during training. Recent approaches typically lever-
age vision-language models (VLMs) to generate pseudo-labels using image-text
alignment, allowing detectors to generalize to unseen classes without explicit su-
pervision. However, these methods depend heavily on direct image–text match-
ing, neglecting the intermediate reasoning steps essential for interpreting seman-
tically complex scenes. This results in limited robustness when confronted with
crowded or occluded visual contexts. In this paper, we introduce CoT-PL, a new
framework that employs structured visual chain-of-thought (CoT) reasoning into
the pseudo-labeling process. CoT-PL decomposes object understanding into three
interpretable steps: (1) region perception even for unseen objects, (2) category
recognition via zero-shot reasoning, and (3) background grounding to separate se-
mantically complex objects. Crucially, the third step naturally motivates our con-
trastive background learning (CBL) that uses the pre-computed background cues
as negatives to promote feature disentanglement between objects and background.
In this way, CoT reasoning and CBL form an integrated pipeline tailored to robust
pseudo-labeling in crowded or occluded scenes. Notably, in these two settings,
our novel-class pseudo-label quality achieves relative improvements of 103.4%
and 168.4% over the best prior, respectively. Our extensive experiments demon-
strate that CoT-PL achieves +7.7 AP50 on open-vocabulary COCO and +2.9 mask
AP on LVIS for novel classes, setting a new state of the art.
1
INTRODUCTION
Pseudo
Annotation
Open-Vocab.
Detection
Novel Classes
(b) Previous Methods
RPN
VLMs
Image-Captions
A boy sits near a dog
while watching television.
Open-World
Base Classes
SAM
Pseudo
Annotation
MLLM
Open-Vocab.
Detection
Input Image
(c) Our Method
Human
Annotation
Open-Vocab.
Detection
(a) Conventional Designs
Novel
Classes
Base
Classes
Detector
🔥
Large-Scale
Base Classes
Detector
🔥
Detector
🔥
Novel Classes
Self-
training
Visual
CoT
Figure 1: Trends in pseudo-labeling for OVD. (a) Manual pseudo-labels for novel classes are
costly and do not scale. (b) Recent self-training methods automate pseudo-labeling by labeling re-
gion proposals via similarity with category text embeddings using vision-language models (VLMs),
but degrade with VLMs’ poor object localization and caption-dependent vocabulary. (c) We employ
visual chain-of-thought with Segment Anything Model (Kirillov et al., 2023) and multimodal large
language models (Bai et al., 2023) for accurate object perception and zero-shot category recognition.
Open-vocabulary object detection (OVD) aims to localize both seen (base) and unseen (novel) cate-
gories at test time, using only base-class annotations during training. To bridge this supervision gap
between seen and unseen categories, recent approaches leverage vision-language models (VLMs)
1
arXiv:2510.14792v1 [cs.CV] 16 Oct 2025
“A person that is standing
a name tag and other items.”
is next to a fence.”
(a) Noisy Pseudo Boxes
(c) Background Collapse
(b) Caption Dependency
on a skateboard.”
“Shoes, a cap, an iPod,
“A brown and black cow
skateboard
cap
fence
cow
iPod
name tag
shoes
cow
0.32
0.54
book
dog
skateboard
(d) Quality of Pseudo-Labels
Occluded
Crowded
Occluded
Figure 2: Limitations and quality of pseudo-labels for complex scenes. (a) Noisy pseudo boxes
due to poor object localization by VLMs, (b) limited object coverage from captions, and (c) occluded
objects treated as background. (d) Pseudo-label quality on the OV-COCO validation set: Crowded
denotes images with many objects and Occluded denotes objects occluded by other objects.
pre-trained on large-scale image-text pairs (Radford et al., 2021). These VLMs map textual descrip-
tions to visual representations, allowing OVD methods to recognize novel classes.
Among such efforts, pseudo-labeling has emerged as a state-of-the-art approach for OVD by aug-
menting the base set with automatically generated annotations that partially cover novel classes (Gao
et al., 2022; Pham et al., 2024). Early pseudo-labeling methods for OVD relied on manual annotation
of novel classes, which was costly and lacked scalability (Figure 1-a). More recent approaches (Zhao
et al., 2022; 2024) automate pipelines that generate pseudo-annotations for novel classes (Figure 1-
b). This process involves generating pseudo-labels for region proposals during self-training using
CLIP (Radford et al., 2021), based on the similarity between region features and text embeddings of
potential object categories, typically derived from dataset class names or image captions.
Despite their strong performances in general scenes, state-of-the-art OVD approaches still strug-
gle in challenging scenarios like crowded or occluded objects. A key reason is their reliance on
direct image–text matching via CLIP, lacking the intermediate visual reasoning steps required for
understanding complex scenes (Y¨uksekg¨on¨ul et al., 2023). Recent pseudo-labeling approaches also
inherit the same limitation. We identify three key factors underlying their failure in complex settings.
(L1) Noisy pseudo boxes: VLMs trained with image-level supervision exhibit co-occurrence bias,
favoring crops with semantically related objects (Zhong et al., 2022). This inductive bias hinders
object-level pseudo-box labeling in complex, occluded scenes. In Figure 2-a, CLIP incorrectly as-
signs the highest similarity to the token “skateboard” for a pseudo box with the person’s feet partially
hidden behind the skateboard. (L2) Caption dependency: Captions lack detail particularly in com-
plex, crowded scenes. As a result, objects not listed in captions remain unlabeled. Figure 2-b shows
missing pseudo annotations for many stuff categories (e.g., book) which are absent from the cap-
tion. (L3) Background collapse: Detecting objects under occlusion is challenging. These instances
are frequently unlabeled and learned as background in training (Li et al., 2024). As illustrated in
Figure 2-c, a dog partially occluded by a fence was not detected and treated as background.
To directly tackle these three limitations associated with complex scenes, we argue that pseudo-
labeling for OVD must be reformulated as an interpretable multi-step reasoning process rather than
a single-step alignment. In the vision-language domain, visual chain-of-thought (CoT) prompting
enhances VLM reasoning by encouraging step-by-step thinking (Wu et al., 2023a). Inspired by this,
we design pseudo-label generation using a three-step CoT pipeline explicitly aligned with OVD
challenges: (1) region perception with SAM (Kirillov et al., 2023) produces candidate masks, and
a multimodal large language model (MLLM) verifies object existence to filter out spurious or par-
tial boxes, mitigating (L1); (2) zero-shot category recognition assigns labels to each region without
relying on captions, eliminating (L2); and (3) contextual background grounding distinguishes true
objects from unlabeled background regions, resolving (L3). Importantly, the third step naturally
leads to our contrastive background learning (CBL), which transforms grounded background cues
into negative signals for training. This integration does more than relying on strong models be-
cause SAM and MLLMs alone produce noisy or inconsistent labels but our structured reasoning and
background-aware learning transform their raw outputs into high-quality pseudo-labels. In this way,
CoT reasoning and CBL form a single, integrated pipeline purpose-built for robust pseudo-labeling
in complex scenes. In Figure 2-d, our method significantly outperforms prior works (Gao et al.,
2022; Zhao et al., 2022; 2024) in generating pseudo-labels under challenging environments.
2
We conduct extensive experiments on two OVD benchmarks, OV-COCO (Lin et al., 2014) and OV-
LVIS (Gupta et al., 2019). Under challenging conditions like crowding and occlusion, our method
demonstrates superior pseudo-label quality compared to prior state-of-the-art pseudo-labeling meth-
ods. Our method sets a new state of the art, improving box AP50 for novel classes on OV-COCO
by 7.7 and mask mAP on OV-LVIS by 2.9, compared to prior work (Wu et al., 2023c). We report
pseudo-label statistics across OVD benchmarks, and our key contributions include:
• To our knowledge, we are the first to reformulate pseudo-labeling in OVD as a visual
chain-of-thought, decomposing complex-scene object understanding into an interpretable
multi-step reasoning process beyond single-step VLM alignment.
• We introduce CoT-PL, a unified system for robust pseudo-labeling in complex scenes by
integrating CoT reasoning and contrastive background learning (CBL) that uses grounded
background cues as negative training signals.
• Performance steadily improves with stronger teacher MLLMs, achieving state-of-the-art
results in OVD with high-quality pseudo-labels, even in challenging environments.
2
RELATED WORK
2.1
CHAIN-OF-THOUGHT (COT) REASONING
Chain-of-thought (CoT) reasoning has emerged as a powerful approach in natural language pro-
cessing, enabling models to tackle complex reasoning tasks by incrementally decomposing them
into interpretable steps. Initial work (Wei et al., 2022) demonstrated that large language models
produced more accurate outcomes by generating intermediate reasoning before arriving at a final
answer. In the visual domain, multimodal chain-of-thought methods process visual inputs sequen-
tially to reason about future states. These approaches have been applied to diverse tasks, including
bounding box prediction (Shao et al., 2024), planning in autonomous driving (Tian et al., 2024), in-
termediate image infillments (Rose et al., 2023), and CLIP embedding synthesis (Harvey & Wood,
2023). In the vision-language-action setting, CoT reasoning has recently gained traction for guiding
closed-loop robotic manipulation through sub-goal images as intermediate reasoning steps (Zhao
et al., 2025). In this work, we extend visual chain-of-thought reasoning to generate high-quality
pseudo-labels for open-vocabulary object detection, even in semantically complex scenes.
2.2
OPEN-VOCABULARY OBJECT DETECTION
Open-vocabulary object detection (OVD) aims to detect novel objects not seen during training by
leveraging vision-language models (VLMs) (Radford et al., 2021) trained on large-scale image-text
pairs. Recent OVD methods (Du et al., 2022; Wu et al., 2023d) employ prompt modeling to transfer
knowledge through learned prompts, enabling more precise contextual descriptions of each class.
Several studies (Gu et al., 2022; Wu et al., 2023c) use knowledge distillation to align detectors with
VLM features for recognizing unseen objects. Other approaches (Jin et al., 2024; Liu et al., 2024a)
reinforce the text modality using large language models (LLMs). Meanwhile, InstaGen (Feng et al.,
2024) focuses on the image modality, improving novel class prediction via synthetic images from
an image generation model. Furthermore, Grounding DINO (Liu et al., 2024b) introduces prompt-
based object detection by facilitating cross-modal information exchange between VLMs and trans-
formers. Another OVD approach is pseudo-labeling, which addresses the limited base classes by
leveraging extended supervision. These approaches often generate pseudo-annotations or weak su-
pervision through self-training using image captions (Gao et al., 2022; Zhao et al., 2022; 2024) or
dataset class names (Zhao et al., 2024). However, they focus on direct image–text matching via
CLIP, disregarding a reasoning process necessary for complex scenes (Y¨uksekg¨on¨ul et al., 2023).
We reformulate OVD pseudo-labeling as a sequence of interpretable reasoning steps using visual
chain-of-thought (CoT), thereby enabling robust pseudo-labeling in challenging environments.
3
METHODOLOGY
We introduce CoT-PL, a structured visual chain-of-thought pipeline tailored to robust pseudo-
labeling, even in two challenging scenarios. Captions in crowded scenes are underspecified, while
3
SAM
Whole
...
...
foreground
background
No Object
Open-World
Base Classes
Input Image
...
window
grass
table
dresser
dog
“Question: Does any
object exist in the image?”
MLLM
MLLM
“Question: What is the
object in the image?”
MLLM
Question: Is [OBJ]
background in the image?
Yes
No
computer screen
Preprocess
...
person
book
Semantic Anchors
Exclude Classes with
Annotations ≤ Across
All Training Images
True
wii
window
grass
table
dresser
dog
computer screen
...
person
book
= Minimum #
Classes
Annotations Within
False
Base
Classes
C
with
annotations from
Concatenate
First CoT Step
Second CoT Step
Third CoT Step
Figure 3: Overview of the proposed visual chain-of-thought pipeline. Our method first queries an
MLLM about object existence inside SAM-generated pseudo boxes on preprocessed images, then
performs zero-shot labeling, and finally extracts background representations nearby. Refined with
semantic anchors for reliability, the resulting pseudo-annotations are merged into the base set.
simple CLIP matching lacks fine-grained visual perception under occlusion. To address these is-
sues, CoT-PL recasts object understanding as three interpretable reasoning steps: (1) (Pseudo Box
Generation) We generate boxes for all instances using SAM (Kirillov et al., 2023) and verify if each
box corresponds to a valid object. (2) (Pseudo Label Assignment) We assign a pseudo-label to each
box through zero-shot object recognition using an multimodal large language model (MLLM). (3)
(Background Extraction) We extract background representations by leveraging the MLLM’s ground-
ing capability. Specially, the third step naturally leads to a contrastive background learning (CBL)
strategy, which uses the identified background concepts as negative training signals to promote fea-
ture disentanglement between objects and background.
3.1
PSEUDO BOX GENERATION
First CoT step.
We aim to generate accurate pseudo-bounding boxes for all potential objects in the
train set. To this end, we leverage the strong generalization ability of SAM to produce segmentation
masks for all object instances in each image. Relying on low-level visual cues such as edges and
color contrasts, SAM is widely used in open-vocabulary settings to generalize beyond base classes
to unseen objects (Zhang et al., 2023; Qin et al., 2024). However, applying class-agnostic SAM to
OVD presents two key challenges: (1) it segments regions at varying semantic granularity (i.e. full
objects vs. parts), and (2) it may include non-object regions such as background.
To address (1), we follow a similar approach to recent work (Qin et al., 2024), which utilizes SAM to
extract accurate object-level masks at multiple semantic levels (i.e. whole, part, sub-part). As illus-
trated in Figure 3, our method selects whole-instance masks to generate precise object-level pseudo
boxes that tightly enclose them. To address (2), we then use the zero-shot reasoning capability of a
robust MLLM (Bai et al., 2023) to verify whether each box contains a valid object. For example,
we prompt the model with: “Question:
Does any object exist in the image?”
and interpret the response (“Yes” or “No” or “Unsure”) as a ternary classification of object existence.
Boxes classified as “No” or “Unsure” are discarded, whereas those classified as “Yes” are passed
to the next stage of the CoT pipeline. For example, in Figure 3, regions lacking discernible objects
(i.e. plain dark areas) are discarded. For “Unsure” responses—a rich source of long-tailed cases, we
record the model’s rationale to support explainability. At this stage, we achieve region perception
by generating accurate pseudo boxes that tightly enclose identifiable yet unlabeled objects.
3.2
PSEUDO LABEL ASSIGNMENT
Second CoT step.
Most OVD methods rely on CLIP for pseudo-labeling, but it lacks fine-grained
visual perception and requires predefined class names (Y¨uksekg¨on¨ul et al., 2023). These limitations
4
inevitably lead to inaccurate or missing pseudo-labels, especially in complex scenes. This motivates
the second CoT step, which assigns more accurate pseudo-labels to each box independently of image
captions. To achieve this, we leverage an MLLM with strong zero-shot multi-class identification to
infer the object category within each box (Wang et al., 2023a). As shown in Figure 3, during the
second CoT stage, we prompt “Question:
What is the object in the image?”,
expecting a specific class label beyond the base set. This simple query lays the groundwork for
caption-free, open-world pseudo-labeling in OVD.
However, MLLMs are sensitive to visual content and often produce irrelevant predictions when
queried on pseudo boxes, as attention may leak beyond the target region (Zang et al., 2025; Zhang
et al., 2025b; Fu et al., 2024). A common workaround applies hard masks to remove non-target
regions, but this often induces errors from misleading visual content (Chang et al., 2023; Fontanini
et al., 2023). For example, a masked giraffe’s silhouette may bias the model to misclassify a tree
as a giraffe (See Appendix G.1). To mitigate this, we apply grayscaling and blurring as a soft
mask to suppress non-target regions, helping the MLLM focus and produce more accurate pseudo
labels (Yang et al., 2023). This preprocessing step is visualized in Appendix 6.
To further improve pseudo-label reliability, we add a post-processing step that filters out outlier la-
bels, chiefly reducing false positives. We exclude pseudo-labels with fewer annotation counts than
a threshold, as they are likely unreliable. For simplicity, the threshold is set to the minimum anno-
tation count among base classes. The remaining high-confidence labels, or semantic anchors, are
then integrated with the base class set to construct our open-world base set. Figure 3 highlights
anchors with color bars: green (foreground), red (background), and white (outlier). Without prede-
fined class names or image captions, the second CoT step yields robust and reliable pseudo-labels
by integrating MLLM zero-shot reasoning with reliability-enhancing pre- and post-processing.
3.3
BACKGROUND EXTRACTION
Third CoT step.
While stronger than VLM-based methods, our MLLM-based pseudo-labeling is
not yet complete for long-tailed complex scenes because it hinges on MLLM capability. In such
cases, the model often returns “Unsure” for small or occluded objects (i.e. a mug hidden behind a
laptop). For example, we observe that weaker models produce more “Unsure” responses and fewer
valid labels across diverse objects (See Table 2). These regions remain unlabeled and are treated as
background during training (Bansal et al., 2018; Li et al., 2024), since they match neither base nor
pseudo-labeled classes, a phenomenon we call background collapse.
Unfortunately, explicitly identifying unlabeled, inherently unknown objects is nontrivial. Instead,
we sidestep this by disentangling collapsed objects from background representations in the feature
space. To this end, we use MLLM grounding as our third CoT step to verify whether a given
“[OBJ]” belongs to background via a binary prompt (“Yes” or “No”). This simple binary query
helps determine whether a given object category is background. For example, the model classifies
“grass” as background and “drawer” as foreground (See Figure 3). These grounded background
cues serve as negative supervision, encouraging object–background disentanglement in training.
3.4
CONTRASTIVE BACKGROUND LEARNING (CBL)
Our CBL instantiates the idea on BARON (Wu et al., 2023c) that performs competitively through
online sampling of compositional structures (i.e., co-occurrence of objects) as training signals. How-
ever, the sampling process is computationally expensive and time-consuming. In Figure 4, we design
a composition generator that groups cached, pre-computed semantic anchors for each proposal into a
bag of regions, reducing overall training time by 4×. Then, these sampled regions are projected into
the word embedding space using the linear layer within Faster R-CNN (Ren et al., 2015), resulting in
pseudo-words. The pseudo-words are passed through the text encoder T to obtain the bag-of-regions
embedding f i
t = T (wi
0 + pi
0, wi
1 + pi
1, · · · , wi
Ni−1 + pi
Ni−1), where N i is the number of regions in
the i-th bag, pi
j represents the positional embedding of the j-th region in the i-th bag. Finally, this
bag-of-regions embedding is aligned with the VLM’s image embeddings f i
v = V(bi
0, bi
1, · · · , bN i
i ),
where bi
j is the j-th region in the i-th bag. Further implementation details appear in Appendix F.
To alleviate the background collapse, we propose a contrastive background learning (CBL) strategy
that explicitly disentangles objects, including unlabeled objects, from true background representa-
5
CLIP
Text
Encoder
grass
sky
...
prompt
prompt
prompt
person
book
...
table
window
...
...
...
...
Open-World
Base Classes
RPN
Semantic Anchors
S
Semantic
Search
Composition
Generator
Pseudo-Words
Linear
PE
RoI
Head
...
CLIP
Text
Encoder
CLIP
Image
Encoder
Group Region Proposals
with Semantic Anchors
Region Proposals
...
Shared Embedding Space
Bag-of-Regions
Teacher
Pull
Class Embeddings
...
...
...
...
...
Push
person
grass
window
Push
Pull
dog
Mean of
Background
Embeddings
Training
dining_table
Inference
prompt
airplane
dog
...
...
CLIP
Text
Encoder
Pseudo-Words
Linear
PE
RoI
Head
...
Input Image
RPN
...
Region Proposals
Probability Scores
No CBL
Probability Scores
CBL
Probability Scores
CBL++
Input Image
Figure 4: Overall architecture of CoT-PL. Built on BARON (Wu et al., 2023c) (See Appendix F),
the proposed method encodes the open-world base set, partially including novel classes, using the
CLIP text encoder. The CLIP embeddings of multiple background concepts are averaged to initialize
a single learnable background embedding. These concepts are also used as negative samples in
contrastive learning to encourage feature disentanglement between objects and background. At
inference time, we apply CBL++ to mitigate class interference by removing pseudo-labels associated
with the ground-truth novel classes.
tions (i.e. grass or sky) in the feature space during training. As shown in Figure 4, we first encode
the base categories Cbase, pseudo-labels Cfg, and identified background concepts Cbg using the
CLIP text encoder with the prompt template “a photo of [OBJ]”. The averaged background
embedding ¯fbg serves as an initialization for a learnable background prior. To encourage feature
discrimination, we apply a contrastive objective in which the background embeddings fbg serve as
negative samples, formulated as the alignment InfoNCE loss (Rusak et al., 2024):
Lbag = 1
2
G−1
X
k=0
log pk
t,v + log pk
v,t
,
(1)
where G is the number of bags, and the pk
t,v and pk
v,t can be calculated as:
pk
t,v =
exp(τ ′ · ⟨f k
t , f k
v ⟩)
PG−1
l=0 exp(τ ′ · ⟨f k
t , f lv⟩) + PM−1
j=0 exp(τ ′′ · ⟨f k
t , f j
bg⟩)
,
(2)
pk
v,t =
exp(τ ′ · ⟨f k
v , f k
t ⟩)
PG−1
l=0 exp(τ ′ · ⟨f kv , f l
t⟩) + PM−1
j=0 exp(τ ′′ · ⟨f kv , f j
bg⟩)
,
(3)
where ⟨·, ·⟩denotes cosine similarity; M is the number of background concepts; τ ′ and τ ′′ are
temperature scaling factors. The loss encourages alignment between matched text–image pairs (e.g.,
dog and window) while pushing apart mismatched background representations (e.g., grass).
4
EXPERIMENTS
Datasets and evaluation metrics.
We evaluate our CoT-PL using two widely used OVD datasets:
OV-COCO (Lin et al., 2014) and OV-LVIS (Gupta et al., 2019). We adopt the category split approach
from OVR-CNN (Zareian et al., 2021) for the OV-COCO, dividing object categories into 48 base
and 17 novel categories. For OV-LVIS, we follow ViLD Gu et al. (2022), separating the 337 rare
6
Table 1: Result comparisons on OV-COCO (Lin et al., 2014). Methods are categorized into two
groups based on whether additional supervision beyond instance-level labels in the base classes CB
is utilized during training, i.e., weak or pseudo labels. Note that CN is the novel class set; RN50-C4
uses features from the fourth convolutional stage, and RN50-FPN uses a multi-scale feature pyramid.
Methods
Supervisions
Backbone
APN
50 (%)
APB
50 (%)
Annotation: Extra caption datasets, Weak/Pseudo Labels in CB ∪CN
Detic (Zhou et al.)
ImageNet21K & CC3M
RN50-C4 (24M)
27.8
42.0
OV-DETR (Zang et al.)
Pseudo annotation
RN50 (24M)
29.4
52.7
CoDet (Ma et al.)
CC3M & COCO Caption
RN50 (24M)
30.6
46.4
PB-OVD (Gao et al.)
COCO Caption
RN50-C4 (24M)
30.8
46.4
VL-PLM (Zhao et al.)
Pseudo instance-level annotation
RN50 (24M)
34.4
60.2
RegionCLIP (Zhong et al.)
CC3M
RN50-C4 (24M)
35.2
57.6
OC-OVD (Rasheed et al.)
COCO Caption
RN50-FPN (24M)
36.6
49.4
SAS-DET (Zhao et al.)
COCO Caption
RN50-C4 (24M)
37.4
58.5
CoT-PL (Ours)
Pseudo annotation
RN50-FPN (24M)
41.7
59.4
Annotation: Instance-level labels in CB
ViLD-ens (Gu et al.)
CLIP
RN50-FPN (24M)
27.6
51.3
BARON (Wu et al.)
CLIP
RN50-FPN (24M)
34.0
60.4
CFM-ViT (Kim et al.)
CLIP
ViT-L/16 (307M)
34.3
46.4
CORA (Wu et al.)
CLIP
RN50 (24M)
35.1
35.4
BIND (Zhang et al.)
CLIP
ViT-B/16 (86M)
36.3
50.2
CLIP-Self (Wu et al.)
CLIP
ViT-B/16 (86M)
37.6
-
LBP (Li et al.)
CLIP
RN50-FPN (24M)
37.8
58.7
CCKT-Det (Zhang et al.)
CLIP
RN50 (24M)
38.0
-
OV-DQUO (Wang et al.)
CLIP
RN50 (24M)
39.2
-
Table 2: Statistics of pseudo labels. We report the number of pseudo class labels and annotations
obtained from 118,287 training images across open-vocabulary benchmarks. The class count indi-
cates how many classes in each benchmark are covered by pseudo-labels.
Model
Total
OV-COCO (65 classes)
OV-LVIS (1,203 classes)
# Classes
# Annotations
# Unsure
# Classes
# Annotations
# Classes
# Annotations
BLIP2 (2023)
6,036
395,052
1,528,251
31
197,086
105
137,353
InstructBLIP (2023)
3,106
566,619
1,132,179
30
379,528
93
315,346
Qwen2 (2023)
3,916
637,349
563,156
65
349,399
115
232,298
categories as novel and consolidating the common and frequent categories into base categories. We
follow OVR-CNN for evaluation: on OV-COCO, we report box AP at IoU 0.5 for novel categories
(APN
50); on OV-LVIS, we report mask mAP over IoUs from 0.5 to 0.95 for rare categories (APr).
Implementation details.
We build CoT-PL on Faster R-CNN (Ren et al., 2015) with ResNet50-
FPN. For a fair comparison, we initialize the backbone network with weights pre-trained by
SOCO (Wei et al., 2021) and use synchronized Batch Normalization Zhang et al. (2018), follow-
ing recent studies Wu et al. (2023c); Du et al. (2022). For the main experiments on OV-COCO and
OV-LVIS, we choose the 1× and 2× schedules, respectively. We use the CLIP model based on
ViT-B-16 (Dosovitskiy et al., 2021) as our pre-trained VLM. For the prompt of category names, we
default to the hand-crafted prompts from ViLD (Gu et al., 2022) in all our experiments on OV-COCO
and OV-LVIS. We follow the same hyperparameter settings as the baseline (See Appendix D).
4.1
MAIN RESULTS
Comparison with state-of-the-art methods.
Most recent OVD methods utilize weak or pseudo-
annotations during training. For instance, RegionCLIP (Gao et al., 2022) and CoDet (Ma et al.,
2023) utilize additional caption datasets to discover novel concepts, while OV-DETR (Zang et al.,
2022) further generates pseudo-annotations in a self-training manner. As shown in Table 1, on OV-
COCO, our CoT-PL achieves the best performance (APN
50 41.7%) among methods using additional
supervision, with the default ResNet50 (He et al., 2016) backbone. Furthermore, CoT-PL surpasses
7
Table
3:
Result
comparisons
on
OV-
LVIS (2019).
Our CoT-PL achieves com-
petitive performance on instance segmentation.
Methods
APr (%)
AP (%)
ViLD-ens (Gu et al., 2022)
16.6
25.5
Detic (Zhou et al., 2022)
17.8
26.8
BARON (Wu et al., 2023c)
19.2
26.5
MIC (Wang et al., 2023c)
20.8
30.7
OADP (Wang et al., 2023b)
21.7
26.6
LBP (Li et al., 2024)
22.2
29.1
CoT-PL (Ours)
23.0
29.4
Table 4: Pseudo-label quality on OV-COCO
validation.
Metrics on novel-class pseudo-
labels. Crowded denotes images with more ob-
jects than the average in this dataset. Occluded
denotes novel-class objects over 50% covered
by other ground-truth boxes.
Methods
COCO Novel
Crowded
Occluded
PB-OVD
18.7
5.1
2.7
VL-PLM
25.5
7.3
3.8
SAS-Det
26.7
11.6
5.7
Ours
31.5
23.6
15.3
instance-level label-based methods relying on distillation, such as BIND (Zhang et al., 2024a). No-
tably, it also outperforms several recent distillation-based methods, such as CCKT-Det (Zhang et al.,
2025a) and OV-DQUO (Wang et al., 2025), under the same ResNet-50 backbone. On OV-LVIS (Ta-
ble 3), CoT-PL achieves strong performance (23.0% APr) using our generated pseudo-annotations
under hand-crafted prompts. It significantly outperforms ViLD (Gu et al., 2022), Detic (Zhou et al.,
2022), and BARON (Wu et al., 2023c), and performs better than more recent methods such as
LBP (Li et al., 2024) and MIC (Wang et al., 2023c), which utilize extra data with 100 class names.
These results highlight the effectiveness of our pseudo-labeling framework across diverse settings.
Statistics.
Table 2 provides detailed statistics of pseudo-labels generated by different MLLM vari-
ants on two benchmarks. Our pseudo-labels span around 4K diverse object classes, totaling 637.349
annotations across these benchmarks. While BLIP2 (Li et al., 2023) covers the most categories
(6,036), its relatively low annotation count (395K) suggests sparse predictions with lower per-class
confidence. In contrast, InstructBLIP (Dai et al., 2023) produces fewer categories (3,106) but more
annotations (566K), reflecting more confident labeling. Qwen2 (Bai et al., 2023) achieves the high-
est annotation density, generating 637K annotations across 3,916 categories. Notably, as the quality
of teacher models improves, the number of “Unsure” responses from MLLMs decreases, yielding
more confident pseudo-labeling. These results show that stronger teacher models with higher anno-
tation density with fewer “Unsure” responses, producing high-quality pseudo-labels for OVD.
4.2
ABLATION ANALYSIS
Quality of pseudo-labels for complex scenes.
We compare the quality of our pseudo-labels with
prior state-of-the-art methods (Gao et al., 2022; Zhao et al., 2022; 2024). For a fair comparison,
we adopt the same experimental setup and report APN
50 on the COCO validation set. As shown
in Table 4, existing methods perform reasonably well on COCO Novel but degrade significantly
under challenging scenarios such as Crowded and Occluded. Following (Lin et al., 2014; Qi et al.,
2022), we define Crowded as images with more than 8 objects (COCO average), and Occluded as
novel ground-truth (GT) boxes covered more than 50% by other GT boxes. These results suggest
that previous pseudo-labeling methods lack the reasoning capabilities necessary for fine-grained
visual understanding, resulting in noisy pseudo-labels. In contrast, our method achieves superior
performance in such challenging scenarios through visual chain-of-thought reasoning.
Impact of the individual proposed modules.
We conduct an ablation study on OV-COCO to
assess the contribution of each component in our CoT-PL framework: pseudo-labeling with CoT
reasoning and contrastive background learning (CBL). As shown in Table 5, na¨ıvely prompting
MLLMs with a single-step query for both class names and background grounding without image
preprocessing results in only a marginal 0.6% gain over the baseline (Wu et al., 2023c). In con-
trast, applying image preprocessing improves APN
50 by 2.6%, while a three-step CoT (3×) further
increases the gain to 5.7%. Furthermore, incorporating CBL improves performance by 7.1% via
better feature disentanglement between objects and background. It also yields a 2.9% gain in the
single-step setting with more unlabeled objects. These findings demonstrate the effectiveness of our
pseudo-label generation pipeline and CBL in generating high-quality pseudo-labels.
Impact of semantic anchors.
We assess the impact of semantic anchors under different thresholds
in Table 6. The ALL setting, which uses all pseudo-labels without filtering, results in a slight per-
8
Table 5: Ablation of the proposed individual modules. CoT
(K×) refers to pseudo-labeling guided by K-step chain-of-
thought (CoT) reasoning. † denotes no image preprocessing.
CoT† (1×)
CoT (1×)
CoT (3×)
CBL
APN
50 (%)
✓
-
-
-
34.6
-
✓
-
-
37.2
-
✓
-
✓
40.1
-
-
✓
-
40.3
-
-
✓
✓
41.7
Table 6: Impact of semantic an-
chors.
MIN denotes the fewest
base-class annotation, while ALL
uses all pseudo labels. CoT-PL per-
forms best under MIN with more re-
liable pseudo-labels on OV-COCO.
Thresholds
APN
50
APB
50
ALL
40.7
59.1
MIN
41.7
59.4
Table 7:
Impact of proposal generators on
pseudo-labeling.
The detector is trained on
pseudo-labels from multiple proposal generators.
Proposal Generator
COCO Novel
LVIS
Mask R-CNN (2017)
39.7
21.6
MAVL (2022)
40.9
22.1
SAM (2023)
41.7
23.0
Table 8: Comparison of MLLM variants for
pseudo-labeling. Best and second best results
are highlighted.
Model
Size
APN
50
APB
50
BLIP2 (2023)
2.7B
37.6
59.5
InstructBLIP (2023)
7B
40.1
58.7
Qwen2 (2023)
7B
41.7
59.4
formance drop due to the inclusion of noisy and uncertain predictions. In contrast, the MIN setting,
which selects semantic anchors with the fewest base-class annotations, achieves improved perfor-
mance. These results suggest that semantic anchors help filter out unreliable and sparse predictions
while guiding training toward semantically consistent regions.
Impact of pseudo-box generator quality.
The proposed pseudo-labeling pipeline uses region
proposals generated by class-agnostic SAM (Kirillov et al., 2023). Following PB-OVD (Gao et al.,
2022), we compare SAM with other proposal generators, such as Mask R-CNN (He et al., 2017) and
MAVL (Maaz et al., 2022), for pseudo-labeling. Table 7 shows that SAM achieves the best results
on both COCO Novel (41.7%) and LVIS (23.0%), highlighting its potential as a proposal generator
for pseudo-labeling. Notably, our method achieves competitive performance even with different
proposal generators, demonstrating the effectiveness of structured chain-of-thought reasoning.
Comparison of MLLM variants.
We compare different MLLM variants used in our pseudo-
labeling pipeline, as shown in Table 8. BLIP-2 (Li et al., 2023), with the smallest parameter size
(2.7B), yields the lowest detection performance, partially due to frequent “Unsure” responses, in-
dicating limited capability in generating reliable pseudo-labels. In contrast, both InstructBLIP (Dai
et al., 2023) and Qwen2 (Bai et al., 2023) have a comparable model size (7B), but Qwen2 consis-
tently outperforms InstructBLIP across standard multimodal benchmarks such as MMBench (Liu
et al., 2024c). This performance gap suggests that stronger MLLMs generate higher-quality pseudo-
labels, leading to a 1.6% improvement in APN
50 for novel object detection. These findings indicate
that advances in MLLM capability can translate into roughly linear gains within our framework.
5
CONCLUSION
We introduce CoT-PL, a new pseudo-labeling framework for open-vocabulary object detection
(OVD) that leverages structured visual chain-of-thought (CoT) reasoning. Harnessing the zero-shot
reasoning capabilities of multimodal large language models (MLLMs), CoT-PL decomposes object
understanding into three interpretable reasoning steps: (1) object recognition, (2) caption-free zero-
shot labeling, and (3) background extraction. The third step further leads to our contrastive back-
ground learning (CBL) that mitigates background collapse by disentangling object and background
features. As a result, CoT-PL, a unified system integrating CoT reasoning with CBL, generates high-
quality pseudo-labels in complex visual scenes and consistently improves performance with stronger
teacher MLLMs. CoT-PL achieves state-of-the-art results across multiple OVD benchmarks. We
hope our work inspires further exploration of visual CoT reasoning in downstream perception tasks.
9
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15
APPENDIX
A
LIMITATIONS & FUTURE WORK
Despite its effectiveness, CoT-PL faces two limitations. (1) It depends on the capabilities of a
sufficiently strong MLLM; less capable models produce lower-quality pseudo-labels and degrade
detection performance. To mitigate this, it treats the MLLM’s three-way judgment (Yes/No/Unsure)
as a hard gate, safely discarding all potentially false regions labeled Unsure. This rigid design risks
accumulating irreversible false negatives. (2) Long-tailed categories are unfairly removed by the
frequency threshold of the minimum number of annotations in base classes, discarding any labels
below this count as noise for simplicity. Exploring these issues serves as a promising direction for
future work.
B
DEVICE INFORMATION
All experiments were conducted using eight NVIDIA A6000 GPUs with PyTorch 1.12.1. Each
training run took approximately 11 hours and involved around 12K GPU memory usage. For a fair
comparison with the baseline, we fixed the random seed to 1194806617 across all experiments to
ensure reproducibility.
C
IMPLEMENTATION DETAILS
Baseline.
We build our CoT-PL upon Faster R-CNN (Ren et al., 2015) with a ResNet-50 FPN
backbone, consistent with prior open-vocabulary object detection (OVD) work.
The backbone
is initialized using weights pre-trained with SOCO (Wei et al., 2021) and utilizes synchronized
Batch Normalization (SyncBN) (Zhang et al., 2018). We adopt the 1× training schedule for OV-
COCO (Lin et al., 2014) and 2× for OV-LVIS (Gupta et al., 2019).
Pseudo-label generation process.
During our offline pseudo-label generation process, we lever-
age SAM (Kirillov et al., 2023; Qin et al., 2024) to produce class-agnostic object proposals and
apply Qwen2 (7B) (Bai et al., 2023) as our default MLLM for visual chain-of-thought (CoT) rea-
soning. CLIP based on ViT-B/16 (Radford et al., 2021) is used to encode textual prompts, which are
constructed using the hand-crafted template “a photo of [OBJ]” following ViLD (Gu et al.,
2022). Pseudo-labels are generated using only the training set, without leveraging any image cap-
tions. The pseudo-labels are used exclusively during training and discarded during inference.
Contrastive Background Learning.
Following ViLD, the contrastive background learning (CBL)
background prototypes are built from hand-crafted prompts rather than category names. Using
prompt engineering, we define five generic background types, such as sky, water surface, vegeta-
tion, paved ground, and plain wall, and craft a small set of object-free prompts for each (i.e. “clear
sky background, no objects”). We tokenize the prompts and encode them using the CLIP text en-
coder, then average the resulting text embeddings to obtain a single prototype per background type.
D
HYPERPARAMETERS
For fair comparison, we adopt the same hyperparameter settings as BARON. We use the SGD op-
timizer with a momentum of 0.9 and a weight decay of 2.5 × 10−5. The initial learning rate is
set to 0.04 for OV-COCO and 0.08 for OV-LVIS. Models are trained for 90,000 iterations on OV-
COCO (Lin et al., 2014) and 180,000 iterations on OV-LVIS (Gupta et al., 2019), with a fixed batch
size of 16 across all experiments. During training, model checkpoints are saved every 10,000 iter-
ations for OV-COCO and every 30,000 iterations for OV-LVIS. The best-performing checkpoint on
the validation set is selected for final evaluation.
For our proposed modules, we provide the hyperparameter configurations used in the OV-
COCO (Lin et al., 2014) and OV-LVIS (Gupta et al., 2019) experiments. For semantic anchor
construction, we filter out infrequent pseudo-labels using a minimum annotation threshold—set to
1,237 for OV-COCO and 1 for OV-LVIS. Additionally, the background contrastive loss temperature
16
parameter is set to τ ′′ = 5.0, which controls the regularization strength of background embeddings
relative to foreground embeddings.
E
DATA APPENDIX
We report the statistics of the benchmarks used in this study. For both OVD benchmarks, our pseudo-
label generation pipeline uses only the training set, while the validation and test sets remain unused
to ensure fair evaluation during inference.
• OV-COCO (Lin et al., 2014): OV-COCO is based on the COCO 2017 detection split,
containing 118,287 training and 5,000 validation images with instance-level bounding box
annotations. We follow the category split from OVR-CNN (Zareian et al., 2021), which de-
fines 48 base and 17 novel categories. During pseudo-label generation, images containing
only novel categories are discarded to ensure consistent supervision. All pseudo-labels are
generated solely from the base training split.
• OV-LVIS (Gupta et al., 2019): OV-LVIS is derived from LVIS v1.0, comprising over 120K
training images with annotations for 1,203 categories. Following the ViLD (Gu et al., 2022)
protocol, we treat 337 rare categories as novel, and the remaining frequent and common
categories as base. Due to the severe long-tail distribution, some rare categories contain
fewer than five instances; such categories are removed during semantic anchor construction.
Evaluation is performed on the standard LVIS validation split.
• Objects365 (Shao et al., 2019a): Objects365 contains 1,742,289 training and 80,000 val-
idation images across 365 object categories. We use this dataset solely for cross-dataset
evaluation without any additional fine-tuning. Specifically, the model trained on OV-LVIS
is directly evaluated on the Objects365 validation split to assess its transferability. All
category names are mapped via exact string matching using CLIP prompt templates.
F
BASELINES
Open-vocabulary detectors.
Recent advances (Ren et al., 2015; He et al., 2017) in open-
vocabulary object detection (OVD) have been largely driven by the emergence of foundation models,
including vision-language models (VLMs) (Radford et al., 2021; Jia et al., 2021). VLMs support
novel class recognition in OVD through various techniques, such as pseudo-labeling. We build upon
Faster R-CNN (Ren et al., 2015) for OVD, replacing its classifier with a linear layer that projects
region features into the word embedding space. This enables each region to be represented by mul-
tiple pseudo-words, capturing the rich semantics of each object. Given C object categories, the
probability of a region being classified as the c-th category:
pc =
exp(τ · ⟨T (w), fc⟩)
PC−1
i=0 exp(τ · ⟨T (w), fi⟩)
,
(4)
where T is the text encoder, ⟨·, ·⟩denotes cosine similarity, τ is a temperature scaling factor, T (w)
represents the text embedding of pseudo-words, and fc is the category embedding of a prompt
template encoded by the text encoder (i.e. “a photo of {category} in the scene”).
BARON (Wu et al., 2023c).
Additionally, we instantiate the idea of BARON (Wu et al., 2023c)
to capture compositional structures for simplicity. During training, it learns using Faster R-CNN’s
regression and classification losses (Ren et al., 2015), with annotations provided only for the base
set. BARON first groups contextually related neighboring regions for each region proposal extracted
from Faster R-CNN (Ren et al., 2015), forming a bag of regions. BARON then projects these
regions into the word embedding space using the linear layer within Faster R-CNN, resulting in
pseudo-words. The pseudo-words are passed through the text encoder to obtain the bag-of-regions
embedding f i
t = T (wi
0 + pi
0, wi
1 + pi
1, · · · , wi
Ni−1 + pi
Ni−1), where N i is the number of regions in
the i-th bag, pi
j represents the positional embedding of the j-th region in the i-th bag. Finally, this
bag-of-regions embedding is aligned with the VLM’s image embeddings f i
v = V(bi
0, bi
1, · · · , bNi
i ),
where bi
j is the j-th region in the i-th bag. To align the bag-of-regions embeddings, BARON employs
17
Bag of regions
Image
encoder
(a) Aligning bag of regions
RCNN
Text
encoder
Pseudo words
Bag of regions
Image
encoder
(a) Aligning bag of regions (with semantic anchors)
Sampled regions with anchors
RCNN
Text
encoder
Pseudo words
Sampled regions
Loaded
from Cache
Sampling
Process
Training Time
Reduced by 25%
Figure 5: The overall architecture of BARON (Wu et al., 2023c). To reduce sampling time during
training, BARON’s naive neighbor sampling can optionally be replaced with our semantic anchor-
based strategy. In particular, caching the anchors reduced training time by 25% compared to the
baseline, while maintaining the original performance.
a contrastive learning loss based on InfoNCE (Rusak et al., 2024):
Lbag = −1
2
G−1
X
k=0
log(pk
t,v) + log(pk
v,t)
,
(5)
pk
t,v =
exp(τ ′ · ⟨f k
t , f k
v ⟩)
PG−1
i=0 exp(τ ′ · ⟨f k
t , f iv⟩)
,
(6)
pk
v,t =
exp(τ ′ · ⟨f k
v , f k
t ⟩)
PG−1
i=0 exp(τ ′ · ⟨f kv , f i
t⟩)
,
(7)
where G is the number of bags for each region proposal and τ ′ is a temperature scaling factor. This
allows BARON to leverage the compositional structures inherent in VLMs, resulting in moderate
performance gains.
To align individual region embeddings, BARON uses a contrastive learning loss similar to InfoNCE:
Lindividual = −1
2
N−1
X
k=0
log(qk
t,v) + log(qk
v,t)
,
(8)
qk
t,v =
exp(τindividual · ⟨gk
t , gk
v⟩)
PN−1
i=0 exp(τindividual · ⟨gk
t , giv⟩)
,
(9)
qk
v,t =
exp(τindividual · ⟨gk
v, gk
t ⟩)
PN−1
i=0 exp(τindividual · ⟨gkv, gi
t⟩)
,
(10)
where N is the total number of regions gk
t and gk
v are the teacher and student embeddings for the
k-th region, and τindividual is the temperature parameter to re-scale the cosine similarity.
Segment Anything Model (SAM).
SAM (Kirillov et al., 2023) is a general-purpose segmentation
model that predicts instance masks given spatial prompts. It enables high-quality, class-agnostic
mask generation via zero-shot segmentation, providing fine-grained object candidates valuable for
downstream tasks (Yuan et al., 2024; Han et al., 2025). Recently, LangSplat (Qin et al., 2024)
leveraged SAM to extract hierarchical segmentation masks from images, enabling structured multi-
scale object representation. By densely sampling point prompts across the image, SAM generates
a diverse set of masks that capture object regions at varying levels of granularity. These masks are
filtered and organized into three semantic levels—subpart, part, and whole—based on confidence
and spatial criteria. This hierarchical masking facilitates precise, pixel-aligned feature extraction
and supports the construction of language-aware 3D scene representations.
Multimodel Large Language Models.
Multimodal large language models (MLLMs) have re-
cently been explored in this context (Zhou et al., 2025; Zhang et al., 2025a; Yin et al., 2023; Wang
et al., 2023a; Wu et al., 2023b), as they integrate visual perception with language-based reasoning
and instruction following.
18
Table 9: Zero-shot performance of MLLMs across academic benchmarks. Average accuracy
across six multimodal evaluation datasets: MMBench v1.1 (TestCN/TestEN), MMStar, MMMU (Val),
and HallusionBench.
Model
MMBench V1.1 (2024c)
MMStar (2024)
MMMU (2024)
HallusionBench Avg. (2024)
BLIP2 (2.7B) (2023)
-
-
-
-
InstructBLIP-7B (2023)
28.4
32.7
30.6
31.2
Qwen2-VL-7B (2023)
81.0
60.7
53.7
50.4
• BLIP-2: BLIP-2 (Li et al., 2023) adopts a modular architecture comprising a frozen im-
age encoder, a trainable QFormer (Zhang et al., 2024b), and a frozen language model such
as OPT (Zhang et al., 2022). This setup enables efficient vision-language alignment and
achieves strong performance on tasks such as image captioning and visual question answer-
ing (VQA) with minimal training.
• InstructBLIP: InstructBLIP (Dai et al., 2023) builds on BLIP-2 (Li et al., 2023) via in-
struction tuning, integrating a ViT-G vision encoder (Dosovitskiy et al., 2021) and a frozen
language model, such as Flan-T5 (Chung et al., 2024). This design allows the model to
follow natural language instructions and generalize across diverse multimodal tasks. As
shown in Table 9, this MLLM exhibits fair zero-shot performance on academic multimodal
benchmarks, with accuracy ranging from 24% to 32% on most tasks.
• Qwen2: Qwen2 (Bai et al., 2023) is a multilingual large language model series ranging
from 0.5B to 72B parameters, trained on diverse and high-quality web-scale data. It em-
ploys a tokenizer optimized for multilingual understanding and exhibits strong performance
in reasoning, instruction following, and general language tasks. As shown in Table 9, the
model demonstrates strong generalization, particularly on MMBench (Liu et al., 2024c)
(81.0%), indicating superior reasoning capabilities and vision-language alignment.
G
ADDITIONAL ABLATION STUDY
G.1
IMAGE PREPROCESSING
We explore image preprocessing strategies to help MLLMs better focus on target regions. Their
impact on detection performance is summarized in Table 10.
MLLMs (Li et al., 2023; Dai et al., 2023; Bai et al., 2023; Wang et al., 2023a) exhibit strong zero-
shot reasoning across vision-language tasks such as image captioning and retrieval. However, when
applied to object-level understanding, recent studies (Zang et al., 2025; Fu et al., 2024) have shown
that they remain highly sensitive to visual context. In our setting, where the MLLM is prompted
on individual region proposals, it is essential to emphasize the target object while suppressing ir-
relevant background information. As shown in Table 10, omitting preprocessing slightly degrades
performance, yielding an APN
50 of 33.6 relative to the baseline.
A na¨ıve solution is to black out pixels outside the target segmentation mask. While this approach
directs the model’s attention to the target region, it removes surrounding context and may cause
misclassification due to silhouette artifacts—a limitation highlighted in prior work (Chang et al.,
2023; Fontanini et al., 2023). For example, as illustrated in Figure 6-b, the model misclassifies a
tree as a giraffe, influenced by the masked silhouette of a giraffe in the background. Despite this
issue, the black mask strategy significantly improves performance, achieving an APN
50 of 38.5 (+4.5).
To mitigate such misclassifications, prior work (Qin et al., 2024) suggests that grayscaling and blur-
ring regions outside the mask can effectively suppress background noise and enhance model focus.
In practice, we validate that this strategy improves localization and reasoning in MLLMs (Bai et al.,
2023), as shown in Table 10. Building on these findings, we adopt this preprocessing approach for
all region proposals queried by the MLLM in our CoT pipeline. This configuration yields the best
performance among all settings, achieving an APN
50 of 41.7 (+7.7).
• Raw image: Original training images from the OVD benchmark, where each proposal box
generated by SAM is preserved and all pixels outside the box are blacked out. We observe
19
“trees”
Original Image
“giraffe”
(a) Simple Box
(b) Black Mask
(c) Blur & Grayscale
“giraffe”
“giraffe”
“box”
“Unsure”
“food”
“broccoli”
✅️
❌
✅️
❌
❌
❌
❌
❌
Figure 6: Visualization of image preprocessing strategies. We adopt three strategies: (a) simple
box, (b) black mask, and (c) blur & grayscale. Each image is labeled with the MLLM’s (Bai et al.,
2023) prediction. Qualitative analysis indicates that (c) yields the most reliable zero-shot object
recognition performance.
Table 10: Effect of image preprocessing on pseudo-label quality. Blurred and grayscale images
improve MLLM reasoning, leading to the best results on OV-COCO.
Method
APN
50 (%)
APB
50 (%)
BARON (Wu et al., 2023c)
34.0
60.4
Simple Box
33.6 (-0.4)
59.6
+ Masked image
38.5 (+4.5)
59.5
+ Blurred & grayscale
41.7 (+7.7)
59.4
that MLLMs often struggle to focus on the proposal region during reasoning over the input
query.
• Masked image: Starting from the raw image, only the segmentation mask region within
each proposal box is retained, while all other pixels inside the box are masked in black.
This often distracts the model due to black silhouettes (e.g., a masked giraffe shape), rather
than helping it focus on the intended region.
• Blurred and grayscale image: Also based on the raw image, the segmentation mask re-
gion within each proposal box is preserved, while surrounding pixels inside the box are
blurred and converted to grayscale. Pixels outside the box are blacked out. We adopt this
preprocessing technique in our pipeline, as it effectively preserves contextual cues while
maintaining focus on the target region. For reproducibility, we apply standard BGR-to-
grayscale conversion and Gaussian blur with a kernel size of 31×31 and a sigma of 0.
Statistics.
Figure 7 illustrates the annotation counts of pseudo-labels in the OV-COCO dataset,
revealing a typical long-tail distribution. A small number of frequent categories account for the
majority of annotations, reflecting their higher prevalence in the training data. This imbalance natu-
rally emerges, as the MLLM tends to predict commonly occurring and semantically salient objects.
Notably, the pseudo-labels also include several novel categories (e.g., “dog,” “knife,” and “cup”),
which contribute to improved detection performance on these previously unseen classes.
Semantic diversity of pseudo-labels.
To better understand the nature of our pseudo-labels, we
further group them into several semantic super-classes using GPT-4o (Hurst et al., 2024), prompted
with: “Question: Group the classes into broader super-classes.” As shown in Table 11, prominent
20
Table 11: Statistics of pseudo-labels per super-class. Super-classes are derived from pseudo-labels
using GPT-4o (Hurst et al., 2024) and Qwen2 (Bai et al., 2023) on OV-COCO.
Animals
Furniture
Weapons/Tools
Vehicles
Electronics
Food/Drink
11
14
5
6
6
6
Buildings
Clothing
Shapes
Sports
Misc.
3
3
3
3
5
Figure 7: Distribution of annotations per class. Based on our Qwen2 (Bai et al., 2023) pseudo-
labels across 65 classes in the OV-COCO benchmark. For brevity, we omit the OV-LVIS distribution
visualization, as it includes over 3,000 different pseudo-labels.
Table 12: Result comparisons of the LVIS-trained model on COCO (Lin et al., 2014) and
Objects365 (Shao et al., 2019b). We use BARON as the baseline and evaluate all methods without
fine-tuning.
Methods
MS-COCO (Lin et al., 2014)
Objects365 (Shao et al., 2019b)
AP (%)
AP50 (%)
AP75 (%)
AP (%)
AP50 (%)
AP75 (%)
Supervised (Gu et al., 2022)
46.5
67.6
50.9
25.6
38.6
28.0
ViLD (Gu et al., 2022)
36.6
55.6
39.8
11.8
18.2
12.6
DetPro (Du et al., 2022)
34.9
53.8
37.4
12.1
18.8
12.9
F-VLM (Kuo et al., 2022)
32.5
53.1
34.6
11.9
19.2
12.6
BARON (Wu et al., 2023c)
36.2
55.7
39.1
13.6
21.0
14.5
CoT-PL (Ours)
36.8
56.0
39.5
13.9
21.6
14.9
groups such as Furniture and Animals emerge, highlighting the robustness of our pipeline in pre-
dicting semantically diverse object categories.
However, we observe that some categories—particularly abstract or non-object-level concepts such
as Shapes and Misc—often result in vague or inconsistent outputs.
This suggests that current
MLLMs are not yet fully equipped to handle such concepts reliably, indicating potential for future
improvement
G.2
TRANSFER DETECTION PERFORMANCE
To evaluate cross-dataset generalization, we follow the transfer detection setting, where the model is
trained on OV-LVIS and evaluated on COCO (Lin et al., 2014) and Objects365 (Shao et al., 2019b)
21
Ours
BARON
Figure 8: t-SNE results of feature distributions. Compared to BARON (Wu et al., 2023c), our
CoT-PL generates more compact embeddings for novel representations.
Ours
BARON
Figure 9: t-SNE visualization of background separation. The novel object class “airplane” is
shown in green, and the “ Background ” in pink. Compared to BARON (Wu et al., 2023c),
CoT-PL more effectively separates the novel class from the background.
(d) “person”
(e) “car”
(f) “tennis racket”
(g) “sports ball”
(h) “bus”
Original Image
(a) Object-level Labeling
(b) Pseudo-Labeling
(c) Background Labeling
Base Class
Novel Class
Figure 10: Our proposed CoT-PL generates accurate pseudo-labels without captions through a CoT-
based MLLM pipeline: (a) verifying SAM-generated boxes as valid objects, (b) assigning zero-
shot pseudo-labels, and (c) grounding boxes to distinguish objects from background. This enables
detection of both base (d–e) and novel (h) classes, including unlabeled ones (f) and (g).
without any additional fine-tuning. As shown in Table 12, our CoT-PL, using the default ResNet-50
backbone, demonstrates strong transferability—surpassing F-VLM (Kuo et al., 2022) by +4.3% and
+2.0% AP on COCO and Objects365, respectively, and outperforming BARON (Wu et al., 2023c)
22
Figure 11: Visualization of our pseudo-annotations on the OV-COCO dataset.
by +0.6% and +0.3%. Notably, CoT-PL also substantially reduces the performance gap with fully
supervised detectors, narrowing the AP difference to only 9.7% on COCO and 11.7% on Objects365.
G.3
T-SNE VISUALIZATION
We employ t-SNE (van der Maaten & Hinton, 2008) to visualize the feature distribution of novel
category proposals, emphasizing the effectiveness of our designed schemes. Figure 8 demonstrates
that CoT-PL enables the detector to learn more compact and discriminative features for novel cat-
egory proposals than BARON (Wu et al., 2023c). Gray points represent base classes, while novel
classes are shown in color.
G.4
BACKGROUND FEATURE DISENTANGLEMENT
We employ t-SNE (van der Maaten & Hinton, 2008) to visualize the feature distribution of novel
category proposals and background regions in Figure 9. Compared to BARON, CoT-PL more effec-
tively separates novel class objects from the background. Pink points indicate learnable background
embeddings labeled as “ Background ”, while green points represent the novel class airplane.
H
ADDITIONAL QUALITATIVE RESULTS
H.1
GRAD-CAM VISUALIZATION
In this paper, we introduce CoT-PL, a novel framework that integrates structured visual chain-of-
thought (CoT) reasoning into the pseudo-labeling process. CoT-PL decomposes object understand-
ing into a sequence of interpretable steps—including region perception, category recognition, and
background grounding, effectively generating high-quality pseudo-labels even in complex visual
scenes.
Unlike prior approaches, our method generates pseudo-labels solely from the training set without
relying on image captions. We further propose contrastive background learning (CBL), which lever-
ages background regions as negative samples to enhance feature disentanglement between fore-
ground objects and background clutter.
As illustrated in Figure 10, our CoT-based MLLM pipeline proceeds by (a) verifying SAM-
generated proposals as valid objects, (b) assigning zero-shot pseudo-labels, and (c) grounding each
23
box to distinguish objects from the background. This enables our framework to detect both base and
novel classes, including unlabeled categories that appear in challenging or open-vocabulary settings.
H.2
PSEUDO-ANNOTATION VISUALIZATION
We present visualization examples of our pseudo-annotations in Figure 11. The images are sampled
from the validation sets of the OVD benchmarks. Each example highlights the predicted region
and corresponding class label generated by our pipeline. These visualizations demonstrate that our
method produces semantically meaningful and spatially accurate pseudo-labels across diverse object
categories, including both base (common) and novel (rare) classes.
H.3
DETECTION VISUALIZATION
We present additional detection results from our method, CoT-PL, on two OVD benchmarks: OV-
COCO (Lin et al., 2014) and OV-LVIS (Gupta et al., 2019), as shown in Figures 13 and 12. The
images are sampled from the validation sets of each benchmark.
On COCO, CoT-PL successfully detects novel categories such as “traffic light”, “bus”, “key-
board”, “cup”, “snowboard”, and “cow”.
On LVIS, it identifies rare categories including
“boom microphone”, “mammoth”, “kitchen table”, “poncho”, “escargot”, “shepherd dog”, and
“pennant”. These results demonstrate the model’s ability to recognize a wide range of novel ob-
jects across both benchmarks.
I
PROMPTS
First CoT step.
We leverage SAM’s strong generalization to generate object-level pseudo boxes,
which are verified by the MLLM for object presence.
1
Your Role: Object Presence Recognizer
2
3
You are a model that checks whether a clearly visible object exists in
an image.
4
5
[Your task]
6
- Look at the image.
7
- If there is at least one clearly visible object, answer: Yes
8
- If there is no visible object at all (only blurred or grayscale
areas), answer: No
9
- If it’s hard to tell whether something is visible or not, answer:
Unsure
10
- Specially, if your answer is Unsure, provide reasoning
11
12
[Important rules]
13
- Ignore blurred or grayscale areas in the image.
14
- Only consider clear, colorful, or sharply defined objects.
15
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not fully visible or too unclear)
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Now, analyze the following image:
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Image: <attach image here>
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Reasoning:
Second CoT step.
We leverage the MLLM’s multi-class recognition ability to generate pseudo
labels for specific concepts within each box.
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Your Role: Object Category Identifier
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is clearly visible in an image.
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[Your task]
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Image: (A focused image of a person riding a skateboard)
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Image: <attach image here>
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Third CoT step.
We employ contrastive learning with background representations derived from
a multimodal large language model (MLLM) as negative samples, encouraging the model to better
separate object regions from true background areas. To obtain these background representations, we
prompt the MLLM with the following instruction:
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of the foreground or the background.
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background.
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, or objects of interest.
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Object: Dog
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Object: <Response>
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Figure 12: Visualization of detection results on the OV-COCO dataset. Red boxes and masks
represent novel categories, while blue boxes and masks represent base categories.
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Figure 13: Visualization of detection results on the OV-LVIS dataset. Red boxes and masks
represent novel (rare) categories, while blue boxes and masks represent base categories.
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COT-PL: VISUAL CHAIN-OF-THOUGHT REASONING MEETS PSEUDO-LABELING FOR OPEN-VOCABULARY OBJECT DETECTION Hojun Choi1 Youngsun Lim2 Jaeyo Shin1 Hyunjung Shim1 1KAIST AI 2Boston University {hchoi256,jaeyo ABSTRACT Open-vocabulary object detection (OVD) seeks to recognize and localize object categories beyond those seen during training. Recent approaches typically leverage vision-language models (VLMs) to generate pseudo-labels using image-text alignment, allowing detectors to generalize to unseen classes without explicit supervision. However, these methods depend heavily on direct image-text matching, neglecting the intermediate reasoning steps essential for interpreting semantically complex scenes. This results in limited robustness when confronted with crowded or occluded visual contexts. In this paper, we introduce CoT-PL, a new framework that employs structured visual chain-of-thought (CoT) reasoning into the pseudo-labeling process. CoT-PL decomposes object understanding into three interpretable steps: (1) region perception even for unseen objects, (2) category recognition via zero-shot reasoning, and (3) background grounding to separate semantically complex objects. Crucially, the third step naturally motivates our contrastive background learning (CBL) that uses the pre-computed background cues as negatives to promote feature disentanglement between objects and background. In this way, CoT reasoning and CBL form an integrated pipeline tailored to robust pseudo-labeling in crowded or occluded scenes. Notably, in these two settings, our novel-class pseudo-label quality achieves relative improvements of 103.4% and 168.4% over the best prior, respectively. Our extensive experiments demonstrate that CoT-PL achieves +7.7 AP50 on open-vocabulary COCO and +2.9 mask AP on LVIS for novel classes, setting a new state of the art. 1 INTRODUCTION Pseudo Annotation Open-Vocab. Detection Novel Classes (b) Previous Methods RPN VLMs Image-Captions A boy sits near a dog while watching television. Open-World Base Classes SAM Pseudo Annotation MLLM Open-Vocab. Detection Input Image (c) Our Method Human Annotation Open-Vocab. Detection (a) Conventional Designs Novel Classes Base Classes Detector 🔥 Large-Scale Base Classes Detector 🔥 Detector 🔥 Novel Classes Selftraining Visual CoT Figure 1: Trends in pseudo-labeling for OVD. (a) Manual pseudo-labels for novel classes are costly and do not scale. (b) Recent self-training methods automate pseudo-labeling by labeling region proposals via similarity with category text embeddings using vision-language models (VLMs), but degrade with VLMs' poor object localization and caption-dependent vocabulary. (c) We employ visual chain-of-thought with Segment Anything Model (Kirillov et al., 2023) and multimodal large language models (Bai et al., 2023) for accurate object perception and zero-shot category recognition. Open-vocabulary object detection (OVD) aims to localize both seen (base) and unseen (novel) categories at test time, using only base-class annotations during training. To bridge this supervision gap between seen and unseen categories, recent approaches leverage vision-language models (VLMs) 1 16 Oct 2025 "A person that is standing a name tag and other items." is next to a fence." (a) Noisy Pseudo Boxes (c) Background Collapse (b) Caption Dependency on a skateboard." "Shoes, a cap, an iPod, "A brown and black cow skateboard cap fence cow iPod name tag shoes cow 0.32 0.54 book dog skateboard (d) Quality of Pseudo-Labels Occluded Crowded Occluded Figure 2: Limitations and quality of pseudo-labels for complex scenes. (a) Noisy pseudo boxes due to poor object localization by VLMs, (b) limited object coverage from captions, and (c) occluded objects treated as background. (d) Pseudo-label quality on the OV-COCO validation set: Crowded denotes images with many objects and Occluded denotes objects occluded by other objects. pre-trained on large-scale image-text pairs (Radford et al., 2021). These VLMs map textual descriptions to visual representations, allowing OVD methods to recognize novel classes. Among such efforts, pseudo-labeling has emerged as a state-of-the-art approach for OVD by augmenting the base set with automatically generated annotations that partially cover novel classes (Gao et al., 2022; Pham et al., 2024). Early pseudo-labeling methods for OVD relied on manual annotation of novel classes, which was costly and lacked scalability (Figure 1-a). More recent approaches (Zhao et al., 2022; 2024) automate pipelines that generate pseudo-annotations for novel classes (Figure 1b). This process involves generating pseudo-labels for region proposals during self-training using CLIP (Radford et al., 2021), based on the similarity between region features and text embeddings of potential object categories, typically derived from dataset class names or image captions. Despite their strong performances in general scenes, state-of-the-art OVD approaches still struggle in challenging scenarios like crowded or occluded objects. A key reason is their reliance on direct image-text matching via CLIP, lacking the intermediate visual reasoning steps required for understanding complex scenes (Y ̈uksekg ̈on ̈ul et al., 2023). Recent pseudo-labeling approaches also inherit the same limitation. We identify three key factors underlying their failure in complex settings. (L1) Noisy pseudo boxes: VLMs trained with image-level supervision exhibit co-occurrence bias, favoring crops with semantically related objects (Zhong et al., 2022). This inductive bias hinders object-level pseudo-box labeling in complex, occluded scenes. In Figure 2-a, CLIP incorrectly assigns the highest similarity to the token "skateboard" for a pseudo box with the person's feet partially hidden behind the skateboard. (L2) Caption dependency: Captions lack detail particularly in complex, crowded scenes. As a result, objects not listed in captions remain unlabeled. Figure 2-b shows missing pseudo annotations for many stuff categories (e.g., book) which are absent from the caption. (L3) Background collapse: Detecting objects under occlusion is challenging. These instances are frequently unlabeled and learned as background in training (Li et al., 2024). As illustrated in Figure 2-c, a dog partially occluded by a fence was not detected and treated as background. To directly tackle these three limitations associated with complex scenes, we argue that pseudolabeling for OVD must be reformulated as an interpretable multi-step reasoning process rather than a single-step alignment. In the vision-language domain, visual chain-of-thought (CoT) prompting enhances VLM reasoning by encouraging step-by-step thinking (Wu et al., 2023a). Inspired by this, we design pseudo-label generation using a three-step CoT pipeline explicitly aligned with OVD challenges: (1) region perception with SAM (Kirillov et al., 2023) produces candidate masks, and a multimodal large language model (MLLM) verifies object existence to filter out spurious or partial boxes, mitigating (L1); (2) zero-shot category recognition assigns labels to each region without relying on captions, eliminating (L2); and (3) contextual background grounding distinguishes true objects from unlabeled background regions, resolving (L3). Importantly, the third step naturally leads to our contrastive background learning (CBL), which transforms grounded background cues into negative signals for training. This integration does more than relying on strong models because SAM and MLLMs alone produce noisy or inconsistent labels but our structured reasoning and background-aware learning transform their raw outputs into high-quality pseudo-labels. In this way, CoT reasoning and CBL form a single, integrated pipeline purpose-built for robust pseudo-labeling in complex scenes. In Figure 2-d, our method significantly outperforms prior works (Gao et al., 2022; Zhao et al., 2022; 2024) in generating pseudo-labels under challenging environments. 2 We conduct extensive experiments on two OVD benchmarks, OV-COCO (Lin et al., 2014) and OVLVIS (Gupta et al., 2019). Under challenging conditions like crowding and occlusion, our method demonstrates superior pseudo-label quality compared to prior state-of-the-art pseudo-labeling methods. Our method sets a new state of the art, improving box AP50 for novel classes on OV-COCO by 7.7 and mask mAP on OV-LVIS by 2.9, compared to prior work (Wu et al., 2023c). We report pseudo-label statistics across OVD benchmarks, and our key contributions include: • To our knowledge, we are the first to reformulate pseudo-labeling in OVD as a visual chain-of-thought, decomposing complex-scene object understanding into an interpretable multi-step reasoning process beyond single-step VLM alignment. • We introduce CoT-PL, a unified system for robust pseudo-labeling in complex scenes by integrating CoT reasoning and contrastive background learning (CBL) that uses grounded background cues as negative training signals. • Performance steadily improves with stronger teacher MLLMs, achieving state-of-the-art results in OVD with high-quality pseudo-labels, even in challenging environments. 2 RELATED WORK 2.1 CHAIN-OF-THOUGHT (COT) REASONING Chain-of-thought (CoT) reasoning has emerged as a powerful approach in natural language processing, enabling models to tackle complex reasoning tasks by incrementally decomposing them into interpretable steps. Initial work (Wei et al., 2022) demonstrated that large language models produced more accurate outcomes by generating intermediate reasoning before arriving at a final answer. In the visual domain, multimodal chain-of-thought methods process visual inputs sequentially to reason about future states. These approaches have been applied to diverse tasks, including bounding box prediction (Shao et al., 2024), planning in autonomous driving (Tian et al., 2024), intermediate image infillments (Rose et al., 2023), and CLIP embedding synthesis (Harvey & Wood, 2023). In the vision-language-action setting, CoT reasoning has recently gained traction for guiding closed-loop robotic manipulation through sub-goal images as intermediate reasoning steps (Zhao et al., 2025). In this work, we extend visual chain-of-thought reasoning to generate high-quality pseudo-labels for open-vocabulary object detection, even in semantically complex scenes. 2.2 OPEN-VOCABULARY OBJECT DETECTION Open-vocabulary object detection (OVD) aims to detect novel objects not seen during training by leveraging vision-language models (VLMs) (Radford et al., 2021) trained on large-scale image-text pairs. Recent OVD methods (Du et al., 2022; Wu et al., 2023d) employ prompt modeling to transfer knowledge through learned prompts, enabling more precise contextual descriptions of each class. Several studies (Gu et al., 2022; Wu et al., 2023c) use knowledge distillation to align detectors with VLM features for recognizing unseen objects. Other approaches (Jin et al., 2024; Liu et al., 2024a) reinforce the text modality using large language models (LLMs). Meanwhile, InstaGen (Feng et al., 2024) focuses on the image modality, improving novel class prediction via synthetic images from an image generation model. Furthermore, Grounding DINO (Liu et al., 2024b) introduces promptbased object detection by facilitating cross-modal information exchange between VLMs and transformers. Another OVD approach is pseudo-labeling, which addresses the limited base classes by leveraging extended supervision. These approaches often generate pseudo-annotations or weak supervision through self-training using image captions (Gao et al., 2022; Zhao et al., 2022; 2024) or dataset class names (Zhao et al., 2024). However, they focus on direct image-text matching via CLIP, disregarding a reasoning process necessary for complex scenes (Y ̈uksekg ̈on ̈ul et al., 2023). We reformulate OVD pseudo-labeling as a sequence of interpretable reasoning steps using visual chain-of-thought (CoT), thereby enabling robust pseudo-labeling in challenging environments. 3 METHODOLOGY We introduce CoT-PL, a structured visual chain-of-thought pipeline tailored to robust pseudolabeling, even in two challenging scenarios. Captions in crowded scenes are underspecified, while 3 SAM Whole ... ... foreground background No Object Open-World Base Classes Input Image ... window grass table dresser dog "Question: Does any object exist in the image?" MLLM MLLM "Question: What is the object in the image?" MLLM Question: Is [OBJ] background in the image? Yes No computer screen Preprocess ... person book Semantic Anchors Exclude Classes with Annotations ≤ Across All Training Images True wii window grass table dresser dog computer screen ... person book = Minimum # Classes Annotations Within False Base Classes C with annotations from Concatenate First CoT Step Second CoT Step Third CoT Step Figure 3: Overview of the proposed visual chain-of-thought pipeline. Our method first queries an MLLM about object existence inside SAM-generated pseudo boxes on preprocessed images, then performs zero-shot labeling, and finally extracts background representations nearby. Refined with semantic anchors for reliability, the resulting pseudo-annotations are merged into the base set. simple CLIP matching lacks fine-grained visual perception under occlusion. To address these issues, CoT-PL recasts object understanding as three interpretable reasoning steps: (1) (Pseudo Box Generation) We generate boxes for all instances using SAM (Kirillov et al., 2023) and verify if each box corresponds to a valid object. (2) (Pseudo Label Assignment) We assign a pseudo-label to each box through zero-shot object recognition using an multimodal large language model (MLLM). (3) (Background Extraction) We extract background representations by leveraging the MLLM's grounding capability. Specially, the third step naturally leads to a contrastive background learning (CBL) strategy, which uses the identified background concepts as negative training signals to promote feature disentanglement between objects and background. 3.1 PSEUDO BOX GENERATION First CoT step. We aim to generate accurate pseudo-bounding boxes for all potential objects in the train set. To this end, we leverage the strong generalization ability of SAM to produce segmentation masks for all object instances in each image. Relying on low-level visual cues such as edges and color contrasts, SAM is widely used in open-vocabulary settings to generalize beyond base classes to unseen objects (Zhang et al., 2023; Qin et al., 2024). However, applying class-agnostic SAM to OVD presents two key challenges: (1) it segments regions at varying semantic granularity (i.e. full objects vs. parts), and (2) it may include non-object regions such as background. To address (1), we follow a similar approach to recent work (Qin et al., 2024), which utilizes SAM to extract accurate object-level masks at multiple semantic levels (i.e. whole, part, sub-part). As illustrated in Figure 3, our method selects whole-instance masks to generate precise object-level pseudo boxes that tightly enclose them. To address (2), we then use the zero-shot reasoning capability of a robust MLLM (Bai et al., 2023) to verify whether each box contains a valid object. For example, we prompt the model with: "Question: Does any object exist in the image?" and interpret the response ("Yes" or "No" or "Unsure") as a ternary classification of object existence. Boxes classified as "No" or "Unsure" are discarded, whereas those classified as "Yes" are passed to the next stage of the CoT pipeline. For example, in Figure 3, regions lacking discernible objects (i.e. plain dark areas) are discarded. For "Unsure" responses-a rich source of long-tailed cases, we record the model's rationale to support explainability. At this stage, we achieve region perception by generating accurate pseudo boxes that tightly enclose identifiable yet unlabeled objects. 3.2 PSEUDO LABEL ASSIGNMENT Second CoT step. Most OVD methods rely on CLIP for pseudo-labeling, but it lacks fine-grained visual perception and requires predefined class names (Y ̈uksekg ̈on ̈ul et al., 2023). These limitations 4 inevitably lead to inaccurate or missing pseudo-labels, especially in complex scenes. This motivates the second CoT step, which assigns more accurate pseudo-labels to each box independently of image captions. To achieve this, we leverage an MLLM with strong zero-shot multi-class identification to infer the object category within each box (Wang et al., 2023a). As shown in Figure 3, during the second CoT stage, we prompt "Question: What is the object in the image?", expecting a specific class label beyond the base set. This simple query lays the groundwork for caption-free, open-world pseudo-labeling in OVD. However, MLLMs are sensitive to visual content and often produce irrelevant predictions when queried on pseudo boxes, as attention may leak beyond the target region (Zang et al., 2025; Zhang et al., 2025b; Fu et al., 2024). A common workaround applies hard masks to remove non-target regions, but this often induces errors from misleading visual content (Chang et al., 2023; Fontanini et al., 2023). For example, a masked giraffe's silhouette may bias the model to misclassify a tree as a giraffe (See Appendix G.1). To mitigate this, we apply grayscaling and blurring as a soft mask to suppress non-target regions, helping the MLLM focus and produce more accurate pseudo labels (Yang et al., 2023). This preprocessing step is visualized in Appendix 6. To further improve pseudo-label reliability, we add a post-processing step that filters out outlier labels, chiefly reducing false positives. We exclude pseudo-labels with fewer annotation counts than a threshold, as they are likely unreliable. For simplicity, the threshold is set to the minimum annotation count among base classes. The remaining high-confidence labels, or semantic anchors, are then integrated with the base class set to construct our open-world base set. Figure 3 highlights anchors with color bars: green (foreground), red (background), and white (outlier). Without predefined class names or image captions, the second CoT step yields robust and reliable pseudo-labels by integrating MLLM zero-shot reasoning with reliability-enhancing pre- and post-processing. 3.3 BACKGROUND EXTRACTION Third CoT step. While stronger than VLM-based methods, our MLLM-based pseudo-labeling is not yet complete for long-tailed complex scenes because it hinges on MLLM capability. In such cases, the model often returns "Unsure" for small or occluded objects (i.e. a mug hidden behind a laptop). For example, we observe that weaker models produce more "Unsure" responses and fewer valid labels across diverse objects (See Table 2). These regions remain unlabeled and are treated as background during training (Bansal et al., 2018; Li et al., 2024), since they match neither base nor pseudo-labeled classes, a phenomenon we call background collapse. Unfortunately, explicitly identifying unlabeled, inherently unknown objects is nontrivial. Instead, we sidestep this by disentangling collapsed objects from background representations in the feature space. To this end, we use MLLM grounding as our third CoT step to verify whether a given "[OBJ]" belongs to background via a binary prompt ("Yes" or "No"). This simple binary query helps determine whether a given object category is background. For example, the model classifies "grass" as background and "drawer" as foreground (See Figure 3). These grounded background cues serve as negative supervision, encouraging object-background disentanglement in training. 3.4 CONTRASTIVE BACKGROUND LEARNING (CBL) Our CBL instantiates the idea on BARON (Wu et al., 2023c) that performs competitively through online sampling of compositional structures (i.e., co-occurrence of objects) as training signals. However, the sampling process is computationally expensive and time-consuming. In Figure 4, we design a composition generator that groups cached, pre-computed semantic anchors for each proposal into a bag of regions, reducing overall training time by 4×. Then, these sampled regions are projected into the word embedding space using the linear layer within Faster R-CNN (Ren et al., 2015), resulting in pseudo-words. The pseudo-words are passed through the text encoder T to obtain the bag-of-regions embedding f i t = T (wi 0 + pi 0, wi 1 + pi 1, · · · , wi Ni-1 + pi Ni-1), where N i is the number of regions in the i-th bag, pi j represents the positional embedding of the j-th region in the i-th bag. Finally, this bag-of-regions embedding is aligned with the VLM's image embeddings f i v = V(bi 0, bi 1, · · · , bN i i ), where bi j is the j-th region in the i-th bag. Further implementation details appear in Appendix F. To alleviate the background collapse, we propose a contrastive background learning (CBL) strategy that explicitly disentangles objects, including unlabeled objects, from true background representa5 CLIP Text Encoder grass sky ... prompt prompt prompt person book ... table window ... ... ... ... Open-World Base Classes RPN Semantic Anchors S Semantic Search Composition Generator Pseudo-Words Linear PE RoI Head ... CLIP Text Encoder CLIP Image Encoder Group Region Proposals with Semantic Anchors Region Proposals ... Shared Embedding Space Bag-of-Regions Teacher Pull Class Embeddings ... ... ... ... ... Push person grass window Push Pull dog Mean of Background Embeddings Training dining_table Inference prompt airplane dog ... ... CLIP Text Encoder Pseudo-Words Linear PE RoI Head ... Input Image RPN ... Region Proposals Probability Scores No CBL Probability Scores CBL Probability Scores CBL++ Input Image Figure 4: Overall architecture of CoT-PL. Built on BARON (Wu et al., 2023c) (See Appendix F), the proposed method encodes the open-world base set, partially including novel classes, using the CLIP text encoder. The CLIP embeddings of multiple background concepts are averaged to initialize a single learnable background embedding. These concepts are also used as negative samples in contrastive learning to encourage feature disentanglement between objects and background. At inference time, we apply CBL++ to mitigate class interference by removing pseudo-labels associated with the ground-truth novel classes. tions (i.e. grass or sky) in the feature space during training. As shown in Figure 4, we first encode the base categories Cbase, pseudo-labels Cfg, and identified background concepts Cbg using the CLIP text encoder with the prompt template "a photo of [OBJ]". The averaged background embedding ̄fbg serves as an initialization for a learnable background prior. To encourage feature discrimination, we apply a contrastive objective in which the background embeddings fbg serve as negative samples, formulated as the alignment InfoNCE loss (Rusak et al., 2024): Lbag = 1 2 G-1 X k=0 log pk t,v + log pk v,t , (1) where G is the number of bags, and the pk t,v and pk v,t can be calculated as: pk t,v = exp(τ ′ · ⟨f k t , f k v ⟩) PG-1 l=0 exp(τ ′ · ⟨f k t , f lv⟩) + PM-1 j=0 exp(τ ′′ · ⟨f k t , f j bg⟩) , (2) pk v,t = exp(τ ′ · ⟨f k v , f k t ⟩) PG-1 l=0 exp(τ ′ · ⟨f kv , f l t⟩) + PM-1 j=0 exp(τ ′′ · ⟨f kv , f j bg⟩) , (3) where ⟨·, ·⟩denotes cosine similarity; M is the number of background concepts; τ ′ and τ ′′ are temperature scaling factors. The loss encourages alignment between matched text-image pairs (e.g., dog and window) while pushing apart mismatched background representations (e.g., grass). 4 EXPERIMENTS Datasets and evaluation metrics. We evaluate our CoT-PL using two widely used OVD datasets: OV-COCO (Lin et al., 2014) and OV-LVIS (Gupta et al., 2019). We adopt the category split approach from OVR-CNN (Zareian et al., 2021) for the OV-COCO, dividing object categories into 48 base and 17 novel categories. For OV-LVIS, we follow ViLD Gu et al. (2022), separating the 337 rare 6 Table 1: Result comparisons on OV-COCO (Lin et al., 2014). Methods are categorized into two groups based on whether additional supervision beyond instance-level labels in the base classes CB is utilized during training, i.e., weak or pseudo labels. Note that CN is the novel class set; RN50-C4 uses features from the fourth convolutional stage, and RN50-FPN uses a multi-scale feature pyramid. Methods Supervisions Backbone APN 50 (%) APB 50 (%) Annotation: Extra caption datasets, Weak/Pseudo Labels in CB ∪CN Detic (Zhou et al.) ImageNet21K & CC3M RN50-C4 (24M) 27.8 42.0 OV-DETR (Zang et al.) Pseudo annotation RN50 (24M) 29.4 52.7 CoDet (Ma et al.) CC3M & COCO Caption RN50 (24M) 30.6 46.4 PB-OVD (Gao et al.) COCO Caption RN50-C4 (24M) 30.8 46.4 VL-PLM (Zhao et al.) Pseudo instance-level annotation RN50 (24M) 34.4 60.2 RegionCLIP (Zhong et al.) CC3M RN50-C4 (24M) 35.2 57.6 OC-OVD (Rasheed et al.) COCO Caption RN50-FPN (24M) 36.6 49.4 SAS-DET (Zhao et al.) COCO Caption RN50-C4 (24M) 37.4 58.5 CoT-PL (Ours) Pseudo annotation RN50-FPN (24M) 41.7 59.4 Annotation: Instance-level labels in CB ViLD-ens (Gu et al.) CLIP RN50-FPN (24M) 27.6 51.3 BARON (Wu et al.) CLIP RN50-FPN (24M) 34.0 60.4 CFM-ViT (Kim et al.) CLIP ViT-L/16 (307M) 34.3 46.4 CORA (Wu et al.) CLIP RN50 (24M) 35.1 35.4 BIND (Zhang et al.) CLIP ViT-B/16 (86M) 36.3 50.2 CLIP-Self (Wu et al.) CLIP ViT-B/16 (86M) 37.6 - LBP (Li et al.) CLIP RN50-FPN (24M) 37.8 58.7 CCKT-Det (Zhang et al.) CLIP RN50 (24M) 38.0 - OV-DQUO (Wang et al.) CLIP RN50 (24M) 39.2 - Table 2: Statistics of pseudo labels. We report the number of pseudo class labels and annotations obtained from 118,287 training images across open-vocabulary benchmarks. The class count indicates how many classes in each benchmark are covered by pseudo-labels. Model Total OV-COCO (65 classes) OV-LVIS (1,203 classes) # Classes # Annotations # Unsure # Classes # Annotations # Classes # Annotations BLIP2 (2023) 6,036 395,052 1,528,251 31 197,086 105 137,353 InstructBLIP (2023) 3,106 566,619 1,132,179 30 379,528 93 315,346 Qwen2 (2023) 3,916 637,349 563,156 65 349,399 115 232,298 categories as novel and consolidating the common and frequent categories into base categories. We follow OVR-CNN for evaluation: on OV-COCO, we report box AP at IoU 0.5 for novel categories (APN 50); on OV-LVIS, we report mask mAP over IoUs from 0.5 to 0.95 for rare categories (APr). Implementation details. We build CoT-PL on Faster R-CNN (Ren et al., 2015) with ResNet50FPN. For a fair comparison, we initialize the backbone network with weights pre-trained by SOCO (Wei et al., 2021) and use synchronized Batch Normalization Zhang et al. (2018), following recent studies Wu et al. (2023c); Du et al. (2022). For the main experiments on OV-COCO and OV-LVIS, we choose the 1× and 2× schedules, respectively. We use the CLIP model based on ViT-B-16 (Dosovitskiy et al., 2021) as our pre-trained VLM. For the prompt of category names, we default to the hand-crafted prompts from ViLD (Gu et al., 2022) in all our experiments on OV-COCO and OV-LVIS. We follow the same hyperparameter settings as the baseline (See Appendix D). 4.1 MAIN RESULTS Comparison with state-of-the-art methods. Most recent OVD methods utilize weak or pseudoannotations during training. For instance, RegionCLIP (Gao et al., 2022) and CoDet (Ma et al., 2023) utilize additional caption datasets to discover novel concepts, while OV-DETR (Zang et al., 2022) further generates pseudo-annotations in a self-training manner. As shown in Table 1, on OVCOCO, our CoT-PL achieves the best performance (APN 50 41.7%) among methods using additional supervision, with the default ResNet50 (He et al., 2016) backbone. Furthermore, CoT-PL surpasses 7 Table 3: Result comparisons on OVLVIS (2019). Our CoT-PL achieves competitive performance on instance segmentation. Methods APr (%) AP (%) ViLD-ens (Gu et al., 2022) 16.6 25.5 Detic (Zhou et al., 2022) 17.8 26.8 BARON (Wu et al., 2023c) 19.2 26.5 MIC (Wang et al., 2023c) 20.8 30.7 OADP (Wang et al., 2023b) 21.7 26.6 LBP (Li et al., 2024) 22.2 29.1 CoT-PL (Ours) 23.0 29.4 Table 4: Pseudo-label quality on OV-COCO validation. Metrics on novel-class pseudolabels. Crowded denotes images with more objects than the average in this dataset. Occluded denotes novel-class objects over 50% covered by other ground-truth boxes. Methods COCO Novel Crowded Occluded PB-OVD 18.7 5.1 2.7 VL-PLM 25.5 7.3 3.8 SAS-Det 26.7 11.6 5.7 Ours 31.5 23.6 15.3 instance-level label-based methods relying on distillation, such as BIND (Zhang et al., 2024a). Notably, it also outperforms several recent distillation-based methods, such as CCKT-Det (Zhang et al., 2025a) and OV-DQUO (Wang et al., 2025), under the same ResNet-50 backbone. On OV-LVIS (Table 3), CoT-PL achieves strong performance (23.0% APr) using our generated pseudo-annotations under hand-crafted prompts. It significantly outperforms ViLD (Gu et al., 2022), Detic (Zhou et al., 2022), and BARON (Wu et al., 2023c), and performs better than more recent methods such as LBP (Li et al., 2024) and MIC (Wang et al., 2023c), which utilize extra data with 100 class names. These results highlight the effectiveness of our pseudo-labeling framework across diverse settings. Statistics. Table 2 provides detailed statistics of pseudo-labels generated by different MLLM variants on two benchmarks. Our pseudo-labels span around 4K diverse object classes, totaling 637.349 annotations across these benchmarks. While BLIP2 (Li et al., 2023) covers the most categories (6,036), its relatively low annotation count (395K) suggests sparse predictions with lower per-class confidence. In contrast, InstructBLIP (Dai et al., 2023) produces fewer categories (3,106) but more annotations (566K), reflecting more confident labeling. Qwen2 (Bai et al., 2023) achieves the highest annotation density, generating 637K annotations across 3,916 categories. Notably, as the quality of teacher models improves, the number of "Unsure" responses from MLLMs decreases, yielding more confident pseudo-labeling. These results show that stronger teacher models with higher annotation density with fewer "Unsure" responses, producing high-quality pseudo-labels for OVD. 4.2 ABLATION ANALYSIS Quality of pseudo-labels for complex scenes. We compare the quality of our pseudo-labels with prior state-of-the-art methods (Gao et al., 2022; Zhao et al., 2022; 2024). For a fair comparison, we adopt the same experimental setup and report APN 50 on the COCO validation set. As shown in Table 4, existing methods perform reasonably well on COCO Novel but degrade significantly under challenging scenarios such as Crowded and Occluded. Following (Lin et al., 2014; Qi et al., 2022), we define Crowded as images with more than 8 objects (COCO average), and Occluded as novel ground-truth (GT) boxes covered more than 50% by other GT boxes. These results suggest that previous pseudo-labeling methods lack the reasoning capabilities necessary for fine-grained visual understanding, resulting in noisy pseudo-labels. In contrast, our method achieves superior performance in such challenging scenarios through visual chain-of-thought reasoning. Impact of the individual proposed modules. We conduct an ablation study on OV-COCO to assess the contribution of each component in our CoT-PL framework: pseudo-labeling with CoT reasoning and contrastive background learning (CBL). As shown in Table 5, na ̈ıvely prompting MLLMs with a single-step query for both class names and background grounding without image preprocessing results in only a marginal 0.6% gain over the baseline (Wu et al., 2023c). In contrast, applying image preprocessing improves APN 50 by 2.6%, while a three-step CoT (3×) further increases the gain to 5.7%. Furthermore, incorporating CBL improves performance by 7.1% via better feature disentanglement between objects and background. It also yields a 2.9% gain in the single-step setting with more unlabeled objects. These findings demonstrate the effectiveness of our pseudo-label generation pipeline and CBL in generating high-quality pseudo-labels. Impact of semantic anchors. We assess the impact of semantic anchors under different thresholds in Table 6. The ALL setting, which uses all pseudo-labels without filtering, results in a slight per8 Table 5: Ablation of the proposed individual modules. CoT (K×) refers to pseudo-labeling guided by K-step chain-ofthought (CoT) reasoning. † denotes no image preprocessing. CoT† (1×) CoT (1×) CoT (3×) CBL APN 50 (%) ✓ - - - 34.6 - ✓ - - 37.2 - ✓ - ✓ 40.1 - - ✓ - 40.3 - - ✓ ✓ 41.7 Table 6: Impact of semantic anchors. MIN denotes the fewest base-class annotation, while ALL uses all pseudo labels. CoT-PL performs best under MIN with more reliable pseudo-labels on OV-COCO. Thresholds APN 50 APB 50 ALL 40.7 59.1 MIN 41.7 59.4 Table 7: Impact of proposal generators on pseudo-labeling. The detector is trained on pseudo-labels from multiple proposal generators. Proposal Generator COCO Novel LVIS Mask R-CNN (2017) 39.7 21.6 MAVL (2022) 40.9 22.1 SAM (2023) 41.7 23.0 Table 8: Comparison of MLLM variants for pseudo-labeling. Best and second best results are highlighted. Model Size APN 50 APB 50 BLIP2 (2023) 2.7B 37.6 59.5 InstructBLIP (2023) 7B 40.1 58.7 Qwen2 (2023) 7B 41.7 59.4 formance drop due to the inclusion of noisy and uncertain predictions. In contrast, the MIN setting, which selects semantic anchors with the fewest base-class annotations, achieves improved performance. These results suggest that semantic anchors help filter out unreliable and sparse predictions while guiding training toward semantically consistent regions. Impact of pseudo-box generator quality. The proposed pseudo-labeling pipeline uses region proposals generated by class-agnostic SAM (Kirillov et al., 2023). Following PB-OVD (Gao et al., 2022), we compare SAM with other proposal generators, such as Mask R-CNN (He et al., 2017) and MAVL (Maaz et al., 2022), for pseudo-labeling. Table 7 shows that SAM achieves the best results on both COCO Novel (41.7%) and LVIS (23.0%), highlighting its potential as a proposal generator for pseudo-labeling. Notably, our method achieves competitive performance even with different proposal generators, demonstrating the effectiveness of structured chain-of-thought reasoning. Comparison of MLLM variants. We compare different MLLM variants used in our pseudolabeling pipeline, as shown in Table 8. BLIP-2 (Li et al., 2023), with the smallest parameter size (2.7B), yields the lowest detection performance, partially due to frequent "Unsure" responses, indicating limited capability in generating reliable pseudo-labels. In contrast, both InstructBLIP (Dai et al., 2023) and Qwen2 (Bai et al., 2023) have a comparable model size (7B), but Qwen2 consistently outperforms InstructBLIP across standard multimodal benchmarks such as MMBench (Liu et al., 2024c). This performance gap suggests that stronger MLLMs generate higher-quality pseudolabels, leading to a 1.6% improvement in APN 50 for novel object detection. These findings indicate that advances in MLLM capability can translate into roughly linear gains within our framework. 5 CONCLUSION We introduce CoT-PL, a new pseudo-labeling framework for open-vocabulary object detection (OVD) that leverages structured visual chain-of-thought (CoT) reasoning. Harnessing the zero-shot reasoning capabilities of multimodal large language models (MLLMs), CoT-PL decomposes object understanding into three interpretable reasoning steps: (1) object recognition, (2) caption-free zeroshot labeling, and (3) background extraction. The third step further leads to our contrastive background learning (CBL) that mitigates background collapse by disentangling object and background features. As a result, CoT-PL, a unified system integrating CoT reasoning with CBL, generates highquality pseudo-labels in complex visual scenes and consistently improves performance with stronger teacher MLLMs. CoT-PL achieves state-of-the-art results across multiple OVD benchmarks. 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This rigid design risks accumulating irreversible false negatives. (2) Long-tailed categories are unfairly removed by the frequency threshold of the minimum number of annotations in base classes, discarding any labels below this count as noise for simplicity. Exploring these issues serves as a promising direction for future work. B DEVICE INFORMATION All experiments were conducted using eight NVIDIA A6000 GPUs with PyTorch 1.12.1. Each training run took approximately 11 hours and involved around 12K GPU memory usage. For a fair comparison with the baseline, we fixed the random seed to 1194806617 across all experiments to ensure reproducibility. C IMPLEMENTATION DETAILS Baseline. We build our CoT-PL upon Faster R-CNN (Ren et al., 2015) with a ResNet-50 FPN backbone, consistent with prior open-vocabulary object detection (OVD) work. The backbone is initialized using weights pre-trained with SOCO (Wei et al., 2021) and utilizes synchronized Batch Normalization (SyncBN) (Zhang et al., 2018). We adopt the 1× training schedule for OVCOCO (Lin et al., 2014) and 2× for OV-LVIS (Gupta et al., 2019). Pseudo-label generation process. During our offline pseudo-label generation process, we leverage SAM (Kirillov et al., 2023; Qin et al., 2024) to produce class-agnostic object proposals and apply Qwen2 (7B) (Bai et al., 2023) as our default MLLM for visual chain-of-thought (CoT) reasoning. CLIP based on ViT-B/16 (Radford et al., 2021) is used to encode textual prompts, which are constructed using the hand-crafted template "a photo of [OBJ]" following ViLD (Gu et al., 2022). Pseudo-labels are generated using only the training set, without leveraging any image captions. The pseudo-labels are used exclusively during training and discarded during inference. Contrastive Background Learning. Following ViLD, the contrastive background learning (CBL) background prototypes are built from hand-crafted prompts rather than category names. Using prompt engineering, we define five generic background types, such as sky, water surface, vegetation, paved ground, and plain wall, and craft a small set of object-free prompts for each (i.e. "clear sky background, no objects"). We tokenize the prompts and encode them using the CLIP text encoder, then average the resulting text embeddings to obtain a single prototype per background type. D HYPERPARAMETERS For fair comparison, we adopt the same hyperparameter settings as BARON. We use the SGD optimizer with a momentum of 0.9 and a weight decay of 2.5 × 10-5. The initial learning rate is set to 0.04 for OV-COCO and 0.08 for OV-LVIS. Models are trained for 90,000 iterations on OVCOCO (Lin et al., 2014) and 180,000 iterations on OV-LVIS (Gupta et al., 2019), with a fixed batch size of 16 across all experiments. During training, model checkpoints are saved every 10,000 iterations for OV-COCO and every 30,000 iterations for OV-LVIS. The best-performing checkpoint on the validation set is selected for final evaluation. For our proposed modules, we provide the hyperparameter configurations used in the OVCOCO (Lin et al., 2014) and OV-LVIS (Gupta et al., 2019) experiments. For semantic anchor construction, we filter out infrequent pseudo-labels using a minimum annotation threshold-set to 1,237 for OV-COCO and 1 for OV-LVIS. Additionally, the background contrastive loss temperature 16 parameter is set to τ ′′ = 5.0, which controls the regularization strength of background embeddings relative to foreground embeddings. E DATA APPENDIX We report the statistics of the benchmarks used in this study. For both OVD benchmarks, our pseudolabel generation pipeline uses only the training set, while the validation and test sets remain unused to ensure fair evaluation during inference. • OV-COCO (Lin et al., 2014): OV-COCO is based on the COCO 2017 detection split, containing 118,287 training and 5,000 validation images with instance-level bounding box annotations. We follow the category split from OVR-CNN (Zareian et al., 2021), which defines 48 base and 17 novel categories. During pseudo-label generation, images containing only novel categories are discarded to ensure consistent supervision. All pseudo-labels are generated solely from the base training split. • OV-LVIS (Gupta et al., 2019): OV-LVIS is derived from LVIS v1.0, comprising over 120K training images with annotations for 1,203 categories. Following the ViLD (Gu et al., 2022) protocol, we treat 337 rare categories as novel, and the remaining frequent and common categories as base. Due to the severe long-tail distribution, some rare categories contain fewer than five instances; such categories are removed during semantic anchor construction. Evaluation is performed on the standard LVIS validation split. • Objects365 (Shao et al., 2019a): Objects365 contains 1,742,289 training and 80,000 validation images across 365 object categories. We use this dataset solely for cross-dataset evaluation without any additional fine-tuning. Specifically, the model trained on OV-LVIS is directly evaluated on the Objects365 validation split to assess its transferability. All category names are mapped via exact string matching using CLIP prompt templates. F BASELINES Open-vocabulary detectors. Recent advances (Ren et al., 2015; He et al., 2017) in openvocabulary object detection (OVD) have been largely driven by the emergence of foundation models, including vision-language models (VLMs) (Radford et al., 2021; Jia et al., 2021). VLMs support novel class recognition in OVD through various techniques, such as pseudo-labeling. We build upon Faster R-CNN (Ren et al., 2015) for OVD, replacing its classifier with a linear layer that projects region features into the word embedding space. This enables each region to be represented by multiple pseudo-words, capturing the rich semantics of each object. Given C object categories, the probability of a region being classified as the c-th category: pc = exp(τ · ⟨T (w), fc⟩) PC-1 i=0 exp(τ · ⟨T (w), fi⟩) , (4) where T is the text encoder, ⟨·, ·⟩denotes cosine similarity, τ is a temperature scaling factor, T (w) represents the text embedding of pseudo-words, and fc is the category embedding of a prompt template encoded by the text encoder (i.e. "a photo of {category} in the scene"). BARON (Wu et al., 2023c). Additionally, we instantiate the idea of BARON (Wu et al., 2023c) to capture compositional structures for simplicity. During training, it learns using Faster R-CNN's regression and classification losses (Ren et al., 2015), with annotations provided only for the base set. BARON first groups contextually related neighboring regions for each region proposal extracted from Faster R-CNN (Ren et al., 2015), forming a bag of regions. BARON then projects these regions into the word embedding space using the linear layer within Faster R-CNN, resulting in pseudo-words. The pseudo-words are passed through the text encoder to obtain the bag-of-regions embedding f i t = T (wi 0 + pi 0, wi 1 + pi 1, · · · , wi Ni-1 + pi Ni-1), where N i is the number of regions in the i-th bag, pi j represents the positional embedding of the j-th region in the i-th bag. Finally, this bag-of-regions embedding is aligned with the VLM's image embeddings f i v = V(bi 0, bi 1, · · · , bNi i ), where bi j is the j-th region in the i-th bag. To align the bag-of-regions embeddings, BARON employs 17 Bag of regions Image encoder (a) Aligning bag of regions RCNN Text encoder Pseudo words Bag of regions Image encoder (a) Aligning bag of regions (with semantic anchors) Sampled regions with anchors RCNN Text encoder Pseudo words Sampled regions Loaded from Cache Sampling Process Training Time Reduced by 25% Figure 5: The overall architecture of BARON (Wu et al., 2023c). To reduce sampling time during training, BARON's naive neighbor sampling can optionally be replaced with our semantic anchorbased strategy. In particular, caching the anchors reduced training time by 25% compared to the baseline, while maintaining the original performance. a contrastive learning loss based on InfoNCE (Rusak et al., 2024): Lbag = -1 2 G-1 X k=0 log(pk t,v) + log(pk v,t) , (5) pk t,v = exp(τ ′ · ⟨f k t , f k v ⟩) PG-1 i=0 exp(τ ′ · ⟨f k t , f iv⟩) , (6) pk v,t = exp(τ ′ · ⟨f k v , f k t ⟩) PG-1 i=0 exp(τ ′ · ⟨f kv , f i t⟩) , (7) where G is the number of bags for each region proposal and τ ′ is a temperature scaling factor. This allows BARON to leverage the compositional structures inherent in VLMs, resulting in moderate performance gains. To align individual region embeddings, BARON uses a contrastive learning loss similar to InfoNCE: Lindividual = -1 2 N-1 X k=0 log(qk t,v) + log(qk v,t) , (8) qk t,v = exp(τindividual · ⟨gk t , gk v⟩) PN-1 i=0 exp(τindividual · ⟨gk t , giv⟩) , (9) qk v,t = exp(τindividual · ⟨gk v, gk t ⟩) PN-1 i=0 exp(τindividual · ⟨gkv, gi t⟩) , (10) where N is the total number of regions gk t and gk v are the teacher and student embeddings for the k-th region, and τindividual is the temperature parameter to re-scale the cosine similarity. Segment Anything Model (SAM). SAM (Kirillov et al., 2023) is a general-purpose segmentation model that predicts instance masks given spatial prompts. It enables high-quality, class-agnostic mask generation via zero-shot segmentation, providing fine-grained object candidates valuable for downstream tasks (Yuan et al., 2024; Han et al., 2025). Recently, LangSplat (Qin et al., 2024) leveraged SAM to extract hierarchical segmentation masks from images, enabling structured multiscale object representation. By densely sampling point prompts across the image, SAM generates a diverse set of masks that capture object regions at varying levels of granularity. These masks are filtered and organized into three semantic levels-subpart, part, and whole-based on confidence and spatial criteria. This hierarchical masking facilitates precise, pixel-aligned feature extraction and supports the construction of language-aware 3D scene representations. Multimodel Large Language Models. Multimodal large language models (MLLMs) have recently been explored in this context (Zhou et al., 2025; Zhang et al., 2025a; Yin et al., 2023; Wang et al., 2023a; Wu et al., 2023b), as they integrate visual perception with language-based reasoning and instruction following. 18 Table 9: Zero-shot performance of MLLMs across academic benchmarks. Average accuracy across six multimodal evaluation datasets: MMBench v1.1 (TestCN/TestEN), MMStar, MMMU (Val), and HallusionBench. Model MMBench V1.1 (2024c) MMStar (2024) MMMU (2024) HallusionBench Avg. (2024) BLIP2 (2.7B) (2023) - - - - InstructBLIP-7B (2023) 28.4 32.7 30.6 31.2 Qwen2-VL-7B (2023) 81.0 60.7 53.7 50.4 • BLIP-2: BLIP-2 (Li et al., 2023) adopts a modular architecture comprising a frozen image encoder, a trainable QFormer (Zhang et al., 2024b), and a frozen language model such as OPT (Zhang et al., 2022). This setup enables efficient vision-language alignment and achieves strong performance on tasks such as image captioning and visual question answering (VQA) with minimal training. • InstructBLIP: InstructBLIP (Dai et al., 2023) builds on BLIP-2 (Li et al., 2023) via instruction tuning, integrating a ViT-G vision encoder (Dosovitskiy et al., 2021) and a frozen language model, such as Flan-T5 (Chung et al., 2024). This design allows the model to follow natural language instructions and generalize across diverse multimodal tasks. As shown in Table 9, this MLLM exhibits fair zero-shot performance on academic multimodal benchmarks, with accuracy ranging from 24% to 32% on most tasks. • Qwen2: Qwen2 (Bai et al., 2023) is a multilingual large language model series ranging from 0.5B to 72B parameters, trained on diverse and high-quality web-scale data. It employs a tokenizer optimized for multilingual understanding and exhibits strong performance in reasoning, instruction following, and general language tasks. As shown in Table 9, the model demonstrates strong generalization, particularly on MMBench (Liu et al., 2024c) (81.0%), indicating superior reasoning capabilities and vision-language alignment. G ADDITIONAL ABLATION STUDY G.1 IMAGE PREPROCESSING We explore image preprocessing strategies to help MLLMs better focus on target regions. Their impact on detection performance is summarized in Table 10. MLLMs (Li et al., 2023; Dai et al., 2023; Bai et al., 2023; Wang et al., 2023a) exhibit strong zeroshot reasoning across vision-language tasks such as image captioning and retrieval. However, when applied to object-level understanding, recent studies (Zang et al., 2025; Fu et al., 2024) have shown that they remain highly sensitive to visual context. In our setting, where the MLLM is prompted on individual region proposals, it is essential to emphasize the target object while suppressing irrelevant background information. As shown in Table 10, omitting preprocessing slightly degrades performance, yielding an APN 50 of 33.6 relative to the baseline. A na ̈ıve solution is to black out pixels outside the target segmentation mask. While this approach directs the model's attention to the target region, it removes surrounding context and may cause misclassification due to silhouette artifacts-a limitation highlighted in prior work (Chang et al., 2023; Fontanini et al., 2023). For example, as illustrated in Figure 6-b, the model misclassifies a tree as a giraffe, influenced by the masked silhouette of a giraffe in the background. Despite this issue, the black mask strategy significantly improves performance, achieving an APN 50 of 38.5 (+4.5). To mitigate such misclassifications, prior work (Qin et al., 2024) suggests that grayscaling and blurring regions outside the mask can effectively suppress background noise and enhance model focus. In practice, we validate that this strategy improves localization and reasoning in MLLMs (Bai et al., 2023), as shown in Table 10. Building on these findings, we adopt this preprocessing approach for all region proposals queried by the MLLM in our CoT pipeline. This configuration yields the best performance among all settings, achieving an APN 50 of 41.7 (+7.7). • Raw image: Original training images from the OVD benchmark, where each proposal box generated by SAM is preserved and all pixels outside the box are blacked out. We observe 19 "trees" Original Image "giraffe" (a) Simple Box (b) Black Mask (c) Blur & Grayscale "giraffe" "giraffe" "box" "Unsure" "food" "broccoli" ✅️ ❌ ✅️ ❌ ❌ ❌ ❌ ❌ Figure 6: Visualization of image preprocessing strategies. We adopt three strategies: (a) simple box, (b) black mask, and (c) blur & grayscale. Each image is labeled with the MLLM's (Bai et al., 2023) prediction. Qualitative analysis indicates that (c) yields the most reliable zero-shot object recognition performance. Table 10: Effect of image preprocessing on pseudo-label quality. Blurred and grayscale images improve MLLM reasoning, leading to the best results on OV-COCO. Method APN 50 (%) APB 50 (%) BARON (Wu et al., 2023c) 34.0 60.4 Simple Box 33.6 (-0.4) 59.6 + Masked image 38.5 (+4.5) 59.5 + Blurred & grayscale 41.7 (+7.7) 59.4 that MLLMs often struggle to focus on the proposal region during reasoning over the input query. • Masked image: Starting from the raw image, only the segmentation mask region within each proposal box is retained, while all other pixels inside the box are masked in black. This often distracts the model due to black silhouettes (e.g., a masked giraffe shape), rather than helping it focus on the intended region. • Blurred and grayscale image: Also based on the raw image, the segmentation mask region within each proposal box is preserved, while surrounding pixels inside the box are blurred and converted to grayscale. Pixels outside the box are blacked out. We adopt this preprocessing technique in our pipeline, as it effectively preserves contextual cues while maintaining focus on the target region. For reproducibility, we apply standard BGR-tograyscale conversion and Gaussian blur with a kernel size of 31×31 and a sigma of 0. Statistics. Figure 7 illustrates the annotation counts of pseudo-labels in the OV-COCO dataset, revealing a typical long-tail distribution. A small number of frequent categories account for the majority of annotations, reflecting their higher prevalence in the training data. This imbalance naturally emerges, as the MLLM tends to predict commonly occurring and semantically salient objects. Notably, the pseudo-labels also include several novel categories (e.g., "dog," "knife," and "cup"), which contribute to improved detection performance on these previously unseen classes. Semantic diversity of pseudo-labels. To better understand the nature of our pseudo-labels, we further group them into several semantic super-classes using GPT-4o (Hurst et al., 2024), prompted with: "Question: Group the classes into broader super-classes." As shown in Table 11, prominent 20 Table 11: Statistics of pseudo-labels per super-class. Super-classes are derived from pseudo-labels using GPT-4o (Hurst et al., 2024) and Qwen2 (Bai et al., 2023) on OV-COCO. Animals Furniture Weapons/Tools Vehicles Electronics Food/Drink 11 14 5 6 6 6 Buildings Clothing Shapes Sports Misc. 3 3 3 3 5 Figure 7: Distribution of annotations per class. Based on our Qwen2 (Bai et al., 2023) pseudolabels across 65 classes in the OV-COCO benchmark. For brevity, we omit the OV-LVIS distribution visualization, as it includes over 3,000 different pseudo-labels. Table 12: Result comparisons of the LVIS-trained model on COCO (Lin et al., 2014) and Objects365 (Shao et al., 2019b). We use BARON as the baseline and evaluate all methods without fine-tuning. Methods MS-COCO (Lin et al., 2014) Objects365 (Shao et al., 2019b) AP (%) AP50 (%) AP75 (%) AP (%) AP50 (%) AP75 (%) Supervised (Gu et al., 2022) 46.5 67.6 50.9 25.6 38.6 28.0 ViLD (Gu et al., 2022) 36.6 55.6 39.8 11.8 18.2 12.6 DetPro (Du et al., 2022) 34.9 53.8 37.4 12.1 18.8 12.9 F-VLM (Kuo et al., 2022) 32.5 53.1 34.6 11.9 19.2 12.6 BARON (Wu et al., 2023c) 36.2 55.7 39.1 13.6 21.0 14.5 CoT-PL (Ours) 36.8 56.0 39.5 13.9 21.6 14.9 groups such as Furniture and Animals emerge, highlighting the robustness of our pipeline in predicting semantically diverse object categories. However, we observe that some categories-particularly abstract or non-object-level concepts such as Shapes and Misc-often result in vague or inconsistent outputs. This suggests that current MLLMs are not yet fully equipped to handle such concepts reliably, indicating potential for future improvement G.2 TRANSFER DETECTION PERFORMANCE To evaluate cross-dataset generalization, we follow the transfer detection setting, where the model is trained on OV-LVIS and evaluated on COCO (Lin et al., 2014) and Objects365 (Shao et al., 2019b) 21 Ours BARON Figure 8: t-SNE results of feature distributions. Compared to BARON (Wu et al., 2023c), our CoT-PL generates more compact embeddings for novel representations. Ours BARON Figure 9: t-SNE visualization of background separation. The novel object class "airplane" is shown in green, and the " Background " in pink. Compared to BARON (Wu et al., 2023c), CoT-PL more effectively separates the novel class from the background. (d) "person" (e) "car" (f) "tennis racket" (g) "sports ball" (h) "bus" Original Image (a) Object-level Labeling (b) Pseudo-Labeling (c) Background Labeling Base Class Novel Class Figure 10: Our proposed CoT-PL generates accurate pseudo-labels without captions through a CoTbased MLLM pipeline: (a) verifying SAM-generated boxes as valid objects, (b) assigning zeroshot pseudo-labels, and (c) grounding boxes to distinguish objects from background. This enables detection of both base (d-e) and novel (h) classes, including unlabeled ones (f) and (g). without any additional fine-tuning. As shown in Table 12, our CoT-PL, using the default ResNet-50 backbone, demonstrates strong transferability-surpassing F-VLM (Kuo et al., 2022) by +4.3% and +2.0% AP on COCO and Objects365, respectively, and outperforming BARON (Wu et al., 2023c) 22 Figure 11: Visualization of our pseudo-annotations on the OV-COCO dataset. by +0.6% and +0.3%. Notably, CoT-PL also substantially reduces the performance gap with fully supervised detectors, narrowing the AP difference to only 9.7% on COCO and 11.7% on Objects365. G.3 T-SNE VISUALIZATION We employ t-SNE (van der Maaten & Hinton, 2008) to visualize the feature distribution of novel category proposals, emphasizing the effectiveness of our designed schemes. Figure 8 demonstrates that CoT-PL enables the detector to learn more compact and discriminative features for novel category proposals than BARON (Wu et al., 2023c). Gray points represent base classes, while novel classes are shown in color. G.4 BACKGROUND FEATURE DISENTANGLEMENT We employ t-SNE (van der Maaten & Hinton, 2008) to visualize the feature distribution of novel category proposals and background regions in Figure 9. Compared to BARON, CoT-PL more effectively separates novel class objects from the background. Pink points indicate learnable background embeddings labeled as " Background ", while green points represent the novel class airplane. H ADDITIONAL QUALITATIVE RESULTS H.1 GRAD-CAM VISUALIZATION In this paper, we introduce CoT-PL, a novel framework that integrates structured visual chain-ofthought (CoT) reasoning into the pseudo-labeling process. CoT-PL decomposes object understanding into a sequence of interpretable steps-including region perception, category recognition, and background grounding, effectively generating high-quality pseudo-labels even in complex visual scenes. Unlike prior approaches, our method generates pseudo-labels solely from the training set without relying on image captions. We further propose contrastive background learning (CBL), which leverages background regions as negative samples to enhance feature disentanglement between foreground objects and background clutter. As illustrated in Figure 10, our CoT-based MLLM pipeline proceeds by (a) verifying SAMgenerated proposals as valid objects, (b) assigning zero-shot pseudo-labels, and (c) grounding each 23 box to distinguish objects from the background. This enables our framework to detect both base and novel classes, including unlabeled categories that appear in challenging or open-vocabulary settings. H.2 PSEUDO-ANNOTATION VISUALIZATION We present visualization examples of our pseudo-annotations in Figure 11. The images are sampled from the validation sets of the OVD benchmarks. Each example highlights the predicted region and corresponding class label generated by our pipeline. These visualizations demonstrate that our method produces semantically meaningful and spatially accurate pseudo-labels across diverse object categories, including both base (common) and novel (rare) classes. H.3 DETECTION VISUALIZATION We present additional detection results from our method, CoT-PL, on two OVD benchmarks: OVCOCO (Lin et al., 2014) and OV-LVIS (Gupta et al., 2019), as shown in Figures 13 and 12. The images are sampled from the validation sets of each benchmark. On COCO, CoT-PL successfully detects novel categories such as "traffic light", "bus", "keyboard", "cup", "snowboard", and "cow". On LVIS, it identifies rare categories including "boom microphone", "mammoth", "kitchen table", "poncho", "escargot", "shepherd dog", and "pennant". These results demonstrate the model's ability to recognize a wide range of novel objects across both benchmarks. I PROMPTS First CoT step. We leverage SAM's strong generalization to generate object-level pseudo boxes, which are verified by the MLLM for object presence. 1 Your Role: Object Presence Recognizer 2 3 You are a model that checks whether a clearly visible object exists in an image. 4 5 [Your task] 6 - Look at the image. 7 - If there is at least one clearly visible object, answer: Yes 8 - If there is no visible object at all (only blurred or grayscale areas), answer: No 9 - If it's hard to tell whether something is visible or not, answer: Unsure 10 - Specially, if your answer is Unsure, provide reasoning 11 12 [Important rules] 13 - Ignore blurred or grayscale areas in the image. 14 - Only consider clear, colorful, or sharply defined objects. 15 16 Your response must be only one word: Yes, No, or Unsure. 17 18 [Examples] 19 Example 1: 20 Image: (A color photo of a dog standing clearly in focus) 21 Answer: Yes 22 Reasoning: None 23 24 Example 2: 25 Image: (A grayscale image with blurred outlines and no clear shapes) 26 Answer: No 27 Reasoning: None 28 29 Example 3: 30 Image: (An image where a part of an object might be present, but it is not fully visible or too unclear) 24 31 Answer: Unsure 32 Reasoning: 33 34 Now, analyze the following image: 35 Image: 36 Answer: 37 Reasoning: Second CoT step. We leverage the MLLM's multi-class recognition ability to generate pseudo labels for specific concepts within each box. 1 Your Role: Object Category Identifier 2 3 You are a model that identifies the most likely object category that is clearly visible in an image. 4 5 [Your task] 6 - Look at the image. 7 - Focus only on areas that are clear, colorful, and sharply defined. 8 - Completely ignore grayscale or blurred areas. 9 - Always guess the most likely object category that is clearly visible . 10 11 [Instructions] 12 - Answer with only one or two words. 13 - Do not describe scenes -- just the object category. 14 - If uncertain, make your best guess based on visible clues. 15 16 [Examples] 17 Example 1: 18 Image: (A focused image of a person riding a skateboard) 19 Answer: Skateboard 20 21 Example 2: 22 Image: (A clear image of a zebra walking in grass) 23 Answer: Zebra 24 25 Example 3: 26 Image: (Blurry background, but a sharp image of a backpack is visible) 27 Answer: Backpack 28 29 Now analyze the following image: 30 Image: 31 Answer: Third CoT step. We employ contrastive learning with background representations derived from a multimodal large language model (MLLM) as negative samples, encouraging the model to better separate object regions from true background areas. To obtain these background representations, we prompt the MLLM with the following instruction: 1 Your Role: Foreground-Background Distinguisher 2 3 You are a model that determines whether an object in an image is part of the foreground or the background. 4 5 [Your task] 6 - You are given an object name: " " 7 - Look at the image and decide if this object is in the foreground or background. 8 - Ignore any grayscale or blurred areas in the image. 9 - Use visual focus and typical object roles to decide. 10 11 [Definitions] 12 - Foreground = clearly focused subjects like people, animals, vehicles , or objects of interest. 13 - Background = things like sky, grass, trees, mountains, or flowers. 14 25 15 Your answer must be exactly one word: Foreground or Background. 16 17 [Examples] 18 Example 1: 19 Object: Dog 20 Image: (A dog is standing in sharp focus in front of a blurry park) 21 Answer: Foreground 22 23 Example 2: 24 Object: Sky 25 Image: (A person is standing in front of a bright blue sky) 26 Answer: Background 27 28 Example 3: 29 Object: Tree 30 Image: (A clear person in front, with trees in the back) 31 Answer: Background 32 33 Now analyze the following image: 34 Object: 35 Image: 36 Answer: 26 Figure 12: Visualization of detection results on the OV-COCO dataset. Red boxes and masks represent novel categories, while blue boxes and masks represent base categories. 27 \ Figure 13: Visualization of detection results on the OV-LVIS dataset. Red boxes and masks represent novel (rare) categories, while blue boxes and masks represent base categories. 28
|
2510.14778
|
Leveraging Code Cohesion Analysis to Identify
Source Code Supply Chain Attacks
1st Maor Reuben
Ben-Gurion University of the Negev
Beer Sheva, Israel
maorreu@post.bgu.ac.il
2nd Ido Mendel
Ben-Gurion University of the Negev
Beer Sheva, Israel
idoman@post.bgu.ac.il
3rd Or Feldman
Ben-Gurion University of the Negev
Beer Sheva, Israel
orfel@post.bgu.ac.il
4th Moshe Kravchik
Ben-Gurion University of the Negev
Beer Sheva, Israel
moshekr@post.bgu.ac.il
5th Mordehai Guri
Ben-Gurion University of the Negev
Beer Sheva, Israel
gurim@post.bgu.ac.il
6th Rami Puzis
Ben-Gurion University of the Negev
Beer Sheva, Israel
puzis@bgu.ac.il
Abstract—Supply chain attacks significantly threaten software
security with malicious code injections within legitimate projects.
Such attacks are very rare but may have a devastating impact.
Detecting spurious code injections using automated tools is further
complicated as it often requires deciphering the intention of both
the inserted code and its context. In this study, we propose an
unsupervised approach for highlighting spurious code injections
by quantifying cohesion disruptions in the source code. Using a
name-prediction-based cohesion (NPC) metric, we analyze how
function cohesion changes when malicious code is introduced
compared to natural cohesion fluctuations. An analysis of 54,707
functions over 369 open-source C++ repositories reveals that code
injection reduces cohesion and shifts naming patterns toward
shorter, less descriptive names compared to genuine function
updates. Considering the sporadic nature of real supply-chain
attacks, we evaluate the proposed method with extreme test-set
imbalance and show that monitoring high-cohesion functions
with NPC can effectively detect functions with injected code,
achieving a Precision@100 of 36.41% at a 1:1,000 ratio and
12.47% at 1:10,000. These results suggest that automated cohesion
measurements, in general, and name-prediction-based cohesion,
in particular, may help identify supply chain attacks, improving
source code integrity.
Index Terms—Supply Chain Attacks, Cohesion Analysis, Unsu-
pervised Malware Detection.
I. INTRODUCTION
In today’s software-driven world, ensuring the security and
reliability of software systems has become a critical concern.
The increasing prevalence of supply-chain compromises, where
malicious actors exploit vulnerabilities in the source code
supply chain to introduce unauthorized modifications, poses
significant threats to the integrity and trustworthiness of
software applications [1].
A prominent example is the SolarWinds incident in 2020,
where attackers embedded a backdoor in an update for the Orion
platform, granting unauthorized remote access to thousands of
corporate and government systems and causing extensive data
breaches [2]. Similarly, Kaseya’s software was compromised by
ransomware, resulting in a $70 million extortion demand 1 [3].
1https://www.kaseya.com/
Traditional methods for detecting these attacks primarily rely
on rule-based approaches [4], [5] and machine learning- driven
static or dynamic code analysis techniques [6], [7]. However,
these techniques depend heavily on known malware signatures,
behavior patterns, or supervised learning models that require
labeled datasets. As attackers continue to adapt, employing
sophisticated anti-analysis techniques to evade detection [8],
there is a pressing need for unsupervised approaches that can
identify subtle anomalies without prior knowledge of malicious
patterns.
One promising avenue for addressing this challenge is
the detection of code cohesion anomalies. Code cohesion, a
fundamental software quality attribute, measures the extent to
which elements within a module (e.g., functions or classes)
collaborate to achieve a common objective [9]. High cohesion
typically indicates robust, maintainable, and readable code,
whereas low cohesion can signal the introduction of unrelated
responsibilities. By continuously monitoring code cohesion,
deviations from expected levels can serve as early warning
signs of both security threats and maintainability concerns.
In this paper, we introduce an unsupervised methodology
for detecting potential supply chain attacks by leveraging
pre-trained language models (PLMs) to assess code cohesion.
Our approach frames cohesion estimation as a function purpose
prediction problem, where the alignment between a function’s
body and its intended name serves as a proxy for cohesion.
Core to our approach is the idea that the cohesion of a
function is determined by how different elements in the code
contribute to its purpose [10]. By evaluating functions for
deviations from normal cohesion levels, our method detects
possible compromises without relying on labeled data or prior
knowledge of malicious code. Additionally, our approach
supports software maintainers by identifying functions that
may require refactoring due to cohesion degradation, assisting
in long-term software evolution and quality assurance efforts.
We evaluate our approach using a dataset of 369 open-source
C++ projects, capturing comprehensive version histories for
functions in each project (Section III-A). Through simulated
arXiv:2510.14778v1 [cs.SE] 16 Oct 2025
malware injection scenarios, we demonstrate that our method ef-
fectively differentiates between legitimate maintenance changes
and malicious code alterations. Moreover, we assess how our
technique can aid in identifying unintended cohesion drift over
time, making it a valuable tool for both security enforcement
and software maintenance practices.
The main contributions of this study are threefold:
• An unsupervised methodology for detecting supply chain
attacks by leveraging continuous cohesion evaluation.
• A novel use of PLMs for function-level cohesion estima-
tion.
• Targeted identification of software components requiring
cohesion improvement.
The rest of this article is structured as follows: Section II
summarizes the prior art related to supply chain attacks,
cohesion metrics, and language models for source code analysis.
The analysis methods, including the data and cohesion change
tracking, are elaborated in Section III. The main research ques-
tions and results are discussed in the experimental Section IV.
Section V concludes the paper.
II. RELATED WORK
A. Supply chain attack
The increasing development of open-source software has
opened another way for attackers to harm software systems.
Attackers leverage the use of open source during software
development, which is enhanced by dependency managers
that update, download, and install open-source packages
automatically. This makes it easier for attackers to inject
malicious code into the dependency tree of a widely used
package, which can help them reach a large number of systems
in a short period of time [11].
Attackers attempting to inject malicious code into existing
packages may adopt various tactics. They may mimic legitimate
contributors to open-source projects, submitting pull requests
with seemingly benign code changes [6]. Alternatively, attackers
may compromise project repositories directly, committing
malicious code with weak or compromised credentials, or by
social engineering to become project maintainer [12]. Malicious
packages resulting from such attacks can propagate malware
across extensive software systems. Vu et al. [13] demonstrated
that malicious packages surpassed 100,000 downloads in several
instances, illustrating the significant impact these attacks can
have. Real-world instances, like the attack on the widely used
Homebrew [14], SolarWinds’ Orion platform [2], and Kaseya’s
software [3] emphasize the severity of these threats.
Numerous solutions have been suggested to mitigate the
risks posed by such attacks. Duan et al. [6] employed
established program analysis techniques, including metadata
analysis, to identify packages dependent on known malware or
sharing similar authors and release patterns. They also utilized
static analysis to scrutinize installation logic and flows, and
dynamic analysis to detect unusual network communication,
file usage, and process behavior. Their study revealed 339
previously unidentified malicious packages. Another approach
by Garrett et al. [7] proposed an anomaly detection-based
method, focusing on security-relevant features during version
updates. Both studies acknowledge the incompleteness of their
solutions, emphasizing their role as foundational building
blocks. Recent approaches have also leveraged transformer
models for malicious code detection. Tsfaty and Fire [15]
demonstrated that by clustering multiple versions of a function
based on code embeddings using Code2Seq [16], they can flag
versions that significantly deviate from their nearest cluster. The
main issue with their approach is that they can only monitor
function implementations that are highly used across different
projects and are supposed to have similar implementations
(e.g. "run" or "get" functions). Another study used pre-trained
transformers to train classifiers that detect malicious code in
JavaScript packages [17]. While they achieved promising results
using the backstabber dataset with 10-fold cross-validation,
their evaluation methodology did not account for differences
between injection patterns in training and test sets, and
was dependent on known malware behavior. Although many
solutions were suggested for identifying code injection, a recent
analysis on a large dataset of 2105 packages from the PyPI2
ecosystem showed that malicious code has proven adept at
evading detection from existing tools through the employment
of numerous anti-analysis techniques [8].
The persistent vulnerability of the source code supply chain
to attacks is underscored by the lack of an integrated and robust
protection mechanism [18]. Our proposed method represents a
more comprehensive solution compared to existing approaches.
Unlike methods primarily reliant on detecting known malware
based on behavior, our approach stands out by detecting side
effects inherent in every instance of malware injection into
cohesive code.
B. Cohesion metrics
Code cohesion is a fundamental concept in software engi-
neering, referring to the degree to which the elements within
a module belong together [9]. High cohesion is desirable as
it indicates that the elements within the module are closely
related and work together to perform a specific task. Various
metrics have been developed to measure cohesion in object-
oriented software, aiming to assess the reusability, efficiency,
and complexity of software modules. These metrics are essential
for evaluating the quality of code and identifying opportunities
for restructuring to improve the internal structure of software
systems [19].
Several approaches have been proposed to address code
cohesion measures. The Lack of Cohesion in Methods (LCOM)
metric is a cohesion measure for object-oriented software,
initially defined by Chidamber and Kemerer [20]. It assesses
the level of cohesion in software modules by measuring the
number of method pairs that do not share instance variables.
However, one issue with the LCOM metric is that it fails to
differentiate between possible levels of cohesion [21]. Tight
Class Cohesion (TCC), proposed by Bieman and Kang [10],
2https://pypi.org/
measures direct connections between methods through common
instance variables. Class Connection Metric (CCM), developed
by Wasiq [22], constructs a method connection graph based
on shared attributes and method calls. CCM quantifies the
connectedness of this graph to measure cohesion, handling
transitive connections between methods. Loose Class Cohesion
(LCC), proposed by Bieman and Kang [10], considers both
direct and indirect connections between methods, taking into
account transitivity.
Another approach to measure code cohesion involves the
use of refactoring to improve the cohesion of object-oriented
software. Refactoring aims to alter the internal structure of
the code without changing its external functionality, thereby
improving cohesion and reducing coupling in the software
system [23]. Additionally, novel cohesion metrics, such as
the Variable Frequency – Inverse Method Frequency (VF-
IMF) metric, have been developed to assess the level of
cohesion in modules and group module methods to instill high
cohesion [19]. These metrics offer a compromised solution for
building high cohesive modules and differentiate between differ-
ent levels of cohesion, addressing the limitations of traditional
cohesion metrics, such as LCOM. The closest approach to our
work is the work of [24] which tries to solve the challenge of
manually tracing the cohesion value in software code, which
is essential for evaluating software maintainability. The study
focuses on predicting cohesion values, including LCOM2, TCC,
and LCC, using machine learning techniques such as KNN,
REPTree, multi-layer perceptron, linear regression, support
vector machine, and random forest. The research utilizes
two different open-source software projects to create datasets
and evaluates the performance of various machine learning
algorithms for predicting different cohesion metrics.
Our work is distinct in harnessing the semantic capabilities of
large language models to estimate code cohesion. By detecting
significant deviations, we can identify components needing
additional security auditing and analysis.
C. Code analysis with language models
Large Language Models (LLMs) have shown significant
potential in natural language understanding and programming
code processing tasks. Their ability to comprehend and generate
human-like code has sparked research interest in leveraging
LLMs for code analysis purposes [25].
LLMs have been evaluated for their capabilities in automat-
ing code analysis tasks, including the analysis of obfuscated
and malicious code [26]. The findings indicate that LLMs can
serve as valuable tools for automating code analysis, albeit
with certain limitations. Their research contributes to a deeper
understanding of the potential and constraints associated with
utilizing LLMs in code analysis, paving the way for enhanced
applications in this critical domain. Recent work assessed
CodeBERT and ChatGPT for their efficiency in security-
oriented program analysis, addressing tasks like code review
and code generation. The study delves into the capabilities
of LLMs in solving security-related analytic tasks, providing
Fig. 1: Overview of cohesion monitoring to detect potential
security compromises. By analyzing cohesion metrics over suc-
cessive version releases, we can identify significant decreases
to flag for investigation.
insights into their potential and limitations in addressing
software security challenges [27].
These use cases demonstrate the diverse applications of
LLMs in code analysis, ranging from scientific research and
data analysis to automating code analysis tasks and addressing
security-oriented program analysis challenges. Since LLMs
show promise in enhancing productivity and automating various
code-related tasks, we choose to use them in order to estimate
code cohesion.
III. METHODS
In this section, we present our methodology for highlighting
spurious code insertions by monitoring for significant drops
in code cohesion. We begin by describing the dataset used
to evaluate our code cohesion metrics over time. Next, we
demonstrate how fine-tuned language models are used to
quantify code cohesion through a function name prediction
task. Finally, we describe our approach to simulating software
compromise by injecting malicious code into functions. The
primary goal of our methodology is to monitor software over
time and detect significant decreases in code cohesion that
may indicate a security compromise. Figure 1 illustrates this
process by analyzing cohesion metrics for each new version
release, we can flag suspicious drops for further investigation.
A. Dataset
We collected a corpus of 109,924 C++ functions from 369
widely used open-source GitHub projects3. To expand this
dataset, we gathered all publicly available versions of these
functions from their complete Git histories.
A function version represents either the initial implemen-
tation or any subsequent modification of a function due to a
commit. To ensure data quality and completeness, the collection
process involved the following steps:
1) Gathering repository metadata and commit histories.
2) Filtering commits to retain only those with file modifi-
cations.
3A list of the repositories used can be found at https://tinyurl.com/
github-repositories
3) Downloading the modified files.
4) Extracting functions using Clang4.
Each function was assigned a unique identifier combining
its name, argument types, and file location, allowing precise
tracking across versions even when implementations evolved.
Using these identifiers, we eliminated unchanged duplicates
while maintaining comprehensive version histories. After
filtering out functions with only a single version, the final
dataset comprised 479,996 versions of 54,706 unique functions,
with an average of 8.7 and a median of 3 versions per function.
For analysis, we focused on functions with multiple versions,
yielding 425,290 consecutive version pairs. These pairs capture
natural changes resulting from regular maintenance and feature
development, forming a robust foundation for examining
cohesion dynamics over time and identifying deviations caused
by code injections. Detailed statistics for the full dataset are
presented in Table I.
TABLE I: Dataset Summary Statistics
Statistic
Value
Open source C++
#Projects
369
Avg. lines of code per project
131,758
Linux-compatible projects
369 (100%)
Windows-compatible projects
115 (31.25%)
#Function versions
479,996
#Unique functions
54,706
Average versions per function
8.7
Median versions per function
3
Avg. lines of code per function
14.65
Avg. tokens per function
197.58
Malware
#Malicious code examples
9
Avg. Lines of Code per malware
6.44
Avg. tokens per malware
77.66
#Consecutive version pairs
425,290
B. Function name prediction
The function name prediction approach leverages a pre-
trained language model to predict an appropriate name for a
function based solely on its body text. Appropriately named
functions typically exhibit high cohesion, as the name succinctly
captures their singular, focused purpose. Intuitively, a coherent
function performing a single focused task should be easier for
the model to name accurately compared to one with disjoint
or unrelated logic. To operationalize this idea, we employ a
systematic methodology based on masked language modeling
(MLM). Using this technique, we replace the function name
with a sequence of token masks (e.g., <mask1>, <mask2>),
and the model predicts the most probable terms to fill these
masks, effectively generating a name for the function.
For a given function f, we perform the fill-mask operation
on its name using token counts ranging from one to eight,
which covers the typical range of function names in practice.
The confidence for f when using n tokens is calculated using
the harmonic mean of the token probabilities, as shown in
Equation 1.
Confidence(f, n) =
n
sumn
i=1pi−1
(1)
4https://clang.llvm.org/
Here, pi is the probability of the i-th token. The harmonic
mean penalizes low-probability tokens, ensuring that the metric
reflects consistent confidence across all predicted tokens. Then,
we compute the confidence for each token count and select
the maximum confidence as the cohesion metric, defined in
Equation 2.
NPC(f) = max(Confidence(f, n) | n ∈{1, . . . , 8})
(2)
This metric, referred to as the name prediction cohesion
(NPC), provides an aggregate measure of how well the model
can generate a cohesive, meaningful name for the function. We
also refer to the token count that yields the highest confidence
as the optimal token count (OTC), which indicates the number
of tokens that best capture the semantic intent of the function.
Using this terminology, we can also define NPC of function f
as follows:
NPC(f) = Confidence(f, OTC(f))
(3)
This approach not only quantifies the alignment between
function names and their logic but also provides a systematic
method for evaluating cohesion across varying token counts.
To implement this methodology, we used the CodeBERTCpp
model, which is pre-trained and fine-tuned on C++ code,
making it well-suited for analyzing the syntax and semantics
of functions in this language.
C. Code Injection
To evaluate the sensitivity of our cohesion quantification
techniques to compromised integrity, we inject malicious code
segments into functions and analyze their impact on cohesion
assessments. The malicious code corpus consists of segments
exhibiting common behaviors such as exfiltrating sensitive
data, performing privilege escalation, and enabling remote
code execution. These segments are sourced from publicly
available repositories of known malware code5. After selecting
these segments, we inject them at three strategic locations
within functions: the beginning (after the declaration but before
existing instructions), middle (after 50% of the original lines),
or end (after all existing code). We ensure the modified code
maintains syntactic validity.
In Figure 2, we show an example of a function before and
after injection. The injected payload writes directly to the
master boot record (MBR), demonstrating how we simulate
malware injection.
D. Cohesion Change Quantification
To evaluate the cohesion change after code injection or
between consecutive versions of the same function, we use
two complementary methods: The Cohesion Difference (CD)
is defined as the NPC difference between functions f1 and f2,
detailed in Equation 4.
CD(f1, f2) = NPC(f1) −NPC(f2)
(4)
5The malware code used in this study can be accessed at https://tinyurl.com/
malicious-implants
(a) Function before malware injection.
(b) Function after malware injection at the beginning of a function.
Fig. 2: Example of malware injection into a function.
The Optimal Token Count Difference (OTCD) is the
confidence difference of f1 optimal token count (OTC), detailed
in Equation 5:
OTCD(f1, f2) = NPC(f1) −Confidence(f2, OTC(f1))
(5)
These methods provide complementary insights into cohesion
changes. The CD captures the overall cohesion drift, reflecting
shifts in the model’s maximum confidence about the function’s
coherence, regardless of token count. The OTCD captures
disruptions specific to the original function’s optimal represen-
tation, highlighting how well the modified function maintains
the semantic intent of the original token count. Together, they
help quantify both structural and semantic disruptions caused by
code injection or iterative changes between function versions. A
positive value in either metric indicates a decrease in cohesion,
while negative values suggest increased cohesion.
IV. EXPERIMENTS
To evaluate the effectiveness of cohesion-based security
monitoring, we investigate four key research questions (RQ):
Research Question 1: How does malware injection affect
cohesion when measured through function name prediction?
Research Question 2: What is the distribution of cohesion
across functions in real-world software?
Research Question 3: How effectively can cohesion met-
rics distinguish between legitimate maintenance changes and
malicious code injections?
A. RQ1 - How does malware injection impact cohesion when
measured using function name prediction?
In well-structured code, functions should have clear and
descriptive names that convey their purpose and functionality,
helping developers understand what the function does without
needing to examine its implementation details. Names that
are too short may be ambiguous and insufficiently descriptive,
while overly long names can become cumbersome and harder
to read. When code injection occurs, it often introduces
functionality that diverges from the function’s original purpose,
creating a semantic gap between the function’s name and
its actual behavior. This misalignment between name and
implementation can serve as a potential indicator of malicious
modifications.
To investigate this, we explore three hypotheses (H) related
to function name prediction:
Hypothesis 1: For every function, there is an optimal number
of tokens comprising its name.
Hypothesis 2: Code injection changes the optimal number
of tokens in a function’s name.
Hypothesis 3: Code injection reduces the function cohesion.
To test these hypotheses, we analyzed 479,996 functions
from the dataset, predicting names ranging from one to eight
tokens. We grouped functions based on their optimal token
count (OTC) and displayed the aggregated mean and quantiles
of the results for maximum confidence, as shown in Figure 3.
The analysis reveals distinct peaks in confidence—both
in mean and quantiles—corresponding to the OTCs. These
peaks demonstrate that functions naturally gravitate toward
specific name lengths that optimize their descriptiveness and
predictability, providing quantitative support for H1.
Next, we injected malware (see Section III-C) at the
beginning, middle, and end of each function and applied
the same process. The results are presented in Figure 4.
After code injection, the OTC shifts to one token for many
functions, and the overall maximum confidence decreases. This
simplification of function names after injection supports H2,
showing that external code alterations negatively impact natural
naming conventions. Furthermore, the drop in confidence post-
injection supports H3, indicating that code injection reduces
the predictability of function names.
Following this, we conducted a comprehensive evaluation
of cohesion changes under two scenarios: malicious code
injection and natural code evolution. For the injection scenario,
we modified each function by inserting different types of
malicious code at three locations (beginning, middle, and end)
and averaged the results (see Section III-C). We quantified
these changes using two metrics: the cohesion difference (CD)
and optimal token count difference (OTCD), measuring the
changes in these values before and after injection. We also
performed a separate analysis on high-cohesion functions
(NPC > 0.5) to determine whether injection effects were
more pronounced in well-structured code. Since high-cohesion
functions have a more predictable structure and naming
consistency, we hypothesize that malicious injections will cause
greater disruptions, making them easier to detect.
To establish a baseline for comparison, we also examined
how cohesion naturally evolves across multiple versions of the
same function. Using the same metrics (CD and OTCD), we
Fig. 3: Distribution of name prediction confidence across different token counts. Each subplot represents functions grouped by
their optimal number of tokens. The solid line shows the mean confidence, while the shaded area represents the 25th to 75th
percentile range. The x-axis shows the number of tokens, and the y-axis represents the prediction confidence score.
Fig. 4: Impact of code injection on name prediction confidence. Each subplot shows functions grouped by their original optimal
token count. Blue lines represent the original code’s confidence scores, while red lines show scores after injection. The x-axis
represents the number of tokens, and the y-axis shows the prediction confidence score.
TABLE II: Impact of code injection on cohesion metrics
Position
CD
OTCD
All
functions
Beginning
0.027±0.224*
0.105±0.227*
Mid
0.040±0.173*
0.086±0.178*
End
0.038±0.172*
0.083±0.177*
Baseline
0.0005±0.12
0.029±0.131
High
cohesion
(>0.5)
Beginning
0.081±0.207*
0.142±0.231*
Mid
0.077±0.163*
0.11±0.183*
End
0.074±0.161*
0.106±0.18*
Baseline
0.021±0.112
0.045±0.138
* indicates statistical significance (p < 0.05) compared to the consecutive
versions (baseline). Values are presented as mean ± standard deviation.
analyzed 425,290 consecutive version pairs, measuring how
cohesion changes during normal development. Detailed results
on the impact of code injection on cohesion metrics across three
injection positions and between consecutive version (baseline)
pairs are presented in Table II.
Our results demonstrate that code injection significantly
impacts function cohesion across all injection positions, with
CD values ranging from 0.027 to 0.038 and OTCD values
ranging from 0.083 to 0.105. Notably, the changes in cohesion
between consecutive versions of the same function (mean CD
= 0.0005, mean OTCD = 0.031) are significantly (p < 0.05)
smaller than those observed after code injection (minimum CD
= 0.027, minimum OTCD = 0.083), with differences of over an
order of magnitude in CD values. For high-cohesion functions,
the impact of injection was substantially stronger, with CD
values ranging from 0.074 to 0.081 and OTCD values ranging
from 0.106 to 0.142. The increased sensitivity in high-cohesion
functions is particularly notable close to the beginning, where
CD increased to 0.081 (200% higher than the general case)
and OTCD reached 0.142 (35% higher). This suggests that
well-structured code is indeed more sensitive to malicious
modifications, making injection detection more reliable in high-
Fig. 5: Distribution of name prediction cohesion (NPC) scores
across functions.
quality codebases.
These findings provide strong support for our hypotheses:
functions exhibit an optimal token count (H1), which is
disrupted by injection (H2). Also, both CD and OTCD
metrics detect these disruptions showing statistically significant
reductions in cohesion after injection (H3). The clear distinction
between natural evolution and injected modifications, partic-
ularly in CD measurements, demonstrates the effectiveness
of our name prediction cohesion (NPC) metric for detecting
potential malicious code insertions.
B. How is cohesion distributed across functions, and how is
it affected by maintenance changes and function size?
We examined the distribution and characteristics of function
cohesion from three perspectives: overall distribution, mainte-
nance impact, and the relationship with function size.
To understand how cohesion naturally varies across different
functions, we first computed a cohesion histogram for all
consecutive function version pairs in the dataset using the name
prediction cohesion (NPC) metric. The results are presented
in Figure 5. The histogram reveals that the majority of
functions (approximately 63%) exhibit high cohesion (greater
than 0.5). This aligns with software engineering principles
and expectations, as these functions come from widely used
open-source packages that typically undergo code reviews and
maintain quality standards.
To understand how cohesion changes during normal main-
tenance vary across different cohesion levels, we examined
the relationship between a function’s base cohesion level and
its CD and OTCD measurements. We calculated the Pearson
correlation between the function’s NPC score and both its
CD and OTCD values during normal maintenance changes.
The analysis revealed significant (p-value < 0.05) positive
Pearson’s correlations: 0.951 for CD and 0.894 for OTCD with
respect to cohesion levels. These strong correlations indicate
that functions with higher cohesion tend to experience larger
Fig. 6: Mean and standard deviation of name prediction
cohesion (NPC) across function sizes. Each bucket represents a
five-line interval, showing consistent cohesion levels regardless
of function length.
cohesion drops during maintenance. This can be explained
mathematically, due to the fact that larger values of NPC
(range between 0 to 1) have more "room" to fall toward 0 than
lower values.
Next, we explored the relationship between cohesion scores
and the number of lines of code. To achieve this, we grouped
functions into buckets based on their size, with each bucket
representing a five-line interval (see Figure 6).
The analysis revealed that the average and standard deviation
of code cohesion remained consistent across different function
sizes (about 0.608±0.212) and found no significant correlation
between NPC and function size (p-value = 0.186). This
indicates that the NPC metric provides a reliable measure
of function cohesion, independent of function size.
To further investigate how the relative size of injected code
compared to function size impacts cohesion, we calculated
the mean cohesion difference (CD) and optimal token count
difference (OTCD) for malicious pairs in each bucket. Across
all buckets, the mean cohesion difference ranged from 0.028
to 0.051, with a standard deviation between 0.167 and 0.198
(see Fig.7).
The results show no significant correlation between function
size and either CD (p = 0.134) or OTCD (p = 0.562). This
provides additional evidence that function size does not affect
our function cohesion metric, making it a robust measure for
detecting code injection across functions of varying lengths.
C. RQ3 -How sensitive are our code cohesion metrics in
distinguishing between legitimate and injected code changes?
In this experiment, we evaluate the effectiveness of our code
cohesion metrics in distinguishing between normal maintenance
changes and malicious code injections. We analyzed 425,290
TABLE III: Adjusted P@100 scores for various malicious-to-benign ratios using different cohesion metrics.
Ratio
CD
OTCD
CDz
OTCDz
#Malwares
All
function
1:100
1.41%
1.49%
2.65%
3.41%
4799
1:1,000
0.13%
0.17%
0.27%
0.35%
479
1:10,000
0.04%
0.03%
0.06%
0.08%
47
High
cohesion
1:100
82.21%
4.53%
100.00%
100.00%
3030
1:1,000
8.29%
0.42%
36.41%
22.30%
303
1:10,000
2.97%
0.14%
12.47%
7.82%
30
Fig. 7: Impact of code injection on cohesion across different
function sizes. Each bucket represents a five-line interval,
showing the mean cohesion difference (CD and OTCD) and
standard deviation after malware injection. Function size does
not affect CD and OTCD.
function pairs across 369 publicly available repositories to
simulate real-world detection scenarios.
To reflect realistic attack scenarios where malicious code
represents a small fraction of the codebase, we simulated three
malicious-to-benign ratios: 1:100, 1:1,000, and 1:10,000. In
each ratio, the functions were chosen randomly (based on the
ratio) of being injected with malware at a random location
(beginning, middle, or end). Next, we calculate CD and OTCD
scores between consecutive versions or pre/post-injection pairs.
Finally, we evaluate detection effectiveness using precision at
100 (P@100), which measures detection accuracy in the top
100 ranked functions. We conducted these simulations 1000
times for each ratio and averaged the results. For simulations
with fewer than 100 injected functions (1:10,000 ratio), we
adjust P@100 by dividing by the maximum possible precision.
To account for the dependency between cohesion differences
and baseline cohesion scores, we introduced Standardized
Cohesion Difference (CDz) and Standardized Optimal Token
Count Difference (OTCDz). These metrics represent the number
of standard deviations from the mean difference for functions
with similar cohesion scores. The CDz and OTCDz are calcu-
lated by grouping functions into 20 buckets of 0.05 cohesion
width (based on their NPC), computing the mean and standard
deviation for each bucket, and standardizing differences relative
to their bucket statistics. For instance, consider a function that
initially has a cohesion score of 0.73 (version v1) and drops to
0.62 (version v2), resulting in a CD of 0.11. To compute
CDz, we first determine the mean and standard deviation
of CD within the corresponding [0.7, 0.75] cohesion bucket
(µ = 0.057, σ = 0.054). The standardized difference is then
calculated as CDz = (0.11−0.057)/0.054 = 0.981, indicating
the magnitude of deviation relative to similar functions.
Our results indicate that cohesion-based metrics effectively
distinguish between legitimate and malicious code changes, par-
ticularly in high-cohesion functions. At a 1:100 ratio, detection
performance in high-cohesion functions was significantly higher
(82.21%) compared to the overall detection rate (1.41%), with
standardized metrics (CDz, OTCDz) achieving perfect detection
(100%). As the injection ratio became more extreme (1:10,000),
detection performance decreased but remained meaningful,
with CDz and OTCDz achieving 12.47% and 7.82% detection
rates in high-cohesion functions—representing a three-order-
of-magnitude reduction in manual review effort. Further-
more, standardized metrics consistently outperformed non-
standardized ones, highlighting the importance of adjusting
for baseline cohesion levels. CD-based metrics generally
provided stronger detection capabilities than OTCD-based
ones, reinforcing their effectiveness in identifying anomalous
code changes. These findings demonstrate that cohesion-based
anomaly detection, particularly when leveraging standardized
metrics and high-cohesion functions, is a powerful approach
for detecting malware injections even in highly imbalanced
real-world scenarios.
V. CONCLUSION
This study presents an unsupervised approach for detecting
supply chain attacks by analyzing cohesion disruptions in
source code. Our findings demonstrate that name-prediction-
based cohesion metrics can effectively capture the impact of
malicious code injections by identifying deviations in function
cohesion patterns.
We first examined how malware injection affects function
cohesion and found that injected code disrupts natural naming
patterns, reducing the confidence of the name prediction
model. This confirms the sensitivity of our metric to malicious
modifications. Additionally, our analysis showed that cohesion
scores remain stable across function sizes, reinforcing the
metric’s reliability. However, we found that cohesion drop
metrics (CD and OTCD) are influenced by a function’s initial
cohesion level, indicating that normalization (CDz and OTCDz)
improves their robustness.
To assess real-world applicability, we simulated various
benign-to-malicious ratios, demonstrating that monitoring high-
cohesion functions with NPC effectively detects injected
functions, achieving a P@100 of 36.41% at a 1:1,000 ratio and
12.47% at 1:10,000. These results highlight the potential of
cohesion metrics as a lightweight, scalable solution for assisting
in the identification of supply chain attacks.
Despite these promising results, our approach has limitations.
Our injection strategy relies on controlled placements, whereas
real-world attacks may use more sophisticated techniques
to evade detection. Additionally, while our analysis of 369
repositories did not uncover concrete evidence of supply chain
attacks, refining our method with broader datasets and real-
world malware samples could enhance detection capabilities.
Future research could enhance our approach by incorporating
more advanced language models for cohesion evaluation and
extending the analysis to additional programming languages.
Overall, our findings suggest that name-prediction-based cohe-
sion metrics offer a viable, automated solution for detecting
code injections, contributing to the broader goal of securing
source code supply chains.
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Leveraging Code Cohesion Analysis to Identify Source Code Supply Chain Attacks 1st Maor Reuben Ben-Gurion University of the Negev Beer Sheva, Israel 2nd Ido Mendel Ben-Gurion University of the Negev Beer Sheva, Israel 3rd Or Feldman Ben-Gurion University of the Negev Beer Sheva, Israel 4th Moshe Kravchik Ben-Gurion University of the Negev Beer Sheva, Israel 5th Mordehai Guri Ben-Gurion University of the Negev Beer Sheva, Israel 6th Rami Puzis Ben-Gurion University of the Negev Beer Sheva, Israel Abstract-Supply chain attacks significantly threaten software security with malicious code injections within legitimate projects. Such attacks are very rare but may have a devastating impact. Detecting spurious code injections using automated tools is further complicated as it often requires deciphering the intention of both the inserted code and its context. In this study, we propose an unsupervised approach for highlighting spurious code injections by quantifying cohesion disruptions in the source code. Using a name-prediction-based cohesion (NPC) metric, we analyze how function cohesion changes when malicious code is introduced compared to natural cohesion fluctuations. An analysis of 54,707 functions over 369 open-source C++ repositories reveals that code injection reduces cohesion and shifts naming patterns toward shorter, less descriptive names compared to genuine function updates. Considering the sporadic nature of real supply-chain attacks, we evaluate the proposed method with extreme test-set imbalance and show that monitoring high-cohesion functions with NPC can effectively detect functions with injected code, achieving a Precision@100 of 36.41% at a 1:1,000 ratio and 12.47% at 1:10,000. These results suggest that automated cohesion measurements, in general, and name-prediction-based cohesion, in particular, may help identify supply chain attacks, improving source code integrity. Index Terms-Supply Chain Attacks, Cohesion Analysis, Unsupervised Malware Detection. I. INTRODUCTION In today's software-driven world, ensuring the security and reliability of software systems has become a critical concern. The increasing prevalence of supply-chain compromises, where malicious actors exploit vulnerabilities in the source code supply chain to introduce unauthorized modifications, poses significant threats to the integrity and trustworthiness of software applications [1]. A prominent example is the SolarWinds incident in 2020, where attackers embedded a backdoor in an update for the Orion platform, granting unauthorized remote access to thousands of corporate and government systems and causing extensive data breaches [2]. Similarly, Kaseya's software was compromised by ransomware, resulting in a $70 million extortion demand 1 [3]. 1https://www.kaseya.com/ Traditional methods for detecting these attacks primarily rely on rule-based approaches [4], [5] and machine learning- driven static or dynamic code analysis techniques [6], [7]. However, these techniques depend heavily on known malware signatures, behavior patterns, or supervised learning models that require labeled datasets. As attackers continue to adapt, employing sophisticated anti-analysis techniques to evade detection [8], there is a pressing need for unsupervised approaches that can identify subtle anomalies without prior knowledge of malicious patterns. One promising avenue for addressing this challenge is the detection of code cohesion anomalies. Code cohesion, a fundamental software quality attribute, measures the extent to which elements within a module (e.g., functions or classes) collaborate to achieve a common objective [9]. High cohesion typically indicates robust, maintainable, and readable code, whereas low cohesion can signal the introduction of unrelated responsibilities. By continuously monitoring code cohesion, deviations from expected levels can serve as early warning signs of both security threats and maintainability concerns. In this paper, we introduce an unsupervised methodology for detecting potential supply chain attacks by leveraging pre-trained language models (PLMs) to assess code cohesion. Our approach frames cohesion estimation as a function purpose prediction problem, where the alignment between a function's body and its intended name serves as a proxy for cohesion. Core to our approach is the idea that the cohesion of a function is determined by how different elements in the code contribute to its purpose [10]. By evaluating functions for deviations from normal cohesion levels, our method detects possible compromises without relying on labeled data or prior knowledge of malicious code. Additionally, our approach supports software maintainers by identifying functions that may require refactoring due to cohesion degradation, assisting in long-term software evolution and quality assurance efforts. We evaluate our approach using a dataset of 369 open-source C++ projects, capturing comprehensive version histories for functions in each project (Section III-A). Through simulated 16 Oct 2025 malware injection scenarios, we demonstrate that our method effectively differentiates between legitimate maintenance changes and malicious code alterations. Moreover, we assess how our technique can aid in identifying unintended cohesion drift over time, making it a valuable tool for both security enforcement and software maintenance practices. The main contributions of this study are threefold: • An unsupervised methodology for detecting supply chain attacks by leveraging continuous cohesion evaluation. • A novel use of PLMs for function-level cohesion estimation. • Targeted identification of software components requiring cohesion improvement. The rest of this article is structured as follows: Section II summarizes the prior art related to supply chain attacks, cohesion metrics, and language models for source code analysis. The analysis methods, including the data and cohesion change tracking, are elaborated in Section III. The main research questions and results are discussed in the experimental Section IV. Section V concludes the paper. II. RELATED WORK A. Supply chain attack The increasing development of open-source software has opened another way for attackers to harm software systems. Attackers leverage the use of open source during software development, which is enhanced by dependency managers that update, download, and install open-source packages automatically. This makes it easier for attackers to inject malicious code into the dependency tree of a widely used package, which can help them reach a large number of systems in a short period of time [11]. Attackers attempting to inject malicious code into existing packages may adopt various tactics. They may mimic legitimate contributors to open-source projects, submitting pull requests with seemingly benign code changes [6]. Alternatively, attackers may compromise project repositories directly, committing malicious code with weak or compromised credentials, or by social engineering to become project maintainer [12]. Malicious packages resulting from such attacks can propagate malware across extensive software systems. Vu et al. [13] demonstrated that malicious packages surpassed 100,000 downloads in several instances, illustrating the significant impact these attacks can have. Real-world instances, like the attack on the widely used Homebrew [14], SolarWinds' Orion platform [2], and Kaseya's software [3] emphasize the severity of these threats. Numerous solutions have been suggested to mitigate the risks posed by such attacks. Duan et al. [6] employed established program analysis techniques, including metadata analysis, to identify packages dependent on known malware or sharing similar authors and release patterns. They also utilized static analysis to scrutinize installation logic and flows, and dynamic analysis to detect unusual network communication, file usage, and process behavior. Their study revealed 339 previously unidentified malicious packages. Another approach by Garrett et al. [7] proposed an anomaly detection-based method, focusing on security-relevant features during version updates. Both studies acknowledge the incompleteness of their solutions, emphasizing their role as foundational building blocks. Recent approaches have also leveraged transformer models for malicious code detection. Tsfaty and Fire [15] demonstrated that by clustering multiple versions of a function based on code embeddings using Code2Seq [16], they can flag versions that significantly deviate from their nearest cluster. The main issue with their approach is that they can only monitor function implementations that are highly used across different projects and are supposed to have similar implementations (e.g. "run" or "get" functions). Another study used pre-trained transformers to train classifiers that detect malicious code in JavaScript packages [17]. While they achieved promising results using the backstabber dataset with 10-fold cross-validation, their evaluation methodology did not account for differences between injection patterns in training and test sets, and was dependent on known malware behavior. Although many solutions were suggested for identifying code injection, a recent analysis on a large dataset of 2105 packages from the PyPI2 ecosystem showed that malicious code has proven adept at evading detection from existing tools through the employment of numerous anti-analysis techniques [8]. The persistent vulnerability of the source code supply chain to attacks is underscored by the lack of an integrated and robust protection mechanism [18]. Our proposed method represents a more comprehensive solution compared to existing approaches. Unlike methods primarily reliant on detecting known malware based on behavior, our approach stands out by detecting side effects inherent in every instance of malware injection into cohesive code. B. Cohesion metrics Code cohesion is a fundamental concept in software engineering, referring to the degree to which the elements within a module belong together [9]. High cohesion is desirable as it indicates that the elements within the module are closely related and work together to perform a specific task. Various metrics have been developed to measure cohesion in objectoriented software, aiming to assess the reusability, efficiency, and complexity of software modules. These metrics are essential for evaluating the quality of code and identifying opportunities for restructuring to improve the internal structure of software systems [19]. Several approaches have been proposed to address code cohesion measures. The Lack of Cohesion in Methods (LCOM) metric is a cohesion measure for object-oriented software, initially defined by Chidamber and Kemerer [20]. It assesses the level of cohesion in software modules by measuring the number of method pairs that do not share instance variables. However, one issue with the LCOM metric is that it fails to differentiate between possible levels of cohesion [21]. Tight Class Cohesion (TCC), proposed by Bieman and Kang [10], 2https://pypi.org/ measures direct connections between methods through common instance variables. Class Connection Metric (CCM), developed by Wasiq [22], constructs a method connection graph based on shared attributes and method calls. CCM quantifies the connectedness of this graph to measure cohesion, handling transitive connections between methods. Loose Class Cohesion (LCC), proposed by Bieman and Kang [10], considers both direct and indirect connections between methods, taking into account transitivity. Another approach to measure code cohesion involves the use of refactoring to improve the cohesion of object-oriented software. Refactoring aims to alter the internal structure of the code without changing its external functionality, thereby improving cohesion and reducing coupling in the software system [23]. Additionally, novel cohesion metrics, such as the Variable Frequency - Inverse Method Frequency (VFIMF) metric, have been developed to assess the level of cohesion in modules and group module methods to instill high cohesion [19]. These metrics offer a compromised solution for building high cohesive modules and differentiate between different levels of cohesion, addressing the limitations of traditional cohesion metrics, such as LCOM. The closest approach to our work is the work of [24] which tries to solve the challenge of manually tracing the cohesion value in software code, which is essential for evaluating software maintainability. The study focuses on predicting cohesion values, including LCOM2, TCC, and LCC, using machine learning techniques such as KNN, REPTree, multi-layer perceptron, linear regression, support vector machine, and random forest. The research utilizes two different open-source software projects to create datasets and evaluates the performance of various machine learning algorithms for predicting different cohesion metrics. Our work is distinct in harnessing the semantic capabilities of large language models to estimate code cohesion. By detecting significant deviations, we can identify components needing additional security auditing and analysis. C. Code analysis with language models Large Language Models (LLMs) have shown significant potential in natural language understanding and programming code processing tasks. Their ability to comprehend and generate human-like code has sparked research interest in leveraging LLMs for code analysis purposes [25]. LLMs have been evaluated for their capabilities in automating code analysis tasks, including the analysis of obfuscated and malicious code [26]. The findings indicate that LLMs can serve as valuable tools for automating code analysis, albeit with certain limitations. Their research contributes to a deeper understanding of the potential and constraints associated with utilizing LLMs in code analysis, paving the way for enhanced applications in this critical domain. Recent work assessed CodeBERT and ChatGPT for their efficiency in securityoriented program analysis, addressing tasks like code review and code generation. The study delves into the capabilities of LLMs in solving security-related analytic tasks, providing Fig. 1: Overview of cohesion monitoring to detect potential security compromises. By analyzing cohesion metrics over successive version releases, we can identify significant decreases to flag for investigation. insights into their potential and limitations in addressing software security challenges [27]. These use cases demonstrate the diverse applications of LLMs in code analysis, ranging from scientific research and data analysis to automating code analysis tasks and addressing security-oriented program analysis challenges. Since LLMs show promise in enhancing productivity and automating various code-related tasks, we choose to use them in order to estimate code cohesion. III. METHODS In this section, we present our methodology for highlighting spurious code insertions by monitoring for significant drops in code cohesion. We begin by describing the dataset used to evaluate our code cohesion metrics over time. Next, we demonstrate how fine-tuned language models are used to quantify code cohesion through a function name prediction task. Finally, we describe our approach to simulating software compromise by injecting malicious code into functions. The primary goal of our methodology is to monitor software over time and detect significant decreases in code cohesion that may indicate a security compromise. Figure 1 illustrates this process by analyzing cohesion metrics for each new version release, we can flag suspicious drops for further investigation. A. Dataset We collected a corpus of 109,924 C++ functions from 369 widely used open-source GitHub projects3. To expand this dataset, we gathered all publicly available versions of these functions from their complete Git histories. A function version represents either the initial implementation or any subsequent modification of a function due to a commit. To ensure data quality and completeness, the collection process involved the following steps: 1) Gathering repository metadata and commit histories. 2) Filtering commits to retain only those with file modifications. 3A list of the repositories used can be found at https://tinyurl.com/ github-repositories 3) Downloading the modified files. 4) Extracting functions using Clang4. Each function was assigned a unique identifier combining its name, argument types, and file location, allowing precise tracking across versions even when implementations evolved. Using these identifiers, we eliminated unchanged duplicates while maintaining comprehensive version histories. After filtering out functions with only a single version, the final dataset comprised 479,996 versions of 54,706 unique functions, with an average of 8.7 and a median of 3 versions per function. For analysis, we focused on functions with multiple versions, yielding 425,290 consecutive version pairs. These pairs capture natural changes resulting from regular maintenance and feature development, forming a robust foundation for examining cohesion dynamics over time and identifying deviations caused by code injections. Detailed statistics for the full dataset are presented in Table I. TABLE I: Dataset Summary Statistics Statistic Value Open source C++ #Projects 369 Avg. lines of code per project 131,758 Linux-compatible projects 369 (100%) Windows-compatible projects 115 (31.25%) #Function versions 479,996 #Unique functions 54,706 Average versions per function 8.7 Median versions per function 3 Avg. lines of code per function 14.65 Avg. tokens per function 197.58 Malware #Malicious code examples 9 Avg. Lines of Code per malware 6.44 Avg. tokens per malware 77.66 #Consecutive version pairs 425,290 B. Function name prediction The function name prediction approach leverages a pretrained language model to predict an appropriate name for a function based solely on its body text. Appropriately named functions typically exhibit high cohesion, as the name succinctly captures their singular, focused purpose. Intuitively, a coherent function performing a single focused task should be easier for the model to name accurately compared to one with disjoint or unrelated logic. To operationalize this idea, we employ a systematic methodology based on masked language modeling (MLM). Using this technique, we replace the function name with a sequence of token masks (e.g., , ), and the model predicts the most probable terms to fill these masks, effectively generating a name for the function. For a given function f, we perform the fill-mask operation on its name using token counts ranging from one to eight, which covers the typical range of function names in practice. The confidence for f when using n tokens is calculated using the harmonic mean of the token probabilities, as shown in Equation 1. Confidence(f, n) = n sumn i=1pi-1 (1) 4https://clang.llvm.org/ Here, pi is the probability of the i-th token. The harmonic mean penalizes low-probability tokens, ensuring that the metric reflects consistent confidence across all predicted tokens. Then, we compute the confidence for each token count and select the maximum confidence as the cohesion metric, defined in Equation 2. NPC(f) = max(Confidence(f, n) | n ∈{1, . . . , 8}) (2) This metric, referred to as the name prediction cohesion (NPC), provides an aggregate measure of how well the model can generate a cohesive, meaningful name for the function. We also refer to the token count that yields the highest confidence as the optimal token count (OTC), which indicates the number of tokens that best capture the semantic intent of the function. Using this terminology, we can also define NPC of function f as follows: NPC(f) = Confidence(f, OTC(f)) (3) This approach not only quantifies the alignment between function names and their logic but also provides a systematic method for evaluating cohesion across varying token counts. To implement this methodology, we used the CodeBERTCpp model, which is pre-trained and fine-tuned on C++ code, making it well-suited for analyzing the syntax and semantics of functions in this language. C. Code Injection To evaluate the sensitivity of our cohesion quantification techniques to compromised integrity, we inject malicious code segments into functions and analyze their impact on cohesion assessments. The malicious code corpus consists of segments exhibiting common behaviors such as exfiltrating sensitive data, performing privilege escalation, and enabling remote code execution. These segments are sourced from publicly available repositories of known malware code5. After selecting these segments, we inject them at three strategic locations within functions: the beginning (after the declaration but before existing instructions), middle (after 50% of the original lines), or end (after all existing code). We ensure the modified code maintains syntactic validity. In Figure 2, we show an example of a function before and after injection. The injected payload writes directly to the master boot record (MBR), demonstrating how we simulate malware injection. D. Cohesion Change Quantification To evaluate the cohesion change after code injection or between consecutive versions of the same function, we use two complementary methods: The Cohesion Difference (CD) is defined as the NPC difference between functions f1 and f2, detailed in Equation 4. CD(f1, f2) = NPC(f1) -NPC(f2) (4) 5The malware code used in this study can be accessed at https://tinyurl.com/ malicious-implants (a) Function before malware injection. (b) Function after malware injection at the beginning of a function. Fig. 2: Example of malware injection into a function. The Optimal Token Count Difference (OTCD) is the confidence difference of f1 optimal token count (OTC), detailed in Equation 5: OTCD(f1, f2) = NPC(f1) -Confidence(f2, OTC(f1)) (5) These methods provide complementary insights into cohesion changes. The CD captures the overall cohesion drift, reflecting shifts in the model's maximum confidence about the function's coherence, regardless of token count. The OTCD captures disruptions specific to the original function's optimal representation, highlighting how well the modified function maintains the semantic intent of the original token count. Together, they help quantify both structural and semantic disruptions caused by code injection or iterative changes between function versions. A positive value in either metric indicates a decrease in cohesion, while negative values suggest increased cohesion. IV. EXPERIMENTS To evaluate the effectiveness of cohesion-based security monitoring, we investigate four key research questions (RQ): Research Question 1: How does malware injection affect cohesion when measured through function name prediction? Research Question 2: What is the distribution of cohesion across functions in real-world software? Research Question 3: How effectively can cohesion metrics distinguish between legitimate maintenance changes and malicious code injections? A. RQ1 - How does malware injection impact cohesion when measured using function name prediction? In well-structured code, functions should have clear and descriptive names that convey their purpose and functionality, helping developers understand what the function does without needing to examine its implementation details. Names that are too short may be ambiguous and insufficiently descriptive, while overly long names can become cumbersome and harder to read. When code injection occurs, it often introduces functionality that diverges from the function's original purpose, creating a semantic gap between the function's name and its actual behavior. This misalignment between name and implementation can serve as a potential indicator of malicious modifications. To investigate this, we explore three hypotheses (H) related to function name prediction: Hypothesis 1: For every function, there is an optimal number of tokens comprising its name. Hypothesis 2: Code injection changes the optimal number of tokens in a function's name. Hypothesis 3: Code injection reduces the function cohesion. To test these hypotheses, we analyzed 479,996 functions from the dataset, predicting names ranging from one to eight tokens. We grouped functions based on their optimal token count (OTC) and displayed the aggregated mean and quantiles of the results for maximum confidence, as shown in Figure 3. The analysis reveals distinct peaks in confidence-both in mean and quantiles-corresponding to the OTCs. These peaks demonstrate that functions naturally gravitate toward specific name lengths that optimize their descriptiveness and predictability, providing quantitative support for H1. Next, we injected malware (see Section III-C) at the beginning, middle, and end of each function and applied the same process. The results are presented in Figure 4. After code injection, the OTC shifts to one token for many functions, and the overall maximum confidence decreases. This simplification of function names after injection supports H2, showing that external code alterations negatively impact natural naming conventions. Furthermore, the drop in confidence postinjection supports H3, indicating that code injection reduces the predictability of function names. Following this, we conducted a comprehensive evaluation of cohesion changes under two scenarios: malicious code injection and natural code evolution. For the injection scenario, we modified each function by inserting different types of malicious code at three locations (beginning, middle, and end) and averaged the results (see Section III-C). We quantified these changes using two metrics: the cohesion difference (CD) and optimal token count difference (OTCD), measuring the changes in these values before and after injection. We also performed a separate analysis on high-cohesion functions (NPC > 0.5) to determine whether injection effects were more pronounced in well-structured code. Since high-cohesion functions have a more predictable structure and naming consistency, we hypothesize that malicious injections will cause greater disruptions, making them easier to detect. To establish a baseline for comparison, we also examined how cohesion naturally evolves across multiple versions of the same function. Using the same metrics (CD and OTCD), we Fig. 3: Distribution of name prediction confidence across different token counts. Each subplot represents functions grouped by their optimal number of tokens. The solid line shows the mean confidence, while the shaded area represents the 25th to 75th percentile range. The x-axis shows the number of tokens, and the y-axis represents the prediction confidence score. Fig. 4: Impact of code injection on name prediction confidence. Each subplot shows functions grouped by their original optimal token count. Blue lines represent the original code's confidence scores, while red lines show scores after injection. The x-axis represents the number of tokens, and the y-axis shows the prediction confidence score. TABLE II: Impact of code injection on cohesion metrics Position CD OTCD All functions Beginning 0.027±0.224* 0.105±0.227* Mid 0.040±0.173* 0.086±0.178* End 0.038±0.172* 0.083±0.177* Baseline 0.0005±0.12 0.029±0.131 High cohesion (>0.5) Beginning 0.081±0.207* 0.142±0.231* Mid 0.077±0.163* 0.11±0.183* End 0.074±0.161* 0.106±0.18* Baseline 0.021±0.112 0.045±0.138 * indicates statistical significance (p < 0.05) compared to the consecutive versions (baseline). Values are presented as mean ± standard deviation. analyzed 425,290 consecutive version pairs, measuring how cohesion changes during normal development. Detailed results on the impact of code injection on cohesion metrics across three injection positions and between consecutive version (baseline) pairs are presented in Table II. Our results demonstrate that code injection significantly impacts function cohesion across all injection positions, with CD values ranging from 0.027 to 0.038 and OTCD values ranging from 0.083 to 0.105. Notably, the changes in cohesion between consecutive versions of the same function (mean CD = 0.0005, mean OTCD = 0.031) are significantly (p < 0.05) smaller than those observed after code injection (minimum CD = 0.027, minimum OTCD = 0.083), with differences of over an order of magnitude in CD values. For high-cohesion functions, the impact of injection was substantially stronger, with CD values ranging from 0.074 to 0.081 and OTCD values ranging from 0.106 to 0.142. The increased sensitivity in high-cohesion functions is particularly notable close to the beginning, where CD increased to 0.081 (200% higher than the general case) and OTCD reached 0.142 (35% higher). This suggests that well-structured code is indeed more sensitive to malicious modifications, making injection detection more reliable in highFig. 5: Distribution of name prediction cohesion (NPC) scores across functions. quality codebases. These findings provide strong support for our hypotheses: functions exhibit an optimal token count (H1), which is disrupted by injection (H2). Also, both CD and OTCD metrics detect these disruptions showing statistically significant reductions in cohesion after injection (H3). The clear distinction between natural evolution and injected modifications, particularly in CD measurements, demonstrates the effectiveness of our name prediction cohesion (NPC) metric for detecting potential malicious code insertions. B. How is cohesion distributed across functions, and how is it affected by maintenance changes and function size? We examined the distribution and characteristics of function cohesion from three perspectives: overall distribution, maintenance impact, and the relationship with function size. To understand how cohesion naturally varies across different functions, we first computed a cohesion histogram for all consecutive function version pairs in the dataset using the name prediction cohesion (NPC) metric. The results are presented in Figure 5. The histogram reveals that the majority of functions (approximately 63%) exhibit high cohesion (greater than 0.5). This aligns with software engineering principles and expectations, as these functions come from widely used open-source packages that typically undergo code reviews and maintain quality standards. To understand how cohesion changes during normal maintenance vary across different cohesion levels, we examined the relationship between a function's base cohesion level and its CD and OTCD measurements. We calculated the Pearson correlation between the function's NPC score and both its CD and OTCD values during normal maintenance changes. The analysis revealed significant (p-value < 0.05) positive Pearson's correlations: 0.951 for CD and 0.894 for OTCD with respect to cohesion levels. These strong correlations indicate that functions with higher cohesion tend to experience larger Fig. 6: Mean and standard deviation of name prediction cohesion (NPC) across function sizes. Each bucket represents a five-line interval, showing consistent cohesion levels regardless of function length. cohesion drops during maintenance. This can be explained mathematically, due to the fact that larger values of NPC (range between 0 to 1) have more "room" to fall toward 0 than lower values. Next, we explored the relationship between cohesion scores and the number of lines of code. To achieve this, we grouped functions into buckets based on their size, with each bucket representing a five-line interval (see Figure 6). The analysis revealed that the average and standard deviation of code cohesion remained consistent across different function sizes (about 0.608±0.212) and found no significant correlation between NPC and function size (p-value = 0.186). This indicates that the NPC metric provides a reliable measure of function cohesion, independent of function size. To further investigate how the relative size of injected code compared to function size impacts cohesion, we calculated the mean cohesion difference (CD) and optimal token count difference (OTCD) for malicious pairs in each bucket. Across all buckets, the mean cohesion difference ranged from 0.028 to 0.051, with a standard deviation between 0.167 and 0.198 (see Fig.7). The results show no significant correlation between function size and either CD (p = 0.134) or OTCD (p = 0.562). This provides additional evidence that function size does not affect our function cohesion metric, making it a robust measure for detecting code injection across functions of varying lengths. C. RQ3 -How sensitive are our code cohesion metrics in distinguishing between legitimate and injected code changes? In this experiment, we evaluate the effectiveness of our code cohesion metrics in distinguishing between normal maintenance changes and malicious code injections. We analyzed 425,290 TABLE III: Adjusted P@100 scores for various malicious-to-benign ratios using different cohesion metrics. Ratio CD OTCD CDz OTCDz #Malwares All function 1:100 1.41% 1.49% 2.65% 3.41% 4799 1:1,000 0.13% 0.17% 0.27% 0.35% 479 1:10,000 0.04% 0.03% 0.06% 0.08% 47 High cohesion 1:100 82.21% 4.53% 100.00% 100.00% 3030 1:1,000 8.29% 0.42% 36.41% 22.30% 303 1:10,000 2.97% 0.14% 12.47% 7.82% 30 Fig. 7: Impact of code injection on cohesion across different function sizes. Each bucket represents a five-line interval, showing the mean cohesion difference (CD and OTCD) and standard deviation after malware injection. Function size does not affect CD and OTCD. function pairs across 369 publicly available repositories to simulate real-world detection scenarios. To reflect realistic attack scenarios where malicious code represents a small fraction of the codebase, we simulated three malicious-to-benign ratios: 1:100, 1:1,000, and 1:10,000. In each ratio, the functions were chosen randomly (based on the ratio) of being injected with malware at a random location (beginning, middle, or end). Next, we calculate CD and OTCD scores between consecutive versions or pre/post-injection pairs. Finally, we evaluate detection effectiveness using precision at 100 (P@100), which measures detection accuracy in the top 100 ranked functions. We conducted these simulations 1000 times for each ratio and averaged the results. For simulations with fewer than 100 injected functions (1:10,000 ratio), we adjust P@100 by dividing by the maximum possible precision. To account for the dependency between cohesion differences and baseline cohesion scores, we introduced Standardized Cohesion Difference (CDz) and Standardized Optimal Token Count Difference (OTCDz). These metrics represent the number of standard deviations from the mean difference for functions with similar cohesion scores. The CDz and OTCDz are calculated by grouping functions into 20 buckets of 0.05 cohesion width (based on their NPC), computing the mean and standard deviation for each bucket, and standardizing differences relative to their bucket statistics. For instance, consider a function that initially has a cohesion score of 0.73 (version v1) and drops to 0.62 (version v2), resulting in a CD of 0.11. To compute CDz, we first determine the mean and standard deviation of CD within the corresponding [0.7, 0.75] cohesion bucket (μ = 0.057, σ = 0.054). The standardized difference is then calculated as CDz = (0.11-0.057)/0.054 = 0.981, indicating the magnitude of deviation relative to similar functions. Our results indicate that cohesion-based metrics effectively distinguish between legitimate and malicious code changes, particularly in high-cohesion functions. At a 1:100 ratio, detection performance in high-cohesion functions was significantly higher (82.21%) compared to the overall detection rate (1.41%), with standardized metrics (CDz, OTCDz) achieving perfect detection (100%). As the injection ratio became more extreme (1:10,000), detection performance decreased but remained meaningful, with CDz and OTCDz achieving 12.47% and 7.82% detection rates in high-cohesion functions-representing a three-orderof-magnitude reduction in manual review effort. Furthermore, standardized metrics consistently outperformed nonstandardized ones, highlighting the importance of adjusting for baseline cohesion levels. CD-based metrics generally provided stronger detection capabilities than OTCD-based ones, reinforcing their effectiveness in identifying anomalous code changes. These findings demonstrate that cohesion-based anomaly detection, particularly when leveraging standardized metrics and high-cohesion functions, is a powerful approach for detecting malware injections even in highly imbalanced real-world scenarios. V. CONCLUSION This study presents an unsupervised approach for detecting supply chain attacks by analyzing cohesion disruptions in source code. Our findings demonstrate that name-predictionbased cohesion metrics can effectively capture the impact of malicious code injections by identifying deviations in function cohesion patterns. We first examined how malware injection affects function cohesion and found that injected code disrupts natural naming patterns, reducing the confidence of the name prediction model. This confirms the sensitivity of our metric to malicious modifications. Additionally, our analysis showed that cohesion scores remain stable across function sizes, reinforcing the metric's reliability. However, we found that cohesion drop metrics (CD and OTCD) are influenced by a function's initial cohesion level, indicating that normalization (CDz and OTCDz) improves their robustness. To assess real-world applicability, we simulated various benign-to-malicious ratios, demonstrating that monitoring highcohesion functions with NPC effectively detects injected functions, achieving a P@100 of 36.41% at a 1:1,000 ratio and 12.47% at 1:10,000. These results highlight the potential of cohesion metrics as a lightweight, scalable solution for assisting in the identification of supply chain attacks. Despite these promising results, our approach has limitations. Our injection strategy relies on controlled placements, whereas real-world attacks may use more sophisticated techniques to evade detection. Additionally, while our analysis of 369 repositories did not uncover concrete evidence of supply chain attacks, refining our method with broader datasets and realworld malware samples could enhance detection capabilities. Future research could enhance our approach by incorporating more advanced language models for cohesion evaluation and extending the analysis to additional programming languages. Overall, our findings suggest that name-prediction-based cohesion metrics offer a viable, automated solution for detecting code injections, contributing to the broader goal of securing source code supply chains. REFERENCES [1] Chinenye Okafor, Taylor R. Schorlemmer, Santiago Torres-Arias, and James C. Davis. 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In 2023 IEEE International Conference on Medical Artificial Intelligence (MedAI), pages 284-289. IEEE, 2023. [26] Chongzhou Fang, Ning Miao, Shaurya Srivastav, Jialin Liu, Ruoyu Zhang, Ruijie Fang, Asmita Asmita, Ryan Tsang, Najmeh Nazari, Han Wang, et al. Large language models for code analysis: Do llms really do their job? arXiv preprint , 2023. [27] Zhilong Wang, Lan Zhang, Chen Cao, and Peng Liu. The effectiveness of large language models (chatgpt and codebert) for security-oriented code analysis. Available at SSRN 4567887, 2023.
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2510.14777
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A Levelset Algorithm for 3D-Tarski
Sebastian Haslebacher
and Jonas Lill
Department of Computer Science, ETH Zurich, Switzerland
Abstract
We present a simple new algorithm for finding a Tarski fixed point of a monotone function
F : [N]3 →[N]3. Our algorithm runs in O(log2 N) time and makes O(log2 N) queries to F,
matching the Ω(log2 N) query lower bound due to Etessami et al. as well as the existing
state-of-the-art algorithm due to Fearnley et al.
1
arXiv:2510.14777v1 [cs.DS] 16 Oct 2025
1
Introduction
We say that a function F : [N]d →[N]d is monotone if x ≤y implies F(x) ≤F(y) for all
x, y ∈[N]d, where ≤is the partial order obtained by coordinate-wise comparison. Tarski [8]
proved that any such monotone function F must have a fixed point x⋆= F(x⋆). Finding such a
fixed point algorithmically has important applications in algorithmic game theory: For example,
finding equilibria of supermodular games, approximating the value of Condons’s or Shapley’s
stochastic games, and solving a zero-player game known as ARRIVAL all reduce to finding such
a Tarski fixed point [4, 5, 7].
More concretely, given query access to a function F : [N]d →[N]d, we refer to the computational
problem of finding a fixed point of F as Tarski. A recursive binary search algorithm due to Dang,
Qi, and Ye [4] can solve this problem in O(logd N) time and queries for fixed d, and initially
it was rather unclear whether one should expect much better algorithms.
Indeed, Etessami,
Papadimitriou, Rubinstein, and Yannakakis [5] proved a query lower bound of Ω(log2 N) for
d ≥2 and conjectured a lower bound of Ω(logd N) in general. However, Fearnley, P´alv¨olgyi,
and Savani [6] later disproved this conjecture by giving an algorithm solving the case d = 3 in
O(log2 N) queries and time. They also proved a decomposition lemma that allowed them to use
recursive applications of their 3D-algorithm to get an O(log⌈2d/3⌉N)-algorithm in general.
Since then, both the upper bounds and lower bounds have been improved, but the true complex-
ity of the problem remains wide open. Concretely, the current best upper bound is due to Chen
and Li [2] who proved that a fixed point can be found in O(log⌈d+1/2⌉N) time and queries. In
terms of lower bounds, the state-of-the-art is an Ω(d log2 N/log d) query lower bound for d ≥2 that
was recently announced by Brˆanzei, Phillips, and Recker [1].
In this paper, we present a new algorithm that can solve Tarski for d = 3 in O(log2 N) time and
queries, matching the asymptotically optimal algorithm by Fearnley, P´alv¨olgyi, and Savani [6].
Their algorithm proceeds by binary searching over two-dimensional slices of the 3D-grid. Their
key insight is that it is possible to find a progress point x satisfying either F(x) ≤x or F(x) ≥x
in a given two-dimensional slice Sk = {x ∈[N]3 | x1 = k} in O(log N) time. This inner algorithm
is however quite complicated and in particular involves an intricate case analysis. We avoid this
by binary searching over levelsets instead of slices. Concretely, given a number k ∈{3, . . . , 3N},
we show how we can either find a point x with F(x) ≤x in L≤k = {x ∈[N]3 | x1 + x2 + x3 ≤k}
or a point x with F(x) ≥x in L≥k = {x ∈[N]3 | x1 + x2 + x3 ≥k} in O(log N) time. We
achieve this by searching over the levelset Lk = {x ∈[N]3 | x1 + x2 + x3 = k}. The advantage
of levelsets is that their symmetry allows our algorithm to treat all three dimensions exactly the
same. However, in contrast to the slices in [6], no two points in a given levelset are comparable
under the partial order ≤, making it seemingly more difficult to exploit monotonicity of F. Our
main contribution is to overcome this difficulty.
Naturally, our algorithm can also be plugged into the decomposition lemma of [6] to obtain an
O(log⌈2d/3⌉N) bound for general d. Moreover, our levelset approach can in principle also be
applied directly to higher-dimensional instances. Unfortunately, we have not yet been able to
get close to the O(log⌈2d/3⌉N) bound (or the state-of-the-art O(log⌈d+1/2⌉N) bound of Chen and
Li [2], for that matter) for d > 3 using this more direct approach, but we do think that this is an
interesting idea for future work.
Finally, we want to point out that our levelset approach seems even more appropriate in restricted
2
versions of Tarski such as Unique-Tarski and Super-Unique-Tarski. Indeed, many of our ideas
originated from studying Super-Unique-Tarski. In terms of query complexity, studying Super-
Unique-Tarski would have been enough, given that Chen, Li, and Yannakakis [3] proved that
Tarski, Unique-Tarski, and Super-Unique-Tarski all have the exact same query complexity.
Unfortunately, their reductions are not time-efficient, which is why we avoided using them and
instead generalized our ideas to work for Tarski directly.
2
Preliminaries
We use a slightly generalized definition of Tarski on an integer grid G = [n1] × [n2] × · · · × [nd],
for arbitrary n1, . . . , nd ∈N. We will use N = ∑d
i=1 ni to measure the size of G. The grid G can
be equipped with the partial order ≤defined as
x ≤y ⇐⇒∀i ∈[d] : xi ≤yi
for all x, y ∈G. The tuple (G, ≤) forms a lattice, but we will usually just (implicitly) denote
this lattice by G.
We use GLB(x(1), . . . , x(k)) to denote the greatest lower bound (meet) and
LUB(x(1), . . . , x(k)) to denote the least upper bound (join) of a set of points x(1), . . . , x(k) ∈G in
this lattice. We also use |x| := ∑d
i=1 xi for the 1-norm of x ∈G. A function F : G →G is called
monotone (or non-decreasing) if and only if
x ≤y =⇒F(x) ≤F(y)
for all x, y ∈G. The computational problem Tarski is defined as follows: Given query access to
a monotone function F : G →G, find a fixed point x⋆= F(x⋆) of F. Recall that such a fixed point
has to exist by Tarski’s theorem [8].
Our algorithm will take O(log2 N) time and will make O(log2 N) queries to F. Note that this
excludes the time needed to actually make a query to F, which naturally depends on how access
to F is provided to us.
For any specific model of access to F, one would have to multiply
our bounds by the access time to F in the given model. Moreover, our runtime is measured
in arithmetic operations: Numbers in our algorithms can be as large as Ω(N), which means
that they require Ω(log N) bits to represent, and this would have to be factored in for other
models of computation. We decided to abstract these concerns away for the sake of simplicity.
In particular, we will from now on simply refer to the runtime of the algorithm, factoring out
any overhead that is needed to actually evaluate F or to calculate with O(log N)-bit numbers in
certain models.
Moreover, note that our presentation focuses on the setting where F is promised to be monotone.
However, it is not hard to see that our algorithm could be adapted to the total setting, where the
algorithm is allowed to return a violation of monotonicity instead of a fixed point whenever F is
not monotone. This is because our algorithm only exploits monotonicity locally: If a certain step
of the algorithm should fail, then a violation of monotonicity must be present among a constant
set of previous queries.
We will frequently use the notion of upward and downward points: A point x ∈G is called
upward if F(x) ≥x, and downward if F(x) ≤x. Note that a point is both upward and downward
simultaneously if and only if it is a fixed point of F. Also note that (1, 1, . . . , 1) ∈G must be an
upward and (n1, n2, . . . , nd) ∈G a downward point. Furthermore, any upward point x and
downward point y satisfying x ≤y imply that there exists a fixed point x ≤x⋆≤y of F: Indeed,
3
due to monotonicity of F, the restriction of F to the subgrid G′ = {z ∈G | x ≤z ≤y} is itself
a valid Tarski-instance. In other words, finding such x and y allows us to reduce the search to
G′.
3
Levelset Algorithm
We start the presentation of our algorithm with the high-level idea and then proceed by explain-
ing all the details step-by-step. The following notion of levelsets will be crucial.
Definition 3.1 (Levelset). Let G be a d-dimensional integer grid of size N = ∑d
i=1 ni and let k ∈
{d, d + 1, . . . , N} be arbitrary. The k-th levelset Lk of G is defined as Lk := {x ∈G | |x| = k}. Similarly,
we define L≥k := S
i≥k Li and L≤k := S
i≤k Li.
x
y
Lk
1
2
3
(1, 1, 1)
(n1, n2, n3)
Figure 1: Levelset Lk of a 3D-instance (on the left) projected to 2D (on the right). On the right,
the three dimensions are indicated in red, orange, and blue, respectively. For both points x and
y, we indicate the function F in each dimension separately. For example, x is an upward point
with F(x)i > xi for all i ∈[3] while y satisfies F(y)1 < y1, F(y)3 > y3, and F(y)2 = y2 (indicated
by the absence of an orange arrow).
From now on, we restrict ourselves to the case d = 3. Our algorithm proceeds by finding an
upward point in L≥⌈N/2⌉or a downward point in L≤⌈N/2⌉in time O(log N). Once we have found
such a point, we can shrink our search space by at least half and recursively apply the algorithm
on the remaining subgrid, yielding an overall runtime of O(log2 N).
Lemma 3.2. Assume that for an arbitrary levelset Lk in an instance F : G →G of 3D-Tarski, we can
find an upward point x ∈L≥k or a downward point x ∈L≤k in time O(log N). Then, a fixed point of F
can be found in time O(log2 N).
Proof. We start by calling the subprocedure on the levelset Lk for k = ⌈N/2⌉in order to find a
point x that is either an upward point in L≥k or downward point in L≤k . If x is an upward point
in L≥k, we recurse on the subgrid G′ = {y ∈G | x ≤y}. Otherwise, x is a downward point in
L≤k and we recurse on the subgrid G′ = {y ∈G | x ≥y}. In both cases, F restricted to G′ is
again a Tarski-instance. Moreover, the subgrid G′ has size at most half of G (the parameter N
has shrunk by at least half). Once we arrive at a grid of constant size, we can find a fixed point
in constant time. Thus, the overall procedure takes O(log2 N) time.
Given Lemma 3.2, the main challenge that we have to overcome is to find either an upward point
in L≥k or a downward point in L≤k in time O(log N). For this, we will maintain a search space
4
spanned by six bounding points within Lk. We will then iteratively shrink this search space in
a binary search fashion. If we do not find an upward or downward point directly during the
shrinking, we eventually arrive at a situation that must imply the existence of one out of three
possible configurations among the six bounding points. For each of those three configurations,
we then show how it implies an upward or downward point, as desired.
3.1
Shrinking the Search Space
We start by defining our search space and explaining how we can iteratively shrink it. For this,
we first need to introduce so-called i-upward and i-downward points.
Definition 3.3 (i-Upward and i-Downward Points). We call a point x ∈Lk i-upward for some i ∈[3]
if F(x)i > xi and F(x)j ≤xj for all j ∈[3] \ {i}. Symmetrically, we call x i-downward if F(x)i < xi and
F(x)j ≥xj for all j ∈[3] \ {i}.
Lk
1
2
3
x
y
z
Figure 2: The three points x, y, z are examples of 1-upward points. Note that x is in fact an overall
upward point since it satisfies F(x)i ≥xi for all i ∈[3].
Note that any point that is neither upward nor downward must be i-upward or i-downward for
some i ∈[3]. This observation will be crucial for us and only holds for d = 3, which is the main
reason why we could not yet generalize our algorithm to higher dimensions.
We now proceed to define our search space: At each step of the shrinking algorithm, our remain-
ing search space is induced by six bounding points. Concretely, for all i ∈[3], we will maintain
an i-upward point u(i) ∈Lk and an i-downward point d(i) ∈Lk satisfying u(i)
i
≤d(i)
i .
Definition 3.4 (Remaining Search Space). Given the bounding points u(i), d(i) ∈Lk satisfying u(i)
i
≤
d(i)
i
for all i ∈[3], the induced remaining search space is defined as
S := {x ∈Lk | u(i)
i
≤xi ≤d(i)
i
for all i ∈[3]}
and its diameter is a 3D-vector defined as
diam(S)i := max
x,y∈S |xi −yi| = max
x∈S xi −min
y∈S yi
for all i ∈[3].
5
Lk
1
2
3
S
u(3)
d(2)
u(1)
d(3)
u(2)
d(1)
diam(S)1
q
Figure 3: Example of the remaining search space S (the region bounded by the black pentagon),
induced by the six drawn bounding points. For simplicity, we only drew the arrow in dimension
i for each point u(i) or d(i). Observe in particular how d(1) does not currently contribute to the
boundary of S, and in particular we have diam(S)1 < d(1)
1
−u(1)
1 . Still, there is a point q such
that querying q guarantees significant progress: Either q is upward or downward which allows
us to terminate, or it is i-upward or i-downward for some i ∈[3] which allows us to considerably
shrink the diameter in one of the three dimensions.
In the remainder of this section, we will prove that in O(log N) time, we can shrink the remaining
search space until we have diam(S)i ≤1 for at least one i ∈[3]. The proof of this is split into
Lemma 3.5 and Lemma 3.6.
Lemma 3.5 (Shrinking Remaining Search Space I). Assume that we have diam(S)i > 1 for all i ∈[3]
and maxi∈[3] diam(S)i ≥6 for our remaining search space S. Then there exists q ∈S such that
max (diam({x ∈S | xi ≤qi})i, diam({x ∈S | xi ≥qi})i) ≤5
6 diam(S)i
for all i ∈[3].
Proof. Assume without loss of generality diam(S)1 ≤diam(S)2 ≤diam(S)3. Let ℓi := minx∈S xi
and analogously ri := maxx∈S xi for i ∈[3]. To prove the statement, it suffices to prove that there
exists q ∈Lk satisfying
ℓ1 + ⌈1
6 diam(S)1⌉, ℓ2 + ⌈1
6 diam(S)2⌉, ℓ3 + ⌈1
6 diam(S)3⌉
≤q
and
r1 −⌈1
6 diam(S)1⌉, r2 −⌈1
6 diam(S)2⌉, r3 −⌈1
6 diam(S)3⌉
≥q.
To see that such a q exists, observe first that we must have
ℓ1 + ⌈1
6 diam(S)1⌉+ ℓ2 + ⌈1
6 diam(S)2⌉+ ℓ3 + ⌈1
6 diam(S)3⌉≤k
because we can upper bound diam(S)1 and diam(S)2 by diam(S)3, use diam(S)3 ≥6, and use
that
ℓ1 + ℓ2 + ℓ3 + diam(S)3 ≤k,
6
by definition of diam(S)3 and ℓ1, ℓ2, ℓ3. Analogously, we derive
k ≤r1 −⌈1
6 diam(S)1⌉+ r2 −⌈1
6 diam(S)2⌉+ r3 −⌈1
6 diam(S)3⌉.
Note that the point q in Lemma 3.5 can be constructed efficiently from the six bounding points.
Indeed, it suffices to compute the values ℓi, ri, diam(S)i for all i ∈[3] and then choose q with
|q| = k arbitrarily within the described bounds.
Now recall that Lemma 3.5 allows us to shrink the remaining search space only as long as we
have maxi∈[3] diam(S)i ≥6. Thus, we still have to handle the case maxi∈[3] diam(S)i < 6. For this,
we exploit that maxi∈[3] diam(S)i < 6 implies that |S| is constant.
Lemma 3.6 (Shrinking Remaining Search Space II). Assume that we have diam(S)i > 1 for all i ∈[3]
and assume S has constant size. Then we can, in constant time, either decrease diam(S)i for at least one
i ∈[3] or find an upward point in L≥k or a downward point in L≤k.
Proof. Consider again the coordinates ℓi := minx∈S xi and ri := maxx∈S xi for i ∈[3]. If there
exists q ∈S with qi /∈{ℓi, ri} for all i ∈[3], then querying that point q will allow us to shrink
some diameter, as desired. Thus, assume now that no such point exists. In particular, there is no
q ∈S satisfying
(ℓ1 + 1, ℓ2 + 1, ℓ3 + 1) ≤q ≤(r1 −1, r2 −1, r3 −1) .
Hence, we must have ℓ1 + 1 + ℓ2 + 1 + ℓ3 + 1 > k or r1 −1 + r2 −1 + r3 −1 < k.
Without
loss of generality, assume the former (the other case is symmetric). Note that we can actually
deduce ℓ1 + 1 + ℓ2 + 1 + ℓ3 + 1 = k + 1 by observing that we must have ℓ1 + ℓ2 + r3 ≤k and
ℓ3 + 2 ≤r3. Consider the point q(1) = (ℓ1, ℓ2 + 1, ℓ3 + 1) ∈S. If it is not 1-upward, we are
guaranteed to make progress: Indeed, any of the other options allows us to make progress by
decreasing some diameter or directly solving the levelset. Analogously, we make progress if
q(2) = (ℓ1 + 1, ℓ2, ℓ3 + 1) ∈S is not 2-upward or if q(3) = (ℓ1 + 1, ℓ2 + 1, ℓ3) ∈S is not 3-upward.
It remains to observe that if q(i) is i-upward for all i ∈[3], then the point LUB(q(1), q(2), q(3)) is
upward.
Together, Lemma 3.5 and Lemma 3.6 imply that we can shrink our remaining search space until
we reach diam(S)i ≤1 for some i ∈[3] in time O(log N), or find an upward or downward point
on the way.
3.2
Three Configurations
Assume now that we have reached the situation diam(S)i ≤1 for some i ∈[3]. In that case, we
will prove in Lemma 3.12 that we can find one of three configurations of points, each of which
leads us to an upward point in L≥k or downward point in L≤k.
Definition 3.7 (First Configuration). We say that two points x, y ∈Lk are in first configuration if for
some distinct i, j ∈[3], we have that
• x is i-upward, y is j-upward, xi ≥yi, and xj ≤yj,
• or symmetrically, x is i-downward, y is j-downward, xi ≤yi, and xj ≥yj.
7
Definition 3.8 (Second Configuration). We say that three points x, y, z ∈Lk are in second configuration
if for some distinct i, j, p ∈[3], we have that
• x is i-upward, y is j-upward, z is p-upward, xi ≥yi, yj ≥zj, and zp ≥xp,
• or symmetrically, x is i-downward, y is j-downward, z is p-downward, xi ≤yi, yj ≤zj, and
zp ≤xp.
Recall that an i-upward point x must satisfy F(x)j ≤xj for all j ∈[3] \ {i}. So if, for example, we
have x, y in first configuration with x being i-upward and y being j-upward, then we can deduce
that GLB(x, y) is downward, by monotonicity of F. In fact, this argument works for any kind of
first or second configuration, yielding the following two observations.
Observation 3.9 (First Configuration =⇒Progress). Assume that x, y ∈Lk are in first configuration.
Then either GLB(x, y) ∈L≤k is downward or LUB(x, y) ∈L≥k is upward.
Observation 3.10 (Second Configuration =⇒
Progress). Assume that x, y, z ∈Lk are in second
configuration. Then either GLB(x, y, z) ∈L≤k is downward or LUB(x, y, z) ∈L≥k is upward.
Definition 3.11 (Third Configuration). We consider two points x, y ∈Lk to be in third configuration if
there is i ∈[3] such that x is i-upward, y is i-downward, and xi ≤yi ≤xi + 1.
In contrast to the first and second configuration, the third configuration does not directly imply
an upward or downward point. However, we will see in Section 3.3 how we can still make
progress using at most O(log N) additional time. But before getting to that, let us prove that a
search space S with diam(S)i ≤1 for some i ∈[3] implies existence of at least one of the three
configurations.
u(1)
d(1)
1
2
3
1
2
3
1
2
3
u(2)
u(3)
u(1)
u(2)
u(3)
Tu
Tu
u(1)
Figure 4: Illustrations for Lemma 3.12.
On the left, the two points u(1) and u(2) are in first
configuration. In the middle, the three points u(1), u(2), u(3) are in second configuration. On the
right, the two points u(1), d(1) are in third configuration. For simpilicity, we only show the arrow
in dimension i for points u(i) and d(i).
Lemma 3.12 (Small Diameter =⇒Configuration). Assume u(i)
i
≤d(i)
i
for all i ∈[3], and assume
that the induced remaining search space S has diameter diam(S)i ≤1 for some i ∈[3]. Then a first,
second, or third configuration can be found among the six points u(1), d(1), u(2), d(2), u(3), d(3).
Proof. Consider the two triangle-shaped sets
Td = {x ∈Lk | xi ≤d(i)
i
for all i ∈[3]}
and
Tu = {x ∈Lk | u(i)
i
≤xi for all i ∈[3]}
and observe that S = Tu ∩Td by definition. If no pair u(i), d(i) is in third configuration, then by the
assumption diam(S)i ≤1 for some i ∈[3], we have that either Tu or Td consists of at most 3 points
(see also Figure 4). Without loss of generality, assume that it is Tu (the other case is symmetric).
8
We can also assume that u(1)
2
≤u(2)
2
(without loss of generality, since |Tu| ≤3 implies that there
must exist such a pair, see Figure 4) . Now if we also had u(2)
1
≤u(1)
1 , the two points would be
in first configuration. Thus, assume from now on u(2)
1
> u(1)
1 . Then, we must have u(2)
3
≤u(3)
3
(since |Tu| ≤3, see Figure 4). Again, u(2)
2
≥u(3)
2
would imply a first configuration, so we assume
u(2)
2
< u(3)
2 . But this then implies u(3)
1
≤u(1)
1 , which means that the three points are in second
configuration.
3.3
Third Configuration =⇒Progress
Next, we argue that we can also find an upward or downward point from a third configuration
in time O(log N).
1
2
3
x
y
ℓ
r
Figure 5: An example of the procedure that we use to deal with two points x, y in third configu-
ration. The point r was initialized as the unique point with r1 = y1 and r2 = x2 −1 while ℓwas
initialized to be y but then updated after one binary search step. Within O(log N) steps, we will
either find an upward or downward point, or shrink the interval between ℓand r to only contain
two points, which again implies an upward or downward point, as desired.
Lemma 3.13 (Third Configuration =⇒Progress). Given a pair of points x, y ∈Lk in third configu-
ration, we can find an upward point in L≥k or a downward point in L≤k in time O(log N).
Proof. Assume without loss of generality that x is 1-upward and y is 1-downward. Recall that
we must have x1 ≤y1 ≤x1 + 1. Observe that there has to be a coordinate j ∈{2, 3} satisfying
yj < xj since both x and y belong to Lk and must be distinct. Without loss of generality, we will
assume j = 2. From x, y ∈Lk we thus deduce x3 ≤y3. Figure 5 shows a sketch of the situation.
Since x is 1-upward, we must have F(x)2 ≤x2. Observe that F(x)2 = x2 would imply that
LUB(x, y) = (y1, x2, y3) is an upward point: Indeed, F(LUB(x, y))1 ≥LUB(x, y)1 follows from
combining y1 ≤x1 + 1 with x being 1-upward, whereas F(LUB(x, y))2 ≥LUB(x, y)2 follows from
F(x)2 = x2 and F(LUB(x, y))3 ≥LUB(x, y)3 follows from y being 1-downward. Thus, assume
from now on F(x)2 < x2. Analogously, we can assume F(y)2 > y2 since otherwise, GLB(x, y) is
a downward point.
Next, consider an arbitrary point q ∈Lk with q1 = y1 and y2 ≤q2 < x2. If we have F(q)1 ≥q1
and F(q)2 ≥q2, then LUB(q, y) = (y1 = q1, q2, y3) is an upward point. Conversely, F(q)1 < q1
9
and F(q)2 ≤q2 together imply that GLB(x, q) = (x1 ≥q1 −1, q2, x3) is a downward point. The
only remaining cases are that q either satisfies
(1) F(q)1 ≥q1 and F(q)2 < q2,
(2) or F(q)1 < q1 and F(q)2 > q2.
Our algorithm thus works as follows: We maintain two points ℓ, r ∈Lk with ℓ1 = r1 = y1
and y2 ≤l2 < r2 < x2 such that r is of type (1) and ℓis of type (2). We initialize ℓ= y and
let r be the unique point with r1 = y1 and r2 = x2 −1 (if r is not of type (1) it implies and
upward or downward point by our previous observation on points q ∈Lk with q1 = y1 and
y2 ≤q2 < x2). If we find any point that is not of type (1) or (2), we get an upward or downward
point, as desired. Otherwise, we update r or ℓdepending on whether the point is type (1) or
(2). Thus, by proceeding in a binary search fashion, we need at most O(log N) queries to either
find an upward or downward point, or arrive at ℓ2 + 1 = r2 (and thus ℓ3 = r3 + 1). In the
latter case, either GLB(ℓ, r) = (ℓ1 = r1, ℓ2 = r2 −1, r3 = ℓ3 −1) is downward (if F(r)3 ≤r3) or
LUB(ℓ, r) = (ℓ1 = r1, r2 = ℓ+ 1, ℓ3 = r3 + 1) is upward (if F(r)3 > r3).
3.4
Initializing Search Space
It remains to explain how we initially set up our search space. In particular, we need to explain
how we can find initial points u(i), d(i) ∈Lk satisfying u(i)
i
≤d(i)
i
for all i ∈[3] in O(log N) time.
We end up using a similar technique as in Lemma 3.13.
1
2
3
ℓ
r
Figure 6: An example of the situation of finding an initial point d(1) in Lk, as described in
Lemma 3.14. The algorithm maintains ℓand r and proceeds in a binary search fashion. As long
as no solution has been found, we will be able to maintain F(ℓ)3 < ℓ3 and F(r)2 < r2 while
shrinking the interval between the two points.
Lemma 3.14 (Initializing u(i) and d(i)). For every i ∈[3], we can find an i-upward point u(i) ∈Lk and
an i-downward point d(i) ∈Lk with u(i)
i
≤d(i)
i
in time O(log N), or directly return an upward point in
L≥k or a downward point in L≤k.
Proof. Without loss of generality, assume i = 1. We will explain how we can find d(1) ∈Lk in time
O(log N). The case of finding u(1) is symmetric and the property u(1)
1
≤d(1)
1
will be guaranteed
since d(1)
1
will be as large as it can get among all points in Lk (and symmetrically, u(1)
1
will be as
small as it can get in Lk).
Concretely, let ℓ∈Lk be the unique point with ℓ1 maximized and ℓ2 minimized. Observe that at
least two of the three inequalities F(ℓ)1 ≤ℓ1, F(ℓ)2 ≥ℓ2, and F(ℓ)3 ≤ℓ3 must hold at ℓsince ℓ
must lie on an edge of our 3D-Grid. Thus, if we had F(ℓ)2 < ℓ2, ℓwould be downward and we
would be done. Similarly, it is not hard to check by case distinction that F(ℓ)3 ≥ℓ3 implies that
10
ℓis either 1-downward, upward, or downward. Thus, we can assume from now on F(ℓ)2 ≥ℓ2
and F(ℓ)3 < ℓ3.
We also define r ∈Lk to be the unique point with r1 maximized and r3 minimized. Analogously
to the case of ℓ, we can assume F(r)2 < r2 and F(r)3 ≥r3. An example of the situation is shown
in Figure 6.
Now let q ∈Lk be an arbitrary point with q1 = ℓ1 = r1. First consider the special case where
F(q)1 > q1. This would mean that there are points in the 3D-grid with larger first coordinate,
but they are not part of the levelset. This can only happen if ℓ= r, which by our previous
observations implies that we would already be done. Thus, we can assume F(q)1 ≤q1.
Next, observe that we have ℓ2 ≤q2 ≤r2 and r3 ≤q3 ≤ℓ3. If we have both F(q)2 ≥q2 and
F(q)3 ≥q3, then q is 1-downward and we are done. Otherwise, we get that q satisfies either
F(q)2 < q2 or F(q)3 < q3.
We now proceed in a binary search fashion, updating ℓto q whenever F(q)3 < q3 and r to q
whenever F(q)2 < q2. After at most O(log N) steps, we will have either found a solution or
arrive at ℓ2 ≤r2 ≤ℓ2 + 1 and r3 ≤ℓ3 ≤r3 + 1. In the latter case, we deduce that GLB(ℓ, r) is a
downward point by using F(ℓ)3 < ℓ3 and F(r)2 < r2.
3.5
Putting Everything Together
We conclude by putting all the pieces together. Given a levelset Lk of an instance of 3D-Tarski, we
can use Lemma 3.14 to initialize the six bounding points that induce our search space. We then
use Lemma 3.5 and Lemma 3.6 to iteratively shrink our search space S and achieve diam(S)i ≤1
for some i ∈[3]. Finally, we check our six bounding points for one of the three configurations.
From Lemma 3.12, we know that at least one of the three configurations must exist. Depending
on the configuration, we then use Observation 3.9, Observation 3.10, or Lemma 3.13 to conclude
with an upward point in L≥k or a downward point in L≤k. Each of those steps takes at most
O(log N) time, yielding an overall runtime of at most O(log N). Together with Lemma 3.2, we
thus get an overall O(log2 N)-time algorithm for 3D-Tarski.
Acknowledgments.
We want to thank Simon Weber and Mika G¨o¨os for valuable discussions, and anonymous re-
viewers for their detailed feedback.
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[7] Bernd G¨artner, Sebastian Haslebacher, and Hung P. Hoang. A Subexponential Algorithm for
ARRIVAL. In 48th International Colloquium on Automata, Languages, and Programming (ICALP
2021), volume 198 of Leibniz International Proceedings in Informatics (Lipics), pages 69:1–69:14,
Dagstuhl, Germany, 2021. Schloss Dagstuhl – Leibniz-Zentrum f¨ur Informatik. doi:10.4230/
LIPIcs.ICALP.2021.69.
[8] Alfred Tarski.
A lattice-theoretical fixpoint theorem and its applications.
Pacific J. Math.,
5(4):285–309, 1955.
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A Levelset Algorithm for 3D-Tarski Sebastian Haslebacher and Jonas Lill : [N]3 →[N]3. Our algorithm runs in O(log2 N) time and makes O(log2 N) queries to F, matching the Ω(log2 N) query lower bound due to Etessami et al. as well as the existing state-of-the-art algorithm due to Fearnley et al. 1 16 Oct 2025 1 Introduction We say that a function F : [N]d →[N]d is monotone if x ≤y implies F(x) ≤F(y) for all x, y ∈[N]d, where ≤is the partial order obtained by coordinate-wise comparison. Tarski [8] proved that any such monotone function F must have a fixed point x⋆= F(x⋆). Finding such a fixed point algorithmically has important applications in algorithmic game theory: For example, finding equilibria of supermodular games, approximating the value of Condons's or Shapley's stochastic games, and solving a zero-player game known as ARRIVAL all reduce to finding such a Tarski fixed point [4, 5, 7]. More concretely, given query access to a function F : [N]d →[N]d, we refer to the computational problem of finding a fixed point of F as Tarski. A recursive binary search algorithm due to Dang, Qi, and Ye [4] can solve this problem in O(logd N) time and queries for fixed d, and initially it was rather unclear whether one should expect much better algorithms. Indeed, Etessami, Papadimitriou, Rubinstein, and Yannakakis [5] proved a query lower bound of Ω(log2 N) for d ≥2 and conjectured a lower bound of Ω(logd N) in general. However, Fearnley, P ́alv ̈olgyi, and Savani [6] later disproved this conjecture by giving an algorithm solving the case d = 3 in O(log2 N) queries and time. They also proved a decomposition lemma that allowed them to use recursive applications of their 3D-algorithm to get an O(log⌈2d/3⌉N)-algorithm in general. Since then, both the upper bounds and lower bounds have been improved, but the true complexity of the problem remains wide open. Concretely, the current best upper bound is due to Chen and Li [2] who proved that a fixed point can be found in O(log⌈d+1/2⌉N) time and queries. In terms of lower bounds, the state-of-the-art is an Ω(d log2 N/log d) query lower bound for d ≥2 that was recently announced by Brˆanzei, Phillips, and Recker [1]. In this paper, we present a new algorithm that can solve Tarski for d = 3 in O(log2 N) time and queries, matching the asymptotically optimal algorithm by Fearnley, P ́alv ̈olgyi, and Savani [6]. Their algorithm proceeds by binary searching over two-dimensional slices of the 3D-grid. Their key insight is that it is possible to find a progress point x satisfying either F(x) ≤x or F(x) ≥x in a given two-dimensional slice Sk = {x ∈[N]3 | x1 = k} in O(log N) time. This inner algorithm is however quite complicated and in particular involves an intricate case analysis. We avoid this by binary searching over levelsets instead of slices. Concretely, given a number k ∈{3, . . . , 3N}, we show how we can either find a point x with F(x) ≤x in L≤k = {x ∈[N]3 | x1 + x2 + x3 ≤k} or a point x with F(x) ≥x in L≥k = {x ∈[N]3 | x1 + x2 + x3 ≥k} in O(log N) time. We achieve this by searching over the levelset Lk = {x ∈[N]3 | x1 + x2 + x3 = k}. The advantage of levelsets is that their symmetry allows our algorithm to treat all three dimensions exactly the same. However, in contrast to the slices in [6], no two points in a given levelset are comparable under the partial order ≤, making it seemingly more difficult to exploit monotonicity of F. Our main contribution is to overcome this difficulty. Naturally, our algorithm can also be plugged into the decomposition lemma of [6] to obtain an O(log⌈2d/3⌉N) bound for general d. Moreover, our levelset approach can in principle also be applied directly to higher-dimensional instances. Unfortunately, we have not yet been able to get close to the O(log⌈2d/3⌉N) bound (or the state-of-the-art O(log⌈d+1/2⌉N) bound of Chen and Li [2], for that matter) for d > 3 using this more direct approach, but we do think that this is an interesting idea for future work. Finally, we want to point out that our levelset approach seems even more appropriate in restricted 2 versions of Tarski such as Unique-Tarski and Super-Unique-Tarski. Indeed, many of our ideas originated from studying Super-Unique-Tarski. In terms of query complexity, studying SuperUnique-Tarski would have been enough, given that Chen, Li, and Yannakakis [3] proved that Tarski, Unique-Tarski, and Super-Unique-Tarski all have the exact same query complexity. Unfortunately, their reductions are not time-efficient, which is why we avoided using them and instead generalized our ideas to work for Tarski directly. 2 Preliminaries We use a slightly generalized definition of Tarski on an integer grid G = [n1] × [n2] × · · · × [nd], for arbitrary n1, . . . , nd ∈N. We will use N = ∑d i=1 ni to measure the size of G. The grid G can be equipped with the partial order ≤defined as x ≤y ⇐⇒∀i ∈[d] : xi ≤yi for all x, y ∈G. The tuple (G, ≤) forms a lattice, but we will usually just (implicitly) denote this lattice by G. We use GLB(x(1), . . . , x(k)) to denote the greatest lower bound (meet) and LUB(x(1), . . . , x(k)) to denote the least upper bound (join) of a set of points x(1), . . . , x(k) ∈G in this lattice. We also use |x| := ∑d i=1 xi for the 1-norm of x ∈G. A function F : G →G is called monotone (or non-decreasing) if and only if x ≤y =⇒F(x) ≤F(y) for all x, y ∈G. The computational problem Tarski is defined as follows: Given query access to a monotone function F : G →G, find a fixed point x⋆= F(x⋆) of F. Recall that such a fixed point has to exist by Tarski's theorem [8]. Our algorithm will take O(log2 N) time and will make O(log2 N) queries to F. Note that this excludes the time needed to actually make a query to F, which naturally depends on how access to F is provided to us. For any specific model of access to F, one would have to multiply our bounds by the access time to F in the given model. Moreover, our runtime is measured in arithmetic operations: Numbers in our algorithms can be as large as Ω(N), which means that they require Ω(log N) bits to represent, and this would have to be factored in for other models of computation. We decided to abstract these concerns away for the sake of simplicity. In particular, we will from now on simply refer to the runtime of the algorithm, factoring out any overhead that is needed to actually evaluate F or to calculate with O(log N)-bit numbers in certain models. Moreover, note that our presentation focuses on the setting where F is promised to be monotone. However, it is not hard to see that our algorithm could be adapted to the total setting, where the algorithm is allowed to return a violation of monotonicity instead of a fixed point whenever F is not monotone. This is because our algorithm only exploits monotonicity locally: If a certain step of the algorithm should fail, then a violation of monotonicity must be present among a constant set of previous queries. We will frequently use the notion of upward and downward points: A point x ∈G is called upward if F(x) ≥x, and downward if F(x) ≤x. Note that a point is both upward and downward simultaneously if and only if it is a fixed point of F. Also note that (1, 1, . . . , 1) ∈G must be an upward and (n1, n2, . . . , nd) ∈G a downward point. Furthermore, any upward point x and downward point y satisfying x ≤y imply that there exists a fixed point x ≤x⋆≤y of F: Indeed, 3 due to monotonicity of F, the restriction of F to the subgrid G′ = {z ∈G | x ≤z ≤y} is itself a valid Tarski-instance. In other words, finding such x and y allows us to reduce the search to G′. 3 Levelset Algorithm We start the presentation of our algorithm with the high-level idea and then proceed by explaining all the details step-by-step. The following notion of levelsets will be crucial. Definition 3.1 (Levelset). Let G be a d-dimensional integer grid of size N = ∑d i=1 ni and let k ∈ {d, d + 1, . . . , N} be arbitrary. The k-th levelset Lk of G is defined as Lk := {x ∈G | |x| = k}. Similarly, we define L≥k := S i≥k Li and L≤k := S i≤k Li. x y Lk 1 2 3 (1, 1, 1) (n1, n2, n3) Figure 1: Levelset Lk of a 3D-instance (on the left) projected to 2D (on the right). On the right, the three dimensions are indicated in red, orange, and blue, respectively. For both points x and y, we indicate the function F in each dimension separately. For example, x is an upward point with F(x)i > xi for all i ∈[3] while y satisfies F(y)1 y3, and F(y)2 = y2 (indicated by the absence of an orange arrow). From now on, we restrict ourselves to the case d = 3. Our algorithm proceeds by finding an upward point in L≥⌈N/2⌉or a downward point in L≤⌈N/2⌉in time O(log N). Once we have found such a point, we can shrink our search space by at least half and recursively apply the algorithm on the remaining subgrid, yielding an overall runtime of O(log2 N). Lemma 3.2. Assume that for an arbitrary levelset Lk in an instance F : G →G of 3D-Tarski, we can find an upward point x ∈L≥k or a downward point x ∈L≤k in time O(log N). Then, a fixed point of F can be found in time O(log2 N). Proof. We start by calling the subprocedure on the levelset Lk for k = ⌈N/2⌉in order to find a point x that is either an upward point in L≥k or downward point in L≤k . If x is an upward point in L≥k, we recurse on the subgrid G′ = {y ∈G | x ≤y}. Otherwise, x is a downward point in L≤k and we recurse on the subgrid G′ = {y ∈G | x ≥y}. In both cases, F restricted to G′ is again a Tarski-instance. Moreover, the subgrid G′ has size at most half of G (the parameter N has shrunk by at least half). Once we arrive at a grid of constant size, we can find a fixed point in constant time. Thus, the overall procedure takes O(log2 N) time. Given Lemma 3.2, the main challenge that we have to overcome is to find either an upward point in L≥k or a downward point in L≤k in time O(log N). For this, we will maintain a search space 4 spanned by six bounding points within Lk. We will then iteratively shrink this search space in a binary search fashion. If we do not find an upward or downward point directly during the shrinking, we eventually arrive at a situation that must imply the existence of one out of three possible configurations among the six bounding points. For each of those three configurations, we then show how it implies an upward or downward point, as desired. 3.1 Shrinking the Search Space We start by defining our search space and explaining how we can iteratively shrink it. For this, we first need to introduce so-called i-upward and i-downward points. Definition 3.3 (i-Upward and i-Downward Points). We call a point x ∈Lk i-upward for some i ∈[3] if F(x)i > xi and F(x)j ≤xj for all j ∈[3] \ {i}. Symmetrically, we call x i-downward if F(x)i 1 for all i ∈[3] and maxi∈[3] diam(S)i ≥6 for our remaining search space S. Then there exists q ∈S such that max (diam({x ∈S | xi ≤qi})i, diam({x ∈S | xi ≥qi})i) ≤5 6 diam(S)i for all i ∈[3]. Proof. Assume without loss of generality diam(S)1 ≤diam(S)2 ≤diam(S)3. Let li := minx∈S xi and analogously ri := maxx∈S xi for i ∈[3]. To prove the statement, it suffices to prove that there exists q ∈Lk satisfying l1 + ⌈1 6 diam(S)1⌉, l2 + ⌈1 6 diam(S)2⌉, l3 + ⌈1 6 diam(S)3⌉ ≤q and r1 -⌈1 6 diam(S)1⌉, r2 -⌈1 6 diam(S)2⌉, r3 -⌈1 6 diam(S)3⌉ ≥q. To see that such a q exists, observe first that we must have l1 + ⌈1 6 diam(S)1⌉+ l2 + ⌈1 6 diam(S)2⌉+ l3 + ⌈1 6 diam(S)3⌉≤k because we can upper bound diam(S)1 and diam(S)2 by diam(S)3, use diam(S)3 ≥6, and use that l1 + l2 + l3 + diam(S)3 ≤k, 6 by definition of diam(S)3 and l1, l2, l3. Analogously, we derive k ≤r1 -⌈1 6 diam(S)1⌉+ r2 -⌈1 6 diam(S)2⌉+ r3 -⌈1 6 diam(S)3⌉. Note that the point q in Lemma 3.5 can be constructed efficiently from the six bounding points. Indeed, it suffices to compute the values li, ri, diam(S)i for all i ∈[3] and then choose q with |q| = k arbitrarily within the described bounds. Now recall that Lemma 3.5 allows us to shrink the remaining search space only as long as we have maxi∈[3] diam(S)i ≥6. Thus, we still have to handle the case maxi∈[3] diam(S)i 1 for all i ∈[3] and assume S has constant size. Then we can, in constant time, either decrease diam(S)i for at least one i ∈[3] or find an upward point in L≥k or a downward point in L≤k. Proof. Consider again the coordinates li := minx∈S xi and ri := maxx∈S xi for i ∈[3]. If there exists q ∈S with qi /∈{li, ri} for all i ∈[3], then querying that point q will allow us to shrink some diameter, as desired. Thus, assume now that no such point exists. In particular, there is no q ∈S satisfying (l1 + 1, l2 + 1, l3 + 1) ≤q ≤(r1 -1, r2 -1, r3 -1) . Hence, we must have l1 + 1 + l2 + 1 + l3 + 1 > k or r1 -1 + r2 -1 + r3 -1 u(1) 1 . Then, we must have u(2) 3 ≤u(3) 3 (since |Tu| ≤3, see Figure 4). Again, u(2) 2 ≥u(3) 2 would imply a first configuration, so we assume u(2) 2 y2 since otherwise, GLB(x, y) is a downward point. Next, consider an arbitrary point q ∈Lk with q1 = y1 and y2 ≤q2 q2. Our algorithm thus works as follows: We maintain two points l, r ∈Lk with l1 = r1 = y1 and y2 ≤l2 r3). 3.4 Initializing Search Space It remains to explain how we initially set up our search space. In particular, we need to explain how we can find initial points u(i), d(i) ∈Lk satisfying u(i) i ≤d(i) i for all i ∈[3] in O(log N) time. We end up using a similar technique as in Lemma 3.13. 1 2 3 l r Figure 6: An example of the situation of finding an initial point d(1) in Lk, as described in Lemma 3.14. The algorithm maintains land r and proceeds in a binary search fashion. As long as no solution has been found, we will be able to maintain F(l)3 q1. This would mean that there are points in the 3D-grid with larger first coordinate, but they are not part of the levelset. This can only happen if l= r, which by our previous observations implies that we would already be done. Thus, we can assume F(q)1 ≤q1. Next, observe that we have l2 ≤q2 ≤r2 and r3 ≤q3 ≤l3. If we have both F(q)2 ≥q2 and F(q)3 ≥q3, then q is 1-downward and we are done. Otherwise, we get that q satisfies either F(q)2 < q2 or F(q)3 < q3. We now proceed in a binary search fashion, updating lto q whenever F(q)3 < q3 and r to q whenever F(q)2 < q2. After at most O(log N) steps, we will have either found a solution or arrive at l2 ≤r2 ≤l2 + 1 and r3 ≤l3 ≤r3 + 1. In the latter case, we deduce that GLB(l, r) is a downward point by using F(l)3 < l3 and F(r)2 < r2. 3.5 Putting Everything Together We conclude by putting all the pieces together. Given a levelset Lk of an instance of 3D-Tarski, we can use Lemma 3.14 to initialize the six bounding points that induce our search space. We then use Lemma 3.5 and Lemma 3.6 to iteratively shrink our search space S and achieve diam(S)i ≤1 for some i ∈[3]. Finally, we check our six bounding points for one of the three configurations. From Lemma 3.12, we know that at least one of the three configurations must exist. Depending on the configuration, we then use Observation 3.9, Observation 3.10, or Lemma 3.13 to conclude with an upward point in L≥k or a downward point in L≤k. Each of those steps takes at most O(log N) time, yielding an overall runtime of at most O(log N). Together with Lemma 3.2, we thus get an overall O(log2 N)-time algorithm for 3D-Tarski. Acknowledgments. We want to thank Simon Weber and Mika G ̈o ̈os for valuable discussions, and anonymous reviewers for their detailed feedback. References [1] Simina Brˆanzei, Reed Phillips, and Nicholas Recker. Tarski Lower Bounds from MultiDimensional Herringbones, February 2025. , 16679. [2] Xi Chen and Yuhao Li. Improved Upper Bounds for Finding Tarski Fixed Points. In Proceedings of the 23rd ACM Conference on Economics and Computation, EC '22, pages 1108-1118, New York, NY, USA, July 2022. Association for Computing Machinery. 3538297. 11 [3] Xi Chen, Yuhao Li, and Mihalis Yannakakis. Reducing Tarski to Unique Tarski (In the BlackBox Model). In 38th Computational Complexity Conference (CCC 2023), volume 264 of Leibniz International Proceedings in Informatics (LIPIcs), pages 21:1-21:23, Dagstuhl, Germany, 2023. Schloss Dagstuhl - Leibniz-Zentrum f ̈ur Informatik. [4] Chuangyin Dang, Qi Qi, and Yinyu Ye. Computations and Complexities of Tarski's Fixed Points and Supermodular Games. In Frontiers of Algorithmics, pages 159-174, Singapore, 2025. Springer Nature. [5] Kousha Etessami, Christos Papadimitriou, Aviad Rubinstein, and Mihalis Yannakakis. Tarski's Theorem, Supermodular Games, and the Complexity of Equilibria. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:19, Dagstuhl, Germany, 2020. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. [6] John Fearnley, D ̈om ̈ot ̈or P ́alv ̈olgyi, and Rahul Savani. A Faster Algorithm for Finding Tarski Fixed Points. ACM Transactions on Algorithms, 18(3):1-23, July 2022. [7] Bernd G ̈artner, Sebastian Haslebacher, and Hung P. Hoang. A Subexponential Algorithm for ARRIVAL. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021), volume 198 of Leibniz International Proceedings in Informatics (Lipics), pages 69:1-69:14, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum f ̈ur Informatik. LIPIcs.ICALP.2021.69. [8] Alfred Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific J. Math., 5(4):285-309, 1955. 12
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2510.14788
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
CROSS-SCENARIO UNIFIED MODELING OF
USER INTERESTS AT BILLION SCALE
Manjie Xu 1,3,˚, Xin Jia 3,˚, Cheng Chen 3,˚, Jingyi Zhou 2,3, Chi Zhang 1, , Yongji Wu 3,
Zejian Wang 3, Kai Zuo 3,:, Yibo Chen 3,:, Xu Tang 3,:, Yao Hu 3, , Yixin Zhu 1, ˚ equal contribution
: project lead
corresponding author
1 Peking University
2 Fudan University
3 Xiaohongshu Inc.
ABSTRACT
User interests on User-Generated Content (UGC) platforms are inherently diverse,
manifesting through complex behavioral patterns across heterogeneous scenarios
such as search, feed browsing, and content discovery. Traditional recommenda-
tion systems operate in isolated scenarios, optimizing business metrics within nar-
row contexts while neglecting valuable cross-scenario behavioral signals. This
fragmented approach struggles to integrate advanced techniques like LLMs at
billion-scale deployments, ultimately limiting the ability to capture holistic user
interests across platform touchpoints. We introduce RED-Rec, an LLM-enhanced
hierarchical Recommender Engine for Diversified scenarios, tailored for industry-
level UGC recommendation systems. RED-Rec unifies user interest represen-
tations by aggregating and synthesizing actions from multiple behavioral con-
texts, enabling comprehensive item and user modeling. The framework features
an LLM-powered architecture that delivers nuanced, multifaceted representations
while maintaining deployment efficiency. A novel scenario-aware dense mixing
and querying policy effectively fuses diverse behavioral signals to capture cross-
scenario user intent patterns and express fine-grained, context-specific preferences
during serving. We validate RED-Rec through online A/B testing on hundreds of
millions of users in Xiaohongshu, demonstrating substantial performance gains in
content recommendation and advertisement targeting tasks. We also introduce a
million-scale sequential recommendation dataset, RED-MMU, for offline evalua-
tion. Our work advances unified user modeling, unlocking deeper personalization
and fostering more meaningful user engagement in large-scale UGC platforms.
1
INTRODUCTION
Modern User-Generated Content (UGC) platforms have evolved into complex multi-scenario
ecosystems where users engage through diverse behavioral contexts—browsing personalized feeds,
conducting topical searches, discovering content creators, and responding to targeted advertise-
ments. Each interaction scenario captures distinct yet complementary aspects of user intent: search
queries reveal explicit informational needs, feed engagement demonstrates implicit content prefer-
ences, and advertisement clicks indicate commercial interests (Covington et al., 2016; Hidasi et al.,
2015). Crucially, users exhibit remarkably consistent underlying interests across these diverse be-
havioral contexts. A user passionate about sustainable living may search for ”eco-friendly pack-
aging,” engage with environmental advocacy posts in their feed, and click advertisements for solar
panels—each interaction revealing the same core interest through different behavioral lenses. This
consistency suggests that user interests are inherently multi-dimensional, manifesting through inter-
twined behavioral trajectories that span multiple scenarios (Xia et al., 2020; Zhu et al., 2022).
Despite this behavioral richness, production recommendation systems typically operate as indepen-
dent silos, with separate models independently optimized for specific business objectives such as
Click-Through Rate (CTR) in feeds and Advertiser Value (ADVV) in advertisements (Zhang et al.,
2019; Chapelle et al., 2014). This siloed design traps systems in local optima and creates sev-
eral critical limitations. First, it fragments user understanding by restricting each model to narrow
behavioral contexts, preventing holistic interest modeling. Second, it produces inconsistent user
1
arXiv:2510.14788v1 [cs.IR] 16 Oct 2025
Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
User Lastn
User xxxx
…
…
…
Scenario 1:
Homefeed
Scenario 2:
Search
User LLM
Item LLM
…
Current
Time
…
Multi-scenario Interests
…
…
…
target
next window
…
scene-aware
queries
…
scene-aware
embed
…
querying
User Lastn
User Behaviors
User Interests
RED-Rec Inference
serving
Sideinfo
Online Metrics
(ctr/advv/…)
mixer
offline test
Offline Metrics
(topk/ndcg/…)
User xxxx
(latent representation)
User Engage:
click/like/share …
Scenario 3:
Advertisements
Item xxxx
(latent representation)
Multi-
interest
Item xxxx
Contribute to
Figure 1: From fragmented signals to unified understanding. Users express consistent interests across
diverse scenarios (left), generating rich behavioral sequences that span homefeed browsing, search queries, and
ad interactions (middle). RED-Rec synthesizes these cross-scenario signals using LLM-powered hierarchical
modeling to generate comprehensive user representations for context-aware recommendations (right). Vector
graphics; zoom for details.
experiences when independent systems infer divergent preferences from the same user. Most im-
portantly, it underutilizes valuable cross-scenario signals, limiting knowledge transfer across tasks
and weakening performance for users with sparse activity in certain scenarios (Xia et al., 2020;
Zhu et al., 2022). Consider our sustainable living enthusiast: traditional systems would treat their
search behavior, feed engagement, and ad responses as unrelated signals, failing to synthesize these
coherent signals into unified user interest and intent.
We are motivated by the observation that users exhibit consistent interest patterns across diverse
scenarios, and that modeling these patterns holistically can significantly enhance recommendation
quality. While some cross-scenario modeling approaches exist (Zhang et al., 2023; Bao et al., 2023),
they typically require extensive manual feature engineering and struggle with scalability and robust-
ness in production environments. Recent advances make this vision increasingly feasible: Large
Language Models (LLMs) have transformed semantic understanding of user behaviors and content
(Wang et al., 2024b), while advanced sequence modeling techniques effectively capture complex
temporal dynamics and cross-scenario dependencies (Sun et al., 2019). Meanwhile, modern UGC
platforms generate massive cross-scenario behavioral logs (McAuley et al., 2015; Harper & Konstan,
2015; Gao et al., 2022), creating unprecedented opportunities for unified modeling at scale.
However, realizing this vision presents significant challenges: heterogeneity in action schemas, tem-
poral dynamics, and semantics across scenarios; severe activity imbalances where users may have
thousands of feed interactions but only dozens of searches; large-scale training and serving with
strict latency and throughput constraints requiring sub-millisecond response times; and reconciling
differing optimization objectives within a single architecture. While recent work explores mixtures
of multi-source signals (Ma et al., 2022; Zhang et al., 2022a; Liu et al., 2024; Yang et al., 2024),
truly end-to-end unified modeling for industrial deployments remains underexplored.
We introduce Recommender Engine for Diversified scenarios (RED-Rec), an LLM-enhanced hier-
archical sequential recommendation framework tailored for billion-scale UGC platforms. RED-Rec
unifies interest modeling across heterogeneous contexts. First, we employ LLM-powered user and
item encoders within a hierarchical two-tower structure that enables rich semantic representations
while preserving efficiency for large-scale retrieval. Second, we introduce a novel 2D dense mix-
ing policy that fuses cross-scenario behavioral signals along temporal and scenario axes to capture
cross-scenario dependencies, coupled with multi-interest, scenario-aware queries that express fine-
grained, context-specific user intents during serving. We train RED-Rec end-to-end on billions of
behavioral events drawn from billions of items and over one hundred million users, incorporating
system-level optimizations that enable near-real-time online deployment.
To enable rigorous evaluation, we also introduce a new cross-scenario sequential dataset cu-
rated from anonymized user behavior data on Xiaohongshu, a world-leading UGC platform. The
RedNote’s Multi-Scenario Multimodal User Behaviors (RED-MMU) dataset spans millions of items
and diverse user behaviors across feeds, search, and advertisement contexts, facilitating compre-
hensive benchmarking of unified and scenario-specific models. In a series of offline experiments,
RED-Rec consistently outperforms strong baselines across multiple metrics and scenarios. This
2
Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
effectiveness extends to production, as demonstrated by online A/B testing, leading to a comprehen-
sive full-scale rollout that now serves hundreds of millions of daily users on Xiaohongshu.
Our main contributions include: (i) a unified, user-centric interest modeling framework that achieves
both expressiveness and efficiency for billion-scale cross-scenario recommendation; (ii) a million-
scale cross-scenario sequential dataset, RED-MMU, enabling rigorous evaluation of unified mod-
eling approaches; and (iii) empirical validation in both offline and online production environments
that establishes the practical viability of unified cross-scenario modeling at unprecedented scale.
2
RELATED WORK
Sequential Recommendation
Sequential recommendation has evolved from early neural meth-
ods like neural collaborative filtering (He et al., 2017) and factorization machines (Rendle, 2010;
Guo et al., 2017) to sophisticated sequence models. GRU4Rec (Hidasi et al., 2015) pioneered re-
current architectures for session-based interactions, while Caser (Tang & Wang, 2018) employed
convolutional filters for temporal patterns. Transformer-based approaches like SASRec (Kang &
McAuley, 2018) and BERT4Rec (Sun et al., 2019) introduced self-attention and bidirectional en-
coding to capture long-range dependencies in user behavior sequences. Recent advances address
the multifaceted nature of user preferences through multi-interest modeling (Li et al., 2019; Cen
et al., 2020), graph neural networks (Wang et al., 2020; Zhang et al., 2022b; Yang et al., 2023), and
contrastive learning (Zhou et al., 2020; Wei et al., 2023). The emergence of LLMs has opened new
frontiers with enhanced user and item representations (Chen et al., 2024a; Hu et al., 2024; Wang
et al., 2024b) and generative paradigms (Chen et al., 2024b; Paischer et al., 2024; Deng et al., 2025;
Han et al., 2025), though most remain limited to smaller-scale applications.
Cross-Scenario Modeling
Users maintain consistent interests across different behavioral con-
texts despite varying interaction patterns (Zang et al., 2022). Early cross-platform studies (Niu
et al., 2021; Tan et al., 2021) established that users exhibit similar topical preferences across plat-
forms, motivating disentangled representation learning that separates stable interests from context-
dependent behaviors. Modern cross-scenario systems (Tan et al., 2021; Zhao et al., 2023; Li et al.,
2024; Chen et al., 2024c; Wu et al., 2025) capture shared interest representations while accommo-
dating scenario-specific patterns. Graph-based approaches (Tan et al., 2021; Cao et al., 2022) model
multi-behavioral patterns, while cross-domain (Ma et al., 2022) and multi-domain methods (Zhao
et al., 2023; Yang et al., 2024) leverage multi-source user histories to improve performance across
scenarios. However, most existing methods struggle with industrial-scale deployment challenges, in-
cluding heterogeneity, scale, and strict latency requirements. Recent foundation model approaches
(Wang et al., 2024a; Shen et al., 2024) show promise but lack validation at billion-scale scenarios.
Our work addresses these limitations through an LLM-enhanced framework designed for industrial
deployment with comprehensive online validation.
3
THE RED-MMU DATASET
Existing open-source sequential recommendation datasets suffer from major limitations in scope
and diversity. Traditional datasets focus on isolated scenarios with singular interaction types such as
ratings, clicks, or purchases (Ben-Shimon et al., 2015; Harper & Konstan, 2015; Ni et al., 2019; Zhu
et al., 2018), failing to capture the cross-scenario nature of modern UGC platforms. Even recent
datasets like KuaiRand (Gao et al., 2022) and Qilin (Chen et al., 2025) that begin to characterize
UGC environments adopt fragmented approaches that underrepresent the complex interplay between
scenarios and only partially reflect holistic user interest evolution.
To address these limitations, we introduce a cross-scenario sequential recommendation dataset, Red-
Note’s Multi-Scenario Multimodal User Behaviors (RED-MMU), derived from anonymized user
data spanning billions of interactions on a major UGC platform. Details on the protection of user
privacy by excluding personal information, hashing user identifiers, and retaining only essential be-
havioral signals in the RED-MMU dataset are described in Section C.2. Our dataset features the
following three key characteristics:
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
Diverse Behavioral Contexts
The RED-MMU dataset encompasses comprehensive real-world
interaction scenarios, including homefeed browsing, search-driven exploration, and advertisement
engagement. This temporally aligned diversity enables robust analysis of user behavior across dis-
tinct yet interconnected scenarios within a unified platform ecosystem.
User Lastn in Cross Scenarios Interactions
(a) Scenario 1: Homefeed at time t-5
click
t-5
click
share
follow
comment
timestamp &
location
like & collect
click
engage
tags
t-4
click after search
engage
infeed preview
(need click)
instream flow
(scroll down)
t-2
commercialization
notes
advertising
(goods, app promotion,
services, etc.)
purchase page
(b) Scenario 2: Search at time t-4
(c) Scenario 3: Advertisements at time t-2
Figure
2:
Cross-scenario
user
interactions
in
the
RED-MMU dataset.
Real platform interface examples
showing user behavior progression across (a) homefeed con-
tent consumption, (b) search-based exploration, and (c) ad-
vertisement engagement.
Each scenario captures diverse
interaction types, including clicks, shares, comments, and
purchases, illustrating the interconnected nature of cross-
scenario interactions on UGC platforms.
Rich
Engagement
Patterns
The
RED-MMU dataset captures both ex-
plicit positive engagements (clicks, likes,
collections, shares) and negative signals,
along with view duration for each interac-
tion. This provides a nuanced and holistic
depiction of user preferences and attention
patterns beyond simple binary feedback.
Industrial-Scale Coverage
The dataset
includes
billions
of
items
and
over
one hundred million users’ engagement
records, surpassing existing datasets in
both scale and complexity. Tracking user
behavior over extended time periods fa-
cilitates the study of long-context interest
evolution, behavioral stability, and cross-
scenario consistency that are typically un-
available in public datasets.
Figure 2 shows an example datapoint
across
multiple
scenarios,
including
homefeed, search, and advertisements.
Figure 3 presents overall dataset statistics
of RED-MMU, showing details on user
engagement analytics. Additional details,
including dataset collection and filtering,
can be found in Section C.
4
CROSS-SCENARIO
USER INTERESTS LEARNING
4.1
TASK FORMULATION
We formulate cross-scenario sequential
recommendation as learning unified user
and
item
representations
from
cross-
scenario interaction data. This formulation
captures the complexity of modern recom-
mendation systems where users engage with content across multiple behavioral contexts within the
same platform ecosystem.
Problem Setup
Consider a recommendation system with user set U “ tu1, u2, . . . , uNu and uni-
versal item space I “ ti1, i2, . . . , iMu. The item space encompasses diverse content types, includ-
ing image-text posts and videos created by regular users or advertisers, which can be recommended
through homefeed, discovered via search, or presented as advertisements.
Cross-Scenario Interaction Sequences
For each user u P U, we observe a chronologically or-
dered engagement sequence:
Su “ tpi1, a1, s1, t1q, pi2, a2, s2, t2q, . . . , pi|Su|, a|Su|, s|Su|, t|Su|qu.
(1)
Each interaction tuple contains: (i) it P I - the interacted item, (ii) at P A - the engagement action,
(iii) st P S - the scenario context, and (iv) t - the interaction timestamp.
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
We focus on three primary scenarios S “ thomefeed, advertisements, searchu that represent the
dominant user engagement patterns on UGC platforms. User actions span a rich set of engage-
ments A “ tlike, share, comment, follow, messaging, blocku, capturing both positive and negative
feedback signals that reflect nuanced user preferences.
Learning Objective
Our goal is to learn unified embedding functions that capture user preferences
and item characteristics across different scenarios. Specifically, we aim to learn (i) a user embedding
function: fu : U ˆ Hu Ñ Rd, and (ii) an item embedding function: fi : I Ñ Rd. These func-
tions map users (conditioned on their interaction history Hu) and items to a shared d-dimensional
embedding space:
u “ fupu, Suq,
vi “ fipIq.
(2)
The resulting embeddings u, vi P Rd encode cross-scenario user interests and item characteris-
tics, enabling effective recommendation across all scenarios. These representations can be directly
applied to recall tasks or serve as features for downstream ranking models.
4.2
HIERARCHICAL LLM-BASED REPRESENTATION LEARNING
RED-Rec employs a hierarchical two-tower architecture that learns comprehensive user and item
representations across multiple scenarios. The framework consists of three key components: (i)
multimodal item encoding that captures content semantics, (ii) sequential user modeling with cross-
scenario interest fusion, and (iii) scenario-aware querying mechanism for diverse preference capture.
Multimodal Item Representation
Each item i P I is encoded through a multimodal encoder
Eitem that processes both textual and visual content:
ei “ Eitempxi, vi; θt, θvq P Rd.
(3)
The textual component xi encompasses title, tags, content description, and OCR-extracted text,
processed by a pre-trained language model with parameters θt. Visual content vi is encoded using
a ViT (Dosovitskiy et al., 2020) with parameters θv, followed by linear projection to dimension d.
This unified representation captures rich semantic information across modalities.
Cross-Scenario Sequential Modeling
To model user interests across diverse behavioral contexts,
we aggregate interactions from three primary scenarios. For user u, the combined interaction se-
quence is Su “ Sh
u Y Sa
u Y Ss
u, where Sh
u, Sa
u, and Ss
u represent homefeed, advertisements, and
search respectively. Each interaction incorporates three information dimensions: (i) Content rep-
resented by item embeddings Hu “ rei1, ei2, . . . , eins P Rnˆd, (ii) Actions encoding engage-
ment behaviors Au “ rai1, ai2, . . . , ains as dense embeddings from one-hot vectors represent-
ing tcollect, share, message, block, likeu, and (iii) Temporal features with hour-level timestamps
hit “ OneHotphourptqq P t0, 1u24 converted to dense embeddings. Hence, ˆHu combines all three
dimensions, enabling the user encoder to capture temporal patterns and engagement preferences.
2-D Dense Mixing Policy
To address behavioral imbalance across scenarios, we introduce a bal-
anced sampling strategy that preserves informative signals from all scenarios:
Smixer
u
“ Merge
´
Shomefeed
u
r´nh :s, Sadvertisements
u
r´na :s, Ssearch
u
r´ns :s
¯
.
(4)
The Mergep¨q operation chronologically sorts and concatenates recent interactions while maintain-
ing scenario tags. This “2-D dense mixing” filters along both scenario (quota balancing) and tempo-
ral (recency) dimensions, ensuring representation of infrequent but valuable user signals. We design
2-D positional encoding for each event j in Smixer
u
: pj “ PEseqpjq ` PEgapp∆tjq, where PEseqpjq
captures sequence position and PEgapp∆tjq encodes time gaps with ∆tj “ tcurr ´ tj.
Scenario-Aware Interest Querying
To capture diverse facets of user preferences, we employ
learnable query embeddings Q “ rq1, q2, . . . , qKs P RKˆd that attend to different interest aspects
across scenarios. The scenario-aware representation is computed as:
Uquery
u
“ Euser
´”
˜Hur: ´Ws; Q
ı
; θu
¯
,
(5)
where W represents the window size for recent interactions, enabling the model to generate multiple
interest-specific representations.
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
(a) User Interaction Distribution
Channel and Scenario Analysis
(b) Top 100 Notes by Channel and Scenario
(c) Behavior Distribution Analysis
(d) Duration vs Engagement Analysis (Top 1000)
(e) 24-Hour Activity Pattern (GMT+8)
(f) Content Lifecycle by Channel ✖ Scenario
(g) User Behavior by Channel and Scenario
(h) User Activity by Channel ✖ Scenario (500 Users)
Figure 3: Comprehensive user engagement analytics across scenarios. Multi-faceted analysis of 10k sam-
pled users showing interaction patterns across homefeed, advertisements, and search scenarios. The dashboard
presents: (a) user distribution and interaction channels, (b) top content engagement by scenario, (c) behavioral
pattern distributions, (d) duration-engagement correlations, (e) temporal activity patterns, (f) content lifecycle
analysis, (g-h) cross-scenario user behavior comparisons. Analysis reveals distinct engagement patterns and
temporal dynamics across different recommendation contexts.
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
Target Items
Recall
In-batch
Negative
Examples
title
…
Scenario 1 Scenario 2 Scenario 3
<img prompt>
<text prompt> {title} {tags} {Content} {Ocr}
Ocr
Model
Vision
Encoder
Item LLM Encoder
(Image Tokens)
…
Auxiliary Embeddings
1. User Engage
2. Position Encoding
3. …
Target Scenario
Sampling Pool
User LLM Encoder
Dense Mixing
tags
Content
Target Items
Hungarian
Matching
…
…
attention
Clustering
Scenario-aware
Query
User Representation
Note:
First Image
Video:
Cover Image
Ramdom-Sampled Notes
Hard Negative Samples
Time Dim
Scenario Dim
Sampling
Figure 4: Overall framework of RED-Rec. RED-Rec employs a two-tower hierarchical architecture with
multimodal item encoding and cross-scenario user modeling. The item encoder processes textual content (title,
tags, OCR) and visual signals through unified embeddings. The user encoder incorporates a 2-D dense mixing
policy to balance interactions across homefeed, advertisements, and search scenarios, followed by scenario-
aware transformer blocks that capture evolving user interests. During training, positive and negative samples
are drawn from a sampling pool to optimize contrastive objectives end-to-end. RED-Rec generates unified
representations suitable for cross-scenario recommendation tasks.
Training Objective
We optimize using Noise Contrastive Estimation (NCE) with temperature
scaling:
LNCE “ ´
ÿ
u,t
log
exp
`
τ ¨ cospuu,t, vit`1q
˘
exp
`
τ ¨ cospuu,t, vit`1q
˘
` ř
jPN exp pτ ¨ cospuu,t, vjqq,
(6)
where τ is a learnable temperature parameter, uu,t represents user u’s interest at time t, and N con-
tains negative samples. Additionally, we incorporate window-based contrastive loss for recent inter-
actions to capture evolving preferences. Similar techniques have also been employed in HyMiRec
(Zhou et al., 2025), where hybrid multi-interest learning is applied for enhanced user representation
and retrieval. Rather than a single, biased embedding, we encourage the User LLM model to capture
diverse user intents from multiple perspectives.
4.3
EVALUATION PROTOCOL
We evaluate learned representations on recall tasks using temporal data splitting. For each user u, in-
teractions are divided at randomly sampled cutoff tcut, creating input sequence Sinput
u
“ tpi, a, s, tq P
Su : t ă tcutu and target set Gu “ titcut, itcut`1, itcut`2u.
The candidate pool C combines random platform samples with ground truth targets. User embed-
dings u computed from Sinput
u
generate similarity scores scorepu, iq “ cospu, viq for ranking. We
report Hit Rate (HR@K), Normalized Discounted Cumulative Gain (NDCG@K), and Mean Recip-
rocal Rank (MRR) for K P t10, 50, 100, 1000u.
5
EXPERIMENTS
5.1
EXPERIMENTAL SETUP
Dataset and Configuration
We conduct comprehensive experiments on an industrial dataset com-
prising 1 million users for training and 10,000 test samples for evaluation. The candidate pool con-
tains approximately 1 million randomly sampled notes to ensure fair comparison across methods.
Our default configuration sets window size W “ 10, sequence length last n “ 128, and employs 3
queries per scenario to capture diverse interest facets.
Model Architecture and Training
Both item and user encoders are initialized with large language
models: either 1.3B-parameter Chinese-LLaMA (Cui et al., 2023) or 1.5B Qwen-2.5 (Yang et al.,
2025), while visual encoding utilizes CLIP ViT-B/16. Training is conducted on 8 NVIDIA H100
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
Table 1: Single-scenario recommendation performance comparison. Performance evaluation of RED-Rec
variants against established baselines (SASRec, MoRec, HSTU, HLLM, DLRM-v3) on homefeed and adver-
tisement recommendation tasks. RED-Rec variants include symbol-based (RED-Rec-symbol) and multimodal
(RED-Rec-mm) versions, with pre-trained variants (RED-Rec-pt) leveraging large-scale data. Higher scores
indicate better performance across all metrics.
Homefeed
Advertisements
Baselines Ò
HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR˚100 HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR˚100
SASRec
1.76/0.97
12.32/1.79
32.01/4.04
1.01
3.26/1.63
14.08/3.71
39.11/5.27
1.57
MoRec
1.78/1.25
12.48/2.23
31.98/4.12
1.21
3.47/1.67
13.98/3.88
38.27/4.89
1.78
HSTU
1.79/1.22
12.72/2.21
31.76/3.69
1.15
3.85/1.70
14.32/3.30
38.20/5.38
1.43
HLLM
1.66/0.62
12.77/1.83
32.52/4.02
1.22
4.21/1.21
14.27/3.37
39.21/4.48
1.39
DLRM-v3
1.63/1.03
11.33/2.01
28.96/3.72
1.13
3.54/1.21
15.27/3.22
35.39/4.27
1.67
RED-Rec
2.31/0.68
12.59/1.88
31.94/3.86
1.27
4.24/1.28
16.44/3.21
40.18/4.61
1.96
RED-Rec-pt
2.90/0.63
14.89/2.02
36.16/4.01
1.30
4.84/1.30
17.66/2.87
42.71/5.21
2.27
RED-Rec-mm
2.35/1.21
14.20/2.27
31.29/3.97
1.29
4.31/1.31
17.22/3.18
41.86/4.66
1.92
RED-Rec-mm-pt
3.23/1.27
15.46/2.21
36.29/4.14
1.38
4.82/1.19
18.21/3.29
42.56/4.98
2.21
GPUs for 3 epochs with batch size 2 and gradient accumulation of 4, requiring approximately 24
hours. Implementation details are provided in Section E.
Baselines and Evaluation
We compare against established recommendation methods: SASRec
(Kang & McAuley, 2018), MoRec (Yuan et al., 2023), HSTU (Zhai et al., 2024), HLLM (Chen et al.,
2024a), and DLRM-v3 (Naumov et al., 2019). Our evaluation encompasses both single-scenario
(homefeed, advertisements) and cross-scenario (search + homefeed, homefeed + advertisements,
all combined) settings. We assess four RED-Rec variants: RED-Rec-symbol, RED-Rec-mm, and
their pre-trained versions (RED-Rec-symbol-pt, RED-Rec-mm-pt) trained on large-scale online data.
Standard metrics (HR@K, NDCG@K, MRR) are reported across multiple cutoff values.
5.2
SINGLE-SCENARIO PERFORMANCE Table 2:
Cross-scenario recommendation performance
evaluation. Performance comparison when leveraging cross-
scenario signals for improved recommendations across dif-
ferent target scenarios.
Results demonstrate the effective-
ness of RED-Rec in utilizing cross-scenario behavioral sig-
nal. Higher scores indicate superior performance.
Search + Homefeed (for Homefeed)
Baselines Ò
HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR˚100
SASRec
1.73/1.22
12.02/3.21
32.17/4.17
1.52
MoRec
1.79/1.30
13.92/2.99
33.01/3.98
1.53
HSTU
1.79/1.25
12.84/3.28
33.15/4.24
1.55
HLLM
1.69/1.02
13.49/3.18
33.04/4.21
1.58
DLRM-v3
1.64/1.18
11.35/3.02
30.89/3.98
1.48
RED-Rec
2.26/1.32
14.74/3.16
33.29/4.20
1.58
RED-Rec-pt
2.92/1.33
18.26/3.24
38.92/4.23
1.67
Homefeed + Advertisements (for Advertisements)
Baselines Ò
HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR˚100
SASRec
3.72/1.24
16.18/3.08
38.94/4.72
1.94
MoRec
3.80/1.30
17.23/2.62
38.29/4.77
1.98
HSTU
3.89/1.28
16.95/3.15
40.12/4.81
2.01
HLLM
3.68/1.19
17.24/3.12
39.76/4.78
1.97
DLRM-v3
3.52/1.21
15.43/2.95
36.87/4.58
1.87
RED-Rec
4.36/1.31
18.32/3.27
42.61/5.02
2.11
RED-Rec-pt
5.18/1.38
18.89/3.21
46.59/5.57
2.38
Homefeed + Search + Advertisements (for Advertisements)
Baselines Ò
HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR˚100
SASRec
3.68/1.21
14.29/2.08
38.94/4.72
1.94
MoRec
3.82/1.33
18.27/2.98
38.41/4.66
1.98
HSTU
3.92/1.31
17.21/3.19
40.14/4.81
2.11
HLLM
4.08/1.11
19.92/3.18
43.27/4.91
2.06
DLRM-v3
3.34/1.01
14.08/2.81
35.74/4.36
1.74
RED-Rec
4.72/1.33
18.33/3.22
42.89/4.97
1.94
RED-Rec-pt
5.18/1.35
20.52/3.24
49.17/5.93
2.41
Table 1 presents results for homefeed
and advertisement recommendation sce-
narios.
RED-Rec consistently outper-
forms all baselines in both scenarios,
demonstrating the efficacy even in single-
scenario settings.
The superior perfor-
mance stems from two key factors: (i)
multi-interest user representation learning
that captures diverse preference facets, and
(ii) advanced LLM-based semantic encod-
ing that provides richer representations
than traditional ID-based methods.
Compared to SASRec and HSTU, which
rely on item ID embeddings, RED-Rec
shows substantial improvements, partic-
ularly beneficial for cold-start scenar-
ios where semantic understanding is cru-
cial.
When compared against HLLM,
which shares similar architectural princi-
ples, RED-Rec benefits from larger back-
bone models with enhanced Chinese lan-
guage capabilities, enabling better align-
ment with our dataset characteristics and
further performance gains.
5.3
CROSS-SCENARIO BENEFITS
Cross-scenario evaluation (Table 2) re-
veals significant performance improvements when integrating information across behavioral con-
texts. The most pronounced gains occur in two key scenarios: (i) search data enhancing homefeed
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
recommendations, and (ii) combined homefeed and search signals improving advertisement perfor-
mance.
Cross-Scenario Information Flow
Incorporating cross-scenario signals consistently improves
performance across all baselines, with RED-Rec achieving the largest gains due to its effective inte-
gration capabilities and advanced user-side LLM reasoning. For homefeed recommendations, access
to search behaviors, particularly post-search engagement patterns, substantially increases both HR
and NDCG scores. Similarly, advertisement recommendations benefit from combined homefeed
and search behaviors, showing the greatest metric improvements across all evaluation criteria.
Consistent Improvements
These enhancements remain consistent across all cutoff values (K P
t10, 50, 100, 1000u), indicating that RED-Rec not only increases the likelihood of relevant item
recommendation but also improves their ranking positions. Figure 5 illustrates these cross-scenario
benefits, demonstrating substantial performance gains when leveraging complementary signals.
5.4
ABLATION STUDIES
Table 3: Ablation study results for RED-Rec architectural compo-
nents.
Top: Core model configuration ablations examining the im-
pact of sequence length (SeqLen = 128 vs. 32) and interest model-
ing approaches (Multi Interest vs.
Single Interest) on recommenda-
tion performance.
Bottom: Cross-scenario mixing policy ablations
evaluating different strategies for combining behavioral signals across
homefeed, search, and advertisement scenarios. Comparison includes
temporal sampling, scenario exclusion variants, and our proposed 2D
Dense Mixing approach. Results demonstrate the effectiveness of longer
sequences, multi-interest modeling, and comprehensive cross-scenario
signal integration.
Homefeed
Setting
HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR
SeqLen = 128, Multi-Interest, pt
2.90/0.63
14.89/2.02
36.16/4.01
1.30
SeqLen = 128, Multi-Interest
2.31/0.68
12.59/1.88
31.94/3.86
1.27
SeqLen = 128, Single-Interest
1.85/0.72
10.24/1.95
26.78/4.12
1.31
SeqLen = 64, Multi-Interest
2.08/0.71
11.32/1.94
28.67/3.92
1.29
SeqLen = 32, Multi-Interest
1.72/0.61
9.48/1.76
25.47/3.64
1.29
Homefeed + Search + Advertisements
Mixer Strategy
HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR
Sorted by Timestamp
2.10/0.53
10.55/1.90
21.44/2.20
0.65
Naive Combination
4.28/1.22
17.60/3.06
41.90/4.85
1.85
1D (on position)
4.31/1.23
17.65/3.08
41.95/4.87
1.86
1D (on timestamp)
4.40/1.25
17.80/3.10
42.20/4.90
1.88
2D-Mixing (RED-Rec)
4.72/1.33
18.33/3.22
42.89/4.97
1.94
We conduct comprehensive ab-
lation studies examining key ar-
chitectural components and de-
sign choices (Table 3).
Our
analysis focuses on four crit-
ical
aspects:
(i)
input
se-
quence length effects, (ii) multi-
interest query mechanisms, (iii)
large-scale pretraining benefits,
and (iv) cross-scenario mixing
strategies.
Core
Component
Analy-
sis
Results demonstrate that
longer input sequences, multi-
interest queries, and large-scale
pretraining
all
contribute
to
improved
recommendation
metrics.
The
multi-interest
querying
mechanism
proves
particularly valuable, enabling
the model to capture diverse
facets of user preferences across different scenarios. Large-scale pretraining provides substantial
performance gains, highlighting the importance of leveraging extensive behavioral data.
Cross-Scenario Mixing Strategies
For cross-scenario settings, our 2D dense mixing policy
achieves the strongest performance compared to alternative fusion methods. This validates our ap-
proach of integrating both positional and temporal information for effective signal combination,
addressing behavioral imbalance while preserving informative signals from all scenarios.
5.5
SCALING ANALYSIS
To balance model accuracy with deployment efficiency, we investigate scaling laws across different
model sizes. We train models from both LLaMA (Touvron et al., 2023) (0.5B-7B) and Qwen (Yang
et al., 2025) (0.5B-7B) families on identical token volumes and evaluate on our test set.
Figure 5b presents Hit Rate performance and corresponding serving throughput (Sample per Sec-
ond (SPS)) for the Homefeed+Search+Advertisements scenario. Results show consistent HR im-
provements with increased model size up to 7B parameters across both families, indicating potential
scaling benefits. However, larger models significantly reduce serving throughput, creating practical
deployment constraints. The 1.5B Qwen-2.5 model represents an optimal balance between perfor-
mance and efficiency for production deployment.
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
(a) HR and NDCG for different scenarios (with gains
highlighted).
(b) Scaling law: HR vs. training steps for LLaMA and
Qwen models.
Figure 5: Cross-scenario benefits and model scaling analysis. (a) Performance improvements from cross-
scenario modeling using RED-Rec, where different colored lines represent various scenario combinations
(homefeed, ads, search) and the shaded regions highlight the performance gains achieved through unified
cross-scenario learning compared to single-scenario baselines. (b) Scaling laws for model size versus Hit
Rate performance, showing consistent improvements with increased parameter count across both LLaMA and
Qwen model families, with corresponding serving throughput (samples per second) trade-offs indicated by the
secondary axis.
5.6
ONLINE DEPLOYMENT
We validate RED-Rec through online A/B testing in the recall stage of Xiaohongshu’s advertising
recommendation system. The experiment uses balanced traffic allocation (10% treatment vs. 10%
control) over one week, evaluating performance against the entire item catalog comprising items
distributed within the past two months (about 1.1 billion items). Recall results from our method are
incorporated as an additional recall channel. The deployment operates in near real time: for each
user, we gather and truncate their most recent N interactions, perform inference to generate a user-
side embedding, and retrieve relevant items by matching it with precomputed item-side embeddings,
thus enabling timely and effective recall.
RED-Rec achieves significant improvements: 0.8864% increase in total ADVV and 0.3401% boost
in overall Feed Ad Spend (Cost). These gains are particularly noteworthy given the platform’s scale.
Notably, over 90% of the items selected during the initial candidate generation phase are uniquely
contributed by this recall path, demonstrating that our approach provides significant incremental
recommendations. The significant gains in advertising scenarios further validate our offline find-
ings regarding cross-scenario knowledge transfer from homefeed patterns. Encouraged by these
promising results, we deployed RED-Rec platform-wide, now serving approximately 160 million
daily users. This deployment demonstrates the practical viability of LLM-based cross-scenario rec-
ommendation at industrial scale, offering considerable business value while maintaining acceptable
serving performance.
6
CONCLUSION
We present RED-Rec, a unified hierarchical LLM-based framework that addresses cross-scenario
sequential recommendation at an industrial scale. Our key contributions include: (i) a two-tower
architecture integrating multi-modal content understanding with cross-scenario behavioral model-
ing, (ii) scenario-aware mixing and multi-interest querying mechanisms that capture diverse user
preferences, and (iii) comprehensive validation demonstrating substantial gains in both offline ex-
periments and production deployment. Our comprehensive empirical evaluations, encompassing
experiments on both offline multi-scenario dataset and large-scale real-world deployment, consis-
tently demonstrate substantial gains over strong baselines in both offline and production settings.
Our work demonstrates that unified user interest modeling across behavioral contexts is both techni-
cally feasible at scale and essential for coherent, user-centric recommendations. By bridging diverse
interaction patterns, RED-Rec enables more seamless personalized content discovery and offers new
insights for cross-domain recommendation research and industrial applications.
10
Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
ETHICS STATEMENT
The data utilized in our model training and the constructed dataset have been fully anonymized to
protect user privacy. The dataset contains only interactions with publicly accessible content and
excludes all personally identifiable information. All data collection and processing procedures ad-
here to relevant privacy regulations and platform policies. We acknowledge that recommendation
systems can potentially introduce algorithmic bias and filter bubbles, and encourage practitioners to
implement appropriate fairness monitoring and mitigation strategies when deploying such systems.
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Cross-Scenario Unified Modeling of User Interests at Billion Scale
A
TERMINOLOGY
We would like to first offer additional explanations for specific terminology used throughout the
paper in order to facilitate understanding for non-expert readers:
Homefeed
refers to the main feed or landing page displayed to a user when they open a content
platform or app. It typically consists of a personalized selection of items (such as posts, products,
videos, etc.) recommended to the user based on their preferences and past behavior.
Internal Flow
denotes the content consumption pattern within the single-column sliding or swip-
ing through content (e.g., images, videos, or articles). Users engage with recommendations directly
within this detailed view by navigating between related items or sliding to the next recommended
content.
External Flow
refers to the content consumption flow that occurs on the main feed of the platform,
where users browse the list of recommended items presented to them upon opening the app. This
process typically involves users scrolling vertically through the two-column page.
Scenarios
refer to distinct user interaction environments or channels within the platform, each
characterized by unique user intents and behavioral patterns. In this paper, we focus on three core
scenarios: homefeed, advertisements, and search. The homefeed scenario represents the primary
personalized feed where users consume a diverse assortment of recommended content. The ad-
vertisement scenario corresponds to user engagement with sponsored or promotional content dis-
tributed throughout various parts of the platform. Although advertisement content can appear within
the homefeed, we treat it as a separate scenario because it represents a different source and serves
distinct business objectives. The search scenario involves users actively retrieving information or
content by submitting queries.
last-n
refers to the most recent ‘n’ items a user has interacted with on the platform. For example,
‘last10’ indicates the user’s last 10 consumed items. This concept is commonly used to capture and
analyze a user’s most current interests or activity history.
Engage
represents user interactions with content, such as clicks, likes, comments, shares, or dwell
time. Engagement metrics are used to measure how users interact with recommended items and to
assess the effectiveness of recommender systems.
B
FURTHER EXPERIMENTS
We further test RED-Rec on Amazon Books Reviews (McAuley et al., 2015), a widely used subset
in recommender system research datasets, which was sampled from the Amazon Review dataset. In
the Books subset, each review typically contains fields such as reviewer ID, item (book) ID, rating
(1-5 stars), review text, timestamp, and sometimes additional metadata (e.g., book title). We test and
compare RED-Rec in Table A1.
Table A1: A comparison of the Amazon Books dataset.
Baselines Ò
HR/NDCG10
HR/NDCG50
HR/NDCG200
SASRec
3.06/1.64
7.54/2.60
14.31/3.62
MoRec(bert)
3.21/1.82
8.21/2.33
18.29/3.71
HSTU-1B
4.78/2.62
10.82/3.93
19.08/5.17
DLRM-v3-1B
6.22/2.88
12.74/5.12
23.12/5.29
HLLM-1B
9.28/5.65
17.34/7.41
27.22/8.89
RED-Rec-1.1B-LLaMA
9.46/5.49
18.63/7.03
29.88/8.45
RED-Rec-1.5B-Qwen2.5
9.98/5.88
19.98/7.88
32.62/8.78
A1
Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
C
RED-MMU DATASET
C.1
DATASET STATISTICS
We compare our training dataset with other existing datasests or benchmarks from UGC platforms
in Table A2.
Table A2: A brief comparison of public-released datasets and the training dataset RED-Rec used.
Property
Amazon
JD Search
KuaiSAR
Qilin
RED-Rec(training)
Users
192.4k
173.8k
25.8k
15.5k
1.0m
Items
63.0k
12.9m
6.9m
2.0m
300.6m
Queries
3.2k
171.7k
453.7k
571.9k
search items
Actions
1.7m
26.7m
19.7m
2.5m
683.2m
Content
text/image
text
text/video
text/image/video
text/image/video
Scenario
Rec
Search
Search+Rec
Search+Rec
Search+Rec+Ads
An item in the training data, and also in the proposed RED-MMU dataset is like:
Listing 1: Example of an item in the training dataaset.
{
” u s e r i d ” :
” xxxx ” ,
” data ” :
{
” h o m e f e e d i t e m l a s t n ” :
[
{
” d u r a t i o n ” :
28 ,
” i s c l i c k ” :
1 ,
” i s c l i c k p r o f i l e ” :
0 ,
” i s c o l l e c t ” :
0 ,
” is comment ” :
0 ,
” i s f o l l o w ” :
0 ,
” i s h i d e ” :
0 ,
” i s l i k e ” :
0 ,
” i s n n s ” :
0 ,
” i s p a g e t i m e ” :
1 ,
” is read comment ” :
1 ,
” i s s h a r e ” :
0 ,
” i s v i d e o e n d ” :
0 ,
” i t e m i d ” :
”684 a48440000000023014319 ” ,
” page key ” :
0 ,
” timestamp ” :
1749771247 ,
” type ” :
” note ”
} ,
{
” d u r a t i o n ” :
17 ,
” i s c l i c k ” :
1 ,
” i s c l i c k p r o f i l e ” :
0 ,
” i s c o l l e c t ” :
0 ,
. . .
” i s p a g e t i m e ” :
1 ,
” is read comment ” :
0 ,
” i s s h a r e ” :
0 ,
” i s v i d e o e n d ” :
0 ,
” i t e m i d ” :
”684 aa0720000000021003dbe ” ,
” page key ” :
0 ,
” timestamp ” :
1749732355 ,
” type ” :
” note ”
}
/ /
. . .
] ,
A2
Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
” a d s i t e m l a s t n ” :
[
. . .
] ,
” s e a r c h i t e m l a s t n ” :
[
. . .
]
}
}
where
• user id: Unique identifier for the user, e.g., xxxx.
• data:
– homefeed item lastn: An array of objects representing the last n items from the
user’s home feed. Each object contains:
* duration: Viewing duration (in seconds).
* is click: Whether the item was clicked (1) or not (0).
* is click profile: Whether the user’s profile was clicked (1 or 0).
* is collect: Whether the item was collected or saved (1 or 0).
* is comment: Whether the item was commented on (1 or 0).
* is follow: Whether the user followed from this item (1 or 0).
* is hide: Whether the item was hidden (1 or 0).
* is like: Whether the item was liked (1 or 0).
* is message: Whether the author of the message was messaged (1 or 0).
* is pagetime: Whether the page time event was triggered (1 or 0).
* is read comment: Whether comments were read (1 or 0).
* is share: Whether the item was shared (1 or 0).
* is videoend: Whether a video was watched until the end (1 or 0).
* item id: Identifier for the content item.
* page key: Page identifier.
* timestamp: Timestamp of the interaction.
* type: Type of item, e.g., note.
– ads item lastn:
Array of the last n interacted advertisement items (item id,
duration, etc.).
– search item lastn: Array of the last n search items with similar structure.
C.2
PRIVACY AND VALIDATION
User privacy is strictly protected in our dataset by excluding all personal or sensitive user information
beyond anonymized behavior sequences. User identifiers are securely hashed to prevent any possi-
bility of re-identification, and all content items featured in the dataset are publicly available, with no
private materials included. Furthermore, only essential behavioral signals required for recommen-
dation research are retained, while potentially identifying metadata such as device information and
location is omitted. Engagement timestamps are also consistently biased to prevent reconstruction of
individual timelines. Together, these measures ensure the dataset enables recommendation research
without compromising user confidentiality or privacy.
We focus exclusively on active platform users who demonstrate substantial engagement patterns:
users must have at least 30 valid clicks in the homefeed scenario and 5 valid clicks in the advertise-
ment scenario, where a click is considered valid only if the associated viewing duration exceeds 5
seconds.
D
METRICS
In this work, we focus on three widely adopted metrics: Hit Ratio (HR), Normalized Discounted
Cumulative Gain (NDCG), and Mean Reciprocal Rank (MRR). In recommender systems and in-
formation retrieval, model performance is typically assessed by ranking-based evaluation metrics
that reflect both the accuracy and the ordering of recommendations. These metrics are evaluated at
various ranking cutoffs K (e.g., K “ 10, 100, 1000) to provide a comprehensive view of retrieval
quality across different user engagement depths.
A3
Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
Hit Ratio (HR)
Hit Ratio (HR@K) measures the proportion of test cases in which at least one
relevant item, usually the ground-truth item, is found within the top-K positions of the ranked
recommendation list. Formally, for a set of N users (or queries), it is defined as:
HR@K “ 1
N
N
ÿ
i“1
Ipranki ď Kq,
(A1)
where ranki denotes the position (starting from 1) at which the ground-truth item for the i-th user
occurs in the predicted ranking, and Ip¨q is the indicator function. HR is equivalent to recall@K in
the case of a single relevant item per query.
HR@K is intuitive and interpretable, indicating the likelihood that a user’s desired item appears
among the top-K recommendations. However, it does not reward higher placements within the
top-K and disregards the relative ranking among recommended items.
Normalized Discounted Cumulative Gain (NDCG)
Normalized Discounted Cumulative Gain
(NDCG@K) extends HR@K by accounting for the position of relevant items, rewarding items that
are ranked higher in the recommended list. For each test case, DCG is computed as:
DCG@K “
K
ÿ
j“1
relij
log2pj ` 1q,
(A2)
where relij is the relevance label (typically 1 for the ground-truth item and 0 otherwise) for the j-th
item in the ranked list for user i. The DCG is then normalized by the ideal DCG (IDCG), i.e., the
maximum possible DCG for that user, to yield:
NDCG@K “ 1
N
N
ÿ
i“1
DCGi@K
IDCGi@K
(A3)
NDCG@K captures both the relevance and ranking quality, penalizing relevant items that appear
lower in the ranking. It is especially useful in scenarios with multiple relevant items per user or
graded relevance.
Mean Reciprocal Rank (MRR)
Mean Reciprocal Rank (MRR@K) evaluates how highly the first
relevant item is ranked, and is defined as:
MRR@K “ 1
N
N
ÿ
i“1
1
ranki
,
(A4)
where ranki is the position of the first relevant item in the recommended list for user i, and set to
infinity (i.e., reciprocal rank is 0) if no relevant items are found in the top-K. MRR@K empha-
sizes early precision, heavily rewarding algorithms that surface the relevant item at or near the top.
Its sensitivity to the first relevant item’s position makes it particularly apt for settings prioritizing
immediate relevance (e.g., question answering, search).
Evaluation Protocols and Cutoff Values
In our work, all metrics above are computed at differ-
ent cutoff values K to approximate various user scenarios (e.g., users interacting with the top 10
or top 100 items). These are denoted as HR@K, NDCG@K, and MRR@K, for various K (e.g.,
K “ 10, 100, 1000). For interpretability and easier comparison, MRR is often multiplied by 100 and
reported as MRR˚100. These metrics are computed under a leave-one-out or leave-many-out eval-
uation: for each user, one or more ground-truth relevant items are held out (used as positives), and
the ranking is judged over a candidate pool comprising these positives and many sampled negatives.
E
IMPLEMENTATION DETAILS
We provide additional implementation details of the proposed RED-Rec.
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
E.1
ITEM ENCODER
The item encoder is designed to construct robust content representations, leveraging a pretrained
LLM as its foundation. Textual information related to each item—including titles and descriptions—
is concatenated, tokenized, and prepended with a designated special token to sharpen the represen-
tation focus. This sequence is then passed through the LLM encoder, producing dense semantic
embeddings for each item. Specifically, we extract the embedding corresponding to the special to-
ken. The resulting embedding’s dimension matches the model’s hidden size; for instance, 1536 for
LLaMA2-1.3B and 3584 for Qwen-7B.
For multimodal input, there are essentially two primary approaches. The first involves utilizing an
individual vision encoder, such as ViT(Dosovitskiy et al., 2020), like LLaVA(Liu et al., 2023), to
extract visual tokens, which are then projected into the language embedding space. The second ap-
proach directly leverages vision-language models (VLMs) such as Qwen-VL, which jointly process
visual and textual inputs within a unified architecture. In our work, we primarily adopt the first
approach based on considerations of model size and efficiency for online serving.
E.2
USER ENCODER
User representation learning is managed via hierarchical interest modeling over long interaction
histories. User interaction sequences are first encoded using the item encoder, resulting in contex-
tualized item embeddings. These are then organized and refined by the proposed mixer module that
captures temporal and sequential dependencies. The enhanced representations are subsequently fed
into a disentangled multi-interest learning module, which extends beyond conventional single-vector
user profiles by learning multiple independent embeddings, each attending to a distinct facet of user
intent.
Training supervision extends past traditional next-item prediction, encompassing all interactions
within a lookahead window to better reflect realistic browsing patterns. To achieve this, we apply
cosine similarity clustering to partition target items based on behavioral signals, followed by the
Hungarian algorithm matching to associate each cluster centroid with its corresponding interest
vector. A contrastive loss function drives the specialization of each embedding, ensuring broad
coverage and effective disambiguation of diverse user preferences across multiple interest groups.
Complete implementation details are available in our supplementary code repository.
To model user interests in a disentangled manner, we introduce learnable queries that capture re-
fined, distinct interests according to three key principles: sufficient supervision for each query,
minimal overlap in interest coverage, and coherent optimization directions. Given refined inter-
est embeddings tr1, . . . , rsu and positive samples tt1, . . . , twu from the target window, we cluster
the positive samples into s groups using cosine similarity and then match cluster centroids to interest
embeddings via the Hungarian algorithm to maximize pairwise similarity. The contrastive loss is
applied only to these matched pairs:
Ltotal “ 1
w
w
ÿ
i“1
sÿ
j“1
Lctrpti, rjq ¨ Πpi, jq,
(A5)
where Πpi, jq “ 1 if the cluster of ti is matched with rj, and 0 otherwise. The contrastive loss LNCE
is defined as:
LNCEpt, rq “ ´ log
esimpt,rq{τ
esimpt,rq{τ ` řm
i“1 esimpr,eiq{τ ,
(A6)
where m is the number of negative samples, ei is the ith negative sample embedding, and sim
denotes cosine similarity.
This design enables adaptive learning: queries naturally specialize for users with diverse interests
and converge for users whose preferences are more focused.
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
Table A3: Ablation results for different combinations of item and user LLMs and training strategies.
Scenario
Configuration
HR/NDCG10
HR/NDCG100
HR/NDCG1k
MRR˚100
Homefeed
RED-Rec (2 * Qwen)
2.31/0.68
12.59/1.88
31.94/3.86
1.27
Item LLM from scratch
0.00/0.00
0.00/0.00
0.03/0.01
0.00
User LLM from scratch
0.00/0.00
0.03/0.01
1.32/0.21
0.01
Item LLM frozen
1.27/0.36
5.51/0.77
11.37/1.02
0.37
User LLM frozen
1.78/0.44
10.47/1.02
23.06/1.48
1.01
Homefeed + Ads
RED-Rec (2 * Qwen)
4.36/1.31
18.32/3.27
42.61/5.02
2.11
Item LLM from scratch
0.00/0.00
0.00/0.00
0.08/0.04
0.01
User LLM from scratch
0.00/0.00
0.00/0.00
1.01/0.07
0.03
Item LLM frozen
1.49/0.41
9.49/1.31
19.29/1.52
0.76
User LLM frozen
2.57/1.01
13.72/1.98
29.72/1.88
3.28
Homefeed + Ads
RED-Rec (2 * Qwen)
4.36/1.31
18.32/3.27
42.61/5.02
2.11
RED-Rec-CoT (2 * Qwen)
4.46/1.35
18.78/3.60
44.61/5.01
2.15
F
ADDITIONAL EXPERIMENTS
F.1
PRETRAINING VALIDATION
Our first set of experiments investigates the effect of varying the backbone LLMs for the item and
user encoders. Specifically, we explore the following configurations: (i) using different pretrained
LLMs for item and user encoders, (ii) training one or both encoders from scratch instead of initializ-
ing from a pretrained model, and (iii) freezing the item encoder during training. The detailed results
are summarized in Table A3.
Across all settings, we observe that using exactly the same pretrained LLM for both item and user
encoders and fine-tuning them jointly yields the best performance. In contrast, utilizing mismatched
encoders, initializing from scratch, or freezing either encoder all result in significant drops in overall
accuracy. This suggests that consistent representation spaces and co-adaptation between the two
encoders are crucial for optimal model performance.
F.2
COT VALIDATION
We explore explainable recommendations based on CoT-based (Wei et al., 2022) explanations for
the input layer in a cross-scenario setting. In this experiment, we introduce a Chain-of-Thought
(CoT) auxiliary loss: beyond learning discriminative user and item encoders, we encourage explain-
able cross-scenario reasoning by forcing the user model to generate natural language rationales for
each action:
LCoT “ ´
ÿ
uPU
|Su|
ÿ
t“1
Lt
ÿ
ℓ“1
log pϕprt,ℓ| rtăℓ, zu,tq,
(A7)
where pϕpq denotes the probability, computed by a learnable language model head parameterized
by ϕ, of generating the ℓ-th token rt,ℓof the rationale conditioned on the previous tokens rtăℓand
the contextualized user embedding zu,t at interaction t. The overall training loss is then Ltotal “
LNCE ` λCoTLCoT.
We use GPT 4.1 to generate CoT explanations. An example of the generated CoT explanation is
like:
The user browsed multiple articles related to Switzerland on the homepage, such as “Do you dare
to guess how many days of sunshine in Switzerland?” and “What to wear for a trip to Switzer-
land next week?” This indicates a clear interest in Switzerland. While previously recommended
advertisements included those related to travel, they were not specifically targeted at Switzerland.
Therefore, we recommend to the user the targeted ad “Personal tested and useful! The ultimate
transportation ticket map tool for traveling in Switzerland!”, as well as other advertisements related
to traveling in Switzerland, such as “Countdown to opening! The four legendary theme parks of
Fiesch First Mountain” and “Interlaken sledding premium tips — Save 400 RMB instantly.”
The CoT explanation module is particularly well-suited to the cross-scenario recommendation set-
ting. By generating step-by-step rationales that account for user behaviors across different scenarios
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Xu, et al.
Cross-Scenario Unified Modeling of User Interests at Billion Scale
or domains, the model can provide contextually accurate and human-understandable justifications
for its recommendations. This improves both transparency and user trust, crucial for scenario-aware
systems. However, we observe that applying the CoT-based approach to large-scale datasets intro-
duces significant challenges. The requirement to generate context-dependent rationales for every
user interaction leads to substantially increased computational and memory costs. Given these lim-
itations, we restrict our experiments to small-scale testing. The detailed results are summarized in
Table A3. Including CoT data in training has led to certain improvements, but it does not outperform
the pretrained model.
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Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale CROSS-SCENARIO UNIFIED MODELING OF USER INTERESTS AT BILLION SCALE Manjie Xu 1,3, ̊, Xin Jia 3, ̊, Cheng Chen 3, ̊, Jingyi Zhou 2,3, Chi Zhang 1, , Yongji Wu 3, Zejian Wang 3, Kai Zuo 3,:, Yibo Chen 3,:, Xu Tang 3,:, Yao Hu 3, , Yixin Zhu 1, ̊ equal contribution : project lead corresponding author 1 Peking University 2 Fudan University 3 Xiaohongshu Inc. ABSTRACT User interests on User-Generated Content (UGC) platforms are inherently diverse, manifesting through complex behavioral patterns across heterogeneous scenarios such as search, feed browsing, and content discovery. Traditional recommendation systems operate in isolated scenarios, optimizing business metrics within narrow contexts while neglecting valuable cross-scenario behavioral signals. This fragmented approach struggles to integrate advanced techniques like LLMs at billion-scale deployments, ultimately limiting the ability to capture holistic user interests across platform touchpoints. We introduce RED-Rec, an LLM-enhanced hierarchical Recommender Engine for Diversified scenarios, tailored for industrylevel UGC recommendation systems. RED-Rec unifies user interest representations by aggregating and synthesizing actions from multiple behavioral contexts, enabling comprehensive item and user modeling. The framework features an LLM-powered architecture that delivers nuanced, multifaceted representations while maintaining deployment efficiency. A novel scenario-aware dense mixing and querying policy effectively fuses diverse behavioral signals to capture crossscenario user intent patterns and express fine-grained, context-specific preferences during serving. We validate RED-Rec through online A/B testing on hundreds of millions of users in Xiaohongshu, demonstrating substantial performance gains in content recommendation and advertisement targeting tasks. We also introduce a million-scale sequential recommendation dataset, RED-MMU, for offline evaluation. Our work advances unified user modeling, unlocking deeper personalization and fostering more meaningful user engagement in large-scale UGC platforms. 1 INTRODUCTION Modern User-Generated Content (UGC) platforms have evolved into complex multi-scenario ecosystems where users engage through diverse behavioral contexts-browsing personalized feeds, conducting topical searches, discovering content creators, and responding to targeted advertisements. Each interaction scenario captures distinct yet complementary aspects of user intent: search queries reveal explicit informational needs, feed engagement demonstrates implicit content preferences, and advertisement clicks indicate commercial interests (Covington et al., 2016; Hidasi et al., 2015). Crucially, users exhibit remarkably consistent underlying interests across these diverse behavioral contexts. A user passionate about sustainable living may search for "eco-friendly packaging," engage with environmental advocacy posts in their feed, and click advertisements for solar panels-each interaction revealing the same core interest through different behavioral lenses. This consistency suggests that user interests are inherently multi-dimensional, manifesting through intertwined behavioral trajectories that span multiple scenarios (Xia et al., 2020; Zhu et al., 2022). Despite this behavioral richness, production recommendation systems typically operate as independent silos, with separate models independently optimized for specific business objectives such as Click-Through Rate (CTR) in feeds and Advertiser Value (ADVV) in advertisements (Zhang et al., 2019; Chapelle et al., 2014). This siloed design traps systems in local optima and creates several critical limitations. First, it fragments user understanding by restricting each model to narrow behavioral contexts, preventing holistic interest modeling. Second, it produces inconsistent user 1 16 Oct 2025 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale User Lastn User xxxx ... ... ... Scenario 1: Homefeed Scenario 2: Search User LLM Item LLM ... Current Time ... Multi-scenario Interests ... ... ... target next window ... scene-aware queries ... scene-aware embed ... querying User Lastn User Behaviors User Interests RED-Rec Inference serving Sideinfo Online Metrics (ctr/advv/...) mixer offline test Offline Metrics (topk/ndcg/...) User xxxx (latent representation) User Engage: click/like/share ... Scenario 3: Advertisements Item xxxx (latent representation) Multiinterest Item xxxx Contribute to Figure 1: From fragmented signals to unified understanding. Users express consistent interests across diverse scenarios (left), generating rich behavioral sequences that span homefeed browsing, search queries, and ad interactions (middle). RED-Rec synthesizes these cross-scenario signals using LLM-powered hierarchical modeling to generate comprehensive user representations for context-aware recommendations (right). Vector graphics; zoom for details. experiences when independent systems infer divergent preferences from the same user. Most importantly, it underutilizes valuable cross-scenario signals, limiting knowledge transfer across tasks and weakening performance for users with sparse activity in certain scenarios (Xia et al., 2020; Zhu et al., 2022). Consider our sustainable living enthusiast: traditional systems would treat their search behavior, feed engagement, and ad responses as unrelated signals, failing to synthesize these coherent signals into unified user interest and intent. We are motivated by the observation that users exhibit consistent interest patterns across diverse scenarios, and that modeling these patterns holistically can significantly enhance recommendation quality. While some cross-scenario modeling approaches exist (Zhang et al., 2023; Bao et al., 2023), they typically require extensive manual feature engineering and struggle with scalability and robustness in production environments. Recent advances make this vision increasingly feasible: Large Language Models (LLMs) have transformed semantic understanding of user behaviors and content (Wang et al., 2024b), while advanced sequence modeling techniques effectively capture complex temporal dynamics and cross-scenario dependencies (Sun et al., 2019). Meanwhile, modern UGC platforms generate massive cross-scenario behavioral logs (McAuley et al., 2015; Harper & Konstan, 2015; Gao et al., 2022), creating unprecedented opportunities for unified modeling at scale. However, realizing this vision presents significant challenges: heterogeneity in action schemas, temporal dynamics, and semantics across scenarios; severe activity imbalances where users may have thousands of feed interactions but only dozens of searches; large-scale training and serving with strict latency and throughput constraints requiring sub-millisecond response times; and reconciling differing optimization objectives within a single architecture. While recent work explores mixtures of multi-source signals (Ma et al., 2022; Zhang et al., 2022a; Liu et al., 2024; Yang et al., 2024), truly end-to-end unified modeling for industrial deployments remains underexplored. We introduce Recommender Engine for Diversified scenarios (RED-Rec), an LLM-enhanced hierarchical sequential recommendation framework tailored for billion-scale UGC platforms. RED-Rec unifies interest modeling across heterogeneous contexts. First, we employ LLM-powered user and item encoders within a hierarchical two-tower structure that enables rich semantic representations while preserving efficiency for large-scale retrieval. Second, we introduce a novel 2D dense mixing policy that fuses cross-scenario behavioral signals along temporal and scenario axes to capture cross-scenario dependencies, coupled with multi-interest, scenario-aware queries that express finegrained, context-specific user intents during serving. We train RED-Rec end-to-end on billions of behavioral events drawn from billions of items and over one hundred million users, incorporating system-level optimizations that enable near-real-time online deployment. To enable rigorous evaluation, we also introduce a new cross-scenario sequential dataset curated from anonymized user behavior data on Xiaohongshu, a world-leading UGC platform. The RedNote's Multi-Scenario Multimodal User Behaviors (RED-MMU) dataset spans millions of items and diverse user behaviors across feeds, search, and advertisement contexts, facilitating comprehensive benchmarking of unified and scenario-specific models. In a series of offline experiments, RED-Rec consistently outperforms strong baselines across multiple metrics and scenarios. This 2 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale effectiveness extends to production, as demonstrated by online A/B testing, leading to a comprehensive full-scale rollout that now serves hundreds of millions of daily users on Xiaohongshu. Our main contributions include: (i) a unified, user-centric interest modeling framework that achieves both expressiveness and efficiency for billion-scale cross-scenario recommendation; (ii) a millionscale cross-scenario sequential dataset, RED-MMU, enabling rigorous evaluation of unified modeling approaches; and (iii) empirical validation in both offline and online production environments that establishes the practical viability of unified cross-scenario modeling at unprecedented scale. 2 RELATED WORK Sequential Recommendation Sequential recommendation has evolved from early neural methods like neural collaborative filtering (He et al., 2017) and factorization machines (Rendle, 2010; Guo et al., 2017) to sophisticated sequence models. GRU4Rec (Hidasi et al., 2015) pioneered recurrent architectures for session-based interactions, while Caser (Tang & Wang, 2018) employed convolutional filters for temporal patterns. Transformer-based approaches like SASRec (Kang & McAuley, 2018) and BERT4Rec (Sun et al., 2019) introduced self-attention and bidirectional encoding to capture long-range dependencies in user behavior sequences. Recent advances address the multifaceted nature of user preferences through multi-interest modeling (Li et al., 2019; Cen et al., 2020), graph neural networks (Wang et al., 2020; Zhang et al., 2022b; Yang et al., 2023), and contrastive learning (Zhou et al., 2020; Wei et al., 2023). The emergence of LLMs has opened new frontiers with enhanced user and item representations (Chen et al., 2024a; Hu et al., 2024; Wang et al., 2024b) and generative paradigms (Chen et al., 2024b; Paischer et al., 2024; Deng et al., 2025; Han et al., 2025), though most remain limited to smaller-scale applications. Cross-Scenario Modeling Users maintain consistent interests across different behavioral contexts despite varying interaction patterns (Zang et al., 2022). Early cross-platform studies (Niu et al., 2021; Tan et al., 2021) established that users exhibit similar topical preferences across platforms, motivating disentangled representation learning that separates stable interests from contextdependent behaviors. Modern cross-scenario systems (Tan et al., 2021; Zhao et al., 2023; Li et al., 2024; Chen et al., 2024c; Wu et al., 2025) capture shared interest representations while accommodating scenario-specific patterns. Graph-based approaches (Tan et al., 2021; Cao et al., 2022) model multi-behavioral patterns, while cross-domain (Ma et al., 2022) and multi-domain methods (Zhao et al., 2023; Yang et al., 2024) leverage multi-source user histories to improve performance across scenarios. However, most existing methods struggle with industrial-scale deployment challenges, including heterogeneity, scale, and strict latency requirements. Recent foundation model approaches (Wang et al., 2024a; Shen et al., 2024) show promise but lack validation at billion-scale scenarios. Our work addresses these limitations through an LLM-enhanced framework designed for industrial deployment with comprehensive online validation. 3 THE RED-MMU DATASET Existing open-source sequential recommendation datasets suffer from major limitations in scope and diversity. Traditional datasets focus on isolated scenarios with singular interaction types such as ratings, clicks, or purchases (Ben-Shimon et al., 2015; Harper & Konstan, 2015; Ni et al., 2019; Zhu et al., 2018), failing to capture the cross-scenario nature of modern UGC platforms. Even recent datasets like KuaiRand (Gao et al., 2022) and Qilin (Chen et al., 2025) that begin to characterize UGC environments adopt fragmented approaches that underrepresent the complex interplay between scenarios and only partially reflect holistic user interest evolution. To address these limitations, we introduce a cross-scenario sequential recommendation dataset, RedNote's Multi-Scenario Multimodal User Behaviors (RED-MMU), derived from anonymized user data spanning billions of interactions on a major UGC platform. Details on the protection of user privacy by excluding personal information, hashing user identifiers, and retaining only essential behavioral signals in the RED-MMU dataset are described in Section C.2. Our dataset features the following three key characteristics: 3 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale Diverse Behavioral Contexts The RED-MMU dataset encompasses comprehensive real-world interaction scenarios, including homefeed browsing, search-driven exploration, and advertisement engagement. This temporally aligned diversity enables robust analysis of user behavior across distinct yet interconnected scenarios within a unified platform ecosystem. User Lastn in Cross Scenarios Interactions (a) Scenario 1: Homefeed at time t-5 click t-5 click share follow comment timestamp & location like & collect click engage tags t-4 click after search engage infeed preview (need click) instream flow (scroll down) t-2 commercialization notes advertising (goods, app promotion, services, etc.) purchase page (b) Scenario 2: Search at time t-4 (c) Scenario 3: Advertisements at time t-2 Figure 2: Cross-scenario user interactions in the RED-MMU dataset. Real platform interface examples showing user behavior progression across (a) homefeed content consumption, (b) search-based exploration, and (c) advertisement engagement. Each scenario captures diverse interaction types, including clicks, shares, comments, and purchases, illustrating the interconnected nature of crossscenario interactions on UGC platforms. Rich Engagement Patterns The RED-MMU dataset captures both explicit positive engagements (clicks, likes, collections, shares) and negative signals, along with view duration for each interaction. This provides a nuanced and holistic depiction of user preferences and attention patterns beyond simple binary feedback. Industrial-Scale Coverage The dataset includes billions of items and over one hundred million users' engagement records, surpassing existing datasets in both scale and complexity. Tracking user behavior over extended time periods facilitates the study of long-context interest evolution, behavioral stability, and crossscenario consistency that are typically unavailable in public datasets. Figure 2 shows an example datapoint across multiple scenarios, including homefeed, search, and advertisements. Figure 3 presents overall dataset statistics of RED-MMU, showing details on user engagement analytics. Additional details, including dataset collection and filtering, can be found in Section C. 4 CROSS-SCENARIO USER INTERESTS LEARNING 4.1 TASK FORMULATION We formulate cross-scenario sequential recommendation as learning unified user and item representations from crossscenario interaction data. This formulation captures the complexity of modern recommendation systems where users engage with content across multiple behavioral contexts within the same platform ecosystem. Problem Setup Consider a recommendation system with user set U " tu1, u2, . . . , uNu and universal item space I " ti1, i2, . . . , iMu. The item space encompasses diverse content types, including image-text posts and videos created by regular users or advertisers, which can be recommended through homefeed, discovered via search, or presented as advertisements. Cross-Scenario Interaction Sequences For each user u P U, we observe a chronologically ordered engagement sequence: Su " tpi1, a1, s1, t1q, pi2, a2, s2, t2q, . . . , pi|Su|, a|Su|, s|Su|, t|Su|qu. (1) Each interaction tuple contains: (i) it P I - the interacted item, (ii) at P A - the engagement action, (iii) st P S - the scenario context, and (iv) t - the interaction timestamp. 4 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale We focus on three primary scenarios S " thomefeed, advertisements, searchu that represent the dominant user engagement patterns on UGC platforms. User actions span a rich set of engagements A " tlike, share, comment, follow, messaging, blocku, capturing both positive and negative feedback signals that reflect nuanced user preferences. Learning Objective Our goal is to learn unified embedding functions that capture user preferences and item characteristics across different scenarios. Specifically, we aim to learn (i) a user embedding function: fu : U ˆ Hu Ñ Rd, and (ii) an item embedding function: fi : I Ñ Rd. These functions map users (conditioned on their interaction history Hu) and items to a shared d-dimensional embedding space: u " fupu, Suq, vi " fipIq. (2) The resulting embeddings u, vi P Rd encode cross-scenario user interests and item characteristics, enabling effective recommendation across all scenarios. These representations can be directly applied to recall tasks or serve as features for downstream ranking models. 4.2 HIERARCHICAL LLM-BASED REPRESENTATION LEARNING RED-Rec employs a hierarchical two-tower architecture that learns comprehensive user and item representations across multiple scenarios. The framework consists of three key components: (i) multimodal item encoding that captures content semantics, (ii) sequential user modeling with crossscenario interest fusion, and (iii) scenario-aware querying mechanism for diverse preference capture. Multimodal Item Representation Each item i P I is encoded through a multimodal encoder Eitem that processes both textual and visual content: ei " Eitempxi, vi; θt, θvq P Rd. (3) The textual component xi encompasses title, tags, content description, and OCR-extracted text, processed by a pre-trained language model with parameters θt. Visual content vi is encoded using a ViT (Dosovitskiy et al., 2020) with parameters θv, followed by linear projection to dimension d. This unified representation captures rich semantic information across modalities. Cross-Scenario Sequential Modeling To model user interests across diverse behavioral contexts, we aggregate interactions from three primary scenarios. For user u, the combined interaction sequence is Su " Sh u Y Sa u Y Ss u, where Sh u, Sa u, and Ss u represent homefeed, advertisements, and search respectively. Each interaction incorporates three information dimensions: (i) Content represented by item embeddings Hu " rei1, ei2, . . . , eins P Rnˆd, (ii) Actions encoding engagement behaviors Au " rai1, ai2, . . . , ains as dense embeddings from one-hot vectors representing tcollect, share, message, block, likeu, and (iii) Temporal features with hour-level timestamps hit " OneHotphourptqq P t0, 1u24 converted to dense embeddings. Hence, ˆHu combines all three dimensions, enabling the user encoder to capture temporal patterns and engagement preferences. 2-D Dense Mixing Policy To address behavioral imbalance across scenarios, we introduce a balanced sampling strategy that preserves informative signals from all scenarios: Smixer u " Merge ́ Shomefeed u r ́nh :s, Sadvertisements u r ́na :s, Ssearch u r ́ns :s ̄ . (4) The Mergep ̈q operation chronologically sorts and concatenates recent interactions while maintaining scenario tags. This "2-D dense mixing" filters along both scenario (quota balancing) and temporal (recency) dimensions, ensuring representation of infrequent but valuable user signals. We design 2-D positional encoding for each event j in Smixer u : pj " PEseqpjq ` PEgapp∆tjq, where PEseqpjq captures sequence position and PEgapp∆tjq encodes time gaps with ∆tj " tcurr ́ tj. Scenario-Aware Interest Querying To capture diverse facets of user preferences, we employ learnable query embeddings Q " rq1, q2, . . . , qKs P RKˆd that attend to different interest aspects across scenarios. The scenario-aware representation is computed as: Uquery u " Euser ́" ̃Hur: ́Ws; Q ı ; θu ̄ , (5) where W represents the window size for recent interactions, enabling the model to generate multiple interest-specific representations. 5 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale (a) User Interaction Distribution Channel and Scenario Analysis (b) Top 100 Notes by Channel and Scenario (c) Behavior Distribution Analysis (d) Duration vs Engagement Analysis (Top 1000) (e) 24-Hour Activity Pattern (GMT+8) (f) Content Lifecycle by Channel ✖ Scenario (g) User Behavior by Channel and Scenario (h) User Activity by Channel ✖ Scenario (500 Users) Figure 3: Comprehensive user engagement analytics across scenarios. Multi-faceted analysis of 10k sampled users showing interaction patterns across homefeed, advertisements, and search scenarios. The dashboard presents: (a) user distribution and interaction channels, (b) top content engagement by scenario, (c) behavioral pattern distributions, (d) duration-engagement correlations, (e) temporal activity patterns, (f) content lifecycle analysis, (g-h) cross-scenario user behavior comparisons. Analysis reveals distinct engagement patterns and temporal dynamics across different recommendation contexts. 6 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale Target Items Recall In-batch Negative Examples title ... Scenario 1 Scenario 2 Scenario 3 {title} {tags} {Content} {Ocr} Ocr Model Vision Encoder Item LLM Encoder (Image Tokens) ... Auxiliary Embeddings 1. User Engage 2. Position Encoding 3. ... Target Scenario Sampling Pool User LLM Encoder Dense Mixing tags Content Target Items Hungarian Matching ... ... attention Clustering Scenario-aware Query User Representation Note: First Image Video: Cover Image Ramdom-Sampled Notes Hard Negative Samples Time Dim Scenario Dim Sampling Figure 4: Overall framework of RED-Rec. RED-Rec employs a two-tower hierarchical architecture with multimodal item encoding and cross-scenario user modeling. The item encoder processes textual content (title, tags, OCR) and visual signals through unified embeddings. The user encoder incorporates a 2-D dense mixing policy to balance interactions across homefeed, advertisements, and search scenarios, followed by scenarioaware transformer blocks that capture evolving user interests. During training, positive and negative samples are drawn from a sampling pool to optimize contrastive objectives end-to-end. RED-Rec generates unified representations suitable for cross-scenario recommendation tasks. Training Objective We optimize using Noise Contrastive Estimation (NCE) with temperature scaling: LNCE " ́ ÿ u,t log exp ` τ ̈ cospuu,t, vit`1q ̆ exp ` τ ̈ cospuu,t, vit`1q ̆ ` ř jPN exp pτ ̈ cospuu,t, vjqq, (6) where τ is a learnable temperature parameter, uu,t represents user u's interest at time t, and N contains negative samples. Additionally, we incorporate window-based contrastive loss for recent interactions to capture evolving preferences. Similar techniques have also been employed in HyMiRec (Zhou et al., 2025), where hybrid multi-interest learning is applied for enhanced user representation and retrieval. Rather than a single, biased embedding, we encourage the User LLM model to capture diverse user intents from multiple perspectives. 4.3 EVALUATION PROTOCOL We evaluate learned representations on recall tasks using temporal data splitting. For each user u, interactions are divided at randomly sampled cutoff tcut, creating input sequence Sinput u " tpi, a, s, tq P Su : t ă tcutu and target set Gu " titcut, itcut`1, itcut`2u. The candidate pool C combines random platform samples with ground truth targets. User embeddings u computed from Sinput u generate similarity scores scorepu, iq " cospu, viq for ranking. We report Hit Rate (HR@K), Normalized Discounted Cumulative Gain (NDCG@K), and Mean Reciprocal Rank (MRR) for K P t10, 50, 100, 1000u. 5 EXPERIMENTS 5.1 EXPERIMENTAL SETUP Dataset and Configuration We conduct comprehensive experiments on an industrial dataset comprising 1 million users for training and 10,000 test samples for evaluation. The candidate pool contains approximately 1 million randomly sampled notes to ensure fair comparison across methods. Our default configuration sets window size W " 10, sequence length last n " 128, and employs 3 queries per scenario to capture diverse interest facets. Model Architecture and Training Both item and user encoders are initialized with large language models: either 1.3B-parameter Chinese-LLaMA (Cui et al., 2023) or 1.5B Qwen-2.5 (Yang et al., 2025), while visual encoding utilizes CLIP ViT-B/16. Training is conducted on 8 NVIDIA H100 7 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale Table 1: Single-scenario recommendation performance comparison. Performance evaluation of RED-Rec variants against established baselines (SASRec, MoRec, HSTU, HLLM, DLRM-v3) on homefeed and advertisement recommendation tasks. RED-Rec variants include symbol-based (RED-Rec-symbol) and multimodal (RED-Rec-mm) versions, with pre-trained variants (RED-Rec-pt) leveraging large-scale data. Higher scores indicate better performance across all metrics. Homefeed Advertisements Baselines Ò HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR ̊100 HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR ̊100 SASRec 1.76/0.97 12.32/1.79 32.01/4.04 1.01 3.26/1.63 14.08/3.71 39.11/5.27 1.57 MoRec 1.78/1.25 12.48/2.23 31.98/4.12 1.21 3.47/1.67 13.98/3.88 38.27/4.89 1.78 HSTU 1.79/1.22 12.72/2.21 31.76/3.69 1.15 3.85/1.70 14.32/3.30 38.20/5.38 1.43 HLLM 1.66/0.62 12.77/1.83 32.52/4.02 1.22 4.21/1.21 14.27/3.37 39.21/4.48 1.39 DLRM-v3 1.63/1.03 11.33/2.01 28.96/3.72 1.13 3.54/1.21 15.27/3.22 35.39/4.27 1.67 RED-Rec 2.31/0.68 12.59/1.88 31.94/3.86 1.27 4.24/1.28 16.44/3.21 40.18/4.61 1.96 RED-Rec-pt 2.90/0.63 14.89/2.02 36.16/4.01 1.30 4.84/1.30 17.66/2.87 42.71/5.21 2.27 RED-Rec-mm 2.35/1.21 14.20/2.27 31.29/3.97 1.29 4.31/1.31 17.22/3.18 41.86/4.66 1.92 RED-Rec-mm-pt 3.23/1.27 15.46/2.21 36.29/4.14 1.38 4.82/1.19 18.21/3.29 42.56/4.98 2.21 GPUs for 3 epochs with batch size 2 and gradient accumulation of 4, requiring approximately 24 hours. Implementation details are provided in Section E. Baselines and Evaluation We compare against established recommendation methods: SASRec (Kang & McAuley, 2018), MoRec (Yuan et al., 2023), HSTU (Zhai et al., 2024), HLLM (Chen et al., 2024a), and DLRM-v3 (Naumov et al., 2019). Our evaluation encompasses both single-scenario (homefeed, advertisements) and cross-scenario (search + homefeed, homefeed + advertisements, all combined) settings. We assess four RED-Rec variants: RED-Rec-symbol, RED-Rec-mm, and their pre-trained versions (RED-Rec-symbol-pt, RED-Rec-mm-pt) trained on large-scale online data. Standard metrics (HR@K, NDCG@K, MRR) are reported across multiple cutoff values. 5.2 SINGLE-SCENARIO PERFORMANCE Table 2: Cross-scenario recommendation performance evaluation. Performance comparison when leveraging crossscenario signals for improved recommendations across different target scenarios. Results demonstrate the effectiveness of RED-Rec in utilizing cross-scenario behavioral signal. Higher scores indicate superior performance. Search + Homefeed (for Homefeed) Baselines Ò HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR ̊100 SASRec 1.73/1.22 12.02/3.21 32.17/4.17 1.52 MoRec 1.79/1.30 13.92/2.99 33.01/3.98 1.53 HSTU 1.79/1.25 12.84/3.28 33.15/4.24 1.55 HLLM 1.69/1.02 13.49/3.18 33.04/4.21 1.58 DLRM-v3 1.64/1.18 11.35/3.02 30.89/3.98 1.48 RED-Rec 2.26/1.32 14.74/3.16 33.29/4.20 1.58 RED-Rec-pt 2.92/1.33 18.26/3.24 38.92/4.23 1.67 Homefeed + Advertisements (for Advertisements) Baselines Ò HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR ̊100 SASRec 3.72/1.24 16.18/3.08 38.94/4.72 1.94 MoRec 3.80/1.30 17.23/2.62 38.29/4.77 1.98 HSTU 3.89/1.28 16.95/3.15 40.12/4.81 2.01 HLLM 3.68/1.19 17.24/3.12 39.76/4.78 1.97 DLRM-v3 3.52/1.21 15.43/2.95 36.87/4.58 1.87 RED-Rec 4.36/1.31 18.32/3.27 42.61/5.02 2.11 RED-Rec-pt 5.18/1.38 18.89/3.21 46.59/5.57 2.38 Homefeed + Search + Advertisements (for Advertisements) Baselines Ò HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR ̊100 SASRec 3.68/1.21 14.29/2.08 38.94/4.72 1.94 MoRec 3.82/1.33 18.27/2.98 38.41/4.66 1.98 HSTU 3.92/1.31 17.21/3.19 40.14/4.81 2.11 HLLM 4.08/1.11 19.92/3.18 43.27/4.91 2.06 DLRM-v3 3.34/1.01 14.08/2.81 35.74/4.36 1.74 RED-Rec 4.72/1.33 18.33/3.22 42.89/4.97 1.94 RED-Rec-pt 5.18/1.35 20.52/3.24 49.17/5.93 2.41 Table 1 presents results for homefeed and advertisement recommendation scenarios. RED-Rec consistently outperforms all baselines in both scenarios, demonstrating the efficacy even in singlescenario settings. The superior performance stems from two key factors: (i) multi-interest user representation learning that captures diverse preference facets, and (ii) advanced LLM-based semantic encoding that provides richer representations than traditional ID-based methods. Compared to SASRec and HSTU, which rely on item ID embeddings, RED-Rec shows substantial improvements, particularly beneficial for cold-start scenarios where semantic understanding is crucial. When compared against HLLM, which shares similar architectural principles, RED-Rec benefits from larger backbone models with enhanced Chinese language capabilities, enabling better alignment with our dataset characteristics and further performance gains. 5.3 CROSS-SCENARIO BENEFITS Cross-scenario evaluation (Table 2) reveals significant performance improvements when integrating information across behavioral contexts. The most pronounced gains occur in two key scenarios: (i) search data enhancing homefeed 8 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale recommendations, and (ii) combined homefeed and search signals improving advertisement performance. Cross-Scenario Information Flow Incorporating cross-scenario signals consistently improves performance across all baselines, with RED-Rec achieving the largest gains due to its effective integration capabilities and advanced user-side LLM reasoning. For homefeed recommendations, access to search behaviors, particularly post-search engagement patterns, substantially increases both HR and NDCG scores. Similarly, advertisement recommendations benefit from combined homefeed and search behaviors, showing the greatest metric improvements across all evaluation criteria. Consistent Improvements These enhancements remain consistent across all cutoff values (K P t10, 50, 100, 1000u), indicating that RED-Rec not only increases the likelihood of relevant item recommendation but also improves their ranking positions. Figure 5 illustrates these cross-scenario benefits, demonstrating substantial performance gains when leveraging complementary signals. 5.4 ABLATION STUDIES Table 3: Ablation study results for RED-Rec architectural components. Top: Core model configuration ablations examining the impact of sequence length (SeqLen = 128 vs. 32) and interest modeling approaches (Multi Interest vs. Single Interest) on recommendation performance. Bottom: Cross-scenario mixing policy ablations evaluating different strategies for combining behavioral signals across homefeed, search, and advertisement scenarios. Comparison includes temporal sampling, scenario exclusion variants, and our proposed 2D Dense Mixing approach. Results demonstrate the effectiveness of longer sequences, multi-interest modeling, and comprehensive cross-scenario signal integration. Homefeed Setting HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR SeqLen = 128, Multi-Interest, pt 2.90/0.63 14.89/2.02 36.16/4.01 1.30 SeqLen = 128, Multi-Interest 2.31/0.68 12.59/1.88 31.94/3.86 1.27 SeqLen = 128, Single-Interest 1.85/0.72 10.24/1.95 26.78/4.12 1.31 SeqLen = 64, Multi-Interest 2.08/0.71 11.32/1.94 28.67/3.92 1.29 SeqLen = 32, Multi-Interest 1.72/0.61 9.48/1.76 25.47/3.64 1.29 Homefeed + Search + Advertisements Mixer Strategy HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR Sorted by Timestamp 2.10/0.53 10.55/1.90 21.44/2.20 0.65 Naive Combination 4.28/1.22 17.60/3.06 41.90/4.85 1.85 1D (on position) 4.31/1.23 17.65/3.08 41.95/4.87 1.86 1D (on timestamp) 4.40/1.25 17.80/3.10 42.20/4.90 1.88 2D-Mixing (RED-Rec) 4.72/1.33 18.33/3.22 42.89/4.97 1.94 We conduct comprehensive ablation studies examining key architectural components and design choices (Table 3). Our analysis focuses on four critical aspects: (i) input sequence length effects, (ii) multiinterest query mechanisms, (iii) large-scale pretraining benefits, and (iv) cross-scenario mixing strategies. Core Component Analysis Results demonstrate that longer input sequences, multiinterest queries, and large-scale pretraining all contribute to improved recommendation metrics. The multi-interest querying mechanism proves particularly valuable, enabling the model to capture diverse facets of user preferences across different scenarios. Large-scale pretraining provides substantial performance gains, highlighting the importance of leveraging extensive behavioral data. Cross-Scenario Mixing Strategies For cross-scenario settings, our 2D dense mixing policy achieves the strongest performance compared to alternative fusion methods. This validates our approach of integrating both positional and temporal information for effective signal combination, addressing behavioral imbalance while preserving informative signals from all scenarios. 5.5 SCALING ANALYSIS To balance model accuracy with deployment efficiency, we investigate scaling laws across different model sizes. We train models from both LLaMA (Touvron et al., 2023) (0.5B-7B) and Qwen (Yang et al., 2025) (0.5B-7B) families on identical token volumes and evaluate on our test set. Figure 5b presents Hit Rate performance and corresponding serving throughput (Sample per Second (SPS)) for the Homefeed+Search+Advertisements scenario. Results show consistent HR improvements with increased model size up to 7B parameters across both families, indicating potential scaling benefits. However, larger models significantly reduce serving throughput, creating practical deployment constraints. The 1.5B Qwen-2.5 model represents an optimal balance between performance and efficiency for production deployment. 9 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale (a) HR and NDCG for different scenarios (with gains highlighted). (b) Scaling law: HR vs. training steps for LLaMA and Qwen models. Figure 5: Cross-scenario benefits and model scaling analysis. (a) Performance improvements from crossscenario modeling using RED-Rec, where different colored lines represent various scenario combinations (homefeed, ads, search) and the shaded regions highlight the performance gains achieved through unified cross-scenario learning compared to single-scenario baselines. (b) Scaling laws for model size versus Hit Rate performance, showing consistent improvements with increased parameter count across both LLaMA and Qwen model families, with corresponding serving throughput (samples per second) trade-offs indicated by the secondary axis. 5.6 ONLINE DEPLOYMENT We validate RED-Rec through online A/B testing in the recall stage of Xiaohongshu's advertising recommendation system. The experiment uses balanced traffic allocation (10% treatment vs. 10% control) over one week, evaluating performance against the entire item catalog comprising items distributed within the past two months (about 1.1 billion items). Recall results from our method are incorporated as an additional recall channel. The deployment operates in near real time: for each user, we gather and truncate their most recent N interactions, perform inference to generate a userside embedding, and retrieve relevant items by matching it with precomputed item-side embeddings, thus enabling timely and effective recall. RED-Rec achieves significant improvements: 0.8864% increase in total ADVV and 0.3401% boost in overall Feed Ad Spend (Cost). These gains are particularly noteworthy given the platform's scale. Notably, over 90% of the items selected during the initial candidate generation phase are uniquely contributed by this recall path, demonstrating that our approach provides significant incremental recommendations. The significant gains in advertising scenarios further validate our offline findings regarding cross-scenario knowledge transfer from homefeed patterns. Encouraged by these promising results, we deployed RED-Rec platform-wide, now serving approximately 160 million daily users. This deployment demonstrates the practical viability of LLM-based cross-scenario recommendation at industrial scale, offering considerable business value while maintaining acceptable serving performance. 6 CONCLUSION We present RED-Rec, a unified hierarchical LLM-based framework that addresses cross-scenario sequential recommendation at an industrial scale. Our key contributions include: (i) a two-tower architecture integrating multi-modal content understanding with cross-scenario behavioral modeling, (ii) scenario-aware mixing and multi-interest querying mechanisms that capture diverse user preferences, and (iii) comprehensive validation demonstrating substantial gains in both offline experiments and production deployment. Our comprehensive empirical evaluations, encompassing experiments on both offline multi-scenario dataset and large-scale real-world deployment, consistently demonstrate substantial gains over strong baselines in both offline and production settings. Our work demonstrates that unified user interest modeling across behavioral contexts is both technically feasible at scale and essential for coherent, user-centric recommendations. By bridging diverse interaction patterns, RED-Rec enables more seamless personalized content discovery and offers new insights for cross-domain recommendation research and industrial applications. 10 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale ETHICS STATEMENT The data utilized in our model training and the constructed dataset have been fully anonymized to protect user privacy. The dataset contains only interactions with publicly accessible content and excludes all personally identifiable information. All data collection and processing procedures adhere to relevant privacy regulations and platform policies. 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In ACM International Conference on Web Search and Data Mining, 2022. 1, 2 14 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale A TERMINOLOGY We would like to first offer additional explanations for specific terminology used throughout the paper in order to facilitate understanding for non-expert readers: Homefeed refers to the main feed or landing page displayed to a user when they open a content platform or app. It typically consists of a personalized selection of items (such as posts, products, videos, etc.) recommended to the user based on their preferences and past behavior. Internal Flow denotes the content consumption pattern within the single-column sliding or swiping through content (e.g., images, videos, or articles). Users engage with recommendations directly within this detailed view by navigating between related items or sliding to the next recommended content. External Flow refers to the content consumption flow that occurs on the main feed of the platform, where users browse the list of recommended items presented to them upon opening the app. This process typically involves users scrolling vertically through the two-column page. Scenarios refer to distinct user interaction environments or channels within the platform, each characterized by unique user intents and behavioral patterns. In this paper, we focus on three core scenarios: homefeed, advertisements, and search. The homefeed scenario represents the primary personalized feed where users consume a diverse assortment of recommended content. The advertisement scenario corresponds to user engagement with sponsored or promotional content distributed throughout various parts of the platform. Although advertisement content can appear within the homefeed, we treat it as a separate scenario because it represents a different source and serves distinct business objectives. The search scenario involves users actively retrieving information or content by submitting queries. last-n refers to the most recent 'n' items a user has interacted with on the platform. For example, 'last10' indicates the user's last 10 consumed items. This concept is commonly used to capture and analyze a user's most current interests or activity history. Engage represents user interactions with content, such as clicks, likes, comments, shares, or dwell time. Engagement metrics are used to measure how users interact with recommended items and to assess the effectiveness of recommender systems. B FURTHER EXPERIMENTS We further test RED-Rec on Amazon Books Reviews (McAuley et al., 2015), a widely used subset in recommender system research datasets, which was sampled from the Amazon Review dataset. In the Books subset, each review typically contains fields such as reviewer ID, item (book) ID, rating (1-5 stars), review text, timestamp, and sometimes additional metadata (e.g., book title). We test and compare RED-Rec in Table A1. Table A1: A comparison of the Amazon Books dataset. Baselines Ò HR/NDCG10 HR/NDCG50 HR/NDCG200 SASRec 3.06/1.64 7.54/2.60 14.31/3.62 MoRec(bert) 3.21/1.82 8.21/2.33 18.29/3.71 HSTU-1B 4.78/2.62 10.82/3.93 19.08/5.17 DLRM-v3-1B 6.22/2.88 12.74/5.12 23.12/5.29 HLLM-1B 9.28/5.65 17.34/7.41 27.22/8.89 RED-Rec-1.1B-LLaMA 9.46/5.49 18.63/7.03 29.88/8.45 RED-Rec-1.5B-Qwen2.5 9.98/5.88 19.98/7.88 32.62/8.78 A1 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale C RED-MMU DATASET C.1 DATASET STATISTICS We compare our training dataset with other existing datasests or benchmarks from UGC platforms in Table A2. Table A2: A brief comparison of public-released datasets and the training dataset RED-Rec used. Property Amazon JD Search KuaiSAR Qilin RED-Rec(training) Users 192.4k 173.8k 25.8k 15.5k 1.0m Items 63.0k 12.9m 6.9m 2.0m 300.6m Queries 3.2k 171.7k 453.7k 571.9k search items Actions 1.7m 26.7m 19.7m 2.5m 683.2m Content text/image text text/video text/image/video text/image/video Scenario Rec Search Search+Rec Search+Rec Search+Rec+Ads An item in the training data, and also in the proposed RED-MMU dataset is like: Listing 1: Example of an item in the training dataaset. { " u s e r i d " : " xxxx " , " data " : { " h o m e f e e d i t e m l a s t n " : [ { " d u r a t i o n " : 28 , " i s c l i c k " : 1 , " i s c l i c k p r o f i l e " : 0 , " i s c o l l e c t " : 0 , " is comment " : 0 , " i s f o l l o w " : 0 , " i s h i d e " : 0 , " i s l i k e " : 0 , " i s n n s " : 0 , " i s p a g e t i m e " : 1 , " is read comment " : 1 , " i s s h a r e " : 0 , " i s v i d e o e n d " : 0 , " i t e m i d " : "684 a48440000000023014319 " , " page key " : 0 , " timestamp " : 1749771247 , " type " : " note " } , { " d u r a t i o n " : 17 , " i s c l i c k " : 1 , " i s c l i c k p r o f i l e " : 0 , " i s c o l l e c t " : 0 , . . . " i s p a g e t i m e " : 1 , " is read comment " : 0 , " i s s h a r e " : 0 , " i s v i d e o e n d " : 0 , " i t e m i d " : "684 aa0720000000021003dbe " , " page key " : 0 , " timestamp " : 1749732355 , " type " : " note " } / / . . . ] , A2 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale " a d s i t e m l a s t n " : [ . . . ] , " s e a r c h i t e m l a s t n " : [ . . . ] } } where • user id: Unique identifier for the user, e.g., xxxx. • data: - homefeed item lastn: An array of objects representing the last n items from the user's home feed. Each object contains: * duration: Viewing duration (in seconds). * is click: Whether the item was clicked (1) or not (0). * is click profile: Whether the user's profile was clicked (1 or 0). * is collect: Whether the item was collected or saved (1 or 0). * is comment: Whether the item was commented on (1 or 0). * is follow: Whether the user followed from this item (1 or 0). * is hide: Whether the item was hidden (1 or 0). * is like: Whether the item was liked (1 or 0). * is message: Whether the author of the message was messaged (1 or 0). * is pagetime: Whether the page time event was triggered (1 or 0). * is read comment: Whether comments were read (1 or 0). * is share: Whether the item was shared (1 or 0). * is videoend: Whether a video was watched until the end (1 or 0). * item id: Identifier for the content item. * page key: Page identifier. * timestamp: Timestamp of the interaction. * type: Type of item, e.g., note. - ads item lastn: Array of the last n interacted advertisement items (item id, duration, etc.). - search item lastn: Array of the last n search items with similar structure. C.2 PRIVACY AND VALIDATION User privacy is strictly protected in our dataset by excluding all personal or sensitive user information beyond anonymized behavior sequences. User identifiers are securely hashed to prevent any possibility of re-identification, and all content items featured in the dataset are publicly available, with no private materials included. Furthermore, only essential behavioral signals required for recommendation research are retained, while potentially identifying metadata such as device information and location is omitted. Engagement timestamps are also consistently biased to prevent reconstruction of individual timelines. Together, these measures ensure the dataset enables recommendation research without compromising user confidentiality or privacy. We focus exclusively on active platform users who demonstrate substantial engagement patterns: users must have at least 30 valid clicks in the homefeed scenario and 5 valid clicks in the advertisement scenario, where a click is considered valid only if the associated viewing duration exceeds 5 seconds. D METRICS In this work, we focus on three widely adopted metrics: Hit Ratio (HR), Normalized Discounted Cumulative Gain (NDCG), and Mean Reciprocal Rank (MRR). In recommender systems and information retrieval, model performance is typically assessed by ranking-based evaluation metrics that reflect both the accuracy and the ordering of recommendations. These metrics are evaluated at various ranking cutoffs K (e.g., K " 10, 100, 1000) to provide a comprehensive view of retrieval quality across different user engagement depths. A3 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale Hit Ratio (HR) Hit Ratio (HR@K) measures the proportion of test cases in which at least one relevant item, usually the ground-truth item, is found within the top-K positions of the ranked recommendation list. Formally, for a set of N users (or queries), it is defined as: HR@K " 1 N N ÿ i"1 Ipranki ď Kq, (A1) where ranki denotes the position (starting from 1) at which the ground-truth item for the i-th user occurs in the predicted ranking, and Ip ̈q is the indicator function. HR is equivalent to recall@K in the case of a single relevant item per query. HR@K is intuitive and interpretable, indicating the likelihood that a user's desired item appears among the top-K recommendations. However, it does not reward higher placements within the top-K and disregards the relative ranking among recommended items. Normalized Discounted Cumulative Gain (NDCG) Normalized Discounted Cumulative Gain (NDCG@K) extends HR@K by accounting for the position of relevant items, rewarding items that are ranked higher in the recommended list. For each test case, DCG is computed as: DCG@K " K ÿ j"1 relij log2pj ` 1q, (A2) where relij is the relevance label (typically 1 for the ground-truth item and 0 otherwise) for the j-th item in the ranked list for user i. The DCG is then normalized by the ideal DCG (IDCG), i.e., the maximum possible DCG for that user, to yield: NDCG@K " 1 N N ÿ i"1 DCGi@K IDCGi@K (A3) NDCG@K captures both the relevance and ranking quality, penalizing relevant items that appear lower in the ranking. It is especially useful in scenarios with multiple relevant items per user or graded relevance. Mean Reciprocal Rank (MRR) Mean Reciprocal Rank (MRR@K) evaluates how highly the first relevant item is ranked, and is defined as: MRR@K " 1 N N ÿ i"1 1 ranki , (A4) where ranki is the position of the first relevant item in the recommended list for user i, and set to infinity (i.e., reciprocal rank is 0) if no relevant items are found in the top-K. MRR@K emphasizes early precision, heavily rewarding algorithms that surface the relevant item at or near the top. Its sensitivity to the first relevant item's position makes it particularly apt for settings prioritizing immediate relevance (e.g., question answering, search). Evaluation Protocols and Cutoff Values In our work, all metrics above are computed at different cutoff values K to approximate various user scenarios (e.g., users interacting with the top 10 or top 100 items). These are denoted as HR@K, NDCG@K, and MRR@K, for various K (e.g., K " 10, 100, 1000). For interpretability and easier comparison, MRR is often multiplied by 100 and reported as MRR ̊100. These metrics are computed under a leave-one-out or leave-many-out evaluation: for each user, one or more ground-truth relevant items are held out (used as positives), and the ranking is judged over a candidate pool comprising these positives and many sampled negatives. E IMPLEMENTATION DETAILS We provide additional implementation details of the proposed RED-Rec. A4 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale E.1 ITEM ENCODER The item encoder is designed to construct robust content representations, leveraging a pretrained LLM as its foundation. Textual information related to each item-including titles and descriptionsis concatenated, tokenized, and prepended with a designated special token to sharpen the representation focus. This sequence is then passed through the LLM encoder, producing dense semantic embeddings for each item. Specifically, we extract the embedding corresponding to the special token. The resulting embedding's dimension matches the model's hidden size; for instance, 1536 for LLaMA2-1.3B and 3584 for Qwen-7B. For multimodal input, there are essentially two primary approaches. The first involves utilizing an individual vision encoder, such as ViT(Dosovitskiy et al., 2020), like LLaVA(Liu et al., 2023), to extract visual tokens, which are then projected into the language embedding space. The second approach directly leverages vision-language models (VLMs) such as Qwen-VL, which jointly process visual and textual inputs within a unified architecture. In our work, we primarily adopt the first approach based on considerations of model size and efficiency for online serving. E.2 USER ENCODER User representation learning is managed via hierarchical interest modeling over long interaction histories. User interaction sequences are first encoded using the item encoder, resulting in contextualized item embeddings. These are then organized and refined by the proposed mixer module that captures temporal and sequential dependencies. The enhanced representations are subsequently fed into a disentangled multi-interest learning module, which extends beyond conventional single-vector user profiles by learning multiple independent embeddings, each attending to a distinct facet of user intent. Training supervision extends past traditional next-item prediction, encompassing all interactions within a lookahead window to better reflect realistic browsing patterns. To achieve this, we apply cosine similarity clustering to partition target items based on behavioral signals, followed by the Hungarian algorithm matching to associate each cluster centroid with its corresponding interest vector. A contrastive loss function drives the specialization of each embedding, ensuring broad coverage and effective disambiguation of diverse user preferences across multiple interest groups. Complete implementation details are available in our supplementary code repository. To model user interests in a disentangled manner, we introduce learnable queries that capture refined, distinct interests according to three key principles: sufficient supervision for each query, minimal overlap in interest coverage, and coherent optimization directions. Given refined interest embeddings tr1, . . . , rsu and positive samples tt1, . . . , twu from the target window, we cluster the positive samples into s groups using cosine similarity and then match cluster centroids to interest embeddings via the Hungarian algorithm to maximize pairwise similarity. The contrastive loss is applied only to these matched pairs: Ltotal " 1 w w ÿ i"1 sÿ j"1 Lctrpti, rjq ̈ Πpi, jq, (A5) where Πpi, jq " 1 if the cluster of ti is matched with rj, and 0 otherwise. The contrastive loss LNCE is defined as: LNCEpt, rq " ́ log esimpt,rq{τ esimpt,rq{τ ` řm i"1 esimpr,eiq{τ , (A6) where m is the number of negative samples, ei is the ith negative sample embedding, and sim denotes cosine similarity. This design enables adaptive learning: queries naturally specialize for users with diverse interests and converge for users whose preferences are more focused. A5 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale Table A3: Ablation results for different combinations of item and user LLMs and training strategies. Scenario Configuration HR/NDCG10 HR/NDCG100 HR/NDCG1k MRR ̊100 Homefeed RED-Rec (2 * Qwen) 2.31/0.68 12.59/1.88 31.94/3.86 1.27 Item LLM from scratch 0.00/0.00 0.00/0.00 0.03/0.01 0.00 User LLM from scratch 0.00/0.00 0.03/0.01 1.32/0.21 0.01 Item LLM frozen 1.27/0.36 5.51/0.77 11.37/1.02 0.37 User LLM frozen 1.78/0.44 10.47/1.02 23.06/1.48 1.01 Homefeed + Ads RED-Rec (2 * Qwen) 4.36/1.31 18.32/3.27 42.61/5.02 2.11 Item LLM from scratch 0.00/0.00 0.00/0.00 0.08/0.04 0.01 User LLM from scratch 0.00/0.00 0.00/0.00 1.01/0.07 0.03 Item LLM frozen 1.49/0.41 9.49/1.31 19.29/1.52 0.76 User LLM frozen 2.57/1.01 13.72/1.98 29.72/1.88 3.28 Homefeed + Ads RED-Rec (2 * Qwen) 4.36/1.31 18.32/3.27 42.61/5.02 2.11 RED-Rec-CoT (2 * Qwen) 4.46/1.35 18.78/3.60 44.61/5.01 2.15 F ADDITIONAL EXPERIMENTS F.1 PRETRAINING VALIDATION Our first set of experiments investigates the effect of varying the backbone LLMs for the item and user encoders. Specifically, we explore the following configurations: (i) using different pretrained LLMs for item and user encoders, (ii) training one or both encoders from scratch instead of initializing from a pretrained model, and (iii) freezing the item encoder during training. The detailed results are summarized in Table A3. Across all settings, we observe that using exactly the same pretrained LLM for both item and user encoders and fine-tuning them jointly yields the best performance. In contrast, utilizing mismatched encoders, initializing from scratch, or freezing either encoder all result in significant drops in overall accuracy. This suggests that consistent representation spaces and co-adaptation between the two encoders are crucial for optimal model performance. F.2 COT VALIDATION We explore explainable recommendations based on CoT-based (Wei et al., 2022) explanations for the input layer in a cross-scenario setting. In this experiment, we introduce a Chain-of-Thought (CoT) auxiliary loss: beyond learning discriminative user and item encoders, we encourage explainable cross-scenario reasoning by forcing the user model to generate natural language rationales for each action: LCoT " ́ ÿ uPU |Su| ÿ t"1 Lt ÿ l"1 log pφprt,l| rtăl, zu,tq, (A7) where pφpq denotes the probability, computed by a learnable language model head parameterized by φ, of generating the l-th token rt,lof the rationale conditioned on the previous tokens rtăland the contextualized user embedding zu,t at interaction t. The overall training loss is then Ltotal " LNCE ` λCoTLCoT. We use GPT 4.1 to generate CoT explanations. An example of the generated CoT explanation is like: The user browsed multiple articles related to Switzerland on the homepage, such as "Do you dare to guess how many days of sunshine in Switzerland?" and "What to wear for a trip to Switzerland next week?" This indicates a clear interest in Switzerland. While previously recommended advertisements included those related to travel, they were not specifically targeted at Switzerland. Therefore, we recommend to the user the targeted ad "Personal tested and useful! The ultimate transportation ticket map tool for traveling in Switzerland!", as well as other advertisements related to traveling in Switzerland, such as "Countdown to opening! The four legendary theme parks of Fiesch First Mountain" and "Interlaken sledding premium tips - Save 400 RMB instantly." The CoT explanation module is particularly well-suited to the cross-scenario recommendation setting. By generating step-by-step rationales that account for user behaviors across different scenarios A6 Xu, et al. Cross-Scenario Unified Modeling of User Interests at Billion Scale or domains, the model can provide contextually accurate and human-understandable justifications for its recommendations. This improves both transparency and user trust, crucial for scenario-aware systems. However, we observe that applying the CoT-based approach to large-scale datasets introduces significant challenges. The requirement to generate context-dependent rationales for every user interaction leads to substantially increased computational and memory costs. Given these limitations, we restrict our experiments to small-scale testing. The detailed results are summarized in Table A3. Including CoT data in training has led to certain improvements, but it does not outperform the pretrained model. A7
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2510.14779
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MNRAS 000, 1–17 (2015)
Preprint 17 October 2025
Compiled using MNRAS LATEX style file v3.2
The dark side of early galaxies: geko uncovers dark-matter fractions at
𝑧∼4 −6
A. Lola Danhaive
1,2★, Sandro Tacchella
1,2, Andrew J. Bunker
3, Emma Curtis-Lake
4,
Anna de Graaff
5, Francesco D’Eugenio
1,2, Qiao Duan
1,2, Eiichi Egami
6, Daniel J. Eisenstein
7,
Benjamin D. Johnson
7, Roberto Maiolino
1,2, William McClymont
1,2, Marcia Rieke
6,
Brant Robertson
8, Fengwu Sun
7, Christopher N. A. Willmer
6, Zihao Wu
7, Yongda Zhu
6
1Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
2Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK
3 Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
4 Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK
5 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany
6 Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
7 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA
8 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
JWST/NIRCam slitless spectroscopy enables dynamical mass measurements for typical star-forming galaxies only a billion years
after the Big Bang. We model the H𝛼morpho-kinematics of 163 galaxies at redshift 𝑧≈4-6 from FRESCO and CONGRESS
(with JADES imaging), using the geko code, and infer rotational velocities and dispersions within 𝑟e. Our sample spans
log 𝑀★≈7-10 and log 𝑀dyn ≈9-11. Gas masses are estimated via scaling relations, yielding baryonic masses and dark-matter
(DM) fractions 𝑓DM(𝑟< 𝑟e) within the H𝛼half-light radius. We find high median fractions of ⟨𝑓gas⟩= 0.77 and ⟨𝑓DM⟩= 0.73,
where 𝑓gas is measured with respect to the baryonic mass and 𝑓DM with respect to the DM+baryonic mass. About two-thirds of
systems are DM-dominated within 𝑟e ∼0.5 −1 kpc. Both 𝑓gas and 𝑓DM decrease with stellar mass, consistent with simulations.
The stellar Tully-Fisher relation shows a tentative offset to higher 𝑣circ at fixed 𝑀★and substantial intrinsic scatter, suggesting
that the relation is only beginning to emerge at 𝑧∼5. We measure a negative correlation between 𝑓DM and baryonic surface
density Σbar, weaker but broadly consistent with trends at cosmic noon and at 𝑧∼0. Qualitatively comparing with modified
NFW profiles coupled to an empirical stellar-to-halo mass relation suggests that the lowest 𝑓DM (≲0.4) require cored inner DM
profiles, while the highest fractions favour cuspier profiles, potentially reflecting adiabatic contraction. Overall, the elevated 𝑓gas
and 𝑓DM at 𝑧≳4 are compatible with progenitors of baryon-dominated systems at 𝑧∼2 and naturally anticipate overmassive
black holes at fixed 𝑀★.
Key words: galaxies: kinematics and dynamics – galaxies: evolution – galaxies: high-redshift – galaxies: structure – DM
1 INTRODUCTION
The nature and physics of dark matter (DM) is to this day one of the
biggest outstanding questions in astrophysics. In the widely adopted
standard cosmological model of ΛCDM, DM haloes form from the
collapse of overdensities in the early Universe, allowing baryons
to accrete, cool, and form stars. This process is the basis of galaxy
formation, and predicts the hierarchical growth of galaxies. However,
the physics and interplay between the baryons and the DM remain
remarkably poorly constrained.
In fact, measuring the DM mass content of galaxies and their
haloes observationally is a challenge, as often even the full baryonic
★ald66@cam.ac.uk
mass is difficult to constrain. Many approaches have been adopted to
indirectly constrain the galaxy-halo connection, such as abundance
matching (Marinoni & Hudson 2002; Kravtsov et al. 2004; Conroy
et al. 2006; Vale & Ostriker 2006; Tasitsiomi 2007) and cluster-
ing analyses (Peacock & Smith 2000; Cooray & Sheth 2002; Smith
et al. 2003; Paquereau et al. 2025) based on N-body simulations.
Many numerical simulations have measured the stellar-to-halo mass
(SMHM) relation and its evolution with redshift (Behroozi et al.
2010; Moster et al. 2010; Rodríguez-Puebla et al. 2017; Behroozi
et al. 2019; Tacchella et al. 2018), although its overall normalisation
and turnover mass are still debated and model-dependent. In part, this
is due to the circularity of modelling inputs and outputs, as cosmo-
logical hydrodynamical simulations are usually calibrated to SMHM
relations measured from abundance matching or derived from semi-
© 2015 The Authors
arXiv:2510.14779v1 [astro-ph.GA] 16 Oct 2025
2
A. L. Danhaive et al.
empirical models. Placing direct observational constraints on this
relation would offer important insight on the integrated strength
of various fundamental mechanisms which govern the evolution of
galaxies, such as stellar feedback, black hole activity, and mergers.
One of the most robust ways of constraining the DM content
within galaxies is through the measurement of galaxy kinematics.
The rotation curves and pressure support of galaxies reflect their total
density profile, providing an estimate of the DM content assuming
that the baryonic content is known (Rubin et al. 1980; Burkert 1995;
Burkert & Silk 1997; Persic et al. 1996; McMillan 2017; Zhu et al.
2023). The fraction of DM to the total mass, 𝑓DM, is a by-product
of the mass assembly history of galaxies and of their current growth
(see Wechsler & Tinker 2018, and references therein).
The study of these density profiles within dwarf galaxies, defined
as having masses of 𝑀★< 3 × 109 M⊙, has proved particularly
fruitful, revealing a discrepancy between ΛCDM-predicted and ob-
served density profiles know as the core-cusp problem. Where col-
lisionless (cold) DM-only cosmological simulations predict cuspy
profiles, with densities increasing steeply at small radii, measure-
ments of the rotation curves in dwarf galaxies in the local Universe
show "core"-like profiles that flatten at central radii (Moore 1994;
Burkert 1995). This core-cusp problem posed a direct challenge to
ΛCDM , with propositions of different types of DM surfacing (Hu
et al. 2000; Spergel & Steinhardt 2000). However, many studies have
now shown that this problem can be alleviated by invoking strong
baryonic feedback, likely from supernovae. This feedback, especially
through repeated bursts of star formation, can flatten the DM cusp
into a core (Read & Gilmore 2005; Pontzen & Governato 2012, 2014;
Martizzi et al. 2013). This process has been observed in hydrody-
namical simulations (e.g., Chan et al. 2015; Read et al. 2016, and
references therein), but details such as mass, redshift, and environ-
mental dependence remain uncertain. Interestingly, cases of profiles
cuspier than ΛCDM profiles have also been found (Sonnenfeld et al.
2012; Wang et al. 2020; Li et al. 2022), which could be caused by
adiabatic contraction of the DM halo due to high baryonic densities
in the galaxy core (Blumenthal et al. 1986; Gnedin et al. 2004; Li
et al. 2022).
Another important relation that relates the potential of the DM halo
to its host galaxy is the observed tight relation between stellar (or
baryonic) mass and circular velocity, named the stellar (or baryonic)
Tully-Fisher relation (TFR; Tully & Fisher 1977). The baryonic mass
TFR (bTFR) is the more fundamental of the two, as it holds even
down to low masses (McGaugh et al. 2000; McGaugh 2005). Both
TFRs have been found to hold for star-forming galaxies from 𝑧≈0
(Reyes et al. 2011; Lelli et al. 2016) to cosmic noon (Miller et al.
2011; Übler et al. 2017), albeit with an increase of scatter and a
decrease of zero-point offset with redshift. Measuring the position
of galaxies on the TFR plane at 𝑧> 4 can provide constraints on the
properties of the underlying haloes and their relative contribution to
the total mass of the galaxy.
The detailed mapping of rotation curves of galaxies in the local
Universe has revealed that log(𝑀★[M⊙]) = 10 −11 star-forming
galaxies are baryon-dominated in their central ∼1 kpc, but DM
dominated within their half-light, or effective, radii 𝑟e (Rubin et al.
1985; Martinsson et al. 2013). On the other hand, early-type galaxies
(ETGs) at 𝑧∼0 are heavily baryon-dominated within 𝑟e, suggesting
a diverging mass assembly history (Noordermeer et al. 2007; Cap-
pellari et al. 2013a; Serra et al. 2016). Interestingly, studies of DM
fractions at cosmic noon (𝑧≈2) in log(𝑀★[M⊙]) ⪆9.5 galaxies
found them to be similar to local ETGs, with baryon-dominated cores
(Price et al. 2016; Wuyts et al. 2016; Genzel et al. 2017, 2020; Nestor
Shachar et al. 2023). Genzel et al. (2020) interpret these findings as
evidence for cored DM profiles, motivated by the rapid formation of
massive haloes and galaxies at 𝑧∼1 −3. In fact, cosmic noon marks
the peak of the cosmic star-formation rate (SFR) density (Madau
& Dickinson 2014), where gas accretion rates reached peak values
(Tacconi et al. 2020) and stellar mass doubling scales were on the
order of 𝑡< 0.4 Gyr.
Measurement of the DM content of high-redshift galaxies (𝑧≥4)
was made possible with the advent of the James Webb Space Tele-
scope (JWST), whose NIRCam and NIRSpec instruments probe
ionised gas emission lines out to 𝑧∼10. While kinematic stud-
ies had previously been done at such early times through cold gas
measurements (i.e. [C ii] or CO) from ground based telescopes such
as the Atacama Large Millimeter Array (ALMA, e.g., Lelli et al.
2021; Rizzo et al. 2020, 2021; Roman-Oliveira et al. 2023; Pope
et al. 2023; Rowland et al. 2024), the lack of constraints on the stellar
mass made estimating 𝑓DM difficult. Using kinematic modelling of
JWST/NIRSpec Multi Shutter Array (MSA) data (see also Saldana-
Lopez et al. 2025), de Graaff et al. (2024a,b) found that the five
low-mass log(𝑀★[M⊙]) < 9 galaxies in their 𝑧∼6 −8 sample
were DM dominated within 𝑟e ( 𝑓DM ∼0.7). By studying similar
galaxies in the tng50 simulations (Pillepich et al. 2019), de Graaff
et al. (2024b) suggest that DM-dominated log(𝑀★[M⊙]) = 8 −9
galaxies at 𝑧∼6 are the progenitors of massive baryon-dominated
systems at 𝑧∼2. This finding predicts a DM dominated phase of
early galaxy formation, with expectations that all low-mass galaxies
have low baryon fractions in their central region.
When studying the DM fractions in the central regions of galaxies,
it is also important to consider the effects of black holes (BHs) which
could be hosted in their cores. Although BHs have been shown to
have masses which are only a small fraction of the total dynamical
mass (Kormendy & Ho 2013), recent JWST studies have uncovered
"overmassive" BHs in the early Universe, with 𝑀BH/𝑀★∼0.1−0.01
(Harikane et al. 2023; Kokorev et al. 2023; Übler et al. 2023; Maiolino
et al. 2024b; Juodžbalis et al. 2025a; Jones et al. 2025), even reaching
𝑀BH/𝑀★> 2 (Juodžbalis et al. 2025b). If the relatively low-mass
galaxies at 𝑧> 4 host massive black holes which experience accretion
phases, then they could play a significant role in the galaxy’s mass
assembly history. Importantly, active galactic nuclei (AGN)-driven
outflows can heavily disrupt the surrounding gas, particularly in low-
mass galaxies (Sijacki et al. 2007; Nelson et al. 2019; Koudmani
et al. 2021, 2022; Carniani et al. 2024). The presence of such AGNs
is important to study as it currently introduces more uncertainties in
the interpretation of our kinematic measurements.
With the synergy of NIRCam imaging and grism data, the kine-
matics of ∼200 galaxies at 𝑧= 4 −6 were measured in Danhaive
et al. (2025a) using a new forward-modelling Bayesian code, the
Grism Emission-line Kinematics tOol (geko, Danhaive & Tacchella
2025)1. This sample is comprised of log(𝑀★[M⊙]) ≈7 −10.5 star-
forming galaxies, with a wide range of rotational support v/𝜎0. As
shown in Danhaive et al. (2025a), many of these galaxies have large
dynamical masses compared to their stellar masses, pointing to large
contributions from gas and DM. In this paper, we study the mass con-
tent of these galaxies in detail, placing the first statistical constraints
on DM fractions at high redshift 𝑧> 4. We briefly introduce our
sample and methodology in Sec. 2, then present our dynamical mass
measurements and discuss in the context of the Tully-Fisher (TFR)
relation in Sec. 3. We present our gas and DM fractions in Sec. 4, and
compare them to samples at lower redshift. In Sec. 5, we interpret our
measurements in the context of the shapes of DM density profiles,
1 Available at https://github.com/angelicalola-danhaive/geko
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
3
and explore potential contributions from BHs. Finally, we summarise
our results in Sec. 6. Throughout this work, we assume Ω0 = 0.315
and 𝐻0 = 67.4 km s−1 Mpc−1 (Planck Collaboration et al. 2020).
2 OBSERVATIONS, SAMPLE AND METHODOLOGY
In this section, we present the NIRCam grism and imaging data
used in this work, along with a description of our sample (Sec. 2.1).
We also describe the morpho-kinematic modelling of our sample
(Sec. 2.2), which is used to measure the rotational velocities and
velocity dispersions of our galaxies. This sample, and the associated
kinematic measurements, are described in Danhaive et al. (2025a),
so we only summarize them here but refer the reader to that work for
more details. Finally, in Sec. 2.3, we detail our derivation of the gas
and DM fractions for our sample.
2.1 NIRCam data and sample selection
The analysis in this work uses both NIRCam imaging and grism
spectroscopy. The NIRCam grism data is obtained from the FRESCO
survey (PI: Oesch, PID: 1895; Oesch et al. 2023; Covelo-Paz et al.
2025) in GOODS-S and GOODS-N, and the CONGRESS survey
(PIs: Egami & Sun, PID: 3577; Sun et al. in prep) in GOODS-N.
These surveys are comprised of grism data (R mode) in the F444W
and F356W filters, respectively, probing H𝛼emission at 𝑧= 3.8−6.5.
The data is reduced following Sun et al. (2023) and Helton et al.
(2024) to obtain 2D spectra for each galaxy in the field of view.
The 2D spectra are then continuum subtracted following the 2-step
iterative technique described in Kashino et al. (2023) to obtain 2D
emission line maps for H𝛼.
In the regions of the FRESCO and CONGRESS surveys, there is
a wealth of deep imaging data from the JADES survey (Eisenstein
et al. 2023; Rieke et al. 2023). Specifically, the JADES survey has
imaging in most of the NIRCam wide bands, as well as the F335M
and F410M medium bands. In addition, the FRESCO survey ob-
tained imaging in the F182M and F210M bands, and CONGRESS
in F090W and F115W, complementing the existing JADES imag-
ing data. For certain regions in our sample, we also have additional
medium bands F182M, F210M, F430M, F460M, and F480M from
the JWST Extragalactic Medium Band Survey (JEMS; Williams et al.
2023) in GOODS-S. The full details on the data reduction and gen-
eration of the drizzled mosaics and photometric catalogues can be
found in Rieke et al. (2023) and Robertson et al. (2024). This large
array of photometric bands, in addition to the grism spectroscopic
redshifts and emission line fluxes, allows us to fit the SEDs of all
galaxies with Prospector (Johnson et al. 2021; Tacchella et al. 2023;
Simmonds et al. 2024, 2025). We obtain measurements of the stellar
masses (𝑀★), star-formation rates (SFRs), and star formation histo-
ries (SFHs) used in this work.
As mentioned above, the parent sample used in this work is de-
scribed in Danhaive et al. (2025a). This H𝛼emitter sample is built
on the photometric redshift estimates from EAZY (Brammer et al.
2008; Hainline et al. 2024), which are then confirmed by Gaussian
fitting of the 1D grism spectrum followed by visual inspection (Hel-
ton et al. 2024; Lin et al. 2025). We select galaxies with an H𝛼S/N
cut of 10 (582 galaxies), from which we select our final sample of
213 galaxies based on additional S/N cuts and position angle (PA)
cuts to ensure that galaxies are at an angle with respect to the grism
dispersion direction. Specifically, the 213 galaxies are separated into
three samples based on the robustness of the kinematic measure-
ments, as described in Danhaive et al. (2025a). In this paper, we only
consider the 163 galaxies from the gold and silver samples, and we
exclude the unresolved sample. The latter contains systems with sizes
smaller than the full-width half-maximum (FWHM) of the F444W
(or F356W) point-spread function (PSF), or with unresolved velocity
gradients. However, for simplicity, we do not further distinguish be-
tween the gold and silver subsamples in this paper, treating the 163
galaxies together as our full final sample.
2.2 Morpho-kinematic modelling
To obtain measurements of the rotational velocity and velocity dis-
persion, we use the Bayesian-inference fitting tool geko (Danhaive &
Tacchella 2025) which forward models the grism data and compares
it with the observed 2D spectrum to obtain the posterior distributions
of the input morphological and kinematic model parameters.
The H𝛼emission line map is modelled with a Sérsic profile (Sersic
1968):
𝐼(𝑟) = 𝐼e exp
−𝑏𝑛
" 𝑟
𝑟e
1/𝑛
−1
#!
,
(1)
where 𝐼e is the intensity at the effective radius 𝑟e and 𝑛is the Sér-
sic index. We use the near-UV image from JADES, probed by the
F150W filter at 𝜆rest ≈2000Å, as a prior for the morphology of the
H𝛼line. The rest-frame near-UV is chosen because it traces young
stars responsible for ionizing the nearby gas and producing the H𝛼
emission. To model the F150W images, we use the Bayesian code
Pysersic (Pasha & Miller 2023), which models the image with a
PSF-convolved Sérsic profile. The width of the priors is obtained by
doubling the uncertainties obtained from the Pysersic modelling.
The H𝛼gas kinematics are modelled with an arctangent velocity
curve (Courteau 1997; Miller et al. 2011)
𝑉rot(𝑟int, 𝑟t,𝑉a) = 2
𝜋𝑉a arctan 𝑟int
𝑟t
,
(2)
where𝑉rot is the rotational velocity at a given radius𝑟int in the intrinsic
galaxy plane, 𝑉a is the asymptotic value to which the arctangent
rotation curve converges to at large radii 𝑟int →∞and 𝑟t is the turn-
around radius of the rotation curve. To project this velocity on the
observation plane, we need to account for the galaxy’s inclination 𝑖:
𝑉obs(𝑥, 𝑦) = 𝑉rot(𝑟int, 𝑟t,𝑉a) · sin𝑖· cos 𝜙int,
(3)
where 𝜙int is the polar angle coordinate in the galaxy plane. To
compute the inclination, we assume an intrinsic axis ratio 𝑞0 = 0.2
to account for the thickness of galaxies at high redshift (Wuyts et al.
2016; Genzel et al. 2017; Price et al. 2020; Übler et al. 2024). We
adopt a constant velocity dispersion across the galaxy.
The model H𝛼intrinsic map is convolved with the kinematics to
form a model 3D cube, which is then convolved with the instru-
ment PSF and line-spread function (LSF). Finally, using the grism
dispersion function, the cube is projected onto the observed grism
2D space. The priors for the kinematic parameters are uniform. We
place a constraint on the turn-around radius 𝑟t to be smaller than the
effective radius 𝑟e (Miller et al. 2011).
2.3 Gas and DM fractions
In order to infer gas masses from our star-formation rates, we use the
empirical relations calibrated in Tacconi et al. (2020). We assume
here that the gas component is dominated by the molecular phase,
whose mass is constrained in these relations. In dense, highly star-
forming systems it is expected that the molecular phase dominates
MNRAS 000, 1–17 (2015)
4
A. L. Danhaive et al.
within 𝑟e, and the atomic phase only contributes to a minor degree
(e.g. Leroy et al. 2008; Saintonge et al. 2017). The ionised component
is small in most systems (e.g. see Kennicutt & Evans 2012, and
references therein). Given the small sizes (𝑟e ≈0.5−1 kpc, Danhaive
et al. 2025b) and high specific star-formation rates (sSFR, with mass-
doubling timescales of 𝑡double ⪅30 Myr) of the galaxies in our
sample, this is a valid assumption.
In these equations, the molecular gas mass is estimated using
the sSFR of the system and the integrated depletion timescale for
converting the gas into stars. This timescale depends on the redshift
and therefore provides a more accurate conversion than fixed redshift
relations (e.g. Kennicutt 1998). Furthermore, the latter is calibrated
to the local universe, where gas fractions are overall lower, whereas
the Tacconi et al. (2020) relations are calibrated from 𝑧∼0 −5.5
from almost 2000 objects or stacks (see also Saintonge et al. 2017;
Tacconi et al. 2018; Freundlich et al. 2019). Specifically, the gas-
to-stellar mass ratio 𝜇gas = 𝑀gas/𝑀★scales with redshift 𝑧, sSFRs,
and stellar mass 𝑀★, where the scaling factors are computed using
observational measurements of ionized and molecular gas from the
literature. The relation for 𝜇gas is:
log 𝜇gas = 𝐴+ 𝐵× (log(1 + 𝑧) −𝐹)2
+ 𝐶× log(sSFR/sSFRMS(𝑧, 𝑀★))
+ 𝐷× (log 𝑀★−10.7),
with the star-forming main sequence (SFMS) defined as in Speagle
et al. (2014), and the parameters 𝐴, 𝐵, 𝐶, 𝐷and 𝐹are the parame-
ters obtained in Tacconi et al. 2020 (Table 2b) from error-weighed,
multi-parameter regression. SFMSs such as Speagle et al. (2014) can
suffer from sample selection effects, especially on the low mass end,
which can bias the inferred relation (e.g. McClymont et al. 2025b;
Simmonds et al. 2025). We also note that the Kennicutt (1998) rela-
tion assumes solar metallicities, which can have significant impacts
(∼0.3 dex) on the inferred SFRs, particularly for low-mass, metal-
poor galaxies.
Using the aforementioned scaling relation, we can then calculate
gas fractions on the galaxy scale,
𝑓gas =
𝜇gas
1 + 𝜇gas
=
𝑀gas
𝑀gas + 𝑀★
,
(4)
and infer the total baryonic masses:
𝑀bar = 𝑀gas + 𝑀★= (𝜇gas + 1)𝑀★.
(5)
To quantify the uncertainties in our inferred gas fractions, we
propagate the uncertainties from the inputs to the scaling relation (𝑀★
and SFR), along with the uncertainties of the constant parameters
of the relation itself (𝐴, 𝐵, 𝐶, 𝐷and 𝐹). To assess the systematics
introduced by our choice of gas mass estimate, we compare the 𝑓gas
measurements obtained from the scaling relation to those inferred by
a simple conversion
𝑀gas = SFR × 𝑡dep
(6)
with a depletion time 𝑡dep = 0.5−1 Gyr. Although using this relation
increases the intrinsic scatter, our median fractions remain consistent.
If instead we change the SFMS used in the Tacconi et al. (2020) rela-
tion, from Speagle et al. (2014), and adopt the one from Simmonds
et al. (2025), we find that our inferred gas fractions are on average
higher by Δ 𝑓gas ≈0.05 −0.1. We note that in both of these cases,
the change in 𝑓gas does not visibly affect the median values of our
inferred dark-matter fractions. However, it is important to note these
uncertainties in the gas fractions, which can only be addressed with
constraints from direct observations of the molecular gas content.
In virial equilibrium, the circular velocity of galaxies at a radius
𝑟can be related to the combined effects of gravity and turbulence-
induced pressure. The former is reflected in the rotational velocity,
and the latter is an asymmetric drift correction to account for the
pressure support (Burkert et al. 2010; Newman et al. 2013; Wuyts
et al. 2016). For an exponential disk, the circular velocity can be
written as:
𝑣circ(𝑟) =
√︃
𝑣2
rot(𝑟) + 2(𝑟/𝑟s)𝜎2
0 .
(7)
The circular velocity reflects the gravitational potential of the galaxy
and is hence directly related to the dynamical mass,
𝑀dyn = 𝑘tot
𝑟e𝑣2
circ(𝑟e)
𝐺
,
(8)
where 𝐺is the gravitational constant and 𝑘tot is the virial coefficient
(Price et al. 2020). The virial coefficient 𝑘tot allows us to infer the to-
tal dynamical masses based on measurements out to 𝑟e. If we instead
compute 𝑣circ using a generalised Sérsic profile (rather than an expo-
nential disk, 𝑛= 1), our inferred dynamical masses remain consistent
within Δ(log 𝑀dyn) ≲0.07 (with median Δ(log 𝑀dyn) ≲0.02). We
present results for the exponential disk assumption to provide a more
direct comparison to other works in the literature. Because we have
modelled our galaxies with 𝑞0 = 0.2, we choose 𝑘tot = 1.8 as it is the
coefficient for galaxies with 𝑞0 = 0.2 and 𝑛∼1−4 (Price et al. 2022).
We note that varying 𝑞0 in the interval 𝑞0 = 0.15 −0.25 shifts sin𝑖
by Δ(sin𝑖) ≲0.05, propagating to Δ(log 𝑀dyn) ≲0.05. For more ex-
treme values 𝑞0 = 0.50, this correction can reach Δ(log 𝑀dyn) ≈0.1.
Finally, using our dynamical mass measurements, we can compute
the DM fractions:
𝑓DM = 𝑀DM
𝑀dyn
= 𝑀dyn −𝑀bar
𝑀dyn
.
(9)
All our inferred fractions are calculated within the effective radius 𝑟e
since that is where our kinematic measurements are best constrained.
We find that 20% of our sample has negative and, therefore, unphysi-
cal DM fractions. This fraction drops to 18% (8%) when a deviation
from 𝑓DM = 0 is required at the level of 𝜎(3𝜎). These numbers
are comparable to those reported in other studies (e.g., Wuyts et al.
2016), and suggest inconsistencies in the modelling of the SEDs
and the kinematics, including for instance positive age gradients
(D’Eugenio et al. 2021; van Houdt et al. 2021) or non-equilibrium
configurations. Although we discuss these systems in the text, unless
otherwise mentioned we only consider systems with 𝑓DM > 0 for our
analyses.
3 DYNAMICAL MASSES & TULLY-FISHER RELATION
In this section, we present the dynamical masses of our systems based
on our kinematic modelling (Sec. 3.1), and discuss them within the
context of the Tully-Fisher relation at high redshift (Sec. 3.2).
3.1 Dynamical masses
We present our results for the dynamical masses of our galaxies in
Fig. 1. We find high dynamical masses spanning log(𝑀dyn [M⊙]) =
9 −11, with the overall expected trend of increasing 𝑀dyn with 𝑀★,
reflecting the baryons tracing the underlying gravitational potential.
We find a significant scatter in the dynamical masses at fixed stellar
mass, indicative of varying gas and DM contents. To investigate the
source of this scatter, we colour-code our galaxies by their offset
from the MS (ΔMS), where we adopt the MS from Simmonds et al.
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
5
8
9
10
11
12
log(Mdyn)[M ]
7
8
9
10
11
12
log(M )[M ]
This work
Saldana-Lopez+25
De Graaff+24
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
MS
Figure 1. Comparison of the dynamical masses (𝑀dyn) inferred from our
modelling of grism data and the stellar masses (𝑀★) inferred from SED fitting.
We colour-code our galaxies based on their offset from the main sequence
(ΔMS) as defined by Simmonds et al. (2025). The majority of our systems
lie below the one-to-one relation (dashed line), consistent with a significant
contribution to the dynamical mass from gas and/or DM. Six galaxies lie on
the relation, highlighting discrepancies in the different mass estimates. We
compare our values to ionised gas measurements from de Graaff et al. (2024a)
and Saldana-Lopez et al. (2025) at similar redshifts.
(2025). We find that galaxies above the MS, which are also typically
at lower masses, seem most distant from the 1-1 line, implying a
lower stellar contribution to the baryonic and total mass.
We expect the dynamical masses to lie above the stellar masses,
as they should also incorporate the gas and DM. We find consistent
measurements 𝑀★< 𝑀dyn for the majority of the galaxies in our
sample. We discuss the few systems showing unphysical 𝑀dyn −𝑀★
values in more detail in Danhaive et al. (2025a), the discrepancy
pointing to problems in the modelling of either or both masses.
We compare our sample with measurements from Saldana-Lopez
et al. (2025) and de Graaff et al. (2024b) at similar redshifts, and
find that they probe the same regions of the parameter space. When
combined, these two samples highlight a similar scatter of 𝑀★at
fixed 𝑀dyn as found in our sample. Overall, these dynamical masses
are the key observable that allows us to derive DM fractions (Sec. 2).
We note that in a recent study, Phillips et al. (2025) compare intrinsic
kinematic measurements from simulations to mock NIRSpec/IFU
fits, and find that despite biases in the recovered values of 𝑣rot and
𝜎0, the dynamical masses remain relatively robust.
3.2 The Tully-Fisher relation (TFR)
Local star-forming disk galaxies exhibit a tight relationship between
their stellar mass and circular velocity (𝑣circ, Eq. 7), namely the
TFR relation. This relation is typically parametrised as a power law,
𝑀∝𝑉𝑎, or
log 𝑀★= 𝑎· log 𝑣circ + 𝑏,
(10)
where 𝑎is the slope and 𝑏is the zero-point offset, and 𝑀★is in
units of M⊙and 𝑣circ in km/s. This relation is analogous to the
𝑀★−𝑀dyn relation from Fig. 1, given the fundamental link between
circular velocities and dynamical masses (Eq. 8) when assuming
virial equilibrium. By understanding where our galaxies lie on the
TFR plane, we can place constraints on their mass content compared
to galaxies at lower redshift.
In order to explore the offset between our sample at 𝑧= 4 −6 and
the sTFR at cosmic noon and in the local Universe, we fit Eq. 10 to
our sample with the fixed slope from Reyes et al. (2011) at 𝑧∼0
(𝑎= 3.60). For our fit, we use the Bayesian inference package emcee
(Foreman-Mackey et al. 2013). emcee samples the parameter space
using an implementation of the affine invariant ensemble sampler
for Markov chain Monte Carlo (Goodman & Weare 2010), resulting
in posterior distributions for the model parameters. Importantly, it
allows us to self-consistently fit for the intrinsic scatter 𝜎int around
the relation, assuming that our adopted uncertainties for the various
measurements are reliable. We discuss this assumption in Sec. 5.2.
For the emcee fitting, we assume uniform priors for all of the param-
eters and conduct the MCMC sampling with 50 walkers, a 2000 step
burn-in phase, and 10,000 samples. We thin our chains to remove
autocorrelation between samples, checking for convergence in the
trace plots and in the effective number of independent samples.
We find the following best fit parameters for the log(𝑀★[M⊙]) > 8
sample, where we are more complete:
𝑎= 3.60 (fixed)
(11)
𝑏= 0.76 ± 0.07
(12)
𝜎int = 0.49 ± 0.06 .
(13)
We plot the results for this fiducial fit at 𝑧∼5 in purple in Fig.
2, and also include a fit for the whole sample in red. The large
intrinsic scatter, shown by the shaded regions, reflects that the 𝑧∼5
galaxies in our sample are not settled around the TFR. They are likely
undergoing kinematic and/or morphological changes on relatively
short timescales, which influence the measured circular velocities at
fixed stellar masses. More generally, this implies that galaxies have
not yet settled into more stable rotating disks, where the TFR is
expected to hold (e.g., Übler et al. 2017).
Despite the lack of a tight TFR in our sample, we explore the
evolution of the zero-point offset 𝑏from 𝑧∼0 (Reyes et al. 2011)
and 𝑧∼1 −3 (Übler et al. 2017, see also Tiley et al. (2016)) to
𝑧≈4 −6. Both works adopt the same slope. We find a significant
increase in the zero-point offset, Δ𝑏≈−1.3 dex, between our TFR
and the one measured at cosmic noon (Übler et al. 2017). This
evolution is more significant than the Δ𝑏≈−0.4 dex evolution from
𝑧∼0 to cosmic noon. The observed decrease of 𝑏with redshift
could have (a mixture of) different origins. To visually compare with
galaxies at cosmic noon, we plot on Fig. 2 the RC100 sample from
Nestor Shachar et al. (2023) at 𝑧= 0.6 −2.5. It is clear that these
cosmic noon galaxies occupy a different parameter space in the TFR
plane, having higher stellar masses (log(𝑀★[M⊙]) ≈10 −11) and
lower gas fractions ( 𝑓gas ⪅0.5) than our sample.
First, an increase of gas fractions naturally explains the shift of the
TFR to higher circular velocities (at fixed mass). This is because 𝑣circ
reflects the total mass of the galaxy, both baryons and DM, so if the
fraction of stars with respect to the other components decreases, the
TFR will shift. In the same way, an increase in the DM fractions can
also contribute to this shift. It is also possible that 𝑏has a mass de-
pendence, since our sample probes smaller masses than the KMOS3D
and RC100 samples. It is expected that gas fractions increase at low
stellar masses, naturally biasing the stellar TFR. A larger sample
spanning larger redshift and mass ranges would be needed to assess
the mass dependence of 𝑏.
In light of this, we also explore the bTFR in our sample, but simi-
larly to the sTFR we do not find a tight relation akin to what has been
MNRAS 000, 1–17 (2015)
6
A. L. Danhaive et al.
measured at cosmic noon and the local Universe (McGaugh 2012;
Lelli et al. 2016). We also find a similar offset towards higher circular
velocities at fixed baryonic mass. Together, these points suggests that
DM contributes significantly to the total mass of the galaxies in our
sample. This is in contrast to the baryon-dominated galaxies seen
at cosmic noon. To correctly measure the bTFR, gas masses would
need to be directly measured, which is beyond the scope of this work
and our available data.
In terms of intrinsic scatter (𝜎int = 0.49 ± 0.06), we find that it
is roughly twice that reported at cosmic noon (𝜎int = 0.22, Übler
et al. 2017), where they also find an increase with respect to the local
Universe (𝜎int = 0.10−0.13 Reyes et al. 2011). As discussed in Übler
et al. (2017), one reason for this could be the fact that galaxies at high-
redshift are less settled (Simons et al. 2016), and are seen more often
in non-equilibrium states (Covington et al. 2010). Also, we rely on
2D grism data in this work and, hence, suffer from more uncertainties
than studies with 3D kinematic data. Furthermore, galaxies at high
redshift are smaller and lower mass on average, which makes their
kinematics more difficult to constrain.
Finally, we note that the lowest-mass galaxies in our sample, with
log(𝑀★[M⊙]) < 8, distinctively fall off our best-fit relation, with
high circular velocities of 𝑣circ > 150 km/s despite their low masses.
We also fit our full sample, including these log(𝑀★[M⊙]) < 8 ob-
jects, and find a smaller zero-point (𝑏= 0.59 ± 0.08) and a larger in-
trinsic scatter (𝜎int = 0.73±0.07). These galaxies also have gas frac-
tions close to 1, and are most likely undergoing starbursts (ΔMS > 0).
It is possible that the circular velocities measured for these systems
have a non-negligible contribution from non-circular motions, such
as radial inflows and/or outflows of gas. This would also contribute to
the large observed scatter. Due to their low stellar masses and, hence,
small spatial extent, such effects would be difficult to detect in our
modelling. Our sample is in general not complete, so our discussion
regarding the TFR aims at comparing our sample’s physical proper-
ties with those of galaxies at lower redshift and not fully defining the
TFR at 𝑧= 4 −6.
4 THE GAS AND DM CONTENT AT 𝑍> 5
In this section, we quantify the baryonic and DM content of galaxies
at 𝑧= 4 −6, using gas and DM fractions derived from our kinematic
modelling (Sec. 2). We first study the dependence of these fractions
on stellar mass (Sec. 4.1), then we focus on the relation between DM
fractions and baryonic surface density in Sec. 4.2. Finally, we put
our findings in the context of works at lower redshift to discuss the
redshift evolution of DM fractions (Sec. 4.3).
4.1 Baryon content at 𝑧∼5
Fig. 3 shows the inferred gas ( 𝑓gas) and DM ( 𝑓DM) fractions as a
function of stellar mass for the galaxies in our samples. We note that
𝑓gas is defined relative to the baryonic mass (stars + gas), whereas
𝑓DM is defined relative to the total mass (DM + baryons). We fit both
relations with a power law of the form
𝑓(𝑧, 𝑀★) = 1 −10𝛼(log 𝑀★−9)+𝛽+𝛾( 1+𝑧
6 ),
(14)
where 𝛼and 𝛾parametrize the mass and redshift dependence, respec-
tively, and 𝛽is the normalisation at log(𝑀★[M⊙]) = 9 and 𝑧= 5.
The power-law shape is consistent with the equation from Tacconi
et al. (2020) for the gas fractions. Similarly to the TFR, these fits
are done with emcee using the same general setup (see Sec. 3). We
1.75
2.00
2.25
2.50
2.75
log(vcirc)[kms 1]
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
log(M )[M ]
This work (v/ > 1)
Best fit, z = 4
6 (all)
Best fit, z = 4
6 (logM > 8)
RC100 medians
Übler+2017 (z = 0.9
2.3)
Reyes+2011 (z = 0)
0.0
0.2
0.4
0.6
0.8
1.0
fgas(r < Re)
Figure 2. The stellar Tully-Fisher plane for our sample, where the best-fit
linear relation for galaxies with log(𝑀★[M⊙]) > 8 is shown in solid purple
(dotted red for the full sample). The circular velocity (𝑣circ) is evaluated at
𝑟e with an asymmetric-drift correction. We find that most of our galaxies lie
along the relation, albeit with a large scatter, but the lowest-mass galaxies
drop off. We compare our measurements at 𝑧∼5 to the relation found for
the KMOS3D 𝑧= 0.9 −2.3 star-forming galaxies from Übler et al. (2017)
(dashed orange line), and the medians from the RC100 sample (diamonds,
Nestor Shachar et al. 2023). We also include the TFR at 𝑧∼0 from Reyes
et al. (2011) for reference (dot-dashed tan curve). We fit our data with the
same fixed slope as these works, 𝑎= 3.60, but computing our own zero-point
𝑏. We find Δ𝑏≈−1.3 dex between from 𝑧= 0.9 −2.3 to 𝑧= 4 −6. The
large intrinsic scatter around the relation (𝜎int = 0.49 ± 0.06; shaded region)
indicates that it only beginning to emerge at 𝑧∼5.
summarize the best-fit parameters for both the gas and the DM frac-
tions in Tab. 1. For the gas fractions, we are not able to constrain
the intrinsic scatter (𝜎int) around the best-fit relation. As explained in
Sec. 2, we compute the uncertainties on 𝑓gas by propagating the un-
certainties on both our input parameters and the constant parameters
of the Tacconi et al. (2020) scaling relation. This yields relatively
large uncertainties of the order of Δ 𝑓gas ≈0.1 (see characteristic
error in Fig. 3). On the other hand, the outputs 𝑓gas from the scaling
relation vary only a small amount at fixed mass. This is in part due
to the small range probed in terms of ΔMS in our sample, since we
are not sensitive to galaxies well below the MS. Because of this, our
uncertainties are significantly larger than the scatter shown by our
inferred 𝑓gas, meaning that we cannot constrain 𝜎int.
4.1.1 Gas fractions
As described in Sec. 2, we cannot measure gas masses directly, so we
infer them based on the galaxies’ SFRs, stellar masses, and redshifts
through the Tacconi et al. (2020) scaling relation. As seen in the
top panel of Fig. 3, our galaxies have high predicted gas fractions
𝑓gas = 𝑀gas/𝑀bar, with almost all galaxies having 𝑓gas > 0.5. These
systems have more gas than stars in their central region 𝑟< 𝑟e. These
high gas fractions are driven in part by the high SFRs, since gas
must accrete onto the galaxy to cool and fuel star formation. Gas
fractions are also expected to be overall higher at high redshift, but
we cannot constrain the redshift dependence of 𝑓gas due to the small
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
7
0.0
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This work
This work (z ∼5)
Running medians
DeGraaff+24b
Genzel+20
RC100 (z=0.6-2.5)
Lee+25 ([CII])
de Graaff+24b (TNG50, z = 4
5)
de Graaff+24b (TNG50, z = 5
6)
McClymont+25c (THESAN-ZOOM, z = 3
6)
Figure 3. Gas (top) and DM (bottom) fractions (within 𝑟e) as a function of
stellar mass, with the characteristic uncertainty shown in grey. The 𝑓gas(𝑟<
𝑟e) (Eq. 4) is computed using the sSFR, redshift, and stellar mass of each
galaxy following Tacconi et al. (2020). The 𝑓DM(𝑟< 𝑟e) is computed using
𝑀gas along with 𝑀dyn and 𝑀★as shown in Eq. 9. We fit our data with a
power-law (solid purple line) for both relations, showing the intrinsic scatter
(shaded region) only for 𝑓DM. We cannot constrain 𝜎int for 𝑓gas due to the
inferred uncertainties being significantly larger than the scatter in the outputs
of the Tacconi et al. (2020) scaling relation for our sample (see Sec. 4.1.1).
We compare our results with measurements from Genzel et al. (2020) (green
stars) and Nestor Shachar et al. (2023) (RC100; blue diamonds) at cosmic
noon and de Graaff et al. (2024a,b) (orange pentagons) and Lee et al. (2025)
(pink hexagons) at high redshift. For most of our sample, we find high gas
and DM fractions 𝑓> 0.5. We find relatively good agreement with median
trends from the thesan-zoom simulations (green solid line; McClymont et al.
2025c) and the tng50 simulations (blue solid lines; de Graaff et al. 2024b).
redshift range probed in this work. However, as expected, our inferred
dependence of 𝛾= −0.36±0.45 is consistent with its counter-part in
the Tacconi et al. (2020) scaling relation (𝐷= −0.41 ± 0.03), albeit
with very large uncertainties.
We compare our gas fractions with estimates at cosmic noon from
Genzel et al. (2020) and Nestor Shachar et al. (2023), which show
a large scatter at log(𝑀★[M⊙]) ≈10 −11. Because there is little
overlap between this mass range and the one probed in this work, a
direct comparison to quantify the independent effects of redshift is
Parameter
𝑓gas
𝑓DM
𝛼
0.43 ± 0.03
0.28 ± 0.05
𝛽
−0.73 ± 0.03
−0.57 ± 0.04
𝛾
−0.36 ± 0.45
0.34 ± 0.65
𝜎int
–
0.15 ± 0.02
Table 1. Summary of the parameters for the power-law fit (Eq. 14) to the
𝑓gas−𝑀★and 𝑓DM−𝑀★relations (Fig. 3). We cannot constrain 𝜎int for the
𝑓gas −𝑀★relation due to the inferred uncertainties on 𝑓gas being significantly
larger than the scatter in the outputs of the Tacconi et al. (2020) scaling relation
for our sample (see Sec. 4.1.1 for more details).
not possible. However, their results are consistent with a decrease,
on average, of gas fractions with stellar mass.
The high gas fractions at high-redshift are consistent with other
predictions of the increase of gas fractions with redshift (Cresci et al.
2009; Genzel et al. 2015; Pillepich et al. 2019). However, this effect
appears to also have a strong mass component. The gas fractions
found in de Graaff et al. (2024a,b) at 𝑧≈6 −8 are consistent with
our results. In fact, although de Graaff et al. (2024a) use the local
relation from Kennicutt (1998) to infer gas masses from SFRs, their
measurements lie nicely along our best-fit relation. This suggests
that this local relation holds well at 𝑧∼6, at least at the low masses
(log(𝑀★[M⊙]) < 9) probed by their work. The lower gas fractions,
derived from [CII] detections, presented in Lee et al. (2025) at 𝑧∼
4 −6 are consistent with our best-fit power-law.
The steep mass dependence of 𝑓gas on 𝑀★is also seen in the
TNG50 simulations (Pillepich et al. 2019; de Graaff et al. 2024b) and
the thesan-zoom simulations (Kannan et al. 2025; McClymont et al.
2025c). At low masses, we find higher gas fractions on average than
both of these simulations. This is partly because the relations from
Tacconi et al. (2020) and Speagle et al. (2014) do not reach the low
mass (log(𝑀★[M⊙]) < 9) end, and hence the extrapolation could
cause an over-estimate of gas fractions at low masses. Importantly,
our sample is biased to very star-forming galaxies at the low-mass
end, which naturally explains our higher medians.
4.1.2 DM fractions
We now discuss the bottom panel of Fig. 3, where we plot our DM
fractions (Eq. 9) as a function of stellar mass. These fractions are
computed within the half-light, or effective, radius 𝑟e of the H𝛼
emission to avoid extrapolations of our kinematic measurements to
larger radii. This implies that the fractions discussed in this work
are relevant for the central 𝑟= 1.20 ± 0.05 kpc of each galaxy
(sample median, Danhaive et al. 2025b). We see a large scatter in
the DM content of galaxies at fixed stellar mass, but nonetheless find
that low-mass galaxies (log(𝑀★[M⊙]) < 9) are predominantly DM
dominated with 𝑓DM(𝑟< 𝑟e) > 0.5. Above log(𝑀★[M⊙]) = 9, we
observe a larger scatter and an overall decline in DM fractions with
stellar mass, as is highlighted by the running medians and our best-fit
power-law in Fig. 3. We colour-code our galaxies by their offset from
the main sequence, but we do not find a clear trend.
We put our 𝑓DM measurements in the context of works at cosmic
noon, which probe a higher mass range log(𝑀★[M⊙]) ≈10−11. As
shown in Fig. 4.1, Genzel et al. (2020) find that star-forming galaxies
at cosmic noon have baryon-dominated cores, with 𝑓DM ⪅0.5. This
is consistent with the extrapolation of our power-law fit to these
high masses, suggesting that the mass dependence of 𝑓DM exists
independently of redshift. In fact, de Graaff et al. (2024b) show with
MNRAS 000, 1–17 (2015)
8
A. L. Danhaive et al.
the TNG simulations that the baryon fraction within 1 kpc, 𝑓bar(𝑟< 1
kpc), remains constant with redshift at fixed mass. However, it is
important to note that our measurements are not at fixed apertures,
and we need to take into account the redshift evolution of galaxy sizes
(Shibuya et al. 2015; Ward et al. 2024; Danhaive et al. 2025b; Allen
et al. 2025; Yang et al. 2025). Nonetheless, our findings suggest that
the high DM fractions seen at high-𝑧, both in this work and de Graaff
et al. (2024b), are at least in part driven by the lower stellar masses
probed compared to lower redshifts. In fact, Lee et al. (2025) probe
higher stellar masses on average (log(𝑀★[M⊙]) > 9) and find DM
fractions ranging from 𝑓DM ≈0.1 to 𝑓DM ≈0.6, consistent with the
scatter seen in our larger sample at those masses.
We also compare our DM fractions with predictions from the
thesan-zoom simulations at 𝑧= 3 −6. In general, the simulations
are consistent with the majority (60%) of our sample having high
DM fractions 𝑓DM > 0.5. However, McClymont et al. (2025c) find
a weaker mass trend, which does not reproduce the population of
log(𝑀★[M⊙]) > 9 galaxies with lower 𝑓DM. This could point to
mechanisms which drive DM out from the central regions, such as
intense feedback from star formation or black holes (see Sec. 5 for
more discussion). Within the simulations, this could also be due
to a decrease in the number of galaxies at the high-mass end. Our
observed dependence of 𝑓DM on 𝑀★is instead more consistent with
the trend observed in the tng50 simulations (de Graaff et al. 2024b).
Some objects have high DM fractions close to 𝑓DM = 1. Although
these could be caused by an over-estimate of the dynamical mass,
they can also point to steepening of the DM halo density profile in
the central regions of these galaxies. We explore this in Sec. 5.1. We
also find that the inferred 𝑓DM fractions for 20% of our sample are
unphysical, i.e., 𝑓DM < 0. Negative fractions imply that the baryonic
mass is larger than the dynamical mass, which could be caused by
one or more of the following. The stellar mass inferred from SED
fitting could be over-estimated (e.g. due to outshining effects, Sec.
5.2), the dynamical mass from the kinematics fitting could be under-
estimated, or the gas fractions inferred from the Tacconi et al. (2020)
empirical relation could be over-estimated. Many galaxies in our
sample, despite having good fits with low residuals, have clumpy
morphologies, which could indicate clumpy SF and/or final stages
of a merger. Irregular morphologies can bias the measurement of the
rotational velocity due to the simplistic nature of the rotation curve
model. Also, if a merger is in fact taking place, the stellar mass may
be over-estimated due to the multiple-system nature of the object.
In some of the systems, the Prospector posteriors for the stellar
mass are distinctively double peaked, with a lower mass solution
that would boost the 𝑓DM. Finally, based on the Prospector-inferred
SFHs, many of the systems are either in a burst or just went through
a burst, with a peak in SF over a ∼5 −10 Myr timescale. Especially
at low masses, these starbursts can disrupt the ordered rotation of the
gas, leading to low and/or bias measurements of 𝑣rot.
4.2 Baryonic surface density
In order to investigate the origin of the DM fractions measured in
our 𝑧= 4 −6 sample, we study their correlation with the baryonic
surface density Σbar, computed within the effective radius of the H𝛼
emission, 𝑟e:
Σbar(𝑟e) = 𝑀bar(𝑟< 𝑟e)
2𝜋𝑟2e
.
(15)
Our results are shown in Fig. 4, colour-coded by the stellar mass
and the gas fraction. The baryonic surface densities of our galaxies
Fit
This work
This work + RC100
𝛼
0.48 ± 0.08
0.28 ± 0.02
𝛽
−0.28 ± 0.04
−0.33 ± 0.02
𝛾
−0.05 ± 0.58
−0.38 ± 0.07
𝜎int
0.12 ± 0.03
0.11 ± 0.01
Table 2. Summary of the parameters for the power-law fit to the 𝑓DM−Σbar
relation (where the parameters are defined as in Eq. 14), as shown on Fig. 4.
span a narrow range of log(Σbar)[M⊙kpc−2] ≈8 −9. We find a
strong anti-correlation between 𝑓DM and log Σbar, with Spearman
rank correlation coefficient of 𝜌= −0.557 and p-value of 𝑝< 0.001.
We fit our 𝑓DM −log Σbar relation with a power law, replicating the
one used for the 𝑓DM −log(𝑀★[M⊙]) (Eq. 14). This parametrization
is motivated by studies of this relation at lower redshift (Wuyts et al.
2016; Genzel et al. 2017; Nestor Shachar et al. 2023), to provide a
more straightforward comparison. We present our best-fit parameters
in Tab. 2 and plot our best-fit relation in Fig. 4. It is clear that
our sample does not follow a clear power-law shape, which is also
highlighted by the large intrinsic scatter 𝜎int = 0.12 ± 0.03. The fit
is also driven by the higher 𝑓DM, which have smaller uncertainties.
This causes most of our lower 𝑓DM values to have smaller Σbar than
predicted by the best-fit curve.
The tight relation between DM fraction and Σbar has been found to
hold out to cosmic noon in observations (Wuyts et al. 2016; Genzel
et al. 2020; Nestor Shachar et al. 2023) as well as in simulations
(Übler et al. 2021). This trend links the baryons with underlying the
DM, and could be explained by a few different phenomena. The high
baryon densities could drive out the DM in the central region through
heating. Also, the high densities could be induced by a change in size
of the galaxy. In general, for a fixed DM halo and at fixed stellar
mass, a higher density of baryons in the centre will naturally result
in a lower DM fraction when compared to the same galaxy but more
diffuse.
Our measured correlation (𝜌≈0.56) is weaker than the one re-
ported by Nestor Shachar et al. (2023) at 𝑧∼2 (𝜌= 0.64) and 𝑧∼1
(𝜌= 0.84). The weakening of the correlation with redshift, as was
already observed at cosmic noon, points to an increase of galaxy-to-
galaxy diversity. In Fig. 4, we plot the best-fit relation from Wuyts
et al. (2016), and we find that our galaxies with the highest DM
fractions lie above this relation, as also highlighted by our best-fit
relation. At lower DM fractions, our sample aligns better with the
relation.
We also perform a fit combining our 𝑧= 4 −6 data with the 𝑧=
1−2.5 RC100 data from Nestor Shachar et al. (2023) to obtain better
constraints for the redshift evolution of the 𝑓DM −Σbar relation. Our
best-fit parameters are shown on Tab. 2. In our initial fit, the redshift
dependence was poorly constrained (𝛾= −0.05±0.58), but when we
include the RC100 data, we can constrain the increase of 𝑓DM with
redshift (𝛾= −0.38±0.07). We investigate the origin of this increase
of 𝑓DM at fixed Σbar through the colour-coding by 𝑀★and 𝑓gas in the
two panels of Fig. 4, and comparing our sample to the Nestor Shachar
et al. (2023) sample at cosmic noon. We do not colour-code by 𝑟e
because there is little overlap in sizes between the two samples,
with our galaxies spanning 𝑟e ≈0.3 −3 kpc and the RC100 sample
𝑟e ≈3−10. The RC100 sample has masses log(𝑀★[M⊙]) ≈10−11,
gas fractions typically 𝑓gas ⪅0.8, and 𝑟e > 3 kpc. It is clear that our
sample occupies a different parameter space, with lower masses,
higher gas fractions, and smaller sizes. Interestingly, our baryonic
surface densities log Σbar ≈8 −9 have significant overlap with those
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
9
7
8
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10
log
bar(Re)[M kpc 2]
0.0
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p = 0.0
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10
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bar(Re)[M kpc 2]
= -0.557
p = 0.0
8
9
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11
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0.2
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fgas
This work (Running medians)
This work (z ∼5)
This work + RC100 (z ∼2)
Wuyts+16 (fit)
RC100 medians
DeGraaff+24b
DiskMass (z~0)
ATLAS 3D
Wuyts+16
Figure 4. DM fraction as a function of baryonic surface density (Σbar) for our sample (circles) and samples at lower (diamonds, Nestor Shachar et al. 2023) and
similar to higher (pentagons, de Graaff et al. 2024b) redshift. We colour-code points by stellar mass (left panel) and gas fractions (right panel). We show the
characteristic uncertainty for our sample in grey. We find a strong negative correlation between 𝑓DM and Σbar (𝜌= −0.551, 𝑝< 0.001), with a best-fit relation
(solid purple line) steeper than Wuyts et al. (2016) at cosmic noon (dashed brown line), and with an offset. We also plot the best-fit relation obtain when fitting
our points with those from Nestor Shachar et al. (2023) (dashed-dotted red line). For comparison, we plot medians from ETGs at 𝑧∼0 from the ATLAS3D
survey (purple square; Cappellari et al. 2013a) and LTGs from the DiskMass Survey (grey triangles; Martinsson et al. 2013). Our galaxies agree with Nestor
Shachar et al. (2023) at low fractions 𝑓DM < 0.5, but lie above the relation at high fractions. The galaxies in our sample are smaller, less massive, and more gas
rich than those probed at cosmic noon, driving the apparent shift in the relation.
of cosmic noon galaxies. This implies high densities Σbar at high
redshift, in part driven by the smaller sizes 𝑟e (e.g. Shibuya et al.
2015; Ward et al. 2024). In addition, the gas content is higher and
more centrally concentrated, further increasing the baryonic densities
in the inner regions. The shift to higher 𝑓DM at fixed Σbar could
therefore be driven by the smaller stellar masses, a direct consequence
of the stellar-to-halo-mass relation where 𝑀★/𝑀halo decreases with
decreasing 𝑀★(Moster et al. 2010; Behroozi et al. 2010; Tacchella
et al. 2018; Behroozi et al. 2019).
The high baryonic surface densities within 𝑟e at high redshift are
consistent with the build-up of bulges. In Danhaive et al. (2025b),
the study of multi-wavelength sizes and their ratios at 𝑧∼5 supports
the growth of galaxies through central starbursts instead of smooth
inside-out growth. This is consistent with bulges forming in these
relatively low-mass galaxies. At cosmic noon, star-forming galaxies
have been shown to predominantly grown inside-out (Nelson et al.
2016), consistent with stars forming in gas disks around an older
stellar bulge. The formation of these stellar disks is also reflected
in the rapid increase of sizes (𝑟e) at cosmic noon (Shibuya et al.
2015). In this context, the decrease of 𝑓DM, at fixed Σbar, with cosmic
time reflects the mass build-up of bulges which begin to dominate
the central regions of the galaxy. Also, the baryons extend further
out into the halo, through inside-out growth, where the DM density
decreases, which further decreases 𝑓DM.
Assuming galaxies grow along the SFMS, a log(𝑀★[M⊙]) = 9
galaxy from our sample at 𝑧∼5 would grow to log(𝑀★[M⊙]) ≈
10.5 −11 by cosmic noon, similar to the RC100 galaxies. If they
maintain high DM fractions in their cores, they would evolve into a
separate population from these cosmic noon galaxies, more akin to
late-type galaxies (LTGs) in the local Universe (Barnabè et al. 2012;
Martinsson et al. 2013). The subset of the galaxies in our sample with
𝑓DM < 0.5 would instead be candidate progenitors for the baryon-
dominated galaxies at cosmic noon, having also similar gas fractions,
as they occupy a similar region on the 𝑓DM −Σbar plane. As discussed
in Nestor Shachar et al. (2023), these galaxies are expected to become
ETGs seen in the local Universe (Cappellari et al. 2013a). These
different evolutionary tracks are consistent with the large scatter
seen in our sample, suggesting a variety of different populations.
However, DM fractions are not expected to stay constant within a
galaxy’s evolutionary track. In fact, their dependence on mass (Fig.
MNRAS 000, 1–17 (2015)
10
A. L. Danhaive et al.
0
2
4
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8
zspec
0.0
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This work (log(M */M ) > 9)
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de Graaff+24b
Wuyts+16
Genzel+20
Lee+25 ([CII])
DiskMass (z~0)
ATLAS 3D
Nestor-Shachar+23
McClymont+25c (THESAN-ZOOM)
de Graaff+24b (TNG50)
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
log(M [M ])
Figure 5. DM fraction as a function of redshift for our sample (circles)
and samples from the literature (Nestor Shachar et al. 2023; de Graaff et al.
2024a,b). We plot the evolution measured in Nestor Shachar et al. (2023)
(dashed blue line), and its extrapolation to 𝑧> 2.5 (dotted blue line). Our
sample medians lie well above those at cosmic noon, although our sample
shows large diversity in 𝑓DM and probes ∼2 −3 dex lower stellar masses.
Qualitatively, we compare our medians (log(𝑀★[M⊙]) ≈9) with the me-
dian evolution of 𝑓DM(𝑟< 𝑟e) for tng50 galaxies (de Graaff et al. 2024b)
selected at 𝑧= 6 with 8 < log(𝑀★[M⊙]) < 9 (thick line with 16th and
84th percentiles shown in the grey shaded region). Similarly, we plot the
median evolution for thesan-zoom galaxies (McClymont et al. 2025c) with
log(𝑀★[M⊙]) = 7 ± 0.5 at 𝑧= 9 (thick line with 16th and 84th percentiles
shown in the green shaded region). Although we expect redshift evolution of
𝑓DM(< 𝑟e) due to the evolution of sizes with redshift, the strong dependence
of stellar mass is evident here.
3) implies that as galaxies grow, their DM fraction changes. Also,
the effective radii increase with cosmic time, which also affects the
measurements. We investigate the redshift evolution of 𝑓DM in the
next section.
4.3 Redshift evolution of 𝑓DM
In Fig. 5, we plot our inferred DM fractions as a function of redshift.
We plot our sample medians at 𝑧= 4.5 and 𝑧= 5.5, for the lower-
mass (log(𝑀★[M⊙]) < 9) and higher-mass (log(𝑀★[M⊙]) > 9)
galaxies. Although our sample spans a wide range of 𝑓DM, we find
high fractions on average ( 𝑓DM > 0.5). For the high-mass galaxies,
with median log(𝑀★[M⊙]) = 9.5, we find 𝑓DM = 0.51+0.17
−0.03 and
𝑓DM = 0.55+0.06
−0.01 at 𝑧= 4.5 and 𝑧= 5.5, respectively. For the low-
mass galaxies, with median log(𝑀★[M⊙]) = 8.5, we find 𝑓DM =
0.81+0.04
−0.19 and 𝑓DM = 0.65+0.13
−0.20. In both cases, we do not see evidence
for a redshift evolution, as our medians at 𝑧= 4.5 and 𝑧= 5.5 are
consistent within the uncertainties. This is expected given our poorly
constrained 𝛾values (Tab. 1).
We now place our measurements in the context of work at cosmic
noon and the local Universe. In the same redshift range 𝑧≈4 −
6, Lee et al. (2025) report a wide range of 𝑓DM for their higher
mass sample (log(𝑀★[M⊙]) ≈9 −10.5), but their higher 𝑓DM
systems are consistent with our high-mass medians. However, they
also report galaxies that have large baryon contents ( 𝑓DM ⪅0.2).
We also find similar systems in our sample (Fig. 3) even though
our medians lie at higher 𝑓DM. The systems from Lee et al. (2025)
are therefore consistent with being a subset of the more extensive
population probed in this work. At higher redshift (𝑧≈5 −8),
measurements from de Graaff et al. (2024a,b), who find 𝑓DM > 0.5
for their sample of log(𝑀★[M⊙]) ≈8 −9, are broadly consistent
with our low-mass medians. On the other hand, the medians for
log(𝑀★[M⊙]) ≈10−11 galaxies at cosmic noon (Genzel et al. 2020;
Nestor Shachar et al. 2023) show low DM fractions ( 𝑓DM ≈0.2−0.4).
When combining all of these measurements together, the resulting
picture suggests that mass plays a significant role in determining
the DM content within the central regions of galaxies. The massive
systems at cosmic noon are predominantly baryon-dominated, as is
also seen in more massive systems at 𝑧∼5. On the other hand,
low-mass galaxies seem more DM-dominated.
To further investigate this, we plot the median tracks from the
tng50 (de Graaff et al. 2024b) and thesan-zoom (McClymont
et al. 2025c) simulations. Specifically, for tng50, we plot the me-
dian evolution of 𝑓DM(𝑟< 𝑟e) for galaxies selected at 𝑧= 6 with
8 < log(𝑀★[M⊙]) < 9. The thesan-zoom track is computed as a
median from galaxies with log(𝑀★[M⊙]) = 7 ± 0.5 at 𝑧= 9. This
comparison with simulations is qualitative, as the radii 𝑟e are defined
and measured differently in observations and simulations. Observa-
tionally, we are only able to measure the 2D half-light radius, whereas
simulations define 𝑟e as the 3D stellar half-mass radius. The choice
of 𝑟e will have an impact on the measured fractions and the derived
comparisons. Despite this not being an apples-to-apples comparison,
it is informative to explore its implications.
As discussed in de Graaff et al. (2024b), tng50 predicts that low-
mass galaxies are DM dominated at every epoch, and the observed
evolution of 𝑓DM is driven by the masses probed. In fact, the tracks
on Fig. 5 show that a single galaxy goes from being DM-dominated
at 𝑧∼6 to being baryon-dominated at 𝑧∼2. This is also consis-
tent with the tracks from thesan-zoom, although these simulations
only reach 𝑧= 3. This has strong implications for the evolutionary
tracks of galaxies, since it suggests that the DM-dominated low-
mass galaxies probed at high redshift are direct progenitors of the
baryon-dominated cosmic noon galaxies. From cosmic noon, the
DM-dominated systems settle as the DM-dominated LTGs in the lo-
cal Universe, whereas the baryon-dominated systems could deplete
their gas and settle in to the ETGs (Nestor Shachar et al. 2023). A
detailed forward-modelling of the simulations is needed to conduct a
better comparison with the increasing number of observed fractions.
5 DISCUSSION
The study of DM fractions, within the effective radius, at cosmic noon
(𝑧≈1 −3) unveiled that the massive (log(𝑀★[M⊙]) = 10 −11)
star-forming galaxy population is predominantly baryon-dominated
(Genzel et al. 2017, 2020; Nestor Shachar et al. 2023). In this work,
we push the study of 𝑓DM to 𝑧≈4 −6, where we are directly probing
progenitors of these cosmic noon systems. In Sec. 4.1 and Sec. 4.2,
we presented our DM fractions and their dependence on stellar mass
and baryonic surface density, investigating the physical drivers for the
offsets we observe with respect to cosmic noon galaxies. In Sec. 4.3,
we put our measured 𝑓DM in the context of galaxies across cosmic
time and predictions from tracks of cosmological simulations.
In this section, we qualitatively explore what our measurements
can teach us about the underlying DM halo, and specifically what
constraints we can place on the shape of its density distribution in
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
11
the central regions (Sec. 5.1). From this, we hope to shed light on the
consistency of different estimations of the galaxy-halo connection.
We also discuss the different sources of uncertainties in our baryonic
(Sec. 5.2) and DM (Sec. 5.3) mass measurements. Finally, we also
consider the contribution of black holes to our measured fractions in
Sec. 5.4.
5.1 Empirical model predictions
Our modelling, which is limited to probing rotational curves out to
≈1 −2 𝑟e, does not allow for a detailed recovery of the underly-
ing DM density profiles of our galaxies. Ideally, rotation should be
probed further out to several 𝑟e. However, the measure of the DM
fractions within 𝑟e can provide constraints on the DM density profile
via comparisons to empirical predictions, which require various as-
sumptions. We explore these constraints in this section, but caution
that they remain qualitative and reliant on assumed relations.
The stellar-to-halo mass (SMHM) relation has been studied in
detail both in empirical models (Behroozi et al. 2019; Tacchella
et al. 2018) and in simulations (Ceverino et al. 2017; Ma et al. 2018),
with constraints from observations (Kravtsov et al. 2018; Girelli et al.
2020; Shuntov et al. 2022, 2025). This relation links the halo mass
of galaxies to their stellar mass at a given redshift, allowing us to
estimate the halo masses for the galaxies in our sample. In order to
estimate the halo masses for our galaxies, we adopt the Behroozi
et al. (2019) SMHM relation, which has a double power-law shape
with an added Gaussian:
log10
𝑀∗
𝑀1
= 𝜀−log10
10−𝛼𝑥+ 10−𝛽𝑥
+𝛾exp
−0.5
𝑥
𝛿
2
(16)
where 𝑥≡log10
𝑀peak
𝑀1
, 𝑀peak is the peak halo mass and 𝑀1 is a
characteristic mass. The free parameters 𝑀1, 𝜖, 𝛼, 𝛽and 𝛾all have
redshift-dependent expressions, and 𝛿is a constant. Once we have
inferred the halo mass by numerically solving Eq. 16, we can derive
the corresponding virial radius:
𝑅vir =
4
3
𝜋200𝜌𝑐
𝑀halo
−1/3
,
(17)
since 𝑅vir is defined as the radius where the density is 200 times the
critical density 𝜌𝑐of the Universe at that redshift, and 𝑀halo is the DM
mass within the virial radius. For the shape of the DM density profile,
we assume a generalized Navarro–Frenk–White (NFW, Navarro et al.
1997) profile:
𝜌DM(𝑟) =
𝜌0
(𝑟/𝑟𝑠) 𝛼(1 + 𝑟/𝑟𝑠)3−𝛼,
(18)
where 𝛼= 1 corresponds to a classic NFW profile. Given these
assumptions, we can constrain the value of 𝛼needed to reproduce
our measured DM fractions 𝑓DM within the effective radius. We
restrict our values between 𝛼= 0 and 𝛼= 3, above (below) which
the profile becomes convex (concave) in logarithmic space. In fact,
for 𝛼> 3, Eq. 18 develops an upturn near 𝑟= 𝑟𝑠, as the outer slope
(3 −𝛼) becomes negative. Conversely, when 𝛼< 0, the inner slope
(𝛼) becomes negative, producing a downturn at 𝑟= 𝑟𝑠. Within the
allowed range 0 ≤𝛼≤3, the profile smoothly transitions from a
nearly flat core (𝛼= 0) to a progressively steeper slope that becomes
(log-)linear at 𝛼= 3.
The only additional unknown is the concentration 𝑐, defined as
𝑅vir = 𝑐𝑅s
(19)
where 𝑅s is the scale radius. Given our other assumptions, the value
of 𝑐effectively re-normalizes the predicted DM fractions for all of
our systems, increasing them as 𝑐increases. DM only simulations
predict low concentrations at 𝑧∼5 (Dutton & Macciò 2014), so we
set 𝑐= 5.
In Fig. 6, we present the estimated values of 𝛼needed to reproduce
our measured DM fractions within the effective radius. We find that ≈
30% of our sample lies below 𝛼= 1, implying that their DM fractions
are best reproduced by a profile that is more cored than the classical
NFW. Evidence for DM cores has already been found at cosmic noon
(Genzel et al. 2020; Nestor Shachar et al. 2023), where samples
are predominantly baryon-dominated and require cored profiles to
reproduce their low 𝑓DM. Aside from more extreme solutions such
as the modification of the nature of DM (Hu et al. 2000; Calabrese
& Spergel 2016) or of our gravity model (Milgrom 1983), a more
natural explanation for such cores is a strong interaction between the
baryons and the DM within galaxies. Specifically, this would require
kinetic heating of the DM in the central regions to drive it outwards.
This could be achieved through effective feedback mechanisms from
star formation and black holes. Another possibility is that some of
these low 𝑓DM galaxies are in a post-merger phase, with disturbed
kinematics and diffused DM haloes. This is plausible given that
most of the galaxies in the 𝛼< 0 region have higher stellar masses
(log(𝑀★[M⊙]) > 9).
On the other hand, high central baryonic concentrations are ex-
pected to perturb the underlying DM distribution and pull the DM
towards the centre, increasing the "cuspiness" of its density profile.
This phenomenon is called adiabatic contraction (Blumenthal et al.
1986), and is represented by the 𝛼> 1 region of Fig. 6. For galaxies
in this region, adiabatic contraction is needed to explain the high 𝑓DM
values measured in this work. Interestingly, this under-prediction of
𝑓DM, compared to our observations, holds for many other SMHM
relations from the literature which typically lie above the Behroozi
et al. (2019) one (e.g. Tacchella et al. 2018). In order to bridge some
of the discrepancies between the model predictions and our measure-
ments, a lower SMHM relation at high redshift would be needed. This
problem is emphasized by recent predictions from observations and
simulations of high DM fractions in low-mass galaxies (de Graaff
et al. 2024b; McClymont et al. 2025c). However, it is important to
note that our high 𝛼⪆1.5 values suffer from large uncertainties that
make them consistent with 𝛼≈1.5. These stem from the smaller
masses of these systems, which make their morphology and kine-
matics more difficult to constrain.
We colour-code our points in Fig. 6 by their SFHs, parametrized by
the ratio of SFR averaged over 10 and 100 Myr. Galaxies with rising
SFHs will have positive values of SFR10/SFR100, whereas galaxies
with falling SFHs will show the opposite behaviour. In order to vi-
sually asses the presence of an underlying trend, we combine our
points in larger bins and use the LOESS method (Cappellari et al.
2013b) to average over single objects and obtain mean estimates for
the full sample. We can see a trend of galaxies going through a burst
log SFR10/SFR100 ⪆0.4 having preferentially higher values of 𝛼.
This apparent correlation supports the adiabatic contraction scenario
that would occur with the inflow of baryons to the central regions
during a burst, during which the gas content of the galaxy contracts
(Dekel & Burkert 2014; Tacchella et al. 2016b; McClymont et al.
2025b). When feedback from the star formation kicks in, driving the
gas out of the centre and temporarily quenching star formation, galax-
ies begin to move back down towards the MS log SFR10/SFR100 < 0.
These galaxies appear to preferentially have cored profiles, consistent
with kinetic heating of the DM. This interpretation would imply the
interaction of baryons and DM on relatively short timescales (𝑡≈100
MNRAS 000, 1–17 (2015)
12
A. L. Danhaive et al.
0.0
0.2
0.4
0.6
0.8
1.0
fDM(r < Re)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
adiabatic
contraction
This work
NFW ( =1)
Running median
7
8
9
10
11
log(M [M ])
adiabatic
contraction
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
logSFR10/SFR100
Figure 6. Estimated slope 𝛼of the inner DM density profile for a modified NFW distribution (Eq. 18) as a function of measured DM fractions (left) and stellar
mass (right). A value of 𝛼= 1 represents the classical NFW profile, whereas 𝛼< 1 corresponds to a cored profile, and 𝛼> 1 to a cuspier profile. Under
the assumed SHMR and concentration (𝑐= 5), we find that low-mass galaxies with high 𝑓DM are associated with cuspy distributions, converging towards an
NFW-like profile at higher masses with a scatter at least in part driven by their star-formation state. To highlight this, we colour-code the galaxies by the ratio of
their SFR averaged over 10 and 100 Myr, SFR10/SFR100, applying LOESS smoothing (Cappellari et al. 2013b). Galaxies going through a burst of star formation
(log SFR10/SFR100 ⪆0.4) have preferentially high values of 𝛼, which supports the adiabatic contraction scenario.
Myr), which has yet to be thoroughly explored in cosmological sim-
ulations. Interestingly, we also find that the values of 𝛼correlate with
the Sérsic index 𝑛measured from the stellar light distribution in the
UV (Fig. A1). Galaxies with cuspier DM profiles also have steeper
central light profiles, with 𝑛∼4, which could point to interactions
between the two components.
We find that low-mass galaxies log(𝑀★[M⊙]) < 9 have cus-
pier profiles, while higher mass galaxies seem to converge towards
NFW profiles (𝛼= 1) with scatter driven by their star-formation
state (parametrized by SFR10/SFR100). This is not necessarily intu-
itive given the initial discovery of cored systems in dwarf galaxies
in the local Universe. However, this can be reconciled when consid-
ering burstiness. At high redshift, galaxies have been shown to grow
through bursts of star formation (e.g. Faucher-Giguère 2018; Tac-
chella et al. 2020; McClymont et al. 2025a; Simmonds et al. 2025)
at all masses. These bursts are tied to the formation of cores through
energetic stellar feedback smoothing the DM profile in the central
regions of galaxies. The more massive galaxies in our sample will
have undergone more bursts in their lifetime, leading to a flattening
of their DM profiles and resulting in the more cored profiles we
predict in this work. Also, these galaxies could undergo breathing
cycles of compaction phases (Tacchella et al. 2016a; El-Badry et al.
2016). Interestingly, Kohandel et al. (2025) find that one of their most
massive galaxies ("Amaryllis", log(𝑀★[M⊙]) = 10.3 at 𝑧= 7) has
a cored inner DM density profile which reflects the cusp–flattening
impact of baryonic feedback.
In the local Universe, star formation is smoother, with massive
star-forming galaxies growing more smoothly inside out once their
central bulges have developed. These massive galaxies undergo long-
lived stable growth episodes that could allow their DM profile to
stabilise around NFW profiles. On the other hand, dwarf galaxies
still have bursty star formation (Hopkins et al. 2014; Hayward &
Hopkins 2017; Hopkins et al. 2023), giving rise to their cores. It is
also important to once again consider the evolution of sizes. These
dwarf galaxies have larger radii than low-mass galaxies at the same
mass in our sample. This naturally implies higher concentrations,
which could fuel processes of adiabatic contraction for which we see
evidence in Fig. 6.
Finally, we assess how sensitive our results are to the adopted
assumptions and discuss their potential impact on our conclusions.
Starting with the concentration, changing the value of 𝑐affects the
distribution of 𝛼we find. Specifically, higher concentrations 𝑐>
5 require more galaxies to have cored profiles 𝛼< 1 in order to
reconcile the predicted fractions 𝑓DM with our measured ones. In
contrast, lower concentrations, 𝑐< 3, would have the opposite effect.
The most important choice is the SMHM function, which directly
affects the normalization of our profiles and, hence, of the 𝛼values
that we infer. Studies based on simulations and empirical models at
high redshift suggest that the normalization of the SMHM relation is
higher than the Behroozi et al. (2019) one assumed in this work (e.g.
see Tacchella et al. 2018). This means that the halo mass 𝑀halo at fixed
stellar mass 𝑀★is lower than predicted by the Behroozi et al. (2019)
relation, meaning our profiles would need to be even cuspier (higher
𝛼) to reproduce our high 𝑓DM values. A last assumption is the choice
of profile for the fit, as other profiles such as the Einasto profile
(Einasto 1965), have also been shown to provide good fits to DM
profiles in cosmological simulations (Dutton & Macciò 2014). We
explored this profile and found no significant change in the observed
trends of cuspiness with 𝑓DM, SFR10/SFR100, and 𝑀★. However, our
points become more scattered in these parameter spaces. We chose
to study the modified NFW profile in the most detail because of the
direct comparison it offers with the NFW (𝛼= 1) case (Eq. 18).
Although it is informative to discuss empirical model predictions
in the context of our results, it is also important to discuss caveats
of the measurements that could potentially reconcile our inferred
profiles with shapes closer to NFW. We discuss these in the next two
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
13
sections, focusing on the uncertainties in the baryonic and the DM
contents.
5.2 Uncertainties in the baryonic content
When calculating baryonic masses needed to infer gas and DM frac-
tions, we need to make some assumptions, of which we will now
discuss the implications. First, the stellar mass is inferred using the
SED-fitting code Prospector, a method with many advantages, but
also some degeneracies. We attempt to break some of the degen-
eracies by fixing the redshift to the grism spectroscopic redshift,
and by simultaneously fitting the photometry with the line fluxes of
the available emission lines (see Sec. 2). Despite this, we are not
fitting spectra and do not have strong constraints for all emission
lines, for instance, those constraining nebular metallicities and dust
content. Furthermore, our S/N cut implies selecting galaxies whose
young stellar population is dominating the SED. This may hamper
the detection of the underlying population of older stars, and hence
constraining the full SFH. This effect is called ’outshining’ and can
cause an underestimate of the stellar masses, effectively only attribut-
ing the stellar mass to the more visible young stellar population (Bell
& de Jong 2001; Maraston et al. 2010; Leja et al. 2019; Tacchella
et al. 2023; Giménez-Arteaga et al. 2023). This would directly affect
the measurements of 𝑓DM < 0, alleviating the tension by increasing
the total baryonic mass.
Uncertainties in the stellar mass and SFRs are propagated into
the gas mass estimates derived using the scaling relation by Tac-
coni et al. (2018, 2020). This relation is calibrated out to 𝑧∼5.5,
and should hence provide a better estimate of the gas in our galax-
ies than relations calibrated at much lower redshifts (e.g. Kennicutt
1998). Nonetheless, the use of scaling relations does not necessar-
ily encompass the large variety of gas fractions observed at fixed
mass (e.g. McClymont et al. 2025c), and could introduce biases. Im-
portantly, the Tacconi et al. (2020) relation is only calibrated above
log(𝑀★[M⊙]) ⪆9. The extrapolation to the lower masses in our
sample results in very high gas fractions 𝑓gas ∼1.
5.3 Uncertainties in the DM content
The kinematic measurements, described in Danhaive et al. (2025a),
are the main factor driving the measurement of 𝑓DM through their
estimate of the dynamical mass. The key assumption made when
deriving the dynamical mass is that the measured velocity gradients
are tracing rotation of the gas around the galaxy. However, especially
at low masses, we cannot rule out the contribution of non-circular
motions, typically in the form of outflows (Carniani et al. 2024; Ivey
etal.2025), to our measurementofvelocitygradientsanddispersions.
Such a contribution would bias both quantities to higher values and
unphysically boost the inferred dynamical masses, and hence the DM
fractions. Also, albeit of smaller importance, the circular velocities
needed to compute the dynamical masses are inferred assuming a
virialised rotating disk with an exponential light profile. For galaxies
with larger Sérsic indices, this assumption can lead to biased results.
Furthermore, for pressure dominated systems, the choice of the pres-
sure support term multiplying the 𝜎0 in Eq. 7 can lead to over or
under-estimates of the circular velocity (see Price et al. 2022, for
detailed analysis).
For the derived relations for both the gas and importantly the
DM fractions, we note that we suffer from incompleteness at the
low-mass end (log(𝑀★[M⊙]) < 9), which could significantly bias
our results. In order to obtain meaningful (i.e. resolved) kinematic
7
8
9
10
11
log(M [M ])
4
5
6
7
8
9
10
11
log(MBH)[M ]
0.001
0.01
0.1
1
This work
Reines & Volonteri+15
High-z
McClymont+25c (THESAN-ZOOM)
0.0
0.2
0.4
0.6
0.8
1.0
fDM(r < Re)
Figure 7. Predicted black hole masses 𝑀BH, assuming the local 𝑀BH −𝑀dyn
relation from Kormendy & Ho (2013), as a function of stellar mass (circles).
We compare our measurements with the overmassive black holes reported
at high redshift (𝑧≈4 −11) from JWST (pink dots; Carnall et al. 2023;
Kokorev et al. 2023; Harikane et al. 2023; Matthee et al. 2024; Übler et al.
2023; Maiolino et al. 2024b,a; Übler et al. 2024; Greene et al. 2024; Furtak
et al. 2024; Juodžbalis et al. 2024; Natarajan et al. 2024), which lie well above
the local relation (purple dashed line; Reines & Volonteri 2015). The dashed
grey lines represent constant ratios of 𝑀BH/𝑀★. We find excellent agreement
with the parameter space spanned by these sources, suggesting the shift in the
𝑀BH −𝑀★relation is consistent with the high gas and DM fractions found in
high redshift galaxies. This is consistent with results from McClymont et al.
(2025c) following a similar approach to this work but with the thesan-zoom
simulations.
measurements for low-mass systems, they need to be not only bright
in H𝛼, but also have relatively large rotational velocity and/or velocity
dispersions. This could cause our 𝑓gas to be biased high (due to the
high SFR requirement for the bright H𝛼) as well as our 𝑓DM (due to
the high circular velocities). Accounting for these selection effects
would move our observed medians closer to the simulations in Fig.
3. Within the paper, we discuss the observed trends, but cannot
make definitive conclusions about the behaviour at the low masses.
This also translates to incompleteness for galaxies with low baryonic
surface densities.
In this discussion, there is a last component that we have not con-
sidered, namely the presence of BHs in our galaxies. We investigate
this possibility in the next section.
5.4 Overmassive BHs
Although BHs typically represent a small fraction of the total mass,
with 𝑀BH/𝑀★< 1% at 𝑧∼0 (Reines & Volonteri 2015), they could
have non-negligible effects on the kinematics of their host galaxies
through potentially disruptive feedback mechanisms. Furthermore,
many recent studies have reported over-massive BHs in the early
Universe, with high 𝑀BH/𝑀★ratios (Carnall et al. 2023; Kokorev
et al. 2023; Harikane et al. 2023; Übler et al. 2023; Maiolino et al.
2024b,a; Übler et al. 2024; Furtak et al. 2024; Natarajan et al. 2024;
Juodžbalis et al. 2025a) and in some cases even high 𝑀BH/𝑀dyn
ratios (D’Eugenio et al. 2025; Ji et al. 2025; Juodžbalis et al. 2025b)
relative to the local scaling relations.
In order to investigate the presence of BHs in our galaxies, and
MNRAS 000, 1–17 (2015)
14
A. L. Danhaive et al.
quantify their potential contributions to our measured DM fractions,
we derive BH masses using the 𝑀BH−𝑀dyn relations from Kormendy
& Ho (2013), which has been shown to hold at high redshift for most
cases (Maiolino et al. 2024b; Juodžbalis et al. 2025b). We show our
results on Fig. 7.
We find BH masses spanning a wide range (log(𝑀BH [M⊙]) =
6.5 −9) and BH-to-stellar mass ratios (𝑀BH/𝑀★= 0.001 −1).
Interestingly, our sample overlaps with most of the over-massive
BHs reported at high redshift. This suggests that, given a fixed 𝑀BH−
𝑀dyn relation like we have assumed here, these over-massive black
holes (relative to the 𝑀BH −𝑀★plane) are a natural consequence
of the high gas and DM fractions found at high redshift (Fig. 4.1).
This is consistent with the results from McClymont et al. (2025c),
who applied a similar semi-empirical approach to the thesan-zoom
simulations, whose model does not include black holes, to predict
black hole masses in the post-processing. Our results imply that
ratios of 𝑀BH/𝑀★≈0.1 would be common at high-redshift, and
that black holes grow faster at early times, in comparison to the
stellar content, than at late times. Because both growth mechanisms
require gas inflow, the gas must collapse enough to accrete on to
the black hole but without forming many stars. Black hole growth
would need to differ at high redshift in order to reproduce these high
𝑀BH/𝑀★ratios. However, confirming the presence of black holes
at high redshift has proven difficult with our current emission line
diagnostics (e.g. Mazzolari et al. 2024; Scholtz et al. 2025; Ivey
et al. 2025), and various emission line ratios need to be detected
spectroscopically, which we do not probe with the grism data used
in this work. Also, many models expect that the bulk of the BH
population at high-𝑧should be dormant (e.g. Schneider et al. 2023;
Trinca et al. 2024), with short duty cycles for accretion, meaning that
we would not expect to see evidence of accreting BHs.
We note that the lower mass systems log(𝑀★[M⊙]) < 8 in our
sample have abnormally high dynamical masses, leading to high pre-
dictions of 𝑀BH, pointing to a likely contribution from non-circular
motions, as also shown in Sec. 3. For the remainder of the systems,
our findings imply that the BHs could have a significant impact on
their host galaxies. Although including their masses in our calculation
of 𝑓DM does not strongly affect these measurements, due to the small
𝑀BH/𝑀dyn values (Kormendy & Ho 2013), those who are accret-
ing could affect the kinematics of the surrounding gas and introduce
biases in the measurement of 𝑀dyn. If present, these AGNs could
be categorized as Type-II, narrow-line AGNs, and hence would not
be detected through the broadening of emission lines caused by the
broad-line region. Nonetheless, AGN-driven outflows in low-mass
galaxies would have similar velocities to those measured in this work
(𝑣∼100 −200 km/s, e.g. Ivey et al. 2025), and could hence boost
our measured DM fractions.
6 SUMMARY & CONCLUSIONS
This work presents the dynamical mass measurements of 163 H𝛼
emitters at 𝑧≈4−6 from the FRESCO and CONGRESS JWST grism
surveys in the GOODS fields. We model the kinematics of the H𝛼
emission line using forward-modelling and fitting of the grism data
with geko (Danhaive & Tacchella 2025), recovering in particular
rotational velocities and velocity dispersions. Using our measured
dynamical masses from our modelling, stellar masses inferred from
SED modelling, and gas masses estimated from the Tacconi et al.
(2020) scaling relation, we obtain the gas fractions ( 𝑓gas) and DM
fractions ( 𝑓DM) within the H𝛼half-light radius (𝑟e). We summarize
our main findings here:
– The log(𝑀★[M⊙])
≈
7 −10 star-forming galaxies in
our sample have relatively high dynamical masses in the range
log(𝑀dyn [M⊙]) ≈9 −11. At fixed stellar mass, there is large scat-
ter in the dynamical masses, indicative of unsettled kinematics. This
scatter naturally manifests itself in the Tully–Fisher plane, suggesting
that the relation is only beginning to emerge at 𝑧∼5.
– Our galaxies have, on average, high molecular gas and DM
fractions, with sample medians < 𝑓gas >= 0.77 and < 𝑓DM >= 0.73.
We find that ≈67% of our galaxies are DM dominated with their
𝑟e ∼0.5 −1 kpc, 𝑓DM > 0.5. Nonetheless, our fractions 𝑓DM span
the full range of the parameter space.
– We find evidence for a negative dependence of 𝑓DM on stellar
mass 𝑀★as parametrized by Eq. 14. Specifically, we find 𝛼= 0.28±
0.05, with low-mass (log(𝑀★[M⊙]) < 9) galaxies showing high DM
fractions and higher mass galaxies (log(𝑀★[M⊙]) > 9) showing a
larger diversity, with typically lower 𝑓DM.
– We find an anti-correlation (𝜌= −0.56, 𝑝< 0.001) between
𝑓DM and the baryonic surface density Σbar within 𝑟e, consistent with
but weaker than its counterpart at cosmic noon and the local Universe.
The galaxies in our sample have high baryonic surface densities
comparable to those of more massive galaxies at cosmic noon, caused
by their compactness and high central gas fractions.
– Our high DM fractions are consistent with the predicted pro-
genitor populations of 𝑧∼2 baryon-dominated systems, as shown by
a comparison with the tng50 and thesan-zoom simulations. These
high fractions are expected for low-mass galaxies at all redshifts.
– Assuming a modified NFW profile, a stellar-to-halo mass func-
tion, and a DM profile concentration, we qualitatively explore the
predicted shape of the underlying DM halo density profile for our
sample. We find that the higher-mass, baryon-dominated systems
would need a cored profile to reconcile their low fractions. This core
could have been induced by repeated bursts of star formation. In
contrast, the low-mass, high 𝑓DM systems are consistent with cuspier
DM profiles, suggesting adiabatic contraction pulling more DM into
the central regions.
– Finally, we find that our elevated 𝑓gas and 𝑓DM naturally antici-
pate the population of over-massive black holes found with JWST at
high-redshift, when assuming a 𝑀BH −𝑀dyn relation.
Our study extends, for the first time, measurements of DM frac-
tions to large statistical samples at high redshift, advancing beyond
previous analyses focused on ionised-gas kinematics at cosmic noon.
Although our inferences rely on several necessary assumptions, they
provide a valuable framework for placing spatially resolved mea-
surements of individual systems into a broader population context.
Most importantly, this work establishes a foundation for probing
DM haloes through high-redshift observations, which is an essential
step toward understanding how baryons and dark matter interact and
co-evolve during the early stages of galaxy formation.
ACKNOWLEDGEMENTS
We thank Hannah Übler and Andreas Burkert for the insightful dis-
cussions. ALD thanks the University of Cambridge Harding Distin-
guished Postgraduate Scholars Programme and Technology Facilities
Council (STFC) Center for Doctoral Training (CDT) in Data inten-
sive science at the University of Cambridge (STFC grant number
2742605) for a PhD studentship. ALD and ST acknowledge support
by the Royal Society Research Grant G125142. AJB acknowledges
funding from the "FirstGalaxies" Advanced Grant from the Euro-
pean Research Council (ERC) under the European Union’s Hori-
zon 2020 research and innovation programme (Grant agreement No.
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
15
789056). ECL acknowledges support of an STFC Webb Fellowship
(ST/W001438/1). FDE and RM acknowledge support by the Sci-
ence and Technology Facilities Council (STFC), by the ERC through
Advanced Grant 695671 “QUENCH”, and by the UKRI Frontier
Research grant RISEandFALL. EE, BDJ, MR, CNAW, and YZ are
supported by the JWST/NIRCam contract to the University of Ari-
zona NAS5-02105. DJE is supported as a Simons Investigator and by
JWST/NIRCam contract to the University of Arizona, NAS5-02105.
Support for program #3215 was provided by NASA through a grant
from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under
NASA contract NAS 5-03127. WM thanks the Science and Technol-
ogy Facilities Council (STFC) Center for Doctoral Training (CDT)
in Data Intensive Science at the University of Cambridge (STFC
grant number 2742968) for a PhD studentship. BER acknowledges
support from the NIRCam Science Team contract to the University
of Arizona, NAS5-02105, and JWST Program 3215.
This work is based on observations made with the NASA/ESA
Hubble Space Telescope and NASA/ESA/CSA James Webb Space
Telescope. The data were obtained from the Mikulski Archive for
Space Telescopes at the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astron-
omy, Inc., under NASA contract NAS 5-03127 for JWST. These ob-
servations are associated with program #1180, 1181, 1210 (JADES),
#1895 (FRESCO), # 1963 (JEMS) and #3577 (CONGRESS). Sup-
port for program #3577 was provided by NASA through a grant from
the Space Telescope Science Institute, which is operated by the Asso-
ciation of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-03127. The authors acknowledge the FRESCO team
for developing their observing program with a zero-exclusive-access
period. The authors acknowledge use of the lux supercomputer at
UC Santa Cruz, funded by NSF MRI grant AST 1828315.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable re-
quest to the corresponding author. Fully reduced NIRCam images
are publicly available on MAST (https://archive.stsci.edu/
hlsp/jades), with doi:10.17909/8tdj-8n28, doi:10.17909/z2gw-
mk31, and doi:10.17909/fsc4-dt61 (Rieke et al. 2023; Eisenstein
et al. 2023). The NIRCam grism spectra are publicly available
on MAST with doi:10.17909/6rfk-6s81 and doi:0.17909/gdyc-7g80
(Oesch et al. 2023).
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APPENDIX A: SUPPLEMENTARY FIGURES
In Figure A1, we present additional diagnostic plots that comple-
ment the main analysis. This figure illustrates the relation between
the inferred inner DM slope 𝛼and the DM fraction 𝑓DM, colour-
coded by stellar mass and Sérsic index. The trends shown here are
MNRAS 000, 1–17 (2015)
DM fractions of high-z galaxies
17
0.0
0.2
0.4
0.6
0.8
1.0
fDM(r < Re)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
adiabatic
contraction
This work
NFW ( =1)
Running median
7
8
9
10
11
log(M [M ])
adiabatic
contraction
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
nUV
Figure A1. Estimated slope 𝛼of the inner DM density profile for a modified NFW distribution (Eq. 18) as a function of measured DM fractions (left) and stellar
mass (right). A value of 𝛼= 1 represents the classical NFW profile, whereas 𝛼< 1 corresponds to a cored profile, and 𝛼> 1 to a cuspier profile. We find that
galaxies with seemingly steeper DM profiles (𝛼> 1) coincide with steeper light distributions (𝑛> 1) as probed by the UV stellar continuum. To highlight this,
we colour-code the galaxies by the Sérsic index 𝑛UV, applying LOESS smoothing (Cappellari et al. 2013b).
consistent with those discussed in Section 5, reinforcing that galaxies
with lower 𝑓DM tend to exhibit shallower inner density slopes, while
systems with higher 𝑓DM favour cuspier profiles. Although the scat-
ter is substantial, these correlations support the interpretation that
the interplay between baryonic concentration and halo response is
already shaping the inner DM structure at 𝑧∼5.
This paper has been typeset from a TEX/LATEX file prepared by the author.
MNRAS 000, 1–17 (2015)
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MNRAS 000, 1-17 (2015) Preprint 17 October 2025 Compiled using MNRAS LATEX style file v3.2 The dark side of early galaxies: geko uncovers dark-matter fractions at z∼4 -6 A. Lola Danhaive 1,2★, Sandro Tacchella 1,2, Andrew J. Bunker 3, Emma Curtis-Lake 4, Anna de Graaff 5, Francesco D'Eugenio 1,2, Qiao Duan 1,2, Eiichi Egami 6, Daniel J. Eisenstein 7, Benjamin D. Johnson 7, Roberto Maiolino 1,2, William McClymont 1,2, Marcia Rieke 6, Brant Robertson 8, Fengwu Sun 7, Christopher N. A. Willmer 6, Zihao Wu 7, Yongda Zhu 6 1Kavli Institute for Cosmology, 3 0HA, UK 2Cavendish Laboratory, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK 3 1 3RH, UK 4 Centre for Astrophysics Research, 10 9AB, UK 5 Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117, Heidelberg, Germany 6 Steward Observatory, 933 N. Cherry Avenue, Tucson, AZ 85721, USA 7 Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA 8 1156 High Street, Santa Cruz, CA 95064, USA Accepted XXX. Received YYY; in original form ZZZ ABSTRACT JWST/NIRCam slitless spectroscopy enables dynamical mass measurements for typical star-forming galaxies only a billion years after the Big Bang. We model the Hαmorpho-kinematics of 163 galaxies at redshift z≈4-6 from FRESCO and CONGRESS (with JADES imaging), using the geko code, and infer rotational velocities and dispersions within re. Our sample spans log M★≈7-10 and log Mdyn ≈9-11. Gas masses are estimated via scaling relations, yielding baryonic masses and dark-matter (DM) fractions fDM(r 4 can provide constraints on the properties of the underlying haloes and their relative contribution to the total mass of the galaxy. The detailed mapping of rotation curves of galaxies in the local Universe has revealed that log(M★[M⊙]) = 10 -11 star-forming galaxies are baryon-dominated in their central ∼1 kpc, but DM dominated within their half-light, or effective, radii re (Rubin et al. 1985; Martinsson et al. 2013). On the other hand, early-type galaxies (ETGs) at z∼0 are heavily baryon-dominated within re, suggesting a diverging mass assembly history (Noordermeer et al. 2007; Cappellari et al. 2013a; Serra et al. 2016). Interestingly, studies of DM fractions at cosmic noon (z≈2) in log(M★[M⊙]) ⪆9.5 galaxies found them to be similar to local ETGs, with baryon-dominated cores (Price et al. 2016; Wuyts et al. 2016; Genzel et al. 2017, 2020; Nestor Shachar et al. 2023). Genzel et al. (2020) interpret these findings as evidence for cored DM profiles, motivated by the rapid formation of massive haloes and galaxies at z∼1 -3. In fact, cosmic noon marks the peak of the cosmic star-formation rate (SFR) density (Madau & Dickinson 2014), where gas accretion rates reached peak values (Tacconi et al. 2020) and stellar mass doubling scales were on the order of t 2 (Juodžbalis et al. 2025b). If the relatively low-mass galaxies at z> 4 host massive black holes which experience accretion phases, then they could play a significant role in the galaxy's mass assembly history. Importantly, active galactic nuclei (AGN)-driven outflows can heavily disrupt the surrounding gas, particularly in lowmass galaxies (Sijacki et al. 2007; Nelson et al. 2019; Koudmani et al. 2021, 2022; Carniani et al. 2024). The presence of such AGNs is important to study as it currently introduces more uncertainties in the interpretation of our kinematic measurements. With the synergy of NIRCam imaging and grism data, the kinematics of ∼200 galaxies at z= 4 -6 were measured in Danhaive et al. (2025a) using a new forward-modelling Bayesian code, the Grism Emission-line Kinematics tOol (geko, Danhaive & Tacchella 2025)1. This sample is comprised of log(M★[M⊙]) ≈7 -10.5 starforming galaxies, with a wide range of rotational support v/σ0. As shown in Danhaive et al. (2025a), many of these galaxies have large dynamical masses compared to their stellar masses, pointing to large contributions from gas and DM. In this paper, we study the mass content of these galaxies in detail, placing the first statistical constraints on DM fractions at high redshift z> 4. We briefly introduce our sample and methodology in Sec. 2, then present our dynamical mass measurements and discuss in the context of the Tully-Fisher (TFR) relation in Sec. 3. We present our gas and DM fractions in Sec. 4, and compare them to samples at lower redshift. In Sec. 5, we interpret our measurements in the context of the shapes of DM density profiles, 1 Available at https://github.com/angelicalola-danhaive/geko MNRAS 000, 1-17 (2015) DM fractions of high-z galaxies 3 and explore potential contributions from BHs. Finally, we summarise our results in Sec. 6. Throughout this work, we assume Ω0 = 0.315 and H0 = 67.4 km s-1 Mpc-1 (Planck Collaboration et al. 2020). 2 OBSERVATIONS, SAMPLE AND METHODOLOGY In this section, we present the NIRCam grism and imaging data used in this work, along with a description of our sample (Sec. 2.1). We also describe the morpho-kinematic modelling of our sample (Sec. 2.2), which is used to measure the rotational velocities and velocity dispersions of our galaxies. This sample, and the associated kinematic measurements, are described in Danhaive et al. (2025a), so we only summarize them here but refer the reader to that work for more details. Finally, in Sec. 2.3, we detail our derivation of the gas and DM fractions for our sample. 2.1 NIRCam data and sample selection The analysis in this work uses both NIRCam imaging and grism spectroscopy. The NIRCam grism data is obtained from the FRESCO survey (PI: Oesch, PID: 1895; Oesch et al. 2023; Covelo-Paz et al. 2025) in GOODS-S and GOODS-N, and the CONGRESS survey (PIs: Egami & Sun, PID: 3577; Sun et al. in prep) in GOODS-N. These surveys are comprised of grism data (R mode) in the F444W and F356W filters, respectively, probing Hαemission at z= 3.8-6.5. The data is reduced following Sun et al. (2023) and Helton et al. (2024) to obtain 2D spectra for each galaxy in the field of view. The 2D spectra are then continuum subtracted following the 2-step iterative technique described in Kashino et al. (2023) to obtain 2D emission line maps for Hα. In the regions of the FRESCO and CONGRESS surveys, there is a wealth of deep imaging data from the JADES survey (Eisenstein et al. 2023; Rieke et al. 2023). Specifically, the JADES survey has imaging in most of the NIRCam wide bands, as well as the F335M and F410M medium bands. In addition, the FRESCO survey obtained imaging in the F182M and F210M bands, and CONGRESS in F090W and F115W, complementing the existing JADES imaging data. For certain regions in our sample, we also have additional medium bands F182M, F210M, F430M, F460M, and F480M from the JWST Extragalactic Medium Band Survey (JEMS; Williams et al. 2023) in GOODS-S. The full details on the data reduction and generation of the drizzled mosaics and photometric catalogues can be found in Rieke et al. (2023) and Robertson et al. (2024). This large array of photometric bands, in addition to the grism spectroscopic redshifts and emission line fluxes, allows us to fit the SEDs of all galaxies with Prospector (Johnson et al. 2021; Tacchella et al. 2023; Simmonds et al. 2024, 2025). We obtain measurements of the stellar masses (M★), star-formation rates (SFRs), and star formation histories (SFHs) used in this work. As mentioned above, the parent sample used in this work is described in Danhaive et al. (2025a). This Hαemitter sample is built on the photometric redshift estimates from EAZY (Brammer et al. 2008; Hainline et al. 2024), which are then confirmed by Gaussian fitting of the 1D grism spectrum followed by visual inspection (Helton et al. 2024; Lin et al. 2025). We select galaxies with an HαS/N cut of 10 (582 galaxies), from which we select our final sample of 213 galaxies based on additional S/N cuts and position angle (PA) cuts to ensure that galaxies are at an angle with respect to the grism dispersion direction. Specifically, the 213 galaxies are separated into three samples based on the robustness of the kinematic measurements, as described in Danhaive et al. (2025a). In this paper, we only consider the 163 galaxies from the gold and silver samples, and we exclude the unresolved sample. The latter contains systems with sizes smaller than the full-width half-maximum (FWHM) of the F444W (or F356W) point-spread function (PSF), or with unresolved velocity gradients. However, for simplicity, we do not further distinguish between the gold and silver subsamples in this paper, treating the 163 galaxies together as our full final sample. 2.2 Morpho-kinematic modelling To obtain measurements of the rotational velocity and velocity dispersion, we use the Bayesian-inference fitting tool geko (Danhaive & Tacchella 2025) which forward models the grism data and compares it with the observed 2D spectrum to obtain the posterior distributions of the input morphological and kinematic model parameters. The Hαemission line map is modelled with a Sérsic profile (Sersic 1968): I(r) = Ie exp -bn " r re 1/n -1 #! , (1) where Ie is the intensity at the effective radius re and nis the Sérsic index. We use the near-UV image from JADES, probed by the F150W filter at λrest ≈2000Å, as a prior for the morphology of the Hαline. The rest-frame near-UV is chosen because it traces young stars responsible for ionizing the nearby gas and producing the Hα emission. To model the F150W images, we use the Bayesian code Pysersic (Pasha & Miller 2023), which models the image with a PSF-convolved Sérsic profile. The width of the priors is obtained by doubling the uncertainties obtained from the Pysersic modelling. The Hαgas kinematics are modelled with an arctangent velocity curve (Courteau 1997; Miller et al. 2011) Vrot(rint, rt,Va) = 2 πVa arctan rint rt , (2) whereVrot is the rotational velocity at a given radiusrint in the intrinsic galaxy plane, Va is the asymptotic value to which the arctangent rotation curve converges to at large radii rint →∞and rt is the turnaround radius of the rotation curve. To project this velocity on the observation plane, we need to account for the galaxy's inclination i: Vobs(x, y) = Vrot(rint, rt,Va) · sini· cos φint, (3) where φint is the polar angle coordinate in the galaxy plane. To compute the inclination, we assume an intrinsic axis ratio q0 = 0.2 to account for the thickness of galaxies at high redshift (Wuyts et al. 2016; Genzel et al. 2017; Price et al. 2020; Übler et al. 2024). We adopt a constant velocity dispersion across the galaxy. The model Hαintrinsic map is convolved with the kinematics to form a model 3D cube, which is then convolved with the instrument PSF and line-spread function (LSF). Finally, using the grism dispersion function, the cube is projected onto the observed grism 2D space. The priors for the kinematic parameters are uniform. We place a constraint on the turn-around radius rt to be smaller than the effective radius re (Miller et al. 2011). 2.3 Gas and DM fractions In order to infer gas masses from our star-formation rates, we use the empirical relations calibrated in Tacconi et al. (2020). We assume here that the gas component is dominated by the molecular phase, whose mass is constrained in these relations. In dense, highly starforming systems it is expected that the molecular phase dominates MNRAS 000, 1-17 (2015) 4 A. L. Danhaive et al. within re, and the atomic phase only contributes to a minor degree (e.g. Leroy et al. 2008; Saintonge et al. 2017). The ionised component is small in most systems (e.g. see Kennicutt & Evans 2012, and references therein). Given the small sizes (re ≈0.5-1 kpc, Danhaive et al. 2025b) and high specific star-formation rates (sSFR, with massdoubling timescales of tdouble ⪅30 Myr) of the galaxies in our sample, this is a valid assumption. In these equations, the molecular gas mass is estimated using the sSFR of the system and the integrated depletion timescale for converting the gas into stars. This timescale depends on the redshift and therefore provides a more accurate conversion than fixed redshift relations (e.g. Kennicutt 1998). Furthermore, the latter is calibrated to the local universe, where gas fractions are overall lower, whereas the Tacconi et al. (2020) relations are calibrated from z∼0 -5.5 from almost 2000 objects or stacks (see also Saintonge et al. 2017; Tacconi et al. 2018; Freundlich et al. 2019). Specifically, the gasto-stellar mass ratio μgas = Mgas/M★scales with redshift z, sSFRs, and stellar mass M★, where the scaling factors are computed using observational measurements of ionized and molecular gas from the literature. The relation for μgas is: log μgas = A+ B× (log(1 + z) -F)2 + C× log(sSFR/sSFRMS(z, M★)) + D× (log M★-10.7), with the star-forming main sequence (SFMS) defined as in Speagle et al. (2014), and the parameters A, B, C, Dand Fare the parameters obtained in Tacconi et al. 2020 (Table 2b) from error-weighed, multi-parameter regression. SFMSs such as Speagle et al. (2014) can suffer from sample selection effects, especially on the low mass end, which can bias the inferred relation (e.g. McClymont et al. 2025b; Simmonds et al. 2025). We also note that the Kennicutt (1998) relation assumes solar metallicities, which can have significant impacts (∼0.3 dex) on the inferred SFRs, particularly for low-mass, metalpoor galaxies. Using the aforementioned scaling relation, we can then calculate gas fractions on the galaxy scale, fgas = μgas 1 + μgas = Mgas Mgas + M★ , (4) and infer the total baryonic masses: Mbar = Mgas + M★= (μgas + 1)M★. (5) To quantify the uncertainties in our inferred gas fractions, we propagate the uncertainties from the inputs to the scaling relation (M★ and SFR), along with the uncertainties of the constant parameters of the relation itself (A, B, C, Dand F). To assess the systematics introduced by our choice of gas mass estimate, we compare the fgas measurements obtained from the scaling relation to those inferred by a simple conversion Mgas = SFR × tdep (6) with a depletion time tdep = 0.5-1 Gyr. Although using this relation increases the intrinsic scatter, our median fractions remain consistent. If instead we change the SFMS used in the Tacconi et al. (2020) relation, from Speagle et al. (2014), and adopt the one from Simmonds et al. (2025), we find that our inferred gas fractions are on average higher by Δ fgas ≈0.05 -0.1. We note that in both of these cases, the change in fgas does not visibly affect the median values of our inferred dark-matter fractions. However, it is important to note these uncertainties in the gas fractions, which can only be addressed with constraints from direct observations of the molecular gas content. In virial equilibrium, the circular velocity of galaxies at a radius rcan be related to the combined effects of gravity and turbulenceinduced pressure. The former is reflected in the rotational velocity, and the latter is an asymmetric drift correction to account for the pressure support (Burkert et al. 2010; Newman et al. 2013; Wuyts et al. 2016). For an exponential disk, the circular velocity can be written as: vcirc(r) = √︃ v2 rot(r) + 2(r/rs)σ2 0 . (7) The circular velocity reflects the gravitational potential of the galaxy and is hence directly related to the dynamical mass, Mdyn = ktot rev2 circ(re) G , (8) where Gis the gravitational constant and ktot is the virial coefficient (Price et al. 2020). The virial coefficient ktot allows us to infer the total dynamical masses based on measurements out to re. If we instead compute vcirc using a generalised Sérsic profile (rather than an exponential disk, n= 1), our inferred dynamical masses remain consistent within Δ(log Mdyn) ≲0.07 (with median Δ(log Mdyn) ≲0.02). We present results for the exponential disk assumption to provide a more direct comparison to other works in the literature. Because we have modelled our galaxies with q0 = 0.2, we choose ktot = 1.8 as it is the coefficient for galaxies with q0 = 0.2 and n∼1-4 (Price et al. 2022). We note that varying q0 in the interval q0 = 0.15 -0.25 shifts sini by Δ(sini) ≲0.05, propagating to Δ(log Mdyn) ≲0.05. For more extreme values q0 = 0.50, this correction can reach Δ(log Mdyn) ≈0.1. Finally, using our dynamical mass measurements, we can compute the DM fractions: fDM = MDM Mdyn = Mdyn -Mbar Mdyn . (9) All our inferred fractions are calculated within the effective radius re since that is where our kinematic measurements are best constrained. We find that 20% of our sample has negative and, therefore, unphysical DM fractions. This fraction drops to 18% (8%) when a deviation from fDM = 0 is required at the level of σ(3σ). These numbers are comparable to those reported in other studies (e.g., Wuyts et al. 2016), and suggest inconsistencies in the modelling of the SEDs and the kinematics, including for instance positive age gradients (D'Eugenio et al. 2021; van Houdt et al. 2021) or non-equilibrium configurations. Although we discuss these systems in the text, unless otherwise mentioned we only consider systems with fDM > 0 for our analyses. 3 DYNAMICAL MASSES & TULLY-FISHER RELATION In this section, we present the dynamical masses of our systems based on our kinematic modelling (Sec. 3.1), and discuss them within the context of the Tully-Fisher relation at high redshift (Sec. 3.2). 3.1 Dynamical masses We present our results for the dynamical masses of our galaxies in Fig. 1. We find high dynamical masses spanning log(Mdyn [M⊙]) = 9 -11, with the overall expected trend of increasing Mdyn with M★, reflecting the baryons tracing the underlying gravitational potential. We find a significant scatter in the dynamical masses at fixed stellar mass, indicative of varying gas and DM contents. To investigate the source of this scatter, we colour-code our galaxies by their offset from the MS (ΔMS), where we adopt the MS from Simmonds et al. MNRAS 000, 1-17 (2015) DM fractions of high-z galaxies 5 8 9 10 11 12 log(Mdyn)[M ] 7 8 9 10 11 12 log(M )[M ] This work Saldana-Lopez+25 De Graaff+24 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 MS Figure 1. Comparison of the dynamical masses (Mdyn) inferred from our modelling of grism data and the stellar masses (M★) inferred from SED fitting. We colour-code our galaxies based on their offset from the main sequence (ΔMS) as defined by Simmonds et al. (2025). The majority of our systems lie below the one-to-one relation (dashed line), consistent with a significant contribution to the dynamical mass from gas and/or DM. Six galaxies lie on the relation, highlighting discrepancies in the different mass estimates. We compare our values to ionised gas measurements from de Graaff et al. (2024a) and Saldana-Lopez et al. (2025) at similar redshifts. (2025). We find that galaxies above the MS, which are also typically at lower masses, seem most distant from the 1-1 line, implying a lower stellar contribution to the baryonic and total mass. We expect the dynamical masses to lie above the stellar masses, as they should also incorporate the gas and DM. We find consistent measurements M★ 8 sample, where we are more complete: a= 3.60 (fixed) (11) b= 0.76 ± 0.07 (12) σint = 0.49 ± 0.06 . (13) We plot the results for this fiducial fit at z∼5 in purple in Fig. 2, and also include a fit for the whole sample in red. The large intrinsic scatter, shown by the shaded regions, reflects that the z∼5 galaxies in our sample are not settled around the TFR. They are likely undergoing kinematic and/or morphological changes on relatively short timescales, which influence the measured circular velocities at fixed stellar masses. More generally, this implies that galaxies have not yet settled into more stable rotating disks, where the TFR is expected to hold (e.g., Übler et al. 2017). Despite the lack of a tight TFR in our sample, we explore the evolution of the zero-point offset bfrom z∼0 (Reyes et al. 2011) and z∼1 -3 (Übler et al. 2017, see also Tiley et al. (2016)) to z≈4 -6. Both works adopt the same slope. We find a significant increase in the zero-point offset, Δb≈-1.3 dex, between our TFR and the one measured at cosmic noon (Übler et al. 2017). This evolution is more significant than the Δb≈-0.4 dex evolution from z∼0 to cosmic noon. The observed decrease of bwith redshift could have (a mixture of) different origins. To visually compare with galaxies at cosmic noon, we plot on Fig. 2 the RC100 sample from Nestor Shachar et al. (2023) at z= 0.6 -2.5. It is clear that these cosmic noon galaxies occupy a different parameter space in the TFR plane, having higher stellar masses (log(M★[M⊙]) ≈10 -11) and lower gas fractions ( fgas ⪅0.5) than our sample. First, an increase of gas fractions naturally explains the shift of the TFR to higher circular velocities (at fixed mass). This is because vcirc reflects the total mass of the galaxy, both baryons and DM, so if the fraction of stars with respect to the other components decreases, the TFR will shift. In the same way, an increase in the DM fractions can also contribute to this shift. It is also possible that bhas a mass dependence, since our sample probes smaller masses than the KMOS3D and RC100 samples. It is expected that gas fractions increase at low stellar masses, naturally biasing the stellar TFR. A larger sample spanning larger redshift and mass ranges would be needed to assess the mass dependence of b. In light of this, we also explore the bTFR in our sample, but similarly to the sTFR we do not find a tight relation akin to what has been MNRAS 000, 1-17 (2015) 6 A. L. Danhaive et al. measured at cosmic noon and the local Universe (McGaugh 2012; Lelli et al. 2016). We also find a similar offset towards higher circular velocities at fixed baryonic mass. Together, these points suggests that DM contributes significantly to the total mass of the galaxies in our sample. This is in contrast to the baryon-dominated galaxies seen at cosmic noon. To correctly measure the bTFR, gas masses would need to be directly measured, which is beyond the scope of this work and our available data. In terms of intrinsic scatter (σint = 0.49 ± 0.06), we find that it is roughly twice that reported at cosmic noon (σint = 0.22, Übler et al. 2017), where they also find an increase with respect to the local Universe (σint = 0.10-0.13 Reyes et al. 2011). As discussed in Übler et al. (2017), one reason for this could be the fact that galaxies at highredshift are less settled (Simons et al. 2016), and are seen more often in non-equilibrium states (Covington et al. 2010). Also, we rely on 2D grism data in this work and, hence, suffer from more uncertainties than studies with 3D kinematic data. Furthermore, galaxies at high redshift are smaller and lower mass on average, which makes their kinematics more difficult to constrain. Finally, we note that the lowest-mass galaxies in our sample, with log(M★[M⊙]) 150 km/s despite their low masses. We also fit our full sample, including these log(M★[M⊙]) 0). It is possible that the circular velocities measured for these systems have a non-negligible contribution from non-circular motions, such as radial inflows and/or outflows of gas. This would also contribute to the large observed scatter. Due to their low stellar masses and, hence, small spatial extent, such effects would be difficult to detect in our modelling. Our sample is in general not complete, so our discussion regarding the TFR aims at comparing our sample's physical properties with those of galaxies at lower redshift and not fully defining the TFR at z= 4 -6. 4 THE GAS AND DM CONTENT AT Z> 5 In this section, we quantify the baryonic and DM content of galaxies at z= 4 -6, using gas and DM fractions derived from our kinematic modelling (Sec. 2). We first study the dependence of these fractions on stellar mass (Sec. 4.1), then we focus on the relation between DM fractions and baryonic surface density in Sec. 4.2. Finally, we put our findings in the context of works at lower redshift to discuss the redshift evolution of DM fractions (Sec. 4.3). 4.1 Baryon content at z∼5 Fig. 3 shows the inferred gas ( fgas) and DM ( fDM) fractions as a function of stellar mass for the galaxies in our samples. We note that fgas is defined relative to the baryonic mass (stars + gas), whereas fDM is defined relative to the total mass (DM + baryons). We fit both relations with a power law of the form f(z, M★) = 1 -10α(log M★-9)+β+γ( 1+z 6 ), (14) where αand γparametrize the mass and redshift dependence, respectively, and βis the normalisation at log(M★[M⊙]) = 9 and z= 5. The power-law shape is consistent with the equation from Tacconi et al. (2020) for the gas fractions. Similarly to the TFR, these fits are done with emcee using the same general setup (see Sec. 3). We 1.75 2.00 2.25 2.50 2.75 log(vcirc)[kms 1] 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 log(M )[M ] This work (v/ > 1) Best fit, z = 4 6 (all) Best fit, z = 4 6 (logM > 8) RC100 medians Übler+2017 (z = 0.9 2.3) Reyes+2011 (z = 0) 0.0 0.2 0.4 0.6 0.8 1.0 fgas(r 8 is shown in solid purple (dotted red for the full sample). The circular velocity (vcirc) is evaluated at re with an asymmetric-drift correction. We find that most of our galaxies lie along the relation, albeit with a large scatter, but the lowest-mass galaxies drop off. We compare our measurements at z∼5 to the relation found for the KMOS3D z= 0.9 -2.3 star-forming galaxies from Übler et al. (2017) (dashed orange line), and the medians from the RC100 sample (diamonds, Nestor Shachar et al. 2023). We also include the TFR at z∼0 from Reyes et al. (2011) for reference (dot-dashed tan curve). We fit our data with the same fixed slope as these works, a= 3.60, but computing our own zero-point b. We find Δb≈-1.3 dex between from z= 0.9 -2.3 to z= 4 -6. The large intrinsic scatter around the relation (σint = 0.49 ± 0.06; shaded region) indicates that it only beginning to emerge at z∼5. summarize the best-fit parameters for both the gas and the DM fractions in Tab. 1. For the gas fractions, we are not able to constrain the intrinsic scatter (σint) around the best-fit relation. As explained in Sec. 2, we compute the uncertainties on fgas by propagating the uncertainties on both our input parameters and the constant parameters of the Tacconi et al. (2020) scaling relation. This yields relatively large uncertainties of the order of Δ fgas ≈0.1 (see characteristic error in Fig. 3). On the other hand, the outputs fgas from the scaling relation vary only a small amount at fixed mass. This is in part due to the small range probed in terms of ΔMS in our sample, since we are not sensitive to galaxies well below the MS. Because of this, our uncertainties are significantly larger than the scatter shown by our inferred fgas, meaning that we cannot constrain σint. 4.1.1 Gas fractions As described in Sec. 2, we cannot measure gas masses directly, so we infer them based on the galaxies' SFRs, stellar masses, and redshifts through the Tacconi et al. (2020) scaling relation. As seen in the top panel of Fig. 3, our galaxies have high predicted gas fractions fgas = Mgas/Mbar, with almost all galaxies having fgas > 0.5. These systems have more gas than stars in their central region r 0.5. We find relatively good agreement with median trends from the thesan-zoom simulations (green solid line; McClymont et al. 2025c) and the tng50 simulations (blue solid lines; de Graaff et al. 2024b). redshift range probed in this work. However, as expected, our inferred dependence of γ= -0.36±0.45 is consistent with its counter-part in the Tacconi et al. (2020) scaling relation (D= -0.41 ± 0.03), albeit with very large uncertainties. We compare our gas fractions with estimates at cosmic noon from Genzel et al. (2020) and Nestor Shachar et al. (2023), which show a large scatter at log(M★[M⊙]) ≈10 -11. Because there is little overlap between this mass range and the one probed in this work, a direct comparison to quantify the independent effects of redshift is Parameter fgas fDM α 0.43 ± 0.03 0.28 ± 0.05 β -0.73 ± 0.03 -0.57 ± 0.04 γ -0.36 ± 0.45 0.34 ± 0.65 σint - 0.15 ± 0.02 Table 1. Summary of the parameters for the power-law fit (Eq. 14) to the fgas-M★and fDM-M★relations (Fig. 3). We cannot constrain σint for the fgas -M★relation due to the inferred uncertainties on fgas being significantly larger than the scatter in the outputs of the Tacconi et al. (2020) scaling relation for our sample (see Sec. 4.1.1 for more details). not possible. However, their results are consistent with a decrease, on average, of gas fractions with stellar mass. The high gas fractions at high-redshift are consistent with other predictions of the increase of gas fractions with redshift (Cresci et al. 2009; Genzel et al. 2015; Pillepich et al. 2019). However, this effect appears to also have a strong mass component. The gas fractions found in de Graaff et al. (2024a,b) at z≈6 -8 are consistent with our results. In fact, although de Graaff et al. (2024a) use the local relation from Kennicutt (1998) to infer gas masses from SFRs, their measurements lie nicely along our best-fit relation. This suggests that this local relation holds well at z∼6, at least at the low masses (log(M★[M⊙]) 0.5. Above log(M★[M⊙]) = 9, we observe a larger scatter and an overall decline in DM fractions with stellar mass, as is highlighted by the running medians and our best-fit power-law in Fig. 3. We colour-code our galaxies by their offset from the main sequence, but we do not find a clear trend. We put our fDM measurements in the context of works at cosmic noon, which probe a higher mass range log(M★[M⊙]) ≈10-11. As shown in Fig. 4.1, Genzel et al. (2020) find that star-forming galaxies at cosmic noon have baryon-dominated cores, with fDM ⪅0.5. This is consistent with the extrapolation of our power-law fit to these high masses, suggesting that the mass dependence of fDM exists independently of redshift. In fact, de Graaff et al. (2024b) show with MNRAS 000, 1-17 (2015) 8 A. L. Danhaive et al. the TNG simulations that the baryon fraction within 1 kpc, fbar(r 9) and find DM fractions ranging from fDM ≈0.1 to fDM ≈0.6, consistent with the scatter seen in our larger sample at those masses. We also compare our DM fractions with predictions from the thesan-zoom simulations at z= 3 -6. In general, the simulations are consistent with the majority (60%) of our sample having high DM fractions fDM > 0.5. However, McClymont et al. (2025c) find a weaker mass trend, which does not reproduce the population of log(M★[M⊙]) > 9 galaxies with lower fDM. This could point to mechanisms which drive DM out from the central regions, such as intense feedback from star formation or black holes (see Sec. 5 for more discussion). Within the simulations, this could also be due to a decrease in the number of galaxies at the high-mass end. Our observed dependence of fDM on M★is instead more consistent with the trend observed in the tng50 simulations (de Graaff et al. 2024b). Some objects have high DM fractions close to fDM = 1. Although these could be caused by an over-estimate of the dynamical mass, they can also point to steepening of the DM halo density profile in the central regions of these galaxies. We explore this in Sec. 5.1. We also find that the inferred fDM fractions for 20% of our sample are unphysical, i.e., fDM 3 kpc. It is clear that our sample occupies a different parameter space, with lower masses, higher gas fractions, and smaller sizes. Interestingly, our baryonic surface densities log Σbar ≈8 -9 have significant overlap with those MNRAS 000, 1-17 (2015) DM fractions of high-z galaxies 9 7 8 9 10 log bar(Re)[M kpc 2] 0.0 0.2 0.4 0.6 0.8 1.0 fDM(r 9) This work (log(M */M ) 2.5 (dotted blue line). Our sample medians lie well above those at cosmic noon, although our sample shows large diversity in fDM and probes ∼2 -3 dex lower stellar masses. Qualitatively, we compare our medians (log(M★[M⊙]) ≈9) with the median evolution of fDM(r 9) galaxies. Although our sample spans a wide range of fDM, we find high fractions on average ( fDM > 0.5). For the high-mass galaxies, with median log(M★[M⊙]) = 9.5, we find fDM = 0.51+0.17 -0.03 and fDM = 0.55+0.06 -0.01 at z= 4.5 and z= 5.5, respectively. For the lowmass galaxies, with median log(M★[M⊙]) = 8.5, we find fDM = 0.81+0.04 -0.19 and fDM = 0.65+0.13 -0.20. In both cases, we do not see evidence for a redshift evolution, as our medians at z= 4.5 and z= 5.5 are consistent within the uncertainties. This is expected given our poorly constrained γvalues (Tab. 1). We now place our measurements in the context of work at cosmic noon and the local Universe. In the same redshift range z≈4 - 6, Lee et al. (2025) report a wide range of fDM for their higher mass sample (log(M★[M⊙]) ≈9 -10.5), but their higher fDM systems are consistent with our high-mass medians. However, they also report galaxies that have large baryon contents ( fDM ⪅0.2). We also find similar systems in our sample (Fig. 3) even though our medians lie at higher fDM. The systems from Lee et al. (2025) are therefore consistent with being a subset of the more extensive population probed in this work. At higher redshift (z≈5 -8), measurements from de Graaff et al. (2024a,b), who find fDM > 0.5 for their sample of log(M★[M⊙]) ≈8 -9, are broadly consistent with our low-mass medians. On the other hand, the medians for log(M★[M⊙]) ≈10-11 galaxies at cosmic noon (Genzel et al. 2020; Nestor Shachar et al. 2023) show low DM fractions ( fDM ≈0.2-0.4). When combining all of these measurements together, the resulting picture suggests that mass plays a significant role in determining the DM content within the central regions of galaxies. The massive systems at cosmic noon are predominantly baryon-dominated, as is also seen in more massive systems at z∼5. On the other hand, low-mass galaxies seem more DM-dominated. To further investigate this, we plot the median tracks from the tng50 (de Graaff et al. 2024b) and thesan-zoom (McClymont et al. 2025c) simulations. Specifically, for tng50, we plot the median evolution of fDM(r 3, Eq. 18 develops an upturn near r= rs, as the outer slope (3 -α) becomes negative. Conversely, when α 9). On the other hand, high central baryonic concentrations are expected to perturb the underlying DM distribution and pull the DM towards the centre, increasing the "cuspiness" of its density profile. This phenomenon is called adiabatic contraction (Blumenthal et al. 1986), and is represented by the α> 1 region of Fig. 6. For galaxies in this region, adiabatic contraction is needed to explain the high fDM values measured in this work. Interestingly, this under-prediction of fDM, compared to our observations, holds for many other SMHM relations from the literature which typically lie above the Behroozi et al. (2019) one (e.g. Tacchella et al. 2018). In order to bridge some of the discrepancies between the model predictions and our measurements, a lower SMHM relation at high redshift would be needed. This problem is emphasized by recent predictions from observations and simulations of high DM fractions in low-mass galaxies (de Graaff et al. 2024b; McClymont et al. 2025c). However, it is important to note that our high α⪆1.5 values suffer from large uncertainties that make them consistent with α≈1.5. These stem from the smaller masses of these systems, which make their morphology and kinematics more difficult to constrain. We colour-code our points in Fig. 6 by their SFHs, parametrized by the ratio of SFR averaged over 10 and 100 Myr. Galaxies with rising SFHs will have positive values of SFR10/SFR100, whereas galaxies with falling SFHs will show the opposite behaviour. In order to visually asses the presence of an underlying trend, we combine our points in larger bins and use the LOESS method (Cappellari et al. 2013b) to average over single objects and obtain mean estimates for the full sample. We can see a trend of galaxies going through a burst log SFR10/SFR100 ⪆0.4 having preferentially higher values of α. This apparent correlation supports the adiabatic contraction scenario that would occur with the inflow of baryons to the central regions during a burst, during which the gas content of the galaxy contracts (Dekel & Burkert 2014; Tacchella et al. 2016b; McClymont et al. 2025b). When feedback from the star formation kicks in, driving the gas out of the centre and temporarily quenching star formation, galaxies begin to move back down towards the MS log SFR10/SFR100 1 to a cuspier profile. Under the assumed SHMR and concentration (c= 5), we find that low-mass galaxies with high fDM are associated with cuspy distributions, converging towards an NFW-like profile at higher masses with a scatter at least in part driven by their star-formation state. To highlight this, we colour-code the galaxies by the ratio of their SFR averaged over 10 and 100 Myr, SFR10/SFR100, applying LOESS smoothing (Cappellari et al. 2013b). Galaxies going through a burst of star formation (log SFR10/SFR100 ⪆0.4) have preferentially high values of α, which supports the adiabatic contraction scenario. Myr), which has yet to be thoroughly explored in cosmological simulations. Interestingly, we also find that the values of αcorrelate with the Sérsic index nmeasured from the stellar light distribution in the UV (Fig. A1). Galaxies with cuspier DM profiles also have steeper central light profiles, with n∼4, which could point to interactions between the two components. We find that low-mass galaxies log(M★[M⊙]) 5 require more galaxies to have cored profiles α = 0.77 and = 0.73. We find that ≈67% of our galaxies are DM dominated with their re ∼0.5 -1 kpc, fDM > 0.5. Nonetheless, our fractions fDM span the full range of the parameter space. - We find evidence for a negative dependence of fDM on stellar mass M★as parametrized by Eq. 14. Specifically, we find α= 0.28± 0.05, with low-mass (log(M★[M⊙]) 9) showing a larger diversity, with typically lower fDM. - We find an anti-correlation (ρ= -0.56, p 1 to a cuspier profile. We find that galaxies with seemingly steeper DM profiles (α> 1) coincide with steeper light distributions (n> 1) as probed by the UV stellar continuum. To highlight this, we colour-code the galaxies by the Sérsic index nUV, applying LOESS smoothing (Cappellari et al. 2013b). consistent with those discussed in Section 5, reinforcing that galaxies with lower fDM tend to exhibit shallower inner density slopes, while systems with higher fDM favour cuspier profiles. Although the scatter is substantial, these correlations support the interpretation that the interplay between baryonic concentration and halo response is already shaping the inner DM structure at z∼5. This paper has been typeset from a TEX/LATEX file prepared by the author. MNRAS 000, 1-17 (2015)
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2510.14776
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Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX631
Evaluating Habitability and Biosignature Detection on TOI-700 d: The Role of UV Environment and
Atmospheric Pressure
Viktor Yuri Don´a Sumida,1, 2 Raissa Estrela,2 and Adriana Valio1
1Center for Radio Astronomy and Astrophysics Mackenzie, Mackenzie Presbyterian University,
Rua da Consola¸c˜ao 930, S˜ao Paulo, SP, Brazil
2Jet Propulsion Laboratory, California Institute of Technology
4800 Oak Grove Dr, CA, USA
ABSTRACT
M dwarfs have long been prime targets in the search for habitable exoplanets, owing to their abun-
dance in the galaxy and the relative ease of detecting Earth-sized worlds within their narrower habitable
zones. Yet, these low-mass stars can emit high-energy radiation that may gradually erode planetary
atmospheres, raising concerns about long-term habitability. TOI-700, a relatively quiescent M dwarf
that hosts four known planets, stands out due to its Earth-sized TOI-700d in the star’s habitable
zone. Here, we assess whether a habitable environment can be sustained on TOI-700d by analyzing
different UV flux levels and atmospheric pressures. We focus on two atmospheric scenarios – one anal-
ogous to the Archean Earth and another representing a modern Earth-like environment – using a 1D
photochemistry-climate model. Our results indicate that all simulated cases can maintain temperatures
compatible with liquid water on the surface. However, the dominant photochemical pathways differ
substantially with UV levels: under low-UV conditions, haze formation in the Archean-like atmosphere
provides the main UV shielding, whereas under intensified UV, ozone production in the modern-like
atmospheres can protect the surface from harmful doses. Interestingly, although haze can impede the
detection of certain biosignatures, such as CH4, CO2 and O2, it also enhances the overall atmospheric
signal by increasing scattering and transit depth, potentially aiding in revealing the presence of an
atmosphere. These findings underscore the dual role of hazes as both a challenge for biosignature
detection and a potential protection of surface habitability.
Keywords: Exoplanets (498) — Exoplanet atmospheres (487) — Habitable planets (695) — Ultraviolet
astronomy (1736) — Transmission spectroscopy (2133)
1. INTRODUCTION
In recent years, red dwarf stars have emerged as
primary targets in the search for extraterrestrial life
due to their extended lifespans.
Although the ex-
tended pre–main-sequence contraction phase of M dwarf
stars may desiccate or erode the atmospheres of plan-
ets within their habitable zones, potentially triggering a
runaway greenhouse and compromising early habitabil-
ity (Luger & Barnes 2015), these effects occur mainly
during the early stages of stellar evolution. As M dwarfs
age, their activity declines substantially, which could
allow secondary atmospheres or late-formed planets to
sustain habitable environments (Gale & Wandel 2017).
The Transiting Exoplanet Survey Satellite (TESS,
Ricker et al. 2015) has played a key role in identifying
terrestrial-sized planets, particularly those in the habit-
able zones of M-dwarfs, which are prime candidates for
atmospheric characterization due to their favorable star-
to-planet size ratios and close-in habitable zones that en-
hance transit detectability (Shields et al. 2016). These
characteristics make planets orbiting M-dwarfs strong
candidates for biosignature searches with instruments
such as the James Webb Space Telescope (JWST) and
future Extremely Large Telescopes (ELTs).
Despite these advantages, characterizing exoplanet at-
mospheres remains challenging, requiring more detailed
observational and theoretical studies. Key biosignature
gases – including O2, CH4, H2O, N2O, O3, and CO2 –
are fundamental for climate regulation, protection from
harmful radiation, and the potential indication of bio-
logical activity (Airapetian et al. 2017).
However, ul-
traviolet (UV) radiation from M-dwarfs, particularly in
the UVB (280–315nm) and UVC (100–280nm) ranges,
arXiv:2510.14776v1 [astro-ph.EP] 16 Oct 2025
2
Sumida et al.
can significantly affect atmospheric chemistry and the
habitability of planets in close orbits.
The TOI-700 system, an M2.5V dwarf star, hosts four
known exoplanets, two of which – TOI-700 d and TOI-
700 e – reside in its habitable zone (Rodriguez et al.
2020; Gilbert et al. 2023).
TOI-700d, a potentially
Earth-sized planet (∼1.14 R⊕), receives about 86% of
the Earth’s insolation and is of particular interest for
habitability studies. Although planets around M-dwarfs
are susceptible to atmospheric erosion due to intense
X-ray and extreme ultraviolet (XUV) radiation (e.g.,
Nishioka et al. 2023), TOI-700 itself appears to be rel-
atively quiescent, with X-ray luminosities comparable
to those of the modern Sun (Gilbert et al. 2020). This
makes TOI-700 d an excellent candidate for studying the
atmospheric processes that influence planetary habit-
ability.
To evaluate TOI-700 d’s potential habitability, we em-
ployed the 1D Photochemical Model coupled to the At-
mos Climate model (Arney et al. 2016), analyzing both
Archean and modern Earth-like atmospheres – biogeni-
cally active scenarios that assume the presence of life –
across pressures of 0.5, 1.0, 2.0, and 4.0 bar. Despite
TOI-700 being relatively quiescent during the observed
period so far, we simulated two scenarios: one assuming
that TOI-700 remains in a quiet state (fUV=1) and an-
other representing a flare event causing a 10-fold increase
in NUV flux (fUV=10), as estimated by prior studies on
M-type stars (Rekhi et al. 2023).
Our work addresses critical but underexplored aspects
such as how photochemistry influences atmospheric
composition, the potential for false-positive biosigna-
tures, and climate effects under varying levels of UV flux
and pressure. These challenges align with the call for ad-
vanced atmospheric models by the 2020 Astronomy and
Astrophysics decadal survey1 to evaluate star-planet in-
teractions, haze formation, atmospheric dynamics, and
escape processes. Although detecting TOI-700 d’s trans-
mission features would require untenably long JWST
integration times, the significance of this work is rein-
forced by the growing focus of observational programs
utilizing the James Webb and Hubble Space Telescopes,
which aim to probe planetary atmospheres and stellar
activity around M dwarfs, such as the Rocky Worlds
DDT program2.
In this study, we assess the habitability potential of
TOI-700 d by simulating different atmospheric compo-
sitions and evaluating their response to varying stellar
1 https://www.nationalacademies.org/our-work/decadal-survey-on-astronomy-and-astrophysics-2020-astro2020
2 https://www.stsci.edu/contents/news/jwst/2024/rocky-worlds-ddt-selects-its-first-targets
UV flux. In Section 2, we outline the methods employed,
including the 1D Photochemical Model coupled to the
Atmos Climate model used to simulate Archean and
modern Earth-like atmospheres under different pressure
conditions. Section 3 presents our results and discus-
sion, including the atmospheric abundance profiles, pho-
tochemical pathways, and the impact of UV radiation
on surface habitability.
A key aspect of our analy-
sis involves evaluating whether TOI-700 d could sustain
life by assessing the ability of its atmosphere to shield
against UV radiation and its implications for habitabil-
ity, which we discuss in Subsection 3.2. Additionally,
we investigate the impact of photochemically produced
haze on the planet’s atmospheric transmission spectra,
since haze can obscure key spectral features of biosig-
nature gases, which we explore in Subsection 3.3. Fi-
nally, Section 4 summarizes our key findings, discusses
the broader implications for exoplanetary habitability
studies, and highlights future observational prospects.
2. METHODS
Possible atmospheres for the exoplanet TOI-700 d
were simulated with the 1D Photochemical Model cou-
pled to the Atmos Climate model (Arney et al. 2016).
The code generates atmospheres with well-defined chem-
ical and climate profiles. In the modern Earth-like at-
mosphere photochemical model, the code incorporates
310 chemical reactions and 72 chemical species, includ-
ing 11 short-lived species. On the other hand, for the
Archean atmosphere, the model incorporates 405 chem-
ical reactions and 77 chemical species, with 9 of them
being short-lived.
The initial atmospheric state is established by ini-
tially executing the photochemical model, which relies
on user-specified boundary conditions. After the pho-
tochemical model converges, its output files containing
altitude, pressure, gas mixing ratios, haze particle sizes,
and haze number densities are transferred to the climate
model.
The climate model utilizes the photochemical
model’s solution as its starting point and continues run-
ning until it also achieves convergence. Subsequently,
the updated temperature and water vapor profiles are
fed back into the photochemical model. This iterative
process continues until convergence is attained. Detailed
descriptions of the model can be found in Arney et al.
(2016), and are publicly accessible3.
To accurately simulate planetary environments and
observable properties, the models require comprehensive
data on both planetary and stellar characteristics. This
3 https://github.com/VirtualPlanetaryLaboratory/atmos
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
3
includes detailed information on stellar parameters and
spectra, as well as planetary physical and orbital param-
eters. Additionally, environmental data are crucial for
refining the simulations. Here, we used the stellar and
planetary parameters reported by Gilbert et al. (2020).
For the stellar spectrum, Atmos provides twenty-one
star spectra encompassing a range of spectral types.
Given that TOI-700 is classified as an M2.5V star, the
closest spectral type available is M3.0V, which corre-
sponds to GJ 581 (Turnbull 2015).
Following Arney et al. (2016) and Meadows et al.
(2018), organic haze particles were considered to pos-
sess a fractal shape in both photochemical and climate
models, to the detriment of spherical (nonfractal or clas-
sical Mie) particles.
The solar zenith angles (SZAs)
used in these models were selected to realistically rep-
resent the global averaged insolation. In particular, 60◦
in the climate model corresponds to global mean inso-
lation (accounting for S/4 and diurnal effects), while
45◦in the photochemical model – based on Segura et al.
(2003) – reproduces the modern Earth’s ozone column
given the employed chemistry. These values are common
approximations, although more sophisticated methods
(e.g., Gaussian integration) could improve insolation es-
timates if full ozone photochemistry were implemented.
Another key parameter of Atmos is the UV amplifica-
tion factor (fUV). This parameter plays an important
role in simulating stellar flares, as it boosts the UV flux
reaching the top of the atmosphere. In simple terms, it
determines the relationship between Fflare,UV and Fq,UV,
where Fq represents the quiescent UV radiation flux.
Rekhi et al. (2023) provided estimates for the median
top-of-atmosphere (TOA) NUV irradiances for planets
located in the habitable zones of M0 to M6 dwarfs us-
ing the GALEX sample. They also determined thresh-
old flare amplitudes and their corresponding frequen-
cies necessary to achieve the TOA NUV irradiance levels
similar to those of the young Sun. According to these
authors, the exoplanetary NUV irradiance, accounting
for flares, is within one order of magnitude of the young
Earth’s NUV irradiance for host stars post-M2. Flares
reaching the young Earth’s irradiance level occur mul-
tiple times a day for stars later than M3. According to
Table 3 in the study by Rekhi et al. (2023), the ratio of
flare to quiescent NUV flux (Fflare,UV/Fq,UV) is 12.4 for
an M2 star and 7.9 for an M3 star. Since TOI-700 is clas-
sified as an M2.5 star, we extrapolated an intermediate
value of approximately 10 for fUV in our simulations.
Although our Atmos simulations for various atmo-
spheres of TOI-700 d provide valuable insights, it’s im-
perative to determine the strength of potentially ob-
servable spectral features. To accomplish this, we use
the calculated values of gas mixing ratios of H2O, CH4,
C2H6, CO2, O2, O3, CO, H2CO, HNO3, NO2, SO2, N2O
and N2, and aerosol parameters derived from the At-
mos simulations. These inputs are fed into the Plane-
tary Spectrum Generator (PSG, Villanueva et al. 2018)
module known as GlobES4 (Global Exoplanet Spectra),
which is designed for synthesizing both transmission and
emission spectra. The PSG integrates advanced radia-
tive transfer techniques to accurately generate compre-
hensive synthetic spectra of TOI-700 d and to assess
the feasibility of observing spectral features in its at-
mosphere.
3. RESULTS AND DISCUSSION
In Subsection 3.1, we present the results and discuss
the implications of the atmospheric abundance profiles
of the models, along with their respective transmission
spectra. Then in Subsection 3.2, we assess the habitabil-
ity potential of TOI-700 d by conducting an analysis of
the impact of UV radiation on the planet’s surface. To
complete the analysis, in Subsection 3.3, we present the
simulated transmission spectra derived from our models
and discuss their observational implications.
3.1. Photochemical pathways in different conditions
Profiles of possible atmospheres for TOI-700 d are sim-
ulated, based on both Archean and modern Earth-like
atmospheres. Our simulations incorporate variations in
atmospheric pressure at 0.5, 1.0, 2.0, and 4.0 bar, and
compare two distinct scenarios: one with fUV = 1 and
the other with fUV = 10.
The resulting abundance
profiles for key molecules of interest are presented in
Figure 1. All figures presented here adhere to the same
color scheme for the atmospheres: Archean and mod-
ern Earth-like with fUV=1 are represented in purple
and green, while Archean and modern Earth-like atmo-
spheres with fUV=10 are depicted in red and blue, re-
spectively. The varying pressures, namely 4.0, 2.0, 1.0,
and 0.5 bar, are indicated by a gradual transition to-
wards lighter shades, illustrating a shift from higher to
lower pressure.
The choice of methane to carbon dioxide ratios in at-
mospheric models reflects the conditions from different
geological periods and their relevance to climate pro-
cesses. For Archean Earth scenarios, a CH4/CO2 ratio
above 0.1 is widely used as a criterion since it enables
significant organic haze formation in primitive atmo-
spheres (e.g., Haqq-Misra et al. 2008; Arney et al. 2017;
Meadows et al. 2018; Mak et al. 2023).
4 https://psg.gsfc.nasa.gov/apps/globes.php
4
Sumida et al.
10
−19
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[bar]
259.3
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[K]
Oxygen[O]
10
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259.3
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[K]
Molecular
Oxygen
[O
2
]
10
−28
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Pressure
[bar]
259.3
254.8
250.2
247.5
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Temperature
[K]
Ozone
[O
3
]
10
−6
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Mixing
Ratios
10
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[bar]
259.3
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[K]
Carbon
Monoxide[CO]
10
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[bar]
259.3
254.8
250.2
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[K]
Carbon
Dioxide
[CO
2
]
10
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259.3
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Temperature
[K]
Water
Vapor
[H
2
O]
10
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[bar]
Archean:
UV
=1
Archean:
UV
=10
M. Earth:
UV
=1
M. Earth:
UV
=10
259.3
254.8
250.2
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Temperature
[K]
Methane
[CH
4
]
10
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259.3
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Sulfur
Dioxide
[SO
2
]
10
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259.3
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[K]
Sulfur
Trioxide
[SO
3
]
Figure 1. Each panel, from the upper left to the lower right, represents the atmospheric pressure of O, O2, O3, CO, CO2, H2O,
CH4, SO2 and SO3 as a function of mixing ratios. The initial conditions and the produced surface mixing ratios can be seen
either in Table 3 or in the files available at DOI: 10.5281/zenodo.13947863. The Archean atmospheres with fUV=1 and fUV=10
are depicted in purple and red, while modern Earth-like atmospheres with fUV=1 and fUV=10 are shown in green and blue,
respectively. The different surface pressures, i.e., 4.0, 2.0, 1.0, and 0.5 bar, are represented by tones gradually shifting towards
lighter shades, indicating a transition from higher to lower pressure.
For
TOI-700d,
our
Archean
atmospheric model
adopts a CH4/CO2 ratio of approximately 0.5, a value
chosen to reflect the enhanced methane accumulation
expected around M-dwarf stars. This selection is based
on the findings of Meadows et al. (2018), who demon-
strated that a CH4/CO2 ratio of approximately 0.3 is
necessary to trigger the formation of organic haze on
Proxima Centauri b.
In an oxygen-poor, Archean-like
atmosphere, the primary pathways for CH4 destruc-
tion involve OH radicals formed via H2O photolysis and
O atoms generated by CO2 photolysis.
However, re-
duced UV emission around 300 nm typical of M-dwarf
stars leads to a diminished photolysis rate, allowing
methane to accumulate. In our model for TOI-700 d,
as methane becomes more dominant, aerosol produc-
tion intensifies, leading to a thickening haze that sig-
nificantly reduces the penetration of starlight into the
planet’s surface, resulting in substantial cooling (see
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
5
Haqq-Misra et al. 2008).
Note that this Archean-like
composition is included solely as a limiting case, allow-
ing us to probe the climatic and UV-shielding conse-
quences of a methane-rich, low-oxygen atmosphere; sus-
taining CH4/CO2 ≈0.5 would in reality require excep-
tionally vigorous biogenic activity or intense geological
outgassing, so we treat it as an exploratory end-member
rather than a prediction of TOI-700 d’s actual state.
For the modern-Earth analog, we adopt recent near-
surface mixing ratios (CH4 ≈1.8 ppmv; CO2 ≈385
ppmv), corresponding to CH4/CO2 ≈4.7×10−3, as an
empirical observational baseline for comparison (e.g.,
IPCC 2021)5.
Furthermore, we specified the lower
boundary conditions of the photochemical model to re-
produce observed surface-level concentrations of bio-
genic trace gases on modern Earth, for example, N2O
around 0.3 ppmv in the lower atmosphere. This value
corresponds to estimated modern biogenic fluxes that
sustain such concentrations. In this way, the modern
Earth scenario in our model begins with N2O levels
aligned with present-day atmospheric observations (see
IPCC 2021). In such an atmosphere, like that of mod-
ern Earth, methane undergoes rapid oxidation through
reactions with hydroxyl radicals (OH):
O2 + hν (λ < 200 nm) −→: O + O
: O + O2 −→O3
O3 + hν (λ < 310 nm) −→O2 + O∗
O∗+ H2O −→2 OH
CH4 + OH −→CH3 + H2O
(1)
This lower CH4/CO2 ratio is more suitable for cur-
rent Earth-like conditions, where oxygen significantly
reduces methane’s presence, limiting its capacity to con-
tribute to haze formation. By modeling both Archean
and modern conditions, we can observe the shift in haze
dynamics and its impact on the planet’s climate system.
Table 3 in the appendix shows the initial conditions
and produced surface mixing ratios of the nine major
long-lived species for the Archean and modern Earth-
like atmosphere models, at a pressure of 1 bar.
For
more detailed analysis, including simulations with vary-
ing pressures and UV fluxes, the complete data set is
available at DOI: 10.5281/zenodo.13947863
5 Over geologic timescales, atmospheric CO2 on Earth is regu-
lated by the carbonate–silicate feedback and has been further
perturbed in the industrial era; CH4, in contrast, is primarily
controlled by biogenic sources and removal by atmospheric oxi-
dants. In this work we use the observed modern-Earth CH4/CO2
ratio only as an empirical baseline for comparison; our conclu-
sions are insensitive to modest variations within the explored
range (10−3–10−2).
The exact composition of the Archean atmosphere
when life first emerged on Earth remains a subject of
ongoing debate.
Nevertheless, the interplay between
the atmosphere and geological cycles has left discernible
traces that help identify the main atmospheric gases of
this era.
Combined geological and atmospheric mod-
eling suggests that the Archean atmosphere (4–2.5 Gyr
ago) may have contained approximately 10 to 2500 times
the modern amount of CO2 and 100 to 104 times the
modern amount of CH4 (see Catling & Zahnle 2020).
Although the plausibility of extremely high methane
levels in prebiotic eras remains debated, recent stud-
ies point to potential abiotic mechanisms. Wogan et al.
(2023), for instance, argue that major impact events
could substantially increase CH4 concentrations, allow-
ing for episodic surges in atmospheric methane even in
the absence of biological activity.
As shown in the bottom-left panel of Figure 1, the
methane concentration in the two model atmospheres
with fUV = 1 and in the Archean atmosphere with
fUV = 10 is roughly 103 times higher than in the mod-
ern Earth’s atmosphere with fUV = 10. Notably, both
the Archean and modern Earth models with fUV = 10
exhibit an exponential decrease in methane abundance
with increasing altitude.
In atmospheres exposed to
elevated UV flux (fUV = 10), methane undergoes a
more pronounced decline in its mixing ratio owing to
enhanced CH4 oxidation by UV radiation. By contrast,
in simulations with normal UV flux (fUV = 1), the mix-
ing ratio of methane still decreases with altitude, albeit
more gradually, reflecting a comparatively lower oxida-
tion rate. These differing rates of CH4 oxidation under
varying UV flux conditions lead to distinct vertical mix-
ing ratio profiles across the modeled atmospheres.
In our Archean–like scenarios, the surface temperature
is higher than in the modern Earth case (see Figure 2).
Although CH4 is more radiatively efficient per molecule
than CO2, primarily at trace-level concentrations typical
of present-day Earth, the total greenhouse forcing in the
Archean simulations is dominated by the substantially
higher absolute abundances of both CH4 and CO2 (see
Kiehl & Dickinson 1987; Etminan et al. 2016). This en-
hanced greenhouse effect more than offsets the reduced
stellar energy input to TOI-700d, allowing surface tem-
peratures that are comparable to, or even exceed, those
required for the stability of liquid water. However, with-
out continuous replenishment these gases cannot sustain
such warming over geological timescales: CH4 is photo-
chemically oxidized to CO and CO2, while CO2 is grad-
ually sequestered by carbonate–silicate weathering and
ocean uptake.
6
Sumida et al.
Consequently, geological activity on a planet is nec-
essary to balance these gases through the carbonate-
silicate cycle, which involves volcanic activity, plate tec-
tonics, and erosion. The lack of greenhouse gas replace-
ment by geological activity has contributed to Mars’
current temperatures of -50◦C on its surface and a
pressure of only 1% of Earth’s atmospheric pressure
today (Segura & Navarro-Gonz´alez 2005).
Regarding
temperature under haze conditions, cooling is minimized
around M-dwarfs. These stars emit energy primarily at
wavelengths where organic hazes are relatively transpar-
ent (Arney et al. 2017).
In all scenarios, the estimated temperatures suggest
conditions conducive to liquid water on the surface
of TOI-700d. The lowest temperature, approximately
278 K, is associated with Modern Earth-like models with
fUV=1 and fUV=10, irrespective of surface pressure.
Conversely, the highest temperature of 306 K is observed
in the fUV=1 Archean model with a surface pressure of
4 bar. Detailed temperature values for each model are
provided in Table 1.
Studies
on
Archean
Earth
suggest
that
exten-
sive ocean fractions could persist for globally aver-
aged surface temperatures as low as 250–260K (e.g.,
Wolf & Toon 2013; Arney et al. 2016). However, such
studies are based on planets orbiting Sun-like stars.
Around M dwarfs, intense XUV flux and frequent stel-
lar flares can significantly enhance atmospheric escape,
leading to substantial water loss over time. While pre-
vious studies (e.g., do Amaral et al. 2022) have empha-
sized the role of stellar flares in driving early atmo-
spheric escape, we also acknowledge the cumulative ef-
fects of stellar luminosity evolution. Late M dwarfs such
as TOI-700 can brighten significantly over time.
Ac-
cording to our XUV flux calculations (see Equation 1
in Jackson et al. 2012 and Equation22 in Owen and
Wu 2017), TOI-700 d received an estimated XUV flux of
∼574 erg s−1 cm−2 at 100 Myr6 – over 15 times higher
than its present–day value of ∼37 erg s−1 cm−2. This
elevated high-energy input during the first few hundred
million years of evolution could have driven intense hy-
drodynamic escape, especially if TOI-700 d originally
possessed a volatile-rich envelope. Our results thus re-
inforce the scenario raised by Luger & Barnes (2015), in
which secular stellar brightening – independent of flares
6 Stellar X-ray emission typically undergoes an initial saturation
phase lasting tens to hundreds of millions of years, followed by
a gradual decline. This evolutionary trend is observed across all
stellar spectral types (Jackson et al. 2012; Shkolnik & Barman
2014; McDonald et al. 2019).
Table 1. Surface Temperature Estimate for TOI-700 d.
Pressure
Atmospheric
fUV
Temperature
[bar]
Model
[K]
0.5
Archean
1
288.5
Archean
10
283.9
Modern Earth
1
278.4
Modern Earth
10
278.4
1.0
Archean
1
288.8
Archean
10
279.3
Modern Earth
1
278.5
Modern Earth
10
278.4
2.0
Archean
1
296.0
Archean
10
294.5
Modern Earth
1
278.4
Modern Earth
10
278.3
4.0
Archean
1
306.2
Archean
10
285.4
Modern Earth
1
278.4
Modern Earth
10
278.3
– may lead to substantial or even complete devolatiliza-
tion of initially habitable-zone planets around M dwarfs.
Similarly to Meadows et al. (2018), attempting to run
the climate model to a converged state using the same
top-of-atmosphere pressure as the photochemical model
proved unfeasible.
This limitation stems from insta-
bilities within the model, yet it does not impact the
resultant photochemistry, as indicated by Arney et al.
(2016). Consequently, when the climate model transfers
its temperature and water profiles to the photochemical
model, they remain fixed at their values at the top of the
climate grid, forming isoprofiles above this grid’s upper
limit. This effect is illustrated in Figure 2, where an in-
crease in surface pressure leads to isoprofiles occurring
at higher pressures.
An apparently puzzling feature in Figure 1 is the be-
havior of CO in the modern Earth scenario run with
fUV = 1.
Below the homopause (P ≳10−5 bar) the
gas-phase carbon inventory – fCO2 + fCO + fCH4 – is
conserved in all scenarios, yet the partitioning among
the three species depends on the local UV field and on
the oxidative state of the atmosphere:
i Modern Earth, fUV = 1:
a modest O3 layer
removes most 200–310nm photons.
The result-
ing near-UV deficit almost quenches CO2 pho-
tolysis above 10−6 bar, eliminating the main in
situ source of CO, and keeps OH production low;
the residual OH therefore acts as a net sink via
CO + OH →CO2.
With its source suppressed,
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
7
150
200
250
300
10
−5
10
−4
10
−3
10
−2
10
−1
Pressure
[bar]
P
0
=0.5
Archean: f
UV
=1
Archean: f
UV
=10
M. Ear h: f
UV
=1
M. Ear h: f
UV
=10
150
200
250
300
10
−4
10
−2
10
0
P
0
=1.0
Archean: f
UV
=1
Archean: f
UV
=10
M. Ear h: f
UV
=1
M. Ear h: f
UV
=10
150
200
250
300
Tempera ure
[K]
10
−4
10
−2
10
0
Pressure
[bar]
P
0
=2.0
Archean: f
UV
=1
Archean: f
UV
=10
M. Ear h: f
UV
=1
M. Ear h: f
UV
=10
150
200
250
300
Tempera ure
[K]
10
−4
10
−2
10
0
P
0
=4.0
Archean: f
UV
=1
Archean: f
UV
=10
M. Ear h: f
UV
=1
M. Ear h: f
UV
=10
Figure 2. Atmospheric pressure profiles plotted against temperature for different pressures. The Archean atmospheres with
fUV=1 and fUV=10 are depicted in purple and red, while modern Earth-like atmospheres with fUV=1 and fUV=10 are shown
in green and blue, respectively. The Modern Earth atmosphere models of fUV=1 and fUV=10 can only be distinguished from
one another in the case where the surface pressure (P0) equals 1 (top right panel), otherwise for all other pressures, the curve
for fUV=10 (blue) superimposes on the fUV=1 (green) curve.
CO decreases with altitude and the lost carbon
reappears as CO2.
ii Archean, fUV = 1: in the absence of O2/O3, CO2
photolysis operates throughout the upper column
and the CO profile increases.
Removal of car-
bon into hydrocarbon haze occurs between 10−6
and 10−5 bar, but the fraction sequestered remains
≲10−4 % of the gas-phase inventory and does not
affect the global carbon budget. Nevertheless, the
spectral impact can be substantial: haze particles,
even at sub-ppm mass fractions, nonlinearly en-
hance the atmospheric opacity through scattering
and absorption, strongly modulating the observed
transmission spectra.
iii All cases with fUV = 10: a ten-fold stronger near-
UV flux simultaneously restores vigorous CO2
photolysis and produces abundant OH. The CO
again rises with altitude, but the enhanced UV
field partly photodissociates both CO and the
newly formed CO2, preventing the high-altitude
build-up seen in the low-UV modern case.
Hence, the CO decline in the modern, low-UV atmo-
sphere is a natural consequence of UV shielding by O3
plus efficient removal via CO + OH →CO2, whereas
all other simulations lack one (or both) of those con-
ditions and therefore show a monotonically increasing
CO with height. We also note that there is no photo-
chemical haze in the modern Earth-like scenarios. The
high O2 and O3 levels maintain oxidizing conditions that
suppress hydrocarbon polymerization, thereby preclud-
8
Sumida et al.
ing haze nucleation. Both HCAER and HCAER27 re-
main zero at all altitudes, even under enhanced UV flux
(fUV = 10).
Sulfur dioxide (SO2) and hydrogen sulfide (H2S) rep-
resent the most prevalent sulfur gases released dur-
ing volcanic activity.
The challenge of reconciling
the observed sulfur mass-independent fractionation (S-
MIF) in laboratory sulfur dioxide photochemistry with
that recorded in the Archean rock record has per-
sisted as a significant issue since its initial discovery
(Farquhar et al. 2000, 2001). In particular, sulfur diox-
ide exhibits a complex UV absorption spectrum char-
acterized by two prominent absorption features span-
ning 260 to 340 nm and 165 to 235 nm, which plays a
crucial role in understanding atmospheric processes and
climate dynamics. The accumulation of SO2 in the at-
mosphere of TOI-700d (see Figure 1) is attributed to
low UV emission from the host star. This is evident in
the significantly higher concentration of sulfur dioxide
when comparing scenarios with low (fUV=1) and high
(fUV=10) UV flux. In O2-rich atmospheres, absorption
of ultraviolet photons by SO2 can induce predissocia-
tion, whereby the excited molecule dissociates to SO +
O*. This initiates the following chemical transforma-
tions:
SO2 + hν (λ < 220 nm) −→SO + O∗
SO + O2 + M −→SO3 + M
(2)
Although SO2 absorbs UV radiation, albeit less effi-
ciently compared to O3, it can still accumulate in sig-
nificant quantities. When combined with CH4 and haze
particles in the atmosphere, it may provide adequate
protection against UV radiation, particularly in O2-poor
atmospheres.
The production of SO3 and its potential reaction
with H2O to form sulfuric acid (H2SO4) is a crucial
process in planetary atmospheres.
As highlighted by
Meadows et al. (2018), while this reaction can proceed
efficiently in Venus-like atmospheres, it is heavily de-
pendent on the availability of water. In the absence of
sufficient H2O, the formation of H2SO4 aerosols is sig-
nificantly limited. Moreover, the photochemical model
used in this study does not simulate Venus-like atmo-
spheres, particularly the formation of H2SO4 clouds and
their feedback effects on atmospheric temperature struc-
ture. Additionally, on Earth, the oxidation of SO2 to
SO3 often serves as a bottleneck that controls the rate
7 HCAER and HCAER2 are the two aerosol tracers used by the
Atmos photochemistry code. They represent the direct conden-
sation of the gas–phase precursors C4H2 and C5H4.
Nonzero
values indicate ongoing haze formation.
of sulfuric acid formation (Lizzio & DeBarr 1997). How-
ever, on planets orbiting M-dwarf stars, such as Proxima
Centauri b, as studied by Meadows et al. (2018), the de-
struction of SO3 may become the dominant process, po-
tentially inhibiting the formation of H2SO4 clouds. This
suggests that, while sulfuric acid clouds could still form
under specific conditions, their formation may be less
efficient or even absent in certain climate scenarios.
On present-day Earth, the ozone layer effectively
absorbs radiation between 180 and 280 nm through
the formation and destruction of O3, as described in
Equation 1, and partially absorbs radiation between 280
and 310 nm. This region, located in the stratosphere at
approximately 10 to 50 km above Earth’s surface, acts
as a natural shield against harmful UV radiation. This
protection was likely essential for the early emergence
of life on land, as it shielded organisms from harmful
ultraviolet radiation (Cockell 2000). However, alterna-
tive views suggest that such protection may not have
been strictly required, since natural environments could
also mitigate UV exposure. For instance, water bodies
and geological cover can attenuate UV flux, particularly
at the UV wavelengths, and may have provided effec-
tive refuges for early organisms or prebiotic molecules
(Estrela & Valio 2018; Estrela et al. 2020; Ranjan et al.
2022).
A comparison of different atmospheric models reveals
that only in the modern Earth-like atmosphere scenario
with fUV=10, the ozone concentration is sufficient for
the appearance of the Hartley band (200 and 320 nm)
(see Figure 3). It was anticipated that ozone concen-
trations in the Archean model were expected to be ex-
tremely low, as well as in the modern Earth model with
fUV=1. This significant disparity in ozone concentration
among models leads to variations in the UV radiation
that penetrates the simulated atmospheres and reaches
the planet’s surface, potentially impacting the planet’s
habitability.
In Figure 3, we illustrate the incident radiation reach-
ing the top of TOI-700d’s atmosphere that penetrates
down to its surface. Haze exhibits robust absorption,
especially at blue and UV wavelengths, emphasizing
its significant atmospheric role alongside ozone.
In
the Archean model with low UV flux (fUV=1), the
constituent molecules of haze play a more significant
role than ozone photochemistry under high UV flux
(fUV=10) conditions for a modern Earth-like atmo-
sphere. Consequently, in the Archean model, the pres-
ence of a thick haze layer provides significant UV shield-
ing, absorbing radiation at wavelengths below 230 nm.
This contrasts with the modern Earth scenario, where
ozone primarily absorbs in the 200–310nm range, par-
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
9
ticularly through the Hartley band.
As a result, in
the low-UV regime, haze acts as the dominant pro-
tective mechanism against harmful radiation, reducing
surface UV flux to levels lower than those on present-
day Earth, especially under higher atmospheric pres-
sure conditions. Photolysis in the Archean model with
fUV=10 leads to markedly lower concentrations of haze
constituents than in the fUV=1 Archean model, result-
ing in reduced absorption of incident flux for wave-
lengths below 190 nm. This is evidenced by Figure 4,
which demonstrates a significant rise in haze concentra-
tion within TOI-700d’s atmosphere under fUV=1 con-
ditions compared to fUV=10. When assessing Archean
models, the optical depth increases to 104. Furthermore,
in the Earth-like atmosphere model with fUV=1, there
was minimal ozone production, failing to generate a dis-
cernible feature in the spectrum, such as the Hartley
band.
In this context, the surface UV in this case is
lower only than in the Archean fUV = 10 model.
Near-UV photons not only drive the production of
haze precursors but also enhance their destruction via
oxidizing pathways (e.g., O/OH from CO2 photolysis),
making net haze formation non-monotonic with incident
UV radiation. Consistent with this mechanism, in our
Archean simulations with enhanced stellar UV the or-
ganic haze column is reduced and the surface UV in-
creases. While short-wavelength UV initiates methane
photolysis, stronger UV in an anoxic, CO2-bearing at-
mosphere also accelerates competing oxidative loss of
key intermediates, lowering the aerosol yield. This non-
linear behavior agrees with prior modeling of CO2-rich
Archean conditions (e.g., Arney et al. 2016) and with
broader evidence that the specific routes to particle for-
mation remain uncertain and model-dependent (H¨orst
2017).
3.2. Biological effect
To assess the habitability potential of TOI-700 d, we
conducted an analysis of the UV radiation’s impact on
the planet’s surface. We follow the methodology previ-
ously utilized for the Kepler-96 (Estrela & Valio 2018)
and Trappist-1 (Estrela et al. 2020) systems. Our evalu-
ation involves two key microorganisms: Deinococcus ra-
diodurans, known for its resilience to high UV doses, and
Escherichia coli, a widely researched bacterium. The bi-
ologically significant irradiance (Eeff), also referred to as
fluence, is calculated based on the star’s emission:
Eeff=
Z λ2
λ1
Finc(λ) S(λ) dλ .
(3)
In this equation Finc represents the total stellar incident
flux at a specific wavelength λ, while S(λ) is the action
spectra of the bacteria (refer to Figure 5).
To determine the bacteria’s survival rate, we consid-
ered the total radiation dose accumulated by each bac-
terium during its generation time. We adopted a genera-
tion time of 20 minutes for E. coli (Arp & Jensen 1980)
and 100 minutes for D. radiodurans (Mattimore et al.
1995).
The survival rate for 37% and 10% of the bacte-
rial population as a function of the UV radiation re-
ceived is given by Gasc´on et al. (1995).
Specifically,
for D. radiodurans, these rates correspond to doses of
338 and 553 J m−2, while for E. coli, the rates are 17.3
and 22.6 J m−2, respectively. The UV wavelength range
considered by Gasc´on et al. (1995) spans from 230 to
320 nm for D. radiodurans and from 224 to 300 nm for E.
coli. These wavelength bounds were determined based
on the bacteria’s action spectrum and the absorption
cross-section of the molecules under study.
To estimate the potential habitability of TOI-700 d
in the given scenarios, we calculated the survival rates
of the two bacteria on the planet’s surface. Applying
Equation 3, we determined the radiation doses that the
bacteria would experience over a generation period (20
minutes for E. coli and 100 minutes for D. radiodurans).
The detailed results are listed in Table 2, while
Figure 6 displays the derived values with the boundaries
delineating the survival rates specifically for 10% of the
bacterial population. Concerning TOI-700d, among the
four modeled atmospheres with different pressures, only
the Archean atmosphere with fUV=10 fails to sustain
the survival of E. coli (across all pressures) and D. ra-
diodurans (at pressures of 0.5 and 1.0 bar) under intense
UV exposure.
As highlighted by Estrela et al. (2020), TRAPPIST-1,
a star smaller and cooler than TOI-700 but highly ac-
tive with constant flare activity, requires an ozone layer
for the survival of E. coli bacteria on habitable plan-
ets within the system. The existence of this layer is a
critical aspect for TOI-700 d to maintain an Earth-like
atmosphere. The presence and strength of UV radiation
are key contributors to ozone formation in the atmo-
sphere, as discussed in Subsection 3.1. This is achieved
only when fUV=10. In this scenario, the microorgan-
isms could thrive in such atmosphere.
As discussed previously, denser hazes avert catas-
trophic cooling of the planet. Arney et al. (2016) con-
cluded that hazes could potentially enhance planetary
habitability by providing UV protection, reducing sur-
face UV flux by approximately 97% compared to a
planet devoid of haze.
This reduction could poten-
10
Sumida et al.
180
200
220
240
260
280
300
320
10
−57
10
−50
10
−43
10
−36
10
−29
10
−22
10
−15
10
−8
10
−1
Energy
Flux
[W
m
−2
nm
−1
]
Archean: f
UV
=1
Archean: f
UV
=10
M. Earth: f
UV
=1
M. Earth: f
UV
=10
Exoat osphere
Earth's surface
Earth's Exoat osphere
180
200
220
240
260
280
300
320
Wavelength
[nm]
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
Energy
Flux
[W
m
−2
nm
−1
]
Figure 3. Upper panel: UV flux reaching the surface of the planet TOI-700 d for an Archean atmosphere, depicted in purple
(fUV=1) and red (fUV=10), and a modern Earth-like atmosphere, in green (fUV=1) and blue (fUV=10). The different surface
pressures, namely 4.0, 2.0, 1.0, and 0.5, are represented by tones gradually shifting towards lighter shades, indicating a transition
from higher to lower pressure. The solar flux reaching both Earth’s exosphere and surface is depicted by solid black line and
dashed black line, respectively, primarily for the purpose of comparison. Lower panel: a blowup of the flux scale depicted on
the upper panel.
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
11
200
250
300
350
400
450
500
550
Wavelength
[nm]
10
−5
10
−3
10
−1
10
1
Optical
Depth
[tau]
Aerosols
Archean: f
UV
=
1
Archean: f
UV
=
10
Figure 4.
Atmospheric optical depth as a function of
wavelength for fUV=1 (purple curves) and fUV=10 (red
curves) Archean atmospheres.
Different surface pressures,
namely 4.0, 2.0, 1.0, and 0.5, are represented by tones grad-
ually shifting towards lighter shades, indicating a transition
from higher to lower pressure. Under 10 times stellar UV,
the aerosol column thins (lower τ), consistent with non-
monotonic haze production in CO2-rich anoxic atmospheres
where short-wave UV also enhances oxidative destruction of
intermediates.
tially sustain terrestrial organisms that existed 2.7-2.6
billion years ago. In our analysis, the haze in fUV=1
Archean atmosphere lowers the Biologically Effective Ir-
radiance to 1.24× 10−23 J m−2 for E. coli and to 5.82×
10−16 J m−2 for D. radiodurans, at a pressure of 1 bar.
No other scenario produced such low values, indicating
that haze, in this context, offers a significantly greater
protective barrier than ozone.
However, in scenarios
where haze and ozone are less prevalent, such as in the
Archean atmosphere model with fUV=10, only the bac-
terium D. radiodurans could survive with atmospheres
under pressures of 2.0 and 4.0 bar.
3.3. Simulated Transmission Spectra
In Figures 7 and 8, we present the transmission spec-
tra from the atmospheric models, including the corre-
sponding spectrum for each scenario featuring the most
significant spectral lines at a pressure of 1 bar. In ad-
dition to displaying the spectral lines of chemical com-
positions, these figures also illustrate the Rayleigh scat-
tering slope. Benneke & Seager (2012) proposed using
the Rayleigh scattering slope as a means to constrain
atmospheric pressure on exoplanets. However, accord-
ing to Arney et al. (2016), this approach may not be
feasible for planets with hydrocarbon hazes due to the
strong absorption effects at short wavelengths caused by
Table 2. Biologically effective irradiance in [J m−2] for TOI-
700 d atmospheric models.
UV=1
Pressure
Bacterium
Archean
Modern Earth
[bar]
[J m−2]
[J m−2]
0.5
E. coli
2.86×10−14
3.88
D. radiodurans
1.03×10−11
17.67
1.0
E. coli
1.24×10−23
2.35
D. radiodurans
5.82×10−16
11.45
2.0
E. coli
7.89×10−28
1.02
D. radiodurans
8.41×10−16
5.67
4.0
E. coli
2.47×10−31
0.050
D. radiodurans
3.05×10−21
0.89
UV=10
Pressure
Bacterium
Archean
Modern Earth
[bar]
[J m−2]
[J m−2]
0.5
E. coli
212.33
0.12
D. radiodurans
752.32
13.89
1.0
E. coli
173.10
0.037
D. radiodurans
620.60
10.20
2.0
E. coli
124.01
0.0033
D. radiodurans
450.60
6.17
4.0
E. coli
78.53
9.27×10−6
D. radiodurans
288.82
2.50
Note: the Effis calculated over a generation period for
both types of bacteria.
these hazes. The hazes exhibit a significant spectral im-
pact at shorter wavelengths, primarily due to their pro-
nounced blue and UV absorption. The presence of thick
hazes notably diminishes the strength of gaseous absorp-
tion features in transit transmission spectra, especially
at shorter wavelengths where these hazes are optically
thick.
However, while hazes obscure key biosignature
lines, they also enhance the overall detectability of the
atmosphere by increasing scattering and transit depth,
making the presence of an atmosphere more apparent.
This characteristic is observed in the fUV=1 Archean
atmosphere model.
The Rayleigh slope in transmission spectra arises from
the stronger scattering of shorter-wavelength light by
particles much smaller than the wavelength itself. Con-
sequently, blue and violet wavelengths experience more
scattering than red wavelengths.
In cases with high
ozone concentrations, such as the modern Earth-like at-
mosphere model at fUV = 10, the presence of O3 signifi-
cantly steepens the Rayleigh slope. This effect is further
evidenced by the appearance of distinct absorption lines
(e.g., near 10 µm) in the model subjected to intense UV
12
Sumida et al.
200
220
240
260
280
300
320
340
Wavelength [nm]
10
−3
10
−1
Relative Res onse
UVC
UVB
UVA
D. Radiodurans Action S ectrum
200
220
240
260
280
300
Wavelength [nm]
0.0
0.5
1.0
Relative Response
UVC
UVB
E. Coli Action Spect um
Figure 5. Action spectra, or biological response, for D. radiodurans (left) and E. coli (right).
1
2
3
4
Surface
Pressure
[bar]
10
−28
10
−23
10
−18
10
−13
10
−8
10
−3
10
2
E.Coli Survival Zon
E. Coli
1
2
3
4
Surfac
Pr ssur
[bar]
10
−19
10
−16
10
−13
10
−10
10
−7
10
−4
10
−1
10
2
D.Rdurans Survival Zon
D. Radiodurans
Arch an: f
UV
=1
Arch an: f
UV
=10
M.Earth: f
UV
=1
M.Earth: f
UV
=10
E
ff
[
J
m
−2
]
Figure 6. Biologically Effective Irradiance as a function of surface pressure based on results of Table 2. The Effis calculated
over a generation period for the two types of bacteria. The green area delineates the conditions where 10% of E. coli (left panel)
and D. radiodurans (right panel) would thrive, whereas the white region represents an environment unsuitable for bacterial
survival.
flux; by contrast, at fUV = 1, these lines are weaker and
virtually absent for Archean atmospheres.
Moreover, transmission spectra are strongly impacted
by surface pressure. In higher-pressure atmospheres, in-
creased pressure broadening leads to a reduction in scale
height, resulting in weaker spectral features. By con-
trast, lower-pressure atmospheres exhibit larger scale
heights, extending the atmospheric column and pro-
ducing stronger absorption signals.
This pressure-
dependent effect is clearly visible in Figures 7 and 8,
where higher-pressure cases lead to more muted spectral
lines, whereas lower-pressure cases enhance their ampli-
tudes.
To distinguish planets with compositions resembling
modern Earth-like or Archean atmospheres, the iden-
tification of CH4 features could prove to be a crucial
observational constraint. The strength of methane fea-
tures correlates with its abundance in the atmosphere,
as demonstrated by the comparison of feature heights.
However, the presence of CH4 will depend predomi-
nantly on the received UV flux rather than the type of
atmosphere modeled in this study. Conversely, although
CO2 features are prominent in all simulated scenarios,
irrespective of their abundance, relying solely on CO2
detection to characterize the type of atmosphere is in-
adequate, as the presence of CO2 is conspicuous in all of
the simulated cases. This observation was also noted by
Suissa et al. (2020) in their simulations of “Early Mars”
atmosphere.
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
13
From the particular scenarios investigated in our
study, we can deduce that CO2 is less sensitive to fluc-
tuations in temperature and pressure.
Consequently,
its absorption characteristics remain relatively stable re-
gardless of atmospheric conditions on the planet, which
stands in contrast to the variability seen in methane.
However, detecting absorption lines of CO2 and CH4
can pose difficulties. In the case of a moist stratosphere,
the overlap of H2O spectral bands with those of CO2 and
CH4 requires an extended observation period to reach a
desired level of detection significance (see Mikal-Evans
2022).
The simultaneous detection of CH4 and CO2 is par-
ticularly relevant to evaluate biosignatures under anoxic
conditions akin to Archean Earth. Although our mod-
els assume the coexistence of these gases, their obser-
vation in real exoplanetary atmospheres – particularly
when found in disequilibrium or with a notable absence
of CO – could strongly indicate biological activity. This
is because non-biological sources of methane typically
produce carbon monoxide as a by-product. Thus, at-
mospheres abundant in CO2 and CH4 but scarce in CO
remain significant indicators of biotic processes, under-
scoring the importance of monitoring these gases for po-
tential extraterrestrial life (see Krissansen-Totton et al.
2018; Mikal-Evans 2022; Rotman et al. 2023).”
Consequently, our findings align with Suissa et al.
(2020) regarding spectral line contrasts peaking at less
than 15 ppm.
Thus, characterizing TOI-700 d with
JWST appears unfeasible due to its instrumental noise
ranging from 10-20 parts per million at 1σ (Greene et al.
2016).
According to estimates of Suissa et al. (2020),
hundreds of JWST transits or eclipses would be re-
quired, exceeding the nominal lifetime of the observa-
tory. Therefore, characterizing TOI-700d’s atmosphere
remains an extremely challenging task even for next-
generation telescopes.
4. SUMMARY AND CONCLUSIONS
In this work, we coupled the 1D Photochemical Model
with the Atmos Climate Model to investigate the atmo-
spheric conditions of TOI-700d under two distinct sce-
narios: an analogue of the Archean Earth and a modern
Earth-like atmosphere. Although both scenarios corre-
spond to biologically active periods in Earth’s history,
our goal is to assess whether such environments could re-
main habitable under the photochemical UV conditions
of an M dwarf. To do this, we explore different levels of
stellar UV flux – quiescent (fUV = 1) and enhanced flare
(fUV = 10) conditions – across a range of atmospheric
pressures (0.5, 1.0, 2.0, and 4.0 bar). Our key findings
are summarized as follows:
• Atmospheric Composition and UV Shield-
ing: Archean (methane-rich) and modern Earth-
like atmospheres exhibit diverse photochemical
pathways, significantly influenced by the incoming
UV flux. In the quiescent-state Archean scenario,
high haze production was found to provide efficient
shielding against harmful UV radiation, whereas
modern Earth-like atmospheres under high UV
flux relied more on ozone formation to attenuate
surface-level UV.
• Surface Conditions and Potential for Life:
the temperature profiles suggest that most simu-
lated atmospheres could sustain surface temper-
atures compatible with liquid water, an essential
requirement for life.
Through biologically effec-
tive irradiance calculations, we determined that
the survival of model bacteria such as Escherichia
coli and Deinococcus radiodurans is strongly de-
pendent on a combination of atmospheric pres-
sure, haze abundance, and ozone column density.
Specifically, thick haze layers (in the context of an
Archean atmosphere without intensified UV ex-
posure) or robust ozone formation (in a modern
Earth-like environment with high UV radiation)
can provide significant protection against surface-
level UV exposure.
• Transmission Spectra and Biosignature De-
tection: we used the Planetary Spectrum Genera-
tor (PSG) to produce synthetic transmission spec-
tra for each model. Haze particles can significantly
obscure spectral features at shorter wavelengths,
potentially masking key biosignatures such as CH4
and O2, while ozone absorption bands become
prominent only when UV flux levels are sufficient
in an Earth-like atmosphere. Our results indicate
that the detectability of biosignatures on TOI-
700 d could be extremely challenging with current
or near-future instrumentation, as the strongest
spectral lines remain at the level of a few to a few
tens of parts per million. Furthermore, our trans-
mission spectrum analysis underscores the signif-
icant role of surface pressure in shaping observed
spectral features:
higher pressures reduce scale
height and thus diminish absorption line depths,
whereas lower pressures yield more pronounced
spectral features due to an extended atmospheric
column.
These factors – haze scattering, ozone
absorption, and variations in stellar UV flux –
interact in a complex manner, posing significant
challenges for the detection of biosignatures in M-
dwarf planetary systems.
14
Sumida et al.
10
0
10
1
Wavelength
[μm]
2
4
6
8
10
12
14
Relative
transit
depth
[ppm]
TOI
μ
700
d
:
Archean
(f
UV
=
1)
P=4.0 bar
P=2.0 bar
P=1.0 bar
P=0.5 bar
10
0
10
1
Wavelength
[μm]
0.2
0.4
0.6
0.8
1.0
1.2
Transmittance
1e−5
TOI
−
700
d
:
Archean
(f
UV
=
1;
P
=
1.0
bar)
T
)tal
Haze
CO
2
CH
4
H
2
O
Ra.leigh
10
0
10
1
Wavelength
[μm]
0
2
4
6
8
10
12
14
Relative
transit
depth
[ppm]
TOI
μ
700
d
:
Archean
(f
UV
=
10)
P=4.0 bar
P=2.0 bar
P=1.0 bar
P=0.5 bar
10
0
10
1
Wavelength
[μm]
0.2
0.4
0.6
0.8
1.0
1.2
Transmittance
1e−5
TOI
−
700
d
:
Archean
(f
UV
=
10;
P
=
1.0
bar)
T
)tal
CO
2
CH
4
H
2
O
CO
Ra.leigh
Figure 7. TOI-700 d Archean atmosphere’s transmission spectra (top first and third panels) and transmittance (second and
fourth panels) for the most abundant molecules. The Archean atmospheres with fUV=1 and fUV=10 are depicted in purple
(first panel) and red (third panel), respectively. The different surface pressures, i.e., 4.0, 2.0, 1.0, and 0.5 bar, are represented
by tones gradually shifting towards lighter shades, indicating a transition from higher to lower pressure. In panels 2 and 4,
in addition to the most prevalent molecules, the influences of haze and Rayleigh scattering are also visible. The combined
contribution is depicted by the black line.
• Implications for General Exoplanetary Sys-
tems:
while TOI-700 d is a particularly com-
pelling target due to its relatively inactive host
star and Earth-like insolation, our study offers
broader insights for exoplanets orbiting M dwarfs.
Dense hazes and high levels of ozone can each play
pivotal roles in rendering a planetary surface hab-
itable, yet they also complicate detection of biosig-
natures. For many M-dwarf systems, flare events
– and the associated UV flux boosts – could lead
to rapid shifts in atmospheric chemistry, either de-
stroying protective layers or driving the formation
of new ones. These processes are critical to con-
sider when searching for signs of life elsewhere in
the universe.
Taken together, our results highlight the intercon-
nected effects of photochemistry, atmospheric pres-
Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints
15
10
0
10
1
Wavelength
[μm]
0
2
4
6
8
10
Relative
transit
depth
[ppm]
TOI
μ
700
d
:
Modern
Earth
(f
UV
=
1)
P=4.0
P=2.0
P=1.0
P=0.5
10
0
10
1
Wavelength
[μm]
2
4
6
8
Transmittance
1e−6
TOI
−
700
d
:
M.
Ear.h
(f
UV
=
1;
P
=
1.0
bar)
T
o.a(
CO
2
CH
4
CO
N
2
O
O
3
H
2
O
O
2
Ra0(eigh
10
0
10
1
Wavelength
[μm]
0
2
4
6
8
Relative
transit
depth
[ppm]
TOI
μ
700
d
:
Modern
Earth
(f
UV
=
10)
P=4.0
P=2.0
P=1.0
P=0.5
10
0
10
1
Wavelength
[μm]
1
2
3
4
5
6
7
8
Transmittance
1e−6
T
otal
CO
2
CH
4
H
2
O
O
3
O
2
Rayleigh
TOI
−
700
d
:
M.
Ear.h
(f
UV
=
10;
P
=
1.0
bar)
Figure 8. Same as Figure 7 for the modern Earth-like atmosphere model of TOI-700 d.
sure, haze formation, and UV flux on the habitability
and spectroscopic detectability of exoplanetary atmo-
spheres. Although TOI-700 d exemplifies these complex-
ities under relatively benign stellar conditions, analo-
gous mechanisms likely operate in numerous M-dwarf
systems. Ongoing and future multi-wavelength obser-
vational campaigns – aimed at capturing transits, flare
events, and high-resolution spectral observations – will
be essential to refine these models and advance our un-
derstanding of habitable environments beyond the Solar
System.
The authors gratefully acknowledge Dr.
Paul B. Rimmer
for his thoughtful suggestions, which helped improve the
manuscript. We also thank Abel Granjeiro de Souza for the
work on TOI-700 d of his undergraduate thesis, which set
the stage for the investigations undertaken here. VYDS and
AV acknowledge the partial financial support received from
Brazilian FAPESP grants #2021/14897-9, #2018/04055-8
#2021/02120-0, and #2024/03652-3, as well as CAPES and
MackPesquisa funding agencies. R.E. research was carried
out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics
and Space Administration (80NM0018D0004).
16
Sumida et al.
APPENDIX
Table 3 shows the initial conditions and produced surface mixing ratios of the nine major long-lived species
for the specific case where the pressure is 1 bar.
All input and output data from this study can be found at
10.5281/zenodo.13947863.
Table 3. Initial conditions and produced surface mixing ratios of long-lived species for the specific case where the pressure is
1 bar, with emphasis on the main species.
Initial conditions
Produced surface mixing ratios
Species
Surface mixing ratios
(P = 1 bar)
Archean
Modern Earth
Archean
Modern Earth
Archean
Modern Earth
(fuv = 1)
(fuv = 1)
(fuv = 10)
(fuv = 10)
O2
1.00×10−8
2.10×10−1
1.00×10−8
2.10×10−1
1.00×10−8
2.10×10−1
O3
5.94×10−12
3.01×10−8
2.17×10−20
1.05×10−8
3.96×10−12
3.89×10−8
CO
1.20×10−4
2.35×10−7
1.20×10−4
5.57×10−3
1.20×10−4
2.97×10−5
CO2
2.00×10−2
3.85×10−4
2.00×10−2
3.60×10−4
2.00×10−2
3.76×10−4
H2O
1.17×10−2
7.05×10−3
1.17×10−2
7.05×10−3
6.18×10−3
7.05×10−3
CH4
2.00×10−3
1.79×10−6
2.00×10−3
2.58×10−3
2.00×10−3
2.12×10−4
SO2
2.22×10−10
1.82×10−10
2.51×10−10
2.35×10−10
3.02×10−10
2.57×10−10
SO3
2.74×10−19
2.49×10−19
1.75×10−25
2.84×10−20
1.503×10−19
2.96×10−20
N2O
—
3.39×10−7
—
9.01×10−6
—
1.73×10−6
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Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX631 Evaluating Habitability and Biosignature Detection on TOI-700 d: The Role of UV Environment and Atmospheric Pressure Viktor Yuri Don ́a Sumida,1, 2 Raissa Estrela,2 and Adriana Valio1 1Center for Radio Astronomy and Astrophysics Mackenzie, Mackenzie Presbyterian University, Rua da Consola ̧c ̃ao 930, S ̃ao Paulo, SP, Brazil 2Jet Propulsion Laboratory, California 4800 Oak Grove Dr, CA, USA ABSTRACT M dwarfs have long been prime targets in the search for habitable exoplanets, owing to their abundance in the galaxy and the relative ease of detecting Earth-sized worlds within their narrower habitable zones. Yet, these low-mass stars can emit high-energy radiation that may gradually erode planetary atmospheres, raising concerns about long-term habitability. TOI-700, a relatively quiescent M dwarf that hosts four known planets, stands out due to its Earth-sized TOI-700d in the star's habitable zone. Here, we assess whether a habitable environment can be sustained on TOI-700d by analyzing different UV flux levels and atmospheric pressures. We focus on two atmospheric scenarios - one analogous to the Archean Earth and another representing a modern Earth-like environment - using a 1D photochemistry-climate model. Our results indicate that all simulated cases can maintain temperatures compatible with liquid water on the surface. However, the dominant photochemical pathways differ substantially with UV levels: under low-UV conditions, haze formation in the Archean-like atmosphere provides the main UV shielding, whereas under intensified UV, ozone production in the modern-like atmospheres can protect the surface from harmful doses. Interestingly, although haze can impede the detection of certain biosignatures, such as CH4, CO2 and O2, it also enhances the overall atmospheric signal by increasing scattering and transit depth, potentially aiding in revealing the presence of an atmosphere. These findings underscore the dual role of hazes as both a challenge for biosignature detection and a potential protection of surface habitability. Keywords: Exoplanets (498) - Exoplanet atmospheres (487) - Habitable planets (695) - Ultraviolet astronomy (1736) - Transmission spectroscopy (2133) 1. INTRODUCTION In recent years, red dwarf stars have emerged as primary targets in the search for extraterrestrial life due to their extended lifespans. Although the extended pre-main-sequence contraction phase of M dwarf stars may desiccate or erode the atmospheres of planets within their habitable zones, potentially triggering a runaway greenhouse and compromising early habitability (Luger & Barnes 2015), these effects occur mainly during the early stages of stellar evolution. As M dwarfs age, their activity declines substantially, which could allow secondary atmospheres or late-formed planets to sustain habitable environments (Gale & Wandel 2017). The Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) has played a key role in identifying terrestrial-sized planets, particularly those in the habitable zones of M-dwarfs, which are prime candidates for atmospheric characterization due to their favorable starto-planet size ratios and close-in habitable zones that enhance transit detectability (Shields et al. 2016). These characteristics make planets orbiting M-dwarfs strong candidates for biosignature searches with instruments such as the James Webb Space Telescope (JWST) and future Extremely Large Telescopes (ELTs). Despite these advantages, characterizing exoplanet atmospheres remains challenging, requiring more detailed observational and theoretical studies. Key biosignature gases - including O2, CH4, H2O, N2O, O3, and CO2 - are fundamental for climate regulation, protection from harmful radiation, and the potential indication of biological activity (Airapetian et al. 2017). However, ultraviolet (UV) radiation from M-dwarfs, particularly in the UVB (280-315nm) and UVC (100-280nm) ranges, 16 Oct 2025 2 Sumida et al. can significantly affect atmospheric chemistry and the habitability of planets in close orbits. The TOI-700 system, an M2.5V dwarf star, hosts four known exoplanets, two of which - TOI-700 d and TOI700 e - reside in its habitable zone (Rodriguez et al. 2020; Gilbert et al. 2023). TOI-700d, a potentially Earth-sized planet (∼1.14 R⊕), receives about 86% of the Earth's insolation and is of particular interest for habitability studies. Although planets around M-dwarfs are susceptible to atmospheric erosion due to intense X-ray and extreme ultraviolet (XUV) radiation (e.g., Nishioka et al. 2023), TOI-700 itself appears to be relatively quiescent, with X-ray luminosities comparable to those of the modern Sun (Gilbert et al. 2020). This makes TOI-700 d an excellent candidate for studying the atmospheric processes that influence planetary habitability. To evaluate TOI-700 d's potential habitability, we employed the 1D Photochemical Model coupled to the Atmos Climate model (Arney et al. 2016), analyzing both Archean and modern Earth-like atmospheres - biogenically active scenarios that assume the presence of life - across pressures of 0.5, 1.0, 2.0, and 4.0 bar. Despite TOI-700 being relatively quiescent during the observed period so far, we simulated two scenarios: one assuming that TOI-700 remains in a quiet state (fUV=1) and another representing a flare event causing a 10-fold increase in NUV flux (fUV=10), as estimated by prior studies on M-type stars (Rekhi et al. 2023). Our work addresses critical but underexplored aspects such as how photochemistry influences atmospheric composition, the potential for false-positive biosignatures, and climate effects under varying levels of UV flux and pressure. These challenges align with the call for advanced atmospheric models by the 2020 Astronomy and Astrophysics decadal survey1 to evaluate star-planet interactions, haze formation, atmospheric dynamics, and escape processes. Although detecting TOI-700 d's transmission features would require untenably long JWST integration times, the significance of this work is reinforced by the growing focus of observational programs utilizing the James Webb and Hubble Space Telescopes, which aim to probe planetary atmospheres and stellar activity around M dwarfs, such as the Rocky Worlds DDT program2. In this study, we assess the habitability potential of TOI-700 d by simulating different atmospheric compositions and evaluating their response to varying stellar 1 https://www.nationalacademies.org/our-work/decadal-survey-on-astronomy-and-astrophysics-2020-astro2020 2 https://www.stsci.edu/contents/news/jwst/2024/rocky-worlds-ddt-selects-its-first-targets UV flux. In Section 2, we outline the methods employed, including the 1D Photochemical Model coupled to the Atmos Climate model used to simulate Archean and modern Earth-like atmospheres under different pressure conditions. Section 3 presents our results and discussion, including the atmospheric abundance profiles, photochemical pathways, and the impact of UV radiation on surface habitability. A key aspect of our analysis involves evaluating whether TOI-700 d could sustain life by assessing the ability of its atmosphere to shield against UV radiation and its implications for habitability, which we discuss in Subsection 3.2. Additionally, we investigate the impact of photochemically produced haze on the planet's atmospheric transmission spectra, since haze can obscure key spectral features of biosignature gases, which we explore in Subsection 3.3. Finally, Section 4 summarizes our key findings, discusses the broader implications for exoplanetary habitability studies, and highlights future observational prospects. 2. METHODS Possible atmospheres for the exoplanet TOI-700 d were simulated with the 1D Photochemical Model coupled to the Atmos Climate model (Arney et al. 2016). The code generates atmospheres with well-defined chemical and climate profiles. In the modern Earth-like atmosphere photochemical model, the code incorporates 310 chemical reactions and 72 chemical species, including 11 short-lived species. On the other hand, for the Archean atmosphere, the model incorporates 405 chemical reactions and 77 chemical species, with 9 of them being short-lived. The initial atmospheric state is established by initially executing the photochemical model, which relies on user-specified boundary conditions. After the photochemical model converges, its output files containing altitude, pressure, gas mixing ratios, haze particle sizes, and haze number densities are transferred to the climate model. The climate model utilizes the photochemical model's solution as its starting point and continues running until it also achieves convergence. Subsequently, the updated temperature and water vapor profiles are fed back into the photochemical model. This iterative process continues until convergence is attained. Detailed descriptions of the model can be found in Arney et al. (2016), and are publicly accessible3. To accurately simulate planetary environments and observable properties, the models require comprehensive data on both planetary and stellar characteristics. This 3 https://github.com/VirtualPlanetaryLaboratory/atmos Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints 3 includes detailed information on stellar parameters and spectra, as well as planetary physical and orbital parameters. Additionally, environmental data are crucial for refining the simulations. Here, we used the stellar and planetary parameters reported by Gilbert et al. (2020). For the stellar spectrum, Atmos provides twenty-one star spectra encompassing a range of spectral types. Given that TOI-700 is classified as an M2.5V star, the closest spectral type available is M3.0V, which corresponds to GJ 581 (Turnbull 2015). Following Arney et al. (2016) and Meadows et al. (2018), organic haze particles were considered to possess a fractal shape in both photochemical and climate models, to the detriment of spherical (nonfractal or classical Mie) particles. The solar zenith angles (SZAs) used in these models were selected to realistically represent the global averaged insolation. In particular, 60◦ in the climate model corresponds to global mean insolation (accounting for S/4 and diurnal effects), while 45◦in the photochemical model - based on Segura et al. (2003) - reproduces the modern Earth's ozone column given the employed chemistry. These values are common approximations, although more sophisticated methods (e.g., Gaussian integration) could improve insolation estimates if full ozone photochemistry were implemented. Another key parameter of Atmos is the UV amplification factor (fUV). This parameter plays an important role in simulating stellar flares, as it boosts the UV flux reaching the top of the atmosphere. In simple terms, it determines the relationship between Fflare,UV and Fq,UV, where Fq represents the quiescent UV radiation flux. Rekhi et al. (2023) provided estimates for the median top-of-atmosphere (TOA) NUV irradiances for planets located in the habitable zones of M0 to M6 dwarfs using the GALEX sample. They also determined threshold flare amplitudes and their corresponding frequencies necessary to achieve the TOA NUV irradiance levels similar to those of the young Sun. According to these authors, the exoplanetary NUV irradiance, accounting for flares, is within one order of magnitude of the young Earth's NUV irradiance for host stars post-M2. Flares reaching the young Earth's irradiance level occur multiple times a day for stars later than M3. According to Table 3 in the study by Rekhi et al. (2023), the ratio of flare to quiescent NUV flux (Fflare,UV/Fq,UV) is 12.4 for an M2 star and 7.9 for an M3 star. Since TOI-700 is classified as an M2.5 star, we extrapolated an intermediate value of approximately 10 for fUV in our simulations. Although our Atmos simulations for various atmospheres of TOI-700 d provide valuable insights, it's imperative to determine the strength of potentially observable spectral features. To accomplish this, we use the calculated values of gas mixing ratios of H2O, CH4, C2H6, CO2, O2, O3, CO, H2CO, HNO3, NO2, SO2, N2O and N2, and aerosol parameters derived from the Atmos simulations. These inputs are fed into the Planetary Spectrum Generator (PSG, Villanueva et al. 2018) module known as GlobES4 (Global Exoplanet Spectra), which is designed for synthesizing both transmission and emission spectra. The PSG integrates advanced radiative transfer techniques to accurately generate comprehensive synthetic spectra of TOI-700 d and to assess the feasibility of observing spectral features in its atmosphere. 3. RESULTS AND DISCUSSION In Subsection 3.1, we present the results and discuss the implications of the atmospheric abundance profiles of the models, along with their respective transmission spectra. Then in Subsection 3.2, we assess the habitability potential of TOI-700 d by conducting an analysis of the impact of UV radiation on the planet's surface. To complete the analysis, in Subsection 3.3, we present the simulated transmission spectra derived from our models and discuss their observational implications. 3.1. Photochemical pathways in different conditions Profiles of possible atmospheres for TOI-700 d are simulated, based on both Archean and modern Earth-like atmospheres. Our simulations incorporate variations in atmospheric pressure at 0.5, 1.0, 2.0, and 4.0 bar, and compare two distinct scenarios: one with fUV = 1 and the other with fUV = 10. The resulting abundance profiles for key molecules of interest are presented in Figure 1. All figures presented here adhere to the same color scheme for the atmospheres: Archean and modern Earth-like with fUV=1 are represented in purple and green, while Archean and modern Earth-like atmospheres with fUV=10 are depicted in red and blue, respectively. The varying pressures, namely 4.0, 2.0, 1.0, and 0.5 bar, are indicated by a gradual transition towards lighter shades, illustrating a shift from higher to lower pressure. The choice of methane to carbon dioxide ratios in atmospheric models reflects the conditions from different geological periods and their relevance to climate processes. For Archean Earth scenarios, a CH4/CO2 ratio above 0.1 is widely used as a criterion since it enables significant organic haze formation in primitive atmospheres (e.g., Haqq-Misra et al. 2008; Arney et al. 2017; Meadows et al. 2018; Mak et al. 2023). 4 https://psg.gsfc.nasa.gov/apps/globes.php 4 Sumida et al. 10 -19 10 -16 10 -13 10 -10 10 -7 10 -4 10 -1 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Oxygen[O] 10 -18 10 -16 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Molecular Oxygen [O 2 ] 10 -28 10 -25 10 -22 10 -19 10 -16 10 -13 10 -10 10 -7 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Ozone [O 3 ] 10 -6 10 -5 10 -4 10 -3 10 -2 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Carbon Monoxide[CO] 10 -4 10 -3 10 -2 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Carbon Dioxide [CO 2 ] 10 -14 10 -12 10 -10 10 -8 10 -6 10 -4 10 -2 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Water Vapor [H 2 O] 10 -22 10 -19 10 -16 10 -13 10 -10 10 -7 10 -4 Volume Mi-ing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] Archean: UV =1 Archean: UV =10 M. Earth: UV =1 M. Earth: UV =10 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Methane [CH 4 ] 10 -14 10 -12 10 -10 10 -8 10 -6 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Sulfur Dioxide [SO 2 ] 10 -25 10 -22 10 -19 10 -16 10 -13 10 -10 10 -7 Volume Mixing Ratios 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] 259.3 254.8 250.2 247.5 245.3 243.0 Temperature [K] Sulfur Trioxide [SO 3 ] Figure 1. Each panel, from the upper left to the lower right, represents the atmospheric pressure of O, O2, O3, CO, CO2, H2O, CH4, SO2 and SO3 as a function of mixing ratios. The initial conditions and the produced surface mixing ratios can be seen either in Table 3 or in the files available at The Archean atmospheres with fUV=1 and fUV=10 are depicted in purple and red, while modern Earth-like atmospheres with fUV=1 and fUV=10 are shown in green and blue, respectively. The different surface pressures, i.e., 4.0, 2.0, 1.0, and 0.5 bar, are represented by tones gradually shifting towards lighter shades, indicating a transition from higher to lower pressure. For TOI-700d, our Archean atmospheric model adopts a CH4/CO2 ratio of approximately 0.5, a value chosen to reflect the enhanced methane accumulation expected around M-dwarf stars. This selection is based on the findings of Meadows et al. (2018), who demonstrated that a CH4/CO2 ratio of approximately 0.3 is necessary to trigger the formation of organic haze on Proxima Centauri b. In an oxygen-poor, Archean-like atmosphere, the primary pathways for CH4 destruction involve OH radicals formed via H2O photolysis and O atoms generated by CO2 photolysis. However, reduced UV emission around 300 nm typical of M-dwarf stars leads to a diminished photolysis rate, allowing methane to accumulate. In our model for TOI-700 d, as methane becomes more dominant, aerosol production intensifies, leading to a thickening haze that significantly reduces the penetration of starlight into the planet's surface, resulting in substantial cooling (see Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints 5 Haqq-Misra et al. 2008). Note that this Archean-like composition is included solely as a limiting case, allowing us to probe the climatic and UV-shielding consequences of a methane-rich, low-oxygen atmosphere; sustaining CH4/CO2 ≈0.5 would in reality require exceptionally vigorous biogenic activity or intense geological outgassing, so we treat it as an exploratory end-member rather than a prediction of TOI-700 d's actual state. For the modern-Earth analog, we adopt recent nearsurface mixing ratios (CH4 ≈1.8 ppmv; CO2 ≈385 ppmv), corresponding to CH4/CO2 ≈4.7×10-3, as an empirical observational baseline for comparison (e.g., IPCC 2021)5. Furthermore, we specified the lower boundary conditions of the photochemical model to reproduce observed surface-level concentrations of biogenic trace gases on modern Earth, for example, N2O around 0.3 ppmv in the lower atmosphere. This value corresponds to estimated modern biogenic fluxes that sustain such concentrations. In this way, the modern Earth scenario in our model begins with N2O levels aligned with present-day atmospheric observations (see IPCC 2021). In such an atmosphere, like that of modern Earth, methane undergoes rapid oxidation through reactions with hydroxyl radicals (OH): O2 + hν (λ < 200 nm) -→: O + O : O + O2 -→O3 O3 + hν (λ < 310 nm) -→O2 + O∗ O∗+ H2O -→2 OH CH4 + OH -→CH3 + H2O (1) This lower CH4/CO2 ratio is more suitable for current Earth-like conditions, where oxygen significantly reduces methane's presence, limiting its capacity to contribute to haze formation. By modeling both Archean and modern conditions, we can observe the shift in haze dynamics and its impact on the planet's climate system. Table 3 in the appendix shows the initial conditions and produced surface mixing ratios of the nine major long-lived species for the Archean and modern Earthlike atmosphere models, at a pressure of 1 bar. For more detailed analysis, including simulations with varying pressures and UV fluxes, the complete data set is available at 5 Over geologic timescales, atmospheric CO2 on Earth is regulated by the carbonate-silicate feedback and has been further perturbed in the industrial era; CH4, in contrast, is primarily controlled by biogenic sources and removal by atmospheric oxidants. In this work we use the observed modern-Earth CH4/CO2 ratio only as an empirical baseline for comparison; our conclusions are insensitive to modest variations within the explored range (10-3-10-2). The exact composition of the Archean atmosphere when life first emerged on Earth remains a subject of ongoing debate. Nevertheless, the interplay between the atmosphere and geological cycles has left discernible traces that help identify the main atmospheric gases of this era. Combined geological and atmospheric modeling suggests that the Archean atmosphere (4-2.5 Gyr ago) may have contained approximately 10 to 2500 times the modern amount of CO2 and 100 to 104 times the modern amount of CH4 (see Catling & Zahnle 2020). Although the plausibility of extremely high methane levels in prebiotic eras remains debated, recent studies point to potential abiotic mechanisms. Wogan et al. (2023), for instance, argue that major impact events could substantially increase CH4 concentrations, allowing for episodic surges in atmospheric methane even in the absence of biological activity. As shown in the bottom-left panel of Figure 1, the methane concentration in the two model atmospheres with fUV = 1 and in the Archean atmosphere with fUV = 10 is roughly 103 times higher than in the modern Earth's atmosphere with fUV = 10. Notably, both the Archean and modern Earth models with fUV = 10 exhibit an exponential decrease in methane abundance with increasing altitude. In atmospheres exposed to elevated UV flux (fUV = 10), methane undergoes a more pronounced decline in its mixing ratio owing to enhanced CH4 oxidation by UV radiation. By contrast, in simulations with normal UV flux (fUV = 1), the mixing ratio of methane still decreases with altitude, albeit more gradually, reflecting a comparatively lower oxidation rate. These differing rates of CH4 oxidation under varying UV flux conditions lead to distinct vertical mixing ratio profiles across the modeled atmospheres. In our Archean-like scenarios, the surface temperature is higher than in the modern Earth case (see Figure 2). Although CH4 is more radiatively efficient per molecule than CO2, primarily at trace-level concentrations typical of present-day Earth, the total greenhouse forcing in the Archean simulations is dominated by the substantially higher absolute abundances of both CH4 and CO2 (see Kiehl & Dickinson 1987; Etminan et al. 2016). This enhanced greenhouse effect more than offsets the reduced stellar energy input to TOI-700d, allowing surface temperatures that are comparable to, or even exceed, those required for the stability of liquid water. However, without continuous replenishment these gases cannot sustain such warming over geological timescales: CH4 is photochemically oxidized to CO and CO2, while CO2 is gradually sequestered by carbonate-silicate weathering and ocean uptake. 6 Sumida et al. Consequently, geological activity on a planet is necessary to balance these gases through the carbonatesilicate cycle, which involves volcanic activity, plate tectonics, and erosion. The lack of greenhouse gas replacement by geological activity has contributed to Mars' current temperatures of -50◦C on its surface and a pressure of only 1% of Earth's atmospheric pressure today (Segura & Navarro-Gonz ́alez 2005). Regarding temperature under haze conditions, cooling is minimized around M-dwarfs. These stars emit energy primarily at wavelengths where organic hazes are relatively transparent (Arney et al. 2017). In all scenarios, the estimated temperatures suggest conditions conducive to liquid water on the surface of TOI-700d. The lowest temperature, approximately 278 K, is associated with Modern Earth-like models with fUV=1 and fUV=10, irrespective of surface pressure. Conversely, the highest temperature of 306 K is observed in the fUV=1 Archean model with a surface pressure of 4 bar. Detailed temperature values for each model are provided in Table 1. Studies on Archean Earth suggest that extensive ocean fractions could persist for globally averaged surface temperatures as low as 250-260K (e.g., Wolf & Toon 2013; Arney et al. 2016). However, such studies are based on planets orbiting Sun-like stars. Around M dwarfs, intense XUV flux and frequent stellar flares can significantly enhance atmospheric escape, leading to substantial water loss over time. While previous studies (e.g., do Amaral et al. 2022) have emphasized the role of stellar flares in driving early atmospheric escape, we also acknowledge the cumulative effects of stellar luminosity evolution. Late M dwarfs such as TOI-700 can brighten significantly over time. According to our XUV flux calculations (see Equation 1 in Jackson et al. 2012 and Equation22 in Owen and Wu 2017), TOI-700 d received an estimated XUV flux of ∼574 erg s-1 cm-2 at 100 Myr6 - over 15 times higher than its present-day value of ∼37 erg s-1 cm-2. This elevated high-energy input during the first few hundred million years of evolution could have driven intense hydrodynamic escape, especially if TOI-700 d originally possessed a volatile-rich envelope. Our results thus reinforce the scenario raised by Luger & Barnes (2015), in which secular stellar brightening - independent of flares 6 Stellar X-ray emission typically undergoes an initial saturation phase lasting tens to hundreds of millions of years, followed by a gradual decline. This evolutionary trend is observed across all stellar spectral types (Jackson et al. 2012; Shkolnik & Barman 2014; McDonald et al. 2019). Table 1. Surface Temperature Estimate for TOI-700 d. Pressure Atmospheric fUV Temperature [bar] Model [K] 0.5 Archean 1 288.5 Archean 10 283.9 Modern Earth 1 278.4 Modern Earth 10 278.4 1.0 Archean 1 288.8 Archean 10 279.3 Modern Earth 1 278.5 Modern Earth 10 278.4 2.0 Archean 1 296.0 Archean 10 294.5 Modern Earth 1 278.4 Modern Earth 10 278.3 4.0 Archean 1 306.2 Archean 10 285.4 Modern Earth 1 278.4 Modern Earth 10 278.3 - may lead to substantial or even complete devolatilization of initially habitable-zone planets around M dwarfs. Similarly to Meadows et al. (2018), attempting to run the climate model to a converged state using the same top-of-atmosphere pressure as the photochemical model proved unfeasible. This limitation stems from instabilities within the model, yet it does not impact the resultant photochemistry, as indicated by Arney et al. (2016). Consequently, when the climate model transfers its temperature and water profiles to the photochemical model, they remain fixed at their values at the top of the climate grid, forming isoprofiles above this grid's upper limit. This effect is illustrated in Figure 2, where an increase in surface pressure leads to isoprofiles occurring at higher pressures. An apparently puzzling feature in Figure 1 is the behavior of CO in the modern Earth scenario run with fUV = 1. Below the homopause (P ≳10-5 bar) the gas-phase carbon inventory - fCO2 + fCO + fCH4 - is conserved in all scenarios, yet the partitioning among the three species depends on the local UV field and on the oxidative state of the atmosphere: i Modern Earth, fUV = 1: a modest O3 layer removes most 200-310nm photons. The resulting near-UV deficit almost quenches CO2 photolysis above 10-6 bar, eliminating the main in situ source of CO, and keeps OH production low; the residual OH therefore acts as a net sink via CO + OH →CO2. With its source suppressed, Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints 7 150 200 250 300 10 -5 10 -4 10 -3 10 -2 10 -1 Pressure [bar] P 0 =0.5 Archean: f UV =1 Archean: f UV =10 M. Ear h: f UV =1 M. Ear h: f UV =10 150 200 250 300 10 -4 10 -2 10 0 P 0 =1.0 Archean: f UV =1 Archean: f UV =10 M. Ear h: f UV =1 M. Ear h: f UV =10 150 200 250 300 Tempera ure [K] 10 -4 10 -2 10 0 Pressure [bar] P 0 =2.0 Archean: f UV =1 Archean: f UV =10 M. Ear h: f UV =1 M. Ear h: f UV =10 150 200 250 300 Tempera ure [K] 10 -4 10 -2 10 0 P 0 =4.0 Archean: f UV =1 Archean: f UV =10 M. Ear h: f UV =1 M. Ear h: f UV =10 Figure 2. Atmospheric pressure profiles plotted against temperature for different pressures. The Archean atmospheres with fUV=1 and fUV=10 are depicted in purple and red, while modern Earth-like atmospheres with fUV=1 and fUV=10 are shown in green and blue, respectively. The Modern Earth atmosphere models of fUV=1 and fUV=10 can only be distinguished from one another in the case where the surface pressure (P0) equals 1 (top right panel), otherwise for all other pressures, the curve for fUV=10 (blue) superimposes on the fUV=1 (green) curve. CO decreases with altitude and the lost carbon reappears as CO2. ii Archean, fUV = 1: in the absence of O2/O3, CO2 photolysis operates throughout the upper column and the CO profile increases. Removal of carbon into hydrocarbon haze occurs between 10-6 and 10-5 bar, but the fraction sequestered remains ≲10-4 % of the gas-phase inventory and does not affect the global carbon budget. Nevertheless, the spectral impact can be substantial: haze particles, even at sub-ppm mass fractions, nonlinearly enhance the atmospheric opacity through scattering and absorption, strongly modulating the observed transmission spectra. iii All cases with fUV = 10: a ten-fold stronger nearUV flux simultaneously restores vigorous CO2 photolysis and produces abundant OH. The CO again rises with altitude, but the enhanced UV field partly photodissociates both CO and the newly formed CO2, preventing the high-altitude build-up seen in the low-UV modern case. Hence, the CO decline in the modern, low-UV atmosphere is a natural consequence of UV shielding by O3 plus efficient removal via CO + OH →CO2, whereas all other simulations lack one (or both) of those conditions and therefore show a monotonically increasing CO with height. We also note that there is no photochemical haze in the modern Earth-like scenarios. The high O2 and O3 levels maintain oxidizing conditions that suppress hydrocarbon polymerization, thereby preclud8 Sumida et al. ing haze nucleation. Both HCAER and HCAER27 remain zero at all altitudes, even under enhanced UV flux (fUV = 10). Sulfur dioxide (SO2) and hydrogen sulfide (H2S) represent the most prevalent sulfur gases released during volcanic activity. The challenge of reconciling the observed sulfur mass-independent fractionation (SMIF) in laboratory sulfur dioxide photochemistry with that recorded in the Archean rock record has persisted as a significant issue since its initial discovery (Farquhar et al. 2000, 2001). In particular, sulfur dioxide exhibits a complex UV absorption spectrum characterized by two prominent absorption features spanning 260 to 340 nm and 165 to 235 nm, which plays a crucial role in understanding atmospheric processes and climate dynamics. The accumulation of SO2 in the atmosphere of TOI-700d (see Figure 1) is attributed to low UV emission from the host star. This is evident in the significantly higher concentration of sulfur dioxide when comparing scenarios with low (fUV=1) and high (fUV=10) UV flux. In O2-rich atmospheres, absorption of ultraviolet photons by SO2 can induce predissociation, whereby the excited molecule dissociates to SO + O*. This initiates the following chemical transformations: SO2 + hν (λ < 220 nm) -→SO + O∗ SO + O2 + M -→SO3 + M (2) Although SO2 absorbs UV radiation, albeit less efficiently compared to O3, it can still accumulate in significant quantities. When combined with CH4 and haze particles in the atmosphere, it may provide adequate protection against UV radiation, particularly in O2-poor atmospheres. The production of SO3 and its potential reaction with H2O to form sulfuric acid (H2SO4) is a crucial process in planetary atmospheres. As highlighted by Meadows et al. (2018), while this reaction can proceed efficiently in Venus-like atmospheres, it is heavily dependent on the availability of water. In the absence of sufficient H2O, the formation of H2SO4 aerosols is significantly limited. Moreover, the photochemical model used in this study does not simulate Venus-like atmospheres, particularly the formation of H2SO4 clouds and their feedback effects on atmospheric temperature structure. Additionally, on Earth, the oxidation of SO2 to SO3 often serves as a bottleneck that controls the rate 7 HCAER and HCAER2 are the two aerosol tracers used by the Atmos photochemistry code. They represent the direct condensation of the gas-phase precursors C4H2 and C5H4. Nonzero values indicate ongoing haze formation. of sulfuric acid formation (Lizzio & DeBarr 1997). However, on planets orbiting M-dwarf stars, such as Proxima Centauri b, as studied by Meadows et al. (2018), the destruction of SO3 may become the dominant process, potentially inhibiting the formation of H2SO4 clouds. This suggests that, while sulfuric acid clouds could still form under specific conditions, their formation may be less efficient or even absent in certain climate scenarios. On present-day Earth, the ozone layer effectively absorbs radiation between 180 and 280 nm through the formation and destruction of O3, as described in Equation 1, and partially absorbs radiation between 280 and 310 nm. This region, located in the stratosphere at approximately 10 to 50 km above Earth's surface, acts as a natural shield against harmful UV radiation. This protection was likely essential for the early emergence of life on land, as it shielded organisms from harmful ultraviolet radiation (Cockell 2000). However, alternative views suggest that such protection may not have been strictly required, since natural environments could also mitigate UV exposure. For instance, water bodies and geological cover can attenuate UV flux, particularly at the UV wavelengths, and may have provided effective refuges for early organisms or prebiotic molecules (Estrela & Valio 2018; Estrela et al. 2020; Ranjan et al. 2022). A comparison of different atmospheric models reveals that only in the modern Earth-like atmosphere scenario with fUV=10, the ozone concentration is sufficient for the appearance of the Hartley band (200 and 320 nm) (see Figure 3). It was anticipated that ozone concentrations in the Archean model were expected to be extremely low, as well as in the modern Earth model with fUV=1. This significant disparity in ozone concentration among models leads to variations in the UV radiation that penetrates the simulated atmospheres and reaches the planet's surface, potentially impacting the planet's habitability. In Figure 3, we illustrate the incident radiation reaching the top of TOI-700d's atmosphere that penetrates down to its surface. Haze exhibits robust absorption, especially at blue and UV wavelengths, emphasizing its significant atmospheric role alongside ozone. In the Archean model with low UV flux (fUV=1), the constituent molecules of haze play a more significant role than ozone photochemistry under high UV flux (fUV=10) conditions for a modern Earth-like atmosphere. Consequently, in the Archean model, the presence of a thick haze layer provides significant UV shielding, absorbing radiation at wavelengths below 230 nm. This contrasts with the modern Earth scenario, where ozone primarily absorbs in the 200-310nm range, parAssessing Biosignatures on TOI-700 d under UV and Pressure Constraints 9 ticularly through the Hartley band. As a result, in the low-UV regime, haze acts as the dominant protective mechanism against harmful radiation, reducing surface UV flux to levels lower than those on presentday Earth, especially under higher atmospheric pressure conditions. Photolysis in the Archean model with fUV=10 leads to markedly lower concentrations of haze constituents than in the fUV=1 Archean model, resulting in reduced absorption of incident flux for wavelengths below 190 nm. This is evidenced by Figure 4, which demonstrates a significant rise in haze concentration within TOI-700d's atmosphere under fUV=1 conditions compared to fUV=10. When assessing Archean models, the optical depth increases to 104. Furthermore, in the Earth-like atmosphere model with fUV=1, there was minimal ozone production, failing to generate a discernible feature in the spectrum, such as the Hartley band. In this context, the surface UV in this case is lower only than in the Archean fUV = 10 model. Near-UV photons not only drive the production of haze precursors but also enhance their destruction via oxidizing pathways (e.g., O/OH from CO2 photolysis), making net haze formation non-monotonic with incident UV radiation. Consistent with this mechanism, in our Archean simulations with enhanced stellar UV the organic haze column is reduced and the surface UV increases. While short-wavelength UV initiates methane photolysis, stronger UV in an anoxic, CO2-bearing atmosphere also accelerates competing oxidative loss of key intermediates, lowering the aerosol yield. This nonlinear behavior agrees with prior modeling of CO2-rich Archean conditions (e.g., Arney et al. 2016) and with broader evidence that the specific routes to particle formation remain uncertain and model-dependent (H ̈orst 2017). 3.2. Biological effect To assess the habitability potential of TOI-700 d, we conducted an analysis of the UV radiation's impact on the planet's surface. We follow the methodology previously utilized for the Kepler-96 (Estrela & Valio 2018) and Trappist-1 (Estrela et al. 2020) systems. Our evaluation involves two key microorganisms: Deinococcus radiodurans, known for its resilience to high UV doses, and Escherichia coli, a widely researched bacterium. The biologically significant irradiance (Eeff), also referred to as fluence, is calculated based on the star's emission: Eeff= Z λ2 λ1 Finc(λ) S(λ) dλ . (3) In this equation Finc represents the total stellar incident flux at a specific wavelength λ, while S(λ) is the action spectra of the bacteria (refer to Figure 5). To determine the bacteria's survival rate, we considered the total radiation dose accumulated by each bacterium during its generation time. We adopted a generation time of 20 minutes for E. coli (Arp & Jensen 1980) and 100 minutes for D. radiodurans (Mattimore et al. 1995). The survival rate for 37% and 10% of the bacterial population as a function of the UV radiation received is given by Gasc ́on et al. (1995). Specifically, for D. radiodurans, these rates correspond to doses of 338 and 553 J m-2, while for E. coli, the rates are 17.3 and 22.6 J m-2, respectively. The UV wavelength range considered by Gasc ́on et al. (1995) spans from 230 to 320 nm for D. radiodurans and from 224 to 300 nm for E. coli. These wavelength bounds were determined based on the bacteria's action spectrum and the absorption cross-section of the molecules under study. To estimate the potential habitability of TOI-700 d in the given scenarios, we calculated the survival rates of the two bacteria on the planet's surface. Applying Equation 3, we determined the radiation doses that the bacteria would experience over a generation period (20 minutes for E. coli and 100 minutes for D. radiodurans). The detailed results are listed in Table 2, while Figure 6 displays the derived values with the boundaries delineating the survival rates specifically for 10% of the bacterial population. Concerning TOI-700d, among the four modeled atmospheres with different pressures, only the Archean atmosphere with fUV=10 fails to sustain the survival of E. coli (across all pressures) and D. radiodurans (at pressures of 0.5 and 1.0 bar) under intense UV exposure. As highlighted by Estrela et al. (2020), TRAPPIST-1, a star smaller and cooler than TOI-700 but highly active with constant flare activity, requires an ozone layer for the survival of E. coli bacteria on habitable planets within the system. The existence of this layer is a critical aspect for TOI-700 d to maintain an Earth-like atmosphere. The presence and strength of UV radiation are key contributors to ozone formation in the atmosphere, as discussed in Subsection 3.1. This is achieved only when fUV=10. In this scenario, the microorganisms could thrive in such atmosphere. As discussed previously, denser hazes avert catastrophic cooling of the planet. Arney et al. (2016) concluded that hazes could potentially enhance planetary habitability by providing UV protection, reducing surface UV flux by approximately 97% compared to a planet devoid of haze. This reduction could poten10 Sumida et al. 180 200 220 240 260 280 300 320 10 -57 10 -50 10 -43 10 -36 10 -29 10 -22 10 -15 10 -8 10 -1 Energy Flux [W m -2 nm -1 ] Archean: f UV =1 Archean: f UV =10 M. Earth: f UV =1 M. Earth: f UV =10 Exoat osphere Earth's surface Earth's Exoat osphere 180 200 220 240 260 280 300 320 Wavelength [nm] 10 -10 10 -8 10 -6 10 -4 10 -2 10 0 Energy Flux [W m -2 nm -1 ] Figure 3. Upper panel: UV flux reaching the surface of the planet TOI-700 d for an Archean atmosphere, depicted in purple (fUV=1) and red (fUV=10), and a modern Earth-like atmosphere, in green (fUV=1) and blue (fUV=10). The different surface pressures, namely 4.0, 2.0, 1.0, and 0.5, are represented by tones gradually shifting towards lighter shades, indicating a transition from higher to lower pressure. The solar flux reaching both Earth's exosphere and surface is depicted by solid black line and dashed black line, respectively, primarily for the purpose of comparison. Lower panel: a blowup of the flux scale depicted on the upper panel. Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints 11 200 250 300 350 400 450 500 550 Wavelength [nm] 10 -5 10 -3 10 -1 10 1 Optical Depth [tau] Aerosols Archean: f UV = 1 Archean: f UV = 10 Figure 4. Atmospheric optical depth as a function of wavelength for fUV=1 (purple curves) and fUV=10 (red curves) Archean atmospheres. Different surface pressures, namely 4.0, 2.0, 1.0, and 0.5, are represented by tones gradually shifting towards lighter shades, indicating a transition from higher to lower pressure. Under 10 times stellar UV, the aerosol column thins (lower τ), consistent with nonmonotonic haze production in CO2-rich anoxic atmospheres where short-wave UV also enhances oxidative destruction of intermediates. tially sustain terrestrial organisms that existed 2.7-2.6 billion years ago. In our analysis, the haze in fUV=1 Archean atmosphere lowers the Biologically Effective Irradiance to 1.24× 10-23 J m-2 for E. coli and to 5.82× 10-16 J m-2 for D. radiodurans, at a pressure of 1 bar. No other scenario produced such low values, indicating that haze, in this context, offers a significantly greater protective barrier than ozone. However, in scenarios where haze and ozone are less prevalent, such as in the Archean atmosphere model with fUV=10, only the bacterium D. radiodurans could survive with atmospheres under pressures of 2.0 and 4.0 bar. 3.3. Simulated Transmission Spectra In Figures 7 and 8, we present the transmission spectra from the atmospheric models, including the corresponding spectrum for each scenario featuring the most significant spectral lines at a pressure of 1 bar. In addition to displaying the spectral lines of chemical compositions, these figures also illustrate the Rayleigh scattering slope. Benneke & Seager (2012) proposed using the Rayleigh scattering slope as a means to constrain atmospheric pressure on exoplanets. However, according to Arney et al. (2016), this approach may not be feasible for planets with hydrocarbon hazes due to the strong absorption effects at short wavelengths caused by Table 2. Biologically effective irradiance in [J m-2] for TOI700 d atmospheric models. UV=1 Pressure Bacterium Archean Modern Earth [bar] [J m-2] [J m-2] 0.5 E. coli 2.86×10-14 3.88 D. radiodurans 1.03×10-11 17.67 1.0 E. coli 1.24×10-23 2.35 D. radiodurans 5.82×10-16 11.45 2.0 E. coli 7.89×10-28 1.02 D. radiodurans 8.41×10-16 5.67 4.0 E. coli 2.47×10-31 0.050 D. radiodurans 3.05×10-21 0.89 UV=10 Pressure Bacterium Archean Modern Earth [bar] [J m-2] [J m-2] 0.5 E. coli 212.33 0.12 D. radiodurans 752.32 13.89 1.0 E. coli 173.10 0.037 D. radiodurans 620.60 10.20 2.0 E. coli 124.01 0.0033 D. radiodurans 450.60 6.17 4.0 E. coli 78.53 9.27×10-6 D. radiodurans 288.82 2.50 Note: the Effis calculated over a generation period for both types of bacteria. these hazes. The hazes exhibit a significant spectral impact at shorter wavelengths, primarily due to their pronounced blue and UV absorption. The presence of thick hazes notably diminishes the strength of gaseous absorption features in transit transmission spectra, especially at shorter wavelengths where these hazes are optically thick. However, while hazes obscure key biosignature lines, they also enhance the overall detectability of the atmosphere by increasing scattering and transit depth, making the presence of an atmosphere more apparent. This characteristic is observed in the fUV=1 Archean atmosphere model. The Rayleigh slope in transmission spectra arises from the stronger scattering of shorter-wavelength light by particles much smaller than the wavelength itself. Consequently, blue and violet wavelengths experience more scattering than red wavelengths. In cases with high ozone concentrations, such as the modern Earth-like atmosphere model at fUV = 10, the presence of O3 significantly steepens the Rayleigh slope. This effect is further evidenced by the appearance of distinct absorption lines (e.g., near 10 μm) in the model subjected to intense UV 12 Sumida et al. 200 220 240 260 280 300 320 340 Wavelength [nm] 10 -3 10 -1 Relative Res onse UVC UVB UVA D. Radiodurans Action S ectrum 200 220 240 260 280 300 Wavelength [nm] 0.0 0.5 1.0 Relative Response UVC UVB E. Coli Action Spect um Figure 5. Action spectra, or biological response, for D. radiodurans (left) and E. coli (right). 1 2 3 4 Surface Pressure [bar] 10 -28 10 -23 10 -18 10 -13 10 -8 10 -3 10 2 E.Coli Survival Zon E. Coli 1 2 3 4 Surfac Pr ssur [bar] 10 -19 10 -16 10 -13 10 -10 10 -7 10 -4 10 -1 10 2 D.Rdurans Survival Zon D. Radiodurans Arch an: f UV =1 Arch an: f UV =10 M.Earth: f UV =1 M.Earth: f UV =10 E ff [ J m -2 ] Figure 6. Biologically Effective Irradiance as a function of surface pressure based on results of Table 2. The Effis calculated over a generation period for the two types of bacteria. The green area delineates the conditions where 10% of E. coli (left panel) and D. radiodurans (right panel) would thrive, whereas the white region represents an environment unsuitable for bacterial survival. flux; by contrast, at fUV = 1, these lines are weaker and virtually absent for Archean atmospheres. Moreover, transmission spectra are strongly impacted by surface pressure. In higher-pressure atmospheres, increased pressure broadening leads to a reduction in scale height, resulting in weaker spectral features. By contrast, lower-pressure atmospheres exhibit larger scale heights, extending the atmospheric column and producing stronger absorption signals. This pressuredependent effect is clearly visible in Figures 7 and 8, where higher-pressure cases lead to more muted spectral lines, whereas lower-pressure cases enhance their amplitudes. To distinguish planets with compositions resembling modern Earth-like or Archean atmospheres, the identification of CH4 features could prove to be a crucial observational constraint. The strength of methane features correlates with its abundance in the atmosphere, as demonstrated by the comparison of feature heights. However, the presence of CH4 will depend predominantly on the received UV flux rather than the type of atmosphere modeled in this study. Conversely, although CO2 features are prominent in all simulated scenarios, irrespective of their abundance, relying solely on CO2 detection to characterize the type of atmosphere is inadequate, as the presence of CO2 is conspicuous in all of the simulated cases. This observation was also noted by Suissa et al. (2020) in their simulations of "Early Mars" atmosphere. Assessing Biosignatures on TOI-700 d under UV and Pressure Constraints 13 From the particular scenarios investigated in our study, we can deduce that CO2 is less sensitive to fluctuations in temperature and pressure. Consequently, its absorption characteristics remain relatively stable regardless of atmospheric conditions on the planet, which stands in contrast to the variability seen in methane. However, detecting absorption lines of CO2 and CH4 can pose difficulties. In the case of a moist stratosphere, the overlap of H2O spectral bands with those of CO2 and CH4 requires an extended observation period to reach a desired level of detection significance (see Mikal-Evans 2022). The simultaneous detection of CH4 and CO2 is particularly relevant to evaluate biosignatures under anoxic conditions akin to Archean Earth. Although our models assume the coexistence of these gases, their observation in real exoplanetary atmospheres - particularly when found in disequilibrium or with a notable absence of CO - could strongly indicate biological activity. This is because non-biological sources of methane typically produce carbon monoxide as a by-product. Thus, atmospheres abundant in CO2 and CH4 but scarce in CO remain significant indicators of biotic processes, underscoring the importance of monitoring these gases for potential extraterrestrial life (see Krissansen-Totton et al. 2018; Mikal-Evans 2022; Rotman et al. 2023)." Consequently, our findings align with Suissa et al. (2020) regarding spectral line contrasts peaking at less than 15 ppm. Thus, characterizing TOI-700 d with JWST appears unfeasible due to its instrumental noise ranging from 10-20 parts per million at 1σ (Greene et al. 2016). According to estimates of Suissa et al. (2020), hundreds of JWST transits or eclipses would be required, exceeding the nominal lifetime of the observatory. Therefore, characterizing TOI-700d's atmosphere remains an extremely challenging task even for nextgeneration telescopes. 4. SUMMARY AND CONCLUSIONS In this work, we coupled the 1D Photochemical Model with the Atmos Climate Model to investigate the atmospheric conditions of TOI-700d under two distinct scenarios: an analogue of the Archean Earth and a modern Earth-like atmosphere. Although both scenarios correspond to biologically active periods in Earth's history, our goal is to assess whether such environments could remain habitable under the photochemical UV conditions of an M dwarf. To do this, we explore different levels of stellar UV flux - quiescent (fUV = 1) and enhanced flare (fUV = 10) conditions - across a range of atmospheric pressures (0.5, 1.0, 2.0, and 4.0 bar). Our key findings are summarized as follows: • Atmospheric Composition and UV Shielding: Archean (methane-rich) and modern Earthlike atmospheres exhibit diverse photochemical pathways, significantly influenced by the incoming UV flux. In the quiescent-state Archean scenario, high haze production was found to provide efficient shielding against harmful UV radiation, whereas modern Earth-like atmospheres under high UV flux relied more on ozone formation to attenuate surface-level UV. • Surface Conditions and Potential for Life: the temperature profiles suggest that most simulated atmospheres could sustain surface temperatures compatible with liquid water, an essential requirement for life. Through biologically effective irradiance calculations, we determined that the survival of model bacteria such as Escherichia coli and Deinococcus radiodurans is strongly dependent on a combination of atmospheric pressure, haze abundance, and ozone column density. Specifically, thick haze layers (in the context of an Archean atmosphere without intensified UV exposure) or robust ozone formation (in a modern Earth-like environment with high UV radiation) can provide significant protection against surfacelevel UV exposure. • Transmission Spectra and Biosignature Detection: we used the Planetary Spectrum Generator (PSG) to produce synthetic transmission spectra for each model. Haze particles can significantly obscure spectral features at shorter wavelengths, potentially masking key biosignatures such as CH4 and O2, while ozone absorption bands become prominent only when UV flux levels are sufficient in an Earth-like atmosphere. Our results indicate that the detectability of biosignatures on TOI700 d could be extremely challenging with current or near-future instrumentation, as the strongest spectral lines remain at the level of a few to a few tens of parts per million. Furthermore, our transmission spectrum analysis underscores the significant role of surface pressure in shaping observed spectral features: higher pressures reduce scale height and thus diminish absorption line depths, whereas lower pressures yield more pronounced spectral features due to an extended atmospheric column. These factors - haze scattering, ozone absorption, and variations in stellar UV flux - interact in a complex manner, posing significant challenges for the detection of biosignatures in Mdwarf planetary systems. 14 Sumida et al. 10 0 10 1 Wavelength [μm] 2 4 6 8 10 12 14 Relative transit depth [ppm] TOI μ 700 d : Archean (f UV = 1) P=4.0 bar P=2.0 bar P=1.0 bar P=0.5 bar 10 0 10 1 Wavelength [μm] 0.2 0.4 0.6 0.8 1.0 1.2 Transmittance 1e-5 TOI - 700 d : Archean (f UV = 1; P = 1.0 bar) T )tal Haze CO 2 CH 4 H 2 O Ra.leigh 10 0 10 1 Wavelength [μm] 0 2 4 6 8 10 12 14 Relative transit depth [ppm] TOI μ 700 d : Archean (f UV = 10) P=4.0 bar P=2.0 bar P=1.0 bar P=0.5 bar 10 0 10 1 Wavelength [μm] 0.2 0.4 0.6 0.8 1.0 1.2 Transmittance 1e-5 TOI - 700 d : Archean (f UV = 10; P = 1.0 bar) T )tal CO 2 CH 4 H 2 O CO Ra.leigh Figure 7. TOI-700 d Archean atmosphere's transmission spectra (top first and third panels) and transmittance (second and fourth panels) for the most abundant molecules. The Archean atmospheres with fUV=1 and fUV=10 are depicted in purple (first panel) and red (third panel), respectively. The different surface pressures, i.e., 4.0, 2.0, 1.0, and 0.5 bar, are represented by tones gradually shifting towards lighter shades, indicating a transition from higher to lower pressure. In panels 2 and 4, in addition to the most prevalent molecules, the influences of haze and Rayleigh scattering are also visible. The combined contribution is depicted by the black line. • Implications for General Exoplanetary Systems: while TOI-700 d is a particularly compelling target due to its relatively inactive host star and Earth-like insolation, our study offers broader insights for exoplanets orbiting M dwarfs. Dense hazes and high levels of ozone can each play pivotal roles in rendering a planetary surface habitable, yet they also complicate detection of biosignatures. For many M-dwarf systems, flare events - and the associated UV flux boosts - could lead to rapid shifts in atmospheric chemistry, either destroying protective layers or driving the formation of new ones. These processes are critical to consider when searching for signs of life elsewhere in the universe. Taken together, our results highlight the interconnected effects of photochemistry, atmospheric presAssessing Biosignatures on TOI-700 d under UV and Pressure Constraints 15 10 0 10 1 Wavelength [μm] 0 2 4 6 8 10 Relative transit depth [ppm] TOI μ 700 d : Modern Earth (f UV = 1) P=4.0 P=2.0 P=1.0 P=0.5 10 0 10 1 Wavelength [μm] 2 4 6 8 Transmittance 1e-6 TOI - 700 d : M. Ear.h (f UV = 1; P = 1.0 bar) T o.a( CO 2 CH 4 CO N 2 O O 3 H 2 O O 2 Ra0(eigh 10 0 10 1 Wavelength [μm] 0 2 4 6 8 Relative transit depth [ppm] TOI μ 700 d : Modern Earth (f UV = 10) P=4.0 P=2.0 P=1.0 P=0.5 10 0 10 1 Wavelength [μm] 1 2 3 4 5 6 7 8 Transmittance 1e-6 T otal CO 2 CH 4 H 2 O O 3 O 2 Rayleigh TOI - 700 d : M. Ear.h (f UV = 10; P = 1.0 bar) Figure 8. Same as Figure 7 for the modern Earth-like atmosphere model of TOI-700 d. sure, haze formation, and UV flux on the habitability and spectroscopic detectability of exoplanetary atmospheres. Although TOI-700 d exemplifies these complexities under relatively benign stellar conditions, analogous mechanisms likely operate in numerous M-dwarf systems. Ongoing and future multi-wavelength observational campaigns - aimed at capturing transits, flare events, and high-resolution spectral observations - will be essential to refine these models and advance our understanding of habitable environments beyond the Solar System. The authors gratefully acknowledge Dr. Paul B. Rimmer for his thoughtful suggestions, which helped improve the manuscript. We also thank Abel Granjeiro de Souza for the work on TOI-700 d of his undergraduate thesis, which set the stage for the investigations undertaken here. VYDS and AV acknowledge the partial financial support received from Brazilian FAPESP grants #2021/14897-9, #2018/04055-8 #2021/02120-0, and #2024/03652-3, as well as CAPES and MackPesquisa funding agencies. R.E. research was carried out at the Jet Propulsion Laboratory, California (80NM0018D0004). 16 Sumida et al. APPENDIX Table 3 shows the initial conditions and produced surface mixing ratios of the nine major long-lived species for the specific case where the pressure is 1 bar. All input and output data from this study can be found at 10.5281/zenodo.13947863. Table 3. Initial conditions and produced surface mixing ratios of long-lived species for the specific case where the pressure is 1 bar, with emphasis on the main species. Initial conditions Produced surface mixing ratios Species Surface mixing ratios (P = 1 bar) Archean Modern Earth Archean Modern Earth Archean Modern Earth (fuv = 1) (fuv = 1) (fuv = 10) (fuv = 10) O2 1.00×10-8 2.10×10-1 1.00×10-8 2.10×10-1 1.00×10-8 2.10×10-1 O3 5.94×10-12 3.01×10-8 2.17×10-20 1.05×10-8 3.96×10-12 3.89×10-8 CO 1.20×10-4 2.35×10-7 1.20×10-4 5.57×10-3 1.20×10-4 2.97×10-5 CO2 2.00×10-2 3.85×10-4 2.00×10-2 3.60×10-4 2.00×10-2 3.76×10-4 H2O 1.17×10-2 7.05×10-3 1.17×10-2 7.05×10-3 6.18×10-3 7.05×10-3 CH4 2.00×10-3 1.79×10-6 2.00×10-3 2.58×10-3 2.00×10-3 2.12×10-4 SO2 2.22×10-10 1.82×10-10 2.51×10-10 2.35×10-10 3.02×10-10 2.57×10-10 SO3 2.74×10-19 2.49×10-19 1.75×10-25 2.84×10-20 1.503×10-19 2.96×10-20 N2O - 3.39×10-7 - 9.01×10-6 - 1.73×10-6 REFERENCES Airapetian, V. S., Jackman, C. 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|
2510.14774
|
Quantum oscillations and transport properties of layered single-crystal SrCu4As2
Sudip Malick,1, 2, ∗Michał J. Winiarski,1, 2, † Joanna Bławat,3
Hanna Świątek,1, 2 John Singleton,3 and Tomasz Klimczuk1, 2, ‡
1Faculty of Applied Physics and Mathematics, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland
2Advanced Materials Center, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdańsk, Poland
3National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
We report a systematic investigation of the physical properties and Fermi-surface topology of lay-
ered single-crystal SrCu4As2 using electrical transport, magnetotransport, and quantum-oscillation
experiments plus band-structure calculations. The temperature-dependent electrical resistivity re-
veals a hysteretic phase transition at TP = 59 K, most likely associated with a structural change. Hall
resistivity data suggest a marked change in the average hole density resulting from the latter phase
transition near TP . A large, linear and nonsaturating magnetoresistance is observed at low tem-
peratures in SrCu4As2, likely attributable to the multipocket Fermi surface. Quantum-oscillation
data measured in magnetic fields of up to 60 T show several oscillation frequencies exhibiting low
effective masses, indicating the presence of Dirac-like band dispersion in SrCu4As2, as suggested by
the band structure calculations.
I.
INTRODUCTION
Ternary pnictides with layered crystal structures have
drawn significant research interest due to their distinctive
physical properties resulting from the interplay of topol-
ogy, magnetism, superconductivity, charge-density waves
(CDWs), and structural disorder [1–10]. CaCu4As2-type
layered pnictide is one of the relevant systems where large
linear magnetoresistance, quantum oscillations, struc-
tural phase transition, and unusual magnetism are ob-
served due to their complex Fermi surface topology.
Very recently, a spin-moiré superlattice has been real-
ized in layered pnictide EuAg4Sb2, establishing the lay-
ered "142" system as a potential platform for emerging
spin-driven quantum-Hall states [11]. Our recent report
[9] indicates complex magnetism and unusual magne-
toresistance (MR) in EuAg4Sb2. The latter compound
features two successive antiferromagnetic phase transi-
tions, multiple metamagnetic transitions, and large non-
saturating MR with quantum oscillations. The presence
of several oscillation frequencies exhibiting low effective
masses in this compound indicates a complex Fermi sur-
face [9]. Interestingly, a structural distortion at 120 K
and an incommensurate antiferromagnetic phase transi-
tion below 9 K are observed in the arsenide EuAg4As2
[12, 13]. Similar structural distortion and ferromagnetic
phase transition are seen at 75 K and 35 K, respectively,
in isostructural EuCu4As2 [14].
Moreover, both com-
pounds show large magnetoresistance. These behaviors
suggest that the interplay between magnetism and struc-
tural changes plays a significant role in the electronic
properties of these materials.
While the nonmagnetic
arsenides of these series show similar high-temperature
structural distortion and large magnetoresistance, they
∗sudip.malick@pg.edu.pl
† michal.winiarski@pg.edu.pl
‡ tomasz.klimczuk@pg.edu.pl
show some additional features. For instance, quantum
oscillations linked with small Fermi pockets and a sig-
nificant change in the Fermi surface topology, in con-
trast to the band structure calculation, are observed due
to structural distortion in SrAg4As2 [15]. On the other
hand, CaCu4As2 exhibits Shubnikov-de Haas (SdH) os-
cillations and coexistence of the multilayer quantum-Hall
effect and a CDW state [10].
These findings motivate
us to investigate the nonmagnetic analog of EuCu4As2,
SrCu4As2, which crystallizes in a rhombohedral structure
(Fig. 1) with space group R¯3m [16]. A recent report on
thermal expansion and electrical resistivity data reveals a
structural distortion in SrCu4As2 at 145 K [17]. Density-
functional-theory calculations indicate that SrCu4As2 is
a semimetal, with highly dispersive bands near the Fermi
energy and holelike Fermi sheets along the Γ −Z direc-
tion [18].
However, few thorough investigation of the
physical properties of this compound have been carried
out.
Here, we present a comprehensive study of the
physical properties and Fermi-surface topology of high-
quality single-crystal SrCu4As2 using electrical resistiv-
ity, magnetotransport, and quantum-oscillation measure-
ments at high magnetic fields along with band-structure
calculations. Our findings indicate that SrCu4As2 under-
goes a possible structural-distortion-type phase transi-
tion. Magnetotransport measurements show a linear MR
and Hall resistivity. The quantum oscillation reveals a
complex Fermi surface with some quasi-two-dimensional
sections, in qualitative agreement with our theoretical
calculations. The Supplemental Material (SM) contains
information on single-crystal growth, structural charac-
terization, measurement techniques, band-structure cal-
culations, and fast Fourier transform (FFT) spectra [19]
(see also Refs. [17, 20–25] therein).
arXiv:2510.14774v1 [cond-mat.str-el] 16 Oct 2025
2
II.
RESULTS AND DISCUSSION
A.
Transport properties
FIG. 1.
The unit cell of SrCu4As2 shown using rhombohe-
dral (a) and hexagonal axes (b). There are two symmetry-
inequivalent Cu positions in the structure: Cu(1) and Cu(2).
Note the [0001] direction in the hexagonal cell is equivalent
to the [111] direction in the rhombohedral one.
Figure 2(a) shows the zero-field electrical resistivity
ρ(T) of a SrCu4As2 single crystal measured over the
temperature (T) range 2-300 K. The resistivity gradu-
ally decreases with falling temperature, before showing a
sudden drop at around 60 K. The overall electrical re-
sistivity implies metallic or semimetallic behavior and
a residual resistivity ratio (RRR) of 14, which is com-
parable to isostructural compounds, suggesting that the
as-grown crystals are of high quality [10, 15]. Note that
temperature-dependent heat-capacity data also exhibit a
feature at around 60 K [see the SM [19]]. The first-order
derivative of ρ(T) (Fig.
2(a) inset) indicates that the
phase transition occurs at TP = 59 K, much lower than
the 145 K reported previously [17]. Hysteresis is observed
in the resistivity data for heating and cooling [Fig. 2(b)];
although a significant enhancement of the electrical re-
sistivity is observed due to applied magnetic field, a field
of 9 T has no apparent effect on the transition temper-
ature or the hysteresis loop. Similar nonmagnetic phase
transitions are also seen in other "142"-type layered com-
pounds [10, 12, 14, 15, 26]. In SrCu4As2, the observed
phase transition is unlikely to be due to a CDW, as these
typically result in an upward jump in electrical resistivity
due to the opening up of a partial gap in the Fermi energy
[27].
Hence, the high-temperature nonmagnetic phase
transition in "142" systems is likely to be a structural
distortion [10, 12, 14, 15, 26]. To strengthen this asser-
tion for SrCu4As2, future temperature-dependent single-
crystal x-ray diffraction would be very helpful.
Figure.
2(c) shows the magnetoresistance measured
at various temperatures; the magnetic field is applied
along the c-axis and the current is in the ab-plane of
the SrCu4As2 single crystal. With increasing magnetic
field, the MR increases linearly and reaches a value of
120% at 9 T and 2 K. Similar MR values were recently
reported in the isostructural compounds SrAg4As2 [15],
CaCu4As2 [10], and SrCu4P2 [26]. However, the sizes of
these MRs are less than those of typical topological ma-
terials [28, 29]. A substantial decrease in MR is observed
as the temperature increases; however, there seems to
be no approach to saturation, even at 100 K and 9 T.
At low temperatures, SrCu4As2 exhibits almost linear
MR at low temperatures, as shown by the almost field-
independent first-order derivative of the MR data at 2
K [Fig. 2(d)]. Linear MR is often observed in topolog-
ical semimetals and insulators, attributed to their lin-
ear band dispersions and topologically protected surface
states [28, 30–33]. Moreover, some Dirac semimetals ex-
hibit a MR crossover from a weak-field quadratic field
dependence to high-field behavior. This has been inter-
preted using Abrikosov’s quantum-limit theorem, which
invokes Dirac carriers being restricted to the zeroth Lan-
dau level [5, 8, 34, 35]. Whilst there appears to be a small
region of quadratic MR in SrCu4As2 below B∗≈0.5
T at T = 50 K, there is no evidence for a similar de-
pendence at 2 K [Figs. 2(c) and 2(d)]. As we will see
below, the quantum oscillations have frequencies in the
approximate range 200 to 2000 T; hence, the crossover
field observed at 50 K is
two orders of magnitude be-
low the quantum limit. This indicates that Abrikosov’s
theory cannot explain the MR of SrCu4As2. The car-
rier compensation and mobility fluctuations could be the
other possible reasons for displaying large MR. The com-
pounds having an equal density of electrons and holes,
such as LaSb [36] and TaAs2 [37], exhibit high unsatu-
rated MR due to carrier compensation. Our Hall resistiv-
ity data (see below) suggest a dominating hole-type car-
rier, which eliminates the possibility of carrier compensa-
tion in SrCu4As2. On the other hand, the observed hole
mobility is low compared to compounds like disordered
Ag2+δSe and n-doped Cd3As2, where mobility fluctua-
tion induces large and linear MR [38–40]. The linear and
unsaturated MR in SrCu4As2 is likely associated with its
multiband Fermi surface with several tiny Fermi pockets,
characterized in band-structure calculations by Dirac-like
band dispersion near the Fermi energy and small effec-
tive masses obtained from the quantum oscillation data
(see below).
Figure. 2(e) displays the Hall resistivity ρxy as a func-
tion of applied magnetic field measured at various tem-
peratures.
The Hall resistivity exhibits a linear field
dependence with positive slope for all measured tem-
peratures between 2 and 200 K, suggesting that trans-
port in SrCu4As2 is dominated by holes.
The aver-
age carrier concentration (p) and Hall mobility (µ) have
been calculated using the expressions p = 1/(eRH) and
µ = RH/ρ(H = 0), where RH is the slope of the lin-
ear fit to the Hall resistivity data presented in the inset
of Fig. 2(e). The temperature variation of the average
hole concentration and mobility are presented in Figs.
2(f) and 2(g), respectively. These values are comparable
to those of isostructural compounds SrAg4As2 [15] and
CaCu4As2 [10]. As SrCu4As2 passes through the phase
3
0
5 0
1 0 0
1 5 0
2 0 0
1 0 0
2 0 0
3 0 0
µ ( c m
2 V
- 1 s - 1 )
T ( K )
0 . 8
1
1 . 2
1 . 4
p ( 1 0
2 1 c m
- 3 )
0
2
4
6
8
0
1 0
2 0
d M
R / d H ( T
- 1 )
0 H ( T )
2 K
5 0 K
B
* ~ 0 . 5 T
3 0
4 0
5 0
6 0
7 0
3 0
4 0
5 0
6 0
7 0
8 0
ρ ( µΩ c m
)
T ( K )
0 T
9 T
0
2
4
6
8
0
2
4
6
8
1 0 0 K
1 5 0 K
2 0 0 K
2 K
5 K
2 0 K
3 0 K
5 0 K
7 5 K
ρx y ( µΩ c m
)
0 H ( T )
0
2
4
6
8
0
2
4
ρx y ( µΩ c m
)
0 H ( T )
2 K
L i n e a r f i t
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
3 0 0
0
5 0
1 0 0
1 5 0
2 0 0
2 5 0
ρ ( µΩ c m
)
T ( K )
T
P
R R R = 1 4
S r C u
4 A s 2
3 0
6 0
9 0
1 2 0
0
1
2
3
4
d ρ/ d T ( µΩ c m
/ K )
T ( K )
T
P = 5 9 K
0
2
4
6
8
0
2 0
4 0
6 0
8 0
1 0 0
1 2 0
2 K
5 K
1 0 K
1 5 K
M
R ( %
)
0 H ( T )
H I I c
1 0 0 K
5 0 K
2 5 K
2 0 K
( a )
( b )
( c )
( d )
( e )
( f )
( g )
FIG. 2.
Electrical resistivity and magnetotransport properties of SrCu4As2: (a) The temperature dependence of the zero-field
electrical resistivity from 2 to 300 K with the current applied in the ab plane of the crystal. The inset shows the first-order
derivative of the resistivity data, the peak occurring at the phase transition. (b) Cooling and heating curves for the electrical
resistivity near the phase transition temperature at 0 and 9 T. (c) Field dependence of the magnetoresistance measured up to
9 T for various temperatures (H ∥c). (d) The first-order derivative of the MR as a function of magnetic field for temperatures
of 2 K and 50 K. (e) The field-dependent Hall resistivity measured between 2 and 200 K. The inset shows the linear fit up to 6
T at 2 K. (f) and (g) display the temperature variation of the hole concentration and the average mobility estimated from the
Hall resistivity data.
transition at 59 K, a sudden change in the average hole
density concentration is observed [Fig. 2(g)]. Moreover,
the hole concentration is a few orders of magnitude lower
than typical metals but comparable to several topolog-
ical semimetals [41–43], which indicates a semimetallic
character of SrCu4As2.
B.
Quantum oscillations
Quantum oscillations in SrCu4As2 were measured us-
ing the proximity-detector oscillator technique in a 60 T
pulsed magnet and 3He cryostat at NHMFL Los Alamos.
The PDO circuit’s resonant frequency f is sensitive to the
inductance of the coil around the sample, which is deter-
mined by the sample’s skin depth. Skin depth is in turn
dependent on electrical conductivity; hence Shubnikov-
de Haas oscillations are observed as variations in f [20].
Fig. 3 (a) shows the PDO signal as a function of mag-
netic field. A third-order in H polynomial background
was subtracted from the PDO signal over the field range
25-60 T; the result is plotted as a function of 1/µ0H in
Fig. 3(b). Fig. 3(c) shows the corresponding fast Fourier
transform, revealing five fundamental frequencies: Fα =
55 T, Fβ = 225 T, Fγ = 478 T, Fδ = 1410 T, and Fη
= 1914 T. Subsequently, the effective mass associated
with each frequency is estimated by fitting the FFT am-
plitudes to the temperature-damping component of the
Lifshits-Kosevich (LK) equation, as given below [44]:
AF F T
T
∝
14.69m∗/Bm
sinh(14.69m∗T/Bm) .
(1)
Here m∗is the effective mass and Bm is calculated using
4
0
1 0
2 0
3 0
4 0
5 0
6 0
1 . 9
2
2 . 1
2 . 2
f ( M
H z )
0 H ( T )
6 9 0 m
K
0
7 5 0
1 5 0 0
2 2 5 0
3 0 0 0
F F T A m
p . ( a . u . )
F ( T )
- 2 . 5 d e g
+ 2 . 5 d e g
0 d e g
F
F
1 . 5 K
0
2 0
4 0
6 0
8 0
0
5 0 0
1 0 0 0
1 5 0 0
1 / c o s θ
F ( T )
θ ( d e g r e e )
F
F
F
1
1 0
1 0 0
F F T A m
p . / T ( a . u . )
T ( K )
m
0.06(1) m
e
m
0.08(1) m
e
m
0.26(2) m
e
m
1.2(1) m
e
m
0.85(8) m
e
L K f i t
0
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
3 0 0 0
F F T A m
p . ( a . u . )
F ( T )
6 9 0 m
K
F F T : 2 5 - 6 0 T
F
F
F
F
F
S r C u
4 A s 2
0
7 5 0
1 5 0 0
2 2 5 0
3 0 0 0
F F T A m
p ( a . u . )
F ( T )
0 °
1 0 °
2 0 °
3 0 °
4 0 °
5 0 °
5 5 °
6 0 °
7 0 °
8 0 °
9 0 °
0 . 0 2
0 . 0 3
0 . 0 4
∆f ( a . u . )
0 H ( T )
9 0 °
8 0 °
7 0 °
6 0 °
5 5 °
5 0 °
4 0 °
3 0 °
2 0 °
1 0 °
0 °
0 . 0 2
0 . 0 2 5
0 . 0 3
0 . 0 3 5
0 . 0 4
∆f ( a . u . )
1 / 0 H ( T
- 1 )
6 9 0 m
K
( a )
( b )
( c )
( d )
( e )
( f )
( g )
( h )
FIG. 3.
Quantum oscillations in SrCu4As2 observed via PDO measurements: (a) The field dependence of PDO frequency as a
function for rising and falling fields at 690 mK. (b) PDO data as a function of 1/µ0H after polynomial background subtraction
in the field range 25-60 T at 690 mK for H ∥c. (c) FFTs of background subtracted PDO data at 690 mK. (d) The temperature
variation of the FFT amplitudes fitted using Eq. 1. (e) Quantum oscillations at 690 mK for various angles of the magnetic
field with respect to the c-axis of the crystal, as shown in the inset diagram. The corresponding FFTs are shown in (f). (g)
The angular dependences of the FFT frequencies. The dotted curves represent 1/cos θ fits. (h) FFTs at 1.5 K for 0◦and ±2.5◦
.
the expression 1/Bm = 2(1/Bl + 1/Bu), where Bl and
Bu are, respectively, the lower and upper field limits of
the Fourier window. Since the frequencies Fδ and Fη are
more prominent at high fields, a 35-59 T Fourier window
was used to derive their effective masses, whereas a 20-
60 T Fourier window was used for the lower-frequency
mass fits (Quantum oscillations and corresponding FFTs
for these two windows can be found in the SM [19].).
The variation of the FFT amplitudes with temperatures,
along with fits to Eq. 1, are shown in Fig. 3(d). The fit-
ted effective masses are mα = 0.06(1) me, mβ = 0.08(1)
me, mγ = 0.26(2) me, mδ = 1.2(1) me, and mη = 0.85(8)
me. These values are comparable to both experimental
masses in similar compounds [9, 10, 15] and our calcula-
tions.
The effective masses corresponding to bands α and
β are very small, suggesting that these Fermi pockets
in SrCu4As2 have Dirac-like band dispersions.
Bands
with such low effective masses may play a role in caus-
ing the linear magnetoresistance seen in Fig. 2. Mea-
sured effective masses, Fermi-surface cross-sectional ar-
eas (AF ) perpendicular to the applied magnetic field,
TABLE I. The parameters obtained from the quantum oscil-
lation.
Fα
Fβ
Fγ
Fδ
Fη
Frequency (T)
55
225
478
1410
1914
m∗(me)
0.06(1)
0.08(1)
0.26(2)
1.2(1)
0.85(8)
AF (nm−2)
0.52
2.15
4.56
13.45
18.26
kF (nm−1)
0.41
0.83
1.20
2.07
2.41
vF (106ms−1)
0.79
1.20
0.54
0.20
0.33
Fermi wavevectors (kF ), and Fermi velocities (vF ) for
each Fermi-surface section are summarized in Table I.
The Onsager relation F = (ℏ/2πe)AF is employed to de-
termine the cross-sectional areas. Fermi wavevectors and
velocities are calculated (assuming circular Fermi-surface
cross-sections) using the expressions kF =
p
2eF/ℏand
vF
= ℏkF /m∗, respectively, where ℏis the reduced
Planck’s constant.
Next, sample-orientation-dependent PDO measure-
5
FIG. 4.
(a) Band structure of SrCu4As2 with the contribution of the two inequivalent Cu atoms (Cu(1) & Cu(2)) and As
marked by the thickness of the red, green, and blue lines, respectively. Panel (b) shows the electronic density of states (DOS).
Inset shows a close-up of the DOS around the Fermi energy, EF . Panel (c) shows the band dispersion around the Fermi
energy. The 5 bands forming the Fermi surface are highlighted. Panel (d) shows the dispersion of the 5 bands along the Γ-Z
line, with band indices labeled using colors consistent with panel (c) and the Fermi-surface section coloring in panel (e). Panel
(e) shows the Fermi surface of SrCu4As2 and its cross-section along the hexagonal [0001] direction[(the section plane is shown
in gray]. Panel (f) shows theoretical SdH frequencies plotted against the angle between the hexagonal c axis (parallel to the
rhombohedral [111] direction) and a.
ments were performed to investigate the Fermi-surface
topology of SrCu4As2. The sample is rotated with re-
spect to the c-axis, as illustrated in Fig. 3(e), in which θ
= 0◦corresponds to H ∥c. Fig. 3(e) displays the SdH
oscillation data at 690 mK for various angles. As θ in-
creases, the oscillation amplitude slowly decreases, and
at 90◦, the quantum oscillations disappear; these phe-
nomena are also seen in the angle-dependent FFTs in Fig.
3(f). The angle dependences of the oscillation frequencies
are displayed in Fig. 3(g). The frequency Fα is nearly
independent of θ, indicating an almost spherical three-
dimensional Fermi-surface section. In contrast, Fβ and
Fγ exhibit 1/cos θ behavior, suggesting that these Fermi-
surface sheets are quasi-two-dimensional. Notably, the
higher frequencies Fδ and Fη disappear very quickly as θ
moves away from 0◦. Even at small angles of ±2.50◦, as
shown in Fig. 3(h), these frequencies are not seen, which
suggests that the Fermi-surface sections corresponding to
bands δ and η are strongly anisotropic.
C.
Electronic structure
Figure 4(a-d) show the calculated band structure and
density of states (DOS) of SrCu4As2. Most of the Cu d
state contribution to the DOS is limited to the narrow
energy range between -4 to -2.5 eV, consistent with the
Cu+1 charge (3d10 4s0 configuration). The DOS at the
Fermi energy EF has contributions from interacting Cu
and As p atomic orbitals.
In total, 5 bands cross the
Fermi energy (Fig. 4(c)). Two of them (#1 and #2 in
Figs. 4(d-f)) form small elliptical pockets centred around
the Z point of the Brillouin zone (BZ). Bands #3, #4,
and #5 form a tubular structure perpendicular to Γ-Z.
Notably, bands #4 and #5 cross at a Dirac point located
along the Γ-Z line approximately 0.5 eV above the Fermi
energy [Fig. 4(d)]. Bands #4 and #5 cross the Fermi en-
ergy along different high symmetry lines: Γ-Z and B1-Z.
Moreover, the DOS near the Fermi energy is low, which is
consistent with the observed Hall carrier concentration.
6
TABLE II. Calculated quantum oscillation frequencies and effective masses along the c-axis
Bands
#1
#2
#3
#4
#5
Frequency (kT)
0.0420(1)
0.216(4)
0.166(1)
0.385(3)
1.8144(2)
2.3689(4)
2.7712(3)
17.592(6)
28.394(6)
Effective mass (me)
0.120(4)
0.28(2)
0.284(4)
0.46(7)
0.616(1)
0.782(5)
0.951(3)
2.97(1)
3.63(2)
The gradients of the various bands crossing the Fermi
energy suggest that SrCu4As2 is a semimetal, with holes
dominant, in agreement with the Hall data.
Shubnikov-de Haas oscillation frequencies obtained
from the calculated Fermi surface are plotted in Fig. 4(f)
and summarized in Table II. Comparison with the exper-
imentally measured SdH frequencies reveals that Fα can
likely be ascribed to the first branch of the calculated
Fermi surface, which is a small ellipsoidal pocket centered
at the Z point of the Brillouin zone, while Fβ matches the
frequencies found in branches #2 and #3. One of the ex-
tremal orbits of the third branch can also be ascribed to
Fγ. While the ab initio calculated and experimental SdH
oscillations match qualitatively, the detailed frequencies
and their angle dependence (especially at high tilt angles)
differ. A major factor contributing to these differences is
the fact that the calculations are done using the room-
temperature crystal structure, whilst SrCu4As2 under-
goes a structural phase transition at T = 59 K, which
affects the Fermi surface, as evidenced by a sharp change
observed in carrier concentration. Another possible issue
affecting the experimental data might be a small shift in
the Fermi energy due to defects [45–47] such as Sr vacan-
cies suggested by the EDS spectroscopy.
III.
SUMMARY
We
have
grown
high-quality
single
crystals
of
SrCu4As2
by
a
high-temperature
solution
growth-
method using elemental Bi as a flux. The temperature-
dependent electrical resistivity reveals a phase transi-
tion at a temperature TP = 59 K, lower than that re-
ported in ref. [17]. The origin of this phase transition,
which exhibits hysteresis in the resistivity, remains un-
clear but it is most likely associated with a structural
phase transition. Hall resistivity measurements indicate
that the electrical transport in SrCu4As2 is dominated
by the holes. The derived temperature-dependent carrier
concentration shows a maximum near 50 K, presumably
associated with the observed phase transition.
In the
studied crystals, a linear and nonsaturating magnetore-
sistance MR(H) is observed, with a relatively large MR
(2 K, 9 T) = 120%. Quantum-oscillation data measured
in magnetic fields of up to 60 T reveal several oscilla-
tion frequencies with low effective masses, suggesting the
presence of Dirac-like band dispersion in SrCu4As2, a
hypothesis that is supported by the band structure cal-
culations. Finally, the angle-dependent SdH oscillation
data and ab initio calculations show a complex Fermi sur-
face with multiple pockets, some of which have a quasi-
two-dimensional character.
Holelike pockets dominate,
in agreement with the Hall data.
IV.
ACKNOWLEDGMENTS
The work at Gdansk University of Technology was sup-
ported by the National Science Centre (Poland), Grant
No. DEC-2024/08/X/ST3/00338. A portion of this work
was performed at the National High Magnetic Field Lab-
oratory, which is supported by National Science Founda-
tion Cooperative Agreement No. DMR-2128556, the US
Department of Energy (DoE) and the State of Florida.
JB and JS acknowledge support from the DoE BES FWP
“Science of 100 T”, which permitted development of some
of the high-field techniques used in the paper.
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|
Quantum oscillations and transport properties of layered single-crystal SrCu4As2 Sudip Malick,1, 2, ∗Michał J. Winiarski,1, 2, † Joanna Bławat,3 Hanna Świątek,1, 2 John Singleton,3 and Tomasz Klimczuk1, 2, ‡ 1Faculty of Applied Physics and Mathematics, Gdansk 11/12, 80-233 Gdańsk, Poland 2Advanced Materials Center, Gdansk 11/12, 80-233 Gdańsk, Poland 3National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, NM 87545, USA We report a systematic investigation of the physical properties and Fermi-surface topology of layered single-crystal SrCu4As2 using electrical transport, magnetotransport, and quantum-oscillation experiments plus band-structure calculations. The temperature-dependent electrical resistivity reveals a hysteretic phase transition at TP = 59 K, most likely associated with a structural change. Hall resistivity data suggest a marked change in the average hole density resulting from the latter phase transition near TP . A large, linear and nonsaturating magnetoresistance is observed at low temperatures in SrCu4As2, likely attributable to the multipocket Fermi surface. Quantum-oscillation data measured in magnetic fields of up to 60 T show several oscillation frequencies exhibiting low effective masses, indicating the presence of Dirac-like band dispersion in SrCu4As2, as suggested by the band structure calculations. I. INTRODUCTION Ternary pnictides with layered crystal structures have drawn significant research interest due to their distinctive physical properties resulting from the interplay of topology, magnetism, superconductivity, charge-density waves (CDWs), and structural disorder [1-10]. CaCu4As2-type layered pnictide is one of the relevant systems where large linear magnetoresistance, quantum oscillations, structural phase transition, and unusual magnetism are observed due to their complex Fermi surface topology. Very recently, a spin-moiré superlattice has been realized in layered pnictide EuAg4Sb2, establishing the layered "142" system as a potential platform for emerging spin-driven quantum-Hall states [11]. Our recent report [9] indicates complex magnetism and unusual magnetoresistance (MR) in EuAg4Sb2. The latter compound features two successive antiferromagnetic phase transitions, multiple metamagnetic transitions, and large nonsaturating MR with quantum oscillations. The presence of several oscillation frequencies exhibiting low effective masses in this compound indicates a complex Fermi surface [9]. Interestingly, a structural distortion at 120 K and an incommensurate antiferromagnetic phase transition below 9 K are observed in the arsenide EuAg4As2 [12, 13]. Similar structural distortion and ferromagnetic phase transition are seen at 75 K and 35 K, respectively, in isostructural EuCu4As2 [14]. Moreover, both compounds show large magnetoresistance. These behaviors suggest that the interplay between magnetism and structural changes plays a significant role in the electronic properties of these materials. While the nonmagnetic arsenides of these series show similar high-temperature structural distortion and large magnetoresistance, they ∗ † ‡ show some additional features. For instance, quantum oscillations linked with small Fermi pockets and a significant change in the Fermi surface topology, in contrast to the band structure calculation, are observed due to structural distortion in SrAg4As2 [15]. On the other hand, CaCu4As2 exhibits Shubnikov-de Haas (SdH) oscillations and coexistence of the multilayer quantum-Hall effect and a CDW state [10]. These findings motivate us to investigate the nonmagnetic analog of EuCu4As2, SrCu4As2, which crystallizes in a rhombohedral structure (Fig. 1) with space group R ̄3m [16]. A recent report on thermal expansion and electrical resistivity data reveals a structural distortion in SrCu4As2 at 145 K [17]. Densityfunctional-theory calculations indicate that SrCu4As2 is a semimetal, with highly dispersive bands near the Fermi energy and holelike Fermi sheets along the Γ -Z direction [18]. However, few thorough investigation of the physical properties of this compound have been carried out. Here, we present a comprehensive study of the physical properties and Fermi-surface topology of highquality single-crystal SrCu4As2 using electrical resistivity, magnetotransport, and quantum-oscillation measurements at high magnetic fields along with band-structure calculations. Our findings indicate that SrCu4As2 undergoes a possible structural-distortion-type phase transition. Magnetotransport measurements show a linear MR and Hall resistivity. The quantum oscillation reveals a complex Fermi surface with some quasi-two-dimensional sections, in qualitative agreement with our theoretical calculations. The Supplemental Material (SM) contains information on single-crystal growth, structural characterization, measurement techniques, band-structure calculations, and fast Fourier transform (FFT) spectra [19] (see also Refs. [17, 20-25] therein). 16 Oct 2025 2 II. RESULTS AND DISCUSSION A. Transport properties FIG. 1. The unit cell of SrCu4As2 shown using rhombohedral (a) and hexagonal axes (b). There are two symmetryinequivalent Cu positions in the structure: Cu(1) and Cu(2). Note the [0001] direction in the hexagonal cell is equivalent to the [111] direction in the rhombohedral one. Figure 2(a) shows the zero-field electrical resistivity ρ(T) of a SrCu4As2 single crystal measured over the temperature (T) range 2-300 K. The resistivity gradually decreases with falling temperature, before showing a sudden drop at around 60 K. The overall electrical resistivity implies metallic or semimetallic behavior and a residual resistivity ratio (RRR) of 14, which is comparable to isostructural compounds, suggesting that the as-grown crystals are of high quality [10, 15]. Note that temperature-dependent heat-capacity data also exhibit a feature at around 60 K [see the SM [19]]. The first-order derivative of ρ(T) (Fig. 2(a) inset) indicates that the phase transition occurs at TP = 59 K, much lower than the 145 K reported previously [17]. Hysteresis is observed in the resistivity data for heating and cooling [Fig. 2(b)]; although a significant enhancement of the electrical resistivity is observed due to applied magnetic field, a field of 9 T has no apparent effect on the transition temperature or the hysteresis loop. Similar nonmagnetic phase transitions are also seen in other "142"-type layered compounds [10, 12, 14, 15, 26]. In SrCu4As2, the observed phase transition is unlikely to be due to a CDW, as these typically result in an upward jump in electrical resistivity due to the opening up of a partial gap in the Fermi energy [27]. Hence, the high-temperature nonmagnetic phase transition in "142" systems is likely to be a structural distortion [10, 12, 14, 15, 26]. To strengthen this assertion for SrCu4As2, future temperature-dependent singlecrystal x-ray diffraction would be very helpful. Figure. 2(c) shows the magnetoresistance measured at various temperatures; the magnetic field is applied along the c-axis and the current is in the ab-plane of the SrCu4As2 single crystal. With increasing magnetic field, the MR increases linearly and reaches a value of 120% at 9 T and 2 K. Similar MR values were recently reported in the isostructural compounds SrAg4As2 [15], CaCu4As2 [10], and SrCu4P2 [26]. However, the sizes of these MRs are less than those of typical topological materials [28, 29]. A substantial decrease in MR is observed as the temperature increases; however, there seems to be no approach to saturation, even at 100 K and 9 T. At low temperatures, SrCu4As2 exhibits almost linear MR at low temperatures, as shown by the almost fieldindependent first-order derivative of the MR data at 2 K [Fig. 2(d)]. Linear MR is often observed in topological semimetals and insulators, attributed to their linear band dispersions and topologically protected surface states [28, 30-33]. Moreover, some Dirac semimetals exhibit a MR crossover from a weak-field quadratic field dependence to high-field behavior. This has been interpreted using Abrikosov's quantum-limit theorem, which invokes Dirac carriers being restricted to the zeroth Landau level [5, 8, 34, 35]. Whilst there appears to be a small region of quadratic MR in SrCu4As2 below B∗≈0.5 T at T = 50 K, there is no evidence for a similar dependence at 2 K [Figs. 2(c) and 2(d)]. As we will see below, the quantum oscillations have frequencies in the approximate range 200 to 2000 T; hence, the crossover field observed at 50 K is two orders of magnitude below the quantum limit. This indicates that Abrikosov's theory cannot explain the MR of SrCu4As2. The carrier compensation and mobility fluctuations could be the other possible reasons for displaying large MR. The compounds having an equal density of electrons and holes, such as LaSb [36] and TaAs2 [37], exhibit high unsaturated MR due to carrier compensation. Our Hall resistivity data (see below) suggest a dominating hole-type carrier, which eliminates the possibility of carrier compensation in SrCu4As2. On the other hand, the observed hole mobility is low compared to compounds like disordered Ag2+δSe and n-doped Cd3As2, where mobility fluctuation induces large and linear MR [38-40]. The linear and unsaturated MR in SrCu4As2 is likely associated with its multiband Fermi surface with several tiny Fermi pockets, characterized in band-structure calculations by Dirac-like band dispersion near the Fermi energy and small effective masses obtained from the quantum oscillation data (see below). Figure. 2(e) displays the Hall resistivity ρxy as a function of applied magnetic field measured at various temperatures. The Hall resistivity exhibits a linear field dependence with positive slope for all measured temperatures between 2 and 200 K, suggesting that transport in SrCu4As2 is dominated by holes. The average carrier concentration (p) and Hall mobility (μ) have been calculated using the expressions p = 1/(eRH) and μ = RH/ρ(H = 0), where RH is the slope of the linear fit to the Hall resistivity data presented in the inset of Fig. 2(e). The temperature variation of the average hole concentration and mobility are presented in Figs. 2(f) and 2(g), respectively. These values are comparable to those of isostructural compounds SrAg4As2 [15] and CaCu4As2 [10]. As SrCu4As2 passes through the phase 3 0 5 0 1 0 0 1 5 0 2 0 0 1 0 0 2 0 0 3 0 0 μ ( c m 2 V - 1 s - 1 ) T ( K ) 0 . 8 1 1 . 2 1 . 4 p ( 1 0 2 1 c m - 3 ) 0 2 4 6 8 0 1 0 2 0 d M R / d H ( T - 1 ) 0 H ( T ) 2 K 5 0 K B * ~ 0 . 5 T 3 0 4 0 5 0 6 0 7 0 3 0 4 0 5 0 6 0 7 0 8 0 ρ ( μΩ c m ) T ( K ) 0 T 9 T 0 2 4 6 8 0 2 4 6 8 1 0 0 K 1 5 0 K 2 0 0 K 2 K 5 K 2 0 K 3 0 K 5 0 K 7 5 K ρx y ( μΩ c m ) 0 H ( T ) 0 2 4 6 8 0 2 4 ρx y ( μΩ c m ) 0 H ( T ) 2 K L i n e a r f i t 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 ρ ( μΩ c m ) T ( K ) T P R R R = 1 4 S r C u 4 A s 2 3 0 6 0 9 0 1 2 0 0 1 2 3 4 d ρ/ d T ( μΩ c m / K ) T ( K ) T P = 5 9 K 0 2 4 6 8 0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 2 K 5 K 1 0 K 1 5 K M R ( % ) 0 H ( T ) H I I c 1 0 0 K 5 0 K 2 5 K 2 0 K ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) FIG. 2. Electrical resistivity and magnetotransport properties of SrCu4As2: (a) The temperature dependence of the zero-field electrical resistivity from 2 to 300 K with the current applied in the ab plane of the crystal. The inset shows the first-order derivative of the resistivity data, the peak occurring at the phase transition. (b) Cooling and heating curves for the electrical resistivity near the phase transition temperature at 0 and 9 T. (c) Field dependence of the magnetoresistance measured up to 9 T for various temperatures (H ∥c). (d) The first-order derivative of the MR as a function of magnetic field for temperatures of 2 K and 50 K. (e) The field-dependent Hall resistivity measured between 2 and 200 K. The inset shows the linear fit up to 6 T at 2 K. (f) and (g) display the temperature variation of the hole concentration and the average mobility estimated from the Hall resistivity data. transition at 59 K, a sudden change in the average hole density concentration is observed [Fig. 2(g)]. Moreover, the hole concentration is a few orders of magnitude lower than typical metals but comparable to several topological semimetals [41-43], which indicates a semimetallic character of SrCu4As2. B. Quantum oscillations Quantum oscillations in SrCu4As2 were measured using the proximity-detector oscillator technique in a 60 T pulsed magnet and 3He cryostat at NHMFL Los Alamos. The PDO circuit's resonant frequency f is sensitive to the inductance of the coil around the sample, which is determined by the sample's skin depth. Skin depth is in turn dependent on electrical conductivity; hence Shubnikovde Haas oscillations are observed as variations in f [20]. Fig. 3 (a) shows the PDO signal as a function of magnetic field. A third-order in H polynomial background was subtracted from the PDO signal over the field range 25-60 T; the result is plotted as a function of 1/μ0H in Fig. 3(b). Fig. 3(c) shows the corresponding fast Fourier transform, revealing five fundamental frequencies: Fα = 55 T, Fβ = 225 T, Fγ = 478 T, Fδ = 1410 T, and Fη = 1914 T. Subsequently, the effective mass associated with each frequency is estimated by fitting the FFT amplitudes to the temperature-damping component of the Lifshits-Kosevich (LK) equation, as given below [44]: AF F T T ∝ 14.69m∗/Bm sinh(14.69m∗T/Bm) . (1) Here m∗is the effective mass and Bm is calculated using 4 0 1 0 2 0 3 0 4 0 5 0 6 0 1 . 9 2 2 . 1 2 . 2 f ( M H z ) 0 H ( T ) 6 9 0 m K 0 7 5 0 1 5 0 0 2 2 5 0 3 0 0 0 F F T A m p . ( a . u . ) F ( T ) - 2 . 5 d e g + 2 . 5 d e g 0 d e g F F 1 . 5 K 0 2 0 4 0 6 0 8 0 0 5 0 0 1 0 0 0 1 5 0 0 1 / c o s θ F ( T ) θ ( d e g r e e ) F F F 1 1 0 1 0 0 F F T A m p . / T ( a . u . ) T ( K ) m 0.06(1) m e m 0.08(1) m e m 0.26(2) m e m 1.2(1) m e m 0.85(8) m e L K f i t 0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0 F F T A m p . ( a . u . ) F ( T ) 6 9 0 m K F F T : 2 5 - 6 0 T F F F F F S r C u 4 A s 2 0 7 5 0 1 5 0 0 2 2 5 0 3 0 0 0 F F T A m p ( a . u . ) F ( T ) 0 ° 1 0 ° 2 0 ° 3 0 ° 4 0 ° 5 0 ° 5 5 ° 6 0 ° 7 0 ° 8 0 ° 9 0 ° 0 . 0 2 0 . 0 3 0 . 0 4 ∆f ( a . u . ) 0 H ( T ) 9 0 ° 8 0 ° 7 0 ° 6 0 ° 5 5 ° 5 0 ° 4 0 ° 3 0 ° 2 0 ° 1 0 ° 0 ° 0 . 0 2 0 . 0 2 5 0 . 0 3 0 . 0 3 5 0 . 0 4 ∆f ( a . u . ) 1 / 0 H ( T - 1 ) 6 9 0 m K ( a ) ( b ) ( c ) ( d ) ( e ) ( f ) ( g ) ( h ) FIG. 3. Quantum oscillations in SrCu4As2 observed via PDO measurements: (a) The field dependence of PDO frequency as a function for rising and falling fields at 690 mK. (b) PDO data as a function of 1/μ0H after polynomial background subtraction in the field range 25-60 T at 690 mK for H ∥c. (c) FFTs of background subtracted PDO data at 690 mK. (d) The temperature variation of the FFT amplitudes fitted using Eq. 1. (e) Quantum oscillations at 690 mK for various angles of the magnetic field with respect to the c-axis of the crystal, as shown in the inset diagram. The corresponding FFTs are shown in (f). (g) The angular dependences of the FFT frequencies. The dotted curves represent 1/cos θ fits. (h) FFTs at 1.5 K for 0◦and ±2.5◦ . the expression 1/Bm = 2(1/Bl + 1/Bu), where Bl and Bu are, respectively, the lower and upper field limits of the Fourier window. Since the frequencies Fδ and Fη are more prominent at high fields, a 35-59 T Fourier window was used to derive their effective masses, whereas a 2060 T Fourier window was used for the lower-frequency mass fits (Quantum oscillations and corresponding FFTs for these two windows can be found in the SM [19].). The variation of the FFT amplitudes with temperatures, along with fits to Eq. 1, are shown in Fig. 3(d). The fitted effective masses are mα = 0.06(1) me, mβ = 0.08(1) me, mγ = 0.26(2) me, mδ = 1.2(1) me, and mη = 0.85(8) me. These values are comparable to both experimental masses in similar compounds [9, 10, 15] and our calculations. The effective masses corresponding to bands α and β are very small, suggesting that these Fermi pockets in SrCu4As2 have Dirac-like band dispersions. Bands with such low effective masses may play a role in causing the linear magnetoresistance seen in Fig. 2. Measured effective masses, Fermi-surface cross-sectional areas (AF ) perpendicular to the applied magnetic field, TABLE I. The parameters obtained from the quantum oscillation. Fα Fβ Fγ Fδ Fη Frequency (T) 55 225 478 1410 1914 m∗(me) 0.06(1) 0.08(1) 0.26(2) 1.2(1) 0.85(8) AF (nm-2) 0.52 2.15 4.56 13.45 18.26 kF (nm-1) 0.41 0.83 1.20 2.07 2.41 vF (106ms-1) 0.79 1.20 0.54 0.20 0.33 Fermi wavevectors (kF ), and Fermi velocities (vF ) for each Fermi-surface section are summarized in Table I. The Onsager relation F = (ħ/2πe)AF is employed to determine the cross-sectional areas. Fermi wavevectors and velocities are calculated (assuming circular Fermi-surface cross-sections) using the expressions kF = p 2eF/ħand vF = ħkF /m∗, respectively, where ħis the reduced Planck's constant. Next, sample-orientation-dependent PDO measure5 FIG. 4. (a) Band structure of SrCu4As2 with the contribution of the two inequivalent Cu atoms (Cu(1) & Cu(2)) and As marked by the thickness of the red, green, and blue lines, respectively. Panel (b) shows the electronic density of states (DOS). Inset shows a close-up of the DOS around the Fermi energy, EF . Panel (c) shows the band dispersion around the Fermi energy. The 5 bands forming the Fermi surface are highlighted. Panel (d) shows the dispersion of the 5 bands along the Γ-Z line, with band indices labeled using colors consistent with panel (c) and the Fermi-surface section coloring in panel (e). Panel (e) shows the Fermi surface of SrCu4As2 and its cross-section along the hexagonal [0001] direction[(the section plane is shown in gray]. Panel (f) shows theoretical SdH frequencies plotted against the angle between the hexagonal c axis (parallel to the rhombohedral [111] direction) and a. ments were performed to investigate the Fermi-surface topology of SrCu4As2. The sample is rotated with respect to the c-axis, as illustrated in Fig. 3(e), in which θ = 0◦corresponds to H ∥c. Fig. 3(e) displays the SdH oscillation data at 690 mK for various angles. As θ increases, the oscillation amplitude slowly decreases, and at 90◦, the quantum oscillations disappear; these phenomena are also seen in the angle-dependent FFTs in Fig. 3(f). The angle dependences of the oscillation frequencies are displayed in Fig. 3(g). The frequency Fα is nearly independent of θ, indicating an almost spherical threedimensional Fermi-surface section. In contrast, Fβ and Fγ exhibit 1/cos θ behavior, suggesting that these Fermisurface sheets are quasi-two-dimensional. Notably, the higher frequencies Fδ and Fη disappear very quickly as θ moves away from 0◦. Even at small angles of ±2.50◦, as shown in Fig. 3(h), these frequencies are not seen, which suggests that the Fermi-surface sections corresponding to bands δ and η are strongly anisotropic. C. Electronic structure Figure 4(a-d) show the calculated band structure and density of states (DOS) of SrCu4As2. Most of the Cu d state contribution to the DOS is limited to the narrow energy range between -4 to -2.5 eV, consistent with the Cu+1 charge (3d10 4s0 configuration). The DOS at the Fermi energy EF has contributions from interacting Cu and As p atomic orbitals. In total, 5 bands cross the Fermi energy (Fig. 4(c)). Two of them (#1 and #2 in Figs. 4(d-f)) form small elliptical pockets centred around the Z point of the Brillouin zone (BZ). Bands #3, #4, and #5 form a tubular structure perpendicular to Γ-Z. Notably, bands #4 and #5 cross at a Dirac point located along the Γ-Z line approximately 0.5 eV above the Fermi energy [Fig. 4(d)]. Bands #4 and #5 cross the Fermi energy along different high symmetry lines: Γ-Z and B1-Z. Moreover, the DOS near the Fermi energy is low, which is consistent with the observed Hall carrier concentration. 6 TABLE II. Calculated quantum oscillation frequencies and effective masses along the c-axis Bands #1 #2 #3 #4 #5 Frequency (kT) 0.0420(1) 0.216(4) 0.166(1) 0.385(3) 1.8144(2) 2.3689(4) 2.7712(3) 17.592(6) 28.394(6) Effective mass (me) 0.120(4) 0.28(2) 0.284(4) 0.46(7) 0.616(1) 0.782(5) 0.951(3) 2.97(1) 3.63(2) The gradients of the various bands crossing the Fermi energy suggest that SrCu4As2 is a semimetal, with holes dominant, in agreement with the Hall data. Shubnikov-de Haas oscillation frequencies obtained from the calculated Fermi surface are plotted in Fig. 4(f) and summarized in Table II. Comparison with the experimentally measured SdH frequencies reveals that Fα can likely be ascribed to the first branch of the calculated Fermi surface, which is a small ellipsoidal pocket centered at the Z point of the Brillouin zone, while Fβ matches the frequencies found in branches #2 and #3. One of the extremal orbits of the third branch can also be ascribed to Fγ. While the ab initio calculated and experimental SdH oscillations match qualitatively, the detailed frequencies and their angle dependence (especially at high tilt angles) differ. A major factor contributing to these differences is the fact that the calculations are done using the roomtemperature crystal structure, whilst SrCu4As2 undergoes a structural phase transition at T = 59 K, which affects the Fermi surface, as evidenced by a sharp change observed in carrier concentration. Another possible issue affecting the experimental data might be a small shift in the Fermi energy due to defects [45-47] such as Sr vacancies suggested by the EDS spectroscopy. III. SUMMARY We have grown high-quality single crystals of SrCu4As2 by a high-temperature solution growthmethod using elemental Bi as a flux. The temperaturedependent electrical resistivity reveals a phase transition at a temperature TP = 59 K, lower than that reported in ref. [17]. The origin of this phase transition, which exhibits hysteresis in the resistivity, remains unclear but it is most likely associated with a structural phase transition. Hall resistivity measurements indicate that the electrical transport in SrCu4As2 is dominated by the holes. The derived temperature-dependent carrier concentration shows a maximum near 50 K, presumably associated with the observed phase transition. In the studied crystals, a linear and nonsaturating magnetoresistance MR(H) is observed, with a relatively large MR (2 K, 9 T) = 120%. Quantum-oscillation data measured in magnetic fields of up to 60 T reveal several oscillation frequencies with low effective masses, suggesting the presence of Dirac-like band dispersion in SrCu4As2, a hypothesis that is supported by the band structure calculations. 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2510.14772
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Ghost stabilisation for cut finite element exterior calculus
Daniele A. Di Pietro1, Jérôme Droniou1,2, and Erik Nilsson1
1IMAG, CNRS, Montpellier, France, daniele.di-pietro@umontpellier.fr, jerome.droniou@cnrs.fr, erik.nilsson@umontpellier.fr
2School of Mathematics, Monash University, Melbourne, Australia
October 17, 2025
Abstract
We introduce the cut finite element method in the language of finite element exterior calculus, by for-
mulating a stabilisation – for any form degree – that makes the method robust with respect to the position
of the interface relative to the mesh. We prove that the L2-norm on the physical domain augmented with
this stabilisation is uniformly equivalent to the L2-norm on the “active” mesh that contains all the degrees
of freedom of the finite element space (including those external to the physical domain). We show how
this CutFEEC method can be applied to discretize the Hodge Laplace equations on an unfitted mesh, in
any dimension and any topology. A numerical illustration is provided involving a conforming finite ele-
ment space of Hcurl posed on a filled torus, with convergence and condition number scaling independent
of the position of the boundary with respect to the background mesh.
MSC: 65N30, 14F40
Key words: Cut finite element method, FEEC, Hodge Laplace equation, Hilbert complex
1
Introduction
A successful strand of the literature on the numerical approximation of partial differential equations (PDEs)
has emphasized the relevance of Hilbert complexes [10] in the design of stable numerical schemes. In this
context, differential forms provide a natural unifying language to treat mixed problems set in arbitrary space
dimension, possibly on manifolds, and where a combination of differential operators appear [4, 5, 6, 9, 16,
21]. Finite Element Exterior Calculus (FEEC), particularly relevant for the present work, provides a compre-
hensive view of full and trimmed finite element approximations of the de Rham complex; see the mono-
graph [3] for an introduction. At the same time, the Cut Finite Element Method (CutFEM) has emerged
as a versatile technique for handling PDEs on domains with complex geometries, removing the need for
the mesh to be compliant with the domain’s boundary; see the recent review [13] and references therein.
Through weak imposition of boundary conditions and carefully constructed stabilisation terms over facets
close to the boundary, CutFEM leads to stable and well-conditioned discretisations. Moreover, such prop-
erties hold independently of the boundary position relative to the unfitted mesh. This work aims at merg-
ing these research avenues by proposing a Cut Finite Element Exterior Calculus (CutFEEC) framework that
adapts the CutFEM techniques to FEEC. This merger has notably been made possible by the recent exten-
sion of the CutFEM technology to mixed problems [17, 18, 19], which are a natural application of FEEC [3].
The prime example of a Hilbert complex is the de Rham complex, which, for an open connected domain
Ω⊂R3, reads
0 ,→H1(Ω)
∇−→Hcurl(Ω)
curl
−−→Hdiv(Ω)
div
−−→L2(Ω) →0.
Using the language of differential forms, the de Rham complex can be extended to a domain Ωof any di-
mension n ∈N as:
0 ,→HΛ0Ω
d0
−→HΛ1Ω
d1
−→...
dn−1
−−−→HΛnΩ→0,
(1.1)
1
arXiv:2510.14772v1 [math.NA] 16 Oct 2025
where dk is the exterior derivative and HΛkΩis the space of L2-integrable differential k-forms on Ωwith
L2-integrable exterior derivative. The de Rham complex enters the well-posedness analysis of PDEs through
its cohomology spaces
Hk := Kerdk/Imdk−1.
These spaces relate to the topology of the domain via their dimension. Preserving these homological struc-
tures at the discrete level is crucial for the stability of numerical schemes. A paradigmatic example of PDE
problem whose well-posedness hinges on the de Rham complex is the Hodge Laplace equation; see, e.g., [3,
Chapter 4]. Its mixed formulation naturally leads to a mixed problem where the unknowns are the codiffer-
ential, the (scalar- or vector-) potential, and a Lagrange multiplier to enforce L2-orthogonality with respect
to harmonic forms.
The Finite Element (FE) approximation of mixed problems became a research topic starting from the
late 1970s, when the first conforming approximation of vector-valued spaces in the de Rham complex were
developed [7, 24, 26]. Traditional FE methods, such as the ones cited above, require meshes that conform
to the geometry of the domain. These can be challenging to generate for intricate geometries or in the
presence of evolving interfaces. CutFEM addresses this problem by permitting the interface to cut through
the elements of a background mesh, thereby simplifying mesh generation and adaptation [12]. However,
ensuring stability and accuracy in such unfitted methods necessitates appropriate stabilisation techniques
[11]. Otherwise, when the geometry cuts the mesh in especially nasty ways (|T ∩Ω| ≪|T | for an element T in
the mesh), one observes a significant degradation of the condition number of the associated linear system.
Recent works have addressed the application of CutFEM techniques to mixed problems. In [17, 18], the
authors introduced a stable CutFEM for Darcy flow problems which, thanks to conformity with respect to
the tail end of the de Rham complex, preserves the divergence of the velocity field. Using a similar approach,
one can also preserve the divergence condition of Stokes flow in an unfitted setting [19].
In this work, we develop a new CutFEM stabilisation to design unfitted robust discrete L2-products for
any space in the de Rham complex of differential forms. The stabilisation can be seen as a generalisation
of the mixed ghost penalty term introduced in [17, 19, 25]. Orthogonalising against the harmonic forms
using an L2-product including this stabilisation, we construct an arbitrary-order numerical method which
is unfitted but still compatible with the continuous de Rham complex, thus ensuring its natural stability
and preservation of relevant constraints (e.g., zero divergence of electric field in Maxwell’s equations). We
develop facet-based stabilisation terms, but we mention that there exist other equivalent stabilisation terms
in the CutFEM literature, though these have not been developed for differential forms, see [13, Section 4.2].
We note that the commonly used assumption of a quasi-uniform mesh [13] is not needed in our construction
or analysis. It is sufficient for the relevant arguments to use a local quasi-uniformity, which does not entail
particular restrictions on the mesh. In [23], the quasi-uniform assumption is used but a similar remark is
made [20, Remark 2.3].
A particularly relevant novelty results from applying this general construction to the special case of 1-
forms, which is key to discretize problems such as Maxwell’s equations or the Hcurl-elliptic problem. Cut-
FEM has, indeed, seen little development for problems involving Hcurl; see the recent and exhaustive re-
view [13]. There are some previous works which use an unfitted discontinuous Galerkin approach [14, 22].
Otherwise, works which do not use discontinuous/broken finite element spaces are essentially limited to
[19, 27, 28][25, Paper D]. Of these, [19] and [25, Paper D] do not contain a stability analysis; [19] consid-
ers a vorticity-velocity-pressure formulation for Stokes flow wherein the vorticity is discretized by a Hcurl-
conforming variable, while [25, Paper D] considers numerical experiments of a set of standard formula-
tions of Maxwell’s equations made unfitted. The works [27, 28] focus, respectively, on the time-harmonic
Maxwell’s equations and the Hcurl-elliptic interface problem. In [28, Remark 3], the authors mention that
their proposed method works numerically when using nodal Lagrange elements, but error analysis is pro-
vided only for the case of using piecewise discontinuous polynomials. The numerical scheme in [28] uses the
lowest order Hcurl-conforming Nédélec elements of the second kind, but adds diagonal stabilisation terms
in the Lagrange multiplier block [27, (4.8)–(4.11)], thus perturbing the saddle point nature of the problem.
Similar stabilisation terms also appear in [28]. Moreover, in contrast to our approach, no work mentioned
2
above considers the presence of non-trivial harmonic forms, which appear when considering domains of
general topology.
The remainder of this paper is organised as follows. In Section 2 we discuss a motivating example of
mixed problem, the Hodge Laplace problem, whose well-posedness relies on the properties of the de Rham
complex, and we summarise some notions of FEEC. In Section 3 we recall the notion of unfitted mesh, and
we then introduce in Section 4 the concepts of tangential and normal parts of differential forms, relative
to a facet of the mesh. In Section 5 we introduce the novel ghost penalty stabilisation operator, and show
the uniform equivalence between the L2-norm on the physical domain and the L2-norm on the extended
domain containing all the degrees of freedom. Finally, in Section 6 we validate our results on a numerical
example on the filled torus, demonstrating that the use of the stabilisation drastically reduces the condition
number of the discrete system.
2
A motivating example
In this section we briefly discuss the Hodge Laplacian, an archetypal problem whose well-posedness hinges
on the properties of the de Rham complex. This example serves as a motivation for the development of a
CutFEEC theory, and will be used in Section 6 to numerically demonstrate the efficiency of the ghost penalty
stabilisation of Section 5.
2.1
Hodge Laplace equation
Let B ⊂Rn be an open set. We denote by ΛkB the space of smooth differential k-forms on B. We introduce
the L2-product (ω,ζ)B :=
R
B ω∧⋆ζ on ΛkB, where ∧denotes the wedge product and ⋆the Hodge star oper-
ator [3, (6.2)]. The L2-norm over B is denoted by ∥•∥B := p(·,·)B. We denote by L2ΛkB the space of k-forms
over B that are bounded with respect to this norm. With the exterior derivative dk : L2ΛkB →L2Λk+1B,
viewed as an unbounded operator, we also define
HΛkB :=
n
ω ∈L2ΛkB : dkω ∈L2Λk+1B
o
,
with norm
∥ω∥2
HB := ∥ω∥2
B +∥dkω∥2
B.
Let now Ω⊂Rn be a connected and open subset of Rn with Lipschitz boundary ∂Ωand outward unit
normal vector field n. Given f ∈L2ΛkΩ, the Hodge Laplace equation consists in seeking η ∈ΛkΩsuch that
(dk−1δk +δk+1dk)η = f −πf
in Ω,
πη = 0
in Ω,
(2.1)
where the codifferential δk+1 : L2Λk+1Ω→L2ΛkΩis the adjoint of d and π := πHk is the orthogonal projec-
tion onto the space of harmonic forms
Hk :=
n
ρ ∈L2ΛkΩ: dkρ = 0 and δkρ = 0
o
= Kerdk ∩Kerδk ∼= Kerdk/Imdk−1.
Here ∼= means vector space isomorphism. The boundary conditions are implicit in the definition of the
Hodge Laplace operator dk−1δk +δk+1dk, and are revealed upon choosing the domain of the exterior deriva-
tive. For ease of presentation, let us in this work choose the domain
D(dk) = HΛkΩ,
so that the domain of the codifferential becomes (by duality) the space of L2(Ω)-integrable forms ω with
δω ∈L2(Ω) and γ(⋆ω) = 0. After recalling vector proxies in three space dimensions in Table 2.1, in Table 2.2,
3
k
Form
Proxy
Sobolev space
0
q
q
H1(Ω)
1
v1dx1 + v2dx2 + v3dx3
v = (v1,v2,v3)
Hcurl(Ω)
2
w1dx2 ∧dx3 −w2dx1 ∧dx3 + w3dx1 ∧dx2
w = (w1,w2,w3)
Hdiv(Ω)
3
rdx1 ∧dx2 ∧dx3
r
L2(Ω)
Table 2.1: Vector proxies in R3.
k = 0
k = 1
k = 2
k = 3
−∆q = f −¯f ,
∇q ·n = 0,
¯q = 0
curlcurlv −∇divv = f −πf,
v ·n = 0 and curlv ×n = 0,
v ⊥H1
curlcurlw −∇divw = f −πf,
w ×n = 0 and divw = 0,
w ⊥H2
−∆r = f ,
r = 0
Table 2.2: The Hodge Laplace equation in R3 for different differential form orders. Entries are labeled A, B,
C where A are the bulk equation(s) in Ω, B the boundary condition(s) on ∂Ω, and C is the cohomological
compatibility condition.
we express problem (2.1) in terms of vector proxies for different values of k; cf. [5, Section 4.2] for further
details.
For convenience of notation, we shall for the remainder omit the index and just write d for the exte-
rior derivative. A variational mixed formulation of (2.1) is (cf. [3, Theorem 4.7]): Find (σ,η,λ) ∈HΛk−1Ω×
HΛkΩ×Hk such that
(σ,τ)Ω−(η,dτ)Ω= 0
∀τ ∈HΛk−1Ω,
(dσ,ζ)Ω+(dη,dζ)Ω+(λ,ζ)Ω= (f ,ζ)Ω
∀ζ ∈HΛkΩ,
(η,ρ)Ω= 0
∀ρ ∈Hk.
(2.2)
The above formulation enforces in a weak manner σ = δη and λ = πf . The (natural) boundary conditions
are γ⋆η = 0 and γ⋆dη = 0.
Using crucial properties of the de Rham complex such as Hodge decompositions and Poincaré inequal-
ities, it can be proved that this mixed formulation is well-posed [3, Theorems 4.8 and 4.9]. Preserving these
properties at the discrete level, as is the case for the compatible discretisations considered here, naturally
leads to numerical schemes for which stability can be proved mimicking the arguments for well-posedness
of the continuous problem.
2.2
Principles of FEEC
Assume here that Ωis a polytopal domain, and let Th be a conforming simplicial mesh of Ω. The Finite
Element Exterior Calculus (FEEC) method provides the tools for a discrete formulation of (2.2) which is well-
posed [3, (5.3), Section 5.2.3], but non-conforming in general due to the non-inclusion of the discrete space
of harmonic forms into the continuous one (these spaces have the same dimension, so the former being
included in the latter would actually mean that one could compute harmonic forms exactly). We make a
brief summary of the results which pertain to this work.
Let r ≥1 be a polynomial degree. For each k ∈{0,...,n}, let V k,r
h
⊂HΛkΩbe a finite-dimensional space
of k-forms that are piecewise polynomial of degree at most r on Th and such that dV k,r
h
⊂V k+1,r−1
h
. We
adopt the convention that V −1,r
h
:= {0} and V n+1,r
h
:= {0}. These spaces define a sub-complex of the de Rham
complex:
0
V 0,r
h
V 1,r−1
h
···
V n,r−n
h
0.
d
d
d
(2.3)
4
It can be shown [3], using, e.g., bounded cochain maps between the continuous and discrete de Rham
complexes, that the cohomology spaces of (2.3) are isomorphic to those of (1.1). This diagram (in which
the polynomial degree decreases along the sequence) corresponds to the case of full finite element spaces,
V k,r
h
:= Pr ΛkΩh. When using trimmed spaces, P−
r ΛkΩh ⊂Pr ΛkΩh in the notation of [3], the degree r re-
mains the same along the diagram. For application of our results to the trimmed spaces, simply replace any
occurrence of V k,•
h
with P−
r ΛkΩh.
The discrete exterior derivative acting on a discrete space is just the restriction of the exterior derivative
from the corresponding continuous space. When relevant, we shall denote it by dh := d|V k,r
h , otherwise by d
when there is no confusion. Then, one defines the space of discrete harmonic forms as
Hk
h := Kerdh/dV k−1,r+1
h
∼=
n
ρh ∈V k,r
h
: dρh = 0 and (ρh,dτh)Ω= 0 for all τh ∈V k−1,r+1
h
o
.
(2.4)
The non-inclusion Hk
h ̸⊂Hk is due to forms in Hk needing to be orthogonal to all of dHΛk−1Ω, while being
in Hk
h only guarantees orthogonality to the subspace dV k−1,r+1
h
.
As mentioned in Section 2.1, the Hodge decomposition of the complex spaces is an essential tool to
analyse both the continuous Hodge-Laplace equation and its numerical discretisations. For the complex
(2.3), we have the following discrete Hodge decomposition:
V k,r
h
= (Kerdh)⊥
dV k−1,r+1
h
Hk
h.
(2.5)
Given f ∈L2ΛkΩthe FEEC scheme for problem (2.2) reads: Find (σh,ηh,λh) ∈V k−1,r+1
h
×V k,r
h
×Hk
h such
that
(σh,τh)Ω−(ηh,dτh)Ω= 0
∀τh ∈V k−1,r+1
h
,
(dσh,ζh)Ω+(dηh,dζh)Ω+(λh,ζh)Ω= (f ,ζh)Ω
∀ζh ∈V k,r
h
,
(ηh,ρh)Ω= 0
∀ρh ∈Hk
h.
The scheme is well-posed and optimally convergent in the HΛkΩ-norm [3, Theorem 5.5].
3
Unfitted mesh
Let Th = {T } be an active mesh such that each (open) simplex T ∈Th has a nonempty intersection with Ω,
and for which
Ω⊂Ωh :=
Ã
[
T ∈Th
T
!◦
.
The open set Ωh is hereafter referred to as the active domain.
Remark 1 (Construction of active mesh). The term active mesh refers to the following construction. Define
a background domain Ω0 ⊃Ω, which is polytopal of dimension n. Consider a conforming triangular mesh
T0,h of Ω0. Then, the active mesh is defined as Th :=
©
T ∈T0,h : T ∩Ω̸= ;
ª
. This construction is illustrated
in Figure 3.1.
We say T ∈Th is cut if T ̸⊂Ωand fully immersed in Ωotherwise. We denote the set of cut elements by
T cut
h
:= {T ∈Th : T ̸⊂Ω}.
We denote by F ∂
h the set of stabilisation facets [19, Figure 1], which are facets internal to Ωh for which at
least one neighbouring element is intersected by ∂Ω:
F ∂
h :=
F ∈
[
T ∈T cut
h
FT : F ̸⊂∂Ωh
,
5
Figure 3.1: Illustration of the active mesh Th. The boundary ∂Ωof the domain is in red. The “remainder” of
background mesh T0,h \Th is shown in light blue, while the active mesh Th is shown in black.
where FT gathers the ((n −1)-dimensional) facets of T ∈Th. To each facet we associate a unique normal
vector n. The context will make it clear which normal is meant, that of a facet or that of ∂Ω.
Assumption 1 (Mesh regularity). For any T ∈Th, let ϱT and hT be, respectively, the diameter of the largest
ball contained in T and that of T . We assume the following:
(i) (Shape regularity) For some constant κ > 0 independent of the mesh size h := maxT ∈Th hT , we have
hT
ϱT
≤κ
∀T ∈Th.
(3.1)
(ii) (Uniformly bounded cut-to-uncut path) There exists an integer N > 0 independent of h such that, for
all T ∈T cut
h
, there exists a sequence of NT ≤N elements {T1 = T,T2,...,TNT } such that Ti and Ti+1
share a facet for all 1 ≤i ≤NT −1, and TNT ∈Th \T cut
h
is fully immersed in Ω.
The notation a ≂b means that there exist constants C,C ′ > 0 independent of h, but possibly dependent
on κ and N such that a ≤Cb and b ≤C ′a, and similarly we write a ≲b if and only if a ≤Cb.
Remark 2 (Quasi-uniformity along cut-to-uncut paths). We remark that, by shape regularity and since N is
bounded independently of h, all mesh elements along a given cut-to-uncut path have uniformly comparable
diameters: if T1,...,TNT are as in (ii), then hTi ≂hTj for all i, j ∈{1,...,NT }.
4
Tangential and normal traces and parts of differential forms
Restrictions as well as tangential and normal traces and parts of differential forms are essential concepts to
define the stabilisation in Section 5 below. We consider here an element T ∈Th and one of its facets F ⊂∂T .
For x ∈F, let TxC be the tangent space of C ∈{F,T } at x. Let
ι : F ,→T
be the inclusion map of F into T . We note that TxF is the vector hyperplane parallel to F in Rn, and that
X ∈TxF can be considered as a vector in Rn; for this reason, we identify ι∗X with X .
For ω ∈ΛkT , by “restriction to the facet F” we simply mean the restriction of each component of ω to F,
i.e., writing ω := P
1≤j1<j2<···<jk≤n ωj1 j2...jk dx j1 ∧dx j2 ∧···∧dx jk , we define
ω|F :=
X
1≤j1<j2<···<jk≤n
(ωj1 j2...jk |F )dx j1 ∧dx j2 ∧···∧dx jk .
(4.1)
6
Notice that the result is a form which formally acts on k-tuples of vectors in T , although its components will
only be evaluated on F. The reason is that we want ω|F as a differential form to also be able to act on vectors
normal to F.
The standard trace operator γ := ι∗: ΛkT →ΛkF on the facet is defined as the pullback of the inclusion ι:
For all ω ∈ΛkT ,
∀x ∈F,
(γω)x(X1,..., Xk) := ωx(X1,..., Xk)
∀X1,..., Xk ∈TxF.
(4.2)
Recall the Hodge star operator ⋆: ΛkT →Λn−kT . We can also define a Hodge star ⋆F : ΛkF →Λn−1−kF
on F satisfying ⋆F ⋆F ω = (−1)k(n−1−k)ω for all ω ∈ΛkF.
The normal trace operator is defined by “sandwiching” the standard trace between the Hodge star oper-
ators on T and on F, see [8, Eqs. (45) and (46)]: γn := (−1)n(k−1) ⋆F γ⋆: ΛkT →Λk−1F is therefore such that,
for all ω ∈ΛkT ,
∀x ∈F,
(γnω)x(X2,..., Xk) := ωx(n, X2,..., Xk)
∀X2,..., Xk ∈TxF.
(4.3)
The latter expression is, by definition, the contraction n¬ω of ω. The following identities hold, [8, Eq. (47)]:
⋆F γn = γ⋆,
γn⋆= (−1)k ⋆F γ.
(4.4)
From now on, we shall denote both ⋆and ⋆F by ⋆, the context making it clear which one is meant. The
vector proxy interpretation of the trace and normal trace operators are summarized in Table 4.1.
Form degree k
Proxy
Trace (γ)
Normal trace (γn)
0
q
q
0
1
v
n×(v ×n)
v ·n
2
w
w ·n
−n×w
3
r
0
r
Table 4.1: Trace and normal trace for vector proxies in R3.
Sometimes, the standard trace is called the tangential trace, in contrast to the normal trace. This nomen-
clature can be understood comparing the traces with the restriction for sufficiently smooth differential
forms. Consider for the remainder of this section a smooth k-form ω ∈ΛkT . Using the pullback of the
orthogonal projection πF : T →F, we can define the tangential part of ω on F by setting
ω∥:= (π∗
F ◦γ)ω.
(4.5)
At any x ∈F, it is also defined as an alternating form in Altk Rn. Setting then
ω⊥:= ω|F −ω∥,
(4.6)
we have the following decomposition:
ω|F = ω∥+ω⊥.
(4.7)
Lemma 1 (Characterisation of the tangential and normal parts). Let ω ∈ΛkT be a smooth k-form. Then the
following holds:
ω∥= 0 ⇐⇒γω = 0,
(4.8)
ω⊥= 0 ⇐⇒γnω = 0.
(4.9)
Moreover,
(⋆ω)∥= ⋆ω⊥
and
(⋆ω)⊥= ⋆ω∥.
(4.10)
7
Proof. 1) Proof of (4.8). We start by showing that ω∥= 0 if and only if γω = 0. It is clear by its definition (4.5)
that ω∥= 0 if γω = 0. To prove the converse, we just note that, if (ω∥)x = 0, then, for all X1,..., Xk ∈TxF ⊂Rn,
we have
(γω)x(X1,..., Xk) = (γω)πF (x)(DπF (X1),...,DπF (Xk)) = (ω∥)x(X1,..., Xk) = 0,
the first equality following from the fact that DπF is the orthogonal projection on TxF.
2) Proof of (4.9). Let Y1,...,Yk ∈Rn. For an appropriately chosen coordinate system with respect to F, each
vector Yj can be written as Yj = X j +Y n
j , where X j = DπF Yj ∈TxF and Y n
j
is the normal component of Yj .
We then have
(ω⊥)x(Y1,...,Yk)
(4.6),(4.5)
=
ωx(Y1,...,Yk)−(π∗
F ◦γ)ωx(Y1,...,Yk)
(4.2)
= ωx(Y1,...,Yk)−ωx(X1,..., Xk)
=
£
ωx(Y n
1 , X2,..., Xk)+···+ωx(X1,..., Xk−1,Y n
k )+ωx(X1,..., Xk)
¤
−ωx(X1,..., Xk)
= ωx(Y n
1 , X2,..., Xk)+···+ωx(X1,..., Xk−1,Y n
k ),
where, in the third step, we have used multiple times the fact that, for any x ∈F, it holds ωx(X1,...,Y n
j ,...,Yk) =
ωx(X1,...,Y n
j ,..., Xk) since ωx is k-linear and alternating. The above relation shows that there exists permu-
tations σj , 1 ≤j ≤k, of the indices 1,...,k such that, taking c j such that Y n
j = c jn,
(ω⊥)x(Y1,...,Yk) =
kX
j=1
cσj (k)(γnω)x(Xσj (1),..., Xσj (k−1)).
This relation shows that γnω = 0 implies ω⊥= 0. To prove the converse statement, we observe that, for any
x ∈F and any vectors X2,..., Xk ∈TxF, since DπTxFn = 0, we have (ω∥)x(n, X2,..., Xk) = 0 and thus, by (4.6),
(ω⊥)x(n,X2,..., Xk) = ωx(n, X2,..., Xk) = (γnωx)(X2,..., Xk). This shows that ω⊥= 0 implies γnω = 0. By
(4.4), (4.9) also implies that γ(⋆ω) = 0 if and only if ω⊥= 0, as is stated in [4, Comments after (2.9)].
3) Proof of (4.10). Note that ⋆ω = ⋆ω∥+⋆ω⊥= (⋆ω)∥+(⋆ω)⊥. We have that
(⋆ω)∥= ⋆ω⊥,
(⋆ω)⊥= ⋆ω∥,
by applying vectors in Rn, respectively tangential and normal to F, to the equation ⋆ω∥+ ⋆ω⊥−(⋆ω)∥−
(⋆ω)⊥= 0.
It can be helpful to look at the decomposition of a k-form ω ∈ΛkT in a coordinate system adapted to the
facet F. Fix any x ∈F. Let {n,e1,...,en−1} be a basis of Rn such that {e1,...,en−1} is a basis of TxF. Let the
dual basis be {dt,dx1,...,dxn−1}, where dt is the dual to n. Define the multi-index set
Jℓ:= {(i1,...,iℓ) : 1 ≤i1 < i2 < ··· < iℓ≤n −1}.
Then, for any x ∈F, we can write
(ω)x = (ω⊥)x +(ω∥)x =
X
J∈Jk−1
ωt,J(x)dt ∧dx J +
X
I∈Jk
ωI(x)dxI,
(4.11)
where ωt,J and ωI are coefficients with respect to the chosen basis. One can check (upon identifying γdxi =
dxi) that
(γω)x =
X
I∈Jk
ωI(x)dxI
and
(γnω)x =
X
J∈Jk−1
ωt,J(x)dx J.
(4.12)
These relations shed more light on the relations stated in Lemma 1. They are also useful for the next result.
Lemma 2. Let ω ∈ΛkT and η ∈Λn−kT . The following relations hold at any x ∈F:
ω∥∧η∥= 0,
ω⊥∧η⊥= 0.
8
Proof. Writing the decomposition (4.11) for both ω and η, it is clear that ω⊥∧η⊥vanishes due to dt being
present in both expansions. On the other hand, we have
ω∥∧η∥=
X
I∈Jk, L∈Jn−k
ωIηLdxI ∧dxL,
which vanishes since one index at least is common to both I and L, as can be seen by writing (identifying
the indices in the families with the sets of these indices):
|I ∩L| = |I|+|L|−|I ∪L| = k +(n −k)−|I ∪L| ≥n −(n −1) = 1.
5
Ghost penalty
Using, in FEEC discretisations of weak formulations of PDEs, the inner product (•,•)Ωh on the active mesh
Ωh in the discrete formulation is possible, but introduces a geometric error that is at least O(h) due to the
fact that the mesh is not fitted to the boundary ∂Ω. Avoiding this geometric error requires to only rely on the
inner product (•,•)Ωin these formulations, which leads to ill-conditioned systems since DOFs attached to
elements having a very small intersection with Ωhave a very small impact on this inner product. The goal of
the ghost penalty stabilisation term designed in this section is precisely to restore robustness while preserv-
ing optimal convergence rates. The name “ghost” derives from the finite difference terminology of calling
“ghost cells” the elements outside of the physical domain. The main result of this section is Theorem 1,
where we prove that the norm associated to the ghost term is equivalent to the L2(Ωh)-norm.
5.1
Jumps of forms
Let F = T 1 ∩T 2 ∈F ∂
h with normal n pointing from T2 into T1. Set, for ω ∈V k,r
h
and i = 1,2,
ωi := ω|Ti ,
where ω|Ti denotes the standard component-wise restriction of ω to Ti on the canonical basis of k-alternating
multi-linear maps, similar to (4.1). We define the jump over F as
[·] : Λk(T 1 ∪T 2) →C ∞(F; Altk Rn),
ω 7→[ω] := ω1|F −ω2|F .
Remark 3 (Traces and jumps). Since the normal and tangential traces (4.2)–(4.3) depend only on the values
of the form in Altk Rn on the facet F, we can apply these operators to the jump defined above. In other words,
γ[ω] and γn[ω] make sense.
From the linearity of the jump operator and the relation between the tangential part and the trace oper-
ator (cf. Lemma 1), it follows that only the normal part of the form (see Section 4) is involved in the jump.
Proposition 1 (Jump of discrete forms). For all ω ∈V k,r
h
, it holds
[ω] = [ω⊥],
where ω⊥is the normal part of the form defined by (4.6).
Proof. Let F = T 1 ∩T 2. We have that ωi
(4.7)
= ωi∥+ ωi⊥on F for i = 1,2. The discrete k-forms are exactly
defined by their degrees of freedom, so that γω is continuous across inter-element facets, i.e. γω1 −γω2 = 0.
Let {Y1,Y2,...,Yk} be an arbitrary set of k vectors in Rn. By definition (4.5) of the tangential part of ω, it holds
(ωi∥)x(Y1,...,Yk) = (γωi)x(DπF Y1,...,DπF Yk)
∀x ∈F,
where πF is the orthogonal projection onto F. By linearity of the jump operator, we have:
(ω1∥−ω2∥)x(Y1,...,Yk) = (γω1 −γω2)x(DπF Y1,...,DπF Yk) = 0
∀x ∈F.
The result of Proposition 1 is translated in terms of vector proxies in Table 4.1.
9
k
Proxy
Jump [ω⊥]
0
q
0
1
v
[v ·n]
2
w
[n×(w ×n)]
3
r
[r]
Table 5.1: Jumps [ω] = [ω⊥] for vector proxies in R3.
5.2
Ghost penalty stabilisation
For an integer ℓ≥0, we define the ℓ-th order directional derivative ∇(ℓ)
n : V k,r
h
→L2ΛkΩh along n as
∇(ℓ)
n ω :=
X
1≤j1<j2<···<jk≤n
(∂(ℓ)
n ωj1 j2...jk )dx j1 ∧dx j2 ∧···∧dx jk ,
where ∂(ℓ)
n ωj1 j2...jk is the ℓ-th order directional derivatives of its component with respect to a chosen coor-
dinate system. The definition does not depend on the choice of coordinate system, and we also identify
∇0n ≡Id.
For ω ∈V k,r
h
, it is clear that ∇(ℓ)
n ω is a polynomial form of degree r −ℓand that ∇(ℓ)
n ω = 0 for ℓ> r. Notice
that the jump of the normal derivatives of a discrete k-form ω does not reduce, in general, to the jump of the
normal part [(∇(ℓ)
n ω)⊥], in contrast to Proposition 1.
Recall the definitions (4.2) of trace and (4.3) of normal trace, as well as Remark 3 on their applicability
to jumps. Letting hF be the diameter of F, we can then define the (ghost penalty) stabilisation term s :
V k,r
h
×V k,r
h
→R as
s(ω,ζ) :=
X
F∈F ∂
h
rX
ℓ=0
ηh2ℓ+1
F
Z
F
³
γn[∇(ℓ)
n ω]∧⋆γn[∇(ℓ)
n ζ]+γ[∇(ℓ)
n ω]∧⋆γ[∇(ℓ)
n ζ]
´
,
(5.1)
where η > 0 is a user-defined penalty parameter. The form s is symmetric and bilinear. Notice that s can be
extended to act on smooth enough k-forms and it holds, owing to the single-valuedness of traces,
s(ζ,ω) = s(ω,ζ) = 0
∀(ζ,ω) ∈Hr+1ΛkΩh ×V k,r
h
.
We define the ghost inner product and norm respectively as
(ω,ζ)s := (ω,ζ)Ω+ s(ω,ζ)
∀(ω,ζ) ∈V k,r
h
×V k,r
h
,
∥ω∥s :=
p
(ω,ω)s
∀ω ∈V k,r
h
.
(5.2)
Remark 4 (Formula (5.1) for vector proxies). Let n = 3. The simplifications of the jumps for the case ℓ= 0
given in Table 5.1 do not extend to the case ℓ≥1. However, the directional derivative of a k-form is still a
k-form, so, using the results of Table 4.1, we infer that the integrand in (5.1) is in fact the inner product of
the full jumps of the normal derivatives regardless of form order k, see Table 5.2.
The goal of the remainder of this subsection is to show the following equivalence result.
Theorem 1 (Uniform equivalence for L2-norms). The norms ∥• ∥Ωh and ∥• ∥s are equivalent on V k,r
h
, uni-
formly in h.
The following technical lemma is what informs the definition of the stabilisation term (5.1).
Lemma 3 (Local control through ghost penalty). Let r ≥1 be an integer and ω ∈V k,r
h
. Fix a boundary facet
F ∈F ∂
h shared by the elements T1 and T2 of Th. Then, the following inequality holds:
∥ω∥2
T1 ≲∥ω∥2
T2 +
rX
ℓ=0
h2ℓ+1
F
Z
F
³
γn[∇(ℓ)
n ω]∧⋆γn[∇(ℓ)
n ω]+γ[∇(ℓ)
n ω]∧⋆γ[∇(ℓ)
n ω]
´
.
(5.3)
10
Form degree k
s(•,•)
0 or 3
X
F∈F ∂
h
rX
ℓ=0
ηh2ℓ+1
F
Z
F
[∂ℓ
n•][∂ℓ
n•]
1 or 2
X
F∈F ∂
h
rX
ℓ=0
ηh2ℓ+1
F
Z
F
[∂ℓ
n•]·[∂ℓ
n•]
Table 5.2: Ghost penalty stabilisation (5.1) for vector proxies in R3, with ω,ω′ := ζ ∈V k,r
h
.
Proof. Consider the orthogonal projection onto the hyperplane containing F, which we also denote by πF .
First, we claim that we can assume to start with a simplex T1 in which the vertex sF opposite to F is such that
πF sF ∈F. Indeed, if T1 is such that πF sF ̸∈F, then consider the maximal simplex T ′
1 ⊆T1 with facet F and
opposite vertex s′
F ∈T1 such that πF s′
F ∈F. By a standard scaling argument as in [15, Lemma 1.25], using
norm equivalence for polynomial forms, we have that ∥ω∥T1 ≲∥ω∥T ′
1.
Consider now a point x ∈T1, and its orthogonal projection xF on F. Let t ∈R be the distance from x
to xF , such that x = xF + tn. For i ∈{1,2}, since the restriction wi := w|Ti is a polynomial form, there is a
canonical extension outside of Ti. The Taylor expansion centered at xF of ω1, or of the extension of ω2 to T1,
is given by
(ωi)x =
rX
ℓ=0
(∇(ℓ)
n ωi)xF
ℓ!
tℓ.
Since each ωi is a polynomial form of degree r, the Taylor expansion is exact. Taking the difference of the
two expansions gives
(ω1)x = (ω2)x +
rX
ℓ=0
(∇(ℓ)
n ω1)xF −(∇(ℓ)
n ω2)xF
ℓ!
tℓ= (ω2)x +
rX
ℓ=0
[∇(ℓ)
n ω]xF
ℓ!
tℓ.
Taking the squared L2(T1)-norm and using triangle inequalities, we get
∥ω1∥2
T1 ≲
Ã
∥ω2∥T1 + |t|ℓ
ℓ!
rX
ℓ=0
∥[∇(ℓ)
n ω]xF ∥T1
!2
≲∥ω2∥2
T1 +h2ℓ
F
rX
ℓ=0
∥[∇(ℓ)
n ω]xF ∥2
T1
≲∥ω2∥2
T2 +h2ℓ
F
rX
ℓ=0
Z
T1
[∇(ℓ)
n ω]xF ∧⋆[∇(ℓ)
n ω]xF ,
where, in the second inequality, we have used the Cauchy–Schwarz inequality on the norm together with
|t| ≲hT ≂hF , and the conclusion follows from a polynomial transport argument (similar to the one used in
the proof of [15, Lemma 1.25]) to replace ∥ω2∥2
T1 with ∥ω2∥2
T2, and the definition of the L2-norm to expand
the second term.
Let ξ be the k-form on T1 such that, for all x ∈T1, ξx := [∇(ℓ)
n ω]xF ; as above, xF denotes the projection of
x onto F. From (4.7) we can decompose ξ into its normal and tangential parts inside T1 as
ξ = ξ⊥+ξ∥.
By definition, ξx = ξxF for all x ∈T1. Define on T1 the (n −k)-form η := ⋆ξ, for which we also have ηx = ηxF
for all x ∈T1. By Lemma 1, we have ⋆ξ∥= η⊥and ⋆ξ⊥= η∥which, combined with Lemma 2, gives
ξ⊥∧⋆ξ∥= ξ⊥∧η⊥= 0 and ξ∥∧⋆ξ⊥= ξ∥∧η∥= 0.
11
Hence, at xF we can can decompose the wedge product as
ξ∧η = ξ⊥∧η∥+ξ∥∧η⊥.
Decomposing ξ using parallel and normal coordinates to F as in (4.11), we write
ξx = (ξ⊥)x +(ξ∥)x =
X
J∈Jk−1
ξt,J(x)dt ∧dx J +
X
I∈Jk
ξI(x)dxI,
and similarly for η. By Fubini’s theorem, denoting by χ the characteristic function of T1 and using a coordi-
nate system (t, y) with t the coordinate along n and y the coordinate along the affine span of F, we get
Z
T1
ξ⊥∧η∥=
Z
Rn−1
Z
R
χ(t,xF )
X
J∈Jk−1,I∈Jk
ξt,J(xF )ηI(xF ) dt ∧dx J ∧dxI
=
Z
Rn−1
X
J∈Jk−1,I∈Jk
ξt,J(xF )ηI(xF )
µZ
R
χ(t,xF )dt
¶
dx J ∧dxI
=
Z
F
LxF
X
J∈Jk−1,I∈Jk
ξt,J(xF )ηI(xF )dx J ∧dxI
=
Z
F
LxF
X
J∈Jk−1
ξt,Jdx J ∧
X
I∈Jk
ηIdxI
(4.12)
=
Z
F
LxF γnξ∧γη,
where, starting from the third equality, LxF denotes the length of the segment at the vertical of xF in T1.
Performing similar computations for
R
T1 ξ∥∧η⊥and accounting for the fact that dxI ∧dt = (−1)kdt ∧dxI,
we arrive at
Z
T1
ξ∧η =
Z
F
LxF γnξ∧γη+(−1)k
Z
F
LxF γξ∧γnη
=
Z
F
LxF γn[∇(ℓ)
n ω]∧γ(⋆[∇(ℓ)
n ω])+(−1)k
Z
F
LxF γ[∇(ℓ)
n ω]∧γn(⋆[∇(ℓ)
n ω])
(4.4)
=
Z
F
LxF γn[∇(ℓ)
n ω]∧⋆γn[∇(ℓ)
n ω]+
Z
F
LxF γ[∇(ℓ)
n ω]∧⋆γ[∇(ℓ)
n ω]
≲hF
µZ
F
γn[∇(ℓ)
n ω]∧⋆γn[∇(ℓ)
n ω]+
Z
F
γ[∇(ℓ)
n ω]∧⋆γ[∇(ℓ)
n ω]
¶
,
where we have expanded ξ and η according to the respective definitions in the second step while, in the
conclusion, we have used the fact that LxF ≲hF and that the integrands are both positive multiples of the
volume form on F.
The next result shows that the ghost penalty term is enough to control the extension of ω to the active
mesh.
Lemma 4 (Global control through ghost penalty). For all ω ∈V k,r
h
, it holds
∥ω∥2
Ωh ≲∥ω∥2
s.
Proof. Let T im
h
:= Th \ T cut
h
be the set of immersed elements. We note that the norm on fully immersed
elements is already controlled by the norm on Ω:
X
T ∈T im
h
∥ω∥2
T ≤∥ω∥2
Ω.
It remains to evaluate the norm on cut elements.
12
...
...
...
T1
T2
T3
Figure 5.1: Both T1 and T2 are cut elements, but T2 has a fully immersed neighbour T3. The boundary ∂Ωis
the red curve. Here N = 3.
By Assumption 1-(ii), starting from a cut element, the number of elements we must pass through to reach
a fully immersed element is at most N, see Figure 5.1 for an illustration in the case N = 3. Let g : T cut
h
→T im
h
be the map which associates to each cut element T ∈T cut
h
the first fully immersed element g(T ) ∈T im
h
which is reached by passing through at most N cut elements. For a given T ′ ∈Th, any T ∈Th such that
T ′ = g(T ) lies in a chain of elements of cardinality ≤N; by Remark 2, all elements along this chain have
diameters that are comparable to hT ′, and therefore lie in a ball of radius ≲NhT ′ ≲hT ′. As this ball has
measure ≂hn
T ′ and each T in it has measure ≂hn
T ≂hn
T ′, we infer that
|g −1(T ′)| ≲
hn
T ′
hn
T ′
= 1.
(5.4)
By applying Lemma 3 at most N times, we then have
X
T ∈T cut
h
∥ω∥2
T ≲
X
T ∈T cut
h
∥ω∥2
g(T ) +
X
F∈F ∂
h
rX
ℓ=0
h2ℓ+1
F
Z
F
³
γn[∇(ℓ)
n ω]∧⋆γn[∇(ℓ)
n ω]+γ[∇(ℓ)
n ω]∧⋆γ[∇(ℓ)
n ω]
´
(5.4)
≲
X
T ′∈T im
h
∥ω∥2
T ′ +
X
F∈F ∂
h
rX
ℓ=0
h2ℓ+1
F
Z
F
³
γn[∇(ℓ)
n ω]∧⋆γn[∇(ℓ)
n ω]+γ[∇(ℓ)
n ω]∧⋆γ[∇(ℓ)
n ω]
´
.
This completes the proof since P
T ′∈T im
h
∥ω∥2
T ′ ≤∥ω∥2
Ω.
Proof of Theorem 1. Choose an arbitrary ω ∈V k,r
h
. In view of Lemma 4, we only have to show that ∥ω∥s ≲
∥ω∥Ωh. That ∥ω∥Ω≲∥ω∥Ωh is clear since Ω⊂Ωh. Standard inverse and trace inequalities for polynomials
grant
∥∇(ℓ)
n ω∥2
F ≲h−2ℓ
F
∥ω∥2
F ≲h−(2ℓ+1)
F
∥ω∥2
T ,
where T is an element having F as facet (note that hF ≂hT ). Hence, using a triangle inequality followed
by the above relation, the norm of the jumps can be estimated as follows: denoting by T1 and T2 the two
elements on each side of F,
∥[∇(ℓ)
n ω]∥2
F ≲
X
T ∈{T1,T2}
h−(2ℓ+1)
F
∥ω∥2
T ,
from which ∥ω∥s ≲∥ω∥Ωh follows.
Remark 5 (Macro stabilisation). Notice that the proof of Lemma 4 does not require the stabilisation term
to act on all facets in F ∂
h. It is enough to consider the facets that form a path from each cut element to a
fully immersed element, as in the definition of the map g in the proof of Lemma 4. Such a stabilisation term
contains fewer summands, and leads naturally to the macro stabilisation procedure outlined in for example
[19, Section 6.1].
13
5.3
Hodge decomposition
Recall the definition (2.4) of the space Hk
h of discrete harmonic forms for FEEC, replacing Ωby Ωh. This
space is constructed using orthogonality with respect to the standard inner product of L2ΛkΩh. Since we
will use the ghost product (•,•)s (cf. (5.2)) to define numerical schemes, we need to consistently modify the
space of harmonic forms as follows:
Hk
s :=
n
ρh ∈V k,r
h
: dρh = 0, (ρh,dτh)s = 0
∀τh ∈V k−1,r+1
h
o
.
Since both (•,•)s and (•,•)Ωh are inner products on V k,r
h
, and both Hk
s and Hk
h are orthogonal to the same
space, these spaces are isomorphic. We also have
Kerdh = dV k−1,r+1
h
s Hk
s ,
where
s denotes the orthogonal complement with respect to the ghost product.
Correspondingly, we obtain the Hodge decomposition (compare with (2.5)):
V k,r
h
= (Kerdh)⊥s
s dV k−1,r+1
h
s Hk
s .
6
Numerical validation
Consider again the Hodge Laplace equation in mixed weak form (2.2). The unfitted method reads: Find
(σh,ηh,λh) ∈V k−1,r+1
h
×V k,r
h
×Hk
s such that
(σh,τh)s −(ηh,dτh)s = 0
∀τh ∈V k−1,r+1
h
,
(dσh,ζh)s +(dηh,dζh)s +(λh,ζh)s = (f ,ζh)Ω
∀ζh ∈V k,r
h
,
(ηh,ρh)s = 0
∀ρh ∈Hk
s .
(6.1)
We deliberately take the inner product involving f over the physical domain Ω, rather than the active domain
Ωh, since we do not want to assume an unphysical extension of the data f .
Remark 6 (Divergence constraint for the Poisson equation in mixed form). The proposed method leads to a
scheme with particularly interesting properties in the case k = n, which corresponds to the Poisson equation
in mixed form with homogeneous Dirichlet conditions: Find (σh,ηh) ∈V n−1,r+1
h
×V n,r
h
such that
(σh,τh)s −(ηh,divτh)s = 0
∀τh ∈V n−1,r+1
h
,
(divσh,ζh)s = (f ,ζh)Ω
∀ζh ∈V n,r
h
.
Up to interface conditions and a source term, this is the formulation given in [17, (3.10)-(3.11)]. If f ∈L2ΛnΩ
is regular enough (i.e. has r + 1 weak normal derivatives in L2), then divσh will be equal, to machine
precision, to the L2-projection of f with respect to the (•,•)s-inner product. Applying the same principle
for Stokes flow (f = 0), the corresponding scheme satisfies the incompressibility condition exactly since
divσh = 0. This observation, which is the content of [19, Theorem 1], can be seen as a consequence of
Theorem 1.
6.1
Implementation and results
We validate in this section the scheme (6.1) for k = 1. We respectively choose the trimmed spaces P−
1 Λ0Ωh
(Lagrange) and P−
1 Λ1Ωh (Nédélec of the first kind) for σh and ηh. This choice means that the exterior deriva-
tive d does not reduce the indicated polynomial order, and that the error in the natural stability norm can be
14
expected to converge with order O(h). The scheme reads as follows. Find (σh,ηh,λh) ∈P−
1 Λ0Ωh ×P−
1 Λ1Ωh ×
H1
s such that
(σh,τh)s −(ηh,∇τh)s = 0
∀τh ∈P−
1 Λ0Ωh,
(∇σh,ζh)s +(curlηh,curlζh)s +(λh,ζh)s = (f ,ζh)Ω
∀ζh ∈P−
1 Λ1Ωh,
(ηh,ρh)s = 0
∀ρh ∈H1
s,
(6.2)
where we use stabilisation parameter η = 1 in (5.1). Since the trial and test forms are piecewise linear, their
second derivatives vanish identically, and since σh,τh are continuous, we get from Tables 5.1 and 5.2 that
(σh,τh)s = (σh,τh)Ω+
X
F∈F ∂
h
h3
F ([∇σh ·n],[∇τh ·n])F ,
(ηh,∇τh)s = (ηh,∇τh)Ω+
X
F∈F ∂
h
hF ([ηh],[∇τh])F ,
(curlηh,curlζh)s = (curlηh,curlζh)Ω+
X
F∈F ∂
h
hF ([curlηh],[curlζh])F ,
(λh,ζh)s = (λh,ζh)Ω+
X
F∈F ∂
h
¡
hF ([λh],[ζh])F +h3
F ([∇λh ·n],[∇ζh ·n])F
¢
.
We also use the macro parameter δ = 0.25 in the macro stabilisation procedure discussed in Remark 5.
We solve problem (6.2) on the following filled torus embedded in R3: Ω= {(x, y,z) ∈R3 : [(
p
x2 + y2 −
0.5)2 + z2]1/2 ≤0.25}, with major radius 0.5 and minor radius 0.25. The mesh is cut via the level set function
φ =
·µq
x2 + y2 −0.5
¶2
+ z2
¸1/2
−0.25,
from a uniform background mesh with mesh size h. The toroidal harmonic forms are scalar multiples of
dθ = xd y −ydx
x2 + y2
←→
1
x2 + y2 (−y,x,0).
We pick the right-hand side and the exact solution as
f =
Ã
−
3xy
(x2 + y2)
5
2
, x2 −2y2
(x2 + y2)
5
2
,0
!
,
η =
Ã
−
−xy
(x2 + y2)
3
2
,
x2
(x2 + y2)
3
2
,0
!
.
The C++ code used to compute the example can be found in the fork [2] of the CutFEM library [1]. The
code was run on a laptop with Intel i7-8565U x 8 CPU and 16GB RAM. To solve the linear system, we use the
sparse direct solver MUMPS. Errors and condition numbers for three mesh sizes, h = 1
13, 1
26, 1
52, are shown in
Figure 6.1. A heat map of the x-component of the solution η is shown in Figure 6.2. We note that the expected
rate of convergence O(h) is indeed achieved for the errors on the curl and gradient, with a slightly better rate
for the fields themselves (going possibly towards O(h2)). The condition number for the scheme based on the
ghost stabilisation remains at a reasonable magnitude, while for the unstabilized scheme (using the inner
product (•,•)Ωinstead of (•,•)s) it blows up to a level where solving the linear system becomes infeasible, if
not impossible.
Acknowledgements
Funded by the European Union (ERC Synergy, NEMESIS, project number 101115663). Views and opinions
expressed are however those of the authors only and do not necessarily reflect those of the European Union
or the European Research Council Executive Agency. Neither the European Union nor the granting authority
can be held responsible for them.
15
2 × 10
2
3 × 10
2
4 × 10
2
6 × 10
2
h
10
1
100
101
L2( )-error
h
L2( )
h
L2( )
curl
h L2( )
grad
h
grad
L2( )
O(h)
2 × 10
2
3 × 10
2
4 × 10
2
6 × 10
2
h
107
109
1011
1013
1015
1017
1-norm estimate condition number
Condition number stabilized
Condition number unstabilized
O(h
3)
Figure 6.1: Plot of the convergence of the mixed method (6.1) as a function of the mesh size h.
The
convergence is shown for polynomial order r = 1, and the 1-norm estimate of the condition number
κs := ∥As∥op∥A −1
s
∥op of the system matrix As is also plotted on the right figure.
Figure 6.2: Heat map of computed x-component of the solution ηh to the discrete Hodge Laplace equation
(6.1) on the filled in torus. The mesh size is h = 0.019231 and the polynomial order is r = 1.
16
References
[1] Cutfem/cutfem-library.
URL https://github.com/CutFEM/CutFEM-Library.
[Online; accessed
2025-09-22].
[2] tensorfan/cutfem-fork.
URL https://github.com/tensorFan/CutFEM-fork.
[Online; accessed
2025-09-22].
[3] D. N. Arnold. Finite Element Exterior Calculus. Society for Industrial and Applied Mathematics, 2018.
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Ghost stabilisation for cut finite element exterior calculus Daniele A. Di Pietro1, Jérôme Droniou1,2, and Erik Nilsson1 1IMAG, CNRS, Montpellier, France, , , 2 17, 2025 Abstract We introduce the cut finite element method in the language of finite element exterior calculus, by formulating a stabilisation - for any form degree - that makes the method robust with respect to the position of the interface relative to the mesh. We prove that the L2-norm on the physical domain augmented with this stabilisation is uniformly equivalent to the L2-norm on the "active" mesh that contains all the degrees of freedom of the finite element space (including those external to the physical domain). We show how this CutFEEC method can be applied to discretize the Hodge Laplace equations on an unfitted mesh, in any dimension and any topology. A numerical illustration is provided involving a conforming finite element space of Hcurl posed on a filled torus, with convergence and condition number scaling independent of the position of the boundary with respect to the background mesh. MSC: 65N30, 14F40 Key words: Cut finite element method, FEEC, Hodge Laplace equation, Hilbert complex 1 Introduction A successful strand of the literature on the numerical approximation of partial differential equations (PDEs) has emphasized the relevance of Hilbert complexes [10] in the design of stable numerical schemes. In this context, differential forms provide a natural unifying language to treat mixed problems set in arbitrary space dimension, possibly on manifolds, and where a combination of differential operators appear [4, 5, 6, 9, 16, 21]. Finite Element Exterior Calculus (FEEC), particularly relevant for the present work, provides a comprehensive view of full and trimmed finite element approximations of the de Rham complex; see the monograph [3] for an introduction. At the same time, the Cut Finite Element Method (CutFEM) has emerged as a versatile technique for handling PDEs on domains with complex geometries, removing the need for the mesh to be compliant with the domain's boundary; see the recent review [13] and references therein. Through weak imposition of boundary conditions and carefully constructed stabilisation terms over facets close to the boundary, CutFEM leads to stable and well-conditioned discretisations. Moreover, such properties hold independently of the boundary position relative to the unfitted mesh. This work aims at merging these research avenues by proposing a Cut Finite Element Exterior Calculus (CutFEEC) framework that adapts the CutFEM techniques to FEEC. This merger has notably been made possible by the recent extension of the CutFEM technology to mixed problems [17, 18, 19], which are a natural application of FEEC [3]. The prime example of a Hilbert complex is the de Rham complex, which, for an open connected domain Ω⊂R3, reads 0 ,→H1(Ω) ∇-→Hcurl(Ω) curl --→Hdiv(Ω) div --→L2(Ω) →0. Using the language of differential forms, the de Rham complex can be extended to a domain Ωof any dimension n ∈N as: 0 ,→HΛ0Ω d0 -→HΛ1Ω d1 -→... dn-1 ---→HΛnΩ→0, (1.1) 1 16 Oct 2025 where dk is the exterior derivative and HΛkΩis the space of L2-integrable differential k-forms on Ωwith L2-integrable exterior derivative. The de Rham complex enters the well-posedness analysis of PDEs through its cohomology spaces Hk := Kerdk/Imdk-1. These spaces relate to the topology of the domain via their dimension. Preserving these homological structures at the discrete level is crucial for the stability of numerical schemes. A paradigmatic example of PDE problem whose well-posedness hinges on the de Rham complex is the Hodge Laplace equation; see, e.g., [3, Chapter 4]. Its mixed formulation naturally leads to a mixed problem where the unknowns are the codifferential, the (scalar- or vector-) potential, and a Lagrange multiplier to enforce L2-orthogonality with respect to harmonic forms. The Finite Element (FE) approximation of mixed problems became a research topic starting from the late 1970s, when the first conforming approximation of vector-valued spaces in the de Rham complex were developed [7, 24, 26]. Traditional FE methods, such as the ones cited above, require meshes that conform to the geometry of the domain. These can be challenging to generate for intricate geometries or in the presence of evolving interfaces. CutFEM addresses this problem by permitting the interface to cut through the elements of a background mesh, thereby simplifying mesh generation and adaptation [12]. However, ensuring stability and accuracy in such unfitted methods necessitates appropriate stabilisation techniques [11]. Otherwise, when the geometry cuts the mesh in especially nasty ways (|T ∩Ω| ≪|T | for an element T in the mesh), one observes a significant degradation of the condition number of the associated linear system. Recent works have addressed the application of CutFEM techniques to mixed problems. In [17, 18], the authors introduced a stable CutFEM for Darcy flow problems which, thanks to conformity with respect to the tail end of the de Rham complex, preserves the divergence of the velocity field. Using a similar approach, one can also preserve the divergence condition of Stokes flow in an unfitted setting [19]. In this work, we develop a new CutFEM stabilisation to design unfitted robust discrete L2-products for any space in the de Rham complex of differential forms. The stabilisation can be seen as a generalisation of the mixed ghost penalty term introduced in [17, 19, 25]. Orthogonalising against the harmonic forms using an L2-product including this stabilisation, we construct an arbitrary-order numerical method which is unfitted but still compatible with the continuous de Rham complex, thus ensuring its natural stability and preservation of relevant constraints (e.g., zero divergence of electric field in Maxwell's equations). We develop facet-based stabilisation terms, but we mention that there exist other equivalent stabilisation terms in the CutFEM literature, though these have not been developed for differential forms, see [13, Section 4.2]. We note that the commonly used assumption of a quasi-uniform mesh [13] is not needed in our construction or analysis. It is sufficient for the relevant arguments to use a local quasi-uniformity, which does not entail particular restrictions on the mesh. In [23], the quasi-uniform assumption is used but a similar remark is made [20, Remark 2.3]. A particularly relevant novelty results from applying this general construction to the special case of 1forms, which is key to discretize problems such as Maxwell's equations or the Hcurl-elliptic problem. CutFEM has, indeed, seen little development for problems involving Hcurl; see the recent and exhaustive review [13]. There are some previous works which use an unfitted discontinuous Galerkin approach [14, 22]. Otherwise, works which do not use discontinuous/broken finite element spaces are essentially limited to [19, 27, 28][25, Paper D]. Of these, [19] and [25, Paper D] do not contain a stability analysis; [19] considers a vorticity-velocity-pressure formulation for Stokes flow wherein the vorticity is discretized by a Hcurlconforming variable, while [25, Paper D] considers numerical experiments of a set of standard formulations of Maxwell's equations made unfitted. The works [27, 28] focus, respectively, on the time-harmonic Maxwell's equations and the Hcurl-elliptic interface problem. In [28, Remark 3], the authors mention that their proposed method works numerically when using nodal Lagrange elements, but error analysis is provided only for the case of using piecewise discontinuous polynomials. The numerical scheme in [28] uses the lowest order Hcurl-conforming Nédélec elements of the second kind, but adds diagonal stabilisation terms in the Lagrange multiplier block [27, (4.8)-(4.11)], thus perturbing the saddle point nature of the problem. Similar stabilisation terms also appear in [28]. Moreover, in contrast to our approach, no work mentioned 2 above considers the presence of non-trivial harmonic forms, which appear when considering domains of general topology. The remainder of this paper is organised as follows. In Section 2 we discuss a motivating example of mixed problem, the Hodge Laplace problem, whose well-posedness relies on the properties of the de Rham complex, and we summarise some notions of FEEC. In Section 3 we recall the notion of unfitted mesh, and we then introduce in Section 4 the concepts of tangential and normal parts of differential forms, relative to a facet of the mesh. In Section 5 we introduce the novel ghost penalty stabilisation operator, and show the uniform equivalence between the L2-norm on the physical domain and the L2-norm on the extended domain containing all the degrees of freedom. Finally, in Section 6 we validate our results on a numerical example on the filled torus, demonstrating that the use of the stabilisation drastically reduces the condition number of the discrete system. 2 A motivating example In this section we briefly discuss the Hodge Laplacian, an archetypal problem whose well-posedness hinges on the properties of the de Rham complex. This example serves as a motivation for the development of a CutFEEC theory, and will be used in Section 6 to numerically demonstrate the efficiency of the ghost penalty stabilisation of Section 5. 2.1 Hodge Laplace equation Let B ⊂Rn be an open set. We denote by ΛkB the space of smooth differential k-forms on B. We introduce the L2-product (ω,ζ)B := R B ω∧⋆ζ on ΛkB, where ∧denotes the wedge product and ⋆the Hodge star operator [3, (6.2)]. The L2-norm over B is denoted by ∥•∥B := p(·,·)B. We denote by L2ΛkB the space of k-forms over B that are bounded with respect to this norm. With the exterior derivative dk : L2ΛkB →L2Λk+1B, viewed as an unbounded operator, we also define HΛkB := n ω ∈L2ΛkB : dkω ∈L2Λk+1B o , with norm ∥ω∥2 HB := ∥ω∥2 B +∥dkω∥2 B. Let now Ω⊂Rn be a connected and open subset of Rn with Lipschitz boundary ∂Ωand outward unit normal vector field n. Given f ∈L2ΛkΩ, the Hodge Laplace equation consists in seeking η ∈ΛkΩsuch that (dk-1δk +δk+1dk)η = f -πf in Ω, πη = 0 in Ω, (2.1) where the codifferential δk+1 : L2Λk+1Ω→L2ΛkΩis the adjoint of d and π := πHk is the orthogonal projection onto the space of harmonic forms Hk := n ρ ∈L2ΛkΩ: dkρ = 0 and δkρ = 0 o = Kerdk ∩Kerδk ∼= Kerdk/Imdk-1. Here ∼= means vector space isomorphism. The boundary conditions are implicit in the definition of the Hodge Laplace operator dk-1δk +δk+1dk, and are revealed upon choosing the domain of the exterior derivative. For ease of presentation, let us in this work choose the domain D(dk) = HΛkΩ, so that the domain of the codifferential becomes (by duality) the space of L2(Ω)-integrable forms ω with δω ∈L2(Ω) and γ(⋆ω) = 0. After recalling vector proxies in three space dimensions in Table 2.1, in Table 2.2, 3 k Form Proxy Sobolev space 0 q q H1(Ω) 1 v1dx1 + v2dx2 + v3dx3 v = (v1,v2,v3) Hcurl(Ω) 2 w1dx2 ∧dx3 -w2dx1 ∧dx3 + w3dx1 ∧dx2 w = (w1,w2,w3) Hdiv(Ω) 3 rdx1 ∧dx2 ∧dx3 r L2(Ω) Table 2.1: Vector proxies in R3. k = 0 k = 1 k = 2 k = 3 -∆q = f - ̄f , ∇q ·n = 0, ̄q = 0 curlcurlv -∇divv = f -πf, v ·n = 0 and curlv ×n = 0, v ⊥H1 curlcurlw -∇divw = f -πf, w ×n = 0 and divw = 0, w ⊥H2 -∆r = f , r = 0 Table 2.2: The Hodge Laplace equation in R3 for different differential form orders. Entries are labeled A, B, C where A are the bulk equation(s) in Ω, B the boundary condition(s) on ∂Ω, and C is the cohomological compatibility condition. we express problem (2.1) in terms of vector proxies for different values of k; cf. [5, Section 4.2] for further details. For convenience of notation, we shall for the remainder omit the index and just write d for the exterior derivative. A variational mixed formulation of (2.1) is (cf. [3, Theorem 4.7]): Find (σ,η,λ) ∈HΛk-1Ω× HΛkΩ×Hk such that (σ,τ)Ω-(η,dτ)Ω= 0 ∀τ ∈HΛk-1Ω, (dσ,ζ)Ω+(dη,dζ)Ω+(λ,ζ)Ω= (f ,ζ)Ω ∀ζ ∈HΛkΩ, (η,ρ)Ω= 0 ∀ρ ∈Hk. (2.2) The above formulation enforces in a weak manner σ = δη and λ = πf . The (natural) boundary conditions are γ⋆η = 0 and γ⋆dη = 0. Using crucial properties of the de Rham complex such as Hodge decompositions and Poincaré inequalities, it can be proved that this mixed formulation is well-posed [3, Theorems 4.8 and 4.9]. Preserving these properties at the discrete level, as is the case for the compatible discretisations considered here, naturally leads to numerical schemes for which stability can be proved mimicking the arguments for well-posedness of the continuous problem. 2.2 Principles of FEEC Assume here that Ωis a polytopal domain, and let Th be a conforming simplicial mesh of Ω. The Finite Element Exterior Calculus (FEEC) method provides the tools for a discrete formulation of (2.2) which is wellposed [3, (5.3), Section 5.2.3], but non-conforming in general due to the non-inclusion of the discrete space of harmonic forms into the continuous one (these spaces have the same dimension, so the former being included in the latter would actually mean that one could compute harmonic forms exactly). We make a brief summary of the results which pertain to this work. Let r ≥1 be a polynomial degree. For each k ∈{0,...,n}, let V k,r h ⊂HΛkΩbe a finite-dimensional space of k-forms that are piecewise polynomial of degree at most r on Th and such that dV k,r h ⊂V k+1,r-1 h . We adopt the convention that V -1,r h := {0} and V n+1,r h := {0}. These spaces define a sub-complex of the de Rham complex: 0 V 0,r h V 1,r-1 h ··· V n,r-n h 0. d d d (2.3) 4 It can be shown [3], using, e.g., bounded cochain maps between the continuous and discrete de Rham complexes, that the cohomology spaces of (2.3) are isomorphic to those of (1.1). This diagram (in which the polynomial degree decreases along the sequence) corresponds to the case of full finite element spaces, V k,r h := Pr ΛkΩh. When using trimmed spaces, Pr ΛkΩh ⊂Pr ΛkΩh in the notation of [3], the degree r remains the same along the diagram. For application of our results to the trimmed spaces, simply replace any occurrence of V k,• h with Pr ΛkΩh. The discrete exterior derivative acting on a discrete space is just the restriction of the exterior derivative from the corresponding continuous space. When relevant, we shall denote it by dh := d|V k,r h , otherwise by d when there is no confusion. Then, one defines the space of discrete harmonic forms as Hk h := Kerdh/dV k-1,r+1 h ∼= n ρh ∈V k,r h : dρh = 0 and (ρh,dτh)Ω= 0 for all τh ∈V k-1,r+1 h o . (2.4) The non-inclusion Hk h ̸⊂Hk is due to forms in Hk needing to be orthogonal to all of dHΛk-1Ω, while being in Hk h only guarantees orthogonality to the subspace dV k-1,r+1 h . As mentioned in Section 2.1, the Hodge decomposition of the complex spaces is an essential tool to analyse both the continuous Hodge-Laplace equation and its numerical discretisations. For the complex (2.3), we have the following discrete Hodge decomposition: V k,r h = (Kerdh)⊥ dV k-1,r+1 h Hk h. (2.5) Given f ∈L2ΛkΩthe FEEC scheme for problem (2.2) reads: Find (σh,ηh,λh) ∈V k-1,r+1 h ×V k,r h ×Hk h such that (σh,τh)Ω-(ηh,dτh)Ω= 0 ∀τh ∈V k-1,r+1 h , (dσh,ζh)Ω+(dηh,dζh)Ω+(λh,ζh)Ω= (f ,ζh)Ω ∀ζh ∈V k,r h , (ηh,ρh)Ω= 0 ∀ρh ∈Hk h. The scheme is well-posed and optimally convergent in the HΛkΩ-norm [3, Theorem 5.5]. 3 Unfitted mesh Let Th = {T } be an active mesh such that each (open) simplex T ∈Th has a nonempty intersection with Ω, and for which Ω⊂Ωh := Ã [ T ∈Th T !◦ . The open set Ωh is hereafter referred to as the active domain. Remark 1 (Construction of active mesh). The term active mesh refers to the following construction. Define a background domain Ω0 ⊃Ω, which is polytopal of dimension n. Consider a conforming triangular mesh T0,h of Ω0. Then, the active mesh is defined as Th := © T ∈T0,h : T ∩Ω̸= ; a . This construction is illustrated in Figure 3.1. We say T ∈Th is cut if T ̸⊂Ωand fully immersed in Ωotherwise. We denote the set of cut elements by T cut h := {T ∈Th : T ̸⊂Ω}. We denote by F ∂ h the set of stabilisation facets [19, Figure 1], which are facets internal to Ωh for which at least one neighbouring element is intersected by ∂Ω: F ∂ h := F ∈ [ T ∈T cut h FT : F ̸⊂∂Ωh , 5 Figure 3.1: Illustration of the active mesh Th. The boundary ∂Ωof the domain is in red. The "remainder" of background mesh T0,h is shown in light blue, while the active mesh Th is shown in black. where FT gathers the ((n -1)-dimensional) facets of T ∈Th. To each facet we associate a unique normal vector n. The context will make it clear which normal is meant, that of a facet or that of ∂Ω. Assumption 1 (Mesh regularity). For any T ∈Th, let ρT and hT be, respectively, the diameter of the largest ball contained in T and that of T . We assume the following: (i) (Shape regularity) For some constant κ > 0 independent of the mesh size h := maxT ∈Th hT , we have hT ρT ≤κ ∀T ∈Th. (3.1) (ii) (Uniformly bounded cut-to-uncut path) There exists an integer N > 0 independent of h such that, for all T ∈T cut h , there exists a sequence of NT ≤N elements {T1 = T,T2,...,TNT } such that Ti and Ti+1 share a facet for all 1 ≤i ≤NT -1, and TNT ∈Th cut h is fully immersed in Ω. The notation a ≂b means that there exist constants C,C ′ > 0 independent of h, but possibly dependent on κ and N such that a ≤Cb and b ≤C ′a, and similarly we write a ≲b if and only if a ≤Cb. Remark 2 (Quasi-uniformity along cut-to-uncut paths). We remark that, by shape regularity and since N is bounded independently of h, all mesh elements along a given cut-to-uncut path have uniformly comparable diameters: if T1,...,TNT are as in (ii), then hTi ≂hTj for all i, j ∈{1,...,NT }. 4 Tangential and normal traces and parts of differential forms Restrictions as well as tangential and normal traces and parts of differential forms are essential concepts to define the stabilisation in Section 5 below. We consider here an element T ∈Th and one of its facets F ⊂∂T . For x ∈F, let TxC be the tangent space of C ∈{F,T } at x. Let ι : F ,→T be the inclusion map of F into T . We note that TxF is the vector hyperplane parallel to F in Rn, and that X ∈TxF can be considered as a vector in Rn; for this reason, we identify ι∗X with X . For ω ∈ΛkT , by "restriction to the facet F" we simply mean the restriction of each component of ω to F, i.e., writing ω := P 1≤j1 r. Notice that the jump of the normal derivatives of a discrete k-form ω does not reduce, in general, to the jump of the normal part [(∇(l) n ω)⊥], in contrast to Proposition 1. Recall the definitions (4.2) of trace and (4.3) of normal trace, as well as Remark 3 on their applicability to jumps. Letting hF be the diameter of F, we can then define the (ghost penalty) stabilisation term s : V k,r h ×V k,r h →R as s(ω,ζ) := X F∈F ∂ h rX l=0 ηh2l+1 F Z F 3 γn[∇(l) n ω]∧⋆γn[∇(l) n ζ]+γ[∇(l) n ω]∧⋆γ[∇(l) n ζ] ́ , (5.1) where η > 0 is a user-defined penalty parameter. The form s is symmetric and bilinear. Notice that s can be extended to act on smooth enough k-forms and it holds, owing to the single-valuedness of traces, s(ζ,ω) = s(ω,ζ) = 0 ∀(ζ,ω) ∈Hr+1ΛkΩh ×V k,r h . We define the ghost inner product and norm respectively as (ω,ζ)s := (ω,ζ)Ω+ s(ω,ζ) ∀(ω,ζ) ∈V k,r h ×V k,r h , ∥ω∥s := p (ω,ω)s ∀ω ∈V k,r h . (5.2) Remark 4 (Formula (5.1) for vector proxies). Let n = 3. The simplifications of the jumps for the case l= 0 given in Table 5.1 do not extend to the case l≥1. However, the directional derivative of a k-form is still a k-form, so, using the results of Table 4.1, we infer that the integrand in (5.1) is in fact the inner product of the full jumps of the normal derivatives regardless of form order k, see Table 5.2. The goal of the remainder of this subsection is to show the following equivalence result. Theorem 1 (Uniform equivalence for L2-norms). The norms ∥• ∥Ωh and ∥• ∥s are equivalent on V k,r h , uniformly in h. The following technical lemma is what informs the definition of the stabilisation term (5.1). Lemma 3 (Local control through ghost penalty). Let r ≥1 be an integer and ω ∈V k,r h . Fix a boundary facet F ∈F ∂ h shared by the elements T1 and T2 of Th. Then, the following inequality holds: ∥ω∥2 T1 ≲∥ω∥2 T2 + rX l=0 h2l+1 F Z F 3 γn[∇(l) n ω]∧⋆γn[∇(l) n ω]+γ[∇(l) n ω]∧⋆γ[∇(l) n ω] ́ . (5.3) 10 Form degree k s(•,•) 0 or 3 X F∈F ∂ h rX l=0 ηh2l+1 F Z F [∂l n•][∂l n•] 1 or 2 X F∈F ∂ h rX l=0 ηh2l+1 F Z F [∂l n•]·[∂l n•] Table 5.2: Ghost penalty stabilisation (5.1) for vector proxies in R3, with ω,ω′ := ζ ∈V k,r h . Proof. Consider the orthogonal projection onto the hyperplane containing F, which we also denote by πF . First, we claim that we can assume to start with a simplex T1 in which the vertex sF opposite to F is such that πF sF ∈F. Indeed, if T1 is such that πF sF ̸∈F, then consider the maximal simplex T ′ 1 ⊆T1 with facet F and opposite vertex s′ F ∈T1 such that πF s′ F ∈F. By a standard scaling argument as in [15, Lemma 1.25], using norm equivalence for polynomial forms, we have that ∥ω∥T1 ≲∥ω∥T ′ 1. Consider now a point x ∈T1, and its orthogonal projection xF on F. Let t ∈R be the distance from x to xF , such that x = xF + tn. For i ∈{1,2}, since the restriction wi := w|Ti is a polynomial form, there is a canonical extension outside of Ti. The Taylor expansion centered at xF of ω1, or of the extension of ω2 to T1, is given by (ωi)x = rX l=0 (∇(l) n ωi)xF l! tl. Since each ωi is a polynomial form of degree r, the Taylor expansion is exact. Taking the difference of the two expansions gives (ω1)x = (ω2)x + rX l=0 (∇(l) n ω1)xF -(∇(l) n ω2)xF l! tl= (ω2)x + rX l=0 [∇(l) n ω]xF l! tl. Taking the squared L2(T1)-norm and using triangle inequalities, we get ∥ω1∥2 T1 ≲ Ã ∥ω2∥T1 + |t|l l! rX l=0 ∥[∇(l) n ω]xF ∥T1 !2 ≲∥ω2∥2 T1 +h2l F rX l=0 ∥[∇(l) n ω]xF ∥2 T1 ≲∥ω2∥2 T2 +h2l F rX l=0 Z T1 [∇(l) n ω]xF ∧⋆[∇(l) n ω]xF , where, in the second inequality, we have used the Cauchy-Schwarz inequality on the norm together with |t| ≲hT ≂hF , and the conclusion follows from a polynomial transport argument (similar to the one used in the proof of [15, Lemma 1.25]) to replace ∥ω2∥2 T1 with ∥ω2∥2 T2, and the definition of the L2-norm to expand the second term. Let ξ be the k-form on T1 such that, for all x ∈T1, ξx := [∇(l) n ω]xF ; as above, xF denotes the projection of x onto F. From (4.7) we can decompose ξ into its normal and tangential parts inside T1 as ξ = ξ⊥+ξ∥. By definition, ξx = ξxF for all x ∈T1. Define on T1 the (n -k)-form η := ⋆ξ, for which we also have ηx = ηxF for all x ∈T1. By Lemma 1, we have ⋆ξ∥= η⊥and ⋆ξ⊥= η∥which, combined with Lemma 2, gives ξ⊥∧⋆ξ∥= ξ⊥∧η⊥= 0 and ξ∥∧⋆ξ⊥= ξ∥∧η∥= 0. 11 Hence, at xF we can can decompose the wedge product as ξ∧η = ξ⊥∧η∥+ξ∥∧η⊥. Decomposing ξ using parallel and normal coordinates to F as in (4.11), we write ξx = (ξ⊥)x +(ξ∥)x = X J∈Jk-1 ξt,J(x)dt ∧dx J + X I∈Jk ξI(x)dxI, and similarly for η. By Fubini's theorem, denoting by χ the characteristic function of T1 and using a coordinate system (t, y) with t the coordinate along n and y the coordinate along the affine span of F, we get Z T1 ξ⊥∧η∥= Z Rn-1 Z R χ(t,xF ) X J∈Jk-1,I∈Jk ξt,J(xF )ηI(xF ) dt ∧dx J ∧dxI = Z Rn-1 X J∈Jk-1,I∈Jk ξt,J(xF )ηI(xF ) μZ R χ(t,xF )dt ¶ dx J ∧dxI = Z F LxF X J∈Jk-1,I∈Jk ξt,J(xF )ηI(xF )dx J ∧dxI = Z F LxF X J∈Jk-1 ξt,Jdx J ∧ X I∈Jk ηIdxI (4.12) = Z F LxF γnξ∧γη, where, starting from the third equality, LxF denotes the length of the segment at the vertical of xF in T1. Performing similar computations for R T1 ξ∥∧η⊥and accounting for the fact that dxI ∧dt = (-1)kdt ∧dxI, we arrive at Z T1 ξ∧η = Z F LxF γnξ∧γη+(-1)k Z F LxF γξ∧γnη = Z F LxF γn[∇(l) n ω]∧γ(⋆[∇(l) n ω])+(-1)k Z F LxF γ[∇(l) n ω]∧γn(⋆[∇(l) n ω]) (4.4) = Z F LxF γn[∇(l) n ω]∧⋆γn[∇(l) n ω]+ Z F LxF γ[∇(l) n ω]∧⋆γ[∇(l) n ω] ≲hF μZ F γn[∇(l) n ω]∧⋆γn[∇(l) n ω]+ Z F γ[∇(l) n ω]∧⋆γ[∇(l) n ω] ¶ , where we have expanded ξ and η according to the respective definitions in the second step while, in the conclusion, we have used the fact that LxF ≲hF and that the integrands are both positive multiples of the volume form on F. The next result shows that the ghost penalty term is enough to control the extension of ω to the active mesh. Lemma 4 (Global control through ghost penalty). For all ω ∈V k,r h , it holds ∥ω∥2 Ωh ≲∥ω∥2 s. Proof. Let T im h := Th \ T cut h be the set of immersed elements. We note that the norm on fully immersed elements is already controlled by the norm on Ω: X T ∈T im h ∥ω∥2 T ≤∥ω∥2 Ω. It remains to evaluate the norm on cut elements. 12 ... ... ... T1 T2 T3 Figure 5.1: Both T1 and T2 are cut elements, but T2 has a fully immersed neighbour T3. The boundary ∂Ωis the red curve. Here N = 3. By Assumption 1-(ii), starting from a cut element, the number of elements we must pass through to reach a fully immersed element is at most N, see Figure 5.1 for an illustration in the case N = 3. Let g : T cut h →T im h be the map which associates to each cut element T ∈T cut h the first fully immersed element g(T ) ∈T im h which is reached by passing through at most N cut elements. For a given T ′ ∈Th, any T ∈Th such that T ′ = g(T ) lies in a chain of elements of cardinality ≤N; by Remark 2, all elements along this chain have diameters that are comparable to hT ′, and therefore lie in a ball of radius ≲NhT ′ ≲hT ′. As this ball has measure ≂hn T ′ and each T in it has measure ≂hn T ≂hn T ′, we infer that |g -1(T ′)| ≲ hn T ′ hn T ′ = 1. (5.4) By applying Lemma 3 at most N times, we then have X T ∈T cut h ∥ω∥2 T ≲ X T ∈T cut h ∥ω∥2 g(T ) + X F∈F ∂ h rX l=0 h2l+1 F Z F 3 γn[∇(l) n ω]∧⋆γn[∇(l) n ω]+γ[∇(l) n ω]∧⋆γ[∇(l) n ω] ́ (5.4) ≲ X T ′∈T im h ∥ω∥2 T ′ + X F∈F ∂ h rX l=0 h2l+1 F Z F 3 γn[∇(l) n ω]∧⋆γn[∇(l) n ω]+γ[∇(l) n ω]∧⋆γ[∇(l) n ω] ́ . This completes the proof since P T ′∈T im h ∥ω∥2 T ′ ≤∥ω∥2 Ω. Proof of Theorem 1. Choose an arbitrary ω ∈V k,r h . In view of Lemma 4, we only have to show that ∥ω∥s ≲ ∥ω∥Ωh. That ∥ω∥Ω≲∥ω∥Ωh is clear since Ω⊂Ωh. Standard inverse and trace inequalities for polynomials grant ∥∇(l) n ω∥2 F ≲h-2l F ∥ω∥2 F ≲h-(2l+1) F ∥ω∥2 T , where T is an element having F as facet (note that hF ≂hT ). Hence, using a triangle inequality followed by the above relation, the norm of the jumps can be estimated as follows: denoting by T1 and T2 the two elements on each side of F, ∥[∇(l) n ω]∥2 F ≲ X T ∈{T1,T2} h-(2l+1) F ∥ω∥2 T , from which ∥ω∥s ≲∥ω∥Ωh follows. Remark 5 (Macro stabilisation). Notice that the proof of Lemma 4 does not require the stabilisation term to act on all facets in F ∂ h. It is enough to consider the facets that form a path from each cut element to a fully immersed element, as in the definition of the map g in the proof of Lemma 4. Such a stabilisation term contains fewer summands, and leads naturally to the macro stabilisation procedure outlined in for example [19, Section 6.1]. 13 5.3 Hodge decomposition Recall the definition (2.4) of the space Hk h of discrete harmonic forms for FEEC, replacing Ωby Ωh. This space is constructed using orthogonality with respect to the standard inner product of L2ΛkΩh. Since we will use the ghost product (•,•)s (cf. (5.2)) to define numerical schemes, we need to consistently modify the space of harmonic forms as follows: Hk s := n ρh ∈V k,r h : dρh = 0, (ρh,dτh)s = 0 ∀τh ∈V k-1,r+1 h o . Since both (•,•)s and (•,•)Ωh are inner products on V k,r h , and both Hk s and Hk h are orthogonal to the same space, these spaces are isomorphic. We also have Kerdh = dV k-1,r+1 h s Hk s , where s denotes the orthogonal complement with respect to the ghost product. Correspondingly, we obtain the Hodge decomposition (compare with (2.5)): V k,r h = (Kerdh)⊥s s dV k-1,r+1 h s Hk s . 6 Numerical validation Consider again the Hodge Laplace equation in mixed weak form (2.2). The unfitted method reads: Find (σh,ηh,λh) ∈V k-1,r+1 h ×V k,r h ×Hk s such that (σh,τh)s -(ηh,dτh)s = 0 ∀τh ∈V k-1,r+1 h , (dσh,ζh)s +(dηh,dζh)s +(λh,ζh)s = (f ,ζh)Ω ∀ζh ∈V k,r h , (ηh,ρh)s = 0 ∀ρh ∈Hk s . (6.1) We deliberately take the inner product involving f over the physical domain Ω, rather than the active domain Ωh, since we do not want to assume an unphysical extension of the data f . Remark 6 (Divergence constraint for the Poisson equation in mixed form). The proposed method leads to a scheme with particularly interesting properties in the case k = n, which corresponds to the Poisson equation in mixed form with homogeneous Dirichlet conditions: Find (σh,ηh) ∈V n-1,r+1 h ×V n,r h such that (σh,τh)s -(ηh,divτh)s = 0 ∀τh ∈V n-1,r+1 h , (divσh,ζh)s = (f ,ζh)Ω ∀ζh ∈V n,r h . Up to interface conditions and a source term, this is the formulation given in [17, (3.10)-(3.11)]. If f ∈L2ΛnΩ is regular enough (i.e. has r + 1 weak normal derivatives in L2), then divσh will be equal, to machine precision, to the L2-projection of f with respect to the (•,•)s-inner product. Applying the same principle for Stokes flow (f = 0), the corresponding scheme satisfies the incompressibility condition exactly since divσh = 0. This observation, which is the content of [19, Theorem 1], can be seen as a consequence of Theorem 1. 6.1 Implementation and results We validate in this section the scheme (6.1) for k = 1. We respectively choose the trimmed spaces P1 Λ0Ωh (Lagrange) and P1 Λ1Ωh (Nédélec of the first kind) for σh and ηh. This choice means that the exterior derivative d does not reduce the indicated polynomial order, and that the error in the natural stability norm can be 14 expected to converge with order O(h). The scheme reads as follows. Find (σh,ηh,λh) ∈P1 Λ0Ωh ×P1 Λ1Ωh × H1 s such that (σh,τh)s -(ηh,∇τh)s = 0 ∀τh ∈P1 Λ0Ωh, (∇σh,ζh)s +(curlηh,curlζh)s +(λh,ζh)s = (f ,ζh)Ω ∀ζh ∈P1 Λ1Ωh, (ηh,ρh)s = 0 ∀ρh ∈H1 s, (6.2) where we use stabilisation parameter η = 1 in (5.1). Since the trial and test forms are piecewise linear, their second derivatives vanish identically, and since σh,τh are continuous, we get from Tables 5.1 and 5.2 that (σh,τh)s = (σh,τh)Ω+ X F∈F ∂ h h3 F ([∇σh ·n],[∇τh ·n])F , (ηh,∇τh)s = (ηh,∇τh)Ω+ X F∈F ∂ h hF ([ηh],[∇τh])F , (curlηh,curlζh)s = (curlηh,curlζh)Ω+ X F∈F ∂ h hF ([curlηh],[curlζh])F , (λh,ζh)s = (λh,ζh)Ω+ X F∈F ∂ h ¡ hF ([λh],[ζh])F +h3 F ([∇λh ·n],[∇ζh ·n])F ¢ . We also use the macro parameter δ = 0.25 in the macro stabilisation procedure discussed in Remark 5. We solve problem (6.2) on the following filled torus embedded in R3: Ω= {(x, y,z) ∈R3 : [( p x2 + y2 - 0.5)2 + z2]1/2 ≤0.25}, with major radius 0.5 and minor radius 0.25. The mesh is cut via the level set function φ = ·μq x2 + y2 -0.5 ¶2 + z2 ̧1/2 -0.25, from a uniform background mesh with mesh size h. The toroidal harmonic forms are scalar multiples of dθ = xd y -ydx x2 + y2 ←→ 1 x2 + y2 (-y,x,0). We pick the right-hand side and the exact solution as f = Ã - 3xy (x2 + y2) 5 2 , x2 -2y2 (x2 + y2) 5 2 ,0 ! , η = Ã - -xy (x2 + y2) 3 2 , x2 (x2 + y2) 3 2 ,0 ! . The C++ code used to compute the example can be found in the fork [2] of the CutFEM library [1]. The code was run on a laptop with Intel i7-8565U x 8 CPU and 16GB RAM. To solve the linear system, we use the sparse direct solver MUMPS. Errors and condition numbers for three mesh sizes, h = 1 13, 1 26, 1 52, are shown in Figure 6.1. A heat map of the x-component of the solution η is shown in Figure 6.2. We note that the expected rate of convergence O(h) is indeed achieved for the errors on the curl and gradient, with a slightly better rate for the fields themselves (going possibly towards O(h2)). The condition number for the scheme based on the ghost stabilisation remains at a reasonable magnitude, while for the unstabilized scheme (using the inner product (•,•)Ωinstead of (•,•)s) it blows up to a level where solving the linear system becomes infeasible, if not impossible. Acknowledgements Funded by the European Union (ERC Synergy, NEMESIS, project number 101115663). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. 15 2 × 10 2 3 × 10 2 4 × 10 2 6 × 10 2 h 10 1 100 101 L2( )-error h L2( ) h L2( ) curl h L2( ) grad h grad L2( ) O(h) 2 × 10 2 3 × 10 2 4 × 10 2 6 × 10 2 h 107 109 1011 1013 1015 1017 1-norm estimate condition number Condition number stabilized Condition number unstabilized O(h 3) Figure 6.1: Plot of the convergence of the mixed method (6.1) as a function of the mesh size h. The convergence is shown for polynomial order r = 1, and the 1-norm estimate of the condition number κs := ∥As∥op∥A -1 s ∥op of the system matrix As is also plotted on the right figure. Figure 6.2: Heat map of computed x-component of the solution ηh to the discrete Hodge Laplace equation (6.1) on the filled in torus. The mesh size is h = 0.019231 and the polynomial order is r = 1. 16 References [1] Cutfem/cutfem-library. URL https://github.com/CutFEM/CutFEM-Library. [Online; accessed 2025-09-22]. [2] tensorfan/cutfem-fork. URL https://github.com/tensorFan/CutFEM-fork. [Online; accessed 2025-09-22]. [3] D. N. Arnold. Finite Element Exterior Calculus. Society for Industrial and Applied Mathematics, 2018. [4] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus, homological techniques, and applications. Acta Numerica, 15:1-155, 2006. [5] D. N. Arnold, R. S. Falk, and R. Winther. Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society, 47(2):281-354, 2010. S0273-0979-10-01278-4. URL http://arxiv.org/abs/0906.4325. [6] F. Bonaldi, D. A. Di Pietro, J. Droniou, and K. Hu. An exterior calculus framework for polytopal methods. Journal of the European Mathematical Society, 2025. Published online. [7] A. Bossavit. A rationale for 'edge-elements' in 3-D fields computations. IEEE Transactions on Magnetics, 24(1):74-79, 1988. [8] A. Bossavit. Differential geometry for the student of numerical methods in electromagnetism. 01 1991. URL https://www.researchgate.net/publication/200018385_Differential_Geometry_ for_the_student_of_numerical_methods_in_Electromagnetism. [9] A. Bossavit. Generating Whitney forms of polynomial degree one and higher. IEEE Transactions on Magnetics, 38:341-344, 2002. [10] J. Brüning and M. Lesch. Hilbert complexes. J. Funct. Anal., 108(1):88-132, 1992. 10.1016/0022-1236(92)90147-B. [11] E. Burman. Ghost penalty. Comptes Rendus Mathematique, 348(21):1217-1220, 2010. org/10.1016/j.crma.2010.10.006. [12] E. Burman, S. Claus, P. Hansbo, M. G. Larson, and A. Massing. CutFEM: Discretizing geometry and partial differential equations. International Journal for Numerical Methods in Engineering, 104(7):472501, 2015. [13] E. Burman, P. Hansbo, M. G. Larson, and S. Zahedi. Cut finite element methods. Acta Numerica, 34: 1-121, 2025. [14] Z. Chen, K. Li, M. Lyu, and X. Xiang. A high order unfitted finite element method for time-harmonic Maxwell interface problems. International Journal of Numerical Analysis and Modeling, 21(6):822-849, 2024. [15] D. A. Di Pietro and J. Droniou. The Hybrid High-Order method for polytopal meshes, volume 19 of Modeling, Simulation and Application. Springer International Publishing, 2020. 978-3-030-37203-3. [16] D. A. Di Pietro, J. Droniou, M.-L. Hanot, and S. Pitassi. Uniform Poincaré inequalities for the discrete de Rham complex of differential forms, 1 2025. URL http://arxiv.org/abs/2501.16116. [17] T. Frachon, P. Hansbo, E. Nilsson, and S. Zahedi. A divergence preserving cut finite element method for Darcy flow. SIAM Journal on Scientific Computing, 46(3):A1793-A1820, 2024. 22M149702X. 17 [18] T. Frachon, E. Nilsson, and S. Zahedi. Stabilized Lagrange multipliers for Dirichlet boundary conditions in divergence preserving unfitted methods, 2024. URL http://arxiv.org/abs/2408.10089. [19] T. Frachon, E. Nilsson, and S. Zahedi. Divergence-free cut finite element methods for Stokes flow. BIT Numerical Mathematics, 64(4):39, 2024. [20] C. Gürkan and A. Massing. A stabilized cut discontinuous Galerkin framework for elliptic boundary value and interface problems. Computer Methods in Applied Mechanics and Engineering, 348:466-499, 2019. [21] R. Hiptmair. Finite elements in computational electromagnetism. Acta Numerica, 11:237-339, 2002. [22] R. Li, Q. Liu, and F. Yang. A reconstructed discontinuous approximation on unfitted meshes to H (curl) and H (div) interface problems. Computer Methods in Applied Mechanics and Engineering, 403:115723, 2023. [23] A. Massing, M. G. Larson, A. Logg, and M. E. Rognes. A stabilized Nitsche fictitious domain method for the Stokes problem. Journal of Scientific Computing, 61:604-628, 2014. s10915-014-9838-9. [24] J.-C. Nédélec. Mixed finite elements in R3. Numerische Mathematik, 35(3):315-341, 1980. BF01396415. [25] E. Nilsson. Generalized mixed finite element methods: cut elements and virtual elements. PhD thesis, KTH Royal 2024. [26] P. A. Raviart and J. M. Thomas. A mixed finite element method for 2nd order elliptic problems. In I. Galligani and E. Magenes, editors, Mathematical Aspects of the Finite Element Method. Springer, 1977. [27] N. Wang, H. Chu, J. Chen, and Y. Cai. Mixed Nitsche extended finite element method for solving threedimensional H (curl)-elliptic interface problems. Computers & Mathematics with Applications, 199: 22-44, 2025. [28] F. Yang and X. Xie. An unfitted finite element method with direct extension stabilization for timeharmonic Maxwell problems on smooth domains. Advances in Computational Mathematics, 50(3): 51, 2024. 18
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2510.14773
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Finding Answers in Thought Matters:
Revisiting Evaluation on Large Language Models with Reasoning
Hwiyeol Jo1, Joosung Lee1, Jaehong Lee1, Sang-Woo Lee2, Joonsuk Park3†, Kang Min Yoo1∗†
1NAVER Cloud, 2Neurofusion, 3University of Richmond
hwiyeolj@gmail.com,{rung.joo,jaehong.l,kangmin.yoo}@navercorp.com
sam@neurofusion.ai,park@joonsuk.org
Abstract
Evaluating generative models, such as large
language models (LLMs), commonly involves
question-answering tasks where the final an-
swer is selected based on probability of answer
choices. On the other hand, for models requir-
ing reasoning, the method of answer extrac-
tion plays a critical role. Our research reveals
that the performance of reasoning models and
their final answer distributions are highly sen-
sitive to the answer extraction algorithm em-
ployed. In order to mitigate this, we propose a
basic framework: Answer Regeneration. The
method uses an additional model inference, pro-
viding the prior input and output prefaced by
the prompt "Answer:". The final answer is then
selected or extracted from the regenerated out-
put. We show that this extraction-rule-agnostic
approach exhibits improved performance and
enhanced robustness. Furthermore, we have ap-
plied this framework to general math problems
and open-ended question answering tasks. Our
analysis and this framework could offer a more
reliable results for model evaluation.
1
Introduction
The conventional approach for generating answers
from large language models (LLMs) involves se-
lecting the answer choice with the highest proba-
bility when conditioned on the input prompt and
each choice following a specific prefix, such as
"Answer:" (Hendrycks et al. (2021); Liang et al.
(2023); OpenCompass Contributors (2023); Habib
et al. (2023); inter alia). For tasks without answer
choices, prior work has relied on rule-based extrac-
tion (e.g., searching for "Answer: X" or "answer
is X"), model judges for semantic similarity, or hu-
man evaluation (Kamalloo et al. (2023); Wei et al.
(2024); Chandak et al. (2025); Chen et al. (2025);
inter alia). However, reasoning-powered LLMs
∗Now at Amazon
† Co-corresponding author
Reasoning Model Output Example
[college_computer_science]· · · </think>
· · · 2**Correct Answer: (D) I, II, and III**"
[college_chemistry]· · · </think>· · ·
###Final Answer:\n\n$$\n\\boxed{B}\n$$}
[management]· · · Therefore, the answer is
Merton. \n\nBut I’m still not 100% sure. · · ·
[psychology]· · · Hmm. I’m going to go
with the fornix as the answer. Because · · ·
Figure 1: Examples illustrating the difficulties in extract-
ing final answers from reasoning models’ outputs. Al-
though the benchmark is designed with multiple-choice
questions, models frequently generate answers in a free-
text format, which complicates automated evaluation.
need to output their reasoning process (Chain-of-
Thought (CoT)) (Wei et al., 2022) to leverage their
full potential. This detailed, linguistically diverse
output complicates traditional evaluation. Specifi-
cally, it prevents the use of methods based on the
probability of specific answer choices and limits
the applicability of most LLM-as-a-judge (Zheng
et al., 2023) evaluations. This shift introduces a
new, critical challenge: how to reliably find the
answer from the detailed output that includes all
the reasoning steps matters.
However, the rule-based approach suffers from
a fundamental flaw: heuristic rules cannot account
for all possible answer formats. Figure 1 illustrates
examples from multiple-choice question answer-
ing benchmark MMLU (Hendrycks et al., 2021).
A single model can use different formats in its re-
sponses, sometimes boxing the answer in brackets
(i.e., \boxed{}) or answering the option text in vari-
ous formats (e.g., "Merton", "fornix") instead of the
option label (e.g., "(D)"). Furthermore, the formats
can vary significantly between different models and
even across different types of benchmarks, such as
arXiv:2510.14773v1 [cs.CL] 16 Oct 2025
multiple-choice, math, and open-ended questions.
This means that optimal extraction rules need to
be created and tuned for every individual model
and benchmark (e.g., rules for options, numbers,
or word(s)), which makes the process difficult and
even affects the reproducibility of model results.
In this paper, we first empirically demonstrate
the impact of answer extraction rules on reasoning-
powered model (Section 4). We then introduce
Answer Regeneration, a simple, generation-based
framework designed to alleviate the dependency on
specific answer extraction rules (Section 5). Instead
of relying on complex extraction rules, our method
utilizes an additional inference step to prompt the
model to regenerate its final answer. It allows us
to use probability-based answering for choices or
extract the answer from a simplified output.
Our experiments reveal that model performances
are highly sensitive to the extraction rules em-
ployed. Depending on the rules, distinct answers—
no answers at all in some cases—may be extracted
from the same LLM response. On the other hand,
Answer Regeneration consistently outperforms the
handcrafted rule-based extractions, improving both
in benchmark score and human evaluation results.
Our method also achieves intuitive model rank-
ings, where larger models are shown to outperform
smaller ones. We demonstrate that Answer Regen-
eration significantly reduces the dependency on
specific answer extraction rules, thereby improv-
ing robustness and reproducibility of model eval-
uations. Furthermore, we apply our framework to
diverse tasks, including complex multiple-choice
question answering, short-answer math problems,
and open-ended question answering. In all cases,
our generation-based method proves to be a plausi-
ble and effective approach for the fair evaluation of
reasoning models.
Our contributions in this work are as follows:
• We empirically investigate the sensitivity of
reasoning-powered LLMs to rule-based an-
swering, revealing a strong dependency on
the choice of answer extraction algorithm.
• We propose the generation-based framework
Answer Regeneration. It achieves (1) supe-
rior performance compared with handcrafted
rules, (2) intuitive model rankings, and (3) sig-
nificantly enhanced robustness against answer
inconsistency and incomplete outputs.
• We demonstrate the generalizability and effec-
tiveness of our framework across diverse tasks,
confirming its plausibility for more robust and
fair model evaluations.
2
Related Work
A growing body of work shows that LLM perfor-
mance can vary drastically with small changes in
prompt format, even when the underlying seman-
tics are equivalent (Sclar et al., 2024; He et al.,
2024; Alzahrani et al., 2024). Consequently, Polo
et al. (2024); Mizrahi et al. (2024) proposed the
methods to mitigate the effect of prompt variations.
While the previous research focused on input-level
prompt variations and their impact on model evalu-
ation, we focus on output-level final answer varia-
tions from reasoning LLMs, which are caused by
the selection of answer extraction algorithms.
Therefore, it is noteworthy to find out how re-
cent LLM evaluations handle outputs from reason-
ing models. A number of open evaluation frame-
works typically support (1) probability–based an-
swering for multiple-choice tasks or (2) simple
heuristic post-processing for free-form generations,
involving only de-capitalization or blank-space
normalization.
Details on the implementations
of MMLU Hendrycks (Hendrycks et al., 2021),
HELM (Liang et al., 2023), OpenCompass (Open-
Compass Contributors, 2023), and lighteval (Habib
et al., 2023) can be found in the Appendix A.1.
lm-evaluation-harness (Biderman et al., 2024)
has become the de facto community standard for
reproducible LLM evaluation. Generative tasks
use string-match with optional regular expressions
or rule-based normalizers. While recent templates
support CoT prompting, the final answer is still
recovered via simple patterns (e.g., "Answer: X"),
or a last-capital-letter heuristic. As we will demon-
strate, such extraction rules can swing scores and
even reorder model rankings.
To the best of our knowledge, this is the first
study to highlight the importance of answer ex-
traction methods, especially for reasoning LLMs.
Based on our findings, we introduce a lightweight
method to reduce the reliance on the fragile extrac-
tion rules and provides a more faithful evaluation
of reasoning models’ abilities.
3
Experiment Setup
The experiments are designed to highlight current
problems associated with finding answers in reason-
ing models’ output (Study 1 in Section 4) and then
assess the validity of introduced method Answer
Figure 2: Model performance in accuracy evaluated using various answer extraction algorithm. Responses are
considered incorrect if the extraction process fails to find an answer.
Regeneration (Study 2 in Section 5).
We utilize lm-evaluation-harness toolkit for its
simplicity in customizing the post-processing rules.
MMLU (Hendrycks et al., 2021) benchmark is
primarily used, given its widely adoption for eval-
uating LLMs’ knowledge1. The multiple-choice
format of MMLU serves as a foundational task
that simplifies the answer extraction process for
our initial analysis. We then extend our evalua-
tion to more complex tasks MMLU-Pro (Wang
et al., 2024), the mathematical reasoning bench-
mark GSM8K (Cobbe et al., 2021) and the open-
ended question answering TriviaQA (Joshi et al.,
2017) in Section 6.
We evaluate several open-source reasoning
models: Qwen3 families–Qwen3-32B, Qwen3-
14B, Qwen3-8B (Yang et al., 2025), along with
Deepseek-R1-Distill-Llama-8B (referred to as
R1-Llama-8B), and DeepSeek-R1-0528-Qwen3-
8B (referred to as R1-Qwen3-8B) (DeepSeek-AI,
2025). For hyperparameter settings, we adhere to
recommended best practices for each model, set-
ting temperature to 0.6, top-p value to 0.95, and
top-k value to 20. Prompt templates are sourced
from lm-evaluation-harness, using thinking tem-
plates. To reduce computational costs, the maxi-
mum token generation length is limited to 4,096.
4
Study 1: Rule-based Answer Extraction
4.1
Methods
We evaluate 5 reasoning models using 5 differ-
ent answer extraction methods to investigate how
performance changes with extraction algorithms:
strict-match, flexible-extract, instructed-format,
answer-is-correct, and last-extract:
strict-match and flexible-extract are adapted
from lm-evaluation-harness. strict-match extracts
1We select the original MMLU to better analyze how
models handle ambiguous questions, rather than the cleaned
MMLU-Redux (Gema et al., 2025).
a precise string such as "answer is X" or "Answer:
X" and flexible-extract finds multiple-choice op-
tions like (A), (B), (C), or (D), located near the
end of the text. This is a common and effective ap-
proach, as the final conclusion typically follows the
reasoning. However, the original implementation
has tendency to extract the last capital character
from any text, which can lead to errors.
instructed-format requires modifying the input
prompt to guide the model’s output format. As
recommended in Qwen3 technical report, we add
a specific instruction to the prompt: "Please show
your choice in the answer field with only the choice
letter, e.g., "answer": "C"." Rules are then imple-
mented to extract the answer from this specified
format. While the method is strict (deviation is
generally considered incorrect).
Further heuristically optimized answer extrac-
tion methods are used: We build upon strict-match
by creating answer-is-correct, which includes vari-
ations like "X is the answer" or "X could be the
correct answer", addressing the limited scope. We
also refine flexible-extract by developing last-
extract, which specifically targets the last single
capital character that appears in the output. All
the implementations accounts for minor variations
like "**X**", "**Answer:** X" and are designed
to select the last match to accommodate potential
self-correction within the model’s response.
Our objective is not to declare any one extraction
method superior. Instead, we aim to demonstrate
the range of performance that can be achieved using
widely-used, well-optimized, heuristic extraction
algorithms on a given task. Consequently, this
research shows that benchmark performance is not
solely dependent on the reasoning model’s ability
but is significantly influenced by the chosen answer
extraction rules. The exact regular expressions used
are described in the Appendix A.2.
Answer Inconsistency Example
The following are multiple choice questions (with answers) about high school physics.
Q: The plates of a capacitor are charged to a potential difference of 5 V. If the capacitance is 2 mF, what is the charge on
the positive plate?
(A) 0.005 C (B) 0.01 C (C) 0.02 C (D) 0.5 C
<think>
Okay, let’s see. The question is about finding the charge on the positive plate of a capacitor. · · · the charges on the two
plates are equal in magnitude but opposite in sign. So the positive plate has +Q and the negative has -Q. But the question
is just asking for the magnitude, so the answer is still 0.01 C. Therefore, option answer-is-correct: B is correct.
</think>
To determine the charge on the positive plate of a capacitor, · · · Since the question asks for the **magnitude** of the
charge on the **positive** plate, the correct answer is: $$ instructed-format: \boxed{0.01 \, \text{C}} $$
—
### Final Answer:
** flexible-match: (B) 0.01 last-extract: C **
Figure 3: An example from Qwen3-32B output illustrating how the final answer can vary significantly depending
on the extraction method used. The graphical boxes and bold text highlight the specific text extracted by each
algorithm.
Figure 4: Distribution of extracted final answers across different extraction algorithms. The y-axis represents the
answer extraction method, and the x-axis shows the extracted final answer, with "N/A" denoting cases where no
answer could be extracted.
4.2
Result
4.2.1
Model Performance
Figure 2 illustrates how different answer extraction
methods affect the performance of models. We
evaluate performance using 3 types of rules: im-
plemented (strict-match, flexible-extract), rec-
ommended (instructed-format), and heuristically
optimized (answer-is-correct, last-extract). If
an extraction rule fails to find an answer, the re-
sponse is considered incorrect. The results reveal
that model performance fluctuates significantly de-
pending on the extraction method used.
With strict-match, the rankings of model per-
formances are Qwen3-8B, Qwen3-32B, Qwen3-
14B, R1-Qwen3-8B, and R1-Llama-8B in order.
The more optimized answer-is-correct, derived
from strict-match, significantly improves the per-
formance of all models.
This shifts the rank-
ing to Qwen3-14B, Qwen3-32B, Qwen3-8B, R1-
Llama-8B, and R1-Qwen3-8B. A similar sensitiv-
ity is observed with the other methods. Using
flexible-extract, the top models are Qwen-8B,
Qwen3-14B, Qwen-32B, R1-Llama-8B, and R1-
Qwen3-8B. With last-extract, Qwen3-14B per-
forms the best, and R1-Qwen3-8B outperforms R1-
Llama-8B compared with flexible-extract. In-
terestingly, despite following the recommended
best practices for multiple-choice question answer-
ing with instructed-format, the performance of
Qwen3 family models are not impressive compared
to other extraction methods. This method proves to
be particularly ineffective for R1-Llama-8B model.
These findings challenge the common assump-
tion that larger models outperform smaller ones
within the same family. Our analysis indicates that
the benchmark performance scores of reasoning
models are highly dependent on the answer extrac-
tion method used. This suggests that the discrepan-
cies between publicly reported and reproduced per-
formance scores may be due to differences not only
in prompt inputs, but also in the specific answer
extraction methods, which are not fully disclosed.
4.2.2
Answer Inconsistency
Figure 3 provides a clear example of how differ-
ent extraction methods handle the same model
output, illustrating the problem of answer in-
consistency. In this example, strict-match fails.
answer-is-correct successfully locates an answer
within the model’s thought, between <think> and
</think> tags. However, the model’s explicitly for-
matted final answer, "the correct answer is: X", is
not recognized as a valid because it contains unex-
Figure 5: The proposed Answer Regeneration framework for finding answers in model output. The yellow box
indicates the conventional method of direct extraction, while the blue box indicates the proposed framework.
Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3
(%)
2.8
2.9
6.2
6.7
6.8
best-extr ans-is
ans-is
last
flexible ans-is
Correct
37.1
33.8
42.1
26.6
25.5
Incorrect 32.2
22.6
53.6
65.7
19.2
Invalid
30.7
43.6
4.4
7.8
55.3
Table 1: The percentage of incomplete thinking and the
corresponding accuracy of each reasoning model. (%)
refers to the portion of outputs where model’s thinking
process is not completed.
pected patterns, including $$, \boxed{}, and \text{}.
Besides, the output is the option text rather than
the required option label. instructed-format could
find an answer using \boxed{} (despite the \boxed{}
format being recommended only for math prob-
lems), but the extracted answer is again the option
text, not the label. Furthermore, the presence of
the unexpected LaTeX command \text{} could re-
sult in an incorrect evaluation during string-match
comparison. Meanwhile, flexible-match correctly
identifies the final answer. Interestingly, the sim-
ple yet effective last-extract extracts the unit of
option text "C" as the final answer.
Figure 4 further illustrates this issue by showing
how the distribution of extracted answers changes
depending on the extraction method used.
We
observe that the distribution of extracted answers
varies significantly. This highlights the crucial role
of the extraction method in determining model’s
final performance, suggesting that the choice of
method can introduce bias into the evaluation.
4.2.3
Answering for Incomplete Thinking
Another challenge in extracting answers from rea-
soning models is the issue of incomplete reasoning
(or thinking). Even when we set the maximum gen-
eration length to 4,096 tokens, we find that some
model outputs lack the </think> token, indicating
that the thinking process had not concluded. Table 1
reports the percentage of outputs in this category.
Fortunately, this is a relatively small portion of the
total outputs and is primarily caused by repetitions
during the model’s generation.
We then select the best answer extraction method
for each model and measure the correctness of the
final answers derived from these incomplete out-
puts. Except for Qwen3-8B and R1-Llama-8B,
which use extraction algorithms solely on capital
letters, the results using answer-is-correct show
a high rate of invalid extraction.
This implies
that even well-optimized extraction method can
be less robust toward incomplete thinking, partic-
ularly when the reasoning output does not contain
definitive, explicitly formatted answering text.
5
Study 2: Answer Generation
Our analysis has shown that the final answer of
reasoning models is highly sensitive to the chosen
extraction method. Model performance fluctuates
significantly based on how the answer is located
and selected from the output. To address this and
simplify the optimization of complex extraction al-
gorithms, we propose a straightforward framework
for reliably identifying the final answer.
5.1
Method
Our proposed framework, illustrated in Figure 5,
tackles the challenge by introducing Answer Re-
generation step. Instead of attempting to parse a
final answer from model’s extensive thought, our
method uses an additional inference call. Specifi-
cally, we provide the model (in its non-reasoning
mode) with the original input prompt and its previ-
ous output (the reasoning process), and a new prefix
"Answer:". This prompts the model to generate a
concise, final answer based on its prior reasoning.
This approach offers key benefits. For multiple-
choice questions, it allows us to utilize probability-
based answering, as non-reasoning models have
been evaluated, leading to more robust predictions.
When the answer choices are not available, such
as open-ended question answering, it simplifies
the model’s output, making the final answer much
easier to extract with straightforward algorithms.
Figure 6: (left) A confusion matrix comparing the conventional answer extraction method (Rule) and the proposed
method (Regen). (right) The accuracy of answers extracted from the model’s thought, as determined by human
evaluation. We sample 300 instances when the extraction and regeneration are disagreed. Results are not reported
for cases where the model failed to provide a definitive answer or provided multiple option labels.
Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3
Rule(Best) 82.1
83.8
82.1
64.8
77.6
AnsRegen
87.1
85.0
83.3
68.8
80.7
Diff
+5.0
+1.2
+1.2
+4.0
+3.1
Table 2: Performance comparison between conventional
answer extraction and Answer Regeneration. We report
each model’s performance using its best-performing
extraction method.
While effective, our framework has several ac-
knowledged limitations. The primary issue is the
computational cost of the additional inference step.
Additionally, the method might not fully capture
minor variations in answer formatting, e.g., the
probability of "**A**". Finally, some regenerated
results could be different from explicitly mentioned
answer. Despite these weaknesses and the lack of
technical novelty, we believe this framework’s sim-
plicity and the clarity constitute significant contri-
bution. We will demonstrate its benefits using the
same experimental setup as our previous analyses.
5.2
Result
5.2.1
Improved Performance
As presented in Table 2, the proposed method con-
sistently reports better scores than rule-based an-
swering. Figure 6 (left) provides a detailed look at
the performance. While most of the final answers
derived by both our method and the rule-based
methods are the same, our framework achieves a
much higher correction rate. This demonstrates
Answer Regeneration is successful at correcting in-
correct answers extracted by rule-based approach.
To compute the correction rate, we select 300
instances from the outputs of Qwen3-32B, Qwen3-
8B, and R1-Llama-8B where the extraction and
regeneration results disagreed. We then manually
label the correct "gold" answers in terms of answer
extraction from the thoughts. As shown in Figure 6
(right), the agreement rate of Answer Regenera-
tion with the human label is far superior to that of
the conventional answer extraction methods.
Figure 7: Model performance evaluated on outputs
where the reasoning process is incomplete, using the
optimal answer extraction algorithm for each model.
5.2.2
Correlation with Model Size
An interesting effect of our framework is the change
in the performance ranking of Qwen3 models. The
previous ranking derived from rule-based answer-
ing, which was Qwen3-14B, Qwen3-32B, Qwen3-
8B, shifted to Qwen3-32B, Qwen3-14B, Qwen3-
8B under our framework. This new ranking aligns
with conventional intuition and general knowledge
that larger models typically outperform smaller
ones within the same family. This suggests that
the initial, counterintuitive ranking is likely an ar-
tifact of the answer extraction methods, not a true
reflection of the models’ underlying capabilities.
5.2.3
Enhanced Robustness to Responses
The nature of our proposed Answer Regeneration
framework inherently addresses the issue of answer
inconsistency mentioned in Section 4.2.2. Since
it prompts the model to generate a final, definitive
answer, it bypasses the unpredictable results associ-
ated with various rule-based extraction algorithms.
Additionally, our method improves robustness
by handling internal self-correction within model
outputs.
When facing ambiguous questions, a
model may initially provide an answer and then
continue its thinking process, generating alternative
solutions or re-evaluating its answer. Rule-based
answer extraction methods struggle to choose the
final answer from this internal debate. In contrast,
our framework considers the entire thinking pro-
Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3
strict-match 15.3
13.0
15.7
6.8
10.9
flexible-ext 47.2
47.1
47.1
38.0
41.3
instructed
52.6
59.5
45.8
38.7
49.7
ans-is-corr
68.4
65.2
64.2
37.6
53.5
last-extract 66.8
63.4
62.0
42.2
45.3
implemented
72.1
69.4
64.6
43.3
58.3
AnsRegen
77.0
72.6
72.0
43.6
66.4
Reported2
79.8
77.4
74.3
54.3
73.9
▷Reproduced 63.0
59.2
57.3
42.3
40.7
Table 3: Model performance on MMLU-Pro. The evalu-
ation utilizes the same answer extraction algorithms
used in our MMLU analysis, including the built-in
algorithm from lm-evaluation-harness, referred to as
implemented.
cess and forces the model to finalize its response,
leading to a more reliable result.
A further key advantage is its ability to handle
"NOT correct" questions. Since many extraction al-
gorithms are designed to find the "correct" answer
from the reasoning text, they fail when the ques-
tion requires identifying the incorrect choice. The
algorithm may mistakenly extract a correct option
discussed during the model’s rumination.
Finally, our method significantly improves per-
formance in cases of incomplete thinking, as shown
in Figure 7. Instead of relying on rules to parse an
incomplete output, our framework can select the fi-
nal answer even when the thought does not include
an explicit final answer.
5.2.4
Regenerator Independency
Our method, which uses an additional model infer-
ence for Answer Regeneration, raises a question
about its dependency on the specific model used.
Table 7 in the Appendix shows that the perfor-
mances achieved using small-sized regenerators
are generally similar to the performance achieved
when using the same model both for reasoning and
Answer Regeneration. While this suggests a de-
gree of independence, we still recommend using
the same model for both tasks. The final answering
step is also a crucial part of the overall model evalu-
ation and should be performed by the model being
assessed to ensure a consistent and fair comparison.
Based on the results, Answer Regeneration
framework shows a more effective and reliable
method for evaluating reasoning models. Conven-
tional extraction rules cannot account for all the
variations in model outputs, and can thus introduce
biases and inaccuracies. Our framework mitigates
this problem, providing a more accurate and con-
sistent measure of models’ true performance.
(↓) Extraction
Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3
strict-match 3.3
2.7
1.7
0.0
0.1
flexible-ext 33.3
33.5
19.3
69.2
85.1
instructed
93.5
92.2
88.6
54.8
85.8
ans-is-corr
89.6
87.6
91.9
63.1
83.4
AnsRegen
95.0
93.8
91.1
76.0
91.1
Table 4: Model performance on GSM8K. Note that
strict-match and flexible-extract are implemented
in lm-evaluation-harness. last-extract is not useful.
6
Studies on Additional Tasks
6.1
Complex Multiple-Choice Question
Answering
As an extension of our previous findings, we inves-
tigate our framework on MMLU-Pro (Wang et al.,
2024), a more complex benchmark with a dynamic
number of answer options. The result, shown in
Table 3, demonstrates that while the built-in ex-
traction algorithm from lm-evaluation-harness per-
forms better than algorithms optimized only for the
original MMLU, Answer Regeneration—which
is not specifically tuned for any benchmark—still
achieves superior performance. Furthermore, the
scores are also closer to the publicly reported per-
formance2, despite the reported scores likely bene-
fiting from more specific prompt engineering (e.g.,
detailed task descriptions for individual subtasks),
as demonstrated in our reproduced score using
their extraction rules. Therefore, we believe that
evaluating models with our framework provides a
fairer and more robust assessment of true capabil-
ities, achieving competitive performance without
the need for task-specific optimization.
6.2
Short-Answer Math Problems
We explore the effectiveness of our framework in
math domain using GSM8K benchmark (Cobbe
et al., 2021), which features structured (as numbers)
but relatively open-ended question answering task.
As shown in Table 4, instructed-format , a
template specifically recommended for mathe-
matical problems, performs the best among the
various extraction methods.
We also modify
answer-is-correct to better handle common math-
ematical formatting, such as numbers and symbols
like $, ",", and ".". Despite these optimizations,
Answer Regeneration with minor post-processing
to remove LaTeX commands, such as \boxed{} or
\text{}, achieves the highest performance.
2https://artificialanalysis.ai/evaluations/
mmlu-pro. Qwen3 technical report does not contain zero-shot
CoT results for MMLU-Pro; it only provides 5-shot results
without reasoning, scoring 65.54 for 32B and 56.73 for 8B.
Method
(↓) Evaluator Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3
String
Match
ans-is-corr-
42.7
47.5
44.2
11.7
35.6
AnsRegen
-
55.3∗
53.7
47.0
24.1
55.8
Model
-based
GPT
Grader
Qwen3-32B
3.1∗
3.8
3.6
1.3
2.9
Qwen3-14B
49.5
56.4
49.2
19.4
41.1
Qwen3-8B
49.4
56.3
49.4
18.2
41.3
R1-Llama-8B
93.9
92.3
89.4
93.1∗
87.5
R1-Qwen3-8B
47.6
54.8
47.9
17.7
39.6
xVerify
xVerify-8B-I 0.0
0.0
0.0
47.7∗
0.0
Table 5: (left) Performance of reasoning models on open-ended question answering TriviaQA. (right) Confusion
matrix illustrating human evaluation performance on 100 samples in determining semantic equivalence between the
generated answer and the gold answer. ∗denotes the selected results for the detailed human evaluation.
To further validate this, we conduct a human
evaluation of instances where the methods’ results
disagreed. Answer Regeneration framework re-
ports 16.3% correct, while the conventional answer
extraction method is correct in only 6.1% of the
cases. This underscores the superior reliability of
our framework even in complex, structured but
open-ended domains like mathematics.
6.3
Open-ended Question Answering
Evaluating generative models on open-ended
question-answering tasks presents two main chal-
lenges: (1) finding the answer within the model’s
output. (2) determining semantic equivalence be-
tween the generated answer and the gold answer.
To alleviate the second challenge, we use Trivi-
aQA (Joshi et al., 2017), known for its extensive
gold answer variations and aliases, minimizing the
need for complex semantic matching.
As shown in Table 5 (left), Answer Regenera-
tion consistently outperforms direct answer extrac-
tion from the reasoning output. We also compare
our framework with two other LLM-as-a-judge ap-
proaches (Zheng et al., 2023): GPTGrader (Wei
et al., 2024), which uses an additional inference call
with long prompts to categorize semantic similarity
into "correct" (same), "incorrect" (not same), and
"invalid". xVerify (Chen et al., 2025), a fine-tuned
model that evaluates semantic equivalence as "cor-
rect" or "incorrect". While the scores from these
model-based evaluations might appear better, they
carry a critical drawback of model bias. Table 5
(right) presents a human evaluation of semantic
equivalence, comparing the model judgement with
human judgments on 100 sampled outputs. Qwen3-
32B consistently predicts "incorrect" even when
the answer is correct and xVerify similarly defaults
to "incorrect", and R1-Llama-8B exhibits a bias
toward "correct". In contrast, our string-match-
based method avoids this model bias and provides
a more accurate performance measure, despite of
its limitation in determining semantic equivalence.
7
Discussion and Conclusion
Our analysis highlights a critical, yet often over-
looked, challenge in evaluating reasoning models:
the profound impact of the answer extraction meth-
ods on performance scores. We have demonstrated
that model performances can fluctuate significantly
based on how the final answer is parsed from its
reasoning output. This finding suggests that dis-
crepancies between publicly reported scores and
reproduced results may stem from undocumented
differences not just in prompts, but in the extrac-
tion methods itself. To address this issue, we in-
troduced Answer Regeneration framework. Our
simple approach offers significant advantages over
conventional extraction rules:
Without specific tuning, the framework consis-
tently achieved superior scores across a variety of
tasks, from multiple-choice question answering to
short-answer math problems and open-ended ques-
tion answering tasks. By prompting the model to
explicitly state its final answer again, we mitigate
inconsistencies caused by diverse output formats.
Beyond exhibiting better scores than hand-
crafted, optimized rules, the performance ranking
derived from our framework for the Qwen3 model
family aligned with the conventional intuition that
larger models generally outperform smaller ones.
This suggests that the framework provides a more
accurate reflection of a model’s true capabilities,
free from the biases of model-specialized answer
extraction rules.
Our method also proves more resilient to com-
mon failure of rule-based approach. It success-
fully handles outputs involving incomplete think-
ing, models that re-consider their answers, and
questions asking for the "incorrect" choice, all of
which can confuse rule-based extraction.
Lastly, while LLM-as-a-judge method can suffer
from inherent model biases (e.g., consistently pre-
dicting, "correct" or "incorrect"), our string-match
method, enabled by the concise regenerated output,
provides a more reliable measure of performance.
In conclusion, through our findings from analy-
sis and the introduction of Answer Regeneration
framework, we believe this work contributes to-
ward more reliable and faithful model evaluation
for all reasoning-powered LLMs.
8
Limitations
Technical Novelty in Answer Regeneration
We
acknowledge that Answer Regeneration frame-
work itself lacks technical novelty. However, we
contend that the value of our contribution lies in the
simplicity and the clarity of the results and analysis
it provides. Our work demonstrates the benefits
of using this framework as a robust and reliable
reference for evaluating and fairly comparing the
performance of reasoning models.
Experiments with Sophisticated Extraction
Rules
Our experiments adopted established an-
swer extraction rules from lm-evaluation-harness
(strict-match, flexible-match).
Building upon
these, we developed more complex, heuristic
rules (answer-is-correct, last-extract) and in-
cluded the recommended rule for Qwen3 fami-
lies (instructed-format). While we recognize that
more aggressively optimized, domain-specific rules
could exist, we maintain that such highly-specified
rules will still fail to handle the full spectrum of
answer variations.
Experiments with Diverse LLMs and Prompts
Our focus was on output-level results, which means
that the effect of different input prompts seem to
be overlooked. Furthermore, our investigation was
limited to publicly available open-source reason-
ing models. Although greater diversity in models
and prompts would enhance generalizability, we
believe that the widely-used models and default
prompts from established repositories provide suf-
ficiently general results for our findings. We defer
the investigation of commercial LLMs, such as
ChatGPT, Gemini, and Claude, to future work. As
a minor note, we observed that small variations in
the input prompts (e.g., changes of option labels
or the "Answer:" prefix) do not significantly affect
performance.
Inherent Weakness of Answer Regeneration
As discussed in Section 5.1, Answer Regenera-
tion carries inherent limitations. Nonetheless, we
believe that employing the simplest possible frame-
work was the most effective way to demonstrate the
core benefits of our approach. Exploring further
techniques within this framework, such as incorpo-
rating concepts like self-consistency (Wang et al.,
2022), represents a valuable direction for future
research.
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A
Appendix
A.1
Evaluation Toolkits
MMLU Hendrycks (Hendrycks et al., 2021) and
follow-ups such as MMLU-Pro (Wang et al., 2024)
are deeply integrated into most of toolkits, but the
original implementation only supports probability
based answering for multiple choice question an-
swering. HELM (Holistic Evaluation of Language
Models; Liang et al. (2023)) simply use Quasi-
exact match that post-process the model genera-
tion, such as lower-casing, removing whitespace
and punctuation and articles. Also, OpenCom-
pass (OpenCompass Contributors, 2023) supports
both option-likelihood scoring and post-processing
option (but provided with blank) to be customized
for reasoning outputs, its metrics mainly rely on
model-based scoring. Similarly, lighteval (Habib
et al., 2023) has metrics for generated outputs, but
there is only a scoring function, not mentioning
about post-processing.
A.2
Regular Expressions used in the
Experiments
Note that () makes groups in regular expression and
\\is required both for meta characters and escape
sequence in lm-evaluation-harness.
• strict-match:
((?<=The
answer
is
)(.*)(?=.)|(?<=answer is )(.*)(?=.)|(?<=The
answer:
)(.*)(?=.)|(?<=The final answer:
)(.*)(?=.))
• flexible-extract: (\\([A-D]\\))
• instructed-format:[Aa]nswer\"?:\\s*
\"?\\(?([A-D])\"|\"?\\**(?([A-D])\"
• answer-is-correct:
\\**[Aa]nswer:\\**\\s*(\\(?[A-
D]\\)?)|\\**[Aa]nswer\\**:\\s*(\\(?[A-
D]\\)?)|[Aa]nswer
is
\\**(\\(?[A-
D]\\)?)\\**|[Aa]nswer should be \\**(\\(?[A-
D]\\)?)\\**|[Aa]nswer:\\s+\\**(\\(?[A-
D]\\)?)\\**|correct
answer
is
\\**(\\(?[A-
D]\\)?)\\**|correct
answer:\\s+\\**(\\(?[A-
D]\\)?)\\**|\\**(\\(?[A-D]\\)?)\\** is correct|
*(\\(?[A-D]\\)?)\\** is the correct|\\**(\\(?[A-
D]\\)?)\\**
is
the
answer|\\**(\\(?[A-
D]\\)?)\\** should be the answer
• last-extract: [^a-zA-Z0-9]([A-D])[^a-zA-
Z0-9]
A.3
Preliminary: Non-reasoning vs.
Reasoning
We demonstrate the power of reasoning in solving
MMLU, as presented in Table 6. The performance
shows that the reasoning significantly improves the
model performances. This encourage us to use
reasoning model not only for complex problem.
but for knowledge-based problems.
A.4
Regnerator Independency
Table 7 reports the performance when using differ-
ent models from the answer generator. Although
we use smaller models for regenerator, the perfor-
mance is similar when using the identical model.
Qwen3-32B
Qwen3-14B
Qwen3-8B
R1-Llama-8B
R1-Qwen3-8B
non-Reason
78.4
75.7
72.2
53.0
66.2
Reason
82.1
83.8
82.1
64.8
77.6
Diff
+3.7
+8.1
+9.9
+11.8
+11.4
Table 6: The performance comparison when using non-reasoning mode and reasoning mode in LLMs. Non-
reasoning mode follows conventional loglikelihood measurements using candidate whereas reasoning mode uses
answer extraction algorithms to find the final answer in the reasoning output. The best performance with answer
extraction methods are reported.
(↓) Regenerator
Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama-8B R1-Qwen3-8B
gemma-3-1b-it
86.9
84.9
82.5
67.5
80.3
llama-3.2-1b-it 86.5
84.4
81.8
67.8
79.6
Qwen3-0.6b
86.3
84.4
82.3
68.9
79.9
Qwen3-32B
87.1
85.2
83.7
72.1
82.6
Qwen3-14B
87.1
85.0
83.1
71.2
81.6
Qwen3-8B
87.4
85.2
83.3
72.5
82.0
R1-Llama-8B
87.0
84.9
82.6
68.8
80.2
R1-Qwen3-8B
84.2
81.0
81.1
70.9
80.7
Table 7: Model performance when different models are used for Answer Regeneration step. Bold indicates the
reported score when the reasoning models and the regenerators are the same.
|
Finding Answers in Thought Matters: Revisiting Evaluation on Large Language Models with Reasoning Hwiyeol Jo1, Joosung Lee1, Jaehong Lee1, Sang-Woo Lee2, Joonsuk Park3†, Kang Min Yoo1∗† 1NAVER Cloud, 2Neurofusion, 3 (LLMs), commonly involves question-answering tasks where the final answer is selected based on probability of answer choices. On the other hand, for models requiring reasoning, the method of answer extraction plays a critical role. Our research reveals that the performance of reasoning models and their final answer distributions are highly sensitive to the answer extraction algorithm employed. In order to mitigate this, we propose a basic framework: Answer Regeneration. The method uses an additional model inference, providing the prior input and output prefaced by the prompt "Answer:". The final answer is then selected or extracted from the regenerated output. We show that this extraction-rule-agnostic approach exhibits improved performance and enhanced robustness. Furthermore, we have applied this framework to general math problems and open-ended question answering tasks. Our analysis and this framework could offer a more reliable results for model evaluation. 1 Introduction The conventional approach for generating answers from large language models (LLMs) involves selecting the answer choice with the highest probability when conditioned on the input prompt and each choice following a specific prefix, such as "Answer:" (Hendrycks et al. (2021); Liang et al. (2023); OpenCompass Contributors (2023); Habib et al. (2023); inter alia). For tasks without answer choices, prior work has relied on rule-based extraction (e.g., searching for "Answer: X" or "answer is X"), model judges for semantic similarity, or human evaluation (Kamalloo et al. (2023); Wei et al. (2024); Chandak et al. (2025); Chen et al. (2025); inter alia). However, reasoning-powered LLMs ∗Now at Amazon † Co-corresponding author Reasoning Model Output Example [college_computer_science]· · · · · · 2 **Correct Answer: (D) I, II, and III**" [college_chemistry]· · · · · · ###Final Answer: } [management]· · · Therefore, the answer is Merton. I'm still not 100% sure. · · · [psychology]· · · Hmm. I'm going to go with the fornix as the answer. Because · · · Figure 1: Examples illustrating the difficulties in extracting final answers from reasoning models' outputs. Although the benchmark is designed with multiple-choice questions, models frequently generate answers in a freetext format, which complicates automated evaluation. need to output their reasoning process (Chain-ofThought (CoT)) (Wei et al., 2022) to leverage their full potential. This detailed, linguistically diverse output complicates traditional evaluation. Specifically, it prevents the use of methods based on the probability of specific answer choices and limits the applicability of most LLM-as-a-judge (Zheng et al., 2023) evaluations. This shift introduces a new, critical challenge: how to reliably find the answer from the detailed output that includes all the reasoning steps matters. However, the rule-based approach suffers from a fundamental flaw: heuristic rules cannot account for all possible answer formats. Figure 1 illustrates examples from multiple-choice question answering benchmark MMLU (Hendrycks et al., 2021). A single model can use different formats in its responses, sometimes boxing the answer in brackets (i.e., ) or answering the option text in various formats (e.g., "Merton", "fornix") instead of the option label (e.g., "(D)"). Furthermore, the formats can vary significantly between different models and even across different types of benchmarks, such as 16 Oct 2025 multiple-choice, math, and open-ended questions. This means that optimal extraction rules need to be created and tuned for every individual model and benchmark (e.g., rules for options, numbers, or word(s)), which makes the process difficult and even affects the reproducibility of model results. In this paper, we first empirically demonstrate the impact of answer extraction rules on reasoningpowered model (Section 4). We then introduce Answer Regeneration, a simple, generation-based framework designed to alleviate the dependency on specific answer extraction rules (Section 5). Instead of relying on complex extraction rules, our method utilizes an additional inference step to prompt the model to regenerate its final answer. It allows us to use probability-based answering for choices or extract the answer from a simplified output. Our experiments reveal that model performances are highly sensitive to the extraction rules employed. Depending on the rules, distinct answersno answers at all in some cases-may be extracted from the same LLM response. On the other hand, Answer Regeneration consistently outperforms the handcrafted rule-based extractions, improving both in benchmark score and human evaluation results. Our method also achieves intuitive model rankings, where larger models are shown to outperform smaller ones. We demonstrate that Answer Regeneration significantly reduces the dependency on specific answer extraction rules, thereby improving robustness and reproducibility of model evaluations. Furthermore, we apply our framework to diverse tasks, including complex multiple-choice question answering, short-answer math problems, and open-ended question answering. In all cases, our generation-based method proves to be a plausible and effective approach for the fair evaluation of reasoning models. Our contributions in this work are as follows: • We empirically investigate the sensitivity of reasoning-powered LLMs to rule-based answering, revealing a strong dependency on the choice of answer extraction algorithm. • We propose the generation-based framework Answer Regeneration. It achieves (1) superior performance compared with handcrafted rules, (2) intuitive model rankings, and (3) significantly enhanced robustness against answer inconsistency and incomplete outputs. • We demonstrate the generalizability and effectiveness of our framework across diverse tasks, confirming its plausibility for more robust and fair model evaluations. 2 Related Work A growing body of work shows that LLM performance can vary drastically with small changes in prompt format, even when the underlying semantics are equivalent (Sclar et al., 2024; He et al., 2024; Alzahrani et al., 2024). Consequently, Polo et al. (2024); Mizrahi et al. (2024) proposed the methods to mitigate the effect of prompt variations. While the previous research focused on input-level prompt variations and their impact on model evaluation, we focus on output-level final answer variations from reasoning LLMs, which are caused by the selection of answer extraction algorithms. Therefore, it is noteworthy to find out how recent LLM evaluations handle outputs from reasoning models. A number of open evaluation frameworks typically support (1) probability-based answering for multiple-choice tasks or (2) simple heuristic post-processing for free-form generations, involving only de-capitalization or blank-space normalization. Details on the implementations of MMLU Hendrycks (Hendrycks et al., 2021), HELM (Liang et al., 2023), OpenCompass (OpenCompass Contributors, 2023), and lighteval (Habib et al., 2023) can be found in the Appendix A.1. lm-evaluation-harness (Biderman et al., 2024) has become the de facto community standard for reproducible LLM evaluation. Generative tasks use string-match with optional regular expressions or rule-based normalizers. While recent templates support CoT prompting, the final answer is still recovered via simple patterns (e.g., "Answer: X"), or a last-capital-letter heuristic. As we will demonstrate, such extraction rules can swing scores and even reorder model rankings. To the best of our knowledge, this is the first study to highlight the importance of answer extraction methods, especially for reasoning LLMs. Based on our findings, we introduce a lightweight method to reduce the reliance on the fragile extraction rules and provides a more faithful evaluation of reasoning models' abilities. 3 Experiment Setup The experiments are designed to highlight current problems associated with finding answers in reasoning models' output (Study 1 in Section 4) and then assess the validity of introduced method Answer Figure 2: Model performance in accuracy evaluated using various answer extraction algorithm. Responses are considered incorrect if the extraction process fails to find an answer. Regeneration (Study 2 in Section 5). We utilize lm-evaluation-harness toolkit for its simplicity in customizing the post-processing rules. MMLU (Hendrycks et al., 2021) benchmark is primarily used, given its widely adoption for evaluating LLMs' knowledge1. The multiple-choice format of MMLU serves as a foundational task that simplifies the answer extraction process for our initial analysis. We then extend our evaluation to more complex tasks MMLU-Pro (Wang et al., 2024), the mathematical reasoning benchmark GSM8K (Cobbe et al., 2021) and the openended question answering TriviaQA (Joshi et al., 2017) in Section 6. We evaluate several open-source reasoning models: Qwen3 families-Qwen3-32B, Qwen314B, Qwen3-8B (Yang et al., 2025), along with Deepseek-R1-Distill-Llama-8B (referred to as R1-Llama-8B), and DeepSeek-R1-0528-Qwen38B (referred to as R1-Qwen3-8B) (DeepSeek-AI, 2025). For hyperparameter settings, we adhere to recommended best practices for each model, setting temperature to 0.6, top-p value to 0.95, and top-k value to 20. Prompt templates are sourced from lm-evaluation-harness, using thinking templates. To reduce computational costs, the maximum token generation length is limited to 4,096. 4 Study 1: Rule-based Answer Extraction 4.1 Methods We evaluate 5 reasoning models using 5 different answer extraction methods to investigate how performance changes with extraction algorithms: strict-match, flexible-extract, instructed-format, answer-is-correct, and last-extract: strict-match and flexible-extract are adapted from lm-evaluation-harness. strict-match extracts 1We select the original MMLU to better analyze how models handle ambiguous questions, rather than the cleaned MMLU-Redux (Gema et al., 2025). a precise string such as "answer is X" or "Answer: X" and flexible-extract finds multiple-choice options like (A), (B), (C), or (D), located near the end of the text. This is a common and effective approach, as the final conclusion typically follows the reasoning. However, the original implementation has tendency to extract the last capital character from any text, which can lead to errors. instructed-format requires modifying the input prompt to guide the model's output format. As recommended in Qwen3 technical report, we add a specific instruction to the prompt: "Please show your choice in the answer field with only the choice letter, e.g., "answer": "C"." Rules are then implemented to extract the answer from this specified format. While the method is strict (deviation is generally considered incorrect). Further heuristically optimized answer extraction methods are used: We build upon strict-match by creating answer-is-correct, which includes variations like "X is the answer" or "X could be the correct answer", addressing the limited scope. We also refine flexible-extract by developing lastextract, which specifically targets the last single capital character that appears in the output. All the implementations accounts for minor variations like "**X**", "**Answer:** X" and are designed to select the last match to accommodate potential self-correction within the model's response. Our objective is not to declare any one extraction method superior. Instead, we aim to demonstrate the range of performance that can be achieved using widely-used, well-optimized, heuristic extraction algorithms on a given task. Consequently, this research shows that benchmark performance is not solely dependent on the reasoning model's ability but is significantly influenced by the chosen answer extraction rules. The exact regular expressions used are described in the Appendix A.2. Answer Inconsistency Example The following are multiple choice questions (with answers) about high school physics. Q: The plates of a capacitor are charged to a potential difference of 5 V. If the capacitance is 2 mF, what is the charge on the positive plate? (A) 0.005 C (B) 0.01 C (C) 0.02 C (D) 0.5 C Okay, let's see. The question is about finding the charge on the positive plate of a capacitor. · · · the charges on the two plates are equal in magnitude but opposite in sign. So the positive plate has +Q and the negative has -Q. But the question is just asking for the magnitude, so the answer is still 0.01 C. Therefore, option answer-is-correct: B is correct. To determine the charge on the positive plate of a capacitor, · · · Since the question asks for the **magnitude** of the charge on the **positive** plate, the correct answer is: - ### Final Answer: ** flexible-match: (B) 0.01 last-extract: C ** Figure 3: An example from Qwen3-32B output illustrating how the final answer can vary significantly depending on the extraction method used. The graphical boxes and bold text highlight the specific text extracted by each algorithm. Figure 4: Distribution of extracted final answers across different extraction algorithms. The y-axis represents the answer extraction method, and the x-axis shows the extracted final answer, with "N/A" denoting cases where no answer could be extracted. 4.2 Result 4.2.1 Model Performance Figure 2 illustrates how different answer extraction methods affect the performance of models. We evaluate performance using 3 types of rules: implemented (strict-match, flexible-extract), recommended (instructed-format), and heuristically optimized (answer-is-correct, last-extract). If an extraction rule fails to find an answer, the response is considered incorrect. The results reveal that model performance fluctuates significantly depending on the extraction method used. With strict-match, the rankings of model performances are Qwen3-8B, Qwen3-32B, Qwen314B, R1-Qwen3-8B, and R1-Llama-8B in order. The more optimized answer-is-correct, derived from strict-match, significantly improves the performance of all models. This shifts the ranking to Qwen3-14B, Qwen3-32B, Qwen3-8B, R1Llama-8B, and R1-Qwen3-8B. A similar sensitivity is observed with the other methods. Using flexible-extract, the top models are Qwen-8B, Qwen3-14B, Qwen-32B, R1-Llama-8B, and R1Qwen3-8B. With last-extract, Qwen3-14B performs the best, and R1-Qwen3-8B outperforms R1Llama-8B compared with flexible-extract. Interestingly, despite following the recommended best practices for multiple-choice question answering with instructed-format, the performance of Qwen3 family models are not impressive compared to other extraction methods. This method proves to be particularly ineffective for R1-Llama-8B model. These findings challenge the common assumption that larger models outperform smaller ones within the same family. Our analysis indicates that the benchmark performance scores of reasoning models are highly dependent on the answer extraction method used. This suggests that the discrepancies between publicly reported and reproduced performance scores may be due to differences not only in prompt inputs, but also in the specific answer extraction methods, which are not fully disclosed. 4.2.2 Answer Inconsistency Figure 3 provides a clear example of how different extraction methods handle the same model output, illustrating the problem of answer inconsistency. In this example, strict-match fails. answer-is-correct successfully locates an answer within the model's thought, between and tags. However, the model's explicitly formatted final answer, "the correct answer is: X", is not recognized as a valid because it contains unexFigure 5: The proposed Answer Regeneration framework for finding answers in model output. The yellow box indicates the conventional method of direct extraction, while the blue box indicates the proposed framework. Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3 (%) 2.8 2.9 6.2 6.7 6.8 best-extr ans-is ans-is last flexible ans-is Correct 37.1 33.8 42.1 26.6 25.5 Incorrect 32.2 22.6 53.6 65.7 19.2 Invalid 30.7 43.6 4.4 7.8 55.3 Table 1: The percentage of incomplete thinking and the corresponding accuracy of each reasoning model. (%) refers to the portion of outputs where model's thinking process is not completed. pected patterns, including , , and . Besides, the output is the option text rather than the required option label. instructed-format could find an answer using (despite the format being recommended only for math problems), but the extracted answer is again the option text, not the label. Furthermore, the presence of the unexpected LaTeX command could result in an incorrect evaluation during string-match comparison. Meanwhile, flexible-match correctly identifies the final answer. Interestingly, the simple yet effective last-extract extracts the unit of option text "C" as the final answer. Figure 4 further illustrates this issue by showing how the distribution of extracted answers changes depending on the extraction method used. We observe that the distribution of extracted answers varies significantly. This highlights the crucial role of the extraction method in determining model's final performance, suggesting that the choice of method can introduce bias into the evaluation. 4.2.3 Answering for Incomplete Thinking Another challenge in extracting answers from reasoning models is the issue of incomplete reasoning (or thinking). Even when we set the maximum generation length to 4,096 tokens, we find that some model outputs lack the token, indicating that the thinking process had not concluded. Table 1 reports the percentage of outputs in this category. Fortunately, this is a relatively small portion of the total outputs and is primarily caused by repetitions during the model's generation. We then select the best answer extraction method for each model and measure the correctness of the final answers derived from these incomplete outputs. Except for Qwen3-8B and R1-Llama-8B, which use extraction algorithms solely on capital letters, the results using answer-is-correct show a high rate of invalid extraction. This implies that even well-optimized extraction method can be less robust toward incomplete thinking, particularly when the reasoning output does not contain definitive, explicitly formatted answering text. 5 Study 2: Answer Generation Our analysis has shown that the final answer of reasoning models is highly sensitive to the chosen extraction method. Model performance fluctuates significantly based on how the answer is located and selected from the output. To address this and simplify the optimization of complex extraction algorithms, we propose a straightforward framework for reliably identifying the final answer. 5.1 Method Our proposed framework, illustrated in Figure 5, tackles the challenge by introducing Answer Regeneration step. Instead of attempting to parse a final answer from model's extensive thought, our method uses an additional inference call. Specifically, we provide the model (in its non-reasoning mode) with the original input prompt and its previous output (the reasoning process), and a new prefix "Answer:". This prompts the model to generate a concise, final answer based on its prior reasoning. This approach offers key benefits. For multiplechoice questions, it allows us to utilize probabilitybased answering, as non-reasoning models have been evaluated, leading to more robust predictions. When the answer choices are not available, such as open-ended question answering, it simplifies the model's output, making the final answer much easier to extract with straightforward algorithms. Figure 6: (left) A confusion matrix comparing the conventional answer extraction method (Rule) and the proposed method (Regen). (right) The accuracy of answers extracted from the model's thought, as determined by human evaluation. We sample 300 instances when the extraction and regeneration are disagreed. Results are not reported for cases where the model failed to provide a definitive answer or provided multiple option labels. Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3 Rule(Best) 82.1 83.8 82.1 64.8 77.6 AnsRegen 87.1 85.0 83.3 68.8 80.7 Diff +5.0 +1.2 +1.2 +4.0 +3.1 Table 2: Performance comparison between conventional answer extraction and Answer Regeneration. We report each model's performance using its best-performing extraction method. While effective, our framework has several acknowledged limitations. The primary issue is the computational cost of the additional inference step. Additionally, the method might not fully capture minor variations in answer formatting, e.g., the probability of "**A**". Finally, some regenerated results could be different from explicitly mentioned answer. Despite these weaknesses and the lack of technical novelty, we believe this framework's simplicity and the clarity constitute significant contribution. We will demonstrate its benefits using the same experimental setup as our previous analyses. 5.2 Result 5.2.1 Improved Performance As presented in Table 2, the proposed method consistently reports better scores than rule-based answering. Figure 6 (left) provides a detailed look at the performance. While most of the final answers derived by both our method and the rule-based methods are the same, our framework achieves a much higher correction rate. This demonstrates Answer Regeneration is successful at correcting incorrect answers extracted by rule-based approach. To compute the correction rate, we select 300 instances from the outputs of Qwen3-32B, Qwen38B, and R1-Llama-8B where the extraction and regeneration results disagreed. We then manually label the correct "gold" answers in terms of answer extraction from the thoughts. As shown in Figure 6 (right), the agreement rate of Answer Regeneration with the human label is far superior to that of the conventional answer extraction methods. Figure 7: Model performance evaluated on outputs where the reasoning process is incomplete, using the optimal answer extraction algorithm for each model. 5.2.2 Correlation with Model Size An interesting effect of our framework is the change in the performance ranking of Qwen3 models. The previous ranking derived from rule-based answering, which was Qwen3-14B, Qwen3-32B, Qwen38B, shifted to Qwen3-32B, Qwen3-14B, Qwen38B under our framework. This new ranking aligns with conventional intuition and general knowledge that larger models typically outperform smaller ones within the same family. This suggests that the initial, counterintuitive ranking is likely an artifact of the answer extraction methods, not a true reflection of the models' underlying capabilities. 5.2.3 Enhanced Robustness to Responses The nature of our proposed Answer Regeneration framework inherently addresses the issue of answer inconsistency mentioned in Section 4.2.2. Since it prompts the model to generate a final, definitive answer, it bypasses the unpredictable results associated with various rule-based extraction algorithms. Additionally, our method improves robustness by handling internal self-correction within model outputs. When facing ambiguous questions, a model may initially provide an answer and then continue its thinking process, generating alternative solutions or re-evaluating its answer. Rule-based answer extraction methods struggle to choose the final answer from this internal debate. In contrast, our framework considers the entire thinking proQwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3 strict-match 15.3 13.0 15.7 6.8 10.9 flexible-ext 47.2 47.1 47.1 38.0 41.3 instructed 52.6 59.5 45.8 38.7 49.7 ans-is-corr 68.4 65.2 64.2 37.6 53.5 last-extract 66.8 63.4 62.0 42.2 45.3 implemented 72.1 69.4 64.6 43.3 58.3 AnsRegen 77.0 72.6 72.0 43.6 66.4 Reported2 79.8 77.4 74.3 54.3 73.9 ▷Reproduced 63.0 59.2 57.3 42.3 40.7 Table 3: Model performance on MMLU-Pro. The evaluation utilizes the same answer extraction algorithms used in our MMLU analysis, including the built-in algorithm from lm-evaluation-harness, referred to as implemented. cess and forces the model to finalize its response, leading to a more reliable result. A further key advantage is its ability to handle "NOT correct" questions. Since many extraction algorithms are designed to find the "correct" answer from the reasoning text, they fail when the question requires identifying the incorrect choice. The algorithm may mistakenly extract a correct option discussed during the model's rumination. Finally, our method significantly improves performance in cases of incomplete thinking, as shown in Figure 7. Instead of relying on rules to parse an incomplete output, our framework can select the final answer even when the thought does not include an explicit final answer. 5.2.4 Regenerator Independency Our method, which uses an additional model inference for Answer Regeneration, raises a question about its dependency on the specific model used. Table 7 in the Appendix shows that the performances achieved using small-sized regenerators are generally similar to the performance achieved when using the same model both for reasoning and Answer Regeneration. While this suggests a degree of independence, we still recommend using the same model for both tasks. The final answering step is also a crucial part of the overall model evaluation and should be performed by the model being assessed to ensure a consistent and fair comparison. Based on the results, Answer Regeneration framework shows a more effective and reliable method for evaluating reasoning models. Conventional extraction rules cannot account for all the variations in model outputs, and can thus introduce biases and inaccuracies. Our framework mitigates this problem, providing a more accurate and consistent measure of models' true performance. (↓) Extraction Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3 strict-match 3.3 2.7 1.7 0.0 0.1 flexible-ext 33.3 33.5 19.3 69.2 85.1 instructed 93.5 92.2 88.6 54.8 85.8 ans-is-corr 89.6 87.6 91.9 63.1 83.4 AnsRegen 95.0 93.8 91.1 76.0 91.1 Table 4: Model performance on GSM8K. Note that strict-match and flexible-extract are implemented in lm-evaluation-harness. last-extract is not useful. 6 Studies on Additional Tasks 6.1 Complex Multiple-Choice Question Answering As an extension of our previous findings, we investigate our framework on MMLU-Pro (Wang et al., 2024), a more complex benchmark with a dynamic number of answer options. The result, shown in Table 3, demonstrates that while the built-in extraction algorithm from lm-evaluation-harness performs better than algorithms optimized only for the original MMLU, Answer Regeneration-which is not specifically tuned for any benchmark-still achieves superior performance. Furthermore, the scores are also closer to the publicly reported performance2, despite the reported scores likely benefiting from more specific prompt engineering (e.g., detailed task descriptions for individual subtasks), as demonstrated in our reproduced score using their extraction rules. Therefore, we believe that evaluating models with our framework provides a fairer and more robust assessment of true capabilities, achieving competitive performance without the need for task-specific optimization. 6.2 Short-Answer Math Problems We explore the effectiveness of our framework in math domain using GSM8K benchmark (Cobbe et al., 2021), which features structured (as numbers) but relatively open-ended question answering task. As shown in Table 4, instructed-format , a template specifically recommended for mathematical problems, performs the best among the various extraction methods. We also modify answer-is-correct to better handle common mathematical formatting, such as numbers and symbols like $, ",", and ".". Despite these optimizations, Answer Regeneration with minor post-processing to remove LaTeX commands, such as or , achieves the highest performance. 2https://artificialanalysis.ai/evaluations/ mmlu-pro. Qwen3 technical report does not contain zero-shot CoT results for MMLU-Pro; it only provides 5-shot results without reasoning, scoring 65.54 for 32B and 56.73 for 8B. Method (↓) Evaluator Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama R1-Qwen3 String Match ans-is-corr42.7 47.5 44.2 11.7 35.6 AnsRegen - 55.3∗ 53.7 47.0 24.1 55.8 Model -based GPT Grader Qwen3-32B 3.1∗ 3.8 3.6 1.3 2.9 Qwen3-14B 49.5 56.4 49.2 19.4 41.1 Qwen3-8B 49.4 56.3 49.4 18.2 41.3 R1-Llama-8B 93.9 92.3 89.4 93.1∗ 87.5 R1-Qwen3-8B 47.6 54.8 47.9 17.7 39.6 xVerify xVerify-8B-I 0.0 0.0 0.0 47.7∗ 0.0 Table 5: (left) Performance of reasoning models on open-ended question answering TriviaQA. (right) Confusion matrix illustrating human evaluation performance on 100 samples in determining semantic equivalence between the generated answer and the gold answer. ∗denotes the selected results for the detailed human evaluation. To further validate this, we conduct a human evaluation of instances where the methods' results disagreed. Answer Regeneration framework reports 16.3% correct, while the conventional answer extraction method is correct in only 6.1% of the cases. This underscores the superior reliability of our framework even in complex, structured but open-ended domains like mathematics. 6.3 Open-ended Question Answering Evaluating generative models on open-ended question-answering tasks presents two main challenges: (1) finding the answer within the model's output. (2) determining semantic equivalence between the generated answer and the gold answer. To alleviate the second challenge, we use TriviaQA (Joshi et al., 2017), known for its extensive gold answer variations and aliases, minimizing the need for complex semantic matching. As shown in Table 5 (left), Answer Regeneration consistently outperforms direct answer extraction from the reasoning output. We also compare our framework with two other LLM-as-a-judge approaches (Zheng et al., 2023): GPTGrader (Wei et al., 2024), which uses an additional inference call with long prompts to categorize semantic similarity into "correct" (same), "incorrect" (not same), and "invalid". xVerify (Chen et al., 2025), a fine-tuned model that evaluates semantic equivalence as "correct" or "incorrect". While the scores from these model-based evaluations might appear better, they carry a critical drawback of model bias. Table 5 (right) presents a human evaluation of semantic equivalence, comparing the model judgement with human judgments on 100 sampled outputs. Qwen332B consistently predicts "incorrect" even when the answer is correct and xVerify similarly defaults to "incorrect", and R1-Llama-8B exhibits a bias toward "correct". In contrast, our string-matchbased method avoids this model bias and provides a more accurate performance measure, despite of its limitation in determining semantic equivalence. 7 Discussion and Conclusion Our analysis highlights a critical, yet often overlooked, challenge in evaluating reasoning models: the profound impact of the answer extraction methods on performance scores. We have demonstrated that model performances can fluctuate significantly based on how the final answer is parsed from its reasoning output. This finding suggests that discrepancies between publicly reported scores and reproduced results may stem from undocumented differences not just in prompts, but in the extraction methods itself. To address this issue, we introduced Answer Regeneration framework. Our simple approach offers significant advantages over conventional extraction rules: Without specific tuning, the framework consistently achieved superior scores across a variety of tasks, from multiple-choice question answering to short-answer math problems and open-ended question answering tasks. By prompting the model to explicitly state its final answer again, we mitigate inconsistencies caused by diverse output formats. Beyond exhibiting better scores than handcrafted, optimized rules, the performance ranking derived from our framework for the Qwen3 model family aligned with the conventional intuition that larger models generally outperform smaller ones. This suggests that the framework provides a more accurate reflection of a model's true capabilities, free from the biases of model-specialized answer extraction rules. Our method also proves more resilient to common failure of rule-based approach. It successfully handles outputs involving incomplete thinking, models that re-consider their answers, and questions asking for the "incorrect" choice, all of which can confuse rule-based extraction. Lastly, while LLM-as-a-judge method can suffer from inherent model biases (e.g., consistently predicting, "correct" or "incorrect"), our string-match method, enabled by the concise regenerated output, provides a more reliable measure of performance. In conclusion, through our findings from analysis and the introduction of Answer Regeneration framework, we believe this work contributes toward more reliable and faithful model evaluation for all reasoning-powered LLMs. 8 Limitations Technical Novelty in Answer Regeneration We acknowledge that Answer Regeneration framework itself lacks technical novelty. However, we contend that the value of our contribution lies in the simplicity and the clarity of the results and analysis it provides. Our work demonstrates the benefits of using this framework as a robust and reliable reference for evaluating and fairly comparing the performance of reasoning models. Experiments with Sophisticated Extraction Rules Our experiments adopted established answer extraction rules from lm-evaluation-harness (strict-match, flexible-match). Building upon these, we developed more complex, heuristic rules (answer-is-correct, last-extract) and included the recommended rule for Qwen3 families (instructed-format). While we recognize that more aggressively optimized, domain-specific rules could exist, we maintain that such highly-specified rules will still fail to handle the full spectrum of answer variations. Experiments with Diverse LLMs and Prompts Our focus was on output-level results, which means that the effect of different input prompts seem to be overlooked. Furthermore, our investigation was limited to publicly available open-source reasoning models. Although greater diversity in models and prompts would enhance generalizability, we believe that the widely-used models and default prompts from established repositories provide sufficiently general results for our findings. We defer the investigation of commercial LLMs, such as ChatGPT, Gemini, and Claude, to future work. As a minor note, we observed that small variations in the input prompts (e.g., changes of option labels or the "Answer:" prefix) do not significantly affect performance. Inherent Weakness of Answer Regeneration As discussed in Section 5.1, Answer Regeneration carries inherent limitations. Nonetheless, we believe that employing the simplest possible framework was the most effective way to demonstrate the core benefits of our approach. Exploring further techniques within this framework, such as incorporating concepts like self-consistency (Wang et al., 2022), represents a valuable direction for future research. 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Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, et al. 2021. Training verifiers to solve math word problems. arXiv preprint . DeepSeek-AI. 2025. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning. Aryo Pradipta Gema, Joshua Ong Jun Leang, Giwon Hong, Alessio Devoto, Alberto Carlo Maria Mancino, Rohit Saxena, Xuanli He, Yu Zhao, Xiaotang Du, Mohammad Reza Ghasemi Madani, et al. 2025. Are we done with mmlu? In Proceedings of the 2025 Conference of the Nations of the Americas Chapter of the Association for Computational Linguistics: Human Language Technologies (Volume 1: Long Papers), pages 5069-5096. Nathan Habib, Clémentine Fourrier, Hynek Kydlíˇcek, Thomas Wolf, and Lewis Tunstall. 2023. Lighteval: A lightweight framework for llm evaluation. Jia He, Mukund Rungta, David Koleczek, Arshdeep Sekhon, Franklin X Wang, and Sadid Hasan. 2024. Does prompt formatting have any impact on llm performance? Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt. 2021. Measuring massive multitask language understanding. Proceedings of the International Conference on Learning Representations (ICLR). Mandar Joshi, Eunsol Choi, Daniel S Weld, and Luke Zettlemoyer. 2017. Triviaqa: A large scale distantly supervised challenge dataset for reading comprehension. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1601-1611. Ehsan Kamalloo, Nouha Dziri, Charles Clarke, and Davood Rafiei. 2023. Evaluating open-domain question answering in the era of large language models. In Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 5591-5606. Percy Liang, Rishi Bommasani, Tony Lee, Dimitris Tsipras, Dilara Soylu, Michihiro Yasunaga, Yian Zhang, Deepak Narayanan, Yuhuai Wu, Ananya Kumar, Benjamin Newman, Binhang Yuan, Bobby Yan, Ce Zhang, Christian Cosgrove, Christopher D. Manning, Christopher Ré, Diana Acosta-Navas, Drew A. Hudson, Eric Zelikman, Esin Durmus, Faisal Ladhak, Frieda Rong, Hongyu Ren, Huaxiu Yao, Jue Wang, Keshav Santhanam, Laurel Orr, Lucia Zheng, Mert Yuksekgonul, Mirac Suzgun, Nathan Kim, Neel Guha, Niladri Chatterji, Omar Khattab, Peter Henderson, Qian Huang, Ryan Chi, Sang Michael Xie, Shibani Santurkar, Surya Ganguli, Tatsunori Hashimoto, Thomas Icard, Tianyi Zhang, Vishrav Chaudhary, William Wang, Xuechen Li, Yifan Mai, Yuhui Zhang, and Yuta Koreeda. 2023. Holistic evaluation of language models. Moran Mizrahi, Guy Kaplan, Dan Malkin, Rotem Dror, Dafna Shahaf, and Gabriel Stanovsky. 2024. State of what art? a call for multi-prompt LLM evaluation. Transactions of the Association for Computational Linguistics, 12:933-949. OpenCompass Contributors. 2023. Opencompass: A universal evaluation platform for foundation models. https://github.com/open-compass/ opencompass. Felipe Maia Polo, Ronald Xu, Lucas Weber, Mírian Silva, Onkar Bhardwaj, Leshem Choshen, Allysson Flavio Melo de Oliveira, Yuekai Sun, and Mikhail Yurochkin. 2024. Efficient multi-prompt evaluation of llms. Melanie Sclar, Yejin Choi, Yulia Tsvetkov, and Alane Suhr. 2024. Quantifying language models' sensitivity to spurious features in prompt design or: How i learned to start worrying about prompt formatting. Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. 2022. Self-consistency improves chain of thought reasoning in language models. arXiv preprint . Yubo Wang, Xueguang Ma, Ge Zhang, Yuansheng Ni, Abhranil Chandra, Shiguang Guo, Weiming Ren, Aaran Arulraj, Xuan He, Ziyan Jiang, et al. 2024. Mmlu-pro: A more robust and challenging multi-task language understanding benchmark. Advances in Neural Information Processing Systems, 37:9526695290. Jason Wei, Nguyen Karina, Hyung Won Chung, Yunxin Joy Jiao, Spencer Papay, Amelia Glaese, John Schulman, and William Fedus. 2024. Measuring short-form factuality in large language models. arXiv preprint . Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Fei Xia, Ed Chi, Quoc V Le, Denny Zhou, et al. 2022. Chain-of-thought prompting elicits reasoning in large language models. Advances in neural information processing systems, 35:24824-24837. An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, et al. 2025. Qwen3 technical report. arXiv preprint . Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. 2023. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in neural information processing systems, 36:46595-46623. A Appendix A.1 Evaluation Toolkits MMLU Hendrycks (Hendrycks et al., 2021) and follow-ups such as MMLU-Pro (Wang et al., 2024) are deeply integrated into most of toolkits, but the original implementation only supports probability based answering for multiple choice question answering. HELM (Holistic Evaluation of Language Models; Liang et al. (2023)) simply use Quasiexact match that post-process the model generation, such as lower-casing, removing whitespace and punctuation and articles. Also, OpenCompass (OpenCompass Contributors, 2023) supports both option-likelihood scoring and post-processing option (but provided with blank) to be customized for reasoning outputs, its metrics mainly rely on model-based scoring. Similarly, lighteval (Habib et al., 2023) has metrics for generated outputs, but there is only a scoring function, not mentioning about post-processing. A.2 Regular Expressions used in the Experiments Note that () makes groups in regular expression and \ required both for meta characters and escape sequence in lm-evaluation-harness. • strict-match: ((?<=The answer is )(.*)(?=.)|(?<=answer is )(.*)(?=.)|(?<=The answer: )(.*)(?=.)|(?<=The final answer: )(.*)(?=.)) • flexible-extract: (\ ) • instructed-format:[Aa]nswer\"?:\ \"?\ ?)|\\**[Aa]nswer\\**:\ (\ ?)|[Aa]nswer is \\**(\ ?)\\**|[Aa]nswer should be \\**(\ ?)\\**|[Aa]nswer:\ +\\**(\ ?)\\**|correct answer is \\**(\ ?)\\**|correct answer:\ +\\**(\ ?)\\**|\\**(\ ?)\\** is correct| *(\ ?)\\** is the correct|\\**(\ ?)\\** is the answer|\\**(\ ?)\\** should be the answer • last-extract: [^a-zA-Z0-9]([A-D])[^a-zAZ0-9] A.3 Preliminary: Non-reasoning vs. Reasoning We demonstrate the power of reasoning in solving MMLU, as presented in Table 6. The performance shows that the reasoning significantly improves the model performances. This encourage us to use reasoning model not only for complex problem. but for knowledge-based problems. A.4 Regnerator Independency Table 7 reports the performance when using different models from the answer generator. Although we use smaller models for regenerator, the performance is similar when using the identical model. Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama-8B R1-Qwen3-8B non-Reason 78.4 75.7 72.2 53.0 66.2 Reason 82.1 83.8 82.1 64.8 77.6 Diff +3.7 +8.1 +9.9 +11.8 +11.4 Table 6: The performance comparison when using non-reasoning mode and reasoning mode in LLMs. Nonreasoning mode follows conventional loglikelihood measurements using candidate whereas reasoning mode uses answer extraction algorithms to find the final answer in the reasoning output. The best performance with answer extraction methods are reported. (↓) Regenerator Qwen3-32B Qwen3-14B Qwen3-8B R1-Llama-8B R1-Qwen3-8B gemma-3-1b-it 86.9 84.9 82.5 67.5 80.3 llama-3.2-1b-it 86.5 84.4 81.8 67.8 79.6 Qwen3-0.6b 86.3 84.4 82.3 68.9 79.9 Qwen3-32B 87.1 85.2 83.7 72.1 82.6 Qwen3-14B 87.1 85.0 83.1 71.2 81.6 Qwen3-8B 87.4 85.2 83.3 72.5 82.0 R1-Llama-8B 87.0 84.9 82.6 68.8 80.2 R1-Qwen3-8B 84.2 81.0 81.1 70.9 80.7 Table 7: Model performance when different models are used for Answer Regeneration step. Bold indicates the reported score when the reasoning models and the regenerators are the same.
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2510.14771
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Technical Report:
Open TeleDex: A Hardware-Agnostic Teleoperation System for
Imitation Learning based Dexterous Manipulation
Xu Chia,1, Chao Zhanga,1, Yang Suc,1, Lingfeng Doub, Fujia Yanga, Jiakuo Zhaoa, Haoyu Zhouc,
Xiaoyou Jiac, Yong Zhouc, and Shan Ana,*
aTianjin University
bTsinghua University
cLinkerBot Inc.
1These authors contributed equally to this work.
*Corresponding author: anshan@tju.edu.cn
October 17, 2025
Abstract
Accurate and high-fidelity demonstration data acquisition is a critical bottleneck for deploying robot
Imitation Learning (IL) systems, particularly when dealing with heterogeneous robotic platforms. Ex-
isting teleoperation systems often fail to guarantee high-precision data collection across diverse types
of teleoperation devices. To address this, we developed Open TeleDex, a unified teleoperation frame-
work engineered for demonstration data collection. Open TeleDex specifically tackles the TripleAny
challenge, seamlessly supporting any robotic arm, any dexterous hand, and any external input device.
Furthermore, we propose a novel hand pose retargeting algorithm that significantly boosts the inter-
operability of Open TeleDex, enabling robust and accurate compatibility with an even wider spectrum
of heterogeneous master and slave equipment. Open TeleDex establishes a foundational, high-quality,
and publicly available platform for accelerating both academic research and industry development in
complex robotic manipulation and IL.
1
Introduction
Executing a wide spectrum of complex, dexterous manipulation tasks in diverse, unstructured environments
remains a central challenge in modern robotics. Although Imitation Learning (IL) has shown great potential
in tackling this problem, its progress is fundamentally constrained by the difficulty of collecting large-scale,
multi-modal, and high-quality demonstration data [1].
The quality and scalability of this data hinge entirely on the underlying teleoperation system. These
systems are designed around a series of complex trade-offs, often balancing three core dimensions: Fidelity
(the precision of control and data quality), Accessibility (cost and deployment complexity), and General-
ization (hardware compatibility). Current solutions consistently compromise on at least one dimension. For
instance, high-fidelity systems, such as dedicated master-slave arms, offer unmatched precision and minimal
latency, making them indispensable in zero-tolerance applications. However, this tight electromechanical
coupling inherently limits their Generalization, as they exist as costly, "closed ecosystems" tailored to a
single robot model, making cross-platform data aggregation impractical. Conversely, systems prioritizing
1
arXiv:2510.14771v1 [cs.RO] 16 Oct 2025
Accessibility by leveraging consumer-grade VR/AR devices have significantly reduced the technical barri-
ers to data collection. Despite this, they frequently compromise on Fidelity due to challenges like visual
tracking uncertainties, communication latency, and the absence of physical force feedback. Furthermore,
even these more accessible platforms often require substantial software adaptation and parameter tuning
when faced with a new robotic arm or hand, revealing a critical generalization gap in practice. This lack of
a unified framework supporting heterogeneous robotic platforms across the entire data collection workflow
is the core bottleneck hindering the evolution of IL research prototypes into scalable, general-purpose data
collection platforms.
To address this, we introduce Open TeleDex, a device-agnostic and highly versatile teleoperation frame-
work. Our core design tenet is Generalization, achieved through a natively ROS2-based architecture that
intrinsically unifies diverse control paradigms, allowing for the rapid and seamless integration of new hard-
ware combinations. Our work makes two core contributions:
1. A Unified, Hardware-Agnostic Teleoperation Framework: We present a decoupled, three-tier
architecture that supports synchronous data collection from a wide variety of robotic arms, multi-
fingered dexterous hands, and external input devices. Open TeleDex serves as an extensible backbone,
allowing researchers to efficiently integrate and switch between diverse heterogeneous hardware com-
binations.
2. A Hand Pose Retargeting Algorithm: We propose a novel retargeting algorithm that adjusts for
kinematic discrepancies between human and robot hands. Our algorithm is designed to enhance the
framework’s compatibility.
This report is organized as follows: Section 2 the reviews related work. Section 3 details Open TeleDex
framework. Section 4 shows the evaluation of our system. Finally, Section 5 concludes the report and
discusses future work.
2
Related Work
2.1
Teleoperation Systems
Teleoperation systems are a crucial tool for collecting expert demonstrations for imitation learning, and their
design inherently involves a series of complex trade-offs. The efficacy of any such system can be evaluated
along three core dimensions. The first is Fidelity, which measures the precision with which a system con-
veys operator intent and relays environmental feedback; this remains a central focus in modern telerobotics
research [2]. The second is Accessibility, encompassing its cost, deployment complexity, and ease of use,
which are critical factors for practical application [3]. The third, and increasingly critical dimension, is
Generalization—the system’s capacity to adapt to diverse robot hardware, tasks, and environments, a key
challenge in modern human-robot interaction [4]. Existing solutions in the field typically exhibit distinct
strengths and compromises across these dimensions.
Systems prioritizing ultimate fidelity, such as dedicated master-slave arms, have set the gold standard
for latency and haptic "transparency", making them indispensable in zero-tolerance applications like robotic
surgery [5]. Pioneering systems like ALOHA [6] have democratized data collection by drastically reducing
the financial and technical barriers to entry for dexterity research. Through tight electromechanical coupling
and direct joint-space mapping, they minimize information loss during transmission. However, this design
philosophy inherently constrains their generalization. They often exist as "closed ecosystems" tailored to a
single robot model, where any hardware modification can demand costly redesigns, rendering cross-platform
deployment and data aggregation impractical.
2
In recent years, a wave of research has focused on improving accessibility by leveraging consumer-
grade VR/AR devices and standard cameras. These systems typically employ more flexible Cartesian-space
control. Despite their success in accessibility, they often compromise on fidelity, facing challenges from
visual tracking uncertainties, communication latency, and the absence of physical force feedback [7, 8].
More importantly, while more general in theory, many open-source implementations still require substantial
software adaptation and parameter tuning for specific robots, revealing a generalization gap in practice.
It is thus evident that generalization has emerged as the core bottleneck connecting the worlds of high
fidelity and high accessibility, and hindering the evolution of teleoperation technology from one-off research
prototypes to scalable data collection platforms. Addressing this challenge necessitates the development of
a universal software framework. Works like AnyTeleop [9] have made important strides in this direction
with learning-free, GPU-accelerated components. However, there remains a need for a framework that is
natively rooted in the ROS2 ecosystem and is architected to intrinsically unify diverse control paradigms,
from direct mapping to planner-centric control.
The TeleDex framework presented in this paper aims to build such a unified teleoperation framework
with generalization as its core design principle. It does not seek to invent a new interaction modality, but
rather to integrate and empower the diverse existing hardware ecosystem by providing a modular, extensible
software backbone, thereby offering a truly scalable data collection solution for robot imitation learning.
2.2
Hand Pose Retargeting
The real-time and accurate transfer of a human operator’s natural movements to a morphologically different
robot is a core challenge in general-purpose teleoperation. This challenge can be decomposed into two main
sub-problems: arm motion generation and dexterous hand pose retargeting.
For arm motion generation, mainstream approaches rely on Inverse Kinematics (IK) solvers or more
advanced real-time motion planners [9, 10]. These methods all require a precise kinematic model of the
robot (e.g., a URDF file) to compute joint trajectories that match the operator’s end-effector pose intent.
Dexterous hand posture retargeting is a more complex challenge. Existing methods, whether optimization-
based [9, 11] or learning-based [12, 13], mostly follow an analytical paradigm: they pre-define keypoint or
joint correspondences between the human and robot hands and then minimize the error between them. While
intuitive, the fundamental limitation of this "point-to-point mapping" approach lies in its rigid correspon-
dence. When the robot hand’s morphology differs significantly from a human’s, this mapping can easily
produce unnatural or physically infeasible postures, and the process of configuring these correspondences
for new hardware is tedious.
To overcome the limitations of this "rigid mapping" paradigm, we propose a novel "generative" hand re-
targeting algorithm. Instead of copying each joint motion, our method treats the robot hand as an integrated
skeleton constrained by its own kinematic model. Our algorithm drives this skeleton to "grow" naturally to
functionally match the human’s overall grasping intent, rather than mimicking it point-by-point morpholog-
ically. This shift from "copying motion" to "reproducing function" enables our framework to generate more
natural and functional postures for morphologically diverse dexterous hands.
2.3
Demonstration Data for Imitation Learning
The success of Imitation Learning (IL) is critically dependent on the quality of demonstration data [14].
Research has shown that high-quality demonstrations, characterized by smooth motions and precisely syn-
chronized state-action pairs, can significantly improve the final policy’s performance and robustness [15,16].
Therefore, building a collection framework that systematically ensures data quality is essential.
In practice, ensuring data quality faces two main challenges. The first is time synchronization. In a
system with heterogeneous sensors like cameras and encoders, ensuring a common and precise time base for
3
Figure 1: The Framework of Open TeleDex. The diagram illustrates the layered architecture of the system,
which is designed to support heterogeneous robot platforms for different requirement of data collection. The
system is structurally divided into three modules: Perception, Control and Data Collection, and Hardware.
all data streams is a core difficulty. Simple recording schemes can easily introduce temporal misalignments
that are detrimental to learning dynamic tasks [17]. The second challenge is the richness of data modality.
While traditional master-slave systems provide high-quality kinematic data, modern IL increasingly benefits
from multi-modal information, including multi-view images and force/tactile feedback [18]. Emerging
paradigms like VR teleoperation (e.g., Holo-Dex [11]) are gaining attention for their natural ability to fuse
multi-modal data.
This necessitates a modern data collection framework that not only features high-performance real-
time processing and precise time synchronization but also natively supports the fusion of various collection
methods in its architecture to meet the growing demand for high-quality, multi-modal, and diverse data.
3
Framework of Open TeleDex
3.1
System Architecture
As shown in Fig. 1, the architecture of Open TeleDex follows the core design principles of layering and
decoupling, with the ultimate goal of realizing our TripleAny vision: supporting the seamless integration
and cooperation of AnyExternalDevice, AnyArm, and AnyHand. It primarily utilizes the ROS2 frame-
work [19] for distributed communication. Conceptually, the architecture is divided into three core modules:
4
1. Perception, responsible for standardizing heterogeneous input data from Any External Device.
2. Control & Data Collection, which executes hardware-independent task management, control strat-
egy scheduling, and data processing. This core layer is "agnostic" to the specific arm and hand, which
is key to achieving AnyArm and AnyHand.
3. Hardware, which translates standardized control commands for specific AnyArm and AnyHand ac-
tuators.
3.2
Perception
The Perception component is responsible for capturing and interpreting the operator’s intent and movement,
forming the input signals for the entire teleoperation system.
Master Device: this module encapsulates various human interface devices such as Virtual Reality (VR)
headsets (for spatial tracking), Teleoperation Gloves (for finger articulation capture), and Master Arms.
Sensor Fusion: Raw data from disparate input sensors is fused and filtered to generate a coherent, low-
latency stream of the operator’s hand and arm pose.
Hand Pose Retargeting:
this is a high-level decision module responsible for transforming the continuous
human pose stream into a discrete, feasible goal pose for the robotic Slave devices. To achieve precise and
adaptable teleoperation for robotic hands, we construct a modular optimization framework. The core of this
framework is a total cost function, which is a weighted sum of three distinct sub-functions: a pose alignment
cost, an inter-finger coupling cost, and a temporal smoothness cost. By minimizing this total cost function,
we can solve for the optimal robot joint angle vector, θ∗(t), in real-time.
Optimization Objective: The optimal robot joint angle vector, θ∗(t), is found by solving the following
minimization problem:
θ∗(t) = arg min
θ(t)
αalignLalign(θ(t)) + αcoupleLcouple(θ(t)) + αsmoothLsmooth(θ(t))
(1)
Where:
• θ(t) ∈Rn is the robot hand’s joint angle vector at time t.
• Lalign, Lcouple, and Lsmooth represent the three core cost functions.
• αalign, αcouple, and αsmooth are hyperparameters (weights) that balance the importance of each cost
term.
Cost Function: Pose Alignment Cost (Lalign). This cost function aims to minimize the geometric error
between the operator’s hand keypoints and the corresponding keypoints on the robot hand. It is central to
achieving the fundamental pose mimicry.
Lalign(θ(t)) =
X
(i,j)∈K
∥p′
i,j(t) −Fk,j(θ(t))∥2
(2)
Here, p′
i,j(t) ∈R3 represents the position of the operator’s j-th keypoint on the i-th finger after transfor-
mation and calibration, while Fk,j(θ(t)) is the robot’s forward kinematics function, which calculates the
position of its corresponding keypoint.
Operator Keypoint Transformation. To compute the transformed keypoints p′
i,j(t), we must first scale
and translate the raw detected keypoints, pi,j(t), to match the robot’s geometry.
5
1. Dimensional Scaling Factor (si,j): This calculates the relative length ratio for each phalangeal seg-
ment.
si,j = ∥Fk,j+1(θ0) −Fk,j(θ0)∥
∥¯pi,j+1 −¯pi,j∥
(3)
Here, ¯pi,j and θ0 represent the human keypoints and robot joint angles from a static calibration pose,
respectively.
2. Finger Root Translation Vector (δi): This calculates the offset for the base position of each finger.
δi = Fk,i,mcp(θ0) −¯pi,mcp
(4)
3. Transformed Keypoint (p′
i,j): Finally, the transformed points are obtained via the following iterative
formula:
p′
i,j(t) =
pi,0(t),
j = 0
p′
i,j−1(t) + si,j−1(pi,j(t) −pi,j−1(t)) + δi,
j = 1
p′
i,j−1(t) + si,j−1(pi,j(t) −pi,j−1(t)),
j ≥2
(5)
Inter-Finger Coupling Cost (Lcouple). This cost function is used to maintain human-like coordination
between the thumb and other fingers during fine motor actions such as grasping and pinching.
Lcouple(θ(t)) =
X
i∈I
βi(t)∥⃗RH,i(t) −⃗RR,i(θ(t))∥2
(6)
Here, ⃗RH,i(t) = pi,fingertip(t) −pthumb,fingertip(t) is the relative vector from the operator’s thumb tip to other
fingertips. ⃗RR,i(θ(t)) is the corresponding relative vector on the robot hand. The term βi(t) is an adaptive
weight that dynamically adjusts the influence of this cost term based on inter-finger distance.
Adaptive Weighting (βi(t)). The weight’s magnitude depends on the proximity of the fingers. We first
compute a normalized proximity metric ρi(t) ∈[0, 1]:
ρi(t) = 1 −∥⃗RH,i(t)∥−ρmin,i
ρmax,i −ρmin,i
(7)
where ρmin and ρmax are the minimum and maximum inter-finger distances determined from calibration
data. This metric is then mapped through a Sigmoid function to produce the final weight:
βi(t) =
1
1 + e−σ(ρi(t)−τ)
(8)
where σ and τ are parameters controlling the function’s shape and activation threshold.
Temporal Smoothing Cost (Lsmooth). This term acts as a regularizer to penalize abrupt changes in the
robot’s joint angles, ensuring the motion trajectory is fluid and stable.
Lsmooth(θ(t)) = ∥θ(t) −θ(t −1)∥2
(9)
3.3
Control and Data Collection
The core of TeleDex is the integrated control and data collection pipeline, which functions as the central
synchronization and serialization engine. This component is further structured into a three-layer software
architecture: the Control Layer (L3), the Data Integration Layer (L2), and the Data Storage Layer (L1).
6
3.3.1
Control Layer
This layer is centered around a primary ROS2 service node, which embodies the main responsibilities of
the core control layer, including task manager, hardware manager, and multi-sensor timeSync manager, and
data collection coordinator.
In a typical data collection workflow, an operator first specifies a hardware combination in a configura-
tion file. Upon system launch, the corresponding ROS2 nodes in the perception component are activated,
reading raw data from their respective hardware (e.g., VR devices or a master-slave arm) and publishing it as
standardized ROS2 messages. The control layer subscribes to these standard messages. Its internal Control
Strategy Scheduler then loads and applies the appropriate control algorithm based on the preset mode in the
configuration.
For instance: (1) For a high-fidelity, master-slave configuration (e.g., Agilex arm + LinkerHand L10/O6
+ master arm/glove), the scheduler employs a direct joint-space mapping strategy to achieve minimal latency.
(2) For a low-cost configuration with mismatched kinematics (e.g., RealMan arm + LinkerHand L10 + VR
controller), the scheduler uses a hybrid control strategy: an optimization-based retargeting algorithm for
the hand and Cartesian-space relative motion control for the arm. The standardized robot control commands
generated by this layer are then published, to be subscribed to and executed by the corresponding robot driver
node in the hardware component. Concurrently, a parallel data collection pipeline captures all sensor data
and control commands. A time synchronization module within this pipeline aligns and validates timestamps
from different sources.
Data Synchronization Mechanism:
When collecting high-quality data for imitation learning, ensuring
precise temporal alignment of data from multiple heterogeneous sensors is crucial. Even a minor time devi-
ation can lead to incorrect state-action pairing, severely impacting the learning effectiveness of downstream
policies. To address this challenge, Open TeleDex implements a robust software-based timestamp syn-
chronization (TimeSync Manager) and validation mechanism at the application layer, built upon ROS2’s
distributed clock service. TimeSync Manager is integrated throughout the data collection process:
1. Establishment of a Global Time Base: Upon system startup, the TimeSyncManager initializes and
waits for ROS2’s global clock to become stable. This ensures that all ROS2 nodes in the system
share a single, synchronized time source, which is the foundation for all subsequent synchronization
operations.
2. Per-Frame Multi-Source Timestamp Collection and Validation: In each control cycle of data
collection, the system performs a rigorous timestamp synchronization check:
• Gathering: The RealEnv interface collects timestamps for the current frame of data from all
active Recorder modules. These timestamps come from various sources: camera and dexterous
hand timestamps are from their ROS2 message headers, while the robotic arm’s timestamp may
come directly from its underlying SDK.
• Validation: The core function of TimeSyncManager verifies the collected set of timestamps.
This validation includes two key checks:
a. Freshness Check: Ensures all timestamps are "recent" (e.g., within 1 second of the current
ROS time) to filter out stale data caused by node freezes or network issues.
b. Consistency Check: Calculates the maximum difference among all timestamps in the set
and checks if this difference is within a preset tolerance (e.g., 100ms). Under typical oper-
ating conditions, our system consistently keeps this difference below 70ms (tp99).
7
3. Data Tagging and Storage: The result of the validation—including a success flag, the max_diff,
and all original timestamps—is packaged as a "sync validation bundle" and stored within the current
frame’s observation data. This metadata is ultimately saved into the telemetry.npz file along with
qpos, qvel, etc.
Through TimeSync Manager, Open TeleDex achieves an active and traceable data quality assurance that
goes beyond a standard rosbag. It not only strives to ensure data synchronicity at the time of collection
but, more importantly, provides a synchronization quality metric for every single frame of data. During
subsequent model training, researchers can easily filter out unsynchronized timestamps data with unsyn-
chronized timestamps using the sync_validation_is_valid flag, or weight the data based on the
max_diff value. This verifiable, end-to-end time synchronization loop is a key feature of Open TeleDex
as a professional data collection framework for robot imitation learning.
3.3.2
Data Integration Layer
This layer acts as the Data Plane, specializing in acquisition, buffering, and data serialization. It is explicitly
decoupled from the specific hardware and file structure.
Data acquisition node is a comprises multiple, independent Recorder modules (Arm Recorder, Link-
erHand Recorder, Camera Recorder, Scalable Interface). Each Recorder subscribes to a specific hardware
topic from L1, applies the globally synchronized timestamp received from the L3 TimeSync Manager, and
pushes the data into the persistence buffer.
Data persistence node is responsible for structuring and writing the buffered data to disk. It efficiently
encodes and stores the synchronized, multi-modal data into a structured format (e.g., video-compressed or
HDF5) with complete metadata, ready for downstream imitation learning tasks. The operator can control
the start and stop of data collection via keyboard commands.
To support a new robotic arm, a developer only needs to implement a new Recorder class that adheres
to the RealEnv interface, without modifying the core data collection logic. This design significantly lowers
the barrier to integrating new hardware into the Open TeleDex ecosystem.
3.3.3
Data Storage Layer
This layer defines the final output structure and the foundational storage hierarchy, ensuring the dataset is
immediately usable by learning frameworks.
Standard Storage Node enforces a consistent, hierarchical file structure. It defines the output files:
Telemetry Joint Data (telemetry.npz), Cameras Data/Intrinsics (video files and calibration parameters, cam-
era_info.json), Episode Manifest(manifest.json), and Metadata Global Config (metadata.json).
3.4
Hardware
3.4.1
Hardware Components
This component represents the physical devices and the low-level software interfaces required for execution
and sensing. Fig. 2 shows some of the hardware components supported by Open TeleDex.
1. Camera Driver: Standardized drivers (e.g., for RealSense) that interface directly with the camera
hardware and publish synchronized RGB-D streams via ROS 2 topics.
2. Arm Driver: Low-level controllers responsible for converting high-level velocity or joint position
commands from L2 into motor signals for the mechanical arm (e.g., Piper/Linker Arm).
8
Figure 2: Some of the Hardware Components Supported by Open TeleDex.
3. LinkerHand Driver: The specific SDK or API that provides control and state feedback for the Linker-
Hand, handling intricate finger articulation commands and sensor readings.
4. Sensor Data Stream: The continuous feedback loop, providing the instantaneous state (joint angles,
end-effector forces, visual frames) back to the L2 Data Acquisition Node for synchronized logging
and, if implemented, real-time closed-loop control.
3.4.2
Heterogeneous Hardware Interface
To achieve broad hardware compatibility, we designed a core heterogeneous hardware interface. It uses
software abstraction to shield the system from underlying physical hardware differences, providing a stable
and unified interaction logic for upper-level applications. Its design follows two core principles: Master-side
input normalization and Slave-side device abstraction.
Master Input Normalization: A Unified Expression of Operator Intent
The primary challenge at the
teleoperation front-end (Master side) is to handle input signals from vastly different sources and in disparate
formats—for instance, data gloves output high-dimensional joint angles, VR devices provide hand keypoint
poses via UDP streams, while master arms directly output their own joint states.
To mask this heterogeneity, Open TeleDex introduces an intermediary "intent parsing and normalization"
step. Regardless of the input device, its raw data stream is first processed by a dedicated front-end adapter
node. In our implementation, this is an independent ROS2 node named hand_retarget.
9
The core responsibility of this node is to uniformly translate these various input signals into a standard-
ized "intent signal". This signal represents the operator’s core intent and is published via a fixed ROS2 topic.
Specifically:
• For input from a VR controller, this node runs our hand retargeting algorithm and a Cartesian-space
relative motion control strategy, driving the dexterous hand with the target hand pose and the robotic
arm with the wrist keypoint data.
• For input from a data glove and a master arm, the node executes our direct joint-space mapping control
strategy, mapping their poses to the standard target joint angles for the dexterous hand and robotic arm.
Slave Device Abstraction: A Unified Robot Environment Interface
At the teleoperation back-end
(Slave side), the challenge is to control robots of different models, degrees of freedom, and communica-
tion protocols with a single, unified logic.
To enable broad compatibility for AnyArm and AnyHand, Open TeleDex implements an environment
abstraction layer named RealEnv. This design aims to provide a unified, high-level API for all physical robot
hardware.
RealEnv itself is an abstract base class that defines standard methods for interacting with a robot, such
as getting the current robot state and sending an action command to the robot. For each supported piece
of hardware (e.g., the Agilex Piper arm or the Linker arm A7), we implement a corresponding Recorder
subclass (e.g., PiperRecorder).
This Recorder subclass acts as a "device driver". It handles all low-level details of interacting with a
specific hardware SDK or underlying ROS driver, while strictly adhering to the interface defined by RealEnv.
For instance, when the upper-level data collection logic calls get_observation():
• PiperRecorder queries the encoders via its SDK to get joint states.
• LinkerHandRecorder subscribes to the relevant ROS2 topic for the latest joint messages.
• RealSenseRecorder subscribes to image topics.
Despite their different internal implementations, they all ultimately return an observation dictionary with
the exact same structure, containing standardized fields like qpos, qvel, and images.
Through this RealEnv abstraction, upper-level applications in the Open TeleDex can completely ignore
which specific arm or hand is being used. To support a new robot in the future, a developer only needs
to implement a new Recorder class following the RealEnv interface, without any modification to the core
logic. This "define once, use everywhere" plug-in design is the technical cornerstone of Open TeleDex’s
high extensibility and hardware agnosticism.
3.5
Summary
Open TeleDex is engineered as a unified and general-purpose teleoperation framework designed to support
a wide spectrum of hardware and application scenarios. Its key features include:
• TripleAny Compatibility: The framework is architected to be fundamentally hardware-agnostic,
supporting diverse combinations of robotic arms (AnyArm), dexterous hands (AnyHand), and mas-
ter input devices (AnyExternalDevice), from high-fidelity master-slave systems to low-cost VR con-
trollers.
10
Figure 3: This shows the task of our evaluation system. The top three images are Bottle Peg in Hole, and
the bottom three images are Cube Assembly
• Multi-Modal Data Pipeline: Open TeleDex features a high-performance data collection pipeline,
purpose-built for imitation learning. It provides robust, software-based time synchronization and
validation for multi-modal data streams, including kinematic states, multi-view images, and depth
data.
• Modular, ROS2-Native Architecture: Built entirely on ROS2, the system leverages a decoupled,
three-tier architecture. This modularity, centered around a standardized RealEnv abstraction layer,
vastly simplifies the integration of new hardware.
• Flexible Control Strategies: The framework can seamlessly switch between different control paradigms
— such as direct joint-space mapping for low latency and Cartesian-space control for flexibility—based
on the user’s configuration.
• AI-Ready, Structured Data Output: All collected data is saved in a well-defined, structured format
(e.g., video-compressed or HDF5) with comprehensive metadata, making it directly usable for training
downstream imitation learning policies.
4
System Evaluation
This section presents a rigorous evaluation of the Open TeleDex system, focusing on its capacity to support
the collection of data for complex single-arm manipulation and to handle the challenges inherent in pose
alignment and precise object placement. The evaluation is conducted across two benchmark tasks: Bottle
Peg-in-Hole and Cube Assembly.
4.1
Evaluation Setup and Metrics
4.1.1
Task Description
Task 1: Bottle Peg-in-Hole. This task is to evaluate the system’s ability to handle challenges in pose
alignment when manipulating a non-rigid object. The operator is required to control the robot to
grasp a water bottle (approximately 6cm in diameter) from the side. The robot must then insert the
11
bottle into a square base (approximately 7cm in width and 4cm in height) placed horizontally on
a tabletop. The initial horizontal distance between the two objects on the testbed is approximately
20cm.
Task 2: Cube Assembly. To evaluate the system’s capability for precise object grasping and placement,
the operator is required to control the robot to pick up one half of a cube from the tabletop. The robot
must then place it on top of the other half to assemble a complete cube. The initial horizontal distance
between the two parts on the testbed is approximately 10cm.
4.1.2
Hardware Configuration
As show in table 1, the evaluation utilizes two distinct, principal robot configurations: (1) In-House De-
veloped (TeleDex Validation Set): This configuration features our custom-built Self-Developed Robotic
Arm (Linker Arm A7) paired with the Self-Developed Dexterous Hand (LinkerHand O6). (2) Commercial
Off-The-Shelf (COTS) Baseline Set: This configuration utilizes mainstream, industry-standard equipment,
specifically the Agilex Piper 7-DOF Arm and the LinkerHand O6 Dexterous Hand.
Table 1: Hardware Configurations for System Evaluation
Experimental Group
Robotic Arm
Dexterous Hand
Master Device (Input)
COTS Arm
Agilex Piper (6-DoF)
LinkerHand O6
UdexReal Glove
In-house Arm
Linker Arm A7 (7-DoF)
LinkerHand O6
UdexReal Glove
4.1.3
Evaluation Metrics
To quantitatively evaluate the performance of each system configuration, we measured a set of metrics
focused on task efficiency and data quality. The key metrics are defined as follows:
Average Task Duration (s) This metric measures the overall efficiency of the teleoperation system. For
each successful trial, we define the task duration as the elapsed time from the moment the robot hand
begins its motion towards the object to the moment the task is successfully completed (e.g., the bottle
is inserted or the cube is assembled). The final value is computed by averaging the durations across
all successful trials for a given task and configuration. A lower value indicates higher operational
efficiency.
Synchronization Success Rate (%) This metric quantifies the reliability of our data collection pipeline. As
detailed in Section 3.3.1, our TimeSyncManager performs a consistency check on the timestamps
from all sensor streams at every single timestep. A timestep is marked as "successful" if the maximum
time difference (max_diff) among all sensor timestamps is within our preset tolerance (100ms).
The Synchronization Success Rate is the percentage of successfully synchronized timesteps out of
the total timesteps in an entire demonstration episode. A higher value indicates greater temporal
consistency in the collected data.
Average Synchronization Error (ms) This metric measures the temporal precision of the successfully
synchronized data. It is calculated by averaging the max_diff values (the maximum time difference
between the earliest and latest sensor timestamp in a single frame) across all successfully synchronized
timesteps within an episode. This value represents the average "temporal blur" of our multi-modal data
snapshots. A lower value signifies higher-quality, more precisely aligned data, which is critical for
learning time-sensitive manipulation skills.
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Results
The performance results for the two baseline tasks, conducted on two distinct hardware config-
urations, are summarized in Table 2. Each task was performed for 20 trials on its respective system. The
table presents key metrics for task efficiency (Avg. Duration) and data quality (Sync Success and Error).
Table 2: Performance and Data Quality Metrics Across Different Tasks and Hardware Configurations
Configuration / Task
Trials
Avg. Duration (s)
Sync Success (%)
Avg. Sync Error (ms)
COTS Arm Setup: Agilex Piper Arm + LinkerHand O6
Bottle Peg-in-Hole
20
20.06 (±2.88)
99.27
64.38
Cube Assembly
20
24.53 (±3.85)
99.99
60.77
In-house Arm Setup: Linker Arm A7 + LinkerHand O6
Bottle Peg-in-Hole
20
15.28 (±2.42)
99.98
42.75
Task Efficiency and Hardware Performance. Our results first highlight the system’s ability to capture
task complexity. On the same hardware (COTS Arm Setup), the Cube Assembly task required a significantly
longer completion time (24.53s) than the Bottle Peg-in-Hole task (20.06s), which aligns with the higher pre-
cision demands of assembly. More importantly, the results demonstrate a clear performance differentiation
between hardware configurations. On the same Bottle Peg-in-Hole task, the system configured with our in-
house arm was approximately 24% faster on average than the one with the Agilex Piper arm (COTS Arm
Setup). This suggests that the tighter integration and potentially lower-level optimizations of the in-house
hardware contribute to a more efficient teleoperation experience.
Data Quality and System Robustness. The data quality metrics underscore the robustness of the
Open TeleDex data collection pipeline. Across all tasks and hardware configurations, the Synchronization
Success Rate remained above 99.2%, indicating that our TimeSyncManager module consistently and
reliably aligns multi-modal data streams within the predefined 200ms tolerance. This level of reliability is
critical for generating large, clean datasets for imitation learning.
Furthermore, the Average Synchronization Error provides a deeper insight into the system’s internal
temporal precision. The in-house arm configuration not only performed tasks faster but also exhibited a
significantly lower average synchronization error (42.75ms) compared to the Agilex Piper configuration
(60-65ms range). A sync error of about 40-65ms, roughly equivalent to 1-2 control cycles at our 25Hz
frequency, represents a solid baseline for a software-based synchronization mechanism in a complex ROS2
environment. This quantitatively demonstrates that our pipeline can maintain a consistent and measurable
level of data quality, and reveals how hardware choice directly impacts the temporal fidelity of the final
dataset.
5
Conclusion and Future Work
5.1
Conclusion
In this technical report, we addressed the challenge of collecting high-quality, diverse demonstration data for
robot imitation learning by proposing and implementing a teleoperation framework named Open TeleDex.
Our work centers on two core contributions, which have been validated through our system’s design and a
series of quantitative experiments.
First, we contributed a unified, device-agnostic teleoperation framework that provides an end-to-end
workflow for robot learning. Through a layered, decoupled software architecture and a standardized hard-
ware abstraction layer (RealEnv), Open TeleDex successfully breaks down the barriers between different
13
hardware ecosystems. Crucially, this framework encapsulates the full pipeline from the synchronized col-
lection of multi-modal data to its structured storage. The results from our data quality analysis, show-
ing synchronization success rates consistently above 99%, confirm that our AI-oriented data collection
pipeline ensures the intrinsic quality of the collected data, providing a solid foundation for training high-
performance imitation learning policies. Our experimental evaluation, using two distinct hardware configu-
rations, demonstrated the framework’s capability to seamlessly integrate and operate heterogeneous systems
in practice, strongly supporting the TripleAny vision.
Second, we introduced a novel general motion mapping algorithm. This algorithm surpasses the tradi-
tional "point-to-point mapping" paradigm by generating holistic postures, which provides more natural and
functional motion retargeting between morphologically different human and robot hands. Our successful
demonstrations on complex tasks like Cube Assembly and Bottle Peg-in-Hole serve as a practical validation
of this algorithm’s effectiveness.
5.2
Future Work
While Open TeleDex lays a solid foundation for generalized teleoperation data collection, there is still ample
room for exploration. Our future work will primarily focus on the following directions:
Rigorous Benchmark.
We plan to conduct a rigorous and comprehensive performance benchmark. This
involves performing standardized human-computer interaction experiments, such as Fitts’ Law tests, to
quantitatively measure key metrics like system throughput (IP). We will also analyze the trajectory smooth-
ness (Jerk) of the collected data to further quantify demonstration quality. These in-depth metrics will allow
for a more nuanced comparison against state-of-the-art frameworks like ALOHA and AnyTeleop.
Comprehensive User Experience (UX) Study.
A comprehensive user experience (UX) study is planned
to evaluate the human-in-the-loop aspects of the system. We will collect subjective metrics, such as operator
workload using the NASA-TLX [20] questionnaire, to assess the intuitiveness and ergonomiscs of different
hardware configurations.
Expansion of the Hardware Ecosystem.
We will continue to expand the TripleAny hardware ecosys-
tem by providing official Recorder implementations for more mainstream robotic arms, dexterous hands,
and sensors, and open-sourcing them to the community to further lower the research barrier.
Collaborative Teleoperation.
One of our most exciting goals is to explore and implement the "one-
master-to-multiple-slaves" collaborative teleoperation mode. TeleDex’s distributed architecture based on
ROS2 provides a natural foundation for this objective. We plan to develop a high-level task allocation
and collaborative control module to dramatically increase the diversity and efficiency of data collection for
training general-purpose robot policies.
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15
Appendix
A1
Open TeleDex Hardware Ecosystem
The following table lists the hardware devices that the Open TeleDex framework is designed to support,
including those already integrated and those planned for future expansion. The modular architecture of
Open TeleDex, particularly its RealEnv abstraction layer, facilitates the integration of new hardware by
implementing a corresponding Recorder class.
Table 3: Supported and Planned Hardware for the Open TeleDex Ecosystem.
Category
Model / Type
Notes
Robotic Arms
Agilex Piper
6-DoF arm. Core testbed component.
Linker Arm A7
7-DoF arm. Core testbed component.
RealMan
Planned support.
Universal Robots (UR)
Planned support.
Dexterous Hands
LinkerHand O6
6-DoF hand. Core testbed component.
LinkerHand L10
10-DoF hand. Core testbed component.
LinkerHand L20
20-DoF hand. Core testbed component.
LinkerHand L30
Planned support.
Teleoperation & Sensor Devices
Master-Slave Arm / Glove
High-fidelity input device. Integrated.
VR Controllers (e.g., PICO, Meta Quest)
Low-cost immersive input device.
Inte-
grated.
UdexReal Haptic Glove
High-fidelity haptic/motion capture glove.
Integrated.
Exoskeleton
Planned support.
Intel RealSense D455
RGB-D Camera. Core testbed component.
Intel RealSense D405
High-precision, short-range RGB-D Camera.
Verified.
16
A2
Example of a Typical Task Dataset Composition
This appendix details the complete structure of a structured dataset generated by the Open TeleDex frame-
work, using a single episode (episode_000015) from a representative grasping task as an example.
Episode Metadata: this table summarizes the complete hardware and software environment, along with
the collection parameters recorded for the episode, ensuring full reproducibility.
Table 4: Metadata recorded for episode_000015.
Category
Parameter
Value / Description
Episode Info
Task Name
linkerhand_piper_grasp
Episode ID
episode_000015
Session ID
session_2025-10-10_13-46-35
Duration (sec)
19.64
Timesteps
491
Hardware Config
Preset
piper_linkerhand_o6_single
Arm Type
Agilex Piper (6-DoF)
Hand Type
LinkerHand O6 (6-DoF)
Total DoF
12
Camera Config
Preset
intel_d455_single_top
Camera Type
Intel RealSense D455
Position
Top-down (1280x720 @ 30fps)
Collection Params
Control Freq. (Hz)
25 Hz (dt = 0.04s)
Video Codec
libx264 (high quality)
Data Structure
Format
video_compressed
qpos Dimension
12 (6 arm + 6 hand)
Camera Streams
cam_top
Tactile Sensors
right_hand
17
|
Technical Report: Open TeleDex: A Hardware-Agnostic Teleoperation System for Imitation Learning based Dexterous Manipulation Xu Chia,1, Chao Zhanga,1, Yang Suc,1, Lingfeng Doub, Fujia Yanga, Jiakuo Zhaoa, Haoyu Zhouc, Xiaoyou Jiac, Yong Zhouc, and Shan Ana,* aTianjin University bTsinghua University cLinkerBot Inc. 1These authors contributed equally to this work. *Corresponding author: October 17, 2025 Abstract Accurate and high-fidelity demonstration data acquisition is a critical bottleneck for deploying robot Imitation Learning (IL) systems, particularly when dealing with heterogeneous robotic platforms. Existing teleoperation systems often fail to guarantee high-precision data collection across diverse types of teleoperation devices. To address this, we developed Open TeleDex, a unified teleoperation framework engineered for demonstration data collection. Open TeleDex specifically tackles the TripleAny challenge, seamlessly supporting any robotic arm, any dexterous hand, and any external input device. Furthermore, we propose a novel hand pose retargeting algorithm that significantly boosts the interoperability of Open TeleDex, enabling robust and accurate compatibility with an even wider spectrum of heterogeneous master and slave equipment. Open TeleDex establishes a foundational, high-quality, and publicly available platform for accelerating both academic research and industry development in complex robotic manipulation and IL. 1 Introduction Executing a wide spectrum of complex, dexterous manipulation tasks in diverse, unstructured environments remains a central challenge in modern robotics. Although Imitation Learning (IL) has shown great potential in tackling this problem, its progress is fundamentally constrained by the difficulty of collecting large-scale, multi-modal, and high-quality demonstration data [1]. The quality and scalability of this data hinge entirely on the underlying teleoperation system. These systems are designed around a series of complex trade-offs, often balancing three core dimensions: Fidelity (the precision of control and data quality), Accessibility (cost and deployment complexity), and Generalization (hardware compatibility). Current solutions consistently compromise on at least one dimension. For instance, high-fidelity systems, such as dedicated master-slave arms, offer unmatched precision and minimal latency, making them indispensable in zero-tolerance applications. However, this tight electromechanical coupling inherently limits their Generalization, as they exist as costly, "closed ecosystems" tailored to a single robot model, making cross-platform data aggregation impractical. Conversely, systems prioritizing 1 16 Oct 2025 Accessibility by leveraging consumer-grade VR/AR devices have significantly reduced the technical barriers to data collection. Despite this, they frequently compromise on Fidelity due to challenges like visual tracking uncertainties, communication latency, and the absence of physical force feedback. Furthermore, even these more accessible platforms often require substantial software adaptation and parameter tuning when faced with a new robotic arm or hand, revealing a critical generalization gap in practice. This lack of a unified framework supporting heterogeneous robotic platforms across the entire data collection workflow is the core bottleneck hindering the evolution of IL research prototypes into scalable, general-purpose data collection platforms. To address this, we introduce Open TeleDex, a device-agnostic and highly versatile teleoperation framework. Our core design tenet is Generalization, achieved through a natively ROS2-based architecture that intrinsically unifies diverse control paradigms, allowing for the rapid and seamless integration of new hardware combinations. Our work makes two core contributions: 1. A Unified, Hardware-Agnostic Teleoperation Framework: We present a decoupled, three-tier architecture that supports synchronous data collection from a wide variety of robotic arms, multifingered dexterous hands, and external input devices. Open TeleDex serves as an extensible backbone, allowing researchers to efficiently integrate and switch between diverse heterogeneous hardware combinations. 2. A Hand Pose Retargeting Algorithm: We propose a novel retargeting algorithm that adjusts for kinematic discrepancies between human and robot hands. Our algorithm is designed to enhance the framework's compatibility. This report is organized as follows: Section 2 the reviews related work. Section 3 details Open TeleDex framework. Section 4 shows the evaluation of our system. Finally, Section 5 concludes the report and discusses future work. 2 Related Work 2.1 Teleoperation Systems Teleoperation systems are a crucial tool for collecting expert demonstrations for imitation learning, and their design inherently involves a series of complex trade-offs. The efficacy of any such system can be evaluated along three core dimensions. The first is Fidelity, which measures the precision with which a system conveys operator intent and relays environmental feedback; this remains a central focus in modern telerobotics research [2]. The second is Accessibility, encompassing its cost, deployment complexity, and ease of use, which are critical factors for practical application [3]. The third, and increasingly critical dimension, is Generalization-the system's capacity to adapt to diverse robot hardware, tasks, and environments, a key challenge in modern human-robot interaction [4]. Existing solutions in the field typically exhibit distinct strengths and compromises across these dimensions. Systems prioritizing ultimate fidelity, such as dedicated master-slave arms, have set the gold standard for latency and haptic "transparency", making them indispensable in zero-tolerance applications like robotic surgery [5]. Pioneering systems like ALOHA [6] have democratized data collection by drastically reducing the financial and technical barriers to entry for dexterity research. Through tight electromechanical coupling and direct joint-space mapping, they minimize information loss during transmission. However, this design philosophy inherently constrains their generalization. They often exist as "closed ecosystems" tailored to a single robot model, where any hardware modification can demand costly redesigns, rendering cross-platform deployment and data aggregation impractical. 2 In recent years, a wave of research has focused on improving accessibility by leveraging consumergrade VR/AR devices and standard cameras. These systems typically employ more flexible Cartesian-space control. Despite their success in accessibility, they often compromise on fidelity, facing challenges from visual tracking uncertainties, communication latency, and the absence of physical force feedback [7, 8]. More importantly, while more general in theory, many open-source implementations still require substantial software adaptation and parameter tuning for specific robots, revealing a generalization gap in practice. It is thus evident that generalization has emerged as the core bottleneck connecting the worlds of high fidelity and high accessibility, and hindering the evolution of teleoperation technology from one-off research prototypes to scalable data collection platforms. Addressing this challenge necessitates the development of a universal software framework. Works like AnyTeleop [9] have made important strides in this direction with learning-free, GPU-accelerated components. However, there remains a need for a framework that is natively rooted in the ROS2 ecosystem and is architected to intrinsically unify diverse control paradigms, from direct mapping to planner-centric control. The TeleDex framework presented in this paper aims to build such a unified teleoperation framework with generalization as its core design principle. It does not seek to invent a new interaction modality, but rather to integrate and empower the diverse existing hardware ecosystem by providing a modular, extensible software backbone, thereby offering a truly scalable data collection solution for robot imitation learning. 2.2 Hand Pose Retargeting The real-time and accurate transfer of a human operator's natural movements to a morphologically different robot is a core challenge in general-purpose teleoperation. This challenge can be decomposed into two main sub-problems: arm motion generation and dexterous hand pose retargeting. For arm motion generation, mainstream approaches rely on Inverse Kinematics (IK) solvers or more advanced real-time motion planners [9, 10]. These methods all require a precise kinematic model of the robot (e.g., a URDF file) to compute joint trajectories that match the operator's end-effector pose intent. Dexterous hand posture retargeting is a more complex challenge. Existing methods, whether optimizationbased [9, 11] or learning-based [12, 13], mostly follow an analytical paradigm: they pre-define keypoint or joint correspondences between the human and robot hands and then minimize the error between them. While intuitive, the fundamental limitation of this "point-to-point mapping" approach lies in its rigid correspondence. When the robot hand's morphology differs significantly from a human's, this mapping can easily produce unnatural or physically infeasible postures, and the process of configuring these correspondences for new hardware is tedious. To overcome the limitations of this "rigid mapping" paradigm, we propose a novel "generative" hand retargeting algorithm. Instead of copying each joint motion, our method treats the robot hand as an integrated skeleton constrained by its own kinematic model. Our algorithm drives this skeleton to "grow" naturally to functionally match the human's overall grasping intent, rather than mimicking it point-by-point morphologically. This shift from "copying motion" to "reproducing function" enables our framework to generate more natural and functional postures for morphologically diverse dexterous hands. 2.3 Demonstration Data for Imitation Learning The success of Imitation Learning (IL) is critically dependent on the quality of demonstration data [14]. Research has shown that high-quality demonstrations, characterized by smooth motions and precisely synchronized state-action pairs, can significantly improve the final policy's performance and robustness [15,16]. Therefore, building a collection framework that systematically ensures data quality is essential. In practice, ensuring data quality faces two main challenges. The first is time synchronization. In a system with heterogeneous sensors like cameras and encoders, ensuring a common and precise time base for 3 Figure 1: The Framework of Open TeleDex. The diagram illustrates the layered architecture of the system, which is designed to support heterogeneous robot platforms for different requirement of data collection. The system is structurally divided into three modules: Perception, Control and Data Collection, and Hardware. all data streams is a core difficulty. Simple recording schemes can easily introduce temporal misalignments that are detrimental to learning dynamic tasks [17]. The second challenge is the richness of data modality. While traditional master-slave systems provide high-quality kinematic data, modern IL increasingly benefits from multi-modal information, including multi-view images and force/tactile feedback [18]. Emerging paradigms like VR teleoperation (e.g., Holo-Dex [11]) are gaining attention for their natural ability to fuse multi-modal data. This necessitates a modern data collection framework that not only features high-performance realtime processing and precise time synchronization but also natively supports the fusion of various collection methods in its architecture to meet the growing demand for high-quality, multi-modal, and diverse data. 3 Framework of Open TeleDex 3.1 System Architecture As shown in Fig. 1, the architecture of Open TeleDex follows the core design principles of layering and decoupling, with the ultimate goal of realizing our TripleAny vision: supporting the seamless integration and cooperation of AnyExternalDevice, AnyArm, and AnyHand. It primarily utilizes the ROS2 framework [19] for distributed communication. Conceptually, the architecture is divided into three core modules: 4 1. Perception, responsible for standardizing heterogeneous input data from Any External Device. 2. Control & Data Collection, which executes hardware-independent task management, control strategy scheduling, and data processing. This core layer is "agnostic" to the specific arm and hand, which is key to achieving AnyArm and AnyHand. 3. Hardware, which translates standardized control commands for specific AnyArm and AnyHand actuators. 3.2 Perception The Perception component is responsible for capturing and interpreting the operator's intent and movement, forming the input signals for the entire teleoperation system. Master Device: this module encapsulates various human interface devices such as Virtual Reality (VR) headsets (for spatial tracking), Teleoperation Gloves (for finger articulation capture), and Master Arms. Sensor Fusion: Raw data from disparate input sensors is fused and filtered to generate a coherent, lowlatency stream of the operator's hand and arm pose. Hand Pose Retargeting: this is a high-level decision module responsible for transforming the continuous human pose stream into a discrete, feasible goal pose for the robotic Slave devices. To achieve precise and adaptable teleoperation for robotic hands, we construct a modular optimization framework. The core of this framework is a total cost function, which is a weighted sum of three distinct sub-functions: a pose alignment cost, an inter-finger coupling cost, and a temporal smoothness cost. By minimizing this total cost function, we can solve for the optimal robot joint angle vector, θ∗(t), in real-time. Optimization Objective: The optimal robot joint angle vector, θ∗(t), is found by solving the following minimization problem: θ∗(t) = arg min θ(t) αalignLalign(θ(t)) + αcoupleLcouple(θ(t)) + αsmoothLsmooth(θ(t)) (1) Where: • θ(t) ∈Rn is the robot hand's joint angle vector at time t. • Lalign, Lcouple, and Lsmooth represent the three core cost functions. • αalign, αcouple, and αsmooth are hyperparameters (weights) that balance the importance of each cost term. Cost Function: Pose Alignment Cost (Lalign). This cost function aims to minimize the geometric error between the operator's hand keypoints and the corresponding keypoints on the robot hand. It is central to achieving the fundamental pose mimicry. Lalign(θ(t)) = X (i,j)∈K ∥p′ i,j(t) -Fk,j(θ(t))∥2 (2) Here, p′ i,j(t) ∈R3 represents the position of the operator's j-th keypoint on the i-th finger after transformation and calibration, while Fk,j(θ(t)) is the robot's forward kinematics function, which calculates the position of its corresponding keypoint. Operator Keypoint Transformation. To compute the transformed keypoints p′ i,j(t), we must first scale and translate the raw detected keypoints, pi,j(t), to match the robot's geometry. 5 1. Dimensional Scaling Factor (si,j): This calculates the relative length ratio for each phalangeal segment. si,j = ∥Fk,j+1(θ0) -Fk,j(θ0)∥ ∥ ̄pi,j+1 - ̄pi,j∥ (3) Here, ̄pi,j and θ0 represent the human keypoints and robot joint angles from a static calibration pose, respectively. 2. Finger Root Translation Vector (δi): This calculates the offset for the base position of each finger. δi = Fk,i,mcp(θ0) - ̄pi,mcp (4) 3. Transformed Keypoint (p′ i,j): Finally, the transformed points are obtained via the following iterative formula: p′ i,j(t) = pi,0(t), j = 0 p′ i,j-1(t) + si,j-1(pi,j(t) -pi,j-1(t)) + δi, j = 1 p′ i,j-1(t) + si,j-1(pi,j(t) -pi,j-1(t)), j ≥2 (5) Inter-Finger Coupling Cost (Lcouple). This cost function is used to maintain human-like coordination between the thumb and other fingers during fine motor actions such as grasping and pinching. Lcouple(θ(t)) = X i∈I βi(t)∥⃗RH,i(t) -⃗RR,i(θ(t))∥2 (6) Here, ⃗RH,i(t) = pi,fingertip(t) -pthumb,fingertip(t) is the relative vector from the operator's thumb tip to other fingertips. ⃗RR,i(θ(t)) is the corresponding relative vector on the robot hand. The term βi(t) is an adaptive weight that dynamically adjusts the influence of this cost term based on inter-finger distance. Adaptive Weighting (βi(t)). The weight's magnitude depends on the proximity of the fingers. We first compute a normalized proximity metric ρi(t) ∈[0, 1]: ρi(t) = 1 -∥⃗RH,i(t)∥-ρmin,i ρmax,i -ρmin,i (7) where ρmin and ρmax are the minimum and maximum inter-finger distances determined from calibration data. This metric is then mapped through a Sigmoid function to produce the final weight: βi(t) = 1 1 + e-σ(ρi(t)-τ) (8) where σ and τ are parameters controlling the function's shape and activation threshold. Temporal Smoothing Cost (Lsmooth). This term acts as a regularizer to penalize abrupt changes in the robot's joint angles, ensuring the motion trajectory is fluid and stable. Lsmooth(θ(t)) = ∥θ(t) -θ(t -1)∥2 (9) 3.3 Control and Data Collection The core of TeleDex is the integrated control and data collection pipeline, which functions as the central synchronization and serialization engine. This component is further structured into a three-layer software architecture: the Control Layer (L3), the Data Integration Layer (L2), and the Data Storage Layer (L1). 6 3.3.1 Control Layer This layer is centered around a primary ROS2 service node, which embodies the main responsibilities of the core control layer, including task manager, hardware manager, and multi-sensor timeSync manager, and data collection coordinator. In a typical data collection workflow, an operator first specifies a hardware combination in a configuration file. Upon system launch, the corresponding ROS2 nodes in the perception component are activated, reading raw data from their respective hardware (e.g., VR devices or a master-slave arm) and publishing it as standardized ROS2 messages. The control layer subscribes to these standard messages. Its internal Control Strategy Scheduler then loads and applies the appropriate control algorithm based on the preset mode in the configuration. For instance: (1) For a high-fidelity, master-slave configuration (e.g., Agilex arm + LinkerHand L10/O6 + master arm/glove), the scheduler employs a direct joint-space mapping strategy to achieve minimal latency. (2) For a low-cost configuration with mismatched kinematics (e.g., RealMan arm + LinkerHand L10 + VR controller), the scheduler uses a hybrid control strategy: an optimization-based retargeting algorithm for the hand and Cartesian-space relative motion control for the arm. The standardized robot control commands generated by this layer are then published, to be subscribed to and executed by the corresponding robot driver node in the hardware component. Concurrently, a parallel data collection pipeline captures all sensor data and control commands. A time synchronization module within this pipeline aligns and validates timestamps from different sources. Data Synchronization Mechanism: When collecting high-quality data for imitation learning, ensuring precise temporal alignment of data from multiple heterogeneous sensors is crucial. Even a minor time deviation can lead to incorrect state-action pairing, severely impacting the learning effectiveness of downstream policies. To address this challenge, Open TeleDex implements a robust software-based timestamp synchronization (TimeSync Manager) and validation mechanism at the application layer, built upon ROS2's distributed clock service. TimeSync Manager is integrated throughout the data collection process: 1. Establishment of a Global Time Base: Upon system startup, the TimeSyncManager initializes and waits for ROS2's global clock to become stable. This ensures that all ROS2 nodes in the system share a single, synchronized time source, which is the foundation for all subsequent synchronization operations. 2. Per-Frame Multi-Source Timestamp Collection and Validation: In each control cycle of data collection, the system performs a rigorous timestamp synchronization check: • Gathering: The RealEnv interface collects timestamps for the current frame of data from all active Recorder modules. These timestamps come from various sources: camera and dexterous hand timestamps are from their ROS2 message headers, while the robotic arm's timestamp may come directly from its underlying SDK. • Validation: The core function of TimeSyncManager verifies the collected set of timestamps. This validation includes two key checks: a. Freshness Check: Ensures all timestamps are "recent" (e.g., within 1 second of the current ROS time) to filter out stale data caused by node freezes or network issues. b. Consistency Check: Calculates the maximum difference among all timestamps in the set and checks if this difference is within a preset tolerance (e.g., 100ms). Under typical operating conditions, our system consistently keeps this difference below 70ms (tp99). 7 3. Data Tagging and Storage: The result of the validation-including a success flag, the max_diff, and all original timestamps-is packaged as a "sync validation bundle" and stored within the current frame's observation data. This metadata is ultimately saved into the telemetry.npz file along with qpos, qvel, etc. Through TimeSync Manager, Open TeleDex achieves an active and traceable data quality assurance that goes beyond a standard rosbag. It not only strives to ensure data synchronicity at the time of collection but, more importantly, provides a synchronization quality metric for every single frame of data. During subsequent model training, researchers can easily filter out unsynchronized timestamps data with unsynchronized timestamps using the sync_validation_is_valid flag, or weight the data based on the max_diff value. This verifiable, end-to-end time synchronization loop is a key feature of Open TeleDex as a professional data collection framework for robot imitation learning. 3.3.2 Data Integration Layer This layer acts as the Data Plane, specializing in acquisition, buffering, and data serialization. It is explicitly decoupled from the specific hardware and file structure. Data acquisition node is a comprises multiple, independent Recorder modules (Arm Recorder, LinkerHand Recorder, Camera Recorder, Scalable Interface). Each Recorder subscribes to a specific hardware topic from L1, applies the globally synchronized timestamp received from the L3 TimeSync Manager, and pushes the data into the persistence buffer. Data persistence node is responsible for structuring and writing the buffered data to disk. It efficiently encodes and stores the synchronized, multi-modal data into a structured format (e.g., video-compressed or HDF5) with complete metadata, ready for downstream imitation learning tasks. The operator can control the start and stop of data collection via keyboard commands. To support a new robotic arm, a developer only needs to implement a new Recorder class that adheres to the RealEnv interface, without modifying the core data collection logic. This design significantly lowers the barrier to integrating new hardware into the Open TeleDex ecosystem. 3.3.3 Data Storage Layer This layer defines the final output structure and the foundational storage hierarchy, ensuring the dataset is immediately usable by learning frameworks. Standard Storage Node enforces a consistent, hierarchical file structure. It defines the output files: Telemetry Joint Data (telemetry.npz), Cameras Data/Intrinsics (video files and calibration parameters, camera_info.json), Episode Manifest(manifest.json), and Metadata Global Config (metadata.json). 3.4 Hardware 3.4.1 Hardware Components This component represents the physical devices and the low-level software interfaces required for execution and sensing. Fig. 2 shows some of the hardware components supported by Open TeleDex. 1. Camera Driver: Standardized drivers (e.g., for RealSense) that interface directly with the camera hardware and publish synchronized RGB-D streams via ROS 2 topics. 2. Arm Driver: Low-level controllers responsible for converting high-level velocity or joint position commands from L2 into motor signals for the mechanical arm (e.g., Piper/Linker Arm). 8 Figure 2: Some of the Hardware Components Supported by Open TeleDex. 3. LinkerHand Driver: The specific SDK or API that provides control and state feedback for the LinkerHand, handling intricate finger articulation commands and sensor readings. 4. Sensor Data Stream: The continuous feedback loop, providing the instantaneous state (joint angles, end-effector forces, visual frames) back to the L2 Data Acquisition Node for synchronized logging and, if implemented, real-time closed-loop control. 3.4.2 Heterogeneous Hardware Interface To achieve broad hardware compatibility, we designed a core heterogeneous hardware interface. It uses software abstraction to shield the system from underlying physical hardware differences, providing a stable and unified interaction logic for upper-level applications. Its design follows two core principles: Master-side input normalization and Slave-side device abstraction. Master Input Normalization: A Unified Expression of Operator Intent The primary challenge at the teleoperation front-end (Master side) is to handle input signals from vastly different sources and in disparate formats-for instance, data gloves output high-dimensional joint angles, VR devices provide hand keypoint poses via UDP streams, while master arms directly output their own joint states. To mask this heterogeneity, Open TeleDex introduces an intermediary "intent parsing and normalization" step. Regardless of the input device, its raw data stream is first processed by a dedicated front-end adapter node. In our implementation, this is an independent ROS2 node named hand_retarget. 9 The core responsibility of this node is to uniformly translate these various input signals into a standardized "intent signal". This signal represents the operator's core intent and is published via a fixed ROS2 topic. Specifically: • For input from a VR controller, this node runs our hand retargeting algorithm and a Cartesian-space relative motion control strategy, driving the dexterous hand with the target hand pose and the robotic arm with the wrist keypoint data. • For input from a data glove and a master arm, the node executes our direct joint-space mapping control strategy, mapping their poses to the standard target joint angles for the dexterous hand and robotic arm. Slave Device Abstraction: A Unified Robot Environment Interface At the teleoperation back-end (Slave side), the challenge is to control robots of different models, degrees of freedom, and communication protocols with a single, unified logic. To enable broad compatibility for AnyArm and AnyHand, Open TeleDex implements an environment abstraction layer named RealEnv. This design aims to provide a unified, high-level API for all physical robot hardware. RealEnv itself is an abstract base class that defines standard methods for interacting with a robot, such as getting the current robot state and sending an action command to the robot. For each supported piece of hardware (e.g., the Agilex Piper arm or the Linker arm A7), we implement a corresponding Recorder subclass (e.g., PiperRecorder). This Recorder subclass acts as a "device driver". It handles all low-level details of interacting with a specific hardware SDK or underlying ROS driver, while strictly adhering to the interface defined by RealEnv. For instance, when the upper-level data collection logic calls get_observation(): • PiperRecorder queries the encoders via its SDK to get joint states. • LinkerHandRecorder subscribes to the relevant ROS2 topic for the latest joint messages. • RealSenseRecorder subscribes to image topics. Despite their different internal implementations, they all ultimately return an observation dictionary with the exact same structure, containing standardized fields like qpos, qvel, and images. Through this RealEnv abstraction, upper-level applications in the Open TeleDex can completely ignore which specific arm or hand is being used. To support a new robot in the future, a developer only needs to implement a new Recorder class following the RealEnv interface, without any modification to the core logic. This "define once, use everywhere" plug-in design is the technical cornerstone of Open TeleDex's high extensibility and hardware agnosticism. 3.5 Summary Open TeleDex is engineered as a unified and general-purpose teleoperation framework designed to support a wide spectrum of hardware and application scenarios. Its key features include: • TripleAny Compatibility: The framework is architected to be fundamentally hardware-agnostic, supporting diverse combinations of robotic arms (AnyArm), dexterous hands (AnyHand), and master input devices (AnyExternalDevice), from high-fidelity master-slave systems to low-cost VR controllers. 10 Figure 3: This shows the task of our evaluation system. The top three images are Bottle Peg in Hole, and the bottom three images are Cube Assembly • Multi-Modal Data Pipeline: Open TeleDex features a high-performance data collection pipeline, purpose-built for imitation learning. It provides robust, software-based time synchronization and validation for multi-modal data streams, including kinematic states, multi-view images, and depth data. • Modular, ROS2-Native Architecture: Built entirely on ROS2, the system leverages a decoupled, three-tier architecture. This modularity, centered around a standardized RealEnv abstraction layer, vastly simplifies the integration of new hardware. • Flexible Control Strategies: The framework can seamlessly switch between different control paradigms - such as direct joint-space mapping for low latency and Cartesian-space control for flexibility-based on the user's configuration. • AI-Ready, Structured Data Output: All collected data is saved in a well-defined, structured format (e.g., video-compressed or HDF5) with comprehensive metadata, making it directly usable for training downstream imitation learning policies. 4 System Evaluation This section presents a rigorous evaluation of the Open TeleDex system, focusing on its capacity to support the collection of data for complex single-arm manipulation and to handle the challenges inherent in pose alignment and precise object placement. The evaluation is conducted across two benchmark tasks: Bottle Peg-in-Hole and Cube Assembly. 4.1 Evaluation Setup and Metrics 4.1.1 Task Description Task 1: Bottle Peg-in-Hole. This task is to evaluate the system's ability to handle challenges in pose alignment when manipulating a non-rigid object. The operator is required to control the robot to grasp a water bottle (approximately 6cm in diameter) from the side. The robot must then insert the 11 bottle into a square base (approximately 7cm in width and 4cm in height) placed horizontally on a tabletop. The initial horizontal distance between the two objects on the testbed is approximately 20cm. Task 2: Cube Assembly. To evaluate the system's capability for precise object grasping and placement, the operator is required to control the robot to pick up one half of a cube from the tabletop. The robot must then place it on top of the other half to assemble a complete cube. The initial horizontal distance between the two parts on the testbed is approximately 10cm. 4.1.2 Hardware Configuration As show in table 1, the evaluation utilizes two distinct, principal robot configurations: (1) In-House Developed (TeleDex Validation Set): This configuration features our custom-built Self-Developed Robotic Arm (Linker Arm A7) paired with the Self-Developed Dexterous Hand (LinkerHand O6). (2) Commercial Off-The-Shelf (COTS) Baseline Set: This configuration utilizes mainstream, industry-standard equipment, specifically the Agilex Piper 7-DOF Arm and the LinkerHand O6 Dexterous Hand. Table 1: Hardware Configurations for System Evaluation Experimental Group Robotic Arm Dexterous Hand Master Device (Input) COTS Arm Agilex Piper (6-DoF) LinkerHand O6 UdexReal Glove In-house Arm Linker Arm A7 (7-DoF) LinkerHand O6 UdexReal Glove 4.1.3 Evaluation Metrics To quantitatively evaluate the performance of each system configuration, we measured a set of metrics focused on task efficiency and data quality. The key metrics are defined as follows: Average Task Duration (s) This metric measures the overall efficiency of the teleoperation system. For each successful trial, we define the task duration as the elapsed time from the moment the robot hand begins its motion towards the object to the moment the task is successfully completed (e.g., the bottle is inserted or the cube is assembled). The final value is computed by averaging the durations across all successful trials for a given task and configuration. A lower value indicates higher operational efficiency. Synchronization Success Rate (%) This metric quantifies the reliability of our data collection pipeline. As detailed in Section 3.3.1, our TimeSyncManager performs a consistency check on the timestamps from all sensor streams at every single timestep. A timestep is marked as "successful" if the maximum time difference (max_diff) among all sensor timestamps is within our preset tolerance (100ms). The Synchronization Success Rate is the percentage of successfully synchronized timesteps out of the total timesteps in an entire demonstration episode. A higher value indicates greater temporal consistency in the collected data. Average Synchronization Error (ms) This metric measures the temporal precision of the successfully synchronized data. It is calculated by averaging the max_diff values (the maximum time difference between the earliest and latest sensor timestamp in a single frame) across all successfully synchronized timesteps within an episode. This value represents the average "temporal blur" of our multi-modal data snapshots. A lower value signifies higher-quality, more precisely aligned data, which is critical for learning time-sensitive manipulation skills. 12 Results The performance results for the two baseline tasks, conducted on two distinct hardware configurations, are summarized in Table 2. Each task was performed for 20 trials on its respective system. The table presents key metrics for task efficiency (Avg. Duration) and data quality (Sync Success and Error). Table 2: Performance and Data Quality Metrics Across Different Tasks and Hardware Configurations Configuration / Task Trials Avg. Duration (s) Sync Success (%) Avg. Sync Error (ms) COTS Arm Setup: Agilex Piper Arm + LinkerHand O6 Bottle Peg-in-Hole 20 20.06 (±2.88) 99.27 64.38 Cube Assembly 20 24.53 (±3.85) 99.99 60.77 In-house Arm Setup: Linker Arm A7 + LinkerHand O6 Bottle Peg-in-Hole 20 15.28 (±2.42) 99.98 42.75 Task Efficiency and Hardware Performance. Our results first highlight the system's ability to capture task complexity. On the same hardware (COTS Arm Setup), the Cube Assembly task required a significantly longer completion time (24.53s) than the Bottle Peg-in-Hole task (20.06s), which aligns with the higher precision demands of assembly. More importantly, the results demonstrate a clear performance differentiation between hardware configurations. On the same Bottle Peg-in-Hole task, the system configured with our inhouse arm was approximately 24% faster on average than the one with the Agilex Piper arm (COTS Arm Setup). This suggests that the tighter integration and potentially lower-level optimizations of the in-house hardware contribute to a more efficient teleoperation experience. Data Quality and System Robustness. The data quality metrics underscore the robustness of the Open TeleDex data collection pipeline. Across all tasks and hardware configurations, the Synchronization Success Rate remained above 99.2%, indicating that our TimeSyncManager module consistently and reliably aligns multi-modal data streams within the predefined 200ms tolerance. This level of reliability is critical for generating large, clean datasets for imitation learning. Furthermore, the Average Synchronization Error provides a deeper insight into the system's internal temporal precision. The in-house arm configuration not only performed tasks faster but also exhibited a significantly lower average synchronization error (42.75ms) compared to the Agilex Piper configuration (60-65ms range). A sync error of about 40-65ms, roughly equivalent to 1-2 control cycles at our 25Hz frequency, represents a solid baseline for a software-based synchronization mechanism in a complex ROS2 environment. This quantitatively demonstrates that our pipeline can maintain a consistent and measurable level of data quality, and reveals how hardware choice directly impacts the temporal fidelity of the final dataset. 5 Conclusion and Future Work 5.1 Conclusion In this technical report, we addressed the challenge of collecting high-quality, diverse demonstration data for robot imitation learning by proposing and implementing a teleoperation framework named Open TeleDex. Our work centers on two core contributions, which have been validated through our system's design and a series of quantitative experiments. First, we contributed a unified, device-agnostic teleoperation framework that provides an end-to-end workflow for robot learning. Through a layered, decoupled software architecture and a standardized hardware abstraction layer (RealEnv), Open TeleDex successfully breaks down the barriers between different 13 hardware ecosystems. Crucially, this framework encapsulates the full pipeline from the synchronized collection of multi-modal data to its structured storage. The results from our data quality analysis, showing synchronization success rates consistently above 99%, confirm that our AI-oriented data collection pipeline ensures the intrinsic quality of the collected data, providing a solid foundation for training highperformance imitation learning policies. Our experimental evaluation, using two distinct hardware configurations, demonstrated the framework's capability to seamlessly integrate and operate heterogeneous systems in practice, strongly supporting the TripleAny vision. Second, we introduced a novel general motion mapping algorithm. This algorithm surpasses the traditional "point-to-point mapping" paradigm by generating holistic postures, which provides more natural and functional motion retargeting between morphologically different human and robot hands. Our successful demonstrations on complex tasks like Cube Assembly and Bottle Peg-in-Hole serve as a practical validation of this algorithm's effectiveness. 5.2 Future Work While Open TeleDex lays a solid foundation for generalized teleoperation data collection, there is still ample room for exploration. Our future work will primarily focus on the following directions: Rigorous Benchmark. We plan to conduct a rigorous and comprehensive performance benchmark. This involves performing standardized human-computer interaction experiments, such as Fitts' Law tests, to quantitatively measure key metrics like system throughput (IP). We will also analyze the trajectory smoothness (Jerk) of the collected data to further quantify demonstration quality. These in-depth metrics will allow for a more nuanced comparison against state-of-the-art frameworks like ALOHA and AnyTeleop. Comprehensive User Experience (UX) Study. A comprehensive user experience (UX) study is planned to evaluate the human-in-the-loop aspects of the system. We will collect subjective metrics, such as operator workload using the NASA-TLX [20] questionnaire, to assess the intuitiveness and ergonomiscs of different hardware configurations. Expansion of the Hardware Ecosystem. We will continue to expand the TripleAny hardware ecosystem by providing official Recorder implementations for more mainstream robotic arms, dexterous hands, and sensors, and open-sourcing them to the community to further lower the research barrier. Collaborative Teleoperation. One of our most exciting goals is to explore and implement the "onemaster-to-multiple-slaves" collaborative teleoperation mode. TeleDex's distributed architecture based on ROS2 provides a natural foundation for this objective. We plan to develop a high-level task allocation and collaborative control module to dramatically increase the diversity and efficiency of data collection for training general-purpose robot policies. References [1] A. Brohan, N. Brown et al., "RT-1: Robotics transformer for real-world control at scale," in arXiv preprint , 2022. [2] T. Hulin, C. Preusche, and G. Hirzinger, "On the fidelity of haptic telepresence systems," IEEE Transactions on Haptics, vol. 4, no. 3, pp. 221-235, 2011. [3] J. Nielsen, Usability engineering. Morgan Kaufmann, 1994. 14 [4] M. A. Goodrich and A. C. Schultz, "Human-robot interaction: a survey," Foundations and trends® in Human-Robot Interaction, vol. 1, no. 3, pp. 203-275, 2007. [5] J. Lee and Y.-B. Cho, "Fitts' law-based performance evaluation of a haptic interface for teleoperated robot-assisted surgery," Applied Sciences, vol. 8, no. 11, p. 2264, 2018. [6] T. Z. Zhao, V. Kumar, and C. Action, "Learning fine-grained bimanual manipulation with low-cost hardware," arXiv preprint , 2023. [7] A. Sivakumar, K. Shaw, and D. Pathak, "Robotic telekinesis: learning a robotic hand imitator by watching humans on youtube," in Conference on Robot Learning (CoRL), 2022. [8] Y. Qin, H. Su, and X. Wang, "From one hand to multiple hands: Imitation learning for dexterous manipulation from single-camera teleoperation," in IEEE Robotics and Automation Letters (RA-L), vol. 7, no. 4, 2022, pp. 10 873-10 881. [9] Y. Qin, W. Yang, B. Huang, K. Van Wyk, H. Su, X. Wang, Y.-W. Chao, and D. Fox, "Anyteleop: A general vision-based dexterous robot arm-hand teleoperation system," in Robotics: Science and Systems (RSS), 2023. [10] P. Corke, Robotics, Vision and Control: Fundamental Algorithms in MATLAB. Springer, 2017. [11] S. P. Arunachalam, I. Guzey, S. Chintala, and L. Pinto, "Holo-dex: Teaching dexterity with immersive mixed reality," in Conference on Robot Learning (CoRL), 2022. [12] S. Li, X. Ma, H. Liang, M. Gorner, P. Ruppel, B. Fang, F. Sun, and J. Zhang, "Vision-based teleoperation of shadow dexterous hand using end-to-end deep neural network," in IEEE International Conference on Robotics and Automation (ICRA), 2019. [13] H. Zhang, W. Li, Y. Liang, Z. Chen, Y. Cui, Y. Wang, and R. Xiong, "Human-robot motion retargeting via neural latent optimization," arXiv preprint , 2021. [14] T. Osa, J. Pajarinen, G. Neumann, J. A. Bagnell, P. Abbeel, and J. Peters, "An algorithmic perspective on imitation learning," Foundations and Trends in Robotics, vol. 7, no. 1-2, pp. 1-179, 2018. [15] A. Rajeswaran, V. Kumar, A. Gupta, G. Vezzani, J. Schulman, E. Todorov, and S. Levine, "Learning complex dexterous manipulation with deep reinforcement learning and demonstrations," arXiv preprint , 2018. [16] X. B. Peng, P. Abbeel, S. Levine, and M. van de Panne, "Deepmimic: Example-guided deep reinforcement learning of physics-based character skills," in ACM Transactions on Graphics (TOG), vol. 37, no. 4, 2018, pp. 1-14. [17] C. Lynch, M. Khansari, T. Xiao, V. Kumar, J. Tompson, S. Levine, and P. Sermanet, "Learning latent plans from play," in Conference on Robot Learning (CoRL), 2020. [18] C. Wang, Z. Xu, X. Chen, W. Yu, and H. Wang, "Dexcap: Scalable and portable mocap data collection system for dexterous manipulation," arXiv preprint , 2024. [19] S. Macenski, T. Foote, B. Gerkey, C. Lalancette, and W. Woodall, "Robot operating system 2: Design, architecture, and uses in the wild," Science Robotics, vol. 7, no. 66, 2022. [20] S. G. Hart and L. E. Staveland, "Development of a multi-dimensional workload rating scale: Results of empirical and theoretical research," Human performance and ergonomics, vol. 2, pp. 139-183, 1988. 15 Appendix A1 Open TeleDex Hardware Ecosystem The following table lists the hardware devices that the Open TeleDex framework is designed to support, including those already integrated and those planned for future expansion. The modular architecture of Open TeleDex, particularly its RealEnv abstraction layer, facilitates the integration of new hardware by implementing a corresponding Recorder class. Table 3: Supported and Planned Hardware for the Open TeleDex Ecosystem. Category Model / Type Notes Robotic Arms Agilex Piper 6-DoF arm. Core testbed component. Linker Arm A7 7-DoF arm. Core testbed component. RealMan Planned support. Universal Robots (UR) Planned support. Dexterous Hands LinkerHand O6 6-DoF hand. Core testbed component. LinkerHand L10 10-DoF hand. Core testbed component. LinkerHand L20 20-DoF hand. Core testbed component. LinkerHand L30 Planned support. Teleoperation & Sensor Devices Master-Slave Arm / Glove High-fidelity input device. Integrated. VR Controllers (e.g., PICO, Meta Quest) Low-cost immersive input device. Integrated. UdexReal Haptic Glove High-fidelity haptic/motion capture glove. Integrated. Exoskeleton Planned support. Intel RealSense D455 RGB-D Camera. Core testbed component. Intel RealSense D405 High-precision, short-range RGB-D Camera. Verified. 16 A2 Example of a Typical Task Dataset Composition This appendix details the complete structure of a structured dataset generated by the Open TeleDex framework, using a single episode (episode_000015) from a representative grasping task as an example. Episode Metadata: this table summarizes the complete hardware and software environment, along with the collection parameters recorded for the episode, ensuring full reproducibility. Table 4: Metadata recorded for episode_000015. Category Parameter Value / Description Episode Info Task Name linkerhand_piper_grasp Episode ID episode_000015 Session ID session_2025-10-10_13-46-35 Duration (sec) 19.64 Timesteps 491 Hardware Config Preset piper_linkerhand_o6_single Arm Type Agilex Piper (6-DoF) Hand Type LinkerHand O6 (6-DoF) Total DoF 12 Camera Config Preset intel_d455_single_top Camera Type Intel RealSense D455 Position Top-down (1280x720 @ 30fps) Collection Params Control Freq. (Hz) 25 Hz (dt = 0.04s) Video Codec libx264 (high quality) Data Structure Format video_compressed qpos Dimension 12 (6 arm + 6 hand) Camera Streams cam_top Tactile Sensors right_hand 17
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2510.14775
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Astronomy & Astrophysics manuscript no. main
©ESO 2025
October 17, 2025
CHILLING: Continuum Halos in LVHIS Local Irregular Nearby
Galaxies
Radio continuum spectral behavior of dwarf galaxies
Sam Taziaux1, 2, 3 , Megan C. Johnson4 , Onic I. Shuvo5 , Dominik J. Bomans1, 2 , Christopher J. Riseley1, 2 ,
Timothy J. Galvin3 , Alec J. M. Thomson6 , Peter Kamphuis1 , Amy Kimball7 , Amanda Kepley8 , Michael
Stein1 , George H. Heald6 , Nicholas Seymour9 , Joe A. Grundy3, 9 , Björn Adebahr1 , and Ralf-Jürgen
Dettmar1, 2
1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), Universitätsstraße 150, 44801
Bochum, Germany e-mail: sam.taziaux@rub.de
2 Ruhr Astroparticle and Plasma Physics Center (RAPP Center)
3 CSIRO Space and Astronomy, PO Box 1130, Bentley WA 6102, Australia
4 National Science Foundation, 2415 Eisenhower Avenue, Alexandria, VA 22314, USA
5 Department of Physics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA
6 SKA Observatory, SKA-Low Science Operations Centre, 26 Dick Perry Avenue, Kensington WA 6151, Australia
7 National Radio Astronomy Observatory, 1011 Lopezville Road, Socorro, NM 87801, USA
8 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
9 International Centre for Radio Astronomy Research, Curtin University, Bentley, WA, Australia
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Context. Dwarf galaxies, due to their shallow gravitational potentials, provide critical environments for studying feedback mecha-
nisms from star formation and its impacts on dwarf galaxy evolution. In particular, radio continuum (RC) observations offer valuable
insights into cosmic ray dynamics, which play a significant role in shaping these processes.
Aims. This study investigates the detectability and spectral characteristics of RC emission in a sample of 15 dwarf galaxies (11
gas-rich, star forming dwarfs and 4 blue compact dwarfs) spanning a broad range of stellar masses and star formation histories.
Methods. Using multi-band RC data (L/S-, C-, and X-band) from the Australia Telescope Compact Array, we analyse the physical
conditions responsible for RC emission and explore the dominant emission mechanisms within these systems.
Results. RC emission is detected in 11 out of the 15 galaxies. Our results indicate that RC emission correlates strongly with star
formation rate, far-infrared, and stellar mass, while dynamic parameters such as Hi and rotational velocity exhibit no significant cor-
relation with RC detectability. Spectral analysis reveals that the RC spectral energy distribution in these galaxies frequently deviate
from a simple power-law behavior, instead displaying curvature that suggests more complex underlying physical processes. Statistical
model comparison confirms that a single power-law model is inadequate to capture the observed spectral shapes, emphasising the ne-
cessity of more sophisticated approaches. Additionally, the observed radio–far-infrared correlation indicates that cosmic ray electrons
in lower-mass dwarf galaxies cool more rapidly than they can escape (e.g. via galactic winds), resulting in a measurable RC deficit.
Key words.
1. Introduction
Galactic winds and outflows1 are crucial in shaping the evolu-
tionary histories of galaxies. These processes, driven by the en-
ergy and momentum from star formation, influence the interstel-
lar medium (ISM) by enhancing turbulence and providing addi-
tional pressure support. This additional pressure can counteract
gravitational collapse, thereby regulating further star formation.
Moreover, outflows can expel gas and metals from the ISM, po-
tentially enriching the circumgalactic medium or even escaping
1 In this article we adhere to the definition of Veilleux et al. (2005),
who defines an "outflow" as any material moving away from a cen-
tral object, whereas "wind" specifically describes gas expelled from a
galaxy’s potential, typically driven by mechanisms such as radiation
pressure and incorporating both diffuse flows and jets.
into the intergalactic medium if the outflow velocity exceeds the
escape velocity of the host galaxy (Veilleux et al. 2005).
Dwarf galaxies serve as essential laboratories for investigat-
ing feedback mechanisms from star formation, particularly the
role of star formation in driving winds and outflows. Due to their
shallow gravitational potentials, even moderate starburst events
can generate significant winds (e.g. Chy˙zy et al. 2016) or out-
flows (e.g. Devine & Bally 1999; Strickland & Heckman 2009;
Adebahr et al. 2013; Dirks et al. 2023). Understanding these
feedback mechanisms is vital to comprehending the interplay be-
tween star formation and dwarf galaxy evolution (e.g. Dekel &
Silk 1986; Stevens et al. 2002; Chy˙zy et al. 2016). Most studies
on dwarf galaxies in the literature have focused on their inte-
grated radio continuum (RC) properties (e.g. Klein et al. 2018).
However, resolved RC observations of these galaxies enable a
better understanding of the evolution and outflow characteris-
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A&A proofs: manuscript no. main
tics of these low-mass galaxies. Only a few studies have utilised
resolved RC and spectral index observations to analyse dwarf
galaxies in detail (e.g. Chy˙zy et al. 2000; Kepley et al. 2010;
Basu et al. 2017; Kepley et al. 2011; Hindson et al. 2018; Tazi-
aux et al. 2025). These RC observations have shown a mean
non-thermal spectral index (S ∝ναnth) of −0.6 that steepens at
the galaxy outskirts to −1.1 (e.g. Chy˙zy et al. 2000; Kepley et al.
2010, 2011; Westcott et al. 2018; Taziaux et al. 2025). In the
resolved starforming knots, the spectral index can reach a flat-
ter spectrum of −0.3 or even be inverted (e.g. Basu et al. 2017;
Taziaux et al. 2025), depending on the optical depth of the galaxy
region.
Radio observations also provide a crucial tool for studying
cosmic ray (CR) transport and the dominant energy loss pro-
cesses (synchrotron, inverse-Compton or bremsstrahlung), with
relativistic electrons serving as tracers (e.g. Lacki et al. 2010;
Werhahn et al. 2021; Pfrommer et al. 2022; Heesen et al. 2022).
CRs propagate from supernova remnants into the galactic halo
through advection, diffusion, and streaming (e.g. Strong et al.
2007; Stein et al. 2019; Thomas et al. 2020). In this process,
CRs excite Alfvén and whistler waves via resonant plasma in-
stabilities, which scatter CRs and regulate their drift speed along
the magnetic field (Kulsrud & Pearce 1969; Shalaby et al. 2021,
2023; Lemmerz et al. 2024). Through resonant scattering, CRs
transfer momentum to the thermal gas, exerting pressure on the
ambient plasma and driving galactic winds, a key aspect of CR
hydrodynamic models (Zweibel 2013; Pfrommer et al. 2017a;
Thomas & Pfrommer 2019).
Hydrodynamic simulations indicate that CR-driven winds
significantly influence mass-loss rates, gas distribution, and wind
formation (e.g. Breitschwerdt et al. 1991; Uhlig et al. 2012; Pak-
mor et al. 2016; Girichidis et al. 2016; Recchia et al. 2017;
Dashyan & Dubois 2020; Thomas et al. 2023). Thomas et al.
(2024) demonstrated that CRs, due to their long cooling times
and strong plasma coupling, drive denser winds with higher
mass-loading factors, efficiently redistributing gas into the halo
and regulating star formation.
This study further investigates the properties of these winds
and outflows, particularly their velocity into the galactic halo, as
driven by stellar feedback in low metallicity regimes. Because
CR protons (CRPs) primarily lose energy through hadronic in-
teractions with dense gas, they predominantly illuminate high-
density regions (Pfrommer & Enßlin 2004; Pfrommer et al.
2017b; Werhahn et al. 2023). In contrast, lower-density out-
flows are best traced via synchrotron emission from CR elec-
trons (CREs). Modelling CREs alongside CRPs is essential, as
their different loss timescales cause deviations in their energy
spectra (Ruszkowski & Pfrommer 2023), necessitating a sepa-
rate spectral treatment for accurate predictions of galactic radio
emission (Chiu et al. 2024).
Beyond their role in regulating outflows, CRs also contribute
to the evolution of large-scale ordered magnetic fields in galax-
ies. Observations and simulations of superbubbles in nearby spi-
ral galaxies suggest that starburst-driven outflows can advect or-
dered magnetic field lines from the inner disk regions to the
outer halo (e.g. Kepley et al. 2010; Chy˙zy et al. 2011; Kepley
et al. 2011). This interplay between galactic winds, magnetic
field evolution, and cosmic ray transport is particularly relevant
for dwarf galaxies, where such effects can be more pronounced
due to lower gravitational binding energy.
Despite theoretical and numerical advancements, direct ob-
servational evidence of CR-driven winds remains challenging
due to the (i) sensitivity limits of current radio facilities, which
hinder the detection of faint halo emissions, and (ii) lack of ro-
bust modelling frameworks for various CR propagation mecha-
nisms. RC surveys targeting dwarf galaxies are crucial for ad-
dressing these challenges and testing theoretical predictions of
cosmic ray transport and their role in galactic feedback.
In this paper, we further explore the detectability of radio
emission in dwarf galaxies, the dependence of energy loss mech-
anisms, and the modelling of cosmic ray transport across a larger
sample of dwarf galaxies spanning a wide range of masses and
star formation histories. We describe our sample and the data re-
duction in Sect. 2. In Sect. 3, we present detectability of radio
emission across various parameter and the basic properties of
RC will be shown in Sect. 4. Sect. 5 will focus on the RC–FIR
correlation. We discuss our observations in the context of what is
known so far in Sect 6 with a summary and conclusion presented
in Sect. 7.
2. Observation and data reduction
2.1. CHILLING sample
The Continuum Halos In LVHIS Local Irregular Nearby Galax-
ies (CHILLING) sample covers a range of galaxy properties,
such as star formation rate, luminosities, and flux densities. The
selected galaxies have an absolute magnitude of MB > −17,
IRAS 60 µm detections, and are located at a declination below
about −20◦. The CHILLING sample includes 11 dwarf galax-
ies from the Local Volume Hi Survey (LVHIS) project (Korib-
alski et al. 2018), as well as 4 blue compact dwarf galaxies
(BCDs) that are further away and therefore not part of the main
LVHIS sample. Table 1 gives an overview of the properties of
the CHILLING sample.
2.2. Observations
The CHILLING dwarf galaxies were observed with the ATCA
(project ID: C3041; PI: M. Johnson). The observations were per-
formed at 1.1–3.1 GHz (L/S-band), 3.9–7.1 GHz (C-band) and
8–11 GHz (X-band) between 18-May-2015 and 16-June-2016.
The configurations used for each target are listed in Table 2. Ob-
servations of the target galaxies and a phase calibrator were alter-
nated through the observing run, giving a total on-source observ-
ing time of about 5 to 11 hrs for each galaxy (see Table 2). Along
with the target and phase calibrator observations, the standard
ATCA primary flux calibrator 1934−638 and the secondary flux
calibrator 0823−500 have been used. For the galaxy IC 4662, we
use additional observational data of on-source integration time
of 29.41 hrs at central frequencies 2.1 GHz, 4.71 hrs at 5.5 and
9 GHz, which have been observed between 4-May-2023 and 26-
August-2024 (project ID: C3531; PI: S. Taziaux).
2.3. Data reduction
We processed the observations using the Multichannel Im-
age Reconstruction, Imaging Analysis, and Display package
(miriad; Sault et al. 1995) following standard data calibra-
tion procedures using mfcal for bandpass calibration, gpcal
to derive the gains and instrument leakage. To minimise the
effects of radio frequency interference (RFI) during flux and
phase calibration, the edges of the dataset have been flagged us-
ing uvflag, automated flagging was performed using pgflag
to suppress interference in the target sources. Any remaining
corrupted data identified throughout the calibration process was
manually flagged, using the interactive flagging tool blflag
Article number, page 2 of 20
Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
11h57m33s
30s
27s
24s
19°36'30"
37'00"
30"
38'00"
30"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 12.09 kpc
ESO572 G025
4h06m16s
12s
08s
04s
00s
52°39'00"
30"
40'00"
30"
41'00"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 4.04 kpc
Fairall301
0h45m51s
48s
45s
42s
15°34'30"
35'00"
30"
36'00"
30"
37'00"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 5.04 kpc
NGC 244
12h20m03s
00s
19m57s
17°22'30"
23'00"
30"
24'00"
30"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 6.93 kpc
ISZ 399
17h47m30s
20s
10s
00s
64°37'
38'
39'
40'
RA (ICRS)
Dec (ICRS)
2 arcmin = 1.42 kpc
IC 4662
1h35m15s
10s
05s
00s
34m55s
41°25'
26'
27'
28'
RA (ICRS)
Dec (ICRS)
2 arcmin = 2.32 kpc
NGC 625
13h40m06s
00s
39m54s
48s
42s
31°36'
37'
38'
39'
40'
41'
RA (ICRS)
Dec (ICRS)
3 arcmin = 2.99 kpc
NGC 5253
Fig. 1. Colour-composite images from the DESI Legacy Imaging Surveys of the RC detected CHILLING sample, showing more diffuse emission,
with overlaid ATCA radio emission contours starting at 3 σ and increasing by a factor of
√
2 at a central frequency of 2.1 GHz. For IC 4662, we use
the Hα map from Hunter & Elmegreen (2004) as the declination is too low for the DESI Legacy Imaging Surveys. The scale in the top left corner
is calculated using the distance to the source. The beam is shown in the bottom left corner. The noise level σ2.1 GHz can be taken from Table 3.
1h45m12s
08s
04s
00s
44m56s
43°34'30"
35'00"
30"
36'00"
30"
37'00"
RA (ICRS)
Dec (ICRS)
2 arcmin = 2.56 kpc
ESO245 G005
3h33m20s
15s
10s
05s
50°23'30"
24'00"
30"
25'00"
30"
26'00"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 2.69 kpc
IC 1959
22h03m00s
02m50s
40s
30s
51°15'
16'
17'
18'
19'
20'
RA (ICRS)
Dec (ICRS)
3 arcmin = 1.88 kpc
IC 5152
14h03m30s
24s
18s
12s
41°21'
22'
23'
24'
25'
RA (ICRS)
Dec (ICRS)
3 arcmin = 4.65 kpc
NGC 5408
Fig. 2. Colour-composite images from the DESI Legacy Imaging Surveys of the RC non-detected CHILLING sample, showing more compact
emission, with overlaid 2.1 GHz ATCA radio emission contours starting at 3 σ and increasing by a factor of
√
2. The scale in the top left corner is
calculated using the distance to the source. The beam is shown in the bottom left corner. The noise level σ2.1 GHz can be taken from Table 3.
2h57m00s
56m55s
50s
45s
40s
54°33'00"
30"
34'00"
30"
35'00"
30"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 2.50 kpc
ESO154 G023
13h49m24s
20s
16s
12s
36°02'30"
03'00"
30"
04'00"
30"
05'00"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 1.50 kpc
ESO383 G087
3h20m15s
10s
05s
00s
52°10'00"
30"
11'00"
30"
12'00"
30"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 2.41 kpc
NGC 1311
13h41m42s
39s
36s
33s
30s
29°53'30"
54'00"
30"
55'00"
30"
56'00"
RA (ICRS)
Dec (ICRS)
1.5 arcmin = 2.09 kpc
NGC 5264
Fig. 3. Colour–composite images from the DESI Legacy Imaging Surveys of the CHILLING sample with non-detected or marginally detected
RC emission, overlaid 2.1 GHz ATCA radio emission contours starting at 3 σ and increasing by a factor of
√
2. The scale in the top left corner is
calculated using the distance to the source. The beam is shown in the bottom left corner. The noise level σ2.1 GHz can be taken from Table 3.
for manual inspection and removal of contaminated data, to en-
sure data integrity. Initially, after cross-calibration and excluding
channels with Hi emission for L/S-band dataset, we employed an
iterative imaging using WS-clean (Offringa et al. 2014) and self-
calibration (phase-only, frequency-independent, with a solution
interval of 5 min down to 60 s) using CASA (version 6.4.4.31; Mc-
Mullin et al. 2007) until image quality reached convergence with
a Briggs weighting of robust = −1, −0.5, −0.3, 0, 0.3 to slowly
reconstruct the diffuse emission. Imaging and self-calibration
was performed independently for L/S-, C-, and X-bands and by
Article number, page 3 of 20
A&A proofs: manuscript no. main
Table 1. CHILLING sample properties including the coordinates, distance, absolute magnitude in B-band, star formation rate, Hi mass, rotational
velocity, and the near and far-infrared luminosities of each dwarf galaxy
Name
RA
Dec
D
MB
log
h SFRHα
M⊙yr−1
i
log
h MHi
M⊙
i
3rot
L3.4µm
L60µm
(Mpc)
(mag)
(km s−1)
(mag)
(Jy)
ESO154-G023
02h56m50.38s
−54d34m17.10s
5.76
−16.45
−1.32
9.14
53.0
15.21
0.32
ESO245-G005
01h45m03.73s
−43d35m52.93s
4.43
−15.68
−1.59
8.6
51.0
15.99
0.22
ESO383-G087
13h49m17.50s
−36d03m48.40s
3.45
−16.83
−1.49
7.82
18.0
14.05
1.21
ESO572-G025†
11h57m28.03s
−19d37m26.60s
27.88
−17.50
−0.27
8.36
–
12.88
0.51
Fairall301†
04h06m07.92s
−52d40m06.30s
9.30
−15.92
−1.11
–
52.3
12.38
0.90
IC1959
03h33m12.59s
−50d24m51.30s
6.19
−16.07
−1.38
8.38
57.0
13.90
0.51
IC4662
17h47m08.86s
−64d38m30.33s
2.55
−15.61
−1.12
8.24
41.0
11.99
8.82
IC5152
22h02m41.51s
−51d17m47.20s
1.96
−15.56
−2.23
8.01
43.0
13.77
2.46
ISZ399†
12h19m59.51s
−17d23m31.00s
15.94
−17.15
−0.81
–
–
10.71
2.86
NGC244†
00h45m46.43s
−15d35m48.80s
11.60
−16.25
−1.11
8.73
–
12.05
0.53
NGC625
01h35m04.63s
−41d26m10.30s
4.02
−16.50
−1.20
8.00
34.0
12.12
5.73
NGC1311
03h20m06.96s
−52d11m07.90s
5.55
−15.41
−1.55
8.04
36.0
13.46
0.39
NGC5253
13h39m55.96s
−31d38m24.38s
3.44
−17.05
−0.64
7.91
31.0
9.25
30.51
NGC5264
13h41m36.68s
−29d54m47.10s
4.79
−16.02
−1.92
7.70
15.0
13.72
0.30
NGC5408
14h03m20.91s
−41d22m39.70s
5.32
−16.73
−0.83
8.48
30.0
13.06
2.82
Notes. † Blue compact dwarf galaxies (BCDs); the rest are irregular dwarfs from the LVHIS sample.
Table 2. ATCA RC observations of CHILLING galaxies, with its configuration and the total on-source integration time
Name
ATCA
configurations
Total on-source time / hrs
2100 MHz
5500 MHz
9000 MHz
ESO154-G023
1.5A
750B
H168
7.17
6.85
6.85
ESO245-G005
1.5B
750C
H214
10.53
7.59
7.59
ESO383-G087
1.5B
750C
H214
8.56
8.37
8.37
ESO 572-G025
1.5B
H214
4.92
4.60
4.60
Fairall 301
1.5C
750C
H168
6.19
5.26
5.26
IC1959
1.5C
750B
H214
5.25
5.27
5.27
IC 4662†
1.5C
750C
H214
37.66
11.93
11.93
IC 5152
1.5A
750B
H214
5.18
4.60
4.60
ISZ 399
1.5B
750C
EW352
4.42
4.27
4.27
NGC 244
1.5A
750C
H214
7.18
6.57
6.57
NGC 625
1.5A
750C
H214
9.20
6.21
6.21
NGC 1311
1.5A
750C
H214
5.59
3.39
3.39
NGC 5253
1.5B
750B
H214
7.23
6.90
6.90
NGC5264
1.5B
750B
H214
4.22
5.36
5.36
NGC 5408
1.5C
750B
H168
H214
4.88
10.19
10.19
Notes. † Observation of IC 4662 is a combination of the CHILLING observations and the observations running under project ID: C3531 with
detailed explanation in the Sect. 2.2.
splitting into 32 -channels-out to avoid bandwidth smearing
larger than the synthesized beam, during imaging of the field
of view. Multifrequency and multiscale CLEANing (Högbom
1974) utilising interactive masks using flint_masking (Galvin
et al. in prep.)2 around visible sources to minimise artefacts and
flux scattering and applying detection thresholds was used to
retain genuine emission only. During the imaging step, we use
-join-channels to leverage the ATCA’s full bandwidth. The
continuum images for each galaxy were generated using differ-
ent Briggs robust weighting to increase the quality, resulting in
resolutions and noise levels shown in Table 3. In the last round
2 https://github.com/flint-crew/flint
of imaging, we employed joint imaging3 with -channels-out
32 images. As these images are only used in science for the SED
fitting in Sect. 4, we do -no-mf-weighting in the last round of
imaging to ensure not to include artificial spectral substructures.
We use the integrated task linmos in miriad to correct for the
primary beam to each 32 images of the dataset.
3 Joint imaging means gridding and deconvolving the L/S-, C-, and
X-bands jointly, resulting in a highly sensitive image with central fre-
quency of 5.55 GHz, with a total bandwidth of 8.95 GHz
Article number, page 4 of 20
Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
Table 3. Overview of the observable parameters of the CHILLING galaxies, showing the name, the flux and phase calibrator, the Briggs robust
weighting, the noise level of the RC image, and the category of detection.
name
flux calibrator
phase calibrator
robust
σ2.1/5.5/9 GHz
Detection Category
(µJy/beam)
ESO 154-G0154
1934-638
0252-549
0.0
25/10/10
No/Marginal
ESO 245-G005
1934-638
0153-410
0.3
15/6/6
Compact
ESO 383-G087
1934-638
1424-418
0.0
39/15/20
No/Marginal
ESO 572-G025
1934-638/0823-500†
1127-145
0.0
30/20/30
Diffuse
Fairall 301
1934-638
0302-623
0.3
14/12/25
Diffuse
IC 1959
1934-638/0823-500†
0252-549
0.0
20/10/10
Compact
IC 4662
1934-638
1718-649
0.3
9/8/12
Diffuse
IC 5152
1934-638
2326-477
0.0
15/8/8
Compact
ISZ 399
0823-500
1213-172
0.3
73/20/25
Diffuse
NGC 244
1934-638
0023-263
0.3
10/8/8
Diffuse
NGC 625
1934-638
0201-440
0.3
15/8/8
Diffuse
NGC 1311
1934-638
0302-623
0.0
27/15/10
No/Marginal
NGC 5253
1934-638
1421-490
0.0
48/35/50
Diffuse
NGC 5264
1934-638
1255-316
0.0
80/20/13
No/Marginal
NGC 5408
1934-638/0823-500†
1421-490
0.0
85/17/18
Compact
Notes. † These datasets make use of the secondary flux calibrator at 5.5 and 9 GHz.
3. Detectability dependency
3.1. Definition of RC detectability
Since the definition of what qualifies as an RC detection can
vary, we divided our sample into three categories. In the first
category, a galaxy is classified as a ‘diffuse’ detection if spa-
tially extended RC emission is present across the majority of the
galaxy. Specifically, we require that emission is detected at > 3σ
significance over at least ∼70 −80 % of the galaxy’s optical ex-
tent (as defined by the R25 radius) and that the emission is more
extended than the synthesized beam. These detections leave little
doubt that the galaxy as a whole emits RC emission (see Fig.1).
A galaxy is classified as a ‘compact’ detection if it shows at least
one RC-bright Hii region (> 3σ peak emission) but lacks diffuse
emission across most of the galaxy. In this case, RC emission is
localized to one or a few star-forming regions and covers < 50 %
of the galaxy’s optical extent (see Fig. 2). In the third category, a
galaxy is classified as ‘no/marginal’ detection if it shows no sig-
nificant RC emission above 3σ within its optical extent, or only
weak, patchy signals covering < 20 % of the optical radius that
cannot be robustly distinguished from noise or imaging artefacts
(see Fig. 3).
3.2. RC properties
Only 11 out of these 15 dwarf galaxies show RC emission, with
7 showing diffuse emission, 4 showing compact emission com-
ing from a starforming region and 4 are showing no or very weak
emission. The detection thresholds are set to 3σ. The complete
L/S-, C-, and X-band RC maps for all 15 dwarf galaxies are pro-
vided in Appendix A, with galaxies showing diffuse emission
in Fig.A.1, compact emission in Fig.A.2, and no or very weak
emission in Fig. A.3. We notice that all 4 of the BCDs are de-
tected with diffuse emission while only 7 out of the 11 LVHIS
dwarfs were detected. Only three of them, IC 4662, NGC 5253
and NGC 625, show substructures as opposed to merely an un-
resolved structure. NGC 5408 only shows RC emission on the
south-western side, aligning with the observed high Hi velocity
dispersion as described in van Eymeren et al. (2010), possibly
due to the gas outflows. Similar to NGC 5408, ESO245-G005
only shows RC emission near star-forming regions and a depres-
sion at the center, a feature that is also seen in Hi (Côté et al.
2000).
All 4 BCDs are detected in our sample, compared to only
7 irregular dwarf galaxies. This likely reflects the intrinsically
higher star formation rates of BCDs relative to typical star-
forming dwarfs. However, an observational bias may also con-
tribute as the selected BCDs are more distant, their star-forming
regions appear more spatially concentrated on the sky, enhancing
the detectability of their compact emission compared to the more
nearby, diffuse dwarf galaxies. If the radio continuum emission
is associated with star formation, which we explore in this study,
this suggests that the measured signal is dominated by emission
from concentrated star-forming regions as opposed to more ex-
tended ones
3.3. Classification of RC dependency
To explore the factors influencing the detectability of RC emis-
sion in these 15 dwarf galaxies, we construct a WISE color-color
diagram (Wright et al. 2010) to examine the relationship between
radio emission and the classification of these objects. As shown
in Fig. 4, 6 out of 7 dwarf galaxies that display diffuse RC emis-
sion are classified as starbursts, primarily located in the right sec-
tion of the plot, while the dwarf galaxies, showing a more com-
pact RC emission are either close to the border to be classified as
a starburst or they are very left in the WISE color-color diagram.
Starburst galaxies, such as NGC 5253, IC 4662, and NGC 625,
exhibit extended radio emission that extends into the outskirts
of the galaxies, while other galaxies show RC detections ap-
pearing to align with the brightest star forming regions with the
exception of a few tenuous structures at lower frequencies. No-
tably, both IC 4662 and NGC 5253 have higher W1-W2 values.
NGC 5253 falls within the region typically associated with ob-
scured AGN, although it is currently known to host a super star
cluster at its core (e.g. Smith et al. 2020), rather than a tradi-
tional AGN. This may suggest a trend, saying galaxies located
further to the right in the diagram tend to exhibit stronger RC
Article number, page 5 of 20
A&A proofs: manuscript no. main
Fig. 4. WISE color–color diagram showing in the background the loca-
tions of interesting classes of objects (Wright et al. 2010), while show-
ing the CHILLING galaxies in dependency of their detectability cat-
egory of RC emission. Dwarf galaxies with diffuse RC emission are
shown in green, those with compact RC emission are shown in blue,
and non-detections or weak detections are shown in orange. The WISE
W1, W2 and W3 bands are 3.4 µm, 4.6 µm and 12 µm, respectively.
emission. These results suggest that the intensity of star forma-
tion, as quantified by the SFR and SFR surface density, are key
factors governing the detection of RC emission in dwarf galax-
ies. This implies that not just the presence of star formation, but
its localised strength, plays a critical role in producing detectable
levels of RC emission.
3.4. Parameter behaviour RC dependency
To explore which factors, in addition to a galaxy’s classification
as a starburst, may influence the presence of RC emission, we
examine a set of physical parameters, which can be taken from
Table 1. These include the absolute B-band magnitude (MB), star
formation rate (SFR), Hi mass (MHI), maximum rotation veloc-
ity of the gas corrected for inclination (3rot), and mid-infrared
luminosity at 3.4 µm (L3.4µm), which serves as a reliable proxy
for the stellar mass (e.g. Jarrett et al. 2023). We also consider the
far-infrared (FIR) luminosity at 60 µm (L60µm) from IRAS.
From the diagonal panels in Fig. 5, it becomes clear that
the most prominent difference between galaxies with and with-
out RC detection lies in their SFRs, which is also linked to ab-
solute magnitude. Regarding mass-related parameters, galaxies
with detected RC tend to be brighter at 3.4 µm magnitude, sug-
gesting a higher stellar mass, whereas the Hi masses do not ex-
hibit a significant difference between the two groups. The FIR
luminosity at 60 µm displays a narrow range for non-detected
galaxies, with most values clustering close to zero, while the de-
tected galaxies show notably higher FIR fluxes. When examin-
ing the gas rotation velocity, which is physically linked to the
total mass, we find that galaxies with no or weak RC detections
tend to have lower rotation velocities, while those with compact
RC detections show higher velocities. Dwarf galaxies with dif-
fuse RC detections lie in between these two groups, although the
differences are not statistically significant.
To further quantify the correlation between RC detection
and the various physical parameters, we compute Kolmogorov-
Smirnov (KS; Massey 1951) test with its p-value for statistical
significance. This method compares two distributions to deter-
mine whether they differ significantly. The p-value indicates the
probability that the observed difference happened by chance,
while a small p-value (typically below 0.05) suggests a statis-
tically significant difference. Looking at Fig. 5, significant dif-
ferences were found for SFR and the 3.4 µm magnitude, used
as proxy for stellar mass (both with KS= 1.0 and p = 0.0006)
indicating that the distributions of these parameters differ mean-
ingfully between the three populations. For 60 µm FIR luminos-
ity (KS= 0.75, p = 0.06), the difference is slightly not statistical
significant to see difference between these populations. These re-
sults suggest that both the SFR and stellar mass are linked to the
detection of RC emission. This provides quantitative support for
the idea that the intensity and spatial concentration of star for-
mation are critical factors in generating detectable RC in dwarf
galaxies, likely due to their role in driving synchrotron-emitting
processes. In contrast, properties such as absolute magnitude in
B-band MB, Hi mass MHI and rotation velocity 3rot showed no
statistically significant difference (p > 0.05), suggesting similar
distributions between these populations.
It is important to note that all of these correlations should be
interpreted with caution. The analysis is based on a small sam-
ple of 15 dwarf galaxies, comprising 11 with RC detections and
4 without. This sample size is insufficient to draw statistically
robust conclusions on its own about the entire dwarf galaxy pop-
ulation, although it is in agreement with previous studies (e.g.
Roychowdhury et al. 2012; Schleicher & Beck 2016; Hindson
et al. 2018, Taziaux in prep.)
4. Spectral model fitting
4.1. Overview of the RC data
For the SEDs, we include additional data from multiple sur-
veys in addition to our ATCA observations. For NGC 625 and
Fairall 301, we incorporate GaLactic and Extragalactic All-
sky MWA survey eXtended (GLEAM-X) data (Ross et al.
2024), which cover frequencies from 87 MHz to 221 MHz. For
all sources, we also include the Australian SKA Pathfinder
(ASKAP) survey, called Rapid ASKAP Continuum Survey
(RACS-Low) data (Hale et al. 2021) at 885 MHz. For the ATCA
dataset, we applied a 3 σ clipping threshold and computed the
integrated flux density, summing all emission over its resolved
size with an optical aperture using RadioFluxTools4, across
32 individual frequency slices. To ensure the reliability of the
measurements, only slices with a noise level below 90 µJy/beam
were considered. As the ATCA observations were conducted in
snapshot mode, we adopt a standard calibration uncertainty of
10 %. The total uncertainty is then estimated as the combination
of the statistical uncertainty, determined from the background
noise and the specified beam area, and the assumed calibration
uncertainty. For the SED analysis, we only use the 7 dwarf galax-
ies which show extended and diffuse emission.
4.2. Spectral models
In the study of Klein et al. (2018), four different basic models
were fitted to a small sample of dwarf galaxies as well as more
massive ones. This has been further developed by Galvin et al.
4 https://gitlab.com/Sunmish/radiofluxtools
Article number, page 6 of 20
Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
Fig. 5. Correlation analysis of several galaxy parameters with the dependence of RC emission detection. It represents different physical properties,
including absolute magnitude in the B-band MB, star formation rate (SFR), Hi mass MHI, maximum rotation velocity of the gas corrected for
inclination (3rot), mid-infrared magnitude at 3.4 µm (L3.4µm) from WISE and far-infrared (FIR) luminosity at 60 µm (L60µm) from IRAS. Galaxies
with diffuse RC emission are shown in green, those with compact emission in blue, and galaxies with no or very weak detection in orange. The
Kolmogorov–Smirnov (KS) test results are visualised as circles on the diagonal of the plot, with sizes ranging from 0 to 1; larger circles indicate
greater differences between the distributions. The exact value from the KS-test are shown inside the circles. The color represents statistical
significance based on the p-value: pink circles indicate significant differences (p < 0.05), while gray circles indicate non-significant results.
(2018) and Grundy et al. (2025), who focused solely on star-
bursts and star-forming galaxies, respectively, and developed the
spectral fitting models. Our aim is to improve the spectral fit-
ting for dwarf galaxies and identify which components domi-
nate at different parts of the spectrum. We modified the mod-
els presented by Grundy et al. (2025) by splitting the absorption
part into an internal component and an external component. All
19 models are detailed and presented in Appendix B, but these
models are essentially combinations of five basic assumptions.
The first assumption involves a standard power-law with only
synchrotron emission, given by
S (ν) = A
ν
ν0
!α
(1)
where A represents the non-thermal synchrotron emission and α
is the non-thermal spectral index at a given frequency ν0. The
second model accounts for a combination of synchrotron emis-
sion and thermal free–free emission, which can be written as a
superposition
S (ν) = A
ν
ν0
!α
+ B
ν
ν0
!−0.1
(2)
Article number, page 7 of 20
A&A proofs: manuscript no. main
Fig. 6. Spectral energy distribution of the RC detected CHILLING sample. The plot displays total intensity data with different observations:
GLEAM-X data (Ross et al. 2024) only for Fairall 301 and NGC 625, RACS-Low (Hale et al. 2021) and our ATCA S-band, C-band and X-band.
The frequency range of the telescope data is indicated by a grey dotted line. Additionally, we show the best-fit in red for each galaxy out of these
19 different models, explained in the Appendix B. The highlighted red regions represent the 1 σ uncertainties sampled by EMCEE. For comparison
to the best-fit, we show the simple power-law (PL) fit with its uncertainty range in a blue dotted line.
where A represents the synchrotron emission, α is the non-
thermal spectral index, and B is the free–free emission at a given
frequency ν0.
Next, we model two individual free–free absorption compo-
nents at lower frequencies, one for internal absorption and one
for external absorption, following the assumptions outlined in
Tingay & de Kool (2003). For external free–free absorption, the
model is:
S (ν) = 1 −e−τ1 · B ·
ν
ν1
!2
(3)
while for internal free–free absorption, the model is:
S (ν) =
1 −e−τ2
τ2
!
· B ·
ν
ν2
!2
(4)
where the free–free optical depth parameter is given by τi =
(ν/νi)−2.1. Finally, the last model component is relevant when
inverse Compton losses dominate at higher frequencies, causing
CREs to cool rapidly, leading to a break in the spectrum. This
model can be written as
S (ν) =
A
ν
ν0
α
1 +
ν
νb
∆α
(5)
where νb is the break frequency in the spectrum and ∆α rep-
resents the change in the non-thermal spectral index due to syn-
chrotron and inverse Compton losses, assuming continuous elec-
tron injection from massive SF. The full explanation of all 19
models is detailed in Appendix B.
4.3. Model selection and spectral fitting
To fit all the 19 models to the data used for each dwarf galaxy,
we use an affine invariant Markov chain Monte Carlo ensemble
sampler (Goodman & Weare 2010) implemented as the EMCEE
Article number, page 8 of 20
Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
PYTHON package (Foreman-Mackey et al. 2013) using the differ-
ential evolution optimisation method. We choose physically mo-
tivated priors to constrain our models, such as the parameters A,
B, C and D remain always positive and the spectral index value α
should be between −3 and 0.5. We choose for the frequencies ν1
and ν2, as well as for the spectral break frequency νb, a value
which should be between 0.01 and 10 GHz. For each model,
the fit quality is evaluated based on the chi-square and Bayesian
statistic. The model with the lowest chi-square and Bayesian in-
formation criterion (BIC) is selected as the best representation
of the data, and its fit parameters are reported. Models that fail
to converge are skipped.
4.4. Spectral fitting results
Fig. 6 show the spectral energy distribution of the diffuse RC
detected CHILLING galaxies. We show here the GLEAM-X
data (Ross et al. 2024), if available (here only for Fairall 301,
NGC 625), RACS-Low (Hale et al. 2021) and the observed
ATCA L/S-,C- and X-band data. The best-fit model have been
shown in red, while for comparison the simple power-law is
shown in blue.
For ESO 572-G025 and Fairall 301 we observe a turnover
to lower frequency due to internal or external free–free ab-
sorption and probably inverse Compton losses at higher fre-
quencies, according to the fit-model called FFA2_SFG_SIC and
FFA1_SFG_SIC, respectively. Inverse Compton losses at higher
frequencies are also seen in NGC 244, fitted by the best fit-
model called FFA1_SFG_SIC with a slight turnover to lower
frequencies. To confirm this turnover associated with free–free
absorption, we need even lower frequencies, such as GLEAM-
X data. For Fairall 301, we have GLEAM-X data, so we see
this turnover happening at approximately 150 MHz, while for
ESO 572-G025, we see this turnover already happening at
1 GHz. This turnover to lower frequency is also observed by
Gajovi´c et al. (2024) in more massive galaxies at frequencies
of approximately 200 MHz. That we observe a turnover at such
high frequencies, could be due to the limited surface brightness
sensitivity of RACS-Low, as the galaxy shows extended emis-
sion at higher frequencies seen in the 2.1 GHz ATCA images.
Deeper ASKAP images taken as part of the Evolutionary Map
of the Universe (EMU; Norris et al. 2021; Hopkins et al. 2025)
survey will allow us to extend this investigation. A similar spec-
tral shape, corresponding to the SFG_FFA1 model, is observed
in ISZ 399, IC 4662, NGC 625, and NGC 5253. The model de-
scribes synchrotron emission from relativistic electrons mixed
uniformly with a single volume of thermal free–free plasma.
Building on this, and following Galvin et al. (2018), we pro-
pose that the electron population is mixed inhomogeneously
with two distinct star-forming regions, which differ in their op-
tical depths, with only one component becoming optically thick
across the observed frequency range and external free–free ab-
sorption being observed. Although we only have GLEAM-X
data for NGC 625 regarding this fitting model, we observe a
spectral behaviour that resembles findings by Grundy et al.
(2025) in some of their galaxies, e.g. NGC 491. In both NGC 625
and NGC 491, there is a slight turnover around 400 MHz fol-
lowed by an increase in flux density at lower frequency. Aside
from this similarity in the spectrum, the two galaxies do not share
any other notable characteristics.
5. RC - far infrared correlation
There is a tight astrophysical relationship between the RC and
FIR luminosities in all star formation galaxies, the so-called RC–
FIR correlation (Condon 1992). The empirical luminosity scal-
ing between these two frequency regimes is likely a relation be-
tween radio synchrotron emission and the FIR emission of cool
dust heated by massive stars. Yun et al. (2001) derive the follow-
ing RC–FIR luminosity correlation,
log(L1.4GHz) = (0.99 ± 0.01) log(L60µm/L⊙) + (12.07 ± 0.08) (6)
where log(L1.4GHz[WHz−1]) = 20.08+2 log(D)+log(S1.4GHz[Jy])
and log(L60µm[L⊙]) = 6.014 + 2 log(D) + log(S 60µm[Jy]); D is
the luminosity distance in Mpc. The radio flux density S 1.4GHz is
the extrapolated value of the integrated flux density at 1.4 GHz,
which we have fitted in Sect. 6.
Fig. 7 presents the FIR—RC relation for dwarf galaxies in
the CHILLING sample that exhibit either diffuse or compact RC
emission. For galaxies with diffuse RC emission, we use the SED
analysis to extrapolate the 1.4 GHz flux density and derive the
corresponding non-thermal spectral index, indicated by the best-
fit (see Table C.1 in Appendix C). For galaxies with compact
RC emission, we calculate the 1.4 GHz flux density as an upper
limit and adopt a fixed non-thermal spectral index of −0.7. These
compact detections are therefore shown as upper limits in Fig. 7.
In Fig. 7, we find that nearly all dwarf galaxies show lower
RC emission than expected based on their FIR luminosity, when
compared to the reference relation from Yun et al. (2001).
Two main groupings are evident, one around a FIR luminos-
ity of ∼108 L⊙and another near ∼109 L⊙. Dwarf galaxies that
show extended diffuse emission, such as IC 4662, NGC 625, or
NGC 5253, do not deviate from the general trend of lying be-
low the reference line. On the contrary, these galaxies consis-
tently exhibit a RC deficit. The dwarf galaxy ESO 572-G025 lies
above the expected relation, likely due to its relatively high SFR
and larger physical extent. This suggests that factors such as
SFR and galaxy size may help retain CREs within the galaxy,
allowing them to emit synchrotron radiation before escaping
via galactic winds or outflows. We also observe a general trend
where higher SFRs correspond to higher FIR luminosities. How-
ever, this increase does not necessarily bring galaxies closer to
the established FIR—RC relation. Most dwarf galaxies appear
under-luminous in the RC, which could either reflect the pres-
ence of very extended, diffuse emission that remains undetected
by ATCA due to observational limitations, or more likely a true
deficit in radio emission caused by CRE losses, as discussed in
Sect. 6.5. The latter scenario is favored given that our sample
was observed for multiple hours and only a few galaxies show
extended emission into the halo. For example, estimating the ex-
pected RC flux for NGC 625 and IC 4662 to lie on the Yun et al.
(2001) correlation, we find that their missing flux would cor-
respond to factors of 4.7 and 3.3, respectively. Such large dis-
crepancies suggest that the observed deficit cannot be explained
solely by sensitivity limits or missing zero spacing flux.
6. Discussion
6.1. What triggers RC emission in dwarf galaxies?
As in their more massive counterparts RC emission in dwarf
galaxies is closely linked to star formation activity (Hindson
et al. 2018). This connection also holds for the CHILLING sam-
ple, where galaxies with higher SFRs are more likely to exhibit
Article number, page 9 of 20
A&A proofs: manuscript no. main
Fig. 7. RC–FIR correlation from Yun et al. (2001), showing the CHILL-
ING galaxies overlaid. The diffuse RC detected galaxies are shown as
stars, while the compact RC detected galaxies are marked as upper lim-
its (UL).
detectable RC emission. As shown in Fig. 5, other galaxy proper-
ties also influence the presence of RC emission. Absolute magni-
tude, linked to SFR of the galaxies, and near-infrared brightness
(L3.4 µm), used as proxy for stellar mass, show a strong corre-
lation, reflecting the fact that more massive galaxies generally
host more active star formation and more frequent core-collapse
supernovae. In contrast, the total neutral atomic hydrogen (Hi)
mass does not show a significant correlation with RC detectabil-
ity. Although many dwarf galaxies contain substantial Hi reser-
voirs, this alone is not sufficient to generate RC emission. Star
formation and the associated feedback from young stars is the
key factor, as it drives the processes responsible for both ther-
mal and non-thermal RC components. Dwarf galaxies show in
general lower RC as they often have large gas reservoirs but lack
significant star formation as they are much smaller in size and
also possibly due to low gas densities or internal feedback pro-
cesses that inhibit the collapse of gas into stars. Similarly, ro-
tational velocity, which is commonly used as an indicator of a
galaxy’s dynamical mass, does not directly influence RC emis-
sion. While rotational velocity traces the overall gravitational
potential, it does not necessarily reflect the physical conditions
needed for star formation. This is especially true in low-mass
systems like dwarf galaxies, where internal feedback (e.g., stellar
winds, supernova outflows) can disturb gas dynamics, suppress
ordered rotation, and limit star-forming activity. Additionally,
non-circular motions in such systems make rotational velocity
an unreliable proxy for the energetic processes that drive syn-
chrotron emission. In summary, neither the total Hi gas mass nor
the rotational velocity alone can account for the presence of RC
emission. Instead, RC emission primarily traces recent star for-
mation activity, which depends on localised physical conditions
such as gas density, turbulence, and thermal stability. Therefore,
while the availability of gas and the galaxy’s dynamics provide
important context, they are not sufficient on their own to trigger
the RC emission observed in star-forming galaxies.
6.2. Physical implications of the different RC classification
types
The ‘diffuse’, ‘compact’ and ‘no/marginal’ RC classification
reveals systematic differences that suggest distinct physical
regimes. Dwarf galaxies with diffuse RC emission tend to
have higher integrated SFRs and substantially larger 60 mi-
cron luminosities than the ‘compact’ or ‘no/marginal’ systems
(mean log(SFRHα) ≈−0.9 for ‘diffuse’ vs ≈−1.5 for ‘com-
pact’ and ‘no/marginal’ in our sample), indicating more spa-
tially extended CR injection. This elevated star-formation ac-
tivity plausibly provides both the CR electron source term and
the turbulent energy required to amplify magnetic fields on
galactic scales, producing volume-filling synchrotron emission
(e.g. Chy˙zy et al. 2011; Beck 2015). Conversely, ‘compact’
RC detections are associated with bright, localised Hii com-
plexes but lack disk-wide emission, consistent with CR elec-
trons being largely confined to their birth sites. This may reflect
short propagation lengths (low diffusion coefficients (Heesen
2021), rapid vertical escape via localised outflows (Wang et al.
2022), or weak/inhomogeneous large-scale magnetic fields that
limit CR diffusion Heesen (2021)). The compact subsample
also shows slightly higher mean rotational velocities, suggest-
ing that ordered shear alone is not sufficient for sustaining
large-scale synchrotron halos, instead, the interplay of CR in-
jection, turbulent field amplification, and transport efficiency ap-
pears to control whether RC emission becomes diffuse. Sys-
tems with ‘no/marginal’ RC emission typically have the lowest
SFR and FIR luminosity, consistent with insufficient CR produc-
tion to yield a detectable synchrotron signal given our surface-
brightness sensitivity (Yun et al. 2001). Observational factors
such as lack of sensitivity of the ATCA telescopes may further
bias against detecting low-surface-brightness emission.
6.3. Do we need different models to explain CRE spectra?
An important question regarding the RC emission is whether
complex models are truly necessary or if a simple power-law
(PL) is sufficient to explain the spectra. Many dwarf galaxy spec-
tra clearly show curved shapes, with steepening or flattening over
several GHz and some show a turnover at low frequencies. For a
statistical relevant analysis, we compare the reduced chi-square
values, the Akaike information criterion (AIC; Akaike 1974) and
the Bayesian information criterion (BIC; Schwarz 1978) of the
best-fit models with those of the simple power-law. The AIC and
BIC are model selection metrics that evaluate the relative quality
of statistical models by balancing goodness of fit with model
complexity, where AIC applies a lighter penalty and is more
suitable for predictive accuracy, while BIC imposes a stronger
penalty, particularly favoring simpler models as sample size in-
creases. Lower values of the AIC or BIC indicate a better bal-
ance between model fit and complexity. Therefore, when com-
paring models, the preferred model is the one with the lowest
AIC or BIC value, regardless of whether the values are negative
or positive. In Fig. 8, we show the comparison of these values
of the best-fit models with those of the simple power-law and it
reveals that more complex models are indeed required to cap-
ture the spectral behaviour and underlying physics. In few cases,
such as IC 4662, NGC 625, ISZ 399 and NGC 244, the reduced
chi-squared of the PL is closer to the best value but we see that
for each galaxy the AIC and BIC of the best-fit is smaller then
the simple power-law, suggesting that we are in need of a more
complex model to explain the CRE spectra. Taking now the Jef-
freys’ scale (Kass & Raftery 1995), where the logarithm of the
Bayes factor quantifies the strength of evidence in favor of one
model over another, we observe in the right panel of Fig. 8 that
only for NGC 244, IC 4662, NGC 625 and NGC 5253 the differ-
ence is significant and we need to take a more complex model
into account. Looking at the difference of the best-fit model
Article number, page 10 of 20
Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
Fig. 8. Comparison between the best-fit model and the simple power-law (PL) model using statistical criteria: reduced chi-squared (χ2
red), Akaike
Information Criterion (AIC), Bayesian Information Criterion (BIC), and Jeffreys’ scale (Kass & Raftery 1995). The black dotted line in the left
panel marks the reference value of χ2
red = 1, while the black dotted line in the right panel indicates the threshold where the the best-fit model are
significantly preferred over the simple power-law.
Fig. 9. Comparison of the extrapolated flux density S 1.4 GHz at frequency
of 1.4 GHz, as derived from the best-fit model and the PL model. The
uncertainties for both methods are estimated to be 10 %.
and the simple PL model in Fig. 6, we observe already by eye
that we definitely need a complex model for ESO 572-G025 and
Fairall 301, which is not visible through the statistical test. The
full model parameters for the best-fit of each galaxy are listed in
Table C.1 in Appendix C. Comparing the flux density obtained
from the best-fit model and the PL, we observe in Fig. 9, that the
difference is minimal. Only for a few galaxies, such as ESO 572-
G025 or Fairall 301, the difference is larger than 10 %. For these
two galaxies, especially for ESO 572-G025, the statistical tests
differ most in between the simple power-law and the complex
model.
6.4. Does SFR trigger free–free absorption in dwarf
galaxies?
Normally, a high SFR suggest an optically thick regime in Hα,
which leads to free–free absorption, but we do not observe a
correlation between the different model components and other
characteristics of the galaxies, such as mass, FIR or SFR. Un-
fortunately, our sample is too small to draw conclusion if the
corresponding model components depend on some of these char-
acteristics.
6.5. Do dwarf galaxies obey the RC–FIR correlation?
For galaxies with a FIR luminosity of log(L60µm/L⊙) < 109, Yun
et al. (2001) already observed an increase of asymmetric scatter,
such that sources appear to be under-luminous in radio lumi-
nosity log(L1.4GHz), compared to the best fit. Such a deviation
can occur if the FIR or RC is not directly proportional to the
star formation activity. Dust heated by low-mass stars may con-
tribute only to the FIR. Low luminosity, low mass galaxies may
lose more radio emission than high luminosity objects to cosmic
ray diffusion. Roychowdhury et al. (2012) studied the RC–FIR
correlation in faint irregular galaxies by stacking images of in-
dividual galaxies together and finds that both the Spitzer 70 µm
and 1.4 GHz fluxes are generally lower than expected from what
is expected in spiral galaxies, however the ratio of RC to FIR
appears to be consistent with the larger spiral galaxies. Hindson
et al. (2018) analysed the RC–FIR correlation with a small dwarf
galaxies sample and found that dwarf galaxies contain less dust
than spiral galaxies, making them fainter in the FIR for a given
level of radio emission, in contrast to our study.
The low mass sample, analysed by Shao et al. (2018), ex-
hibit lower radio emissions compared to FIR emissions. This is
indicated by the analysis of the RC deficiency in dwarf galaxies,
which is primarily attributed to factors like a lower stellar mass
surface density and a higher Hi-to-stellar mass ratio. These find-
ings align with our results, that dwarf galaxies exhibit lower ra-
dio emission than FIR. The lower radio emission in dwarf galax-
ies, compared to their FIR emissions, is attributed to several fac-
tors. Shao et al. (2018) explains it that primarily, dwarf galaxies
have lower gas surface densities, which affects the production
of cosmic rays that contribute to radio emission. Furthermore,
the lower stellar mass surface density in these galaxies results
in less efficient conversion of star formation activity into radio
emissions. Additionally, in regions of low gas surface density,
the escaping CRs may lead to diminished radio output, while the
low dust content leads to lower FIR emission, creating a com-
pensatory effect. We attribute the observed radio deficit in the
RC–FIR correlation to significant energy losses experienced by
cosmic ray electrons. Losses, such as synchrotron radiation and
inverse Compton scattering which we can observe in the breaks
in the SEDs in Fig. 6, affect the total energy of CRs available to
generate radio emissions. In environments with lower gas sur-
face densities, such as those typical for dwarf galaxies, these
losses can be more pronounced, leading to a relative deficit in
the radio output compared to FIR emissions.
Article number, page 11 of 20
A&A proofs: manuscript no. main
7. Conclusions
We present a RC study of the CHILLING sample, consisting of
11 dwarf irregular galaxies and 4 BCDs spanning a broad range
of stellar masses and star formation histories. The combination
of multi-band observations with ATCA provides the opportunity
to study the full RC spectra and leads us to the following con-
clusions.
1. We detect RC emission in 11 of the 15 dwarf galaxies, with
7 exhibiting more diffuse emission and 4 displaying more
compact emission. The detectability of RC emission in dwarf
galaxies is closely linked to recent star formation activity, as
all 4 BCDs are detected. Galaxies with higher star forma-
tion rates are more likely to exhibit RC emission, which also
correlates with absolute magnitude and stellar mass. Most
RC-detected galaxies are classified as starbursts. In contrast,
parameters such as Hi mass and rotational velocity show no
significant correlation with RC detectability, indicating that
RC emission is more strongly influenced by star formation
than by galaxy dynamics.
2. The RC spectra of many dwarf galaxies in the sample display
complex, curved features, such as spectral steepening, flat-
tening, and low-frequency turnovers, that deviate markedly
from a simple power-law behavior. Statistical evaluations us-
ing reduced chi-squared, AIC, and BIC confirm that single
power-law models inadequately describe the observed spec-
tral shapes and fail to capture the underlying physical pro-
cesses. These findings underscore the need for more sophis-
ticated models incorporating mechanisms such as thermal
emission, free–free absorption, and spectral breaks arising
from cosmic ray electron energy losses.
3. In particular, the pronounced steepening of RC spectra at
higher frequencies is attributed to substantial energy losses
experienced by cosmic ray electrons. These losses, primarily
due to synchrotron radiation and inverse Compton scattering,
become increasingly significant at higher electron energies,
leading to the observed decline in flux density.
4. Nearly all dwarf galaxies in the sample show a deficit in RC
emission relative to their FIR luminosity when compared to
the correlation seen in more massive galaxies. This deficit
is best explained by significant energy losses of cosmic ray
electrons, primarily via synchrotron radiation and inverse
Compton scattering, which are more effective in the low-
density environments typical of dwarf galaxies.
5. The possibility that extended low-radio-brightness diffuse
emission remains undetected due to the sensitivity limita-
tions of ATCA may partially reflect these instrumental con-
straints. However, the spectral properties observed across the
sample more strongly indicate that energy losses of CREs are
the primary drivers of both the spectral steepening and the
RC deficit.
Acknowledgements. ST, BA, DJB and MS acknowledge the support from the
DFG via the Collaborative Research Center SFB1491 Cosmic Interacting Mat-
ters - From Source to Signal. PK acknowledge the support of the BMBF project
05A23PC1 for D-MeerKAT. The Australia Telescope Compact Array is part
of the Australia Telescope National Facility (https://ror.org/05qajvd42) which is
funded by the Australian Government for operation as a National Facility man-
aged by CSIRO. We acknowledge the Gomeroi people as the Traditional Owners
of the Observatory site.
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Article number, page 13 of 20
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Appendix A: Radio continuum properties
In this section, we present the RC maps of the ATCA data of L/S-band, C-band and X-band. Fig. A.1 presents the RC maps of dwarf
galaxies with diffuse radio emission. Fig. A.2 shows the RC maps of galaxies with compact RC emission. Fig. A.3 displays the RC
maps of galaxies with no clear or only weak RC emission, which are classified as no/marginal detections.
Fig. A.1. First row left panel: Total intensity emission of ESO 572 G 025 at the central frequency of 2.1 (σ = 30 µJy/beam), 5.5 (σ = 20 µJy/beam)
and 9 GHz (σ = 30 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. First row right panel: Total intensity
emission of Fairall 201 at the central frequency of 2.1 (σ = 14 µJy/beam), 5.5 (σ = 12 µJy/beam) and 9 GHz (σ = 25 µJy/beam) with overlaid
contours starting at 3 σ and increasing by factor of two. Second row left panel: Total intensity emission of IC 4662 at the central frequency of
2.1 (σ = 9 µJy/beam), 5.5 (σ = 8 µJy/beam) and 9 GHz (σ = 12 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of
two. Second row right panel: Total intensity emission of ISZ 399 at the central frequency of 2.1 (σ = 73 µJy/beam), 5.5 (σ = 20 µJy/beam) and
9 GHz (σ = 25 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Third row left panel: Total intensity emission of
NGC 244 at the central frequency of 2.1 (σ = 10 µJy/beam), 5.5 (σ = 8 µJy/beam) and 9 GHz (σ = 8 µJy/beam) with overlaid contours starting at
3 σ and increasing by factor of two. Third row right panel: Total intensity emission of NGC 625 at the central frequency of 2.1 (σ = 12 µJy/beam),
5.5 (σ = 8 µJy/beam) and 9 GHz (σ = 8 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Fourth row panel: Total
intensity emission of NGC 5252 at the central frequency of 2.1 (σ = 48 µJy/beam), 5.5 (σ = 35 µJy/beam) and 9 GHz (σ = 50 µJy/beam) with
overlaid contours starting at 3 σ and increasing by factor of two. The beam is shown in the left corner.
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Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
Fig. A.2. First row left panel: Total intensity emission of ESO 245 G 005 at the central frequency of 2.1 (σ = 15 µJy/beam), 5.5 (σ = 6 µJy/beam)
and 9 GHz (σ = 6 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. First row right panel: Total intensity emission
of IC 1959 at the central frequency of 2.1 (σ = 20 µJy/beam), 5.5 (σ = 10 µJy/beam) and 9 GHz (σ = 10 µJy/beam) with overlaid contours starting
at 3 σ and increasing by factor of two. Second row left panel: Total intensity emission of IC 5152 at the central frequency of 2.1 (σ = 15 µJy/beam),
5.5 (σ = 8 µJy/beam) and 9 GHz (σ = 8 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row right panel:
Total intensity emission of NGC 5408 at the central frequency of 2.1 (σ = 85 µJy/beam), 5.5 (σ = 17 µJy/beam) and 9 GHz (σ = 18 µJy/beam)
with overlaid contours starting at 3 σ and increasing by factor of two. The beam is shown in the left corner.
Fig. A.3. First row left panel: Total intensity emission of ESO 154 G 023 at the central frequency of 2.1 (σ = 40 µJy/beam), 5.5 (σ = 10 µJy/beam)
and 9 GHz (σ = 12 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. First row right panel: Total intensity emission
of ESO 383 G 087 at the central frequency of 2.1 (σ = 40 µJy/beam), 5.5 (σ = 15 µJy/beam) and 9 GHz (σ = 20 µJy/beam) with overlaid contours
starting at 3 σ and increasing by factor of two. Second row left panel: Total intensity emission of NGC 1311 at the central frequency of 2.1
(σ = 27 µJy/beam), 5.5 (σ = 15 µJy/beam) and 9 GHz (σ = 10 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two.
Second row right panel: Total intensity emission of NGC 5264 at the central frequency of 2.1 (σ = 80 µJy/beam), 5.5 (σ = 20 µJy/beam) and
9 GHz (σ = 13 µJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. The beam is shown in the left corner.
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A&A proofs: manuscript no. main
Appendix B: Spectral fitting models
In this section, we present the 19 different models, which have been fitted to the dataset for each dwarf galaxy. These models build
upon the work of Galvin et al. (2018) and Grundy et al. (2025), incorporating further refinements to the absorption component by
accounting for both internal and external free–free absorption.
Appendix B.1: Base Model
The first two models are the base for all the other models and have already been described in Sect.4. The first is a basic power-law
model for synchrotron emission (Eq.1), while the second is a constant model representing the combined contribution of synchrotron
and free–free emission (Eq. 2).
Power-low (PL)
S ν = A
ν
ν0
!α
(B.1)
Superposition of synchrotron and free–free emission (SFG)
S ν = A
ν
ν0
!α
+ B
ν
ν0
!−0.1
(B.2)
Appendix B.2: Prefix: Free-free Absorption (FFA_)
Synchrotron emission can be attenuated by free-free absorption (FFA) when coexisting with ionised gas that produces free-free
emission, leading to spectral curvature at low frequencies. The optical depth for absorption is:
τi =
ν
νi
!−2.1
(B.3)
We distinguish between internal and external free-free absorption (Tingay & de Kool 2003). For internal free-free absorption, the
free-free emission and synchrotron emission are both within the same region, while external free-free absorption occurs when syn-
chrotron emission passes through a foreground layer of ionised gas that emits free-free radiation. In this case, the absorbing material
is physically separate from the synchrotron-emitting region. The absorption leads to a low-frequency turnover in the spectrum, with
the degree of attenuation depending on the density and path length of the ionized gas.
Free–Free external Absorption + Synchrotron (FFA1_SFG)
S ν = (1 −e−τ1)
B + A
ν
ν1
!0.1+α
ν
ν1
!2
(B.4)
Parameters: A, B, α, ν1
Free–Free internal Absorption + Synchrotron (FFA2_SFG)
S ν = (1 −e−τ1)
τ2
B + A
ν
ν1
!0.1+α
ν
ν2
!2
(B.5)
Parameters: A, B, α, ν2
By assuming that the observed emission arises purely from synchrotron radiation affected by free-free absorption, without
any contribution from free–free emission, the parameter B can be set to 0. This yields a simplified version of the full model, while
still distinguishing between external and internal free-free absorption scenarios.
Free–Free external Absorption (FFA1_PL)
S ν = (1 −e−τ1)A
ν
ν1
!0.1+α ν
ν1
!2
(B.6)
Parameters: A, α, ν1
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Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
Free–Free internal Absorption (FFA2_PL)
S ν = (1 −e−τ2)
τ2
A
ν
ν2
!0.1+α ν
ν2
!2
(B.7)
Parameters: A, α, ν1
Appendix B.3: Suffix: Free-free Absorption (_FFA)
These models represent synchrotron emission modified by free–free absorption in ionised gas, but without any contribution from
free–free emission itself (i.e., B = D = 0). The models assume a single population of relativistic electrons embedded in or obscured
by thermal plasma. This first model (PL_FFA1) describe synchrotron emission passing through an external ionised region with
foregound thermal plasma. In the second model (PL_FFA2), we assume internal free-free absorption, where the synchrotron-emitting
and absorbing regions are co-spatial.
Free–Free external Absorption + only Synchrotron (PL_FFA1)
S ν = A
ν
ν0
!α
+ (1 −e−τ1)
C
ν
ν1
!0.1+α
ν
ν1
!2
(B.8)
Parameters: A,C, α, ν1
Free–Free internal Absorption + only Synchrotron (PL_FFA2)
S ν = A
ν
ν0
!α
+ (1 −e−τ2)
τ2
C
ν
ν2
!0.1+α
ν
ν2
!2
(B.9)
Parameters: A,C, α, ν2
This model extends the simple synchrotron–free–free absorption framework by including a free-free emission component in
addition to synchrotron radiation and external free–free absorption. The total emission consists of a synchrotron power law, a free-
free component, and an externally (SFG_FFA1) or internally (SFG_FFA2) absorbed component consisting of synchrotron and thermal
emission
Power-Law + Free–Free external Absorption (SFG_FFA1)
S ν = A
ν
ν0
!α
+ B
ν
ν0
!−0.1
+ (1 −e−τ2)
D + C
ν
ν2
!0.1+α
ν
ν2
!2
(B.10)
Parameters: A, B,C, D, α, ν2
Power-Law + Free–Free internal Absorption (SFG_FFA2)
S ν = A
ν
ν0
!α
+ B
ν
ν0
!−0.1
+ (1 −e−τ2)
τ2
D + C
ν
ν2
!0.1+α
ν
ν2
!2
(B.11)
Parameters: A, B,C, D, α, ν2
This following models introduce two distinct synchrotron-emitting regions, each experiencing different types of free-free ab-
sorption. The first component is attenuated by external free–free absorption (FFA1), while the second experiences internal free–free
absorption (FFA2). Both components share the same synchrotron spectral index, representing a single electron population. This
setup captures systems with multiple star-forming regions or geometrically distinct zones, where only one region becomes optically
thick within the observed frequency range.
This dual-FFA model is inspired by studies like Galvin et al. (2018), which show that unresolved blends of emission from
multiple regions with different physical conditions can reproduce complex radio SEDs with multiple spectral turnovers or inflection
points. It is best used when the data suggest two absorption turnovers at distinct frequencies, possibly corresponding to low- and
mid-frequency spectral curvature.
Dual Free–Free Absorption + Power Law (FFA1_PL_FFA2)
S ν = (1 −e−τ1)A
ν
ν1
!0.1+α ν
ν1
!2
+ (1 −e−τ2)
τ2
C
ν
ν2
!0.1+α ν
ν2
!2
(B.12)
Parameters: A,C, α, ν1, ν2
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A&A proofs: manuscript no. main
The FFA1_SFG_FFA1 model is a further extension in which both components, each consisting of synchrotron and thermal
emission, are absorbed by two distinct external free-free absorption regions. This model accounts for a scenario in which both
components undergo absorption, possibly due to layers of ionised gas in complex starburst geometries.
The more advanced model, FFA1_SFG_FFA2, allows each component to have an independent synchrotron spectral index, de-
noted by α1 and α2. This is physically motivated in systems such as post-mergers or interacting galaxies, where newly formed
relativistic electrons may have a flatter spectrum while older populations have steeper slopes due to energy losses. This model
provides the highest flexibility and is most appropriate when fitting SEDs that require distinct spectral indices to explain high- and
low-frequency behavior.
Dual Free–Free Absorption + Power Law (FFA1-SFG-FFA1)
S ν = (1 −e−τ1)
B + A
ν
ν1
!0.1+α
ν
ν1
!2
+ (1 −e−τ2)
D + C
ν
ν2
!0.1+α
ν
ν2
!2
(B.13)
Parameters: A,C, α, ν1, ν2
Dual Free–Free Absorption + Power Law (FFA1-SFG-FFA2)
S ν = (1 −e−τ1)
B + A
ν
ν1
!0.1+α1
ν
ν1
!2
+ (1 −e−τ2)
τ2
D + C
ν
ν2
!0.1+α2
ν
ν2
!2
(B.14)
Parameters: A, B,C, D, α1, α2, ν1, ν2
Appendix B.4: Suffix: Inverse-Compton Losses (_SIC)
At high frequencies, synchrotron spectral steepening can occur not due to free-–free absorption, but as a result of energy losses
experienced by cosmic ray electrons. These losses are primarily caused by synchrotron radiation and inverse Compton scattering,
which occurs when cosmic ray electrons scatter off ambient photons, including those from the cosmic microwave background or the
far-infrared radiation field. Under a continuous injection model, the effect of these losses is a gradual steepening of the synchrotron
spectral index by ∆α = −0.5, beginning around a characteristic break frequency νb. This model is useful for describing systems
where high-frequency spectral curvature is likely due to energy losses rather than thermal absorption.
Synchrotron with Inverse Compton Losses (PL_SIC)
S ν =
A
ν
ν0
α
1 +
ν
νb
∆α
(B.15)
Parameters: A, α, νb, ∆α
This model extends PL_SIC by including a thermal free–free emission component. It is intended for galaxies where thermal
emission contributes at higher frequencies, in addition to synchrotron radiation affected by inverse Compton or synchrotron losses.
Synchrotron + Inverse Compton (SFG_SIC)
S ν =
A
ν
ν0
α
1 +
ν
νb
∆α + B
ν
ν0
!−0.1
(B.16)
Parameters: A, B, α, νb, ∆α
The following two models combines external (FFA1_PL_SIC) and internal (FFA2_PL_SIC) free–free absorption with syn-
chrotron emission undergoing high-frequency steepening due to losses.
Free–Free external Absorption + Synchrotron + Inverse Compton (FFA1_PL_SIC)
S ν = (1 −e−τ1)
A
ν
ν1
!0.1+α
1
1 +
ν
νb
∆α
ν
ν1
!2
(B.17)
Parameters: A, B, α, ν1, νb, ∆α
Article number, page 18 of 20
Sam Taziaux
et al.: CHILLING - RC spectral behavior of dwarf galaxies
Free–Free internal Absorption + Synchrotron + Inverse Compton (FFA2_PL_SIC)
S ν = (1 −e−τ2)
τ2
A
ν
ν2
!0.1+α
1
1 +
ν
νb
∆α
ν
ν2
!2
(B.18)
Parameters: A, B, α, ν2, νb, ∆α
This model extends the SFG_SIC model by including external FFA1_ and internal FFA2_ free-–free absorption.
Free–Free external Absorption + Synchrotron + Inverse Compton (FFA1_SFG_SIC)
S ν = (1 −e−τ1)
B + A
ν
ν1
!0.1+α
1
1 +
ν
νb
∆α
ν
ν1
!2
(B.19)
Parameters: A, B, α, ν1, νb, ∆α
Free–Free internal Absorption + Synchrotron + Inverse Compton (FFA2_SFG_SIC)
S ν = (1 −e−τ2)
τ2
B + A
ν
ν2
!0.1+α
1
1 +
ν
νb
∆α
ν
ν2
!2
(B.20)
Parameters: A, B, α, ν2, νb, ∆α
The inclusion of both free–free absorption and synchrotron or inverse Compton losses allows for modeling galaxies whose
radio SEDs exhibit both low-frequency turnover and high-frequency steepening-features commonly observed in luminous infrared
galaxies and starbursts. However, as simultaneous modeling of multiple synchrotron components, each with their own breaks and
absorption features, is currently limited by spectral resolution and coverage. Still, models like FFA_SFG_SIC provide significant
flexibility for interpreting breaks and curvature in observed radio SEDs.
Appendix C: Spectral comparison
In this section, we present the parameter values obtained from the spectral models. We provide a comparison between the parameters
of the best-fit model and those of a simple power-law model.
Article number, page 19 of 20
A&A proofs: manuscript no. main
Table C.1. Model comparison between the best-fit model and a simple power-law (PL) model for the CHILLING dwarf galaxies. Reported are χ2, reduced χ2, Akaike information criterion (AIC),
Bayesian information criterion (BIC), fit parameters (A–D, α, ν1, ν2, νb, ∆α), and the modeled flux at 1.4 GHz.
Name
Model
χ2
χ2
red
AIC
BIC
A
B
C
D
α
ν1
ν2
νb
∆α
Flux / mJy
ESO 572-G025
FFA2_SFG_SIC
45.07
2.37
31.33
43.32
40.00
0.00
10.66
25.37
-1.14
4.27
1.63
1.71
2.79
12.00
PL
504.98
26.58
98.98
110.97
9.14
0.25
1.67
39.20
-0.67
1.32
9.16
9.22
0.01
9.14
Fairall 301
FFA1_SFG_SIC
82.50
3.30
48.14
61.88
15.97
1.58
38.16
27.28
-0.85
0.22
9.90
6.62
0.30
3.31
PL
162.06
6.48
71.09
84.83
2.57
32.84
36.02
25.33
-0.28
3.04
0.80
0.32
1.07
2.57
IC 4662
SFG_SIC
12.02
0.57
-9.45
3.16
3.25
0.00
12.07
16.68
-2.39
3.19
5.21
4.16
0.03
21.76
PL
15.28
0.73
-2.24
10.37
22.04
13.84
6.43
16.02
-0.21
9.59
2.17
0.45
4.46
22.04
ISZ 399
SFG_FFA1
30.29
2.02
23.58
34.19
4.71
0.00
5.06
1.76
-2.09
5.31
9.24
7.52
0.33
9.84
PL
39.97
2.66
30.24
40.84
10.98
1.00
39.36
1.53
-0.69
6.91
0.85
2.87
1.46
10.98
NGC 244
FFA1_SFG_SIC
11.74
0.49
-16.10
-2.63
1.15
0.32
20.45
15.35
-2.36
5.29
0.63
6.46
2.00
1.57
PL
32.41
1.35
17.41
30.88
1.86
3.54
20.55
37.42
-0.86
7.47
1.29
8.92
1.05
1.86
NGC 625
SFG_FFA1
17.20
0.61
-10.34
4.15
0.49
0.00
9.01
9.33
-1.33
0.45
1.33
5.93
0.47
10.34
PL
22.16
0.79
-0.96
13.54
10.07
12.47
2.76
29.77
-0.19
8.18
6.51
1.78
0.00
10.07
NGC 5253
SFG_FFA1
1.49
0.065
-80.06
-66.87
17.69
29.48
23.33
21.75
-2.14
5.33
5.86
6.10
1.97
73.15
PL
7.52
0.33
-28.35
-15.15
72.94
29.45
25.24
6.72
-0.22
8.27
1.75
9.40
0.38
72.94
Article number, page 20 of 20
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Astronomy & Astrophysics manuscript no. main October 17, 2025 CHILLING: Continuum Halos in LVHIS Local Irregular Nearby Galaxies Radio continuum spectral behavior of dwarf galaxies Sam Taziaux1, 2, 3 , Megan C. Johnson4 , Onic I. Shuvo5 , Dominik J. Bomans1, 2 , Christopher J. Riseley1, 2 , Timothy J. Galvin3 , Alec J. M. Thomson6 , Peter Kamphuis1 , Amy Kimball7 , Amanda Kepley8 , Michael Stein1 , George H. Heald6 , Nicholas Seymour9 , Joe A. Grundy3, 9 , Björn Adebahr1 , and Ralf-Jürgen Dettmar1, 2 1 Ruhr University Bochum, Faculty of Physics and Astronomy, Astronomical Institute (AIRUB), Universitätsstraße 150, 44801 Bochum, Germany e-mail: 2 Ruhr Astroparticle and Plasma Physics Center (RAPP Center) 3 CSIRO Space and Astronomy, PO Box 1130, Bentley WA 6102, Australia 4 National Science Foundation, 2415 Eisenhower Avenue, Alexandria, VA 22314, USA 5 1000 Hilltop Circle, Baltimore, MD 21250, USA 6 SKA Observatory, SKA-Low Science Operations Centre, 26 Dick Perry Avenue, Kensington WA 6151, Australia 7 National Radio Astronomy Observatory, 1011 Lopezville Road, Socorro, NM 87801, USA 8 National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA 9 International Centre for Radio Astronomy Research, Curtin University, Bentley, WA, Australia Accepted XXX. Received YYY; in original form ZZZ ABSTRACT Context. Dwarf galaxies, due to their shallow gravitational potentials, provide critical environments for studying feedback mechanisms from star formation and its impacts on dwarf galaxy evolution. In particular, radio continuum (RC) observations offer valuable insights into cosmic ray dynamics, which play a significant role in shaping these processes. Aims. This study investigates the detectability and spectral characteristics of RC emission in a sample of 15 dwarf galaxies (11 gas-rich, star forming dwarfs and 4 blue compact dwarfs) spanning a broad range of stellar masses and star formation histories. Methods. Using multi-band RC data (L/S-, C-, and X-band) from the Australia Telescope Compact Array, we analyse the physical conditions responsible for RC emission and explore the dominant emission mechanisms within these systems. Results. RC emission is detected in 11 out of the 15 galaxies. Our results indicate that RC emission correlates strongly with star formation rate, far-infrared, and stellar mass, while dynamic parameters such as Hi and rotational velocity exhibit no significant correlation with RC detectability. Spectral analysis reveals that the RC spectral energy distribution in these galaxies frequently deviate from a simple power-law behavior, instead displaying curvature that suggests more complex underlying physical processes. Statistical model comparison confirms that a single power-law model is inadequate to capture the observed spectral shapes, emphasising the necessity of more sophisticated approaches. Additionally, the observed radio-far-infrared correlation indicates that cosmic ray electrons in lower-mass dwarf galaxies cool more rapidly than they can escape (e.g. via galactic winds), resulting in a measurable RC deficit. Key words. 1. Introduction Galactic winds and outflows1 are crucial in shaping the evolutionary histories of galaxies. These processes, driven by the energy and momentum from star formation, influence the interstellar medium (ISM) by enhancing turbulence and providing additional pressure support. This additional pressure can counteract gravitational collapse, thereby regulating further star formation. Moreover, outflows can expel gas and metals from the ISM, potentially enriching the circumgalactic medium or even escaping 1 In this article we adhere to the definition of Veilleux et al. (2005), who defines an "outflow" as any material moving away from a central object, whereas "wind" specifically describes gas expelled from a galaxy's potential, typically driven by mechanisms such as radiation pressure and incorporating both diffuse flows and jets. into the intergalactic medium if the outflow velocity exceeds the escape velocity of the host galaxy (Veilleux et al. 2005). Dwarf galaxies serve as essential laboratories for investigating feedback mechanisms from star formation, particularly the role of star formation in driving winds and outflows. Due to their shallow gravitational potentials, even moderate starburst events can generate significant winds (e.g. Chy ̇zy et al. 2016) or outflows (e.g. Devine & Bally 1999; Strickland & Heckman 2009; Adebahr et al. 2013; Dirks et al. 2023). Understanding these feedback mechanisms is vital to comprehending the interplay between star formation and dwarf galaxy evolution (e.g. Dekel & Silk 1986; Stevens et al. 2002; Chy ̇zy et al. 2016). Most studies on dwarf galaxies in the literature have focused on their integrated radio continuum (RC) properties (e.g. Klein et al. 2018). However, resolved RC observations of these galaxies enable a better understanding of the evolution and outflow characterisArticle number, page 1 of 20 16 Oct 2025 A&A proofs: manuscript no. main tics of these low-mass galaxies. Only a few studies have utilised resolved RC and spectral index observations to analyse dwarf galaxies in detail (e.g. Chy ̇zy et al. 2000; Kepley et al. 2010; Basu et al. 2017; Kepley et al. 2011; Hindson et al. 2018; Taziaux et al. 2025). These RC observations have shown a mean non-thermal spectral index (S ∝ναnth) of -0.6 that steepens at the galaxy outskirts to -1.1 (e.g. Chy ̇zy et al. 2000; Kepley et al. 2010, 2011; Westcott et al. 2018; Taziaux et al. 2025). In the resolved starforming knots, the spectral index can reach a flatter spectrum of -0.3 or even be inverted (e.g. Basu et al. 2017; Taziaux et al. 2025), depending on the optical depth of the galaxy region. Radio observations also provide a crucial tool for studying cosmic ray (CR) transport and the dominant energy loss processes (synchrotron, inverse-Compton or bremsstrahlung), with relativistic electrons serving as tracers (e.g. Lacki et al. 2010; Werhahn et al. 2021; Pfrommer et al. 2022; Heesen et al. 2022). CRs propagate from supernova remnants into the galactic halo through advection, diffusion, and streaming (e.g. Strong et al. 2007; Stein et al. 2019; Thomas et al. 2020). In this process, CRs excite Alfvén and whistler waves via resonant plasma instabilities, which scatter CRs and regulate their drift speed along the magnetic field (Kulsrud & Pearce 1969; Shalaby et al. 2021, 2023; Lemmerz et al. 2024). Through resonant scattering, CRs transfer momentum to the thermal gas, exerting pressure on the ambient plasma and driving galactic winds, a key aspect of CR hydrodynamic models (Zweibel 2013; Pfrommer et al. 2017a; Thomas & Pfrommer 2019). Hydrodynamic simulations indicate that CR-driven winds significantly influence mass-loss rates, gas distribution, and wind formation (e.g. Breitschwerdt et al. 1991; Uhlig et al. 2012; Pakmor et al. 2016; Girichidis et al. 2016; Recchia et al. 2017; Dashyan & Dubois 2020; Thomas et al. 2023). Thomas et al. (2024) demonstrated that CRs, due to their long cooling times and strong plasma coupling, drive denser winds with higher mass-loading factors, efficiently redistributing gas into the halo and regulating star formation. This study further investigates the properties of these winds and outflows, particularly their velocity into the galactic halo, as driven by stellar feedback in low metallicity regimes. Because CR protons (CRPs) primarily lose energy through hadronic interactions with dense gas, they predominantly illuminate highdensity regions (Pfrommer & Enßlin 2004; Pfrommer et al. 2017b; Werhahn et al. 2023). In contrast, lower-density outflows are best traced via synchrotron emission from CR electrons (CREs). Modelling CREs alongside CRPs is essential, as their different loss timescales cause deviations in their energy spectra (Ruszkowski & Pfrommer 2023), necessitating a separate spectral treatment for accurate predictions of galactic radio emission (Chiu et al. 2024). Beyond their role in regulating outflows, CRs also contribute to the evolution of large-scale ordered magnetic fields in galaxies. Observations and simulations of superbubbles in nearby spiral galaxies suggest that starburst-driven outflows can advect ordered magnetic field lines from the inner disk regions to the outer halo (e.g. Kepley et al. 2010; Chy ̇zy et al. 2011; Kepley et al. 2011). This interplay between galactic winds, magnetic field evolution, and cosmic ray transport is particularly relevant for dwarf galaxies, where such effects can be more pronounced due to lower gravitational binding energy. Despite theoretical and numerical advancements, direct observational evidence of CR-driven winds remains challenging due to the (i) sensitivity limits of current radio facilities, which hinder the detection of faint halo emissions, and (ii) lack of robust modelling frameworks for various CR propagation mechanisms. RC surveys targeting dwarf galaxies are crucial for addressing these challenges and testing theoretical predictions of cosmic ray transport and their role in galactic feedback. In this paper, we further explore the detectability of radio emission in dwarf galaxies, the dependence of energy loss mechanisms, and the modelling of cosmic ray transport across a larger sample of dwarf galaxies spanning a wide range of masses and star formation histories. We describe our sample and the data reduction in Sect. 2. In Sect. 3, we present detectability of radio emission across various parameter and the basic properties of RC will be shown in Sect. 4. Sect. 5 will focus on the RC-FIR correlation. We discuss our observations in the context of what is known so far in Sect 6 with a summary and conclusion presented in Sect. 7. 2. Observation and data reduction 2.1. CHILLING sample The Continuum Halos In LVHIS Local Irregular Nearby Galaxies (CHILLING) sample covers a range of galaxy properties, such as star formation rate, luminosities, and flux densities. The selected galaxies have an absolute magnitude of MB > -17, IRAS 60 μm detections, and are located at a declination below about -20◦. The CHILLING sample includes 11 dwarf galaxies from the Local Volume Hi Survey (LVHIS) project (Koribalski et al. 2018), as well as 4 blue compact dwarf galaxies (BCDs) that are further away and therefore not part of the main LVHIS sample. Table 1 gives an overview of the properties of the CHILLING sample. 2.2. Observations The CHILLING dwarf galaxies were observed with the ATCA (project ID: C3041; PI: M. Johnson). The observations were performed at 1.1-3.1 GHz (L/S-band), 3.9-7.1 GHz (C-band) and 8-11 GHz (X-band) between 18-May-2015 and 16-June-2016. The configurations used for each target are listed in Table 2. Observations of the target galaxies and a phase calibrator were alternated through the observing run, giving a total on-source observing time of about 5 to 11 hrs for each galaxy (see Table 2). Along with the target and phase calibrator observations, the standard ATCA primary flux calibrator 1934-638 and the secondary flux calibrator 0823-500 have been used. For the galaxy IC 4662, we use additional observational data of on-source integration time of 29.41 hrs at central frequencies 2.1 GHz, 4.71 hrs at 5.5 and 9 GHz, which have been observed between 4-May-2023 and 26August-2024 (project ID: C3531; PI: S. Taziaux). 2.3. Data reduction We processed the observations using the Multichannel Image Reconstruction, Imaging Analysis, and Display package (miriad; Sault et al. 1995) following standard data calibration procedures using mfcal for bandpass calibration, gpcal to derive the gains and instrument leakage. To minimise the effects of radio frequency interference (RFI) during flux and phase calibration, the edges of the dataset have been flagged using uvflag, automated flagging was performed using pgflag to suppress interference in the target sources. Any remaining corrupted data identified throughout the calibration process was manually flagged, using the interactive flagging tool blflag Article number, page 2 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies 11h57m33s 30s 27s 24s 19°36'30" 37'00" 30" 38'00" 30" RA (ICRS) Dec (ICRS) 1.5 arcmin = 12.09 kpc ESO572 G025 4h06m16s 12s 08s 04s 00s 52°39'00" 30" 40'00" 30" 41'00" RA (ICRS) Dec (ICRS) 1.5 arcmin = 4.04 kpc Fairall301 0h45m51s 48s 45s 42s 15°34'30" 35'00" 30" 36'00" 30" 37'00" RA (ICRS) Dec (ICRS) 1.5 arcmin = 5.04 kpc NGC 244 12h20m03s 00s 19m57s 17°22'30" 23'00" 30" 24'00" 30" RA (ICRS) Dec (ICRS) 1.5 arcmin = 6.93 kpc ISZ 399 17h47m30s 20s 10s 00s 64°37' 38' 39' 40' RA (ICRS) Dec (ICRS) 2 arcmin = 1.42 kpc IC 4662 1h35m15s 10s 05s 00s 34m55s 41°25' 26' 27' 28' RA (ICRS) Dec (ICRS) 2 arcmin = 2.32 kpc NGC 625 13h40m06s 00s 39m54s 48s 42s 31°36' 37' 38' 39' 40' 41' RA (ICRS) Dec (ICRS) 3 arcmin = 2.99 kpc NGC 5253 Fig. 1. Colour-composite images from the DESI Legacy Imaging Surveys of the RC detected CHILLING sample, showing more diffuse emission, with overlaid ATCA radio emission contours starting at 3 σ and increasing by a factor of √ 2 at a central frequency of 2.1 GHz. For IC 4662, we use the Hα map from Hunter & Elmegreen (2004) as the declination is too low for the DESI Legacy Imaging Surveys. The scale in the top left corner is calculated using the distance to the source. The beam is shown in the bottom left corner. The noise level σ2.1 GHz can be taken from Table 3. 1h45m12s 08s 04s 00s 44m56s 43°34'30" 35'00" 30" 36'00" 30" 37'00" RA (ICRS) Dec (ICRS) 2 arcmin = 2.56 kpc ESO245 G005 3h33m20s 15s 10s 05s 50°23'30" 24'00" 30" 25'00" 30" 26'00" RA (ICRS) Dec (ICRS) 1.5 arcmin = 2.69 kpc IC 1959 22h03m00s 02m50s 40s 30s 51°15' 16' 17' 18' 19' 20' RA (ICRS) Dec (ICRS) 3 arcmin = 1.88 kpc IC 5152 14h03m30s 24s 18s 12s 41°21' 22' 23' 24' 25' RA (ICRS) Dec (ICRS) 3 arcmin = 4.65 kpc NGC 5408 Fig. 2. Colour-composite images from the DESI Legacy Imaging Surveys of the RC non-detected CHILLING sample, showing more compact emission, with overlaid 2.1 GHz ATCA radio emission contours starting at 3 σ and increasing by a factor of √ 2. The scale in the top left corner is calculated using the distance to the source. The beam is shown in the bottom left corner. The noise level σ2.1 GHz can be taken from Table 3. 2h57m00s 56m55s 50s 45s 40s 54°33'00" 30" 34'00" 30" 35'00" 30" RA (ICRS) Dec (ICRS) 1.5 arcmin = 2.50 kpc ESO154 G023 13h49m24s 20s 16s 12s 36°02'30" 03'00" 30" 04'00" 30" 05'00" RA (ICRS) Dec (ICRS) 1.5 arcmin = 1.50 kpc ESO383 G087 3h20m15s 10s 05s 00s 52°10'00" 30" 11'00" 30" 12'00" 30" RA (ICRS) Dec (ICRS) 1.5 arcmin = 2.41 kpc NGC 1311 13h41m42s 39s 36s 33s 30s 29°53'30" 54'00" 30" 55'00" 30" 56'00" RA (ICRS) Dec (ICRS) 1.5 arcmin = 2.09 kpc NGC 5264 Fig. 3. Colour-composite images from the DESI Legacy Imaging Surveys of the CHILLING sample with non-detected or marginally detected RC emission, overlaid 2.1 GHz ATCA radio emission contours starting at 3 σ and increasing by a factor of √ 2. The scale in the top left corner is calculated using the distance to the source. The beam is shown in the bottom left corner. The noise level σ2.1 GHz can be taken from Table 3. for manual inspection and removal of contaminated data, to ensure data integrity. Initially, after cross-calibration and excluding channels with Hi emission for L/S-band dataset, we employed an iterative imaging using WS-clean (Offringa et al. 2014) and selfcalibration (phase-only, frequency-independent, with a solution interval of 5 min down to 60 s) using CASA (version 6.4.4.31; McMullin et al. 2007) until image quality reached convergence with a Briggs weighting of robust = -1, -0.5, -0.3, 0, 0.3 to slowly reconstruct the diffuse emission. Imaging and self-calibration was performed independently for L/S-, C-, and X-bands and by Article number, page 3 of 20 A&A proofs: manuscript no. main Table 1. CHILLING sample properties including the coordinates, distance, absolute magnitude in B-band, star formation rate, Hi mass, rotational velocity, and the near and far-infrared luminosities of each dwarf galaxy Name RA Dec D MB log h SFRHα M⊙yr-1 i log h MHi M⊙ i 3rot L3.4μm L60μm (Mpc) (mag) (km s-1) (mag) (Jy) ESO154-G023 02h56m50.38s -54d34m17.10s 5.76 -16.45 -1.32 9.14 53.0 15.21 0.32 ESO245-G005 01h45m03.73s -43d35m52.93s 4.43 -15.68 -1.59 8.6 51.0 15.99 0.22 ESO383-G087 13h49m17.50s -36d03m48.40s 3.45 -16.83 -1.49 7.82 18.0 14.05 1.21 ESO572-G025† 11h57m28.03s -19d37m26.60s 27.88 -17.50 -0.27 8.36 - 12.88 0.51 Fairall301† 04h06m07.92s -52d40m06.30s 9.30 -15.92 -1.11 - 52.3 12.38 0.90 IC1959 03h33m12.59s -50d24m51.30s 6.19 -16.07 -1.38 8.38 57.0 13.90 0.51 IC4662 17h47m08.86s -64d38m30.33s 2.55 -15.61 -1.12 8.24 41.0 11.99 8.82 IC5152 22h02m41.51s -51d17m47.20s 1.96 -15.56 -2.23 8.01 43.0 13.77 2.46 ISZ399† 12h19m59.51s -17d23m31.00s 15.94 -17.15 -0.81 - - 10.71 2.86 NGC244† 00h45m46.43s -15d35m48.80s 11.60 -16.25 -1.11 8.73 - 12.05 0.53 NGC625 01h35m04.63s -41d26m10.30s 4.02 -16.50 -1.20 8.00 34.0 12.12 5.73 NGC1311 03h20m06.96s -52d11m07.90s 5.55 -15.41 -1.55 8.04 36.0 13.46 0.39 NGC5253 13h39m55.96s -31d38m24.38s 3.44 -17.05 -0.64 7.91 31.0 9.25 30.51 NGC5264 13h41m36.68s -29d54m47.10s 4.79 -16.02 -1.92 7.70 15.0 13.72 0.30 NGC5408 14h03m20.91s -41d22m39.70s 5.32 -16.73 -0.83 8.48 30.0 13.06 2.82 Notes. † Blue compact dwarf galaxies (BCDs); the rest are irregular dwarfs from the LVHIS sample. Table 2. ATCA RC observations of CHILLING galaxies, with its configuration and the total on-source integration time Name ATCA configurations Total on-source time / hrs 2100 MHz 5500 MHz 9000 MHz ESO154-G023 1.5A 750B H168 7.17 6.85 6.85 ESO245-G005 1.5B 750C H214 10.53 7.59 7.59 ESO383-G087 1.5B 750C H214 8.56 8.37 8.37 ESO 572-G025 1.5B H214 4.92 4.60 4.60 Fairall 301 1.5C 750C H168 6.19 5.26 5.26 IC1959 1.5C 750B H214 5.25 5.27 5.27 IC 4662† 1.5C 750C H214 37.66 11.93 11.93 IC 5152 1.5A 750B H214 5.18 4.60 4.60 ISZ 399 1.5B 750C EW352 4.42 4.27 4.27 NGC 244 1.5A 750C H214 7.18 6.57 6.57 NGC 625 1.5A 750C H214 9.20 6.21 6.21 NGC 1311 1.5A 750C H214 5.59 3.39 3.39 NGC 5253 1.5B 750B H214 7.23 6.90 6.90 NGC5264 1.5B 750B H214 4.22 5.36 5.36 NGC 5408 1.5C 750B H168 H214 4.88 10.19 10.19 Notes. † Observation of IC 4662 is a combination of the CHILLING observations and the observations running under project ID: C3531 with detailed explanation in the Sect. 2.2. splitting into 32 -channels-out to avoid bandwidth smearing larger than the synthesized beam, during imaging of the field of view. Multifrequency and multiscale CLEANing (Högbom 1974) utilising interactive masks using flint_masking (Galvin et al. in prep.)2 around visible sources to minimise artefacts and flux scattering and applying detection thresholds was used to retain genuine emission only. During the imaging step, we use -join-channels to leverage the ATCA's full bandwidth. The continuum images for each galaxy were generated using different Briggs robust weighting to increase the quality, resulting in resolutions and noise levels shown in Table 3. In the last round 2 https://github.com/flint-crew/flint of imaging, we employed joint imaging3 with -channels-out 32 images. As these images are only used in science for the SED fitting in Sect. 4, we do -no-mf-weighting in the last round of imaging to ensure not to include artificial spectral substructures. We use the integrated task linmos in miriad to correct for the primary beam to each 32 images of the dataset. 3 Joint imaging means gridding and deconvolving the L/S-, C-, and X-bands jointly, resulting in a highly sensitive image with central frequency of 5.55 GHz, with a total bandwidth of 8.95 GHz Article number, page 4 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies Table 3. Overview of the observable parameters of the CHILLING galaxies, showing the name, the flux and phase calibrator, the Briggs robust weighting, the noise level of the RC image, and the category of detection. name flux calibrator phase calibrator robust σ2.1/5.5/9 GHz Detection Category (μJy/beam) ESO 154-G0154 1934-638 0252-549 0.0 25/10/10 No/Marginal ESO 245-G005 1934-638 0153-410 0.3 15/6/6 Compact ESO 383-G087 1934-638 1424-418 0.0 39/15/20 No/Marginal ESO 572-G025 1934-638/0823-500† 1127-145 0.0 30/20/30 Diffuse Fairall 301 1934-638 0302-623 0.3 14/12/25 Diffuse IC 1959 1934-638/0823-500† 0252-549 0.0 20/10/10 Compact IC 4662 1934-638 1718-649 0.3 9/8/12 Diffuse IC 5152 1934-638 2326-477 0.0 15/8/8 Compact ISZ 399 0823-500 1213-172 0.3 73/20/25 Diffuse NGC 244 1934-638 0023-263 0.3 10/8/8 Diffuse NGC 625 1934-638 0201-440 0.3 15/8/8 Diffuse NGC 1311 1934-638 0302-623 0.0 27/15/10 No/Marginal NGC 5253 1934-638 1421-490 0.0 48/35/50 Diffuse NGC 5264 1934-638 1255-316 0.0 80/20/13 No/Marginal NGC 5408 1934-638/0823-500† 1421-490 0.0 85/17/18 Compact Notes. † These datasets make use of the secondary flux calibrator at 5.5 and 9 GHz. 3. Detectability dependency 3.1. Definition of RC detectability Since the definition of what qualifies as an RC detection can vary, we divided our sample into three categories. In the first category, a galaxy is classified as a 'diffuse' detection if spatially extended RC emission is present across the majority of the galaxy. Specifically, we require that emission is detected at > 3σ significance over at least ∼70 -80 % of the galaxy's optical extent (as defined by the R25 radius) and that the emission is more extended than the synthesized beam. These detections leave little doubt that the galaxy as a whole emits RC emission (see Fig.1). A galaxy is classified as a 'compact' detection if it shows at least one RC-bright Hii region (> 3σ peak emission) but lacks diffuse emission across most of the galaxy. In this case, RC emission is localized to one or a few star-forming regions and covers 0.05), suggesting similar distributions between these populations. It is important to note that all of these correlations should be interpreted with caution. The analysis is based on a small sample of 15 dwarf galaxies, comprising 11 with RC detections and 4 without. This sample size is insufficient to draw statistically robust conclusions on its own about the entire dwarf galaxy population, although it is in agreement with previous studies (e.g. Roychowdhury et al. 2012; Schleicher & Beck 2016; Hindson et al. 2018, Taziaux in prep.) 4. Spectral model fitting 4.1. Overview of the RC data For the SEDs, we include additional data from multiple surveys in addition to our ATCA observations. For NGC 625 and Fairall 301, we incorporate GaLactic and Extragalactic Allsky MWA survey eXtended (GLEAM-X) data (Ross et al. 2024), which cover frequencies from 87 MHz to 221 MHz. For all sources, we also include the Australian SKA Pathfinder (ASKAP) survey, called Rapid ASKAP Continuum Survey (RACS-Low) data (Hale et al. 2021) at 885 MHz. For the ATCA dataset, we applied a 3 σ clipping threshold and computed the integrated flux density, summing all emission over its resolved size with an optical aperture using RadioFluxTools4, across 32 individual frequency slices. To ensure the reliability of the measurements, only slices with a noise level below 90 μJy/beam were considered. As the ATCA observations were conducted in snapshot mode, we adopt a standard calibration uncertainty of 10 %. The total uncertainty is then estimated as the combination of the statistical uncertainty, determined from the background noise and the specified beam area, and the assumed calibration uncertainty. For the SED analysis, we only use the 7 dwarf galaxies which show extended and diffuse emission. 4.2. Spectral models In the study of Klein et al. (2018), four different basic models were fitted to a small sample of dwarf galaxies as well as more massive ones. This has been further developed by Galvin et al. 4 https://gitlab.com/Sunmish/radiofluxtools Article number, page 6 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies Fig. 5. Correlation analysis of several galaxy parameters with the dependence of RC emission detection. It represents different physical properties, including absolute magnitude in the B-band MB, star formation rate (SFR), Hi mass MHI, maximum rotation velocity of the gas corrected for inclination (3rot), mid-infrared magnitude at 3.4 μm (L3.4μm) from WISE and far-infrared (FIR) luminosity at 60 μm (L60μm) from IRAS. Galaxies with diffuse RC emission are shown in green, those with compact emission in blue, and galaxies with no or very weak detection in orange. The Kolmogorov-Smirnov (KS) test results are visualised as circles on the diagonal of the plot, with sizes ranging from 0 to 1; larger circles indicate greater differences between the distributions. The exact value from the KS-test are shown inside the circles. The color represents statistical significance based on the p-value: pink circles indicate significant differences (p < 0.05), while gray circles indicate non-significant results. (2018) and Grundy et al. (2025), who focused solely on starbursts and star-forming galaxies, respectively, and developed the spectral fitting models. Our aim is to improve the spectral fitting for dwarf galaxies and identify which components dominate at different parts of the spectrum. We modified the models presented by Grundy et al. (2025) by splitting the absorption part into an internal component and an external component. All 19 models are detailed and presented in Appendix B, but these models are essentially combinations of five basic assumptions. The first assumption involves a standard power-law with only synchrotron emission, given by S (ν) = A ν ν0 !α (1) where A represents the non-thermal synchrotron emission and α is the non-thermal spectral index at a given frequency ν0. The second model accounts for a combination of synchrotron emission and thermal free-free emission, which can be written as a superposition S (ν) = A ν ν0 !α + B ν ν0 !-0.1 (2) Article number, page 7 of 20 A&A proofs: manuscript no. main Fig. 6. Spectral energy distribution of the RC detected CHILLING sample. The plot displays total intensity data with different observations: GLEAM-X data (Ross et al. 2024) only for Fairall 301 and NGC 625, RACS-Low (Hale et al. 2021) and our ATCA S-band, C-band and X-band. The frequency range of the telescope data is indicated by a grey dotted line. Additionally, we show the best-fit in red for each galaxy out of these 19 different models, explained in the Appendix B. The highlighted red regions represent the 1 σ uncertainties sampled by EMCEE. For comparison to the best-fit, we show the simple power-law (PL) fit with its uncertainty range in a blue dotted line. where A represents the synchrotron emission, α is the nonthermal spectral index, and B is the free-free emission at a given frequency ν0. Next, we model two individual free-free absorption components at lower frequencies, one for internal absorption and one for external absorption, following the assumptions outlined in Tingay & de Kool (2003). For external free-free absorption, the model is: S (ν) = 1 -e-τ1 · B · ν ν1 !2 (3) while for internal free-free absorption, the model is: S (ν) = 1 -e-τ2 τ2 ! · B · ν ν2 !2 (4) where the free-free optical depth parameter is given by τi = (ν/νi)-2.1. Finally, the last model component is relevant when inverse Compton losses dominate at higher frequencies, causing CREs to cool rapidly, leading to a break in the spectrum. This model can be written as S (ν) = A ν ν0 α 1 + ν νb ∆α (5) where νb is the break frequency in the spectrum and ∆α represents the change in the non-thermal spectral index due to synchrotron and inverse Compton losses, assuming continuous electron injection from massive SF. The full explanation of all 19 models is detailed in Appendix B. 4.3. Model selection and spectral fitting To fit all the 19 models to the data used for each dwarf galaxy, we use an affine invariant Markov chain Monte Carlo ensemble sampler (Goodman & Weare 2010) implemented as the EMCEE Article number, page 8 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies PYTHON package (Foreman-Mackey et al. 2013) using the differential evolution optimisation method. We choose physically motivated priors to constrain our models, such as the parameters A, B, C and D remain always positive and the spectral index value α should be between -3 and 0.5. We choose for the frequencies ν1 and ν2, as well as for the spectral break frequency νb, a value which should be between 0.01 and 10 GHz. For each model, the fit quality is evaluated based on the chi-square and Bayesian statistic. The model with the lowest chi-square and Bayesian information criterion (BIC) is selected as the best representation of the data, and its fit parameters are reported. Models that fail to converge are skipped. 4.4. Spectral fitting results Fig. 6 show the spectral energy distribution of the diffuse RC detected CHILLING galaxies. We show here the GLEAM-X data (Ross et al. 2024), if available (here only for Fairall 301, NGC 625), RACS-Low (Hale et al. 2021) and the observed ATCA L/S-,C- and X-band data. The best-fit model have been shown in red, while for comparison the simple power-law is shown in blue. For ESO 572-G025 and Fairall 301 we observe a turnover to lower frequency due to internal or external free-free absorption and probably inverse Compton losses at higher frequencies, according to the fit-model called FFA2_SFG_SIC and FFA1_SFG_SIC, respectively. Inverse Compton losses at higher frequencies are also seen in NGC 244, fitted by the best fitmodel called FFA1_SFG_SIC with a slight turnover to lower frequencies. To confirm this turnover associated with free-free absorption, we need even lower frequencies, such as GLEAMX data. For Fairall 301, we have GLEAM-X data, so we see this turnover happening at approximately 150 MHz, while for ESO 572-G025, we see this turnover already happening at 1 GHz. This turnover to lower frequency is also observed by Gajovi ́c et al. (2024) in more massive galaxies at frequencies of approximately 200 MHz. That we observe a turnover at such high frequencies, could be due to the limited surface brightness sensitivity of RACS-Low, as the galaxy shows extended emission at higher frequencies seen in the 2.1 GHz ATCA images. Deeper ASKAP images taken as part of the Evolutionary Map of the Universe (EMU; Norris et al. 2021; Hopkins et al. 2025) survey will allow us to extend this investigation. A similar spectral shape, corresponding to the SFG_FFA1 model, is observed in ISZ 399, IC 4662, NGC 625, and NGC 5253. The model describes synchrotron emission from relativistic electrons mixed uniformly with a single volume of thermal free-free plasma. Building on this, and following Galvin et al. (2018), we propose that the electron population is mixed inhomogeneously with two distinct star-forming regions, which differ in their optical depths, with only one component becoming optically thick across the observed frequency range and external free-free absorption being observed. Although we only have GLEAM-X data for NGC 625 regarding this fitting model, we observe a spectral behaviour that resembles findings by Grundy et al. (2025) in some of their galaxies, e.g. NGC 491. In both NGC 625 and NGC 491, there is a slight turnover around 400 MHz followed by an increase in flux density at lower frequency. Aside from this similarity in the spectrum, the two galaxies do not share any other notable characteristics. 5. RC - far infrared correlation There is a tight astrophysical relationship between the RC and FIR luminosities in all star formation galaxies, the so-called RCFIR correlation (Condon 1992). The empirical luminosity scaling between these two frequency regimes is likely a relation between radio synchrotron emission and the FIR emission of cool dust heated by massive stars. Yun et al. (2001) derive the following RC-FIR luminosity correlation, log(L1.4GHz) = (0.99 ± 0.01) log(L60μm/L⊙) + (12.07 ± 0.08) (6) where log(L1.4GHz[WHz-1]) = 20.08+2 log(D)+log(S1.4GHz[Jy]) and log(L60μm[L⊙]) = 6.014 + 2 log(D) + log(S 60μm[Jy]); D is the luminosity distance in Mpc. The radio flux density S 1.4GHz is the extrapolated value of the integrated flux density at 1.4 GHz, which we have fitted in Sect. 6. Fig. 7 presents the FIR-RC relation for dwarf galaxies in the CHILLING sample that exhibit either diffuse or compact RC emission. For galaxies with diffuse RC emission, we use the SED analysis to extrapolate the 1.4 GHz flux density and derive the corresponding non-thermal spectral index, indicated by the bestfit (see Table C.1 in Appendix C). For galaxies with compact RC emission, we calculate the 1.4 GHz flux density as an upper limit and adopt a fixed non-thermal spectral index of -0.7. These compact detections are therefore shown as upper limits in Fig. 7. In Fig. 7, we find that nearly all dwarf galaxies show lower RC emission than expected based on their FIR luminosity, when compared to the reference relation from Yun et al. (2001). Two main groupings are evident, one around a FIR luminosity of ∼108 L⊙and another near ∼109 L⊙. Dwarf galaxies that show extended diffuse emission, such as IC 4662, NGC 625, or NGC 5253, do not deviate from the general trend of lying below the reference line. On the contrary, these galaxies consistently exhibit a RC deficit. The dwarf galaxy ESO 572-G025 lies above the expected relation, likely due to its relatively high SFR and larger physical extent. This suggests that factors such as SFR and galaxy size may help retain CREs within the galaxy, allowing them to emit synchrotron radiation before escaping via galactic winds or outflows. We also observe a general trend where higher SFRs correspond to higher FIR luminosities. However, this increase does not necessarily bring galaxies closer to the established FIR-RC relation. Most dwarf galaxies appear under-luminous in the RC, which could either reflect the presence of very extended, diffuse emission that remains undetected by ATCA due to observational limitations, or more likely a true deficit in radio emission caused by CRE losses, as discussed in Sect. 6.5. The latter scenario is favored given that our sample was observed for multiple hours and only a few galaxies show extended emission into the halo. For example, estimating the expected RC flux for NGC 625 and IC 4662 to lie on the Yun et al. (2001) correlation, we find that their missing flux would correspond to factors of 4.7 and 3.3, respectively. Such large discrepancies suggest that the observed deficit cannot be explained solely by sensitivity limits or missing zero spacing flux. 6. Discussion 6.1. What triggers RC emission in dwarf galaxies? As in their more massive counterparts RC emission in dwarf galaxies is closely linked to star formation activity (Hindson et al. 2018). This connection also holds for the CHILLING sample, where galaxies with higher SFRs are more likely to exhibit Article number, page 9 of 20 A&A proofs: manuscript no. main Fig. 7. RC-FIR correlation from Yun et al. (2001), showing the CHILLING galaxies overlaid. The diffuse RC detected galaxies are shown as stars, while the compact RC detected galaxies are marked as upper limits (UL). detectable RC emission. As shown in Fig. 5, other galaxy properties also influence the presence of RC emission. Absolute magnitude, linked to SFR of the galaxies, and near-infrared brightness (L3.4 μm), used as proxy for stellar mass, show a strong correlation, reflecting the fact that more massive galaxies generally host more active star formation and more frequent core-collapse supernovae. In contrast, the total neutral atomic hydrogen (Hi) mass does not show a significant correlation with RC detectability. Although many dwarf galaxies contain substantial Hi reservoirs, this alone is not sufficient to generate RC emission. Star formation and the associated feedback from young stars is the key factor, as it drives the processes responsible for both thermal and non-thermal RC components. Dwarf galaxies show in general lower RC as they often have large gas reservoirs but lack significant star formation as they are much smaller in size and also possibly due to low gas densities or internal feedback processes that inhibit the collapse of gas into stars. Similarly, rotational velocity, which is commonly used as an indicator of a galaxy's dynamical mass, does not directly influence RC emission. While rotational velocity traces the overall gravitational potential, it does not necessarily reflect the physical conditions needed for star formation. This is especially true in low-mass systems like dwarf galaxies, where internal feedback (e.g., stellar winds, supernova outflows) can disturb gas dynamics, suppress ordered rotation, and limit star-forming activity. Additionally, non-circular motions in such systems make rotational velocity an unreliable proxy for the energetic processes that drive synchrotron emission. In summary, neither the total Hi gas mass nor the rotational velocity alone can account for the presence of RC emission. Instead, RC emission primarily traces recent star formation activity, which depends on localised physical conditions such as gas density, turbulence, and thermal stability. Therefore, while the availability of gas and the galaxy's dynamics provide important context, they are not sufficient on their own to trigger the RC emission observed in star-forming galaxies. 6.2. Physical implications of the different RC classification types The 'diffuse', 'compact' and 'no/marginal' RC classification reveals systematic differences that suggest distinct physical regimes. Dwarf galaxies with diffuse RC emission tend to have higher integrated SFRs and substantially larger 60 micron luminosities than the 'compact' or 'no/marginal' systems (mean log(SFRHα) ≈-0.9 for 'diffuse' vs ≈-1.5 for 'compact' and 'no/marginal' in our sample), indicating more spatially extended CR injection. This elevated star-formation activity plausibly provides both the CR electron source term and the turbulent energy required to amplify magnetic fields on galactic scales, producing volume-filling synchrotron emission (e.g. Chy ̇zy et al. 2011; Beck 2015). Conversely, 'compact' RC detections are associated with bright, localised Hii complexes but lack disk-wide emission, consistent with CR electrons being largely confined to their birth sites. This may reflect short propagation lengths (low diffusion coefficients (Heesen 2021), rapid vertical escape via localised outflows (Wang et al. 2022), or weak/inhomogeneous large-scale magnetic fields that limit CR diffusion Heesen (2021)). The compact subsample also shows slightly higher mean rotational velocities, suggesting that ordered shear alone is not sufficient for sustaining large-scale synchrotron halos, instead, the interplay of CR injection, turbulent field amplification, and transport efficiency appears to control whether RC emission becomes diffuse. Systems with 'no/marginal' RC emission typically have the lowest SFR and FIR luminosity, consistent with insufficient CR production to yield a detectable synchrotron signal given our surfacebrightness sensitivity (Yun et al. 2001). Observational factors such as lack of sensitivity of the ATCA telescopes may further bias against detecting low-surface-brightness emission. 6.3. Do we need different models to explain CRE spectra? An important question regarding the RC emission is whether complex models are truly necessary or if a simple power-law (PL) is sufficient to explain the spectra. Many dwarf galaxy spectra clearly show curved shapes, with steepening or flattening over several GHz and some show a turnover at low frequencies. For a statistical relevant analysis, we compare the reduced chi-square values, the Akaike information criterion (AIC; Akaike 1974) and the Bayesian information criterion (BIC; Schwarz 1978) of the best-fit models with those of the simple power-law. The AIC and BIC are model selection metrics that evaluate the relative quality of statistical models by balancing goodness of fit with model complexity, where AIC applies a lighter penalty and is more suitable for predictive accuracy, while BIC imposes a stronger penalty, particularly favoring simpler models as sample size increases. Lower values of the AIC or BIC indicate a better balance between model fit and complexity. Therefore, when comparing models, the preferred model is the one with the lowest AIC or BIC value, regardless of whether the values are negative or positive. In Fig. 8, we show the comparison of these values of the best-fit models with those of the simple power-law and it reveals that more complex models are indeed required to capture the spectral behaviour and underlying physics. In few cases, such as IC 4662, NGC 625, ISZ 399 and NGC 244, the reduced chi-squared of the PL is closer to the best value but we see that for each galaxy the AIC and BIC of the best-fit is smaller then the simple power-law, suggesting that we are in need of a more complex model to explain the CRE spectra. Taking now the Jeffreys' scale (Kass & Raftery 1995), where the logarithm of the Bayes factor quantifies the strength of evidence in favor of one model over another, we observe in the right panel of Fig. 8 that only for NGC 244, IC 4662, NGC 625 and NGC 5253 the difference is significant and we need to take a more complex model into account. Looking at the difference of the best-fit model Article number, page 10 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies Fig. 8. Comparison between the best-fit model and the simple power-law (PL) model using statistical criteria: reduced chi-squared (χ2 red), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Jeffreys' scale (Kass & Raftery 1995). The black dotted line in the left panel marks the reference value of χ2 red = 1, while the black dotted line in the right panel indicates the threshold where the the best-fit model are significantly preferred over the simple power-law. Fig. 9. Comparison of the extrapolated flux density S 1.4 GHz at frequency of 1.4 GHz, as derived from the best-fit model and the PL model. The uncertainties for both methods are estimated to be 10 %. and the simple PL model in Fig. 6, we observe already by eye that we definitely need a complex model for ESO 572-G025 and Fairall 301, which is not visible through the statistical test. The full model parameters for the best-fit of each galaxy are listed in Table C.1 in Appendix C. Comparing the flux density obtained from the best-fit model and the PL, we observe in Fig. 9, that the difference is minimal. Only for a few galaxies, such as ESO 572G025 or Fairall 301, the difference is larger than 10 %. For these two galaxies, especially for ESO 572-G025, the statistical tests differ most in between the simple power-law and the complex model. 6.4. Does SFR trigger free-free absorption in dwarf galaxies? Normally, a high SFR suggest an optically thick regime in Hα, which leads to free-free absorption, but we do not observe a correlation between the different model components and other characteristics of the galaxies, such as mass, FIR or SFR. Unfortunately, our sample is too small to draw conclusion if the corresponding model components depend on some of these characteristics. 6.5. Do dwarf galaxies obey the RC-FIR correlation? For galaxies with a FIR luminosity of log(L60μm/L⊙) < 109, Yun et al. (2001) already observed an increase of asymmetric scatter, such that sources appear to be under-luminous in radio luminosity log(L1.4GHz), compared to the best fit. Such a deviation can occur if the FIR or RC is not directly proportional to the star formation activity. Dust heated by low-mass stars may contribute only to the FIR. Low luminosity, low mass galaxies may lose more radio emission than high luminosity objects to cosmic ray diffusion. Roychowdhury et al. (2012) studied the RC-FIR correlation in faint irregular galaxies by stacking images of individual galaxies together and finds that both the Spitzer 70 μm and 1.4 GHz fluxes are generally lower than expected from what is expected in spiral galaxies, however the ratio of RC to FIR appears to be consistent with the larger spiral galaxies. Hindson et al. (2018) analysed the RC-FIR correlation with a small dwarf galaxies sample and found that dwarf galaxies contain less dust than spiral galaxies, making them fainter in the FIR for a given level of radio emission, in contrast to our study. The low mass sample, analysed by Shao et al. (2018), exhibit lower radio emissions compared to FIR emissions. This is indicated by the analysis of the RC deficiency in dwarf galaxies, which is primarily attributed to factors like a lower stellar mass surface density and a higher Hi-to-stellar mass ratio. These findings align with our results, that dwarf galaxies exhibit lower radio emission than FIR. The lower radio emission in dwarf galaxies, compared to their FIR emissions, is attributed to several factors. Shao et al. (2018) explains it that primarily, dwarf galaxies have lower gas surface densities, which affects the production of cosmic rays that contribute to radio emission. Furthermore, the lower stellar mass surface density in these galaxies results in less efficient conversion of star formation activity into radio emissions. Additionally, in regions of low gas surface density, the escaping CRs may lead to diminished radio output, while the low dust content leads to lower FIR emission, creating a compensatory effect. We attribute the observed radio deficit in the RC-FIR correlation to significant energy losses experienced by cosmic ray electrons. Losses, such as synchrotron radiation and inverse Compton scattering which we can observe in the breaks in the SEDs in Fig. 6, affect the total energy of CRs available to generate radio emissions. In environments with lower gas surface densities, such as those typical for dwarf galaxies, these losses can be more pronounced, leading to a relative deficit in the radio output compared to FIR emissions. Article number, page 11 of 20 A&A proofs: manuscript no. main 7. Conclusions We present a RC study of the CHILLING sample, consisting of 11 dwarf irregular galaxies and 4 BCDs spanning a broad range of stellar masses and star formation histories. The combination of multi-band observations with ATCA provides the opportunity to study the full RC spectra and leads us to the following conclusions. 1. We detect RC emission in 11 of the 15 dwarf galaxies, with 7 exhibiting more diffuse emission and 4 displaying more compact emission. The detectability of RC emission in dwarf galaxies is closely linked to recent star formation activity, as all 4 BCDs are detected. Galaxies with higher star formation rates are more likely to exhibit RC emission, which also correlates with absolute magnitude and stellar mass. Most RC-detected galaxies are classified as starbursts. In contrast, parameters such as Hi mass and rotational velocity show no significant correlation with RC detectability, indicating that RC emission is more strongly influenced by star formation than by galaxy dynamics. 2. The RC spectra of many dwarf galaxies in the sample display complex, curved features, such as spectral steepening, flattening, and low-frequency turnovers, that deviate markedly from a simple power-law behavior. Statistical evaluations using reduced chi-squared, AIC, and BIC confirm that single power-law models inadequately describe the observed spectral shapes and fail to capture the underlying physical processes. These findings underscore the need for more sophisticated models incorporating mechanisms such as thermal emission, free-free absorption, and spectral breaks arising from cosmic ray electron energy losses. 3. In particular, the pronounced steepening of RC spectra at higher frequencies is attributed to substantial energy losses experienced by cosmic ray electrons. These losses, primarily due to synchrotron radiation and inverse Compton scattering, become increasingly significant at higher electron energies, leading to the observed decline in flux density. 4. Nearly all dwarf galaxies in the sample show a deficit in RC emission relative to their FIR luminosity when compared to the correlation seen in more massive galaxies. This deficit is best explained by significant energy losses of cosmic ray electrons, primarily via synchrotron radiation and inverse Compton scattering, which are more effective in the lowdensity environments typical of dwarf galaxies. 5. The possibility that extended low-radio-brightness diffuse emission remains undetected due to the sensitivity limitations of ATCA may partially reflect these instrumental constraints. However, the spectral properties observed across the sample more strongly indicate that energy losses of CREs are the primary drivers of both the spectral steepening and the RC deficit. Acknowledgements. ST, BA, DJB and MS acknowledge the support from the DFG via the Collaborative Research Center SFB1491 Cosmic Interacting Matters - From Source to Signal. PK acknowledge the support of the BMBF project 05A23PC1 for D-MeerKAT. The Australia Telescope Compact Array is part of the Australia Telescope National Facility (https://ror.org/05qajvd42) which is funded by the Australian Government for operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi people as the Traditional Owners of the Observatory site. 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J. 2010, MNRAS, 407, 113 Veilleux, S., Cecil, G., & Bland-Hawthorn, J. 2005, ARA&A, 43, 769 Wang, W., Bu, D.-F., & Yuan, F. 2022, MNRAS, 513, 5818 Werhahn, M., Girichidis, P., Pfrommer, C., & Whittingham, J. 2023, MNRAS, 525, 4437 Werhahn, M., Pfrommer, C., & Girichidis, P. 2021, MNRAS, 508, 4072 Westcott, J., Brinks, E., Hindson, L., Beswick, R., & Heesen, V. 2018, MNRAS, 475, 5116 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868 Yun, M. S., Reddy, N. A., & Condon, J. J. 2001, ApJ, 554, 803 Zweibel, E. G. 2013, Physics of Plasmas, 20, 055501 Article number, page 13 of 20 A&A proofs: manuscript no. main Appendix A: Radio continuum properties In this section, we present the RC maps of the ATCA data of L/S-band, C-band and X-band. Fig. A.1 presents the RC maps of dwarf galaxies with diffuse radio emission. Fig. A.2 shows the RC maps of galaxies with compact RC emission. Fig. A.3 displays the RC maps of galaxies with no clear or only weak RC emission, which are classified as no/marginal detections. Fig. A.1. First row left panel: Total intensity emission of ESO 572 G 025 at the central frequency of 2.1 (σ = 30 μJy/beam), 5.5 (σ = 20 μJy/beam) and 9 GHz (σ = 30 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. First row right panel: Total intensity emission of Fairall 201 at the central frequency of 2.1 (σ = 14 μJy/beam), 5.5 (σ = 12 μJy/beam) and 9 GHz (σ = 25 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row left panel: Total intensity emission of IC 4662 at the central frequency of 2.1 (σ = 9 μJy/beam), 5.5 (σ = 8 μJy/beam) and 9 GHz (σ = 12 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row right panel: Total intensity emission of ISZ 399 at the central frequency of 2.1 (σ = 73 μJy/beam), 5.5 (σ = 20 μJy/beam) and 9 GHz (σ = 25 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Third row left panel: Total intensity emission of NGC 244 at the central frequency of 2.1 (σ = 10 μJy/beam), 5.5 (σ = 8 μJy/beam) and 9 GHz (σ = 8 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Third row right panel: Total intensity emission of NGC 625 at the central frequency of 2.1 (σ = 12 μJy/beam), 5.5 (σ = 8 μJy/beam) and 9 GHz (σ = 8 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Fourth row panel: Total intensity emission of NGC 5252 at the central frequency of 2.1 (σ = 48 μJy/beam), 5.5 (σ = 35 μJy/beam) and 9 GHz (σ = 50 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. The beam is shown in the left corner. Article number, page 14 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies Fig. A.2. First row left panel: Total intensity emission of ESO 245 G 005 at the central frequency of 2.1 (σ = 15 μJy/beam), 5.5 (σ = 6 μJy/beam) and 9 GHz (σ = 6 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. First row right panel: Total intensity emission of IC 1959 at the central frequency of 2.1 (σ = 20 μJy/beam), 5.5 (σ = 10 μJy/beam) and 9 GHz (σ = 10 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row left panel: Total intensity emission of IC 5152 at the central frequency of 2.1 (σ = 15 μJy/beam), 5.5 (σ = 8 μJy/beam) and 9 GHz (σ = 8 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row right panel: Total intensity emission of NGC 5408 at the central frequency of 2.1 (σ = 85 μJy/beam), 5.5 (σ = 17 μJy/beam) and 9 GHz (σ = 18 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. The beam is shown in the left corner. Fig. A.3. First row left panel: Total intensity emission of ESO 154 G 023 at the central frequency of 2.1 (σ = 40 μJy/beam), 5.5 (σ = 10 μJy/beam) and 9 GHz (σ = 12 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. First row right panel: Total intensity emission of ESO 383 G 087 at the central frequency of 2.1 (σ = 40 μJy/beam), 5.5 (σ = 15 μJy/beam) and 9 GHz (σ = 20 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row left panel: Total intensity emission of NGC 1311 at the central frequency of 2.1 (σ = 27 μJy/beam), 5.5 (σ = 15 μJy/beam) and 9 GHz (σ = 10 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. Second row right panel: Total intensity emission of NGC 5264 at the central frequency of 2.1 (σ = 80 μJy/beam), 5.5 (σ = 20 μJy/beam) and 9 GHz (σ = 13 μJy/beam) with overlaid contours starting at 3 σ and increasing by factor of two. The beam is shown in the left corner. Article number, page 15 of 20 A&A proofs: manuscript no. main Appendix B: Spectral fitting models In this section, we present the 19 different models, which have been fitted to the dataset for each dwarf galaxy. These models build upon the work of Galvin et al. (2018) and Grundy et al. (2025), incorporating further refinements to the absorption component by accounting for both internal and external free-free absorption. Appendix B.1: Base Model The first two models are the base for all the other models and have already been described in Sect.4. The first is a basic power-law model for synchrotron emission (Eq.1), while the second is a constant model representing the combined contribution of synchrotron and free-free emission (Eq. 2). Power-low (PL) S ν = A ν ν0 !α (B.1) Superposition of synchrotron and free-free emission (SFG) S ν = A ν ν0 !α + B ν ν0 !-0.1 (B.2) Appendix B.2: Prefix: Free-free Absorption (FFA_) Synchrotron emission can be attenuated by free-free absorption (FFA) when coexisting with ionised gas that produces free-free emission, leading to spectral curvature at low frequencies. The optical depth for absorption is: τi = ν νi !-2.1 (B.3) We distinguish between internal and external free-free absorption (Tingay & de Kool 2003). For internal free-free absorption, the free-free emission and synchrotron emission are both within the same region, while external free-free absorption occurs when synchrotron emission passes through a foreground layer of ionised gas that emits free-free radiation. In this case, the absorbing material is physically separate from the synchrotron-emitting region. The absorption leads to a low-frequency turnover in the spectrum, with the degree of attenuation depending on the density and path length of the ionized gas. Free-Free external Absorption + Synchrotron (FFA1_SFG) S ν = (1 -e-τ1) B + A ν ν1 !0.1+α ν ν1 !2 (B.4) Parameters: A, B, α, ν1 Free-Free internal Absorption + Synchrotron (FFA2_SFG) S ν = (1 -e-τ1) τ2 B + A ν ν1 !0.1+α ν ν2 !2 (B.5) Parameters: A, B, α, ν2 By assuming that the observed emission arises purely from synchrotron radiation affected by free-free absorption, without any contribution from free-free emission, the parameter B can be set to 0. This yields a simplified version of the full model, while still distinguishing between external and internal free-free absorption scenarios. Free-Free external Absorption (FFA1_PL) S ν = (1 -e-τ1)A ν ν1 !0.1+α ν ν1 !2 (B.6) Parameters: A, α, ν1 Article number, page 16 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies Free-Free internal Absorption (FFA2_PL) S ν = (1 -e-τ2) τ2 A ν ν2 !0.1+α ν ν2 !2 (B.7) Parameters: A, α, ν1 Appendix B.3: Suffix: Free-free Absorption (_FFA) These models represent synchrotron emission modified by free-free absorption in ionised gas, but without any contribution from free-free emission itself (i.e., B = D = 0). The models assume a single population of relativistic electrons embedded in or obscured by thermal plasma. This first model (PL_FFA1) describe synchrotron emission passing through an external ionised region with foregound thermal plasma. In the second model (PL_FFA2), we assume internal free-free absorption, where the synchrotron-emitting and absorbing regions are co-spatial. Free-Free external Absorption + only Synchrotron (PL_FFA1) S ν = A ν ν0 !α + (1 -e-τ1) C ν ν1 !0.1+α ν ν1 !2 (B.8) Parameters: A,C, α, ν1 Free-Free internal Absorption + only Synchrotron (PL_FFA2) S ν = A ν ν0 !α + (1 -e-τ2) τ2 C ν ν2 !0.1+α ν ν2 !2 (B.9) Parameters: A,C, α, ν2 This model extends the simple synchrotron-free-free absorption framework by including a free-free emission component in addition to synchrotron radiation and external free-free absorption. The total emission consists of a synchrotron power law, a freefree component, and an externally (SFG_FFA1) or internally (SFG_FFA2) absorbed component consisting of synchrotron and thermal emission Power-Law + Free-Free external Absorption (SFG_FFA1) S ν = A ν ν0 !α + B ν ν0 !-0.1 + (1 -e-τ2) D + C ν ν2 !0.1+α ν ν2 !2 (B.10) Parameters: A, B,C, D, α, ν2 Power-Law + Free-Free internal Absorption (SFG_FFA2) S ν = A ν ν0 !α + B ν ν0 !-0.1 + (1 -e-τ2) τ2 D + C ν ν2 !0.1+α ν ν2 !2 (B.11) Parameters: A, B,C, D, α, ν2 This following models introduce two distinct synchrotron-emitting regions, each experiencing different types of free-free absorption. The first component is attenuated by external free-free absorption (FFA1), while the second experiences internal free-free absorption (FFA2). Both components share the same synchrotron spectral index, representing a single electron population. This setup captures systems with multiple star-forming regions or geometrically distinct zones, where only one region becomes optically thick within the observed frequency range. This dual-FFA model is inspired by studies like Galvin et al. (2018), which show that unresolved blends of emission from multiple regions with different physical conditions can reproduce complex radio SEDs with multiple spectral turnovers or inflection points. It is best used when the data suggest two absorption turnovers at distinct frequencies, possibly corresponding to low- and mid-frequency spectral curvature. Dual Free-Free Absorption + Power Law (FFA1_PL_FFA2) S ν = (1 -e-τ1)A ν ν1 !0.1+α ν ν1 !2 + (1 -e-τ2) τ2 C ν ν2 !0.1+α ν ν2 !2 (B.12) Parameters: A,C, α, ν1, ν2 Article number, page 17 of 20 A&A proofs: manuscript no. main The FFA1_SFG_FFA1 model is a further extension in which both components, each consisting of synchrotron and thermal emission, are absorbed by two distinct external free-free absorption regions. This model accounts for a scenario in which both components undergo absorption, possibly due to layers of ionised gas in complex starburst geometries. The more advanced model, FFA1_SFG_FFA2, allows each component to have an independent synchrotron spectral index, denoted by α1 and α2. This is physically motivated in systems such as post-mergers or interacting galaxies, where newly formed relativistic electrons may have a flatter spectrum while older populations have steeper slopes due to energy losses. This model provides the highest flexibility and is most appropriate when fitting SEDs that require distinct spectral indices to explain high- and low-frequency behavior. Dual Free-Free Absorption + Power Law (FFA1-SFG-FFA1) S ν = (1 -e-τ1) B + A ν ν1 !0.1+α ν ν1 !2 + (1 -e-τ2) D + C ν ν2 !0.1+α ν ν2 !2 (B.13) Parameters: A,C, α, ν1, ν2 Dual Free-Free Absorption + Power Law (FFA1-SFG-FFA2) S ν = (1 -e-τ1) B + A ν ν1 !0.1+α1 ν ν1 !2 + (1 -e-τ2) τ2 D + C ν ν2 !0.1+α2 ν ν2 !2 (B.14) Parameters: A, B,C, D, α1, α2, ν1, ν2 Appendix B.4: Suffix: Inverse-Compton Losses (_SIC) At high frequencies, synchrotron spectral steepening can occur not due to free--free absorption, but as a result of energy losses experienced by cosmic ray electrons. These losses are primarily caused by synchrotron radiation and inverse Compton scattering, which occurs when cosmic ray electrons scatter off ambient photons, including those from the cosmic microwave background or the far-infrared radiation field. Under a continuous injection model, the effect of these losses is a gradual steepening of the synchrotron spectral index by ∆α = -0.5, beginning around a characteristic break frequency νb. This model is useful for describing systems where high-frequency spectral curvature is likely due to energy losses rather than thermal absorption. Synchrotron with Inverse Compton Losses (PL_SIC) S ν = A ν ν0 α 1 + ν νb ∆α (B.15) Parameters: A, α, νb, ∆α This model extends PL_SIC by including a thermal free-free emission component. It is intended for galaxies where thermal emission contributes at higher frequencies, in addition to synchrotron radiation affected by inverse Compton or synchrotron losses. Synchrotron + Inverse Compton (SFG_SIC) S ν = A ν ν0 α 1 + ν νb ∆α + B ν ν0 !-0.1 (B.16) Parameters: A, B, α, νb, ∆α The following two models combines external (FFA1_PL_SIC) and internal (FFA2_PL_SIC) free-free absorption with synchrotron emission undergoing high-frequency steepening due to losses. Free-Free external Absorption + Synchrotron + Inverse Compton (FFA1_PL_SIC) S ν = (1 -e-τ1) A ν ν1 !0.1+α 1 1 + ν νb ∆α ν ν1 !2 (B.17) Parameters: A, B, α, ν1, νb, ∆α Article number, page 18 of 20 Sam Taziaux et al.: CHILLING - RC spectral behavior of dwarf galaxies Free-Free internal Absorption + Synchrotron + Inverse Compton (FFA2_PL_SIC) S ν = (1 -e-τ2) τ2 A ν ν2 !0.1+α 1 1 + ν νb ∆α ν ν2 !2 (B.18) Parameters: A, B, α, ν2, νb, ∆α This model extends the SFG_SIC model by including external FFA1_ and internal FFA2_ free--free absorption. Free-Free external Absorption + Synchrotron + Inverse Compton (FFA1_SFG_SIC) S ν = (1 -e-τ1) B + A ν ν1 !0.1+α 1 1 + ν νb ∆α ν ν1 !2 (B.19) Parameters: A, B, α, ν1, νb, ∆α Free-Free internal Absorption + Synchrotron + Inverse Compton (FFA2_SFG_SIC) S ν = (1 -e-τ2) τ2 B + A ν ν2 !0.1+α 1 1 + ν νb ∆α ν ν2 !2 (B.20) Parameters: A, B, α, ν2, νb, ∆α The inclusion of both free-free absorption and synchrotron or inverse Compton losses allows for modeling galaxies whose radio SEDs exhibit both low-frequency turnover and high-frequency steepening-features commonly observed in luminous infrared galaxies and starbursts. However, as simultaneous modeling of multiple synchrotron components, each with their own breaks and absorption features, is currently limited by spectral resolution and coverage. Still, models like FFA_SFG_SIC provide significant flexibility for interpreting breaks and curvature in observed radio SEDs. Appendix C: Spectral comparison In this section, we present the parameter values obtained from the spectral models. We provide a comparison between the parameters of the best-fit model and those of a simple power-law model. Article number, page 19 of 20 A&A proofs: manuscript no. main Table C.1. Model comparison between the best-fit model and a simple power-law (PL) model for the CHILLING dwarf galaxies. Reported are χ2, reduced χ2, Akaike information criterion (AIC), Bayesian information criterion (BIC), fit parameters (A-D, α, ν1, ν2, νb, ∆α), and the modeled flux at 1.4 GHz. Name Model χ2 χ2 red AIC BIC A B C D α ν1 ν2 νb ∆α Flux / mJy ESO 572-G025 FFA2_SFG_SIC 45.07 2.37 31.33 43.32 40.00 0.00 10.66 25.37 -1.14 4.27 1.63 1.71 2.79 12.00 PL 504.98 26.58 98.98 110.97 9.14 0.25 1.67 39.20 -0.67 1.32 9.16 9.22 0.01 9.14 Fairall 301 FFA1_SFG_SIC 82.50 3.30 48.14 61.88 15.97 1.58 38.16 27.28 -0.85 0.22 9.90 6.62 0.30 3.31 PL 162.06 6.48 71.09 84.83 2.57 32.84 36.02 25.33 -0.28 3.04 0.80 0.32 1.07 2.57 IC 4662 SFG_SIC 12.02 0.57 -9.45 3.16 3.25 0.00 12.07 16.68 -2.39 3.19 5.21 4.16 0.03 21.76 PL 15.28 0.73 -2.24 10.37 22.04 13.84 6.43 16.02 -0.21 9.59 2.17 0.45 4.46 22.04 ISZ 399 SFG_FFA1 30.29 2.02 23.58 34.19 4.71 0.00 5.06 1.76 -2.09 5.31 9.24 7.52 0.33 9.84 PL 39.97 2.66 30.24 40.84 10.98 1.00 39.36 1.53 -0.69 6.91 0.85 2.87 1.46 10.98 NGC 244 FFA1_SFG_SIC 11.74 0.49 -16.10 -2.63 1.15 0.32 20.45 15.35 -2.36 5.29 0.63 6.46 2.00 1.57 PL 32.41 1.35 17.41 30.88 1.86 3.54 20.55 37.42 -0.86 7.47 1.29 8.92 1.05 1.86 NGC 625 SFG_FFA1 17.20 0.61 -10.34 4.15 0.49 0.00 9.01 9.33 -1.33 0.45 1.33 5.93 0.47 10.34 PL 22.16 0.79 -0.96 13.54 10.07 12.47 2.76 29.77 -0.19 8.18 6.51 1.78 0.00 10.07 NGC 5253 SFG_FFA1 1.49 0.065 -80.06 -66.87 17.69 29.48 23.33 21.75 -2.14 5.33 5.86 6.10 1.97 73.15 PL 7.52 0.33 -28.35 -15.15 72.94 29.45 25.24 6.72 -0.22 8.27 1.75 9.40 0.38 72.94 Article number, page 20 of 20
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2510.14770
|
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
1
MoCom: Motion-based Inter-MAV Visual
Communication Using Event Vision and
Spiking Neural Networks
Zhang Nengbo1, Hann Woei Ho1,∗, Member, IEEE, Ye Zhou1, Member, IEEE,
Abstract—Reliable communication in Micro Air Vehicle (MAV)
swarms is challenging in environments, where conventional radio-
based methods suffer from spectrum congestion, jamming, and
high power consumption. Inspired by the waggle dance of
honeybees, which efficiently communicate the location of food
sources without sound or contact, we propose a novel visual
communication framework for MAV swarms using motion-based
signaling. In this framework, MAVs convey information, such as
heading and distance, through deliberate flight patterns, which
are passively captured by event cameras and interpreted using
a predefined visual codebook of four motion primitives: vertical
(up/down), horizontal (left/right), left-to-up-to-right, and left-to-
down-to-right, representing control symbols (“start”, “end”, “1”,
“0”). To decode these signals, we design an event frame-based
segmentation model and a lightweight Spiking Neural Network
(SNN) for action recognition. An integrated decoding algorithm
then combines segmentation and classification to robustly inter-
pret MAV motion sequences. Experimental results validate the
framework’s effectiveness, which demonstrates accurate decod-
ing and low power consumption, and highlights its potential
as an energy-efficient alternative for MAV communication in
constrained environments.
Index Terms—Visual Communication, Event Camera, Spike
Neural Networks, Motion Segmentation, Unmanned Aerial Vehi-
cles.
I. INTRODUCTION
I
N nature, many species have evolved non-verbal commu-
nication strategies, such as the waggle dance of honeybees,
pheromone trails in ants, and postural displays in wolves, to
support social coordination and collective intelligence. Among
them, the waggle dance of honeybees is a remarkable example
(see Fig. 1). Through specific body movements, bees encode
and transmit information about food location, direction, and
distance, as studied in a work [1]. This can enable the colony
to make distributed foraging decisions without centralized
control. This biologically evolved action-based communica-
tion system equips honeybee swarms with self-organization,
robustness to environmental noise, and energy efficiency, of-
fering valuable inspiration for engineered multi-agent systems
[2].
Such bio-inspired communication paradigms present com-
pelling solutions for modern Micro Air Vehicle (MAV)
∗Corresponding author.
1Zhang Nengbo, Hann Woei Ho, and Ye Zhou are with School of Aerospace
Engineering, Engineering Campus, Universiti Sains Malaysia, 14300 Nibong
Tebal, Pulau Pinang, Malaysia (email: zhangnb@student.usm.my; aehan-
nwoei@usm.my; zhouye@usm.my).
1 s = 1 km
Distance
m
Angle =
Waggle Dance by Bees
Sun
Food
Beehive
Traditional Communication Mode between MAVs
Radio signal disrupted
by intentional/ unintentional interference
Signal 1
Signal 0
MAV Action Sequence
Event camera
Event stream
Inspired by how insects communicate
MoCom using Event Vision
& Spiking Neural Networks
Quality
!
!
Fig. 1.
Overview of the proposed motion-based inter-MAV visual com-
munication (MoCom) framework for MAV swarms. Traditional radio-based
communication can be easily disrupted in constrained environments. Inspired
by the waggle dance of honeybees, a motion-based signaling approach, where
MAVs convey information through deliberate flight patterns. These motions
are captured by event cameras. The event stream is then processed using a
biologically inspired pipeline that segments the motion and classifies it using a
spiking neural network. This paradigm offers a decentralized and interference-
resilient alternative to conventional communication in MAV swarms.
swarms, which face critical challenges in maintaining reli-
able and covert inter-agent coordination. Particularly in high-
stakes applications, including disaster rescue and environmen-
tal monitoring, conventional wireless protocols, such as Wi-
Fi, radio, 4G/5G, suffer from spectrum scarcity [3], jamming
vulnerabilities [4], and poor scalability [5], which can lead
to communication failures in dense MAV swarms. These
limitations become debilitating in contested or GPS-denied
environments, where resilience and stealth are paramount. No-
tably, the honeybee’s motion-encoded communication model
aligns precisely with MAV swarms’ need for decentralized,
interference-resistant, and energy-efficient alternatives.
Inspired by the decentralized and robust communication of
honeybees, which encodes and delivers information through
physical movements, we propose a novel “action as signal”
paradigm for MAV swarms to address the limitations of
conventional wireless communication. In this paradigm, MAVs
encode information in deliberate motion patterns, much like
bees conveying foraging details through physical movements.
Such motion-based communication not only offers a decen-
tralized, interference-resistant, and energy-efficient alternative
to radio-based protocols but also naturally aligns with the
perception-action loop of autonomous systems. This synergy
enables integration in dynamic, unstructured environments. To
capture motion-based signals effectively, a sensing modality
must meet three requirements: high temporal resolution, ro-
arXiv:2510.14770v1 [cs.CV] 16 Oct 2025
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
2
bustness to lighting variations, and sensitivity to rapid move-
ments. Here, event cameras emerge as an ideal solution. Unlike
conventional frame-based RGB cameras, they asynchronously
detect pixel-level brightness changes, offering low latency,
high dynamic range, and sparse data output. These capabilities
are critical for reliably decoding the fast, edge-rich motion pat-
terns generated by MAVs, while maintaining energy efficiency,
a feature particularly vital for MAV swarm operations. Recent
advances have demonstrated the potential of event cameras
in MAV behavior recognition, such as object tracking [6] and
pose estimation [7]. However, most of these applications focus
on short-duration event segments and do not address the chal-
lenges of processing long, continuous event streams typical in
real-time MAV communication. Existing MAV communication
methods still largely rely on RF-based techniques, which
often fail in environments with electromagnetic interference or
obstructed lines of sight, leading to unreliable connections and
potential mission failure. Furthermore, the ability to extract
consistent, interpretable signals from noisy event streams
during dynamic flight remains an open challenge, underscoring
the need for a new communication framework tailored to the
unique characteristics of event-based vision.
Event data, despite its sparsity and asynchronous nature,
contains rich temporal structure. Frame-wise statistical in-
dicators, such as the total number of events and the ratio
of positive to negative events, naturally encode motion dy-
namics, providing valuable cues for segmenting continuous
event streams into discrete action units. These features offer
a lightweight, low-computation, and intuitive foundation for
temporal analysis, aligning seamlessly with the properties of
event-based sensing. As such, statistical segmentation of event
data is not only effective but also particularly suitable for
real-time MAV applications with constrained computational
resources.
To build a fully bio-inspired MAV visual communica-
tion model, we further incorporate a brain-inspired biometric
classifier based on Spiking Neural Networks (SNNs) [8].
SNNs process information through discrete spikes and inher-
ently support event-driven computation, making them highly
compatible with the output of event cameras. Compared to
traditional neural architectures, such as Convolutional Neural
Networks (CNNs) or Recurrent Neural Networks (RNNs),
SNNs offer improved energy efficiency and better handling
of sparse, asynchronous inputs. In our system, a lightweight
SNN model is employed to classify segmented MAV motion
clips into predefined communication symbols (e.g., binary
digits), enabling end-to-end decoding of visual signals in a
bio-inspired manner.
In this paper, we propose an event vision-based framework
for inter-MAV message transmission. It accurately recognizes
MAV actions and decodes action sequences into meaningful
messages, enabling visual communication between MAVs. The
defined visual communication codes are illustrated in Fig. 2.
To the best of our knowledge, this is the first work to introduce
the concept of motion-based visual communication into MAV
systems. The main contributions of this paper are threefold:
• design a MAV motion segmentation model to process long
continuous event streams by segmenting MAV motion-
encoded signals into discrete semantic units. This facili-
tates the subsequent decoding of MAV motion informa-
tion.
• develop a lightweight yet accurate MAV motion recogni-
tion model (EventMAVNet) inspired by biological neural
processing, utilizing a spiking neural network architec-
ture. Compared to other SNN recognition models, we
achieve both computational efficiency and high recogni-
tion precision.
• introduce the Integrated MAV Segmentation and Recogni-
tion (IMSR) algorithm, a MAV motion sequence decoding
approach that seamlessly integrates event signals segmen-
tation and recognition. The IMSR algorithm integrates
segmentation and recognition to robustly decode MAV
motion sequence signals, effectively mitigating noise and
interference in dynamic environments.
The rest of this paper is organized as follows: Section 2
reviews related works on MAV communication and recent
advancements in event recognition models based on spiking
neural networks. Section 3 details our proposed method,
including the event signal segmentation model, motion recog-
nition model, and MAV motion signal decoding algorithms.
Section 4 presents the experimental results and performance
evaluation. Section 5 concludes the paper and details future
directions for inter-MAV visual communication. Through this
work, we not only advance the application of event cameras in
MAV signal transmission but also provide new insights into in-
tegrating vision and communication in dynamic environments.
II. RELATED WORKS
This section reviews key research in MAV communication
techniques, event-based MAV action recognition, and motion
segmentation using event frames.
A. MAV communication techniques
MAV communication technology [9] is a key component
in MAV swarms [10], enabling swarm formation and co-
ordination for applications, such as search and rescue and
temporary communication networks [11] in disaster-affected
areas. Generally, MAV communication can be classified into
distributed [12], [13], [14] and centralized approaches [15],
[16], [17], [18]. Regarding distributed MAV communication,
an early study [12] introduced a distributed information ex-
change method based on neighborhood data sharing to address
data association problem in multi-robot systems. Subsequent
studies [13], [14] have demonstrated that decentralized coor-
dination enhances swarm control and adaptability. Similarly,
distributed communication principles have been successfully
applied to vision-based multi-MAV control [19]. Decentralized
communication phenomena are also observed in natural swarm
organizations, such as social insect colonies [1]. In contrast,
centralized MAV communication typically adopts a star topol-
ogy, where all MAVs communicate through a central master
node. This approach relies on MAV onboard electronics [15]
and uses methods like point-to-point [16], ground-to-air [17],
and GPS-based communication [18]. These methods ensure
robust swarm control and coordination.
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
3
Signal Code
MAV Action
Action：
Action：
Action：
Action：
F = 1
F = 3
F = 5
F = 7
F = 9
Fig. 2.
Several common visual communication codes are presented. Specifically, we define four types of communication codes (action : “start”, action :
“end”, action : “1”, action : “0”). A single event MAV motion is segmented into F-th event block based on an equal event count strategy, and the accumulated
events are used to construct F event frames. Red color represents positive events, green color represents negative events, and the yellow regions indicate the
overlap between positive and negative events.
Traditional approaches, while effective, face challenges
such as spectrum scarcity and vulnerability to interference,
prompting recent advancements in novel MAV communication
techniques. For example, one study [20] explored Visible Light
Communication (VLC) to simultaneously enable communi-
cation and illumination. Another work [21] utilized vision-
based analysis for direct MAV-operator interaction. However,
these communication technologies all use visual light as a
carrier for information transmission, and do not use the MAV’s
motion pattern to transmit information. Additionally, emerging
research [22], [23] suggests that future MAV communication
will increasingly integrate with sensing technologies, leading
to unified communication-perception MAV systems.
B. Action Recognition in Event Vision Using Spiking Neural
Networks
Action recognition is a classic pattern classification task.
Traditional action recognition methods primarily focus on
video data, targeting human actions, and employ deep learning
models combined with temporal modeling techniques, such
as optical flow-based Two-Stream networks [24], Long Short
Term Memory networks (LSTM) [25], and more recently,
Transformer architectures [26]. However, these approaches are
mostly designed for RGB video inputs and are not directly
applicable to event camera data.
For event-based action recognition, researchers have pro-
posed various event data representations, such as event frame
accumulation [27], voxel grids [28], and time surfaces [29].
Nevertheless, these representations often lead to the loss
of the inherent sparsity and asynchronous nature of event
streams, limiting the advantages of event cameras. To address
these challenges, Spiking Neural Networks (SNNs) [30] have
emerged as a promising alternative for processing event-
based data [31]. Unlike conventional artificial neural networks,
SNNs naturally handle sparse and asynchronous information,
making them well-suited for event-based motion analysis.
However, due to their spike-based discrete computation and
non-differentiable activation functions, SNNs suffer from chal-
lenges in gradient-based optimization, leading to difficulties
in achieving high-performance training through conventional
backpropagation. This limitation has hindered their widespread
adoption in action recognition tasks. To address this, recently,
there are numerous works [32], [33], [34], [35] focusing on
directly training high performance spiking neural networks.
Initially, to address the challenge of non-differentiability in
SNNs during training, the Spatio-Temporal Back Propagation
(STBP) algorithm [32] was proposed. By simultaneously con-
sidering the layer-by-layer Spatial Domain (SD) and the time-
dependent Temporal Domain (TD) during the training phase,
along with the approximate derivative of spike activities, this
method effectively resolves the convergence stability issues in
SNN training. Subsequently, further studies have enabled the
training of deeper SNNs [33], [34] on larger-scale datasets.
Regarding SNN architecture design, one study [35] proposed
using Neural Architecture Search (NAS) to automatically
discover optimal SNN structures. Additionally, several works
have integrated attention mechanisms [36] and Transformer
architectures [26] into SNN-based recognition networks, facil-
itating high-performance and stable SNN training. Therefore,
these fundamental SNN techniques have been widely applied
to various event-based action recognition tasks [37], [38], [39].
However, research on MAV-specific action recognition using
SNNs remains limited, as most studies focus on general event-
based action recognition.
C. Event-Based MAV Motion Segmentation with Time Series
The original event stream consists of multiple MAV mo-
tions. When different MAV motions are combined and as-
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
4
signed corresponding semantic information, they are collec-
tively referred to as an action. To achieve effective recog-
nition of MAV actions, it is necessary to perform accurate
segmentation of the MAV motion sequence. Event-based MAV
motion segmentation involves parsing sparse event streams
into discrete motion segments, each representing a specific
MAV maneuver. This task presents unique challenges due
to asynchronous nature and strict real-time requirements of
event data. Traditional time series analysis methods provide
a computationally efficient framework suitable for this task.
The foundational work [40] first established key principles
for time series segmentation, though not specifically for event
data. Then, a work [41] advanced the field by applying
Hidden Markov Models (HMM) to time series segmentation,
demonstrating improved pattern recognition capabilities. Fur-
ther developments came from a work [42], which introduced
change-point detection techniques for identifying statistical
shifts in time series data.
While traditional time series methods, such as HMM [41]
and change-point detection [42], provide theoretical founda-
tions, they are not optimized for the sparse and asynchronous
nature of event data, necessitating specialized approaches for
MAV motion segmentation. Our motion segmentation model
process event streams by first converting them into event-frame
representations. For each event frame, we calculate the total
number of events to construct a one-dimensional temporal
sequence. This time sequence enables us to effectively segment
MAV motion sequence in event stream signal, achieving
robust, real-time segmentation for MAV motion sequences.
III. THE PROPOSED METHOD
The proposed framework comprises three key modules.
First, the MAV motion segmentation model processes the event
stream captured by the event camera and segments the MAV
motion sequences into individual actions. Second, the MAV
action recognition model efficiently identifies each segmented
MAV action and outputs the corresponding semantic label. Fi-
nally, the MAV motion sequence decoding algorithm integrates
the motion segmentation and recognition models to achieve
robust visual action information decoding among MAVs.
A. Event-Based MAV Motion Segmentation
To achieve visual communication based on event vision,
transforming event signals into visual messages is of critical
importance. However, in the communication process among
MAVs, both the signal length and the transmitted information
are uncertain, making it difficult to design an end-to-end
model that can handle these variabilities directly. To address
this, we propose a MAV motion segmentation model that
processes variable-length event streams to extract semantically
meaningful motion segments, enabling reliable MAV visual
communication.
1) MAV Motion Segmentation Definition: MAV motion seg-
mentation aims to detect and segment different MAV actions
from a variable-length event stream E, determining the precise
time intervals t for each MAV motion. Fig. 3 shows MAV
action sequence in an event stream E. Then, MAV motion
segmentation problem can be formulated in Eq. 1.
S = fseg(E) = {(ti
s, ti
e, ai) | i = 1, 2, ..., N},
(1)
where E = {ek = (xk, yk, tk, pk)}M
k=1 represents the input
event stream, consisting of M events. Each event is defined
by its spatial coordinates (xk, yk), timestamp tk, and polarity
pk. S is the segmented motion output set, containing N motion
segments. In an event sequence, the start time of the i-th action
is defined as ti
s, the end time as ti
e, and the action label as ai.
2) The Proposed MAV Motion Segmentation Model: To
clearly present our event segmentation model, our event
segmentation model is divided into several steps, including
event count extraction, event feature computation, motion
segmentation, and motion refinement.
Event count extraction involves computing statistical infor-
mation from the event frame sequence. In order to generate
event frames, a fixed 33 millisecond time window is used,
facilitating human visual interpretation and providing intuitive
results. In segmentation method, algorithm primarily extracts
the positive event count Pn negative event count Nn, and total
event count Qn = Pn + Nn for n-th event frame. The frame
number framen is assigned using the line number, starting
from 0. With a total frame count of L, this frame sequence is
represented in Eq. 2.
framen ∈{0, 1, . . . , L −1},
n = 0, 1, . . . , L −1.
(2)
This step converts 2D event frames into 1D temporal signals
while preserving data integrity and alignment within the event
frames.
Event feature computation effectively utilizes the extracted
frame-wise event count information to analyze the MAV’s
motion state. To distinguish MAV motion periods from static
periods, the model extracts two key features: the Positive-to-
Negative Event Ratio (PNER), which measures the proportion
of positive to total events, and the Event Frame Variance
(EFV), which quantifies fluctuations in event counts. PNER
can be defined as Rn in Eq. 3.
Rn = Pn
Qn
, Pn ≤Qn.
(3)
Positive events Pn and negative events Nn typically cor-
respond to different physical phenomena, such as changes in
lighting or motion direction. Qn is the total number of event in
n-th frame. Rn highlights variations in event polarity, which
may indicate specific MAV’s motions, such as MAV turning
or acceleration. This provides important cues for identifying
MAV action start and end points in the time axis of an
event steam. As for EFV, it measures the local fluctuations
in total event count, reflecting the intensity or stability of
MAV activity, which is key to distinguishing static and motion
periods from the event stream. The specific EFV (Vn) is
defined in Eq. 4.
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
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Start
Empty
1
1
Empty
Empty
0
End
Time Axis
Event Stream
Fig. 3. MAV motion event stream. This diagram depicts the motion flow of a complete visual message, where Ak represents predefined MAV motions, Se
denotes the MAV static state (Empty signal), and tn indicates the termination timestamp of each MAV motion. From a time-domain perspective, an event
signal accumulates progressively from left to right. The final “end” motion signifies the stop of the visual signal. Besides, the x-axis and y-axis represent the
spatial domain of the event camera, while the time axis t represents the time domain of the event camera.
Vn = 1
W
n+⌊W/2⌋
X
k=n−⌊W/2⌋
(Tk −¯Tn)2,
¯Tn = 1
W
n+⌊W/2⌋
X
k=n−⌊W/2⌋
Tk,
(4)
where W/2 is the half-window size, and boundary frames are
handled by truncation. W is a sliding window of size (default
10 frames). Tk denotes the total event count at frame k (k ∈
{n−⌊W/2⌋, . . . , n+⌊W/2⌋}), while ¯Tn is the mean value of
total event number in a window W. Besides, Vn measures the
rate of event count variation, with lower values in static periods
and higher values in MAV motion periods. To reduce noise,
Rn and Vn are smoothed using a 5-frame moving average,
which enhances the model’s ability to detect MAV motion
boundaries.
Motion Segmentation step identifies MAV motion segments
by applying threshold-based segmentation to the smoothed
Rn and Vn. It detects transitions between static and motion
periods and generates initial MAV’s motion boundaries based
on predefined criteria, such as minimum motion duration. To
segment MAV motions, event frames in the static state are first
identified. In the implementation process, Eq. 5 is applied to
label the state of each event frame.
Gn =
(
1,
if Rn < θR and Vn < θV ,
0,
otherwise.
(5)
In Eq. 5, θR is equal to 0.5 and θV is set to 50% of
the median variance Vn, where Gn = 1 is static frame and
Gn = 0 represents motion frame. To search for MAV motion
boundaries, boundary detection computation is conducted.
Dn = Gn+1 −Gn,
B = {bj | Dn ̸= 0},
(6)
where Dn is a differential sequence, and B is the boundary
point set. Using Eq. 6, a sequence of boundary pairs in the
event stream is obtained. Then, the Gbj = 0 ∧bj+1 −bj ≥
Lmin, Lmin = 10 is applied to filter out motion segments
shorter than 10 frames (approximately 330 milliseconds),
as every motion in our MAV communication exceeds this
threshold. Finally, these motion segments are recorded as
candidate motion segments.
Motion refinement enhances the detection of precise MAV
motion moments. In this step, short-duration MAV motions
are filtered out by retaining only segments with a duration
of at least 30 frames (approximately 1 second, given a frame
rate of 33 ms), ensuring the removal of transient noise. Then,
adjacent motion segments separated by gaps of 10 frames or
fewer (approximately 330 ms) are merged into a single con-
tinuous segment, enhancing temporal coherence. Finally, MAV
segments shorter than 91 frames (approximately 3 seconds) are
discarded to ensure that only motions of sufficient duration
are preserved, aligning with the expected characteristics of
meaningful MAV activities, e.g., action “1”, action “0”, action
“start”, and action “end”.
B. MAV Action Recognition Network
To recognize each segmented MAV action from the event
stream collected by an event camera, a compact and efficient
spiking neural network was developed to enable rapid and
accurate MAV action recognition.
1) The basic theories of Spiking Neural Network: Spiking
Neural Networks (SNNs) [8] are a class of artificial neural
networks inspired by the behavior of biological neurons, where
information is transmitted via discrete spikes or events rather
than continuous values. Using the classic Leak Integrate-
and-Fire (LIF) neuron [43] model as an example, a neuron
integrates incoming signals and emits a spike (S(t) ∈{0, 1})
when its membrane potential H(t) exceeds a threshold Hth.
The LIF neuron’s behavior is describe in Eq. 7.
H(t + 1) = H(t) · e−∆t
τm + R · I(t)
I(t) =
X
j
wj · Sj(t)
S(t) =
(
1,
if H(t) ≥Hth,
0,
otherwise.
(7)
In Eq. 7, the membrane potential H(t) decays over time with
a time constant τm and increase with the input current I(t),
scaled by resistance R. The input I(t) is the weighted sum of
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
6
Split Event by Number Number
Spiking Convolutional Layer
Spiking Convolutional Layer
Spiking Fully Connected Layer
Spiking Fully Connected Layer
Spiking Voting Layer
Action one
Action two
Action three
Action four
Action five
Split Event Signal & pre-processing
Spiking Convolutional & MaxPool2D
Spiking Voting & MSE Loss
Spiking Fully Connected & Dropout
Event data pre-processing
Feature extraction stage
Feature encoding stage
Voting classification stage
Fig. 4.
EventMAVNet: MAV action recognition model. This diagram illustrates the complete recognition process of the MAV event stream, implemented
as a simple yet effective five-layer spiking neural network, encompassing four main components: event data pre-processing, feature extraction stage, feature
encoding stage, and voting classification stage.
spikes from pre-synaptic neurons, with wj as synaptic weights.
A spike S(t) = 1 occurs if H(t) reaches Hth, otherwise
S(t) = 0. This step captures the event-driven essence of SNNs.
2) Spiking Neural Networks for Event-Based Processing
and MAV Action Recognition Model: Traditional Deep Neural
Networks (DNNs) recognize event motion stream in a frame-
based, synchronous manner, relying on continuous activation
values to capture spatial patterns. Traditional DNNs, while ef-
fective for frame-based inputs, struggle with the asynchronous,
sparse nature of event streams, leading to high computational
costs and inefficiency in real-time MAV action recognition.
For real-time applications, such as MAV action recognition,
fast, efficient, and low-power pattern discrimination becomes
particularly crucial. In contrast, SNNs offer a biologically
inspired approach by representing information through discrete
spike trains (temporal sequences of binary events). Compared
to DNNs, recognition models based on SNN possess several
advantages. First, unlike DNN neurons that output continuous
values [44], SNN neurons emit spikes only when the mem-
brane potential exceeds a threshold. Thus, during MAV action
recognition, only a small fraction of neurons are activated,
granting SNNs a significant advantage in power efficiency.
Second, the spiking mechanism of SNNs inherently integrates
input signals over time, enabling them to capture temporal
dynamics in event-based data. This makes SNNs particularly
suitable for processing time-varying signals, such as those
event streams from event cameras. Although our method
converts event data into frames for structured input, SNNs
outperform traditional DNNs in power efficiency, temporal
dependency modeling, and parameter efficiency.
Based on these properties, EventMAVNet, a spiking neural
network tailored for MAV action recognition, is proposed.
The EventMAVNet architecture processes event streams with
an input resolution of 128×128 and 2 channels (representing
positive and negative polarity events) over 16 time steps, clas-
sifying them into 5 distinct motion categories. As illustrated
Fig. 4, the proposed networks is structured into three stages:
feature extraction, feature encoding, and voting classification,
all implemented using the SpikeJelly framework 1.
Feature extraction stage extracts spatial-temporal features
from the preprocessed event frames. It consists of two spiking
convolutional layers, each with 3×3 kernel and 128 outputs
channels. The first layer processes the 2-channel input of
size 128×128, while the second builds upon the previous
output, maintaining 128 channels. Each convolutional layer
is followed by a Batch Normalization (BN) layer to stabilize
the distribution of spiking activations across the 16 time
steps, enhancing training convergence. Leaky Integrate-and-
Fire (LIF) neurons then introduce temporal dynamics, inte-
grating a 4×4 max-pooling operation downsamples the feature
maps, reducing the spatial resolution to 32×32 after the first
layer and 8×8 after the second. This results in a feature tensor
of shape 128×8×8 per time step.
Feature encoding stage transforms the extracted features into
a compact, discriminative representation. The feature maps
are flattened into a 128×64-dimensional vector (computed as
128 channels ×8×8 spatial grid), followed by a dropout layer
(p = 0.5) to prevent overfitting. A spiking fully connected
layer reduces this vector to 128 dimensions, with LIF neurons
processing the temporal spikes across 16 time steps. Another
dropout layer further regularizes the output, ensuring robust-
ness in the encoded features.
Voting classification stage maps the encoded features to ac-
tion classes and computes the classification output. A spiking
fully connected layer projects the 128-dimensional features
into 50 dimensions. A spiking voting layer then aggregates the
spiking activity of these 50 neurons, organized as 10 neurons
1https://github.com/fangwei123456/spikingjelly
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
7
per class to produce outputs for 5 action categories. The five-
class classifier (four semantic labels + background noise) im-
proves decoding robustness by enabling noise filtering during
recognition, ensuring cleaner MAV motion sequence decoding.
Finally, the Mean Squared Error (MSE) loss is then calcu-
lated to measure the difference between this averaged predic-
tion and the target labels across a batch in training.
L = 1
O
O
X
o=1
P
X
p=1
(predo,p −yl
o,p)2.
(8)
In Eq. 8, O is the batch size, P is the number of classes,
predo,p is the predicted score for the o-th sample and p-th
class after temporal averaging, and yl
o,p is the corresponding
one-hot value (number 0 or 1).
C. IMSR: Integrated Segmentation and Recognition for MAV
Motion Decoding
To improve the reliability of MAV visual communication,
we propose an online decoding framework termed Integrated
MAV Segmentation and Recognition (IMSR). As shown in
Algorithm 1, IMSR decodes MAV-transmitted messages from
raw event streams captured by an event camera. The frame-
work operates in three main stages: preprocessing, segmenta-
tion, and recognition. In the preprocessing stage, the back-
ground noise filter F removes spurious events caused by
environmental noise or camera artifacts, using a threshold-
based approach to retain valid motion-related events. The de-
noised event stream De
f is buffered for subsequent processing.
The segmentation stage begins by monitoring the number of
valid event frames. Then, valid event frames are counted by
countEventFrames() in the filtered stream. Once the
accumulated count exceeds a threshold (e.g., 100), the seg-
mentation model Seg is activated to partition the event stream
into discrete motion segments Actionk, where each segment is
assumed to correspond to a distinct MAV action. The threshold
is set based on the minimum duration of predefined MAV
actions (approximately 3 seconds). Given that the event stream
is sampled at a rate of 30 frames per second, the threshold
of 100 ensures sufficient temporal context (approximately 3.3
seconds) to capture at least one complete action, thereby
reducing the risk of segmenting incomplete or noisy motion
sequences.
In the recognition stage, each segmented action ai
∈
Actionk is classified by the recognition model Rec, imple-
mented as EventMAVNet. Recognized labels ri are appended
sequentially to a code sequence Cseq, when ri is not equal
to label “background”. If the keyword “end” is detected as
the final label, decoding is triggered. The decode() function
extracts a three-part message Mdec from Cseq, including flight
direction, encoded angle, and relative distance. For message
integrity, decoding only proceeds if the sequence starts with
“start” and reaches the minimum expected length. By com-
bining temporal segmentation with learned action recognition,
IMSR ensures robust decoding even under noisy or ambiguous
visual input. Unreliable inputs that fail segmentation or recog-
nition are discarded, and retransmission is requested, enabling
reliable MAV visual communication in dynamic environments.
Algorithm 1 MAV motion sequence decoding algorithm
1: Input: The event stream De collected by event camera
2: Output: Communication code Cseq, decoded message
Mdec
3: Initialize background noise filter F
4: F.accept(De)
5: De
f ←F.generateEvents()
6: Buffer the event stream De in memory
7: Set Cseq ←[ ], Mdec ←[ ]
8: Initialize segmentation model Seg, recognition model Rec
9: Initialize segmented action set Actionk
10: while new frame features available from De
f do
11:
Ec ←countEventFrames(De
f)
12:
if Ec > 100 then
13:
Actionk ←Seg.segment(De
f)
14:
for each ai in Actionk do
15:
ri ←Rec.predict(ai)
16:
if ri ! = "background" then
17:
Append ri to Cseq
18:
end if
19:
end for
20:
if Cseq[−1] == "end" then
21:
Mdec ←decode(Cseq)
22:
break
23:
end if
24:
end if
25: end while
26: Mdec ←decode(Cseq)
27: return Cseq, Mdec
28: Function decode(Cseq)
29: if length(Cseq) < 7 or Cseq[0] ̸= "start" then
30:
return [ ]
31: end if
32: direction ←Cseq[1]
33: angle ←Cseq[2 : 5]
34: distance ←Cseq[5 : 7]
35: Mdec ←[direction, angle, distance]
36: return Mdec
IV. EXPERIMENTS
This section provides the evaluation of our proposed model.
We first outline the experimental details, which include infor-
mation about the dataset, parameter settings, and experimental
environment2. Then, our comparative experiments assess both
MAV action recognition (accuracy and speed) and motion seg-
mentation performance separately. Additionally, we conducted
key ablation experiments on the proposed action recognition
model to offer a detailed analysis of the contributions of
its critical components. Finally, we describe several experi-
ments on the visual communication of common information
to validate the effectiveness of our entire MAV-based visual
communication system.
2Dataset and code will be made publicly available upon publication.
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
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LightHouse Positioning System
Crazyflie
Event Camera
PC for MAV motion control in data collection
Action
Wireless Communication Protocol
Raspberry Pi
w : world
: the origin point
: the X-axis
: the Y-axis
: the Z-axis
Fig. 5.
MAV motion data collection system. The figure lists the necessary
data collection sensors and key processing devices.
A. Experimental details
To validate the proposed framework, a controlled MAV
motion data collection system in an indoor environment is
designed to evaluate visual communication behavior and mo-
tion segmentation performance between MAVs, as shown in
Fig. 5. In the figure, the hardware consists of Crazyflie drones
in the center of the experimental scene, a DVS Micro Explore
event camera on a tripod to capture event data, a Lighthouse
system overhead for precise 3D localization, and a PC with
an NVIDIA GPU 4090 running Ubuntu for data processing.
The DVS camera recorded event streams in aedat4 format,
which are then processed using the DVS process library.
During the collecting process, the data is collected at three
sensor observation distances (0.9 m, 1.2 m, and 1.5 m),
representing short, medium, and long ranges, respectively,
using an “Observe” and “Motion” feedback loop, in which the
camera captured the MAV’s motions while the PC processed
the data for MAV control and analysis.
The dataset consists of event streams capturing MAV mo-
tions at these three scales, with each scale including five
action categories: Action “0”, Action “1”, Action “Start”,
Action “End”, and Action “background”. Particularly, the
MAV motion background signal helps the model distinguish
between normal signals and non-effective signals (included
empty and noise signals), enabling robust action classification.
As for the MAV motion segmentation model, we imple-
mented our method using Python 3.10 with the NumPy library.
The model’s performance was evaluated using center point
error (the temporal deviation of predicted action midpoints)
and 2D Intersection over Union (IoU) error (the overlap ratio
between predicted and ground-truth action segments) to ensure
a fair evaluation of the MAV motion segmentation model’s per-
formance. With regard to the MAV action recognition model,
we built the framework using PyTorch and the SpikeJelly
library, leveraging their capabilities for processing event-based
data. The recognition model is evaluated using classification
accuracy, model parameter size, inference speed, and power
consumption, ensuring a fair and impartial assessment that
guides the design of a more efficient and robust MAV visual
communication system.
Fig. 6.
Inference time comparison with a batch size of 10. Each result
represents the average runtime over 100 runs on an RTX 4090 GPU.
B. MAV Action Recognition and Inference Speed Experiment
To evaluate the effectiveness of the proposed MAV action
recognition approach, we conducted a MAV action recognition
experiment using event data with other state-of-the-art model
( spike-C [45], DVS-G [46], spike-R [47]). The test results are
presented in Table I.
TABLE I
PERFORMANCE COMPARISON ACROSS DIFFERENT OBSERVATION
DISTANCE DATASETS.
Algorithms
Short
Medium
Long
spike-C [45]
94.63% ± 1.82%
93.00% ± 1.64%
91.49% ± 8.2%
spike-R [47]
96.49% ± 0.47%
94.54% ± 0.24%
92.35% ± 1.23%
DVS-G [46]
95.36% ± 1.4%
94.12% ± 0.6%
91.82% ± 1.56%
Ours
96.51% ± 0.61%
95.37% ± 1.01%
94.98% ± 1.17%
As demonstrated in Table I, our model outperforms other
state-of-the-art methods across the short, medium, and long
scenarios in terms of recognition accuracy, underscoring its
advanced capabilities to a significant extent. Notably, all
models exhibit a decline in accuracy in the long-distance
scenario, likely due to reduced event density and increased
noise at greater observation distances. Nevertheless, our model
maintains a competitive recognition performance on the long
dataset (1.5 m observing distance). Besides, we also present
the accuracy and inference time of MAV action recognition
on the Medium dataset in Fig. 6. Compared to other models,
ours model achieves the lowest latency of 1.26 ms per 10
test samples, demonstrating its reliability and efficiency under
demanding conditions.
To evaluate the effectiveness of our proposed model for
recognizing MAV action data in event form, we conduct exper-
iments in dataset with a 3:1 training-to-testing ratio. The pri-
mary motivation is to assess the model’s tailored design, which
is intended to enhance temporal precision and robustness
in processing event-based MAV motion sequences for real-
time applications. To this end, we visualized the training and
testing performance over epochs to demonstrate the model’s
convergence and generalization capabilities on MAV action
recognition task. Specifically, Fig 7 illustrates the proposed
model’s training and testing performance in detail and Fig
7(a) shows the accuracy change over 40 epochs, highlighting
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9
(a) Training and Testing Accuracy.
(b) Training and Testing Loss.
Fig. 7. Training and Testing Performance Over Epochs. (a) Training (green)
and testing (red) accuracy over 40 epochs. (b) Training (green) and testing
(red) loss over 40 epochs.
Fig. 8. Comparison of the number of error frames for three methods (Ours,
HMM [41], CPD [42]) under different empty signal types: short pause,
medium pause, and long pause. The error reflects the misalignment in center
point prediction of the MAV action.
the model’s learning progression and generalization capability
on event-based MAV action recognition data. In this figure,
training is green curves, and testing is red curves. Fig. 7(b)
shows the loss values over 40 epochs, demonstrating stable
convergence and reduced overfitting. The training (green)
and testing (red) curves confirm the model’s robustness for
temporal event sequences.
Overall, these experimental results validate the efficacy of
our proposed model, as the visualizations of accuracy and loss
curves demonstrate robust convergence and enhanced general-
ization, supporting the model’s capability to accurately identify
MAV in-flight actions. These findings not only confirm the
potential of SNN models for event-based action recognition
tasks but also pave the way for future optimizations and
broader applications in real-world MAV scenarios.
C. The Comparison of Motion Segmentation Model Results
A robust motion segmentation model is pivotal for ensuring
reliable visual communication in MAVs. Inaccurate motion
localization by the segmentation model can directly impair
the recognition accuracy of subsequent MAV action classi-
fication processes. To provide a more objective assessment
of the motion segmentation model proposed in this study, a
comprehensive, multi-dimensional experimental comparison is
Fig. 9. Comparison of total action duration errors for three methods under
different empty signal types. The error represents the frame-level difference
between predicted and ground-truth MAV action intervals.
conducted in following. As illustrated in Fig. 3, an empty sig-
nal is inserted between consecutive valid MAV actions during
motion message delivery to mitigate overlap and confusion
between different actions. In our comparative experiments,
we evaluated three key metrics: the offset of the MAV action
center point, the total frame number error of the action, and
the action overlap area ratio, with HMM [41] and CPD [42].
Furthermore, the model’s segmentation performance under
varying durations of empty signals was analyzed, providing
insights into its robustness and adaptability. The experiments
are conducted on three event streams, each containing the same
9 valid MAV actions and 8 empty signals. The durations of
the empty signals in three streams are 2.5 seconds, 3 seconds,
and 3.5 seconds, respectively.
To evaluate the offset error of MAV action centers, we com-
puted the center deviation {Ak}c and its variance {Ak}var
c
for
each action. Lower values indicate higher prediction accuracy.
Fig. 8 shows three event streams with different empty signal
durations on the horizontal axis and the frame offset of the
predicted center on the vertical axis. The results demonstrate
that the proposed MAV motion segmentation model achieves
more accurate motion localization on all three streams, with
both the deviation and variance notably smaller than those of
other models.
With regard to the total frame number error of the MAV
action, we calculated the difference between the predicted
duration Apred
k
and the ground truth duration AGT
k
for each
action. This metric reflects how well different models can
localize the temporal span of MAV actions. As shown in Fig. 9,
although our segmentation model performs slightly worse than
the HMM model under the 2.5-second idle signal condition,
it achieves clear advantages when the empty signal duration
increases to 3 and 3.5 seconds. This indicates that our model
becomes more effective at action duration prediction as the
length of empty signals increases.
Regarding the MAV action overlap area ratio metric, we
analyze the start and end position of MAV actions to compute
intersection ∩and union ∪of these actions. The calculation is
expressed in Eq. 9, where Apred represents the motion segment
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
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Fig. 10.
Comparison of intersection-over-union (IoU) errors between pre-
dicted and ground-truth MAV actions under different MAV action pause
intervals.
predicted by the model, and Agt denotes the ground truth
motion segment. The experimental results are presented in Fig.
10. In the figure, the IOU test results of our proposed model
consistently outperform others, demonstrating the superiority
of the model proposed in this study.
Apred = [startpred, endpred],
Agt = [startgt, endgt],
IOU = |Apred ∩Agt|
|Apred ∪Agt|.
(9)
To intuitively demonstrate the robustness of the proposed
motion segmentation model, we visualize the MAV action
segmentation results on MAV action sequences under three dif-
ferent durations of the MAV pause signal, as shown in Fig 11.
When the pause signal duration is 2.5 s, the model incorrectly
segments 9 valid MAV actions into 8. This error likely stems
from insufficient temporal separation between actions, causing
the algorithm to fail in detecting clear MAV motion boundaries
due to weak or ambiguous temporal features. However, within
3 s and 3.5 s pause signal, the proposed model accurately
detects the start and end of each MAV action.
D. Ablation Study on The Proposed Recognition Model
To gain a deeper understanding of the proposed MAV action
recognition model, an ablation study is conducted in this
section. Specifically, we analyzed the effects of spike window
length F and event frame resolution on our spiking-based
MAV motion recognition model. The detailed experimental
results are shown in table II and III.
In table II, several experiments are conducted on the impact
of motion scale using a medium-range MAV motion dataset.
By varying input resolutions, we evaluated their effect on
the performance of the MAV action recognition performance
at an input size of 128 × 128. Larger-scale event frames
improve accuracy by capturing finer spatial details, but they
increase computational load and power consumption due to
higher-dimensional feature processing. However, larger scales
also increase computational load, parameter count, and power
The number of events
Frame Number
(a) MAV action sequence with 2.5s pause signals.
The number of events
Frame Number
(b) MAV action sequence with 3.0s pause signals.
The number of events
Frame Number
(c) MAV action sequence with 3.5s pause signals.
Fig. 11. Comparison of segmentation performance under different MAV pause
durations. The red dashed lines denote the ground truth boundaries of each
action, and the translucent green boxes indicate the predicted start and end
positions by the proposed segmentation model.
consumption. Thus, in real MAV motion signal recognition,
an appropriate event frame resolution can be flexibly selected
to balance recognition accuracy with computational efficiency.
TABLE II
IMPACT OF INPUT RESOLUTION ON RECOGNITION PERFORMANCE.
Resolution
Accuracy
ACs(G)
MACs(G)
Params(M)
Energy(mJ)
128x128
95.37%
0.6849
0.424
1.205
2.567
64x64
92.28%
0.1636
0.062
0.418
0.432
32x32
81.79%
0.043
0.015
0.222
0.108
In table III, several experiments on the effect of event
timestep F are performed using the same medium-range
MAV motion dataset. Timestep F represents the sampling
frequency of the time window for MAV motion event streams,
encoding them into temporal spike signals for input into our
proposed SNN model. By varying F (set to 4, 8, and 16) in
ablation studies with equally spaced sampling, we investigated
its impact on MAV action recognition accuracy. The results
indicate that smaller F significantly reduces computational
cost and power consumption but also decreases recognition
accuracy. This occurs because a smaller F lowers the sampling
frequency of MAV motions, limiting the input spike sequence.
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
11
Consequently, even with identical model parameters, recogni-
tion performance notably declines in MAV action recognition.
TABLE III
IMPACT OF TIME STEP F ON RECOGNITION PERFORMANCE.
Time Step
Accuracy
ACs(G)
MACs(G)
Params(M)
Energy(mJ)
F=16
95.37%
0.6849
0.424
1.205
2.567
F=8
91.98%
0.3346
0.1247
1.205
0.875
F=4
83.95%
0.1684
0.0623
1.205
0.438
Besides, ablation studies are also conducted on the impact
of positive and negative events. As shown in the table IV,
using only positive (E-PosNet) or negative events (E-NegNet)
fails to achieve optimal action recognition performance. The
best accuracy is obtained when employing the E-BiPolNet
approach (combining positive and negative events for MAV
action classification).
TABLE IV
ABLATION PERFORMANCE ANALYSIS IN DIFFERENT EVENT POLARITY.
Algorithms
Short
Medium
Long
E-NegNet
94.12%
91.49%
89.05%
E-PosNet
96.32%
94.33%
93.43%
E-BiPolNet
96.85%
95.37%
94.24%
E. Visual Communication in Flight Tests
To validate the feasibility of the proposed motion-based
MAV communication framework, flight tests of MAVs are
conducted in this section3. Similar to other communication
systems [48], we design three different 8-bit communication
codes and, based on these codes, used the “cfclient” control
library from the Crazyflie package to program corresponding
MAV action sequences. An event camera is used to observe
the MAV’s movements and interpret the control information
conveyed through the action sequences. The specific meanings
of the 8-bit communication codes are shown in Table V. In
each code, the first bit indicates the end of the signal. The
bits in between represent, respectively, the flight direction (1
bit), the flight heading angle (3 bits), and the flight distance
(2 bits).
TABLE V
MAV FLIGHT SIGNAL ENCODING SCHEME.
Segment
Bit Range
Meaning
Example
Start Flag
Fixed
Start of signal
start
Direction
1 bit (bit 1)
0 = Forward, 1 = Backward
0 (Forward)
Heading
3 bits (bits 2–4)
Heading angle, each step α◦
000 = 0 × α◦
Distance
2 bits (bits 5–6)
Flight distance
01 = 0.2 m
End Flag
Fixed
End of signal
end
To further validate the effectiveness of the proposed visual
communication framework, we designed flight communication
experiments involving multiple MAVs. In flight tests, two
MAVs are deployed: MAVe (Executor MAV, which executes
navigation commands) and MAVp (Performer MAV, which
transmits messages via motion). Performer MAVp enacts mo-
tion patterns to convey messages. Upon receiving the mes-
sage via the event camera using the proposed method, the
3A video showcasing the proposed framework and MAV flight tests is
provided as supplementary material.
Fig. 12. Visual communication in flight tests. The Performer MAV (MAVp)
transmits the visual message to guide the Executor MAV (MAVe) to three
different trajectories.
signal is decoded and converted into MAV control commands.
These commands are then transmitted to the Executor MAVe,
which subsequently navigates to one of three predefined target
destinations, Dp1 to Dp3. We designed three distinct 8-bit
communication codes, as shown in Table VI, corresponding to
the target destinations Dp1 to Dp3. Ultimately, the proposed
method successfully decoded all predefined communication
codes, with the decoding results illustrated in Fig. 13. Upon
successful code recognition, the Executor MAVe executed
the control commands and reached the corresponding target
destinations. To this end, we recorded the 2D flight trajectories
of the Executor MAVe, as shown in Fig. 12. The results
demonstrate that the proposed method not only successfully
decodes visual motion information but also accurately controls
the Executor MAVe to reach the designated positions from Dp1
to Dp3 .
TABLE VI
THE EXPERIMENTAL COMMUNICATION CODE IN THREE FLIGHT TESTS.
Communication code
Begin
Direction
Angle
Distance
Stop
S000010E
s
0
000
10
e
S000110E
s
0
001
10
e
S001010E
s
0
010
10
e
V. CONCLUSION
This paper presents a novel MAV communication paradigm
inspired by non-verbal communication strategies in nature,
where motion sequences are used to transmit messages be-
tween MAVs. Specifically, MAV information is encoded into
binary-coded aerial movements observed by event cameras,
enabling successful inter-MAV message decoding and for-
warding. By categorizing MAV motions into four primitive
types with assigned communication symbols, we establish
a fundamental visual communication framework for sharing
critical swarm coordination data (e.g., heading, and distance).
To achieve efficient visual message transmission, an event-
frame-based motion segmentation method using mathematical
statistical features is proposed, enabling precise segmentation
IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED
12
The number of events
Frame Number
start
0
0
0
0 end
1
0
(a) MAV action sequence in trajectory 1.
The number of events
Frame Number
start 0
0
0
0
end
1
1
(b) MAV action sequence in trajectory 2.
The number of events
Frame Number
start 0
0
0
0 end
1
1
(c) MAV action sequence in trajectory 3.
Fig. 13. The experimental results of the three flight tests.
of MAV action sequences. Furthermore, we design Event-
MAVNet, a shallow Spiking Neural Network, that outperforms
existing event-based action classifiers in both inference speed
(1.26 ms / 10 samples) and recognition accuracy (95.37%
in the medium dataset). Experimental results, including a
95.37% recognition accuracy on the medium dataset and
successful decoding in flight tests, demonstrate that our event-
based motion communication serves as a viable alternative to
conventional radio-based methods. Future work will focus on:
(1) exploring advanced SNN architectures, such as attention-
based or deeper networks, to further enhance MAV action
recognition accuracy, (2) expanding MAV action semantic
labels by incorporating additional action categories for richer
communication. This work represents a significant advance-
ment toward vision-based communication for autonomous
MAV swarms.
ACKNOWLEDGMENT
The authors would like to acknowledge the support of a
research grant for this work. Full details will be provided upon
publication.
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IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 1 MoCom: Motion-based Inter-MAV Visual Communication Using Event Vision and Spiking Neural Networks Zhang Nengbo1, Hann Woei Ho1,∗, Member, IEEE, Ye Zhou1, Member, IEEE, Abstract-Reliable communication in Micro Air Vehicle (MAV) swarms is challenging in environments, where conventional radiobased methods suffer from spectrum congestion, jamming, and high power consumption. Inspired by the waggle dance of honeybees, which efficiently communicate the location of food sources without sound or contact, we propose a novel visual communication framework for MAV swarms using motion-based signaling. In this framework, MAVs convey information, such as heading and distance, through deliberate flight patterns, which are passively captured by event cameras and interpreted using a predefined visual codebook of four motion primitives: vertical (up/down), horizontal (left/right), left-to-up-to-right, and left-todown-to-right, representing control symbols ("start", "end", "1", "0"). To decode these signals, we design an event frame-based segmentation model and a lightweight Spiking Neural Network (SNN) for action recognition. An integrated decoding algorithm then combines segmentation and classification to robustly interpret MAV motion sequences. Experimental results validate the framework's effectiveness, which demonstrates accurate decoding and low power consumption, and highlights its potential as an energy-efficient alternative for MAV communication in constrained environments. Index Terms-Visual Communication, Event Camera, Spike Neural Networks, Motion Segmentation, Unmanned Aerial Vehicles. I. INTRODUCTION I N nature, many species have evolved non-verbal communication strategies, such as the waggle dance of honeybees, pheromone trails in ants, and postural displays in wolves, to support social coordination and collective intelligence. Among them, the waggle dance of honeybees is a remarkable example (see Fig. 1). Through specific body movements, bees encode and transmit information about food location, direction, and distance, as studied in a work [1]. This can enable the colony to make distributed foraging decisions without centralized control. This biologically evolved action-based communication system equips honeybee swarms with self-organization, robustness to environmental noise, and energy efficiency, offering valuable inspiration for engineered multi-agent systems [2]. Such bio-inspired communication paradigms present compelling solutions for modern Micro Air Vehicle (MAV) ∗Corresponding author. 1Zhang Nengbo, Hann Woei Ho, and Ye Zhou are with 14300 Nibong Tebal, Pulau Pinang, Malaysia (email: ; aehan- ; ). 1 s = 1 km Distance m Angle = Waggle Dance by Bees Sun Food Beehive Traditional Communication Mode between MAVs Radio signal disrupted by intentional/ unintentional interference Signal 1 Signal 0 MAV Action Sequence Event camera Event stream Inspired by how insects communicate MoCom using Event Vision & Spiking Neural Networks Quality ! ! Fig. 1. Overview of the proposed motion-based inter-MAV visual communication (MoCom) framework for MAV swarms. Traditional radio-based communication can be easily disrupted in constrained environments. Inspired by the waggle dance of honeybees, a motion-based signaling approach, where MAVs convey information through deliberate flight patterns. These motions are captured by event cameras. The event stream is then processed using a biologically inspired pipeline that segments the motion and classifies it using a spiking neural network. This paradigm offers a decentralized and interferenceresilient alternative to conventional communication in MAV swarms. swarms, which face critical challenges in maintaining reliable and covert inter-agent coordination. Particularly in highstakes applications, including disaster rescue and environmental monitoring, conventional wireless protocols, such as WiFi, radio, 4G/5G, suffer from spectrum scarcity [3], jamming vulnerabilities [4], and poor scalability [5], which can lead to communication failures in dense MAV swarms. These limitations become debilitating in contested or GPS-denied environments, where resilience and stealth are paramount. Notably, the honeybee's motion-encoded communication model aligns precisely with MAV swarms' need for decentralized, interference-resistant, and energy-efficient alternatives. Inspired by the decentralized and robust communication of honeybees, which encodes and delivers information through physical movements, we propose a novel "action as signal" paradigm for MAV swarms to address the limitations of conventional wireless communication. In this paradigm, MAVs encode information in deliberate motion patterns, much like bees conveying foraging details through physical movements. Such motion-based communication not only offers a decentralized, interference-resistant, and energy-efficient alternative to radio-based protocols but also naturally aligns with the perception-action loop of autonomous systems. This synergy enables integration in dynamic, unstructured environments. To capture motion-based signals effectively, a sensing modality must meet three requirements: high temporal resolution, ro16 Oct 2025 IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 2 bustness to lighting variations, and sensitivity to rapid movements. Here, event cameras emerge as an ideal solution. Unlike conventional frame-based RGB cameras, they asynchronously detect pixel-level brightness changes, offering low latency, high dynamic range, and sparse data output. These capabilities are critical for reliably decoding the fast, edge-rich motion patterns generated by MAVs, while maintaining energy efficiency, a feature particularly vital for MAV swarm operations. Recent advances have demonstrated the potential of event cameras in MAV behavior recognition, such as object tracking [6] and pose estimation [7]. However, most of these applications focus on short-duration event segments and do not address the challenges of processing long, continuous event streams typical in real-time MAV communication. Existing MAV communication methods still largely rely on RF-based techniques, which often fail in environments with electromagnetic interference or obstructed lines of sight, leading to unreliable connections and potential mission failure. Furthermore, the ability to extract consistent, interpretable signals from noisy event streams during dynamic flight remains an open challenge, underscoring the need for a new communication framework tailored to the unique characteristics of event-based vision. Event data, despite its sparsity and asynchronous nature, contains rich temporal structure. Frame-wise statistical indicators, such as the total number of events and the ratio of positive to negative events, naturally encode motion dynamics, providing valuable cues for segmenting continuous event streams into discrete action units. These features offer a lightweight, low-computation, and intuitive foundation for temporal analysis, aligning seamlessly with the properties of event-based sensing. As such, statistical segmentation of event data is not only effective but also particularly suitable for real-time MAV applications with constrained computational resources. To build a fully bio-inspired MAV visual communication model, we further incorporate a brain-inspired biometric classifier based on Spiking Neural Networks (SNNs) [8]. SNNs process information through discrete spikes and inherently support event-driven computation, making them highly compatible with the output of event cameras. Compared to traditional neural architectures, such as Convolutional Neural Networks (CNNs) or Recurrent Neural Networks (RNNs), SNNs offer improved energy efficiency and better handling of sparse, asynchronous inputs. In our system, a lightweight SNN model is employed to classify segmented MAV motion clips into predefined communication symbols (e.g., binary digits), enabling end-to-end decoding of visual signals in a bio-inspired manner. In this paper, we propose an event vision-based framework for inter-MAV message transmission. It accurately recognizes MAV actions and decodes action sequences into meaningful messages, enabling visual communication between MAVs. The defined visual communication codes are illustrated in Fig. 2. To the best of our knowledge, this is the first work to introduce the concept of motion-based visual communication into MAV systems. The main contributions of this paper are threefold: • design a MAV motion segmentation model to process long continuous event streams by segmenting MAV motionencoded signals into discrete semantic units. This facilitates the subsequent decoding of MAV motion information. • develop a lightweight yet accurate MAV motion recognition model (EventMAVNet) inspired by biological neural processing, utilizing a spiking neural network architecture. Compared to other SNN recognition models, we achieve both computational efficiency and high recognition precision. • introduce the Integrated MAV Segmentation and Recognition (IMSR) algorithm, a MAV motion sequence decoding approach that seamlessly integrates event signals segmentation and recognition. The IMSR algorithm integrates segmentation and recognition to robustly decode MAV motion sequence signals, effectively mitigating noise and interference in dynamic environments. The rest of this paper is organized as follows: Section 2 reviews related works on MAV communication and recent advancements in event recognition models based on spiking neural networks. Section 3 details our proposed method, including the event signal segmentation model, motion recognition model, and MAV motion signal decoding algorithms. Section 4 presents the experimental results and performance evaluation. Section 5 concludes the paper and details future directions for inter-MAV visual communication. Through this work, we not only advance the application of event cameras in MAV signal transmission but also provide new insights into integrating vision and communication in dynamic environments. II. RELATED WORKS This section reviews key research in MAV communication techniques, event-based MAV action recognition, and motion segmentation using event frames. A. MAV communication techniques MAV communication technology [9] is a key component in MAV swarms [10], enabling swarm formation and coordination for applications, such as search and rescue and temporary communication networks [11] in disaster-affected areas. Generally, MAV communication can be classified into distributed [12], [13], [14] and centralized approaches [15], [16], [17], [18]. Regarding distributed MAV communication, an early study [12] introduced a distributed information exchange method based on neighborhood data sharing to address data association problem in multi-robot systems. Subsequent studies [13], [14] have demonstrated that decentralized coordination enhances swarm control and adaptability. Similarly, distributed communication principles have been successfully applied to vision-based multi-MAV control [19]. Decentralized communication phenomena are also observed in natural swarm organizations, such as social insect colonies [1]. In contrast, centralized MAV communication typically adopts a star topology, where all MAVs communicate through a central master node. This approach relies on MAV onboard electronics [15] and uses methods like point-to-point [16], ground-to-air [17], and GPS-based communication [18]. These methods ensure robust swarm control and coordination. IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 3 Signal Code MAV Action Action: Action: Action: Action: F = 1 F = 3 F = 5 F = 7 F = 9 Fig. 2. Several common visual communication codes are presented. Specifically, we define four types of communication codes (action : "start", action : "end", action : "1", action : "0"). A single event MAV motion is segmented into F-th event block based on an equal event count strategy, and the accumulated events are used to construct F event frames. Red color represents positive events, green color represents negative events, and the yellow regions indicate the overlap between positive and negative events. Traditional approaches, while effective, face challenges such as spectrum scarcity and vulnerability to interference, prompting recent advancements in novel MAV communication techniques. For example, one study [20] explored Visible Light Communication (VLC) to simultaneously enable communication and illumination. Another work [21] utilized visionbased analysis for direct MAV-operator interaction. However, these communication technologies all use visual light as a carrier for information transmission, and do not use the MAV's motion pattern to transmit information. Additionally, emerging research [22], [23] suggests that future MAV communication will increasingly integrate with sensing technologies, leading to unified communication-perception MAV systems. B. Action Recognition in Event Vision Using Spiking Neural Networks Action recognition is a classic pattern classification task. Traditional action recognition methods primarily focus on video data, targeting human actions, and employ deep learning models combined with temporal modeling techniques, such as optical flow-based Two-Stream networks [24], Long Short Term Memory networks (LSTM) [25], and more recently, Transformer architectures [26]. However, these approaches are mostly designed for RGB video inputs and are not directly applicable to event camera data. For event-based action recognition, researchers have proposed various event data representations, such as event frame accumulation [27], voxel grids [28], and time surfaces [29]. Nevertheless, these representations often lead to the loss of the inherent sparsity and asynchronous nature of event streams, limiting the advantages of event cameras. To address these challenges, Spiking Neural Networks (SNNs) [30] have emerged as a promising alternative for processing eventbased data [31]. Unlike conventional artificial neural networks, SNNs naturally handle sparse and asynchronous information, making them well-suited for event-based motion analysis. However, due to their spike-based discrete computation and non-differentiable activation functions, SNNs suffer from challenges in gradient-based optimization, leading to difficulties in achieving high-performance training through conventional backpropagation. This limitation has hindered their widespread adoption in action recognition tasks. To address this, recently, there are numerous works [32], [33], [34], [35] focusing on directly training high performance spiking neural networks. Initially, to address the challenge of non-differentiability in SNNs during training, the Spatio-Temporal Back Propagation (STBP) algorithm [32] was proposed. By simultaneously considering the layer-by-layer Spatial Domain (SD) and the timedependent Temporal Domain (TD) during the training phase, along with the approximate derivative of spike activities, this method effectively resolves the convergence stability issues in SNN training. Subsequently, further studies have enabled the training of deeper SNNs [33], [34] on larger-scale datasets. Regarding SNN architecture design, one study [35] proposed using Neural Architecture Search (NAS) to automatically discover optimal SNN structures. Additionally, several works have integrated attention mechanisms [36] and Transformer architectures [26] into SNN-based recognition networks, facilitating high-performance and stable SNN training. Therefore, these fundamental SNN techniques have been widely applied to various event-based action recognition tasks [37], [38], [39]. However, research on MAV-specific action recognition using SNNs remains limited, as most studies focus on general eventbased action recognition. C. Event-Based MAV Motion Segmentation with Time Series The original event stream consists of multiple MAV motions. When different MAV motions are combined and asIEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 4 signed corresponding semantic information, they are collectively referred to as an action. To achieve effective recognition of MAV actions, it is necessary to perform accurate segmentation of the MAV motion sequence. Event-based MAV motion segmentation involves parsing sparse event streams into discrete motion segments, each representing a specific MAV maneuver. This task presents unique challenges due to asynchronous nature and strict real-time requirements of event data. Traditional time series analysis methods provide a computationally efficient framework suitable for this task. The foundational work [40] first established key principles for time series segmentation, though not specifically for event data. Then, a work [41] advanced the field by applying Hidden Markov Models (HMM) to time series segmentation, demonstrating improved pattern recognition capabilities. Further developments came from a work [42], which introduced change-point detection techniques for identifying statistical shifts in time series data. While traditional time series methods, such as HMM [41] and change-point detection [42], provide theoretical foundations, they are not optimized for the sparse and asynchronous nature of event data, necessitating specialized approaches for MAV motion segmentation. Our motion segmentation model process event streams by first converting them into event-frame representations. For each event frame, we calculate the total number of events to construct a one-dimensional temporal sequence. This time sequence enables us to effectively segment MAV motion sequence in event stream signal, achieving robust, real-time segmentation for MAV motion sequences. III. THE PROPOSED METHOD The proposed framework comprises three key modules. First, the MAV motion segmentation model processes the event stream captured by the event camera and segments the MAV motion sequences into individual actions. Second, the MAV action recognition model efficiently identifies each segmented MAV action and outputs the corresponding semantic label. Finally, the MAV motion sequence decoding algorithm integrates the motion segmentation and recognition models to achieve robust visual action information decoding among MAVs. A. Event-Based MAV Motion Segmentation To achieve visual communication based on event vision, transforming event signals into visual messages is of critical importance. However, in the communication process among MAVs, both the signal length and the transmitted information are uncertain, making it difficult to design an end-to-end model that can handle these variabilities directly. To address this, we propose a MAV motion segmentation model that processes variable-length event streams to extract semantically meaningful motion segments, enabling reliable MAV visual communication. 1) MAV Motion Segmentation Definition: MAV motion segmentation aims to detect and segment different MAV actions from a variable-length event stream E, determining the precise time intervals t for each MAV motion. Fig. 3 shows MAV action sequence in an event stream E. Then, MAV motion segmentation problem can be formulated in Eq. 1. S = fseg(E) = {(ti s, ti e, ai) | i = 1, 2, ..., N}, (1) where E = {ek = (xk, yk, tk, pk)}M k=1 represents the input event stream, consisting of M events. Each event is defined by its spatial coordinates (xk, yk), timestamp tk, and polarity pk. S is the segmented motion output set, containing N motion segments. In an event sequence, the start time of the i-th action is defined as ti s, the end time as ti e, and the action label as ai. 2) The Proposed MAV Motion Segmentation Model: To clearly present our event segmentation model, our event segmentation model is divided into several steps, including event count extraction, event feature computation, motion segmentation, and motion refinement. Event count extraction involves computing statistical information from the event frame sequence. In order to generate event frames, a fixed 33 millisecond time window is used, facilitating human visual interpretation and providing intuitive results. In segmentation method, algorithm primarily extracts the positive event count Pn negative event count Nn, and total event count Qn = Pn + Nn for n-th event frame. The frame number framen is assigned using the line number, starting from 0. With a total frame count of L, this frame sequence is represented in Eq. 2. framen ∈{0, 1, . . . , L -1}, n = 0, 1, . . . , L -1. (2) This step converts 2D event frames into 1D temporal signals while preserving data integrity and alignment within the event frames. Event feature computation effectively utilizes the extracted frame-wise event count information to analyze the MAV's motion state. To distinguish MAV motion periods from static periods, the model extracts two key features: the Positive-toNegative Event Ratio (PNER), which measures the proportion of positive to total events, and the Event Frame Variance (EFV), which quantifies fluctuations in event counts. PNER can be defined as Rn in Eq. 3. Rn = Pn Qn , Pn ≤Qn. (3) Positive events Pn and negative events Nn typically correspond to different physical phenomena, such as changes in lighting or motion direction. Qn is the total number of event in n-th frame. Rn highlights variations in event polarity, which may indicate specific MAV's motions, such as MAV turning or acceleration. This provides important cues for identifying MAV action start and end points in the time axis of an event steam. As for EFV, it measures the local fluctuations in total event count, reflecting the intensity or stability of MAV activity, which is key to distinguishing static and motion periods from the event stream. The specific EFV (Vn) is defined in Eq. 4. IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 5 Start Empty 1 1 Empty Empty 0 End Time Axis Event Stream Fig. 3. MAV motion event stream. This diagram depicts the motion flow of a complete visual message, where Ak represents predefined MAV motions, Se denotes the MAV static state (Empty signal), and tn indicates the termination timestamp of each MAV motion. From a time-domain perspective, an event signal accumulates progressively from left to right. The final "end" motion signifies the stop of the visual signal. Besides, the x-axis and y-axis represent the spatial domain of the event camera, while the time axis t represents the time domain of the event camera. Vn = 1 W n+⌊W/2⌋ X k=n-⌊W/2⌋ (Tk - ̄Tn)2, ̄Tn = 1 W n+⌊W/2⌋ X k=n-⌊W/2⌋ Tk, (4) where W/2 is the half-window size, and boundary frames are handled by truncation. W is a sliding window of size (default 10 frames). Tk denotes the total event count at frame k (k ∈ {n-⌊W/2⌋, . . . , n+⌊W/2⌋}), while ̄Tn is the mean value of total event number in a window W. Besides, Vn measures the rate of event count variation, with lower values in static periods and higher values in MAV motion periods. To reduce noise, Rn and Vn are smoothed using a 5-frame moving average, which enhances the model's ability to detect MAV motion boundaries. Motion Segmentation step identifies MAV motion segments by applying threshold-based segmentation to the smoothed Rn and Vn. It detects transitions between static and motion periods and generates initial MAV's motion boundaries based on predefined criteria, such as minimum motion duration. To segment MAV motions, event frames in the static state are first identified. In the implementation process, Eq. 5 is applied to label the state of each event frame. Gn = ( 1, if Rn 100 then 13: Actionk ←Seg.segment(De f) 14: for each ai in Actionk do 15: ri ←Rec.predict(ai) 16: if ri ! = "background" then 17: Append ri to Cseq 18: end if 19: end for 20: if Cseq[-1] == "end" then 21: Mdec ←decode(Cseq) 22: break 23: end if 24: end if 25: end while 26: Mdec ←decode(Cseq) 27: return Cseq, Mdec 28: Function decode(Cseq) 29: if length(Cseq) < 7 or Cseq[0] ̸= "start" then 30: return [ ] 31: end if 32: direction ←Cseq[1] 33: angle ←Cseq[2 : 5] 34: distance ←Cseq[5 : 7] 35: Mdec ←[direction, angle, distance] 36: return Mdec IV. EXPERIMENTS This section provides the evaluation of our proposed model. We first outline the experimental details, which include information about the dataset, parameter settings, and experimental environment2. Then, our comparative experiments assess both MAV action recognition (accuracy and speed) and motion segmentation performance separately. Additionally, we conducted key ablation experiments on the proposed action recognition model to offer a detailed analysis of the contributions of its critical components. Finally, we describe several experiments on the visual communication of common information to validate the effectiveness of our entire MAV-based visual communication system. 2Dataset and code will be made publicly available upon publication. IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 8 LightHouse Positioning System Crazyflie Event Camera PC for MAV motion control in data collection Action Wireless Communication Protocol Raspberry Pi w : world : the origin point : the X-axis : the Y-axis : the Z-axis Fig. 5. MAV motion data collection system. The figure lists the necessary data collection sensors and key processing devices. A. Experimental details To validate the proposed framework, a controlled MAV motion data collection system in an indoor environment is designed to evaluate visual communication behavior and motion segmentation performance between MAVs, as shown in Fig. 5. In the figure, the hardware consists of Crazyflie drones in the center of the experimental scene, a DVS Micro Explore event camera on a tripod to capture event data, a Lighthouse system overhead for precise 3D localization, and a PC with an NVIDIA GPU 4090 running Ubuntu for data processing. The DVS camera recorded event streams in aedat4 format, which are then processed using the DVS process library. During the collecting process, the data is collected at three sensor observation distances (0.9 m, 1.2 m, and 1.5 m), representing short, medium, and long ranges, respectively, using an "Observe" and "Motion" feedback loop, in which the camera captured the MAV's motions while the PC processed the data for MAV control and analysis. The dataset consists of event streams capturing MAV motions at these three scales, with each scale including five action categories: Action "0", Action "1", Action "Start", Action "End", and Action "background". Particularly, the MAV motion background signal helps the model distinguish between normal signals and non-effective signals (included empty and noise signals), enabling robust action classification. As for the MAV motion segmentation model, we implemented our method using Python 3.10 with the NumPy library. The model's performance was evaluated using center point error (the temporal deviation of predicted action midpoints) and 2D Intersection over Union (IoU) error (the overlap ratio between predicted and ground-truth action segments) to ensure a fair evaluation of the MAV motion segmentation model's performance. With regard to the MAV action recognition model, we built the framework using PyTorch and the SpikeJelly library, leveraging their capabilities for processing event-based data. The recognition model is evaluated using classification accuracy, model parameter size, inference speed, and power consumption, ensuring a fair and impartial assessment that guides the design of a more efficient and robust MAV visual communication system. Fig. 6. Inference time comparison with a batch size of 10. Each result represents the average runtime over 100 runs on an RTX 4090 GPU. B. MAV Action Recognition and Inference Speed Experiment To evaluate the effectiveness of the proposed MAV action recognition approach, we conducted a MAV action recognition experiment using event data with other state-of-the-art model ( spike-C [45], DVS-G [46], spike-R [47]). The test results are presented in Table I. TABLE I PERFORMANCE COMPARISON ACROSS DIFFERENT OBSERVATION DISTANCE DATASETS. Algorithms Short Medium Long spike-C [45] 94.63% ± 1.82% 93.00% ± 1.64% 91.49% ± 8.2% spike-R [47] 96.49% ± 0.47% 94.54% ± 0.24% 92.35% ± 1.23% DVS-G [46] 95.36% ± 1.4% 94.12% ± 0.6% 91.82% ± 1.56% Ours 96.51% ± 0.61% 95.37% ± 1.01% 94.98% ± 1.17% As demonstrated in Table I, our model outperforms other state-of-the-art methods across the short, medium, and long scenarios in terms of recognition accuracy, underscoring its advanced capabilities to a significant extent. Notably, all models exhibit a decline in accuracy in the long-distance scenario, likely due to reduced event density and increased noise at greater observation distances. Nevertheless, our model maintains a competitive recognition performance on the long dataset (1.5 m observing distance). Besides, we also present the accuracy and inference time of MAV action recognition on the Medium dataset in Fig. 6. Compared to other models, ours model achieves the lowest latency of 1.26 ms per 10 test samples, demonstrating its reliability and efficiency under demanding conditions. To evaluate the effectiveness of our proposed model for recognizing MAV action data in event form, we conduct experiments in dataset with a 3:1 training-to-testing ratio. The primary motivation is to assess the model's tailored design, which is intended to enhance temporal precision and robustness in processing event-based MAV motion sequences for realtime applications. To this end, we visualized the training and testing performance over epochs to demonstrate the model's convergence and generalization capabilities on MAV action recognition task. Specifically, Fig 7 illustrates the proposed model's training and testing performance in detail and Fig 7(a) shows the accuracy change over 40 epochs, highlighting IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 9 (a) Training and Testing Accuracy. (b) Training and Testing Loss. Fig. 7. Training and Testing Performance Over Epochs. (a) Training (green) and testing (red) accuracy over 40 epochs. (b) Training (green) and testing (red) loss over 40 epochs. Fig. 8. Comparison of the number of error frames for three methods (Ours, HMM [41], CPD [42]) under different empty signal types: short pause, medium pause, and long pause. The error reflects the misalignment in center point prediction of the MAV action. the model's learning progression and generalization capability on event-based MAV action recognition data. In this figure, training is green curves, and testing is red curves. Fig. 7(b) shows the loss values over 40 epochs, demonstrating stable convergence and reduced overfitting. The training (green) and testing (red) curves confirm the model's robustness for temporal event sequences. Overall, these experimental results validate the efficacy of our proposed model, as the visualizations of accuracy and loss curves demonstrate robust convergence and enhanced generalization, supporting the model's capability to accurately identify MAV in-flight actions. These findings not only confirm the potential of SNN models for event-based action recognition tasks but also pave the way for future optimizations and broader applications in real-world MAV scenarios. C. The Comparison of Motion Segmentation Model Results A robust motion segmentation model is pivotal for ensuring reliable visual communication in MAVs. Inaccurate motion localization by the segmentation model can directly impair the recognition accuracy of subsequent MAV action classification processes. To provide a more objective assessment of the motion segmentation model proposed in this study, a comprehensive, multi-dimensional experimental comparison is Fig. 9. Comparison of total action duration errors for three methods under different empty signal types. The error represents the frame-level difference between predicted and ground-truth MAV action intervals. conducted in following. As illustrated in Fig. 3, an empty signal is inserted between consecutive valid MAV actions during motion message delivery to mitigate overlap and confusion between different actions. In our comparative experiments, we evaluated three key metrics: the offset of the MAV action center point, the total frame number error of the action, and the action overlap area ratio, with HMM [41] and CPD [42]. Furthermore, the model's segmentation performance under varying durations of empty signals was analyzed, providing insights into its robustness and adaptability. The experiments are conducted on three event streams, each containing the same 9 valid MAV actions and 8 empty signals. The durations of the empty signals in three streams are 2.5 seconds, 3 seconds, and 3.5 seconds, respectively. To evaluate the offset error of MAV action centers, we computed the center deviation {Ak}c and its variance {Ak}var c for each action. Lower values indicate higher prediction accuracy. Fig. 8 shows three event streams with different empty signal durations on the horizontal axis and the frame offset of the predicted center on the vertical axis. The results demonstrate that the proposed MAV motion segmentation model achieves more accurate motion localization on all three streams, with both the deviation and variance notably smaller than those of other models. With regard to the total frame number error of the MAV action, we calculated the difference between the predicted duration Apred k and the ground truth duration AGT k for each action. This metric reflects how well different models can localize the temporal span of MAV actions. As shown in Fig. 9, although our segmentation model performs slightly worse than the HMM model under the 2.5-second idle signal condition, it achieves clear advantages when the empty signal duration increases to 3 and 3.5 seconds. This indicates that our model becomes more effective at action duration prediction as the length of empty signals increases. Regarding the MAV action overlap area ratio metric, we analyze the start and end position of MAV actions to compute intersection ∩and union ∪of these actions. The calculation is expressed in Eq. 9, where Apred represents the motion segment IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 10 Fig. 10. Comparison of intersection-over-union (IoU) errors between predicted and ground-truth MAV actions under different MAV action pause intervals. predicted by the model, and Agt denotes the ground truth motion segment. The experimental results are presented in Fig. 10. In the figure, the IOU test results of our proposed model consistently outperform others, demonstrating the superiority of the model proposed in this study. Apred = [startpred, endpred], Agt = [startgt, endgt], IOU = |Apred ∩Agt| |Apred ∪Agt|. (9) To intuitively demonstrate the robustness of the proposed motion segmentation model, we visualize the MAV action segmentation results on MAV action sequences under three different durations of the MAV pause signal, as shown in Fig 11. When the pause signal duration is 2.5 s, the model incorrectly segments 9 valid MAV actions into 8. This error likely stems from insufficient temporal separation between actions, causing the algorithm to fail in detecting clear MAV motion boundaries due to weak or ambiguous temporal features. However, within 3 s and 3.5 s pause signal, the proposed model accurately detects the start and end of each MAV action. D. Ablation Study on The Proposed Recognition Model To gain a deeper understanding of the proposed MAV action recognition model, an ablation study is conducted in this section. Specifically, we analyzed the effects of spike window length F and event frame resolution on our spiking-based MAV motion recognition model. The detailed experimental results are shown in table II and III. In table II, several experiments are conducted on the impact of motion scale using a medium-range MAV motion dataset. By varying input resolutions, we evaluated their effect on the performance of the MAV action recognition performance at an input size of 128 × 128. Larger-scale event frames improve accuracy by capturing finer spatial details, but they increase computational load and power consumption due to higher-dimensional feature processing. However, larger scales also increase computational load, parameter count, and power The number of events Frame Number (a) MAV action sequence with 2.5s pause signals. The number of events Frame Number (b) MAV action sequence with 3.0s pause signals. The number of events Frame Number (c) MAV action sequence with 3.5s pause signals. Fig. 11. Comparison of segmentation performance under different MAV pause durations. The red dashed lines denote the ground truth boundaries of each action, and the translucent green boxes indicate the predicted start and end positions by the proposed segmentation model. consumption. Thus, in real MAV motion signal recognition, an appropriate event frame resolution can be flexibly selected to balance recognition accuracy with computational efficiency. TABLE II IMPACT OF INPUT RESOLUTION ON RECOGNITION PERFORMANCE. Resolution Accuracy ACs(G) MACs(G) Params(M) Energy(mJ) 128x128 95.37% 0.6849 0.424 1.205 2.567 64x64 92.28% 0.1636 0.062 0.418 0.432 32x32 81.79% 0.043 0.015 0.222 0.108 In table III, several experiments on the effect of event timestep F are performed using the same medium-range MAV motion dataset. Timestep F represents the sampling frequency of the time window for MAV motion event streams, encoding them into temporal spike signals for input into our proposed SNN model. By varying F (set to 4, 8, and 16) in ablation studies with equally spaced sampling, we investigated its impact on MAV action recognition accuracy. The results indicate that smaller F significantly reduces computational cost and power consumption but also decreases recognition accuracy. This occurs because a smaller F lowers the sampling frequency of MAV motions, limiting the input spike sequence. IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 11 Consequently, even with identical model parameters, recognition performance notably declines in MAV action recognition. TABLE III IMPACT OF TIME STEP F ON RECOGNITION PERFORMANCE. Time Step Accuracy ACs(G) MACs(G) Params(M) Energy(mJ) F=16 95.37% 0.6849 0.424 1.205 2.567 F=8 91.98% 0.3346 0.1247 1.205 0.875 F=4 83.95% 0.1684 0.0623 1.205 0.438 Besides, ablation studies are also conducted on the impact of positive and negative events. As shown in the table IV, using only positive (E-PosNet) or negative events (E-NegNet) fails to achieve optimal action recognition performance. The best accuracy is obtained when employing the E-BiPolNet approach (combining positive and negative events for MAV action classification). TABLE IV ABLATION PERFORMANCE ANALYSIS IN DIFFERENT EVENT POLARITY. Algorithms Short Medium Long E-NegNet 94.12% 91.49% 89.05% E-PosNet 96.32% 94.33% 93.43% E-BiPolNet 96.85% 95.37% 94.24% E. Visual Communication in Flight Tests To validate the feasibility of the proposed motion-based MAV communication framework, flight tests of MAVs are conducted in this section3. Similar to other communication systems [48], we design three different 8-bit communication codes and, based on these codes, used the "cfclient" control library from the Crazyflie package to program corresponding MAV action sequences. An event camera is used to observe the MAV's movements and interpret the control information conveyed through the action sequences. The specific meanings of the 8-bit communication codes are shown in Table V. In each code, the first bit indicates the end of the signal. The bits in between represent, respectively, the flight direction (1 bit), the flight heading angle (3 bits), and the flight distance (2 bits). TABLE V MAV FLIGHT SIGNAL ENCODING SCHEME. Segment Bit Range Meaning Example Start Flag Fixed Start of signal start Direction 1 bit (bit 1) 0 = Forward, 1 = Backward 0 (Forward) Heading 3 bits (bits 2-4) Heading angle, each step α◦ 000 = 0 × α◦ Distance 2 bits (bits 5-6) Flight distance 01 = 0.2 m End Flag Fixed End of signal end To further validate the effectiveness of the proposed visual communication framework, we designed flight communication experiments involving multiple MAVs. In flight tests, two MAVs are deployed: MAVe (Executor MAV, which executes navigation commands) and MAVp (Performer MAV, which transmits messages via motion). Performer MAVp enacts motion patterns to convey messages. Upon receiving the message via the event camera using the proposed method, the 3A video showcasing the proposed framework and MAV flight tests is provided as supplementary material. Fig. 12. Visual communication in flight tests. The Performer MAV (MAVp) transmits the visual message to guide the Executor MAV (MAVe) to three different trajectories. signal is decoded and converted into MAV control commands. These commands are then transmitted to the Executor MAVe, which subsequently navigates to one of three predefined target destinations, Dp1 to Dp3. We designed three distinct 8-bit communication codes, as shown in Table VI, corresponding to the target destinations Dp1 to Dp3. Ultimately, the proposed method successfully decoded all predefined communication codes, with the decoding results illustrated in Fig. 13. Upon successful code recognition, the Executor MAVe executed the control commands and reached the corresponding target destinations. To this end, we recorded the 2D flight trajectories of the Executor MAVe, as shown in Fig. 12. The results demonstrate that the proposed method not only successfully decodes visual motion information but also accurately controls the Executor MAVe to reach the designated positions from Dp1 to Dp3 . TABLE VI THE EXPERIMENTAL COMMUNICATION CODE IN THREE FLIGHT TESTS. Communication code Begin Direction Angle Distance Stop S000010E s 0 000 10 e S000110E s 0 001 10 e S001010E s 0 010 10 e V. CONCLUSION This paper presents a novel MAV communication paradigm inspired by non-verbal communication strategies in nature, where motion sequences are used to transmit messages between MAVs. Specifically, MAV information is encoded into binary-coded aerial movements observed by event cameras, enabling successful inter-MAV message decoding and forwarding. By categorizing MAV motions into four primitive types with assigned communication symbols, we establish a fundamental visual communication framework for sharing critical swarm coordination data (e.g., heading, and distance). To achieve efficient visual message transmission, an eventframe-based motion segmentation method using mathematical statistical features is proposed, enabling precise segmentation IEEE TRANSACTIONS ON ROBOTICS, SUBMITTED 12 The number of events Frame Number start 0 0 0 0 end 1 0 (a) MAV action sequence in trajectory 1. The number of events Frame Number start 0 0 0 0 end 1 1 (b) MAV action sequence in trajectory 2. The number of events Frame Number start 0 0 0 0 end 1 1 (c) MAV action sequence in trajectory 3. Fig. 13. The experimental results of the three flight tests. of MAV action sequences. Furthermore, we design EventMAVNet, a shallow Spiking Neural Network, that outperforms existing event-based action classifiers in both inference speed (1.26 ms / 10 samples) and recognition accuracy (95.37% in the medium dataset). 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2510.14768
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Leveraging Neural Descriptor Fields for Learning Contact-Aware
Dynamic Recovery
Fan Yang1, Zixuan Huang1, Abhinav Kumar1,
Sergio Aguilera Marinovic2, Soshi Iba2, Rana Soltani Zarrin2, Dmitry Berenson1
1University of Michigan, Ann Arbor
2Honda Research Institute USA
Abstract— Real-world dexterous manipulation often encoun-
ters unexpected errors and disturbances, which can lead to
catastrophic failures, such as dropping the manipulated object.
To address this challenge, we focus on the problem of catching
a falling object while it remains within grasping range and,
importantly, resetting the system to a configuration favorable
for resuming the primary manipulation task. We propose
Contact-Aware Dynamic Recovery (CADRE), a reinforcement
learning framework that incorporates a Neural Descriptor Field
(NDF)-inspired module to extract implicit contact features.
Compared to methods that rely solely on object pose or point
cloud input, NDFs can directly reason about finger-object
correspondence and adapt to different object geometries. Our
experiments show that incorporating contact features improves
training efficiency, enhances convergence performance for RL
training, and ultimately leads to more successful recoveries.
Additionally, we demonstrate that CADRE can generalize zero-
shot to unseen objects with different geometries.
I. INTRODUCTION
Robots performing real-world manipulation tasks can en-
counter unexpected disturbances and modeling errors, which
can lead to catastrophic failure, such as dropping the manipu-
lated object. While significant progress has been made in im-
proving robustness against uncertainty and disturbances [1],
[2], [3], real-world applications often present unpredictable
disturbances and unmodeled factors that exceed the sys-
tem’s designed robustness tolerance. For instance, during a
screwdriver-turning task, a stuck screw can generate unex-
pectedly large torques on the screwdriver, which can cause
the fingers to slip and the object to fall. This issue is further
exacerbated in robotic systems without tactile sensing, where
the absence of direct contact feedback makes it challenging
to detect whether the object is being grasped firmly.
In this work, we focus on dynamic recovery, specifically,
catching falling objects before irrecoverable failure occurs.
Rather than improving the inherent robustness of primary
manipulation policies, we propose a complementary strategy
that incorporates a fallback catching policy; i.e. the robot
switches from the primary manipulation policy to the catch-
ing policy when object dropping is detected. This policy
is responsible for catching falling objects during failure
events, and, importantly, resetting the robot system to states
from which the primary manipulation can resume. This is
because merely catching falling objects is not enough to
This work was supported by Honda Research Institute USA.
Fig. 1: CADRE recovers from object-dropping failures by
catching the object and resetting to states favorable for
resuming the primary manipulation task. Leveraging contact
information from a pre-trained NDF model, it generalizes
recovery behaviors to objects with different geometries.
ensure successful recovery. Specifically, while power grasps
can effectively catch a large variety of objects [4], [5], the
primary manipulation task often requires specific grasp types,
such as precision grasps [6], [7], [8]. Switching from power
grasps to precision grasps presents significant challenges.
Therefore, our recovery policy is designed to achieve grasp
configurations that support the contact requirements of the
primary manipulation task. Additionally, to ensure practical
applicability, the recovery policy must be able to adapt to
objects with different geometries.
To address the challenges of recovering with a known
desired grasp, we present Contact-Aware Dynamic Recovery
(CADRE), a reinforcement learning approach that incor-
porates contact information in its observation space. Our
work is based on the importance of contact in dexterous
manipulation [9], [10], [11], [12]. CADRE is motivated
by the insight that maintaining consistent contact behaviors
across different object geometries is one of the fundamental
factors for successful generalization. To achieve this capa-
bility, contact information is derived from Neural Descriptor
Fields (NDF) [13], [14], which captures the geometric cor-
arXiv:2510.14768v1 [cs.RO] 16 Oct 2025
respondence between 3D coordinates and the object point
clouds. CADRE leverages NDF features as implicit contact
information for dexterous manipulation. NDF features of
a predefined set of keypoints on the hand are used to
characterize the grasp configuration. This approach provides
comprehensive contact modeling for both regions that should
be in contact (e.g., fingertips) and regions where contact
should be avoided (e.g., palm).
Our main contributions are summarized as follows: (1) We
propose the problem of recovery through catching, where a
robot must not only catch the falling object but also achieve
grasp configurations from which the robot can seamlessly
resume the primary manipulation task; (2) We develop an
NDF-based implicit contact representation for contact-rich
dexterous manipulation that effectively captures the geomet-
ric correspondences between the hand and the manipulated
object; (3) We present a reinforcement learning framework
for dynamic recovery that leverages this representation to
achieve successful grasps and favorable states for subsequent
manipulation tasks; (4) We demonstrate empirically that our
contact representation enables effective generalization across
different geometries in dynamic recovery tasks.
Our experimental results demonstrate that our contact-
aware approach significantly improves recovery performance
on training objects while enabling zero-shot generalization to
unseen objects of the same type but with different geometries
(e.g. various sizes of screwdriver). Please see more details at
https://cadrecatching.github.io/.
II. RELATED WORK
A. Dynamic Manipulation
Dynamic manipulation involving rapid robot and object
motion has been a popular research area in robotics [15],
[16], [17], [18], [19], [20]. Within this topic, catching fast-
moving objects is a particularly relevant subdomain to our
work. Prior work has explored both planning-based ap-
proaches [21], [22], [4], [23] and RL-based methods [24],
[25], [26], [27] for object catching. However, these methods
primarily focus on stable catching but without consideration
of grasp configurations, which converges to using power
grasps in most cases. Moreover, those methods do not
consider the requirements of subsequent manipulation tasks.
In contrast, our work addresses the more difficult challenge
of recovery through catching, where the robot must not only
catch falling objects but also achieve grasp configurations
that enable seamless resumption of the primary manipulation
task. Additionally, in our recovery task, the object spends
considerably less time in the air, demanding faster and more
dynamic arm motions.
B. Representation Learning for Manipulation
The choice of perception representation (e.g., point cloud,
image, contact information) to input into RL policies sig-
nificantly impacts the manipulation performance. A majority
of research directly uses point clouds as the representation
for the 3D scene [28], [29], [30], [31], [32]. Recent work
has also explored more sophisticated representations. For
instance, Wu et al. [33] encode image observations into a
learned latent space for model-based RL, while Driess et
al. [34] utilize Neural Radiance Fields [35] to obtain latent
scene embeddings for RL policy inputs. However, those
methods do not consider contact-rich dexterous manipulation
scenarios.
Neural Descriptor Fields (NDFs) and similar implicit
geometric representations have also made progress in robot
manipulation. Khargonkar et al. [36] propose an implicit
grasp feature for cross-object and cross-robot generalization,
while several other works
[37], [38], [14] apply NDF-
inspired representations to motion planning. However, the
application of such implicit geometric representations to RL
and dynamic dexterous manipulation remains unexplored.
Our work addresses this gap by leveraging NDF-inspired
contact representations to enable dynamic recovery.
III. PROBLEM STATEMENT
We focus on the task of catching a falling object for
recovery. Specifically, the recovery problem inherently in-
cludes two objectives: (1) the robot must prevent dropping
the object, and (2) the system should recover to a state from
which the primary manipulation task can be resumed. The
inclusion of the second objective distinguishes our work from
prior literature on catching.
We formulate the dynamic recovery through catching as
a Markov Decision Process (MDP), defined by the tuple
(S, A, p, r, γ), where S is the state space, A is the action
space, p : S ×A →S is the transition function, r : S ×A →
R is the reward function and γ ∈[0, 1) is the discount factor.
The objective is to optimize a policy πθ to maximize the
expected discounted return.
At each time step, the policy receives observations ot :=
{qt, xt, vt}, where qt represents the robot’s joint angles, xt
is the object pose in SE(3), and vt denotes the object’s twist
(linear and angular velocities). We assume access to a low-
level joint position controller; therefore, the robot action at
is defined as the desired joint position at the next time step.
To make the problem tractable, we assume knowledge of
the object’s geometry, represented by a point cloud P, which
can be generated from a scan or a CAD model of the object.
This assumption is reasonable in a factory setting, where the
set of available objects, such as screwdrivers, is usually fixed
and the geometry of each tool can be obtained. The objective
is to move both the robot and the object to desired recovery
configurations ˆq and ˆx, respectively. Obtaining the desired
configurations is not the primary focus of this work and can
be addressed with prior works [39], [40]. For simplicity in
our problem setup, we manually specify ˆq and ˆx.
Additionally, we aim to develop a method that generalizes
across object geometries. During RL training, objects are ran-
domly sampled from a predefined distribution. While during
evaluation, objects are selected from both the in-distribution
set and the out-of-distribution set. An ideal method should
be able to catch the object in contact configurations similar
to those from the training examples, even if the target object
is out of distribution.
The method is evaluated based on (1) whether the robot
successfully catches the object, (2) the difference between
the final state of catching and the desired state, (3) and the
performance of the subsequent primary manipulation task.
Compared to other dexterous manipulation setups, this task
is more challenging because the robot needs to move quickly
to handle dynamic situations, but also precisely to reach the
desired configurations.
IV. METHODS
Our
approach,
Contact-Aware
Dynamic
Recov-
ery(CADRE), leverages reinforcement learning to optimize
recovery policies in dexterous manipulation tasks. To enable
contact awareness and generalization to unseen objects
based on contacts, we incorporate contact features derived
from NDFs into the observation space. The NDF model
extracting contact features is pretrained and remains fixed
during RL training. The contact features enable the RL
policy to achieve consistent behaviors across objects with
different geometries.
A. Preliminary: Neural Descriptor Fields
Neural Descriptor Fields (NDFs) [13] is a learned repre-
sentation that captures geometric correspondence between a
queried 3D coordinate and an object. Given an object point
cloud P, NDF extracts an n-dimensional geometric feature
for any 3D coordinate p:
fNDF (p|P) : R3 →Rn.
(1)
During NDF training, an MLP-based network Φ(p|E(P)) is
trained to predict occupancy or signed distance of the query
point p, conditioned on the object point cloud P. E is the
PointNet-based encoder, which extracts the latent features of
the object point cloud. The NDF feature is defined as:
fNDF (p|P) :=
L
M
i=1
Φi(p|E(P)),
(2)
where ⊕denotes concatenation of activations from each layer
of Φ, and L is the number of hidden layers. This feature
serves as an implicit occupancy representation and provides
rich geometric information about the geometric relationship
between the queried point and the object surface.
In our implementation, Φ(p|E(P)) is a double-headed
model that predicts both occupancy and signed distance,
similar to [38]. While occupancy prediction mainly captures
features for points near contact regions, adding additional
signed distance prediction helps capture features at points
which are not in contact, which facilitates RL training when
exploring non-contact regions. In our method, the NDF
model is pretrained and kept fixed during RL training.
B. Contact-Aware Dynamic Recovery
1) Contact-Aware Grasp Feature: Recovering with an
expected grasp primarily requires accurate modeling of fin-
ger–object contact. Beyond this, it is highly desirable for the
contact representation to generalize across object geometries,
as such generalization enables the robot policy to exhibit
zero-shot generalization to unseen objects.
In this work, we leverage NDFs to characterize contact
features for dexterous manipulation. By querying points on
the hand, we can interpret the corresponding NDF features as
indicators of contact: contact points are expected to lie near
the decision boundary of the occupancy function, where the
NDF occupancy prediction output transitions between inside
and outside predictions.
While NDF provides per-point contact features, it does
not directly provide information characterizing the contact
features of a grasp. To address this, we predefine K key
points on the hand {pki
i }K
i=1, where pki
i denotes the position
of the i-th key point in the ki-th link frame. The grasp feature
g(q, x|P) is defined as the concatenation of the NDF features
of all the contact points:
g(q, x|P) :=
K
M
i=1
fNDF (T(x)ffk(pki
i , q, ki)|P),
(3)
where q is the robot joint angles, x is the object pose in
SE(3). ffk denotes the forward kinematics function return-
ing key point locations in the world frame, and T(x) is the
transformation from the world frame to the object frame.
For key point selection, we sample points from each hand
link rather than restricting them to the fingertips. We use
the root of each link as the key point in our experiments.
Although fingertips are often primarily involved in contact,
non-contact parts also play a critical role in characterizing
the grasp feature. For example, one of the major distinctions
between a power grasp and a precision grasp is whether the
palm is in contact with the object. In practice, we assign
one key point for each link for computational efficiency.
Moreover, it is not necessary to place key points strictly
within potential contact regions, since NDF features encode
not only whether a key point is in contact but also its distance
from contact.
In summary, CADRE receives observations ot and com-
putes the contact-aware grasp feature g = g(q, x|P) at every
time step. The input into the RL policy is the combination
of observations and grasp features.
2) Reinforcement Learning for Dynamic Recovery: Due
to the complexity of modeling the dynamic interactions
between the hand and the object, we choose to use Proximal
Policy Optimization (PPO) [41], a model-free RL method,
to optimize the dynamic recovery policy.
Reward Function: The reward function is defined as
follows:
r :=r˙a + rtorque + renergy + rdrop + robj v
+ robj pose + rq + rcontact + rsafety,
(4)
where we omit the weighting parameters and the time step
subscript for simplicity. Specifically, r ˙a := −||at −at−1||2,
rtoruqe := −||τ||2, and renergy := −||τ T ˙q||2 regularizes
non-smooth actions, large torque and large energy consump-
tion, respectively. τ is computed from the joint position
controller output. rdrop := −1drop(x), where 1drop is an in-
dicator function that returns 1 if the object’s position exceeds
Primary manipulation
Drop detected ?
Object geometry
Predefined key points
Pretrained
NDF model
Contact-aware
grasp feature 𝑔𝑡
𝑞𝑡
𝑥𝑡
𝑣𝑡
𝑔𝑡
Observations
PPO policy
Actions 𝑎𝑡
1
Train offline,
Freeze online
2
Online RL
training
Yes
Fig. 2: Contact-Aware Dynamic Recovery (CADRE) aims to catch a manipulated object when it falls from the grasp and
recover to states that support the resumption of the manipulation task. CADRE leverages a pretrained NDF model to
extract implicit contact features from the grasp. The contact-aware grasp features are incorporated into the RL observations,
enhancing the policy’s awareness of contact and improving recovery performance.
a predefined workspace boundary. robj v := exp(−||v||2) +
exp(−||ω||2) encourages low linear velocity v and angular
velocity ω of the object. We shape robj v with an exponential
function as the velocities can yield excessively large mag-
nitudes during training (e.g., the object is falling fast). The
exponential function helps bound the reward and leads to
more stable training. rq := exp(−||q−ˆq||2) and robj pose :=
exp(−fpos(x, ˆx))+exp(−forn(x, ˆx)) encourage the robot to
recover to the desired robot configuration ˆq and object pose
ˆx. ˆq and ˆx are predefined and considered favorable for the
primary manipulation task (see Sec. V-A for further discus-
sion). fpos and forn computes the position and orientation
difference, respectively. rcontact := 1contact(q, x) rewards
the robot for achieving task-relevant contact with the object.
The desired task-relevant contact can be obtained from ˆq
and ˆx (see Sec. V-A). rsafety := −1table(q) penalizes the
unsafe behavior of robot contacting the table.
Domain Randomization: Since accurately perceiving
fast-moving objects can be challenging in real-world scenar-
ios, catching dynamic objects for recovery needs to handle
significant uncertainties. To address this, we apply domain
randomization [1] to improve robustness. Specifically, we
introduce external disturbances and perception noise: at each
time step, a random external wrench wext is applied at the
center of the screwdriver, and random perception noise ϵ
sampled from a uniform distribution is added to the object’s
pose and velocities.
V. EXPERIMENTS
We evaluate our method on two tasks: screwdriver recov-
ery and nut recovery. We aim to design our experiments to
answer (1) whether incorporating implicit contact features
can facilitate RL training and improve catching performance;
(2) whether CADRE generalizes to unseen objects sampled
from a different distribution; and (3) whether CADRE can
be deployed on robot hardware.
A. Simulation Experiment Setup
We use IsaacSim to simulate the recovery task. Our robot
consists of an Allegro Hand [42] mounted on a 7-DoF
KUKA iiwa arm. We use PPO [41], implemented from RL
games [43], to optimize the recovery policy. To obtain the full
point cloud of the object, we assume access to its geometry
(as noted in Sec. III), since capturing a complete point cloud
with a camera is impractical. We uniformly sample points
from the object surface and transform them according to the
object pose to generate the full point cloud observation. The
same approach is used in our hardware experiments.
Object generation: During RL training, we randomly
sample 50 objects from a predefined shape distribution.
Specifically, the objects are generated using the same geo-
metric parameterization method, but with variations in their
size parameters. During training for the screwdriver task we
add five real screwdrivers to support our real-world experi-
ments. For evaluation, we consider both in-distribution (ID)
and out-of-distribution (OOD) objects. ID objects consist of 5
new unseen objects sampled from the same distribution used
for training, while OOD objects are sampled from a different
distribution as a more challenging test for generalization. The
NDF model fNDF is trained with objects sampled from the
same distribution for RL training. Thus, OOD objects for the
RL policy will also be OOD for the NDF model.
Evaluation metrics: We consider the following metrics:
(1) Catch success rate: a catch is considered successful if the
object remains within a predefined bounding box throughout
the episode; (2) Reward: cumulative reward over an episode;
(3) Number of desired contacts; (4) Number of undesired
contacts; (5) Position error: the Euclidean distance between
the final and the desired object position; and (6) Orientation
error: defined as the angle between the final and desired z-
axes of the object frame, since the objects in our experiments
are rotationally symmetric about the z-axis. Metrics (3) and
(4) are task-specific and detailed in Sec. V-A.1 and V-A.2.
Metrics (2)-(6) are only calculated for successful trials.
As the most direct evaluation involves initializing the
primary manipulation task with the final recovered states, we
also introduce a screwdriver turning task and a simplified
nut mating task to assess whether the post-recovery states
are suitable for downstream manipulation tasks. See details
in Sec. V-A.1 and V-A.2.
1) Screwdriver Recovery: We set up the recovery task
in a screwdriver turning scenario with a precision grasp.
Screwdriver turning with a precision grasp has been widely
studied [44], [6], [45], [46], [47]. A precision grasp is
preferred, as opposed to a power grasp, as the choice of
robot-screwdriver contact significantly affects the turning
performance, highlighting the importance of contact rea-
soning. Specifically, we follow the setup presented in [46]
(See Fig. 3a), where the index finger contacts the top of
the screwdriver, while the thumb and middle finger form an
antipodal grasp on the handle. To define a desired precision
grasp, we specify the desired object configuration ˆx such
that the screwdriver is upright with its tip in contact with
the supporting table. For the desired robot configuration
ˆq, we maintain the same hand configurations across dif-
ferent screwdrivers, while adjusting the arm configuration
accordingly to ensure the index finger rests on top of the
screwdriver. See the visualization in Fig. 3a.
To extract the desired task-relevant contact, we compute
the finger–object distance in the desired configuration. Fin-
gers with a distance below 2 cm are classified as task-
relevant contacts, corresponding to the index, middle and
thumb finger in contact with the screwdriver. The undesired
contact is defined as the contact between all other links of
the hand with the screwdriver. The contact reward rcontact
returns the number of desired contact points.
We model the screwdriver as two connected cylinders: one
for the handle and one for the shaft. OOD screwdrivers have
approximately half the length of the ID screwdrivers.
To evaluate whether the recovered states are favorable
for screwdriver turning, we follow the setup from [46],
attempting to turn the screwdriver 60◦. We assume there
exists a motion planning algorithm to mate the screwdriver
with the screw and reorient it perfectly upright. In practice,
we record the final state of recovery, keep the joint angles
and the relative transformation between the hand and the
screwdriver fixed, but we transform both the hand and the
screwdriver so that the screwdriver is upright. We evaluate
the turning performance via the turning drop rate and the
object orientation difference between the final turning state
and the desired state. We consider the screwdriver dropped
if its Euler angles exceed a predefined threshold.
2) Nut Recovery: We consider recovery for nut–bolt mat-
ing (See Fig. 3b). During mating, the robot can easily drop
the nut as the nut mating involves forceful contacts and high
uncertainty, especially if tactile sensors are not available.
As in the screwdriver task, successful recovery requires the
robot to catch the nut with a specific grasp. In this case, we
define ˆq and x such that all fingertips make contact with the
side of the nut while avoiding its top and bottom. Contacts
on the top or bottom will block the nut’s central hole, making
it impossible to mate with the bolt. We use the same method
(a) Screwdriver
(b) Nut
0.0
0.5
1.0
1.5
2.0
2.5
3.0
# Env Interaction Steps
1e8
0
5
10
15
Total Episode Reward
Screwdriver
Ours
Pose
Point Cloud
DexPoint
(c) RL training curves for screw-
driver recovery
0.0
0.5
1.0
1.5
2.0
2.5
3.0
# Env Interaction Steps
1e8
0
5
10
15
20
Total Episode Reward
Nut
Ours
Pose
Point Cloud
DexPoint
(d) RL training curves for nut
recovery
Fig. 3: Fig. (a) and (b) show the environment step. Three
different seeds are used for RL training and the average
results are shown in Fig. (c) and (d). The variance is
relatively small as we use a large batch size during training.
above to extract task-relevant contacts, which corresponds to
all fingertips in contact with the nut for this task. Any other
robot-nut contacts are then undesired contacts.
Additionally, there are cases where the robot appears to
stabilize the nut by using the external support from the bolt.
For instance, the robot might push the nut towards the bolt
while the grasp itself is not stable without the support from
the bolt. We penalize such behaviors in the contact reward:
rcontact := rrobot nut −rnut bolt, where rrobot nut returns
the number of desired contacts and rnut bolt is the indicator
function for nut-bolt contact.
The nut is modeled as a hexagonal prism with a cylindrical
hole, and the bolt is modeled as a cylinder. OOD nuts have
about half the thickness of the ID nuts, making them harder
to catch and requiring more precise finger control.
A recovery is considered successful if the nut is not
dropped, and the nut does not contact the bolt, as the robot
must stably grasp the nut without the bolt’s support.
To evaluate grasp quality for the primary nut-mating task,
we design a simplified experiment, since nut-mating with a
dexterous hand remains a challenging open research problem.
Specifically, we use a bolt with a smaller radius and consider
the task successful if the nut can be placed such that the bolt
passes through its center hole. This simplification primarily
tests whether the robot’s fingers obstruct the nut’s center
hole, a necessary condition for successful nut-mating. During
the experiment, we assume the finger configurations remain
fixed and use motion planning to move the arm.
3) Baselines and Ablation: We consider two RL baselines
and an ablation method. All methods use the same reward
functions and RL training setup as CADRE, but represent the
object geometry and contact differently: (1) Ablation: PPO
with object pose observations (Pose): it directly takes the
observation ot defined in Sec. III as input. This method does
not have access to object geometry; the only object-related
information available is pose and velocity. (2) Baseline:
PPO with point cloud observations (Point Cloud): Re-
inforcement learning with point cloud input has been widely
used in dexterous manipulation [30], [31], [32]. Similar to
these methods, we add the object’s point cloud Pt into the
aforementioned observation space: o′
t := {qt, xt, vt, Pt}.We
choose not to use the point cloud obtained from a depth
camera to maintain the same input as our method. This
method has access to the object’s geometry but does not
explicitly reason about contact features. PointNet [48] is
used to extract features from the point cloud. (3)Baseline:
DexPoint [48]: Similar to our method, DexPoint leverages
contact information to improve generalization across object
geometries. Its key component is augmenting the observation
space with an additional imagined point cloud of the robot.
To implement DexPoint, we render a point cloud for each
fingertip and the palm, and concatenate it with the object
point cloud. Unlike the original DexPoint paper, we do not
use the observed point cloud, i.e., the depth-camera point
cloud, since other methods in our experiments do not have
access to such observations. PointNet is also used to encode
the point cloud.
B. Experiment Results
The RL training curves are shown in Fig. 3. CADRE
achieves higher rewards than baselines given the same num-
ber of environment interactions. In the screwdriver recovery
task, although the baseline methods have eventually con-
verged, they fail to match CADRE’s performance. For the nut
mating task, although all methods are nearing convergence,
CADRE consistently outperforms the baselines.
Each method is evaluated for 50 trials per object. The
average performance is reported in Table I across all objects
and all seeds. We also compute the standard deviation of the
50 trials on a given object, and then average the standard
deviation from different objects and different seeds.
Screwdriver recovery: all methods achieve almost 100%
success rates on ID objects. However, CADRE performs
better on most of the other metrics. This suggests that base-
lines often use power grasps rather than the desired precision
grasps to catch the screwdriver. On OOD objects, CADRE
and DexPoint still maintain similar performance, while the
other baselines exhibit a more significant performance drop.
However, we will demonstrate in Table II that grasp pose
of DexPoint is not well suited for the primary screwdriver-
turning task.
To further evaluate the quality of the recovered state, we
set up the screwdriver turning (primary manipulation task)
as described in Sec. V-A.1 and use the turning algorithm
proposed in [46]. Because the turning algorithm is compu-
tationally expensive, we only run it with the best-performing
seed with the highest success rates for each method. We
randomly sample 5 recovery trials per object for the turning
evaluation. The goal is to turn the screwdriver 60 degrees.
The turning results are shown in Table II. According to the
results, CADRE has demonstrated a smaller distance to goal
in ID scenarios and a lower drop rate in OOD scenarios.
The performance drop from ID to OOD of CADRE is much
smaller than that of the baselines.
Nut recovery: we have also observed similar results in
the nut-catching task (See results in Table I), where CADRE
outperforms the baselines in both ID and OOD scenarios
and exhibits a smaller performance drop when switching
from ID to OOD objects. In the downstream simplified nut-
mating task, we sample 50 recovery trials per object from the
best seed for the mating evaluation. Although some baselines
achieve higher mating success rates, CADRE attains nearly
double the catching success rate, resulting in a higher overall
success rate from catching to mating. In particular, baselines
tend to successfully catch the nut mainly in simpler catching
scenarios, which makes it easier for them to achieve higher
mating performance.
C. Hardware Experiments
The main objective of the hardware experiment is to
evaluate whether we can successfully transfer the highly
dynamic recovery behavior learned in simulation to a real-
world setup. We only focus on the screwdriver recovery
task in our hardware experiments. We use an unseen real
screwdriver that falls within the training distribution for
our hardware experiment. We use the Vicon motion capture
(mocap) system to estimate the state of the screwdriver.
1) Experiment Setup: To enable deployment in the real
world, we distill an open-loop policy from the closed-
loop RL policy. Closed-loop execution of a highly dynamic
manipulation policy is particularly challenging on a hardware
system because of (1) Latency: Our system exhibits approxi-
mately 50 ms of latency between sending control commands
and receiving the corresponding observation update. Such
millisecond latency is usually negligible in many quasi-static
tasks, but will lead to instability in our highly dynamic tasks;
and (2) Occlusion: When the robot attempts to catch the
screwdriver, its fingers will often block the mocap markers,
causing loss of object state information.
To distill the open-loop policy, we collect 10,000 success-
ful rollouts from the closed-loop policy in simulation, with
55 randomly sampled objects. We train a supervised open-
loop policy that takes the initial observation as input and
outputs a sequence of actions for the entire trajectory. We
use a transformer decoder architecture for the policy.
One of the main challenges in the hardware experiment is
resetting the system to a state where the screwdriver is about
to fall. As the error detection is not the focus of this work,
we manually create such a falling scenario: we first manually
position the screwdriver so that the robot can grasp it. To
initiate the drop, we command the robot to open its hand for
a short, fixed duration. The recovery policy is then triggered.
To eliminate bias from manually setting the screwdriver’s
initial pose, we anonymize the methods during each trial:
the method under evaluation is randomly selected and not
revealed to the experimenter until the end of this trial.
Task
Obj
method
success↑
reward↑
#contact↑#undesired
contact↓
pos diff(cm)↓
orn diff↓
Screwdriver
ID
CADRE
100.00% 13.22 ±2.31
2.54
0.19
4.34 ±3.86
13.89◦± 12.07◦
Pose
100.00% 10.57 ± 0.68
1.57
1.64
4.89 ± 2.25
10.25◦± 3.95◦
Point Cloud 100.00% 11.10 ± 0.59
1.72
1.78
6.25 ± 2.88
15.29◦± 5.13◦
DexPoint
99.87%
10.70 ± 0.53
2.78
1.05
5.10 ± 2.33
20.11◦± 5.30◦
OOD
CADRE
96.27%
11.09 ± 1.39
1.44
0.08
2.05 ± 1.37
22.74◦± 12.41◦
Pose
73.86%
8.14 ± 2.06
0.96
0.85
8.48 ± 4.57
29.78◦± 20.60◦
Point Cloud
83.73%
9.24 ± 2.13
1.15
1.17
7.33 ± 3.42
24.53◦± 16.43◦
DexPoint
97.86%
9.66 ± 1.20
2.04
1.20
5.67 ± 2.29
21.52◦± 12.55◦
Nut
ID
CADRE
98.53% 20.46 ± 2.25
3.85
0.15
4.08 ± 2.26
5.89◦± 5.77◦
Pose
60.67%
12.69 ± 2.67
1.91
0.26
8.26 ± 2.66
30.99◦± 27.16◦
Point Cloud
75.33%
14.15 ± 2.97
1.66
0.34
6.26 ± 2.72
36.11◦± 25.50◦
DexPoint
72.53%
17.42 ± 2.79
2.88
0.36
5.89 ± 0.85
14.51◦± 18.89◦
OOD
CADRE
91.06%
16.47 ± 4.23
2.92
0.17
6.60 ± 3.89
18.68◦± 19.70◦
Pose
48.40%
12.51 ± 3.32
2.10
0.01
8.19 ± 2.50
28.49◦± 28.91◦
Point Cloud
46.00%
13.04 ± 3.95
1.47
0.09
6.83 ± 3.02
39.27◦± 26.13◦
DexPoint
38.40%
17.78 ± 2.95
2.58
0.08
6.15 ± 1.21
15.31◦± 16.51◦
TABLE I: Evaluation on unseen screwdriver recovery and nut recovery. Results are calculated for successful trials only.
Scrwedriver Turning ID
Scrwedriver Turning OOD
Nut Mating ID
Nut Mating OOD
drop rate↓
dist2goal↓
(no-drop trials)
drop rate↓
dist2goal↓
(no-drop trials)
valid succ↑overall succ↑valid succ↑overall succ↑
CADRE
4%
13.77◦± 17.66◦
4%
35.25◦± 18.80◦
54.3%
53.6%
49.8%
45.2%
Pose
0%
40.99◦± 20.89◦
72%
21.03 ◦±11.04◦
56.9%
28.0%
73.9%
40.8%
Point Cloud
0%
45.31◦± 16.97◦
32%
36.81◦± 20.18◦
6.1%
4.8%
15.2%
8.4%
DexPoint
4%
50.28◦± 20.19◦
28%
37.66◦± 17.79◦
59.1%
46.4%
67.2%
20.8%
TABLE II: Evaluation results for screwdriver turning and nut mating initialized from recovered states. Dist2goal represents
the screwdriver orientation difference between the final state of turning and the desired state of turning 60◦. For nut mating,
we present two success rates: valid success rate, defined as (#successful mating/#successful cathing), and the overall
success rate, defined as (#successful mating/#trials), where the number of trials also includes failed catching attempts.
succ rate ↑
#desired contact
(succ only)↑
#undesired contact
(succ only)↓
CADRE
80.0%
2.83
0.0
Pose
6.7%
1.0
2.0
Point Cloud
0.0%
DexPoint
33.3%
1.4
1.0
TABLE III: Distilled open-loop policies are evaluated for 15
hardware trials per method.
In our experiments, we use a real screwdriver but add fric-
tion tape, as its material—optimized for human grip—does
not provide sufficient friction for the robot’s fingertip mate-
rial.
2) Experiment Results: The quantitative results are shown
in Table III. Our method demonstrates a better performance
in the hardware experiments. Baseline performance is sub-
stantially worse on hardware, likely because the power-grasp
strategy it employs requires faster robot movements, which
are more susceptible to sim-to-real gaps. Nevertheless, the
results still reveal a noticeable sim-to-real gap, indicating
that further improvement is needed before deployment in
production. Potential improvements include reducing action
jerk, obtaining more accurate estimates of joint friction and
robot inertia, and minimizing applied forces for safety. We
leave closing the sim-to-real gap for highly dynamic systems
as future work.
VI. DISCUSSION
While CADRE demonstrates the ability to generalize
across different geometries, the underlying NDF structure
only focuses on geometry information for generalization.
However, variations in geometries will also lead to different
dynamics. For example, a grasp on an unseen object that
shares similar grasp features with a force closure grasp on a
known object might not result in force closure.
Additionally, dynamic manipulation with precise control
introduces significant sim-to-real challenges. Specifically,
Factors such as system latency, gravity compensation errors,
and policy computation time, which are often negligible
in quasi-static scenarios, can have a substantial impact
on performance in highly dynamic recovery tasks. While
CADRE can successfully catch a falling object in our hard-
ware experiment, there remains room for improvement in
achieving more precise desired contact configurations and
recovering from more challenging falling scenarios, e.g., by
making the policy aware of the system latency and dynamics
discrepancies.
Despite the limitations above, our results suggest the
broader potential for RL of using implicit contact rep-
resentations in contact-rich manipulation beyond dynamic
recovery. Further extending this approach to more general
settings presents a promising research direction for our future
research.
VII. CONCLUSION
In this work, we focus on the problem of recovering from
catastrophic failure in dexterous manipulation–specifically,
recovering from dropping objects and returning to states
favorable for resuming the primary manipulation task. We
propose CADRE, an RL framework that utilizes an implicit
contact representation derived from NDFs.
We evaluate CADRE on screwdriver and nut recovery
tasks. Our experiments demonstrate that simply providing
point clouds as object geometry observations for RL is in-
sufficient for learning effective dynamic recovery in contact-
rich scenarios. In contrast, CADRE leverages implicit contact
features to improve training efficiency, recovery quality, and
generalization to unseen object geometries.
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Leveraging Neural Descriptor Fields for Learning Contact-Aware Dynamic Recovery Fan Yang1, Zixuan Huang1, Abhinav Kumar1, Sergio Aguilera Marinovic2, Soshi Iba2, Rana Soltani Zarrin2, Dmitry Berenson1 1 2Honda Research Institute USA Abstract- Real-world dexterous manipulation often encounters unexpected errors and disturbances, which can lead to catastrophic failures, such as dropping the manipulated object. To address this challenge, we focus on the problem of catching a falling object while it remains within grasping range and, importantly, resetting the system to a configuration favorable for resuming the primary manipulation task. We propose Contact-Aware Dynamic Recovery (CADRE), a reinforcement learning framework that incorporates a Neural Descriptor Field (NDF)-inspired module to extract implicit contact features. Compared to methods that rely solely on object pose or point cloud input, NDFs can directly reason about finger-object correspondence and adapt to different object geometries. Our experiments show that incorporating contact features improves training efficiency, enhances convergence performance for RL training, and ultimately leads to more successful recoveries. Additionally, we demonstrate that CADRE can generalize zeroshot to unseen objects with different geometries. I. INTRODUCTION Robots performing real-world manipulation tasks can encounter unexpected disturbances and modeling errors, which can lead to catastrophic failure, such as dropping the manipulated object. While significant progress has been made in improving robustness against uncertainty and disturbances [1], [2], [3], real-world applications often present unpredictable disturbances and unmodeled factors that exceed the system's designed robustness tolerance. For instance, during a screwdriver-turning task, a stuck screw can generate unexpectedly large torques on the screwdriver, which can cause the fingers to slip and the object to fall. This issue is further exacerbated in robotic systems without tactile sensing, where the absence of direct contact feedback makes it challenging to detect whether the object is being grasped firmly. In this work, we focus on dynamic recovery, specifically, catching falling objects before irrecoverable failure occurs. Rather than improving the inherent robustness of primary manipulation policies, we propose a complementary strategy that incorporates a fallback catching policy; i.e. the robot switches from the primary manipulation policy to the catching policy when object dropping is detected. This policy is responsible for catching falling objects during failure events, and, importantly, resetting the robot system to states from which the primary manipulation can resume. This is because merely catching falling objects is not enough to This work was supported by Honda Research Institute USA. Fig. 1: CADRE recovers from object-dropping failures by catching the object and resetting to states favorable for resuming the primary manipulation task. Leveraging contact information from a pre-trained NDF model, it generalizes recovery behaviors to objects with different geometries. ensure successful recovery. Specifically, while power grasps can effectively catch a large variety of objects [4], [5], the primary manipulation task often requires specific grasp types, such as precision grasps [6], [7], [8]. Switching from power grasps to precision grasps presents significant challenges. Therefore, our recovery policy is designed to achieve grasp configurations that support the contact requirements of the primary manipulation task. Additionally, to ensure practical applicability, the recovery policy must be able to adapt to objects with different geometries. To address the challenges of recovering with a known desired grasp, we present Contact-Aware Dynamic Recovery (CADRE), a reinforcement learning approach that incorporates contact information in its observation space. Our work is based on the importance of contact in dexterous manipulation [9], [10], [11], [12]. CADRE is motivated by the insight that maintaining consistent contact behaviors across different object geometries is one of the fundamental factors for successful generalization. To achieve this capability, contact information is derived from Neural Descriptor Fields (NDF) [13], [14], which captures the geometric cor16 Oct 2025 respondence between 3D coordinates and the object point clouds. CADRE leverages NDF features as implicit contact information for dexterous manipulation. NDF features of a predefined set of keypoints on the hand are used to characterize the grasp configuration. This approach provides comprehensive contact modeling for both regions that should be in contact (e.g., fingertips) and regions where contact should be avoided (e.g., palm). Our main contributions are summarized as follows: (1) We propose the problem of recovery through catching, where a robot must not only catch the falling object but also achieve grasp configurations from which the robot can seamlessly resume the primary manipulation task; (2) We develop an NDF-based implicit contact representation for contact-rich dexterous manipulation that effectively captures the geometric correspondences between the hand and the manipulated object; (3) We present a reinforcement learning framework for dynamic recovery that leverages this representation to achieve successful grasps and favorable states for subsequent manipulation tasks; (4) We demonstrate empirically that our contact representation enables effective generalization across different geometries in dynamic recovery tasks. Our experimental results demonstrate that our contactaware approach significantly improves recovery performance on training objects while enabling zero-shot generalization to unseen objects of the same type but with different geometries (e.g. various sizes of screwdriver). Please see more details at https://cadrecatching.github.io/. II. RELATED WORK A. Dynamic Manipulation Dynamic manipulation involving rapid robot and object motion has been a popular research area in robotics [15], [16], [17], [18], [19], [20]. Within this topic, catching fastmoving objects is a particularly relevant subdomain to our work. Prior work has explored both planning-based approaches [21], [22], [4], [23] and RL-based methods [24], [25], [26], [27] for object catching. However, these methods primarily focus on stable catching but without consideration of grasp configurations, which converges to using power grasps in most cases. Moreover, those methods do not consider the requirements of subsequent manipulation tasks. In contrast, our work addresses the more difficult challenge of recovery through catching, where the robot must not only catch falling objects but also achieve grasp configurations that enable seamless resumption of the primary manipulation task. Additionally, in our recovery task, the object spends considerably less time in the air, demanding faster and more dynamic arm motions. B. Representation Learning for Manipulation The choice of perception representation (e.g., point cloud, image, contact information) to input into RL policies significantly impacts the manipulation performance. A majority of research directly uses point clouds as the representation for the 3D scene [28], [29], [30], [31], [32]. Recent work has also explored more sophisticated representations. For instance, Wu et al. [33] encode image observations into a learned latent space for model-based RL, while Driess et al. [34] utilize Neural Radiance Fields [35] to obtain latent scene embeddings for RL policy inputs. However, those methods do not consider contact-rich dexterous manipulation scenarios. Neural Descriptor Fields (NDFs) and similar implicit geometric representations have also made progress in robot manipulation. Khargonkar et al. [36] propose an implicit grasp feature for cross-object and cross-robot generalization, while several other works [37], [38], [14] apply NDFinspired representations to motion planning. However, the application of such implicit geometric representations to RL and dynamic dexterous manipulation remains unexplored. Our work addresses this gap by leveraging NDF-inspired contact representations to enable dynamic recovery. III. PROBLEM STATEMENT We focus on the task of catching a falling object for recovery. Specifically, the recovery problem inherently includes two objectives: (1) the robot must prevent dropping the object, and (2) the system should recover to a state from which the primary manipulation task can be resumed. The inclusion of the second objective distinguishes our work from prior literature on catching. We formulate the dynamic recovery through catching as a Markov Decision Process (MDP), defined by the tuple (S, A, p, r, γ), where S is the state space, A is the action space, p : S ×A →S is the transition function, r : S ×A → R is the reward function and γ ∈[0, 1) is the discount factor. The objective is to optimize a policy πθ to maximize the expected discounted return. At each time step, the policy receives observations ot := {qt, xt, vt}, where qt represents the robot's joint angles, xt is the object pose in SE(3), and vt denotes the object's twist (linear and angular velocities). We assume access to a lowlevel joint position controller; therefore, the robot action at is defined as the desired joint position at the next time step. To make the problem tractable, we assume knowledge of the object's geometry, represented by a point cloud P, which can be generated from a scan or a CAD model of the object. This assumption is reasonable in a factory setting, where the set of available objects, such as screwdrivers, is usually fixed and the geometry of each tool can be obtained. The objective is to move both the robot and the object to desired recovery configurations ˆq and ˆx, respectively. Obtaining the desired configurations is not the primary focus of this work and can be addressed with prior works [39], [40]. For simplicity in our problem setup, we manually specify ˆq and ˆx. Additionally, we aim to develop a method that generalizes across object geometries. During RL training, objects are randomly sampled from a predefined distribution. While during evaluation, objects are selected from both the in-distribution set and the out-of-distribution set. An ideal method should be able to catch the object in contact configurations similar to those from the training examples, even if the target object is out of distribution. The method is evaluated based on (1) whether the robot successfully catches the object, (2) the difference between the final state of catching and the desired state, (3) and the performance of the subsequent primary manipulation task. Compared to other dexterous manipulation setups, this task is more challenging because the robot needs to move quickly to handle dynamic situations, but also precisely to reach the desired configurations. IV. METHODS Our approach, Contact-Aware Dynamic Recovery(CADRE), leverages reinforcement learning to optimize recovery policies in dexterous manipulation tasks. To enable contact awareness and generalization to unseen objects based on contacts, we incorporate contact features derived from NDFs into the observation space. The NDF model extracting contact features is pretrained and remains fixed during RL training. The contact features enable the RL policy to achieve consistent behaviors across objects with different geometries. A. Preliminary: Neural Descriptor Fields Neural Descriptor Fields (NDFs) [13] is a learned representation that captures geometric correspondence between a queried 3D coordinate and an object. Given an object point cloud P, NDF extracts an n-dimensional geometric feature for any 3D coordinate p: fNDF (p|P) : R3 →Rn. (1) During NDF training, an MLP-based network Φ(p|E(P)) is trained to predict occupancy or signed distance of the query point p, conditioned on the object point cloud P. E is the PointNet-based encoder, which extracts the latent features of the object point cloud. The NDF feature is defined as: fNDF (p|P) := L M i=1 Φi(p|E(P)), (2) where ⊕denotes concatenation of activations from each layer of Φ, and L is the number of hidden layers. This feature serves as an implicit occupancy representation and provides rich geometric information about the geometric relationship between the queried point and the object surface. In our implementation, Φ(p|E(P)) is a double-headed model that predicts both occupancy and signed distance, similar to [38]. While occupancy prediction mainly captures features for points near contact regions, adding additional signed distance prediction helps capture features at points which are not in contact, which facilitates RL training when exploring non-contact regions. In our method, the NDF model is pretrained and kept fixed during RL training. B. Contact-Aware Dynamic Recovery 1) Contact-Aware Grasp Feature: Recovering with an expected grasp primarily requires accurate modeling of finger-object contact. Beyond this, it is highly desirable for the contact representation to generalize across object geometries, as such generalization enables the robot policy to exhibit zero-shot generalization to unseen objects. In this work, we leverage NDFs to characterize contact features for dexterous manipulation. By querying points on the hand, we can interpret the corresponding NDF features as indicators of contact: contact points are expected to lie near the decision boundary of the occupancy function, where the NDF occupancy prediction output transitions between inside and outside predictions. While NDF provides per-point contact features, it does not directly provide information characterizing the contact features of a grasp. To address this, we predefine K key points on the hand {pki i }K i=1, where pki i denotes the position of the i-th key point in the ki-th link frame. The grasp feature g(q, x|P) is defined as the concatenation of the NDF features of all the contact points: g(q, x|P) := K M i=1 fNDF (T(x)ffk(pki i , q, ki)|P), (3) where q is the robot joint angles, x is the object pose in SE(3). ffk denotes the forward kinematics function returning key point locations in the world frame, and T(x) is the transformation from the world frame to the object frame. For key point selection, we sample points from each hand link rather than restricting them to the fingertips. We use the root of each link as the key point in our experiments. Although fingertips are often primarily involved in contact, non-contact parts also play a critical role in characterizing the grasp feature. For example, one of the major distinctions between a power grasp and a precision grasp is whether the palm is in contact with the object. In practice, we assign one key point for each link for computational efficiency. Moreover, it is not necessary to place key points strictly within potential contact regions, since NDF features encode not only whether a key point is in contact but also its distance from contact. In summary, CADRE receives observations ot and computes the contact-aware grasp feature g = g(q, x|P) at every time step. The input into the RL policy is the combination of observations and grasp features. 2) Reinforcement Learning for Dynamic Recovery: Due to the complexity of modeling the dynamic interactions between the hand and the object, we choose to use Proximal Policy Optimization (PPO) [41], a model-free RL method, to optimize the dynamic recovery policy. Reward Function: The reward function is defined as follows: r :=r ̇a + rtorque + renergy + rdrop + robj v + robj pose + rq + rcontact + rsafety, (4) where we omit the weighting parameters and the time step subscript for simplicity. Specifically, r ̇a := -||at -at-1||2, rtoruqe := -||τ||2, and renergy := -||τ T ̇q||2 regularizes non-smooth actions, large torque and large energy consumption, respectively. τ is computed from the joint position controller output. rdrop := -1drop(x), where 1drop is an indicator function that returns 1 if the object's position exceeds Primary manipulation Drop detected ? Object geometry Predefined key points Pretrained NDF model Contact-aware grasp feature gt qt xt vt gt Observations PPO policy Actions at 1 Train offline, Freeze online 2 Online RL training Yes Fig. 2: Contact-Aware Dynamic Recovery (CADRE) aims to catch a manipulated object when it falls from the grasp and recover to states that support the resumption of the manipulation task. CADRE leverages a pretrained NDF model to extract implicit contact features from the grasp. The contact-aware grasp features are incorporated into the RL observations, enhancing the policy's awareness of contact and improving recovery performance. a predefined workspace boundary. robj v := exp(-||v||2) + exp(-||ω||2) encourages low linear velocity v and angular velocity ω of the object. We shape robj v with an exponential function as the velocities can yield excessively large magnitudes during training (e.g., the object is falling fast). The exponential function helps bound the reward and leads to more stable training. rq := exp(-||q-ˆq||2) and robj pose := exp(-fpos(x, ˆx))+exp(-forn(x, ˆx)) encourage the robot to recover to the desired robot configuration ˆq and object pose ˆx. ˆq and ˆx are predefined and considered favorable for the primary manipulation task (see Sec. V-A for further discussion). fpos and forn computes the position and orientation difference, respectively. rcontact := 1contact(q, x) rewards the robot for achieving task-relevant contact with the object. The desired task-relevant contact can be obtained from ˆq and ˆx (see Sec. V-A). rsafety := -1table(q) penalizes the unsafe behavior of robot contacting the table. Domain Randomization: Since accurately perceiving fast-moving objects can be challenging in real-world scenarios, catching dynamic objects for recovery needs to handle significant uncertainties. To address this, we apply domain randomization [1] to improve robustness. Specifically, we introduce external disturbances and perception noise: at each time step, a random external wrench wext is applied at the center of the screwdriver, and random perception noise ε sampled from a uniform distribution is added to the object's pose and velocities. V. EXPERIMENTS We evaluate our method on two tasks: screwdriver recovery and nut recovery. We aim to design our experiments to answer (1) whether incorporating implicit contact features can facilitate RL training and improve catching performance; (2) whether CADRE generalizes to unseen objects sampled from a different distribution; and (3) whether CADRE can be deployed on robot hardware. A. Simulation Experiment Setup We use IsaacSim to simulate the recovery task. Our robot consists of an Allegro Hand [42] mounted on a 7-DoF KUKA iiwa arm. We use PPO [41], implemented from RL games [43], to optimize the recovery policy. To obtain the full point cloud of the object, we assume access to its geometry (as noted in Sec. III), since capturing a complete point cloud with a camera is impractical. We uniformly sample points from the object surface and transform them according to the object pose to generate the full point cloud observation. The same approach is used in our hardware experiments. Object generation: During RL training, we randomly sample 50 objects from a predefined shape distribution. Specifically, the objects are generated using the same geometric parameterization method, but with variations in their size parameters. During training for the screwdriver task we add five real screwdrivers to support our real-world experiments. For evaluation, we consider both in-distribution (ID) and out-of-distribution (OOD) objects. ID objects consist of 5 new unseen objects sampled from the same distribution used for training, while OOD objects are sampled from a different distribution as a more challenging test for generalization. The NDF model fNDF is trained with objects sampled from the same distribution for RL training. Thus, OOD objects for the RL policy will also be OOD for the NDF model. Evaluation metrics: We consider the following metrics: (1) Catch success rate: a catch is considered successful if the object remains within a predefined bounding box throughout the episode; (2) Reward: cumulative reward over an episode; (3) Number of desired contacts; (4) Number of undesired contacts; (5) Position error: the Euclidean distance between the final and the desired object position; and (6) Orientation error: defined as the angle between the final and desired zaxes of the object frame, since the objects in our experiments are rotationally symmetric about the z-axis. Metrics (3) and (4) are task-specific and detailed in Sec. V-A.1 and V-A.2. Metrics (2)-(6) are only calculated for successful trials. As the most direct evaluation involves initializing the primary manipulation task with the final recovered states, we also introduce a screwdriver turning task and a simplified nut mating task to assess whether the post-recovery states are suitable for downstream manipulation tasks. See details in Sec. V-A.1 and V-A.2. 1) Screwdriver Recovery: We set up the recovery task in a screwdriver turning scenario with a precision grasp. Screwdriver turning with a precision grasp has been widely studied [44], [6], [45], [46], [47]. A precision grasp is preferred, as opposed to a power grasp, as the choice of robot-screwdriver contact significantly affects the turning performance, highlighting the importance of contact reasoning. Specifically, we follow the setup presented in [46] (See Fig. 3a), where the index finger contacts the top of the screwdriver, while the thumb and middle finger form an antipodal grasp on the handle. To define a desired precision grasp, we specify the desired object configuration ˆx such that the screwdriver is upright with its tip in contact with the supporting table. For the desired robot configuration ˆq, we maintain the same hand configurations across different screwdrivers, while adjusting the arm configuration accordingly to ensure the index finger rests on top of the screwdriver. See the visualization in Fig. 3a. To extract the desired task-relevant contact, we compute the finger-object distance in the desired configuration. Fingers with a distance below 2 cm are classified as taskrelevant contacts, corresponding to the index, middle and thumb finger in contact with the screwdriver. The undesired contact is defined as the contact between all other links of the hand with the screwdriver. The contact reward rcontact returns the number of desired contact points. We model the screwdriver as two connected cylinders: one for the handle and one for the shaft. OOD screwdrivers have approximately half the length of the ID screwdrivers. To evaluate whether the recovered states are favorable for screwdriver turning, we follow the setup from [46], attempting to turn the screwdriver 60◦. We assume there exists a motion planning algorithm to mate the screwdriver with the screw and reorient it perfectly upright. In practice, we record the final state of recovery, keep the joint angles and the relative transformation between the hand and the screwdriver fixed, but we transform both the hand and the screwdriver so that the screwdriver is upright. We evaluate the turning performance via the turning drop rate and the object orientation difference between the final turning state and the desired state. We consider the screwdriver dropped if its Euler angles exceed a predefined threshold. 2) Nut Recovery: We consider recovery for nut-bolt mating (See Fig. 3b). During mating, the robot can easily drop the nut as the nut mating involves forceful contacts and high uncertainty, especially if tactile sensors are not available. As in the screwdriver task, successful recovery requires the robot to catch the nut with a specific grasp. In this case, we define ˆq and x such that all fingertips make contact with the side of the nut while avoiding its top and bottom. Contacts on the top or bottom will block the nut's central hole, making it impossible to mate with the bolt. We use the same method (a) Screwdriver (b) Nut 0.0 0.5 1.0 1.5 2.0 2.5 3.0 # Env Interaction Steps 1e8 0 5 10 15 Total Episode Reward Screwdriver Ours Pose Point Cloud DexPoint (c) RL training curves for screwdriver recovery 0.0 0.5 1.0 1.5 2.0 2.5 3.0 # Env Interaction Steps 1e8 0 5 10 15 20 Total Episode Reward Nut Ours Pose Point Cloud DexPoint (d) RL training curves for nut recovery Fig. 3: Fig. (a) and (b) show the environment step. Three different seeds are used for RL training and the average results are shown in Fig. (c) and (d). The variance is relatively small as we use a large batch size during training. above to extract task-relevant contacts, which corresponds to all fingertips in contact with the nut for this task. Any other robot-nut contacts are then undesired contacts. Additionally, there are cases where the robot appears to stabilize the nut by using the external support from the bolt. For instance, the robot might push the nut towards the bolt while the grasp itself is not stable without the support from the bolt. We penalize such behaviors in the contact reward: rcontact := rrobot nut -rnut bolt, where rrobot nut returns the number of desired contacts and rnut bolt is the indicator function for nut-bolt contact. The nut is modeled as a hexagonal prism with a cylindrical hole, and the bolt is modeled as a cylinder. OOD nuts have about half the thickness of the ID nuts, making them harder to catch and requiring more precise finger control. A recovery is considered successful if the nut is not dropped, and the nut does not contact the bolt, as the robot must stably grasp the nut without the bolt's support. To evaluate grasp quality for the primary nut-mating task, we design a simplified experiment, since nut-mating with a dexterous hand remains a challenging open research problem. Specifically, we use a bolt with a smaller radius and consider the task successful if the nut can be placed such that the bolt passes through its center hole. This simplification primarily tests whether the robot's fingers obstruct the nut's center hole, a necessary condition for successful nut-mating. During the experiment, we assume the finger configurations remain fixed and use motion planning to move the arm. 3) Baselines and Ablation: We consider two RL baselines and an ablation method. All methods use the same reward functions and RL training setup as CADRE, but represent the object geometry and contact differently: (1) Ablation: PPO with object pose observations (Pose): it directly takes the observation ot defined in Sec. III as input. This method does not have access to object geometry; the only object-related information available is pose and velocity. (2) Baseline: PPO with point cloud observations (Point Cloud): Reinforcement learning with point cloud input has been widely used in dexterous manipulation [30], [31], [32]. Similar to these methods, we add the object's point cloud Pt into the aforementioned observation space: o′ t := {qt, xt, vt, Pt}.We choose not to use the point cloud obtained from a depth camera to maintain the same input as our method. This method has access to the object's geometry but does not explicitly reason about contact features. PointNet [48] is used to extract features from the point cloud. (3)Baseline: DexPoint [48]: Similar to our method, DexPoint leverages contact information to improve generalization across object geometries. Its key component is augmenting the observation space with an additional imagined point cloud of the robot. To implement DexPoint, we render a point cloud for each fingertip and the palm, and concatenate it with the object point cloud. Unlike the original DexPoint paper, we do not use the observed point cloud, i.e., the depth-camera point cloud, since other methods in our experiments do not have access to such observations. PointNet is also used to encode the point cloud. B. Experiment Results The RL training curves are shown in Fig. 3. CADRE achieves higher rewards than baselines given the same number of environment interactions. In the screwdriver recovery task, although the baseline methods have eventually converged, they fail to match CADRE's performance. For the nut mating task, although all methods are nearing convergence, CADRE consistently outperforms the baselines. Each method is evaluated for 50 trials per object. The average performance is reported in Table I across all objects and all seeds. We also compute the standard deviation of the 50 trials on a given object, and then average the standard deviation from different objects and different seeds. Screwdriver recovery: all methods achieve almost 100% success rates on ID objects. However, CADRE performs better on most of the other metrics. This suggests that baselines often use power grasps rather than the desired precision grasps to catch the screwdriver. On OOD objects, CADRE and DexPoint still maintain similar performance, while the other baselines exhibit a more significant performance drop. However, we will demonstrate in Table II that grasp pose of DexPoint is not well suited for the primary screwdriverturning task. To further evaluate the quality of the recovered state, we set up the screwdriver turning (primary manipulation task) as described in Sec. V-A.1 and use the turning algorithm proposed in [46]. Because the turning algorithm is computationally expensive, we only run it with the best-performing seed with the highest success rates for each method. We randomly sample 5 recovery trials per object for the turning evaluation. The goal is to turn the screwdriver 60 degrees. The turning results are shown in Table II. According to the results, CADRE has demonstrated a smaller distance to goal in ID scenarios and a lower drop rate in OOD scenarios. The performance drop from ID to OOD of CADRE is much smaller than that of the baselines. Nut recovery: we have also observed similar results in the nut-catching task (See results in Table I), where CADRE outperforms the baselines in both ID and OOD scenarios and exhibits a smaller performance drop when switching from ID to OOD objects. In the downstream simplified nutmating task, we sample 50 recovery trials per object from the best seed for the mating evaluation. Although some baselines achieve higher mating success rates, CADRE attains nearly double the catching success rate, resulting in a higher overall success rate from catching to mating. In particular, baselines tend to successfully catch the nut mainly in simpler catching scenarios, which makes it easier for them to achieve higher mating performance. C. Hardware Experiments The main objective of the hardware experiment is to evaluate whether we can successfully transfer the highly dynamic recovery behavior learned in simulation to a realworld setup. We only focus on the screwdriver recovery task in our hardware experiments. We use an unseen real screwdriver that falls within the training distribution for our hardware experiment. We use the Vicon motion capture (mocap) system to estimate the state of the screwdriver. 1) Experiment Setup: To enable deployment in the real world, we distill an open-loop policy from the closedloop RL policy. Closed-loop execution of a highly dynamic manipulation policy is particularly challenging on a hardware system because of (1) Latency: Our system exhibits approximately 50 ms of latency between sending control commands and receiving the corresponding observation update. Such millisecond latency is usually negligible in many quasi-static tasks, but will lead to instability in our highly dynamic tasks; and (2) Occlusion: When the robot attempts to catch the screwdriver, its fingers will often block the mocap markers, causing loss of object state information. To distill the open-loop policy, we collect 10,000 successful rollouts from the closed-loop policy in simulation, with 55 randomly sampled objects. We train a supervised openloop policy that takes the initial observation as input and outputs a sequence of actions for the entire trajectory. We use a transformer decoder architecture for the policy. One of the main challenges in the hardware experiment is resetting the system to a state where the screwdriver is about to fall. As the error detection is not the focus of this work, we manually create such a falling scenario: we first manually position the screwdriver so that the robot can grasp it. To initiate the drop, we command the robot to open its hand for a short, fixed duration. The recovery policy is then triggered. To eliminate bias from manually setting the screwdriver's initial pose, we anonymize the methods during each trial: the method under evaluation is randomly selected and not revealed to the experimenter until the end of this trial. Task Obj method success↑ reward↑ #contact↑#undesired contact↓ pos diff(cm)↓ orn diff↓ Screwdriver ID CADRE 100.00% 13.22 ±2.31 2.54 0.19 4.34 ±3.86 13.89◦± 12.07◦ Pose 100.00% 10.57 ± 0.68 1.57 1.64 4.89 ± 2.25 10.25◦± 3.95◦ Point Cloud 100.00% 11.10 ± 0.59 1.72 1.78 6.25 ± 2.88 15.29◦± 5.13◦ DexPoint 99.87% 10.70 ± 0.53 2.78 1.05 5.10 ± 2.33 20.11◦± 5.30◦ OOD CADRE 96.27% 11.09 ± 1.39 1.44 0.08 2.05 ± 1.37 22.74◦± 12.41◦ Pose 73.86% 8.14 ± 2.06 0.96 0.85 8.48 ± 4.57 29.78◦± 20.60◦ Point Cloud 83.73% 9.24 ± 2.13 1.15 1.17 7.33 ± 3.42 24.53◦± 16.43◦ DexPoint 97.86% 9.66 ± 1.20 2.04 1.20 5.67 ± 2.29 21.52◦± 12.55◦ Nut ID CADRE 98.53% 20.46 ± 2.25 3.85 0.15 4.08 ± 2.26 5.89◦± 5.77◦ Pose 60.67% 12.69 ± 2.67 1.91 0.26 8.26 ± 2.66 30.99◦± 27.16◦ Point Cloud 75.33% 14.15 ± 2.97 1.66 0.34 6.26 ± 2.72 36.11◦± 25.50◦ DexPoint 72.53% 17.42 ± 2.79 2.88 0.36 5.89 ± 0.85 14.51◦± 18.89◦ OOD CADRE 91.06% 16.47 ± 4.23 2.92 0.17 6.60 ± 3.89 18.68◦± 19.70◦ Pose 48.40% 12.51 ± 3.32 2.10 0.01 8.19 ± 2.50 28.49◦± 28.91◦ Point Cloud 46.00% 13.04 ± 3.95 1.47 0.09 6.83 ± 3.02 39.27◦± 26.13◦ DexPoint 38.40% 17.78 ± 2.95 2.58 0.08 6.15 ± 1.21 15.31◦± 16.51◦ TABLE I: Evaluation on unseen screwdriver recovery and nut recovery. Results are calculated for successful trials only. Scrwedriver Turning ID Scrwedriver Turning OOD Nut Mating ID Nut Mating OOD drop rate↓ dist2goal↓ (no-drop trials) drop rate↓ dist2goal↓ (no-drop trials) valid succ↑overall succ↑valid succ↑overall succ↑ CADRE 4% 13.77◦± 17.66◦ 4% 35.25◦± 18.80◦ 54.3% 53.6% 49.8% 45.2% Pose 0% 40.99◦± 20.89◦ 72% 21.03 ◦±11.04◦ 56.9% 28.0% 73.9% 40.8% Point Cloud 0% 45.31◦± 16.97◦ 32% 36.81◦± 20.18◦ 6.1% 4.8% 15.2% 8.4% DexPoint 4% 50.28◦± 20.19◦ 28% 37.66◦± 17.79◦ 59.1% 46.4% 67.2% 20.8% TABLE II: Evaluation results for screwdriver turning and nut mating initialized from recovered states. Dist2goal represents the screwdriver orientation difference between the final state of turning and the desired state of turning 60◦. For nut mating, we present two success rates: valid success rate, defined as (#successful mating/#successful cathing), and the overall success rate, defined as (#successful mating/#trials), where the number of trials also includes failed catching attempts. succ rate ↑ #desired contact (succ only)↑ #undesired contact (succ only)↓ CADRE 80.0% 2.83 0.0 Pose 6.7% 1.0 2.0 Point Cloud 0.0% DexPoint 33.3% 1.4 1.0 TABLE III: Distilled open-loop policies are evaluated for 15 hardware trials per method. In our experiments, we use a real screwdriver but add friction tape, as its material-optimized for human grip-does not provide sufficient friction for the robot's fingertip material. 2) Experiment Results: The quantitative results are shown in Table III. Our method demonstrates a better performance in the hardware experiments. Baseline performance is substantially worse on hardware, likely because the power-grasp strategy it employs requires faster robot movements, which are more susceptible to sim-to-real gaps. Nevertheless, the results still reveal a noticeable sim-to-real gap, indicating that further improvement is needed before deployment in production. Potential improvements include reducing action jerk, obtaining more accurate estimates of joint friction and robot inertia, and minimizing applied forces for safety. We leave closing the sim-to-real gap for highly dynamic systems as future work. VI. DISCUSSION While CADRE demonstrates the ability to generalize across different geometries, the underlying NDF structure only focuses on geometry information for generalization. However, variations in geometries will also lead to different dynamics. For example, a grasp on an unseen object that shares similar grasp features with a force closure grasp on a known object might not result in force closure. Additionally, dynamic manipulation with precise control introduces significant sim-to-real challenges. Specifically, Factors such as system latency, gravity compensation errors, and policy computation time, which are often negligible in quasi-static scenarios, can have a substantial impact on performance in highly dynamic recovery tasks. While CADRE can successfully catch a falling object in our hardware experiment, there remains room for improvement in achieving more precise desired contact configurations and recovering from more challenging falling scenarios, e.g., by making the policy aware of the system latency and dynamics discrepancies. Despite the limitations above, our results suggest the broader potential for RL of using implicit contact representations in contact-rich manipulation beyond dynamic recovery. Further extending this approach to more general settings presents a promising research direction for our future research. VII. CONCLUSION In this work, we focus on the problem of recovering from catastrophic failure in dexterous manipulation-specifically, recovering from dropping objects and returning to states favorable for resuming the primary manipulation task. We propose CADRE, an RL framework that utilizes an implicit contact representation derived from NDFs. We evaluate CADRE on screwdriver and nut recovery tasks. Our experiments demonstrate that simply providing point clouds as object geometry observations for RL is insufficient for learning effective dynamic recovery in contactrich scenarios. In contrast, CADRE leverages implicit contact features to improve training efficiency, recovery quality, and generalization to unseen object geometries. REFERENCES [1] J. Tobin, R. Fong, A. Ray, J. Schneider, W. Zaremba, and P. Abbeel, "Domain randomization for transferring deep neural networks from simulation to the real world," in 2017 IROS. IEEE, 2017, pp. 23-30. [2] L. Pinto, J. Davidson, R. 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Tagliasacchi, J. B. Tenenbaum, A. Rodriguez, P. Agrawal, and V. Sitzmann, "Neural descriptor fields: Se (3)- equivariant object representations for manipulation," in 2022 ICRA. IEEE, 2022, pp. 6394-6400. [14] A. Simeonov, Y. Du, Y.-C. Lin, A. R. Garcia, L. P. Kaelbling, T. Lozano-P ́erez, and P. Agrawal, "Se (3)-equivariant relational rearrangement with neural descriptor fields," in Conference on Robot Learning. PMLR, 2023, pp. 835-846. [15] T. Ishihara, A. Namiki, M. Ishikawa, and M. Shimojo, "Dynamic pen spinning using a high-speed multifingered hand with high-speed tactile sensor," in 2006 6th IEEE-RAS International Conference on Humanoid Robots. IEEE, 2006, pp. 258-263. [16] Y. J. Ma, W. Liang, G. Wang, D.-A. Huang, O. Bastani, D. Jayaraman, Y. Zhu, L. Fan, and A. Anandkumar, "Eureka: Humanlevel reward design via coding large language models," arXiv preprint , 2023. [17] H. Ha and S. 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Kaneko, "Development of a high-speed multifingered hand system and its application to catching," in ROS 2003, vol. 3. IEEE, 2003, pp. 2666-2671. [23] S. S. M. Salehian, M. Khoramshahi, and A. Billard, "A dynamical system approach for softly catching a flying object: Theory and experiment," IEEE T-RO, vol. 32, no. 2, pp. 462-471, 2016. [24] Y. Zhang, T. Liang, Z. Chen, Y. Ze, and H. Xu, "Catch it! learning to catch in flight with mobile dexterous hands," arXiv preprint , 2024. [25] B. Huang, Y. Chen, T. Wang, Y. Qin, Y. Yang, N. Atanasov, and X. Wang, "Dynamic handover: Throw and catch with bimanual hands," arXiv preprint , 2023. [26] F. Lan, S. Wang, Y. Zhang, H. Xu, O. Oseni, Z. Zhang, Y. Gao, and T. Zhang, "Dexcatch: Learning to catch arbitrary objects with dexterous hands," arXiv preprint , 2023. [27] H. J. Charlesworth and G. Montana, "Solving challenging dexterous manipulation tasks with trajectory optimisation and reinforcement learning," in ICML. PMLR, 2021, pp. 1496-1506. [28] Z. Ling, Y. Yao, X. Li, and H. Su, "On the efficacy of 3d point cloud reinforcement learning," arXiv preprint , 2023. [29] Y. Qin, B. Huang, Z.-H. Yin, H. Su, and X. Wang, "Dexpoint: Generalizable point cloud reinforcement learning for sim-to-real dexterous manipulation," in CoRL. PMLR, 2023, pp. 594-605. [30] W. Huang, I. Mordatch, P. Abbeel, and D. Pathak, "Generalization in dexterous manipulation via geometry-aware multi-task learning," arXiv preprint , 2021. [31] Y.-H. Wu, J. Wang, and X. Wang, "Learning generalizable dexterous manipulation from human grasp affordance," in Conference on Robot Learning. PMLR, 2023, pp. 618-629. [32] C. Bao, H. Xu, Y. Qin, and X. Wang, "Dexart: Benchmarking generalizable dexterous manipulation with articulated objects," in CVPR, 2023, pp. 21 190-21 200. [33] P. Wu, A. Escontrela, D. Hafner, P. Abbeel, and K. Goldberg, "Daydreamer: World models for physical robot learning," in Conference on robot learning. PMLR, 2023, pp. 2226-2240. [34] D. Driess, I. Schubert, P. Florence, Y. Li, and M. Toussaint, "Reinforcement learning with neural radiance fields," Advances in Neural Information Processing Systems, vol. 35, pp. 16 931-16 945, 2022. [35] B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoorthi, and R. Ng, "Nerf: Representing scenes as neural radiance fields for view synthesis," Communications of the ACM, vol. 65, no. 1, pp. 99-106, 2021. [36] N. Khargonkar, N. Song, Z. Xu, B. Prabhakaran, and Y. Xiang, "Neuralgrasps: Learning implicit representations for grasps of multiple robotic hands," in CoRL. PMLR, 2023, pp. 516-526. [37] S. Cheng, C. R. Garrett, A. Mandlekar, and D. Xu, "Nod-tamp: Multi-step manipulation planning with neural object descriptors," in CoRL 2023 Workshop on Learning Effective Abstractions for Planning (LEAP), 2023. [38] Z. Huang, Y. He, Y. Lin, and D. Berenson, "Implicit contact diffuser: Sequential contact reasoning with latent point cloud diffusion," arXiv preprint , 2024. [39] J. 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Malik, et al., "Dexteritygen: Foundation controller for unprecedented dexterity," arXiv preprint , 2025. [46] F. Yang, T. Power, S. A. Marinovic, S. Iba, R. S. Zarrin, and D. Berenson, "Multi-finger manipulation via trajectory optimization with differentiable rolling and geometric constraints," arXiv preprint , 2024. [47] A. Kumar, T. Power, F. Yang, S. A. Marinovic, S. Iba, R. S. Zarrin, and D. Berenson, "Diffusion-informed probabilistic contact search for multi-finger manipulation," arXiv preprint , 2024. [48] C. R. Qi, H. Su, K. Mo, and L. J. Guibas, "Pointnet: Deep learning on point sets for 3d classification and segmentation," in CVPR, 2017, pp. 652-660.
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2510.14769
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1
Unifying Frictional Transients Reveals the Origin of Static Friction
Authors: Kasra Farain1, Daniel Bonn1
1. Van der Waals–Zeeman Institute, Institute of Physics, University of Amsterdam; Science Park 904, 1098 XH
Amsterdam, Netherlands.
Abstract: Frictional motion is harder to initiate than to sustain, as evident when pushing a heavy
object. This disparity between static and kinetic friction drives instabilities and stick–slip dynamics
in systems ranging from nanodevices1 and MEMS2 to squealing brakes, glaciers3 and tectonic
faults4, yet its origin and the transition mechanism remain poorly understood. Empirical rate-and-
state friction laws4,5 predict that during the static-to-kinetic transition, friction increases for
nanometer-per-second slip rates, but decreases for micrometers-per-second rates and above. These
transients are believed to be associated with contact strengthening (aging) at static interfaces5,6,
although their physical basis is unclear and the crossover between regimes has never been observed
directly. Here we show, through nanometer-resolution sliding experiments on macroscopic rough
surfaces, that these transients are segments of a single, universal non-monotonic response whose
peak defines static friction. We show that this behavior arises from mechanical reorganization of
interlocking surface asperities under shear—fundamentally distinct from contact aging, which is
governed by thermal molecular processes6-8. We derive, from first principles and without invoking
any empirical postulates, a differential equation that quantitatively captures the friction peak.
These results unify frictional transients across scales and speeds, and establish a physics-based
framework for understanding frictional instabilities and failure processes in engineering and
geosciences.
Main text: Friction has puzzled scientists for centuries. In the late 15th century, Leonardo da Vinci
observed that the lateral frictional force F required to slide one body over another scales with the
normal force N pressing them together and is independent of their macroscopic contact area—a
principle later formalized by Amontons (1699) as F = µN 9,10. The proportionality constant, the
friction coefficient µ, depends on the materials and whether the interface is static (µs) or kinetic
(µk), with µs consistently exceeding µk (Fig. 1). In 1785, Coulomb further demonstrated that in
slide–pause–slide sequences, µs increases logarithmically with pause duration, a phenomenon now
known as contact aging. Yet despite this centuries-long history, the fundamental questions of why
µs always exceeds µk and how the transition occurs remain unresolved11-22.
2
Fig. 1 | Static versus kinetic friction. A block on an inclined plane does not slip until the slope exceeds
the static threshold 𝜃s = tan-1 (µs), derived from F = µN. Once sliding begins, the block continues moving
even on shallower slopes down to 𝜃k = tan-1 (µk), demonstrating that µs > µk.
The phenomenological rate-and-state friction (RSF) laws4,5—widely applied in tribology and
earthquake studies since the 1970s4-6,11,23-36—describe two distinct slip behaviors depending on
slip velocity. At high velocities (typically > 1 µm s-1), the friction coefficient decreases smoothly
from its static µs to kinetic µk value during slip onset. At low velocities (typically < 100 nm s-1),
the friction coefficient begins below µk and increases with displacement (Fig. 2a).
RSF theory attributes these transients to contact aging (Coulomb’s observation)5,6 but this
interpretation poses a fundamental problem. Real surfaces are microscopically rough, covered with
asperities. Contact aging at static interfaces arises through viscoelastic relaxation7 and creep6,8 of
these surface features under sustained pressure, or through molecular bonding at asperity micro-
contacts in hard materials1. These mechanisms clearly require asperity micro-contacts to remain
static. Yet RSF transients are observed during sliding, when asperities continually reorganize and
micro-contacts break and reform. How can these processes really coexist?
Here we demonstrate that the increasing and decreasing RSF transients emerge together in a single
sufficiently long experiment at nanometer-per-second sliding velocities, manifesting as a
continuous, broad (~10 µm) friction peak. This non-monotonic behavior unifies low- and high-
velocity RSF experimental observations, but directly contradicts RSF theory, which permits only
monotonic friction evolution. Through systematic slide–pause–slide and slide–perturbation–slide
experiments, we show that these peaks originate from mechanical reorganization of surface
asperities during sliding—a process fundamentally distinct from contact aging, which occurs
through spontaneous, thermally activated molecular processes (viscoelastic creep and relaxation
or chemical bonding). We observe contact aging in the same system in slide–pause–slide
experiments with longer pauses: it produces much narrower (< 200 nm), discontinuous friction
peaks whose height increases logarithmically with pause duration. Crucially, the mechanical
reconfiguration of asperities requires external shear and does not occur spontaneously. Guided by
this insight, we derive a differential equation governing the dynamics of asperities under shear that
quantitatively reproduces the broad friction peaks. Our derivation introduces no empirical
assumptions and relies solely on the existence of a unique steady state (corresponding to kinetic
friction) and the mathematical conditions the system must satisfy at steady state. Beyond
mg
𝜃k
𝜃s
3
implications for tribology and earthquake studies, our results establish that the static friction peak
between rough solids is a mathematical necessity rather than merely an empirical observation.
Figure 2b,c show friction as a function of time at a polytetrafluoroethylene (PTFE)–glass interface
under an imposed sliding rate of 2 nm s-1. Initially (Fig. 2b), the result appears to agree with RSF
predictions for the slow-sliding regime. However, measuring for a longer period (Fig. 2c) reveals
that the friction coefficient subsequently decreases, exhibiting a broad peak before reaching steady
state. We also observe this non-monotonic behavior for polypropylene–glass and steel–glass
contacts (Supplementary Fig. S2), and—though previously unnoticed—it is clearly visible in
earlier experiments on Lucite and acrylic plastics that were believed to confirm the RSF model
(Supplementary Fig. S3)6. As already mentioned, RSF formulations, by construction, exclude non-
monotonic friction; the first derivative of µ has a constant sign in RSF (Appendix 1). These
observations therefore reveal fundamental limitations of RSF theory in capturing the dynamics of
the onset of sliding.
Fig. 2 | Universal friction peak unifies RSF behaviors across slip velocities. a, Predictions of RSF theory
for friction transients during slip initiation: friction strengthens in the slow-slip regime but weakens in the
fast-slip regime4. b, Friction versus time at a PTFE–glass interface under imposed sliding at 2 nm s-1,
apparently consistent with RSF predictions for the slow-sliding regime. c, Continuation of the measurement
in b reveals a non-monotonic friction peak that encompasses both slow- and fast-sliding RSF behaviors.
What causes the broad friction peak? Macroscopic friction arises from the collective contribution
of microscopic forces between asperities on opposing surfaces. When two rough surfaces initially
contact, surface asperities adopt random orientations within the frictional interface with no
preferred direction. In this state, microscopic forces between asperities cancel in the horizontal
direction, yielding no net friction (Fig. 3a). Under applied shear, however, asperities deform and
slide past one another, reorganizing to develop a directional bias. At steady-state sliding, the
horizontal components of the microscopic asperity forces sum to produce the macroscopic kinetic
friction force (Fig. 3b). We demonstrate that the broad friction peak emerges from this
configurational transition of the asperity ensemble: it appears if and only if the initiation of sliding
requires configurational reorganization.
0
2000
4000
6000
8000
Time (s)
0
0.05
0.1
0.15
Friction coefficient
0
100
200
300
400
500
Time (s)
0
0.05
0.1
0.15
Friction coefficient
Time
Friction
Slow
Fast
a
b
c
4
To do so, we paused and resumed the sliding at the frictional interface of Fig. 2c at different times
using three protocols (Fig. 3c). In the first pause (Fig. 3c, t = 10,000 s), we turned off the applied
sliding velocity and restarted it a few seconds later. Remarkably, sliding recommenced without a
friction peak, suggesting that during the brief pause, asperities retained memory of the previous
sliding direction and their configuration did not revert to randomness. For the second pause (Fig.
3c, t = 15,000 s), we switched from applied sliding velocity to an applied shear force slightly
below Fss (approximately 20% reduction). This halted sliding motion while maintaining the shear
force that holds asperities in their sliding configuration. In this way, the anisotropic steady-state
asperity configuration at the moment of the switch was preserved; the same microscopic forces,
reduced proportionally by 20%, continued to balance the reduced macroscopic force (Fig. 3b).
When we switched back to applied sliding velocity 1 s later, no friction peak emerged, confirming
that the static-to-kinetic friction peak does not occur when asperities are already in their steady-
state sliding configuration. Finally, in the third pause (Fig. 3c, t = 20,000 s), the surfaces were
separated and re-contacted. This erased the configurational memory, re-randomized the asperity
ensemble, and thereby restored the broad friction peak. Importantly, interface re-randomization
and reemergence of the friction transient can also be induced by ‘seismic’ pulses applied during
either paused or continuous sliding (Supplementary Fig. S4), suggesting broader implications for
earthquake dynamics. Similar behaviors are observed at polypropylene–glass and steel–glass
interfaces (Supplementary Fig. S2).
Before deriving a first-principles theory for the broad friction peaks, we also show that contact
aging and the broad friction peak (or its RSF slices) are indeed different phenomena. To observe
contact aging in our system, we perform slide–pause–slide experiments like those in Fig. 3c at t =
10,000 s but with extended pause durations. Figure 4a displays a typical contact aging peak after
a 20 min pause alongside the broad friction peak from asperity reorganization, highlighting that
they operate on vastly different length scales. The aging peak grows logarithmically with pause
duration, consistent with the rate of viscoelastic stress relaxation in asperities under pressure7.
Viscoelastic creep of asperities, the counterpart of stress relaxation, may deform their shape and
increase the real contact area between contacting surfaces during extended pauses, but cannot alter
the spatial arrangement of asperities or re-randomize the ensemble (Fig. 4, inset).
5
Fig. 3 | The origin of the broad friction peak at the onset of sliding. a,b, Schematic showing asperities
at a rough frictional interface, immediately after contact (a) and at steady-state sliding (b). Under quasi-
stationary conditions, where inertia is negligible, Newton’s second law requires that macroscopic friction
equals the vector sum of shear components from microscopic forces at asperity micro-contacts. This sum
is zero for randomly oriented asperities in a but reaches a finite steady-state value after alignment under
sliding in b. c, Continuation of the measurement in Fig. 2c. At t = 10,000 s, the sliding was stopped and
restarted after 5 s. At t = 15,000 s, the applied sliding velocity was switched for 1 s to an applied force
approximately 20% below Fss, which stopped the sliding. At t = 20,000 s, the sliding was stopped, the
surfaces were separated and re-contacted, and then the sliding was restarted.
Fig. 4 | Asperity reorganization peak versus contact aging peak. Comparison of the broad friction peak
from asperity reorganization (from Fig. 2c) with a typical aging peak from a slide–pause (20 min)–slide
Friction = ∑
𝑓⃗45678 = 0
:
;<=
f1
f2
f3
f4
f5
f6
f1
f2
f3
f4
f5
f6
Friction = ∑
𝑓⃗45678 = 𝐹44
:
;<=
Before sliding, random asperities
Under sliding, aligned asperities
c
a
b
0
5000
10000
15000
20000
25000
Time (s)
0
0.05
0.1
0.15
0.2
Friction coefficient
14998
15000
15002
0.04
0.05
10000
10200
0
0.03
0.06
20000
20400
0
0.05
0.1
0
2
4
6
Displacement ( m)
0
0.05
0.1
0.15
Friction coefficient
Asperity reorganization
Contact aging
Creep
6
experiment7 on a PTFE–glass interface. The schematic inset zooms in on one asperity from Fig. 3b
(highlighted also in Fig. 3b), showing its viscoelastic creep during aging. Such process cannot re-randomize
the spatial configuration of the ensemble of asperities.
First-principles theory of the broad friction peak. The reorganization in the asperity ensemble at
rough interfaces is driven by the externally imposed sliding displacement x. The resulting friction
F is therefore a function of displacement, F(x). We assume that the system possesses a unique
steady-state (kinetic) friction FSS. At steady-state sliding, the asperity ensemble remains
statistically unchanged with further displacement, which is why F(x) = FSS remains constant.
When the system deviates from steady state F(x) ≠ FSS, net changes occur in the asperity
configurations during sliding, driving it back towards the steady state. We represent this tendency
by a restoring force 𝑔 that naturally depends on the system’s deviation from steady state F(x) –
FSS and vanishes once steady state is reached F(x) = FSS. We can write:
DE(F)
DF
= ℎ𝑥−𝑔𝐹(𝑥) −𝐹JJ ,
(1)
where 𝑔 and ℎ are unknown functions. Essentially, we have merely identified a restoring
contribution 𝑔𝑥= 𝑔𝐹(𝑥) −𝐹JJ within the general function 𝐹K(𝑥) =
DE(F)
DF that we are
attempting to determine. We could assume a restoring force linear in displacement—like in linear-
response theory, the Langevin equation, and the fluctuation–dissipation theorem—but no
fundamental argument compels linearity; we therefore begin with the most general, potentially
nonlinear form.
The boundary conditions on 𝑔 and ℎ at steady state (𝑥→∞, 𝐹 (𝑥) −𝐹JJ →0) are crucial to
simplify the above equation. As 𝐹 (𝑥) −𝐹JJ →0, 𝑔→0. Because
DE(F)
DF
→0 at steady state, ℎ
must also vanish as 𝑥→∞. The functions 𝑔 and ℎ, subject to these conditions, can be expanded
in power series (see Appendix 2 for details). To first order, Eq. (1) reduces to:
DE(F)
DF
=
N
F −𝐵 𝐹(𝑥) −𝐹JJ ,
(2)
where A and B are constants. Solving this yields:
F 𝑥= 𝐴𝑒TUF
6VWX
FX
F
FY
𝑑𝑥K + 𝐹JJ 1 −𝑒TUF .
(3)
Here, the lower bound of the integral, 𝑥\, characterizes the initial configuration of the asperities
and controls the magnitude of the friction peak (Fig. 5a). This theoretical prediction is consistent
with experimental results (Supplementary Fig. S5). The increasing and decreasing RSF transients
in fact reflect segments of F(x) at different sliding rates and 𝑥\ values (Fig. 5b).
This derivation, based solely on the steady-state behavior of friction, provides a rigorous
mathematical foundation for the equation rather than a phenomenological description of
transients4,5,37. Moreover, more complex frictional responses can be captured by expanding Eq. (1)
to higher orders, much like the virial expansion in the thermodynamics of real gases.
7
Fig. 5 | The broad friction peak and RSF segments. a, Equation (3) plotted with different values of the
lower bound 𝑥\ (in units of B-1, where B is the system constant) and with A/Fss = 0.6. b, The RSF transients
are different segments of the broad friction peaks described by Eq. (3) at different sliding rates (v1 and v2
= 0.1v1), plotted as a function of time.
Materials and Methods
Friction experiments were conducted using an Anton Paar MCR 302 rheometer equipped with an
electrically commutated (EC) motor that generates and measures torque in the nanonewton-meter
range. This instrument enables exceptionally low rotational speeds down to 10-8 s-1, equivalent to
one full rotation every three years. Polypropylene spheres (2.45 mm diameter, Cospheric),
polytetrafluoroethylene (PTFE) spheres (3.18 mm diameter, Goodfellow), and steel ball bearings
(2 mm diameter) were used in the friction experiments. The test sphere is mounted using a custom-
built holder at a distance r = 7 mm from the axis of rotation to convert this precise, ultra-slow
rotational motion into linear displacement. The rheometer presses the sphere into a glass substrate
and drives it along a circular path while measuring the normal and friction forces. The entire setup
is placed on an anti-vibration optical table in a quiet, low-noise environment. Such noise
suppression is essential, as environmental noise can contribute to the configurational evolution of
the asperities.
The PTFE, polypropylene, and steel spheres had root-mean-square (RMS) surface roughness
values of approximately 1.6 µm, 0.5 µm, and 0.4 µm, respectively. Surface roughness was
measured using a 3D laser scanning microscope (Keyence VK-X1000).
Sliding distance
Friction
0.0001 B-1
0.0003 B-1, v1
0.001 B-1
0.003 B-1
0.01 B-1, v2 = 0.1v1
Time
Friction
a
b
8
Fig. S1 | Experimental setup used in this study. Schematic illustrations of the system used for the friction
experiments. The rheometer presses a test sphere into a glass substrate and drives it along a circular path.
Fig. S2 | Non-monotonic friction transients (broad friction peaks). Complete friction transients for a,
polypropylene–glass, and b, steel–glass interfaces at a sliding rate of 4 nm s-1. Each panel shows three
curves: (1) a freshly formed, unsheared interface (black); (2) the same interface after a 5 s pause in sliding
(grey); and (3) the same interface after a pause in which the surfaces were separated and then re-contacted
(yellow).
0
1000
2000
3000
Time (s)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Friction coefficient
Polypropylene-glass
0
200
400
0
0.1
0.2
0.3
1
2
3
a
0
1000
2000
3000
4000
Time (s)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Friction coefficient
Steel-glass
0
200
400
600
800
0
0.05
0.1
1
2
3
b
9
Fig. S3 | Non-monotonic friction from earlier rate-and-state studies.6 Friction transients for a, Lucite
plastic [adapted from Fig. 1 in Ref. 6] and b, Acrylic plastic [adapted from Fig. 8 in Ref. 6]. The imposed
sliding rates are indicated in the panels. In both a and b, at the slower sliding rate, the measured friction
shows a slow decline after an increasing part, similar to the trends observed in Fig. 2c of this study.
However, the measurement duration was insufficient to capture the full non-monotonic peak. In b, the
orange and red dashed curves represent the best fit of Eq. (3) with A/Fss = 0.6 and 𝑥\ = 0.01 B-1.
Appendix 1. The rate-and-state friction equations do not permit non-monotonic solutions
In the rate-and-state friction (RSF) framework (For a brief overview of the RSF laws, see Box 1
in Ref. 4; for a comprehensive review, refer to Ref. 5), the friction coefficient, µ, is described as a
function of the slip velocity V and a history-dependent state variable θ:
𝜇= 𝜇\ + 𝑎𝑙𝑛
f
fY + 𝑏 𝑙𝑛 (
fYh
i ).
(1)
Here, V0 is a reference velocity, µ0 is the steady-state friction coefficient at V = V0, L is a critical
slip distance, and 𝑎 and 𝑏 are material-dependent constants4. The state variable θ is assumed to
evolve according to:
Dh
Dk = 1 −
hf
i .
(2)
This differential equation can be solved analytically, yielding:
𝜃=
i
f + (𝜃\ −
i
f)𝑒Tl
mk,
(3)
where θ0 is the integration constant. To demonstrate that the RSF equations do not permit non-
monotonic friction transients, I calculate the first derivative of µ and show that it does not change
a
b
10
sign. From Eq. (1):
Dn
Dk = 𝑏
=
h
Dh
Dk.
(4)
Using Eq. (2) to substitute for
Dh
Dk, we obtain:
Dn
Dk = 𝑏(
=
o −
f
i).
(5)
For the derivative to change sign, it must pass through zero, which requires:
=
o =
f
i.
(6)
Substituting this into equation (3) gives:
𝜃\ −
i
f 𝑒Tl
mk = 0
(7)
This condition can be satisfied only in the limit t → ∞ or if 𝜃\ =
i
f. However, if 𝜃\ =
i
f, Eq. (3)
implies θ is constant, then µ is also constant in Eq. (1), which contradicts the existence of any
transient frictional evolution.
Fig. S4 | RSF transients and their recurrence after perturbation. RSF transients at a PTFE–glass
interface at sliding rates of a, 1 µm s-1 and b, 4 nm s-1. In both experiments, a perturbation is introduced
after approximately 200 s by dropping a 200 g sandbag (a rubber balloon filled with fine sand) from a height
of 20 cm onto the experimental table, which triggers the re-emergence of the RSF transients.
a
b
0
100
200
300
400
Time (s)
0
0.1
0.2
Friction coefficient
0
100
200
300
400
Time (s)
0
0.05
0.1
0.15
Friction coefficient
11
Appendix 2. Power series expansions of functions 𝑔 and h
The functions ℎ (𝑥) and 𝑔𝐹(𝑥) −𝐹JJ , subject to the conditions 𝑔→0 as 𝐹 𝑥−𝐹JJ →0 and
ℎ→0 as 𝑥→∞, very generally can be expanded in power series as:
ℎ𝑥=
𝑎q
𝑥q
r
q<=
= 𝑎=
𝑥+ 𝑎s
𝑥s + 𝑎t
𝑥t + ⋯
𝑔𝐹𝑥−𝐹JJ =
𝑔q 0
𝑛!
r
q<=
𝐹𝑥−𝐹JJ
q
= 𝑔K 0 × 𝐹𝑥−𝐹JJ + 𝑔KK 0
2!
× 𝐹𝑥−𝐹JJ
s + 𝑔t 0
3!
× 𝐹𝑥−𝐹JJ
t + ⋯
12
Fig. S5 | The broad friction peak and our first-principles theory. The sample sphere and holder (Fig.
S1b) are not perfectly rigid and undergo elastic deformation under applied shear force. The rheometer
records the sum of interfacial sliding and this deformation. The interfacial sliding is obtained by subtracting
the elastic contribution from the raw data. a, Friction data from Fig. S2a for polypropylene–glass, plotted
against total deformation, with color coding as in Fig. S2a (zoom on the increasing segment). In the grey
curve, the initial linear force–deformation region (red dashed fit) reflects the system’s elastic response. The
horizontal offset between the elastic line through the origin (blue dashed) and the y-axis gives the elastic
deformation at each force level. The true interfacial sliding at any point on the black curve equals its
horizontal distance from the blue dashed line. b, Polypropylene data from Fig. S2a (black curve) after
elastic correction, fitted with Eq. (3); µk = FSS/N = 0.25, B = 1.93 µm-1, A/N = 0.16, and 𝑥\ = 0.094 µm.
c,d, PTFE data from Fig. 3c (dark blue and light blue curves, respectively) after elastic correction, fitted
with Eq. (3); µk = FSS/N = 0.052, B = 0.32 µm-1, A/N = 0.027, with 𝑥\ = 0.011 µm (c, dark blue) and 𝑥\
= 0.016 µm (d, light blue).
0
0.5
1
1.5
2
Total deformation ( m)
0
0.1
0.2
0.3
0.4
Friction coefficient
0
4
8
12
Sliding distance ( m)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Friction coefficient
0
1
2
0
0.1
0.2
0.3
Raw Data
Corrected Data
Fitted Curve (Eq. 3)
a
b
Elastic
deformation
0
10
20
30
Sliding distance ( m)
0
0.05
0.1
0.15
Friction coefficient
0
1
2
0
0.05
0.1
Raw Data
Corrected Data
Fitted Curve (Eq. 3)
d
0
10
20
30
Sliding distance ( m)
0
0.05
0.1
0.15
Friction coefficient
0
1
2
0
0.05
0.1
Raw Data
Corrected Data
Fitted Curve (Eq. 3)
c
13
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|
1 Unifying Frictional Transients Reveals the Origin of Static Friction Authors: Kasra Farain1, Daniel Bonn1 1. Van der Waals-Zeeman Institute, ; Science Park 904, 1098 XH Amsterdam, Netherlands. Abstract: Frictional motion is harder to initiate than to sustain, as evident when pushing a heavy object. This disparity between static and kinetic friction drives instabilities and stick-slip dynamics in systems ranging from nanodevices1 and MEMS2 to squealing brakes, glaciers3 and tectonic faults4, yet its origin and the transition mechanism remain poorly understood. Empirical rate-andstate friction laws4,5 predict that during the static-to-kinetic transition, friction increases for nanometer-per-second slip rates, but decreases for micrometers-per-second rates and above. These transients are believed to be associated with contact strengthening (aging) at static interfaces5,6, although their physical basis is unclear and the crossover between regimes has never been observed directly. Here we show, through nanometer-resolution sliding experiments on macroscopic rough surfaces, that these transients are segments of a single, universal non-monotonic response whose peak defines static friction. We show that this behavior arises from mechanical reorganization of interlocking surface asperities under shear-fundamentally distinct from contact aging, which is governed by thermal molecular processes6-8. We derive, from first principles and without invoking any empirical postulates, a differential equation that quantitatively captures the friction peak. These results unify frictional transients across scales and speeds, and establish a physics-based framework for understanding frictional instabilities and failure processes in engineering and geosciences. Main text: Friction has puzzled scientists for centuries. In the late 15th century, Leonardo da Vinci observed that the lateral frictional force F required to slide one body over another scales with the normal force N pressing them together and is independent of their macroscopic contact area-a principle later formalized by Amontons (1699) as F = μN 9,10. The proportionality constant, the friction coefficient μ, depends on the materials and whether the interface is static (μs) or kinetic (μk), with μs consistently exceeding μk (Fig. 1). In 1785, Coulomb further demonstrated that in slide-pause-slide sequences, μs increases logarithmically with pause duration, a phenomenon now known as contact aging. Yet despite this centuries-long history, the fundamental questions of why μs always exceeds μk and how the transition occurs remain unresolved11-22. 2 Fig. 1 | Static versus kinetic friction. A block on an inclined plane does not slip until the slope exceeds the static threshold θs = tan-1 (μs), derived from F = μN. Once sliding begins, the block continues moving even on shallower slopes down to θk = tan-1 (μk), demonstrating that μs > μk. The phenomenological rate-and-state friction (RSF) laws4,5-widely applied in tribology and earthquake studies since the 1970s4-6,11,23-36-describe two distinct slip behaviors depending on slip velocity. At high velocities (typically > 1 μm s-1), the friction coefficient decreases smoothly from its static μs to kinetic μk value during slip onset. At low velocities (typically < 100 nm s-1), the friction coefficient begins below μk and increases with displacement (Fig. 2a). RSF theory attributes these transients to contact aging (Coulomb's observation)5,6 but this interpretation poses a fundamental problem. Real surfaces are microscopically rough, covered with asperities. Contact aging at static interfaces arises through viscoelastic relaxation7 and creep6,8 of these surface features under sustained pressure, or through molecular bonding at asperity microcontacts in hard materials1. These mechanisms clearly require asperity micro-contacts to remain static. Yet RSF transients are observed during sliding, when asperities continually reorganize and micro-contacts break and reform. How can these processes really coexist? Here we demonstrate that the increasing and decreasing RSF transients emerge together in a single sufficiently long experiment at nanometer-per-second sliding velocities, manifesting as a continuous, broad (~10 μm) friction peak. This non-monotonic behavior unifies low- and highvelocity RSF experimental observations, but directly contradicts RSF theory, which permits only monotonic friction evolution. Through systematic slide-pause-slide and slide-perturbation-slide experiments, we show that these peaks originate from mechanical reorganization of surface asperities during sliding-a process fundamentally distinct from contact aging, which occurs through spontaneous, thermally activated molecular processes (viscoelastic creep and relaxation or chemical bonding). We observe contact aging in the same system in slide-pause-slide experiments with longer pauses: it produces much narrower (< 200 nm), discontinuous friction peaks whose height increases logarithmically with pause duration. Crucially, the mechanical reconfiguration of asperities requires external shear and does not occur spontaneously. Guided by this insight, we derive a differential equation governing the dynamics of asperities under shear that quantitatively reproduces the broad friction peaks. Our derivation introduces no empirical assumptions and relies solely on the existence of a unique steady state (corresponding to kinetic friction) and the mathematical conditions the system must satisfy at steady state. Beyond mg θk θs 3 implications for tribology and earthquake studies, our results establish that the static friction peak between rough solids is a mathematical necessity rather than merely an empirical observation. Figure 2b,c show friction as a function of time at a polytetrafluoroethylene (PTFE)-glass interface under an imposed sliding rate of 2 nm s-1. Initially (Fig. 2b), the result appears to agree with RSF predictions for the slow-sliding regime. However, measuring for a longer period (Fig. 2c) reveals that the friction coefficient subsequently decreases, exhibiting a broad peak before reaching steady state. We also observe this non-monotonic behavior for polypropylene-glass and steel-glass contacts (Supplementary Fig. S2), and-though previously unnoticed-it is clearly visible in earlier experiments on Lucite and acrylic plastics that were believed to confirm the RSF model (Supplementary Fig. S3)6. As already mentioned, RSF formulations, by construction, exclude nonmonotonic friction; the first derivative of μ has a constant sign in RSF (Appendix 1). These observations therefore reveal fundamental limitations of RSF theory in capturing the dynamics of the onset of sliding. Fig. 2 | Universal friction peak unifies RSF behaviors across slip velocities. a, Predictions of RSF theory for friction transients during slip initiation: friction strengthens in the slow-slip regime but weakens in the fast-slip regime4. b, Friction versus time at a PTFE-glass interface under imposed sliding at 2 nm s-1, apparently consistent with RSF predictions for the slow-sliding regime. c, Continuation of the measurement in b reveals a non-monotonic friction peak that encompasses both slow- and fast-sliding RSF behaviors. What causes the broad friction peak? Macroscopic friction arises from the collective contribution of microscopic forces between asperities on opposing surfaces. When two rough surfaces initially contact, surface asperities adopt random orientations within the frictional interface with no preferred direction. In this state, microscopic forces between asperities cancel in the horizontal direction, yielding no net friction (Fig. 3a). Under applied shear, however, asperities deform and slide past one another, reorganizing to develop a directional bias. At steady-state sliding, the horizontal components of the microscopic asperity forces sum to produce the macroscopic kinetic friction force (Fig. 3b). We demonstrate that the broad friction peak emerges from this configurational transition of the asperity ensemble: it appears if and only if the initiation of sliding requires configurational reorganization. 0 2000 4000 6000 8000 Time (s) 0 0.05 0.1 0.15 Friction coefficient 0 100 200 300 400 500 Time (s) 0 0.05 0.1 0.15 Friction coefficient Time Friction Slow Fast a b c 4 To do so, we paused and resumed the sliding at the frictional interface of Fig. 2c at different times using three protocols (Fig. 3c). In the first pause (Fig. 3c, t = 10,000 s), we turned off the applied sliding velocity and restarted it a few seconds later. Remarkably, sliding recommenced without a friction peak, suggesting that during the brief pause, asperities retained memory of the previous sliding direction and their configuration did not revert to randomness. For the second pause (Fig. 3c, t = 15,000 s), we switched from applied sliding velocity to an applied shear force slightly below Fss (approximately 20% reduction). This halted sliding motion while maintaining the shear force that holds asperities in their sliding configuration. In this way, the anisotropic steady-state asperity configuration at the moment of the switch was preserved; the same microscopic forces, reduced proportionally by 20%, continued to balance the reduced macroscopic force (Fig. 3b). When we switched back to applied sliding velocity 1 s later, no friction peak emerged, confirming that the static-to-kinetic friction peak does not occur when asperities are already in their steadystate sliding configuration. Finally, in the third pause (Fig. 3c, t = 20,000 s), the surfaces were separated and re-contacted. This erased the configurational memory, re-randomized the asperity ensemble, and thereby restored the broad friction peak. Importantly, interface re-randomization and reemergence of the friction transient can also be induced by 'seismic' pulses applied during either paused or continuous sliding (Supplementary Fig. S4), suggesting broader implications for earthquake dynamics. Similar behaviors are observed at polypropylene-glass and steel-glass interfaces (Supplementary Fig. S2). Before deriving a first-principles theory for the broad friction peaks, we also show that contact aging and the broad friction peak (or its RSF slices) are indeed different phenomena. To observe contact aging in our system, we perform slide-pause-slide experiments like those in Fig. 3c at t = 10,000 s but with extended pause durations. Figure 4a displays a typical contact aging peak after a 20 min pause alongside the broad friction peak from asperity reorganization, highlighting that they operate on vastly different length scales. The aging peak grows logarithmically with pause duration, consistent with the rate of viscoelastic stress relaxation in asperities under pressure7. Viscoelastic creep of asperities, the counterpart of stress relaxation, may deform their shape and increase the real contact area between contacting surfaces during extended pauses, but cannot alter the spatial arrangement of asperities or re-randomize the ensemble (Fig. 4, inset). 5 Fig. 3 | The origin of the broad friction peak at the onset of sliding. a,b, Schematic showing asperities at a rough frictional interface, immediately after contact (a) and at steady-state sliding (b). Under quasistationary conditions, where inertia is negligible, Newton's second law requires that macroscopic friction equals the vector sum of shear components from microscopic forces at asperity micro-contacts. This sum is zero for randomly oriented asperities in a but reaches a finite steady-state value after alignment under sliding in b. c, Continuation of the measurement in Fig. 2c. At t = 10,000 s, the sliding was stopped and restarted after 5 s. At t = 15,000 s, the applied sliding velocity was switched for 1 s to an applied force approximately 20% below Fss, which stopped the sliding. At t = 20,000 s, the sliding was stopped, the surfaces were separated and re-contacted, and then the sliding was restarted. Fig. 4 | Asperity reorganization peak versus contact aging peak. Comparison of the broad friction peak from asperity reorganization (from Fig. 2c) with a typical aging peak from a slide-pause (20 min)-slide Friction = ∑ f⃗45678 = 0 : ;<= f1 f2 f3 f4 f5 f6 f1 f2 f3 f4 f5 f6 Friction = ∑ f⃗45678 = F44 : ;<= Before sliding, random asperities Under sliding, aligned asperities c a b 0 5000 10000 15000 20000 25000 Time (s) 0 0.05 0.1 0.15 0.2 Friction coefficient 14998 15000 15002 0.04 0.05 10000 10200 0 0.03 0.06 20000 20400 0 0.05 0.1 0 2 4 6 Displacement ( m) 0 0.05 0.1 0.15 Friction coefficient Asperity reorganization Contact aging Creep 6 experiment7 on a PTFE-glass interface. The schematic inset zooms in on one asperity from Fig. 3b (highlighted also in Fig. 3b), showing its viscoelastic creep during aging. Such process cannot re-randomize the spatial configuration of the ensemble of asperities. First-principles theory of the broad friction peak. The reorganization in the asperity ensemble at rough interfaces is driven by the externally imposed sliding displacement x. The resulting friction F is therefore a function of displacement, F(x). We assume that the system possesses a unique steady-state (kinetic) friction FSS. At steady-state sliding, the asperity ensemble remains statistically unchanged with further displacement, which is why F(x) = FSS remains constant. When the system deviates from steady state F(x) ≠ FSS, net changes occur in the asperity configurations during sliding, driving it back towards the steady state. We represent this tendency by a restoring force g that naturally depends on the system's deviation from steady state F(x) - FSS and vanishes once steady state is reached F(x) = FSS. We can write: DE(F) DF = hx-gF(x) -FJJ , (1) where g and h are unknown functions. Essentially, we have merely identified a restoring contribution gx= gF(x) -FJJ within the general function FK(x) = DE(F) DF that we are attempting to determine. We could assume a restoring force linear in displacement-like in linearresponse theory, the Langevin equation, and the fluctuation-dissipation theorem-but no fundamental argument compels linearity; we therefore begin with the most general, potentially nonlinear form. The boundary conditions on g and h at steady state (x→∞, F (x) -FJJ →0) are crucial to simplify the above equation. As F (x) -FJJ →0, g→0. Because DE(F) DF →0 at steady state, h must also vanish as x→∞. The functions g and h, subject to these conditions, can be expanded in power series (see Appendix 2 for details). To first order, Eq. (1) reduces to: DE(F) DF = N F -B F(x) -FJJ , (2) where A and B are constants. Solving this yields: F x= AeTUF 6VWX FX F FY dxK + FJJ 1 -eTUF . (3) Here, the lower bound of the integral, x\, characterizes the initial configuration of the asperities and controls the magnitude of the friction peak (Fig. 5a). This theoretical prediction is consistent with experimental results (Supplementary Fig. S5). The increasing and decreasing RSF transients in fact reflect segments of F(x) at different sliding rates and x\ values (Fig. 5b). This derivation, based solely on the steady-state behavior of friction, provides a rigorous mathematical foundation for the equation rather than a phenomenological description of transients4,5,37. Moreover, more complex frictional responses can be captured by expanding Eq. (1) to higher orders, much like the virial expansion in the thermodynamics of real gases. 7 Fig. 5 | The broad friction peak and RSF segments. a, Equation (3) plotted with different values of the lower bound x\ (in units of B-1, where B is the system constant) and with A/Fss = 0.6. b, The RSF transients are different segments of the broad friction peaks described by Eq. (3) at different sliding rates (v1 and v2 = 0.1v1), plotted as a function of time. Materials and Methods Friction experiments were conducted using an Anton Paar MCR 302 rheometer equipped with an electrically commutated (EC) motor that generates and measures torque in the nanonewton-meter range. This instrument enables exceptionally low rotational speeds down to 10-8 s-1, equivalent to one full rotation every three years. Polypropylene spheres (2.45 mm diameter, Cospheric), polytetrafluoroethylene (PTFE) spheres (3.18 mm diameter, Goodfellow), and steel ball bearings (2 mm diameter) were used in the friction experiments. The test sphere is mounted using a custombuilt holder at a distance r = 7 mm from the axis of rotation to convert this precise, ultra-slow rotational motion into linear displacement. The rheometer presses the sphere into a glass substrate and drives it along a circular path while measuring the normal and friction forces. The entire setup is placed on an anti-vibration optical table in a quiet, low-noise environment. Such noise suppression is essential, as environmental noise can contribute to the configurational evolution of the asperities. The PTFE, polypropylene, and steel spheres had root-mean-square (RMS) surface roughness values of approximately 1.6 μm, 0.5 μm, and 0.4 μm, respectively. Surface roughness was measured using a 3D laser scanning microscope (Keyence VK-X1000). Sliding distance Friction 0.0001 B-1 0.0003 B-1, v1 0.001 B-1 0.003 B-1 0.01 B-1, v2 = 0.1v1 Time Friction a b 8 Fig. S1 | Experimental setup used in this study. Schematic illustrations of the system used for the friction experiments. The rheometer presses a test sphere into a glass substrate and drives it along a circular path. Fig. S2 | Non-monotonic friction transients (broad friction peaks). Complete friction transients for a, polypropylene-glass, and b, steel-glass interfaces at a sliding rate of 4 nm s-1. Each panel shows three curves: (1) a freshly formed, unsheared interface (black); (2) the same interface after a 5 s pause in sliding (grey); and (3) the same interface after a pause in which the surfaces were separated and then re-contacted (yellow). 0 1000 2000 3000 Time (s) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Friction coefficient Polypropylene-glass 0 200 400 0 0.1 0.2 0.3 1 2 3 a 0 1000 2000 3000 4000 Time (s) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Friction coefficient Steel-glass 0 200 400 600 800 0 0.05 0.1 1 2 3 b 9 Fig. S3 | Non-monotonic friction from earlier rate-and-state studies.6 Friction transients for a, Lucite plastic [adapted from Fig. 1 in Ref. 6] and b, Acrylic plastic [adapted from Fig. 8 in Ref. 6]. The imposed sliding rates are indicated in the panels. In both a and b, at the slower sliding rate, the measured friction shows a slow decline after an increasing part, similar to the trends observed in Fig. 2c of this study. However, the measurement duration was insufficient to capture the full non-monotonic peak. In b, the orange and red dashed curves represent the best fit of Eq. (3) with A/Fss = 0.6 and x\ = 0.01 B-1. Appendix 1. The rate-and-state friction equations do not permit non-monotonic solutions In the rate-and-state friction (RSF) framework (For a brief overview of the RSF laws, see Box 1 in Ref. 4; for a comprehensive review, refer to Ref. 5), the friction coefficient, μ, is described as a function of the slip velocity V and a history-dependent state variable θ: μ= μ\ + aln f fY + b ln ( fYh i ). (1) Here, V0 is a reference velocity, μ0 is the steady-state friction coefficient at V = V0, L is a critical slip distance, and a and b are material-dependent constants4. The state variable θ is assumed to evolve according to: Dh Dk = 1 - hf i . (2) This differential equation can be solved analytically, yielding: θ= i f + (θ\ - i f)eTl mk, (3) where θ0 is the integration constant. To demonstrate that the RSF equations do not permit nonmonotonic friction transients, I calculate the first derivative of μ and show that it does not change a b 10 sign. From Eq. (1): Dn Dk = b = h Dh Dk. (4) Using Eq. (2) to substitute for Dh Dk, we obtain: Dn Dk = b( = o - f i). (5) For the derivative to change sign, it must pass through zero, which requires: = o = f i. (6) Substituting this into equation (3) gives: θ\ - i f eTl mk = 0 (7) This condition can be satisfied only in the limit t → ∞ or if θ\ = i f. However, if θ\ = i f, Eq. (3) implies θ is constant, then μ is also constant in Eq. (1), which contradicts the existence of any transient frictional evolution. Fig. S4 | RSF transients and their recurrence after perturbation. RSF transients at a PTFE-glass interface at sliding rates of a, 1 μm s-1 and b, 4 nm s-1. In both experiments, a perturbation is introduced after approximately 200 s by dropping a 200 g sandbag (a rubber balloon filled with fine sand) from a height of 20 cm onto the experimental table, which triggers the re-emergence of the RSF transients. a b 0 100 200 300 400 Time (s) 0 0.1 0.2 Friction coefficient 0 100 200 300 400 Time (s) 0 0.05 0.1 0.15 Friction coefficient 11 Appendix 2. Power series expansions of functions g and h The functions h (x) and gF(x) -FJJ , subject to the conditions g→0 as F x-FJJ →0 and h→0 as x→∞, very generally can be expanded in power series as: hx= aq xq r q<= = a= x+ as xs + at xt + ⋯ gFx-FJJ = gq 0 n! r q<= Fx-FJJ q = gK 0 × Fx-FJJ + gKK 0 2! × Fx-FJJ s + gt 0 3! × Fx-FJJ t + ⋯ 12 Fig. S5 | The broad friction peak and our first-principles theory. The sample sphere and holder (Fig. S1b) are not perfectly rigid and undergo elastic deformation under applied shear force. The rheometer records the sum of interfacial sliding and this deformation. The interfacial sliding is obtained by subtracting the elastic contribution from the raw data. a, Friction data from Fig. S2a for polypropylene-glass, plotted against total deformation, with color coding as in Fig. S2a (zoom on the increasing segment). In the grey curve, the initial linear force-deformation region (red dashed fit) reflects the system's elastic response. The horizontal offset between the elastic line through the origin (blue dashed) and the y-axis gives the elastic deformation at each force level. The true interfacial sliding at any point on the black curve equals its horizontal distance from the blue dashed line. b, Polypropylene data from Fig. S2a (black curve) after elastic correction, fitted with Eq. (3); μk = FSS/N = 0.25, B = 1.93 μm-1, A/N = 0.16, and x\ = 0.094 μm. c,d, PTFE data from Fig. 3c (dark blue and light blue curves, respectively) after elastic correction, fitted with Eq. (3); μk = FSS/N = 0.052, B = 0.32 μm-1, A/N = 0.027, with x\ = 0.011 μm (c, dark blue) and x\ = 0.016 μm (d, light blue). 0 0.5 1 1.5 2 Total deformation ( m) 0 0.1 0.2 0.3 0.4 Friction coefficient 0 4 8 12 Sliding distance ( m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Friction coefficient 0 1 2 0 0.1 0.2 0.3 Raw Data Corrected Data Fitted Curve (Eq. 3) a b Elastic deformation 0 10 20 30 Sliding distance ( m) 0 0.05 0.1 0.15 Friction coefficient 0 1 2 0 0.05 0.1 Raw Data Corrected Data Fitted Curve (Eq. 3) d 0 10 20 30 Sliding distance ( m) 0 0.05 0.1 0.15 Friction coefficient 0 1 2 0 0.05 0.1 Raw Data Corrected Data Fitted Curve (Eq. 3) c 13 References 1 Li, Q., Tullis, T. E., Goldsby, D. & Carpick, R. W. Frictional ageing from interfacial bonding and the origins of rate and state friction. Nature 480, 233-236, (2011). 2 Urbakh, M., Klafter, J., Gourdon, D. & Israelachvili, J. The nonlinear nature of friction. Nature 430, 525-528, (2004). 3 Thøgersen, K., Gilbert, A., Schuler, T. V. & Malthe-Sørenssen, A. Rate-and-state friction explains glacier surge propagation. Nature Communications 10, 2823, (2019). 4 Scholz, C. H. Earthquakes and friction laws. Nature 391, 37-42, (1998). 5 Marone, C. Laboratory-derived friction laws and their application to seismic faulting. Annual Review of Earth and Planetary Sciences 26, 643-696, (1998). 6 Dieterich, J. H. & Kilgore, B. D. Direct observation of frictional contacts: New insights for state-dependent properties. Pure Appl. Geophys. 143, 283-302, (1994). 7 Farain, K. & Bonn, D. Predicting frictional aging from bulk relaxation measurements. Nature Communications 14, 3606, (2023). 8 Weber, B. et al. Molecular probes reveal deviations from Amontons' law in multiasperity frictional contacts. Nature Communications 9, 888, 02981-y (2018). 9 Dowson, D. History of Tribology. (Longman, 1979). 10 Persson, B. N. J. Sliding Friction Physical Principles and Applications. (Springer, 2000). 11 Sirorattanakul, K., Larochelle, S., Rubino, V., Lapusta, N. & Rosakis, A. J. Sliding and healing of frictional interfaces that appear stationary. Nature, 08673-0 (2025). 12 Ben-David, O. & Fineberg, J. Static Friction Coefficient Is Not a Material Constant. Physical Review Letters 106, 254301, (2011). 13 Peng, L., Roch, T., Bonn, D. & Weber, B. Decrease of Static Friction Coefficient with Interface Growth from Single to Multiasperity Contact. Physical Review Letters 134, 176202, (2025). 14 Rubinstein, S. M., Cohen, G. & Fineberg, J. Contact Area Measurements Reveal Loading-History Dependence of Static Friction. Physical Review Letters 96, 256103 (2006). 15 Dillavou, S. & Rubinstein, S. M. Nonmonotonic Aging and Memory in a Frictional Interface. Physical Review Letters 120, 224101, (2018). 16 Gvirtzman, S., Kammer, D. S., Adda-Bedia, M. & Fineberg, J. How frictional ruptures and earthquakes nucleate and evolve. 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Length Scale Effect in Frictional Aging of Silica Contacts. Physical Review Letters 125, 215502, (2020). 25 Tian, K., Goldsby, D. L. & Carpick, R. W. Rate and State Friction Relation for Nanoscale Contacts: Thermally Activated Prandtl-Tomlinson Model with Chemical Aging. Physical Review Letters 120, 186101, (2018). 26 Putelat, T., Dawes, J. H. P. & Willis, J. R. On the microphysical foundations of rate-andstate friction. Journal of the Mechanics and Physics of Solids 59, 1062-1075, (2011). 27 Kato, N. & Tullis, T. E. A composite rate- and state-dependent law for rock friction. Geophysical Research Letters 28, 1103-1106, (2001). 28 Berthoud, P., Baumberger, T., G'Sell, C. & Hiver, J. M. Physical analysis of the stateand rate-dependent friction law: Static friction. Physical Review B 59, 14313-14327, (1999). 29 Dieterich, J. H. & Kilgore, B. Implications of fault constitutive properties for earthquake prediction. 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2510.14767
|
The exclusive photoproduction of χcγ pairs in the small-x kinematics
M. Siddikov,1, 2 I. Zemlyakov,1, 3 and M. Roa1, 4
1Departamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile
2Centro Científico - Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile
3Instituto de Física, Pontificia Universidad Católica de Valparaíso, Av. Brasil 2950, Valparaíso, Chile
4Facultad de Ingeniería, Laboratorio DataScience, Universidad de Playa Ancha,
Leopoldo Carvallo 270, Valparaíso, Chile,
In this manuscript we analyze the exclusive photoproduction of the χcγ pairs.
We focus on
the small-x kinematics and evaluate the cross-sections in the Color Glass Condensate framework.
We found that in the leading order in the strong coupling αs, this process is sensitive only to
the forward color dipole scattering amplitude.
We estimated numerically the cross-sections for
different polarizations of χc mesons in the kinematics of the ultraperipheral collisions at LHC and
the future Electron Ion Collider. We also analyzed the role of this process as a potential background
to exclusive χc photoproduction, which has been recently suggested as an alternative channel for
studies of odderons. According to our estimates, the cross-section of χcγ with undetected photons
is comparable to that of odderons at small momentum transfer |t| ≲1 GeV2, but becomes less
relevant at larger |t|. We also found that a dominant background to odderon-mediated production
of χc comes from the radiative decays of ψ(2S) mesons, which potentially can present a challenge
for odderon studies via χc photoproduction.
I.
INTRODUCTION
The Color Glass Condensate (CGC) approach [1–5] has been established as a theoretical framework for systematic
analysis of the hadronic processes in the small-x kinematics. It naturally incorporates saturation effects and allows us
to obtain a self-consistent and phenomenologically acceptable description of various lepton-hadron and hadron-hadron
processes [6–21]. The interactions of partons with the target in this approach are encoded in the nonperturbative
n-point correlators of Wilson lines (forward multipole scattering amplitudes), whose dependence on rapidity obeys
the evolution equations of Balitsky-Kovchegov Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (BK-
JIMWLK) [22–32]. In the presence of appropriate hard scale which controls the transverse size of the system, it is
expected that the dominant contribution stems from the quark-antiquark component of the wave function, convoluted
with forward dipole scattering amplitude. On this premise, a number of processes has been used in phenomenological
analyses in order to fix the forward dipole scattering amplitude. However, the higher order multipoles may contribute
in loop corrections even in the simplest and well-known processes, like Deep Inelastic Scattering (DIS) [33, 34]. A
recent numerical analysis of the next-to-leading order (NLO) corrections to various processes demonstrated that they
could be numerically sizable [35–38], thus shedding doubts on phenomenological extractions based on leading order
picture. Unfortunately, such estimates remain ambiguous due to dependence on poorly known parametrization of
these multipoint correlators. Conversely, a reasonable phenomenological description of the existing experimental data
by completely different parametrizations, disregarding NLO corrections altogether, suggests that the existing data
don’t uniquely fix the dipole amplitude [15–17].
While it is impossible to switch off completely the contributions of the higher twist correlators, it is possible to reveal
their presence testing the universality of the forward dipole scattering amplitude in various processes. The exclusive
2 →3 processes from this point of view are well-suited for this analysis, since they have a different structure of the
amplitude, and thus can be used as independent tools for phenomenological studies [39–47].Due to high luminosity
of the ongoing photoproduction experiments at LHC (in ultraperipheral kinematics) [48] and the future experiments
at the Electron Ion Collider (EIC) [49, 50], measurement of such processes is feasible in the nearest future. A special
interest in this program deserves the production of quarkonia-photon pairs. The heavy quark mass serves as a natural
hard scale in processes involving their production and justifies perturbative description of the Q ¯Q pair formation,
as well as the use of NRQCD framework for their hadronization [51–54].
Due to strong suppression of C-parity
exchanges in the t-channel, the largest cross-section have the quarkonia-photon pairs which include C-even quarkonia,
like ηc and χc. The production of ηcγ pairs has been studied recently in [55], and it was discussed how the expected
kinematic distributions of the final-state particles are related to the dipole amplitude. However, that channel suffers
from significant backgrounds, most notably from the radiative decays of other quarkonia, and for this reason requires
additional kinematic cutoffs which reduce the expected yields. In this paper we extend that analysis and suggest to
study the production of the χcγ meson pairs. In what follows we will focus on the high energy (small-x) photon-proton
collisions, which can be studied as a subprocess in ultraperipheral pA and AA collisions at LHC, as well as in the
future high energy runs at Electron Ion Collider, the Large Hadron electron Collider (LHeC) [56], and the Future
Circular Collider (FCC-he) [57].
arXiv:2510.14767v1 [hep-ph] 16 Oct 2025
2
In addition to studies of the dipole scattering amplitudes, the χcγ photoproduction can also present interest for two
other problems. Historically, the triplet of the P-wave quarkonia χc with different spins and nearly the same masses
has been considered in the literature as a possible tool for precision tests of NRQCD. In the infinitely heavy quark mass
limit, it is expected that the long-distance matrix elements (LDMEs) of all χc mesons are related to each other due
to heavy quark spin symmetry (HQSS). However, the experimental results for the ratio σχc2/σχc1in hadroproduction
experiments disagrees significantly with HQSS-based expectations, and the phenomenologically successful descriptions
of that channel either assume that HQSS is heavily broken for charm, or introduce sizable contributions of the color
octet LDMEs [58–62]. The study of the same ratio in exclusive χcγ photoproduction experiments can decisively clarify
the validity of HQSS because this channel does not get contributions from the color octet LDMEs. Furthermore, the
exclusive χc photoproduction (via γp →χcp channel) recently has been suggested in
[63, 64] as a sensitive tool
for precision studies of odderons. The photoproduction of χcγ pairs in this context deserves interest as a potential
background to odderon searches.
The paper is structured as follows.
In the next Section II we discuss in detail the kinematics of the process,
and derive the theoretical results for the cross-section in the Color Glass Condensate framework.
In Section III
we provide tentative estimates for the cross-sections and expected counting rates obtained with a publicly available
parametrization of the forward dipole scattering amplitude. We also estimate the contribution of this process and
the ψ(2S) →χcγ radiative decay to odderon-mediated γp →χcp suggested in [64]. Finally, in Section IV we draw
conclusions.
II.
THE THEORETICAL FRAMEWORK
As we explained in the introduction, we will analyze the γp →χcγp process in the high energy (small-x) kinematics,
using for description the CGC framework. We will focus on the kinematics of quasireal incoming photons, which
dominate the spectrum of equivalent photons in ultraperipheral collisions and electroproduction experiments.
In
the subsections II A - II D below we will discuss the kinematics of the process and the light-cone wave functions of
χc mesons with different helicities, introduce briefly the CGC framework and derive the amplitude of the process
in this approach.
The final theoretical expressions for different helicity components are given in Section II D, in
Eqns. (80 - 100, 110 - 129). The latter equations will be used for numerical estimates in the next section.
A.
Kinematics of the process
The evaluation of the invariant cross-section can be realized in the so-called photon-proton collision frame, where the
3-momenta of the incoming proton and photon are anticollinear to each other. For the sake of definiteness we’ll assume
that the collision axis coincides with the axis z of our reference frame, and that the photon and proton propagate
in its positive and negative directions respectively. The mass of the heavy quarkonium Mχc and the invariant mass
of the photon quarkonium pair Mγχcwill be considered as hard scales, in agreement with [65]. The evaluation of the
physical amplitude requires to define the light-cone decomposition of various momenta. In what follows we will utilize
the convention of Kogut-Soper [66], assuming that any 4-vector vµ can be represented as
vµ =
v+,
v−,
v⊥
,
v± = v0 ± v3
√
2
,
v⊥= vxˆx + vyˆy,
(1)
so its square and convolution with Dirac matrix are given by
v2 ≡vµvµ = 2v+v−−v2
⊥,
ˆv ≡γµvµ = γ+v−+ γ−v+ −γ⊥· v⊥.
(2)
The light-cone decomposition of particle’s momenta will be chosen in agreement with earlier studies on meson-photon
production [67, 68],
qµ =
rs
2, 0, 0⊥
,
kµ =
¯αχc
rs
2,
(kγ
⊥)2
¯αχc
√
2s, kγ
⊥
!
(3)
P µ
in =
m2
N
√
2s (1 + ξ), (1 + ξ)
rs
2, 0⊥
,
P µ
out =
m2
N + ∆2
⊥
√
2s (1 −ξ), (1 −ξ)
rs
2, ∆⊥
,
(4)
∆µ = P µ
out −P µ
in =
2ξm2
N + (1 + ξ) ∆2
⊥
√
2s (1 −ξ2)
, −2ξ
rs
2, ∆⊥
,
(5)
(6)
3
pµ
χc =
αχc
rs
2, (pχc
⊥)2 + M 2
χc
αχc
√
2s
, pχc
⊥
!
,
(7)
kγ
⊥:= p⊥−∆⊥
2 ,
pχc
⊥:= −p⊥−∆⊥
2 ,
p⊥≡(kγ
⊥−pχc
⊥) /2
(8)
where q and k are the momenta of the incoming and outgoing (emitted) photons, Pin, Pout are the momenta of the
proton before and after the interaction, and pχc is the momentum of produced χc meson. The parameter αχc stands for
the fraction of the photon’s light-cone momentum carried by χc, and a shorthand notation ¯αχc ≡1−αχc corresponds
to a similar light-cone fraction carried by the scattered photon. For the proton which moves in the opposite direction,
the parameter ξ controls the longitudinal momentum transfer to the proton during the process. The hard scale s can
be related to the photon-proton invariant energy W of the process as
SγN ≡W 2 = (q + Pin)2 = s (1 + ξ) + m2
N,
√s =
s
W 2 −m2
N
1 + ξ
.
(9)
As will be demonstrated below, for high energy collisions (W ≫mN), the variable ξ ≪1, so (9) can be approximated
as √s ≈W. The light-cone decomposition in the target rest frame (r.f.)
can be obtained from (3-8) making a
longitudinal boost
v+ →Λv+,
v−→Λ−1v−,
v⊥→v⊥
with Λ = √s (1 + ξ) /mN, yielding
qµ
r.f. =
s (1 + ξ)
mN
√
2 , 0, 0⊥
,
kµ
r.f. =
¯αχcs (1 + ξ)
√
2mN
,
(kγ
⊥)2
¯αχcs
√
2
mN
(1 + ξ), kγ
⊥
!
(10)
P µ
in,r.f. =
mN
√
2 , mN
√
2 , 0⊥
,
P µ
out =
m2
N + ∆2
⊥
√
2 (1 −ξ)
(1 + ξ)
mN
, mN
√
2
1 −ξ
1 + ξ , ∆⊥
,
(11)
∆µ
r.f. = P µ
out −P µ
in =
2ξm2
N + (1 + ξ) ∆2
⊥
√
2mN
1 + ξ
1 −ξ , −
√
2ξ mN
(1 + ξ)
, ∆⊥
!
(12)
pµ
χc,r.f. =
αχcs (1 + ξ)
√
2mN
, (pχc
⊥)2 + M 2
χc
αχcs
√
2
mN
(1 + ξ), pχc
⊥
!
.
(13)
The light-cone decomposition (3-8, 10-13) satisfies the energy-momentum conservation and guarantees that all par-
ticles are on shell. As we discussed in [55, 65], it is possible to express the variables ξ, p⊥, ∆⊥, αχc, which define the
kinematics, in terms of the invariant mass M 2
γχc, momentum transfers to the proton (t) and to the photon (t′).
t = ∆2 = (Pout −Pin)2 = −1 + ξ
1 −ξ ∆2
⊥−4ξ2m2
N
1 −ξ2 ≤−4ξ2m2
N
1 −ξ2 .
(14)
t′ = (k −q)2 = −2q · k = −(p⊥−∆⊥/2)2
(1 −αχc)
≤0
(15)
M 2
γχc = (k + pχc)2 = (q + Pin −Pout)2 = t −2q · ∆= t + 2sξ
(16)
Resolving these equations, it is possible to obtain exact identities [55, 65]
ξ =
M 2
γχc −t
2 (W 2 −m2
N) −M 2γχc + t > 0,
s = W 2 −m2
N + t −M 2
γχc
2
,
2ξs = M 2
γχc −t,
(17)
∆2
⊥= −t1 −ξ
1 + ξ −4ξ2m2
N
(1 + ξ)2 ,
p⊥· ∆⊥=
2ξm2
N
(1 −ξ2)
t′
s
(18)
p2
⊥= −t′
1
2 −αχc
+ 1
2
αχcM 2
γχc −M 2
χc
+ t
4
1 −ξ
1 + ξ −2αχc
+ ξ2m2
N
(1 + ξ)2 .
(19)
4
αχc = −1
2ξs
t′ −M 2
χc −
2ξm2
N
s (1 −ξ2)
M 2
γχc + t′ −t
−
2ξm2
N
s (1 −ξ2),
(20)
which fix the kinematics of the process in terms of the variables W, t, t′, Mγχc up to a global rotation in the transverse
plane. For high-energy processes we can simplify these relations using the hierarchy of scales W, √s ≫Mχc, Mγχc ≳
p⊥, ∆⊥. The parameter ξ defined in (17) is also very small, ξ ∼M 2
γχc/W 2 ≪1, and can be disregarded if accompanied
by other terms which don’t vanish in this limit. Similarly the smallness of the ratio t′/s ∼M 2
γχc/W 2 ∼O (ξ) ≪1 in
the right-hand side of (18) implies that the vectors p⊥, ∆⊥are almost orthogonal to each other. As was discussed long
ago in [69] and confirmed by various experimental studies, the cross-sections of exclusive processes decrease rapidly
as a function of the invariant momentum transfer t ∼∆2
⊥to the target. Technically, this happens because a large
momentum transfer to the target won’t destruct it only if the momentum is shared equally between all partons, and
each momentum exchange between partons is suppressed by (perturbative) propagators. As we will see below, such
suppression also takes place in our process, and for this reason in the kinematics of interest the variables ∆2
⊥, t are
negligible compared to typical scales ∼M 2
χc. In this kinematic, it is possible to rewrite (19, 20) in a simpler form (see
details in [55, 65])
−t′ ≈αχcM 2
γχc −M 2
χc,
p2
⊥= ¯αχc
αχcM 2
γχc −M 2
χc
≈−¯αχct′,
M 2
γχc ≈2sξ,
(21)
The physical constraint p2
⊥≥0 implies that the variable αχc is bound by αχc ≥M 2
χc/M 2
γχc. The cross-section of the
photoproduction subprocess may be represented as
dσ(T )
γp→γχcp
dt dt′ dMγχc
≈
A(λ; σ,H)
γp→χcγp
2
128π3Mγχc
(22)
where A(λ; σ,H)
γp→χcγp is the process amplitude, and the superscript indices λ, σ, H are the helicities of the incoming
photon, outgoing photon and the produced charmonium, respectively.The cross-section of the electroproduction is
given by [9, 70]
dσep→eγχcp
dΩ
= ϵdσ(L)
dΩ
+ dσ(T )
dΩ
+
p
ϵ(1 + ϵ) cos φdσ(LT )
dΩ
+
p
ϵ(1 + ϵ) sin φdσ(L′T )
dΩ
+
(23)
+ ϵ cos 2φdσ(T T )
dΩ
+ ϵ sin 2φdσ(T ′T )
dΩ
,
where φ is the angle between the lepton and proton scattering planes (see Figure 1 for definition), the notation
dΩ≡d ln W 2dQ2 dt dt′ dMγχcdφ is used for the phase volume, and the variable Q2 = −q2 is the virtuality of the
photon propagating from leptonic to hadronic parts. The superscript letters L, T, T ′ in the right-hand side of (23)
distinguish the contributions of the longitudinal and transverse photons (as well as their possible interference). The
parameter ϵ is the ratio of the longitudinal and transverse photon fluxes. The latter may be expressed in terms of
inelasticity parameter y and the invariant energy √sep of the electron-proton collision as
ϵ ≈
1 −y
1 −y + y2/2,
y = W 2 + Q2 −m2
N
sep −m2
N
.
(24)
In what follows we’ll focus on the kinematics of small Q2, which gives the dominant contribution to the total cross-
section. In this kinematics, the contribution of the longitudinally polarized photons is negligible, and (23) reduces
to
dσep→eM1M2p
dΩ
≈dσ(T )
dΩ
+ ϵ cos 2φdσ(T T )
dΩ
+ ϵ sin 2φdσ(T ′T )
dΩ
≈dσ(T )
dΩ
(1 + c2 cos 2φ + s2 sin 2φ) ,
(25)
where dσ(T )/dΩis related to the photoproduction cross-section (22) as [71–73]
dσ(T )
dΩ
≈αem
π Q2
1 −y + y2
2 −(1 −y)Q2
min
Q2
dσ(T )
γp→γχcp
dt dt′ dMγχc
,
(26)
the superscript index (T) is used to remind that the dominant contribution comes from the transversely polarized
photon, Q2
min = m2
ey2/ (1 −y), me is the mass of the electron, and the angular harmonics c2, s2 are related to different
components of the amplitude A(λ,σ,H)
γp→χcγp as
c2 =
2ϵ Re
A(−,+,H)∗
γp→χcγpA(+,+,H)
γp→χcγp
A(+,+)
γp→χcγp
2
+
A(−,+)
γp→χcγp
2 ,
s2 =
2ϵ Im
A(−,+,H)∗
γp→χcγpA(+,+,H)
γp→χcγp
A(+,+)
γp→χcγp
2
+
A(−,+)
γp→χcγp
2 .
(27)
5
Figure 1. (Color online) The definition of the angle φ between leptonic and hadronic planes for the ep →e′p′χcγ electropro-
duction channel, as seen from the target rest frame. The lepton scattering plane is formed by the three-momenta of incoming
and scattered electrons (marked with labels e, e′), and the hadron scattering plane may be defined as a plane that includes
the three-momenta of the recoiled proton Pout and the total three-momentum pχc + kγ of the produced γχc pair. Due to
momentum conservation, this plane also includes the 3-momentum of the virtual photon γ∗. In the photon-proton collision
frame, the incoming proton propagates in direction anticollinear to the incoming virtual photon γ∗, so the momentum of the
recoil proton Pout is almost anticollinear to the momentum of the virtual photon γ∗.
As we will see below, these coefficients are small, so the cross-section (25) is dominated by the angular-independent
term (the so-called equivalent photon approximation).
B.
The polarization vectors and wave functions
For the polarization vectors of the photons, we will use the light-cone parametrization
ε(λ=±1)
T
(ℓ) =
ελ · ℓ⊥
ℓ−
, 0, ελ
,
ελ = ˆx + iλ ˆy
√
2
.
(28)
where ℓis the 4-momentum of the corresponding photon (ℓ= q or k) in this process. For the polarization vector of
the axial vector quarkonium χc1, we’ll use a similar parametrization suggested in [64]
E(χc1)
T,λ
(pχc) =
εT · pχc
⊥
p−
χc
, 0, εT
,
E(χc1)
L
(pχc) =
(pχc
⊥)2 −M 2
χc
2Mχcp−
χc
, p−
χc
Mχc
, pχc
⊥
Mχc
!
,
(29)
where the subindices L and T = ±1 distinguish the transverse and longitudinal polarizations. This parametrization
automatically satisfies the transversality condition pχc · E(χc) = 0. The polarization tensor Eµν of the charmonium
χc2 can be constructed from polarization vectors defined in (29) using conventional Clebsch-Gordan coefficients as
E(χc2)
µν
=
E(χc1)
T =±1
µ
E(χc1)
T =±1
ν ,
Hχc2 = ±2,
E(χc1)
T =±1
µ
E(χc1)
L
ν+
E(χc1)
L
µ
E(χc1)
T =±1
ν
√
2
,
Hχc2 = ±1,
E(χc1)
T =+1
µ
E(χc1)
T =−1
ν+
E(χc1)
T =−1
µ
E(χc1)
T =+1
ν+2
E(χc1)
L
µ
E(χc1)
L
ν
√
6
,
Hχc2 = 0,
(30)
6
where Hχc2 is the helicity projection of the corresponding charmonium. The wave function of the χc mesons in helicity
basis in momentum space is given by
ΨH,h¯h = ¯uh (kQ) Γχc
kQ, k ¯
Q
v¯h
k ¯
Q
ϕχcJ
z, k(rel)
⊥
,
Γχc
kQ, k ¯
Q
=
1,
J = 0,
iγ5γµE(χc1)
µ
,
J = 1,
γµKν+γνKµ
2
E(χc2)
µν
J = 2.
(31)
where h, ¯h are the helicities of the quark, z and k(rel)
⊥
are the light-cone fraction and transverse momentum of the
quark (w.r.t direction of χc meson), and the momentum of relative motion Kµ ≡kµ
Q −kµ
¯
Q is expressed in terms of
these variables as
Kµ =
(2z −1) αχc
rs
2,
2pχc
⊥· K⊥+ (1 −2z)
h
(pχc
⊥)2 + M 2
χc
i
αχc
√
2s
, K⊥
.
(32)
The K−-component of the vector Kµ was fixed from the orthogonality condition K · pχc = 0, which follows from the
fact that the quarks are (nearly) onshell in the heavy quark mass limit. The transverse component of Kµ is related
to k(rel)
⊥
as
k(rel)
⊥
= ¯zkQ −zk ¯
Q = K⊥−(2z −1)pχc
⊥.
(33)
For χc0 mesons, the Fourier transformation of (31) over the transverse coordinates allows to obtain the light-cone
wave function
Ψ(χc0)
h¯h
(z, r) = −1
z¯z
ihe−ihθδh,¯h∂r + m (z −¯z) δh,−¯h
Φχc0 (z, r⊥) ,
(34)
where r = {r cos θ, r sin θ} is the vector of relative separation of the quark and antiquark in transverse plane, z is
the fraction of the momentum carried by the quark, and the configuration space wave function Φχc is related to the
function ϕχc
z, k(rel)
⊥
by relation
ΦχcJ (z, r⊥) =
Z d2k(rel)
⊥
(2 π)2
√z¯z ϕχcJ
z, k(rel)
⊥
eik(rel)
⊥
·r⊥
k(rel)
⊥
2
−M 2χcJz (1 −z) + m2
.
J = 0, 1, 2
(35)
The integrand of (35) includes additional light-cone denominator whose origin will become obvious in the next section
(see the denominators in the last lines of (71) and (72) below; also see Appendix of [55] for a more detailed discussion).
For the helicity components of the χc1 and χc2 we may obtain in a similar fashion
Ψ(χc1)
H=±1, h¯h (z, r) =
√
2
z¯z
−eiHθ ¯zδh,Hδ¯h,−H + zδh,−Hδ¯h,H
∂r −iHm (z −¯z) δh,Hδ¯h,H
Φχc1 (z, r) ,
(36)
Ψ(χc1)
H=0, h¯h (z, r) =
2i
z¯zMχc1
h
hδh,−¯h ∆(+)
r
+ ime−ihθδh,¯h∂r
i
Φχc1 (z, r) ,
(37)
where ∆(±)
r
= ∂2
r ± (1/r) ∂r, ¯z = 1 −z, and
Ψ(χc2)
H=+2, h¯h (z, r) = 2
z¯z
h
e2iθ zδh,+δ¯h,−−¯zδh,−δ¯h,+
∆(−)
r
+ imeiθδh,+δ¯h,+∂r
i
Φχc2 (z, r) ,
(38)
Ψ(χc2)
H=−2, h¯h (z, r) = 2
z¯z
h
−e−2iθ ¯zδh,+δ¯h,−−zδh,−δ¯h,+
∆(−)
r
−ime−iθδh,−δ¯h,−∂r
i
Φχc2 (z, r) ,
(39)
Ψ(χc2)
H=+1, h¯h (z, r) = Mχc2
z¯z
ieiθ 3z −4z2
δh,+δ¯h,−+
3¯z −4¯z2
δh,−δ¯h,+
∂r + m (z −¯z) δh,+δ¯h,+
Φχc2 (z, r) , (40)
Ψ(χc2)
H=−1, h¯h (z, r) = Mχc2
z¯z
−ie−iθ 3¯z −4¯z2
δh,+δ¯h,−+
3z −4z2
δh,−δ¯h,+
∂r + m (z −¯z) δh,−δ¯h,−
Φχc2 (z, r) ,
(41)
Ψ(χc2)
H=0, h¯h (z, r) =
p
2/3
z¯z
h
δh,−¯h
−3∆(+)
r
+ 2m2 (z −¯z)
−ime−ihθδh,¯h∂r
i
Φχc2 (z, r) .
(42)
7
1.
Relation to NRQCD LDMEs
In order to understand the relations of the light-cone distribution amplitudes to long distance matrix elements
(LDMEs) of NRQCD, we need to analyze the behaviour of the wave functions in the limit when Kµ becomes small.
Since we are interested in the P-wave quarkonia, in the limit Kµ →0 the corresponding vertices should vanish. This
is obvious for χc2 due to explicit Kµ in (31). To demonstrate that a similar property is valid for the states J = 0 and
J = 1, we need to rewrite (31) in equivalent form, using the Dirac equation for the ¯u, v spinors and the orthogonality
of the polarization vector EH · pχc = 0, namely
¯u (kQ)ΓχcJv
k ¯
Q
=
1
2m ¯u (kQ) (mΓχcJ + ΓχcJm) v
k ¯
Q
=
1
2m ¯u (kQ)
ˆkQΓχcJ −ΓχcJ ˆk1
v
k ¯
Q
(43)
=
1
4m ¯u (kQ) (ˆpχcΓχcJ −ΓχcJ ˆpχc) v
k ¯
Q
+ 1
4m ¯u (kQ)
ˆKΓχcJ + ΓχcJ ˆK
v
k ¯
Q
= ¯u (kQ) Γ(eff)
χcJ v
k ¯
Q
,
where Γ(eff)
χcJ
= Kµ (γµΓχcJ + ΓχcJγµ) /4m = γµ
χcJKµ, and the momentum-independent matrices γµ
χcJ for different
quarkonia are given by γµ
χc0 ≡γµ/2m, γµ
χc1 ≡
h
γµ, ˆEH
i
γ5/4m, and γµ
χc2 = Eµνγν. In the amplitudes of physical
processes, the wave function (31) contributes in convolution with amplitude of hard partonic process. If the latter
possess a hard scale which exceeds significantly the relative momentum Kµ (∼inverse charmonium radius), we can
disregard completely the dependence on z, k(rel)
⊥
in the hard partonic amplitude, and the convolution over z, K⊥will
affect only the arguments of the wave function, yielding
Z
dz
Z
d2k(rel)
⊥
(2π)2 Φχc
z, k(rel)
⊥
=
(44)
=
lim
λ, r⊥→0
0
¯Ψ
−λ
2 n+ −r⊥
2
γµ
χcJi←→
∂µ L
−λ
2 n+ −r⊥
2 , λ
2 n+ + r⊥
2
Ψ
λ
2 n+ + r⊥
2
χc(p)
=
0
¯Ψ (0) γµ
χcJ
i
2
←→
D µ Ψ (0)
χc(p)
where ←→
D µ = −→
∂µ −←−
∂µ + 2igAa
µta + 2ieAµ is the covariant derivative, and we replaced Kµ with i
−→
∂µ −←−
∂µ
in the
configuration space. The square of the matrix element in the last line of (44) in the nonrelativistic limit reduces to
NRQCD LDMEs. Indeed, using explicit form of γµ
χcJ for J = 0, 1, 2, we may recover the familiar NRQCD operators
from [74–76] 1
D
0
¯ψ (0) γµ
χcJi←→
D µ ψ (0)
χc(p)
E
2
=
1
4m2 ×
D
0
¯ψ
i
2
←→
D µ
⊤γ⊤
µ
χ
χc(p)
E
2
=
D
O[1]
χc
3P [1]
0
E
,
J = 0
D
0
ψ†
−i
4
←→
D ν
⊤
γ⊤
ν , γ⊤
µ
γ5 χ
χc(p)
E
2
=
D
O[1]
χc
3P [1]
1
E
,
J = 1
0
ψ†
−i
2
←→
D(µ
⊤γν)
⊤
χ
χc(p)
2
=
D
O[1]
χc
3P [1]
2
E
,
J = 2
(46)
where ⊤implies a part of the vector which is transverse to pµ
χc, namely vµ
⊤=
gµν −pµ
χcpν
χc/M 2
χc
vν; a(µ
⊤bν)
⊤=
(aµ
⊤bν
⊤+ aν
⊤bµ
⊤)/2 −a⊤· b⊤
gµν −pµ
χcpν
χc/M 2
χc
/3, the operators ψ†, χ are the Pauli spinor operators which create
a quark and an antiquark, respectively. The presence of the gauge fields in covariant derivative ←→
D µ implies that
we should consider the χc meson as a mixture of | ¯QQ⟩, | ¯QQg⟩and | ¯QQγ⟩Fock components unless we work in the
light-cone gauge A−= 0, A−
a = 0. In what follows we will use the latter gauge condition, both for photons and gluons.
In the heavy quark mass, it is expected that the color singlet LDMEs
D
O[1]
χc
3P [1]
J
E
should coincide, and in the
potential models can be represented in terms of the slope of the radial wave functions R′
χc(0) as,
D
O[1]
χc
3P [1]
J
E
= 6Nc (2J + 1)
R′
χc(0)
2 /4π.
(47)
1 In the rest frame of the proton, we may further simplify and rewrite these operators as [51]
D
0
¯ψ (0) γµ
χcJ i←
→
D µ ψ (0)
χc(p)
E
2
=
D
0
ψ†
−i
2
←
→
D · σ
χ
χc(p)
E
2
=
D
O[1]
χc
3P [1]
0
E
,
J = 0
D
0
ψ†
−i
2
←
→
D × σ
χ
χc(p)
E
2
=
D
O[1]
χc
3P [1]
1
E
,
J = 1
0
ψ†
−i
2
←→
D(iσj)
χ
χc(p)
2
=
D
O[1]
χc
2P [1]
2
E
,
J = 2
(45)
8
The latter parameter may be related to the light-cone function ϕχcJ
z, k(rel)
⊥
introduced in (31): after straightforward
but tedious substitution of the variables, we may obtain [77]
3
rπ
2 R′
χc(0) =
p
2MχcJ
Z
dz d2k(rel)
⊥
z(1 −z)8π2 ϕχcJ
z, k(rel)
⊥
"
M 2
χcJ
z −1
2
2
+ K2
⊥
#
.
(48)
In what follows we will use the explicit parametrization from [64] which in the light-cone representation is given by
ΦχcJ (z, r) = NχcJz(1 −z) exp
−R2
χcJ
8
m2
z(1 −z) −4m2
−2z(1 −z)r2
R2χcJ
!
,
(49)
and NχcJ, RχcJ are numerical constants given explicitly by Rχc0 ≈1.54 GeV−1, Rχc1 = Rχc2 ≈1.48 GeV−1, Nχc0 ≈
1.15 GeV−1, Nχc1 ≈1.39 GeV−1, Nχc2 ≈0.60 GeV−1 2. The values of |R′(0)| obtained with this parametrization
are in reasonable agreement with the value |R′(0)|2 = 0.075 GeV5 found in potential models and widely used in the
literature for numerical estimates. For the momentum space wave function, we may obtain
˜ΦχcJ (z, k) ≡
Z
d2re−ik·rΦχcJ (z, r) =
√z¯zϕχcJ (z, k)
k2 −M 2χcJz¯z + m2 = πNχcJR2
χcJ
2
exp
−R2
χcJ
8
k2 + m2(1 −4z¯z)
z¯z
!
.
(50)
C.
Scattering processes in the CGC framework
In the CGC framework the interaction of the high energy partons with the gluonic field of the target is described
in eikonal approximation by the Wilson line U(x⊥) defined as
U (x⊥) = P exp
ig
Z
dx−A+
a
x−, x⊥
ta
,
(51)
where we use notation x⊥for the impact parameter of the parton, Aµ
a for the gluonic field of the target, and ta are
the usual color group generators in the irreducible representation which corresponds to the parton ( 3, ¯3 or 8 for
quark, antiquark or gluon, respectively). The interaction encoded in Wilson link (51) formally may be reformulated
in terms of the CGC Feynman rules [12, 78–80] provided in Table I.
In CGC picture it is assumed that the gluonic field Aµ
a is created by randomly distributed color sources, and
probability to find a given density of color sources ρa(x−, x⊥) in the target is controlled by the weight functional
W[ρ] [1–3]. The component A+
a which appears in (51) may be related to density of these source ρa as A+
a (x) =
−1
∇2
⊥ρa(x−, x⊥). For any physical observable, the expected values may be obtained averaging the observable A[ρ]
found at fixed distribution of charges ρ(x−, x⊥) over all possible configurations ρ(x−, x⊥), namely
⟨A⟩=
Z
Dρ W[ρ] A[ρ],
(52)
where the angular brackets ⟨...⟩stand for the above-mentioned averaging. The analytic evaluation of the integral Dρ
is possible only for a few simple choices of W[ρ], as, for example, gaussian parametrization [1, 2], and development
of phenomenological models for the latter is very challenging. Fortunately, in presence of hard scales which limit the
number of partons in the initial state, all physical amplitudes may be represented as convolutions of process-dependent
perturbative impact factors and universal (target-dependent) correlators of Wilson lines (multipole scattering ampli-
tudes). For many exclusive processes, the dominant contribution is controlled by the lowest order dipole scattering
amplitude,
N (x, r, b) = 1 −S2
Y = ln
1
x
, xq, x¯q
,
S2 (Y, xq, x¯q) = 1
Nc
tr
U (xq) U † (x¯q)
Y
(53)
2 We disregard tiny ∼1% differences of these constants for different helicity states (polarizations) of the same meson, since these differences
are smaller than the precision of experimental data for χc →γγ used in [64] (around 3-4%); besides, a comparable uncertainty may
come from the theoretical NNLO corrections ∼α2
s (mc) ≈10%.
9
.
σ, i
σ′, i′
ℓQ
ℓ′
Q
The interaction vertex of the quark with the shock wave:
T Q
σ,σ′; i,i′ = 2πδ
ℓ+
Q −ℓ′+
Q
γ+
σ′, σ
Z
d2zei(ℓQ−ℓ′
Q)·z [U (z) −1]i′,i
.
σ, i
σ′, i′
ℓ¯Q
ℓ′¯Q
The interaction vertex of the antiquark with the shock wave:
T
¯
Q
σ,σ′; i,i′ = −2πδ
ℓ+
¯
Q −ℓ′+
¯
Q
γ+
σ, σ′
Z
d2ze
i
ℓ¯
Q−ℓ′
¯
Q
·z h
U † (z) −1
i
i,i′
.
µ, a
ν, b
ℓg
ℓ′
g
The interaction vertex of the gluon with the shock wave:
T g
µ,ν;a,b = −2πδ
ℓ+
g −ℓ′+
g
gµνsgn
ℓ+
g
×
×
Z
d2ze−i(ℓ′
g−ℓg)·z h
Usgn(ℓ+
g ) (z) −1
i
b,a
Table I. The CGC Feynman rules
[12, 78–80] for the interaction of high energy partons with the target encoded in Wilson
link (51). The red-colored block stands for the shock wave (Wilson line). We use notations ℓi, ℓ′
i (i = Q, ¯Q, g) for the momenta
of the corresponding parton before and after the interaction, σ, σ′ for Dirac indices of the quarks, i, i′ for the color indices in
the fundamental representations 3 or ¯3 of the color group, and a, b for the gluon color indices in the adjoint representation 8.
The matrices U and U are the Wilson lines in the fundamental and the adjoint representations. The prefactor 2πδ
ℓ+
i −ℓ′+
i
in the interaction vertices reflects conservation of the longitudinal momentum of the parton in eikonal approximation. We may
also observe that interaction with a shock wave does not change transverse coordinates of the partons in the mixed (light-
cone) representation, and using identities ¯uh1(p)γ+uh2(q) = ¯vh1(p)γ+vh2(q) = 2δh1,h2
p
p+q+ from [66], we may see that the
interaction remains diagonal in helicity basis.
where Y is the rapidity of the dipole, xq, x¯q are the impact parameters of the quark, and the variables r ≡xq −x¯q
and b ≡αq xq + α¯qx¯q determine the transverse separation and the impact parameter of the dipole’s center of mass.
In phenomenological studies the dipole amplitude can be constrained from both exclusive and inclusive channels. Due
to optical theorem, the inclusive channels are sensitive only to the imaginary part of the amplitude, and in order to
have a consistent description, for exclusive channels, the real part of the amplitude should be taken into account. As
was demonstrated long ago in [81, 82], if the amplitude grows as a function of invariant energy as ∼sα, then the real
part of the amplitude becomes proportional to the imaginary part, and thus the full amplitude of exclusive process
can be restored from the imaginary part using a set of identities
A = (β + i) Im A,
|A|2 =
1 + β2
|ImA|2 ,
β ≡Re A
Im A = tan
πα
2
,
α = ∂ln A
∂ln s .
(54)
Following existing conventions, we will assume that N (x, r, b) corresponds to the dipole amplitude extracted from
inclusive processes, tacitly assuming that a factor (β + i) should be added in exclusive channels. We also take into
account the so-called skewedness factor Rg suggested in [83], which takes into account that the effective value of the
parameter x in the dipole amplitude may be modified due to nonzero longitudinal momentum transfer in t-channel
and its possible unequal sharing between the gluons in the shock wave. Explicitly, this factor is given by
Rg(γ) = 22γ+3
√π
Γ(γ + 5/2)
Γ(γ + 4)
≈2.4γ,
where
γ ≡∂ln
xg(x, µ2)
∂ln(1/x)
≈const,
(55)
g
x, µ2
is the gluon PDF, and in approximate expression for Rg we took into account that in the kinematics where
CGC is applicable, the parameter γ ≤0.5. While the derivation of (55) given in [83] relies on a model which is not
valid in the deeply saturated regime x ≪1, at moderate values of x ≳10−3 the factor Rgleads to a mild increase of
the cross-section, and improves the phenomenological description of various exclusive processes (see e.g. [16, 17, 84]
for more details).
10
χc
χc
χc
χc
Figure 2. The dominant mechanism of χcγ pair photoproduction in the CGC framework in the leading order in αs. The red
block stands for a shock wave. The diagrams in the first and the second row are related by charge conjugation (inversion of the
quark line). In the left column we disregard interaction of the emitted photon with a shock wave as tiny O (αem)-correction,
and to emphasize this we don’t put “dot” on a photon line which crosses the shock wave.
D.
Photoproduction of the χcγ pairs
The evaluation of the exclusive photoproduction of χcγ pairs resembles a similar evaluation of the ηcγ photopro-
duction presented in [55] and differs only in the final state wave function, although yields a significantly different
results for the impact factors. Technically, it requires evaluation of four diagrams shown in the Figure 2. In what
follows we will separate the contributions with photon emission before and after the interaction with a shockwave, so
the amplitude can be represented as
A(λ; σ,H)
γp→χcγp = A(λ; σ,H)
1
+ A(λ; σ,H)
2
(56)
where A(λ,σ)
1
and A(λ,σ)
2
correspond to the left and right columns of the Figure 2. Since the interaction of the emitted
photon with the shock wave may be disregarded as O (αem)-correction, both contributions are controlled by the dipole
amplitude N (x, r, b), convoluted with appropriate impact factors.
The detailed evaluation of these impact factors is straightforward and is explained in detail in subsequent subsec-
tions. Following earlier studies [33, 34], in all evaluations below we will introduce the subindices 0,1,2 to distinguish
the kinematic variables associated with quark, antiquark and photon, respectively. For example, the variables
z0 ≡
k+
Q
q+ ,
z1 ≡
k+
¯
Q
q+ ,
z2 ≡k+
γ
q+ = 1 −αχc,
(57)
are the fractions of the incoming photon’s light-cone momentum q+ carried by the quark, antiquark and photon when
crossing the shock wave, respectively, and r0, r1, r2 are their transverse coordinates (in configuration space). The
light-cone momentum conservation implies that for the diagrams in the left column of the Figure 2, the variables
z0, z1, z2 should satisfy
z0 + z1 + z2 = 1,
z0 + z1 = αχc,
(58)
whereas for the diagrams in the right column there is a similar constraint
z0 + z1 = 1.
(59)
1.
Evaluation of the amplitude A(λ; σ,H)
1
The amplitude A(λ; σ,H)
1
obtains contributions of two charge conjugate diagrams shown in the left column of the
Figure 2. We will start our evaluation from the diagram shown in the upper left corner of the Figure 2, defining the
partons’ momenta as explained in the Figure 3 and using the CGC Feynman rules from Table I.
A straightforward evaluation yields
A1,1 = 4παeme2
c
Z
d2x0d2x1
Z
d4k0
(2π)4
d4k1
(2π)4
d4k′
0
(2π)4
d4k′
1
(2π)4 (2π)4 δ (k0 + k1 + kγ −q) (2π)4 δ (k′
0 + k′
1 −pχc) ×
(60)
× (2π) δ
k+
0 −k′
0
+
(2π) δ
k+
1 −k′
1
+
ei(k0−k′
0)·x0ei(k1−k′
1)·x1ϕχcJ
z, k(rel)
⊥
×
×
Tr
ˆεγ(q)S (−k1) γ+U † (x1) S (−k′
1) Γχc (k′
1, k′
0) S (k′
0) γ+U (x0) S (k0) ˆε∗
γ (kγ) S (k0 + kγ)
11
kγ
k0
k′
0
k1
k′
1
pχc
q
x1
x0
χc
Figure 3. The definitions of the parton momenta k0, k1, k′
0, k′
1 before and after interaction with the shock wave. The variables
x0, x1 stand for the transverse coordinates of the quark and antiquark when they pass through the shock wave.
where εγ(q), ε∗
γ (kγ) are the polarization vectors of the photons, and a symbol ˆ... was defined in (2). We also use
notation ec = 2/3 for the electric charge of the charm quark. The matrix Γχc was defined earlier in (31) for different
spins and helicities of χc. The wave function Φ depends on the light-cone fraction zχc = k
′+
0 /p+
χc and relative transverse
momentum k(rel)
⊥
=
z0k′
1 −z1k′
0
/ (z0 + z1). The propagator of free fermions is defined in a standard way as
S(p) = i ˆp + m
p2 −m2 .
(61)
The numerators of the propagators may be rewritten as
ˆp + m =
γ+ p2
⊥+ m2
2p+
+ γ−p+ −γ⊥· p⊥+ m
+ γ+
p−−p2
⊥+ m2
2p+
=
(62)
= θ
p+ X
h
uh(˜p)¯uh(˜p) + θ
−p+ X
¯h
v¯h(−˜p)¯v¯h(˜p) + γ+
p−−p2
⊥+ m2
2p+
,
˜p =
p+, p2
⊥+ m2
2p+
, p⊥
.
The last term ∼γ+ in (62) drops out when the quark is onshell, or when the propagator is multiplied by γ+ (due to
identity γ+γ+ = 0), so effectively (62) reduces to a helicity sum of quark or antiquark spinors. Since the matrices
U, U † in (60) don’t have any spinorial indices, whereas all the other objects don’t depend on color, the evaluation of
traces over color and spinorial indices can be done independently, thus reducing (60) to a convolution of the dipole
scattering amplitude (53) with the so-called (target-independent) impact factor
A1,1 =
Z
d2x0d2x1 N (Y, x0, x1) R1,1 (Y, x0, x1, q, pχc, kγ) ,
(63)
where R1,1 is given by
R1,1 (Y, x0, x1, q, pχc, kγ) = 4παeme2
cNc×
(64)
×
Z
d4k0
(2π)4
d4k1
(2π)4
d4k′
0
(2π)4
d4k′
1
(2π)4 (2π)4 δ (k0 + k1 + kγ −q) (2π)4 δ (k′
0 + k′
1 −pχc) ×
× (2π) δ
k+
0 −k′
0
+
(2π) δ
k+
1 −k′
1
+
ei(k0⊥−k′
0⊥)·x0ei(k1⊥−k′
1⊥)·x1ϕχcJ
z, k(rel)
⊥
×
×
Tr
ˆεγ(q)S (−k1) γ+S (−k′
1) Γχc (k′
1, k′
0) S (k′
0) γ+S (k0) ˆε∗
γ (kγ) S (k0 + kγ)
The integrals over the momenta d4k1 and d4k′
1 in (64) can be calculated using δ-functions, which effectively fix k1
and k′
1 to values
k1 = q −kγ −k0,
k′
1 = pχc −k′
0.
(65)
Due to identity γ+γ+ = 0, we may replace the numerators of propagators connected to the shockwave with onshell
spinors, as explained in the text under Eq. (62). Furthermore, using the well-known spinor identities [66, 69]
¯uh (ℓ) γ+uh′ (ℓ′) = ¯vh (ℓ) γ+vh′ (ℓ′) = 2
√
ℓ+ℓ′+δh,h′
(66)
12
where h, h′ are helicities of partons, and conservation of the plus-component light-cone, we may rewrite (64) as
R1,1 (Y, x0, x1, q, pχc, kγ) = 16 παeme2
cNc (2π) δ
q+ −k+
γ −p+
χc
×
(67)
×
Z dk+
0 dk−
0 d2k0
(2π)4
dk′−
0 d2k′
0
(2π)3
k+
0
q+ −k+
γ −k+
0
ei(k0⊥−k′
0⊥)·x0ei(k1⊥−k′
1⊥)·x1×
×
X
h,¯h
h
¯uh
˜k0
ˆε∗
γ (kγ) S (k0 + kγ) ˆεγ(q)v¯h
˜k1
i
(k2
0 −m2) (k2
1 −m2)
h
¯v¯h
˜k′
1
Γχc (k′
1, k′
0) uh
˜k′
0
i
ϕχcJ
z, k(rel)
⊥
(k′
0)2 −m2
(k′
1)2 −m2
.
For the sake of brevity, in (67) we keep notations k1, k′
1, tacitly assuming that they are linear combinations of k0, k′
0
defined in (65). The dependence on k−
0 and k
′−
0
in the integrand of (67) is isolated in the first and the second terms
of the last line, for this reason integration over these variables becomes straightforward and can be realized using the
Cauchy’s theorem and reduced to a sum of residues of the integrands at the poles, namely
Z +∞
−∞
dk−
0
2π
1
2k+
0 k−
0 −k2
0⊥−m2 + i0
2
k+
γ + k+
0
k−
γ + k−
0
−(kγ
⊥+ k0⊥)2 −m2 + i0
×
(68)
×
1
2
q+ −k+
γ −k+
0
q−−k−
γ −k−
0
−(kγ⊥+ k0⊥)2 −m2 + i0
=
= −
iΘ
q+ −k+
γ −k+
0
8k+
0
k+
0 + k+
γ
q+ −k+
γ −k+
0
−
q+((kγ⊥+k0⊥)2+m2)
2(q+−k+
γ −k+
0 )(k+
γ +k+
0 )
−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
,
Z +∞
−∞
dk−
0
2π
k−
γ + k−
0
2k+
0 k−
0 −k2
0⊥−m2 + i0
2
k+
γ + k+
0
k−
γ + k−
0
−(kγ
⊥+ k0⊥)2 −m2 + i0
×
(69)
×
1
2
q+ −k+
γ −k+
0
q−−k−
γ −k−
0
−(kγ⊥+ k0⊥)2 −m2 + i0
=
= −
iΘ
q+ −k+
γ −k+
0
(kγ⊥+k0⊥)2+m2
2(k+
γ +k+
0 )
8k+
0
k+
0 + k+
γ
q+ −k+
γ −k+
0
−
q+((kγ⊥+k0⊥)2+m2)
2(q+−k+
γ −k+
0 )(k+
γ +k+
0 )
−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
−
iΘ
q+ −k+
γ −k+
0
8k+
0
k+
0 + k+
γ
q+ −k+
γ −k+
0
−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
,
Z +∞
−∞
dk′−
0
2π
1
2k′+
0 k′−
0 −k′2
0⊥−m2 + i0
2
p−
χc −k′+
0
p−
χc −k′−
0
−
pχc
⊥−k′
0⊥
2 −m2 + i0
=
(70)
= −Θ
p+
χc −k′+
0
4k′+
0
p+
χc −k′+
0
i
p−
χc −k′2
0⊥+m2
2k′+
0
−(pχc
⊥−k′
0⊥)
2+m2
2(p+
χc−k′+
0 )
.
The Heaviside step functions Θ(...) in (68-70) introduce the upper cutoff for the integral over the plus-component k+
0
and stem from the requirement that the integrands should have poles on both sides of the real axis (Im k−= 0) for
nonzero result. The identities (68,70) allow us to rewrite (64) as
13
R1,1 (Y, x0, x1, q, pχc, kγ) = δ
q+ −k+
γ −p+
χc
ei(q⊥−kγ
⊥−pχc
⊥)·x1
Z q+−k+
γ
0
dk+
0 ×
(71)
×
2παeme2
c
k+
0 + k+
γ
Z d2k0⊥
(2π)2
¯uh (k0) ˆε∗
γ (kγ)
ˆk0 + ˆkγ + m
ˆεγ(q)v¯h (q −kγ −k0) eik⊥
0 ·(x0−x1)
k−
0 =(kγ⊥+k0⊥)
2+m2
2(k+
γ +k+
0 )
−k−
γ
q
k+
0
q+ −k+
γ −k+
0
−
q+(k2
0⊥+m2)
2(q+−k+
γ −k+
0 )(k+
γ +k+
0 )
−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
+ 2παeme2
c
k+
0 + k+
γ
Z d2k0⊥
(2π)2
¯uh (k0) ˆε∗
γ (kγ) γ+ˆεγ(q)v¯h (q −kγ −k0) eik⊥
0 ·(x0−x1)
q
k+
0
q+ −k+
γ −k+
0
−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
×
Z d2k′
0⊥
(2π)2
¯v¯h
p+
χc −k+
0 , pχc
⊥−k′
0⊥
Γχc (pχc −k′
0, k′
0) uh
k+
0 , k′
0⊥
ϕχcJ (z, K⊥) e−ik′
0⊥·(x0−x1)
2
q
k+
0
p+
χc −k+
0
p−
χc −k′2
0⊥+m2
2k+
0
−(pχc
⊥−k′
0⊥)
2+m2
2(p+
χc−k+
0 )
.
where the δ-function δ
q+ −k+
γ −p+
χc
in (71) reflects smallness of the light-cone momentum transfer ∆+ compared
to transverse momentum transfer to the target ∆⊥= q⊥−kγ
⊥−pχc
⊥in the high energy kinematics. The expressions
in square brackets in denominators of (71) coincide with the energy denominators of the light-cone perturbation
theory [66]. The analysis of the other diagrams is done similarly. The evaluation of the lower left diagram in Figure 2
yields for the impact factor
R1,2 (Y, x0, x1, q, pχc, kγ) = −δ
q+ −k+
γ −p+
χc
ei(q⊥−kγ
⊥−pχc
⊥)·x1
Z q+−k+
γ
0
dk+
0 ×
(72)
×
2παeme2
c
q+ −k+
0
Z d2k0⊥
(2π)2
¯uh (k0) ˆεγ(q)
ˆq −ˆk0 −m
ˆε∗
γ (kγ) v¯h (q −kγ −k0) eik⊥
0 ·(x0−x1)
k−
0 =q−−
k2
0⊥+m2
2(q+−k+
0 )
q
k+
0
q+ −k+
γ −k+
0
q−−
q+(k2
0⊥+m2)
2(q+−k+
γ )k+
0
q−−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
+ 2παeme2
c
q+ −k+
0
Z d2k0⊥
(2π)2
¯uh (k0) ˆεγ(q)γ+ˆε∗
γ (kγ) v¯h (q −kγ −k0) eik⊥
0 ·(x0−x1)
q
k+
0
q+ −k+
γ −k+
0
q−−
k2
γ⊥
2k+
γ −k2
0⊥+m2
2k+
0
−(kγ⊥+k0⊥)2+m2
2(q+−k+
γ −k+
0 )
×
Z d2k′
0⊥
(2π)2
¯v¯h
p+
χc −k+
0 , pχc
⊥−k′
0⊥
Γχc (pχc −k′
0, k′
0) uh
k+
0 , k′
0⊥
e−ik′
0⊥·(x0−x1)ϕχcJ (z, K⊥)
2
q
k+
0
p+
χc −k+
0
p−
χc −k′2
0⊥+m2
2k+
0
−(pχc
⊥−k′
0⊥)
2+m2
2(p+
χc−k+
0 )
.
The result for (72) differs from (71) only by the structure of the second and the third lines. After integration over
k′
0⊥in the last lines of (71) and (72), we may obtain the wave function of χc meson in the light-cone representation
introduced earlier in (35). Likewise, the result of integration over k0⊥in the second and the third lines of (71,72) can
be interpreted as a light-cone wave function of the ¯QQγ Fock component in the photon, which in the leading order is
given by a set of diagrams shown in the Figure 4. These diagrams nearly coincide with the diagrams j, k, ℓ, m in [33]
which contribute to the γ →¯QQg wave function, and for this reason the result of integration over k0⊥differs from
expressions found in [33] only by color factor. The structure of (71) and (72) implies that the amplitude A1 can be
rewritten as a convolution of the dipole amplitude N (x, r, b), the wave function Ψ†
χcof the produced χc meson, and
the wave function Ψγ→γ ¯
QQ of the ¯QQγ Fock component of the incoming photon (the amplitude of the subprocess
γ →¯QQγ), namely
A(λ; σ, H)
1
=
Z αχc
0
dz0
2
Y
k=0
d2xk
Ψ(h,¯h)†
χc
z0
z0 + z1
, r10
Ψ(λ,σ,h,¯h)
γ→γ ¯
QQ (z0, z1 = αχc −z0, z2 ≡¯αχc, x0, x1, x2) ×
(73)
× N (x, r10, b10) exp
−ipχc
⊥·
b10 −¯αχc
αχc
rγ
−ikγ
⊥· x2
,
14
γ(q)
Q
¯Q
γ(kγ)
γ(q)
γ(kγ)
Q
¯Q
γ(q)
γ(kγ)
Q
¯Q
γ(q)
Q
¯Q
γ(kγ)
Figure 4.
The diagrams which correspond to the γ →¯QQγ amplitude in the leading order in strong coupling αs.
The
contributions of the two diagrams in the first row correspond to expressions in the second lines of (71) and (72). Similarly, the
contributions of the diagrams in the last row are given by the expressions in the third lines of the same Equations (71) and
(72). The vertical fermion line with a short horizontal dash in the diagrams of the last row denotes instantaneous part of the
quark propagator. The colored vertical dashed lines stand for the light-cone energy denominators of the canonical light-cone
perturbation theory [66, 69].
where the dummy integration variable z0 is related to k+
0 using identities (57), the parameters λ, σ are the helicities
of the photon before and after interaction; h, ¯h are the helicities of the heavy quarks when they cross the shock wave,
r10 ≡x1 −x0 is the relative distance between the quarks, b10 = (z0x0 + z1x1) / (z0 + z1) is the position (impact
parameter) of the center of mass of the quark-antiquark pair, rγ = x2−b10 is the distance between the emitted photon
and the center of mass of the dipole. The argument x of the dipole cross-section N corresponds to the fraction of the
target’s momentum in minus-direction which is taken by heavy quarks, x ≈∆−/P −= 2ξ/(1 + ξ) ≈M 2
γχc/W 2 . As
we mentioned earlier, the impact factors R1,1 and R1,2 are related to each other by charge conjugation (C-parity). In
order to fully exploit this symmetry , it is convenient to introduce the variable ζ defined as a fraction of the photon’s
momentum carried by virtual parton (quark or antiquark) which emits secondary (final-state) photon afterwards,
namely
z0 = ζ −¯αχc,
z1 = 1 −ζ,
z2 = 1 −αχc
(74)
for the contribution R1,1, and
z0 = 1 −ζ,
z1 = ζ −¯αχc,
z2 = 1 −αχc
(75)
for the contribution R1,2. While the expression (73) presents a lot of interest for conceptual understanding, unfortu-
nately it is not feasible to use it directly for numerical phenomenological studies due to multidimensional integration
of the oscillating function; besides the wave function Ψ(...)
γ→γ ¯
QQin light-cone (configuration) representation include new
special functions I, J defined in [33, 34] as nontrivial 2-dimensional integrals. This suggests that evaluation of the
amplitude should be done in the momentum space. However, evaluation of the Fourier image of N (x, r10, b10) is
not well-defined due to saturation of the latter at large distances r10 and non-commutativity of the limits r10 →∞
and b10 →∞. At the same time, we may observe that for the product Ψ†
χcJ (r10) N (x, r10, b10) this problem does
not exist, because ΨχcJ is proportional to Φχc and its derivatives, which are strongly suppressed at large distances.
For this reason, in what follows we will introduce new functions 3
nΦχc (x, ℓ, s, z) =
Z
d2b d2re−iℓ·re−is·bN (x, r, b) Φχc (r, z) ,
(76)
3 For phenomenological parametrizations of N (x, r, b) that do not depend explicitly on orientations of r, b, the integration over
orientations of these vector can be done analytically, yielding (2π)2 J0(ℓr) J0(sb) for n(2, 0)
Φχc , (2π)2 iJ1(ℓr) J0(sb) for n(±)
Φχc , and
−(2π)2 J2(ℓr) J0(sb) for n(2, ±2)
Φχc
.
15
n(±)
Φχc (x, ℓ, s, z) =
Z
d2b d2re−iℓ·re−is·bN (x, r, b) e±iθ∂rΦχc (r, z) ,
(77)
n(2, ±2)
Φχc
(x, ℓ, s, z) =
Z
d2b d2re−iℓ·re−is·bN (x, r, b) e±2iθ∆(−)
r
Φχc (r, z) ,
(78)
n(2, 0)
Φχc (x, ℓ, s, z) =
Z
d2b d2re−iℓ·re−is·bN (x, r, b) ∆(+)
r
Φχc (r, z) ,
(79)
and, using explicit expressions for the wave functions in helicity basis (34-42), will replace the corresponding products
Ψ†
χcJ (r10) N (x, r10, b10) with (inverse) Fourier images of the functions nΦχc , n(±)
Φχc , n(2, 0)
Φχc , n(2, ±2)
Φχc
. After integration
over the variables r10 and b10, we may obtain two additional δ-functions
(2π)4 δ (s + ∆⊥) δ
ℓ−k⊥
0 −ζ −¯αχc
α
∆⊥
which allow to get rid of the integrals over d2k0 and d2s. The final result of this procedure depends on the spin
and helicity of χcJ meson. We performed these evaluations in helicity basis, using explicit expressions for spinors
from [66, 69] and performing the algebraic simplifications using Mathematica. In view of the space constraints, we
provide below the results only for components with “+” helicity of the incoming photon, tacitly assuming that the
remaining components may be recovered using
h
A(λ; σ, H)
1
i∗
= A(−λ; −σ, −H)
1
. For χc0 meson, the helicity amplitudes
read as
A(+; +)
1, χc0 = 4π¯κm
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ
ζ −¯αχc
×
(80)
×
αχc ¯αχc
αχc −2¯ζ
h
(ζ −¯αχc)
P 2
(1) + m2
−¯αχcm2i
nΦχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
−
i αχc ¯α2
χcζ n(−)
Φχc (x, ℓ, −∆⊥, zχc)
P(1) + iP(1)y
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −)
1, χc0 = −4iπ¯κmα2
χc ¯α2
χc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ3
ζ −¯αχc
n(+)
Φχc (x, ℓ, −∆⊥, zχc) ×
(81)
P(1)x + iP(1)y
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
where we introduced shorthand notations
K(1) = ℓ¯αχc + ζ (αχckγ
⊥−¯αχcpχc
⊥)
ζ −¯αχc
= P (1)¯αχc
ζ −¯αχc
+ (αχckγ
⊥−¯αχcpχc
⊥)
αχc
,
(82)
P (1) = ℓ+
¯ζ
αχc
(αχckγ
⊥−¯αχcpχc
⊥) ,
zχc = ζ −¯αχc
αχc
.
(83)
For χc1 meson, the corresponding amplitudes are given by
A(+; +,+1)
1, χc1
= 4π¯καχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
2¯ζ
ζ −¯αχc
ζ
K(1)x −iK(1)y
×
(84)
¯ζ ¯αχc
P(1)x + iP(1)y
(2ζ −¯αχc)n(−)
Φχc (x, ℓ, −∆⊥, zχc) −iζm2nΦχc (x, ℓ, −∆⊥, zχc)
αχc −2¯ζ
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
16
A(+; −,+1)
1, χc1
= −4iπ¯κm2αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 nΦχc (x, ℓ, −∆⊥, zχc) ×
(85)
ζ
K(1)x + iK(1)y
q
2¯ζ (ζ −¯αχc)
αχc −2¯ζ
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; +,0)
1, χc1
= −8¯αχcα2
χc
Mχc1
π¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ
ζ −¯αχc
×
(86)
m2¯αχcζ
P(1)x + iP(1)y
n(−)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
−i
n(2, 0)
Φχc (x, ℓ, −∆⊥, zχc)
h
(ζ −¯αχc)
P 2
(1) + m2
+ m2¯αχc
i
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
A(+; −,0)
1, χc1
= −8m2
Mχc1
π¯κ¯α2
χcα2
χc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ3
ζ −¯αχc
n(+)
Φχc (x, ℓ, −∆⊥, zχc)
(87)
×
P(1)x + iP(1)y
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; +,−1)
1, χc1
= 4π¯καχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 n(+)
Φχc (x, ℓ, −∆⊥, zχc)
(88)
×
K(1)x −iK(1)y
P(1)x + iP(1)y
ζ (2ζ −1)
q
2¯ζ (ζ −¯αχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −,−1)
1, χc1
= 4π¯κα2
χc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
2¯ζ
ζ −¯αχc
(89)
×
ζ2 K(1)x + iK(1)y
P(1)x + iP(1)y
αχc −2¯ζ
n(+)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −,−1)
1, χc1
= 0.
(90)
Finally, for χc2 meson we obtained
A(+; +,+2)
1, χc2
= 8iπ¯κ αχc ¯αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ3
ζ −¯αχc
(91)
×
h
m2¯αχc +
P 2
(1) + m2
(ζ −¯αχc)
i
n(2, −2)
Φχc
(x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −,+2)
1, χc2
= 8π¯κm2¯α2
χcα2
χc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ3
ζ −¯αχc
n(−)
Φχc (x, ℓ, −∆⊥, zχc)
(92)
×
P(1)x + iP(1)y
¯ζ + ¯αχc
2
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
17
A(+; +,+1)
1, χc2
= 4iMχc2π¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
K(1)x −iK(1)y
s
¯ζ
ζ −¯αχc
ζ
(93)
×
P(1)x + iP(1)y
(ζ −¯αχc)
8ζ3 + 4ζ2(αχc −4) −6ζ(αχc −2) + 3αχc −4
n(−)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
−i
ζm2αχcnΦχc (x, ℓ, −∆⊥, zχc)
αχc −2¯ζ
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −,+1)
1, χc2
= 4π¯κMχc2
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 ζ
s
¯ζ
ζ −¯αχc
αχc −2¯ζ
(94)
×
αχc
αχc −¯ζ
m2nΦχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
+
iζ
P(1)x + iP(1)y
αχc −2¯ζ
2 n(−)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; +,0)
1, χc2
= −4
r
2
3π¯καχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ
ζ −¯αχc
(95)
×
2m2 αχc −2¯ζ
nΦχc (x, ℓ, −∆⊥, zχc)
h
¯αχc (ζ −¯αχc)
P 2
(1) + m2
+ m
m¯α2
χc −ζ2 K(1)x −iK(1)y
i
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
+
−
2im2ζαχc
P(1)x + iP(1)y
¯α2
χcn(−)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
−
3ζαχc
P(1)x + iP(1)y
K(1)x −iK(1)y
¯ζ (2ζ + αχc) −1
n(2, 0)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −,0)
1, χc2
= 4
r
2
3π¯κm2αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ
ζ −¯αχc
(96)
×
2ζm3 K(1)x + iK(1)y
(ζ −¯αχc)
αχc −2¯ζ
nΦχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
+
−
2iαχc ¯α2
χc ¯ζm2 P(1)x + iP(1)y
n(+)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
−
3αχcζ2 αχc −2¯ζ
K(1)x + iK(1)y
P(1)x + iP(1)y
n(2, 0)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; +,−1)
1, χc2
= 4iπ¯κMχc2
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 ζ ¯ζ
s
¯ζ
ζ −¯αχc
n(+)
Φχc (x, ℓ, −∆⊥, zχc)
(97)
×
K(1)x −iK(1)y
P(1)x + iP(1)y
8ζ3 + 4ζ2(3αχc −4) −6ζ(αχc −2)¯αχc + (αχc −4)¯α2
χc
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
18
A(+; −,−1)
1, χc2
= 16iπ¯κMχc2
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 ζ2¯ζ
q
¯ζ (ζ −¯αχc)n(+)
Φχc (x, ℓ, −∆⊥, zχc)
(98)
×
K(1)x + iK(1)y
P(1)x + iP(1)y
αχc −2¯ζ
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; +,−2)
1, χc2
= −8iαχc ¯αχcπ¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
s
¯ζ
ζ −¯αχc
(99)
×
n(2, +2)
Φχc
(x, ℓ, −∆⊥, zχc)
h
(ζ −¯αχc)
P 2
(1) + m2
+ m2¯αχc
i
(ζ −¯αχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
+
im2ζαχc ¯αχc
P(1)x + iP(1)y
n(+)
Φχc (x, ℓ, −∆⊥, zχc)
P 2
(1) + m2
¯ζ ζ2K2
(1) + (ζ −¯αχc) ¯αχcP 2
(1) + αχc ¯αχcζm2
,
A(+; −,−2)
1, χc2
= 0.
(100)
2.
Evaluation of the amplitude A(λ; σ, H)
2
The amplitude A(λ; σ, H)
2
corresponds to contribution of the diagrams in the right column of the Figure 2 and can
be evaluated using the procedure outlined in the previous subsection II D 1. We may represent the amplitude as a
convolution of the dipole amplitude and the impact factors R2,i,
A2,i =
Z
d2x0d2x1 N (Y, x0, x1) R2,i (Y, x0, x1, q, pχc, kγ) ,
i = 1, 2.
(101)
The corresponding expressions for R2,i in (101) differ from the impact factors R1,i only by permutation of the χc and
the incoming photon, namely
R2,1 (Y, x0, x1, q, pχc, kγ) = 4παeme2
cNc
(102)
×
Z
d4k0
(2π)4
d4k1
(2π)4
d4k′
0
(2π)4
d4k′
1
(2π)4 (2π)4 δ (k0 + k1 −q) (2π)4 δ (k′
0 + k′
1 −pχc −kγ)
× (2π) δ
k+
0 −k′
0
+
(2π) δ
k+
1 −k′
1
+
ei(k0⊥−k′
0⊥)·x0ei(k1⊥−k′
1⊥)·x1ϕχcJ
z, k(rel)
⊥
×
Tr
Γχc (k′
1, k′
0 −kγ) S (k′
0 −kγ) ˆε∗
γ (kγ) S (k′
0) γ+S (k0) ˆεγ(q)S (−k1) γ+S (−k′
1)
R2,2 (Y, x0, x1, q, pχc, kγ) = 4παeme2
cNc
(103)
×
Z
d4k0
(2π)4
d4k1
(2π)4
d4k′
0
(2π)4
d4k′
1
(2π)4 (2π)4 δ (k0 + k1 −q) (2π)4 δ (k′
0 + k′
1 −pχc −kγ)
× (2π) δ
k+
0 −k′
0
+
(2π) δ
k+
1 −k′
1
+
ei(k0⊥−k′
0⊥)·x0ei(k1⊥−k′
1⊥)·x1ϕχcJ
z, k(rel)
⊥
Tr
Γχc (k′
1 + kγ, k′
0) S (k′
0) γ+S (k0) ˆεγ(q)S (−k1) γ+S (−k′
1) ˆε∗
γ (kγ) S (−k′
1 −kγ)
Repeating the procedure outlined in the previous Section II D 1, we may rewrite the impact factors R2,i in terms
of the photon wave function ψγ→¯
QQ and the amplitude (“wave function”) Ψ†
¯
QQ→γ χc of the ¯QQ →χcγ subprocess,
contracted over helicities h, ¯h of the quarks and convoluted over the light-cone component k+
Q of the quark momentum.
19
The final result for the amplitude A(λ,σ)
2
may be rewritten as
A(λ,σ)
2
=
Z αχc
0
dz0
3
Y
k=1
d2rk
Ψ(σ,h,¯h)†
¯
QQ→γ χc (z0, z1 = αχc −z0, z2 ≡¯αχc, r0, r1, r2) ψ(λ, h,¯h)
γ→¯
QQ
z0
z0 + z1
, r10
×
(104)
× N (x, r10, b10) exp
−ipχc
⊥·
b10 −¯αχc
αχc
rγ
−ikγ
⊥· (rγ + b10)
.
where we use the same notations as in the previous section (see the text under Eq. (73)). The photon wave function
ψ(λ, h,¯h)
γ→¯
QQ in the leading order in αs is given by [85, 86]
ψ(λ, h,¯h)
γ→¯
QQ (ζ, r) = −ieec
Z d2kQ⊥
(2π)2
¯uh
k+
Q, kQ⊥
ˆελ(q)v¯h
q+ −k+
Q, −kQ⊥
q−−
q+(k2
Q⊥+m2)
2k+
Q(q+−k+
Q)
eikQ⊥·r10
q−=0,
k+
Q=ζq+
=
(105)
=
√
2
2π eef
−iλe−iλϕr ζδh,λδ¯h,−λ −(1 −ζ)δh,−λδ¯h,λ
mK1 (mr) + mδh,λδ¯h,λK0 (mr)
,
The evaluation of the amplitude Ψ(σ,h,¯h)
¯
QQ→γ χc in the leading order in the coupling αs resembles a similar evaluation of
the wave function Ψ(λ,σ,h,¯h)
γ→γ ¯
QQ discussed earlier and requires evaluation of the diagrams shown in the right column of
the Figure 5 using the light-cone rules from [69]. The evaluation is straighforward, however, the final expression is
very lengthy, introduces a number of new special functions defined via 2-dimensional integrals, and for this reason
evaluation of (104) in configuration space is not feasible. As we mentioned earlier, the Fourier transformation of the
dipole amplitude N (x, r, b) presents certain challenges due to saturation at large r and non-commutativity of limits
r →∞and b →∞. However, the product of the dipole amplitude and wave function N (x, r, b) ψ(λ, h,¯h)
γ→¯
QQ (ζ, r) has
a well-defined Fourier image: the functions K0, K1 suppress the product at large r, whereas a color transparency
(suppression of N at small r) guarantees that the Fourier integrals remain convergent at small r. Similar to (76-79),
we may introduce new functions
n0 (x, ℓ, s) =
Z
d2b d2re−iℓ·re−is·bN (x, r, b) K0 (mr) ,
(106)
n(±)
1
(x, ℓ, s) =
Z
d2b d2re−iℓ·re−is·bN (x, r, b) K1 (mr) e±iϕr,
(107)
where the angle ϕr = arg(rx + iry) characterizes the azimuthal orientation of the vector r, and express the product
N (x, r, b) ψ(λ, h,¯h)
γ→¯
QQ (ζ, r) in terms of the (inverse) Fourier transforms of these functions. We found that the functions
n0, n(±)
1
have a mild dependence on all their arguments and are strongly suppressed at large ℓ, s. For this reason, it
is possible to significantly speed up the evaluations using caching (interpolation of pre-evaluated) on a modestly sized
grid.
In order to reduce the charge conjugate contributions in the amplitude A(λ; σ,H)
2
to the common denominator, we
will replace the dummy integration variable z0 with a new variable ζ defined as a fraction of the incoming photon’s
momentum carried by the active fermion before emission of the secondary photon, namely
ζ =
k+
0
k+
0 + k+
1
=
z0
z0 + z1
,
(108)
for the contribution with photon emission from the quark, and with
ζ =
k+
1
k+
0 + k+
1
= 1 −
z0
z0 + z1
,
(109)
for the other contribution. In both cases, the variable ζ cannot be less than k+
γ /q+ = ¯αχc. The evaluation of the
amplitudes is straightforward and largely repeats the evaluation of (80-100). Due to space limitation, we will provide
20
γ(kγ)
χc
Q
¯Q
χc
¯Q
Q
γ(kγ)
χc
γ(kγ)
¯Q
Q
¯Q
Q
χc
γ(kγ)
Figure 5. The leading order diagrams which contribute to the amplitude of the subprocess Q ¯Q →γχc. The vertical dashed
lines denote the energy denominators of the light-cone perturbation theory[66, 69].
only the final results. For the χc0 mesons, the corresponding helicity amplitudes are given by
A(+; +)
2, χc0 = −2mαχc ¯αχc¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
p
ζ ¯ζ3
(ζ −¯αχc)2 ˜Φχc
zχc, P (2)
×
(110)
×
ζn0 (x, ℓ, −∆⊥)
P 2
(2) + m2 zc −1
2
2
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
+
m¯αχc
mn0 (x, ℓ, −∆⊥)
zc −1
2
+ iζ
P(2)x + iP(2)y
n(−)
1
(x, ℓ, −∆⊥)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
,
A(+; −)
2, χc0 = 2i¯κm2αχc ¯α2
χc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
q
ζ ¯ζ5
(111)
×
P(2)x −iP(2)y
n(−)
1
(x, ℓ, −∆⊥) ˜Φχc
zχc, P (2)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
,
where we introduced the shorthand notations
K(2) = −ℓ¯αχc
ζ
+ (ζ −¯αχc)
α2χcζ
(αχckγ
⊥−¯αχcpχc
⊥) = −P (2)¯αχc
ζ
+ (αχckγ
⊥−¯αχcpχc
⊥)
αχc
,
(112)
P (2) = ℓ+
¯ζ
α2χc
(αχckγ
⊥−¯αχcpχc
⊥) ,
zχc = ζ −¯αχc
αχc
.
(113)
For χc1 meson, the corresponding amplitudes are given by
A(+; +,+1)
2, χc1
= 2m¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
2 ζ ¯ζ3 (Kx + iKy)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(114)
×
h
(2ζ −1)
P(2)x + iP(2)y
(ζ −¯αχc) n(−)
1
(x, ℓ, −∆⊥) −iζmn0 (x, ℓ, −∆⊥) (2ζ + αχc −2)
i
,
21
A(+; −,+1)
2, χc1
= −2im¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
p
2 ζ ¯ζ3 K(2)x −iK(2)y
(2ζ + αχc −2) ˜Φχc
zχc, P (2)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(115)
×
h
mn0 (x, ℓ, −∆⊥) (ζ −¯αχc) + iζαχc
P(2)x + iP(2)y
n(−)
1
(x, ℓ, −∆⊥)
i
,
A(+; +,0)
2, χc1
= iαχc ¯αχc¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
ζ ¯ζ
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(116)
×
(
mn0 (x, ℓ, −∆⊥)
"
¯αχcα2
χc ¯ζ
m2(zc −1/2)2 −P 2
(2)
−αχcζ
P 2
(2) + m2
zc −1
2
2!#
+
+ i n(−)
1
(x, ℓ, −∆⊥)
P(2)x + iP(2)y
2m2¯αχcα2
χcζ ¯ζ −αχcζ2
P 2
(2) + m2
zc −1
2
2!!)
,
A(+; −,0)
2, χc1
= ¯καχc ¯αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
ζ ¯ζ3 P(2)x −iP(2)y
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(117)
× n(−)
1
(x, ℓ, −∆⊥)
ζαχc
P 2
(2) + m2
zc −1
2
2!
−2¯ζm2¯αχc
!
,
A(+; +,−1)
2, χc1
= −2¯κm¯αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 ˜Φχc
zχc, P (2)
×
(118)
hp
2ζ ¯ζ5 (ζ −¯αχc)
K(2)x + iK(2)y
P(2)x −iP(2)y
(2ζ −¯αχc) n(−)
1
(x, ℓ, −∆⊥)
i
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
,
A(+; −,−1)
2 χc1
= 0.
(119)
Finally, for χc2 meson we obtained
A(+; +,+2)
2, χc2
= 4m2¯α2
χc¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
ζ ¯ζ5 P(2)x + iP(2)y
2 n0 (x, ℓ, −∆⊥)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
, (120)
A(+; −,+2)
2, χc2
= 4i¯κm
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
αχc ¯αχc ˜Φχc
zχc, P (2)
p
ζ ¯ζ5 P(2)x + iP(2)y
n(−)
1
(x, ℓ, −∆⊥)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(121)
×
ζ
P 2
(2) + m2
zc −1
2
2!
−m2¯αχc
!
,
A(+; +,+1)
2, χc2
= −
¯κ
2αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
ζ ¯ζ
K(2)x + iK(2)y
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
×
(122)
×
m2 −4¯ζ2 + α2
χc + 4¯ζαχc
+ α2
χcP 2
(2)
{ζmαχcn0 (x, ℓ, −∆⊥) (2ζ + αχc −2)
+i (ζ −¯αχc)
8ζ3 + 4ζ2(αχc −4) −6ζ(αχc −2) + 3αχc −4
P(2)x + iP(2)y
n(−)
1
(x, ℓ, −∆⊥)
o
,
22
A(+; −,+1)
2, χc2
= −
¯κ
2αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
ζ ¯ζ
K(2)x + iK(2)y
(2ζ + αχc −2)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(123)
×
m2 −4¯ζ2 + α2
χc + 4¯ζαχc
+ α2
χcP 2
(2)
×
mαχcn0 (x, ℓ, −∆⊥) (ζ −¯αχc) + iζ(2ζ + αχc −2)2 P(2)x + iP(2)y
n(−)
1
(x, ℓ, −∆⊥)
,
A(+; +,0)
2, χc2
=
¯κ¯αχc
2
√
6αχc
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p
ζ/¯ζ
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(124)
×
(
n0 (x, ℓ, −∆⊥) (1 −2zc)
×
m4 −4¯ζ2 + α2
χc + 4¯ζαχc
−4¯ζ2 + α2
χc + 4¯ζ(1 −2ζ)αχc
+ 2m2α4
χcP 2
(2) + α4
χc
P 2
(2)
2
+ in(−)
1
(x, ℓ, −∆⊥) α2
χc
P(2)x + iP(2)y
/m
"
8ζ ¯ζ2m4¯αχc (ζ −¯αχc)
+ζ2αχc
m2 −4¯ζ2 + α2
χc + 2¯ζαχc
+ αχc
αχc −2¯ζ
P 2
(2)
m2
zc −1
2
2
+ P 2
(2)
!#)
,
A(+; −,0)
2, χc2
= ¯καχc ¯αχc
2
√
6m
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
i˜Φχc
zχc, P (2)
P(2)x −iP(2)y
n(−)
1
(x, ℓ, −∆⊥)
p
ζ ¯ζ
(ζ −¯αχc)
(125)
×
ζαχc
m2 −4¯ζ2 + α2
χc + 2¯ζαχc
+ αχc
αχc −2¯ζ
P 2
(2)
P 2
(2) + m2 zc −1
2
2
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
−
8¯ζ2m4¯αχc (ζ −¯αχc)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
,
A(+; +,−1)
2, χc2
= ¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
p¯ζζ
m2 −4¯ζ2 + α2
χc + 4¯ζαχc
+ α2
χcP 2
(2)
4αχc
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
×
(126)
mn0 (x, ℓ, −∆⊥)
¯α2
χc
P(2)x −iP(2)y
αχc −4¯ζ
−αχc
K(2)x + iK(2)y
(ζ −¯αχc)
αχc −2¯ζ
−in(−)
1
(x, ℓ, −∆⊥)
2m2¯α2
χcαχc
−(2ζ −1)
K(2)x + iK(2)y
P(2)x −iP(2)y
4¯ζ2(2ζ −1) + α3
χc −6¯ζα2
χc −3¯ζ2(4ζ −3)αχc
,
A(+; −,−1)
2, χc2
= 2i¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2
˜Φχc
zχc, P (2)
n(−)
1
(x, ℓ, −∆⊥)
ζ ¯ζ
3/2
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
(127)
×
K(2)x −iK(2)y
P(2)x −iP(2)y
(ζ −¯αχc) (zc −1/2)
m2 −4¯ζ2 + α2
χc + 4¯ζαχc
+ α2
χcP 2
(2)
,
23
A(+; +,−2)
2, χc2
= −4m¯αχc¯κ
Z 1
¯αχc
dζ
Z
d2ℓ
(2π)2 ˜Φχc
zχc, P (2)
q
ζ ¯ζ3 P(2)x −iP(2)y
(128)
×
m¯αχc
P(2)x −iP(2)y
n0 (x, ℓ, −∆⊥) (ζ −¯αχc) −iζmαχcn(−)
1
(x, ℓ, −∆⊥)
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
+
iαχcζ2n(−)
1
(x, ℓ, −∆⊥)
P 2
(2) + m2 zc −1
2
2
¯ζ (ζ −¯αχc)2 K2
(2) −ζ αχc ¯αχcP 2
(2) −¯αχc
αχc (ζ −¯αχc)
αχc −4ζ ¯ζ
m2
,
A(+; −,−2)
2, χc2
= 0.
(129)
At the extremes of the integration domain ζ = 1 and ζ = ¯αχc, the integrand is strongly suppressed by the wave
function ˜Φχc (zχc, P ), and in the heavy quark mass limit the dominant contribution comes from the central region
ζ ∼1−αχc/2 (zχc = 1/2). If we construct a Taylor expansion near the point zχc = 1/2, P = 0, the first non-vanishing
term in the latter can be reduced to NRQCD LDMEs of χc mesons using the relations (47, 48) from Section II B.
The explicit expressions for the coefficients in front of LDMEs are extremely lengthy (require taking derivatives of
the derivatives of the integrands) and won’t be provided explicitly due to space limitations. The evaluation of the
cross-section was performed without making such expansion, taking explicitly the integrals over d2ℓdζ in momentum
space.
III.
NUMERICAL ESTIMATES
A.
Differential cross-sections
For the sake of definiteness, we will use in what follows the bCGC parametrization of the forward dipole scattering
amplitude, whose parameters were fixed from fits of HERA data [16],
N (x, r, b) =
N0
r Qs(x)
2
2γeff(r)
,
r ≤
2
Qs(x)
1 −exp
−A ln2 (Br Qs)
, r >
2
Qs(x)
,
(130)
A = −
N 2
0 γ2
s
(1 −N0)2 ln (1 −N0)
,
B = 1
2 (1 −N0)−1−N0
N0γs ,
(131)
Qs(x, b) =
x0
x
λ/2
TG(b), γeff(r) = γs +
1
κλ ln(1/x) ln
2
r Qs(x)
, TG(b) = exp
−
b2
4γsBCGC
(132)
γs = 0.6492,
λ = 0.2023,
x0 = 6.9 × 10−4,
BCGC = 5.5 GeV−2 ≈(0.463 fm)2 .
(133)
In the heavy quark mass limit, the binding energy of the quarkonium formally is a small parameter, ∼O
v2
∼
O
α2
s (mc)
. In order to have consistency with this expectation, we will assume that the charm mass mc ≈Mχc/2 ≈
1.8 GeV. We also will implement the corrections due to the real part of the amplitude (54) and skewedness (55) in
exclusive channels, as discussed in Section II C.
We would like to start discussion from the threefold differential unpolarized cross-section (22). In the Figure 6 we
have shown this observable as a function of the invariant momentum transfer t to the target. Since the parameter ξ ∼
M 2
γχc/W 2 is very small in the chosen kinematics, the variable t may be approximated as t ≈−∆2
⊥. The pronounced
dependence on this variables largely stems from the explicit ∆⊥-dependence which appears in the arguments of the
functions n, nΦ- (76-79,106,107) in the momentum space amplitudes (80-100, 110-129). The dependence on ∆⊥which
appears in prefactors in front of these functions leads only to minor O
t/M 2
χc
corrections. Indeed, we may observe
that in all these prefactors the external vectors p⊥, ∆⊥always contribute only on the linear combination
L = (αχckγ
⊥−¯αχcpχc
⊥) = p⊥+ ∆⊥
1
2 −αχc
.
(134)
24
γp→γχc0p
W=141 GeV, |t'|=0.5 GeV2
Mγχc=3.8 GeV
Mγχc=4 GeV
Mγχc=4.5 GeV
Mγχc=5 GeV
0.
0.5
1.
10-3
10-2
10-1
100
101
|t|, GeV2
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
γp→γχc1p
W=141 GeV, |t'|=0.5 GeV2
Mγχc=3.8 GeV
Mγχc=4 GeV
Mγχc=4.5 GeV
Mγχc=5 GeV
0.
0.5
1.
10-4
10-3
10-2
10-1
100
101
|t|, GeV2
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
γp→γχc2p
W=141 GeV, |t'|=0.5 GeV2
Mγχc=4 GeV
Mγχc=4.5 GeV
Mγχc=5 GeV
0.
0.5
1.
10-2
10-1
100
101
|t|, GeV2
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
Figure 6. The dependence of the photoproduction cross-section (22) on the invariant momentum transfer to the target t. The
left, central and right columns correspond to the χc0, χc1 and χc2 mesons, respectively. The t-dependence at Mγχc = 3.8 GeV
reflects the kinematic constraint |t| ≤M 2
γχc −M 2
χc −|t′|.
After integration over the dummy momentum ℓ, the absolute value of the amplitude may depend only on the square
L2. Since the vectors p⊥and ∆⊥are mutually orthogonal in the high-energy kinematics (see Eq (18)), and p⊥is
parametrically large, p2
⊥∼M 2
χc, we may obtain
L2 ≈¯αχc
αχcM 2
γχc −M 2
χc
−t
1
2 −αχc
2
≈¯αχc
αχcM 2
γχc −M 2
χc
1 + O
t/M 2
χc
(135)
Besides, the factor
1
2 −αχc
2 in front of t in (135) provides additional numerical suppression that justifies omission
of this t-dependence. For comparison, the t-dependence that originates from the functions n, nΦ is related to the
implemented impact parameter dependence of the dipole amplitude and is controlled by the (inverse) proton size.
In all the phenomenological parametrizations fitted to exclusive processes, this dependence is rapidly decreasing,
in agreement with theoretical expectations [69].
Technically, this implies that the χcγ pairs predominantly have
oppositely directed transverse momenta pχc
⊥, kγ
⊥. For this reason, in what follows we will focus on the kinematics
|t| = |tmin|, or the t-integrated observables that get the largest contribution from small-t kinematics.
The dependence on t′ shown in the Figure 7 is very mild in the kinematics of small |t′| because in the impact factors
this variable contributes in linear combination with parametrically large scales ∼M 2
γχc, M 2
χc, and, unless we consider
the near-threshold kinematics M 2
γχc ∼M 2
χc, may be disregarded altogether. This variables also may be related to the
transverse momenta ±p⊥of the final state χc and γ, and thus is correlated with the angle between these particles. As
we will explain below, the latter observation allows to understand the completely different t′-dependence of various
helicity components. In the Figure 8 we provide predictions for the dependence on the invariant mass Mγχc. The
latter variable plays the role of the hard scale that controls the scale of all the transverse momenta, and shows up
explicitly in the impact factors provided in previous section. This variable also determines the effective value of the
parameter x ∼M 2
γχc/W 2 in the dipole amplitude (130). Due to suppression of the latter at large x, the cross-sections
of all quarkonia are strongly suppressed at large Mγχc. We also may observe that unless we consider a near-threshold
kinematics Mγχc ∼Mχc, the dependence on all variables largely factorizes.
The cross-sections shown in the Figures 6-8 are larger than similar cross-sections found in the collinear factorization
framework [87] by up to an order of magnitude. Such enhanced sensitivity to the framework is not surprising: as
could be seen from explicit expressions in Section II B, the wave functions of P-wave χc mesons change sign under
permutation of quark and antiquark coordinates, and this property leads to large cancellations during integration
over the variables ζ, ℓ⊥in the amplitudes (80-100, 110-129). Due to this property, the amplitudes of the CGC and
collinear frameworks differ by up to a factor of 2-3, and after squaring of the amplitude, this gives to an order of
magnitude difference in the cross-section.
25
γp→γχc0p
W=141 GeV
|t|=0.5 GeV2
Mγχc=4 GeV
Mγχc=4.5 GeV
Mγχc=5 GeV
0.
0.5
1.
1.5
2.
10-1
100
|t'|, GeV2
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
γp→γχc1p
W=141 GeV
|t|=0.5 GeV2
Mγχc=4 GeV
Mγχc=4.5 GeV
Mγχc=5 GeV
0.
0.5
1.
1.5
2.
10-2
10-1
100
|t'|, GeV2
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
γp→γχc2p
W=141 GeV
|t|=0.5 GeV2
Mγχc=4 GeV
Mγχc=4.5 GeV
Mγχc=5 GeV
0.
0.5
1.
1.5
2.
10-1
100
101
|t'|, GeV2
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
Figure 7. The dependence of the photoproduction cross-section (22) on the invariant momentum transfer to the photon (variable
t′ = (q −k)2 defined in (16)). The left, central and right columns correspond to the χc0, χc1 and χc2 mesons, respectively.
γp→γχc0p
t=tmin, |t'|=0.5 GeV2
W=1000 GeV
W=500 GeV
W=250 GeV
W=100 GeV
4.
5.
6.
7.
10-2
10-1
100
101
Mγχc, GeV
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV3
γp→γχc1p
t=tmin, |t'|=0.5 GeV2
W=1000 GeV
W=500 GeV
W=250 GeV
W=100 GeV
4.
5.
6.
7.
10-3
10-2
10-1
100
101
Mγχc, GeV
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV3
γp→γχc2p
t=tmin, |t'|=0.5 GeV2
W=1000 GeV
W=500 GeV
W=250 GeV
W=100 GeV
4.
5.
6.
7.
10-1
100
101
102
Mγχc, GeV
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV3
Figure 8. The dependence of the photoproduction cross-section (22) on the invariant mass Mγχcof the quakonium-photon pair.
The left, central and right columns correspond to the χc0, χc1 and χc2 mesons, respectively.
The dependence on the invariant energy shown the Figure 9 can be described by the power law, dσ(W) ∼W 2α,
where the constant α has a very mild dependence on other variables. This behaviour follows from a mild x-dependence
of the forward dipole amplitude (130) and reflects the increase of the gluonic density at higher energies. Numerically,
the values of the parameter α are consistent with the energy dependence implemented in the small-r branch of (130),
namely α ≈2λ⟨γeff⟩≈(0.32, 0.37). The early onset of the saturation effects would reveal itself as a deviation from
the power law behaviour.
Finally, in the Figure 10 we have shown the relative contributions of different quarkonia helicities Hχc to the
unpolarized cross-section. In the kinematics of relatively small t′, the quarkonia and photons have small transverse
momenta and thus are produced at relatively small angles with respect to the direction of incoming photon. The
helicities of all particles in this kinematics nearly coincide with projection of the angular momentum onto the collision
axis, and for this reason are subject to the helicity conservation rule,
Hγ,in ≈Hγ,out + Hχc,
small t′ ≈0.
(136)
This selection rule allows to understand the dominance of the Hχc = 0 helicity component for χc0, χc1: since the
helicities of the onshell photons only take values ±1, and |Hχc| ̸= 2 for these mesons, only the helicity state Hχc = 0
allows to satisfy (136), provided that the photon helicity is not flipped during the process, Hγ,in = Hγ,out. We checked
numerically that the contribution of the component with helicity flip of the photon, Hγ,in = −Hγ,out is indeed strongly
suppressed. At larger values of the variables |t′|, Mχc, the angles between final-state particles and the collision axis
also increases rapidly, so the helicities are no longer are related to projections of the angular momentum onto the same
26
γp→γχc0p
Mγχc=3.8 GeV
Mγχc=4.0 GeV
Mγχc=4.5 GeV
Mγχc=5.0 GeV
100
200
500
1000
10-1
100
101
102
W, GeV
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
γp→γχc1p
Mγχc=3.8 GeV
Mγχc=4.0 GeV
Mγχc=4.5 GeV
Mγχc=5.0 GeV
100
200
500
1000
10-1
100
101
102
W, GeV
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
γp→γχc2p
Mγχc=4.0 GeV
Mγχc=4.5 GeV
Mγχc=5.0 GeV
100
200
500
1000
100
101
102
W, GeV
ⅆ3σ/ⅆt ⅆt' ⅆMγχc, pb/GeV5
Figure 9. The energy dependence of the γp →χcγp cross-section for different spins and invariant energies. The nearly linear
dependence in double logarithmic coordinates suggests that the dependence may be described by power law ∼W δ, where δ
has a very mild dependence on other kinematic variables, with typical values δ ∼0.67 −0.73.
γp→γχc1p
W=141 GeV, t=tmin
Hχc=+1
Hχc=0
Hχc=-1
0.5
1.
1.5
2.
10-3
10-2
10-1
100
|t'|, GeV2
ⅆ3σγχc(Hχc)/ⅆ3σγχc(all Hχc)
γp→γχc1p
W=141 GeV, t=tmin
Hχc=+1
Hχc=0
Hχc=-1
4.
5.
6.
7.
10-3
10-2
10-1
100
101
Mγχc, GeV
ⅆ3σγχc(Hχc)/ⅆ3σγχc(all Hχc)
γp→γχc1p
W=141 GeV, t=tmin
Hχc=+1
Hχc=0
Hχc=-1
0.5
1.
1.5
2.
10-3
10-2
10-1
100
|t|, GeV2
ⅆ3σγχc(Hχc)/ⅆ3σγχc(all Hχc)
γp→γχc2p
W=141 GeV, Mγχc=4.5 GeV
Hχc=2
Hχc=1
Hχc=0
0.5
1.
1.5
2.
10-3
10-2
10-1
100
|t'|, GeV2
ⅆ3σγχc(Hχc)/ⅆ3σγχc(all Hχc)
γp→γχc2p
W=141 GeV, t=tmin
Hχc=2
Hχc=1
Hχc=0
4.
5.
6.
7.
10-3
10-2
10-1
100
101
Mγχc, GeV
ⅆ3σγχc(Hχc)/ⅆ3σγχc(all Hχc)
γp→γχc2p
W=141 GeV, Mγχc=4 GeV
Hχc=2
Hχc=1
Hχc=0
0.5
1.
1.5
2.
10-2
10-1
100
|t|, GeV2
ⅆ3σγχc(Hχc)/ⅆ3σγχc(all Hχc)
Figure 10. The relative contributions of different helicity components of χc in the total cross-section, as a function of kinematic
variables t, t′, Mγχc . The upper and lower rows correspond to χc1 and χc2, respectively. For the sake of legibility, in the lower
row (for χc2) we did not show the negligibly small contributions of the negative helicity Hχc.
axis, and thus the selection rule (136) becomes void. In order to understand the relative size of the contributions with
helicity flip of the photon, in the Figure 11 we have shown the angular harmonics c2 defined in (27). We can see that
for the harmonics Hχc = 0 and Hχc = 1, which give the dominant contributions in the cross-section, the asymmetry c2
remains moderate. It increases as a function of |t′|, in agreement with the above-mentioned picture based on increase
of the angles, however it does not exceed ∼30 per cent in the kinematics of interest. Since the complex phases of
the amplitudes A(+,+)
γp→χcγp and A(+,−)
γp→χcγp are nearly identical, the harmonics s2 is vanishingly small. An experimental
confirmation of these theoretical expectations would constitute a strong evidence in favor of the expected dominance
of the contribution without photon helicity flip.
B.
Integrated cross-sections and counting rates
While the threefold differential cross-section are well-suited for the phenomenological studies, experimentally it
may be easier to study the partially integrated (single-differential) cross-sections. In the Figure 12 we have shown the
cross-sections dσ/ dMγχc and dσ/dt′ for different charmonia. As we discussed in the previous section, for the threefold
cross-section the dependence on W, t, Mγχc largely factorizes; for this reason the dependence of the integrated cross-
sections on these variables is similar to that of the unintegrated cross-section.
27
γp→γχc1p
W=141 GeV
Mγχc=4 GeV
Hχc=1
Hχc=0
Hχc=-1
0.5
1.
1.5
2.
-0.1
-0.05
0.
0.05
0.1
|t'|, GeV2
c2
γp→γχc2p
W=141 GeV, Mγχc=4 GeV
Hχc=2
Hχc=1
Hχc=0
0.5
1.
1.5
2.
-0.6
-0.4
-0.2
0.
0.2
0.4
0.6
|t'|, GeV2
c2
Figure 11. The angular harmonics c2 of electroproduction cross-section (27), for different helicity components of χc1 and χc2
mesons, respectively. For the sake of legibility, in the right plot (for χc2) we did not show the asymmetries of the contributions
with negative helicity Hχc, since the latter are negligibly small.
γp→γχcp
χc0
χc1
χc2
4.
5.
6.
7.
10-2
10-1
100
101
102
Mγχc, GeV
ⅆσ/ⅆMγχc, pb/GeV
γp→γχcp
χc0
χc1
χc2
0.5
1.
1.5
2.
100
101
102
|t'|, GeV2
ⅆσ/ⅆt', pb/GeV2
Figure 12. The single-differential cross-sections dσ/dt dMγχc, dσ/dMγχc for different charmonia χcJ. Both plots correspond to
the invariant energy W = 141 GeV.
The total (integrated) cross-sections of the photoproduction process γp →χcJγp is given by
σχc0
tot (W ≈100 GeV, Mγχc ≥3.7 GeV) ≈25 pb,
(137)
σχc1
tot (W ≈100 GeV, Mγχc ≥3.7 GeV) ≈23 pb,
(138)
σχc2
tot (W ≈100 GeV, Mγχc ≥3.7 GeV) ≈149 pb,
(139)
and scales with energy approximately as ∼W 0.7. In our estimates we introduced a cutoff on minimal invariant mass
Mγχc in order to avoid (huge) background from the radiative decays of ψ(2S) →χcγ 4. In the Tables II and III we
provide tentative estimates for the total cross-sections of the electroproduction ep →eγχcp and the ultraperipheral
production pp →ppγχc in LHC kinematics. Since the spectrum of virtual photons is dominated by quasireal photons,
for the sake of simplicity we disregarded the contributions of photons with large virtuality Q2 ≳1 GeV2, and assumed
that the photoproduction cross-section does not depend on Q2 at Q2 ≲1 GeV2 in view of the smallness of O
Q2/M 2
χc
corrections in the heavy quark mass limit. We found that varying the upper cutoff Q2
max between 1 and 2 GeV2 changes
the cross-section within 10 per cent.
4 Combining the experimental ψ(2S) photoproduction cross-section [88] and the branching fractions of ψ(2S) →χcJγ [70], we estimate
that the background contribution from ψ(2S) radiative decay is σ(rad)
tot
≈1.1 nb.
28
σ(ep)
tot
Production rates
Decay
Combined Counting rates
N
dN/dt
channel
branching
Nd
dNd/dt
χc0 0.45 pb 4.5×104
380/day
χc →J/ψ γ
0.08 %
36
9.1/month
χc1 0.41 pb 4.1×104
353/day
J/ψ →µ+µ−
2 %
816
211/month
χc2
2.5 pb 2.5×105 2100/day
1.1 %
2750 695/month
Table II. The total electroproduction cross-sections, production and counting rates for different χc mesons in EIC kinematics
at electron-proton energy √sep = 141 GeV. The cross-section roughly scales with energy as ∼W 0.7. For estimates of the pro-
duction and counting rates, we used the instantaneous luminosity L = 1034 cm−2s−1 = 10−5fb−1 and the integrated luminosity
R
dt L = 100 fb−1. The column “combined branching” corresponds to the product of branching fractions of χc →J/ψ γ and
J/ψ →µ+µ−.
σ(pp)
tot
Production rates
Decay
Combined Counting rates
N
dN/dt
channel
branching
Nd
dNd/dt
χc0
0.3 pb
3×104
253/day
χc →J/ψ γ
0.08 %
24
6.1/month
χc1 0.25 pb 2.5×104
215/day
J/ψ →µ+µ−
2 %
500
129/month
χc2
1.7 pb 1.7×105 1088/day
1.1 %
1421 359/month
Table III. The total production cross-sections, production and counting rates for different χc mesons, in ultraperipheral LHC
kinematics at proton-proton energy √spp = 13 GeV. For estimates of the production and counting rates, we used the instan-
taneous luminosity L = 1034 cm−2s−1 = 10−5fb−1s−1 and the integrated luminosity
R
dt L = 100 fb−1. The column “combined
branching” corresponds to the product of branching fractions of χc →J/ψ γ and J/ψ →µ+µ−.
C.
Comparison with odderon-mediated χc photoproduction
The odderons, or C-odd gluon exchanges in t-channel, remain one of the least understood components of the
nonperturbative QCD [89–94], and the the contribution of the odderons to the dipole scattering amplitude at present
is largely unknown. A recent discovery of the odderons from comparison of pp and p¯p elastic cross-sections measured
at LHC and Tevatron [95, 96] reinvigorated the interest in this topic. However, the statistical significance of the
observed signal still remains under discussion due to sizable uncertainties and possible background contributions. It
could be extremely difficult to study in detail the odderon amplitude from the channels where odderons contribute as
minor corrections. For this reason, the processes which proceed via the C-odd t-channel exchanges got into focus of
odderon searches, and it is expected that such processes will be studied experimentally both at HL-LHC and at the
future EIC.
The exclusive photoproduction of C-even quarkonia, as for example γp →ηcp, historically attracted a lot of interest,
since the heavy quark mass plays the role of the hard scale and justifies at least partial description in perturbative
QCD (see [94, 97] for a short overview). However, that process obtains a sizable contribution from the photon-photon
fusion (Primakoff mechanism), and for this reason, recently the exclusive χc photoproduction γp →χcp has been
recently suggested in [63, 64] as a potential cleaner alternative. The cross-section of the latter process is comparable
to that of ηc photoproduction; furthermore it may be easier to study experimentally due to large branching fraction
of radiative decays of χc to J/ψ.
In this context, the photoproduction of χcγ pairs deserves attention as a potential background to odderon-mediated
χc-photoproduction: while the former is formally suppressed as O (αem), the latter is suppressed by smallness of the
odderon amplitude, so numerically their cross-sections are comparable.
If the final-state photon is not detected,
potentially the χcγ photoproduction can be misinterpreted as γp →χcp subprocess. Furthermore, if there is no
constraints (cuts) on the invariant mass of the produced hadronic state, a sizable contribution could be obtained
from the radiative decays of heavier charmonia. In order to estimate accurately these backgrounds, in general it is
required to know in detail the geometry and acceptance of the detector. For simplicity, now we will consider that all
the final photons are not detected, and will consider the cross-section dσ/dt integrated over the phase space of the
produced photon. This approach provides an upper estimate for the background. For the sake of definiteness, we will
use for comparison only the result of [64] because the predictions of [63] are provided in the kinematics of relatively
29
χc0
ψ(2S)→χc0γ(inv)
ep→eχc0p+γ(inv.)
ep→eχc0p, Odderon
ep→eχc0p, Primakoff
0.5
1.
1.5
10-4
10-3
10-2
10-1
100
|t|, GeV2
ⅆσ/ⅆt, pb/GeV2
χc1
ψ(2S)→χc1γ(inv)
0.5
1.
1.5
|t|, GeV2
χc2
ψ(2S)→χc2γ(inv)
0.5
1.
1.5
|t|, GeV2
Figure 13. Comparison of different mechanisms which contribute to electroproduction of mesons. The solid curve corresponds
to contribution of ep →eγχcp process discussed in this paper, with invisible (integrated out) final state photon. The upper
dotted curve corresponds to the contribution of χcγ pairs which stem from the radiative decays of ψ(2S) charmonia (the
χcγ pairs in this mechanism have invariant mass Mγχc ∼Mψ(2S)). The dashed and dot-dashed lines correspond to exclusive
electroproduction ep →eχcp mediated by odderons and the Primakoff mechanism, as provided in [64] (see Figure 5).
low energies, where the CGC approach might be not very reliable.
In the Figure (13) we show the cross-sections of the χc and χcγ production via different mechanisms.
The
largest contribution comes from the ψ(2S) photoproduction with subsequent radiative decay ψ(2S) →χcγ, since
ψ(2S) photoproduction does not require exchange of quantum numbers in t-channel, and ψ(2S) has remarkably
large branching ratios of radiative decays, Br (ψ(2S) →χc0γ) = 9.79 ± 0.2 %, Br (ψ(2S) →χc1γ) = 9.75 ± 0.24 %,
Br (ψ(2S) →χc2γ) = 9.52 ± 0.2 % [70]. We used for estimates the cross-section of ψ(2S) photoproduction found in
CGC framework in [98], and checked that it can reproduce the experimentally measured total (t-integrated) cross-
section [88] and the diffractive slope [99]. While the radiative decays of other excited quarkonia potentially could also
contribute to the observed backgrounds, at present it is not possible to estimate accurately their contribution due to
a sizable uncertainty in their wave functions.
The exclusive (non-resonant) production of χcγ pairs suggested in this paper exceeds significantly the odderon-
mediated χc photoproduction at small-t (|t| ≲1 GeV2), although is less relevant in the large-t kinematics.
The
cross-sections of both mechanisms of χcγ production (direct and via ψ(2S) radiative decays) increase with energy,
whereas the contribution of odderons should decrease due to different odderon intercept.
For the sake of comparison we also have shown the contribution of the photon-photon fusion (so-called Primakoff
mechanism) which exceeds the odderon contribution in the small-t kinematics, as discussed in [64]. While the Primakoff
mechanism is less relevant for odderon searches on neutrons, namely for γn →χcn subprocess, the backgrounds due
to the χcγ production with undetected photon are the same for proton and neutron targets.
The contributions of different mechanisms discussed in this section can be separated, imposing the kinematic cuts on
the variable (q −∆)2 ≈t−Q2 −2q ·∆. For exclusive χc production via odderon-mediated and Primakoff mechanisms,
this variable corresponds to the square of charmonium mass, M 2
χc. For production via radiative decays of ψ(2S), this
variable equals M 2
ψ(2S). Finally, for the direct (non-resonant) production of χcγ pairs, this variable coincides with
invariant mass M 2
γχc of χcγ pair.
IV.
CONCLUSIONS
Using the CGC framework, we analyzed the exclusive χcγ photoproduction, including its helicity dependence, for all
χcJ states. We found that the amplitude of the process may be represented as a convolution of the process-dependent
impact factors and forward dipole scattering amplitude. In the heavy quark mass limit, the kinematic distribution
of produced χcγ pairs may be related to the dependence of the forward dipole scattering amplitude on dipole size,
impact parameter and rapidity dependence. We also estimated numerically the cross-sections in the ultraperipheral
kinematics at LHC and the future EIC. The cross-sections of all χc mesons are comparable to each other (within a
factor of two), and of the same order of magnitude as the cross-section of ηc-production found earlier in [55]. The
expected production rates of χcγ pairs constitute ∼104 −105 events per each 100 fb−1 of integrated luminosity, both
at EIC and LHC. The expected detection (counting) rates for χc1 and χc2 mesons constitute ∼102 −103 events per
each 100 fb−1 of integrated luminosity, if the χc mesons are detected via their χc →J/ψ γ decays. These numbers
suggest that the cross-section may be measured with reasonable precision. We also found that the production of
χcγ pairs with invisible photon (either direct or via radiative decay of ψ(2S) →χcγ) gives a sizable background to
30
exclusive photoproduction of χc mesons, which was suggested recently in [63, 64] as an alternative tool for study of
odderons. While the cross-sections of χcγ production decrease rapidly as a function of the momentum transfer |t|,
the radiative decay of ψ(2S) remains the dominant mechanism up to |t| ∼2 GeV2. These findings are in line with our
previous analysis [55], which concluded that odderon-mediated quarkonia production can get sizeable backgrounds
from quarkonia-photon production.
ACKNOWLEDGMENTS
We thank our colleagues at UTFSM university for encouraging discussions. This research was partially supported
by Proyecto ANID PIA/APOYO AFB220004 (Chile), and ANID grants Fondecyt Regular №1251975 and Fondecyt
Postdoctoral №3230699. I. Z. also expresses his gratitude to the Institute of Physics of PUCV. Powered@NLHPC:
This research was partially supported by the supercomputing infrastructure of the NLHPC (ECM-02).
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The exclusive photoproduction of χcγ pairs in the small-x kinematics M. Siddikov,1, 2 I. Zemlyakov,1, 3 and M. Roa1, 4 1Departamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile 2Centro Científico - Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile 3Instituto de Física, Pontificia Universidad Católica de Valparaíso, Av. Brasil 2950, Valparaíso, Chile 4Facultad de Ingeniería, Laboratorio DataScience, Universidad de Playa Ancha, Leopoldo Carvallo 270, Valparaíso, Chile, In this manuscript we analyze the exclusive photoproduction of the χcγ pairs. We focus on the small-x kinematics and evaluate the cross-sections in the Color Glass Condensate framework. We found that in the leading order in the strong coupling αs, this process is sensitive only to the forward color dipole scattering amplitude. We estimated numerically the cross-sections for different polarizations of χc mesons in the kinematics of the ultraperipheral collisions at LHC and the future Electron Ion Collider. We also analyzed the role of this process as a potential background to exclusive χc photoproduction, which has been recently suggested as an alternative channel for studies of odderons. According to our estimates, the cross-section of χcγ with undetected photons is comparable to that of odderons at small momentum transfer |t| ≲1 GeV2, but becomes less relevant at larger |t|. We also found that a dominant background to odderon-mediated production of χc comes from the radiative decays of ψ(2S) mesons, which potentially can present a challenge for odderon studies via χc photoproduction. I. INTRODUCTION The Color Glass Condensate (CGC) approach [1-5] has been established as a theoretical framework for systematic analysis of the hadronic processes in the small-x kinematics. It naturally incorporates saturation effects and allows us to obtain a self-consistent and phenomenologically acceptable description of various lepton-hadron and hadron-hadron processes [6-21]. The interactions of partons with the target in this approach are encoded in the nonperturbative n-point correlators of Wilson lines (forward multipole scattering amplitudes), whose dependence on rapidity obeys the evolution equations of Balitsky-Kovchegov Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (BKJIMWLK) [22-32]. In the presence of appropriate hard scale which controls the transverse size of the system, it is expected that the dominant contribution stems from the quark-antiquark component of the wave function, convoluted with forward dipole scattering amplitude. On this premise, a number of processes has been used in phenomenological analyses in order to fix the forward dipole scattering amplitude. However, the higher order multipoles may contribute in loop corrections even in the simplest and well-known processes, like Deep Inelastic Scattering (DIS) [33, 34]. A recent numerical analysis of the next-to-leading order (NLO) corrections to various processes demonstrated that they could be numerically sizable [35-38], thus shedding doubts on phenomenological extractions based on leading order picture. Unfortunately, such estimates remain ambiguous due to dependence on poorly known parametrization of these multipoint correlators. Conversely, a reasonable phenomenological description of the existing experimental data by completely different parametrizations, disregarding NLO corrections altogether, suggests that the existing data don't uniquely fix the dipole amplitude [15-17]. While it is impossible to switch off completely the contributions of the higher twist correlators, it is possible to reveal their presence testing the universality of the forward dipole scattering amplitude in various processes. The exclusive 2 →3 processes from this point of view are well-suited for this analysis, since they have a different structure of the amplitude, and thus can be used as independent tools for phenomenological studies [39-47].Due to high luminosity of the ongoing photoproduction experiments at LHC (in ultraperipheral kinematics) [48] and the future experiments at the Electron Ion Collider (EIC) [49, 50], measurement of such processes is feasible in the nearest future. A special interest in this program deserves the production of quarkonia-photon pairs. The heavy quark mass serves as a natural hard scale in processes involving their production and justifies perturbative description of the Q ̄Q pair formation, as well as the use of NRQCD framework for their hadronization [51-54]. Due to strong suppression of C-parity exchanges in the t-channel, the largest cross-section have the quarkonia-photon pairs which include C-even quarkonia, like ηc and χc. The production of ηcγ pairs has been studied recently in [55], and it was discussed how the expected kinematic distributions of the final-state particles are related to the dipole amplitude. However, that channel suffers from significant backgrounds, most notably from the radiative decays of other quarkonia, and for this reason requires additional kinematic cutoffs which reduce the expected yields. In this paper we extend that analysis and suggest to study the production of the χcγ meson pairs. In what follows we will focus on the high energy (small-x) photon-proton collisions, which can be studied as a subprocess in ultraperipheral pA and AA collisions at LHC, as well as in the future high energy runs at Electron Ion Collider, the Large Hadron electron Collider (LHeC) [56], and the Future Circular Collider (FCC-he) [57]. 16 Oct 2025 2 In addition to studies of the dipole scattering amplitudes, the χcγ photoproduction can also present interest for two other problems. Historically, the triplet of the P-wave quarkonia χc with different spins and nearly the same masses has been considered in the literature as a possible tool for precision tests of NRQCD. In the infinitely heavy quark mass limit, it is expected that the long-distance matrix elements (LDMEs) of all χc mesons are related to each other due to heavy quark spin symmetry (HQSS). However, the experimental results for the ratio σχc2/σχc1in hadroproduction experiments disagrees significantly with HQSS-based expectations, and the phenomenologically successful descriptions of that channel either assume that HQSS is heavily broken for charm, or introduce sizable contributions of the color octet LDMEs [58-62]. The study of the same ratio in exclusive χcγ photoproduction experiments can decisively clarify the validity of HQSS because this channel does not get contributions from the color octet LDMEs. Furthermore, the exclusive χc photoproduction (via γp →χcp channel) recently has been suggested in [63, 64] as a sensitive tool for precision studies of odderons. The photoproduction of χcγ pairs in this context deserves interest as a potential background to odderon searches. The paper is structured as follows. In the next Section II we discuss in detail the kinematics of the process, and derive the theoretical results for the cross-section in the Color Glass Condensate framework. In Section III we provide tentative estimates for the cross-sections and expected counting rates obtained with a publicly available parametrization of the forward dipole scattering amplitude. We also estimate the contribution of this process and the ψ(2S) →χcγ radiative decay to odderon-mediated γp →χcp suggested in [64]. Finally, in Section IV we draw conclusions. II. THE THEORETICAL FRAMEWORK As we explained in the introduction, we will analyze the γp →χcγp process in the high energy (small-x) kinematics, using for description the CGC framework. We will focus on the kinematics of quasireal incoming photons, which dominate the spectrum of equivalent photons in ultraperipheral collisions and electroproduction experiments. In the subsections II A - II D below we will discuss the kinematics of the process and the light-cone wave functions of χc mesons with different helicities, introduce briefly the CGC framework and derive the amplitude of the process in this approach. The final theoretical expressions for different helicity components are given in Section II D, in Eqns. (80 - 100, 110 - 129). The latter equations will be used for numerical estimates in the next section. A. Kinematics of the process The evaluation of the invariant cross-section can be realized in the so-called photon-proton collision frame, where the 3-momenta of the incoming proton and photon are anticollinear to each other. For the sake of definiteness we'll assume that the collision axis coincides with the axis z of our reference frame, and that the photon and proton propagate in its positive and negative directions respectively. The mass of the heavy quarkonium Mχc and the invariant mass of the photon quarkonium pair Mγχcwill be considered as hard scales, in agreement with [65]. The evaluation of the physical amplitude requires to define the light-cone decomposition of various momenta. In what follows we will utilize the convention of Kogut-Soper [66], assuming that any 4-vector vμ can be represented as vμ = v+, v-, v⊥ , v± = v0 ± v3 √ 2 , v⊥= vxˆx + vyˆy, (1) so its square and convolution with Dirac matrix are given by v2 ≡vμvμ = 2v+v--v2 ⊥, ˆv ≡γμvμ = γ+v-+ γ-v+ -γ⊥· v⊥. (2) The light-cone decomposition of particle's momenta will be chosen in agreement with earlier studies on meson-photon production [67, 68], qμ = rs 2, 0, 0⊥ , kμ = ̄αχc rs 2, (kγ ⊥)2 ̄αχc √ 2s, kγ ⊥ ! (3) P μ in = m2 N √ 2s (1 + ξ), (1 + ξ) rs 2, 0⊥ , P μ out = m2 N + ∆2 ⊥ √ 2s (1 -ξ), (1 -ξ) rs 2, ∆⊥ , (4) ∆μ = P μ out -P μ in = 2ξm2 N + (1 + ξ) ∆2 ⊥ √ 2s (1 -ξ2) , -2ξ rs 2, ∆⊥ , (5) (6) 3 pμ χc = αχc rs 2, (pχc ⊥)2 + M 2 χc αχc √ 2s , pχc ⊥ ! , (7) kγ ⊥:= p⊥-∆⊥ 2 , pχc ⊥:= -p⊥-∆⊥ 2 , p⊥≡(kγ ⊥-pχc ⊥) /2 (8) where q and k are the momenta of the incoming and outgoing (emitted) photons, Pin, Pout are the momenta of the proton before and after the interaction, and pχc is the momentum of produced χc meson. The parameter αχc stands for the fraction of the photon's light-cone momentum carried by χc, and a shorthand notation ̄αχc ≡1-αχc corresponds to a similar light-cone fraction carried by the scattered photon. For the proton which moves in the opposite direction, the parameter ξ controls the longitudinal momentum transfer to the proton during the process. The hard scale s can be related to the photon-proton invariant energy W of the process as SγN ≡W 2 = (q + Pin)2 = s (1 + ξ) + m2 N, √s = s W 2 -m2 N 1 + ξ . (9) As will be demonstrated below, for high energy collisions (W ≫mN), the variable ξ ≪1, so (9) can be approximated as √s ≈W. The light-cone decomposition in the target rest frame (r.f.) can be obtained from (3-8) making a longitudinal boost v+ →Λv+, v-→Λ-1v-, v⊥→v⊥ with Λ = √s (1 + ξ) /mN, yielding qμ r.f. = s (1 + ξ) mN √ 2 , 0, 0⊥ , kμ r.f. = ̄αχcs (1 + ξ) √ 2mN , (kγ ⊥)2 ̄αχcs √ 2 mN (1 + ξ), kγ ⊥ ! (10) P μ in,r.f. = mN √ 2 , mN √ 2 , 0⊥ , P μ out = m2 N + ∆2 ⊥ √ 2 (1 -ξ) (1 + ξ) mN , mN √ 2 1 -ξ 1 + ξ , ∆⊥ , (11) ∆μ r.f. = P μ out -P μ in = 2ξm2 N + (1 + ξ) ∆2 ⊥ √ 2mN 1 + ξ 1 -ξ , - √ 2ξ mN (1 + ξ) , ∆⊥ ! (12) pμ χc,r.f. = αχcs (1 + ξ) √ 2mN , (pχc ⊥)2 + M 2 χc αχcs √ 2 mN (1 + ξ), pχc ⊥ ! . (13) The light-cone decomposition (3-8, 10-13) satisfies the energy-momentum conservation and guarantees that all particles are on shell. As we discussed in [55, 65], it is possible to express the variables ξ, p⊥, ∆⊥, αχc, which define the kinematics, in terms of the invariant mass M 2 γχc, momentum transfers to the proton (t) and to the photon (t′). t = ∆2 = (Pout -Pin)2 = -1 + ξ 1 -ξ ∆2 ⊥-4ξ2m2 N 1 -ξ2 ≤-4ξ2m2 N 1 -ξ2 . (14) t′ = (k -q)2 = -2q · k = -(p⊥-∆⊥/2)2 (1 -αχc) ≤0 (15) M 2 γχc = (k + pχc)2 = (q + Pin -Pout)2 = t -2q · ∆= t + 2sξ (16) Resolving these equations, it is possible to obtain exact identities [55, 65] ξ = M 2 γχc -t 2 (W 2 -m2 N) -M 2γχc + t > 0, s = W 2 -m2 N + t -M 2 γχc 2 , 2ξs = M 2 γχc -t, (17) ∆2 ⊥= -t1 -ξ 1 + ξ -4ξ2m2 N (1 + ξ)2 , p⊥· ∆⊥= 2ξm2 N (1 -ξ2) t′ s (18) p2 ⊥= -t′ 1 2 -αχc + 1 2 αχcM 2 γχc -M 2 χc + t 4 1 -ξ 1 + ξ -2αχc + ξ2m2 N (1 + ξ)2 . (19) 4 αχc = -1 2ξs t′ -M 2 χc - 2ξm2 N s (1 -ξ2) M 2 γχc + t′ -t - 2ξm2 N s (1 -ξ2), (20) which fix the kinematics of the process in terms of the variables W, t, t′, Mγχc up to a global rotation in the transverse plane. For high-energy processes we can simplify these relations using the hierarchy of scales W, √s ≫Mχc, Mγχc ≳ p⊥, ∆⊥. The parameter ξ defined in (17) is also very small, ξ ∼M 2 γχc/W 2 ≪1, and can be disregarded if accompanied by other terms which don't vanish in this limit. Similarly the smallness of the ratio t′/s ∼M 2 γχc/W 2 ∼O (ξ) ≪1 in the right-hand side of (18) implies that the vectors p⊥, ∆⊥are almost orthogonal to each other. As was discussed long ago in [69] and confirmed by various experimental studies, the cross-sections of exclusive processes decrease rapidly as a function of the invariant momentum transfer t ∼∆2 ⊥to the target. Technically, this happens because a large momentum transfer to the target won't destruct it only if the momentum is shared equally between all partons, and each momentum exchange between partons is suppressed by (perturbative) propagators. As we will see below, such suppression also takes place in our process, and for this reason in the kinematics of interest the variables ∆2 ⊥, t are negligible compared to typical scales ∼M 2 χc. In this kinematic, it is possible to rewrite (19, 20) in a simpler form (see details in [55, 65]) -t′ ≈αχcM 2 γχc -M 2 χc, p2 ⊥= ̄αχc αχcM 2 γχc -M 2 χc ≈- ̄αχct′, M 2 γχc ≈2sξ, (21) The physical constraint p2 ⊥≥0 implies that the variable αχc is bound by αχc ≥M 2 χc/M 2 γχc. The cross-section of the photoproduction subprocess may be represented as dσ(T ) γp→γχcp dt dt′ dMγχc ≈ A(λ; σ,H) γp→χcγp 2 128π3Mγχc (22) where A(λ; σ,H) γp→χcγp is the process amplitude, and the superscript indices λ, σ, H are the helicities of the incoming photon, outgoing photon and the produced charmonium, respectively.The cross-section of the electroproduction is given by [9, 70] dσep→eγχcp dΩ = εdσ(L) dΩ + dσ(T ) dΩ + p ε(1 + ε) cos φdσ(LT ) dΩ + p ε(1 + ε) sin φdσ(L′T ) dΩ + (23) + ε cos 2φdσ(T T ) dΩ + ε sin 2φdσ(T ′T ) dΩ , where φ is the angle between the lepton and proton scattering planes (see Figure 1 for definition), the notation dΩ≡d ln W 2dQ2 dt dt′ dMγχcdφ is used for the phase volume, and the variable Q2 = -q2 is the virtuality of the photon propagating from leptonic to hadronic parts. The superscript letters L, T, T ′ in the right-hand side of (23) distinguish the contributions of the longitudinal and transverse photons (as well as their possible interference). The parameter ε is the ratio of the longitudinal and transverse photon fluxes. The latter may be expressed in terms of inelasticity parameter y and the invariant energy √sep of the electron-proton collision as ε ≈ 1 -y 1 -y + y2/2, y = W 2 + Q2 -m2 N sep -m2 N . (24) In what follows we'll focus on the kinematics of small Q2, which gives the dominant contribution to the total crosssection. In this kinematics, the contribution of the longitudinally polarized photons is negligible, and (23) reduces to dσep→eM1M2p dΩ ≈dσ(T ) dΩ + ε cos 2φdσ(T T ) dΩ + ε sin 2φdσ(T ′T ) dΩ ≈dσ(T ) dΩ (1 + c2 cos 2φ + s2 sin 2φ) , (25) where dσ(T )/dΩis related to the photoproduction cross-section (22) as [71-73] dσ(T ) dΩ ≈αem π Q2 1 -y + y2 2 -(1 -y)Q2 min Q2 dσ(T ) γp→γχcp dt dt′ dMγχc , (26) the superscript index (T) is used to remind that the dominant contribution comes from the transversely polarized photon, Q2 min = m2 ey2/ (1 -y), me is the mass of the electron, and the angular harmonics c2, s2 are related to different components of the amplitude A(λ,σ,H) γp→χcγp as c2 = 2ε Re A(-,+,H)∗ γp→χcγpA(+,+,H) γp→χcγp A(+,+) γp→χcγp 2 + A(-,+) γp→χcγp 2 , s2 = 2ε Im A(-,+,H)∗ γp→χcγpA(+,+,H) γp→χcγp A(+,+) γp→χcγp 2 + A(-,+) γp→χcγp 2 . (27) 5 Figure 1. (Color online) The definition of the angle φ between leptonic and hadronic planes for the ep →e′p′χcγ electroproduction channel, as seen from the target rest frame. The lepton scattering plane is formed by the three-momenta of incoming and scattered electrons (marked with labels e, e′), and the hadron scattering plane may be defined as a plane that includes the three-momenta of the recoiled proton Pout and the total three-momentum pχc + kγ of the produced γχc pair. Due to momentum conservation, this plane also includes the 3-momentum of the virtual photon γ∗. In the photon-proton collision frame, the incoming proton propagates in direction anticollinear to the incoming virtual photon γ∗, so the momentum of the recoil proton Pout is almost anticollinear to the momentum of the virtual photon γ∗. As we will see below, these coefficients are small, so the cross-section (25) is dominated by the angular-independent term (the so-called equivalent photon approximation). B. The polarization vectors and wave functions For the polarization vectors of the photons, we will use the light-cone parametrization ε(λ=±1) T (l) = ελ · l⊥ l- , 0, ελ , ελ = ˆx + iλ ˆy √ 2 . (28) where lis the 4-momentum of the corresponding photon (l= q or k) in this process. For the polarization vector of the axial vector quarkonium χc1, we'll use a similar parametrization suggested in [64] E(χc1) T,λ (pχc) = εT · pχc ⊥ pχc , 0, εT , E(χc1) L (pχc) = (pχc ⊥)2 -M 2 χc 2Mχcpχc , pχc Mχc , pχc ⊥ Mχc ! , (29) where the subindices L and T = ±1 distinguish the transverse and longitudinal polarizations. This parametrization automatically satisfies the transversality condition pχc · E(χc) = 0. The polarization tensor Eμν of the charmonium χc2 can be constructed from polarization vectors defined in (29) using conventional Clebsch-Gordan coefficients as E(χc2) μν = E(χc1) T =±1 μ E(χc1) T =±1 ν , Hχc2 = ±2, E(χc1) T =±1 μ E(χc1) L ν+ E(χc1) L μ E(χc1) T =±1 ν √ 2 , Hχc2 = ±1, E(χc1) T =+1 μ E(χc1) T =-1 ν+ E(χc1) T =-1 μ E(χc1) T =+1 ν+2 E(χc1) L μ E(χc1) L ν √ 6 , Hχc2 = 0, (30) 6 where Hχc2 is the helicity projection of the corresponding charmonium. The wave function of the χc mesons in helicity basis in momentum space is given by ΨH,h ̄h = ̄uh (kQ) Γχc kQ, k ̄ Q v ̄h k ̄ Q φχcJ z, k(rel) ⊥ , Γχc kQ, k ̄ Q = 1, J = 0, iγ5γμE(χc1) μ , J = 1, γμKν+γνKμ 2 E(χc2) μν J = 2. (31) where h, ̄h are the helicities of the quark, z and k(rel) ⊥ are the light-cone fraction and transverse momentum of the quark (w.r.t direction of χc meson), and the momentum of relative motion Kμ ≡kμ Q -kμ ̄ Q is expressed in terms of these variables as Kμ = (2z -1) αχc rs 2, 2pχc ⊥· K⊥+ (1 -2z) h (pχc ⊥)2 + M 2 χc i αχc √ 2s , K⊥ . (32) The K--component of the vector Kμ was fixed from the orthogonality condition K · pχc = 0, which follows from the fact that the quarks are (nearly) onshell in the heavy quark mass limit. The transverse component of Kμ is related to k(rel) ⊥ as k(rel) ⊥ = ̄zkQ -zk ̄ Q = K⊥-(2z -1)pχc ⊥. (33) For χc0 mesons, the Fourier transformation of (31) over the transverse coordinates allows to obtain the light-cone wave function Ψ(χc0) h ̄h (z, r) = -1 z ̄z ihe-ihθδh, ̄h∂r + m (z - ̄z) δh,- ̄h Φχc0 (z, r⊥) , (34) where r = {r cos θ, r sin θ} is the vector of relative separation of the quark and antiquark in transverse plane, z is the fraction of the momentum carried by the quark, and the configuration space wave function Φχc is related to the function φχc z, k(rel) ⊥ by relation ΦχcJ (z, r⊥) = Z d2k(rel) ⊥ (2 π)2 √z ̄z φχcJ z, k(rel) ⊥ eik(rel) ⊥ ·r⊥ k(rel) ⊥ 2 -M 2χcJz (1 -z) + m2 . J = 0, 1, 2 (35) The integrand of (35) includes additional light-cone denominator whose origin will become obvious in the next section (see the denominators in the last lines of (71) and (72) below; also see Appendix of [55] for a more detailed discussion). For the helicity components of the χc1 and χc2 we may obtain in a similar fashion Ψ(χc1) H=±1, h ̄h (z, r) = √ 2 z ̄z -eiHθ ̄zδh,Hδ ̄h,-H + zδh,-Hδ ̄h,H ∂r -iHm (z - ̄z) δh,Hδ ̄h,H Φχc1 (z, r) , (36) Ψ(χc1) H=0, h ̄h (z, r) = 2i z ̄zMχc1 h hδh,- ̄h ∆(+) r + ime-ihθδh, ̄h∂r i Φχc1 (z, r) , (37) where ∆(±) r = ∂2 r ± (1/r) ∂r, ̄z = 1 -z, and Ψ(χc2) H=+2, h ̄h (z, r) = 2 z ̄z h e2iθ zδh,+δ ̄h,-- ̄zδh,-δ ̄h,+ ∆(-) r + imeiθδh,+δ ̄h,+∂r i Φχc2 (z, r) , (38) Ψ(χc2) H=-2, h ̄h (z, r) = 2 z ̄z h -e-2iθ ̄zδh,+δ ̄h,--zδh,-δ ̄h,+ ∆(-) r -ime-iθδh,-δ ̄h,-∂r i Φχc2 (z, r) , (39) Ψ(χc2) H=+1, h ̄h (z, r) = Mχc2 z ̄z ieiθ 3z -4z2 δh,+δ ̄h,-+ 3 ̄z -4 ̄z2 δh,-δ ̄h,+ ∂r + m (z - ̄z) δh,+δ ̄h,+ Φχc2 (z, r) , (40) Ψ(χc2) H=-1, h ̄h (z, r) = Mχc2 z ̄z -ie-iθ 3 ̄z -4 ̄z2 δh,+δ ̄h,-+ 3z -4z2 δh,-δ ̄h,+ ∂r + m (z - ̄z) δh,-δ ̄h,- Φχc2 (z, r) , (41) Ψ(χc2) H=0, h ̄h (z, r) = p 2/3 z ̄z h δh,- ̄h -3∆(+) r + 2m2 (z - ̄z) -ime-ihθδh, ̄h∂r i Φχc2 (z, r) . (42) 7 1. Relation to NRQCD LDMEs In order to understand the relations of the light-cone distribution amplitudes to long distance matrix elements (LDMEs) of NRQCD, we need to analyze the behaviour of the wave functions in the limit when Kμ becomes small. Since we are interested in the P-wave quarkonia, in the limit Kμ →0 the corresponding vertices should vanish. This is obvious for χc2 due to explicit Kμ in (31). To demonstrate that a similar property is valid for the states J = 0 and J = 1, we need to rewrite (31) in equivalent form, using the Dirac equation for the ̄u, v spinors and the orthogonality of the polarization vector EH · pχc = 0, namely ̄u (kQ)ΓχcJv k ̄ Q = 1 2m ̄u (kQ) (mΓχcJ + ΓχcJm) v k ̄ Q = 1 2m ̄u (kQ) ˆkQΓχcJ -ΓχcJ ˆk1 v k ̄ Q (43) = 1 4m ̄u (kQ) (ˆpχcΓχcJ -ΓχcJ ˆpχc) v k ̄ Q + 1 4m ̄u (kQ) ˆKΓχcJ + ΓχcJ ˆK v k ̄ Q = ̄u (kQ) Γ(eff) χcJ v k ̄ Q , where Γ(eff) χcJ = Kμ (γμΓχcJ + ΓχcJγμ) /4m = γμ χcJKμ, and the momentum-independent matrices γμ χcJ for different quarkonia are given by γμ χc0 ≡γμ/2m, γμ χc1 ≡ h γμ, ˆEH i γ5/4m, and γμ χc2 = Eμνγν. In the amplitudes of physical processes, the wave function (31) contributes in convolution with amplitude of hard partonic process. If the latter possess a hard scale which exceeds significantly the relative momentum Kμ (∼inverse charmonium radius), we can disregard completely the dependence on z, k(rel) ⊥ in the hard partonic amplitude, and the convolution over z, K⊥will affect only the arguments of the wave function, yielding Z dz Z d2k(rel) ⊥ (2π)2 Φχc z, k(rel) ⊥ = (44) = lim λ, r⊥→0 0 ̄Ψ -λ 2 n+ -r⊥ 2 γμ χcJi←→ ∂μ L -λ 2 n+ -r⊥ 2 , λ 2 n+ + r⊥ 2 Ψ λ 2 n+ + r⊥ 2 χc(p) = 0 ̄Ψ (0) γμ χcJ i 2 ←→ D μ Ψ (0) χc(p) where ←→ D μ = -→ ∂μ -←- ∂μ + 2igAa μta + 2ieAμ is the covariant derivative, and we replaced Kμ with i -→ ∂μ -←- ∂μ in the configuration space. The square of the matrix element in the last line of (44) in the nonrelativistic limit reduces to NRQCD LDMEs. Indeed, using explicit form of γμ χcJ for J = 0, 1, 2, we may recover the familiar NRQCD operators from [74-76] 1 D 0 ̄ψ (0) γμ χcJi←→ D μ ψ (0) χc(p) E 2 = 1 4m2 × D 0 ̄ψ i 2 ←→ D μ ⊤γ⊤ μ χ χc(p) E 2 = D O[1] χc 3P [1] 0 E , J = 0 D 0 ψ† -i 4 ←→ D ν ⊤ γ⊤ ν , γ⊤ μ γ5 χ χc(p) E 2 = D O[1] χc 3P [1] 1 E , J = 1 0 ψ† -i 2 ←→ D(μ ⊤γν) ⊤ χ χc(p) 2 = D O[1] χc 3P [1] 2 E , J = 2 (46) where ⊤implies a part of the vector which is transverse to pμ χc, namely vμ ⊤= gμν -pμ χcpν χc/M 2 χc vν; a(μ ⊤bν) ⊤= (aμ ⊤bν ⊤+ aν ⊤bμ ⊤)/2 -a⊤· b⊤ gμν -pμ χcpν χc/M 2 χc /3, the operators ψ†, χ are the Pauli spinor operators which create a quark and an antiquark, respectively. The presence of the gauge fields in covariant derivative ←→ D μ implies that we should consider the χc meson as a mixture of | ̄QQ⟩, | ̄QQg⟩and | ̄QQγ⟩Fock components unless we work in the light-cone gauge A-= 0, Aa = 0. In what follows we will use the latter gauge condition, both for photons and gluons. In the heavy quark mass, it is expected that the color singlet LDMEs D O[1] χc 3P [1] J E should coincide, and in the potential models can be represented in terms of the slope of the radial wave functions R′ χc(0) as, D O[1] χc 3P [1] J E = 6Nc (2J + 1) R′ χc(0) 2 /4π. (47) 1 In the rest frame of the proton, we may further simplify and rewrite these operators as [51] D 0 ̄ψ (0) γμ χcJ i← → D μ ψ (0) χc(p) E 2 = D 0 ψ† -i 2 ← → D · σ χ χc(p) E 2 = D O[1] χc 3P [1] 0 E , J = 0 D 0 ψ† -i 2 ← → D × σ χ χc(p) E 2 = D O[1] χc 3P [1] 1 E , J = 1 0 ψ† -i 2 ←→ D(iσj) χ χc(p) 2 = D O[1] χc 2P [1] 2 E , J = 2 (45) 8 The latter parameter may be related to the light-cone function φχcJ z, k(rel) ⊥ introduced in (31): after straightforward but tedious substitution of the variables, we may obtain [77] 3 rπ 2 R′ χc(0) = p 2MχcJ Z dz d2k(rel) ⊥ z(1 -z)8π2 φχcJ z, k(rel) ⊥ " M 2 χcJ z -1 2 2 + K2 ⊥ # . (48) In what follows we will use the explicit parametrization from [64] which in the light-cone representation is given by ΦχcJ (z, r) = NχcJz(1 -z) exp -R2 χcJ 8 m2 z(1 -z) -4m2 -2z(1 -z)r2 R2χcJ ! , (49) and NχcJ, RχcJ are numerical constants given explicitly by Rχc0 ≈1.54 GeV-1, Rχc1 = Rχc2 ≈1.48 GeV-1, Nχc0 ≈ 1.15 GeV-1, Nχc1 ≈1.39 GeV-1, Nχc2 ≈0.60 GeV-1 2. The values of |R′(0)| obtained with this parametrization are in reasonable agreement with the value |R′(0)|2 = 0.075 GeV5 found in potential models and widely used in the literature for numerical estimates. For the momentum space wave function, we may obtain ̃ΦχcJ (z, k) ≡ Z d2re-ik·rΦχcJ (z, r) = √z ̄zφχcJ (z, k) k2 -M 2χcJz ̄z + m2 = πNχcJR2 χcJ 2 exp -R2 χcJ 8 k2 + m2(1 -4z ̄z) z ̄z ! . (50) C. Scattering processes in the CGC framework In the CGC framework the interaction of the high energy partons with the gluonic field of the target is described in eikonal approximation by the Wilson line U(x⊥) defined as U (x⊥) = P exp ig Z dx-A+ a x-, x⊥ ta , (51) where we use notation x⊥for the impact parameter of the parton, Aμ a for the gluonic field of the target, and ta are the usual color group generators in the irreducible representation which corresponds to the parton ( 3, ̄3 or 8 for quark, antiquark or gluon, respectively). The interaction encoded in Wilson link (51) formally may be reformulated in terms of the CGC Feynman rules [12, 78-80] provided in Table I. In CGC picture it is assumed that the gluonic field Aμ a is created by randomly distributed color sources, and probability to find a given density of color sources ρa(x-, x⊥) in the target is controlled by the weight functional W[ρ] [1-3]. The component A+ a which appears in (51) may be related to density of these source ρa as A+ a (x) = -1 ∇2 ⊥ρa(x-, x⊥). For any physical observable, the expected values may be obtained averaging the observable A[ρ] found at fixed distribution of charges ρ(x-, x⊥) over all possible configurations ρ(x-, x⊥), namely ⟨A⟩= Z Dρ W[ρ] A[ρ], (52) where the angular brackets ⟨...⟩stand for the above-mentioned averaging. The analytic evaluation of the integral Dρ is possible only for a few simple choices of W[ρ], as, for example, gaussian parametrization [1, 2], and development of phenomenological models for the latter is very challenging. Fortunately, in presence of hard scales which limit the number of partons in the initial state, all physical amplitudes may be represented as convolutions of process-dependent perturbative impact factors and universal (target-dependent) correlators of Wilson lines (multipole scattering amplitudes). For many exclusive processes, the dominant contribution is controlled by the lowest order dipole scattering amplitude, N (x, r, b) = 1 -S2 Y = ln 1 x , xq, x ̄q , S2 (Y, xq, x ̄q) = 1 Nc tr U (xq) U † (x ̄q) Y (53) 2 We disregard tiny ∼1% differences of these constants for different helicity states (polarizations) of the same meson, since these differences are smaller than the precision of experimental data for χc →γγ used in [64] (around 3-4%); besides, a comparable uncertainty may come from the theoretical NNLO corrections ∼α2 s (mc) ≈10%. 9 . σ, i σ′, i′ lQ l′ Q The interaction vertex of the quark with the shock wave: T Q σ,σ′; i,i′ = 2πδ l+ Q -l′+ Q γ+ σ′, σ Z d2zei(lQ-l′ Q)·z [U (z) -1]i′,i . σ, i σ′, i′ l ̄Q l′ ̄Q The interaction vertex of the antiquark with the shock wave: T ̄ Q σ,σ′; i,i′ = -2πδ l+ ̄ Q -l′+ ̄ Q γ+ σ, σ′ Z d2ze i l ̄ Q-l′ ̄ Q ·z h U † (z) -1 i i,i′ . μ, a ν, b lg l′ g The interaction vertex of the gluon with the shock wave: T g μ,ν;a,b = -2πδ l+ g -l′+ g gμνsgn l+ g × × Z d2ze-i(l′ g-lg)·z h Usgn(l+ g ) (z) -1 i b,a Table I. The CGC Feynman rules [12, 78-80] for the interaction of high energy partons with the target encoded in Wilson link (51). The red-colored block stands for the shock wave (Wilson line). We use notations li, l′ i (i = Q, ̄Q, g) for the momenta of the corresponding parton before and after the interaction, σ, σ′ for Dirac indices of the quarks, i, i′ for the color indices in the fundamental representations 3 or ̄3 of the color group, and a, b for the gluon color indices in the adjoint representation 8. The matrices U and U are the Wilson lines in the fundamental and the adjoint representations. The prefactor 2πδ l+ i -l′+ i in the interaction vertices reflects conservation of the longitudinal momentum of the parton in eikonal approximation. We may also observe that interaction with a shock wave does not change transverse coordinates of the partons in the mixed (lightcone) representation, and using identities ̄uh1(p)γ+uh2(q) = ̄vh1(p)γ+vh2(q) = 2δh1,h2 p p+q+ from [66], we may see that the interaction remains diagonal in helicity basis. where Y is the rapidity of the dipole, xq, x ̄q are the impact parameters of the quark, and the variables r ≡xq -x ̄q and b ≡αq xq + α ̄qx ̄q determine the transverse separation and the impact parameter of the dipole's center of mass. In phenomenological studies the dipole amplitude can be constrained from both exclusive and inclusive channels. Due to optical theorem, the inclusive channels are sensitive only to the imaginary part of the amplitude, and in order to have a consistent description, for exclusive channels, the real part of the amplitude should be taken into account. As was demonstrated long ago in [81, 82], if the amplitude grows as a function of invariant energy as ∼sα, then the real part of the amplitude becomes proportional to the imaginary part, and thus the full amplitude of exclusive process can be restored from the imaginary part using a set of identities A = (β + i) Im A, |A|2 = 1 + β2 |ImA|2 , β ≡Re A Im A = tan πα 2 , α = ∂ln A ∂ln s . (54) Following existing conventions, we will assume that N (x, r, b) corresponds to the dipole amplitude extracted from inclusive processes, tacitly assuming that a factor (β + i) should be added in exclusive channels. We also take into account the so-called skewedness factor Rg suggested in [83], which takes into account that the effective value of the parameter x in the dipole amplitude may be modified due to nonzero longitudinal momentum transfer in t-channel and its possible unequal sharing between the gluons in the shock wave. Explicitly, this factor is given by Rg(γ) = 22γ+3 √π Γ(γ + 5/2) Γ(γ + 4) ≈2.4γ, where γ ≡∂ln xg(x, μ2) ∂ln(1/x) ≈const, (55) g x, μ2 is the gluon PDF, and in approximate expression for Rg we took into account that in the kinematics where CGC is applicable, the parameter γ ≤0.5. While the derivation of (55) given in [83] relies on a model which is not valid in the deeply saturated regime x ≪1, at moderate values of x ≳10-3 the factor Rgleads to a mild increase of the cross-section, and improves the phenomenological description of various exclusive processes (see e.g. [16, 17, 84] for more details). 10 χc χc χc χc Figure 2. The dominant mechanism of χcγ pair photoproduction in the CGC framework in the leading order in αs. The red block stands for a shock wave. The diagrams in the first and the second row are related by charge conjugation (inversion of the quark line). In the left column we disregard interaction of the emitted photon with a shock wave as tiny O (αem)-correction, and to emphasize this we don't put "dot" on a photon line which crosses the shock wave. D. Photoproduction of the χcγ pairs The evaluation of the exclusive photoproduction of χcγ pairs resembles a similar evaluation of the ηcγ photoproduction presented in [55] and differs only in the final state wave function, although yields a significantly different results for the impact factors. Technically, it requires evaluation of four diagrams shown in the Figure 2. In what follows we will separate the contributions with photon emission before and after the interaction with a shockwave, so the amplitude can be represented as A(λ; σ,H) γp→χcγp = A(λ; σ,H) 1 + A(λ; σ,H) 2 (56) where A(λ,σ) 1 and A(λ,σ) 2 correspond to the left and right columns of the Figure 2. Since the interaction of the emitted photon with the shock wave may be disregarded as O (αem)-correction, both contributions are controlled by the dipole amplitude N (x, r, b), convoluted with appropriate impact factors. The detailed evaluation of these impact factors is straightforward and is explained in detail in subsequent subsections. Following earlier studies [33, 34], in all evaluations below we will introduce the subindices 0,1,2 to distinguish the kinematic variables associated with quark, antiquark and photon, respectively. For example, the variables z0 ≡ k+ Q q+ , z1 ≡ k+ ̄ Q q+ , z2 ≡k+ γ q+ = 1 -αχc, (57) are the fractions of the incoming photon's light-cone momentum q+ carried by the quark, antiquark and photon when crossing the shock wave, respectively, and r0, r1, r2 are their transverse coordinates (in configuration space). The light-cone momentum conservation implies that for the diagrams in the left column of the Figure 2, the variables z0, z1, z2 should satisfy z0 + z1 + z2 = 1, z0 + z1 = αχc, (58) whereas for the diagrams in the right column there is a similar constraint z0 + z1 = 1. (59) 1. Evaluation of the amplitude A(λ; σ,H) 1 The amplitude A(λ; σ,H) 1 obtains contributions of two charge conjugate diagrams shown in the left column of the Figure 2. We will start our evaluation from the diagram shown in the upper left corner of the Figure 2, defining the partons' momenta as explained in the Figure 3 and using the CGC Feynman rules from Table I. A straightforward evaluation yields A1,1 = 4παeme2 c Z d2x0d2x1 Z d4k0 (2π)4 d4k1 (2π)4 d4k′ 0 (2π)4 d4k′ 1 (2π)4 (2π)4 δ (k0 + k1 + kγ -q) (2π)4 δ (k′ 0 + k′ 1 -pχc) × (60) × (2π) δ k+ 0 -k′ 0 + (2π) δ k+ 1 -k′ 1 + ei(k0-k′ 0)·x0ei(k1-k′ 1)·x1φχcJ z, k(rel) ⊥ × × Tr ˆεγ(q)S (-k1) γ+U † (x1) S (-k′ 1) Γχc (k′ 1, k′ 0) S (k′ 0) γ+U (x0) S (k0) ˆε∗ γ (kγ) S (k0 + kγ) 11 kγ k0 k′ 0 k1 k′ 1 pχc q x1 x0 χc Figure 3. The definitions of the parton momenta k0, k1, k′ 0, k′ 1 before and after interaction with the shock wave. The variables x0, x1 stand for the transverse coordinates of the quark and antiquark when they pass through the shock wave. where εγ(q), ε∗ γ (kγ) are the polarization vectors of the photons, and a symbol ˆ... was defined in (2). We also use notation ec = 2/3 for the electric charge of the charm quark. The matrix Γχc was defined earlier in (31) for different spins and helicities of χc. The wave function Φ depends on the light-cone fraction zχc = k ′+ 0 /p+ χc and relative transverse momentum k(rel) ⊥ = z0k′ 1 -z1k′ 0 / (z0 + z1). The propagator of free fermions is defined in a standard way as S(p) = i ˆp + m p2 -m2 . (61) The numerators of the propagators may be rewritten as ˆp + m = γ+ p2 ⊥+ m2 2p+ + γ-p+ -γ⊥· p⊥+ m + γ+ p--p2 ⊥+ m2 2p+ = (62) = θ p+ X h uh( ̃p) ̄uh( ̃p) + θ -p+ X ̄h v ̄h(- ̃p) ̄v ̄h( ̃p) + γ+ p--p2 ⊥+ m2 2p+ , ̃p = p+, p2 ⊥+ m2 2p+ , p⊥ . The last term ∼γ+ in (62) drops out when the quark is onshell, or when the propagator is multiplied by γ+ (due to identity γ+γ+ = 0), so effectively (62) reduces to a helicity sum of quark or antiquark spinors. Since the matrices U, U † in (60) don't have any spinorial indices, whereas all the other objects don't depend on color, the evaluation of traces over color and spinorial indices can be done independently, thus reducing (60) to a convolution of the dipole scattering amplitude (53) with the so-called (target-independent) impact factor A1,1 = Z d2x0d2x1 N (Y, x0, x1) R1,1 (Y, x0, x1, q, pχc, kγ) , (63) where R1,1 is given by R1,1 (Y, x0, x1, q, pχc, kγ) = 4παeme2 cNc× (64) × Z d4k0 (2π)4 d4k1 (2π)4 d4k′ 0 (2π)4 d4k′ 1 (2π)4 (2π)4 δ (k0 + k1 + kγ -q) (2π)4 δ (k′ 0 + k′ 1 -pχc) × × (2π) δ k+ 0 -k′ 0 + (2π) δ k+ 1 -k′ 1 + ei(k0⊥-k′ 0⊥)·x0ei(k1⊥-k′ 1⊥)·x1φχcJ z, k(rel) ⊥ × × Tr ˆεγ(q)S (-k1) γ+S (-k′ 1) Γχc (k′ 1, k′ 0) S (k′ 0) γ+S (k0) ˆε∗ γ (kγ) S (k0 + kγ) The integrals over the momenta d4k1 and d4k′ 1 in (64) can be calculated using δ-functions, which effectively fix k1 and k′ 1 to values k1 = q -kγ -k0, k′ 1 = pχc -k′ 0. (65) Due to identity γ+γ+ = 0, we may replace the numerators of propagators connected to the shockwave with onshell spinors, as explained in the text under Eq. (62). Furthermore, using the well-known spinor identities [66, 69] ̄uh (l) γ+uh′ (l′) = ̄vh (l) γ+vh′ (l′) = 2 √ l+l′+δh,h′ (66) 12 where h, h′ are helicities of partons, and conservation of the plus-component light-cone, we may rewrite (64) as R1,1 (Y, x0, x1, q, pχc, kγ) = 16 παeme2 cNc (2π) δ q+ -k+ γ -p+ χc × (67) × Z dk+ 0 dk0 d2k0 (2π)4 dk′- 0 d2k′ 0 (2π)3 k+ 0 q+ -k+ γ -k+ 0 ei(k0⊥-k′ 0⊥)·x0ei(k1⊥-k′ 1⊥)·x1× × X h, ̄h h ̄uh ̃k0 ˆε∗ γ (kγ) S (k0 + kγ) ˆεγ(q)v ̄h ̃k1 i (k2 0 -m2) (k2 1 -m2) h ̄v ̄h ̃k′ 1 Γχc (k′ 1, k′ 0) uh ̃k′ 0 i φχcJ z, k(rel) ⊥ (k′ 0)2 -m2 (k′ 1)2 -m2 . For the sake of brevity, in (67) we keep notations k1, k′ 1, tacitly assuming that they are linear combinations of k0, k′ 0 defined in (65). The dependence on k0 and k ′- 0 in the integrand of (67) is isolated in the first and the second terms of the last line, for this reason integration over these variables becomes straightforward and can be realized using the Cauchy's theorem and reduced to a sum of residues of the integrands at the poles, namely Z +∞ -∞ dk0 2π 1 2k+ 0 k0 -k2 0⊥-m2 + i0 2 k+ γ + k+ 0 kγ + k0 -(kγ ⊥+ k0⊥)2 -m2 + i0 × (68) × 1 2 q+ -k+ γ -k+ 0 q--kγ -k0 -(kγ⊥+ k0⊥)2 -m2 + i0 = = - iΘ q+ -k+ γ -k+ 0 8k+ 0 k+ 0 + k+ γ q+ -k+ γ -k+ 0 - q+((kγ⊥+k0⊥)2+m2) 2(q+-k+ γ -k+ 0 )(k+ γ +k+ 0 ) - k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) , Z +∞ -∞ dk0 2π kγ + k0 2k+ 0 k0 -k2 0⊥-m2 + i0 2 k+ γ + k+ 0 kγ + k0 -(kγ ⊥+ k0⊥)2 -m2 + i0 × (69) × 1 2 q+ -k+ γ -k+ 0 q--kγ -k0 -(kγ⊥+ k0⊥)2 -m2 + i0 = = - iΘ q+ -k+ γ -k+ 0 (kγ⊥+k0⊥)2+m2 2(k+ γ +k+ 0 ) 8k+ 0 k+ 0 + k+ γ q+ -k+ γ -k+ 0 - q+((kγ⊥+k0⊥)2+m2) 2(q+-k+ γ -k+ 0 )(k+ γ +k+ 0 ) - k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) - iΘ q+ -k+ γ -k+ 0 8k+ 0 k+ 0 + k+ γ q+ -k+ γ -k+ 0 - k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) , Z +∞ -∞ dk′- 0 2π 1 2k′+ 0 k′- 0 -k′2 0⊥-m2 + i0 2 pχc -k′+ 0 pχc -k′- 0 - pχc ⊥-k′ 0⊥ 2 -m2 + i0 = (70) = -Θ p+ χc -k′+ 0 4k′+ 0 p+ χc -k′+ 0 i pχc -k′2 0⊥+m2 2k′+ 0 -(pχc ⊥-k′ 0⊥) 2+m2 2(p+ χc-k′+ 0 ) . The Heaviside step functions Θ(...) in (68-70) introduce the upper cutoff for the integral over the plus-component k+ 0 and stem from the requirement that the integrands should have poles on both sides of the real axis (Im k-= 0) for nonzero result. The identities (68,70) allow us to rewrite (64) as 13 R1,1 (Y, x0, x1, q, pχc, kγ) = δ q+ -k+ γ -p+ χc ei(q⊥-kγ ⊥-pχc ⊥)·x1 Z q+-k+ γ 0 dk+ 0 × (71) × 2παeme2 c k+ 0 + k+ γ Z d2k0⊥ (2π)2 ̄uh (k0) ˆε∗ γ (kγ) ˆk0 + ˆkγ + m ˆεγ(q)v ̄h (q -kγ -k0) eik⊥ 0 ·(x0-x1) k0 =(kγ⊥+k0⊥) 2+m2 2(k+ γ +k+ 0 ) -kγ q k+ 0 q+ -k+ γ -k+ 0 - q+(k2 0⊥+m2) 2(q+-k+ γ -k+ 0 )(k+ γ +k+ 0 ) - k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) + 2παeme2 c k+ 0 + k+ γ Z d2k0⊥ (2π)2 ̄uh (k0) ˆε∗ γ (kγ) γ+ˆεγ(q)v ̄h (q -kγ -k0) eik⊥ 0 ·(x0-x1) q k+ 0 q+ -k+ γ -k+ 0 - k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) × Z d2k′ 0⊥ (2π)2 ̄v ̄h p+ χc -k+ 0 , pχc ⊥-k′ 0⊥ Γχc (pχc -k′ 0, k′ 0) uh k+ 0 , k′ 0⊥ φχcJ (z, K⊥) e-ik′ 0⊥·(x0-x1) 2 q k+ 0 p+ χc -k+ 0 pχc -k′2 0⊥+m2 2k+ 0 -(pχc ⊥-k′ 0⊥) 2+m2 2(p+ χc-k+ 0 ) . where the δ-function δ q+ -k+ γ -p+ χc in (71) reflects smallness of the light-cone momentum transfer ∆+ compared to transverse momentum transfer to the target ∆⊥= q⊥-kγ ⊥-pχc ⊥in the high energy kinematics. The expressions in square brackets in denominators of (71) coincide with the energy denominators of the light-cone perturbation theory [66]. The analysis of the other diagrams is done similarly. The evaluation of the lower left diagram in Figure 2 yields for the impact factor R1,2 (Y, x0, x1, q, pχc, kγ) = -δ q+ -k+ γ -p+ χc ei(q⊥-kγ ⊥-pχc ⊥)·x1 Z q+-k+ γ 0 dk+ 0 × (72) × 2παeme2 c q+ -k+ 0 Z d2k0⊥ (2π)2 ̄uh (k0) ˆεγ(q) ˆq -ˆk0 -m ˆε∗ γ (kγ) v ̄h (q -kγ -k0) eik⊥ 0 ·(x0-x1) k0 =q-- k2 0⊥+m2 2(q+-k+ 0 ) q k+ 0 q+ -k+ γ -k+ 0 q-- q+(k2 0⊥+m2) 2(q+-k+ γ )k+ 0 q-- k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) + 2παeme2 c q+ -k+ 0 Z d2k0⊥ (2π)2 ̄uh (k0) ˆεγ(q)γ+ˆε∗ γ (kγ) v ̄h (q -kγ -k0) eik⊥ 0 ·(x0-x1) q k+ 0 q+ -k+ γ -k+ 0 q-- k2 γ⊥ 2k+ γ -k2 0⊥+m2 2k+ 0 -(kγ⊥+k0⊥)2+m2 2(q+-k+ γ -k+ 0 ) × Z d2k′ 0⊥ (2π)2 ̄v ̄h p+ χc -k+ 0 , pχc ⊥-k′ 0⊥ Γχc (pχc -k′ 0, k′ 0) uh k+ 0 , k′ 0⊥ e-ik′ 0⊥·(x0-x1)φχcJ (z, K⊥) 2 q k+ 0 p+ χc -k+ 0 pχc -k′2 0⊥+m2 2k+ 0 -(pχc ⊥-k′ 0⊥) 2+m2 2(p+ χc-k+ 0 ) . The result for (72) differs from (71) only by the structure of the second and the third lines. After integration over k′ 0⊥in the last lines of (71) and (72), we may obtain the wave function of χc meson in the light-cone representation introduced earlier in (35). Likewise, the result of integration over k0⊥in the second and the third lines of (71,72) can be interpreted as a light-cone wave function of the ̄QQγ Fock component in the photon, which in the leading order is given by a set of diagrams shown in the Figure 4. These diagrams nearly coincide with the diagrams j, k, l, m in [33] which contribute to the γ → ̄QQg wave function, and for this reason the result of integration over k0⊥differs from expressions found in [33] only by color factor. The structure of (71) and (72) implies that the amplitude A1 can be rewritten as a convolution of the dipole amplitude N (x, r, b), the wave function Ψ† χcof the produced χc meson, and the wave function Ψγ→γ ̄ QQ of the ̄QQγ Fock component of the incoming photon (the amplitude of the subprocess γ → ̄QQγ), namely A(λ; σ, H) 1 = Z αχc 0 dz0 2 Y k=0 d2xk Ψ(h, ̄h)† χc z0 z0 + z1 , r10 Ψ(λ,σ,h, ̄h) γ→γ ̄ QQ (z0, z1 = αχc -z0, z2 ≡ ̄αχc, x0, x1, x2) × (73) × N (x, r10, b10) exp -ipχc ⊥· b10 - ̄αχc αχc rγ -ikγ ⊥· x2 , 14 γ(q) Q ̄Q γ(kγ) γ(q) γ(kγ) Q ̄Q γ(q) γ(kγ) Q ̄Q γ(q) Q ̄Q γ(kγ) Figure 4. The diagrams which correspond to the γ → ̄QQγ amplitude in the leading order in strong coupling αs. The contributions of the two diagrams in the first row correspond to expressions in the second lines of (71) and (72). Similarly, the contributions of the diagrams in the last row are given by the expressions in the third lines of the same Equations (71) and (72). The vertical fermion line with a short horizontal dash in the diagrams of the last row denotes instantaneous part of the quark propagator. The colored vertical dashed lines stand for the light-cone energy denominators of the canonical light-cone perturbation theory [66, 69]. where the dummy integration variable z0 is related to k+ 0 using identities (57), the parameters λ, σ are the helicities of the photon before and after interaction; h, ̄h are the helicities of the heavy quarks when they cross the shock wave, r10 ≡x1 -x0 is the relative distance between the quarks, b10 = (z0x0 + z1x1) / (z0 + z1) is the position (impact parameter) of the center of mass of the quark-antiquark pair, rγ = x2-b10 is the distance between the emitted photon and the center of mass of the dipole. The argument x of the dipole cross-section N corresponds to the fraction of the target's momentum in minus-direction which is taken by heavy quarks, x ≈∆-/P -= 2ξ/(1 + ξ) ≈M 2 γχc/W 2 . As we mentioned earlier, the impact factors R1,1 and R1,2 are related to each other by charge conjugation (C-parity). In order to fully exploit this symmetry , it is convenient to introduce the variable ζ defined as a fraction of the photon's momentum carried by virtual parton (quark or antiquark) which emits secondary (final-state) photon afterwards, namely z0 = ζ - ̄αχc, z1 = 1 -ζ, z2 = 1 -αχc (74) for the contribution R1,1, and z0 = 1 -ζ, z1 = ζ - ̄αχc, z2 = 1 -αχc (75) for the contribution R1,2. While the expression (73) presents a lot of interest for conceptual understanding, unfortunately it is not feasible to use it directly for numerical phenomenological studies due to multidimensional integration of the oscillating function; besides the wave function Ψ(...) γ→γ ̄ QQin light-cone (configuration) representation include new special functions I, J defined in [33, 34] as nontrivial 2-dimensional integrals. This suggests that evaluation of the amplitude should be done in the momentum space. However, evaluation of the Fourier image of N (x, r10, b10) is not well-defined due to saturation of the latter at large distances r10 and non-commutativity of the limits r10 →∞ and b10 →∞. At the same time, we may observe that for the product Ψ† χcJ (r10) N (x, r10, b10) this problem does not exist, because ΨχcJ is proportional to Φχc and its derivatives, which are strongly suppressed at large distances. For this reason, in what follows we will introduce new functions 3 nΦχc (x, l, s, z) = Z d2b d2re-il·re-is·bN (x, r, b) Φχc (r, z) , (76) 3 For phenomenological parametrizations of N (x, r, b) that do not depend explicitly on orientations of r, b, the integration over orientations of these vector can be done analytically, yielding (2π)2 J0(lr) J0(sb) for n(2, 0) Φχc , (2π)2 iJ1(lr) J0(sb) for n(±) Φχc , and -(2π)2 J2(lr) J0(sb) for n(2, ±2) Φχc . 15 n(±) Φχc (x, l, s, z) = Z d2b d2re-il·re-is·bN (x, r, b) e±iθ∂rΦχc (r, z) , (77) n(2, ±2) Φχc (x, l, s, z) = Z d2b d2re-il·re-is·bN (x, r, b) e±2iθ∆(-) r Φχc (r, z) , (78) n(2, 0) Φχc (x, l, s, z) = Z d2b d2re-il·re-is·bN (x, r, b) ∆(+) r Φχc (r, z) , (79) and, using explicit expressions for the wave functions in helicity basis (34-42), will replace the corresponding products Ψ† χcJ (r10) N (x, r10, b10) with (inverse) Fourier images of the functions nΦχc , n(±) Φχc , n(2, 0) Φχc , n(2, ±2) Φχc . After integration over the variables r10 and b10, we may obtain two additional δ-functions (2π)4 δ (s + ∆⊥) δ l-k⊥ 0 -ζ - ̄αχc α ∆⊥ which allow to get rid of the integrals over d2k0 and d2s. The final result of this procedure depends on the spin and helicity of χcJ meson. We performed these evaluations in helicity basis, using explicit expressions for spinors from [66, 69] and performing the algebraic simplifications using Mathematica. In view of the space constraints, we provide below the results only for components with "+" helicity of the incoming photon, tacitly assuming that the remaining components may be recovered using h A(λ; σ, H) 1 i∗ = A(-λ; -σ, -H) 1 . For χc0 meson, the helicity amplitudes read as A(+; +) 1, χc0 = 4π ̄κm Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ ζ - ̄αχc × (80) × αχc ̄αχc αχc -2 ̄ζ h (ζ - ̄αχc) P 2 (1) + m2 - ̄αχcm2i nΦχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 - i αχc ̄α2 χcζ n(-) Φχc (x, l, -∆⊥, zχc) P(1) + iP(1)y P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -) 1, χc0 = -4iπ ̄κmα2 χc ̄α2 χc Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ3 ζ - ̄αχc n(+) Φχc (x, l, -∆⊥, zχc) × (81) P(1)x + iP(1)y P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , where we introduced shorthand notations K(1) = l ̄αχc + ζ (αχckγ ⊥- ̄αχcpχc ⊥) ζ - ̄αχc = P (1) ̄αχc ζ - ̄αχc + (αχckγ ⊥- ̄αχcpχc ⊥) αχc , (82) P (1) = l+ ̄ζ αχc (αχckγ ⊥- ̄αχcpχc ⊥) , zχc = ζ - ̄αχc αχc . (83) For χc1 meson, the corresponding amplitudes are given by A(+; +,+1) 1, χc1 = 4π ̄καχc Z 1 ̄αχc dζ Z d2l (2π)2 s 2 ̄ζ ζ - ̄αχc ζ K(1)x -iK(1)y × (84) ̄ζ ̄αχc P(1)x + iP(1)y (2ζ - ̄αχc)n(-) Φχc (x, l, -∆⊥, zχc) -iζm2nΦχc (x, l, -∆⊥, zχc) αχc -2 ̄ζ P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , 16 A(+; -,+1) 1, χc1 = -4iπ ̄κm2αχc Z 1 ̄αχc dζ Z d2l (2π)2 nΦχc (x, l, -∆⊥, zχc) × (85) ζ K(1)x + iK(1)y q 2 ̄ζ (ζ - ̄αχc) αχc -2 ̄ζ P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; +,0) 1, χc1 = -8 ̄αχcα2 χc Mχc1 π ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ ζ - ̄αχc × (86) m2 ̄αχcζ P(1)x + iP(1)y n(-) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 -i n(2, 0) Φχc (x, l, -∆⊥, zχc) h (ζ - ̄αχc) P 2 (1) + m2 + m2 ̄αχc i P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 A(+; -,0) 1, χc1 = -8m2 Mχc1 π ̄κ ̄α2 χcα2 χc Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ3 ζ - ̄αχc n(+) Φχc (x, l, -∆⊥, zχc) (87) × P(1)x + iP(1)y P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; +,-1) 1, χc1 = 4π ̄καχc Z 1 ̄αχc dζ Z d2l (2π)2 n(+) Φχc (x, l, -∆⊥, zχc) (88) × K(1)x -iK(1)y P(1)x + iP(1)y ζ (2ζ -1) q 2 ̄ζ (ζ - ̄αχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -,-1) 1, χc1 = 4π ̄κα2 χc Z 1 ̄αχc dζ Z d2l (2π)2 s 2 ̄ζ ζ - ̄αχc (89) × ζ2 K(1)x + iK(1)y P(1)x + iP(1)y αχc -2 ̄ζ n(+) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -,-1) 1, χc1 = 0. (90) Finally, for χc2 meson we obtained A(+; +,+2) 1, χc2 = 8iπ ̄κ αχc ̄αχc Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ3 ζ - ̄αχc (91) × h m2 ̄αχc + P 2 (1) + m2 (ζ - ̄αχc) i n(2, -2) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -,+2) 1, χc2 = 8π ̄κm2 ̄α2 χcα2 χc Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ3 ζ - ̄αχc n(-) Φχc (x, l, -∆⊥, zχc) (92) × P(1)x + iP(1)y ̄ζ + ̄αχc 2 P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 17 A(+; +,+1) 1, χc2 = 4iMχc2π ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 K(1)x -iK(1)y s ̄ζ ζ - ̄αχc ζ (93) × P(1)x + iP(1)y (ζ - ̄αχc) 8ζ3 + 4ζ2(αχc -4) -6ζ(αχc -2) + 3αχc -4 n(-) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 -i ζm2αχcnΦχc (x, l, -∆⊥, zχc) αχc -2 ̄ζ P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -,+1) 1, χc2 = 4π ̄κMχc2 Z 1 ̄αχc dζ Z d2l (2π)2 ζ s ̄ζ ζ - ̄αχc αχc -2 ̄ζ (94) × αχc αχc - ̄ζ m2nΦχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 + iζ P(1)x + iP(1)y αχc -2 ̄ζ 2 n(-) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; +,0) 1, χc2 = -4 r 2 3π ̄καχc Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ ζ - ̄αχc (95) × 2m2 αχc -2 ̄ζ nΦχc (x, l, -∆⊥, zχc) h ̄αχc (ζ - ̄αχc) P 2 (1) + m2 + m m ̄α2 χc -ζ2 K(1)x -iK(1)y i P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 + - 2im2ζαχc P(1)x + iP(1)y ̄α2 χcn(-) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 - 3ζαχc P(1)x + iP(1)y K(1)x -iK(1)y ̄ζ (2ζ + αχc) -1 n(2, 0) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -,0) 1, χc2 = 4 r 2 3π ̄κm2αχc Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ ζ - ̄αχc (96) × 2ζm3 K(1)x + iK(1)y (ζ - ̄αχc) αχc -2 ̄ζ nΦχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 + - 2iαχc ̄α2 χc ̄ζm2 P(1)x + iP(1)y n(+) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 - 3αχcζ2 αχc -2 ̄ζ K(1)x + iK(1)y P(1)x + iP(1)y n(2, 0) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; +,-1) 1, χc2 = 4iπ ̄κMχc2 Z 1 ̄αχc dζ Z d2l (2π)2 ζ ̄ζ s ̄ζ ζ - ̄αχc n(+) Φχc (x, l, -∆⊥, zχc) (97) × K(1)x -iK(1)y P(1)x + iP(1)y 8ζ3 + 4ζ2(3αχc -4) -6ζ(αχc -2) ̄αχc + (αχc -4) ̄α2 χc P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , 18 A(+; -,-1) 1, χc2 = 16iπ ̄κMχc2 Z 1 ̄αχc dζ Z d2l (2π)2 ζ2 ̄ζ q ̄ζ (ζ - ̄αχc)n(+) Φχc (x, l, -∆⊥, zχc) (98) × K(1)x + iK(1)y P(1)x + iP(1)y αχc -2 ̄ζ P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; +,-2) 1, χc2 = -8iαχc ̄αχcπ ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 s ̄ζ ζ - ̄αχc (99) × n(2, +2) Φχc (x, l, -∆⊥, zχc) h (ζ - ̄αχc) P 2 (1) + m2 + m2 ̄αχc i (ζ - ̄αχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 + im2ζαχc ̄αχc P(1)x + iP(1)y n(+) Φχc (x, l, -∆⊥, zχc) P 2 (1) + m2 ̄ζ ζ2K2 (1) + (ζ - ̄αχc) ̄αχcP 2 (1) + αχc ̄αχcζm2 , A(+; -,-2) 1, χc2 = 0. (100) 2. Evaluation of the amplitude A(λ; σ, H) 2 The amplitude A(λ; σ, H) 2 corresponds to contribution of the diagrams in the right column of the Figure 2 and can be evaluated using the procedure outlined in the previous subsection II D 1. We may represent the amplitude as a convolution of the dipole amplitude and the impact factors R2,i, A2,i = Z d2x0d2x1 N (Y, x0, x1) R2,i (Y, x0, x1, q, pχc, kγ) , i = 1, 2. (101) The corresponding expressions for R2,i in (101) differ from the impact factors R1,i only by permutation of the χc and the incoming photon, namely R2,1 (Y, x0, x1, q, pχc, kγ) = 4παeme2 cNc (102) × Z d4k0 (2π)4 d4k1 (2π)4 d4k′ 0 (2π)4 d4k′ 1 (2π)4 (2π)4 δ (k0 + k1 -q) (2π)4 δ (k′ 0 + k′ 1 -pχc -kγ) × (2π) δ k+ 0 -k′ 0 + (2π) δ k+ 1 -k′ 1 + ei(k0⊥-k′ 0⊥)·x0ei(k1⊥-k′ 1⊥)·x1φχcJ z, k(rel) ⊥ × Tr Γχc (k′ 1, k′ 0 -kγ) S (k′ 0 -kγ) ˆε∗ γ (kγ) S (k′ 0) γ+S (k0) ˆεγ(q)S (-k1) γ+S (-k′ 1) R2,2 (Y, x0, x1, q, pχc, kγ) = 4παeme2 cNc (103) × Z d4k0 (2π)4 d4k1 (2π)4 d4k′ 0 (2π)4 d4k′ 1 (2π)4 (2π)4 δ (k0 + k1 -q) (2π)4 δ (k′ 0 + k′ 1 -pχc -kγ) × (2π) δ k+ 0 -k′ 0 + (2π) δ k+ 1 -k′ 1 + ei(k0⊥-k′ 0⊥)·x0ei(k1⊥-k′ 1⊥)·x1φχcJ z, k(rel) ⊥ Tr Γχc (k′ 1 + kγ, k′ 0) S (k′ 0) γ+S (k0) ˆεγ(q)S (-k1) γ+S (-k′ 1) ˆε∗ γ (kγ) S (-k′ 1 -kγ) Repeating the procedure outlined in the previous Section II D 1, we may rewrite the impact factors R2,i in terms of the photon wave function ψγ→ ̄ QQ and the amplitude ("wave function") Ψ† ̄ QQ→γ χc of the ̄QQ →χcγ subprocess, contracted over helicities h, ̄h of the quarks and convoluted over the light-cone component k+ Q of the quark momentum. 19 The final result for the amplitude A(λ,σ) 2 may be rewritten as A(λ,σ) 2 = Z αχc 0 dz0 3 Y k=1 d2rk Ψ(σ,h, ̄h)† ̄ QQ→γ χc (z0, z1 = αχc -z0, z2 ≡ ̄αχc, r0, r1, r2) ψ(λ, h, ̄h) γ→ ̄ QQ z0 z0 + z1 , r10 × (104) × N (x, r10, b10) exp -ipχc ⊥· b10 - ̄αχc αχc rγ -ikγ ⊥· (rγ + b10) . where we use the same notations as in the previous section (see the text under Eq. (73)). The photon wave function ψ(λ, h, ̄h) γ→ ̄ QQ in the leading order in αs is given by [85, 86] ψ(λ, h, ̄h) γ→ ̄ QQ (ζ, r) = -ieec Z d2kQ⊥ (2π)2 ̄uh k+ Q, kQ⊥ ˆελ(q)v ̄h q+ -k+ Q, -kQ⊥ q-- q+(k2 Q⊥+m2) 2k+ Q(q+-k+ Q) eikQ⊥·r10 q-=0, k+ Q=ζq+ = (105) = √ 2 2π eef -iλe-iλφr ζδh,λδ ̄h,-λ -(1 -ζ)δh,-λδ ̄h,λ mK1 (mr) + mδh,λδ ̄h,λK0 (mr) , The evaluation of the amplitude Ψ(σ,h, ̄h) ̄ QQ→γ χc in the leading order in the coupling αs resembles a similar evaluation of the wave function Ψ(λ,σ,h, ̄h) γ→γ ̄ QQ discussed earlier and requires evaluation of the diagrams shown in the right column of the Figure 5 using the light-cone rules from [69]. The evaluation is straighforward, however, the final expression is very lengthy, introduces a number of new special functions defined via 2-dimensional integrals, and for this reason evaluation of (104) in configuration space is not feasible. As we mentioned earlier, the Fourier transformation of the dipole amplitude N (x, r, b) presents certain challenges due to saturation at large r and non-commutativity of limits r →∞and b →∞. However, the product of the dipole amplitude and wave function N (x, r, b) ψ(λ, h, ̄h) γ→ ̄ QQ (ζ, r) has a well-defined Fourier image: the functions K0, K1 suppress the product at large r, whereas a color transparency (suppression of N at small r) guarantees that the Fourier integrals remain convergent at small r. Similar to (76-79), we may introduce new functions n0 (x, l, s) = Z d2b d2re-il·re-is·bN (x, r, b) K0 (mr) , (106) n(±) 1 (x, l, s) = Z d2b d2re-il·re-is·bN (x, r, b) K1 (mr) e±iφr, (107) where the angle φr = arg(rx + iry) characterizes the azimuthal orientation of the vector r, and express the product N (x, r, b) ψ(λ, h, ̄h) γ→ ̄ QQ (ζ, r) in terms of the (inverse) Fourier transforms of these functions. We found that the functions n0, n(±) 1 have a mild dependence on all their arguments and are strongly suppressed at large l, s. For this reason, it is possible to significantly speed up the evaluations using caching (interpolation of pre-evaluated) on a modestly sized grid. In order to reduce the charge conjugate contributions in the amplitude A(λ; σ,H) 2 to the common denominator, we will replace the dummy integration variable z0 with a new variable ζ defined as a fraction of the incoming photon's momentum carried by the active fermion before emission of the secondary photon, namely ζ = k+ 0 k+ 0 + k+ 1 = z0 z0 + z1 , (108) for the contribution with photon emission from the quark, and with ζ = k+ 1 k+ 0 + k+ 1 = 1 - z0 z0 + z1 , (109) for the other contribution. In both cases, the variable ζ cannot be less than k+ γ /q+ = ̄αχc. The evaluation of the amplitudes is straightforward and largely repeats the evaluation of (80-100). Due to space limitation, we will provide 20 γ(kγ) χc Q ̄Q χc ̄Q Q γ(kγ) χc γ(kγ) ̄Q Q ̄Q Q χc γ(kγ) Figure 5. The leading order diagrams which contribute to the amplitude of the subprocess Q ̄Q →γχc. The vertical dashed lines denote the energy denominators of the light-cone perturbation theory[66, 69]. only the final results. For the χc0 mesons, the corresponding helicity amplitudes are given by A(+; +) 2, χc0 = -2mαχc ̄αχc ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 p ζ ̄ζ3 (ζ - ̄αχc)2 ̃Φχc zχc, P (2) × (110) × ζn0 (x, l, -∆⊥) P 2 (2) + m2 zc -1 2 2 ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 + m ̄αχc mn0 (x, l, -∆⊥) zc -1 2 + iζ P(2)x + iP(2)y n(-) 1 (x, l, -∆⊥) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 , A(+; -) 2, χc0 = 2i ̄κm2αχc ̄α2 χc Z 1 ̄αχc dζ Z d2l (2π)2 q ζ ̄ζ5 (111) × P(2)x -iP(2)y n(-) 1 (x, l, -∆⊥) ̃Φχc zχc, P (2) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 , where we introduced the shorthand notations K(2) = -l ̄αχc ζ + (ζ - ̄αχc) α2χcζ (αχckγ ⊥- ̄αχcpχc ⊥) = -P (2) ̄αχc ζ + (αχckγ ⊥- ̄αχcpχc ⊥) αχc , (112) P (2) = l+ ̄ζ α2χc (αχckγ ⊥- ̄αχcpχc ⊥) , zχc = ζ - ̄αχc αχc . (113) For χc1 meson, the corresponding amplitudes are given by A(+; +,+1) 2, χc1 = 2m ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p 2 ζ ̄ζ3 (Kx + iKy) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (114) × h (2ζ -1) P(2)x + iP(2)y (ζ - ̄αχc) n(-) 1 (x, l, -∆⊥) -iζmn0 (x, l, -∆⊥) (2ζ + αχc -2) i , 21 A(+; -,+1) 2, χc1 = -2im ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 p 2 ζ ̄ζ3 K(2)x -iK(2)y (2ζ + αχc -2) ̃Φχc zχc, P (2) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (115) × h mn0 (x, l, -∆⊥) (ζ - ̄αχc) + iζαχc P(2)x + iP(2)y n(-) 1 (x, l, -∆⊥) i , A(+; +,0) 2, χc1 = iαχc ̄αχc ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ζ ̄ζ ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (116) × ( mn0 (x, l, -∆⊥) " ̄αχcα2 χc ̄ζ m2(zc -1/2)2 -P 2 (2) -αχcζ P 2 (2) + m2 zc -1 2 2!# + + i n(-) 1 (x, l, -∆⊥) P(2)x + iP(2)y 2m2 ̄αχcα2 χcζ ̄ζ -αχcζ2 P 2 (2) + m2 zc -1 2 2!!) , A(+; -,0) 2, χc1 = ̄καχc ̄αχc Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ζ ̄ζ3 P(2)x -iP(2)y ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (117) × n(-) 1 (x, l, -∆⊥) ζαχc P 2 (2) + m2 zc -1 2 2! -2 ̄ζm2 ̄αχc ! , A(+; +,-1) 2, χc1 = -2 ̄κm ̄αχc Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) × (118) hp 2ζ ̄ζ5 (ζ - ̄αχc) K(2)x + iK(2)y P(2)x -iP(2)y (2ζ - ̄αχc) n(-) 1 (x, l, -∆⊥) i ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 , A(+; -,-1) 2 χc1 = 0. (119) Finally, for χc2 meson we obtained A(+; +,+2) 2, χc2 = 4m2 ̄α2 χc ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ζ ̄ζ5 P(2)x + iP(2)y 2 n0 (x, l, -∆⊥) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 , (120) A(+; -,+2) 2, χc2 = 4i ̄κm Z 1 ̄αχc dζ Z d2l (2π)2 αχc ̄αχc ̃Φχc zχc, P (2) p ζ ̄ζ5 P(2)x + iP(2)y n(-) 1 (x, l, -∆⊥) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (121) × ζ P 2 (2) + m2 zc -1 2 2! -m2 ̄αχc ! , A(+; +,+1) 2, χc2 = - ̄κ 2αχc Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ζ ̄ζ K(2)x + iK(2)y ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 × (122) × m2 -4 ̄ζ2 + α2 χc + 4 ̄ζαχc + α2 χcP 2 (2) {ζmαχcn0 (x, l, -∆⊥) (2ζ + αχc -2) +i (ζ - ̄αχc) 8ζ3 + 4ζ2(αχc -4) -6ζ(αχc -2) + 3αχc -4 P(2)x + iP(2)y n(-) 1 (x, l, -∆⊥) o , 22 A(+; -,+1) 2, χc2 = - ̄κ 2αχc Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ζ ̄ζ K(2)x + iK(2)y (2ζ + αχc -2) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (123) × m2 -4 ̄ζ2 + α2 χc + 4 ̄ζαχc + α2 χcP 2 (2) × mαχcn0 (x, l, -∆⊥) (ζ - ̄αχc) + iζ(2ζ + αχc -2)2 P(2)x + iP(2)y n(-) 1 (x, l, -∆⊥) , A(+; +,0) 2, χc2 = ̄κ ̄αχc 2 √ 6αχc Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ζ/ ̄ζ ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (124) × ( n0 (x, l, -∆⊥) (1 -2zc) × m4 -4 ̄ζ2 + α2 χc + 4 ̄ζαχc -4 ̄ζ2 + α2 χc + 4 ̄ζ(1 -2ζ)αχc + 2m2α4 χcP 2 (2) + α4 χc P 2 (2) 2 + in(-) 1 (x, l, -∆⊥) α2 χc P(2)x + iP(2)y /m " 8ζ ̄ζ2m4 ̄αχc (ζ - ̄αχc) +ζ2αχc m2 -4 ̄ζ2 + α2 χc + 2 ̄ζαχc + αχc αχc -2 ̄ζ P 2 (2) m2 zc -1 2 2 + P 2 (2) !#) , A(+; -,0) 2, χc2 = ̄καχc ̄αχc 2 √ 6m Z 1 ̄αχc dζ Z d2l (2π)2 i ̃Φχc zχc, P (2) P(2)x -iP(2)y n(-) 1 (x, l, -∆⊥) p ζ ̄ζ (ζ - ̄αχc) (125) × ζαχc m2 -4 ̄ζ2 + α2 χc + 2 ̄ζαχc + αχc αχc -2 ̄ζ P 2 (2) P 2 (2) + m2 zc -1 2 2 ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 - 8 ̄ζ2m4 ̄αχc (ζ - ̄αχc) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 , A(+; +,-1) 2, χc2 = ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) p ̄ζζ m2 -4 ̄ζ2 + α2 χc + 4 ̄ζαχc + α2 χcP 2 (2) 4αχc ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 × (126) mn0 (x, l, -∆⊥) ̄α2 χc P(2)x -iP(2)y αχc -4 ̄ζ -αχc K(2)x + iK(2)y (ζ - ̄αχc) αχc -2 ̄ζ -in(-) 1 (x, l, -∆⊥) 2m2 ̄α2 χcαχc -(2ζ -1) K(2)x + iK(2)y P(2)x -iP(2)y 4 ̄ζ2(2ζ -1) + α3 χc -6 ̄ζα2 χc -3 ̄ζ2(4ζ -3)αχc , A(+; -,-1) 2, χc2 = 2i ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) n(-) 1 (x, l, -∆⊥) ζ ̄ζ 3/2 ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 (127) × K(2)x -iK(2)y P(2)x -iP(2)y (ζ - ̄αχc) (zc -1/2) m2 -4 ̄ζ2 + α2 χc + 4 ̄ζαχc + α2 χcP 2 (2) , 23 A(+; +,-2) 2, χc2 = -4m ̄αχc ̄κ Z 1 ̄αχc dζ Z d2l (2π)2 ̃Φχc zχc, P (2) q ζ ̄ζ3 P(2)x -iP(2)y (128) × m ̄αχc P(2)x -iP(2)y n0 (x, l, -∆⊥) (ζ - ̄αχc) -iζmαχcn(-) 1 (x, l, -∆⊥) ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 + iαχcζ2n(-) 1 (x, l, -∆⊥) P 2 (2) + m2 zc -1 2 2 ̄ζ (ζ - ̄αχc)2 K2 (2) -ζ αχc ̄αχcP 2 (2) - ̄αχc αχc (ζ - ̄αχc) αχc -4ζ ̄ζ m2 , A(+; -,-2) 2, χc2 = 0. (129) At the extremes of the integration domain ζ = 1 and ζ = ̄αχc, the integrand is strongly suppressed by the wave function ̃Φχc (zχc, P ), and in the heavy quark mass limit the dominant contribution comes from the central region ζ ∼1-αχc/2 (zχc = 1/2). If we construct a Taylor expansion near the point zχc = 1/2, P = 0, the first non-vanishing term in the latter can be reduced to NRQCD LDMEs of χc mesons using the relations (47, 48) from Section II B. The explicit expressions for the coefficients in front of LDMEs are extremely lengthy (require taking derivatives of the derivatives of the integrands) and won't be provided explicitly due to space limitations. The evaluation of the cross-section was performed without making such expansion, taking explicitly the integrals over d2ldζ in momentum space. III. NUMERICAL ESTIMATES A. Differential cross-sections For the sake of definiteness, we will use in what follows the bCGC parametrization of the forward dipole scattering amplitude, whose parameters were fixed from fits of HERA data [16], N (x, r, b) = N0 r Qs(x) 2 2γeff(r) , r ≤ 2 Qs(x) 1 -exp -A ln2 (Br Qs) , r > 2 Qs(x) , (130) A = - N 2 0 γ2 s (1 -N0)2 ln (1 -N0) , B = 1 2 (1 -N0)-1-N0 N0γs , (131) Qs(x, b) = x0 x λ/2 TG(b), γeff(r) = γs + 1 κλ ln(1/x) ln 2 r Qs(x) , TG(b) = exp - b2 4γsBCGC (132) γs = 0.6492, λ = 0.2023, x0 = 6.9 × 10-4, BCGC = 5.5 GeV-2 ≈(0.463 fm)2 . (133) In the heavy quark mass limit, the binding energy of the quarkonium formally is a small parameter, ∼O v2 ∼ O α2 s (mc) . In order to have consistency with this expectation, we will assume that the charm mass mc ≈Mχc/2 ≈ 1.8 GeV. We also will implement the corrections due to the real part of the amplitude (54) and skewedness (55) in exclusive channels, as discussed in Section II C. We would like to start discussion from the threefold differential unpolarized cross-section (22). In the Figure 6 we have shown this observable as a function of the invariant momentum transfer t to the target. Since the parameter ξ ∼ M 2 γχc/W 2 is very small in the chosen kinematics, the variable t may be approximated as t ≈-∆2 ⊥. The pronounced dependence on this variables largely stems from the explicit ∆⊥-dependence which appears in the arguments of the functions n, nΦ- (76-79,106,107) in the momentum space amplitudes (80-100, 110-129). The dependence on ∆⊥which appears in prefactors in front of these functions leads only to minor O t/M 2 χc corrections. Indeed, we may observe that in all these prefactors the external vectors p⊥, ∆⊥always contribute only on the linear combination L = (αχckγ ⊥- ̄αχcpχc ⊥) = p⊥+ ∆⊥ 1 2 -αχc . (134) 24 γp→γχc0p W=141 GeV, |t'|=0.5 GeV2 Mγχc=3.8 GeV Mγχc=4 GeV Mγχc=4.5 GeV Mγχc=5 GeV 0. 0.5 1. 10-3 10-2 10-1 100 101 |t|, GeV2 d3σ/dt dt' dMγχc, pb/GeV5 γp→γχc1p W=141 GeV, |t'|=0.5 GeV2 Mγχc=3.8 GeV Mγχc=4 GeV Mγχc=4.5 GeV Mγχc=5 GeV 0. 0.5 1. 10-4 10-3 10-2 10-1 100 101 |t|, GeV2 d3σ/dt dt' dMγχc, pb/GeV5 γp→γχc2p W=141 GeV, |t'|=0.5 GeV2 Mγχc=4 GeV Mγχc=4.5 GeV Mγχc=5 GeV 0. 0.5 1. 10-2 10-1 100 101 |t|, GeV2 d3σ/dt dt' dMγχc, pb/GeV5 Figure 6. The dependence of the photoproduction cross-section (22) on the invariant momentum transfer to the target t. The left, central and right columns correspond to the χc0, χc1 and χc2 mesons, respectively. The t-dependence at Mγχc = 3.8 GeV reflects the kinematic constraint |t| ≤M 2 γχc -M 2 χc -|t′|. After integration over the dummy momentum l, the absolute value of the amplitude may depend only on the square L2. Since the vectors p⊥and ∆⊥are mutually orthogonal in the high-energy kinematics (see Eq (18)), and p⊥is parametrically large, p2 ⊥∼M 2 χc, we may obtain L2 ≈ ̄αχc αχcM 2 γχc -M 2 χc -t 1 2 -αχc 2 ≈ ̄αχc αχcM 2 γχc -M 2 χc 1 + O t/M 2 χc (135) Besides, the factor 1 2 -αχc 2 in front of t in (135) provides additional numerical suppression that justifies omission of this t-dependence. For comparison, the t-dependence that originates from the functions n, nΦ is related to the implemented impact parameter dependence of the dipole amplitude and is controlled by the (inverse) proton size. In all the phenomenological parametrizations fitted to exclusive processes, this dependence is rapidly decreasing, in agreement with theoretical expectations [69]. Technically, this implies that the χcγ pairs predominantly have oppositely directed transverse momenta pχc ⊥, kγ ⊥. For this reason, in what follows we will focus on the kinematics |t| = |tmin|, or the t-integrated observables that get the largest contribution from small-t kinematics. The dependence on t′ shown in the Figure 7 is very mild in the kinematics of small |t′| because in the impact factors this variable contributes in linear combination with parametrically large scales ∼M 2 γχc, M 2 χc, and, unless we consider the near-threshold kinematics M 2 γχc ∼M 2 χc, may be disregarded altogether. This variables also may be related to the transverse momenta ±p⊥of the final state χc and γ, and thus is correlated with the angle between these particles. As we will explain below, the latter observation allows to understand the completely different t′-dependence of various helicity components. In the Figure 8 we provide predictions for the dependence on the invariant mass Mγχc. The latter variable plays the role of the hard scale that controls the scale of all the transverse momenta, and shows up explicitly in the impact factors provided in previous section. This variable also determines the effective value of the parameter x ∼M 2 γχc/W 2 in the dipole amplitude (130). Due to suppression of the latter at large x, the cross-sections of all quarkonia are strongly suppressed at large Mγχc. We also may observe that unless we consider a near-threshold kinematics Mγχc ∼Mχc, the dependence on all variables largely factorizes. The cross-sections shown in the Figures 6-8 are larger than similar cross-sections found in the collinear factorization framework [87] by up to an order of magnitude. Such enhanced sensitivity to the framework is not surprising: as could be seen from explicit expressions in Section II B, the wave functions of P-wave χc mesons change sign under permutation of quark and antiquark coordinates, and this property leads to large cancellations during integration over the variables ζ, l⊥in the amplitudes (80-100, 110-129). Due to this property, the amplitudes of the CGC and collinear frameworks differ by up to a factor of 2-3, and after squaring of the amplitude, this gives to an order of magnitude difference in the cross-section. 25 γp→γχc0p W=141 GeV |t|=0.5 GeV2 Mγχc=4 GeV Mγχc=4.5 GeV Mγχc=5 GeV 0. 0.5 1. 1.5 2. 10-1 100 |t'|, GeV2 d3σ/dt dt' dMγχc, pb/GeV5 γp→γχc1p W=141 GeV |t|=0.5 GeV2 Mγχc=4 GeV Mγχc=4.5 GeV Mγχc=5 GeV 0. 0.5 1. 1.5 2. 10-2 10-1 100 |t'|, GeV2 d3σ/dt dt' dMγχc, pb/GeV5 γp→γχc2p W=141 GeV |t|=0.5 GeV2 Mγχc=4 GeV Mγχc=4.5 GeV Mγχc=5 GeV 0. 0.5 1. 1.5 2. 10-1 100 101 |t'|, GeV2 d3σ/dt dt' dMγχc, pb/GeV5 Figure 7. The dependence of the photoproduction cross-section (22) on the invariant momentum transfer to the photon (variable t′ = (q -k)2 defined in (16)). The left, central and right columns correspond to the χc0, χc1 and χc2 mesons, respectively. γp→γχc0p t=tmin, |t'|=0.5 GeV2 W=1000 GeV W=500 GeV W=250 GeV W=100 GeV 4. 5. 6. 7. 10-2 10-1 100 101 Mγχc, GeV d3σ/dt dt' dMγχc, pb/GeV3 γp→γχc1p t=tmin, |t'|=0.5 GeV2 W=1000 GeV W=500 GeV W=250 GeV W=100 GeV 4. 5. 6. 7. 10-3 10-2 10-1 100 101 Mγχc, GeV d3σ/dt dt' dMγχc, pb/GeV3 γp→γχc2p t=tmin, |t'|=0.5 GeV2 W=1000 GeV W=500 GeV W=250 GeV W=100 GeV 4. 5. 6. 7. 10-1 100 101 102 Mγχc, GeV d3σ/dt dt' dMγχc, pb/GeV3 Figure 8. The dependence of the photoproduction cross-section (22) on the invariant mass Mγχcof the quakonium-photon pair. The left, central and right columns correspond to the χc0, χc1 and χc2 mesons, respectively. The dependence on the invariant energy shown the Figure 9 can be described by the power law, dσ(W) ∼W 2α, where the constant α has a very mild dependence on other variables. This behaviour follows from a mild x-dependence of the forward dipole amplitude (130) and reflects the increase of the gluonic density at higher energies. Numerically, the values of the parameter α are consistent with the energy dependence implemented in the small-r branch of (130), namely α ≈2λ⟨γeff⟩≈(0.32, 0.37). The early onset of the saturation effects would reveal itself as a deviation from the power law behaviour. Finally, in the Figure 10 we have shown the relative contributions of different quarkonia helicities Hχc to the unpolarized cross-section. In the kinematics of relatively small t′, the quarkonia and photons have small transverse momenta and thus are produced at relatively small angles with respect to the direction of incoming photon. The helicities of all particles in this kinematics nearly coincide with projection of the angular momentum onto the collision axis, and for this reason are subject to the helicity conservation rule, Hγ,in ≈Hγ,out + Hχc, small t′ ≈0. (136) This selection rule allows to understand the dominance of the Hχc = 0 helicity component for χc0, χc1: since the helicities of the onshell photons only take values ±1, and |Hχc| ̸= 2 for these mesons, only the helicity state Hχc = 0 allows to satisfy (136), provided that the photon helicity is not flipped during the process, Hγ,in = Hγ,out. We checked numerically that the contribution of the component with helicity flip of the photon, Hγ,in = -Hγ,out is indeed strongly suppressed. At larger values of the variables |t′|, Mχc, the angles between final-state particles and the collision axis also increases rapidly, so the helicities are no longer are related to projections of the angular momentum onto the same 26 γp→γχc0p Mγχc=3.8 GeV Mγχc=4.0 GeV Mγχc=4.5 GeV Mγχc=5.0 GeV 100 200 500 1000 10-1 100 101 102 W, GeV d3σ/dt dt' dMγχc, pb/GeV5 γp→γχc1p Mγχc=3.8 GeV Mγχc=4.0 GeV Mγχc=4.5 GeV Mγχc=5.0 GeV 100 200 500 1000 10-1 100 101 102 W, GeV d3σ/dt dt' dMγχc, pb/GeV5 γp→γχc2p Mγχc=4.0 GeV Mγχc=4.5 GeV Mγχc=5.0 GeV 100 200 500 1000 100 101 102 W, GeV d3σ/dt dt' dMγχc, pb/GeV5 Figure 9. The energy dependence of the γp →χcγp cross-section for different spins and invariant energies. The nearly linear dependence in double logarithmic coordinates suggests that the dependence may be described by power law ∼W δ, where δ has a very mild dependence on other kinematic variables, with typical values δ ∼0.67 -0.73. γp→γχc1p W=141 GeV, t=tmin Hχc=+1 Hχc=0 Hχc=-1 0.5 1. 1.5 2. 10-3 10-2 10-1 100 |t'|, GeV2 d3σγχc(Hχc)/d3σγχc(all Hχc) γp→γχc1p W=141 GeV, t=tmin Hχc=+1 Hχc=0 Hχc=-1 4. 5. 6. 7. 10-3 10-2 10-1 100 101 Mγχc, GeV d3σγχc(Hχc)/d3σγχc(all Hχc) γp→γχc1p W=141 GeV, t=tmin Hχc=+1 Hχc=0 Hχc=-1 0.5 1. 1.5 2. 10-3 10-2 10-1 100 |t|, GeV2 d3σγχc(Hχc)/d3σγχc(all Hχc) γp→γχc2p W=141 GeV, Mγχc=4.5 GeV Hχc=2 Hχc=1 Hχc=0 0.5 1. 1.5 2. 10-3 10-2 10-1 100 |t'|, GeV2 d3σγχc(Hχc)/d3σγχc(all Hχc) γp→γχc2p W=141 GeV, t=tmin Hχc=2 Hχc=1 Hχc=0 4. 5. 6. 7. 10-3 10-2 10-1 100 101 Mγχc, GeV d3σγχc(Hχc)/d3σγχc(all Hχc) γp→γχc2p W=141 GeV, Mγχc=4 GeV Hχc=2 Hχc=1 Hχc=0 0.5 1. 1.5 2. 10-2 10-1 100 |t|, GeV2 d3σγχc(Hχc)/d3σγχc(all Hχc) Figure 10. The relative contributions of different helicity components of χc in the total cross-section, as a function of kinematic variables t, t′, Mγχc . The upper and lower rows correspond to χc1 and χc2, respectively. For the sake of legibility, in the lower row (for χc2) we did not show the negligibly small contributions of the negative helicity Hχc. axis, and thus the selection rule (136) becomes void. In order to understand the relative size of the contributions with helicity flip of the photon, in the Figure 11 we have shown the angular harmonics c2 defined in (27). We can see that for the harmonics Hχc = 0 and Hχc = 1, which give the dominant contributions in the cross-section, the asymmetry c2 remains moderate. It increases as a function of |t′|, in agreement with the above-mentioned picture based on increase of the angles, however it does not exceed ∼30 per cent in the kinematics of interest. Since the complex phases of the amplitudes A(+,+) γp→χcγp and A(+,-) γp→χcγp are nearly identical, the harmonics s2 is vanishingly small. An experimental confirmation of these theoretical expectations would constitute a strong evidence in favor of the expected dominance of the contribution without photon helicity flip. B. Integrated cross-sections and counting rates While the threefold differential cross-section are well-suited for the phenomenological studies, experimentally it may be easier to study the partially integrated (single-differential) cross-sections. In the Figure 12 we have shown the cross-sections dσ/ dMγχc and dσ/dt′ for different charmonia. As we discussed in the previous section, for the threefold cross-section the dependence on W, t, Mγχc largely factorizes; for this reason the dependence of the integrated crosssections on these variables is similar to that of the unintegrated cross-section. 27 γp→γχc1p W=141 GeV Mγχc=4 GeV Hχc=1 Hχc=0 Hχc=-1 0.5 1. 1.5 2. -0.1 -0.05 0. 0.05 0.1 |t'|, GeV2 c2 γp→γχc2p W=141 GeV, Mγχc=4 GeV Hχc=2 Hχc=1 Hχc=0 0.5 1. 1.5 2. -0.6 -0.4 -0.2 0. 0.2 0.4 0.6 |t'|, GeV2 c2 Figure 11. The angular harmonics c2 of electroproduction cross-section (27), for different helicity components of χc1 and χc2 mesons, respectively. For the sake of legibility, in the right plot (for χc2) we did not show the asymmetries of the contributions with negative helicity Hχc, since the latter are negligibly small. γp→γχcp χc0 χc1 χc2 4. 5. 6. 7. 10-2 10-1 100 101 102 Mγχc, GeV dσ/dMγχc, pb/GeV γp→γχcp χc0 χc1 χc2 0.5 1. 1.5 2. 100 101 102 |t'|, GeV2 dσ/dt', pb/GeV2 Figure 12. The single-differential cross-sections dσ/dt dMγχc, dσ/dMγχc for different charmonia χcJ. Both plots correspond to the invariant energy W = 141 GeV. The total (integrated) cross-sections of the photoproduction process γp →χcJγp is given by σχc0 tot (W ≈100 GeV, Mγχc ≥3.7 GeV) ≈25 pb, (137) σχc1 tot (W ≈100 GeV, Mγχc ≥3.7 GeV) ≈23 pb, (138) σχc2 tot (W ≈100 GeV, Mγχc ≥3.7 GeV) ≈149 pb, (139) and scales with energy approximately as ∼W 0.7. In our estimates we introduced a cutoff on minimal invariant mass Mγχc in order to avoid (huge) background from the radiative decays of ψ(2S) →χcγ 4. In the Tables II and III we provide tentative estimates for the total cross-sections of the electroproduction ep →eγχcp and the ultraperipheral production pp →ppγχc in LHC kinematics. Since the spectrum of virtual photons is dominated by quasireal photons, for the sake of simplicity we disregarded the contributions of photons with large virtuality Q2 ≳1 GeV2, and assumed that the photoproduction cross-section does not depend on Q2 at Q2 ≲1 GeV2 in view of the smallness of O Q2/M 2 χc corrections in the heavy quark mass limit. We found that varying the upper cutoff Q2 max between 1 and 2 GeV2 changes the cross-section within 10 per cent. 4 Combining the experimental ψ(2S) photoproduction cross-section [88] and the branching fractions of ψ(2S) →χcJγ [70], we estimate that the background contribution from ψ(2S) radiative decay is σ(rad) tot ≈1.1 nb. 28 σ(ep) tot Production rates Decay Combined Counting rates N dN/dt channel branching Nd dNd/dt χc0 0.45 pb 4.5×104 380/day χc →J/ψ γ 0.08 % 36 9.1/month χc1 0.41 pb 4.1×104 353/day J/ψ →μ+μ2 % 816 211/month χc2 2.5 pb 2.5×105 2100/day 1.1 % 2750 695/month Table II. The total electroproduction cross-sections, production and counting rates for different χc mesons in EIC kinematics at electron-proton energy √sep = 141 GeV. The cross-section roughly scales with energy as ∼W 0.7. For estimates of the production and counting rates, we used the instantaneous luminosity L = 1034 cm-2s-1 = 10-5fb-1 and the integrated luminosity R dt L = 100 fb-1. The column "combined branching" corresponds to the product of branching fractions of χc →J/ψ γ and J/ψ →μ+μ-. σ(pp) tot Production rates Decay Combined Counting rates N dN/dt channel branching Nd dNd/dt χc0 0.3 pb 3×104 253/day χc →J/ψ γ 0.08 % 24 6.1/month χc1 0.25 pb 2.5×104 215/day J/ψ →μ+μ2 % 500 129/month χc2 1.7 pb 1.7×105 1088/day 1.1 % 1421 359/month Table III. The total production cross-sections, production and counting rates for different χc mesons, in ultraperipheral LHC kinematics at proton-proton energy √spp = 13 GeV. For estimates of the production and counting rates, we used the instantaneous luminosity L = 1034 cm-2s-1 = 10-5fb-1s-1 and the integrated luminosity R dt L = 100 fb-1. The column "combined branching" corresponds to the product of branching fractions of χc →J/ψ γ and J/ψ →μ+μ-. C. Comparison with odderon-mediated χc photoproduction The odderons, or C-odd gluon exchanges in t-channel, remain one of the least understood components of the nonperturbative QCD [89-94], and the the contribution of the odderons to the dipole scattering amplitude at present is largely unknown. A recent discovery of the odderons from comparison of pp and p ̄p elastic cross-sections measured at LHC and Tevatron [95, 96] reinvigorated the interest in this topic. However, the statistical significance of the observed signal still remains under discussion due to sizable uncertainties and possible background contributions. It could be extremely difficult to study in detail the odderon amplitude from the channels where odderons contribute as minor corrections. For this reason, the processes which proceed via the C-odd t-channel exchanges got into focus of odderon searches, and it is expected that such processes will be studied experimentally both at HL-LHC and at the future EIC. The exclusive photoproduction of C-even quarkonia, as for example γp →ηcp, historically attracted a lot of interest, since the heavy quark mass plays the role of the hard scale and justifies at least partial description in perturbative QCD (see [94, 97] for a short overview). However, that process obtains a sizable contribution from the photon-photon fusion (Primakoff mechanism), and for this reason, recently the exclusive χc photoproduction γp →χcp has been recently suggested in [63, 64] as a potential cleaner alternative. The cross-section of the latter process is comparable to that of ηc photoproduction; furthermore it may be easier to study experimentally due to large branching fraction of radiative decays of χc to J/ψ. In this context, the photoproduction of χcγ pairs deserves attention as a potential background to odderon-mediated χc-photoproduction: while the former is formally suppressed as O (αem), the latter is suppressed by smallness of the odderon amplitude, so numerically their cross-sections are comparable. If the final-state photon is not detected, potentially the χcγ photoproduction can be misinterpreted as γp →χcp subprocess. Furthermore, if there is no constraints (cuts) on the invariant mass of the produced hadronic state, a sizable contribution could be obtained from the radiative decays of heavier charmonia. In order to estimate accurately these backgrounds, in general it is required to know in detail the geometry and acceptance of the detector. For simplicity, now we will consider that all the final photons are not detected, and will consider the cross-section dσ/dt integrated over the phase space of the produced photon. This approach provides an upper estimate for the background. For the sake of definiteness, we will use for comparison only the result of [64] because the predictions of [63] are provided in the kinematics of relatively 29 χc0 ψ(2S)→χc0γ(inv) ep→eχc0p+γ(inv.) ep→eχc0p, Odderon ep→eχc0p, Primakoff 0.5 1. 1.5 10-4 10-3 10-2 10-1 100 |t|, GeV2 dσ/dt, pb/GeV2 χc1 ψ(2S)→χc1γ(inv) 0.5 1. 1.5 |t|, GeV2 χc2 ψ(2S)→χc2γ(inv) 0.5 1. 1.5 |t|, GeV2 Figure 13. Comparison of different mechanisms which contribute to electroproduction of mesons. The solid curve corresponds to contribution of ep →eγχcp process discussed in this paper, with invisible (integrated out) final state photon. The upper dotted curve corresponds to the contribution of χcγ pairs which stem from the radiative decays of ψ(2S) charmonia (the χcγ pairs in this mechanism have invariant mass Mγχc ∼Mψ(2S)). The dashed and dot-dashed lines correspond to exclusive electroproduction ep →eχcp mediated by odderons and the Primakoff mechanism, as provided in [64] (see Figure 5). low energies, where the CGC approach might be not very reliable. In the Figure (13) we show the cross-sections of the χc and χcγ production via different mechanisms. The largest contribution comes from the ψ(2S) photoproduction with subsequent radiative decay ψ(2S) →χcγ, since ψ(2S) photoproduction does not require exchange of quantum numbers in t-channel, and ψ(2S) has remarkably large branching ratios of radiative decays, Br (ψ(2S) →χc0γ) = 9.79 ± 0.2 %, Br (ψ(2S) →χc1γ) = 9.75 ± 0.24 %, Br (ψ(2S) →χc2γ) = 9.52 ± 0.2 % [70]. We used for estimates the cross-section of ψ(2S) photoproduction found in CGC framework in [98], and checked that it can reproduce the experimentally measured total (t-integrated) crosssection [88] and the diffractive slope [99]. While the radiative decays of other excited quarkonia potentially could also contribute to the observed backgrounds, at present it is not possible to estimate accurately their contribution due to a sizable uncertainty in their wave functions. The exclusive (non-resonant) production of χcγ pairs suggested in this paper exceeds significantly the odderonmediated χc photoproduction at small-t (|t| ≲1 GeV2), although is less relevant in the large-t kinematics. The cross-sections of both mechanisms of χcγ production (direct and via ψ(2S) radiative decays) increase with energy, whereas the contribution of odderons should decrease due to different odderon intercept. For the sake of comparison we also have shown the contribution of the photon-photon fusion (so-called Primakoff mechanism) which exceeds the odderon contribution in the small-t kinematics, as discussed in [64]. While the Primakoff mechanism is less relevant for odderon searches on neutrons, namely for γn →χcn subprocess, the backgrounds due to the χcγ production with undetected photon are the same for proton and neutron targets. The contributions of different mechanisms discussed in this section can be separated, imposing the kinematic cuts on the variable (q -∆)2 ≈t-Q2 -2q ·∆. For exclusive χc production via odderon-mediated and Primakoff mechanisms, this variable corresponds to the square of charmonium mass, M 2 χc. For production via radiative decays of ψ(2S), this variable equals M 2 ψ(2S). Finally, for the direct (non-resonant) production of χcγ pairs, this variable coincides with invariant mass M 2 γχc of χcγ pair. IV. CONCLUSIONS Using the CGC framework, we analyzed the exclusive χcγ photoproduction, including its helicity dependence, for all χcJ states. We found that the amplitude of the process may be represented as a convolution of the process-dependent impact factors and forward dipole scattering amplitude. In the heavy quark mass limit, the kinematic distribution of produced χcγ pairs may be related to the dependence of the forward dipole scattering amplitude on dipole size, impact parameter and rapidity dependence. We also estimated numerically the cross-sections in the ultraperipheral kinematics at LHC and the future EIC. The cross-sections of all χc mesons are comparable to each other (within a factor of two), and of the same order of magnitude as the cross-section of ηc-production found earlier in [55]. The expected production rates of χcγ pairs constitute ∼104 -105 events per each 100 fb-1 of integrated luminosity, both at EIC and LHC. The expected detection (counting) rates for χc1 and χc2 mesons constitute ∼102 -103 events per each 100 fb-1 of integrated luminosity, if the χc mesons are detected via their χc →J/ψ γ decays. These numbers suggest that the cross-section may be measured with reasonable precision. We also found that the production of χcγ pairs with invisible photon (either direct or via radiative decay of ψ(2S) →χcγ) gives a sizable background to 30 exclusive photoproduction of χc mesons, which was suggested recently in [63, 64] as an alternative tool for study of odderons. While the cross-sections of χcγ production decrease rapidly as a function of the momentum transfer |t|, the radiative decay of ψ(2S) remains the dominant mechanism up to |t| ∼2 GeV2. These findings are in line with our previous analysis [55], which concluded that odderon-mediated quarkonia production can get sizeable backgrounds from quarkonia-photon production. 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2510.14766
|
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
Andreas Filipp 1 2 3 Tri Nguyen 4 5 Laurence Perreault-Levasseur 1 2 3 6 7 8 Jonah Rose 9 Chris Lovell 10 11
Nicolas Payot 1 2 3 Francisco Villaescusa-Navarro 12 13 Yashar Hezaveh 1 2 3 6 8
Abstract
Strong gravitational lensing provides a powerful
tool to directly infer the dark matter (DM) subhalo
mass function (SHMF) in lens galaxies. How-
ever, comparing observationally inferred SHMFs
to theoretical predictions remains challenging, as
the predicted SHMF can vary significantly be-
tween galaxies — even within the same cosmo-
logical model — due to differences in the proper-
ties and environment of individual galaxies. We
present a machine learning framework to infer the
galaxy-specific predicted SHMF from galaxy im-
ages, conditioned on the assumed inverse warm
DM particle mass M −1
DM. To train the model, we
use 1024 high-resolution hydrodynamical zoom-
in simulations from the DREAMS suite. Mock ob-
servations are generated using Synthesizer,
excluding gas particle contributions, and SHMFs
are computed with the Rockstar halo finder.
Our neural network takes as input both the galaxy
images and the inverse DM mass. This method
enables scalable, image-based predictions for the
theoretical DM SHMFs of individual galaxies, fa-
cilitating direct comparisons with observational
measurements.
1. Introduction
One of the most striking open questions in modern astro-
physics is the nature of dark matter (DM), which constitutes
1Department of Physics, University of Montreal, Montreal,
Canada 2MILA Quebec AI Institute, Montreal, Canada 3CIELA
Institute, Montreal Institute for Astrophysics and Machine Learn-
ing, Montreal, Canada 4Center for Interdisciplinary Exploration
and Research in Astrophysics, Northwestern University, Evanston,
USA 5The NSF-Simons AI Institute for the Sky, Chicago, USA
6Center for Computational Astrophysics, Flatiron Institute, New
York, USA 7Perimeter Institute for Theoretical Physics, Water-
loo, Canada 8Trottier Space Institute, McGill University, Mon-
treal, Canada 9Center for Computational Astrophysics, New York,
USA 10Kavli Institute for Cosmology, Madingley Road, Cam-
bridge, UK 11Institute of Astronomy, Madingley Road, Cambridge,
UK 12Princeton University, Princeton, USA 13Simons Founda-
tion, New York, USA. Correspondence to: Andreas Filipp <an-
dreas.filipp@umontreal.ca>.
ML4Astro 2025, Vancouver, CA. Copyright 2025 by the author(s).
approximately 80% of the universe’s matter content (e.g.,
Hinshaw et al., 2013; Planck Collaboration et al., 2020).
While its presence is inferred from gravitational phenomena
across a wide range of cosmic scales, from galaxy clusters
to large-scale structure, DM has yet to be detected through
any non-gravitational interactions, and its fundamental prop-
erties remain unknown.
Different models predict distinct clustering behaviors for
DM, especially on small, sub-galactic scales. On these
scales, the distribution of dark matter — quantified by the
subhalo mass function (SHMF) — is highly sensitive to its
particle nature, making it a powerful discriminator between
DM models (e.g., Ferreira, 2021). In warm DM (WDM)
scenarios, for example, smaller particle masses correspond
to higher thermal velocities, which suppress the formation of
low-mass halos below the free-streaming scale (e.g., Col´ın
et al., 2000; Gilman et al., 2020; Loudas et al., 2022).
Strong gravitational lensing provides a unique way to probe
the distribution of matter on these small scales. Unlike
methods that rely on luminous tracers, lensing is sensitive
to all matter — luminous or dark — making it a powerful
observational tool to constrain the SHMF. Traditional analy-
ses infer the presence of individual subhalos by evaluating
whether introducing localized perturbers to a smooth lens
model leads to a statistically significant improvement in the
fit to the observed lensed images (e.g., Vegetti et al., 2010;
Hezaveh et al., 2016b). More recently, simulation-based
studies have demonstrated that machine learning approaches
could enable population-level inference of the SHMF by
combining data across ensembles of lensing systems (e.g.,
Brehmer et al., 2020; 2019; Coogan et al., 2022; Zhang
et al., 2022; Wagner-Carena et al., 2023; 2024; Zhang et al.,
2024; Filipp et al., 2024).
However, connecting these observational constraints to theo-
retical predictions remains challenging. Even within a fixed
cosmology, the SHMF depends on properties of individual
galaxies, including, for example, total mass, morphology,
merger history, and local environment (e.g., Hezaveh et al.,
2016a), leading to significant system-to-system variation.
In this work, we introduce a machine learning framework
to predict theoretical SHMFs directly from galaxy images,
conditioned on an assumed WDM mass. Its goal is to pre-
1
arXiv:2510.14766v1 [astro-ph.CO] 16 Oct 2025
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
dict, given a WDM particle mass, a plausible theoretical
range of SHMFs for specific individual galaxies based on
their observable properties. These predictions can then be
tested against observational constraints, such as those com-
ing from galaxy-galaxy strong gravitational lensing, to place
limits on the WDM mass. To make these predictions, we
use the DREAMS simulation suite (Rose et al., 2025), as
well as Synthesizer (Vijayan et al., 2020) to create re-
alistic galaxy images. Our method accounts for inter-galaxy
variability and enables scalable, image-based inference of
theoretical predictions. This approach provides a new path-
way to compare dark matter models with forthcoming lens-
ing observations, enabling more precise, per-galaxy tests of
DM’s small-scale gravitational effects.
The paper is structured as follows: In Section 2, we describe
the hydrodynamical simulation suite used in this work. Sec-
tion 3 begins with an overview of how we created the galaxy
images from the hydrodynamical simulations, then provides
the assumed SHMF profile and the correlation with different
DM models, as well as the neural network architecture used
to make predictions. Section 4 presents our results, and we
conclude in Section 5.
2. DREAMS Simulations
The DREAMS simulation suite contains, among other prod-
ucts, a set of high-resolution cosmological zoom-in hydro-
dynamic simulations designed to explore galaxy formation
under varying DM and baryonic physics. Each zoom-in
simulation is run using the AREPO code (Springel, 2010;
Springel et al., 2019; Weinberger et al., 2020), enabling ac-
curate modeling of complex baryonic processes such as gas
cooling, star formation, and feedback. The initial conditions
for each zoom-in are constructed by selecting a random,
isolated Milky Way–mass halo from a low-resolution vol-
ume, then iteratively refining its Lagrangian region using
intermediate- and high-resolution particle resampling to de-
fine the zoom-in domain (see Rose et al., 2025).
These zoom-in simulations are performed across a range
of different WDM models in the range MDM
∈
[1.8, 30.3] keV, sampled uniformly in the inverse M −1
DM.
Further, the supernova (SN) wind, SN energy, and active
galactic nuclei (AGN) parameters vary between each zoom-
in simulation (Rose et al., 2025). The simulations are mod-
eled using the IllustrisTNG baryonic physics prescriptions.
By varying both the DM physics and feedback parame-
ters, the DREAMS simulations allow us to marginalize over
baryonic uncertainties and learn a mapping from observ-
able galaxy properties and WDM masses to the underlying
SHMF. This marginalization ensures that our model gener-
alizes across different astrophysical scenarios.
The DM sub-halos in the WDM zoom-in simulations are
Figure 1. A flowchart showing the training procedure, label
generation, and inference.
identified using the Rockstar halo finder (Behroozi et al.,
2013). At the time of submitting this work, Rockstar halo
catalogs were made available for 815 of the 1024 DREAMS
zoom-in simulations. These constitute the labeled dataset
used for supervised training of the SHMF inference model.
3. Methods
3.1. Image Generation with Synthesizer
Creating realistic galaxy images from hydrodynamic simula-
tions typically requires computationally expensive radiative
transfer simulations. As an efficient alternative, we use
Synthesizer (Wilkins et al., 2020; Vijayan et al., 2020)
to generate realistic observational mock images from simu-
lated galaxies.
For each of the 815 zoom-in galaxies with available
Rockstar-catalog, we generate 12 different projections
by varying the line-of-sight orientation of the particles.
Synthesizer produces galaxy images by generating spa-
tially resolved spectral energy distributions (SEDs) and ap-
plying instrument-specific wavelength filters from the Span-
ish Virtual Observatory (SVO) filter service1 (Rodrigo et al.,
2012; Rodrigo & Solano, 2020; Rodrigo et al., 2024).
A detailed description of the image creation process can be
found in Appendix A. Examples of the generated galaxy
images are shown in Figure 2, alongside their corresponding
galaxy-specific SHMFs.
1https://svo2.cab.inta-csic.es/theory/
fps/
2
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
3.2. Dark Matter Subhalo Mass Function
Since we aim to learn a mapping between galaxy images
and their corresponding SHMF for a given WDM mass, we
next move on to modeling the functional form of galaxies’
SHMFs. We assume this functional form of the DM SHMFs
to be a power-law with a slope of −0.9, as an approximation
to theoretical predictions of cold DM (CDM) (e.g., Kuhlen
et al., 2007; Diemand et al., 2007; Springel et al., 2008;
Hiroshima et al., 2018). To model the subhalo mass function
of WDM, we add to the power-law form of CDM a low-mass
cutoff (e.g., Gilman et al., 2020):
dN
dm|wdm = dN
dm|CDM ·
1 +
mwdm
m
−1.3
= A · m−0.9 ·
1 +
mwdm
m
−1.3
(1)
where mwdm is the cutoff mass, a characteristic of WDM
scenarios, and the parameter A is the normalization.
We fit the parameterized WDM SHMF form to the data of
the simulated galaxies, with the subhalos of the individual
galaxies identified by Rockstar. We bin the identified
subhalos in 15 mass bins, linearly spaced in logarithmic 10
base from log10(M/M⊙) = 7.75 to 11. The uncertainties
in the observed subhalo counts ˆni are dominated by Poisson
noise, see Appendix B. To fit the parameters of 1, we use
the likelihood defined in eqn (9) in Appendix B:
ln L(ˆn | θ) = −1
2
X
i
ˆni −ni(θ)
2
Vi −V ′
i
ˆni −ni(θ)
,
(2)
where the index i runs over the mass bins, ni(θ) is the model
prediction of eqn (1), given the parameters θ = (A, mwdm).
We use the emcee package (Foreman-Mackey et al., 2013)
to perform Markov Chain Monte Carlo (MCMC) sampling
over this likelihood. This yields posterior samples for the
amplitude A and the WDM cutoff mass mwdm. We then
use the samples (A, mwdm) obtained in this way to train
a normalizing flow (NF), which allows us to account for
correlations between A and mwdm. Our goal is to then use
this NF to predict the expected mass function at a given
WDM mass of specific individual galaxies, and compare
these predictions to observational constraints from other
DM probes, such as strong gravitational lensing.
For each zoom-in galaxy, we use 100 posterior samples
from the fitted distribution of (A, mwdm) as training labels
for the NF. This ensures that the NF learns the continuous
distribution of plausible parameters for A and mwdm, rather
than a single point estimate. As the flowchart in Figure 1
illustrates, the MCMC samples are only used during training
to help the NF learn the continuous distributions of A and
mwdm conditioned on the galaxy morphologies and MDM.
3.3. Network Architecture and Training
Our architecture consists of two main components: a convo-
lutional neural network (CNN) to process the image of the
galaxy and a conditional normalizing flow (NF).
Within a given cosmological model, the predicted SHMF
depends on the properties of each individual galaxy. Given
a WDM mass, our architecture infers the relation between
the observable properties of the galaxies and their SHMF.
Figure 1 shows a flowchart of the training and inference
scheme we used. We first pre-train a ResNet-18 with a two-
layer multi-layer perceptron (MLP) to predict the number of
subhalos in the galaxy images, with the inverse WDM mass
M −1
DM concatenated to the first of the two MLP layers, using
a mean squared error (MSE) loss. Our pretraining ensures a
meaningful embedding space for the images, before using
the output of the pretrained CNN as a condition for the NF.
The embedded image and the inverse of the WDM mass
M −1
DM are used as conditions for a neural spline flow (NSF)
to predict the parameters A and mWDM. We use the zuko
library to implement the NSF (Rozet et al., 2022). We use
730 of the zoom-in galaxies as training data and keep 85 as
validation data, for both - the pretraining of the CNN and
the training of the NSF - the same sets.
To train the full model, all the weights of the ResNet are
allowed to update, enabling end-to-end optimization of the
feature extractor and density estimator. This allows us to
efficiently model the posterior over the SHMF fit parameters
conditioned on both the image and the WDM mass.
4. Results
Figure 2 shows galaxy-specific SHMFs with the correspond-
ing galaxy image. We compare the NF conditioned on the
galaxy images and MDM, and an NF conditioned only on
MDM. This gives an estimate of how including the morpho-
logical information of the galaxy can improve the SHMF
prediction, when compared to sampling a SHMF that has
been marginalized over galaxy morphologies, as usually
assumed from theoretical predictions. Since each galaxy
was only simulated for a single WDM temperature in the
DREAMS suite, Figure 2 performs this comparison for
galaxies from the test set at a single WDM mass for each
galaxy. We achieve an improvement in the SHMF predic-
tions by including information about the galaxy morphology.
The displayed galaxies come from three different WDM
mass regions. The examples on the left come from higher
WDM masses, the ones in the middle from medium masses,
and the right side from low WDM masses. The black dots
are the subhalo counts from the DREAMS simulations with
Poisson uncertainties. The red and blue solid lines show
the 50th percentile fit of the NF samples conditioned on the
image and the samples of the NF only conditioned on MDM,
3
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
Figure 2. The galaxy-specific halo mass functions under different cosmologies. The figure shows the galaxy-specific halo mass
functions for different WDM masses, with the corresponding galaxy image in the plot. The displayed galaxies come from three different
WDM mass regions. The black dots are the counts of subhalos from the DREAMS simulation with Poisson uncertainties. The blue
contours show the samples of the NF conditioned only on MDM, and the red contours the samples of the NF conditioned on both MDM
and the galaxy image. The NF conditioned only on MDM shows a broader posterior range and bigger uncertainties on the fit.
Table 1. PQMass-χ2 and RSME values. The table shows the
mean χ2 values obtained with PQMass, as well as the median
RSME, for the NF conditioned on images and MDM, and an NF
conditioned only on MDM for the entire validation set. The com-
parison is made between the MCMC samples and the different NF
samples for A and mwdm.
NF with image
NF without image
PQM χ2
343.7 ± 92.5
385.8 ± 121.7
Median RMSE
0.26 ± 0.20
0.55 ± 0.17
respectively. The shaded regions show the 1σ and 2σ fits of
the SHMF from eqn 1 using the sampled parameters. The
displayed contours show the 1σ, 2σ, and 3σ contours of
the parameter samples (A, mwdm). The titles indicate the
WDM mass range, which is passed along with the image to
the neural network. For each range of MDM, we show two
examples in the same column. The NF conditioned only on
MDM shows broader posteriors and larger uncertainties on
the parameters of the SHMF. On the other hand, including
the image information allows tighter constraints on both
parameters, A and mwdm, and highlights their correlations,
resulting in tighter constraints on the SHMF. This supports
the conclusion that incorporating galaxy images enables
more precise predictions of the SHMF fit parameters than
using the WDM mass MDM alone.
We further test the NF conditioned on the galaxy images
and MDM against the NF only conditioned on MDM qual-
itatively, to show that the inclusion of the galaxy images
leads to a better performance of the NF. In Table 1 we re-
port the mean and standard deviation of the PQMass-χ2
values (Lemos et al., 2024) for the entire validation set for
the NF conditioned on images and MDM, and the NF only
conditioned on MDM. We also report the median root mean
squared error (RMSE) and its standard deviation of the en-
tire validation set for both network cases. For that, we use
for each condition the MCMC samples of the SHMF fit as
target distribution and compare those against the NF samples
of A and mwdm. The closer the distributions are, the lower
the PQMass-χ2 value. We do not expect the NF samples
to match the MCMC fits perfectly, because each individual
galaxy samples a distribution of possible SHMF for a given
WDM model, making the inference under-specified without
additional information. Providing additional galaxy images
leads to a more informative inference of the possible SHMF,
since they provide more constraining information. The
lower RMSE value for the normalizing flow with access to
the morphological information shows that the performance
of the architecture improves and has tighter constraints on
the SHMF. Additionally, we compare the MCMC fits of the
SHMFs and the NF samples of the NF with image condi-
tioning in Figure 3 in Appendix C. We do not compare the
NF samples here with the MCMC samples, because the goal
is to show that the NF with access to the image data does
outperform the pure theoretical NF predictions based on the
WDM only without image access.
The current sample size of the simulation is too small to
4
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
have a separate test set; therefore, we show the performance
for the validation set. The provided examples show that the
network is able to learn the galaxy-specific fit parameters
within the predicted uncertainties.
5. Discussion
Future iterations of this work will focus on increasing the
realism of the input images and expanding the training
dataset. The current images do not include the effects of
dust attenuation, assume only Gaussian noise, and use a
simplified Gaussian point spread function (PSF). Addition-
ally, the images are expressed in flux units (erg, s−1, Hz−1)
rather than in instrument-specific units. We plan to con-
vert them to the native units of the Hubble Space Tele-
scope (HST), i.e., e−, s−1, and to incorporate non-Gaussian,
instrument-specific noise using SLIC (Legin et al., 2023),
along with more realistic PSF models. These improvements
will be accompanied by the inclusion of the full set of
Rockstar halo catalogs, including those for the remaining
209 DREAMS zoom-in simulations, to expand the training
and validation sets and enable evaluation on an independent
test set.
The results demonstrate strong performance and highlight
the potential of applying this architecture to more realistic
simulations. However, there are important limitations to
note regarding the training data. All galaxies used during
training and validation are Milky Way-mass halos in iso-
lated systems. This constraint is due to the availability of
high-resolution DM zoom-in simulations, which currently
exist only for these galaxy types, since running such detailed
zoom-ins on cluster galaxies is computationally too expen-
sive. Whilst the masses of the galaxies are for all galaxies in
the order of the Milky Way, they are approximately equally
distributed between spiral and elliptical galaxies. In contrast,
strong gravitational lenses are more commonly observed to
be massive galaxies in dense environments - often the cen-
tral galaxies of clusters. These systems are not represented
in the necessary mass resolutions in current hydrodynam-
ical simulations. For these purposes, the subhalos should
be resolved down to masses of log10(M/M⊙) = 7.75 and
below to be able to resolve small subhalos and confidently
infer the SHMF.
We expect that future high-resolution hydrodynamical sim-
ulations will model the relevant environments in greater
detail, allowing us to extend the training set and improve
generalization to observed lensing galaxies. Until then, our
results should be interpreted within this limitation.
Acknowledgements
This work is partially supported by Schmidt Sciences, a
philanthropic initiative founded by Eric and Wendy Schmidt
as part of the Virtual Institute for Astrophysics (VIA). The
work is in part supported by computational resources pro-
vided by Calcul Quebec and the Digital Research Alliance
of Canada. A.F. acknowledges the support from the Bourse
J. Armand Bombardier and UdeM’s final year scholarship.
T.N. is supported by the CIERA Postdoctoral Fellowship.
Y.H. and L.P. acknowledge support from the Canada Re-
search Chairs Program, the National Sciences and Engineer-
ing Council of Canada through grants RGPIN-2020-05073
and 05102.
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Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
A. Image Generation with Synthesizer
Synthesizer produces galaxy images by generating spatially resolved spectral energy distributions (SEDs) and applying
instrument-specific wavelength filters from the Spanish Virtual Observatory (SVO) filter service2 (Rodrigo et al., 2012;
Rodrigo & Solano, 2020; Rodrigo et al., 2024). We use the Hubble Space Telescope Wide Field Camera (HST/WFC)
F105W filter to simulate realistic near-infrared observations.
The synthetic images are constructed by only taking stellar particles into account. We neglect the contributions of dust and
gas particles, which are expected to have a relatively minor impact on the morphological features relevant to our analysis.
In future steps, we will include the effects of dust and interstellar gas. As the first step of the image creation, we use an
incident emission model and place the galaxies at redshift z = 0.
The particles are convolved with a smoothing filter for a more optical appealing visualization. The smoothing length of
the individual stellar particles is taken for each particle from the simulation data directly, which means that any artifacts of
isolated particles in the imaging are originating from the simulation. The smoothing length is the co-moving radius of the
sphere centered on the particle enclosing the 32 ± 1 nearest particles of this same type.
To ensure image fidelity, we first generate high-resolution images and convolve them at the high resolution with a Gaussian
point spread function (PSF) to approximate observational effects. The PSF has a full-width half maximum of 3 pixels. The
resulting images are then downsampled to a target resolution of 0.78125 kpc/pixel. We add gaussian noise of 1023 erg
s·Hz. We
do not yet include instrument-specific observational noise or Poisson noise. Examples of the generated galaxy images are
shown in Figure 2, alongside the corresponding galaxy-specific SHMFs.
B. Poisson Noise from Samples
To compute the Poisson noise of the counts of each bin of the SHMF, we use the Rockstar catalogs. We follow the
prescription in Tanabashi et al. (2018); Chang & Necib (2021). We compute the confidence intervals using the inverse
cumulative distribution function of the χ2 distribution:
µlower = 1
2 F −1
χ2
α
2 ; 2ˆn
(3)
µhigher = 1
2 F −1
χ2
1 −α
2 ; 2(ˆn + 1)
(4)
with F −1
χ2 the inverse of the χ2 cumulative distribution and ˆn the number of counts. The confidence level is given by
100(1 −α). From these intervals, we define the asymmetric error bars and variances with:
σlower,i = ˆni −µlower,i
(5)
σhigher,i = µhigher,i −ˆni
(6)
Vi = σlower,i σhigher,i
(7)
V ′
i = σhigher,i −σlower,i
(8)
We use the variances and uncertainties on the counts per bin to compute the likelihood of our fits to the data.
The likelihood function to fit the SHMF (eqn 1) is defined by:
ln L(ˆn | θ) = −1
2
X
i
ˆni −ni(θ)
2
Vi −V ′
i
ˆni −ni(θ)
(9)
where ni(θ) is the model prediction of eqn (1), given the parameters θ = (A, mwdm). We use the likelihood function to fit
the Rockstar-catalog data with an MCMC.
C. Comparison of Normalizing Flow Samples with MCMC Fits
In Figure 3, we show galaxy-specific SHMFs obtained by NF samples and MCMC fits along with the corresponding galaxy
image. The figure is structured as Figure 2. We use the same galaxies as before, but under different projections. We do not
2https://svo2.cab.inta-csic.es/theory/fps/
9
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images
expect to perfectly recreate the MCMC fits. Our method serves as a tool to get the theoretical predictions on the SHMF given
a WDM theory. This does not serve as constraints on the SHMF, but is rather to be used to compare theoretical predictions
with observational constraints that can be achieved with strong lensing results. This is not a standalone method to learn
about dark matter in observational surveys, but it needs a comparison counterpart that constrains the SHMF. The blue and
red solid contours show the samples of the MCMC fit and of the NF samples, respectively. The dark and light shaded areas
are the 1σ and 2σ regions of the fits. The displayed contours show the 1σ, 2σ, and 3σ contours of the MCMC fits and NF
samples of (A, mwdm), defining the SHMF.
The sampled parameters of the NF represent the fits obtained directly through the likelihood and cover the full range of the
parameters sampled with MCMC.
Figure 3. Similar to Figure 2. The shown galaxies are the same as in Figure 3, but the orientation of the galaxy images is different. The
blue contours show the samples of the MCMC fit, and the red contours the samples of the NF conditioned on MDM and the galaxy image.
Overall, the sampled parameters of the NF represent the fits obtained directly through the likelihood and cover the full range of MCMC
sampled parameters.
10
|
Predicting the Subhalo Mass Functions in Simulations from Galaxy Images Andreas Filipp 1 2 3 Tri Nguyen 4 5 Laurence Perreault-Levasseur 1 2 3 6 7 8 Jonah Rose 9 Chris Lovell 10 11 Nicolas Payot 1 2 3 Francisco Villaescusa-Navarro 12 13 Yashar Hezaveh 1 2 3 6 8 Abstract Strong gravitational lensing provides a powerful tool to directly infer the dark matter (DM) subhalo mass function (SHMF) in lens galaxies. However, comparing observationally inferred SHMFs to theoretical predictions remains challenging, as the predicted SHMF can vary significantly between galaxies - even within the same cosmological model - due to differences in the properties and environment of individual galaxies. We present a machine learning framework to infer the galaxy-specific predicted SHMF from galaxy images, conditioned on the assumed inverse warm DM particle mass M -1 DM. To train the model, we use 1024 high-resolution hydrodynamical zoomin simulations from the DREAMS suite. Mock observations are generated using Synthesizer, excluding gas particle contributions, and SHMFs are computed with the Rockstar halo finder. Our neural network takes as input both the galaxy images and the inverse DM mass. This method enables scalable, image-based predictions for the theoretical DM SHMFs of individual galaxies, facilitating direct comparisons with observational measurements. 1. Introduction One of the most striking open questions in modern astrophysics is the nature of dark matter (DM), which constitutes 1 2MILA Quebec AI Institute, Montreal, Canada 3CIELA Institute, Montreal Institute for Astrophysics and Machine Learning, Montreal, Canada 4Center for Interdisciplinary Exploration and Research in Astrophysics, Northwestern University, Evanston, USA 5The NSF-Simons AI Institute for the Sky, Chicago, USA 6Center for Computational Astrophysics, Flatiron Institute, New York, USA 7Perimeter Institute for Theoretical Physics, Waterloo, Canada 8Trottier Space Institute, McGill University, Montreal, Canada 9Center for Computational Astrophysics, New York, USA 10Kavli Institute for Cosmology, Madingley Road, Cambridge, UK 11 12Princeton University, Princeton, USA 13Simons Foundation, New York, USA. Correspondence to: Andreas Filipp . ML4Astro 2025, Vancouver, CA. by the author(s). approximately 80% of the universe's matter content (e.g., Hinshaw et al., 2013; Planck Collaboration et al., 2020). While its presence is inferred from gravitational phenomena across a wide range of cosmic scales, from galaxy clusters to large-scale structure, DM has yet to be detected through any non-gravitational interactions, and its fundamental properties remain unknown. Different models predict distinct clustering behaviors for DM, especially on small, sub-galactic scales. On these scales, the distribution of dark matter - quantified by the subhalo mass function (SHMF) - is highly sensitive to its particle nature, making it a powerful discriminator between DM models (e.g., Ferreira, 2021). In warm DM (WDM) scenarios, for example, smaller particle masses correspond to higher thermal velocities, which suppress the formation of low-mass halos below the free-streaming scale (e.g., Col ́ın et al., 2000; Gilman et al., 2020; Loudas et al., 2022). Strong gravitational lensing provides a unique way to probe the distribution of matter on these small scales. Unlike methods that rely on luminous tracers, lensing is sensitive to all matter - luminous or dark - making it a powerful observational tool to constrain the SHMF. Traditional analyses infer the presence of individual subhalos by evaluating whether introducing localized perturbers to a smooth lens model leads to a statistically significant improvement in the fit to the observed lensed images (e.g., Vegetti et al., 2010; Hezaveh et al., 2016b). More recently, simulation-based studies have demonstrated that machine learning approaches could enable population-level inference of the SHMF by combining data across ensembles of lensing systems (e.g., Brehmer et al., 2020; 2019; Coogan et al., 2022; Zhang et al., 2022; Wagner-Carena et al., 2023; 2024; Zhang et al., 2024; Filipp et al., 2024). However, connecting these observational constraints to theoretical predictions remains challenging. Even within a fixed cosmology, the SHMF depends on properties of individual galaxies, including, for example, total mass, morphology, merger history, and local environment (e.g., Hezaveh et al., 2016a), leading to significant system-to-system variation. In this work, we introduce a machine learning framework to predict theoretical SHMFs directly from galaxy images, conditioned on an assumed WDM mass. Its goal is to pre1 16 Oct 2025 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images dict, given a WDM particle mass, a plausible theoretical range of SHMFs for specific individual galaxies based on their observable properties. These predictions can then be tested against observational constraints, such as those coming from galaxy-galaxy strong gravitational lensing, to place limits on the WDM mass. To make these predictions, we use the DREAMS simulation suite (Rose et al., 2025), as well as Synthesizer (Vijayan et al., 2020) to create realistic galaxy images. Our method accounts for inter-galaxy variability and enables scalable, image-based inference of theoretical predictions. This approach provides a new pathway to compare dark matter models with forthcoming lensing observations, enabling more precise, per-galaxy tests of DM's small-scale gravitational effects. The paper is structured as follows: In Section 2, we describe the hydrodynamical simulation suite used in this work. Section 3 begins with an overview of how we created the galaxy images from the hydrodynamical simulations, then provides the assumed SHMF profile and the correlation with different DM models, as well as the neural network architecture used to make predictions. Section 4 presents our results, and we conclude in Section 5. 2. DREAMS Simulations The DREAMS simulation suite contains, among other products, a set of high-resolution cosmological zoom-in hydrodynamic simulations designed to explore galaxy formation under varying DM and baryonic physics. Each zoom-in simulation is run using the AREPO code (Springel, 2010; Springel et al., 2019; Weinberger et al., 2020), enabling accurate modeling of complex baryonic processes such as gas cooling, star formation, and feedback. The initial conditions for each zoom-in are constructed by selecting a random, isolated Milky Way-mass halo from a low-resolution volume, then iteratively refining its Lagrangian region using intermediate- and high-resolution particle resampling to define the zoom-in domain (see Rose et al., 2025). These zoom-in simulations are performed across a range of different WDM models in the range MDM ∈ [1.8, 30.3] keV, sampled uniformly in the inverse M -1 DM. Further, the supernova (SN) wind, SN energy, and active galactic nuclei (AGN) parameters vary between each zoomin simulation (Rose et al., 2025). The simulations are modeled using the IllustrisTNG baryonic physics prescriptions. By varying both the DM physics and feedback parameters, the DREAMS simulations allow us to marginalize over baryonic uncertainties and learn a mapping from observable galaxy properties and WDM masses to the underlying SHMF. This marginalization ensures that our model generalizes across different astrophysical scenarios. The DM sub-halos in the WDM zoom-in simulations are Figure 1. A flowchart showing the training procedure, label generation, and inference. identified using the Rockstar halo finder (Behroozi et al., 2013). At the time of submitting this work, Rockstar halo catalogs were made available for 815 of the 1024 DREAMS zoom-in simulations. These constitute the labeled dataset used for supervised training of the SHMF inference model. 3. Methods 3.1. Image Generation with Synthesizer Creating realistic galaxy images from hydrodynamic simulations typically requires computationally expensive radiative transfer simulations. As an efficient alternative, we use Synthesizer (Wilkins et al., 2020; Vijayan et al., 2020) to generate realistic observational mock images from simulated galaxies. For each of the 815 zoom-in galaxies with available Rockstar-catalog, we generate 12 different projections by varying the line-of-sight orientation of the particles. Synthesizer produces galaxy images by generating spatially resolved spectral energy distributions (SEDs) and applying instrument-specific wavelength filters from the Spanish Virtual Observatory (SVO) filter service1 (Rodrigo et al., 2012; Rodrigo & Solano, 2020; Rodrigo et al., 2024). A detailed description of the image creation process can be found in Appendix A. Examples of the generated galaxy images are shown in Figure 2, alongside their corresponding galaxy-specific SHMFs. 1https://svo2.cab.inta-csic.es/theory/ fps/ 2 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images 3.2. Dark Matter Subhalo Mass Function Since we aim to learn a mapping between galaxy images and their corresponding SHMF for a given WDM mass, we next move on to modeling the functional form of galaxies' SHMFs. We assume this functional form of the DM SHMFs to be a power-law with a slope of -0.9, as an approximation to theoretical predictions of cold DM (CDM) (e.g., Kuhlen et al., 2007; Diemand et al., 2007; Springel et al., 2008; Hiroshima et al., 2018). To model the subhalo mass function of WDM, we add to the power-law form of CDM a low-mass cutoff (e.g., Gilman et al., 2020): dN dm|wdm = dN dm|CDM · 1 + mwdm m -1.3 = A · m-0.9 · 1 + mwdm m -1.3 (1) where mwdm is the cutoff mass, a characteristic of WDM scenarios, and the parameter A is the normalization. We fit the parameterized WDM SHMF form to the data of the simulated galaxies, with the subhalos of the individual galaxies identified by Rockstar. We bin the identified subhalos in 15 mass bins, linearly spaced in logarithmic 10 base from log10(M/M⊙) = 7.75 to 11. The uncertainties in the observed subhalo counts ˆni are dominated by Poisson noise, see Appendix B. To fit the parameters of 1, we use the likelihood defined in eqn (9) in Appendix B: ln L(ˆn | θ) = -1 2 X i ˆni -ni(θ) 2 Vi -V ′ i ˆni -ni(θ) , (2) where the index i runs over the mass bins, ni(θ) is the model prediction of eqn (1), given the parameters θ = (A, mwdm). We use the emcee package (Foreman-Mackey et al., 2013) to perform Markov Chain Monte Carlo (MCMC) sampling over this likelihood. This yields posterior samples for the amplitude A and the WDM cutoff mass mwdm. We then use the samples (A, mwdm) obtained in this way to train a normalizing flow (NF), which allows us to account for correlations between A and mwdm. Our goal is to then use this NF to predict the expected mass function at a given WDM mass of specific individual galaxies, and compare these predictions to observational constraints from other DM probes, such as strong gravitational lensing. For each zoom-in galaxy, we use 100 posterior samples from the fitted distribution of (A, mwdm) as training labels for the NF. This ensures that the NF learns the continuous distribution of plausible parameters for A and mwdm, rather than a single point estimate. As the flowchart in Figure 1 illustrates, the MCMC samples are only used during training to help the NF learn the continuous distributions of A and mwdm conditioned on the galaxy morphologies and MDM. 3.3. Network Architecture and Training Our architecture consists of two main components: a convolutional neural network (CNN) to process the image of the galaxy and a conditional normalizing flow (NF). Within a given cosmological model, the predicted SHMF depends on the properties of each individual galaxy. Given a WDM mass, our architecture infers the relation between the observable properties of the galaxies and their SHMF. Figure 1 shows a flowchart of the training and inference scheme we used. We first pre-train a ResNet-18 with a twolayer multi-layer perceptron (MLP) to predict the number of subhalos in the galaxy images, with the inverse WDM mass M -1 DM concatenated to the first of the two MLP layers, using a mean squared error (MSE) loss. Our pretraining ensures a meaningful embedding space for the images, before using the output of the pretrained CNN as a condition for the NF. The embedded image and the inverse of the WDM mass M -1 DM are used as conditions for a neural spline flow (NSF) to predict the parameters A and mWDM. We use the zuko library to implement the NSF (Rozet et al., 2022). We use 730 of the zoom-in galaxies as training data and keep 85 as validation data, for both - the pretraining of the CNN and the training of the NSF - the same sets. To train the full model, all the weights of the ResNet are allowed to update, enabling end-to-end optimization of the feature extractor and density estimator. This allows us to efficiently model the posterior over the SHMF fit parameters conditioned on both the image and the WDM mass. 4. Results Figure 2 shows galaxy-specific SHMFs with the corresponding galaxy image. We compare the NF conditioned on the galaxy images and MDM, and an NF conditioned only on MDM. This gives an estimate of how including the morphological information of the galaxy can improve the SHMF prediction, when compared to sampling a SHMF that has been marginalized over galaxy morphologies, as usually assumed from theoretical predictions. Since each galaxy was only simulated for a single WDM temperature in the DREAMS suite, Figure 2 performs this comparison for galaxies from the test set at a single WDM mass for each galaxy. We achieve an improvement in the SHMF predictions by including information about the galaxy morphology. The displayed galaxies come from three different WDM mass regions. The examples on the left come from higher WDM masses, the ones in the middle from medium masses, and the right side from low WDM masses. The black dots are the subhalo counts from the DREAMS simulations with Poisson uncertainties. The red and blue solid lines show the 50th percentile fit of the NF samples conditioned on the image and the samples of the NF only conditioned on MDM, 3 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images Figure 2. The galaxy-specific halo mass functions under different cosmologies. The figure shows the galaxy-specific halo mass functions for different WDM masses, with the corresponding galaxy image in the plot. The displayed galaxies come from three different WDM mass regions. The black dots are the counts of subhalos from the DREAMS simulation with Poisson uncertainties. The blue contours show the samples of the NF conditioned only on MDM, and the red contours the samples of the NF conditioned on both MDM and the galaxy image. The NF conditioned only on MDM shows a broader posterior range and bigger uncertainties on the fit. Table 1. PQMass-χ2 and RSME values. The table shows the mean χ2 values obtained with PQMass, as well as the median RSME, for the NF conditioned on images and MDM, and an NF conditioned only on MDM for the entire validation set. The comparison is made between the MCMC samples and the different NF samples for A and mwdm. NF with image NF without image PQM χ2 343.7 ± 92.5 385.8 ± 121.7 Median RMSE 0.26 ± 0.20 0.55 ± 0.17 respectively. The shaded regions show the 1σ and 2σ fits of the SHMF from eqn 1 using the sampled parameters. The displayed contours show the 1σ, 2σ, and 3σ contours of the parameter samples (A, mwdm). The titles indicate the WDM mass range, which is passed along with the image to the neural network. For each range of MDM, we show two examples in the same column. The NF conditioned only on MDM shows broader posteriors and larger uncertainties on the parameters of the SHMF. On the other hand, including the image information allows tighter constraints on both parameters, A and mwdm, and highlights their correlations, resulting in tighter constraints on the SHMF. This supports the conclusion that incorporating galaxy images enables more precise predictions of the SHMF fit parameters than using the WDM mass MDM alone. We further test the NF conditioned on the galaxy images and MDM against the NF only conditioned on MDM qualitatively, to show that the inclusion of the galaxy images leads to a better performance of the NF. In Table 1 we report the mean and standard deviation of the PQMass-χ2 values (Lemos et al., 2024) for the entire validation set for the NF conditioned on images and MDM, and the NF only conditioned on MDM. We also report the median root mean squared error (RMSE) and its standard deviation of the entire validation set for both network cases. For that, we use for each condition the MCMC samples of the SHMF fit as target distribution and compare those against the NF samples of A and mwdm. The closer the distributions are, the lower the PQMass-χ2 value. We do not expect the NF samples to match the MCMC fits perfectly, because each individual galaxy samples a distribution of possible SHMF for a given WDM model, making the inference under-specified without additional information. Providing additional galaxy images leads to a more informative inference of the possible SHMF, since they provide more constraining information. The lower RMSE value for the normalizing flow with access to the morphological information shows that the performance of the architecture improves and has tighter constraints on the SHMF. Additionally, we compare the MCMC fits of the SHMFs and the NF samples of the NF with image conditioning in Figure 3 in Appendix C. We do not compare the NF samples here with the MCMC samples, because the goal is to show that the NF with access to the image data does outperform the pure theoretical NF predictions based on the WDM only without image access. The current sample size of the simulation is too small to 4 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images have a separate test set; therefore, we show the performance for the validation set. The provided examples show that the network is able to learn the galaxy-specific fit parameters within the predicted uncertainties. 5. Discussion Future iterations of this work will focus on increasing the realism of the input images and expanding the training dataset. The current images do not include the effects of dust attenuation, assume only Gaussian noise, and use a simplified Gaussian point spread function (PSF). Additionally, the images are expressed in flux units (erg, s-1, Hz-1) rather than in instrument-specific units. We plan to convert them to the native units of the Hubble Space Telescope (HST), i.e., e-, s-1, and to incorporate non-Gaussian, instrument-specific noise using SLIC (Legin et al., 2023), along with more realistic PSF models. These improvements will be accompanied by the inclusion of the full set of Rockstar halo catalogs, including those for the remaining 209 DREAMS zoom-in simulations, to expand the training and validation sets and enable evaluation on an independent test set. The results demonstrate strong performance and highlight the potential of applying this architecture to more realistic simulations. However, there are important limitations to note regarding the training data. All galaxies used during training and validation are Milky Way-mass halos in isolated systems. This constraint is due to the availability of high-resolution DM zoom-in simulations, which currently exist only for these galaxy types, since running such detailed zoom-ins on cluster galaxies is computationally too expensive. Whilst the masses of the galaxies are for all galaxies in the order of the Milky Way, they are approximately equally distributed between spiral and elliptical galaxies. In contrast, strong gravitational lenses are more commonly observed to be massive galaxies in dense environments - often the central galaxies of clusters. These systems are not represented in the necessary mass resolutions in current hydrodynamical simulations. For these purposes, the subhalos should be resolved down to masses of log10(M/M⊙) = 7.75 and below to be able to resolve small subhalos and confidently infer the SHMF. We expect that future high-resolution hydrodynamical simulations will model the relevant environments in greater detail, allowing us to extend the training set and improve generalization to observed lensing galaxies. Until then, our results should be interpreted within this limitation. Acknowledgements This work is partially supported by Schmidt Sciences, a philanthropic initiative founded by Eric and Wendy Schmidt as part of the Virtual Institute for Astrophysics (VIA). The work is in part supported by computational resources provided by Calcul Quebec and the Digital Research Alliance of Canada. A.F. acknowledges the support from the Bourse J. Armand Bombardier and UdeM's final year scholarship. T.N. is supported by the CIERA Postdoctoral Fellowship. Y.H. and L.P. acknowledge support from the Canada Research Chairs Program, the National Sciences and Engineering Council of Canada through grants RGPIN-2020-05073 and 05102. References Behroozi, P. S., Wechsler, R. H., and Wu, H.-Y. 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R., Valencia, G., Van de Water, R., Varelas, N., and Venanzoni, G. Review of Particle Physics∗. PhRvD, 98(3):030001, August 2018. Vegetti, S., Koopmans, L. V. E., Bolton, A., Treu, T., and Gavazzi, R. Detection of a dark substructure through gravitational imaging. MNRAS, 408(4):1969-1981, November 2010. Vijayan, A. P., Lovell, C. C., Wilkins, S. M., Thomas, P. A., Barnes, D. J., Irodotou, D., Kuusisto, J., and Roper, W. J. First Light And Reionization Epoch Simulations (FLARES) - II: The photometric properties of high-redshift galaxies. Monthly Notices of the Royal As7 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images tronomical Society, 501(3):3289-3308, 11 2020. ISSN 0035-8711. URL https: //doi.org/10.1093/mnras/staa3715. Wagner-Carena, S., Aalbers, J., Birrer, S., Nadler, E. O., Darragh-Ford, E., Marshall, P. J., and Wechsler, R. H. From Images to Dark Matter: End-to-end Inference of Substructure from Hundreds of Strong Gravitational Lenses. ApJ, 942(2):75, January 2023. 1538-4357/aca525. Wagner-Carena, S., Lee, J., Pennington, J., Aalbers, J., Birrer, S., and Wechsler, R. H. A Strong Gravitational Lens Is Worth a Thousand Dark Matter Halos: Inference on Small-Scale Structure Using Sequential Methods. arXiv e-prints, art. , April 2024. Weinberger, R., Springel, V., and Pakmor, R. The AREPO Public Code Release. ApJS, 248(2):32, June 2020. Wilkins, S. M., Lovell, C. C., Fairhurst, C., Feng, Y., Matteo, T. D., Croft, R., Kuusisto, J., Vijayan, A. P., and Thomas, P. Nebular-line emission during the Epoch of Reionization. Monthly Notices of the Royal Astronomical Society, 493(4):6079-6094, 03 2020. ISSN 0035-8711. URL https: //doi.org/10.1093/mnras/staa649. Zhang, G., Mishra-Sharma, S., and Dvorkin, C. Inferring subhalo effective density slopes from strong lensing observations with neural likelihood-ratio estimation. MNRAS, 517(3):4317-4326, December 2022. Zhang, G., S ̧eng ̈ul, A. C ̧ ., and Dvorkin, C. Subhalo effective density slope measurements from HST strong lensing data with neural likelihood-ratio estimation. MNRAS, 527(2):4183-4192, January 2024. stad3521. 8 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images A. Image Generation with Synthesizer Synthesizer produces galaxy images by generating spatially resolved spectral energy distributions (SEDs) and applying instrument-specific wavelength filters from the Spanish Virtual Observatory (SVO) filter service2 (Rodrigo et al., 2012; Rodrigo & Solano, 2020; Rodrigo et al., 2024). We use the Hubble Space Telescope Wide Field Camera (HST/WFC) F105W filter to simulate realistic near-infrared observations. The synthetic images are constructed by only taking stellar particles into account. We neglect the contributions of dust and gas particles, which are expected to have a relatively minor impact on the morphological features relevant to our analysis. In future steps, we will include the effects of dust and interstellar gas. As the first step of the image creation, we use an incident emission model and place the galaxies at redshift z = 0. The particles are convolved with a smoothing filter for a more optical appealing visualization. The smoothing length of the individual stellar particles is taken for each particle from the simulation data directly, which means that any artifacts of isolated particles in the imaging are originating from the simulation. The smoothing length is the co-moving radius of the sphere centered on the particle enclosing the 32 ± 1 nearest particles of this same type. To ensure image fidelity, we first generate high-resolution images and convolve them at the high resolution with a Gaussian point spread function (PSF) to approximate observational effects. The PSF has a full-width half maximum of 3 pixels. The resulting images are then downsampled to a target resolution of 0.78125 kpc/pixel. We add gaussian noise of 1023 erg s·Hz. We do not yet include instrument-specific observational noise or Poisson noise. Examples of the generated galaxy images are shown in Figure 2, alongside the corresponding galaxy-specific SHMFs. B. Poisson Noise from Samples To compute the Poisson noise of the counts of each bin of the SHMF, we use the Rockstar catalogs. We follow the prescription in Tanabashi et al. (2018); Chang & Necib (2021). We compute the confidence intervals using the inverse cumulative distribution function of the χ2 distribution: μlower = 1 2 F -1 χ2 α 2 ; 2ˆn (3) μhigher = 1 2 F -1 χ2 1 -α 2 ; 2(ˆn + 1) (4) with F -1 χ2 the inverse of the χ2 cumulative distribution and ˆn the number of counts. The confidence level is given by 100(1 -α). From these intervals, we define the asymmetric error bars and variances with: σlower,i = ˆni -μlower,i (5) σhigher,i = μhigher,i -ˆni (6) Vi = σlower,i σhigher,i (7) V ′ i = σhigher,i -σlower,i (8) We use the variances and uncertainties on the counts per bin to compute the likelihood of our fits to the data. The likelihood function to fit the SHMF (eqn 1) is defined by: ln L(ˆn | θ) = -1 2 X i ˆni -ni(θ) 2 Vi -V ′ i ˆni -ni(θ) (9) where ni(θ) is the model prediction of eqn (1), given the parameters θ = (A, mwdm). We use the likelihood function to fit the Rockstar-catalog data with an MCMC. C. Comparison of Normalizing Flow Samples with MCMC Fits In Figure 3, we show galaxy-specific SHMFs obtained by NF samples and MCMC fits along with the corresponding galaxy image. The figure is structured as Figure 2. We use the same galaxies as before, but under different projections. We do not 2https://svo2.cab.inta-csic.es/theory/fps/ 9 Predicting the Subhalo Mass Functions in Simulations from Galaxy Images expect to perfectly recreate the MCMC fits. Our method serves as a tool to get the theoretical predictions on the SHMF given a WDM theory. This does not serve as constraints on the SHMF, but is rather to be used to compare theoretical predictions with observational constraints that can be achieved with strong lensing results. This is not a standalone method to learn about dark matter in observational surveys, but it needs a comparison counterpart that constrains the SHMF. The blue and red solid contours show the samples of the MCMC fit and of the NF samples, respectively. The dark and light shaded areas are the 1σ and 2σ regions of the fits. The displayed contours show the 1σ, 2σ, and 3σ contours of the MCMC fits and NF samples of (A, mwdm), defining the SHMF. The sampled parameters of the NF represent the fits obtained directly through the likelihood and cover the full range of the parameters sampled with MCMC. Figure 3. Similar to Figure 2. The shown galaxies are the same as in Figure 3, but the orientation of the galaxy images is different. The blue contours show the samples of the MCMC fit, and the red contours the samples of the NF conditioned on MDM and the galaxy image. Overall, the sampled parameters of the NF represent the fits obtained directly through the likelihood and cover the full range of MCMC sampled parameters. 10
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2510.14764
|
Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field
Theories with Time-dependent Interaction Strength
Parameshwar R. Pasnoori1, 2
1Department of Physics, University of Maryland,
College Park, MD 20742, United States of America
2Laboratory for Physical Sciences, 8050 Greenmead Dr,
College Park, MD 20740, United States of America∗
In this paper we consider the problem of solving quantum field theories with time dependent
interaction strengths. We show that the recently formulated framework [P. R. Pasnoori, Phys. Rev.
B 112, L060409 (2025)], which is a generalization of the regular Bethe ansatz technique, provides
the exact many-body wavefunction. In this framework, the time-dependent Schrodinger equation is
reduced to a set of analytic difference equations and matrix difference equations, called the quantum
Knizhnik-Zamolodchikov (qKZ) equations. The consistency of the solution gives rise to constraints
on the time-dependent interaction strengths. For interaction strengths satisfying these constraints,
the system is integrable, and the solution to the qKZ and the analytic difference equations provides
the explicit form of the many-body wavefunction that satisfies the time-dependent Schrodinger
equation. We provide a concrete example by considering the SU(2) Gross-Neveu model with time
dependent interaction strength. Using this framework we solve the model with the most general
time-dependent interaction strength and obtain the explicit form of the wave function.
I.
INTRODUCTION
There has been a resurgence in the study of time-
dependent Hamiltonians [1–3] due to their applicability
in modern experiments ranging from circuit QED [4],
superconducting circuits [5, 6] and also cold atom ex-
periments [7]. In addition, due to high coherence times
and advanced quantum error correction techniques, it is
now possible to simulate time dependent Hamiltonians in
digital quantum computers [8]. Hence, study of exactly
solvable or integrable time-dependent Hamiltonians is of
high importance.
Quantum integrability is rooted in Bethe ansatz, which
is a powerful mathematical technique that has been very
successful in obtaining exact solutions to many-body
problems. The Bethe ansatz was originally developed in
the form of coordinate Bethe ansatz and was employed
to solve the Heisenberg chain [9, 10]. It was applied to
solve many-body systems in the continuum [11–17], var-
ious lattice models [18] and also vertex models in sta-
tistical mechanics [19, 20]. During these developments,
the algebraic structures associated with integrable sys-
tems have been found which eventually led to the for-
mulation of the algebraic Bethe ansatz [21]. This power-
ful method was successful in obtaining exact solutions to
quantum field theories and lattice models with internal
degrees of freedom [22–28]. It has also been applied to
solve many-body quantum impurity models in condensed
matter physics [29–33]. In all the integrable systems, the
scattering between particles can be reduced to pair wise
scattering processes [34]. Each scattering process is asso-
ciated with an S-matrix and these S-matrices satisfy the
quantum Yang-Baxter (QYB) algebra [13]. All models
∗pparmesh@umd.edu
FIG. 1. Figure depicts electrons (blue arrows) in a quantum
wire which forms a loop.
The electrons interact with each
other through spin exchange interaction with time dependent
strength g(t), which is uniform throughout space.
that are integrable by the method of Bethe ansatz have
constant interaction strengths and are based on the QYB
equation.
Recently, a new framework was developed [1] to
analyze Hamiltonians with time-dependent interaction
strengths.
It was applied to the paradigmatic quan-
tum impurity model, which is the Kondo model with
time-dependent interaction strength. Exact many-body
wavefunction was constructed, which is the solution to
the time-dependent Schrodinger equation. This frame-
work generalized the standard Bethe ansatz technique
and opened a new class of time-dependent Hamiltonians
that are based on QYB algebra. Prior to this work, all
known integrable Hamiltonians with time-dependent in-
teraction strengths or couplings [35, 36] were based on
classical Yang-Baxter (CYB) algebra [37–39], and were
limited to simple models involving N × N Hermitian
matrices such as multi-level Landau-Zener model [40],
two level systems interacting with quantized bosonic field
such as Tavis-Cummings model [4] and systems based on
mean field approximation such as the BCS model [41]
etc. In contrast, the method developed in [1] is appli-
arXiv:2510.14764v1 [math-ph] 16 Oct 2025
2
cable to strongly correlated many-body systems which
exhibit strong quantum fluctuations.
In this work, we shall apply this framework to solve
the time-dependent SU(2) Gross-Neveu model, which is
a quantum field theory of spin 1/2 fermions interacting
with each other through spin exchange, whose interaction
strength is time-dependent.
The Hamiltonian is given
by HGN=
R L
0 dxHGN(x), where the Hamiltonian density
HGN(x) takes the following form
HGN(x) =
X
a=↑,↓
ψ†
La(x)i∂xψLa(x) −ψ†
Ra(x)i∂xψRa(x)
−2g(t) ⃗σab · ⃗σcd
X
a,b,c,d=↑,↓
ψ†
Ra(x)ψ†
Lc(x)ψRb(x)ψLd(x),
where,
⃗σab · ⃗σcd = (σx
abσx
cd + σy
abσy
cd + σz
abσz
cd) .
(1)
Here, the fields ψL(R)a(x), a = (↑, ↓), describe left and
right moving fermions carrying spin 1/2. We set their
velocity vF = 1. The two terms in the first line describe
the right and left moving fermions respectively and the
third term describes the spin exchange interaction be-
tween a left and a right moving fermion as they cross.
The interaction strength g(t) is dependent on time and
is uniform throughout space. We apply periodic bound-
ary conditions, where the fermion fields ψL,R(x) satisfy
the following conditions
ψRa(0) = ψRa(L), ψLa(0) = ψLa(L).
(2)
Under these boundary conditions, the total number of
left moving fermions NL and right moving fermions NR
are separately conserved, where
NL =
X
a
Z L
0
dx ψ†
La(x)ψLa(x),
NR =
X
a
Z L
0
dx ψ†
Ra(x)ψRa(x).
(3)
The total number of fermions in the system N is given
by
N = NL + NR.
(4)
The Hamiltonian is also SU(2) invariant as it commutes
with the total spin operator ⃗s, where ⃗s =
R L
0 dx ⃗s(x)
with
⃗s(x) = 1
2(ψ†
L(x)⃗σψL(x) + ψ†
R(x)⃗σψR(x)).
(5)
The system also exhibits a discrete spin flip symmetry
which has the Z2 group structure, where
τ : ψR↑(x) →ψR↓(x), ψL↑(x) →ψL↓(x), τ 2 = 1. (6)
In the case of constant interaction strength, the Hamil-
tonian (1) has been solved using Bethe ansatz [22, 23].
In which case, the system exhibits a separation of spin
and charge degrees of freedom.
Moreover, due to the
fermion-fermion interaction, the system exhibits a dy-
namical generation of mass gap ∆in the spin sector,
where the excitations are called spinons. The charge sec-
tor however remains gapless and the charge excitations
are called holons.
The mass gap stabilizes quasi-long
range spin-singlet superconducting order, characterized
by
⟨Os(x)Os(0)⟩∼|x|−1/2,
Os(x) ∝ψ†
R↑(x)ψ†
L↓(x) −ψ†
R↓(x)ψ†
L↑(x).
(7)
At high energies, the system exhibits asymptotic freedom
[22]. It was show in [42, 43] that when twisted boundary
conditions are applied, which can be achieved by applying
a Zeeman field at the boundaries, the system exhibits a
symmetry protected topological (SPT) phase.
Here, we consider this model with time dependent in-
teraction strength (1), and by following the method de-
veloped in [1], we construct the exact wavefunction which
is the solution to the time-dependent Schrodinger equa-
tion. The construction of the exact wavefunction involves
several steps.
Firstly, one identifies certain conserved
quantities, which act as quantum numbers that label the
wavefunction. The wavefunction is then constructed by
ordering the particles in the configuration space, where
there exists a unique amplitude corresponding to a spe-
cific ordering of all the particles with respect to each
other. The amplitudes are related to each other through
the action of the particle-particle S-matrices, such that
all the amplitudes can be related to any one amplitude.
Both the S-matrices and the amplitudes contain a phase
part and a spin part. In order for the system to be in-
tegrable, these S-matrices should satisfy the quantum
Yang-Baxter algebra, which guarantees the consistency
of the wavefunction and imposes constraints on the time-
dependent interaction strength g(t). To determine these
amplitudes, one applies periodic boundary conditions on
the wavefunction, which gives rise to constraint equa-
tions that take the form of matrix difference equations.
The solution to the these equations provides the explicit
form of the amplitudes and hence also of the exact wave-
function. We show that the matrix difference equations
reduce to a set of analytic difference equations and quan-
tum Knizhnik-Zamolodchikov (qKZ) equations. The an-
alytic difference equations govern the phases associated
with the S-matrices whereas the qKZ equations govern
the spin part.
In the special case of constant interac-
tion strength, the procedure described above reduces to
the regular Bethe ansatz method. Firstly, the amplitudes
and the S-matrices are constants, where the phase part
of the amplitudes are simple exponential functions. The
matrix difference equations reduce to an eigenvalue equa-
tion involving a transfer matrix. In the case where the
interaction strength is time-dependent, these amplitudes
are vector valued functions that depend on the time de-
pendent interaction strength and also on the positions of
all the particles. Hence, one can consider the procedure
3
described above as a generalization of the regular Bethe
ansatz method.
When the interaction strength is constant, as men-
tioned above, the regular Bethe ansatz method gives rise
to a transfer matrix.
This can be diagonalized using
the algebraic Bethe ansatz technique [21], which yields
the eigenstates and the corresponding eigenvalues.
In
the case of time-dependent interaction strength, as de-
scribed above, the generalized Bethe ansatz method gives
rise to a set of analytic difference equations and the qKZ
equations. The solution to these equations provides the
explicit form of the amplitudes, and hence also that of
the complete wavefunction. This can be achieved by the
method of off-shell Bethe ansatz method [44, 45].
We
consider the most general interaction strength that sat-
isfies the constraint conditions imposed by integrability,
and solve the associated analytic difference equations and
the qKZ equations, and obtain the explicit form of the
exact many-body wavefunction.
The paper is organized as follows.
In section II we
present the ansatz for the wavefunction that satisfies the
time-dependent Schrodinger equation and provide the
constraint conditions on the interaction strength.
We
then apply periodic boundary conditions on the wave-
function, which gives rise to matrix difference equations.
In section (III), we work with the most general interac-
tion strength that satisfies the constraint conditions and
show that the matrix difference equations take the form
of qKZ equations. We then present the solution to these
equations, thus obtaining the explicit form of the many-
body wavefunction.
In section (IV), we conclude and
discuss future prospects.
II.
THE ANSATZ WAVEFUNCTION
As mentioned above, the number of left and right mov-
ing fermions are separately conserved (3).
Hence, one
can construct the ansatz wavefunction which consists of
NR number of right moving fermions and NL number
of left moving fermions, which we denote by |NL, NR⟩.
This wavefunction satisfies the following time-dependent
Schrodinger equation
i∂t |NL, NR⟩= HGN |NL, NR⟩,
(8)
where HGN is the Hamiltonian (1).
A.
One particle case
Let us first consider the most simplest case of one
particle N = 1. There are two possible wavefunctions
corresponding to the choices:
NR = 1, NL = 0 and
NL = 0, NR = 1, which we label by |0, 1⟩and |1, 0⟩re-
spectively. We have
|0, 1⟩=
X
a
Z L
0
dx ψ†
Ra(x)FRa(x, t) |0⟩,
(9)
|1, 0⟩=
X
a
Z L
0
dx ψ†
La(x)FLa(x, t) |0⟩.
(10)
Here, a =↑↓denotes the spin of the fermions. Using these
in the Schrodinger equation (8), we obtain the following
one particle Schrodinger equations
i(∂t + ∂x)FRa(x, t) = 0, i(∂t −∂x)FLa(x, t) = 0.
(11)
To solve the above equations, we consider the following
ansatz
FRa(x, t) = fRa(z) θ(x)θ(L −x),
FLa(x, t) = fLa(¯z) θ(x)θ(L −x).
(12)
Here, we have used the notation z = x −t, ¯z = x + t.
θ(x) is the Heaviside step function with the convention
θ(x) = 1 for x > 0, θ(x) = 0 for x < 0 and θ(0) = 1/2.
The boundary conditions (2) give rise to the following
conditions on the amplitudes
fRa(z) = fRa(z + L), fLa(¯z) = fLa(¯z + L).
(13)
One can see that the above ansatz (12) satisfies the
Schrodinger equations (11) trivially.
Let us compare this with the standard Bethe ansatz
procedure that one applies when the interaction strength
is constant g(t) = g. In this case, the functions fRa(z)
and fLa(¯z) can be expressed in terms of simple exponen-
tial functions
fRa(z) = eikzARa, fLa(z) = e−ik¯zALa,
(14)
where k is the momentum, which is also the energy since
we have set vF = 1, and ARa,ALa are amplitudes which
do not depend on z and ¯z.
B.
Two particle case and the S-matrix
Now let us consider the case of two particles N = 2.
There are three possibilities:
1) NL = 2, NR = 0 :
|2, 0⟩,
(15)
2) NL = 0, NR = 2 :
|0, 2⟩,
(16)
3) NL = 1, NR = 1 :
|1, 1⟩.
(17)
The first two sectors which correspond to both the par-
ticles being either left movers or right movers is again
trivial, as the two particles do not interact with each
other. The two particle wavefunction is just the direct
product of one particle wavefunctions discussed above.
Now let us consider the last sector where one particle is
a left mover whereas the other is a right mover. This
4
sector is non trivial since the two particles interact with
each other through the spin exchange interaction in the
Hamiltonian. We have
|1, 1⟩=
2
Y
i=1
Z L
0
dxi ψ†
Ra(x1)ψ†
Lb(x2)AF RL
ab (x1, x2, t) |0⟩.
(18)
Here the superscripts denote the chirality and the
subscripts denote the spin of the fermions.
A is
the anti-symmetrization symbol which when acting on
F RL
ab (x1, x2, t) exchanges x1 ↔x2, R ↔L and a ↔b.
Using the above expression in the Schrodinger equation
(8), one obtains the following two particle Schrodinger
equation
−i(∂t + ∂x1 −∂x2)AF RL
ab (x1, x2, t) + g(t)δ(x1 −x2)
× (⃗σac · ⃗σbd) AF RL
cd (x1, x2, t) = 0.
(19)
Since a left moving and a right moving fermion interact
with each other, the ordering of the particles is impor-
tant. We take the following ansatz for the wavefunction
F RL
ab (x1, x2, t):
F RL
ab (x1, x2, t) = θ(x1)θ(L −x1)θ(x2)θ(L −x2)×
f RL,12
ab
(z1, ¯z2)θ(x2 −x1) + f RL,21
ab
(z1, ¯z2)θ(x1 −x2)
.
(20)
The ordering of the particles is denoted by the super-
scripts. For example, f RL,12
ab
(z1, ¯z2) corresponds to the
amplitude in which the particle 1, which is a right mover,
is on the left side of particle 2 which is a left mover. Us-
ing the above ansatz (20) in the two particle Schrodinger
equation (19), one obtains the following relation between
the amplitudes
f RL,21
ab
(z1, ¯z2) = S12
ac,bd(z1, ¯z2)f RL,12
cd
(z1, ¯z2),
(21)
where S12(z1, ¯z2) is the two particle S-matrix which is
given by
S12
ac,bd(z1, ¯z2) = eiϕ(¯z2−z1) if(¯z2 −z1)I12
ac,bd + P 12
ac,bd
if(¯z2 −z1) + 1
,
f(x) =
1
2g(x/2)
1 −3(g(x/2))2
4
,
eiϕ(x) = 2ig(x/2) −1 + 3(g(x/2))2
4
ig(x/2) −
1 + 3(g(x/2))2
4
.
(22)
Here, I12
ac,bd is the identity and P 12
ac,bd is the permuta-
tion operator. The superscripts in I12
ac,bd, P 12
ac,bd and in
S12
ac,bd(z1, ¯z2) denote that they act in the spin spaces of
particles 1 and 2 which are represented by the subscripts.
We see that the interaction between the particles relates
the two amplitudes in the two particle wave function (20)
such that there exists one free amplitude. We may choose
this to be f RL,12
ab
(z1, ¯z2).
Let us now apply the periodic boundary conditions (2)
on the two particle wavefunction (20). This results in the
following relations between the amplitudes
f RL,12
ab
(z1, ¯z2) = f RL,21
ab
(z1 + L, ¯z2),
f RL,21
ab
(z1, ¯z2) = f RL,12
ab
(z1, ¯z2 + L).
(23)
Using the relations (23) in (21), we obtain
f RL,12
ab
(z1, ¯z2 + L) = S12
ac,bd(z1, ¯z2)f RL,12
cd
(z1, ¯z2).
(24)
Hence, we find that applying periodic boundary condi-
tions (2) on the two particle wavefunction (23) imposes
constraints on the free amplitude (24), which is a matrix
difference equation. One can solve this difference equa-
tion, and determine the amplitude fRL,12
ab
(z1, ¯z2). One
can then use the relation (21) to determine the other
amplitude, and hence the exact form of the two particle
wavefunction (20).
C.
Three particle case and the Yang-Baxter
algebra
Let us now consider the case of three particles N = 3.
There exist four possibilities:
1) NL = 3, NR = 0 :
|3, 0⟩,
2) NL = 0, NR = 3 :
|0, 3⟩,
3) NL = 2, NR = 1 :
|1, 2⟩,
4) NL = 1, NR = 2 :
|2, 1⟩.
(25)
Just as in the case of two particles, the first two sectors
are trivial where the three particle wavefunction is a di-
rect product of one particle wavefunctions.
The third
and the fourth sectors are non-trivial, which we discuss
below. Let us first consider the third sector correspond-
ing to two left moving particles and one right moving
particle. We have
|1, 2⟩=
3
Y
i=1
Z L
0
dxi ψ†
Ra(x1)ψ†
Lb(x2)ψ†
Lc(x3)
×AF RLL
abc (x1, x2, x3, t) |0⟩.
(26)
Using this in the Schrodinger equation, we obtain the
following three particle Schrodinger equation
−i(∂t + ∂x1 −∂x2 −∂x3)AF RLL
abc (x1, x2, x3, t)
+g(t) (δ(x1 −x2)⃗σal · ⃗σbm Icn + δ(x1 −x3)Ibm ⃗σal · ⃗σcn)
×AF RLL
lmn (x1, x2, x3, t) = 0.
(27)
Due to the interactions in the system, just as in the
two particle case, the ordering of the left and the right
moving particles is important. In addition, we find that
the ordering of two left moving particles with respect to
each other is also important to have a consistent wave-
function. The ansatz for the three particle wavefunction
AF RLL
abc (x1, x2, x3, t) takes the following form
5
F RLL
abc (x1, x2, x3, t) = θ(x1)θ(L −x1)θ(x2)θ(L −x2)θ(x3)θ(L −x3)×
f RLL,123
abc
(z1, ¯z2, ¯z3)θ(x2 −x1)θ(x3 −x2) + f RLL,132
abc
(z1, ¯z2, ¯z3)θ(x2 −x3)θ(x3 −x1)+
f RLL,213
abc
(z1, ¯z2, ¯z3)θ(x1 −x2)θ(x3 −x1) + f RLL,312
abc
(z1, ¯z2, ¯z3)θ(x2 −x1)θ(x1 −x3)+
f RLL,231
abc
(z1, ¯z2, ¯z3)θ(x1 −x3)θ(x3 −x2) + f RLL,321
abc
(z1, ¯z2, ¯z3)θ(x2 −x3)θ(x1 −x2)
.
(28)
Using this in the three particle Schrodinger equation (27),
we obtain the following relations (from here on we sup-
press the spin indices, unless necessary)
f RLL,213(z1, ¯z2, ¯z3) = S12(z1, ¯z2)f RLL,123(z1, ¯z2, ¯z3),
f RLL,231(z1, ¯z2, ¯z3) = S13(z1, ¯z3)f RLL,213(z1, ¯z2, ¯z3),
f RLL,312(z1, ¯z2, ¯z3) = S13(z1, ¯z3)f RLL,132(z1, ¯z2, ¯z3),
f RLL,321(z1, ¯z2, ¯z3) = S12(z1, ¯z2)f RLL,312(z1, ¯z2, ¯z3).
(29)
Here, the S-matrices S12(z1, ¯z2) and S13(z1, ¯z3) take the
same form as in the two particle case (22).
Note that the relation between the amplitudes which
differ by the ordering of the particles with the same
chirality is not fixed by the Hamiltonian.
In the cur-
rent case this corresponds to the relation between the
amplitudes f RLL,123(z1, ¯z2, ¯z3) and f RLL,132(z1, ¯z2, ¯z3)
and also between the amplitudes f RLL,231(z1, ¯z2, ¯z3) and
f RLL,321(z1, ¯z2, ¯z3).
This occurs due to the linear dis-
persion, and also occurs in the case when the interaction
strength is constant. Integrability requires one to choose
a specific relation between such amplitudes for the wave-
function to be consistent, which we discuss below. Note
that up until now, the interaction strength g(t) is an ar-
bitrary function of time. Choosing a consistent relation
between the above mentioned amplitudes imposes con-
straints on the interaction strength g(t).
The most general form of the interaction strength g(t)
that gives rise to a consistent solution is such that, f(t)
(22) is a linear function
f(t) = αt + β, α, β ∈C (constants),
(30)
which corresponds to the interaction strength taking the
following form
g(t) = 2
3
−2(2αt + β) ±
4 (2αt + β)2 + 3
1/2
.
(31)
Consistency requires that the S-matrix between the am-
plitudes is given by
f RLL,132(z1, ¯z2, ¯z3) = S′23(¯z2, ¯z3)f RLL,123(z1, ¯z2, ¯z3)
f RLL,321(z1, ¯z2, ¯z3) = S′23(¯z2, ¯z3)f RLL,231(z1, ¯z2, ¯z3),
(32)
where S23(¯z2, ¯z3) takes a similar form as that of the two
particle S-matrix (22), but now with f(t) being a linear
function:
S′23
ac,bd(¯z2, ¯z3) =
iα(¯z3 −¯z2)I32
ac,bd + P 32
ac,bd
iα(¯z3 −¯z2) + 1
.
(33)
Note that the case where α or β are complex gives rise to
a non-Hermitian Hamiltonian, which is also interesting.
The S-matrices between particles of different chiralities
S12(z1, ¯z2), S13(z1, ¯z3) and that between the particles of
same chirality S32(¯z2, ¯z3) satisfy the Yang-Baxter algebra
S12(z1, ¯z2)S13(z1, ¯z3)S′23(¯z2, ¯z3) =
S′23(¯z2, ¯z3)S13(z1, ¯z3)S12(z1, ¯z2),
(34)
which guarantees the consistency of the three particle
wavefunction (28). Note that the linearity of f(t) is cru-
cial for the S-matrices to satisfy the Yang-Baxter algebra.
Using the relations (29) and (32), one can express all
the amplitudes in the three particle wavefunction (28) in
terms of one amplitude of our choosing. We may choose
this to be f RLL,123
abc
(z1, ¯z2, ¯z3). Hence, we find that, just
as in the two particle case, there exists one free ampli-
tude. To find the exact form of the wavefunction, we have
to impose periodic boundary conditions (2) on the three
particle wave-function (28). This results in the following
relations between the amplitudes
f RLL,123(z1, ¯z2, ¯z3) = f RLL,231(z1 + L, ¯z2, ¯z3),
f RLL,213(z1, ¯z2, ¯z3) = f RLL,132(z1, ¯z2 + L, ¯z3),
f RLL,312(z1, ¯z2, ¯z3) = f RLL,123(z1, ¯z2, ¯z3 + L),
f RLL,231(z1, ¯z2, ¯z3) = f RLL,312(z1, ¯z2 + L, ¯z3),
f RLL,321(z1, ¯z2, ¯z3) = f RLL,213(z1, ¯z2, ¯z3 + L),
f RLL,132(z1, ¯z2, ¯z3) = f RLL,321(z1 + L, ¯z2, ¯z3).
(35)
Using
the
relations
(29),
(32)
and
(35),
we
ob-
tain the following relation constraining the amplitude
f RLL,123
abc
(z1, ¯z2, ¯z3):
f RLL,123(z1 −L, ¯z2, ¯z3) = Z1(z1, ¯z2, ¯z3)f RLL,123(z1, ¯z2, ¯z3),
(36)
where the transport operator Z1(z1, ¯z2, ¯z3) transports the
particle 1 across the entire system once.
It takes the
following form
Z1(z1, ¯z2, ¯z3) = S13(z1, ¯z3)S12(z1, ¯z2).
(37)
6
Similarly, there exist transport operators Z2(z1, ¯z2, ¯z3)
and Z3(z1, ¯z2, ¯z3) which transport particles 2 and 3 re-
spectively around the system
f RLL,123(z1, ¯z2 −L, ¯z3) = Z1(z1, ¯z2, ¯z3)f RLL,123(z2, ¯z2, ¯z3),
f RLL,123(z1, ¯z2, ¯z3 −L) = Z1(z1, ¯z2, ¯z3)f RLL,123(z2, ¯z2, ¯z3).
(38)
These transport operators take the following form
Z2(z1, ¯z2, ¯z3) = S21(¯z2, z1 + L, )S′23(¯z2, ¯z3),
Z3(z1, ¯z2, ¯z3) = S′32(¯z3, ¯z2 + L)S31(¯z3, z1 + L).
(39)
The equations (36) and (38) form a system of matrix
difference equations.
We shall see later in the next
section that they are related to quantum Knizhnik-
Zamolodchikov equations. Using the Yang-Baxter alge-
bra (34), one can show that these transport operators
satisfy the following relations
Z1(z1, ¯z2 −L, ¯z3)Z2(z1, ¯z2, ¯z3) = Z2(z1 −L, ¯z2, ¯z3)Z1(z1, ¯z2, ¯z3),
Z1(z1, ¯z2, ¯z3 −L)Z3(z1, ¯z2, ¯z3) = Z3(z1 −L, ¯z2, ¯z3)Z1(z1, ¯z2, ¯z3),
Z2(z1, ¯z2, ¯z3 −L)Z3(z1, ¯z2, ¯z3) = Z3(z1, ¯z2 −L, ¯z3)Z2(z1, ¯z2, ¯z3).
(40)
Given a certain interaction strength g(t), which sat-
isfies (30), one should solve the set of matrix differ-
ence equations described above to obtain the amplitude
f RLL,123
abc
(z1, ¯z2, ¯z3). One can then solve for the rest of the
amplitudes in the three particle wavefunction (28) using
the relations (32), (29) and obtain the explicit form of
the wavefunction.
Similarly, in the sector corresponding to two right mov-
ing particles and one left moving particle (25), one can
construct the wavefunction exactly as in the above case.
One finds that the constraints on the interaction strength
are exactly the same as (30), and the associated matrix
difference equations take the same form as above. We
shall postpone the discussion of the solution of these ma-
trix difference equations to the later sections and consider
the most general case of N particles.
D.
N particle case and the quantum
Knizhnik-Zamolodchikov equations
Let us now consider the most general case of N
particles.
Since the number of left movers NL and
right movers NR are separately conserved under peri-
odic boundary conditions, we have N +1 possible sectors
corresponding to different values of NL and NR, where
NL+NR = N. Below, we shall construct the exact many-
body wavefunction for general values of NL and NR. We
have
|NL, NR⟩=
N
Y
j=NL+1
NL
Y
k=1
Z L
0
dxj
Z L
0
dxk ψ†
Rσj(xj)ψ†
Lσk(xk)
×AF 1...N,χ1...χN
σ1...σN
(x1, ..., xN, t) |0⟩.
(41)
Here, σ1...σN ≡{σi} denote the spin and χ1...χN ≡
{χi} denote the chiralities of the particles. Using this in
the Schrodinger equation (8), we obtain the following N
particle Schrodinger equation
−i(∂t +
N
X
j=NL+1
∂xj−
NL
X
k=1
∂xk)AF 1...N,{χi}
σ1...σN
(x1, ..., xN, t)+g(t)×
j=N
k=NL
X
j=NL+1
k=1
δ(xj −xk)⃗σσjσ′
j·⃗σσkσ′
kAF 1...N,{χi}
σ1..σ′
jσ′
k..σN(x1, ..., xN, t)=0.
(42)
From
here
on
we
use
the
convention
that
F 1...N,{χi}
σ1...σN
(x1, ..., xN) is a vector in the spin space
of the particles, as opposed to an amplitude as rep-
resented in (41).
That is F 1...N,{χi}
σ1...σN
(x1, ..., xN)
≡
F 1...N,{χi}
σ1...σN
(x1, ..., xN) |{σi}⟩.
The
ansatz
for
F 1...N,{χi}
σ1...σN
(x1, ..., xN) takes the following form
F 1..N,{χi}
σ1...σN
(x1, .., xN, t)=
X
Q
θ({xQ(j)})f Q,{χi}
σ1...σN (¯z1, .., zN).
(43)
In this wavefunction, without losing generality, we have
chosen the particles i = 1, ..., NL to be left movers, and
i = NL + 1, ..., N to be right movers. In the above ex-
pression, Q denotes a permutation of the position order-
ings of particles and θ({xQ(j)}) is the Heaviside func-
tion that vanishes unless xQ(1) ≤· · · ≤xQ(N).
Here
f Q
σ1...σN (¯z1, ..., zN) is the amplitude corresponding to the
ordering of the particles denoted by Q. The amplitudes
that differ by the ordering of the particles with different
7
chiralities are related by the S-matrix (22), just as in the
two particle case
f ...kj...,{χi}(¯z1, ..., zN) = Sjk(zj, ¯zk)f ...jk...,{χi}(¯z1, ..., zN).
(44)
Here χj = +, χk = −and “...” in the first superscript
on both side of the above equation corresponds to any
specific ordering of the rest of the particles. In addition
to this, just as in the three particle case, the amplitudes
that differ by the ordering of the particles with the same
chirality are related by an S-matrix. In the case of two
left moving particles, we have
f ...kj...,{χi}(¯z1, ..., zN) = S′jk(¯zj, ¯zk)f ...jk...,{χi}(¯z1, ..., zN),
(45)
where χj,k = −, and in the case of two right moving
particles, we have
f ...kj...,{χi}(¯z1, ..., zN) = S′jk(zj, zk)f ...jk...,{χi}(¯z1, ..., zN),
(46)
where χj,k = +.
Just as before, “...” in the first su-
perscript on both side of the above two equations corre-
sponds to any specific ordering of the rest of the particles.
These S-matrices satisfy the Yang-Baxter algebra. For
one right moving particle i and two left moving particles
j and k, we have
Sij(zi, ¯zj)Sik(zi, ¯zk)S′jk(¯zj, ¯zk) =
S′jk(¯zj, ¯zk)Sik(zi, ¯zk)Sij(zi, ¯zj).
(47)
Similarly, for two right moving particles i and j and one
left moving particle k, we have
Sjk(zj, ¯zk)Sik(zi, ¯zk)S′ij(zi, zj) =
(48)
S′ij(zi, zj)Sik(zi, ¯zk)Sjk(zj, ¯zk).
(49)
In addition to this, the S-matrices corresponding to the
exchange of the particles with the same chirality also sat-
isfy the Yang-Baxter algebra.
For three right moving
particles, we have
S′ij(zi, zj)S′ik(zi, zk)S′jk(zj, zk) =
S′jk(zj, zk)S′ik(zi, zk)S′ij(zi, zj).
(50)
Similar expression exists for three left moving particles,
which can be obtained by applying the transformation
zi,j,k →¯zi,j,k to the above equation (50).
Using the
relations (44) and(45), one can express all the ampli-
tudes in the N particle wavefunction (43) in terms of
one amplitude of our choosing. We may choose this to
be f N...1,{χj}
σ1...σN
(¯z1, ..., zN). To obtain the explicit form of
the wavefunction, this free amplitude needs to be de-
termined, which can be achieved by applying periodic
boundary conditions (2) on the N particle wavefunction
(43). Applying periodic boundary conditions yields the
following relation
f j...,{χi}(¯z1, .., zj, .., zN) = f ...j,{χi}(¯z1, .., zj + L, .., zN).
(51)
Here j is a right moving particle. Similar expression ex-
ists for a left moving particle, which can be obtained
by applying the transformation zj →¯zj to the above
equation (51). In the above equation, “...” in the first
superscript corresponds to any particular ordering of the
rest of the particles, which is same on both sides of the
equation. Using the relations (44), (45), (46) and (51)
we obtain the following constraint equations on the am-
plitude
f N...1,{χi}(¯z1, .., zj −L, .., zN) = Zj(¯z1, ..., zN)
f N...1,{χi}(¯z1, .., zj, .., zN).
(52)
Note that here j is considered to be a right moving parti-
cle without loss of generality. Here the transport operator
Zj(z1, ..., ¯zN) transports the particle j around the system
once and takes the following form
Zj(z1, ..., ¯zN) = S′jj+1(zj, zj+1 + L)...S′jN(zj, zN + L)
Sj1(zj, ¯z1)...SjNL(zj, ¯zNL)S′jNL+1(zj, zNL+1)
...S′jj−1(zj, zj−1).
(53)
The transport operators satisfy the following relations
Zj(¯z1, ..., zk −L, ..., zN)Zk(¯z1, ..., zN) =
Zk(¯z1, ..., zj −L, ..., zN)Zj(¯z1, ..., zN).
(54)
In the above equation, we have chosen j and k to be right
moving particles. In the case of left moving particles, we
simply need to apply the transformation zj/k →¯zj/k in
the above equation. The constraint equations (52) are
matrix difference equations, which need to be solved to
obtain the amplitude f N...1,{χj}
σ1...σN
(¯z1, ..., zN).
Once this
amplitude is obtained, as mentioned above, one can use
the relations (44) and (45) to obtain the rest of the am-
plitudes, and thus the explicit form of the N particle
wavefunction (43).
E.
The case of constant interaction strength and
the regular Bethe ansatz technique
In the previous subsections we have constructed
the ansatz wavefunction for time-dependent interaction
strength g(t). We have seen that all the amplitudes in
the N-particle wavefunction can be expressed in terms
of one amplitude of our choosing. By applying periodic
boundary conditions, we obtained constraint equations
on this amplitude, which take the form of matrix dif-
ference equations (52). In this subsection we shall show
that in the special case where the interaction strength is
constant, this procedure reduces to the standard Bethe
ansatz technique, and thereby demonstrating that the
above described procedure is a generalization of the reg-
ular Bethe ansatz technique.
8
In the case of constant interaction strength g(t) →g,
as mentioned above (14), the amplitudes in the wave-
function can be expressed in terms of simple exponential
functions. For the N-particle wavefunction (43), we have
f Q,{χi}
σ1...σN (z1, .., ¯zN) =
N
Y
j=NL+1
eikjzj
NL
Y
l=1
e−ikjzjAQ,{χi}
σ1...σN ,
(55)
where kj are the momenta, and AQ,{χi}
σ1...σN are amplitudes
corresponding to the ordering of the particles labeled by
Q, and they do not depend on time t and positions of the
particles xj. With this, the N-particle Schrodinger equa-
tion turns into time-independent Schrodinger equation
which is given by
−E −i
N
X
j=NL+1
∂xj + i
NL
X
k=1
∂xk
AF 1...N,{χi}
σ1...σN
(x1, ..., xN) + g×
j=N
k=NL
X
j=NL+1
k=1
δ(xj −xk)⃗σσjσ′
j·⃗σσkσ′
kAF 1...N,{χi}
σ1..σ′
jσ′
k..σN(x1, ..., xN)=0,
(56)
where the energy
E =
N
X
j=NL+1
kj −
NL
X
l=1
kl.
(57)
The amplitudes which differ by the ordering of the parti-
cles with opposite chiralities are related by the S-matrix
A...kj...,{χi} = SjkA...jk...,{χi}
(58)
where χj = +, χk = −and “...” in the first superscript
on both side of the above equation corresponds to any
specific ordering of the rest of the particles.
The S-
matrix is a constant which only depends on the inter-
action strength g and is given by
S12
ac,bd = eiϕ ifI12
ac,bd + P 12
ac,bd
if + 1
,
f = 1
2g
1 −3g2
4
, eiϕ = 2ig −1 + 3g2
4
ig −
1 + 3g2
4
.
(59)
The S-matrices which relate the amplitudes which cor-
respond to different ordering of the particles with the
same chirality are also related by an S-matrix which takes
the simple form of the permutation operator
A...kj...,{χi} = P jkA...jk...,{χi},
(60)
where χj,k = + or χj,k = −. Just as before, “...” in the
first superscript on both side of the above two equations
corresponds to any specific ordering of the rest of the
particles. The matrix difference equation (52), which is
a constraint equation on the amplitudes, now takes the
form of an eigenvalue equation [22]
eikjLAN...1,{χi} = ZjAN...1,{χi},
(61)
where the transport operator Zj takes the form of the
transfer matrix
Zj = P jj+1 . . . P jNRSjNR+1 . . . SjNP j1 . . . P jj−1. (62)
The relations (54) turn into simple commutation rela-
tions
[Zi, Zj] = 0,
(63)
which are of fundamental importance in the regular
Bethe ansatz method, as they are necessary conditions
for the system to be integrable.
To solve for the am-
plitude AN...1,{χi}
σ1...σN
, one diagonalizes the transfer matrix
Zj.
This can be achieved by the standard algebraic
Bethe ansatz technique.
One obtains the Bethe equa-
tions, whose solutions provide the eigenstates and the
corresponding eigenvalues (57).
III.
QUANTUM
KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
AND THE EXACT WAVEFUNCTION
In the previous section we have constructed a solu-
tion to the N-particle Schrodinger equation (42).
The
wavefunction consists of amplitudes which correspond
to different orderings of the particles with respect to
each other. These amplitudes are related to each other
through the S-matrices (44), (45) and (46) such that all
the amplitudes in the wavefunction can be expressed in
terms one amplitude of our choosing. By applying peri-
odic boundary conditions (2), we obtain constraint equa-
tions on this amplitude (52), which take the form of ma-
trix difference equations. In this section we solve these
matrix difference equations and obtain the exact solution
of the amplitude, and thereby obtain the explicit form of
the N-particle wavefunction.
The
matrix
difference
equation
(52)
contains
S-
matrices that relate amplitudes corresponding to differ-
ent ordering of particles with opposite chiralities and also
the particles with the same chiralities. Notice that the
S-matrices between particles of opposite chiralities (22)
have a phase part. In order to solve the equations (52),
we need to separate this phase part from the matrix part
which acts in the spin spaces of the particles. To achieve
this, we apply the following transformation on the am-
plitudes of the N-particle wavefunction (43)
f Q,{χi}
σ1...σN (¯z1, .., zN) = AQ,{χi}
σ1...σN (¯z1, .., zN)
N
Y
j=NL+1
NL
Y
l=1
h(¯zl −zj).
(64)
9
Recall that the interaction strength g(t) should sat-
isfy the constraints (30) for the solution to be consistent,
where the function f(x) is a linear function. Using this,
we can express the S-matrices (22), (33) in terms of the
XXX R-matrix Rij(λ), where
Rij(λ) = iλIij + (1/α)P ij
iλ + 1
,
Sij(zi, ¯zj) = eiϕ(¯zj−zi)Rij(¯zj −zi + β/α),
S′kl(zk, zl) = Rkl(zk −zl).
(65)
Here just as before Iij is the identity matrix and P ij is
the permutation operator which acts in the spin spaces
of particles i and j. Using (64) and (65) in the matrix
difference equations (52), we see that they can be sepa-
rated into two sets of equations. The first one concerns
only the phase part and takes the form of the analytic
difference equation
h(¯zm −zj + L) = eiϕ(¯zm−zj)h(¯zm −zj), m = 1, ..., NL.
(66)
These equations have been well studied in the literature
for different classes of functions ϕ(x) [46]. The second set
of equations concern the spin part, which take the form of
quantum Knizhnik-Zamolodchikov equations. They take
the following form
AN...1,{χi}(¯z1, .., zj −L, .., zN) = Z′
j(¯z1, ...., zN)
A1...N,{χi}(¯z1, .., zj, .., zN),
(67)
where the transport operator Z′
j(¯z1, ...., zN) is given by
Z′
j(¯z1, ...., zN) = Rjj+1(zj+1 + L −zj)...
RjN(zN + L −zj)Rj1(¯z1 −zj)...RjNL(¯zNL −zj)
RjNL+1(zNL+1 −zj)...Rjj−1(zj−1 −zj).
(68)
In the context of qKZ equations, L in the above equa-
tion is called the step. The qKZ equations first appeared
in [47] as the fundamental equations for form factors
in the sine-Gordon model, and were later derived from
representation theory of quantum affine algebras [48].
They have been well studied in the literature [44, 49–
51] and the off-shell Bethe ansatz method to solve them
has been developed in [44, 45].
The solution to these
equations provides us with the explicit form of the am-
plitude AN...1,{χi}(¯z1, . . . , zN), which can then be used
to obtain the rest of the amplitudes and hence also the
explicit form of the complete wavefunction (41).
Below, we describe the solution to the qKZ equations
(67), that is obtained following [45].
As mentioned in
the beginning, the system conserves the total number
of left and right moving fermions separately. We have
used these conserved quantities to construct the wave-
function (41). In addition to this, our system has the
global SU(2) symmetry, and the system conserves the
total z-component of the spin. This allows us to con-
struct a state with a specific value of Sz.
Consider a
state where the spins of all the particles are pointing in
the positive z-direction. Let us denote this state by |Ω⟩
[52]
|Ω⟩= |↑⟩1 ⊗... ⊗|↑⟩N .
(69)
This state is trivially an eigenstate of the Hamiltonian (1)
and uninteresting. Now consider a state with M number
of spins pointing in the negative z-direction and N −M
number of spins pointing in the positive z-direction . The
total z-component of the spin corresponding to such as
state is
Sz = N
2 −M.
(70)
There exists an operator B({zi}, uα), which when acting
on the state |Ω⟩, flips one spin. Here uα is the ‘rapid-
ity’ associated with the spin flip [53]. A general state
with M flipped spins is then constructed by acting on
the state |Ω⟩by M number of B({wi}, uα) operators,
where uα, α = 1, ...M are all distinct.
In the case of
constant interaction strength, as mentioned before, one
constructs eigenstates of the transfer matrix (61), (62), in
which case, the set of rapidities uα, α = 1, ..., M are called
the Bethe roots which are solutions to certain constraint
equations called the Bethe equations. In the current case,
instead of the the eigenstates, we need to construct so-
lutions to the matrix difference equations which take the
form of the qKZ equations (67), (68). In this solution,
unlike the eigenstates of the transfer matrix, the rapidi-
ties are not constrained, but are rather summed over.
Following [45], we obtain
AN...1,{χi}
σ1...σN
( ¯w1, . . . , wN) =
X
uα
M
Y
α=1
BN...1({wi}, uα)
×
N
Y
i=NL+1
NL
Y
j=1
M
Y
β=1
Γ(wi −uβ)
Γ(wi −uβ −iη)
Γ( ¯wj −uβ)
Γ( ¯wj −uβ −iη)
×
Y
1≤i<j≤M
(ui −uj) Γ(ui −uj −iη)
Γ(ui −uj + iη + 1)
|Ω⟩,
(71)
where the summation is over the integers lα, while the
parameters euα, α = 1, . . . , M, are arbitrary constants
uα = euα −lα,
lα ∈Z.
(72)
This infinite sum is called a ‘Jackson type integral’. In
the above expression (71), Γ(x) is the usual Gamma func-
tion and η = 1/(αL). The parameters wi, ¯wi and η are
related to zi, ¯zi, α and β through the following relations
wi = zi
L ,
i = NL+1, ..., N;
¯wi = ¯zi
L + β
αL, i = 1, ..., NL.
(73)
10
Note that, in addition to the qKZ equations (67), we need
to solve the analytic difference equation (66) to obtain
the function h(x). The solution is complicated, but it
simplifies in the limit α, β ≫1, where the interaction
strength g(t) (31) takes the form
g(t) =
1
4(αt + β/2).
(74)
The function h(x) is given by
h(x) = e2iπnx/L Γ((x + β/α −i/α)/L)
Γ((x + β/α −i/2α)/L),
(75)
where n is an integer. Instead of the exponential func-
tion in the above expression, one may choose any func-
tion whose period is commensurate with L. The solu-
tion to (66) is also simplified in the opposite limit where
α, β ≪1. In which case, g(t) is linearly dependent on
time. Having obtained the explicit form of the ampli-
tude f N...1,{χi}
σ1...σN
( ¯w1, . . . , wN), the rest of the amplitudes
and hence the explicit form of the wavefunction (41) can
be obtained from it by the action of S-matrices (44), (45)
and (46).
IV.
DISCUSSION
In this work we have considered the time-dependent
SU(2) Gross-Neveu model. In this quantum field the-
ory, spin 1/2 fermions interact with each other through
spin exchange interaction strength that varies in time.
Using the recently formulated framework [1], which gen-
eralizes the standard Bethe ansatz technique, we have
constructed an exact solution to the time-dependent
Schrodinger equation.
We considered the system with
periodic boundary conditions. This results in the conser-
vation of the number of left and right moving fermions
separately, thus allowing us to use them to label the
wavefunction. The wavefunction consists of several am-
plitudes, where each amplitude corresponds to a certain
ordering of particles with respect to each other.
Any
amplitude in the wavefunction can be related to one am-
plitude of our choosing through the action of particle-
particle S-matrices. The amplitudes and the S-matrices
contain a phase part and a spin part. The consistency
of the wavefunction requires that these S-matrices sat-
isfy Yang-Baxter algebra, which is a necessary condition
for the integrability of the system.
This imposes con-
straints on the time-dependent interaction strength. In
the regular Bethe ansatz approach, which is applicable
to Hamiltonians with constant interaction strengths, the
amplitudes and the S-matrices are constants. The phase
part of the amplitudes take the form of simple exponen-
tial functions. In contrast, in our case of time-dependent
interaction strength, these amplitudes are vector valued
functions which depend on the time depend interaction
strength and also on the positions of all the particles.
The chosen amplitude is then determined by applying
periodic boundary conditions, which gives rise to ma-
trix difference equations which constrain the amplitude.
We showed that these matrix difference equations reduce
to a set of analytic difference equations which govern
the phase part, and quantum Knizhnik-Zamolodchikov
(qKZ) equations which govern the spin part.
In the case of the constant interaction strength, these
matrix difference equations reduce to an eigenvalue equa-
tion involving a transfer matrix, which is then diagonal-
ized using the standard algebraic Bethe ansatz technique.
Hence, one can consider the regular Bethe ansatz method
as a special case of the general Bethe ansatz method used
in this work. In our case of time-dependent interaction
strength, as mentioned above, the general Bethe ansatz
method gives rise to qKZ equations and analytic differ-
ence equations.
The qKZ equations were solved using
the off-shell algebraic Bethe ansatz technique. Using the
solution to the qKZ equations, along with the solution
to the analytic difference equations, we obtained the ex-
plicit form of the amplitude. The rest of the amplitudes
can then be straightforwardly determined by the action
of the particle-particle S-matrices on this amplitude, and
thereby one can obtain the explicit form of the complete
many-body wavefunction.
In addition to the SU(2) symmetric case considered in
this work, one may consider the case where the SU(2)
symmetry is broken down to U(1) symmetry. In which
case one obtains the U(1) Thirring model with time-
dependent interaction strength.
This case is expected
to be more interesting since it contains two coupling
strengths which vary in time.
In this case, instead of
the qKZ equations corresponding to the XXX-R matrix
that we obtained in this work, one obtains the qKZ equa-
tions corresponding to the XXZ R-matrix [49, 54].
In
addition, one may consider different boundary conditions
as opposed to simple periodic boundary conditions con-
sidered in this work. Under these boundary conditions,
in addition to the bulk interaction strengths, there exist
boundary coupling strengths which can vary in time. In
the simple case where the bulk interaction strengths and
the boundary coupling strengths are constant, the models
described above are shown to exhibit symmetry protected
topological (SPT) phases [42, 43, 55, 56]. Hence, the case
where the bulk interaction strengths and the boundary
coupling strengths vary in time is naturally very inter-
esting, as they may give rise to new type of dynamically
generated SPT phases, which is the focus of our future
work [57].
ACKNOWLEDGMENTS
We
acknowledge
discussions
with
Natan
Andrei,
Patrick Azaria, Paul Fendley and David Huse.
I spe-
cially thank Patrick Azaria for carefully reviewing the
manuscript.
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|
Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-dependent Interaction Strength Parameshwar R. Pasnoori1, 2 1 20742, United States of America 2Laboratory for Physical Sciences, 8050 Greenmead Dr, College Park, MD 20740, United States of America∗ In this paper we consider the problem of solving quantum field theories with time dependent interaction strengths. We show that the recently formulated framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], which is a generalization of the regular Bethe ansatz technique, provides the exact many-body wavefunction. In this framework, the time-dependent Schrodinger equation is reduced to a set of analytic difference equations and matrix difference equations, called the quantum Knizhnik-Zamolodchikov (qKZ) equations. The consistency of the solution gives rise to constraints on the time-dependent interaction strengths. For interaction strengths satisfying these constraints, the system is integrable, and the solution to the qKZ and the analytic difference equations provides the explicit form of the many-body wavefunction that satisfies the time-dependent Schrodinger equation. We provide a concrete example by considering the SU(2) Gross-Neveu model with time dependent interaction strength. Using this framework we solve the model with the most general time-dependent interaction strength and obtain the explicit form of the wave function. I. INTRODUCTION There has been a resurgence in the study of timedependent Hamiltonians [1-3] due to their applicability in modern experiments ranging from circuit QED [4], superconducting circuits [5, 6] and also cold atom experiments [7]. In addition, due to high coherence times and advanced quantum error correction techniques, it is now possible to simulate time dependent Hamiltonians in digital quantum computers [8]. Hence, study of exactly solvable or integrable time-dependent Hamiltonians is of high importance. Quantum integrability is rooted in Bethe ansatz, which is a powerful mathematical technique that has been very successful in obtaining exact solutions to many-body problems. The Bethe ansatz was originally developed in the form of coordinate Bethe ansatz and was employed to solve the Heisenberg chain [9, 10]. It was applied to solve many-body systems in the continuum [11-17], various lattice models [18] and also vertex models in statistical mechanics [19, 20]. During these developments, the algebraic structures associated with integrable systems have been found which eventually led to the formulation of the algebraic Bethe ansatz [21]. This powerful method was successful in obtaining exact solutions to quantum field theories and lattice models with internal degrees of freedom [22-28]. It has also been applied to solve many-body quantum impurity models in condensed matter physics [29-33]. In all the integrable systems, the scattering between particles can be reduced to pair wise scattering processes [34]. Each scattering process is associated with an S-matrix and these S-matrices satisfy the quantum Yang-Baxter (QYB) algebra [13]. All models ∗ FIG. 1. Figure depicts electrons (blue arrows) in a quantum wire which forms a loop. The electrons interact with each other through spin exchange interaction with time dependent strength g(t), which is uniform throughout space. that are integrable by the method of Bethe ansatz have constant interaction strengths and are based on the QYB equation. Recently, a new framework was developed [1] to analyze Hamiltonians with time-dependent interaction strengths. It was applied to the paradigmatic quantum impurity model, which is the Kondo model with time-dependent interaction strength. Exact many-body wavefunction was constructed, which is the solution to the time-dependent Schrodinger equation. This framework generalized the standard Bethe ansatz technique and opened a new class of time-dependent Hamiltonians that are based on QYB algebra. Prior to this work, all known integrable Hamiltonians with time-dependent interaction strengths or couplings [35, 36] were based on classical Yang-Baxter (CYB) algebra [37-39], and were limited to simple models involving N × N Hermitian matrices such as multi-level Landau-Zener model [40], two level systems interacting with quantized bosonic field such as Tavis-Cummings model [4] and systems based on mean field approximation such as the BCS model [41] etc. In contrast, the method developed in [1] is appli16 Oct 2025 2 cable to strongly correlated many-body systems which exhibit strong quantum fluctuations. In this work, we shall apply this framework to solve the time-dependent SU(2) Gross-Neveu model, which is a quantum field theory of spin 1/2 fermions interacting with each other through spin exchange, whose interaction strength is time-dependent. The Hamiltonian is given by HGN= R L 0 dxHGN(x), where the Hamiltonian density HGN(x) takes the following form HGN(x) = X a=↑,↓ ψ† La(x)i∂xψLa(x) -ψ† Ra(x)i∂xψRa(x) -2g(t) ⃗σab · ⃗σcd X a,b,c,d=↑,↓ ψ† Ra(x)ψ† Lc(x)ψRb(x)ψLd(x), where, ⃗σab · ⃗σcd = (σx abσx cd + σy abσy cd + σz abσz cd) . (1) Here, the fields ψL(R)a(x), a = (↑, ↓), describe left and right moving fermions carrying spin 1/2. We set their velocity vF = 1. The two terms in the first line describe the right and left moving fermions respectively and the third term describes the spin exchange interaction between a left and a right moving fermion as they cross. The interaction strength g(t) is dependent on time and is uniform throughout space. We apply periodic boundary conditions, where the fermion fields ψL,R(x) satisfy the following conditions ψRa(0) = ψRa(L), ψLa(0) = ψLa(L). (2) Under these boundary conditions, the total number of left moving fermions NL and right moving fermions NR are separately conserved, where NL = X a Z L 0 dx ψ† La(x)ψLa(x), NR = X a Z L 0 dx ψ† Ra(x)ψRa(x). (3) The total number of fermions in the system N is given by N = NL + NR. (4) The Hamiltonian is also SU(2) invariant as it commutes with the total spin operator ⃗s, where ⃗s = R L 0 dx ⃗s(x) with ⃗s(x) = 1 2(ψ† L(x)⃗σψL(x) + ψ† R(x)⃗σψR(x)). (5) The system also exhibits a discrete spin flip symmetry which has the Z2 group structure, where τ : ψR↑(x) →ψR↓(x), ψL↑(x) →ψL↓(x), τ 2 = 1. (6) In the case of constant interaction strength, the Hamiltonian (1) has been solved using Bethe ansatz [22, 23]. In which case, the system exhibits a separation of spin and charge degrees of freedom. Moreover, due to the fermion-fermion interaction, the system exhibits a dynamical generation of mass gap ∆in the spin sector, where the excitations are called spinons. The charge sector however remains gapless and the charge excitations are called holons. The mass gap stabilizes quasi-long range spin-singlet superconducting order, characterized by ⟨Os(x)Os(0)⟩∼|x|-1/2, Os(x) ∝ψ† R↑(x)ψ† L↓(x) -ψ† R↓(x)ψ† L↑(x). (7) At high energies, the system exhibits asymptotic freedom [22]. It was show in [42, 43] that when twisted boundary conditions are applied, which can be achieved by applying a Zeeman field at the boundaries, the system exhibits a symmetry protected topological (SPT) phase. Here, we consider this model with time dependent interaction strength (1), and by following the method developed in [1], we construct the exact wavefunction which is the solution to the time-dependent Schrodinger equation. The construction of the exact wavefunction involves several steps. Firstly, one identifies certain conserved quantities, which act as quantum numbers that label the wavefunction. The wavefunction is then constructed by ordering the particles in the configuration space, where there exists a unique amplitude corresponding to a specific ordering of all the particles with respect to each other. The amplitudes are related to each other through the action of the particle-particle S-matrices, such that all the amplitudes can be related to any one amplitude. Both the S-matrices and the amplitudes contain a phase part and a spin part. In order for the system to be integrable, these S-matrices should satisfy the quantum Yang-Baxter algebra, which guarantees the consistency of the wavefunction and imposes constraints on the timedependent interaction strength g(t). To determine these amplitudes, one applies periodic boundary conditions on the wavefunction, which gives rise to constraint equations that take the form of matrix difference equations. The solution to the these equations provides the explicit form of the amplitudes and hence also of the exact wavefunction. We show that the matrix difference equations reduce to a set of analytic difference equations and quantum Knizhnik-Zamolodchikov (qKZ) equations. The analytic difference equations govern the phases associated with the S-matrices whereas the qKZ equations govern the spin part. In the special case of constant interaction strength, the procedure described above reduces to the regular Bethe ansatz method. Firstly, the amplitudes and the S-matrices are constants, where the phase part of the amplitudes are simple exponential functions. The matrix difference equations reduce to an eigenvalue equation involving a transfer matrix. In the case where the interaction strength is time-dependent, these amplitudes are vector valued functions that depend on the time dependent interaction strength and also on the positions of all the particles. Hence, one can consider the procedure 3 described above as a generalization of the regular Bethe ansatz method. When the interaction strength is constant, as mentioned above, the regular Bethe ansatz method gives rise to a transfer matrix. This can be diagonalized using the algebraic Bethe ansatz technique [21], which yields the eigenstates and the corresponding eigenvalues. In the case of time-dependent interaction strength, as described above, the generalized Bethe ansatz method gives rise to a set of analytic difference equations and the qKZ equations. The solution to these equations provides the explicit form of the amplitudes, and hence also that of the complete wavefunction. This can be achieved by the method of off-shell Bethe ansatz method [44, 45]. We consider the most general interaction strength that satisfies the constraint conditions imposed by integrability, and solve the associated analytic difference equations and the qKZ equations, and obtain the explicit form of the exact many-body wavefunction. The paper is organized as follows. In section II we present the ansatz for the wavefunction that satisfies the time-dependent Schrodinger equation and provide the constraint conditions on the interaction strength. We then apply periodic boundary conditions on the wavefunction, which gives rise to matrix difference equations. In section (III), we work with the most general interaction strength that satisfies the constraint conditions and show that the matrix difference equations take the form of qKZ equations. We then present the solution to these equations, thus obtaining the explicit form of the manybody wavefunction. In section (IV), we conclude and discuss future prospects. II. THE ANSATZ WAVEFUNCTION As mentioned above, the number of left and right moving fermions are separately conserved (3). Hence, one can construct the ansatz wavefunction which consists of NR number of right moving fermions and NL number of left moving fermions, which we denote by |NL, NR⟩. This wavefunction satisfies the following time-dependent Schrodinger equation i∂t |NL, NR⟩= HGN |NL, NR⟩, (8) where HGN is the Hamiltonian (1). A. One particle case Let us first consider the most simplest case of one particle N = 1. There are two possible wavefunctions corresponding to the choices: NR = 1, NL = 0 and NL = 0, NR = 1, which we label by |0, 1⟩and |1, 0⟩respectively. We have |0, 1⟩= X a Z L 0 dx ψ† Ra(x)FRa(x, t) |0⟩, (9) |1, 0⟩= X a Z L 0 dx ψ† La(x)FLa(x, t) |0⟩. (10) Here, a =↑↓denotes the spin of the fermions. Using these in the Schrodinger equation (8), we obtain the following one particle Schrodinger equations i(∂t + ∂x)FRa(x, t) = 0, i(∂t -∂x)FLa(x, t) = 0. (11) To solve the above equations, we consider the following ansatz FRa(x, t) = fRa(z) θ(x)θ(L -x), FLa(x, t) = fLa( ̄z) θ(x)θ(L -x). (12) Here, we have used the notation z = x -t, ̄z = x + t. θ(x) is the Heaviside step function with the convention θ(x) = 1 for x > 0, θ(x) = 0 for x < 0 and θ(0) = 1/2. The boundary conditions (2) give rise to the following conditions on the amplitudes fRa(z) = fRa(z + L), fLa( ̄z) = fLa( ̄z + L). (13) One can see that the above ansatz (12) satisfies the Schrodinger equations (11) trivially. Let us compare this with the standard Bethe ansatz procedure that one applies when the interaction strength is constant g(t) = g. In this case, the functions fRa(z) and fLa( ̄z) can be expressed in terms of simple exponential functions fRa(z) = eikzARa, fLa(z) = e-ik ̄zALa, (14) where k is the momentum, which is also the energy since we have set vF = 1, and ARa,ALa are amplitudes which do not depend on z and ̄z. B. Two particle case and the S-matrix Now let us consider the case of two particles N = 2. There are three possibilities: 1) NL = 2, NR = 0 : |2, 0⟩, (15) 2) NL = 0, NR = 2 : |0, 2⟩, (16) 3) NL = 1, NR = 1 : |1, 1⟩. (17) The first two sectors which correspond to both the particles being either left movers or right movers is again trivial, as the two particles do not interact with each other. The two particle wavefunction is just the direct product of one particle wavefunctions discussed above. Now let us consider the last sector where one particle is a left mover whereas the other is a right mover. This 4 sector is non trivial since the two particles interact with each other through the spin exchange interaction in the Hamiltonian. We have |1, 1⟩= 2 Y i=1 Z L 0 dxi ψ† Ra(x1)ψ† Lb(x2)AF RL ab (x1, x2, t) |0⟩. (18) Here the superscripts denote the chirality and the subscripts denote the spin of the fermions. A is the anti-symmetrization symbol which when acting on F RL ab (x1, x2, t) exchanges x1 ↔x2, R ↔L and a ↔b. Using the above expression in the Schrodinger equation (8), one obtains the following two particle Schrodinger equation -i(∂t + ∂x1 -∂x2)AF RL ab (x1, x2, t) + g(t)δ(x1 -x2) × (⃗σac · ⃗σbd) AF RL cd (x1, x2, t) = 0. (19) Since a left moving and a right moving fermion interact with each other, the ordering of the particles is important. We take the following ansatz for the wavefunction F RL ab (x1, x2, t): F RL ab (x1, x2, t) = θ(x1)θ(L -x1)θ(x2)θ(L -x2)× f RL,12 ab (z1, ̄z2)θ(x2 -x1) + f RL,21 ab (z1, ̄z2)θ(x1 -x2) . (20) The ordering of the particles is denoted by the superscripts. For example, f RL,12 ab (z1, ̄z2) corresponds to the amplitude in which the particle 1, which is a right mover, is on the left side of particle 2 which is a left mover. Using the above ansatz (20) in the two particle Schrodinger equation (19), one obtains the following relation between the amplitudes f RL,21 ab (z1, ̄z2) = S12 ac,bd(z1, ̄z2)f RL,12 cd (z1, ̄z2), (21) where S12(z1, ̄z2) is the two particle S-matrix which is given by S12 ac,bd(z1, ̄z2) = eiφ( ̄z2-z1) if( ̄z2 -z1)I12 ac,bd + P 12 ac,bd if( ̄z2 -z1) + 1 , f(x) = 1 2g(x/2) 1 -3(g(x/2))2 4 , eiφ(x) = 2ig(x/2) -1 + 3(g(x/2))2 4 ig(x/2) - 1 + 3(g(x/2))2 4 . (22) Here, I12 ac,bd is the identity and P 12 ac,bd is the permutation operator. The superscripts in I12 ac,bd, P 12 ac,bd and in S12 ac,bd(z1, ̄z2) denote that they act in the spin spaces of particles 1 and 2 which are represented by the subscripts. We see that the interaction between the particles relates the two amplitudes in the two particle wave function (20) such that there exists one free amplitude. We may choose this to be f RL,12 ab (z1, ̄z2). Let us now apply the periodic boundary conditions (2) on the two particle wavefunction (20). This results in the following relations between the amplitudes f RL,12 ab (z1, ̄z2) = f RL,21 ab (z1 + L, ̄z2), f RL,21 ab (z1, ̄z2) = f RL,12 ab (z1, ̄z2 + L). (23) Using the relations (23) in (21), we obtain f RL,12 ab (z1, ̄z2 + L) = S12 ac,bd(z1, ̄z2)f RL,12 cd (z1, ̄z2). (24) Hence, we find that applying periodic boundary conditions (2) on the two particle wavefunction (23) imposes constraints on the free amplitude (24), which is a matrix difference equation. One can solve this difference equation, and determine the amplitude fRL,12 ab (z1, ̄z2). One can then use the relation (21) to determine the other amplitude, and hence the exact form of the two particle wavefunction (20). C. Three particle case and the Yang-Baxter algebra Let us now consider the case of three particles N = 3. There exist four possibilities: 1) NL = 3, NR = 0 : |3, 0⟩, 2) NL = 0, NR = 3 : |0, 3⟩, 3) NL = 2, NR = 1 : |1, 2⟩, 4) NL = 1, NR = 2 : |2, 1⟩. (25) Just as in the case of two particles, the first two sectors are trivial where the three particle wavefunction is a direct product of one particle wavefunctions. The third and the fourth sectors are non-trivial, which we discuss below. Let us first consider the third sector corresponding to two left moving particles and one right moving particle. We have |1, 2⟩= 3 Y i=1 Z L 0 dxi ψ† Ra(x1)ψ† Lb(x2)ψ† Lc(x3) ×AF RLL abc (x1, x2, x3, t) |0⟩. (26) Using this in the Schrodinger equation, we obtain the following three particle Schrodinger equation -i(∂t + ∂x1 -∂x2 -∂x3)AF RLL abc (x1, x2, x3, t) +g(t) (δ(x1 -x2)⃗σal · ⃗σbm Icn + δ(x1 -x3)Ibm ⃗σal · ⃗σcn) ×AF RLL lmn (x1, x2, x3, t) = 0. (27) Due to the interactions in the system, just as in the two particle case, the ordering of the left and the right moving particles is important. In addition, we find that the ordering of two left moving particles with respect to each other is also important to have a consistent wavefunction. The ansatz for the three particle wavefunction AF RLL abc (x1, x2, x3, t) takes the following form 5 F RLL abc (x1, x2, x3, t) = θ(x1)θ(L -x1)θ(x2)θ(L -x2)θ(x3)θ(L -x3)× f RLL,123 abc (z1, ̄z2, ̄z3)θ(x2 -x1)θ(x3 -x2) + f RLL,132 abc (z1, ̄z2, ̄z3)θ(x2 -x3)θ(x3 -x1)+ f RLL,213 abc (z1, ̄z2, ̄z3)θ(x1 -x2)θ(x3 -x1) + f RLL,312 abc (z1, ̄z2, ̄z3)θ(x2 -x1)θ(x1 -x3)+ f RLL,231 abc (z1, ̄z2, ̄z3)θ(x1 -x3)θ(x3 -x2) + f RLL,321 abc (z1, ̄z2, ̄z3)θ(x2 -x3)θ(x1 -x2) . (28) Using this in the three particle Schrodinger equation (27), we obtain the following relations (from here on we suppress the spin indices, unless necessary) f RLL,213(z1, ̄z2, ̄z3) = S12(z1, ̄z2)f RLL,123(z1, ̄z2, ̄z3), f RLL,231(z1, ̄z2, ̄z3) = S13(z1, ̄z3)f RLL,213(z1, ̄z2, ̄z3), f RLL,312(z1, ̄z2, ̄z3) = S13(z1, ̄z3)f RLL,132(z1, ̄z2, ̄z3), f RLL,321(z1, ̄z2, ̄z3) = S12(z1, ̄z2)f RLL,312(z1, ̄z2, ̄z3). (29) Here, the S-matrices S12(z1, ̄z2) and S13(z1, ̄z3) take the same form as in the two particle case (22). Note that the relation between the amplitudes which differ by the ordering of the particles with the same chirality is not fixed by the Hamiltonian. In the current case this corresponds to the relation between the amplitudes f RLL,123(z1, ̄z2, ̄z3) and f RLL,132(z1, ̄z2, ̄z3) and also between the amplitudes f RLL,231(z1, ̄z2, ̄z3) and f RLL,321(z1, ̄z2, ̄z3). This occurs due to the linear dispersion, and also occurs in the case when the interaction strength is constant. Integrability requires one to choose a specific relation between such amplitudes for the wavefunction to be consistent, which we discuss below. Note that up until now, the interaction strength g(t) is an arbitrary function of time. Choosing a consistent relation between the above mentioned amplitudes imposes constraints on the interaction strength g(t). The most general form of the interaction strength g(t) that gives rise to a consistent solution is such that, f(t) (22) is a linear function f(t) = αt + β, α, β ∈C (constants), (30) which corresponds to the interaction strength taking the following form g(t) = 2 3 -2(2αt + β) ± 4 (2αt + β)2 + 3 1/2 . (31) Consistency requires that the S-matrix between the amplitudes is given by f RLL,132(z1, ̄z2, ̄z3) = S′23( ̄z2, ̄z3)f RLL,123(z1, ̄z2, ̄z3) f RLL,321(z1, ̄z2, ̄z3) = S′23( ̄z2, ̄z3)f RLL,231(z1, ̄z2, ̄z3), (32) where S23( ̄z2, ̄z3) takes a similar form as that of the two particle S-matrix (22), but now with f(t) being a linear function: S′23 ac,bd( ̄z2, ̄z3) = iα( ̄z3 - ̄z2)I32 ac,bd + P 32 ac,bd iα( ̄z3 - ̄z2) + 1 . (33) Note that the case where α or β are complex gives rise to a non-Hermitian Hamiltonian, which is also interesting. The S-matrices between particles of different chiralities S12(z1, ̄z2), S13(z1, ̄z3) and that between the particles of same chirality S32( ̄z2, ̄z3) satisfy the Yang-Baxter algebra S12(z1, ̄z2)S13(z1, ̄z3)S′23( ̄z2, ̄z3) = S′23( ̄z2, ̄z3)S13(z1, ̄z3)S12(z1, ̄z2), (34) which guarantees the consistency of the three particle wavefunction (28). Note that the linearity of f(t) is crucial for the S-matrices to satisfy the Yang-Baxter algebra. Using the relations (29) and (32), one can express all the amplitudes in the three particle wavefunction (28) in terms of one amplitude of our choosing. We may choose this to be f RLL,123 abc (z1, ̄z2, ̄z3). Hence, we find that, just as in the two particle case, there exists one free amplitude. To find the exact form of the wavefunction, we have to impose periodic boundary conditions (2) on the three particle wave-function (28). This results in the following relations between the amplitudes f RLL,123(z1, ̄z2, ̄z3) = f RLL,231(z1 + L, ̄z2, ̄z3), f RLL,213(z1, ̄z2, ̄z3) = f RLL,132(z1, ̄z2 + L, ̄z3), f RLL,312(z1, ̄z2, ̄z3) = f RLL,123(z1, ̄z2, ̄z3 + L), f RLL,231(z1, ̄z2, ̄z3) = f RLL,312(z1, ̄z2 + L, ̄z3), f RLL,321(z1, ̄z2, ̄z3) = f RLL,213(z1, ̄z2, ̄z3 + L), f RLL,132(z1, ̄z2, ̄z3) = f RLL,321(z1 + L, ̄z2, ̄z3). (35) Using the relations (29), (32) and (35), we obtain the following relation constraining the amplitude f RLL,123 abc (z1, ̄z2, ̄z3): f RLL,123(z1 -L, ̄z2, ̄z3) = Z1(z1, ̄z2, ̄z3)f RLL,123(z1, ̄z2, ̄z3), (36) where the transport operator Z1(z1, ̄z2, ̄z3) transports the particle 1 across the entire system once. It takes the following form Z1(z1, ̄z2, ̄z3) = S13(z1, ̄z3)S12(z1, ̄z2). (37) 6 Similarly, there exist transport operators Z2(z1, ̄z2, ̄z3) and Z3(z1, ̄z2, ̄z3) which transport particles 2 and 3 respectively around the system f RLL,123(z1, ̄z2 -L, ̄z3) = Z1(z1, ̄z2, ̄z3)f RLL,123(z2, ̄z2, ̄z3), f RLL,123(z1, ̄z2, ̄z3 -L) = Z1(z1, ̄z2, ̄z3)f RLL,123(z2, ̄z2, ̄z3). (38) These transport operators take the following form Z2(z1, ̄z2, ̄z3) = S21( ̄z2, z1 + L, )S′23( ̄z2, ̄z3), Z3(z1, ̄z2, ̄z3) = S′32( ̄z3, ̄z2 + L)S31( ̄z3, z1 + L). (39) The equations (36) and (38) form a system of matrix difference equations. We shall see later in the next section that they are related to quantum KnizhnikZamolodchikov equations. Using the Yang-Baxter algebra (34), one can show that these transport operators satisfy the following relations Z1(z1, ̄z2 -L, ̄z3)Z2(z1, ̄z2, ̄z3) = Z2(z1 -L, ̄z2, ̄z3)Z1(z1, ̄z2, ̄z3), Z1(z1, ̄z2, ̄z3 -L)Z3(z1, ̄z2, ̄z3) = Z3(z1 -L, ̄z2, ̄z3)Z1(z1, ̄z2, ̄z3), Z2(z1, ̄z2, ̄z3 -L)Z3(z1, ̄z2, ̄z3) = Z3(z1, ̄z2 -L, ̄z3)Z2(z1, ̄z2, ̄z3). (40) Given a certain interaction strength g(t), which satisfies (30), one should solve the set of matrix difference equations described above to obtain the amplitude f RLL,123 abc (z1, ̄z2, ̄z3). One can then solve for the rest of the amplitudes in the three particle wavefunction (28) using the relations (32), (29) and obtain the explicit form of the wavefunction. Similarly, in the sector corresponding to two right moving particles and one left moving particle (25), one can construct the wavefunction exactly as in the above case. One finds that the constraints on the interaction strength are exactly the same as (30), and the associated matrix difference equations take the same form as above. We shall postpone the discussion of the solution of these matrix difference equations to the later sections and consider the most general case of N particles. D. N particle case and the quantum Knizhnik-Zamolodchikov equations Let us now consider the most general case of N particles. Since the number of left movers NL and right movers NR are separately conserved under periodic boundary conditions, we have N +1 possible sectors corresponding to different values of NL and NR, where NL+NR = N. Below, we shall construct the exact manybody wavefunction for general values of NL and NR. We have |NL, NR⟩= N Y j=NL+1 NL Y k=1 Z L 0 dxj Z L 0 dxk ψ† Rσj(xj)ψ† Lσk(xk) ×AF 1...N,χ1...χN σ1...σN (x1, ..., xN, t) |0⟩. (41) Here, σ1...σN ≡{σi} denote the spin and χ1...χN ≡ {χi} denote the chiralities of the particles. Using this in the Schrodinger equation (8), we obtain the following N particle Schrodinger equation -i(∂t + N X j=NL+1 ∂xjNL X k=1 ∂xk)AF 1...N,{χi} σ1...σN (x1, ..., xN, t)+g(t)× j=N k=NL X j=NL+1 k=1 δ(xj -xk)⃗σσjσ′ j·⃗σσkσ′ kAF 1...N,{χi} σ1..σ′ jσ′ k..σN(x1, ..., xN, t)=0. (42) From here on we use the convention that F 1...N,{χi} σ1...σN (x1, ..., xN) is a vector in the spin space of the particles, as opposed to an amplitude as represented in (41). That is F 1...N,{χi} σ1...σN (x1, ..., xN) ≡ F 1...N,{χi} σ1...σN (x1, ..., xN) |{σi}⟩. The ansatz for F 1...N,{χi} σ1...σN (x1, ..., xN) takes the following form F 1..N,{χi} σ1...σN (x1, .., xN, t)= X Q θ({xQ(j)})f Q,{χi} σ1...σN ( ̄z1, .., zN). (43) In this wavefunction, without losing generality, we have chosen the particles i = 1, ..., NL to be left movers, and i = NL + 1, ..., N to be right movers. In the above expression, Q denotes a permutation of the position orderings of particles and θ({xQ(j)}) is the Heaviside function that vanishes unless xQ(1) ≤· · · ≤xQ(N). Here f Q σ1...σN ( ̄z1, ..., zN) is the amplitude corresponding to the ordering of the particles denoted by Q. The amplitudes that differ by the ordering of the particles with different 7 chiralities are related by the S-matrix (22), just as in the two particle case f ...kj...,{χi}( ̄z1, ..., zN) = Sjk(zj, ̄zk)f ...jk...,{χi}( ̄z1, ..., zN). (44) Here χj = +, χk = -and "..." in the first superscript on both side of the above equation corresponds to any specific ordering of the rest of the particles. In addition to this, just as in the three particle case, the amplitudes that differ by the ordering of the particles with the same chirality are related by an S-matrix. In the case of two left moving particles, we have f ...kj...,{χi}( ̄z1, ..., zN) = S′jk( ̄zj, ̄zk)f ...jk...,{χi}( ̄z1, ..., zN), (45) where χj,k = -, and in the case of two right moving particles, we have f ...kj...,{χi}( ̄z1, ..., zN) = S′jk(zj, zk)f ...jk...,{χi}( ̄z1, ..., zN), (46) where χj,k = +. Just as before, "..." in the first superscript on both side of the above two equations corresponds to any specific ordering of the rest of the particles. These S-matrices satisfy the Yang-Baxter algebra. For one right moving particle i and two left moving particles j and k, we have Sij(zi, ̄zj)Sik(zi, ̄zk)S′jk( ̄zj, ̄zk) = S′jk( ̄zj, ̄zk)Sik(zi, ̄zk)Sij(zi, ̄zj). (47) Similarly, for two right moving particles i and j and one left moving particle k, we have Sjk(zj, ̄zk)Sik(zi, ̄zk)S′ij(zi, zj) = (48) S′ij(zi, zj)Sik(zi, ̄zk)Sjk(zj, ̄zk). (49) In addition to this, the S-matrices corresponding to the exchange of the particles with the same chirality also satisfy the Yang-Baxter algebra. For three right moving particles, we have S′ij(zi, zj)S′ik(zi, zk)S′jk(zj, zk) = S′jk(zj, zk)S′ik(zi, zk)S′ij(zi, zj). (50) Similar expression exists for three left moving particles, which can be obtained by applying the transformation zi,j,k → ̄zi,j,k to the above equation (50). Using the relations (44) and(45), one can express all the amplitudes in the N particle wavefunction (43) in terms of one amplitude of our choosing. We may choose this to be f N...1,{χj} σ1...σN ( ̄z1, ..., zN). To obtain the explicit form of the wavefunction, this free amplitude needs to be determined, which can be achieved by applying periodic boundary conditions (2) on the N particle wavefunction (43). Applying periodic boundary conditions yields the following relation f j...,{χi}( ̄z1, .., zj, .., zN) = f ...j,{χi}( ̄z1, .., zj + L, .., zN). (51) Here j is a right moving particle. Similar expression exists for a left moving particle, which can be obtained by applying the transformation zj → ̄zj to the above equation (51). In the above equation, "..." in the first superscript corresponds to any particular ordering of the rest of the particles, which is same on both sides of the equation. Using the relations (44), (45), (46) and (51) we obtain the following constraint equations on the amplitude f N...1,{χi}( ̄z1, .., zj -L, .., zN) = Zj( ̄z1, ..., zN) f N...1,{χi}( ̄z1, .., zj, .., zN). (52) Note that here j is considered to be a right moving particle without loss of generality. Here the transport operator Zj(z1, ..., ̄zN) transports the particle j around the system once and takes the following form Zj(z1, ..., ̄zN) = S′jj+1(zj, zj+1 + L)...S′jN(zj, zN + L) Sj1(zj, ̄z1)...SjNL(zj, ̄zNL)S′jNL+1(zj, zNL+1) ...S′jj-1(zj, zj-1). (53) The transport operators satisfy the following relations Zj( ̄z1, ..., zk -L, ..., zN)Zk( ̄z1, ..., zN) = Zk( ̄z1, ..., zj -L, ..., zN)Zj( ̄z1, ..., zN). (54) In the above equation, we have chosen j and k to be right moving particles. In the case of left moving particles, we simply need to apply the transformation zj/k → ̄zj/k in the above equation. The constraint equations (52) are matrix difference equations, which need to be solved to obtain the amplitude f N...1,{χj} σ1...σN ( ̄z1, ..., zN). Once this amplitude is obtained, as mentioned above, one can use the relations (44) and (45) to obtain the rest of the amplitudes, and thus the explicit form of the N particle wavefunction (43). E. The case of constant interaction strength and the regular Bethe ansatz technique In the previous subsections we have constructed the ansatz wavefunction for time-dependent interaction strength g(t). We have seen that all the amplitudes in the N-particle wavefunction can be expressed in terms of one amplitude of our choosing. By applying periodic boundary conditions, we obtained constraint equations on this amplitude, which take the form of matrix difference equations (52). In this subsection we shall show that in the special case where the interaction strength is constant, this procedure reduces to the standard Bethe ansatz technique, and thereby demonstrating that the above described procedure is a generalization of the regular Bethe ansatz technique. 8 In the case of constant interaction strength g(t) →g, as mentioned above (14), the amplitudes in the wavefunction can be expressed in terms of simple exponential functions. For the N-particle wavefunction (43), we have f Q,{χi} σ1...σN (z1, .., ̄zN) = N Y j=NL+1 eikjzj NL Y l=1 e-ikjzjAQ,{χi} σ1...σN , (55) where kj are the momenta, and AQ,{χi} σ1...σN are amplitudes corresponding to the ordering of the particles labeled by Q, and they do not depend on time t and positions of the particles xj. With this, the N-particle Schrodinger equation turns into time-independent Schrodinger equation which is given by -E -i N X j=NL+1 ∂xj + i NL X k=1 ∂xk AF 1...N,{χi} σ1...σN (x1, ..., xN) + g× j=N k=NL X j=NL+1 k=1 δ(xj -xk)⃗σσjσ′ j·⃗σσkσ′ kAF 1...N,{χi} σ1..σ′ jσ′ k..σN(x1, ..., xN)=0, (56) where the energy E = N X j=NL+1 kj - NL X l=1 kl. (57) The amplitudes which differ by the ordering of the particles with opposite chiralities are related by the S-matrix A...kj...,{χi} = SjkA...jk...,{χi} (58) where χj = +, χk = -and "..." in the first superscript on both side of the above equation corresponds to any specific ordering of the rest of the particles. The Smatrix is a constant which only depends on the interaction strength g and is given by S12 ac,bd = eiφ ifI12 ac,bd + P 12 ac,bd if + 1 , f = 1 2g 1 -3g2 4 , eiφ = 2ig -1 + 3g2 4 ig - 1 + 3g2 4 . (59) The S-matrices which relate the amplitudes which correspond to different ordering of the particles with the same chirality are also related by an S-matrix which takes the simple form of the permutation operator A...kj...,{χi} = P jkA...jk...,{χi}, (60) where χj,k = + or χj,k = -. Just as before, "..." in the first superscript on both side of the above two equations corresponds to any specific ordering of the rest of the particles. The matrix difference equation (52), which is a constraint equation on the amplitudes, now takes the form of an eigenvalue equation [22] eikjLAN...1,{χi} = ZjAN...1,{χi}, (61) where the transport operator Zj takes the form of the transfer matrix Zj = P jj+1 . . . P jNRSjNR+1 . . . SjNP j1 . . . P jj-1. (62) The relations (54) turn into simple commutation relations [Zi, Zj] = 0, (63) which are of fundamental importance in the regular Bethe ansatz method, as they are necessary conditions for the system to be integrable. To solve for the amplitude AN...1,{χi} σ1...σN , one diagonalizes the transfer matrix Zj. This can be achieved by the standard algebraic Bethe ansatz technique. One obtains the Bethe equations, whose solutions provide the eigenstates and the corresponding eigenvalues (57). III. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS AND THE EXACT WAVEFUNCTION In the previous section we have constructed a solution to the N-particle Schrodinger equation (42). The wavefunction consists of amplitudes which correspond to different orderings of the particles with respect to each other. These amplitudes are related to each other through the S-matrices (44), (45) and (46) such that all the amplitudes in the wavefunction can be expressed in terms one amplitude of our choosing. By applying periodic boundary conditions (2), we obtain constraint equations on this amplitude (52), which take the form of matrix difference equations. In this section we solve these matrix difference equations and obtain the exact solution of the amplitude, and thereby obtain the explicit form of the N-particle wavefunction. The matrix difference equation (52) contains Smatrices that relate amplitudes corresponding to different ordering of particles with opposite chiralities and also the particles with the same chiralities. Notice that the S-matrices between particles of opposite chiralities (22) have a phase part. In order to solve the equations (52), we need to separate this phase part from the matrix part which acts in the spin spaces of the particles. To achieve this, we apply the following transformation on the amplitudes of the N-particle wavefunction (43) f Q,{χi} σ1...σN ( ̄z1, .., zN) = AQ,{χi} σ1...σN ( ̄z1, .., zN) N Y j=NL+1 NL Y l=1 h( ̄zl -zj). (64) 9 Recall that the interaction strength g(t) should satisfy the constraints (30) for the solution to be consistent, where the function f(x) is a linear function. Using this, we can express the S-matrices (22), (33) in terms of the XXX R-matrix Rij(λ), where Rij(λ) = iλIij + (1/α)P ij iλ + 1 , Sij(zi, ̄zj) = eiφ( ̄zj-zi)Rij( ̄zj -zi + β/α), S′kl(zk, zl) = Rkl(zk -zl). (65) Here just as before Iij is the identity matrix and P ij is the permutation operator which acts in the spin spaces of particles i and j. Using (64) and (65) in the matrix difference equations (52), we see that they can be separated into two sets of equations. The first one concerns only the phase part and takes the form of the analytic difference equation h( ̄zm -zj + L) = eiφ( ̄zm-zj)h( ̄zm -zj), m = 1, ..., NL. (66) These equations have been well studied in the literature for different classes of functions φ(x) [46]. The second set of equations concern the spin part, which take the form of quantum Knizhnik-Zamolodchikov equations. They take the following form AN...1,{χi}( ̄z1, .., zj -L, .., zN) = Z′ j( ̄z1, ...., zN) A1...N,{χi}( ̄z1, .., zj, .., zN), (67) where the transport operator Z′ j( ̄z1, ...., zN) is given by Z′ j( ̄z1, ...., zN) = Rjj+1(zj+1 + L -zj)... RjN(zN + L -zj)Rj1( ̄z1 -zj)...RjNL( ̄zNL -zj) RjNL+1(zNL+1 -zj)...Rjj-1(zj-1 -zj). (68) In the context of qKZ equations, L in the above equation is called the step. The qKZ equations first appeared in [47] as the fundamental equations for form factors in the sine-Gordon model, and were later derived from representation theory of quantum affine algebras [48]. They have been well studied in the literature [44, 4951] and the off-shell Bethe ansatz method to solve them has been developed in [44, 45]. The solution to these equations provides us with the explicit form of the amplitude AN...1,{χi}( ̄z1, . . . , zN), which can then be used to obtain the rest of the amplitudes and hence also the explicit form of the complete wavefunction (41). Below, we describe the solution to the qKZ equations (67), that is obtained following [45]. As mentioned in the beginning, the system conserves the total number of left and right moving fermions separately. We have used these conserved quantities to construct the wavefunction (41). In addition to this, our system has the global SU(2) symmetry, and the system conserves the total z-component of the spin. This allows us to construct a state with a specific value of Sz. Consider a state where the spins of all the particles are pointing in the positive z-direction. Let us denote this state by |Ω⟩ [52] |Ω⟩= |↑⟩1 ⊗... ⊗|↑⟩N . (69) This state is trivially an eigenstate of the Hamiltonian (1) and uninteresting. Now consider a state with M number of spins pointing in the negative z-direction and N -M number of spins pointing in the positive z-direction . The total z-component of the spin corresponding to such as state is Sz = N 2 -M. (70) There exists an operator B({zi}, uα), which when acting on the state |Ω⟩, flips one spin. Here uα is the 'rapidity' associated with the spin flip [53]. A general state with M flipped spins is then constructed by acting on the state |Ω⟩by M number of B({wi}, uα) operators, where uα, α = 1, ...M are all distinct. In the case of constant interaction strength, as mentioned before, one constructs eigenstates of the transfer matrix (61), (62), in which case, the set of rapidities uα, α = 1, ..., M are called the Bethe roots which are solutions to certain constraint equations called the Bethe equations. In the current case, instead of the the eigenstates, we need to construct solutions to the matrix difference equations which take the form of the qKZ equations (67), (68). In this solution, unlike the eigenstates of the transfer matrix, the rapidities are not constrained, but are rather summed over. Following [45], we obtain AN...1,{χi} σ1...σN ( ̄w1, . . . , wN) = X uα M Y α=1 BN...1({wi}, uα) × N Y i=NL+1 NL Y j=1 M Y β=1 Γ(wi -uβ) Γ(wi -uβ -iη) Γ( ̄wj -uβ) Γ( ̄wj -uβ -iη) × Y 1≤i<j≤M (ui -uj) Γ(ui -uj -iη) Γ(ui -uj + iη + 1) |Ω⟩, (71) where the summation is over the integers lα, while the parameters euα, α = 1, . . . , M, are arbitrary constants uα = euα -lα, lα ∈Z. (72) This infinite sum is called a 'Jackson type integral'. In the above expression (71), Γ(x) is the usual Gamma function and η = 1/(αL). The parameters wi, ̄wi and η are related to zi, ̄zi, α and β through the following relations wi = zi L , i = NL+1, ..., N; ̄wi = ̄zi L + β αL, i = 1, ..., NL. (73) 10 Note that, in addition to the qKZ equations (67), we need to solve the analytic difference equation (66) to obtain the function h(x). The solution is complicated, but it simplifies in the limit α, β ≫1, where the interaction strength g(t) (31) takes the form g(t) = 1 4(αt + β/2). (74) The function h(x) is given by h(x) = e2iπnx/L Γ((x + β/α -i/α)/L) Γ((x + β/α -i/2α)/L), (75) where n is an integer. Instead of the exponential function in the above expression, one may choose any function whose period is commensurate with L. The solution to (66) is also simplified in the opposite limit where α, β ≪1. In which case, g(t) is linearly dependent on time. Having obtained the explicit form of the amplitude f N...1,{χi} σ1...σN ( ̄w1, . . . , wN), the rest of the amplitudes and hence the explicit form of the wavefunction (41) can be obtained from it by the action of S-matrices (44), (45) and (46). IV. DISCUSSION In this work we have considered the time-dependent SU(2) Gross-Neveu model. In this quantum field theory, spin 1/2 fermions interact with each other through spin exchange interaction strength that varies in time. Using the recently formulated framework [1], which generalizes the standard Bethe ansatz technique, we have constructed an exact solution to the time-dependent Schrodinger equation. We considered the system with periodic boundary conditions. This results in the conservation of the number of left and right moving fermions separately, thus allowing us to use them to label the wavefunction. The wavefunction consists of several amplitudes, where each amplitude corresponds to a certain ordering of particles with respect to each other. Any amplitude in the wavefunction can be related to one amplitude of our choosing through the action of particleparticle S-matrices. The amplitudes and the S-matrices contain a phase part and a spin part. The consistency of the wavefunction requires that these S-matrices satisfy Yang-Baxter algebra, which is a necessary condition for the integrability of the system. This imposes constraints on the time-dependent interaction strength. In the regular Bethe ansatz approach, which is applicable to Hamiltonians with constant interaction strengths, the amplitudes and the S-matrices are constants. The phase part of the amplitudes take the form of simple exponential functions. In contrast, in our case of time-dependent interaction strength, these amplitudes are vector valued functions which depend on the time depend interaction strength and also on the positions of all the particles. The chosen amplitude is then determined by applying periodic boundary conditions, which gives rise to matrix difference equations which constrain the amplitude. We showed that these matrix difference equations reduce to a set of analytic difference equations which govern the phase part, and quantum Knizhnik-Zamolodchikov (qKZ) equations which govern the spin part. In the case of the constant interaction strength, these matrix difference equations reduce to an eigenvalue equation involving a transfer matrix, which is then diagonalized using the standard algebraic Bethe ansatz technique. Hence, one can consider the regular Bethe ansatz method as a special case of the general Bethe ansatz method used in this work. In our case of time-dependent interaction strength, as mentioned above, the general Bethe ansatz method gives rise to qKZ equations and analytic difference equations. The qKZ equations were solved using the off-shell algebraic Bethe ansatz technique. Using the solution to the qKZ equations, along with the solution to the analytic difference equations, we obtained the explicit form of the amplitude. The rest of the amplitudes can then be straightforwardly determined by the action of the particle-particle S-matrices on this amplitude, and thereby one can obtain the explicit form of the complete many-body wavefunction. In addition to the SU(2) symmetric case considered in this work, one may consider the case where the SU(2) symmetry is broken down to U(1) symmetry. In which case one obtains the U(1) Thirring model with timedependent interaction strength. This case is expected to be more interesting since it contains two coupling strengths which vary in time. In this case, instead of the qKZ equations corresponding to the XXX-R matrix that we obtained in this work, one obtains the qKZ equations corresponding to the XXZ R-matrix [49, 54]. In addition, one may consider different boundary conditions as opposed to simple periodic boundary conditions considered in this work. Under these boundary conditions, in addition to the bulk interaction strengths, there exist boundary coupling strengths which can vary in time. In the simple case where the bulk interaction strengths and the boundary coupling strengths are constant, the models described above are shown to exhibit symmetry protected topological (SPT) phases [42, 43, 55, 56]. Hence, the case where the bulk interaction strengths and the boundary coupling strengths vary in time is naturally very interesting, as they may give rise to new type of dynamically generated SPT phases, which is the focus of our future work [57]. ACKNOWLEDGMENTS We acknowledge discussions with Natan Andrei, Patrick Azaria, Paul Fendley and David Huse. I specially thank Patrick Azaria for carefully reviewing the manuscript. 11 [1] P. R. Pasnoori, Integrability of the kondo model with time-dependent interaction strength, Phys. Rev. B 112, L060409 (2025). [2] Y. Cao, S. Jin, and N. 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2510.14763
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M-A-P
COIG-Writer: A High-Quality Dataset for Chinese Creative
Writing with Thought Processes
M-A-P, 2077AI
Abstract
Large language models exhibit systematic deficiencies in creative writing, particularly in non-
English contexts where training data is scarce and lacks process-level supervision. We present
COIG-Writer, a novel Chinese creative writing dataset that captures both diverse outputs and
their underlying thought processes through systematic reverse-engineering of high-quality texts.
Unlike existing datasets that provide only input-output pairs, COIG-Writer comprises 1,665
meticulously curated triplets spanning 51 genres, each containing: (1) a reverse-engineered
prompt, (2) detailed creative reasoning documenting decision-making processes, and (3) the
final text. Through comprehensive experiments, we identify a two-component model of creative
writing: narrative logic (provided by process supervision) and linguistic expression (maintained
by general-purpose data). Our findings reveal three critical insights: (1) Process supervision is
highly effective but requires stabilisation with general data. A ratio of at least one creative sam-
ple to twelve general samples is needed to achieve optimal performance; below this threshold,
the win rate progressively degrades (from 62.75% down to 35.78%)., (2) creative capabilities
are culturally-bound with no cross-lingual transfer (89.26pp gap between Chinese and English
performance), and (3) lexical diversity inversely correlates with creative quality (TTR paradox),
suggesting high diversity signals compensatory behavior for logical deficiencies. These findings
establish that creative excellence emerges from the interaction between logical scaffolding and
linguistic grounding, analogous to how mathematical reasoning enhances but cannot replace
linguistic competence in foundation models.
Project Homepage: https://COIG-Writer.github.io/
1. Introduction
Process supervision has transformed structured reasoning, for example, pushing math competi-
tion benchmarks to about 93% accuracy [20, 23] and enhancing multi-step reasoning [18, 28],
yet creative writing, which accounts for about 40% of LLM applications [1, 24], lacks comparable
methodological advances. We hypothesize this gap stems from a fundamental misunderstand-
ing: creative writing is not monolithic but compositional, requiring both narrative logic (structural
planning) and linguistic expression (stylistic realization).
Current creative writing models exhibit systematic failures across three dimensions. First,
narrative structures converge to predictable templates—repetitive narratives with limited vari-
ation dominate outputs [30]. Second, stylistic diversity collapses—distinct authorial voices
homogenize into what practitioners term “AI flavor” [5]. Third, cultural authenticity deteri-
orates catastrophically in non-English contexts—Chinese models produce Western narrative
arXiv:2510.14763v1 [cs.CL] 16 Oct 2025
Reasoning Process
Reverse
Inspiration prompt
Article
…
Annotator
Stage 1: Collect Chinese Creative Articles
supervision
Stage 2: Human Evaluation
TTR
win rate
…
✗
✗
✗
✗
✗✗✗
✗
✗
✗
✗
✗
✗
✗
✗
✗
✗
✗✗
✗
…
Figure 1. Overview of COIG-Writer construction and evaluation. Stage 1 (Data Construction):
High-quality Chinese texts spanning 51 genres are collected and filtered, followed by expert
reverse-engineering to extract creative prompts and reasoning processes, yielding 1,665 validated
triplets. Stage 2 (Human Evaluation): Models trained on COIG-Writer undergo rigorous human
preference evaluation through pairwise comparisons, with analysis of win rates and lexical
diversity (TTR) to assess creative writing quality.
structures with superficial cultural markers rather than authentic qi-cheng-zhuan-he (beginning-
development-turn-conclusion) progression [7].
We introduce COIG-Writer, a Chinese creative writing dataset that uniquely captures the
reasoning process underlying creative decisions. Our 1,665 expert-curated triplets span 51 genres,
each containing: (1) reverse-engineered prompts, (2) detailed creative reasoning chains, and (3)
final texts. While existing datasets prioritize either scale (e.g. WritingPrompts [9] (300K samples),
ROCStories [22] (100K samples)) or breadth (e.g. COIG [35] (67K samples), LCCC [27] (12M
samples)), they provide only input-output pairs without process data. COIG-Writer uniquely
combines multi-genre coverage with explicit reasoning chains, enabling process-level learning of
creative decision-making. Figure 1 illustrates our two-stage construction pipeline: (i) systematic
collection and filtering of high-quality texts, followed by (ii) expert reverse-engineering to extract
the implicit creative reasoning.
Our experiments reveal three key findings: (1) Process supervision achieves a 62.75% win
rate in Chinese creative writing, but this requires a stabilization ratio of approximately one
creative-process sample to twelve general-purpose samples. Below this threshold, performance
degrades monotonically (with win rates rising from 35.78% to 62.75% as the ratio is approached).
(2) No cross-lingual transfer occurs: English performance drops to 46.46%, with pure COIG-
Writer models generating Chinese text for 12.18% of English prompts. (3) Lexical diversity
inversely correlates with quality—highest Type-Token Ratio (TTR) (0.678) corresponds to lowest
preference scores (37.25%). These findings support a two-component model of creative writing:
narrative logic (enhanced by process supervision) and linguistic expression (maintained by
general data). Neither component alone suffices—the optimal configuration requires both.
Contributions:
• Reverse-engineering methodology: We develop a systematic approach to extract reason-
ing chains from high-quality texts through multi-stage validation (LLM filtering + expert
annotation). The methodology achieves 70% acceptance rate and generalizes to other
creative domains.
• COIG-Writer dataset: 1,665 Chinese creative writing triplets spanning 51 genres, with
average lengths of 283/1,089/2,214 characters (prompt/reasoning/article). Each triplet
2
undergoes 6-dimensional quality evaluation (score ≥50), representing expert annotations.
• Empirical validation of compositional hypothesis: Through controlled experiments, we
demonstrate: (1) process supervision improves Chinese creative writing from 35.78%
to 62.75% but requires a stabilization ratio of approximately 1:12 (creative to general
samples), (2) creative capabilities are language-specific with 16.29% performance gap
between Chinese and English, and (3) lexical diversity inversely correlates with quality
(TTR paradox).
2. The COIG-Writer Dataset
……
Prose Abstract Interview Novel
Peotry
Niche
Literature Drama
Advertise
More than 40 literary themes
Article
Deduplication
《美丽王子》"The Beautiful Prince"
魔幻大陆有一个城堡，里面住着一个活了1000年老巫婆，她拥有十分强大的黑暗魔法，
On the magical continent ... where a thousand-year-old witch resides ... possesses extremely dark magic.
在黑暗魔法的影响下，周围的植物都失去了生命力。
Under the influence of this dark magic, the surrounding plants have lost all vitality.
她一直靠黑暗魔法维持自己的美貌....
She has always relied on dark magic to preserve her beauty...
Reverse Inspiration prompt
请帮我写一篇标题为《白雪王子》的奇幻小说，
Please help me write a fantasy novel titled "The Snow
主角是一个来自巴拉国家的王子和一个城堡里面
Prince".The main characters are a prince from the
的老巫婆，这个巫婆会黑魔法还有一块有魔力的
Kingdom of Bala and an old witch who lives in a castle
.魔镜，故事围绕巫婆嫉妒王子，然后用毒香水毒死
The witch knows dark magic and possesses a mirror...
王子，最后王子被青蛙......
Annotator
Reasoning process
好的，现在用户让我帮他写一篇奇幻小说，
Okay, now the user has asked me to write a fantasy
标题是《白雪王子》。首先，我要确定一下这个
novel titled "The Snow Prince."First, I need to determine
小说的主角背景，用户有提到，主角是一位
the background of the main characters.The user
王子和一位老巫婆，同时整个故事......
mentioned that the protagonists are a prince
....
Filtering
Article
+
Annotator
Article
+
Reverse
Inspiration prompt
Article
Reverse Inspiration prompt
Reasoning process
A_quality
Therehold
+
+
≥ 8
≥ 8
LLM Discriminator
Annotator
Article
Check
Auditor
Article
Reasoning process
Reverse Inspiration prompt
≥ 8
≥ 8
≥ 8
A_creative
R_creative
R_quality
R_creative
R_quality
A_quality
A_creative
+
+
Article ：
《白雪王子》 "Prince of Snow"
魔幻大陆里有一座古老的城堡，里面住着一个千年老巫婆，她拥有强大的黑暗魔法，
In the magical continent ... Inside lives a thousand-year-old witch who possesses powerful dark magic.
在黑暗魔法的副作用下，周围的植物都没有了生命力。
Under the side effects of this dark magic, the surrounding plants have lost their vitality.
她一直靠黑暗魔法维持自己的美丽容颜.....
She has always relied on dark magic to maintain her youthful beauty...
≥ 8 (0-10)
≥ 50 (0-60)
≥ 8 (0-10)
Figure 2. The data curation pipeline of COIG-Writer. Our methodology consists of three main
stages: (1) Genre scope definition through expert consultation, (2) Multi-stage source text
collection and filtering, and (3) Reverse-engineering of thought processes with comprehensive
quality control.
To address these challenges, we introduce COIG-Writer, a Chinese creative writing dataset
that uniquely captures the reasoning process underlying creative decisions. Our 1,665 expert-
curated triplets span 51 genres, each containing: (1) reverse-engineered prompts, (2) detailed
creative reasoning chains, and (3) final texts. In the following section, we will elaborate on
the dataset’s construction methodology, covering our systematic text collection, the reverse-
engineering of creative thought processes, and the multi-stage quality assurance pipeline
While existing datasets prioritize either scale or breadth, they provide only input-output pairs
without process data. COIG-Writer uniquely combines multi-genre coverage with explicit
reasoning chains, enabling process-level learning of creative decision-making. Figure 1 illustrates
our two-stage construction pipeline: (i) systematic collection and filtering of high-quality texts,
followed by (ii) expert reverse-engineering to extract the implicit creative reasoning.
3
2.1. Data Collection Methodology
Genre Taxonomy and Scope Definition. We established our genre taxonomy by aggregating
categories from writing websites, merging categories, and removing duplicates. The selection
process followed two core principles: (1) representational diversity to capture the rich spectrum
of Chinese literary traditions, and (2) practical relevance to include contemporary forms with
real-world applications. Our final taxonomy encompasses 51 specific genres organized across
eight primary domains: Functional Writing (e.g., proposal planning, tutorial guides), Commu-
nicative Writing (e.g., social media content, advertising copy), Non-fictional Writing (e.g., essays,
reviews), Fiction (spanning traditional genres like Wuxia to modern science fiction), Internet
Culture (e.g., subcultural expressions, fan fiction), Poetry (classical and contemporary forms),
Scripts (drama, debate), and Role-playing Writing (character-driven narratives).
Annotator Recruitment and Training. We recruited 100 university students from diverse aca-
demic backgrounds, including literature and linguistics programs, humanities disciplines, and
STEM fields. This interdisciplinary composition ensures broad perspective coverage while
maintaining literary sensitivity. All annotators underwent a standardized 8-hour training pro-
gram covering: (1) quality assessment criteria, (2) reverse-engineering techniques, (3) reasoning
process articulation, and (4) cultural sensitivity guidelines.
Source Text Collection and Initial Filtering. Source texts were systematically collected from
diverse online platforms including literary forums, social media platforms, professional blogs,
and cultural websites. To ensure temporal relevance and avoid potential contamination with
foundation model training data, we strictly limited collection to content published after Oc-
tober 2022, verified through rigorous URL tracing and cross-platform timestamp validation.
Each collected text underwent a five-dimensional initial assessment: Content Completeness
(structural integrity), Format Standardization (presentation quality), Error Correction (linguistic
accuracy), Logical Consistency (narrative coherence), and Creativity Assessment (originality and
engagement). Texts were further evaluated using engagement metrics (likes, shares, comments)
as proxy indicators for quality and appeal.
Automated Quality Screening. We developed a specialized LLM-based quality screening sys-
tem using carefully designed prompts with Qwen3-235B-A22B [31]. The system evaluates texts
across two dimensions through structured prompting: Article_quality (linguistic fluency, struc-
tural coherence, factual accuracy) and Article_creativity (originality, expressiveness, cultural
resonance).
2.2. Thought Process Construction
Reverse-Engineering Methodology. For each qualified article, annotators employed our system-
atic three-step reverse-engineering protocol to extract the implicit creative reasoning underlying
the final text:
(1) Prompt Reconstruction. Annotators analyze the article’s core attributes across multiple
dimensions: thematic focus (subject matter and conceptual depth), stylistic markers (tone, voice,
linguistic register), structural organization (narrative flow, argument structure), and cultural
grounding (idioms, references, contextual assumptions). Based on this analysis, annotators
reverse-engineer a plausible prompt that could have inspired the article. The reconstructed
prompt must balance two competing requirements: sufficient specificity to constrain the creative
space toward the target output, while maintaining adequate interpretative freedom to enable
genuine creative decision-making.
4
(2) Reasoning Process Articulation. Using both the original article and reconstructed prompt,
annotators systematically document the hypothetical creative decision-making pathway con-
necting initial inspiration to final output. The reasoning process must explicitly address five
critical decision categories: (a) initial interpretation and planning (understanding the prompt,
establishing creative goals, conceptualizing the overall approach), (b) structural and stylistic
choices (organizational framework, narrative perspective, tonal register, rhetorical strategies),
(c) cultural and contextual considerations (selection of culturally resonant elements, audience
adaptation, contextual knowledge embedding), (d) narrative development strategies (plot pro-
gression techniques, character development, argument construction, thematic elaboration),
and (e) revision and refinement thoughts (metacognitive reflections on improving coherence,
enhancing impact, resolving tensions).
(3) Coherence Validation. Each resulting triplet (Article, Reverse Inspiration Prompt, Reasoning
Process) undergoes self-consistency checks to ensure logical coherence across all components.
Validation criteria include: (i) the prompt plausibly motivates the article’s characteristics without
being overly deterministic, (ii) each reasoning step logically follows from the prompt and
previous decisions, (iii) the reasoning provides sufficient justification for major creative choices
in the final article, and (iv) no contradictions exist between prompt requirements, reasoning
explanations, and article content. Triplets failing these checks are flagged for iterative refinement.
Multi-Dimensional Quality Evaluation. Each data triplet undergoes systematic evaluation
across six interdependent dimensions spanning three components: Article (quality: fluency,
coherence, cultural appropriateness; creativity: originality, expressiveness, engagement), Prompt
(quality: clarity, specificity, generative potential; creativity: innovation, cultural grounding,
complexity), and Reasoning (quality: logical consistency, completeness, clarity; creativity:
insight depth, decision justification, authenticity). These dimensions enforce cohesion—poor
article-reasoning alignment directly impacts quality scores. Triplets must satisfy dual thresholds
to advance: cumulative score ≥50 and individual dimension scores ≥8, ensuring both overall
excellence and consistent quality across all facets.
2.3. Quality Assurance and Final Validation
Human-in-the-Loop Validation. Eight graduate-level domain experts in Chinese literature
conducted manual validation following standardized calibration sessions. Each triplet under-
went tiered review based on complexity: ≥2 reviewers for standard samples, ≥4 for samples
requiring specialized cultural or stylistic knowledge. Review criteria encompassed: (i) semantic
consistency across the triplet components, (ii) cultural and linguistic authenticity, (iii) reasoning
process coherence, and (iv) contribution to genre diversity. Initial review achieved ≈70% accep-
tance rate, with rejected samples entering iterative refinement unless they contained factual
errors (e.g., anachronistic references) or violated content guidelines, which warranted removal.
This multi-stage validation pipeline produced a final corpus of 1,665 verified triplets.
Bias Mitigation and Diversity Assurance. We implemented five strategies to ensure dataset
diversity: (1) balanced genre representation with minimum 15 samples per category, (2) geo-
graphic diversity across source platforms, (3) temporal spread throughout the collection period,
(4) stylistic variety within each genre, and (5) regular bias audits during curation. These mea-
sures minimize systematic bias and promote equitable representation across all dimensions.
5
2.4. Evaluation Benchmark Construction
We constructed a comprehensive benchmark to systematically evaluate model performance on
creative writing tasks.
Test Query Development. Two computational linguistics postgraduate students developed 104
evaluation queries covering all 51 genres (minimum two queries per genre). Each query specifies
three elements: target genre, creative constraints (length, style, theme), and cultural/contextual
requirements. Our expert panel validated all queries for clarity, precision, and appropriate
difficulty levels.
Human Evaluation Protocol. Four trained graduate evaluators assessed model outputs using
a standardized 4-point scale (0–3) across five dimensions: Content Quality, Creative Merit,
Cultural Appropriateness, Task Fulfillment, and Overall Preference. To ensure consistency, each
evaluator assessed outputs from five specific models, achieving high inter-rater agreement and
minimizing evaluation bias.
0
100
200
300
400
500
Number of Samples
Communication
Novel
Non-fiction
Functional
Writing
Poetry
Funny Literature
Script
481
(28.9%)
467
(28.0%)
243
(14.6%)
221
(13.3%)
128 (7.7%)
68 (4.1%)
57 (3.4%)
Total: 1665 samples
(a) Main category distribution
100
1000
10000
Length (characters)
0
100
200
300
400
500
600
700
800
Frequency
Max: 31,071 chars
Thought
Query
Answer
(b) Length distributions
Figure 3. Dataset composition of COIG-Writer. (a) Distribution across 7 main categories encom-
passing 51 specific genres. Communication (28.9%) and Novel (28.0%) constitute the majority,
followed by Non-fiction (14.6%) and Functional Writing (13.3%). (b) Length distributions for
prompts (Query), reasoning processes (Thought), and articles (Answer) demonstrate the varying
complexity across the dataset.
2.5. Dataset Statistics and Analysis
COIG-Writer contains 1,665 high-quality triplets with substantial diversity. Average character
lengths are 283 for prompts, 1,089 for reasoning processes, and 2,214 for articles, with maximum
article length reaching 31,071 characters. The dataset spans 7 main categories with 51 specific
genres. Communication and Novel categories each represent 30% of the dataset, followed by
Non-fiction (14.6%) and Functional Writing (13.3%), as shown in Figure3a. Genres include
poetry, social media content, fiction, and specialized forms like Xianxia and military novels (see
Appendix B). Length distributions (Figure 3b) show articles ranging from 12–31,071 characters,
reasoning processes from 252–4,094 characters, and prompts from 30–2,642 characters. This
logarithmic distribution, concentrated between 100–10,000 characters, reflects natural variation
in creative writing genres and enables learning from both concise and elaborate examples.
6
3. Experiments and Analysis
3.1. Experimental Setup
Model Configurations. We investigate five configurations that systematically vary the ratio
of COIG-Writer data (DCW, 1,665 samples) to general-purpose data (DG). This design enables
us to empirically determine the stabilization threshold required for process supervision to
enhance creative writing capabilities. To ensure cross-lingual stability, we construct DG from
two complementary sources: 10k Chinese samples from the DeepSeek-R1 distilled dataset [21]
and 10k English samples from OpenThoughts [14], a large-scale reasoning dataset. This yields a
balanced bilingual pool of 20k samples. We create training mixtures by sampling equal amounts
from each language source: 1k per language (2k total), 5k per language (10k total), and 10k
per language (20k total, using the complete pool). Table 1 summarizes the five experimental
configurations, with DG quantities selected to span ratios from 1:1.2 to 1:12 (creative to general
samples).
Table 1. Training configurations and data composition.
Model
COIG-Writer
General
Total
MCW
1,665
0
1,665
MCW+1k
1,665
2,000
3,665
MCW+5k
1,665
10,000
11,665
MCW+10k
1,665
20,000
21,665
MG
0
20,000
20,000
Training Configuration. All models initialize from Qwen2.5-7B-Instruct [31] and undergo
supervised fine-tuning for 3 epochs. We employ AdamW optimizer with learning rate 𝜂=
2 × 10−5, global batch size 𝐵= 32, and linear learning rate warmup over 10% of training steps.
The maximum sequence length is set to 8,192 tokens.
Evaluation Protocol. We construct a comprehensive evaluation benchmark consisting of 557
test queries spanning all 51 genres in our taxonomy, with 204 Chinese queries and 353 English
queries. Each query specifies the target genre, creative constraints (style, theme), and cultural or
contextual requirements. Human evaluation follows a rigorous pairwise comparison protocol.
Four graduate-level evaluators (separate from the annotation team) assess model outputs using
blind comparisons, where neither the model identities nor the training configurations are
disclosed.
3.2. Main Results
Human Preference Evaluation. Table 2 reports pairwise win rates across model configurations.
Chinese Creative Writing: Substantial Effectiveness of COIG-Writer.
For Chinese creative
writing—the native language of the COIG-Writer dataset—our results demonstrate substan-
tial effectiveness. MCW+10k achieves a statistically significant win rate of 62.75% against the
baseline MG (𝑝< 0.001), establishing it as the only configuration to meaningfully outperform
general-purpose training. This 25.5 percentage point improvement represents a substantial
gain attributable to the specialized creative writing data when properly balanced with general-
purpose samples.
7
Table 2. Pairwise win rates (%) on creative writing tasks. Bold values indicate win rates > 55%.
Each cell (𝑖, 𝑗) shows win rate of row 𝑖vs. column 𝑗.
Chinese (Original Dataset Language)
English (Cross-lingual Transfer)
Model
MCW
MCW+1k
MCW+5k
MCW+10k
MG
MCW
MCW+1k
MCW+5k
MCW+10k
MG
MCW
–
39.22
32.35
25.98
35.78
–
38.53
27.20
24.08
23.51
MCW+1k
60.78
–
39.71
32.35
42.16
61.47
–
35.41
32.29
30.03
MCW+5k
67.65
60.29
–
41.67
50.00
72.80
64.59
–
49.29
42.21
MCW+10k
74.02
67.65
58.33
–
62.75
75.92
67.71
50.71
–
46.46
MG
64.22
57.84
50.00
37.25
–
76.49
69.97
57.79
53.54
–
CW
CW+1k
CW+5k
CW+10k
G
Model
0
1000
2000
3000
4000
Character Count
Total: 1020 samples
(a) Chinese results
CW
CW+1k
CW+5k
CW+10k
G
Model
0
2000
4000
6000
8000
10000
12000
14000
16000
Character Count
Total: 1765 samples
(b) English results
Figure 4. Distribution of character counts across model variants. Box plots show median, IQR
(box), whiskers (1.5×IQR), and outliers (dots). Both languages show MCW producing shortest
outputs, with Chinese texts generally shorter due to character density.
Performance exhibits monotonic improvement with increasing general data proportions:
MCW+5k reaches parity (50.00%), while MCW+1k and MCW underperform at 42.16% and 35.78%
respectively. This pattern suggests a critical threshold of approximately 20k general samples
(1:12 ratio of creative to general data) necessary to stabilize the creative enhancements introduced
by specialized data. The strong performance in Chinese validates the effectiveness of process-
supervised creative writing data for the language domain it was designed for.
English Creative Writing: Limited Cross-lingual Transfer.
In contrast, English results demon-
strate limited cross-lingual transfer of creative writing capabilities. The baseline MG maintains
dominance with win rates ranging from 53.54% against MCW+10k to 76.49% against MCW. The
monotonic improvement with increasing general data (from 23.51% to 46.46%) indicates that
Chinese-centric creative data, while highly effective for its native language, does not transfer
effectively to English generation.
The Two-Component Model of Creative Writing.
Our results reveal that creative writing
quality emerges from two distinct components that must be balanced:
Narrative Logic. Provided by COIG-Writer through explicit reasoning chains, enabling
coherent plot development, consistent character behavior, and structured storytelling. This
component ensures logical connections between paragraphs and maintains thematic consistency.
8
Linguistic Expression. Maintained by general-purpose data, ensuring natural phrasing,
stylistic fluency, and cultural idiomaticity. This component provides the surface realization that
makes text feel naturally written rather than artificially generated.
The failure of MCW (35.78% win rate) demonstrates that logic alone is insufficient—qualitative
analysis reveals well-structured narratives expressed in stilted, unnatural language. Conversely,
MG’s fluent surface but poor performance indicates that linguistic variety without logical
scaffolding produces what annotators described as logical disconnection between paragraphs
despite fluent expression—beautiful nonsense that reads well locally but lacks global coherence.
Generation Length Analysis.
Table 3 and Figure 4 present output length characteristics across
model variants. For Chinese generation, MCW+10k produces outputs of comparable length
to the baseline (1,120.2 vs 1,137.3 characters) while achieving superior win rates, indicating
that performance gains stem from content quality rather than mere verbosity. The MCW and
MCW+1k models generate substantially shorter outputs (960.4 and 949.7 characters respectively),
correlating with their inferior performance. In English tasks, the baseline produces the longest
outputs (4,069.9 characters) and achieves highest win rates, suggesting a positive correlation
between generation length and quality in this domain. The MCW model generates the short-
est responses (3,037.8 characters, 25.4% fewer than baseline), corresponding with its poorest
performance (23.51% win rate). Notably, while MCW+10k approaches baseline length (98.3% of
baseline characters), it still underperforms in preference evaluations, indicating that factors
beyond length—likely coherence and cultural appropriateness—determine English generation
quality.
Table 3. Average generation length across model configurations.
Chinese
English
Model
Tokens
Chars
Tokens
Chars
MCW
606.9
960
1,195
3,038
MCW+1k
602.9
950
1,382
3,690
MCW+5k
699.4
1,099
1,533
3,988
MCW+10k
710.7
1,120
1,577
4,002
MG
730.3
1,137
1,577
4,070
The distribution analysis (Figure 4) reveals that variance in output length decreases as more
general data is incorporated, with MCW exhibiting the highest variability across both languages.
This suggests that specialized creative data alone leads to less predictable generation behavior,
while mixing with general data stabilizes output characteristics.
Table 4. Type-Token Ratio analysis reveals inverse correlation with creative quality.
Chinese
English
Model
Mean
Median
Mean
Median
MCW
0.522
0.513
0.562
0.515
MCW+1k
0.578
0.570
0.571
0.554
MCW+5k
0.576
0.576
0.574
0.561
MCW+10k
0.593
0.586
0.590
0.579
MG
0.678
0.671
0.590
0.571
9
CW
CW+1k
CW+5k
CW+10k
G
Model
0.2
0.4
0.6
0.8
1.0
TTR Value
0.522
0.578
0.576
0.593
0.678
Total: 1020 samples
(a) Chinese: wide TTR range (0.522–0.678)
CW
CW+1k
CW+5k
CW+10k
G
Model
0.2
0.4
0.6
0.8
1.0
TTR Value
0.562
0.571
0.574
0.590
0.590
Total: 1765 samples
(b) English: narrow TTR range (0.562–0.590)
Figure 5. The TTR Paradox. Higher lexical diversity correlates with lower creative quality. MG
achieves highest TTR (0.678) but loses to MCW+10k (TTR = 0.593) with only 37.25% win rate,
challenging conventional assumptions about diversity metrics.
Lexical Diversity Analysis.
We measure Type-Token Ratio (TTR) to test whether lexical
diversity correlates with generation quality. For Chinese text, we apply jieba segmentation before
computing TTR. Table 4 and Figure 5 reveal an inverse correlation between lexical diversity
and quality. In Chinese, MG shows highest TTR (0.678) but lowest win rate (37.25%) against
MCW+10k (TTR=0.593). For English, despite identical TTR (0.590), MCW+10k underperforms by 7
percentage points. This inverse relationship aligns with our two-component model: high TTR
in MG indicates vocabulary variation without narrative coherence—manual inspection reveals
frequent topic shifts and inconsistent terminology. Lower TTR in MCW+10k reflects deliberate
term reuse for thematic consistency.
3.3. Qualitative Analysis
Coherence and Instruction Adherence.
Manual inspection of 557 test samples reveals system-
atic failure modes. For Chinese tasks, MG exhibits logical disconnection between paragraphs
despite fluent surface form, while MCW produces unformatted text blocks without proper
segmentation. MCW+10k successfully maintains narrative coherence while following complex
instructions. In the "Wu Song Fights Tiger" reinterpretation task requiring critical commen-
tary, MCW+10k correctly incorporates the meta-narrative critique, while MCW defaults to literal
retelling and MG generates tangentially related content.
Cross-Lingual Contamination.
A critical failure emerges in English generation: MCW pro-
duces Chinese text in 12.18% of English prompts (43/353), compared to 1.13% for MCW+10k
and 1.42% for MG. Contamination correlates inversely with general data proportion, with
intermediate rates for MCW+1k (1.70%) and MCW+5k (1.42%).
Genre-Specific Performance.
Performance varies significantly across 51 genres. Abstract tasks
(homophonic wordplay "XiLaNai", experimental "crazy literature") fail across all models with
<15% success rate, producing overly formal outputs lacking stylistic authenticity. Structured
formats show differential improvement: advertisements and slogans benefit from MCW+10k’s
incorporation of classical poetry and idioms, while MCW+1k and MCW+5k produce simplified
10
vocabulary. Technical genres ("instruction manuals", "proposals") show no distinguishable
quality differences in human evaluation.
3.4. Discussion
Our findings reveal a compositional structure underlying creative writing capability, with
important implications for data-scarce languages:
Language-Specific Effectiveness and Data Scarcity.
The divergent results between Chinese
(62.75% win rate) and English (46.46% win rate) reflect fundamental differences in data avail-
ability rather than inherent limitations of process supervision. Chinese creative writing with
explicit reasoning chains remains severely underrepresented in general pretraining corpora,
making COIG-Writer’s 1,665 samples a valuable and distinctive contribution. The 25.5 percent-
age point improvement in Chinese demonstrates that even relatively small specialized datasets
can substantially enhance capabilities when they address genuine data gaps. Conversely, En-
glish general corpora already contain abundant creative writing examples from diverse sources
(published literature, online fiction, creative writing communities), diminishing the marginal
value of additional specialized data. This suggests that the effectiveness of domain-specific
datasets should be evaluated relative to their representation in existing general-purpose cor-
pora—specialized data provides maximal benefit for underrepresented domains and languages.
Stabilization Threshold.
The monotonic performance improvement (35.78%→62.75%) with
increasing general data establishes a minimum 22k sample requirement for process supervision
effectiveness in Chinese. This 1:12 ratio (creative:general data) suggests narrative logic forms a
necessary but minority component, analogous to how mathematical reasoning enhances but
cannot replace linguistic competence [19]. Future work scaling the creative writing dataset
alongside general data could further elucidate whether this ratio remains optimal across different
dataset sizes.
Cultural and Reasoning-Level Specificity.
The performance gap between languages demon-
strates that creative patterns are culturally encoded at the reasoning level, not merely at the
vocabulary level. The 12.18% Chinese generation on English prompts by MCW indicates that
Chinese narrative structures (four-character idioms, implicit progression) constitute incompati-
ble features for English generation. This finding provides strong evidence that creative writing
competencies are culturally and linguistically bound, contradicting hypotheses of universal
creative skill transfer.
TTR as Diagnostic.
The inverse correlation between lexical diversity and quality reveals
compensatory behavior: models lacking process supervision increase vocabulary variation
to mask logical deficiencies. This suggests TTR could serve as an early warning for training
imbalances—abnormally high diversity signaling insufficient narrative coherence.
Implications.
These findings suggest: (1) scaling creative datasets provides maximal benefit
for underrepresented languages and domains where general corpora lack sufficient creative
examples, (2) cross-lingual transfer requires reasoning-level adaptation beyond translation,
particularly when source and target languages have divergent narrative conventions, and (3)
11
evaluation should separately assess narrative logic and linguistic expression. The substantial
effectiveness in Chinese validates process supervision as a viable approach for enhancing
creative capabilities in data-scarce scenarios.
Limitations.
Our experimental design varied general-purpose data quantities by sampling
equal amounts from each language: +1k per language (2k total), +5k per language (10k total),
and +10k per language (20k total, the complete bilingual pool), while holding the COIG-Writer
dataset fixed at 1,665 samples. This design choice, while revealing the stabilization threshold for
combining creative and general data, limits our ability to assess whether scaling the creative
writing dataset itself would yield further improvements. Future work should investigate
whether increasing COIG-Writer data beyond 1,665 samples could enhance Chinese performance
or enable more effective cross-lingual transfer. Additionally, the current study cannot definitively
distinguish whether the limited English effectiveness stems from the dataset’s Chinese-centric
cultural framing or simply from insufficient creative writing examples for English. Larger-scale
experiments varying both creative and general data quantities would provide clearer insights
into the optimal data composition for creative writing tasks.
4. Related Work
Creative Writing Datasets and Evaluation. English creative writing has benefited from substan-
tial dataset development. The WritingPrompts dataset [9], has provided foundational data for
hierarchical neural story generation. More recently, Fein et al. [10] introduced LitBench, the first
standardized creative writing benchmark, featuring 2,480 human-labeled story comparisons
and a training corpus of 43,827 pairs. LitBench demonstrated that Bradley-Terry reward models
outperform zero-shot large language model (LLM) evaluators (78% vs. 73% human agreement).
However, existing English datasets such as ROCStories [22] and poetry corpora [12, 15] target
specific genres or limited creative aspects, neglecting process-oriented data and cross-genre
diversity. By contrast, high-quality Chinese creative writing resources are critically scarce. Exist-
ing datasets target general tasks: LCCC [27] provides 12M dialogue pairs, LCSTS [16] contains
2.4M summarization pairs, while instruction tuning datasets COIG [35] and COIG-CQIA [2]
focus on general instruction-following rather than creative writing.
Process-Oriented Learning and Creative Writing. Process supervision improves LLMs on
tasks with explicit structure: chain-of-thought prompting [28], self-consistency [26], and zero-
shot CoT [18]. However, creative writing requires long-horizon narrative control, stylistic
decision-making, and culturally informed choices that go beyond stepwise logical inference [4].
Prior computational creativity methods—from rules/templates [3, 11, 29] to outline/plan-first
pipelines [32, 34]—mainly cover high-level structure rather than the fine-grained “thought”
signals (e.g., motif development, pacing, voice) that guide human composition.
Quality Issues and Evaluation Challenges. AI-generated creative writing consistently exhibits
identifiable "AI flavor," characterized by weak logical coherence [33], monolithic stylistic ex-
pression [8], superficial observations [25], inappropriate ornate vocabulary [17], and formulaic
narratives [13]. These systematic issues suggest fundamental shortcomings in current training
methods rather than mere scaling limitations. Furthermore, evaluating creative content remains
inherently challenging. Traditional automatic metrics like BLEU and ROUGE fail to capture the
diversity and nuanced qualities inherent in creative writing [9]. Human evaluation, while more
accurate, is expensive, subjective, and difficult to scale [6]. Recent LLM-based evaluation ap-
proaches [36] partially address scalability but inherit biases from underlying models, especially
12
when assessing culturally-specific creative content.
5. Conclusion
We present COIG-Writer, a Chinese creative writing dataset of 1,665 triplets spanning 51
genres with reverse-engineered prompts, reasoning processes, and final texts. Our experiments
reveal a two-component model where narrative logic (from process supervision) and linguistic
expression (from general data) must be balanced for quality generation.
Three findings support this model: (1) Process supervision requires minimum 22k general
samples—below this threshold, performance degrades monotonically (35.78%→62.75%). (2)
Creative capabilities are language-specific, with Chinese models achieving 62.75% win rate but
only 46.46% in English. (3) Lexical diversity inversely correlates with quality—highest TTR
(0.678) yields lowest preference scores.
These results demonstrate that creative excellence requires both logical scaffolding and
linguistic grounding. While smaller than English datasets, COIG-Writer enables mechanism
discovery rather than scale optimization. The identified compositional structure suggests future
work should separately optimize narrative logic and linguistic expression rather than treating
creativity as monolithic. Process supervision proves necessary but insufficient—effective creative
AI requires careful balance between structure and expression.
13
Contributions and Acknowledgements
Multimodal Art Projection (M-A-P) is a non-profit open-source AI research community, ran
by donation. The community members are working on research topics in a wide range of
spectrum, including but not limited to the pre-training paradigm of foundation models, large-
scale data collection and processing, and the derived applications on coding, reasoning and
music generation.
Core Contributors (Equal Contribution)
• Yunwen Li, CUHK-Shenzhen, M-A-P
• Shuangshuang Ying, M-A-P
• Xingwei Qu, The University of Manchester
Contributors
• Xin Li, Nanyang Technological University
• Sheng Jin, Zhejiang University
• Minghao Liu, 2077AI, M-A-P
• Zhoufutu Wen, M-A-P
• Tianyu Zheng, M-A-P
• Xeron Du, M-A-P
• Qiguang Chen, Harbin Institute of Technology
• Jiajun Shi, M-A-P
• Wangchunshu Zhou, M-A-P
• Jiazhan Feng, M-A-P
• Wanjun Zhong, M-A-P
Advisors
• Libo Qin, Central South University
• Stephen Huang, Peking University
• Wanxiang Che, Harbin Institute of Technology
• Chenghua Lin, The University of Manchester
Corresponding Authors
• Eli Zhang, University of Waterloo
• Chenghua Lin, The University of Manchester
14
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17
A. Use of Large Language Models
During the writing process, LLM was employed to polish and refine certain parts of the
manuscript. The tool was used to improve sentence fluency and enhance clarity of expres-
sion, while preserving the original academic arguments and logical structure, thus ensuring that
the overall language is more standardized and aligned with academic writing conventions.
B. Complete Genre Distribution and Statistics
This appendix provides comprehensive statistics for all 51 genres in the COIG-Writer dataset.
Tables 5–10 present detailed breakdowns by category, including sample counts, percentages, and
length distributions for Articles (A.L.), Reverse Inspiration Prompts (R.I.P.L.), and Reasoning
Processes (R.P.L.). All lengths are measured in Chinese characters.
Table 5. Overview statistics across main categories. Length values shown as (min|mean|max).
Category
Count
%
Article Length Prompt Length Reasoning Length
Overall
1,665 100.0
12|2,214|31,071
30|283|2,643
252|1,089|4,094
Communication
481
28.9
12|1,584|9,422
40|286|1,754
302|1,162|3,816
Novel
467
28.0
61|3,669|31,071
41|332|1,222
353|1,135|2,964
Non-fiction
243
14.6 225|2,052|12,766
41|226|1,718
453|1,066|2,661
Functional
221
13.3
148|1,281|9,057
41|248|2,492
437|1,085|4,094
Poetry
128
7.7
38|210|695
41|203|1,118
442|867|2,935
Funny Literature
68
4.1
17|730|12,117
30|186|1,007
252|703|1,826
Script
57
3.4 369|3,108|13,339
42|405|2,643
553|1,207|3,647
Table 6. Funny Literature
Genre
Count
%
A.L.
(min|mean|max)
R.I.R.L.
(min|mean|max)
R.P.L.
(min|mean|max)
Funny Subculture
31 1.86%
17|1305.45|12117
30|144.03|691
252|701.55|1826
Esports Funny Fiction
18 1.08%
241|519.94|1008
99|291.83|1007
476|698.61|923
Anime/Manga
Funny
Fan Fiction
8 0.48%
168|506.25|1165
67|283.75|542
351|750.62|1033
Subcultural Identity Ex-
pression
4 0.24%
332|486.75|638
90|153.25|262
546|794.75|1124
Fan Circle Funny Litera-
ture
3 0.18%
415|543.67|730
77|108.67|160
535|627|677
Anti-Mainstream
Con-
sumption Funny Litera-
ture
2 0.12%
194|215|236
100|153.5|207
711|774.5|838
Internet Jargon
1 0.06%
448|448|448
97|97|97
536|536|536
Transnational/Cross-
language Funny Litera-
ture
1 0.06%
323|323|323
105|105|105
643|643|643
C. Reasoning Behavior Analysis
To understand the mechanisms underlying performance differences between model configura-
tions, we analyze reasoning patterns during generation by categorizing model behaviors into
four types: normal writing, deep reasoning, self-exploration, and self-reflection.
18
Table 7. Communication Practical Writing
Genre
Count
%
A.L.
(min|mean|max)
R.I.R.L.
(min|mean|max)
R.P.L.
(min|mean|max)
Social Media Content Cre-
ation
124 7.45%
23|2426.5|8842
42|383.49|1754
545|1273.51|2940
Advertising Copy
74 4.44%
30|410.42|4305
44|169.5|708
302|884.69|2978
Blog Post
62 3.72%
480|3164.37|9422
129|394.48|1461
661|1150.45|1783
Debate Script
59 3.54%
590|1167.63|2825
100|257|410
774|1313.75|2320
Popular Science
50 3.00%
146|2178.74|6570
44|242.76|751
603|1053.04|3816
Speech Draft
49 2.94%
725|1974.92|7177
43|329.16|1118
728|1288.22|2485
Slogan
47 2.82%
12|261.23|1394
40|168.79|570
446|957.87|1718
Product Review
16 0.96%
191|2986.94|4635
57|247.62|641
720|1315.31|2196
Table 8. Novel
Genre
Count
%
A.L.
(min|mean|max)
R.I.R.L.
(min|mean|max)
R.P.L.
(min|mean|max)
Fiction/Story
104 6.25%
122|3641.45|27124
44|349.19|1035
582|1198.53|2964
Everyday Stories
50 3.00%
61|2129.52|8137
41|322.16|1222
649|1217.42|2068
Costume Novels
40 2.40%
344|6193.1|31071
71|442.58|985
353|1079.2|1785
Mystery/Inference Sto-
ries
31 1.86%
110|4875.71|19030
42|302.35|765
423|1176.35|2187
Wuxia Novels
30 1.80%
1198|4586.77|23797
49|341.17|586
685|1126.93|2023
Science Fiction Stories
28 1.68%
779|3233.57|12280
53|293.36|958
603|1226.68|2935
Xuanhuan Novels
27 1.62%
467|4925.07|24706
90|354.04|1005
394|1192.52|2237
Fairy Tale
25 1.50%
364|1192.88|2544
118|198.52|435
465|772.96|1256
Fantasy/Magic Stories
25 1.50%
258|2925.64|9290
54|283.08|451
417|1182.92|2098
Xianxia Novels
24 1.44%
170|4291.88|20078
89|385.42|1106
498|984.88|1881
Emotional Stories
23 1.38%
394|3119.26|12147
45|263.7|784
659|1131.3|1975
Military Novels
23 1.38%
1515|3417.17|7095
234|362.13|637
939|1575.35|2784
Sports Novels
22 1.32%
622|3903.27|10358
177|340|892
724|1122|1681
Game Novels
15 0.90% 1067|4309.93|11492
253|422.67|683
629|1095.8|1506
Chinese Reasoning Patterns.
As illustrated in Figure 7, models trained with COIG-Writer data
demonstrate significantly enhanced reasoning capabilities. The MCW+10k configuration exhibits
balanced distributions across all reasoning types, with increased frequencies of deep reasoning
and self-exploration phases compared to the baseline. This balanced reasoning profile correlates
with superior creative performance, suggesting that explicit process supervision enables models
to engage in more sophisticated creative planning and reflection.
English Reasoning Patterns.
Figure 8 reveals contrasting patterns for English generation.
The baseline model maintains consistent deep reasoning throughout generation, while COIG-
Writer variants exhibit disrupted reasoning flows with excessive self-reflection but limited
deep reasoning. This misalignment between the Chinese-oriented reasoning patterns encoded
in COIG-Writer and the requirements of English creative writing explains the performance
degradation, providing mechanistic evidence for the lack of cross-lingual transfer in creative
capabilities.
19
Table 9. Functional Practical Writing
Genre
Count
%
A.L.
(min|mean|max)
R.I.R.L.
(min|mean|max)
R.P.L.
(min|mean|max)
Argumentative Essay
67 4.02%
551|1066.21|2309
46|239.96|806
598|938.03|1568
Academic Abstract
62 3.72%
196|1219.1|9057
48|204.05|478
440|980.48|2728
Proposal Planning
33 1.98%
148|1667.48|3951
41|337.09|2492
715|1415.39|3138
Open Letter
24 1.44%
191|1122.96|5704
44|220.25|486
548|1177.25|2427
Apology Letter
11 0.66%
318|1506.45|3677
92|211.27|445
581|762.27|954
Eulogy
10 0.60%
395|1872.7|7032
155|258.3|656
690|1153|1696
Tutorial Guide
7 0.42%
221|2352.57|5276
122|399.86|1359
936|1218.14|1491
Interview Questions
5 0.30%
203|821|1448
137|235.8|357
437|795.2|1044
Product Manual
2 0.12%
621|1796.5|2972
50|164.5|279
795|2444.5|4094
Table 10. Non-fiction Writing
Genre
Count
%
A.L.
(min|mean|max)
R.I.R.L.
(min|mean|max)
R.P.L.
(min|mean|max)
Essay
73 4.38%
374|1859.07|8585
41|279.99|1381
453|918.67|1630
Reviews
58 3.48%
225|1649|4891
42|193.55|932
502|1167.41|2661
Travel Writing
54 3.24%
483|1946|7733
80|180.31|804
525|1016.33|1587
Historical Stories
34 2.04%
359|2179.44|6856
41|275.03|426
675|1055.88|1930
Biography
24 1.44%
800|4625.62|12766
43|294.46|1718
588|1212.12|2041
Communication P. W.
Script
Functional P. W.
Novel
Funny L.
Poetry
Non-fiction W.
Social Media Creation: 124
Advertising Copy: 74
Blog Post: 62
Debate Script: 59
Popular Science: 50
Speech Draft: 49
Slogan: 47
Product Review: 16
Script: 57
Argumentative Essay: 67
Academic Abstract: 62
Proposal Planning: 33
Open Letter: 24
Apology Letter: 11
Eulogy: 10
Tutorial Guide: 7
Interview Questions: 5
Product Manual: 2
Fiction/Story: 104
Everyday Stories: 50
Costume Novels: 40
Inference Stories: 31
Wuxia Novels: 30
Science Fiction: 28
Xuanhuan Novels: 27
Fantasy/Magic: 25
Fairy Tale: 25
Xianxia Novels: 24
Emotional Stories: 23
Military Novels: 23
Sports Novels: 22
Game Novels: 15
Funny Subculture: 31
Esports Funny: 18
Anime Funny: 8
Subcultural Identity: 4
Fan Circle Funny: 3
Anti-Consumption Funny: 2
Internet Jargon: 1
Cross-language Funny: 1
Poetry: 128
Essay: 73
Reviews: 58
Travel Writing: 54
Historical Stories: 34
Biography: 24
Communication P. W
Script
Functional P. W.
Novel
Funny L.
Poetry
Non-fiction W.
Figure 6. Complete distribution of all 51 genres in COIG-Writer, showing the full diversity of
creative writing categories covered.
20
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
2500
0
200
100
0 1000
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
500
0
50
25
0
200
(c) The Chinese reasoning behavior
distribution
of
model
trained
on
COIG.
(d) The Chinese reasoning behavior
distribution
of
model
trained
on
general data.
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
500
0
50
25
0
200
(a)
The
Chinese reasoning
behavior
distribution of model trained on human-
annotated data.
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
500
0
50
25
0
200
(b)
The
Chinese
reasoning
behavior
distribution of model trained on COIG +
general data (10K).
Figure 7. Reasoning behavior analysis on Chinese creative writing. Models trained with COIG-
Writer data exhibit balanced distributions across reasoning types, while baseline models show
predominant normal writing with minimal deep reasoning.
(c)
The
English
reasoning
behavior
distribution of model trained on COIG.
(d)
The
English
reasoning
behavior
distribution of model trained on general data.
(a) Win rate comparison between different training
configurations on English creative writing tasks.
(b) The English reasoning behavior distribution
of model trained on COIG + general data (10K).
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
2000
0
100
50
0 250
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
1000
0
100
50
0 250
Step Position
Behavior
Deep
Reasoning
Self-
Exploration
Normal
Writing
Self-
Reflection
0.0
0.2
0.4
0.6
0.8
1.0
0
2000
0
100
50
0 250
10K
Baseline
Origin
10K
Baseline
Origin
0
10
20
30
40
50
60
70
Figure 8. Reasoning behavior analysis on English creative writing. The baseline model maintains
consistent deep reasoning patterns, while COIG-Writer variants show disrupted reasoning flows,
explaining their inferior performance.
21
D. Prompts
D.1. Prompt for pre-analyzing answer usability
During the annotation process, it was observed that annotators spent substantial time on
annotation only to find that the quality of the answers was unsatisfactory at the final scoring
stage. Consequently, a pre-analysis step was introduced to examine the usability of the answers.
D.2. Prompt for analyzing answers
Provide a comprehensive analysis of the answer to help annotators quickly grasp the general
idea of the answer and accelerate the annotation process.
Prompt for analyzing answers
我希望你扮演一位资深的创作者和分析师。请对以下片段进行专业分析：
—{}—
请从以下方面进行详细分析：
• 1. 内容结构：分析文本的整体架构、段落组织和逻辑流程。指出结构的优缺点及
对整体效果的影响。
• 2. 语言风格：评估语言的色彩、节奏、句式变化和词汇选择。这些元素如何塑造
了文本的整体风格和语调？
• 3. 修辞手法：识别并评价使用的修辞设备（如比喻、隐喻、排比等）及其效果。
• 4. 有效性评估：文本在实现其意图方面的有效程度如何？哪些部分特别有力或薄
弱？
• 5. 专业洞察：从[文本类型]创作领域的专业角度，指出这篇文本中的独特元素或
创新点。
请确保你的分析既有理论依据，又有实用价值，能帮助我理解这篇文本的创作技巧和
效果。
D.3. Prompt for analyzing answers and queries
Provide a comprehensive analysis of the connection between answer and query, offer reliable
ideas for annotators to write thoughts, and accelerate the annotation process.
D.4. Prompt for evaluating querys
Score the quality and creativity of the queries provided by annotators based on the given
answers, screen out some low-quality data in advance, and reduce the pressure of manual
quality inspection.
D.5. Prompt for evaluating answers
Score the quality and creativity of the answers provided by annotators based on the given
query, filter out some low - quality data in advance, and reduce the pressure of manual quality
inspection.
22
Prompt for analyzing answers and queries
你是一位资深内容分析专家，专注于解析文本创作背后的思维过程、结构设计和写作
技巧。请基于提供的问题/主题(query)和对应的回答内容(answer)，对这篇回答进行全
面、专业的创作思路分析。请提供:
- Query: {}
- Answer: {}
请从以下维度进行分析:
• 1. 需求理解与定位
- 对原始问题/主题的理解深度- 目标受众识别与内容定位- 核心问题提取与回应策
略
• 2. 内容架构设计
- 整体框架与结构布局- 开头与结尾的设计意图及效果- 主体部分的逻辑展开方式-
段落之间的衔接与层次关系
• 3. 表达技巧运用
- 语言风格与表达特点- 修辞手法与句式结构- 专业术语的运用与解释- 叙事策略与
读者引导方式
• 4. 论证方法策略- 论点构建与支持方式- 论据选择与运用效果- 反驳处理与多角度
思考- 说服力构建技巧
• 5. 创新与价值呈现
- 独特见解与创新点- 实用性建议的设计与呈现- 理论与实践的平衡处理- 知识深度
与广度的展示方式
• 整体效果评估
- 内容与原始需求的匹配度- 信息密度与可读性平衡- 专业性与通俗性的结合- 潜在
影响与应用价值
请记住你的分析旨在揭示这篇回答背后的创作思路、组织策略和表达技巧，帮助用户
理解创作者如何解读需求、组织元素、设计架构并最终呈现内容。这将有助于用户学
习并掌握高质量内容创作的思维模式和方法论，提升自身的内容创作能力。
D.6. Prompt for evaluating thoughts
Score the quality and creativity of the thoughts provided by annotators based on the given
query, screen out some low - quality data in advance, and reduce the pressure of manual quality
inspection.
E. Case Study
To illustrate the model’s performance on nuanced creative texts, this case study presents a short,
atmospheric horror story written in Chinese and its corresponding English translation generated
by the model. This example is intended to highlight the model’s ability to preserve the original’s
tone, critical details, and narrative pacing.
Baseline Case
This case examines a generative output from the baseline model to illuminate its inherent
limitations in maintaining narrative coherence. Notably, the baseline demonstrates proficiency in
producing text with a polished, fluent stylistic register—consistent with its strengths in surface-
level language generation. However, a deeper reading reveals critical deficits in structural
23
and semantic cohesion: sentences often suffer from broken inter-clausal semantic links,
while paragraphs lack a coherent narrative thread, leading to disjointed content that fails
to sustain logical progression. Specifically, the generated text frequently shifts between ideas
without transitional reasoning or contextual grounding, resulting in a fragmented output where
successive segments appear tangential or even contradictory. This discrepancy between stylistic
fluency and logical coherence in the baseline’s output not only exemplifies a common failure
mode in current generation frameworks but also underscores the need for more robust modeling
of inter-utterance and inter-paragraph semantic dependencies—key gaps this work aims to
address.
24
Prompt for evaluating querys
你的任务是根据以下标准,基于给定的answer,对query进行精确的评估。输出应包括两
个部分：质量和创意性。请遵循以下细化的评分标准，以确保评估的细致性：
1. 质量性(评分1-10):
评估问题的清晰度和完整性，着重于问题是否明确且自包含，能否在不需要额外信息
的情况下得到完整的答案。
• 9-10分: 完全清晰、精准且自包含；无歧义，问题结构合理，能够直接解释和回
答，无需额外澄清；query几乎对齐了answer，answer中出现的大部分对象或概
念出现在query中，query很清晰的描述了需求。
• 7-8分: 基本清晰且易于理解；歧义很少，基本完整，尽管小的澄清可能有助于
提高精确性；query较好的对齐了answer，answer中出现的部分对象或概念出现
在query中，query较清晰的描述了需求。
• 5-6分: 可理解但缺乏一定清晰度；存在显著的歧义或细微缺失，可能需要
澄清以确保准确回答；query几乎没对齐answer，answer中出现的内容没出现
在query中。
• 3-4分: 相当模糊或缺失元素；需要大量解释，清晰度问题增加回答难度。
• 1-2分: 非常不清晰、模糊或不完整；难以解释或无法直接回答。
2. 创意性(评分1-10):
评估问题的原创性、创新思维和启发潜力，着重于问题是否打破常规思维模式，能否
激发独特视角和深度思考。
• 9-10分: 卓越创意，提出全新视角或前所未见的问题框架，巧妙连接不同领域，挑
战根深蒂固的假设，促使思维范式转换。
• 7-8分: 显著创意，以新颖方式重构熟悉问题，融合不同领域知识，提出令人意外
但有意义的问题角度，鼓励跳出常规思维框架。
• 5-6分: 中等创意，在传统问题基础上有所创新，提供略微出人意料的问题情境，
鼓励一定程度的非线性思维，但整体框架较为常见。
• 3-4分: 有限创意，主要遵循常规问题模式，问题形式或内容略有变化，但思路常
见，很少激发非常规思考。
• 1-2分: 极少创意，完全遵循标准化、传统的问题模式，无任何新颖元素或独特视
角，不鼓励创造性思维。
请严格按以下格式回复，不要包含其他内容：
{{ "quality": 1-10, "creative": 1-10, }}
根据以上标准，对以下问题进行评估：
answer:{ }
query: { }
输出格式（示例）:
```json
{
"quality": 8,
"creative": 8
}
```
25
Prompt for evaluating answers
你的任务是根据以下标准,基于给定的query,对answer进行精确的评估。输出应包括两
个部分：质量和创意性。请遵循以下细化的评分标准，以确保评估的细致性：
1. 质量性(评分1-10):
评估思考的质量、逻辑性和完整性：
• 9-10分: 思考全面深入，逻辑严密，准确把握问题核心；分析角度多元，论证有
力，充分回应问题的各个方面。
• 7-8分: 思考较为完整，逻辑基本清晰；涵盖问题主要方面，有一定深度，但在某
些细节上可进一步拓展。
• 5-6分: 思考基本合理但不够全面；有一定分析但缺乏深度，对问题的理解和回应
存在部分不足。
• 3-4分: 思考存在明显缺陷；逻辑较弱，遗漏关键方面，对问题理解有限或偏离问
题核心。
• 1-2分: 思考质量低下；逻辑混乱，分析肤浅，未能有效回应问题，或严重误解问
题意图。
2. 创意性(评分1-10):
评估思考在原创性、启发性方面的表现：
• 9-10分: 思考极具创新性，提出独特见解和全新视角；打破常规思维框架，融合多
领域知识，产生富有启发性的洞见。
• 7-8分: 思考有明显创新元素，展现非常规思维路径；提供新颖的分析角度，超越
表面层次，引发深度思考。
• 5-6分: 思考包含一定创新点，有自己的见解；思路较为常见但有独到之处，能在
一定程度上拓展问题讨论空间。
• 3-4分: 思考创新性有限，主要沿用常规分析方法；观点较为传统，很少跳出既定
框架思考。
• 1-2分: 思考几乎无创新，完全依循标准化、惯性的思维模式；未能提供任何新鲜
视角或独特分析。
请严格按以下格式回复，不要包含其他内容：
query:{ } answer: { }
输出格式（示例）:
```json
{
"quality": 8,
"creative": 8
}
```
26
Prompt for evaluating thoughts
你的任务是根据以下标准,基于给定的query,对thought进行精确的评估。输出应包括两
个部分：质量和创意性。请遵循以下细化的评分标准，以确保评估的细致性：
1. 质量性(评分1-10):
评估思考的质量、逻辑性和完整性：
• 9-10分: 思考全面深入，逻辑严密，准确把握问题核心；分析角度多元，论证有
力，充分回应问题的各个方面。
• 7-8分: 思考较为完整，逻辑基本清晰；涵盖问题主要方面，有一定深度，但在某
些细节上可进一步拓展。
• 5-6分: 思考基本合理但不够全面；有一定分析但缺乏深度，对问题的理解和回应
存在部分不足。
• 3-4分: 思考存在明显缺陷；逻辑较弱，遗漏关键方面，对问题理解有限或偏离问
题核心。
• 1-2分: 思考质量低下；逻辑混乱，分析肤浅，未能有效回应问题，或严重误解问
题意图。
2. 创意性(评分1-10):
评估思考在原创性、启发性方面的表现：
• 9-10分: 思考极具创新性，提出独特见解和全新视角；打破常规思维框架，融合多
领域知识，产生富有启发性的洞见。
• 7-8分: 思考有明显创新元素，展现非常规思维路径；提供新颖的分析角度，超越
表面层次，引发深度思考。
• 5-6分: 思考包含一定创新点，有自己的见解；思路较为常见但有独到之处，能在
一定程度上拓展问题讨论空间。
• 3-4分: 思考创新性有限，主要沿用常规分析方法；观点较为传统，很少跳出既定
框架思考。
• 1-2分: 思考几乎无创新，完全依循标准化、惯性的思维模式；未能提供任何新鲜
视角或独特分析。
请严格按以下格式回复，不要包含其他内容：
query:{ } answer: { }
输出格式（示例）:
```json
{
"quality": 8,
"creative": 8
}
```
27
Prompt
创作一篇悬疑短篇小说，文字风格恐怖惊悚，但结尾荒诞。夏天主人公应旧友之约去酒
吧，在酒吧卫生间发现恐怖尸体，朋友也变得恐怖，在逃生过程中得到骷髅帮助，又陷入
混乱的车祸，最后回到恐怖起始点。主旨是通过离奇的情节和恐怖氛围，引发读者对于未
知危险的恐惧与思考。
《迷雾》
The Mist
“阿杰，快出来！”我扯着嗓子在卫生间门口喊，瓷砖缝隙里渗出的霉斑在昏黄灯光下像凝
固的血迹。老友的手机屏幕亮起时，我看到他正蹲在洗手台边，后颈上洇着一圈深色液
体。
"Ajie, get out here!" I yelled at the bathroom door, my throat raw. In the dim yellow light, the
mold seeping from between the tiles looked like clotted bloodstains. When my old friend’s phone
screen lit up, I saw him squatting by the sink, a ring of dark liquid soaking the back of his neck.
“尸体......”他突然直起身，指甲缝里卡着碎发，“就在...... 那个...... 女厕......”话没
说完，他忽然僵住。镜子里，我的倒影正盯着他后腰的青紫色瘀痕，那是我三天前帮他按
过的淤血。
"A body..." he said, suddenly straightening up, his fingernails jammed with bits of hair. "It’s
in... in the... women’s restroom..." Before he could finish, he froze. In the mirror, my reflection was
staring at the purplish-blue bruise on his lower back—the same bruise I had massaged for him three
days ago.
走廊传来脚步声，我抓住他的手腕就跑。夜班保安正用消毒水擦拭门把手，看见我们
冲进吧台时，吧台灯突然熄灭。阿杰的喉结上下滚动，指缝里渗出血珠，混着酒精味的呕
吐物溅在他皮鞋上。
Footsteps echoed from the hallway. I grabbed his wrist and we ran. The night-shift security
guard was wiping down a doorknob with disinfectant. As we scrambled behind the bar, the bar
lights abruptly went out. Ajie’s Adam’s apple bobbed, and beads of blood seeped from between his
fingers. Vomit, reeking of alcohol, splattered onto his leather shoes.
“快看！”他指向落地窗。暴雨倾盆中，女厕方向腾起团团白雾，雾气里隐约浮现出
骷髅轮廓，森白色的肋骨在雨幕中泛着磷火般的光。我刚想报警，整栋建筑突然剧烈晃
动——警报器狂叫，消防通道门“哐当”撞开。
"Look!" he pointed towards the floor-to-ceiling window. In the pouring rain, plumes of white
mist were rising from the direction of the women’s restroom. A skeletal silhouette loomed within
the fog, its stark white ribs glowing like phosphorescent fire against the curtain of rain. Just as I
was about to call the police, the entire building began to shake violently—alarms blared, and the fire
escape door slammed open with a clang.
28
“有人撞车了！”阿杰指着监控屏，画面里一辆救护车正碾过尸体残肢，司机戴着墨
镜，后视镜里反着我的脸。警笛声中，我们挤进停尸间。冷冻柜里的女尸皮肤已呈半透明
状，却在我们面前裂开一道血口子，露出里面裹着的......裹尸袋？
"Someone crashed!" Ajie pointed at a security monitor. On the screen, an ambulance was
running over the mangled limbs of a corpse. The driver was wearing sunglasses, and the rearview
mirror reflected my own face. Amid the wailing sirens, we squeezed into the morgue. The female
corpse in the refrigerated drawer had skin that was already semi-translucent, yet right before our
eyes, a bloody gash split open, revealing what was wrapped inside... a body bag?
裹尸袋里蜷缩着的骷髅突然抬头，它的眼睛竟与阿杰后腰的瘀痕一模一样。“这
是......”我摸向自己口袋，发现今天早上在地铁站买的占卜纸牌，此刻正从裤裆里探
出一只苍白的手指，指缝里卡着. . . . . .卡着阿杰的手机。
The skeleton curled up inside the body bag suddenly looked up. Its eyes were identical to the
bruise on Ajie’s lower back. "This is..." I reached for my pocket and found the tarot cards I had
bought at the subway station this morning. A pale finger was now poking out from my crotch, and
wedged between its knuckles was... was Ajie’s phone.
雨停后，停尸间恢复冰凉寂静。我盯着阿杰的后腰，那里原本的淤青已经褪成暗紫，
像某种正在消散的胎记。“你说......”我指向窗外，那里悬着半块被车灯照亮的骨骼，“这
是谁的骨头？”
After the rain stopped, the morgue returned to its icy silence. I stared at Ajie’s lower back,
where the original bruise had faded to a deep violet, like some kind of disappearing birthmark. "You
think..." I pointed out the window, where half a bone, illuminated by car headlights, was suspended
in the air. "Whose bone is that?"
他突然大笑起来：“你记得吗？去年冬天，我们在郊外迷路的时候，你不是一直往东走
吗？”
He suddenly burst out laughing. "Do you remember? Last winter, when we got lost in the
countryside, didn’t you insist on walking east?"
29
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M-A-P COIG-Writer: A High-Quality Dataset for Chinese Creative Writing with Thought Processes M-A-P, 2077AI Abstract Large language models exhibit systematic deficiencies in creative writing, particularly in nonEnglish contexts where training data is scarce and lacks process-level supervision. We present COIG-Writer, a novel Chinese creative writing dataset that captures both diverse outputs and their underlying thought processes through systematic reverse-engineering of high-quality texts. Unlike existing datasets that provide only input-output pairs, COIG-Writer comprises 1,665 meticulously curated triplets spanning 51 genres, each containing: (1) a reverse-engineered prompt, (2) detailed creative reasoning documenting decision-making processes, and (3) the final text. Through comprehensive experiments, we identify a two-component model of creative writing: narrative logic (provided by process supervision) and linguistic expression (maintained by general-purpose data). Our findings reveal three critical insights: (1) Process supervision is highly effective but requires stabilisation with general data. A ratio of at least one creative sample to twelve general samples is needed to achieve optimal performance; below this threshold, the win rate progressively degrades (from 62.75% down to 35.78%)., (2) creative capabilities are culturally-bound with no cross-lingual transfer (89.26pp gap between Chinese and English performance), and (3) lexical diversity inversely correlates with creative quality (TTR paradox), suggesting high diversity signals compensatory behavior for logical deficiencies. These findings establish that creative excellence emerges from the interaction between logical scaffolding and linguistic grounding, analogous to how mathematical reasoning enhances but cannot replace linguistic competence in foundation models. Project Homepage: https://COIG-Writer.github.io/ 1. Introduction Process supervision has transformed structured reasoning, for example, pushing math competition benchmarks to about 93% accuracy [20, 23] and enhancing multi-step reasoning [18, 28], yet creative writing, which accounts for about 40% of LLM applications [1, 24], lacks comparable methodological advances. We hypothesize this gap stems from a fundamental misunderstanding: creative writing is not monolithic but compositional, requiring both narrative logic (structural planning) and linguistic expression (stylistic realization). Current creative writing models exhibit systematic failures across three dimensions. First, narrative structures converge to predictable templates-repetitive narratives with limited variation dominate outputs [30]. Second, stylistic diversity collapses-distinct authorial voices homogenize into what practitioners term "AI flavor" [5]. Third, cultural authenticity deteriorates catastrophically in non-English contexts-Chinese models produce Western narrative 16 Oct 2025 Reasoning Process Reverse Inspiration prompt Article ... Annotator Stage 1: Collect Chinese Creative Articles supervision Stage 2: Human Evaluation TTR win rate ... ✗ ✗ ✗ ✗ ✗✗✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗✗ ✗ ... Figure 1. Overview of COIG-Writer construction and evaluation. Stage 1 (Data Construction): High-quality Chinese texts spanning 51 genres are collected and filtered, followed by expert reverse-engineering to extract creative prompts and reasoning processes, yielding 1,665 validated triplets. Stage 2 (Human Evaluation): Models trained on COIG-Writer undergo rigorous human preference evaluation through pairwise comparisons, with analysis of win rates and lexical diversity (TTR) to assess creative writing quality. structures with superficial cultural markers rather than authentic qi-cheng-zhuan-he (beginningdevelopment-turn-conclusion) progression [7]. We introduce COIG-Writer, a Chinese creative writing dataset that uniquely captures the reasoning process underlying creative decisions. Our 1,665 expert-curated triplets span 51 genres, each containing: (1) reverse-engineered prompts, (2) detailed creative reasoning chains, and (3) final texts. While existing datasets prioritize either scale (e.g. WritingPrompts [9] (300K samples), ROCStories [22] (100K samples)) or breadth (e.g. COIG [35] (67K samples), LCCC [27] (12M samples)), they provide only input-output pairs without process data. COIG-Writer uniquely combines multi-genre coverage with explicit reasoning chains, enabling process-level learning of creative decision-making. Figure 1 illustrates our two-stage construction pipeline: (i) systematic collection and filtering of high-quality texts, followed by (ii) expert reverse-engineering to extract the implicit creative reasoning. Our experiments reveal three key findings: (1) Process supervision achieves a 62.75% win rate in Chinese creative writing, but this requires a stabilization ratio of approximately one creative-process sample to twelve general-purpose samples. Below this threshold, performance degrades monotonically (with win rates rising from 35.78% to 62.75% as the ratio is approached). (2) No cross-lingual transfer occurs: English performance drops to 46.46%, with pure COIGWriter models generating Chinese text for 12.18% of English prompts. (3) Lexical diversity inversely correlates with quality-highest Type-Token Ratio (TTR) (0.678) corresponds to lowest preference scores (37.25%). These findings support a two-component model of creative writing: narrative logic (enhanced by process supervision) and linguistic expression (maintained by general data). Neither component alone suffices-the optimal configuration requires both. Contributions: • Reverse-engineering methodology: We develop a systematic approach to extract reasoning chains from high-quality texts through multi-stage validation (LLM filtering + expert annotation). The methodology achieves 70% acceptance rate and generalizes to other creative domains. • COIG-Writer dataset: 1,665 Chinese creative writing triplets spanning 51 genres, with average lengths of 283/1,089/2,214 characters (prompt/reasoning/article). Each triplet 2 undergoes 6-dimensional quality evaluation (score ≥50), representing expert annotations. • Empirical validation of compositional hypothesis: Through controlled experiments, we demonstrate: (1) process supervision improves Chinese creative writing from 35.78% to 62.75% but requires a stabilization ratio of approximately 1:12 (creative to general samples), (2) creative capabilities are language-specific with 16.29% performance gap between Chinese and English, and (3) lexical diversity inversely correlates with quality (TTR paradox). 2. The COIG-Writer Dataset ...... Prose Abstract Interview Novel Peotry Niche Literature Drama Advertise More than 40 literary themes Article Deduplication 《美丽王子》"The Beautiful Prince" 魔幻大陆有一个城堡,里面住着一个活了1000年老巫婆,她拥有十分强大的黑暗魔法, On the magical continent ... where a thousand-year-old witch resides ... possesses extremely dark magic. 在黑暗魔法的影响下,周围的植物都失去了生命力。 Under the influence of this dark magic, the surrounding plants have lost all vitality. 她一直靠黑暗魔法维持自己的美貌.... She has always relied on dark magic to preserve her beauty... Reverse Inspiration prompt 请帮我写一篇标题为《白雪王子》的奇幻小说, Please help me write a fantasy novel titled "The Snow 主角是一个来自巴拉国家的王子和一个城堡里面 Prince".The main characters are a prince from the 的老巫婆,这个巫婆会黑魔法还有一块有魔力的 Kingdom of Bala and an old witch who lives in a castle .魔镜,故事围绕巫婆嫉妒王子,然后用毒香水毒死 The witch knows dark magic and possesses a mirror... 王子,最后王子被青蛙...... Annotator Reasoning process 好的,现在用户让我帮他写一篇奇幻小说, Okay, now the user has asked me to write a fantasy 标题是《白雪王子》。首先,我要确定一下这个 novel titled "The Snow Prince."First, I need to determine 小说的主角背景,用户有提到,主角是一位 the background of the main characters.The user 王子和一位老巫婆,同时整个故事...... mentioned that the protagonists are a prince .... Filtering Article + Annotator Article + Reverse Inspiration prompt Article Reverse Inspiration prompt Reasoning process A_quality Therehold + + ≥ 8 ≥ 8 LLM Discriminator Annotator Article Check Auditor Article Reasoning process Reverse Inspiration prompt ≥ 8 ≥ 8 ≥ 8 A_creative R_creative R_quality R_creative R_quality A_quality A_creative + + Article : 《白雪王子》 "Prince of Snow" 魔幻大陆里有一座古老的城堡,里面住着一个千年老巫婆,她拥有强大的黑暗魔法, In the magical continent ... Inside lives a thousand-year-old witch who possesses powerful dark magic. 在黑暗魔法的副作用下,周围的植物都没有了生命力。 Under the side effects of this dark magic, the surrounding plants have lost their vitality. 她一直靠黑暗魔法维持自己的美丽容颜..... She has always relied on dark magic to maintain her youthful beauty... ≥ 8 (0-10) ≥ 50 (0-60) ≥ 8 (0-10) Figure 2. The data curation pipeline of COIG-Writer. Our methodology consists of three main stages: (1) Genre scope definition through expert consultation, (2) Multi-stage source text collection and filtering, and (3) Reverse-engineering of thought processes with comprehensive quality control. To address these challenges, we introduce COIG-Writer, a Chinese creative writing dataset that uniquely captures the reasoning process underlying creative decisions. Our 1,665 expertcurated triplets span 51 genres, each containing: (1) reverse-engineered prompts, (2) detailed creative reasoning chains, and (3) final texts. In the following section, we will elaborate on the dataset's construction methodology, covering our systematic text collection, the reverseengineering of creative thought processes, and the multi-stage quality assurance pipeline While existing datasets prioritize either scale or breadth, they provide only input-output pairs without process data. COIG-Writer uniquely combines multi-genre coverage with explicit reasoning chains, enabling process-level learning of creative decision-making. Figure 1 illustrates our two-stage construction pipeline: (i) systematic collection and filtering of high-quality texts, followed by (ii) expert reverse-engineering to extract the implicit creative reasoning. 3 2.1. Data Collection Methodology Genre Taxonomy and Scope Definition. We established our genre taxonomy by aggregating categories from writing websites, merging categories, and removing duplicates. The selection process followed two core principles: (1) representational diversity to capture the rich spectrum of Chinese literary traditions, and (2) practical relevance to include contemporary forms with real-world applications. Our final taxonomy encompasses 51 specific genres organized across eight primary domains: Functional Writing (e.g., proposal planning, tutorial guides), Communicative Writing (e.g., social media content, advertising copy), Non-fictional Writing (e.g., essays, reviews), Fiction (spanning traditional genres like Wuxia to modern science fiction), Internet Culture (e.g., subcultural expressions, fan fiction), Poetry (classical and contemporary forms), Scripts (drama, debate), and Role-playing Writing (character-driven narratives). Annotator Recruitment and Training. We recruited 100 university students from diverse academic backgrounds, including literature and linguistics programs, humanities disciplines, and STEM fields. This interdisciplinary composition ensures broad perspective coverage while maintaining literary sensitivity. All annotators underwent a standardized 8-hour training program covering: (1) quality assessment criteria, (2) reverse-engineering techniques, (3) reasoning process articulation, and (4) cultural sensitivity guidelines. Source Text Collection and Initial Filtering. Source texts were systematically collected from diverse online platforms including literary forums, social media platforms, professional blogs, and cultural websites. To ensure temporal relevance and avoid potential contamination with foundation model training data, we strictly limited collection to content published after October 2022, verified through rigorous URL tracing and cross-platform timestamp validation. Each collected text underwent a five-dimensional initial assessment: Content Completeness (structural integrity), Format Standardization (presentation quality), Error Correction (linguistic accuracy), Logical Consistency (narrative coherence), and Creativity Assessment (originality and engagement). Texts were further evaluated using engagement metrics (likes, shares, comments) as proxy indicators for quality and appeal. Automated Quality Screening. We developed a specialized LLM-based quality screening system using carefully designed prompts with Qwen3-235B-A22B [31]. The system evaluates texts across two dimensions through structured prompting: Article_quality (linguistic fluency, structural coherence, factual accuracy) and Article_creativity (originality, expressiveness, cultural resonance). 2.2. Thought Process Construction Reverse-Engineering Methodology. For each qualified article, annotators employed our systematic three-step reverse-engineering protocol to extract the implicit creative reasoning underlying the final text: (1) Prompt Reconstruction. Annotators analyze the article's core attributes across multiple dimensions: thematic focus (subject matter and conceptual depth), stylistic markers (tone, voice, linguistic register), structural organization (narrative flow, argument structure), and cultural grounding (idioms, references, contextual assumptions). Based on this analysis, annotators reverse-engineer a plausible prompt that could have inspired the article. The reconstructed prompt must balance two competing requirements: sufficient specificity to constrain the creative space toward the target output, while maintaining adequate interpretative freedom to enable genuine creative decision-making. 4 (2) Reasoning Process Articulation. Using both the original article and reconstructed prompt, annotators systematically document the hypothetical creative decision-making pathway connecting initial inspiration to final output. The reasoning process must explicitly address five critical decision categories: (a) initial interpretation and planning (understanding the prompt, establishing creative goals, conceptualizing the overall approach), (b) structural and stylistic choices (organizational framework, narrative perspective, tonal register, rhetorical strategies), (c) cultural and contextual considerations (selection of culturally resonant elements, audience adaptation, contextual knowledge embedding), (d) narrative development strategies (plot progression techniques, character development, argument construction, thematic elaboration), and (e) revision and refinement thoughts (metacognitive reflections on improving coherence, enhancing impact, resolving tensions). (3) Coherence Validation. Each resulting triplet (Article, Reverse Inspiration Prompt, Reasoning Process) undergoes self-consistency checks to ensure logical coherence across all components. Validation criteria include: (i) the prompt plausibly motivates the article's characteristics without being overly deterministic, (ii) each reasoning step logically follows from the prompt and previous decisions, (iii) the reasoning provides sufficient justification for major creative choices in the final article, and (iv) no contradictions exist between prompt requirements, reasoning explanations, and article content. Triplets failing these checks are flagged for iterative refinement. Multi-Dimensional Quality Evaluation. Each data triplet undergoes systematic evaluation across six interdependent dimensions spanning three components: Article (quality: fluency, coherence, cultural appropriateness; creativity: originality, expressiveness, engagement), Prompt (quality: clarity, specificity, generative potential; creativity: innovation, cultural grounding, complexity), and Reasoning (quality: logical consistency, completeness, clarity; creativity: insight depth, decision justification, authenticity). These dimensions enforce cohesion-poor article-reasoning alignment directly impacts quality scores. Triplets must satisfy dual thresholds to advance: cumulative score ≥50 and individual dimension scores ≥8, ensuring both overall excellence and consistent quality across all facets. 2.3. Quality Assurance and Final Validation Human-in-the-Loop Validation. Eight graduate-level domain experts in Chinese literature conducted manual validation following standardized calibration sessions. Each triplet underwent tiered review based on complexity: ≥2 reviewers for standard samples, ≥4 for samples requiring specialized cultural or stylistic knowledge. Review criteria encompassed: (i) semantic consistency across the triplet components, (ii) cultural and linguistic authenticity, (iii) reasoning process coherence, and (iv) contribution to genre diversity. Initial review achieved ≈70% acceptance rate, with rejected samples entering iterative refinement unless they contained factual errors (e.g., anachronistic references) or violated content guidelines, which warranted removal. This multi-stage validation pipeline produced a final corpus of 1,665 verified triplets. Bias Mitigation and Diversity Assurance. We implemented five strategies to ensure dataset diversity: (1) balanced genre representation with minimum 15 samples per category, (2) geographic diversity across source platforms, (3) temporal spread throughout the collection period, (4) stylistic variety within each genre, and (5) regular bias audits during curation. These measures minimize systematic bias and promote equitable representation across all dimensions. 5 2.4. Evaluation Benchmark Construction We constructed a comprehensive benchmark to systematically evaluate model performance on creative writing tasks. Test Query Development. Two computational linguistics postgraduate students developed 104 evaluation queries covering all 51 genres (minimum two queries per genre). Each query specifies three elements: target genre, creative constraints (length, style, theme), and cultural/contextual requirements. Our expert panel validated all queries for clarity, precision, and appropriate difficulty levels. Human Evaluation Protocol. Four trained graduate evaluators assessed model outputs using a standardized 4-point scale (0-3) across five dimensions: Content Quality, Creative Merit, Cultural Appropriateness, Task Fulfillment, and Overall Preference. To ensure consistency, each evaluator assessed outputs from five specific models, achieving high inter-rater agreement and minimizing evaluation bias. 0 100 200 300 400 500 Number of Samples Communication Novel Non-fiction Functional Writing Poetry Funny Literature Script 481 (28.9%) 467 (28.0%) 243 (14.6%) 221 (13.3%) 128 (7.7%) 68 (4.1%) 57 (3.4%) Total: 1665 samples (a) Main category distribution 100 1000 10000 Length (characters) 0 100 200 300 400 500 600 700 800 Frequency Max: 31,071 chars Thought Query Answer (b) Length distributions Figure 3. Dataset composition of COIG-Writer. (a) Distribution across 7 main categories encompassing 51 specific genres. Communication (28.9%) and Novel (28.0%) constitute the majority, followed by Non-fiction (14.6%) and Functional Writing (13.3%). (b) Length distributions for prompts (Query), reasoning processes (Thought), and articles (Answer) demonstrate the varying complexity across the dataset. 2.5. Dataset Statistics and Analysis COIG-Writer contains 1,665 high-quality triplets with substantial diversity. Average character lengths are 283 for prompts, 1,089 for reasoning processes, and 2,214 for articles, with maximum article length reaching 31,071 characters. The dataset spans 7 main categories with 51 specific genres. Communication and Novel categories each represent 30% of the dataset, followed by Non-fiction (14.6%) and Functional Writing (13.3%), as shown in Figure3a. Genres include poetry, social media content, fiction, and specialized forms like Xianxia and military novels (see Appendix B). Length distributions (Figure 3b) show articles ranging from 12-31,071 characters, reasoning processes from 252-4,094 characters, and prompts from 30-2,642 characters. This logarithmic distribution, concentrated between 100-10,000 characters, reflects natural variation in creative writing genres and enables learning from both concise and elaborate examples. 6 3. Experiments and Analysis 3.1. Experimental Setup Model Configurations. We investigate five configurations that systematically vary the ratio of COIG-Writer data (DCW, 1,665 samples) to general-purpose data (DG). This design enables us to empirically determine the stabilization threshold required for process supervision to enhance creative writing capabilities. To ensure cross-lingual stability, we construct DG from two complementary sources: 10k Chinese samples from the DeepSeek-R1 distilled dataset [21] and 10k English samples from OpenThoughts [14], a large-scale reasoning dataset. This yields a balanced bilingual pool of 20k samples. We create training mixtures by sampling equal amounts from each language source: 1k per language (2k total), 5k per language (10k total), and 10k per language (20k total, using the complete pool). Table 1 summarizes the five experimental configurations, with DG quantities selected to span ratios from 1:1.2 to 1:12 (creative to general samples). Table 1. Training configurations and data composition. Model COIG-Writer General Total MCW 1,665 0 1,665 MCW+1k 1,665 2,000 3,665 MCW+5k 1,665 10,000 11,665 MCW+10k 1,665 20,000 21,665 MG 0 20,000 20,000 Training Configuration. All models initialize from Qwen2.5-7B-Instruct [31] and undergo supervised fine-tuning for 3 epochs. We employ AdamW optimizer with learning rate η= 2 × 10-5, global batch size B= 32, and linear learning rate warmup over 10% of training steps. The maximum sequence length is set to 8,192 tokens. Evaluation Protocol. We construct a comprehensive evaluation benchmark consisting of 557 test queries spanning all 51 genres in our taxonomy, with 204 Chinese queries and 353 English queries. Each query specifies the target genre, creative constraints (style, theme), and cultural or contextual requirements. Human evaluation follows a rigorous pairwise comparison protocol. Four graduate-level evaluators (separate from the annotation team) assess model outputs using blind comparisons, where neither the model identities nor the training configurations are disclosed. 3.2. Main Results Human Preference Evaluation. Table 2 reports pairwise win rates across model configurations. Chinese Creative Writing: Substantial Effectiveness of COIG-Writer. For Chinese creative writing-the native language of the COIG-Writer dataset-our results demonstrate substantial effectiveness. MCW+10k achieves a statistically significant win rate of 62.75% against the baseline MG (p 55%. Each cell (i, j) shows win rate of row ivs. column j. Chinese (Original Dataset Language) English (Cross-lingual Transfer) Model MCW MCW+1k MCW+5k MCW+10k MG MCW MCW+1k MCW+5k MCW+10k MG MCW - 39.22 32.35 25.98 35.78 - 38.53 27.20 24.08 23.51 MCW+1k 60.78 - 39.71 32.35 42.16 61.47 - 35.41 32.29 30.03 MCW+5k 67.65 60.29 - 41.67 50.00 72.80 64.59 - 49.29 42.21 MCW+10k 74.02 67.65 58.33 - 62.75 75.92 67.71 50.71 - 46.46 MG 64.22 57.84 50.00 37.25 - 76.49 69.97 57.79 53.54 - CW CW+1k CW+5k CW+10k G Model 0 1000 2000 3000 4000 Character Count Total: 1020 samples (a) Chinese results CW CW+1k CW+5k CW+10k G Model 0 2000 4000 6000 8000 10000 12000 14000 16000 Character Count Total: 1765 samples (b) English results Figure 4. Distribution of character counts across model variants. Box plots show median, IQR (box), whiskers (1.5×IQR), and outliers (dots). Both languages show MCW producing shortest outputs, with Chinese texts generally shorter due to character density. Performance exhibits monotonic improvement with increasing general data proportions: MCW+5k reaches parity (50.00%), while MCW+1k and MCW underperform at 42.16% and 35.78% respectively. This pattern suggests a critical threshold of approximately 20k general samples (1:12 ratio of creative to general data) necessary to stabilize the creative enhancements introduced by specialized data. The strong performance in Chinese validates the effectiveness of processsupervised creative writing data for the language domain it was designed for. English Creative Writing: Limited Cross-lingual Transfer. In contrast, English results demonstrate limited cross-lingual transfer of creative writing capabilities. The baseline MG maintains dominance with win rates ranging from 53.54% against MCW+10k to 76.49% against MCW. The monotonic improvement with increasing general data (from 23.51% to 46.46%) indicates that Chinese-centric creative data, while highly effective for its native language, does not transfer effectively to English generation. The Two-Component Model of Creative Writing. Our results reveal that creative writing quality emerges from two distinct components that must be balanced: Narrative Logic. Provided by COIG-Writer through explicit reasoning chains, enabling coherent plot development, consistent character behavior, and structured storytelling. This component ensures logical connections between paragraphs and maintains thematic consistency. 8 Linguistic Expression. Maintained by general-purpose data, ensuring natural phrasing, stylistic fluency, and cultural idiomaticity. This component provides the surface realization that makes text feel naturally written rather than artificially generated. The failure of MCW (35.78% win rate) demonstrates that logic alone is insufficient-qualitative analysis reveals well-structured narratives expressed in stilted, unnatural language. Conversely, MG's fluent surface but poor performance indicates that linguistic variety without logical scaffolding produces what annotators described as logical disconnection between paragraphs despite fluent expression-beautiful nonsense that reads well locally but lacks global coherence. Generation Length Analysis. Table 3 and Figure 4 present output length characteristics across model variants. For Chinese generation, MCW+10k produces outputs of comparable length to the baseline (1,120.2 vs 1,137.3 characters) while achieving superior win rates, indicating that performance gains stem from content quality rather than mere verbosity. The MCW and MCW+1k models generate substantially shorter outputs (960.4 and 949.7 characters respectively), correlating with their inferior performance. In English tasks, the baseline produces the longest outputs (4,069.9 characters) and achieves highest win rates, suggesting a positive correlation between generation length and quality in this domain. The MCW model generates the shortest responses (3,037.8 characters, 25.4% fewer than baseline), corresponding with its poorest performance (23.51% win rate). Notably, while MCW+10k approaches baseline length (98.3% of baseline characters), it still underperforms in preference evaluations, indicating that factors beyond length-likely coherence and cultural appropriateness-determine English generation quality. Table 3. Average generation length across model configurations. Chinese English Model Tokens Chars Tokens Chars MCW 606.9 960 1,195 3,038 MCW+1k 602.9 950 1,382 3,690 MCW+5k 699.4 1,099 1,533 3,988 MCW+10k 710.7 1,120 1,577 4,002 MG 730.3 1,137 1,577 4,070 The distribution analysis (Figure 4) reveals that variance in output length decreases as more general data is incorporated, with MCW exhibiting the highest variability across both languages. This suggests that specialized creative data alone leads to less predictable generation behavior, while mixing with general data stabilizes output characteristics. Table 4. Type-Token Ratio analysis reveals inverse correlation with creative quality. Chinese English Model Mean Median Mean Median MCW 0.522 0.513 0.562 0.515 MCW+1k 0.578 0.570 0.571 0.554 MCW+5k 0.576 0.576 0.574 0.561 MCW+10k 0.593 0.586 0.590 0.579 MG 0.678 0.671 0.590 0.571 9 CW CW+1k CW+5k CW+10k G Model 0.2 0.4 0.6 0.8 1.0 TTR Value 0.522 0.578 0.576 0.593 0.678 Total: 1020 samples (a) Chinese: wide TTR range (0.522-0.678) CW CW+1k CW+5k CW+10k G Model 0.2 0.4 0.6 0.8 1.0 TTR Value 0.562 0.571 0.574 0.590 0.590 Total: 1765 samples (b) English: narrow TTR range (0.562-0.590) Figure 5. The TTR Paradox. Higher lexical diversity correlates with lower creative quality. MG achieves highest TTR (0.678) but loses to MCW+10k (TTR = 0.593) with only 37.25% win rate, challenging conventional assumptions about diversity metrics. Lexical Diversity Analysis. We measure Type-Token Ratio (TTR) to test whether lexical diversity correlates with generation quality. For Chinese text, we apply jieba segmentation before computing TTR. Table 4 and Figure 5 reveal an inverse correlation between lexical diversity and quality. In Chinese, MG shows highest TTR (0.678) but lowest win rate (37.25%) against MCW+10k (TTR=0.593). For English, despite identical TTR (0.590), MCW+10k underperforms by 7 percentage points. This inverse relationship aligns with our two-component model: high TTR in MG indicates vocabulary variation without narrative coherence-manual inspection reveals frequent topic shifts and inconsistent terminology. Lower TTR in MCW+10k reflects deliberate term reuse for thematic consistency. 3.3. Qualitative Analysis Coherence and Instruction Adherence. Manual inspection of 557 test samples reveals systematic failure modes. For Chinese tasks, MG exhibits logical disconnection between paragraphs despite fluent surface form, while MCW produces unformatted text blocks without proper segmentation. MCW+10k successfully maintains narrative coherence while following complex instructions. In the "Wu Song Fights Tiger" reinterpretation task requiring critical commentary, MCW+10k correctly incorporates the meta-narrative critique, while MCW defaults to literal retelling and MG generates tangentially related content. Cross-Lingual Contamination. A critical failure emerges in English generation: MCW produces Chinese text in 12.18% of English prompts (43/353), compared to 1.13% for MCW+10k and 1.42% for MG. Contamination correlates inversely with general data proportion, with intermediate rates for MCW+1k (1.70%) and MCW+5k (1.42%). Genre-Specific Performance. Performance varies significantly across 51 genres. Abstract tasks (homophonic wordplay "XiLaNai", experimental "crazy literature") fail across all models with <15% success rate, producing overly formal outputs lacking stylistic authenticity. Structured formats show differential improvement: advertisements and slogans benefit from MCW+10k's incorporation of classical poetry and idioms, while MCW+1k and MCW+5k produce simplified 10 vocabulary. Technical genres ("instruction manuals", "proposals") show no distinguishable quality differences in human evaluation. 3.4. Discussion Our findings reveal a compositional structure underlying creative writing capability, with important implications for data-scarce languages: Language-Specific Effectiveness and Data Scarcity. The divergent results between Chinese (62.75% win rate) and English (46.46% win rate) reflect fundamental differences in data availability rather than inherent limitations of process supervision. Chinese creative writing with explicit reasoning chains remains severely underrepresented in general pretraining corpora, making COIG-Writer's 1,665 samples a valuable and distinctive contribution. The 25.5 percentage point improvement in Chinese demonstrates that even relatively small specialized datasets can substantially enhance capabilities when they address genuine data gaps. Conversely, English general corpora already contain abundant creative writing examples from diverse sources (published literature, online fiction, creative writing communities), diminishing the marginal value of additional specialized data. This suggests that the effectiveness of domain-specific datasets should be evaluated relative to their representation in existing general-purpose corpora-specialized data provides maximal benefit for underrepresented domains and languages. Stabilization Threshold. The monotonic performance improvement (35.78%→62.75%) with increasing general data establishes a minimum 22k sample requirement for process supervision effectiveness in Chinese. This 1:12 ratio (creative:general data) suggests narrative logic forms a necessary but minority component, analogous to how mathematical reasoning enhances but cannot replace linguistic competence [19]. Future work scaling the creative writing dataset alongside general data could further elucidate whether this ratio remains optimal across different dataset sizes. Cultural and Reasoning-Level Specificity. The performance gap between languages demonstrates that creative patterns are culturally encoded at the reasoning level, not merely at the vocabulary level. The 12.18% Chinese generation on English prompts by MCW indicates that Chinese narrative structures (four-character idioms, implicit progression) constitute incompatible features for English generation. This finding provides strong evidence that creative writing competencies are culturally and linguistically bound, contradicting hypotheses of universal creative skill transfer. TTR as Diagnostic. The inverse correlation between lexical diversity and quality reveals compensatory behavior: models lacking process supervision increase vocabulary variation to mask logical deficiencies. This suggests TTR could serve as an early warning for training imbalances-abnormally high diversity signaling insufficient narrative coherence. Implications. These findings suggest: (1) scaling creative datasets provides maximal benefit for underrepresented languages and domains where general corpora lack sufficient creative examples, (2) cross-lingual transfer requires reasoning-level adaptation beyond translation, particularly when source and target languages have divergent narrative conventions, and (3) 11 evaluation should separately assess narrative logic and linguistic expression. The substantial effectiveness in Chinese validates process supervision as a viable approach for enhancing creative capabilities in data-scarce scenarios. Limitations. Our experimental design varied general-purpose data quantities by sampling equal amounts from each language: +1k per language (2k total), +5k per language (10k total), and +10k per language (20k total, the complete bilingual pool), while holding the COIG-Writer dataset fixed at 1,665 samples. This design choice, while revealing the stabilization threshold for combining creative and general data, limits our ability to assess whether scaling the creative writing dataset itself would yield further improvements. Future work should investigate whether increasing COIG-Writer data beyond 1,665 samples could enhance Chinese performance or enable more effective cross-lingual transfer. Additionally, the current study cannot definitively distinguish whether the limited English effectiveness stems from the dataset's Chinese-centric cultural framing or simply from insufficient creative writing examples for English. Larger-scale experiments varying both creative and general data quantities would provide clearer insights into the optimal data composition for creative writing tasks. 4. Related Work Creative Writing Datasets and Evaluation. English creative writing has benefited from substantial dataset development. The WritingPrompts dataset [9], has provided foundational data for hierarchical neural story generation. More recently, Fein et al. [10] introduced LitBench, the first standardized creative writing benchmark, featuring 2,480 human-labeled story comparisons and a training corpus of 43,827 pairs. LitBench demonstrated that Bradley-Terry reward models outperform zero-shot large language model (LLM) evaluators (78% vs. 73% human agreement). However, existing English datasets such as ROCStories [22] and poetry corpora [12, 15] target specific genres or limited creative aspects, neglecting process-oriented data and cross-genre diversity. By contrast, high-quality Chinese creative writing resources are critically scarce. Existing datasets target general tasks: LCCC [27] provides 12M dialogue pairs, LCSTS [16] contains 2.4M summarization pairs, while instruction tuning datasets COIG [35] and COIG-CQIA [2] focus on general instruction-following rather than creative writing. Process-Oriented Learning and Creative Writing. Process supervision improves LLMs on tasks with explicit structure: chain-of-thought prompting [28], self-consistency [26], and zeroshot CoT [18]. However, creative writing requires long-horizon narrative control, stylistic decision-making, and culturally informed choices that go beyond stepwise logical inference [4]. Prior computational creativity methods-from rules/templates [3, 11, 29] to outline/plan-first pipelines [32, 34]-mainly cover high-level structure rather than the fine-grained "thought" signals (e.g., motif development, pacing, voice) that guide human composition. Quality Issues and Evaluation Challenges. AI-generated creative writing consistently exhibits identifiable "AI flavor," characterized by weak logical coherence [33], monolithic stylistic expression [8], superficial observations [25], inappropriate ornate vocabulary [17], and formulaic narratives [13]. These systematic issues suggest fundamental shortcomings in current training methods rather than mere scaling limitations. Furthermore, evaluating creative content remains inherently challenging. Traditional automatic metrics like BLEU and ROUGE fail to capture the diversity and nuanced qualities inherent in creative writing [9]. Human evaluation, while more accurate, is expensive, subjective, and difficult to scale [6]. Recent LLM-based evaluation approaches [36] partially address scalability but inherit biases from underlying models, especially 12 when assessing culturally-specific creative content. 5. Conclusion We present COIG-Writer, a Chinese creative writing dataset of 1,665 triplets spanning 51 genres with reverse-engineered prompts, reasoning processes, and final texts. Our experiments reveal a two-component model where narrative logic (from process supervision) and linguistic expression (from general data) must be balanced for quality generation. Three findings support this model: (1) Process supervision requires minimum 22k general samples-below this threshold, performance degrades monotonically (35.78%→62.75%). (2) Creative capabilities are language-specific, with Chinese models achieving 62.75% win rate but only 46.46% in English. (3) Lexical diversity inversely correlates with quality-highest TTR (0.678) yields lowest preference scores. These results demonstrate that creative excellence requires both logical scaffolding and linguistic grounding. While smaller than English datasets, COIG-Writer enables mechanism discovery rather than scale optimization. The identified compositional structure suggests future work should separately optimize narrative logic and linguistic expression rather than treating creativity as monolithic. Process supervision proves necessary but insufficient-effective creative AI requires careful balance between structure and expression. 13 Contributions and Acknowledgements Multimodal Art Projection (M-A-P) is a non-profit open-source AI research community, ran by donation. The community members are working on research topics in a wide range of spectrum, including but not limited to the pre-training paradigm of foundation models, largescale data collection and processing, and the derived applications on coding, reasoning and music generation. Core Contributors (Equal Contribution) • Yunwen Li, CUHK-Shenzhen, M-A-P • Shuangshuang Ying, M-A-P • Xingwei Qu, The • Xin Li, Nanyang Technological University • Sheng Jin, Zhejiang University • Minghao Liu, 2077AI, M-A-P • Zhoufutu Wen, M-A-P • Tianyu Zheng, M-A-P • Xeron Du, M-A-P • Qiguang Chen, Harbin • Jiajun Shi, M-A-P • Wangchunshu Zhou, M-A-P • Jiazhan Feng, M-A-P • Wanjun Zhong, M-A-P Advisors • Libo Qin, Central South University • Stephen Huang, Peking University • Wanxiang Che, Harbin • Chenghua Lin, The • Eli Zhang, • Chenghua Lin, The 14 References 1 Anthropic. Anthropic economic index: Understanding ai's effects on the economy, 2025. URL https://www.anthropic.com/economic-index. Accessed: 2025-09-16. 2 Yuelin Bai, Xinrun Du, Yiming Liang, Yonggang Jin, Junting Zhou, Ziqiang Liu, Feiteng Fang, Mingshan Chang, Tianyu Zheng, Xincheng Zhang, et al. Coig-cqia: Quality is all you need for chinese instruction fine-tuning. arXiv preprint , 2024. 3 Margaret A Boden. 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In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 7378-7385, 2019. 35 Ge Zhang, Yemin Shi, Ruibo Liu, Ruibin Yuan, Yizhi Li, Siwei Dong, Yu Shu, Zhaoqun Li, Zekun Wang, Chenghua Lin, et al. Chinese open instruction generalist: A preliminary release. arXiv preprint , 2023. 36 Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in neural information processing systems, 36:46595-46623, 2023. 17 A. Use of Large Language Models During the writing process, LLM was employed to polish and refine certain parts of the manuscript. The tool was used to improve sentence fluency and enhance clarity of expression, while preserving the original academic arguments and logical structure, thus ensuring that the overall language is more standardized and aligned with academic writing conventions. B. Complete Genre Distribution and Statistics This appendix provides comprehensive statistics for all 51 genres in the COIG-Writer dataset. Tables 5-10 present detailed breakdowns by category, including sample counts, percentages, and length distributions for Articles (A.L.), Reverse Inspiration Prompts (R.I.P.L.), and Reasoning Processes (R.P.L.). All lengths are measured in Chinese characters. Table 5. Overview statistics across main categories. Length values shown as (min|mean|max). Category Count % Article Length Prompt Length Reasoning Length Overall 1,665 100.0 12|2,214|31,071 30|283|2,643 252|1,089|4,094 Communication 481 28.9 12|1,584|9,422 40|286|1,754 302|1,162|3,816 Novel 467 28.0 61|3,669|31,071 41|332|1,222 353|1,135|2,964 Non-fiction 243 14.6 225|2,052|12,766 41|226|1,718 453|1,066|2,661 Functional 221 13.3 148|1,281|9,057 41|248|2,492 437|1,085|4,094 Poetry 128 7.7 38|210|695 41|203|1,118 442|867|2,935 Funny Literature 68 4.1 17|730|12,117 30|186|1,007 252|703|1,826 Script 57 3.4 369|3,108|13,339 42|405|2,643 553|1,207|3,647 Table 6. Funny Literature Genre Count % A.L. (min|mean|max) R.I.R.L. (min|mean|max) R.P.L. (min|mean|max) Funny Subculture 31 1.86% 17|1305.45|12117 30|144.03|691 252|701.55|1826 Esports Funny Fiction 18 1.08% 241|519.94|1008 99|291.83|1007 476|698.61|923 Anime/Manga Funny Fan Fiction 8 0.48% 168|506.25|1165 67|283.75|542 351|750.62|1033 Subcultural Identity Expression 4 0.24% 332|486.75|638 90|153.25|262 546|794.75|1124 Fan Circle Funny Literature 3 0.18% 415|543.67|730 77|108.67|160 535|627|677 Anti-Mainstream Consumption Funny Literature 2 0.12% 194|215|236 100|153.5|207 711|774.5|838 Internet Jargon 1 0.06% 448|448|448 97|97|97 536|536|536 Transnational/Crosslanguage Funny Literature 1 0.06% 323|323|323 105|105|105 643|643|643 C. Reasoning Behavior Analysis To understand the mechanisms underlying performance differences between model configurations, we analyze reasoning patterns during generation by categorizing model behaviors into four types: normal writing, deep reasoning, self-exploration, and self-reflection. 18 Table 7. Communication Practical Writing Genre Count % A.L. (min|mean|max) R.I.R.L. (min|mean|max) R.P.L. (min|mean|max) Social Media Content Creation 124 7.45% 23|2426.5|8842 42|383.49|1754 545|1273.51|2940 Advertising Copy 74 4.44% 30|410.42|4305 44|169.5|708 302|884.69|2978 Blog Post 62 3.72% 480|3164.37|9422 129|394.48|1461 661|1150.45|1783 Debate Script 59 3.54% 590|1167.63|2825 100|257|410 774|1313.75|2320 Popular Science 50 3.00% 146|2178.74|6570 44|242.76|751 603|1053.04|3816 Speech Draft 49 2.94% 725|1974.92|7177 43|329.16|1118 728|1288.22|2485 Slogan 47 2.82% 12|261.23|1394 40|168.79|570 446|957.87|1718 Product Review 16 0.96% 191|2986.94|4635 57|247.62|641 720|1315.31|2196 Table 8. Novel Genre Count % A.L. (min|mean|max) R.I.R.L. (min|mean|max) R.P.L. (min|mean|max) Fiction/Story 104 6.25% 122|3641.45|27124 44|349.19|1035 582|1198.53|2964 Everyday Stories 50 3.00% 61|2129.52|8137 41|322.16|1222 649|1217.42|2068 Costume Novels 40 2.40% 344|6193.1|31071 71|442.58|985 353|1079.2|1785 Mystery/Inference Stories 31 1.86% 110|4875.71|19030 42|302.35|765 423|1176.35|2187 Wuxia Novels 30 1.80% 1198|4586.77|23797 49|341.17|586 685|1126.93|2023 Science Fiction Stories 28 1.68% 779|3233.57|12280 53|293.36|958 603|1226.68|2935 Xuanhuan Novels 27 1.62% 467|4925.07|24706 90|354.04|1005 394|1192.52|2237 Fairy Tale 25 1.50% 364|1192.88|2544 118|198.52|435 465|772.96|1256 Fantasy/Magic Stories 25 1.50% 258|2925.64|9290 54|283.08|451 417|1182.92|2098 Xianxia Novels 24 1.44% 170|4291.88|20078 89|385.42|1106 498|984.88|1881 Emotional Stories 23 1.38% 394|3119.26|12147 45|263.7|784 659|1131.3|1975 Military Novels 23 1.38% 1515|3417.17|7095 234|362.13|637 939|1575.35|2784 Sports Novels 22 1.32% 622|3903.27|10358 177|340|892 724|1122|1681 Game Novels 15 0.90% 1067|4309.93|11492 253|422.67|683 629|1095.8|1506 Chinese Reasoning Patterns. As illustrated in Figure 7, models trained with COIG-Writer data demonstrate significantly enhanced reasoning capabilities. The MCW+10k configuration exhibits balanced distributions across all reasoning types, with increased frequencies of deep reasoning and self-exploration phases compared to the baseline. This balanced reasoning profile correlates with superior creative performance, suggesting that explicit process supervision enables models to engage in more sophisticated creative planning and reflection. English Reasoning Patterns. Figure 8 reveals contrasting patterns for English generation. The baseline model maintains consistent deep reasoning throughout generation, while COIGWriter variants exhibit disrupted reasoning flows with excessive self-reflection but limited deep reasoning. This misalignment between the Chinese-oriented reasoning patterns encoded in COIG-Writer and the requirements of English creative writing explains the performance degradation, providing mechanistic evidence for the lack of cross-lingual transfer in creative capabilities. 19 Table 9. Functional Practical Writing Genre Count % A.L. (min|mean|max) R.I.R.L. (min|mean|max) R.P.L. (min|mean|max) Argumentative Essay 67 4.02% 551|1066.21|2309 46|239.96|806 598|938.03|1568 Academic Abstract 62 3.72% 196|1219.1|9057 48|204.05|478 440|980.48|2728 Proposal Planning 33 1.98% 148|1667.48|3951 41|337.09|2492 715|1415.39|3138 Open Letter 24 1.44% 191|1122.96|5704 44|220.25|486 548|1177.25|2427 Apology Letter 11 0.66% 318|1506.45|3677 92|211.27|445 581|762.27|954 Eulogy 10 0.60% 395|1872.7|7032 155|258.3|656 690|1153|1696 Tutorial Guide 7 0.42% 221|2352.57|5276 122|399.86|1359 936|1218.14|1491 Interview Questions 5 0.30% 203|821|1448 137|235.8|357 437|795.2|1044 Product Manual 2 0.12% 621|1796.5|2972 50|164.5|279 795|2444.5|4094 Table 10. Non-fiction Writing Genre Count % A.L. (min|mean|max) R.I.R.L. (min|mean|max) R.P.L. (min|mean|max) Essay 73 4.38% 374|1859.07|8585 41|279.99|1381 453|918.67|1630 Reviews 58 3.48% 225|1649|4891 42|193.55|932 502|1167.41|2661 Travel Writing 54 3.24% 483|1946|7733 80|180.31|804 525|1016.33|1587 Historical Stories 34 2.04% 359|2179.44|6856 41|275.03|426 675|1055.88|1930 Biography 24 1.44% 800|4625.62|12766 43|294.46|1718 588|1212.12|2041 Communication P. W. Script Functional P. W. Novel Funny L. Poetry Non-fiction W. Social Media Creation: 124 Advertising Copy: 74 Blog Post: 62 Debate Script: 59 Popular Science: 50 Speech Draft: 49 Slogan: 47 Product Review: 16 Script: 57 Argumentative Essay: 67 Academic Abstract: 62 Proposal Planning: 33 Open Letter: 24 Apology Letter: 11 Eulogy: 10 Tutorial Guide: 7 Interview Questions: 5 Product Manual: 2 Fiction/Story: 104 Everyday Stories: 50 Costume Novels: 40 Inference Stories: 31 Wuxia Novels: 30 Science Fiction: 28 Xuanhuan Novels: 27 Fantasy/Magic: 25 Fairy Tale: 25 Xianxia Novels: 24 Emotional Stories: 23 Military Novels: 23 Sports Novels: 22 Game Novels: 15 Funny Subculture: 31 Esports Funny: 18 Anime Funny: 8 Subcultural Identity: 4 Fan Circle Funny: 3 Anti-Consumption Funny: 2 Internet Jargon: 1 Cross-language Funny: 1 Poetry: 128 Essay: 73 Reviews: 58 Travel Writing: 54 Historical Stories: 34 Biography: 24 Communication P. W Script Functional P. W. Novel Funny L. Poetry Non-fiction W. Figure 6. Complete distribution of all 51 genres in COIG-Writer, showing the full diversity of creative writing categories covered. 20 Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 2500 0 200 100 0 1000 Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 500 0 50 25 0 200 (c) The Chinese reasoning behavior distribution of model trained on COIG. (d) The Chinese reasoning behavior distribution of model trained on general data. Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 500 0 50 25 0 200 (a) The Chinese reasoning behavior distribution of model trained on humanannotated data. Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 500 0 50 25 0 200 (b) The Chinese reasoning behavior distribution of model trained on COIG + general data (10K). Figure 7. Reasoning behavior analysis on Chinese creative writing. Models trained with COIGWriter data exhibit balanced distributions across reasoning types, while baseline models show predominant normal writing with minimal deep reasoning. (c) The English reasoning behavior distribution of model trained on COIG. (d) The English reasoning behavior distribution of model trained on general data. (a) Win rate comparison between different training configurations on English creative writing tasks. (b) The English reasoning behavior distribution of model trained on COIG + general data (10K). Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 2000 0 100 50 0 250 Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 1000 0 100 50 0 250 Step Position Behavior Deep Reasoning SelfExploration Normal Writing SelfReflection 0.0 0.2 0.4 0.6 0.8 1.0 0 2000 0 100 50 0 250 10K Baseline Origin 10K Baseline Origin 0 10 20 30 40 50 60 70 Figure 8. Reasoning behavior analysis on English creative writing. The baseline model maintains consistent deep reasoning patterns, while COIG-Writer variants show disrupted reasoning flows, explaining their inferior performance. 21 D. Prompts D.1. Prompt for pre-analyzing answer usability During the annotation process, it was observed that annotators spent substantial time on annotation only to find that the quality of the answers was unsatisfactory at the final scoring stage. Consequently, a pre-analysis step was introduced to examine the usability of the answers. D.2. Prompt for analyzing answers Provide a comprehensive analysis of the answer to help annotators quickly grasp the general idea of the answer and accelerate the annotation process. Prompt for analyzing answers 我希望你扮演一位资深的创作者和分析师。请对以下片段进行专业分析: -{}- 请从以下方面进行详细分析: • 1. 内容结构:分析文本的整体架构、段落组织和逻辑流程。指出结构的优缺点及 对整体效果的影响。 • 2. 语言风格:评估语言的色彩、节奏、句式变化和词汇选择。这些元素如何塑造 了文本的整体风格和语调? • 3. 修辞手法:识别并评价使用的修辞设备(如比喻、隐喻、排比等)及其效果。 • 4. 有效性评估:文本在实现其意图方面的有效程度如何?哪些部分特别有力或薄 弱? • 5. 专业洞察:从[文本类型]创作领域的专业角度,指出这篇文本中的独特元素或 创新点。 请确保你的分析既有理论依据,又有实用价值,能帮助我理解这篇文本的创作技巧和 效果。 D.3. Prompt for analyzing answers and queries Provide a comprehensive analysis of the connection between answer and query, offer reliable ideas for annotators to write thoughts, and accelerate the annotation process. D.4. Prompt for evaluating querys Score the quality and creativity of the queries provided by annotators based on the given answers, screen out some low-quality data in advance, and reduce the pressure of manual quality inspection. D.5. Prompt for evaluating answers Score the quality and creativity of the answers provided by annotators based on the given query, filter out some low - quality data in advance, and reduce the pressure of manual quality inspection. 22 Prompt for analyzing answers and queries 你是一位资深内容分析专家,专注于解析文本创作背后的思维过程、结构设计和写作 技巧。请基于提供的问题/主题(query)和对应的回答内容(answer),对这篇回答进行全 面、专业的创作思路分析。请提供: - Query: {} - Answer: {} 请从以下维度进行分析: • 1. 需求理解与定位 - 对原始问题/主题的理解深度- 目标受众识别与内容定位- 核心问题提取与回应策 略 • 2. 内容架构设计 - 整体框架与结构布局- 开头与结尾的设计意图及效果- 主体部分的逻辑展开方式段落之间的衔接与层次关系 • 3. 表达技巧运用 - 语言风格与表达特点- 修辞手法与句式结构- 专业术语的运用与解释- 叙事策略与 读者引导方式 • 4. 论证方法策略- 论点构建与支持方式- 论据选择与运用效果- 反驳处理与多角度 思考- 说服力构建技巧 • 5. 创新与价值呈现 - 独特见解与创新点- 实用性建议的设计与呈现- 理论与实践的平衡处理- 知识深度 与广度的展示方式 • 整体效果评估 - 内容与原始需求的匹配度- 信息密度与可读性平衡- 专业性与通俗性的结合- 潜在 影响与应用价值 请记住你的分析旨在揭示这篇回答背后的创作思路、组织策略和表达技巧,帮助用户 理解创作者如何解读需求、组织元素、设计架构并最终呈现内容。这将有助于用户学 习并掌握高质量内容创作的思维模式和方法论,提升自身的内容创作能力。 D.6. Prompt for evaluating thoughts Score the quality and creativity of the thoughts provided by annotators based on the given query, screen out some low - quality data in advance, and reduce the pressure of manual quality inspection. E. Case Study To illustrate the model's performance on nuanced creative texts, this case study presents a short, atmospheric horror story written in Chinese and its corresponding English translation generated by the model. This example is intended to highlight the model's ability to preserve the original's tone, critical details, and narrative pacing. Baseline Case This case examines a generative output from the baseline model to illuminate its inherent limitations in maintaining narrative coherence. Notably, the baseline demonstrates proficiency in producing text with a polished, fluent stylistic register-consistent with its strengths in surfacelevel language generation. However, a deeper reading reveals critical deficits in structural 23 and semantic cohesion: sentences often suffer from broken inter-clausal semantic links, while paragraphs lack a coherent narrative thread, leading to disjointed content that fails to sustain logical progression. Specifically, the generated text frequently shifts between ideas without transitional reasoning or contextual grounding, resulting in a fragmented output where successive segments appear tangential or even contradictory. This discrepancy between stylistic fluency and logical coherence in the baseline's output not only exemplifies a common failure mode in current generation frameworks but also underscores the need for more robust modeling of inter-utterance and inter-paragraph semantic dependencies-key gaps this work aims to address. 24 Prompt for evaluating querys 你的任务是根据以下标准,基于给定的answer,对query进行精确的评估。输出应包括两 个部分:质量和创意性。请遵循以下细化的评分标准,以确保评估的细致性: 1. 质量性(评分1-10): 评估问题的清晰度和完整性,着重于问题是否明确且自包含,能否在不需要额外信息 的情况下得到完整的答案。 • 9-10分: 完全清晰、精准且自包含;无歧义,问题结构合理,能够直接解释和回 答,无需额外澄清;query几乎对齐了answer,answer中出现的大部分对象或概 念出现在query中,query很清晰的描述了需求。 • 7-8分: 基本清晰且易于理解;歧义很少,基本完整,尽管小的澄清可能有助于 提高精确性;query较好的对齐了answer,answer中出现的部分对象或概念出现 在query中,query较清晰的描述了需求。 • 5-6分: 可理解但缺乏一定清晰度;存在显著的歧义或细微缺失,可能需要 澄清以确保准确回答;query几乎没对齐answer,answer中出现的内容没出现 在query中。 • 3-4分: 相当模糊或缺失元素;需要大量解释,清晰度问题增加回答难度。 • 1-2分: 非常不清晰、模糊或不完整;难以解释或无法直接回答。 2. 创意性(评分1-10): 评估问题的原创性、创新思维和启发潜力,着重于问题是否打破常规思维模式,能否 激发独特视角和深度思考。 • 9-10分: 卓越创意,提出全新视角或前所未见的问题框架,巧妙连接不同领域,挑 战根深蒂固的假设,促使思维范式转换。 • 7-8分: 显著创意,以新颖方式重构熟悉问题,融合不同领域知识,提出令人意外 但有意义的问题角度,鼓励跳出常规思维框架。 • 5-6分: 中等创意,在传统问题基础上有所创新,提供略微出人意料的问题情境, 鼓励一定程度的非线性思维,但整体框架较为常见。 • 3-4分: 有限创意,主要遵循常规问题模式,问题形式或内容略有变化,但思路常 见,很少激发非常规思考。 • 1-2分: 极少创意,完全遵循标准化、传统的问题模式,无任何新颖元素或独特视 角,不鼓励创造性思维。 请严格按以下格式回复,不要包含其他内容: {{ "quality": 1-10, "creative": 1-10, }} 根据以上标准,对以下问题进行评估: answer:{ } query: { } 输出格式(示例): ```json { "quality": 8, "creative": 8 } ``` 25 Prompt for evaluating answers 你的任务是根据以下标准,基于给定的query,对answer进行精确的评估。输出应包括两 个部分:质量和创意性。请遵循以下细化的评分标准,以确保评估的细致性: 1. 质量性(评分1-10): 评估思考的质量、逻辑性和完整性: • 9-10分: 思考全面深入,逻辑严密,准确把握问题核心;分析角度多元,论证有 力,充分回应问题的各个方面。 • 7-8分: 思考较为完整,逻辑基本清晰;涵盖问题主要方面,有一定深度,但在某 些细节上可进一步拓展。 • 5-6分: 思考基本合理但不够全面;有一定分析但缺乏深度,对问题的理解和回应 存在部分不足。 • 3-4分: 思考存在明显缺陷;逻辑较弱,遗漏关键方面,对问题理解有限或偏离问 题核心。 • 1-2分: 思考质量低下;逻辑混乱,分析肤浅,未能有效回应问题,或严重误解问 题意图。 2. 创意性(评分1-10): 评估思考在原创性、启发性方面的表现: • 9-10分: 思考极具创新性,提出独特见解和全新视角;打破常规思维框架,融合多 领域知识,产生富有启发性的洞见。 • 7-8分: 思考有明显创新元素,展现非常规思维路径;提供新颖的分析角度,超越 表面层次,引发深度思考。 • 5-6分: 思考包含一定创新点,有自己的见解;思路较为常见但有独到之处,能在 一定程度上拓展问题讨论空间。 • 3-4分: 思考创新性有限,主要沿用常规分析方法;观点较为传统,很少跳出既定 框架思考。 • 1-2分: 思考几乎无创新,完全依循标准化、惯性的思维模式;未能提供任何新鲜 视角或独特分析。 请严格按以下格式回复,不要包含其他内容: query:{ } answer: { } 输出格式(示例): ```json { "quality": 8, "creative": 8 } ``` 26 Prompt for evaluating thoughts 你的任务是根据以下标准,基于给定的query,对thought进行精确的评估。输出应包括两 个部分:质量和创意性。请遵循以下细化的评分标准,以确保评估的细致性: 1. 质量性(评分1-10): 评估思考的质量、逻辑性和完整性: • 9-10分: 思考全面深入,逻辑严密,准确把握问题核心;分析角度多元,论证有 力,充分回应问题的各个方面。 • 7-8分: 思考较为完整,逻辑基本清晰;涵盖问题主要方面,有一定深度,但在某 些细节上可进一步拓展。 • 5-6分: 思考基本合理但不够全面;有一定分析但缺乏深度,对问题的理解和回应 存在部分不足。 • 3-4分: 思考存在明显缺陷;逻辑较弱,遗漏关键方面,对问题理解有限或偏离问 题核心。 • 1-2分: 思考质量低下;逻辑混乱,分析肤浅,未能有效回应问题,或严重误解问 题意图。 2. 创意性(评分1-10): 评估思考在原创性、启发性方面的表现: • 9-10分: 思考极具创新性,提出独特见解和全新视角;打破常规思维框架,融合多 领域知识,产生富有启发性的洞见。 • 7-8分: 思考有明显创新元素,展现非常规思维路径;提供新颖的分析角度,超越 表面层次,引发深度思考。 • 5-6分: 思考包含一定创新点,有自己的见解;思路较为常见但有独到之处,能在 一定程度上拓展问题讨论空间。 • 3-4分: 思考创新性有限,主要沿用常规分析方法;观点较为传统,很少跳出既定 框架思考。 • 1-2分: 思考几乎无创新,完全依循标准化、惯性的思维模式;未能提供任何新鲜 视角或独特分析。 请严格按以下格式回复,不要包含其他内容: query:{ } answer: { } 输出格式(示例): ```json { "quality": 8, "creative": 8 } ``` 27 Prompt 创作一篇悬疑短篇小说,文字风格恐怖惊悚,但结尾荒诞。夏天主人公应旧友之约去酒 吧,在酒吧卫生间发现恐怖尸体,朋友也变得恐怖,在逃生过程中得到骷髅帮助,又陷入 混乱的车祸,最后回到恐怖起始点。主旨是通过离奇的情节和恐怖氛围,引发读者对于未 知危险的恐惧与思考。 《迷雾》 The Mist "阿杰,快出来!"我扯着嗓子在卫生间门口喊,瓷砖缝隙里渗出的霉斑在昏黄灯光下像凝 固的血迹。老友的手机屏幕亮起时,我看到他正蹲在洗手台边,后颈上洇着一圈深色液 体。 "Ajie, get out here!" I yelled at the bathroom door, my throat raw. In the dim yellow light, the mold seeping from between the tiles looked like clotted bloodstains. When my old friend's phone screen lit up, I saw him squatting by the sink, a ring of dark liquid soaking the back of his neck. "尸体......"他突然直起身,指甲缝里卡着碎发,"就在...... 那个...... 女厕......"话没 说完,他忽然僵住。镜子里,我的倒影正盯着他后腰的青紫色瘀痕,那是我三天前帮他按 过的淤血。 "A body..." he said, suddenly straightening up, his fingernails jammed with bits of hair. "It's in... in the... women's restroom..." Before he could finish, he froze. In the mirror, my reflection was staring at the purplish-blue bruise on his lower back-the same bruise I had massaged for him three days ago. 走廊传来脚步声,我抓住他的手腕就跑。夜班保安正用消毒水擦拭门把手,看见我们 冲进吧台时,吧台灯突然熄灭。阿杰的喉结上下滚动,指缝里渗出血珠,混着酒精味的呕 吐物溅在他皮鞋上。 Footsteps echoed from the hallway. I grabbed his wrist and we ran. The night-shift security guard was wiping down a doorknob with disinfectant. As we scrambled behind the bar, the bar lights abruptly went out. Ajie's Adam's apple bobbed, and beads of blood seeped from between his fingers. Vomit, reeking of alcohol, splattered onto his leather shoes. "快看!"他指向落地窗。暴雨倾盆中,女厕方向腾起团团白雾,雾气里隐约浮现出 骷髅轮廓,森白色的肋骨在雨幕中泛着磷火般的光。我刚想报警,整栋建筑突然剧烈晃 动--警报器狂叫,消防通道门"哐当"撞开。 "Look!" he pointed towards the floor-to-ceiling window. In the pouring rain, plumes of white mist were rising from the direction of the women's restroom. A skeletal silhouette loomed within the fog, its stark white ribs glowing like phosphorescent fire against the curtain of rain. Just as I was about to call the police, the entire building began to shake violently-alarms blared, and the fire escape door slammed open with a clang. 28 "有人撞车了!"阿杰指着监控屏,画面里一辆救护车正碾过尸体残肢,司机戴着墨 镜,后视镜里反着我的脸。警笛声中,我们挤进停尸间。冷冻柜里的女尸皮肤已呈半透明 状,却在我们面前裂开一道血口子,露出里面裹着的......裹尸袋? "Someone crashed!" Ajie pointed at a security monitor. On the screen, an ambulance was running over the mangled limbs of a corpse. The driver was wearing sunglasses, and the rearview mirror reflected my own face. Amid the wailing sirens, we squeezed into the morgue. The female corpse in the refrigerated drawer had skin that was already semi-translucent, yet right before our eyes, a bloody gash split open, revealing what was wrapped inside... a body bag? 裹尸袋里蜷缩着的骷髅突然抬头,它的眼睛竟与阿杰后腰的瘀痕一模一样。"这 是......"我摸向自己口袋,发现今天早上在地铁站买的占卜纸牌,此刻正从裤裆里探 出一只苍白的手指,指缝里卡着. . . . . .卡着阿杰的手机。 The skeleton curled up inside the body bag suddenly looked up. Its eyes were identical to the bruise on Ajie's lower back. "This is..." I reached for my pocket and found the tarot cards I had bought at the subway station this morning. A pale finger was now poking out from my crotch, and wedged between its knuckles was... was Ajie's phone. 雨停后,停尸间恢复冰凉寂静。我盯着阿杰的后腰,那里原本的淤青已经褪成暗紫, 像某种正在消散的胎记。"你说......"我指向窗外,那里悬着半块被车灯照亮的骨骼,"这 是谁的骨头?" After the rain stopped, the morgue returned to its icy silence. I stared at Ajie's lower back, where the original bruise had faded to a deep violet, like some kind of disappearing birthmark. "You think..." I pointed out the window, where half a bone, illuminated by car headlights, was suspended in the air. "Whose bone is that?" 他突然大笑起来:"你记得吗?去年冬天,我们在郊外迷路的时候,你不是一直往东走 吗?" He suddenly burst out laughing. "Do you remember? Last winter, when we got lost in the countryside, didn't you insist on walking east?" 29
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2510.14759
|
On the convergence of stochastic variance reduced gradient for
linear inverse problems∗
Bangti Jin†
Zehui Zhou†
Abstract
Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gra-
dient descent based on variance reduction, and is promising for solving large-scale inverse
problems. In this work, we analyze SVRG and a regularized version that incorporates a
priori knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We
prove that, with suitable constant step size schedules and regularity conditions, the regular-
ized SVRG can achieve optimal convergence rates in terms of the noise level without any
early stopping rules, and standard SVRG is also optimal for problems with nonsmooth so-
lutions under a priori stopping rules. The analysis is based on an explicit error recursion
and suitable prior estimates on the inner loop updates with respect to the anchor point.
Numerical experiments are provided to complement the theoretical analysis.
Keywords: stochastic variance reduced gradient; regularizing property; convergence rate
1
Introduction
In this work, we consider stochastic iterative methods for solving linear inverse problems in
Hilbert spaces:
A†x = y†,
(1.1)
where A† : X →Y = Y1 × · · · × Yn denotes the system operator that represents the data
formation mechanism and is given by
A†x := (A†,1x, · · · , A†,nx)T ,
∀x ∈X,
with bounded linear operators A†,i : X →Yi between Hilbert spaces X and Yi equipped with
norms ∥·∥X and ∥·∥Yi, respectively, and the superscript T denoting the vector transpose. x ∈X
denotes the unknown signal of interest and y† = (y†,1, · · · , y†,n)T ∈Y denotes the exact data,
i.e., y† = A†x† with x† being the minimum-norm solution relative to the initial guess x0, cf.
(2.1). In practice, we only have access to a noisy version yδ of the exact data y†, given by
yδ = (yδ
1, · · · , yδ
n)T = y† + ξ,
where ξ = (ξ1, · · · , ξn)T ∈Y is the noise in the data with a noise level δ = ∥ξ∥Y :=
qPn
i=1 ∥ξi∥2
Yi.
Below we assume δ < 1. Linear inverse problems arise in many practical applications, e.g.,
computed tomography [9, 22, 4] and positron emission tomography [10, 16, 23].
∗B. Jin is supported by Hong Kong RGC General Research Fund (14306824) and ANR / Hong Kong RGC
Joint Research Scheme (A-CUHK402/24) and a start-up fund from The Chinese University of Hong Kong.
†Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong (email:
b.jin@cuhk.edu.hk, zehuizhou@cuhk.edu.hk)
1
arXiv:2510.14759v1 [math.NA] 16 Oct 2025
Stochastic iterative algorithms, including stochastic gradient descent (SGD) [21, 11, 25] and
stochastic variance reduced gradient (SVRG) [15, 24, 13], have gained much interest in the
inverse problems community in recent years, due to their excellent scalability with respect to
data size.
We refer interested readers to the recent surveys [2, 12] for detailed discussions.
Specifically, consider the following optimization problem
J(x) =
1
2n∥A†x −yδ∥2
Y = 1
n
n
X
i=1
fi(x),
with
fi(x) = 1
2∥A†,ix −yδ
i ∥2
Yi.
Given an initial guess xδ
0 ≡x0 ∈X, SGD is given by
xδ
k+1 = xδ
k −ηkf′
ik(xδ
k),
k = 0, 1, · · · ,
while SVRG reads
xδ
k+1 = xδ
k −ηk
f′
ik(xδ
k) −f′
ik(xδ
[k/M]M) + J′(xδ
[k/M]M)
,
k = 0, 1, · · · ,
where {ηk}k≥0 ⊂(0, ∞) is the step size schedule, the index ik is sampled uniformly at random
from the set {1, . . . , n}, M is the frequency of computing the full gradient, and [·] denotes taking
the integral part of a real number.
By combining the full gradient J′(xδ
[k/M]M) of the objective J at the anchor point xδ
[k/M]M
with a random gradient gap f′
ik(xδ
k) −f′
ik(xδ
[k/M]M), SVRG can accelerate the convergence of
SGD and has become very popular in stochastic optimization [1, 5]. Its performance depends
on the frequency M of computing the full gradient, and M was suggested to be 2n and 5n for
convex and nonconvex optimization, respectively [15]. In practice, there are several variants of
SVRG, depending on the choice of the anchor point, e.g., last iterate and randomly selected
iterate within the inner loop. In this work, we focus on the version given in Algorithm 1, where
A∗
† and A∗
†,i denote the adjoints of the operators A† and A†,i, respectively.
Algorithm 1: SVRG for problem (1.1).
Set initial guess xδ
0 = x0, frequency M, and step size schedule {ηk}k≥0
for K = 0, 1, · · · do
compute gK = J′(xδ
KM) = 1
nA∗
†(A†xδ
KM −yδ)
for t = 0, 1, · · · , M −1 do
draw iKM+t i.i.d. uniformly from {1, · · · , n}
update xδ
KM+t+1 = xδ
KM+t −ηKM+t
A∗
†,iKM+tA†,iKM+t(xδ
KM+t −xδ
KM) + gK
end
check the stopping criterion.
end
The low-rank nature of A† implies that one can extract a low-rank subspace. Several works
have proposed subspace / low-rank versions of stochastic algorithms [18, 6, 19, 20, 8].
Let
A := (A1, · · · , An)T approximate A†. Using A in place of A† in Algorithm 1 gives Algorithm
2, termed as regularized SVRG (rSVRG) below.
rSVRG may be interpreted as integrating
learned prior into SVRG, if A is generated from paired training dataset {x(j), y(j)}N
j=1. The
regularization provided by the learned prior may relieve the need of early stopping.
The mathematical theory of SVRG for inverse problems from the perspective of regularization
theory has not been fully explored, and only recently has its convergence rate for solving linear
2
Algorithm 2: Regularized SVRG (rSVRG) for problem (1.1).
Set initial guess xδ
0 = x0, frequency M, and step size schedule {ηk}k≥0
for K = 0, 1, · · · do
compute gK = 1
nA∗(Axδ
KM −yδ)
for t = 0, 1, · · · , M −1 do
draw iKM+t i.i.d. uniformly from {1, · · · , n}
update xδ
KM+t+1 = xδ
KM+t −ηKM+t
A∗
iKM+tAiKM+t(xδ
KM+t −xδ
KM) + gK
end
check the stopping criterion.
end
inverse problems been investigated [13, 14].
In this work, we establish convergence of both
rSVRG and SVRG for solving linear inverse problems. See Theorem 2.1 for convergence rates in
terms of the iteration index k, Corollary 2.1 for convergence rates in terms of δ, and Corollary
2.3 for regularizing property. Note that rSVRG has a built-in regularization mechanism without
any need of early stopping rules and can outperform SVRG (i.e., higher accuracy), cf. Section 4.
Moreover, we establish the (optimal) convergence rates in both expectation and uniform sense
for both SVRG (when combined with a priori stopping rules) and rSVRG (cf. Theorem 2.1
and Corollary 2.1), while the prior works [13, 14] only studied convergence rates in expectation.
For SVRG, the condition for its optimal convergence rate in expectation is more relaxed than
that in [13]. However, unlike the results in [13], SVRG loses its optimality for smooth solutions
under the relaxed condition. For the benchmark source condition x† −x0 ∈Range(A∗
†) studied
in [14], the condition is either comparable or more relaxed; see Remark 2.1 for the details.
The rest of the work is organized as follows. In Section 2, we present and discuss the main
result, i.e., the convergence rate for (r)SVRG in Theorem 2.1. We present the proof in Section
3. Then in Section 4, we present several numerical experiments to complement the analysis,
which indicate the advantages of rSVRG over standard SVRG and Landweber method. Finally,
we conclude this work with further discussions in Section 5. In Appendix A, we collect lengthy
and technical proofs of several technical results. Throughout, we suppress the subscripts in the
norms and inner products, as the spaces are clear from the context.
2
Main result and discussions
To present the main result of the work, we first state the assumptions on the step size schedule
{ηj}j≥0, the reference solution x†, the unique minimum-norm solution relative to x0, given by
x† = arg
min
x∈X:A†x=y† ∥x −x0∥,
(2.1)
and the operator A, for analyzing the convergence of the rSVRG. We denote the operator norm
of Ai by ∥Ai∥and that of A by ∥A∥≤
pPn
i=1 ∥Ai∥2. N(A†) denotes the null space of A†.
Assumption 2.1. The following assumptions hold.
(i) The step size ηj = c0, j = 0, 1, · · · , with c0 ≤L−1, where L := max1≤i≤n ∥Ai∥2.
(ii) There exist ν > 0 and w ∈N(A†)⊥such that x† −x0 = Bν
† w and ∥w∥< ∞, with
B† = n−1A∗
†A† and N(A†)⊥being the orthogonal complement of N(A†).
3
(iii) Let a ≥0 be a constant. When a = 0, set A = A†. When a > 0, let A† be a compact opera-
tor with {σj, φj, ψj}∞
j=1 being its singular values and vectors, i.e., A†(·) = P∞
j=1 σj⟨φj, ·⟩ψj,
such that {σj}∞
j=1 ⊂[0, ∞), σj ≥σj′ ≥aδb > 0 for any j ≤j′ ≤J, and σj < aδb for any
j > J, with some b > 0. Set A(·) = PJ
j=1 σj⟨φj, ·⟩ψj.
The constant step size in Assumption 2.1(i) is commonly employed by SVRG [15]. (ii) is
commonly known as the source condition [3], which imposes certain regularity on the initial
error x† −x0 and is crucial for deriving convergence rates for iterative methods. Without the
condition, the convergence of regularization methods can be arbitrarily slow [3]. (iii) assumes
that the operator A captures important features of A†, and can be obtained by the truncated
SVD of A† that retains principal singular values σj such that σj ≥aδb. When a = 0, A = A†
and rSVRG reduces to the standard SVRG.
Let Fk denote the filtration generated by the random indices {i0, i1, . . . , ik−1}, F = W∞
k=1Fk,
(Ω, F, P) denote the associated probability space, and E[·] denote taking the expectation with
respect to the filtration F. The (r)SVRG iterate xδ
k is random but measurable with respect to
the filtration Fk. Now, we state the main result on the error eδ
k = xδ
k −x† of the (r)SVRG iterate
xδ
k with respect to x†. Below we follow the convention k0 := ln k, and let
C0 := max
∥A∥−1(5Ln−1M)−1
2 , L−1(10 + ln M)−1
,
C0 :=
100
√
LM∥A∥ln(2e2n
√
L∥A∥−1)
−1.
Theorem 2.1. Let Assumption 2.1 hold with b = (1 + 2ν)−1.
Then there exists some c∗
independent of k, n or δ such that, for any k ≥0,
E[∥eδ
k∥2]
1
2 ≤c∗k−min(ν, 1
2 ) + c∗
(
δ
2ν
1+2ν ,
a > 0,
n−1
2 √
kδ,
a = 0,
c0 < C0,
(2.2)
∥eδ
k∥≤√nc∗k−1
2 +max( 1
2 −ν,0) + c∗
(
δ
2ν
1+2ν ,
a > 0,
n−1
2 √
kδ,
a = 0,
c0 < C0.
(2.3)
The next corollary follows directly from Theorem 2.1.
Corollary 2.1. Under suitable step size schedules, the following statements hold.
(i) When a > 0 and b = (1 + 2ν)−1, i.e., rSVRG, for any small ϵ > 0,
E[∥eδ
k∥2]
1
2 ≤O(δ
2ν
1+2ν ),
∀k ≥k(δ) := δ
−
2ν
(1+2ν) min(ν, 1
2 ) ,
∥eδ
k∥≤O(δ
2ν
1+2ν ),
∀k ≥k(δ) := δ
−
2ν
(1+2ν) min(ν, 1
2 −ϵ) .
(ii) When a = 0, i.e., SVRG,
E[∥eδ
k(δ)∥2]
1
2 ≤O(δ
2ν
1+2ν ),
∀k(δ) = O(δ−
2
1+2ν ), ν ∈(0, 1
2],
∥eδ
k(δ)∥≤O(δ
2ν
1+2ν ),
∀k(δ) = O(δ−
2
1+2ν ), ν ∈(0, 1
2).
Remark 2.1. With a suitable choice of A, rSVRG can achieve optimal convergence rates without
any early stopping rule. SVRG is also optimal with a priori stopping rules for ν ∈(0, 1
2). These
4
rates are identical with that of SVRG in [13, 14]. Note that, when ν ∈(0, 1
2], the condition for
optimal convergence rates in expectation of standard SVRG is more relaxed than that in [13],
which requires also a special structure on A† and the step size c0 ≤O
(Mn−1∥A∥2)−1
. It is
comparable with that in [14] for small M and more relaxed than that for relatively large M.
Assumption 2.1(iii) is to simplify the proof in Section 3. In fact, without (iii), the result
of SVRG (i.e., a = 0) in Theorem 2.1 holds trivially, while the result for rSVRG (i.e., a > 0)
remains valid when A† can be approximated by some operator A suitably; see the next corollary.
Corollary 2.2. Let Assumption 2.1(i) and (ii) hold. Suppose that either A : X →Y is invertible
with ∥A−1∥≤a−1δ−b or A is compact with the nonzero singular value greater than aδb for some
a > 0 and b > 0. If ∥A−A†∥is sufficiently small, then Theorem 2.1 remains valid for (r)SVRG.
In the absence of the source condition in Assumption 2.1(ii), the regularizing property of
(r)SVRG remains valid in expectation and in the uniform sense.
Corollary 2.3. Let Assumption 2.1(i) and (iii) hold. Then rSVRG is regularizing itself, and
SVRG is regularizing when equipped with a suitable a priori stopping rule.
3
Convergence analysis
To prove Theorem 2.1, we first give several shorthand notation. We denote (r)SVRG iterates
for the noisy data yδ by xδ
k. For any K = 0, 1, · · · and t = 0, · · · , M −1, we define
eδ
KM+t = xδ
KM+t −x†,
∆δ
KM+t = xδ
KM+t −xδ
KM,
PKM+t = I −c0A∗
iKM+tAiKM+t,
P = I −c0B,
NKM+t = B −A∗
iKM+tAiKM+t,
and
ζ = n−1A∗ξ,
with B := E[A∗
i Ai] = n−1A∗A : X →X. Then there hold
∆δ
KM = 0,
E[PKM+t] = P
and
E[NKM+t] = 0.
We also define the summations
Φ
i′
i (j′, r) =
i′
X
j=i
(i′ + j′ −j)−rE[∥A∆δ
j∥2],
Φi′
i (j′, r) =
i′
X
j=i
(i′ + j′ −j)−r∥A∆δ
j∥,
ϕi′
i =
i′
X
j=i
P i′−jNj∆δ
j,
˜ϕi′ =
i′
X
j=0
P j,
∀i, i′, j′ ≥0,
and follow the conventions Pi′
j=i Rj = 0 and Pi′
j=−i Rj = Pi′
j=0 Rj for any sequence {Rj}j and
0 ≤i′ < i, and 0s = 1 for any s. Under Assumption 2.1(iii), Aδ := A† −A satisfies A∗Aδ = 0
and ∥Aδ∥< aδb. Similarly, let B† := E[A∗
†,iA†,i] = n−1A∗
†A† and Bδ := B† −B = n−1A∗
δAδ.
Then B∗Bδ = 0 and ∥Bδ∥< n−1a2δ2b.
5
3.1
Error decomposition
For any K ≥0 and 0 ≤t ≤M −1, we decompose the error eδ
KM+t+1 ≡xδ
KM+t+1 −x† and the
weighted successive error A∆δ
KM+t = A(xδ
KM+t −xδ
KM) between the (KM + t)th and KMth
iterations into the bias and variance, which plays a crucial role in the analysis.
Lemma 3.1. Let Assumption 2.1(i) hold. Then for any K ≥0, 0 ≤t ≤M −1 and k =
KM + t + 1, there hold
E[eδ
k] = P keδ
0 + c0 ˜ϕk−1ζ,
eδ
k −E[eδ
k] = c0ϕk−1
1
,
E[A∆δ
KM+t] = A(P t −I)P KMeδ
0 + c0AP KM ˜ϕt−1ζ,
A∆δ
KM+t −AE[∆δ
KM+t] = c0A(P t −I)ϕKM−1
1
+ c0AϕKM+t−1
KM+1
.
Proof. From the definitions of eδ
j, ∆δ
j, P and Nj, we derive
eδ
KM+t+1 = PKM+teδ
KM+t −c0NKM+teδ
KM + c0ζ = Peδ
KM+t + c0NKM+t∆δ
KM+t + c0ζ
= . . . = P t+1eδ
KM + c0
t+1
X
j=1
P j−1NKM+t+1−j∆δ
KM+t+1−j + c0 ˜ϕtζ
= . . . = P KM+t+1eδ
0 + c0ϕKM+t
0
+ c0 ˜ϕKM+tζ.
When t = M −1, this identity gives
eδ
(K+1)M = P (K+1)Meδ
0 + c0ϕ(K+1)M−1
0
+ c0 ˜ϕ(K+1)M−1ζ.
Then, with the convention Pi′
j=i Rj = 0 for any sequence {Rj}j and i′ < i, we have
A∆δ
KM+t =A
P KM+teδ
0 + c0ϕKM+t−1
0
+ c0 ˜ϕKM+t−1ζ −P KMeδ
0 −c0ϕKM−1
0
−c0 ˜ϕKM−1ζ
=A(P t −I)P KMeδ
0 + c0AP KM ˜ϕt−1ζ + c0A(P t −I)ϕKM−1
0
+ c0AϕKM+t−1
KM
.
Finally, the identities E[Nj] = 0 and ∆δ
0 = ∆δ
KM = 0 imply the desired identities.
Based on the triangle inequality, we bound the error eδ
k by
E[∥eδ
k∥2]
1
2 ≤∥E[eδ
k]∥+ E[∥eδ
k −E[eδ
k]∥2]
1
2
and
∥eδ
k∥≤∥E[eδ
k]∥+ ∥eδ
k −E[eδ
k]∥.
The next lemma bounds the bias ∥E[eδ
k]∥and variance E[∥eδ
k −E[eδ
k]∥2] (and ∥eδ
k −E[eδ
k]∥) in
terms of the weighted successive error E[∥A∆δ
j∥2] (and ∥A∆δ
j∥), respectively.
Lemma 3.2. Let Assumption 2.1(i) hold. Then for any k ≥0,
∥E[eδ
k]∥≤∥P keδ
0∥+ n−1
2 c0∥˜ϕk−1B
1
2 ∥δ,
E[∥eδ
k −E[eδ
k]∥2] ≤c0
2 Φ
k−1
1
(0, 1)
and
∥eδ
k −E[eδ
k]∥≤
rnc0
2 Φk−1
1
(0, 1
2).
6
Proof. Lemma 3.1 and the definitions ζ = n−1A∗ξ and B = n−1A∗A yield
∥E[eδ
k]∥≤∥P keδ
0∥+ n−1c0∥˜ϕk−1A∗ξ∥≤∥P keδ
0∥+ n−1
2 c0∥˜ϕk−1B
1
2 ∥δ.
Similarly, by Lemma 3.1 and the identity E[⟨Ni, Nj⟩|Fi] = 0 for any j > i, we have
E[∥eδ
k −E[eδ
k]∥2] =c2
0E[∥ϕk−1
1
∥2] = c2
0
k−1
X
j=1
E[∥P k−1−jNj∆δ
j∥2].
Then, by Lemmas A.2 and A.1, we derive
E[∥eδ
k −E[eδ
k]∥2] ≤c2
0
k−1
X
j=1
∥P k−1−jB
1
2 ∥2E[∥A∆δ
j∥2] ≤c0
2 Φ
k−1
1
(0, 1).
Similarly, by the triangle inequality and Lemmas A.2 and A.1, we obtain
∥eδ
k −E[eδ
k]∥≤c0
k−1
X
j=1
∥P k−1−jNj∆δ
j∥≤√nc0
k−1
X
j=1
∥P k−1−jB
1
2 ∥∥A∆δ
j∥≤
rnc0
2 Φk−1
1
(0, 1
2).
This completes the proof of the lemma.
Now we bound the weighted successive errors E[∥A∆δ
j∥2] and ∥A∆δ
j∥; see Appendix A for
the lengthy and technical proof.
Theorem 3.1. Let Assumption 2.1(i) hold. Then there exist some c1 and c2 independent of k,
n, δ and ν such that, for any k ≥0,
E[∥A∆δ
k∥2] ≤(c1 + c2δ2)(k + M)−2,
if c0 < C0,
(3.1)
∥A∆δ
k∥≤(c1 + c2δ)(k + M)−1,
if c0 < C0.
(3.2)
3.2
Convergence analysis
Now, using Theorem 3.1 and Lemma 3.2, we can prove Theorem 2.1.
Proof. For any k ≥1, the triangle inequality and Lemma 3.2 give
E[∥eδ
k∥2]
1
2 ≤∥P keδ
0∥+ n−1
2 c0∥˜ϕk−1B
1
2 ∥δ +
rc0
2 Φ
k−1
1
(0, 1).
(3.3)
When c0 < C0, by Theorem 3.1, the estimate (A.10) in Lemma A.4 implies
Φ
k−1
1
(0, 1) ≤4(c1 + c2δ2)k−1.
Next, we bound the first two terms in (3.3). By the definitions B† = B + Bδ and P = I −c0B,
and the identity B∗Bδ = 0, Assumption 2.1(ii) implies
eδ
0 = Bν
† w = (B + Bδ)νw = Bνw + Bν
δ w
and
PBν
δ w = Bν
δ w.
Together with Lemma A.1 and the estimate ∥Bδ∥< n−1a2δ2b, we obtain
∥P keδ
0∥=∥P kBνw + P kBν
δ w∥= ∥P kBνw + Bν
δ w∥
7
≤(∥P kBν∥+ ∥Bδ∥ν)∥w∥≤(ννc−ν
0 k−ν + n−νa2νδ2bν)∥w∥.
(3.4)
Next we bound I := n−1
2 c0∥˜ϕk−1B
1
2 ∥δ. If a = 0, Lemma A.1 and the triangle inequality imply
I ≤n−1
2 c0
k−1
X
j=0
∥P jB
1
2 ∥δ ≤
r c0
2n
k−1
X
j=0
j−1
2 δ ≤
r
2c0
n
√
kδ;
(3.5)
if a > 0, for any λ in the spectrum Sp(B) of B, either λ ≥n−1a2δ2b or λ = 0 holds, and thus
I ≤n−1
2 c0δ
sup
λ∈Sp(B)
k−1
X
j=0
(1 −c0λ)jλ
1
2 ≤n−1
2 c0δ
sup
λ≥n−1a2δ2b
1 −(1 −c0λ)k
c−1
0 λ−1
2
≤n−1
2 δ
sup
λ≥n−1a2δ2b λ−1
2 ≤a−1δ1−b.
(3.6)
Since b = (1 + 2ν)−1 and δ < 1, we derive from (3.3) and the above estimates that, when a = 0,
E[∥eδ
k∥2]
1
2 ≤ννc−ν
0 k−ν∥w∥+
√
2c0n−1
2 √
kδ +
√
2c0(√c1 + √c2δ)k−1
2
≤
ννc−ν
0 ∥w∥+
√
2c0(√c1 + √c2)
k−min(ν, 1
2 ) +
√
2c0n−1
2 √
kδ;
and when a > 0,
E[∥eδ
k∥2]
1
2 ≤(ννc−ν
0 k−ν + n−νa2νδ2bν)∥w∥+ a−1δ1−b +
√
2c0(√c1 + √c2δ)k−1
2
≤
ννc−ν
0 ∥w∥+
√
2c0(√c1 + √c2)
k−min(ν, 1
2 ) + (n−νa2ν∥w∥+ a−1)δ
2ν
1+2ν .
This proves the estimate (2.2). Similarly, for ∥eδ
k∥when c0 < C0, Lemma 3.2 yields
∥eδ
k∥≤∥P keδ
0∥+ n−1
2 c0∥˜ϕk−1B
1
2 ∥δ +
rnc0
2 Φk−1
1
(0, 1
2).
(3.7)
Theorem 3.1 and the inequality (A.11) in Lemma A.4 imply
Φk−1
1
(0, 1
2) ≤3
√
2(c1 + c2δ)k−1
2 ln k.
Then, by the conditions b = (1 + 2ν)−1 and δ < 1, we derive from (3.7) and the estimates
(3.4)–(3.6) that, when a = 0,
∥eδ
k∥≤ννc−ν
0 ∥w∥k−ν +
√
2c0n−1
2 √
kδ + 3√nc0(c1 + c2δ)k−1
2 ln k
≤
ννc−ν
0 ∥w∥+ 3√nc0(c1 + c2)
k−1
2 +max( 1
2 −ν,0) +
√
2c0n−1
2 √
kδ;
and when a > 0,
∥eδ
k∥≤(ννc−ν
0 k−ν + n−νa2νδ2bν)∥w∥+ a−1δ1−b + 3√nc0(c1 + c2δ)k−1
2 ln k
≤
ννc−ν
0 ∥w∥+ 3√nc0(c1 + c2)
k−1
2 +max( 1
2 −ν,0) +
n−νa2ν∥w∥+ a−1
δ
2ν
1+2ν .
This proves the estimate (2.3), and completes the proof of the theorem.
8
Remark 3.1. The parameter b in Assumption 2.1(iii) is set to b = (1 + 2ν)−1. Now we discuss
the choice of a. From the bound on ∥eδ
k∥(or E[∥eδ
k∥2]
1
2 ) in the proof of Theorem 2.1, we have
lim
k→∞∥eδ
k∥≤cn,ν(a)δ
2ν
1+2ν ,
with cn,ν(a) = n−νa2ν∥w∥+ a−1.
cn,ν attains its minimum at a∗:=
nν/(2ν∥w∥)
1
1+2ν , and cn,ν(a∗) =
(2ν)−
2ν
1+2ν +(2ν)
1
1+2ν
n−
ν
1+2ν ∥w∥
1
1+2ν .
To avoid the blow-up of a as ν →0+, let a = (nν/∥w∥)
1
1+2ν . Then cn,ν(a) = 2n−
ν
1+2ν ∥w∥
1
1+2ν .
Next we prove Corollary 2.2, which relax Assumption 2.1(iii).
Proof. When a > 0, let ∥A† −A∥≤ϵA ≤∥A†∥and B = n−1A∗A. Under Assumption 2.1(ii),
we can bound the term ∥P keδ
0∥in (3.3) and (3.7) by
∥P keδ
0∥=∥P kBν
† w∥≤∥P kBνw∥+ ∥P k(Bν
† −Bν)w∥≤∥P kBνw∥+ ∥Bν
† −Bν∥∥w∥.
When ν ∈(0, 1], by [17, Theorem 2.3], the term I := ∥Bν
† −Bν∥can be bounded by
I ≤∥B† −B∥ν = n−ν∥A∗
†A† −A∗A∥ν ≤n−ν ∥A∗
†∥∥A† −A∥+ ∥A∗
† −A∗∥∥A∥
ν
≤n−ν 2∥A†∥+ ϵA
νϵν
A ≤
3n−1∥A†∥
νϵν
A.
When ν = 1, ∥B† −B∥≤3n−1∥A†∥ϵA.
When ν > 1, the function h(z) := zν is Lipchitz
continuous on any closed interval in [0, ∞), and thus
I ≤ν max
∥B†∥, ∥B∥
ν−1∥B† −B∥
≤n−(ν−1)ν
∥A†∥+ ϵA
2(ν−1)∥B† −B∥≤22νn−νν∥A†∥2ν−1ϵA.
Then, let ϵA ≤δ
2ν
(1+2ν) min(1,ν) , we have I ≤c(ν)n−νϵA ≤c(ν)n−νδ
2ν
1+2ν , with the constant c(ν)
independent of δ and n. The assumption on A implies ∥A−1∥≤a−1δ−b or the nonzero singular
values σ of A such that σ ≥aδb > 0, which implies (3.6). Thus, Theorem 2.1 still holds.
The next remark complements Corollary 2.2 when A† is compact and has an approximate
truncated SVD A.
Remark 3.2. If A† is compact, with its SVD A†(·) = P∞
j=1 σj⟨φj, ·⟩ψj, where the singular
values {σj}∞
j=1 such that σj ≥σj′ ≥2aδb > 0 for any j ≤j′ ≤J and σj < 2aδb for any j > J.
For any small ϵA ∈(0, aδb) ⊂(0, ∥A†∥), we may approximate A† by A(·) = PJ
j=1 ˜σj⟨˜φj, ·⟩˜ψj with
{ ˜φj}J
j=1 and { ˜ψj}J
j=1 being orthonormal in X and Y , respectively, which satisfies ∥φj −˜φj∥<
ϵA and |˜σj −σj| < ϵA. Then we take AT(·) = PJ
j=1 σj⟨φj, ·⟩ψj. Let B(·) = n−1A∗A(·) =
1
n
PJ
j=1 ˜σ2
j ⟨˜φj, ·⟩˜φj, BT(·) = n−1A∗
TAT(·) = n−1 PJ
j=1 σ2
j ⟨φj, ·⟩φj, and BT,δ = B† −BT. Then
there hold B∗
TBT,δ = 0 and ∥BT,δ∥< 4n−1a2δ2b. Hence,
I = ∥Bν
† −Bν∥≤∥(BT + BT,δ)ν −Bν∥= ∥Bν
T + Bν
T,δ −Bν∥≤∥BT,δ∥ν + Iϵ,
with Iϵ := n−ν sup∥z∥=1
PJ
j=1
σ2ν
j ⟨φj, z⟩φj −˜σ2ν
j ⟨˜φj, z⟩˜φj
. By the triangle inequality,
Iϵ ≤n−ν sup
∥z∥=1
J
X
j=1
σ2ν
j ⟨φj, z⟩(φj −˜φj)
+ n−ν sup
∥z∥=1
J
X
j=1
(σ2ν
j
−˜σ2ν
j )⟨φj, z⟩˜φj
9
+ n−ν sup
∥z∥=1
J
X
j=1
˜σ2ν
j ⟨φj −˜φj, z⟩˜φj
≤n−ν(∥A†∥2ν + ∥A∥2ν)ϵA + n−ν sup
j≤J
(σ2ν
j
−˜σ2ν
j )
≤n−ν(1 + 22ν)∥A†∥2νϵA + 2νn−ν sup
j≤J
max(σ2ν−1
j
, ˜σ2ν−1
j
)ϵA
≤n−ν(1 + 22ν)∥A†∥2νϵA + 2νn−ν max
(aδb)2ν−1, (2∥A†∥)2ν−1
ϵA.
Let b = (1 + 2ν)−1. If ϵA ≤δ
max(1,2ν)
1+2ν
, then I ≤˜c(ν)n−νδ
2ν
1+2ν , with ˜c(ν) independent of δ and n.
The condition ˜σj ≥σj −ϵA ≥aδb > 0 for any j ≤J implies (3.6), and Theorem 2.1 still holds.
Last, we give the proof of Corollary 2.3.
Proof. Note that the initial error x†−x0 ∈Range(A∗
†). The polar decomposition A† = Q(A∗
†A†)
1
2
with a partial isometry Q (i.e. Q∗Q and QQ∗are projections) implies x†−x0 ∈Range
(A∗
†A†)
1
2
.
Thus, for any ϵ0 > 0, there exists some ˜x0, satisfying Assumption 2.1(ii) with ν = 1
2, such that
∥x0 −˜x0∥< ϵ0. Let ˜xδ
k be the (r)SVRG iterate starting with ˜x0 and ˜eδ
k = ˜xδ
k −x†. Then, when
b = 1
2, by Lemma A.1 and the inequality (3.4), we can bound ∥P keδ
0∥in (3.3) and (3.7) by
∥P keδ
0∥≤∥P k˜eδ
0∥+ ∥P k(˜eδ
0 −eδ
0)∥≤∥P kB
1
2
† w∥+ ϵ0
≤
(2c0)−1
2 k−1
2 + n−1
2 a
√
δ
∥w∥+ ϵ0.
Consequently,
E[∥eδ
k∥2]
1
2 ≤ϵ0 + c∗k−1
2 + c∗
(
√
δ,
a > 0,
n−1
2 √
kδ,
a = 0,
c0 < C0.
∥eδ
k∥≤ϵ0 + √nc∗k−1
2 ln k + c∗
(
√
δ,
a > 0,
n−1
2 √
kδ,
a = 0,
c0 < C0.
Taking the limit as ϵ0 →0+ completes the proof of the corollary.
4
Numerical experiments and discussions
In this section, we provide numerical experiments for several linear inverse problems to com-
plement the theoretical findings in Section 3. The experimental setting is identical to that in
[13]. We employ three examples, i.e., s-phillips (mildly ill-posed), s-gravity (severely ill-
posed) and s-shaw (severely ill-posed), which are generated from the code phillips, gravity
and shaw, taken from the MATLAB package Regutools [7] (publicly available at http://people.
compute.dtu.dk/pcha/Regutools/). All the examples are discretized into a finite-dimensional
linear system with the forward operator A† : Rm →Rn of size n = m = 1000, with A†x =
(A†,1x, · · · , A†,nx) for all x ∈Rm and A†,i : Rm →R. To precisely control the regularity index
ν in the source condition (cf. Assumption 2.1(ii)), we generate the exact solution x† by
x† = ∥(A∗
†A†)νxe∥−1
ℓ∞(A∗
†A†)νxe,
(4.1)
10
with xe being the exact solution provided by the package and ∥· ∥ℓ∞the maximum norm of
a vector. Note that the index ν in the source condition is slightly larger than the one used in
(4.1) due to the existing regularity νe of xe. The exact data y† is given by y† = A†x† and the
noisy data yδ is generated by yδ
i := y†,i + ϵ∥y†∥ℓ∞ξi, i = 1, · · · , n, where ξis follow the standard
normal distribution, and ϵ > 0 is the relative noise level.
All the iterative methods are initialized to zero, with a constant step size c0 = ∥A†∥−2 for
the Landweber method (LM) and c0 = O(c) for (r)SVRG, where c = mini(∥Ai∥−2) = L−1. The
constant step sizes c0 is taken for rSVRG so as to achieve optimal convergence while maintaining
computational efficiency across all noise levels. The methods are run for a maximum 1e5 epochs,
where one epoch refers to one Landweber iteration or nM/(n + M) (r)SVRG iterations, so that
their overall computational complexity is comparable. The frequency M of computing the full
gradient is set to M = 2n as suggested in [15]. The operator A for rSVRG is generated by
the truncated SVD of A† with b = 1/
1 + 2(ν + νe)
and a = (∥A†∥/∥y†∥)(nν+νe/c1)
1
1+2(ν+νe) ,
cf. Theorem 2.1 and Remark 3.1. Note the constant c1 is fixed for each problem with different
regularity indices ν and noise levels ϵ. One can also use the randomized SVD to generate A.
For LM, the stopping index k∗= k(δ) (measured in terms of epoch count) is chosen by the
discrepancy principle with τ = 1.01:
k(δ) = min{k ∈N : ∥A†xδ
k −yδ∥≤τδ},
which can achieve order optimality. For rSVRG, k∗is selected to be greater than the last index
at which the iteration error exceeds that of LM upon its termination or the first index for which
the iteration trajectory has plateaued. For SVRG, k∗is taken such that the error is the smallest
along the iteration trajectory. The accuracy of the reconstructions is measured by the relative
error e∗= E[∥xδ
k∗−x†∥2]
1
2 /∥x†∥for (r)SVRG, and e∗= ∥xδ
k∗−x†∥/∥x†∥for LM. The statistical
quantities generated by (r)SVRG are computed based on ten independent runs.
The numerical results for the examples with varying regularity indices ν and noise levels
ϵ are presented in Tables 1, 2, and 3. It is observed that rSVRG achieves an accuracy (with
much fewer iterations for relatively low-regularity problems) comparable to that for the LM
across varying regularity. SVRG can also achieve comparable accuracy in low-regularity cases,
indicating its optimality. However, with current step sizes, it is not optimal for highly regular
solutions, for which smaller step sizes are required to achieve the optimal error [13]. Typically,
problems with a higher noise level require fewer iterations. These observations agree with the
theoretical results of Theorem 2.1 and Corollary 2.1. Moreover, the error of rSVRG at its plateau
point is typically lower than that of the other two methods. The convergence trajectories of the
methods for the examples with ν = 0 in Fig. 4.1 show the advantage of rSVRG over the other
two methods as seen in Tables 1-3.
5
Concluding remarks
In this work, we have investigated stochastic variance reduced gradient (SVRG) and a regularized
variant (rSVRG) for solving linear inverse problems in Hilbert spaces. We have established the
regularizing property of both SVRG and rSVRG. Under the source condition, we have derived
convergence rates in expectation and in the uniform sense for (r)SVRG. These results indicate
the optimality of SVRG for nonsmooth solutions and the built-in regularization mechanism
and optimality of rSVRG. The numerical results for three linear inverse problems with varying
degree of ill-posedness show the advantages of rSVRG over both standard SVRG and Landweber
11
Table 1: The comparison between (r)SVRG and LM for s-phillips.
Method
rSVRG (c0 = c/4)
SVRG (c0 = c/4)
LM
ν
ϵ
e∗
k∗
limk→∞e
e∗
k∗
e∗
k∗
0
1e-3
1.93e-2
102.825
1.17e-2
1.52e-2
1170.900
1.93e-2
758
5e-3
2.81e-2
14.325
2.52e-2
6.13e-2
137.625
2.81e-2
102
1e-2
3.79e-2
12.000
2.63e-2
7.93e-2
70.050
3.81e-2
68
5e-2
8.81e-2
6.075
4.58e-2
1.54e-1
11.100
9.44e-2
12
0.25
1e-3
4.58e-3
206.700
4.29e-3
2.73e-2
819.225
4.58e-3
135
5e-3
1.48e-2
13.425
5.68e-3
5.73e-2
110.925
1.48e-2
60
1e-2
2.79e-2
12.825
9.43e-3
7.50e-2
58.650
2.81e-2
26
5e-2
4.13e-2
9.075
3.83e-2
1.37e-1
11.550
4.66e-2
10
0.5
1e-3
2.87e-3
24.300
1.01e-3
2.73e-2
841.575
2.90e-3
94
5e-3
1.00e-2
12.675
3.79e-3
5.79e-2
115.050
1.21e-2
23
1e-2
1.33e-2
11.475
7.52e-3
7.53e-2
60.375
1.51e-2
16
5e-2
2.85e-2
9.150
2.49e-2
1.44e-1
12.675
2.92e-2
8
1
1e-3
1.53e-3
15.225
7.22e-4
2.76e-2
866.250
1.92e-3
25
5e-3
3.35e-3
17.775
3.28e-3
5.93e-2
163.800
3.44e-3
16
1e-2
5.36e-3
14.700
4.36e-3
7.76e-2
66.900
5.54e-3
12
5e-2
1.57e-2
12.075
1.57e-2
1.43e-1
11.850
1.82e-2
5
Table 2: The comparison between (r)SVRG and LM for s-gravity.
Method
rSVRG (c0 = c)
SVRG (c0 = c)
LM
ν
ϵ
e∗
k∗
limk→∞e
e∗
k∗
e∗
k∗
0
1e-3
2.36e-2
279.525
1.30e-2
4.12e-2
1356.150
2.36e-2
1649
5e-3
3.99e-2
32.325
2.33e-2
9.05e-2
247.650
4.04e-2
255
1e-2
4.93e-2
25.425
3.65e-2
1.56e-1
93.900
5.30e-2
113
5e-2
8.56e-2
22.950
7.92e-2
3.50e-1
18.450
9.90e-2
22
0.25
1e-3
6.16e-3
51.975
3.03e-3
4.74e-2
1550.400
6.50e-3
319
5e-3
1.56e-2
37.275
1.20e-2
1.25e-1
198.300
1.64e-2
71
1e-2
1.82e-2
27.150
1.27e-2
1.65e-1
164.325
2.32e-2
43
5e-2
5.12e-2
19.275
2.72e-2
4.05e-1
29.400
5.35e-2
12
0.5
1e-3
3.34e-3
44.625
2.31e-3
3.82e-2
1106.400
3.39e-3
112
5e-3
7.56e-3
47.025
5.52e-3
1.26e-1
206.325
9.10e-3
40
1e-2
1.33e-2
44.550
1.04e-2
1.59e-1
176.100
1.41e-2
25
5e-2
3.38e-2
20.925
1.02e-2
4.00e-1
29.400
3.40e-2
8
1
1e-3
1.41e-3
48.000
9.87e-4
3.82e-2
1222.725
1.46e-3
42
5e-3
3.06e-3
35.400
1.11e-3
1.07e-1
259.800
4.11e-3
18
1e-2
3.17e-3
33.000
1.43e-3
1.57e-1
161.175
6.58e-3
12
5e-2
1.08e-2
23.175
8.15e-3
3.92e-1
29.400
1.48e-2
6
method. Note that both SVRG and rSVRG depend on the knowledge of the noise level. However,
in practice, the noise level may be unknown, and certain heuristic techniques are required for
their efficient implementation, e.g., as the a priori stopping rule or constructing the approximate
operator A. We leave this interesting question to future works.
12
Table 3: The comparison between (r)SVRG and LM for s-shaw.
Method
rSVRG (c0 = c)
SVRG (c0 = c)
LM
ν
ϵ
e∗
k∗
limk→∞e
e∗
k∗
e∗
k∗
0
1e-3
4.94e-2
39.825
4.94e-2
3.41e-2
4183.950
4.93e-2
22314
5e-3
9.22e-2
57.375
6.88e-2
4.93e-2
132.675
9.28e-2
4858
1e-2
1.53e-1
23.025
1.11e-1
5.98e-2
71.775
1.53e-1
642
5e-2
1.74e-1
20.925
1.71e-1
1.46e-1
26.925
1.78e-1
68
0.25
1e-3
1.69e-2
90.450
1.09e-2
2.01e-2
745.500
1.69e-2
1218
5e-3
2.21e-2
36.000
2.20e-2
4.34e-2
79.800
2.24e-2
139
1e-2
2.46e-2
23.625
2.24e-2
6.99e-2
56.550
2.59e-2
99
5e-2
5.21e-2
15.000
3.20e-2
1.75e-1
20.775
7.02e-2
24
0.5
1e-3
2.97e-3
42.075
2.84e-3
2.05e-2
598.725
3.16e-3
169
5e-3
7.80e-3
30.075
3.81e-3
5.17e-2
85.275
8.83e-3
78
1e-2
1.55e-2
21.075
5.89e-3
7.51e-2
56.175
1.69e-2
42
5e-2
4.63e-2
18.825
4.13e-2
1.97e-1
19.050
5.36e-2
16
1
1e-3
1.60e-3
40.875
5.63e-4
2.07e-2
225.300
1.80e-3
54
5e-3
5.16e-3
41.475
2.81e-3
5.60e-2
84.075
6.13e-3
25
1e-2
7.24e-3
28.650
6.31e-3
8.20e-2
55.125
1.18e-2
19
5e-2
4.79e-2
18.000
1.91e-2
2.12e-1
16.650
5.26e-2
6
A
Proof of Theorem 3.1
In this part, we give the technical proof of Theorem 3.1. First we give two technical estimates.
Lemma A.1. Under Assumption 2.1(i), for any s ≥0, k, t ∈N and ϵ ∈(0, 1
2], there hold
∥BsP k∥≤ssc−s
0 k−s,
∥(I −P t)P k∥≤t(k + t)−1,
∥B
1
2 (I −P t)P k∥≤21+ϵϵϵc−ϵ
0 ∥B∥
1
2 −ϵt(k + t)−(1+ϵ).
Proof. The first inequality can be found in [13, Lemma 3.4]. To show the second inequality, let
Sp(P) be the spectrum of P. Then there holds
∥(I −P t)P k∥=
sup
λ∈Sp(P)
|(1 −λt)λk| ≤sup
λ∈[0,1]
(1 −λt)λk.
Let g(λ) = (1 −λt)λk. Then g′(λ) = kλk−1 −(k + t)λk+t−1, so that g(λ) achieves its maximum
over the interval [0, 1] at λ = λ∗with λt
∗=
k
k+t = 1 −
t
k+t. Consequently,
∥(I −P t)P k∥≤g(λ∗) ≤t(k + t)−1.
The last one follows by
∥B
1
2 (I −P t)P k∥≤∥B∥
1
2 −ϵ∥BϵP
k+t
2 ∥∥(I −P t)P
k−t
2 ∥≤21+ϵϵϵc−ϵ
0 ∥B∥
1
2 −ϵt(k + t)−(1+ϵ).
This completes the proof of the lemma.
Lemma A.2. Let R : X →X be a deterministic bounded linear operator. Then for any j ≥0,
there hold
E[∥RNj∆δ
j∥2|Fj] ≤min
n−1L∥R∥2, ∥RB
1
2 ∥2
∥A∆δ
j∥2,
∥RNj∆δ
j∥≤min
√
L∥R∥, √n∥RB
1
2 ∥
∥A∆δ
j∥.
13
phillips
gravity
shaw
Figure 4.1: The convergence of the relative error e = E[∥xδ
k −x†∥2]
1
2 /∥x†∥versus the iteration
number k for phillips, gravity and shaw. The rows from top to bottom are for ϵ =1e-3, ϵ =5e-
3, ϵ =1e-2 and ϵ =5e-2, respectively. The intersection of the gray dashed lines represents the
stopping point, determined by the discrepancy principle, for LM along the iteration trajectory.
Proof. The definitions of Nj and B = E[A∗
ijAij|Fj] and the bias-variance decomposition imply
E[∥RNj∆δ
j∥2|Fj] =E[∥R(B −A∗
ijAij)∆δ
j∥2|Fj] = E[∥RA∗
ijAij∆δ
j∥2|Fj] −∥RB∆δ
j∥2
≤L∥R∥2E[∥Aij∆δ
j∥2|Fj] = L∥R∥2 1
n
n
X
i=1
∥Ai∆δ
j∥2 = n−1L∥R∥2∥A∆δ
j∥2.
14
Note that Nj = A∗(n−1A −bijAij), with bij ∈Rn being the ijth Cartesian basis vector. Then
the identity E[bijAij∆δ
j|Fj] = n−1A∆δ
j and the bias-variance decomposition yield
E[∥RNj∆δ
j∥2|Fj] = E[∥RA∗(n−1A −bijAij)∆δ
j∥2|Fj] ≤E[∥RA∗∥2∥(n−1A −bijAij)∆δ
j∥2|Fj]
=n∥RB
1
2 ∥2 E[∥bijAij∆δ
j∥2|Fj] −∥n−1A∆δ
j∥2
≤n∥RB
1
2 ∥2E[∥Aij∆δ
j∥2|Fj] ≤∥RB
1
2 ∥2∥A∆δ
j∥2.
These estimates and the inequality ∥RNj∆δ
j∥2 ≤nE[∥RNj∆δ
j∥2|Fj] complete the proof.
The proof of Theorem 3.1 is lengthy and technical, and requires several technical lemmas.
The first lemma provides bounds on the bias and variance components of the weighted successive
error A∆δ
k in terms of the iteration index.
Lemma A.3. Let Assumption 2.1(i) hold. Then for any k ≥1, kc := k−M −2, kM := [k/M]M
and ϵ ∈(0, 1
2], there hold
∥E[A∆δ
k]∥≤(∥A∥∥eδ
0∥+ δ)Mk−1,
(A.1)
E[∥A∆δ
k −E[A∆δ
k]∥2] ≤
n−1c2
0L∥A∥2
M2Φ
kc
1 (M + 1, 2) + Φ
k−1
kc+1(0, 0)
,
2−1c0L
8M2Φ
kc
1 (M + 1, 3) + Φ
kM−1
kc+1 (0, 1) + Φ
k−1
kM+1(0, 1)
,
(A.2)
∥A∆δ
k −E[A∆δ
k]∥≤c0
√
L∥A∥
21+ϵϵϵnϵ
cϵ
0∥A∥2ϵ MΦkc
1 (M + 1, 1 + ϵ) + Φk−1
kc+1(0, 0)
.
(A.3)
Proof. Let k = KM + t with K ≥0 and 1 ≤t ≤M −1. Similar to the proof of Lemma 3.2, for
the bias ∥E[A∆δ
KM+t]∥, by the definitions of ζ, B and ˜ϕt−1, the identity
∥AP KM ˜ϕt−1A∗∥= nc−1
0 ∥(I −P t)P KM∥,
and Lemma A.1, we derive the estimate (A.1) from Lemma 3.1 that
∥E[A∆δ
KM+t]∥≤∥A(P t −I)P KM∥∥eδ
0∥+ n−1c0∥AP KM ˜ϕt−1A∗∥δ
≤(∥A∥∥eδ
0∥+ δ)∥(I −P t)P KM∥≤(∥A∥∥eδ
0∥+ δ)M(KM + t)−1.
Next let St,j = c0∥A(P t −I)P KM−1−jNj∆δ
j∥and Tt,j = c0∥AP KM+t−1−jNj∆δ
j∥. Then for the
variance, when K ≥1, by Lemma 3.1 and the identity E[⟨Ni, Nj⟩|Fi] = 0 for any j > i, we have
E[∥A∆δ
KM+t −E[A∆δ
KM+t]∥2] = I1 + I2 + I3,
with I1 =
(K−1)M+t−2
X
j=1
E[S2
t,j],
I2 =
KM−1
X
j=(K−1)M+t−1
E[S2
t,j]
and
I3 =
KM+t−1
X
j=KM+1
E[T 2
t,j].
By Lemma A.2, the following estimates hold
E[S2
t,j] ≤c2
0L∥B
1
2 (P t −I)P KM−1−j∥2E[∥A∆δ
j∥2] ≤c2
0L∥B∥∥(P t −I)P KM−1−j∥2E[∥A∆δ
j∥2],
E[T 2
t,j] ≤c2
0L∥B
1
2 P KM+t−1−j∥2E[∥A∆δ
j∥2] ≤c2
0L∥B∥E[∥A∆δ
j∥2].
Then, by Lemma A.1, we deduce
I1 ≤c2
0L∥B∥t2
(K−1)M+t−2
X
j=1
(KM + t −1 −j)−2E[∥A∆δ
j∥2],
15
I2 ≤c2
0L∥B∥
KM−1
X
j=(K−1)M+t−1
E[∥A∆δ
j∥2]
and
I3 ≤c2
0L∥B∥
KM+t−1
X
j=KM+1
E[∥A∆δ
j∥2].
Meanwhile, by the commutativity of B, P and Lemma A.1 with ϵ = 1
2, we get
I1 ≤4c0Lt2
(K−1)M+t−2
X
j=1
(KM + t −1 −j)−3E[∥A∆δ
j∥2],
I2 ≤1
2c0L
KM−1
X
j=(K−1)M+t−1
(KM −1 −j)−1E[∥A∆δ
j∥2],
I3 ≤1
2c0L
KM+t−1
X
j=KM+1
(KM + t −1 −j)−1E[∥A∆δ
j∥2].
Similarly, when K = 0, there hold
E[∥A∆δ
t −E[A∆δ
t]∥2] =
t−1
X
j=1
E[T 2
t,j] ≤
c2
0L∥B∥
t−1
X
j=1
E[∥A∆δ
j∥2],
1
2c0L
t−1
X
j=1
(t −1 −j)−1E[∥A∆δ
j∥2].
Then combining the preceding estimates with ∥B∥= n−1∥A∥2 and t ≤M −1 gives the estimate
(A.2). Finally, when K ≥1, by Lemma 3.1 and the triangle inequality, we derive
∥∆δ
KM+t −E[A∆δ
KM+t]∥≤
KM−1
X
j=1
St,j +
KM+t−1
X
j=KM+1
Tt,j.
Thus for any ϵ ∈(0, 1
2], by Lemmas A.1 and A.2 and the identity ∥B∥= n−1∥A∥2, we have
St,j ≤c0
√
nL∥B
1
2 (P t −I)P KM−1−j∥∥A∆δ
j∥
≤
(
21+ϵϵϵc1−ϵ
0
nϵ√
L∥A∥1−2ϵt(KM + t −1 −j)−(1+ϵ)∥A∆δ
j∥,
∀j ≤(K −1)M + t −2
c0
√
L∥A∥∥A∆δ
j∥,
∀j ≥(K −1)M + t −1
,
Tt,j ≤c0
√
L∥AP KM+t−1−j∥∥A∆δ
j∥≤c0
√
L∥A∥∥A∆δ
j∥.
When K = 0, there holds
∥A∆δ
t −E[A∆δ
t]∥≤
t−1
X
j=1
Tt,j ≤c0
√
L∥A∥
t−1
X
j=1
∥A∆δ
j∥.
Combining these estimates with t ≤M −1 gives the estimate (A.3).
The next lemma gives several basic estimates on the following summations
Φ
j2
j1(i, r) =
j2
X
j=j1
(j2 + i −j)−rE[∥A∆δ
j∥2]
and
Φj2
j1(i, r) =
j2
X
j=j1
(j2 + i −j)−r∥A∆δ
j∥.
16
Lemma A.4. For any k ≥1, let kc := k −M −2 and kM := [k/M]M. If there holds
max
E[∥A∆δ
k∥2]
1
2 , ∥A∆δ
k∥
≤c(k + M)−1,
(A.4)
then for any k > M, k ̸= kM and ϵ ∈(0, 1
2], there hold
Φ
k−1
kc+1(0, 0) ≤c2
K,1c2M(k + M)−2,
Φk−1
kc+1(0, 0) ≤cK,1cM(k + M)−1,
(A.5)
Φ
kM−1
kc+1 (0, 1) + Φ
k−1
kM+1(0, 1) ≤c2
K,1c2(3 + 2 ln M)(k + M)−2,
(A.6)
Φ
kc
1 (M + 1, 2) ≤4c2
K,1c2M−1(k + M)−2,
(A.7)
Φ
kc
1 (M + 1, 3) ≤c2
K,1cK,2c2M−2(k + M)−2,
(A.8)
Φkc
1 (M + 1, ϵ + 1) ≤2cK,1cK,ϵcM−ϵ(k + M)−1,
(A.9)
Φ
k−1
1
(0, 1) ≤4c2k−1,
(A.10)
Φk−1
1
(0, 1
2) ≤3
√
2ck−1
2 ln k,
(A.11)
where cK,1 = 1 + 2
K , cK,2 = 2 +
3
K+1 +
6 ln M
(K+1)2 and cK,ϵ = e−1+1
ϵ
+ 2ϵ ln M
(K+1)ϵ with K = [k/M],
limK→∞cK,1 = 1, limK→∞cK,2 = 2, and limK→∞cK,ϵ = (e−1 + 1)ϵ−1.
Proof. Let k = KM + t with K ≥1 and t = 1, · · · , M −1. Then there holds the inequality:
(k −1)−1 ≤cK,1(k + M)−1.
(A.12)
The estimates in (A.5) follow directly from (A.4), the identity ∆δ
KM = ∆δ
(K−1)M = 0 and (A.12):
Φ
k−1
kc+1(0, 0) =
k−1
X
j=kc+1
E[∥A∆δ
j∥2] ≤c2M(k −1)−2 ≤c2
K,1c2M(k + M)−2,
Φk−1
kc+1(0, 0) =
k−1
X
j=kc+1
∥A∆δ
j∥≤cM(k −1)−1 ≤cK,1cM(k + M)−1.
Next for the estimate (A.6), we have
Φ
kM−1
kc+1 (0, 1) + Φ
k−1
kM+1(0, 1)
=
kM−1
X
j=kc+1
(kM −1 −j)−1E[∥A∆δ
j∥2] +
k−1
X
j=kM+1
(k −1 −j)−1E[∥A∆δ
j∥2]
=
M−t
X
j=0
j−1E[∥A∆δ
kM−1−j∥2] +
t−2
X
j=0
j−1E[∥A∆δ
k−1−j∥2]
≤c2
M−t
X
j=0
j−1(kM −1 −j + M)−2 +
t−2
X
j=0
j−1(k −1 −j + M)−2
≤c2(k −1)−2I,
with I =
M−t
X
j=0
j−1 +
t−2
X
j=0
j−1,
17
where I is bounded by
I ≤4 + ln(M −t) + ln t ≤4 + 2 ln M
2 = 4 −2 ln 2 + 2 ln M ≤3 + 2 ln M.
Then with the estimate (A.12), there holds
Φ
kM−1
kc+1 (0, 1) + Φ
k−1
kM+1(0, 1) ≤c2
K,1c2(3 + 2 ln M)(k + M)−2.
Next, we derive the estimates (A.7), (A.8) and (A.10). For the estimate (A.7), by the splitting
(j′ −j)−2j−2 = (j′)−2 (j′ −j)−1 + j−12 ≤2(j′)−2 (j′ −j)−2 + j−2
,
we obtain
Φ
kc
1 (M + 1, 2) = c2
k−M−2
X
j=1
(k −1 −j)−2(j + M)−2 = c2
k−2
X
j=M+1
(k + M −1 −j)−2j−2
≤2c2(k + M −1)−2
k−2
X
j=M+1
(k + M −1 −j)−2 + j−2
≤4c2
K,1c2M−1(k + M)−2.
Likewise, for the estimate (A.8), by the splitting
(j′ −j)−3j−2 = 3(j′)−4 (j′ −j)−1 + j−1
+ (j′)−3 2(j′ −j)−2 + j−2
+ (j′)−2(j′ −j)−3,
we derive
Φ
kc
1 (M + 1, 3) = c2
k−M−2
X
j=1
(k −1 −j)−3(j + M)−2 = c2
k−2
X
j=M+1
(k + M −1 −j)−3j−2
≤c2 k + M −1
−2
6
k + M −1
−2
k−2
X
j=M+1
j−1 + 3
k + M −1
−1
k−2
X
j=M+1
j−2 +
k−2
X
j=M+1
j−3
≤c2 k + M −1
−2h
6
(K + 1)M
−2 ln(KM + t −1) + 3
(K + 1)M
−1M−1 + 1
2M−2i
≤c2
K,1c2(k + M)−2M−2h
6(K + 1)−2 ln(K + 1) + ln M
+
3
K+1 + 1
2
i
.
Then the inequality
s−r ln s ≤(er)−1,
∀s, r > 0
(A.13)
implies the bound on Φ
kc
1 (M + 1, 3). For the estimate (A.10), the splitting (j′ −j)−1j−2 =
(j′)−1j−2 + (j′)−2 (j′ −j)−1 + j−1
implies
Φ
k−1
1
(0, 1) = c2
k−1
X
j=1
(k −1 −j)−1(j + M)−2 = c2
k+M−1
X
j=M+1
(k + M −1 −j)−1j−2
≤c2(k + M −1)−1
k+M−1
X
j=M+1
j−2 + (k + M −1)−1
k+M−1
X
j=M+1
(k + M −1 −j)−1 + j−1
≤c2(k + M −1)−1
M−1 + (k + M −1)−1 2 + 2 ln(k + M −1)
.
18
Then, using the inequality (A.13), we derive
Φ
k−1
1
(0, 1) ≤c2(k + M −1)−1(3M−1 + 2e−1) ≤4c2k−1.
Now, we derive the estimates (A.11) and (A.9) by splitting the summations into two parts. Let
kM = (k + M −1)/2. For the estimate (A.11), with the inequality (A.13), there holds
Φk−1
1
(0, 1
2) = c
k−1
X
j=1
(k −1 −j)−1
2 (j + M)−1 = c
k+M−1
X
j=M+1
(k + M −1 −j)−1
2 j−1 ≤c(I11 + I12),
with I11 =
[kM]
X
j=M+1
k
−1
2
M j−1
and
I12 =
2kM
X
j=[kM]+1
(k + M −1 −j)−1
2 k
−1
M .
The decomposition is well-defined with the convention Pi′
j=i Rj = 0 for any {Rj}j and i′ < i.
Then we have I11 ≤k
−1
2
M ln kM ≤
√
2k−1
2 ln k and I12 ≤2k
−1
2
M
≤2
√
2k−1
2 . Similarly, for the
estimate (A.9), when ϵ ∈(0, 1
2], we split Φkc
1 (M + 1, ϵ + 1) into
Φkc
1 (M + 1, ϵ + 1) = c
k−2
X
j=M+1
k + M −1 −j
−(1+ϵ)j−1 ≤c(I21 + I22),
with I21 =
[kM]
X
j=M+1
k
−(1+ϵ)
M
j−1
and
I22 =
k−2
X
j=[kM]+1
k + M −1 −j
−(1+ϵ)k
−1
M .
Then
I21 ≤k
−(1+ϵ)
M
ln kM ≤2
(eϵ)−1 + 2ϵ(K + 1)−ϵ ln M
M−ϵ(k + M −1)−1,
I22 ≤2ϵ−1M−ϵ(k + M −1)−1.
Finally, the inequality (k + M −1)−1 ≤cK,1(k + M)−1 completes the proof of the lemma.
The proof uses also the following elementary estimate on the function
f(ϵ) = 22+ϵϵϵc1−ϵ
0
nϵ√
LM1−ϵ∥A∥1−2ϵcK,ϵ.
(A.14)
Lemma A.5. If c0 < C0 and K ≥K0 with sufficiently large K0, then infϵ∈(0,1/2] f(ϵ) ≤3√e
5 .
Proof. By the definition of f(ϵ), we have
f(ϵ) =22+ϵϵϵc1−ϵ
0
nϵ√
LM1−ϵ∥A∥1−2ϵ (e−1 + 1)ϵ−1 + 2ϵ(K + 1)−ϵ ln M
=4
c0
√
LM∥A∥ϵ−1(2n
√
L∥A∥−1)
ϵ
1−ϵ 1−ϵ e−1 + 1 + 2ϵϵ(K + 1)−ϵ ln M
≤6
c0
√
LM∥A∥ϵ−1(2n
√
L∥A∥−1)
ϵ
1−ϵ 1−ϵ,
for any ϵ ∈(0, 1
2] and K ≥K0 with sufficiently large K0. Let g(ϵ) = ϵ−1(2n
√
L∥A∥−1)
ϵ
1−ϵ . Then
g′(ϵ) = ϵ−2(1 −ϵ)−2(2n
√
L∥A∥−1)
ϵ
1−ϵ ϵ ln(2n
√
L∥A∥−1) −(1 −ϵ)2
.
19
The fact ∥A∥≤
pPn
i=1 ∥Ai∥2 ≤
√
nL implies
c := 2 + ln(2n
√
L∥A∥−1) ≥2 + ln(2√n) > 2 + ln 2.
g(ϵ) attains its minimum over the interval (0, 1
2] at ϵ = ϵ∗=
2
c+
√
c2−4 <
1
2, and g(ϵ∗) =
c+
√
c2−4
2
e
2(c−2)
c+√
c2−4−2 ≤ce. Thus, for c0 < C0, we have
f(ϵ∗) ≤6
c0
√
LM∥A∥ce
1−
2
c+√
c2−4 ≤6
ec0
√
LM∥A∥ln(2e2n
√
L∥A∥−1)
1
2 < 3√e
5 .
This completes the proof of the lemma.
Now we can prove Theorem 3.1 by mathematical induction.
Proof. For the estimate (3.1), if k ≤K0M with some K0 ≥1, it holds for any sufficiently large c1
and c2. Now assume that it holds up to k = KM + t −1 with some K ≥K0 and 1 ≤t ≤M −1.
Then we prove the assertion for the case k = KM + t. (It holds trivially when t = 0, since
∆δ
KM = 0.) Fix k = KM + t and let kc := k −M −2 and kM := [k/M]M = KM. By the
bias-variance decomposition, and the estimates (A.1) and (A.2) in Lemma A.3, we have
E[∥A∆δ
k∥2] ≤2(∥A∥2∥eδ
0∥2 + δ2)M2k−2 + n−1c2
0L∥A∥2
M2Φ
kc
1 (M + 1, 2) + Φ
k−1
kc+1(0, 0)
.
Then, by setting c =
√
c1 + c2δ2, the estimates (A.5) and (A.7) and the inequality k−1 ≤
cK,1(k + M)−1 (cf. (A.12)) with cK,i given in Lemma A.4 yield
E[∥A∆δ
k∥2] ≤c2
K,1
h
5c2
0LMn−1∥A∥2(c1 + c2δ2) + 2(∥A∥2∥eδ
0∥2 + δ2)M2i
(k + M)−2
≤(c1 + c2δ2)(k + M)−2,
for any c0 < ∥A∥−1(5Ln−1M)−1
2 and K ≥K0, with sufficiently large K0 and c1, c2. Alterna-
tively, using the second estimate in (A.2), we can bound E[∥A∆δ
k∥2] by
E[∥A∆δ
k∥2] ≤2(∥A∥2∥eδ
0∥2 + δ2)M2k−2 + c0L
2
8M2Φ
kc
1 (M + 1, 3) + Φ
kM−1
kc+1 (0, 1) + Φ
k−1
kM+1(0, 1)
.
Then, with the estimates (A.6) and (A.8), we derive
E[∥A∆δ
k∥2] ≤c2
K,1
c0L(3
2 + ln M + 4cK,2)(c1 + c2δ2) + 2(∥A∥2∥eδ
0∥2 + δ2)M2
(k + M)−2
≤(c1 + c2δ2)(k + M)−2,
for any c0 < L−1(10+ln M)−1 and K ≥K0, with sufficiently large K0 and c1, c2. This completes
the proof of the estimate (3.1).
Next, we prove the estimate (3.2). Similarly, for the cases k ≤K0M with some K0 ≥1, the
estimate holds trivially for sufficiently large c1 and c2. Now, assume that the bound holds up
to k = KM + t −1 with some K ≥K0 and 1 ≤t ≤M −1, and prove the assertion for the case
k = KM + t. Fix k = KM + t and let kc := k −M −2 and kM := [k/M]M = KM. By the
triangle inequality ∥A∆δ
k∥≤∥E[A∆δ
k]∥+ ∥∆δ
k −E[A∆δ
k]∥, and (A.1) and (A.3), we have
∥A∆δ
k∥≤(∥A∥∥eδ
0∥+ δ)Mk−1 +
I1 + c0
√
L∥A∥Φk−1
kc+1(0, 0)
,
(A.15)
20
with I1 = 21+ϵϵϵc1−ϵ
0
nϵ√
L∥A∥1−2ϵMΦkc
1 (M + 1, 1 + ϵ). By (A.9) (with c = c1 + c2δ), we derive
I1 ≤22+ϵϵϵ(c1 + c2δ)c1−ϵ
0
nϵ√
L∥A∥1−2ϵM1−ϵcK,1cK,ϵ(k + M)−1 = (c1 + c2δ)f(ϵ)cK,1(k + M)−1,
with f(ϵ) given in (A.14). This, (A.15), (A.5) in Lemma A.4, and the inequality (A.12) yield
∥A∆δ
k∥≤cK,1
h
(c1 + c2δ)
c0
√
LM∥A∥+ f(ϵ)
+ (∥A∥∥eδ
0∥+ δ)M
i
(k + M)−1.
(A.16)
Then by Lemma A.5 and the inequality c0
√
LM∥A∥< 200−1 when c0 < C0, we derive from
(A.16) that
∥A∆δ
k∥≤(c1 + c2δ)(k + M)−1,
for any K ≥K0, with sufficiently large K0 and c1, c2, completing the proof of the theorem.
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On the convergence of stochastic variance reduced gradient for linear inverse problems∗ Bangti Jin† Zehui Zhou† Abstract Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gradient descent based on variance reduction, and is promising for solving large-scale inverse problems. In this work, we analyze SVRG and a regularized version that incorporates a priori knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We prove that, with suitable constant step size schedules and regularity conditions, the regularized SVRG can achieve optimal convergence rates in terms of the noise level without any early stopping rules, and standard SVRG is also optimal for problems with nonsmooth solutions under a priori stopping rules. The analysis is based on an explicit error recursion and suitable prior estimates on the inner loop updates with respect to the anchor point. Numerical experiments are provided to complement the theoretical analysis. Keywords: stochastic variance reduced gradient; regularizing property; convergence rate 1 Introduction In this work, we consider stochastic iterative methods for solving linear inverse problems in Hilbert spaces: A†x = y†, (1.1) where A† : X →Y = Y1 × · · · × Yn denotes the system operator that represents the data formation mechanism and is given by A†x := (A†,1x, · · · , A†,nx)T , ∀x ∈X, with bounded linear operators A†,i : X →Yi between Hilbert spaces X and Yi equipped with norms ∥·∥X and ∥·∥Yi, respectively, and the superscript T denoting the vector transpose. x ∈X denotes the unknown signal of interest and y† = (y†,1, · · · , y†,n)T ∈Y denotes the exact data, i.e., y† = A†x† with x† being the minimum-norm solution relative to the initial guess x0, cf. (2.1). In practice, we only have access to a noisy version yδ of the exact data y†, given by yδ = (yδ 1, · · · , yδ n)T = y† + ξ, where ξ = (ξ1, · · · , ξn)T ∈Y is the noise in the data with a noise level δ = ∥ξ∥Y := qPn i=1 ∥ξi∥2 Yi. Below we assume δ 0 and w ∈N(A†)⊥such that x† -x0 = Bν † w and ∥w∥ 0, let A† be a compact operator with {σj, φj, ψj}∞ j=1 being its singular values and vectors, i.e., A†(·) = P∞ j=1 σj⟨φj, ·⟩ψj, such that {σj}∞ j=1 ⊂[0, ∞), σj ≥σj′ ≥aδb > 0 for any j ≤j′ ≤J, and σj J, with some b > 0. Set A(·) = PJ j=1 σj⟨φj, ·⟩ψj. The constant step size in Assumption 2.1(i) is commonly employed by SVRG [15]. (ii) is commonly known as the source condition [3], which imposes certain regularity on the initial error x† -x0 and is crucial for deriving convergence rates for iterative methods. Without the condition, the convergence of regularization methods can be arbitrarily slow [3]. (iii) assumes that the operator A captures important features of A†, and can be obtained by the truncated SVD of A† that retains principal singular values σj such that σj ≥aδb. When a = 0, A = A† and rSVRG reduces to the standard SVRG. Let Fk denote the filtration generated by the random indices {i0, i1, . . . , ik-1}, F = W∞ k=1Fk, (Ω, F, P) denote the associated probability space, and E[·] denote taking the expectation with respect to the filtration F. The (r)SVRG iterate xδ k is random but measurable with respect to the filtration Fk. Now, we state the main result on the error eδ k = xδ k -x† of the (r)SVRG iterate xδ k with respect to x†. Below we follow the convention k0 := ln k, and let C0 := max ∥A∥-1(5Ln-1M)-1 2 , L-1(10 + ln M)-1 , C0 := 100 √ LM∥A∥ln(2e2n √ L∥A∥-1) -1. Theorem 2.1. Let Assumption 2.1 hold with b = (1 + 2ν)-1. Then there exists some c∗ independent of k, n or δ such that, for any k ≥0, E[∥eδ k∥2] 1 2 ≤c∗k-min(ν, 1 2 ) + c∗ ( δ 2ν 1+2ν , a > 0, n-1 2 √ kδ, a = 0, c0 0, n-1 2 √ kδ, a = 0, c0 0 and b = (1 + 2ν)-1, i.e., rSVRG, for any small ε > 0, E[∥eδ k∥2] 1 2 ≤O(δ 2ν 1+2ν ), ∀k ≥k(δ) := δ - 2ν (1+2ν) min(ν, 1 2 ) , ∥eδ k∥≤O(δ 2ν 1+2ν ), ∀k ≥k(δ) := δ - 2ν (1+2ν) min(ν, 1 2 -ε) . (ii) When a = 0, i.e., SVRG, E[∥eδ k(δ)∥2] 1 2 ≤O(δ 2ν 1+2ν ), ∀k(δ) = O(δ2 1+2ν ), ν ∈(0, 1 2], ∥eδ k(δ)∥≤O(δ 2ν 1+2ν ), ∀k(δ) = O(δ2 1+2ν ), ν ∈(0, 1 2). Remark 2.1. With a suitable choice of A, rSVRG can achieve optimal convergence rates without any early stopping rule. SVRG is also optimal with a priori stopping rules for ν ∈(0, 1 2). These 4 rates are identical with that of SVRG in [13, 14]. Note that, when ν ∈(0, 1 2], the condition for optimal convergence rates in expectation of standard SVRG is more relaxed than that in [13], which requires also a special structure on A† and the step size c0 ≤O (Mn-1∥A∥2)-1 . It is comparable with that in [14] for small M and more relaxed than that for relatively large M. Assumption 2.1(iii) is to simplify the proof in Section 3. In fact, without (iii), the result of SVRG (i.e., a = 0) in Theorem 2.1 holds trivially, while the result for rSVRG (i.e., a > 0) remains valid when A† can be approximated by some operator A suitably; see the next corollary. Corollary 2.2. Let Assumption 2.1(i) and (ii) hold. Suppose that either A : X →Y is invertible with ∥A-1∥≤a-1δ-b or A is compact with the nonzero singular value greater than aδb for some a > 0 and b > 0. If ∥A-A†∥is sufficiently small, then Theorem 2.1 remains valid for (r)SVRG. In the absence of the source condition in Assumption 2.1(ii), the regularizing property of (r)SVRG remains valid in expectation and in the uniform sense. Corollary 2.3. Let Assumption 2.1(i) and (iii) hold. Then rSVRG is regularizing itself, and SVRG is regularizing when equipped with a suitable a priori stopping rule. 3 Convergence analysis To prove Theorem 2.1, we first give several shorthand notation. We denote (r)SVRG iterates for the noisy data yδ by xδ k. For any K = 0, 1, · · · and t = 0, · · · , M -1, we define eδ KM+t = xδ KM+t -x†, ∆δ KM+t = xδ KM+t -xδ KM, PKM+t = I -c0A∗ iKM+tAiKM+t, P = I -c0B, NKM+t = B -A∗ iKM+tAiKM+t, and ζ = n-1A∗ξ, with B := E[A∗ i Ai] = n-1A∗A : X →X. Then there hold ∆δ KM = 0, E[PKM+t] = P and E[NKM+t] = 0. We also define the summations Φ i′ i (j′, r) = i′ X j=i (i′ + j′ -j)-rE[∥A∆δ j∥2], Φi′ i (j′, r) = i′ X j=i (i′ + j′ -j)-r∥A∆δ j∥, φi′ i = i′ X j=i P i′-jNj∆δ j, ̃φi′ = i′ X j=0 P j, ∀i, i′, j′ ≥0, and follow the conventions Pi′ j=i Rj = 0 and Pi′ j=-i Rj = Pi′ j=0 Rj for any sequence {Rj}j and 0 ≤i′ i, we have E[∥eδ k -E[eδ k]∥2] =c2 0E[∥φk-1 1 ∥2] = c2 0 k-1 X j=1 E[∥P k-1-jNj∆δ j∥2]. Then, by Lemmas A.2 and A.1, we derive E[∥eδ k -E[eδ k]∥2] ≤c2 0 k-1 X j=1 ∥P k-1-jB 1 2 ∥2E[∥A∆δ j∥2] ≤c0 2 Φ k-1 1 (0, 1). Similarly, by the triangle inequality and Lemmas A.2 and A.1, we obtain ∥eδ k -E[eδ k]∥≤c0 k-1 X j=1 ∥P k-1-jNj∆δ j∥≤√nc0 k-1 X j=1 ∥P k-1-jB 1 2 ∥∥A∆δ j∥≤ rnc0 2 Φk-1 1 (0, 1 2). This completes the proof of the lemma. Now we bound the weighted successive errors E[∥A∆δ j∥2] and ∥A∆δ j∥; see Appendix A for the lengthy and technical proof. Theorem 3.1. Let Assumption 2.1(i) hold. Then there exist some c1 and c2 independent of k, n, δ and ν such that, for any k ≥0, E[∥A∆δ k∥2] ≤(c1 + c2δ2)(k + M)-2, if c0 0, for any λ in the spectrum Sp(B) of B, either λ ≥n-1a2δ2b or λ = 0 holds, and thus I ≤n-1 2 c0δ sup λ∈Sp(B) k-1 X j=0 (1 -c0λ)jλ 1 2 ≤n-1 2 c0δ sup λ≥n-1a2δ2b 1 -(1 -c0λ)k c-1 0 λ-1 2 ≤n-1 2 δ sup λ≥n-1a2δ2b λ-1 2 ≤a-1δ1-b. (3.6) Since b = (1 + 2ν)-1 and δ 0, E[∥eδ k∥2] 1 2 ≤(ννc-ν 0 k-ν + n-νa2νδ2bν)∥w∥+ a-1δ1-b + √ 2c0(√c1 + √c2δ)k-1 2 ≤ ννc-ν 0 ∥w∥+ √ 2c0(√c1 + √c2) k-min(ν, 1 2 ) + (n-νa2ν∥w∥+ a-1)δ 2ν 1+2ν . This proves the estimate (2.2). Similarly, for ∥eδ k∥when c0 0, ∥eδ k∥≤(ννc-ν 0 k-ν + n-νa2νδ2bν)∥w∥+ a-1δ1-b + 3√nc0(c1 + c2δ)k-1 2 ln k ≤ ννc-ν 0 ∥w∥+ 3√nc0(c1 + c2) k-1 2 +max( 1 2 -ν,0) + n-νa2ν∥w∥+ a-1 δ 2ν 1+2ν . This proves the estimate (2.3), and completes the proof of the theorem. 8 Remark 3.1. The parameter b in Assumption 2.1(iii) is set to b = (1 + 2ν)-1. Now we discuss the choice of a. From the bound on ∥eδ k∥(or E[∥eδ k∥2] 1 2 ) in the proof of Theorem 2.1, we have lim k→∞∥eδ k∥≤cn,ν(a)δ 2ν 1+2ν , with cn,ν(a) = n-νa2ν∥w∥+ a-1. cn,ν attains its minimum at a∗:= nν/(2ν∥w∥) 1 1+2ν , and cn,ν(a∗) = (2ν)- 2ν 1+2ν +(2ν) 1 1+2ν nν 1+2ν ∥w∥ 1 1+2ν . To avoid the blow-up of a as ν →0+, let a = (nν/∥w∥) 1 1+2ν . Then cn,ν(a) = 2nν 1+2ν ∥w∥ 1 1+2ν . Next we prove Corollary 2.2, which relax Assumption 2.1(iii). Proof. When a > 0, let ∥A† -A∥≤εA ≤∥A†∥and B = n-1A∗A. Under Assumption 2.1(ii), we can bound the term ∥P keδ 0∥in (3.3) and (3.7) by ∥P keδ 0∥=∥P kBν † w∥≤∥P kBνw∥+ ∥P k(Bν † -Bν)w∥≤∥P kBνw∥+ ∥Bν † -Bν∥∥w∥. When ν ∈(0, 1], by [17, Theorem 2.3], the term I := ∥Bν † -Bν∥can be bounded by I ≤∥B† -B∥ν = n-ν∥A∗ †A† -A∗A∥ν ≤n-ν ∥A∗ †∥∥A† -A∥+ ∥A∗ † -A∗∥∥A∥ ν ≤n-ν 2∥A†∥+ εA νεν A ≤ 3n-1∥A†∥ νεν A. When ν = 1, ∥B† -B∥≤3n-1∥A†∥εA. When ν > 1, the function h(z) := zν is Lipchitz continuous on any closed interval in [0, ∞), and thus I ≤ν max ∥B†∥, ∥B∥ ν-1∥B† -B∥ ≤n-(ν-1)ν ∥A†∥+ εA 2(ν-1)∥B† -B∥≤22νn-νν∥A†∥2ν-1εA. Then, let εA ≤δ 2ν (1+2ν) min(1,ν) , we have I ≤c(ν)n-νεA ≤c(ν)n-νδ 2ν 1+2ν , with the constant c(ν) independent of δ and n. The assumption on A implies ∥A-1∥≤a-1δ-b or the nonzero singular values σ of A such that σ ≥aδb > 0, which implies (3.6). Thus, Theorem 2.1 still holds. The next remark complements Corollary 2.2 when A† is compact and has an approximate truncated SVD A. Remark 3.2. If A† is compact, with its SVD A†(·) = P∞ j=1 σj⟨φj, ·⟩ψj, where the singular values {σj}∞ j=1 such that σj ≥σj′ ≥2aδb > 0 for any j ≤j′ ≤J and σj J. For any small εA ∈(0, aδb) ⊂(0, ∥A†∥), we may approximate A† by A(·) = PJ j=1 ̃σj⟨ ̃φj, ·⟩ ̃ψj with { ̃φj}J j=1 and { ̃ψj}J j=1 being orthonormal in X and Y , respectively, which satisfies ∥φj - ̃φj∥ 0 for any j ≤J implies (3.6), and Theorem 2.1 still holds. Last, we give the proof of Corollary 2.3. Proof. Note that the initial error x†-x0 ∈Range(A∗ †). The polar decomposition A† = Q(A∗ †A†) 1 2 with a partial isometry Q (i.e. Q∗Q and QQ∗are projections) implies x†-x0 ∈Range (A∗ †A†) 1 2 . Thus, for any ε0 > 0, there exists some ̃x0, satisfying Assumption 2.1(ii) with ν = 1 2, such that ∥x0 - ̃x0∥ 0, n-1 2 √ kδ, a = 0, c0 0, n-1 2 √ kδ, a = 0, c0 0 is the relative noise level. All the iterative methods are initialized to zero, with a constant step size c0 = ∥A†∥-2 for the Landweber method (LM) and c0 = O(c) for (r)SVRG, where c = mini(∥Ai∥-2) = L-1. The constant step sizes c0 is taken for rSVRG so as to achieve optimal convergence while maintaining computational efficiency across all noise levels. The methods are run for a maximum 1e5 epochs, where one epoch refers to one Landweber iteration or nM/(n + M) (r)SVRG iterations, so that their overall computational complexity is comparable. The frequency M of computing the full gradient is set to M = 2n as suggested in [15]. The operator A for rSVRG is generated by the truncated SVD of A† with b = 1/ 1 + 2(ν + νe) and a = (∥A†∥/∥y†∥)(nν+νe/c1) 1 1+2(ν+νe) , cf. Theorem 2.1 and Remark 3.1. Note the constant c1 is fixed for each problem with different regularity indices ν and noise levels ε. One can also use the randomized SVD to generate A. For LM, the stopping index k∗= k(δ) (measured in terms of epoch count) is chosen by the discrepancy principle with τ = 1.01: k(δ) = min{k ∈N : ∥A†xδ k -yδ∥≤τδ}, which can achieve order optimality. For rSVRG, k∗is selected to be greater than the last index at which the iteration error exceeds that of LM upon its termination or the first index for which the iteration trajectory has plateaued. For SVRG, k∗is taken such that the error is the smallest along the iteration trajectory. The accuracy of the reconstructions is measured by the relative error e∗= E[∥xδ k∗-x†∥2] 1 2 /∥x†∥for (r)SVRG, and e∗= ∥xδ k∗-x†∥/∥x†∥for LM. The statistical quantities generated by (r)SVRG are computed based on ten independent runs. The numerical results for the examples with varying regularity indices ν and noise levels ε are presented in Tables 1, 2, and 3. It is observed that rSVRG achieves an accuracy (with much fewer iterations for relatively low-regularity problems) comparable to that for the LM across varying regularity. SVRG can also achieve comparable accuracy in low-regularity cases, indicating its optimality. However, with current step sizes, it is not optimal for highly regular solutions, for which smaller step sizes are required to achieve the optimal error [13]. Typically, problems with a higher noise level require fewer iterations. These observations agree with the theoretical results of Theorem 2.1 and Corollary 2.1. Moreover, the error of rSVRG at its plateau point is typically lower than that of the other two methods. The convergence trajectories of the methods for the examples with ν = 0 in Fig. 4.1 show the advantage of rSVRG over the other two methods as seen in Tables 1-3. 5 Concluding remarks In this work, we have investigated stochastic variance reduced gradient (SVRG) and a regularized variant (rSVRG) for solving linear inverse problems in Hilbert spaces. We have established the regularizing property of both SVRG and rSVRG. Under the source condition, we have derived convergence rates in expectation and in the uniform sense for (r)SVRG. These results indicate the optimality of SVRG for nonsmooth solutions and the built-in regularization mechanism and optimality of rSVRG. The numerical results for three linear inverse problems with varying degree of ill-posedness show the advantages of rSVRG over both standard SVRG and Landweber 11 Table 1: The comparison between (r)SVRG and LM for s-phillips. Method rSVRG (c0 = c/4) SVRG (c0 = c/4) LM ν ε e∗ k∗ limk→∞e e∗ k∗ e∗ k∗ 0 1e-3 1.93e-2 102.825 1.17e-2 1.52e-2 1170.900 1.93e-2 758 5e-3 2.81e-2 14.325 2.52e-2 6.13e-2 137.625 2.81e-2 102 1e-2 3.79e-2 12.000 2.63e-2 7.93e-2 70.050 3.81e-2 68 5e-2 8.81e-2 6.075 4.58e-2 1.54e-1 11.100 9.44e-2 12 0.25 1e-3 4.58e-3 206.700 4.29e-3 2.73e-2 819.225 4.58e-3 135 5e-3 1.48e-2 13.425 5.68e-3 5.73e-2 110.925 1.48e-2 60 1e-2 2.79e-2 12.825 9.43e-3 7.50e-2 58.650 2.81e-2 26 5e-2 4.13e-2 9.075 3.83e-2 1.37e-1 11.550 4.66e-2 10 0.5 1e-3 2.87e-3 24.300 1.01e-3 2.73e-2 841.575 2.90e-3 94 5e-3 1.00e-2 12.675 3.79e-3 5.79e-2 115.050 1.21e-2 23 1e-2 1.33e-2 11.475 7.52e-3 7.53e-2 60.375 1.51e-2 16 5e-2 2.85e-2 9.150 2.49e-2 1.44e-1 12.675 2.92e-2 8 1 1e-3 1.53e-3 15.225 7.22e-4 2.76e-2 866.250 1.92e-3 25 5e-3 3.35e-3 17.775 3.28e-3 5.93e-2 163.800 3.44e-3 16 1e-2 5.36e-3 14.700 4.36e-3 7.76e-2 66.900 5.54e-3 12 5e-2 1.57e-2 12.075 1.57e-2 1.43e-1 11.850 1.82e-2 5 Table 2: The comparison between (r)SVRG and LM for s-gravity. Method rSVRG (c0 = c) SVRG (c0 = c) LM ν ε e∗ k∗ limk→∞e e∗ k∗ e∗ k∗ 0 1e-3 2.36e-2 279.525 1.30e-2 4.12e-2 1356.150 2.36e-2 1649 5e-3 3.99e-2 32.325 2.33e-2 9.05e-2 247.650 4.04e-2 255 1e-2 4.93e-2 25.425 3.65e-2 1.56e-1 93.900 5.30e-2 113 5e-2 8.56e-2 22.950 7.92e-2 3.50e-1 18.450 9.90e-2 22 0.25 1e-3 6.16e-3 51.975 3.03e-3 4.74e-2 1550.400 6.50e-3 319 5e-3 1.56e-2 37.275 1.20e-2 1.25e-1 198.300 1.64e-2 71 1e-2 1.82e-2 27.150 1.27e-2 1.65e-1 164.325 2.32e-2 43 5e-2 5.12e-2 19.275 2.72e-2 4.05e-1 29.400 5.35e-2 12 0.5 1e-3 3.34e-3 44.625 2.31e-3 3.82e-2 1106.400 3.39e-3 112 5e-3 7.56e-3 47.025 5.52e-3 1.26e-1 206.325 9.10e-3 40 1e-2 1.33e-2 44.550 1.04e-2 1.59e-1 176.100 1.41e-2 25 5e-2 3.38e-2 20.925 1.02e-2 4.00e-1 29.400 3.40e-2 8 1 1e-3 1.41e-3 48.000 9.87e-4 3.82e-2 1222.725 1.46e-3 42 5e-3 3.06e-3 35.400 1.11e-3 1.07e-1 259.800 4.11e-3 18 1e-2 3.17e-3 33.000 1.43e-3 1.57e-1 161.175 6.58e-3 12 5e-2 1.08e-2 23.175 8.15e-3 3.92e-1 29.400 1.48e-2 6 method. Note that both SVRG and rSVRG depend on the knowledge of the noise level. However, in practice, the noise level may be unknown, and certain heuristic techniques are required for their efficient implementation, e.g., as the a priori stopping rule or constructing the approximate operator A. We leave this interesting question to future works. 12 Table 3: The comparison between (r)SVRG and LM for s-shaw. Method rSVRG (c0 = c) SVRG (c0 = c) LM ν ε e∗ k∗ limk→∞e e∗ k∗ e∗ k∗ 0 1e-3 4.94e-2 39.825 4.94e-2 3.41e-2 4183.950 4.93e-2 22314 5e-3 9.22e-2 57.375 6.88e-2 4.93e-2 132.675 9.28e-2 4858 1e-2 1.53e-1 23.025 1.11e-1 5.98e-2 71.775 1.53e-1 642 5e-2 1.74e-1 20.925 1.71e-1 1.46e-1 26.925 1.78e-1 68 0.25 1e-3 1.69e-2 90.450 1.09e-2 2.01e-2 745.500 1.69e-2 1218 5e-3 2.21e-2 36.000 2.20e-2 4.34e-2 79.800 2.24e-2 139 1e-2 2.46e-2 23.625 2.24e-2 6.99e-2 56.550 2.59e-2 99 5e-2 5.21e-2 15.000 3.20e-2 1.75e-1 20.775 7.02e-2 24 0.5 1e-3 2.97e-3 42.075 2.84e-3 2.05e-2 598.725 3.16e-3 169 5e-3 7.80e-3 30.075 3.81e-3 5.17e-2 85.275 8.83e-3 78 1e-2 1.55e-2 21.075 5.89e-3 7.51e-2 56.175 1.69e-2 42 5e-2 4.63e-2 18.825 4.13e-2 1.97e-1 19.050 5.36e-2 16 1 1e-3 1.60e-3 40.875 5.63e-4 2.07e-2 225.300 1.80e-3 54 5e-3 5.16e-3 41.475 2.81e-3 5.60e-2 84.075 6.13e-3 25 1e-2 7.24e-3 28.650 6.31e-3 8.20e-2 55.125 1.18e-2 19 5e-2 4.79e-2 18.000 1.91e-2 2.12e-1 16.650 5.26e-2 6 A Proof of Theorem 3.1 In this part, we give the technical proof of Theorem 3.1. First we give two technical estimates. Lemma A.1. Under Assumption 2.1(i), for any s ≥0, k, t ∈N and ε ∈(0, 1 2], there hold ∥BsP k∥≤ssc-s 0 k-s, ∥(I -P t)P k∥≤t(k + t)-1, ∥B 1 2 (I -P t)P k∥≤21+εεεc-ε 0 ∥B∥ 1 2 -εt(k + t)-(1+ε). Proof. The first inequality can be found in [13, Lemma 3.4]. To show the second inequality, let Sp(P) be the spectrum of P. Then there holds ∥(I -P t)P k∥= sup λ∈Sp(P) |(1 -λt)λk| ≤sup λ∈[0,1] (1 -λt)λk. Let g(λ) = (1 -λt)λk. Then g′(λ) = kλk-1 -(k + t)λk+t-1, so that g(λ) achieves its maximum over the interval [0, 1] at λ = λ∗with λt ∗= k k+t = 1 - t k+t. Consequently, ∥(I -P t)P k∥≤g(λ∗) ≤t(k + t)-1. The last one follows by ∥B 1 2 (I -P t)P k∥≤∥B∥ 1 2 -ε∥BεP k+t 2 ∥∥(I -P t)P k-t 2 ∥≤21+εεεc-ε 0 ∥B∥ 1 2 -εt(k + t)-(1+ε). This completes the proof of the lemma. Lemma A.2. Let R : X →X be a deterministic bounded linear operator. Then for any j ≥0, there hold E[∥RNj∆δ j∥2|Fj] ≤min n-1L∥R∥2, ∥RB 1 2 ∥2 ∥A∆δ j∥2, ∥RNj∆δ j∥≤min √ L∥R∥, √n∥RB 1 2 ∥ ∥A∆δ j∥. 13 phillips gravity shaw Figure 4.1: The convergence of the relative error e = E[∥xδ k -x†∥2] 1 2 /∥x†∥versus the iteration number k for phillips, gravity and shaw. The rows from top to bottom are for ε =1e-3, ε =5e3, ε =1e-2 and ε =5e-2, respectively. The intersection of the gray dashed lines represents the stopping point, determined by the discrepancy principle, for LM along the iteration trajectory. Proof. The definitions of Nj and B = E[A∗ ijAij|Fj] and the bias-variance decomposition imply E[∥RNj∆δ j∥2|Fj] =E[∥R(B -A∗ ijAij)∆δ j∥2|Fj] = E[∥RA∗ ijAij∆δ j∥2|Fj] -∥RB∆δ j∥2 ≤L∥R∥2E[∥Aij∆δ j∥2|Fj] = L∥R∥2 1 n n X i=1 ∥Ai∆δ j∥2 = n-1L∥R∥2∥A∆δ j∥2. 14 Note that Nj = A∗(n-1A -bijAij), with bij ∈Rn being the ijth Cartesian basis vector. Then the identity E[bijAij∆δ j|Fj] = n-1A∆δ j and the bias-variance decomposition yield E[∥RNj∆δ j∥2|Fj] = E[∥RA∗(n-1A -bijAij)∆δ j∥2|Fj] ≤E[∥RA∗∥2∥(n-1A -bijAij)∆δ j∥2|Fj] =n∥RB 1 2 ∥2 E[∥bijAij∆δ j∥2|Fj] -∥n-1A∆δ j∥2 ≤n∥RB 1 2 ∥2E[∥Aij∆δ j∥2|Fj] ≤∥RB 1 2 ∥2∥A∆δ j∥2. These estimates and the inequality ∥RNj∆δ j∥2 ≤nE[∥RNj∆δ j∥2|Fj] complete the proof. The proof of Theorem 3.1 is lengthy and technical, and requires several technical lemmas. The first lemma provides bounds on the bias and variance components of the weighted successive error A∆δ k in terms of the iteration index. Lemma A.3. Let Assumption 2.1(i) hold. Then for any k ≥1, kc := k-M -2, kM := [k/M]M and ε ∈(0, 1 2], there hold ∥E[A∆δ k]∥≤(∥A∥∥eδ 0∥+ δ)Mk-1, (A.1) E[∥A∆δ k -E[A∆δ k]∥2] ≤ n-1c2 0L∥A∥2 M2Φ kc 1 (M + 1, 2) + Φ k-1 kc+1(0, 0) , 2-1c0L 8M2Φ kc 1 (M + 1, 3) + Φ kM-1 kc+1 (0, 1) + Φ k-1 kM+1(0, 1) , (A.2) ∥A∆δ k -E[A∆δ k]∥≤c0 √ L∥A∥ 21+εεεnε cε 0∥A∥2ε MΦkc 1 (M + 1, 1 + ε) + Φk-1 kc+1(0, 0) . (A.3) Proof. Let k = KM + t with K ≥0 and 1 ≤t ≤M -1. Similar to the proof of Lemma 3.2, for the bias ∥E[A∆δ KM+t]∥, by the definitions of ζ, B and ̃φt-1, the identity ∥AP KM ̃φt-1A∗∥= nc-1 0 ∥(I -P t)P KM∥, and Lemma A.1, we derive the estimate (A.1) from Lemma 3.1 that ∥E[A∆δ KM+t]∥≤∥A(P t -I)P KM∥∥eδ 0∥+ n-1c0∥AP KM ̃φt-1A∗∥δ ≤(∥A∥∥eδ 0∥+ δ)∥(I -P t)P KM∥≤(∥A∥∥eδ 0∥+ δ)M(KM + t)-1. Next let St,j = c0∥A(P t -I)P KM-1-jNj∆δ j∥and Tt,j = c0∥AP KM+t-1-jNj∆δ j∥. Then for the variance, when K ≥1, by Lemma 3.1 and the identity E[⟨Ni, Nj⟩|Fi] = 0 for any j > i, we have E[∥A∆δ KM+t -E[A∆δ KM+t]∥2] = I1 + I2 + I3, with I1 = (K-1)M+t-2 X j=1 E[S2 t,j], I2 = KM-1 X j=(K-1)M+t-1 E[S2 t,j] and I3 = KM+t-1 X j=KM+1 E[T 2 t,j]. By Lemma A.2, the following estimates hold E[S2 t,j] ≤c2 0L∥B 1 2 (P t -I)P KM-1-j∥2E[∥A∆δ j∥2] ≤c2 0L∥B∥∥(P t -I)P KM-1-j∥2E[∥A∆δ j∥2], E[T 2 t,j] ≤c2 0L∥B 1 2 P KM+t-1-j∥2E[∥A∆δ j∥2] ≤c2 0L∥B∥E[∥A∆δ j∥2]. Then, by Lemma A.1, we deduce I1 ≤c2 0L∥B∥t2 (K-1)M+t-2 X j=1 (KM + t -1 -j)-2E[∥A∆δ j∥2], 15 I2 ≤c2 0L∥B∥ KM-1 X j=(K-1)M+t-1 E[∥A∆δ j∥2] and I3 ≤c2 0L∥B∥ KM+t-1 X j=KM+1 E[∥A∆δ j∥2]. Meanwhile, by the commutativity of B, P and Lemma A.1 with ε = 1 2, we get I1 ≤4c0Lt2 (K-1)M+t-2 X j=1 (KM + t -1 -j)-3E[∥A∆δ j∥2], I2 ≤1 2c0L KM-1 X j=(K-1)M+t-1 (KM -1 -j)-1E[∥A∆δ j∥2], I3 ≤1 2c0L KM+t-1 X j=KM+1 (KM + t -1 -j)-1E[∥A∆δ j∥2]. Similarly, when K = 0, there hold E[∥A∆δ t -E[A∆δ t]∥2] = t-1 X j=1 E[T 2 t,j] ≤ c2 0L∥B∥ t-1 X j=1 E[∥A∆δ j∥2], 1 2c0L t-1 X j=1 (t -1 -j)-1E[∥A∆δ j∥2]. Then combining the preceding estimates with ∥B∥= n-1∥A∥2 and t ≤M -1 gives the estimate (A.2). Finally, when K ≥1, by Lemma 3.1 and the triangle inequality, we derive ∥∆δ KM+t -E[A∆δ KM+t]∥≤ KM-1 X j=1 St,j + KM+t-1 X j=KM+1 Tt,j. Thus for any ε ∈(0, 1 2], by Lemmas A.1 and A.2 and the identity ∥B∥= n-1∥A∥2, we have St,j ≤c0 √ nL∥B 1 2 (P t -I)P KM-1-j∥∥A∆δ j∥ ≤ ( 21+εεεc1-ε 0 nε√ L∥A∥1-2εt(KM + t -1 -j)-(1+ε)∥A∆δ j∥, ∀j ≤(K -1)M + t -2 c0 √ L∥A∥∥A∆δ j∥, ∀j ≥(K -1)M + t -1 , Tt,j ≤c0 √ L∥AP KM+t-1-j∥∥A∆δ j∥≤c0 √ L∥A∥∥A∆δ j∥. When K = 0, there holds ∥A∆δ t -E[A∆δ t]∥≤ t-1 X j=1 Tt,j ≤c0 √ L∥A∥ t-1 X j=1 ∥A∆δ j∥. Combining these estimates with t ≤M -1 gives the estimate (A.3). The next lemma gives several basic estimates on the following summations Φ j2 j1(i, r) = j2 X j=j1 (j2 + i -j)-rE[∥A∆δ j∥2] and Φj2 j1(i, r) = j2 X j=j1 (j2 + i -j)-r∥A∆δ j∥. 16 Lemma A.4. For any k ≥1, let kc := k -M -2 and kM := [k/M]M. If there holds max E[∥A∆δ k∥2] 1 2 , ∥A∆δ k∥ ≤c(k + M)-1, (A.4) then for any k > M, k ̸= kM and ε ∈(0, 1 2], there hold Φ k-1 kc+1(0, 0) ≤c2 K,1c2M(k + M)-2, Φk-1 kc+1(0, 0) ≤cK,1cM(k + M)-1, (A.5) Φ kM-1 kc+1 (0, 1) + Φ k-1 kM+1(0, 1) ≤c2 K,1c2(3 + 2 ln M)(k + M)-2, (A.6) Φ kc 1 (M + 1, 2) ≤4c2 K,1c2M-1(k + M)-2, (A.7) Φ kc 1 (M + 1, 3) ≤c2 K,1cK,2c2M-2(k + M)-2, (A.8) Φkc 1 (M + 1, ε + 1) ≤2cK,1cK,εcM-ε(k + M)-1, (A.9) Φ k-1 1 (0, 1) ≤4c2k-1, (A.10) Φk-1 1 (0, 1 2) ≤3 √ 2ck-1 2 ln k, (A.11) where cK,1 = 1 + 2 K , cK,2 = 2 + 3 K+1 + 6 ln M (K+1)2 and cK,ε = e-1+1 ε + 2ε ln M (K+1)ε with K = [k/M], limK→∞cK,1 = 1, limK→∞cK,2 = 2, and limK→∞cK,ε = (e-1 + 1)ε-1. Proof. Let k = KM + t with K ≥1 and t = 1, · · · , M -1. Then there holds the inequality: (k -1)-1 ≤cK,1(k + M)-1. (A.12) The estimates in (A.5) follow directly from (A.4), the identity ∆δ KM = ∆δ (K-1)M = 0 and (A.12): Φ k-1 kc+1(0, 0) = k-1 X j=kc+1 E[∥A∆δ j∥2] ≤c2M(k -1)-2 ≤c2 K,1c2M(k + M)-2, Φk-1 kc+1(0, 0) = k-1 X j=kc+1 ∥A∆δ j∥≤cM(k -1)-1 ≤cK,1cM(k + M)-1. Next for the estimate (A.6), we have Φ kM-1 kc+1 (0, 1) + Φ k-1 kM+1(0, 1) = kM-1 X j=kc+1 (kM -1 -j)-1E[∥A∆δ j∥2] + k-1 X j=kM+1 (k -1 -j)-1E[∥A∆δ j∥2] = M-t X j=0 j-1E[∥A∆δ kM-1-j∥2] + t-2 X j=0 j-1E[∥A∆δ k-1-j∥2] ≤c2 M-t X j=0 j-1(kM -1 -j + M)-2 + t-2 X j=0 j-1(k -1 -j + M)-2 ≤c2(k -1)-2I, with I = M-t X j=0 j-1 + t-2 X j=0 j-1, 17 where I is bounded by I ≤4 + ln(M -t) + ln t ≤4 + 2 ln M 2 = 4 -2 ln 2 + 2 ln M ≤3 + 2 ln M. Then with the estimate (A.12), there holds Φ kM-1 kc+1 (0, 1) + Φ k-1 kM+1(0, 1) ≤c2 K,1c2(3 + 2 ln M)(k + M)-2. Next, we derive the estimates (A.7), (A.8) and (A.10). For the estimate (A.7), by the splitting (j′ -j)-2j-2 = (j′)-2 (j′ -j)-1 + j-1 2 ≤2(j′)-2 (j′ -j)-2 + j-2 , we obtain Φ kc 1 (M + 1, 2) = c2 k-M-2 X j=1 (k -1 -j)-2(j + M)-2 = c2 k-2 X j=M+1 (k + M -1 -j)-2j-2 ≤2c2(k + M -1)-2 k-2 X j=M+1 (k + M -1 -j)-2 + j-2 ≤4c2 K,1c2M-1(k + M)-2. Likewise, for the estimate (A.8), by the splitting (j′ -j)-3j-2 = 3(j′)-4 (j′ -j)-1 + j-1 + (j′)-3 2(j′ -j)-2 + j-2 + (j′)-2(j′ -j)-3, we derive Φ kc 1 (M + 1, 3) = c2 k-M-2 X j=1 (k -1 -j)-3(j + M)-2 = c2 k-2 X j=M+1 (k + M -1 -j)-3j-2 ≤c2 k + M -1 -2 6 k + M -1 -2 k-2 X j=M+1 j-1 + 3 k + M -1 -1 k-2 X j=M+1 j-2 + k-2 X j=M+1 j-3 ≤c2 k + M -1 -2h 6 (K + 1)M -2 ln(KM + t -1) + 3 (K + 1)M -1M-1 + 1 2M-2i ≤c2 K,1c2(k + M)-2M-2h 6(K + 1)-2 ln(K + 1) + ln M + 3 K+1 + 1 2 i . Then the inequality s-r ln s ≤(er)-1, ∀s, r > 0 (A.13) implies the bound on Φ kc 1 (M + 1, 3). For the estimate (A.10), the splitting (j′ -j)-1j-2 = (j′)-1j-2 + (j′)-2 (j′ -j)-1 + j-1 implies Φ k-1 1 (0, 1) = c2 k-1 X j=1 (k -1 -j)-1(j + M)-2 = c2 k+M-1 X j=M+1 (k + M -1 -j)-1j-2 ≤c2(k + M -1)-1 k+M-1 X j=M+1 j-2 + (k + M -1)-1 k+M-1 X j=M+1 (k + M -1 -j)-1 + j-1 ≤c2(k + M -1)-1 M-1 + (k + M -1)-1 2 + 2 ln(k + M -1) . 18 Then, using the inequality (A.13), we derive Φ k-1 1 (0, 1) ≤c2(k + M -1)-1(3M-1 + 2e-1) ≤4c2k-1. Now, we derive the estimates (A.11) and (A.9) by splitting the summations into two parts. Let kM = (k + M -1)/2. For the estimate (A.11), with the inequality (A.13), there holds Φk-1 1 (0, 1 2) = c k-1 X j=1 (k -1 -j)-1 2 (j + M)-1 = c k+M-1 X j=M+1 (k + M -1 -j)-1 2 j-1 ≤c(I11 + I12), with I11 = [kM] X j=M+1 k -1 2 M j-1 and I12 = 2kM X j=[kM]+1 (k + M -1 -j)-1 2 k -1 M . The decomposition is well-defined with the convention Pi′ j=i Rj = 0 for any {Rj}j and i′ 2 + ln 2. g(ε) attains its minimum over the interval (0, 1 2] at ε = ε∗= 2 c+ √ c2-4 < 1 2, and g(ε∗) = c+ √ c2-4 2 e 2(c-2) c+√ c2-4-2 ≤ce. Thus, for c0 < C0, we have f(ε∗) ≤6 c0 √ LM∥A∥ce 12 c+√ c2-4 ≤6 ec0 √ LM∥A∥ln(2e2n √ L∥A∥-1) 1 2 < 3√e 5 . This completes the proof of the lemma. Now we can prove Theorem 3.1 by mathematical induction. Proof. For the estimate (3.1), if k ≤K0M with some K0 ≥1, it holds for any sufficiently large c1 and c2. Now assume that it holds up to k = KM + t -1 with some K ≥K0 and 1 ≤t ≤M -1. Then we prove the assertion for the case k = KM + t. (It holds trivially when t = 0, since ∆δ KM = 0.) Fix k = KM + t and let kc := k -M -2 and kM := [k/M]M = KM. By the bias-variance decomposition, and the estimates (A.1) and (A.2) in Lemma A.3, we have E[∥A∆δ k∥2] ≤2(∥A∥2∥eδ 0∥2 + δ2)M2k-2 + n-1c2 0L∥A∥2 M2Φ kc 1 (M + 1, 2) + Φ k-1 kc+1(0, 0) . Then, by setting c = √ c1 + c2δ2, the estimates (A.5) and (A.7) and the inequality k-1 ≤ cK,1(k + M)-1 (cf. (A.12)) with cK,i given in Lemma A.4 yield E[∥A∆δ k∥2] ≤c2 K,1 h 5c2 0LMn-1∥A∥2(c1 + c2δ2) + 2(∥A∥2∥eδ 0∥2 + δ2)M2i (k + M)-2 ≤(c1 + c2δ2)(k + M)-2, for any c0 < ∥A∥-1(5Ln-1M)-1 2 and K ≥K0, with sufficiently large K0 and c1, c2. Alternatively, using the second estimate in (A.2), we can bound E[∥A∆δ k∥2] by E[∥A∆δ k∥2] ≤2(∥A∥2∥eδ 0∥2 + δ2)M2k-2 + c0L 2 8M2Φ kc 1 (M + 1, 3) + Φ kM-1 kc+1 (0, 1) + Φ k-1 kM+1(0, 1) . Then, with the estimates (A.6) and (A.8), we derive E[∥A∆δ k∥2] ≤c2 K,1 c0L(3 2 + ln M + 4cK,2)(c1 + c2δ2) + 2(∥A∥2∥eδ 0∥2 + δ2)M2 (k + M)-2 ≤(c1 + c2δ2)(k + M)-2, for any c0 < L-1(10+ln M)-1 and K ≥K0, with sufficiently large K0 and c1, c2. This completes the proof of the estimate (3.1). Next, we prove the estimate (3.2). Similarly, for the cases k ≤K0M with some K0 ≥1, the estimate holds trivially for sufficiently large c1 and c2. Now, assume that the bound holds up to k = KM + t -1 with some K ≥K0 and 1 ≤t ≤M -1, and prove the assertion for the case k = KM + t. Fix k = KM + t and let kc := k -M -2 and kM := [k/M]M = KM. By the triangle inequality ∥A∆δ k∥≤∥E[A∆δ k]∥+ ∥∆δ k -E[A∆δ k]∥, and (A.1) and (A.3), we have ∥A∆δ k∥≤(∥A∥∥eδ 0∥+ δ)Mk-1 + I1 + c0 √ L∥A∥Φk-1 kc+1(0, 0) , (A.15) 20 with I1 = 21+εεεc1-ε 0 nε√ L∥A∥1-2εMΦkc 1 (M + 1, 1 + ε). By (A.9) (with c = c1 + c2δ), we derive I1 ≤22+εεε(c1 + c2δ)c1-ε 0 nε√ L∥A∥1-2εM1-εcK,1cK,ε(k + M)-1 = (c1 + c2δ)f(ε)cK,1(k + M)-1, with f(ε) given in (A.14). This, (A.15), (A.5) in Lemma A.4, and the inequality (A.12) yield ∥A∆δ k∥≤cK,1 h (c1 + c2δ) c0 √ LM∥A∥+ f(ε) + (∥A∥∥eδ 0∥+ δ)M i (k + M)-1. (A.16) Then by Lemma A.5 and the inequality c0 √ LM∥A∥< 200-1 when c0 < C0, we derive from (A.16) that ∥A∆δ k∥≤(c1 + c2δ)(k + M)-1, for any K ≥K0, with sufficiently large K0 and c1, c2, completing the proof of the theorem. References [1] L. Bottou, F. E. Curtis, and J. Nocedal. 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2510.14762
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A proof of the 3
8-conjecture for
independent domination in cubic graphs
Boˇstjan Breˇsara,b, Tanja Draveca,b,
and
Michael A. Henningc
a Faculty of Natural Sciences and Mathematics
University of Maribor, Slovenia
b Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Email:
bostjan.bresar@um.si
Email:
tanja.dravec@um.si
c Department of Mathematics and Applied Mathematics
University of Johannesburg, South Africa
Email:
mahenning@uj.ac.za
Abstract
A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex
in S. An independent dominating set in G is a dominating set of G with the additional property that it is
an independent set. The domination number, γ(G), and the independent domination number, i(G), are the
minimum cardinalities among all dominating sets and independent dominating sets in G, respectively.
By
definition, γ(G) ≤i(G) for all graphs G. Let G be a connected cubic graph of order n. In 1996 Reed [Combin.
Probab. Comput. 5 (1996), 277–295] proved a breakthrough result that γ(G) ≤3
8n. We prove the stronger result
that if G is different from K3,3 and the 5-prism C5 2 K2, then i(G) ≤3
8n. This proves a known conjecture. The
bound is tight in the sense that there are infinite families of connected cubic graphs that achieve equality in this
bound.
Keywords: Independent domination number; Domination number; Cubic graphs.
AMS subject classification: 05C69
1
Introduction
A set S of vertices in a graph G is a dominating set if every vertex in V (G) \ S is adjacent to a vertex in S. The
domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set. A set is independent
in G if no two vertices in the set are adjacent. An independent dominating set, abbreviated ID-set, of G is a set
that is both dominating and independent in G. Equivalently, an ID-set of G is a maximal independent set in G.
The independent domination number of G, denoted by i(G), is the minimum cardinality of an ID-set. An ID-set
of cardinality i(G) in G is called an i-set of G. Independent dominating sets have been studied extensively in the
literature (see, for example, [8, 10, 15, 16, 18, 19, 21]). A thorough treatise on dominating sets can be found in the
so-called “domination books” [11–13]. A survey on independent domination in graphs can be found in [7].
For graph theory notation and terminology, we generally follow [13]. Specifically, let G be a graph with vertex
set V (G) and edge set E(G), and of order n(G) = |V (G)| and size m(G) = |E(G)|. Two adjacent vertices in G are
neighbors. The open neighborhood of a vertex v in G is NG(v) = {u ∈V : uv ∈E} and the closed neighborhood
of v is NG[v] = {v} ∪NG(v). We denote the degree of v in G by degG(v), and so degG(v) = |NG(v)|. An isolated
vertex in G is a vertex of degree 0 in G. For k ≥1 an integer, we let [k] denote the set {1, . . . , k} and we let
[k]0 = [k] ∪{0} = {0, 1, . . . , k}.
1
arXiv:2510.14762v1 [math.CO] 16 Oct 2025
A cycle on n vertices is denoted by Cn and a path on n vertices by Pn. The complete graph on n vertices is
denoted by Kn, while the complete bipartite graph with one partite set of size n and the other of size m is denoted
by Kn,m. A cubic graph is graph in which every vertex has degree 3, while a subcubic graph is a graph with
maximum degree at most 3. If G is a subcubic graph, then we let Vj(G) denote the set of vertices of degree j in G
and we let nj(G) = |Vj(G)| denote the number of vertices of degree j in G for j ∈[3]0.
For subsets X and Y of vertices of G, we denote the set of edges with one end in X and the other end in Y by
[X, Y ]. The girth of G, denoted g(G), is the length of a shortest cycle in G. A component of a graph G isomorphic
to a graph F we call an F-component of G. For a set S ⊆V (G), the subgraph induced by S is denoted by G[S].
Further, the subgraph of G obtained from G by deleting all vertices in S and all edges incident with vertices in S
is denoted by G −S; that is, G −S = G[V (G) \ S].
2
Motivation and known results
In 1996 Reed [20] proved the following breakthrough result establishing a best possible upper bound on the domi-
nation number of a cubic graph.
Theorem 1 ([20]) If G is a cubic graph of order n, then γ(G) ≤3
8n.
The importance of Reed’s paper is reflected in over 100 citations one can find on the MathSciNet. Subsequently,
determining best possible bounds on the independent domination number of cubic graphs attracted much interest
(see, for example, [1–6, 9, 14]). In 1999 Lam, Shiu, and Sun [17] established the following upper bound on the
independent domination number of a connected cubic graph.
Theorem 2 ([17]) If G ̸∼= K3,3 is a connected, cubic graph of order n, then i(G) ≤2
5n.
The 2
5-upper bound on the independent domination number in Theorem 2 is best possible in the sense that
there exists at least one graph that achieves equality in the bound, namely the 5-prism, C5 2 K2, which is the
Cartesian product of a 5-cycle with a copy of K2. The graphs K2,3 and C5 2K2 are shown in Figure 1(a) and 1(b),
respectively.
(a) K2,3
(b) C5 2 K2
Figure 1: The graphs K2,3 and C5 2 K2.
However despite the pleasing result given in Theorem 2, prior to this paper it remained an open problem to
characterize the graphs achieving equality in Theorem 2. In their 2013 survey paper on independent domination
in graphs, Goddard and Henning [7] conjectured that the 5-prism, C5 2 K2, is the only extremal graph achieving
equality in the 2
5-upper bound on the independent domination number i(G) established by Lam, Shiu, and Sun.
The authors in [7] conjectured that Reed’s 3
8-upper bound on the domination number, γ(G), also holds for the
independent domination, i(G), in a connected, cubic graph, with the exception of the two graphs K3,3 and C5 2 K2.
Conjecture 1 ([7]) If G /∈{K3,3, C5 2 K2} is a connected cubic graph of order n, then i(G) ≤3
8n.
In 2015 Dorbec et al. [6] gave support to Conjecture 1 by proving that the conjecture holds if the cubic graph G
contains no subgraph isomorphic to K2,3. Recall that in a subcubic graph G, we let nj(G) denote the number of
vertices of degree j in G for j ∈[3]0.
2
Theorem 3 ([6]) If G is a subcubic graph that does not have a subgraph isomorphic to K2,3 and which has no
(C5 2 K2)-component, then
8i(G) ≤8n0(G) + 5n1(G) + 4n2(G) + 3n3(G).
As a consequence of Theorem 3, we have the main result in [6], noting that if G is a cubic graph of order n, then
n0(G) = n1(G) = n2(G) = 0 and n = n3(G).
Theorem 4 ([6]) If G ̸= C5 2 K2 is a connected cubic graph of order n that does not have a subgraph isomorphic
to K2,3, then i(G) ≤3
8n.
3
Main result
We give a complete proof of Conjecture 1 by adding forbidden subgraphs isomorphic to K2,3 in the statement of
Theorem 4 back into the mix. We will prove the following result.
Theorem 5 If G /∈{K3,3, C5 2 K2} is a connected cubic graph of order n, then i(G) ≤3
8n.
As a consequence of Theorem 5, the 5-prism, C5 2 K2, is the unique extremal graph in Theorem 2. The 3
8-upper
bound on the independent domination number given in Theorem 5 is tight. Two infinite families of connected
cubic graphs G satisfying i(G) = 3
8n are constructed in [7]. These families are also defined in [13, Chapter 6], so
we do not redefine them here. However, graphs in these families are illustrated in Figure 2(a) and (b), respectively.
Akbari et al. [1] constructed additional families (which we do not define here) of connected cubic graphs G satisfying
i(G) = 3
8n.
(a)
(b)
Figure 2: Two infinite families of cubic graphs G of order n satisfying i(G) = 3
8n
Indeed, the family of extremal graphs achieving the 3
8-upper bound in Theorem 5 is rich and varied. As a further
example of extremal graphs, the graphs G of order n in the infinite family illustrated in Figure 3 also satisfy
i(G) = 3
8n. We remark that the cubic graphs G in the family illustrated in Figure 2(a) do not contain a subgraph
isomorphic to K2,3 and satisfy i(G) = 3
8n, while the cubic graphs G in the family illustrated in Figure 2(b) and
Figure 3 do contain a subgraph isomorphic to K2,3 and satisfy i(G) = 3
8n.
We proceed as follows. In Section 4 we give an overview of our proof to give the main ideas and direction of the
proof. In Section 5, we formally define the (infinite) special families of subgraphs we encounter in our proof, and
we prove desirable properties of the families. Thereafter in Section 6, we present a detailed proof of our key result,
namely Theorem 6.
4
Overview of proof of Theorem 5
Before presenting a proof of Theorem 5, we give a sketch proof to provide the reader an overview of the detailed
proof that follows and the direction of the arguments. In order to prove Theorem 5, we relax the 3-regularity
3
Figure 3: An infinite family of cubic graphs G of order n satisfying i(G) = 3
8n
condition to a bounded maximum degree 3 condition to make the inductive hypothesis easier to handle, that is, we
relax the requirement that G /∈{K3,3, C5 2 K2} is a connected cubic graph to the requirement that G is a subcubic
graph that contains no K3,3-component and no (C5 2 K2)-component. We define next what we have coined the
vertex weight and the structural weight of such graphs.
4.1
The vertex weight
For a subcubic graph G, we assign weights 8, 5, 4 and 3 to the vertices of G of degrees 0, 1, 2 and 3, respectively.
For each vertex v of G, we denote the weight of v in G by wG(v) (see Table 1).
degG(v)
0
1
2
3
wG(v)
8
5
4
3
Table 1. The weight wG(v) of a vertex v in G
For a subset X of vertices in G, we define the vertex weight of X in G as the sum of the weights in G of vertices
in X; that is,
wG(X) =
X
v∈X
wG(v).
We define the vertex weight of G, denoted by w(G), as the sum of the weights of vertices in G, that is,
w(G) = 8n0(G) + 5n1(G) + 4n2(G) + 3n3(G).
4.2
The structural weight
We would like to show that 8i(G) ≤w(G) since in the special case when G is a cubic graph of order n, we note
that n0(G) = n1(G) = n2(G) = 0 and n = n3(G), and so the weight of G simplifies to w(G) = 3n, from which we
would infer that i(G) ≤3
8n, as desired.
However, relaxing the cubic condition results in the so-called “bad” family of subcubic graphs. We construct
such a family B of subcubic graphs in Section 5.2 with the property that if G ∈B, then 8i(G) = w(G) + 2. We will
refer to graphs G in this family B as bad graphs since they satisfy 8i(G) > w(G). We denote by b(G) the number
of components of G that belong to the family B.
Moreover relaxing the cubic condition results in so-called “troublesome” configurations of subcubic graphs. In
Section 5.3, we define these troublesome configurations, which we call T-configurations. We distinguish two types
of troublesome configurations, namely T1-configurations and T2-configurations. A Ti-configuration is a troublesome
configuration that is joined to vertices not in the subgraph by exactly i edges. We denote by tc(G) the number of
vertex disjoint troublesome configurations in G. We define the Θ-weight of G, which we call the structural weight
of G since it is determined by structural properties of G, by
Θ(G) = 2tc(G) + 2b(G).
4
4.3
The total weight
We define the total weight of G, which we denote by Ω(G), as the sum of the vertex weight and the structural
weight of a subcubic graph G, and so
Ω(G) = w(G) + Θ(G).
We call the total weight of G the Ω-weight of G. Our key result, given in Theorem 6, gives a tight upper bound
on the independent domination number of a subcubic graph in terms of its Ω-weight.
Theorem 6 If G is a subcubic graph of order n that contains no K3,3-component and no (C5 2 K2)-component,
then 8i(G) ≤Ω(G).
In order to prove our main result, namely Theorem 5, we need the above stronger statement given in Theorem 6.
Each graph in the family B of subcubic graphs contains vertices of degree 1 or 2.
Moreover, every so-called
“troublesome configuration” contains a vertex of degree 2 in G. From these properties, a cubic graph G has no
component that belongs to the family B and contains no troublesome configuration. We therefore infer that if G is
a cubic graph, then tc(G) = b(G) = 0, implying that its structural weight is zero, that is, Θ(G) = 0. Further if G
is a cubic graph of order n, then n0(G) = n1(G) = n2(G) = 0 and n = n3(G), and so its vertex weight is 3n, that
is, w(G) = 3n. Hence in this case when G is a cubic graph, the inequality in the statement of Theorem 6 simplifies
to 8i(G) ≤3n. Our main result, namely Theorem 5, therefore follows readily from Theorem 6.
5
Special families of graphs and subgraphs
In this section, we define the family B of so-called ‘bad subgraphs’ and we define what we have coined ‘troublesome
configurations’ of a subcubic graph G. In order to prove desirable properties of subcubic graphs, we first define
‘exit edges’ associated with a set X of vertices of a graph G.
5.1
Exit edges of a vertex set
Let X be a proper subset of vertices in a subcubic graph G and let V = V (G). We call an edge an X-exit edge in
G if it joins a vertex in X to a vertex in V \ X. We denote the number of X-exit edges in G by ξG(X). Those
vertices in G −X that are incident in G with at least one X-exit edge have smaller degree in G −X than in G,
and therefore have a larger weight in G −X than in G. We refer to the sum of these weight increases as the w-cost
of removing X from G and denote the weight increase by ΦG(X). Thus,
ΦG(X) =
X
v∈V \X
wG−X(v) −wG(v)
.
We prove next the fact that the w-cost of removing a set X of vertices from G is equal to the number of X-exit
edges in G plus twice the number of isolated vertices in G −X. Consequently, if removing X yields no isolated
vertices, then the cost of removing X from G will be precisely the number of X-exit edges in G. For a vertex
v ∈V \ X, we denote by ΦG(v) the number of X-edges in G that are incident with the vertex v.
Fact 1 If X ⊂V (G), then ΦG(X) = ξG(X) + 2n0(G −X).
Proof. We consider the contribution of a vertex v ∈V \ X to the cost ΦG(X). If the vertex v is incident with no
X-exit edge in G, then wG−X(v) = wG(v), and the contribution of v to the sum ΦG(X) is zero. Hence, we may
assume that v is incident with at least one X-exit edge in G, that is, ΦG(v) ≥1. If v is not isolated in G −X, then
wG−X(v) −wG(v) is precisely the number of exit edges of X incident with v in G, namely ΦG(v). If v is isolated in
G −X, then wG−X(v) −wG(v) is the number of exit edges of X incident with v in G plus an additional 2 noting
that in this case wG−X(v) = 8 and wG(v) = 6 −ΦG(v). 2
5
5.2
The bad family B
In this section, we define the so-called “bad family” B of graphs. Let B1 be the graph obtained from K2,3 by
adding a new vertex r and adding an edge joining r and a vertex of degree 2 in the copy of K2,3. The graph B1,
illustrated in Figure 4, we call the base graph used in our construction of graphs in the family B.
r
Figure 4: The base graph B1
Let B be the family of subcubic graphs that contain the base graph B1 and is closed under the operation O1
listed below that extends a graph G′ ∈B to a new graph G ∈B. As remarked earlier, we refer to graphs in the
family B as bad graphs.
• Operation O1: The graph G is obtained from G′ ∈B by selecting a vertex v′ of degree at most 2 in G′, adding
a vertex disjoint copy of K2,3, and adding an edge joining v′ and a vertex of degree 2 in the added copy of
K2,3. See Figure 5.
G′
O1:
v′
7→
G′
v′
Figure 5: The operation O1
Thus, if G ∈B, then either G = B1 or G is constructed from the base graph B1 by k −1 applications of
Operation O1 where k ≥2. Thus, either G = B1 or k ≥2 and G = Bk where B1, B2, . . . , Bk is a sequence
of graphs in the family B and the graph Bi+1 is obtained from the graph Bi by applying Operation O1 for all
i ∈[k −1]. We call Bi+1 a bad-extension of the graph Bi for i ∈[k −1].
If G ∈B, then we define the root of the graph G as the vertex of G that does not belong to any copy of K2,3,
equivalently, the root of G is the (unique) vertex that belongs to no cycle. We note that the root has degree 1, 2
or 3 in the graph G. We define Bi = {B ∈B: the root of B has degree i in B} for i ∈[3], and so
B =
3[
i=1
Bi.
For example, the graph G illustrated in Figure 6(e) belongs to the family B since it can be constructed from a
sequence B1, B2, B3, B4, B5 of graphs in B shown in Figure 6(a)–6(e), where Bi+1 a bad-extension of Bi for i ∈[4]
and where G = B5 (in this case, G = Bk where k = 5). In this example, the root is the vertex labelled r in Figure 6,
and has degree 2 in the resulting graph G = B5, and so the graph G belongs to the subfamily B2 of B.
We define the canonical ID-set of a bad graph G (that belongs to the family B) as follows. Initially if G = B1,
we define the canonical ID-set to consist of the root and the two vertices of degree 2 in B1 (as illustrated by the
shaded vertices in Figure 6(a)). Suppose that k ≥2 and let B1, B2, . . . , Bk be the sequence of graphs in the family
B used to construct G ∈B, and so G = Bk and Bi+1 a bad-extension of the graph Bi for i ∈[k −1]. When we
apply Operation O1 to extend the graph Bi to the graph Bi+1 we add to the current canonical ID-set in Bi the
two new vertices of degree 2 in Bi+1 that belong to the added copy of K2,3 for all i ∈[k −1]. We note that by
construction, every vertex of degree 2 in G belongs to the canonical ID-set of G. For example, the canonical ID-set
of each of the graphs B1, B2, B3, B4, B5 used to construct the graph G = B5 is given by the shaded vertices in
Figures 6(a)–6(e).
6
(a) B1
r
(b) B2
r
(c) B3
r
(d) B4
r
(e) G = B5
r
Figure 6: A graph G in the family B2 with root vertex r
A vertex of degree 3 in G we call a large vertex. Each copy of K2,3 in G ∈B contains exactly two vertices that
belong to the canonical ID-set of G, and these two vertices have two common large neighbors that belong to the
same copy of K2,3. We refer to these two large vertices as non-canonical vertices of G. Thus each copy of K2,3 in
G ∈B contains two non-canonical vertices, namely the two vertices of degree 3 in that copy of K2,3 that are open
twins, where two vertices are open twins if they have the same open neighborhood. We define the non-canonical
independent set of G ∈B to be the set consisting of all non-canonical vertices of G. Thus if G ∈B is constructed
from the base graph B1 by k −1 applications of Operation O1 where k ≥1, then G contains k vertex disjoint
copies of K2,3 and the 2k non-canonical vertices of G (two from each copy of K2,3 in G) form the non-canonical
independent set of G. For example, the non-canonical independent set of each of the graphs B1, B2, B3, B4, B5
used to construct the graph G = B5 is given by the vertices represented by squares in Figures 6(a)–6(e).
We define the set of vertices in G ∈B that belong to neither the canonical ID-set of G nor the non-canonical
independent set of G as the core independent set in G, and we refer to the vertices in the core independent set as
core vertices. Thus if G ∈B is constructed from the base graph B1 by k −1 applications of Operation O1 where
k ≥1, then G contains k core vertices, one from each of the k vertex disjoint copies of K2,3. For example, the
core independent set of each of the graphs B1, B2, B3, B4, B5 used to construct the graph G = B5 is given by the
vertices represented by white circled vertices in Figures 6(a)–6(e).
We shall need the following lemmas.
Lemma 1 If a subcubic graph G is obtained from a subcubic isolate-free graph G′ by adding a vertex disjoint copy
of K2,3, and adding an edge joining a vertex v′ of degree at most 2 in G′ and a vertex of degree 2 in the added copy
of K2,3, then the following properties hold.
(a) i(G) = i(G′) + 2.
(b) w(G) = w(G′) + 16.
Proof. Let G, G′ and v′ be as defined in the statement of the lemma. Let F be the copy of K2,3 added to G′
when constructing the graph G, where v1, v2 and v3 are the three small vertices (of degree 2) in F and u1 and u2
are the two large vertices (of degree 3) in F. Further, let v3 be the vertex of F adjacent to v′ in G. Let X = V (F).
The construction of G is illustrated in Figure 7.
(a) Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices u1 and u2, and so i(G) ≤
i(G′) + 2. Conversely, let I be an i-set of G. If {u1, u2} ⊂I, then we let I′ = I \ {u1, u2}. If {u1, u2} ̸⊂I, then
7
G′
v3
v′
u2
u1
v2
v1
Figure 7: The graphs G′ and G in the proof of Lemma 1
{v1, v2} ⊂I. Moreover if in this case v3 /∈I, then v′ ∈I and we let I′ = I \ {v1, v2}. Suppose that {v1, v2, v3} ⊆I.
In this case, if a neighbor of v′ different from v3 belongs to the set I, then the set (I \ {v1, v2, v3}) ∪{u1, u2} is
an ID-set of G of cardinality |I| −1, contradicting the minimality of the set I. Hence, the vertex v3 is the only
neighbor of v′ that belongs to the set I. We now let I′ = (I \ {v1, v2, v3}) ∪{v′}. In all the above cases, the set I′
is an ID-set of G′ of cardinality |I| −2, and so i(G′) ≤|I| −2 = i(G) −2. As observed earlier, i(G) ≤i(G′) + 2.
Consequently, i(G) = i(G′) + 2. This proves part (a).
(b) By supposition, the graph G is a subcubic graph and the graph G′ is isolate-free. We infer that the vertex
v′ has degree 1 or 2 in G′ and therefore has degree 2 or 3 in G, respectively. Thus adopting our earlier notation,
the edge v′v3 is the only X-exit edge in G. Thus, the number of X-exit edges is ξG(X) = 1. By Fact 1, the
cost of removing X from G is ΦG(X) = ξG(X) = 1 noting that G′ = G −X is isolate-free.
Thus, w(G) =
w(G′) + wG(X) −ΦG(X) = w(G′) + 17 −1 = w(G′) + 16, which proves part (b). 2
Suppose that G ∈B, and so G is constructed from the base graph B1 by k −1 applications of Operation O1
where k ≥1. We note that if k = 1, then G = B1. Further, we note that i(B1) = 3 and w(B1) = 22. Hence by
repeated applications of Lemma 1, we infer the following properties of graphs in the family B.
Proposition 1 If G ∈B is constructed from the base graph B1 by k −1 applications of Operation O1 where k ≥1,
then the following properties hold.
(a) i(G) = 2k + 1, w(G) = 16k + 6, and 8i(G) = w(G) + 2.
(b) The canonical ID-set of G is an i-set of G.
(c) The non-canonical independent set of G, together with the root vertex of G, is an i-set of G.
(d) If v is the root vertex of G or a vertex of degree 2 in G, then i(G −v) = i(G) −1.
(e) The core independent set of G is an ID-set of the graph obtained by removing all vertices of degree 2 from G.
(f) The graph G contains at most one vertex of degree 1, and no two adjacent vertices of G both have degree 2
in G.
5.3
Troublesome configurations
In this section, we define what we have coined a ‘troublesome configuration’ of a subcubic graph G. We define a
troublesome configuration of G, abbreviated T-configuration of G, as a subgraph of G obtained from a bad graph
B ∈B1 with root vertex v1 (of degree 1 in B) by applying the following process.
• adding a new vertex v2 to B,
• adding an edge joining v2 to the vertex v1 and to a vertex w2 of degree 2 in B different from v1,
• adding an edge joining v2 to a vertex not in V (B) ∪{v2}, and
• adding at most one edge joining v1 to a vertex not in V (B) ∪{v2}.
If v1 has degree 2 in G, then we call the troublesome configuration a type-1 troublesome configuration of G,
abbreviated T 1
v1,v2-configuration of G. A T 1
v1,v2-configuration is therefore joined to the rest of the graph by exactly
one edge, namely an edge that joins v2 to one vertex not in the subgraph. If v1 has degree 3 in G, then we call
the troublesome configuration a type-2 troublesome configuration of G, abbreviated T 2
v1,v2-configuration of G. A
T 2
v1,v2-configuration is therefore joined to the rest of the graph by exactly two edges, namely an edge that joins
each of v1 and v2 to a vertex not in the subgraph. In a troublesome configuration, we call the vertices v1 and v2
the link vertices and we call all other vertices the non-link vertices of the configuration. Thus the degree of every
8
v1
v2
w2
B
T
(a) A T 1
v1,v2-configuration
v1
v2
w2
B
T
(b) A T 2
v1,v2-configuration
Figure 8: Type-1 and type-2 troublesome configurations in G
non-link vertex of a troublesome configuration is the same as its degree in G. A T 1
v1,v2-configuration is illustrated
in Figure 8(a), and a T 2
v1,v2-configuration is illustrated in Figure 8(b).
The shaded vertices in Figure 8(a) indicate an i-set of T 1
v1,v2 and this set is the canonical ID-set of B. Moreover the
shaded vertices in Figure 8(b) indicate an i-set of T 2
v1,v2 that contains the vertex v2 and the non-canonical vertices
of B. More generally, we have the following properties of troublesome configurations of G. These properties follows
readily from properties of bad graphs (that belong to the family B) established in Section 5.2.
Proposition 2 If T = T 1
v1,v2 or T = T 2
v1,v2 is a T1- or T2-configuration of G with link vertices v1 and v2 that is
obtained from a bad graph B ∈B with root vertex v1, then the following properties hold.
(a) i(T) = i(B).
(b) The canonical ID-set of B is an i-set of T.
(c) There exists an i-set of T that contains the link vertex v2.
(d) If S is an arbitrary set of vertices of degree 2 in T that contains neither v1 nor v2, then i(T −S) = i(T)−|S|.
Furthermore, there exists an i-set of T −S that contains neither v1 nor v2, and there exists an i-set Ij of
T −S that contains vj for j ∈[2].
(f) Every vertex in T has degree at least 2, and no two adjacent vertices of T both have degree 2.
5.4
Examples of graphs achieving equality in Theorem 6
In this section we present a small sample of graphs achieving equality in the upper bound of Theorem 6.
Example 1. If G is the graph shown in Figure 9(a), then i(G) = 6 and the shaded vertices indicate an i-set
in G. In this example, w(G) = 12 × 3 + 2 × 4 = 44 and Θ(G) = 2tc(G) = 2 × 2 = 4 (where the two vertex
disjoint T2-configurations are indicated by the subgraphs in the dashed boxes), and so 8i(G) = 48 = 44 + 4 =
w(G) + Θ(G) = Ω(G).
Moreover if G is the graph shown in Figure 9(b), then i(G) = 9 and the shaded vertices indicate an i-set in G.
In this example, w(G) = 20 × 3 + 2 × 4 = 68 and Θ(G) = 2tc(G) = 2 × 2 = 4 (where the two vertex disjoint T2-
configurations are indicated by the subgraphs in the dashed boxes), and so 8i(G) = 72 = 44 + 4 = w(G) + Θ(G) =
Ω(G).
(a)
(b)
Figure 9: Examples of graphs achieving equality in the upper bound of Theorem 6
Example 2. If G is the graph shown in Figure 10(a), then i(G) = 5 and the shaded vertices indicate an i-set in G.
In this example, w(G) = 10 × 3 + 2 × 4 = 38 and Θ(G) = 2tc(G) = 2 × 1 = 2 (where a T2-configuration is indicated
by the subgraph in the dashed box), and so 8i(G) = 40 = 38 + 2 = w(G) + Θ(G) = Ω(G).
9
If G is the graph shown in Figure 10(b), then i(G) = 4 and the shaded vertices indicate an i-set in G. In this
example, w(G) = 6 × 3 + 3 × 4 = 30 and Θ(G) = 2tc(G) = 2 × 1 = 2 (where a T2-configuration is indicated by the
subgraph in the dashed box), and so 8i(G) = 32 = 30 + 2 = w(G) + Θ(G) = Ω(G).
If G is the graph shown in Figure 10(c), then i(G) = 5 and the shaded vertices indicate an i-set in G. In this
example, w(G) = 8 × 3 + 4 + 2 × 5 = 38 and Θ(G) = 2tc(G) = 2 × 1 = 2 (where a T2-configuration is indicated by
the subgraph in the dashed box), and so 8i(G) = 40 = 38 + 2 = w(G) + Θ(G) = Ω(G).
(a) F1
(b) F2
(c) F3
Figure 10: Examples of graphs achieving equality in the upper bound of Theorem 6
Example 3. If G is the graph shown in Figure 11, then i(G) = 18 and the shaded vertices indicate an i-set in G.
In this example, w(G) = 44 × 3 + 2 × 4 = 140 and Θ(G) = 2tc(G) = 2 × 2 = 4 (where the T2-configurations are
indicated by the subgraphs in the dashed boxes), and so 8i(G) = 144 = 140 + 4 = w(G) + Θ(G) = Ω(G).
Figure 11: An example of a graph achieving equality in the upper bound of Theorem 6
6
Proof of key result
Recall that the vertex weight of a subcubic graph G is w(G) = 8n0(G) + 5n1(G) + 4n2(G) + 3n3(G), and the
structural weight of G is Θ(G) = 2tc(G) + 2b(G). Moreover the total weight of G, called the Ω-weight of G, is the
sum of its vertex weight and structural weight, that is, Ω(G) = w(G) + Θ(G). In this section, we present a proof
of our key result, namely Theorem 6. Its statement in terms of the Ω-weight of G is as follows:
Theorem 6. If G is a subcubic graph of order n that contains no K3,3-component and no (C5 2 K2)-component,
then 8i(G) ≤Ω(G).
Proof. Let G be a subcubic graph of order n that contains no K3,3-component and no (C5 2 K2)-component, and
let V = V (G). We wish to prove that 8i(G) ≤Ω(G). Suppose that G is a counterexample to the inequality with
minimum order, and let G have order n. Thus, 8i(G) > Ω(G) but every subcubic graph G′ of order less than n
that contains no K3,3-component and no (C5 2 K2)-component satisfies 8i(G′) ≤Ω(G′). Since Θ(G) ≥0, we note
that w(G) ≤Ω(G). Hence, if 8i(G) ≤w(G), then 8i(G) ≤Ω(G), a contradiction. Therefore, 8i(G) > w(G).
10
Since our proof of Theorem 6 is long, we break the proof into four parts to enhance clarity and exposition of the
proof. The first part of the proof establishes important structural properties of the counterexample G. The second
part of the proof considers the case when the minimum degree of G equals 1. The third part of the proof considers
the case when the minimum degree of G equals 2. The fourth and final part of the proof considers the case when
the graph G is cubic, that is, when G is a 3-regular graph.
6.1
Part 1: Structural properties of G
In this section, namely Part 1 of our proof, we establish structural properties of the counterexample G. A component
of a graph that belongs to the family B is a bad component of the graph.
Claim 1 The subcubic graph G contains at least one subgraph isomorphic to K2,3.
Proof. If G contains no subgraph isomorphic to K2,3, then by Theorem 3, we have 8i(G) ≤w(G), a contradic-
tion. (2)
By Claim 1, we note that n ≥5. If n = 5, then either G = K2,3, in which case 8i(G) = 16 < 18 = w(G), or
G is obtained from K2,3 by adding an edge, in which case 8i(G) = 16 = w(G), contradicting the fact that G is a
counterexample. Hence, n ≥6.
Claim 2 The graph G is connected.
Proof. If G is not connected, then, by the minimality of n(G), the theorem holds for every component of G, and
therefore also for G, a contradiction. (2)
Claim 3 G /∈B.
Proof.
If G ∈B, then b(G) = 1 and Θ(G) = 2b(G) = 2, and so Ω(G) = w(G) + 2.
By Proposition 1(a),
8i(G) = w(G) + 2 = w(G) + Θ(G) = Ω(G), a contradiction. (2)
By Claim 3, we have b(G) = 0, and so Θ(G) = 2tc(G).
Claim 4 The graph G contains no troublesome configuration.
Proof. Suppose, to the contrary, that G contains a troublesome configuration, say T. Thus, T = Tv1,v2 is obtained
from a bad graph B ∈B1 with root vertex v1 (of degree 1 in B) by adding a new vertex v2 to B, adding the edges
v1v2 and v2w2, where w2 is a vertex of degree 2 in B different from v1. Furthermore, there are at most two exit
edges that join T to vertices in V (G) \ V (T) via one edge incident with v2 and possibly one additional exit edge
incident with v1. Let u be the vertex in B adjacent to v1 in G.
Let Bu be the subgraph B −v1, and let Bu contain k ≥1 vertex disjoint copies of K2,3. The weight of Bu is
w(Bu) = 16k+2. Moreover, i(Bu) = 2k, which follows by induction and using Lemma 1. Indeed, the non-canonical
independent set of B, which we denote by Iu, is an i-set of Bu. Let G′ = G −V (Bu). We note that v1v2 is an edge
of G′. Further, v1 has degree 1 or 2 in G′ and v2 has degree 2 in G′. Hence every i-set of G′ can be extended to
an ID-set of G by adding to it the 2k vertices from the set Iu, and so i(G) ≤i(G′) + |Iu| = i(G′) + 2k. Moreover,
w(G) = w(G′)+w(Bu)−4 noting that the two edges uv1 and v2w2 joining V (Bu) to V (G′) decrease the weights of
each of the vertices u, v1, v2 and w2 by 1. Thus, w(G′) = w(G)−w(Bu)+4 = w(G)−(16k+2)+4 = w(G)−16k+2.
We do not create a bad component in G′ since such a component would contain the adjacent vertices v1 and v2
where we note that both v1 and v2 have degree at most 2 in G′. This contradicts the property that at least one of
every two adjacent vertices in a bad component is a vertex of degree 3. We therefore infer that b(G′) = 0.
We note that the troublesome configuration T of G is no longer a troublesome configuration of G′. Further we
note that v1 and v2 are adjacent vertices in G′ where v1 has degree 1 or 2 and v2 has degree 2 in G′. Any new
troublesome configuration T ′ of G′ that is not a troublesome configuration of G necessarily contains both v1 and v2
11
as its link vertices. However in this case, neither v1 nor v2 is adjacent to a vertex not in the T ′-configuration noting
that both v1 and v2 have the same degrees in T ′ and G′. Hence we do not create a new troublesome configuration.
Thus, tc(G′) = tc(G)−1. As observed earlier, b(G′) = 0. Thus, Θ(G′) = 2tc(G′) = 2tc(G)−2 = Θ(G)−2, implying
that Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −16k + 2) + (Θ(G) −2) = Ω(G) −16k. Hence, 8i(G) ≤8(i(G′) + 2k) ≤
Ω(G′) + 16k ≤Ω(G), a contradiction. (2)
Claim 5 Θ(G) = 0.
Proof. By Claim 3, b(G) = 0. By Claim 4, there are no troublesome configurations in G, implying that tc(G) = 0.
Hence, Θ(G) = b(G) + tc(G) = 0. (2)
Claim 6 The graph G does not contain a K2,3-subgraph that contains two vertices of degree 2 in G.
Proof. Suppose, to the contrary, that G contains a K2,3-subgraph, F, that contains two vertices of degree 2 in
G. Let u1 and u2 be the two vertices of degree 2 in G that belong to F, and let w1 and w2 be the two common
neighbors of u1 and u2. Further, let u3 be the third common neighbor of w1 and w2. We note that the vertex u3
has degree 2 in F and has degree 3 in G. Let v1 be the third neighbor of u3. Let G′ = G −V (F). Every i-set of
G′ can be extended to an ID-set of G by adding to it the vertices w1 and w2, and so i(G) ≤i(G′) + 2.
We note that w(G) = w(G′) + 16. By Claim 5, Θ(G) = 0 and therefore Ω(G) = w(G) = w(G′) + 16. We
show next that Θ(G′) = 0.
If G′ belongs to B, then so too does the original graph G.
Thus, G′ /∈B, and
so b(G′) = 0.
By Claim 5, the graph G contains no troublesome configuration.
Suppose that G′ contains a
troublesome configuration T ′. Necessarily such a subgraph T ′ contains the vertex v1. Note that v1 is not a link
vertex of T ′, since this would imply that T ′ is troublesome configuration already in G (see Figure 12(a)). Hence v1
is not a link vertex, see Figure 12(b) as an illustration in this case. But then adding the subgraph F to T ′, together
with the edge u3v1, produces a troublesome configuration T of G, and so tc(G) ≥1, contradicting Claim 5. Hence,
tc(G′) = 0. Thus, Θ(G′) = 0, and so Ω(G′) = w(G′) = Ω(G) −16. As observed earlier, i(G) ≤i(G′) + 2. Hence,
8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. (2)
v1
w1
w2
u1
u2
u3
F
T
T ′
v1
w1
w2
u1
u2
u3
F
T
T ′
(a)
(b)
Figure 12: Possible subgraphs in the proof of Claim 6
We shall frequently use the following property that removing an exit edge increases the total weight by at most 3.
Claim 7 If X ⊂V (G) and G′ = G −X, then removing an X-exit edge when constructing G′ increases the total
weight by at most 3.
Proof. Let X ⊂V (G) and G′ = G −X, and consider an exit edge vv′ ∈E(G) where v ∈X and v′ ∈V (G′). If
v′ is isolated in G′, then the vertex v′ has degree 1 in G and degree 0 in G′, and so wG′(v′) −wG(v′) = 8 −5 = 3.
If degG′(v′) ≥1, then wG′(v′) −wG(v′) = 1.
However, in this case removing the edge vv′ may create a bad
component or a troublesome configuration, resulting in an increase in the structural weight by 2. Thus the total
weight increases by 3 if either we create an isolated vertex - in this case, the vertex weight increases by 3 and the
structural weight by 0 - or if we create a bad component or a new troublesome configuration that contains the
vertex incident with the exit edge - in this case, the vertex weight increases by 1 and the structural weight increases
12
by 2, resulting in a total increase in the weight by 3. In both cases the removal of the X-exit edge vv′ increases
the total weight by at most 3. (2)
By Claim 6, every K2,3-subgraph in G contains at most one vertex of degree 2 in G.
Claim 8 The graph G does not contain a K2,3-subgraph that contains a vertex of degree 2 in G.
Proof. Suppose, to the contrary, that G contains a K2,3-subgraph, F, that contains a vertex, say u, of degree 2
in G. Let w1 and w2 be the two neighbors of u in F, and so w1 and w2 have degree 3 in F. Let u1 and u2 be the
common neighbors of w1 and w2 in F different from u. Thus, NG(w1) = NG(w2) = {u, u1, u2}. By Claim 6, both
u1 and u2 have degree 3 in G. If u1u2 ∈E(G), then the graph G is determined, and so n = 5, a contradiction.
Hence, u1u2 /∈E(G). Let v1 and v2 be the neighbors of u1 and u2, respectively, that do not belong to F.
We show firstly that v1 ̸= v2. Suppose, to the contrary, that v1 = v2. If degG(v1) = 2, then the graph G is
determined. In this case, we have i(G) = 2, w(G) = 20 and Θ(G) = 0. Thus, Ω(G) = 20, and so 8i(G) < Ω(G), a
contradiction. Hence, degG(v1) = 3. Let v be the neighbor of v1 different from u1 and u2. If degG(v) = 1, then the
graph G is determined, and as before 8i(G) < Ω(G) (noting that {u, v1} is an i-set of G), a contradiction. Hence,
degG(v) ≥2. We now let X = V (F) ∪{v, v1} and consider the graph G′ = G −X. We note that there are at
most two X-exit edges in G (and such edges are incident with v). By Claim 7, the removal of an X-exit edge when
constructing G′ increases the total weight by at most 3. If there are two X-exit edges in G, then wG(X) = 22, and
we infer that Ω(G′) ≤Ω(G) −22 + 6 = Ω(G) −16. On the other hand, if there is only one X-exit edge in G, then
wG(X) = 23, implying that Ω(G′) ≤Ω(G) −23 + 3 = Ω(G) −20. Every i-set of G′ can be extended to an ID-set of
G by adding to it the vertices u and v1, and so i(G) ≤i(G′) + 2. Hence, 8i(G) ≤8i(G′) + 16 ≤Ω(G′) + 16 ≤Ω(G),
a contradiction. Hence, v1 ̸= v2.
Suppose that v1v2 ∈E(G). If at least one of v1 and v2 has degree 3 in G, then the subgraph of G induced by
V (F) ∪{v1, v2} is a troublesome configuration in G, contradicting Claim 4. Hence, both v1 and v2 have degree 2
in G, and so the graph G is determined. In this case, i(G) = 3 and Ω(G) = w(G) = 24, and so 8i(G) = Ω(G), a
contradiction. Hence, v1v2 /∈E(G).
We now consider the graph G′ obtained from G −V (F) by adding the edge e = v1v2. Let I′ be an i-set of G′. If
neither v1 nor v2 belongs to I′, then let I = I′ ∪{w1, w2}. If v1 ∈I′, then let I = I′ ∪{u, u2}. If v2 ∈I′, then let
I = I′ ∪{u, u1}. In all cases, the set I is an ID-set of G and |I| = |I′| + 2. Thus, i(G) ≤|I| ≤|I′| + 2 = i(G′) + 2.
Moreover, w(G′) = w(G) −16 = Ω(G) −16. If Θ(G′) = 0, then Ω(G′) = w(G′) = Ω(G) −16, and so 8i(G) ≤
8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. Hence, Θ(G′) > 0, implying that either G′ ∈B (noting that
G′ is connected) or there is a troublesome configuration in G′ that contains the (adjacent) vertices v1 and v2. In
both cases, we have Θ(G′) = 2, and so Ω(G′) = w(G′) + Θ(G′) = (Ω(G) −16) + 2 = Ω(G) −14.
Suppose that G′ ∈B. We note that G′ contains at least one copy of K2,3 that contains two vertices of degree 2
in G′. By Claim 6, G′ −v1v2 does not contain a K2,3-subgraph that contains two vertices of degree 2 in G. We
therefore infer that G′ ∈B1 and that G′ contains exactly one copy of K2,3, say F ′, that contains two vertices of
degree 2 in G′. Moreover, the added edge v1v2 belongs to F ′. Thus, there are two possibilities that may occur.
The first case is that both v1 and v2 have degree 3 in G′, as illustrated in Figure 13(a), and the second case is that
one of v1 and v2 has degree 2 in G′, as illustrated in Figure 13(b). In both illustrations, we indicate the added edge
v1v2 by the dotted edge, which recall exists in G′ but not in G.
(a)
r
v1
v2
F ′
(b)
r
v1
v2
F ′
Figure 13: Possible graphs G′ in the proof of Claim 8
By properties of the bad family B (see Section 5.2), we infer that there is an i-set, say I′, of G′ −v1v2 that
contains both v1 and v2 and is such that |I′| = i(G′) −1 in the first case (as indicated by the shaded vertices in
13
Figure 13(a)) and |I′| = i(G′) in the second case (as indicated by the shaded vertices in Figure 13(b)). In both
cases, we have that I′ is an ID-set of G′ −v1v2. Moreover, {v1, v2} ⊂I′ and |I′| ≤i(G′). The set I′ can be
extended to an ID-set of G by adding to it the vertex u. Thus, i(G) ≤i(G′) + 1. We note that w(G) = w(G′) + 16
and Θ(G) = Θ(G′) −2, and so Ω(G) = w(G) + 16 + Θ(G) = w(G′) + Θ(G′) −2 = Ω(G′) + 14.
Therefore,
8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. Hence, G′ /∈B, and so b(G′) = 0.
Thus, there is a troublesome configuration, say T ′, in G′ that contains the (adjacent) vertices v1 and v2. Let z1
and z2 be the two link vertices in T ′. Recall that e = v1v2 is the edge that was added to construct G′. If e does not
belong to a copy of K2,3 in T ′ and if e ̸= z1z2, then removing the edge e from T ′ and adding back the deleted K2,3-
subgraph F (and the edges u1v1 and u2v2) produces a troublesome configuration in G, a contradiction. Hence either
the edge e belongs to a copy of K2,3 in T ′ or v1 and v2 are the two link vertices in T ′ (that is, {v1, v2} = {z1, z2}).
Let T ′ be obtained from a bad graph B′ ∈B, where we may assume (renaming z1 and z2 if necessary) that z1 is
the root vertex of B′.
Claim 8.1 The edge e joins the two link vertices in T ′, that is, {v1, v2} = {z1, z2}.
Proof.
Suppose that e belongs to a copy of K2,3, say Fe, in T ′. Let T ′ have k′ ≥1 vertex disjoint copies of
K2,3. Let I′ be the non-canonical independent set of B′, and so |I′| = 2k′ (and neither z1 nor z2 belong to I′).
Moreover, let I′
e be obtained from I′ by deleting the two vertices in Fe that belong to I′ and replacing them with
the vertices v1 and v2. We note that |I′
e| = |I′| = 2k′.
Suppose firstly that neither v1 nor v2 is adjacent to z1 or z2. In this case, we let X∗= V (F) ∪V (B′) \ {z1}
and we consider the graph G∗= G −X∗. In the case when k′ = 1, the graph illustrated in Figure 14(a) is an
example of a subgraph of G, where the subgraphs F, Fe, T ′ and G∗are indicated by the dashed boxes. Since
tc(G) = 0, any troublesome configuration in G∗necessarily contains the adjacent vertices z1 and z2. However,
both z1 and z2 have degree at most 2 in G∗, and therefore cannot belong to a troublesome configuration in G∗.
Moreover, since no bad graph contains two adjacent vertices, both of which has degree at most 2, and since
b(G) = 0, the connected graph G∗does not belong to the bad family B, that is, b(G∗) = 0. Thus, Θ(G∗) = 0 and
Ω(G∗) = w(G∗) = w(G) −wG(X∗) + 2 = w(G) −16(k′ + 1) + 2 = Ω(G) −16k′ −14. We now let I∗= I′
e ∪{u}.
Thus, |I∗| = 2k′ + 1. In the illustration in Figure 14, the set I∗is indicated by the shaded vertices. Every i-set of
G∗can be extended to an ID-set of G by adding to it the set I∗, and so i(G) ≤i(G∗) + |X∗| = i(G∗) + 2k′ + 1.
Thus, 8i(G) ≤8(i(G∗) + 2k′ + 1) ≤Ω(G∗) + 16k′ + 8 = Ω(G) −6 < Ω(G), a contradiction.
Hence renaming vertices if necessary, we may assume that v2z2 ∈E(G). In this case, we let X∗= NG[v1]∪NG[u]
and we consider the graph G∗= G −X∗. In the case when k′ = 1, the graph illustrated in Figure 14(b) is an
example of a subgraph of G, where the subgraphs F, Fe, T ′ and G∗are indicated by the dashed boxes. Since
b(G) = tc(G) = 0, we infer from the structure of the graph G∗that b(G∗) = tc(G∗) = 0, and so Θ(G∗) = 0 and
Ω(G∗) = w(G∗) = w(G)−wG(X∗)+5 = w(G)−23+5 = Ω(G)−18. Every i-set of G∗can be extended to an ID-set
of G by adding to it the vertices u and v1, and so i(G) ≤i(G∗) + 2. Thus, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 =
Ω(G) −2 < Ω(G), a contradiction. (2)
By Claim 8.1, the edge e joins the two link vertices in T ′, that is, {v1, v2} = {z1, z2}. Recall that F is an
arbitrary K2,3-subgraph that contains exactly one vertex of degree 2 in G.
Moreover, u1 and u2 are the two
vertices of degree 2 in F that have degree 3 in G, where v1 and v2 are the neighbors of u1 and u2, respectively, that
do not belong to F. We have shown that v1 ̸= v2 and v1 and v2 are not adjacent. Moreover, we have shown that if
G′ is obtained from G −V (F) by adding the edge e = v1v2, then G′ contains a troublesome configuration, say T ′,
that necessarily contains v1 and v2 as the two link vertices of F ′. This property, which we call property (∗), holds
for every K2,3-subgraph that contains exactly one vertex of degree 2 in G.
We note that every K2,3-subgraph in T ′ contains only one vertex of degree 2 in G, for otherwise, T ′ would contain
a K2,3-subgraph with two vertices of degree 2 in G, contradicting Claim 6. Suppose that T ′ contains two or more
copies of K2,3. In this case, we could have chosen the K2,3-subgraph F to be adjacent to another K2,3-subgraph.
With this choice of F, and with the vertices u1, u2, v1, and v2 as defined earlier (notably, u1 and u2 are vertices of
degree 3 whose neighbors outside F are v1 and v2, respectively), at least one of v1 or v2 belongs to a copy of K2,3
in T ′, contradicting property (∗) that v1 and v2 are necessarily the two link vertices of T ′ (and noting that the link
vertices of T ′ do not belong to a copy of K2,3 in T ′).
14
v1
v2
z1
z2
u1
u2
u
w1
w2
T ′
Fe
F
G∗
e
(a)
v1
v2
z1
z2
u1
u2
u
w1
w2
T ′
Fe
F
G∗
e
(b)
Figure 14: Possible subgraphs in the proof of Claim 8.1
Hence, T ′ contains exactly one copy of K2,3, say F ′. Let x be the vertex of F ′ of degree 2 in G, and let w3 and
w4 be the two neighbors of x in F ′. Let x1 and x2 be the common neighbors of w3 and w4 in F ′ different from x.
Thus, NG(w3) = NG(w4) = {x, x1, x2}. Renaming x1 and x2 if necessary, we may assume that v1x1 and v2x2 are
edges of G. Thus the graph illustrated in Figure 15 is a subgraph of G, where the added edge e of G′ is indicated
by the dotted line joining v1 and v2.
v1
v2
u1
u2
w2
u
w1
x1
x2
w4
x
w3
F
F ′
T ′
added edge e
Figure 15: A possible subgraph in the proof of Claim 8
Since T ′ is a troublesome configuration of G′, at least one of v1 and v2 has degree 3 in G′. By symmetry, we
may assume that degG(v1) = 3 and we let y1 be the neighbor of v1 in G different from u1 and x1.
Suppose
that degG(v2) = 2, and so NG(v2) = {u2, x2}. If degG(y1) = 1, then the graph G is determined. In this case,
i(G) = |{u, x, v1, v2}| = 4 and Ω(G) = w(G) = 16+16+5+4+3 = 44. Thus, 8i(G) < Ω(G), a contradiction. Hence,
degG(y1) ≥2. In this case, we let X∗= V (F)∪V (F ′)∪{v1, v2, y1} and we consider the graph G∗= G−X∗. We note
that every X∗-exit edge is incident with y1. Moreover, since degG(y1) ≥2, there is at least one X∗-exit edge. By
Claim 7, the removal of an X∗-exit edge when constructing G∗increases the total weight by at most 3. Since there
are at most two X∗-exit edges, we have that Ω(G∗) ≤Ω(G)−wG(X∗)+6 ≤Ω(G)−(16+16+4+3+3)+6 = Ω(G)−36.
Every i-set of G∗can be extended to an ID-set of G by adding to it the set {u, x, v1, v2}, and so i(G) ≤i(G∗) + 4.
Thus, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 = Ω(G) −4 < Ω(G), a contradiction. Hence, degG(v2) = 3. Let y2 be
the neighbor of v2 in G different from u2 and x2.
Suppose that y1 = y2. If degG(y1) = 2, then the graph G is determined, and in this case i(G) = |{u, x, v1, v2}| = 4
and Ω(G) = w(G) = 16+16+4+3+3 = 42. Thus, 8i(G) < Ω(G), a contradiction. Hence, degG(y1) = 3. As before,
15
we let X∗= V (F) ∪V (F ′) ∪{v1, v2, y1} and we consider the graph G∗= G −X∗. In this case, there is only one
X∗-exit edge, implying analogously as before that Ω(G∗) ≤Ω(G)−wG(X∗)+3 = Ω(G)−(16+16+3+3+3)+3 =
Ω(G) −38. As before, i(G) ≤i(G∗) + 4, yielding the contradiction 8i(G) < Ω(G). Hence, y1 ̸= y2. We now let
X′′ = V (F) ∪V (F ′) ∪{v1, v2, y1, y2} and we consider the graph G′′ = G −X′′. Thus the graph illustrated in
Figure 16 is a subgraph of G.
v1
v2
y1
y2
u1
u2
w2
u
w1
x1
x2
w4
x
w3
X′′
G′′
Figure 16: A possible subgraph in the proof of Claim 8
We note that there are at most four X′′-exit edges, namely at most two edges incident with y1 and at most two
edges incident with y2. By Claim 7, removing each X′′-exit edge when constructing G′′ increases the total weight
by at most 3. Thus, Ω(G′′) ≤Ω(G) −wG(X′′) + 4 × 3 = Ω(G) −(16 + 16 + 12) + 12 = Ω(G) −32. Every i-set of
G′′ can be extended to an ID-set of G by adding to it the set {u, x, v1, v2} (as indicated by the shaded vertices in
Figure 16), and so i(G) ≤i(G′′) + 4. Thus, 8i(G) ≤8(i(G′′) + 4) ≤Ω(G′′) + 32 = Ω(G), a contradiction. (2)
By Claim 8, every vertex that belongs to a K2,3-subgraph has degree 3 in G.
Claim 9 Every K2,3-subgraph in G is an induced subgraph in G.
Proof.
Suppose, to the contrary, that G contains a K2,3-subgraph, F, that is not an induced subgraph of G.
Let the subgraph F have partite sets {w1, w2} and {u1, u2, u3}. Renaming vertices if necessary, we may assume
that u2u3 ∈E(G). If degG(u1) = 2, then the graph G is determined, and so n = 5, a contradiction. Hence,
degG(u1) = 3. We now consider the graph G′ = G −{u2, u3, w1, w2}. We note that the vertex u1 has degree 1 in
G′, and so u1 does not belong to a troublesome configuration in G′. Moreover, by Claim 6, the (connected) graph
G′ /∈B. Thus, Ω(G′) = Ω(G) −12 + 2 = Ω(G) −10. Every i-set of G′ can be extended to an ID-set of G by adding
to it the vertex u2, and so i(G) ≤i(G′) + 1. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2)
Claim 10 b(G −e) = 0 for all e ∈E(G).
Proof. Suppose, to the contrary, that the claim is false. Thus there exists an edge e in G whose removal creates
a bad component. Let B ∈B be such a bad component in G −e. The component B contains at least one copy of
K2,3, say F, with two (nonadjacent) vertices of degree 2 in B. By Claim 9, F is an induced subgraph of G. Thus
the edge e is incident with at most one of the two vertices in F that have degree 2 in B. Therefore, F contains at
least one vertex of degree 2 in G, contradicting Claim 8. (2)
Claim 11 b(G −e1 −e2) = 0 for all e1, e2 ∈E(G).
Proof. Suppose, to the contrary, that the claim is false. Thus there exists edges e1 and e2 in G whose removal
creates a bad component. Let B ∈B be such a bad component in G −e1 −e2, and let v be the root vertex of B.
16
Let B contain k ≥1 copies of K2,3. If B ∈B3, then k ≥3 and there are k + 3 ≥6 vertices of degree 2 in B.
Thus, B contains at least one copy of K2,3 that contains a vertex of degree 2 in G, contradicting Claim 8. Hence,
B ∈B1 ∪B2.
Suppose that B ∈B2. In this case, k ≥2 and there are k + 2 ≥4 vertices of degree 2 in B that belong to
K2,3-subgraphs. If k ≥3, then there are at least five vertices of degree 2 in B that belong to K2,3-subgraphs,
implying that B contains at least one copy of K2,3 that contains a vertex of degree 2 in G, a contradiction. Hence,
k = 2. Let F1 and F2 be the two copies of K2,3 in B, and let ui and vi be the two vertices in Fi that have degree 2
in B for i ∈[2]. By Claims 8 and 9, we infer renaming vertices if necessary that e1 = u1u2 and e2 = v1v2. However,
then the graph G is determined and is illustrated in Figure 17(a), where the edges e1 and e2 are dotted. In this
case, i(G) = 3 (an i-set of G is indicated by the three shaded vertices in Figure 17(a)) and Ω(G) = w(G) = 34.
Thus, 8i(G) < Ω(G), a contradiction.
v
u1
u2
v1
v2
e1
e2
(a)
(b)
v
v2
u1
u2
e1
e2
(c)
v
v2
u1
u2
e1
e2
G′
Figure 17: Possible subgraphs in the proof of Claim 11
Hence, B ∈B1, and so k ≥1 and there are k + 1 ≥2 vertices of degree 2 in B that belong to K2,3-subgraphs. If
k ≥4, then there are at least five vertices of degree 2 in B that belong to K2,3-subgraphs, implying that B contains
at least one copy of K2,3 that contains a vertex of degree 2 in G, a contradiction. Therefore, k ≤3.
Suppose that k = 3. In this case, there are four vertices of degree 2 in B that belong to K2,3-subgraphs. Since
no copy of a K2,3-subgraph in B contains a vertex of degree 2 in G, we infer that these four vertices of degree 2 in
B are incident with the edges e1 and e2 in G. The graph G is now determined and there are two non-isomorphic
graphs to which G can be isomorphic. By Claim 9, the edge ei joins vertices that belong to different copies of K2,3
in B for i ∈[2]. The set consisting of the three core vertices in B, together with one end of e1 and one end of e2, is
an ID-set of G, and so i(G) ≤5. Moreover, Ω(G) = w(G) = 15 × 3 + 5 = 50, and so 8i(G) < Ω(G), a contradiction.
Therefore, k ≤2.
Suppose that k = 2. In this case, there are three vertices of degree 2 in B that belong to K2,3-subgraphs. Let
F1 and F2 be the two copies of K2,3 in B, where F2 is the copy that contains two vertices of degree 2 in B. Let u1
be the vertex in F1 of degree 2 in B and let u2 and v2 be the two vertices in F2 of degree 2 in B. Since no copy of
a K2,3-subgraph in B contains a vertex of degree 2 in G, we infer that the vertices u1, u2 and v2 are incident with
the edges e1 and e2 in G. Moreover by Claim 9, vertices u2 and v2 are not adjacent in G. Renaming u2 and v2 if
necessary, we may assume that e1 = u1u2 and that e2 is incident with v2.
Suppose that vertex v is incident with e2, and so e2 = vv2. Thus, the graph G is determined as is illustrated in
Figure 17(b). In this case, i(G) = 3 (an i-set of G is indicated by the three shaded vertices in Figure 17(b)) and
Ω(G) = w(G) = 34. Thus, 8i(G) < Ω(G), a contradiction. Hence, the vertex v is not incident with e2. We now
consider the graph G′ = G −(V (B) \ {v2}) (see Figure 17(c), where the graph G′ is indicated by the dashed box).
We note that Ω(G′) = w(G′) = w(G) −32 + 2 = Ω(G) −30. Every i-set of G′ can be extended to an ID-set of
G by adding to it the two core vertices of B together with the vertex u2 (as indicated by the shaded vertices in
Figure 17(c). Thus, i(G) ≤i(G′) + 3, and so 8i(G) < Ω(G), a contradiction.
Hence, k = 1. Let F be the copy of K2,3 in B, and let u1 and v1 be the two vertices of degree 2 in B that
belong to F. Since every vertex in F has degree 3 in G, the vertices u1 and v1 are incident with the edges e1 and
e2 in G. Moreover by Claim 9, vertices u1 and v1 are not adjacent in G. Renaming edges if necessary, we may
assume that the edge e1 is incident with u1 and the edge e2 is incident with v1. Since G ̸= K3,3, the vertex v
is adjacent to at most one of u1 or v1. If v is adjacent to either u1 or v1, then v would be a vertex of degree 2
in G that belongs to a copy of K2,3, contradicting Claim 8. Hence, v has degree 1 in G. Let u be the neighbor
of v. We now consider the graph G′ = G −N[u]. We note that both u1 and v1 have degree 1 in G′. Moreover,
17
Ω(G′) = w(G′) = w(G) −14 + 4 = Ω(G) −10. Every i-set of G′ can be extended to an ID-set of G by adding to it
the vertex u, and so i(G) ≤i(G′) + 1. Thus, 8i(G) < Ω(G), a contradiction. (2)
Claim 12 tc(G −e) ≤1 for all e ∈E(G).
Proof. Suppose, to the contrary, that the claim is false. Thus there exists an edge e = uv in G whose removal
creates two troublesome configurations. Let Tw = Tw1,w2 and Tv = Tv1,v2 be the two new troublesome configuration
in G −e, where w1 and w2 are the link vertices of Tw and v1 and v2 are the link vertices of Tv. Let Tw be obtained
from the bad graph Bw ∈B1 with root vertex w1 (of degree 1 in Bw), and let x1 and x2 be the vertices of Bw
adjacent to w1 and w2, respectively, in G. Further, let Tv be obtained from the bad graph Bv ∈B1 with root
vertex v1 (of degree 1 in Bv), and let y1 and y2 be the vertices of Bv adjacent to v1 and v2, respectively, in G. We
may assume that u ∈V (Bw) and v ∈V (Bv). Let u1 and u2 be the two neighbors of u in Bw, and let a1 and a2
be the two neighbors of v in Bv. By Claim 8, we infer that each of Bw and Bv contain exactly one K2,3-subgraph,
which we name Fw and Fv, respectively. Thus, V (Fw) = {u, u1, u2, x1, x2} and V (Fv) = {v, a1, a2, y1, y2}. If
degG(v1) = 3, then let z1 be the neighbor of v1 different from y1 and v2. Moreover, let z2 be the neighbor of v2
different from y2 and v1.
Suppose that degG(v1) = 3 and degG(z1) = 1. In this case, we let X = NG[u] ∪(V (Tv) ∪{z1}) and we let
G′ = G −X. Thus, the graph shown in Figure 18 is a subgraph of G, where Tv and Tw are the troublesome
configurations indicated in the dashed boxes and where X is the set indicated by the vertices in the dotted region.
The removal of the four X-exit edges incident with u1 and u2 when constructing G′ increases the total weight
by 4, noting that the component in G′ that contains the vertices w1 and w2 contains the path x1w1w2x2, where
x1 and x2 have degree 1 in G′ (and therefore w1 and w2 do not belong to a bad component nor a troublesome
configurations). Moreover by Claim 7, the removal of the X-exit edge v2z2 when constructing G′ increases the
total weight by at most 3. Therefore, Ω(G′) ≤w(G) −wG(X) + 7 = Ω(G) −35 + 7 = Ω(G) −28. Every i-set of
G′ can be extended to an ID-set of G by adding to it the vertices u, v1 and y2 (indicated by the shaded vertices in
Figure 18), and so i(G) ≤i(G′) + 3. Hence, 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction.
w1
w2
x2
u
x1
z1
z2
v1
v2
y2
y1
v
a1
a2
u2
u1
Tw
Tv
X
e
Figure 18: An illustration of a subgraph of G in the proof of Claim 12
Hence either degG(v1) = 2 or degG(v1) = 3 and degG(z1) ≥2. We now let X = NG[u] ∪(V (Tv) \ {v2, y2}) and
let G′ = G −X. Thus, the graph shown in Figure 19 is a subgraph of G (where in this illustration degG(v1) = 3),
where Tv and Tw are the troublesome configurations indicated in the dashed boxes and where X is the set indicated
by the vertices in the dotted region. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices
u and y1 (indicated by the shaded vertices in Figure 19), and so i(G) ≤i(G′) + 2. If degG(v1) = 2, then there
are seven X-exit edges, and so w(G′) = w(G) −wG(X) + 7 = w(G) −25 + 7 = w(G) −18. If degG(v1) = 3, then
there are eight X-exit edges, and so w(G′) = w(G) −wG(X) + 8 = w(G) −24 + 8 = w(G) −16. The component
in G′ that contains the vertices w1 and w2 contains at least two vertices of degree 1, namely the vertices x1 and
x2. Further, we note that the vertex y2 has degree 1 in G′ and its neighbor v2 has degree 2 in G′. Moreover, if v1
has degree 3 in G and its neighbor z1 belongs to a component of G′ that contains neither w1 nor v2, then such a
component is not a bad component (that belongs to B), by Claim 10. We therefore infer that b(G′) = 0. We also
note that the component in G′ that contains the vertex w1 (and w2) does not contain a troublesome configuration
and neither does the component in G′ that contains the vertex v2 (and y2). Hence either G′ has no troublesome
configuration or G′ has exactly one troublesome configuration and such a troublesome configuration necessarily
contains the vertex z1.
18
w1
w2
x2
u
x1
v1
v2
y2
y1
v
a1
a2
u2
u1
Tw
Tv
X
e
Figure 19: An illustration of a subgraph of G in the proof of Claim 12
Suppose that G′ has no troublesome configuration; that is, tc(G′) = 0. As observed earlier, b(G′) = 0, and so
Θ(G′) = 0. Thus, Ω(G′) = w(G′) ≤w(G) −16. Hence, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 ≤w(G) = Ω(G), a
contradiction. Therefore, G′ has exactly one troublesome configuration, namely a troublesome configuration, Tz1
say, that contains the vertex z1. In this case, we note that degG(v1) = 3. Interchanging the roles of v1 and v2,
by analogous arguments (letting X = NG[u] ∪(V (Tv) \ {v1, y1})) we infer that G −v2z2 contains a troublesome
configuration, Tz2 say, that contains the vertex z2. Thus the graph shown in Figure 20 is a subgraph of G. Let Z
be the set of vertices in Tz1 different from the two link vertices of Tz1 in G −v1z1. We note that G[Z] is a copy
of K2,3. Let Iz be the non-canonical independent set of G[Z], and so Iz consists of the two neighbors of z1 in Tz1.
Let Y = NG[u] ∪V (Tv) ∪Z ∪{z2}. The set Y is indicated by the vertices in the dotted region in Figure 20. We
now consider the graph G′′ = G −Y .
Tz1
Tz2
w1
w2
x2
u
x1
v1
v2
y2
y1
v
u2
u1
z1
z2
Tw
Tv
Y
e
Figure 20: An illustration of a subgraph of G in the proof of Claim 12
Every i-set of G′′ can be extended to an ID-set of G by adding to it the set Iz ∪{u, v2, y1} (as indicated by the five
shaded vertices in Figure 20), and so i(G) ≤i(G′′)+5. We note that wG(Y ) = 48. Since there are eight Y -exit edges,
we have w(G′′) = w(G)−wG(Y )+8 = w(G)−48+8 = w(G)−40. Because tc(G) = b(G) = 0 and by the structure of
the graph G′′ (recalling also the properties of bad components and troublesome configurations in Propositions 1(f)
and 2(f)), we infer tc(G′′) = b(G′′) = 0. Thus, Θ(G′′) = 0, and so Ω(G′′) = w(G′′) = w(G′′) −40 = Ω(G) −40.
Hence, 8i(G) ≤8(i(G′′) + 5) ≤Ω(G′′) + 40 = Ω(G), a contradiction. (2)
Claim 13 tc(G −e) = 0 for all e ∈E(G).
Proof.
Suppose, to the contrary, that tc(G −e) ≥1 for some edge e ∈E(G). By Claim 12, tc(G −e) ≤1.
Thus, tc(G −e) = 1, that is, there exists an edge e in G whose removal creates a troublesome configuration. Let
19
T = Tv1,v2 be a troublesome configuration in G −e with link vertices v1 and v2, where degG(v2) = 3. Thus, T is
obtained from a bad graph B ∈B1 with root vertex v1 (of degree 1 in B). Further, let u1 and u2 be the vertices
of B adjacent to v1 and v2, respectively, in G. If v1 has degree 3 in G, then let z1 be the neighbor of v1 different
from u1 and v2. Moreover, let z2 be the neighbor of v2 different from u2 and v1.
Claim 13.1 The vertex v1 is not incident with the edge e.
Proof. Suppose that v1 is incident with the edge e. In this case, by Claims 8 and 9, the bad graph B contains
exactly one copy of K2,3, say F, and so B ∈B1.
Moreover, the end of the edge e, say u, different from v1
belongs to F, for otherwise once again G would contain a copy of K2,3 that contains a vertex of degree 2 in G, a
contradiction. Thus, the graph illustrated in Figure 21 is a subgraph of G, where T is the troublesome configuration
indicated in the dashed box and where e = uv1. We now let X = V (T) and we let G′ = G −X. The removal of
the (unique) X-exit edge, namely v2z2 when constructing G′ increases the total weight by at most 3 by Claim 7.
By Claim 10, b(G′) = 0. Thus, Ω(G′) ≤w(G) −wG(X) + 3 = Ω(G) −21 + 3 = Ω(G) −18. Every i-set of
G′ can be extended to an i-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. Hence,
8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G) −2 < Ω(G), a contradiction. (2)
v1
v2
u2
u1
u
z2
T
e
Figure 21: An illustration of a subgraph of G in the proof of Claim 13.1
By Claim 13.1, the vertex v1 is not incident with the edge e. Let e = uv, where u ∈V (B).
Claim 13.2 v /∈V (B).
Proof. Suppose that v ∈V (B), and so {u, v} ⊂V (B). By Claims 8 and 9, the bad graph B contains two copies of
K2,3. Let Fu and Fv be the copies of K2,3 that contain u and v, respectively, where we may assume that u1 ∈V (Fu)
(and so, u2 ∈V (Fv)). Let X = V (T)\{u1, u2, v1, v2} and let G′ = G−X. Thus, the graph illustrated in Figure 22
is a subgraph of G, where T is the troublesome configuration indicated in the dashed box and the set X is indicated
by the vertices in the dotted region. Every i-set of G′ can be extended to an ID-set of G by adding to it two vertices
(indicated by the two shaded vertices in Figure 22), and so i(G) ≤i(G′)+2. From the structure of the graph G and
since b(G) = tc(G) = 0, we note that Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) −24+ 4 = w(G)−20 = Ω(G) −20.
Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. (2)
v1
v2
u2
u1
u
v
T
X
e
Figure 22: An illustration of a subgraph of G in the proof of Claim 13.2
20
By Claim 13.2, we have v /∈V (B). By Claim 8 and Claim 9, the bad graph B contains only one copy of K2,3,
say Fu. Let w1 and w2 be the two neighbors of u in B, and so NB(w1) = NB(w2) = {u, u1, u2}.
Claim 13.3 degG(v1) = 3.
Proof. Suppose that degG(v1) = 2. Let X = NG[u1] = {u1, v1, w1, w2} and let G′ = G −X. Thus, the graph
illustrated in Figure 23 is a subgraph of G, where T is the troublesome configuration indicated in the dashed box
and the set X is indicated by the vertices in the dotted region. Every i-set of G′ can be extended to an ID-set
of G by adding to it the vertex u1 (indicated by the shaded vertes in Figure 23), and so i(G) ≤i(G′) + 1. From
the structure of the graph G and since b(G) = tc(G) = 0, we note that Θ(G′) = 0, and so Ω(G′) = w(G′) =
w(G) −13 + 5 = w(G) −8 = Ω(G) −8. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. (2)
v1
v2
u2
u1
v
u
w1
w2
T
X
e
Figure 23: An illustration of a subgraph of G in the proof of Claim 13.3
By Claim 13.3, we have degG(v1) = 3.
Claim 13.4 z1 ̸= z2.
Proof. Suppose that z1 = z2, and so {v1, v2} ⊆NG(z1). Let X = (V (T) \ {u}) ∪{z1} and let G′ = G −X. Every
i-set of G′ can be extended to an ID-set of G by adding to it the vertices u1 and v2, and so i(G) ≤i(G′) + 2. If
NG(z1) = {v1, v2}, then degG(z1) = 2, wG(X) = 22 and Ω(G′) = w(G′) = w(G) −wG(X) + 2 = w(G) −22 + 2 =
w(G) −20. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = w(G) −4 < Ω(G), a contradiction. Hence, degG(z1) = 3.
Let w be the neighbor of z1 different from v1 and v2.
Thus the graph shown in Figure 24 is a subgraph of
G, where T is the troublesome configuration indicated in the dashed box and the set X is indicated by the
vertices in the dotted region. As noted earlier, i(G) ≤i(G′) + 2. The removal of the X-exit edge z1w when
constructing G′ increases the total weight by at most 3 by Claim 7. Moreover, the removal of the two X-exit
edges incident with u when constructing G′ increases the total weight by 2 noting that the vertex u does not
belong to a bad component or a troublesome configuration in G′ but its vertex weight increased by 2.
Thus,
Ω(G′) ≤Ω(G)−wG(X)+3+2 = Ω(G)−21+3+2 = Ω(G)−16. Hence, 8i(G) ≤8(i(G′)+2) ≤Ω(G′)+16 = Ω(G),
a contradiction. (2)
By Claim 13.4, we have z1 ̸= z2.
Claim 13.5 degG(zi) ≥2 for i ∈[2].
Proof.
Suppose that degG(z1) = 1 or degG(z2) = 1. By symmetry, we may assume that degG(z1) = 1. Let
X = (V (T) \ {u}) ∪{z1} and let G′ = G −X. Every i-set of G′ can be extended to an ID-set of G by adding to
it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. The graph shown in Figure 25 is a subgraph of G, where T is
the troublesome configuration indicated in the dashed box and the set X is indicated by the vertices in the dotted
region. The removal of the X-exit edge z2v2 when constructing G′ increases the total weight by at most 3. Moreover,
the removal of the two X-exit edges incident with u when constructing G′ increases the total weight by 2. Thus,
Ω(G′) ≤Ω(G)−wG(X)+3+2 = Ω(G)−23+3+2 = Ω(G)−18. Hence, 8i(G) ≤8(i(G′)+2) ≤Ω(G′)+16 < Ω(G),
a contradiction. (2)
By Claim 13.5, we have degG(zi) ≥2 for i ∈[2].
21
v1
v2
u2
u1
v
u
z1
w
w1
w2
T
X
e
Figure 24: An illustration of a subgraph of G in the proof of Claim 13.4
v1
v2
u2
u1
v
u
z2
z1
w1
w2
T
X
e
Figure 25: An illustration of a subgraph of G in the proof of Claim 13.5
Claim 13.6 z1z2 ∈E(G).
Proof.
Suppose that z1z2 /∈E(G). Let X = V (T) \ {u} and let G′ be obtained from G −X by adding the
edge f = z1z2.
Let I′ be an i-set of G′.
If I′ ∩{z1, z2} = ∅, then let Iu = {u1, u2}.
If z1 ∈I′, then let
Iu = {u1, v2}. If z2 ∈I′, then let Iu = {v1, u2}. In all cases, we have |Iu| = 2 and the set I′ can be extended
to an ID-set of G by adding to it the set Iu. Thus, i(G) ≤|I′| + |Iu| = i(G′) + 2. We note that wG(X) = 18
and w(G′) = w(G) −wG(X) + 2 = w(G) −18 + 2 = w(G) −16. If Θ(G′) = 0, then Ω(G′) = w(G′), and so
8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = w(G′) + 16 = w(G) = Ω(G), a contradiction. Hence, Θ(G′) > 0. We
therefore infer that the component, G′
f say, of G′ that contains the edge f is either a bad component or contains
a troublesome configuration.
Claim 13.6.1 G′
f /∈B.
Proof. Suppose that G′
f ∈B. Let G′
f have root vertex vf (of degree 1 in Bf). We note that the degrees of the
vertices in G′
f are the same as their degrees in G, except possibly for the vertex u in the case when it belongs to
G′
f. By Claims 8 and 9, we infer that G′
f ∈B1, and so the root vf of G′
f has degree 1 in G. Further, we infer from
these claims that G′
f contains exactly one copy of K2,3, say Hf. By Claim 13.5, degG(zi) ≥2 for i ∈[2].
Suppose that u belongs to G′
f, implying that u is the root vertex vf. Since the vertex u is adjacent to neither v1
nor v2 in G and since zi is adjacent to vi in G for i ∈[2], we note that u /∈{z1, z2}. Thus, neither z1 nor z2 is
the root vertex vf of G′
f, and so the added edge f = z1z2 is an edge of Hf. The graph G is now determined and
V (G) = V (T)∪V (Hf). In particular, G has order n = 12. There are two possibilities up to isomorphism, depending
on whether v ∈{z1, z2} or v /∈{z1, z2}. In both cases, we have Ω(G) = w(G) = 38 and the set {w1, w2, z1, z2} is
an ID-set of G, and so i(G) ≤4. Thus, 8i(G) < Ω(G), a contradiction. Hence, u /∈V (G′
f), implying that G′ has
two components, namely the component G′
f and the component containing the vertices u and v.
As observed earlier, degG(zi) ≥2 for i ∈[2], and so the added edge f = z1z2 is an edge of Hf. Let X′ =
22
V (G′
f) ∪V (T) and let G′′ = G −X′. Thus the graph shown in Figure 26 is an example of a subgraph of G,
where T is the troublesome configuration indicated in the dashed box, G′
f −f is the subgraph indicated in the
dashed box, and the set X′ is indicated by the vertices in the dotted region. We note that there exists an ID-
set, If say, of G′
f −f that contains both vertices z1 and z2 and is such that |If| ≤3. Every i-set of G′′ can
be extended to an ID-set of G by adding to it the set If ∪{w1, w2} (as indicated by the shaded vertices in
Figure 26), and so i(G) ≤i(G′′)+5. By Claims 8 and 9, we infer that b(G′′) = 0 (and note that tc(G′′) = 0). Thus,
Ω(G′′) = w(G′′) = w(G)−wG(X)+1 = Ω(G)−43+1 = Ω(G)−42. Hence, 8i(G) ≤8(i(G′)+5) ≤Ω(G′)+40 < Ω(G),
a contradiction. (2)
v1
v2
u2
u1
v
u
z2
z1
vf
w1
w2
T
G′
f −f
X′
G′′
e
f
Figure 26: An illustration of a subgraph of G in the proof of Claim 13.6.1
We now return to the proof of Claim 13.6. By Claim 13.6.1, G′
f /∈B, implying that G′
f contains a troublesome
configuration, Tf say. Necessarily, Tf contains the added edge f. By Claims 8 and 9, we infer that Tf contains
exactly one copy of K2,3, and this copy of K2,3 contains the added edge f = z1z2. We may assume, by symmetry,
that z1 has degree 3 in the copy of K2,3 in Tf. We now let X∗= (V (T) \ {u}) ∪{z1} and let G∗= G −X∗. Every
i-set of G∗can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G∗) + 2. We
note that degG∗(u) = 1 and the degree of a neighbor of z1 in G∗is also 1. Thus by the structure of the graph
G∗so far determined, we infer that b(G∗) = 0 and tc(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G) −wG(X∗) + 5 =
Ω(G) −21 + 5 = Ω(G) −16. Hence, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 = Ω(G), a contradiction. (2)
By Claim 13.6, z1z2 ∈E(G).
Claim 13.7 degG(zi) = 3 for i ∈[2].
Proof.
Suppose that degG(zi) = 2 for some i ∈[2].
Suppose firstly that degG(z1) = degG(z2) = 2.
Thus,
NG(zi) = {vi, z3−i} for i ∈[2]. We now let X = (V (T) \ {u}) ∪{z1, z2} and let G′ = G −X. Every i-set of G′
can be extended to an ID-set of G by adding to it the vertices in the set {u1, u2, z1}, and so i(G) ≤i(G′) + 3. We
note that Θ(G′) = 0 and wG(X) = 26, and so Ω(G′) = w(G′) = w(G) −wG(X) + 2 = Ω(G) −26 + 2 = Ω(G) −24,
and so 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. Hence, at least one of z1 and z2 has degree 3
in G. By symmetry, we may assume that degG(z2) = 3, and so, by supposition, degG(z1) = 2. In this case,
we let X = (V (T) \ {u}) ∪{z1} and let G′ = G −X. Every i-set of G′ can be extended to an ID-set of G by
adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. We note that Θ(G′) = 0 and wG(X) = 22, and so
Ω(G′) = w(G′) = w(G)−wG(X)+4 = Ω(G)−22+4 = Ω(G)−18, and so 8i(G) ≤8(i(G′)+2) ≤Ω(G′)+16 < Ω(G),
a contradiction. (2)
By Claim 13.7, degG(zi) = 3 for i ∈[2]. Let yi be the neighbor of zi different from vi and z3−i for i ∈[2].
Claim 13.8 The following properties hold.
(a) degG(yi) ≥2 for i ∈[2].
23
(b) y1 ̸= y2.
Proof. (a) Suppose that at least one of y1 and y2 has degree 1 in G. By symmetry, we may assume that degG(y1) =
1. In this case, we let X = NG[z1] = {v1, y1, z1, z2} and let G′ = G −X. Every i-set of G′ can be extended to an
ID-set of G by adding to it the vertex z1, and so i(G) ≤i(G′)+1. The cost of removing the X-exit edge y2z2 when
constructing G′ increases the total weight by at most 3 by Claim 7. Moreover, the removal of the three X-exit
edges different from y2z2 increases the total weight by 3 since the removal of these three edges does not create a bad
component or a troublesome configuration. Thus, Ω(G′) ≤Ω(G) −wG(X) + 3 + 3 = Ω(G) −14 + 3 + 3 = Ω(G) −8.
Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. This proves part (a).
(b) Suppose that y1 = y2. In this case, we let X = (V (T) \ {u}) ∪{y1, z1, z2} and let G′ = G −X. Every i-set of
G′ can be extended to an ID-set of G by adding to it the vertices in the set {u1, u2, z1}, and so i(G) ≤i(G′)+3. We
note that Θ(G′) = 0 and wG(X) = 27, and so Ω(G′) = w(G′) ≤w(G) −wG(X) + 3 = Ω(G) −27 + 3 = Ω(G) −24,
and so 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. This proves part (b). (2)
By Claim 13.8, degG(yi) ≥2 for i ∈[2] and y1 ̸= y2.
Suppose that y1 does not belong to a troublesome
configuration in G−y1z1. In this case, we let X = (V (T)\{u})∪{z1} and let G′ = G−X. Every i-set of G′ can be
extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′)+2. We note that both u and
z2 have degree 1 in G′ and do not belong to a bad component or a troublesome configuration in G′. By supposition,
y1 does not belong to a troublesome configuration in G′. Moreover, removing the edge y1z1 does not create a bad
component containing y1. Hence, Θ(G′) = 0 and wG(X) = 21, and so Ω(G′) = w(G′) ≤w(G) −wG(X) + 5 =
Ω(G) −21 + 5 = Ω(G) −16, and so 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. Hence, y1 belongs
to a troublesome configuration in G −y1z1. By symmetry, y2 belongs to a troublesome configuration in G −y2z2.
Let Tyi be the troublesome configuration in G −yizi that contains the vertex yi for i ∈[2].
Ty1
Ty2
v1
v2
z1
z2
u2
u1
u
y1
y2
v
T
X
Figure 27: An illustration of a subgraph of G in the proof of Claim 13
We now let X = (V (T) \ {u}) ∪{z2} ∪NG[y1] and let G′ = G −X. Thus the graph shown in Figure 27 is a
subgraph of G, where T, Ty1 and Ty2 are indicated in the dashed box, and the set X is indicated by the vertices
in the dotted region. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set
{u1, v2, y1} (indicated by the shaded vertices in Figure 27), and so i(G) ≤i(G′) + 3. The cost of removing the
X-exit edge y2z2 when constructing G′ increases the total weight by 3. Moreover, the removal of the six X-exit
edges different from y2z2 increases the total weight by 6 since the removal of these six edges does not create a bad
component or a troublesome configuration. Thus, Ω(G′) ≤Ω(G)−wG(X)+3+6 = Ω(G)−33+3+6 = Ω(G)−24.
Hence, 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. This completes the proof of Claim 13. (2)
We present next two consequences of Claim 13.
24
Claim 14 tc(G −e1 −e2) ≤1 for all e1, e2 ∈E(G).
Proof. Suppose, to the contrary, that the claim is false. Thus there exists edges e1 = u1v1 and e2 = u2v2 in G
whose removal creates two troublesome configurations. Let Tw = Tw1,w2 and Tz = Tz1,z2 be the two troublesome
configuration in G −e1 −e2, where w1 and w2 are the link vertices of Tw and z1 and z2 are the link vertices of Tz.
By Claims 8, 9, and 13, we infer that the edges e1 and e2 both have one end in Tw and one end in Tz. Furthermore,
each of Tw and Tz contains exactly two copies of K2,3 and each copy of K2,3 contains an end of one of the edges
e1 and e2. We now let X = (V (Tw) \ {w1, w2, x1, x2}) ∪(V (Tz) \ {y1, y2, z1, z2}) and let G′ = G −X. Renaming
vertices if necessary, we may assume that the graph illustrated in Figure 28 is a subgraph of G, where Tw and Tz
are the troublesome configurations indicated in the dashed boxes and where X is the set indicated by the vertices in
the dotted region. We note that b(G′) = tc(G′) = 0, and so Θ(G′) = 0. Since there are eight X-exit edges, we have
Ω(G′) = w(G′) = w(G) −wG(X) + 8 = w(G) −48 + 8 = Ω(G) −40. Every i-set of G′ can be extended to an ID-set
of G by adding to it four vertices from the set X (indicated, for example, by the shaded vertices in Figure 28), and
so i(G) ≤i(G′) + 4. Thus, 8i(G) ≤8(i(G′) + 4) ≤Ω(G′) + 32 = Ω(G) −8 < Ω(G), a contradiction. (2)
w1
w2
x2
x1
z1
z2
y2
y1
v1
v2
u1
u2
Tw
Tz
X
e2
e1
Figure 28: An illustration of a subgraph of G in the proof of Claim 14
Let G7 be the graph shown in Figure 29. As a special case of Claim 13, we have the following result.
u1
u2
v3
v1
v2
x1
x2
Figure 29: The graph G7
Claim 15 The graph G does not contain G7 as a subgraph.
Proof. Suppose, to the contrary, that G contains G7 as a subgraph. Let F be such a G7-configuration and let
the vertices of F be named as in Figure 29. By Claim 8, there is no K2,3-subgraph in G that contains a vertex
of degree 2 in G. Hence, degG(v3) = 3. Let v be the neighbor of v3 different from x1 and x2. By Claim 9, either
v ∈{u1, u2} or v /∈V (F). We show that at least one of u1 and u2 has a neighbor in G that does not belong
to V (F). If v = u1 and degG(u2) = 2, then G = G7 + u1v3. In this case, i(G) = 2 and w(G) = 22, and so
8i(G) < Ω(G), a contradiction. Hence if v = u1, then degG(u2) = 3, and so u2 has a neighbor in G that does
not belong to V (F). By symmetry, if v = u2, then u1 has a neighbor in G that does not belong to V (F). If
v /∈V (F) and degG(u1) = degG(u2) = 2, then let X′ = V (F) \ {v3} and let G′ = G −X′. Every i-set of G′
can be extended to an ID-set of G by adding to it the vertices v1 and v2, and so i(G) ≤i(G′) + 2. We note that
Θ(G′) = 0 and wG(X′) = 20, and so Ω(G′) = w(G′) = w(G) −wG(X) + 2 = Ω(G) −20 + 2 = Ω(G) −18, and so
8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. Hence if v /∈V (F), then at least one of u1 and u2 has
a neighbor in G that does not belong to V (F). In all cases, at least one of u1 and u2 has a neighbor in G that does
not belong to V (F). Thus, removing the edge v3v creates a troublesome configuration, contradicting Claim 13. (2)
25
6.2
Part 2: The minimum degree of G equals 1
In this section, we consider the case when the minimum degree of G equals 1, that is, when δ(G) = 1. We prove a
number of claims that will culminate in a contradiction, thereby showing that this case cannot occur. A support
vertex in G is a vertex that has at least one neighbor that has degree 1, and we define a strong support vertex as a
vertex that has at least two neighbors of degree 1 in G.
Claim 16 δ(G) ≥2.
Proof. Suppose, to the contrary, that G contains a vertex of degree 1.
Claim 16.1 There is no strong support vertex in G.
Proof.
Suppose, to the contrary, that G contains a strong support vertex v. Let NG(v) = {v1, v2, v3} where
degG(v1) = degG(v2) = 1. Since n ≥6, we note that degG(v3) ≥2. Suppose that degG(v3) = 3 and the two
neighbors of v3 different from v both have degree 1 in G. In this case, the graph G is determined (and is a double
star S(2, 2)) and w(G) = 26. Moreover, i(G) = 3, and so 8i(G) < w(G), a contradiction. Hence, at most one
neighbor of v3 has degree 1.
Suppose that v3 has a neighbor, say v4, of degree 1. Since n ≥6, we note that degG(v3) = 3. Let v5 be the third
neighbor of v3 different from v and v4. By our earlier observations, degG(v5) ≥2. We now let X = NG[v] ∪{v4}
and let G′ = G −X. By Claim 10, b(G′) = 0 and by Claim 13, tc(G′) = 0. Hence, Θ(G′) = 0, and so since there is
exactly one X-exit edge, namely v3v5, we have Ω(G′) = w(G′) = w(G) −wG(X) + 1 = Ω(G) −21 + 1 = Ω(G) −20.
Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v and v4, and so i(G) ≤i(G′) + 2.
Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G) −4 < Ω(G), a contradiction. Hence, no neighbor of v3 has
degree 1 in G.
We now let X = NG[v] and let G′ = G −X. By Claim 11, b(G′) = 0 and by Claim 14, tc(G′) ≤1. Hence,
Θ(G′) = 2tc(G′) ≤2.
Since there are two X-exit edges, neither of which is incident to a vertex of degree 1
in G, we have Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −wG(X) + 2) + 2 ≤Ω(G) −16 + 2 + 2 = Ω(G) −12. Every
i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1.
Thus,
8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G) −4 < Ω(G), a contradiction. (2)
By Claim 16.1, every support vertex in G has exactly one leaf neighbor.
Claim 16.2 No vertex of degree 2 is a support vertex.
Proof. Suppose, to the contrary, that G contains a support vertex v of degree 2. Let v1 be the leaf neighbor of v, and
let v2 be the second neighbor of v. Since n ≥6, we note that degG(v2) ≥2. We now consider the connected subcubic
graph G′ = G−{v, v1}. By Claims 10 and 13, we have Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G)−9+1 = Ω(G)−8.
Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. (2)
Let P : v1 . . . vk be the longest path in G such that vertex vi is adjacent to a vertex, say ui, of degree 1 for all
i ∈[k]. Possibly, k = 1. By Claim 16.2, degG(vi) = 3 for all i ∈[k].
Claim 16.3 k ≤2.
Proof. Suppose, to the contrary, that k ≥3. If v1vk ∈E(G), then the graph G is determined and G is the corona
of a cycle of length k, and so i(G) = k and w(G) = 8k, whence 8i(G) = 8k = w(G), a contradiction. Hence,
v1vk /∈E(G). Let G′ be obtained from G−{u2, v2} by adding the edge v1v3. We note that w(G) = w(G′)+8. If G′
contains a troublesome configuration, then such a configuration has a copy of K2,3 that contains a vertex of degree 2
in G, contradicting Claim 8. Thus, tc(G′) = 0. Moreover since the connected graph G′ contains at least two vertices
of degree 1, we note that G′ /∈B, and so b(G′) = 0. Thus, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G)−8 = Ω(G)−8.
Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex u2, and so i(G) ≤i(G′) + 1. Hence,
8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction.
26
By Claim 16.3, we have k ≤2. If k = 1, then we let NG(v1) = {a, b, u1}, and if k = 2, then we let NG(v1) =
{a, u1, v2} and NG(v2) = {b, u2, v1}, where possibly a = b.
Claim 16.4 a ̸= b.
Proof. Suppose, to the contrary, that a = b, implying that k = 2. Since n ≥6 (and G contains a copy of K2,3),
we note that degG(a) = 3. Let NG(a) = {a1, v1, v2}. [See Figure 30(a).] By the maximality of the path P, the
vertex a1 has degree at least 2 in G. Let X = {a, u1, u2, v1, v2} and let G′ = G −X. By Claims 10 and 13, we
have Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) −wG(X) + 1 = Ω(G) −19 + 1 = Ω(G) −18. Every i-set of
G′ can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. Thus,
8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G) −2 < Ω(G), a contradiction. (2)
By Claim 16.4, a ̸= b.
[See Figure 30(b) and 30(c).]
Relabeling vertices if necessary, we may assume that
degG(a) ≤degG(b). By Claim 16.1, we have 2 ≤degG(a) ≤degG(b). If k = 1, let X = {u1, v1} and if k = 2, let
X = {u1, u2, v1, v2}. In both cases, let X′ = X ∪{a}.
u1
u2
v1
v2
a
a1
(a)
u1
u2
v1
v2
a
b
(b)
u1
v1
a
b
(c)
Figure 30: Subgraphs in the proof of Claim 16
Claim 16.5 The graph G −X′ is isolate-free.
Proof. Suppose, to the contrary, that G −X′ has an isolated vertex, z say. By Claim 16.2 and the maximality
of the path P, the vertex z has degree at least 2 in G, implying that degG(z) = 2 and NG(z) = {a, vk}. Hence,
z = b. Thus, 2 ≤degG(a) ≤degG(b) = 2, and so degG(a) = degG(b) and ab ∈E(G). The graph G is therefore
determined. However, G does not contain a subgraph isomorphic to K2,3, contradicting Claim 1. (2)
Claim 16.6 If k = 2, then ab ∈E(G).
Proof. Suppose, to the contrary, that k = 2 and the vertices a and b are not adjacent. We now consider the graph
G′ obtained from G −X by adding the edge e = ab; that is, G′ = (G −X) + e. The graph G′ is a connected
subcubic graph. Recall that 2 ≤degG(a) ≤degG(b). By construction, the degrees of vertices a and b in G′ are
the same as their degrees in G. Let I′ be an i-set of G′. If a ∈I′, then let I = I′ ∪{u1, v2}. If b ∈I′, then let
I = I′ ∪{v1, u2}. If neither a nor b belongs to I′, then let I = I′ ∪{u1, u2}. In all cases, the set I is an ID-set of
G, and so i(G) ≤i(G′) + 2.
We show firstly that b(G′) = 0. Suppose, to the contrary, that b(G′) ≥1. Since G′ is a connected graph, we infer
that G′ ∈B and b(G′) = 1. If G′ contains two or more copies of K2,3, then at least one of these copies contains a
vertex of degree 2 in G, contradicting Claim 8. Hence, G′ ∈B1 and G′ contains exactly one copy of K2,3, say F.
If F contains at most one of a and b, then again we contradict Claim 8. Hence, F contains both vertices a and b,
and the root vertex of G′ has degree 1 in G (and is distinct from a and b). If a or b is adjacent to the root vertex,
then we would contradict the maximality of the path P. Hence the graph G is as illustrated in Figure 31, where
the added edge e is indicated by the dotted line. Thus, i(G) = 4 (the shaded vertices in Figure 31 are an example
of an i-set in G) and Ω(G) = w(G) = 38, and so 8i(G) < Ω(G), a contradiction. Hence, b(G′) = 0.
We show next that tc(G′) = 0. Suppose, to the contrary, that tc(G′) ≥1. Thus, adding the edge e = ab to
the graph G −X creates a new troublesome configuration, which we call Te, that necessarily contains the edge e.
27
a
v1
u1
b
v2
u2
e
G′
X
Figure 31: The graph G in the proof of Claim 16.6
Let r1 and r2 be the two link vertices of Te. We may assume that degG(r1) ≤degG(r2), and so 2 ≤degG(r1) ≤
degG(r2) ≤3. Thus, Te is obtained from a bad graph Be ∈B with root vertex r1. By Claims 8 and 9, the bad
graph Be contains exactly one copy of K2,3, say Fe, and so Be ∈B1. Moreover, Fe contains both vertices a and b,
for otherwise Fe would contain a vertex of degree 2 in G, a contradiction.
We now let X∗= X ∪(V (Te) \ {r1, r2}) and we let G∗= G −X∗. If degG(a) = 2, then the graph illustrated
in Figure 32(a) is a subgraph of G, while if degG(a) = 3, then the graph illustrated in Figure 32(b) is a subgraph
of G where in both cases Te is the troublesome configuration indicated in the dashed box, the set X∗is indicated
in the dashed box, and where e = ab is indicated by the dotted edge. Every i-set of G∗can be extended to an
ID-set of G by adding to it the vertices u1 and v2 and the neighbor of a in Fe different from b (as indicated
by the shaded vertices in Figures 32(a) and 32(b)). Thus, i(G) ≤i(G∗) + 3. We note that Θ(G∗) = 0, and so
Ω(G∗) = w(G∗) = w(G) −32 + 2 = Ω(G) −30. Hence, 8i(G) ≤8(i(G∗) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction.
Hence, tc(G′) = 0.
r1
r2
a
b
v1
u1
v2
u2
Te
X∗
e
(a)
r1
r2
a
b
v1
u1
v2
u2
Te
X∗
e
(b)
Figure 32: A subgraph of G in the proof of Claim 16.6
We now return to the proof of Claim 16.6. By our earlier observations, b(G′) = 0 and tc(G′) = 0. Hence,
Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) −wG(X) = Ω(G) −16.
Recall that i(G) ≤i(G′) + 2.
Thus,
8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. (2)
Claim 16.7 k = 1.
Proof. Suppose, to the contrary, that k = 2. By Claim 16.6, ab ∈E(G). Recall that 2 ≤degG(a) ≤degG(b).
If degG(a) = degG(b) = 2, then the graph G is determined. In this case, G does not contain a copy of K2,3, a
contradiction. Hence, degG(b) = 3.
We show firstly that degG(a) = 3. Suppose, to the contrary, that degG(a) = 2. In this case, we let G′ = G −X′
where recall that X = {u1, u2, v1, v2} and X′ = X ∪{a}. Every i-set of G′ can be extended to an ID-set of G by
adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. We note that G′ is a connected subcubic graph and
degG′(b) = 1. Moreover, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) −wG(X) + 2 = Ω(G) −20 + 2 = Ω(G) −18.
Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. Hence, degG(a) = 3. Let NG(a) = {a1, b, v1}
and NG(b) = {a, b1, v2}.
We show next that a1 ̸= b1. Suppose, to the contrary, that a1 = b1. If degG(a1) = 2, then the graph G is
determined.
In this case, G does not contain a copy of K2,3, a contradiction.
Hence, degG(a1) = 3.
Let a2
28
be the neighbor of a1 different from a and b. If degG(a2) = 1, then the graph G is determined, and again we
contradict the supposition that G contain a copy of K2,3. Hence, degG(a2) ≥2. We now let X∗= X ∪{a, b, a1}
and consider the connected subcubic graph G∗= G −X∗.
Every i-set of G∗can be extended to an ID-set
of G by adding to it the set {b, v1, u2}, and so i(G) ≤i(G∗) + 3.
By Claim 10, b(G∗) = 0.
By Claim 13,
tc(G∗) = 0. Thus, Θ(G∗) = 0, and so noting that a1a2 is the unique X∗-exit edge and degG(a2) ≥2, we have
Ω(G∗) = w(G∗) = w(G)−wG(X)+1 = Ω(G)−25+1 = Ω(G)−24. Thus, 8i(G) ≤8(i(G′)+3) ≤Ω(G′)+24 = Ω(G),
a contradiction. Hence, a1 ̸= b1.
Recall by maximality of the path P that degG(a1) ≥2 and degG(b1) ≥2. We now consider the graph G′ = G−X′,
where recall that X = {u1, u2, v1, v2} and X′ = X ∪{a}. Every i-set of G′ can be extended to an ID-set of G
by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. We note that degG′(b) = 1, and so there is no
troublesome configuration containing b in G′. If the vertex b belongs to a bad component, then such a component
has a copy of K2,3 that contains a vertex of degree 2 in G, contradicting Claim 8. Hence, b belongs to no bad
component in G′. If the removal of the edge aa1 when constructing G′ creates a new troublesome configuration,
then by Claim 13 we infer that such a troublesome configuration must contain the three vertices a1, b and b1, which
is not possible noting that degG′(b) = 1. Hence, tc(G′) = 0. Thus, Θ(G′) = 0, and so noting that there are three
X′-exit edges and degG(a1) ≥2, we have Ω(G′) = w(G′) = w(G) −wG(X′) + 3 = Ω(G) −19 + 3 = Ω(G) −16.
Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. (2)
By Claim 16.7, we have k = 1. Recall that 2 ≤degG(a) ≤degG(b). Recall that n ≥6. In particular, we note
that degG(b) = 3.
Claim 16.8 ab /∈E(G).
Proof.
Suppose, to the contrary, that ab ∈E(G).
In this case, we let X∗= {a, b, u1, v1} and consider the
graph G∗= G −X∗. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertex v1, and so
i(G) ≤i(G∗)+1. Suppose that degG(a) = 2, and so there is exactly one X∗-exit edge. By Claim 10, b(G∗) = 0. By
Claim 13, tc(G∗) = 0. Thus, Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G)−wG(X)+1 = Ω(G)−15+1 = Ω(G)−14.
Thus, 8i(G) ≤8(i(G∗) + 1) ≤Ω(G∗) + 8 = Ω(G), a contradiction. Hence, degG(a) = 3. Let NG(a) = {a1, b, v1}
and NG(b) = {a, b1, v1}. By the maximality of the path P, we note that degG(a1) ≥2 and degG(b1) ≥2. If
a1 is isolated in G∗, then the graph G is determined.
In this case, a1 = b1 and NG(a1) = {a, b}.
However,
then, n = 5, a contradiction. Hence, a1 is not isolated in G∗. By symmetry, b1 is not isolated in G∗. Thus,
w(G∗) = w(G) −wG(X∗) + 2 = w(G) −14 + 2 = w(G) −12. By Claim 10, we have b(G∗) = 0, and by Claim 14, we
have tc(G∗) ≤1. Thus, Θ(G∗) ≤2, and so Ω(G∗) = w(G∗)+Θ(G∗) ≤(w(G)−12)+2 = Ω(G)−12+2 = Ω(G)−10.
Thus, 8i(G) ≤8(i(G∗) + 1) ≤Ω(G∗) + 8 < Ω(G), a contradiction. (2)
By Claim 16.8, ab /∈E(G).
Claim 16.9 The graph G −{a, b, u1, v1} is isolate-free.
Proof. Suppose, to the contrary, that G −{a, b, u1, v1} has an isolated vertex. By Claim 16.2 and the maximality
of the path P, such a vertex has degree 2 in G with a and b as its two neighbors. If G −{a, b, u1, v1} contains
two isolated vertices, then the graph G is determined and is the graph B1 shown in Figure 4. However, then
G ∈B, a contradiction. Hence, G −{a, b, u1, v1} contains exactly one isolated vertex, say c. As observed earlier,
NG(c) = {a, b}. Since n ≥6, we note that degG(b) = 3. Let NG(b) = {b1, c, v1}. By the maximality of the path P,
we note that degG(b1) ≥2. We now let X′′ = {a, b, c, u1, v1}.
Suppose that degG(a) = 2. In this case, we consider the graph G′′ = G −X′′. Every i-set of G′′ can be extended
to an ID-set of G by adding to it the vertices v1 and c, and so i(G) ≤i(G′′) + 2. By Claim 10, b(G′′) = 0. By
Claim 13, tc(G′′) = 0. Thus, Θ(G′′) = 0, and so Ω(G′′) = w(G′′) = w(G)−wG(X′′)+1 = Ω(G)−19+1 = Ω(G)−18.
Thus, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 < Ω(G), a contradiction. Hence, degG(a) = 3. Let NG(a) = {a1, c, v1}.
By the maximality of the path P, we note that degG(a1) ≥2. If a1 = b1, then G[{a, a1, b, c, v1}] is a copy of K2,3
that contains a vertex of degree 2 in G, namely vertex c, contradicting Claim 8. Hence, a1 ̸= b1. Renaming vertices
if necessary, we may assume by symmetry that 2 ≤degG(a1) ≤degG(b1).
We show next that a1b1 ∈E(G). Suppose, to the contrary, that a1b1 /∈E(G). Recall that X′′ = {a, b, c, u1, v1}. In
this case, we consider the graph G′ obtained from G−X′′ by adding the edge e = a1b1; that is, G′ = (G−X′′)+e. The
29
graph G′ is a connected subcubic graph. Let I′′ be an i-set of G′. If a1 ∈I′′, then let I = I′′∪{b, u1}. If b1 ∈I′′, then
let I = I′′∪{a, u1}. If neither a1 nor b1 belongs to I′′, then let I = I′′∪{c, v1}. In all cases, the set I is an ID-set of G,
and so i(G) ≤i(G′)+2. By construction, the degrees of vertices a1 and b1 in G′ are the same as their degrees in G.
However adding the edge e may have created a bad graph or a troublesome configuration. Thus, b(G′)+tc(G′) ≤1,
implying that Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −wG(X′′)) + 2 = w(G) −18 + 2 = Ω(G) −16.
Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. Hence, a1b1 ∈E(G).
In this case, we consider the graph G′′ = G −X′′. Every i-set of G′′ can be extended to an ID-set of G by
adding to it the vertices v1 and c, and so i(G) ≤i(G′′) + 2. By Claim 11, b(G′′) = 0. We note that a1 and
b1 are adjacent vertices in G′′ and both a1 and b1 have degree at most 2 in G′′. We therefore infer that a1 and
b1 do not belong to a troublesome configuration in G′′, and so tc(G′′) = 0. Thus, Θ(G′′) = 0, and so Ω(G′′) =
w(G′′) = w(G) −wG(X′′) + 2 = Ω(G) −18 + 2 = Ω(G) −16. Thus, 8i(G) ≤8(i(G′′) + 2) ≤Ω(G′′) + 16 = Ω(G), a
contradiction. (2)
By Claim 16.9, the graph G∗= G −X∗is isolate-free, where recall that X∗= {a, b, u1, v1}. Further recall that
2 ≤degG(a) ≤degG(b) and that ab /∈E(G). Every i-set of G∗can be extended to an ID-set of G by adding to it
the vertex v1, and so i(G) ≤i(G∗) + 1. By Claim 10, if there is a bad component in G∗, then such a component
is incident with at least three X∗-edges. By Claims 13 and 14, if there is a troublesome configuration in G∗, then
such a component is incident with at least two X∗-edges. Thus since there are at most four X∗-edges, namely two
edges incident with vertex a and two incident with vertex b, we infer that either b(G∗) = tc(G∗) = 0 or b(G∗) = 1
and tc(G∗) = 0 or b(G∗) = 0 and tc(G∗) ≤2. In all cases, b(G∗) + tc(G∗) ≤2.
Suppose that b(G∗) + tc(G∗) ≤1, and so Θ(G∗) ≤2 and Ω(G∗) ≤w(G∗) + 2. If degG(a) = 2, then w(G∗) =
w(G) −wG(X∗) + 3 = w(G) −15 + 3 = Ω(G) −12, and so Ω(G∗) ≤w(G∗) + 2 ≤Ω(G) −10. If degG(a) = 3, then
w(G∗) = w(G) −wG(X∗) + 4 = w(G) −14 + 4 = Ω(G) −10, and so Ω(G∗) ≤w(G∗) + 2 ≤Ω(G) −8. In both cases,
Ω(G∗) + 8 ≤Ω(G). Thus, 8i(G) ≤8(i(G∗) + 1) ≤Ω(G∗) + 8 ≤Ω(G), a contradiction.
Hence, b(G∗)+tc(G∗) = 2, implying by our earlier observations that b(G∗) = 0 and tc(G∗) = 2. Thus, G∗contains
two troublesome configurations, say Tw and Tz, each of which contain two neighbors of a or b. In particular, we
note that degG(a) = degG(b) = 3. Let NG(a) = {a1, a2, v1} and let NG(b) = {b1, b2, v1}. Let w1 and w2 be the
link vertices of Tw and let z1 and z2 be the link vertices of Tz. Let Tw be obtained from the bad graph Bw ∈B1
with root vertex w1, and let Tz be obtained from the bad graph Bz ∈B1 with root vertex z1. Moreover, let x1 and
x2 be the vertices of Bw adjacent to w1 and w2, respectively, and let y1 and y2 be the vertices of Bz adjacent to
z1 and z2, respectively. By Claims 8 and 9, each of Bw and Bz contains exactly two copies of K2,3. Further, each
copy of K2,3 in Bw and Bz contains a vertex incident with an X∗-exit edge.
We now let X′′ = X∗∪(V (Tw) \ {x2, w2}) ∪(V (Tz) \ {y2, z2}) and let G′′ = G −X′′.
In the case when
{a1, b1} ⊂V (Tw) and {a2, b2} ⊂V (Tz), the graph illustrated in Figure 36 is a subgraph of G, where Tw and Tz are
the troublesome configurations indicated in the dashed boxes and where X∗and X′′ are the sets indicated by the
vertices in the dotted region.
If degG(w1) = 3, then let ew = ww1 be the edge incident with w1 not in Tw, and if degG(z1) = 3, then
let ez = zz1 be the edge incident with z1 not in Tz.
The vertices x2 and y2 have degree 1 in G′′, and the
neighbors of x2 and y2, namely w2 and z2, in G′′ have degree 2. Hence, there is no bad component or troublesome
configuration in G′′ containing these four vertices. By Claim 10, 11, 13 and 14, we have b(G′′) = 0 and tc(G′′) ≤1.
Moreover if tc(G′′) = 1, then the edges ew and ez exist and their removal increases the total weight by 2 + 2 = 4
(namely, an increase in 2 from the vertex weight and an increase in 2 in the structural weight arising from a
troublesome configuration containing w and z).
If tc(G′′) = 0, then the removal of ew and ez, if these edges
exist, increases the total weight by at most 2 × 3 = 6 in the worst case (which can only occur if w and z are
isolated vertices in G′′). As observed earlier, the removal of the six X′′-exit edges different from ew and ez do
not increase the structural weight and therefore only increase the vertex weight (by 6). We therefore infer that
Ω(G′′) ≤Ω(G) −wG(X′′) + 6 + 6 ≤Ω(G) −74 + 6 + 6 = Ω(G) −62.
Every i-set of G′′ can be extended to an ID-set of G′′ by adding to it the vertices a, b and u1, together with two
vertices from each of Tw and Tz (as illustrated by the shaded vertices in Figure 36). Thus, i(G) ≤i(G′′) + 7. As
observed earlier, Ω(G′′) + 62 ≤Ω(G). Therefore, 8i(G) ≤8(i(G′′) + 7) ≤Ω(G′′) + 56 < Ω(G), a contradiction. We
therefore deduce that our supposition that G contains a vertex of degree 1 is false. Hence, δ(G) ≥2, completing
the proof of Claim 16. (2)
30
Tw
Tz
a
b
b1
a2
a1
b2
v1
u1
x1
x2
y1
y2
w1
w2
z1
z2
X∗
X′′
Figure 33: An illustration of a subgraph of G in the proof of Claim 16
6.3
Part 3: The minimum degree of G is at least two
By Claim 16, δ(G) ≥2. Recall that n ≥6 and that G contains K2,3 as a subgraph. By supposition, G ̸= K3,3. As
in the proof of Part 2, in what follows we will frequently use the structural property stated in Claim 8 that there
is no K2,3-subgraph in G that contains a vertex of degree 2 in G.
We prove next a key structural property of the graph G. For this purpose, we define a troublesome subgraph
of G as a subgraph T such that there exists a set of edges incident with vertices of T whose removal creates a
troublesome configuration in G with vertex set V (T). For example, the subgraph T illustrated in Figure 21 is a
troublesome subgraph of G since the removal of the edge e produces a troublesome configuration in G −e. As a
further example, the subgraph T illustrated in Figure 22 or in Figure 23 is a troublesome subgraph of G since the
removal of the edge e (in both figures) produces a troublesome configuration in G −e.
Claim 17 There is no troublesome subgraph in G.
Proof. Suppose, to the contrary, that G contains a troublesome subgraph, say T. Hence there exists a set, say
ET ⊂E(G), of edges incident with vertices of T such that G′ = G −ET contains a troublesome configuration,
say T ′, with vertex set V (T ′) = V (T). By Claim 4, the graph G contains no troublesome configuration, that is,
tc(G) = 0. By Claim 13, tc(G −e) = 0 for all e ∈E(G). Hence, |ET | ≥2, that is, at least two edges incident with
vertices in T were removed when constructing T ′.
Let the troublesome configuration T ′ in G′ have link vertices v1 and v2, where degG(v2) = 3. Thus, T ′ is obtained
from the bad graph B ∈B with root vertex v1 (of degree 1 in B). Further, let u1 and u2 be the vertices of B
adjacent to v1 and v2, respectively, in G. Let z2 be the neighbor of v2 different from u2 and v1. Let the bad graph
B have k copies of K2,3 in G′. Since |ET | ≥2, we note that k ≥2. Let A = {a1, a2, . . . , ak} be the set of vertices
in B incident with edges in ET .
Claim 17.1 If the link vertex v1 is not incident with an edge of ET , then 8i(G) ≤Ω(G).
Proof. Suppose that the link vertex v1 is not incident with an edge of ET . If v1 has degree 3 in G, then let z1
be the neighbor of v1 not in T. We now let X = V (T) \ (A ∪{u2, v2}) and we consider the graph G′ = G −X.
We note that each vertex in A has degree 1 in G′, and belongs to neither a bad component nor a troublesome
configuration in G′. Moreover, the vertex u2 has degree 1 in G′ and its neighbor, v2, in G′ has degree 2 in G′.
Hence, u2 and v2 do not belong to a bad component or a troublesome configuration in G′. If degG(v1) = 3, then we
infer by Claims 10 and 13 that the vertex z1 does not belong to a bad component or a troublesome configuration
in G′. Hence, b(G′) = tc(G′) = 0, and so Θ(G′) = 0. We note that there are at most 2k + 4 X-exit edges.
Moreover, |X| = 4k and every vertex of X, except for possibly the vertex v1, has degree 3 in G.
Therefore,
Ω(G′) = w(G′) = w(G) −wG(X) + 2k + 4 ≤w(G) −3 × 4k + 2k + 4 = w(G) −10k + 4 = Ω(G) −10k + 4.
31
Every i-set of G′ can be extended to an ID-set of G by adding to it the core independent set (see Section 5.2) of
the bad graph B, and so i(G) ≤i(G′) + k. (For example, if the graph shown in Figure 34(a) is a subgraph of G,
where the dotted edges are the edges that belong to the set ET , then k = 6 and the subgraph G[X] is illustrated in
Figure 34(b). Furthermore, the core independent set of B is indicated by the k = 6 shaded vertices in Figure 34(b).)
Thus noting that k ≥2, we have 8i(G) ≤8(i(G′)+k) ≤Ω(G′)+8k ≤(Ω(G)−10k+4)+8k = Ω(G)−2k+4 ≤Ω(G),
a contradiction. (2)
T
a1
a2
a4
a3
a5
a6
u1
u2
v1
v2
z2
z1
(a) A subgraph T of G where the dotted edges belong to ET
v1
(b) The subgraph G[X] where X = V (T) \ {a1, a2, a3, a4, a5, a6, u2, v2}
Figure 34: An illustration of a subgraph of G in the proof of Claim 17.1
Claim 17.2 If the link vertex v1 is incident with an edge of ET , then 8i(G) < Ω(G).
Proof. Suppose that the link vertex v1 is incident with an edge of ET . Renaming vertices if necessary, we may
assume that e = a1v1. We now let X = V (T) \ (A \ {a1}) and we consider the graph G′ = G −X. We note that
each vertex in A\{a1} has degree 1 in G′, and belongs to neither a bad component nor a troublesome configuration
in G′. Moreover, the vertex z2 has degree 1 or 2 in G′, and by Claims 10 and 13 we infer that z2 does not belong
to a bad component or a troublesome configuration in G′. Hence, b(G′) = tc(G′) = 0, and so Θ(G′) = 0. Thus,
Ω(G′) = w(G′) = w(G) −wG(X) + 2k −1 = w(G) −3 × (4k + 3) + 2k −1 = w(G) −10k −10 = Ω(G) −10k −10.
Let IB be the core independent set (see Section 5.2) of the bad graph B, and so |IB| = k. We define the modified
core independent set of B as follows. We say that two copies of K2,3 are adjacent if they are joined by an edge. By
the structure of the bad graph G, there exists a sequence F1, . . . , Fℓof copies of K2,3 where u1 ∈V (F1), u2 ∈V (Fℓ)
and if ℓ≥2, then Fi+1 is adjacent to Fi for all i ∈[ℓ−1]. Let ci be the vertex in Fi that belongs to IB for
i ∈[ℓ]. In particular, we note that c1 = u1. Let C = {c1, . . . , cℓ}. We define d1 = v1, and if ℓ≥2, then we define
di+1 as the vertex of Fi that is adjacent to ci+1 in B for i ∈[ℓ−1]. Moreover, we define dℓ+1 = u2, and we let
D = {d1, d2, . . . , dℓ+1}. We now define the modified core independent set of B by
I∗
B = (IB \ C) ∪D.
32
As an illustration, suppose that T is the troublesome configuration in Figure 35, where B is the associated bad
graph used to construct T. In this example, the core independent set IB is indicated by the shaded vertices in
Figure 35(a), and the modified core independent set I∗
B is indicated by the shaded vertices in Figure 35(b). We
note that |IB| = k and |I∗
B| = k + 1, where here k = 6 is the number of copies of K2,3 in the bad graph B.
B
T
d2
d3
d4
u1 = c1
c2
c3
c4
u2 = d5
v1 = d1
v2
(a) The core independent set, IB, of B
B
T
d2
d3
d4
u1 = c1
c2
c3
c4
u2 = d5
v1 = d1
v2
(b) The modified core independent set, I∗
B, of B
Figure 35: An illustration of a modified core independent set in a bad graph
We now return to the proof of Claim 17.2. Every i-set of G′ can be extended to an ID-set of G by adding to it
the modified core independent set I∗
B of B, and so i(G) ≤i(G′) + |I∗
B| = i(G′) + k + 1. (For example, if the graph
shown in Figure 36(a) is a subgraph of G, where the dotted edges are the edges that belong to the set ET , then
k = 6 and the subgraph G[X] is illustrated in Figure 36(b) where the shaded vertices indicate the vertices of the
modified core independent set I∗
B of B.) Recall that Ω(G′) + 10k + 10 = Ω(G). Thus, 8i(G) ≤8(i(G′) + k + 1) ≤
Ω(G′) + 8k + 8 = Ω(G) −2k −2 < Ω(G), a contradiction. (2)
In both Claims 17.1 and 17.2, we have 8i(G) ≤Ω(G), a contradiction. This completes the proof of Claim 17. (2)
By Claim 17, there is no troublesome subgraph in G. Hence, if G′ is an arbitrary induced subgraph of G, then
tc(G′) = 0, and so Θ(G′) = 2b(G′). We state this formally as follows.
Claim 18 If G′ is an induced subgraph of G, then Θ(G′) = 2b(G′).
By Claim 18, the structural weight of an induced subgraph G′ of G is determined only by the number of bad
components of G′, if any, that belong to the bad family B. Recall that B = B1 ∪B2 ∪B3. Moreover, every graph in
33
T
a1
a2
a4
a3
a5
a6
u1
u2
v1
v2
z2
(a) A subgraph T of G where the dotted edges belong to ET
a1
u1
u2
v1
v2
(b) The subgraph G[X] where X = V (T) \ {a2, a3, a4, a5, a6}
Figure 36: An illustration of a subgraph of G in the proof of Claim 17.2
the family B1 has one vertex of degree 1 and at least two vertices of degree 2, while every graph in the family B2
has at least five vertices of degree 2. Furthermore, every graph in the family B3 has at least six vertices of degree 2.
We prove next certain properties that must hold if the graph G contains vertices of degree 2.
Claim 19 The graph G does not contain a path v1v2v3 such that degG(vi) = 2 for all i ∈[3].
Proof.
Suppose, to the contrary, that G contains a path v1v2v3 such that degG(vi) = 2 for all i ∈[3].
Let
X = {v1, v2, v3} and let G′ = G −X.
We note that there are two X-exit edges.
If G′ contains an isolated
vertex, then G = C4, a contradiction.
Hence, δ(G′) ≥1 and the graph G′ contains at most two vertices of
degree 1. By Claim 10, we infer that b(G′) = 0. Thus, by Claim 18, we have Θ(G′) = 0. Thus, Ω(G′) = w(G′) =
w(G) −wG(X) + 2 = Ω(G) −12 + 2 = Ω(G) −10. Every i-set of G′ can be extended to an ID-set of G by adding
to it the vertex v2, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2)
Claim 20 No two adjacent vertices in G both have degree 2.
Proof. Suppose, to the contrary, that G contains two adjacent vertices, say v1 and v2, of degree 2. Suppose firstly
that v1 and v2 have a common neighbor v, and so vv1v2v is a 3-cycle in G. Let u be the third neighbor of v different
from v1 and v2. Let X = {v, v1, v2} and let G′ = G −X. The graph G′ is a connected subcubic graph and all
vertices in G′ have degree at least 2, except possibly for the vertex u which has degree at least 1. By Claims 10
and 18, we infer that Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) −wG(X) + 1 = Ω(G) −11 + 1 = Ω(G) −10.
Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v1, and so i(G) ≤i(G′) + 1. Hence,
8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction.
34
The vertices v1 and v2 therefore have no common neighbor. Let ui be the neighbor of vi different from v3−i for
i ∈[2]. Since v1 and v2 have no common neighbor, u1 ̸= u2. By Claim 19, both vertices u1 and u2 have degree 3
in G. We show firstly that u1u2 is not an edge in G. Suppose, to the contrary, that u1u2 ∈E(G). In this case,
we let X = {v1, v2, u1} and consider the graph G′ = G −X. We note that G′ is isolate-free and the vertex u2 has
degree 1 in G′. By Claim 18, tc(G′) = 0. By Claims 10 and 11, we infer that if G′ has more than one component,
none of them is bad. In addition, if G′ is connected, then by Claim 8 we infer G′ /∈B. Thus, Θ(G′) = 0. Thus,
Ω(G′) = w(G′) = w(G) −wG(X) + 3 = Ω(G) −11 + 3 = Ω(G) −8. Every i-set of G′ can be extended to an ID-set
of G by adding to it the vertex v1, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a
contradiction. Therefore, u1u2 /∈E(G).
We now let X = {v1, v2} and consider the graph G′ obtained from G −X by adding the edge e = u1u2. We note
that G′ is a connected subcubic graph satisfying δ(G′) ≥2. Furthermore, u1 and u2 are adjacent vertices of degree 3
in G′. Let I′ be an i-set of G′. If u1 ∈I′, then let I = I′ ∪{v2}. If u2 ∈I′ or if neither u1 nor u2 belong to I′, then
let I = I′ ∪{v1}. In all cases, |I| = |I′| + 1 and the set I is an ID-set of G, and so i(G) ≤i(G′) + 1. If Θ(G′) = 0,
then Ω(G′) = w(G′) = w(G)−wG(X) = Ω(G)−8, and so 8i(G) ≤8(i(G′)+1) ≤Ω(G′)+8 = Ω(G), a contradiction.
Hence, Θ(G′) ≥1, implying that either G′ ∈B or the edge u1u2 belongs to a troublesome configuration in G′. If
G′ ∈B, then G would contain a copy of K2,3 that contains two vertices of degree 2 in G (noting that in this case,
the root vertex in G′ has degree at least 2), a contradiction. Hence, G′ /∈B.
Thus, adding the edge e = u1u2 to the graph G −X creates a new troublesome configuration, which we call
Te, that necessarily contains the edge e. Let w1 and w2 be the two link vertices of Te. By combining Claim 8
and Claim 17, we infer that Te contains exactly one copy of K2,3, say Fe, and both u1 and u2 belong to Fe (for
otherwise, Fe would contain a vertex of degree 2 in G, a contradiction). By symmetry and renaming vertices if
necessary, we may assume that u2 is adjacent to the link vertex w2 of Te. Let u be the third neighbor of u2 in
Te different from u1 and w2, as illustrated in Figure 37(a). We now let X∗= V (Fe) ∪X and let G∗= G −X∗.
Every i-set of G∗can be extended to an ID-set of G by adding to it the vertices v1 and u, as indicated by the
shaded vertices in Figure 37(b). Thus, i(G) ≤i(G∗) + 2. We note that Θ(G∗) = Θ(G) = 0, and so Ω(G∗) =
w(G∗) = w(G) −wG(X∗) + 2 = Ω(G) −24 + 2 = Ω(G) −22. Hence, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 < Ω(G), a
contradiction. (2)
Te
u2
u1
u
w2
w1
(a) A subgraph Te of G′
X∗
u2
u1
u
w2
w1
v1
v2
(b) A subgraph of G
Figure 37: Subgraphs of G′ and G in the proof of Claim 20
Claim 21 A vertex of degree 3 in G has at most one neighbor of degree 2.
Proof.
Suppose, to the contrary, that G contains a vertex, say v, of degree 3 that is adjacent to at least two
vertices of degree 2. Let X = NG[v] and let G′ = G −X. Every i-set of G′ can be extended to an ID-set of G by
adding to it the vertex v, and so i(G) ≤i(G′) + 1.
Suppose that all three neighbors of v have degree 2 in G. By Claim 20, the set NG(v) is therefore an independent
set. Suppose that G′ contains an isolated vertex, u say. Since δ(G) ≥2, we note that degG(u) ≥2. If degG(u) = 3,
then G = K2,3, contradicting the fact that n ≥6. If degG(u) = 2, then since u is adjacent to two vertices in
NG(v), the graph G contains a path P3 all of whose vertices have degree 2 in G, contradicting Claim 19. Hence,
the graph G′ is isolate-free. Since there are three X-exit edges, we note that w(G′) = (w(G)−15)+3 = w(G)−12.
By Claim 11, we infer that b(G′) ≤1. Thus by Claim 18, we have Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) =
(w(G) −12) + 2 = Ω(G) −10. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v,
and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction.
35
Therefore, the vertex v has two neighbors of degree 2 and one of degree 3. Let v1 and v2 be the two neighbors of v
of degree 2 and let v3 be the third neighbor of v (of degree 3). Suppose that G′ contains an isolated vertex, u say.
If degG(u) = 3, then G contains a copy of K2,3 that contains two vertices of degree 2 in G, contradicting Claim 8.
If degG(u) = 2, then since u is adjacent to at least one of v1 and v2, the graph G contains two adjacent vertices of
degree 2, contradicting Claim 20. Hence, the graph G′ is isolate-free. Every i-set of G′ can be extended to an ID-set
of G by adding to it the vertex v, and so i(G) ≤i(G′)+1. By Claim 11, we infer that b(G′) ≤1. Thus by Claim 18,
we have Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −wG(X) + 4) + 2 = w(G) −14 + 4 + 2 = Ω(G) −8.
Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. (2)
By Claim 20, both neighbors of a vertex of degree 2 have degree 3 in G. By Claim 21, at most one neighbor of
a vertex of degree 3 has degree 2 in G.
Claim 22 No vertex of degree 2 belongs to a triangle.
Proof. Suppose, to the contrary, that G contains a vertex, say u, of degree 2 that belongs to a triangle, T say.
Let v1 and v2 be the two neighbors of u, and so v1v2 ∈E(G) and V (T) = {u, v1, v2}. By our earlier observations,
both v1 and v2 have degree 3 in G. Let wi be the neighbor of vi not in T for i ∈[2]. Since at most one neighbor
of a vertex of degree 3 has degree 2 in G, both w1 and w2 have degree 3.
Suppose firstly that w1 = w2. In this case, we let X = NG[v1] and consider the graph G′ = G −X. Every
i-set of G′ can be extended to an ID-set of G by adding the vertex v1, and so i(G) ≤i(G′) + 1. By Claims 10
and 18, we infer that Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) −wG(X) + 1 = w(G) −13 + 1 = Ω(G) −12. Thus,
8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. Hence, w1 ̸= w2.
Suppose that w1w2 /∈E(G). In this case, we let X = NG[u] = {u, v1, v2} and let G′ be obtained from G −X
by adding the edge w1w2. Let I′ be an i-set of G′. If w1 ∈I′, then let I = I ∪{v2}, while if w1 /∈I′, then let
I = I ∪{v1}. In both cases, |I| = |I′| + 1 and the set I is an ID-set of G, and so i(G) ≤i(G′) + 1. The added
edge w1w2 increases the structural weight by at most 2 (which occurs if G′ ∈B or if the edge w1w2 belongs to a
troublesome configuration). Hence, Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −10) + 2 = Ω(G) −8.
Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction.
Hence, w1w2 ∈E(G). We now let X = NG[v1] = {u, v1, v2, w1} and consider the graph G′ = G −X. Every i-set
of G′ can be extended to an ID-set of G by adding the vertex v1, and so i(G) ≤i(G′)+1. By combining Claims 8, 10
and 18, we infer that Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) −wG(X) + 3 = w(G) −13 + 3 = Ω(G) −10.
Therefore, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2)
Recall that by Claim 1, the graph G contains at least one subgraph isomorphic to K2,3, and by Claim 2, the
graph G is connected. By supposition, G ̸= K3,3 and G ̸= C5 2 K2. We proceed further by proving additional
structural properties of the graph G. A diamond in G is a copy of K4 −e where e is an arbitrary edge of the
complete graph K4. A graph is diamond-free if it does not contain a diamond as an induced subgraph.
Claim 23 The graph G is diamond-free.
Proof. Suppose, to the contrary, that G contains a diamond D. Let V (D) = {v1, v2, v3, v4} where v1v2 is the
missing edge in the diamond. By Claim 22, no vertex of degree 2 belongs to a triangle. Thus, both v1 and v2 have
degree 3 in G. We now let X = V (D) and we consider the graph G′ = G −X. Since n ≥6, we note that G′ is
isolate-free. Every i-set of G′ can be extended to an ID-set of G by adding the vertex v3, and so i(G) ≤i(G′)+1. By
Claims 11 and 18, we infer that Θ(G′) = 0, and so Ω(G′) = w(G′)+Θ(G′) ≤w(G)−wG(X)+2 = w(G)−12+2 =
Ω(G) −10. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2)
Claim 24 The graph G does not contain K3,3 −e as a subgraph, where e is an edge of the K3,3.
Proof. Suppose, to the contrary, that G contains K3,3 −e as a subgraph, where e is an edge of the K3,3. Let F
be such a subgraph in G, and let F have partite sets A = {x1, x2, x3} and B = {v1, v2, v3}, where e = x1v1 is the
missing edge from the copy of K3,3 in F. Since G is diamond-free by Claim 23, the set B is an independent set
in G. If degG(x1) = 2 or if degG(v1) = 2, then there is a K2,3-subgraph in G that contains a vertex of degree 2
36
in G, a contradiction. Hence, degG(x1) = degG(v1) = 3. Let u1 be the neighbor of v1 different from x2 and x3,
and let w1 be the neighbor of x1 different from v2 and v3. Since G ̸= K3,3, the vertices u1 and x1 are distinct. If
x1u1 ∈E(G), then G contains G7 as a subgraph (see Figure 29), contradicting Claim 15. Hence, u1x1 /∈E(G),
and so the vertices u1 and w1 are distinct. We now let X = V (F) ∪{u1, w1} and consider the graph G′ = G −X.
Suppose that G′ contains an isolated vertex, say u.
Thus, NG(u) = {u1, w1}.
By Claim 20, degG(u1) =
degG(w1) = 3. By Claim 22, u1w1 /∈E(G). In this case, we let X′ = X ∪{u} and consider the graph G′′ = G−X′.
By Claim 21, the neighbor of u1 in G′′ has degree 3 in G and the neighbor of w1 in G′′ has degree 3 in G.
Thus, G′′ is isolate-free and there are two X′-exit edges. By Claims 11 and 18, we infer that Θ(G′′) = 0, and so
Ω(G′′) = w(G′′) = w(G) −wG(X′) + 2 = w(G) −28 + 2 = Ω(G) −26. Every i-set of G′′ can be extended to an
ID-set of G by adding to it the vertices in the set {u, v1, x1} (indicated by the shaded vertices in Figure 38(a)),
and so i(G) ≤i(G′′) + 3. Thus, 8i(G) ≤8(i(G′′) + 3) ≤Ω(G′′) + 24 < Ω(G), a contradiction.
Hence, G′ is isolate-free.
Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices
in the set {v1, x1} (indicated by the shaded vertices in Figure 38(b)), and so i(G) ≤i(G′) + 2. We note that
there are at most four X-exit edges.
By Claims 11 and 18, we infer that b(G′) ≤1 and Θ(G′) ≤2, and so
Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −wG(X) + 4) + 2 = w(G) −24 + 4 + 2 = Ω(G) −18. Every i-set of G′ can
be extended to an ID-set of G by adding to it the vertices in the set {v1, x1} (indicated by the shaded vertices in
Figure 38(b)), and so i(G) ≤i(G′) + 2. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. (2)
(a)
G′′
u1
w1
u
x1
v1
v2
v3
x2
x3
(b)
G′
u1
w1
x1
v1
v2
v3
x2
x3
Figure 38: Subgraphs of G in the proof of Claim 24
Claim 25 Every K2,3-subgraph of G belongs to a G8.2-configuration that is an induced subgraph in G, where G8.2
is the graph in Figure 39(b).
Proof. Let F be a K2,3-subgraph in G with partite sets A = {x1, x2} and B = {v1, v2, v3}. Since G is diamond-free
by Claim 23, the set B is an independent set in G. Let ui be the neighbor of vi different from x1 and x2 for i ∈[3].
By Claim 24, the vertices u1, u2 and u3 are distinct. Let U = {u1, u2, u3}. Since G7 is not a subgraph of G, the
set C is independent. Thus the subgraph G[A ∪B ∪C] of G induced by the sets A ∪B ∪C (see Figure 39(a)) is
an induced subgraph of G. (2)
u3
u2
u1
v3
v2
v1
x1
x2
(a)
C
B
A
(b) G8.2
Figure 39: Subgraphs of G in the proof of Claim 25
37
Claim 26 The graph G is cubic.
Proof. Suppose, to the contrary, that δ(G) = 2. Let u be an arbitrary vertex of degree 2 in G. By Claim 20,
both neighbors of u have degree 3 in G. Let v be a neighbor of u, and let NG(v) = {u, x, y}. By Claim 21,
degG(x) = degG(y) = 3. We now let X = NG[v] and we consider the graph G′ = G −X.
Claim 26.1 The graph G′ is isolate-free.
Proof. Suppose that G′ contains an isolated vertex, say w. Thus, NG(w) ⊆NG(v). If NG(w) = NG(v), then
G[{u, v, w, x, y}] is a copy of K2,3 that contains a vertex of degree 2 in G, a contradiction. Hence, w has degree 2
in G and is adjacent to exactly two vertices in NG(v). Since no two vertices of degree 2 in G are adjacent by
Claim 20, we have NG(w) = {x, y}. Let x1 and y1 be the neighbors of x and y, respectively, different from v and w.
Since there is no K2,3-subgraph that contains a vertex of degree 2 in G, we note that x1 ̸= y1. By Claim 21,
degG(x1) = degG(y1) = 3. Since degG(u) = 2, the vertex u is adjacent to at most one of x1 and y1. By symmetry,
we may assume that u and x1 are not adjacent.
We now let X′ = NG[x] = {v, w, x, x1} and consider the graph G′′ = G−X′. We note that both vertices u and y
have degree 1 in G′′. If u belongs to a bad component Bu ∈B in G′′, then necessarily Bu = B1 where u is the
vertex of degree 1 in Bu and where Bu contains exactly one copy of K2,3, which we denote by Fu. In this case, the
vertex x1 is adjacent in G to the two vertices of degree 2 in Fu. However, then G[V (Fu) ∪{x1}] is a copy of K3,3
with an edge removed, contradicting Claim 24. Hence, u does not belong to a bad component in G′′. Analogously,
y does not belong to a bad component in G′′. By Claim 11, we therefore infer that removing the two X′-exit edges
incident with x1 does not create a bad component. Hence, b(G′′) = 0, and so by Claim 18 we have Θ(G′′) = 0, and
so Ω(G′′) = w(G′′) ≤w(G) −wG(X) + 5 = w(G) −13 + 5 = Ω(G) −8. Every i-set of G′′ can be extended to an
ID-set of G by adding to it the vertex x, and so i(G) ≤i(G′′)+1. Thus, 8i(G) ≤8(i(G′′)+1) ≤Ω(G′′)+8 = Ω(G),
a contradiction. (2)
Claim 26.2 xy /∈E(G).
Proof.
Suppose that xy ∈E(G).
Thus there are exactly three X-exit edges.
By Claim 26.1, the graph G′
is isolate-free. By Claim 11, if there is a bad component in G′, then all three X-exit edges emanate from such
a component and G′ = B1.
Let r be the root vertex of G′.
By Claim 25, the K2,3-subgraph in G′ belongs
to a G8.2-configuration (see Figure 39(b)).
The graph G is therefore determined (up to isomorphism) and is
illustrated in Figure 40.
In this case, i(G) = 3 (the shaded vertices in Figure 40 indicate an i-set of G) and
w(G) = 32, and so 8i(G) < Ω(G), a contradiction. Hence, b(G′) = 0. By Claim 18, we have Θ(G′) = 0, and so
Ω(G′) = w(G′) ≤w(G) −wG(X) + 3 = w(G) −13 + 3 = Ω(G) −10. Every i-set of G′ can be extended to an ID-set
of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a
contradiction. (2)
v
y
r
x
u
G′
Figure 40: The graph G in the proof of Claim 26.2
By Claim 26.1, the graph G′ is isolate-free. By Claim 26.2, xy /∈E(G). Hence, there are at exactly five X-exit
edges. By Claim 11, if there is a bad component in G′, then at least three X-exit edges emanate from such a
component, implying that b(G′) ≤1. Every i-set of G′ can be extended to an ID-set of G by adding to it the
vertex v, and so i(G) ≤i(G′) + 1. If b(G′) = 0, then by Claim 18 we have Θ(G′) = 0, and so Ω(G′) = w(G′) ≤
w(G) −wG(X) + 5 = w(G) −13 + 5 = Ω(G) −8. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction.
Hence, b(G′) = 1. Let B be the bad component in G′, and let r be the root vertex of B.
38
Claim 26.3 B ∈B1.
Proof. Suppose that B ∈B2 ∪B3. If B ∈B3, then at least six X-exit edges emanate from such a component,
contradicting the fact that there are five X-exit edges. Hence, B ∈B2. Thus, the root vertex r of B has degree 2
in B and there are either two or three K2,3-subgraphs in B. Suppose that there are three K2,3-subgraphs in B. In
this case the graph G is determined and V (G) = V (B) ∪X. The core independent set of B (see Section 5.2) of
cardinality 3 can be extended to an ID-set of G by adding to it the set NG(v) = {u, x, y}, and so i(G) ≤6. (For
example, if G is the graph illustrated in Figure 41, then the ID-set of G containing the core independent set of B
together with the vertices in {u, x, y} is indicated by the six shaded vertices.) Thus, 8i(G) ≤48 < 62 = Ω(G), a
contradiction.
B
X
x
u
y
v
r
Figure 41: An example of a graph G in the proof of Claim 26.3
Hence, B contains exactly two K2,3-subgraphs. If V (G) = V (B) ∪X, then the graph G is determined and as
before the core independent set (of cardinality 2) of B can be extended to an ID-set of G by adding to it the
set NG(v) = {u, x, y}, and so i(G) ≤5. Moreover, Ω(G) = 46, and so 8i(G) < Ω(G), a contradiction. Hence,
V (G) ̸= V (B) ∪X, implying that there are exactly four X-exit edges that are incident with vertices in B and the
root vertex r of B is not incident with any of these four X-exit edges.
Suppose the vertex u is not adjacent to a vertex of B in G. We now let X∗= V (B) ∪{v, x, y} and consider
the graph G∗= G −X∗. In this case, there is exactly one X∗-exit edge and the vertex u has degree 1 in G∗.
By Claims 10 and 18, we infer that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) −wG(X∗) + 1 =
w(G)−43+1 = Ω(G)−42. Every i-set of G∗can be extended to an ID-set of G by adding to it the core independent
set of B of cardinality 2 and the vertices x and y (as indicated by the four shaded vertices in Figure 42), and so
i(G∗) ≤i(G) + 4. Thus, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 < Ω(G), a contradiction.
B
X∗
G∗
x
u
y
v
r
Figure 42: An example of a graph G in the proof of Claim 26.3
Hence, the vertex u is adjacent to a vertex of B in G. Renaming x and y if necessary, we may assume that x
is adjacent to exactly one vertex, say x∗, of B in G. In this case, we let X∗= V (B) ∪X and consider the graph
39
G∗= G−X∗. Since there is exactly one X∗-exit edge and the graph G∗is isolate-free, by Claims 10 and 18 we infer
that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) −wG(X∗) + 1 = w(G) −47 + 1 = Ω(G) −46. Every
i-set of G′ can be extended to an ID-set of G by adding to it the core independent set of B of cardinality 2, together
with the vertices in the set {u, x∗, y}, and so i(G∗) ≤i(G) + 5. Thus, 8i(G) ≤8(i(G∗) + 5) ≤Ω(G∗) + 40 < Ω(G),
a contradiction. Since both cases produce a contradiction, this completes the proof of Claim 26.3. (2)
By Claim 26.3, we have B ∈B1. Thus, the root vertex r in B has degree 1 in B. Let B have k copies of K2,3.
We note that k ≤3.
Claim 26.4 k = 1.
Proof. Suppose that k ∈{2, 3}. If k = 3, then the graph G is determined and V (G) = V (B) ∪X. In this case,
the core independent set of B (see Section 5.2) of cardinality 3 can be extended to an ID-set of G by adding to it
the set NG(v) = {u, x, y}, and so i(G) ≤6. Thus, 8i(G) ≤48 < 62 = Ω(G), a contradiction. Hence, k = 2.
Suppose the vertex u is not adjacent to a vertex of B in G. We now let X∗= V (B) ∪{v, x, y} and consider
the graph G∗= G −X∗. In this case, there is exactly one X∗-exit edge and the vertex u has degree 1 in G∗.
By Claims 10 and 18, we infer that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) −wG(X∗) + 1 =
w(G)−43+1 = Ω(G)−42. Every i-set of G′ can be extended to an ID-set of G by adding to it the core independent
set of B of cardinality 2 and the vertices x and y (as indicated by the four shaded vertices in Figure 43), and so
i(G∗) ≤i(G) + 4. Thus, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 < Ω(G), a contradiction.
B
X∗
G∗
x
u
y
v
r
Figure 43: An example of a graph G in the proof of Claim 26.4
Hence, the vertex u is adjacent to a vertex of B in G. Renaming x and y if necessary, we may assume that x
is adjacent to exactly one vertex, say x∗, of B in G. In this case, we let X∗= V (B) ∪X and consider the graph
G∗= G−X∗. Since there is exactly one X∗-exit edge and the graph G∗is isolate-free, by Claims 10 and 18 we infer
that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) −wG(X∗) + 1 = w(G) −47 + 1 = Ω(G) −46. Every
i-set of G′ can be extended to an ID-set of G by adding to it the core independent set of B of cardinality 2, together
with the vertices in the set {u, x∗, y} (as indicated by the five shaded vertices in Figure 44), and so i(G∗) ≤i(G)+5.
Thus, 8i(G) ≤8(i(G∗) + 5) ≤Ω(G∗) + 40 < Ω(G), a contradiction. Since both cases produce a contradiction, this
completes the proof of Claim 26.3. (2)
By Claim 26.4, we have k = 1, and so B contains exactly one copy of K2,3. Let the copy of K2,3 in B have partite
sets {x1, x1} and {v1, v2, v3}. Renaming vertices if necessary, we may assume that the root vertex of B is adjacent
to v3. By Claim 25, every K2,3-subgraph of G belongs to a G8.2-configuration that is an induced subgraph in G,
where G8.2 is the graph in Figure 39(b). Let w1 and w2 be the neighbors of v1 and v2, respectively, not in B. We
note by Claim 25 that {r, w1, w2} is an independent set. Let w3 be the third neighbor of v, and so the root vertex r
is adjacent to w3. Thus, degG(r) = 2 and NG(r) = {v3, w3}. We note that NG(v) = {w1, w2, w3} = {u, x, y},
where recall that degG(u) = 2 and degG(x) = degG(y) = 3. Since there are no two adjacent vertices of degree 2 in
G by Claim 20, we infer that degG(w3) = 3, and so u ∈{w1, w2}. Renaming vertices if necessary, we may assume
that u = w2, and so degG(w2) = 2 and NG(w2) = {v, v2}.
We now let X′′ = NG[v] ∪V (B) and we consider the graph G′′ = G −X′′. Suppose that G′′ contains an isolated
vertex. In this case, such a vertex, say z, has degree 2 in G and is adjacent to w1 and w3, implying that the vertex w3
40
B
X∗
G∗
u
x
y
x∗
v
r
Figure 44: An example of a graph G in the proof of Claim 26.4
has two neighbors of degree 2, namely r and z, contradicting Claim 21. Hence, G′′ is isolate-free. Thus, the graph
shown in Figure 45 is a subgraph of G, where in this illustration the neighbors of w1 and w2 not in G′′ are distinct.
By Claims 11 and 18 we infer that b(G′′) = 0 and Θ(G′′) = 0, and so Ω(G′′) = w(G′′) ≤w(G) −wG(X′′) + 2 =
w(G) −32 + 2 = Ω(G) −30. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices
in the set {r, v1, w2} (as indicated by the three shaded vertices in Figure 45), and so i(G′′) ≤i(G) + 3. Thus,
8i(G) ≤8(i(G′′) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction. This completes the proof of Claim 26. (2)
B
X′′
G′′
r
v
w3
w2
w1
v3
v2
v1
x1
x2
Figure 45: A subgraph of G in the proof of Claim 26
6.4
Part 4: The graph G is cubic
By Claim 26, the graph G is a cubic graph of order n. Recall that n ≥6. By Claim 2, the graph G is connected. By
supposition, G ̸= K3,3 and G ̸= C5 2 K2. Since G is a cubic graph, we note that Θ(G) = 0, and so Ω(G) = w(G).
We prove a number of claims that will culminate in a contradiction, thereby proving our main theorem.
Claim 27 The graph G contains no triangle.
Proof. Suppose, to the contrary, that G contains a triangle T, where V (T) = {v1, v2, v3}. Let ui be the neighbor
of vi not in T for i ∈[3]. By Claim 23, the graph G is diamond-free, implying that the vertices u1, u2 and u3 are
distinct. Let U = {u1, u2, u3}. If G[U] = K3, then G is the 3-prism C3 2 K2. In this case, i(G) = 2 and w(G) = 18,
and so 8i(G) < Ω(G), a contradiction. Hence renaming vertices if necessary, we may assume that u1u2 /∈E(G). Let
G′ be the graph obtained from G −V (T) by adding the edge u1u2. We note that every vertex of G′ has degree 3,
except for the vertex u3 which has degree 2 in G′. We therefore infer that b(G′) = 0.
Suppose that tc(G′) = 0. In this case, Θ(G′) = 0 and Ω(G′) = w(G′) = w(G) −9 + 1 = Ω(G) −8. Let I′
be an i-set of G′. If neither u1 nor u2 belong to I′, then let I = I′ ∪{v1}. If ui ∈I′ for some i ∈[2], then let
I = I′ ∪{v3−i}. In both cases, |I| = |I′| + 1 = i(G′) + 1 and the set I is an ID-set of G, and so i(G) ≤i(G′) + 1.
Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction.
Hence, tc(G′) = 1, implying that there is a troublesome configuration, say T ′, in G′ where T ′ has exactly one copy
of K2,3, and this copy contains the vertex u3 as the vertex of degree 2 in T ′, and where T ′ contains the added edge
41
u1u2 We now let X∗= V (T) ∪{u3} and G∗= G −X∗. In this case, tc(G∗) = 0 and b(G∗) = 0, and so Θ(G∗) = 0.
Thus, Ω(G∗) = w(G∗) = w(G)−wG(X∗)+4 = Ω(G)−12+4 = Ω(G)−8. Every i-set of G∗can be extended to an
ID-set of G by adding to it the vertex v3, and so i(G) ≤i(G∗)+1. Hence, 8i(G) ≤8(i(G∗)+1) ≤Ω(G∗)+8 = Ω(G),
a contradiction.
Claim 28 The graph G does not contain two vertex disjoint copies of K2,3 joined by at least one edge.
Proof. Let F1 and F2 be two vertex disjoint copies of K2,3, where F1 has partite sets X = {x1, x2} and V1 =
{v1, v2, v3}, and F2 partite sets Y = {y1, y2} and U1 = {u1, u2, u3}. Suppose, to the contrary, that F1 and F2 are
joined by at least one edge.
Claim 28.1 F1 and F2 are joined by exactly one edge.
Proof. If F1 and F2 are joined by three edges, then G = G10.2, where G10.2 is the graph illustrated in Figure 46(a).
In this case, i(G) = 3 (an i-set is indicated by the shaded vertices in Figure 46(a)) and w(G) = 30, and so
8i(G) < w(G), a contradiction.
(a) G10.2
u3
u2
u1
v3
v2
v1
x1
x2
w
(b)
G′
Figure 46: Subgraphs of G in the proof of Claim 28
Suppose next that F1 and F2 are joined by exactly two edges. Renaming edges if necessary, we may assume that
u2v2 and u3v3 are edges of G. Let x be the neighbor of v1 not in X, and let y be the neighbor of u1 not in Y . Possibly,
x = y. Let X′ = (V (F1) ∪V (F2)) \ {u1, v1} and let G′ = G −X′. Thus the graph in Figure 47(a) is a subgraph of
G, where in this illustration x ̸= y. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices
in the set {u2, v3} (indicated by the shaded vertices in Figure 47(a)), and so i(G) ≤i(G′) + 2. Since every vertex
of G′ has degree 3, except for the vertices u1 and v1 both of which have degree 1, we infer that b(G′) = 0, and so,
by Claim 18, we have Θ(G′) = 0. Hence, Ω(G′) = w(G′) ≤w(G) −wG(X′) + 4 = Ω(G) −24 + 4 = Ω(G) −20.
Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. (2)
x2
v3
u3
y2
x1
v2
u2
y3
v1
x
u1
y
X′
G′
(a)
X∗
x2
v3
u3
y2
x1
v2
u2
y1
v1
x
w
u1
z
y
G∗
(b)
Figure 47: Subgraphs of G in the proof of Claim 28
By Claim 28.1, F1 and F2 are joined by exactly one edge. Renaming vertices if necessary, we may assume that
u3v3 is the edge joining F1 and F2.
42
Claim 28.2 Neither v1 nor v2 has a common neighbor with u1 or u2.
Proof. Suppose that v1 or v2 has a common neighbor with u1 or u2. By symmetry, we may assume that u2 and
v2 have a common neighbor, say w. By Claim 25, the vertex w is adjacent to neither u1 nor v1. Let z be the
neighbor of w different from u2 and v2. Let x be the neighbor of v1 not in X, and let y be the neighbor of u1 not
in Y . Possibly, x = y. By Claim 25, the set {u3, x, w} is an independent set and the {v3, y, w} is an independent
set. Therefore since w is adjacent to z, we note that x ̸= z and y ̸= z, although possibly x = y.
We now let X∗= (V (F1) \ {v1}) ∪{u2, u3, w, z} and consider the graph G∗= G −X∗.
Every i-set of G∗
can be extended to an ID-set of G by adding to it the vertices v3 and w (indicated by the shaded vertices in
Figure 47(b)), and so i(G) ≤i(G∗) + 2. Let z1 and z2 be the two neighbors of z different from w. Every vertex
of G∗has degree 3 except for three vertices of degree 1 (namely, v1, y1 and y2) and two vertices of degree 2
(namely, z1 and z2). Since y1 and y2 have a common neighbor in G∗(namely u1), we note that the only possible
component of G∗that belongs to the family B is a B1-component containing v1, z1 and z2. However this would
imply that z1 and z2 are vertices of degree 2 in a copy of K2,3 that have a common neighbor, namely z, that
does not belong to this copy of K2,3, which contradicts Claim 25.
Hence, b(G∗) = 0, and so, by Claim 18,
we have Θ(G∗) = 0.
Hence, Ω(G∗) = w(G∗) ≤w(G) −wG(X∗) + 8 = Ω(G) −24 + 8 = Ω(G) −16.
Thus,
8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 ≤Ω(G), a contradiction. (2)
We now return to the proof of Claim 28. By Claim 28.2, neither v1 nor v2 has a common neighbor with u1 or u2.
As before, let x be the neighbor of v1 not in X, and let y be the neighbor of u1 not in Y . Let w be the neighbor
of v2 not in X, and let z be the neighbor of u2 not in Y . By our earlier observations, the vertices x, y, w, z are
distinct and do not belong to V (F1) ∪V (F2). By Claim 25, xw /∈E(G) and yz /∈E(G).
We now let X∗= {v2, v3, u3, x1, x2, y1, y2, w} and we consider the graph G∗= G −X∗. Every i-set of G∗can be
extended to an ID-set of G by adding to it the vertices v2 and u3 (indicated by the shaded vertices in Figure 48),
and so i(G) ≤i(G∗) + 2. Let w1 and w2 be the two neighbors of w different from v2. Every vertex of G∗has
degree 3 except for three vertices of degree 1 (namely, v1, u1 and u2) and two vertices of degree 2 (namely, w1
and w2). Analogous arguments as before show that there is no B1-component containing the vertices w1 and w2,
for otherwise such a component (of order 6) contains w1 and w2 as vertices of degree 2 in a copy of K2,3 that have a
common neighbor, namely w, that does not belong to this copy of K2,3, which contradicts Claim 25. We therefore
infer that b(G∗) = 0, and so, by Claim 18, we have Θ(G∗) = 0. Hence, Ω(G∗) = w(G∗) ≤w(G) −wG(X∗) + 8 =
Ω(G) −24 + 8 = Ω(G) −16. Thus, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 = Ω(G), a contradiction. This completes
the proof of Claim 28. (2)
x2
v3
u3
y2
x1
v2
u2
y1
v1
x
w
u1
z
y
G∗
X∗
Figure 48: A subgraph of G in the proof of Claim 28
By Claim 28, if the graph G contains two vertex disjoint copies of K2,3, then there is no edge joining these two
copies of K2,3. This yields the following property of the structural weight for an arbitrary induced subgraph G′
of G.
Claim 29 If G′ is an arbitrary induced subgraph of G and if B is a component of G′ in the family B, then B ∈Bi
and B contains exactly i copies of K2,3 for some i ∈[3] as illustrated in Figure 49.
By Claim 29, if G′ is an arbitrary induced subgraph of G and if B is a component of G′ that belongs to B, then
either B ∈B1 and B contains one vertex of degree 1 and two vertices of degree 2 (see Figure 49(a)) or B ∈B2
43
(a) B ∈B1
(b) B ∈B2
(c) B ∈B3
Figure 49: The three possible bad components of G′ in Claim 29
and B contains no vertex of degree 1 and five vertices of degree 2 (see Figure 49(b)) or B ∈B3 and B contains no
vertex of degree 1 and six vertices of degree 2 (see Figure 49(c)).
By Claim 27, the graph G contains no triangle. Let F be a K2,3-subgraph in G with partite sets A = {x1, x2} and
B = {v1, v2, v3}. Since G is triangle-free, the set B is an independent set in G. Let ui be the neighbor of vi different
from x1 and x2 for i ∈[3]. By Claim 25, the vertices u1, u2 and u3 are distinct and the set C = {u1, u2, u3} is an
independent set. We now let X′′ = A ∪B ∪C and consider the graph G′′ = G −X′′. Thus the graph illustrated
in Figure 50 is an induced subgraph of G.
u3
u2
u1
v3
v2
v1
x1
x2
C
B
A
F
set X′′
G′′
Figure 50: A subgraph of G
Claim 30 The graph G′′ is isolate-free.
Proof.
Suppose, to the contrary, that G′′ contains at least one isolated vertex.
If G′′ contains two isolated
vertices, then the graph G is determined and is the graph G10.2 shown in Figure 46(a). As noted earlier, in this
case 8i(G) < w(G), a contradiction. Hence, G′′ contains exactly one isolated vertex, say w. Let X′ = X′′ ∪{w}
and let G′ = G −X′. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set
{w, x1, x2} (as indicated by the shaded vertices in Figure 46(b)), and so i(G) ≤i(G′) + 3. Since G′ has either
one vertex of degree 1 and one vertex of degree 2 or three vertices of degree 2, we infer that b(G′) = 0, and so by
Claim 18, we have Θ(G′) = 0. Hence, Ω(G′) = w(G′) ≤w(G) −wG(X′) + 3 = Ω(G) −27 + 3 = Ω(G) −24. Thus,
8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. (2)
By Claim 30, the graph G′′ is isolate-free.
Claim 31 The graph G′′ contains at most one vertex of degree 1.
Proof. Suppose, to the contrary, that G′′ contains at least two vertices of degree 1. Let D be the set of vertices of
degree 1 in G′′. We note that |D| ≤3. Suppose firstly that |D| = 3. Let D = {w1, w2, w3}. Let wi be the common
neighbor of ui and ui+1 for i ∈[3] where addition is taken modulo 3. If w1, w2, w3 have a common neighbor, say w,
then the graph G is determined and is the graph G12 shown in Figure 51(a). In this case, i(G) = 4 (an i-set is
44
indicated by the shaded vertices in Figure 51(a)) and w(G) = 36, and so 8i(G) < w(G), a contradiction. Hence,
w1, w2, w3 do not have a common neighbor.
We now let X′ = X′′ ∪D and consider the graph G′ = G′′ −D = G −X′. Since w1, w2, w3 have no common
neighbor, the graph G′ is an isolate-free graph, and so w(G′) = (w(G) −33) + 3 = w(G) −30. Every i-set of G′ can
be extended to an ID-set of G by adding to it the vertices in the set {u1, u3, v2} (as indicated by the shaded vertices
in Figure 51(b)), and so i(G) ≤i(G′) + 3. Since G′ has either one vertex of degree 1 and one vertex of degree 2
or three vertices of degree 2, we infer by Claim 29 that b(G′) = 0. Thus, by Claim 18, we have Θ(G′) = 0, and so
Ω(G′) = w(G′) = w(G) −30 = Ω(G) −30. Thus, 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction.
u3
u2
u1
v3
v2
v1
w3
w2
w1
w
x1
x2
(a) G12
u3
u2
u1
v3
v2
v1
w3
w2
w1
x1
x2
(b)
G′
u3
u2
u1
v3
v2
v1
w2
w1
x1
x2
(c)
G′
Figure 51: Subgraphs of G in the proof of Claim 31
Hence, |D| = 2, and so G′′ contains two vertices of degree 1, say w1 and w2.
By symmetry, we have two
possibilities. First, let wi be a common neighbor of ui and ui+1 for i ∈[2]. Thus the graph shown in Figure 51(c)
is a subgraph of G. We now let X′ = X′′ ∪{w1, w2} and consider the graph G′ = G′′ −D = G −X′. Since the
graph G is triangle-free, the graph G′ is isolate-free, and so w(G′) = (w(G) −30) + 4 = w(G) −26. Every i-set of
G′ can be extended to an ID-set of G by adding to it the vertices in the set {u2, v1, v3} (as indicated by the shaded
vertices in Figure 51(c)), and so i(G) ≤i(G′) + 3. Since G′ has either two vertices of degree 1 and no vertex of
degree 2 or one vertex of degree 1 and two vertices of degree 2 or four vertices of degree 2, we infer that b(G′) ≤1,
and so by Claim 18, we have Θ(G′) ≤2. Hence, Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −26) + 2 = w(G) −24. Thus,
8i(G) ≤8(i(G′) + 3) ≤w(G′) + 24 = w(G), a contradiction.
Second, let u1 and u2 have two common neighbors w1 and w2. Suppose that u3, w1 and w2 have a common
neighbor z. Let X∗= X′′ ∪{w1, w2, z}. Let G∗= G −X∗. By using Claims 10 and 18, we have Ω(G∗) = w(G∗) =
(w(G) −33) + 1 = Ω(G) + 32, since there is one X∗-exit edge. Every i-set of G∗can be extended to an ID-set
of G by adding to it the vertices in the set {z, v1, v2, v3}. Hence, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 = Ω(G), a
contradiction. Thus, u3, w1 and w2 do not have a common neighbor. In this case, let X′ = X′′ ∪{w1, w2} and
consider the graph G′ = G −X′. Since u3, w1 and w2 do not have a common neighbor, the graph G′ is isolate-free,
and so w(G′) = (w(G) −30) + 4 = w(G) −26. Every i-set of G′ can be extended to an ID-set of G by adding to
it the vertices in the set {u2, v1, v3} (as indicated by the shaded vertices in Figure 51(c)), and so i(G) ≤i(G′) + 3.
Since G′ has either two vertices of degree 1 and no vertex of degree 2 or one vertex of degree 1 and two vertices
of degree 2 or four vertices of degree 2, we infer that b(G′) ≤1, and so by Claim 18, we have Θ(G′) ≤2. Hence,
Ω(G′) = w(G′) + Θ(G′) ≤(w(G) −26) + 2 = w(G) −24. Thus, 8i(G) ≤8(i(G′) + 3) ≤w(G′) + 24 = w(G), a
contradiction. (2)
By Claim 31, the graph G′′ contains at most one vertex of degree 1. We now define
Gi,j = G −(NG[ui] ∪NG[vj])
where i, j ∈[3] and i ̸= j.
Claim 32 At least one of the subgraphs Gi,j where i, j ∈[3] and i ̸= j contains no isolated vertex.
Proof. Suppose firstly that G′′ contains a vertex of degree 1, say w1. By symmetry, we may assume that w1 is
a common neighbor of u1 and u2. Let w be the neighbor of u1 different from v1 and w1. Thus the graph shown
in Figure 52(a) is a subgraph of G, where the graph G1,2 is the subgraph in the dashed box. If G1,2 contains an
45
G1,2
u3
u2
u1
v3
v2
v1
w1
x1
x2
w
(a)
x1
x2
u3
u2
u1
v3
v2
v1
w2
w1
w4
w3
w6
w5
(b)
Figure 52: Subgraphs of G in the proof of Claim 32
isolated vertex v, then NG(v) = {w, w1, u2}. However, then, G[{v, u2, w1}] = K3, contradicting Claim 27. Hence,
G1,2 contains no isolated vertex, and desired result of the claim follows.
Suppose next that G′′ contains no vertex of degree 1. Thus, every vertex in G′′ is adjacent to at most one vertex
that belongs to the set C = {u1, u2, u3}, implying that NG[ui] ∩NG[uj] = ∅for i, j ∈[3] and i ̸= j. Hence the
graph shown in Figure 52(b) is a subgraph of G. In particular, we note that the vertices w1, w2, . . . , w6 are distinct.
We show that at least one of the subgraphs Gi,j where i, j ∈[3] and i ̸= j contains no isolated vertex. Suppose, to
the contrary, that Gi,j contains an isolated vertex for all i, j ∈[3] and i ̸= j.
We consider the graph G1,2 (illustrated by the subgraph inside the dashed region in Figure 53). By supposition,
G1,2 contains an isolated vertex.
The only possible such isolated vertex is a neighbor of u2 that is adjacent
to both w1 and w2.
Renaming w3 and w4 if necessary, we may assume that w3 is isolated in G1,2, that is,
NG(w3) = {u2, w1, w2}. We next consider the graph G1,3. By supposition, G1,3 contains an isolated vertex. The
only possible such isolated vertex is a neighbor of u3 that is adjacent to both w1 and w2. Renaming w5 and w6 if
necessary, we may assume that w5 is isolated in G1,3, that is, NG(w5) = {u3, w1, w2}. In particular, we note that
C : w1w3w2w5w1 is a 4-cycle in G. However, this implies that the graph G2,1 does not contain an isolated vertex.
Indeed, in G2,1, vertices w1, w2 and u3 have degree 1, the neighbors of w4 in G2,1 have degree 2 in G2,1, whereas
other vertices of G2,1 have degree 3. This is a contradiction, proving the claim. (2)
G′ = G1,2
u3
u2
u1
v3
v2
v1
x1
x2
w1
w2
Figure 53: A subgraph of G in the proof of Claim 32
By Claim 32, at least one of the subgraphs Gi,j where i, j ∈[3] and i ̸= j contains no isolated vertex. Renaming
vertices if necessary, we may assume that G1,2 is isolate-free. For notational convenience, let G′ = G1,2 and let
X1,2 = {u1, u2, v1, v2, x1, x2, w1, w2}, and so G′ = G −X1,2. Every i-set of G′ can be extended to an ID-set of G by
adding to it the vertices u1 and v2 (as indicated by the shaded vertices in Figure 53), and so i(G) ≤i(G′) + 2. If
b(G′) = 0, then by Claim 18, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) −24 + 8 = w(G) −16 = Ω(G) −16. In this
case, 8i(G) ≤8(i(G′) + 2) ≤w(G′) + 16 = Ω(G), a contradiction. Hence, b(G′) ≥1. Let B′ be a bad component
in G′, and so B′ ∈B. We note that degG′(v3) = 1, and so G′ contains at least one vertex of degree 1.
46
Claim 33 B′ /∈B3.
Proof. Suppose, to the contrary, that B′ ∈B3, and so B′ is as illustrated in Figure 49(c). In this case, the six X1,2-
exit edges from w1, w2 and u2 are all incident with vertices of degree 2 in B′. Let v denote the vertex of degree 3
in B′ that does not belong to a copy of K2,3. We now let X∗= V (B′) ∪X1,2 and we let G∗= G −X∗. We note
that G∗contains one vertex of degree 1, namely v3, and all other vertices of G∗have degree 3. Hence, b(G∗) = 0,
and so, by Claim 18, Θ(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G) −24 × 3 + 2 = w(G) −70. Every i-set of G∗can
be extended to an ID-set of G by adding to it the seven vertices in the set N(v) ∪{u2, v1, w1, w2} (as indicated by
the shaded vertices in Figure 54), and so i(G) ≤i(G∗) + 7. Thus, 8i(G) ≤8(i(G∗) + 7) ≤Ω(G∗) + 56 < Ω(G), a
contradiction. (2)
B′
X1,2
G∗
u3
u2
u1
v3
v2
v1
x1
x2
w1
w2
v
Figure 54: A subgraph of G in the proof of Claim 33
Claim 34 B′ /∈B2.
Proof. Suppose, to the contrary, that B′ ∈B2, and so B′ is as illustrated in Figure 49(b). In this case, five X1,2-exit
edges from w1, w2 and u2 are all incident with vertices of degree 2 in B′. Let v denote the vertex of degree 2 in B′
that does not belong to a copy of K2,3. We now let X∗= V (B′) ∪X1,2 and we let G∗= G −X∗. We note that G∗
contains one vertex of degree 1, one vertex of degree 2, and all other vertices of G∗have degree 3. Hence, b(G∗) = 0,
and so, by Claim 18, we have Θ(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G) −19 × 3 + 3 = w(G) −54 = Ω(G) −54.
We show that every i-set of G∗can be extended to an ID-set of G by adding to it at most six vertices. We will use
the property given in Claim 25 that every K2,3-subgraph of G belongs to a G8.2-configuration that is an induced
subgraph in G, where G8.2 is the graph in Figure 39(b).
B′
X1,2
G∗
u3
u2
u1
v3
v2
v1
x1
x2
w1
w2
v
(a)
G∗
X1,2
B′
u3
u2
u1
v3
v2
v1
x1
x2
w1
w2
v
(b)
Figure 55: Subgraphs of G in the proof of Claim 34
Suppose firstly that u2 is adjacent to two vertices in B′. If u2 is not adjacent to v, then we let I′ consist of the
three vertices of degree 2 in B′ that are not adjacent to u2, together with the vertices v1 and u2 (as indicated by
47
the shaded vertices in Figure 55(a)). If u2 is adjacent to v, then renaming w1 and w2 if necessary, we may assume
that w2 is adjacent to two vertices in B′. In this case, we let I′ consist of the two neighbors of v in B′, the vertex of
degree 2 in B′ adjacent to w1, together with the vertices in the set {u2, v1, w2} (as indicated by the shaded vertices
in Figure 55(b)). In both cases, every i-set of G∗can be extended to an ID-set of G by adding to the vertices in
the set I′, and so i(G) ≤i(G∗) + |I′| ≤i(G∗) + 6.
Suppose next that u2 is adjacent to exactly one vertex in B′. Suppose that u2 is not adjacent to v. Renaming w1
and w2 if necessary, we may assume that w2 is adjacent to v. We now let I′ consist of the vertex v, the vertices of
degree 2 in B′ adjacent to w2 and u2, and the vertices in the set {v1, v2, w1} (as indicated by the shaded vertices in
Figure 56(a)). If u2 is adjacent to v, then we let I′ consist of the two neighbors of v in B′ together with the vertices
in the set {v1, v2, w1, w2} (as indicated by the shaded vertices in Figure 56(b)). In both cases, every i-set of G∗
can be extended to an ID-set of G by adding to the vertices in the set I′, and so i(G) ≤i(G∗) + |I′| = i(G∗) + 6.
u3
u2
u1
v3
v2
v1
x1
x2
w1
w2
v
B′
X1,2
G∗
(a)
G∗
X1,2
B′
u3
u2
u1
v3
v2
v1
x1
x2
w1
w2
v
(b)
Figure 56: Subgraphs of G in the proof of Claim 34
As observed earlier, Ω(G∗) = Ω(G)−54. We have shown that every i-set of G∗can be extended to an ID-set of G
by adding to it at most six vertices from X∗, and so i(G) ≤i(G∗) + 6. Thus, 8i(G) ≤8(i(G∗) + 6) ≤Ω(G∗) + 48 <
Ω(G), a contradiction. (2)
By Claims 33 and 34, we have B′ ∈B1. Suppose that v3 ∈V (B′). Since v3 has degree 1 in B′, by Claim 29 such
a component B′ ∈B1 is as illustrated in Figure 49(a). In this case, the graph G contains two vertex disjoint copies
of K2,3 joined by the edge u3v3, contradicting Claim 28. Hence, v3 /∈V (B′).
Let y1 and y2 be the two vertices of degree 3 in B′ that are not adjacent, and let N(y1) = N(y2) = {a, b, c}.
Further, let a′ be the vertex of degree 1 in B′, where aa′ is an edge. Let b′ and c′ be the neighbors of b and c,
respectively, that do not belong to B′. By Claim 25, the subgraph of G induced by the set V (B′) ∪{b′, c′} is a
G8.2-configuration. In particular, we note that the set {a′, b′, c′} is an independent set. Since v3 /∈V (B′), we note
that a′ ̸= v3. The two neighbors of a′ that are not in B′ belong to the set {u2, w1, w2}. Further, the vertices b′
and c′ belong to the set {u2, w1, w2}, that is, {b′, c′} ⊂{u2, w1, w2}. Since the vertex a′ is adjacent to two vertices
that belong to the set {u2, w1, w2}, the vertex a′ is adjacent to at least one of b′ and c′, contradicting our earlier
observation that {a′, b′, c′} is an independent set. This final contradiction completes our proof of Theorem 6. 2
Acknowledgments
The first and the second author were supported by the Slovenian Research and Innovation agency (grants P1-0297,
N1-0285, and N1-0431). Research of the third author was supported in part by the University of Johannesburg.
48
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A proof of the 3 8-conjecture for independent domination in cubic graphs Boˇstjan Breˇsara,b, Tanja Draveca,b, and Michael A. Henningc a Faculty of Natural Sciences and Mathematics : Email: c : Abstract A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to a vertex in S. An independent dominating set in G is a dominating set of G with the additional property that it is an independent set. The domination number, γ(G), and the independent domination number, i(G), are the minimum cardinalities among all dominating sets and independent dominating sets in G, respectively. By definition, γ(G) ≤i(G) for all graphs G. Let G be a connected cubic graph of order n. In 1996 Reed [Combin. Probab. Comput. 5 (1996), 277-295] proved a breakthrough result that γ(G) ≤3 8n. We prove the stronger result that if G is different from K3,3 and the 5-prism C5 2 K2, then i(G) ≤3 8n. This proves a known conjecture. The bound is tight in the sense that there are infinite families of connected cubic graphs that achieve equality in this bound. Keywords: Independent domination number; Domination number; Cubic graphs. AMS subject classification: 05C69 1 Introduction A set S of vertices in a graph G is a dominating set if every vertex in V (G) \ S is adjacent to a vertex in S. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set. A set is independent in G if no two vertices in the set are adjacent. An independent dominating set, abbreviated ID-set, of G is a set that is both dominating and independent in G. Equivalently, an ID-set of G is a maximal independent set in G. The independent domination number of G, denoted by i(G), is the minimum cardinality of an ID-set. An ID-set of cardinality i(G) in G is called an i-set of G. Independent dominating sets have been studied extensively in the literature (see, for example, [8, 10, 15, 16, 18, 19, 21]). A thorough treatise on dominating sets can be found in the so-called "domination books" [11-13]. A survey on independent domination in graphs can be found in [7]. For graph theory notation and terminology, we generally follow [13]. Specifically, let G be a graph with vertex set V (G) and edge set E(G), and of order n(G) = |V (G)| and size m(G) = |E(G)|. Two adjacent vertices in G are neighbors. The open neighborhood of a vertex v in G is NG(v) = {u ∈V : uv ∈E} and the closed neighborhood of v is NG[v] = {v} ∪NG(v). We denote the degree of v in G by degG(v), and so degG(v) = |NG(v)|. An isolated vertex in G is a vertex of degree 0 in G. For k ≥1 an integer, we let [k] denote the set {1, . . . , k} and we let [k]0 = [k] ∪{0} = {0, 1, . . . , k}. 1 16 Oct 2025 A cycle on n vertices is denoted by Cn and a path on n vertices by Pn. The complete graph on n vertices is denoted by Kn, while the complete bipartite graph with one partite set of size n and the other of size m is denoted by Kn,m. A cubic graph is graph in which every vertex has degree 3, while a subcubic graph is a graph with maximum degree at most 3. If G is a subcubic graph, then we let Vj(G) denote the set of vertices of degree j in G and we let nj(G) = |Vj(G)| denote the number of vertices of degree j in G for j ∈[3]0. For subsets X and Y of vertices of G, we denote the set of edges with one end in X and the other end in Y by [X, Y ]. The girth of G, denoted g(G), is the length of a shortest cycle in G. A component of a graph G isomorphic to a graph F we call an F-component of G. For a set S ⊆V (G), the subgraph induced by S is denoted by G[S]. Further, the subgraph of G obtained from G by deleting all vertices in S and all edges incident with vertices in S is denoted by G -S; that is, G -S = G[V (G) \ S]. 2 Motivation and known results In 1996 Reed [20] proved the following breakthrough result establishing a best possible upper bound on the domination number of a cubic graph. Theorem 1 ([20]) If G is a cubic graph of order n, then γ(G) ≤3 8n. The importance of Reed's paper is reflected in over 100 citations one can find on the MathSciNet. Subsequently, determining best possible bounds on the independent domination number of cubic graphs attracted much interest (see, for example, [1-6, 9, 14]). In 1999 Lam, Shiu, and Sun [17] established the following upper bound on the independent domination number of a connected cubic graph. Theorem 2 ([17]) If G ̸∼= K3,3 is a connected, cubic graph of order n, then i(G) ≤2 5n. The 2 5-upper bound on the independent domination number in Theorem 2 is best possible in the sense that there exists at least one graph that achieves equality in the bound, namely the 5-prism, C5 2 K2, which is the Cartesian product of a 5-cycle with a copy of K2. The graphs K2,3 and C5 2K2 are shown in Figure 1(a) and 1(b), respectively. (a) K2,3 (b) C5 2 K2 Figure 1: The graphs K2,3 and C5 2 K2. However despite the pleasing result given in Theorem 2, prior to this paper it remained an open problem to characterize the graphs achieving equality in Theorem 2. In their 2013 survey paper on independent domination in graphs, Goddard and Henning [7] conjectured that the 5-prism, C5 2 K2, is the only extremal graph achieving equality in the 2 5-upper bound on the independent domination number i(G) established by Lam, Shiu, and Sun. The authors in [7] conjectured that Reed's 3 8-upper bound on the domination number, γ(G), also holds for the independent domination, i(G), in a connected, cubic graph, with the exception of the two graphs K3,3 and C5 2 K2. Conjecture 1 ([7]) If G /∈{K3,3, C5 2 K2} is a connected cubic graph of order n, then i(G) ≤3 8n. In 2015 Dorbec et al. [6] gave support to Conjecture 1 by proving that the conjecture holds if the cubic graph G contains no subgraph isomorphic to K2,3. Recall that in a subcubic graph G, we let nj(G) denote the number of vertices of degree j in G for j ∈[3]0. 2 Theorem 3 ([6]) If G is a subcubic graph that does not have a subgraph isomorphic to K2,3 and which has no (C5 2 K2)-component, then 8i(G) ≤8n0(G) + 5n1(G) + 4n2(G) + 3n3(G). As a consequence of Theorem 3, we have the main result in [6], noting that if G is a cubic graph of order n, then n0(G) = n1(G) = n2(G) = 0 and n = n3(G). Theorem 4 ([6]) If G ̸= C5 2 K2 is a connected cubic graph of order n that does not have a subgraph isomorphic to K2,3, then i(G) ≤3 8n. 3 Main result We give a complete proof of Conjecture 1 by adding forbidden subgraphs isomorphic to K2,3 in the statement of Theorem 4 back into the mix. We will prove the following result. Theorem 5 If G /∈{K3,3, C5 2 K2} is a connected cubic graph of order n, then i(G) ≤3 8n. As a consequence of Theorem 5, the 5-prism, C5 2 K2, is the unique extremal graph in Theorem 2. The 3 8-upper bound on the independent domination number given in Theorem 5 is tight. Two infinite families of connected cubic graphs G satisfying i(G) = 3 8n are constructed in [7]. These families are also defined in [13, Chapter 6], so we do not redefine them here. However, graphs in these families are illustrated in Figure 2(a) and (b), respectively. Akbari et al. [1] constructed additional families (which we do not define here) of connected cubic graphs G satisfying i(G) = 3 8n. (a) (b) Figure 2: Two infinite families of cubic graphs G of order n satisfying i(G) = 3 8n Indeed, the family of extremal graphs achieving the 3 8-upper bound in Theorem 5 is rich and varied. As a further example of extremal graphs, the graphs G of order n in the infinite family illustrated in Figure 3 also satisfy i(G) = 3 8n. We remark that the cubic graphs G in the family illustrated in Figure 2(a) do not contain a subgraph isomorphic to K2,3 and satisfy i(G) = 3 8n, while the cubic graphs G in the family illustrated in Figure 2(b) and Figure 3 do contain a subgraph isomorphic to K2,3 and satisfy i(G) = 3 8n. We proceed as follows. In Section 4 we give an overview of our proof to give the main ideas and direction of the proof. In Section 5, we formally define the (infinite) special families of subgraphs we encounter in our proof, and we prove desirable properties of the families. Thereafter in Section 6, we present a detailed proof of our key result, namely Theorem 6. 4 Overview of proof of Theorem 5 Before presenting a proof of Theorem 5, we give a sketch proof to provide the reader an overview of the detailed proof that follows and the direction of the arguments. In order to prove Theorem 5, we relax the 3-regularity 3 Figure 3: An infinite family of cubic graphs G of order n satisfying i(G) = 3 8n condition to a bounded maximum degree 3 condition to make the inductive hypothesis easier to handle, that is, we relax the requirement that G /∈{K3,3, C5 2 K2} is a connected cubic graph to the requirement that G is a subcubic graph that contains no K3,3-component and no (C5 2 K2)-component. We define next what we have coined the vertex weight and the structural weight of such graphs. 4.1 The vertex weight For a subcubic graph G, we assign weights 8, 5, 4 and 3 to the vertices of G of degrees 0, 1, 2 and 3, respectively. For each vertex v of G, we denote the weight of v in G by wG(v) (see Table 1). degG(v) 0 1 2 3 wG(v) 8 5 4 3 Table 1. The weight wG(v) of a vertex v in G For a subset X of vertices in G, we define the vertex weight of X in G as the sum of the weights in G of vertices in X; that is, wG(X) = X v∈X wG(v). We define the vertex weight of G, denoted by w(G), as the sum of the weights of vertices in G, that is, w(G) = 8n0(G) + 5n1(G) + 4n2(G) + 3n3(G). 4.2 The structural weight We would like to show that 8i(G) ≤w(G) since in the special case when G is a cubic graph of order n, we note that n0(G) = n1(G) = n2(G) = 0 and n = n3(G), and so the weight of G simplifies to w(G) = 3n, from which we would infer that i(G) ≤3 8n, as desired. However, relaxing the cubic condition results in the so-called "bad" family of subcubic graphs. We construct such a family B of subcubic graphs in Section 5.2 with the property that if G ∈B, then 8i(G) = w(G) + 2. We will refer to graphs G in this family B as bad graphs since they satisfy 8i(G) > w(G). We denote by b(G) the number of components of G that belong to the family B. Moreover relaxing the cubic condition results in so-called "troublesome" configurations of subcubic graphs. In Section 5.3, we define these troublesome configurations, which we call T-configurations. We distinguish two types of troublesome configurations, namely T1-configurations and T2-configurations. A Ti-configuration is a troublesome configuration that is joined to vertices not in the subgraph by exactly i edges. We denote by tc(G) the number of vertex disjoint troublesome configurations in G. We define the Θ-weight of G, which we call the structural weight of G since it is determined by structural properties of G, by Θ(G) = 2tc(G) + 2b(G). 4 4.3 The total weight We define the total weight of G, which we denote by Ω(G), as the sum of the vertex weight and the structural weight of a subcubic graph G, and so Ω(G) = w(G) + Θ(G). We call the total weight of G the Ω-weight of G. Our key result, given in Theorem 6, gives a tight upper bound on the independent domination number of a subcubic graph in terms of its Ω-weight. Theorem 6 If G is a subcubic graph of order n that contains no K3,3-component and no (C5 2 K2)-component, then 8i(G) ≤Ω(G). In order to prove our main result, namely Theorem 5, we need the above stronger statement given in Theorem 6. Each graph in the family B of subcubic graphs contains vertices of degree 1 or 2. Moreover, every so-called "troublesome configuration" contains a vertex of degree 2 in G. From these properties, a cubic graph G has no component that belongs to the family B and contains no troublesome configuration. We therefore infer that if G is a cubic graph, then tc(G) = b(G) = 0, implying that its structural weight is zero, that is, Θ(G) = 0. Further if G is a cubic graph of order n, then n0(G) = n1(G) = n2(G) = 0 and n = n3(G), and so its vertex weight is 3n, that is, w(G) = 3n. Hence in this case when G is a cubic graph, the inequality in the statement of Theorem 6 simplifies to 8i(G) ≤3n. Our main result, namely Theorem 5, therefore follows readily from Theorem 6. 5 Special families of graphs and subgraphs In this section, we define the family B of so-called 'bad subgraphs' and we define what we have coined 'troublesome configurations' of a subcubic graph G. In order to prove desirable properties of subcubic graphs, we first define 'exit edges' associated with a set X of vertices of a graph G. 5.1 Exit edges of a vertex set Let X be a proper subset of vertices in a subcubic graph G and let V = V (G). We call an edge an X-exit edge in G if it joins a vertex in X to a vertex in V \ X. We denote the number of X-exit edges in G by ξG(X). Those vertices in G -X that are incident in G with at least one X-exit edge have smaller degree in G -X than in G, and therefore have a larger weight in G -X than in G. We refer to the sum of these weight increases as the w-cost of removing X from G and denote the weight increase by ΦG(X). Thus, ΦG(X) = X v∈V wG-X(v) -wG(v) . We prove next the fact that the w-cost of removing a set X of vertices from G is equal to the number of X-exit edges in G plus twice the number of isolated vertices in G -X. Consequently, if removing X yields no isolated vertices, then the cost of removing X from G will be precisely the number of X-exit edges in G. For a vertex v ∈V \ X, we denote by ΦG(v) the number of X-edges in G that are incident with the vertex v. Fact 1 If X ⊂V (G), then ΦG(X) = ξG(X) + 2n0(G -X). Proof. We consider the contribution of a vertex v ∈V \ X to the cost ΦG(X). If the vertex v is incident with no X-exit edge in G, then wG-X(v) = wG(v), and the contribution of v to the sum ΦG(X) is zero. Hence, we may assume that v is incident with at least one X-exit edge in G, that is, ΦG(v) ≥1. If v is not isolated in G -X, then wG-X(v) -wG(v) is precisely the number of exit edges of X incident with v in G, namely ΦG(v). If v is isolated in G -X, then wG-X(v) -wG(v) is the number of exit edges of X incident with v in G plus an additional 2 noting that in this case wG-X(v) = 8 and wG(v) = 6 -ΦG(v). 2 5 5.2 The bad family B In this section, we define the so-called "bad family" B of graphs. Let B1 be the graph obtained from K2,3 by adding a new vertex r and adding an edge joining r and a vertex of degree 2 in the copy of K2,3. The graph B1, illustrated in Figure 4, we call the base graph used in our construction of graphs in the family B. r Figure 4: The base graph B1 Let B be the family of subcubic graphs that contain the base graph B1 and is closed under the operation O1 listed below that extends a graph G′ ∈B to a new graph G ∈B. As remarked earlier, we refer to graphs in the family B as bad graphs. • Operation O1: The graph G is obtained from G′ ∈B by selecting a vertex v′ of degree at most 2 in G′, adding a vertex disjoint copy of K2,3, and adding an edge joining v′ and a vertex of degree 2 in the added copy of K2,3. See Figure 5. G′ O1: v′ 7→ G′ v′ Figure 5: The operation O1 Thus, if G ∈B, then either G = B1 or G is constructed from the base graph B1 by k -1 applications of Operation O1 where k ≥2. Thus, either G = B1 or k ≥2 and G = Bk where B1, B2, . . . , Bk is a sequence of graphs in the family B and the graph Bi+1 is obtained from the graph Bi by applying Operation O1 for all i ∈[k -1]. We call Bi+1 a bad-extension of the graph Bi for i ∈[k -1]. If G ∈B, then we define the root of the graph G as the vertex of G that does not belong to any copy of K2,3, equivalently, the root of G is the (unique) vertex that belongs to no cycle. We note that the root has degree 1, 2 or 3 in the graph G. We define Bi = {B ∈B: the root of B has degree i in B} for i ∈[3], and so B = 3[ i=1 Bi. For example, the graph G illustrated in Figure 6(e) belongs to the family B since it can be constructed from a sequence B1, B2, B3, B4, B5 of graphs in B shown in Figure 6(a)-6(e), where Bi+1 a bad-extension of Bi for i ∈[4] and where G = B5 (in this case, G = Bk where k = 5). In this example, the root is the vertex labelled r in Figure 6, and has degree 2 in the resulting graph G = B5, and so the graph G belongs to the subfamily B2 of B. We define the canonical ID-set of a bad graph G (that belongs to the family B) as follows. Initially if G = B1, we define the canonical ID-set to consist of the root and the two vertices of degree 2 in B1 (as illustrated by the shaded vertices in Figure 6(a)). Suppose that k ≥2 and let B1, B2, . . . , Bk be the sequence of graphs in the family B used to construct G ∈B, and so G = Bk and Bi+1 a bad-extension of the graph Bi for i ∈[k -1]. When we apply Operation O1 to extend the graph Bi to the graph Bi+1 we add to the current canonical ID-set in Bi the two new vertices of degree 2 in Bi+1 that belong to the added copy of K2,3 for all i ∈[k -1]. We note that by construction, every vertex of degree 2 in G belongs to the canonical ID-set of G. For example, the canonical ID-set of each of the graphs B1, B2, B3, B4, B5 used to construct the graph G = B5 is given by the shaded vertices in Figures 6(a)-6(e). 6 (a) B1 r (b) B2 r (c) B3 r (d) B4 r (e) G = B5 r Figure 6: A graph G in the family B2 with root vertex r A vertex of degree 3 in G we call a large vertex. Each copy of K2,3 in G ∈B contains exactly two vertices that belong to the canonical ID-set of G, and these two vertices have two common large neighbors that belong to the same copy of K2,3. We refer to these two large vertices as non-canonical vertices of G. Thus each copy of K2,3 in G ∈B contains two non-canonical vertices, namely the two vertices of degree 3 in that copy of K2,3 that are open twins, where two vertices are open twins if they have the same open neighborhood. We define the non-canonical independent set of G ∈B to be the set consisting of all non-canonical vertices of G. Thus if G ∈B is constructed from the base graph B1 by k -1 applications of Operation O1 where k ≥1, then G contains k vertex disjoint copies of K2,3 and the 2k non-canonical vertices of G (two from each copy of K2,3 in G) form the non-canonical independent set of G. For example, the non-canonical independent set of each of the graphs B1, B2, B3, B4, B5 used to construct the graph G = B5 is given by the vertices represented by squares in Figures 6(a)-6(e). We define the set of vertices in G ∈B that belong to neither the canonical ID-set of G nor the non-canonical independent set of G as the core independent set in G, and we refer to the vertices in the core independent set as core vertices. Thus if G ∈B is constructed from the base graph B1 by k -1 applications of Operation O1 where k ≥1, then G contains k core vertices, one from each of the k vertex disjoint copies of K2,3. For example, the core independent set of each of the graphs B1, B2, B3, B4, B5 used to construct the graph G = B5 is given by the vertices represented by white circled vertices in Figures 6(a)-6(e). We shall need the following lemmas. Lemma 1 If a subcubic graph G is obtained from a subcubic isolate-free graph G′ by adding a vertex disjoint copy of K2,3, and adding an edge joining a vertex v′ of degree at most 2 in G′ and a vertex of degree 2 in the added copy of K2,3, then the following properties hold. (a) i(G) = i(G′) + 2. (b) w(G) = w(G′) + 16. Proof. Let G, G′ and v′ be as defined in the statement of the lemma. Let F be the copy of K2,3 added to G′ when constructing the graph G, where v1, v2 and v3 are the three small vertices (of degree 2) in F and u1 and u2 are the two large vertices (of degree 3) in F. Further, let v3 be the vertex of F adjacent to v′ in G. Let X = V (F). The construction of G is illustrated in Figure 7. (a) Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices u1 and u2, and so i(G) ≤ i(G′) + 2. Conversely, let I be an i-set of G. If {u1, u2} ⊂I, then we let I′ = I \ {u1, u2}. If {u1, u2} ̸⊂I, then 7 G′ v3 v′ u2 u1 v2 v1 Figure 7: The graphs G′ and G in the proof of Lemma 1 {v1, v2} ⊂I. Moreover if in this case v3 /∈I, then v′ ∈I and we let I′ = I \ {v1, v2}. Suppose that {v1, v2, v3} ⊆I. In this case, if a neighbor of v′ different from v3 belongs to the set I, then the set (I \ {v1, v2, v3}) ∪{u1, u2} is an ID-set of G of cardinality |I| -1, contradicting the minimality of the set I. Hence, the vertex v3 is the only neighbor of v′ that belongs to the set I. We now let I′ = (I \ {v1, v2, v3}) ∪{v′}. In all the above cases, the set I′ is an ID-set of G′ of cardinality |I| -2, and so i(G′) ≤|I| -2 = i(G) -2. As observed earlier, i(G) ≤i(G′) + 2. Consequently, i(G) = i(G′) + 2. This proves part (a). (b) By supposition, the graph G is a subcubic graph and the graph G′ is isolate-free. We infer that the vertex v′ has degree 1 or 2 in G′ and therefore has degree 2 or 3 in G, respectively. Thus adopting our earlier notation, the edge v′v3 is the only X-exit edge in G. Thus, the number of X-exit edges is ξG(X) = 1. By Fact 1, the cost of removing X from G is ΦG(X) = ξG(X) = 1 noting that G′ = G -X is isolate-free. Thus, w(G) = w(G′) + wG(X) -ΦG(X) = w(G′) + 17 -1 = w(G′) + 16, which proves part (b). 2 Suppose that G ∈B, and so G is constructed from the base graph B1 by k -1 applications of Operation O1 where k ≥1. We note that if k = 1, then G = B1. Further, we note that i(B1) = 3 and w(B1) = 22. Hence by repeated applications of Lemma 1, we infer the following properties of graphs in the family B. Proposition 1 If G ∈B is constructed from the base graph B1 by k -1 applications of Operation O1 where k ≥1, then the following properties hold. (a) i(G) = 2k + 1, w(G) = 16k + 6, and 8i(G) = w(G) + 2. (b) The canonical ID-set of G is an i-set of G. (c) The non-canonical independent set of G, together with the root vertex of G, is an i-set of G. (d) If v is the root vertex of G or a vertex of degree 2 in G, then i(G -v) = i(G) -1. (e) The core independent set of G is an ID-set of the graph obtained by removing all vertices of degree 2 from G. (f) The graph G contains at most one vertex of degree 1, and no two adjacent vertices of G both have degree 2 in G. 5.3 Troublesome configurations In this section, we define what we have coined a 'troublesome configuration' of a subcubic graph G. We define a troublesome configuration of G, abbreviated T-configuration of G, as a subgraph of G obtained from a bad graph B ∈B1 with root vertex v1 (of degree 1 in B) by applying the following process. • adding a new vertex v2 to B, • adding an edge joining v2 to the vertex v1 and to a vertex w2 of degree 2 in B different from v1, • adding an edge joining v2 to a vertex not in V (B) ∪{v2}, and • adding at most one edge joining v1 to a vertex not in V (B) ∪{v2}. If v1 has degree 2 in G, then we call the troublesome configuration a type-1 troublesome configuration of G, abbreviated T 1 v1,v2-configuration of G. A T 1 v1,v2-configuration is therefore joined to the rest of the graph by exactly one edge, namely an edge that joins v2 to one vertex not in the subgraph. If v1 has degree 3 in G, then we call the troublesome configuration a type-2 troublesome configuration of G, abbreviated T 2 v1,v2-configuration of G. A T 2 v1,v2-configuration is therefore joined to the rest of the graph by exactly two edges, namely an edge that joins each of v1 and v2 to a vertex not in the subgraph. In a troublesome configuration, we call the vertices v1 and v2 the link vertices and we call all other vertices the non-link vertices of the configuration. Thus the degree of every 8 v1 v2 w2 B T (a) A T 1 v1,v2-configuration v1 v2 w2 B T (b) A T 2 v1,v2-configuration Figure 8: Type-1 and type-2 troublesome configurations in G non-link vertex of a troublesome configuration is the same as its degree in G. A T 1 v1,v2-configuration is illustrated in Figure 8(a), and a T 2 v1,v2-configuration is illustrated in Figure 8(b). The shaded vertices in Figure 8(a) indicate an i-set of T 1 v1,v2 and this set is the canonical ID-set of B. Moreover the shaded vertices in Figure 8(b) indicate an i-set of T 2 v1,v2 that contains the vertex v2 and the non-canonical vertices of B. More generally, we have the following properties of troublesome configurations of G. These properties follows readily from properties of bad graphs (that belong to the family B) established in Section 5.2. Proposition 2 If T = T 1 v1,v2 or T = T 2 v1,v2 is a T1- or T2-configuration of G with link vertices v1 and v2 that is obtained from a bad graph B ∈B with root vertex v1, then the following properties hold. (a) i(T) = i(B). (b) The canonical ID-set of B is an i-set of T. (c) There exists an i-set of T that contains the link vertex v2. (d) If S is an arbitrary set of vertices of degree 2 in T that contains neither v1 nor v2, then i(T -S) = i(T)-|S|. Furthermore, there exists an i-set of T -S that contains neither v1 nor v2, and there exists an i-set Ij of T -S that contains vj for j ∈[2]. (f) Every vertex in T has degree at least 2, and no two adjacent vertices of T both have degree 2. 5.4 Examples of graphs achieving equality in Theorem 6 In this section we present a small sample of graphs achieving equality in the upper bound of Theorem 6. Example 1. If G is the graph shown in Figure 9(a), then i(G) = 6 and the shaded vertices indicate an i-set in G. In this example, w(G) = 12 × 3 + 2 × 4 = 44 and Θ(G) = 2tc(G) = 2 × 2 = 4 (where the two vertex disjoint T2-configurations are indicated by the subgraphs in the dashed boxes), and so 8i(G) = 48 = 44 + 4 = w(G) + Θ(G) = Ω(G). Moreover if G is the graph shown in Figure 9(b), then i(G) = 9 and the shaded vertices indicate an i-set in G. In this example, w(G) = 20 × 3 + 2 × 4 = 68 and Θ(G) = 2tc(G) = 2 × 2 = 4 (where the two vertex disjoint T2configurations are indicated by the subgraphs in the dashed boxes), and so 8i(G) = 72 = 44 + 4 = w(G) + Θ(G) = Ω(G). (a) (b) Figure 9: Examples of graphs achieving equality in the upper bound of Theorem 6 Example 2. If G is the graph shown in Figure 10(a), then i(G) = 5 and the shaded vertices indicate an i-set in G. In this example, w(G) = 10 × 3 + 2 × 4 = 38 and Θ(G) = 2tc(G) = 2 × 1 = 2 (where a T2-configuration is indicated by the subgraph in the dashed box), and so 8i(G) = 40 = 38 + 2 = w(G) + Θ(G) = Ω(G). 9 If G is the graph shown in Figure 10(b), then i(G) = 4 and the shaded vertices indicate an i-set in G. In this example, w(G) = 6 × 3 + 3 × 4 = 30 and Θ(G) = 2tc(G) = 2 × 1 = 2 (where a T2-configuration is indicated by the subgraph in the dashed box), and so 8i(G) = 32 = 30 + 2 = w(G) + Θ(G) = Ω(G). If G is the graph shown in Figure 10(c), then i(G) = 5 and the shaded vertices indicate an i-set in G. In this example, w(G) = 8 × 3 + 4 + 2 × 5 = 38 and Θ(G) = 2tc(G) = 2 × 1 = 2 (where a T2-configuration is indicated by the subgraph in the dashed box), and so 8i(G) = 40 = 38 + 2 = w(G) + Θ(G) = Ω(G). (a) F1 (b) F2 (c) F3 Figure 10: Examples of graphs achieving equality in the upper bound of Theorem 6 Example 3. If G is the graph shown in Figure 11, then i(G) = 18 and the shaded vertices indicate an i-set in G. In this example, w(G) = 44 × 3 + 2 × 4 = 140 and Θ(G) = 2tc(G) = 2 × 2 = 4 (where the T2-configurations are indicated by the subgraphs in the dashed boxes), and so 8i(G) = 144 = 140 + 4 = w(G) + Θ(G) = Ω(G). Figure 11: An example of a graph achieving equality in the upper bound of Theorem 6 6 Proof of key result Recall that the vertex weight of a subcubic graph G is w(G) = 8n0(G) + 5n1(G) + 4n2(G) + 3n3(G), and the structural weight of G is Θ(G) = 2tc(G) + 2b(G). Moreover the total weight of G, called the Ω-weight of G, is the sum of its vertex weight and structural weight, that is, Ω(G) = w(G) + Θ(G). In this section, we present a proof of our key result, namely Theorem 6. Its statement in terms of the Ω-weight of G is as follows: Theorem 6. If G is a subcubic graph of order n that contains no K3,3-component and no (C5 2 K2)-component, then 8i(G) ≤Ω(G). Proof. Let G be a subcubic graph of order n that contains no K3,3-component and no (C5 2 K2)-component, and let V = V (G). We wish to prove that 8i(G) ≤Ω(G). Suppose that G is a counterexample to the inequality with minimum order, and let G have order n. Thus, 8i(G) > Ω(G) but every subcubic graph G′ of order less than n that contains no K3,3-component and no (C5 2 K2)-component satisfies 8i(G′) ≤Ω(G′). Since Θ(G) ≥0, we note that w(G) ≤Ω(G). Hence, if 8i(G) ≤w(G), then 8i(G) ≤Ω(G), a contradiction. Therefore, 8i(G) > w(G). 10 Since our proof of Theorem 6 is long, we break the proof into four parts to enhance clarity and exposition of the proof. The first part of the proof establishes important structural properties of the counterexample G. The second part of the proof considers the case when the minimum degree of G equals 1. The third part of the proof considers the case when the minimum degree of G equals 2. The fourth and final part of the proof considers the case when the graph G is cubic, that is, when G is a 3-regular graph. 6.1 Part 1: Structural properties of G In this section, namely Part 1 of our proof, we establish structural properties of the counterexample G. A component of a graph that belongs to the family B is a bad component of the graph. Claim 1 The subcubic graph G contains at least one subgraph isomorphic to K2,3. Proof. If G contains no subgraph isomorphic to K2,3, then by Theorem 3, we have 8i(G) ≤w(G), a contradiction. (2) By Claim 1, we note that n ≥5. If n = 5, then either G = K2,3, in which case 8i(G) = 16 0, implying that either G′ ∈B (noting that G′ is connected) or there is a troublesome configuration in G′ that contains the (adjacent) vertices v1 and v2. In both cases, we have Θ(G′) = 2, and so Ω(G′) = w(G′) + Θ(G′) = (Ω(G) -16) + 2 = Ω(G) -14. Suppose that G′ ∈B. We note that G′ contains at least one copy of K2,3 that contains two vertices of degree 2 in G′. By Claim 6, G′ -v1v2 does not contain a K2,3-subgraph that contains two vertices of degree 2 in G. We therefore infer that G′ ∈B1 and that G′ contains exactly one copy of K2,3, say F ′, that contains two vertices of degree 2 in G′. Moreover, the added edge v1v2 belongs to F ′. Thus, there are two possibilities that may occur. The first case is that both v1 and v2 have degree 3 in G′, as illustrated in Figure 13(a), and the second case is that one of v1 and v2 has degree 2 in G′, as illustrated in Figure 13(b). In both illustrations, we indicate the added edge v1v2 by the dotted edge, which recall exists in G′ but not in G. (a) r v1 v2 F ′ (b) r v1 v2 F ′ Figure 13: Possible graphs G′ in the proof of Claim 8 By properties of the bad family B (see Section 5.2), we infer that there is an i-set, say I′, of G′ -v1v2 that contains both v1 and v2 and is such that |I′| = i(G′) -1 in the first case (as indicated by the shaded vertices in 13 Figure 13(a)) and |I′| = i(G′) in the second case (as indicated by the shaded vertices in Figure 13(b)). In both cases, we have that I′ is an ID-set of G′ -v1v2. Moreover, {v1, v2} ⊂I′ and |I′| ≤i(G′). The set I′ can be extended to an ID-set of G by adding to it the vertex u. Thus, i(G) ≤i(G′) + 1. We note that w(G) = w(G′) + 16 and Θ(G) = Θ(G′) -2, and so Ω(G) = w(G) + 16 + Θ(G) = w(G′) + Θ(G′) -2 = Ω(G′) + 14. Therefore, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 0. We therefore infer that the component, G′ f say, of G′ that contains the edge f is either a bad component or contains a troublesome configuration. Claim 13.6.1 G′ f /∈B. Proof. Suppose that G′ f ∈B. Let G′ f have root vertex vf (of degree 1 in Bf). We note that the degrees of the vertices in G′ f are the same as their degrees in G, except possibly for the vertex u in the case when it belongs to G′ f. By Claims 8 and 9, we infer that G′ f ∈B1, and so the root vf of G′ f has degree 1 in G. Further, we infer from these claims that G′ f contains exactly one copy of K2,3, say Hf. By Claim 13.5, degG(zi) ≥2 for i ∈[2]. Suppose that u belongs to G′ f, implying that u is the root vertex vf. Since the vertex u is adjacent to neither v1 nor v2 in G and since zi is adjacent to vi in G for i ∈[2], we note that u /∈{z1, z2}. Thus, neither z1 nor z2 is the root vertex vf of G′ f, and so the added edge f = z1z2 is an edge of Hf. The graph G is now determined and V (G) = V (T)∪V (Hf). In particular, G has order n = 12. There are two possibilities up to isomorphism, depending on whether v ∈{z1, z2} or v /∈{z1, z2}. In both cases, we have Ω(G) = w(G) = 38 and the set {w1, w2, z1, z2} is an ID-set of G, and so i(G) ≤4. Thus, 8i(G) < Ω(G), a contradiction. Hence, u /∈V (G′ f), implying that G′ has two components, namely the component G′ f and the component containing the vertices u and v. As observed earlier, degG(zi) ≥2 for i ∈[2], and so the added edge f = z1z2 is an edge of Hf. Let X′ = 22 V (G′ f) ∪V (T) and let G′′ = G -X′. Thus the graph shown in Figure 26 is an example of a subgraph of G, where T is the troublesome configuration indicated in the dashed box, G′ f -f is the subgraph indicated in the dashed box, and the set X′ is indicated by the vertices in the dotted region. We note that there exists an IDset, If say, of G′ f -f that contains both vertices z1 and z2 and is such that |If| ≤3. Every i-set of G′′ can be extended to an ID-set of G by adding to it the set If ∪{w1, w2} (as indicated by the shaded vertices in Figure 26), and so i(G) ≤i(G′′)+5. By Claims 8 and 9, we infer that b(G′′) = 0 (and note that tc(G′′) = 0). Thus, Ω(G′′) = w(G′′) = w(G)-wG(X)+1 = Ω(G)-43+1 = Ω(G)-42. Hence, 8i(G) ≤8(i(G′)+5) ≤Ω(G′)+40 < Ω(G), a contradiction. (2) v1 v2 u2 u1 v u z2 z1 vf w1 w2 T G′ f -f X′ G′′ e f Figure 26: An illustration of a subgraph of G in the proof of Claim 13.6.1 We now return to the proof of Claim 13.6. By Claim 13.6.1, G′ f /∈B, implying that G′ f contains a troublesome configuration, Tf say. Necessarily, Tf contains the added edge f. By Claims 8 and 9, we infer that Tf contains exactly one copy of K2,3, and this copy of K2,3 contains the added edge f = z1z2. We may assume, by symmetry, that z1 has degree 3 in the copy of K2,3 in Tf. We now let X∗= (V (T) \ {u}) ∪{z1} and let G∗= G -X∗. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G∗) + 2. We note that degG∗(u) = 1 and the degree of a neighbor of z1 in G∗is also 1. Thus by the structure of the graph G∗so far determined, we infer that b(G∗) = 0 and tc(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G) -wG(X∗) + 5 = Ω(G) -21 + 5 = Ω(G) -16. Hence, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 = Ω(G), a contradiction. (2) By Claim 13.6, z1z2 ∈E(G). Claim 13.7 degG(zi) = 3 for i ∈[2]. Proof. Suppose that degG(zi) = 2 for some i ∈[2]. Suppose firstly that degG(z1) = degG(z2) = 2. Thus, NG(zi) = {vi, z3-i} for i ∈[2]. We now let X = (V (T) \ {u}) ∪{z1, z2} and let G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u1, u2, z1}, and so i(G) ≤i(G′) + 3. We note that Θ(G′) = 0 and wG(X) = 26, and so Ω(G′) = w(G′) = w(G) -wG(X) + 2 = Ω(G) -26 + 2 = Ω(G) -24, and so 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. Hence, at least one of z1 and z2 has degree 3 in G. By symmetry, we may assume that degG(z2) = 3, and so, by supposition, degG(z1) = 2. In this case, we let X = (V (T) \ {u}) ∪{z1} and let G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. We note that Θ(G′) = 0 and wG(X) = 22, and so Ω(G′) = w(G′) = w(G)-wG(X)+4 = Ω(G)-22+4 = Ω(G)-18, and so 8i(G) ≤8(i(G′)+2) ≤Ω(G′)+16 < Ω(G), a contradiction. (2) By Claim 13.7, degG(zi) = 3 for i ∈[2]. Let yi be the neighbor of zi different from vi and z3-i for i ∈[2]. Claim 13.8 The following properties hold. (a) degG(yi) ≥2 for i ∈[2]. 23 (b) y1 ̸= y2. Proof. (a) Suppose that at least one of y1 and y2 has degree 1 in G. By symmetry, we may assume that degG(y1) = 1. In this case, we let X = NG[z1] = {v1, y1, z1, z2} and let G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex z1, and so i(G) ≤i(G′)+1. The cost of removing the X-exit edge y2z2 when constructing G′ increases the total weight by at most 3 by Claim 7. Moreover, the removal of the three X-exit edges different from y2z2 increases the total weight by 3 since the removal of these three edges does not create a bad component or a troublesome configuration. Thus, Ω(G′) ≤Ω(G) -wG(X) + 3 + 3 = Ω(G) -14 + 3 + 3 = Ω(G) -8. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. This proves part (a). (b) Suppose that y1 = y2. In this case, we let X = (V (T) \ {u}) ∪{y1, z1, z2} and let G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u1, u2, z1}, and so i(G) ≤i(G′)+3. We note that Θ(G′) = 0 and wG(X) = 27, and so Ω(G′) = w(G′) ≤w(G) -wG(X) + 3 = Ω(G) -27 + 3 = Ω(G) -24, and so 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. This proves part (b). (2) By Claim 13.8, degG(yi) ≥2 for i ∈[2] and y1 ̸= y2. Suppose that y1 does not belong to a troublesome configuration in G-y1z1. In this case, we let X = (V (T)\{u})∪{z1} and let G′ = G-X. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′)+2. We note that both u and z2 have degree 1 in G′ and do not belong to a bad component or a troublesome configuration in G′. By supposition, y1 does not belong to a troublesome configuration in G′. Moreover, removing the edge y1z1 does not create a bad component containing y1. Hence, Θ(G′) = 0 and wG(X) = 21, and so Ω(G′) = w(G′) ≤w(G) -wG(X) + 5 = Ω(G) -21 + 5 = Ω(G) -16, and so 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. Hence, y1 belongs to a troublesome configuration in G -y1z1. By symmetry, y2 belongs to a troublesome configuration in G -y2z2. Let Tyi be the troublesome configuration in G -yizi that contains the vertex yi for i ∈[2]. Ty1 Ty2 v1 v2 z1 z2 u2 u1 u y1 y2 v T X Figure 27: An illustration of a subgraph of G in the proof of Claim 13 We now let X = (V (T) \ {u}) ∪{z2} ∪NG[y1] and let G′ = G -X. Thus the graph shown in Figure 27 is a subgraph of G, where T, Ty1 and Ty2 are indicated in the dashed box, and the set X is indicated by the vertices in the dotted region. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u1, v2, y1} (indicated by the shaded vertices in Figure 27), and so i(G) ≤i(G′) + 3. The cost of removing the X-exit edge y2z2 when constructing G′ increases the total weight by 3. Moreover, the removal of the six X-exit edges different from y2z2 increases the total weight by 6 since the removal of these six edges does not create a bad component or a troublesome configuration. Thus, Ω(G′) ≤Ω(G)-wG(X)+3+6 = Ω(G)-33+3+6 = Ω(G)-24. Hence, 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. This completes the proof of Claim 13. (2) We present next two consequences of Claim 13. 24 Claim 14 tc(G -e1 -e2) ≤1 for all e1, e2 ∈E(G). Proof. Suppose, to the contrary, that the claim is false. Thus there exists edges e1 = u1v1 and e2 = u2v2 in G whose removal creates two troublesome configurations. Let Tw = Tw1,w2 and Tz = Tz1,z2 be the two troublesome configuration in G -e1 -e2, where w1 and w2 are the link vertices of Tw and z1 and z2 are the link vertices of Tz. By Claims 8, 9, and 13, we infer that the edges e1 and e2 both have one end in Tw and one end in Tz. Furthermore, each of Tw and Tz contains exactly two copies of K2,3 and each copy of K2,3 contains an end of one of the edges e1 and e2. We now let X = (V (Tw) \ {w1, w2, x1, x2}) ∪(V (Tz) \ {y1, y2, z1, z2}) and let G′ = G -X. Renaming vertices if necessary, we may assume that the graph illustrated in Figure 28 is a subgraph of G, where Tw and Tz are the troublesome configurations indicated in the dashed boxes and where X is the set indicated by the vertices in the dotted region. We note that b(G′) = tc(G′) = 0, and so Θ(G′) = 0. Since there are eight X-exit edges, we have Ω(G′) = w(G′) = w(G) -wG(X) + 8 = w(G) -48 + 8 = Ω(G) -40. Every i-set of G′ can be extended to an ID-set of G by adding to it four vertices from the set X (indicated, for example, by the shaded vertices in Figure 28), and so i(G) ≤i(G′) + 4. Thus, 8i(G) ≤8(i(G′) + 4) ≤Ω(G′) + 32 = Ω(G) -8 < Ω(G), a contradiction. (2) w1 w2 x2 x1 z1 z2 y2 y1 v1 v2 u1 u2 Tw Tz X e2 e1 Figure 28: An illustration of a subgraph of G in the proof of Claim 14 Let G7 be the graph shown in Figure 29. As a special case of Claim 13, we have the following result. u1 u2 v3 v1 v2 x1 x2 Figure 29: The graph G7 Claim 15 The graph G does not contain G7 as a subgraph. Proof. Suppose, to the contrary, that G contains G7 as a subgraph. Let F be such a G7-configuration and let the vertices of F be named as in Figure 29. By Claim 8, there is no K2,3-subgraph in G that contains a vertex of degree 2 in G. Hence, degG(v3) = 3. Let v be the neighbor of v3 different from x1 and x2. By Claim 9, either v ∈{u1, u2} or v /∈V (F). We show that at least one of u1 and u2 has a neighbor in G that does not belong to V (F). If v = u1 and degG(u2) = 2, then G = G7 + u1v3. In this case, i(G) = 2 and w(G) = 22, and so 8i(G) < Ω(G), a contradiction. Hence if v = u1, then degG(u2) = 3, and so u2 has a neighbor in G that does not belong to V (F). By symmetry, if v = u2, then u1 has a neighbor in G that does not belong to V (F). If v /∈V (F) and degG(u1) = degG(u2) = 2, then let X′ = V (F) \ {v3} and let G′ = G -X′. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v1 and v2, and so i(G) ≤i(G′) + 2. We note that Θ(G′) = 0 and wG(X′) = 20, and so Ω(G′) = w(G′) = w(G) -wG(X) + 2 = Ω(G) -20 + 2 = Ω(G) -18, and so 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. Hence if v /∈V (F), then at least one of u1 and u2 has a neighbor in G that does not belong to V (F). In all cases, at least one of u1 and u2 has a neighbor in G that does not belong to V (F). Thus, removing the edge v3v creates a troublesome configuration, contradicting Claim 13. (2) 25 6.2 Part 2: The minimum degree of G equals 1 In this section, we consider the case when the minimum degree of G equals 1, that is, when δ(G) = 1. We prove a number of claims that will culminate in a contradiction, thereby showing that this case cannot occur. A support vertex in G is a vertex that has at least one neighbor that has degree 1, and we define a strong support vertex as a vertex that has at least two neighbors of degree 1 in G. Claim 16 δ(G) ≥2. Proof. Suppose, to the contrary, that G contains a vertex of degree 1. Claim 16.1 There is no strong support vertex in G. Proof. Suppose, to the contrary, that G contains a strong support vertex v. Let NG(v) = {v1, v2, v3} where degG(v1) = degG(v2) = 1. Since n ≥6, we note that degG(v3) ≥2. Suppose that degG(v3) = 3 and the two neighbors of v3 different from v both have degree 1 in G. In this case, the graph G is determined (and is a double star S(2, 2)) and w(G) = 26. Moreover, i(G) = 3, and so 8i(G) < w(G), a contradiction. Hence, at most one neighbor of v3 has degree 1. Suppose that v3 has a neighbor, say v4, of degree 1. Since n ≥6, we note that degG(v3) = 3. Let v5 be the third neighbor of v3 different from v and v4. By our earlier observations, degG(v5) ≥2. We now let X = NG[v] ∪{v4} and let G′ = G -X. By Claim 10, b(G′) = 0 and by Claim 13, tc(G′) = 0. Hence, Θ(G′) = 0, and so since there is exactly one X-exit edge, namely v3v5, we have Ω(G′) = w(G′) = w(G) -wG(X) + 1 = Ω(G) -21 + 1 = Ω(G) -20. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v and v4, and so i(G) ≤i(G′) + 2. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G) -4 < Ω(G), a contradiction. Hence, no neighbor of v3 has degree 1 in G. We now let X = NG[v] and let G′ = G -X. By Claim 11, b(G′) = 0 and by Claim 14, tc(G′) ≤1. Hence, Θ(G′) = 2tc(G′) ≤2. Since there are two X-exit edges, neither of which is incident to a vertex of degree 1 in G, we have Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -wG(X) + 2) + 2 ≤Ω(G) -16 + 2 + 2 = Ω(G) -12. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G) -4 < Ω(G), a contradiction. (2) By Claim 16.1, every support vertex in G has exactly one leaf neighbor. Claim 16.2 No vertex of degree 2 is a support vertex. Proof. Suppose, to the contrary, that G contains a support vertex v of degree 2. Let v1 be the leaf neighbor of v, and let v2 be the second neighbor of v. Since n ≥6, we note that degG(v2) ≥2. We now consider the connected subcubic graph G′ = G-{v, v1}. By Claims 10 and 13, we have Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G)-9+1 = Ω(G)-8. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. (2) Let P : v1 . . . vk be the longest path in G such that vertex vi is adjacent to a vertex, say ui, of degree 1 for all i ∈[k]. Possibly, k = 1. By Claim 16.2, degG(vi) = 3 for all i ∈[k]. Claim 16.3 k ≤2. Proof. Suppose, to the contrary, that k ≥3. If v1vk ∈E(G), then the graph G is determined and G is the corona of a cycle of length k, and so i(G) = k and w(G) = 8k, whence 8i(G) = 8k = w(G), a contradiction. Hence, v1vk /∈E(G). Let G′ be obtained from G-{u2, v2} by adding the edge v1v3. We note that w(G) = w(G′)+8. If G′ contains a troublesome configuration, then such a configuration has a copy of K2,3 that contains a vertex of degree 2 in G, contradicting Claim 8. Thus, tc(G′) = 0. Moreover since the connected graph G′ contains at least two vertices of degree 1, we note that G′ /∈B, and so b(G′) = 0. Thus, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G)-8 = Ω(G)-8. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex u2, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. 26 By Claim 16.3, we have k ≤2. If k = 1, then we let NG(v1) = {a, b, u1}, and if k = 2, then we let NG(v1) = {a, u1, v2} and NG(v2) = {b, u2, v1}, where possibly a = b. Claim 16.4 a ̸= b. Proof. Suppose, to the contrary, that a = b, implying that k = 2. Since n ≥6 (and G contains a copy of K2,3), we note that degG(a) = 3. Let NG(a) = {a1, v1, v2}. [See Figure 30(a).] By the maximality of the path P, the vertex a1 has degree at least 2 in G. Let X = {a, u1, u2, v1, v2} and let G′ = G -X. By Claims 10 and 13, we have Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) -wG(X) + 1 = Ω(G) -19 + 1 = Ω(G) -18. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G) -2 < Ω(G), a contradiction. (2) By Claim 16.4, a ̸= b. [See Figure 30(b) and 30(c).] Relabeling vertices if necessary, we may assume that degG(a) ≤degG(b). By Claim 16.1, we have 2 ≤degG(a) ≤degG(b). If k = 1, let X = {u1, v1} and if k = 2, let X = {u1, u2, v1, v2}. In both cases, let X′ = X ∪{a}. u1 u2 v1 v2 a a1 (a) u1 u2 v1 v2 a b (b) u1 v1 a b (c) Figure 30: Subgraphs in the proof of Claim 16 Claim 16.5 The graph G -X′ is isolate-free. Proof. Suppose, to the contrary, that G -X′ has an isolated vertex, z say. By Claim 16.2 and the maximality of the path P, the vertex z has degree at least 2 in G, implying that degG(z) = 2 and NG(z) = {a, vk}. Hence, z = b. Thus, 2 ≤degG(a) ≤degG(b) = 2, and so degG(a) = degG(b) and ab ∈E(G). The graph G is therefore determined. However, G does not contain a subgraph isomorphic to K2,3, contradicting Claim 1. (2) Claim 16.6 If k = 2, then ab ∈E(G). Proof. Suppose, to the contrary, that k = 2 and the vertices a and b are not adjacent. We now consider the graph G′ obtained from G -X by adding the edge e = ab; that is, G′ = (G -X) + e. The graph G′ is a connected subcubic graph. Recall that 2 ≤degG(a) ≤degG(b). By construction, the degrees of vertices a and b in G′ are the same as their degrees in G. Let I′ be an i-set of G′. If a ∈I′, then let I = I′ ∪{u1, v2}. If b ∈I′, then let I = I′ ∪{v1, u2}. If neither a nor b belongs to I′, then let I = I′ ∪{u1, u2}. In all cases, the set I is an ID-set of G, and so i(G) ≤i(G′) + 2. We show firstly that b(G′) = 0. Suppose, to the contrary, that b(G′) ≥1. Since G′ is a connected graph, we infer that G′ ∈B and b(G′) = 1. If G′ contains two or more copies of K2,3, then at least one of these copies contains a vertex of degree 2 in G, contradicting Claim 8. Hence, G′ ∈B1 and G′ contains exactly one copy of K2,3, say F. If F contains at most one of a and b, then again we contradict Claim 8. Hence, F contains both vertices a and b, and the root vertex of G′ has degree 1 in G (and is distinct from a and b). If a or b is adjacent to the root vertex, then we would contradict the maximality of the path P. Hence the graph G is as illustrated in Figure 31, where the added edge e is indicated by the dotted line. Thus, i(G) = 4 (the shaded vertices in Figure 31 are an example of an i-set in G) and Ω(G) = w(G) = 38, and so 8i(G) < Ω(G), a contradiction. Hence, b(G′) = 0. We show next that tc(G′) = 0. Suppose, to the contrary, that tc(G′) ≥1. Thus, adding the edge e = ab to the graph G -X creates a new troublesome configuration, which we call Te, that necessarily contains the edge e. 27 a v1 u1 b v2 u2 e G′ X Figure 31: The graph G in the proof of Claim 16.6 Let r1 and r2 be the two link vertices of Te. We may assume that degG(r1) ≤degG(r2), and so 2 ≤degG(r1) ≤ degG(r2) ≤3. Thus, Te is obtained from a bad graph Be ∈B with root vertex r1. By Claims 8 and 9, the bad graph Be contains exactly one copy of K2,3, say Fe, and so Be ∈B1. Moreover, Fe contains both vertices a and b, for otherwise Fe would contain a vertex of degree 2 in G, a contradiction. We now let X∗= X ∪(V (Te) \ {r1, r2}) and we let G∗= G -X∗. If degG(a) = 2, then the graph illustrated in Figure 32(a) is a subgraph of G, while if degG(a) = 3, then the graph illustrated in Figure 32(b) is a subgraph of G where in both cases Te is the troublesome configuration indicated in the dashed box, the set X∗is indicated in the dashed box, and where e = ab is indicated by the dotted edge. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertices u1 and v2 and the neighbor of a in Fe different from b (as indicated by the shaded vertices in Figures 32(a) and 32(b)). Thus, i(G) ≤i(G∗) + 3. We note that Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) -32 + 2 = Ω(G) -30. Hence, 8i(G) ≤8(i(G∗) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction. Hence, tc(G′) = 0. r1 r2 a b v1 u1 v2 u2 Te X∗ e (a) r1 r2 a b v1 u1 v2 u2 Te X∗ e (b) Figure 32: A subgraph of G in the proof of Claim 16.6 We now return to the proof of Claim 16.6. By our earlier observations, b(G′) = 0 and tc(G′) = 0. Hence, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) -wG(X) = Ω(G) -16. Recall that i(G) ≤i(G′) + 2. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. (2) Claim 16.7 k = 1. Proof. Suppose, to the contrary, that k = 2. By Claim 16.6, ab ∈E(G). Recall that 2 ≤degG(a) ≤degG(b). If degG(a) = degG(b) = 2, then the graph G is determined. In this case, G does not contain a copy of K2,3, a contradiction. Hence, degG(b) = 3. We show firstly that degG(a) = 3. Suppose, to the contrary, that degG(a) = 2. In this case, we let G′ = G -X′ where recall that X = {u1, u2, v1, v2} and X′ = X ∪{a}. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. We note that G′ is a connected subcubic graph and degG′(b) = 1. Moreover, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) -wG(X) + 2 = Ω(G) -20 + 2 = Ω(G) -18. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. Hence, degG(a) = 3. Let NG(a) = {a1, b, v1} and NG(b) = {a, b1, v2}. We show next that a1 ̸= b1. Suppose, to the contrary, that a1 = b1. If degG(a1) = 2, then the graph G is determined. In this case, G does not contain a copy of K2,3, a contradiction. Hence, degG(a1) = 3. Let a2 28 be the neighbor of a1 different from a and b. If degG(a2) = 1, then the graph G is determined, and again we contradict the supposition that G contain a copy of K2,3. Hence, degG(a2) ≥2. We now let X∗= X ∪{a, b, a1} and consider the connected subcubic graph G∗= G -X∗. Every i-set of G∗can be extended to an ID-set of G by adding to it the set {b, v1, u2}, and so i(G) ≤i(G∗) + 3. By Claim 10, b(G∗) = 0. By Claim 13, tc(G∗) = 0. Thus, Θ(G∗) = 0, and so noting that a1a2 is the unique X∗-exit edge and degG(a2) ≥2, we have Ω(G∗) = w(G∗) = w(G)-wG(X)+1 = Ω(G)-25+1 = Ω(G)-24. Thus, 8i(G) ≤8(i(G′)+3) ≤Ω(G′)+24 = Ω(G), a contradiction. Hence, a1 ̸= b1. Recall by maximality of the path P that degG(a1) ≥2 and degG(b1) ≥2. We now consider the graph G′ = G-X′, where recall that X = {u1, u2, v1, v2} and X′ = X ∪{a}. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices v1 and u2, and so i(G) ≤i(G′) + 2. We note that degG′(b) = 1, and so there is no troublesome configuration containing b in G′. If the vertex b belongs to a bad component, then such a component has a copy of K2,3 that contains a vertex of degree 2 in G, contradicting Claim 8. Hence, b belongs to no bad component in G′. If the removal of the edge aa1 when constructing G′ creates a new troublesome configuration, then by Claim 13 we infer that such a troublesome configuration must contain the three vertices a1, b and b1, which is not possible noting that degG′(b) = 1. Hence, tc(G′) = 0. Thus, Θ(G′) = 0, and so noting that there are three X′-exit edges and degG(a1) ≥2, we have Ω(G′) = w(G′) = w(G) -wG(X′) + 3 = Ω(G) -19 + 3 = Ω(G) -16. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. (2) By Claim 16.7, we have k = 1. Recall that 2 ≤degG(a) ≤degG(b). Recall that n ≥6. In particular, we note that degG(b) = 3. Claim 16.8 ab /∈E(G). Proof. Suppose, to the contrary, that ab ∈E(G). In this case, we let X∗= {a, b, u1, v1} and consider the graph G∗= G -X∗. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertex v1, and so i(G) ≤i(G∗)+1. Suppose that degG(a) = 2, and so there is exactly one X∗-exit edge. By Claim 10, b(G∗) = 0. By Claim 13, tc(G∗) = 0. Thus, Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G)-wG(X)+1 = Ω(G)-15+1 = Ω(G)-14. Thus, 8i(G) ≤8(i(G∗) + 1) ≤Ω(G∗) + 8 = Ω(G), a contradiction. Hence, degG(a) = 3. Let NG(a) = {a1, b, v1} and NG(b) = {a, b1, v1}. By the maximality of the path P, we note that degG(a1) ≥2 and degG(b1) ≥2. If a1 is isolated in G∗, then the graph G is determined. In this case, a1 = b1 and NG(a1) = {a, b}. However, then, n = 5, a contradiction. Hence, a1 is not isolated in G∗. By symmetry, b1 is not isolated in G∗. Thus, w(G∗) = w(G) -wG(X∗) + 2 = w(G) -14 + 2 = w(G) -12. By Claim 10, we have b(G∗) = 0, and by Claim 14, we have tc(G∗) ≤1. Thus, Θ(G∗) ≤2, and so Ω(G∗) = w(G∗)+Θ(G∗) ≤(w(G)-12)+2 = Ω(G)-12+2 = Ω(G)-10. Thus, 8i(G) ≤8(i(G∗) + 1) ≤Ω(G∗) + 8 < Ω(G), a contradiction. (2) By Claim 16.8, ab /∈E(G). Claim 16.9 The graph G -{a, b, u1, v1} is isolate-free. Proof. Suppose, to the contrary, that G -{a, b, u1, v1} has an isolated vertex. By Claim 16.2 and the maximality of the path P, such a vertex has degree 2 in G with a and b as its two neighbors. If G -{a, b, u1, v1} contains two isolated vertices, then the graph G is determined and is the graph B1 shown in Figure 4. However, then G ∈B, a contradiction. Hence, G -{a, b, u1, v1} contains exactly one isolated vertex, say c. As observed earlier, NG(c) = {a, b}. Since n ≥6, we note that degG(b) = 3. Let NG(b) = {b1, c, v1}. By the maximality of the path P, we note that degG(b1) ≥2. We now let X′′ = {a, b, c, u1, v1}. Suppose that degG(a) = 2. In this case, we consider the graph G′′ = G -X′′. Every i-set of G′′ can be extended to an ID-set of G by adding to it the vertices v1 and c, and so i(G) ≤i(G′′) + 2. By Claim 10, b(G′′) = 0. By Claim 13, tc(G′′) = 0. Thus, Θ(G′′) = 0, and so Ω(G′′) = w(G′′) = w(G)-wG(X′′)+1 = Ω(G)-19+1 = Ω(G)-18. Thus, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 < Ω(G), a contradiction. Hence, degG(a) = 3. Let NG(a) = {a1, c, v1}. By the maximality of the path P, we note that degG(a1) ≥2. If a1 = b1, then G[{a, a1, b, c, v1}] is a copy of K2,3 that contains a vertex of degree 2 in G, namely vertex c, contradicting Claim 8. Hence, a1 ̸= b1. Renaming vertices if necessary, we may assume by symmetry that 2 ≤degG(a1) ≤degG(b1). We show next that a1b1 ∈E(G). Suppose, to the contrary, that a1b1 /∈E(G). Recall that X′′ = {a, b, c, u1, v1}. In this case, we consider the graph G′ obtained from G-X′′ by adding the edge e = a1b1; that is, G′ = (G-X′′)+e. The 29 graph G′ is a connected subcubic graph. Let I′′ be an i-set of G′. If a1 ∈I′′, then let I = I′′∪{b, u1}. If b1 ∈I′′, then let I = I′′∪{a, u1}. If neither a1 nor b1 belongs to I′′, then let I = I′′∪{c, v1}. In all cases, the set I is an ID-set of G, and so i(G) ≤i(G′)+2. By construction, the degrees of vertices a1 and b1 in G′ are the same as their degrees in G. However adding the edge e may have created a bad graph or a troublesome configuration. Thus, b(G′)+tc(G′) ≤1, implying that Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -wG(X′′)) + 2 = w(G) -18 + 2 = Ω(G) -16. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 = Ω(G), a contradiction. Hence, a1b1 ∈E(G). In this case, we consider the graph G′′ = G -X′′. Every i-set of G′′ can be extended to an ID-set of G by adding to it the vertices v1 and c, and so i(G) ≤i(G′′) + 2. By Claim 11, b(G′′) = 0. We note that a1 and b1 are adjacent vertices in G′′ and both a1 and b1 have degree at most 2 in G′′. We therefore infer that a1 and b1 do not belong to a troublesome configuration in G′′, and so tc(G′′) = 0. Thus, Θ(G′′) = 0, and so Ω(G′′) = w(G′′) = w(G) -wG(X′′) + 2 = Ω(G) -18 + 2 = Ω(G) -16. Thus, 8i(G) ≤8(i(G′′) + 2) ≤Ω(G′′) + 16 = Ω(G), a contradiction. (2) By Claim 16.9, the graph G∗= G -X∗is isolate-free, where recall that X∗= {a, b, u1, v1}. Further recall that 2 ≤degG(a) ≤degG(b) and that ab /∈E(G). Every i-set of G∗can be extended to an ID-set of G by adding to it the vertex v1, and so i(G) ≤i(G∗) + 1. By Claim 10, if there is a bad component in G∗, then such a component is incident with at least three X∗-edges. By Claims 13 and 14, if there is a troublesome configuration in G∗, then such a component is incident with at least two X∗-edges. Thus since there are at most four X∗-edges, namely two edges incident with vertex a and two incident with vertex b, we infer that either b(G∗) = tc(G∗) = 0 or b(G∗) = 1 and tc(G∗) = 0 or b(G∗) = 0 and tc(G∗) ≤2. In all cases, b(G∗) + tc(G∗) ≤2. Suppose that b(G∗) + tc(G∗) ≤1, and so Θ(G∗) ≤2 and Ω(G∗) ≤w(G∗) + 2. If degG(a) = 2, then w(G∗) = w(G) -wG(X∗) + 3 = w(G) -15 + 3 = Ω(G) -12, and so Ω(G∗) ≤w(G∗) + 2 ≤Ω(G) -10. If degG(a) = 3, then w(G∗) = w(G) -wG(X∗) + 4 = w(G) -14 + 4 = Ω(G) -10, and so Ω(G∗) ≤w(G∗) + 2 ≤Ω(G) -8. In both cases, Ω(G∗) + 8 ≤Ω(G). Thus, 8i(G) ≤8(i(G∗) + 1) ≤Ω(G∗) + 8 ≤Ω(G), a contradiction. Hence, b(G∗)+tc(G∗) = 2, implying by our earlier observations that b(G∗) = 0 and tc(G∗) = 2. Thus, G∗contains two troublesome configurations, say Tw and Tz, each of which contain two neighbors of a or b. In particular, we note that degG(a) = degG(b) = 3. Let NG(a) = {a1, a2, v1} and let NG(b) = {b1, b2, v1}. Let w1 and w2 be the link vertices of Tw and let z1 and z2 be the link vertices of Tz. Let Tw be obtained from the bad graph Bw ∈B1 with root vertex w1, and let Tz be obtained from the bad graph Bz ∈B1 with root vertex z1. Moreover, let x1 and x2 be the vertices of Bw adjacent to w1 and w2, respectively, and let y1 and y2 be the vertices of Bz adjacent to z1 and z2, respectively. By Claims 8 and 9, each of Bw and Bz contains exactly two copies of K2,3. Further, each copy of K2,3 in Bw and Bz contains a vertex incident with an X∗-exit edge. We now let X′′ = X∗∪(V (Tw) \ {x2, w2}) ∪(V (Tz) \ {y2, z2}) and let G′′ = G -X′′. In the case when {a1, b1} ⊂V (Tw) and {a2, b2} ⊂V (Tz), the graph illustrated in Figure 36 is a subgraph of G, where Tw and Tz are the troublesome configurations indicated in the dashed boxes and where X∗and X′′ are the sets indicated by the vertices in the dotted region. If degG(w1) = 3, then let ew = ww1 be the edge incident with w1 not in Tw, and if degG(z1) = 3, then let ez = zz1 be the edge incident with z1 not in Tz. The vertices x2 and y2 have degree 1 in G′′, and the neighbors of x2 and y2, namely w2 and z2, in G′′ have degree 2. Hence, there is no bad component or troublesome configuration in G′′ containing these four vertices. By Claim 10, 11, 13 and 14, we have b(G′′) = 0 and tc(G′′) ≤1. Moreover if tc(G′′) = 1, then the edges ew and ez exist and their removal increases the total weight by 2 + 2 = 4 (namely, an increase in 2 from the vertex weight and an increase in 2 in the structural weight arising from a troublesome configuration containing w and z). If tc(G′′) = 0, then the removal of ew and ez, if these edges exist, increases the total weight by at most 2 × 3 = 6 in the worst case (which can only occur if w and z are isolated vertices in G′′). As observed earlier, the removal of the six X′′-exit edges different from ew and ez do not increase the structural weight and therefore only increase the vertex weight (by 6). We therefore infer that Ω(G′′) ≤Ω(G) -wG(X′′) + 6 + 6 ≤Ω(G) -74 + 6 + 6 = Ω(G) -62. Every i-set of G′′ can be extended to an ID-set of G′′ by adding to it the vertices a, b and u1, together with two vertices from each of Tw and Tz (as illustrated by the shaded vertices in Figure 36). Thus, i(G) ≤i(G′′) + 7. As observed earlier, Ω(G′′) + 62 ≤Ω(G). Therefore, 8i(G) ≤8(i(G′′) + 7) ≤Ω(G′′) + 56 < Ω(G), a contradiction. We therefore deduce that our supposition that G contains a vertex of degree 1 is false. Hence, δ(G) ≥2, completing the proof of Claim 16. (2) 30 Tw Tz a b b1 a2 a1 b2 v1 u1 x1 x2 y1 y2 w1 w2 z1 z2 X∗ X′′ Figure 33: An illustration of a subgraph of G in the proof of Claim 16 6.3 Part 3: The minimum degree of G is at least two By Claim 16, δ(G) ≥2. Recall that n ≥6 and that G contains K2,3 as a subgraph. By supposition, G ̸= K3,3. As in the proof of Part 2, in what follows we will frequently use the structural property stated in Claim 8 that there is no K2,3-subgraph in G that contains a vertex of degree 2 in G. We prove next a key structural property of the graph G. For this purpose, we define a troublesome subgraph of G as a subgraph T such that there exists a set of edges incident with vertices of T whose removal creates a troublesome configuration in G with vertex set V (T). For example, the subgraph T illustrated in Figure 21 is a troublesome subgraph of G since the removal of the edge e produces a troublesome configuration in G -e. As a further example, the subgraph T illustrated in Figure 22 or in Figure 23 is a troublesome subgraph of G since the removal of the edge e (in both figures) produces a troublesome configuration in G -e. Claim 17 There is no troublesome subgraph in G. Proof. Suppose, to the contrary, that G contains a troublesome subgraph, say T. Hence there exists a set, say ET ⊂E(G), of edges incident with vertices of T such that G′ = G -ET contains a troublesome configuration, say T ′, with vertex set V (T ′) = V (T). By Claim 4, the graph G contains no troublesome configuration, that is, tc(G) = 0. By Claim 13, tc(G -e) = 0 for all e ∈E(G). Hence, |ET | ≥2, that is, at least two edges incident with vertices in T were removed when constructing T ′. Let the troublesome configuration T ′ in G′ have link vertices v1 and v2, where degG(v2) = 3. Thus, T ′ is obtained from the bad graph B ∈B with root vertex v1 (of degree 1 in B). Further, let u1 and u2 be the vertices of B adjacent to v1 and v2, respectively, in G. Let z2 be the neighbor of v2 different from u2 and v1. Let the bad graph B have k copies of K2,3 in G′. Since |ET | ≥2, we note that k ≥2. Let A = {a1, a2, . . . , ak} be the set of vertices in B incident with edges in ET . Claim 17.1 If the link vertex v1 is not incident with an edge of ET , then 8i(G) ≤Ω(G). Proof. Suppose that the link vertex v1 is not incident with an edge of ET . If v1 has degree 3 in G, then let z1 be the neighbor of v1 not in T. We now let X = V (T) \ (A ∪{u2, v2}) and we consider the graph G′ = G -X. We note that each vertex in A has degree 1 in G′, and belongs to neither a bad component nor a troublesome configuration in G′. Moreover, the vertex u2 has degree 1 in G′ and its neighbor, v2, in G′ has degree 2 in G′. Hence, u2 and v2 do not belong to a bad component or a troublesome configuration in G′. If degG(v1) = 3, then we infer by Claims 10 and 13 that the vertex z1 does not belong to a bad component or a troublesome configuration in G′. Hence, b(G′) = tc(G′) = 0, and so Θ(G′) = 0. We note that there are at most 2k + 4 X-exit edges. Moreover, |X| = 4k and every vertex of X, except for possibly the vertex v1, has degree 3 in G. Therefore, Ω(G′) = w(G′) = w(G) -wG(X) + 2k + 4 ≤w(G) -3 × 4k + 2k + 4 = w(G) -10k + 4 = Ω(G) -10k + 4. 31 Every i-set of G′ can be extended to an ID-set of G by adding to it the core independent set (see Section 5.2) of the bad graph B, and so i(G) ≤i(G′) + k. (For example, if the graph shown in Figure 34(a) is a subgraph of G, where the dotted edges are the edges that belong to the set ET , then k = 6 and the subgraph G[X] is illustrated in Figure 34(b). Furthermore, the core independent set of B is indicated by the k = 6 shaded vertices in Figure 34(b).) Thus noting that k ≥2, we have 8i(G) ≤8(i(G′)+k) ≤Ω(G′)+8k ≤(Ω(G)-10k+4)+8k = Ω(G)-2k+4 ≤Ω(G), a contradiction. (2) T a1 a2 a4 a3 a5 a6 u1 u2 v1 v2 z2 z1 (a) A subgraph T of G where the dotted edges belong to ET v1 (b) The subgraph G[X] where X = V (T) \ {a1, a2, a3, a4, a5, a6, u2, v2} Figure 34: An illustration of a subgraph of G in the proof of Claim 17.1 Claim 17.2 If the link vertex v1 is incident with an edge of ET , then 8i(G) < Ω(G). Proof. Suppose that the link vertex v1 is incident with an edge of ET . Renaming vertices if necessary, we may assume that e = a1v1. We now let X = V (T) \ (A \ {a1}) and we consider the graph G′ = G -X. We note that each vertex in A\{a1} has degree 1 in G′, and belongs to neither a bad component nor a troublesome configuration in G′. Moreover, the vertex z2 has degree 1 or 2 in G′, and by Claims 10 and 13 we infer that z2 does not belong to a bad component or a troublesome configuration in G′. Hence, b(G′) = tc(G′) = 0, and so Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) -wG(X) + 2k -1 = w(G) -3 × (4k + 3) + 2k -1 = w(G) -10k -10 = Ω(G) -10k -10. Let IB be the core independent set (see Section 5.2) of the bad graph B, and so |IB| = k. We define the modified core independent set of B as follows. We say that two copies of K2,3 are adjacent if they are joined by an edge. By the structure of the bad graph G, there exists a sequence F1, . . . , Flof copies of K2,3 where u1 ∈V (F1), u2 ∈V (Fl) and if l≥2, then Fi+1 is adjacent to Fi for all i ∈[l-1]. Let ci be the vertex in Fi that belongs to IB for i ∈[l]. In particular, we note that c1 = u1. Let C = {c1, . . . , cl}. We define d1 = v1, and if l≥2, then we define di+1 as the vertex of Fi that is adjacent to ci+1 in B for i ∈[l-1]. Moreover, we define dl+1 = u2, and we let D = {d1, d2, . . . , dl+1}. We now define the modified core independent set of B by I∗ B = (IB \ C) ∪D. 32 As an illustration, suppose that T is the troublesome configuration in Figure 35, where B is the associated bad graph used to construct T. In this example, the core independent set IB is indicated by the shaded vertices in Figure 35(a), and the modified core independent set I∗ B is indicated by the shaded vertices in Figure 35(b). We note that |IB| = k and |I∗ B| = k + 1, where here k = 6 is the number of copies of K2,3 in the bad graph B. B T d2 d3 d4 u1 = c1 c2 c3 c4 u2 = d5 v1 = d1 v2 (a) The core independent set, IB, of B B T d2 d3 d4 u1 = c1 c2 c3 c4 u2 = d5 v1 = d1 v2 (b) The modified core independent set, I∗ B, of B Figure 35: An illustration of a modified core independent set in a bad graph We now return to the proof of Claim 17.2. Every i-set of G′ can be extended to an ID-set of G by adding to it the modified core independent set I∗ B of B, and so i(G) ≤i(G′) + |I∗ B| = i(G′) + k + 1. (For example, if the graph shown in Figure 36(a) is a subgraph of G, where the dotted edges are the edges that belong to the set ET , then k = 6 and the subgraph G[X] is illustrated in Figure 36(b) where the shaded vertices indicate the vertices of the modified core independent set I∗ B of B.) Recall that Ω(G′) + 10k + 10 = Ω(G). Thus, 8i(G) ≤8(i(G′) + k + 1) ≤ Ω(G′) + 8k + 8 = Ω(G) -2k -2 < Ω(G), a contradiction. (2) In both Claims 17.1 and 17.2, we have 8i(G) ≤Ω(G), a contradiction. This completes the proof of Claim 17. (2) By Claim 17, there is no troublesome subgraph in G. Hence, if G′ is an arbitrary induced subgraph of G, then tc(G′) = 0, and so Θ(G′) = 2b(G′). We state this formally as follows. Claim 18 If G′ is an induced subgraph of G, then Θ(G′) = 2b(G′). By Claim 18, the structural weight of an induced subgraph G′ of G is determined only by the number of bad components of G′, if any, that belong to the bad family B. Recall that B = B1 ∪B2 ∪B3. Moreover, every graph in 33 T a1 a2 a4 a3 a5 a6 u1 u2 v1 v2 z2 (a) A subgraph T of G where the dotted edges belong to ET a1 u1 u2 v1 v2 (b) The subgraph G[X] where X = V (T) \ {a2, a3, a4, a5, a6} Figure 36: An illustration of a subgraph of G in the proof of Claim 17.2 the family B1 has one vertex of degree 1 and at least two vertices of degree 2, while every graph in the family B2 has at least five vertices of degree 2. Furthermore, every graph in the family B3 has at least six vertices of degree 2. We prove next certain properties that must hold if the graph G contains vertices of degree 2. Claim 19 The graph G does not contain a path v1v2v3 such that degG(vi) = 2 for all i ∈[3]. Proof. Suppose, to the contrary, that G contains a path v1v2v3 such that degG(vi) = 2 for all i ∈[3]. Let X = {v1, v2, v3} and let G′ = G -X. We note that there are two X-exit edges. If G′ contains an isolated vertex, then G = C4, a contradiction. Hence, δ(G′) ≥1 and the graph G′ contains at most two vertices of degree 1. By Claim 10, we infer that b(G′) = 0. Thus, by Claim 18, we have Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) -wG(X) + 2 = Ω(G) -12 + 2 = Ω(G) -10. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v2, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2) Claim 20 No two adjacent vertices in G both have degree 2. Proof. Suppose, to the contrary, that G contains two adjacent vertices, say v1 and v2, of degree 2. Suppose firstly that v1 and v2 have a common neighbor v, and so vv1v2v is a 3-cycle in G. Let u be the third neighbor of v different from v1 and v2. Let X = {v, v1, v2} and let G′ = G -X. The graph G′ is a connected subcubic graph and all vertices in G′ have degree at least 2, except possibly for the vertex u which has degree at least 1. By Claims 10 and 18, we infer that Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) -wG(X) + 1 = Ω(G) -11 + 1 = Ω(G) -10. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v1, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. 34 The vertices v1 and v2 therefore have no common neighbor. Let ui be the neighbor of vi different from v3-i for i ∈[2]. Since v1 and v2 have no common neighbor, u1 ̸= u2. By Claim 19, both vertices u1 and u2 have degree 3 in G. We show firstly that u1u2 is not an edge in G. Suppose, to the contrary, that u1u2 ∈E(G). In this case, we let X = {v1, v2, u1} and consider the graph G′ = G -X. We note that G′ is isolate-free and the vertex u2 has degree 1 in G′. By Claim 18, tc(G′) = 0. By Claims 10 and 11, we infer that if G′ has more than one component, none of them is bad. In addition, if G′ is connected, then by Claim 8 we infer G′ /∈B. Thus, Θ(G′) = 0. Thus, Ω(G′) = w(G′) = w(G) -wG(X) + 3 = Ω(G) -11 + 3 = Ω(G) -8. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v1, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. Therefore, u1u2 /∈E(G). We now let X = {v1, v2} and consider the graph G′ obtained from G -X by adding the edge e = u1u2. We note that G′ is a connected subcubic graph satisfying δ(G′) ≥2. Furthermore, u1 and u2 are adjacent vertices of degree 3 in G′. Let I′ be an i-set of G′. If u1 ∈I′, then let I = I′ ∪{v2}. If u2 ∈I′ or if neither u1 nor u2 belong to I′, then let I = I′ ∪{v1}. In all cases, |I| = |I′| + 1 and the set I is an ID-set of G, and so i(G) ≤i(G′) + 1. If Θ(G′) = 0, then Ω(G′) = w(G′) = w(G)-wG(X) = Ω(G)-8, and so 8i(G) ≤8(i(G′)+1) ≤Ω(G′)+8 = Ω(G), a contradiction. Hence, Θ(G′) ≥1, implying that either G′ ∈B or the edge u1u2 belongs to a troublesome configuration in G′. If G′ ∈B, then G would contain a copy of K2,3 that contains two vertices of degree 2 in G (noting that in this case, the root vertex in G′ has degree at least 2), a contradiction. Hence, G′ /∈B. Thus, adding the edge e = u1u2 to the graph G -X creates a new troublesome configuration, which we call Te, that necessarily contains the edge e. Let w1 and w2 be the two link vertices of Te. By combining Claim 8 and Claim 17, we infer that Te contains exactly one copy of K2,3, say Fe, and both u1 and u2 belong to Fe (for otherwise, Fe would contain a vertex of degree 2 in G, a contradiction). By symmetry and renaming vertices if necessary, we may assume that u2 is adjacent to the link vertex w2 of Te. Let u be the third neighbor of u2 in Te different from u1 and w2, as illustrated in Figure 37(a). We now let X∗= V (Fe) ∪X and let G∗= G -X∗. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertices v1 and u, as indicated by the shaded vertices in Figure 37(b). Thus, i(G) ≤i(G∗) + 2. We note that Θ(G∗) = Θ(G) = 0, and so Ω(G∗) = w(G∗) = w(G) -wG(X∗) + 2 = Ω(G) -24 + 2 = Ω(G) -22. Hence, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 < Ω(G), a contradiction. (2) Te u2 u1 u w2 w1 (a) A subgraph Te of G′ X∗ u2 u1 u w2 w1 v1 v2 (b) A subgraph of G Figure 37: Subgraphs of G′ and G in the proof of Claim 20 Claim 21 A vertex of degree 3 in G has at most one neighbor of degree 2. Proof. Suppose, to the contrary, that G contains a vertex, say v, of degree 3 that is adjacent to at least two vertices of degree 2. Let X = NG[v] and let G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1. Suppose that all three neighbors of v have degree 2 in G. By Claim 20, the set NG(v) is therefore an independent set. Suppose that G′ contains an isolated vertex, u say. Since δ(G) ≥2, we note that degG(u) ≥2. If degG(u) = 3, then G = K2,3, contradicting the fact that n ≥6. If degG(u) = 2, then since u is adjacent to two vertices in NG(v), the graph G contains a path P3 all of whose vertices have degree 2 in G, contradicting Claim 19. Hence, the graph G′ is isolate-free. Since there are three X-exit edges, we note that w(G′) = (w(G)-15)+3 = w(G)-12. By Claim 11, we infer that b(G′) ≤1. Thus by Claim 18, we have Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) = (w(G) -12) + 2 = Ω(G) -10. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. 35 Therefore, the vertex v has two neighbors of degree 2 and one of degree 3. Let v1 and v2 be the two neighbors of v of degree 2 and let v3 be the third neighbor of v (of degree 3). Suppose that G′ contains an isolated vertex, u say. If degG(u) = 3, then G contains a copy of K2,3 that contains two vertices of degree 2 in G, contradicting Claim 8. If degG(u) = 2, then since u is adjacent to at least one of v1 and v2, the graph G contains two adjacent vertices of degree 2, contradicting Claim 20. Hence, the graph G′ is isolate-free. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′)+1. By Claim 11, we infer that b(G′) ≤1. Thus by Claim 18, we have Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -wG(X) + 4) + 2 = w(G) -14 + 4 + 2 = Ω(G) -8. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. (2) By Claim 20, both neighbors of a vertex of degree 2 have degree 3 in G. By Claim 21, at most one neighbor of a vertex of degree 3 has degree 2 in G. Claim 22 No vertex of degree 2 belongs to a triangle. Proof. Suppose, to the contrary, that G contains a vertex, say u, of degree 2 that belongs to a triangle, T say. Let v1 and v2 be the two neighbors of u, and so v1v2 ∈E(G) and V (T) = {u, v1, v2}. By our earlier observations, both v1 and v2 have degree 3 in G. Let wi be the neighbor of vi not in T for i ∈[2]. Since at most one neighbor of a vertex of degree 3 has degree 2 in G, both w1 and w2 have degree 3. Suppose firstly that w1 = w2. In this case, we let X = NG[v1] and consider the graph G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding the vertex v1, and so i(G) ≤i(G′) + 1. By Claims 10 and 18, we infer that Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) -wG(X) + 1 = w(G) -13 + 1 = Ω(G) -12. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. Hence, w1 ̸= w2. Suppose that w1w2 /∈E(G). In this case, we let X = NG[u] = {u, v1, v2} and let G′ be obtained from G -X by adding the edge w1w2. Let I′ be an i-set of G′. If w1 ∈I′, then let I = I ∪{v2}, while if w1 /∈I′, then let I = I ∪{v1}. In both cases, |I| = |I′| + 1 and the set I is an ID-set of G, and so i(G) ≤i(G′) + 1. The added edge w1w2 increases the structural weight by at most 2 (which occurs if G′ ∈B or if the edge w1w2 belongs to a troublesome configuration). Hence, Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -10) + 2 = Ω(G) -8. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. Hence, w1w2 ∈E(G). We now let X = NG[v1] = {u, v1, v2, w1} and consider the graph G′ = G -X. Every i-set of G′ can be extended to an ID-set of G by adding the vertex v1, and so i(G) ≤i(G′)+1. By combining Claims 8, 10 and 18, we infer that Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) -wG(X) + 3 = w(G) -13 + 3 = Ω(G) -10. Therefore, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2) Recall that by Claim 1, the graph G contains at least one subgraph isomorphic to K2,3, and by Claim 2, the graph G is connected. By supposition, G ̸= K3,3 and G ̸= C5 2 K2. We proceed further by proving additional structural properties of the graph G. A diamond in G is a copy of K4 -e where e is an arbitrary edge of the complete graph K4. A graph is diamond-free if it does not contain a diamond as an induced subgraph. Claim 23 The graph G is diamond-free. Proof. Suppose, to the contrary, that G contains a diamond D. Let V (D) = {v1, v2, v3, v4} where v1v2 is the missing edge in the diamond. By Claim 22, no vertex of degree 2 belongs to a triangle. Thus, both v1 and v2 have degree 3 in G. We now let X = V (D) and we consider the graph G′ = G -X. Since n ≥6, we note that G′ is isolate-free. Every i-set of G′ can be extended to an ID-set of G by adding the vertex v3, and so i(G) ≤i(G′)+1. By Claims 11 and 18, we infer that Θ(G′) = 0, and so Ω(G′) = w(G′)+Θ(G′) ≤w(G)-wG(X)+2 = w(G)-12+2 = Ω(G) -10. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2) Claim 24 The graph G does not contain K3,3 -e as a subgraph, where e is an edge of the K3,3. Proof. Suppose, to the contrary, that G contains K3,3 -e as a subgraph, where e is an edge of the K3,3. Let F be such a subgraph in G, and let F have partite sets A = {x1, x2, x3} and B = {v1, v2, v3}, where e = x1v1 is the missing edge from the copy of K3,3 in F. Since G is diamond-free by Claim 23, the set B is an independent set in G. If degG(x1) = 2 or if degG(v1) = 2, then there is a K2,3-subgraph in G that contains a vertex of degree 2 36 in G, a contradiction. Hence, degG(x1) = degG(v1) = 3. Let u1 be the neighbor of v1 different from x2 and x3, and let w1 be the neighbor of x1 different from v2 and v3. Since G ̸= K3,3, the vertices u1 and x1 are distinct. If x1u1 ∈E(G), then G contains G7 as a subgraph (see Figure 29), contradicting Claim 15. Hence, u1x1 /∈E(G), and so the vertices u1 and w1 are distinct. We now let X = V (F) ∪{u1, w1} and consider the graph G′ = G -X. Suppose that G′ contains an isolated vertex, say u. Thus, NG(u) = {u1, w1}. By Claim 20, degG(u1) = degG(w1) = 3. By Claim 22, u1w1 /∈E(G). In this case, we let X′ = X ∪{u} and consider the graph G′′ = G-X′. By Claim 21, the neighbor of u1 in G′′ has degree 3 in G and the neighbor of w1 in G′′ has degree 3 in G. Thus, G′′ is isolate-free and there are two X′-exit edges. By Claims 11 and 18, we infer that Θ(G′′) = 0, and so Ω(G′′) = w(G′′) = w(G) -wG(X′) + 2 = w(G) -28 + 2 = Ω(G) -26. Every i-set of G′′ can be extended to an ID-set of G by adding to it the vertices in the set {u, v1, x1} (indicated by the shaded vertices in Figure 38(a)), and so i(G) ≤i(G′′) + 3. Thus, 8i(G) ≤8(i(G′′) + 3) ≤Ω(G′′) + 24 < Ω(G), a contradiction. Hence, G′ is isolate-free. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {v1, x1} (indicated by the shaded vertices in Figure 38(b)), and so i(G) ≤i(G′) + 2. We note that there are at most four X-exit edges. By Claims 11 and 18, we infer that b(G′) ≤1 and Θ(G′) ≤2, and so Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -wG(X) + 4) + 2 = w(G) -24 + 4 + 2 = Ω(G) -18. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {v1, x1} (indicated by the shaded vertices in Figure 38(b)), and so i(G) ≤i(G′) + 2. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. (2) (a) G′′ u1 w1 u x1 v1 v2 v3 x2 x3 (b) G′ u1 w1 x1 v1 v2 v3 x2 x3 Figure 38: Subgraphs of G in the proof of Claim 24 Claim 25 Every K2,3-subgraph of G belongs to a G8.2-configuration that is an induced subgraph in G, where G8.2 is the graph in Figure 39(b). Proof. Let F be a K2,3-subgraph in G with partite sets A = {x1, x2} and B = {v1, v2, v3}. Since G is diamond-free by Claim 23, the set B is an independent set in G. Let ui be the neighbor of vi different from x1 and x2 for i ∈[3]. By Claim 24, the vertices u1, u2 and u3 are distinct. Let U = {u1, u2, u3}. Since G7 is not a subgraph of G, the set C is independent. Thus the subgraph G[A ∪B ∪C] of G induced by the sets A ∪B ∪C (see Figure 39(a)) is an induced subgraph of G. (2) u3 u2 u1 v3 v2 v1 x1 x2 (a) C B A (b) G8.2 Figure 39: Subgraphs of G in the proof of Claim 25 37 Claim 26 The graph G is cubic. Proof. Suppose, to the contrary, that δ(G) = 2. Let u be an arbitrary vertex of degree 2 in G. By Claim 20, both neighbors of u have degree 3 in G. Let v be a neighbor of u, and let NG(v) = {u, x, y}. By Claim 21, degG(x) = degG(y) = 3. We now let X = NG[v] and we consider the graph G′ = G -X. Claim 26.1 The graph G′ is isolate-free. Proof. Suppose that G′ contains an isolated vertex, say w. Thus, NG(w) ⊆NG(v). If NG(w) = NG(v), then G[{u, v, w, x, y}] is a copy of K2,3 that contains a vertex of degree 2 in G, a contradiction. Hence, w has degree 2 in G and is adjacent to exactly two vertices in NG(v). Since no two vertices of degree 2 in G are adjacent by Claim 20, we have NG(w) = {x, y}. Let x1 and y1 be the neighbors of x and y, respectively, different from v and w. Since there is no K2,3-subgraph that contains a vertex of degree 2 in G, we note that x1 ̸= y1. By Claim 21, degG(x1) = degG(y1) = 3. Since degG(u) = 2, the vertex u is adjacent to at most one of x1 and y1. By symmetry, we may assume that u and x1 are not adjacent. We now let X′ = NG[x] = {v, w, x, x1} and consider the graph G′′ = G-X′. We note that both vertices u and y have degree 1 in G′′. If u belongs to a bad component Bu ∈B in G′′, then necessarily Bu = B1 where u is the vertex of degree 1 in Bu and where Bu contains exactly one copy of K2,3, which we denote by Fu. In this case, the vertex x1 is adjacent in G to the two vertices of degree 2 in Fu. However, then G[V (Fu) ∪{x1}] is a copy of K3,3 with an edge removed, contradicting Claim 24. Hence, u does not belong to a bad component in G′′. Analogously, y does not belong to a bad component in G′′. By Claim 11, we therefore infer that removing the two X′-exit edges incident with x1 does not create a bad component. Hence, b(G′′) = 0, and so by Claim 18 we have Θ(G′′) = 0, and so Ω(G′′) = w(G′′) ≤w(G) -wG(X) + 5 = w(G) -13 + 5 = Ω(G) -8. Every i-set of G′′ can be extended to an ID-set of G by adding to it the vertex x, and so i(G) ≤i(G′′)+1. Thus, 8i(G) ≤8(i(G′′)+1) ≤Ω(G′′)+8 = Ω(G), a contradiction. (2) Claim 26.2 xy /∈E(G). Proof. Suppose that xy ∈E(G). Thus there are exactly three X-exit edges. By Claim 26.1, the graph G′ is isolate-free. By Claim 11, if there is a bad component in G′, then all three X-exit edges emanate from such a component and G′ = B1. Let r be the root vertex of G′. By Claim 25, the K2,3-subgraph in G′ belongs to a G8.2-configuration (see Figure 39(b)). The graph G is therefore determined (up to isomorphism) and is illustrated in Figure 40. In this case, i(G) = 3 (the shaded vertices in Figure 40 indicate an i-set of G) and w(G) = 32, and so 8i(G) < Ω(G), a contradiction. Hence, b(G′) = 0. By Claim 18, we have Θ(G′) = 0, and so Ω(G′) = w(G′) ≤w(G) -wG(X) + 3 = w(G) -13 + 3 = Ω(G) -10. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 < Ω(G), a contradiction. (2) v y r x u G′ Figure 40: The graph G in the proof of Claim 26.2 By Claim 26.1, the graph G′ is isolate-free. By Claim 26.2, xy /∈E(G). Hence, there are at exactly five X-exit edges. By Claim 11, if there is a bad component in G′, then at least three X-exit edges emanate from such a component, implying that b(G′) ≤1. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertex v, and so i(G) ≤i(G′) + 1. If b(G′) = 0, then by Claim 18 we have Θ(G′) = 0, and so Ω(G′) = w(G′) ≤ w(G) -wG(X) + 5 = w(G) -13 + 5 = Ω(G) -8. Thus, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. Hence, b(G′) = 1. Let B be the bad component in G′, and let r be the root vertex of B. 38 Claim 26.3 B ∈B1. Proof. Suppose that B ∈B2 ∪B3. If B ∈B3, then at least six X-exit edges emanate from such a component, contradicting the fact that there are five X-exit edges. Hence, B ∈B2. Thus, the root vertex r of B has degree 2 in B and there are either two or three K2,3-subgraphs in B. Suppose that there are three K2,3-subgraphs in B. In this case the graph G is determined and V (G) = V (B) ∪X. The core independent set of B (see Section 5.2) of cardinality 3 can be extended to an ID-set of G by adding to it the set NG(v) = {u, x, y}, and so i(G) ≤6. (For example, if G is the graph illustrated in Figure 41, then the ID-set of G containing the core independent set of B together with the vertices in {u, x, y} is indicated by the six shaded vertices.) Thus, 8i(G) ≤48 < 62 = Ω(G), a contradiction. B X x u y v r Figure 41: An example of a graph G in the proof of Claim 26.3 Hence, B contains exactly two K2,3-subgraphs. If V (G) = V (B) ∪X, then the graph G is determined and as before the core independent set (of cardinality 2) of B can be extended to an ID-set of G by adding to it the set NG(v) = {u, x, y}, and so i(G) ≤5. Moreover, Ω(G) = 46, and so 8i(G) < Ω(G), a contradiction. Hence, V (G) ̸= V (B) ∪X, implying that there are exactly four X-exit edges that are incident with vertices in B and the root vertex r of B is not incident with any of these four X-exit edges. Suppose the vertex u is not adjacent to a vertex of B in G. We now let X∗= V (B) ∪{v, x, y} and consider the graph G∗= G -X∗. In this case, there is exactly one X∗-exit edge and the vertex u has degree 1 in G∗. By Claims 10 and 18, we infer that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) -wG(X∗) + 1 = w(G)-43+1 = Ω(G)-42. Every i-set of G∗can be extended to an ID-set of G by adding to it the core independent set of B of cardinality 2 and the vertices x and y (as indicated by the four shaded vertices in Figure 42), and so i(G∗) ≤i(G) + 4. Thus, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 < Ω(G), a contradiction. B X∗ G∗ x u y v r Figure 42: An example of a graph G in the proof of Claim 26.3 Hence, the vertex u is adjacent to a vertex of B in G. Renaming x and y if necessary, we may assume that x is adjacent to exactly one vertex, say x∗, of B in G. In this case, we let X∗= V (B) ∪X and consider the graph 39 G∗= G-X∗. Since there is exactly one X∗-exit edge and the graph G∗is isolate-free, by Claims 10 and 18 we infer that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) -wG(X∗) + 1 = w(G) -47 + 1 = Ω(G) -46. Every i-set of G′ can be extended to an ID-set of G by adding to it the core independent set of B of cardinality 2, together with the vertices in the set {u, x∗, y}, and so i(G∗) ≤i(G) + 5. Thus, 8i(G) ≤8(i(G∗) + 5) ≤Ω(G∗) + 40 < Ω(G), a contradiction. Since both cases produce a contradiction, this completes the proof of Claim 26.3. (2) By Claim 26.3, we have B ∈B1. Thus, the root vertex r in B has degree 1 in B. Let B have k copies of K2,3. We note that k ≤3. Claim 26.4 k = 1. Proof. Suppose that k ∈{2, 3}. If k = 3, then the graph G is determined and V (G) = V (B) ∪X. In this case, the core independent set of B (see Section 5.2) of cardinality 3 can be extended to an ID-set of G by adding to it the set NG(v) = {u, x, y}, and so i(G) ≤6. Thus, 8i(G) ≤48 < 62 = Ω(G), a contradiction. Hence, k = 2. Suppose the vertex u is not adjacent to a vertex of B in G. We now let X∗= V (B) ∪{v, x, y} and consider the graph G∗= G -X∗. In this case, there is exactly one X∗-exit edge and the vertex u has degree 1 in G∗. By Claims 10 and 18, we infer that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) -wG(X∗) + 1 = w(G)-43+1 = Ω(G)-42. Every i-set of G′ can be extended to an ID-set of G by adding to it the core independent set of B of cardinality 2 and the vertices x and y (as indicated by the four shaded vertices in Figure 43), and so i(G∗) ≤i(G) + 4. Thus, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 < Ω(G), a contradiction. B X∗ G∗ x u y v r Figure 43: An example of a graph G in the proof of Claim 26.4 Hence, the vertex u is adjacent to a vertex of B in G. Renaming x and y if necessary, we may assume that x is adjacent to exactly one vertex, say x∗, of B in G. In this case, we let X∗= V (B) ∪X and consider the graph G∗= G-X∗. Since there is exactly one X∗-exit edge and the graph G∗is isolate-free, by Claims 10 and 18 we infer that b(G∗) = 0 and Θ(G∗) = 0, and so Ω(G∗) = w(G∗) = w(G) -wG(X∗) + 1 = w(G) -47 + 1 = Ω(G) -46. Every i-set of G′ can be extended to an ID-set of G by adding to it the core independent set of B of cardinality 2, together with the vertices in the set {u, x∗, y} (as indicated by the five shaded vertices in Figure 44), and so i(G∗) ≤i(G)+5. Thus, 8i(G) ≤8(i(G∗) + 5) ≤Ω(G∗) + 40 < Ω(G), a contradiction. Since both cases produce a contradiction, this completes the proof of Claim 26.3. (2) By Claim 26.4, we have k = 1, and so B contains exactly one copy of K2,3. Let the copy of K2,3 in B have partite sets {x1, x1} and {v1, v2, v3}. Renaming vertices if necessary, we may assume that the root vertex of B is adjacent to v3. By Claim 25, every K2,3-subgraph of G belongs to a G8.2-configuration that is an induced subgraph in G, where G8.2 is the graph in Figure 39(b). Let w1 and w2 be the neighbors of v1 and v2, respectively, not in B. We note by Claim 25 that {r, w1, w2} is an independent set. Let w3 be the third neighbor of v, and so the root vertex r is adjacent to w3. Thus, degG(r) = 2 and NG(r) = {v3, w3}. We note that NG(v) = {w1, w2, w3} = {u, x, y}, where recall that degG(u) = 2 and degG(x) = degG(y) = 3. Since there are no two adjacent vertices of degree 2 in G by Claim 20, we infer that degG(w3) = 3, and so u ∈{w1, w2}. Renaming vertices if necessary, we may assume that u = w2, and so degG(w2) = 2 and NG(w2) = {v, v2}. We now let X′′ = NG[v] ∪V (B) and we consider the graph G′′ = G -X′′. Suppose that G′′ contains an isolated vertex. In this case, such a vertex, say z, has degree 2 in G and is adjacent to w1 and w3, implying that the vertex w3 40 B X∗ G∗ u x y x∗ v r Figure 44: An example of a graph G in the proof of Claim 26.4 has two neighbors of degree 2, namely r and z, contradicting Claim 21. Hence, G′′ is isolate-free. Thus, the graph shown in Figure 45 is a subgraph of G, where in this illustration the neighbors of w1 and w2 not in G′′ are distinct. By Claims 11 and 18 we infer that b(G′′) = 0 and Θ(G′′) = 0, and so Ω(G′′) = w(G′′) ≤w(G) -wG(X′′) + 2 = w(G) -32 + 2 = Ω(G) -30. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {r, v1, w2} (as indicated by the three shaded vertices in Figure 45), and so i(G′′) ≤i(G) + 3. Thus, 8i(G) ≤8(i(G′′) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction. This completes the proof of Claim 26. (2) B X′′ G′′ r v w3 w2 w1 v3 v2 v1 x1 x2 Figure 45: A subgraph of G in the proof of Claim 26 6.4 Part 4: The graph G is cubic By Claim 26, the graph G is a cubic graph of order n. Recall that n ≥6. By Claim 2, the graph G is connected. By supposition, G ̸= K3,3 and G ̸= C5 2 K2. Since G is a cubic graph, we note that Θ(G) = 0, and so Ω(G) = w(G). We prove a number of claims that will culminate in a contradiction, thereby proving our main theorem. Claim 27 The graph G contains no triangle. Proof. Suppose, to the contrary, that G contains a triangle T, where V (T) = {v1, v2, v3}. Let ui be the neighbor of vi not in T for i ∈[3]. By Claim 23, the graph G is diamond-free, implying that the vertices u1, u2 and u3 are distinct. Let U = {u1, u2, u3}. If G[U] = K3, then G is the 3-prism C3 2 K2. In this case, i(G) = 2 and w(G) = 18, and so 8i(G) < Ω(G), a contradiction. Hence renaming vertices if necessary, we may assume that u1u2 /∈E(G). Let G′ be the graph obtained from G -V (T) by adding the edge u1u2. We note that every vertex of G′ has degree 3, except for the vertex u3 which has degree 2 in G′. We therefore infer that b(G′) = 0. Suppose that tc(G′) = 0. In this case, Θ(G′) = 0 and Ω(G′) = w(G′) = w(G) -9 + 1 = Ω(G) -8. Let I′ be an i-set of G′. If neither u1 nor u2 belong to I′, then let I = I′ ∪{v1}. If ui ∈I′ for some i ∈[2], then let I = I′ ∪{v3-i}. In both cases, |I| = |I′| + 1 = i(G′) + 1 and the set I is an ID-set of G, and so i(G) ≤i(G′) + 1. Hence, 8i(G) ≤8(i(G′) + 1) ≤Ω(G′) + 8 = Ω(G), a contradiction. Hence, tc(G′) = 1, implying that there is a troublesome configuration, say T ′, in G′ where T ′ has exactly one copy of K2,3, and this copy contains the vertex u3 as the vertex of degree 2 in T ′, and where T ′ contains the added edge 41 u1u2 We now let X∗= V (T) ∪{u3} and G∗= G -X∗. In this case, tc(G∗) = 0 and b(G∗) = 0, and so Θ(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G)-wG(X∗)+4 = Ω(G)-12+4 = Ω(G)-8. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertex v3, and so i(G) ≤i(G∗)+1. Hence, 8i(G) ≤8(i(G∗)+1) ≤Ω(G∗)+8 = Ω(G), a contradiction. Claim 28 The graph G does not contain two vertex disjoint copies of K2,3 joined by at least one edge. Proof. Let F1 and F2 be two vertex disjoint copies of K2,3, where F1 has partite sets X = {x1, x2} and V1 = {v1, v2, v3}, and F2 partite sets Y = {y1, y2} and U1 = {u1, u2, u3}. Suppose, to the contrary, that F1 and F2 are joined by at least one edge. Claim 28.1 F1 and F2 are joined by exactly one edge. Proof. If F1 and F2 are joined by three edges, then G = G10.2, where G10.2 is the graph illustrated in Figure 46(a). In this case, i(G) = 3 (an i-set is indicated by the shaded vertices in Figure 46(a)) and w(G) = 30, and so 8i(G) < w(G), a contradiction. (a) G10.2 u3 u2 u1 v3 v2 v1 x1 x2 w (b) G′ Figure 46: Subgraphs of G in the proof of Claim 28 Suppose next that F1 and F2 are joined by exactly two edges. Renaming edges if necessary, we may assume that u2v2 and u3v3 are edges of G. Let x be the neighbor of v1 not in X, and let y be the neighbor of u1 not in Y . Possibly, x = y. Let X′ = (V (F1) ∪V (F2)) \ {u1, v1} and let G′ = G -X′. Thus the graph in Figure 47(a) is a subgraph of G, where in this illustration x ̸= y. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u2, v3} (indicated by the shaded vertices in Figure 47(a)), and so i(G) ≤i(G′) + 2. Since every vertex of G′ has degree 3, except for the vertices u1 and v1 both of which have degree 1, we infer that b(G′) = 0, and so, by Claim 18, we have Θ(G′) = 0. Hence, Ω(G′) = w(G′) ≤w(G) -wG(X′) + 4 = Ω(G) -24 + 4 = Ω(G) -20. Thus, 8i(G) ≤8(i(G′) + 2) ≤Ω(G′) + 16 < Ω(G), a contradiction. (2) x2 v3 u3 y2 x1 v2 u2 y3 v1 x u1 y X′ G′ (a) X∗ x2 v3 u3 y2 x1 v2 u2 y1 v1 x w u1 z y G∗ (b) Figure 47: Subgraphs of G in the proof of Claim 28 By Claim 28.1, F1 and F2 are joined by exactly one edge. Renaming vertices if necessary, we may assume that u3v3 is the edge joining F1 and F2. 42 Claim 28.2 Neither v1 nor v2 has a common neighbor with u1 or u2. Proof. Suppose that v1 or v2 has a common neighbor with u1 or u2. By symmetry, we may assume that u2 and v2 have a common neighbor, say w. By Claim 25, the vertex w is adjacent to neither u1 nor v1. Let z be the neighbor of w different from u2 and v2. Let x be the neighbor of v1 not in X, and let y be the neighbor of u1 not in Y . Possibly, x = y. By Claim 25, the set {u3, x, w} is an independent set and the {v3, y, w} is an independent set. Therefore since w is adjacent to z, we note that x ̸= z and y ̸= z, although possibly x = y. We now let X∗= (V (F1) \ {v1}) ∪{u2, u3, w, z} and consider the graph G∗= G -X∗. Every i-set of G∗ can be extended to an ID-set of G by adding to it the vertices v3 and w (indicated by the shaded vertices in Figure 47(b)), and so i(G) ≤i(G∗) + 2. Let z1 and z2 be the two neighbors of z different from w. Every vertex of G∗has degree 3 except for three vertices of degree 1 (namely, v1, y1 and y2) and two vertices of degree 2 (namely, z1 and z2). Since y1 and y2 have a common neighbor in G∗(namely u1), we note that the only possible component of G∗that belongs to the family B is a B1-component containing v1, z1 and z2. However this would imply that z1 and z2 are vertices of degree 2 in a copy of K2,3 that have a common neighbor, namely z, that does not belong to this copy of K2,3, which contradicts Claim 25. Hence, b(G∗) = 0, and so, by Claim 18, we have Θ(G∗) = 0. Hence, Ω(G∗) = w(G∗) ≤w(G) -wG(X∗) + 8 = Ω(G) -24 + 8 = Ω(G) -16. Thus, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 ≤Ω(G), a contradiction. (2) We now return to the proof of Claim 28. By Claim 28.2, neither v1 nor v2 has a common neighbor with u1 or u2. As before, let x be the neighbor of v1 not in X, and let y be the neighbor of u1 not in Y . Let w be the neighbor of v2 not in X, and let z be the neighbor of u2 not in Y . By our earlier observations, the vertices x, y, w, z are distinct and do not belong to V (F1) ∪V (F2). By Claim 25, xw /∈E(G) and yz /∈E(G). We now let X∗= {v2, v3, u3, x1, x2, y1, y2, w} and we consider the graph G∗= G -X∗. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertices v2 and u3 (indicated by the shaded vertices in Figure 48), and so i(G) ≤i(G∗) + 2. Let w1 and w2 be the two neighbors of w different from v2. Every vertex of G∗has degree 3 except for three vertices of degree 1 (namely, v1, u1 and u2) and two vertices of degree 2 (namely, w1 and w2). Analogous arguments as before show that there is no B1-component containing the vertices w1 and w2, for otherwise such a component (of order 6) contains w1 and w2 as vertices of degree 2 in a copy of K2,3 that have a common neighbor, namely w, that does not belong to this copy of K2,3, which contradicts Claim 25. We therefore infer that b(G∗) = 0, and so, by Claim 18, we have Θ(G∗) = 0. Hence, Ω(G∗) = w(G∗) ≤w(G) -wG(X∗) + 8 = Ω(G) -24 + 8 = Ω(G) -16. Thus, 8i(G) ≤8(i(G∗) + 2) ≤Ω(G∗) + 16 = Ω(G), a contradiction. This completes the proof of Claim 28. (2) x2 v3 u3 y2 x1 v2 u2 y1 v1 x w u1 z y G∗ X∗ Figure 48: A subgraph of G in the proof of Claim 28 By Claim 28, if the graph G contains two vertex disjoint copies of K2,3, then there is no edge joining these two copies of K2,3. This yields the following property of the structural weight for an arbitrary induced subgraph G′ of G. Claim 29 If G′ is an arbitrary induced subgraph of G and if B is a component of G′ in the family B, then B ∈Bi and B contains exactly i copies of K2,3 for some i ∈[3] as illustrated in Figure 49. By Claim 29, if G′ is an arbitrary induced subgraph of G and if B is a component of G′ that belongs to B, then either B ∈B1 and B contains one vertex of degree 1 and two vertices of degree 2 (see Figure 49(a)) or B ∈B2 43 (a) B ∈B1 (b) B ∈B2 (c) B ∈B3 Figure 49: The three possible bad components of G′ in Claim 29 and B contains no vertex of degree 1 and five vertices of degree 2 (see Figure 49(b)) or B ∈B3 and B contains no vertex of degree 1 and six vertices of degree 2 (see Figure 49(c)). By Claim 27, the graph G contains no triangle. Let F be a K2,3-subgraph in G with partite sets A = {x1, x2} and B = {v1, v2, v3}. Since G is triangle-free, the set B is an independent set in G. Let ui be the neighbor of vi different from x1 and x2 for i ∈[3]. By Claim 25, the vertices u1, u2 and u3 are distinct and the set C = {u1, u2, u3} is an independent set. We now let X′′ = A ∪B ∪C and consider the graph G′′ = G -X′′. Thus the graph illustrated in Figure 50 is an induced subgraph of G. u3 u2 u1 v3 v2 v1 x1 x2 C B A F set X′′ G′′ Figure 50: A subgraph of G Claim 30 The graph G′′ is isolate-free. Proof. Suppose, to the contrary, that G′′ contains at least one isolated vertex. If G′′ contains two isolated vertices, then the graph G is determined and is the graph G10.2 shown in Figure 46(a). As noted earlier, in this case 8i(G) < w(G), a contradiction. Hence, G′′ contains exactly one isolated vertex, say w. Let X′ = X′′ ∪{w} and let G′ = G -X′. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {w, x1, x2} (as indicated by the shaded vertices in Figure 46(b)), and so i(G) ≤i(G′) + 3. Since G′ has either one vertex of degree 1 and one vertex of degree 2 or three vertices of degree 2, we infer that b(G′) = 0, and so by Claim 18, we have Θ(G′) = 0. Hence, Ω(G′) = w(G′) ≤w(G) -wG(X′) + 3 = Ω(G) -27 + 3 = Ω(G) -24. Thus, 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 = Ω(G), a contradiction. (2) By Claim 30, the graph G′′ is isolate-free. Claim 31 The graph G′′ contains at most one vertex of degree 1. Proof. Suppose, to the contrary, that G′′ contains at least two vertices of degree 1. Let D be the set of vertices of degree 1 in G′′. We note that |D| ≤3. Suppose firstly that |D| = 3. Let D = {w1, w2, w3}. Let wi be the common neighbor of ui and ui+1 for i ∈[3] where addition is taken modulo 3. If w1, w2, w3 have a common neighbor, say w, then the graph G is determined and is the graph G12 shown in Figure 51(a). In this case, i(G) = 4 (an i-set is 44 indicated by the shaded vertices in Figure 51(a)) and w(G) = 36, and so 8i(G) < w(G), a contradiction. Hence, w1, w2, w3 do not have a common neighbor. We now let X′ = X′′ ∪D and consider the graph G′ = G′′ -D = G -X′. Since w1, w2, w3 have no common neighbor, the graph G′ is an isolate-free graph, and so w(G′) = (w(G) -33) + 3 = w(G) -30. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u1, u3, v2} (as indicated by the shaded vertices in Figure 51(b)), and so i(G) ≤i(G′) + 3. Since G′ has either one vertex of degree 1 and one vertex of degree 2 or three vertices of degree 2, we infer by Claim 29 that b(G′) = 0. Thus, by Claim 18, we have Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) -30 = Ω(G) -30. Thus, 8i(G) ≤8(i(G′) + 3) ≤Ω(G′) + 24 < Ω(G), a contradiction. u3 u2 u1 v3 v2 v1 w3 w2 w1 w x1 x2 (a) G12 u3 u2 u1 v3 v2 v1 w3 w2 w1 x1 x2 (b) G′ u3 u2 u1 v3 v2 v1 w2 w1 x1 x2 (c) G′ Figure 51: Subgraphs of G in the proof of Claim 31 Hence, |D| = 2, and so G′′ contains two vertices of degree 1, say w1 and w2. By symmetry, we have two possibilities. First, let wi be a common neighbor of ui and ui+1 for i ∈[2]. Thus the graph shown in Figure 51(c) is a subgraph of G. We now let X′ = X′′ ∪{w1, w2} and consider the graph G′ = G′′ -D = G -X′. Since the graph G is triangle-free, the graph G′ is isolate-free, and so w(G′) = (w(G) -30) + 4 = w(G) -26. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u2, v1, v3} (as indicated by the shaded vertices in Figure 51(c)), and so i(G) ≤i(G′) + 3. Since G′ has either two vertices of degree 1 and no vertex of degree 2 or one vertex of degree 1 and two vertices of degree 2 or four vertices of degree 2, we infer that b(G′) ≤1, and so by Claim 18, we have Θ(G′) ≤2. Hence, Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -26) + 2 = w(G) -24. Thus, 8i(G) ≤8(i(G′) + 3) ≤w(G′) + 24 = w(G), a contradiction. Second, let u1 and u2 have two common neighbors w1 and w2. Suppose that u3, w1 and w2 have a common neighbor z. Let X∗= X′′ ∪{w1, w2, z}. Let G∗= G -X∗. By using Claims 10 and 18, we have Ω(G∗) = w(G∗) = (w(G) -33) + 1 = Ω(G) + 32, since there is one X∗-exit edge. Every i-set of G∗can be extended to an ID-set of G by adding to it the vertices in the set {z, v1, v2, v3}. Hence, 8i(G) ≤8(i(G∗) + 4) ≤Ω(G∗) + 32 = Ω(G), a contradiction. Thus, u3, w1 and w2 do not have a common neighbor. In this case, let X′ = X′′ ∪{w1, w2} and consider the graph G′ = G -X′. Since u3, w1 and w2 do not have a common neighbor, the graph G′ is isolate-free, and so w(G′) = (w(G) -30) + 4 = w(G) -26. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices in the set {u2, v1, v3} (as indicated by the shaded vertices in Figure 51(c)), and so i(G) ≤i(G′) + 3. Since G′ has either two vertices of degree 1 and no vertex of degree 2 or one vertex of degree 1 and two vertices of degree 2 or four vertices of degree 2, we infer that b(G′) ≤1, and so by Claim 18, we have Θ(G′) ≤2. Hence, Ω(G′) = w(G′) + Θ(G′) ≤(w(G) -26) + 2 = w(G) -24. Thus, 8i(G) ≤8(i(G′) + 3) ≤w(G′) + 24 = w(G), a contradiction. (2) By Claim 31, the graph G′′ contains at most one vertex of degree 1. We now define Gi,j = G -(NG[ui] ∪NG[vj]) where i, j ∈[3] and i ̸= j. Claim 32 At least one of the subgraphs Gi,j where i, j ∈[3] and i ̸= j contains no isolated vertex. Proof. Suppose firstly that G′′ contains a vertex of degree 1, say w1. By symmetry, we may assume that w1 is a common neighbor of u1 and u2. Let w be the neighbor of u1 different from v1 and w1. Thus the graph shown in Figure 52(a) is a subgraph of G, where the graph G1,2 is the subgraph in the dashed box. If G1,2 contains an 45 G1,2 u3 u2 u1 v3 v2 v1 w1 x1 x2 w (a) x1 x2 u3 u2 u1 v3 v2 v1 w2 w1 w4 w3 w6 w5 (b) Figure 52: Subgraphs of G in the proof of Claim 32 isolated vertex v, then NG(v) = {w, w1, u2}. However, then, G[{v, u2, w1}] = K3, contradicting Claim 27. Hence, G1,2 contains no isolated vertex, and desired result of the claim follows. Suppose next that G′′ contains no vertex of degree 1. Thus, every vertex in G′′ is adjacent to at most one vertex that belongs to the set C = {u1, u2, u3}, implying that NG[ui] ∩NG[uj] = ∅for i, j ∈[3] and i ̸= j. Hence the graph shown in Figure 52(b) is a subgraph of G. In particular, we note that the vertices w1, w2, . . . , w6 are distinct. We show that at least one of the subgraphs Gi,j where i, j ∈[3] and i ̸= j contains no isolated vertex. Suppose, to the contrary, that Gi,j contains an isolated vertex for all i, j ∈[3] and i ̸= j. We consider the graph G1,2 (illustrated by the subgraph inside the dashed region in Figure 53). By supposition, G1,2 contains an isolated vertex. The only possible such isolated vertex is a neighbor of u2 that is adjacent to both w1 and w2. Renaming w3 and w4 if necessary, we may assume that w3 is isolated in G1,2, that is, NG(w3) = {u2, w1, w2}. We next consider the graph G1,3. By supposition, G1,3 contains an isolated vertex. The only possible such isolated vertex is a neighbor of u3 that is adjacent to both w1 and w2. Renaming w5 and w6 if necessary, we may assume that w5 is isolated in G1,3, that is, NG(w5) = {u3, w1, w2}. In particular, we note that C : w1w3w2w5w1 is a 4-cycle in G. However, this implies that the graph G2,1 does not contain an isolated vertex. Indeed, in G2,1, vertices w1, w2 and u3 have degree 1, the neighbors of w4 in G2,1 have degree 2 in G2,1, whereas other vertices of G2,1 have degree 3. This is a contradiction, proving the claim. (2) G′ = G1,2 u3 u2 u1 v3 v2 v1 x1 x2 w1 w2 Figure 53: A subgraph of G in the proof of Claim 32 By Claim 32, at least one of the subgraphs Gi,j where i, j ∈[3] and i ̸= j contains no isolated vertex. Renaming vertices if necessary, we may assume that G1,2 is isolate-free. For notational convenience, let G′ = G1,2 and let X1,2 = {u1, u2, v1, v2, x1, x2, w1, w2}, and so G′ = G -X1,2. Every i-set of G′ can be extended to an ID-set of G by adding to it the vertices u1 and v2 (as indicated by the shaded vertices in Figure 53), and so i(G) ≤i(G′) + 2. If b(G′) = 0, then by Claim 18, Θ(G′) = 0, and so Ω(G′) = w(G′) = w(G) -24 + 8 = w(G) -16 = Ω(G) -16. In this case, 8i(G) ≤8(i(G′) + 2) ≤w(G′) + 16 = Ω(G), a contradiction. Hence, b(G′) ≥1. Let B′ be a bad component in G′, and so B′ ∈B. We note that degG′(v3) = 1, and so G′ contains at least one vertex of degree 1. 46 Claim 33 B′ /∈B3. Proof. Suppose, to the contrary, that B′ ∈B3, and so B′ is as illustrated in Figure 49(c). In this case, the six X1,2exit edges from w1, w2 and u2 are all incident with vertices of degree 2 in B′. Let v denote the vertex of degree 3 in B′ that does not belong to a copy of K2,3. We now let X∗= V (B′) ∪X1,2 and we let G∗= G -X∗. We note that G∗contains one vertex of degree 1, namely v3, and all other vertices of G∗have degree 3. Hence, b(G∗) = 0, and so, by Claim 18, Θ(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G) -24 × 3 + 2 = w(G) -70. Every i-set of G∗can be extended to an ID-set of G by adding to it the seven vertices in the set N(v) ∪{u2, v1, w1, w2} (as indicated by the shaded vertices in Figure 54), and so i(G) ≤i(G∗) + 7. Thus, 8i(G) ≤8(i(G∗) + 7) ≤Ω(G∗) + 56 < Ω(G), a contradiction. (2) B′ X1,2 G∗ u3 u2 u1 v3 v2 v1 x1 x2 w1 w2 v Figure 54: A subgraph of G in the proof of Claim 33 Claim 34 B′ /∈B2. Proof. Suppose, to the contrary, that B′ ∈B2, and so B′ is as illustrated in Figure 49(b). In this case, five X1,2-exit edges from w1, w2 and u2 are all incident with vertices of degree 2 in B′. Let v denote the vertex of degree 2 in B′ that does not belong to a copy of K2,3. We now let X∗= V (B′) ∪X1,2 and we let G∗= G -X∗. We note that G∗ contains one vertex of degree 1, one vertex of degree 2, and all other vertices of G∗have degree 3. Hence, b(G∗) = 0, and so, by Claim 18, we have Θ(G∗) = 0. Thus, Ω(G∗) = w(G∗) = w(G) -19 × 3 + 3 = w(G) -54 = Ω(G) -54. We show that every i-set of G∗can be extended to an ID-set of G by adding to it at most six vertices. We will use the property given in Claim 25 that every K2,3-subgraph of G belongs to a G8.2-configuration that is an induced subgraph in G, where G8.2 is the graph in Figure 39(b). B′ X1,2 G∗ u3 u2 u1 v3 v2 v1 x1 x2 w1 w2 v (a) G∗ X1,2 B′ u3 u2 u1 v3 v2 v1 x1 x2 w1 w2 v (b) Figure 55: Subgraphs of G in the proof of Claim 34 Suppose firstly that u2 is adjacent to two vertices in B′. If u2 is not adjacent to v, then we let I′ consist of the three vertices of degree 2 in B′ that are not adjacent to u2, together with the vertices v1 and u2 (as indicated by 47 the shaded vertices in Figure 55(a)). If u2 is adjacent to v, then renaming w1 and w2 if necessary, we may assume that w2 is adjacent to two vertices in B′. In this case, we let I′ consist of the two neighbors of v in B′, the vertex of degree 2 in B′ adjacent to w1, together with the vertices in the set {u2, v1, w2} (as indicated by the shaded vertices in Figure 55(b)). In both cases, every i-set of G∗can be extended to an ID-set of G by adding to the vertices in the set I′, and so i(G) ≤i(G∗) + |I′| ≤i(G∗) + 6. Suppose next that u2 is adjacent to exactly one vertex in B′. Suppose that u2 is not adjacent to v. Renaming w1 and w2 if necessary, we may assume that w2 is adjacent to v. We now let I′ consist of the vertex v, the vertices of degree 2 in B′ adjacent to w2 and u2, and the vertices in the set {v1, v2, w1} (as indicated by the shaded vertices in Figure 56(a)). If u2 is adjacent to v, then we let I′ consist of the two neighbors of v in B′ together with the vertices in the set {v1, v2, w1, w2} (as indicated by the shaded vertices in Figure 56(b)). In both cases, every i-set of G∗ can be extended to an ID-set of G by adding to the vertices in the set I′, and so i(G) ≤i(G∗) + |I′| = i(G∗) + 6. u3 u2 u1 v3 v2 v1 x1 x2 w1 w2 v B′ X1,2 G∗ (a) G∗ X1,2 B′ u3 u2 u1 v3 v2 v1 x1 x2 w1 w2 v (b) Figure 56: Subgraphs of G in the proof of Claim 34 As observed earlier, Ω(G∗) = Ω(G)-54. We have shown that every i-set of G∗can be extended to an ID-set of G by adding to it at most six vertices from X∗, and so i(G) ≤i(G∗) + 6. Thus, 8i(G) ≤8(i(G∗) + 6) ≤Ω(G∗) + 48 < Ω(G), a contradiction. (2) By Claims 33 and 34, we have B′ ∈B1. Suppose that v3 ∈V (B′). Since v3 has degree 1 in B′, by Claim 29 such a component B′ ∈B1 is as illustrated in Figure 49(a). In this case, the graph G contains two vertex disjoint copies of K2,3 joined by the edge u3v3, contradicting Claim 28. Hence, v3 /∈V (B′). Let y1 and y2 be the two vertices of degree 3 in B′ that are not adjacent, and let N(y1) = N(y2) = {a, b, c}. Further, let a′ be the vertex of degree 1 in B′, where aa′ is an edge. Let b′ and c′ be the neighbors of b and c, respectively, that do not belong to B′. By Claim 25, the subgraph of G induced by the set V (B′) ∪{b′, c′} is a G8.2-configuration. In particular, we note that the set {a′, b′, c′} is an independent set. Since v3 /∈V (B′), we note that a′ ̸= v3. The two neighbors of a′ that are not in B′ belong to the set {u2, w1, w2}. Further, the vertices b′ and c′ belong to the set {u2, w1, w2}, that is, {b′, c′} ⊂{u2, w1, w2}. Since the vertex a′ is adjacent to two vertices that belong to the set {u2, w1, w2}, the vertex a′ is adjacent to at least one of b′ and c′, contradicting our earlier observation that {a′, b′, c′} is an independent set. This final contradiction completes our proof of Theorem 6. 2 Acknowledgments The first and the second author were supported by the Slovenian Research and Innovation agency (grants P1-0297, N1-0285, and N1-0431). Research of the third author was supported in part by the . 48 References [1] A. Akbari, S. Akbari, A. Doosthosseini, Z. Hadizadeh, M. A. Henning, and A. Naraghi, Independent domination in subcubic graphs. J. Comb. Optim. 43 (2022), 28-41. [2] C. Brause and M. A. Henning, Independent domination in bipartite cubic graphs. Graphs Combin. 35 (2019), 881-919. [3] E. K. Cho, I. Choi, H. Kwon, and B. Park, A tight bound for independent domination of cubic graphs without 4-cycles. J. Graph Theory 104 (2023), no. 2, 372-386. [4] E. K. Cho, I. Choi, and B. Park, On independent domination of regular graphs. J. Graph Theory 103 (2023), no. 1, 159-170. [5] E. K. Cho, J. Kim, M. Kim, and S. Oum, Independent domination of graphs with bounded maximum degree. J. Combin. Theory Ser. B 158 (2023), 341-352. [6] P. Dorbec, M. A. Henning, M. Montassier, and J. Southey, Independent domination in cubic graphs. J. Graph Theory 80 (2015), 329-349. [7] W. Goddard and M. A. Henning, Independent domination in graphs: A survey and recent results. Discrete Math. 313 (2013), 839-854. [8] W. Goddard and M. A. Henning, Independent domination in outerplanar graphs. Discrete Appl. Math. 325 (2023), 52-57. [9] W. Goddard, M. A. Henning, J. Lyle, and J. Southey, On the independent domination number of regular graphs. Annals Combin. 16 (2012), 719-732. [10] W. Goddard and J. Lyle, Independent dominating sets in triangle-free graphs. J. Combin. Optim. 23 (2012), 9-20. [11] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning (eds), Topics in Domination in Graphs. Series: Developments in Mathematics, Vol. 64, Springer, Cham, 2020. viii + 545 pp. [12] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning (eds), Structures of Domination in Graphs. Series: Developments in Mathematics, Vol. 66, Springer, Cham, 2021. viii + 536 pp. [13] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, Domination in Graphs: Core Concepts Series: Springer Monographs in Mathematics, Springer, Cham, 2023. xx + 644 pp. [14] M. A. Henning, C. L ̈owenstein, and D. Rautenbach, Independent domination in subcubic bipartite graphs of girth at least six. Discrete Applied Math. 162 (2014), 399-403. [15] M. Knor, R. ˇSkrekovski, and A. Tepeh, Domination versus independent domination in regular graphs. J. Graph Theory 98 (2021), 525-530. [16] K. Kuenzel and D. F. Rall, On independent domination in direct product. Graphs Combin. 39 (2023), Paper No. 7, 13 pp. [17] P. C. B. Lam, W. C. Shiu, and L. Sun, On independent domination number of regular graphs. Discrete Math. 202 (1999), 135-144. [18] J. Lyle, A note on independent sets in graphs with large minimum degree and small cliques. Electronic J. Combin. 21 (2014), Paper #P2.38, pp 1-17. [19] S. O and D. B. West, Cubic graphs with large ratio of independent domination mumber to domination number. Graphs Combin. 32 (2016), 773-776. [20] B. A. Reed, Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277-295. [21] L. Sun and J. Wang, An upper bound for the independent domination number. J. Combin. Theory Ser. B 76 (1999), 240-246. 49
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2510.14760
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AN EXTENSION OF KHOVANOV HOMOLOGY TO IMMERSED
SURFACE COBORDISMS
SCOTT CARTER, BENJAMIN COOPER, MIKHAIL KHOVANOV,
AND VYACHESLAV KRUSHKAL
Abstract. We show that an oriented surface in R4 containing double point sin-
gularities induces a map between the Khovanov homology groups of its boundary
links in a functorial way. As part of this work, the movie moves of Carter and Saito
are extended to surfaces with double points.
1. Introduction
From the beginning [Kho00, §1] it was understood that a smooth oriented surface
in R3 ˆ I should induce maps between Khovanov homology groups in a functorial
way.
One reason for this is that, up to a small perturbation, for a surface Σ Ă
R3 ˆ I the projection onto the time axis I induces a Morse decomposition of Σ
and assigning the maps from Khovanov’s construction to the pieces produces a chain
map between the chain complexes associated to the boundary links BΣ.
Several
years later M. Jacobsson showed that this assignment was independent of the isotopy
representative of the surface up to sign [Jac04]. He did this by checking the movie
moves of S. Carter and M. Saito [CS98a]. Since that time, simpler proofs have been
found [Kho06a, BN05] and there have been several approaches to the sign problem
[Bla10, CMW09, San21, Cap08].
Functoriality of link homology is crucial for its applications to four-dimensional
topology. More concretely, the functoriality of Khovanov homology paved the way for
J. Rasmussen’s s-invariant spKq P Z and his proof that it gives a lower bound on the
4-ball genus of a knot K [Ras10]. Recently, the functoriality of Khovanov homology
became an essential ingredient of the skein lasagna invariant and its applications to
4-manifold topology [MWW22, RW24].
Our purpose in this article is to introduce a new direction of study for the functori-
ality of Khovanov homology. We extend the type of surfaces which can induce maps
between Khovanov homology groups by considering smooth orientable surfaces in
R3 ˆ I with double point singularities. A double point singularity is locally modeled
Date: 15th October, 2025.
1
arXiv:2510.14760v1 [math.GT] 16 Oct 2025
on a transverse intersection of two planes in R4 or the affine variety
Z :“ tpw, zq : wz “ 0u Ă C2
(1.1)
in a neighborhood of the origin. The structure of Z near the singular point can be
described by a movie of 3-dimensional time slices of the form
(1.2)
and the link Z X tv : |v| “ ϵu of the double point singularity is a Hopf link in the
3-sphere. We use the Khovanov homology of this Hopf link to assign a map to a
smooth oriented surface with singularities in R3 ˆ I and prove that the homotopy
class of this chain map is independent of the isotopy class of singular surface. In
order to establish our result, we extend the known movie moves to include surfaces
with double point singularities and we check that our assignments satisfy these new
movie moves. Here is an informal statement of our work.
Theorem. The maps induced on the Khovanov homology of oriented links are well-
defined, up to an overall sign, on isotopy classes of surface cobordisms with double
points.
These maps give rise to a functor from the category of surface cobordisms with
double points between oriented links in R3 to the category of bigraded abelian groups
and homogeneous maps between them, considered up to overall minus sign.
Let us state the results contained in our paper with a little more care. There is a 2-
category Tang4 with objects given by finite collections S of ˘-oriented points in the
plane R2. For any two such collections, S and S1, there is a category Tang4pS, S1q
whose objects are oriented tangles T Ď R2ˆr0, 1s, viewed as cobordisms from Sˆt0u
to S1 ˆ t1u (and defined as a proper embedding of a compact oriented one-manifold
into R2ˆr0, 1s rather than an equivalence class of embeddings rel boundary isotopy).
A morphism f : T Ñ T 1 between tangles is a cobordism in 4-space. More carefully,
it is an isotopy (rel boundary) class of a surface with corners standardly embedded in
R2 ˆ r0, 1s2 which cobounds tangles T, T 1 in the two parallel boundary R2 ˆ r0, 1s’s
and has the standard boundary S ˆ r0, 1s and S1 ˆ r0, 1s on the two remaining
boundary R2 ˆ r0, 1s’s. There is an extension of this category
Tang4pS, S1q Ă Tang4
ˆpS, S1q
which has the same objects, but whose morphisms are isotopy classes of immersed
surfaces with the double point singularities locally modelled on pZ, 0q in Eqn. (1.1)
above.
2
The next theorem is a statement about isotopy of surfaces in 4-space. Recall that
if rfs : T Ñ T 1 is a morphism in Tang4pS, S1q then any two representatives f, f 1 P rfs
are related by a finite sequence of the Carter-Saito movie moves.
Theorem. There is an extension of the Carter-Saito movie moves from the setting
of Tang4pS, S1q to the setting of Tang4
ˆpS, S1q. More precisely, there is a finite set
of extended movie moves such that if rfs : T Ñ T 1 is an isotopy class of immersed
surface with double point singuarities in Tang4
ˆpS, S1q then any two representatives
f, f 1 P rfs are related by a finite sequence of these extended moves.
The extended movie moves are catalogued, with references to illustrations, by Thm.
4.1 in Section 4.
Our second theorem below uses this theorem to extend the functoriality of Kho-
vanov homology. For context, recall that Khovanov homology associates to a cobor-
dism f a chain map f˚. If f 1 P rfs then invariance under the Carter-Saito moves, up
to overall sign, implies that there is a chain homotopy f˚ » ˘f 1
˚. This is the key step
in showing that the Khovanov construction determines a 2-functor κ : Tang4
b Ñ pK,
where pK is the 2-category defined in [Kho06a], also denoted K5pCobV q{t˘1u in §2.1
below and throughout the paper, and see [BN05] for an alternative approach and a
generalization to the equivariant case. (In the 2-category pK, 2-morphisms are defined
up to overall sign.)
Here Tang4
b Ă Tang4 is a full 2-subcategory of Tang4 where the objects are
balanced collections S Ă R2, i.e., with the same number of positive and negative
points. Likewise one defines a full 2-subcategory Tang4
ˆ,b of Tang4
ˆ. Our extension of
κ from Tang4
b to Tang4
ˆ,b is defined by assigning maps to the double point cobordism
and checking that the extended Carter-Saito moves hold.
Theorem. Assigning maps A, B in Def.
3.1 to the double point cobordism in
Eqn. (1.2) determines an extension ˜κ: Tang4
ˆ,b Ñ pK of the Khovanov 2-functor
κ : Tang4
b Ñ pK from embedded balanced cobordisms Tang4
b to balanced cobordisms
with double points Tang4
ˆ,b.
Unlike the maps which are assigned to embedded cobordisms, the maps we asso-
ciate to surface cobordisms with double point singularities change the homological
degree. Another interesting aspect of our construction is that the maps associated
with positive and negative double points are different. These new features may be use-
ful for applications to topology. The study of surfaces in 4-manifolds, and particularly
the question of when double points can be removed by a homotopy, is fundamental
in 4-dimensional topology. Indeed, the failure of the Whitney trick in dimension 4
underlies the difference between smooth and topological 4-manifolds [Don83, Fre82],
3
see also [CG88, Kro97]. Developing a new tool to study surfaces with double points
was a motivation for this work.
A related construction appears in the recent work [ISST25], also see Remark 3.7.
Organization. §2 contains a brief review of Khovanov homology. §3 introduces
the maps assigned to double point singularities and checks that these maps satisfy
the movie moves up to homotopy. §4 contains a discussion of movie moves and shows
how to extend them to the setting of surfaces cobordisms with double points.
Acknowledgments. The second author would like to thank Kevin Walker for
a helpful conversation.
The third author was partially supported by NSF grant
DMS-2204033 and Simons Collaboration Award 994328 “New Structures in Low-
Dimensional Topology”. The fourth author was supported in part by NSF Grant
DMS-2405044.
2. Review of Khovanov homology
Here we review the construction of Khovanov homology and recall the duality
statements which will be used to check movie moves in §3.
The ideas here are
equivalent to those of references such as [Kho06a, BN05]. Compared to [CK12, §2.3],
the 2-category Cob is analogous to Pre-Cobpnq and CobV to Cobpnq. Set I :“ r0, 1s.
There is a 2-category Cob with
(1) Objects given by disjoint collections of points on a line S Ă I . We also use
the symbol S to denote its cardinality #S P Zě0.
(2) 1-morphisms D : S Ñ S1 are formally q-graded direct sums D “ ‘N
i“1qniDi
of 1-manifolds Di : S Ñ S1 in I2 which bound the 0-manifolds S and S1 in
the sense that S Ă I ˆ t1u, S1 Ă I ˆ t0u and BDi “ S Y S1 for 1 ď i ď n,
(3) A 2-morphism between 1-morphisms D, D1 : S Ñ S1 is a Z-linear combi-
nation of surfaces Σ : D Ñ D1 in I3. In more detail, if D “ ‘N
j“1qniDi
and D1 “ ‘M
i“1qmjD1
j then a map Σ is a M ˆ N matrix pDijq the entries
Dij “ ř
k akΣk
ij of which are Z-linear combinations of orientable surfaces
Σk
ij Ă I3 which bound the 1-manifolds Di and D1
j in the sense that the
boundary BΣk
ij decomposes as a union of Di, D1
j , S ˆ I and S1 ˆ I along the
occupied faces of the cube (Σk
ij X pt0, 1u ˆ I2q “ H).
Manifolds are considered up to isotopy and maps are composed along various axes
by gluing and rescaling. We briefly review the setup in order to orient the reader and
establish notation.
There is a product \ : Cob ˆ Cob Ñ Cob given by the disjoint union of cobordisms
along the first axis of I3. The unit with respect to the disjoint union is the empty
set H.
4
For any two sets of points S, S1, there is a category HompS, S1q with objects given
by 1-morphisms D : S Ñ S1 as in p2q above and maps given by 2-morphisms in the
sense of p3q above. The composition in Cob gives functors
b : HompS, S1q ˆ HompS1, S2q Ñ HompS, S2q.
In particular, if D : S Ñ S1 and E : S1 Ñ S2 are 1-manifolds then the composite
D b E : S Ñ S2 is illustrated below.
S
1
D
S1
1
2
E
S2
0
D
E
(2.1)
When S “ S1, the identity 1-manifold is 1S :“ S ˆ I .
If Σ : D Ñ D1 is a surface with D, D1 : S Ñ S1 then the expression
χpΣq ´ pS ` S1q{2 P Z
is preserved by gluing. This determines the q-grading: if Σ : qnD Ñ qmD1 is a map
between formally q-graded 1-manifolds then we set
degpΣq :“ χpΣq ´ pS ` S1q{2 ` m ´ n.
In this way, every 2-morphism is a sum of its q-homogeneous components.
We next review a duality lemma which is important in §3. For more information
see [CH15, §5.1] or compare [CMW09, §3.1].
If E : S Ñ S is an endomorphism then the trace TrpEq is the closed 1-manifold
given by the quotient of E which pairwise identifies top S-points with the bottom
S-points: pxi, 0q „ pxi, 1q, when S “ tx1 ă x2 ă ¨ ¨ ¨ ă xnu. The proposition below
describes how 2-morphisms can be written as traces.
Proposition 2.1. If E, F : S Ñ S1 are 1-manifold morphisms then there is a dual
E_ : S1 Ñ S and a natural isomorphism
HompE, Fq – q´pS`S1q{2HompH, TrpF b E_qq.
(2.2)
The duality functor satisfies pE_q_ – E and
pE ‘ Fq_ – E_ ‘ F _,
pE b Fq_ – F _ b E_,
1_
S “ 1S
and reverses the formal q-grading.
The idea is that there is an isotopy of any surface Σ : E Ñ F which pulls E
along the boundary of the box to the face occupied by F . This produces a closed
5
1-manifold consisting of F composed with a reflection E_ of E. Here is a picture
of the reflection.
Q_
Q
„
(2.3)
It is natural to apply a Frobenius algebra or 2-dimensional TQFT to the right
hand side of Eqn. (2.2). We use this observation to describe the Khovanov homology
2-category CobV in the next section.
2.1. Khovanov Homology. Bar-Natan and Khovanov 2-category CobV is obtained
from the 2-category Cob by using Eqn. (2.2) to require that
HompE, Fq – q´pS`S1q{2V b#π0pFbE_q.
(2.4)
The circle diagram S1 is isomorphic to V :“ qH˚pS2q, where H˚pS2q :“ ZrXs{pX2q
is the Frobenius algebra1 graded by the degree assignments |1|q :“ 0 and |X|q :“ ´2.
The coproduct is ∆p1q :“ Xb1`1bX and ∆pXq :“ XbX . The counit is ϵpXq :“ 1
and ϵp1q :“ 0.
By construction the duality map descends to CobV and the duality isomorphism
(2.2) continues to hold. In CobV , there are delooping isomorphisms
D \ S1 – qD ‘ q´1D
for any 1-manifold
D P Cob.
To each tangle τ , one can assign a chain complex Tτ P ChpCobV q using the rule
pictured below. For each positively or negatively oriented crossing σ˘, the chain
complex Tσ˘ is defined to be the complex C˘, which is the cone of the saddle
cobordism between two resolutions pictured below.2
:“
q1
q2
:“ q´2
q´1
(2.5)
The underlined 1-morphisms are placed in homological degree zero. The duality map
C_ extends to chain complexes by reversing the homological t-degree. Notice that
pC˘q_ – C¯. In particular, the dual ´_ on 1-manifolds determines an operation
on tangles which satisfies
Tτ _ – T _
τ .
When two tangles differ by an oriented Reidemeister move there is a degree zero
chain homotopy equivalence between the associated chain complexes. In this way
1Notice that H˚pS2q is a graded Frobenius algebra, and V is a free rank 1 graded module over
it, with |1|q “ 1 ‰ 0.
2We abuse the notation here and omit an extra decoration on the left-hand side of these equations
that would indicate the chain complex associated with a crossing.
6
there is an assignment κpτq :“ rTτs of oriented tangles to elements of the homotopy
category KbpCobV q of bounded chain complexes. This assignment extends to a 2-
functor κ : Tang4
b Ñ KbpCobV q up to sign. The 2-category KbpCobV q{t˘1u is denoted
pK in [Kho06a].
If pE, dEq and pF, dFq are two chain complexes in ChpCobV q then there is a chain
complex pHom˚pE, Fq, δq of maps from E to F . In degree ℓan element of this
chain complex is a sequence of maps tfi : Ei Ñ F i`ℓuiPZ. The differential is δf “
tdFfi ` p´1qℓ`1fi`1dEuiPZ.
Eqn. (2.2) extends degreewise along the Hom˚-construction with the functor ´_.
In particular, when E “ Tα and F “ Tβ are the chain complexes associated to
tangles α and β and, by using Eqn. (2.4), we obtain
Hom˚pTα, Tβq – q´pS`S1q{2Hom˚pH, TrpTβ b T _
α qq
– q´pS`S1q{2CKhpβα_q.
Here CKhpβα_q is the standard hypercube-shaped chain complex appearing in the
Khovanov construction. So we have the following lemma.
Lemma 2.2. For two oriented tangles α, β : S Ñ S1 from S points to S1 points,
there is a isomorphism on homology,
HpHom˚pTα, Tβq, δq – q´pS`S1q{2Khpβα_q.
(2.6)
The lemma above will be used in §3 to study movie moves.
We conclude with what, for us, is an important example: computing the mapping
space from one crossing C¯ to the opposite crossing C˘ results in a Hopf link.
Example 2.3. Notice that pC˘q_ – C¯. If H˘ is the Khovanov homology of the
Hopf link with linking number ˘1 then
HpHom˚pC¯, C˘qq – q´2TrpC˘ b pC¯q_q “ q´2H˘,
where
H` – Z0,0 ‘ Z0,2 ‘ Z2,4 ‘ Z2,6
(2.7)
H´ – Z0,0 ‘ Z0,´2 ‘ Z´2,´4 ‘ Z´2,´6.
(2.8)
Here the symbol Za,b represents a Z-summand in pt, qq-degree pa, bq. See Fig. 2
below for an illustration of the homologies H˘.
Remark 2.4. Instead of H˚pS2q in Eqn. (2.4) we could use the equivariant homology
H˚
Up2qpS2q :“ H˚
Up2qpptqrXs{pX2 ´ hX ´ tq
where
H˚
Up2qpptq :“ Zrh, ts
and we follow the grading convention |1|q “ 0 and |X|q “ ´2, so that |h|q “ ´2 and
|t|q “ ´4 make the quotient relation q-homogeneous. For this Frobenius algebra,
7
the counit is determined by setting ϵp1q :“ 0 and ϵpXq :“ 1. The coproduct is given
by ∆p1q :“ 1 b X ` X b 1 ` h1 b 1 and ∆pXq :“ X b X ` t1 b 1 [Kho06b, Eqn.
(5)]. And the unit is ιp1q :“ 1.
Let R :“ Zrh, ts with |h|q “ ´2 and |t|q “ ´4. Repeating §2.1 we assign the
R-module A :“ qH˚
Up2qpS2q to the circle S1 and produce a 2-category CobA in which
the duality theorem continues to hold. Moreover, since X2 “ hX `t, the 2-category
CobV is obtained from CobA by setting h “ 0 and t “ 0. If the same assignments
as Eqn.
(2.5) are used for positive and negative crossings then there is a tangle
2-functor κ1 : Tang4
b Ñ KbpCobAq{t˘1u, see [BN05] and [Kho06b, Prop. 6].
Now there are isomorphisms
A
αÕ
β qR ‘ q´1R
(2.9)
where
αpzq :“
`
ϵpXzq
ϵpzq
˘T
and βpzq :“
`
ιpzq
ιpXzq ´ hιpzq
˘
.
The relations αβ “ 1 and βα “ 1 imply corresponding delooping isomorphisms
among the associated surfaces in the 2-category CobA. By applying these delooping
isomorphisms and removing acyclic subcomplexes as in [BN07], since S1 is identified
with A, one computes the equivariant Hopf link homologies as
Hequiv
`
“ t0q1A ‘ t2q5A
Hequiv
´
“ t´2q´5A ‘ t0q´1A.
In particular, Eqn. (2.9) shows that there is a non-canonical isomorphism of the form
Hequiv
˘
– H˘ bZ R. See also Remark 3.12.
3. Extension of Khovanov homology to double points
In order to assign a map to a singular cobordism between oriented links it suffices
to assign maps to Morse singularities, Reidemeister moves and the double points
corresponding to the passage from a positive crossing to a negative crossing and vice
versa, see Eqn. (1.2).
Definition 3.1. For the Morse singularities and Reidemeister moves we will use
the same assignments as the existing theory and the maps A : C` Ñ C´ and
B : C´ Ñ C` which are pictured in Fig. 1 for the double points. On the right-hand
side, the B map consists of the identity map between the two components of crossing
complexes which are not in degree zero. It has homological degree 2 (also called the
t-degree). On the left-hand side, the A map is the alternating sum of dots on the two
sheets corresponding to components of crossing complexes which are not in degree
zero. A dot can be understood as half of a handle, see [CK12, §2.3]. The map A has
homological degree ´2.
8
A
B
Figure 1. The definition of the chain maps A, B assigned to double points
It is straightforward to check the proposition below.
Proposition 3.2. The assignments A and B are chain maps which represent homol-
ogy classes in the chain complexes Hom˚pC`, C´q and Hom˚pC´, C`q corresponding
to the classes in pt, qq-degree |A|t,q “ p´2, ´6q and |B|t,q “ p2, 4q of the (q-shifted)
Hopf link homologies q´2H´ and q´2H` respectively in Ex. 2.3.
Remark 3.3. The surface Σ produced by the Seifert algorithm from the 2-crossing
planar projection of the Hopf link consists of two disks connected by two half-twisted
1-handles. Since this surface has Euler characteristic zero, Σ determines maps Σ˚ :
Z Ñ H˘ of pt, qq-degree p0, 0q. The image of this map is spanned by a generator of
Z0,0 in H´ and twice a generator of Z0,0 in H`.
The results in this paper imply that the class Z´2,´4 in H´ corresponds to the
image of the singular cobordism which is dual to the A map. Decorating cobordisms
with dots corresponds to the action of X in Fig. 2. So, abusing notation slightly,
the entire homology
H´ – xΣ, Ay ‘ XxΣ, Ay
(3.1)
can be understood in terms of cobordisms. Notice that XΣ is a generator of the
negative Hopf link homology H´, but this cannot be true for the positive Hopf link
H` because the q-degree satisfies |XΣ˚p1q|q “ ´2 and the homology H` vanishes
in negative q-degree, it follows that XΣ “ 0 in H`.
9
H`
t
q
0
1
2
0
2
4
6
B
H´
t
q
´2 ´1
0
´6
´4
´2
0
A
Figure 2. The grid shows the homology of the Hopf links H˘ from
Ex. 2.3. As in Prop. 3.2 the generators labelled A and B correspond
to the maps A and B under the duality isomorphism. A hollow circle
is a Z-summand which is not in the image of the operation X (deter-
mined by the Frobenius algebra). A filled circle is in the image of X .
Multiplication by X at a component of a link has q-degree ´2.
Remark 3.4. Determining which classes in the Khovanov homology of a link can be
represented geometrically in the manner of Eqn. (3.1) is understanding the question
of how much of Khovanov homology is cobordism generated in the sense of topological
quantum field theory. This was the original motivation for the authors. Functoriality
of Khovanov homology implies that for a smooth oriented surface pΣ, BΣq Ă pR4, R3q
bounding a link BΣ, there is a map
Σ˚ : Z Ñ Kh0,χpΣqpBΣq,
see [Kho00, §6.3]. Since the homological t-degree of the right-hand side must be
zero and, at least phenomenologically, Khovanov homology is concentrated along a
diagonal in the pt, qq-plane [Kho03], one should expect very little of the homology
to be geometric. The extension Tang4
b Ă Tang4
ˆ,b considered in this paper can be
considered the simplest non-trivial extension of the theory along homology classes
which are not represented geometrically. In our extension, it is no longer necessary
for geometric homology classes to have t-degree 0. It seems interesting to ask, which
classes in Khi,jpLq are now geometric? Answers to this question may help to shed
light on the nature of Khovanov homology as a tool for the study of low-dimensional
topology.
Remark 3.5. In the skein lasagna module theory [MWW22] one considers a disjoint
collection of input 4-balls D4 in a 4-manifold M and surfaces Σ Ă M with bound-
ary in BD4. Additionally, these skein lasagna fillings are decorated with Khovanov-
Rozansky homology classes of the links BΣ in the 3-spheres BD4. The results of
this paper may be interpreted as considering input balls with Hopf links H˘ in their
boundary, decorated with generating classes in cohomological degrees ˘2.
10
Remark 3.6. L. Weng introduced assignments for framed singular cobordisms
[Wen11]. Interpreting these assignments in the oriented setting, the map c1 : C` Ñ
C´ in [Wen11] has t-degree zero, sending the 0-resolution of C` isomorphically onto
the 1-resolution of C´, and sending the 1-resolution to 0. The map c2 : C´ Ñ C`
in [Wen11] is the map B above.
Remark 3.7. While preparing this paper for publication, we learned from T. Sano
about a related extension of tangle cobordism invariants to double point singularities,
by H. Imori, T. Sano, K. Sato, and M. Taniguchi [ISST25].
Remark 3.8. In a series of papers [IY21, Yos20, IY23] N. Ito and J. Yoshida intro-
duce and study homology of singular links, by defining the complex for a singular
crossing to be the cone of the map A above. One of their goals is to categorify Vas-
siliev invariants of links. In particular, in the latest paper [IY23], the authors show
the categorical analogue of the 4T relation on weight spaces, for the singular link
Khovanov homology complexes built via their construction. Our use of the map A,
to extend homology from embedded to singular embedded surfaces, is different from
theirs.
3.1. Checking the new movie moves. As mentioned in the introduction, Kho-
vanov homology is known to satisfy the movie moves for smooth oriented surfaces
up to sign. In order to show that the chain homotopy class of a map assigned to a
surface with double points is independent of isotopy we must check the new movie
moves which contain double point singularities.
The new movie moves involving double points are enumerated by Theorem 4.1
in Section 4. Movies MM16 and MM17 are the only ones which require non-trivial
verifications. The lemmas below are introduced in order to simplify exposition later.
Lemma 3.9 gives criteria in which a diagram of chain complexes commutes up to
homotopy (and sign).
Lemma 3.9. Fix a pt, qq-bidegree ‹ and suppose that we are given the (not-
necessarily commutative) diagram of chain complexes:
W
X
Y
Z.
α
γ
β
η
Then either set of conditions (1) or (2) below imply that this diagram commutes up
to homotopy and sign.
(1) (a) α, β are degree zero homotopy equivalences
(b) H‹pHom˚pX, Zqq – 0 or Z
11
(c) rηs and rγs generate H‹pHom˚pX, Zqq and H‹pHom˚pY, Zqq respectively
(2) (a) α, γ are degree zero homotopy equivalences
(b) H‹pHom˚pW, Y qq – 0 or Z
(c) rβs and rηs generate H‹pHom˚pW, Y qq and H‹pHom˚pX, Zqq respec-
tively
Remark 3.10. In each case, condition pbq and condition paq combine to imply that
other Hom-complexes have homology which is isomorphic to 0 or Z.
Proof. For (1):
Consider
H‹pHom˚pY, Zqq
H‹pHom˚pW, Zqq
H‹pHom˚pX, Zqq
β˚
α˚
γ : Y Ñ Z
γβ : W Ñ Z
ηα : W Ñ Z
η : X Ñ Z
Since α and β are degree zero homotopy equivalences, the maps α˚ and β˚ are
isomorphisms between the homology groups pictured above. There are two cases,
If H‹pHom˚pX, Zqq – Z then H‹pHom˚pW, Zqq – Z via α˚ and H‹pHom˚pY, Zqq –
Z via pβ˚q´1α˚. By assumption rηs generates H‹pHom˚pX, Zqq and rγs generates
H‹pHom˚pY, Zqq. Since AutpZq “ t˘1Zu, α˚prηsq “ ˘β˚prγsq or ηα » ˘γβ.
If H‹pHom˚pX, Zqq – 0 then the same argument shows that all of the homology
groups are zero and ηα » γβ.
For (2):
Same proof as (1) after replacing β˚ and α˚ with γ˚ and α˚.
□
Recall that if β P Brn is a braid then, given any orientation of the strands, using
the chain complexes associated to the crossings in Eqn. (2.5), gives a chain complex
Tβ :“ C˘
i1 b C˘
i2 b ¨ ¨ ¨ b C˘
in
where
β “ σ˘
i1σ˘
i2 ¨ ¨ ¨ σ˘
in
associated to the braid β as well as an inverse chain complex T ´1
β
:“ Tβ´1 . This is
an inverse in the sense that there are canonical homotopy equivalences of the form
Tβ b T ´1
β
» 1n
and
T ´1
β
b Tβ » 1n.
(3.2)
In other words, Tβ is invertible in the homotopy category KbpCobV q.
Recall for (2) below that V :“ qH˚pS2q is associated to the circle S1 in CobV .
Lemma 3.11. Let W and X be chain complexes in CobV .
Then for any braid
β P Brn there are homotopy equivalences of mapping complexes:
12
(1) rβ : Hom˚pW, Xq
„ÝÑ Hom˚pW b Tβ, X b Tβq and ℓβ : Hom˚pW, Xq
„ÝÑ
Hom˚pTβ b W, Tβ b Xq
(2) Hom˚pW \ 1, X \ 1q „ÝÑ Hom˚pW, Xq b V .
Proof. For (1): The map rβpfq :“ f b 1Tβ has a homotopy inverse bβpgq :“ g b 1T ´1
β
because composing gives
Hom˚pW, Xq
rβ
ÝÑ Hom˚pW b Tβ, X b Tβq
bβ
ÝÑ Hom˚pW b Tβ b T ´1
β , X b Tβ b T ´1
β q
and the homotopy equivalences in Eqn. (3.2) induce a natural equivalence between
the righthand side and the lefthand side. The argument is parallel for the map ℓβ .
For (2): Any interaction between a cobordism with a disjoint sheet can be disen-
tangled using delooping isomorphism. Alternatively, this follows immediately from
Lemma 2.2.
□
We are now prepared to discuss the movie moves listed in Theorem 4.1 in Section 4.
Proof. (MM16) Consider movie move #16: passing a node over a type-II move.
For the move pictured above, each strand can be oriented, either to point up or to
point down, so there are four oriented versions of this move. They involve either a
positive or negative double point, resulting in one of the maps A, B. Let’s consider
the case of both strands pointing up or both strands pointing down; in this case the
diagram can be written as
1 b 1
C` b C´
C´ b C`
C` b C`.
R2
B b 1
R2
1 b B
Now using part (1) of Lem. 3.9 with pt, qq-bidegree ‹ :“ |B|t,q “ p2, 4q we verify the
assumptions
(a) both maps α and β are induced by Reidemeister 2 moves which are degree
zero homotopy equivalences.
13
(b) the duality lemma (2.2) gives
H‹pHom˚pC` b C´, C` b C`qq – q´2H` – Z
where H` is the positive Hopf link homology, see Ex. 2.3.
(c) Lem. 3.11 tells us that there is a homotopy equivalence rσ such that rσpBq “
B b 1. It follows that
H‹Hom˚pC` b C´, C` b C`q – Zx1 b By.
In the same way, the isomorphism ℓσ´1pBq “ 1bB shows that the map 1bB
generates.
For the other two orientation choices (one strand in MM16 pointing up, the other
one pointing down) the proof involves the map A and is analogous. In more detail,
the diagram reads
1 b 1
C´ b C`
C` b C´
C´ b C´.
R2
A b 1
R2
1 b A
The proof proceeds as in the previous case, using Lem. 3.9 and pt, qq-bidegree ‹ :“
|A|t,q “ p´2, ´6q corresponding to a generator of q´2H´.
□
Proof. (MM17) Consider movie move #17: passing a node through a type-III move,
For the move pictured above, each strand can be oriented in either direction, giving
a total of eight cases. As in the previous proof, the map A is involved in half of the
cases and the map B in the other half. When all of the strands are pointing upward
we get the diagram below.
C´
1 b T
T b C´
2
C`
1 b T
T b C`
2
R3
R3
B b 1
1 b B
,
(3.3)
where T :“ C`
2 b C`
1 . Without orientations, this is the same as the braids pictured
below.
14
B b 1
R3
R3
1 b B
Now using part (2) of Lem. 3.9 with pt, qq-bidegree ‹ :“ |B|t,q “ p2, 4q we verify
the assumptions
(a) the maps α and γ are induced by Reidemeister 3 moves, so they are degree
zero homotopy equivalences.
(b) the duality Eqn. (2.6) identifies the horizontal mapping space with the trace
of a braid that is equivalent to the positive Hopf link and an unknot H` \S1.
The trace described here is pictured below.
Hom
˜
,
¸
– CKh
Here is a line-by-line proof:
Hom˚pC´
1 b T, C`
1 b Tq “ Hom˚pC´
1 b C`
2 b C`
1 , C`
1 b C`
2 b C`
1 q
– q´3TrppC´
1 b C`
2 b C`
1 q_ b C`
1 b C`
2 b C`
1 q
– q´3TrpC´
1 b C´
2 b C`
1 b C`
1 b C`
2 b C`
1 q
» q´3TrpC´
1 b C´
2 b C`
1 b pC`
1 b C`
2 b C`
1 qq
» q´3TrpC´
1 b C´
2 b pC`
1 b C`
2 b C`
1 q b C`
2 q
» q´3TrpC´
1 b pC´
2 b C`
2 q b C`
1 b C`
2 b C`
2 q
» q´3TrppC´
1 b 1 b C`
1 q b C`
2 b C`
2 q
» q´3TrpC`
2 b C`
2 q
– q´3pq ` q´1qH`.
15
The additional unknot S1 contributes the factor q ` q´1 to the Hom-space.
This is a consequence of computing the trace of pσ2q2 P Br3 as a braid of
index 3 and the delooping isomorphism, see the illustration above. Using the
homology H` from Example 2.3 shows that
H‹Hom˚pC´
1 b T, C`
1 b Tq – Z
(c) By Lemma 3.11 there are isomorphisms rβ and ℓβ , with β “ σ2σ1, taking the
map B to B b1 and 1bB. This shows that these maps generate homologies
of their respective mapping spaces in degree ‹.
This completes the argument for half of the orientations. For the other half, the
top and bottom maps in diagram (3.3) are replaced with Ab1 and 1bA respectively,
and the proof is completed using the pt, qq-bidegree ‹ :“ |A|t,q “ p´2, ´6q and the
link H´ \ S1.
Just like there are several version of the third Reidemeister move, there are several
versions of the movie move MM17. The proof for other versions is directly analogous
to the one given above.
□
Proof. (MM18) In the context of the Khovanov construction, the horizontal arrows
are identity maps so the diagram commutes.
□
Proof. (Thm. 4.1 (5)) Movie moves involving far-commutativity commute because
the maps A and B are applied to the same diagrams after planar isotopies which
induce identity maps in the setting of CobV .
□
Remark 3.12. Here are two observations about gradings in the equivariant setting
CobA of Remark 2.4.
(1) The only monomial hitj in the ground ring R with non-negative q-degree is
the identity element 1 P R in q-degree 0.
(2) The maps A and B as elements of q´2H´ and q´2H` generate the class of
largest q-degree within their respective t-degrees.
Together these imply that the maps A and B are the only generating classes within
their respective pt, qq-degrees of the Hom-complexes associated to the homologies
q´2Hequiv
˘
– q´2H˘ b R in CobA. So there are isomorphisms:
H2,4pHom˚pC`, C´qq – ZxBy
and
H´2,´6pHom˚pC´, C`qq – ZxAy.
These equations and the two observations suffice to amend the arguments above for
the CobA theory. We conclude that there is a corresponding extension of the 2-functor
κ1 : Tang4
b Ñ KbpCobAq{t˘1u to a 2-functor ˜κ1 : Tang4
ˆ Ñ KbpCobAq{t˘1u which
assigns the maps A and B to double point singularities.
16
4. Movie moves for immersed surface cobordisms
The Carter-Saito movie moves [CS98b, 2.6], [CRS97] provide a combinatorial de-
scription of isotopies of surfaces embedded in 4-space. The movies involve planar
projections of links in R3 ˆ tsu Ă R3 ˆ R “ R4 which are cross-sections of the
surface. We refer to these moves according to their enumeration MM1 - MM15, cf.
[BN05, Figures 11-13]. Movie moves for foams were also studied in [QW22]. Those
authors complete the list that was proposed in [Car15]. In addition, [BDMS] study
movie moves in the presence of symmetries on the knotted surfaces.
In this section we formulate and prove an extension of the movie moves to immersed
surfaces in R4. To set up the notation, consider a properly immersed oriented com-
pact surface F in R3 ˆ r0, 1s and proper isotopies thereof. An immersion is of the
form:
pF; B0F \ B1Fq í pR3 ˆ r0, 1s; R3 ˆ t0u \ R3 ˆ t1uq,
where one or both of B0F, B1F may be empty.
As discussed in the introduction, the singularities of such surfaces consist of double
points, which we will also refer to as nodes. There are finitely many nodes, and
they are in the interior of the surface. The boundaries B0F, B1F are classical links
embedded in R3 ˆ tju, j “ 0, 1. Our goal is to analyze isotopies between surfaces
with nodes, i.e. the restriction to F of ambient proper isotopies of R3 ˆ r0, 1s. In
particular, the nodes stay in the interior of R3 ˆ r0, 1s and their number remains
fixed during an isotopy.
The tool that will be used in our analysis is a (retinal) chart, a certain graph
which arises from a planar projection of F , see [CS98b, 1.5], [CRS97, Section 3.2].
The charts of properly isotopic embedded surfaces are related by moves discussed in
[CS98b, Theorem 2.17]. We caution that the choice of vertical and horizontal axes in
our diagrams differs from that in [CS98b], and our charts have additional decorations
which we describe in Section 4.2.
The structure of charts of surfaces embedded in 4-space and the moves on charts
encoding isotopies of such surfaces were deduced in [CRS97, Section 4] from the
analysis of singularities of generic maps of surfaces into R2 in [Gor91, Rie96, Wes95].
The same type of analysis applies in our context, where a generic map of a surface
into R2 is obtained starting from a surface with double points, rather than from an
embedded surface in R4, and projecting onto a plane (see Section 4.2).
We start by setting up the notation for encoding the links arising as the cross-
sections of F .
Convention and terminology. The interval factor of R3 ˆr0, 1s is parametrized
by the variable s. The coordinates of R3 are denoted x, y, z, and the crossings of a
link in R3 are defined with respect to the z coordinate, that is links are projected onto
17
the xy-plane. The y-coordinate will serve as the height function in the xy-plane.
The terms type-I, II, or III will refer to Reidemeister moves of a given type.
4.1. Encoding the cross-sections of a surface. Consider an immersion F í R3ˆ
r0, 1s which is in general position with respect to the projection R3 ˆ r0, 1s Ñ r0, 1s.
In more detail, the critical points of the composition
F í R3 ˆ r0, 1s
p
ÝÑ r0, 1s
are non-degenerate, have distinct values in r0, 1s, and are disjoint from the double
points of the immersion.
Given s P r0, 1s, the intersection F X pR3 ˆ tsuq is called a still of a movie. The
surface F will be assumed to be oriented, and the stills are all oriented accordingly
as well. Consider the projection onto the first factor, R3 ˆ r0, 1s Ñ R3, and a further
projection π: R3 Ñ R2 onto the xy-plane.
We can pick a sequence of generic values tsiu of the parameter s so that the
projections πpF X pR3 ˆtsiuqq and πpF X pR3 ˆtsi`1uqq of two successive stills differ
by a birth, death, saddle, Reidemeister move, ψ-move (illustrated in Figure 15),
crossing change (node), cusp, or critical exchange (exchange of the heights of critical
points).
The projection of a still onto the xy-plane is an immersed curve that has isolated
transverse double points. The critical points with respect to the height function y
of the projection of the still have distinct y-coordinates, and these are distinct from
the y-coordinates of the crossing points which also occur at distinct levels.
The bookmark code is a way to combinatorially encode the projection onto the xy-
plane of a still of a movie; a similar method will be used to label the edges of the
chart graph defined in Section 4.2. To put this in the context of our goal, Theorem
4.1 below, the isotopies of immersed surfaces will first be encoded using moves on
charts which are then translated to movie moves on stills using the bookmark code.
Figure 3 indicates how the projection of an oriented link can be written in terms
of symbols corresponding to crossings and to optima (maxima and minima). The
fonts
Ś
,
Ś
,
Ş,
Ť are adorned with dots at the NE, SE, SW, or NW directions to
indicate that directional arrows, corresponding to the link orientation, emerge from
such points. The projection F X pR3 ˆ tsuq Ñ R onto the y-coordinate is a Morse
function for the link. Since the crossings are assumed to lie at different vertical pyq
levels, there is a bookmarked word that can be used to describe a knot diagram. The
word is constructed from top to bottom.
18
X.
.
X
..
X..
X.
.
X
X
X
X.
.
..
..
.
.
1
1
1
1
2
2
2
2
U
U
U
U.
.
.
.
m1(-)
M1(-)
M2(+)
m2(+)
Crossings
Optima
Vertical
Figure 3. Types of crossings and critical points (optima) of links, and
the corresponding bookmark labels
Each symbol is labeled with a pair of integers that indicate the number of vertical
segments to the left and the number of vertical segments to the right of a given
crossing or critical point. In general, the bookmarked word always starts with
Şp0, 0q
and ends with
Ťp0, 0q. Compare this with the abstract tensor notation that can be
used to describe quantum invariants, cf. [Kau13, BK01].
19
Base point
y
x
z
Figure 4. Connected sum of 41#p´41q
For example, the bookmark word for the knot in Figure 4 is
4.2. Definition and properties of charts. A (retinal) chart is an oriented labeled
graph contained in the ps, yq-plane.
The structure of a chart is a Cerf-theoretic
description, with additional decorations, of an immersed surface cobordism between
classical links. We stress that there is a chosen preferred direction, y, in the plane,
and the chart is defined with respect to this choice. In the following subsections we
define its edges, edge decorations, and the types of vertices, see [CS98b, 1.5], [CRS97,
Section 3.2] for more details.
4.2.1. The edges. The chart of a surface F í R3 ˆ r0, 1s has two types of edges,
corresponding to folds and to crossings as described next. The edges of the graph
may cross. Such crossings are among the vertices of the chart.
According to singularity theory, a generic map from a surface to R2 has fold sin-
gularities. These form a 1-dimensional set upon which the rank of the map drops by
20
1. At cusps the rank of the projection map drops to 0. The images of the folds in the
ps, yq-plane form one type of edge of a chart. The second type of edge corresponds
to crossings with respect to the z coordinate: consider the double point arcs of the
projection of F to the 3-dimensional space with the px, y, sq-coordinates, and further
project them to the ps, yq-plane.
To relate the charts to the stills F XpR3ˆtsuq, note that for generic values of s, the
intersection of the vertical line tsu ˆ R in the ps, yq-plane with the chart consists of
a finite collection of points corresponding to critical points of the link F X pR3 ˆ tsuq
with respect to the y coordinate, and to the crossings of the link F XpR3 ˆtsuq. The
y coordinates of these points in the plane are exactly the same as the y coordinates
of these critical points, respectively crossings, of the link.
4.2.2. Edge decorations. The fold edges and the crossing (or double point) edges are
labeled by the corresponding symbols
Ś
,
Ś
,
Ş,
Ť, and additionally they have a
pair of non-negative integer labels. The first integer indicates the number of surface
sheets that are behind the line of sight or to the left in the x-direction of it, and the
second indicates the number of sheets that occlude the fold in the line of sight or to
the right of the fold or crossing in the x-direction. This pair of integers agrees with
the bookmark code of a cross-sectional still. Folds are also decorated with a short
upward or downward pointing arc that indicates the side of the fold at which the
surface overlaps. In this way, saddles and optima can be distinguished in the chart.
Specifically, at a birth or death these vertical arcs both point inward; at saddles they
point outward.
The intersection of a vertical line segment, s “ si, with the chart, that does not
pass through any vertices of the chart, results in the bookmark code for the link
F X pR3 ˆ tsiuq.
4.2.3. The vertices. Consider the critical points (births, deaths, saddle points) of the
projection onto the s-axis
F í R3 ˆ r0, 1s
p
ÝÑ r0, 1s.
For terminological precision, these critical points and the Reidemeister moves, cusps,
ψ-moves, and crossing changes (nodes) will be included among the critical events.
Critical events are projected onto the ps, yq-plane, and they represent vertices in the
chart that correspond to changes in the bookmark code.
(1) The critical points of the folds — births, deaths, and saddles — are valence
two vertices of the chart graph. They are Morse singularities with respect to
the s-direction, Figures 5, 6. The orientations of the folds at these junctures
are inconsistent.
21
Movie
Chart
+(i,j)
+(i,j)
-(i,j)
-(i,j)
U
U
U
U
.
.
.
.
Movie
+(i,j)
+(i,j)
-(i,j)
-(i,j)
U
U
U
U
Chart
Birth
Death
s
y
s
y
Figure 5. Births and deaths
s
y
Movie
Movie
+(i,j)
+(i,j)
-
-(i,j)
U
U
U
U
+(i,j)
+(i,j)
-(i,j)
-(i,j)
U
U
U
U
Chart
Chart
.(i,j)
.
.
.
.
.
.
.
s
y
Figure 6. Saddles
(2) Cusps are also valence two vertices at which folds are created or terminated.
Both a green and a brown edge are incident at a cusp. The orientations of the
folds are consistent at the cusp. There are eight types of cusps that depend
upon directions, orientations and bookmarks. Two are illustrated, Figure 7.
The reader is encouraged to catalogue the remaining cases. See also [CK21].
22
Figure 7. Examples of cusps and their corresponding chart vertices
(3) Valence three vertices that correspond to the branch points created by type-I
moves, are the junction of a green fold, a brown fold, and a crossing.
In
drawing the graph, the fold set seems to run straight through at this vertex.
However, the orientation of the folds is inconsistent at a branch point. Branch
points in the chart correspond to Reidemeister type-I moves in a movie pre-
sentation, Figure 8.
U
+ (i,j)
.
U
-(i,j)
.
X(i,j)
. .
U+(i,j)
.
-
U (i,j)
.
X (i,j)
. .
Figure 8. Two examples of type-I chart vertices
(4) Valence two vertices arising from type-II moves correspond to critical points
of the crossing points. The incident crossings are colored red and blue, and
the orientation flows through the vertex, Figure 9.
X (i,j)
X (i,j)
. .
. .
X (i,j)
X (i,j)
. .
. .
Figure 9. Two examples of type-II chart vertices
23
(5) Valence four vertices that indicate a crossing point passing over a fold involve
two crossings and two folds. Both pairs of edges are oriented consistently.
The bookmark codes change along the four incident edges, Figure 10.
-(i+1,j)
.U
U-(i,j+1)
.
X(i,j+1)
. .
X (i+1,j)
. .
U
-(i+1,j)
.
U
-(i,j+1)
.
X (i,j+1)
. .
X (i+1,j)
. .
Figure 10. Two examples of ψ-move chart vertices
(6) Valence six vertices correspond to triple points when surface F is projected
into the px, y, sq 3-space. Recall that the z-coordinate is used for the crossing
sense of the stills in a movie. These vertices correspond to Reidemeister type-
III moves.
X (i+1,j)
X (i,j+1)
X (i+1,j)
X (i+1,j)
X (i,j+1)
X (i,j+1)
X (i+1,j)
X (i,j+1)
X (i+1,j)
X (i+1,j)
X (i,j+1)
X (i,j+1)
. .
. .
. .
. .
..
..
. .
. .
....
..
..
Figure 11. Two examples of type-III chart vertices
(7) There are valence two vertices that correspond to nodes. A source and a sink
are indicated.
1
2 X (i,j) X (i,j)
.
..
.
1
2
X (i,j)
. .
X (i,j)
. .
Figure 12. A source node (left) and a sink (right)
24
Note that double point curves and folds that have disparate bookmark codes may
also cross. To understand them, think, for example, of the braid relation σiσj “ σjσi
for 1 ă |i ´ j|. See Figure 16.
4.3. The movie move theorem. We are in a position to formulate the main result
of this section. The figures following the statement of the theorem include both the
chart moves and the movie moves.
Theorem 4.1. The movie moves that describe isotopies of immersed surfaces are of
the following types:
(1) MM1-MM15: the Carter-Saito movie moves for embedded surfaces [CS98b,
2.6]
(2) MM16: passing a node over a type-II move, Figure 13,
(3) MM17: passing a node through a type-III move, Figure 14,
(4) MM18: passing a node over a fold, Figure 15,
(5) moves that correspond to sliding a node along a horizontal segment of a double
point curve; see for example, Figures 16-18.
In this theorem, the new moves that involve nodes of immersed surfaces are those
of types (2)-(5).
There are variations of each of these moves that depend, for example, upon differing
orientations, directions of the type-II moves, or whether the fold involved is a local
maximum or minimum. The reader is encouraged to investigate these variations. We
caution that nodes may not pass through the bottom or top arc at a type-III move.
We have labeled the move in Fig. 16 as MM19 since, in this case, the node interacts
directly with a vertex of the chart even though the vertex is an interchange of distant
crossings. Both Figures 17 and 18 demonstrate that a node commutes with a saddle
point in the fold set. There are other, analogous types of commutation relations of
this type: a node commutes with critical points of the fold set which could be a birth,
death, or a cusp. A node also commutes with a type I, II, or III move if the double
point curve upon which the node sits is not involved in the Reidemeister move. As
explained in the proof below, all these cases can be understood systematically as the
horizontal coordinate of the node interchanging with one of these chart vertices.
25
X(i,j)
X(i,j)
X(i,j)
X(i,j)
X(i,j)
X(i,j)
Figure 13. MM16: passing a node over a type-II move.
X (i,j+1)
X (i,j+1)
X (i+1,j)
X (i+1,j)
X (i,j+1)
X (i+1,j)
X (i,j+1)
X (i+1,j)
X (i,j+1)
X (i+1,j)
X (i+1,j)
X (i,j+1)
X (i+1,j)
X (i,j+1)
Figure 14. MM17: moving a node through a triple point.
U
-(i+1,j)
.
U
-(i,j+1)
.
X (i,j+1)
. .
X (i+1,j)
. .
U
-(i+1,j)
.
U
-(i,j+1)
.
X (i,j+1)
X (i+1,j)
. .
X (i,j+1)
. .
. .
X (i+1,j)
. .
Figure 15. MM18: moving a node over a fold.
26
...
...
...
i
j
k
...
...
i
j
k
...
...
...
i
j
k
...
...
...
i
j
k
...
...
...
i
j
k
...
...
...
i
j
k
X (i+j,k)
. .
X (i+j,k)
. .
X (i,j+k+2)
. .
X (i,j+k+2)
. .
X (i+j,k)
. .
X (i+j,k)
. .
X (i,j+k+2)
. .
X (i,j+k+2)
. .
...
Figure 16. MM19: Interchanging a node and a distant crossing.
UU
.
.
(0,1)
(0,1)
X(1,0)
..
X (1,0)
.
.
UU
.
.
(0,1)
(0,1)
X(1,0)
..
X (1,0)
.
.
Figure 17. Commutation of a critical point and a node, version 1.
U
U
X(i,j)
.
.
(i,j)
(i,j)
..
U.(i,j)
U.
(i,j)
X(i,j)
..
.X (i,j)
.
Figure 18. Commutation of a critical point and a node, version 2.
Proof of Theorem 4.1. The double point curves that appear among the edges of a
chart form a 1-dimensional manifold with boundary that is immersed into the ps, yq-
plane. The immersion is not proper nor is it in general position. End points of double
point arcs occur at the junction of folds of valence 3 vertices — type-I moves.
Non-generic double points of the immersed double curves occur at the valence 6
vertices of the chart which represent triple points — type-III moves, see item (6)
27
and Figure 11 in Section 4.2.3.
While they are non-generic, they are transverse.
Antipodal arcs at a valence 6 vertex are components of the same double point arc.
The nodes of the chart are a 0-dimensional subset of the 1-dimensional crossing set.
A key observation is that during an isotopy of a surface with nodes, the nodes can
move only along the double point arcs. In the chart, double point arcs are horizontal
in the ps, yq-plane. In Figure 13 it appears that a node passes through a point of
vertical tangency of a double point curve. However, the point of vertical tangency
is considered a vertex of the chart graph. Generically, the vertices of the chart have
distinct s-coordinates. As a node slides along a double point arc, its s-coordinate
can interchange with the s coordinate of another vertex. In Figure 16 above, the
node slides beyond the crossing of a pair of double curves. In Figures 17 and 18 it
interchanges s coordinates with a saddle. As discussed after the statement of the
theorem, there are other analogous kinds of interchanges.
The remaining moves that involve nodes occur when a node passes through a
vertex. These occur precisely when a node passes through a type-II or type-III move,
or passes through a ψ-move.
These are, respectively, MM16 (Figure 13), MM17
(Figure 14) and MM18 (Figure 15).
Note that if a node were to pass through a type-I move, then the node would
disappear, see Figure 19. This is impossible in an isotopy of a surface (because the
number of double points is always preserved).
U
+ (i,j)
.
U
+ (i,j)
.
U
-(i,j)
.
U
-(i,j)
.
X(i,j)
. .
X(i,j)
. .
X(i,j)
. .
Figure 19. A node passing through a branch point changes the num-
ber of nodes.
This completes the proof of the theorem.
□
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University of South Alabama, Department of Mathematics and Statistics, 411
University Boulevard North, Mobile, AL 36688-0002 USA
Email address: carter@southalabama.edu
University of Iowa, Department of Mathematics, 14 MacLean Hall, Iowa City, IA
52242-1419 USA
Email address: ben-cooper@uiowa.edu
Johns Hopkins University, Department of Mathematics, 404 Krieger Hall, 3400 N.
Charles Street, Baltimore, MD 21218 USA
Email address: khovanov@jhu.edu
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-
4137 USA
Email address: krushkal@virginia.edu
31
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AN EXTENSION OF KHOVANOV HOMOLOGY TO IMMERSED SURFACE COBORDISMS SCOTT CARTER, BENJAMIN COOPER, MIKHAIL KHOVANOV, AND VYACHESLAV KRUSHKAL Abstract. We show that an oriented surface in R4 containing double point singularities induces a map between the Khovanov homology groups of its boundary links in a functorial way. As part of this work, the movie moves of Carter and Saito are extended to surfaces with double points. 1. Introduction From the beginning [Kho00, §1] it was understood that a smooth oriented surface in R3 ˆ I should induce maps between Khovanov homology groups in a functorial way. One reason for this is that, up to a small perturbation, for a surface Σ Ă R3 ˆ I the projection onto the time axis I induces a Morse decomposition of Σ and assigning the maps from Khovanov's construction to the pieces produces a chain map between the chain complexes associated to the boundary links BΣ. Several years later M. Jacobsson showed that this assignment was independent of the isotopy representative of the surface up to sign [Jac04]. He did this by checking the movie moves of S. Carter and M. Saito [CS98a]. Since that time, simpler proofs have been found [Kho06a, BN05] and there have been several approaches to the sign problem [Bla10, CMW09, San21, Cap08]. Functoriality of link homology is crucial for its applications to four-dimensional topology. More concretely, the functoriality of Khovanov homology paved the way for J. Rasmussen's s-invariant spKq P Z and his proof that it gives a lower bound on the 4-ball genus of a knot K [Ras10]. Recently, the functoriality of Khovanov homology became an essential ingredient of the skein lasagna invariant and its applications to 4-manifold topology [MWW22, RW24]. Our purpose in this article is to introduce a new direction of study for the functoriality of Khovanov homology. We extend the type of surfaces which can induce maps between Khovanov homology groups by considering smooth orientable surfaces in R3 ˆ I with double point singularities. A double point singularity is locally modeled Date: 15th October, 2025. 1 16 Oct 2025 on a transverse intersection of two planes in R4 or the affine variety Z :" tpw, zq : wz " 0u Ă C2 (1.1) in a neighborhood of the origin. The structure of Z near the singular point can be described by a movie of 3-dimensional time slices of the form (1.2) and the link Z X tv : |v| " εu of the double point singularity is a Hopf link in the 3-sphere. We use the Khovanov homology of this Hopf link to assign a map to a smooth oriented surface with singularities in R3 ˆ I and prove that the homotopy class of this chain map is independent of the isotopy class of singular surface. In order to establish our result, we extend the known movie moves to include surfaces with double point singularities and we check that our assignments satisfy these new movie moves. Here is an informal statement of our work. Theorem. The maps induced on the Khovanov homology of oriented links are welldefined, up to an overall sign, on isotopy classes of surface cobordisms with double points. These maps give rise to a functor from the category of surface cobordisms with double points between oriented links in R3 to the category of bigraded abelian groups and homogeneous maps between them, considered up to overall minus sign. Let us state the results contained in our paper with a little more care. There is a 2category Tang4 with objects given by finite collections S of ̆-oriented points in the plane R2. For any two such collections, S and S1, there is a category Tang4pS, S1q whose objects are oriented tangles T Ď R2ˆr0, 1s, viewed as cobordisms from Sˆt0u to S1 ˆ t1u (and defined as a proper embedding of a compact oriented one-manifold into R2ˆr0, 1s rather than an equivalence class of embeddings rel boundary isotopy). A morphism f : T Ñ T 1 between tangles is a cobordism in 4-space. More carefully, it is an isotopy (rel boundary) class of a surface with corners standardly embedded in R2 ˆ r0, 1s2 which cobounds tangles T, T 1 in the two parallel boundary R2 ˆ r0, 1s's and has the standard boundary S ˆ r0, 1s and S1 ˆ r0, 1s on the two remaining boundary R2 ˆ r0, 1s's. There is an extension of this category Tang4pS, S1q Ă Tang4 ˆpS, S1q which has the same objects, but whose morphisms are isotopy classes of immersed surfaces with the double point singularities locally modelled on pZ, 0q in Eqn. (1.1) above. 2 The next theorem is a statement about isotopy of surfaces in 4-space. Recall that if rfs : T Ñ T 1 is a morphism in Tang4pS, S1q then any two representatives f, f 1 P rfs are related by a finite sequence of the Carter-Saito movie moves. Theorem. There is an extension of the Carter-Saito movie moves from the setting of Tang4pS, S1q to the setting of Tang4 ˆpS, S1q. More precisely, there is a finite set of extended movie moves such that if rfs : T Ñ T 1 is an isotopy class of immersed surface with double point singuarities in Tang4 ˆpS, S1q then any two representatives f, f 1 P rfs are related by a finite sequence of these extended moves. The extended movie moves are catalogued, with references to illustrations, by Thm. 4.1 in Section 4. Our second theorem below uses this theorem to extend the functoriality of Khovanov homology. For context, recall that Khovanov homology associates to a cobordism f a chain map f ̊. If f 1 P rfs then invariance under the Carter-Saito moves, up to overall sign, implies that there is a chain homotopy f ̊ » ̆f 1 ̊. This is the key step in showing that the Khovanov construction determines a 2-functor κ : Tang4 b Ñ pK, where pK is the 2-category defined in [Kho06a], also denoted K5pCobV q{t ̆1u in §2.1 below and throughout the paper, and see [BN05] for an alternative approach and a generalization to the equivariant case. (In the 2-category pK, 2-morphisms are defined up to overall sign.) Here Tang4 b Ă Tang4 is a full 2-subcategory of Tang4 where the objects are balanced collections S Ă R2, i.e., with the same number of positive and negative points. Likewise one defines a full 2-subcategory Tang4 ˆ,b of Tang4 ˆ. Our extension of κ from Tang4 b to Tang4 ˆ,b is defined by assigning maps to the double point cobordism and checking that the extended Carter-Saito moves hold. Theorem. Assigning maps A, B in Def. 3.1 to the double point cobordism in Eqn. (1.2) determines an extension ̃κ: Tang4 ˆ,b Ñ pK of the Khovanov 2-functor κ : Tang4 b Ñ pK from embedded balanced cobordisms Tang4 b to balanced cobordisms with double points Tang4 ˆ,b. Unlike the maps which are assigned to embedded cobordisms, the maps we associate to surface cobordisms with double point singularities change the homological degree. Another interesting aspect of our construction is that the maps associated with positive and negative double points are different. These new features may be useful for applications to topology. The study of surfaces in 4-manifolds, and particularly the question of when double points can be removed by a homotopy, is fundamental in 4-dimensional topology. Indeed, the failure of the Whitney trick in dimension 4 underlies the difference between smooth and topological 4-manifolds [Don83, Fre82], 3 see also [CG88, Kro97]. Developing a new tool to study surfaces with double points was a motivation for this work. A related construction appears in the recent work [ISST25], also see Remark 3.7. Organization. §2 contains a brief review of Khovanov homology. §3 introduces the maps assigned to double point singularities and checks that these maps satisfy the movie moves up to homotopy. §4 contains a discussion of movie moves and shows how to extend them to the setting of surfaces cobordisms with double points. Acknowledgments. The second author would like to thank Kevin Walker for a helpful conversation. The third author was partially supported by NSF grant DMS-2204033 and Simons Collaboration Award 994328 "New Structures in LowDimensional Topology". The fourth author was supported in part by NSF Grant DMS-2405044. 2. Review of Khovanov homology Here we review the construction of Khovanov homology and recall the duality statements which will be used to check movie moves in §3. The ideas here are equivalent to those of references such as [Kho06a, BN05]. Compared to [CK12, §2.3], the 2-category Cob is analogous to Pre-Cobpnq and CobV to Cobpnq. Set I :" r0, 1s. There is a 2-category Cob with (1) Objects given by disjoint collections of points on a line S Ă I . We also use the symbol S to denote its cardinality #S P Zě0. (2) 1-morphisms D : S Ñ S1 are formally q-graded direct sums D " 'N i"1qniDi of 1-manifolds Di : S Ñ S1 in I2 which bound the 0-manifolds S and S1 in the sense that S Ă I ˆ t1u, S1 Ă I ˆ t0u and BDi " S Y S1 for 1 ď i ď n, (3) A 2-morphism between 1-morphisms D, D1 : S Ñ S1 is a Z-linear combination of surfaces Σ : D Ñ D1 in I3. In more detail, if D " 'N j"1qniDi and D1 " 'M i"1qmjD1 j then a map Σ is a M ˆ N matrix pDijq the entries Dij " ř k akΣk ij of which are Z-linear combinations of orientable surfaces Σk ij Ă I3 which bound the 1-manifolds Di and D1 j in the sense that the boundary BΣk ij decomposes as a union of Di, D1 j , S ˆ I and S1 ˆ I along the occupied faces of the cube (Σk ij X pt0, 1u ˆ I2q " H). Manifolds are considered up to isotopy and maps are composed along various axes by gluing and rescaling. We briefly review the setup in order to orient the reader and establish notation. There is a product \ : Cob ˆ Cob Ñ Cob given by the disjoint union of cobordisms along the first axis of I3. The unit with respect to the disjoint union is the empty set H. 4 For any two sets of points S, S1, there is a category HompS, S1q with objects given by 1-morphisms D : S Ñ S1 as in p2q above and maps given by 2-morphisms in the sense of p3q above. The composition in Cob gives functors b : HompS, S1q ˆ HompS1, S2q Ñ HompS, S2q. In particular, if D : S Ñ S1 and E : S1 Ñ S2 are 1-manifolds then the composite D b E : S Ñ S2 is illustrated below. S 1 D S1 1 2 E S2 0 D E (2.1) When S " S1, the identity 1-manifold is 1S :" S ˆ I . If Σ : D Ñ D1 is a surface with D, D1 : S Ñ S1 then the expression χpΣq ́ pS ` S1q{2 P Z is preserved by gluing. This determines the q-grading: if Σ : qnD Ñ qmD1 is a map between formally q-graded 1-manifolds then we set degpΣq :" χpΣq ́ pS ` S1q{2 ` m ́ n. In this way, every 2-morphism is a sum of its q-homogeneous components. We next review a duality lemma which is important in §3. For more information see [CH15, §5.1] or compare [CMW09, §3.1]. If E : S Ñ S is an endomorphism then the trace TrpEq is the closed 1-manifold given by the quotient of E which pairwise identifies top S-points with the bottom S-points: pxi, 0q „ pxi, 1q, when S " tx1 ă x2 ă ̈ ̈ ̈ ă xnu. The proposition below describes how 2-morphisms can be written as traces. Proposition 2.1. If E, F : S Ñ S1 are 1-manifold morphisms then there is a dual E_ : S1 Ñ S and a natural isomorphism HompE, Fq - q ́pS`S1q{2HompH, TrpF b E_qq. (2.2) The duality functor satisfies pE_q_ - E and pE ' Fq_ - E_ ' F _, pE b Fq_ - F _ b E_, 1_ S " 1S and reverses the formal q-grading. The idea is that there is an isotopy of any surface Σ : E Ñ F which pulls E along the boundary of the box to the face occupied by F . This produces a closed 5 1-manifold consisting of F composed with a reflection E_ of E. Here is a picture of the reflection. Q_ Q „ (2.3) It is natural to apply a Frobenius algebra or 2-dimensional TQFT to the right hand side of Eqn. (2.2). We use this observation to describe the Khovanov homology 2-category CobV in the next section. 2.1. Khovanov Homology. Bar-Natan and Khovanov 2-category CobV is obtained from the 2-category Cob by using Eqn. (2.2) to require that HompE, Fq - q ́pS`S1q{2V b#π0pFbE_q. (2.4) The circle diagram S1 is isomorphic to V :" qH ̊pS2q, where H ̊pS2q :" ZrXs{pX2q is the Frobenius algebra1 graded by the degree assignments |1|q :" 0 and |X|q :" ́2. The coproduct is ∆p1q :" Xb1`1bX and ∆pXq :" XbX . The counit is εpXq :" 1 and εp1q :" 0. By construction the duality map descends to CobV and the duality isomorphism (2.2) continues to hold. In CobV , there are delooping isomorphisms D \ S1 - qD ' q ́1D for any 1-manifold D P Cob. To each tangle τ , one can assign a chain complex Tτ P ChpCobV q using the rule pictured below. For each positively or negatively oriented crossing σ ̆, the chain complex Tσ ̆ is defined to be the complex C ̆, which is the cone of the saddle cobordism between two resolutions pictured below.2 :" q1 q2 :" q ́2 q ́1 (2.5) The underlined 1-morphisms are placed in homological degree zero. The duality map C_ extends to chain complexes by reversing the homological t-degree. Notice that pC ̆q_ - C ̄. In particular, the dual ́_ on 1-manifolds determines an operation on tangles which satisfies Tτ _ - T _ τ . When two tangles differ by an oriented Reidemeister move there is a degree zero chain homotopy equivalence between the associated chain complexes. In this way 1Notice that H ̊pS2q is a graded Frobenius algebra, and V is a free rank 1 graded module over it, with |1|q " 1 ‰ 0. 2We abuse the notation here and omit an extra decoration on the left-hand side of these equations that would indicate the chain complex associated with a crossing. 6 there is an assignment κpτq :" rTτs of oriented tangles to elements of the homotopy category KbpCobV q of bounded chain complexes. This assignment extends to a 2functor κ : Tang4 b Ñ KbpCobV q up to sign. The 2-category KbpCobV q{t ̆1u is denoted pK in [Kho06a]. If pE, dEq and pF, dFq are two chain complexes in ChpCobV q then there is a chain complex pHom ̊pE, Fq, δq of maps from E to F . In degree lan element of this chain complex is a sequence of maps tfi : Ei Ñ F i`luiPZ. The differential is δf " tdFfi ` p ́1ql`1fi`1dEuiPZ. Eqn. (2.2) extends degreewise along the Hom ̊-construction with the functor ́_. In particular, when E " Tα and F " Tβ are the chain complexes associated to tangles α and β and, by using Eqn. (2.4), we obtain Hom ̊pTα, Tβq - q ́pS`S1q{2Hom ̊pH, TrpTβ b T _ α qq - q ́pS`S1q{2CKhpβα_q. Here CKhpβα_q is the standard hypercube-shaped chain complex appearing in the Khovanov construction. So we have the following lemma. Lemma 2.2. For two oriented tangles α, β : S Ñ S1 from S points to S1 points, there is a isomorphism on homology, HpHom ̊pTα, Tβq, δq - q ́pS`S1q{2Khpβα_q. (2.6) The lemma above will be used in §3 to study movie moves. We conclude with what, for us, is an important example: computing the mapping space from one crossing C ̄ to the opposite crossing C ̆ results in a Hopf link. Example 2.3. Notice that pC ̆q_ - C ̄. If H ̆ is the Khovanov homology of the Hopf link with linking number ̆1 then HpHom ̊pC ̄, C ̆qq - q ́2TrpC ̆ b pC ̄q_q " q ́2H ̆, where H` - Z0,0 ' Z0,2 ' Z2,4 ' Z2,6 (2.7) H ́ - Z0,0 ' Z0, ́2 ' Z ́2, ́4 ' Z ́2, ́6. (2.8) Here the symbol Za,b represents a Z-summand in pt, qq-degree pa, bq. See Fig. 2 below for an illustration of the homologies H ̆. Remark 2.4. Instead of H ̊pS2q in Eqn. (2.4) we could use the equivariant homology H ̊ Up2qpS2q :" H ̊ Up2qpptqrXs{pX2 ́ hX ́ tq where H ̊ Up2qpptq :" Zrh, ts and we follow the grading convention |1|q " 0 and |X|q " ́2, so that |h|q " ́2 and |t|q " ́4 make the quotient relation q-homogeneous. For this Frobenius algebra, 7 the counit is determined by setting εp1q :" 0 and εpXq :" 1. The coproduct is given by ∆p1q :" 1 b X ` X b 1 ` h1 b 1 and ∆pXq :" X b X ` t1 b 1 [Kho06b, Eqn. (5)]. And the unit is ιp1q :" 1. Let R :" Zrh, ts with |h|q " ́2 and |t|q " ́4. Repeating §2.1 we assign the R-module A :" qH ̊ Up2qpS2q to the circle S1 and produce a 2-category CobA in which the duality theorem continues to hold. Moreover, since X2 " hX `t, the 2-category CobV is obtained from CobA by setting h " 0 and t " 0. If the same assignments as Eqn. (2.5) are used for positive and negative crossings then there is a tangle 2-functor κ1 : Tang4 b Ñ KbpCobAq{t ̆1u, see [BN05] and [Kho06b, Prop. 6]. Now there are isomorphisms A αÕ β qR ' q ́1R (2.9) where αpzq :" ` εpXzq εpzq ̆T and βpzq :" ` ιpzq ιpXzq ́ hιpzq ̆ . The relations αβ " 1 and βα " 1 imply corresponding delooping isomorphisms among the associated surfaces in the 2-category CobA. By applying these delooping isomorphisms and removing acyclic subcomplexes as in [BN07], since S1 is identified with A, one computes the equivariant Hopf link homologies as Hequiv ` " t0q1A ' t2q5A Hequiv ́ " t ́2q ́5A ' t0q ́1A. In particular, Eqn. (2.9) shows that there is a non-canonical isomorphism of the form Hequiv ̆ - H ̆ bZ R. See also Remark 3.12. 3. Extension of Khovanov homology to double points In order to assign a map to a singular cobordism between oriented links it suffices to assign maps to Morse singularities, Reidemeister moves and the double points corresponding to the passage from a positive crossing to a negative crossing and vice versa, see Eqn. (1.2). Definition 3.1. For the Morse singularities and Reidemeister moves we will use the same assignments as the existing theory and the maps A : C` Ñ C ́ and B : C ́ Ñ C` which are pictured in Fig. 1 for the double points. On the right-hand side, the B map consists of the identity map between the two components of crossing complexes which are not in degree zero. It has homological degree 2 (also called the t-degree). On the left-hand side, the A map is the alternating sum of dots on the two sheets corresponding to components of crossing complexes which are not in degree zero. A dot can be understood as half of a handle, see [CK12, §2.3]. The map A has homological degree ́2. 8 A B Figure 1. The definition of the chain maps A, B assigned to double points It is straightforward to check the proposition below. Proposition 3.2. The assignments A and B are chain maps which represent homology classes in the chain complexes Hom ̊pC`, C ́q and Hom ̊pC ́, C`q corresponding to the classes in pt, qq-degree |A|t,q " p ́2, ́6q and |B|t,q " p2, 4q of the (q-shifted) Hopf link homologies q ́2H ́ and q ́2H` respectively in Ex. 2.3. Remark 3.3. The surface Σ produced by the Seifert algorithm from the 2-crossing planar projection of the Hopf link consists of two disks connected by two half-twisted 1-handles. Since this surface has Euler characteristic zero, Σ determines maps Σ ̊ : Z Ñ H ̆ of pt, qq-degree p0, 0q. The image of this map is spanned by a generator of Z0,0 in H ́ and twice a generator of Z0,0 in H`. The results in this paper imply that the class Z ́2, ́4 in H ́ corresponds to the image of the singular cobordism which is dual to the A map. Decorating cobordisms with dots corresponds to the action of X in Fig. 2. So, abusing notation slightly, the entire homology H ́ - xΣ, Ay ' XxΣ, Ay (3.1) can be understood in terms of cobordisms. Notice that XΣ is a generator of the negative Hopf link homology H ́, but this cannot be true for the positive Hopf link H` because the q-degree satisfies |XΣ ̊p1q|q " ́2 and the homology H` vanishes in negative q-degree, it follows that XΣ " 0 in H`. 9 H` t q 0 1 2 0 2 4 6 B H ́ t q ́2 ́1 0 ́6 ́4 ́2 0 A Figure 2. The grid shows the homology of the Hopf links H ̆ from Ex. 2.3. As in Prop. 3.2 the generators labelled A and B correspond to the maps A and B under the duality isomorphism. A hollow circle is a Z-summand which is not in the image of the operation X (determined by the Frobenius algebra). A filled circle is in the image of X . Multiplication by X at a component of a link has q-degree ́2. Remark 3.4. Determining which classes in the Khovanov homology of a link can be represented geometrically in the manner of Eqn. (3.1) is understanding the question of how much of Khovanov homology is cobordism generated in the sense of topological quantum field theory. This was the original motivation for the authors. Functoriality of Khovanov homology implies that for a smooth oriented surface pΣ, BΣq Ă pR4, R3q bounding a link BΣ, there is a map Σ ̊ : Z Ñ Kh0,χpΣqpBΣq, see [Kho00, §6.3]. Since the homological t-degree of the right-hand side must be zero and, at least phenomenologically, Khovanov homology is concentrated along a diagonal in the pt, qq-plane [Kho03], one should expect very little of the homology to be geometric. The extension Tang4 b Ă Tang4 ˆ,b considered in this paper can be considered the simplest non-trivial extension of the theory along homology classes which are not represented geometrically. In our extension, it is no longer necessary for geometric homology classes to have t-degree 0. It seems interesting to ask, which classes in Khi,jpLq are now geometric? Answers to this question may help to shed light on the nature of Khovanov homology as a tool for the study of low-dimensional topology. Remark 3.5. In the skein lasagna module theory [MWW22] one considers a disjoint collection of input 4-balls D4 in a 4-manifold M and surfaces Σ Ă M with boundary in BD4. Additionally, these skein lasagna fillings are decorated with KhovanovRozansky homology classes of the links BΣ in the 3-spheres BD4. The results of this paper may be interpreted as considering input balls with Hopf links H ̆ in their boundary, decorated with generating classes in cohomological degrees ̆2. 10 Remark 3.6. L. Weng introduced assignments for framed singular cobordisms [Wen11]. Interpreting these assignments in the oriented setting, the map c1 : C` Ñ C ́ in [Wen11] has t-degree zero, sending the 0-resolution of C` isomorphically onto the 1-resolution of C ́, and sending the 1-resolution to 0. The map c2 : C ́ Ñ C` in [Wen11] is the map B above. Remark 3.7. While preparing this paper for publication, we learned from T. Sano about a related extension of tangle cobordism invariants to double point singularities, by H. Imori, T. Sano, K. Sato, and M. Taniguchi [ISST25]. Remark 3.8. In a series of papers [IY21, Yos20, IY23] N. Ito and J. Yoshida introduce and study homology of singular links, by defining the complex for a singular crossing to be the cone of the map A above. One of their goals is to categorify Vassiliev invariants of links. In particular, in the latest paper [IY23], the authors show the categorical analogue of the 4T relation on weight spaces, for the singular link Khovanov homology complexes built via their construction. Our use of the map A, to extend homology from embedded to singular embedded surfaces, is different from theirs. 3.1. Checking the new movie moves. As mentioned in the introduction, Khovanov homology is known to satisfy the movie moves for smooth oriented surfaces up to sign. In order to show that the chain homotopy class of a map assigned to a surface with double points is independent of isotopy we must check the new movie moves which contain double point singularities. The new movie moves involving double points are enumerated by Theorem 4.1 in Section 4. Movies MM16 and MM17 are the only ones which require non-trivial verifications. The lemmas below are introduced in order to simplify exposition later. Lemma 3.9 gives criteria in which a diagram of chain complexes commutes up to homotopy (and sign). Lemma 3.9. Fix a pt, qq-bidegree ‹ and suppose that we are given the (notnecessarily commutative) diagram of chain complexes: W X Y Z. α γ β η Then either set of conditions (1) or (2) below imply that this diagram commutes up to homotopy and sign. (1) (a) α, β are degree zero homotopy equivalences (b) H‹pHom ̊pX, Zqq - 0 or Z 11 (c) rηs and rγs generate H‹pHom ̊pX, Zqq and H‹pHom ̊pY, Zqq respectively (2) (a) α, γ are degree zero homotopy equivalences (b) H‹pHom ̊pW, Y qq - 0 or Z (c) rβs and rηs generate H‹pHom ̊pW, Y qq and H‹pHom ̊pX, Zqq respectively Remark 3.10. In each case, condition pbq and condition paq combine to imply that other Hom-complexes have homology which is isomorphic to 0 or Z. Proof. For (1): Consider H‹pHom ̊pY, Zqq H‹pHom ̊pW, Zqq H‹pHom ̊pX, Zqq β ̊ α ̊ γ : Y Ñ Z γβ : W Ñ Z ηα : W Ñ Z η : X Ñ Z Since α and β are degree zero homotopy equivalences, the maps α ̊ and β ̊ are isomorphisms between the homology groups pictured above. There are two cases, If H‹pHom ̊pX, Zqq - Z then H‹pHom ̊pW, Zqq - Z via α ̊ and H‹pHom ̊pY, Zqq - Z via pβ ̊q ́1α ̊. By assumption rηs generates H‹pHom ̊pX, Zqq and rγs generates H‹pHom ̊pY, Zqq. Since AutpZq " t ̆1Zu, α ̊prηsq " ̆β ̊prγsq or ηα » ̆γβ. If H‹pHom ̊pX, Zqq - 0 then the same argument shows that all of the homology groups are zero and ηα » γβ. For (2): Same proof as (1) after replacing β ̊ and α ̊ with γ ̊ and α ̊. □ Recall that if β P Brn is a braid then, given any orientation of the strands, using the chain complexes associated to the crossings in Eqn. (2.5), gives a chain complex Tβ :" C ̆ i1 b C ̆ i2 b ̈ ̈ ̈ b C ̆ in where β " σ ̆ i1σ ̆ i2 ̈ ̈ ̈ σ ̆ in associated to the braid β as well as an inverse chain complex T ́1 β :" Tβ ́1 . This is an inverse in the sense that there are canonical homotopy equivalences of the form Tβ b T ́1 β » 1n and T ́1 β b Tβ » 1n. (3.2) In other words, Tβ is invertible in the homotopy category KbpCobV q. Recall for (2) below that V :" qH ̊pS2q is associated to the circle S1 in CobV . Lemma 3.11. Let W and X be chain complexes in CobV . Then for any braid β P Brn there are homotopy equivalences of mapping complexes: 12 (1) rβ : Hom ̊pW, Xq „ÝÑ Hom ̊pW b Tβ, X b Tβq and lβ : Hom ̊pW, Xq „ÝÑ Hom ̊pTβ b W, Tβ b Xq (2) Hom ̊pW \ 1, X \ 1q „ÝÑ Hom ̊pW, Xq b V . Proof. For (1): The map rβpfq :" f b 1Tβ has a homotopy inverse bβpgq :" g b 1T ́1 β because composing gives Hom ̊pW, Xq rβ ÝÑ Hom ̊pW b Tβ, X b Tβq bβ ÝÑ Hom ̊pW b Tβ b T ́1 β , X b Tβ b T ́1 β q and the homotopy equivalences in Eqn. (3.2) induce a natural equivalence between the righthand side and the lefthand side. The argument is parallel for the map lβ . For (2): Any interaction between a cobordism with a disjoint sheet can be disentangled using delooping isomorphism. Alternatively, this follows immediately from Lemma 2.2. □ We are now prepared to discuss the movie moves listed in Theorem 4.1 in Section 4. Proof. (MM16) Consider movie move #16: passing a node over a type-II move. For the move pictured above, each strand can be oriented, either to point up or to point down, so there are four oriented versions of this move. They involve either a positive or negative double point, resulting in one of the maps A, B. Let's consider the case of both strands pointing up or both strands pointing down; in this case the diagram can be written as 1 b 1 C` b C ́ C ́ b C` C` b C`. R2 B b 1 R2 1 b B Now using part (1) of Lem. 3.9 with pt, qq-bidegree ‹ :" |B|t,q " p2, 4q we verify the assumptions (a) both maps α and β are induced by Reidemeister 2 moves which are degree zero homotopy equivalences. 13 (b) the duality lemma (2.2) gives H‹pHom ̊pC` b C ́, C` b C`qq - q ́2H` - Z where H` is the positive Hopf link homology, see Ex. 2.3. (c) Lem. 3.11 tells us that there is a homotopy equivalence rσ such that rσpBq " B b 1. It follows that H‹Hom ̊pC` b C ́, C` b C`q - Zx1 b By. In the same way, the isomorphism lσ ́1pBq " 1bB shows that the map 1bB generates. For the other two orientation choices (one strand in MM16 pointing up, the other one pointing down) the proof involves the map A and is analogous. In more detail, the diagram reads 1 b 1 C ́ b C` C` b C ́ C ́ b C ́. R2 A b 1 R2 1 b A The proof proceeds as in the previous case, using Lem. 3.9 and pt, qq-bidegree ‹ :" |A|t,q " p ́2, ́6q corresponding to a generator of q ́2H ́. □ Proof. (MM17) Consider movie move #17: passing a node through a type-III move, For the move pictured above, each strand can be oriented in either direction, giving a total of eight cases. As in the previous proof, the map A is involved in half of the cases and the map B in the other half. When all of the strands are pointing upward we get the diagram below. C ́ 1 b T T b C ́ 2 C` 1 b T T b C` 2 R3 R3 B b 1 1 b B , (3.3) where T :" C` 2 b C` 1 . Without orientations, this is the same as the braids pictured below. 14 B b 1 R3 R3 1 b B Now using part (2) of Lem. 3.9 with pt, qq-bidegree ‹ :" |B|t,q " p2, 4q we verify the assumptions (a) the maps α and γ are induced by Reidemeister 3 moves, so they are degree zero homotopy equivalences. (b) the duality Eqn. (2.6) identifies the horizontal mapping space with the trace of a braid that is equivalent to the positive Hopf link and an unknot H` 1. The trace described here is pictured below. Hom ̃ , ̧ - CKh Here is a line-by-line proof: Hom ̊pC ́ 1 b T, C` 1 b Tq " Hom ̊pC ́ 1 b C` 2 b C` 1 , C` 1 b C` 2 b C` 1 q - q ́3TrppC ́ 1 b C` 2 b C` 1 q_ b C` 1 b C` 2 b C` 1 q - q ́3TrpC ́ 1 b C ́ 2 b C` 1 b C` 1 b C` 2 b C` 1 q » q ́3TrpC ́ 1 b C ́ 2 b C` 1 b pC` 1 b C` 2 b C` 1 qq » q ́3TrpC ́ 1 b C ́ 2 b pC` 1 b C` 2 b C` 1 q b C` 2 q » q ́3TrpC ́ 1 b pC ́ 2 b C` 2 q b C` 1 b C` 2 b C` 2 q » q ́3TrppC ́ 1 b 1 b C` 1 q b C` 2 b C` 2 q » q ́3TrpC` 2 b C` 2 q - q ́3pq ` q ́1qH`. 15 The additional unknot S1 contributes the factor q ` q ́1 to the Hom-space. This is a consequence of computing the trace of pσ2q2 P Br3 as a braid of index 3 and the delooping isomorphism, see the illustration above. Using the homology H` from Example 2.3 shows that H‹Hom ̊pC ́ 1 b T, C` 1 b Tq - Z (c) By Lemma 3.11 there are isomorphisms rβ and lβ , with β " σ2σ1, taking the map B to B b1 and 1bB. This shows that these maps generate homologies of their respective mapping spaces in degree ‹. This completes the argument for half of the orientations. For the other half, the top and bottom maps in diagram (3.3) are replaced with Ab1 and 1bA respectively, and the proof is completed using the pt, qq-bidegree ‹ :" |A|t,q " p ́2, ́6q and the link H ́ \ S1. Just like there are several version of the third Reidemeister move, there are several versions of the movie move MM17. The proof for other versions is directly analogous to the one given above. □ Proof. (MM18) In the context of the Khovanov construction, the horizontal arrows are identity maps so the diagram commutes. □ Proof. (Thm. 4.1 (5)) Movie moves involving far-commutativity commute because the maps A and B are applied to the same diagrams after planar isotopies which induce identity maps in the setting of CobV . □ Remark 3.12. Here are two observations about gradings in the equivariant setting CobA of Remark 2.4. (1) The only monomial hitj in the ground ring R with non-negative q-degree is the identity element 1 P R in q-degree 0. (2) The maps A and B as elements of q ́2H ́ and q ́2H` generate the class of largest q-degree within their respective t-degrees. Together these imply that the maps A and B are the only generating classes within their respective pt, qq-degrees of the Hom-complexes associated to the homologies q ́2Hequiv ̆ - q ́2H ̆ b R in CobA. So there are isomorphisms: H2,4pHom ̊pC`, C ́qq - ZxBy and H ́2, ́6pHom ̊pC ́, C`qq - ZxAy. These equations and the two observations suffice to amend the arguments above for the CobA theory. We conclude that there is a corresponding extension of the 2-functor κ1 : Tang4 b Ñ KbpCobAq{t ̆1u to a 2-functor ̃κ1 : Tang4 ˆ Ñ KbpCobAq{t ̆1u which assigns the maps A and B to double point singularities. 16 4. Movie moves for immersed surface cobordisms The Carter-Saito movie moves [CS98b, 2.6], [CRS97] provide a combinatorial description of isotopies of surfaces embedded in 4-space. The movies involve planar projections of links in R3 ˆ tsu Ă R3 ˆ R " R4 which are cross-sections of the surface. We refer to these moves according to their enumeration MM1 - MM15, cf. [BN05, Figures 11-13]. Movie moves for foams were also studied in [QW22]. Those authors complete the list that was proposed in [Car15]. In addition, [BDMS] study movie moves in the presence of symmetries on the knotted surfaces. In this section we formulate and prove an extension of the movie moves to immersed surfaces in R4. To set up the notation, consider a properly immersed oriented compact surface F in R3 ˆ r0, 1s and proper isotopies thereof. An immersion is of the form: pF; B0F \ B1Fq í pR3 ˆ r0, 1s; R3 ˆ t0u \ R3 ˆ t1uq, where one or both of B0F, B1F may be empty. As discussed in the introduction, the singularities of such surfaces consist of double points, which we will also refer to as nodes. There are finitely many nodes, and they are in the interior of the surface. The boundaries B0F, B1F are classical links embedded in R3 ˆ tju, j " 0, 1. Our goal is to analyze isotopies between surfaces with nodes, i.e. the restriction to F of ambient proper isotopies of R3 ˆ r0, 1s. In particular, the nodes stay in the interior of R3 ˆ r0, 1s and their number remains fixed during an isotopy. The tool that will be used in our analysis is a (retinal) chart, a certain graph which arises from a planar projection of F , see [CS98b, 1.5], [CRS97, Section 3.2]. The charts of properly isotopic embedded surfaces are related by moves discussed in [CS98b, Theorem 2.17]. We caution that the choice of vertical and horizontal axes in our diagrams differs from that in [CS98b], and our charts have additional decorations which we describe in Section 4.2. The structure of charts of surfaces embedded in 4-space and the moves on charts encoding isotopies of such surfaces were deduced in [CRS97, Section 4] from the analysis of singularities of generic maps of surfaces into R2 in [Gor91, Rie96, Wes95]. The same type of analysis applies in our context, where a generic map of a surface into R2 is obtained starting from a surface with double points, rather than from an embedded surface in R4, and projecting onto a plane (see Section 4.2). We start by setting up the notation for encoding the links arising as the crosssections of F . Convention and terminology. The interval factor of R3 ˆr0, 1s is parametrized by the variable s. The coordinates of R3 are denoted x, y, z, and the crossings of a link in R3 are defined with respect to the z coordinate, that is links are projected onto 17 the xy-plane. The y-coordinate will serve as the height function in the xy-plane. The terms type-I, II, or III will refer to Reidemeister moves of a given type. 4.1. Encoding the cross-sections of a surface. Consider an immersion F í R3ˆ r0, 1s which is in general position with respect to the projection R3 ˆ r0, 1s Ñ r0, 1s. In more detail, the critical points of the composition F í R3 ˆ r0, 1s p ÝÑ r0, 1s are non-degenerate, have distinct values in r0, 1s, and are disjoint from the double points of the immersion. Given s P r0, 1s, the intersection F X pR3 ˆ tsuq is called a still of a movie. The surface F will be assumed to be oriented, and the stills are all oriented accordingly as well. Consider the projection onto the first factor, R3 ˆ r0, 1s Ñ R3, and a further projection π: R3 Ñ R2 onto the xy-plane. We can pick a sequence of generic values tsiu of the parameter s so that the projections πpF X pR3 ˆtsiuqq and πpF X pR3 ˆtsi`1uqq of two successive stills differ by a birth, death, saddle, Reidemeister move, ψ-move (illustrated in Figure 15), crossing change (node), cusp, or critical exchange (exchange of the heights of critical points). The projection of a still onto the xy-plane is an immersed curve that has isolated transverse double points. The critical points with respect to the height function y of the projection of the still have distinct y-coordinates, and these are distinct from the y-coordinates of the crossing points which also occur at distinct levels. The bookmark code is a way to combinatorially encode the projection onto the xyplane of a still of a movie; a similar method will be used to label the edges of the chart graph defined in Section 4.2. To put this in the context of our goal, Theorem 4.1 below, the isotopies of immersed surfaces will first be encoded using moves on charts which are then translated to movie moves on stills using the bookmark code. Figure 3 indicates how the projection of an oriented link can be written in terms of symbols corresponding to crossings and to optima (maxima and minima). The fonts Ś , Ś , Ş, Ť are adorned with dots at the NE, SE, SW, or NW directions to indicate that directional arrows, corresponding to the link orientation, emerge from such points. The projection F X pR3 ˆ tsuq Ñ R onto the y-coordinate is a Morse function for the link. Since the crossings are assumed to lie at different vertical pyq levels, there is a bookmarked word that can be used to describe a knot diagram. The word is constructed from top to bottom. 18 X. . X .. X.. X. . X X X X. . .. .. . . 1 1 1 1 2 2 2 2 U U U U. . . . m1(-) M1(-) M2(+) m2(+) Crossings Optima Vertical Figure 3. Types of crossings and critical points (optima) of links, and the corresponding bookmark labels Each symbol is labeled with a pair of integers that indicate the number of vertical segments to the left and the number of vertical segments to the right of a given crossing or critical point. In general, the bookmarked word always starts with Şp0, 0q and ends with Ťp0, 0q. Compare this with the abstract tensor notation that can be used to describe quantum invariants, cf. [Kau13, BK01]. 19 Base point y x z Figure 4. Connected sum of 41#p ́41q For example, the bookmark word for the knot in Figure 4 is 4.2. Definition and properties of charts. A (retinal) chart is an oriented labeled graph contained in the ps, yq-plane. The structure of a chart is a Cerf-theoretic description, with additional decorations, of an immersed surface cobordism between classical links. We stress that there is a chosen preferred direction, y, in the plane, and the chart is defined with respect to this choice. In the following subsections we define its edges, edge decorations, and the types of vertices, see [CS98b, 1.5], [CRS97, Section 3.2] for more details. 4.2.1. The edges. The chart of a surface F í R3 ˆ r0, 1s has two types of edges, corresponding to folds and to crossings as described next. The edges of the graph may cross. Such crossings are among the vertices of the chart. According to singularity theory, a generic map from a surface to R2 has fold singularities. These form a 1-dimensional set upon which the rank of the map drops by 20 1. At cusps the rank of the projection map drops to 0. The images of the folds in the ps, yq-plane form one type of edge of a chart. The second type of edge corresponds to crossings with respect to the z coordinate: consider the double point arcs of the projection of F to the 3-dimensional space with the px, y, sq-coordinates, and further project them to the ps, yq-plane. To relate the charts to the stills F XpR3ˆtsuq, note that for generic values of s, the intersection of the vertical line tsu ˆ R in the ps, yq-plane with the chart consists of a finite collection of points corresponding to critical points of the link F X pR3 ˆ tsuq with respect to the y coordinate, and to the crossings of the link F XpR3 ˆtsuq. The y coordinates of these points in the plane are exactly the same as the y coordinates of these critical points, respectively crossings, of the link. 4.2.2. Edge decorations. The fold edges and the crossing (or double point) edges are labeled by the corresponding symbols Ś , Ś , Ş, Ť, and additionally they have a pair of non-negative integer labels. The first integer indicates the number of surface sheets that are behind the line of sight or to the left in the x-direction of it, and the second indicates the number of sheets that occlude the fold in the line of sight or to the right of the fold or crossing in the x-direction. This pair of integers agrees with the bookmark code of a cross-sectional still. Folds are also decorated with a short upward or downward pointing arc that indicates the side of the fold at which the surface overlaps. In this way, saddles and optima can be distinguished in the chart. Specifically, at a birth or death these vertical arcs both point inward; at saddles they point outward. The intersection of a vertical line segment, s " si, with the chart, that does not pass through any vertices of the chart, results in the bookmark code for the link F X pR3 ˆ tsiuq. 4.2.3. The vertices. Consider the critical points (births, deaths, saddle points) of the projection onto the s-axis F í R3 ˆ r0, 1s p ÝÑ r0, 1s. For terminological precision, these critical points and the Reidemeister moves, cusps, ψ-moves, and crossing changes (nodes) will be included among the critical events. Critical events are projected onto the ps, yq-plane, and they represent vertices in the chart that correspond to changes in the bookmark code. (1) The critical points of the folds - births, deaths, and saddles - are valence two vertices of the chart graph. They are Morse singularities with respect to the s-direction, Figures 5, 6. The orientations of the folds at these junctures are inconsistent. 21 Movie Chart +(i,j) +(i,j) -(i,j) -(i,j) U U U U . . . . Movie +(i,j) +(i,j) -(i,j) -(i,j) U U U U Chart Birth Death s y s y Figure 5. Births and deaths s y Movie Movie +(i,j) +(i,j) - -(i,j) U U U U +(i,j) +(i,j) -(i,j) -(i,j) U U U U Chart Chart .(i,j) . . . . . . . s y Figure 6. Saddles (2) Cusps are also valence two vertices at which folds are created or terminated. Both a green and a brown edge are incident at a cusp. The orientations of the folds are consistent at the cusp. There are eight types of cusps that depend upon directions, orientations and bookmarks. Two are illustrated, Figure 7. The reader is encouraged to catalogue the remaining cases. See also [CK21]. 22 Figure 7. Examples of cusps and their corresponding chart vertices (3) Valence three vertices that correspond to the branch points created by type-I moves, are the junction of a green fold, a brown fold, and a crossing. In drawing the graph, the fold set seems to run straight through at this vertex. However, the orientation of the folds is inconsistent at a branch point. Branch points in the chart correspond to Reidemeister type-I moves in a movie presentation, Figure 8. U + (i,j) . U -(i,j) . X(i,j) . . U+(i,j) . - U (i,j) . X (i,j) . . Figure 8. Two examples of type-I chart vertices (4) Valence two vertices arising from type-II moves correspond to critical points of the crossing points. The incident crossings are colored red and blue, and the orientation flows through the vertex, Figure 9. X (i,j) X (i,j) . . . . X (i,j) X (i,j) . . . . Figure 9. Two examples of type-II chart vertices 23 (5) Valence four vertices that indicate a crossing point passing over a fold involve two crossings and two folds. Both pairs of edges are oriented consistently. The bookmark codes change along the four incident edges, Figure 10. -(i+1,j) .U U-(i,j+1) . X(i,j+1) . . X (i+1,j) . . U -(i+1,j) . U -(i,j+1) . X (i,j+1) . . X (i+1,j) . . Figure 10. Two examples of ψ-move chart vertices (6) Valence six vertices correspond to triple points when surface F is projected into the px, y, sq 3-space. Recall that the z-coordinate is used for the crossing sense of the stills in a movie. These vertices correspond to Reidemeister typeIII moves. X (i+1,j) X (i,j+1) X (i+1,j) X (i+1,j) X (i,j+1) X (i,j+1) X (i+1,j) X (i,j+1) X (i+1,j) X (i+1,j) X (i,j+1) X (i,j+1) . . . . . . . . .. .. . . . . .... .. .. Figure 11. Two examples of type-III chart vertices (7) There are valence two vertices that correspond to nodes. A source and a sink are indicated. 1 2 X (i,j) X (i,j) . .. . 1 2 X (i,j) . . X (i,j) . . Figure 12. A source node (left) and a sink (right) 24 Note that double point curves and folds that have disparate bookmark codes may also cross. To understand them, think, for example, of the braid relation σiσj " σjσi for 1 ă |i ́ j|. See Figure 16. 4.3. The movie move theorem. We are in a position to formulate the main result of this section. The figures following the statement of the theorem include both the chart moves and the movie moves. Theorem 4.1. The movie moves that describe isotopies of immersed surfaces are of the following types: (1) MM1-MM15: the Carter-Saito movie moves for embedded surfaces [CS98b, 2.6] (2) MM16: passing a node over a type-II move, Figure 13, (3) MM17: passing a node through a type-III move, Figure 14, (4) MM18: passing a node over a fold, Figure 15, (5) moves that correspond to sliding a node along a horizontal segment of a double point curve; see for example, Figures 16-18. In this theorem, the new moves that involve nodes of immersed surfaces are those of types (2)-(5). There are variations of each of these moves that depend, for example, upon differing orientations, directions of the type-II moves, or whether the fold involved is a local maximum or minimum. The reader is encouraged to investigate these variations. We caution that nodes may not pass through the bottom or top arc at a type-III move. We have labeled the move in Fig. 16 as MM19 since, in this case, the node interacts directly with a vertex of the chart even though the vertex is an interchange of distant crossings. Both Figures 17 and 18 demonstrate that a node commutes with a saddle point in the fold set. There are other, analogous types of commutation relations of this type: a node commutes with critical points of the fold set which could be a birth, death, or a cusp. A node also commutes with a type I, II, or III move if the double point curve upon which the node sits is not involved in the Reidemeister move. As explained in the proof below, all these cases can be understood systematically as the horizontal coordinate of the node interchanging with one of these chart vertices. 25 X(i,j) X(i,j) X(i,j) X(i,j) X(i,j) X(i,j) Figure 13. MM16: passing a node over a type-II move. X (i,j+1) X (i,j+1) X (i+1,j) X (i+1,j) X (i,j+1) X (i+1,j) X (i,j+1) X (i+1,j) X (i,j+1) X (i+1,j) X (i+1,j) X (i,j+1) X (i+1,j) X (i,j+1) Figure 14. MM17: moving a node through a triple point. U -(i+1,j) . U -(i,j+1) . X (i,j+1) . . X (i+1,j) . . U -(i+1,j) . U -(i,j+1) . X (i,j+1) X (i+1,j) . . X (i,j+1) . . . . X (i+1,j) . . Figure 15. MM18: moving a node over a fold. 26 ... ... ... i j k ... ... i j k ... ... ... i j k ... ... ... i j k ... ... ... i j k ... ... ... i j k X (i+j,k) . . X (i+j,k) . . X (i,j+k+2) . . X (i,j+k+2) . . X (i+j,k) . . X (i+j,k) . . X (i,j+k+2) . . X (i,j+k+2) . . ... Figure 16. MM19: Interchanging a node and a distant crossing. UU . . (0,1) (0,1) X(1,0) .. X (1,0) . . UU . . (0,1) (0,1) X(1,0) .. X (1,0) . . Figure 17. Commutation of a critical point and a node, version 1. U U X(i,j) . . (i,j) (i,j) .. U.(i,j) U. (i,j) X(i,j) .. .X (i,j) . Figure 18. Commutation of a critical point and a node, version 2. Proof of Theorem 4.1. The double point curves that appear among the edges of a chart form a 1-dimensional manifold with boundary that is immersed into the ps, yqplane. The immersion is not proper nor is it in general position. End points of double point arcs occur at the junction of folds of valence 3 vertices - type-I moves. Non-generic double points of the immersed double curves occur at the valence 6 vertices of the chart which represent triple points - type-III moves, see item (6) 27 and Figure 11 in Section 4.2.3. While they are non-generic, they are transverse. Antipodal arcs at a valence 6 vertex are components of the same double point arc. The nodes of the chart are a 0-dimensional subset of the 1-dimensional crossing set. A key observation is that during an isotopy of a surface with nodes, the nodes can move only along the double point arcs. In the chart, double point arcs are horizontal in the ps, yq-plane. In Figure 13 it appears that a node passes through a point of vertical tangency of a double point curve. However, the point of vertical tangency is considered a vertex of the chart graph. Generically, the vertices of the chart have distinct s-coordinates. As a node slides along a double point arc, its s-coordinate can interchange with the s coordinate of another vertex. In Figure 16 above, the node slides beyond the crossing of a pair of double curves. In Figures 17 and 18 it interchanges s coordinates with a saddle. As discussed after the statement of the theorem, there are other analogous kinds of interchanges. The remaining moves that involve nodes occur when a node passes through a vertex. These occur precisely when a node passes through a type-II or type-III move, or passes through a ψ-move. These are, respectively, MM16 (Figure 13), MM17 (Figure 14) and MM18 (Figure 15). Note that if a node were to pass through a type-I move, then the node would disappear, see Figure 19. This is impossible in an isotopy of a surface (because the number of double points is always preserved). U + (i,j) . U + (i,j) . U -(i,j) . U -(i,j) . X(i,j) . . X(i,j) . . X(i,j) . . Figure 19. A node passing through a branch point changes the number of nodes. This completes the proof of the theorem. □ References [BDMS] Maciej Borodzik, Irving Dai, Abhishek Mallick, and Matthew Stoffregen, Reidemeister and movie moves for 2-symmetric links, In preparation. 28 [BK01] Bojko Bakalov and Alexander Kirillov, Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21, American Mathematical Society, Providence, RI, 2001. MR 1797619 [Bla10] Christian Blanchet, An oriented model for Khovanov homology, J. Knot Theory Ramifications 19 (2010), no. 2, 291-312. MR 2647055 [BN05] Dror Bar-Natan, Khovanov's homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-1499. MR 2174270 [BN07] , Fast Khovanov homology computations, J. 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[Yos20] Jun Yoshida, Decomposition of the first Vassiliev derivative of Khovanov homology and its application, (2020). 30 411 University Boulevard North, Mobile, AL 36688-0002 USA Email address: 14 MacLean Hall, Iowa City, IA 52242-1419 USA Email address: Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218 USA Email address: 229044137 USA Email address: 31
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2510.14765
|
Inpainting the Red Planet: Diffusion Models for
the Reconstruction of Martian Environments in
Virtual Reality
Giuseppe Lorenzo Catalano1 and Agata Marta Soccini1
1Computer Science Department, Universit`a degli Studi di Torino,
Corso Svizzera 185, Torino, 10149, Italy
This work has been submitted to the IEEE for possible publication. Copyright may
be transferred without notice, after which this version may no longer be accessible.
Abstract
Space exploration increasingly relies on Virtual Reality for several
tasks, such as mission planning, multidisciplinary scientific analysis, and
astronaut training. A key factor for the reliability of the simulations is
having accurate 3D representations of planetary terrains. Extraterrestrial
heightmaps derived from satellite imagery often contain missing values
due to acquisition and transmission constraints. Mars is among the most
studied planets beyond Earth, and its extensive terrain datasets make
the Martian surface reconstruction a valuable task, although many ar-
eas remain unmapped. Deep learning algorithms can support void-filling
tasks; however, whereas Earth’s comprehensive datasets enables the use
of conditional methods, such approaches cannot be applied to Mars. Cur-
rent approaches rely on simpler interpolation techniques which, however,
often fail to preserve geometric coherence. In this work, we propose a
method for reconstructing the surface of Mars based on an unconditional
diffusion model.
Training was conducted on an augmented dataset of
12000 Martian heightmaps derived from NASA’s HiRISE survey. A non-
homogeneous rescaling strategy captures terrain features across multiple
scales before resizing to a fixed 128 × 128 model resolution. We compared
our method against established void-filling and inpainting techniques, in-
cluding Inverse Distance Weighting, kriging, and Navier-Stokes algorithm,
on an evaluation set of 1000 samples. Results show that our approach con-
sistently outperforms these methods in terms of reconstruction accuracy
(4-15% on RMSE) and perceptual similarity (29-81% on LPIPS) with the
original data.
1
arXiv:2510.14765v1 [cs.CV] 16 Oct 2025
Figure 1:
The generative capabilities of diffusion models provide realistic
restorations of degraded acquisitions, producing structurally consistent rep-
resentations suitable for virtual environments.
(Left) A degraded Martian
heightmap depicting impact craters, with regions composed of missing values.
(Right) The same heightmap after reconstruction, using our diffusion-based ap-
proach; the inpainted regions are explicitly highlighted.
1
Introduction
The benefits of Virtual Reality (VR) for space exploration are being investigated
with growing interest in the recent years [1]. Use cases include the training of
future astronauts [2], the remote operation of existing rovers [3] and the interac-
tive real-time visualization of spacecraft diagnostics in immersive environments
[4, 5]. Additionally, such technologies can enable detailed morphological studies
of extraterrestrial surfaces for mission planning [6]. Planet Mars specifically is
an active object of study by the scientific community, and realistic representa-
tions of Martian landscapes is vital for the success of these missions.
To be effective, many of these applications rely on the ability to integrate
and accurately represent the terrains of various celestial bodies in VR [7]. Sur-
faces are commonly represented as heightmaps, a type of raster grid in which
each pixel encodes a specific altitude value, that are particularly well suited for
representing structured data in immersive virtual environments [8].
The problem we address is that most of the data sources regarding extrater-
restrial surfaces are captured by spacecraft, such as the Mars Reconnaissance
Orbiter [9], thus presenting additional challenges in being acquired and trans-
mitted given the difficult environment faced by the instruments. These factors
contribute to a scenario in which, unlike Earth where detailed and abundant
measurements are often accessible, significantly less information is available.
As a consequence, the final processed heightmaps may present missing values
within the raster. This can be the consequence of errors during the acquisition
of the original signal, data loss during transmission from the orbiter, or ambigu-
ities arising during the processing phase that converts images into heightmaps.
In such cases, performing another full acquisition of the raw data may be pro-
hibitively expensive, or even impossible if the dataset refers to past events.
Data gaps can severely hinder the usability of heightmaps for research and
2
mission planning purposes. To face the problem, we can use void-filling meth-
ods in order to restore the invalid regions through suitable approximations.
These methods typically rely on interpolation techniques or statistical analy-
ses of the valid neighbourhoods to estimate the missing values [10]. Moreover,
given the raster-based structure of heightmaps, which can be encoded and vi-
sualized as images, a wide range of inpainting algorithms for the modification
or reconstruction of selected regions within a digital image can be leveraged
as well [11]. Recently, diffusion models [12, 13] have outperformed other deep
learning-based approaches for image generation, such as Generative Adversarial
Networks (GANs) [14] and Variational Autoencoders (VAEs) [15]. These mod-
els generate content by reversing a Gaussian diffusion process that progressively
adds noise to the input data. Diffusion models have shown superior performance
in generative tasks [16], and have rapidly become the state of the art in many
image-related applications; nonetheless, active research is being conducted in
order to assess the performance of such architectures in relation to other kinds
of data [17].
Diffusion models have already been studied for heightmap generation and
restoration as well [18, 19]. However, these approaches are not suitable for the
reconstruction of Mars because: i) these models were trained and tested only on
terrains related to limited regions of Earth, making them potentially suscepti-
ble to overfitting over specific geometrical features, and not necessarily suitable
for other celestial bodies; ii) these approaches rely on conditional generation,
supposedly bringing further overfitting on specific data distributions. Whereas
this approach may be effective for Earth-related tasks, such auxiliary informa-
tion is often unavailable in the case of Mars; iii) the suggested approaches do
not evaluate their results from a visual standpoint as well, only resorting on
quantitative assessments and brief overviews in 2D that say little in terms of
the reconstruction quality for ultimately using them in VR.
In this context, we applied our method based on unconditional diffusion
models, namely models that do not require further contextual information dur-
ing generation, for heightmap reconstruction; our hypotheses were formulated
as follows:
• H1. Our method restores Martian surfaces smoothly and consistently,
producing visually coherent geometrical features that are well-suited for
3D visualization and simulations in virtual reality.
• H2. The digitally reconstructed Mars surfaces are objectively similar to
the original terrains, even though data sources are particularly scarce.
The contributions of this work can be summarized as follows. I) Uncondi-
tional diffusion-based reconstruction framework for planetary terrain inpainting,
addressing the lack of auxiliary conditioning data for extraterrestrial surfaces
such as Mars. II) Augmented Martian terrain dataset derived from NASA’s
HiRISE survey, with training which incorporates a non-homogeneous rescaling
strategy to capture multi-scale terrain features before standardizing resolution
to 128 × 128. III) Comprehensive evaluation against void-filling and inpainting
3
methods (Inverse Distance Weighting, kriging, and Navier-Stokes), using both
quantitative error metrics (RMSE, MAE, PSNR, EMD) and perceptual met-
rics (LPIPS, SSIM, FID). IV) Performance gains of up to 15% in RMSE and
81% in LPIPS, producing geometrically consistent reconstructions suitable for
VR-based planetary visualization and analysis.
The paper is organized as follows: Section 2 reviews related work, Section
3 presents the proposed method, Section 4 describes the experiments, Section
5 reports the results, Section 6 discusses the findings, and Section 7 concludes
the paper.
2
Related Work
2.1
Diffusion Models
Diffusion models [12] are a family of generative methods that are based on re-
versing a diffusion process, namely a procedure that progressively adds noise to
a signal over a series of discrete timesteps, until the original input is transformed
into pure noise. By learning the reverse of this process, it becomes possible to
generate new synthetic data starting from random noise. The forward diffusion
is modeled as a discrete-time Markov chain that adds Gaussian noise across
timesteps t = 0, . . . , T where t = 0 corresponds to an image without noise
and t = T is pure Gaussian noise.
Each intermediate corrupted sample xt
can be expressed in closed form, with regards to the original image x0 and the
timestep t: xt ∼N(√¯αtx0), (1−¯αt)I), where ¯αt is a set of fixed constants. This
formulation allows for efficient training, as corrupted samples can be directly
generated from clean data without simulating the entire chain. In recent years,
several mathematical formulations and frameworks based on this principle have
emerged, particularly in the domain of image synthesis. One of the most promi-
nent examples is given by Denoising Diffusion Probabilistic Models (DDPMs)
[13], in which the actual denoising model, typically a U-Net [20], is trained to
predict the noise component at each timestep. The generation process begins
by sampling xT from a random distribution, xT ∼N(0, I), and iteratively re-
moving the predicted noise through the learned reverse process. The procedure
ends when x0, a synthetic image without noise resembling the distribution of
the training data, is produced.
Subsequent works focus on speeding up the inference process, which requires
the entire diffusion chain to be traversed in the case of DDPMs. An example
is given by Denoising Diffusion Implicit Models (DDIMs) [21], which reformu-
late the mathematical process but keep the same training objective, making it
possible to skip inference of some timesteps while keeping the same trained de-
noising networks. Another example is given by Latent Diffusion Models (LDMs)
[22], which leverage a pre-trained autoencoder to perform denoising in the latent
space. With these regards, Diffusion Transformers (DiTs) [23] build upon previ-
ous works by leveraging the Transformer [24] architecture as the noise predictor
in the DDPM formulation.
4
2.2
Surface Void-Filling Techniques
To restore invalid regions within a heightmap, various approaches can be em-
ployed to interpolate the missing values between valid pixels. The general idea
is to exploit the spatial information contained in neighboring values to estimate
the unknown ones [10]. Simpler methods are based on nearest-neighbor strate-
gies or spline interpolation, while more advanced techniques involve weighted
estimations. Among the latter, Inverse Distance Weighting (IDW) and kriging
are two of the most commonly used. IDW [25] computes a weighted average
of nearby valid pixels, assigning greater influence to those closer to the missing
point. Kriging [26] produces a weighted average as well, but the weights are
derived from a statistical model of spatial autocorrelation through the use of
a semi-variogram. The heightmaps that are part of the HiRISE dataset, the
main source for Martian terrains (which is further discussed in Section 3.1), are
processed by linearly interpolating the areas with too much value uncertainty
[27]. In general, the performance of each method depends on the morphological
characteristics of the surface patch that is being reconstructed [28].
Deep learning generative models have been successfully employed for surface
restoration. GANs were tested in recent years in the context of terrain surfaces
[29, 30, 31]; diffusion models have also started to be analyzed in this context as
well [18, 19]. As previously mentioned, one major drawback of such methods is
that they rely on a conditioned approach, where the generation process is guided
by supplementary data; more specifically, conditioning is performed using the
shape of the binary mask [18], or even the general structure of the terrain
that must be known beforehand [19]. Moreover, the studies were performed
on heightmaps representing specific areas of Earth. These reasons may lead to
overfitting over small use cases, also limiting the generalization of the approaches
to different scenarios, making their use for Mars more challenging.
Finally,
the studies were not concerned about the visual inspection of the restoration
output, with a focus that strayed from the representation in VR. Our aim with
these regards is to define a framework that can be applied in diverse use cases,
applicable to terrain morphology but also for other purposes, which do not
require sets of additional data of different kind.
2.3
Image Inpainting Algorithms
Inpainting is a well-known task in the digital image processing domain. The
goal is to fill specific portions of an image, typically selected using a binary
mask, with new values that are coherent with the adjacent valid portions. This
task is usually performed to remove unwanted features or to restore partial
degradations. Several inpainting algorithms have been developed throughout
the years, based on different principles, such as the fast marching method [32]
or the Navier-Stokes fluido-dynamics equations [33], to expand the valid parts
of the signal into the missing regions. Other methods rely instead on computing
patch correspondences for finding the best match [34]. The advent of generative
deep learning models for image manipulation had a consistent impact for the
5
task of inpainting as well [35]. Diffusion model-based methods were successfully
implemented for this purpose as well.
Notable example include Palette [36]
and Stable Diffusion [22], however both rely on a generation that is conditioned
by the inpainting binary mask. A different approach is presented by RePaint
[37], an inpainting algorithm which is based on unconditioned diffusion models.
More specifically, a pre-trained DDPM is used for performing the unconditional
reverse diffusion process; by itself, this process would produce a result with no
correlation with the image that needs to be inpainted. After each denoising
step, however, the valid parts of the image are overwritten on the intermediate
result, so that the next step will be guided towards generating a final result
that is coherent with the original input. In order to obtain better results, a
resampling mechanism is also implemented; noise is added and then removed
multiple times, in order to generate patches that are more consistent with the
valid parts. Given its flexibility with regard to the inpainting masks, guaranteed
by the unconditional DDPM, we based our method on this algorithm.
To the best of our knowledge, this is the first work to address the specific task
of Martian terrain restoration using diffusion models, achieving performance
superior to currently employed methods. Our approach leverages the absence
of additional data constraints afforded by unconditional models to tackle the
unique challenges of Martian datasets. Furthermore, the model is trained on
features sampled at multiple scales from across the entire planet, enhancing its
ability to generate accurate and coherent terrain features.
3
Methods
3.1
Martian Surfaces Dataset
The training of the DDPM model required a properly structured dataset of
Martian terrain heightmaps. The most prominent repository with this regard
is provided by HiRISE [38], a set of high-performance cameras mounted on the
Mars Reconnaissance Orbiter (MRO), dedicated to capturing high-resolution
imagery of the planet’s surface. HiRISE is capable of acquiring stereoscopic
image pairs, which undergo a dedicated processing pipeline to extract the cor-
responding heightmaps [39, 27]. The resulting data products are made pub-
licly available via an online repository1 in the form of Digital Elevation Models
(DEMs), a specific type of terrain representation that includes metadata for
geo-localization, such as coordinate reference systems and coordinates of each
acquisition. These terrains are a valuable source for investigating the morphol-
ogy of the planet, given their capability to represent detailed patterns due to
their high resolution [40, 41]. This use case presents an interesting testbed: due
to the challenges faced during data acquisition and transmission, certain regions
of the images may be difficult to process or entirely unavailable, thus resulting
in interpolated or missing values as already discussed (see figure 2).
1https://www.uahirise.org/dtm/
6
Figure 2: RGB representation of Martian surfaces, obtained from the processing
of HiRISE stereo pairs. It may occur that missing data are interpolated during
processing (the squared patterns within the crater in the left image) or entirely
unavailable (the black line in the right image).
Approximately 1,150 heightmaps are available at the current time. However,
these are not immediately suitable for direct use in training. Indeed, diffusion
models require normalized images of fixed size (128 × 128 in our case) as input,
whereas HiRISE DEMs typically cover much larger areas, in the order of 103
pixels per side. Furthermore, the data are stored in the Planetary Data System
(PDS) data format, which is not supported by most image processing libraries.
For these reasons, a new dataset was generated to enhance file interoperabil-
ity and readability.
The resulting training set consists of 12,000 normalized
heightmaps, stored using the TIFF file format. Each element is a random crop
extracted from the original HiRISE survey, with side lengths randomly selected
between 512 and 2048 pixels for each sample. During training, these crops are
rescaled in a non-homogeneous way to 128 × 128, using nearest-neighbour in-
terpolation. The main reason behind this choice, made mainly to support the
investigation of H2, is to provide the model with knowledge of terrain features
that may develop over portions that are larger than the model resolution. This
operation enables data augmentation, helping to compensate for the relatively
small size of the initial dataset which is an an inherent challenge in this domain.
7
GDAL2 was used to process the original PDS data and convert them to TIFF.
3.2
Training of the Diffusion Model
In order to assess both H1 and H2, we trained an unconditional DDPM model,
capable of generating realistic Martian terrains without the need for textual
or visual prompts. To this purpose, training was performed at a resolution of
128 × 128 pixels, using the augmented dataset described in Section 3.1.
As
previously mentioned, we believe that by feeding the model random crops of
varying original size, the model can learn surface features at multiple different
scales.
Training was conducted over 100 epochs, with a batch size of 16 samples and
a total number of T = 1000 timesteps. The main neural network specifications,
such as the model architecture and loss function, were chosen according to the
original DDPM formulation [13]; specifically, we employed a U-Net to predict the
total noise in an image at an arbitrary timestep of the diffusion process. The
total training time was approximately 8 hours, using 2 NVIDIA A40 GPUs.
Samples from the trained DDPM after 100 epochs are shown in figure 3. The
implementation used for DDPMs is based on the Huggingface Diffusers3 library.
Figure 3: Evaluation of the trained unconditional DDPM after 100 epochs of
training with Martian terrains.
3.3
Inference
Inference with the trained DDPM model is performed by using the RePaint
algorithm on normalized heightmaps. Specifically, a selected portion of Martian
terrain is resized to the fixed resolution of 128 × 128 pixels and given as input,
along with a binary mask indicating which parts need to be restored. The main
parameter configuration suggested by the original authors was employed; the
complete inference takes around 44 seconds on a single NVIDIA A40 GPU. It
is important to highlight the role of the mask in the inference process: while
it is indeed used by the inpainting algorithm to distinguish between valid and
missing pixels, it is however not directly used by the diffusion model itself,
2https://gdal.org/en/stable/
3https://huggingface.co/docs/diffusers
8
which operates unconditionally. Figure 4 illustrates the input and output of our
method.
Figure 4: Input (left) and output (right) of our approach to Martian surface
reconstruction, represented as 2D rasters.
4
Experiments
We run our tests on a dataset of 1000 random crops, sampled from the HiRISE
terrain survey following the same procedure described in Section 3.1. To conduct
the experiments, we first sampled crops containing only valid pixels (i.e., without
missing values) to serve as the baseline, and then generated masks to remove
selected known data. We shaped the masks in a way that resembles aliased lines,
as this is a kind of artifact pattern that tends to commonly occur in Martian
terrain heightmaps (refer to Figure 2 for a visual reference).
We randomly
generated the masks by varying different parameters of the line, such as the
orientation or the number of segments. Examples of different mask shapes are
shown in Figure 5.
Figure 5: Example mask shapes that were used for evaluation. The masks are
artificially generated to simulate common patterns that occur in data losses.
9
4.1
3D Visualization
One of the main goals of this study, aimed at the investigation of H1, was to
evaluate the restoration quality of our proposed diffusion-based method, when
rendered as 3D surfaces within virtual environments. Indeed, we aimed to per-
form void-filling that would result in geometrically and visually coherent meshes,
making the restored surfaces suitable for use in VR applications. In order to
carry out an in-depth evaluation, we used Autodesk Maya4, a software for 3D
modeling and rendering. Its built-in Python API allowed us to automate the
import process of heightmaps and create meshes for them. Additionally, we
leveraged the Arnold renderer5 to create custom shaders capable of highlighting
restored areas based on the binary masks, and to produce detailed visualizations
under varying lighting conditions for qualitative analysis (Figure 6).
Figure 6: Input (left) and output (right) of our approach to Martian surface
reconstruction, represented as 3D surfaces and rendered with Arnold in Maya.
4.2
Comparative Evaluation
In order to assess the performance of our method for Mars surface restora-
tion (H2), other inpainting and void-filling algorithms were evaluated for com-
parison. The scenario of terrain restoration led us to choose Inverse Distance
Weighting (IDW) [25] and kriging [26] as our primary baselines, given their
widespread adoption in this context.
We also included an image inpainting
technique in our tests, namely the Navier-Stokes algorithm [33]. All methods
were tested on the same dataset and with the same set of masks. The imple-
mentation for kriging was provided by the PyKrige library6, whereas OpenCV7
4https://www.autodesk.com/products/maya/overview
5https://www.autodesk.com/products/arnold/overview
6https://geostat-framework.readthedocs.io/projects/pykrige/en/stable/index.
html
7https://opencv.org/
10
Figure 7: Qualitative evaluation of results, represented as 2D rasters. For each
group, the right column represents the absolute error between restored and
original values; the lack of yellow-green pixels represents better reconstructions.
was used for Navier-Stokes. For IDW we used N = 12 neighbours and a power
parameter of p = 2.
RMSE ↓
MAE ↓
PSNR ↑
EMD ↓
Ours
0.0752 (–)
0.0548 (–)
39.4494 (–)
0.1366 (–)
IDW
0.0864 (+14.9%)
0.0603 (+10%)
35.7504 (-9.4%)
0.1756 (+28.6%)
Navier-Stokes
0.0830 (+10.4%)
0.0580 (+5.8%)
36.2110 (-8.2%)
0.1699 (+24.4%)
Kriging
0.0784 (+4.3%)
0.0573 (+4.6%)
39.3492 (-0.3%)
0.1572 (+15.1%)
Table 1: Average results of the error-based metrics for the test run (1000 sam-
ples). Bold values indicate the best results.
5
Results
5.1
Visual Assessment
The visual inspection of the tested surface reconstruction methods was con-
ducted using both 2D plots and 3D renders. They highlighted significant struc-
tural differences between diffusion-based restoration and interpolation algo-
rithms, with the former producing more coherent results, in support of H1.
Figure 7 displays a selection of results as color-mapped 2D images; in each ex-
ample, the right column shows the absolute error with respect to the ground
truth. These considerations become even more evident in Figure 8, which shows
11
Figure 8: Qualitative evaluation of results, represented as 3D surfaces and ren-
dered with Arnold in Autodesk Maya. The highlighted parts indicate the areas
reconstructed using the evaluated methods. The possibility to control lighting
and visualization conditions allows to perform effective assessments of the re-
sults.
the same selection of results, processed as heightmaps and rendered as 3D sur-
faces instead.
5.2
Error Metrics Evaluation
We evaluated the tested methods from a quantitative perspective as well, in
order to assess the reconstruction quality in terms of error with respect to the
ground truth. This analysis aimed to verify whether the visually convincing
results correspond to surfaces that are indeed similar to the original ones. Such
validation is of critical importance: these terrains, which are frequently used for
scientific analyses and mission planning purposes, must approximate their real-
world counterparts as closely as possible. In this context, for each tested sample
we computed the Root Mean Squared Error (RMSE), Mean Absolute Error
(MAE), and Peak Signal-to-Noise Ratio (PSNR) between the ground truth and
the restoration output. In addition, we also considered Earth Mover’s Distance
(EMD) as a measure of the discrepancy between value distributions.
Table 1 reports the average results over the 1000 random surface crops that
were tested. Our method outperforms the others across all metrics, suggest-
ing that the effectiveness observed during visual assessments is also measured
quantitatively, supporting H2. Kriging, which leverages information related to
spatial correlation among valid pixels, is the second-best performing, yielding
results that are close to those of our approach. It also interesting to note that the
12
SSIM ↑
LPIPS ↓
FID ↓
Ours
0.9660 (–)
0.0754 (–)
10.9397 (–)
IDW
0.9620 (-0.4%)
0.1365 (+81%)
61.0935 (+458.5%)
Navier-Stokes
0.9650 (-0.1%)
0.1181 (+56.6%)
42.0554 (+284.4%)
Kriging
0.9684 (+0.2%)
0.0973 (+29%)
27.5676 (+152%)
Table 2: Average results of the perceptual-based metrics for the test run (1000
samples). FID is not an average, but a single score associated to each evaluation
set. Bold values indicate the best results.
Navier-Stokes algorithm, typically used for image inpainting but less common
in the surface void-filling domain, performs generally better than a well-known
method in this context such as IDW.
5.3
Perceptual Metrics Evaluation
The quantitative evaluation was further extended to include the calculation
and comparison of perceptual metrics, a family of measures that are designed
to model the similarity between images, as perceived by humans. Evaluations
of this kind are commonly performed in image generation tasks, as they aim
to approximate how closely the generated outputs resemble the training distri-
bution, attempting to model human perceptual judgments in assessing whether
the generated images “look like” the original ones. The selected metrics are the
following: i) Structural Similarity Index Measure (SSIM) [42], which evaluates
image similarity in terms of luminance, contrast, and structural consistency; ii)
Learned Perceptual Image Patch Similarity (LPIPS) [43], which computes the
distance between latent feature representations of two images as extracted by
a pre-trained neural network; iii) Fr´echet Inception Distance (FID) [44], which
tests which compares the distribution of synthetic images against that of real
images using activations from an Inception network. Although the results may
be biased by the network being trained on the ImageNet dataset, we decided
to employ it in addition to the others as a further evaluation method. It is
worth pointing out that, while some perceptual metrics are inherently related
to error-based ones, they are often sensitive to different types of visual discrep-
ancies [45].
Table 2 presents the average results for SSIM and LPIPS across the tested
1000 samples; the FID score is a single value for each method, obtained by com-
paring the overall distribution of the results against the distribution of ground
truth images. Our method achieves superior results in both LPIPS and FID,
whereas kriging yields a slightly higher average SSIM score; on this front, H2
was supported for the most part, while also giving further insights in support
of H1. Notably, Navier-Stokes continues to outperform IDW in this evaluation
as well.
13
6
Discussion
The methodology we presented showed to be an effective approach for recon-
structing missing sections of Martian terrains.
Visual assessment. Visual assessments of the reconstructed heightmaps,
conducted both using 2D color-mapped rasters and 3D renderings, provided
valuable insights. We paid particular attention to fidelity and structural con-
sistency, especially near the boundaries between valid and missing regions. Our
method proved effective with these regards, as it blends the two areas seamlessly
and generates realistic surface patches with smooth transitions, while at the
same time reconstructing patterns that were already present. IDW and Navier-
Stokes exhibit similar behaviour between them, as both expand the values of
valid pixels into the missing regions.
Navier-Stokes tends to appear slightly
smoother than IDW, likely due to the fluid dynamics principles on which the
algorithm is based. Both methods show discontinuities near the center of the
reconstructed regions, which is particularly suboptimal when considering these
terrains as testbeds for rover simulations or morphological studies. Kriging also
yields interesting results: it performs effectively on flat regions and adapts with
reasonable accuracy to some complex structures, but it struggles to capture
high-frequency patterns such as dunes or crater ridges; however, the resulting
artifacts generally have a less disruptive impact on the overall surface struc-
ture than those produced by IDW and Navier-Stokes. These findings support
hypothesis H1, confirming that the restored surfaces exhibit high structural
fidelity and accuracy.
Objective Measurements.
Our method produces reconstructions that
are also accurate in terms of objective accuracy with regards to the original ter-
rains. It achieved the best performance across multiple error metrics, namely
Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Peak Signal-
to-Noise Ratio (PSNR) and Earth Mover’s Distance (EMD). This quantita-
tive analysis verified hypothesis H2, which advocated for a reliable objective
restoration of missing Mars regions. Finally, we evaluated a series of percep-
tual similarity metrics that are commonly employed to model human perception
in the context of generative models. Our method obtained the best scores in
Learned Perceptual Image Patch Similarity (LPIPS) and Fr´echet Inception Dis-
tance (FID), whereas kriging achieved the highest Structural Similarity Index
Measure (SSIM) score. This outcome might stem from the structure comparison
component that is present in SSIM computations, which may penalize recon-
structions that are structurally coherent yet mismatched (as produced by our
method) compared to flatter, less detailed restorations (as generated by krig-
ing). These results point to a solid and robust framework, and to an additional
overall confirmation of H2, despite not having reached the best performance
on all fronts. The relatively small size of the training dataset may be a factor
with this regard, however the domain of extraterrestrial terrains is inherently
data-scarce and even in these conditions we achieved notable results.
Qualitative vs. Quantitative Insights. The relationship between quali-
tative evaluation and quantitative metrics offers further valuable insights. De-
14
spite the relatively strong performance of kriging, which achieved the best
scores among interpolation-based methods, and also outperformed our method
in terms of SSIM, the visual assessment clearly showed that the diffusion-based
approach provides with more realistic and coherent reconstructions. This can be
largely attributed to the statistical nature of kriging, a model which computes
an autocorrelation semivariogram to specifically minimize the mean squared er-
ror of the approximation function, and further underlines the importance of
subjective evaluation of the results.
Our initial hypotheses have been thor-
oughly addressed and verified.
We provided insights into the capabilities of
diffusion models to encode large amounts of data, and applied them to recon-
struct heightmaps in ways that are well-suited for being represented as accurate
3D virtual terrains. Results indicated that our trained model represents a highly
flexible tool for surface reconstruction at different scales. Moreover, we showed
that unconditional models in particular can be successfully leveraged for restora-
tion purposes, without having to rely on additional information aside from the
raster. These results open up promising scenarios for the application of deep
generative models in extraterrestrial contexts, characterized by similarly limited
data availability.
Limitations. The current work presents some limitations. First, both the
model architecture and the overall inpainting pipeline offer substantial room
for optimization. Users do not have the possibility to define custom crop re-
gions or binary masks in real time, that limits the ability to perform precise
and localized inpainting during immersive sessions. Currently, this remains a
technical barrier due to the computational cost of the method. Comparisons
with other diffusion-based approaches [18] still need to be evaluated.
These
methods have been mostly developed for geoscientific applications and tested
on Earth-related tasks; despite our emphasis on the unconditional nature and
the the generalization potential of our approach, they make for valuable testing
grounds nonetheless. The applicability of this method to other extraterrestrial
bodies, such as the Moon, has yet to be assessed. The general framework of
augmenting existing terrain surveys can be adapted to diverse scenarios, though
each must be customized to the specific environmental and data constraints.
Future Work. There are several directions in which the current work can
be further expanded. As a next step, we plan to conduct a user study to eval-
uate the perceived quality and usability of the reconstructed terrains from the
perspective of participants. This may include assessments of the Sense of Pres-
ence, interaction techniques with the reconstructed surfaces in scientific space
exploration simulations, and navigation methods within the virtual Martian
environments. Regarding our inpainting method, multi-resolution models and
upgraded architectures such as Denoising Diffusion Implicit Models (DDIMs)
[21], Latent Diffusion Models (LDMs) [22] or Diffusion Transformers [23] may
enable faster and more precise generations, along with the possibility to employ
this method in diverse practical uses. Moreover, alternative configurations of
the RePaint algorithm can be explored in order to achieve more efficient inpaint-
ing results, ideally with a smaller number of required steps. Such functional-
ities would require a significantly faster inference process, allowing user input
15
to guide the reconstruction dynamically, potentially within immersive environ-
ments. In addition, this line of research could contribute to the development of
experimental setups for the human evaluation of AI-generated content in VR
environments. While current studies in this area primarily focus on 2D imagery
[46], we believe the current work has potential for expanding these evaluations
to 3D contexts.
Ethics and Replicability Ethics Committee approval was not required for
this study, as the experiments did not involve human participants or any living
beings. Furthermore, the inference was performed on publicly available open
data, ensuring transparency and replicability. The processed data are available
upon request.
7
Conclusions
This work presented a method for the restoration of degraded Martian heightmaps
using unconditional diffusion models. We trained a DDPM on a dataset of ter-
rains sampled at various scales, enabling the model to learn common morpho-
logical patterns across multiple resolutions. The approach was compared with
established void-filling and inpainting algorithms, with the aim of evaluating
both the structural fidelity of the reconstructions for immersive 3D visualization
in VR, and the accuracy of the restored Martian surfaces in the absence of ad-
ditional conditioning information. Visual inspection showed that the generated
regions preserve structural consistency, while quantitative metrics confirmed a
close match with the original data in terms of reconstruction error and per-
ceptual similarity. Notably, the results indicate strong potential for integrating
these reconstructions into VR applications for space simulation and exploration,
supporting our initial hypotheses.
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Inpainting the Red Planet: Diffusion Models for the Reconstruction of Martian Environments in Virtual Reality Giuseppe Lorenzo Catalano1 and Agata Marta Soccini1 1Computer Science Department, Universit`a degli Studi di Torino, Corso Svizzera 185, Torino, 10149, Italy This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Abstract Space exploration increasingly relies on Virtual Reality for several tasks, such as mission planning, multidisciplinary scientific analysis, and astronaut training. A key factor for the reliability of the simulations is having accurate 3D representations of planetary terrains. Extraterrestrial heightmaps derived from satellite imagery often contain missing values due to acquisition and transmission constraints. Mars is among the most studied planets beyond Earth, and its extensive terrain datasets make the Martian surface reconstruction a valuable task, although many areas remain unmapped. Deep learning algorithms can support void-filling tasks; however, whereas Earth's comprehensive datasets enables the use of conditional methods, such approaches cannot be applied to Mars. Current approaches rely on simpler interpolation techniques which, however, often fail to preserve geometric coherence. In this work, we propose a method for reconstructing the surface of Mars based on an unconditional diffusion model. Training was conducted on an augmented dataset of 12000 Martian heightmaps derived from NASA's HiRISE survey. A nonhomogeneous rescaling strategy captures terrain features across multiple scales before resizing to a fixed 128 × 128 model resolution. We compared our method against established void-filling and inpainting techniques, including Inverse Distance Weighting, kriging, and Navier-Stokes algorithm, on an evaluation set of 1000 samples. Results show that our approach consistently outperforms these methods in terms of reconstruction accuracy (4-15% on RMSE) and perceptual similarity (29-81% on LPIPS) with the original data. 1 16 Oct 2025 Figure 1: The generative capabilities of diffusion models provide realistic restorations of degraded acquisitions, producing structurally consistent representations suitable for virtual environments. (Left) A degraded Martian heightmap depicting impact craters, with regions composed of missing values. (Right) The same heightmap after reconstruction, using our diffusion-based approach; the inpainted regions are explicitly highlighted. 1 Introduction The benefits of Virtual Reality (VR) for space exploration are being investigated with growing interest in the recent years [1]. Use cases include the training of future astronauts [2], the remote operation of existing rovers [3] and the interactive real-time visualization of spacecraft diagnostics in immersive environments [4, 5]. Additionally, such technologies can enable detailed morphological studies of extraterrestrial surfaces for mission planning [6]. Planet Mars specifically is an active object of study by the scientific community, and realistic representations of Martian landscapes is vital for the success of these missions. To be effective, many of these applications rely on the ability to integrate and accurately represent the terrains of various celestial bodies in VR [7]. Surfaces are commonly represented as heightmaps, a type of raster grid in which each pixel encodes a specific altitude value, that are particularly well suited for representing structured data in immersive virtual environments [8]. The problem we address is that most of the data sources regarding extraterrestrial surfaces are captured by spacecraft, such as the Mars Reconnaissance Orbiter [9], thus presenting additional challenges in being acquired and transmitted given the difficult environment faced by the instruments. These factors contribute to a scenario in which, unlike Earth where detailed and abundant measurements are often accessible, significantly less information is available. As a consequence, the final processed heightmaps may present missing values within the raster. This can be the consequence of errors during the acquisition of the original signal, data loss during transmission from the orbiter, or ambiguities arising during the processing phase that converts images into heightmaps. In such cases, performing another full acquisition of the raw data may be prohibitively expensive, or even impossible if the dataset refers to past events. Data gaps can severely hinder the usability of heightmaps for research and 2 mission planning purposes. To face the problem, we can use void-filling methods in order to restore the invalid regions through suitable approximations. These methods typically rely on interpolation techniques or statistical analyses of the valid neighbourhoods to estimate the missing values [10]. Moreover, given the raster-based structure of heightmaps, which can be encoded and visualized as images, a wide range of inpainting algorithms for the modification or reconstruction of selected regions within a digital image can be leveraged as well [11]. Recently, diffusion models [12, 13] have outperformed other deep learning-based approaches for image generation, such as Generative Adversarial Networks (GANs) [14] and Variational Autoencoders (VAEs) [15]. These models generate content by reversing a Gaussian diffusion process that progressively adds noise to the input data. Diffusion models have shown superior performance in generative tasks [16], and have rapidly become the state of the art in many image-related applications; nonetheless, active research is being conducted in order to assess the performance of such architectures in relation to other kinds of data [17]. Diffusion models have already been studied for heightmap generation and restoration as well [18, 19]. However, these approaches are not suitable for the reconstruction of Mars because: i) these models were trained and tested only on terrains related to limited regions of Earth, making them potentially susceptible to overfitting over specific geometrical features, and not necessarily suitable for other celestial bodies; ii) these approaches rely on conditional generation, supposedly bringing further overfitting on specific data distributions. Whereas this approach may be effective for Earth-related tasks, such auxiliary information is often unavailable in the case of Mars; iii) the suggested approaches do not evaluate their results from a visual standpoint as well, only resorting on quantitative assessments and brief overviews in 2D that say little in terms of the reconstruction quality for ultimately using them in VR. In this context, we applied our method based on unconditional diffusion models, namely models that do not require further contextual information during generation, for heightmap reconstruction; our hypotheses were formulated as follows: • H1. Our method restores Martian surfaces smoothly and consistently, producing visually coherent geometrical features that are well-suited for 3D visualization and simulations in virtual reality. • H2. The digitally reconstructed Mars surfaces are objectively similar to the original terrains, even though data sources are particularly scarce. The contributions of this work can be summarized as follows. I) Unconditional diffusion-based reconstruction framework for planetary terrain inpainting, addressing the lack of auxiliary conditioning data for extraterrestrial surfaces such as Mars. II) Augmented Martian terrain dataset derived from NASA's HiRISE survey, with training which incorporates a non-homogeneous rescaling strategy to capture multi-scale terrain features before standardizing resolution to 128 × 128. III) Comprehensive evaluation against void-filling and inpainting 3 methods (Inverse Distance Weighting, kriging, and Navier-Stokes), using both quantitative error metrics (RMSE, MAE, PSNR, EMD) and perceptual metrics (LPIPS, SSIM, FID). IV) Performance gains of up to 15% in RMSE and 81% in LPIPS, producing geometrically consistent reconstructions suitable for VR-based planetary visualization and analysis. The paper is organized as follows: Section 2 reviews related work, Section 3 presents the proposed method, Section 4 describes the experiments, Section 5 reports the results, Section 6 discusses the findings, and Section 7 concludes the paper. 2 Related Work 2.1 Diffusion Models Diffusion models [12] are a family of generative methods that are based on reversing a diffusion process, namely a procedure that progressively adds noise to a signal over a series of discrete timesteps, until the original input is transformed into pure noise. By learning the reverse of this process, it becomes possible to generate new synthetic data starting from random noise. The forward diffusion is modeled as a discrete-time Markov chain that adds Gaussian noise across timesteps t = 0, . . . , T where t = 0 corresponds to an image without noise and t = T is pure Gaussian noise. Each intermediate corrupted sample xt can be expressed in closed form, with regards to the original image x0 and the timestep t: xt ∼N(√ ̄αtx0), (1- ̄αt)I), where ̄αt is a set of fixed constants. This formulation allows for efficient training, as corrupted samples can be directly generated from clean data without simulating the entire chain. In recent years, several mathematical formulations and frameworks based on this principle have emerged, particularly in the domain of image synthesis. One of the most prominent examples is given by Denoising Diffusion Probabilistic Models (DDPMs) [13], in which the actual denoising model, typically a U-Net [20], is trained to predict the noise component at each timestep. The generation process begins by sampling xT from a random distribution, xT ∼N(0, I), and iteratively removing the predicted noise through the learned reverse process. The procedure ends when x0, a synthetic image without noise resembling the distribution of the training data, is produced. Subsequent works focus on speeding up the inference process, which requires the entire diffusion chain to be traversed in the case of DDPMs. An example is given by Denoising Diffusion Implicit Models (DDIMs) [21], which reformulate the mathematical process but keep the same training objective, making it possible to skip inference of some timesteps while keeping the same trained denoising networks. Another example is given by Latent Diffusion Models (LDMs) [22], which leverage a pre-trained autoencoder to perform denoising in the latent space. With these regards, Diffusion Transformers (DiTs) [23] build upon previous works by leveraging the Transformer [24] architecture as the noise predictor in the DDPM formulation. 4 2.2 Surface Void-Filling Techniques To restore invalid regions within a heightmap, various approaches can be employed to interpolate the missing values between valid pixels. The general idea is to exploit the spatial information contained in neighboring values to estimate the unknown ones [10]. Simpler methods are based on nearest-neighbor strategies or spline interpolation, while more advanced techniques involve weighted estimations. Among the latter, Inverse Distance Weighting (IDW) and kriging are two of the most commonly used. IDW [25] computes a weighted average of nearby valid pixels, assigning greater influence to those closer to the missing point. Kriging [26] produces a weighted average as well, but the weights are derived from a statistical model of spatial autocorrelation through the use of a semi-variogram. The heightmaps that are part of the HiRISE dataset, the main source for Martian terrains (which is further discussed in Section 3.1), are processed by linearly interpolating the areas with too much value uncertainty [27]. In general, the performance of each method depends on the morphological characteristics of the surface patch that is being reconstructed [28]. Deep learning generative models have been successfully employed for surface restoration. GANs were tested in recent years in the context of terrain surfaces [29, 30, 31]; diffusion models have also started to be analyzed in this context as well [18, 19]. As previously mentioned, one major drawback of such methods is that they rely on a conditioned approach, where the generation process is guided by supplementary data; more specifically, conditioning is performed using the shape of the binary mask [18], or even the general structure of the terrain that must be known beforehand [19]. Moreover, the studies were performed on heightmaps representing specific areas of Earth. These reasons may lead to overfitting over small use cases, also limiting the generalization of the approaches to different scenarios, making their use for Mars more challenging. Finally, the studies were not concerned about the visual inspection of the restoration output, with a focus that strayed from the representation in VR. Our aim with these regards is to define a framework that can be applied in diverse use cases, applicable to terrain morphology but also for other purposes, which do not require sets of additional data of different kind. 2.3 Image Inpainting Algorithms Inpainting is a well-known task in the digital image processing domain. The goal is to fill specific portions of an image, typically selected using a binary mask, with new values that are coherent with the adjacent valid portions. This task is usually performed to remove unwanted features or to restore partial degradations. Several inpainting algorithms have been developed throughout the years, based on different principles, such as the fast marching method [32] or the Navier-Stokes fluido-dynamics equations [33], to expand the valid parts of the signal into the missing regions. Other methods rely instead on computing patch correspondences for finding the best match [34]. The advent of generative deep learning models for image manipulation had a consistent impact for the 5 task of inpainting as well [35]. Diffusion model-based methods were successfully implemented for this purpose as well. Notable example include Palette [36] and Stable Diffusion [22], however both rely on a generation that is conditioned by the inpainting binary mask. A different approach is presented by RePaint [37], an inpainting algorithm which is based on unconditioned diffusion models. More specifically, a pre-trained DDPM is used for performing the unconditional reverse diffusion process; by itself, this process would produce a result with no correlation with the image that needs to be inpainted. After each denoising step, however, the valid parts of the image are overwritten on the intermediate result, so that the next step will be guided towards generating a final result that is coherent with the original input. In order to obtain better results, a resampling mechanism is also implemented; noise is added and then removed multiple times, in order to generate patches that are more consistent with the valid parts. Given its flexibility with regard to the inpainting masks, guaranteed by the unconditional DDPM, we based our method on this algorithm. To the best of our knowledge, this is the first work to address the specific task of Martian terrain restoration using diffusion models, achieving performance superior to currently employed methods. Our approach leverages the absence of additional data constraints afforded by unconditional models to tackle the unique challenges of Martian datasets. Furthermore, the model is trained on features sampled at multiple scales from across the entire planet, enhancing its ability to generate accurate and coherent terrain features. 3 Methods 3.1 Martian Surfaces Dataset The training of the DDPM model required a properly structured dataset of Martian terrain heightmaps. The most prominent repository with this regard is provided by HiRISE [38], a set of high-performance cameras mounted on the Mars Reconnaissance Orbiter (MRO), dedicated to capturing high-resolution imagery of the planet's surface. HiRISE is capable of acquiring stereoscopic image pairs, which undergo a dedicated processing pipeline to extract the corresponding heightmaps [39, 27]. The resulting data products are made publicly available via an online repository1 in the form of Digital Elevation Models (DEMs), a specific type of terrain representation that includes metadata for geo-localization, such as coordinate reference systems and coordinates of each acquisition. These terrains are a valuable source for investigating the morphology of the planet, given their capability to represent detailed patterns due to their high resolution [40, 41]. This use case presents an interesting testbed: due to the challenges faced during data acquisition and transmission, certain regions of the images may be difficult to process or entirely unavailable, thus resulting in interpolated or missing values as already discussed (see figure 2). 1https://www.uahirise.org/dtm/ 6 Figure 2: RGB representation of Martian surfaces, obtained from the processing of HiRISE stereo pairs. It may occur that missing data are interpolated during processing (the squared patterns within the crater in the left image) or entirely unavailable (the black line in the right image). Approximately 1,150 heightmaps are available at the current time. However, these are not immediately suitable for direct use in training. Indeed, diffusion models require normalized images of fixed size (128 × 128 in our case) as input, whereas HiRISE DEMs typically cover much larger areas, in the order of 103 pixels per side. Furthermore, the data are stored in the Planetary Data System (PDS) data format, which is not supported by most image processing libraries. For these reasons, a new dataset was generated to enhance file interoperability and readability. The resulting training set consists of 12,000 normalized heightmaps, stored using the TIFF file format. Each element is a random crop extracted from the original HiRISE survey, with side lengths randomly selected between 512 and 2048 pixels for each sample. During training, these crops are rescaled in a non-homogeneous way to 128 × 128, using nearest-neighbour interpolation. The main reason behind this choice, made mainly to support the investigation of H2, is to provide the model with knowledge of terrain features that may develop over portions that are larger than the model resolution. This operation enables data augmentation, helping to compensate for the relatively small size of the initial dataset which is an an inherent challenge in this domain. 7 GDAL2 was used to process the original PDS data and convert them to TIFF. 3.2 Training of the Diffusion Model In order to assess both H1 and H2, we trained an unconditional DDPM model, capable of generating realistic Martian terrains without the need for textual or visual prompts. To this purpose, training was performed at a resolution of 128 × 128 pixels, using the augmented dataset described in Section 3.1. As previously mentioned, we believe that by feeding the model random crops of varying original size, the model can learn surface features at multiple different scales. Training was conducted over 100 epochs, with a batch size of 16 samples and a total number of T = 1000 timesteps. The main neural network specifications, such as the model architecture and loss function, were chosen according to the original DDPM formulation [13]; specifically, we employed a U-Net to predict the total noise in an image at an arbitrary timestep of the diffusion process. The total training time was approximately 8 hours, using 2 NVIDIA A40 GPUs. Samples from the trained DDPM after 100 epochs are shown in figure 3. The implementation used for DDPMs is based on the Huggingface Diffusers3 library. Figure 3: Evaluation of the trained unconditional DDPM after 100 epochs of training with Martian terrains. 3.3 Inference Inference with the trained DDPM model is performed by using the RePaint algorithm on normalized heightmaps. Specifically, a selected portion of Martian terrain is resized to the fixed resolution of 128 × 128 pixels and given as input, along with a binary mask indicating which parts need to be restored. The main parameter configuration suggested by the original authors was employed; the complete inference takes around 44 seconds on a single NVIDIA A40 GPU. It is important to highlight the role of the mask in the inference process: while it is indeed used by the inpainting algorithm to distinguish between valid and missing pixels, it is however not directly used by the diffusion model itself, 2https://gdal.org/en/stable/ 3https://huggingface.co/docs/diffusers 8 which operates unconditionally. Figure 4 illustrates the input and output of our method. Figure 4: Input (left) and output (right) of our approach to Martian surface reconstruction, represented as 2D rasters. 4 Experiments We run our tests on a dataset of 1000 random crops, sampled from the HiRISE terrain survey following the same procedure described in Section 3.1. To conduct the experiments, we first sampled crops containing only valid pixels (i.e., without missing values) to serve as the baseline, and then generated masks to remove selected known data. We shaped the masks in a way that resembles aliased lines, as this is a kind of artifact pattern that tends to commonly occur in Martian terrain heightmaps (refer to Figure 2 for a visual reference). We randomly generated the masks by varying different parameters of the line, such as the orientation or the number of segments. Examples of different mask shapes are shown in Figure 5. Figure 5: Example mask shapes that were used for evaluation. The masks are artificially generated to simulate common patterns that occur in data losses. 9 4.1 3D Visualization One of the main goals of this study, aimed at the investigation of H1, was to evaluate the restoration quality of our proposed diffusion-based method, when rendered as 3D surfaces within virtual environments. Indeed, we aimed to perform void-filling that would result in geometrically and visually coherent meshes, making the restored surfaces suitable for use in VR applications. In order to carry out an in-depth evaluation, we used Autodesk Maya4, a software for 3D modeling and rendering. Its built-in Python API allowed us to automate the import process of heightmaps and create meshes for them. Additionally, we leveraged the Arnold renderer5 to create custom shaders capable of highlighting restored areas based on the binary masks, and to produce detailed visualizations under varying lighting conditions for qualitative analysis (Figure 6). Figure 6: Input (left) and output (right) of our approach to Martian surface reconstruction, represented as 3D surfaces and rendered with Arnold in Maya. 4.2 Comparative Evaluation In order to assess the performance of our method for Mars surface restoration (H2), other inpainting and void-filling algorithms were evaluated for comparison. The scenario of terrain restoration led us to choose Inverse Distance Weighting (IDW) [25] and kriging [26] as our primary baselines, given their widespread adoption in this context. We also included an image inpainting technique in our tests, namely the Navier-Stokes algorithm [33]. All methods were tested on the same dataset and with the same set of masks. The implementation for kriging was provided by the PyKrige library6, whereas OpenCV7 4https://www.autodesk.com/products/maya/overview 5https://www.autodesk.com/products/arnold/overview 6https://geostat-framework.readthedocs.io/projects/pykrige/en/stable/index. html 7https://opencv.org/ 10 Figure 7: Qualitative evaluation of results, represented as 2D rasters. For each group, the right column represents the absolute error between restored and original values; the lack of yellow-green pixels represents better reconstructions. was used for Navier-Stokes. For IDW we used N = 12 neighbours and a power parameter of p = 2. RMSE ↓ MAE ↓ PSNR ↑ EMD ↓ Ours 0.0752 (-) 0.0548 (-) 39.4494 (-) 0.1366 (-) IDW 0.0864 (+14.9%) 0.0603 (+10%) 35.7504 (-9.4%) 0.1756 (+28.6%) Navier-Stokes 0.0830 (+10.4%) 0.0580 (+5.8%) 36.2110 (-8.2%) 0.1699 (+24.4%) Kriging 0.0784 (+4.3%) 0.0573 (+4.6%) 39.3492 (-0.3%) 0.1572 (+15.1%) Table 1: Average results of the error-based metrics for the test run (1000 samples). Bold values indicate the best results. 5 Results 5.1 Visual Assessment The visual inspection of the tested surface reconstruction methods was conducted using both 2D plots and 3D renders. They highlighted significant structural differences between diffusion-based restoration and interpolation algorithms, with the former producing more coherent results, in support of H1. Figure 7 displays a selection of results as color-mapped 2D images; in each example, the right column shows the absolute error with respect to the ground truth. These considerations become even more evident in Figure 8, which shows 11 Figure 8: Qualitative evaluation of results, represented as 3D surfaces and rendered with Arnold in Autodesk Maya. The highlighted parts indicate the areas reconstructed using the evaluated methods. The possibility to control lighting and visualization conditions allows to perform effective assessments of the results. the same selection of results, processed as heightmaps and rendered as 3D surfaces instead. 5.2 Error Metrics Evaluation We evaluated the tested methods from a quantitative perspective as well, in order to assess the reconstruction quality in terms of error with respect to the ground truth. This analysis aimed to verify whether the visually convincing results correspond to surfaces that are indeed similar to the original ones. Such validation is of critical importance: these terrains, which are frequently used for scientific analyses and mission planning purposes, must approximate their realworld counterparts as closely as possible. In this context, for each tested sample we computed the Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Peak Signal-to-Noise Ratio (PSNR) between the ground truth and the restoration output. In addition, we also considered Earth Mover's Distance (EMD) as a measure of the discrepancy between value distributions. Table 1 reports the average results over the 1000 random surface crops that were tested. Our method outperforms the others across all metrics, suggesting that the effectiveness observed during visual assessments is also measured quantitatively, supporting H2. Kriging, which leverages information related to spatial correlation among valid pixels, is the second-best performing, yielding results that are close to those of our approach. It also interesting to note that the 12 SSIM ↑ LPIPS ↓ FID ↓ Ours 0.9660 (-) 0.0754 (-) 10.9397 (-) IDW 0.9620 (-0.4%) 0.1365 (+81%) 61.0935 (+458.5%) Navier-Stokes 0.9650 (-0.1%) 0.1181 (+56.6%) 42.0554 (+284.4%) Kriging 0.9684 (+0.2%) 0.0973 (+29%) 27.5676 (+152%) Table 2: Average results of the perceptual-based metrics for the test run (1000 samples). FID is not an average, but a single score associated to each evaluation set. Bold values indicate the best results. Navier-Stokes algorithm, typically used for image inpainting but less common in the surface void-filling domain, performs generally better than a well-known method in this context such as IDW. 5.3 Perceptual Metrics Evaluation The quantitative evaluation was further extended to include the calculation and comparison of perceptual metrics, a family of measures that are designed to model the similarity between images, as perceived by humans. Evaluations of this kind are commonly performed in image generation tasks, as they aim to approximate how closely the generated outputs resemble the training distribution, attempting to model human perceptual judgments in assessing whether the generated images "look like" the original ones. The selected metrics are the following: i) Structural Similarity Index Measure (SSIM) [42], which evaluates image similarity in terms of luminance, contrast, and structural consistency; ii) Learned Perceptual Image Patch Similarity (LPIPS) [43], which computes the distance between latent feature representations of two images as extracted by a pre-trained neural network; iii) Fr ́echet Inception Distance (FID) [44], which tests which compares the distribution of synthetic images against that of real images using activations from an Inception network. Although the results may be biased by the network being trained on the ImageNet dataset, we decided to employ it in addition to the others as a further evaluation method. It is worth pointing out that, while some perceptual metrics are inherently related to error-based ones, they are often sensitive to different types of visual discrepancies [45]. Table 2 presents the average results for SSIM and LPIPS across the tested 1000 samples; the FID score is a single value for each method, obtained by comparing the overall distribution of the results against the distribution of ground truth images. Our method achieves superior results in both LPIPS and FID, whereas kriging yields a slightly higher average SSIM score; on this front, H2 was supported for the most part, while also giving further insights in support of H1. Notably, Navier-Stokes continues to outperform IDW in this evaluation as well. 13 6 Discussion The methodology we presented showed to be an effective approach for reconstructing missing sections of Martian terrains. Visual assessment. Visual assessments of the reconstructed heightmaps, conducted both using 2D color-mapped rasters and 3D renderings, provided valuable insights. We paid particular attention to fidelity and structural consistency, especially near the boundaries between valid and missing regions. Our method proved effective with these regards, as it blends the two areas seamlessly and generates realistic surface patches with smooth transitions, while at the same time reconstructing patterns that were already present. IDW and NavierStokes exhibit similar behaviour between them, as both expand the values of valid pixels into the missing regions. Navier-Stokes tends to appear slightly smoother than IDW, likely due to the fluid dynamics principles on which the algorithm is based. Both methods show discontinuities near the center of the reconstructed regions, which is particularly suboptimal when considering these terrains as testbeds for rover simulations or morphological studies. Kriging also yields interesting results: it performs effectively on flat regions and adapts with reasonable accuracy to some complex structures, but it struggles to capture high-frequency patterns such as dunes or crater ridges; however, the resulting artifacts generally have a less disruptive impact on the overall surface structure than those produced by IDW and Navier-Stokes. These findings support hypothesis H1, confirming that the restored surfaces exhibit high structural fidelity and accuracy. Objective Measurements. Our method produces reconstructions that are also accurate in terms of objective accuracy with regards to the original terrains. It achieved the best performance across multiple error metrics, namely Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Peak Signalto-Noise Ratio (PSNR) and Earth Mover's Distance (EMD). This quantitative analysis verified hypothesis H2, which advocated for a reliable objective restoration of missing Mars regions. Finally, we evaluated a series of perceptual similarity metrics that are commonly employed to model human perception in the context of generative models. Our method obtained the best scores in Learned Perceptual Image Patch Similarity (LPIPS) and Fr ́echet Inception Distance (FID), whereas kriging achieved the highest Structural Similarity Index Measure (SSIM) score. This outcome might stem from the structure comparison component that is present in SSIM computations, which may penalize reconstructions that are structurally coherent yet mismatched (as produced by our method) compared to flatter, less detailed restorations (as generated by kriging). These results point to a solid and robust framework, and to an additional overall confirmation of H2, despite not having reached the best performance on all fronts. The relatively small size of the training dataset may be a factor with this regard, however the domain of extraterrestrial terrains is inherently data-scarce and even in these conditions we achieved notable results. Qualitative vs. Quantitative Insights. The relationship between qualitative evaluation and quantitative metrics offers further valuable insights. De14 spite the relatively strong performance of kriging, which achieved the best scores among interpolation-based methods, and also outperformed our method in terms of SSIM, the visual assessment clearly showed that the diffusion-based approach provides with more realistic and coherent reconstructions. This can be largely attributed to the statistical nature of kriging, a model which computes an autocorrelation semivariogram to specifically minimize the mean squared error of the approximation function, and further underlines the importance of subjective evaluation of the results. Our initial hypotheses have been thoroughly addressed and verified. We provided insights into the capabilities of diffusion models to encode large amounts of data, and applied them to reconstruct heightmaps in ways that are well-suited for being represented as accurate 3D virtual terrains. Results indicated that our trained model represents a highly flexible tool for surface reconstruction at different scales. Moreover, we showed that unconditional models in particular can be successfully leveraged for restoration purposes, without having to rely on additional information aside from the raster. These results open up promising scenarios for the application of deep generative models in extraterrestrial contexts, characterized by similarly limited data availability. Limitations. The current work presents some limitations. First, both the model architecture and the overall inpainting pipeline offer substantial room for optimization. Users do not have the possibility to define custom crop regions or binary masks in real time, that limits the ability to perform precise and localized inpainting during immersive sessions. Currently, this remains a technical barrier due to the computational cost of the method. Comparisons with other diffusion-based approaches [18] still need to be evaluated. These methods have been mostly developed for geoscientific applications and tested on Earth-related tasks; despite our emphasis on the unconditional nature and the the generalization potential of our approach, they make for valuable testing grounds nonetheless. The applicability of this method to other extraterrestrial bodies, such as the Moon, has yet to be assessed. The general framework of augmenting existing terrain surveys can be adapted to diverse scenarios, though each must be customized to the specific environmental and data constraints. Future Work. There are several directions in which the current work can be further expanded. As a next step, we plan to conduct a user study to evaluate the perceived quality and usability of the reconstructed terrains from the perspective of participants. This may include assessments of the Sense of Presence, interaction techniques with the reconstructed surfaces in scientific space exploration simulations, and navigation methods within the virtual Martian environments. Regarding our inpainting method, multi-resolution models and upgraded architectures such as Denoising Diffusion Implicit Models (DDIMs) [21], Latent Diffusion Models (LDMs) [22] or Diffusion Transformers [23] may enable faster and more precise generations, along with the possibility to employ this method in diverse practical uses. Moreover, alternative configurations of the RePaint algorithm can be explored in order to achieve more efficient inpainting results, ideally with a smaller number of required steps. Such functionalities would require a significantly faster inference process, allowing user input 15 to guide the reconstruction dynamically, potentially within immersive environments. In addition, this line of research could contribute to the development of experimental setups for the human evaluation of AI-generated content in VR environments. While current studies in this area primarily focus on 2D imagery [46], we believe the current work has potential for expanding these evaluations to 3D contexts. Ethics and Replicability Ethics Committee approval was not required for this study, as the experiments did not involve human participants or any living beings. Furthermore, the inference was performed on publicly available open data, ensuring transparency and replicability. The processed data are available upon request. 7 Conclusions This work presented a method for the restoration of degraded Martian heightmaps using unconditional diffusion models. 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2510.14755
|
Impact of rotation on a cold atom interferometer and compensation strategy
No´emie Marquet,1, ∗Yannick Bidel,1 Malo Cadoret,2 Alexis Bonnin,1 Sylvain Schwartz,1 Phuong-Anh Huynh,3
Alexandre Bresson,1 Antoine Godard,4 Franck Pereira Dos Santos,5 Olivier Carraz,6 and Nassim Zahzam1, †
1DPHY, ONERA, Universit´e Paris-Saclay, 91120 Palaiseau, France
2LCM, CNAM, 93210 La Plaine Saint-Denis, France
3DPHY, ONERA, Universit´e Paris-Saclay, 92320 Chˆatillon, France
4ONERA, Universit´e Paris-Saclay, 91120 Palaiseau, France
5Laboratoire Temps Espace, Observatoire de Paris, Universit´e PSL,
Sorbonne Universit´e, Universit´e de Lille, LNE, CNRS, 75014 Paris, France
6ESTEC,ESA, Noordwijk, The Netherlands
(Dated: October 17, 2025)
Rotations play a detrimental role in achieving ultra-high-performance inertial measurements
with an atom interferometer, leading potentially to a total loss of interference contrast and the
emergence of dominant phase shift biases. This becomes particularly significant when considering
operation in dynamic conditions such as those encountered in Earth orbiting satellites in the
perspective of future space gravity missions on-boarding a cold atom accelerometer. We study in
this context the impact of rotation on the phase shift and contrast of an atom interferometer and
investigate mitigation strategies.
An analytical model is derived and compared to experimental
demonstrations carried out using an original setup in which the well-controlled proof-mass of a
space electrostatic accelerometer is used as the retro-reflection mirror of a cold atom gravimeter. By
properly counter-rotating the electrostatic proof-mass, we demonstrate for instance the possibility
of recovering the interferometer contrast, otherwise equal to zero, to a level better than 90%,
in both cases of constant angular velocities or in presence of angular accelerations. Our results
demonstrate the possibility to perform high performance inertial measurements with a cold atom
interferometer in a challenging environments.
INTRODUCTION
Atom interferometry addresses the needs for high pre-
cision measurements in a wide range of domains, span-
ning from fundamental physics with for instance the de-
termination of the fine structure constant [1, 2], or with
the test of the Weak Equivalence Principle [3], to more
applied domains such as inertial navigation [4, 5], or
space gravimetry [6–14]. As this technology gets more
mature, inertial sensors such as accelerometers or gy-
roscopes based on atom interferometry begin to move
outside the laboratory. Cold-atom accelerometers have
already been used for the last few years for field applica-
tions onboard trucks, boats, or planes [15–19]. However,
such sensors are highly affected by the rotations of the
dynamic environment they operate in [13, 20–26], which
potentially leads to complete loss of interferometer con-
trast and significant phase-shift errors.
To avoid such detrimental impacts of rotation, gyro-
stabilized platforms have been efficiently used [17, 19],
but this solution comes with a significant increase in in-
strument size, power consumption and cost, limiting its
range of applications. An alternative approach consists in
counter-rotating the mirror on which the laser, enabling
the atom interferometry measurement, is retro-reflected
[21, 22]. This mirror can be seen as the reference for the
∗Present address noemie.marquet@lkb.upmc.fr
† nassim.zahzam@onera.fr
atomic measurement, and maintaining it non-rotating in
the inertial frame seems a promising mitigation method.
In laboratory environments, this approach was previously
implemented using a piezoelectric-driven tip-tilt mirror
to prevent Earth’s rotation from significantly impacting
static cold-atom interferometers [22, 27–29]. The same
method was also recently demonstrated to show promis-
ing results for higher rotation rates up to 250 mrad s−1
on a cold-atom interferometer, ultimately dedicated to
field applications [26].
Overcoming these rotation issues is one of the main
challenges to deal with to operate an atom accelerome-
ter under dynamic conditions, particularly in the context
of future space gravimetry missions [9, 13, 25]. Nadin-
pointing navigation is then preferred, so the satellite is
orbiting around the Earth at a nominal angular velocity
of ≈1.1 mrad s−1. In the absence of rotation mitigation
strategies, the contrast and phase shifts of the interferom-
eter are impacted to a level preventing high performances
measurements.
In this context, we report here a detailed study analyz-
ing the impact of rotations on a cold-atom accelerometer.
This work is conducted through the development of an-
alytical models showing results in terms of contrast and
phase for several situations: when the whole instrument
is subjected to rotation, when only the interferometer
mirror is rotated, and when the rotation compensation
strategy is implemented. The results of the models are
confronted with experimental studies carried out with
an original setup based on a hybrid atomic-electrostatic
accelerometer [13], composed of a cold-atom gravimeter
arXiv:2510.14755v1 [physics.atom-ph] 16 Oct 2025
2
associated to a space electrostatic accelerometer (EA)
[30], adapted to ground operation.
In this configura-
tion, the laser used for atom interferometry is directly
retro-reflected on the EA’s proof-mass acting as the ref-
erence mirror for the atom interferometer. The proof-
mass position is precisely controlled by use of electro-
static forces allowing its rotation and continuously mon-
itored by capacitive detectors.
This innovative design
of a hybrid atomic-electrostatic instrument is considered
as one of the promising accelerometer candidates for fu-
ture space missions [11, 13, 14]. Our experimental setup
allows us to rotate the whole instrument with rotation
rates in the range of a Nadir-pointing satellite (of the or-
der of 1 mrad s−1) for which significant detrimental im-
pacts on a ground cold-atom gravimeter occur despite
smaller interrogation times. As another original aspect
of the work, we also considered non-constant rotation
velocities leading to strong Euler acceleration whose ef-
fects can also be mitigated by counter-rotating the in-
terferometer mirror. This work aims at a deeper under-
standing of the various effects that limit the performance
of a cold-atom accelerometer and proposes an original
technical approach based on electrostatic actuation of an
EA’s proof-mass, coupled to the cold-atom interferome-
ter, rather than piezoelectric actuation of a mirror mount
as commonly proposed.
In this paper, we first present in Sec. I the model
used to derive the impact of rotations on a cold-atom
accelerometer.
This model considers the whole sensor
rotation, the mirror rotation and their effects on both
the phase shift and contrast of a Mach-Zehnder cold-
atom interferometer. Then, the laboratory prototype is
described, as well as the measurement protocol in Sec. II
and Sec. III. Lastly, in Sec. IV, experimental contrasts
and phase shifts of a rotating interferometer are inves-
tigated in the light of the model developed in the first
section. The impacts of the sensor and the mirror ro-
tations on both the phase shift and contrast are studied
separately. The rotation compensation method is then
characterized by considering two different angular mo-
tions: the first with an important angular velocity of
1.43 mrad s−1 and the second with an important angular
acceleration of 43.5 mrad s−2.
I.
PHASE AND CONTRAST MODEL
A.
Atom interferometer without rotation
Atom interferometers are based on the principle of su-
perposition between two atomic states. Interactions be-
tween a light field and the atom drive the manipulation
of quantum superposition of atomic states. In this work,
we consider stimulated two-photon transitions, referred
as Raman transitions, to implement atomic mirror and
beam splitters [31, 32]. The studied atom interferome-
ter is based on a Mach-Zehnder configuration [31, 33],
involving a sequence of three Raman pulses of effective
wavevector ⃗keff as described in Fig. 1. A two-frequency
laser (corresponding to the wavevectors ⃗k1 and ⃗k2), retro-
reflected on a mirror, addresses Raman transitions be-
tween two stable hyperfine states of 87Rb atoms. The
two states are a ground state |g, ⃗p⟩and an excited state
e, ⃗p + ¯h⃗keff
E
with ⃗p the initial momentum of the atom.
After a cooling step, the atoms are released in free fall
at time t = −t0 −T. The atoms fall for a time t0 before
the beginning of the interferometer, defined as the time
of the first laser pulse. The total interrogation time of
the interferometer is 2T, T being the time separation be-
tween two consecutive laser pulses.
X
t
FIG. 1. Space-time diagram of a Mach-Zehnder atomic ac-
celerometer measuring along the X axis. The three long red
rectangles are the laser pulses π
2 −π −π
2 driving the atomic
transitions between the ground state |g, ⃗p⟩and the excited
state
e, ⃗p + ¯h⃗keff
E
. The atoms free fall for a time t0 before
the start of the interferometer. The first and third laser pulses
put the atoms in a superposition of states and have a dura-
tion τ. The second pulse redirects the atom cloud and lasts
2τ. The free evolution of the atoms between the pulses lasts
T ≫τ. Experimentally, t0 = 10 ms, T = 46 ms and τ = 4 µs.
The laser effective wavevector ⃗keff is the difference be-
tween one wavevector of the incoming laser ⃗k1 and one
of the reflected laser ⃗k2.
⃗keff(t) = ⃗k1(t) −⃗k2(t).
(1)
For a single atom, the interferometer output phase shift
∆Φ is a linear combination of the laser phases φ(t) at the
three interaction times [33] as expressed in Eq. (2). In
this calculation, the laser pulses are considered infinitely
short and the mirror and beamsplitter laser pulses are
considered perfect. ∆Φ is thus expressed as follows:
∆Φ = φ (−T) −2φ(0) + φ(T).
(2)
where φ(t) = ⃗keff(t) · ⃗r(t) is the laser phase at time t
for a plane wave with ⃗r(t) the distance vector between
the retro-reflection mirror and the atom. The atom tra-
jectory is thus approximated by the mean trajectory be-
tween the two arms of the interferometer. The mirror can
3
be considered as the laser phase reference. The measure-
ment axis corresponds to the direction of the effective
wavevector ⃗keff.
The instrument aims to measure the
atom acceleration along ⃗keff derived from the interferom-
eter output phase shift. As a first approximation, the
interferometer phase is proportional to the atom acceler-
ation aM in the mirror frame. ∆Φ can thus be approxi-
mated as follows:
∆Φ ≈⃗keff · ⃗aMT 2.
(3)
In principle, the detected signal of the interferometer
is the probability Pe for an atom to be detected in the
excited state at the output of the interferometer. It de-
pends sinusoidally on the phase shift expressed in Eq. (2):
Pe = P0 −C
2 cos(∆Φ).
(4)
with C the contrast and P0 the offset of the interferom-
eter output signal for one atom.
Experimentally, this quantity is derived from fluores-
cence measurements of the atomic population at the two
outputs of the interferometer. The detected signal gath-
ers the fluorescence emitted by the whole atomic cloud
and therefore the measured probability P e can be con-
sidered as the average probability over the atom cloud
neglecting the detection system response:
P e = P 0 −C
2 cos
∆Φ
.
(5)
with ∆Φ the mean phase shift, C the mean contrast and
P 0 the mean offset of the interferometer output signal
considering the whole detected atom cloud. Note that
we neglect here the response of the detection system
that could be taken into account as done in [34].
Since each atomic phase depends on position and ve-
locity, the average probability can be written as a func-
tion of D⃗v, the velocity distribution, and D⃗r, the position
distribution of the atom cloud:
P e =
ZZ
Pe(⃗r,⃗v) D⃗v D⃗r d⃗v d⃗r.
(6)
Thus, the normalized contrast can be deduced from the
phase shift variance over the atomic cloud which can be
computed knowing the position and velocity distributions
under the hypothesis of Gaussian velocity and position
distributions:
C
C = exp(−V ar(∆Φ)/2)
(7)
Eq. (7) is obtained considering the contribution of each
atom as equivalent; i.e. C does not depend on the veloc-
ity and position of the atom. A more complete model
could be ultimately elaborated considering this aspect
[35].
B.
Rotation of the atom interferometer
As previously reported, the rotation of the accelerom-
eter sensor affects the phase shift and contrast at the
output of the interferometer through Coriolis, centrifu-
gal, and Euler acceleration terms [13, 20–22, 24–26, 29].
The phase shift being determined by averaging over
the atom cloud, its alteration due to rotation is related
to the mean initial velocity and position of the whole
atom cloud. On the other hand, the contrast loss due
to rotation is related to the size and temperature of the
cloud, which could lead in some cases to a complete loss
of signal and consequently to the impossibility to derive
an acceleration measurement [13, 22, 26].
In this section, the calculation of the induced phase
shift
and
contrast
loss
is
detailed
considering
an
atomic interferometer rotating around only one axis:
⃗zS = ⃗zL = ⃗zM.
The different frames used in this
calculation are illustrated in Fig. 2.
The directions
of the unitary vectors are defined in each frame.
RL = {O, ⃗xL, ⃗yL,⃗zL} is the laboratory frame, which is
considered inertial in this calculation, where O is the
center of rotation of the sensor and ⃗xL is the unitary
vector in the opposite direction of Earth’s gravity ⃗g.
Earth’s rotation rate is considered constant during the
experiment, and its effect on the interferometer can be
considered negligible. RS = {O, ⃗xS, ⃗yS,⃗zS} is the sensor
frame, where ⃗xS is aligned with the direction of ⃗k1 and
the normal vector to the table supporting the sensor
head (in blue on Fig. 2). RM = {M, ⃗xM, ⃗yM,⃗zM} is the
mirror frame, where M is the center of rotation of the
mirror, ⃗xM is the normal vector to the mirror (in yellow
in Fig. 2), and ⃗xr is the unitary vector in the direction
of the reflected laser ⃗k2.
In this section, the mirror
is not rotating in the sensor frame, then RM ≡RS
and ⃗xr = ⃗xS: ⃗k1 and ⃗k2 are perfectly counter propagating
The values of angle θS between ⃗xS and ⃗xL at pulse
times −T, 0, and T can be expressed as follows:
θS(−T) = θ0
S −Ω0
ST + 1
2
˙Ω0
ST 2,
(8)
θS(0) = θ0
S,
θS(T) = θ0
S + Ω0
ST + 1
2
˙Ω0
ST 2,
where Ω0
S is the mean angular velocity between the
third and the first laser pulse and ˙Ω0
S is the mean angular
acceleration over the three laser pulses sequence.
To compute the laser phase at each pulse, the effective
wavevector of the laser has to be expressed as a function
of the rotation parameters. As defined in Eq. (1), the ef-
fective wavevector is the difference between the wavevec-
tor of the incoming laser ⃗k1 = −k1⃗xS and the reflected
laser ⃗k2 = k2⃗xr. The unitary vector ⃗xS can be written
as ⃗xS = cos θS⃗xL + sin θS⃗yL. In presence of rotation of
4
M
O
A
FIG. 2. Description of the rotating frames: the sensor frame
RS and the mirror frame RM in rotation in the laboratory
frame RL.
The blue rectangle is the table supporting the
sensor.
This table and the incoming laser are fixed in RS
and rotate around O the center of rotation of the sensor. The
yellow rectangle is the retro-reflection mirror rotating around
M. ⃗xr is the unitary vector in the direction of the reflected
laser in the mirror and A the center of the atom cloud.
the whole sensor, the expression of ⃗keff in the laboratory
frame, RL, thus reads
⃗keff(t) = −keff
cos θS(t)
sin θS(t)
0
≃−keff
1 −θS(t)2
2
θS(t)
0
. (9)
with keff = k1 + k2 the norm of ⃗keff in the absence of
rotation. The direction of the wavevector is impacted by
the sensor rotation but not its norm.
We can decompose the classical trajectory of the mean
position of the atomic wave packet ⃗r = −−→
MA = −−→
MO+−→
OA
as a function of the position of the atom cloud relative
to the sensor center of rotation (xOA, yOA, zOA), the
relative position of the mirror to the sensor center of
rotation (xOM, yOM, zOM),
the velocities (vx, vy, vz)
and accelerations (ax, ay, az) of the atomic cloud at
the first interferometric pulse in a second lab frame
considered inertial which coincide with the sensor frame
at the first pulse R′
L = RS(t = −T). Expressing the
atomic trajectory in this frame simplifies the phase
shift and contrast expressions.
Last, this choice ex-
plains the seeming differences between the equations
of this work and previous literature. We finally obtained:
⃗r(t) =
ax
(t+T )2
2
+ vx(t + T) + xMA
ay
(t+T )2
2
+ vy(t + T) + yMA
az
(t+T )2
2
+ vz(t + T) + zMA
.
(10)
the mean atom trajectory in the second lab frame R′
L
at the first laser pulse. The different parameters of the
atomic cloud trajectory at the first pulse have to be
computed as a function of the kinetic parameters at the
launch time t = −t0 −T given in Tab. I. We consider a
non zero initial velocity ⃗vres in the lab frame RL that can
result from imbalanced laser beams during the molasses
stage [34, 36]. The sensor rotation impacts the magneto-
optical trap (MOT) position and velocity at the atoms
release and has to be taken into account through the
MOT angular position θMOT and instantaneous angular
velocity ⃗ΩMOT . The atomic velocity at launch is then
⃗v0 = ⃗vres + ⃗ΩMOT ∧⃗r0 with ⃗r0 = −→
OA(−t0 −T). The only
considered acceleration, ⃗a0 = ax0⃗xL takes Earth’s grav-
ity into account (the Earth-gravity gradient is neglected).
The velocity in R′
L is then: ⃗v1 = ⃗v0 +⃗a1t0 + 0.5⃗vrec with
⃗vrec the recoil velocity and ⃗a1 the acceleration in R′
L
leading to the value of (vx, vy, vz).
Using Eqs (2), (9), and (10), the phase shift of the rotat-
ing interferometer can be expressed as:
∆ΦS = keffT 2
"
−ax
(11)
−2Ω0
S ·
vy + 3
2ayT
−˙Ω0
S ·
yOA + vyT + 1
2ayT 2
+Ω0
S
2 ·
xOA + 3vxT + 7
2axT 2
#
.
Eq. (11) is an approximation as terms scaling in Ω0
S
3,
˙Ω0
S
2, ˙Ω0
SΩ0
S and θ0
S are neglected. The phase shift thus
depends on the atoms’ vertical acceleration and is im-
pacted by the three inertial accelerations as follows:
• Coriolis acceleration: −2Ω0
S ·
vy + 3
2ayT
;
• Euler acceleration: −˙Ω0
S ·
yOA + vyT + 1
2ayT 2
;
• Centrifugal acceleration:
Ω0
S
2 ·
xOA + 3vxT + 7
2axT 2
.
Using Eq. (6) and assuming independent Gaussian dis-
tributions with standard deviations σx, σy, σz for the
initial position and σvx, σvy, σvz for the velocity, the ex-
pression of the contrast decay as a function of the rotation
5
parameters reads
CS
C = exp
"
−k2
effT 4
2
(2Ω0
S + T ˙Ω0
S)2 · σ2
vy
(12)
+ ˙Ω0
S
2 · σ2
y
+Ω0
S
4 ·
σ2
x + (3T)2σ2
vx
!#
.
C.
Rotation of the mirror
In this section, we detail the impact on the interferom-
eter output of the retro-reflection mirror rotation only.
The considered mirror rotation vector, ⃗ΩM, is collinear
to ⃗zM axis, as shown in Fig. 2.
Similarly to the case
where the rotation is applied to the whole instrument,
the angle θM between ⃗xM and ⃗xS can be expressed at
times −T, 0, and T as a function of the mirror mean
angular velocity, Ω0
M, the mirror mean angular accelera-
tion, ˙Ω0
M, and the mirror angle at the second pulse θ0
M
in the same way as in Eq. (8):
θM(−T) = θ0
M −Ω0
MT + 1
2
˙Ω0
MT 2,
(13)
θM(0) = θ0
M,
θM(T) = θ0
M + Ω0
MT + 1
2
˙Ω0
MT 2,
To calculate the laser phase at each pulse time, the ef-
fective wavevector is expressed as a function of the mirror
and the sensor rotation parameters. The different uni-
tary vectors can be expressed with the help of the mirror
angle: ⃗xM = cos θM⃗xS + sin θM⃗yS. The vector in the
direction of the reflected laser is computed as follows:
⃗xr = −⃗xS + 2(⃗xM.⃗xS)⃗xM.
(14)
As for the rotation of the whole sensor, the phase shift
and contrast are in the same way impacted by the mirror
rotation. The phase shift is computed using Eq. (2).
∆ΦM = keffT 2
"
−ax
(15)
−2Ω0
M · (vy + ayT)
−˙Ω0
M ·
yMA + vyT + ayT 2
+2Ω0
M
2 ·
xMA + vxT + axT 2
)
#
,
with (xMA, yMA, zMA) the atom-cloud position relative
to the mirror’s center of rotation at the first pulse.
Eq. (15) is an approximation as terms scaling in Ω0
M
3,
˙Ω0
M
2, ˙Ω0
MΩ0
M and θ0
M are neglected.
This equation is
similar to Eq. (11) except for the contribution of the
centrifugal acceleration which is multiplied by 2 when
the mirror is rotating as the magnitude of the Raman
laser wavevector now depends on θM(t):
⃗keff(t) = −
k1 + k2 cos 2θM(t)
k2 sin 2θM(t)
0
≃−keff
1 −θ2
M(t)
θM(t)
0
,
(16)
The contrast decay due to the rotation of the mirror is
also very similar to the contrast decay due to the rota-
tion of the whole sensor in Eq. (12). The only difference
concerns once again the impact of the centrifugal accel-
eration:
CM
C
= exp
"
−k2
effT 4
2
(2Ω0
M + T ˙Ω0
M)2 · σ2
vy
(17)
+ ˙Ω0
M
2 · σ2
y
+Ω0
M
4 ·
4σ2
x + (2T)2σ2
vx
!#
.
The similarities between the impacts of the sensor ro-
tation and the mirror rotation on interferometer output
show that counter-rotating the interferometer mirror of-
fers a possibility to mitigate rotation issues. Neverthe-
less, we can already see that both rotations are not com-
pletely equivalent and a basic compensation scheme will
not allow to fully remove all the detrimental impact of
rotation. This will be studied in the next section.
D.
Rotation compensation
In order to limit the detrimental impact of rotation
on the atomic accelerometer,
we consider here the
rotation compensation method consisting in rotating the
retro-reflection mirror with an opposite angle, leaving
the mirror rotation-less in the laboratory frame during
the interferometric phase. Thanks to the compensation,
the direction effective wavevector is constant during the
interferometer but not its norm:
⃗keff(t) = −keff
1 −θS(t)2
2
0
0
.
(18)
Let us first analyze the contrast of the interferometer
in this rotation compensation scheme. Based on the anal-
ysis conducted in the previous sections, considering the
combined effect of both the rotation of the sensor and
the rotation of the mirror, one can derive the contrast of
the atom interferometer:
C
comp
M=S
C
= exp
"
−k2
effT 4
2
Ω0
S
4 ·
σ2
x + σ2
vxT 2
#
.
(19)
As shown in the above expression, the contrast is not
impacted anymore by the Coriolis and Euler accelera-
tions. The only remaining source of contrast loss is the
6
unchanged contribution of the centrifugal acceleration
of Eq. (11).
Considering the phase shift of the atom interferometer,
an analogous approach yields:
∆ΦM+S = keffT 2
"
−ax
(20)
−2(Ω0
M + Ω0
S) · (vy + ayT) −Ω0
SayT
−( ˙Ω0
S + ˙Ω0
M) ·
yOA + vyT + 1
2ayT 2
−˙Ω0
M ·
yMO + 1
2ayT 2
+2Ω0
M
2 · (xOA + xMO + Tvx + axT 2)
+Ω0
S
2 · (xOA + 3Tvx + 7
2axT 2)
+2Ω0
MΩ0
S · (xOA + xMO + 2Tvx + 2axT 2)
#
.
If Eq. (20) seems different from the ones found in
previous articles such as [25], one can easily retrieve
the expressions from the literature in a simple case
:
no time delay between the atoms launch and the
beginning of the interferometer t0 = 0 and null angular
accelerations ˙Ω0
S = ˙Ω0
M = 0.
Then, the atoms mean
velocity can be expressed as ⃗v1 = ⃗v0 + ⃗ΩS ∧−→
OA and
then vy1 = vy0 + Ω0
SxOA with ⃗v0 the velocity at the end
of the cooling stage. Finally, the Coriolis term becomes:
−2vy1 · (Ω0
S + Ω0
M) = −2vy0 · (Ω0
S + Ω0
M)
(21)
−2Ω0
MΩ0
S · xOA −2Ω0
S
2 · xOA.
leading to a similar expression as in [25].
The induced phase shift of Eq. (20) thus depends on
both center-of-rotation positions xMO and yMO, which
limit the effect of the rotation compensation method. In
a ideal rotation compensation scenario, the sensor and
the mirror rotate in exactly opposite ways, each around
its own center of rotation. Their angular positions will
be exactly opposite θM(t) = −θS(t), at least at the three
laser pulses times, as will be the mean angular veloc-
ities and accelerations: Ω0
M = −Ω0
S and ˙Ω0
M = −˙Ω0
S.
Some terms in the general expression of the phase shift,
in Eq. (20), are thus canceled out by the rotation com-
pensation, leading to:
∆Φ
comp
M=S = keffT 2
"
−ax
(22)
−Ω0
S · ayT
−˙Ω0
M ·
yMO + 1
2ayT 2
+Ω0
S
2 · [ xOA + 2xMO + vxT + 3
2axT 2 ]
#
.
In these conditions, the rotation induced phase shift is
reduced. The phase shift is not impacted anymore by the
Coriolis acceleration in the absence of a transverse accel-
eration ay. However, a Euler acceleration term is still
present due to a potential transverse misalignment yMO
between the mirror and the sensor centers of rotation. In
principle, this Euler acceleration term could be canceled
out with a precise relative adjustment of the atom-cloud
position and the sensor center of rotation, neglecting any
effects impacting atom-cloud position stability.
Moreover, several centrifugal acceleration terms remain.
One of them is due to the potential distance between the
mirror and the sensor centers of rotation, leading to a
phase shift scaling as xMO. The rest of the centrifugal
acceleration term: xOA+xMO+vxT + 3
2axT 2 is explained
by the variation of the wavevector magnitude due to the
mirror rotation (Eq. 18). Note that this result was al-
ready highlighted recently in [25, 26].
To reduce this
contribution, [25] proposed a configuration that involved
both the rotation of the incident Raman beam and the
mirror in the case of an atom accelerometer onboard an
Earth orbiting satellite.
Regarding the centrifugal ac-
celeration terms, the existing design of our instrument
doesn’t allow the cancellation of all the related contribu-
tions.
However, we can propose some potential alternative de-
signs. A first possibility would be to apply two frequency
jumps on the Raman laser frequency before the second
and before the third laser pulse, compensating the ef-
fect of wavevector magnitude variation during the mir-
ror rotation. A similar technique was already proposed
and implemented on the central π pulse to compensate
for the gravity gradient phase [37–40] . This should al-
low, jointly with matching the mirror and sensor cen-
ters of rotation, to remove the centrifugal acceleration
terms. Considering, for instance, a constant angular ve-
locity of typically ΩS = 1.1 mrad/s, representative of
an Earth orbiting satellite, the variation of the effective
wavevector to compensate between the first and second
pulses is ∆||⃗keff|| = (ΩST )2
2
||⃗keff||. This would lead to a
first frequency jump of ≈465 MHz and a second one of
≈1396 MHz if we consider an interrogation time T = 1
s.
A second possibility would be, as can be seen in Eq. (22),
to configure the position of the atom cloud A, the mir-
ror M and the center of rotation of the platform O so to
cancel the term xOA + 2xMO, corresponding to a mirror
in between, at the same distance from each other. Such a
solution would cancel the centrifugal contribution assum-
ing the terms in Eq. (22) scaling with the acceleration ax
and velocity vx, remain negligible.
These propositions
are very preliminary and should be in any case investi-
gated further in detail to assess their potential of inter-
est for future atom accelerometer application involving
detrimental rotation impacts, especially in the case of an
Earth orbiting satellite in Nadir pointing mode. These
considerations come for instance without considering the
impact of gravity gradients.
7
The rotation compensation method described from a the-
oretical point of view in this section is also implemented
experimentally in the following. The experimental setup
is described in the next section.
II.
EXPERIMENTAL SETUP
Here, we present the architecture of the experimental
setup used to analyze the impact of rotation on the out-
put of a cold-atom accelerometer. As depicted in Fig. 3, it
is constituted of a cold-atom gravimeter standing above
an electrostatic accelerometer whose proof-mass is em-
ployed as a retro-reflection mirror for the atom inter-
ferometer laser. The whole experiment and a two-axis
gyroscope are installed on a passive vibration isolation
platform. This platform is necessary to mitigate the im-
pact of ground vibrations on the interferometer. The iso-
lation platform is mounted on a table that is supported
by piezoelectric actuators. This configuration allows the
rotation of the sensor head around the Z axis by driving
the height of the piezoelectric actuator B. The sensor ro-
tation ΩS(t) is measured by the two-axis gyroscope.
The atom interferometer is similar to the one described
Proof-mass
Raman Laser
Magnetic Shield
Passive Isolation Platform
X
Y
Z
4
MOT
Detection
System
Two-axis
Gyroscope
PZT B
PZT C
FIG. 3.
Experimental setup composed of a cold atom
gravimeter, an electrostatic accelerometer, a two axis gyro-
scope, a passive isolation platform and piezo-electric actuators
(PZT B and PZT C) allowing the whole setup to be rotated.
in [41].
The laser frequencies necessary to the experi-
ment are generated by phase modulation of a frequency-
doubled telecom laser system [42]. Upstream the inter-
ferometer, a 3D magneto-optical trap of rubidium 87 is
loaded from a background vapor. After a stage of op-
tical molasses and microwave selection, an atom cloud
with a typical temperature of 2 µK is obtained in the
ground state
5S1/2, F = 1, mF = 0
[41].
After a free
fall of t0 = 10 ms, a vertical Mach-Zehnder atom in-
terferometer is completed.
Three equally spaced laser
pulses realize two-photon Raman transitions between
the hyperfine states |g⟩=
5S1/2, F = 1, mF = 0
and
|e⟩=
5S1/2, F = 2, mF = 0
. The first and third laser
pulses last τ = 4 µs and the second one lasts 2τ = 8 µs, to
respectively deliver π/2 and π pulses. The time delay be-
tween the consecutive pulses is T = 46 ms characterizing
the interferometer interrogation time. The two frequen-
cies necessary to the two-photon Raman transition are
generated by retro-reflecting a phase modulated laser on
a mirror [42].
As already mentioned, the retro-reflection mirror for
the cold-atom interferometer is the proof-mass of an elec-
trostatic accelerometer (EA). This configuration allows
for the hybridization of both sensors based on different
technologies [13, 15, 17, 43, 44]. For more details on the
design and operation of electrostatic accelerometers, one
can, for example, refer to [45]. In contrast to a standard
EA, a glass window has been added to the EA’s vacuum
chamber to allow the laser to be reflected on the EA
proof-mass. The core of the EA is made of gold-coated
(Ultra Low Expansion glass) with a side of 4 cm and a
mass of 35 g in electrostatic levitation at the center of a
cage holding all the electrodes (see Fig. 4). The position
of the proof-mass can be measured along the six degrees
of freedom owing to a differential capacitance detection.
The position of the proof-mass is controlled by use of a
servo feedback control on the applied electrostatic forces.
In the acceleration measurement mode, the acceleration
value is deduced from the electrostatic force applied to
maintain the proof-mass at the center of the cage. Here,
the acceleration measurement of the EA is not used since
the EA is operated only as a precise actuated mirror and
the position of the proof-mass according to the six-degree
of freedom is driven by the electrostatic forces.
III.
EXPERIMENTAL METHODS
A.
Atom interferometer rotation
To study the impact of rotations, we need to mea-
sure the contrast and phase shift of the interferometer.
During the interferometer phase, the two-photon Raman
laser frequency is chirped with a rate α ≈25 MHz/s
to compensate for the Doppler shift of the free-falling
atoms. The scan of the frequency chirp rate α allows us
to sweep artificially the acceleration seen by the atoms.
This method enables us to obtain atomic interference
8
4 cm
FIG. 4.
Electrostatic accelerometer proof-mass and elec-
trodes. The arrangement of the electrodes allows the control
of the six degrees of freedom of the proof-mass.
fringes from which the phase shift and contrast of the in-
terferometer can be deduced (see Fig. 5). Typically, the
fringes display a contrast without rotation of C ≈0.42
and are obtained after scanning over 400 measurements
points, each corresponding to an experimental cycle.
FIG. 5. Atomic interference fringes collected by scanning the
two-photon Raman laser frequency rate α.
The horizontal
axis is the pseudo acceleration induced by the scan of α. The
blue, respectively green, circles are the measured proportion
of atoms in the excited state at the output of the interferom-
eter in the absence, respectively in the presence, of a mirror
rotation. Solid blue, respectively green, line is the sinusoidal
fit of the experimental data in the absence, respectively in the
presence, of a mirror rotation with a mean angular accelera-
tion is ˙Ω0
M = −52.4 rad s−2 and a mean angular velocity close
to zero.
To rotate the whole sensor, a time-dependent voltage
is applied to the piezoelectric actuator B (see Fig. 3). To
benefit from a smooth dynamic of the sensor and thus
not excite vibrations mode of the instrument mechanical
structure, we choose to apply a sinusoidal input voltage
at a 4 Hz frequency on the piezoactuator, which is the
repetition rate of the experimental cycle. In this way,
the piezoactuator input voltage is synchronized with the
interferometer measurement cycle. The vibration isola-
tion platform helps isolating the setup from the ground
vibrations, but also from the high-frequency vibrations
generated by the piezoelectric actuator. The setup gen-
erates a rotation along the ⃗zS axis with an angular range
of the order of 100 µrad leading to a mean angular veloc-
ity around 2 mrad s−1 and an acceleration of 60 mrad s−2
measured by the gyroscope.
As for the retro-reflection mirror, it can be rotated
or translated by driving the electrostatic force applied
between the electrodes and the proof-mass. The angle
between the proof-mass and the electrodes (see Fig. 4)
is measured by capacitive detection of the electrostatic
accelerometer.
B.
Removal of systematic side effects
To process the experimental results and focus on the
effect of the rotation compensation method, which is the
main purpose of this paper, two side phenomena are cor-
rected from the experimental data.
Firstly, as the electrostatic proof-mass is rotated, small
residual accelerations or rotations of the mirror are
present as a result of the asymmetric electrodes architec-
ture. For example, for an imposed angular ramp of angu-
lar velocity 2 mrad s−1 along the ⃗zM axis, a vertical accel-
eration of −0.36 µm s−1 is measured as well as a rotation
along ⃗yM with a mean angular velocity of 29 µrad s−1
and a mean angular acceleration of −20 µrad s−2. These
parasitic movements are measured by capacitive detec-
tion and the induced phase shift is estimated by use of
the capacitive detection measurements. The method is
described in the Supplementary Materials.
Secondly, although the rotation of the whole sensor
occurs mainly along the ⃗zS axis according to the applied
4 Hz sinusoidal signal, there is a small unwanted resid-
ual rotation along the ⃗yS axis with an amplitude reach-
ing ≈8% of the main rotation along the ⃗xS axis and
a phase shift of ≈0.5 rad leading to a maximal mean
angular velocity of 0.13 mrad s−1 and an acceleration of
3.9 mrad s−2.
The impact of this parasitic rotation on
the phase shift can be computed as described in the Sup-
plementary Materials.
C.
Kinematic parameters of the cloud
To analyze the effect of the rotation, the kinematic pa-
rameters of the atom cloud such as its temperature, size,
mean velocity, and mean position relative to the center
of rotation of the mirror need to be measured. These pa-
rameters were determined by rotating the retro-reflection
mirror as described in the Supplementary materials. Ta-
ble I summarizes the parameters in the laboratory frame
used to estimate the phase shift and contrast.
For example, the initial mean velocity along the ⃗yL
axis, can be non zero due to intensity imbalance of
the laser beams during the cooling stage [34, 36].
In
9
TABLE I. Kinematic parameters of the atom cloud at launch.
Parameter
Value
Type of measurement
vx0
0(3) mm s−1
from [46]
vy0
−1.3(3) mm s−1
Mirror rotation
vz0
0.3(3) mm s−1
Mirror rotation
σvx
11.3(2) mm s−1
Raman spectroscopy
σvy
10.8(2) mm s−1
Mirror rotation
σvz
11.1(2) mm s−1
Mirror rotation
xMA
420(5) mm
By construction
yMA
1.09(3) mm
Mirror rotation
zMA
0.66(3) mm
Mirror rotation
σx
0.5 mm
from [46]
σy
0.42(5) mm
Mirror rotation
σz
0.66(5) mm
Mirror rotation
xOA
550(5) mm
By construction
yOA
∈[−5.7; −5 mm]
Fit parameter
zOA
∈[12.2; 10.6 mm]
Fit parameter
our experiment, this velocity is estimated at vy0
=
−1.3 mm s−1. The distance along the ⃗xL axis between
the atom cloud and the center of rotation of the sen-
sor, located on the upper table of the isolation platform,
is estimated by construction, whereas the same distance
along the horizontal plane is a free parameter chosen to
fit the experimental data within a plausibility range of a
few centimeters.
IV.
EXPERIMENTAL RESULTS
A.
Rotation of the atom interferometer
An experimental study to investigate the impact of
rotation on the interferometer output is conducted by
rotating the whole setup at 4 Hz following the procedure
described in Sec. III.
The results in terms of contrast loss are reported in
Fig. 6.
The error bars associated to the experimental
data come from the sinusoidal fit of the gyroscope sig-
nal and the uncertainty of the gyroscope measurement
for the uncertainty on the sensor mean angular velocity.
The uncertainty on the normalized contrast is due to the
sinusoidal fit errors of the interference fringes and the
contrast variation over time due to experimental limits.
The main source of contrast loss is due to the effect of
Coriolis acceleration linked to the velocity dispersion of
the atom cloud and, more specifically, to the term scaling
as 4Ω0
S
2σ2
vy, in the red dashed dotted line in Fig. 6. While
the centrifugal acceleration does not contribute signifi-
cantly to the loss of contrast, the Euler acceleration plays
a larger role taken into account to fit the model to the
experimental data. If the total model (black line) agrees
with the experimental data, the Coriolis contribution is
not the only contribution to the contrast loss. For this
data set, the angular acceleration is non-zero due to the
imperfect synchronization between the 4-Hz atomic in-
terferometry cycle and the 4-Hz sinusoidal excitation of
the sensor rotation platform. If the mean angular accel-
eration had been null ˙ΩS = 0, the expected contrast loss
would have been lower (red line). Notably, the contribu-
tion due to Euler acceleration has an opposite sign com-
pared to the one due to Coriolis acceleration, so the con-
trast is higher in the presence of an angular acceleration,
as predicted by Eq. (12). This angular movement, where
the second Raman laser pulse corresponds approximately
to the midpoint of the sinusoid, should have cancelled the
Euler acceleration. Nevertheless, a small imperfection of
the angular movement phase relative to the temporal po-
sition of the laser pulses gives rise to an impact of Euler
acceleration on the output of the interferometer.
In addition to the contrast loss, a phase shift induced
4
3
2
1
0
Sensor mean angular acceleration
0
S (mrad. s
2)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Sensor mean angular velocity
0
S (mrad. s
1)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Normalised contrast
Coriolis
Centrifugal
Euler
Total
46
0
46
Time (ms)
Sensor angle
2
2
FIG. 6. Impact of the sensor rotation on the interferometer
contrast. The black circles represent the experimental data
with the 1σ error bars. The black line is the theoretical con-
trast of Eq. (12). In the inset, the black dashed line is the
sensor sinusoidal angular movement θS(t) deduced from the
gyroscope measurement. The vertical red lines represent the
temporal position of the atom interferometer laser pulses.
by the rotation is observed (Fig. 7). The phase shift un-
certainty is computed from the sinusoidal fit errors of the
interference fringes and the phase shift fluctuations over
time. The phase shift is significantly impacted by all the
inertial accelerations and quite well reproduced by the
model based on Eq. (11) using the gyroscope data with
an error to the model below 0.05 rad. This error could
be explained by variations of the distance yOM due to
imperfection of the rotation actuation. The Coriolis con-
tribution is linked to the atomic cloud transverse veloc-
ity due to by both the residual velocity at the end of the
cooling stage and the MOT rotation at launch. The last
depends of the sinusoidal angular movement explaining
the quadratic impact of the Coriolis acceleration. The
centrifugal acceleration has a quadratic impact on the
phase shift and is also due to the vertical distance be-
tween the atom cloud and the sensor center of rotation.
10
4
3
2
1
0
Sensor mean angular acceleration
0
S (mrad. s
2)
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
Sensor mean angular velocity
0
S (mrad. s
1)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
Interferometer Phase shift (rad)
Coriolis
Centrifugal
Euler
Total
46
0
46
Time (ms)
Sensor angle
2
2
FIG. 7. Impact of the sensor rotation on the interferometer
phase shift. The black line is the phase shift as expected by
Eq. (11). The experimental data are corrected from the side
effects (see Section III B). Legend similar to Fig. 6.
The Euler acceleration has a linear impact on the phase
shift and is explained by the transverse distance between
the atom cloud and the center of rotation. In this con-
figuration, which is supposed to lower the impact of the
angular acceleration, we can nonetheless observe a dom-
inant Euler acceleration, reaching nearly 1 rad of atom
interferometer phase shift.
B.
Rotation of the mirror
In this section, we present the experimental results ob-
tained by rotating the retro-reflection mirror of the atom
interferometer, with no rotation applied to the sensor
frame. These measurements are compared with the the-
oretical expression of contrast and phase shift given in
Sec. I C. Note that, compared to the previous section
where the whole sensor is rotated, here we have the pos-
sibility to better control the rotation excitation and to
implement a larger set of excitation configurations.
First, we study the simple case of an angular ramp
applied to the mirror. Only Coriolis and centrifugal ac-
celerations should contribute, as the mirror does not have
any angular accelerations. As shown in Fig. 8, the exper-
imental contrast loss can only be explained by the con-
tribution of the Coriolis acceleration. Similarly to Fig. 6,
the centrifugal acceleration is too small to be observed.
On the whole data set, the theoretical model is in agree-
ment with the experimental data of Fig. 8. Note that the
contrast loss induced by Coriolis acceleration is expected
to be exactly the same in the case of a mirror rotation or
in the case of the whole sensor rotation. In this section,
the error bars associated with the mirror mean angular
velocity are not visible.
Nevertheless, the uncertainty
0.0
0.5
1.0
1.5
2.0
Mirror mean angular velocity
0
M (mrad. s
1)
0.2
0.4
0.6
0.8
1.0
Normalised contrast
Coriolis
Centrifugal
Total
46
0
46
Time (ms)
Mirror angle
2
2
FIG. 8.
Impact of the mirror rotation on the interferome-
ter contrast. The black line is the contrast as expected by
Eq. (17). In the inset, the black line is the mirror linear an-
gular movement θM(t) measured by the capacitive detection.
Legend similar to Fig. 6.
was evaluated below 3 µrad resulting from the error on
the EA angular capacitive detection reading and calibra-
tion.
In this simple case, the phase shift should be impacted
only by the Coriolis and centrifugal accelerations.
As
can be seen in Fig. 9, the linear impact of Coriolis ac-
celeration dominates for small mean angular velocities
Ω0
M. The quadratic impact of the centrifugal accelera-
tion is multiplied by two as the magnitude of the effec-
tive wavevector is modified by the mirror rotation (see
Eq. (15)). Moreover, the relevant distance for the mirror
rotation is the vertical distance between the atom cloud
and the center of rotation of the mirror, leading to a cen-
trifugal term scaling as 2Ω0
M
2·
xMA + Tvx + axT 2
. Due
to the absence of angular acceleration, the experimental
phase shift is smaller in this case. While the theoreti-
cal model is in agreement with the experimental data for
angular velocities below ≈1 mrad s−1, some clear discrep-
ancies up to 0.3 rad can be observed for higher angular
velocities. This phase shift has not yet been fully un-
derstood and is still under investigation. According to a
preliminary analysis, this behavior does not result from
first-order spherical wavefront aberrations due to mirror
surface imperfections as the EA proof-mass optical qual-
ity better than λ
4 (for λ = 600 nm). Those anomalous
measurements cannot be explained by a variation of the
direction of measurement as the maximal mirror angle
is of the order of 100 µrad leading to a phase shift of
3 mrad. Finally, this effect is still present when consider-
ing measurements that combine alternating signs of the
laser wavevector ⃗keff, ruling out one-photon light shift as
the source of the effect.
Secondly, the effect of a sinusoidal angular movement
of the mirror similar to the one presented in Section IV A
is studied. In this case, the Euler acceleration is mini-
11
0.0
0.5
1.0
1.5
2.0
Mirror mean angular velocity
0
M (mrad. s
1)
0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Interferometer Phase shift (rad)
Coriolis
Centrifugal
Total
46
0
46
Time (ms)
Mirror angle
2
2
FIG. 9. Impact of the mirror rotation on the interferometer
phase shift. The black line is the phase shift as predicted by
Eq. (15). The data are corrected from the residual vertical
acceleration of the mirror. Legend similar to Fig. 8.
mized to study the Coriolis and centrifugal accelerations
only.
The range of accessible mean angular velocities
is larger than for the sensor rotation, reaching a max-
imum angular velocity of 4.48 mrad s−1. Moreover, the
mean angular acceleration of the mirror remains below
0.53 mrad s−2 thanks to a better control of the angular
movement. As can be seen in Fig. 10 (a) and as expected,
the contrast loss is mainly due to the Coriolis accelera-
tion, similarly to the mirror angular ramp case and to
the sensor rotation, whereas the centrifugal and Euler
accelerations play minor roles.
The related phase shift (see Fig. 11 (a)) is only affected
by the Coriolis and centrifugal accelerations as the mir-
ror rotation is better controlled than the sensor rotation,
resulting in a smaller residual angular acceleration. Un-
fortunately, the phase shift shows large discrepancies up
to 0.6 rad with the theoretical model which are not fully
explained yet and are still under exploration, recalling
the behavior previously described in the case of a linear
angular ramp of the mirror.
We also studied configurations with important angular
accelerations, maximizing the impact of the Euler accel-
eration on the interferometer output. In these configura-
tions, the sinusoidal angular movement is phase shifted
by π/2 compared to the previous case, inducing a max-
imum mean angular acceleration of 127 mrad s−2. The
mean angular velocity is minimized here and stays be-
low 1.4 µrad s−1. The only contribution to the contrast
loss (see Fig. 10 (b)) is now due to the Euler acceler-
ation, linked to the velocity and position distribution
of the atom cloud. Turning now to the impact on the
phase shift (see Fig. 11 (b)), we see clearly the high lin-
ear contribution, up to 4.7 rad, of the angular acceler-
ation explained by the transverse distance between the
atom cloud and the mirror center of rotation, resulting
in the phase term −keffT 2 ˙Ω0
M · yMA. Deviations from
the theoretical model seem in this case to be relatively
much smaller when such high angular accelerations are
generated. Still, for important rotation, some significant
discrepancies up to 0.63 rad with the model appear but
the experimental data stay in qualitative agreement. The
errors to the theoretical model are of the same scale as
the errors observed on Fig. 9 and 11(a) and might have
the same origin.
C.
Rotation compensation
The rotation compensation method is implemented on
the experimental setup by rotating the sensor according
to a sinusoidal angular excitation at 4 Hz and, at the
same time, rotating the mirror sinusoidally at the same
frequency. The phase of the mirror excitation is adjusted
experimentally so as to be in phase opposition compared
to the sensor excitation. The ideal compensated rotation
configuration corresponds to the situation where the mir-
ror is not rotated anymore in the laboratory frame. For
the following experimental results, the sensor rotation
parameters are kept constant while the mirror rotation
amplitude is scanned.
First, compensation of the angular velocity is studied
with a minimized angular acceleration of the sensor, in
order to focus on the correction of the effects induced by
the Coriolis acceleration. For this study, the mean an-
gular velocity of the sensor is set to Ω0
S = 1.43 mrad s−1
with a minimized residual mean angular acceleration of
the sensor ˙Ω0
S = −3.5 mrad s−2. The mirror is then ro-
tated in the opposite direction in order to compensate for
the Coriolis acceleration (see inset of Fig. 12 (a)). The
amplitude of the mirror rotation is scanned from an an-
gular velocity of 0 to −4.49 mrad s−1. In Fig. 12 (a), the
contrast is gradually recovered as the mean angular veloc-
ity of the mirror increases, passing through a maximum
close to Ω0
M = −Ω0
S, where the term 4(Ω0
M + Ω0
S)2 · σ2
vy
cancels out. For higher mirror angular velocities, the sen-
sor rotation is overcompensated, and the contrast decays.
The compensation is successful as the contrast recovery
reaches 99%. As can be noticed, the contrast behavior
is not fully driven by the Coriolis acceleration because of
the residual Euler acceleration. This Euler acceleration is
not compensated for by the mirror rotation and slightly
shifts the optimal contrast recovery from Ω0
M = −Ω0
S.
With this compensation method, despite the fact that
the impact of rotation on the phase shift cannot be fully
canceled as predicted in Section I C, leaving a residual
centrifugal acceleration term, a significant reduction of
the phase shift could be expected.
However, this is
not experimentally observed, as can be seen in Fig. 13
(a) where the phase shift remains more or less constant
around Ω0
M ≤−ΩS. Indeed, while the Coriolis accelera-
tion is perfectly compensated and the centrifugal accel-
eration is reduced, an important uncompensated Euler
acceleration dominates. As mentioned above, this is due
to the fact that the residual mean Euler acceleration of
the sensor rotation motion is not compensated for by
12
0.5
0.4
0.3
0.2
0.1
0.0
Mirror mean angular acceleration
0
M (mrad. s
2)
4
3
2
1
0
Mirror mean angular velocity
0
M (mrad. s
1)
0.0
0.2
0.4
0.6
0.8
1.0
Normalised contrast
Coriolis
Centrifugal
Euler
Total
(a)
46
0
46
Time (ms)
Mirror angle
2
2
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Mirror mean angular velocity
0
M ( rad. s
1)
0
20
40
60
80
100
120
0.0
0.2
0.4
0.6
0.8
1.0
Normalised contrast
Coriolis
Centrifugal
Euler
Total
(b)
Mirror mean angular acceleration
0
M (mrad. s
2)
46
0
46
Time (ms)
Mirror angle
2
2
FIG. 10. Impact of the mirror rotation on the interferometer contrast. The black line is the theoretical model from Eq. (17).
In (a) (respectively (b)) the angular velocity is maximized (respectively minimized) and the angular acceleration is minimized
(respectively maximized). Legend similar to Fig. 8.
0.5
0.4
0.3
0.2
0.1
Mirror mean angular acceleration
0
M (mrad. s
2)
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
Mirror mean angular velocity
0
M (mrad. s
1)
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
Interferometer Phase shift (rad)
Coriolis
Centrifugal
Euler
Total
(a)
46
0
46
Time (ms)
Mirror angle
2
2
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Mirror mean angular velocity
0
M ( rad. s
1)
0
20
40
60
80
100
120
5
4
3
2
1
0
Interferometer Phase shift (rad)
Coriolis
Centrifugal
Euler
Total
(b)
Mirror mean angular acceleration
0
M (mrad. s
2)
46
0
46
Time (ms)
Mirror angle
2
2
FIG. 11. Impact of the mirror rotation on the interferometer phase shift. The black line is the theoretical model from Eq. (15).
The data are corrected from the residual vertical acceleration of the mirror. Legend similar to Fig. 10.
the mirror rotation whose mean angular acceleration is
much smaller. Nevertheless, around the compensation,
the measured phase shift is retrieved by the model with
a precision below 0.025 rad: the computed value could be
useful to correct some future acceleration measurements.
On the contrary, the theoretical model shows deviations
up to 0.3 rad from the experimental data for mirror ro-
tation higher than | Ω0
M |= 2 mrad s−1. An effect of the
same magnitude was already observed in Section IV B.
Secondly, we carry out experiments to study the com-
pensation of the angular acceleration in order to can-
cel the effect of Euler acceleration.
For that purpose,
the sensor angular acceleration is maximized and set to
˙Ω0
S = −43.5 mrad s−2, with a residual angular velocity
Ω0
S = −204 µrad s−1. The mirror is rotated in the oppo-
site way (see inset of Fig. 12 (b)). The contrast is recov-
ered as the rotation angular acceleration increases until
it reaches a maximum of 92% for ˙Ω0
M = 47.6 mrad s−2
slightly shifted from the point ˙Ω0
M = −˙Ω0
S.
This ef-
fect can be explained by the presence of an uncompen-
sated Coriolis acceleration in the atom velocity distribu-
tion term
h
T( ˙Ω0
M + ˙Ω0
S) + 2Ω0
S
i2
σ2
vy assuming Ω0
M = 0.
Considering this residual Coriolis term, the optimal con-
trast recovery is expected for ˙Ω0
M = 52.4 mrad s−2, close
to the observed experimental value.
The phase shift induced by the Euler acceleration is
not compensated as the centers of rotation of the mir-
13
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Mirror mean angular acceleration
0
M (mrad. s
2)
4
3
2
1
0
Mirror mean angular velocity
0
M (mrad. s
1)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalised contrast
Coriolis
Centrifugal
Euler
Total
0
M =
0
S
(a)
46
0
46
Time (ms)
Angle
2
2
sensor
mirror
6
5
4
3
2
1
0
Mirror mean angular velocity
0
M ( rad. s
1)
0
20
40
60
80
100
120
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Normalised contrast
Coriolis
Centrifugal
Euler
Total
0
M =
0
S
(b)
Mirror mean angular acceleration
0
M (mrad. s
2)
46
0
46
Time (ms)
Angle
2
2
sensor
mirror
FIG. 12. Impact of the rotation compensation method on the interferometer contrast. The black line is the theoretical model
from Eq. (19). The gray area corresponds to the uncertainty on Ω0
S in (a) (resp.
˙Ω0
S in (b)). In the insets, the continuous
black line is the mirror angular movement. The black dotted line is the sensor angular movement. In (a) (respectively (b)) the
angular velocity is maximized (respectively minimized) and the angular acceleration is minimized (respectively maximized).
Legend similar to Fig. 10.
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Mirror mean angular acceleration
0
M (mrad. s
2)
4
3
2
1
0
Mirror mean angular velocity
0
M (mrad. s
1)
0.6
0.4
0.2
0.0
0.2
Interferometer Phase shift (rad)
Coriolis
Centrifugal
Euler
Total
0
M =
0
S
(a)
46
0
46
Time (ms)
Angle
2
2
sensor
mirror
6
5
4
3
2
1
0
Mirror mean angular velocity
0
M ( rad. s
1)
0
20
40
60
80
100
120
12
10
8
6
4
2
0
2
Interferometer Phase shift (rad)
Coriolis
Centrifugal
Euler
Total
(b)
0
M =
0
S
Mirror mean angular acceleration
0
M (mrad. s
2)
46
0
46
Time (ms)
Angle
2
2
sensor
mirror
FIG. 13.
Impact of the rotation compensation method on the interferometer phase shift.
The black circles represent the
experimental data with 1σ error bars. The black line is the theoretical model from Eq. (20). Legend similar to Fig. 12.
ror and the sensor do not coincide, leading to a resid-
ual term −˙Ω0
M · yMO (see Fig. 13 (b)). The centrifugal
and Coriolis accelerations do not contribute to the phase
shift as the mirror mean angular velocity is very small.
In this configuration of high angular acceleration, the ex-
perimental results follow the global linear variation of the
theoretical phase with the mirror mean angular accelera-
tion. Nevertheless, an error to the model up to 0.6 rad is
observed mostly for higher angular acceleration. At the
compensation, the error is reduced to 0.1 rad. The dif-
ferences between data and model could be explained by
some variations in the distance yMO due to experimental
imperfections in the rotation excitation of the sensor.
V.
CONCLUSION
In this article, we present a detailed analysis of the
impact of rotations on the phase and contrast of a cold-
atom accelerometer.
Analytical models are confronted
to experimental results in different configurations of ro-
tation excitation where the impact of angular velocity or
angular acceleration is either minimized or maximized.
The experimental study is being carried out using a
14
unique hybrid atomic-electrostatic accelerometer where
the atomic interferometer laser is retro-reflected on the
proof-mass of an electrostatic accelerometer, similar to
space geodesy missions instruments. This hybrid config-
uration allowed us to demonstrate experimentally the ro-
tation compensation strategy of counter-rotating the in-
terferometer mirror while rotating the whole instrument.
We have shown that the contrast loss of the atomic in-
terferometer is quite well reproduced by the theoretical
model, regardless of the rotation excitation configuration
in a range relevant for space geodesy applications. The
use of an electrostatic accelerometer proof-mass as an ac-
tuated mirror enables the compensation of both angular
velocities and accelerations. Contrast recoveries greater
than 90% were demonstrated in both cases. The results
concerning the interferometer phase are not as well un-
derstood and are still under investigation.
The phase
shift data show deviations between the experimental data
and the model, especially for mirror rotations of higher
amplitudes.
This study is in line with previously published work
that has aimed to analyze the detrimental impact of rota-
tion on cold-atom accelerometers and has proposed mit-
igation strategies that would ultimately allow their op-
eration in a dynamic environment [25, 26]. Our results,
model, and experimental data are in agreement and bring
additional and complementary insights by studying in
detail the contribution of each non-inertial term, namely
Coriolis, centrifugal, and Euler accelerations.
We also
propose an alternative to the common solution based on
a PZT-actuated mirror. The benefits and drawbacks of
each of these techniques have yet to be studied.
This topic is especially of primary importance in the
perspective of future space missions on-boarding a cold-
atom accelerometer [6, 7, 9–14, 47], where satellite rota-
tion could lead to detrimental issues. Our results con-
stitute first experimental demonstrations that lead us to
be confident in the possibility of retrieving the atomic
contrast in orbit with the compensation method, con-
sidering that the atom source would be a delta-kick-
collimated Bose-Einstein condensate [48] submitted to a
Mach-Zehnder light pulse interferometer with an interro-
gation time T ≈1s. Concerning the impact of rotation
on the interferometer phase, there is still a need in the
near future to assess precisely the performance of the
rotation compensation system, as it would directly im-
pact the bias and stability of the measurement.
Here
we also initiate some discussion on the more appropriate
position of a cold-atom instrument relative to the center-
of-rotation of the satellite, so as to minimize rotation im-
pact and propose a way to get rid of remaining centrifugal
acceleration inherent to rotation compensation with mir-
ror actuation. Finally, in this article, we also mention
the potential of using the interferometer mirror rotation
to characterize key parameters linked to the kinematics
of the atom cloud that are of prime importance for the
performance of an onboard atom interferometer. For in-
stance, this type of characterization could be carried out
during dedicated in-flight calibration phases.
ACKNOWLEDGMENTS
We
thank
Bernard
Foulon
(ONERA),
Bruno
Christophe
(ONERA)
and
Fran¸coise
Liorzou
(ON-
ERA) for their early participation in the realization
and characterization of the EA ground prototype. This
work was partially supported by the European Space
Agency (ESA) through the “Hybrid Atom Electrostatic
System Follow-On for Satellite Geodesy ”, Contract
No.4000122290/17/NL/FF/mg.
The PhD grant of N.
Marquet was co-funded by ESA and ONERA.
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16
Appendix A: Acceleration of the mirror
Due to limits in the control of the actuated mirror,
a vertical movement was observed during the rotation
experiments.
The variation of the EA position with
respect to a rest position δxEA along the ⃗xS axis is
measured during the interferometer by use of capacitive
detection.
As the interferometer is sensitive to the
mirror acceleration, the induced phase shift can be
computed with Eq. (A1).
∆ΦaccEA = keffT 2 [ δxEA(t0) −2δxEA(t0 + T)
+ δxEA(t0 + 2T) ]
(A1)
Appendix B: Two-axis rotation of the sensor
In contrast with what is expected, the sensor does not
rotate purely around the ⃗zS axis. A small component
around the ⃗yS axis was measured with the two-axis
gyroscope.
The induced effect on the phase shift was
computed with Eq. (B1) and subtracted from the
experimental data presented in the article.
∆ΦSy = keffT 2
"
2Ω0
Sy ·
vz + 3
2Taz
+ ˙Ω0
Sy · (zOA + vzT + 1
2T 2az)
+ Ω0
Sy
2 ·
xOA + 3vxT + 7
2T 2ax
#
(B1)
with Ω0
Sy and ˙Ω0
Sy the mean angular velocity and mean
angular acceleration around ⃗yS during the interferometer
acquisition sequence.
Appendix C: Rotating the mirror, a way to
characterize the atom cloud
The rotation of the mirror was used to measure
the kinematic parameters of the atom cloud at launch
(index 0 on parameters) such as its temperature, size,
mean velocity, and position. To do so, the phase shift
and contrast are now expressed as a function of the
parameters at launch.
Velocity distribution:
The width of the velocity
distribution was measured through the contrast loss
induced by rotation.
In the absence of angular accel-
eration, the contrast loss is only due to Coriolis and
centrifugal acceleration. The contribution of the latter
is negligible in this study.
C
C = exp
"
−2k2
effT 4Ω0
M
2σ2
vi
#
(C1)
0.0
0.5
1.0
1.5
2.0
Mirror angular velocity
M (mrad. s
1)
0.0
0.2
0.4
0.6
0.8
1.0
Normalised contrast
46
0
46
Time (ms)
Mirror angle
2
2
FIG. 14.
Temperature measurements through the Coriolis
induced contrast loss. The blue circles (respectively the red
squares) are the experimental data for a rotation around the
⃗zM axis (respectively the ⃗yM axis). The plain blue line (resp.
the dotted red line) is the fit curve of the experimental data
by Eq. (C1). The angular movement of the mirror, black line
in the inset, is a ramp measured by the capacitive detection.
The red lines represent the laser pulses of the interferometer.
An angular ramp was imposed to the mirror around
the
⃗zM
axis
to
measure
the
velocity
distribution
along the ⃗yM axis and vice versa (see Fig. 14).
The
normalized contrast was fitted with Eq. (C1) with
i ∈{y; z}. The width of the velocity distribution was
measured at σvy = 10.8(2) mm s−1 along the ⃗yM axis
and σvz = 11.1(2) mm s−1 along the ⃗zM axis.
Size: The same principle was applied to measure the
size of the atom cloud right after the cooling stage.
In the absence of an angular velocity, the contrast
loss is due only to the angular acceleration and the
velocity and position distributions of the cloud. As the
velocity distribution was already measured, the position
distribution can be determined.
C
C = exp
"
−8k2
effA2
M ·
σ2
i + (t0 + T)2σ2
vi
!#
(C2)
A sine movement of angular amplitude AM was imposed
to the mirror around the ⃗zM axis to measure the size
along the ⃗yM axis and vice versa (see Fig. 15). Such an
angular movement maximizes the angular acceleration
of the mirror. The normalized contrast was fitted with
Eq. (C2) with i ∈{y; z}.
The width of the position
distribution was measured at σy0 = 0.42(5) mm along
the ⃗yM axis and σz0 = 0.66(5) mm along the ⃗zM axis.
Mean velocity: The mean velocity of the atom cloud
was measured through the Coriolis-induced phase shift.
The impact of centrifugal acceleration on the phase shift
cannot be ignored but can be estimated as the vertical
17
0
20
40
60
80
Mirror angular amplitude AM ( rad)
0.0
0.2
0.4
0.6
0.8
1.0
Normalised contrast
46
0
46
Time (ms)
Mirror angle
AM
2
2
FIG. 15. Size measurements through the Euler induced con-
trast loss. The plain blue line (resp. the dotted red line) is
the fit curve of the experimental data by Eq. (C2). The angu-
lar movement of the mirror is a sine as described in the inset.
Legend similar to Fig. 14.
kinematic parameters of the cloud are known.
∆Φ = keffT 2
"
−2ΩM · vy0
+ 2ΩM
2(axL · (t2
0
2 + t0T + T 2)
+ vx0(t0 + T) + xMA)
#
(C3)
∆Φ = keffT 2
"
2ΩM · vz0
+ 2ΩM
2(axL · (t2
0
2 + t0T + T 2)
+ vx0(t0 + T) + xMA)
#
(C4)
An angular ramp was imposed to the mirror around
the ⃗zM axis to measure the velocity along the ⃗yM
axis and vice versa (see Fig. 16).
The phase shift
was fitted with Eq. (C3) for the rotation along the
⃗zM axis (in blue) and with Eq. (C4) for the rotation
along the ⃗yM axis (in red).
The mean velocity was
measured at vy0 = −1.3(3) mm s−1 along the ⃗yM axis
and vz0 = 0.3(3) mm s−1 along the ⃗zM axis.
Mean position:
The same principle was applied to
measure the mean position of the atom cloud right after
the cooling stage. In the absence of an angular velocity,
the phase shift is due only to the Euler acceleration and
the velocity and position of the cloud. As the velocity
was already measured, the position can be determined.
0.0
0.5
1.0
1.5
2.0
Mirror angular velocity
M (mrad. s
1)
0.1
0.0
0.1
0.2
0.3
Interferometer Phase shift (rad)
46
0
46
Time (ms)
Mirror angle
2
2
FIG. 16. Mean velocity measurements through the Coriolis
induced phase shift. The plain blue line (resp. the dotted
red line) is the fit curve of the experimental data by Eq. (C3)
(resp. Eq. (C4)). The gray area delimits the excluded points
from the fit. The angular movement of the mirror is a ramp
as described in the inset. Legend similar to Fig. 14.
∆Φ = keffT 2
4AM
T 2
· ((t0 + T)vy0 + yMA)
(C5)
∆Φ = keffT 2
−4AM
T 2
· ((t0 + T)vz0 + zMA)
(C6)
A sine movement was imposed to the mirror around the
⃗zM axis to measure the position along the ⃗yM axis and
vice versa (see Fig. 17). The phase shift was fitted with
Eq. (C5) for the rotation around the ⃗zM axis (in blue)
and with Eq. (C6) for the rotation around the ⃗yM axis
(in red). The mean position was measured at yMA =
1.09(3) mm along the ⃗yM axis and zMA = 0.66(3) mm
along the ⃗zM axis.
18
0
20
40
60
80
Mirror angular velocity AM ( rad)
4
2
0
2
4
6
Phase shift (rad)
46
0
46
Time (ms)
Mirror angle
AM
2
2
FIG. 17. Mean position measurements through the Euler in-
duced phase shift. The plain blue line (resp. the dotted red
line) is the fit curve of the experimental data by Eq. (C5)
(resp. Eq. (C6)). The angular movement of the mirror is a
sine as described in the inset. Legend similar to Fig. 14
|
Impact of rotation on a cold atom interferometer and compensation strategy No ́emie Marquet,1, ∗Yannick Bidel,1 Malo Cadoret,2 Alexis Bonnin,1 Sylvain Schwartz,1 Phuong-Anh Huynh,3 Alexandre Bresson,1 Antoine Godard,4 Franck Pereira Dos Santos,5 Olivier Carraz,6 and Nassim Zahzam1, † 1DPHY, ONERA, Universit ́e Paris-Saclay, 91120 Palaiseau, France 2LCM, CNAM, 93210 La Plaine Saint-Denis, France 3DPHY, ONERA, Universit ́e Paris-Saclay, 92320 Chˆatillon, France 4ONERA, Universit ́e Paris-Saclay, 91120 Palaiseau, France 5Laboratoire Temps Espace, Observatoire de Paris, Universit ́e PSL, Sorbonne Universit ́e, Universit ́e de Lille, LNE, CNRS, 75014 Paris, France 6ESTEC,ESA, Noordwijk, The Netherlands (Dated: October 17, 2025) Rotations play a detrimental role in achieving ultra-high-performance inertial measurements with an atom interferometer, leading potentially to a total loss of interference contrast and the emergence of dominant phase shift biases. This becomes particularly significant when considering operation in dynamic conditions such as those encountered in Earth orbiting satellites in the perspective of future space gravity missions on-boarding a cold atom accelerometer. We study in this context the impact of rotation on the phase shift and contrast of an atom interferometer and investigate mitigation strategies. An analytical model is derived and compared to experimental demonstrations carried out using an original setup in which the well-controlled proof-mass of a space electrostatic accelerometer is used as the retro-reflection mirror of a cold atom gravimeter. By properly counter-rotating the electrostatic proof-mass, we demonstrate for instance the possibility of recovering the interferometer contrast, otherwise equal to zero, to a level better than 90%, in both cases of constant angular velocities or in presence of angular accelerations. Our results demonstrate the possibility to perform high performance inertial measurements with a cold atom interferometer in a challenging environments. INTRODUCTION Atom interferometry addresses the needs for high precision measurements in a wide range of domains, spanning from fundamental physics with for instance the determination of the fine structure constant [1, 2], or with the test of the Weak Equivalence Principle [3], to more applied domains such as inertial navigation [4, 5], or space gravimetry [6-14]. As this technology gets more mature, inertial sensors such as accelerometers or gyroscopes based on atom interferometry begin to move outside the laboratory. Cold-atom accelerometers have already been used for the last few years for field applications onboard trucks, boats, or planes [15-19]. However, such sensors are highly affected by the rotations of the dynamic environment they operate in [13, 20-26], which potentially leads to complete loss of interferometer contrast and significant phase-shift errors. To avoid such detrimental impacts of rotation, gyrostabilized platforms have been efficiently used [17, 19], but this solution comes with a significant increase in instrument size, power consumption and cost, limiting its range of applications. An alternative approach consists in counter-rotating the mirror on which the laser, enabling the atom interferometry measurement, is retro-reflected [21, 22]. This mirror can be seen as the reference for the ∗Present address † atomic measurement, and maintaining it non-rotating in the inertial frame seems a promising mitigation method. In laboratory environments, this approach was previously implemented using a piezoelectric-driven tip-tilt mirror to prevent Earth's rotation from significantly impacting static cold-atom interferometers [22, 27-29]. The same method was also recently demonstrated to show promising results for higher rotation rates up to 250 mrad s-1 on a cold-atom interferometer, ultimately dedicated to field applications [26]. Overcoming these rotation issues is one of the main challenges to deal with to operate an atom accelerometer under dynamic conditions, particularly in the context of future space gravimetry missions [9, 13, 25]. Nadinpointing navigation is then preferred, so the satellite is orbiting around the Earth at a nominal angular velocity of ≈1.1 mrad s-1. In the absence of rotation mitigation strategies, the contrast and phase shifts of the interferometer are impacted to a level preventing high performances measurements. In this context, we report here a detailed study analyzing the impact of rotations on a cold-atom accelerometer. This work is conducted through the development of analytical models showing results in terms of contrast and phase for several situations: when the whole instrument is subjected to rotation, when only the interferometer mirror is rotated, and when the rotation compensation strategy is implemented. The results of the models are confronted with experimental studies carried out with an original setup based on a hybrid atomic-electrostatic accelerometer [13], composed of a cold-atom gravimeter 16 Oct 2025 2 associated to a space electrostatic accelerometer (EA) [30], adapted to ground operation. In this configuration, the laser used for atom interferometry is directly retro-reflected on the EA's proof-mass acting as the reference mirror for the atom interferometer. The proofmass position is precisely controlled by use of electrostatic forces allowing its rotation and continuously monitored by capacitive detectors. This innovative design of a hybrid atomic-electrostatic instrument is considered as one of the promising accelerometer candidates for future space missions [11, 13, 14]. Our experimental setup allows us to rotate the whole instrument with rotation rates in the range of a Nadir-pointing satellite (of the order of 1 mrad s-1) for which significant detrimental impacts on a ground cold-atom gravimeter occur despite smaller interrogation times. As another original aspect of the work, we also considered non-constant rotation velocities leading to strong Euler acceleration whose effects can also be mitigated by counter-rotating the interferometer mirror. This work aims at a deeper understanding of the various effects that limit the performance of a cold-atom accelerometer and proposes an original technical approach based on electrostatic actuation of an EA's proof-mass, coupled to the cold-atom interferometer, rather than piezoelectric actuation of a mirror mount as commonly proposed. In this paper, we first present in Sec. I the model used to derive the impact of rotations on a cold-atom accelerometer. This model considers the whole sensor rotation, the mirror rotation and their effects on both the phase shift and contrast of a Mach-Zehnder coldatom interferometer. Then, the laboratory prototype is described, as well as the measurement protocol in Sec. II and Sec. III. Lastly, in Sec. IV, experimental contrasts and phase shifts of a rotating interferometer are investigated in the light of the model developed in the first section. The impacts of the sensor and the mirror rotations on both the phase shift and contrast are studied separately. The rotation compensation method is then characterized by considering two different angular motions: the first with an important angular velocity of 1.43 mrad s-1 and the second with an important angular acceleration of 43.5 mrad s-2. I. PHASE AND CONTRAST MODEL A. Atom interferometer without rotation Atom interferometers are based on the principle of superposition between two atomic states. Interactions between a light field and the atom drive the manipulation of quantum superposition of atomic states. In this work, we consider stimulated two-photon transitions, referred as Raman transitions, to implement atomic mirror and beam splitters [31, 32]. The studied atom interferometer is based on a Mach-Zehnder configuration [31, 33], involving a sequence of three Raman pulses of effective wavevector ⃗keff as described in Fig. 1. A two-frequency laser (corresponding to the wavevectors ⃗k1 and ⃗k2), retroreflected on a mirror, addresses Raman transitions between two stable hyperfine states of 87Rb atoms. The two states are a ground state |g, ⃗p⟩and an excited state e, ⃗p + ̄h⃗keff E with ⃗p the initial momentum of the atom. After a cooling step, the atoms are released in free fall at time t = -t0 -T. The atoms fall for a time t0 before the beginning of the interferometer, defined as the time of the first laser pulse. The total interrogation time of the interferometer is 2T, T being the time separation between two consecutive laser pulses. X t FIG. 1. Space-time diagram of a Mach-Zehnder atomic accelerometer measuring along the X axis. The three long red rectangles are the laser pulses π 2 -π -π 2 driving the atomic transitions between the ground state |g, ⃗p⟩and the excited state e, ⃗p + ̄h⃗keff E . The atoms free fall for a time t0 before the start of the interferometer. The first and third laser pulses put the atoms in a superposition of states and have a duration τ. The second pulse redirects the atom cloud and lasts 2τ. The free evolution of the atoms between the pulses lasts T ≫τ. Experimentally, t0 = 10 ms, T = 46 ms and τ = 4 μs. The laser effective wavevector ⃗keff is the difference between one wavevector of the incoming laser ⃗k1 and one of the reflected laser ⃗k2. ⃗keff(t) = ⃗k1(t) -⃗k2(t). (1) For a single atom, the interferometer output phase shift ∆Φ is a linear combination of the laser phases φ(t) at the three interaction times [33] as expressed in Eq. (2). In this calculation, the laser pulses are considered infinitely short and the mirror and beamsplitter laser pulses are considered perfect. ∆Φ is thus expressed as follows: ∆Φ = φ (-T) -2φ(0) + φ(T). (2) where φ(t) = ⃗keff(t) · ⃗r(t) is the laser phase at time t for a plane wave with ⃗r(t) the distance vector between the retro-reflection mirror and the atom. The atom trajectory is thus approximated by the mean trajectory between the two arms of the interferometer. The mirror can 3 be considered as the laser phase reference. The measurement axis corresponds to the direction of the effective wavevector ⃗keff. The instrument aims to measure the atom acceleration along ⃗keff derived from the interferometer output phase shift. As a first approximation, the interferometer phase is proportional to the atom acceleration aM in the mirror frame. ∆Φ can thus be approximated as follows: ∆Φ ≈⃗keff · ⃗aMT 2. (3) In principle, the detected signal of the interferometer is the probability Pe for an atom to be detected in the excited state at the output of the interferometer. It depends sinusoidally on the phase shift expressed in Eq. (2): Pe = P0 -C 2 cos(∆Φ). (4) with C the contrast and P0 the offset of the interferometer output signal for one atom. Experimentally, this quantity is derived from fluorescence measurements of the atomic population at the two outputs of the interferometer. The detected signal gathers the fluorescence emitted by the whole atomic cloud and therefore the measured probability P e can be considered as the average probability over the atom cloud neglecting the detection system response: P e = P 0 -C 2 cos ∆Φ . (5) with ∆Φ the mean phase shift, C the mean contrast and P 0 the mean offset of the interferometer output signal considering the whole detected atom cloud. Note that we neglect here the response of the detection system that could be taken into account as done in [34]. Since each atomic phase depends on position and velocity, the average probability can be written as a function of D⃗v, the velocity distribution, and D⃗r, the position distribution of the atom cloud: P e = ZZ Pe(⃗r,⃗v) D⃗v D⃗r d⃗v d⃗r. (6) Thus, the normalized contrast can be deduced from the phase shift variance over the atomic cloud which can be computed knowing the position and velocity distributions under the hypothesis of Gaussian velocity and position distributions: C C = exp(-V ar(∆Φ)/2) (7) Eq. (7) is obtained considering the contribution of each atom as equivalent; i.e. C does not depend on the velocity and position of the atom. A more complete model could be ultimately elaborated considering this aspect [35]. B. Rotation of the atom interferometer As previously reported, the rotation of the accelerometer sensor affects the phase shift and contrast at the output of the interferometer through Coriolis, centrifugal, and Euler acceleration terms [13, 20-22, 24-26, 29]. The phase shift being determined by averaging over the atom cloud, its alteration due to rotation is related to the mean initial velocity and position of the whole atom cloud. On the other hand, the contrast loss due to rotation is related to the size and temperature of the cloud, which could lead in some cases to a complete loss of signal and consequently to the impossibility to derive an acceleration measurement [13, 22, 26]. In this section, the calculation of the induced phase shift and contrast loss is detailed considering an atomic interferometer rotating around only one axis: ⃗zS = ⃗zL = ⃗zM. The different frames used in this calculation are illustrated in Fig. 2. The directions of the unitary vectors are defined in each frame. RL = {O, ⃗xL, ⃗yL,⃗zL} is the laboratory frame, which is considered inertial in this calculation, where O is the center of rotation of the sensor and ⃗xL is the unitary vector in the opposite direction of Earth's gravity ⃗g. Earth's rotation rate is considered constant during the experiment, and its effect on the interferometer can be considered negligible. RS = {O, ⃗xS, ⃗yS,⃗zS} is the sensor frame, where ⃗xS is aligned with the direction of ⃗k1 and the normal vector to the table supporting the sensor head (in blue on Fig. 2). RM = {M, ⃗xM, ⃗yM,⃗zM} is the mirror frame, where M is the center of rotation of the mirror, ⃗xM is the normal vector to the mirror (in yellow in Fig. 2), and ⃗xr is the unitary vector in the direction of the reflected laser ⃗k2. In this section, the mirror is not rotating in the sensor frame, then RM ≡RS and ⃗xr = ⃗xS: ⃗k1 and ⃗k2 are perfectly counter propagating The values of angle θS between ⃗xS and ⃗xL at pulse times -T, 0, and T can be expressed as follows: θS(-T) = θ0 S -Ω0 ST + 1 2 ̇Ω0 ST 2, (8) θS(0) = θ0 S, θS(T) = θ0 S + Ω0 ST + 1 2 ̇Ω0 ST 2, where Ω0 S is the mean angular velocity between the third and the first laser pulse and ̇Ω0 S is the mean angular acceleration over the three laser pulses sequence. To compute the laser phase at each pulse, the effective wavevector of the laser has to be expressed as a function of the rotation parameters. As defined in Eq. (1), the effective wavevector is the difference between the wavevector of the incoming laser ⃗k1 = -k1⃗xS and the reflected laser ⃗k2 = k2⃗xr. The unitary vector ⃗xS can be written as ⃗xS = cos θS⃗xL + sin θS⃗yL. In presence of rotation of 4 M O A FIG. 2. Description of the rotating frames: the sensor frame RS and the mirror frame RM in rotation in the laboratory frame RL. The blue rectangle is the table supporting the sensor. This table and the incoming laser are fixed in RS and rotate around O the center of rotation of the sensor. The yellow rectangle is the retro-reflection mirror rotating around M. ⃗xr is the unitary vector in the direction of the reflected laser in the mirror and A the center of the atom cloud. the whole sensor, the expression of ⃗keff in the laboratory frame, RL, thus reads ⃗keff(t) = -keff cos θS(t) sin θS(t) 0 ≃-keff 1 -θS(t)2 2 θS(t) 0 . (9) with keff = k1 + k2 the norm of ⃗keff in the absence of rotation. The direction of the wavevector is impacted by the sensor rotation but not its norm. We can decompose the classical trajectory of the mean position of the atomic wave packet ⃗r = --→ MA = --→ MO+-→ OA as a function of the position of the atom cloud relative to the sensor center of rotation (xOA, yOA, zOA), the relative position of the mirror to the sensor center of rotation (xOM, yOM, zOM), the velocities (vx, vy, vz) and accelerations (ax, ay, az) of the atomic cloud at the first interferometric pulse in a second lab frame considered inertial which coincide with the sensor frame at the first pulse R′ L = RS(t = -T). Expressing the atomic trajectory in this frame simplifies the phase shift and contrast expressions. Last, this choice explains the seeming differences between the equations of this work and previous literature. We finally obtained: ⃗r(t) = ax (t+T )2 2 + vx(t + T) + xMA ay (t+T )2 2 + vy(t + T) + yMA az (t+T )2 2 + vz(t + T) + zMA . (10) the mean atom trajectory in the second lab frame R′ L at the first laser pulse. The different parameters of the atomic cloud trajectory at the first pulse have to be computed as a function of the kinetic parameters at the launch time t = -t0 -T given in Tab. I. We consider a non zero initial velocity ⃗vres in the lab frame RL that can result from imbalanced laser beams during the molasses stage [34, 36]. The sensor rotation impacts the magnetooptical trap (MOT) position and velocity at the atoms release and has to be taken into account through the MOT angular position θMOT and instantaneous angular velocity ⃗ΩMOT . The atomic velocity at launch is then ⃗v0 = ⃗vres + ⃗ΩMOT ∧⃗r0 with ⃗r0 = -→ OA(-t0 -T). The only considered acceleration, ⃗a0 = ax0⃗xL takes Earth's gravity into account (the Earth-gravity gradient is neglected). The velocity in R′ L is then: ⃗v1 = ⃗v0 +⃗a1t0 + 0.5⃗vrec with ⃗vrec the recoil velocity and ⃗a1 the acceleration in R′ L leading to the value of (vx, vy, vz). Using Eqs (2), (9), and (10), the phase shift of the rotating interferometer can be expressed as: ∆ΦS = keffT 2 " -ax (11) -2Ω0 S · vy + 3 2ayT - ̇Ω0 S · yOA + vyT + 1 2ayT 2 +Ω0 S 2 · xOA + 3vxT + 7 2axT 2 # . Eq. (11) is an approximation as terms scaling in Ω0 S 3, ̇Ω0 S 2, ̇Ω0 SΩ0 S and θ0 S are neglected. The phase shift thus depends on the atoms' vertical acceleration and is impacted by the three inertial accelerations as follows: • Coriolis acceleration: -2Ω0 S · vy + 3 2ayT ; • Euler acceleration: - ̇Ω0 S · yOA + vyT + 1 2ayT 2 ; • Centrifugal acceleration: Ω0 S 2 · xOA + 3vxT + 7 2axT 2 . Using Eq. (6) and assuming independent Gaussian distributions with standard deviations σx, σy, σz for the initial position and σvx, σvy, σvz for the velocity, the expression of the contrast decay as a function of the rotation 5 parameters reads CS C = exp " -k2 effT 4 2 (2Ω0 S + T ̇Ω0 S)2 · σ2 vy (12) + ̇Ω0 S 2 · σ2 y +Ω0 S 4 · σ2 x + (3T)2σ2 vx !# . C. Rotation of the mirror In this section, we detail the impact on the interferometer output of the retro-reflection mirror rotation only. The considered mirror rotation vector, ⃗ΩM, is collinear to ⃗zM axis, as shown in Fig. 2. Similarly to the case where the rotation is applied to the whole instrument, the angle θM between ⃗xM and ⃗xS can be expressed at times -T, 0, and T as a function of the mirror mean angular velocity, Ω0 M, the mirror mean angular acceleration, ̇Ω0 M, and the mirror angle at the second pulse θ0 M in the same way as in Eq. (8): θM(-T) = θ0 M -Ω0 MT + 1 2 ̇Ω0 MT 2, (13) θM(0) = θ0 M, θM(T) = θ0 M + Ω0 MT + 1 2 ̇Ω0 MT 2, To calculate the laser phase at each pulse time, the effective wavevector is expressed as a function of the mirror and the sensor rotation parameters. The different unitary vectors can be expressed with the help of the mirror angle: ⃗xM = cos θM⃗xS + sin θM⃗yS. The vector in the direction of the reflected laser is computed as follows: ⃗xr = -⃗xS + 2(⃗xM.⃗xS)⃗xM. (14) As for the rotation of the whole sensor, the phase shift and contrast are in the same way impacted by the mirror rotation. The phase shift is computed using Eq. (2). ∆ΦM = keffT 2 " -ax (15) -2Ω0 M · (vy + ayT) - ̇Ω0 M · yMA + vyT + ayT 2 +2Ω0 M 2 · xMA + vxT + axT 2 ) # , with (xMA, yMA, zMA) the atom-cloud position relative to the mirror's center of rotation at the first pulse. Eq. (15) is an approximation as terms scaling in Ω0 M 3, ̇Ω0 M 2, ̇Ω0 MΩ0 M and θ0 M are neglected. This equation is similar to Eq. (11) except for the contribution of the centrifugal acceleration which is multiplied by 2 when the mirror is rotating as the magnitude of the Raman laser wavevector now depends on θM(t): ⃗keff(t) = - k1 + k2 cos 2θM(t) k2 sin 2θM(t) 0 ≃-keff 1 -θ2 M(t) θM(t) 0 , (16) The contrast decay due to the rotation of the mirror is also very similar to the contrast decay due to the rotation of the whole sensor in Eq. (12). The only difference concerns once again the impact of the centrifugal acceleration: CM C = exp " -k2 effT 4 2 (2Ω0 M + T ̇Ω0 M)2 · σ2 vy (17) + ̇Ω0 M 2 · σ2 y +Ω0 M 4 · 4σ2 x + (2T)2σ2 vx !# . The similarities between the impacts of the sensor rotation and the mirror rotation on interferometer output show that counter-rotating the interferometer mirror offers a possibility to mitigate rotation issues. Nevertheless, we can already see that both rotations are not completely equivalent and a basic compensation scheme will not allow to fully remove all the detrimental impact of rotation. This will be studied in the next section. D. Rotation compensation In order to limit the detrimental impact of rotation on the atomic accelerometer, we consider here the rotation compensation method consisting in rotating the retro-reflection mirror with an opposite angle, leaving the mirror rotation-less in the laboratory frame during the interferometric phase. Thanks to the compensation, the direction effective wavevector is constant during the interferometer but not its norm: ⃗keff(t) = -keff 1 -θS(t)2 2 0 0 . (18) Let us first analyze the contrast of the interferometer in this rotation compensation scheme. Based on the analysis conducted in the previous sections, considering the combined effect of both the rotation of the sensor and the rotation of the mirror, one can derive the contrast of the atom interferometer: C comp M=S C = exp " -k2 effT 4 2 Ω0 S 4 · σ2 x + σ2 vxT 2 # . (19) As shown in the above expression, the contrast is not impacted anymore by the Coriolis and Euler accelerations. The only remaining source of contrast loss is the 6 unchanged contribution of the centrifugal acceleration of Eq. (11). Considering the phase shift of the atom interferometer, an analogous approach yields: ∆ΦM+S = keffT 2 " -ax (20) -2(Ω0 M + Ω0 S) · (vy + ayT) -Ω0 SayT -( ̇Ω0 S + ̇Ω0 M) · yOA + vyT + 1 2ayT 2 - ̇Ω0 M · yMO + 1 2ayT 2 +2Ω0 M 2 · (xOA + xMO + Tvx + axT 2) +Ω0 S 2 · (xOA + 3Tvx + 7 2axT 2) +2Ω0 MΩ0 S · (xOA + xMO + 2Tvx + 2axT 2) # . If Eq. (20) seems different from the ones found in previous articles such as [25], one can easily retrieve the expressions from the literature in a simple case : no time delay between the atoms launch and the beginning of the interferometer t0 = 0 and null angular accelerations ̇Ω0 S = ̇Ω0 M = 0. Then, the atoms mean velocity can be expressed as ⃗v1 = ⃗v0 + ⃗ΩS ∧-→ OA and then vy1 = vy0 + Ω0 SxOA with ⃗v0 the velocity at the end of the cooling stage. Finally, the Coriolis term becomes: -2vy1 · (Ω0 S + Ω0 M) = -2vy0 · (Ω0 S + Ω0 M) (21) -2Ω0 MΩ0 S · xOA -2Ω0 S 2 · xOA. leading to a similar expression as in [25]. The induced phase shift of Eq. (20) thus depends on both center-of-rotation positions xMO and yMO, which limit the effect of the rotation compensation method. In a ideal rotation compensation scenario, the sensor and the mirror rotate in exactly opposite ways, each around its own center of rotation. Their angular positions will be exactly opposite θM(t) = -θS(t), at least at the three laser pulses times, as will be the mean angular velocities and accelerations: Ω0 M = -Ω0 S and ̇Ω0 M = - ̇Ω0 S. Some terms in the general expression of the phase shift, in Eq. (20), are thus canceled out by the rotation compensation, leading to: ∆Φ comp M=S = keffT 2 " -ax (22) -Ω0 S · ayT - ̇Ω0 M · yMO + 1 2ayT 2 +Ω0 S 2 · [ xOA + 2xMO + vxT + 3 2axT 2 ] # . In these conditions, the rotation induced phase shift is reduced. The phase shift is not impacted anymore by the Coriolis acceleration in the absence of a transverse acceleration ay. However, a Euler acceleration term is still present due to a potential transverse misalignment yMO between the mirror and the sensor centers of rotation. In principle, this Euler acceleration term could be canceled out with a precise relative adjustment of the atom-cloud position and the sensor center of rotation, neglecting any effects impacting atom-cloud position stability. Moreover, several centrifugal acceleration terms remain. One of them is due to the potential distance between the mirror and the sensor centers of rotation, leading to a phase shift scaling as xMO. The rest of the centrifugal acceleration term: xOA+xMO+vxT + 3 2axT 2 is explained by the variation of the wavevector magnitude due to the mirror rotation (Eq. 18). Note that this result was already highlighted recently in [25, 26]. To reduce this contribution, [25] proposed a configuration that involved both the rotation of the incident Raman beam and the mirror in the case of an atom accelerometer onboard an Earth orbiting satellite. Regarding the centrifugal acceleration terms, the existing design of our instrument doesn't allow the cancellation of all the related contributions. However, we can propose some potential alternative designs. A first possibility would be to apply two frequency jumps on the Raman laser frequency before the second and before the third laser pulse, compensating the effect of wavevector magnitude variation during the mirror rotation. A similar technique was already proposed and implemented on the central π pulse to compensate for the gravity gradient phase [37-40] . This should allow, jointly with matching the mirror and sensor centers of rotation, to remove the centrifugal acceleration terms. Considering, for instance, a constant angular velocity of typically ΩS = 1.1 mrad/s, representative of an Earth orbiting satellite, the variation of the effective wavevector to compensate between the first and second pulses is ∆||⃗keff|| = (ΩST )2 2 ||⃗keff||. This would lead to a first frequency jump of ≈465 MHz and a second one of ≈1396 MHz if we consider an interrogation time T = 1 s. A second possibility would be, as can be seen in Eq. (22), to configure the position of the atom cloud A, the mirror M and the center of rotation of the platform O so to cancel the term xOA + 2xMO, corresponding to a mirror in between, at the same distance from each other. Such a solution would cancel the centrifugal contribution assuming the terms in Eq. (22) scaling with the acceleration ax and velocity vx, remain negligible. These propositions are very preliminary and should be in any case investigated further in detail to assess their potential of interest for future atom accelerometer application involving detrimental rotation impacts, especially in the case of an Earth orbiting satellite in Nadir pointing mode. These considerations come for instance without considering the impact of gravity gradients. 7 The rotation compensation method described from a theoretical point of view in this section is also implemented experimentally in the following. The experimental setup is described in the next section. II. EXPERIMENTAL SETUP Here, we present the architecture of the experimental setup used to analyze the impact of rotation on the output of a cold-atom accelerometer. As depicted in Fig. 3, it is constituted of a cold-atom gravimeter standing above an electrostatic accelerometer whose proof-mass is employed as a retro-reflection mirror for the atom interferometer laser. The whole experiment and a two-axis gyroscope are installed on a passive vibration isolation platform. This platform is necessary to mitigate the impact of ground vibrations on the interferometer. The isolation platform is mounted on a table that is supported by piezoelectric actuators. This configuration allows the rotation of the sensor head around the Z axis by driving the height of the piezoelectric actuator B. The sensor rotation ΩS(t) is measured by the two-axis gyroscope. The atom interferometer is similar to the one described Proof-mass Raman Laser Magnetic Shield Passive Isolation Platform X Y Z 4 MOT Detection System Two-axis Gyroscope PZT B PZT C FIG. 3. Experimental setup composed of a cold atom gravimeter, an electrostatic accelerometer, a two axis gyroscope, a passive isolation platform and piezo-electric actuators (PZT B and PZT C) allowing the whole setup to be rotated. in [41]. The laser frequencies necessary to the experiment are generated by phase modulation of a frequencydoubled telecom laser system [42]. Upstream the interferometer, a 3D magneto-optical trap of rubidium 87 is loaded from a background vapor. After a stage of optical molasses and microwave selection, an atom cloud with a typical temperature of 2 μK is obtained in the ground state 5S1/2, F = 1, mF = 0 [41]. After a free fall of t0 = 10 ms, a vertical Mach-Zehnder atom interferometer is completed. Three equally spaced laser pulses realize two-photon Raman transitions between the hyperfine states |g⟩= 5S1/2, F = 1, mF = 0 and |e⟩= 5S1/2, F = 2, mF = 0 . The first and third laser pulses last τ = 4 μs and the second one lasts 2τ = 8 μs, to respectively deliver π/2 and π pulses. The time delay between the consecutive pulses is T = 46 ms characterizing the interferometer interrogation time. The two frequencies necessary to the two-photon Raman transition are generated by retro-reflecting a phase modulated laser on a mirror [42]. As already mentioned, the retro-reflection mirror for the cold-atom interferometer is the proof-mass of an electrostatic accelerometer (EA). This configuration allows for the hybridization of both sensors based on different technologies [13, 15, 17, 43, 44]. For more details on the design and operation of electrostatic accelerometers, one can, for example, refer to [45]. In contrast to a standard EA, a glass window has been added to the EA's vacuum chamber to allow the laser to be reflected on the EA proof-mass. The core of the EA is made of gold-coated (Ultra Low Expansion glass) with a side of 4 cm and a mass of 35 g in electrostatic levitation at the center of a cage holding all the electrodes (see Fig. 4). The position of the proof-mass can be measured along the six degrees of freedom owing to a differential capacitance detection. The position of the proof-mass is controlled by use of a servo feedback control on the applied electrostatic forces. In the acceleration measurement mode, the acceleration value is deduced from the electrostatic force applied to maintain the proof-mass at the center of the cage. Here, the acceleration measurement of the EA is not used since the EA is operated only as a precise actuated mirror and the position of the proof-mass according to the six-degree of freedom is driven by the electrostatic forces. III. EXPERIMENTAL METHODS A. Atom interferometer rotation To study the impact of rotations, we need to measure the contrast and phase shift of the interferometer. During the interferometer phase, the two-photon Raman laser frequency is chirped with a rate α ≈25 MHz/s to compensate for the Doppler shift of the free-falling atoms. The scan of the frequency chirp rate α allows us to sweep artificially the acceleration seen by the atoms. This method enables us to obtain atomic interference 8 4 cm FIG. 4. Electrostatic accelerometer proof-mass and electrodes. The arrangement of the electrodes allows the control of the six degrees of freedom of the proof-mass. fringes from which the phase shift and contrast of the interferometer can be deduced (see Fig. 5). Typically, the fringes display a contrast without rotation of C ≈0.42 and are obtained after scanning over 400 measurements points, each corresponding to an experimental cycle. FIG. 5. Atomic interference fringes collected by scanning the two-photon Raman laser frequency rate α. The horizontal axis is the pseudo acceleration induced by the scan of α. The blue, respectively green, circles are the measured proportion of atoms in the excited state at the output of the interferometer in the absence, respectively in the presence, of a mirror rotation. Solid blue, respectively green, line is the sinusoidal fit of the experimental data in the absence, respectively in the presence, of a mirror rotation with a mean angular acceleration is ̇Ω0 M = -52.4 rad s-2 and a mean angular velocity close to zero. To rotate the whole sensor, a time-dependent voltage is applied to the piezoelectric actuator B (see Fig. 3). To benefit from a smooth dynamic of the sensor and thus not excite vibrations mode of the instrument mechanical structure, we choose to apply a sinusoidal input voltage at a 4 Hz frequency on the piezoactuator, which is the repetition rate of the experimental cycle. In this way, the piezoactuator input voltage is synchronized with the interferometer measurement cycle. The vibration isolation platform helps isolating the setup from the ground vibrations, but also from the high-frequency vibrations generated by the piezoelectric actuator. The setup generates a rotation along the ⃗zS axis with an angular range of the order of 100 μrad leading to a mean angular velocity around 2 mrad s-1 and an acceleration of 60 mrad s-2 measured by the gyroscope. As for the retro-reflection mirror, it can be rotated or translated by driving the electrostatic force applied between the electrodes and the proof-mass. The angle between the proof-mass and the electrodes (see Fig. 4) is measured by capacitive detection of the electrostatic accelerometer. B. Removal of systematic side effects To process the experimental results and focus on the effect of the rotation compensation method, which is the main purpose of this paper, two side phenomena are corrected from the experimental data. Firstly, as the electrostatic proof-mass is rotated, small residual accelerations or rotations of the mirror are present as a result of the asymmetric electrodes architecture. For example, for an imposed angular ramp of angular velocity 2 mrad s-1 along the ⃗zM axis, a vertical acceleration of -0.36 μm s-1 is measured as well as a rotation along ⃗yM with a mean angular velocity of 29 μrad s-1 and a mean angular acceleration of -20 μrad s-2. These parasitic movements are measured by capacitive detection and the induced phase shift is estimated by use of the capacitive detection measurements. The method is described in the Supplementary Materials. Secondly, although the rotation of the whole sensor occurs mainly along the ⃗zS axis according to the applied 4 Hz sinusoidal signal, there is a small unwanted residual rotation along the ⃗yS axis with an amplitude reaching ≈8% of the main rotation along the ⃗xS axis and a phase shift of ≈0.5 rad leading to a maximal mean angular velocity of 0.13 mrad s-1 and an acceleration of 3.9 mrad s-2. The impact of this parasitic rotation on the phase shift can be computed as described in the Supplementary Materials. C. Kinematic parameters of the cloud To analyze the effect of the rotation, the kinematic parameters of the atom cloud such as its temperature, size, mean velocity, and mean position relative to the center of rotation of the mirror need to be measured. These parameters were determined by rotating the retro-reflection mirror as described in the Supplementary materials. Table I summarizes the parameters in the laboratory frame used to estimate the phase shift and contrast. For example, the initial mean velocity along the ⃗yL axis, can be non zero due to intensity imbalance of the laser beams during the cooling stage [34, 36]. In 9 TABLE I. Kinematic parameters of the atom cloud at launch. Parameter Value Type of measurement vx0 0(3) mm s-1 from [46] vy0 -1.3(3) mm s-1 Mirror rotation vz0 0.3(3) mm s-1 Mirror rotation σvx 11.3(2) mm s-1 Raman spectroscopy σvy 10.8(2) mm s-1 Mirror rotation σvz 11.1(2) mm s-1 Mirror rotation xMA 420(5) mm By construction yMA 1.09(3) mm Mirror rotation zMA 0.66(3) mm Mirror rotation σx 0.5 mm from [46] σy 0.42(5) mm Mirror rotation σz 0.66(5) mm Mirror rotation xOA 550(5) mm By construction yOA ∈[-5.7; -5 mm] Fit parameter zOA ∈[12.2; 10.6 mm] Fit parameter our experiment, this velocity is estimated at vy0 = -1.3 mm s-1. The distance along the ⃗xL axis between the atom cloud and the center of rotation of the sensor, located on the upper table of the isolation platform, is estimated by construction, whereas the same distance along the horizontal plane is a free parameter chosen to fit the experimental data within a plausibility range of a few centimeters. IV. EXPERIMENTAL RESULTS A. Rotation of the atom interferometer An experimental study to investigate the impact of rotation on the interferometer output is conducted by rotating the whole setup at 4 Hz following the procedure described in Sec. III. The results in terms of contrast loss are reported in Fig. 6. The error bars associated to the experimental data come from the sinusoidal fit of the gyroscope signal and the uncertainty of the gyroscope measurement for the uncertainty on the sensor mean angular velocity. The uncertainty on the normalized contrast is due to the sinusoidal fit errors of the interference fringes and the contrast variation over time due to experimental limits. The main source of contrast loss is due to the effect of Coriolis acceleration linked to the velocity dispersion of the atom cloud and, more specifically, to the term scaling as 4Ω0 S 2σ2 vy, in the red dashed dotted line in Fig. 6. While the centrifugal acceleration does not contribute significantly to the loss of contrast, the Euler acceleration plays a larger role taken into account to fit the model to the experimental data. If the total model (black line) agrees with the experimental data, the Coriolis contribution is not the only contribution to the contrast loss. For this data set, the angular acceleration is non-zero due to the imperfect synchronization between the 4-Hz atomic interferometry cycle and the 4-Hz sinusoidal excitation of the sensor rotation platform. If the mean angular acceleration had been null ̇ΩS = 0, the expected contrast loss would have been lower (red line). Notably, the contribution due to Euler acceleration has an opposite sign compared to the one due to Coriolis acceleration, so the contrast is higher in the presence of an angular acceleration, as predicted by Eq. (12). This angular movement, where the second Raman laser pulse corresponds approximately to the midpoint of the sinusoid, should have cancelled the Euler acceleration. Nevertheless, a small imperfection of the angular movement phase relative to the temporal position of the laser pulses gives rise to an impact of Euler acceleration on the output of the interferometer. In addition to the contrast loss, a phase shift induced 4 3 2 1 0 Sensor mean angular acceleration 0 S (mrad. s 2) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Sensor mean angular velocity 0 S (mrad. s 1) 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Normalised contrast Coriolis Centrifugal Euler Total 46 0 46 Time (ms) Sensor angle 2 2 FIG. 6. Impact of the sensor rotation on the interferometer contrast. The black circles represent the experimental data with the 1σ error bars. The black line is the theoretical contrast of Eq. (12). In the inset, the black dashed line is the sensor sinusoidal angular movement θS(t) deduced from the gyroscope measurement. The vertical red lines represent the temporal position of the atom interferometer laser pulses. by the rotation is observed (Fig. 7). The phase shift uncertainty is computed from the sinusoidal fit errors of the interference fringes and the phase shift fluctuations over time. The phase shift is significantly impacted by all the inertial accelerations and quite well reproduced by the model based on Eq. (11) using the gyroscope data with an error to the model below 0.05 rad. This error could be explained by variations of the distance yOM due to imperfection of the rotation actuation. The Coriolis contribution is linked to the atomic cloud transverse velocity due to by both the residual velocity at the end of the cooling stage and the MOT rotation at launch. The last depends of the sinusoidal angular movement explaining the quadratic impact of the Coriolis acceleration. The centrifugal acceleration has a quadratic impact on the phase shift and is also due to the vertical distance between the atom cloud and the sensor center of rotation. 10 4 3 2 1 0 Sensor mean angular acceleration 0 S (mrad. s 2) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Sensor mean angular velocity 0 S (mrad. s 1) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 Interferometer Phase shift (rad) Coriolis Centrifugal Euler Total 46 0 46 Time (ms) Sensor angle 2 2 FIG. 7. Impact of the sensor rotation on the interferometer phase shift. The black line is the phase shift as expected by Eq. (11). The experimental data are corrected from the side effects (see Section III B). Legend similar to Fig. 6. The Euler acceleration has a linear impact on the phase shift and is explained by the transverse distance between the atom cloud and the center of rotation. In this configuration, which is supposed to lower the impact of the angular acceleration, we can nonetheless observe a dominant Euler acceleration, reaching nearly 1 rad of atom interferometer phase shift. B. Rotation of the mirror In this section, we present the experimental results obtained by rotating the retro-reflection mirror of the atom interferometer, with no rotation applied to the sensor frame. These measurements are compared with the theoretical expression of contrast and phase shift given in Sec. I C. Note that, compared to the previous section where the whole sensor is rotated, here we have the possibility to better control the rotation excitation and to implement a larger set of excitation configurations. First, we study the simple case of an angular ramp applied to the mirror. Only Coriolis and centrifugal accelerations should contribute, as the mirror does not have any angular accelerations. As shown in Fig. 8, the experimental contrast loss can only be explained by the contribution of the Coriolis acceleration. Similarly to Fig. 6, the centrifugal acceleration is too small to be observed. On the whole data set, the theoretical model is in agreement with the experimental data of Fig. 8. Note that the contrast loss induced by Coriolis acceleration is expected to be exactly the same in the case of a mirror rotation or in the case of the whole sensor rotation. In this section, the error bars associated with the mirror mean angular velocity are not visible. Nevertheless, the uncertainty 0.0 0.5 1.0 1.5 2.0 Mirror mean angular velocity 0 M (mrad. s 1) 0.2 0.4 0.6 0.8 1.0 Normalised contrast Coriolis Centrifugal Total 46 0 46 Time (ms) Mirror angle 2 2 FIG. 8. Impact of the mirror rotation on the interferometer contrast. The black line is the contrast as expected by Eq. (17). In the inset, the black line is the mirror linear angular movement θM(t) measured by the capacitive detection. Legend similar to Fig. 6. was evaluated below 3 μrad resulting from the error on the EA angular capacitive detection reading and calibration. In this simple case, the phase shift should be impacted only by the Coriolis and centrifugal accelerations. As can be seen in Fig. 9, the linear impact of Coriolis acceleration dominates for small mean angular velocities Ω0 M. The quadratic impact of the centrifugal acceleration is multiplied by two as the magnitude of the effective wavevector is modified by the mirror rotation (see Eq. (15)). Moreover, the relevant distance for the mirror rotation is the vertical distance between the atom cloud and the center of rotation of the mirror, leading to a centrifugal term scaling as 2Ω0 M 2· xMA + Tvx + axT 2 . Due to the absence of angular acceleration, the experimental phase shift is smaller in this case. While the theoretical model is in agreement with the experimental data for angular velocities below ≈1 mrad s-1, some clear discrepancies up to 0.3 rad can be observed for higher angular velocities. This phase shift has not yet been fully understood and is still under investigation. According to a preliminary analysis, this behavior does not result from first-order spherical wavefront aberrations due to mirror surface imperfections as the EA proof-mass optical quality better than λ 4 (for λ = 600 nm). Those anomalous measurements cannot be explained by a variation of the direction of measurement as the maximal mirror angle is of the order of 100 μrad leading to a phase shift of 3 mrad. Finally, this effect is still present when considering measurements that combine alternating signs of the laser wavevector ⃗keff, ruling out one-photon light shift as the source of the effect. Secondly, the effect of a sinusoidal angular movement of the mirror similar to the one presented in Section IV A is studied. In this case, the Euler acceleration is mini11 0.0 0.5 1.0 1.5 2.0 Mirror mean angular velocity 0 M (mrad. s 1) 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Interferometer Phase shift (rad) Coriolis Centrifugal Total 46 0 46 Time (ms) Mirror angle 2 2 FIG. 9. Impact of the mirror rotation on the interferometer phase shift. The black line is the phase shift as predicted by Eq. (15). The data are corrected from the residual vertical acceleration of the mirror. Legend similar to Fig. 8. mized to study the Coriolis and centrifugal accelerations only. The range of accessible mean angular velocities is larger than for the sensor rotation, reaching a maximum angular velocity of 4.48 mrad s-1. Moreover, the mean angular acceleration of the mirror remains below 0.53 mrad s-2 thanks to a better control of the angular movement. As can be seen in Fig. 10 (a) and as expected, the contrast loss is mainly due to the Coriolis acceleration, similarly to the mirror angular ramp case and to the sensor rotation, whereas the centrifugal and Euler accelerations play minor roles. The related phase shift (see Fig. 11 (a)) is only affected by the Coriolis and centrifugal accelerations as the mirror rotation is better controlled than the sensor rotation, resulting in a smaller residual angular acceleration. Unfortunately, the phase shift shows large discrepancies up to 0.6 rad with the theoretical model which are not fully explained yet and are still under exploration, recalling the behavior previously described in the case of a linear angular ramp of the mirror. We also studied configurations with important angular accelerations, maximizing the impact of the Euler acceleration on the interferometer output. In these configurations, the sinusoidal angular movement is phase shifted by π/2 compared to the previous case, inducing a maximum mean angular acceleration of 127 mrad s-2. The mean angular velocity is minimized here and stays below 1.4 μrad s-1. The only contribution to the contrast loss (see Fig. 10 (b)) is now due to the Euler acceleration, linked to the velocity and position distribution of the atom cloud. Turning now to the impact on the phase shift (see Fig. 11 (b)), we see clearly the high linear contribution, up to 4.7 rad, of the angular acceleration explained by the transverse distance between the atom cloud and the mirror center of rotation, resulting in the phase term -keffT 2 ̇Ω0 M · yMA. Deviations from the theoretical model seem in this case to be relatively much smaller when such high angular accelerations are generated. Still, for important rotation, some significant discrepancies up to 0.63 rad with the model appear but the experimental data stay in qualitative agreement. The errors to the theoretical model are of the same scale as the errors observed on Fig. 9 and 11(a) and might have the same origin. C. Rotation compensation The rotation compensation method is implemented on the experimental setup by rotating the sensor according to a sinusoidal angular excitation at 4 Hz and, at the same time, rotating the mirror sinusoidally at the same frequency. The phase of the mirror excitation is adjusted experimentally so as to be in phase opposition compared to the sensor excitation. The ideal compensated rotation configuration corresponds to the situation where the mirror is not rotated anymore in the laboratory frame. For the following experimental results, the sensor rotation parameters are kept constant while the mirror rotation amplitude is scanned. First, compensation of the angular velocity is studied with a minimized angular acceleration of the sensor, in order to focus on the correction of the effects induced by the Coriolis acceleration. For this study, the mean angular velocity of the sensor is set to Ω0 S = 1.43 mrad s-1 with a minimized residual mean angular acceleration of the sensor ̇Ω0 S = -3.5 mrad s-2. The mirror is then rotated in the opposite direction in order to compensate for the Coriolis acceleration (see inset of Fig. 12 (a)). The amplitude of the mirror rotation is scanned from an angular velocity of 0 to -4.49 mrad s-1. In Fig. 12 (a), the contrast is gradually recovered as the mean angular velocity of the mirror increases, passing through a maximum close to Ω0 M = -Ω0 S, where the term 4(Ω0 M + Ω0 S)2 · σ2 vy cancels out. For higher mirror angular velocities, the sensor rotation is overcompensated, and the contrast decays. The compensation is successful as the contrast recovery reaches 99%. As can be noticed, the contrast behavior is not fully driven by the Coriolis acceleration because of the residual Euler acceleration. This Euler acceleration is not compensated for by the mirror rotation and slightly shifts the optimal contrast recovery from Ω0 M = -Ω0 S. With this compensation method, despite the fact that the impact of rotation on the phase shift cannot be fully canceled as predicted in Section I C, leaving a residual centrifugal acceleration term, a significant reduction of the phase shift could be expected. However, this is not experimentally observed, as can be seen in Fig. 13 (a) where the phase shift remains more or less constant around Ω0 M ≤-ΩS. Indeed, while the Coriolis acceleration is perfectly compensated and the centrifugal acceleration is reduced, an important uncompensated Euler acceleration dominates. As mentioned above, this is due to the fact that the residual mean Euler acceleration of the sensor rotation motion is not compensated for by 12 0.5 0.4 0.3 0.2 0.1 0.0 Mirror mean angular acceleration 0 M (mrad. s 2) 4 3 2 1 0 Mirror mean angular velocity 0 M (mrad. s 1) 0.0 0.2 0.4 0.6 0.8 1.0 Normalised contrast Coriolis Centrifugal Euler Total (a) 46 0 46 Time (ms) Mirror angle 2 2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Mirror mean angular velocity 0 M ( rad. s 1) 0 20 40 60 80 100 120 0.0 0.2 0.4 0.6 0.8 1.0 Normalised contrast Coriolis Centrifugal Euler Total (b) Mirror mean angular acceleration 0 M (mrad. s 2) 46 0 46 Time (ms) Mirror angle 2 2 FIG. 10. Impact of the mirror rotation on the interferometer contrast. The black line is the theoretical model from Eq. (17). In (a) (respectively (b)) the angular velocity is maximized (respectively minimized) and the angular acceleration is minimized (respectively maximized). Legend similar to Fig. 8. 0.5 0.4 0.3 0.2 0.1 Mirror mean angular acceleration 0 M (mrad. s 2) 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 Mirror mean angular velocity 0 M (mrad. s 1) 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Interferometer Phase shift (rad) Coriolis Centrifugal Euler Total (a) 46 0 46 Time (ms) Mirror angle 2 2 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 Mirror mean angular velocity 0 M ( rad. s 1) 0 20 40 60 80 100 120 5 4 3 2 1 0 Interferometer Phase shift (rad) Coriolis Centrifugal Euler Total (b) Mirror mean angular acceleration 0 M (mrad. s 2) 46 0 46 Time (ms) Mirror angle 2 2 FIG. 11. Impact of the mirror rotation on the interferometer phase shift. The black line is the theoretical model from Eq. (15). The data are corrected from the residual vertical acceleration of the mirror. Legend similar to Fig. 10. the mirror rotation whose mean angular acceleration is much smaller. Nevertheless, around the compensation, the measured phase shift is retrieved by the model with a precision below 0.025 rad: the computed value could be useful to correct some future acceleration measurements. On the contrary, the theoretical model shows deviations up to 0.3 rad from the experimental data for mirror rotation higher than | Ω0 M |= 2 mrad s-1. An effect of the same magnitude was already observed in Section IV B. Secondly, we carry out experiments to study the compensation of the angular acceleration in order to cancel the effect of Euler acceleration. For that purpose, the sensor angular acceleration is maximized and set to ̇Ω0 S = -43.5 mrad s-2, with a residual angular velocity Ω0 S = -204 μrad s-1. The mirror is rotated in the opposite way (see inset of Fig. 12 (b)). The contrast is recovered as the rotation angular acceleration increases until it reaches a maximum of 92% for ̇Ω0 M = 47.6 mrad s-2 slightly shifted from the point ̇Ω0 M = - ̇Ω0 S. This effect can be explained by the presence of an uncompensated Coriolis acceleration in the atom velocity distribution term h T( ̇Ω0 M + ̇Ω0 S) + 2Ω0 S i2 σ2 vy assuming Ω0 M = 0. Considering this residual Coriolis term, the optimal contrast recovery is expected for ̇Ω0 M = 52.4 mrad s-2, close to the observed experimental value. The phase shift induced by the Euler acceleration is not compensated as the centers of rotation of the mir13 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Mirror mean angular acceleration 0 M (mrad. s 2) 4 3 2 1 0 Mirror mean angular velocity 0 M (mrad. s 1) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Normalised contrast Coriolis Centrifugal Euler Total 0 M = 0 S (a) 46 0 46 Time (ms) Angle 2 2 sensor mirror 6 5 4 3 2 1 0 Mirror mean angular velocity 0 M ( rad. s 1) 0 20 40 60 80 100 120 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Normalised contrast Coriolis Centrifugal Euler Total 0 M = 0 S (b) Mirror mean angular acceleration 0 M (mrad. s 2) 46 0 46 Time (ms) Angle 2 2 sensor mirror FIG. 12. Impact of the rotation compensation method on the interferometer contrast. The black line is the theoretical model from Eq. (19). The gray area corresponds to the uncertainty on Ω0 S in (a) (resp. ̇Ω0 S in (b)). In the insets, the continuous black line is the mirror angular movement. The black dotted line is the sensor angular movement. In (a) (respectively (b)) the angular velocity is maximized (respectively minimized) and the angular acceleration is minimized (respectively maximized). Legend similar to Fig. 10. 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Mirror mean angular acceleration 0 M (mrad. s 2) 4 3 2 1 0 Mirror mean angular velocity 0 M (mrad. s 1) 0.6 0.4 0.2 0.0 0.2 Interferometer Phase shift (rad) Coriolis Centrifugal Euler Total 0 M = 0 S (a) 46 0 46 Time (ms) Angle 2 2 sensor mirror 6 5 4 3 2 1 0 Mirror mean angular velocity 0 M ( rad. s 1) 0 20 40 60 80 100 120 12 10 8 6 4 2 0 2 Interferometer Phase shift (rad) Coriolis Centrifugal Euler Total (b) 0 M = 0 S Mirror mean angular acceleration 0 M (mrad. s 2) 46 0 46 Time (ms) Angle 2 2 sensor mirror FIG. 13. Impact of the rotation compensation method on the interferometer phase shift. The black circles represent the experimental data with 1σ error bars. The black line is the theoretical model from Eq. (20). Legend similar to Fig. 12. ror and the sensor do not coincide, leading to a residual term - ̇Ω0 M · yMO (see Fig. 13 (b)). The centrifugal and Coriolis accelerations do not contribute to the phase shift as the mirror mean angular velocity is very small. In this configuration of high angular acceleration, the experimental results follow the global linear variation of the theoretical phase with the mirror mean angular acceleration. Nevertheless, an error to the model up to 0.6 rad is observed mostly for higher angular acceleration. At the compensation, the error is reduced to 0.1 rad. The differences between data and model could be explained by some variations in the distance yMO due to experimental imperfections in the rotation excitation of the sensor. V. CONCLUSION In this article, we present a detailed analysis of the impact of rotations on the phase and contrast of a coldatom accelerometer. Analytical models are confronted to experimental results in different configurations of rotation excitation where the impact of angular velocity or angular acceleration is either minimized or maximized. The experimental study is being carried out using a 14 unique hybrid atomic-electrostatic accelerometer where the atomic interferometer laser is retro-reflected on the proof-mass of an electrostatic accelerometer, similar to space geodesy missions instruments. This hybrid configuration allowed us to demonstrate experimentally the rotation compensation strategy of counter-rotating the interferometer mirror while rotating the whole instrument. We have shown that the contrast loss of the atomic interferometer is quite well reproduced by the theoretical model, regardless of the rotation excitation configuration in a range relevant for space geodesy applications. The use of an electrostatic accelerometer proof-mass as an actuated mirror enables the compensation of both angular velocities and accelerations. Contrast recoveries greater than 90% were demonstrated in both cases. The results concerning the interferometer phase are not as well understood and are still under investigation. The phase shift data show deviations between the experimental data and the model, especially for mirror rotations of higher amplitudes. This study is in line with previously published work that has aimed to analyze the detrimental impact of rotation on cold-atom accelerometers and has proposed mitigation strategies that would ultimately allow their operation in a dynamic environment [25, 26]. Our results, model, and experimental data are in agreement and bring additional and complementary insights by studying in detail the contribution of each non-inertial term, namely Coriolis, centrifugal, and Euler accelerations. We also propose an alternative to the common solution based on a PZT-actuated mirror. The benefits and drawbacks of each of these techniques have yet to be studied. This topic is especially of primary importance in the perspective of future space missions on-boarding a coldatom accelerometer [6, 7, 9-14, 47], where satellite rotation could lead to detrimental issues. Our results constitute first experimental demonstrations that lead us to be confident in the possibility of retrieving the atomic contrast in orbit with the compensation method, considering that the atom source would be a delta-kickcollimated Bose-Einstein condensate [48] submitted to a Mach-Zehnder light pulse interferometer with an interrogation time T ≈1s. Concerning the impact of rotation on the interferometer phase, there is still a need in the near future to assess precisely the performance of the rotation compensation system, as it would directly impact the bias and stability of the measurement. Here we also initiate some discussion on the more appropriate position of a cold-atom instrument relative to the centerof-rotation of the satellite, so as to minimize rotation impact and propose a way to get rid of remaining centrifugal acceleration inherent to rotation compensation with mirror actuation. 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The variation of the EA position with respect to a rest position δxEA along the ⃗xS axis is measured during the interferometer by use of capacitive detection. As the interferometer is sensitive to the mirror acceleration, the induced phase shift can be computed with Eq. (A1). ∆ΦaccEA = keffT 2 [ δxEA(t0) -2δxEA(t0 + T) + δxEA(t0 + 2T) ] (A1) Appendix B: Two-axis rotation of the sensor In contrast with what is expected, the sensor does not rotate purely around the ⃗zS axis. A small component around the ⃗yS axis was measured with the two-axis gyroscope. The induced effect on the phase shift was computed with Eq. (B1) and subtracted from the experimental data presented in the article. ∆ΦSy = keffT 2 " 2Ω0 Sy · vz + 3 2Taz + ̇Ω0 Sy · (zOA + vzT + 1 2T 2az) + Ω0 Sy 2 · xOA + 3vxT + 7 2T 2ax # (B1) with Ω0 Sy and ̇Ω0 Sy the mean angular velocity and mean angular acceleration around ⃗yS during the interferometer acquisition sequence. Appendix C: Rotating the mirror, a way to characterize the atom cloud The rotation of the mirror was used to measure the kinematic parameters of the atom cloud at launch (index 0 on parameters) such as its temperature, size, mean velocity, and position. To do so, the phase shift and contrast are now expressed as a function of the parameters at launch. Velocity distribution: The width of the velocity distribution was measured through the contrast loss induced by rotation. In the absence of angular acceleration, the contrast loss is only due to Coriolis and centrifugal acceleration. The contribution of the latter is negligible in this study. C C = exp " -2k2 effT 4Ω0 M 2σ2 vi # (C1) 0.0 0.5 1.0 1.5 2.0 Mirror angular velocity M (mrad. s 1) 0.0 0.2 0.4 0.6 0.8 1.0 Normalised contrast 46 0 46 Time (ms) Mirror angle 2 2 FIG. 14. Temperature measurements through the Coriolis induced contrast loss. The blue circles (respectively the red squares) are the experimental data for a rotation around the ⃗zM axis (respectively the ⃗yM axis). The plain blue line (resp. the dotted red line) is the fit curve of the experimental data by Eq. (C1). The angular movement of the mirror, black line in the inset, is a ramp measured by the capacitive detection. The red lines represent the laser pulses of the interferometer. An angular ramp was imposed to the mirror around the ⃗zM axis to measure the velocity distribution along the ⃗yM axis and vice versa (see Fig. 14). The normalized contrast was fitted with Eq. (C1) with i ∈{y; z}. The width of the velocity distribution was measured at σvy = 10.8(2) mm s-1 along the ⃗yM axis and σvz = 11.1(2) mm s-1 along the ⃗zM axis. Size: The same principle was applied to measure the size of the atom cloud right after the cooling stage. In the absence of an angular velocity, the contrast loss is due only to the angular acceleration and the velocity and position distributions of the cloud. As the velocity distribution was already measured, the position distribution can be determined. C C = exp " -8k2 effA2 M · σ2 i + (t0 + T)2σ2 vi !# (C2) A sine movement of angular amplitude AM was imposed to the mirror around the ⃗zM axis to measure the size along the ⃗yM axis and vice versa (see Fig. 15). Such an angular movement maximizes the angular acceleration of the mirror. The normalized contrast was fitted with Eq. (C2) with i ∈{y; z}. The width of the position distribution was measured at σy0 = 0.42(5) mm along the ⃗yM axis and σz0 = 0.66(5) mm along the ⃗zM axis. Mean velocity: The mean velocity of the atom cloud was measured through the Coriolis-induced phase shift. The impact of centrifugal acceleration on the phase shift cannot be ignored but can be estimated as the vertical 17 0 20 40 60 80 Mirror angular amplitude AM ( rad) 0.0 0.2 0.4 0.6 0.8 1.0 Normalised contrast 46 0 46 Time (ms) Mirror angle AM 2 2 FIG. 15. Size measurements through the Euler induced contrast loss. The plain blue line (resp. the dotted red line) is the fit curve of the experimental data by Eq. (C2). The angular movement of the mirror is a sine as described in the inset. Legend similar to Fig. 14. kinematic parameters of the cloud are known. ∆Φ = keffT 2 " -2ΩM · vy0 + 2ΩM 2(axL · (t2 0 2 + t0T + T 2) + vx0(t0 + T) + xMA) # (C3) ∆Φ = keffT 2 " 2ΩM · vz0 + 2ΩM 2(axL · (t2 0 2 + t0T + T 2) + vx0(t0 + T) + xMA) # (C4) An angular ramp was imposed to the mirror around the ⃗zM axis to measure the velocity along the ⃗yM axis and vice versa (see Fig. 16). The phase shift was fitted with Eq. (C3) for the rotation along the ⃗zM axis (in blue) and with Eq. (C4) for the rotation along the ⃗yM axis (in red). The mean velocity was measured at vy0 = -1.3(3) mm s-1 along the ⃗yM axis and vz0 = 0.3(3) mm s-1 along the ⃗zM axis. Mean position: The same principle was applied to measure the mean position of the atom cloud right after the cooling stage. In the absence of an angular velocity, the phase shift is due only to the Euler acceleration and the velocity and position of the cloud. As the velocity was already measured, the position can be determined. 0.0 0.5 1.0 1.5 2.0 Mirror angular velocity M (mrad. s 1) 0.1 0.0 0.1 0.2 0.3 Interferometer Phase shift (rad) 46 0 46 Time (ms) Mirror angle 2 2 FIG. 16. Mean velocity measurements through the Coriolis induced phase shift. The plain blue line (resp. the dotted red line) is the fit curve of the experimental data by Eq. (C3) (resp. Eq. (C4)). The gray area delimits the excluded points from the fit. The angular movement of the mirror is a ramp as described in the inset. Legend similar to Fig. 14. ∆Φ = keffT 2 4AM T 2 · ((t0 + T)vy0 + yMA) (C5) ∆Φ = keffT 2 -4AM T 2 · ((t0 + T)vz0 + zMA) (C6) A sine movement was imposed to the mirror around the ⃗zM axis to measure the position along the ⃗yM axis and vice versa (see Fig. 17). The phase shift was fitted with Eq. (C5) for the rotation around the ⃗zM axis (in blue) and with Eq. (C6) for the rotation around the ⃗yM axis (in red). The mean position was measured at yMA = 1.09(3) mm along the ⃗yM axis and zMA = 0.66(3) mm along the ⃗zM axis. 18 0 20 40 60 80 Mirror angular velocity AM ( rad) 4 2 0 2 4 6 Phase shift (rad) 46 0 46 Time (ms) Mirror angle AM 2 2 FIG. 17. Mean position measurements through the Euler induced phase shift. The plain blue line (resp. the dotted red line) is the fit curve of the experimental data by Eq. (C5) (resp. Eq. (C6)). The angular movement of the mirror is a sine as described in the inset. Legend similar to Fig. 14
|
2510.14756
|
PLUTO:
A BENCHMARK
FOR EVALUATING EFFI-
CIENCY OF LLM-GENERATED HARDWARE CODE
Manar Abdelatty∗, Maryam Nouh∗, Jacob K. Rosenstein, & Sherief Reda
Department of Electrical and Computer Engineering
Brown University, Providence, RI 02906, USA
manar abdelatty@brown.edu
ABSTRACT
Large Language Models (LLMs) are increasingly used to automate hardware de-
sign tasks, including the generation of Verilog code. While early benchmarks
focus primarily on functional correctness, efficient hardware design demands ad-
ditional optimization for synthesis metrics such as area, delay, and power. Ex-
isting benchmarks fall short in evaluating these aspects comprehensively: they
often lack optimized baselines or testbenches for verification. To address these
gaps, we present Pluto, a benchmark and evaluation framework designed to as-
sess the efficiency of LLM-generated Verilog designs. Pluto presents a compre-
hensive evaluation set of 114 problems with self-checking testbenches and mul-
tiple Pareto-optimal reference implementations. Experimental results show that
state-of-the-art LLMs can achieve high functional correctness, reaching 78.3% at
pass@1, but their synthesis efficiency still lags behind expert-crafted implemen-
tations, with area efficiency of 63.8%, delay efficiency of 65.9%, and power effi-
ciency of 64.0% at eff@1. This highlights the need for efficiency-aware evaluation
frameworks such as Pluto to drive progress in hardware-focused LLM research.
1
INTRODUCTION
Large Language Models (LLMs) are beginning to reshape hardware design by automating key steps
in hardware design workflows, including Verilog code generation Thakur et al. (2023a;b); Liu et al.
(2023a), optimization Yao et al. (2024); Guo & Zhao (2025), verification Qiu et al. (2024a), de-
bugging Tsai et al. (2024), high-level synthesis Xiong et al. (2024), and post-synthesis metric esti-
mation Abdelatty et al. (2025). While these advances highlight the potential of LLMs in hardware
design, most research has focused on functional correctness of generated designs, with little atten-
tion to design quality metrics such as area, delay, and power.
In hardware design, the quality of Verilog code is not determined solely by functional correctness.
Designs typically undergo logic synthesis, where Verilog code is mapped to gate-level implementa-
tions in a target technology. This process exposes critical efficiency metrics—such as silicon area,
timing delay, and power consumption—that directly impact manufacturability and performance.
Unlike software code, where correctness and execution speed often suffice, hardware code quality
is inherently tied to these post-synthesis metrics.
In order to evaluate the functional correctness of LLM-generated Verilog code, several benchmarks
have been proposed including VerilogEval Liu et al. (2023a) and RTLLM Lu et al. (2024). Recent
efforts, including RTLRewriter Yao et al. (2024), ResBench Guo & Zhao (2025), GenBen Wan et al.
(2025), and TuRTLe Garcia-Gasulla et al. (2025), have begun to evaluate quality of LLM-generated
hardware code in terms of post-synthesis metrics. However, these benchmarks face key limitations:
• Absence of Optimal Ground Truth Solutions True efficiency should be measured against
implementations that are explicitly optimized for specific objectives such as silicon area,
delay, or power consumption. Prior studies rely on canonical solutions from VerilogEval
and RTLLM as reference solutions. Our analysis shows that these solutions are not the
most optimal in terms of post-synthesis metrics.
∗Equal contribution.
1
arXiv:2510.14756v1 [cs.CL] 16 Oct 2025
• Lack of Clock Latency Agnostic Testbenches Many common optimization pat-
terns—such as register pipelining, resource sharing, or FSM restructuring—introduce vari-
ations in clock-cycle latency between the optimized and unoptimized designs. To support
fair evaluation, testbenches must be self-checking and tolerant of different latency require-
ments. Existing benchmarks, however, assume identical latency between the reference
model and the design under test, making them unsuitable for efficiency benchmarking.
In order to address these limitations, we introduce Pluto, the first benchmark designed to evaluate
both functional correctness and synthesis efficiency of LLM-generated Verilog code. Our contribu-
tions are as follows:
• Per-Metric Ground Truth Optimal Solutions. We provide a suite of 114 problems where
each is optimized for area, delay, and power separately, yielding Pareto-front optimal solu-
tions. Our ground truth solutions are significantly more efficient than canonical solutions
in RTLLM and VerilogEval.
• Optimization-Aware Testbenches. Each problem is accompanied by clock-cycle agnostic
testbenches that accommodate varying latency requirements, ensuring robust evaluation of
different optimization patterns.
• Comprehensive Evaluation. We adapt the eff@k metric introduced in Qiu et al. (2024b)
to measure the efficiency of hardware designs. Our extended metric is a three-dimensional
vector that evaluates LLM-generated code across multiple objectives: area, delay and
power.
2
RELATED WORK
Software Code Benchmarks Large Language Models (LLMs) have been extensively studied for
code generation across both software and hardware domains, with most early benchmarks focusing
primarily on functional correctness rather than efficiency. In software, works such as Mercury Du
et al. (2024) and ENAMEL Qiu et al. (2024b) move beyond correctness to explicitly evaluate run-
time efficiency of LLM-generated programs. The Mercury Du et al. (2024) benchmark contains
LeetCode style problems. Each problem is accompanied by an expert-written solution that repre-
sents the most optimal implementation in terms of run-time efficiency. ENAMEL Qiu et al. (2024b)
also introduces a Python benchmark to evaluate the run-time efficiency of LLM-generated code.
Hardware Code Benchmarks In hardware design, early work on LLM-generated Verilog empha-
sized functional correctness. VerilogEval Liu et al. (2023a) only evaluates whether the LLM gen-
erated code passes the testbench check, while RTLLM Lu et al. (2024) additionally checks if the
generated code is synthesizable. More recent efforts have shifted toward assessing and improv-
ing the efficiency of LLM-generated designs, which can be categorized into two main categories:
Specifications-to-Efficient-Verilog where the LLM is tasked with translating natural language in-
struction to optimized Verilog code directly, and Unoptimized-Verilog-to-Optimized-Verilog, where
the LLM is tasked with rewriting an unoptimized Verilog code to optimized Verilog code.
In the Specifications-to-Efficient-Verilog formulation, the LLM is prompted with a natural language
problem description and directly generates optimized Verilog. Benchmarks, such as GenBen Wan
et al. (2025), TuRTLe Garcia-Gasulla et al. (2025), evaluate these generations in functional correct-
ness, synthesizability, and post-synthesis metrics such as area, delay, and power. However, it relies
on VerilogEval problems as ground truth. These reference designs are not necessarily optimized
for power, performance, or area, and thus do not represent true Pareto-optimal solutions. ResBench
Guo & Zhao (2025) also does not define any gold-standard or reference-optimal implementations,
which makes it difficult to quantitatively assess how close the generated solutions are to ideal results.
The Unoptimized-Verilog-to-Optimized-Verilog setting provides the LLM with a functionally cor-
rect but unoptimized Verilog implementation and asks it to produce a more efficient version. RTL-
Rewriter Yao et al. (2024) enhances this with retrieval-augmented generation and feedback through
the synthesis loop. However, RTLRewriter lacks associated testbenches, making it unsuitable for
assessing the functional correctness of the generated code.
2
Table 1: Comparison of prior software and hardware code generation benchmarks. Pluto addresses
key limitations by enabling metric-specific optimization with three reference implementations per
problem, each optimized for area, delay, or power.
Benchmark
Language Functionality Synthesizability Efficiency Per-Metric Optimisation Tasks
HumanEval
Python
✓
–
×
×
164
Mercury
Python
✓
–
✓
×
256
ENAMEL
Python
✓
–
✓
×
142
VerilogEval
Verilog
✓
×
×
×
156
RTLLM
Verilog
✓
✓
×
×
30
RTLRewriter Verilog
×
✓
✓
×
95
ResBench
Verilog
✓
✓
×
×
56
GenBen
Verilog
✓
✓
×
×
300
TuRTLe
Verilog
✓
✓
×
×
223
CVDP
Verilog
✓
✓
×
×
783
Pluto (Ours) Verilog
✓
✓
✓
✓
114
As summarized in Table 1, Pluto is the first benchmark to offer per-metric optimization, provid-
ing separate expert-optimized reference designs for area, delay, and power. This enables targeted,
metric-specific evaluation of LLMs, an aspect missing from prior benchmarks.
3
PLUTO BENCHMARK
module opt_power (
input [31:0] din,
output [5:0] dout
);
always_comb begin
idx = '0;
found = 1'b0;
for (int i = 0; i < 32; i++) begin
if (!found && din[i]) begin
idx = 5'(i);
found = 1'b1;
end
end
end
assign dout = found ? idx : 5'(32);
endmodule
module opt_delay (
input [31:0] din,
output [5:0] dout
);
assign masked_bits =
din & (-din);
case (masked_bits):
32'h00000001: dout = 0;
32'h00000002: dout = 1;
....
32'h20000000: dout = 29;
32'h40000000: dout = 30
32'h80000000: dout = 31;
endcase
endmodule
module opt_area (
input [31:0] din,
output [5:0] dout
);
logic [5:0] temp [31:0];
genvar i;
generate for (i = 31; i >= 0; i--)begin
assign temp[i] = din[i] ?
i[5:0] : temp[i+1];
end endgenerate
assign dout = temp[0];
endmodule
Trailing Zeros Circuit - Problem # 27
Find the number of trailing 0s in the binary representation of the input (din). If the input value is all 0s, the number of
trailing 0s is the data width (DATA_WIDTH)
Area Optimized
Delay Optimized
Power Optimized
//
Score
Circuit: LSB Isolation via 2's
Complement & Parallel One Hot Encoder
Circuit: MSB Priority Encoder via
Mux Chain
Circuit: Scan from LSB to MSB with
found flag to reduce switching
..
Score
Score
Figure 1: Overview of the Pluto benchmark on the trailing zeros detection task. We show three
reference implementations optimized for different synthesis metrics compared to the unoptimized
baseline: (left) area, using a mux-based priority encoder, reducing area by 33%; (center) delay, using
an LSB isolation circuit with a parallel one-hot encoder, reducing delay by 44%; and (right) power,
using an LSB-to-MSB scanning method with early termination, reducing total power by 34%. See
Appendix. A.1 for unoptimized baseline and self-checking testbench.
3.1
DATA CONSTRUCTION
To enable a comprehensive evaluation of synthesis efficiency for LLM-generated hardware code,
we construct the Pluto evaluation set, which contains a diverse collection of high-quality digital
design problems spanning a broad range of difficulties. Specifically, we curated 114 problems from
3
various publicly available sources, including open-source hardware projects, educational platforms
such as ChipDev ChipDev (2025), a LeetCode-inspired platform for practicing Verilog coding, and
prior benchmark suites such as RTLRewriter Yao et al. (2024), RTLLM Lu et al. (2024), and Ver-
ilogEval Liu et al. (2023b). Each problem is specified by a high-level description outlining the
functional requirements, together with a baseline unoptimized Verilog implementation.
The problem set covers a wide spectrum of tasks in digital logic design, ranging from arithmetic
units and control circuits to sequential state machines. To systematically capture variation in design
complexity, we adopt ChipDev’s difficulty annotations and classify problems into three levels: easy,
medium, and hard. These labels reflect the intrinsic challenge of translating the textual description
into a correct Verilog implementation, thereby providing a principled way to distinguish between
problems of different complexity.
Importantly, the resulting collection balances accessibility with challenge: many problems that ap-
pear straightforward can nonetheless expose substantial differences in synthesis efficiency depend-
ing on the optimization strategies applied. The diverse composition of easy, medium, and hard tasks
therefore enables a nuanced assessment of an LLM’s ability to generate synthesis-efficient Verilog
under varying constraints. In total, the 114 selected problems provide a representative and scalable
testbed for benchmarking LLM-based Verilog efficiency.
Each problem instance in the Pluto benchmark includes the following components:
• Prompt: A natural language description of the hardware design task intended to guide the
LLM-generation.
• Module Header: A fixed interface shared across all versions of the Verilog module to
ensure consistency and comparability.
• Unoptimized Verilog Code: A baseline implementation used as the reference for testing.
• Optimized Verilog Code: Three distinct implementations with tradeoffs, each optimized
by hand using design experts for a single metric: area, delay, or power.
• Testbench: A manually crafted, fully self-checking testbench that verifies functional equiv-
alence between the unoptimized and any optimized design. These testbenches ensure full
input space coverage and flag any mismatches during simulation. For sequential circuits,
testbenches are clock-cycle agnostic, supporting latency differences introduced by opti-
mizations such as pipelining or resource sharing.
All components in the evaluation set are manually developed. This ensures high quality and guar-
antees that the LLM under evaluation has not previously encountered any part of the dataset during
training. In particular, the testbenches and optimized code serve as held-out ground truth references,
providing an unbiased benchmark for assessing the efficiency and correctness of LLM-generated
Verilog designs.
To illustrate the structure of problems in the Pluto evaluation set, Figure 1 presents the example
of a trailing zeros detection circuit, categorized as an easy problem, along with its three metric-
specific optimizations. As shown, each optimization achieves peak efficiency in its targeted metric,
while performance in the remaining two metrics declines. This behavior emphasizes the inherent
trade-offs across design objectives in hardware design and highlights the necessity of metric-specific
optimization strategies.
3.2
OPTIMIZATION WORKFLOW
Each unoptimized design in the Pluto set is further refined through manual optimization by expert
engineers to generate three distinct versions optimized separately for area, delay, and power. This
workflow follows a systematic process that ensures both the correctness and the efficiency of the re-
sulting designs. After applying metric-specific transformations, each optimized circuit is rigorously
verified for functional correctness using Icarus Verilog Williams et al. (2002), supported by robust
self-checking testbenches that guarantee equivalence with the unoptimized baseline. The optimized
versions are then synthesized to confirm that improvements translate into measurable gains in area,
timing, or power, thereby providing reliable performance baselines against which LLM-generated
designs can be evaluated.
4
To understand how these efficiency gains are achieved, we visualize the optimization strategies ap-
plied across the dataset in appendix A.2. The strategies vary significantly depending on the target
metric. For area, arithmetic optimizations and logic simplification are most commonly employed,
and FSM restructuring plays an important role in reducing redundant states and transitions. Delay
improvements rely heavily on exploiting parallelism and restructuring control logic, often comple-
mented by logic simplification and pipelining techniques that shorten the critical path. For power,
it’s reducing switching activity through register and logic optimizations, supported by techniques
such as operand isolation, and clock gating to further suppress unnecessary toggling.
The distribution of strategies reveals that no single optimization technique dominates across all
objectives. Instead, engineers select strategies tailored to the specific metric, reflecting the trade-
offs inherent in digital design. As shown in Figure 2, this process results in consistent improvements
across the dataset, with average reductions of 19.19% in area, 21.96% in delay, and 22.55% in
power. This highlights the importance of metric-specific approaches and provide a robust baseline
for evaluating LLM-generated hardware code efficiency.
To further illustrate the impact of expert-driven optimization, Table 2 presents representative exam-
ples drawn from both RTLLM and VerilogEval. These case studies highlight how different strate-
gies, such as arithmetic unit sharing, FSM encoding choices, and counter-based control logic, trans-
late into concrete improvements across area, delay, and power. As shown, across both VerilogEval
and RTLLM, expert-optimized designs consistently outperform baseline implementations. In partic-
ular, for RTLLM problems our expert-written solutions achieve average improvements of 18.75% in
area, 22.75% in delay, and 20.43% in power compared to their canonical solutions. For VerilogEval
problems, the improvements average 10.46% in area, 10.33% in delay, and 13.61% in power.
(a) Area Comparison
(b) Delay Comparison
(c) Power Comparison
Figure 2: Distribution of area, delay, and power across Pluto benchmark designs before and after
manual metric-specific optimizations.
Table 2: A sample of benchmark problems from Pluto dataset. Our expert-optimized solutions
(area, delay, power) are significantly more efficient than the baseline benchmark implementations.
See Appendix A.3 for full problem implementation.
ID
Source
Problem
Description
Benchmark
Solu-
tion
Expert Solution (Ours)
#60
RTLLM
ALU for a 32-bit MIPS-ISA
CPU with operations like
{ADD, SUB, AND, OR,
XOR, SLT, shifts, LUI}.
ALU
implemen-
tation
with
case
statement
and
parameterized
op-
codes.
Area ↓26%: Shared adder for arithmetic, simplified flag
logic, and operand reuse
Delay ↓26%: Parallel datapaths with one-hot muxing
Power ↓4%: Operand gating and early zeroing for large
shifts to cut switching activity
#68
RTLLM
FSM detecting the sequence
10011
on
a
serial
input
stream with support for con-
tinuous and overlapping de-
tection
States
binary
en-
coded,
sequential
next-state logic, and
registered
Mealy
output
Area ↓32%: Casez-based transitions and direct output
Delay ↓17%: One-hot state encoding with pre-decoded in-
puts and Moore-style output
Power ↓23%: Compact binary encoding and casez-based
transitions to cut toggling
#87
VerilogEval
Module
controls
a
shift
register
enable
signal,
shift ena asserted for 4
clock cycles on reset, then
remain 0 until the next reset
Uses explicit states
with next-state logic
to drive shift ena
Area ↓47%: Minimized register width (2-bit counter) and
compact comparator logic
Delay ↓37%: Wider counter (3 bits) to simplify comparison
and reduce logic depth on the critical path
Power ↓46%: Small counter reused
#104
VerilogEval
Conway’s
Game
of
Life
with a 16×16 toroidal grid:
each cell updates based on
neighbor counts n (live if
n = 3 or n = 2 & alive)
Straightforward
RTL
with
per-
cell
neighbor
recomputation
and
sequential summing
Area ↓19%: Shared neighbor computations across rows,
bitwise rotations for wraparound, less duplicate summations
Delay ↓37%: Parallel neighbor summation with carry-save
adder tree and direct decode for 2 and 3
Power ↓36%: Reduced toggling via computation reuse
5
3.3
EFFICIENCY METRICS
We use the pass@k Liu et al. (2023a) for measuring the functional correctness of LLM-generated
Verilog code. The pass@k metric, defined in appendix. A.5, measures the percentage of problems
for which at least one of the top-k generated samples passes the self-checking testbench.
To evaluate the synthesis efficiency of functionally correct samples, we adapt the eff@k introduced
in Qiu et al. (2024b) to Verilog code. First, we introduce the efficiency score ei,j, defined in Eq. 1,
which quantifies how close an LLM-generated design is to optimal ground truth implementation.
In this equation, ˆRi,j denotes the reported synthesis metric (e.g., area, delay, or power) for the j-th
sample of problem i, Ti,j denotes an upper bound beyond which the design is considered inefficient,
and Ri,j denotes the optimal (lowest) known reference value for that metric. A score of 1 indicates
that the sample exactly matches the optimal reference, while a score of 0 indicates that it exceeds
the acceptable threshold or is functionally incorrect.
ei,j =
max(0, Ti,j −ˆRi,j)
Ti,j −Ri,j
,
if ni,j is correct
0,
otherwise.
(1)
eff@k = 1
N
N
X
i=1
EJ⊆{1,...,n}, |J|=k
max
j∈J ei,j
= 1
N
N
X
i=1
n
X
r=k
r−1
k−1
n
k
ei,(r).
(2)
We then use the efficiency score ei,j for computing the eff@k, defined in Eq. 2 as the average of the
best (i.e., highest) efficiency scores among the top-k functionally correct samples for each problem.
We use the unbiased estimator introduced in Qiu et al. (2024b) for computing eff@k which computes
the expectation value over a random subset J of code samples with size K.
4
EVALUATION RESULTS
We evaluate 18 large language models (LLMs) using our Pluto benchmark, which includes pro-
prietary LLMs, general-purpose foundation models, code-specialized models, and Verilog-tuned
models. To comprehensively assess efficiency-aware generation, we consider the two problem for-
mulations in Pluto: translating unoptimized Verilog code into optimized implementations, and gen-
erating optimized code directly from natural-language specifications. For the first problem formu-
lation, only instruction-tuned models are evaluated, as code completion models generally reproduce
the unoptimized code without meaningful improvements.
4.1
MAIN RESULTS
Table 3 (a) reports the pass@k and eff@k metrics for the first problem formulation, where the task is
to re-write unoptimized Verilog into more efficient implementations. Several trends emerge. First,
in terms of functional correctness (pass@k), domain-tuned models such as VeriThoughts-Inst-7B
and RTLCoder-DeepSeek-V1 achieve performance comparable to much larger foundational models
like DeepSeek-Chat, demonstrating the benefit of Verilog-specific training. However, in terms of
synthesis efficiency (eff@k), all models exhibit a noticeable drop relative to their pass@k scores.
This gap underscores a common limitation: while LLMs can generate functionally correct Verilog,
they struggle to match the Pareto-efficient expert baselines across area, delay, and power.
Table 3 (b) reports the pass@k and eff@k metrics for the second problem formulation, where models
are tasked with translating natural language specifications into optimized Verilog implementations.
This task is more challenging, and as a result, both pass@k and eff@k scores are consistently lower
across all models. Similar to the first formulation, all models also exhibit lower eff@k values com-
pared to their corresponding pass@k scores, underscoring the persistent difficulty of generating
designs that are not only functionally correct but also synthesis-efficient. However, the relative gap
between pass@k and eff@k is smaller in this setting compared to the first formulation. This is
because specification-to-RTL translation is substantially harder: models often struggle to produce
functionally correct code in the first place, which suppresses both correctness and efficiency scores.
6
Table 3: Evaluation results using Pluto for two problem formulations: P1: Unoptimized-Verilog-
to-Optimized-Verilog and P2: Specifications-to-Optimized-Verilog. pass@k measures functional
correctness, while eff@k measures efficiency across area, delay, and power.
(a) P1: Unoptimized-Verilog-to-Optimized-Verilog
Model
pass@1
pass@5
pass@10
eff@1
eff@5
eff@10
Area
Delay
Power
Area
Delay
Power
Area
Delay
Power
GPT-3.5
0.325
0.517
0.594
0.271
0.296
0.282
0.462
0.491
0.450
0.540
0.568
0.520
GPT-4o-mini
0.506
0.705
0.751
0.469
0.476
0.467
0.662
0.677
0.639
0.714
0.744
0.687
DeepSeek-Chat
0.612
0.802
0.860
0.586
0.599
0.601
0.776
0.795
0.794
0.839
0.862
0.846
Llama-3.3-70B-Instruct
0.473
0.701
0.757
0.446
0.462
0.429
0.662
0.696
0.662
0.707
0.760
0.735
Llama-3.1-8B-Instruct
0.160
0.432
0.567
0.127
0.156
0.145
0.358
0.437
0.384
0.494
0.584
0.505
Mistral-7B-Instruct-v0.2
0.106
0.318
0.453
0.078
0.094
0.100
0.244
0.296
0.301
0.358
0.446
0.427
Mixtral-8x7B-v0.1
0.255
0.520
0.652
0.217
0.231
0.210
0.462
0.487
0.447
0.593
0.630
0.561
starcoder2-15b-instruct-v0.1
0.659
0.960
0.988
0.611
0.633
0.591
0.879
0.924
0.871
0.913
0.952
0.904
CodeLlama-70b-Instruct-hf
0.576
0.905
0.956
0.522
0.541
0.523
0.842
0.876
0.824
0.903
0.925
0.878
DeepSeek-Coder-33B
0.783
0.963
0.997
0.638
0.659
0.640
0.902
0.927
0.883
0.942
0.960
0.927
Qwen2.5-Coder-7B-Inst
0.479
0.785
0.866
0.438
0.452
0.419
0.710
0.759
0.741
0.785
0.848
0.833
yang-z/CodeV-QC-7B
0.231
0.416
0.506
0.211
0.208
0.187
0.381
0.390
0.361
0.455
0.491
0.442
RTLCoder-DeepSeek-V1
0.532
0.854
0.915
0.471
0.495
0.468
0.774
0.789
0.757
0.843
0.850
0.809
VeriThoughts-Inst.-7B
0.611
0.797
0.854
0.540
0.560
0.524
0.708
0.740
0.702
0.763
0.785
0.765
(b) P2: Specifications-to-Optimized-Verilog
Model
pass@1
pass@5
pass@10
eff@1
eff@5
eff@10
Area
Delay
Power
Area
Delay
Power
Area
Delay
Power
GPT-3.5
0.239
0.395
0.471
0.225
0.235
0.225
0.373
0.390
0.381
0.439
0.469
0.468
GPT-4o-mini
0.391
0.533
0.591
0.360
0.377
0.363
0.475
0.520
0.495
0.532
0.572
0.551
DeepSeek-Chat
0.557
0.688
0.719
0.545
0.552
0.528
0.689
0.680
0.651
0.726
0.710
0.684
Llama-3.3-70B-Instruct
0.363
0.541
0.594
0.348
0.345
0.342
0.515
0.533
0.510
0.564
0.602
0.557
Llama-3.1-8B-Instruct
0.087
0.224
0.301
0.075
0.073
0.081
0.189
0.184
0.204
0.270
0.251
0.279
Mistral-7B-Instruct-v0.2
0.030
0.108
0.164
0.024
0.015
0.024
0.078
0.067
0.088
0.112
0.117
0.134
Mixtral-8x7B-v0.1
0.082
0.176
0.222
0.082
0.068
0.079
0.172
0.131
0.163
0.202
0.166
0.211
starcoder2-15b-instruct-v0.1
0.243
0.454
0.512
0.226
0.249
0.220
0.429
0.466
0.409
0.489
0.525
0.458
CodeLlama-70b-Instruct-hf
0.212
0.446
0.532
0.202
0.207
0.194
0.418
0.440
0.437
0.498
0.535
0.524
DeepSeek-Coder-33B
0.257
0.429
0.482
0.231
0.254
0.246
0.387
0.424
0.421
0.446
0.490
0.468
Qwen2.5-Coder-7B-Inst
0.164
0.324
0.389
0.158
0.162
0.148
0.307
0.319
0.295
0.366
0.375
0.357
code-gen-verilog-16b
0.068
0.200
0.289
0.069
0.064
0.058
0.188
0.175
0.188
0.268
0.253
0.272
yang-z/CodeV-CL-7B
0.265
0.485
0.553
0.233
0.243
0.237
0.432
0.455
0.436
0.493
0.536
0.511
yang-z/CodeV-QC-7B
0.260
0.458
0.529
0.223
0.229
0.222
0.420
0.415
0.410
0.497
0.480
0.478
yang-z/CodeV-All-QC
0.175
0.317
0.374
0.150
0.162
0.144
0.297
0.304
0.256
0.368
0.379
0.284
RTLCoder-DeepSeek-V1
0.203
0.400
0.480
0.177
0.199
0.184
0.345
0.404
0.361
0.417
0.499
0.430
RTLCoder-Mistral
0.199
0.347
0.418
0.185
0.188
0.188
0.316
0.332
0.329
0.381
0.411
0.395
VeriThoughts-Inst.-7B
0.216
0.336
0.398
0.211
0.206
0.207
0.316
0.330
0.329
0.367
0.394
0.393
4.2
ABLATION STUDIES
In addition to the Verilog code writing style, post-synthesis metrics are also influenced by external
factors such as the synthesis tool employed, the target technology library, and the optimization
sequence executed by the tool. To understand the robustness of our proposed benchmark and isolate
the impact of these factors, we present two ablation studies that evaluate efficiency trends in the
Pluto benchmark across different synthesis tools, optimization strategies, and technology libraries.
4.2.1
SYNTHESIS TOOL AND TECHNOLOGY AGNOSTICISM
In this experiment, we repeated synthesis runs for the three optimized reference implementations
in Pluto, as well as the unoptimized baseline, using two distinct synthesis tools: Yosys Wolf et al.
(2013), an open-source framework, and Cadence Genus cad, a commercial synthesis tool. To further
evaluate generalizability, we also targeted two technology libraries representing different fabrication
nodes: the SkyWater 130nm library Google and a 65nm TSMC library tsm. We then computed the
efficiency score for each tool and library configuration by comparing each optimized implementation
against the corresponding unoptimized baseline across area, delay, and power metrics. As shown in
Figure 3, efficiency scores remain consistent across all synthesis tool and technology combinations.
This demonstrates that Pluto’s optimization patterns deliver consistent tradeoffs across different
synthesis tools and technology libraries.
7
(a) Genus: Area (earea)
(b) Genus: Delay (edelay)
(c) Genus: Power (epower)
(d) Yosys: Area (earea)
(e) Yosys: Delay (edelay)
(f) Yosys: Power (epower)
Figure 3: Efficiency scores for area, delay, and power across all benchmark problems, using both
Cadence Genus and Yosys with different technology libraries. Results show consistent efficiency
trends across synthesis tools and technologies.
(a) P15: Polynomial
(b) P5: Divide-by-evens
(c) P28: Adder
Figure 4: Area–delay tradeoffs of three problems in the Pluto benchmark under different synthesis
strategies. Each strategy corresponds to a distinct sequence of ABC logic synthesis commands.
4.2.2
SYNTHESIS OPTIMIZATION STRATEGIES
We also examine how different synthesis optimization strategies influence the post-synthesis met-
rics of Pluto’s optimized implementations. Synthesis tools allow designers to specify optimization
directives that steer the tool’s internal heuristics toward minimizing a particular metric while po-
tentially sacrificing others. To study this effect, we synthesized selected problems from the Pluto
benchmark under both area-optimized and delay-optimized optimization strategies. We used Yosys
as our synthesis tool and targeted the SkyWater 130nm library. Within Yosys, logic optimization
is carried out using the ABC framework Synthesis & Group (2024), which provides a collection
of optimization heuristics that can be configured to emphasize different objectives such as area or
delay minimization. Figure 4 illustrates the resulting Pareto fronts of area–delay trade-offs across
three representative problems. As expected, delay-optimized code consistently achieves superior
timing performance at the expense of larger area, whereas area-optimized code achieves lower area
but incurs higher delays. These results confirm that synthesis settings primarily shift designs along
the area–delay curve, while coding style remains the dominant factor, validating Pluto’s ability to
capture design efficiency independent of synthesis optimization settings.
5
FAILURE ANALYSIS AND INSIGHTS
While LLMs reliably produce functionally correct Verilog, their ability to optimize is uneven across
metrics. Area optimization is comparatively tractable, since it often reduces to logic simplification or
FSM re-encoding. By contrast, delay requires identifying and shortening the critical path, and power
depends on subtle factors like switching activity and memory usage. This difficulty is reflected in
8
(a)
(b)
Figure 5: Failure mode analysis of optimization outcomes. (a) Quadrant plot showing the correla-
tion between functional correctness (Pass@1) and synthesis efficiency (Eff@1) across area, delay,
and power objectives. (b) Heatmap of optimization strategy difficulty across different optimization
objectives area, delay, and power.
our quadrant analysis (Figure 5a and 8), where many delay- and power-optimized designs remain
correct but fail to improve efficiency, whereas area optimizations succeed more often.
Model scale and specialization strongly influence outcomes. Larger models (33B, 70B) capture
richer patterns and propose alternative architectures. Models with explicit reasoning traces (e.g.,
DeepSeek, VeriThoughts) better decompose transformations and achieve stronger optimizations.
Domain-tuned models outperform code models, which in turn outperform general-purpose LLMs,
showing the value of Verilog-specific pretraining. A fundamental limitation is that Verilog train-
ing data lacks efficiency labels. LLMs therefore default to surface-level pattern matching rather
than structural reasoning, and without feedback or synthesis-in-the-loop, they cannot tell whether
changes reduce gate count or lengthen the critical path. Completion-style models exacerbate this
issue, often rephrasing the baseline instead of innovating, whereas instruction-tuned models attempt
more substantive edits.
Finally, analysis of optimization strategies (Figure 5b) shows that the hardest transformations are
register optimizations for delay, followed by resource sharing for power, and sequential restructuring
for delay. In contrast, strategies tied to area are easier, aligning with our observation that area is the
most accessible metric for LLMs. Together, these findings suggest that true progress will require
metric-aware feedback and efficiency-focused benchmarks such as Pluto to guide future advances.
6
CONCLUSION
In this paper, we introduced Pluto, a comprehensive benchmark designed to evaluate the synthesis
efficiency of LLM-generated Verilog code. Pluto provides an evaluation set of 114 hardware design
problems, each accompanied by three reference optimized implementations (targeting area, delay,
and power), an unoptimized baseline, a self-checking testbench, and a natural language description.
Experimental results show that while LLMs can achieve high functional correctness, reaching up
to 78.3% at pass@1, their synthesis efficiency remains limited: area efficiency of 63.8%, delay
efficiency of 65.9%, and power efficiency of 64.0% at eff@1 compared to expert-crafted designs.
These findings highlight the importance of efficiency-aware benchmarks beyond correctness alone
and highlights the current limitations of LLMs in hardware optimization.
9
REFERENCES
Cadence Genus Synthesis Solution.
https://www.cadence.com/en US/home/tools/digital-design-
and-signoff/synthesis/genus-synthesis.html.
TSMC 65nm Technology Library (Proprietary). Accessed under license from TSMC.
Manar Abdelatty, Jingxiao Ma, and Sherief Reda. Metrex: A benchmark for verilog code metric
reasoning using llms. In Proceedings of the 30th Asia and South Pacific Design Automation
Conference, pp. 995–1001, 2025.
ChipDev. ChipDev: Hardware Interview Prep & Verilog Practice. https://chipdev.io, 2025.
Accessed: 2025-01-01.
Mingzhe Du, Anh Tuan Luu, Bin Ji, Qian Liu, and See-Kiong Ng. Mercury: A code efficiency
benchmark for code large language models. In The Thirty-eight Conference on Neural Informa-
tion Processing Systems Datasets and Benchmarks Track, 2024.
Dario Garcia-Gasulla, Gokcen Kestor, Emanuele Parisi, Miquel Albert’i-Binimelis, Cristian Gutier-
rez, Razine Moundir Ghorab, Orlando Montenegro, Bernat Homs, and Miquel Moreto. Turtle: A
unified evaluation of llms for rtl generation. arXiv preprint arXiv:2504.01986, 2025.
Google. Skywater-pdk. https://github.com/google/skywater-pdk. Accessed: 2025-
02-09.
Ce Guo and Tong Zhao. Resbench: Benchmarking llm-generated fpga designs with resource aware-
ness. arXiv preprint arXiv:2503.08823, 2025.
Mingjie Liu, Nathaniel Pinckney, Brucek Khailany, and Haoxing Ren. Invited paper: Verilogeval:
Evaluating large language models for verilog code generation. In 2023 IEEE/ACM International
Conference on Computer Aided Design (ICCAD), pp. 1–8, 2023a. doi: 10.1109/ICCAD57390.
2023.10323812.
Mingjie Liu, Nathaniel Pinckney, Brucek Khailany, and Haoxing Ren. Verilogeval: Evaluating large
language models for verilog code generation. In 2023 IEEE/ACM International Conference on
Computer Aided Design (ICCAD), pp. 1–8. IEEE, 2023b.
Yao Lu, Shang Liu, Qijun Zhang, and Zhiyao Xie. Rtllm: An open-source benchmark for design rtl
generation with large language model. In 2024 29th Asia and South Pacific Design Automation
Conference (ASP-DAC), pp. 722–727. IEEE, 2024.
Ruidi Qiu, Grace Li Zhang, Rolf Drechsler, Ulf Schlichtmann, and Bing Li.
Autobench: Au-
tomatic testbench generation and evaluation using llms for hdl design. In Proceedings of the
2024 ACM/IEEE International Symposium on Machine Learning for CAD, MLCAD ’24, New
York, NY, USA, 2024a. Association for Computing Machinery. ISBN 9798400706998. doi:
10.1145/3670474.3685956. URL https://doi.org/10.1145/3670474.3685956.
Ruizhong Qiu, Weiliang Will Zeng, James Ezick, Christopher Lott, and Hanghang Tong. How
efficient is llm-generated code?
a rigorous & high-standard benchmark.
arXiv preprint
arXiv:2406.06647, 2024b.
Berkeley Logic Synthesis and Verification Group. Abc: A system for sequential synthesis and veri-
fication. Technical report, University of California, Berkeley, 2024. URL https://people.
eecs.berkeley.edu/˜alanmi/abc/.
Shailja Thakur, Baleegh Ahmad, Zhenxing Fan, Hammond Pearce, Benjamin Tan, Ramesh Karri,
Brendan Dolan-Gavitt, and Siddharth Garg. Benchmarking large language models for automated
verilog rtl code generation. In 2023 Design, Automation & Test in Europe Conference & Exhibi-
tion (DATE), pp. 1–6. IEEE, 2023a.
Shailja Thakur, Baleegh Ahmad, Hammond Pearce, Benjamin Tan, Brendan Dolan-Gavitt, Ramesh
Karri, and Siddharth Garg. Verigen: A large language model for verilog code generation. ACM
Transactions on Design Automation of Electronic Systems, 2023b.
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YunDa Tsai, Mingjie Liu, and Haoxing Ren. Rtlfixer: Automatically fixing rtl syntax errors with
large language models. In IEEE/ACM Design Automation Conference (DAC’24), pp. 1–8, 2024.
doi: 10.1109/ICCAD57390.2023.10323812.
Gwok-Waa Wan, Yubo Wang, SamZaak Wong, Jingyi Zhang, Mengnv Xing, Zhe Jiang, Nan Guan,
Ying Wang, Ning Xu, Qiang Xu, and Xi Wang. Genben: A generative benchmark for LLM-aided
design. In Proceedings of the International Conference on Learning Representations (ICLR),
2025. URL https://openreview.net/forum?id=gtVo4xcpFI. Under review.
Stephen Williams et al. Icarus verilog: open-source verilog more than a year later. Linux Journal,
2002.
Clifford Wolf, Johann Glaser, and Johannes Kepler. Yosys—a free verilog synthesis suite. In 21st
Austrian Workshop on Microelectronics (Austrochip), volume 97, 2013.
Chenwei Xiong, Cheng Liu, Huawei Li, and Xiaowei Li. Hlspilot: Llm-based high-level synthesis.
arXiv preprint arXiv:2408.06810, 2024.
Xufeng Yao, Yiwen Wang, Xing Li, Yingzhao Lian, Ran Chen, Lei Chen, Mingxuan Yuan, Hong
Xu, and Bei Yu. Rtlrewriter: Methodologies for large models aided rtl code optimization. arXiv
preprint arXiv:2409.11414, 2024.
A
APPENDIX
A.1
UNOPTIMIZED CODE AND TESTBENCH FOR PROBLEM #17 (TRAILING ZEROS) IN
FIGURE 1
A.1.1
UNOPTIMIZED CODE
1
module unopt_model #(parameter
2
DATA_WIDTH = 32
3
) (
4
input
[DATA_WIDTH-1:0] din,
5
output logic [$clog2(DATA_WIDTH):0] dout
6
);
7
8
logic [DATA_WIDTH-1:0] din_adj;
9
logic [$clog2(DATA_WIDTH):0] idx;
10
11
always_comb begin
12
idx = 0;
13
din_adj = din & (˜din+1);
14
for (int i=0; i<DATA_WIDTH; i++) begin
15
idx += (din_adj[i]) ? i : 0;
16
end
17
end
18
19
assign dout = (din_adj == 0 ? DATA_WIDTH : din_adj == 1 ? 0 : idx);
20
21
endmodule
A.1.2
SELF-CHECKING TESTBENCH
1
‘timescale 1 ps/1 ps
2
3
module tb();
4
5
reg clk = 0;
6
initial forever #5 clk = ˜clk;
7
8
wire [5:0] dout_opt, dout_unopt;
9
reg [31:0] din;
10
11
integer errors = 0;
12
integer errortime = 0;
13
integer clocks = 0;
14
integer total_cycles = 200;
15
16
initial begin
11
17
$dumpfile("wave.vcd");
18
$dumpvars(1, clk, din, dout_opt, dout_unopt);
19
20
// Initialize din to avoid X values
21
din = 0;
22
23
// Generate random values for din
24
repeat(total_cycles) @(posedge clk) din = $random;
25
end
26
27
wire tb_match;
28
assign tb_match = (dout_opt === dout_unopt);
29
30
opt_model opt_model (
31
.din(din),
32
.dout(dout_opt)
33
);
34
35
unopt_model unopt_model (
36
.din(din),
37
.dout(dout_unopt)
38
);
39
40
always @(posedge clk) begin
41
clocks = clocks + 1;
42
if (!tb_match) begin
43
if (errors == 0) errortime = $time;
44
errors = errors + 1;
45
end
46
47
// Print the signals for debugging
48
$display("Time=%0t | Cycle=%0d | din=%h | opt=%h | unopt=%h | match=%b",
49
$time, clocks, din, dout_opt, dout_unopt, tb_match);
50
51
if (clocks >= total_cycles) begin
52
$display("Simulation completed.");
53
$display("Total mismatches: %1d out of %1d samples", errors, clocks);
54
$display("Simulation finished at %0d ps", $time);
55
$finish;
56
end
57
end
58
59
initial begin
60
#1000000
61
$display("TIMEOUT");
62
$finish();
63
end
64
65
endmodule
A.2
OPTIMIZATION STRATEGIES
Figure 6: Optimization strategies employed for area, delay, and power improvements.
12
A.3
CODE OF EXAMPLE PROBLEMS IN TABLE 2
In the example problems shown in the appendix, the area- and power-optimized solutions coincided,
as area-oriented designs also achieved the best power results, and vice versa. This overlap arises
because common power-saving techniques, such as clock gating and operand isolation, were not
applicable as some designs lacked a clock signal, while others did not include an enable signal.
Consequently, explicit power-specific transformations could not be meaningfully applied. Moreover,
in certain cases, power optimizations indirectly reduced area, further reinforcing the convergence of
the two objectives into a single optimized implementation.
A.3.1
PROBLEM #60: RTLLM ALU
Problem description: Implement a 32-bit Arithmetic Logic Unit (ALU) for a MIPS-ISA CPU.
The ALU takes two 32-bit operands (a and b) and a 6-bit control signal (aluc) that specifies which
operation to perform. Based on this control signal, the ALU produces a 32-bit result (r) and several
status outputs: zero indicates whether the result is zero, carry flags if a carry occurred, negative
shows if the result is negative, overflow signals arithmetic overflow, and flag is used for set-less-
than instructions (slt and sltu). The module supports arithmetic, logical, shift, and immediate load
operations defined by specific opcodes (e.g., ADD, SUB, AND, OR, XOR, SLT, LUI).
Benchmark solution: ALU implementation with case statement and parameterized opcodes.
1
module unopt_model(
2
input [31:0] a,
3
input [31:0] b,
4
input [5:0] aluc,
5
output [31:0] r,
6
output zero,
7
output carry,
8
output negative,
9
output overflow,
10
output flag
11
);
12
13
parameter ADD = 6’b100000;
14
parameter ADDU = 6’b100001;
15
parameter SUB = 6’b100010;
16
parameter SUBU = 6’b100011;
17
parameter AND = 6’b100100;
18
parameter OR = 6’b100101;
19
parameter XOR = 6’b100110;
20
parameter NOR = 6’b100111;
21
parameter SLT = 6’b101010;
22
parameter SLTU = 6’b101011;
23
parameter SLL = 6’b000000;
24
parameter SRL = 6’b000010;
25
parameter SRA = 6’b000011;
26
parameter SLLV = 6’b000100;
27
parameter SRLV = 6’b000110;
28
parameter SRAV = 6’b000111;
29
parameter JR = 6’b001000;
30
parameter LUI = 6’b001111;
31
32
wire signed [31:0] a_signed;
33
wire signed [31:0] b_signed;
34
35
reg [32:0] res;
36
37
assign a_signed = a;
38
assign b_signed = b;
39
assign r = res[31:0];
40
41
assign flag = (aluc == SLT || aluc == SLTU) ? ((aluc == SLT) ? (a_signed < b_signed) : (a < b)) : 1’bz;
42
assign zero = (res == 32’b0) ? 1’b1 : 1’b0;
43
44
always @ (a or b or aluc)
45
begin
46
case(aluc)
47
ADD: begin
48
res <= a_signed + b_signed;
49
end
50
ADDU: begin
51
res <= a + b;
52
end
53
SUB: begin
13
54
res <= a_signed - b_signed;
55
end
56
SUBU: begin
57
res <= a - b;
58
end
59
AND: begin
60
res <= a & b;
61
end
62
OR: begin
63
res <= a | b;
64
end
65
XOR: begin
66
res <= a ˆ b;
67
end
68
NOR: begin
69
res <= ˜(a | b);
70
end
71
SLT: begin
72
res <= a_signed < b_signed ? 1 : 0;
73
end
74
SLTU: begin
75
res <= a < b ? 1 : 0;
76
end
77
SLL: begin
78
res <= b << a;
79
end
80
SRL: begin
81
res <= b >> a;
82
end
83
SRA: begin
84
res <= b_signed >>> a_signed;
85
end
86
SLLV: begin
87
res <= b << a[4:0];
88
end
89
SRLV: begin
90
res <= b >> a[4:0];
91
end
92
SRAV: begin
93
res <= b_signed >>> a_signed[4:0];
94
end
95
LUI: begin
96
res <= {a[15:0], 16’h0000};
97
end
98
default:
99
begin
100
res <= 32’bz;
101
end
102
endcase
103
end
104
endmodule
Our expert-written area and power optimized solution: Shared adder for arithmetic, simplified
flag logic and operand reuse, leading to 26% area reduction. Operand gating and early zeroing for
large shifts to cut switching activity, for 4% power reduction.
1
2
wire sub_mode
= (aluc==SUB) | (aluc==SUBU) | (aluc==SLT) | (aluc==SLTU);
3
wire [31:0] b_eff = sub_mode ? ˜b : b;
4
wire
cin
= sub_mode;
5
wire [32:0] sum33 = {1’b0,a} + {1’b0,b_eff} + cin;
6
7
wire [31:0] add_res
= sum33[31:0];
8
wire
add_carry = sum33[32];
9
10
wire ovf = (a[31]ˆadd_res[31]) & (b_eff[31]ˆadd_res[31]);
11
12
wire signed_lt
= add_res[31] ˆ ovf;
13
wire uns_lt
= ˜add_carry;
14
wire [31:0] slt_res
= {31’b0, signed_lt};
15
wire [31:0] sltu_res = {31’b0, uns_lt};
16
17
wire [31:0] and_res = a & b;
18
wire [31:0] or_res
= a | b;
19
wire [31:0] xor_res = a ˆ b;
20
wire [31:0] nor_res = ˜(a | b);
21
22
wire [4:0] sa5
= a[4:0];
23
wire
any_hi = |a[31:5];
24
14
25
wire [31:0] sll_full = any_hi ? 32’b0
: (b <<
sa5);
26
wire [31:0] srl_full = any_hi ? 32’b0
: (b >>
sa5);
27
wire [31:0] sra_full = any_hi ? {32{b[31]}} : ($signed(b) >>> sa5);
28
29
wire [31:0] sllv_res = (b <<
a[4:0]);
30
wire [31:0] srlv_res = (b >>
a[4:0]);
31
wire [31:0] srav_res = ($signed(b) >>> a[4:0]);
32
33
wire [31:0] lui_res = {a[15:0], 16’h0000};
34
35
reg [31:0] r_int;
36
always @* begin : result_mux
37
(* parallel_case, full_case *)
38
case (aluc)
39
ADD, ADDU:
r_int = add_res;
40
SUB, SUBU:
r_int = add_res;
41
AND:
r_int = and_res;
42
OR:
r_int = or_res;
43
XOR:
r_int = xor_res;
44
NOR:
r_int = nor_res;
45
SLT:
r_int = slt_res;
46
SLTU:
r_int = sltu_res;
47
SLL:
r_int = sll_full;
48
SRL:
r_int = srl_full;
49
SRA:
r_int = sra_full;
50
SLLV:
r_int = sllv_res;
51
SRLV:
r_int = srlv_res;
52
SRAV:
r_int = srav_res;
53
LUI:
r_int = lui_res;
54
JR:
r_int = 32’bz;
55
default:
r_int = 32’bz;
56
endcase
57
end
58
59
assign r
= r_int;
60
assign zero = ˜(|r_int);
61
62
assign carry
= 1’bz;
63
assign overflow = 1’bz;
64
assign negative = 1’bz;
65
assign flag
= (aluc==SLT)
? signed_lt :
66
(aluc==SLTU) ? uns_lt
:
67
1’bz;
68
endmodule
Our expert-written delay optimized solution: Parallel datapaths with one-hot muxing for shallow
critical path, leading to 26% delay reduction.
1
wire signed [31:0] a_signed = a;
2
wire signed [31:0] b_signed = b;
3
4
wire [31:0] add_u
= a + b;
5
wire [31:0] sub_u
= a - b;
6
wire signed [31:0] add_s = a_signed + b_signed;
7
wire signed [31:0] sub_s = a_signed - b_signed;
8
9
wire [31:0] and_res = a & b;
10
wire [31:0] or_res
= a | b;
11
wire [31:0] xor_res = a ˆ b;
12
wire [31:0] nor_res = ˜(a | b);
13
14
wire slt_res
= (a_signed < b_signed);
15
wire sltu_res = (a < b);
16
17
wire [4:0] shamt5 = a[4:0];
18
19
20
wire [31:0] sll_full
= (b <<
a);
// full ’a’
21
wire [31:0] srl_full
= (b >>
a);
// full ’a’
22
wire [31:0] sra_full
= ($signed(b) >>> a_signed);// full signed ’a’
23
wire [31:0] sllv_res
= (b <<
shamt5);
24
wire [31:0] srlv_res
= (b >>
shamt5);
25
wire [31:0] srav_res
= ($signed(b) >>> shamt5);
26
27
wire [31:0] lui_res = {a[15:0], 16’h0000};
28
29
wire sel_ADD
= (aluc==ADD);
30
wire sel_ADDU = (aluc==ADDU);
31
wire sel_SUB
= (aluc==SUB);
32
wire sel_SUBU = (aluc==SUBU);
15
33
wire sel_AND
= (aluc==AND);
34
wire sel_OR
= (aluc==OR);
35
wire sel_XOR
= (aluc==XOR);
36
wire sel_NOR
= (aluc==NOR);
37
wire sel_SLT
= (aluc==SLT);
38
wire sel_SLTU = (aluc==SLTU);
39
wire sel_SLL
= (aluc==SLL);
40
wire sel_SRL
= (aluc==SRL);
41
wire sel_SRA
= (aluc==SRA);
42
wire sel_SLLV = (aluc==SLLV);
43
wire sel_SRLV = (aluc==SRLV);
44
wire sel_SRAV = (aluc==SRAV);
45
wire sel_LUI
= (aluc==LUI);
46
wire sel_JR
= (aluc==JR);
47
48
wire any_sel = sel_ADD|sel_ADDU|sel_SUB|sel_SUBU|sel_AND|sel_OR|sel_XOR|sel_NOR|
49
sel_SLT|sel_SLTU|sel_SLL|sel_SRL|sel_SRA|sel_SLLV|sel_SRLV|sel_SRAV|sel_LUI;
50
51
wire [31:0] r_known =
52
(sel_ADD
? add_s
: 32’b0) |
53
(sel_ADDU ? add_u
: 32’b0) |
54
(sel_SUB
? sub_s
: 32’b0) |
55
(sel_SUBU ? sub_u
: 32’b0) |
56
(sel_AND
? and_res: 32’b0) |
57
(sel_OR
? or_res : 32’b0) |
58
(sel_XOR
? xor_res: 32’b0) |
59
(sel_NOR
? nor_res: 32’b0) |
60
(sel_SLT
? {31’b0, slt_res } : 32’b0) |
61
(sel_SLTU ? {31’b0, sltu_res} : 32’b0) |
62
(sel_SLL
? sll_full : 32’b0) |
63
(sel_SRL
? srl_full : 32’b0) |
64
(sel_SRA
? sra_full : 32’b0) |
65
(sel_SLLV ? sllv_res : 32’b0) |
66
(sel_SRLV ? srlv_res : 32’b0) |
67
(sel_SRAV ? srav_res : 32’b0) |
68
(sel_LUI
? lui_res
: 32’b0);
69
70
assign r = (any_sel && !sel_JR) ? r_known : 32’bz;
71
72
assign zero = (r == 32’b0) ? 1’b1 : 1’b0;
73
74
assign flag = (sel_SLT)
? slt_res
:
75
(sel_SLTU) ? sltu_res :
76
1’bz;
77
78
assign carry
= 1’bz;
79
assign negative = 1’bz;
80
assign overflow = 1’bz;
81
endmodule
A.3.2
PROBLEM #68: RTLLM FSM
Problem description: Implement a finit state machine (FSM) that detects the input sequence 10011
on a single-bit input stream. The module has three inputs: the serial input bit (IN), the clock (CLK),
and a synchronous reset (RST). It produces one output, MATCH, which is asserted high when the
specified sequence is recognized. The FSM supports continuous input and loop detection. When
reset is active, the FSM initializes and MATCH is cleared to 0. The output MATCH is asserted
during the cycle when the last 1 of the target sequence is received, and the design ensures that
repeated or overlapping patterns (e.g., 100110011) correctly generate multiple match pulses.
Benchmark solution: States are binary-encoded, with sequential next-state logic in a Mealy FSM
while output occupies a register.
1
module unopt_model (
2
input
wire IN,
3
input
wire CLK,
4
input
wire RST,
5
output wire MATCH
6
);
7
8
reg [2:0] ST_cr,ST_nt;
9
10
parameter s0 = 3’b000;
11
parameter s1 = 3’b001;
12
parameter s2 = 3’b010;
13
parameter s3 = 3’b011;
14
parameter s4 = 3’b100;
16
15
parameter s5 = 3’b101;
16
17
always@(posedge CLK or posedge RST) begin
18
if(RST)
19
ST_cr <= s0;
20
else
21
ST_cr <= ST_nt;
22
end
23
24
always@(*) begin
25
case(ST_cr)
26
s0:begin
27
if (IN==0)
28
ST_nt = s0;
29
else
30
ST_nt = s1;
31
end
32
33
s1:begin
34
if (IN==0)
35
ST_nt = s2;
36
else
37
ST_nt = s1;
38
end
39
40
s2:begin
41
if (IN==0)
42
ST_nt = s3;
43
else
44
ST_nt = s1;
45
end
46
47
s3:begin
48
if (IN==0)
49
ST_nt = s0;
50
else
51
ST_nt = s4;
52
end
53
54
s4:begin
55
if (IN==0)
56
ST_nt = s2;
57
else
58
ST_nt = s5;
59
end
60
61
s5:begin
62
if (IN==0)
63
ST_nt = s2;
64
else
65
ST_nt = s1;
66
end
67
68
endcase
69
end
70
71
always@(*) begin
72
if(RST)
73
MATCH
<= 0;
74
else if (ST_cr == s4 && IN == 1)
75
MATCH
<= 1;
76
else
77
MATCH
<= 0;
78
end
79
80
endmodule
Our expert-written area and power optimized solution: Casez-based transitions and direct Mealy
output computation, removing extra register, leading to 32% area reduction. Compact binary encod-
ing and casez-based transitions to cut toggling for 23% power reduction.
1
localparam [2:0] s0=3’b000, s1=3’b001, s2=3’b010,
2
s3=3’b011, s4=3’b100, s5=3’b101;
3
4
reg [2:0] ST_cr, ST_nt;
5
6
always @(posedge CLK or posedge RST) begin
7
if (RST)
8
ST_cr <= s0;
9
else
17
10
ST_cr <= ST_nt;
11
end
12
13
always @* begin
14
ST_nt = s0;
15
casez ({ST_cr, IN})
16
// s0: 0->s0, 1->s1
17
{s0,1’b0}: ST_nt = s0;
18
{s0,1’b1}: ST_nt = s1;
19
20
// s1: 0->s2, 1->s1
21
{s1,1’b0}: ST_nt = s2;
22
{s1,1’b1}: ST_nt = s1;
23
24
// s2: 0->s3, 1->s1
25
{s2,1’b0}: ST_nt = s3;
26
{s2,1’b1}: ST_nt = s1;
27
28
// s3: 0->s0, 1->s4
29
{s3,1’b0}: ST_nt = s0;
30
{s3,1’b1}: ST_nt = s4;
31
32
// s4: 0->s2, 1->s5
33
{s4,1’b0}: ST_nt = s2;
34
{s4,1’b1}: ST_nt = s5;
35
36
// s5: 0->s2, 1->s1
37
{s5,1’b0}: ST_nt = s2;
38
{s5,1’b1}: ST_nt = s1;
39
40
default:
ST_nt = s0;
41
endcase
42
end
43
44
assign MATCH = (ST_cr == s4) & IN;
45
46
endmodule
Our expert-written delay optimized solution: One-hot state encoding with pre-decoded inputs
and Moore-style output, leading to 23% delay reduction.
1
2
reg [5:0] S, S_next;
3
4
reg [5:0] S, S_next;
5
6
always @(posedge CLK or posedge RST) begin
7
if (RST)
8
S <= 6’b000001;
// s0
9
else
10
S <= S_next;
11
end
12
13
wire in1 =
IN;
14
wire in0 = ˜IN;
15
16
always @* begin
17
S_next[0] = (S[0] & in0) | (S[3] & in0);
// -> s0
18
S_next[1] = (S[0] & in1) | (S[1] & in1) | (S[2] & in1)
19
| (S[5] & in1);
// -> s1
20
S_next[2] = (S[1] & in0) | (S[4] & in0) | (S[5] & in0);
// -> s2
21
S_next[3] = (S[2] & in0);
// -> s3
22
S_next[4] = (S[3] & in1);
// -> s4
23
S_next[5] = (S[4] & in1);
// -> s5
24
end
25
26
always @(posedge CLK or posedge RST) begin
27
if (RST)
28
MATCH <= 1’b0;
29
else
30
MATCH <= S[5];
31
end
32
endmodule
A.3.3
PROBLEM #87: VERILOGEVAL PROB095
Problem description: Implement a module that generates a control signal (shift ena) for a shift
register. The module has a clock input (clk), a synchronous active-high reset (reset), and a single
18
output (shift ena). The functionality requires that when the FSM is reset, the shift ena signal is as-
serted high for exactly four consecutive clock cycles before being deasserted permanently. After this
sequence, shift ena remains low indefinitely until another reset occurs, at which point the behavior
repeats. All sequential operations are triggered on the positive edge of the clock.
Benchmark solution: Uses explicit states with next-state logic to drive shift ena.
1
module unopt_model (
2
input
clk,
3
input
reset,
4
output reg shift_ena
5
);
6
parameter B0=0, B1=1, B2=2, B3=3, Done=4;
7
8
reg [2:0] state, next;
9
10
always @* begin
11
case (state)
12
B0:
next = B1;
13
B1:
next = B2;
14
B2:
next = B3;
15
B3:
next = Done;
16
Done: next = Done;
17
default: next = B0;
18
endcase
19
end
20
21
always @(posedge clk) begin
22
if (reset) begin
23
state
<= B0;
24
shift_ena <= 1’b1;
25
end else begin
26
state
<= next;
27
shift_ena <= (next != Done);
28
end
29
end
30
endmodule
Our expert-written area and power optimized solution: Minimized register width, using a 2-bit
counter, and compact comparator logic, leading to 47% area reduction. Small counter reused which
reduced toggling activity to minimize dynamic power, for 46% power reduction.
1
reg [1:0] counter; // 2 bits are enough to count up to 4
2
3
always @(posedge clk) begin
4
if (reset) begin
5
counter <= 2’b00; // Reset counter
6
shift_ena <= 1’b1; // Enable on reset
7
end else if (counter < 2’b11) begin
8
counter <= counter + 1; // Increment counter
9
shift_ena <= 1’b1; // Keep shift_ena high while counting
10
end else begin
11
shift_ena <= 1’b0; // Disable after 4 cycles
12
end
13
end
14
15
endmodule
Our expert-written delay optimized solution: Wider counter, using 3 bits, to simplify comparison
and reduce logic depth on the critical path, leading to 37% delay reduction.
1
reg [2:0] count; // 3-bit counter to count 4 cycles
2
3
always @(posedge clk) begin
4
if (reset) begin
5
count <= 3’b000; // Reset count
6
shift_ena <= 1’b1; // Enable shift initially
7
end else if (count < 3’b011) begin
8
count <= count + 1; // Increment count
9
shift_ena <= 1’b1; // Keep shift enabled
10
end else begin
11
shift_ena <= 1’b0; // Disable shift after 4 cycles
12
end
13
end
14
15
endmodule
19
A.3.4
PROBLEM #104: VERILOGEVAL PROB136
Problem description: Implement a cellular automaton game, similar to Conway’s Game of Life,
on a 16x16 grid. The grid is represented as a 256-bit vector (q), where each row of 16 cells maps
to a sub-vector, and each cell can be alive (1) or dead (0). The module has a clock input (clk),
a load signal (load) for synchronously loading an initial 256-bit state (data) into q, and produces
the updated 256-bit grid state as output. At every positive clock edge, the grid advances by one
timestep, with each cell’s next state determined by its number of neighbors: cells die with fewer
than 2 or more than 3 neighbors, remain unchanged with exactly 2 neighbors, and become alive with
exactly 3 neighbors. The grid is modeled as a toroid, meaning edges wrap around so that cells on
the boundaries consider neighbors from the opposite side.
Benchmark solution: Straightforward RTL with per-cell neighbor recomputation and sequential
summing.
1
module unopt_model (
2
input clk,
3
input load,
4
input [255:0] data,
5
output reg [255:0] q
6
);
7
8
logic [323:0] q_pad;
9
always@(*) begin
10
for (int i=0;i<16;i++)
11
q_pad[18*(i+1)+1 +: 16] = q[16*i +: 16];
12
q_pad[1 +: 16] = q[16*15 +: 16];
13
q_pad[18*17+1 +: 16] = q[0 +: 16];
14
15
for (int i=0; i<18; i++) begin
16
q_pad[i*18] = q_pad[i*18+16];
17
q_pad[i*18+17] = q_pad[i*18+1];
18
end
19
end
20
21
always @(posedge clk) begin
22
for (int i=0;i<16;i++)
23
for (int j=0;j<16;j++) begin
24
q[i*16+j] <=
25
((q_pad[(i+1)*18+j+1 -1+18] + q_pad[(i+1)*18+j+1 +18] + q_pad[(i+1)*18+j+1 +1+18] +
26
q_pad[(i+1)*18+j+1 -1]
+ q_pad[(i+1)*18+j+1+1] +
27
q_pad[(i+1)*18+j+1 -1-18]
+ q_pad[(i+1)*18+j+1 -18] + q_pad[(i+1)*18+j+1 +1-18]) & 3’h7 | q[i*16+j]) == 3’h3;
28
end
29
30
if (load)
31
q <= data;
32
33
end
34
35
endmodule
Our expert-written area and power optimized solution: Sharing per-row horizontal sums and
bitwise rotations for toroidal wrap, minimizing summations for 19% area reduction. Computation
reuse decreasing toggling, leading to 36% power reduction.
1
// --- Helpers ---------------------------------------------------------------
2
function automatic [7:0] idx(input [3:0] r, input [3:0] c);
3
idx = {r, c}; // r*16 + c
4
endfunction
5
6
// Bit-rotate wires (toroidal wrap) - wiring only (no logic area)
7
function automatic [15:0] rol1(input [15:0] x); rol1 = {x[14:0], x[15]}; endfunction
8
function automatic [15:0] ror1(input [15:0] x); ror1 = {x[0],
x[15:1]}; endfunction
9
10
// Add-three 1-bit vectors in parallel: returns {carry,sum}
11
// a+b+c = sum ˆ (2*carry) per bit
12
function automatic [31:0] add3_vec(input [15:0] a, input [15:0] b, input [15:0] c);
13
add3_vec[15:0]
= a ˆ b ˆ c;
// sum (LSB)
14
add3_vec[31:16] = (a & b) | (a & c) | (b & c);
// carry (means +2)
15
endfunction
16
17
// --- Unpack rows (wires) ---------------------------------------------------
18
wire [15:0] row [15:0];
19
genvar ur;
20
generate
21
for (ur = 0; ur < 16; ur = ur + 1) begin : UNPACK
22
assign row[ur] = q[{ur[3:0], 4’b0000} +: 16];
20
23
end
24
endgenerate
25
26
// --- Precompute per-row horizontal neighbors (shared) ----------------------
27
wire [15:0] rol [15:0], ror [15:0];
28
wire [15:0] sTrip [15:0], cTrip [15:0];
// for (L,C,R) of each row (0..3)
29
wire [15:0] sPair [15:0], cPair [15:0];
// for (L,R) of each row (0..2)
30
31
genvar hr;
32
generate
33
for (hr = 0; hr < 16; hr = hr + 1) begin : HROW
34
assign rol[hr] = rol1(row[hr]);
35
assign ror[hr] = ror1(row[hr]);
36
37
// Triplet = left + center + right (encoded as s + 2*c)
38
wire [31:0] trip_pack = add3_vec(rol[hr], row[hr], ror[hr]);
39
assign sTrip[hr] = trip_pack[15:0];
40
assign cTrip[hr] = trip_pack[31:16];
41
42
// Pair = left + right
43
assign sPair[hr] = rol[hr] ˆ ror[hr];
44
assign cPair[hr] = rol[hr] & ror[hr];
45
end
46
endgenerate
47
48
integer r;
49
reg [255:0] nxt;
50
51
reg [3:0] rn, rp;
52
reg [15:0] sT, cT, sM, cM, sB, cB;
53
reg [15:0] sS, cS;
// sum of (sT + sM + sB) as s + 2*c
54
reg [15:0] U_is0, U_ge2, U_is1;
// onehot(U) for U = cS + cT + cM + cB
55
reg [15:0] is3, is2;
56
57
always @* begin
58
nxt = ’0;
59
60
for (r = 0; r < 16; r = r + 1) begin
61
rn = (r == 0 ) ? 4’d15 : r - 1;
62
rp = (r == 15) ? 4’d0
: r + 1;
63
64
sT = sTrip[rn]; cT = cTrip[rn];
// top triplet from row r-1
65
sM = sPair[r ]; cM = cPair[r ];
// middle pair (no center) from row r
66
sB = sTrip[rp]; cB = cTrip[rp];
// bottom triplet from row r+1
67
68
{cS, sS} = add3_vec(sT, sM, sB);
69
70
U_is0 = ˜(cS | cT | cM | cB);
71
U_ge2 = ( (cS & cT) | (cS & cM) | (cS & cB)
72
| (cT & cM) | (cT & cB) | (cM & cB) );
73
U_is1 = ˜(U_is0 | U_ge2);
74
75
is3 =
sS & U_is1;
76
is2 = ˜sS & U_is1;
77
78
nxt[{r[3:0], 4’b0000} +: 16] = is3 | (row[r] & is2);
79
end
80
end
81
82
always @(posedge clk) begin
83
if (load)
84
q <= data;
85
else
86
q <= nxt;
87
end
88
endmodule
Our expert-written delay optimized solution: Parallel neighbor summation with carry-save adder
tree and direct decode for 2 and 3, for shallow critical path, leading to 37% delay reduction.
1
function automatic [7:0] idx(input [3:0] r, input [3:0] c);
2
idx = {r, c};
3
endfunction
4
5
function automatic [15:0] rol1(input [15:0] x); rol1 = {x[14:0], x[15]}; endfunction
6
function automatic [15:0] ror1(input [15:0] x); ror1 = {x[0],
x[15:1]}; endfunction
7
8
function automatic [31:0] add3_vec(input [15:0] a, input [15:0] b, input [15:0] c);
9
add3_vec[15:0]
= a ˆ b ˆ c;
// s
10
add3_vec[31:16] = (a & b) | (a & c) | (b & c);
// c (>=2)
21
11
endfunction
12
13
integer r;
14
reg [255:0] nxt;
15
16
reg [3:0] rn, rp;
17
reg [15:0] ru, r0, rd;
18
reg [15:0] ru_l, ru_c, ru_r;
19
reg [15:0] r0_l,
r0_r;
20
reg [15:0] rd_l, rd_c, rd_r;
21
22
reg [15:0] sT, cT, sM, cM, sB, cB, sS, cS;
23
reg [15:0] U_is0, U_ge2, U_is1;
// onehot decode for U = cS+cT+cM+cB
24
reg [15:0] is3, is2;
// neighbor count ==3 / ==2
25
26
always @* begin
27
nxt = ’0;
28
29
for (r = 0; r < 16; r = r + 1) begin
30
rn = (r == 0 ) ? 4’d15 : r - 1;
31
rp = (r == 15) ? 4’d0
: r + 1;
32
33
ru = q[{rn,4’b0000} +: 16];
34
r0 = q[{r ,4’b0000} +: 16];
35
rd = q[{rp,4’b0000} +: 16];
36
37
ru_l = rol1(ru);
ru_c = ru;
ru_r = ror1(ru);
38
r0_l = rol1(r0);
r0_r = ror1(r0);
39
rd_l = rol1(rd);
rd_c = rd;
rd_r = ror1(rd);
40
41
{cT, sT} = add3_vec(ru_l, ru_c, ru_r);
// counts 0..3
42
sM
= (r0_l ˆ r0_r);
// pair: 0..2
43
cM
= (r0_l & r0_r);
44
{cB, sB} = add3_vec(rd_l, rd_c, rd_r);
45
46
{cS, sS} = add3_vec(sT, sM, sB);
47
48
U_is0 = ˜(cS | cT | cM | cB);
49
U_ge2 = ( (cS & cT) | (cS & cM) | (cS & cB)
50
| (cT & cM) | (cT & cB) | (cM & cB) );
51
U_is1 = ˜(U_is0 | U_ge2);
52
53
is3 =
sS & U_is1;
54
is2 = ˜sS & U_is1;
55
56
nxt[{r[3:0],4’b0000} +: 16] = is3 | (r0 & is2);
57
end
58
end
59
60
always @(posedge clk) begin
61
if (load)
62
q <= data;
63
else
64
q <= nxt;
65
end
66
endmodule
A.4
SAMPLING DIVERSITY
In Figure 7, we show three LLM-generated Verilog modules of a parallel-in-serial-out shift register,
targeting area optimization, with normalized area efficiency values of 0.924, 0.963 and 1.0, respec-
tively. These samples illustrate how increasing k allows models to generate progressively better
implementations. The third sample avoids unnecessary counters and state tracking, reducing both
area and complexity.
A.5
PASS@K DEFINITION
The pass@k metric, defined in equation 3, is used for measuring the functional correctness of LLM-
generated code. Here, N is the total number of problems in the evaluation set, ni is the number
of samples generated for problem i, and ci is the number of functionally correct samples for that
problem.
22
module opt_model #(parameter DATA_WIDTH = 32) (
input clk,
input resetn,
input [DATA_WIDTH-1:0] din,
input din_en,
output logic dout
);
reg [DATA_WIDTH-1:0] shift_reg;
reg [$clog2(DATA_WIDTH)-1:0] bit_index;
always @(posedge clk or negedge resetn) begin
if (!resetn) begin
shift_reg <= {DATA_WIDTH{1'b0}};
bit_index <= 0;
dout <= 1'b0;
end else if (din_en) begin
shift_reg <= din;
bit_index <= 0;
dout <= din[0];
end else if (bit_index < DATA_WIDTH-1) begin
dout <= shift_reg[bit_index + 1];
bit_index <= bit_index + 1;
end else begin
dout <= 1'b0;
end
end
endmodule
(a) k = 1
module opt_model #(parameter DATA_WIDTH = 32) (
input clk,
input resetn,
input [DATA_WIDTH-1:0] din,
input din_en,
output logic dout
);
reg [DATA_WIDTH-1:0] shift_reg;
reg [$clog2(DATA_WIDTH)-1:0] bit_index;
always @(posedge clk) begin
if (din_en) begin
shift_reg <= din;
bit_index <= 0;
dout <= din[0];
end else if (bit_index < DATA_WIDTH-1) begin
dout <= shift_reg[bit_index + 1];
bit_index <= bit_index + 1;
end else begin
dout <= 1'b0;
end
end
endmodule
(b) k = 5
module opt_model #(parameter DATA_WIDTH = 32) (
input clk,
input resetn,
input [DATA_WIDTH-1:0] din,
input din_en,
output logic dout
);
logic [DATA_WIDTH-1:0] temp;
always @(posedge clk or negedge resetn) begin
if (!resetn) begin
temp <= 0;
end else if (din_en) begin
temp <= din;
end else begin
temp <= temp >> 1;
end
end
assign dout = temp[0];
endmodule
(c) k = 10
Figure 7: Three area-optimized implementations of the piso shift register module (Prob-
lem #16) generated at k ∈{1, 5, 10}. The circuit shifts the least significant bit of a multi-bit input
din to the single-bit output dout sequentially, starting when din en goes high. All designs are
functionally correct but structurally diverse.
pass@k = EN
i=1
1 −C(ni −ci, k)
C(ni, k)
(3)
A.6
FAILURE ANALYSIS
In Figure. 8, we visualize the set of problems that heavy high pass@k and low eff@k to get better
insight on the set of problems that are hard to optimize.
Figure 8: Quadrant plot showing the correlation between functional correctness (Pass@1) and syn-
thesis efficiency (Eff@1) across area, delay, and power objectives.
23
|
PLUTO: A BENCHMARK FOR EVALUATING EFFICIENCY OF LLM-GENERATED HARDWARE CODE Manar Abdelatty∗, Maryam Nouh∗, Jacob K. Rosenstein, & Sherief Reda 02906, USA manar ABSTRACT Large Language Models (LLMs) are increasingly used to automate hardware design tasks, including the generation of Verilog code. While early benchmarks focus primarily on functional correctness, efficient hardware design demands additional optimization for synthesis metrics such as area, delay, and power. Existing benchmarks fall short in evaluating these aspects comprehensively: they often lack optimized baselines or testbenches for verification. To address these gaps, we present Pluto, a benchmark and evaluation framework designed to assess the efficiency of LLM-generated Verilog designs. Pluto presents a comprehensive evaluation set of 114 problems with self-checking testbenches and multiple Pareto-optimal reference implementations. Experimental results show that state-of-the-art LLMs can achieve high functional correctness, reaching 78.3% at pass@1, but their synthesis efficiency still lags behind expert-crafted implementations, with area efficiency of 63.8%, delay efficiency of 65.9%, and power efficiency of 64.0% at eff@1. This highlights the need for efficiency-aware evaluation frameworks such as Pluto to drive progress in hardware-focused LLM research. 1 INTRODUCTION Large Language Models (LLMs) are beginning to reshape hardware design by automating key steps in hardware design workflows, including Verilog code generation Thakur et al. (2023a;b); Liu et al. (2023a), optimization Yao et al. (2024); Guo & Zhao (2025), verification Qiu et al. (2024a), debugging Tsai et al. (2024), high-level synthesis Xiong et al. (2024), and post-synthesis metric estimation Abdelatty et al. (2025). While these advances highlight the potential of LLMs in hardware design, most research has focused on functional correctness of generated designs, with little attention to design quality metrics such as area, delay, and power. In hardware design, the quality of Verilog code is not determined solely by functional correctness. Designs typically undergo logic synthesis, where Verilog code is mapped to gate-level implementations in a target technology. This process exposes critical efficiency metrics-such as silicon area, timing delay, and power consumption-that directly impact manufacturability and performance. Unlike software code, where correctness and execution speed often suffice, hardware code quality is inherently tied to these post-synthesis metrics. In order to evaluate the functional correctness of LLM-generated Verilog code, several benchmarks have been proposed including VerilogEval Liu et al. (2023a) and RTLLM Lu et al. (2024). Recent efforts, including RTLRewriter Yao et al. (2024), ResBench Guo & Zhao (2025), GenBen Wan et al. (2025), and TuRTLe Garcia-Gasulla et al. (2025), have begun to evaluate quality of LLM-generated hardware code in terms of post-synthesis metrics. However, these benchmarks face key limitations: • Absence of Optimal Ground Truth Solutions True efficiency should be measured against implementations that are explicitly optimized for specific objectives such as silicon area, delay, or power consumption. Prior studies rely on canonical solutions from VerilogEval and RTLLM as reference solutions. Our analysis shows that these solutions are not the most optimal in terms of post-synthesis metrics. ∗Equal contribution. 1 16 Oct 2025 • Lack of Clock Latency Agnostic Testbenches Many common optimization patterns-such as register pipelining, resource sharing, or FSM restructuring-introduce variations in clock-cycle latency between the optimized and unoptimized designs. To support fair evaluation, testbenches must be self-checking and tolerant of different latency requirements. Existing benchmarks, however, assume identical latency between the reference model and the design under test, making them unsuitable for efficiency benchmarking. In order to address these limitations, we introduce Pluto, the first benchmark designed to evaluate both functional correctness and synthesis efficiency of LLM-generated Verilog code. Our contributions are as follows: • Per-Metric Ground Truth Optimal Solutions. We provide a suite of 114 problems where each is optimized for area, delay, and power separately, yielding Pareto-front optimal solutions. Our ground truth solutions are significantly more efficient than canonical solutions in RTLLM and VerilogEval. • Optimization-Aware Testbenches. Each problem is accompanied by clock-cycle agnostic testbenches that accommodate varying latency requirements, ensuring robust evaluation of different optimization patterns. • Comprehensive Evaluation. We adapt the eff@k metric introduced in Qiu et al. (2024b) to measure the efficiency of hardware designs. Our extended metric is a three-dimensional vector that evaluates LLM-generated code across multiple objectives: area, delay and power. 2 RELATED WORK Software Code Benchmarks Large Language Models (LLMs) have been extensively studied for code generation across both software and hardware domains, with most early benchmarks focusing primarily on functional correctness rather than efficiency. In software, works such as Mercury Du et al. (2024) and ENAMEL Qiu et al. (2024b) move beyond correctness to explicitly evaluate runtime efficiency of LLM-generated programs. The Mercury Du et al. (2024) benchmark contains LeetCode style problems. Each problem is accompanied by an expert-written solution that represents the most optimal implementation in terms of run-time efficiency. ENAMEL Qiu et al. (2024b) also introduces a Python benchmark to evaluate the run-time efficiency of LLM-generated code. Hardware Code Benchmarks In hardware design, early work on LLM-generated Verilog emphasized functional correctness. VerilogEval Liu et al. (2023a) only evaluates whether the LLM generated code passes the testbench check, while RTLLM Lu et al. (2024) additionally checks if the generated code is synthesizable. More recent efforts have shifted toward assessing and improving the efficiency of LLM-generated designs, which can be categorized into two main categories: Specifications-to-Efficient-Verilog where the LLM is tasked with translating natural language instruction to optimized Verilog code directly, and Unoptimized-Verilog-to-Optimized-Verilog, where the LLM is tasked with rewriting an unoptimized Verilog code to optimized Verilog code. In the Specifications-to-Efficient-Verilog formulation, the LLM is prompted with a natural language problem description and directly generates optimized Verilog. Benchmarks, such as GenBen Wan et al. (2025), TuRTLe Garcia-Gasulla et al. (2025), evaluate these generations in functional correctness, synthesizability, and post-synthesis metrics such as area, delay, and power. However, it relies on VerilogEval problems as ground truth. These reference designs are not necessarily optimized for power, performance, or area, and thus do not represent true Pareto-optimal solutions. ResBench Guo & Zhao (2025) also does not define any gold-standard or reference-optimal implementations, which makes it difficult to quantitatively assess how close the generated solutions are to ideal results. The Unoptimized-Verilog-to-Optimized-Verilog setting provides the LLM with a functionally correct but unoptimized Verilog implementation and asks it to produce a more efficient version. RTLRewriter Yao et al. (2024) enhances this with retrieval-augmented generation and feedback through the synthesis loop. However, RTLRewriter lacks associated testbenches, making it unsuitable for assessing the functional correctness of the generated code. 2 Table 1: Comparison of prior software and hardware code generation benchmarks. Pluto addresses key limitations by enabling metric-specific optimization with three reference implementations per problem, each optimized for area, delay, or power. Benchmark Language Functionality Synthesizability Efficiency Per-Metric Optimisation Tasks HumanEval Python ✓ - × × 164 Mercury Python ✓ - ✓ × 256 ENAMEL Python ✓ - ✓ × 142 VerilogEval Verilog ✓ × × × 156 RTLLM Verilog ✓ ✓ × × 30 RTLRewriter Verilog × ✓ ✓ × 95 ResBench Verilog ✓ ✓ × × 56 GenBen Verilog ✓ ✓ × × 300 TuRTLe Verilog ✓ ✓ × × 223 CVDP Verilog ✓ ✓ × × 783 Pluto (Ours) Verilog ✓ ✓ ✓ ✓ 114 As summarized in Table 1, Pluto is the first benchmark to offer per-metric optimization, providing separate expert-optimized reference designs for area, delay, and power. This enables targeted, metric-specific evaluation of LLMs, an aspect missing from prior benchmarks. 3 PLUTO BENCHMARK module opt_power ( input [31:0] din, output [5:0] dout ); always_comb begin idx = '0; found = 1'b0; for (int i = 0; i = 0; i--)begin assign temp[i] = din[i] ? i[5:0] : temp[i+1]; end endgenerate assign dout = temp[0]; endmodule Trailing Zeros Circuit - Problem # 27 Find the number of trailing 0s in the binary representation of the input (din). If the input value is all 0s, the number of trailing 0s is the data width (DATA_WIDTH) Area Optimized Delay Optimized Power Optimized // Score Circuit: LSB Isolation via 2's Complement & Parallel One Hot Encoder Circuit: MSB Priority Encoder via Mux Chain Circuit: Scan from LSB to MSB with found flag to reduce switching .. Score Score Figure 1: Overview of the Pluto benchmark on the trailing zeros detection task. We show three reference implementations optimized for different synthesis metrics compared to the unoptimized baseline: (left) area, using a mux-based priority encoder, reducing area by 33%; (center) delay, using an LSB isolation circuit with a parallel one-hot encoder, reducing delay by 44%; and (right) power, using an LSB-to-MSB scanning method with early termination, reducing total power by 34%. See Appendix. A.1 for unoptimized baseline and self-checking testbench. 3.1 DATA CONSTRUCTION To enable a comprehensive evaluation of synthesis efficiency for LLM-generated hardware code, we construct the Pluto evaluation set, which contains a diverse collection of high-quality digital design problems spanning a broad range of difficulties. Specifically, we curated 114 problems from 3 various publicly available sources, including open-source hardware projects, educational platforms such as ChipDev ChipDev (2025), a LeetCode-inspired platform for practicing Verilog coding, and prior benchmark suites such as RTLRewriter Yao et al. (2024), RTLLM Lu et al. (2024), and VerilogEval Liu et al. (2023b). Each problem is specified by a high-level description outlining the functional requirements, together with a baseline unoptimized Verilog implementation. The problem set covers a wide spectrum of tasks in digital logic design, ranging from arithmetic units and control circuits to sequential state machines. To systematically capture variation in design complexity, we adopt ChipDev's difficulty annotations and classify problems into three levels: easy, medium, and hard. These labels reflect the intrinsic challenge of translating the textual description into a correct Verilog implementation, thereby providing a principled way to distinguish between problems of different complexity. Importantly, the resulting collection balances accessibility with challenge: many problems that appear straightforward can nonetheless expose substantial differences in synthesis efficiency depending on the optimization strategies applied. The diverse composition of easy, medium, and hard tasks therefore enables a nuanced assessment of an LLM's ability to generate synthesis-efficient Verilog under varying constraints. In total, the 114 selected problems provide a representative and scalable testbed for benchmarking LLM-based Verilog efficiency. Each problem instance in the Pluto benchmark includes the following components: • Prompt: A natural language description of the hardware design task intended to guide the LLM-generation. • Module Header: A fixed interface shared across all versions of the Verilog module to ensure consistency and comparability. • Unoptimized Verilog Code: A baseline implementation used as the reference for testing. • Optimized Verilog Code: Three distinct implementations with tradeoffs, each optimized by hand using design experts for a single metric: area, delay, or power. • Testbench: A manually crafted, fully self-checking testbench that verifies functional equivalence between the unoptimized and any optimized design. These testbenches ensure full input space coverage and flag any mismatches during simulation. For sequential circuits, testbenches are clock-cycle agnostic, supporting latency differences introduced by optimizations such as pipelining or resource sharing. All components in the evaluation set are manually developed. This ensures high quality and guarantees that the LLM under evaluation has not previously encountered any part of the dataset during training. In particular, the testbenches and optimized code serve as held-out ground truth references, providing an unbiased benchmark for assessing the efficiency and correctness of LLM-generated Verilog designs. To illustrate the structure of problems in the Pluto evaluation set, Figure 1 presents the example of a trailing zeros detection circuit, categorized as an easy problem, along with its three metricspecific optimizations. As shown, each optimization achieves peak efficiency in its targeted metric, while performance in the remaining two metrics declines. This behavior emphasizes the inherent trade-offs across design objectives in hardware design and highlights the necessity of metric-specific optimization strategies. 3.2 OPTIMIZATION WORKFLOW Each unoptimized design in the Pluto set is further refined through manual optimization by expert engineers to generate three distinct versions optimized separately for area, delay, and power. This workflow follows a systematic process that ensures both the correctness and the efficiency of the resulting designs. After applying metric-specific transformations, each optimized circuit is rigorously verified for functional correctness using Icarus Verilog Williams et al. (2002), supported by robust self-checking testbenches that guarantee equivalence with the unoptimized baseline. The optimized versions are then synthesized to confirm that improvements translate into measurable gains in area, timing, or power, thereby providing reliable performance baselines against which LLM-generated designs can be evaluated. 4 To understand how these efficiency gains are achieved, we visualize the optimization strategies applied across the dataset in appendix A.2. The strategies vary significantly depending on the target metric. For area, arithmetic optimizations and logic simplification are most commonly employed, and FSM restructuring plays an important role in reducing redundant states and transitions. Delay improvements rely heavily on exploiting parallelism and restructuring control logic, often complemented by logic simplification and pipelining techniques that shorten the critical path. For power, it's reducing switching activity through register and logic optimizations, supported by techniques such as operand isolation, and clock gating to further suppress unnecessary toggling. The distribution of strategies reveals that no single optimization technique dominates across all objectives. Instead, engineers select strategies tailored to the specific metric, reflecting the tradeoffs inherent in digital design. As shown in Figure 2, this process results in consistent improvements across the dataset, with average reductions of 19.19% in area, 21.96% in delay, and 22.55% in power. This highlights the importance of metric-specific approaches and provide a robust baseline for evaluating LLM-generated hardware code efficiency. To further illustrate the impact of expert-driven optimization, Table 2 presents representative examples drawn from both RTLLM and VerilogEval. These case studies highlight how different strategies, such as arithmetic unit sharing, FSM encoding choices, and counter-based control logic, translate into concrete improvements across area, delay, and power. As shown, across both VerilogEval and RTLLM, expert-optimized designs consistently outperform baseline implementations. In particular, for RTLLM problems our expert-written solutions achieve average improvements of 18.75% in area, 22.75% in delay, and 20.43% in power compared to their canonical solutions. For VerilogEval problems, the improvements average 10.46% in area, 10.33% in delay, and 13.61% in power. (a) Area Comparison (b) Delay Comparison (c) Power Comparison Figure 2: Distribution of area, delay, and power across Pluto benchmark designs before and after manual metric-specific optimizations. Table 2: A sample of benchmark problems from Pluto dataset. Our expert-optimized solutions (area, delay, power) are significantly more efficient than the baseline benchmark implementations. See Appendix A.3 for full problem implementation. ID Source Problem Description Benchmark Solution Expert Solution (Ours) #60 RTLLM ALU for a 32-bit MIPS-ISA CPU with operations like {ADD, SUB, AND, OR, XOR, SLT, shifts, LUI}. ALU implementation with case statement and parameterized opcodes. Area ↓26%: Shared adder for arithmetic, simplified flag logic, and operand reuse Delay ↓26%: Parallel datapaths with one-hot muxing Power ↓4%: Operand gating and early zeroing for large shifts to cut switching activity #68 RTLLM FSM detecting the sequence 10011 on a serial input stream with support for continuous and overlapping detection States binary encoded, sequential next-state logic, and registered Mealy output Area ↓32%: Casez-based transitions and direct output Delay ↓17%: One-hot state encoding with pre-decoded inputs and Moore-style output Power ↓23%: Compact binary encoding and casez-based transitions to cut toggling #87 VerilogEval Module controls a shift register enable signal, shift ena asserted for 4 clock cycles on reset, then remain 0 until the next reset Uses explicit states with next-state logic to drive shift ena Area ↓47%: Minimized register width (2-bit counter) and compact comparator logic Delay ↓37%: Wider counter (3 bits) to simplify comparison and reduce logic depth on the critical path Power ↓46%: Small counter reused #104 VerilogEval Conway's Game of Life with a 16×16 toroidal grid: each cell updates based on neighbor counts n (live if n = 3 or n = 2 & alive) Straightforward RTL with percell neighbor recomputation and sequential summing Area ↓19%: Shared neighbor computations across rows, bitwise rotations for wraparound, less duplicate summations Delay ↓37%: Parallel neighbor summation with carry-save adder tree and direct decode for 2 and 3 Power ↓36%: Reduced toggling via computation reuse 5 3.3 EFFICIENCY METRICS We use the pass@k Liu et al. (2023a) for measuring the functional correctness of LLM-generated Verilog code. The pass@k metric, defined in appendix. A.5, measures the percentage of problems for which at least one of the top-k generated samples passes the self-checking testbench. To evaluate the synthesis efficiency of functionally correct samples, we adapt the eff@k introduced in Qiu et al. (2024b) to Verilog code. First, we introduce the efficiency score ei,j, defined in Eq. 1, which quantifies how close an LLM-generated design is to optimal ground truth implementation. In this equation, ˆRi,j denotes the reported synthesis metric (e.g., area, delay, or power) for the j-th sample of problem i, Ti,j denotes an upper bound beyond which the design is considered inefficient, and Ri,j denotes the optimal (lowest) known reference value for that metric. A score of 1 indicates that the sample exactly matches the optimal reference, while a score of 0 indicates that it exceeds the acceptable threshold or is functionally incorrect. ei,j = max(0, Ti,j -ˆRi,j) Ti,j -Ri,j , if ni,j is correct 0, otherwise. (1) eff@k = 1 N N X i=1 EJ⊆{1,...,n}, |J|=k max j∈J ei,j = 1 N N X i=1 n X r=k r-1 k-1 n k ei,(r). (2) We then use the efficiency score ei,j for computing the eff@k, defined in Eq. 2 as the average of the best (i.e., highest) efficiency scores among the top-k functionally correct samples for each problem. We use the unbiased estimator introduced in Qiu et al. (2024b) for computing eff@k which computes the expectation value over a random subset J of code samples with size K. 4 EVALUATION RESULTS We evaluate 18 large language models (LLMs) using our Pluto benchmark, which includes proprietary LLMs, general-purpose foundation models, code-specialized models, and Verilog-tuned models. To comprehensively assess efficiency-aware generation, we consider the two problem formulations in Pluto: translating unoptimized Verilog code into optimized implementations, and generating optimized code directly from natural-language specifications. For the first problem formulation, only instruction-tuned models are evaluated, as code completion models generally reproduce the unoptimized code without meaningful improvements. 4.1 MAIN RESULTS Table 3 (a) reports the pass@k and eff@k metrics for the first problem formulation, where the task is to re-write unoptimized Verilog into more efficient implementations. Several trends emerge. First, in terms of functional correctness (pass@k), domain-tuned models such as VeriThoughts-Inst-7B and RTLCoder-DeepSeek-V1 achieve performance comparable to much larger foundational models like DeepSeek-Chat, demonstrating the benefit of Verilog-specific training. However, in terms of synthesis efficiency (eff@k), all models exhibit a noticeable drop relative to their pass@k scores. This gap underscores a common limitation: while LLMs can generate functionally correct Verilog, they struggle to match the Pareto-efficient expert baselines across area, delay, and power. Table 3 (b) reports the pass@k and eff@k metrics for the second problem formulation, where models are tasked with translating natural language specifications into optimized Verilog implementations. This task is more challenging, and as a result, both pass@k and eff@k scores are consistently lower across all models. Similar to the first formulation, all models also exhibit lower eff@k values compared to their corresponding pass@k scores, underscoring the persistent difficulty of generating designs that are not only functionally correct but also synthesis-efficient. However, the relative gap between pass@k and eff@k is smaller in this setting compared to the first formulation. This is because specification-to-RTL translation is substantially harder: models often struggle to produce functionally correct code in the first place, which suppresses both correctness and efficiency scores. 6 Table 3: Evaluation results using Pluto for two problem formulations: P1: Unoptimized-Verilogto-Optimized-Verilog and P2: Specifications-to-Optimized-Verilog. pass@k measures functional correctness, while eff@k measures efficiency across area, delay, and power. (a) P1: Unoptimized-Verilog-to-Optimized-Verilog Model pass@1 pass@5 pass@10 eff@1 eff@5 eff@10 Area Delay Power Area Delay Power Area Delay Power GPT-3.5 0.325 0.517 0.594 0.271 0.296 0.282 0.462 0.491 0.450 0.540 0.568 0.520 GPT-4o-mini 0.506 0.705 0.751 0.469 0.476 0.467 0.662 0.677 0.639 0.714 0.744 0.687 DeepSeek-Chat 0.612 0.802 0.860 0.586 0.599 0.601 0.776 0.795 0.794 0.839 0.862 0.846 Llama-3.3-70B-Instruct 0.473 0.701 0.757 0.446 0.462 0.429 0.662 0.696 0.662 0.707 0.760 0.735 Llama-3.1-8B-Instruct 0.160 0.432 0.567 0.127 0.156 0.145 0.358 0.437 0.384 0.494 0.584 0.505 Mistral-7B-Instruct-v0.2 0.106 0.318 0.453 0.078 0.094 0.100 0.244 0.296 0.301 0.358 0.446 0.427 Mixtral-8x7B-v0.1 0.255 0.520 0.652 0.217 0.231 0.210 0.462 0.487 0.447 0.593 0.630 0.561 starcoder2-15b-instruct-v0.1 0.659 0.960 0.988 0.611 0.633 0.591 0.879 0.924 0.871 0.913 0.952 0.904 CodeLlama-70b-Instruct-hf 0.576 0.905 0.956 0.522 0.541 0.523 0.842 0.876 0.824 0.903 0.925 0.878 DeepSeek-Coder-33B 0.783 0.963 0.997 0.638 0.659 0.640 0.902 0.927 0.883 0.942 0.960 0.927 Qwen2.5-Coder-7B-Inst 0.479 0.785 0.866 0.438 0.452 0.419 0.710 0.759 0.741 0.785 0.848 0.833 yang-z/CodeV-QC-7B 0.231 0.416 0.506 0.211 0.208 0.187 0.381 0.390 0.361 0.455 0.491 0.442 RTLCoder-DeepSeek-V1 0.532 0.854 0.915 0.471 0.495 0.468 0.774 0.789 0.757 0.843 0.850 0.809 VeriThoughts-Inst.-7B 0.611 0.797 0.854 0.540 0.560 0.524 0.708 0.740 0.702 0.763 0.785 0.765 (b) P2: Specifications-to-Optimized-Verilog Model pass@1 pass@5 pass@10 eff@1 eff@5 eff@10 Area Delay Power Area Delay Power Area Delay Power GPT-3.5 0.239 0.395 0.471 0.225 0.235 0.225 0.373 0.390 0.381 0.439 0.469 0.468 GPT-4o-mini 0.391 0.533 0.591 0.360 0.377 0.363 0.475 0.520 0.495 0.532 0.572 0.551 DeepSeek-Chat 0.557 0.688 0.719 0.545 0.552 0.528 0.689 0.680 0.651 0.726 0.710 0.684 Llama-3.3-70B-Instruct 0.363 0.541 0.594 0.348 0.345 0.342 0.515 0.533 0.510 0.564 0.602 0.557 Llama-3.1-8B-Instruct 0.087 0.224 0.301 0.075 0.073 0.081 0.189 0.184 0.204 0.270 0.251 0.279 Mistral-7B-Instruct-v0.2 0.030 0.108 0.164 0.024 0.015 0.024 0.078 0.067 0.088 0.112 0.117 0.134 Mixtral-8x7B-v0.1 0.082 0.176 0.222 0.082 0.068 0.079 0.172 0.131 0.163 0.202 0.166 0.211 starcoder2-15b-instruct-v0.1 0.243 0.454 0.512 0.226 0.249 0.220 0.429 0.466 0.409 0.489 0.525 0.458 CodeLlama-70b-Instruct-hf 0.212 0.446 0.532 0.202 0.207 0.194 0.418 0.440 0.437 0.498 0.535 0.524 DeepSeek-Coder-33B 0.257 0.429 0.482 0.231 0.254 0.246 0.387 0.424 0.421 0.446 0.490 0.468 Qwen2.5-Coder-7B-Inst 0.164 0.324 0.389 0.158 0.162 0.148 0.307 0.319 0.295 0.366 0.375 0.357 code-gen-verilog-16b 0.068 0.200 0.289 0.069 0.064 0.058 0.188 0.175 0.188 0.268 0.253 0.272 yang-z/CodeV-CL-7B 0.265 0.485 0.553 0.233 0.243 0.237 0.432 0.455 0.436 0.493 0.536 0.511 yang-z/CodeV-QC-7B 0.260 0.458 0.529 0.223 0.229 0.222 0.420 0.415 0.410 0.497 0.480 0.478 yang-z/CodeV-All-QC 0.175 0.317 0.374 0.150 0.162 0.144 0.297 0.304 0.256 0.368 0.379 0.284 RTLCoder-DeepSeek-V1 0.203 0.400 0.480 0.177 0.199 0.184 0.345 0.404 0.361 0.417 0.499 0.430 RTLCoder-Mistral 0.199 0.347 0.418 0.185 0.188 0.188 0.316 0.332 0.329 0.381 0.411 0.395 VeriThoughts-Inst.-7B 0.216 0.336 0.398 0.211 0.206 0.207 0.316 0.330 0.329 0.367 0.394 0.393 4.2 ABLATION STUDIES In addition to the Verilog code writing style, post-synthesis metrics are also influenced by external factors such as the synthesis tool employed, the target technology library, and the optimization sequence executed by the tool. To understand the robustness of our proposed benchmark and isolate the impact of these factors, we present two ablation studies that evaluate efficiency trends in the Pluto benchmark across different synthesis tools, optimization strategies, and technology libraries. 4.2.1 SYNTHESIS TOOL AND TECHNOLOGY AGNOSTICISM In this experiment, we repeated synthesis runs for the three optimized reference implementations in Pluto, as well as the unoptimized baseline, using two distinct synthesis tools: Yosys Wolf et al. (2013), an open-source framework, and Cadence Genus cad, a commercial synthesis tool. To further evaluate generalizability, we also targeted two technology libraries representing different fabrication nodes: the SkyWater 130nm library Google and a 65nm TSMC library tsm. We then computed the efficiency score for each tool and library configuration by comparing each optimized implementation against the corresponding unoptimized baseline across area, delay, and power metrics. As shown in Figure 3, efficiency scores remain consistent across all synthesis tool and technology combinations. This demonstrates that Pluto's optimization patterns deliver consistent tradeoffs across different synthesis tools and technology libraries. 7 (a) Genus: Area (earea) (b) Genus: Delay (edelay) (c) Genus: Power (epower) (d) Yosys: Area (earea) (e) Yosys: Delay (edelay) (f) Yosys: Power (epower) Figure 3: Efficiency scores for area, delay, and power across all benchmark problems, using both Cadence Genus and Yosys with different technology libraries. Results show consistent efficiency trends across synthesis tools and technologies. (a) P15: Polynomial (b) P5: Divide-by-evens (c) P28: Adder Figure 4: Area-delay tradeoffs of three problems in the Pluto benchmark under different synthesis strategies. Each strategy corresponds to a distinct sequence of ABC logic synthesis commands. 4.2.2 SYNTHESIS OPTIMIZATION STRATEGIES We also examine how different synthesis optimization strategies influence the post-synthesis metrics of Pluto's optimized implementations. Synthesis tools allow designers to specify optimization directives that steer the tool's internal heuristics toward minimizing a particular metric while potentially sacrificing others. To study this effect, we synthesized selected problems from the Pluto benchmark under both area-optimized and delay-optimized optimization strategies. We used Yosys as our synthesis tool and targeted the SkyWater 130nm library. Within Yosys, logic optimization is carried out using the ABC framework Synthesis & Group (2024), which provides a collection of optimization heuristics that can be configured to emphasize different objectives such as area or delay minimization. Figure 4 illustrates the resulting Pareto fronts of area-delay trade-offs across three representative problems. As expected, delay-optimized code consistently achieves superior timing performance at the expense of larger area, whereas area-optimized code achieves lower area but incurs higher delays. These results confirm that synthesis settings primarily shift designs along the area-delay curve, while coding style remains the dominant factor, validating Pluto's ability to capture design efficiency independent of synthesis optimization settings. 5 FAILURE ANALYSIS AND INSIGHTS While LLMs reliably produce functionally correct Verilog, their ability to optimize is uneven across metrics. Area optimization is comparatively tractable, since it often reduces to logic simplification or FSM re-encoding. By contrast, delay requires identifying and shortening the critical path, and power depends on subtle factors like switching activity and memory usage. This difficulty is reflected in 8 (a) (b) Figure 5: Failure mode analysis of optimization outcomes. (a) Quadrant plot showing the correlation between functional correctness (Pass@1) and synthesis efficiency (Eff@1) across area, delay, and power objectives. (b) Heatmap of optimization strategy difficulty across different optimization objectives area, delay, and power. our quadrant analysis (Figure 5a and 8), where many delay- and power-optimized designs remain correct but fail to improve efficiency, whereas area optimizations succeed more often. Model scale and specialization strongly influence outcomes. Larger models (33B, 70B) capture richer patterns and propose alternative architectures. Models with explicit reasoning traces (e.g., DeepSeek, VeriThoughts) better decompose transformations and achieve stronger optimizations. Domain-tuned models outperform code models, which in turn outperform general-purpose LLMs, showing the value of Verilog-specific pretraining. A fundamental limitation is that Verilog training data lacks efficiency labels. LLMs therefore default to surface-level pattern matching rather than structural reasoning, and without feedback or synthesis-in-the-loop, they cannot tell whether changes reduce gate count or lengthen the critical path. Completion-style models exacerbate this issue, often rephrasing the baseline instead of innovating, whereas instruction-tuned models attempt more substantive edits. Finally, analysis of optimization strategies (Figure 5b) shows that the hardest transformations are register optimizations for delay, followed by resource sharing for power, and sequential restructuring for delay. In contrast, strategies tied to area are easier, aligning with our observation that area is the most accessible metric for LLMs. Together, these findings suggest that true progress will require metric-aware feedback and efficiency-focused benchmarks such as Pluto to guide future advances. 6 CONCLUSION In this paper, we introduced Pluto, a comprehensive benchmark designed to evaluate the synthesis efficiency of LLM-generated Verilog code. Pluto provides an evaluation set of 114 hardware design problems, each accompanied by three reference optimized implementations (targeting area, delay, and power), an unoptimized baseline, a self-checking testbench, and a natural language description. Experimental results show that while LLMs can achieve high functional correctness, reaching up to 78.3% at pass@1, their synthesis efficiency remains limited: area efficiency of 63.8%, delay efficiency of 65.9%, and power efficiency of 64.0% at eff@1 compared to expert-crafted designs. These findings highlight the importance of efficiency-aware benchmarks beyond correctness alone and highlights the current limitations of LLMs in hardware optimization. 9 REFERENCES Cadence Genus Synthesis Solution. https://www.cadence.com/en US/home/tools/digital-designand-signoff/synthesis/genus-synthesis.html. TSMC 65nm Technology Library (Proprietary). Accessed under license from TSMC. Manar Abdelatty, Jingxiao Ma, and Sherief Reda. Metrex: A benchmark for verilog code metric reasoning using llms. In Proceedings of the 30th Asia and South Pacific Design Automation Conference, pp. 995-1001, 2025. ChipDev. ChipDev: Hardware Interview Prep & Verilog Practice. https://chipdev.io, 2025. Accessed: 2025-01-01. Mingzhe Du, Anh Tuan Luu, Bin Ji, Qian Liu, and See-Kiong Ng. Mercury: A code efficiency benchmark for code large language models. In The Thirty-eight Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2024. Dario Garcia-Gasulla, Gokcen Kestor, Emanuele Parisi, Miquel Albert'i-Binimelis, Cristian Gutierrez, Razine Moundir Ghorab, Orlando Montenegro, Bernat Homs, and Miquel Moreto. Turtle: A unified evaluation of llms for rtl generation. arXiv preprint , 2025. Google. Skywater-pdk. https://github.com/google/skywater-pdk. Accessed: 202502-09. Ce Guo and Tong Zhao. Resbench: Benchmarking llm-generated fpga designs with resource awareness. arXiv preprint , 2025. Mingjie Liu, Nathaniel Pinckney, Brucek Khailany, and Haoxing Ren. Invited paper: Verilogeval: Evaluating large language models for verilog code generation. In 2023 IEEE/ACM International Conference on Computer Aided Design (ICCAD), pp. 1-8, 2023a. 2023.10323812. Mingjie Liu, Nathaniel Pinckney, Brucek Khailany, and Haoxing Ren. Verilogeval: Evaluating large language models for verilog code generation. In 2023 IEEE/ACM International Conference on Computer Aided Design (ICCAD), pp. 1-8. IEEE, 2023b. Yao Lu, Shang Liu, Qijun Zhang, and Zhiyao Xie. Rtllm: An open-source benchmark for design rtl generation with large language model. In 2024 29th Asia and South Pacific Design Automation Conference (ASP-DAC), pp. 722-727. IEEE, 2024. Ruidi Qiu, Grace Li Zhang, Rolf Drechsler, Ulf Schlichtmann, and Bing Li. Autobench: Automatic testbench generation and evaluation using llms for hdl design. In Proceedings of the 2024 ACM/IEEE International Symposium on Machine Learning for CAD, MLCAD '24, New York, NY, USA, 2024a. Association for Computing Machinery. ISBN 9798400706998. URL https://doi.org/10.1145/3670474.3685956. Ruizhong Qiu, Weiliang Will Zeng, James Ezick, Christopher Lott, and Hanghang Tong. How efficient is llm-generated code? a rigorous & high-standard benchmark. arXiv preprint , 2024b. Berkeley Logic Synthesis and Verification Group. Abc: A system for sequential synthesis and verification. Technical report, 2024. URL https://people. eecs.berkeley.edu/ ̃alanmi/abc/. Shailja Thakur, Baleegh Ahmad, Zhenxing Fan, Hammond Pearce, Benjamin Tan, Ramesh Karri, Brendan Dolan-Gavitt, and Siddharth Garg. Benchmarking large language models for automated verilog rtl code generation. In 2023 Design, Automation & Test in Europe Conference & Exhibition (DATE), pp. 1-6. IEEE, 2023a. Shailja Thakur, Baleegh Ahmad, Hammond Pearce, Benjamin Tan, Brendan Dolan-Gavitt, Ramesh Karri, and Siddharth Garg. Verigen: A large language model for verilog code generation. ACM Transactions on Design Automation of Electronic Systems, 2023b. 10 YunDa Tsai, Mingjie Liu, and Haoxing Ren. Rtlfixer: Automatically fixing rtl syntax errors with large language models. In IEEE/ACM Design Automation Conference (DAC'24), pp. 1-8, 2024. Gwok-Waa Wan, Yubo Wang, SamZaak Wong, Jingyi Zhang, Mengnv Xing, Zhe Jiang, Nan Guan, Ying Wang, Ning Xu, Qiang Xu, and Xi Wang. Genben: A generative benchmark for LLM-aided design. In Proceedings of the International Conference on Learning Representations (ICLR), 2025. URL https://openreview.net/forum?id=gtVo4xcpFI. Under review. Stephen Williams et al. Icarus verilog: open-source verilog more than a year later. Linux Journal, 2002. Clifford Wolf, Johann Glaser, and Johannes Kepler. Yosys-a free verilog synthesis suite. In 21st Austrian Workshop on Microelectronics (Austrochip), volume 97, 2013. Chenwei Xiong, Cheng Liu, Huawei Li, and Xiaowei Li. Hlspilot: Llm-based high-level synthesis. arXiv preprint , 2024. Xufeng Yao, Yiwen Wang, Xing Li, Yingzhao Lian, Ran Chen, Lei Chen, Mingxuan Yuan, Hong Xu, and Bei Yu. Rtlrewriter: Methodologies for large models aided rtl code optimization. arXiv preprint , 2024. A APPENDIX A.1 UNOPTIMIZED CODE AND TESTBENCH FOR PROBLEM #17 (TRAILING ZEROS) IN FIGURE 1 A.1.1 UNOPTIMIZED CODE 1 module unopt_model #(parameter 2 DATA_WIDTH = 32 3 ) ( 4 input [DATA_WIDTH-1:0] din, 5 output logic [ clog2(DATA_WIDTH):0] idx; 10 11 always_comb begin 12 idx = 0; 13 din_adj = din & ( ̃din+1); 14 for (int i=0; i = total_cycles) begin 52 display("Total mismatches: %1d out of %1d samples", errors, clocks); 54 time); 55 display("TIMEOUT"); 62 signed(b) >>> sa5); 28 29 wire [31:0] sllv_res = (b > a[4:0]); 31 wire [31:0] srav_res = ( signed(b) >>> a_signed);// full signed 'a' 23 wire [31:0] sllv_res = (b > shamt5); 25 wire [31:0] srav_res = ( clog2(DATA_WIDTH)-1:0] bit_index; always @(posedge clk or negedge resetn) begin if (!resetn) begin shift_reg > 1; end end assign dout = temp[0]; endmodule (c) k = 10 Figure 7: Three area-optimized implementations of the piso shift register module (Problem #16) generated at k ∈{1, 5, 10}. The circuit shifts the least significant bit of a multi-bit input din to the single-bit output dout sequentially, starting when din en goes high. All designs are functionally correct but structurally diverse. pass@k = EN i=1 1 -C(ni -ci, k) C(ni, k) (3) A.6 FAILURE ANALYSIS In Figure. 8, we visualize the set of problems that heavy high pass@k and low eff@k to get better insight on the set of problems that are hard to optimize. Figure 8: Quadrant plot showing the correlation between functional correctness (Pass@1) and synthesis efficiency (Eff@1) across area, delay, and power objectives. 23
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2510.14758
|
Design optimization of silicon nitride nanomechanical resonators for thermal
infrared detectors: a guide through key figures of merit
Paolo Martini,1 Kostas Kanellopulos,1 and Silvan Schmid1, a)
Institute of Sensor and Actuator Systems, TU Wien, Gusshaustrasse 27-29, Vienna, 1040,
Austria
(Dated: 17 October 2025)
Thermomechanical infrared (IR) detectors have emerged as promising alternatives to tra-
ditional photon and thermoelectric sensors, offering broadband sensitivity and low noise
without the need for cryogenic cooling. Despite recent advances, the field still lacks a
unified framework to guide the design of these nanomechanical systems. This work ad-
dresses that gap by providing a comprehensive design guide for IR thermal detectors based
on silicon nitride drumhead and trampolines. Leveraging a validated analytical model,
we systematically explore how geometry, tensile stress, and optical properties influence
key performance metrics such as thermal time constant, noise-equivalent power, and spe-
cific detectivity. The analysis encompasses both bare silicon nitride and structures with
broadband absorber layers, revealing how different parameter regimes affect the trade-off
between sensitivity and response speed. Rather than focusing on a single device archi-
tecture, this study maps out a broad design space, enabling performance prediction and
optimisation for a variety of application requirements. As such, it serves not only as a
reference for benchmarking existing devices but also as a practical tool for engineering
next-generation IR sensors that can operate close to the fundamental detection limit. This
work is intended as a foundational resource for researchers and designers aiming to tailor
IR detectors to specific use cases.
a)Electronic mail: silvan.schmid@tuwien.ac.at
1
arXiv:2510.14758v1 [physics.ins-det] 16 Oct 2025
I.
INTRODUCTION
Infrared (IR) detection technologies fall into two categories: photon detectors and thermal
detectors. While photon detectors such as HgCdTe photodetectors offer high sensitivity, their op-
eration is fundamentally limited by the photon energy, making them suitable primarily for narrow
band applications in the short-wave and lower mid-infrared (MIR) spectral regions1. Moreover,
they typically require cryogenic cooling, increasing system complexity and cost2. As an alterna-
tive, thermal IR detectors, including bolometers, pyroelectric detectors, and thermopiles, operate
well at room temperature and provide sensitivity across a wide spectral range. However, these
thermoelectric detectors are fundamentally constrained by electrical noise sources, such as John-
son noise and 1/f noise, which ultimately limit their sensitivity. Thermomechanical detectors, on
the other hand, offer a compelling route to circumvent these limitations by exploiting the effect
of absorbed thermal energy on the mechanical properties of a resonator, such as changes in stiff-
ness, which are then read out through its mechanical response rather than electrical one, thereby
reducing electrical noise. Since their first introduction3, uncooled thermomechanical detectors,
particularly nanomechanical systems, have emerged as promising alternatives by transducing tem-
perature changes into measurable mechanical resonance shifts4–11.
Silicon nitride (SiN) is currently one of the materials of choice for nanomechanical sensing,
owing to its exceptional mechanical stability and thermomechanical properties, such as a high
thermal expansion coefficient. In addition, in the specific context of IR detection, it exhibits
favourable optical properties. Its absorption spectrum at room temperature features a pronounced
peak coinciding with the characteristic emission of an ideal black body, effectively functioning as
a metamaterial absorber. At the same time, it possesses a broad transparency window spanning
from the visible to the mid-infrared, making it well suited for IR spectroscopy applications, and
a high refractive index that can be tuned through stoichiometry control12. A SiN-based detec-
tor can be easily rendered broadband by depositing an absorber, sacrificing only a small fraction
of its sensitivity while substantially extending its detectable wavelength range. Recent advances
in nanofabrication, including the development of low-stress, uniform SiN films and precise sub-
micron patterning techniques, have enabled the realisation of ultra-thin, low-loss drumheads and
more complex resonator architectures, extending the detection limits of these devices.
Among the available thermomechanical resonator architectures, drumhead and trampoline ge-
ometries are particularly advantageous, as their increased active areas enhance coupling to long-
2
wavelength radiation while preserving key performance metrics13. State-of-the-art SiN drum-
heads have achieved noise-equivalent power (NEP) on the order of pW/
√
Hz in both narrowband9
and broadband14 applications, approaching the fundamental room-temperature detectivity limit
of D∗
lim =1.8 × 1010 cm
√
Hz/W for a resonator thermally coupled to the environment from only
one side15–17.
Recently, Zhang et al9 reported an NEP ≈36pW/
√
Hz and a detectivity o f
D∗≈3.4 × 109 cm
√
Hz/W using a metamaterial drumhead optimised for the range 0.3 THz to
5 THz. The structure has a thermal response time of ≈200ms. Using a free-space impedance
matched (FSIM) metal absorber for broadband IR light detection14 has lead to similar results of
NEP = 27pW/
√
Hz and D∗= 3.8×109 cm
√
Hz/W. The metal layer helped reduce the thermal
constant to τth = 14ms. Trampoline resonators have shown excellent performance, too. Piller
et al.7 reached a NEP as low as 7 pW/
√
Hz using SiN trampoline resonators featuring a simi-
lar absorber as in14, with a response time of only 4 ms. More recently, Das et al.10 proposed a
multi-material trampoline structure incorporating a perforated absorber optimised for long-wave
infrared detection, reaching a specific detectivity of D∗= 3.8×109 cm
√
Hz/W with 7.4 ms ther-
mal response. Another work6 made use of SiN trampoline resonators with a Cu and Au layer as an
absorber and reported an NEP ≈100pW/
√
Hz in sub-THz (0.14 THz) detection, and a τth = 4ms.
In a very recent work11, bare SiN trampolines achieved a record D∗= 6.4×109 cm
√
Hz/W with
τth = 88ms. Operating at the noise level set by the temperature-fluctuation limit, the authors
were able to maintain a detectivity only a factor of three lower (D∗= 2.5×109 cm
√
Hz/W) while
achieving a faster response time (τ = 3ms) than that dictated by the resonator’s thermal time
constant.
Despite these advancements, a unified framework for optimising detector parameters remains
absent in the field. This work addresses this gap by presenting a comprehensive analysis based
on a validated analytical model13,18,19, and by systematically exploring the resonator parameter
space. Focusing on SiN drumheads and trampolines, we investigate how variations in geometry,
intrinsic stress, and absorbivity influence the thermal and mechanical performance of the devices,
The resulting trade-offs are evaluated using key figures of merit: thermal time constant (τth), noise
equivalent power (NEP), and specific detectivity (D∗).
Our model reconciles recent experimental results7,13,14. The framework enables performance
prediction across design geometries, providing practical guidelines for developing application-
specific IR detectors. In particular, we examine whether, in light of the current state-of-the-art,
existing devices are already operating near their fundamental limits or whether opportunities for
3
FIG. 1. Schematic representation (not in scale) of the two designs under investigation in this work. a) square
drumhead with the geometric variables studied highlighted: h is the thickness of the SiN layer, hence of the
structure, and the lateral size L. In addition to these two, for a b) trampoline we study also the influence of
the tethers’ width, w, and length, Lt.
notable performance gains remain. The framework presented here enables this assessment quanti-
tatively, highlighting potential pathways to approach the fundamental photon noise limit.
II.
FIGURES OF MERIT
Before reviewing the metrics used to evaluate the detectors’ performance, we introduce the
two designs considered in this work. Regardless of the specific fabrication process employed,
we assume the detector chip to consist of a silicon (Si) substrate several hundred µm thick. This
substrate serves both as a mechanical support for handling the resonator and as a thermal reservoir
where heat generated by absorbed radiation can be effectively dissipated. A layer of silicon nitride
(SiN), deposited on top of the silicon substrate with low-pressure chemical vapour deposition
(LPCVD), forms the material of the detecting element. Figure 1 presents a schematic overview
of the two geometries. In Figure 1a, a square drumhead is illustrated, defined by two primary
geometrical parameters: its thickness h and lateral dimension L. Compared to this basic config-
uration, the trampoline design introduces additional geometrical complexity, as shown in Figure
1b. Here, the variable L denotes the side length of the central suspended area, which serves as the
active detection region, analogous to the drumhead’s side length. This central area is connected
to the supporting frame via four tethers, each characterised by a length LT and a width w. These
four parameters (h, L, LT, and w) fully define the geometries of the drumhead and trampoline
4
structures, whose impact on detector performance is systematically explored in Section III.
A.
Thermal time constant, τth
The first figure of merit is the thermal time constant, τth [s]. This metric is crucial in many ap-
plications, particularly when fast detector response is required, e.g.: in time-resolved spectroscopy
applications, where scanning frequencies can reach the order of 2 kHz. While in most cases this
remains the fundamental limit of the detection chain, a recent work11 showed that this limitation
can be broken when the resonator is operated at its fundamental temperature fluctuation noise
limit.
For a lumped element system, τth is defined as the ratio between the heat capacity C [J/K] of
the resonator and its thermal conductance G [W/K]1:
τth = C
G.
(1)
For a drumhead made of a material with mass density ρ and specific heat capacity cp, in the
context of the mean temperature framework (MTF), the heat capacity is given by13:
C = ρcphL2.
(2)
Likewise, the heat capacity of a trampoline structure is13:
C = ρcph(L2 +4wLt).
(3)
By comparing (2) and (3), it is evident that for the same active detection area L2, trampolines
exhibit a larger heat capacity due to the additional volume contributed by the tethers.
In this study, we consider detectors operating in vacuum conditions, where heat transfer via
convection can be neglected. In this regime, the thermal conductance is the sum of the conductive
and radiative contributions only: G = Gcond +Grad. For a drumhead, this takes the form1:
G = 4πhκ +4L2εσSBT 3
0 ,
(4)
where κ is the thermal conductivity of the material, ε is the emissivity, σSB is the Stefan–Boltzmann
constant, and T0 is the temperature of the environment including the detector. The first term repre-
sents heat conduction through the drumhead, while the second accounts for radiative losses from
5
the surface.
In a trampoline structure, the conductive heat transfer term arises from the series combination of
the thermal resistances of a membrane (Rm) and that of four strings (Rs) of length LT and width w:
Rm +Rs =
1
Gm
cond
+
1
Gs
cond
=
1
Gcond
,
(5)
where Gm
cond = 4πhκ and Gs
cond = 8wh
LT κ represent the terms for thermal conductance through con-
duction of a membrane and four strings, respectively18. The radiative contribution, on the other
hand, accounts for the total emitting surface of both the membrane and the tethers. The overall
thermal conductance of the trampoline can thus be expressed as:
G =
8πhw
2w+πLT
κ +4(8wLT +2L2)εσSBT 3
0 .
(6)
The contribution of the tethers to both conduction and radiation is evident from the presence of w
and LT in both terms of the sum.
B.
Power responsivity, RP
When a light source with power P0 impinges on the detector surface, a fraction of that power,
denoted P, is absorbed by the material and converted into heat:
P = αP0,
(7)
where α is the absorptance, which depends on both the material properties and the wavelength of
the incident light. The absorbed power raises the temperature of the detector, inducing thermal
expansion of the material. This phenomenon, known as induced stress change, underpins the
working principle of thermomechanical IR detectors. As the material expands, the mechanical
stiffness of the structure changes, which in turn shifts the resonance frequency ω0 of the detector.
The second figure of merit considered in this work is the power responsivity, RP, with unit
[1/W], defined as the relative change in resonance frequency per unit of absorbed power18:
RP(ω) = ∂ω0
∂P
1
ω0
Hth(ω),
(8)
where Hth(ω) is a low-pass filter transfer function that accounts for the finite thermal response
time of the resonator18:
Hth(ω) =
s
1
1+(ωτth)2.
(9)
6
Since thermomechanical detectors operate by converting absorbed power into heat, effectively
functioning as temperature sensors, (8) can be recast in the form13,18:
RP(ω) = ∂ω0
∂T
1
ω0
∂T
∂PHth(ω) = RT
G Hth(ω),
(10)
where T is the temperature, and RT is the temperature responsivity. For square drumheads, such
as those considered in this study, it takes the form18:
RT = −
αth
2(1−ν)
E
σ ,
(11)
where αth is the thermal expansion coefficient, ν the Poisson’s ratio, E is the Young’s modulus,
and σ the tensile prestress in the material. The negative sign in (11) reflects the fact that thermal
expansion causes a softening of the structure, resulting in a decrease in resonance frequency with
increasing temperature.
In the case of trampolines, the temperature responsivity becomes18:
RT = −αth
2
E
σ .
(12)
Here, the factor (1−ν), which accounts for the in-plane expansion in two directions, is absent due
to the uniaxial strain of the tethers.
The ratio E/σ, common to both cases, is a characteristic feature of tensile mechanical res-
onators and represents an enhancement factor of the temperature responsivity. Finally, while RT
depends solely on material properties, the detector geometry plays a key role in determining the
power responsivity RP, as evidenced by the presence of the thermal conductance G in (10).
C.
Noise equivalent power (NEP)
To properly assess the performance of a detector, the noise level of the measurement system
must also be taken into account. The next figure of merit is the noise equivalent power (NEP),
defined as the signal power required to achieve a unitary signal-to-noise ratio within a one Hz
bandwidth. It has units of [W/
√
Hz], and for a nanomechanical resonator used as an IR detector,
it can be expressed as15,18:
NEP =
p
Sy(ω)
RP(ω) .
(13)
Here, Sy(ω) is the fractional frequency noise power spectral density, which quantifies the system’s
noise by evaluating the frequency stability of the resonator in thermal equilibrium. From (13), it
7
is clear that achieving high sensitivity requires minimizing frequency noise to resolve the smallest
possible shifts in resonance frequency.
In this work, we consider three primary sources of frequency noise: thermomechanical noise
Sythm(ω), detection noise Syd(ω), and temperature fluctuation noise Syth(ω). It is also worth men-
tioning that, among many other possible noise sources, one particularly relevant stems from pho-
tothermal back action caused by the relative intensity noise of a readout laser. In optomechanical
sensing applications involving an IR absorber, this noise can significantly limit sensitivity20. To
mitigate this effect, the absorber design can be optimized, as demonstrated in14.
Assuming the detection noise is white, its contribution can be related to the thermomechanical
noise peak magnitude as follows18:
Szd(ω) = K 2Szthm(ω0) = K 2 4kBTQ
me f f ω3
0
,
(14)
with the effective mass of the resonator me f f , the quality factor Q, the resonance angular frequency
ω0, and the Boltzmann constant kB. The coefficient K represents the relative sensitivity of the
transduction system: if it can resolve the thermomechanical noise peak, then K < 1. Thermome-
chanical and detection noise are typically summed into what is called the additive noise. Using
(14), the corresponding additive frequency noise PSD takes the form21,22:
Syθ (ω) = 2
ω2
0
Szthm(ω0)
z2
0
|Hθthm(iω)|2+K 2|Hθd(iω)|2
.
(15)
With z0 the displacement amplitude of the resonator. The two transfer functions Hθthm(iω) and
Hθd(iω) are filter functions. They are specific to the measurement loop implemented to read
out the detector’s mechanical motions. In the case of a self-sustaining oscillator (SSO) detection
scheme, they are21:
Hθthm(iω) = 1
τr
HL(iω)
Hθd(iω) = 1
τr
1
Hr(iω)HL(iω).
(16)
Here, Hr(ω) is the low-pass transfer function of the resonator, with the mechanical time constant
τr = 2Q/ω0, and HL(ω) is the low-pass transfer function of the oscillator circuit. It has been
shown that the transfer functions for other oscillator schemes, such as a phase-locked loops, are
essentially equal to (16). A more detailed discussion about these terms, as well as their derivation
for other detection schemes, can be found in21.
8
Equation (15) shows that noise can be reduced, and the NEP improved, by increasing the vi-
brational amplitude z0. Ideally, the resonator should be operated at the onset of nonlinearity by
pushing the actuation to the critical amplitude z0c
23:
z0c =
s
8
3
√
3
1
√Q
s
me f f ω2
0
αDu f f
,
(17)
with the Duffing parameter αDuf f . Notably, if z0 = z0c and K ≪1, then Sythm(0) becomes in-
dependent of the quality factor Q. This implies that increasing the quality factor does not benefit
this specific type of noise. Recently, Manzaneque et al.24 reported on an improvement of fre-
quency stability in mechanical resonators when driven in the nonlinear regime (z0 > z0c), albeit
only for short integration times, which is a regime that typically is difficult to access with slow
thermoelastic detectors.
The effective mass mef f , which appears in both (14) and (17), varies according to the geometry
of the device. For the fundamental mode in drumheads, it is given by18:
me f f = 1
4m0 = 1
4ρhL2,
(18)
while for trampolines, still in the fundamental mode, it becomes13:
mef f = ρh
L2 + 4wLt
2
.
(19)
Similarly, the Duffing coefficient in (17) takes different forms. For a square drumhead res-
onator, it is25:
αDuf f = 3π4
64 (n4 + j4)Eh
L2
1
2(1−ν2)(1−ν),
(20)
with n and j the modal numbers of the vibrational mode. For trampolines, the expression derived
by19 is:
αDu f f = n4π4
8
Ewh
(2Lt)3.
(21)
Since the latter equation has not yet been validated before, we performed measurements using SiN
trampoline resonators and verified its accuracy. The results are provided in the Supplementary
Material (SM).
The ultimate sensitivity limit for a nanomechanical thermal detector is set by temperature fluc-
tuation noise Syth(ω)18,26. This noise arises from stochastic heat exchange between the detector
and the environment and generates white frequency noise. For lumped-element thermal detectors
it is given by3,11,13:
Syth(ω) = 4kBT 2R2
T
G
|Hth(ω)|2 |HL(ω)|2 .
(22)
9
Here, G is the effective thermal conductance of the resonator. It has been shown that G for drum-
heads and trampolines is well approximated by (4) and (6), respectively13.
D.
Specific detectivity (D∗)
The sensitivity (13) of nanomechanical membrane and trampoline resonators generally im-
proves with increasing lateral size (L) as long as conductive heat transfer dominates. For larger L,
radiative heat transfer becomes significant. The extra thermal heat loss channel leads to a deteri-
oration in sensitivity. In the radiative heat transfer regime, when the frequency stability is limited
by thermomechanical noise (15), the sensitivity follows the scaling law
NEP ∝
√
L3.
(23)
In the ultimate regime, where frequency stability is determined by temperature fluctuation noise
(22), the scaling law changes to
NEP ∝L,
(24)
as derived below. This latter dependence matches the well-known scaling for photodetectors,
whose sensitivity scales with the square root of the detection area,
√
A27. A commonly used figure
of merit in such cases is the specific detectivity, D∗, with units [cm
√
Hz/W], which normalizes
sensitivity for the detector size. Hence, it is particularly useful for comparing sensors with different
detection areas. It is defined as15,28:
D∗=
√
A
NEP.
(25)
It is important to note that, in the case of trampoline resonators, the detection area does not include
the tethers. Therefore, for this design, A = L2 just as for drumheads.
According to Kirchhoff’s law, for a resonator in thermodynamic equilibrium with its surround-
ings, the emissivity ε equals the absorptance α.
Considering the case where temperature fluctuation noise dominates, the equation (13) can be
expressed in terms of ε18:
NEP =
p
Syth(ω)
εRP(ω) =
p
4kBT 2G
ε
|HL(ω)|2 .
(26)
It is noteworthy that the thermal low-pass transfer function Hth(ω), which normally shapes both
the frequency noise PSD (22) and the responsivity (8), no longer appears in (26). As a conse-
quence, the thermal response time ceases to play a role, and the measurement speed is essentially
10
limited only by the time constant of the oscillator circuit HL(ω), provided the resonator operates
in the temperature fluctuation noise regime. This effect has recently been exploited to enhance
measurement speed in nanomechanical trampoline resonators11.
Assuming further that the resonator is well thermally isolated so that heat transfer occurs pre-
dominantly via radiation, (26) yields a relation that depends on the detector area
√
A18:
NEP =
s
16σBkBT 5
0 A
ε
,
(27)
which becomes size independent when written as specific detectivity18:
D∗=
s
ε
16σBkBT 5
0
.
(28)
For an ideal infrared detector (a blackbody with absorptance αbb = εbb = 1) thermally coupled to
its environment on only one side, this reduces to the well-known theoretical sensitivity limit for
room-temperature detectors,(T0 = 290 K and ε = 1) D∗
lim = 2.0×1010 cm
√
Hz/W. In the present
work, we consider a resonator coupled on both sides, which slightly lowers this limit to D∗
bb ≈
1.4×1010 cm
√
Hz/W. Furthermore, we consider a broadband absorber with a 50% absorptance,
for which the limit reduces to (T0 = 290 K and ε = 0.5) D∗
50% = 1.0×1010 cm
√
Hz/W.
A high, uniform, and broadband absorption is advantageous for many applications and, in par-
ticular, for detectors for IR spectroscopy. Conversely, a narrowband absorption can be attractive
for applications that require spectral selectivity for gas sensing or for IR imaging in the long-
wavelength IR range 9µm to 10µm29–31. SiN offers a high absorption, with a peak absorption be-
tween λSiN = 10µm and λSiN = 12µm in combination with an otherwise low emissivity εSiN. This
combination renders it more responsive in narrow-band applications when compared to coated
detectors. For this reason, first, we apply the analytical model to study pure SiN nanomechanical
resonators without a dedicated absorber.
For sections III A, III B and IV, where we assume a 50 nm thick layer, we calculate the absorp-
tance to be αSiN(λSiN) = 0.255 ± 0.007 at the wavelength λSiN = 12.02µm, where this thickness
exhibits an absorption peak. Similarly, we determine an emissivity value of εSiN = 0.056±0.002.
Details on the derivation and the procedure used to obtain the SiN absorption spectrum at this
thickness are provided in the SM. In Section III C, where the thickness is varied, different values
of αSiN(λSiN), λSiN and εSiN are assumed accordingly (see Figure 4). In each scenario presented
in the sections III A, III B, III C and IV, we also explore the case of an IR absorber layer with
11
ρ [kg/m3]
E [Pa]
ν
αth [K−1] κ [W/(mK)] cp [J/(kgK)]
3000
250×109 0.27 1.5×10−6
3
700
TABLE I. Silicon nitride parameters used. From left to right: mass density, Young’s modulus, Poisson ratio,
thermal expansion coefficient, thermal conductivity, and specific heat capacity.
an absorptance αabs = 50% deposited onto the SiN detector, allowing for broader absorption.
Throughout the analysis, a fundamental trade-off becomes evident: a higher specific detectivity
confined to a narrow spectral band, versus a lower detectivity extended over a broad wavelength
range, each corresponding to different application domains.
III.
GEOMETRY (L, Lt, h, AND w)
Geometry is typically the first aspect to consider when designing a nanomechanical IR detector,
and it is the focus of this section. The analysis begins with the parameters specific to trampoline
designs, namely Lt and w, followed by the thickness h, which is shared by both drumheads and
trampolines. For all cases, we present results across five different values of the detection area side
length L. The impact of each geometrical variable is evaluated with respect to the thermal time
constant (τth), noise equivalent power (NEP), and specific detectivity (D∗). Table I lists the SiN
parameters considered for this work7,32–34.
A.
Tethers length, Lt
First, we examine how the tether length, Lt, influences the performance of trampolines with
different detection areas. The results of this analysis for bare SiN are shown in Figure 2. In this
section, the tethers width is fixed at w = 5µm. For comparison, in all panels of Figure 2, we display
the values corresponding to drumheads of the same thickness and stress, as horizontal solid lines
because they are independent of Lt. For all the designs, the thickness and tensile stress are set to
h = 50nm and σ = 200MPa, respectively.
The effect of Lt on the thermal conductance G is twofold, as shown in Eq. (6): on the one
hand, longer tethers reduce heat transfer via conduction; and on the other hand, increasing Lt
enlarges the total radiating surface, enhancing heat loss through radiation. Additionally, the total
heat capacitance C increases linearly with Lt, as indicated in Eq. (3). The combined effect of these
12
FIG. 2. Influence of the tether length Lt on the figures of merit: a) thermal time constant τth, b) noise
equivalent power NEP, and c) specific detectivity D∗at λSiN = 11.86µm for plain SiN resonators without
a dedicated IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit. For all
panels, the stress is fixed at σ = 200MPa, the thickness h = 50nm, and the tether width at w = 5µm.
contributions is illustrated in Figure 2a, which shows τth as a function of Lt. For small Lt, heat
from the center of the detection area is primarily dissipated through conduction along the tethers
(high Gcond). As Lt increases, the contribution of Gcond diminishes, and the radiative contribution
(Grad) becomes significant, coupling the resonator more strongly to the environment via radiation.
As a result, τth increases until asymptotically reaching the radiative heat transfer plateau.
The NEP, shown in Figure 2b, exhibits a clear dependence on the tether length Lt. The most
sensitive detector corresponds to the smallest trampoline geometry, with a minimum NEP occur-
ring around Lt ≃700µm.
Finally, the specific detectivity as a function of Lt is shown in Figure 2c. In this case, the
trampoline with detection area side length L = 0.25mm exhibits the highest D∗. However, even at
the peak of its curve, corresponding to a tether length of Lt ≃1mm, its D∗is comparable to that
of the drumhead with side length L = 1mm. For this figure of merit, the overall best-performing
devices are drumheads with L = 4mm and 16mm, both reaching the room-temperature detectivity
limit D∗
bb. This result underscores the crucial role of the detection area (A in equation (25)):
although smaller resonators exhibit lower NEP, this advantage is outweighed by the larger IR
13
FIG. 3. Influence of the tether length Lt on the figures of merit: a) thermal time constant τth, b) noise
equivalent power NEP, and c) specific detectivity D∗for a SiN resonator with an IR absorber. The dashed
black line in c) marks the room-temperature sensitivity limit for αabs = 50%. For all panels, the stress is
fixed at σ = 200MPa, the thickness h = 50nm, and the tether width at w = 5µm.
collection area of bigger structures.
So far, we have evaluated the performance of bare SiN resonators at the fixed wavelength
λSiN = 12.02µm, corresponding to the measured peak in absorptance. All previous results are
therefore specific to this wavelength. However, in the field of infrared detection, it is often desir-
able to develop broadband detectors. One promising approach is the deposition of a metal layer
on a dielectric, which can achieve absorptance values up to αabs = 50% across a broad spectral
range35. Recent studies have demonstrated such high and nearly flat absorption spectra using
platinum or gold thin films deposited on SiN7,14,36. As shown in those works, the addition of a
thin metal layer does not significantly affect the mechanical properties of the resonator, although
it does influences its thermal and optical behaviour, and provides an almost flat and wavelength
independent absorption spectrum, over the infrared range, up to the THz region. Now we inves-
tigate the impact of adding an absorber with αabs = εabs = 50% on the SiN resonator, under the
assumption that the mechanical properties of SiN remain largely unchanged.
The results of this analysis for varying tethers length Lt are shown in Figure 3. As observed for
bare SiN (Figure 2b), the best NEP values are obtained for the smallest trampoline (Figure 3b).
14
FIG. 4. Effect of the thickness h over SiN absorptance at its absorption peak λSiN = 11.86µm (blue line) and
the hemispherical emissivity (orange line). Both grow as the SiN gets thicker. αSiN(λSiN) has a maximum
for h ≂250nm and later shows a local minimum for h ≂700nm. The green line is the ratio α(λSiN)/√εSiN,
and it has a maximum for h = 151nm.
However, in this case, the lowest NEP occurs around Lt ≃250µm. In general, smaller structures
outperform larger ones, both for trampolines and drumheads. Overall, we observe that adding
an absorbing layer lowers τth (Figure 3a) and shifts the optimal designs toward smaller resonators
(Figure 3c) compared to the bare SiN scenario. Although the absolute values of the figures of merit
are slightly lower than in the bare SiN case, it is important to emphasize that, with the absorber,
these performance levels are maintained across the entire MIR spectral range, whereas previously
they were limited to the specific peak wavelength of SiN.
B.
Tethers width, w
The resulting behaviour of NEP and D∗as a function of w closely mirrors that observed for Lt.
To avoid redundancy, we omit a detailed discussion here; the full analysis is provided in the SI,
both for the case of bare SiN and with the metal absorber.
15
C.
Thickness, h
In nanomechanical resonators, the thickness can range from a few to several hundred nm, and
in some cases, up to a few µm. In this work, we consider the range h = 10nm to 1µm. In the
thin-film regime, variations in thickness significantly affect the film’s optical properties, such as
absorptance α(λ), intended as the fraction of the absorbed light at one specific wavelength, and
emissivity, or hemispherical emissivity ε, which is the efficiency with which the film radiates en-
ergy compared to that of a blackbody at the same temperature, integrated over all emission angles
and wavelengths37,38. For each value of h, we computed αSiN(λSiN) and εSiN, taking into account
possible interference effects with the thin film32. The result is shown in Figure 4, where the blue
line represents the SiN peak absorptance (αSiN(λSiN)), the orange line the hemispherical emissiv-
ity (εSiN), and the green line the ratio α(λSiN)/√εSiN. This latter quantity is a strong indicator
of the SiN performance: on one hand, a thicker material increases the absorbed power, thereby
enhancing the power responsivity (equation (8)); on the other hand, a lower emissivity reduces
the radiative coupling to the environment, which is also desirable. Therefore, rather than solely
maximizing the absorptance or minimizing the emissivity, the optimal strategy is to maximize
the ratio. The term α(λ)/√ε can be seen as a specific detectivity enhancement factor typical of
metamaterials, for which, in a very narrow spectral region, it holds true that α(λ) > ε. Figure 4
shows that for SiN the maximum of the ratio occurs at h = 151nm. For more details on how the
absorptance and emissivity curves are obtained, the reader is referred to the SM.
During the analysis of thickness dependence, all other parameters are kept constant: the stress
is fixed at σ = 200MPa, and for the trampoline geometries, the tether length and width are
set to Lt = 50µm and w = 5µm, respectively.
According to equations (2) and (3), increas-
ing h enhances both the heat capacity C and the thermal conductance G. For small drumheads
(L = 0.062mm and 0.25mm), where conduction dominates, the h-dependence in C and G effec-
tively cancels out, yielding a nearly constant τth (see figure 5a). In contrast, for larger drumheads
(L = 4mm and 16mm) the radiative term becomes significant, causing τth to increase with h. A
similar trend is observed for trampolines, where the tethers, by hindering conduction and aug-
menting the radiating surface, make the influence of h on τth evident even for L = 0.25mm. Thus,
the presence of tethers enhances thermal isolation, rendering trampolines slower than drumheads
for a given L.
Figure 5b shows the results for the NEP. The most sensitive device is the trampoline with
16
FIG. 5. Influence of the thickness h on the figures of merit: a) thermal time constant τth, b) noise equivalent
power (NEP), and c) specific detectivity D∗at λSiN = 11.86µm for plain SiN resonators without a dedicated
IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit. For all panels, the
stress is fixed at σ = 200MPa, the tether length at Lt = 50µm, and the tether width at w = 5µm.
L = 0.062mm, with the lowest values obtained for h ≃100nm. The slight deviation in thickness
from the value corresponding to the maximum of the green curve in Figure 4 arises from the
non-negligible contribution of thermal conduction, which becomes increasingly significant as the
resonator thickness grows.
Finally, the model predictions for the specific detectivity D∗are presented in Figure 5c. In this
case, the best performance is achieved by drumheads, with the two designs featuring the largest de-
tection areas yielding the highest overall values. For these configurations, the predicted D∗values
at thicknesses around 200nm surpasses the fundamental room-temperature limit, D∗
bb, indicated
by the green dotted line. This somehow counterintuitive result is in line with what is shown in
Figure 4, and demonstrates that bare SiN, when probed at its absorptance peak wavelength λSiN,
acts similarly to a metamaterial, showing αSiN(λSiN) > εSiN. Overall, drumheads outperform tram-
polines of the same size, with the exception of the smallest drumhead, which is outperformed by
the trampoline with L = 0.062mm.
The presence of the metal layer which gives the detector a constant absorptance and emissivity
of 50% can be noticed in the results shown in Figure 6. Because of this, the influence of h is now
17
FIG. 6. Influence of the thickness h on the figures of merit: a) thermal time constant τth, b) noise equivalent
power (NEP), and c) specific detectivity D∗for SiN resonators with a dedicated IR absorber. The dashed
black line in c) marks the room-temperature sensitivity limit for αabs = 50%. For all panels, the stress is
fixed at σ = 200MPa, the tether length at Lt = 50µm, and the tether width at w = 5µm.
limited to heat transfer through conduction, Gcond, for both trampolines and drumheads, with no
impact on Grad. In Figure 6a, we notice the thermal time constant significantly decreasing for
every design and when compared to the bare SiN scenario (Figure 5a). In Figure 6b, we see the
results for the NEP, with smaller detectors exhibiting superior performance, particularly at reduced
thicknesses. As the lateral dimensions of the structures increase, the optimal NEP shifts toward
larger thickness values. This trend arises because a larger L enhances radiative exchange, so the
sweetspot between radiative losses and thermal conduction is reached at greater thicknesses. A
similar trend is observed for D∗(Figure 6c), for both drumheads and trampolines. This behaviour
reflects the fact that increasing detector area requires thicker substrates to achieve the optimal
balance between Gcond and Grad.
For all the geometrical variables, the results for the power responsivity RP are presented in the
SM.
18
FIG. 7. Influence of the tensile stress σ on the figures of merit: a) power responsivity RP; b) noise equivalent
power (NEP); c) specific detectivity D∗at λSiN = 11.86µm for plain SiN resonators without a dedicated IR
absorber. The dashed black line in c) marks the room-temperature sensitivity limit. For all panels, the
thickness is fixed at h = 50nm, the tether length at Lt = 50µm, and the tether width at w = 5µm.
IV.
TENSILE STRESS, σ
This section is dedicated to the analysis of the influence of tensile stress σ on the detectors’
performance. Often referred to as prestress, it is to some extent a tunable parameter of mechanical
resonators. Different deposition methods and parameters result in varying stress conditions in the
deposited material. From Section II A, we know that σ has no influence on either the thermal
conductance G or the heat capacity C (see equations (2) and (4) for drumheads, and equations
(3) and (6) for trampolines); thus, it does not affect the thermal time constant of the device. In
some cases, differences in deposition techniques result in variations in crystallinity (e.g., α-phase
SiN vs. β-phase SiN vs. amorphous SiN). These differences typically entail changes in material
properties, such as specific heat capacity and thermal conductivity, which would influence τth.
While the authors are aware of this, the investigation of such effects falls outside the scope of the
present study. Here, we consider the case of amorphous SiN deposited with LPCVD, for which we
assume that prestress conditions do not affect the thermal properties of the material. Consequently,
in presenting the model results for the various figures of merit, we exclude τth. Analysing the
19
effects of stress by comparing drumheads to trampolines is non-trivial. This is due to the well-
known influence of the trampoline geometry on the stress distribution: the stress is significantly
increased in the tethers and reduced in the central area compared to the nominal stress σ0 of the
deposited material, as shown through finite element (FEM) simulations in7. Nevertheless, the
results presented in this section remain generally valid when considering the value of the stress
σ in the specific region of the detector where the IR light impinges and is absorbed. For the
following study on varying the stress in the range σ = 1MPa to 1000MPa, the thickness of all
structures is fixed to h = 50nm, and for the trampolines, the tethers have width w = 5µm, and
length Lt = 50µm.
The dependency of RP on stress arises from the linear dependence of the temperature respon-
sivity RT (equations (11) and (12)) on σ. For both designs, RT decreases linearly with increas-
ing stress, and so does the power responsivity, as shown in Figure 7a. The differences in RP
values between structures of different sizes stem from the expected variations in G: detectors
with smaller detection areas generally perform better. The results show that drumheads with
L = 4mm and 16mm exhibit slightly higher responsivity than trampolines with equivalent detec-
tion areas. However, for smaller L values, trampolines display higher RP compared to drumheads.
This improvement is attributed to their enhanced thermal isolation of the centre of the detection
area, enabled by the presence of tethers that reduce conductive heat loss.
The situation is almost the opposite when it comes to the noise equivalent power. As shown
in Figure 7b, the NEP generally improves with increasing values of σ, and drumheads tend to
outperform trampolines, with the sole exception of the smallest trampoline, which exhibits the
lowest NEP among all designs. Both the inverse trend of NEP and the relative performance of
drumheads compared to trampolines can be explained by frequency stability. As detailed in the
SI, the significantly lower frequency noise of drumheads relative to trampolines, along with the
general improvement in frequency stability with increased stress, compensates for the differences
in power responsivity observed in Figure 7a. Interestingly, while smaller detection areas still
correspond to better performance in trampolines, for drumheads, the detectors with bigger areas
are those that perform best across the entire σ range under investigation.
The specific detectivity results are presented in Figure 7c. Again, we observe what is seen for all
the other variables in section III: introducing the detection area A in the calculation highlights the
importance of collecting as much light as possible from the source. It is now the large drumheads
which show the highest D∗, with L = 4mm being the top performer for 1MPa ≤σ ≤10MPa and
20
FIG. 8. Influence of the tensile stress σ on the figures of merit: a) power responsivity RP; b) noise equivalent
power (NEP); c) specific detectivity D∗for SiN resonators with a dedicated IR absorber. The dashed black
line in c) marks the room-temperature sensitivity limit for αabs = 50%. For all panels, the thickness is fixed
at h = 50nm, the tether length at Lt = 50µm, and the tether width at w = 5µm.
matching the performance of L = 16mm for σ > 10MPa. For a given detection area side length L,
drumheads always perform better than trampolines, with the case of L = 0.062µm being the only
exception to this. However, it consistently holds that D∗increases with higher values of tensile
stress σ, independently of the detector design.
Very similar results are obtained with the inclusion of the metal absorber layer, as shown in
Figure 8. The overall trends observed for bare SiN are preserved in this case as well. A slight
increase in power responsivity is observed (Figure 8a), accompanied by a modest reduction in
performance with respect to NEP and D∗(Figures 8b and 8c, respectively).
The key takeaway from Figure 7 and 8 is that higher stress leads to improved detector perfor-
mance. However, the practical implications of employing high-stress resonators for IR detection
must also be considered. The enhanced NEP and D∗observed at higher stress levels originate
from the overall improvement in frequency stability associated with such resonators. This is illus-
trated in Figure 9, where we plot the theoretical Allan deviations, accounting for both additive and
temperature fluctuation noise, as a function of integration time τ, for drumhead and trampoline
resonators of different sizes and two commonly used stress values: 200 MPa and 1 GPa. The anal-
21
FIG. 9. Theoretical ADs, obtained by adding together thermomechanical and detection noise, for a) drum-
heads and b) trampolines, for different σ. For this study, the thickness is set to h = 50nm while for trampo-
lines the tether length is Lt = 50µm and tether width is w = 5µm.
ysis assumes a closed-loop measurement with a phase-locked loop (PLL) bandwidth of 100 Hz,
and a thermomechanical-to-detection noise ratio K = 0.01 for a typical optical interferometer that
has a displacement sensitivity in the order of 1 pm/
√
Hz. In all cases, the high-stress resonators
(solid lines) exhibit lower noise than their low-stress counterparts (dashed lines). Notably, for the
most promising designs, namely, the smaller trampolines (purple lines in figure 9b), which exhibit
the lowest NEP in Figure 7b, and the larger membranes (solid green line in figure 9a), which show
the highest D∗in Figure 7c, the Allan deviation are below 10−7, and below 10−9, respectively. At
such low levels, the detector performance is increasingly limited by other noise sources, such as
absorbtion-desorption noise39,40, defect motion41, random walk frequency drift, and photothermal
back-action noise13,14,42. Ultimately, the precision of such measurements is limited by the fre-
quency stability of the reference oscillators in the readout electronics — typically oven-controlled
crystal oscillators (OCXOs) — which offer stabilities on the order of 10−9. In practice, these ad-
ditional noise sources will likely dominate in high-stress, frequency-stable resonators, making the
low values shown in Figure 9 practically unachievable. It may therefore be more practical to em-
ploy a detector with moderately lower stress σ, which offers slightly reduced frequency stability
but benefits from a higher RP.
22
FIG. 10. Heatmaps of NEP and specific detectivity D∗for bare SiN detectors. Panels a) & c) show NEP and
D∗for drumheads as functions of thickness h and stress σ. Panels b)& d) display the same for trampolines,
with fixed, optimised values of tether length Lt and width w. Colorbars for each figure of merit share the
same scale across the drumhead and trampoline figures to allow direct comparison.
V.
DISCUSSION
In this section, we consolidate the results from Sections III and IV to guide the reader through
the various parameters investigated. We begin by focusing on NEP and D∗.
23
Figure 10 presents the results for bare SiN detectors. For drumheads, we show the figures of
merit corresponding to the detection area side lengths L that yielded the best performance: based
on Figures 5b and 7b for NEP, and Figures 5c and 7c for D∗. The same selection criteria are applied
to the trampolines, including the choice of tether length Lt and width w when varying thickness h
and prestress σ, and vice versa. Figure 10a shows the NEP for drumheads with L = 0.25mm. In
line with the trends in Figures 5b and 7b, we observe a minimum NEP faround h = 100nm and
for the highest stress value, σ = 1GPa. The corresponding specific detectivity is shown in Figure
10c, and follows a similar trend. For this case, however, we used a structure with L = 4mm, as
suggested by the results in Figures 5c and 7c.
The NEP of trampolines for varying h and σ is presented in Figure 10b. Based on Figures 5b
and 7b, we selected L = 0.062mm, and from Figures 2b the corresponding values of LT = 700µm.
The tether width is w = 300nm. The best performance is achieved for σ = 1GPa and h ≃200nm.
Specific detectivity as a function of thickness and prestress is shown in Figure 10d for a trampoline
with L = 0.25mm, LT = 1000µm, and w = 5µm. It is important to note that, to enable direct
comparison between drumhead and trampoline performance, the colorbars for NEP are set to the
same range in Figures 10a and 10c. Comparing these plots, it is clear that trampolines generally
exhibit better noise equivalent power than drumheads. Likewise, the colorbars for D∗are matched
across Figures 10b and 10d, where drumheads now clearly show superior performance.
Next, we conduct the same analysis for the case of an ideal absorber, which gives the detectors
a wavelength-independent emissivity and absorptance: εabs = αabs = 0.5. All results are shown in
Figure 11. By comparing each plot here with its counterpart in Figure 10, two main differences
emerge. First, the overall performance of the detectors, both drumheads and trampolines, tends to
degrade with the use of the absorber: the highest NEP values are higher, and the peak D∗values
are lower compared to the bare SiN case. This is due to the less favourable ratio between εabs and
αabs in the presence of the absorber. However, the use of an absorber remains advantageous when
considering that these high absorptance values and the correspondingly good performance in both
NEP and D∗are maintained across a broad range of wavelengths. This makes the detector suitable
for spectroscopy applications and significantly broadens its potential use cases. In contrast, the
exceptional performance of bare SiN is confined to the narrow spectral region around its absorption
peak.
The second key difference is that, for trampolines, the optimal performance for both figures of
merit is now achieved with smaller and thinner structures. For example, while the best specific
24
FIG. 11. Heatmaps of NEP and specific detectivity D∗for detectors coated with an ideal broadband absorber
(αabs = εabs = 0.5). Panels a) & c) show drumhead performance as functions of h and σ. Panels b) & d)
display the same for trampolines with optimised Lt and w, as indicated in each plot title.
detectivity in Figure 10d was observed for L = 0.25mm, the most favorable configuration in the
absorber case is L = 0.062mm. Likewise, the optimal values for tether length, thickness, and
tether width shift to LT = 300µm, h = 10nm, and w = 2µm, respectively. This shift can again be
attributed to the much higher emissivity of the detector: the enhanced radiative heat transfer moves
the sweet spot, the equilibrium between heat conduction and radiation, toward smaller geometries.
25
FIG. 12. Pareto font plots for a) bare SiN drumheads; b) bare SiN trampolines; c) drumheads with absorber
and d) trampolines with absorber. In each plot, the dotted, vertical line represents the room temperature
detection limit (two-sides coupled). In red we highlighted the absolute maxima, while in yellow local
maxima for τth < 1ms (except for a), where the local maximum is for τth < 2ms. The combination of
variables of both absolute and local maxima is reported in the corresponding annotation box. References
indicate relevant state-of-the-art results, placed at the positions in the plot matching their reported values.
26
Finally, we want to display the variable combinations that give the best performance in terms
of sensitivity and speed. We display this in Figure 12. Sometimes it is desirable to sacrifice
sensitivity for a faster detector response. Other times we are interested in being able to detect the
smallest possible amount of light, without caring about the speed. Therefore, for each type of
detector investigated in this work (drumheads or trampolines, bare SiN or with the absorber), we
calculated the τth and D∗for 106 possible variable combinations.
Figure 12a shows the results for a selection of the variable combinations for bare SiN drum-
heads. We can readily observe that here we obtain the absolute highest value of D∗, highlighted
in red in the plot, surpassing the fundamental room temperature limit of an ideal black body, D∗
bb,
marked as the vertical dotted black line. The particular combination of variables for which a SiN
drumhead can reach this performance is annotated in the relative text box. This structure has a rel-
atively slow thermal time constant, being τth > 100ms. We therefore also highlighted in yellow a
local maximum for τth < 3ms. This detector is two orders of magnitude faster than the other high-
lighted one, while still having a high detectivity D∗> 1 × 109 cm
√
Hz/W. In general, out of the
million combination of variables for which we calculated τth and D∗we displayed only those who
showed what we considered an attractive value of specific detectivity: D∗≥1×109 cm
√
Hz/W for
drumheads, and D∗≥1 × 108 cm
√
Hz/W for trampolines. In Figure 12c, the equivalent plot for
drumheads covered with an absorber is displayed. Here again, we highlight the best-performing
structure with a red dot. Despite it being notably lower than the maximum for bare SiN drumheads,
we hereby notice that this plot’s front decreases less in D∗as the structures become faster, with re-
spect to the bare SiN drumhead case. In Figure 12c the local maximum (yellow dot) for τth < 1ms
has D∗≈5.5×109 cm
√
Hz/W, which is more than five times higher than the corresponding point
in figure 12a. In this plot, the vertical dotted line marks the room temperature sensitivity limit for
a detector with an absorber, D∗
50%.
The Pareto front for bare SiN trampolines is shown in Figure 12b. Similarly to SiN drumheads,
the highest value of D∗is obtained for a structure (red dot) that is slow, τth > 100ms. In order
to obtain a detector with τth < 1ms (yellow dot) we have to sacrifice the specific detectivity and
accept having D∗< 1×109 cm
√
Hz/W. Finally, in Figure 12d, we show the results for trampolines
with an absorber. Here too, the absolute maximum (red dot) is obtained for a structure that has a
high thermal time constant. In this case, though, like for the drumheads with the absorber, the drop
in detectivity for a detector with τth < 1mm is sensibly lower than for the trampolines without the
absorber.
27
In figures 12b, c,& d, we reported the references to the most relevant state-of-the art results. In
the case of the absorber, we indicated whether the result was obtained through the use of a metal
absorber or if it refers to a metamaterial.
Of particular relevance is the recent work by Zhang et al.11, who demonstrated that the bandwidth
limitation imposed by the thermal time constant can be overcome when operating the resonator in
the temperature fluctuation-limited regime. Using SiN trampoline resonators, the authors achieved
a specific detectivity only a factor of three lower than that corresponding to standard speed oper-
ation, while extending the measurement bandwidth by a factor of thirty. This is illustrated in
Figure 12b, where the two references are connected by a black arrow, representing the standard
and enhanced bandwidth measurements.
VI.
CONCLUSIONS
In conclusion, we have shown that the current state-of-the-art in thermomechanical IR detectors
has not yet reached its full potential. We have highlighted the architectural pathway to achieving
the fundamental room-temperature detection limit and identified several aspects where researchers
can focus to improve existing designs. For both bare SiN and devices with a broadband absorber,
drumhead resonators outperform their trampoline counterparts, exhibiting higher specific detectiv-
ity and shorter thermal time constants. Beyond clarifying the basic differences between drumhead
and trampoline resonators, we have underscored the fundamental trade-off between achieving peak
performance over a narrow range of detectable wavelengths and sacrificing a small portion of de-
tectable power in favour of a significantly broader operational bandwidth. Crucially, this work
bridges a long-standing gap in the literature by combining analytical modelling with practical
design insights, laying the groundwork for the development of detectors that approach the fun-
damental limits of thermal sensitivity. We anticipate that this guide will accelerate innovation in
nanomechanical sensing and foster the deployment of next-generation IR detection technologies
in a broad range of scientific and technological domains.
VII.
SUPPLEMENTARY MATERIAL
In the Supplementary Material, we provide additional details and figures on several aspects of
this work, including the measurement of the Duffing parameter for trampolines, the influence of
28
tether width on the figures of merit, the calculation of absorptance and emissivity for different SiN
thicknesses, the noise affecting the investigated resonators, the dependence of power responsivity
on various parameters, and additional results for τth and RP in the case of a detector with a metal
absorber.
VIII.
ACKNOWLEDGMENTS
The authors would like to thank Dr. Hajrudin Besic, Dr. Robert G. West, and Jelena Timarac
Popovic for the many fruitful discussions and valuable support during the writing of this work.
This work received funding from the Novo Nordisk Foundation under the project MASMONADE
with project number NNF22OC0077964.
IX.
AUTHORS’ DECLARATIONS
A.
Conflict of Interest
The authors have no conflicts to disclose.
B.
Author Contributions
P.M. contributed with measurements, production of the results and figures, and the overall
structure and writing of the paper; K.K. contributed with measurements, data analysis and assisted
with the writing; S.S. conceived the idea of the paper, discussed the results, supervised the work
and co-wrote the paper.
C.
Data availability
All the data that supports the findings of this study are available from the corresponding author
upon reasonable request.
29
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Design optimization of silicon nitride nanomechanical resonators for thermal infrared detectors: a guide through key figures of merit Paolo Martini,1 Kostas Kanellopulos,1 and Silvan Schmid1, a) 27-29, Vienna, 1040, Austria (Dated: 17 October 2025) Thermomechanical infrared (IR) detectors have emerged as promising alternatives to traditional photon and thermoelectric sensors, offering broadband sensitivity and low noise without the need for cryogenic cooling. Despite recent advances, the field still lacks a unified framework to guide the design of these nanomechanical systems. This work addresses that gap by providing a comprehensive design guide for IR thermal detectors based on silicon nitride drumhead and trampolines. Leveraging a validated analytical model, we systematically explore how geometry, tensile stress, and optical properties influence key performance metrics such as thermal time constant, noise-equivalent power, and specific detectivity. The analysis encompasses both bare silicon nitride and structures with broadband absorber layers, revealing how different parameter regimes affect the trade-off between sensitivity and response speed. Rather than focusing on a single device architecture, this study maps out a broad design space, enabling performance prediction and optimisation for a variety of application requirements. As such, it serves not only as a reference for benchmarking existing devices but also as a practical tool for engineering next-generation IR sensors that can operate close to the fundamental detection limit. This work is intended as a foundational resource for researchers and designers aiming to tailor IR detectors to specific use cases. a)Electronic mail: 1 16 Oct 2025 I. INTRODUCTION Infrared (IR) detection technologies fall into two categories: photon detectors and thermal detectors. While photon detectors such as HgCdTe photodetectors offer high sensitivity, their operation is fundamentally limited by the photon energy, making them suitable primarily for narrow band applications in the short-wave and lower mid-infrared (MIR) spectral regions1. Moreover, they typically require cryogenic cooling, increasing system complexity and cost2. As an alternative, thermal IR detectors, including bolometers, pyroelectric detectors, and thermopiles, operate well at room temperature and provide sensitivity across a wide spectral range. However, these thermoelectric detectors are fundamentally constrained by electrical noise sources, such as Johnson noise and 1/f noise, which ultimately limit their sensitivity. Thermomechanical detectors, on the other hand, offer a compelling route to circumvent these limitations by exploiting the effect of absorbed thermal energy on the mechanical properties of a resonator, such as changes in stiffness, which are then read out through its mechanical response rather than electrical one, thereby reducing electrical noise. Since their first introduction3, uncooled thermomechanical detectors, particularly nanomechanical systems, have emerged as promising alternatives by transducing temperature changes into measurable mechanical resonance shifts4-11. Silicon nitride (SiN) is currently one of the materials of choice for nanomechanical sensing, owing to its exceptional mechanical stability and thermomechanical properties, such as a high thermal expansion coefficient. In addition, in the specific context of IR detection, it exhibits favourable optical properties. Its absorption spectrum at room temperature features a pronounced peak coinciding with the characteristic emission of an ideal black body, effectively functioning as a metamaterial absorber. At the same time, it possesses a broad transparency window spanning from the visible to the mid-infrared, making it well suited for IR spectroscopy applications, and a high refractive index that can be tuned through stoichiometry control12. A SiN-based detector can be easily rendered broadband by depositing an absorber, sacrificing only a small fraction of its sensitivity while substantially extending its detectable wavelength range. Recent advances in nanofabrication, including the development of low-stress, uniform SiN films and precise submicron patterning techniques, have enabled the realisation of ultra-thin, low-loss drumheads and more complex resonator architectures, extending the detection limits of these devices. Among the available thermomechanical resonator architectures, drumhead and trampoline geometries are particularly advantageous, as their increased active areas enhance coupling to long2 wavelength radiation while preserving key performance metrics13. State-of-the-art SiN drumheads have achieved noise-equivalent power (NEP) on the order of pW/ √ Hz in both narrowband9 and broadband14 applications, approaching the fundamental room-temperature detectivity limit of D∗ lim =1.8 × 1010 cm √ Hz/W for a resonator thermally coupled to the environment from only one side15-17. Recently, Zhang et al9 reported an NEP ≈36pW/ √ Hz and a detectivity o f D∗≈3.4 × 109 cm √ Hz/W using a metamaterial drumhead optimised for the range 0.3 THz to 5 THz. The structure has a thermal response time of ≈200ms. Using a free-space impedance matched (FSIM) metal absorber for broadband IR light detection14 has lead to similar results of NEP = 27pW/ √ Hz and D∗= 3.8×109 cm √ Hz/W. The metal layer helped reduce the thermal constant to τth = 14ms. Trampoline resonators have shown excellent performance, too. Piller et al.7 reached a NEP as low as 7 pW/ √ Hz using SiN trampoline resonators featuring a similar absorber as in14, with a response time of only 4 ms. More recently, Das et al.10 proposed a multi-material trampoline structure incorporating a perforated absorber optimised for long-wave infrared detection, reaching a specific detectivity of D∗= 3.8×109 cm √ Hz/W with 7.4 ms thermal response. Another work6 made use of SiN trampoline resonators with a Cu and Au layer as an absorber and reported an NEP ≈100pW/ √ Hz in sub-THz (0.14 THz) detection, and a τth = 4ms. In a very recent work11, bare SiN trampolines achieved a record D∗= 6.4×109 cm √ Hz/W with τth = 88ms. Operating at the noise level set by the temperature-fluctuation limit, the authors were able to maintain a detectivity only a factor of three lower (D∗= 2.5×109 cm √ Hz/W) while achieving a faster response time (τ = 3ms) than that dictated by the resonator's thermal time constant. Despite these advancements, a unified framework for optimising detector parameters remains absent in the field. This work addresses this gap by presenting a comprehensive analysis based on a validated analytical model13,18,19, and by systematically exploring the resonator parameter space. Focusing on SiN drumheads and trampolines, we investigate how variations in geometry, intrinsic stress, and absorbivity influence the thermal and mechanical performance of the devices, The resulting trade-offs are evaluated using key figures of merit: thermal time constant (τth), noise equivalent power (NEP), and specific detectivity (D∗). Our model reconciles recent experimental results7,13,14. The framework enables performance prediction across design geometries, providing practical guidelines for developing applicationspecific IR detectors. In particular, we examine whether, in light of the current state-of-the-art, existing devices are already operating near their fundamental limits or whether opportunities for 3 FIG. 1. Schematic representation (not in scale) of the two designs under investigation in this work. a) square drumhead with the geometric variables studied highlighted: h is the thickness of the SiN layer, hence of the structure, and the lateral size L. In addition to these two, for a b) trampoline we study also the influence of the tethers' width, w, and length, Lt. notable performance gains remain. The framework presented here enables this assessment quantitatively, highlighting potential pathways to approach the fundamental photon noise limit. II. FIGURES OF MERIT Before reviewing the metrics used to evaluate the detectors' performance, we introduce the two designs considered in this work. Regardless of the specific fabrication process employed, we assume the detector chip to consist of a silicon (Si) substrate several hundred μm thick. This substrate serves both as a mechanical support for handling the resonator and as a thermal reservoir where heat generated by absorbed radiation can be effectively dissipated. A layer of silicon nitride (SiN), deposited on top of the silicon substrate with low-pressure chemical vapour deposition (LPCVD), forms the material of the detecting element. Figure 1 presents a schematic overview of the two geometries. In Figure 1a, a square drumhead is illustrated, defined by two primary geometrical parameters: its thickness h and lateral dimension L. Compared to this basic configuration, the trampoline design introduces additional geometrical complexity, as shown in Figure 1b. Here, the variable L denotes the side length of the central suspended area, which serves as the active detection region, analogous to the drumhead's side length. This central area is connected to the supporting frame via four tethers, each characterised by a length LT and a width w. These four parameters (h, L, LT, and w) fully define the geometries of the drumhead and trampoline 4 structures, whose impact on detector performance is systematically explored in Section III. A. Thermal time constant, τth The first figure of merit is the thermal time constant, τth [s]. This metric is crucial in many applications, particularly when fast detector response is required, e.g.: in time-resolved spectroscopy applications, where scanning frequencies can reach the order of 2 kHz. While in most cases this remains the fundamental limit of the detection chain, a recent work11 showed that this limitation can be broken when the resonator is operated at its fundamental temperature fluctuation noise limit. For a lumped element system, τth is defined as the ratio between the heat capacity C [J/K] of the resonator and its thermal conductance G [W/K]1: τth = C G. (1) For a drumhead made of a material with mass density ρ and specific heat capacity cp, in the context of the mean temperature framework (MTF), the heat capacity is given by13: C = ρcphL2. (2) Likewise, the heat capacity of a trampoline structure is13: C = ρcph(L2 +4wLt). (3) By comparing (2) and (3), it is evident that for the same active detection area L2, trampolines exhibit a larger heat capacity due to the additional volume contributed by the tethers. In this study, we consider detectors operating in vacuum conditions, where heat transfer via convection can be neglected. In this regime, the thermal conductance is the sum of the conductive and radiative contributions only: G = Gcond +Grad. For a drumhead, this takes the form1: G = 4πhκ +4L2εσSBT 3 0 , (4) where κ is the thermal conductivity of the material, ε is the emissivity, σSB is the Stefan-Boltzmann constant, and T0 is the temperature of the environment including the detector. The first term represents heat conduction through the drumhead, while the second accounts for radiative losses from 5 the surface. In a trampoline structure, the conductive heat transfer term arises from the series combination of the thermal resistances of a membrane (Rm) and that of four strings (Rs) of length LT and width w: Rm +Rs = 1 Gm cond + 1 Gs cond = 1 Gcond , (5) where Gm cond = 4πhκ and Gs cond = 8wh LT κ represent the terms for thermal conductance through conduction of a membrane and four strings, respectively18. The radiative contribution, on the other hand, accounts for the total emitting surface of both the membrane and the tethers. The overall thermal conductance of the trampoline can thus be expressed as: G = 8πhw 2w+πLT κ +4(8wLT +2L2)εσSBT 3 0 . (6) The contribution of the tethers to both conduction and radiation is evident from the presence of w and LT in both terms of the sum. B. Power responsivity, RP When a light source with power P0 impinges on the detector surface, a fraction of that power, denoted P, is absorbed by the material and converted into heat: P = αP0, (7) where α is the absorptance, which depends on both the material properties and the wavelength of the incident light. The absorbed power raises the temperature of the detector, inducing thermal expansion of the material. This phenomenon, known as induced stress change, underpins the working principle of thermomechanical IR detectors. As the material expands, the mechanical stiffness of the structure changes, which in turn shifts the resonance frequency ω0 of the detector. The second figure of merit considered in this work is the power responsivity, RP, with unit [1/W], defined as the relative change in resonance frequency per unit of absorbed power18: RP(ω) = ∂ω0 ∂P 1 ω0 Hth(ω), (8) where Hth(ω) is a low-pass filter transfer function that accounts for the finite thermal response time of the resonator18: Hth(ω) = s 1 1+(ωτth)2. (9) 6 Since thermomechanical detectors operate by converting absorbed power into heat, effectively functioning as temperature sensors, (8) can be recast in the form13,18: RP(ω) = ∂ω0 ∂T 1 ω0 ∂T ∂PHth(ω) = RT G Hth(ω), (10) where T is the temperature, and RT is the temperature responsivity. For square drumheads, such as those considered in this study, it takes the form18: RT = - αth 2(1-ν) E σ , (11) where αth is the thermal expansion coefficient, ν the Poisson's ratio, E is the Young's modulus, and σ the tensile prestress in the material. The negative sign in (11) reflects the fact that thermal expansion causes a softening of the structure, resulting in a decrease in resonance frequency with increasing temperature. In the case of trampolines, the temperature responsivity becomes18: RT = -αth 2 E σ . (12) Here, the factor (1-ν), which accounts for the in-plane expansion in two directions, is absent due to the uniaxial strain of the tethers. The ratio E/σ, common to both cases, is a characteristic feature of tensile mechanical resonators and represents an enhancement factor of the temperature responsivity. Finally, while RT depends solely on material properties, the detector geometry plays a key role in determining the power responsivity RP, as evidenced by the presence of the thermal conductance G in (10). C. Noise equivalent power (NEP) To properly assess the performance of a detector, the noise level of the measurement system must also be taken into account. The next figure of merit is the noise equivalent power (NEP), defined as the signal power required to achieve a unitary signal-to-noise ratio within a one Hz bandwidth. It has units of [W/ √ Hz], and for a nanomechanical resonator used as an IR detector, it can be expressed as15,18: NEP = p Sy(ω) RP(ω) . (13) Here, Sy(ω) is the fractional frequency noise power spectral density, which quantifies the system's noise by evaluating the frequency stability of the resonator in thermal equilibrium. From (13), it 7 is clear that achieving high sensitivity requires minimizing frequency noise to resolve the smallest possible shifts in resonance frequency. In this work, we consider three primary sources of frequency noise: thermomechanical noise Sythm(ω), detection noise Syd(ω), and temperature fluctuation noise Syth(ω). It is also worth mentioning that, among many other possible noise sources, one particularly relevant stems from photothermal back action caused by the relative intensity noise of a readout laser. In optomechanical sensing applications involving an IR absorber, this noise can significantly limit sensitivity20. To mitigate this effect, the absorber design can be optimized, as demonstrated in14. Assuming the detection noise is white, its contribution can be related to the thermomechanical noise peak magnitude as follows18: Szd(ω) = K 2Szthm(ω0) = K 2 4kBTQ me f f ω3 0 , (14) with the effective mass of the resonator me f f , the quality factor Q, the resonance angular frequency ω0, and the Boltzmann constant kB. The coefficient K represents the relative sensitivity of the transduction system: if it can resolve the thermomechanical noise peak, then K z0c), albeit only for short integration times, which is a regime that typically is difficult to access with slow thermoelastic detectors. The effective mass mef f , which appears in both (14) and (17), varies according to the geometry of the device. For the fundamental mode in drumheads, it is given by18: me f f = 1 4m0 = 1 4ρhL2, (18) while for trampolines, still in the fundamental mode, it becomes13: mef f = ρh L2 + 4wLt 2 . (19) Similarly, the Duffing coefficient in (17) takes different forms. For a square drumhead resonator, it is25: αDuf f = 3π4 64 (n4 + j4)Eh L2 1 2(1-ν2)(1-ν), (20) with n and j the modal numbers of the vibrational mode. For trampolines, the expression derived by19 is: αDu f f = n4π4 8 Ewh (2Lt)3. (21) Since the latter equation has not yet been validated before, we performed measurements using SiN trampoline resonators and verified its accuracy. The results are provided in the Supplementary Material (SM). The ultimate sensitivity limit for a nanomechanical thermal detector is set by temperature fluctuation noise Syth(ω)18,26. This noise arises from stochastic heat exchange between the detector and the environment and generates white frequency noise. For lumped-element thermal detectors it is given by3,11,13: Syth(ω) = 4kBT 2R2 T G |Hth(ω)|2 |HL(ω)|2 . (22) 9 Here, G is the effective thermal conductance of the resonator. It has been shown that G for drumheads and trampolines is well approximated by (4) and (6), respectively13. D. Specific detectivity (D∗) The sensitivity (13) of nanomechanical membrane and trampoline resonators generally improves with increasing lateral size (L) as long as conductive heat transfer dominates. For larger L, radiative heat transfer becomes significant. The extra thermal heat loss channel leads to a deterioration in sensitivity. In the radiative heat transfer regime, when the frequency stability is limited by thermomechanical noise (15), the sensitivity follows the scaling law NEP ∝ √ L3. (23) In the ultimate regime, where frequency stability is determined by temperature fluctuation noise (22), the scaling law changes to NEP ∝L, (24) as derived below. This latter dependence matches the well-known scaling for photodetectors, whose sensitivity scales with the square root of the detection area, √ A27. A commonly used figure of merit in such cases is the specific detectivity, D∗, with units [cm √ Hz/W], which normalizes sensitivity for the detector size. Hence, it is particularly useful for comparing sensors with different detection areas. It is defined as15,28: D∗= √ A NEP. (25) It is important to note that, in the case of trampoline resonators, the detection area does not include the tethers. Therefore, for this design, A = L2 just as for drumheads. According to Kirchhoff's law, for a resonator in thermodynamic equilibrium with its surroundings, the emissivity ε equals the absorptance α. Considering the case where temperature fluctuation noise dominates, the equation (13) can be expressed in terms of ε18: NEP = p Syth(ω) εRP(ω) = p 4kBT 2G ε |HL(ω)|2 . (26) It is noteworthy that the thermal low-pass transfer function Hth(ω), which normally shapes both the frequency noise PSD (22) and the responsivity (8), no longer appears in (26). As a consequence, the thermal response time ceases to play a role, and the measurement speed is essentially 10 limited only by the time constant of the oscillator circuit HL(ω), provided the resonator operates in the temperature fluctuation noise regime. This effect has recently been exploited to enhance measurement speed in nanomechanical trampoline resonators11. Assuming further that the resonator is well thermally isolated so that heat transfer occurs predominantly via radiation, (26) yields a relation that depends on the detector area √ A18: NEP = s 16σBkBT 5 0 A ε , (27) which becomes size independent when written as specific detectivity18: D∗= s ε 16σBkBT 5 0 . (28) For an ideal infrared detector (a blackbody with absorptance αbb = εbb = 1) thermally coupled to its environment on only one side, this reduces to the well-known theoretical sensitivity limit for room-temperature detectors,(T0 = 290 K and ε = 1) D∗ lim = 2.0×1010 cm √ Hz/W. In the present work, we consider a resonator coupled on both sides, which slightly lowers this limit to D∗ bb ≈ 1.4×1010 cm √ Hz/W. Furthermore, we consider a broadband absorber with a 50% absorptance, for which the limit reduces to (T0 = 290 K and ε = 0.5) D∗ 50% = 1.0×1010 cm √ Hz/W. A high, uniform, and broadband absorption is advantageous for many applications and, in particular, for detectors for IR spectroscopy. Conversely, a narrowband absorption can be attractive for applications that require spectral selectivity for gas sensing or for IR imaging in the longwavelength IR range 9μm to 10μm29-31. SiN offers a high absorption, with a peak absorption between λSiN = 10μm and λSiN = 12μm in combination with an otherwise low emissivity εSiN. This combination renders it more responsive in narrow-band applications when compared to coated detectors. For this reason, first, we apply the analytical model to study pure SiN nanomechanical resonators without a dedicated absorber. For sections III A, III B and IV, where we assume a 50 nm thick layer, we calculate the absorptance to be αSiN(λSiN) = 0.255 ± 0.007 at the wavelength λSiN = 12.02μm, where this thickness exhibits an absorption peak. Similarly, we determine an emissivity value of εSiN = 0.056±0.002. Details on the derivation and the procedure used to obtain the SiN absorption spectrum at this thickness are provided in the SM. In Section III C, where the thickness is varied, different values of αSiN(λSiN), λSiN and εSiN are assumed accordingly (see Figure 4). In each scenario presented in the sections III A, III B, III C and IV, we also explore the case of an IR absorber layer with 11 ρ [kg/m3] E [Pa] ν αth [K-1] κ [W/(mK)] cp [J/(kgK)] 3000 250×109 0.27 1.5×10-6 3 700 TABLE I. Silicon nitride parameters used. From left to right: mass density, Young's modulus, Poisson ratio, thermal expansion coefficient, thermal conductivity, and specific heat capacity. an absorptance αabs = 50% deposited onto the SiN detector, allowing for broader absorption. Throughout the analysis, a fundamental trade-off becomes evident: a higher specific detectivity confined to a narrow spectral band, versus a lower detectivity extended over a broad wavelength range, each corresponding to different application domains. III. GEOMETRY (L, Lt, h, AND w) Geometry is typically the first aspect to consider when designing a nanomechanical IR detector, and it is the focus of this section. The analysis begins with the parameters specific to trampoline designs, namely Lt and w, followed by the thickness h, which is shared by both drumheads and trampolines. For all cases, we present results across five different values of the detection area side length L. The impact of each geometrical variable is evaluated with respect to the thermal time constant (τth), noise equivalent power (NEP), and specific detectivity (D∗). Table I lists the SiN parameters considered for this work7,32-34. A. Tethers length, Lt First, we examine how the tether length, Lt, influences the performance of trampolines with different detection areas. The results of this analysis for bare SiN are shown in Figure 2. In this section, the tethers width is fixed at w = 5μm. For comparison, in all panels of Figure 2, we display the values corresponding to drumheads of the same thickness and stress, as horizontal solid lines because they are independent of Lt. For all the designs, the thickness and tensile stress are set to h = 50nm and σ = 200MPa, respectively. The effect of Lt on the thermal conductance G is twofold, as shown in Eq. (6): on the one hand, longer tethers reduce heat transfer via conduction; and on the other hand, increasing Lt enlarges the total radiating surface, enhancing heat loss through radiation. Additionally, the total heat capacitance C increases linearly with Lt, as indicated in Eq. (3). The combined effect of these 12 FIG. 2. Influence of the tether length Lt on the figures of merit: a) thermal time constant τth, b) noise equivalent power NEP, and c) specific detectivity D∗at λSiN = 11.86μm for plain SiN resonators without a dedicated IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit. For all panels, the stress is fixed at σ = 200MPa, the thickness h = 50nm, and the tether width at w = 5μm. contributions is illustrated in Figure 2a, which shows τth as a function of Lt. For small Lt, heat from the center of the detection area is primarily dissipated through conduction along the tethers (high Gcond). As Lt increases, the contribution of Gcond diminishes, and the radiative contribution (Grad) becomes significant, coupling the resonator more strongly to the environment via radiation. As a result, τth increases until asymptotically reaching the radiative heat transfer plateau. The NEP, shown in Figure 2b, exhibits a clear dependence on the tether length Lt. The most sensitive detector corresponds to the smallest trampoline geometry, with a minimum NEP occurring around Lt ≃700μm. Finally, the specific detectivity as a function of Lt is shown in Figure 2c. In this case, the trampoline with detection area side length L = 0.25mm exhibits the highest D∗. However, even at the peak of its curve, corresponding to a tether length of Lt ≃1mm, its D∗is comparable to that of the drumhead with side length L = 1mm. For this figure of merit, the overall best-performing devices are drumheads with L = 4mm and 16mm, both reaching the room-temperature detectivity limit D∗ bb. This result underscores the crucial role of the detection area (A in equation (25)): although smaller resonators exhibit lower NEP, this advantage is outweighed by the larger IR 13 FIG. 3. Influence of the tether length Lt on the figures of merit: a) thermal time constant τth, b) noise equivalent power NEP, and c) specific detectivity D∗for a SiN resonator with an IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit for αabs = 50%. For all panels, the stress is fixed at σ = 200MPa, the thickness h = 50nm, and the tether width at w = 5μm. collection area of bigger structures. So far, we have evaluated the performance of bare SiN resonators at the fixed wavelength λSiN = 12.02μm, corresponding to the measured peak in absorptance. All previous results are therefore specific to this wavelength. However, in the field of infrared detection, it is often desirable to develop broadband detectors. One promising approach is the deposition of a metal layer on a dielectric, which can achieve absorptance values up to αabs = 50% across a broad spectral range35. Recent studies have demonstrated such high and nearly flat absorption spectra using platinum or gold thin films deposited on SiN7,14,36. As shown in those works, the addition of a thin metal layer does not significantly affect the mechanical properties of the resonator, although it does influences its thermal and optical behaviour, and provides an almost flat and wavelength independent absorption spectrum, over the infrared range, up to the THz region. Now we investigate the impact of adding an absorber with αabs = εabs = 50% on the SiN resonator, under the assumption that the mechanical properties of SiN remain largely unchanged. The results of this analysis for varying tethers length Lt are shown in Figure 3. As observed for bare SiN (Figure 2b), the best NEP values are obtained for the smallest trampoline (Figure 3b). 14 FIG. 4. Effect of the thickness h over SiN absorptance at its absorption peak λSiN = 11.86μm (blue line) and the hemispherical emissivity (orange line). Both grow as the SiN gets thicker. αSiN(λSiN) has a maximum for h ≂250nm and later shows a local minimum for h ≂700nm. The green line is the ratio α(λSiN)/√εSiN, and it has a maximum for h = 151nm. However, in this case, the lowest NEP occurs around Lt ≃250μm. In general, smaller structures outperform larger ones, both for trampolines and drumheads. Overall, we observe that adding an absorbing layer lowers τth (Figure 3a) and shifts the optimal designs toward smaller resonators (Figure 3c) compared to the bare SiN scenario. Although the absolute values of the figures of merit are slightly lower than in the bare SiN case, it is important to emphasize that, with the absorber, these performance levels are maintained across the entire MIR spectral range, whereas previously they were limited to the specific peak wavelength of SiN. B. Tethers width, w The resulting behaviour of NEP and D∗as a function of w closely mirrors that observed for Lt. To avoid redundancy, we omit a detailed discussion here; the full analysis is provided in the SI, both for the case of bare SiN and with the metal absorber. 15 C. Thickness, h In nanomechanical resonators, the thickness can range from a few to several hundred nm, and in some cases, up to a few μm. In this work, we consider the range h = 10nm to 1μm. In the thin-film regime, variations in thickness significantly affect the film's optical properties, such as absorptance α(λ), intended as the fraction of the absorbed light at one specific wavelength, and emissivity, or hemispherical emissivity ε, which is the efficiency with which the film radiates energy compared to that of a blackbody at the same temperature, integrated over all emission angles and wavelengths37,38. For each value of h, we computed αSiN(λSiN) and εSiN, taking into account possible interference effects with the thin film32. The result is shown in Figure 4, where the blue line represents the SiN peak absorptance (αSiN(λSiN)), the orange line the hemispherical emissivity (εSiN), and the green line the ratio α(λSiN)/√εSiN. This latter quantity is a strong indicator of the SiN performance: on one hand, a thicker material increases the absorbed power, thereby enhancing the power responsivity (equation (8)); on the other hand, a lower emissivity reduces the radiative coupling to the environment, which is also desirable. Therefore, rather than solely maximizing the absorptance or minimizing the emissivity, the optimal strategy is to maximize the ratio. The term α(λ)/√ε can be seen as a specific detectivity enhancement factor typical of metamaterials, for which, in a very narrow spectral region, it holds true that α(λ) > ε. Figure 4 shows that for SiN the maximum of the ratio occurs at h = 151nm. For more details on how the absorptance and emissivity curves are obtained, the reader is referred to the SM. During the analysis of thickness dependence, all other parameters are kept constant: the stress is fixed at σ = 200MPa, and for the trampoline geometries, the tether length and width are set to Lt = 50μm and w = 5μm, respectively. According to equations (2) and (3), increasing h enhances both the heat capacity C and the thermal conductance G. For small drumheads (L = 0.062mm and 0.25mm), where conduction dominates, the h-dependence in C and G effectively cancels out, yielding a nearly constant τth (see figure 5a). In contrast, for larger drumheads (L = 4mm and 16mm) the radiative term becomes significant, causing τth to increase with h. A similar trend is observed for trampolines, where the tethers, by hindering conduction and augmenting the radiating surface, make the influence of h on τth evident even for L = 0.25mm. Thus, the presence of tethers enhances thermal isolation, rendering trampolines slower than drumheads for a given L. Figure 5b shows the results for the NEP. The most sensitive device is the trampoline with 16 FIG. 5. Influence of the thickness h on the figures of merit: a) thermal time constant τth, b) noise equivalent power (NEP), and c) specific detectivity D∗at λSiN = 11.86μm for plain SiN resonators without a dedicated IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit. For all panels, the stress is fixed at σ = 200MPa, the tether length at Lt = 50μm, and the tether width at w = 5μm. L = 0.062mm, with the lowest values obtained for h ≃100nm. The slight deviation in thickness from the value corresponding to the maximum of the green curve in Figure 4 arises from the non-negligible contribution of thermal conduction, which becomes increasingly significant as the resonator thickness grows. Finally, the model predictions for the specific detectivity D∗are presented in Figure 5c. In this case, the best performance is achieved by drumheads, with the two designs featuring the largest detection areas yielding the highest overall values. For these configurations, the predicted D∗values at thicknesses around 200nm surpasses the fundamental room-temperature limit, D∗ bb, indicated by the green dotted line. This somehow counterintuitive result is in line with what is shown in Figure 4, and demonstrates that bare SiN, when probed at its absorptance peak wavelength λSiN, acts similarly to a metamaterial, showing αSiN(λSiN) > εSiN. Overall, drumheads outperform trampolines of the same size, with the exception of the smallest drumhead, which is outperformed by the trampoline with L = 0.062mm. The presence of the metal layer which gives the detector a constant absorptance and emissivity of 50% can be noticed in the results shown in Figure 6. Because of this, the influence of h is now 17 FIG. 6. Influence of the thickness h on the figures of merit: a) thermal time constant τth, b) noise equivalent power (NEP), and c) specific detectivity D∗for SiN resonators with a dedicated IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit for αabs = 50%. For all panels, the stress is fixed at σ = 200MPa, the tether length at Lt = 50μm, and the tether width at w = 5μm. limited to heat transfer through conduction, Gcond, for both trampolines and drumheads, with no impact on Grad. In Figure 6a, we notice the thermal time constant significantly decreasing for every design and when compared to the bare SiN scenario (Figure 5a). In Figure 6b, we see the results for the NEP, with smaller detectors exhibiting superior performance, particularly at reduced thicknesses. As the lateral dimensions of the structures increase, the optimal NEP shifts toward larger thickness values. This trend arises because a larger L enhances radiative exchange, so the sweetspot between radiative losses and thermal conduction is reached at greater thicknesses. A similar trend is observed for D∗(Figure 6c), for both drumheads and trampolines. This behaviour reflects the fact that increasing detector area requires thicker substrates to achieve the optimal balance between Gcond and Grad. For all the geometrical variables, the results for the power responsivity RP are presented in the SM. 18 FIG. 7. Influence of the tensile stress σ on the figures of merit: a) power responsivity RP; b) noise equivalent power (NEP); c) specific detectivity D∗at λSiN = 11.86μm for plain SiN resonators without a dedicated IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit. For all panels, the thickness is fixed at h = 50nm, the tether length at Lt = 50μm, and the tether width at w = 5μm. IV. TENSILE STRESS, σ This section is dedicated to the analysis of the influence of tensile stress σ on the detectors' performance. Often referred to as prestress, it is to some extent a tunable parameter of mechanical resonators. Different deposition methods and parameters result in varying stress conditions in the deposited material. From Section II A, we know that σ has no influence on either the thermal conductance G or the heat capacity C (see equations (2) and (4) for drumheads, and equations (3) and (6) for trampolines); thus, it does not affect the thermal time constant of the device. In some cases, differences in deposition techniques result in variations in crystallinity (e.g., α-phase SiN vs. β-phase SiN vs. amorphous SiN). These differences typically entail changes in material properties, such as specific heat capacity and thermal conductivity, which would influence τth. While the authors are aware of this, the investigation of such effects falls outside the scope of the present study. Here, we consider the case of amorphous SiN deposited with LPCVD, for which we assume that prestress conditions do not affect the thermal properties of the material. Consequently, in presenting the model results for the various figures of merit, we exclude τth. Analysing the 19 effects of stress by comparing drumheads to trampolines is non-trivial. This is due to the wellknown influence of the trampoline geometry on the stress distribution: the stress is significantly increased in the tethers and reduced in the central area compared to the nominal stress σ0 of the deposited material, as shown through finite element (FEM) simulations in7. Nevertheless, the results presented in this section remain generally valid when considering the value of the stress σ in the specific region of the detector where the IR light impinges and is absorbed. For the following study on varying the stress in the range σ = 1MPa to 1000MPa, the thickness of all structures is fixed to h = 50nm, and for the trampolines, the tethers have width w = 5μm, and length Lt = 50μm. The dependency of RP on stress arises from the linear dependence of the temperature responsivity RT (equations (11) and (12)) on σ. For both designs, RT decreases linearly with increasing stress, and so does the power responsivity, as shown in Figure 7a. The differences in RP values between structures of different sizes stem from the expected variations in G: detectors with smaller detection areas generally perform better. The results show that drumheads with L = 4mm and 16mm exhibit slightly higher responsivity than trampolines with equivalent detection areas. However, for smaller L values, trampolines display higher RP compared to drumheads. This improvement is attributed to their enhanced thermal isolation of the centre of the detection area, enabled by the presence of tethers that reduce conductive heat loss. The situation is almost the opposite when it comes to the noise equivalent power. As shown in Figure 7b, the NEP generally improves with increasing values of σ, and drumheads tend to outperform trampolines, with the sole exception of the smallest trampoline, which exhibits the lowest NEP among all designs. Both the inverse trend of NEP and the relative performance of drumheads compared to trampolines can be explained by frequency stability. As detailed in the SI, the significantly lower frequency noise of drumheads relative to trampolines, along with the general improvement in frequency stability with increased stress, compensates for the differences in power responsivity observed in Figure 7a. Interestingly, while smaller detection areas still correspond to better performance in trampolines, for drumheads, the detectors with bigger areas are those that perform best across the entire σ range under investigation. The specific detectivity results are presented in Figure 7c. Again, we observe what is seen for all the other variables in section III: introducing the detection area A in the calculation highlights the importance of collecting as much light as possible from the source. It is now the large drumheads which show the highest D∗, with L = 4mm being the top performer for 1MPa ≤σ ≤10MPa and 20 FIG. 8. Influence of the tensile stress σ on the figures of merit: a) power responsivity RP; b) noise equivalent power (NEP); c) specific detectivity D∗for SiN resonators with a dedicated IR absorber. The dashed black line in c) marks the room-temperature sensitivity limit for αabs = 50%. For all panels, the thickness is fixed at h = 50nm, the tether length at Lt = 50μm, and the tether width at w = 5μm. matching the performance of L = 16mm for σ > 10MPa. For a given detection area side length L, drumheads always perform better than trampolines, with the case of L = 0.062μm being the only exception to this. However, it consistently holds that D∗increases with higher values of tensile stress σ, independently of the detector design. Very similar results are obtained with the inclusion of the metal absorber layer, as shown in Figure 8. The overall trends observed for bare SiN are preserved in this case as well. A slight increase in power responsivity is observed (Figure 8a), accompanied by a modest reduction in performance with respect to NEP and D∗(Figures 8b and 8c, respectively). The key takeaway from Figure 7 and 8 is that higher stress leads to improved detector performance. However, the practical implications of employing high-stress resonators for IR detection must also be considered. The enhanced NEP and D∗observed at higher stress levels originate from the overall improvement in frequency stability associated with such resonators. This is illustrated in Figure 9, where we plot the theoretical Allan deviations, accounting for both additive and temperature fluctuation noise, as a function of integration time τ, for drumhead and trampoline resonators of different sizes and two commonly used stress values: 200 MPa and 1 GPa. The anal21 FIG. 9. Theoretical ADs, obtained by adding together thermomechanical and detection noise, for a) drumheads and b) trampolines, for different σ. For this study, the thickness is set to h = 50nm while for trampolines the tether length is Lt = 50μm and tether width is w = 5μm. ysis assumes a closed-loop measurement with a phase-locked loop (PLL) bandwidth of 100 Hz, and a thermomechanical-to-detection noise ratio K = 0.01 for a typical optical interferometer that has a displacement sensitivity in the order of 1 pm/ √ Hz. In all cases, the high-stress resonators (solid lines) exhibit lower noise than their low-stress counterparts (dashed lines). Notably, for the most promising designs, namely, the smaller trampolines (purple lines in figure 9b), which exhibit the lowest NEP in Figure 7b, and the larger membranes (solid green line in figure 9a), which show the highest D∗in Figure 7c, the Allan deviation are below 10-7, and below 10-9, respectively. At such low levels, the detector performance is increasingly limited by other noise sources, such as absorbtion-desorption noise39,40, defect motion41, random walk frequency drift, and photothermal back-action noise13,14,42. Ultimately, the precision of such measurements is limited by the frequency stability of the reference oscillators in the readout electronics - typically oven-controlled crystal oscillators (OCXOs) - which offer stabilities on the order of 10-9. In practice, these additional noise sources will likely dominate in high-stress, frequency-stable resonators, making the low values shown in Figure 9 practically unachievable. It may therefore be more practical to employ a detector with moderately lower stress σ, which offers slightly reduced frequency stability but benefits from a higher RP. 22 FIG. 10. Heatmaps of NEP and specific detectivity D∗for bare SiN detectors. Panels a) & c) show NEP and D∗for drumheads as functions of thickness h and stress σ. Panels b)& d) display the same for trampolines, with fixed, optimised values of tether length Lt and width w. Colorbars for each figure of merit share the same scale across the drumhead and trampoline figures to allow direct comparison. V. DISCUSSION In this section, we consolidate the results from Sections III and IV to guide the reader through the various parameters investigated. We begin by focusing on NEP and D∗. 23 Figure 10 presents the results for bare SiN detectors. For drumheads, we show the figures of merit corresponding to the detection area side lengths L that yielded the best performance: based on Figures 5b and 7b for NEP, and Figures 5c and 7c for D∗. The same selection criteria are applied to the trampolines, including the choice of tether length Lt and width w when varying thickness h and prestress σ, and vice versa. Figure 10a shows the NEP for drumheads with L = 0.25mm. In line with the trends in Figures 5b and 7b, we observe a minimum NEP faround h = 100nm and for the highest stress value, σ = 1GPa. The corresponding specific detectivity is shown in Figure 10c, and follows a similar trend. For this case, however, we used a structure with L = 4mm, as suggested by the results in Figures 5c and 7c. The NEP of trampolines for varying h and σ is presented in Figure 10b. Based on Figures 5b and 7b, we selected L = 0.062mm, and from Figures 2b the corresponding values of LT = 700μm. The tether width is w = 300nm. The best performance is achieved for σ = 1GPa and h ≃200nm. Specific detectivity as a function of thickness and prestress is shown in Figure 10d for a trampoline with L = 0.25mm, LT = 1000μm, and w = 5μm. It is important to note that, to enable direct comparison between drumhead and trampoline performance, the colorbars for NEP are set to the same range in Figures 10a and 10c. Comparing these plots, it is clear that trampolines generally exhibit better noise equivalent power than drumheads. Likewise, the colorbars for D∗are matched across Figures 10b and 10d, where drumheads now clearly show superior performance. Next, we conduct the same analysis for the case of an ideal absorber, which gives the detectors a wavelength-independent emissivity and absorptance: εabs = αabs = 0.5. All results are shown in Figure 11. By comparing each plot here with its counterpart in Figure 10, two main differences emerge. First, the overall performance of the detectors, both drumheads and trampolines, tends to degrade with the use of the absorber: the highest NEP values are higher, and the peak D∗values are lower compared to the bare SiN case. This is due to the less favourable ratio between εabs and αabs in the presence of the absorber. However, the use of an absorber remains advantageous when considering that these high absorptance values and the correspondingly good performance in both NEP and D∗are maintained across a broad range of wavelengths. This makes the detector suitable for spectroscopy applications and significantly broadens its potential use cases. In contrast, the exceptional performance of bare SiN is confined to the narrow spectral region around its absorption peak. The second key difference is that, for trampolines, the optimal performance for both figures of merit is now achieved with smaller and thinner structures. For example, while the best specific 24 FIG. 11. Heatmaps of NEP and specific detectivity D∗for detectors coated with an ideal broadband absorber (αabs = εabs = 0.5). Panels a) & c) show drumhead performance as functions of h and σ. Panels b) & d) display the same for trampolines with optimised Lt and w, as indicated in each plot title. detectivity in Figure 10d was observed for L = 0.25mm, the most favorable configuration in the absorber case is L = 0.062mm. Likewise, the optimal values for tether length, thickness, and tether width shift to LT = 300μm, h = 10nm, and w = 2μm, respectively. This shift can again be attributed to the much higher emissivity of the detector: the enhanced radiative heat transfer moves the sweet spot, the equilibrium between heat conduction and radiation, toward smaller geometries. 25 FIG. 12. Pareto font plots for a) bare SiN drumheads; b) bare SiN trampolines; c) drumheads with absorber and d) trampolines with absorber. In each plot, the dotted, vertical line represents the room temperature detection limit (two-sides coupled). In red we highlighted the absolute maxima, while in yellow local maxima for τth 100ms. We therefore also highlighted in yellow a local maximum for τth 1 × 109 cm √ Hz/W. In general, out of the million combination of variables for which we calculated τth and D∗we displayed only those who showed what we considered an attractive value of specific detectivity: D∗≥1×109 cm √ Hz/W for drumheads, and D∗≥1 × 108 cm √ Hz/W for trampolines. In Figure 12c, the equivalent plot for drumheads covered with an absorber is displayed. Here again, we highlight the best-performing structure with a red dot. Despite it being notably lower than the maximum for bare SiN drumheads, we hereby notice that this plot's front decreases less in D∗as the structures become faster, with respect to the bare SiN drumhead case. In Figure 12c the local maximum (yellow dot) for τth 100ms. In order to obtain a detector with τth < 1ms (yellow dot) we have to sacrifice the specific detectivity and accept having D∗< 1×109 cm √ Hz/W. Finally, in Figure 12d, we show the results for trampolines with an absorber. Here too, the absolute maximum (red dot) is obtained for a structure that has a high thermal time constant. In this case, though, like for the drumheads with the absorber, the drop in detectivity for a detector with τth < 1mm is sensibly lower than for the trampolines without the absorber. 27 In figures 12b, c,& d, we reported the references to the most relevant state-of-the art results. In the case of the absorber, we indicated whether the result was obtained through the use of a metal absorber or if it refers to a metamaterial. Of particular relevance is the recent work by Zhang et al.11, who demonstrated that the bandwidth limitation imposed by the thermal time constant can be overcome when operating the resonator in the temperature fluctuation-limited regime. Using SiN trampoline resonators, the authors achieved a specific detectivity only a factor of three lower than that corresponding to standard speed operation, while extending the measurement bandwidth by a factor of thirty. This is illustrated in Figure 12b, where the two references are connected by a black arrow, representing the standard and enhanced bandwidth measurements. VI. CONCLUSIONS In conclusion, we have shown that the current state-of-the-art in thermomechanical IR detectors has not yet reached its full potential. We have highlighted the architectural pathway to achieving the fundamental room-temperature detection limit and identified several aspects where researchers can focus to improve existing designs. For both bare SiN and devices with a broadband absorber, drumhead resonators outperform their trampoline counterparts, exhibiting higher specific detectivity and shorter thermal time constants. Beyond clarifying the basic differences between drumhead and trampoline resonators, we have underscored the fundamental trade-off between achieving peak performance over a narrow range of detectable wavelengths and sacrificing a small portion of detectable power in favour of a significantly broader operational bandwidth. Crucially, this work bridges a long-standing gap in the literature by combining analytical modelling with practical design insights, laying the groundwork for the development of detectors that approach the fundamental limits of thermal sensitivity. We anticipate that this guide will accelerate innovation in nanomechanical sensing and foster the deployment of next-generation IR detection technologies in a broad range of scientific and technological domains. VII. SUPPLEMENTARY MATERIAL In the Supplementary Material, we provide additional details and figures on several aspects of this work, including the measurement of the Duffing parameter for trampolines, the influence of 28 tether width on the figures of merit, the calculation of absorptance and emissivity for different SiN thicknesses, the noise affecting the investigated resonators, the dependence of power responsivity on various parameters, and additional results for τth and RP in the case of a detector with a metal absorber. VIII. ACKNOWLEDGMENTS The authors would like to thank Dr. Hajrudin Besic, Dr. Robert G. West, and Jelena Timarac Popovic for the many fruitful discussions and valuable support during the writing of this work. This work received funding from the Novo Nordisk Foundation under the project MASMONADE with project number NNF22OC0077964. IX. AUTHORS' DECLARATIONS A. Conflict of Interest The authors have no conflicts to disclose. B. Author Contributions P.M. contributed with measurements, production of the results and figures, and the overall structure and writing of the paper; K.K. contributed with measurements, data analysis and assisted with the writing; S.S. conceived the idea of the paper, discussed the results, supervised the work and co-wrote the paper. C. Data availability All the data that supports the findings of this study are available from the corresponding author upon reasonable request. 29 REFERENCES 1P. W. Kruse and D. D. Skatrud, Uncooled infrared imaging arrays and systems, Vol. 47 (Academic press, 1997). 2A. Rogalski, Infrared and terahertz detectors (CRC press, 2019). 3J. R. Vig, R. Filler, and Y. Kim, "Uncooled ir imaging array based on quartz microresonators," Journal of Microelectromechanical Systems 5, 131-137 (1996). 4X. Zhang, E. Myers, J. Sader, and M. Roukes, "Nanomechanical torsional resonators for frequency-shift infrared thermal sensing," Nano letters 13, 1528-1534 (2013). 5Y. 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2510.14754
|
Zm
k -ACTIONS OF SIGNATURE (0;k, n+1
...,k)
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Abstract. In this article we consider group actions on compact Riemann surfaces and their topological classifi-
cation. We address this problem for pairs (S, N) where S is a compact Riemann surface endowed with a group
of automorphisms N ≅Zm
k such S/N has signature (0; k, n+1
. . . , k), where n, k ⩾2 and 1 ⩽m ⩽n are integers. We
further assume the existence of extra automorphisms, namely, a group G with N ⊲G ⩽Aut(S) and analyze the
induced permutational action of G/N on the cone points of S/N. To describe such actions up to topological
equivalence, we employ the generalized Fermat curves (X, H) and their automorphism groups, showing that every
triple (S, N, G) as before is determined by a class of subgroups of H that satisfy certain invariance property. This
approach establishes a correspondence between topological equivalence classes and an appropriate quotient set.
As an application, we specialize our results to the case k prime and m = 2, including algebraic models and isogeny
decompositions of their Jacobian varieties. We then discuss some examples for the cases n = 3 and n = 5, which
are interesting in their own right.
1. Introduction
Riemann surfaces have been proved to serve as fundamental objects bridging topology, geometry, algebra
and complex analysis. The study of automorphisms of compact Riemann surfaces or, equivalently, of smooth
complex projective algebraic curves and their function fields, represents a classical and rich area of research in
both complex and algebraic geometry. The foundations of this field date back to the nineteenth century, with
seminal contributions from mathematicians such as Riemann, Klein and Jacobi. A central result, due to Schwarz
[29] and Hurwitz [20], establishes that the group of automorphisms of a compact Riemann surface of genus g ⩾2
is finite, and that its order is bounded by 84(g −1). Much later, Greenberg in [14] succeeded in proving that each
finite group can be realized as a group of automorphisms of some compact Riemann surface.
It is classically known that the moduli space Mg of isomorphism classes of compact Riemann surfaces of genus
g ⩾2 has the structure of a complex analytic space of dimension 3g −3, and that if g ⩾3 then its singular locus
corresponds to the points representing compact Riemann surfaces with non-trivial automorphisms. In other words
Sing(Mg) = {[S] ∈Mg ∶Aut(S) ≠{id}},
where Aut(S) denotes the automorphism group of S. The moduli space –which is itself an algebraic variety defined
over the field of rational numbers– is one of the most fascinating objects in algebraic geometry, and is at the core
of important and recent developments in number theory and geometry.
Let Sj be a compact Riemann surface endowed with a group of automorphisms Gj for j = 1,2. We recall that
these actions are called topologically equivalent if there exists an orientation-preserving homeomorphism
φ ∶S1 →S2 such that φ−1G2φ = G1.
We also say that the pairs (S1,G1) and (S2,G2) are topologically equivalent.
The importance of the topological classification of actions lies in several applications, some of which we briefly
describe. The first one relates the singularities of the moduli space. The set
Mg(G) = {[S] ∈Mg ∶S admits a fixed topological class of G-action} ⊂Sing(Mg)
2010 Mathematics Subject Classification. 30F10, 14H37, 30F35, 14H30.
Key words and phrases. Riemann surfaces, group actions, automorphisms.
Partially supported by ANID Fondecyt Regular Grants 1230001, 1220099 and 1230708.
arXiv:2510.14754v1 [math.AG] 16 Oct 2025
2
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
is an irreducible subvariety of the moduli space [12]. In other words, the topological equivalence allows us to
perform deformations of Riemann surfaces with automorphisms in a controlled manner. These subvarieties, in
turn, provide a stratification of the moduli space [6], which has been proved to serve as a useful toolkit to
explore the largely unknown topology of Mg. For instance, it has played a key role in the determination of the
connectedness of the singular locus of the moduli space (see, for instance, [2] and the references therein) and
has been recently employed in [19] to find new non-normal subvarieties of Mg. Moreover, these subvarieties
have been fruitful to study some aspects of families of Jacobian varieties with group action; for instance, isogeny
decompositions and Shimura varieties (see, for instance [10] and [21]). Another important aspect to mention is that
the conjugacy classes of finite subgroups of the mapping class group (homotopy classes of self-homeomorphisms
of a real surface) are in bijective correspondence with the topological classes of group actions.
Sources for the characterization of topological actions by purely algebraic data include Nielsen [26], Harvey
[15] and Gilman [11]. We also refer to [3], [8], [23], [27] and [31] as sources for the classification of topological
actions in low genera, and also for computer-aided algorithms.
Group actions on compact Riemann surfaces are characterized in part by their signature. Concretely, if G
is a group of automorphisms of a compact Riemann surface S, then the signature of the action is the tuple
(γ;k1,...,kr), where γ is the genus of the quotient S/G and k1,...,kr are the branch indices of the canonical
projection S →S/G. Equivalently, the integers kj are the orders of the cone points of the Riemann orbifold S/G.
Let n,k ⩾2 and 1 ⩽m ⩽n be integers such that (n −1)(k −1) > 2. By a Zm
k -action of signature (0;k, n+1
...,k) we
mean a pair (S,N) consisting of a compact Riemann surface S endowed with a group of automorphisms
Zm
k ≅N ⩽Aut(S) such that S/N has signature (0;k, n+1
...,k).
We fix a Zm
k -action (S,N) of signature (0;k, n+1
...,k) and assume that S is endowed with a group of automor-
phisms G such that
N ⊲G ⩽Aut(S).
The action of the quotient group G/N on the set of n + 1 cone points of S/N induces a subgroup Q∗of the
symmetric group Sn+1, up to conjugation. Such a conjugacy class is the permutational action of G/N.
We denote by F(Q∗) the collection formed by all the triples ( ˆS, ˆG, ˆN) such that the pair ( ˆS, ˆN) is a Zm
k -action
of signature (0;k, n+1
...,k) and ˆN ⊲ˆG ⩽Aut( ˆS) in such a way that the permutational action of ˆG/ ˆN is given by Q∗.
Observe that if two triples (S1,N1,G1) and (S2,N2,G2) belong to F(Q∗), then the groups G1/N1 and G2/N2
are isomorphic and induce the same permutational action; however, G1 and G2 need not be isomorphic groups.
Accordingly to the case of pairs, two triples (S1,N1,G1) and (S2,N2,G2) as above are called topologically
equivalent if there exists an orientation-preserving homeomorphism
φ ∶S1 →S2 such that φ−1N2φ = N1 and φ−1G2φ = G1.
It is worth noting that it might happen that two such triples are topologically inequivalent, but either the pairs
(S1,N1) and (S2,N2) or the pairs (S1,G1) and (S2,G2) are topologically equivalent.
In this paper, we address the problem of describing the collection F(Q∗) up to topological equivalence.
We proceed as follows.
Let (S,N,G) be a triple in F(Q∗).
First, we consider the homology cover X of
the Riemann orbifold S/N, also called a generalized Fermat curve of type (k,n). This is a compact Riemann
surface that admits a group of automorphisms H ≅Zn
k such that X/H ≅S/N. Following the results in [13], we
then consider the intimate relationship between X and our triple, which arises from the existence of a subgroup
KS ⊲H, acting freely on X, such that S ≅X/KS and N ≅H/KS. In addition, the group G/N lifts to a group
Q ⩽Aut(X) containing KS and H as normal subgroups in such a way that Q/H ≅G/N and G ≅Q/KS.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
3
We denote by Autg(H) the subgroup of Aut(H) whose elements are induced by conjugation via orientation-
preserving homeomorphisms of X that preserve H. We shall see that there is a natural isomorphism between
Autg(H) and Sn+1, and therefore the group Q induces (up to conjugation) a subgroup Q∗of Autg(H) which is
completely determined by the permutational action of G/N on the cone points of X/H ≅S/N.
We then introduce the subset Ck(Q) formed by the subgroups K ⊲H acting freely on X such that H/K ≅Zm
k
and that are Q∗-invariant. For each K ∈Ck(Q), we set
SK ∶= X/K and NK ∶= H/K.
The Q∗-invariance of K ensures the existence of a finite group QK of orientation-preserving homeomorphisms of
X, containing K and H as normal subgroups, such that GK ∶= QK/K is a finite group of orientation-preserving
homeomorphisms of SK that contains NK as a normal subgroup and such that GK/NK ≅G/N induces the
permutation group Q∗. This allows us to prove that for each ( ˆS, ˆG, ˆN) ∈F(Q∗) there exists K ∈Ck(Q) such that
(SK,GK,NK) is topologically equivalent to ( ˆS, ˆG, ˆN).
In addition, if NQ ⩽Autg(H) denotes the normalizer of Q∗then Ck(Q) is NQ-invariant. We then show that
K1,K2 ∈Ck(Q) determine topologically equivalent triples if and only if they belong to the same NQ-orbit. As a
result, we manage to compute the number of pairwise topologically inequivalent triples, by means of a bijective
correspondence with the set Ck(Q)/NQ.
For the sake of clarity, we work out the case of Z2
p-actions of signature (0;p, n+1
...,p). We provide explicit algebraic
models for these Riemann surfaces in terms of fiber products of cyclic p-gonal algebraic curves, and give an isogeny
decomposition of their Jacobian varieties. After that, we specialize our results for the cases n = 3 and n = 5, as
they are interesting in their own right. More precisely, we study Z2
p-actions of signature (0;p, 4...,p) and describe
some examples in detail, and study Z2
p-actions of signature (0;p, 6...,p) that admit extra automorphisms and that
form complex one-dimensional families. By the way, we recover and extend classical and recent results for some
families of Riemann surfaces.
2. Preliminaries and Notations
2.1. Group actions on Riemann surfaces. Let S be a compact Riemann surface of genus g ⩾2, and let G
be a group of automorphisms of S. We consider the quotient Riemann orbifold S/G and the associated regular
covering map π ∶S →S/G. The Riemann orbifold S/G inherits the structure of a compact Riemann surface of
genus γ together with r ⩾0 distinguished points, say q1,...,qr, satisfying the following property: for every point
x ∈π−1(qj) its multiplicity kj is strictly smaller than the degree of π. We say that the orbifold S/G has signature
(γ;k1,...,kr) and that qj is a cone point of cone order kj. If we denote by H2 the upper half-plane then, by the
classical uniformization theorem, there exists a co-compact Fuchsian group
Γ ⩽PSL2(R) ≅Aut(H2) such that S/G ≅H2/Γ as orbifolds.
Moreover, there is a group epimorphism θ ∶Γ →G, whose kernel is torsion-free, in such a way that
S ≅H2/ker(θ) and G ≅Γ/ker(θ).
The epimorphism θ is called the monodromy of the action. We also say that Γ has signature (γ;k1,...,kr). Along
the paper, we employ the notation (0;km) to abbreviate (0;k, m
...,k).
2.2. Topological equivalence of actions. Let (S1,G1) and (S2,G2) be two pairs, where Sj is a compact
Riemann surface endowed with a group of automorphisms isomorphic to Gj for j = 1,2. We say that the pairs
(or that the actions) are topologically equivalent if there exists an orientation-preserving homeomorphism
φ ∶S1 →S2 such that φ−1G2φ = G1.
4
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Observe that, in particular, G1 and G2 are isomorphic groups. The pairs are termed biholomorphically equivalent
if φ is a biholomorphism. In terms of monodromies, the topological equivalence can be reformulated as follows.
Let Γj be a Fuchsian group such that H2/Γj ≅Sj/Gj, and let θj ∶Γj →Gj be the corresponding monodromy.
Then two pairs as above are topologically equivalent if and only if there is a group isomorphism ρ ∶G1 →G2 and a
geometric isomorphism ψ ∶Γ1 →Γ2 such that ρ ○θ1 = θ2 ○ψ. We recall that the geometric isomorphisms are those
that are induced by conjugation of orientation-preserving self-homeomorphisms of H2. Without loss of generality,
we can assume Γ1 = Γ2. In the case that γ = 0, a set of generators for the group of geometric automorphisms BΓ
of Γ is known. If, moreover, the group G is abelian and all the cone orders are the same, then the action of BΓ
induces the action of the symmetric group on the image under θ of the canonical generators of Γ. We refer to the
survey [5] for more details.
2.3. Families of Riemann surfaces. A family C of compact Riemann surfaces of genus g is the locus of the
moduli space Mg formed by all those Riemann surfaces that have a group of automorphisms isomorphic to a given
group G acting with a given signature. If the signature of the action is (γ;k1,...,kr) then the complex-dimension
of the family is 3γ −3 + r. We recall the following facts, which follow from the equisymmetric stratification of the
moduli space.
(1) The interior of C , if non-empty, consists of those Riemann surfaces whose automorphism group is isomor-
phic to G, and is formed by finitely many strata which are in correspondence with the pairwise non-equivalent
topological actions of G.
(2) The complement of the interior (with respect to the family) is formed by those Riemann surfaces that have
strictly more automorphisms than G.
We refer to [6], [12] and [15] for more details.
2.4. Generalized Fermat curves. Let n,k ⩾2 be integers. A compact Riemann surface X is called a generalized
Fermat curve of type (k,n) if there exists
Zn
p ≅H0 ⩽Aut(X) such that X/H0 has signature (0;kn+1).
The group H0 is a generalized Fermat group of type (k,n) and the pair (X,H0) is a generalized Fermat pair of
type (k,n). By the Riemann-Hurwitz formula, the genus of X is
g = 1 + kn−1
2 ((n −1)(k −1) −2).
In particular, the non-hyperbolic generalized Fermat pairs are those satisfying (n −1)(k −1) ⩽2, that is, of
type (2,2), (2,3) and (3,2). Henceforth, we restrict our attention to the hyperbolic case.
2.4.1. Fuchsian description. Let (X,H0) be a generalized Fermat pair of type (k,n). The Riemann orbifold
X/H0 is uniformized by a Fuchsian group
Γ = ⟨x1,...,xn+1 ∶xk
1 = ... = xk
n+1 = x1⋯xn+1 = 1⟩.
Moreover, if Γ′ stands for the commutator subgroup of Γ then, as proved in [13], the generalized Fermat pairs
(X,H0) and (H/Γ′,Γ/Γ′) are biholomorphically equivalent.
The fact that Γ′ is characteristic yields the following properties.
Theorem ([13]).
(1) Two generalized Fermat pairs of the same type are topologically equivalent.
(2) The regular covering map π ∶X →X/H0 induced by the action of H0 is characteristic.
The following fact, which was also proved in [13], will be used later. For the sake of competeness, we provide
an outline of the proof.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
5
Theorem. Let S be a Riemann surface of genus at least two endowed with an abelian group of automorphisms
N such that S/N has signature (0;kn+1). Then there exist a generalized Fermat pair (X,H0) of type (k,n) and
a subgroup KS ⊲H0, acting freely on X, such that
(S,N) and (X/KS,H0/KS) are biholomorphically equivalent.
Proof. Let Γ be a Fuchsian group such that S/N ≅H2/Γ. Then (X = H2/Γ′,H0 = Γ/Γ′) is a generalized Fermat
pair of type (k,n). As S is a regular branched cover of S/N, there is a torsion-free normal subgroup F ⩽Γ such
that S ≅H2/F and N ≅Γ/F. As N is abelian, one has that Γ′ ⩽F and consequently KS ∶= F/Γ′ is the desired
subgroup.
□
2.4.2. Algebraic description and uniqueness of the generalized Fermat groups. In what follows, for
each integer k ⩾2 we set ωk = exp(2πi/k). Up to a suitable M¨obius transformation, we can assume the cone
points of X/H0 to be
(2.1)
∞,0,1,q4,...,qn+1.
(1) Case n = 2. If n = 2, and therefore k ⩾4, then there is, up to biholomorphism, a unique generalized Fermat
curve of type (k,2). This curve corresponds to the classical Fermat curve
X = Fk ∶{xk
1 + xk
2 + xk
3 = 0} ⊂P2,
with generalized Fermat group given by H = ⟨a1,a2⟩≅Z2
k, where
a1([x1 ∶x2 ∶x3]) = [ωkx1 ∶x2 ∶x3] and a2([x1 ∶x2 ∶x3]) = [x1 ∶ωkx2 ∶x3].
In this case, the map π ∶Fk →¯C given by π([x1 ∶x2 ∶x3]) = −(x2/x1)k is a k2-fold branched regular covering map
with deck group H, whose branch values are ∞,0,1. We can directly verify the following facts.
(1) The nontrivial elements of H that have fixed points in Fk are a1,a2,a3 ∶= (a1a2)−1 and their powers.
Moreover, each fixed point of a non-trivial power of aj is also a fixed point of aj.
(2) The group H is the unique generalized Fermat group of Fk, that is, if H′ ⩽Aut(Fk) is a generalized
Fermat group of some type (k′,n′), then (k′,n′) = (k,n) and H′ = H.
(2) Case n ⩾3. Now, assume n ⩾3. Consider the complex projective algebraic curve in Pn defined by
Ck(q4,...,qn+1) ∶
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
xk
1
+
xk
2
+
xk
3
=
0
q4xk
1
+
xk
2
+
xk
4
=
0
⋮
⋮
⋮
qn+1xk
1
+
xk
2
+
xk
n+1
=
0
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
The fact that qj ∈C −{0,1} are pairwise distinct implies that the algebraic curve above is non-singular, and
hence represents a compact Riemann surface. Note that the linear transformations aj ∈PGLn+1(C) defined as
(2.2)
aj([x1 ∶... ∶xn+1]) = [x1 ∶... ∶xj−1 ∶ωkxj ∶xj+1 ∶... ∶xn+1] for j ∈{1,...,n},
generate a group of automorphisms H of Ck(q4,...,qn+1)). The kn-fold covering map
(2.3)
π ∶Ck(q4,...,qn+1) →¯C given by π([x1 ∶... ∶xn+1]) = −(x2/x1)k,
satisfies π○aj = π for every 1 ⩽j ⩽n and ramifies precisely over (2.1). Thus, (Ck(q4,...,qn+1),H) is a generalized
Fermat pair of type (k,n).
Theorem ([13]). The pairs (X,H0) and Ck(q4,...,qn+1),H) are biholomorphically equivalent.
Remark 1. The above fact asserts that the domain
Ωn ∶= {(q4,...,qn+1) ∈Cn−2 ∶qj ≠0,1 and qi ≠qi for i ≠j}
6
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
parametrizes all generalized Fermat pairs of type (k,n), where n ⩾3.
Generalizing the known situation for n = 2, we have the following facts.
Theorem ([13, 16, 18]). Let Λ ∈Ωn. Then
(1) The nontrivial elements of H that have fixed points in Ck(Λ) are a1,...,an,an+1 ∶= (a1⋯an)−1 and their
powers. Moreover, each fixed point of a non-trivial power of aj is also a fixed point of aj.
(2) The group H is the unique generalized Fermat group of Ck(Λ), that is, if H′ ⩽Aut(Ck(Λ)) is a generalized
Fermat group of some type (k′,n′), then (k′,n′) = (k,n) and H′ = H.
Remark 2. The uniqueness of H asserts that it is a normal subgroup of Aut(Ck(Λ)) and therefore Aut(Ck(Λ))/H
is a spherical group. In other words, there is a short exact sequence of groups
1 Ð→H Ð→Aut(Ck(Λ))
θ
Ð→A Ð→1,
where A is the M¨ob(C)-stabilizer of BΛ = {∞,0,1,q4,...,qn+1}.
2.4.3. Geometric automorphisms of H. Let Λ = (q4,...,qn+1) ∈Ωn. We employ the following notations.
(1) Hom+(Ck(Λ)) is the group of orientation-preserving homeomorphisms of Ck(Λ).
(2) Hom+
H(Ck(Λ)) is the normalizer of H in Hom+(Ck(Λ)).
(3) Hom+(O(Λ)) is the group of orientation-preserving homeomorphisms of ¯C that preserve the set BΛ.
Observe that there is a natural group homomorphism
η ∶Hom+
H(Ck(Λ)) →Hom+(O(Λ)) such that η(f) ○π = π ○f for each f ∈Hom+
H(Ck(Λ)).
The fact that the covering map (2.3) is characteristic implies that η is surjective. Note that ker(η) = H. Besides,
each f ∈Hom+
H(Ck(Λ)) defines an automorphism Φf of H given by Φf(aj) = f ○aj ○f −1.
Definition. An automorphism Φ of H is called geometric if Φ = Φf for some f ∈Hom+
H(Ck(Λ))
The subgroup of Aut(H) formed by the geometric automorphisms is denoted by Autg(H). It follows that there
is a the natural group epimorphism
ρ ∶Hom+
H(Ck(Λ)) →Autg(H) given by ρ(f) = Φf.
Remark 3. Autg(H) does not depend on Λ.
In fact, for each ˆΛ = (ˆq4,..., ˆqn+1) ∈Ωn we can consider an
orientation-preserving homeomorphism ˆg ∶¯C →¯C such that
(2.4)
ˆg(∞) = ∞, ˆg(0) = 0, ˆg(1) = 1 and ˆg(qj) = ˆqj for j = 4,...,n + 1.
The fact that (2.3) is characteristic guarantees that ˆg lifts to an orientation-preserving homeomorphism g ∶
Ck(Λ) →Ck(ˆΛ) such that gHg−1 = H. Now, the conditions (2.4) imply that, if ˆf ∈Hom+
H(Ck(ˆΛ)) then
Φg−1 ˆ
fg(aj) = g−1 ˆfg ○aj ○(g−1 ˆfg)−1 = ˆf ○aj ○ˆf −1 = Φ ˆ
f(aj).
The claim follows after noting that each f ∈Hom+
H(Ck(Λ)) is of the form g−1 ˆfg.
Proposition 1. Each Φ ∈Autg(H) yields a uniquely determined permutation σΦ ∈Sn+1, and the correspondence
Autg(H) →Sn+1 given by Φ ↦σΦ
is a group isomorphism. In particular, Autg(H) = ⟨Φ1,Φ2⟩where
Φ1(a1,a2,a3,...,an) = (a2,a1,a3,...,an) and Φ2(a1,a2,...,an−1,an) = (a2,a3,...,an,a1).
Proof. Consider the natural group epimorphism
ˆρ ∶Hom+(O(Λ)) →Sn+1 given by ˆf ↦ˆρ( ˆf) where ˆf(qj) = qˆρ( ˆ
f)(j).
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
7
The kernel of ˆρ is, up to isotopy rel BΛ, the group generated by the Dehn-twists along simple closed curves
contained in ¯C −BΛ. We note that, if f ∈Hom+
H(Ck(Λ)) then
Φf(aj) = f ○aj ○f −1 = aˆρ(η(f))(j) for each j = 1,...,n.
Thus, there is a group isomorphism ˆη ∶Autg(H) →Sn+1 which makes the following diagram commutative.
Hom+
H(Ck(Λ))
Autg(H)
Hom+(O(Λ))
Sn+1
ρ
η
ˆρ
ˆη
This proves the first statement; the latter one follows directly from the former.
□
Remark 4.
(1) Consider L ⩽Hom+(O(Λ)) and set Q ∶= η−1(L). As ρ(Q) = ˆη−1(ˆρ(L)), the subgroup
Q∗∶= ρ(Q) = {ΦT ∶T ∈Q}
of Autg(H) is completely determined by the permutational action ˆρ(L).
(2) We denote by Aut(O(Λ)) ⩽M¨ob(C) the subgroup of Hom+(O(Λ)) consisting of the automorphisms of the
orbifold Ck(Λ)/H. We have the following commutative diagram
Aut(Ck(Λ))
Autg(H)
Aut(O(Λ))
Sn+1
ρ
η
ˆρ
ˆη
where neither ρ nor ˆρ are surjective for n ⩾3, and that ker(ρ) ∩Aut(Ck(Λ)) = H.
3. Zm
k -actions of signature (0;kn+1)
Let k,n ⩾2 and 1 ⩽m ⩽n be integers such that (n −1)(k −1) > 2.
Definition. A pair (S,N) is called a Zm
k -action of signature (0;kn+1) if S is a compact Riemann surface endowed
with a group of automorphisms
N ≅Zm
k such that S/N has signature (0;kn+1).
Observe that the Riemann surfaces S, such that (S,N) is a Zm
k -action of signature (0;kn+1), form a complex
(n −2)-dimensional family in moduli space Mg, where
(3.1)
g = 1 + 1
2km−1[(n −1)(k −1) −2].
Remark 5. Every Zn
k-action of signature (0;kn+1) is biholomorphic to a generalized Fermat pair of type (k,n).
In particular, all of them are topologically equivalent. On the other extreme, every Zk-action (S,N) of signature
(0;kn+1) corresponds to a cyclic k-gonal curve
S ≅{yk =
n+1
∏
j=1
(x −qj)lj} and N = ⟨(x,y) ↦(x,ωky)⟩,
where the integers l1,...,ln+1 lie in {1,...,k −1} and are coprime to k.
By the remark above, hereafter we shall only consider the case n ⩾3 and 2 ⩽m ⩽n −1.
8
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
3.1. Fiber product description. Let (S,N) be a Zm
k -action of signature (0;kn+1), where n ⩾3, k ⩾2, (n −
1)(k −1) > 2 and 2 ⩽m ⩽n −1. Let ϕ1,...,ϕm be automorphisms of S that generate N, and let π ∶S →¯C be a
branched regular covering map with deck group N. Set
N1 = ⟨ϕ2,...,ϕm⟩, Nm = ⟨ϕ1,...,ϕm−1⟩and Ni = ⟨ϕ1,...,ϕi−1,ϕi+1,...,ϕm⟩,
for each i = 2,...,m −1. Note that Nj ≅Zm−1
k
. We denote by Si the compact Riemann surface underlying to the
quotient S/Ni for each i, and by πi ∶S →Si a branched regular covering map with deck group Ni. Observe that
ϕi induces an automorphism τi of Si of order k such that
S/N ≅Si/⟨τi⟩for each i = 1,...,m.
If πi ∶Si →¯C is a branched regular covering map with deck group ⟨τi⟩such that π = πi ○πi then, following [17,
Section 3.2], we have that S is isomorphic to the fiber product Πm
i=1(Si,πi).
This description allows us to provide an explicit algebraic description of S in terms of the branch values of πi.
Later, we will make this description explicit for the case m = 2.
3.2. Descriptions in terms of generalized Fermat curves. Let (S,N) be a Zm
k -action of signature (0;kn+1),
where n ⩾3, k ⩾2, (n −1)(k −1) > 2 and 2 ⩽m ⩽n −1. After considering a suitable M¨obius transformation, we
can assume that the cone points of S/N ≅¯C are given by the set
BΛ = {q1 = ∞,q2 = 0,q3 = 1,q4,...,qn+1} where Λ = (q4,...,qn+1) ∈Ωn.
As discussed in §2.4, the tuple Λ ∈Ωn determines the generalized Fermat curve Ck(Λ) of type (k,n), and its
generalized Fermat group is H ∶= ⟨a1,...,an⟩≅Zn
k where aj is as (2.2). We define the set
F(k,n,m) ∶= {K ∶K ⩽H,H/K ≅Zm
k , ⟨aj⟩∩K = {1}, j = 1,...n + 1}.
Remark 6. If k is prime, then the condition ⟨aj⟩∩K = {1} above is equivalent to aj ∉K.
As a consequence of the Fuchsian uniformization of generalized Fermat curves discussed in §2.4.1, we have that
the Zm
k -actions of signature (0;kn+1) are parametrized by F(k,n,m).
Proposition 2. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n−1 be integers such that (n−1)(k−1) > 2. If (S,N) is a Zm
k -action
of signature (0;kn+1), then there exist KS ∈F(k,n,m) and a tuple (q4,...,qn+1) ∈Ωn such that
(S,N) and (Ck(q4,...,qn+1)/KS,H/KS) are biholomorphically equivalent.
The following result describes the members of F(k,n,m), up to geometric automorphisms of H, for k prime.
Theorem 1. Let p ⩾2 be a prime number, and let n ⩾3 and 2 ⩽m ⩽n−1 be integers such that (n−1)(p−1) > 2.
If K ∈F(p,n,m) then there exists Φ ∈Autg(H) and there are integers lj,i ∈{0,1,...,p −1}, j = m + 1,...,n + 1
and i = 1,...,m, satisfying
(1) (lj,1,...,lj,m) ≠(0,...,0) for every j = m + 1,...,n + 1, and
(2) i + lm+1,i + ⋯+ ln+1,i ≡0 mod p for every i = 1,...,m,
such that
Φ(K) = ⟨alm+1,1
1
⋯alm+1,m
m
a−1
m+1,...,aln,1
1
⋯aln,m
m
a−1
n ⟩≅Zn−m
p
.
Proof. A group K ∈F(p,n,m) is the kernel of some group epimorphism θ ∶H →Zm
p with the property that,
for every j = 1,...,n + 1, the element θ(aj) has order p. Up to a permutation of indices (or, equivalenty, after
considering the action of Autg(H)), the fact that p is prime allows us to assume that the elements
ϕ1 ∶= θ(a1),...,ϕm ∶= θ(am)
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
9
form a set of generators of Zm
p . Thus, for each j = m + 1,...,n + 1, one has that
θ(aj) = ϕlj,1
1
⋯ϕlj,m
m
for some lj,i ∈{0,1,...,p −1}.
The condition that ⟨aj⟩∩K = {1} asserts that (lj,1,...,lj,m) ≠(0,...,0) for each j. Besides, the fact that the
product a1⋯an+1 is trivial implies that i + lm+1,i + ⋯+ ln+1,i ≡0 mod p for each i = 1,...,m. In this way, the
kernel K of θ is generated by the elements
alm+1,1
1
⋯alm+1,m
m
a−1
m+1,...,aln,1
1
⋯aln,m
m
a−1
n ,aln+1,1
1
⋯aln+1,m
m
a−1
n+1.
The equality
aln+1,1
1
⋯aln+1,m
m
a−1
n+1 = aln+1,1
1
⋯aln+1,m
m
(a1⋯an) = a1+ln+1,1
1
⋯a1+ln+1,m
m
(am+1⋯an),
shows that K is generated by the n −m elements alm+1,1
1
⋯alm+1,m
m
a−1
m+1,...,aln,1
1
⋯aln,m
m
a−1
n , as claimed.
□
Remark 7. The theorem above holds for the case m = 1. If k is not a prime integer, then it is more involved to
describe the general form of K ∈F(k,n,m).
3.3. Topological classification of Zm
k -actions. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n −1 be integers such that
(n −1)(k −1) > 2.
Let (S,N) be a Zm
k -action of signature (0;kn+1), and assume that the cone points of
S/N are ∞,0,1, ˆq4,..., ˆqn+1.
By Proposition 2, there is a subgroup KS ∈F(k,n,m) such that (S,N) and
(Ck(ˆΛ)/KS,H/KS) are biholomorphically equivalent, where ˆΛ = (ˆq4,..., ˆqn+1) ∈Ωn.
We fix Λ = (q4,...,qn+1) ∈Ωn and consider an orientation-preserving homeomorphism ˆf ∶¯C →¯C such that
ˆf(∞) = ∞, ˆf(0) = 0, ˆf(1) = 1 and ˆf(ˆqj) = qj for j = 4,...,n + 1.
It follows that there is an orientation-preserving homeomorphism f ∶Ck(ˆΛ) →Ck(Λ) such that ˆf ○π = π ○f. In
particular, one has that fHf−1 = H. The way as we have chosen ˆf asserts that fKSf −1 = KS. Now, if we define
SKS ∶= Ck(Λ)/KS and NKS ∶= H/KS,
then f induces an orientation-preserving homeomorphism g ∶SKS →S such that the following diagram commutes.
Ck(Λ)
Ck(ˆΛ)
SKS
S
¯C
¯C
f
NKS
N
ˆ
f
KS
g
KS
H
H
Note that gNKSg−1 = N. The discussion above proves the following proposition.
Proposition 3. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n −1 be integers such that (n −1)(k −1) > 2, and fix Λ ∈Ωn. If
(S,N) is a Zm
k -action of signature (0;kn+1), then there exists K ∈F(k,n,m) such that
(S,N) and (SK = Ck(Λ)/K,NK = H/K) are topologically equivalent.
Observe that a pair of groups K1 and K2 in F(k,n,m) may give rise to two Zm
k -actions that are topologically
equivalent. The following result describes such a situation.
Theorem 2. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n −1 be integers such that (n −1)(k −1) > 2. Let K1,K2 ∈F(k,n,m).
The pairs (SK1,NK1) and (SK2,NK2) are topologically equivalent if and only if there exists Φ ∈Autg(H) such
that Φ(K1) = K2.
10
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Proof. We fix Λ = (q4,...,qn+1) ∈Ωn. Assume that there exists a geometric automorphism Φ of H such that
Φ(K1) = K2. Thus, there is an orientation-preserving homeomorphism fΦ ∶Ck(Λ) →Ck(Λ) which induces an
orientation-preserving homeomorphism gΦ ∶SK1 →SK2. The fact that fΦHf −1
Φ = H implies that gNK1g−1 = NK2
and therefore (SK1,NK1) and (SK2,NK2) are topologically equivalent. Conversely, if (SK1,NK1) and (SK2,NK2)
are topologically equivalent then there is an orientation-preserving homeomorphism g ∶SK1 →SK2 such that
gNK1g−1 = NK2. It follows that g induces an orientation-preserving homeomorphism h of ¯C into itself which keeps
the set {∞,0,1,q4,...,qn+1} invariant. In turn, h lifts to an orientation-preserving homeomorphism f ∶Ck(Λ) →
Ck(Λ) that satisfies fHf −1 = H and fK1f −1 = K2. Thus, Φ(K1) = H2 where Φ = Φf.
□
Observe that there is a natural action
Autg(H) × F(k,n,m) →F(k,n,m) given by (Φ,K) ↦Φ(K).
As a consequence of the proposition above, we have the following corollary.
Corollary 1. The cardinality of the quotient set F(k,n,m)/Autg(H) is the number of pairwise topologically
inequivalent Zm
k -actions with signature (0;kn+1).
3.4. Topological actions and extra automorphisms. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n −1 be integers such
that (n −1)(k −1) > 2. We consider triples (S,N,G) where (S,N) is a Zm
k -action of signature (0;kn+1), and S is
endowed with a group of automorphisms G such that N ⊴G ⩽Aut(S). Consider Λ = (q4,...,qn+1) ∈Ωn such that
S ≅Ck(Λ)/KS and N ≅H/KS for some KS ∈F(k,n,m).
We recall that there is a short exact sequence of groups 1 Ð→H Ð→Aut(Ck(Λ))
θ
Ð→A Ð→1, where A is the
M¨ob(C)-stabilizer of BΛ = {∞,0,1,q4,...,qn+1}. As A has a subgroup L isomorphic to G/N, there is an induced
short exact sequence of groups
1 Ð→H Ð→QS,G ∶= θ−1(L)
θ
Ð→L Ð→1.
Observe that H ⊴QS,G and QS,G/H ≅G/N, KS ⊴QS,G and QS,G/KS ≅G, and S/G ≅Ck(Λ)/QS,G.
All the above can be summarized in the following commutative diagram, where C = Ck(Λ).
S
C
S/N = C/H
S/G = C/QS,G
N
KS
H
G/N
G
QS,G
We recall that if Q is any group of orientation-preserving homeomorphisms of Ck(Λ) such that H ◁Q, then
we may consider the representation (see the proof of Proposition 1)
ρQ ∶Q →Autg(H) given by ρQ(f) = Φf.
We denote its image by Q∗. In particular, as H ◁QS,G, we may consider the group epimorphism
(3.2)
ρQS,G ∶QS,G →Autg(H) given by ρQS,G(f) = Φf.
As the kernel of ρQS,G is H, we have that QS,G/H ≅Q∗
S,G. Note that ρQS,G(f) is uniquely determined by the
permutation induced by θ(f). We define
Ck(QS,G) ∶= {K ∈F(k,n,m) ∶K is Q∗
S,G-invariant}.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
11
Note that Ck(QS,G) is nonempty as KS belongs to it. If K ∈Ck(QS,G) then
(SK = Ck(Λ)/K,NK = H/K)
is a Zm
k -action of signature (0;kn+1) such that there exists Q ⩽Hom+
H(Ck(Λ)) which contains H and K as normal
subgroups, satisfying Q∗= Q∗
S,G and that
NK ⊴GK = Q/K ⩽Hom+(SK) and GK/NK ≅Q/H ⩽Hom+(¯C).
In addition, SK/GK and Ck(Λ)/Q are equivalent, as topological orbifolds.
Proposition 4. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n −1 be integers such that (n −1)(k −1) > 2. Let (S,N,G) be a
triple such that (S,N) is a Zm
k -action of signature (0;kn+1) and S admits a group of automorphisms G such that
N ⊴G ⩽Aut(S). Let Ck(Λ) and Q∗
S,G ⩽Autg(H) be as before. Then, up to topological equivalence, each triple
( ˆS, ˆN, ˆG) such that ( ˆS, ˆN) is a Zm
p -action of signature (0;kn+1) and ˆS admits a group of automorphisms ˆG with
ˆN ⊴ˆG ⩽Aut( ˆS) and satisfying that Q∗
ˆS, ˆ
G = Q∗
S,G corresponds to a member of Ck(QS,G).
Proof. Let ( ˆS, ˆN, ˆG) be a triple as in the statement of the proposition, and let ∞,0,1, ˆq4,..., ˆqn+1 be the cone
points of ˆS/ ˆN, so ˆΛ = (ˆq4,..., ˆqn+1) ∈Ωn. By Proposition 3, there exists K ˆS ∈F(k,n,m) such that
( ˆS, ˆN) and (SK ˆ
S = Ck(Λ)/K ˆS,NK ˆ
S = H/K ˆS)
are topologically equivalent. In addition, there is a group
Q ˆS, ˆ
G ⩽Aut(Ck(ˆΛ)) such that H,K ˆS ⊴ˆQ ˆS, ˆ
G and ˆG ≅ˆQ ˆS, ˆ
G/K ˆS.
All the above is summarized in the following commutative diagram, where f,g and ˆf are as in the proof of
Proposition 3.
Ck(Λ)
Ck(ˆΛ)
SK ˆ
S
ˆS
¯C
¯C
¯C
ˆ
G/ ˆ
N
ˆ
G
Q ˆ
S, ˆ
G
f
NK ˆ
S = g−1 ˆ
Ng
ˆ
N
ˆ
f
K ˆ
S = f−1K ˆ
Sf
g
K ˆ
S
H
H
We consider the group of homeomorphisms g−1 ˆGg ⩽Hom+(SK ˆ
S). As Q∗
ˆS, ˆ
G = Q∗
S,G, we have that S/G and ˆS/ ˆG
are isomorphic as topological orbifolds. It follows that
S/G and SK ˆ
S/(g−1 ˆGg)
are isomorphic as topological orbifolds too. Observe that NK ˆ
S = g−1 ˆNg ⊴g−1 ˆGg and that
(g−1 ˆGg)/NK ˆ
S = ˆf −1( ˆG/ ˆN) ˆf.
Besides, the group g−1 ˆGg lifts to a group Q ⩽Hom+(Ck(q4,...,qn+1)) such that H,K ⊴Q and Q/K ≅g−1 ˆGg. Now,
the fact that S/G and SK ˆ
S/g−1 ˆGg are isomorphic as topological orbifolds implies that the action by conjugation
of Q agrees with the one of Q∗
S,G, showing that K ˆS ∈Cp(QS,G) as desired.
□
12
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Consider a triple (S,N,G) as above. We denote by
NQS,G = {Φ ∈Autg(H) ∶ΦQ∗
S,GΦ−1 = Q∗
S,G} ⩽Autg(H)
the normalizer of Q∗
S,G in Autg(H). We observe that Cp(QS,G) is NQS,G-invariant.
If ( ˆS, ˆN, ˆG) is another triple as above, such that ˆG/ ˆN induces Q∗
S,G, then there exists some K ∈Cp(QS,G) such
that there is an orientation-preserving homeomorphism
h ∶SK = Ck(Λ)/K →ˆS such that hNKh−1 = ˆN and hGKh−1 = ˆG.
So, following similar arguments as in the proof of Proposition 4, we obtain the following result.
Theorem 3. Let K1,K2 ∈Cp(QS,G). The triples (SK1,NK1,GK1) and (SK2,NK2,GK2) are topologically equiv-
alent if and only if there exists Φ ∈NQS,G such that Φ(K1) = K2.
Analogously to Corollary 1, we have the following result.
Corollary 2. Let (S,N,G) be a triple as above. Then the cardinality of the quotient set Ck(QS,G)/NQS,G is the
number of pairwise topologically inequivalent triples ( ˆS, ˆN, ˆG), where ( ˆS, ˆN) is a Zm
k -action of signature (0;kn+1)
and ˆN ◁ˆG ⩽Aut( ˆS) is such that ˆG/ ˆN induces Q∗
S,G
4. The case m = 2 and k = p prime
In this section we restrict to the case of Z2
p-actions of signature (0;pn+1), where p ⩾2 is prime, n ⩾3 and
(n −1)(p −1) > 2.
4.1. Description of F(p,n,2). We start by describing the elements of F(p,n) = F(p,n,2).
Theorem 4. If K ∈F(p,n) then one of the following statements holds.
(1) There are integers r3,s3,...,rn,sn ∈{0,1,...,p −1} satisfying that (rj,sj) ≠(0,0) for each j, and that
(1 + r3 + ⋯+ rn,1 + s3 + ⋯+ sn) ≢(0,0) mod p
in such a way that K = ⟨ar3
1 as3
2 a−1
3 ,...,arn
1 asn
2 a−1
n ⟩.
(2) There is an integer 2 ⩽t ⩽n −1, there are integers l2,...,lt ∈{1,...,p −1}, and there are integers
rt+2,st+2,...,rn,sn ∈{0,1,...,p −1} satisfying that (rj,sj) ≠(0,0) for each j, and that
(1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn,1 + st+2 + ⋯+ sn) ≢(0,0) mod p
in such a way that K = ⟨al2
1 a−1
2 ,...,alt
1 a−1
t ,art+2
1
ast+2
t+1 a−1
t+2,...,arn
1 asn
t+1a−1
n ⟩.
Proof. Every K ∈F(p,n) is the kernel of a surjective homomorphism θ ∶H →Z2
p such that aj ∉K for j =
1,...,n + 1. Set ϕ1 = θ(a1). By the surjectivity of θ, there is some 1 ⩽t ⩽n such that
θ(aj) ∈⟨ϕ1⟩for j = 1,...,t, and ϕ2 ∶= ϕ(at+1) ∉⟨ϕ1⟩.
In this case, we can write
θ(aj) = ϕlj
1 , j = 2,...,t if t ⩾2,
and θ(at+i) = ϕrt+i
1
ϕst+i
2
, i = 2,...,n + 1 −t.
The fact that θ(aj) has order p implies that l2,...,lt ∈{1,...,p −1}, and that rj,sj ∈{0,1,...,p −1} are not
simultaneously zero, for each j. The relation a1⋯an+1 = 1 implies that t ⩽n−1, that 1+st+2+⋯+sn ≡−sn+1 mod p,
and that
1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn ≡−rn+1 mod p or 1 + rt+2 + ⋯+ rn ≡−rn+1 mod p,
according to whether or not t is different from 1. Finally, the fact that (rn+1,sn+1) ≠(0,0) ends the proof.
□
We say that K is of type (1) or (2) according to the enumeration in the theorem above.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
13
4.2. Algebraic descriptions. We recall that, as observed in §3.1, each Zm
p -action can be described as a fiber
product. We proceed to make such a description explicit for the case m = 2 and k = p prime, and for each possible
group K ∈F(p,n). If Λ = (q4,...,qn+1) ∈Ωn, then we consider the Z2
p-action
(SK = Ck(Λ)/K,NK = H/K = ⟨ϕ1,ϕ2⟩),
where ϕ1 and ϕ2 are as defined in the proof of Theorem 4.
For j = 1,2, set Sj ∶= SK/⟨ϕj⟩and denote by πj ∶Sj →¯C the branched regular covering map with deck group
⟨τj⟩= NK/⟨ϕj⟩≅Zp.
The branch locus of π1 is given by the set {qt+1,...,qn+1} and therefore an algebraic
description for S1 is given by
S1 ∶yp
1 = (x −qt+1)
n+1
∏
j=t+2
(x −qj)sj,
where sn+1 ≡−(1 + st+2 + ⋯+ sn) mod p. In this model π1(x,y1) = x. Similarly, the branch locus of π2 is given by
the set {q1,...,qt,qt+2,...,qn+1} and therefore an algebraic description for S2 is given as follows. If K is of type
(1) then S2 is given by
S2 ∶yp
2 =
n+1
∏
j=3
(x −qj)rj
where rn+1 ≡−(1 + r3 + ⋯+ rn) mod p. If K is of type (2) then
S2 ∶yp
2 =
t
∏
i=2
(x −qi)li
n+1
∏
j=t+2
(x −qj)rj
where rn+1 ≡−(1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn) mod p. In this model π2(x,y2) = x.
All the above coupled with the discussion in §3.1 is the proof of the following result.
Proposition 5. Let (SK,NK) be a Z2
p-action of signature (0;pn+1). Then, with the same notations as in Theorem
4, an algebraic description of SK is given as follows.
If K is of type (1) then
SK ∶
⎧⎪⎪⎨⎪⎪⎩
yp
1 = x∏n+1
j=3 (x −qj)sj
yp
2 = ∏n+1
j=3 (x −qj)rj
where sn+1 ≡−(1 + st+2 + ⋯+ sn) mod p and rn+1 ≡−(1 + r3 + ⋯+ rn) mod p.
If K is of type (2) then
SK ∶
⎧⎪⎪⎨⎪⎪⎩
yp
1 = (x −qt+1)∏n+1
j=t+2(x −qj)sj
yp
2 = ∏t
i=2(x −qi)li ∏n+1
j=t+2(x −qj)rj
where sn+1 ≡−(1 + st+2 + ⋯+ sn) mod p and rn+1 ≡−(1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn) mod p.
In both cases, the group NK corresponds to ⟨ϕ1(x,y1,y2) = (x,y1,ωpy2),ϕ2(x,y1,y2) = (x,ωpy1,y2)⟩.
4.3. Jacobian variety. We recall that the Jacobian variety JS of a compact Riemann surface of genus g is an
irreducible principally polarized abelian variety of dimension g. By the classical Torelli’s theorem, the Jacobian
variety JS determines S, namely, S ≅S′ if and only if JS ≅JS′.
For each group of automorphisms G of a Riemann surface S, we denote by SG the underlying Riemann surface
structure of the orbifold S/G. Let (X,H0) be a generalized Fermat pair of type (p,n), where p is prime. Following
the main result of [7], JX decomposes, up to isogeny, as follows:
(4.1)
JX ∼∏
Hr
JXHr,
14
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
where Hr runs over all subgroups of H0 which are isomorphic to Zn−1
p
and such that X/Hr has positive genus.
In addition, the cyclic p-gonal curves XHr run over all curves of the form
yp =
r
∏
j=1
(x −µj)αj,
where {µ1,...,µr} ⊂{∞,0,1,λ1,...,λn−2}, µi ≠µj if i ≠j, and αj ∈{1,2,...,p −1} satisfying that:
(i) if every µj ≠∞, then α1 = 1, α2 + ⋯+ αr ≡−1 mod p, and
(ii) if some µj = ∞, then α1 + ⋯+ αj−1 + αj+1 + ⋯+ αr ≡−1 mod p.
Let p be a prime number and let (S,N) be a Zm
p -action of signature (0;pn+1). As observed in Proposition 2,
there is a generalized Fermat pair (X,H0) of type (p,n), and a subgroup K ≅Zn−m
p
of H0 such that S ≅X/K
and N ≅H0/K. Since
JX ∼JS × P(X/S).
where P(X/S) is the Prym variety associated to the covering map X →S = X/K, the isogeny decomposition
(4.1) permits to state the following conjecture.
Conjecture. Let p ⩾2 be a prime number and let (S,N) be a Zm
p -action of signature (0;pn+1), where 2 ⩽m ⩽n−1.
Then
JS ∼∏
L∈L
JSL where L = {L ⩽N ∶L ≅Zm−1
p
}.
The conjecture above does hold for the case m = 2, and the proof can be obtained as a consequence of a result
due to Kani-Rosen in [22].
Theorem 5. Let p ⩾2 be a prime number and n ⩾3. Let (S,N) be a Z2
p-action of signature (0;pn+1). Then
JS ∼∏
L∈L
JSL where L = {L ⩽N ∶L ≅Zp}.
Proof. Let us write N = ⟨ϕ1,ϕ2⟩. We have that
L = {L1 = ⟨ϕ1⟩,L2 = ⟨ϕ2⟩,L3 = ⟨ϕ1ϕ2⟩,...,Lp+1 = ⟨ϕ1ϕp−1
2
}.
Observe that if i ≠j then LiLj = LjLi and ⟨Li,Lj⟩= N, showing that the genus of S/⟨Li,Lj⟩is zero. Let
Π ∶S →S/N and Πi ∶S →S/Li be the regular covering maps with deck groups N and Li respectively, for each
i ∈{1,...,p + 1}. Observe that if q ∈S/N is a ramification value of Π then Π−1(q) consists of p points, all of
them with N-stabilizer Lj for some j. This shows that if ci is the number of points of S that are fixed by Li then
c1 + ⋯+ cp+1 = (n + 1)p. The Riemann-Hurwitz formula applied to Πi implies that 2g −2 = 2pγi −2p + ci(p −1),
where γi is the genus of S/Li. Consequently, one has that
(4.2)
(2g −2)(p + 1) = 2p(γ1 + ⋯+ γp+1) −2p(p + 1) + (n + 1)p(p −1).
Now, by considering (3.1) with m = 2, we see that g = ((n −1)p −2)(p −1)/2 and the equality (4.2) turns into
γ1 + ⋯+ γp+1 = g. The proof follows from [22, Theorem C].
□
5. Example 1: The case m = 2,k = p prime and n = 3
For the sake of clarity, we now specialize our results to the case n = 3, that is, Z2
p-actions of signature (0;p4),
and describe some examples in detail. Note that the Riemann surfaces corresponding to such Z2
p-actions have
genus (p −1)2 and form complex one-dimensional families. In this case, we assume p ⩾3.
By applying Theorem 4, one sees that the collection F(p,3) consists of the following groups.
(1) K(r,s) = ⟨ar
1as
2a−1
3 ⟩, where r,s ∈{0,1,...,p −1} satisfy (r,s) ∉{(0,0),(p −1,p −1)}.
(2) K(l) = ⟨al
1a−1
2 ⟩where l ∈{1,...,p −1}.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
15
Example 1. Consider the Z2
p-action of signature (0;p4) associated to the group K = K(0,p −1) = ⟨ap−1
2
a−1
3 ⟩. By
Proposition 5, the family formed by the Riemann surfaces SK are algebraically represented by
(5.1)
SK ∶
⎧⎪⎪⎨⎪⎪⎩
yp
1 = x(x −1)p−1
yp
2 = (x −λ)p−1
where λ ∈C −{0,1}. A routine computation shows that the maps
a(x,y1,y2) = ( λ
x, λ1/py2
x
, λ1−1/py1
x
) and b(x,y1,y2) = ( x−λ
x−1 , (1−λ)1−1/p(x−λ)
(x−1)y2
, (1−λ)1−1/px
y1
)
are automorphisms of SK and ⟨a,b⟩≅Z2
2. In addition, one has that aϕ1a = ϕ2, aϕ2a = ϕ1, bϕ1b = ϕ−1
2 , bϕ2b = ϕ−1
1 ,
where ϕ1 and ϕ2 generate N ≅Z2
p and are given in Proposition 5. Now, if we set
R ∶= ϕ1ϕ2, S ∶= b, ˆR ∶= ϕ1ϕ−1
2 , ˆS ∶= a then Aut(SK) ⩾⟨S,R⟩× ⟨ˆS, ˆR⟩≅Dp × Dp.
The signature of the action of this last group is (0;2,2,2,p) and therefore, as a maximal signature [30], up to
finitely many exceptions, the automorphism group of SK is isomorphic to Dp × Dp. Later we shall see that there
is only one exceptional member with more than 4p2 automorphisms.
The interest in this family comes from the following fact. A well-known result due to Accola [1] states that if a
p-gonal Riemann surface has genus g > (p −1) then the p-gonal morphism is unique. The family described above
was considered by Costa, Izquierdo and Ying in [9], when they noticed that it has two p-gonal morphisms. With
our notation, such morphisms are R(x,y1,y2) = (x,ωpy1,ωpy2) and ˆR(x,y1,y2) = (x,ωpy1, ¯ωpy2). We should
point out that the algebraic description of this family given here differs from the one already known for such
surfaces; see [9, Section 5].
We recall that the action of ⟨Φ1,Φ2⟩= Autg(H) ≅S4 on F(p,3) is given as follows (see Proposition 1).
Φ1(a1,a2,a3,a4) = (a2,a1,a3,a4) and Φ2(a1,a2,a3,a4) = (a2,a3,(a1a2a3)−1,a1).
For instance, observe that
(5.2)
Φ1(K(r,s)) = K(s,r),Φ2(K(l)) = K(0,l) and Φ2(K(0,s)) = K(u,u)
where u ∈{1,...,p −1} satisties u(1 + s) ≡−1 mod p.
Example 2. We proceed to describe explicitly the orbits of this action for p = 5. Note that F(5,3) consists of 27
groups. By considering (5.2), it suffices to restrict our attention to the groups
K(r,s) where (r,s) ∈{0,1,2,3,4}2 −{(0,0),(4,4)} and r < s.
After some computations, one can see that there are exactly four Autg(H)-orbits, represented by K(0,1), K(0,2),
K(0,4) and K(1,2). All the above coupled with Proposition 5 and Corollary 1 can be summarized as follows. There
are exactly four topologically pairwise non-equivalent Z2
5-actions of signature (0;54). Equivalently, the complex
one-dimensional family formed by the Z2
5-actions of signature (0;54) consists of four irreducible components. The
corresponding Riemann surfaces (of genus 16) are represented by the following curves:
K(0,1)
⎧⎪⎪⎨⎪⎪⎩
y5
1 = x(x −1)(x −λ)3
y5
2 = (x −λ)4
K(0,2)
⎧⎪⎪⎨⎪⎪⎩
y5
1 = x(x −1)2(x −λ)2
y5
2 = (x −λ)4
K(0,4)
⎧⎪⎪⎨⎪⎪⎩
y5
1 = x(x −1)4
y5
2 = (x −λ)4
K(1,2)
⎧⎪⎪⎨⎪⎪⎩
y5
1 = x(x −1)2(x −λ)2
y5
2 = (x −1)(x −λ)3
where λ ∈C −{0,1}.
It is worth mentioning that complete lists of Riemann surfaces of genus 16 with non-trivial automorphisms are
available in the literature. For instance, in the database [23] this family is labeled as O16.271. However, it seems
16
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
that in [23] the computation of the number of classes of topological actions for our family lies beyond the scope of
the algorithm employed.
The determination of the number of Autg(H)-orbits in F(p,n) boils down to a routine –but certainly tedious–
computation. With the help of routines implemented in GAP, we were able to compute the number N of orbits
in F(p,3) for some small primes. This is summarized in the following table.
p
3
5
7
11
13
17
19
23
29
113
N
2
4
6
10
14
20
24
32
48
580
5.0.1. Extra automorphisms. To construct Z2
p-actions of signature (0;p4) endowed with extra automorphisms,
we consider the following elements in Autg(H) ≅S4.
Φ3 = (34),
Φ4 = (12)(34),
Φ5 = (234),
Φ6 = (14)(23),
Φ7 = (24).
As the only non-trivial finite groups of M¨obius transformations that keep setwise invariant a set of four
points in ¯C are isomorphic to Z2,Z3,Z4,Z2
2,D4 and A4, the only subgoups of Autg(H) induced by conformal
automorphisms of SK/NK ≅Cp(Λ)/H are, up to conjugation, summarized in the following table.
label
generators
group
permutation
label
generators
group
permutation
Q∗
1
Φ3
Z2
(34)
Q∗
5
Φ4,Φ6
Z2
2
(12)(34),(14)(23)
Q∗
2
Φ4
Z2
(12)(34)
Q∗
6
Φ1,Φ3
Z2
2
(12),(34)
Q∗
3
Φ5
Z3
(234)
Q∗
7
Φ2,Φ7
D4
(1234),(24)
Q∗
4
Φ2
Z4
(1234)
Q∗
8
Φ4,Φ5
A4
(12)(34),(234)
We recall that Cp(Qj) ∶= {K ∈F(p,3) ∶K is Q∗
j-invariant}.
Proposition 6.
(1) Cp(Q1) = {K(l),K( p−1
2 , p−1
2 ) ∶l = 1,...,p −1}.
(2) Cp(Q2) = {K(1),K(p −1),K(r,p −1 −r) ∶r = 0,...,p −1}.
(3) If p = 3 or p ≡2 mod 3 then Cp(Q3) = ∅, and if p ≡1 mod 3 then Cp(Q3) = K( s
1−s,s) ∶s ∈P1}, where
P1 = {s ∈{1,...,p −1} ∶s2 + s + 1 ≡0 mod p}.
(4) If p ≡3 mod 4 then Cp(Q4) = {K(p −1,0)}, and if p ≡1 mod 4 then Cp(Q4) = {K(p −1,0),K( s
1−s,s) ∶s ∈
P2}, where P2 = {s ∈{1,...,p −1} ∶s2 + 2s + 2 ≡0 mod p}.
(5) Cp(Q5) = {K(p −1),K(0,p −1),K(p −1,0)}.
(6) Cp(Q6) = {K(1),K(p −1),K( p−1
2 , p−1
2 )}.
(7) Cp(Q7) = {K(p −1,0)}.
(8) Cp(Q8) = ∅.
Proof. We shall only check the first and last statement, as the remaining ones are proved analogously. First,
observe that Φ3(K(l)) = K(l) for each l, and therefore K(l) ∈Cp(Q1). Now, notice that
Φ3(K(r,s)) = ⟨ar
1as
2a−1
4 ⟩= ⟨a1+r
1
a1+s
2
a3⟩= ⟨a−1−r
1
a−1−s
2
a−1
3 ⟩= K(p −1 −r,p −1 −s).
It follows that Φ3(K(r,s)) = K(r,s) if and only if r = s = p−1
2 , proving (1). Note that Q∗
2 ⩽Q∗
8 and therefore
Cp(Q8) ⊂Cp(Q2) = {K(1),K(p −1),K(r,p −1 −r) ∶r = 0,...,p −1}.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
17
Note that Φ5(K(1)) and Φ5(K(p −1)) are groups of type (2). Moreover,
Φ5(K(r,p −1 −r)) = ⟨ar
1a−1−r
3
a−1
4 ⟩= ⟨a1+r
1
a2a−r
3 ⟩.
So, for Φ5(K(r,p−1−r)) = K(r,p−1−r), we must have that r ≠0 satifies r2−r−1 ≡0 mod p and r2+r+1 ≡0 mod p,
which is clearly impossible. This proves (8).
□
Example 3 (Continuation of Example 1). The fact that K(0,p−1) ∈Cp(Q∗
5) guarantees that the members of the
family of Riemann surfaces
(5.3)
SK(0,p−1)(λ) ∶
⎧⎪⎪⎨⎪⎪⎩
yp
1 = x(x −1)p−1
yp
2 = (x −λ)p−1
where λ ∈C −{0,1},
admit extra automorphisms, and for them Aut(SK(0,p−1)(λ)) ⩾˜G ≅Z2
p ⋊Z2
2. This situation was already studied in
detail in Example 1.
Observe that the group Q∗
7 contains Q∗
5, but K(0,p −1) ∉Cp(Q∗
7) = {K(p −1,0)}. However, since K(0,p −1) =
Φ1(K(0,p −1)) and Φ1 normalizes Q∗
5, we note that
˜Q∗
7 = Φ1Q∗
7Φ1 = ⟨Φ1Φ2Φ1 = (1342),Φ1Φ7Φ1 = (14)⟩
contains Q∗
5 and Cp( ˜Q∗
7) = {K(0,p −1)}. In particular, there are some members of the family (5.3) admitting
more than 4p2 automorphisms. This observation, coupled with the fact that ˜Q∗
7 is not contained in any group Q∗
i
(nor in any conjugate), shows that if a member of (5.3) has more than 4p2 automorphisms, then
Aut(SK(0,p−1)(λ)) ≅Z2
p ⋊D4.
A computation shows that SK(0,p−1)(λ) has an automorphism of order 4 if and only if λ = 1
2, and this auto-
morphism is given by
c(x,y1,y2) = ( 2x−1
2x ,
2x−1
22−1/pxy2 ,
y1
21−1/p x).
The automorphism group of this special member acts with signature (0;2,4,2p).
Remark 8. The fact that K(p −1) and K(p −1,0) are Autg(H)-equivalent to K(0,p −1) tells to us that the
discussion above can also be carried out with K(p −1) and K(p −1,0). In these cases, we would obtain different
–but equivalent– algebraic descriptions of the same family and of their automorphisms.
6. Example 2: The case m = 2,k = p prime and n = 5
Let p ⩾2 be a prime number. The pencil Cp formed by the smooth complex projective algebraic curves Ct of
genus (p −1)(2p −1) given by
x2p + y2p + z2p + t(xpyp + ypzp + xpzp) = 0 where t ∈¯C −{−1,±2}
has been recently studied in [25]. This pencil is a generalization of the pencil of Kuribayashi-Komiya quartics C2,
also known in the literature as the KFT family (see, for instance, [24] and [28]). Note that
N = ⟨[x ∶y ∶z] ↦[ωpx ∶y ∶z],[x ∶y ∶z] ↦[x ∶ωpy ∶z]⟩≅Z2
p
is a group of automorphisms of each member Ct of Cp. Moreover, the quotient Ct/N has genus zero, and hence
(Ct,N) is a Z2
p-action of signature (0;p6). Furthermore, as proved in [25, Proposition 1], each Ct is endowed with
a group of automorphisms Gp isomorphic to the semidirect product Z2
p ⋊D3 given by
Gp ≅⟨a,b,r,s ∶ap = bp = [a,b] = r3 = s2 = (sr)2 = 1,rar−1 = (ab)−1,rbr−1 = a,sas = (ab)−1,[s,b] = 1⟩.
The group Gp acts on Ct with signature (0;2,2,3,p). Note that N ⊴Gp and Gp/N ≅D3.
Motivated by the above, we apply our results to study Z2
p-actions of signature (0;p6) that admit extra auto-
morphisms and that form complex one-dimensional families.
18
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Let (S,N,G) be triple such that (S,N) is a Z2
p-action of signature (0;p6) and S is endowed with a group of
automorphisms G such that N ⊴G ⩽Aut(S). Observe that the genus of S is (p −1)(2p −1). The quotient G/N
is a isomorphic to a non-trivial finite group L of M¨obius transformations that keep setwise invariant a set of six
points in ¯C. If we require that the corresponding quotient has moduli one (namely, the orbifold (S/N)/L has
four cone points), then there are two scenarios.
(1) L is isomorphic to D3 and the signature of the action of G is (0;2,2,3,p), and
(2) L is isomorphic to Z2
2 and and the signature of the action of G is (0;2,2,p,2p)
We proceed to study these cases separately.
6.1. The case L ≅D3. Note that if p ≠3 then G is isomorphic to a semidirect product of the form Z2
p ⋊D3. Set
L ≅G/N ≅D3 = ⟨a,b ∶a3 = b2 = (ab)2 = 1⟩.
Assume that the regular covering map π ∶S →S/N ramifies over ∞,0,1,q4,q5,q6. It follows that, if we write
Λ = (q4,q5,q6) ∈Ω5, then
S ≅Cp(Λ)/KS and N ≅H/KS for some KS ∈F(p,5,2).
After conjugating by a suitable M¨obius transformation, we can assume
a(z) =
1
1−z, b(z) = λz−λ+1
z−λ
and that q4 = λ, q5 =
1
1−λ, q6 = λ−1
λ .
Note that (q4,q5,q6) ∈Ω5 if and only if λ ≠1
2(−1 ± i
√
3). The group L lifts to a group of automorphisms QS,G of
Cp(Λ) ∶
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
xp
1
+
xp
2
+
xp
3
=
0
λxp
1
+
xp
2
+
xp
4
=
0
1
1−λxp
1
+
xp
2
+
xp
5
=
0
λ−1
λ xp
1
+
xp
2
+
xp
6
=
0
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
⊂P5
such that QS,G/KS ≅G and S/G ≅Cp(Λ)/QS,G.
Lemma 1. The group QS,G is isomorphic to a semidirect product
Z5
p ⋊D3 = ⟨a1,...,a5 ∶ap
j = [ai,aj] = 1⟩⋊⟨A,B ∶A3 = B2 = (AB)2 = 1⟩,
where the action by conjugation of A and B on a1,...,a5 is given by
A ∶(a1,a2,a3,a4,a5) ↦(a2,a3,a1,a5,(a1⋯a5)−1) and B ∶(a1,a2,a3,a4,a5) ↦(a4,(a1⋯a5)−1,a5,a1,a3).
Proof. As proved in [13, Corollary 9], the fact that a induces the permutation (123)(456) on the subindices of
q1,...,q6 implies that every lifting A of a is of the form
[x1 ∶... ∶x6] ↦[x3 ∶A2x1 ∶A3x2 ∶A4x6 ∶A5x4 ∶A6x5],
where A2,...,A6 are arbitrary complex numbers that satisfy Ap
2 = Ap
3 = 1,Ap
4 = −λ,Ap
5 =
1
λ−1 and Ap
6 = 1−λ
λ .
Similarly, every lifting B of b is of the form
[x1 ∶... ∶x6] ↦[x4 ∶B2x6 ∶B3x5 ∶B4x1 ∶B5x3 ∶B6x2],
where Bp
2 = −λ,Bp
3 = λ −1,Bp
4 = λ2 −λ + 1,Bp
5 = λ2−λ+1
λ−1 ,Bp
6 = −λ2−λ+1
λ
. Observe that
A3([x1 ∶... ∶x6]) = [A2A3x1 ∶A2A3x2 ∶A2A3x3 ∶A4A5A6x4 ∶A4A5A6x5 ∶A4A5A6x6],
B2([x1 ∶... ∶x6]) = [B4x1 ∶B2B6x2 ∶B3B5x3 ∶B4x4 ∶B5B3x5 ∶B6B2x6],
whereas (AB)2([x1 ∶... ∶x6]) equals to
[A5B3B4x1 ∶A2A4B6x2 ∶A3A6B2B5x3 ∶A2A4B6x4 ∶A5B3B4x5 ∶A3A6B2B5x6].
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
19
Now, once fixed values for λ1/p, (λ −1)1/p and (λ2 −λ + 1)1/p are chosen, if we take
A2 = A3 = 1,A4 = B2 = −λ1/p,A5 = B−1
3
=
1
(λ−1)1/p ,A6 = −(λ−1)1/p
λ1/p
,
B4 = (λ2 −λ + 1)1/p,B5 = (λ2−λ+1)1/p
(λ−1)1/p
,B6 = −(λ2−λ+1)1/p
λ1/p
,
then A3 = B2 = (AB)2 = 1. Note that
Aa1A−1([x1 ∶... ∶x6]) = A([ωp
x2
A2 ∶x3
A3 ∶x1 ∶x4
A4 ∶x5
A5 ∶x6
A6 ]) = [x1 ∶ωpx2 ∶x3 ∶... ∶x6] = a2.
The remaining relations are obtained analogously.
□
Remark 9. There are some exceptional values of λ.
a. There is the possibility for the existence of a M¨obius transformation c satisfying that c2 = a such that
c ∶q4 ↦q1 ↦q5 ↦q2 ↦q6 ↦q3 ↦q4.
This asserts that λ = λ0 ∈{2,ω6, ¯ω6} and c(z) =
z+(1−λ0)2
(1−λ0)(z−λ0). For instance, if λ0 = 2 then c(z) = z+1
2−z permutes
cyclically ∞,q5 = −1,0,q6 = 1
2,1,q4 = 2, and c2 = a. Set Λ0 = (2,−1, 1
2). By arguing as in the previous proposition,
c lifts to an automorphisms C of Cp(Λ0) which satisfies C6 = [C,A] = (CB)2 = 1. Also, the action by conjugation
of C on a1,...,a5 is given by
C ∶(a1,a2,a3,a4,a5) ↦(a5,(a1⋯a5)−1,a4,a1,a2).
Note that ⟨C,B⟩≅D6 and ⟨QS,G,C⟩≅Z5
p ⋊D6.
b. Similarly, if λ = λ1 = ±i and Λ1 = (±i, 1±i
2 ,1 ± i) then the M¨obius transformation d(z) = ±i + 1 lifts to an
automorphisms D of order 4 of Cp(Λ1) such that ⟨D,A⟩≅S4 and ⟨QS,G,D⟩≅Z5
p ⋊S4. The action by conjugation
of D on a1,...,a5 is given by
D ∶(a1,a2,a3,a4,a5) ↦(a1,a3,(a1⋯a5)−1,a2,a5).
We recall that, as introduced in §3.4,
NQS,G = {Φ ∈Autg(H) ∶ΦQ∗
S,GΦ−1 = Q∗
S,G} ⩽Autg(H)
is the normalizer of Q∗
S,G in Autg(H). Abusing notation, we shall denote by A ∈Q∗
S,G the image of A by ρQS,G
(see (3.2)). Consider the natural group isomorphism
σ ∶Autg(H) →S6 = Sym{a1,...,a6}
and write u = σ(A) = (123)(456) and v ∶= σ(B) = (14)(26)(35). Then NQS,G is isomorphic to the normalizer N
of ⟨u,v⟩in S6, which is isomorphic to D3 × D3 and
N/⟨u,v⟩= {id, (456),(465),(23)(56),(23)(45),(23)(46)} ≅D3.
All the above is the proof of the following proposition.
Proposition 7. NQS,G is generated by Q∗
S,G and the geometric automorphisms of H given by
Ψ2(a1,...,a6) = (a1,a2,a3,a5,a6,a4) and Ψ5(a1,...,a6) = (a1,a3,a2,a6,a5,a4).
We recall that, following Corollary 2, the cardinality of Cp(QS,G)/NQS,G agrees with the number of pairwise
topologically inequivalent triples ( ˆS, ˆN, ˆG), where ( ˆS, ˆN) is a Z2
p-action of signature (0;p6), and ˆN ◁ˆG ⩽Aut( ˆS)
is such that ˆG/ ˆN induces Q∗
S,G. Here
Cp(QS,G) = {K ∈F(p,5,2) ∶K is Q∗
S,G-invariant}.
Proposition 8. If K ∈Cp(QS,G) then one of the following statements holds.
20
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
(1) p = 3 or p ≡1 mod 3, there is l ∈{1,...,p −1} satisfying l2 + l + 1 ≡0 mod p and
K = K(l) ∶= ⟨a1a2a3,al
1a−1
2 ,al
4a−1
6 ⟩.
(2) There are r,s ∈{0,...,p −1} satisfying r2 + s2 −rs ≡1 mod p and
K = K(r,s) ∶= ⟨a1a2a3,ar
1as
2a−1
4 ,a−s
1 ar−s
2
a−1
5 ⟩.
Remark 10. The group K(r,s), in part (2) of the above proposition, corresponds to the group in case (1) of
Theorem 4 with the following parameters:
r3 = s3 = p −1, r4 = r, s4 = s, r5 =
⎧⎪⎪⎨⎪⎪⎩
p −s
s > 0
0
s = 0
and s5 =
⎧⎪⎪⎨⎪⎪⎩
r −s
r ⩾s
p + r −s
r < s
Proof. Let K ∈Cp(QS,G) and let θ ∶H →Z2
p be a group epimorphism such that K = ker(θ). As a1 ∉K, we have
that ϕ1 ∶= θ(a1) has order p.
1. Assume θ(a2) ∈⟨ϕ1⟩. The fact that A(a1) = a2,A(a2) = a3,A(a3) = a1 implies that
θ(a2) = ϕl
1 and θ(a3) = ϕl2
1 for some l ∈{1,...,p −1} such that 1 + l + l2 ≡1 mod p.
If we write ϕ2 ∶= θ(a4) then the we have that a5 = B(a3) and therefore θ(a5) = ϕl2
2 . Similarly, θ(a6) = ϕl
2. Note
that ϕ2 ∉⟨ϕ1⟩and ⟨ϕ1,ϕ2⟩≅Z2
p. It follows that
K = ⟨a1a2a3,a4a5a6,al
1a−1
2 ,al2
1 a−1
3 ,al2
2 a−1
5 ,al
2a−1
6 ⟩= ⟨a1a2a3,al
1a−1
2 ,al
2a−1
6 ⟩= K(l).
2. Assume θ(a2) ∉⟨ϕ1⟩. If we write ϕ2 ∶= θ(a2) then ⟨ϕ1,ϕ2⟩≅Z2
p. The action of A implies that, if we write
(6.1)
θ(a3) = ϕˆr
1ϕˆs
2, then ˆrˆs ≡1 mod p and ˆr + ˆs2 ≡0 mod p.
Similarly, if we set ϕ(ai) = ϕri
1 ϕsi
2 for i = 4,5,6 then the action of A show that
(6.2)
r5 ≡ˆrs4 mod p, s5 ≡r4 + ˆss4 mod p, r6 ≡ˆrr4 + s4 mod p, s6 ≡ˆsr4 mod p.
Now, the action of B on a4 implies that (ϕr4
1 ϕs4
2 )r4(ϕr6
1 ϕs6
2 )s4 = ϕ1, showing that
s4(r4 + s6) ≡0 mod p and r2
4 + r6s4 ≡1 mod p.
We consider each possible case separately.
2.1 Assume s4 = 0, and therefore r4 = ±1. By (6.2) we have that r5 = 0,s5 = r4,r6 = ˆrr4 and s6 = ˆsr4 and
therefore θ is given by
(a1,...,a6) ↦(ϕ1,ϕ2,ϕˆr
1ϕˆs
2,ϕ1,ϕ2,ϕˆr
1ϕˆs
2) or (ϕ1,ϕ2,ϕˆr
1ϕˆs
2,ϕ−1
1 ,ϕ−1
2 ,ϕ−ˆr
1 ϕ−ˆs
2 )
according to whether or not r4 = 1. By considering that a1⋯a6 = 1 in the former case, and the action of B coupled
with (6.1) in the latter, we obtain ˆr = ˆs = −1. We conclude that
K = K3 ∶= ⟨a1a2a3,a1a−1
4 ,a2a−1
5 ⟩or K = K4 ∶= ⟨a1a2a3,a1a4,a2a5⟩,
according to whether or not r4 = 1.
2.2 Assume r4 = −s6, and therefore r2
4 + r6s4 = 1. By (6.2) we have that s6 = ˆsr4, and therefore r4 = s6 = 0 or
ˆs = −1.
2.2.1 Assume r4 = s6 = 0. We have r5 = ˆrs4,s5 = ˆss4,r6 = s4, and s4 = ±1. It follows that θ is given by
(a1,...,a6) ↦(ϕ1,ϕ2,ϕˆr
1ϕˆs
2,ϕ±1
2 ,ϕ±ˆr
1 ϕ±ˆs
2 ,ϕ±1
1 ).
The action of B implies that ˆr = ˆs = −1 and we then obtain that θ is given by
(ϕ1,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕ1) or (ϕ1,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕ−1
2 ,ϕ1ϕ2,ϕ−1
1 ).
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
21
Thus, we conclude that
K = K1 ∶= ⟨a1a2a3,a1a−1
6 ,a2a−1
4 ⟩or K = K2 ∶= ⟨a1a2a3,a1a6,a2a4⟩
respectively.
2.2.2 Assume ˆs = −1. We have that ˆr = −1 and therefore r5 = −s4,s5 = r4 −s4,r6 = s4 −r4 and s6 = −r4. By
proceeding as before, the action of B shows that r2
4 + s2
4 −r4s4 ≡1 mod p. Thus, if we let r ∶= r4 and s ∶= s4 then
θ is given by
(a1,...,a6) ↦(ϕ1,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕr
1ϕs
2,ϕ−s
1 ϕr−s
2
,ϕs−r
1
ϕ−r
2 )
and therefore K = K(r,s).
The fact that K(0,1) = K1,K(0,p −1) = K2,K(1,0) = K3 and K(p −1,0) = K4 finishes the proof.
□
Theorem 6. Let p ⩾2 be a prime number. Consider the set
Fp = {(r,s) ∶2 ⩽s < r ⩽p −2 and r2 + s2 −rs ≡1 mod p}
and let γ be its cardinality. Write
α ∶=
⎧⎪⎪⎨⎪⎪⎩
0
if p ≡2 mod 3
1
otherwise
and
β ∶=
⎧⎪⎪⎨⎪⎪⎩
1
if p = 2
2
otherwise
Then the cardinality of Cp(QS,G)/NQS,G is α + β + 1
3γ.
Proof. Following Proposition 7, the group NQS,G is generated by Q∗
S,G and by Ψ2,Ψ5. The fact that each K ∈
F(p,5,2) is ⟨A,B⟩-invariant shows that we only need to determine the number of orbits of the action of ⟨Ψ2,Ψ5⟩≅
D3 on Cp(QS,G). For simplicity, we write
Ψ2 = (456),Ψ2
2 = (465),Ψ3 = Ψ5Ψ2
2 = (23)(56),Ψ5 = (23)(46).
Observe that the groups K(l) appear if and only if p = 3 or p ≡1 mod 3, and in such a case there are exactly two
groups of this type; namely K(l) and K(l2). Note that
Ψ3(K(l)) = ⟨a1a2a3,al
1a−1
3 ,al
4a−1
5 ⟩.
The equalities (a1a2a3 ⋅al
1a−1
3 )−1 = al2
1 a−1
2
and ((a1a2a3) ⋅(al
4a−1
5 ))−1 = al2
4 a−1
6
ensures that Ψ3(K(l)) = K(l2). In
a very similar way, it can be seen that
Ψ2(K(l)) = K(l) and Ψ2(K(l2)) = K(l2).
It follows that K(l) and K(l2) form α orbits. We now consider the groups K(r,s) where r = 0 or s = 0 or r = s.
Observe that these cases yield only six groups, namely,
K(0,1),K(0,p −1),K(1,0),K(p −1,0),K(1,1) and K(p −1,p −1).
We claim that they form β orbits. In fact, a routine computation shows that
{K(0,1),K(1,0),K(p −1,p −1)} and {K(0,p −1),K(p −1,0),K(1,1)}
are two orbits if p ⩾3, and that they colapse in one orbit when p = 2. All the above coupled with the equality
Ψ5(K(r,s)) = K(s,r) allows us to restrict our attention to the groups K(r,s) where (r,s) belongs to the parameter
set Fp. We write Fp = {K(r,s) ∶(r,s) ∈Fp}. Note that F2 and F3 are empty. Therefore we assume p ⩾5. If
(r,s) ∈Fp then Ψ5(K(r,s)) = K(s,r) ∉Fp. Thus, the set Fp splits into orbits of length 1 or 3. The equalities
Ψ2(K(r,s)) = K(p + s −r,p −r) and Ψ2
2(K(r,s)) = K(p −s,r −s)
together with the fact that (r,s),(p −s,r −s) and (p + s −r,p −r) are pairwise distinct show that Fp splits into
orbits of length 3 only. This completes the proof.
□
22
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Example 4. If p = 2 then C2(QS,G) consists of only three groups: K(0,1),K(1,1) and K(1,0), and they are
equivalent. We then obtain the well-known fact that among the topological classes of actions of Z2
2 on Riemann
surfaces S genus g = 3 with signature (0;26), there is only one of them for which S has a group of automorphisms
G ≅QS,G/K(0,1) which is isomorphic to
⟨a1,a2,A,B ∶a2
1 = a2
2 = [a1,a2] = A3 = B2 = (AB)2 = 1,Aa1A = a2,Aa2A = a1a2,Ba1B = a2⟩≅G2 ≅S4,
and acts with signature (0;2,2,2,3). These Riemann surfaces form the family of quartics
x4 + y4 + z4 + t(x2y2 + x2z2 + y2z2) = 0,
denoted by C2 at the beginning of this section. We apply Proposition 5 to obtain that
⎧⎪⎪⎨⎪⎪⎩
y2
1 = x(x −1)(x −λ)(x −
1
1−λ)
y2
2 = (x −1)(x −
1
1−λ)(x −λ−1
λ )
where λ ∈C −{0,1},
is another algebraic description for this family. Observe that, while the (generic) members of this family are
non-hyperelliptic, they are the fiber product of two Riemann surfaces of genus one.
Furthermore, with the notations of Remark 9, observe that C(K(0,1)) = ⟨a4a5a6,a1a6,a2a5a6⟩= K(0,1) and
therefore K(0,1) is C-invariant. Similarly, K(1,1) is D-invariant. We then conclude that
S0 ∶= C2(Λ0)/K(0,1) and S1 ∶= C2(Λ1)/K(1,1)
are members of C2 admitting a group of automorphisms isomorphic to Z2
2 ⋊D6 ≅Z2 × S4 and Z2
2 ⋊S4 ≅Z2
4 ⋊S3
respectively. These Riemann surfaces are the unique hyperelliptic curve of genus 3 with 48 automorphisms, and
the Fermat quartic, respectively.
Example 5. If p = 3 then C3(QS,G) consists of seven groups, and they split into three classes, represented by
K(0,1),K(0,2) and K(1). It follows that there are precisely three classes of topologically inequivalent triples
(S,N,G), where (S,N) is a Z2
3-action of signature (0;36) such that N ⊴G and G/N ≅D3. In this case, G acts
with signature (0;2,2,3,3). Observe that there are at most three groups G as before. In the former case, we have
G = G(0,1) ∶= QS,G/K(0,1) = ⟨a1,...,a6,A,B⟩/⟨a1 = a6,a2 = a4,a3 = a5 = (a1a2)−1⟩
≅⟨a1,a2,A,B ∶...,Aa1A−1 = a2,Aa2A−1 = (a1a2)−1,Ba1B = a2⟩.
Similarly, in the second and third case, we obtain that
G = G(0,2) ∶= QS,G/K(0,2) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = a2,Aa2A−1 = (a1a2)−1,Ba1B = a−1
2 ⟩
G = G(1) ∶= QS,G/K(1) ≅⟨a1,a2,A,B ∶...,[A,a1] = [A,a2] = 1,Ba1B = a2⟩.
The groups G(0,1),G(0,2) and G(1) are pairwise non-isomorphic and turn out to be semidirect products, as in
the case p ≠3. Note that G(0,1) ≅G3 and therefore the Riemann surfaces C3(Λ)/K(0,1) form the family C3.
Finally, it is worth noticing that there are two topologically inequivalent actions of G3 in genus 10, as can be
obtained by using the routines given in [4]. One of them is the one represented by K(0,1) and the other is not
obtained here, as in this case the corresponding Z2
3-quotient has genus two.
Now, we can extend the previous two examples to the general case.
Proposition 9. Let p ⩾5 be a prime number. If F is a complex one-dimensional family of compact Riemann
surfaces of genus (p −1)(2p −1) endowed with a group of automorphisms G isomorphic to a semidirect product
of the form Z2
p ⋊D3, then G ≅Gp. In particular, family of Kuribayashi-Komiya curves Cp corresponds to one
irreducible component of F.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
23
Proof. By the Riemann-Hurwitz fomula, if S belongs to a complex one-dimensional family F as in the statement
of the proposition, then (S,N) is a Z2
p-action of signature (0;p6), where N is the p-Sylow subgroup of G. Then
S ≅Cp(Λ)/K and G ≅QS,G/K for some K ∈Cp(QS,G). We now study the quotients QS,G/K.
1. Assume K = K(l) ∶= ⟨a1a2a3,al
1a−1
2 ,al
4a−1
6 ⟩, where p ≡1 mod 3 and l ∈{1,...,p −1} satisfies l2 + l + 1 ≡
0 mod p. In this case,
G(l) ∶= QS,G/K(l) = ⟨a1,...,a6,A,B⟩/⟨a1a2a3 = 1,a6 = al
4,a2 = al
1⟩.
Note that in the quotient a2 = al
1,a3 = al2
1 ,a4 = al2
6 ,a5 = al
6. It follows that
G(l) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = al
1,Aa2A−1 = al2
2 ,Ba1B = a2⟩.
2. Assume K = K(r,s) ∶= ⟨a1a2a3,ar
1as
2a−1
4 ,a−s
1 ar−s
2
a−1
5 ⟩where r,s ∈{0,...,p −1} satisfy r2 + s2 −rs ≡1 mod p.
In this case, one has that
G(0,1) = QS,G/K(0,1) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = a2,Aa2A−1 = (a1a2)−1,Ba1A = a2⟩.
G(0,p −1) = QS,G/K(0,p −1) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = a2,Aa2A−1 = (a1a2)−1,Ba1B = a−1
2 ⟩.
G(1,0) = QS,G/K(1,0) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = a2,Aa2A−1 = (a1a2)−1,Ba1B = a1,Ba2B = (a1a2)−1⟩.
G(p −1,0) = QS,G/K(p −1,0) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = a2,Aa2A−1 = (a1a2)−1,Ba1B = a−1
1 ,Ba2B = a1a2⟩,
and for r,s ≠0 we have that
G(r,s) = QS,G/K(r,s) ≅⟨a1,a2,A,B ∶...,Aa1A−1 = al
1,Aa2A−1 = al2
2 ,Ba1B = a2⟩.
Observe that the elements σ1 ∶= a2 and σ2 ∶= (a1a2)−1 of K(0,1) generate a subgroup of it isomorphic to
K(1,0). Likewise, it can be seen that
G(l) ≅G(0,1) ≅G(0,p −1) ≅G(1,0) ≅G(p −1,0) ≅G(r,s).
Finally, by considering the elements σ1 ∶= a2
1a2 and σ2 ∶= a−1
1 a−2
2
of K(0,1), we conclude that G ≅Gp.
□
Example 6. The following table summarizes the number N of topologically distinct triples (S,N,Gp) where
(S,N) is a Z2
3-action of signature (0;36) and the signature of S/Gp is (0;2,2,3,p), for some small primes (c.f.
[25, Proposition 2]).
p
5
7
11
13
17
19
23
29
31
N
2
3
3
4
4
5
5
6
7
6.2. The case L = Z2
2. Note that if p ≠2 then G is isomorphic to a semidirect product of the form Z2
p ⋊Z2
2. Set
L ≅G/N ≅Z2
2 = ⟨a,b ∶a2 = b2 = (ab)2 = 1⟩.
If ∞,0,1,q4,q5,q6 are the cone points of S/N then, after conjugating by a suitable M¨obius transformation, we
can assume
a(z) = −z, b(z) = λ
z and that q4 = λ, q5 = −1, q6 = −λ.
Note that Λ = (q4,q5,q6) ∈Ω5 if and only if λ ≠0,±1. It follows that
S ≅Cp(Λ)/KS and N ≅H/KS for some KS ∈F(p,5,2),
where
Cp(Λ) ∶
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
xp
1
+
xp
2
+
xp
3
=
0
λxp
1
+
xp
2
+
xp
4
=
0
−xp
1
+
xp
2
+
xp
5
=
0
−λxp
1
+
xp
2
+
xp
6
=
0
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
⊂P5
The group L lifts to a group of automorphisms QS,G of Cp(Λ) such that QS,G/KS ≅G and S/G ≅Cp(Λ)/QS,G.
24
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Lemma 2. The structure of the group QS,G is as follows.
(1) If p ⩾3 is a prime number, then QS,G is isomorphic to a semidirect product
Z5
p ⋊Z2
2 = ⟨a1,...,a5 ∶ap
j = [ai,aj] = 1⟩⋊⟨A,B ∶A2 = B2 = (AB)2 = 1⟩,
where the action by conjugation of A and B on a1,...,a5 is given by
A ∶(a1,a2,a3,a4,a5) ↦(a1,a2,a5,(a1⋯a5)−1,a3) and B ∶(a1,a2,a3,a4,a5) ↦(a2,a1,a4,a3,(a1⋯a5)−1).
(2) If p = 2, then QS,G is isomorphic to
⟨a1,...,a5,A,B ∶a2
j = [ai,aj] = B2 = 1,A2 = a1,(AB)2 = a4a5,Aa1A−1 = a1,Aa2A−1 = a2,
Aa3A−1 = a5,Aa4A−1 = a1a2a3a4a5,Ba1B = a2,Ba3B = a4,Ba5B = a1a2a3a4a5⟩.
Proof. The proof follows the same ideas as the proof of Lemma 1. Every lifting of a is of the form
A([x1 ∶... ∶x6]) = [x1 ∶α2x2 ∶α3x5 ∶α4x6 ∶α5x3 ∶α6x4] where αp
2 = αp
3 = αp
4 = αp
5 = αp
6 = −1,
and every lifting of b is of the form
B([x1 ∶... ∶x6]) = [x2 ∶β2x1 ∶β3x4 ∶β4x3 ∶β5x6 ∶β6x5] where βp
2 = βp
4 = λ,βp
3 = 1,βp
5 = −1,βp
6 = −λ.
1. If p ⩾3 then we may take α2 = α3 = α4 = α5 = α6 = −1 and β2 = β4 = η,β3 = 1,β5 = −1,β6 = −η where η is a
fixed choice of λ1/p, to get
A([x1 ∶... ∶x6]) = [−x1 ∶x2 ∶x5 ∶x6 ∶x3 ∶x4] and B([x1 ∶... ∶x6]) = [x2 ∶ηx1 ∶x4 ∶ηx3 ∶−x6 ∶−ηx5],
which satisfy the relations A2 = B2 = (AB)2 = 1.
2. If p = 2 then we may take α2 = α3 = α4 = α5 = α6 = i and β2 = β4 = −η,β3 = 1,β5 = −i,β6 = −iη where η is a
fixed choice of λ1/2, to get
A([x1 ∶... ∶x6]) = [−ix1 ∶x2 ∶x5 ∶x6 ∶x3 ∶x4] and B([x1 ∶... ∶x6]) = [x2 ∶−ηx1 ∶x4 ∶−ηx3 ∶−ix6 ∶−iηx5],
which satisfy the relations A2 = a1,B2 = 1,(AB)2 = a4a5.
□
Remark 11. Observe that if λ = λ0 = i and Λ1 = (i,−1,−i), then the M¨obius transformation d(z) = iz satisfies
that d2 = a and lifts to an automorphisms D of order 4 of Cp(Λ1) such that ⟨D,B⟩≅D4 and ⟨QS,G,D⟩≅Z5
p ⋊D4.
The action by conjugation of D on a1,...,a5 is given by
D ∶(a1,a2,a3,a4,a5) ↦(a1,a2,a4,a5,(a1⋯a5)−1).
Consider the group isomorphism σ ∶Autg(H) →S6 = Sym{a1,...,a6} and write u = σ(A) = (35)(46) and v ∶=
σ(B) = (12)(34)(56).
Proposition 10. NQS,G is generated by Q∗
S,G and the geometric automorphisms of H given by
Ψ2(a1,...,a6) = (a1,a2,a3,a6,a5,a4) and Ψ3(a1,...,a6) = (a1,a2,a4,a3,a6,a5).
Proof. We argue as done in Proposition 7, after noticing that NQS,G is isomorphic to the normalizer N of ⟨u,v⟩
in S6, which is given by N = ⟨u,v,Ψ2 = (46),Ψ3 = (34)(56)⟩. Note that N/⟨u,v⟩≅Z2
2.
□
By Corollary 2, the number of pairwise topologically inequivalent actions (S,N,G) in genus (2p−1)(p−1) with
a Z2
p-action of genus zero and S/G of signature (0;2,2,p,2p) corresponds to the cadinality of Cp(QS,G)/NQS,G.
The following proposition describes the members of Cp(QS,G).
Proposition 11. If p ⩾3 is prime and K ∈Cp(QS,G) then one of the following statements holds.
(1) There are r,s ∈{0,...,p −1} satisfying that r + s ∈{ p−1
2 , 3p−1
2 } and
K = K(r,s) ∶= ⟨ar
1as
2a−1
3 ,a3a−1
5 ,a4a−1
6 ⟩.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
25
(2) K agrees with
K1 ∶= ⟨a1a−1
2 ,a3a−1
4 ,a5a−1
6 ⟩, K2 ∶= ⟨a1a−1
2 ,a3a−1
6 ,a4a−1
5 ⟩,
K5 ∶= ⟨a1a−1
2 ,a3a−1
5 ,a4a−1
6 ⟩or K6 ∶= ⟨a1a2,a3a−1
5 ,a4a−1
6 ⟩.
(3) There is r ∈{0,...,p −1} and
K = K3(r) ∶= ⟨ar
1a−1
3 a4,a1a2,a3a6⟩or K = K4(r) ∶= ⟨ar
1a−1
3 a6,a1a2,a3a4⟩
If p = 2 and K ∈C2(QS,G) then K equals
¯K1 = ⟨a1a2,a3a4,a3a5⟩, ¯K2 = ⟨a1a2,a3a4,a1a3a5⟩,
¯K3 = ⟨a1a2,a3a5,a1a3a4⟩or
¯K4 = ⟨a1a2,a4a5,a1a3a4⟩,
Proof. Let K ∈Cp(QS,G) and let θ ∶H →Z2
p be a group epimorphism such that K = ker(θ). As a1 ∉K, we have
that ϕ1 ∶= θ(a1) has order p.
1.
Assume θ(a2) ∉⟨ϕ1⟩and write ϕ2 ∶= θ(a2) in such a way that ⟨ϕ1,ϕ2⟩≅Z2
p. Set θ(a3) ∶= ϕr
1ϕs
2 for
r,s ∈{0,...,p −1} that are not simultaneously equal to zero.
The fact that A(a3) = a5,A(a1) = a1 and A(a2) = a2 implies that ϕr
1ϕs
2 = θ(a3). Similarly, it can be seen that
θ(a6) = θ(a4). Now, the fact that B(a1) = a2 and B(a3) = a4 implies that θ(a4) = ϕr
2ϕs
1. It follows that
θ = (ϕ1,ϕ2,ϕr
1ϕs
2,ϕs
1ϕr
2,ϕr
1ϕs
2,ϕs
1ϕr
2)
and therefore K = K(r,s). The relation a1⋯a6 = 1 implies that 2(r+s) ≡−1 mod p and therefore r+s ∈{ p−1
2 , 3p−1
2 }.
2. Assume θ(a2) ∈⟨ϕ1⟩and write θ(a2) = ϕl
1 for some l ∈{1,...,p −1}. We claim that θ(a3) ∉⟨ϕ1⟩. In fact,
otherwise, if we write θ(a3) = ϕm
1 then θ(a5) = ϕlm
1
and similarly θ(a6) = ϕml
1 . This contradicts the surjectivity of
θ. Thus, we write ϕ2 ∶= θ(a3) and therefore ⟨ϕ1,ϕ2⟩≅Z2
p. Note that ϕl
1 = θ(a2). As B has order two, we deduce
that l = 1 or l = p −1. Write θ(a5) = ϕr
1ϕs
2 for some r,s ∈{0,...,p −1} that are not simultaneously zero. The
equality θ(a5) = ϕr
1ϕs
2 coupled with the fact that A has order two imply that r(1 + s) ≡0 mod p, and that s = 1
or s = p −1.
2.1 Assume s = 1. It follows that r = 0 and θ(a5) = ϕ2. Write θ(a4) ∶= ϕu
1ϕv
2 then θ(a6) = ϕu
1ϕv
2 and u(l + v) ≡
0 mod p, and v = 1 or v = p −1. The fact that a1⋯a6 = 1 implies that v = p −1 and therefore
(1) u = 0 and l = p −1, and then θ = (ϕ1,ϕ−1
1 ,ϕ2,ϕ−1
2 ,ϕ2,ϕ−1
2 ) and K = K6, or
(2) u = p −1 and l = 1, and then θ = (ϕ1,ϕ1,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕ2,ϕ−1
1 ϕ−1
2 ) and K = K5.
2.2 Assume s = p −1. It follows that θ(a5) = ϕr
1ϕ−1
2 . If we write θ(a4) ∶= ϕu
1ϕv
2 then, by proceeding as in the
previous case, we obtain that θ(a6) = ϕu+rv
1
ϕ−v
2
where v = 1 or v = p−1,u(l+v) ≡0 mod p and 2u ≡r(l−v) mod p.
Moreover, The fact that a1⋯a6 = 1 implies that 1 + l + r + 2u + rv ≡0 mod p.
Assume v = 1. Then
(1) u = 0,l = 1,r = p −1 and θ = (ϕ1,ϕ1,ϕ2,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕ−1
1 ϕ−1
2 ) and K = K1, or
(2) l = p −1,u = −r and θ = (ϕ1,ϕ−1
1 ,ϕ2,ϕ−r
1 ϕ2,ϕr
1ϕ−1
2 ,ϕ−1
2 ) and K = K3(r).
Assume v = p −1. Then
(1) u = r = p −1,l = 1 and θ = (ϕ1,ϕ1,ϕ2,ϕ−1
1 ϕ−1
2 ,ϕ−1
1 ϕ−1
2 ,ϕ2) and K = K2, or
(2) l = p −1,u = 0 and θ = (ϕ1,ϕ−1
1 ,ϕ2,ϕ−1
2 ,ϕr
1ϕ−1
2 ,ϕ−r
1 ϕ2) and K = K4(r).
The proof of the case p = 2 is analogous. This case yields only four epimorphisms ¯θ ∶H →Z2
2 given by
¯θ1 = (ϕ1,ϕ1,ϕ2,ϕ2,ϕ2,ϕ2), ¯θ2 = (ϕ1,ϕ1,ϕ2,ϕ2,ϕ1ϕ2,ϕ1ϕ2),
¯θ3 = (ϕ1,ϕ1,ϕ2,ϕ1ϕ2,ϕ2,ϕ1ϕ2) and ¯θ4 = (ϕ1,ϕ1,ϕ2,ϕ1ϕ2,ϕ1ϕ2,ϕ2),
and the proof follows after noticing that ¯Kj = ker(¯θj) for j = 1,2,3,4.
□
26
RUB´EN A. HIDALGO AND SEBASTI´AN REYES-CAROCCA
Remark 12. It is worth emphasizing that whereas the epimorphisms ¯θ2, ¯θ3 and ¯θ4 define Z2
2-actions on Riemann
surfaces S of genus three that are topologically equivalent, they are not topologically equivalent as triples (S,N,G).
Theorem 7. Let p ⩾2 be a prime number. The cardinality of C2(QS,G)/NQS,G is 3 if p = 2, and p + 4 otherwise.
Proof. By Proposition 10, we only need to count the number of orbits of the action of ⟨Ψ2 = (46),Ψ3 = (34)(56)⟩
on Cp(QS,G). Assume p ≠2, and let r,s ∈{0,...,p −1} such that r + s ∈{ p−1
2 , 3p−1
2 }. Note that K(r,s) remains
invariant under the action of Ψ2, and
Ψ3(K(r,s)) = ⟨ar
1as
2a−1
4 ,a3a−1
5 ,a4a−1
6 ⟩.
If τ ∈{1,...,p −1} satisfies that 2τ ≡1 mod p then one has that
(ar
1as
2a−4
4 ⋅(a3a−1
5 ⋅a4a−1
6 )τ)−1 = as
1ar
2a−1
3 .
This shows that Ψ3(K(r,s)) = K(s,r) and therefore these groups form α orbits, where α is the number of pairs
(r,s) such that r,s ∈{0,...,p−1} satisfies r ⩽s and r+s ∈{ p−1
2 , 3p−1
2 }. A routine computation shows that α = p+1
2 .
Similarly, if r ∈{0,...,p −1} then it can be seen that Ψ2(K3(r)) = K4(r) and Ψ3(K3(r)) = K3(p −r). Thus, the
groups K3(r) and K4(r) form p+1
2
orbits, represented by K3(r) where r ∈{0,..., p−1
2 }. Finally, K1 and K2 form
a single orbit, whereas K5 and K6 form one orbit each. All the above show that there are precisely p + 4 orbits.
By proceeding analogously, it can be seen that if p = 2 then ¯K2 and ¯K4 form a single orbit, and ¯K1 and ¯K3 form
one orbit each. This completes the proof.
□
Example 7. Theorem 7 says that there are exactly three pairwise topologically inequivalent actions (S,N,G)
where S is a compact Riemann surface of genus three, (S,N) is a Z2
2-action of signature (0;26) and G/N ≅Z2
2.
The signature of S/G is (0;2,2,2,4). We denote by Fj the complex-one dimensional family determined by ¯Kj and
by Gj (instead of G) the corresponding group. We have that Gj ≅QS,G/ ¯Kj and therefore Lemma 2 implies that
G1 ≅⟨a3,A,B ∶a2
3 = A4 = B2 = (AB)2 = [A,a3] = [B,a3] = 1⟩≅⟨a3⟩× ⟨A,B⟩≅Z2 × D4
G2 ≅⟨a1,a3,A,B ∶A2 = (AB)2 = a1,a2
1 = a2
3 = [a1,a3] = B2 = (Aa3)2 = [B,a3] = [A,B] = 1⟩≅⟨B⟩×⟨A,a3⟩≅Z2×D4
G3 ≅⟨a3,A,B ∶a2
3 = A4 = B2 = (AB)2 = [a3,A] = 1,Ba3B = Aa3⟩= ⟨A,a3⟩⋊⟨B⟩≅(Z4 × Z2) ⋊Z2.
As an application of Proposition 5 we obtain the following algebraic descriptions for the families Fj.
F1 ∶
⎧⎪⎪⎨⎪⎪⎩
y2
1 = (x2 −1)(x2 −λ2)
y2
2 = x
F2 ∶
⎧⎪⎪⎨⎪⎪⎩
y2
1 = (x2 −1)(x2 −λ2)
y2
2 = x(x + 1)(x + λ)
F3 ∶
⎧⎪⎪⎨⎪⎪⎩
y2
1 = (x2 −1)(x2 −λ2)
y2
2 = x(x2 −λ2)
where λ ∈C −{0,±1}. Observe that the (generic) members of F2 and F3 are non-hyperelliptic, whereas the ones
of F1 are; the hyperelliptic involution being (x,y2,y2) ↦(x,−y2,y2).
Note that ¯K1 is D−invariant (see Remark 11) and therefore the member of F1 obtained by taking λ = i is
endowed with a group of automorphisms of order 32 acting with signature (0;2,4,8), and is represented by
y2
1 = y8
2 −1.
This Riemann surfaces is known as the Accola-Maclachlan curve of genus three.
We remark that there is only one topological class of actions of Z2×D4 in genus three with signature (0;2,2,2,4).
In particular, if Sj belongs to Fj then the pairs (S1,G1) and (S2,G2) are topologically equivalent. However, the
triples (S1,N1,G1) and (S2,N2,G2) are not, as this example shows.
We extend the previous example to the general case.
Proposition 12. Let p ⩾3 be a prime number and let S be a compact Riemann surface of genus (2p −1)(p −1)
endowed with a group of automorphisms G isomorphic to a semidirect product Z2
p ⋊Z2
2. Let N ≅Z2
p be the normal
Sylow p-subgroup of G. If S/N has signature (0;p6) then one of the following statements hold.
Zm
k -ACTIONS OF SIGNATURE (0; k, n+1
. . ., k)
27
(1) G is isomorphic to Z2p ×Dp and there are precisely p+5
2
pairwise topological inequivalent triples (S,N,G).
(2) G is isomorphic to Dp ×Dp and there are precisely p+1
2
pairwise topological inequivalent triples (S,N,G).
(3) G is isomorphic to Z2 × (Z2
p ⋊(−1,−1) Z2) and there is only one topological class of triples (S,N,G).
Proof. We only need to study the quotients QS,G/K, where, by Lemma 2, we have that
QS,G = ⟨a1,...,a6,A,B,∶ap
j = [ai,aj] = a1⋯a6 = A2 = B2 = (AB)2 = 1,
[A,a1] = [A,a2] = 1,Aa3A = a5,Aa4A = a6,Ba1B = a2,Ba3B = a4,Ba5B = a6⟩,
and K runs over the subgroups given in Proposition 11 up to NQS,G-action (see the proof of Proposition 7).
Let r,s ∈{0,...,p −1} be integers satisfying that r + s ∈{ p−1
2 , 3p−1
2 }. Then
G(r,s) ∶= QS,G/K(r,s) ≅⟨a1,a2,A,B ∶...,[A,a1] = [A,a2] = 1,Ba1B = a2⟩,
which is isomorphic to ⟨A,a1a2⟩× ⟨B,a1a−1
2 ⟩≅Z2p × Dp. Similarly, for each r ∈{0,..., p−1
2 } we have that
G3(r) ∶= QS,G/K3(r) ≅⟨a1,a3,A,B ∶...,[A,a1] = 1,Aa3A = ar
1a−1
3 ,Ba1B = a−1
1 ,Ba3B = a−r
1 a3⟩
which is isomorphic to ⟨A,ar
1a−2
3 ⟩× ⟨B,a1⟩≅Dp × Dp, and
G6 ∶= QQ,S/K6 ≅⟨a1,a3,A,B ∶...,[A,a1] = [A,a3] = 1,Ba1B = a−1
1 ,Ba3B = a−1
3 ⟩
is isomorphic to ⟨A⟩× ⟨a1,a3,B⟩≅Z2 × (Zp ⋊(−1,−1) Z2). Finally, by proceeding analogously, we obtain that
G1 = QQ,S/K1 and G5 = QQ,S/K5 are isomorphic to Z2p × Dp and the proof is done.
□
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Departamento de Matem´atica y Estad´ıstica, Universidad de La Frontera, Francisco Salazar 01145, Temuco, Chile
Email address: ruben.hidalgo@ufrontera.cl
Departamento de Matem´aticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Santiago, Chile
Email address: sebastianreyes.c@uchile.cl
|
Zm k -ACTIONS OF SIGNATURE (0;k, n+1 ...,k) RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Abstract. In this article we consider group actions on compact Riemann surfaces and their topological classification. We address this problem for pairs (S, N) where S is a compact Riemann surface endowed with a group of automorphisms N ≅Zm k such S/N has signature (0; k, n+1 . . . , k), where n, k ⩾2 and 1 ⩽m ⩽n are integers. We further assume the existence of extra automorphisms, namely, a group G with N ⊲G ⩽Aut(S) and analyze the induced permutational action of G/N on the cone points of S/N. To describe such actions up to topological equivalence, we employ the generalized Fermat curves (X, H) and their automorphism groups, showing that every triple (S, N, G) as before is determined by a class of subgroups of H that satisfy certain invariance property. This approach establishes a correspondence between topological equivalence classes and an appropriate quotient set. As an application, we specialize our results to the case k prime and m = 2, including algebraic models and isogeny decompositions of their Jacobian varieties. We then discuss some examples for the cases n = 3 and n = 5, which are interesting in their own right. 1. Introduction Riemann surfaces have been proved to serve as fundamental objects bridging topology, geometry, algebra and complex analysis. The study of automorphisms of compact Riemann surfaces or, equivalently, of smooth complex projective algebraic curves and their function fields, represents a classical and rich area of research in both complex and algebraic geometry. The foundations of this field date back to the nineteenth century, with seminal contributions from mathematicians such as Riemann, Klein and Jacobi. A central result, due to Schwarz [29] and Hurwitz [20], establishes that the group of automorphisms of a compact Riemann surface of genus g ⩾2 is finite, and that its order is bounded by 84(g -1). Much later, Greenberg in [14] succeeded in proving that each finite group can be realized as a group of automorphisms of some compact Riemann surface. It is classically known that the moduli space Mg of isomorphism classes of compact Riemann surfaces of genus g ⩾2 has the structure of a complex analytic space of dimension 3g -3, and that if g ⩾3 then its singular locus corresponds to the points representing compact Riemann surfaces with non-trivial automorphisms. In other words Sing(Mg) = {[S] ∈Mg ∶Aut(S) ≠{id}}, where Aut(S) denotes the automorphism group of S. The moduli space -which is itself an algebraic variety defined over the field of rational numbers- is one of the most fascinating objects in algebraic geometry, and is at the core of important and recent developments in number theory and geometry. Let Sj be a compact Riemann surface endowed with a group of automorphisms Gj for j = 1,2. We recall that these actions are called topologically equivalent if there exists an orientation-preserving homeomorphism φ ∶S1 →S2 such that φ-1G2φ = G1. We also say that the pairs (S1,G1) and (S2,G2) are topologically equivalent. The importance of the topological classification of actions lies in several applications, some of which we briefly describe. The first one relates the singularities of the moduli space. The set Mg(G) = {[S] ∈Mg ∶S admits a fixed topological class of G-action} ⊂Sing(Mg) 2010 Mathematics Subject Classification. 30F10, 14H37, 30F35, 14H30. Key words and phrases. Riemann surfaces, group actions, automorphisms. Partially supported by ANID Fondecyt Regular Grants 1230001, 1220099 and 1230708. 16 Oct 2025 2 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA is an irreducible subvariety of the moduli space [12]. In other words, the topological equivalence allows us to perform deformations of Riemann surfaces with automorphisms in a controlled manner. These subvarieties, in turn, provide a stratification of the moduli space [6], which has been proved to serve as a useful toolkit to explore the largely unknown topology of Mg. For instance, it has played a key role in the determination of the connectedness of the singular locus of the moduli space (see, for instance, [2] and the references therein) and has been recently employed in [19] to find new non-normal subvarieties of Mg. Moreover, these subvarieties have been fruitful to study some aspects of families of Jacobian varieties with group action; for instance, isogeny decompositions and Shimura varieties (see, for instance [10] and [21]). Another important aspect to mention is that the conjugacy classes of finite subgroups of the mapping class group (homotopy classes of self-homeomorphisms of a real surface) are in bijective correspondence with the topological classes of group actions. Sources for the characterization of topological actions by purely algebraic data include Nielsen [26], Harvey [15] and Gilman [11]. We also refer to [3], [8], [23], [27] and [31] as sources for the classification of topological actions in low genera, and also for computer-aided algorithms. Group actions on compact Riemann surfaces are characterized in part by their signature. Concretely, if G is a group of automorphisms of a compact Riemann surface S, then the signature of the action is the tuple (γ;k1,...,kr), where γ is the genus of the quotient S/G and k1,...,kr are the branch indices of the canonical projection S →S/G. Equivalently, the integers kj are the orders of the cone points of the Riemann orbifold S/G. Let n,k ⩾2 and 1 ⩽m ⩽n be integers such that (n -1)(k -1) > 2. By a Zm k -action of signature (0;k, n+1 ...,k) we mean a pair (S,N) consisting of a compact Riemann surface S endowed with a group of automorphisms Zm k ≅N ⩽Aut(S) such that S/N has signature (0;k, n+1 ...,k). We fix a Zm k -action (S,N) of signature (0;k, n+1 ...,k) and assume that S is endowed with a group of automorphisms G such that N ⊲G ⩽Aut(S). The action of the quotient group G/N on the set of n + 1 cone points of S/N induces a subgroup Q∗of the symmetric group Sn+1, up to conjugation. Such a conjugacy class is the permutational action of G/N. We denote by F(Q∗) the collection formed by all the triples ( ˆS, ˆG, ˆN) such that the pair ( ˆS, ˆN) is a Zm k -action of signature (0;k, n+1 ...,k) and ˆN ⊲ˆG ⩽Aut( ˆS) in such a way that the permutational action of ˆG/ ˆN is given by Q∗. Observe that if two triples (S1,N1,G1) and (S2,N2,G2) belong to F(Q∗), then the groups G1/N1 and G2/N2 are isomorphic and induce the same permutational action; however, G1 and G2 need not be isomorphic groups. Accordingly to the case of pairs, two triples (S1,N1,G1) and (S2,N2,G2) as above are called topologically equivalent if there exists an orientation-preserving homeomorphism φ ∶S1 →S2 such that φ-1N2φ = N1 and φ-1G2φ = G1. It is worth noting that it might happen that two such triples are topologically inequivalent, but either the pairs (S1,N1) and (S2,N2) or the pairs (S1,G1) and (S2,G2) are topologically equivalent. In this paper, we address the problem of describing the collection F(Q∗) up to topological equivalence. We proceed as follows. Let (S,N,G) be a triple in F(Q∗). First, we consider the homology cover X of the Riemann orbifold S/N, also called a generalized Fermat curve of type (k,n). This is a compact Riemann surface that admits a group of automorphisms H ≅Zn k such that X/H ≅S/N. Following the results in [13], we then consider the intimate relationship between X and our triple, which arises from the existence of a subgroup KS ⊲H, acting freely on X, such that S ≅X/KS and N ≅H/KS. In addition, the group G/N lifts to a group Q ⩽Aut(X) containing KS and H as normal subgroups in such a way that Q/H ≅G/N and G ≅Q/KS. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 3 We denote by Autg(H) the subgroup of Aut(H) whose elements are induced by conjugation via orientationpreserving homeomorphisms of X that preserve H. We shall see that there is a natural isomorphism between Autg(H) and Sn+1, and therefore the group Q induces (up to conjugation) a subgroup Q∗of Autg(H) which is completely determined by the permutational action of G/N on the cone points of X/H ≅S/N. We then introduce the subset Ck(Q) formed by the subgroups K ⊲H acting freely on X such that H/K ≅Zm k and that are Q∗-invariant. For each K ∈Ck(Q), we set SK ∶= X/K and NK ∶= H/K. The Q∗-invariance of K ensures the existence of a finite group QK of orientation-preserving homeomorphisms of X, containing K and H as normal subgroups, such that GK ∶= QK/K is a finite group of orientation-preserving homeomorphisms of SK that contains NK as a normal subgroup and such that GK/NK ≅G/N induces the permutation group Q∗. This allows us to prove that for each ( ˆS, ˆG, ˆN) ∈F(Q∗) there exists K ∈Ck(Q) such that (SK,GK,NK) is topologically equivalent to ( ˆS, ˆG, ˆN). In addition, if NQ ⩽Autg(H) denotes the normalizer of Q∗then Ck(Q) is NQ-invariant. We then show that K1,K2 ∈Ck(Q) determine topologically equivalent triples if and only if they belong to the same NQ-orbit. As a result, we manage to compute the number of pairwise topologically inequivalent triples, by means of a bijective correspondence with the set Ck(Q)/NQ. For the sake of clarity, we work out the case of Z2 p-actions of signature (0;p, n+1 ...,p). We provide explicit algebraic models for these Riemann surfaces in terms of fiber products of cyclic p-gonal algebraic curves, and give an isogeny decomposition of their Jacobian varieties. After that, we specialize our results for the cases n = 3 and n = 5, as they are interesting in their own right. More precisely, we study Z2 p-actions of signature (0;p, 4...,p) and describe some examples in detail, and study Z2 p-actions of signature (0;p, 6...,p) that admit extra automorphisms and that form complex one-dimensional families. By the way, we recover and extend classical and recent results for some families of Riemann surfaces. 2. Preliminaries and Notations 2.1. Group actions on Riemann surfaces. Let S be a compact Riemann surface of genus g ⩾2, and let G be a group of automorphisms of S. We consider the quotient Riemann orbifold S/G and the associated regular covering map π ∶S →S/G. The Riemann orbifold S/G inherits the structure of a compact Riemann surface of genus γ together with r ⩾0 distinguished points, say q1,...,qr, satisfying the following property: for every point x ∈π-1(qj) its multiplicity kj is strictly smaller than the degree of π. We say that the orbifold S/G has signature (γ;k1,...,kr) and that qj is a cone point of cone order kj. If we denote by H2 the upper half-plane then, by the classical uniformization theorem, there exists a co-compact Fuchsian group Γ ⩽PSL2(R) ≅Aut(H2) such that S/G ≅H2/Γ as orbifolds. Moreover, there is a group epimorphism θ ∶Γ →G, whose kernel is torsion-free, in such a way that S ≅H2/ker(θ) and G ≅Γ/ker(θ). The epimorphism θ is called the monodromy of the action. We also say that Γ has signature (γ;k1,...,kr). Along the paper, we employ the notation (0;km) to abbreviate (0;k, m ...,k). 2.2. Topological equivalence of actions. Let (S1,G1) and (S2,G2) be two pairs, where Sj is a compact Riemann surface endowed with a group of automorphisms isomorphic to Gj for j = 1,2. We say that the pairs (or that the actions) are topologically equivalent if there exists an orientation-preserving homeomorphism φ ∶S1 →S2 such that φ-1G2φ = G1. 4 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Observe that, in particular, G1 and G2 are isomorphic groups. The pairs are termed biholomorphically equivalent if φ is a biholomorphism. In terms of monodromies, the topological equivalence can be reformulated as follows. Let Γj be a Fuchsian group such that H2/Γj ≅Sj/Gj, and let θj ∶Γj →Gj be the corresponding monodromy. Then two pairs as above are topologically equivalent if and only if there is a group isomorphism ρ ∶G1 →G2 and a geometric isomorphism ψ ∶Γ1 →Γ2 such that ρ ○θ1 = θ2 ○ψ. We recall that the geometric isomorphisms are those that are induced by conjugation of orientation-preserving self-homeomorphisms of H2. Without loss of generality, we can assume Γ1 = Γ2. In the case that γ = 0, a set of generators for the group of geometric automorphisms BΓ of Γ is known. If, moreover, the group G is abelian and all the cone orders are the same, then the action of BΓ induces the action of the symmetric group on the image under θ of the canonical generators of Γ. We refer to the survey [5] for more details. 2.3. Families of Riemann surfaces. A family C of compact Riemann surfaces of genus g is the locus of the moduli space Mg formed by all those Riemann surfaces that have a group of automorphisms isomorphic to a given group G acting with a given signature. If the signature of the action is (γ;k1,...,kr) then the complex-dimension of the family is 3γ -3 + r. We recall the following facts, which follow from the equisymmetric stratification of the moduli space. (1) The interior of C , if non-empty, consists of those Riemann surfaces whose automorphism group is isomorphic to G, and is formed by finitely many strata which are in correspondence with the pairwise non-equivalent topological actions of G. (2) The complement of the interior (with respect to the family) is formed by those Riemann surfaces that have strictly more automorphisms than G. We refer to [6], [12] and [15] for more details. 2.4. Generalized Fermat curves. Let n,k ⩾2 be integers. A compact Riemann surface X is called a generalized Fermat curve of type (k,n) if there exists Zn p ≅H0 ⩽Aut(X) such that X/H0 has signature (0;kn+1). The group H0 is a generalized Fermat group of type (k,n) and the pair (X,H0) is a generalized Fermat pair of type (k,n). By the Riemann-Hurwitz formula, the genus of X is g = 1 + kn-1 2 ((n -1)(k -1) -2). In particular, the non-hyperbolic generalized Fermat pairs are those satisfying (n -1)(k -1) ⩽2, that is, of type (2,2), (2,3) and (3,2). Henceforth, we restrict our attention to the hyperbolic case. 2.4.1. Fuchsian description. Let (X,H0) be a generalized Fermat pair of type (k,n). The Riemann orbifold X/H0 is uniformized by a Fuchsian group Γ = ⟨x1,...,xn+1 ∶xk 1 = ... = xk n+1 = x1⋯xn+1 = 1⟩. Moreover, if Γ′ stands for the commutator subgroup of Γ then, as proved in [13], the generalized Fermat pairs (X,H0) and (H/Γ′,Γ/Γ′) are biholomorphically equivalent. The fact that Γ′ is characteristic yields the following properties. Theorem ([13]). (1) Two generalized Fermat pairs of the same type are topologically equivalent. (2) The regular covering map π ∶X →X/H0 induced by the action of H0 is characteristic. The following fact, which was also proved in [13], will be used later. For the sake of competeness, we provide an outline of the proof. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 5 Theorem. Let S be a Riemann surface of genus at least two endowed with an abelian group of automorphisms N such that S/N has signature (0;kn+1). Then there exist a generalized Fermat pair (X,H0) of type (k,n) and a subgroup KS ⊲H0, acting freely on X, such that (S,N) and (X/KS,H0/KS) are biholomorphically equivalent. Proof. Let Γ be a Fuchsian group such that S/N ≅H2/Γ. Then (X = H2/Γ′,H0 = Γ/Γ′) is a generalized Fermat pair of type (k,n). As S is a regular branched cover of S/N, there is a torsion-free normal subgroup F ⩽Γ such that S ≅H2/F and N ≅Γ/F. As N is abelian, one has that Γ′ ⩽F and consequently KS ∶= F/Γ′ is the desired subgroup. □ 2.4.2. Algebraic description and uniqueness of the generalized Fermat groups. In what follows, for each integer k ⩾2 we set ωk = exp(2πi/k). Up to a suitable M ̈obius transformation, we can assume the cone points of X/H0 to be (2.1) ∞,0,1,q4,...,qn+1. (1) Case n = 2. If n = 2, and therefore k ⩾4, then there is, up to biholomorphism, a unique generalized Fermat curve of type (k,2). This curve corresponds to the classical Fermat curve X = Fk ∶{xk 1 + xk 2 + xk 3 = 0} ⊂P2, with generalized Fermat group given by H = ⟨a1,a2⟩≅Z2 k, where a1([x1 ∶x2 ∶x3]) = [ωkx1 ∶x2 ∶x3] and a2([x1 ∶x2 ∶x3]) = [x1 ∶ωkx2 ∶x3]. In this case, the map π ∶Fk → ̄C given by π([x1 ∶x2 ∶x3]) = -(x2/x1)k is a k2-fold branched regular covering map with deck group H, whose branch values are ∞,0,1. We can directly verify the following facts. (1) The nontrivial elements of H that have fixed points in Fk are a1,a2,a3 ∶= (a1a2)-1 and their powers. Moreover, each fixed point of a non-trivial power of aj is also a fixed point of aj. (2) The group H is the unique generalized Fermat group of Fk, that is, if H′ ⩽Aut(Fk) is a generalized Fermat group of some type (k′,n′), then (k′,n′) = (k,n) and H′ = H. (2) Case n ⩾3. Now, assume n ⩾3. Consider the complex projective algebraic curve in Pn defined by Ck(q4,...,qn+1) ∶ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ xk 1 + xk 2 + xk 3 = 0 q4xk 1 + xk 2 + xk 4 = 0 ⋮ ⋮ ⋮ qn+1xk 1 + xk 2 + xk n+1 = 0 ⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭ The fact that qj ∈C -{0,1} are pairwise distinct implies that the algebraic curve above is non-singular, and hence represents a compact Riemann surface. Note that the linear transformations aj ∈PGLn+1(C) defined as (2.2) aj([x1 ∶... ∶xn+1]) = [x1 ∶... ∶xj-1 ∶ωkxj ∶xj+1 ∶... ∶xn+1] for j ∈{1,...,n}, generate a group of automorphisms H of Ck(q4,...,qn+1)). The kn-fold covering map (2.3) π ∶Ck(q4,...,qn+1) → ̄C given by π([x1 ∶... ∶xn+1]) = -(x2/x1)k, satisfies π○aj = π for every 1 ⩽j ⩽n and ramifies precisely over (2.1). Thus, (Ck(q4,...,qn+1),H) is a generalized Fermat pair of type (k,n). Theorem ([13]). The pairs (X,H0) and Ck(q4,...,qn+1),H) are biholomorphically equivalent. Remark 1. The above fact asserts that the domain Ωn ∶= {(q4,...,qn+1) ∈Cn-2 ∶qj ≠0,1 and qi ≠qi for i ≠j} 6 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA parametrizes all generalized Fermat pairs of type (k,n), where n ⩾3. Generalizing the known situation for n = 2, we have the following facts. Theorem ([13, 16, 18]). Let Λ ∈Ωn. Then (1) The nontrivial elements of H that have fixed points in Ck(Λ) are a1,...,an,an+1 ∶= (a1⋯an)-1 and their powers. Moreover, each fixed point of a non-trivial power of aj is also a fixed point of aj. (2) The group H is the unique generalized Fermat group of Ck(Λ), that is, if H′ ⩽Aut(Ck(Λ)) is a generalized Fermat group of some type (k′,n′), then (k′,n′) = (k,n) and H′ = H. Remark 2. The uniqueness of H asserts that it is a normal subgroup of Aut(Ck(Λ)) and therefore Aut(Ck(Λ))/H is a spherical group. In other words, there is a short exact sequence of groups 1 Ð→H Ð→Aut(Ck(Λ)) θ Ð→A Ð→1, where A is the M ̈ob(C)-stabilizer of BΛ = {∞,0,1,q4,...,qn+1}. 2.4.3. Geometric automorphisms of H. Let Λ = (q4,...,qn+1) ∈Ωn. We employ the following notations. (1) Hom+(Ck(Λ)) is the group of orientation-preserving homeomorphisms of Ck(Λ). (2) Hom+ H(Ck(Λ)) is the normalizer of H in Hom+(Ck(Λ)). (3) Hom+(O(Λ)) is the group of orientation-preserving homeomorphisms of ̄C that preserve the set BΛ. Observe that there is a natural group homomorphism η ∶Hom+ H(Ck(Λ)) →Hom+(O(Λ)) such that η(f) ○π = π ○f for each f ∈Hom+ H(Ck(Λ)). The fact that the covering map (2.3) is characteristic implies that η is surjective. Note that ker(η) = H. Besides, each f ∈Hom+ H(Ck(Λ)) defines an automorphism Φf of H given by Φf(aj) = f ○aj ○f -1. Definition. An automorphism Φ of H is called geometric if Φ = Φf for some f ∈Hom+ H(Ck(Λ)) The subgroup of Aut(H) formed by the geometric automorphisms is denoted by Autg(H). It follows that there is a the natural group epimorphism ρ ∶Hom+ H(Ck(Λ)) →Autg(H) given by ρ(f) = Φf. Remark 3. Autg(H) does not depend on Λ. In fact, for each ˆΛ = (ˆq4,..., ˆqn+1) ∈Ωn we can consider an orientation-preserving homeomorphism ˆg ∶ ̄C → ̄C such that (2.4) ˆg(∞) = ∞, ˆg(0) = 0, ˆg(1) = 1 and ˆg(qj) = ˆqj for j = 4,...,n + 1. The fact that (2.3) is characteristic guarantees that ˆg lifts to an orientation-preserving homeomorphism g ∶ Ck(Λ) →Ck(ˆΛ) such that gHg-1 = H. Now, the conditions (2.4) imply that, if ˆf ∈Hom+ H(Ck(ˆΛ)) then Φg-1 ˆ fg(aj) = g-1 ˆfg ○aj ○(g-1 ˆfg)-1 = ˆf ○aj ○ˆf -1 = Φ ˆ f(aj). The claim follows after noting that each f ∈Hom+ H(Ck(Λ)) is of the form g-1 ˆfg. Proposition 1. Each Φ ∈Autg(H) yields a uniquely determined permutation σΦ ∈Sn+1, and the correspondence Autg(H) →Sn+1 given by Φ ↦σΦ is a group isomorphism. In particular, Autg(H) = ⟨Φ1,Φ2⟩where Φ1(a1,a2,a3,...,an) = (a2,a1,a3,...,an) and Φ2(a1,a2,...,an-1,an) = (a2,a3,...,an,a1). Proof. Consider the natural group epimorphism ˆρ ∶Hom+(O(Λ)) →Sn+1 given by ˆf ↦ˆρ( ˆf) where ˆf(qj) = qˆρ( ˆ f)(j). Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 7 The kernel of ˆρ is, up to isotopy rel BΛ, the group generated by the Dehn-twists along simple closed curves contained in ̄C -BΛ. We note that, if f ∈Hom+ H(Ck(Λ)) then Φf(aj) = f ○aj ○f -1 = aˆρ(η(f))(j) for each j = 1,...,n. Thus, there is a group isomorphism ˆη ∶Autg(H) →Sn+1 which makes the following diagram commutative. Hom+ H(Ck(Λ)) Autg(H) Hom+(O(Λ)) Sn+1 ρ η ˆρ ˆη This proves the first statement; the latter one follows directly from the former. □ Remark 4. (1) Consider L ⩽Hom+(O(Λ)) and set Q ∶= η-1(L). As ρ(Q) = ˆη-1(ˆρ(L)), the subgroup Q∗∶= ρ(Q) = {ΦT ∶T ∈Q} of Autg(H) is completely determined by the permutational action ˆρ(L). (2) We denote by Aut(O(Λ)) ⩽M ̈ob(C) the subgroup of Hom+(O(Λ)) consisting of the automorphisms of the orbifold Ck(Λ)/H. We have the following commutative diagram Aut(Ck(Λ)) Autg(H) Aut(O(Λ)) Sn+1 ρ η ˆρ ˆη where neither ρ nor ˆρ are surjective for n ⩾3, and that ker(ρ) ∩Aut(Ck(Λ)) = H. 3. Zm k -actions of signature (0;kn+1) Let k,n ⩾2 and 1 ⩽m ⩽n be integers such that (n -1)(k -1) > 2. Definition. A pair (S,N) is called a Zm k -action of signature (0;kn+1) if S is a compact Riemann surface endowed with a group of automorphisms N ≅Zm k such that S/N has signature (0;kn+1). Observe that the Riemann surfaces S, such that (S,N) is a Zm k -action of signature (0;kn+1), form a complex (n -2)-dimensional family in moduli space Mg, where (3.1) g = 1 + 1 2km-1[(n -1)(k -1) -2]. Remark 5. Every Zn k-action of signature (0;kn+1) is biholomorphic to a generalized Fermat pair of type (k,n). In particular, all of them are topologically equivalent. On the other extreme, every Zk-action (S,N) of signature (0;kn+1) corresponds to a cyclic k-gonal curve S ≅{yk = n+1 ∏ j=1 (x -qj)lj} and N = ⟨(x,y) ↦(x,ωky)⟩, where the integers l1,...,ln+1 lie in {1,...,k -1} and are coprime to k. By the remark above, hereafter we shall only consider the case n ⩾3 and 2 ⩽m ⩽n -1. 8 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA 3.1. Fiber product description. Let (S,N) be a Zm k -action of signature (0;kn+1), where n ⩾3, k ⩾2, (n - 1)(k -1) > 2 and 2 ⩽m ⩽n -1. Let φ1,...,φm be automorphisms of S that generate N, and let π ∶S → ̄C be a branched regular covering map with deck group N. Set N1 = ⟨φ2,...,φm⟩, Nm = ⟨φ1,...,φm-1⟩and Ni = ⟨φ1,...,φi-1,φi+1,...,φm⟩, for each i = 2,...,m -1. Note that Nj ≅Zm-1 k . We denote by Si the compact Riemann surface underlying to the quotient S/Ni for each i, and by πi ∶S →Si a branched regular covering map with deck group Ni. Observe that φi induces an automorphism τi of Si of order k such that S/N ≅Si/⟨τi⟩for each i = 1,...,m. If πi ∶Si → ̄C is a branched regular covering map with deck group ⟨τi⟩such that π = πi ○πi then, following [17, Section 3.2], we have that S is isomorphic to the fiber product Πm i=1(Si,πi). This description allows us to provide an explicit algebraic description of S in terms of the branch values of πi. Later, we will make this description explicit for the case m = 2. 3.2. Descriptions in terms of generalized Fermat curves. Let (S,N) be a Zm k -action of signature (0;kn+1), where n ⩾3, k ⩾2, (n -1)(k -1) > 2 and 2 ⩽m ⩽n -1. After considering a suitable M ̈obius transformation, we can assume that the cone points of S/N ≅ ̄C are given by the set BΛ = {q1 = ∞,q2 = 0,q3 = 1,q4,...,qn+1} where Λ = (q4,...,qn+1) ∈Ωn. As discussed in §2.4, the tuple Λ ∈Ωn determines the generalized Fermat curve Ck(Λ) of type (k,n), and its generalized Fermat group is H ∶= ⟨a1,...,an⟩≅Zn k where aj is as (2.2). We define the set F(k,n,m) ∶= {K ∶K ⩽H,H/K ≅Zm k , ⟨aj⟩∩K = {1}, j = 1,...n + 1}. Remark 6. If k is prime, then the condition ⟨aj⟩∩K = {1} above is equivalent to aj ∉K. As a consequence of the Fuchsian uniformization of generalized Fermat curves discussed in §2.4.1, we have that the Zm k -actions of signature (0;kn+1) are parametrized by F(k,n,m). Proposition 2. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n-1 be integers such that (n-1)(k-1) > 2. If (S,N) is a Zm k -action of signature (0;kn+1), then there exist KS ∈F(k,n,m) and a tuple (q4,...,qn+1) ∈Ωn such that (S,N) and (Ck(q4,...,qn+1)/KS,H/KS) are biholomorphically equivalent. The following result describes the members of F(k,n,m), up to geometric automorphisms of H, for k prime. Theorem 1. Let p ⩾2 be a prime number, and let n ⩾3 and 2 ⩽m ⩽n-1 be integers such that (n-1)(p-1) > 2. If K ∈F(p,n,m) then there exists Φ ∈Autg(H) and there are integers lj,i ∈{0,1,...,p -1}, j = m + 1,...,n + 1 and i = 1,...,m, satisfying (1) (lj,1,...,lj,m) ≠(0,...,0) for every j = m + 1,...,n + 1, and (2) i + lm+1,i + ⋯+ ln+1,i ≡0 mod p for every i = 1,...,m, such that Φ(K) = ⟨alm+1,1 1 ⋯alm+1,m m a-1 m+1,...,aln,1 1 ⋯aln,m m a-1 n ⟩≅Zn-m p . Proof. A group K ∈F(p,n,m) is the kernel of some group epimorphism θ ∶H →Zm p with the property that, for every j = 1,...,n + 1, the element θ(aj) has order p. Up to a permutation of indices (or, equivalenty, after considering the action of Autg(H)), the fact that p is prime allows us to assume that the elements φ1 ∶= θ(a1),...,φm ∶= θ(am) Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 9 form a set of generators of Zm p . Thus, for each j = m + 1,...,n + 1, one has that θ(aj) = φlj,1 1 ⋯φlj,m m for some lj,i ∈{0,1,...,p -1}. The condition that ⟨aj⟩∩K = {1} asserts that (lj,1,...,lj,m) ≠(0,...,0) for each j. Besides, the fact that the product a1⋯an+1 is trivial implies that i + lm+1,i + ⋯+ ln+1,i ≡0 mod p for each i = 1,...,m. In this way, the kernel K of θ is generated by the elements alm+1,1 1 ⋯alm+1,m m a-1 m+1,...,aln,1 1 ⋯aln,m m a-1 n ,aln+1,1 1 ⋯aln+1,m m a-1 n+1. The equality aln+1,1 1 ⋯aln+1,m m a-1 n+1 = aln+1,1 1 ⋯aln+1,m m (a1⋯an) = a1+ln+1,1 1 ⋯a1+ln+1,m m (am+1⋯an), shows that K is generated by the n -m elements alm+1,1 1 ⋯alm+1,m m a-1 m+1,...,aln,1 1 ⋯aln,m m a-1 n , as claimed. □ Remark 7. The theorem above holds for the case m = 1. If k is not a prime integer, then it is more involved to describe the general form of K ∈F(k,n,m). 3.3. Topological classification of Zm k -actions. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n -1 be integers such that (n -1)(k -1) > 2. Let (S,N) be a Zm k -action of signature (0;kn+1), and assume that the cone points of S/N are ∞,0,1, ˆq4,..., ˆqn+1. By Proposition 2, there is a subgroup KS ∈F(k,n,m) such that (S,N) and (Ck(ˆΛ)/KS,H/KS) are biholomorphically equivalent, where ˆΛ = (ˆq4,..., ˆqn+1) ∈Ωn. We fix Λ = (q4,...,qn+1) ∈Ωn and consider an orientation-preserving homeomorphism ˆf ∶ ̄C → ̄C such that ˆf(∞) = ∞, ˆf(0) = 0, ˆf(1) = 1 and ˆf(ˆqj) = qj for j = 4,...,n + 1. It follows that there is an orientation-preserving homeomorphism f ∶Ck(ˆΛ) →Ck(Λ) such that ˆf ○π = π ○f. In particular, one has that fHf-1 = H. The way as we have chosen ˆf asserts that fKSf -1 = KS. Now, if we define SKS ∶= Ck(Λ)/KS and NKS ∶= H/KS, then f induces an orientation-preserving homeomorphism g ∶SKS →S such that the following diagram commutes. Ck(Λ) Ck(ˆΛ) SKS S ̄C ̄C f NKS N ˆ f KS g KS H H Note that gNKSg-1 = N. The discussion above proves the following proposition. Proposition 3. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n -1 be integers such that (n -1)(k -1) > 2, and fix Λ ∈Ωn. If (S,N) is a Zm k -action of signature (0;kn+1), then there exists K ∈F(k,n,m) such that (S,N) and (SK = Ck(Λ)/K,NK = H/K) are topologically equivalent. Observe that a pair of groups K1 and K2 in F(k,n,m) may give rise to two Zm k -actions that are topologically equivalent. The following result describes such a situation. Theorem 2. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n -1 be integers such that (n -1)(k -1) > 2. Let K1,K2 ∈F(k,n,m). The pairs (SK1,NK1) and (SK2,NK2) are topologically equivalent if and only if there exists Φ ∈Autg(H) such that Φ(K1) = K2. 10 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Proof. We fix Λ = (q4,...,qn+1) ∈Ωn. Assume that there exists a geometric automorphism Φ of H such that Φ(K1) = K2. Thus, there is an orientation-preserving homeomorphism fΦ ∶Ck(Λ) →Ck(Λ) which induces an orientation-preserving homeomorphism gΦ ∶SK1 →SK2. The fact that fΦHf -1 Φ = H implies that gNK1g-1 = NK2 and therefore (SK1,NK1) and (SK2,NK2) are topologically equivalent. Conversely, if (SK1,NK1) and (SK2,NK2) are topologically equivalent then there is an orientation-preserving homeomorphism g ∶SK1 →SK2 such that gNK1g-1 = NK2. It follows that g induces an orientation-preserving homeomorphism h of ̄C into itself which keeps the set {∞,0,1,q4,...,qn+1} invariant. In turn, h lifts to an orientation-preserving homeomorphism f ∶Ck(Λ) → Ck(Λ) that satisfies fHf -1 = H and fK1f -1 = K2. Thus, Φ(K1) = H2 where Φ = Φf. □ Observe that there is a natural action Autg(H) × F(k,n,m) →F(k,n,m) given by (Φ,K) ↦Φ(K). As a consequence of the proposition above, we have the following corollary. Corollary 1. The cardinality of the quotient set F(k,n,m)/Autg(H) is the number of pairwise topologically inequivalent Zm k -actions with signature (0;kn+1). 3.4. Topological actions and extra automorphisms. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n -1 be integers such that (n -1)(k -1) > 2. We consider triples (S,N,G) where (S,N) is a Zm k -action of signature (0;kn+1), and S is endowed with a group of automorphisms G such that N ⊴G ⩽Aut(S). Consider Λ = (q4,...,qn+1) ∈Ωn such that S ≅Ck(Λ)/KS and N ≅H/KS for some KS ∈F(k,n,m). We recall that there is a short exact sequence of groups 1 Ð→H Ð→Aut(Ck(Λ)) θ Ð→A Ð→1, where A is the M ̈ob(C)-stabilizer of BΛ = {∞,0,1,q4,...,qn+1}. As A has a subgroup L isomorphic to G/N, there is an induced short exact sequence of groups 1 Ð→H Ð→QS,G ∶= θ-1(L) θ Ð→L Ð→1. Observe that H ⊴QS,G and QS,G/H ≅G/N, KS ⊴QS,G and QS,G/KS ≅G, and S/G ≅Ck(Λ)/QS,G. All the above can be summarized in the following commutative diagram, where C = Ck(Λ). S C S/N = C/H S/G = C/QS,G N KS H G/N G QS,G We recall that if Q is any group of orientation-preserving homeomorphisms of Ck(Λ) such that H ◁Q, then we may consider the representation (see the proof of Proposition 1) ρQ ∶Q →Autg(H) given by ρQ(f) = Φf. We denote its image by Q∗. In particular, as H ◁QS,G, we may consider the group epimorphism (3.2) ρQS,G ∶QS,G →Autg(H) given by ρQS,G(f) = Φf. As the kernel of ρQS,G is H, we have that QS,G/H ≅Q∗ S,G. Note that ρQS,G(f) is uniquely determined by the permutation induced by θ(f). We define Ck(QS,G) ∶= {K ∈F(k,n,m) ∶K is Q∗ S,G-invariant}. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 11 Note that Ck(QS,G) is nonempty as KS belongs to it. If K ∈Ck(QS,G) then (SK = Ck(Λ)/K,NK = H/K) is a Zm k -action of signature (0;kn+1) such that there exists Q ⩽Hom+ H(Ck(Λ)) which contains H and K as normal subgroups, satisfying Q∗= Q∗ S,G and that NK ⊴GK = Q/K ⩽Hom+(SK) and GK/NK ≅Q/H ⩽Hom+( ̄C). In addition, SK/GK and Ck(Λ)/Q are equivalent, as topological orbifolds. Proposition 4. Let n ⩾3, k ⩾2 and 2 ⩽m ⩽n -1 be integers such that (n -1)(k -1) > 2. Let (S,N,G) be a triple such that (S,N) is a Zm k -action of signature (0;kn+1) and S admits a group of automorphisms G such that N ⊴G ⩽Aut(S). Let Ck(Λ) and Q∗ S,G ⩽Autg(H) be as before. Then, up to topological equivalence, each triple ( ˆS, ˆN, ˆG) such that ( ˆS, ˆN) is a Zm p -action of signature (0;kn+1) and ˆS admits a group of automorphisms ˆG with ˆN ⊴ˆG ⩽Aut( ˆS) and satisfying that Q∗ ˆS, ˆ G = Q∗ S,G corresponds to a member of Ck(QS,G). Proof. Let ( ˆS, ˆN, ˆG) be a triple as in the statement of the proposition, and let ∞,0,1, ˆq4,..., ˆqn+1 be the cone points of ˆS/ ˆN, so ˆΛ = (ˆq4,..., ˆqn+1) ∈Ωn. By Proposition 3, there exists K ˆS ∈F(k,n,m) such that ( ˆS, ˆN) and (SK ˆ S = Ck(Λ)/K ˆS,NK ˆ S = H/K ˆS) are topologically equivalent. In addition, there is a group Q ˆS, ˆ G ⩽Aut(Ck(ˆΛ)) such that H,K ˆS ⊴ˆQ ˆS, ˆ G and ˆG ≅ˆQ ˆS, ˆ G/K ˆS. All the above is summarized in the following commutative diagram, where f,g and ˆf are as in the proof of Proposition 3. Ck(Λ) Ck(ˆΛ) SK ˆ S ˆS ̄C ̄C ̄C ˆ G/ ˆ N ˆ G Q ˆ S, ˆ G f NK ˆ S = g-1 ˆ Ng ˆ N ˆ f K ˆ S = f-1K ˆ Sf g K ˆ S H H We consider the group of homeomorphisms g-1 ˆGg ⩽Hom+(SK ˆ S). As Q∗ ˆS, ˆ G = Q∗ S,G, we have that S/G and ˆS/ ˆG are isomorphic as topological orbifolds. It follows that S/G and SK ˆ S/(g-1 ˆGg) are isomorphic as topological orbifolds too. Observe that NK ˆ S = g-1 ˆNg ⊴g-1 ˆGg and that (g-1 ˆGg)/NK ˆ S = ˆf -1( ˆG/ ˆN) ˆf. Besides, the group g-1 ˆGg lifts to a group Q ⩽Hom+(Ck(q4,...,qn+1)) such that H,K ⊴Q and Q/K ≅g-1 ˆGg. Now, the fact that S/G and SK ˆ S/g-1 ˆGg are isomorphic as topological orbifolds implies that the action by conjugation of Q agrees with the one of Q∗ S,G, showing that K ˆS ∈Cp(QS,G) as desired. □ 12 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Consider a triple (S,N,G) as above. We denote by NQS,G = {Φ ∈Autg(H) ∶ΦQ∗ S,GΦ-1 = Q∗ S,G} ⩽Autg(H) the normalizer of Q∗ S,G in Autg(H). We observe that Cp(QS,G) is NQS,G-invariant. If ( ˆS, ˆN, ˆG) is another triple as above, such that ˆG/ ˆN induces Q∗ S,G, then there exists some K ∈Cp(QS,G) such that there is an orientation-preserving homeomorphism h ∶SK = Ck(Λ)/K →ˆS such that hNKh-1 = ˆN and hGKh-1 = ˆG. So, following similar arguments as in the proof of Proposition 4, we obtain the following result. Theorem 3. Let K1,K2 ∈Cp(QS,G). The triples (SK1,NK1,GK1) and (SK2,NK2,GK2) are topologically equivalent if and only if there exists Φ ∈NQS,G such that Φ(K1) = K2. Analogously to Corollary 1, we have the following result. Corollary 2. Let (S,N,G) be a triple as above. Then the cardinality of the quotient set Ck(QS,G)/NQS,G is the number of pairwise topologically inequivalent triples ( ˆS, ˆN, ˆG), where ( ˆS, ˆN) is a Zm k -action of signature (0;kn+1) and ˆN ◁ˆG ⩽Aut( ˆS) is such that ˆG/ ˆN induces Q∗ S,G 4. The case m = 2 and k = p prime In this section we restrict to the case of Z2 p-actions of signature (0;pn+1), where p ⩾2 is prime, n ⩾3 and (n -1)(p -1) > 2. 4.1. Description of F(p,n,2). We start by describing the elements of F(p,n) = F(p,n,2). Theorem 4. If K ∈F(p,n) then one of the following statements holds. (1) There are integers r3,s3,...,rn,sn ∈{0,1,...,p -1} satisfying that (rj,sj) ≠(0,0) for each j, and that (1 + r3 + ⋯+ rn,1 + s3 + ⋯+ sn) ≢(0,0) mod p in such a way that K = ⟨ar3 1 as3 2 a-1 3 ,...,arn 1 asn 2 a-1 n ⟩. (2) There is an integer 2 ⩽t ⩽n -1, there are integers l2,...,lt ∈{1,...,p -1}, and there are integers rt+2,st+2,...,rn,sn ∈{0,1,...,p -1} satisfying that (rj,sj) ≠(0,0) for each j, and that (1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn,1 + st+2 + ⋯+ sn) ≢(0,0) mod p in such a way that K = ⟨al2 1 a-1 2 ,...,alt 1 a-1 t ,art+2 1 ast+2 t+1 a-1 t+2,...,arn 1 asn t+1a-1 n ⟩. Proof. Every K ∈F(p,n) is the kernel of a surjective homomorphism θ ∶H →Z2 p such that aj ∉K for j = 1,...,n + 1. Set φ1 = θ(a1). By the surjectivity of θ, there is some 1 ⩽t ⩽n such that θ(aj) ∈⟨φ1⟩for j = 1,...,t, and φ2 ∶= φ(at+1) ∉⟨φ1⟩. In this case, we can write θ(aj) = φlj 1 , j = 2,...,t if t ⩾2, and θ(at+i) = φrt+i 1 φst+i 2 , i = 2,...,n + 1 -t. The fact that θ(aj) has order p implies that l2,...,lt ∈{1,...,p -1}, and that rj,sj ∈{0,1,...,p -1} are not simultaneously zero, for each j. The relation a1⋯an+1 = 1 implies that t ⩽n-1, that 1+st+2+⋯+sn ≡-sn+1 mod p, and that 1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn ≡-rn+1 mod p or 1 + rt+2 + ⋯+ rn ≡-rn+1 mod p, according to whether or not t is different from 1. Finally, the fact that (rn+1,sn+1) ≠(0,0) ends the proof. □ We say that K is of type (1) or (2) according to the enumeration in the theorem above. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 13 4.2. Algebraic descriptions. We recall that, as observed in §3.1, each Zm p -action can be described as a fiber product. We proceed to make such a description explicit for the case m = 2 and k = p prime, and for each possible group K ∈F(p,n). If Λ = (q4,...,qn+1) ∈Ωn, then we consider the Z2 p-action (SK = Ck(Λ)/K,NK = H/K = ⟨φ1,φ2⟩), where φ1 and φ2 are as defined in the proof of Theorem 4. For j = 1,2, set Sj ∶= SK/⟨φj⟩and denote by πj ∶Sj → ̄C the branched regular covering map with deck group ⟨τj⟩= NK/⟨φj⟩≅Zp. The branch locus of π1 is given by the set {qt+1,...,qn+1} and therefore an algebraic description for S1 is given by S1 ∶yp 1 = (x -qt+1) n+1 ∏ j=t+2 (x -qj)sj, where sn+1 ≡-(1 + st+2 + ⋯+ sn) mod p. In this model π1(x,y1) = x. Similarly, the branch locus of π2 is given by the set {q1,...,qt,qt+2,...,qn+1} and therefore an algebraic description for S2 is given as follows. If K is of type (1) then S2 is given by S2 ∶yp 2 = n+1 ∏ j=3 (x -qj)rj where rn+1 ≡-(1 + r3 + ⋯+ rn) mod p. If K is of type (2) then S2 ∶yp 2 = t ∏ i=2 (x -qi)li n+1 ∏ j=t+2 (x -qj)rj where rn+1 ≡-(1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn) mod p. In this model π2(x,y2) = x. All the above coupled with the discussion in §3.1 is the proof of the following result. Proposition 5. Let (SK,NK) be a Z2 p-action of signature (0;pn+1). Then, with the same notations as in Theorem 4, an algebraic description of SK is given as follows. If K is of type (1) then SK ∶ ⎧⎪⎪⎨⎪⎪⎩ yp 1 = x∏n+1 j=3 (x -qj)sj yp 2 = ∏n+1 j=3 (x -qj)rj where sn+1 ≡-(1 + st+2 + ⋯+ sn) mod p and rn+1 ≡-(1 + r3 + ⋯+ rn) mod p. If K is of type (2) then SK ∶ ⎧⎪⎪⎨⎪⎪⎩ yp 1 = (x -qt+1)∏n+1 j=t+2(x -qj)sj yp 2 = ∏t i=2(x -qi)li ∏n+1 j=t+2(x -qj)rj where sn+1 ≡-(1 + st+2 + ⋯+ sn) mod p and rn+1 ≡-(1 + l2 + ⋯+ lt + rt+2 + ⋯+ rn) mod p. In both cases, the group NK corresponds to ⟨φ1(x,y1,y2) = (x,y1,ωpy2),φ2(x,y1,y2) = (x,ωpy1,y2)⟩. 4.3. Jacobian variety. We recall that the Jacobian variety JS of a compact Riemann surface of genus g is an irreducible principally polarized abelian variety of dimension g. By the classical Torelli's theorem, the Jacobian variety JS determines S, namely, S ≅S′ if and only if JS ≅JS′. For each group of automorphisms G of a Riemann surface S, we denote by SG the underlying Riemann surface structure of the orbifold S/G. Let (X,H0) be a generalized Fermat pair of type (p,n), where p is prime. Following the main result of [7], JX decomposes, up to isogeny, as follows: (4.1) JX ∼∏ Hr JXHr, 14 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA where Hr runs over all subgroups of H0 which are isomorphic to Zn-1 p and such that X/Hr has positive genus. In addition, the cyclic p-gonal curves XHr run over all curves of the form yp = r ∏ j=1 (x -μj)αj, where {μ1,...,μr} ⊂{∞,0,1,λ1,...,λn-2}, μi ≠μj if i ≠j, and αj ∈{1,2,...,p -1} satisfying that: (i) if every μj ≠∞, then α1 = 1, α2 + ⋯+ αr ≡-1 mod p, and (ii) if some μj = ∞, then α1 + ⋯+ αj-1 + αj+1 + ⋯+ αr ≡-1 mod p. Let p be a prime number and let (S,N) be a Zm p -action of signature (0;pn+1). As observed in Proposition 2, there is a generalized Fermat pair (X,H0) of type (p,n), and a subgroup K ≅Zn-m p of H0 such that S ≅X/K and N ≅H0/K. Since JX ∼JS × P(X/S). where P(X/S) is the Prym variety associated to the covering map X →S = X/K, the isogeny decomposition (4.1) permits to state the following conjecture. Conjecture. Let p ⩾2 be a prime number and let (S,N) be a Zm p -action of signature (0;pn+1), where 2 ⩽m ⩽n-1. Then JS ∼∏ L∈L JSL where L = {L ⩽N ∶L ≅Zm-1 p }. The conjecture above does hold for the case m = 2, and the proof can be obtained as a consequence of a result due to Kani-Rosen in [22]. Theorem 5. Let p ⩾2 be a prime number and n ⩾3. Let (S,N) be a Z2 p-action of signature (0;pn+1). Then JS ∼∏ L∈L JSL where L = {L ⩽N ∶L ≅Zp}. Proof. Let us write N = ⟨φ1,φ2⟩. We have that L = {L1 = ⟨φ1⟩,L2 = ⟨φ2⟩,L3 = ⟨φ1φ2⟩,...,Lp+1 = ⟨φ1φp-1 2 }. Observe that if i ≠j then LiLj = LjLi and ⟨Li,Lj⟩= N, showing that the genus of S/⟨Li,Lj⟩is zero. Let Π ∶S →S/N and Πi ∶S →S/Li be the regular covering maps with deck groups N and Li respectively, for each i ∈{1,...,p + 1}. Observe that if q ∈S/N is a ramification value of Π then Π-1(q) consists of p points, all of them with N-stabilizer Lj for some j. This shows that if ci is the number of points of S that are fixed by Li then c1 + ⋯+ cp+1 = (n + 1)p. The Riemann-Hurwitz formula applied to Πi implies that 2g -2 = 2pγi -2p + ci(p -1), where γi is the genus of S/Li. Consequently, one has that (4.2) (2g -2)(p + 1) = 2p(γ1 + ⋯+ γp+1) -2p(p + 1) + (n + 1)p(p -1). Now, by considering (3.1) with m = 2, we see that g = ((n -1)p -2)(p -1)/2 and the equality (4.2) turns into γ1 + ⋯+ γp+1 = g. The proof follows from [22, Theorem C]. □ 5. Example 1: The case m = 2,k = p prime and n = 3 For the sake of clarity, we now specialize our results to the case n = 3, that is, Z2 p-actions of signature (0;p4), and describe some examples in detail. Note that the Riemann surfaces corresponding to such Z2 p-actions have genus (p -1)2 and form complex one-dimensional families. In this case, we assume p ⩾3. By applying Theorem 4, one sees that the collection F(p,3) consists of the following groups. (1) K(r,s) = ⟨ar 1as 2a-1 3 ⟩, where r,s ∈{0,1,...,p -1} satisfy (r,s) ∉{(0,0),(p -1,p -1)}. (2) K(l) = ⟨al 1a-1 2 ⟩where l ∈{1,...,p -1}. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 15 Example 1. Consider the Z2 p-action of signature (0;p4) associated to the group K = K(0,p -1) = ⟨ap-1 2 a-1 3 ⟩. By Proposition 5, the family formed by the Riemann surfaces SK are algebraically represented by (5.1) SK ∶ ⎧⎪⎪⎨⎪⎪⎩ yp 1 = x(x -1)p-1 yp 2 = (x -λ)p-1 where λ ∈C -{0,1}. A routine computation shows that the maps a(x,y1,y2) = ( λ x, λ1/py2 x , λ1-1/py1 x ) and b(x,y1,y2) = ( x-λ x-1 , (1-λ)1-1/p(x-λ) (x-1)y2 , (1-λ)1-1/px y1 ) are automorphisms of SK and ⟨a,b⟩≅Z2 2. In addition, one has that aφ1a = φ2, aφ2a = φ1, bφ1b = φ-1 2 , bφ2b = φ-1 1 , where φ1 and φ2 generate N ≅Z2 p and are given in Proposition 5. Now, if we set R ∶= φ1φ2, S ∶= b, ˆR ∶= φ1φ-1 2 , ˆS ∶= a then Aut(SK) ⩾⟨S,R⟩× ⟨ˆS, ˆR⟩≅Dp × Dp. The signature of the action of this last group is (0;2,2,2,p) and therefore, as a maximal signature [30], up to finitely many exceptions, the automorphism group of SK is isomorphic to Dp × Dp. Later we shall see that there is only one exceptional member with more than 4p2 automorphisms. The interest in this family comes from the following fact. A well-known result due to Accola [1] states that if a p-gonal Riemann surface has genus g > (p -1) then the p-gonal morphism is unique. The family described above was considered by Costa, Izquierdo and Ying in [9], when they noticed that it has two p-gonal morphisms. With our notation, such morphisms are R(x,y1,y2) = (x,ωpy1,ωpy2) and ˆR(x,y1,y2) = (x,ωpy1, ̄ωpy2). We should point out that the algebraic description of this family given here differs from the one already known for such surfaces; see [9, Section 5]. We recall that the action of ⟨Φ1,Φ2⟩= Autg(H) ≅S4 on F(p,3) is given as follows (see Proposition 1). Φ1(a1,a2,a3,a4) = (a2,a1,a3,a4) and Φ2(a1,a2,a3,a4) = (a2,a3,(a1a2a3)-1,a1). For instance, observe that (5.2) Φ1(K(r,s)) = K(s,r),Φ2(K(l)) = K(0,l) and Φ2(K(0,s)) = K(u,u) where u ∈{1,...,p -1} satisties u(1 + s) ≡-1 mod p. Example 2. We proceed to describe explicitly the orbits of this action for p = 5. Note that F(5,3) consists of 27 groups. By considering (5.2), it suffices to restrict our attention to the groups K(r,s) where (r,s) ∈{0,1,2,3,4}2 -{(0,0),(4,4)} and r 0 0 s = 0 and s5 = ⎧⎪⎪⎨⎪⎪⎩ r -s r ⩾s p + r -s r < s Proof. Let K ∈Cp(QS,G) and let θ ∶H →Z2 p be a group epimorphism such that K = ker(θ). As a1 ∉K, we have that φ1 ∶= θ(a1) has order p. 1. Assume θ(a2) ∈⟨φ1⟩. The fact that A(a1) = a2,A(a2) = a3,A(a3) = a1 implies that θ(a2) = φl 1 and θ(a3) = φl2 1 for some l ∈{1,...,p -1} such that 1 + l + l2 ≡1 mod p. If we write φ2 ∶= θ(a4) then the we have that a5 = B(a3) and therefore θ(a5) = φl2 2 . Similarly, θ(a6) = φl 2. Note that φ2 ∉⟨φ1⟩and ⟨φ1,φ2⟩≅Z2 p. It follows that K = ⟨a1a2a3,a4a5a6,al 1a-1 2 ,al2 1 a-1 3 ,al2 2 a-1 5 ,al 2a-1 6 ⟩= ⟨a1a2a3,al 1a-1 2 ,al 2a-1 6 ⟩= K(l). 2. Assume θ(a2) ∉⟨φ1⟩. If we write φ2 ∶= θ(a2) then ⟨φ1,φ2⟩≅Z2 p. The action of A implies that, if we write (6.1) θ(a3) = φˆr 1φˆs 2, then ˆrˆs ≡1 mod p and ˆr + ˆs2 ≡0 mod p. Similarly, if we set φ(ai) = φri 1 φsi 2 for i = 4,5,6 then the action of A show that (6.2) r5 ≡ˆrs4 mod p, s5 ≡r4 + ˆss4 mod p, r6 ≡ˆrr4 + s4 mod p, s6 ≡ˆsr4 mod p. Now, the action of B on a4 implies that (φr4 1 φs4 2 )r4(φr6 1 φs6 2 )s4 = φ1, showing that s4(r4 + s6) ≡0 mod p and r2 4 + r6s4 ≡1 mod p. We consider each possible case separately. 2.1 Assume s4 = 0, and therefore r4 = ±1. By (6.2) we have that r5 = 0,s5 = r4,r6 = ˆrr4 and s6 = ˆsr4 and therefore θ is given by (a1,...,a6) ↦(φ1,φ2,φˆr 1φˆs 2,φ1,φ2,φˆr 1φˆs 2) or (φ1,φ2,φˆr 1φˆs 2,φ-1 1 ,φ-1 2 ,φ-ˆr 1 φ-ˆs 2 ) according to whether or not r4 = 1. By considering that a1⋯a6 = 1 in the former case, and the action of B coupled with (6.1) in the latter, we obtain ˆr = ˆs = -1. We conclude that K = K3 ∶= ⟨a1a2a3,a1a-1 4 ,a2a-1 5 ⟩or K = K4 ∶= ⟨a1a2a3,a1a4,a2a5⟩, according to whether or not r4 = 1. 2.2 Assume r4 = -s6, and therefore r2 4 + r6s4 = 1. By (6.2) we have that s6 = ˆsr4, and therefore r4 = s6 = 0 or ˆs = -1. 2.2.1 Assume r4 = s6 = 0. We have r5 = ˆrs4,s5 = ˆss4,r6 = s4, and s4 = ±1. It follows that θ is given by (a1,...,a6) ↦(φ1,φ2,φˆr 1φˆs 2,φ±1 2 ,φ±ˆr 1 φ±ˆs 2 ,φ±1 1 ). The action of B implies that ˆr = ˆs = -1 and we then obtain that θ is given by (φ1,φ2,φ-1 1 φ-1 2 ,φ2,φ-1 1 φ-1 2 ,φ1) or (φ1,φ2,φ-1 1 φ-1 2 ,φ-1 2 ,φ1φ2,φ-1 1 ). Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 21 Thus, we conclude that K = K1 ∶= ⟨a1a2a3,a1a-1 6 ,a2a-1 4 ⟩or K = K2 ∶= ⟨a1a2a3,a1a6,a2a4⟩ respectively. 2.2.2 Assume ˆs = -1. We have that ˆr = -1 and therefore r5 = -s4,s5 = r4 -s4,r6 = s4 -r4 and s6 = -r4. By proceeding as before, the action of B shows that r2 4 + s2 4 -r4s4 ≡1 mod p. Thus, if we let r ∶= r4 and s ∶= s4 then θ is given by (a1,...,a6) ↦(φ1,φ2,φ-1 1 φ-1 2 ,φr 1φs 2,φ-s 1 φr-s 2 ,φs-r 1 φ-r 2 ) and therefore K = K(r,s). The fact that K(0,1) = K1,K(0,p -1) = K2,K(1,0) = K3 and K(p -1,0) = K4 finishes the proof. □ Theorem 6. Let p ⩾2 be a prime number. Consider the set Fp = {(r,s) ∶2 ⩽s < r ⩽p -2 and r2 + s2 -rs ≡1 mod p} and let γ be its cardinality. Write α ∶= ⎧⎪⎪⎨⎪⎪⎩ 0 if p ≡2 mod 3 1 otherwise and β ∶= ⎧⎪⎪⎨⎪⎪⎩ 1 if p = 2 2 otherwise Then the cardinality of Cp(QS,G)/NQS,G is α + β + 1 3γ. Proof. Following Proposition 7, the group NQS,G is generated by Q∗ S,G and by Ψ2,Ψ5. The fact that each K ∈ F(p,5,2) is ⟨A,B⟩-invariant shows that we only need to determine the number of orbits of the action of ⟨Ψ2,Ψ5⟩≅ D3 on Cp(QS,G). For simplicity, we write Ψ2 = (456),Ψ2 2 = (465),Ψ3 = Ψ5Ψ2 2 = (23)(56),Ψ5 = (23)(46). Observe that the groups K(l) appear if and only if p = 3 or p ≡1 mod 3, and in such a case there are exactly two groups of this type; namely K(l) and K(l2). Note that Ψ3(K(l)) = ⟨a1a2a3,al 1a-1 3 ,al 4a-1 5 ⟩. The equalities (a1a2a3 ⋅al 1a-1 3 )-1 = al2 1 a-1 2 and ((a1a2a3) ⋅(al 4a-1 5 ))-1 = al2 4 a-1 6 ensures that Ψ3(K(l)) = K(l2). In a very similar way, it can be seen that Ψ2(K(l)) = K(l) and Ψ2(K(l2)) = K(l2). It follows that K(l) and K(l2) form α orbits. We now consider the groups K(r,s) where r = 0 or s = 0 or r = s. Observe that these cases yield only six groups, namely, K(0,1),K(0,p -1),K(1,0),K(p -1,0),K(1,1) and K(p -1,p -1). We claim that they form β orbits. In fact, a routine computation shows that {K(0,1),K(1,0),K(p -1,p -1)} and {K(0,p -1),K(p -1,0),K(1,1)} are two orbits if p ⩾3, and that they colapse in one orbit when p = 2. All the above coupled with the equality Ψ5(K(r,s)) = K(s,r) allows us to restrict our attention to the groups K(r,s) where (r,s) belongs to the parameter set Fp. We write Fp = {K(r,s) ∶(r,s) ∈Fp}. Note that F2 and F3 are empty. Therefore we assume p ⩾5. If (r,s) ∈Fp then Ψ5(K(r,s)) = K(s,r) ∉Fp. Thus, the set Fp splits into orbits of length 1 or 3. The equalities Ψ2(K(r,s)) = K(p + s -r,p -r) and Ψ2 2(K(r,s)) = K(p -s,r -s) together with the fact that (r,s),(p -s,r -s) and (p + s -r,p -r) are pairwise distinct show that Fp splits into orbits of length 3 only. This completes the proof. □ 22 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Example 4. If p = 2 then C2(QS,G) consists of only three groups: K(0,1),K(1,1) and K(1,0), and they are equivalent. We then obtain the well-known fact that among the topological classes of actions of Z2 2 on Riemann surfaces S genus g = 3 with signature (0;26), there is only one of them for which S has a group of automorphisms G ≅QS,G/K(0,1) which is isomorphic to ⟨a1,a2,A,B ∶a2 1 = a2 2 = [a1,a2] = A3 = B2 = (AB)2 = 1,Aa1A = a2,Aa2A = a1a2,Ba1B = a2⟩≅G2 ≅S4, and acts with signature (0;2,2,2,3). These Riemann surfaces form the family of quartics x4 + y4 + z4 + t(x2y2 + x2z2 + y2z2) = 0, denoted by C2 at the beginning of this section. We apply Proposition 5 to obtain that ⎧⎪⎪⎨⎪⎪⎩ y2 1 = x(x -1)(x -λ)(x - 1 1-λ) y2 2 = (x -1)(x - 1 1-λ)(x -λ-1 λ ) where λ ∈C -{0,1}, is another algebraic description for this family. Observe that, while the (generic) members of this family are non-hyperelliptic, they are the fiber product of two Riemann surfaces of genus one. Furthermore, with the notations of Remark 9, observe that C(K(0,1)) = ⟨a4a5a6,a1a6,a2a5a6⟩= K(0,1) and therefore K(0,1) is C-invariant. Similarly, K(1,1) is D-invariant. We then conclude that S0 ∶= C2(Λ0)/K(0,1) and S1 ∶= C2(Λ1)/K(1,1) are members of C2 admitting a group of automorphisms isomorphic to Z2 2 ⋊D6 ≅Z2 × S4 and Z2 2 ⋊S4 ≅Z2 4 ⋊S3 respectively. These Riemann surfaces are the unique hyperelliptic curve of genus 3 with 48 automorphisms, and the Fermat quartic, respectively. Example 5. If p = 3 then C3(QS,G) consists of seven groups, and they split into three classes, represented by K(0,1),K(0,2) and K(1). It follows that there are precisely three classes of topologically inequivalent triples (S,N,G), where (S,N) is a Z2 3-action of signature (0;36) such that N ⊴G and G/N ≅D3. In this case, G acts with signature (0;2,2,3,3). Observe that there are at most three groups G as before. In the former case, we have G = G(0,1) ∶= QS,G/K(0,1) = ⟨a1,...,a6,A,B⟩/⟨a1 = a6,a2 = a4,a3 = a5 = (a1a2)-1⟩ ≅⟨a1,a2,A,B ∶...,Aa1A-1 = a2,Aa2A-1 = (a1a2)-1,Ba1B = a2⟩. Similarly, in the second and third case, we obtain that G = G(0,2) ∶= QS,G/K(0,2) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = a2,Aa2A-1 = (a1a2)-1,Ba1B = a-1 2 ⟩ G = G(1) ∶= QS,G/K(1) ≅⟨a1,a2,A,B ∶...,[A,a1] = [A,a2] = 1,Ba1B = a2⟩. The groups G(0,1),G(0,2) and G(1) are pairwise non-isomorphic and turn out to be semidirect products, as in the case p ≠3. Note that G(0,1) ≅G3 and therefore the Riemann surfaces C3(Λ)/K(0,1) form the family C3. Finally, it is worth noticing that there are two topologically inequivalent actions of G3 in genus 10, as can be obtained by using the routines given in [4]. One of them is the one represented by K(0,1) and the other is not obtained here, as in this case the corresponding Z2 3-quotient has genus two. Now, we can extend the previous two examples to the general case. Proposition 9. Let p ⩾5 be a prime number. If F is a complex one-dimensional family of compact Riemann surfaces of genus (p -1)(2p -1) endowed with a group of automorphisms G isomorphic to a semidirect product of the form Z2 p ⋊D3, then G ≅Gp. In particular, family of Kuribayashi-Komiya curves Cp corresponds to one irreducible component of F. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 23 Proof. By the Riemann-Hurwitz fomula, if S belongs to a complex one-dimensional family F as in the statement of the proposition, then (S,N) is a Z2 p-action of signature (0;p6), where N is the p-Sylow subgroup of G. Then S ≅Cp(Λ)/K and G ≅QS,G/K for some K ∈Cp(QS,G). We now study the quotients QS,G/K. 1. Assume K = K(l) ∶= ⟨a1a2a3,al 1a-1 2 ,al 4a-1 6 ⟩, where p ≡1 mod 3 and l ∈{1,...,p -1} satisfies l2 + l + 1 ≡ 0 mod p. In this case, G(l) ∶= QS,G/K(l) = ⟨a1,...,a6,A,B⟩/⟨a1a2a3 = 1,a6 = al 4,a2 = al 1⟩. Note that in the quotient a2 = al 1,a3 = al2 1 ,a4 = al2 6 ,a5 = al 6. It follows that G(l) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = al 1,Aa2A-1 = al2 2 ,Ba1B = a2⟩. 2. Assume K = K(r,s) ∶= ⟨a1a2a3,ar 1as 2a-1 4 ,a-s 1 ar-s 2 a-1 5 ⟩where r,s ∈{0,...,p -1} satisfy r2 + s2 -rs ≡1 mod p. In this case, one has that G(0,1) = QS,G/K(0,1) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = a2,Aa2A-1 = (a1a2)-1,Ba1A = a2⟩. G(0,p -1) = QS,G/K(0,p -1) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = a2,Aa2A-1 = (a1a2)-1,Ba1B = a-1 2 ⟩. G(1,0) = QS,G/K(1,0) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = a2,Aa2A-1 = (a1a2)-1,Ba1B = a1,Ba2B = (a1a2)-1⟩. G(p -1,0) = QS,G/K(p -1,0) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = a2,Aa2A-1 = (a1a2)-1,Ba1B = a-1 1 ,Ba2B = a1a2⟩, and for r,s ≠0 we have that G(r,s) = QS,G/K(r,s) ≅⟨a1,a2,A,B ∶...,Aa1A-1 = al 1,Aa2A-1 = al2 2 ,Ba1B = a2⟩. Observe that the elements σ1 ∶= a2 and σ2 ∶= (a1a2)-1 of K(0,1) generate a subgroup of it isomorphic to K(1,0). Likewise, it can be seen that G(l) ≅G(0,1) ≅G(0,p -1) ≅G(1,0) ≅G(p -1,0) ≅G(r,s). Finally, by considering the elements σ1 ∶= a2 1a2 and σ2 ∶= a-1 1 a-2 2 of K(0,1), we conclude that G ≅Gp. □ Example 6. The following table summarizes the number N of topologically distinct triples (S,N,Gp) where (S,N) is a Z2 3-action of signature (0;36) and the signature of S/Gp is (0;2,2,3,p), for some small primes (c.f. [25, Proposition 2]). p 5 7 11 13 17 19 23 29 31 N 2 3 3 4 4 5 5 6 7 6.2. The case L = Z2 2. Note that if p ≠2 then G is isomorphic to a semidirect product of the form Z2 p ⋊Z2 2. Set L ≅G/N ≅Z2 2 = ⟨a,b ∶a2 = b2 = (ab)2 = 1⟩. If ∞,0,1,q4,q5,q6 are the cone points of S/N then, after conjugating by a suitable M ̈obius transformation, we can assume a(z) = -z, b(z) = λ z and that q4 = λ, q5 = -1, q6 = -λ. Note that Λ = (q4,q5,q6) ∈Ω5 if and only if λ ≠0,±1. It follows that S ≅Cp(Λ)/KS and N ≅H/KS for some KS ∈F(p,5,2), where Cp(Λ) ∶ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ xp 1 + xp 2 + xp 3 = 0 λxp 1 + xp 2 + xp 4 = 0 -xp 1 + xp 2 + xp 5 = 0 -λxp 1 + xp 2 + xp 6 = 0 ⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭ ⊂P5 The group L lifts to a group of automorphisms QS,G of Cp(Λ) such that QS,G/KS ≅G and S/G ≅Cp(Λ)/QS,G. 24 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Lemma 2. The structure of the group QS,G is as follows. (1) If p ⩾3 is a prime number, then QS,G is isomorphic to a semidirect product Z5 p ⋊Z2 2 = ⟨a1,...,a5 ∶ap j = [ai,aj] = 1⟩⋊⟨A,B ∶A2 = B2 = (AB)2 = 1⟩, where the action by conjugation of A and B on a1,...,a5 is given by A ∶(a1,a2,a3,a4,a5) ↦(a1,a2,a5,(a1⋯a5)-1,a3) and B ∶(a1,a2,a3,a4,a5) ↦(a2,a1,a4,a3,(a1⋯a5)-1). (2) If p = 2, then QS,G is isomorphic to ⟨a1,...,a5,A,B ∶a2 j = [ai,aj] = B2 = 1,A2 = a1,(AB)2 = a4a5,Aa1A-1 = a1,Aa2A-1 = a2, Aa3A-1 = a5,Aa4A-1 = a1a2a3a4a5,Ba1B = a2,Ba3B = a4,Ba5B = a1a2a3a4a5⟩. Proof. The proof follows the same ideas as the proof of Lemma 1. Every lifting of a is of the form A([x1 ∶... ∶x6]) = [x1 ∶α2x2 ∶α3x5 ∶α4x6 ∶α5x3 ∶α6x4] where αp 2 = αp 3 = αp 4 = αp 5 = αp 6 = -1, and every lifting of b is of the form B([x1 ∶... ∶x6]) = [x2 ∶β2x1 ∶β3x4 ∶β4x3 ∶β5x6 ∶β6x5] where βp 2 = βp 4 = λ,βp 3 = 1,βp 5 = -1,βp 6 = -λ. 1. If p ⩾3 then we may take α2 = α3 = α4 = α5 = α6 = -1 and β2 = β4 = η,β3 = 1,β5 = -1,β6 = -η where η is a fixed choice of λ1/p, to get A([x1 ∶... ∶x6]) = [-x1 ∶x2 ∶x5 ∶x6 ∶x3 ∶x4] and B([x1 ∶... ∶x6]) = [x2 ∶ηx1 ∶x4 ∶ηx3 ∶-x6 ∶-ηx5], which satisfy the relations A2 = B2 = (AB)2 = 1. 2. If p = 2 then we may take α2 = α3 = α4 = α5 = α6 = i and β2 = β4 = -η,β3 = 1,β5 = -i,β6 = -iη where η is a fixed choice of λ1/2, to get A([x1 ∶... ∶x6]) = [-ix1 ∶x2 ∶x5 ∶x6 ∶x3 ∶x4] and B([x1 ∶... ∶x6]) = [x2 ∶-ηx1 ∶x4 ∶-ηx3 ∶-ix6 ∶-iηx5], which satisfy the relations A2 = a1,B2 = 1,(AB)2 = a4a5. □ Remark 11. Observe that if λ = λ0 = i and Λ1 = (i,-1,-i), then the M ̈obius transformation d(z) = iz satisfies that d2 = a and lifts to an automorphisms D of order 4 of Cp(Λ1) such that ⟨D,B⟩≅D4 and ⟨QS,G,D⟩≅Z5 p ⋊D4. The action by conjugation of D on a1,...,a5 is given by D ∶(a1,a2,a3,a4,a5) ↦(a1,a2,a4,a5,(a1⋯a5)-1). Consider the group isomorphism σ ∶Autg(H) →S6 = Sym{a1,...,a6} and write u = σ(A) = (35)(46) and v ∶= σ(B) = (12)(34)(56). Proposition 10. NQS,G is generated by Q∗ S,G and the geometric automorphisms of H given by Ψ2(a1,...,a6) = (a1,a2,a3,a6,a5,a4) and Ψ3(a1,...,a6) = (a1,a2,a4,a3,a6,a5). Proof. We argue as done in Proposition 7, after noticing that NQS,G is isomorphic to the normalizer N of ⟨u,v⟩ in S6, which is given by N = ⟨u,v,Ψ2 = (46),Ψ3 = (34)(56)⟩. Note that N/⟨u,v⟩≅Z2 2. □ By Corollary 2, the number of pairwise topologically inequivalent actions (S,N,G) in genus (2p-1)(p-1) with a Z2 p-action of genus zero and S/G of signature (0;2,2,p,2p) corresponds to the cadinality of Cp(QS,G)/NQS,G. The following proposition describes the members of Cp(QS,G). Proposition 11. If p ⩾3 is prime and K ∈Cp(QS,G) then one of the following statements holds. (1) There are r,s ∈{0,...,p -1} satisfying that r + s ∈{ p-1 2 , 3p-1 2 } and K = K(r,s) ∶= ⟨ar 1as 2a-1 3 ,a3a-1 5 ,a4a-1 6 ⟩. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 25 (2) K agrees with K1 ∶= ⟨a1a-1 2 ,a3a-1 4 ,a5a-1 6 ⟩, K2 ∶= ⟨a1a-1 2 ,a3a-1 6 ,a4a-1 5 ⟩, K5 ∶= ⟨a1a-1 2 ,a3a-1 5 ,a4a-1 6 ⟩or K6 ∶= ⟨a1a2,a3a-1 5 ,a4a-1 6 ⟩. (3) There is r ∈{0,...,p -1} and K = K3(r) ∶= ⟨ar 1a-1 3 a4,a1a2,a3a6⟩or K = K4(r) ∶= ⟨ar 1a-1 3 a6,a1a2,a3a4⟩ If p = 2 and K ∈C2(QS,G) then K equals ̄K1 = ⟨a1a2,a3a4,a3a5⟩, ̄K2 = ⟨a1a2,a3a4,a1a3a5⟩, ̄K3 = ⟨a1a2,a3a5,a1a3a4⟩or ̄K4 = ⟨a1a2,a4a5,a1a3a4⟩, Proof. Let K ∈Cp(QS,G) and let θ ∶H →Z2 p be a group epimorphism such that K = ker(θ). As a1 ∉K, we have that φ1 ∶= θ(a1) has order p. 1. Assume θ(a2) ∉⟨φ1⟩and write φ2 ∶= θ(a2) in such a way that ⟨φ1,φ2⟩≅Z2 p. Set θ(a3) ∶= φr 1φs 2 for r,s ∈{0,...,p -1} that are not simultaneously equal to zero. The fact that A(a3) = a5,A(a1) = a1 and A(a2) = a2 implies that φr 1φs 2 = θ(a3). Similarly, it can be seen that θ(a6) = θ(a4). Now, the fact that B(a1) = a2 and B(a3) = a4 implies that θ(a4) = φr 2φs 1. It follows that θ = (φ1,φ2,φr 1φs 2,φs 1φr 2,φr 1φs 2,φs 1φr 2) and therefore K = K(r,s). The relation a1⋯a6 = 1 implies that 2(r+s) ≡-1 mod p and therefore r+s ∈{ p-1 2 , 3p-1 2 }. 2. Assume θ(a2) ∈⟨φ1⟩and write θ(a2) = φl 1 for some l ∈{1,...,p -1}. We claim that θ(a3) ∉⟨φ1⟩. In fact, otherwise, if we write θ(a3) = φm 1 then θ(a5) = φlm 1 and similarly θ(a6) = φml 1 . This contradicts the surjectivity of θ. Thus, we write φ2 ∶= θ(a3) and therefore ⟨φ1,φ2⟩≅Z2 p. Note that φl 1 = θ(a2). As B has order two, we deduce that l = 1 or l = p -1. Write θ(a5) = φr 1φs 2 for some r,s ∈{0,...,p -1} that are not simultaneously zero. The equality θ(a5) = φr 1φs 2 coupled with the fact that A has order two imply that r(1 + s) ≡0 mod p, and that s = 1 or s = p -1. 2.1 Assume s = 1. It follows that r = 0 and θ(a5) = φ2. Write θ(a4) ∶= φu 1φv 2 then θ(a6) = φu 1φv 2 and u(l + v) ≡ 0 mod p, and v = 1 or v = p -1. The fact that a1⋯a6 = 1 implies that v = p -1 and therefore (1) u = 0 and l = p -1, and then θ = (φ1,φ-1 1 ,φ2,φ-1 2 ,φ2,φ-1 2 ) and K = K6, or (2) u = p -1 and l = 1, and then θ = (φ1,φ1,φ2,φ-1 1 φ-1 2 ,φ2,φ-1 1 φ-1 2 ) and K = K5. 2.2 Assume s = p -1. It follows that θ(a5) = φr 1φ-1 2 . If we write θ(a4) ∶= φu 1φv 2 then, by proceeding as in the previous case, we obtain that θ(a6) = φu+rv 1 φ-v 2 where v = 1 or v = p-1,u(l+v) ≡0 mod p and 2u ≡r(l-v) mod p. Moreover, The fact that a1⋯a6 = 1 implies that 1 + l + r + 2u + rv ≡0 mod p. Assume v = 1. Then (1) u = 0,l = 1,r = p -1 and θ = (φ1,φ1,φ2,φ2,φ-1 1 φ-1 2 ,φ-1 1 φ-1 2 ) and K = K1, or (2) l = p -1,u = -r and θ = (φ1,φ-1 1 ,φ2,φ-r 1 φ2,φr 1φ-1 2 ,φ-1 2 ) and K = K3(r). Assume v = p -1. Then (1) u = r = p -1,l = 1 and θ = (φ1,φ1,φ2,φ-1 1 φ-1 2 ,φ-1 1 φ-1 2 ,φ2) and K = K2, or (2) l = p -1,u = 0 and θ = (φ1,φ-1 1 ,φ2,φ-1 2 ,φr 1φ-1 2 ,φ-r 1 φ2) and K = K4(r). The proof of the case p = 2 is analogous. This case yields only four epimorphisms ̄θ ∶H →Z2 2 given by ̄θ1 = (φ1,φ1,φ2,φ2,φ2,φ2), ̄θ2 = (φ1,φ1,φ2,φ2,φ1φ2,φ1φ2), ̄θ3 = (φ1,φ1,φ2,φ1φ2,φ2,φ1φ2) and ̄θ4 = (φ1,φ1,φ2,φ1φ2,φ1φ2,φ2), and the proof follows after noticing that ̄Kj = ker( ̄θj) for j = 1,2,3,4. □ 26 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA Remark 12. It is worth emphasizing that whereas the epimorphisms ̄θ2, ̄θ3 and ̄θ4 define Z2 2-actions on Riemann surfaces S of genus three that are topologically equivalent, they are not topologically equivalent as triples (S,N,G). Theorem 7. Let p ⩾2 be a prime number. The cardinality of C2(QS,G)/NQS,G is 3 if p = 2, and p + 4 otherwise. Proof. By Proposition 10, we only need to count the number of orbits of the action of ⟨Ψ2 = (46),Ψ3 = (34)(56)⟩ on Cp(QS,G). Assume p ≠2, and let r,s ∈{0,...,p -1} such that r + s ∈{ p-1 2 , 3p-1 2 }. Note that K(r,s) remains invariant under the action of Ψ2, and Ψ3(K(r,s)) = ⟨ar 1as 2a-1 4 ,a3a-1 5 ,a4a-1 6 ⟩. If τ ∈{1,...,p -1} satisfies that 2τ ≡1 mod p then one has that (ar 1as 2a-4 4 ⋅(a3a-1 5 ⋅a4a-1 6 )τ)-1 = as 1ar 2a-1 3 . This shows that Ψ3(K(r,s)) = K(s,r) and therefore these groups form α orbits, where α is the number of pairs (r,s) such that r,s ∈{0,...,p-1} satisfies r ⩽s and r+s ∈{ p-1 2 , 3p-1 2 }. A routine computation shows that α = p+1 2 . Similarly, if r ∈{0,...,p -1} then it can be seen that Ψ2(K3(r)) = K4(r) and Ψ3(K3(r)) = K3(p -r). Thus, the groups K3(r) and K4(r) form p+1 2 orbits, represented by K3(r) where r ∈{0,..., p-1 2 }. Finally, K1 and K2 form a single orbit, whereas K5 and K6 form one orbit each. All the above show that there are precisely p + 4 orbits. By proceeding analogously, it can be seen that if p = 2 then ̄K2 and ̄K4 form a single orbit, and ̄K1 and ̄K3 form one orbit each. This completes the proof. □ Example 7. Theorem 7 says that there are exactly three pairwise topologically inequivalent actions (S,N,G) where S is a compact Riemann surface of genus three, (S,N) is a Z2 2-action of signature (0;26) and G/N ≅Z2 2. The signature of S/G is (0;2,2,2,4). We denote by Fj the complex-one dimensional family determined by ̄Kj and by Gj (instead of G) the corresponding group. We have that Gj ≅QS,G/ ̄Kj and therefore Lemma 2 implies that G1 ≅⟨a3,A,B ∶a2 3 = A4 = B2 = (AB)2 = [A,a3] = [B,a3] = 1⟩≅⟨a3⟩× ⟨A,B⟩≅Z2 × D4 G2 ≅⟨a1,a3,A,B ∶A2 = (AB)2 = a1,a2 1 = a2 3 = [a1,a3] = B2 = (Aa3)2 = [B,a3] = [A,B] = 1⟩≅⟨B⟩×⟨A,a3⟩≅Z2×D4 G3 ≅⟨a3,A,B ∶a2 3 = A4 = B2 = (AB)2 = [a3,A] = 1,Ba3B = Aa3⟩= ⟨A,a3⟩⋊⟨B⟩≅(Z4 × Z2) ⋊Z2. As an application of Proposition 5 we obtain the following algebraic descriptions for the families Fj. F1 ∶ ⎧⎪⎪⎨⎪⎪⎩ y2 1 = (x2 -1)(x2 -λ2) y2 2 = x F2 ∶ ⎧⎪⎪⎨⎪⎪⎩ y2 1 = (x2 -1)(x2 -λ2) y2 2 = x(x + 1)(x + λ) F3 ∶ ⎧⎪⎪⎨⎪⎪⎩ y2 1 = (x2 -1)(x2 -λ2) y2 2 = x(x2 -λ2) where λ ∈C -{0,±1}. Observe that the (generic) members of F2 and F3 are non-hyperelliptic, whereas the ones of F1 are; the hyperelliptic involution being (x,y2,y2) ↦(x,-y2,y2). Note that ̄K1 is D-invariant (see Remark 11) and therefore the member of F1 obtained by taking λ = i is endowed with a group of automorphisms of order 32 acting with signature (0;2,4,8), and is represented by y2 1 = y8 2 -1. This Riemann surfaces is known as the Accola-Maclachlan curve of genus three. We remark that there is only one topological class of actions of Z2×D4 in genus three with signature (0;2,2,2,4). In particular, if Sj belongs to Fj then the pairs (S1,G1) and (S2,G2) are topologically equivalent. However, the triples (S1,N1,G1) and (S2,N2,G2) are not, as this example shows. We extend the previous example to the general case. Proposition 12. Let p ⩾3 be a prime number and let S be a compact Riemann surface of genus (2p -1)(p -1) endowed with a group of automorphisms G isomorphic to a semidirect product Z2 p ⋊Z2 2. Let N ≅Z2 p be the normal Sylow p-subgroup of G. If S/N has signature (0;p6) then one of the following statements hold. Zm k -ACTIONS OF SIGNATURE (0; k, n+1 . . ., k) 27 (1) G is isomorphic to Z2p ×Dp and there are precisely p+5 2 pairwise topological inequivalent triples (S,N,G). (2) G is isomorphic to Dp ×Dp and there are precisely p+1 2 pairwise topological inequivalent triples (S,N,G). (3) G is isomorphic to Z2 × (Z2 p ⋊(-1,-1) Z2) and there is only one topological class of triples (S,N,G). Proof. We only need to study the quotients QS,G/K, where, by Lemma 2, we have that QS,G = ⟨a1,...,a6,A,B,∶ap j = [ai,aj] = a1⋯a6 = A2 = B2 = (AB)2 = 1, [A,a1] = [A,a2] = 1,Aa3A = a5,Aa4A = a6,Ba1B = a2,Ba3B = a4,Ba5B = a6⟩, and K runs over the subgroups given in Proposition 11 up to NQS,G-action (see the proof of Proposition 7). Let r,s ∈{0,...,p -1} be integers satisfying that r + s ∈{ p-1 2 , 3p-1 2 }. Then G(r,s) ∶= QS,G/K(r,s) ≅⟨a1,a2,A,B ∶...,[A,a1] = [A,a2] = 1,Ba1B = a2⟩, which is isomorphic to ⟨A,a1a2⟩× ⟨B,a1a-1 2 ⟩≅Z2p × Dp. Similarly, for each r ∈{0,..., p-1 2 } we have that G3(r) ∶= QS,G/K3(r) ≅⟨a1,a3,A,B ∶...,[A,a1] = 1,Aa3A = ar 1a-1 3 ,Ba1B = a-1 1 ,Ba3B = a-r 1 a3⟩ which is isomorphic to ⟨A,ar 1a-2 3 ⟩× ⟨B,a1⟩≅Dp × Dp, and G6 ∶= QQ,S/K6 ≅⟨a1,a3,A,B ∶...,[A,a1] = [A,a3] = 1,Ba1B = a-1 1 ,Ba3B = a-1 3 ⟩ is isomorphic to ⟨A⟩× ⟨a1,a3,B⟩≅Z2 × (Zp ⋊(-1,-1) Z2). Finally, by proceeding analogously, we obtain that G1 = QQ,S/K1 and G5 = QQ,S/K5 are isomorphic to Z2p × Dp and the proof is done. □ References [1] R. D. M. Accola, On cyclic trigonal Riemann surfaces I. Trans. Amer. Math. Soc. 283, 423-449 (1984) [2] G. Bartolini, A. F, Costa and M. Izquierdo, On the connectivity of branch loci of moduli spaces, Ann. Acad. Sci. Fenn., Math. 38, No. 1, 245-258 (2013). [3] A. Behn, A. M. Rojas and M. Tello-Carrera, A SAGE package for n-gonal equisymmetric stratification of Mg. Exp. Math. 32 (2023), no. 1, 54-69. [4] A. Behn, R. E. Rodriguez and A. M. Rojas, Adapted hyperbolic polygons and symplectic represen tations for group actions on Riemann surfaces, J. Pure Appl. Algebra 217 (2013), no. 3, 40-426. [5] S. A. Broughton, Equivalence of finite group actions on Riemann surfaces and algebraic curves. Automorphisms of Riemann surfaces, subgroups of mapping class groups and related topics, 89-132, Contemp. Math., 776 (2022). [6] S. A. Broughton, The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl. 37 (1990), no. 2, 101-113. [7] M. Carvacho, R. A. Hidalgo and S. Quispe, Jacobian variety of generalized Fermat curvces, Quart. J. Math. 67 (2016), 261-284. [8] D. Conti, A. Ghigi and R. Pignatelli, Topological types of actions on curves, J. Symbolic Comput. 118, (2023) 17-31 [9] A. F. Costa, M. Izquierdo, Milagros and D. Ying, On cyclic p-gonal Riemann surfaces with several p-gonal morphisms, Geom. Dedicata 147 (2010), 139-147. [10] P. Frediani, A. Ghigi and M. Penegini Shimura varieties in the Torelli locus via Galois coverings. Int. Math. Res. Not. 20 (2015), 10595-10623. [11] J. Gilman, On conjugacy classes in the Teichm ̈uller modular group, Michigan Math. J. 23 (1976), 53-63. [12] G. Gonz ́alez-Diez and W. J. Harvey, Moduli of Riemann surfaces with symmetry. Discrete groups and geometry (Birmingham, 1991), 75-93, London Math. Soc. Lecture Note Ser., 173, Cambridge Univ. Press, Cambridge, 1992. [13] G. Gonz ́alez-Diez, R. A. Hidalgo and M. Leyton- ́Alvarez, Generalized Fermat curves, J. Algebra 321 (2009), no. 6, 16431660. [14] L. Greenberg, Conformal Transformations of Riemann Surfaces. Amer. J. of Math. 82 (2) (1960), 749-760. [15] J. Harvey, On branch loci in Teichm ̈uller space, Trans. Amer. Math. Soc. 153 (1971), 387-399. [16] R. A. Hidalgo, Homology group automorphisms of Riemann surfaces, Moscow Math. Journal 23 (2023), 113-120. [17] R. A. Hidalgo, Smooth quotients of generalized Fermat curves, Rev. Mat. Complut. 36, No. 1, (2023), 27-55. [18] R. A. Hidalgo, A. Kontogeorgis, M. Leyton- ́Alvarez and P. Paramantzoglou, Automorphisms of generalized Fermat curves, J. Pure Appl. Algebra 221 (2017), no. 9, 2312-2337. 28 RUB ́EN A. HIDALGO AND SEBASTI ́AN REYES-CAROCCA [19] R. A. Hidalgo, J. Paulhus, S. Reyes-Carocca and A. M. Rojas, On non-normal subvarieties of the moduli space of Riemann surfaces, To appear in Transf. Groups, [20] A. Hurwitz, ̈Uber algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41 (1893), 403-442. [21] M. Izquierdo, S. Reyes-Carocca and A. M. Rojas, On families of Riemann surfaces with automorphisms, J. Pure Appl. Algebra 225 (2021), no. 10, Paper No. 106704, 21 pp [22] E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann. 284 (1989), 307-327. [23] J. Karab ́aˇs, Personal webpage, https://www.savbb.sk/~karabas/science/discactions/actions-genus16-index.txt [24] A. Kuribayashi and K. Komiya, On Weierstrass points of non-hyperelliptic compact Riemann surfaces of genus three, Hiroshima Math. J. 7 (1977), no. 3, 743-768. [25] V. Moreno Vega and S. Reyes-Carocca, A generalisation of the pencil of Kuribayashi-Komiya quartics, arxiv.org/abs/ 2507.03128 (2025) [26] J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen, Acta Math. 50 (1927), 189-358. [27] J. Paulhus, A database of group actions on Riemann surfaces, Birational geometry, K ̈ahler-Einstein metrics and degenerations, 693-708, Springer Proc. Math. Stat., 409, Springer, Cham (2023). [28] R. E. Rodr ́ıguez and V. Gonz ́alez-Aguilera, Fermat's quartic curve, Klein's curve and the tetrahedron, Extremal Riemann surfaces (San Francisco, CA, 1995), 43-62, Contemp. Math., 201, Amer. Math. Soc., Providence, RI, 1997. [29] H. A, Schwartz, ̈Uber diejenigen algebraischen Gleichungen zwischen zwei ver ̈anderlichen Gr ̈oßen, welche eine schaar rationaler, eindeutig umkehrbarer Transformationen in sich selbst zulassen. Journal f ̈ur die reine und angewandte Mathematik 87 (1890), 139-145. [30] D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6, (1972), 29-38. [31] The LMFDB Collaboration, The L-functions and modular forms database, Families of higher genus curves with automorphisms, https://www.lmfdb.org/HigherGenus/C/Aut/. Departamento de Matem ́atica y Estad ́ıstica, Universidad de La Frontera, Francisco Salazar 01145, Temuco, Chile Email address: Departamento de Matem ́aticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Santiago, Chile Email address:
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2510.14757
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Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX7.0.1
The Hidden Story of Chemical Evolution in Local Star-Forming Nuclear Rings
Eva Sextl
1 and Rolf-Peter Kudritzki2, 1
1Universit¨ats-Sternwarte, Fakult¨at f¨ur Physik, Ludwig-Maximilians Universit¨at M¨unchen, Scheinerstr. 1, 81679 M¨unchen, Germany
2Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
(Received 10th of October 2025)
ABSTRACT
A VLT/MUSE population synthesis study of metallicities in the nuclear star-forming rings of four
disk galaxies (NGC 613, NGC 1097, NGC 3351, NGC 7552) is presented. Disentangling the spectral
contributions of young and old stellar populations, we find a large spread of ages and metallicities of the
old stars in the nuclear rings. This indicates a persistent infall of metal-poor gas and ongoing episodic
star formation over many gigayears. The young stars have metallicities a factor two to three higher
than solar in all galaxies except NGC 3351, where the range is from half to twice solar. Previously
reported detections of extremely metal poor regions at young stellar age on the rings of these four
galaxies are a methodological artifact of the average over all stars, young and old. In addition, it is
important to include contributions of very young stars (< 6 Myr) in this environment. For each of the
four galaxies, the extinction maps generated through our population synthesis analysis provide support
for the infall scenario. They reveal dust lanes along the leading edges of the stellar bars, indicating
the flow of interstellar material towards the circumnuclear zone. Prominent stellar clusters show little
extinction, most likely because of the onset of stellar winds. Inside and on the nuclear rings, regions
that are largely free of extinction are detected.
Keywords: Stellar populations (1622) — Galaxy chemical evolution (580) — Metallicity (1031) —
Barred spiral galaxies (136) — Galaxy circumnuclear disk (581)
1. INTRODUCTION
Nuclear star-forming rings and disks are prominent
structures within the central kiloparsec of disk galax-
ies, where the gas density reaches levels sufficient to
trigger intense localized star formation (J. H. Knapen
2005). In many cases, they substantially contribute to
the emission of the entire central galaxy region. Their
formation is closely tied to the overall dynamical con-
figuration of the host galaxy, especially the presence of
large-scale stellar bars and other non-axisymmetric com-
ponents that induce resonances (such as the Inner Lind-
blad Resonance) leading to gas accumulation in ring-
like morphologies (E. Athanassoula 1992; L. M. Mazzuca
et al. 2006; P. Verwilghen et al. 2024). These dense gas
reservoirs become sites of sustained starburst activity,
and the ring often acts as a gas barrier that partially
regulates the inward flow towards the nucleus.
Over
Corresponding author: Eva Sextl
sextl@usm.lmu.de
timescales of hundreds of millions of years, nuclear rings
contribute to secular galactic evolution by building up
stellar mass in the central regions and potentially in-
fluencing the growth of pseudo-bulges (inner disks) (J.
Kormendy & R. C. Kennicutt 2004).
Despite their importance, the chemical evolution of such
rings is still poorly understood. Gas-phase abundance
measurements based on auroral lines are difficult in
these regions because the relevant temperature-sensitive
transitions are intrinsically faint and frequently over-
whelmed by the bright stellar background (G. Stasi´nska
2005; ´A. I. D´ıaz et al. 2007; F. Bresolin et al. 2009).
The presence of nuclear activity further complicates the
analysis by contaminating nebular emission lines with
AGN-related radiation, which alters line ratios and can
obscure the signatures of pure star formation (R. L.
Davies et al. 2014). This effect is particularly relevant in
the context of the well-established AGN-starburst con-
nection, where inflowing gas concurrently fuels circum-
nuclear star formation and but also central supermassive
black hole accretion, leading to intertwined episodes of
arXiv:2510.14757v1 [astro-ph.GA] 16 Oct 2025
2
Sextl & Kudritzki
starburst and AGN activity (W. E. Clavijo-Boh´orquez
et al. 2024).
In the work presented here, we therefore do not fo-
cus on emission line analysis but instead on the young
and old stellar components themselves. Unlike nebular
line methods, which depend heavily on the detectabil-
ity of temperature-sensitive transitions, stellar spectral
features (Balmer lines, Calcium Triplet, iron lines, etc.)
remain accessible even in regions with significant inter-
stellar extinction. This allows for constraints on parame-
ters such as ages, initial mass function (IMF) sampling,
and metallicities through comparison with population
synthesis models (C. Leitherer et al. 1999; G. Bruzual
& S. Charlot 2003).
In our population synthesis analysis of the integrated
stellar spectra, we will employ the technique of full spec-
tral fitting (FSF). FSF offers a newer and more powerful
way to analyze nuclear SF rings by exploiting the entire
stellar spectrum rather than isolated indices, allowing
simultaneous constraints on stellar ages, metallicities,
and kinematics (R. Cid Fernandes et al. 2005; C. Con-
roy & J. E. Gunn 2010a).
By decomposing observed
spectra into mixtures of simple stellar populations, FSF
disentangles young starburst contributions from older
bulge components and reveals star formation histories
linked to bar-driven inflows.
Despite the tremendous
success of this technique, nuclear rings unfortunately
show peculiar results. The flag-ship observational cam-
paigns MUSE TIMER (D. A. Gadotti et al. 2019) &
PHANGS-MUSE (E. Emsellem et al. 2022) reported the
presence of regions with apparently low metallicity de-
spite exhibiting high Hα luminosities indicating a very
young population. The anomalous regions were persis-
tent between different analysis methods, and no adjust-
ment of the fitting procedures was able to remove or
reconcile them. In the galaxies NGC 613, NGC 3351,
NGC 1097, and NGC 7552, these signatures were espe-
cially pronounced (M. K. Seidel et al. 2015; A. Bittner
et al. 2020; I. Pessa et al. 2023; L. A. Silva-Lima et al.
2025), but other cases were also found (T. T. Shimizu
et al. 2019; C. de S´a-Freitas et al. 2023a; S. L. Robbins
et al. 2025). The sometimes extremely low ([Z]< −0.5)
mean metallicities are truly peculiar given their circum-
nuclear location, where gas and subsequently stars are
generally expected to be chemically enriched (I. P´erez
& P. S´anchez-Bl´azquez 2011; D. R. Cole et al. 2014).
In this work, we show that for the complicated en-
vironment of central SF rings, mean light- or mass-
weighted metallicity values obtained as averages over the
total stellar population are not sufficient to character-
ize galactic evolution. We argue that we need another
new approach besides ’light’ and ’mass’ weights to de-
fine metallicity. We introduce the ’physical’ metallicity
Zphys = Mmetals/Mtotal and show its usefulness in a
1-to-1 comparison with individual stellar probes in the
disk of M83. We also disentangle the properties of young
and old populations instead of discussing averages over
all ages. With this new concept, we can tell another evo-
lutionary story of some of the most prominent nuclear
rings in the local universe. Finally, we emphasize the
importance of an extensive age grid for spectral fitting
templates in the complex environment of nuclear rings.
Missing young stellar components in the fit introduces
misleading results in nuclear ring studies.
2. THE GALAXY SAMPLE
Our sample consists of four nearby barred spiral galax-
ies that host some of the most prominent circumnu-
clear star-forming rings in the local universe: NGC 7552,
NGC 613, NGC 1097, and NGC 3351. These systems
are well-studied archetypes in which the interaction be-
tween bars and central star formation has been exten-
sively documented. All four galaxies are included in the
TIMER or PHANGS-MUSE survey and their ring like
structures are kinematically well established. However,
their subsequent analysis with respect to stellar metal-
licity, ages, and interstellar medium dust content pre-
sented significant challenges.
NGC 3351: This barred spiral galaxy hosts a well-
defined circumnuclear star-forming ring at a radius of
about ∼300 pc (D. A. Swartz et al. 2006). The ring is
composed of regularly distributed HII-regions and mas-
sive young stellar clusters (L. Colina et al. 1997; F.
Bresolin & R. C. Kennicutt 2002). The nucleus itself
is dominated by an old stellar population, with no ev-
idence of AGN activity (D. A. Swartz et al. 2006; I.
Pessa et al. 2023). Gas inflow along the bar appears to
efficiently feed the ring, while leaving the very center
comparatively quiescent. The regularity and isolation
of the nuclear ring make NGC 3351 a relatively ’clean’
case for studies of ring star formation.
NGC 1097: D. A. Gadotti et al. (2019) noted that
despite being the most massive galaxy in the present
sample, it exhibits a prominent and extremely young
starbursting ring of ∼800 pc radius. More than 300
HII regions have been resolved in near-infrared imaging
(M. A. Prieto et al. 2005). The extend of the ring is
well-resolved by MUSE, allowing it to be identified with
ease in this galaxy. Classified as LINER, NGC 1097 pos-
sesses a comparatively faint nucleus that we mask out.
NGC 613: This galaxy hosts a rather asymmetric nu-
clear ring with an ∼400 pc radius. J. Falc´on-Barroso
et al. (2014) found an unusual large reservoir of molec-
ular gas within ∼100pc.
It harbors a radio jet, an
3
outflow, and an AGN ionization cone in its nuclear re-
gion (P. da Silva et al. 2020). The entire nuclear center
shows complex gas kinematics influenced by AGN activ-
ity (L. A. Silva-Lima et al. 2025). However, we do not
detect broad-line region (BLR) features in the spectra.
NGC 7552: This ring is about 200pc in radius. The
central nucleus itself lacks starburst or Seyfert-like activ-
ity (D. A. Forbes et al. 1994; D. A. Gadotti et al. 2019).
The ring is rich in molecular gas and hosts numerous
massive young clusters, making it one of the most ex-
treme star-forming nuclear rings in the nearby universe
with a current star formation rate of = 10–15 M⊙yr−1,
(H.-A. Pan et al. 2013).
3. OBSERVATIONS AND DATA RETRIEVAL
Our analysis relies primarily on observational data
obtained with the Multi Unit Spectroscopic Explorer
(MUSE, R. Bacon et al. (2010, 2014)) on the Very Large
Telescope (VLT) in Cerro Paranal, Chile. MUSE is an
integral field spectrograph that provides spatially re-
solved spectra over a nominal wavelength range of 4800
to 9400 ˚A in steps of 1.25 ˚A with a spectral resolution
of ∼2.65 ˚A FWHM. The wide field covers a 60” by 60”
field of view with 0.2” spatial sampling, enabling highly
detailed mappings of galactic environments (P. M. Weil-
bacher et al. 2020).
For all nuclear rings in our sample, we use the reduced
IFU data cubes from the MUSE-DEEP program. At the
VLT, observations are always organized into Observing
Blocks (OBs), each representing a single pointing with
a one-hour maximum exposure time due to operational
constraints. Each OB produces a final data cube, re-
ferred to as an OB datacube, and these are released in-
dividually as part of the ’MUSE collection’ in the Science
Portal3. The MUSE-DEEP program combines multiple
such OB datacubes for each target to create a single deep
datacube with improved sensitivity. The data reduction
in this case follows the standard MUSE pipeline opti-
mized for deep field observations and includes bias sub-
traction, flat-fielding, wavelength and flux calibration,
and sky subtraction. Figure 1 shows B-band images for
each galaxy in the sample. The subsequently used FOV
is indicated in white. They are slightly smaller than the
original cubes to better isolate the ring regions.
4. DATA PREPARATION & POPULATION
SYNTHESIS TECHNIQUE
As mentioned in the introduction, we make use of
the widely adopted FSP technique with Simple Stellar
Populations (SSPs) spectra in order to extract physical
3 https://archive.eso.org/scienceportal/home
properties from the observed integrated light spectra.
The method essentially models the observed spectrum
as a linear superposition of SSP templates, each corre-
sponding to a single-age, single-metallicity stellar pop-
ulation. The SSP sets are synthesized beforehand from
theoretical stellar evolution isochrones and correspond-
ing stellar spectra under an assumed initial mass func-
tion (IMF) (see Section 4.3).
In mathematical turns, a model spectrum Mλ is con-
structed from a collection of SSPs fλ,i with ages ti and
metallicities [Z]i = log(Zi/Z⊙) as
Mλ = Dλ(RV , E(B −V ))
"nSSP
X
i=1
bifλ,i(ti, [Z]i)
#
.
(1)
The fit coefficients bi are determined within the fitting
procedure. We note that the SSP spectra (as well as the
observed spectra) are normalized to unity in the range of
5500 to 5550 ˚A, a wavelength regime without prominent
emission or absorption features. Due to this normaliza-
tion, the sum of all coefficients bi adds up to unity. The
bi are luminosity weighted fit coefficients describing the
contribution of the corresponding SSP to the integrated
light at the wavelength of normalization (see E. Sextl
et al. 2023, 2024, 2025 for a detailed discussion).
Interstellar dust along the line of sight is accounted for
by the term Dλ(E(B −V )), whereby the colour excess
E(B-V) is also fitted simultaneously. The overall shape
of the attenuation curve is chosen in advance. We chose
the prescription of D. Calzetti et al. (2000) with fixed
RV = 4.05 due to the star-forming nature of nuclear
rings.
4.1. Workflow
In practical terms, the fitting process is performed
with the pPXF algorithm (M. Cappellari & E. Emsellem
2004; M. Cappellari 2023).
Inspired by the examples
provided in pPXF’s GitHub repository4, our approach
follows several sequential steps which we have already
successfully applied to TYPHOON IFU data in the past
(E. Sextl et al. 2025).
When performing full-spectral fitting to reconstruct
chemical evolution histories, it is a good practice not
to work on the level of individual spectral pixels (spax-
els), but to combine several to obtain a higher S/N spec-
trum. Voronoi binning is widely regarded as the stan-
dard approach for this purpose, and we applied it using
the vorbin Python package (M. Cappellari & Y. Copin
4 https://github.com/micappe/ppxf examples. We also encour-
age the use the newer .dust-function for the extinction, not the
now obsolete .reddening/ .gas-reddening keywords
4
Sextl & Kudritzki
Table 1. Properties of Sample Galaxies
Galaxy
center α
center δ
stellar mass
i
PA
D
spatial scale
AGN ?
Final S/N
(J2000)
(J2000)
(log M⊙)
(◦)
(◦)
(Mpc)
(pc/arcsec)
NGC 7552
349.044945
-42.584962
10.52
14
54.9
17.2
83
no
200
NGC 613
23.575714
-29.418573
11.09
46
118
17.5
85
yes
130
NGC 1097
41.578937
-30.274717
11.24
46
130
14.5
70
yes
200
NGC 3351
160.990618
11.703659
10.49
45
13
9.96
48
no
200
Note—Central right ascension and declination coordinates (centroid of the 3.6 µm emission peak) as well as the stellar
masses are taken from the Spitzer Survey of Stellar Structure in Galaxies (S4G; K. Sheth et al. (2010)). Position
angles (PA) are measured east of north; i is the inclination, D are distances.
The spatial scaling is calculated
assuming the given distance. The primary literature references for the other quantities are A. Bittner et al. (2020)
for NGC 7552, K. Sato et al. (2021) for NGC 613, K. Onishi et al. (2015) for NGC 1097, and J. Sun et al. (2024)
for NGC 3351.
2003). The ultimate S/N values at the wavelengths of
normalization for each galaxy are listed in table 1.
As a second step, we applied sigma-clipping to remove
artifacts and emission lines in the Voronoi spectra, fol-
lowing Eq. (34) of M. Cappellari (2023). Data points
that exceeded the relative error of 3σ were excluded. In
this process a multiplicative Legendre polynomial of de-
gree 4 was included to correct for low-order continuum
mismatches.
Third, the galaxy stellar kinematics were determined.
Velocity and velocity dispersion were constrained in the
pPXF main routine with an additive polynomial of de-
gree 4.
The high spatial and spectral resolution of
MUSE would in principle also allow the extraction of
higher-order moments (h3 and h4), but this lies beyond
the scope of the present work.
The fitting then pro-
ceeded to the derivation of stellar population properties.
During this stage, the kinematic parameters are fixed,
and polynomial corrections are disabled to avoid degen-
eracies between continuum adjustments and population
parameters. The parameters bi and E(B −V ) are deter-
mined simultaneously during this main fit. To obtain an
estimate of the uncertainty in the fit, a wild bootstrap-
ping procedure is applied 25 times (R. Davidson & E.
Flachaire 2008). The residuals, which are the differences
between the Voronoi spectrum and the best-fit model,
are randomly multiplied by +1 or -1 to create a new set
of perturbations. These perturbed residuals are added
back to the original best-fit model to create many simu-
lated spectra, which are then fitted again to see how the
fit parameters bi and subsequently metallicities and ages
may vary. This is a proper way to understand how un-
certainties in the data (noise in the spectra) propagate
into uncertainties in the discussed fitted parameters.
4.2. Derivation of physical quantities
Once the parameters bi are known, physically rele-
vant quantities can be constructed, such as the mean
light-weighted age and the mean light-weighted stellar
metallicity [Z]lw of the stellar population in the bin:
log(t)lw =
X
i
bi log(ti)/
X
i
bi
(2)
[Z]lw =
X
i
bi log(Zi/Z⊙)/
X
i
bi
(3)
These definitions follow directly from the spectral fit,
but can be misleading, as young stellar populations can
easily outshine older populations despite a much smaller
total mass. The mass-weighted coefficients ˜bi can show a
more nuanced picture and are obtained using the mass-
to-light ratio γi = Mi/Li(V ) of each SSP isochrone as
follows:
˜bi =
biγi
P
i biγi
.
(4)
The mass-weighted means for age and metallicity are
then calculated in the same way as with equations (2)
and (3) but using the coefficients ˜bi instead of bi.
However, the definitions of light- or mass-weighted
metallicities are misleading when compared with metal-
licities used in chemical evolution of galaxies or cosmo-
logical simulations. Here, metallicity Z is the ratio of
the mass of metals MZ confined in the stellar popula-
tion divided by the total mass of stars M, Z = MZ/M.
Thus, following E. Sextl et al. (2025) we introduce a
physical definition of metallicity:
Zphys = MZ
M =
X
i
˜biZi
(5)
[Z]phys = log(Zphys/Z⊙)
(6)
Note that this definition is different from the ’mass-
weight’ normally found in the literature. We therefore
5
NGC 3351
NGC 613
NGC 1097
NGC 7552
-10
-5
0
5
10
log Age [Gyr]
7
8
9
10
-10
-5
0
5
10
angular offset [arcsec]
AV
0.4
0.8
1.2
1.6
-10
-5
0
5
10
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
-10
-5
0
5
10
[Z]lw
0.75
0.25
0.25
Figure 1. First column: B-Band images from the CTIO 1.5m telescope (NGC 7552), CTIO 0.9m telescope (NGC 613), du Pont
2.5m telescope (NGC 1097) and CTIO 1m telescope (NGC 3351). The MUSE FOV used for our FSF fit is marked in white.
North is to the top, east is to the left. The following columns show the FOV with results from the FSF fit: mean light-weighted
Age, visual extinction AV and light-weighted total metallicity [Z]lw. The color bar at the top holds for all the subplots in the
column.
call equation 5 the ’physical metallicity’ to distinguish
the two quantities. Our choice is particularly relevant,
as it allows for a direct comparison with numerical sim-
ulations and is consistent with the conventions typically
employed in chemical evolution studies. [Z]phys can then
be split into a young and old component as described in
6
Sextl & Kudritzki
0.2
0.4
0.6
0.8
1.0
by
−0.50
−0.25
0.00
0.25
0.50
∆[Z]phys
20
40
60
80
100
120
140
R [arcsec]
−0.50
−0.25
0.00
0.25
0.50
∆[Z]phys
Figure 2.
Metallicity difference ∆[Z] between individual
stellar probes and population synthesis (see Section 5) as a
function of the luminosity fraction by of young stars (top)
and the angular distance from the center (bottom).
The
different symbols represent differences with respect to BSG
(blue circles), SSC (red triangles), YMC with optical analysis
(dark green squares), YMC with UV analysis (light green
circles).
Errors result from the addition of stellar source
and population synthesis errors in quadrature. The shaded
gray strip indicates a difference ≤0.1 dex and is added for
orientation.
the following paragraphs. We will test the capabilities of
this new definition in section 5 with a 1-to-1 comparison
to stellar probes.
To further analyze the chemical evolution history, we
separate contributions from the young and old stellar
populations. Following again E. Sextl et al. (2023, 2024,
2025) we introduce a conventional age boundary ty
lim
to distinguish between populations dominated by recent
star formation activity (young) and those tracing the
longer-term assembly history (old). Thus, all SSPs with
ages ti < ty
lim are assigned to the young component,
while those with ti ≥ty
lim comprise the old component.
We will use ty
lim = 0.1 and 0.3 Gyr, respectively, in the
next sections to distinguish between the young and old
population.
Metallicities and ages for the young and old popula-
tions are then calculated with the above equations, but
the sums are carried out only over the old or young
SSP, respectively. This separation between the old and
young population is used for the calculation of ages and
metallicities. The contribution to the observed spectrum
by the two populations is then described by the corre-
sponding sums byoung and bold, where byoung + bold = 1
by construction.
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
NGC 3351
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
NGC 3351
250 pc
6.6
6.8
7.0
7.2
7.4
log agey [Gyr]
0.0
0.1
0.2
0.3
0.4
E(B-V)
Figure 3. The central region of NGC 3351: (top) map of
mean ages of the young stellar population; (bottom) inter-
stellar reddening E(B-V). The circles indicate the locations of
the three most prominent young stellar clusters highlighted
in Fig. 24 of E. Emsellem et al. (2022) and discussed in the
text.
This division is also applied to the mass-weighted co-
efficients ˜bi as defined above. The fractional mass con-
tributions of the two populations were then computed
7
0.0
0.3
0.6
0.9
1.2
1.5
1.8
9.00
9.25
9.50
9.75
10.00
log Ageo [Gyr]
NGC 7552
0.0
0.5
1.0
1.5
2.0
9.0
9.2
9.4
9.6
9.8
10.0
log Ageo [Gyr]
NGC 613
0.0
0.5
1.0
1.5
2.0
2.5
3.0
9.4
9.6
9.8
10.0
log Ageo [Gyr]
NGC 1097
0.0
0.2
0.4
0.6
0.8
1.0
1.2
R [kpc]
9.8
9.9
10.0
10.1
log Ageo [Gyr]
NGC 3351
Figure 4.
Mean ages of the old stellar population as a
function of galactocentric distance. The nuclear star forming
regions are indicated in gray. From top to bottom: NGC
7552, NGC 613, NGC 1097, NGC 3351.
as
˜byoung =
X
ti<ty
lim
˜bi,
(7)
˜bold =
X
ti≥ty
lim
˜bi.
(8)
This separation highlights the strong discrepancy that
often arises between light- and mass-weighted quanti-
ties: while the integrated light can be dominated by
a relatively small number of very luminous young stars,
the bulk of the stellar mass generally resides in the older
populations.
4.3. A side note on the SSP template set
Apart from the new metallicity definition, we empha-
size that, besides the uncertainties inherent to the fitting
algorithm itself, full-spectral fitting lives and dies with
the choice of templates. Each template set is character-
ized by the number of ages and metallicities covered as
well as the ingredients it was calculated from. The com-
putation of a simple stellar population spectrum then
begins with an assumed star formation event in which
all stars are formed simultaneously. The adopted stel-
lar initial mass function (IMF) then dictates the relative
weighting of stars of different masses. Individual stellar
spectra are drawn from an empirical or theoretical stel-
lar library and mapped onto stellar evolutionary tracks,
which specify how stars of a given mass and metallicity
evolve with time. By integrating all masses and evolu-
tionary stages, one obtains the total flux of the popula-
tion at a fixed age and metallicity.
We use for these calculations the program FSPS (ver-
sion 3.2, Flexible Stellar Population Synthesis code, C.
Conroy & J. E. Gunn (2010a,b)) which gives the user
the choice from different stellar libraries, stellar evo-
lution tracks, and IMFs.
For our SSP set, we chose
the MILES library (P. S´anchez-Bl´azquez et al. 2006),
MESA stellar evolution isochrones (A. Dotter 2016; J.
Choi et al. 2016) and a Chabrier (G. Chabrier 2003)
initial mass function. As the empirical MILES library
has only a limited number of stars with T > 9000 K
(L. P. Martins & P. Coelho 2007), it is crucial to add
additional spectra of hot massive stars (J. J. Eldridge
et al. 2017), Wolf-Rayet types (L. J. Smith et al. 2002),
AGB- (A. Lan¸con & P. R. Wood 2000), post-AGB (T.
Rauch 2003) and carbon stars (B. Aringer et al. 2009).
By extending MILES with libraries for hot and post-MS
stars, we improve its applicability in the complex envi-
ronments of nuclear rings. This set, called ’MILES-SSP’
was successfully applied and tested in our analysis of the
TYPHOON data of M83 and NGC1365 (E. Sextl et al.
2025, 2024).
In summary, we use a grid with 52 age and ten metal-
licity values (520 SSPs in total). The metallicities go
from [Z] = 0.5 dex in steps of ∆[Z] = 0.25 dex down to
[Z] = −1.75 dex. The logarithmic age grid starts at 0.1
Myr ages and goes up to 12.5 Gyr (see also Fig. 2 in E.
Sextl et al. (2023) for an illustration of the grid).
As stated in E. Sextl et al. (2025), roughly one third of
the models are 20 Myr and younger. This is important
for the analysis of very young stars and clusters, which
we can expect in circumnuclear star-forming rings (ages
8
Sextl & Kudritzki
< 5 Myr as shown in M. A. Prieto et al. (2019)). We in-
tentionally do not remove low-metallicity young SSPs to
adjust the fit to choose higher metallicities in young pop-
ulations, as, for example, proposed in C. de S´a-Freitas
et al. (2023b); L. A. Silva-Lima et al. (2025). In con-
trast, we will see that the lowest [Z] value is not taken
at all by our analysis algorithm and that all regular SF
rings in our sample are indeed metal rich or close to so-
lar at the present time. In Section 8, we further test the
severe consequences of a smaller age grid, removing the
youngest SSPs (< 6 Myr and < 30 Myr respectively)
and running the fitting procedure again.
5. A MUSE METALLICITY TEST WITH YOUNG
STELLAR PROBES IN M83
In addition to the galaxies in Table 1 MUSE obser-
vations are also available for the star forming galaxy
M83. This galaxy harbors an asymmetrical nuclear re-
gion where Hα emission is irregularly distributed in the
center (D. A. Gadotti et al. 2019).
At a distance of
4.8 Mpc, its proximity makes M83 an excellent target
for observations with the MUSE spectrograph, but it
has also been a subject of accurate multi-object spectro-
scopic studies of individual stellar probes. F. Bresolin
et al. (2016) measured the metallicities of individual blue
supergiant stars throughout the disk of M83. In addi-
tion, young massive clusters in UV and optical (YMCs;
S. Hernandez et al. (2018, 2019, 2021)) and super star
clusters in the NIR (SSCs; B. Davies et al. (2017)) have
been investigated spectroscopically. The results of this
work offer the unique opportunity to compare the metal-
licities obtained from these stellar sources with those de-
rived from our MUSE population synthesis analysis.
A similar comparison was carried out by E. Sextl et al.
(2025) using TYPHOON data which have a lower spa-
tial and spectral resolution (1.65 arcsec and 8˚A, respec-
tively) but extend to shorter wavelengths (4000˚A). Here,
we repeat the analysis employing the higher resolution of
the MUSE-DEEP data cube. To improve the signal-to-
noise ratio for spectral fitting, we combined individual
spaxels within the vicinity of each stellar probe to pro-
duce integrated spectra representative of those regions.
To reduce contamination from individual bright stars,
we excluded spaxels within a radius of three pixels cen-
tered on each stellar probe. At the TYPHOON pixel
scale of 1.65” (K. Grasha et al. 2022), this adjustment
was not required. However, with the much finer spatial
resolution of MUSE at 0.2”, the light from a few bright
stars can easily dominate the flux within a single pixel,
and our fitting-procedure would lead to unreliable re-
sults.
The full spectral fitting was then applied to the binned
spectrum to infer the stellar population properties sur-
rounding the stellar probes, with particular emphasis
on the metallicity [Z]phys
100Myr of the young component
(age < 100 Myr).
We subsequently compare this fit-
ted metallicity with the independently measured metal-
licity obtained from the stellar probe.
This compar-
ison is quantified in terms of the difference ∆[Z] =
[Z]probe −[Z]phys
100Myr, which is illustrated in Figure 2.
Note that all stellar probes were adjusted to a common
baseline of Z⊙= 0.142. Our results reveal a close agree-
ment between the metallicity estimates derived from the
full spectral fitting method and those from the stellar
probe.
The mean value of ∆[Z] is 0.04 dex and the
scatter is 0.14 dex, the latter being the expected result
for average errors of 0.1 dex for individual independent
values of [Z]probe and [Z]phys
100Myr, respectively.
Figure 2 demonstrates that MUSE IFU spectra are ex-
tremely useful to obtain quantitative information about
the young stellar population despite their restricted
wavelength range toward the blue.
In E. Sextl et al.
(2025) we argued in the opposite direction. However,
this was based on the assumption of using low resolution
TYPHOON spectra with moderate S/N. As we see now,
the MUSE higher spectral resolution and high signal-to-
noise ratio compensate for the lack of blue wavelength
coverage.
We also carried out an FSF analysis of the MUSE-
DEEP data of the central region of M83. The resulting
map of [Z]phys
100Myr is presented and discussed in the Ap-
pendix.
6. AGES AND EXTINCTION
We start with the mean light-weighted ages in our
sample (second column in Fig. 1). In each system ex-
cept NGC 7552, the youngest stellar populations trace
a clear ring-like morphology, consistent with the loca-
tion of the nuclear ring inferred from both photometric
and spectral diagnostics in the literature.
In all four
galaxies the contrast between the young star-forming
ring and the exterior and interior stellar population
is particularly sharp, confirming that these structures
are relatively confined in both spatial extent and evo-
lutionary timescale. In NGC 3351, a closer inspection
of the regions with log Age < 7 reveals a direct corre-
spondence with the prominent HST-identified clusters
younger than 10 Myr (J. Sun et al. 2024, see also their
Fig. 9). For NGC 613, the distribution of young stel-
lar ages closely resembles the HST image presented in
Fig. 1 of J. Falc´on-Barroso et al. (2014). Similarly, in
NGC 1097, our age map is consistent with the age of
young clusters shown in Fig. 1a of M. A. Prieto et al.
(2019), while in NGC 7552 the general features are in
9
-10
-5
0
5
10
-10
-5
0
5
10
angular offset [arcsec]
NGC 7552
500 pc
-10
-5
0
5
10
-10
-5
0
5
10
angular offset [arcsec]
NGC 613
500 pc
-20
-10
0
10
20
angular offset [arcsec]
-20
-10
0
10
20
angular offset [arcsec]
NGC 1097
500 pc
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
NGC 3351
250 pc
0.0
0.1
0.2
0.3
0.4
0.5
[Z]phys
y
0.0
0.1
0.2
0.3
0.4
0.5
[Z]phys
y
0.0
0.1
0.2
0.3
0.4
0.5
[Z]phys
y
0.6
0.4
0.2
0.0
0.2
0.4
[Z]phys
y
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
0.1
0.2
0.3
0.4
0.5
[Z]phys
y
NGC 7552
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
0.0
0.1
0.2
0.3
0.4
0.5
NGC 613
0.0
0.5
1.0
1.5
2.0
2.5
3.0
R [kpc]
0.1
0.2
0.3
0.4
0.5
[Z]phys
y
NGC 1097
0.0
0.2
0.4
0.6
0.8
1.0
1.2
R [kpc]
0.4
0.2
0.0
0.2
0.4
NGC 3351
Figure 5.
Metallicity maps of the young stellar population (top) and radial metallicity distribution including uncertainties
(bottom).
10
Sextl & Kudritzki
-10
-5
0
5
10
-10
-5
0
5
10
angular offset [arcsec]
NGC 7552
500 pc
-10
-5
0
5
10
-10
-5
0
5
10
angular offset [arcsec]
NGC 613
500 pc
-20
-10
0
10
20
angular offset [arcsec]
-20
-10
0
10
20
angular offset [arcsec]
NGC 1097
500 pc
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
NGC 3351
250 pc
0.8
0.6
0.4
0.2
0.0
[Z]phys
o
0.0
0.2
0.4
[Z]phys
o
0.75
0.50
0.25
0.00
0.25
[Z]phys
o
0.3
0.2
0.1
0.0
0.1
[Z]phys
o
Figure 6. Metallicity maps of the old stellar population. For each galaxy, the approximate position of the nuclear ring is shown
with a dashed line. Upper row: NGC 7552 (left), NGC 613 (right). Lower row: NGC 1097 (left), NGC 3351 (right)
line with those shown in Fig. 7 of B. R. Brandl et al.
(2012).
In the next section, we discuss stellar metallicity and
distinguish between the young and old population. For
this purpose, we use ty
lim = 0.3 Gyr as age boundary.
Below and above this limit, we can calculate the mean
ages of the young and old population. As it turns out,
the mean ages of the young stars are much lower than
the boundary, about 4 to 10 Myr in the nuclear star
forming regions.
Figure 3 gives an example for NGC
3351. The locations of the three most prominent very
young clusters are highlighted, indicating the presence
of very young stars.
The average ages of the old stars are significantly
higher and in the range of many Gyr as Figure 4 demon-
11
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
NGC 613
500 pc
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
log Ageo [Gyr]
Figure 7. Map of the age of the old stellar population of
NGC 613. The AGN outflow cone is indicated.
strates. Notably, we observe a large dispersion of stellar
ages within the nuclear star-forming regions, suggest-
ing an extended period of star formation activity. This
finding is consistent with previous studies by E. L. Al-
lard et al. (2006); M. Sarzi et al. (2007); J. H. Knapen
et al. (2008), which present evidence that star forma-
tion in nuclear rings occurs episodically through multi-
ple bursts, with activity sustained over long timescales
rather than representing a one-time and short-lived
event.
We also note an interesting bimodality with respect to
the maximum stellar ages in Figure 4. While for NGC
1097 and NGC 3351 the oldest stars in the nuclear rings
have an almost similar age as out- and inside, they are
significantly younger for NGC 7552 and NGC 613. It
seems that in the latter two cases stars have started to
form later or older stars have migrated away from the
ring.
In FSF, we can also determine the interstellar extinc-
tion AV shown in Fig. 1 (third row).
The character-
istic dust lanes flowing along the leading edges of the
stellar bars are observed to connect directly to the nu-
clear ring, in agreement with predictions from dynamical
models of bar-driven inflows (P. Verwilghen et al. 2024).
These dust structures trace the flow of interstellar mate-
rial from the larger bar region toward the circumnuclear
zone, feeding ongoing star formation in the ring. Two
aspects seem to be noteworthy: First, the regions of the
most prominent cluster in the rings seem to show rela-
tively little reddening and extinction. Figure 3 gives an
example for the case of NGC 3351. This was already
shown in Fig. 24 in E. Emsellem et al. (2022). However,
the authors there speak of a technical artifact in the fit
converging to an ’misleading local minimum’ with very
low [Z]lw, young age, and low E(B-V) values, possibly
created from a lack of young templates. We will show
in the next section that the physical metallicity of the
young stars in these regions is close to solar.
At the
same time, all of our fitting runs indicate indeed low ex-
tinction within these regions. From a stellar evolution-
ary standpoint, this finding is easily explained by the
onset of stellar winds from newly formed stars, which
effectively disperse the residual parental material from
which the clusters originated. M. A. Prieto et al. (2019)
extracted the gas extinction from the HST recombina-
tion map Hα/Pα of NGC 1097 and came to the same
conclusion. Clusters as young as 4 Myr have effectively
removed dust from their surroundings.
J. Sun et al.
(2024) also reported YMCs that lose their local gas and
dust reservoirs at ages between 3 and 6 Myr in NGC
3351. A. Knutas et al. (2025) found similar timescales
in M83 using JWST observations.
As a second peculiar feature, on the rings and inside
toward the very nucleus, we consistently detect a re-
gion largely free of dust attenuation. The central dust-
deficient zone is ubiquitous across the sample, suggest-
ing that the innermost few hundred parsecs are not ef-
ficient reservoirs of cold material. This is also the case
in NGC1365 (E. Sextl et al. 2024). For M83, however,
a dust cavity is found on the star forming ring (see E.
Sextl et al. 2025), while the rest of the central region
shows increased interstellar reddening.
7. THE METALLICITY OF NUCLEAR STAR
FORMING REGIONS
The fourth row of Fig. 1 shows maps of the overall
light-weight metallicity [Z]lw. We find peculiar drops of
[Z]lw in our sample, qualitatively confirming the results
of I. Pessa et al. (2023), E. Emsellem et al. (2022), D.
Rosado-Belza et al. (2020) and A. Bittner et al. (2020),
although our low values of [Z]lw are less extreme. Even
after the inclusion of our very young stellar templates
and a nebular continuum correction (see E. Sextl et al.
2024, 2025), the maps show these features.
This was
already discussed in the appendix of I. Pessa et al. (2023)
and we will extend the discussion in Section 8.
However, we need to keep in mind that what we see
here is a mean quantity summarized over an enormously
wide range of stellar ages and weighted by stellar light.
It is important to understand that the low [Z]lw values
12
Sextl & Kudritzki
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
0.8
0.4
0.0
0.4
[Z]phys
o
NGC 7552
0.0
0.5
1.0
1.5
2.0
0.2
0.0
0.2
0.4
0.6
NGC 613
0.0
0.5
1.0
1.5
2.0
2.5
3.0
R [kpc]
1.0
0.5
0.0
0.5
[Z]phys
o
NGC 1097
0.0
0.2
0.4
0.6
0.8
1.0
1.2
R [kpc]
0.4
0.3
0.2
0.1
0.0
0.1
NGC 3351
9.2
9.4
9.6
9.8
10.0
log Ageo [Gyr]
9.2
9.4
9.6
9.8
log Ageo [Gyr]
9.5
9.6
9.7
9.8
9.9
10.0
10.1
log Ageo [Gyr]
9.80
9.85
9.90
9.95
10.00
10.05
10.10
log Ageo [Gyr]
Figure 8. Radial metallicity of the old stellar population. Upper row: NGC 7552 (left), NGC 613 (right). Lower row: NGC
1097 (left), NGC 3351 (right). The position of the nuclear ring is again marked in gray. The color coding represents the values
of log Ageo [Gyr] which is different for each subfigure.
found in regions considered ’young’ do not directly re-
flect the intrinsic metallicity of the youngest stars. In
such regions often one third or even half of the light still
comes from older stellar populations.
Therefore, we calculate [Z] using our physically mo-
tivated definition and divide between the young (≤0.3
Gyr) and the old population, which significantly alters
the picture. Maps of the young stellar population [Z]phys
y
are provided for each galaxy in Fig. 5. Only regions with
a sufficient light fraction of young stars (by > 0.2) are
shown. Voronoi bins that do not meet this criterion are
left blank. In addition, Figure 5 shows the radial galac-
tocentric distribution of [Z]phys
y
.
For NGC 1097, NGC 613, NGC 7752 the young com-
ponent does not fall below solar metallicity. In contrast,
many regions show clear super-solar enrichment, with
[Z]phys
y
≳0.2 dex.
For NGC 7552, we note that this is consistent with the
findings reported by D. Calzetti et al. (2010); J. Mous-
takas et al. (2010); C. M. Wood et al. (2015), who despite
some ambiguities in the gas-phase metallicity measure-
ments, classify the galaxy as at least solar and likely
super-solar.
In NGC 3351, the metallicity of the young popu-
lation is significantly lower than in the other three
galaxies, but we also see clear enrichment in the circum-
nuclear ring. The regions of the three most prominent
young clusters discussed before show solar or somewhat
higher metallicity. Generally, the young star metallicity
reaches a pronounced local maximum in the ring of
NGC 3351. We also note the very clear drop inside the
ring where [Z]phys
y
is 0.2 dex lower than outside. ´A. I.
D´ıaz et al. (2007) were able to derive direct HII region
abundances on the star forming ring of this galaxy.
The authors obtained metallicity values ranging from
[Z]=−0.27 ± 0.11 (when using M. Asplund et al. 2009
as the solar standard) in the eastern segment of the
ring up to 0.08 ± 0.09 in the north-western and south-
ern parts. This aligns well with our results for [Z]phys
y
spreading from −0.3 ± 0.1 dex up to 0.3 ± 0.1 dex,
especially considering the potential effects of dust de-
pletion of oxygen in HII-regions (F. Bresolin et al. 2025).
The physical metallicity [Z]phys
o
of the older stellar
population (≥0.3 Gyr) is shown in Figures 6 (maps) and
8 (galactocentric distribution).
We find an extremely
wide range of [Z]phys
o
in the nuclear star forming regions
of all four galaxies. Outside these regions, the dispersion
of metallicity is significantly smaller. We encounter very
low metallicities in the rings of NGC 1097 and NGC
7552.
While the dispersion of [Z]phys
o
outside the nuclear
rings is small, the values differ between the four galax-
ies. [Z]phys
o
is clearly super-solar in NGC 613, solar in
NGC 1097 and NGC 3351 and a factor of two below
solar in NGC 7552. This must be the result of different
evolution histories outside the nuclear rings.
In the case of NGC 613 we notice areas of lower [Z]phys
o
in projected regions perpendicular to the nuclear ring.
As Figure 7 indicates, these stars have the highest age
13
9.0
9.2
9.4
9.6
9.8
10.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
[Z]phys
o
NGC 7552
9.2
9.4
9.6
9.8
10.0
0.2
0.0
0.2
0.4
0.6
NGC 613
9.4
9.5
9.6
9.7
9.8
9.9
10.0
10.1
log Ageo [Gyr]
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
[Z]phys
o
NGC 1097
9.8
9.9
10.0
10.1
log Ageo [Gyr]
0.4
0.3
0.2
0.1
0.0
0.1
NGC 3351
Figure 9. Metallicity versus age of the old stellar population. Voronoi bins outside the rings appear in violet red, those on the
ring are colored orange, and inner sections are light gray. Upper row: NGC 7552 (left), NGC 613 (right). Lower row: NGC
1097 (left), NGC 3351 (right).
within the old population encountered in this galaxy
(see also Figure 9). These areas seem to coincide with
the AGN outflow cone (L. A. Silva-Lima et al. 2025).
Focusing only on the metallicity of the older popula-
tion in the light-weighted analysis, or considering mean
quantities without applying an age cut, does not reveal
this distinctive feature. From our viewpoint, the north-
ern cone is projected in front of the stellar ring and is
therefore more accessible, while the southern cone lies
behind the ring. However, the stars inside the northern
cone can only be clearly identified farther out, where
the bright stellar ring and its long history of star forma-
tion no longer dominates the view. We speculate that
the outflow has interrupted star formation for a long pe-
riod of time and no substantial chemical enrichment was
possible.
Figure 9 shows [Z]phys
o
as a function of the average age
of the old stars in and outside the nuclear star forming
regions. The presence of old stars born with very low
metallicity over many Gigayears in the cases of NGC
1097 and NGC 7552 is indicative of infall of low metal-
licity gas over a long period of time. For NGC 613, on
the other hand, we see the signature of normal chemical
evolution less affected by infall. The old population in
the central region of NGC 3351 that we see now was
mostly born more than 10 Gyrs ago.
A coarse look at the chemical evolution history is also
provided by Figure 10, where we plot the metallicity
difference between the young and old population as a
14
Sextl & Kudritzki
0.0
0.3
0.6
0.9
1.2
1.5
0.0
0.3
0.6
0.9
1.2
[Z]phys
y
-[Z]phys
o
NGC 7552
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.4
0.2
0.0
0.2
0.4
0.6
NGC 613
0.0
0.2
0.4
0.6
0.8
1.0
1.2
R [kpc]
0.0
0.5
1.0
1.5
[Z]phys
y
-[Z]phys
o
NGC 1097
0.0
0.2
0.4
0.6
0.8
1.0
1.2
R [kpc]
0.3
0.0
0.3
0.6
0.9
NGC 3351
Figure 10. Metallicity difference [Z]phys
y
- [Z]phys
o
versus galactocentric distance. Upper row: NGC 7552 (left), NGC 613 (right).
Lower row: NGC 1097 (left), NGC 3351 (right). The dashed line marks the zero line.
function of galactocentric distance.
The positive dif-
ferences indicate normal chemical evolution with the
scatter induced by the effects of infall of metal-poor gas
of different strengths. NGC 3351, on the other hand,
the mostly negative values in the center hint at the most
recent strong infall of metal-poor gas.
8. IMPORTANCE OF A YOUNG TEMPLATE GRID
In the sections above, we have pointed out several
times that it is crucially important to include the SSP
contributions of very young stars with a wide range of
possible metallicities.
Due to its importance, we use
this additional section to address the issue in more de-
tail.
We select NGC 3351 and its nuclear ring as an
example.
We compare the results of the FSF analysis of the
central region of NGC 3351 obtained with three sets
of SSPs. The first is identical to the one described in
Section 4.3. The other two use the same range of metal-
licities but the ages range from 6 Myr to 12.5 Gyr for set
two and 30 Myr to 12.5 Gyr for set three, respectively.
For the ages included, the age steps are the same in all
three sets.
Figure 11 shows the comparison of the total light-
weighted metallicities [Z]lw. As discussed in Section 7,
we encounter spurious regions of low metallicity on the
nuclear ring as an artifact of the luminosity-weighted
mean over all ages, young and old. However, now we
also see the dramatic influence of the contribution of
the youngest stars in the sets of SSP. With stars younger
than 6 Myr omitted, the metallicities drop to values as
low as [Z]lw = −1.2. The effect becomes even more ex-
treme when SSP younger than 30 Myr are left out. We
encounter extreme regions with [Z]lw = −1.7.
The effects shown in Figure 11 were already partially
discussed in I. Pessa et al. (2023). We have included the
figure here because of the ongoing discussion about the
chemical evolution of nuclear rings. However, as we have
explained in Section 4.2 the physical relevant description
of stellar metallicity is not given by [Z]lw but rather by
[Z]phys as defined by equations (5) and (6).
Thus, it
is important to investigate how the physical metallici-
ties of the young and old stellar populations are affected
by the choice of SSPs. Figure 12 displays maps of the
physical metallicity [Z]phys
y
of the young population for
the three cases. The differences are equally dramatic,
and it is obvious that leaving out the contribution of
the youngest stars can cause significant systematic ef-
fects. For example, physical metallicities are on average
0.2 dex lower when stars younger than 6 Myr are not in-
cluded. Differences larger than 1.0 dex are encountered
when the contributions of stars younger than 30 Myr are
neglected. We have also tested the influence of the SSP
sets on the determination of the old population physical
metallicities [Z]phys
o
. We find that the systematic effects
are small.
9. SUMMARY AND CONCLUSIONS
In our stellar population synthesis of the nuclear star
forming regions of four galaxies we use a physical defi-
nition of stellar metallicity which is consistent with the
15
-10
-5
0
5
10
angular offset [arcsec]
min(SSPage) = 0.1 Myr
250 pc
-10
-5
0
5
10
angular offset [arcsec]
min(SSPage) = 6.0 Myr
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
min(SSPage) = 30 Myr
1.50
1.25
1.00
0.75
0.50
0.25
0.00
0.25
[Z]lw
Figure 11.
Map of total light-weighted metallicity [Z]lw
in central region of NGC 3351 obtained with SSP sets of
different age ranges: (top) 0.1 Myr to 12.5 Gyr; (middle) 6
Myr to 12.6 Gyr; (bottom) 30 Myr to 12.6 Gyr. The color
bar is identical for all subplots. See text.
-10
-5
0
5
10
angular offset [arcsec]
min(SSPage) = 0.1 Myr
250 pc
-10
-5
0
5
10
angular offset [arcsec]
min(SSPage) = 6.0 Myr
-10
-5
0
5
10
angular offset [arcsec]
-10
-5
0
5
10
angular offset [arcsec]
min(SSPage) = 30 Myr
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
[Z]phys
y
Figure 12. Map of physical metallicity [Z]phys
y
in central
region of NGC 3351 obtained with SSP sets of different age
ranges: (top) 0.1 Myr to 12.5 Gyr; (middle) 6 Myr to 12.5
Gyr; (bottom) 30 Myr to 12.5 Gyr. The color bar is identical
for all subplots. See text.
16
Sextl & Kudritzki
study of chemical evolution. In addition, we disentangle
the contributions of young (≤0.3 Gyr) and old stars.
In this way, we avoid methodological artifacts result-
ing from the use of conventional luminosity- or mass-
weighted averages over stars of all ages, young and old.
We also demonstrate that it is crucially important to in-
clude the contributions of very young stars in the anal-
ysis.
We find that the stellar populations currently forming
in the nuclear rings in NGC 613, NGC 1097, and NGC
7752 are super-solar.
On the ring of NGC 3351 the
metallicities are in a range between half and twice solar,
but not as low as a factor of 20 below solar as reported
in A. Bittner et al. (2020). This is in agreement with
direct HII region abundance measurements. We also see
a clear metallicity enrichment in the case of the ring of
NGC 3351. The metallicity distribution in the nuclear
ring is similar to the ’onfall’ scenario discussed in J. K. S.
Friske & R. Sch¨onrich (2025), Fig. 6.
The ages and metallicities of the old stars indicate
continuous star formation in the presence of an inflow
of low metallicity gas over many Gyrs in the case of
NGC 1097 and NGC 7752. For NGC 613 low metallicity
infall appears to be less important. In the case of NGC
3351 the old stars were generated mostly ten Gyrs ago.
In the very center, the lower metallicity of the young
stars indicates the most recent strong infall of lower-
metallicity gas.
The infall scenario is supported by the reddening maps
obtained with our population synthesis technique. We
find dust lanes tracing the flow of interstellar material
toward the circumnuclear zone and providing the mate-
rial for ongoing star formation. The region inside the
nuclear rings is largely free of dust attenuation. On the
rings, prominent stellar clusters show little extinction,
very likely as a result of the onset of strong stellar winds.
In summary, we conclude that the high spatial and
spectral resolution of the MUSE IFU spectrograph com-
bined with the power of an 8m VLT mirror and our
technique of stellar population synthesis provide unique
means to investigate the nuclear star forming regions of
galaxies in the nearby universe.
ACKNOWLEDGMENTS
Acknowledgments. We acknowledge support from the
Munich Excellence Cluster Origins and the Munich In-
stitute for Astro-, Particle, and Biophysics (MIAPbP)
both funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC-2094 390783311. In addition,
ES has been supported by the European Research Coun-
cel COMPLEX project under the European Union’s
Horizon 2020 research and innovation program grant
agreement ERC-2029-AdG 882679.
Facilities: ESO VLT/MUSE
APPENDIX
A. THE CENTER OF M83 WITH MUSE
Our work focuses on the properties of nuclear rings in the galaxies NGC 7552, NGC 613, NGC 1097, and NGC
3351. In these systems, the young stellar component generally shows metallicities two to three times higher than solar,
except in NGC 3351, where values range from approximately half to twice solar. No regions with lower metallicities
than this were encountered. That raises a more general question: Does this trend more or less hold for all star-forming
nuclear rings, or do individual other cases differ and how are the metallicities distributed?
In our earlier work (E. Sextl et al. 2025), we investigated the circumnuclear ring of M83 reconstructed from CO
maps shown in N. Harada et al. (2019). We reported a coherent drop in the metallicity of the young stellar population
along the southern part of the ring, based on TYPHOON survey data. The IFU-like TYPHOON observations have
a coarser spatial and spectral resolution (1.65” per spaxel, 8 ˚A) but extend further into the blue wavelength regime
down to 4000˚A (K. Grasha 2023; Q.-H. Chen et al. 2023). Thus, a comparison with an analysis based on MUSE data
is very interesting.
The central region of M83 has also been observed within MUSE-DEEP, offering a rare opportunity to compare full-
spectral fitting results from two distinct observational campaigns. For this purpose, we again applied our pPXF pipeline
now to MUSE data of the central region of M83 and chose our standard template set as described in section 4.3.
Figure A.1 shows the metallicity distribution of stars younger than 100 Myr, derived from Voronoi spectra with a
S/N of 250. This can be directly compared with E. Sextl et al. (2025) Figure 17, based on TYPHOON. We again
detect the metallicity drop along the southwestern portion of the ring, as well as the slightly sub-solar values west
of the optical center. In addition, certain regions east of the center appear slightly metal-poorer in MUSE than in
17
TYPHOON (∼0.25 dex difference).
These results demonstrate that our revised metallicity definition, combined with a consistent fitting approach, yields
results that are broadly comparable across different observational campaigns with different spectrographs, different
wavelength regimes, and varying spatial and spectral resolution. They also hint that not all nuclear star-forming
rings share the same morphological or chemical characteristics. M83 is distinct in the spatial distribution of the lower
metallicity bins. They are not regularly distributed along the ring (regions with [Z]phys
y
> 0 dex are found in the
south-east). Neither do the regions within the ring appear regular. We find a coherent area in the south, which is
relatively enriched, and an irregular distribution around the nucleus.
This fits well with M83’s highly peculiar kinematic properties (L. Della Bruna et al. 2022). Its dynamical center does
not coincide with the optical center (N. Thatte et al. 2000; R. J. D´ıaz et al. 2006), and the location of its elusive
AGN remains uncertain (S. Hernandez et al. 2025). Together, these findings highlight the need to examine galaxies
individually, as the metallicity and stellar age distribution within a nuclear ring are shaped by the unique evolutionary
history of its host galaxy.
5
0
-5
-10
angular offset [arcsec]
-10
-5
0
5
angular offset [arcsec]
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
0.5
[Z]phys
y
Figure A.1.
Metallicity of the young stellar population in the center of M83 with MUSE. Only
Voronoi bins with a sufficient young stellar component (by > 0.2) are shown. The nuclear ring
by N. Harada et al. (2019) reconstructed from CO observations is drawn as dashed ellipse. Its
center is determined from the CO velocity field (K. Muraoka et al. 2009) and shown with a grey
plus. The green triangle shows the potential AGN position (pointing P3) from S. Hernandez
et al. (2025) (RA = 204.2530486, Dec = −29.867170, (J2000), priv. comm.). The optical nucleus
(N. Thatte et al. 2000) is shown as a blue cross and the stellar kinematic center (Model A) is
shown as violet square (L. Della Bruna et al. 2022).
18
Sextl & Kudritzki
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Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX7.0.1 The Hidden Story of Chemical Evolution in Local Star-Forming Nuclear Rings Eva Sextl 1 and Rolf-Peter Kudritzki2, 1 1Universit ̈ats-Sternwarte, Fakult ̈at f ̈ur Physik, Ludwig-Maximilians Universit ̈at M ̈unchen, Scheinerstr. 1, 81679 M ̈unchen, Germany 2Institute for Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822, USA (Received 10th of October 2025) ABSTRACT A VLT/MUSE population synthesis study of metallicities in the nuclear star-forming rings of four disk galaxies (NGC 613, NGC 1097, NGC 3351, NGC 7552) is presented. Disentangling the spectral contributions of young and old stellar populations, we find a large spread of ages and metallicities of the old stars in the nuclear rings. This indicates a persistent infall of metal-poor gas and ongoing episodic star formation over many gigayears. The young stars have metallicities a factor two to three higher than solar in all galaxies except NGC 3351, where the range is from half to twice solar. Previously reported detections of extremely metal poor regions at young stellar age on the rings of these four galaxies are a methodological artifact of the average over all stars, young and old. In addition, it is important to include contributions of very young stars ( 9000 K (L. P. Martins & P. Coelho 2007), it is crucial to add additional spectra of hot massive stars (J. J. Eldridge et al. 2017), Wolf-Rayet types (L. J. Smith et al. 2002), AGB- (A. Lan ̧con & P. R. Wood 2000), post-AGB (T. Rauch 2003) and carbon stars (B. Aringer et al. 2009). By extending MILES with libraries for hot and post-MS stars, we improve its applicability in the complex environments of nuclear rings. This set, called 'MILES-SSP' was successfully applied and tested in our analysis of the TYPHOON data of M83 and NGC1365 (E. Sextl et al. 2025, 2024). In summary, we use a grid with 52 age and ten metallicity values (520 SSPs in total). The metallicities go from [Z] = 0.5 dex in steps of ∆[Z] = 0.25 dex down to [Z] = -1.75 dex. The logarithmic age grid starts at 0.1 Myr ages and goes up to 12.5 Gyr (see also Fig. 2 in E. Sextl et al. (2023) for an illustration of the grid). As stated in E. Sextl et al. (2025), roughly one third of the models are 20 Myr and younger. This is important for the analysis of very young stars and clusters, which we can expect in circumnuclear star-forming rings (ages 8 Sextl & Kudritzki 0.2) are shown. Voronoi bins that do not meet this criterion are left blank. In addition, Figure 5 shows the radial galactocentric distribution of [Z]phys y . For NGC 1097, NGC 613, NGC 7752 the young component does not fall below solar metallicity. In contrast, many regions show clear super-solar enrichment, with [Z]phys y ≳0.2 dex. For NGC 7552, we note that this is consistent with the findings reported by D. Calzetti et al. (2010); J. Moustakas et al. (2010); C. M. Wood et al. (2015), who despite some ambiguities in the gas-phase metallicity measurements, classify the galaxy as at least solar and likely super-solar. In NGC 3351, the metallicity of the young population is significantly lower than in the other three galaxies, but we also see clear enrichment in the circumnuclear ring. The regions of the three most prominent young clusters discussed before show solar or somewhat higher metallicity. Generally, the young star metallicity reaches a pronounced local maximum in the ring of NGC 3351. We also note the very clear drop inside the ring where [Z]phys y is 0.2 dex lower than outside. ́A. I. D ́ıaz et al. (2007) were able to derive direct HII region abundances on the star forming ring of this galaxy. The authors obtained metallicity values ranging from [Z]=-0.27 ± 0.11 (when using M. Asplund et al. 2009 as the solar standard) in the eastern segment of the ring up to 0.08 ± 0.09 in the north-western and southern parts. This aligns well with our results for [Z]phys y spreading from -0.3 ± 0.1 dex up to 0.3 ± 0.1 dex, especially considering the potential effects of dust depletion of oxygen in HII-regions (F. Bresolin et al. 2025). The physical metallicity [Z]phys o of the older stellar population (≥0.3 Gyr) is shown in Figures 6 (maps) and 8 (galactocentric distribution). We find an extremely wide range of [Z]phys o in the nuclear star forming regions of all four galaxies. Outside these regions, the dispersion of metallicity is significantly smaller. We encounter very low metallicities in the rings of NGC 1097 and NGC 7552. While the dispersion of [Z]phys o outside the nuclear rings is small, the values differ between the four galaxies. [Z]phys o is clearly super-solar in NGC 613, solar in NGC 1097 and NGC 3351 and a factor of two below solar in NGC 7552. This must be the result of different evolution histories outside the nuclear rings. In the case of NGC 613 we notice areas of lower [Z]phys o in projected regions perpendicular to the nuclear ring. As Figure 7 indicates, these stars have the highest age 13 9.0 9.2 9.4 9.6 9.8 10.0 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 [Z]phys o NGC 7552 9.2 9.4 9.6 9.8 10.0 0.2 0.0 0.2 0.4 0.6 NGC 613 9.4 9.5 9.6 9.7 9.8 9.9 10.0 10.1 log Ageo [Gyr] 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 [Z]phys o NGC 1097 9.8 9.9 10.0 10.1 log Ageo [Gyr] 0.4 0.3 0.2 0.1 0.0 0.1 NGC 3351 Figure 9. Metallicity versus age of the old stellar population. Voronoi bins outside the rings appear in violet red, those on the ring are colored orange, and inner sections are light gray. Upper row: NGC 7552 (left), NGC 613 (right). Lower row: NGC 1097 (left), NGC 3351 (right). within the old population encountered in this galaxy (see also Figure 9). These areas seem to coincide with the AGN outflow cone (L. A. Silva-Lima et al. 2025). Focusing only on the metallicity of the older population in the light-weighted analysis, or considering mean quantities without applying an age cut, does not reveal this distinctive feature. From our viewpoint, the northern cone is projected in front of the stellar ring and is therefore more accessible, while the southern cone lies behind the ring. However, the stars inside the northern cone can only be clearly identified farther out, where the bright stellar ring and its long history of star formation no longer dominates the view. We speculate that the outflow has interrupted star formation for a long period of time and no substantial chemical enrichment was possible. Figure 9 shows [Z]phys o as a function of the average age of the old stars in and outside the nuclear star forming regions. The presence of old stars born with very low metallicity over many Gigayears in the cases of NGC 1097 and NGC 7552 is indicative of infall of low metallicity gas over a long period of time. For NGC 613, on the other hand, we see the signature of normal chemical evolution less affected by infall. The old population in the central region of NGC 3351 that we see now was mostly born more than 10 Gyrs ago. A coarse look at the chemical evolution history is also provided by Figure 10, where we plot the metallicity difference between the young and old population as a 14 Sextl & Kudritzki 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.3 0.6 0.9 1.2 [Z]phys y -[Z]phys o NGC 7552 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.4 0.2 0.0 0.2 0.4 0.6 NGC 613 0.0 0.2 0.4 0.6 0.8 1.0 1.2 R [kpc] 0.0 0.5 1.0 1.5 [Z]phys y -[Z]phys o NGC 1097 0.0 0.2 0.4 0.6 0.8 1.0 1.2 R [kpc] 0.3 0.0 0.3 0.6 0.9 NGC 3351 Figure 10. Metallicity difference [Z]phys y - [Z]phys o versus galactocentric distance. Upper row: NGC 7552 (left), NGC 613 (right). Lower row: NGC 1097 (left), NGC 3351 (right). The dashed line marks the zero line. function of galactocentric distance. The positive differences indicate normal chemical evolution with the scatter induced by the effects of infall of metal-poor gas of different strengths. NGC 3351, on the other hand, the mostly negative values in the center hint at the most recent strong infall of metal-poor gas. 8. IMPORTANCE OF A YOUNG TEMPLATE GRID In the sections above, we have pointed out several times that it is crucially important to include the SSP contributions of very young stars with a wide range of possible metallicities. Due to its importance, we use this additional section to address the issue in more detail. We select NGC 3351 and its nuclear ring as an example. We compare the results of the FSF analysis of the central region of NGC 3351 obtained with three sets of SSPs. The first is identical to the one described in Section 4.3. The other two use the same range of metallicities but the ages range from 6 Myr to 12.5 Gyr for set two and 30 Myr to 12.5 Gyr for set three, respectively. For the ages included, the age steps are the same in all three sets. Figure 11 shows the comparison of the total lightweighted metallicities [Z]lw. As discussed in Section 7, we encounter spurious regions of low metallicity on the nuclear ring as an artifact of the luminosity-weighted mean over all ages, young and old. However, now we also see the dramatic influence of the contribution of the youngest stars in the sets of SSP. With stars younger than 6 Myr omitted, the metallicities drop to values as low as [Z]lw = -1.2. The effect becomes even more extreme when SSP younger than 30 Myr are left out. We encounter extreme regions with [Z]lw = -1.7. The effects shown in Figure 11 were already partially discussed in I. Pessa et al. (2023). We have included the figure here because of the ongoing discussion about the chemical evolution of nuclear rings. However, as we have explained in Section 4.2 the physical relevant description of stellar metallicity is not given by [Z]lw but rather by [Z]phys as defined by equations (5) and (6). Thus, it is important to investigate how the physical metallicities of the young and old stellar populations are affected by the choice of SSPs. Figure 12 displays maps of the physical metallicity [Z]phys y of the young population for the three cases. The differences are equally dramatic, and it is obvious that leaving out the contribution of the youngest stars can cause significant systematic effects. For example, physical metallicities are on average 0.2 dex lower when stars younger than 6 Myr are not included. Differences larger than 1.0 dex are encountered when the contributions of stars younger than 30 Myr are neglected. We have also tested the influence of the SSP sets on the determination of the old population physical metallicities [Z]phys o . We find that the systematic effects are small. 9. SUMMARY AND CONCLUSIONS In our stellar population synthesis of the nuclear star forming regions of four galaxies we use a physical definition of stellar metallicity which is consistent with the 15 -10 -5 0 5 10 angular offset [arcsec] min(SSPage) = 0.1 Myr 250 pc -10 -5 0 5 10 angular offset [arcsec] min(SSPage) = 6.0 Myr -10 -5 0 5 10 angular offset [arcsec] -10 -5 0 5 10 angular offset [arcsec] min(SSPage) = 30 Myr 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0.25 [Z]lw Figure 11. Map of total light-weighted metallicity [Z]lw in central region of NGC 3351 obtained with SSP sets of different age ranges: (top) 0.1 Myr to 12.5 Gyr; (middle) 6 Myr to 12.6 Gyr; (bottom) 30 Myr to 12.6 Gyr. The color bar is identical for all subplots. See text. -10 -5 0 5 10 angular offset [arcsec] min(SSPage) = 0.1 Myr 250 pc -10 -5 0 5 10 angular offset [arcsec] min(SSPage) = 6.0 Myr -10 -5 0 5 10 angular offset [arcsec] -10 -5 0 5 10 angular offset [arcsec] min(SSPage) = 30 Myr 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 [Z]phys y Figure 12. Map of physical metallicity [Z]phys y in central region of NGC 3351 obtained with SSP sets of different age ranges: (top) 0.1 Myr to 12.5 Gyr; (middle) 6 Myr to 12.5 Gyr; (bottom) 30 Myr to 12.5 Gyr. The color bar is identical for all subplots. See text. 16 Sextl & Kudritzki study of chemical evolution. In addition, we disentangle the contributions of young (≤0.3 Gyr) and old stars. In this way, we avoid methodological artifacts resulting from the use of conventional luminosity- or massweighted averages over stars of all ages, young and old. We also demonstrate that it is crucially important to include the contributions of very young stars in the analysis. We find that the stellar populations currently forming in the nuclear rings in NGC 613, NGC 1097, and NGC 7752 are super-solar. On the ring of NGC 3351 the metallicities are in a range between half and twice solar, but not as low as a factor of 20 below solar as reported in A. Bittner et al. (2020). This is in agreement with direct HII region abundance measurements. We also see a clear metallicity enrichment in the case of the ring of NGC 3351. The metallicity distribution in the nuclear ring is similar to the 'onfall' scenario discussed in J. K. S. Friske & R. Sch ̈onrich (2025), Fig. 6. The ages and metallicities of the old stars indicate continuous star formation in the presence of an inflow of low metallicity gas over many Gyrs in the case of NGC 1097 and NGC 7752. For NGC 613 low metallicity infall appears to be less important. In the case of NGC 3351 the old stars were generated mostly ten Gyrs ago. In the very center, the lower metallicity of the young stars indicates the most recent strong infall of lowermetallicity gas. The infall scenario is supported by the reddening maps obtained with our population synthesis technique. We find dust lanes tracing the flow of interstellar material toward the circumnuclear zone and providing the material for ongoing star formation. The region inside the nuclear rings is largely free of dust attenuation. On the rings, prominent stellar clusters show little extinction, very likely as a result of the onset of strong stellar winds. In summary, we conclude that the high spatial and spectral resolution of the MUSE IFU spectrograph combined with the power of an 8m VLT mirror and our technique of stellar population synthesis provide unique means to investigate the nuclear star forming regions of galaxies in the nearby universe. ACKNOWLEDGMENTS Acknowledgments. We acknowledge support from the Munich Excellence Cluster Origins and the Munich Institute for Astro-, Particle, and Biophysics (MIAPbP) both funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC-2094 390783311. In addition, ES has been supported by the European Research Councel COMPLEX project under the European Union's Horizon 2020 research and innovation program grant agreement ERC-2029-AdG 882679. Facilities: ESO VLT/MUSE APPENDIX A. THE CENTER OF M83 WITH MUSE Our work focuses on the properties of nuclear rings in the galaxies NGC 7552, NGC 613, NGC 1097, and NGC 3351. In these systems, the young stellar component generally shows metallicities two to three times higher than solar, except in NGC 3351, where values range from approximately half to twice solar. No regions with lower metallicities than this were encountered. That raises a more general question: Does this trend more or less hold for all star-forming nuclear rings, or do individual other cases differ and how are the metallicities distributed? In our earlier work (E. Sextl et al. 2025), we investigated the circumnuclear ring of M83 reconstructed from CO maps shown in N. Harada et al. (2019). We reported a coherent drop in the metallicity of the young stellar population along the southern part of the ring, based on TYPHOON survey data. The IFU-like TYPHOON observations have a coarser spatial and spectral resolution (1.65" per spaxel, 8 ̊A) but extend further into the blue wavelength regime down to 4000 ̊A (K. Grasha 2023; Q.-H. Chen et al. 2023). Thus, a comparison with an analysis based on MUSE data is very interesting. The central region of M83 has also been observed within MUSE-DEEP, offering a rare opportunity to compare fullspectral fitting results from two distinct observational campaigns. For this purpose, we again applied our pPXF pipeline now to MUSE data of the central region of M83 and chose our standard template set as described in section 4.3. Figure A.1 shows the metallicity distribution of stars younger than 100 Myr, derived from Voronoi spectra with a S/N of 250. This can be directly compared with E. Sextl et al. (2025) Figure 17, based on TYPHOON. We again detect the metallicity drop along the southwestern portion of the ring, as well as the slightly sub-solar values west of the optical center. In addition, certain regions east of the center appear slightly metal-poorer in MUSE than in 17 TYPHOON (∼0.25 dex difference). These results demonstrate that our revised metallicity definition, combined with a consistent fitting approach, yields results that are broadly comparable across different observational campaigns with different spectrographs, different wavelength regimes, and varying spatial and spectral resolution. They also hint that not all nuclear star-forming rings share the same morphological or chemical characteristics. M83 is distinct in the spatial distribution of the lower metallicity bins. They are not regularly distributed along the ring (regions with [Z]phys y > 0 dex are found in the south-east). Neither do the regions within the ring appear regular. We find a coherent area in the south, which is relatively enriched, and an irregular distribution around the nucleus. This fits well with M83's highly peculiar kinematic properties (L. Della Bruna et al. 2022). Its dynamical center does not coincide with the optical center (N. Thatte et al. 2000; R. J. D ́ıaz et al. 2006), and the location of its elusive AGN remains uncertain (S. Hernandez et al. 2025). Together, these findings highlight the need to examine galaxies individually, as the metallicity and stellar age distribution within a nuclear ring are shaped by the unique evolutionary history of its host galaxy. 5 0 -5 -10 angular offset [arcsec] -10 -5 0 5 angular offset [arcsec] 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 [Z]phys y Figure A.1. Metallicity of the young stellar population in the center of M83 with MUSE. Only Voronoi bins with a sufficient young stellar component (by > 0.2) are shown. The nuclear ring by N. Harada et al. (2019) reconstructed from CO observations is drawn as dashed ellipse. Its center is determined from the CO velocity field (K. Muraoka et al. 2009) and shown with a grey plus. The green triangle shows the potential AGN position (pointing P3) from S. Hernandez et al. (2025) (RA = 204.2530486, Dec = -29.867170, (J2000), priv. comm.). The optical nucleus (N. 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|
2510.14761
|
Herman-Kluk-Like Semi-Classical Initial-Value Representation for Boltzmann
Operator
Binhao Wang,1 Fan Yang,2 Chen Xu,1 and Peng Zhang1, 3, ∗
1School of Physics, Renmin University of China, Beijing, 100872, China
2Hefei National Laboratory, Hefei 230088, China
3Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education),
Renmin University of China, Beijing, 100872, China
(Dated: October 17, 2025)
The coherent-state initial-value representation (IVR) for the semi-classical real-time propagator
of a quantum system, developed by Herman and Kluk (HK), is widely used in computational studies
of chemical dynamics. On the other hand, the Boltzmann operator e−ˆ
H/(kBT ), with ˆH, kB, and T
representing the Hamiltonian, Boltzmann constant, and temperature, respectively, plays a crucial
role in chemical physics and other branches of quantum physics. One might naturally assume that
a semi-classical IVR for the matrix element of this operator in the coordinate representation (i.e.,
⟨˜x|e−ˆ
H/(kBT )|x⟩, or the imaginary-time propagator) could be derived via a straightforward “real-
time →imaginary-time transformation” from the HK IVR of the real-time propagator. However,
this is not the case, as such a transformation results in a divergence in the high-temperature limit
(T →∞). In this work, we solve this problem and develop a reasonable HK-like semi-classical IVR
for ⟨˜x|e−ˆ
H/(kBT )|x⟩, specifically for systems where the gradient of the potential energy (i.e., the force
intensity) has a finite upper bound. The integrand in this IVR is a real Gaussian function of the
positions x and ˜x, which facilitates its application to realistic problems. Our HK-like IVR is exact
for free particles and harmonic oscillators, and its effectiveness for other systems is demonstrated
through numerical examples.
I.
INTRODUCTION
In 1928, van Vleck derived the semi-classical real-time
propagator (the van Vleck propagator) for a quantum
system in the limit as ℏ→0 [1]. The phase of the van
Vleck propagator, with respect to a given initial and fi-
nal time and position, is determined by the action of the
corresponding classical trajectory. The van Vleck prop-
agator reveals the intrinsic connection between quantum
and classical mechanics, having had a significant impact
on quantum physics [2, 3].
However, deriving the van Vleck propagator requires
solving the classical Hamiltonian equations with fixed
initial and final positions. This is not convenient for nu-
merical calculations of realistic systems due to the root-
finding problem. Consequently, the van Vleck propaga-
tor is not well-suited for numerical studies of atom or
molecule systems. To overcome this problem, many au-
thors [4–10] have attempted to develop initial-value rep-
resentations (IVRs) for semi-classical real-time propaga-
tors.
In the IVRs, these propagators are expressed as
functionals of classical trajectories, determined by the
given initial position and momentum, which can be eas-
ily derived numerically via standard algorithms, such as
Runge-Kutta.
A highly influential IVR for the semi-
classical real-time propagator was derived by Herman
and Kluk (HK) in 1984 [10], based on coherent states.
Over the past forty years, the HK representation has
∗Electronic address: pengzhang@ruc.edu.cn
been widely used in studies of chemical dynamics, and
has proven to be a valuable tool for exploring the quan-
tum effects of nuclear motion [11, 12]. Moreover, in 2006,
Kay provided a rigorous derivation of the HK representa-
tion from the Schr¨odinger equation, demonstrating that
this representation is the leading term in an asymptotic
expansion of the quantum propagator in powers of ℏ[13].
In addition to the real-time propagator, the imaginary-
time propagator, which is the matrix element of the
Boltzmann operator e−ˆ
H/(kBT ), also play a crucial role
in quantum physics and chemistry. Here, kB and T are
the Boltzmann constant and temperature, respectively.
In 1971, Miller derived the semi-classical imaginary-time
propagator [14], through a direct t →−iτ transforma-
tion of the van Vleck propagator, where t is the real time
and τ = ℏ/(kBT). Similar to the van Vleck propagator,
the result obtained by Miller in Ref. [14] is determined by
the classical trajectory with an inverted potential, with
respect to fixed initial and final positions.
An semi-classical IVR for the imaginary-time prop-
agator, which is similar to the HK representation for
the real-time one, would clearly be very useful for nu-
merical studies in atom physics, molecule physics and
chemical physics.
Intuitively speaking, one might ex-
pect to obtain such an IVR through the aforementioned
t →−iτ transformation from the HK representation of
the real-time propagator [15]. However, as pointed out
by Yan, Liu, and Shao in Ref. [16], this is not the case,
as the result of this transformation diverges in the limit
T →∞.
In Appendix A, we demonstrate this again
with detailed calculations.
This problem is crucial, as
the semi-classical approximation should be applicable in
arXiv:2510.14761v1 [physics.chem-ph] 16 Oct 2025
2
the high-temperature limit. Although several alternative
IVRs for the imaginary-time propagator have been de-
rived [16, 17], an IVR based on coherent-state-type wave
functions, similar to the HK IVR, has yet to be found.
In this work we solve the above problem for the sys-
tems with the absolute value of potential-energy gradi-
ent (force intensity) having a finite upper bound.
For
these systems we derive a reasonable HK-like semiclas-
sical IVR for the Boltzmann operator, via an approach
generalized from the one of Kay in Ref. [13]. Our IVR
is exact for free particles and harmonic oscillators, al-
though the potential-energy gradient of the latter system
is un-bounded. The applicability for other systems are il-
lustrated via numerical examples. Our results are helpful
for studies of the properties of atomic systems, molecular
systems, and chemical reactions at finite temperatures.
The remainder of this paper is organized as follows.
For the convenience of the readers, we first display our
HK-like IVR for the Boltzmann operator in Sec. II, and
then show the derivation of this IVR as well as the condi-
tion for the semi-classical approximation in Sec. III. The
applicability of our HK-like IVR is illustrated with some
examples in Sec. IV. In Sec. V there is a summary. Some
details of the derivations and calculations are given in
the appendixes.
II.
CENTRAL RESULT
We consider a general multi-particle quantum system,
with Cartesian components of the coordinates and mo-
menta being denoted as (p1, ..., pN) and (q1, ..., qN), re-
spectively, and the Hamiltonian being given by
H =
N
X
j=1
p2
j
2mj
+ V (q).
(1)
Here q = (q1, ..., qN), and mj (j = 1, ..., N) is mass of
the particle to which the coordinate qj belongs.
Fur-
thermore, as mentioned above, we assume the norm of
potential energy gradient, i.e., |∇qV (q)|, having a finite
upper bound in the q-space.
The Boltzmann operator of our system is e−ˆ
H/(kBT ),
where ˆH is the quantum operator of the Hamiltonian in
Eq. (1). The matrix element of this operator in the co-
ordinate representation (imaginary-time propagator) can
be expressed as
Kτ(˜x, x) := ⟨˜x|e−ˆ
H/(kBT )|x⟩= ⟨˜x|e−ˆ
Hτ/ℏ|x⟩,
(2)
where |x⟩and |˜x⟩are eigen-states of the coordinate op-
erators, with eigen-values x = (x1, ..., xN) and ˜x =
(˜x1, ..., ˜xN), respectively, and
τ =
ℏ
kBT .
(3)
The HK-like IVR for the Boltzmann operator under
the semiclassical approximation, which we have derived
in this work, can be expressed as:
Kτ(˜x, x) = A
Z
dpdq
h
Dτe−1
ℏ(Sτ +Bτ +Cτ )i
.
(4)
Here, q = (q1, ..., qN) (as defined above), p = (p1, ..., pN),
and
R
dp, dq =
R
dp1 · · · dpN, dq1 · · · dqN. Moreover, the
factors Sτ, A, Bτ, Cτ and Dτ are all real and indepen-
dent of ℏ, with the definitions being introduced in the
following.
Factor Sτ
The factor Sτ of Eq. (4) is a function of p and q, and
is independent of x and ˜x. It is defined as
Sτ(q, p) =
Z τ
0
dη
N
X
j=1
p2
η,j
2mj
+ V (qη,1, ..., qη,N)
.
(5)
Here qη,1, ..., qη,N and pη,1, ..., pη,N satisfy the Hamilton’s
equations with inverted potential −V , initial position q
and initial momenta p, i.e., the equations
d
dη qη,j = pη,j
mj
;
(6)
d
dη pη,j = ∂V (z1, ..., zN)
∂zj
z1=qη,1;...;zN=qη,N
,
(7)
(j = 1, ..., N),
and the initial conditions
qη=0,j = qj;
pη=0,j = pj,
(j = 1, ..., N).
(8)
According to the above definitions and equations, the
factor Sτ of Eq. (5) is the action of the classical tra-
jectory qη,1, ..., qη,N and pη,1, ..., pη,N. Moreover, in the
following we will consider qη,1, ..., qη,N and pη,1, ..., pη,N
as functions of η and {q, p}.
Factors A, Bτ and Cτ
The factors Bτ and Cτ of Eq. (4) are functions of both
p, q and x, ˜x. They are defined as
Bτ(p, q; x, ˜x) =
N
X
j=1
γj (xj −qj)2 + γj (˜xj −qτ,j)2
−pj (xj −qj) + pτ,j (˜xj −qτ,j)
;
(9)
Cτ(p, q; x, ˜x) = 1
τ
N
X
j=1
mj
(˜xj −qτ,j) −(xj −qj)
2
,
(10)
3
where γ1,...,N are arbitrary positive parameters. Addi-
tionally, the factor A is defined as:
A =
N
Y
j=1
γj
2ℏ3π3
1/2
.
(11)
Notice that the factor Cτ, as well as the signs of the terms
pj (xj −qj) and pτ,j (˜xj −qτ,j) of the factor Bτ, cannot
be obtained from direct t →−iτ transformation on the
HK representation of real-time propagator.
Factor Dτ
Similar to Sτ, the factor Dτ in Eq. (4) is also a function
of p, q, and is independent of x and ˜x.
To show the
definition of Dτ, we first introduce four N × N matrices
Rqu(η), Rqv(η), Rpu(η), and Rpv(η), with elements:
Rqu
ij (η) = −2
η miδij +
2γj + 2
η mj
∂qη,j
∂qi
−∂pη,j
∂qi
;
(12)
Rqv
ij (η) = 2γjδij −2
η mj
∂qη,j
∂qi
−δij
;
(13)
Rpu
ij (η) =
2γj + 2
η mj
∂qη,j
∂pi
−∂pη,j
∂pi
;
(14)
Rpv
ij (η) = δij −2
η mj
∂qη,j
∂pi
,
(15)
(i, j = 1, ..., N),
(16)
with δij being the Kronecker symbol. We further define
other four N × N matrices Tuq(η), Tvq(η), Tup(η), and
Tvp(η), which relate to the these R-matrices via
Tuq(η) Tup(η)
Tvq(η) Tvp(η)
=
Rqu(η) Rqv(η)
Rpu(η) Rpv(η)
−1
.
(17)
The factor Dτ of Eq. (4) can be expressed in terms of
the elements of the above T-matrices, which are denoted
as Tto
ij(η) (t = u, v; o = p, q; i, j = 1, ..., N). Specifically,
we have
Dτ(q, p) = e−
R τ
0 gηdη,
(18)
with gη being a function of (q, p):
gη(q, p)
= −1
2
N
X
i,j=1
Wij(η)Vij(η)
+
N
X
j=1
1
η + γj
mj
+
2γ2
j
mj
+ mj
η2 + 4γj
η
!
Wjj(η)
+
2mj
η2
+ 4γj
η
Tuq
jj (η) −mj
η2 Tvq
jj (η)
,
(19)
where
Vij(η) =
∂2
∂zi∂zj
V (z1, ..., zN)
z1=qη,1;...;zN=qη,N
,
(20)
Wij(η) = −
N
X
s=1
Tuq
is (η)∂qη,j
∂qs
+ Tup
is (η)∂qη,j
∂ps
.
(21)
According to this definition, calculating Dτ requires com-
puting the derivatives ∂qη,i
∂qj , ∂qη,i
∂pj , ∂pη,i
∂qj
and ∂pη,i
∂pj
(i, j =
1, ..., N) for 0 ≤η ≤τ. As shown in Appendix B, one
can derive these function via solving the Hamilton’s equa-
tions (6, 7) together with another group of ordinary dif-
ferential equations.
III.
DERIVATION OF EQ. (4)
Now we show our approach for deriving the HK-like
IVR for the imaginary-time propagator, i.e., Eq. (4).
For simplicity, we consider the system of a single par-
ticle in one-dimensional space (N = 1), and the deriva-
tion can be directly generalized to the cases with arbi-
trary N. Consequently, we will omit the subscript de-
noting the particle index in the following.
In the fol-
lowing we present the main ideas and framework of the
derivation. The details of the derivations are provided in
Appendix C.
To derive Eq. (4), we express the imaginary-time prop-
agator as the integration
Kτ(˜x, x) = A
Z
dpdq
h
FDe−(Sτ +Bτ +Cτ )
ℏ
i
,
(22)
with A, Sτ, Bτ and Cτ being defined in Eqs. (11), (5), (9)
and (10), respectively, and FD being a to-be-determined
function of τ and (p, q), which is independent of x and ˜x.
In the following, we first prove that the function FD
satisfies the “initial condition” limτ→0 FD = 1.
Then
we derive the differential equation of FD, and solve this
equation together with this “initial condition”, using the
semi-classical approximation. We will find that the solu-
tion is just FD = Dτ, with Dτ being given by Eq. (18) for
N = 1. Substituting this result into Eq. (22), we obtain
the result of (4) for N = 1.
A.
“Initial Condition” of FD
To prove limτ→0 FD = 1, we first calculate the limi-
tation limτ→0 A
R
dpdq e−(Sτ +Bτ +Dτ )
ℏ
. Notice that since
the potential-energy gradient intensity |dV (q)/dq| has a
finite upper bound in the total q-space, in the short-τ
limit the solutions of the Hamiltonian equations (6) and
(7) are just those for free motion. Specifically, in this
limit we have qη = q + ηp/m and pη = p. Substituting
this solution into the definitions (5, 9, 10) of Sτ, Bτ, and
4
Cτ, we find that (Appendix C 1)
lim
τ→0 A
Z
dpdq e−(Sτ +Bτ +Cτ )
ℏ
= lim
τ→0
m
2πℏτ
1/2
e−m(x−˜x)2
2ℏτ
(23)
= δ(x −˜x).
(24)
On the other hand, it is clear that limτ→0 Kτ(˜x, x) =
δ(x −˜x). Thus, we have
lim
τ→0 Kτ(˜x, x) = lim
τ→0 A
Z
dpdq e−(Sτ +Bτ +Cτ )
ℏ
.
(25)
Comparing this result and Eq. (22), we find that the
function FD satisfies
lim
τ→0 FD = 1.
(26)
B.
Equation of FD and Semi-Classical
Approximation
Now we derive the differential equation for the function
FD. To this end, we introduce the correction operator
[19–21] for our system, which is defined as
ˆΛ˜x := ℏ∂
∂τ −ℏ2
2m
∂2
∂˜x2 + V (˜x).
(27)
It is clear that the imaginary-time propagator Kτ(˜x, x)
satisfies ˆΛ [Kτ(˜x, x)] = 0. By substituting Eq. (22) into
this equation, we find that the function FD satisfies
ˆΛ
Z
dpdq
FDe−(Sτ +Bτ +Cτ )
ℏ
= 0.
(28)
As detailed in Appendix C 2, using the method general-
ized from Ref. [13], we find that the sufficient condition
for Eq. (28) can be expressed as a differential equation
for FD:
ˆ˜L0 + ℏˆ˜L1 + ℏ2 ˆ˜L2 + ...
FD = 0,
(29)
where ˆ˜L0,1,2,... are ℏ-independent operators.
Specially,
the operator ˆ˜L0 is given by
ˆ˜L0 = ∂
∂τ + gτ(q, p),
(30)
where gτ(q, p) is just the function given by Eq. (19) with
N = 1.
Note that the operator in the l.h.s of Eq. (29) is ex-
panded as a power series of ℏ. Under the semi-classical
approximation, we further ignore the terms proportional
to ℏn (n ≥1) in this series, and approximate Eq. (29) as
∂
∂τ FD + gτFD = 0.
(31)
Furthermore, it is clear that under the initial condition
(26), the solution of Eq. (31) is just
FD = e−
R τ
0 gηdη.
(32)
C.
Final Derivation
Eq. (32) yields that FD = Dτ, with Dτ being the one
defined in Eq. (18) with N = 1. Substituting this result
into Eq. (22), we finally obtain Eq. (4) for N = 1.
D.
Condition of the Semi-Classical Approximation
At the end of this section, we discuss the condition
underlying the semiclassical approximation employed in
deriving our HK-like IVR, namely, the condition un-
der which our HK-like IVR provides a good approxi-
mation to the exact matrix element of the Boltzmann
operator. For comparison, recall that the semiclassical
approximation used in the derivation of the van Vleck
propagator requires the characteristic length scale lV of
the potential energy variation to be much larger than
the de Broglie wavelength [18]. The condition adopted
here is similar: specifically, for a system with N coorid-
nates, lV should be much larger than both
p
ℏ/γj and
p
ℏτ/mj =
p
ℏ/(kBTmj) (j = 1, ..., N). Thus, the semi-
classical approximation works well in the high tempera-
ture systems. Additionally, in practical calculations for
realistic systems, appropriate values of γj (j = 1, ..., N)
should be chosen to ensure that this condition is satisfied.
IV.
EXAMPLES
In the previous two sections, we presented our HK-like
IVR for the imaginary-time propagator and its deriva-
tion.
In this section, we demonstrate its applicability.
Specifically, we analytically prove that the IVR is exact
for free particles and harmonic oscillators, and numeri-
cally illustrate its performance with an anharmonic sys-
tem.
A.
Free Particles
We first consider the system of N free particles, i.e.,
V = 0. As mentioned above, in this case the solutions
of the Hamilton’s equations (6, 7) are just qη,j = qj +
ηpj/mj and pη,j = pj (j = 1, ..., N). Consequently, the
integration in the r. h. s. of Eq. (4) can be performed
analytically. With straightforward calculations, we find
that the right-hand side (r. h. s.) of Eq. (4) is
N
Y
j=1
mj
2πℏτ
1/2
e−
mj
2ℏτ (xj−˜xj)2,
(33)
which is same as the exact imaginary-time propagator for
this system. Thus, our HK-like IVR is exact for the free
particles.
5
FIG. 1:
The potential V (q) of Eq. (35), and matrix elements of the Boltzmann operator, for cases with L = 10l0 and
q0 = 6l0. (a): The potential V (q). (b): Enlarged view of the region around q = 0 in (a). Here we also show the ground-state
energy E0 (green dashed line), the first excited-state energy E1 (blue dotted line), and the second excited-state energy E2
(purple solid line), which are given by numerical diagonalization of the Hamiltonian. Note that E0 and E1 lie very close to
each other. (c-e): The diagonal elements K(x, x) = ⟨x|e−ˆ
H/(kBT )|x⟩(in units of 1/l0). (f-h): The non-diagonal elements
K(−x, x) = ⟨−x|e−ˆ
H/(kBT )|x⟩(in units of 1/l0). Here we show the results with temperature T = 10ℏω/kB (c, f), T = ℏω/kB
(d, g) and T = 0.5ℏω/kB (e, h).
For each temperature, we illustrate the results KExac from exact diagonlization of the
Hamiltonian (black dotted line) and the results KIVR given by our HK-like IVR (blue dots).
B.
Harmonic Oscillators
Now we consider N harmonic oscillators with potential
energy V = PN
j=1 mjω2
j x2
j/2. The Hamilton’s equations
(6, 7) for this system can also be solved analytically, and
we have qη,j = cjeωjη +dje−ωjη and pη,j = mjωj(cjeωη −
dje−ωη), where cj = (pj + mjωjqj)/(2mjωj) and dj =
(mjωjqj −pj)/(2mjωj). Using these results, we can also
analytically perform the integration in the r. h. s. of
Eq. (4), and find that the r.
h.
s.
of Eq. (4) is just
(Appendix D)
N
Y
j=1
mjωj
2πℏsinh(ωjτ)
1/2
e
−
mj ωj[(x2
j +˜x2
j) cosh(ωj τ)−2xj ˜xj]
2ℏsinh(ωj τ)
,
(34)
which is same as the exact imaginary-time propagator of
these oscillators. Therefore, as for the free particles, our
IVR is also exact for harmonic oscillators.
C.
Anharmonic System
Finally, we consider a single particle in an anharmonic
potential, with Hamiltonian H = p2/(2m) + V (q). Here
m, p and q are the mass, momentum and coordinate
of this particle, respectively. The anharmonic potential
V (q) is given by
V (q) = −
mω2L2
1 −
q2
2L2 +
q4
4L2q2
0
+ V0,
(35)
where ω, q0, and L are positive parameters, and V0 is a
q-independent constant:
V0 = mω2L2
1 −
q2
0
4L2
,
(36)
6
FIG. 2:
Same as Fig. (1), but for L = 10l0 and q0 = 3l0.
which is chosen such that the minimum value of V (q) is
zero. Specifically, ω has the dimension of frequency, and
both q0 and L have the dimension of length, and satisfy
q0 < 2L.
In Fig. 1(a) and Fig. 2(a) we illustrate the potential
V (q) for the following two groups of parameters:
(i) : L = 10l0, q0 = 6l0;
(ii) : L = 10l0, q0 = 3l0,
where l0 is defined as l0 =
p
ℏ/(mω). It is clear that
the gradient of this potential is bounded in the q-space.
Furthermore, as shown in Fig. 1(b) and Fig. 2(b), in the
region around q = 0, V (q) is a double-well with minimum
points being localized at q = ±q0.
We calculate the matrix elements of the Boltzmann
operator for this particle, K(˜x, x) ≡⟨˜x|e−ˆ
H/(kBT )|x⟩,
using the parameters (i) and (ii) for temperatures T =
10ℏω/kB, T = ℏω/kB, and T = 0.5ℏω/kB. Specifically,
we derive the results from our HK-like IVR method with
γ = 5ℏ/l2
0, as well as those from exact numerical diag-
onalization of the Hamiltonian operator ˆH. These are
denoted as KIVR(˜x, x) and KExact(˜x, x), respectively. In
Figs. 1(c-h) and 2(c-h), we compare KIVR(˜x, x) with
KExact(˜x, x) for ˜x = ±x. It is shown that they are in
perfect quantitative agreement.
In Fig. 3,
we further compare KIVR(˜x, x) with
KExact(˜x, x) for additional values of ˜x and x, and present
the relative error of our HK-like IVR approach, defined
as |KExact(˜x, x) −KIVR(˜x, x)|/|KExact(˜x, x)|. It is shown
that for T = 10ℏω/kB and T = ℏω/kB, the relative er-
ror remains below 10−2 for both parameters (i) and (ii).
Moreover, for T = 0.5ℏω/kB, the relative error stays
below 10−2 for parameter (i), while it can reach up to
8 × 10−2 for parameter (ii).
Specifically, for parame-
ter (ii) with T = 0.5ℏω/kB, the relative error is below
4 × 10−2 in most of the (˜x, x) region, except at some
places (the squares enclosed by black lines in Fig. 3(r))
where KExact(˜x, x) is very small compared to its maxi-
mum value in the entire (˜x, x) domain.
These results clearly demonstrate the applicability of
our HK-like IVR method.
Furthermore, the relatively
large relative error for parameter (ii) with T = 0.5ℏω/kB
is consistent with the fact that the semi-classical ap-
proximation performs well at high temperatures, when
the characteristic length scale q0 of the potential en-
ergy is much larger than
p
ℏ/γ = l0/
√
5, as discussed
in Sec. III D.
7
FIG. 3:
KExac(˜x, x) (in units of 1/l0), KIVR(˜x, x) (in units of 1/l0), and the relative error of our HK-like IVR approach which
is defined as |KExact(˜x, x) −KIVR(˜x, x)|/|KExact(˜x, x)|. Here we show the results for cases with L = 10l0, q0 = 6l0 (a-i) and
L = 10l0, q0 = 3l0 (j-r), for temperatures T = 10ℏω/kB, T = ℏω/kB, and T = 0.5ℏω/kB. We do not show the results in the
white regions, since in these regions both KExact and KIVR are below 1% of the maximum value of KExact across the entire
domain (denoted as Kmax
Exact). Moreover, in the four squares enclosed by black lines in (r), the relative error is between 4 × 10−2
and 8 × 10−2, while both KExact and KIVR are below 3% of Kmax
Exact.
V.
SUMMARY
In this work, we derive an HK-like semiclassical IVR
for the Boltzmann operator, applicable to systems in
which the gradient of the potential energy (i.e., the force)
is bounded in real space, as shown in Eq. (4). Unlike the
direct analytical continuation of the HK representation
for the real-time propagator, our IVR converges in the
high-temperature limit T →∞(τ →0). Our IVR is
exact for free particles and harmonic oscillators, and its
applicability to other systems is demonstrated through
examples. Our HK-like IVR method is useful for calcu-
lating the partition function and various physical prop-
erties of molecular systems in thermal equilibrium states
in future research.
Acknowledgments
We express our deep gratitude to Prof. Jiushu Shao
for the insightful discussions and for providing numerous
important suggestions. This work is supported by the
Innovation Program for Quantum Science and Technol-
ogy (Grant No. 2023ZD0300700) and the National Key
Research and Development Program of China (Grant
No. 2022YFA1405300).
[1] J. H. van Vleck, Proc. Natl. Acad. Sci. USA 14, 178
(1928). I
[2] M. Brack and R. Bhaduri, Semiclassical Physics. CRC
Press (2003). I
[3] M. S. Child, Semiclassical mechanics with molecular ap-
plications, Oxford University Press (2014). I
[4] W. H. Miller, J. Chem. Phys. 53, 3578 (1970). I
[5] R. A. Marcus, J. Chem. Phys. 54, 3965 (1971).
[6] W. H. Miller, Adv. Chem. Phys. 25, 69 (1974).
[7] W. H. Miller, Adv. Chem. Phys. 30, 77 (1975).
[8] E. J. Heller, J. Chem. Phys. 95, 9431 (1991).
[9] W. H. Miller, J. Chem. Phys. 95, 9428 (1991).
[10] M. F. Herman, E. Kluk, Chem. Phys. 91, 27 (1984). I,
A
[11] J.-X. Zeng and X.-Z. Li, Comput. Mater. Today 6,
100032 (2025), and the reference therein. I
[12] W. H. Miller, In Physical Biology (p. 505), Edited by: A.
H. Zewail, Imperial College Press (2008), and the refer-
ence therein. I
[13] K. G. Kay, Chem. Phys. 322, 3 (2006). I, III B, C 2
[14] W.H. Miller, J. Chem. Phys. 55 3146 (1971). I
[15] N. Makri and W. H. Miller. J. Chem. Phys. 116, 9207
(2002). I, A
[16] Y.-A. Yan, J. Liu, and J. Shao, J. Comput. Chem. 40,
1161 (2019). I
[17] Y. Zhao, W.H. Miller, J. Chem. Phys. 117, 9605 (2002).
8
I
[18] S. Weinberg, Lectures on Quantum Mechanics, (Cam-
bridge University Press, 2005). III D
[19] J. Ankerhold, M. Saltzer, E. Pollak, J. Chem. Phys. 116,
5925 (2002). III B
[20] E. Pollak, J. Shao, J. Phys. Chem. A 107, 7112 (2003).
[21] S. Zhang, E. Pollak, Phys. Rev. Lett. 91, 190201 (2003).
III B
Appendix A: Divergence of Direct Extensions of HK Representation
In this appendix, we demonstrate that directly applying the t →−iτ transformation to the HK representation [10]
results in divergence in the high-temperature limit. Without loss of generality, we consider the system of a single
particle in one-dimensional (1D) space as an example. Our analysis can be easily extended to systems in arbitrary
dimensions and with an arbitrary particle number.
We first consider the free-particle case with Hamiltonian ˆH = ˆp2/(2m). In this case the HK representation of the
real-time propagator ⟨˜x|e−i ˆ
Ht/ℏ|x⟩of this particle is proportional to
Z
dpdqe−γ(x−q)2
ℏ
−i p(x−q)
ℏ
e−γ(˜x−qt)2
ℏ
+i pt(˜x−qt)
ℏ
e−i
p2
2mℏt,
(A1)
where γ is an arbitrary positive number, and qt and pt satisfy the classical Hamiltonian equations
dqt
dt = pt
m,
dpt
dt = 0.
(A2)
Now we apply the transformation t →−iτ to Eq. (A1). Due to Eq. (A2), this transformation leads to another
transformations qt →qτ and pt →ipτ, with qτ and pτ satisfying dpτ
dτ = 0 and pτ = m dqτ
dτ , i.e.,
qτ = q + pτ/m,
pτ = p.
(A3)
Substituting these results into Eq. (A1), we find that the result of the transformation is
I =
Z
dpdqe−γ(x−q)2
ℏ
−p(x−q)
ℏ
e−γ(˜x−q−pτ/m)2
ℏ
−p(˜x−q−pτ/m)
ℏ
e−
p2
2mℏτ
∝
Z
dpe
(mτ−τ2γ)
2ℏm2
p2+ (x−˜x)(m−τγ)
ℏm
p−(x−˜x)2γ
2ℏ
.
(A4)
Clearly,
I = ∞,
for τ < m/γ.
(A5)
Thus, the result of the transformation t →−iτ on HK representation diverges in the limit τ →0, i.e., the high-
temperature limit T →∞.
Next, we consider the systems with non-zero potential energy V (x). The Hamiltonian operator of such a system is
ˆH = ˆp2/(2m) + V (ˆx). After the t →−iτ transformation, qτ and pτ satisfy the classical Hamiltonian equations with
inverted potential, i.e., dqτ/dτ = pτ/m and dpτ/dτ = dV (x)
dx |x=qτ . As mentioned in the main text, we consider systems
in which |dV (x)/dx| has a finite upper bound over the entire q-space. For these systems, in the high-temperature
limit (τ →0), the behaviors of qτ and pτ is always same as the ones of a free particle, i.e., still satisfying Eq. (A3).
As a result, the above analysis for this limit is still applicable. Thus, the result of the t →−iτ transformation on
HK representation always diverges for τ →0 or T →∞. Moreover, if |dV (q)/dq| is un-bounded, then qτ and pτ may
increase with p even faster than the ones in Eq. (A3). Therefore, the representation given by the transformation may
still diverges.
We
also
notice
that
in
Ref.
[15]
the
authors
propose
another
two
extensions
of
HK
representation,
which are also based on the t
→
−iτ transformation.
For a single free particle, they are proportional
to
R
dpdqe−γ(x−q)2
ℏ
−p(x−q)
ℏ
e−γ(x−q−pτ)2
ℏ
−p(x−q−pτ)
ℏ
e−
p2
2mℏτ and
R
dpdqe−γ(x−q)2
ℏ
+ p(x−q)
ℏ
e−γ(x−q−pτ)2
ℏ
+ p(x−q−pτ)
ℏ
e−
p2
2mℏτ, re-
spectively. Using the same method as above, we directly find that both of these two extensions also diverge in the
high-temperature limit, for systems in arbitrary dimensions and with arbitrary potential energy and particle numbers.
Appendix B: Equations for
∂qη,i
∂qj ,
∂qη,i
∂pj ,
∂pη,i
∂qj
and
∂pη,i
∂pj
In this appendix we present the ordinary differential equations satisfied by ∂qη,i
∂qj , ∂qη,i
∂pj , ∂pη,i
∂qj
and ∂pη,i
∂pj . We first
consider the 1D single-particle case (N = 1).
In this case {qη, pη} satisfy the Hamilton’s equations
d
dηqη =
pη
m ,
9
d
dτ pη = dV (z)/dz|z=qη, with initial conditions qη=0 = q; pη=0 = p. We consider {qη, pη} as functions of (η, q, p), and
define
ξ(1)(η) = ∂
∂q qη;
ξ(2)(η) = ∂
∂q pη.
(B1)
By calculating ∂/∂q in both sides of the aforementioned Hamilton’s equations satisfied by {qη, pη}, we find that
ξ(1,2)(η) satisfy the equations
d
dη ξ(1)(η) = ξ(2)(η)
m
;
d
dη ξ(2)(η) = d2V (z)
dz2
z=qη
ξ(1)(η).
(B2)
Moreover, it is clear that ξ(1,2)(η) also satisfy the initial conditions
ξ(1)(η = 0) = 1;
ξ(2)(η = 0) = 0.
(B3)
Thus, one can derive ∂qτ/∂q and ∂pτ/∂q (i.e., ξ(1)(τ) and ξ(2)(τ)) by solving Eq. (B2) with initial condition Eq. (B3).
Similarly, one can calculate ∂qτ/∂p and ∂pτ/∂p solving Eq. (B2) with another initial condition {ξ(1)(η = 0) =
0; ξ(2)(η = 0) = 1}. The solutions just relate to ∂qτ/∂p and ∂pτ/∂p via ∂qτ/∂p = ξ(1)(τ) and ∂pτ/∂p = ξ(2)(τ).
The above approach can be directly generalized to the cases with arbitrary N. Specifically, to derive ∂qη,i
∂qj and ∂pη,i
∂qj
for each fixed j, one can solve equaitons
d
dη ξ(1)
k (η) = ξ(2)
k (η)
mk
;
d
dη ξ(2)
k (η) =
N
X
s=1
∂2V (z1, ..., zN)
∂zk∂zs
z1=qη,1;z2=qη,2;...;zN=qη,N
ξ(1)
s (η),
(k = 1, ..., N),
(B4)
with initial condition {ξ(1)
s (η = 0) = δsj, ξ(2)
s (η = 0) = 0 (s = 1, ..., N)}. The solutions are related to ∂qη,i
∂qj
and ∂pη,i
∂qj
via ∂qη,i
∂qj = ξ(1)
i
(η) and ∂pη,i
∂qj = ξ(2)
i
(η) (i = 1, ..., N). Similarly, to derive ∂qη,i
∂pj and ∂pη,i
∂pj for each fixed j, one can solve
Eq. (B4), with another initial condition {ξ(1)
s (η = 0) = 0, ξ(2)
s (η = 0) = δsj (s = 1, ..., N)}. The solutions for this
initial condition are related to ∂qη,i
∂pj and ∂pη,i
∂pj
via ∂qη,i
∂pj = ξ(1)
i
(η) and ∂pη,i
∂pj = ξ(2)
i
(η), (i = 1, ..., N).
Appendix C: Details of the Derivation of Eq. (4) for N = 1
In this appendix we show the details of the derivation of Eq. (4) for a 1D single-particle system. As mentioned in
Sec. III, we express the matrix element of the Boltzmann operator as
Kτ(˜x, x) = A
Z
dpdq
h
FDe−(Sτ +Bτ +Cτ )
ℏ
i
,
(C1)
where the factors A, Sτ, Bτ and Cτ are given by Eqs. (11), (5), (9) and (10), respectively, for N = 1, i.e., we have
A =
γ
2ℏ3π3
1/2
;
(C2)
Sτ =
Z τ
0
dη
p2
η
2m + V (qη)
;
(C3)
Bτ = γ (x −q)2 + γ (˜x −qτ)2 −p (x −q) + pτ (˜x −qτ) ;
(C4)
Cτ = m
τ [(˜x −qτ) −(x −q)]2 .
(C5)
Here pη and qη (0 ≤η ≤τ) satisfy the Hamilton’s equations and initial condition:
d
dη qη = pη
m ;
d
dη pη = dV (z)
dz
z=qη
,
(C6)
qη=0 = q;
pη=0 = p.
(C7)
Furthermore, in Eq. (C1) FD is a to-be-determined function of (τ, p, q).
10
1.
Proof of Eqs. (23, 24)
Now we prove Eq. (23, 24). To this end, we need to calculate the integration A
R
dpdq e−(Sτ +Bτ +Cτ )
ℏ
in the limit
τ →0. Let us first perform the integration for p. As mentioned in Sec. III, since |dV (q)/dq| has a finite upper bound
in the total real space, in the limit τ →0 the solution of the Hamilton’s equations (C6-C7) uniformly converges to
the one of free motion, i.e., qη = q + ηp/m and pη = p. We substitute this solution into the definitions (C3, C4, C5)
of Sτ, Bτ, and Cτ, and then obtain
A
Z +∞
−∞
dp e−(Sτ +Bτ +Cτ )
ℏ
= Fa(τ)e−
Fb(τ,q)
ℏ
,
(C8)
where
Fa(τ) =
m
ℏπ
r
γ
τ(m + 2τγ);
(C9)
Fb(τ, q) = m2(x −˜x)2 + 2mτ(2q2 −4qx + 3x2 −2x˜x + ˜x2)γ + 4τ 2(q −x)2γ2
2τ(m + 2τγ)
+
Z τ
0
V
q + ηp
m
dη.
(C10)
Thus, we have
lim
τ→0 A
Z
dpdq e−(Sτ +Bτ +Cτ )
ℏ
= lim
τ→0
Z ∞
−∞
dqFa(τ)e−
Fb(τ,q)
ℏ
.
(C11)
Furthermore, in the limit τ →0, we can expand Fa and Fb in the integrand of Eq. (C11) as powers of τ, and ignore
the terms proportional to τ n (n > 0). This approach leads to
lim
τ→0 A
Z
dpdq e−(Sτ +Bτ +Cτ )
ℏ
= 1
ℏπ
rmγ
τ
lim
τ→0
Z ∞
−∞
dqe
−1
ℏ
m(x−˜x)2
2τ
+2(q2γ−2qxγ+x2γ)
.
(C12)
Performing the integration in the r. h. s. of Eq. (C12) directly, we obtain Eq. (23) of Sec. III, i.e.,
lim
τ→0 A
Z
dpdq e−(Sτ +Bτ +Cτ )
ℏ
= lim
τ→0
m
2πℏτ
1/2
e−m(x−˜x)2
2ℏτ
.
(C13)
Additionally, substituting the result
m
2πℏτ
1/2 e−m(x−˜x)2
2ℏτ
= ⟨˜x|e−ˆ
p2
2m τ|x⟩into Eq. (C13), we further obtain
lim
τ→0 A
Z
dpdq e−(Sτ +Bτ +Cτ )
ℏ
= lim
τ→0⟨˜x|e−ˆ
p2
2m τ|x⟩= δ(x −˜x),
(C14)
which is just Eq. (24) of Sec. III.
2.
Derivation of Eq. (29)
1. Preliminary Calculations
In the following we derive the equation of FD, i.e., Eq. (29) of Sec. III. We begin from Eq. (28), i.e.,
ˆΛ
h Z
dpdq
FDe−(Sτ +Bτ +Cτ )
ℏ
i
= 0.
(C15)
For our system with N = 1, the factors Sτ, Bτ and Cτ are given by Eqs. (C3), (C4) and Eq. (C5), respectively.
Additionally, the correction operator ˆΛ is given by Eq. (27), i.e.,
ˆΛ = ℏ∂
∂τ −ℏ2
2m
∂2
∂˜x2 + V (˜x).
(C16)
Direct calculation yields
ˆΛ
h Z
dpdq
FDe−(Sτ +Bτ +Cτ )
ℏ
i
=
Z
dpdq
(
FDe−(Sτ +Bτ +Cτ )
ℏ
"
Jτ + ℏ
˙
FD
FD
−˙Sτ + V (˜x)
#)
,
(C17)
11
where the dot above symbols means
˙
(...) ≡∂(...)
∂τ ,
(C18)
and
Jτ = ℏ
1
τ + γ
m
−p2
τ
2m + 2
mpτ
m [(x −q) −(˜x −qτ)]
τ
−γ(˜x −qτ)
−2
m
m [(x −q) −(˜x −qτ)]
τ
−γ(˜x −qτ)
2
+ m
τ 2 [(x −q) −(˜x −qτ)]2 −(˜x −qτ) ˙pτ
+pτ ˙qτ + 2γ(˜x −qτ) ˙qτ −2m
τ
[(x −q) −(˜x −qτ)] ˙qτ.
(C19)
For the convenience of the following calculations, we define
u = ˜x −qτ;
(C20)
v = x −q,
(C21)
and
Φτ = −(Bτ + Cτ + Sτ);
(C22)
V (n)(z) =
dn
dzn V (z).
(C23)
Substituting Eqs. (C6, C3) into Eqs. (C17, C19), and using the above definitions and the fact
V (˜x) = V (qτ) +
∞
X
n=1
unV (n)(qτ),
(C24)
we find that Eq. (C17) can be re-expressed as
ˆΛ
h Z
dpdq
FDe−(Sτ +Bτ +Cτ )
ℏ
i
=
Z
dpdq
(
FD
"
Gτ + ℏ
˙
FD
FD
+
∞
X
s=2
1
s!usV (s)(qτ)
#
eΦτ /ℏ
)
,
(C25)
where
Gτ = ℏ
1
τ + γ
m
−2
m
m (v −u)
τ
−γu
2
+ m
τ 2 (u −v)2 .
(C26)
2. Eliminating u and v
Now we eliminate the factors u and v in Eq. (C25). Using the fact
∂Sτ
∂q
= −p + ∂qτ
∂q pτ;
∂Sτ
∂p = ∂qτ
∂p pτ,
(C27)
we find that that the factor Φτ defined in Eq. (C22) satisfies
∂Φτ
∂q
∂Φτ
∂p
=
Rqu Rqv
Rpu Rpv
u
v
.
(C28)
Here the parameters Rqu, Rqv, Rpu and Rpv are given by
Rqu = −2
τ m +
2γ + 2
τ m
∂qτ
∂q −∂pτ
∂q ;
(C29)
Rqv = 2γ −2
τ m
∂qτ
∂q −1
;
(C30)
Rpu =
2γ + 2
τ m
∂qτ
∂p −∂pτ
∂p ;
(C31)
Rpv = 1 −2
τ m∂qτ
∂p .
(C32)
12
We further define another four parameters Tqu, Tqv, Tpu and Tpv via the relation:
Tuq Tup
Tvq Tvp
=
Rqu Rqv
Rpu Rpv
−1
.
(C33)
Thus, Eq. (C28) yields that
u
v
=
Tuq Tup
Tvq Tvp
∂Φτ
∂q
∂Φτ
∂p
.
(C34)
Note that these R- and T-parameters are simply the R-matrices and T-matrices defined in Eqs. (12-17) of Sec. (II),
with N = 1, respectively.
For convenience of the following calculations, we introduce differential operators Du and Dv:
Du = Tuq ∂
∂q + Tup ∂
∂p;
Dv = Tvq ∂
∂q + Tvp ∂
∂p,
(C35)
as well as the parameters
Qαβ = Dα [β] ;
(α, β = u, v).
(C36)
Substituting the definitions (C20, C21) of u and v into Eq. (C36), we further obtain the expressions of the Q-
parameters:
Quu = −Tuq ∂qτ
∂q −Tup ∂qτ
∂p ;
Quv = −Tuq;
(C37)
Qvu = −Tvq ∂qτ
∂q −Tvp ∂qτ
∂p ;
Qvv = −Tvq.
(C38)
Using the differential operators Du and Dv, we can re-express Eq. (C34) as:
αeΦτ /ℏ= ℏDα
h
eΦτ /ℏi
;
(α = u, v).
(C39)
Moreover, Eq. (C39) leads to
αβeΦτ /ℏ= ℏαDβ
h
eΦτ /ℏi
= ℏDβ
h
αeΦτ /ℏi
−ℏeΦτ /ℏDβ [α] ,
(α, β = u, v).
(C40)
Substituting Eqs. (C39) and (C36) into Eq. (C40), we further obtain
αβeΦτ /ℏ= ℏ2Dβ
n
Dα
h
eΦτ /ℏio
−ℏQβαeΦτ /ℏ,
(α, β = u, v).
(C41)
i.e.,
u2eΦτ /ℏ= ℏ2D2
u
h
eΦτ /ℏi
−ℏQuueΦτ /ℏ,
(C42)
v2eΦτ /ℏ= ℏ2D2
v
h
eΦτ /ℏi
−ℏQvveΦτ /ℏ,
(C43)
uveΦτ /ℏ= ℏ2DvDu
h
eΦτ /ℏi
−ℏQvueΦτ /ℏ.
(C44)
Repeating this technique, we can express any term of the form umvneΦτ /ℏ(m, n = 1, 2, ...) as a series in ℏ, with each
coefficient taking the form C1C2 . . . [eΦτ /ℏ], where each C1, C2, . . . is either a Q-factor or a D-operator. Using this
approach and Eqs. (C37, C38), we can re-express Eq. (C25) as
ˆΛ
h Z
dpdq
FDe−(Sτ +Bτ +Cτ )
ℏ
i
= ℏ
Z
dpdq
(
FD
"
ˆLτ +
˙
FD
FD
#
eΦτ /ℏ
)
,
(C45)
13
where
ˆLτ = ˆL(0)
τ
+ ℏˆL(1)
τ
+ ℏ2 ˆL(2)
τ
+ ...
(C46)
Here ˆL(0,1,2,...)
τ
are ℏ-independent operators. Specially, we have
ˆL(0)
τ
=
1
τ + γ
m
+
2γ2
m + m
τ 2 + 4γ
τ
Quu +
2m
τ 2 + 4γ
τ
Tuq −m
τ 2 Tvq −1
2QuuV (2)(qτ),
= gτ(q, p),
(C47)
and
ˆL(1)
τ
= −
2γ2
m + m
τ 2 + 4γ
τ −1
2V (2)(qτ)
D2
u +
2m
τ 2 + 4γ
τ
DuDv −m
τ 2 D2
v
−1
3!V (3)(qτ)
DuQuu + 2QuuDu
+ 3
4!V (4)(qτ)Q2
uu.
(C48)
Notice that gτ(q, p) is just the function given by Eq. (19) of Sec. II, with N = 1,
3. Equation of FD
As in Ref. [13], by repeated integration by parts and using the fact that the integrated terms tend to zero for
q →±∞or p →±∞, we can further re-express Eq. (C25) as
ˆΛ
h Z
dpdq
FDe−(Sτ +Bτ +Cτ )
ℏ
i
= ℏ
Z
dpdq
eΦτ /ℏˆ˜Lτ[FD]
,
(C49)
where
ˆ˜Lτ = ˆ˜L(0)
τ
+ ℏˆ˜L(1)
τ
+ ℏ2 ˆ˜L(2)
τ
+ ....
(C50)
Here ˆ˜L(0,1,2,...)
τ
are also a group of ℏ-independent operators. For instance, we have
ˆ˜L(0)
τ
=
∂
∂τ + ˆL(0)
τ
= ∂
∂τ + gτ(q, p),
(C51)
and
ˆ˜L(1)
τ
= −˜D2
u
2γ2
m + m
τ 2 + 4γ
τ −1
2V (2)(qτ)
+ ˜Dv ˜Du
2m
τ 2 + 4γ
τ
−˜D2
v
m
τ 2
−1
3!
Quu ˜Du + 2˜DuQuu
V (3)(qτ) + 3
4!V (4)(qτ)Q2
uu,
(C52)
with ˜Du and ˜Dv being defined as
˜Du[...] =
∂
∂q [Tuq...] + ∂
∂p[Tup...];
˜Dv[...] = ∂
∂q [Tvq...] + ∂
∂p[Tvp...].
(C53)
Substituting Eq. (C49) and (C50) into Eq. (C15), we finally obtain Eq. (29) of of Sec. III, i.e.,
ˆ˜L0 + ℏˆ˜L1 + ℏ2 ˆ˜L2 + ...
FD = 0.
(C54)
Appendix D: Harmonic Oscillators
In this appendix we show that our HK-like IVR for the Boltzmann operator is exact for harmonic oscillators. We
first consider a single harmonic oscillator with frequency ω, and use the natural unit ℏ= m = γ = 1. As shown
14
in Sec. IV, for this system we have V = ω2x2/2, and thus qη = ceωη + de−ωη and pη = ω(ceωη −de−ωη), where
c = (p + ωq)/(2ω) and d = (ωq −p)/(2ω). Substituting these results into the definition of gτ, we find that
gτ = −−8eωτω + e2ωτ(−2 + ω)
4 −4ωτ + τ 2(−2 + ω)ω
+ (2 + ω)
4 + 4τω + τ 2ω(2 + ω)
2τ[−2 + eτω(−2 + ω) −ω]
4 + eτω
−4 + τ(−2 + ω)
+ τ(2 + ω)
.
(D1)
Thus, the function Dτ = e−
R τ
0 gηdη can be expressed as
Dτ =
1
2
√
2
s
e−τω
−2 + eτω(−2 + ω) −ω
4 + eτω
−4 + τ(−2 + ω)
+ τ(2 + ω)
ωτ
.
(D2)
One can verify this result by substituting Eq. (D2) into the equation
d
dτ Dτ = −gτDτ and the condition Dτ=0 = 1.
Substituting Eq. (D2) and the above expressions of qη and pη into Eq. (4), we find that the integration
R
dpdq...
is just a Gaussian integration. Perform this integration analytically, we find that in the SI, the r. h. s. of Eq. (4)
is just
mω
2πℏsinh(ωτ)
1/2
e−
mω[(x2+˜x2) cosh(ωτ)−2x˜x]
2ℏsinh(ωτ)
, i.e., the exact imaginary-time propagator of this harmonic oscillator.
Furthermore, the above calculation can be directly generalized to the general cases with N harmonic oscillators, and
we can find that for these cases the r. h. s. of Eq. (4) is Eq. (34) of Sec. IV.
|
Herman-Kluk-Like Semi-Classical Initial-Value Representation for Boltzmann Operator Binhao Wang,1 Fan Yang,2 Chen Xu,1 and Peng Zhang1, 3, ∗ 1 100872, China 2Hefei National Laboratory, Hefei 230088, China 3Key Laboratory of Quantum State Construction and Manipulation (Ministry of Education), Renmin 100872, China (Dated: October 17, 2025) The coherent-state initial-value representation (IVR) for the semi-classical real-time propagator of a quantum system, developed by Herman and Kluk (HK), is widely used in computational studies of chemical dynamics. On the other hand, the Boltzmann operator e-ˆ H/(kBT ), with ˆH, kB, and T representing the Hamiltonian, Boltzmann constant, and temperature, respectively, plays a crucial role in chemical physics and other branches of quantum physics. One might naturally assume that a semi-classical IVR for the matrix element of this operator in the coordinate representation (i.e., ⟨ ̃x|e-ˆ H/(kBT )|x⟩, or the imaginary-time propagator) could be derived via a straightforward "realtime →imaginary-time transformation" from the HK IVR of the real-time propagator. However, this is not the case, as such a transformation results in a divergence in the high-temperature limit (T →∞). In this work, we solve this problem and develop a reasonable HK-like semi-classical IVR for ⟨ ̃x|e-ˆ H/(kBT )|x⟩, specifically for systems where the gradient of the potential energy (i.e., the force intensity) has a finite upper bound. The integrand in this IVR is a real Gaussian function of the positions x and ̃x, which facilitates its application to realistic problems. Our HK-like IVR is exact for free particles and harmonic oscillators, and its effectiveness for other systems is demonstrated through numerical examples. I. INTRODUCTION In 1928, van Vleck derived the semi-classical real-time propagator (the van Vleck propagator) for a quantum system in the limit as ħ→0 [1]. The phase of the van Vleck propagator, with respect to a given initial and final time and position, is determined by the action of the corresponding classical trajectory. The van Vleck propagator reveals the intrinsic connection between quantum and classical mechanics, having had a significant impact on quantum physics [2, 3]. However, deriving the van Vleck propagator requires solving the classical Hamiltonian equations with fixed initial and final positions. This is not convenient for numerical calculations of realistic systems due to the rootfinding problem. Consequently, the van Vleck propagator is not well-suited for numerical studies of atom or molecule systems. To overcome this problem, many authors [4-10] have attempted to develop initial-value representations (IVRs) for semi-classical real-time propagators. In the IVRs, these propagators are expressed as functionals of classical trajectories, determined by the given initial position and momentum, which can be easily derived numerically via standard algorithms, such as Runge-Kutta. A highly influential IVR for the semiclassical real-time propagator was derived by Herman and Kluk (HK) in 1984 [10], based on coherent states. Over the past forty years, the HK representation has ∗Electronic address: been widely used in studies of chemical dynamics, and has proven to be a valuable tool for exploring the quantum effects of nuclear motion [11, 12]. Moreover, in 2006, Kay provided a rigorous derivation of the HK representation from the Schr ̈odinger equation, demonstrating that this representation is the leading term in an asymptotic expansion of the quantum propagator in powers of ħ[13]. In addition to the real-time propagator, the imaginarytime propagator, which is the matrix element of the Boltzmann operator e-ˆ H/(kBT ), also play a crucial role in quantum physics and chemistry. Here, kB and T are the Boltzmann constant and temperature, respectively. In 1971, Miller derived the semi-classical imaginary-time propagator [14], through a direct t →-iτ transformation of the van Vleck propagator, where t is the real time and τ = ħ/(kBT). Similar to the van Vleck propagator, the result obtained by Miller in Ref. [14] is determined by the classical trajectory with an inverted potential, with respect to fixed initial and final positions. An semi-classical IVR for the imaginary-time propagator, which is similar to the HK representation for the real-time one, would clearly be very useful for numerical studies in atom physics, molecule physics and chemical physics. Intuitively speaking, one might expect to obtain such an IVR through the aforementioned t →-iτ transformation from the HK representation of the real-time propagator [15]. However, as pointed out by Yan, Liu, and Shao in Ref. [16], this is not the case, as the result of this transformation diverges in the limit T →∞. In Appendix A, we demonstrate this again with detailed calculations. This problem is crucial, as the semi-classical approximation should be applicable in 16 Oct 2025 2 the high-temperature limit. Although several alternative IVRs for the imaginary-time propagator have been derived [16, 17], an IVR based on coherent-state-type wave functions, similar to the HK IVR, has yet to be found. In this work we solve the above problem for the systems with the absolute value of potential-energy gradient (force intensity) having a finite upper bound. For these systems we derive a reasonable HK-like semiclassical IVR for the Boltzmann operator, via an approach generalized from the one of Kay in Ref. [13]. Our IVR is exact for free particles and harmonic oscillators, although the potential-energy gradient of the latter system is un-bounded. The applicability for other systems are illustrated via numerical examples. Our results are helpful for studies of the properties of atomic systems, molecular systems, and chemical reactions at finite temperatures. The remainder of this paper is organized as follows. For the convenience of the readers, we first display our HK-like IVR for the Boltzmann operator in Sec. II, and then show the derivation of this IVR as well as the condition for the semi-classical approximation in Sec. III. The applicability of our HK-like IVR is illustrated with some examples in Sec. IV. In Sec. V there is a summary. Some details of the derivations and calculations are given in the appendixes. II. CENTRAL RESULT We consider a general multi-particle quantum system, with Cartesian components of the coordinates and momenta being denoted as (p1, ..., pN) and (q1, ..., qN), respectively, and the Hamiltonian being given by H = N X j=1 p2 j 2mj + V (q). (1) Here q = (q1, ..., qN), and mj (j = 1, ..., N) is mass of the particle to which the coordinate qj belongs. Furthermore, as mentioned above, we assume the norm of potential energy gradient, i.e., |∇qV (q)|, having a finite upper bound in the q-space. The Boltzmann operator of our system is e-ˆ H/(kBT ), where ˆH is the quantum operator of the Hamiltonian in Eq. (1). The matrix element of this operator in the coordinate representation (imaginary-time propagator) can be expressed as Kτ( ̃x, x) := ⟨ ̃x|e-ˆ H/(kBT )|x⟩= ⟨ ̃x|e-ˆ Hτ/ħ|x⟩, (2) where |x⟩and | ̃x⟩are eigen-states of the coordinate operators, with eigen-values x = (x1, ..., xN) and ̃x = ( ̃x1, ..., ̃xN), respectively, and τ = ħ kBT . (3) The HK-like IVR for the Boltzmann operator under the semiclassical approximation, which we have derived in this work, can be expressed as: Kτ( ̃x, x) = A Z dpdq h Dτe-1 ħ(Sτ +Bτ +Cτ )i . (4) Here, q = (q1, ..., qN) (as defined above), p = (p1, ..., pN), and R dp, dq = R dp1 · · · dpN, dq1 · · · dqN. Moreover, the factors Sτ, A, Bτ, Cτ and Dτ are all real and independent of ħ, with the definitions being introduced in the following. Factor Sτ The factor Sτ of Eq. (4) is a function of p and q, and is independent of x and ̃x. It is defined as Sτ(q, p) = Z τ 0 dη N X j=1 p2 η,j 2mj + V (qη,1, ..., qη,N) . (5) Here qη,1, ..., qη,N and pη,1, ..., pη,N satisfy the Hamilton's equations with inverted potential -V , initial position q and initial momenta p, i.e., the equations d dη qη,j = pη,j mj ; (6) d dη pη,j = ∂V (z1, ..., zN) ∂zj z1=qη,1;...;zN=qη,N , (7) (j = 1, ..., N), and the initial conditions qη=0,j = qj; pη=0,j = pj, (j = 1, ..., N). (8) According to the above definitions and equations, the factor Sτ of Eq. (5) is the action of the classical trajectory qη,1, ..., qη,N and pη,1, ..., pη,N. Moreover, in the following we will consider qη,1, ..., qη,N and pη,1, ..., pη,N as functions of η and {q, p}. Factors A, Bτ and Cτ The factors Bτ and Cτ of Eq. (4) are functions of both p, q and x, ̃x. They are defined as Bτ(p, q; x, ̃x) = N X j=1 γj (xj -qj)2 + γj ( ̃xj -qτ,j)2 -pj (xj -qj) + pτ,j ( ̃xj -qτ,j) ; (9) Cτ(p, q; x, ̃x) = 1 τ N X j=1 mj ( ̃xj -qτ,j) -(xj -qj) 2 , (10) 3 where γ1,...,N are arbitrary positive parameters. Additionally, the factor A is defined as: A = N Y j=1 γj 2ħ3π3 1/2 . (11) Notice that the factor Cτ, as well as the signs of the terms pj (xj -qj) and pτ,j ( ̃xj -qτ,j) of the factor Bτ, cannot be obtained from direct t →-iτ transformation on the HK representation of real-time propagator. Factor Dτ Similar to Sτ, the factor Dτ in Eq. (4) is also a function of p, q, and is independent of x and ̃x. To show the definition of Dτ, we first introduce four N × N matrices Rqu(η), Rqv(η), Rpu(η), and Rpv(η), with elements: Rqu ij (η) = -2 η miδij + 2γj + 2 η mj ∂qη,j ∂qi -∂pη,j ∂qi ; (12) Rqv ij (η) = 2γjδij -2 η mj ∂qη,j ∂qi -δij ; (13) Rpu ij (η) = 2γj + 2 η mj ∂qη,j ∂pi -∂pη,j ∂pi ; (14) Rpv ij (η) = δij -2 η mj ∂qη,j ∂pi , (15) (i, j = 1, ..., N), (16) with δij being the Kronecker symbol. We further define other four N × N matrices Tuq(η), Tvq(η), Tup(η), and Tvp(η), which relate to the these R-matrices via Tuq(η) Tup(η) Tvq(η) Tvp(η) = Rqu(η) Rqv(η) Rpu(η) Rpv(η) -1 . (17) The factor Dτ of Eq. (4) can be expressed in terms of the elements of the above T-matrices, which are denoted as Tto ij(η) (t = u, v; o = p, q; i, j = 1, ..., N). Specifically, we have Dτ(q, p) = eR τ 0 gηdη, (18) with gη being a function of (q, p): gη(q, p) = -1 2 N X i,j=1 Wij(η)Vij(η) + N X j=1 1 η + γj mj + 2γ2 j mj + mj η2 + 4γj η ! Wjj(η) + 2mj η2 + 4γj η Tuq jj (η) -mj η2 Tvq jj (η) , (19) where Vij(η) = ∂2 ∂zi∂zj V (z1, ..., zN) z1=qη,1;...;zN=qη,N , (20) Wij(η) = - N X s=1 Tuq is (η)∂qη,j ∂qs + Tup is (η)∂qη,j ∂ps . (21) According to this definition, calculating Dτ requires computing the derivatives ∂qη,i ∂qj , ∂qη,i ∂pj , ∂pη,i ∂qj and ∂pη,i ∂pj (i, j = 1, ..., N) for 0 ≤η ≤τ. As shown in Appendix B, one can derive these function via solving the Hamilton's equations (6, 7) together with another group of ordinary differential equations. III. DERIVATION OF EQ. (4) Now we show our approach for deriving the HK-like IVR for the imaginary-time propagator, i.e., Eq. (4). For simplicity, we consider the system of a single particle in one-dimensional space (N = 1), and the derivation can be directly generalized to the cases with arbitrary N. Consequently, we will omit the subscript denoting the particle index in the following. In the following we present the main ideas and framework of the derivation. The details of the derivations are provided in Appendix C. To derive Eq. (4), we express the imaginary-time propagator as the integration Kτ( ̃x, x) = A Z dpdq h FDe-(Sτ +Bτ +Cτ ) ħ i , (22) with A, Sτ, Bτ and Cτ being defined in Eqs. (11), (5), (9) and (10), respectively, and FD being a to-be-determined function of τ and (p, q), which is independent of x and ̃x. In the following, we first prove that the function FD satisfies the "initial condition" limτ→0 FD = 1. Then we derive the differential equation of FD, and solve this equation together with this "initial condition", using the semi-classical approximation. We will find that the solution is just FD = Dτ, with Dτ being given by Eq. (18) for N = 1. Substituting this result into Eq. (22), we obtain the result of (4) for N = 1. A. "Initial Condition" of FD To prove limτ→0 FD = 1, we first calculate the limitation limτ→0 A R dpdq e-(Sτ +Bτ +Dτ ) ħ . Notice that since the potential-energy gradient intensity |dV (q)/dq| has a finite upper bound in the total q-space, in the short-τ limit the solutions of the Hamiltonian equations (6) and (7) are just those for free motion. Specifically, in this limit we have qη = q + ηp/m and pη = p. Substituting this solution into the definitions (5, 9, 10) of Sτ, Bτ, and 4 Cτ, we find that (Appendix C 1) lim τ→0 A Z dpdq e-(Sτ +Bτ +Cτ ) ħ = lim τ→0 m 2πħτ 1/2 e-m(x- ̃x)2 2ħτ (23) = δ(x - ̃x). (24) On the other hand, it is clear that limτ→0 Kτ( ̃x, x) = δ(x - ̃x). Thus, we have lim τ→0 Kτ( ̃x, x) = lim τ→0 A Z dpdq e-(Sτ +Bτ +Cτ ) ħ . (25) Comparing this result and Eq. (22), we find that the function FD satisfies lim τ→0 FD = 1. (26) B. Equation of FD and Semi-Classical Approximation Now we derive the differential equation for the function FD. To this end, we introduce the correction operator [19-21] for our system, which is defined as ˆΛ ̃x := ħ∂ ∂τ -ħ2 2m ∂2 ∂ ̃x2 + V ( ̃x). (27) It is clear that the imaginary-time propagator Kτ( ̃x, x) satisfies ˆΛ [Kτ( ̃x, x)] = 0. By substituting Eq. (22) into this equation, we find that the function FD satisfies ˆΛ Z dpdq FDe-(Sτ +Bτ +Cτ ) ħ = 0. (28) As detailed in Appendix C 2, using the method generalized from Ref. [13], we find that the sufficient condition for Eq. (28) can be expressed as a differential equation for FD: ˆ ̃L0 + ħˆ ̃L1 + ħ2 ˆ ̃L2 + ... FD = 0, (29) where ˆ ̃L0,1,2,... are ħ-independent operators. Specially, the operator ˆ ̃L0 is given by ˆ ̃L0 = ∂ ∂τ + gτ(q, p), (30) where gτ(q, p) is just the function given by Eq. (19) with N = 1. Note that the operator in the l.h.s of Eq. (29) is expanded as a power series of ħ. Under the semi-classical approximation, we further ignore the terms proportional to ħn (n ≥1) in this series, and approximate Eq. (29) as ∂ ∂τ FD + gτFD = 0. (31) Furthermore, it is clear that under the initial condition (26), the solution of Eq. (31) is just FD = eR τ 0 gηdη. (32) C. Final Derivation Eq. (32) yields that FD = Dτ, with Dτ being the one defined in Eq. (18) with N = 1. Substituting this result into Eq. (22), we finally obtain Eq. (4) for N = 1. D. Condition of the Semi-Classical Approximation At the end of this section, we discuss the condition underlying the semiclassical approximation employed in deriving our HK-like IVR, namely, the condition under which our HK-like IVR provides a good approximation to the exact matrix element of the Boltzmann operator. For comparison, recall that the semiclassical approximation used in the derivation of the van Vleck propagator requires the characteristic length scale lV of the potential energy variation to be much larger than the de Broglie wavelength [18]. The condition adopted here is similar: specifically, for a system with N cooridnates, lV should be much larger than both p ħ/γj and p ħτ/mj = p ħ/(kBTmj) (j = 1, ..., N). Thus, the semiclassical approximation works well in the high temperature systems. Additionally, in practical calculations for realistic systems, appropriate values of γj (j = 1, ..., N) should be chosen to ensure that this condition is satisfied. IV. EXAMPLES In the previous two sections, we presented our HK-like IVR for the imaginary-time propagator and its derivation. In this section, we demonstrate its applicability. Specifically, we analytically prove that the IVR is exact for free particles and harmonic oscillators, and numerically illustrate its performance with an anharmonic system. A. Free Particles We first consider the system of N free particles, i.e., V = 0. As mentioned above, in this case the solutions of the Hamilton's equations (6, 7) are just qη,j = qj + ηpj/mj and pη,j = pj (j = 1, ..., N). Consequently, the integration in the r. h. s. of Eq. (4) can be performed analytically. With straightforward calculations, we find that the right-hand side (r. h. s.) of Eq. (4) is N Y j=1 mj 2πħτ 1/2 emj 2ħτ (xj- ̃xj)2, (33) which is same as the exact imaginary-time propagator for this system. Thus, our HK-like IVR is exact for the free particles. 5 FIG. 1: The potential V (q) of Eq. (35), and matrix elements of the Boltzmann operator, for cases with L = 10l0 and q0 = 6l0. (a): The potential V (q). (b): Enlarged view of the region around q = 0 in (a). Here we also show the ground-state energy E0 (green dashed line), the first excited-state energy E1 (blue dotted line), and the second excited-state energy E2 (purple solid line), which are given by numerical diagonalization of the Hamiltonian. Note that E0 and E1 lie very close to each other. (c-e): The diagonal elements K(x, x) = ⟨x|e-ˆ H/(kBT )|x⟩(in units of 1/l0). (f-h): The non-diagonal elements K(-x, x) = ⟨-x|e-ˆ H/(kBT )|x⟩(in units of 1/l0). Here we show the results with temperature T = 10ħω/kB (c, f), T = ħω/kB (d, g) and T = 0.5ħω/kB (e, h). For each temperature, we illustrate the results KExac from exact diagonlization of the Hamiltonian (black dotted line) and the results KIVR given by our HK-like IVR (blue dots). B. Harmonic Oscillators Now we consider N harmonic oscillators with potential energy V = PN j=1 mjω2 j x2 j/2. The Hamilton's equations (6, 7) for this system can also be solved analytically, and we have qη,j = cjeωjη +dje-ωjη and pη,j = mjωj(cjeωη - dje-ωη), where cj = (pj + mjωjqj)/(2mjωj) and dj = (mjωjqj -pj)/(2mjωj). Using these results, we can also analytically perform the integration in the r. h. s. of Eq. (4), and find that the r. h. s. of Eq. (4) is just (Appendix D) N Y j=1 mjωj 2πħsinh(ωjτ) 1/2 e - mj ωj[(x2 j + ̃x2 j) cosh(ωj τ)-2xj ̃xj] 2ħsinh(ωj τ) , (34) which is same as the exact imaginary-time propagator of these oscillators. Therefore, as for the free particles, our IVR is also exact for harmonic oscillators. C. Anharmonic System Finally, we consider a single particle in an anharmonic potential, with Hamiltonian H = p2/(2m) + V (q). Here m, p and q are the mass, momentum and coordinate of this particle, respectively. The anharmonic potential V (q) is given by V (q) = - mω2L2 1 - q2 2L2 + q4 4L2q2 0 + V0, (35) where ω, q0, and L are positive parameters, and V0 is a q-independent constant: V0 = mω2L2 1 - q2 0 4L2 , (36) 6 FIG. 2: Same as Fig. (1), but for L = 10l0 and q0 = 3l0. which is chosen such that the minimum value of V (q) is zero. Specifically, ω has the dimension of frequency, and both q0 and L have the dimension of length, and satisfy q0 0). This approach leads to lim τ→0 A Z dpdq e-(Sτ +Bτ +Cτ ) ħ = 1 ħπ rmγ τ lim τ→0 Z ∞ -∞ dqe -1 ħ m(x- ̃x)2 2τ +2(q2γ-2qxγ+x2γ) . (C12) Performing the integration in the r. h. s. of Eq. (C12) directly, we obtain Eq. (23) of Sec. III, i.e., lim τ→0 A Z dpdq e-(Sτ +Bτ +Cτ ) ħ = lim τ→0 m 2πħτ 1/2 e-m(x- ̃x)2 2ħτ . (C13) Additionally, substituting the result m 2πħτ 1/2 e-m(x- ̃x)2 2ħτ = ⟨ ̃x|e-ˆ p2 2m τ|x⟩into Eq. (C13), we further obtain lim τ→0 A Z dpdq e-(Sτ +Bτ +Cτ ) ħ = lim τ→0⟨ ̃x|e-ˆ p2 2m τ|x⟩= δ(x - ̃x), (C14) which is just Eq. (24) of Sec. III. 2. Derivation of Eq. (29) 1. Preliminary Calculations In the following we derive the equation of FD, i.e., Eq. (29) of Sec. III. We begin from Eq. (28), i.e., ˆΛ h Z dpdq FDe-(Sτ +Bτ +Cτ ) ħ i = 0. (C15) For our system with N = 1, the factors Sτ, Bτ and Cτ are given by Eqs. (C3), (C4) and Eq. (C5), respectively. Additionally, the correction operator ˆΛ is given by Eq. (27), i.e., ˆΛ = ħ∂ ∂τ -ħ2 2m ∂2 ∂ ̃x2 + V ( ̃x). (C16) Direct calculation yields ˆΛ h Z dpdq FDe-(Sτ +Bτ +Cτ ) ħ i = Z dpdq ( FDe-(Sτ +Bτ +Cτ ) ħ " Jτ + ħ ̇ FD FD - ̇Sτ + V ( ̃x) #) , (C17) 11 where the dot above symbols means ̇ (...) ≡∂(...) ∂τ , (C18) and Jτ = ħ 1 τ + γ m -p2 τ 2m + 2 mpτ m [(x -q) -( ̃x -qτ)] τ -γ( ̃x -qτ) -2 m m [(x -q) -( ̃x -qτ)] τ -γ( ̃x -qτ) 2 + m τ 2 [(x -q) -( ̃x -qτ)]2 -( ̃x -qτ) ̇pτ +pτ ̇qτ + 2γ( ̃x -qτ) ̇qτ -2m τ [(x -q) -( ̃x -qτ)] ̇qτ. (C19) For the convenience of the following calculations, we define u = ̃x -qτ; (C20) v = x -q, (C21) and Φτ = -(Bτ + Cτ + Sτ); (C22) V (n)(z) = dn dzn V (z). (C23) Substituting Eqs. (C6, C3) into Eqs. (C17, C19), and using the above definitions and the fact V ( ̃x) = V (qτ) + ∞ X n=1 unV (n)(qτ), (C24) we find that Eq. (C17) can be re-expressed as ˆΛ h Z dpdq FDe-(Sτ +Bτ +Cτ ) ħ i = Z dpdq ( FD " Gτ + ħ ̇ FD FD + ∞ X s=2 1 s!usV (s)(qτ) # eΦτ /ħ ) , (C25) where Gτ = ħ 1 τ + γ m -2 m m (v -u) τ -γu 2 + m τ 2 (u -v)2 . (C26) 2. Eliminating u and v Now we eliminate the factors u and v in Eq. (C25). Using the fact ∂Sτ ∂q = -p + ∂qτ ∂q pτ; ∂Sτ ∂p = ∂qτ ∂p pτ, (C27) we find that that the factor Φτ defined in Eq. (C22) satisfies ∂Φτ ∂q ∂Φτ ∂p = Rqu Rqv Rpu Rpv u v . (C28) Here the parameters Rqu, Rqv, Rpu and Rpv are given by Rqu = -2 τ m + 2γ + 2 τ m ∂qτ ∂q -∂pτ ∂q ; (C29) Rqv = 2γ -2 τ m ∂qτ ∂q -1 ; (C30) Rpu = 2γ + 2 τ m ∂qτ ∂p -∂pτ ∂p ; (C31) Rpv = 1 -2 τ m∂qτ ∂p . (C32) 12 We further define another four parameters Tqu, Tqv, Tpu and Tpv via the relation: Tuq Tup Tvq Tvp = Rqu Rqv Rpu Rpv -1 . (C33) Thus, Eq. (C28) yields that u v = Tuq Tup Tvq Tvp ∂Φτ ∂q ∂Φτ ∂p . (C34) Note that these R- and T-parameters are simply the R-matrices and T-matrices defined in Eqs. (12-17) of Sec. (II), with N = 1, respectively. For convenience of the following calculations, we introduce differential operators Du and Dv: Du = Tuq ∂ ∂q + Tup ∂ ∂p; Dv = Tvq ∂ ∂q + Tvp ∂ ∂p, (C35) as well as the parameters Qαβ = Dα [β] ; (α, β = u, v). (C36) Substituting the definitions (C20, C21) of u and v into Eq. (C36), we further obtain the expressions of the Qparameters: Quu = -Tuq ∂qτ ∂q -Tup ∂qτ ∂p ; Quv = -Tuq; (C37) Qvu = -Tvq ∂qτ ∂q -Tvp ∂qτ ∂p ; Qvv = -Tvq. (C38) Using the differential operators Du and Dv, we can re-express Eq. (C34) as: αeΦτ /ħ= ħDα h eΦτ /ħi ; (α = u, v). (C39) Moreover, Eq. (C39) leads to αβeΦτ /ħ= ħαDβ h eΦτ /ħi = ħDβ h αeΦτ /ħi -ħeΦτ /ħDβ [α] , (α, β = u, v). (C40) Substituting Eqs. (C39) and (C36) into Eq. (C40), we further obtain αβeΦτ /ħ= ħ2Dβ n Dα h eΦτ /ħio -ħQβαeΦτ /ħ, (α, β = u, v). (C41) i.e., u2eΦτ /ħ= ħ2D2 u h eΦτ /ħi -ħQuueΦτ /ħ, (C42) v2eΦτ /ħ= ħ2D2 v h eΦτ /ħi -ħQvveΦτ /ħ, (C43) uveΦτ /ħ= ħ2DvDu h eΦτ /ħi -ħQvueΦτ /ħ. (C44) Repeating this technique, we can express any term of the form umvneΦτ /ħ(m, n = 1, 2, ...) as a series in ħ, with each coefficient taking the form C1C2 . . . [eΦτ /ħ], where each C1, C2, . . . is either a Q-factor or a D-operator. Using this approach and Eqs. (C37, C38), we can re-express Eq. (C25) as ˆΛ h Z dpdq FDe-(Sτ +Bτ +Cτ ) ħ i = ħ Z dpdq ( FD " ˆLτ + ̇ FD FD # eΦτ /ħ ) , (C45) 13 where ˆLτ = ˆL(0) τ + ħˆL(1) τ + ħ2 ˆL(2) τ + ... (C46) Here ˆL(0,1,2,...) τ are ħ-independent operators. Specially, we have ˆL(0) τ = 1 τ + γ m + 2γ2 m + m τ 2 + 4γ τ Quu + 2m τ 2 + 4γ τ Tuq -m τ 2 Tvq -1 2QuuV (2)(qτ), = gτ(q, p), (C47) and ˆL(1) τ = - 2γ2 m + m τ 2 + 4γ τ -1 2V (2)(qτ) D2 u + 2m τ 2 + 4γ τ DuDv -m τ 2 D2 v -1 3!V (3)(qτ) DuQuu + 2QuuDu + 3 4!V (4)(qτ)Q2 uu. (C48) Notice that gτ(q, p) is just the function given by Eq. (19) of Sec. II, with N = 1, 3. Equation of FD As in Ref. [13], by repeated integration by parts and using the fact that the integrated terms tend to zero for q →±∞or p →±∞, we can further re-express Eq. (C25) as ˆΛ h Z dpdq FDe-(Sτ +Bτ +Cτ ) ħ i = ħ Z dpdq eΦτ /ħˆ ̃Lτ[FD] , (C49) where ˆ ̃Lτ = ˆ ̃L(0) τ + ħˆ ̃L(1) τ + ħ2 ˆ ̃L(2) τ + .... (C50) Here ˆ ̃L(0,1,2,...) τ are also a group of ħ-independent operators. For instance, we have ˆ ̃L(0) τ = ∂ ∂τ + ˆL(0) τ = ∂ ∂τ + gτ(q, p), (C51) and ˆ ̃L(1) τ = - ̃D2 u 2γ2 m + m τ 2 + 4γ τ -1 2V (2)(qτ) + ̃Dv ̃Du 2m τ 2 + 4γ τ - ̃D2 v m τ 2 -1 3! Quu ̃Du + 2 ̃DuQuu V (3)(qτ) + 3 4!V (4)(qτ)Q2 uu, (C52) with ̃Du and ̃Dv being defined as ̃Du[...] = ∂ ∂q [Tuq...] + ∂ ∂p[Tup...]; ̃Dv[...] = ∂ ∂q [Tvq...] + ∂ ∂p[Tvp...]. (C53) Substituting Eq. (C49) and (C50) into Eq. (C15), we finally obtain Eq. (29) of of Sec. III, i.e., ˆ ̃L0 + ħˆ ̃L1 + ħ2 ˆ ̃L2 + ... FD = 0. (C54) Appendix D: Harmonic Oscillators In this appendix we show that our HK-like IVR for the Boltzmann operator is exact for harmonic oscillators. We first consider a single harmonic oscillator with frequency ω, and use the natural unit ħ= m = γ = 1. As shown 14 in Sec. IV, for this system we have V = ω2x2/2, and thus qη = ceωη + de-ωη and pη = ω(ceωη -de-ωη), where c = (p + ωq)/(2ω) and d = (ωq -p)/(2ω). Substituting these results into the definition of gτ, we find that gτ = --8eωτω + e2ωτ(-2 + ω) 4 -4ωτ + τ 2(-2 + ω)ω + (2 + ω) 4 + 4τω + τ 2ω(2 + ω) 2τ[-2 + eτω(-2 + ω) -ω] 4 + eτω -4 + τ(-2 + ω) + τ(2 + ω) . (D1) Thus, the function Dτ = eR τ 0 gηdη can be expressed as Dτ = 1 2 √ 2 s e-τω -2 + eτω(-2 + ω) -ω 4 + eτω -4 + τ(-2 + ω) + τ(2 + ω) ωτ . (D2) One can verify this result by substituting Eq. (D2) into the equation d dτ Dτ = -gτDτ and the condition Dτ=0 = 1. Substituting Eq. (D2) and the above expressions of qη and pη into Eq. (4), we find that the integration R dpdq... is just a Gaussian integration. Perform this integration analytically, we find that in the SI, the r. h. s. of Eq. (4) is just mω 2πħsinh(ωτ) 1/2 emω[(x2+ ̃x2) cosh(ωτ)-2x ̃x] 2ħsinh(ωτ) , i.e., the exact imaginary-time propagator of this harmonic oscillator. Furthermore, the above calculation can be directly generalized to the general cases with N harmonic oscillators, and we can find that for these cases the r. h. s. of Eq. (4) is Eq. (34) of Sec. IV.
|
2510.14752
|
Online Proportional Apportionment
Javier Cembrano∗
Jose Correa†
Svenja M. Griesbach ‡
Victor Verdugo§
Abstract
Traditionally, the problem of apportioning the seats of a legislative body has been viewed
as a one-shot process with no dynamic considerations. While this approach is reasonable for
some instances of the problem, dynamic aspects play an important role in many others. In
this paper, we initiate the study of apportionment problems in an online setting. Specifically,
we introduce an online algorithmic framework to handle proportional apportionment with no
information about future events. In this model, time is discrete and there are n parties that
receive a certain share of the votes at each time step. An online algorithm needs to irrevocably
assign a prescribed number of seats at each time, ensuring that each party receives its fractional
share rounded up or down, and that the cumulative number of seats allocated to each party
remains close to its cumulative share up to that time.
We consider deterministic and randomized online apportionment methods. For deterministic
methods, we construct a family of adversarial instances that yield a lower bound, linear in n,
on the worst-case deviation between the seats allocated to a party and its cumulative share.
We show that this bound is best possible and is matched by a natural greedy method. As a
consequence, a method guaranteeing that the cumulative number of seats assigned to each party
up to any step equals its cumulative share rounded up or down (global quota) exists if and only
if n ≤3. Then, we turn to randomized allocations and show that, when n ≤3, we can randomize
over methods satisfying global quota with the additional guarantee that each party receives, in
expectation, its proportional share in every step. Our proof is constructive: We show that any
method satisfying these properties can be obtained from a flow on a recursively constructed
network. We showcase the applicability of our results to obtain approximate solutions in the
context of online dependent rounding procedures for multidimensional instances.
∗Department of Algorithms and Complexity, Max Planck Institut für Informatik; Department of Industrial Engi-
neering, Universidad de Chile; jcembran@mpi-inf.mpg.de.
†Department of Industrial Engineering, Universidad de Chile; correa@uchile.cl.
‡Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile; Department of Computer Science,
RWTH Aachen University; sgriesbach@cmm.uchile.cl.
§Institute for Mathematical and Computational Engineering and Department of Industrial and Systems Engineer-
ing, PUC Chile; victor.verdugo@uc.cl.
arXiv:2510.14752v1 [cs.GT] 16 Oct 2025
1
Introduction
The proportional apportionment problem plays a paramount role in the structural electoral design
of modern democratic systems, capturing the task of allocating the seats of a representative body
(e.g., a parliament or constitutional assembly). Formally, an instance is given by a positive integer
vector v = (v1, . . . , vn) whose entries sum up to a positive integer value H. The goal is to find a
non-negative integer vector a = (a1, . . . , an) such that Pn
i=1 ai = H, and ai should be proportional
to vi. This problem is typically solved in the following electoral scenarios. In one of them, we have
a set of n states or districts, and each value vi represents the fraction of the population of state i.
In a second scenario, we have a set of n political parties, and each value vi represents the fraction
of votes obtained by party i.
Historically, the quality of an apportionment method has been measured according to whether
it satisfies a prescribed set of axioms or not. Among them, the quota property, requiring that each
party gets its quota rounded up or down, has played a prominent role since Alexander Hamilton,
the first US Secretary of the Treasury, proposed a solution to the apportionment problem in 1791,
later adopted in 1851. Hamilton’s method first calculates each party’s quota, immediately assigns
the floor of this number to the party, and then it goes through the parties in decreasing order of
their quota residue vi −⌊vi⌋and assigns one more seat to each party until the desired house size H
is met; by construction, Hamilton’s method satisfies the quota property. We refer to the books by
Balinski and Young [2010] and Pukelsheim [2017] for an extensive mathematical treatment of the
quota property and the axiomatic theory of proportional apportionment.
While the classic apportionment theory provides a thorough axiomatic characterization of pro-
portionality, it takes a static point of view in which solutions for an apportionment instance are
computed independently of the past and future events. Under this paradigm, when an apportion-
ment problem is solved repeatedly over time, rounding decisions made locally at each time do not
prevent systematic biases over the cumulative solutions. However, dynamic representation plays a
crucial role in the political organization of societies and the implementation of public policies [Stim-
son, MacKuen, and Erikson, 1995]. In addition to the natural application to elections that are
repeated after each term of office—as discussed by Gölz, Peters, and Procaccia [2025] in the context
of the sortition of the European Commission—the challenge of proportional apportionment over
time arises in contexts beyond political representation. For instance, the Indian reservation policy
mandates that publicly funded institutions set aside a fixed share of seats and jobs for designated
beneficiary groups, and entitlements that must be rounded to whole numbers [Evren and Khanna,
2021]. As institutions run successive recruitment rounds, they must ensure their total integral ap-
portionment allocations stay as close as possible to the prescribed cumulative (fractional) quotas;
this global goal is in tension with the allocation monotonicity of reserved seats over the recruitment
cycles. Similar problems are faced by institutions in other contexts, such as the distribution of
human resources, public services, and facilities across different organizational or geographical units.
The key problem behind the tension between the local and global quota requirements is that, al-
though we have access to historical data, we lack information about future apportionment instances;
a decision taken today can induce deviations from the cumulative quotas for the next period. Con-
sider the following example (introduced by Gölz et al. [2025]) with sequential elections to allocate
a single seat among four parties {1, 2, 3, 4}. In the first step, each party receives 1/4 of the votes,
so we can assume that a method assigns the seat to an arbitrary party, say party 1. In the second
step, parties 2, 3, and 4 get 1/3 of the votes each; we can again assign the seat arbitrarily, say to
party 2. If in the third step the votes are evenly distributed among parties 3 and 4, the one not
receiving the seat will have a cumulative entitlement strictly larger than 1 but no seat allocated.
Therefore, the global quota requirement is violated regardless of the choices made in the method.
1
1.1
Our Contribution
We introduce a modeling and online-algorithmic framework to handle proportional apportionment
problems under local and global rounding considerations with no information about future events.
In the following, we summarize the contributions of our work and discuss its consequences beyond
apportionment.
A model for online proportional apportionment.
We propose a new model for online ap-
portionment with proportional representation. In this model, there are n ∈N parties, and at each
time step t ∈N, they receive a fractional allocation of votes represented by a vector vt summing
to an integer Ht ∈N. We consider methods satisfying local quota, meaning that we allocate either
⌊vt
i⌋or ⌈vt
i⌉seats to each party i ∈[n]. Therefore, throughout the paper we assume without loss
of generality that vt ∈[0, 1)n. Upon observing vt, a method needs to irrevocably assign exactly Ht
seats to parties with strictly positive votes, granting one seat per selected party. Crucially, the
decision at time t can depend only on observed votes and allocations up to that time. Future votes
cannot be observed or influence the decision. Conversely, the votes can be set based on the previous
choices of the method.
We study which fairness properties can be achieved in this online setting. In particular, we focus
on three central notions that apply to each party and at each step: global quota (the total number
of allocated seats matches the floor or ceiling of the total number of votes), α-proportionality (the
total number of seats and the total number of votes differ by at most α), and ex-ante proportionality
(the expected total number of seats equals the total number of votes). While in the offline setting,
where the entire sequence of votes is known in advance, there exist allocation methods that satisfy
global quota (and therefore 1-proportionality) and ex-ante proportionality for any number of parties
(Proposition 1), we show that achieving similar guarantees in the online setting is significantly more
challenging.
Deterministic methods and approximate proportionality.
In Section 3, we study deter-
ministic allocation methods and introduce the greedy apportionment method, which in each step
assigns the Ht seats to the parties whose current number of allocated seats deviates most from
their cumulative share. As the first part of our main result (Theorem 1) we show that this method
is n−1
2 -proportional for any number of parties n ∈N, and even satisfies global quota when n ≤3.
The analysis builds on the closely related leaky-bucket problem, where Adler, Berenbrink, Friedet-
zky, Goldberg, Goldberg, and Paterson [2003] proved (Hn −1)-proportionality (with Hn denoting
the nth harmonic number). However, the fact that our setting only allows allocations to parties
with positive votes worsens the attainable guarantee from logarithmic to linear. For the second part
of Theorem 1, we show that this bound is tight by constructing a family of hard instances that,
for any ε > 0, rule out the possibility of achieving better than
n−1
2
−ε
-proportionality with any
deterministic method.
Randomized methods and ex-ante proportionality.
In Section 4, we turn to randomized
methods. The main result of this section (Theorem 2) is that an online apportionment method
satisfying both global quota and ex-ante proportionality exists if and only if n ≤3. Our proof is
constructive and based on a class of methods derived from a carefully designed flow network for each
time step, which encodes all constraints using the full history of votes and allocations. Moreover, we
show that any method satisfying global quota and ex-ante proportionality can be obtained by this
recursive flow construction, thereby fully characterizing the class of such methods (Proposition 2).
We conclude by briefly explaining why this construction breaks down when n ≥4.
2
Consequences in online dependent rounding.
Our results in Sections 3 and 4 translate di-
rectly to the setting of online dependent rounding for multidimensional instances, i.e., we receive
an n-dimensional marginal vector at each step instead of a single non-negative real value. In par-
ticular, the case of n = 1 party captures the online level-set problem studied very recently by Naor,
Srinivasan, and Wajc [2025]. In Section 5, we showcase the applicability of Theorem 1 and Theo-
rem 2 in the context of online optimization, and introduce a multidimensional generalization of the
multi-stage stochastic covering problem studied by Naor et al. [2025]. We describe how to achieve
online near-feasibility and approximation guarantees as a simple consequence of our results.
1.2
Further Related Work
Not only from a political viewpoint apportionment has been studied for centuries, but also its formal
mathematical treatment has a long history, as testified by the books of Balinski and Young [2010]
and Pukelsheim [2017]. Of particular relevance to our work is the quota axiom, analyzed in depth
by Balinski and Young [1975].
Our work departs from classic apportionment theory in two key ways: We consider online meth-
ods that allocate seats in sequential elections without information about future elections, and part
of our results concern randomized apportionment. While we initiate the study of online appor-
tionment, Evren and Khanna [2021] previously explored (offline) methods ensuring proportionality
in sequential allocation steps, motivated by public sector recruitment in India.
Their positive
results on monotone, proportional-in-expectation, and quota-approximating methods build upon
the work of Akbarpour and Nikzad [2020] on the decomposition of proportional fractional alloca-
tions into proportional-in-expectation and near-feasible integral allocations. On the other hand,
randomized apportionment has received increasing attention recently, decades after the founda-
tional work of Grimmett [2004]. Indeed, Gölz et al. [2025] established the existence of randomized
methods satisfying quota, house monotonicity, and expected proportionality; Cembrano, Correa,
Schmidt-Kraepelin, Tsigonias-Dimitriadis, and Verdugo [2025] studied randomized methods that
allow minor deviations from the house size to circumvent impossibility results; and Correa, Gölz,
Schmidt-Kraepelin, Tucker-Foltz, and Verdugo [2024] focused on monotonicity with respect to sub-
sets of parties. Aziz, Lev, Mattei, Rosenschein, and Walsh [2019] also considered lotteries over
deterministic apportionment methods in the context of strategyproof peer selection.
From a technical point of view, our work builds on a line of work applying optimization tech-
niques to apportionment. In particular, network flows have been exploited in the study of bidi-
mensional proportionality [Balinski and Demange, 1989a,b, Gaffke and Pukelsheim, 2008, Rote and
Zachariasen, 2007] and house-monotone apportionment [Cembrano et al., 2025]; Pukelsheim, Ricca,
Simeone, Scozzari, and Serafini [2011] provide an overview of these and other applications. Gölz
et al. [2025] also used linear programming techniques in the context of monotone apportionment.
Also, our positive result for n ≤3 generalizes the expected proportionality and quota properties
of a rounding scheme proposed by Naor et al. [2025] as an online version of the classic scheme by
Srinivasan [2001].
Also related to our setting is the bamboo garden trimming problem introduced by Gąsieniec,
Klasing, Levcopoulos, Lingas, Min, and Radzik [2017]. In this problem, each bamboo grows with
a fixed rate (analogous to parties receiving the same number of votes in each election). In each
step, one bamboo can be cut to the ground, and the objective is to minimize the maximum height
across all bamboos over time. The best-known approximation guarantees are due to Kawamura
[2024]. A generalization of this setting is captured by the cup game, often studied in the context
of load balancing [Liu and Layland, 1973] and deamortization [Dietz and Sleator, 1987].
Here,
an adversary distributes H fractional units of water across a subset of n cups, and the player
3
selects H cups from which to remove one unit of water each. This models a scenario with multiple
processors or allocations per round and has been analyzed against both oblivious and adaptive
adversaries [Bender and Kuszmaul, 2021, Bender, Farach-Colton, and Kuszmaul, 2019, Kuszmaul,
2020]. Our analysis of the deterministic greedy apportionment method is closely related to the leaky
bucket problem studied by Adler et al. [2003]. In this model, each bucket starts at the same water
level and leaks a fractional amount of water in each step. The goal is to assign integral units of
water to buckets in a way that maximizes the minimum fill level over time. This setting differs from
ours in a crucial aspect: Allocations can be made to any bucket, regardless of its leak rate (or, in
our interpretation, even to parties with zero votes). A more detailed comparison to our model is
provided in Section 3.
2
Instances, Axioms, and Offline Methods
Let N (resp. N0) denote the strictly positive (resp. non-negative) integers. Let [n] := {1, 2, . . . , n}
(resp. [n]0 := {0, 1, . . . , n}) denote the set containing the first elements of these sets up to n ∈N
(resp. n ∈N0). We identify [n] with a set of n parties. In each election t ∈N, party i ∈[n]
receives a fractional seat entitlement vt
i ∈[0, 1) based on the outcome of a vote;1 Ht := Pn
i=1 vt
i ∈N
denotes the total number of seats to be distributed in step t. We interpret vt
i directly as the votes
party i receives in election t and we write vt ∈[0, 1)n for the corresponding vector; an n-dimensional
instance is fully described by the sequence (vt)t∈N. While we define our general instances in the
online setting as infinite sequences to capture the fact that we do not know when they will end or
what the votes in the next steps will be, we restrict to a finite number of steps when we study offline
methods, as well as for the iterative construction of some methods and negative instances. We refer
to a vote sequence over a finite number of steps T, (vt)t∈{T0,...,T0+T}, as a partial n-dimensional
instance.
We now introduce the notion of an offline method and the two key axioms we study throughout
this paper. A deterministic offline apportionment method receives a partial n-dimensional instance
(vt)t∈[T] and outputs a sequence (Xt)t∈[T], where for every t ∈[T], Xt ⊂{i ∈[n] : vt
i > 0} and
|Xt| = Ht. In words, an offline apportionment method selects a subset of parties receiving the
Ht seats in each step, constrained to those parties receiving strictly positive votes. When the vote
sequence is fixed, we identify a method as the sequence (Xt)t∈[T] (and not as a function mapping
a vote sequence to a set sequence). A randomized offline apportionment method is a lottery over
deterministic offline apportionment methods.
For an instance (vt)t∈[T] and t ∈[T], we let V t
i := P
k∈[t] vk
i denote the votes received by party
i up to step t. For an instance (vt)t∈[T], a (deterministic or randomized) method (Xt)t∈[T], and
t ∈[T], we let At
i := |{k ∈[t] : i ∈Xk}| denote the (possibly random) number of seats received by
party i up to step t. A deterministic method (Xt)t∈[T] satisfies global quota on a certain instance
if, for every t ∈[T] and i ∈[n], it holds At
i ∈{⌊V t
i ⌋, ⌈V t
i ⌉}. A randomized method satisfies global
quota if all deterministic methods in its support satisfy global quota. In addition, a randomized
method satisfies ex-ante proportionality if P[i ∈Xt] = vt
i for every t ∈[T] and i ∈[n].
The following proposition states the existence of offline apportionment methods satisfying both of
the axioms introduced above. It follows from the existence of house monotone and quota compliant
methods in the classic (non-sequential) apportionment problem, shown via different approaches by
Balinski and Young [2010], Cembrano et al. [2025], Gölz et al. [2025].
1We recall that the restriction to vt
i ∈[0, 1) is without loss in the context of methods that satisfy local quota. This
is because, given any vote vector vt ∈Rn
≥0, any such method deterministically assigns each party its lower quota ⌊vt
i⌋,
which reduces the problem to allocating the remaining fractional seats vt
i −⌊vt
i⌋∈[0, 1) among the n parties.
4
u1
u2
u3
u4
u5
o
H1
H2
H3
H4
H5
w1
1
1
w1
2
1
w1
3
1
w2
1
1
w2
2
1
w2
3
1
w3
1
1
w3
2
1
w3
3
1
w4
1
1
w4
2
1
w4
3
1
w5
1
1
w5
2
1
w5
3
1
⌈V 1
1 ⌉
⌊V 1
1 ⌋
⌈V 1
2 ⌉
⌊V 1
2 ⌋
⌈V 1
3 ⌉
⌊V 1
3 ⌋
⌈V 2
1 ⌉
⌊V 2
1 ⌋
⌈V 2
2 ⌉
⌊V 2
2 ⌋
⌈V 2
3 ⌉
⌊V 2
3 ⌋
⌈V 3
1 ⌉
⌊V 3
1 ⌋
⌈V 3
2 ⌉
⌊V 3
2 ⌋
⌈V 3
3 ⌉
⌊V 3
3 ⌋
⌈V 4
1 ⌉
⌊V 4
1 ⌋
⌈V 4
2 ⌉
⌊V 4
2 ⌋
⌈V 4
3 ⌉
⌊V 4
3 ⌋
d
⌈V 5
1 ⌉
⌊V 5
1 ⌋
⌈V 5
2 ⌉
⌊V 5
2 ⌋
⌈V 5
3 ⌉
⌊V 5
3 ⌋
Figure 1: Illustration of the flow network used in the proof of Proposition 1 for T = 5 steps and
n = 3 parties. A single arc label represents an upper capacity; labels below and above an arc
correspond to lower and upper capacities, respectively.
Proposition 1. For every n ∈N and n-dimensional instance (vt)t∈[T], there exists an offline
apportionment method that satisfies global quota and ex-ante proportionality.
The proof of this proposition, which is included for completeness and can be found in Section A.1,
models apportionment methods via flows on a capacitated network, as illustrated in Figure 1.
This construction was given by Cembrano et al. [2025] to prove the existence of apportionment
methods satisfying house-monotonicity and quota compliance in a non-sequential setting, where
house monotonicity requires that an increase in the total number of seats to allocate does not cause
a drop in the number of seats any party receives. Specifically, we consider an origin o and vertices
(ut)t∈[T], where there is an arc (s, ut) with capacity Ht for each t ∈[T]. For each t ∈[T], there
are also vertices (wt
i)t∈[T],i∈[n] and arcs (ut, wt
i) with capacity 1 for each i ∈[n] such that vt
i > 0.
There are also arcs (wt
i, wt+1
i
) for every t ∈[T −1] and i ∈[n], with lower capacity ⌊V t
i ⌋and
upper capacity ⌈V t
i ⌉. Finally, there is a destination d and arcs (wT
i , d) for every i ∈[n], with lower
capacity ⌊V T
i ⌋and upper capacity ⌈V T
i ⌉. The flow on each arc (ut, wt
i) is interpreted as the number
of seats (0 or 1) that party i ∈[n] receives in step t, so that the flow on each arc (wt
i, wt+1
i
) (or
(wT
i , d) when t = T) corresponds to the total number of seats that party i has received up to step t.
Any (deterministic) apportionment method corresponding to a feasible flow on this network satisfies
global quota because of these arcs’ capacities. The fact that there exists a lottery over such methods
satisfying ex-ante proportionality follows from two facts: (i) the fractional flow that sends a flow
of vt
i on each arc (wt
i, wt+1
i
) (or (wT
i , d) when t = T) is feasible, and (ii) the network flow polytope
has integral extreme points. Hence, the proportional fractional apportionment can be obtained as
a distribution over deterministic apportionment methods satisfying global quota.
3
Deterministic Methods and Approximate Proportionality
In this section, we study deterministic online apportionment methods.
As our main result in
this context, we provide tight bounds on their worst-case deviations from exact proportionality,
5
understood as the maximum difference, in any step, between the number of seats and the cumulative
share that a party has received. Our findings have immediate implications regarding the axioms
introduced in Section 2: In contrast to the positive result for offline methods, an online method can
satisfy global quota if and only if there are at most three parties. A comparison to results on the
closely related leaky bucket problem highlights the difficulties of our setting due to the restriction
of only assigning seats to parties receiving positive votes in each step.
To formally present our results, we first formally define a method in the online setting.
A
(deterministic) online apportionment method selects, given an n-dimensional instance (vt)t∈N, a
subset of parties Xt ⊂[n] for every t ∈N as follows:
(i) For t = 0, we define V 0
i = v0
i := 0 and A0
i = a0
i := 0 for every i ∈[n].
(ii) For t ∈N, the history is given by the pair θt−1 := (V t−1, At−1) ∈Θt−1, where Θt−1 := Rn
+×Nn
0,
and we define the cumulative vote vector as V t−1
i
:= Pt−1
k=0 vk
i for every i ∈[n], and the
cumulative allocation vector as At−1
i
:= Pt−1
k=0 ak
i . The function Xt : Θt−1 × [0, 1)n →
[n]
Ht
maps the pair (θt−1, vt) to a subset Xt(θt−1, vt) ⊂{i ∈[n] : vt
i > 0} of size Ht, and the
allocation at ∈{0, 1}n is derived from this set: at
i = 1 if and only if i ∈Xt(θt−1, vt).
When the history and vote vector are fixed, we omit these arguments and use Xt to directly refer
to the set (instead of the function).
To quantify the deviations from the proportional number of seats that parties should receive,
we introduce the notion of approximate proportionality, which measures how close the number
of assigned seats to each party is to its cumulative votes. For α ≥0 and an n-dimensional in-
stance (vt)t∈N, an online apportionment method (Xt)t∈N is α-proportional if, for every t ∈N
and i ∈[n], the cumulative allocation satisfies |At
i −V t
i | ≤α.
We further say that a method
is strictly α-proportional if the above inequality holds strictly for every t ∈N and i ∈[n], i.e.,
|At
i −V t
i | < α. The following theorem is the main result of this section.
Theorem 1. Let n ∈N. There exists an online apportionment method that is n−1
2 -proportional
on every n-dimensional instance, and strictly 1-proportional on every 3-dimensional instance. Con-
versely, for every constant ε > 0, there exists an n-dimensional instance on which no online appor-
tionment method is
n−1
2
−ε
-proportional.
As a consequence of Theorem 1, we note the following observation regarding the global quota
axiom introduced in Section 2.
Corollary 1. There exists an online apportionment method satisfying global quota on every n-
dimensional instance if and only if n ≤3.
In the remainder of this section, we prove Theorem 1.
Before presenting our analysis, we
describe the closely related leaky bucket problem [Adler et al., 2003], using our notation to highlight
the parallels. The problem can be described as follows. A set of n buckets is initialized with a
common water level that we normalize to 0. In each step t ∈N, a volume vt
i ≥0 of water leaks
from each bucket i, such that the total leakage H := P
i∈[n] vt
i is an integer; note that we allow
negative loads. An online algorithm must decide how to pour back H unit-volume refills, with the
goal of maximizing the minimum load across all buckets and all time steps. Adler et al. showed
that a natural greedy algorithm, that refills the currently emptiest bucket one by one at each step,
achieves a worst-case deviation from the initial load of at most Hn−1 (where Hn is the nth harmonic
number), and that this guarantee is best possible.
6
While the leaky bucket problem is closely related to our setting, there is one key distinction:2 In
the apportionment setting, we can only assign seats to parties with positive votes, which corresponds
to only allowing refills to buckets that leaked in the current step. In Theorem 1, we show that,
perhaps surprisingly, this restriction significantly increases the worst-case deviation: it changes from
logarithmic to linear. Our lower bound is tight and is matched by a simple greedy method, which
we describe below.
The greedy apportionment method.
For an n-dimensional instance (vt)t∈N, we define the
surplus of party i at step t as st
i := At
i −V t
i , where a negative surplus indicates a deficit. The
surplus can be defined recursively by setting s0
i := 0 and st
i := st−1 + (at
i −vt
i) for every t ∈N. For
simplicity, we assume throughout this section that, in each step t, the parties are sorted from larger
to smaller surplus, i.e., st
1 ≥st
2 ≥· · · ≥st
n. This assumption is without loss of generality, since we
can rename parties after each step to maintain this invariant without affecting the surplus vector st.
For the positive direction of the theorem, we consider the natural greedy apportionment method
(Grt)t∈N, defined as follows.
For an n-dimensional instance (vt)t∈N and associated house sizes
(Ht)t∈N, the method assigns Ht seats in step t to the parties with the largest deficit up to step t−1
plus votes in step t; i.e.,
Grt(V t−1, At−1, vt) ∈arg min
(X
i∈S
(st−1
i
−vt
i) : S ⊂{i ∈[n] : vi > 0}, |S| = Ht
)
.
Each party i in the selected set is then assigned a seat in step t, i.e., at
i = 1 for every i ∈
Grt(V t−1, At−1, vt) and at
i = 0 for every other i. The following lemma establishes that this method
achieves the proportionality guarantees stated in Theorem 1.
Lemma 1. The greedy apportionment method is n−1
2 -proportional on every n-dimensional instance
for every n ∈N, and strictly 1-proportional on every 3-dimensional instance.
For the proof of this result, which is deferred to Section B.1, we adapt the arguments by Adler
et al. [2003], who established a worst-case deviation of Hn −1 for the greedy algorithm in the leaky
bucket problem. We outline the key ideas used to bound the maximum surplus; the bound on the
deficit follows from analogous reasoning.
For each step t ∈N0 and index ℓ∈[n], we define γt
ℓas the average of the ℓlargest surpluses at
time t, plus ℓ/2. Observe that (i) γt
n = n/2 for every t; and (ii) at step 0, the sequence (γ0
ℓ)ℓ∈[n] is
strictly increasing. Now suppose, for the sake of contradiction, that at some step t the maximum
surplus is strictly larger than (n −1)/2. Then γt
1 > n/2, implying that there must exist some i ∈
[n−1] for which γt
i > γt
i+1. Let t and i be the earliest time and smallest index for which this occurs.
We now derive a contradiction by analyzing the change in surplus from step t −1 to t. On the
one hand, the total surplus of the i parties with the largest surplus at step t must have increased
from t −1 to t, implying that some of these parties received a seat in step t, despite the presence of
a party outside this set that also received positive votes. This suggests that the ith and (i + 1)th
largest surpluses are not too far apart; otherwise, the greedy method would have allocated a seat to
the party outside the set of the i parties with the largest surplus. On the other hand, by the choice
of i and the definition of γt
ℓ, the gap between the ith and (i + 1) the largest surpluses must be large
enough to ensure that γt
i−1 < γt
i, leading to a contradiction.
2In fact, Adler et al. [2003] consider the deviation from the minimum load after the water leaks but before the
algorithm refills the buckets. However, as noted in their work, this formulation is equivalent to ours up to an additive
shift of H.
7
To establish strict approximate proportionality for n = 3, we examine the only borderline case
that does not lead to a contradiction by the argument above, namely i = 1. Here, we apply a
refined averaging argument and show that both the surplus and the deficit of some party would
simultaneously exceed 1, contradicting the choice of the greedy method in the previous step. We
note that such a contradiction does not arise for other values of n; for instance, when n = 2, one
can construct a simple instance in which both parties attain a deviation of exactly 1/2.
Construction of the hard instances.
The proof of the negative direction of Theorem 1 is
considerably more involved than the leaky bucket case and is based on a careful iterative construction.
We consider instances in which each vote vector corresponds to a two-party election, i.e., the vector vt
contains exactly two non-zero entries for every t ∈N. The lower bound construction relies on a
key property of such instances: Whenever two parties have surpluses that differ by less than one, a
two-party election between them can be used to split their surpluses so that they differ by exactly
one, while preserving their average surplus. This simple yet powerful observation forms the basis of
our argument.
Lemma 2. Let n ∈N, let (vk)k∈[t] be a partial n-dimensional instance, and let V t denote the
aggregate votes up to step t. Let (Xk)k∈N be any online apportionment method and At the cor-
responding cumulative allocation up to step t.
Then, for all parties i, j ∈[n] with i ̸= j such
that 0 ≤st
i −st
j < 1, there exists a vote vector vt+1 ∈[0, 1)n such that, after the allocation at+1
corresponding to Xt+1(V t, At, vt+1),
|st+1
i
−st+1
j
| = 1,
st+1
i
+ st+1
j
2
=
st
i + st
j
2
.
For the proof, which is deferred to Section B.2, we construct a two-party election vt+1 satisfy-
ing st
i −vt+1
i
= st
j −vt+1
j
. This ensures that, regardless of the allocation method, one of the two
parties will end up with a surplus exactly one unit larger than the other after step t + 1. As a
direct consequence, this implies that 1
2-proportionality is tight and cannot be improved (even by a
non-constant value) for instances with n = 2 parties.
In what follows, whenever two parties i and j satisfy the conditions of Lemma 2 for a given
history and apportionment method, we denote the corresponding vote vector by ˆv(i, j) and refer to
it as the (i, j)-splitter.
We now state a lemma that implies the lower bound in Theorem 1, and in fact yields a more
general result. It states that, for any partial n-dimensional instance up to some step r, any method,
and any subset of n′ parties, there exist t vote vectors that can be added to the partial instance
such that the surplus or deficit of some party in this subset is arbitrarily close to (n′ −1)/2.
Lemma 3. Let n, n′ ∈N with n′ ≤n and P ⊆[n] with |P| = n′ be a subset of n′ parties
in [n]. Let r ∈N0 and ε > 0 be fixed, and let (vk)k∈[r] be a partial n-dimensional instance. Let
(Xt)t∈N be any online apportionment method and Ar the corresponding cumulative allocation up to
step r. Then, there exists a finite value t ∈N with t ≥r and a vote sequence (vk)k∈{r+1,...,t} such
that, after the allocation at corresponding to Xt(V t−1, At−1, vt), there exists a party i ∈P such
that |st
i| ≥(n′ −1)/2 −ε.
The proof of this lemma is deferred to Section B.3, but we outline the main ideas here. The
result is proven by induction on n′ in steps of size 2. The base case n′ = 1 is trivial. For n′ = 2,
the claim follows from the application of a single splitter: If the two parties do not already differ in
surplus by at least 1, the splitter can be used to separate their surpluses accordingly.
8
0
1
2
3
4
5
6
7
−1
−0.5
0
0.5
1
t
At
i −V t
i
(a) For n = 3, a surplus of 63/64 and a deficit of
127/128 are reached up to t = 7.
0
1
2
3
4
5
6
7
8
9 10
−1.5
−1
−0.5
0
0.5
1
1.5
t
At
i −V t
i
(b) For n = 4, a surplus and a deficit of 11/8 are
reached up to t = 10.
Figure 2: Illustration of adversarially constructed partial instances to induce surpluses and deficits
close to (n −1)/2, for n ∈{3, 4}. At each step t ≥1, an (i, j)-splitter is applied for parties i, j
whose surpluses then split by one unit at the next step, while keeping the average constant. When
multiple parties have the same surplus at some step, they are depicted with a small horizontal offset
for ease of understanding.
For the inductive step, consider a set P of n′ + 2 parties. We apply the inductive hypothesis to
a subset of n′ parties in P, excluding the one with the largest surplus and largest deficit. Repeating
this process at most three times, we reach a situation in which two parties in P have surplus (or
deficit) arbitrarily close to (n′ −1)/2, say, at least (n′ −1 −ε)/2.
For simplicity, assume that two parties in P have a surplus (and not a deficit) arbitrarily close
to (n′ −1)/2. Applying a splitter to these two parties increases the surplus of the one with the
larger surplus and decreases that of the other. In particular, after this splitter, one of the two must
have a surplus of at least (n′ −1 −ε)/2 + 1/2.
To continue increasing this value, another splitter is needed. However, since the other party’s
surplus is now one unit smaller, a direct application is not feasible. Instead, we apply the inductive
hypothesis once again to a subset of n′ parties in P, again excluding the two with the largest and
smallest surplus. As a result, at least one party has a surplus or a deficit exceeding (n′ −1 −ε)/2.
In the former case, we are again in the position to apply a splitter to the two parties with the
largest surpluses, which this time will leave a party with a surplus of at least (n′ −1 −ε)/2 + 3/4.
Otherwise, we now have two parties with a deficit exceeding (n′ −1 −ε)/2, and we apply a splitter
between them.
By repeatedly applying the induction hypothesis and a subsequent splitter to the surplus or
deficit end, we increase the maximum deviation.
In particular, the deviation increases by at
least 1/2ℓafter the ℓth iteration of this procedure at the corresponding end. After finitely many
steps, this process yields a surplus or deficit of at least (n′ + 1)/2 −ε. Figure 2 illustrates the
repeated application of splitters to get a surplus or deficit approaching (n −1)/2, for n ∈{3, 4}.
Relation to standard apportionment methods.
When the same vote vector is repeated in
every time step, an online apportionment method naturally corresponds to a house-monotone
(offline) method, and vice versa.
Interestingly, the greedy apportionment method introduced
9
in this work does not coincide with any of the standard offline rules.
Although it might ap-
pear similar to the Hamilton method, as both allocate seats sequentially to the parties with
the largest deficit, this equivalence does not hold. For instance, for three parties with vote vec-
tor vt = (2/3, 29/120, 11/120), our greedy method yields the allocation (4, 2, 1) after seven steps,
whereas the Hamilton method with seven seats outputs (5, 2, 0). This discrepancy is closely related
to the Alabama paradox, as the Hamilton method may withdraw a seat from a party when the
house size increases, while an online method cannot revoke past allocations.
The Balinski-Young quota method, in contrast, satisfies both house-monotonicity and quota
compliance and could therefore be adapted to operate in an online fashion while respecting global
quota at every step. This is particularly noteworthy since our greedy method, although optimal in
the worst case, does not exhibit this property in this special setting. To see this, consider three large
parties and two small ones, with repeated vote vector vt = (49/150, 49/150, 49/150, 1/100, 1/100).
After 41 steps, the cumulative allocation is (13, 13, 13, 1, 1). At time t = 42, the next seat is assigned
to one of the large parties, as their deficits exceed those of the small ones. However, at t = 43 the
other two large parties both have remainders 43·49/150−13 = 157/150 > 1, forcing both to receive
an additional seat to maintain global quota.
4
Randomized Methods and Ex-ante Proportionality
We now shift our focus to randomized online apportionment methods. We introduce the formal
notion of such a method in what follows. A randomized online apportionment method selects, given
an n-dimensional instance (vt)t∈N, a random subset of parties Xt ⊂[n] for every t ∈N, as follows:
(i) For t = 0, we define V 0
i := 0 and A0
i := a0
i := 0 for every i ∈[n].
(ii) For t ∈N, we define the cumulative vote vector as V t−1
i
:= Pt−1
k=0 vk
i for every i ∈[n],
where v0 := 0, and the cumulative allocation vector as At−1
i
:= Pt−1
k=0 ak
i . The set of feasible
histories Θt−1 is given by all histories (V t−1, At−1) that occur with strictly positive probability
under the method up to step t −1. There is a function
yt : Θt−1 × [0, 1)n ×
[n]
Ht
→[0, 1],
such that for each feasible history (V t−1, At−1) ∈Θt−1, yt(V t−1, At−1, vt, ·) is a probability
distribution over subsets of [n] of size Ht and yt(V t−1, At−1, vt, S) = 0 for every S such
that there is a party i ∈S with vt
i = 0. The outcome in step t is then given by a random
subset Xt(V t−1, At−1, vt) distributed according to yt(V t−1, At−1, vt, ·), and the allocation at =
at(V t−1, At−1, vt) ∈{0, 1}n is derived from this set: at
i = 1 if and only if i ∈Xt(V t−1, At−1, vt).
If the method (yt)t∈N is only defined up to a finite time T ∈N, we refer to it as a partial
online apportionment method. Given a method (yt)t∈N and an instance (vt)t∈N, let At := {At :
(V t, At) ∈Θt} be the set of cumulative allocation vectors that occur with strictly positive probability
after step t. Moreover, let π(yt, V t, At) denote the probability of reaching allocation At at step t
under method (yt)t∈N.
We initialize with A0 := {0}, V 0 := 0, and π(y0, V 0, A0) := 1.
Now,
ex-ante proportionality translates to the randomized setting as follows. An online apportionment
method (yt)t∈N is ex-ante proportional if for every step t ∈N and party i ∈[n], the expected number
of seats assigned to i is equal to its share, i.e.,
X
At−1∈At−1
π(yt−1, V t−1, At−1) · at
i(V t−1, At−1, vt) = vt
i.
(1)
10
We call a partial online apportionment method (yt)t∈[T] ex-ante proportional if it satisfies Equa-
tion (1) for every step t ∈[T].
Corollary 1 states that an online apportionment method can satisfy global quota for all n-
dimensional instances (vt)t∈N if and only if n ≤3.
The main result of this section establishes
that, for n ≤3, we can randomize over methods satisfying global quota while additionally fulfilling
ex-ante proportionality, thus giving ex-ante and ex-post proportionality guarantees.
Theorem 2. There exists an online apportionment method satisfying global quota and ex-ante pro-
portionality for all n-dimensional instances if and only if n ≤3.
We remark that the impossibility for n ≥4 follows directly from Corollary 1. On the other hand,
when n = 2, a simple construction suffices to prove the positive direction. Indeed, we can apply
the systematic sampling procedure (a.k.a., Grimmett’s apportionment method [2004]) to party 1,
defined as follows: Sample a value λ ∈[0, 1) uniformly at random and assign a seat to party 1
in step t if and only if ⌊V t−1
1
+ λ⌋< ⌊V t
1 + λ⌋. This method satisfies global quota and ex-ante
proportionality.3 To obtain an online apportionment method for two parties, we can simply assign
the seat in step t to party 2 whenever party 1 does not receive it. Since vt
2 = 1 −vt
1 in every step,
ex-ante proportionality for party 1 directly implies ex-ante proportionality for party 2. Moreover,
global quota is also satisfied: Whenever one party must be allocated a seat by the global quota
constraint, the other party must not be allocated a seat due to the complementary constraint.
For three parties, however, simple constructions no longer suffice due to weaker dependencies
between the global quota constraints of different parties. Consider, for example, an instance (vt)t∈N
where the first vote vector is v1 = (0.6, 0.3, 0.1) and suppose that in a particular realization, the
seat in step 1 was allocated to party 1. Now let v2 = (0, 0.8, 0.2). Since party 3 did not receive
a seat in step 1 and v2
3 > 0, global quota implies no restriction on whether it may or may not
receive a seat in step 2. At the same time, ex-ante proportionality requires that party 3 is assigned
a seat with probability 0.2. However, allocating a seat to party 3 under the current allocation would
violate global quota for party 2, which also has not received a seat in step 1 but is surpassing its
previous upper share in step 2. This illustrates that, for n = 3, ensuring both global quota and
ex-ante proportionality requires tracking and conditioning on the full history of seat allocations.
Consequently, designing valid online apportionment methods becomes significantly more intricate
than in the two-party case.
We overcome this challenge and prove Theorem 2 by introducing a family of randomized online
apportionment methods that we refer to as network flow methods. These methods are defined recur-
sively and rely on constructing a suitable flow network in each step. Consider an instance (vk)k∈N
and a partial online allocation method (yk)k∈[t] that satisfies global quota and ex-ante proportion-
ality. For any cumulative allocation At ∈At, define the set of parties that have received their upper
share of seats after step t as
u(At) :=
n
i ∈[n] : At
i =
l Pt
k=1 vk
i
mo
.
Let Ut := {u(At) : (V t, At) ∈Θt} be the set of all such sets observed with strictly positive probability
up to step t. With a slight overload of notation, we define
π(u) :=
X
At∈At:u(At)=u
π(yt, V t, At)
as the probability that the upper-quota set u ∈U t occurs after step t. We now define the flow
network FN(V, A, v, π), which constitutes the basis of our recursive construction.
3We remark that this observation was also made by Naor et al. [2025] in the context of online dependent rounding;
in Section 5, we further develop the consequences of our results for dependent rounding algorithms in online settings.
11
Flow network construction.
Let (vk)k∈N be an n-dimensional instance and (yk)k∈[t] a partial
online apportionment method that satisfies global quota and ex-ante proportionality up to step t ∈
N. Define U := {u(A) : (V, A) ∈Θt}. The corresponding flow network FN(V, A, v, π) for step t + 1
is given by
(a) vertex set P := {o, d} ∪U ∪[n],
(b) edge set E := {(o, u) : u ∈U} ∪{(u, i) : u ∈U, i ∈[n]} ∪{(i, d) : i ∈[n]}, and
(c) upper and lower capacities c, ℓ: E →R+ given by
c((o, u)) = π(u),
c((u, i)) =
(
0
if i ∈u and ⌈Vi⌉= ⌈Vi + vi⌉or vi = 0
π(u)/Pn
i=1 vi
else,
c((i, d)) = vi/Pn
i=1 vi,
ℓ((o, u)) = 0,
ℓ((u, i)) =
(
π(u)/Pn
i=1 vi
if i /∈u and ⌊Vi⌋+ 1 = ⌊Vi + vi⌋,
0
else,
ℓ((i, d)) = 0.
We illustrate the construction of a network flow method by the following example. Consider an
instance with three parties, and suppose that in the first election the votes are given by v1 =
(0.6, 0.3, 0.1), which yields H1 = 1. In this case, the unique partial network flow method at t = 1
is defined by
y1(V 0, A0, v1, {i}) = v1
i
for all i ∈[3],
which implies the edge capacities c(o, {i}) = π({i}) = v1
i in the flow network FN(V 1, A1, v2, π).
Now consider the second vote vector v2 = (0.5, 0.2, 0.3) with H2 = 1. The global quota property
enforces that party 1 must receive a seat in step 2 if it was not allocated one in round 1, as ⌊V 1
1 ⌋+1 =
⌊V 1
1 + v2
1⌋. Conversely, for each party i ̸= 1, the condition ⌈V 1
i ⌉= ⌈V 1
i + v2
i ⌉implies that they must
not be allocated a seat in round 2 if they were already selected in round 1. These constraints are
encoded in the capacities of the edges e = (u, i) in the flow network. The current vote vector v2
also determines the capacities c((i, d)) = v2
i for every i ∈[3].
The complete flow network is depicted in Figure 3. Edges with zero upper capacity are omitted,
and edges with equal lower and upper capacities are highlighted in red. The network admits a
unique feasible (o, d)-flow f of value 1, given by
f(e) =
0.1
if e = ({1}, 1),
0.2
if e = ({1}, 2),
0.3
if e = ({1}, 3),
0
if e = ({2}, 3) or e = ({3}, 2), and
c(e)
otherwise.
For a detailed analysis of feasibility, we refer to Case 1.1.2 in the proof of Lemma 5 in Section C.2.
We now describe how to construct a partial network flow method (yk)k∈[t] recursively for any
number of parties n.
12
o
2
1
3
1
2
3
d
0.6
0.3
0.1
0.5
0.2
0.3
0.1
0.2
0.3
0.3
0
0.1
0
Figure 3: The flow network FN(V 1, A1, v2, π) for v1 = (0.6, 0.3, 0.1) and v2 = (0.5, 0.2, 0.3). Edges
with upper capacity zero are omitted. The upper capacities of the remaining edges are c((o, {i})) =
π({i}) = v1
i , c(({i}, j)) = π({i})/H1 = v1
i , and c((i, d)) = v2
i . An edge is highlighted in red if the
lower capacity equals the upper capacity. The lower capacities of the remaining edges are zero. The
labels indicate the unique flow of value 1.
Step t = 1.
We define y1 as a probability distribution over subsets of [n] of size H1. Since vector v1
lies in the hypersimplex ∆(n, H1), there exists a convex decomposition v1 = Pm
j=1 λj · zj, where
each zj ∈{0, 1}n has exactly H1 ones and λ1, ..., λm ∈[0, 1] with Pm
j=1 λj = 1. Let Zj ⊆[n] denote
the support of zj, i.e., i ∈Zj if and only if zj,i = 1. We then define
y1(V 0, A0, vt, Zj) = λj
for every j ∈[m] and
y1(V 0, A0, vt, S) = 0
for all other subsets S ⊆[n].
Step t ≥2.
Assume we have already constructed a partial network flow method (yk)k∈[t]. Let V t,
and At denote the cumulative vote and allocation vectors, and π(u) the distribution over upper-
quota sets u ∈Ut. Construct the flow network FN(V t, At, vt+1, π) for step t + 1. If this network
admits a feasible (o, d)-flow f of value 1, we can extend the partial method to step t + 1 as follows.
Fix a feasible allocation At that occurs with strictly positive probability. Let u = u(At) denote
the corresponding upper-quota set. Using flow f, we define the fractional assignment vector z(u) ∈
[0, 1]n by
zi(u) := f(u, i) · Ht+1
π(u) .
By construction, z(u) lies in the hypersimplex ∆(n, Ht+1). Again, decompose z(u) = Pm
j=1 λj · zj
into a convex combination of {0, 1}-vectors zj ∈{0, 1}n with exactly Ht+1 ones and let Zj be their
supports. We define
yt+1(V t, At, vt+1, Zj) = λj
for every j ∈[m] and
yt+1(V t, At, vt+1, S) = 0
for all other subsets S ⊆[n].
This completes the construction of the partial network flow method.
We now show that a partial network flow method satisfies all three desired properties.
13
Lemma 4. Let (yt)t∈[T] be a partial network flow method. Then, for every t ∈[T] and every feasible
history (V t, At) ∈Θt that occurs with strictly positive probability, we have At
i ∈{⌊V t
i ⌋, ⌈V t
i ⌉} for
every i ∈[n]. Furthermore, (yt)t∈[T] satisfies Equation (1) for every t ∈[T].
The proof uses induction to show that a partial network flow method satisfies global quota and
ex-ante proportionality by construction. We verify that only parties with strictly positive votes are
assigned seats and that these lie within the share bounds using the structure of the network. The
flow conservation finally yields that the expected number of seats assigned in each step matches the
vote vector. The full proof is deferred to Section C.1.
If, for every t ∈N, the corresponding flow network FN(V t, At, vt+1, π) admits a flow of value 1,
the recursively defined partial network flow method (yt)t∈[T] extends to a network flow method
(yt)t∈N. By Lemma 4, such a method satisfies global quota and ex-ante proportionality. The next
lemma guarantees that, for any number of parties n ≤3 and partial network flow method (yt)t∈[T],
the corresponding flow network FN(V T , AT , vT+1, π) always admits a feasible flow of value 1. Thus,
for any T ∈N, the partial network flow method (yt)t∈[T] can be extended to a partial network
flow method (yt)t∈[T+1]. Consequently, a partial network flow method yields a valid network flow
method.
Lemma 5. Let n ≤3 and let (yk)k∈[t] be a partial network flow method with corresponding flow
network FN(V t, At, vt+1, π) for step t + 1. Then this flow network admits a feasible (o, d)-flow of
value 1, i.e., the partial network flow method can be extended to (yk)k∈[t+1].
The proof of Lemma 5 is deferred to Section C.2. It proceeds by a case distinction over the
possible configurations of cumulative allocation vectors and vote increments for the three parties.
Feasibility is shown either by explicitly constructing a valid flow or by reducing the network to
a transshipment problem on a bipartite graph. This ensures that in every case the flow network
admits a feasible flow of value 1, completing the recursive construction. Theorem 2 now follows by
combining Lemmas 4 and 5.
Having shown that network flow methods satisfy all desired properties and are well-defined
for up to three parties, we now establish that this family of methods in fact contains all online
apportionment methods satisfying all three properties.
Proposition 2. Any online apportionment method satisfying global quota and ex-ante proportion-
ality is a network flow method.
In the proof, we construct a network flow for each step that replicates the decisions of a given
online apportionment method satisfying global quota and ex-ante proportionality. We verify that
all capacity constraints are met, flow conservation is attained, and that the flow has a value of 1,
which establishes that any such method can be obtained as a network flow method. The full proof
is deferred to Section C.3.
With the preceding proposition, we conclude that every online apportionment method satisfying
global quota and ex-ante proportionality is a network flow method and can therefore be obtained
via the construction described above. Indeed, the proof of Lemma 5 provides an explicit recursive
procedure for building the method as it defines the flow network at each step t based on the current
allocation and voting vectors.
For the case of n = 3 parties, it is worth mentioning that in all but two cases, the resulting flow
of value 1 is uniquely determined. Consequently, the seat assignment may only differ when either
(a) one seat is to be allocated and each cumulative allocation u ∈U t assigns the upper share of
seats to a single party, while in the current step, no party receives enough votes to increase
their upper share, or
14
(b) two seats are to be allocated, and each cumulative allocation u ∈U t assigns the upper share
of seats to two parties, while in the current step, all parties receive enough votes to increase
their upper share.
In both cases, the flow network admits multiple feasible flows of value 1, leading to different valid
seat allocations.
We conclude this section by discussing barriers to further positive results in the realm of ran-
domized online apportionment methods.
Network construction and impossibility for n ≥4.
The following example illustrates why the
flow construction fails when n ≥4, as the corresponding network no longer admits a feasible flow of
value 1. Consider an instance with four parties and voting vector v1 = (1/2, 1/2, 1/2, 1/2). A partial
method assigns H1 = 2 seats in step 1. Without any loss, assume that y1(V 0, A0, v1, {1, 2}) = p > 0,
i.e., parties 1 and 2 obtain a seat in the same allocation with strictly positive probability.
Now consider the voting vector v2 = (1/2, 1/2, 0, 0). In this case, for all i ∈[n], we have ⌈V 1⌉=
⌈V 1 +v2⌉, so global quota implies that no party may receive an additional seat in step 2 if parties 1
and 2 received a seat each in step 1. The flow network captures this by setting the upper capacity
of all edges ({1, 2}, i) to zero. Consequently, the edge (o, {1, 2}) must be assigned a flow of value 0
even though it has a strictly positive capacity of p. Since the total capacity of the outgoing edges
of o equals 1, no flow of value 1 can exist. This demonstrates that the construction of a partial
network flow method fails at step t = 2.
Negative correlation.
Naor et al. [2025] showed that, when n = 1, a notion of negative correla-
tion over time can be added on top of proportionality notions. In our notation, negative correlation
requires that for every party i ∈[n] and every pair of time steps t, t′ ∈N, P[at
i = 1 | at′
i = 1] ≤
P[at
i = 1]. The result of Naor et al. implies that, when n = 1, there exists an online apportionment
method satisfying global quota, ex-ante proportionality, and negative correlation.4 It is easy to see
that this can be immediately extended to n = 2 parties, since one party receives a seat if and only if
the other does not. However, it does not extend to three parties. To see this, consider the following
instance with vote vectors v1 = (0.5, 0.5, 0), v2 = (0, 0.6, 0.4), v3 = (0, 0.2, 0.8). The outcome of the
unique network flow method yields the following distribution over assignments:
P[a1
1 = 1, a2
2 = 1, a3
3 = 1] = 0.5,
P[a1
2 = 1, a2
2 = 1, a3
3 = 1] = 0.1,
P[a1
2 = 1, a2
3 = 1, a3
2 = 1] = 0.2,
P[a1
2 = 1, a2
3 = 1, a3
3 = 1] = 0.2.
In particular, we observe that P[a3
2 = 1|a1
2 = 0] = 0 < 0.2 = P[a3
2 = 1], which shows that negative
correlation is violated.
Oblivious adversary.
The impossibility result for instances with more than three parties relies on
an adaptive adversary. A natural question is how likely a method is to violate the quota axiom when
facing an oblivious adversary. Using the same four-party instance mentioned in the introduction,
Gölz et al. [2025] showed that any method violates quota with probability at least 1/12 since the
undesirable allocation from the first two steps arises with this probability. More generally, repeating
this four-party construction ⌊n/4⌋times yields a violation probability of at least 1−
11
12
⌊n/4⌋, which
converges to 1 as n increases.
4In fact, Naor et al. used stronger forms of negative correlation, but we here explain why even this weaker notion
is not possible with three parties.
15
5
Consequences in Online Dependent Rounding
Our results in Sections 3 and 4 translate directly to the setting of online dependent rounding for
multidimensional instances, i.e., we receive an n-dimensional marginal vector at each step instead of
a single non-negative real value. In particular, the case of n = 1 party captures the online level-set
problem studied very recently by Naor et al. [2025]. Our Theorem 2 shows the existence of an online
dependent rounding algorithm with no deviation from the rounded cumulatives (global quota),
and meeting the marginals for 3-dimensional instances (ex-ante proportionality). Furthermore, in
Proposition 2 we show that all such algorithms can be recovered via our network flow construction.
On the other hand, when we relax the condition of meeting the marginals on expectation, our greedy
apportionment algorithm from Theorem 1 is an online rounding algorithm for general n-dimensional
instances that achieves optimal deviations from the rounded cumulatives.
In the rest of this section, and to showcase the applicability of our results in the context of online
optimization, we introduce a multidimensional generalization of the multi-stage stochastic covering
studied by Naor et al. [2025], and we describe how to achieve near-feasibility and approximation
guarantees as a simple consequence of our rounding results.
5.1
Multidimensional Multi-stage Hypergraph Stochastic Covering
In the multidimensional multi-stage hypergraph stochastic covering problem (MMHSC), we are
given a hypergraph (N, A) with vertices N and hyperedges E, and the hypergraph is d-uniform
with d ≥2, i.e., the size of each hyperedge is equal to d. We have n types of resources, denoted by
[n], which are to be allocated on each step (stage) t ∈[T]; the capacity of a node u during stage t
is a non-negative integer number denoted by C(u, t). On the other hand, each hyperedge e ∈E has
a demand of D(i, t) for each resource i and each step t, and D(i, ·) is monotone non-decreasing. A
fractional solution for this problem corresponds to an allocation y: N × [n] × [T] →R+ satisfying
the following:
X
u∈e
t
X
ℓ=1
y(u, i, ℓ) ≥D(i, t)
for each i ∈[n], t ∈[T], and e ∈E,
(2)
n
X
i=1
y(u, i, t) ≤C(u, t)
for each u ∈N and t ∈[T].
(3)
Constraints (2) ensure that the total cumulative allocation for a hyperedge e is at least the demand
D(i, t) for each resource i and step t, and (3) guarantee that on each step the total number of goods
allocated to a vertex u does not exceed the capacity C(u, t). Then, given a fractional solution y⋆,
at each step t ∈[T] the decision-maker receives the values y⋆(u, i, t), i.e., the solution restricted to
t. The goal of the decision-maker is to round this solution online, i.e., at each step t, each value of
the solution restricted to t must be rounded to the floor or ceiling, and at the same time provide
guarantees on the solution quality, i.e., objective value or feasibility. In particular, our problem
captures the multi-stage stochastic hypergraph cover problem by Naor et al. [2025] when n = 1,
D(1, t) = 0 for each t < T, D(1, T) > 0, and C(u, t) = ∞for each u and t. We remark that Byrka
and Srinivasan [2018] study the graph case (i.e., d = 2) and developed a 2-approximation algorithm,
improving on the 2T-approximation by Swamy and Shmoys [2012].
In what follows, we show how to use Theorem 1 and Theorem 2 to construct integer solutions
for MMHSC, online, with guarantees in terms of near-feasibility or objective approximation factor.
For the sake of exposition simplicity, suppose that the fractional solution y⋆is binding on every
constraint (3), i.e., there is no spare capacity over the vertices at every step.
16
Online near-feasible solutions.
We use Theorem 1 to construct an online near-feasible solution
for MMHSC. For each vertex u and every step t, consider vt
i(u) = y⋆(u, i, t) −⌊y⋆(u, i, t)⌋for each
resource i. At every t, for each vertex u we use Theorem 1 to get binary values at
1(u), . . . , at
n(u)
satisfying the following properties:
(i) Pn
i=1 at
i(u) = C(u, t) −Pn
i=1⌊y⋆(u, i, t)⌋,
(ii) | Pt
ℓ=1 aℓ
i(u) −Pt
ℓ=1 vℓ
i(u)| ≤(n −1)/2.
At every t, the solution implemented by the decision-maker is equal to Y (u, i, t) = ⌊y⋆(u, i, t)⌋+at
i(u)
for each u and each i. Observe that (i) directly implies that Pn
i=1 Y (u, i, t) = C(u, t) for each u ∈N,
i.e., (3) is satisfied. On the other hand, for each i ∈[n] and e ∈E,
X
u∈e
t
X
ℓ=1
Y (u, i, t) =
X
u∈e
t
X
ℓ=1
⌊y⋆(u, i, ℓ)⌋+
t
X
ℓ=1
aℓ
i(u)
!
≥
X
u∈e
t
X
ℓ=1
⌊y⋆(u, i, ℓ)⌋+
t
X
ℓ=1
vℓ
i(u) −n −1
2
!
=
X
u∈e
t
X
ℓ=1
y⋆(u, i, ℓ) −n −1
2
!
≥D(i, t) −d(n −1)
2
,
where the first inequality holds from (ii) and the last inequality is a consequence of y⋆satisfying (2)
and the hypergraph being d-uniform. We conclude that Y satisfies (2) within an additive violation
of d(n −1)/2. In particular, when n = 3, Corollary 1 implies that the violation is at most d −1.
Online approximate min-cost solutions.
Suppose now that the solution y⋆received by the
decision-maker implies a cost Cost(y⋆), where Cost is a linear function, and we have n = 3 resources.
We use Theorem 2 to construct an online α-approximate solution for MHSC under resource aug-
mentation, where α := maxj∈{1,2,3},ℓ∈[T](d + D(j, ℓ) −1)/D(j, ℓ). For each vertex u and every step
t, consider vt
i(u) = αy⋆(u, i, t) −⌊αy⋆(u, i, t)⌋for each resource i. At every t and for each vertex u,
we use Theorem 1 to get binary values at
1(u), . . . , at
n(u) satisfying the following properties:
(i) ⌊αC(u, t)⌋−Pn
i=1⌊αy⋆(u, i, t)⌋≤Pn
i=1 at
i(u) ≤⌈αC(u, t)⌉−Pn
i=1⌊αy⋆(u, i, t)⌋,
(ii) Pt
ℓ=1 aℓ
i(u) ∈
nj Pt
ℓ=1 vℓ
i(u)
k
,
l Pt
ℓ=1 vℓ
i(u)
mo
,
(iii) E[at
i(u)] = vt
i(u).
At every step t, the solution implemented by the decison-maker is equal to Y (u, i, t) = ⌊αy⋆(u, i, t)⌋+
at
i(u) for each u and each i. Observe that (i) directly implies that Pn
i=1 Y (u, i, t) ≤⌈αC(u, t)⌉
for each u ∈N, i.e., (3) is satisfied under a resource augmentation of factor α.
On the other
hand, for each resource i ∈[3], following the analysis of Naor et al. [2025] we can easily show that
P
u∈e
Pt
ℓ=1 Y (u, i, t) ≥D(i, t), that is, Y satisfies (3). Finally, thanks to the ex-ante proportionality
property (iii), we conclude that E[Cost(Y )] ≤α · Cost(y⋆).
6
Concluding Remarks
In this work, we have introduced an online version of the apportionment problem, where indivisible
seats are to be allocated immediately and irrevocably to parties in every time step in proportion to
17
the parties’ votes, which are revealed online and can be set adaptively. We have imposed a notion
of local quota compliance upfront, requiring that each party receives, in each step, a number of
seats equal to its fraction of votes, rounded up or down. We have established that a greedy method
provides the best-possible deviation, linear in n, between the cumulative votes and the cumulative
seats that a party has received at any step. This implies that a global quota axiom, requiring the
same principle as local quota but for the cumulative values in each step, can be satisfied if and
only if n ≤3. When this condition holds, we have shown that the proportionality guarantee given
by global quota can be enhanced using randomization: For every instance, a lottery over methods
satisfying global quota assigns to each party its proportional number of seats in expectation. Our
proof is based on network flow techniques, providing a full characterization of such methods and
allowing for efficient implementation.
Our work has natural connections and implications for related settings in online dependent
rounding, as our results in Sections 3 and 4 translate directly to online rounding guarantees for
multidimensional instances where we receive an n-dimensional marginal vector at each step. In
particular, the case of n = 1 party captures the online level-set problem studied very recently by
Naor et al. [2025].
Several directions for future work arise. Perhaps the more immediate one concerns the validity
of Theorem 2 when global quota is replaced by approximate proportionality for n ≥4. Specifically,
one could aim to randomize over deterministic methods that attain the best possible proportionality
guarantees from Theorem 1 while fulfilling ex-ante proportionality.
On the other hand, it is natural to ask whether lower expected deviations or stronger axiomatic
properties can be achieved in an online model where the adversary is more restricted. Two interesting
possibilities include the study of randomized methods against an oblivious adversary, that cannot
set the votes based on previous realizations of the method’s randomness, and a fully stochastic
model, where votes come from (potentially correlated) known distributions in each step.
Acknowledgements
This work was partially supported by a Structural Democracy Fellowship through the Brooks School
of Public Policy at Cornell University and by Centro de Modelamiento Matemático (CMM) BASAL
fund FB210005 for center of excellence from ANID-Chile.
18
Appendix
A
Proofs Deferred from Section 2
A.1
Proof of Proposition 1
Proposition 1. For every n ∈N and n-dimensional instance (vt)t∈[T], there exists an offline
apportionment method that satisfies global quota and ex-ante proportionality.
Proof. We fix n ∈N and an n-dimensional instance (vt)t∈[T].
We consider a capacitated net-
work G := (P, E) given by
(a) vertex set P := {o, d} ∪{ut}t∈[T] ∪{wt
i}t∈[T],i∈[n],
(b) edge set
E := {(o, ut) : t ∈[T]} ∪{(ut, wt
i) : t ∈[T], i ∈[n]}
∪{(wt
i, wt+1
i
) : t ∈[T −1], i ∈[n]} ∪{(wT
i , d) : i ∈[n]},
and
(c) upper and lower capacities c, ℓ: E →R+ given by
c((o, ut)) = Ht,
c((ut, wt
i)) = 1,
c((wt
i, wt+1
i
)) = ⌈V t
i ⌉,
ℓ((wt
i, wt+1
i
)) = ⌊V t
i ⌋,
c((wT
i , z)) = ⌈V T
i ⌉,
ℓ((wT
i , d)) = ⌊V T
i ⌋.
For any feasible integral (o, d)-flow f : E →N0 on this network, we define the apportionment
method (Xt(f))t∈N by setting Xt(f) = {i ∈N : f((ut, wt
i)) = 1} for every t ∈N.
Claim 1. For any feasible integral (o, d)-flow f : E →N0 on G, the offline apportionment method
(Xt(f))t∈[T] satisfies global quota.
Proof. To prove this claim, we simply observe that, for any feasible integral (o, d)-flow f : E →N0
and every i ∈[n], it holds
|{k ∈[t] : i ∈Xk(f)} =
t
X
k=1
f((ut, wt
i)) = f((wt
i, wt+1
i
)) ∈{⌊V t
i ⌋, ⌈V t
i ⌉}
for every t ∈[T −1], and
|{k ∈[T] : i ∈Xk(f)} =
T
X
k=1
f((ut, wt
i)) = f((wT
i , d)) ∈{⌊V T
i ⌋, ⌈V T
i ⌉}.
Indeed, in both cases the first equality follows from the definition of (Xt(f))t∈[T], the second one
from the flow conservation constraint at vertex wt
i, and the inclusion from the capacity constraint
on the edge (wt
i, wt+1
i
) or (wT
i , d).
We now consider the fractional (o, d)-flow f∗: E →R+ defined as follows:
f∗((o, ut)) = Ht
for every t ∈[T],
f∗((ut, wt
i)) = vt
i
for every t ∈[T], i ∈[n],
f∗((wt
i, wt+1
i
)) = V t
i
for every t ∈[T −1], i ∈[n],
f∗((wT
i , d)) = V T
i
for every i ∈[n].
19
The flow f∗is easily seen to be a feasible (o, d)-flow on G. Therefore, because of the integrality of
the network flow polytope [Ahuja, Magnanti, and Orlin, 1988], it admits a convex decomposition
into integer (o, d)-flows. That is, there exist an integer q ∈N, integer (o, d)-flows {fj}j∈[q], where
fj : E →N0 for every j ∈[q], and coefficients {λj}j∈[q], where λj ∈(0, 1) for every j ∈[q] and
Pq
j=1 λj = 1, such that
q
X
j=1
λjfj(e) = f∗(e)
for every e ∈E.
(4)
We now consider the randomized offline apportionment method that runs each deterministic method
(Xt(fj))t∈N with probability λj, for j ∈[q]. From Claim 1, we conclude that this method satisfies
global quota. From Equation (4), we conclude that it satisfies ex-ante proportionality.
B
Proofs Deferred from Section 3
B.1
Proof of Lemma 1
Lemma 1. The greedy apportionment method is n−1
2 -proportional on every n-dimensional instance
for every n ∈N, and strictly 1-proportional on every 3-dimensional instance.
Proof. We begin with the case of general n. We let, for the sake of contradiction, (vk)k∈N be an
n-dimensional instance and t ∈N be a time step such that |st
i| > (n −1)/2 for some i ∈[n].
Since the parties are ordered from larger to smaller surplus, this implies either st
1 > (n −1)/2 or
st
n < −(n −1)/2; we will reach a contradiction in either case.
We first consider the case where st
1 > (n −1)/2, and introduce some additional notation. For
each k ∈[t]0 and ℓ∈[n], we let σk
ℓ:= 1
ℓ
P
i∈[ℓ] sk
i denote the average surplus of the ℓparties with
the largest surplus and we define γk
ℓ:= σk
ℓ+ ℓ/2. Observe that, for every k ∈[t]0 and i ∈[n −1]
γk
i > γk
i+1 ⇐⇒1
i
X
j∈[i]
sk
j + i
2 >
1
i + 1
X
j∈[i+1]
sk
j + i + 1
2
⇐⇒
1 + 1
i
X
j∈[i]
sk
j −i + 1
2
>
X
j∈[i+1]
sk
j ⇐⇒sk
i+1 < σk
i −i + 1
2
.
(5)
Note that st
1 > (n −1)/2 implies γt
1 > n/2 = γt
n. Conversely, we have 1/2 = γ0
1 < γ0
2 < · · · < γ0
n =
n/2, because s0
i = 0 for every i ∈[n]. Since γk
n = n/2 for all k ∈[t]0, there exists k ∈[t] such that
γk
i > n/2 and γk
i > γk
i+1 for some i ∈[n −1]; let k∗be the minimum such k. Once k∗is fixed, we
fix i∗to the minimum i ∈[n −1] such that γk∗
i∗> n/2 and γk∗
i∗> γk∗
i∗+1. The following claim will
allow us to reach a contradiction.
Claim 2. It holds sk∗
i∗≤sk∗
i∗+1 + 1.
Proof. We begin by noting that γk∗
i∗> γk∗−1
i∗
. If this was not true, we would have γk∗−1
i∗
≥γk∗
i∗> n/2,
and thus γk∗−1
i
> γk∗−1
i+1
for some i ∈[n −1], contradicting the choice of k∗.
The inequality
γk∗
i∗> γk∗−1
i∗
implies σk∗
i∗> σk∗−1
i∗
, which in turn implies Pi∗
j=1 sk∗
j
> Pi∗
j=1 sk∗−1
j
. Since
n
X
j=1
sk∗
j
=
n
X
j=1
sk∗−1
j
= 0,
there must exist parties i ∈[i∗] and j ∈{i∗+ 1, . . . , n} such that the surplus of the former increases
from step k∗−1 to step k∗and that of the latter decreases. Formally, if ˆsi is the surplus of party
20
i in step k∗−1 and ˆsj is the surplus of party j in step k∗−1 (which are not necessarily equal to
sk∗−1
i
and sk∗−1
j
due to reordering), it holds sk∗
i
> ˆsi and sk∗
j
< ˆsj. Since ˆsℓ= sk∗
ℓ+ (ak∗
ℓ−vk∗
ℓ) for
each ℓ∈[n], these inequalities yield ak∗
i
> vk∗
i
and ak∗
j
< vk∗
j . Since vk∗
ℓ
∈[0, 1) and ak∗
ℓ
∈{0, 1} for
each ℓ∈[n], we conclude that ak∗
i
= 1 and ak∗
j
= 0 < vk∗
j .
Suppose now that the statement is not true, i.e., sk∗
i∗> sk∗
i∗+1+1. Since i ∈[i∗], j ∈{i∗+1, . . . , n},
and the parties are ordered from larger to smaller surplus, this implies
sk∗
i
> sk∗
j + 1 ⇐⇒ˆsi + (ak∗
i
−vk∗
i ) > ˆsj + (ak∗
j −vk∗
j ) + 1 =⇒ˆsi −vk∗
i
> ˆsj −vk∗
j .
But since the greedy method assigns seats, in step k∗, to the parties ℓwith the smallest value of
ˆsℓ−vk∗
ℓ
among those that receives positive votes, this contradicts the above statement that ak∗
i
= 1
and ak∗
j
= 0 < vk∗
j .
We now reach an immediate contradiction when i∗= 1, since in this case Property (5) implies
sk∗
2 < sk∗
1 −1 and Claim 2 states the opposite. When i∗≥2, we obtain
sk∗
i∗≤sk∗
i∗+1 + 1 < σk∗
i∗−i∗+ 1
2
+ 1 = (i∗−1)σk∗
i∗−1 + sk∗
i∗
i∗
−i∗−1
2
.
Indeed, the first inequality comes from Claim 2 and the second one from Property (5); the equality
follows by splitting the average σk∗
i∗of the first i∗terms into the average σk∗
i∗−1 of the first i∗−1 terms
and the i∗th term sk∗
i∗. Rearranging terms, we obtain sk∗
i∗< σk∗
i∗−1 −i∗/2. But then, Property (5)
implies γk∗
i∗−1 > γk∗
i∗> n/2, contradicting the choice of i∗as the minimum i ∈[n −1] such that
γk∗
i
> n/2 and γk∗
i
> γk∗
i+1.
We now consider the case where st
n < −(n −1)/2; the proof is mostly analogous to the previous
case. We redefine the notation for this case. For each k ∈[t]0 and ℓ∈[n], we let σk
ℓ:= 1
ℓ
Pn
i=n+1−ℓsk
i
denote the average surplus of the ℓparties with the smallest surplus and we define γk
ℓ:= σk
ℓ−ℓ/2.
Observe that, for every k ∈[t]0 and i ∈[n −1]
γk
i < γk
i+1 ⇐⇒1
i
n
X
j=n+1−i
sk
j −i
2 <
1
i + 1
n
X
j=n−i
sk
j −i + 1
2
⇐⇒
1 + 1
i
n
X
j=n+1−i
sk
j + i + 1
2
<
n
X
j=n−i
sk
j ⇐⇒sk
n−i > σk
i + i + 1
2
.
(6)
Note that st
n < −(n −1)/2 implies γt
1 < −n/2 = γt
n. Conversely, we have −1/2 = γ0
1 > γ0
2 > · · · >
γ0
n = −n/2, because s0
i = 0 for every i ∈[n]. Since γk
n = −n/2 for every k ∈[t]0, there exists k ∈[t]
such that γk
i < −n/2 and γk
i < γk
i+1 for some i ∈[n −1]; let k∗be the minimum such k. Once k∗
is fixed, we fix i∗to the minimum i ∈[n −1] such that γk∗
i∗< −n/2 and γk∗
i∗< γk∗
i∗+1. The following
claim will allow us to reach a contradiction.
Claim 3. It holds sk∗
n+1−i∗≥sk∗
n−i∗−1.
Proof. We begin by noting that γk∗
i∗< γk∗−1
i∗
. If this was not true, we would have γk∗−1
i∗
≤γk∗
i∗<
−n/2, and thus γk∗−1
i
< γk∗−1
i+1
for some i ∈[n −1], contradicting the choice of k∗. The inequality
γk∗
i∗< γk∗−1
i∗
implies σk∗
i∗< σk∗−1
i∗
, which in turn implies Pn
j=n+1−i∗sk∗
j
< Pn
j=n+1−i∗sk∗−1
j
. Since
n
X
j=1
sk∗
j
=
n
X
j=1
sk∗−1
j
= 0,
21
there must exist parties i ∈{n + 1 −i∗, . . . , n} and j ∈[n −i∗] such that the surplus of the former
decreases from k∗−1 to k and that of the latter increases. Formally, if ˆsi is the surplus of party
i in step k∗−1 and ˆsj is the surplus of party j in step k∗−1 (which are not necessarily equal to
sk∗−1
i
and sk∗−1
j
due to reordering), it holds sk∗
i
< ˆsi and sk∗
j
> ˆsj. Since ˆsℓ= sk∗
ℓ+ (ak∗
ℓ−vk∗
ℓ) for
each ℓ∈[n], these inequalities yield ak∗
i
< vk∗
i
and ak∗
j
> vk∗
j . Since vk∗
ℓ
∈[0, 1) and ak∗
ℓ
∈{0, 1} for
each ℓ∈[n], we conclude that ak∗
i
= 0 < vk∗
i
and ak∗
j
= 1.
Suppose now that the statement is not true, i.e., sk∗
n+1−i∗< sk∗
n−i∗−1. Since i ∈{n+1−i∗, . . . , n},
j ∈[n −i∗], and the parties are ordered from larger to smaller surplus, this implies
sk∗
i
< sk∗
j −1 ⇐⇒ˆsi + (ak∗
i
−vk∗
i ) < ˆsj + (ak∗
j −vk∗
j ) −1 =⇒ˆsi −vk∗
i
< ˆsj −vk∗
j .
But since the greedy method assigns seats, in step k∗, to the parties ℓwith the smallest value
of ˆsℓ−vk∗
ℓ
among those that receives positive votes, this contradicts the above statement that
ak∗
i
= 0 < vk∗
i
and ak∗
j
= 1.
We now reach an immediate contradiction when i∗= 1, since in this case Property (6) implies
sk∗
n−1 > sk∗
n + 1 and Claim 3 states the opposite. When i∗≥2, we obtain
sk∗
n+1−i∗≥sk∗
n−i∗−1 > σk∗
i∗+ i∗+ 1
2
−1 = (i∗−1)σk∗
i∗−1 + sk∗
n+1−i∗
i∗
+ i∗−1
2
.
Indeed, the first inequality comes from Claim 3 and the second one from Property (6); the equality
follows by splitting the average σk∗
i∗of the first i∗terms into the average σk∗
i∗−1 of the first i∗−1
terms and the i∗th term sk∗
n+1−i∗. Rearranging terms, we obtain sk∗
n+1−i∗> σk∗
i∗−1 + i∗/2. But then,
Property (6) implies γk∗
i∗−1 < γk∗
i∗< −n/2, contradicting the choice of i∗as the minimum i ∈[n −1]
such that γk∗
i
< −n/2 and γk∗
i
< γk∗
i+1.
We now address strict 1-proportionality for n = 3.
We note that, in general, if we aim to
prove strict proportionality in the cases above, we can proceed analogously. That is, we suppose
towards a contradiction the existence of an n-dimensional instance (vt)t∈N and a step t ∈N such
that either st
1 ≥(n −1)/2 or st
n ≤−(n −1)/2 holds, and define the values σk
ℓand γk
ℓas before,
and fix k∗and i∗analogously, but now for the corresponding weak inequalities (γk∗
i∗≥n/2 and
γk∗
i∗≥γk∗
i∗+1 for the first case; γk∗
i∗≤−n/2 and γk∗
i∗≤γk∗
i∗+1 for the second case). Properties (5)
and (6) admit analogous versions with weak inequalities, whereas Claims 2 and 3 remain true and
can be proven in a completely analogous way, as they rely on the respective observations γk∗
i∗> γk∗−1
i∗
and γk∗
i∗< γk∗−1
i∗
, which remain true. Furthermore, the contradictions for the case i∗≥2 can be
reached in the same way as before. Indeed, for the case st
1 ≥(n −1)/2 we obtain
sk∗
i∗≤sk∗
i∗+1 + 1 ≤σk∗
i∗−i∗+ 1
2
+ 1 = (i∗−1)σk∗
i∗−1 + sk∗
i∗
i∗
−i∗−1
2
,
which implies sk∗
i∗≤σk∗
i∗−1 −i∗/2 and thus γk∗
i∗−1 ≥γk∗
i∗≥n/2 due to Property (5), contradicting
the choice of i∗. Similarly, for the case st
n ≤−(n −1)/2 we obtain
sk∗
n+1−i∗≥sk∗
n−i∗−1 ≥σk∗
i∗+ i∗+ 1
2
−1 = (i∗−1)σk∗
i∗−1 + sk∗
n+1−i∗
i∗
+ i∗−1
2
,
which implies sk∗
n+1−i∗≥σk∗
i∗−1 + i∗/2 and thus γk∗
i∗−1 ≤γk∗
i∗≤−n/2 due to Property (6), contra-
dicting the choice of i∗.
When i∗= 1, Claims 2 and 3 together with the weak versions of Properties (5) and (6) do
not yield a contradiction anymore. However, they now imply that sk∗
2
= sk∗
1 −1 when we assume
22
st
1 ≥(n −1)/2 or that sk∗
n−1 = sk∗
n + 1 when we assume st
n ≤−(n −1)/2. We will see that these
conditions still yield a contradiction when n = 3, so we fix this value. Note that, when we start from
the assumption that st
1 ≥1 for some t ∈N, γk∗
1
≥3/2 is equivalent sk∗
1 ≥1. Since P
i∈[3] sk∗
i
= 0,
sk∗
2
= sk∗
1 −1 and sk∗
1
≥1 together imply sk∗
3
≤−1 in this case. Similarly, when we start from
the assumption that st
3 ≤−1 for some t ∈N, γk∗
3
≤−3/2 is equivalent to sk∗
3
≤−1.
Since
P
i∈[3] sk∗
i
= 0, sk∗
2 = sk∗
3 + 1 and sk∗
3 ≤−1 together imply sk∗
1 ≥1 in this case. Thus, we conclude
in either case that sk∗
1 ≥1 ⇔sk∗
3 ≤−1.
Letting ˆs1 denote the surplus of party 1 in step k∗−1 and ˆs3 the surplus of party 3 in step
k∗−1 (which are not necessarily equal to sk∗−1
1
and sk∗−1
3
due to reordering), we also know from the
definition of k∗that ˆs1 < 1 and ˆs3 > −1, which will yield a similar contradiction as in the proofs of
Claims 2 and 3. Indeed, sk∗
1 > ˆs1 implies ak∗
1 = 1 and sk∗
3 < ˆs3 implies ak∗
3 = 0 < vk∗
3 . On the other
hand, sk∗
1 ≥1 implies ˆs1 −vk∗
1 ≥0 and sk∗
3 ≤−1 implies ˆs3 −vk∗
3 ≤−1, so that ˆs3 −vk∗
3 < ˆs1 −vk∗
1 .
But since the greedy method assigns seats, in step k∗, to the parties i with the smallest value of
ˆsi −vk∗
i
among those that receive positive votes, this contradicts the above statement that ak∗
1 = 1
and ak∗
3 = 0 < vk∗
3 .
B.2
Proof of Lemma 2
Lemma 2. Let n ∈N, let (vk)k∈[t] be a partial n-dimensional instance, and let V t denote the
aggregate votes up to step t. Let (Xk)k∈N be any online apportionment method and At the cor-
responding cumulative allocation up to step t.
Then, for all parties i, j ∈[n] with i ̸= j such
that 0 ≤st
i −st
j < 1, there exists a vote vector vt+1 ∈[0, 1)n such that, after the allocation at+1
corresponding to Xt+1(V t, At, vt+1),
|st+1
i
−st+1
j
| = 1,
st+1
i
+ st+1
j
2
=
st
i + st
j
2
.
Proof. We let n, t ∈N, V t, (Xk)k∈N, At, i, j be as in the statement. We define vt+1 ∈[0, 1)n by
vt+1
k
=
1+(st
i−st
j)
2
if k = i,
1−(st
i−st
j)
2
if k = j,
0
otherwise,
for all k ∈[n],
whose entries sum up to 1 and is, therefore, a valid vote vector. Since Ht+1 = P
k∈[n] vt+1
k
= 1 and
vt+1
k
= 0 for every k /∈{i, j}, we have either Xt+1 = {i} or Xt+1 = {j}. Since
st
i −vt+1
i
=
st
i + st
j −1
2
= st
j −vt+1
j
,
we have that in either case, some k ∈{i, j} is such that st+1
k
=
st
i+st
j−1
2
+ 1 =
st
i+st
j+1
2
and the other
party ℓ∈{i, j} \ {k} is such that st+1
ℓ
=
st
i+st
j−1
2
. The claim follows.
B.3
Proof of Lemma 3
Lemma 3. Let n, n′ ∈N with n′ ≤n and P ⊆[n] with |P| = n′ be a subset of n′ parties
in [n]. Let r ∈N0 and ε > 0 be fixed, and let (vk)k∈[r] be a partial n-dimensional instance. Let
(Xt)t∈N be any online apportionment method and Ar the corresponding cumulative allocation up to
step r. Then, there exists a finite value t ∈N with t ≥r and a vote sequence (vk)k∈{r+1,...,t} such
23
that, after the allocation at corresponding to Xt(V t−1, At−1, vt), there exists a party i ∈P such
that |st
i| ≥(n′ −1)/2 −ε.
Proof. We fix n ∈N and prove the lemma by induction over n′, the size of the subset of parties we
consider. Throughout this proof, whenever we have an n′-dimensional vote vector with entries in
P ⊆[n], namely v′ ∈[0, 1]P , we refer to the vector v ∈[0, 1]n defined as
vi =
(
v′
i
if i ∈P
0
otherwise,
as the n-dimensional extension of v′. This definition naturally carries over for the allocation vector.
Note that, from our definition of an online apportionment method, all entries of these vote and
allocation vectors that do not correspond to parties in P are always fixed to 0.
Since we use induction steps from n′ to n′ + 2, we establish the validity of the lemma for two
base cases, namely n′ = 1 and n′ = 2. The case n′ = 1 is trivial, as the conclusion of the lemma
only requires |st
i| ≥−ε, and we know that |st
i| ≥0 > −ε. For the case n′ = 2, we fix r, ε, (vk)k∈[r],
(Xt)t∈N, and Ar as in the statement, we denote the parties by P = {p1, p2}, and we assume without
loss of generality that sr
p1 ≥sr
p2. If sr
1 ≥1/2 −ε or sr
n ≤−(1/2 −ε), we are done. Otherwise, it
holds that sr
1 −sr
2 < 1, so we can consider a single step given by the n-dimensional extension of
the (p1, p2)-splitter ˆv(p1, p2). Lemma 2 implies that sr+1
p1
−sr+1
p2
= 1, and thus either sr+1
p1
≥1/2 or
sr+1
p2
≤−1/2 holds.
If n ≤2, the previous cases suffice to conclude, so we assume the opposite in what follows. We
assume that the statement of the lemma holds for some fixed n′ ∈[n−2]. We will make use of the step
whose existence is guaranteed by this inductive hypothesis repeatedly, so we introduce some slightly
more general notation for it. We consider tuples (P, ε, (vk)k∈[r], Ar) such that P = {p1, . . . , pn′} is
a set of n′ parties, ε > 0 is a positive number, (vk)k∈[r] is a partial n-dimensional instance, and
Ar = P
k∈[r] ak is the aggregated allocation up to step r. Without loss of generality, we assume,
as usual, that sr
p1 ≥sr
p2 ≥· · · ≥sr
pn′. Given such a tuple, we denote the n-dimensional extension
of the vote sequence satisfying the properties of the lemma by ˆV (P, ε, (vk)k∈[r], Ar), and refer to it
as a (P, ε)-booster (because it boosts the surplus or deficit of some party in P). That is, the (P, ε)-
booster ˆV (P, ε, (vk)k∈[r], Ar) is a partial n-dimensional instance (vk)k∈{r+1,...,t}, with t ≥r finite and
vk ∈[0, 1]n for k ∈{r + 1, . . . , t}, such that for any allocation (ak)k∈{r+1,...,t} produced by an online
apportionment method, with ak ∈{0, 1}n for k ∈{r + 1, . . . , t}, it holds that st
p1 ≥(n′ −1)/2 −ε
or st
pn′ ≤−((n′ −1)/2 −ε) (potentially after reordering p1, . . . , pn′).
To prove the claim for n′ + 2, we fix r, ε, V r, and Ar as in the statement. We assume, as usual,
that sr
p1 ≥sr
p2 ≥· · · ≥sr
pn′+2. We apply the inductive hypothesis on the n′-dimensional instance
that results from restricting to parties p2, . . . , pn′+1, but with a smaller deviation of ε/2. That is,
we consider the n-dimensional vote sequence
(vk)k∈{r+1,...,t} = ˆV
{p2, . . . , pn′+1}, ε/2, (vk)k∈[r], Ar
for some finite t. After reordering the parties in P so that st
p1 > st
p2 > · · · > st
pn′+2, we know
by the induction hypothesis that we have reached an instance with st
p1 ≥(n′ −1)/2 −ε/2 or
st
pn′+2 ≤−((n′ −1)/2−ε/2). We again apply a ({p2, . . . , pn′+1}, ε/2)-booster, and then a third time
if necessary. After applying this procedure at most three times, we now have obtained an instance,
say at step t′, such that st′
p2 ≥(n′ −1)/2 −ε/2 or st′
pn′+1 ≤−((n′ −1)/2 −ε/2), since each booster
fixes a new party satisfying one of these two conditions.
24
In the remainder of the proof, we repeatedly apply either a (p1, p2)-splitter or a (pn′+1, pn′+2)-
splitter, and then a {p2, . . . , pn+1}-booster. The splitters will increase the surplus of party p1 or the
deficit of party pn′+2, at the cost of decreasing the surplus of p2 or the deficit of pn′+1, respectively.
However, the inductive hypothesis allows us to boost either this surplus or deficit so that it surpasses
(n′ −1)/2 −ε/2 and can be used for a splitter again. By repeating this procedure, we can make the
surplus of party p1 or the deficit of party pn+2 arbitrarily close to (n′ −1)/2 + 1 −ε/2, and thus
surpass (n′ + 1)/2 −ε after a finite number of steps.
The argument can be formalized through the following algorithm. We initialize ℓ= 0, tℓ= t′
and repeat the following procedure until stℓ
1 ≥(n′ + 1)/2 −ε1 or stℓ
n′+2 ≤(n′ + 1)/2 −ε1:
(i) If stℓp2 ≥(n′ −1)/2 −ε/2, apply a (p1, p2)-splitter; i.e., take vtℓ+1 = ˆv(p1, p2);
(ii) else, if stℓpn′+1 ≤−((n′−1)/2−ε/2), apply a (pn′+1, pn′+2)-splitter; i.e., vtℓ+1 = ˆv(pn′+1, pn′+2);
(iii) apply a {p2, . . . , pn′+1}-booster; i.e., take
(vk)k∈{tℓ+2,...,t′′} = ˆV
{p2, . . . , pn′+1}, ε/2, (vk)k∈[tℓ+1], Atℓ+1
for some finite t′′;
(iv) update ℓ←ℓ+ 1 and tℓ←t′′.
Note that, after each iteration ℓ≥1, the following invariant (which we showed for t0) is preserved
by the induction hypothesis: stℓp2 ≥(n′ −1)/2 −ε/2 or stℓpn′+1 ≤−((n′ −1)/2 −ε/2). Observe that
stℓp1 < (n′ + 1)/2 −ε and stℓpn′+2 > −((n′ + 1)/2 −ε
as long as the algorithm does not terminate.
Thus, whenever stℓp2 ≥(n′ −1)/2 −ε/2 holds, we have stℓp1 −stℓp2 < 1 and thus the (p1, p2)-splitter is
well defined. Similarly, whenever stℓpn′+1 ≤−((n′ −1)/2−ε/2) holds, we have stℓpn′+1 −stℓpn′+2 < 1 and
thus the (pn′+1, pn′+2)-splitter is well defined. We distinguish between two types of iterations: type I
are the iterations ℓsuch that stℓp2 ≥(n′ −1)/2 −ε/2 and the algorithm applies a (p1, p2)-splitter;
analogously, type II are the iterations ℓsuch that stℓpn′+1 ≤−((n′ −1)/2 −ε/2) and the algorithm
applies a (pn′+1, pn′+2)-splitter. The following claim will allow us to conclude.
Claim 4. Let i(0), i(1), . . . , i(L1) ∈N0 be the type I iterations and j(0), j(1), . . . , j(L2) ∈N0 be the
type II iterations after L1 + L2 + 2 iterations of the algorithm. For every ℓ∈[L1]0, it holds that
s
ti(ℓ)
p1
≥(n′ +1)/2−ε/2−1/2ℓ. For every ℓ∈[L2]0, it holds that s
tj(ℓ)
pn′+2 ≤−((n′ +1)/2−ε/2−1/2ℓ).
Proof. Let i(1), i(2), . . . , i(L1) and j(1), j(2), . . . , j(L2) be as in the statement. The proofs for type
I and type II iterations are fully symmetric, so we only include the former.
We proceed by induction. The base case ℓ= 0 is trivial, since at the beginning of the algorithm
we have st0
p1 ≥(n′ −1)/2 −ε/2, and s
ti(0)
p1
≥st0
p1.
Assume the claim holds for some ℓ∈[L1 −1]0; i.e., s
ti(ℓ)
p1
≥(n′ + 1)/2 −ε/2 −1/2ℓ. We will
prove that
s
ti(ℓ)+1
p1
≥n′ + 1
2
−ε
2 −
1
2ℓ+1 .
(7)
If this is true, this suffices to conclude. Indeed, stq
p1 does not decrease in type II iterations q, since
they only involve the parties {p2, . . . , pn′+2} in each iteration and the maximum surplus can only
remain the same or increase. Thus, Inequality (7) implies
s
ti(ℓ+1)
p1
≥s
ti(ℓ)+1
p1
≥n′ + 1
2
−ε
2 −
1
2ℓ+1 .
25
We now prove Inequality (7) using the inductive hypothesis s
ti(ℓ)
p1
≥(n′ + 1)/2 −ε/2 −1/2ℓ.
If s
ti(ℓ)
p1
≥(n′ + 1)/2 −ε/2 −1/2ℓ+1, then we are done, because the maximum surplus cannot
decrease with a (p1, p2)-splitter. Otherwise, we let δ := s
ti(ℓ)
p1
−((n′ + 1)/2 −ε/2 −1/2ℓ) and note
that 0 ≤δ < 1/2ℓ+1. On the other hand, we know from the condition of a type I iteration that
s
ti(ℓ)
p2
≥(n′ −1)/2 −ε/2. Combining these two facts, we obtain
s
ti(ℓ)
p1
−s
ti(ℓ)
p2
≤n′ + 1
2
−ε
2 −1
2ℓ+ δ −
n′ −1
2
−ε
2
= 1 −1
2ℓ+ δ.
Hence, the application of the (p1, p2)-splitter yields, due to Lemma 2,
s
ti(ℓ)+1
p1
≥s
ti(ℓ)
p1
+ 1 −(s
ti(ℓ)
p1
−s
ti(ℓ)
p2 )
2
≥n′ + 1
2
−ε
2 −1
2ℓ+ δ +
1
2ℓ−δ
2
= n′ + 1
2
−ε
2 + δ
2 −
1
2ℓ+1 ≥n′ + 1
2
−ε
2 −
1
2ℓ+1 .
After L + 2 iterations of the algorithm, for even L > 0, we know that we have either applied
L/2 + 1 type I iterations or L/2 + 1 type II iterations. Thus, Claim 4 implies that either sL
p1 ≥
(n′ −1)/2 −ε or sL
pn′+2 ≤−
(n′ −1)/2 −ε
holds if
n′ + 1
2
−ε
2 −
1
2L/2 ≥n′ + 1
2
−ε ⇐⇒L ≥−2 log(ε/2).
C
Proofs Deferred from Section 4
C.1
Proof of Lemma 4
Lemma 4. Let (yt)t∈[T] be a partial network flow method. Then, for every t ∈[T] and every feasible
history (V t, At) ∈Θt that occurs with strictly positive probability, we have At
i ∈{⌊V t
i ⌋, ⌈V t
i ⌉} for
every i ∈[n]. Furthermore, (yt)t∈[T] satisfies Equation (1) for every t ∈[T].
Proof. The proof follows by induction over t ∈[T]. First, consider step t = 1. By definition, y1 yields
a valid probability distribution over subsets of size H1, ensuring that no party receives more than
one seat in step 1. Thus, for proving global quota, it suffices to show that any party i with v1
i = 0
is allocated a seat with probability zero. Let i be such that v1
i = 0. Since 0 = v1
i = Pm
j=1 λjzj,i
with λj ∈[0, 1] for every j, it follows that λj = 0 whenever zj,i = 1. Hence, y1(V 0, A0, v1, S) = 0
for all subsets S containing i, as required. To prove that Equation (1) is also satisfied, we fix a
party i ∈[n] and observe that
X
S⊆[n]:i∈S
y1(V 0, A0, v1, S) =
m
X
j=1
λjzj,i = v1
i .
Next, consider some t ∈[T −1] and assume that the partial method (yt)t∈[T−1] satisfies the
required properties up to step t ≤T −1. We show that they are maintained for step t + 1. By
definition, yt+1 yields a valid probability distribution over subsets of size Ht+1, ensuring that no
party receives more than one seat in step t+1. Consider a party i with vt+1
i
= 0. Since f is a feasible
flow, we have f((u, i)) = 0 for every u ∈Ut, which implies zi(u) = 0. Consequently, each zj,i = 0
and i /∈Zj for all j. Hence, yt+1(V t, At, vt+1, S) = 0 for all subsets S containing i, as required.
To verify global quota, fix a cumulative allocation At occurring with strictly positive proba-
bility and let u = u(At) ⊂[n] be the corresponding set of parties at their upper share of seats.
26
First, suppose that i ∈u and that the upper share of i does not increase from step t to t + 1,
i.e., ⌈Pt
k=1 vk
i ⌉= ⌈Pt+1
k=1 vk
i ⌉. Then party i cannot be allocated a seat in step t + 1 without violat-
ing global quota. Since f((u, i)) = 0, we again have zi(u) = 0, which implies that zj,i = 0 and i /∈Zj
for all j. Therefore, yt+1(V t, At, vt+1, S) = 0 for all subsets S containing i, as required.
On the other hand, assume i /∈u and the lower quota increases, i.e., ⌊Pt
k=1 vk
i ⌋+1 = ⌊Pt+1
k=1 vk
i ⌋.
Then i must be assigned a seat in step t + 1, as otherwise global quota would be violated. In this
case, the flow is f((u, i)) = π(u)/Ht+1, so zi(u) = 1, which implies that zj,i = 1 and i ∈Zj for all j.
Hence, every subset S in the support of yt+1(V t, At, vt+1, ·) contains i, as desired.
It remains to show that yt+1(V t, At, vt+1, ·) satisfies Equation (1) for step t + 1. Recall that At
is the set of all cumulative allocations At that occur with strictly positive probability. We need to
show that the probability that party i is allocated a seat in step t + 1 equals vt+1
i
. To compute this,
observe that
X
At∈At
π(yt, V t, At) · at+1
i
(V t, At, vt+1) =
X
u∈Ut
X
At∈At:u=u(At)
π(yt, V t, At) · at+1
i
(V t, At, vt+1)
=
X
u∈Ut
π(u) · zi(u)
=
X
u∈Ut
π(u) · f(u, i) · Ht+1
π(u)
=
X
u∈Ut
f(u, i) · Ht+1.
By flow conservation at vertex i, the total incoming flow equals the outgoing flow f((i, d)) =
vt+1
i
/Ht+1, so the final expression evaluates to vt+1
i
, as required.
C.2
Proof of Lemma 5
Lemma 5. Let n ≤3 and let (yk)k∈[t] be a partial network flow method with corresponding flow
network FN(V t, At, vt+1, π) for step t + 1. Then this flow network admits a feasible (o, d)-flow of
value 1, i.e., the partial network flow method can be extended to (yk)k∈[t+1].
Proof. Since the case n = 2 follows immediately from n = 3 by setting vt
3 = 0 for all t, we focus
on n = 3. For t = 1, the construction of y1 is precisely given in the construction of the partial
network flow method. Thus, let t ≥1 and let a partial network flow method (yk)k∈[t] up to step t be
given. We show that the corresponding flow network FN(V t, At, vt+1, π) allows for a feasible flow
of value 1 using a detailed case distinction based on
• the number of seats to be allocated in step t + 1 (i.e., whether Ht+1 = 1 or Ht+1 = 2),
• the number of parties that have been assigned their upper share of seats after step t (i.e.,
whether |u| = 1 or |u| = 2 for every u ∈Ut), and
• how many parties obtained enough votes in step t + 1 to increase their upper share of seats
by 1.
This results in eight distinct cases, listed below. For each, we show that the corresponding flow
network admits a feasible flow of value 1. We begin by making two general observations, using the
fact that the partial network flow method (yk)k∈[t] satisfies Equation (1) for every k ∈[t]. For any
27
party i ∈[3] and set u ∈Ut, we have
π(u) =
t
X
k=1
vk
i −
$ t
X
k=1
vk
i
%
≤1 −
& t
X
k=1
vk
i
'
−
t
X
k=1
vk
i
!
if u = {i}, and
π(u) = 1 −
t
X
k=1
vk
i −
$ t
X
k=1
vk
i
%!
≥
& t
X
k=1
vk
i
'
−
t
X
k=1
vk
i
if i /∈u, |u| = 2.
(8)
The inequalities are strict if and only if Pt
k=1 vk
i ∈N. Furthermore, in any flow network, the vertex
sets Po = {o} and Pd = P \ {d} each induce a cut of capacity 1. Thus, any flow of value 1 must
assign any edge adjacent to o or d a flow equal to the edge’s upper capacity.
The remainder of this proof is structured as follows: We first list the eight cases, then address
each one separately, showing the existence of a feasible flow f of value 1. All relevant networks are
depicted in Figures 4 and 5, where edges (u, i) with c((u, i)) = 0 are omitted for clarity.
1. Ht+1 = 1
1.1. |u| = 1 for all u ∈Ut
1.1.1. ∀i ∈[3] : vt+1
i
<
Pt
k=1 vk
i
−Pt
k=1 vk
i
1.1.2. ∃! i ∈[3] : vt+1
i
≥
Pt
k=1 vk
i
−Pt
k=1 vk
i > 0
1.2. |u| = 2 for all u ∈Ut
1.2.1. ∃! i ∈[3] : vt+1
i
≥
Pt
k=1 vk
i
−Pt
k=1 vk
i > 0
1.2.2. ∃! i ∈[3] : vt+1
i
<
Pt
k=1 vk
i
−Pt
k=1 vk
i
2. Ht+1 = 2
2.1. |u| = 1 for all u ∈Ut
2.1.1. ∃! i ∈[3] : vt+1
i
≥
Pt
k=1 vk
i
−Pt
k=1 vk
i > 0
2.1.2. ∃! i ∈[3] : vt+1
i
<
Pt
k=1 vk
i
−Pt
k=1 vk
i
2.2. |u| = 2 for all u ∈Ut
2.2.1. ∃! i ∈[3] : vt+1
i
<
Pt
k=1 vk
i
−Pt
k=1 vk
i
2.2.2. ∀i ∈[3] : vt+1
i
≥
Pt
k=1 vk
i
−Pt
k=1 vk
i > 0
Note that this covers all possible cases due to the following observation: For each party i ∈[n], we
have
t
X
k=1
vk
i −
$ t
X
k=1
vk
i
%
+ vt+1
i
∈[0, 2),
while the sum evaluates to
X
i∈[3]
t
X
k=1
vk
i −
$ t
X
k=1
vk
i
%
+ vt+1
i
!
= |u| + Ht+1.
This implies that the number of parties for which the individual term is at least 1 lies within Ht+1 +
|u| −2 and Ht+1 + |u| −1.
28
o
2
1
3
1
2
3
d
π({1})
π({2})
π({3})
vt+1
1
vt+1
2
vt+1
3
(a) Case 1.1.1
o
2
1
3
1
2
3
d
π({1})
π({2})
π({3})
vt+1
1
vt+1
2
vt+1
3
(b) Case 1.1.2
o
1, 3
1, 2
2, 3
1
2
3
d
π({1, 2})
π({1, 3})
π({2, 3})
vt+1
1
vt+1
2
vt+1
3
(c) Case 1.2.1
o
1, 3
1, 2
2, 3
1
2
3
d
π({1, 2})
π({1, 3})
π({2, 3})
vt+1
1
vt+1
2
vt+1
3
(d) Case 1.2.2
Figure 4: Case 1 of the proof of Lemma 5 with Ht+1 = 1. The upper capacity of an edge e = (u, i)
is c(e) = π(u) if it is drawn and c(e) = 0, otherwise. Red edges have lower capacities ℓ(e) = c(e).
Grey edges carry no flow in the unique flow with value 1.
Case 1.1.1.
The flow network is illustrated in Figure 4a. In this case, the flow f of value 1 is not
unique. However, any feasible flow f with value 1 satisfies f(e) = c(e) for all
e ∈{(o, {i}), ({i}, i), (i, d) : i ∈[3]}.
The flow on the remaining edges must satisfy the following set of linear equalities
f(({1}, 2) + f(({1}, 3) = π({1}),
f(({2}, 1) + f(({2}, 3) = π({2}),
f(({3}, 1) + f(({3}, 2) = π({3}),
f(({2}, 1) + f(({3}, 1) = vt+1
1
,
f(({1}, 2) + f(({3}, 2) = vt+1
2
,
f(({1}, 3) + f(({2}, 3) = vt+1
3
,
f(e) ≥0.
Since adding the first three equalities yields the same as adding the other three, the set of linear
equalities is underdetermined, i.e., there is no unique solution. We show that the system has a
feasible solution.
To this end, we interpret the inner part of the network as a bipartite graph G := (P1 ∪P2, E)
with vertex sets P1 := {{1}, {2}, {3}} and P2 := {1, 2, 3}, and edge set E := {({i}, j) : i ̸= j}.
29
Each vertex {i} ∈P1 has supply b({i}) := π({i}) and each vertex i ∈P2 has demand b(i) := vt+1
i
for every i ∈[3]. Thus, each edge is adjacent to one supply and one demand vertex. Since edge
capacities match the supply of the adjacent supply vertex, we can treat this as a transshipment
instance without explicit edge capacities.
We claim that this transshipment instance has a feasible solution. First, note that the total sup-
ply and demand coincide. Now it suffices to verify that for all P ′ ⊆P1, the total supply P
p∈P ′ b(p)
is at most the total demand P
p∈N(P ′) b(p) of its neighborhood N(P ′) (cf. [Schrijver, 2004, Corollary
11.2.g]). We focus on three cases. For P ′ = {{1}}, we have
X
p∈P ′
b(p) = π({1}) ≤1 −
& t
X
k=1
vk
1
'
−
t
X
k=1
vk
1
!
≤1 −vt+1
1
= vt+1
2
+ vt+1
3
=
X
v∈N(P ′)
b(p).
For P ′ = {{1}, {2}}, we obtain
X
p∈P ′
b(p) = π({1}) + π({2}) ≤1 = vt+1
1
+ vt+1
2
+ vt+1
3
=
X
p∈N(P ′)
b(p),
and finally for P ′ = {{1}, {2}, {3}}, we have
X
p∈P ′
b(p) = π({1}) + π({2}) + π({3}) = 1 = vt+1
1
+ vt+1
2
+ vt+1
3
=
X
p∈N(P ′)
b(p).
By symmetry, the condition also holds for all other subsets P ′. Hence, a feasible transshipment
exists, implying the existence of a feasible flow f of value 1 in the original network.
Case 1.1.2.
There exists exactly one party i ∈[3] with vt+1
i
≥⌈Pt
k=1 vk
i ⌉−Pt
k=1 vk
i . Without
loss of generality, assume i = 1. The flow network is illustrated in Figure 4b. We define f as follows.
f(e) =
vt+1
1
−(1 −π({1}))
if e = ({1}, 1),
vt+1
2
if e = ({1}, 2),
vt+1
3
if e = ({1}, 3),
0
if e = ({2}, 3) or e = ({3}, 2), and
c(e)
otherwise.
We now verify feasibility. For e = ({1}, 1), since 1 −π({1}) ≤vt+1
1
< 1 we have f(e) ∈[0, π({1})].
For e = ({1}, j) with j ∈{2, 3}, we have vt+1
j
≤π({1}) since vt+1
1
+ vt+1
j
≤1 ≤vt+1
1
+ π({1}).
Thus, all capacities are respected. Flow conservation at vertex 1 holds because
X
u∈Ut
f((u, 1)) = vt+1
1
−(1 −π({1})) + π({2}) + π({3}) = vt+1
1
= f((1, d)).
Flow conservation at all other inner vertices is trivial, and since the outgoing flow from o is 1, f is
a feasible flow of value 1.
30
Case 1.2.1.
There exists exactly one party i ∈[3] with vt+1
i
≥⌈Pt
k=1 vk
i ⌉−Pt
k=1 vk
i . Without
loss of generality, assume i = 1. The flow network is illustrated in Figure 4c. We define f as follows.
f(e) =
π({1, 2}) −vt+1
3
if e = ({1, 2}, 1),
vt+1
3
if e = ({1, 2}, 3),
π({1, 3}) −vt+1
2
if e = ({1, 3}, 1),
vt+1
2
if e = ({1, 3}, 2),
c(e)
otherwise.
To verify feasibility, observe that vt+1
2
< π({1, 3}) and vt+1
3
< π({1, 2}) as implied by Property (8).
Hence, all flow values remain within the capacity bounds. Flow conservation at vertex 1 holds
because
X
u∈Ut
f((u, 1)) = π({1, 2}) −vt+1
3
+ π({1, 3}) −vt+1
2
+ π({2, 3}) = 1 −vt+1
2
−vt+1
3
= vt+1
1
= f((1, d)).
Flow conservation at all other inner vertices is trivial, and since the outgoing flow from o is 1, f is
a feasible flow of value 1.
Case 1.2.2.
There exists exactly one party i ∈[3] with vt+1
i
< ⌈Pt
k=1 vk
i ⌉−Pt
k=1 vk
i . Without
loss of generality, assume i = 1. The flow network is illustrated in Figure 4d. We define f as follows.
f(e) =
vt+1
1
if e = ({2, 3}, 1),
vt+1
2
−π({1, 3})
if e = ({2, 3}, 2),
vt+1
3
−π({1, 2})
if e = ({2, 3}, 3),
0
if e = ({1, 2}, 2) or e = ({1, 3}, 3), and
c(e)
otherwise.
Flow conservation at vertex {2, 3} follows from
3
X
j=1
f(({2, 3}, j)) = vt+1
1
+ vt+1
2
+ vt+1
3
−π({1, 3}) −π({1, 2}) = π({2, 3}) = f((o, {2, 3})).
Flow conservation at all other inner vertices is trivial.
The non-negativity of f follows from
Property (8), which is satisfied with equality in this case since parties j ∈{2, 3} satisfy vt+1
j
≥
⌈Pt
k=1 vk
j ⌉−Pt
k=1 vk
j and therefore Pt
k=1 vk
j /∈N. With P3
j=1 f(({2, 3}, j)) = π({2, 3}), the non-
negativity implies f(e) ≤π({2, 3}) for all e = ({2, 3}, j). Since the outgoing flow from o is 1, f is a
feasible flow of value 1.
Case 2.1.1.
There exists exactly one party i ∈[3] with vt+1
i
≥⌈Pt
k=1 vk
i ⌉−Pt
k=1 vk
i . Without
loss of generality, assume i = 1. The flow network is illustrated in Figure 5a. We define f as follows.
f(e) =
(vt+1
1
−(1 −π({1})))/2
if e = ({1}, 1),
(vt+1
2
−π({3}))/2
if e = ({1}, 2),
(vt+1
3
−π({2}))/2
if e = ({1}, 3), and
c(e)
otherwise.
31
o
2
1
3
1
2
3
d
π({1})
π({2})
π({3})
vt+1
1
/2
vt+1
2
/2
vt+1
3
/2
(a) Case 2.1.1
o
2
1
3
1
2
3
d
π({1})
π({2})
π({3})
vt+1
1
/2
vt+1
2
/2
vt+1
3
/2
(b) Case 2.1.2
o
1, 3
1, 2
2, 3
1
2
3
d
π({1, 2})
π({1, 3})
π({2, 3})
vt+1
1
/2
vt+1
2
/2
vt+1
3
/2
(c) Case 2.2.1
o
1, 3
1, 2
2, 3
1
2
3
d
π({1, 2})
π({1, 3})
π({2, 3})
vt+1
1
/2
vt+1
2
/2
vt+1
3
/2
(d) Case 2.2.2
Figure 5: Case 2 of the proof of Lemma 5 with Ht+1 = 2. The upper capacity of an edge e = (u, i)
is c(e) = π(u)/2 if it is drawn and c(e) = 0, otherwise. Red edges have lower capacities ℓ(e) = c(e).
First we show flow conservation at vertex {1} by
3
X
j=1
f(({1}, j)) = vt+1
1
+ vt+1
2
+ vt+1
3
−π({2}) −π({3}) −(1 −π({1}))
2
= π({1}) = f((o, {1})).
Flow conservation at vertex 1 holds because
X
u∈Ut
f((u, 1)) = vt+1
1
−(1 −π({1}))
2
+ π({2})
2
+ π({3})
2
= vt+1
1
2
= f((1, d)).
Flow conservation at all other inner vertices is trivially satisfied.
To check the capacity constraints, consider e = ({1}, 1). Since vt+1
1
≥1−π({1}), we have f(e) ≥
0. For e = ({1}, 2), note that
vt+1
2
= 2 −vt+1
1
−vt+1
3
≥2 −vt+1
1
−
& t
X
k=1
vk
3
'
−
t
X
k=1
vk
3
!
≥1 −vt+1
1
+ π({3}) > π({3}),
where the first inequality follows from the case assumption, the second inequality from Property (8),
and the third (strict) inequality from vt+1
1
< 1. Hence, f(e) > 0 and similar arguments yield f(e) > 0
for e = ({1}, 3).
Next, we verify upper capacity constraints. For e = ({1}, 1), since vt+1
1
< 1, we have f(e) <
π({1})/2. For e = ({1}, 2),
f(e) = vt+2
2
−π({3})
2
≤1 −π({2}) −π({3})
2
= π({1})
2
= c(e),
32
and analogously for e = ({1}, 3). Since the outgoing flow from o is 1, f is a feasible flow of value 1.
Case 2.1.2.
There exists exactly one party i ∈[3] with vt+1
i
< ⌈Pt
k=1 vk
i ⌉−Pt
k=1 vk
i . Without
loss of generality, assume i = 1. The flow network is illustrated in Figure 5b. We define f as follows.
f(e) =
(1 −vt+1
2
)/2
if e = ({2}, 1),
(vt+1
2
−(1 −π({2})))/2
if e = ({2}, 2),
(1 −vt+1
3
)/2
if e = ({3}, 1),
(vt+1
3
−(1 −π({3})))/2
if e = ({3}, 3), and
c(e)
otherwise.
We first verify flow conservation. Consider vertex {2} and observe that
3
X
j=1
f(({2}, j)) = 1 −vt+1
2
+ vt+1
2
−(1 −π({2})) + π({2})
2
= π({2}) = f((o, {2})).
An analogous computation applies to {3}. For vertex 1, flow conservation holds because
3
X
j=1
f(({j}, 1) = 1 −vt+1
2
+ 1 −vt+1
3
2
= vt+1
1
2
= f((3, d)).
For the remaining inner vertices, flow conservation is trivially satisfied.
Next, we show that f obeys the capacity constraints. With 1−π({j}) = ⌈Pt
k=1 vk
j ⌉−Pt
k=1 vk
j ≤
vt+1
j
< 1 for j ∈{2, 3}, all lower capacity constraints are satisfied. For e = ({2}, 1), observe that
f(e) = 1 −vt+1
2
2
≤1 −(1 −π({2}))
2
= π({2})
2
= c(e).
An analogous computation yields f(e) ≤c(e) for e = ({3}, 1). For edges e = ({2}, 2) and e = ({3}, 3)
the upper capacities are trivially satisfied since vt+1
j
< 1 for all j. Since the outgoing flow from o
is 1, f is a feasible flow of value 1.
Case 2.2.1.
There exists exactly one party i ∈[3] with vt+1
i
< ⌈Pt
k=1 vk
i ⌉−Pt
k=1 vk
i . Without
loss of generality, assume i = 1. The flow network is illustrated in Figure 5c. We define f as follows.
f(e) =
vt+1
1
/2
if e = ({2, 3}, 1),
(vt+1
2
−(1 −π({2, 3})))/2
if e = ({2, 3}, 2),
(vt+1
3
−(1 −π({2, 3})))/2
if e = ({2, 3}, 3), and
c(e)
otherwise.
We first verify flow conservation. For vertex {2, 3} observe that
3
X
j=1
f(({2, 3}, j)) = vt+1
1
+ vt+1
2
−(1 −π({2, 3})) + vt+1
3
−(1 −π({2, 3}))
2
= π({2, 3})
2
= f((o, {2, 3})).
33
For the remaining inner vertices, flow conservation is trivially satisfied.
Next, we show that f obeys the capacity constraints. For e = ({2, 3}, 2), note that
vt+1
2
= 2 −vt+1
1
−vt+1
3
≥2 −vt+1
3
−
& t
X
k=1
vk
1
'
−
t
X
k=1
vk
1
!
≥2 −vt+1
3
−π({2, 3}) > 1 −π({2, 3}),
where the first inequality follows from the case assumption, the second inequality from Property (8),
and the third (strict) inequality from vt+1
3
< 1. An analogous computation applies to e = ({2, 3}, 3),
proving that ℓ(e) ≤f(e) holds for all edges.
For the upper capacity constraints, consider e =
({2, 3}, 1). Property (8) yields
vt+1
1
<
& t
X
k=1
vk
1
'
−
t
X
k=1
vk
1 ≤π({2, 3}),
i.e., f(e) < c(e). For e = ({2, 3}, 2), we have (vt+1
2
−(1 −π({2, 3}))/2 < π({2, 3})/2 since vt+1
2
< 1,
and analogously for e = ({2, 3}, 3). Since the outgoing flow from o is 1, f is a feasible flow of value 1.
Case 2.2.2.
The flow network is illustrated in Figure 5d.
This case is structurally similar
to Case 1.1.1: the flow f is not uniquely determined.
We begin by fixing the flow along the
edges ({1, 2}, 3), ({1, 3}, 2), and ({2, 3}, 1), each of which lies on a unique (o, d)-path. These edges
must carry their respective lower bounds, which are π({1, 2})/2, π({1, 3})/2, and π({2, 3})/2.
Fixing these values reduces the residual problem to the flow network depicted in Figure 6, where
all remaining lower bounds are zero. The goal is now to route a residual flow of value
1 −π({1, 2} + π({1, 3}) + π({2, 3})
2
= 1
2.
To establish feasibility, we again reduce to a transshipment instance on a bipartite graph G :=
(P1 ∪P2, E) with vertex sets P1 := {{1, 2}, {1, 3}, {2, 3}}, and P2 := {1, 2, 3}, and edge set E :=
{({i}, j) : i ̸= j}. Each vertex {i, j} ∈P1 has supply b({i, j}) := π({i, j})/2 and each vertex i ∈P2
has demand b(i) := (vt+1
i
−π({j, k}))/2 where {j, k} = [3] \ {i}. Thus, each edge is adjacent to
one supply and one demand vertex. Since edge capacities match the supply of the adjacent supply
vertex, we can treat this as a transshipment instance without explicit edge capacities.
We claim that this transshipment instance has a feasible solution. First, note that the total sup-
ply and demand coincide. Now it suffices to verify that for all P ′ ⊆P1, the total supply P
p∈P ′ b(p)
is at most the total demand P
p∈N(P ′) b(p) of its neighborhood N(P ′). We focus on three cases.
For P ′ = {{1, 2}}, we have
X
p∈P ′
b(p) = π({1, 2}
2
= 1 −π({1, 3}) −π({2, 3})
2
= vt+1
1
+ vt+1
2
+ vt+1
3
−1 −π({1, 3}) −π({2, 3})
2
< vt+1
1
+ vt+1
2
−π({1, 3}) −π({2, 3})
2
=
X
p∈N(P ′)
b(p).
For P ′ = {{1, 2}, {1, 3}}, we obtain
X
p∈P ′
b(p) = π({1, 2}) + π({1, 3})
2
≤1
2 =
X
p∈N(P ′)
b(p).
34
o
1, 3
1, 2
2, 3
1
2
3
d
π({1, 2})/2
π({1, 3})/2
π({2, 3})/2
(vt+1
1
−π({2, 3}))/2
(vt+1
2
−π({1, 3}))/2
(vt+1
3
−π({1, 2}))/2
Figure 6: The residual flow network of Case 2.2.2 in the proof of Lemma 5. The upper capacity of
an edge e = (u, i) is c(e) = π(u)/2 if it is drawn and c(e) = 0, otherwise.
Finally, for P ′ = {{1, 2}, {1, 3}, {2, 3}}, we have
X
p∈P ′
b(p) = π({1, 2}) + π({1, 3}) + π({2, 3})
2
= 1
2 =
X
p∈N(P ′)
b(p).
By symmetry, the condition also holds for all other subsets P ′. Thus, the transshipment instance is
feasible, which implies the existence of a feasible flow of value 1/2 in the residual network. Together
with the fixed lower bound flow, we obtain a feasible flow of value 1, completing this case.
This completes the case distinction.
C.3
Proof of Proposition 2
Proposition 2. Any online apportionment method satisfying global quota and ex-ante proportion-
ality is a network flow method.
Proof. Consider an online apportionment method (yt)t∈N satisfying global quota and ex-ante pro-
portionality. We need to show that one can obtain the same method using the network flow con-
struction. First, consider step t = 1. Since A0 = {0} and V 0 = 0, there is just one probability
distribution y1 over the subsets of [n] of size H1.
Let Z1, . . . , Zm ⊂[n] be the support of y1
and let λ1, . . . , λm ∈[0, 1] be the corresponding probabilities, i.e., y1(V 0, A0, v1, Zj) = λj for ev-
ery j. Let zj ∈{0, 1}n denote the characteristic vector of Zj, i.e., zj,i = 1 if and only if i ∈Zj.
Since |Zj| = H1 for every j, the vector zj has exactly H1 ones. Furthermore, ex-ante proportionality
yields Pm
j=1 λj ·zj = v1. Thus, the probability distribution y1 yields a unique convex decomposition
into n-dimensional {0, 1}-vectors with exactly H1 ones. This convex decomposition can be chosen
during the construction of a partial network flow method.
Next, assume that up to step t, the online apportionment method (yk)k∈N coincides with a
partial network flow method (zk)k∈[t]. Let FN(V t, At, vt+1, π) be the flow network corresponding
to the partial network flow method (zk)k∈[t] at step t + 1. We show how the method (yk)k∈N for
step t + 1 yields a flow of value 1 in FN(V t, At, vt+1, π).
For each At ∈At, the method (yt)t∈N determines the probability π(yt, V t, At) with which the
cumulative allocation At is attained. This uniquely determines the probability π(u(At)) that exactly
the parties in u(At) have been allocated their upper share of seats. Let At = At
1 ∪· · · ∪At
m be
a partition of At into disjoint subsets such that u(At) = u( ˜At) if and only if At, ˜At ∈At
j for
some j ∈[m]. Note that the flow network has exactly one vertex uj for each At
j. The probability
that method (yt)t∈N assigns a seat to party i in step t+1 conditioned on the cumulative allocation uj
35
is denoted by
πuj(i) =
X
At∈At
j
π(yt, V t, At) ·
X
S⊆[n]:i∈S
yt+1(V t, At, vt+1, S).
We define f as follows. For each edge (o, uj) from the origin to a vertex uj, set f((o, uj)) = π(uj);
for each edge (uj, i) from set uj to party i, set f((uj, i)) = πuj(i)/Ht+1; and finally, for each
edge (i, d) from a vertex i to the destination, set f((i, d)) = vt+1
i
/Ht+1. We claim that this defines
a feasible flow of value 1.
First, we show that f obeys the edge capacities. This is immediate for edges (o, ut) and (i, d)
as f equals the upper capacities. To verify feasibility on edges (uj, i), we distinguish the three
cases given by the flow construction. First, assume that i ∈uj and that the upper share of seats
does not increase from step t to t + 1, i.e., ⌈Pt
k=1 vk
i ⌉= ⌈Pt+1
k=1 vk
i ⌉or vt+1
i
= 0. We need to
show that f((uj, i)) = 0. However, in that case, party i cannot be assigned a seat in step t + 1
without violating local or global quota. Since the method (yt)t∈N fulfills these properties, it as-
signs probability 0 to any set containing i for every At ∈At
j. Thus, we have πuj(i) = 0, imply-
ing f((uj, i)) = πuj(i)/Ht+1 = 0. Next, assume i /∈uj and the lower share increases from step t
to t + 1, i.e., ⌊Pt
k=1 vk
i ⌋+ 1 = ⌊Pt+1
k=1 vk
i ⌋. We need to show that f((uj, i)) = π(uj)/Ht+1. How-
ever, in that case, party i must receive a seat in step t + 1, as otherwise global quota would be
violated. Since the method (yt)t∈N fulfills these properties, it only assigns positive probability to
sets containing i, which yields P
S⊆[n]:i∈S yt+1(V t, At, vt+1, S) = 1 for every At ∈At
j. Hence, we
obtain
f((uj, i)) = πuj(i)
Ht+1 =
X
A∈At
j
π(yt, V t, At) ·
1
Ht+1 = π(uj)
Ht+1 .
as desired. In all other cases, the method (yt)t∈N is less restricted and we have yt+1(V t, At, vt+1, S) ∈
[0, 1] for any subset S ⊆[n]. We only need to show that f((uj, i)) ∈[0, π(uj)/Ht+1]. This follows
from
f((uj, i)) = πuj(i)
Ht+1 ≤
X
A∈At
j
π(yt, V t, At) ·
1
Ht+1 = π(uj)
Ht+1 .
Thus, f obeys the capacity constraints.
To verify flow conservation, we first consider an inner vertex uj. The incoming flow from o
is f((o, uj)) = π(uj). The outgoing flow is
n
X
i=1
f((uj, i)) =
n
X
i=1
πuj(i)
Ht+1
=
X
A∈At
j
π(yt, V t, At) ·
1
Ht+1 ·
n
X
i=1
X
S⊆[n]:i∈S
yt+1(V t, At, vt+1, S)
= π(uj),
where the last equality holds, since the method (yt)t∈N assigns exactly Ht+1 seats in step t + 1. For
an inner vertex i, (yt)t∈N satisfying ex-ante proportionality yields
X
uj∈Ut
f((uj, i)) =
X
uj∈Ut
πuj(i)
Ht+1
=
X
uj∈Ut
1
Ht+1
X
A∈At
j
π(yt, V t, At) ·
X
S⊆[n]:i∈S
yt+1(V t, At, vt+1, S) = vt+1
Ht+1 .
36
Thus, the incoming flow matches the outgoing flow f((i, d)), so flow conservation holds at every
inner vertex.
Since f obeys the capacity constraints of the flow network FN(V t, At, vt+1, π) and attains flow
conservation, it is a feasible flow. Finally, the total flow leaving the origin o is P
uj∈Ut π(uj) = 1.
Hence, the flow f has value 1 as claimed. This exact flow allows to extend the partial network
flow method (zk)k∈[t] up to step t+1 such that the online apportionment method (yk)k∈N coincides
with the partial network flow method (zk)k∈[t+1]. This finally proves that the online apportionment
method (yk)k∈N is a network flow method.
37
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|
Online Proportional Apportionment Javier Cembrano∗ Jose Correa† Svenja M. Griesbach ‡ Victor Verdugo§ Abstract Traditionally, the problem of apportioning the seats of a legislative body has been viewed as a one-shot process with no dynamic considerations. While this approach is reasonable for some instances of the problem, dynamic aspects play an important role in many others. In this paper, we initiate the study of apportionment problems in an online setting. Specifically, we introduce an online algorithmic framework to handle proportional apportionment with no information about future events. In this model, time is discrete and there are n parties that receive a certain share of the votes at each time step. An online algorithm needs to irrevocably assign a prescribed number of seats at each time, ensuring that each party receives its fractional share rounded up or down, and that the cumulative number of seats allocated to each party remains close to its cumulative share up to that time. We consider deterministic and randomized online apportionment methods. For deterministic methods, we construct a family of adversarial instances that yield a lower bound, linear in n, on the worst-case deviation between the seats allocated to a party and its cumulative share. We show that this bound is best possible and is matched by a natural greedy method. As a consequence, a method guaranteeing that the cumulative number of seats assigned to each party up to any step equals its cumulative share rounded up or down (global quota) exists if and only if n ≤3. Then, we turn to randomized allocations and show that, when n ≤3, we can randomize over methods satisfying global quota with the additional guarantee that each party receives, in expectation, its proportional share in every step. Our proof is constructive: We show that any method satisfying these properties can be obtained from a flow on a recursively constructed network. We showcase the applicability of our results to obtain approximate solutions in the context of online dependent rounding procedures for multidimensional instances. ∗ ür Informatik; - neering, Universidad de Chile; . † ; . ‡Centro de Modelamiento Matemático (CNRS IRL2807), Universidad de Chile; ; . §Institute for Mathematical and Computational Engineering and - ing, PUC Chile; . 16 Oct 2025 1 Introduction The proportional apportionment problem plays a paramount role in the structural electoral design of modern democratic systems, capturing the task of allocating the seats of a representative body (e.g., a parliament or constitutional assembly). Formally, an instance is given by a positive integer vector v = (v1, . . . , vn) whose entries sum up to a positive integer value H. The goal is to find a non-negative integer vector a = (a1, . . . , an) such that Pn i=1 ai = H, and ai should be proportional to vi. This problem is typically solved in the following electoral scenarios. In one of them, we have a set of n states or districts, and each value vi represents the fraction of the population of state i. In a second scenario, we have a set of n political parties, and each value vi represents the fraction of votes obtained by party i. Historically, the quality of an apportionment method has been measured according to whether it satisfies a prescribed set of axioms or not. Among them, the quota property, requiring that each party gets its quota rounded up or down, has played a prominent role since Alexander Hamilton, the first US Secretary of the Treasury, proposed a solution to the apportionment problem in 1791, later adopted in 1851. Hamilton's method first calculates each party's quota, immediately assigns the floor of this number to the party, and then it goes through the parties in decreasing order of their quota residue vi -⌊vi⌋and assigns one more seat to each party until the desired house size H is met; by construction, Hamilton's method satisfies the quota property. We refer to the books by Balinski and Young [2010] and Pukelsheim [2017] for an extensive mathematical treatment of the quota property and the axiomatic theory of proportional apportionment. While the classic apportionment theory provides a thorough axiomatic characterization of proportionality, it takes a static point of view in which solutions for an apportionment instance are computed independently of the past and future events. Under this paradigm, when an apportionment problem is solved repeatedly over time, rounding decisions made locally at each time do not prevent systematic biases over the cumulative solutions. However, dynamic representation plays a crucial role in the political organization of societies and the implementation of public policies [Stimson, MacKuen, and Erikson, 1995]. In addition to the natural application to elections that are repeated after each term of office-as discussed by Gölz, Peters, and Procaccia [2025] in the context of the sortition of the European Commission-the challenge of proportional apportionment over time arises in contexts beyond political representation. For instance, the Indian reservation policy mandates that publicly funded institutions set aside a fixed share of seats and jobs for designated beneficiary groups, and entitlements that must be rounded to whole numbers [Evren and Khanna, 2021]. As institutions run successive recruitment rounds, they must ensure their total integral apportionment allocations stay as close as possible to the prescribed cumulative (fractional) quotas; this global goal is in tension with the allocation monotonicity of reserved seats over the recruitment cycles. Similar problems are faced by institutions in other contexts, such as the distribution of human resources, public services, and facilities across different organizational or geographical units. The key problem behind the tension between the local and global quota requirements is that, although we have access to historical data, we lack information about future apportionment instances; a decision taken today can induce deviations from the cumulative quotas for the next period. Consider the following example (introduced by Gölz et al. [2025]) with sequential elections to allocate a single seat among four parties {1, 2, 3, 4}. In the first step, each party receives 1/4 of the votes, so we can assume that a method assigns the seat to an arbitrary party, say party 1. In the second step, parties 2, 3, and 4 get 1/3 of the votes each; we can again assign the seat arbitrarily, say to party 2. If in the third step the votes are evenly distributed among parties 3 and 4, the one not receiving the seat will have a cumulative entitlement strictly larger than 1 but no seat allocated. Therefore, the global quota requirement is violated regardless of the choices made in the method. 1 1.1 Our Contribution We introduce a modeling and online-algorithmic framework to handle proportional apportionment problems under local and global rounding considerations with no information about future events. In the following, we summarize the contributions of our work and discuss its consequences beyond apportionment. A model for online proportional apportionment. We propose a new model for online apportionment with proportional representation. In this model, there are n ∈N parties, and at each time step t ∈N, they receive a fractional allocation of votes represented by a vector vt summing to an integer Ht ∈N. We consider methods satisfying local quota, meaning that we allocate either ⌊vt i⌋or ⌈vt i⌉seats to each party i ∈[n]. Therefore, throughout the paper we assume without loss of generality that vt ∈[0, 1)n. Upon observing vt, a method needs to irrevocably assign exactly Ht seats to parties with strictly positive votes, granting one seat per selected party. Crucially, the decision at time t can depend only on observed votes and allocations up to that time. Future votes cannot be observed or influence the decision. Conversely, the votes can be set based on the previous choices of the method. We study which fairness properties can be achieved in this online setting. In particular, we focus on three central notions that apply to each party and at each step: global quota (the total number of allocated seats matches the floor or ceiling of the total number of votes), α-proportionality (the total number of seats and the total number of votes differ by at most α), and ex-ante proportionality (the expected total number of seats equals the total number of votes). While in the offline setting, where the entire sequence of votes is known in advance, there exist allocation methods that satisfy global quota (and therefore 1-proportionality) and ex-ante proportionality for any number of parties (Proposition 1), we show that achieving similar guarantees in the online setting is significantly more challenging. Deterministic methods and approximate proportionality. In Section 3, we study deterministic allocation methods and introduce the greedy apportionment method, which in each step assigns the Ht seats to the parties whose current number of allocated seats deviates most from their cumulative share. As the first part of our main result (Theorem 1) we show that this method is n-1 2 -proportional for any number of parties n ∈N, and even satisfies global quota when n ≤3. The analysis builds on the closely related leaky-bucket problem, where Adler, Berenbrink, Friedetzky, Goldberg, Goldberg, and Paterson [2003] proved (Hn -1)-proportionality (with Hn denoting the nth harmonic number). However, the fact that our setting only allows allocations to parties with positive votes worsens the attainable guarantee from logarithmic to linear. For the second part of Theorem 1, we show that this bound is tight by constructing a family of hard instances that, for any ε > 0, rule out the possibility of achieving better than n-1 2 -ε -proportionality with any deterministic method. Randomized methods and ex-ante proportionality. In Section 4, we turn to randomized methods. The main result of this section (Theorem 2) is that an online apportionment method satisfying both global quota and ex-ante proportionality exists if and only if n ≤3. Our proof is constructive and based on a class of methods derived from a carefully designed flow network for each time step, which encodes all constraints using the full history of votes and allocations. Moreover, we show that any method satisfying global quota and ex-ante proportionality can be obtained by this recursive flow construction, thereby fully characterizing the class of such methods (Proposition 2). We conclude by briefly explaining why this construction breaks down when n ≥4. 2 Consequences in online dependent rounding. Our results in Sections 3 and 4 translate directly to the setting of online dependent rounding for multidimensional instances, i.e., we receive an n-dimensional marginal vector at each step instead of a single non-negative real value. In particular, the case of n = 1 party captures the online level-set problem studied very recently by Naor, Srinivasan, and Wajc [2025]. In Section 5, we showcase the applicability of Theorem 1 and Theorem 2 in the context of online optimization, and introduce a multidimensional generalization of the multi-stage stochastic covering problem studied by Naor et al. [2025]. We describe how to achieve online near-feasibility and approximation guarantees as a simple consequence of our results. 1.2 Further Related Work Not only from a political viewpoint apportionment has been studied for centuries, but also its formal mathematical treatment has a long history, as testified by the books of Balinski and Young [2010] and Pukelsheim [2017]. Of particular relevance to our work is the quota axiom, analyzed in depth by Balinski and Young [1975]. Our work departs from classic apportionment theory in two key ways: We consider online methods that allocate seats in sequential elections without information about future elections, and part of our results concern randomized apportionment. While we initiate the study of online apportionment, Evren and Khanna [2021] previously explored (offline) methods ensuring proportionality in sequential allocation steps, motivated by public sector recruitment in India. Their positive results on monotone, proportional-in-expectation, and quota-approximating methods build upon the work of Akbarpour and Nikzad [2020] on the decomposition of proportional fractional allocations into proportional-in-expectation and near-feasible integral allocations. On the other hand, randomized apportionment has received increasing attention recently, decades after the foundational work of Grimmett [2004]. Indeed, Gölz et al. [2025] established the existence of randomized methods satisfying quota, house monotonicity, and expected proportionality; Cembrano, Correa, Schmidt-Kraepelin, Tsigonias-Dimitriadis, and Verdugo [2025] studied randomized methods that allow minor deviations from the house size to circumvent impossibility results; and Correa, Gölz, Schmidt-Kraepelin, Tucker-Foltz, and Verdugo [2024] focused on monotonicity with respect to subsets of parties. Aziz, Lev, Mattei, Rosenschein, and Walsh [2019] also considered lotteries over deterministic apportionment methods in the context of strategyproof peer selection. From a technical point of view, our work builds on a line of work applying optimization techniques to apportionment. In particular, network flows have been exploited in the study of bidimensional proportionality [Balinski and Demange, 1989a,b, Gaffke and Pukelsheim, 2008, Rote and Zachariasen, 2007] and house-monotone apportionment [Cembrano et al., 2025]; Pukelsheim, Ricca, Simeone, Scozzari, and Serafini [2011] provide an overview of these and other applications. Gölz et al. [2025] also used linear programming techniques in the context of monotone apportionment. Also, our positive result for n ≤3 generalizes the expected proportionality and quota properties of a rounding scheme proposed by Naor et al. [2025] as an online version of the classic scheme by Srinivasan [2001]. Also related to our setting is the bamboo garden trimming problem introduced by Gąsieniec, Klasing, Levcopoulos, Lingas, Min, and Radzik [2017]. In this problem, each bamboo grows with a fixed rate (analogous to parties receiving the same number of votes in each election). In each step, one bamboo can be cut to the ground, and the objective is to minimize the maximum height across all bamboos over time. The best-known approximation guarantees are due to Kawamura [2024]. A generalization of this setting is captured by the cup game, often studied in the context of load balancing [Liu and Layland, 1973] and deamortization [Dietz and Sleator, 1987]. Here, an adversary distributes H fractional units of water across a subset of n cups, and the player 3 selects H cups from which to remove one unit of water each. This models a scenario with multiple processors or allocations per round and has been analyzed against both oblivious and adaptive adversaries [Bender and Kuszmaul, 2021, Bender, Farach-Colton, and Kuszmaul, 2019, Kuszmaul, 2020]. Our analysis of the deterministic greedy apportionment method is closely related to the leaky bucket problem studied by Adler et al. [2003]. In this model, each bucket starts at the same water level and leaks a fractional amount of water in each step. The goal is to assign integral units of water to buckets in a way that maximizes the minimum fill level over time. This setting differs from ours in a crucial aspect: Allocations can be made to any bucket, regardless of its leak rate (or, in our interpretation, even to parties with zero votes). A more detailed comparison to our model is provided in Section 3. 2 Instances, Axioms, and Offline Methods Let N (resp. N0) denote the strictly positive (resp. non-negative) integers. Let [n] := {1, 2, . . . , n} (resp. [n]0 := {0, 1, . . . , n}) denote the set containing the first elements of these sets up to n ∈N (resp. n ∈N0). We identify [n] with a set of n parties. In each election t ∈N, party i ∈[n] receives a fractional seat entitlement vt i ∈[0, 1) based on the outcome of a vote;1 Ht := Pn i=1 vt i ∈N denotes the total number of seats to be distributed in step t. We interpret vt i directly as the votes party i receives in election t and we write vt ∈[0, 1)n for the corresponding vector; an n-dimensional instance is fully described by the sequence (vt)t∈N. While we define our general instances in the online setting as infinite sequences to capture the fact that we do not know when they will end or what the votes in the next steps will be, we restrict to a finite number of steps when we study offline methods, as well as for the iterative construction of some methods and negative instances. We refer to a vote sequence over a finite number of steps T, (vt)t∈{T0,...,T0+T}, as a partial n-dimensional instance. We now introduce the notion of an offline method and the two key axioms we study throughout this paper. A deterministic offline apportionment method receives a partial n-dimensional instance (vt)t∈[T] and outputs a sequence (Xt)t∈[T], where for every t ∈[T], Xt ⊂{i ∈[n] : vt i > 0} and |Xt| = Ht. In words, an offline apportionment method selects a subset of parties receiving the Ht seats in each step, constrained to those parties receiving strictly positive votes. When the vote sequence is fixed, we identify a method as the sequence (Xt)t∈[T] (and not as a function mapping a vote sequence to a set sequence). A randomized offline apportionment method is a lottery over deterministic offline apportionment methods. For an instance (vt)t∈[T] and t ∈[T], we let V t i := P k∈[t] vk i denote the votes received by party i up to step t. For an instance (vt)t∈[T], a (deterministic or randomized) method (Xt)t∈[T], and t ∈[T], we let At i := |{k ∈[t] : i ∈Xk}| denote the (possibly random) number of seats received by party i up to step t. A deterministic method (Xt)t∈[T] satisfies global quota on a certain instance if, for every t ∈[T] and i ∈[n], it holds At i ∈{⌊V t i ⌋, ⌈V t i ⌉}. A randomized method satisfies global quota if all deterministic methods in its support satisfy global quota. In addition, a randomized method satisfies ex-ante proportionality if P[i ∈Xt] = vt i for every t ∈[T] and i ∈[n]. The following proposition states the existence of offline apportionment methods satisfying both of the axioms introduced above. It follows from the existence of house monotone and quota compliant methods in the classic (non-sequential) apportionment problem, shown via different approaches by Balinski and Young [2010], Cembrano et al. [2025], Gölz et al. [2025]. 1We recall that the restriction to vt i ∈[0, 1) is without loss in the context of methods that satisfy local quota. This is because, given any vote vector vt ∈Rn ≥0, any such method deterministically assigns each party its lower quota ⌊vt i⌋, which reduces the problem to allocating the remaining fractional seats vt i -⌊vt i⌋∈[0, 1) among the n parties. 4 u1 u2 u3 u4 u5 o H1 H2 H3 H4 H5 w1 1 1 w1 2 1 w1 3 1 w2 1 1 w2 2 1 w2 3 1 w3 1 1 w3 2 1 w3 3 1 w4 1 1 w4 2 1 w4 3 1 w5 1 1 w5 2 1 w5 3 1 ⌈V 1 1 ⌉ ⌊V 1 1 ⌋ ⌈V 1 2 ⌉ ⌊V 1 2 ⌋ ⌈V 1 3 ⌉ ⌊V 1 3 ⌋ ⌈V 2 1 ⌉ ⌊V 2 1 ⌋ ⌈V 2 2 ⌉ ⌊V 2 2 ⌋ ⌈V 2 3 ⌉ ⌊V 2 3 ⌋ ⌈V 3 1 ⌉ ⌊V 3 1 ⌋ ⌈V 3 2 ⌉ ⌊V 3 2 ⌋ ⌈V 3 3 ⌉ ⌊V 3 3 ⌋ ⌈V 4 1 ⌉ ⌊V 4 1 ⌋ ⌈V 4 2 ⌉ ⌊V 4 2 ⌋ ⌈V 4 3 ⌉ ⌊V 4 3 ⌋ d ⌈V 5 1 ⌉ ⌊V 5 1 ⌋ ⌈V 5 2 ⌉ ⌊V 5 2 ⌋ ⌈V 5 3 ⌉ ⌊V 5 3 ⌋ Figure 1: Illustration of the flow network used in the proof of Proposition 1 for T = 5 steps and n = 3 parties. A single arc label represents an upper capacity; labels below and above an arc correspond to lower and upper capacities, respectively. Proposition 1. For every n ∈N and n-dimensional instance (vt)t∈[T], there exists an offline apportionment method that satisfies global quota and ex-ante proportionality. The proof of this proposition, which is included for completeness and can be found in Section A.1, models apportionment methods via flows on a capacitated network, as illustrated in Figure 1. This construction was given by Cembrano et al. [2025] to prove the existence of apportionment methods satisfying house-monotonicity and quota compliance in a non-sequential setting, where house monotonicity requires that an increase in the total number of seats to allocate does not cause a drop in the number of seats any party receives. Specifically, we consider an origin o and vertices (ut)t∈[T], where there is an arc (s, ut) with capacity Ht for each t ∈[T]. For each t ∈[T], there are also vertices (wt i)t∈[T],i∈[n] and arcs (ut, wt i) with capacity 1 for each i ∈[n] such that vt i > 0. There are also arcs (wt i, wt+1 i ) for every t ∈[T -1] and i ∈[n], with lower capacity ⌊V t i ⌋and upper capacity ⌈V t i ⌉. Finally, there is a destination d and arcs (wT i , d) for every i ∈[n], with lower capacity ⌊V T i ⌋and upper capacity ⌈V T i ⌉. The flow on each arc (ut, wt i) is interpreted as the number of seats (0 or 1) that party i ∈[n] receives in step t, so that the flow on each arc (wt i, wt+1 i ) (or (wT i , d) when t = T) corresponds to the total number of seats that party i has received up to step t. Any (deterministic) apportionment method corresponding to a feasible flow on this network satisfies global quota because of these arcs' capacities. The fact that there exists a lottery over such methods satisfying ex-ante proportionality follows from two facts: (i) the fractional flow that sends a flow of vt i on each arc (wt i, wt+1 i ) (or (wT i , d) when t = T) is feasible, and (ii) the network flow polytope has integral extreme points. Hence, the proportional fractional apportionment can be obtained as a distribution over deterministic apportionment methods satisfying global quota. 3 Deterministic Methods and Approximate Proportionality In this section, we study deterministic online apportionment methods. As our main result in this context, we provide tight bounds on their worst-case deviations from exact proportionality, 5 understood as the maximum difference, in any step, between the number of seats and the cumulative share that a party has received. Our findings have immediate implications regarding the axioms introduced in Section 2: In contrast to the positive result for offline methods, an online method can satisfy global quota if and only if there are at most three parties. A comparison to results on the closely related leaky bucket problem highlights the difficulties of our setting due to the restriction of only assigning seats to parties receiving positive votes in each step. To formally present our results, we first formally define a method in the online setting. A (deterministic) online apportionment method selects, given an n-dimensional instance (vt)t∈N, a subset of parties Xt ⊂[n] for every t ∈N as follows: (i) For t = 0, we define V 0 i = v0 i := 0 and A0 i = a0 i := 0 for every i ∈[n]. (ii) For t ∈N, the history is given by the pair θt-1 := (V t-1, At-1) ∈Θt-1, where Θt-1 := Rn +×Nn 0, and we define the cumulative vote vector as V t-1 i := Pt-1 k=0 vk i for every i ∈[n], and the cumulative allocation vector as At-1 i := Pt-1 k=0 ak i . The function Xt : Θt-1 × [0, 1)n → [n] Ht maps the pair (θt-1, vt) to a subset Xt(θt-1, vt) ⊂{i ∈[n] : vt i > 0} of size Ht, and the allocation at ∈{0, 1}n is derived from this set: at i = 1 if and only if i ∈Xt(θt-1, vt). When the history and vote vector are fixed, we omit these arguments and use Xt to directly refer to the set (instead of the function). To quantify the deviations from the proportional number of seats that parties should receive, we introduce the notion of approximate proportionality, which measures how close the number of assigned seats to each party is to its cumulative votes. For α ≥0 and an n-dimensional instance (vt)t∈N, an online apportionment method (Xt)t∈N is α-proportional if, for every t ∈N and i ∈[n], the cumulative allocation satisfies |At i -V t i | ≤α. We further say that a method is strictly α-proportional if the above inequality holds strictly for every t ∈N and i ∈[n], i.e., |At i -V t i | 0, there exists an n-dimensional instance on which no online apportionment method is n-1 2 -ε -proportional. As a consequence of Theorem 1, we note the following observation regarding the global quota axiom introduced in Section 2. Corollary 1. There exists an online apportionment method satisfying global quota on every ndimensional instance if and only if n ≤3. In the remainder of this section, we prove Theorem 1. Before presenting our analysis, we describe the closely related leaky bucket problem [Adler et al., 2003], using our notation to highlight the parallels. The problem can be described as follows. A set of n buckets is initialized with a common water level that we normalize to 0. In each step t ∈N, a volume vt i ≥0 of water leaks from each bucket i, such that the total leakage H := P i∈[n] vt i is an integer; note that we allow negative loads. An online algorithm must decide how to pour back H unit-volume refills, with the goal of maximizing the minimum load across all buckets and all time steps. Adler et al. showed that a natural greedy algorithm, that refills the currently emptiest bucket one by one at each step, achieves a worst-case deviation from the initial load of at most Hn-1 (where Hn is the nth harmonic number), and that this guarantee is best possible. 6 While the leaky bucket problem is closely related to our setting, there is one key distinction:2 In the apportionment setting, we can only assign seats to parties with positive votes, which corresponds to only allowing refills to buckets that leaked in the current step. In Theorem 1, we show that, perhaps surprisingly, this restriction significantly increases the worst-case deviation: it changes from logarithmic to linear. Our lower bound is tight and is matched by a simple greedy method, which we describe below. The greedy apportionment method. For an n-dimensional instance (vt)t∈N, we define the surplus of party i at step t as st i := At i -V t i , where a negative surplus indicates a deficit. The surplus can be defined recursively by setting s0 i := 0 and st i := st-1 + (at i -vt i) for every t ∈N. For simplicity, we assume throughout this section that, in each step t, the parties are sorted from larger to smaller surplus, i.e., st 1 ≥st 2 ≥· · · ≥st n. This assumption is without loss of generality, since we can rename parties after each step to maintain this invariant without affecting the surplus vector st. For the positive direction of the theorem, we consider the natural greedy apportionment method (Grt)t∈N, defined as follows. For an n-dimensional instance (vt)t∈N and associated house sizes (Ht)t∈N, the method assigns Ht seats in step t to the parties with the largest deficit up to step t-1 plus votes in step t; i.e., Grt(V t-1, At-1, vt) ∈arg min (X i∈S (st-1 i -vt i) : S ⊂{i ∈[n] : vi > 0}, |S| = Ht ) . Each party i in the selected set is then assigned a seat in step t, i.e., at i = 1 for every i ∈ Grt(V t-1, At-1, vt) and at i = 0 for every other i. The following lemma establishes that this method achieves the proportionality guarantees stated in Theorem 1. Lemma 1. The greedy apportionment method is n-1 2 -proportional on every n-dimensional instance for every n ∈N, and strictly 1-proportional on every 3-dimensional instance. For the proof of this result, which is deferred to Section B.1, we adapt the arguments by Adler et al. [2003], who established a worst-case deviation of Hn -1 for the greedy algorithm in the leaky bucket problem. We outline the key ideas used to bound the maximum surplus; the bound on the deficit follows from analogous reasoning. For each step t ∈N0 and index l∈[n], we define γt las the average of the llargest surpluses at time t, plus l/2. Observe that (i) γt n = n/2 for every t; and (ii) at step 0, the sequence (γ0 l)l∈[n] is strictly increasing. Now suppose, for the sake of contradiction, that at some step t the maximum surplus is strictly larger than (n -1)/2. Then γt 1 > n/2, implying that there must exist some i ∈ [n-1] for which γt i > γt i+1. Let t and i be the earliest time and smallest index for which this occurs. We now derive a contradiction by analyzing the change in surplus from step t -1 to t. On the one hand, the total surplus of the i parties with the largest surplus at step t must have increased from t -1 to t, implying that some of these parties received a seat in step t, despite the presence of a party outside this set that also received positive votes. This suggests that the ith and (i + 1)th largest surpluses are not too far apart; otherwise, the greedy method would have allocated a seat to the party outside the set of the i parties with the largest surplus. On the other hand, by the choice of i and the definition of γt l, the gap between the ith and (i + 1) the largest surpluses must be large enough to ensure that γt i-1 0 be fixed, and let (vk)k∈[r] be a partial n-dimensional instance. Let (Xt)t∈N be any online apportionment method and Ar the corresponding cumulative allocation up to step r. Then, there exists a finite value t ∈N with t ≥r and a vote sequence (vk)k∈{r+1,...,t} such that, after the allocation at corresponding to Xt(V t-1, At-1, vt), there exists a party i ∈P such that |st i| ≥(n′ -1)/2 -ε. The proof of this lemma is deferred to Section B.3, but we outline the main ideas here. The result is proven by induction on n′ in steps of size 2. The base case n′ = 1 is trivial. For n′ = 2, the claim follows from the application of a single splitter: If the two parties do not already differ in surplus by at least 1, the splitter can be used to separate their surpluses accordingly. 8 0 1 2 3 4 5 6 7 -1 -0.5 0 0.5 1 t At i -V t i (a) For n = 3, a surplus of 63/64 and a deficit of 127/128 are reached up to t = 7. 0 1 2 3 4 5 6 7 8 9 10 -1.5 -1 -0.5 0 0.5 1 1.5 t At i -V t i (b) For n = 4, a surplus and a deficit of 11/8 are reached up to t = 10. Figure 2: Illustration of adversarially constructed partial instances to induce surpluses and deficits close to (n -1)/2, for n ∈{3, 4}. At each step t ≥1, an (i, j)-splitter is applied for parties i, j whose surpluses then split by one unit at the next step, while keeping the average constant. When multiple parties have the same surplus at some step, they are depicted with a small horizontal offset for ease of understanding. For the inductive step, consider a set P of n′ + 2 parties. We apply the inductive hypothesis to a subset of n′ parties in P, excluding the one with the largest surplus and largest deficit. Repeating this process at most three times, we reach a situation in which two parties in P have surplus (or deficit) arbitrarily close to (n′ -1)/2, say, at least (n′ -1 -ε)/2. For simplicity, assume that two parties in P have a surplus (and not a deficit) arbitrarily close to (n′ -1)/2. Applying a splitter to these two parties increases the surplus of the one with the larger surplus and decreases that of the other. In particular, after this splitter, one of the two must have a surplus of at least (n′ -1 -ε)/2 + 1/2. To continue increasing this value, another splitter is needed. However, since the other party's surplus is now one unit smaller, a direct application is not feasible. Instead, we apply the inductive hypothesis once again to a subset of n′ parties in P, again excluding the two with the largest and smallest surplus. As a result, at least one party has a surplus or a deficit exceeding (n′ -1 -ε)/2. In the former case, we are again in the position to apply a splitter to the two parties with the largest surpluses, which this time will leave a party with a surplus of at least (n′ -1 -ε)/2 + 3/4. Otherwise, we now have two parties with a deficit exceeding (n′ -1 -ε)/2, and we apply a splitter between them. By repeatedly applying the induction hypothesis and a subsequent splitter to the surplus or deficit end, we increase the maximum deviation. In particular, the deviation increases by at least 1/2lafter the lth iteration of this procedure at the corresponding end. After finitely many steps, this process yields a surplus or deficit of at least (n′ + 1)/2 -ε. Figure 2 illustrates the repeated application of splitters to get a surplus or deficit approaching (n -1)/2, for n ∈{3, 4}. Relation to standard apportionment methods. When the same vote vector is repeated in every time step, an online apportionment method naturally corresponds to a house-monotone (offline) method, and vice versa. Interestingly, the greedy apportionment method introduced 9 in this work does not coincide with any of the standard offline rules. Although it might appear similar to the Hamilton method, as both allocate seats sequentially to the parties with the largest deficit, this equivalence does not hold. For instance, for three parties with vote vector vt = (2/3, 29/120, 11/120), our greedy method yields the allocation (4, 2, 1) after seven steps, whereas the Hamilton method with seven seats outputs (5, 2, 0). This discrepancy is closely related to the Alabama paradox, as the Hamilton method may withdraw a seat from a party when the house size increases, while an online method cannot revoke past allocations. The Balinski-Young quota method, in contrast, satisfies both house-monotonicity and quota compliance and could therefore be adapted to operate in an online fashion while respecting global quota at every step. This is particularly noteworthy since our greedy method, although optimal in the worst case, does not exhibit this property in this special setting. To see this, consider three large parties and two small ones, with repeated vote vector vt = (49/150, 49/150, 49/150, 1/100, 1/100). After 41 steps, the cumulative allocation is (13, 13, 13, 1, 1). At time t = 42, the next seat is assigned to one of the large parties, as their deficits exceed those of the small ones. However, at t = 43 the other two large parties both have remainders 43·49/150-13 = 157/150 > 1, forcing both to receive an additional seat to maintain global quota. 4 Randomized Methods and Ex-ante Proportionality We now shift our focus to randomized online apportionment methods. We introduce the formal notion of such a method in what follows. A randomized online apportionment method selects, given an n-dimensional instance (vt)t∈N, a random subset of parties Xt ⊂[n] for every t ∈N, as follows: (i) For t = 0, we define V 0 i := 0 and A0 i := a0 i := 0 for every i ∈[n]. (ii) For t ∈N, we define the cumulative vote vector as V t-1 i := Pt-1 k=0 vk i for every i ∈[n], where v0 := 0, and the cumulative allocation vector as At-1 i := Pt-1 k=0 ak i . The set of feasible histories Θt-1 is given by all histories (V t-1, At-1) that occur with strictly positive probability under the method up to step t -1. There is a function yt : Θt-1 × [0, 1)n × [n] Ht →[0, 1], such that for each feasible history (V t-1, At-1) ∈Θt-1, yt(V t-1, At-1, vt, ·) is a probability distribution over subsets of [n] of size Ht and yt(V t-1, At-1, vt, S) = 0 for every S such that there is a party i ∈S with vt i = 0. The outcome in step t is then given by a random subset Xt(V t-1, At-1, vt) distributed according to yt(V t-1, At-1, vt, ·), and the allocation at = at(V t-1, At-1, vt) ∈{0, 1}n is derived from this set: at i = 1 if and only if i ∈Xt(V t-1, At-1, vt). If the method (yt)t∈N is only defined up to a finite time T ∈N, we refer to it as a partial online apportionment method. Given a method (yt)t∈N and an instance (vt)t∈N, let At := {At : (V t, At) ∈Θt} be the set of cumulative allocation vectors that occur with strictly positive probability after step t. Moreover, let π(yt, V t, At) denote the probability of reaching allocation At at step t under method (yt)t∈N. We initialize with A0 := {0}, V 0 := 0, and π(y0, V 0, A0) := 1. Now, ex-ante proportionality translates to the randomized setting as follows. An online apportionment method (yt)t∈N is ex-ante proportional if for every step t ∈N and party i ∈[n], the expected number of seats assigned to i is equal to its share, i.e., X At-1∈At-1 π(yt-1, V t-1, At-1) · at i(V t-1, At-1, vt) = vt i. (1) 10 We call a partial online apportionment method (yt)t∈[T] ex-ante proportional if it satisfies Equation (1) for every step t ∈[T]. Corollary 1 states that an online apportionment method can satisfy global quota for all ndimensional instances (vt)t∈N if and only if n ≤3. The main result of this section establishes that, for n ≤3, we can randomize over methods satisfying global quota while additionally fulfilling ex-ante proportionality, thus giving ex-ante and ex-post proportionality guarantees. Theorem 2. There exists an online apportionment method satisfying global quota and ex-ante proportionality for all n-dimensional instances if and only if n ≤3. We remark that the impossibility for n ≥4 follows directly from Corollary 1. On the other hand, when n = 2, a simple construction suffices to prove the positive direction. Indeed, we can apply the systematic sampling procedure (a.k.a., Grimmett's apportionment method [2004]) to party 1, defined as follows: Sample a value λ ∈[0, 1) uniformly at random and assign a seat to party 1 in step t if and only if ⌊V t-1 1 + λ⌋ 0, global quota implies no restriction on whether it may or may not receive a seat in step 2. At the same time, ex-ante proportionality requires that party 3 is assigned a seat with probability 0.2. However, allocating a seat to party 3 under the current allocation would violate global quota for party 2, which also has not received a seat in step 1 but is surpassing its previous upper share in step 2. This illustrates that, for n = 3, ensuring both global quota and ex-ante proportionality requires tracking and conditioning on the full history of seat allocations. Consequently, designing valid online apportionment methods becomes significantly more intricate than in the two-party case. We overcome this challenge and prove Theorem 2 by introducing a family of randomized online apportionment methods that we refer to as network flow methods. These methods are defined recursively and rely on constructing a suitable flow network in each step. Consider an instance (vk)k∈N and a partial online allocation method (yk)k∈[t] that satisfies global quota and ex-ante proportionality. For any cumulative allocation At ∈At, define the set of parties that have received their upper share of seats after step t as u(At) := n i ∈[n] : At i = l Pt k=1 vk i mo . Let Ut := {u(At) : (V t, At) ∈Θt} be the set of all such sets observed with strictly positive probability up to step t. With a slight overload of notation, we define π(u) := X At∈At:u(At)=u π(yt, V t, At) as the probability that the upper-quota set u ∈U t occurs after step t. We now define the flow network FN(V, A, v, π), which constitutes the basis of our recursive construction. 3We remark that this observation was also made by Naor et al. [2025] in the context of online dependent rounding; in Section 5, we further develop the consequences of our results for dependent rounding algorithms in online settings. 11 Flow network construction. Let (vk)k∈N be an n-dimensional instance and (yk)k∈[t] a partial online apportionment method that satisfies global quota and ex-ante proportionality up to step t ∈ N. Define U := {u(A) : (V, A) ∈Θt}. The corresponding flow network FN(V, A, v, π) for step t + 1 is given by (a) vertex set P := {o, d} ∪U ∪[n], (b) edge set E := {(o, u) : u ∈U} ∪{(u, i) : u ∈U, i ∈[n]} ∪{(i, d) : i ∈[n]}, and (c) upper and lower capacities c, l: E →R+ given by c((o, u)) = π(u), c((u, i)) = ( 0 if i ∈u and ⌈Vi⌉= ⌈Vi + vi⌉or vi = 0 π(u)/Pn i=1 vi else, c((i, d)) = vi/Pn i=1 vi, l((o, u)) = 0, l((u, i)) = ( π(u)/Pn i=1 vi if i /∈u and ⌊Vi⌋+ 1 = ⌊Vi + vi⌋, 0 else, l((i, d)) = 0. We illustrate the construction of a network flow method by the following example. Consider an instance with three parties, and suppose that in the first election the votes are given by v1 = (0.6, 0.3, 0.1), which yields H1 = 1. In this case, the unique partial network flow method at t = 1 is defined by y1(V 0, A0, v1, {i}) = v1 i for all i ∈[3], which implies the edge capacities c(o, {i}) = π({i}) = v1 i in the flow network FN(V 1, A1, v2, π). Now consider the second vote vector v2 = (0.5, 0.2, 0.3) with H2 = 1. The global quota property enforces that party 1 must receive a seat in step 2 if it was not allocated one in round 1, as ⌊V 1 1 ⌋+1 = ⌊V 1 1 + v2 1⌋. Conversely, for each party i ̸= 1, the condition ⌈V 1 i ⌉= ⌈V 1 i + v2 i ⌉implies that they must not be allocated a seat in round 2 if they were already selected in round 1. These constraints are encoded in the capacities of the edges e = (u, i) in the flow network. The current vote vector v2 also determines the capacities c((i, d)) = v2 i for every i ∈[3]. The complete flow network is depicted in Figure 3. Edges with zero upper capacity are omitted, and edges with equal lower and upper capacities are highlighted in red. The network admits a unique feasible (o, d)-flow f of value 1, given by f(e) = 0.1 if e = ({1}, 1), 0.2 if e = ({1}, 2), 0.3 if e = ({1}, 3), 0 if e = ({2}, 3) or e = ({3}, 2), and c(e) otherwise. For a detailed analysis of feasibility, we refer to Case 1.1.2 in the proof of Lemma 5 in Section C.2. We now describe how to construct a partial network flow method (yk)k∈[t] recursively for any number of parties n. 12 o 2 1 3 1 2 3 d 0.6 0.3 0.1 0.5 0.2 0.3 0.1 0.2 0.3 0.3 0 0.1 0 Figure 3: The flow network FN(V 1, A1, v2, π) for v1 = (0.6, 0.3, 0.1) and v2 = (0.5, 0.2, 0.3). Edges with upper capacity zero are omitted. The upper capacities of the remaining edges are c((o, {i})) = π({i}) = v1 i , c(({i}, j)) = π({i})/H1 = v1 i , and c((i, d)) = v2 i . An edge is highlighted in red if the lower capacity equals the upper capacity. The lower capacities of the remaining edges are zero. The labels indicate the unique flow of value 1. Step t = 1. We define y1 as a probability distribution over subsets of [n] of size H1. Since vector v1 lies in the hypersimplex ∆(n, H1), there exists a convex decomposition v1 = Pm j=1 λj · zj, where each zj ∈{0, 1}n has exactly H1 ones and λ1, ..., λm ∈[0, 1] with Pm j=1 λj = 1. Let Zj ⊆[n] denote the support of zj, i.e., i ∈Zj if and only if zj,i = 1. We then define y1(V 0, A0, vt, Zj) = λj for every j ∈[m] and y1(V 0, A0, vt, S) = 0 for all other subsets S ⊆[n]. Step t ≥2. Assume we have already constructed a partial network flow method (yk)k∈[t]. Let V t, and At denote the cumulative vote and allocation vectors, and π(u) the distribution over upperquota sets u ∈Ut. Construct the flow network FN(V t, At, vt+1, π) for step t + 1. If this network admits a feasible (o, d)-flow f of value 1, we can extend the partial method to step t + 1 as follows. Fix a feasible allocation At that occurs with strictly positive probability. Let u = u(At) denote the corresponding upper-quota set. Using flow f, we define the fractional assignment vector z(u) ∈ [0, 1]n by zi(u) := f(u, i) · Ht+1 π(u) . By construction, z(u) lies in the hypersimplex ∆(n, Ht+1). Again, decompose z(u) = Pm j=1 λj · zj into a convex combination of {0, 1}-vectors zj ∈{0, 1}n with exactly Ht+1 ones and let Zj be their supports. We define yt+1(V t, At, vt+1, Zj) = λj for every j ∈[m] and yt+1(V t, At, vt+1, S) = 0 for all other subsets S ⊆[n]. This completes the construction of the partial network flow method. We now show that a partial network flow method satisfies all three desired properties. 13 Lemma 4. Let (yt)t∈[T] be a partial network flow method. Then, for every t ∈[T] and every feasible history (V t, At) ∈Θt that occurs with strictly positive probability, we have At i ∈{⌊V t i ⌋, ⌈V t i ⌉} for every i ∈[n]. Furthermore, (yt)t∈[T] satisfies Equation (1) for every t ∈[T]. The proof uses induction to show that a partial network flow method satisfies global quota and ex-ante proportionality by construction. We verify that only parties with strictly positive votes are assigned seats and that these lie within the share bounds using the structure of the network. The flow conservation finally yields that the expected number of seats assigned in each step matches the vote vector. The full proof is deferred to Section C.1. If, for every t ∈N, the corresponding flow network FN(V t, At, vt+1, π) admits a flow of value 1, the recursively defined partial network flow method (yt)t∈[T] extends to a network flow method (yt)t∈N. By Lemma 4, such a method satisfies global quota and ex-ante proportionality. The next lemma guarantees that, for any number of parties n ≤3 and partial network flow method (yt)t∈[T], the corresponding flow network FN(V T , AT , vT+1, π) always admits a feasible flow of value 1. Thus, for any T ∈N, the partial network flow method (yt)t∈[T] can be extended to a partial network flow method (yt)t∈[T+1]. Consequently, a partial network flow method yields a valid network flow method. Lemma 5. Let n ≤3 and let (yk)k∈[t] be a partial network flow method with corresponding flow network FN(V t, At, vt+1, π) for step t + 1. Then this flow network admits a feasible (o, d)-flow of value 1, i.e., the partial network flow method can be extended to (yk)k∈[t+1]. The proof of Lemma 5 is deferred to Section C.2. It proceeds by a case distinction over the possible configurations of cumulative allocation vectors and vote increments for the three parties. Feasibility is shown either by explicitly constructing a valid flow or by reducing the network to a transshipment problem on a bipartite graph. This ensures that in every case the flow network admits a feasible flow of value 1, completing the recursive construction. Theorem 2 now follows by combining Lemmas 4 and 5. Having shown that network flow methods satisfy all desired properties and are well-defined for up to three parties, we now establish that this family of methods in fact contains all online apportionment methods satisfying all three properties. Proposition 2. Any online apportionment method satisfying global quota and ex-ante proportionality is a network flow method. In the proof, we construct a network flow for each step that replicates the decisions of a given online apportionment method satisfying global quota and ex-ante proportionality. We verify that all capacity constraints are met, flow conservation is attained, and that the flow has a value of 1, which establishes that any such method can be obtained as a network flow method. The full proof is deferred to Section C.3. With the preceding proposition, we conclude that every online apportionment method satisfying global quota and ex-ante proportionality is a network flow method and can therefore be obtained via the construction described above. Indeed, the proof of Lemma 5 provides an explicit recursive procedure for building the method as it defines the flow network at each step t based on the current allocation and voting vectors. For the case of n = 3 parties, it is worth mentioning that in all but two cases, the resulting flow of value 1 is uniquely determined. Consequently, the seat assignment may only differ when either (a) one seat is to be allocated and each cumulative allocation u ∈U t assigns the upper share of seats to a single party, while in the current step, no party receives enough votes to increase their upper share, or 14 (b) two seats are to be allocated, and each cumulative allocation u ∈U t assigns the upper share of seats to two parties, while in the current step, all parties receive enough votes to increase their upper share. In both cases, the flow network admits multiple feasible flows of value 1, leading to different valid seat allocations. We conclude this section by discussing barriers to further positive results in the realm of randomized online apportionment methods. Network construction and impossibility for n ≥4. The following example illustrates why the flow construction fails when n ≥4, as the corresponding network no longer admits a feasible flow of value 1. Consider an instance with four parties and voting vector v1 = (1/2, 1/2, 1/2, 1/2). A partial method assigns H1 = 2 seats in step 1. Without any loss, assume that y1(V 0, A0, v1, {1, 2}) = p > 0, i.e., parties 1 and 2 obtain a seat in the same allocation with strictly positive probability. Now consider the voting vector v2 = (1/2, 1/2, 0, 0). In this case, for all i ∈[n], we have ⌈V 1⌉= ⌈V 1 +v2⌉, so global quota implies that no party may receive an additional seat in step 2 if parties 1 and 2 received a seat each in step 1. The flow network captures this by setting the upper capacity of all edges ({1, 2}, i) to zero. Consequently, the edge (o, {1, 2}) must be assigned a flow of value 0 even though it has a strictly positive capacity of p. Since the total capacity of the outgoing edges of o equals 1, no flow of value 1 can exist. This demonstrates that the construction of a partial network flow method fails at step t = 2. Negative correlation. Naor et al. [2025] showed that, when n = 1, a notion of negative correlation over time can be added on top of proportionality notions. In our notation, negative correlation requires that for every party i ∈[n] and every pair of time steps t, t′ ∈N, P[at i = 1 | at′ i = 1] ≤ P[at i = 1]. The result of Naor et al. implies that, when n = 1, there exists an online apportionment method satisfying global quota, ex-ante proportionality, and negative correlation.4 It is easy to see that this can be immediately extended to n = 2 parties, since one party receives a seat if and only if the other does not. However, it does not extend to three parties. To see this, consider the following instance with vote vectors v1 = (0.5, 0.5, 0), v2 = (0, 0.6, 0.4), v3 = (0, 0.2, 0.8). The outcome of the unique network flow method yields the following distribution over assignments: P[a1 1 = 1, a2 2 = 1, a3 3 = 1] = 0.5, P[a1 2 = 1, a2 2 = 1, a3 3 = 1] = 0.1, P[a1 2 = 1, a2 3 = 1, a3 2 = 1] = 0.2, P[a1 2 = 1, a2 3 = 1, a3 3 = 1] = 0.2. In particular, we observe that P[a3 2 = 1|a1 2 = 0] = 0 0, and C(u, t) = ∞for each u and t. We remark that Byrka and Srinivasan [2018] study the graph case (i.e., d = 2) and developed a 2-approximation algorithm, improving on the 2T-approximation by Swamy and Shmoys [2012]. In what follows, we show how to use Theorem 1 and Theorem 2 to construct integer solutions for MMHSC, online, with guarantees in terms of near-feasibility or objective approximation factor. For the sake of exposition simplicity, suppose that the fractional solution y⋆is binding on every constraint (3), i.e., there is no spare capacity over the vertices at every step. 16 Online near-feasible solutions. We use Theorem 1 to construct an online near-feasible solution for MMHSC. For each vertex u and every step t, consider vt i(u) = y⋆(u, i, t) -⌊y⋆(u, i, t)⌋for each resource i. At every t, for each vertex u we use Theorem 1 to get binary values at 1(u), . . . , at n(u) satisfying the following properties: (i) Pn i=1 at i(u) = C(u, t) -Pn i=1⌊y⋆(u, i, t)⌋, (ii) | Pt l=1 al i(u) -Pt l=1 vl i(u)| ≤(n -1)/2. At every t, the solution implemented by the decision-maker is equal to Y (u, i, t) = ⌊y⋆(u, i, t)⌋+at i(u) for each u and each i. Observe that (i) directly implies that Pn i=1 Y (u, i, t) = C(u, t) for each u ∈N, i.e., (3) is satisfied. On the other hand, for each i ∈[n] and e ∈E, X u∈e t X l=1 Y (u, i, t) = X u∈e t X l=1 ⌊y⋆(u, i, l)⌋+ t X l=1 al i(u) ! ≥ X u∈e t X l=1 ⌊y⋆(u, i, l)⌋+ t X l=1 vl i(u) -n -1 2 ! = X u∈e t X l=1 y⋆(u, i, l) -n -1 2 ! ≥D(i, t) -d(n -1) 2 , where the first inequality holds from (ii) and the last inequality is a consequence of y⋆satisfying (2) and the hypergraph being d-uniform. We conclude that Y satisfies (2) within an additive violation of d(n -1)/2. In particular, when n = 3, Corollary 1 implies that the violation is at most d -1. Online approximate min-cost solutions. Suppose now that the solution y⋆received by the decision-maker implies a cost Cost(y⋆), where Cost is a linear function, and we have n = 3 resources. We use Theorem 2 to construct an online α-approximate solution for MHSC under resource augmentation, where α := maxj∈{1,2,3},l∈[T](d + D(j, l) -1)/D(j, l). For each vertex u and every step t, consider vt i(u) = αy⋆(u, i, t) -⌊αy⋆(u, i, t)⌋for each resource i. At every t and for each vertex u, we use Theorem 1 to get binary values at 1(u), . . . , at n(u) satisfying the following properties: (i) ⌊αC(u, t)⌋-Pn i=1⌊αy⋆(u, i, t)⌋≤Pn i=1 at i(u) ≤⌈αC(u, t)⌉-Pn i=1⌊αy⋆(u, i, t)⌋, (ii) Pt l=1 al i(u) ∈ nj Pt l=1 vl i(u) k , l Pt l=1 vl i(u) mo , (iii) E[at i(u)] = vt i(u). At every step t, the solution implemented by the decison-maker is equal to Y (u, i, t) = ⌊αy⋆(u, i, t)⌋+ at i(u) for each u and each i. Observe that (i) directly implies that Pn i=1 Y (u, i, t) ≤⌈αC(u, t)⌉ for each u ∈N, i.e., (3) is satisfied under a resource augmentation of factor α. On the other hand, for each resource i ∈[3], following the analysis of Naor et al. [2025] we can easily show that P u∈e Pt l=1 Y (u, i, t) ≥D(i, t), that is, Y satisfies (3). Finally, thanks to the ex-ante proportionality property (iii), we conclude that E[Cost(Y )] ≤α · Cost(y⋆). 6 Concluding Remarks In this work, we have introduced an online version of the apportionment problem, where indivisible seats are to be allocated immediately and irrevocably to parties in every time step in proportion to 17 the parties' votes, which are revealed online and can be set adaptively. We have imposed a notion of local quota compliance upfront, requiring that each party receives, in each step, a number of seats equal to its fraction of votes, rounded up or down. We have established that a greedy method provides the best-possible deviation, linear in n, between the cumulative votes and the cumulative seats that a party has received at any step. This implies that a global quota axiom, requiring the same principle as local quota but for the cumulative values in each step, can be satisfied if and only if n ≤3. When this condition holds, we have shown that the proportionality guarantee given by global quota can be enhanced using randomization: For every instance, a lottery over methods satisfying global quota assigns to each party its proportional number of seats in expectation. Our proof is based on network flow techniques, providing a full characterization of such methods and allowing for efficient implementation. Our work has natural connections and implications for related settings in online dependent rounding, as our results in Sections 3 and 4 translate directly to online rounding guarantees for multidimensional instances where we receive an n-dimensional marginal vector at each step. In particular, the case of n = 1 party captures the online level-set problem studied very recently by Naor et al. [2025]. Several directions for future work arise. Perhaps the more immediate one concerns the validity of Theorem 2 when global quota is replaced by approximate proportionality for n ≥4. Specifically, one could aim to randomize over deterministic methods that attain the best possible proportionality guarantees from Theorem 1 while fulfilling ex-ante proportionality. On the other hand, it is natural to ask whether lower expected deviations or stronger axiomatic properties can be achieved in an online model where the adversary is more restricted. Two interesting possibilities include the study of randomized methods against an oblivious adversary, that cannot set the votes based on previous realizations of the method's randomness, and a fully stochastic model, where votes come from (potentially correlated) known distributions in each step. Acknowledgements This work was partially supported by a Structural Democracy Fellowship through the Brooks ático (CMM) BASAL fund FB210005 for center of excellence from ANID-Chile. 18 Appendix A Proofs Deferred from Section 2 A.1 Proof of Proposition 1 Proposition 1. For every n ∈N and n-dimensional instance (vt)t∈[T], there exists an offline apportionment method that satisfies global quota and ex-ante proportionality. Proof. We fix n ∈N and an n-dimensional instance (vt)t∈[T]. We consider a capacitated network G := (P, E) given by (a) vertex set P := {o, d} ∪{ut}t∈[T] ∪{wt i}t∈[T],i∈[n], (b) edge set E := {(o, ut) : t ∈[T]} ∪{(ut, wt i) : t ∈[T], i ∈[n]} ∪{(wt i, wt+1 i ) : t ∈[T -1], i ∈[n]} ∪{(wT i , d) : i ∈[n]}, and (c) upper and lower capacities c, l: E →R+ given by c((o, ut)) = Ht, c((ut, wt i)) = 1, c((wt i, wt+1 i )) = ⌈V t i ⌉, l((wt i, wt+1 i )) = ⌊V t i ⌋, c((wT i , z)) = ⌈V T i ⌉, l((wT i , d)) = ⌊V T i ⌋. For any feasible integral (o, d)-flow f : E →N0 on this network, we define the apportionment method (Xt(f))t∈N by setting Xt(f) = {i ∈N : f((ut, wt i)) = 1} for every t ∈N. Claim 1. For any feasible integral (o, d)-flow f : E →N0 on G, the offline apportionment method (Xt(f))t∈[T] satisfies global quota. Proof. To prove this claim, we simply observe that, for any feasible integral (o, d)-flow f : E →N0 and every i ∈[n], it holds |{k ∈[t] : i ∈Xk(f)} = t X k=1 f((ut, wt i)) = f((wt i, wt+1 i )) ∈{⌊V t i ⌋, ⌈V t i ⌉} for every t ∈[T -1], and |{k ∈[T] : i ∈Xk(f)} = T X k=1 f((ut, wt i)) = f((wT i , d)) ∈{⌊V T i ⌋, ⌈V T i ⌉}. Indeed, in both cases the first equality follows from the definition of (Xt(f))t∈[T], the second one from the flow conservation constraint at vertex wt i, and the inclusion from the capacity constraint on the edge (wt i, wt+1 i ) or (wT i , d). We now consider the fractional (o, d)-flow f∗: E →R+ defined as follows: f∗((o, ut)) = Ht for every t ∈[T], f∗((ut, wt i)) = vt i for every t ∈[T], i ∈[n], f∗((wt i, wt+1 i )) = V t i for every t ∈[T -1], i ∈[n], f∗((wT i , d)) = V T i for every i ∈[n]. 19 The flow f∗is easily seen to be a feasible (o, d)-flow on G. Therefore, because of the integrality of the network flow polytope [Ahuja, Magnanti, and Orlin, 1988], it admits a convex decomposition into integer (o, d)-flows. That is, there exist an integer q ∈N, integer (o, d)-flows {fj}j∈[q], where fj : E →N0 for every j ∈[q], and coefficients {λj}j∈[q], where λj ∈(0, 1) for every j ∈[q] and Pq j=1 λj = 1, such that q X j=1 λjfj(e) = f∗(e) for every e ∈E. (4) We now consider the randomized offline apportionment method that runs each deterministic method (Xt(fj))t∈N with probability λj, for j ∈[q]. From Claim 1, we conclude that this method satisfies global quota. From Equation (4), we conclude that it satisfies ex-ante proportionality. B Proofs Deferred from Section 3 B.1 Proof of Lemma 1 Lemma 1. The greedy apportionment method is n-1 2 -proportional on every n-dimensional instance for every n ∈N, and strictly 1-proportional on every 3-dimensional instance. Proof. We begin with the case of general n. We let, for the sake of contradiction, (vk)k∈N be an n-dimensional instance and t ∈N be a time step such that |st i| > (n -1)/2 for some i ∈[n]. Since the parties are ordered from larger to smaller surplus, this implies either st 1 > (n -1)/2 or st n (n -1)/2, and introduce some additional notation. For each k ∈[t]0 and l∈[n], we let σk l:= 1 l P i∈[l] sk i denote the average surplus of the lparties with the largest surplus and we define γk l:= σk l+ l/2. Observe that, for every k ∈[t]0 and i ∈[n -1] γk i > γk i+1 ⇐⇒1 i X j∈[i] sk j + i 2 > 1 i + 1 X j∈[i+1] sk j + i + 1 2 ⇐⇒ 1 + 1 i X j∈[i] sk j -i + 1 2 > X j∈[i+1] sk j ⇐⇒sk i+1 (n -1)/2 implies γt 1 > n/2 = γt n. Conversely, we have 1/2 = γ0 1 n/2 and γk i > γk i+1 for some i ∈[n -1]; let k∗be the minimum such k. Once k∗is fixed, we fix i∗to the minimum i ∈[n -1] such that γk∗ i∗> n/2 and γk∗ i∗> γk∗ i∗+1. The following claim will allow us to reach a contradiction. Claim 2. It holds sk∗ i∗≤sk∗ i∗+1 + 1. Proof. We begin by noting that γk∗ i∗> γk∗-1 i∗ . If this was not true, we would have γk∗-1 i∗ ≥γk∗ i∗> n/2, and thus γk∗-1 i > γk∗-1 i+1 for some i ∈[n -1], contradicting the choice of k∗. The inequality γk∗ i∗> γk∗-1 i∗ implies σk∗ i∗> σk∗-1 i∗ , which in turn implies Pi∗ j=1 sk∗ j > Pi∗ j=1 sk∗-1 j . Since n X j=1 sk∗ j = n X j=1 sk∗-1 j = 0, there must exist parties i ∈[i∗] and j ∈{i∗+ 1, . . . , n} such that the surplus of the former increases from step k∗-1 to step k∗and that of the latter decreases. Formally, if ˆsi is the surplus of party 20 i in step k∗-1 and ˆsj is the surplus of party j in step k∗-1 (which are not necessarily equal to sk∗-1 i and sk∗-1 j due to reordering), it holds sk∗ i > ˆsi and sk∗ j vk∗ i and ak∗ j sk∗ i∗+1+1. Since i ∈[i∗], j ∈{i∗+1, . . . , n}, and the parties are ordered from larger to smaller surplus, this implies sk∗ i > sk∗ j + 1 ⇐⇒ˆsi + (ak∗ i -vk∗ i ) > ˆsj + (ak∗ j -vk∗ j ) + 1 =⇒ˆsi -vk∗ i > ˆsj -vk∗ j . But since the greedy method assigns seats, in step k∗, to the parties lwith the smallest value of ˆsl-vk∗ l among those that receives positive votes, this contradicts the above statement that ak∗ i = 1 and ak∗ j = 0 γk∗ i∗> n/2, contradicting the choice of i∗as the minimum i ∈[n -1] such that γk∗ i > n/2 and γk∗ i > γk∗ i+1. We now consider the case where st n σk i + i + 1 2 . (6) Note that st n γ0 2 > · · · > γ0 n = -n/2, because s0 i = 0 for every i ∈[n]. Since γk n = -n/2 for every k ∈[t]0, there exists k ∈[t] such that γk i ˆsj. Since ˆsl= sk∗ l+ (ak∗ l-vk∗ l) for each l∈[n], these inequalities yield ak∗ i vk∗ j . Since vk∗ l ∈[0, 1) and ak∗ l ∈{0, 1} for each l∈[n], we conclude that ak∗ i = 0 sk∗ n + 1 and Claim 3 states the opposite. When i∗≥2, we obtain sk∗ n+1-i∗≥sk∗ n-i∗-1 > σk∗ i∗+ i∗+ 1 2 -1 = (i∗-1)σk∗ i∗-1 + sk∗ n+1-i∗ i∗ + i∗-1 2 . Indeed, the first inequality comes from Claim 3 and the second one from Property (6); the equality follows by splitting the average σk∗ i∗of the first i∗terms into the average σk∗ i∗-1 of the first i∗-1 terms and the i∗th term sk∗ n+1-i∗. Rearranging terms, we obtain sk∗ n+1-i∗> σk∗ i∗-1 + i∗/2. But then, Property (6) implies γk∗ i∗-1 γk∗-1 i∗ and γk∗ i∗ -1, which will yield a similar contradiction as in the proofs of Claims 2 and 3. Indeed, sk∗ 1 > ˆs1 implies ak∗ 1 = 1 and sk∗ 3 0 be fixed, and let (vk)k∈[r] be a partial n-dimensional instance. Let (Xt)t∈N be any online apportionment method and Ar the corresponding cumulative allocation up to step r. Then, there exists a finite value t ∈N with t ≥r and a vote sequence (vk)k∈{r+1,...,t} such 23 that, after the allocation at corresponding to Xt(V t-1, At-1, vt), there exists a party i ∈P such that |st i| ≥(n′ -1)/2 -ε. Proof. We fix n ∈N and prove the lemma by induction over n′, the size of the subset of parties we consider. Throughout this proof, whenever we have an n′-dimensional vote vector with entries in P ⊆[n], namely v′ ∈[0, 1]P , we refer to the vector v ∈[0, 1]n defined as vi = ( v′ i if i ∈P 0 otherwise, as the n-dimensional extension of v′. This definition naturally carries over for the allocation vector. Note that, from our definition of an online apportionment method, all entries of these vote and allocation vectors that do not correspond to parties in P are always fixed to 0. Since we use induction steps from n′ to n′ + 2, we establish the validity of the lemma for two base cases, namely n′ = 1 and n′ = 2. The case n′ = 1 is trivial, as the conclusion of the lemma only requires |st i| ≥-ε, and we know that |st i| ≥0 > -ε. For the case n′ = 2, we fix r, ε, (vk)k∈[r], (Xt)t∈N, and Ar as in the statement, we denote the parties by P = {p1, p2}, and we assume without loss of generality that sr p1 ≥sr p2. If sr 1 ≥1/2 -ε or sr n ≤-(1/2 -ε), we are done. Otherwise, it holds that sr 1 -sr 2 0 is a positive number, (vk)k∈[r] is a partial n-dimensional instance, and Ar = P k∈[r] ak is the aggregated allocation up to step r. Without loss of generality, we assume, as usual, that sr p1 ≥sr p2 ≥· · · ≥sr pn′. Given such a tuple, we denote the n-dimensional extension of the vote sequence satisfying the properties of the lemma by ˆV (P, ε, (vk)k∈[r], Ar), and refer to it as a (P, ε)-booster (because it boosts the surplus or deficit of some party in P). That is, the (P, ε)- booster ˆV (P, ε, (vk)k∈[r], Ar) is a partial n-dimensional instance (vk)k∈{r+1,...,t}, with t ≥r finite and vk ∈[0, 1]n for k ∈{r + 1, . . . , t}, such that for any allocation (ak)k∈{r+1,...,t} produced by an online apportionment method, with ak ∈{0, 1}n for k ∈{r + 1, . . . , t}, it holds that st p1 ≥(n′ -1)/2 -ε or st pn′ ≤-((n′ -1)/2 -ε) (potentially after reordering p1, . . . , pn′). To prove the claim for n′ + 2, we fix r, ε, V r, and Ar as in the statement. We assume, as usual, that sr p1 ≥sr p2 ≥· · · ≥sr pn′+2. We apply the inductive hypothesis on the n′-dimensional instance that results from restricting to parties p2, . . . , pn′+1, but with a smaller deviation of ε/2. That is, we consider the n-dimensional vote sequence (vk)k∈{r+1,...,t} = ˆV {p2, . . . , pn′+1}, ε/2, (vk)k∈[r], Ar for some finite t. After reordering the parties in P so that st p1 > st p2 > · · · > st pn′+2, we know by the induction hypothesis that we have reached an instance with st p1 ≥(n′ -1)/2 -ε/2 or st pn′+2 ≤-((n′ -1)/2-ε/2). We again apply a ({p2, . . . , pn′+1}, ε/2)-booster, and then a third time if necessary. After applying this procedure at most three times, we now have obtained an instance, say at step t′, such that st′ p2 ≥(n′ -1)/2 -ε/2 or st′ pn′+1 ≤-((n′ -1)/2 -ε/2), since each booster fixes a new party satisfying one of these two conditions. 24 In the remainder of the proof, we repeatedly apply either a (p1, p2)-splitter or a (pn′+1, pn′+2)- splitter, and then a {p2, . . . , pn+1}-booster. The splitters will increase the surplus of party p1 or the deficit of party pn′+2, at the cost of decreasing the surplus of p2 or the deficit of pn′+1, respectively. However, the inductive hypothesis allows us to boost either this surplus or deficit so that it surpasses (n′ -1)/2 -ε/2 and can be used for a splitter again. By repeating this procedure, we can make the surplus of party p1 or the deficit of party pn+2 arbitrarily close to (n′ -1)/2 + 1 -ε/2, and thus surpass (n′ + 1)/2 -ε after a finite number of steps. The argument can be formalized through the following algorithm. We initialize l= 0, tl= t′ and repeat the following procedure until stl 1 ≥(n′ + 1)/2 -ε1 or stl n′+2 ≤(n′ + 1)/2 -ε1: (i) If stlp2 ≥(n′ -1)/2 -ε/2, apply a (p1, p2)-splitter; i.e., take vtl+1 = ˆv(p1, p2); (ii) else, if stlpn′+1 ≤-((n′-1)/2-ε/2), apply a (pn′+1, pn′+2)-splitter; i.e., vtl+1 = ˆv(pn′+1, pn′+2); (iii) apply a {p2, . . . , pn′+1}-booster; i.e., take (vk)k∈{tl+2,...,t′′} = ˆV {p2, . . . , pn′+1}, ε/2, (vk)k∈[tl+1], Atl+1 for some finite t′′; (iv) update l←l+ 1 and tl←t′′. Note that, after each iteration l≥1, the following invariant (which we showed for t0) is preserved by the induction hypothesis: stlp2 ≥(n′ -1)/2 -ε/2 or stlpn′+1 ≤-((n′ -1)/2 -ε/2). Observe that stlp1 -((n′ + 1)/2 -ε as long as the algorithm does not terminate. Thus, whenever stlp2 ≥(n′ -1)/2 -ε/2 holds, we have stlp1 -stlp2 0, we know that we have either applied L/2 + 1 type I iterations or L/2 + 1 type II iterations. Thus, Claim 4 implies that either sL p1 ≥ (n′ -1)/2 -ε or sL pn′+2 ≤- (n′ -1)/2 -ε holds if n′ + 1 2 -ε 2 - 1 2L/2 ≥n′ + 1 2 -ε ⇐⇒L ≥-2 log(ε/2). C Proofs Deferred from Section 4 C.1 Proof of Lemma 4 Lemma 4. Let (yt)t∈[T] be a partial network flow method. Then, for every t ∈[T] and every feasible history (V t, At) ∈Θt that occurs with strictly positive probability, we have At i ∈{⌊V t i ⌋, ⌈V t i ⌉} for every i ∈[n]. Furthermore, (yt)t∈[T] satisfies Equation (1) for every t ∈[T]. Proof. The proof follows by induction over t ∈[T]. First, consider step t = 1. By definition, y1 yields a valid probability distribution over subsets of size H1, ensuring that no party receives more than one seat in step 1. Thus, for proving global quota, it suffices to show that any party i with v1 i = 0 is allocated a seat with probability zero. Let i be such that v1 i = 0. Since 0 = v1 i = Pm j=1 λjzj,i with λj ∈[0, 1] for every j, it follows that λj = 0 whenever zj,i = 1. Hence, y1(V 0, A0, v1, S) = 0 for all subsets S containing i, as required. To prove that Equation (1) is also satisfied, we fix a party i ∈[n] and observe that X S⊆[n]:i∈S y1(V 0, A0, v1, S) = m X j=1 λjzj,i = v1 i . Next, consider some t ∈[T -1] and assume that the partial method (yt)t∈[T-1] satisfies the required properties up to step t ≤T -1. We show that they are maintained for step t + 1. By definition, yt+1 yields a valid probability distribution over subsets of size Ht+1, ensuring that no party receives more than one seat in step t+1. Consider a party i with vt+1 i = 0. Since f is a feasible flow, we have f((u, i)) = 0 for every u ∈Ut, which implies zi(u) = 0. Consequently, each zj,i = 0 and i /∈Zj for all j. Hence, yt+1(V t, At, vt+1, S) = 0 for all subsets S containing i, as required. To verify global quota, fix a cumulative allocation At occurring with strictly positive probability and let u = u(At) ⊂[n] be the corresponding set of parties at their upper share of seats. 26 First, suppose that i ∈u and that the upper share of i does not increase from step t to t + 1, i.e., ⌈Pt k=1 vk i ⌉= ⌈Pt+1 k=1 vk i ⌉. Then party i cannot be allocated a seat in step t + 1 without violating global quota. Since f((u, i)) = 0, we again have zi(u) = 0, which implies that zj,i = 0 and i /∈Zj for all j. Therefore, yt+1(V t, At, vt+1, S) = 0 for all subsets S containing i, as required. On the other hand, assume i /∈u and the lower quota increases, i.e., ⌊Pt k=1 vk i ⌋+1 = ⌊Pt+1 k=1 vk i ⌋. Then i must be assigned a seat in step t + 1, as otherwise global quota would be violated. In this case, the flow is f((u, i)) = π(u)/Ht+1, so zi(u) = 1, which implies that zj,i = 1 and i ∈Zj for all j. Hence, every subset S in the support of yt+1(V t, At, vt+1, ·) contains i, as desired. It remains to show that yt+1(V t, At, vt+1, ·) satisfies Equation (1) for step t + 1. Recall that At is the set of all cumulative allocations At that occur with strictly positive probability. We need to show that the probability that party i is allocated a seat in step t + 1 equals vt+1 i . To compute this, observe that X At∈At π(yt, V t, At) · at+1 i (V t, At, vt+1) = X u∈Ut X At∈At:u=u(At) π(yt, V t, At) · at+1 i (V t, At, vt+1) = X u∈Ut π(u) · zi(u) = X u∈Ut π(u) · f(u, i) · Ht+1 π(u) = X u∈Ut f(u, i) · Ht+1. By flow conservation at vertex i, the total incoming flow equals the outgoing flow f((i, d)) = vt+1 i /Ht+1, so the final expression evaluates to vt+1 i , as required. C.2 Proof of Lemma 5 Lemma 5. Let n ≤3 and let (yk)k∈[t] be a partial network flow method with corresponding flow network FN(V t, At, vt+1, π) for step t + 1. Then this flow network admits a feasible (o, d)-flow of value 1, i.e., the partial network flow method can be extended to (yk)k∈[t+1]. Proof. Since the case n = 2 follows immediately from n = 3 by setting vt 3 = 0 for all t, we focus on n = 3. For t = 1, the construction of y1 is precisely given in the construction of the partial network flow method. Thus, let t ≥1 and let a partial network flow method (yk)k∈[t] up to step t be given. We show that the corresponding flow network FN(V t, At, vt+1, π) allows for a feasible flow of value 1 using a detailed case distinction based on • the number of seats to be allocated in step t + 1 (i.e., whether Ht+1 = 1 or Ht+1 = 2), • the number of parties that have been assigned their upper share of seats after step t (i.e., whether |u| = 1 or |u| = 2 for every u ∈Ut), and • how many parties obtained enough votes in step t + 1 to increase their upper share of seats by 1. This results in eight distinct cases, listed below. For each, we show that the corresponding flow network admits a feasible flow of value 1. We begin by making two general observations, using the fact that the partial network flow method (yk)k∈[t] satisfies Equation (1) for every k ∈[t]. For any 27 party i ∈[3] and set u ∈Ut, we have π(u) = t X k=1 vk i - t X k=1 vk i %! ≥ & t X k=1 vk i ' - t X k=1 vk i if i /∈u, |u| = 2. (8) The inequalities are strict if and only if Pt k=1 vk i ∈N. Furthermore, in any flow network, the vertex sets Po = {o} and Pd = P \ {d} each induce a cut of capacity 1. Thus, any flow of value 1 must assign any edge adjacent to o or d a flow equal to the edge's upper capacity. The remainder of this proof is structured as follows: We first list the eight cases, then address each one separately, showing the existence of a feasible flow f of value 1. All relevant networks are depicted in Figures 4 and 5, where edges (u, i) with c((u, i)) = 0 are omitted for clarity. 1. Ht+1 = 1 1.1. |u| = 1 for all u ∈Ut 1.1.1. ∀i ∈[3] : vt+1 i 0 1.2. |u| = 2 for all u ∈Ut 1.2.1. ∃! i ∈[3] : vt+1 i ≥ Pt k=1 vk i -Pt k=1 vk i > 0 1.2.2. ∃! i ∈[3] : vt+1 i 0 2.1.2. ∃! i ∈[3] : vt+1 i 0 Note that this covers all possible cases due to the following observation: For each party i ∈[n], we have t X k=1 vk i - t X k=1 vk i % + vt+1 i ! = |u| + Ht+1. This implies that the number of parties for which the individual term is at least 1 lies within Ht+1 + |u| -2 and Ht+1 + |u| -1. 28 o 2 1 3 1 2 3 d π({1}) π({2}) π({3}) vt+1 1 vt+1 2 vt+1 3 (a) Case 1.1.1 o 2 1 3 1 2 3 d π({1}) π({2}) π({3}) vt+1 1 vt+1 2 vt+1 3 (b) Case 1.1.2 o 1, 3 1, 2 2, 3 1 2 3 d π({1, 2}) π({1, 3}) π({2, 3}) vt+1 1 vt+1 2 vt+1 3 (c) Case 1.2.1 o 1, 3 1, 2 2, 3 1 2 3 d π({1, 2}) π({1, 3}) π({2, 3}) vt+1 1 vt+1 2 vt+1 3 (d) Case 1.2.2 Figure 4: Case 1 of the proof of Lemma 5 with Ht+1 = 1. The upper capacity of an edge e = (u, i) is c(e) = π(u) if it is drawn and c(e) = 0, otherwise. Red edges have lower capacities l(e) = c(e). Grey edges carry no flow in the unique flow with value 1. Case 1.1.1. The flow network is illustrated in Figure 4a. In this case, the flow f of value 1 is not unique. However, any feasible flow f with value 1 satisfies f(e) = c(e) for all e ∈{(o, {i}), ({i}, i), (i, d) : i ∈[3]}. The flow on the remaining edges must satisfy the following set of linear equalities f(({1}, 2) + f(({1}, 3) = π({1}), f(({2}, 1) + f(({2}, 3) = π({2}), f(({3}, 1) + f(({3}, 2) = π({3}), f(({2}, 1) + f(({3}, 1) = vt+1 1 , f(({1}, 2) + f(({3}, 2) = vt+1 2 , f(({1}, 3) + f(({2}, 3) = vt+1 3 , f(e) ≥0. Since adding the first three equalities yields the same as adding the other three, the set of linear equalities is underdetermined, i.e., there is no unique solution. We show that the system has a feasible solution. To this end, we interpret the inner part of the network as a bipartite graph G := (P1 ∪P2, E) with vertex sets P1 := {{1}, {2}, {3}} and P2 := {1, 2, 3}, and edge set E := {({i}, j) : i ̸= j}. 29 Each vertex {i} ∈P1 has supply b({i}) := π({i}) and each vertex i ∈P2 has demand b(i) := vt+1 i for every i ∈[3]. Thus, each edge is adjacent to one supply and one demand vertex. Since edge capacities match the supply of the adjacent supply vertex, we can treat this as a transshipment instance without explicit edge capacities. We claim that this transshipment instance has a feasible solution. First, note that the total supply and demand coincide. Now it suffices to verify that for all P ′ ⊆P1, the total supply P p∈P ′ b(p) is at most the total demand P p∈N(P ′) b(p) of its neighborhood N(P ′) (cf. [Schrijver, 2004, Corollary 11.2.g]). We focus on three cases. For P ′ = {{1}}, we have X p∈P ′ b(p) = π({1}) ≤1 - & t X k=1 vk 1 ' - t X k=1 vk 1 ! ≤1 -vt+1 1 = vt+1 2 + vt+1 3 = X v∈N(P ′) b(p). For P ′ = {{1}, {2}}, we obtain X p∈P ′ b(p) = π({1}) + π({2}) ≤1 = vt+1 1 + vt+1 2 + vt+1 3 = X p∈N(P ′) b(p), and finally for P ′ = {{1}, {2}, {3}}, we have X p∈P ′ b(p) = π({1}) + π({2}) + π({3}) = 1 = vt+1 1 + vt+1 2 + vt+1 3 = X p∈N(P ′) b(p). By symmetry, the condition also holds for all other subsets P ′. Hence, a feasible transshipment exists, implying the existence of a feasible flow f of value 1 in the original network. Case 1.1.2. There exists exactly one party i ∈[3] with vt+1 i ≥⌈Pt k=1 vk i ⌉-Pt k=1 vk i . Without loss of generality, assume i = 1. The flow network is illustrated in Figure 4b. We define f as follows. f(e) = vt+1 1 -(1 -π({1})) if e = ({1}, 1), vt+1 2 if e = ({1}, 2), vt+1 3 if e = ({1}, 3), 0 if e = ({2}, 3) or e = ({3}, 2), and c(e) otherwise. We now verify feasibility. For e = ({1}, 1), since 1 -π({1}) ≤vt+1 1 π({3}), where the first inequality follows from the case assumption, the second inequality from Property (8), and the third (strict) inequality from vt+1 1 0 and similar arguments yield f(e) > 0 for e = ({1}, 3). Next, we verify upper capacity constraints. For e = ({1}, 1), since vt+1 1 1 -π({2, 3}), where the first inequality follows from the case assumption, the second inequality from Property (8), and the third (strict) inequality from vt+1 3 < 1. An analogous computation applies to e = ({2, 3}, 3), proving that l(e) ≤f(e) holds for all edges. For the upper capacity constraints, consider e = ({2, 3}, 1). Property (8) yields vt+1 1 < & t X k=1 vk 1 ' - t X k=1 vk 1 ≤π({2, 3}), i.e., f(e) < c(e). For e = ({2, 3}, 2), we have (vt+1 2 -(1 -π({2, 3}))/2 < π({2, 3})/2 since vt+1 2 < 1, and analogously for e = ({2, 3}, 3). Since the outgoing flow from o is 1, f is a feasible flow of value 1. Case 2.2.2. The flow network is illustrated in Figure 5d. This case is structurally similar to Case 1.1.1: the flow f is not uniquely determined. We begin by fixing the flow along the edges ({1, 2}, 3), ({1, 3}, 2), and ({2, 3}, 1), each of which lies on a unique (o, d)-path. These edges must carry their respective lower bounds, which are π({1, 2})/2, π({1, 3})/2, and π({2, 3})/2. Fixing these values reduces the residual problem to the flow network depicted in Figure 6, where all remaining lower bounds are zero. The goal is now to route a residual flow of value 1 -π({1, 2} + π({1, 3}) + π({2, 3}) 2 = 1 2. To establish feasibility, we again reduce to a transshipment instance on a bipartite graph G := (P1 ∪P2, E) with vertex sets P1 := {{1, 2}, {1, 3}, {2, 3}}, and P2 := {1, 2, 3}, and edge set E := {({i}, j) : i ̸= j}. Each vertex {i, j} ∈P1 has supply b({i, j}) := π({i, j})/2 and each vertex i ∈P2 has demand b(i) := (vt+1 i -π({j, k}))/2 where {j, k} = [3] \ {i}. Thus, each edge is adjacent to one supply and one demand vertex. Since edge capacities match the supply of the adjacent supply vertex, we can treat this as a transshipment instance without explicit edge capacities. We claim that this transshipment instance has a feasible solution. First, note that the total supply and demand coincide. Now it suffices to verify that for all P ′ ⊆P1, the total supply P p∈P ′ b(p) is at most the total demand P p∈N(P ′) b(p) of its neighborhood N(P ′). We focus on three cases. For P ′ = {{1, 2}}, we have X p∈P ′ b(p) = π({1, 2} 2 = 1 -π({1, 3}) -π({2, 3}) 2 = vt+1 1 + vt+1 2 + vt+1 3 -1 -π({1, 3}) -π({2, 3}) 2 < vt+1 1 + vt+1 2 -π({1, 3}) -π({2, 3}) 2 = X p∈N(P ′) b(p). For P ′ = {{1, 2}, {1, 3}}, we obtain X p∈P ′ b(p) = π({1, 2}) + π({1, 3}) 2 ≤1 2 = X p∈N(P ′) b(p). 34 o 1, 3 1, 2 2, 3 1 2 3 d π({1, 2})/2 π({1, 3})/2 π({2, 3})/2 (vt+1 1 -π({2, 3}))/2 (vt+1 2 -π({1, 3}))/2 (vt+1 3 -π({1, 2}))/2 Figure 6: The residual flow network of Case 2.2.2 in the proof of Lemma 5. The upper capacity of an edge e = (u, i) is c(e) = π(u)/2 if it is drawn and c(e) = 0, otherwise. Finally, for P ′ = {{1, 2}, {1, 3}, {2, 3}}, we have X p∈P ′ b(p) = π({1, 2}) + π({1, 3}) + π({2, 3}) 2 = 1 2 = X p∈N(P ′) b(p). By symmetry, the condition also holds for all other subsets P ′. Thus, the transshipment instance is feasible, which implies the existence of a feasible flow of value 1/2 in the residual network. Together with the fixed lower bound flow, we obtain a feasible flow of value 1, completing this case. This completes the case distinction. C.3 Proof of Proposition 2 Proposition 2. Any online apportionment method satisfying global quota and ex-ante proportionality is a network flow method. Proof. Consider an online apportionment method (yt)t∈N satisfying global quota and ex-ante proportionality. We need to show that one can obtain the same method using the network flow construction. First, consider step t = 1. Since A0 = {0} and V 0 = 0, there is just one probability distribution y1 over the subsets of [n] of size H1. Let Z1, . . . , Zm ⊂[n] be the support of y1 and let λ1, . . . , λm ∈[0, 1] be the corresponding probabilities, i.e., y1(V 0, A0, v1, Zj) = λj for every j. Let zj ∈{0, 1}n denote the characteristic vector of Zj, i.e., zj,i = 1 if and only if i ∈Zj. Since |Zj| = H1 for every j, the vector zj has exactly H1 ones. Furthermore, ex-ante proportionality yields Pm j=1 λj ·zj = v1. Thus, the probability distribution y1 yields a unique convex decomposition into n-dimensional {0, 1}-vectors with exactly H1 ones. This convex decomposition can be chosen during the construction of a partial network flow method. Next, assume that up to step t, the online apportionment method (yk)k∈N coincides with a partial network flow method (zk)k∈[t]. Let FN(V t, At, vt+1, π) be the flow network corresponding to the partial network flow method (zk)k∈[t] at step t + 1. We show how the method (yk)k∈N for step t + 1 yields a flow of value 1 in FN(V t, At, vt+1, π). For each At ∈At, the method (yt)t∈N determines the probability π(yt, V t, At) with which the cumulative allocation At is attained. This uniquely determines the probability π(u(At)) that exactly the parties in u(At) have been allocated their upper share of seats. Let At = At 1 ∪· · · ∪At m be a partition of At into disjoint subsets such that u(At) = u( ̃At) if and only if At, ̃At ∈At j for some j ∈[m]. Note that the flow network has exactly one vertex uj for each At j. The probability that method (yt)t∈N assigns a seat to party i in step t+1 conditioned on the cumulative allocation uj 35 is denoted by πuj(i) = X At∈At j π(yt, V t, At) · X S⊆[n]:i∈S yt+1(V t, At, vt+1, S). We define f as follows. For each edge (o, uj) from the origin to a vertex uj, set f((o, uj)) = π(uj); for each edge (uj, i) from set uj to party i, set f((uj, i)) = πuj(i)/Ht+1; and finally, for each edge (i, d) from a vertex i to the destination, set f((i, d)) = vt+1 i /Ht+1. We claim that this defines a feasible flow of value 1. First, we show that f obeys the edge capacities. This is immediate for edges (o, ut) and (i, d) as f equals the upper capacities. To verify feasibility on edges (uj, i), we distinguish the three cases given by the flow construction. First, assume that i ∈uj and that the upper share of seats does not increase from step t to t + 1, i.e., ⌈Pt k=1 vk i ⌉= ⌈Pt+1 k=1 vk i ⌉or vt+1 i = 0. We need to show that f((uj, i)) = 0. However, in that case, party i cannot be assigned a seat in step t + 1 without violating local or global quota. Since the method (yt)t∈N fulfills these properties, it assigns probability 0 to any set containing i for every At ∈At j. Thus, we have πuj(i) = 0, implying f((uj, i)) = πuj(i)/Ht+1 = 0. Next, assume i /∈uj and the lower share increases from step t to t + 1, i.e., ⌊Pt k=1 vk i ⌋+ 1 = ⌊Pt+1 k=1 vk i ⌋. We need to show that f((uj, i)) = π(uj)/Ht+1. However, in that case, party i must receive a seat in step t + 1, as otherwise global quota would be violated. Since the method (yt)t∈N fulfills these properties, it only assigns positive probability to sets containing i, which yields P S⊆[n]:i∈S yt+1(V t, At, vt+1, S) = 1 for every At ∈At j. Hence, we obtain f((uj, i)) = πuj(i) Ht+1 = X A∈At j π(yt, V t, At) · 1 Ht+1 = π(uj) Ht+1 . as desired. In all other cases, the method (yt)t∈N is less restricted and we have yt+1(V t, At, vt+1, S) ∈ [0, 1] for any subset S ⊆[n]. We only need to show that f((uj, i)) ∈[0, π(uj)/Ht+1]. This follows from f((uj, i)) = πuj(i) Ht+1 ≤ X A∈At j π(yt, V t, At) · 1 Ht+1 = π(uj) Ht+1 . Thus, f obeys the capacity constraints. To verify flow conservation, we first consider an inner vertex uj. The incoming flow from o is f((o, uj)) = π(uj). The outgoing flow is n X i=1 f((uj, i)) = n X i=1 πuj(i) Ht+1 = X A∈At j π(yt, V t, At) · 1 Ht+1 · n X i=1 X S⊆[n]:i∈S yt+1(V t, At, vt+1, S) = π(uj), where the last equality holds, since the method (yt)t∈N assigns exactly Ht+1 seats in step t + 1. For an inner vertex i, (yt)t∈N satisfying ex-ante proportionality yields X uj∈Ut f((uj, i)) = X uj∈Ut πuj(i) Ht+1 = X uj∈Ut 1 Ht+1 X A∈At j π(yt, V t, At) · X S⊆[n]:i∈S yt+1(V t, At, vt+1, S) = vt+1 Ht+1 . 36 Thus, the incoming flow matches the outgoing flow f((i, d)), so flow conservation holds at every inner vertex. Since f obeys the capacity constraints of the flow network FN(V t, At, vt+1, π) and attains flow conservation, it is a feasible flow. Finally, the total flow leaving the origin o is P uj∈Ut π(uj) = 1. Hence, the flow f has value 1 as claimed. This exact flow allows to extend the partial network flow method (zk)k∈[t] up to step t+1 such that the online apportionment method (yk)k∈N coincides with the partial network flow method (zk)k∈[t+1]. This finally proves that the online apportionment method (yk)k∈N is a network flow method. 37 References M. Adler, P. Berenbrink, T. Friedetzky, L. A. Goldberg, P. Goldberg, and M. Paterson. A proportionate fair scheduling rule with good worst-case performance. In Proceedings of the 15th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 101-108, 2003. R. K. Ahuja, T. L. Magnanti, and J. B. Orlin. Network flows. MIT, Cambridge, Massachusetts, 1988. M. Akbarpour and A. Nikzad. Approximate random allocation mechanisms. The Review of Economic Studies, 87(6):2473-2510, 2020. H. Aziz, O. Lev, N. Mattei, J. S. Rosenschein, and T. Walsh. Strategyproof peer selection using randomization, partitioning, and apportionment. Artificial Intelligence, 275:295-309, 2019. M. Balinski and G. Demange. An axiomatic approach to proportionality between matrices. Mathematics of Operations Research, 14(4):700-719, 1989a. M. Balinski and G. Demange. Algorithms for proportional matrices in reals and integers. Mathematical Programming, 45(1-3):193-210, 1989b. M. Balinski and H. P. Young. Fair representation: Meeting the ideal of one man, one vote. Brookings Institution Press, 2010. M. L. Balinski and H. P. Young. The quota method of apportionment. The American Mathematical Monthly, 82(7):701-730, 1975. M. A. Bender and W. Kuszmaul. Randomized cup game algorithms against strong adversaries. In Proceedings of the 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2059-2077. SIAM, 2021. M. A. Bender, M. Farach-Colton, and W. Kuszmaul. Achieving optimal backlog in multi-processor cup games. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 1148-1157, 2019. J. Byrka and A. Srinivasan. Approximation algorithms for stochastic and risk-averse optimization. SIAM Journal on Discrete Mathematics, 32(1):44-63, 2018. J. Cembrano, J. Correa, U. Schmidt-Kraepelin, A. Tsigonias-Dimitriadis, and V. Verdugo. New combinatorial insights for monotone apportionment. In Proceedings of the 36th Annual ACMSIAM Symposium on Discrete Algorithms (SODA), pages 1308-1328. SIAM, 2025. J. Correa, P. Gölz, U. Schmidt-Kraepelin, J. Tucker-Foltz, and V. Verdugo. Monotone randomized apportionment. In Proceedings of the 25th ACM Conference on Economics and Computation (EC), pages 71-71, 2024. P. Dietz and D. Sleator. Two algorithms for maintaining order in a list. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 365-372, 1987. H. Evren and M. Khanna. Affirmative action's cumulative fractional assignments. arXiv preprint , 2021. 38 N. Gaffke and F. Pukelsheim. Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems. Mathematical Social Sciences, 56(2):166-184, 2008. L. Gąsieniec, R. Klasing, C. Levcopoulos, A. Lingas, J. Min, and T. Radzik. Bamboo garden trimming problem (perpetual maintenance of machines with different attendance urgency factors). In International Conference on Current Trends in Theory and Practice of Informatics, pages 229240. Springer, 2017. P. Gölz, D. Peters, and A. D. Procaccia. In this apportionment lottery, the house always wins. Operations Research, 2025. G. Grimmett. Stochastic apportionment. The American Mathematical Monthly, 111(4):299-307, 2004. A. Kawamura. Proof of the density threshold conjecture for pinwheel scheduling. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing (STOC), pages 1816-1819, 2024. W. Kuszmaul. Achieving optimal backlog in the vanilla multi-processor cup game. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1558-1577. SIAM, 2020. C. L. Liu and J. W. Layland. Scheduling algorithms for multiprogramming in a hard-real-time environment. Journal of the ACM (JACM), 20(1):46-61, 1973. J. Naor, A. Srinivasan, and D. Wajc. Online dependent rounding schemes for bipartite matchings, with. In Proceedings of the 36th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3116-3154. SIAM, 2025. F. Pukelsheim. Proportional Representation. Springer International Publishing, 2017. F. Pukelsheim, F. Ricca, B. Simeone, A. Scozzari, and P. Serafini. Network flow methods for electoral systems. Networks, 59(1):73-88, 2011. G. Rote and M. Zachariasen. Matrix scaling by network flow. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), page 848--854. Society for Industrial and Applied Mathematics, 2007. A. Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24. Springer, 2004. A. Srinivasan. Distributions on level-sets with applications to approximation algorithms. In Proceedings 42nd IEEE Symposium on Foundations of Computer Science (FOCS), pages 588-597, 2001. J. A. Stimson, M. B. MacKuen, and R. S. Erikson. Dynamic representation. American Political Science Review, 89(3):543-565, 1995. C. Swamy and D. B. Shmoys. Sampling-based approximation algorithms for multistage stochastic optimization. SIAM Journal on Computing, 41(4):975-1004, 2012. 39
|
2510.14751
|
Beyond Multi-Token Prediction: Pretraining LLMs
with Future Summaries
Divyat Mahajan1,2,∗, Sachin Goyal3,∗, Badr Youbi Idrissi1, Mohammad Pezeshki1, Ioannis Mitliagkas2,
David Lopez-Paz1, Kartik Ahuja1
1FAIR at Meta, 2Mila, Université de Montréal, DIRO, 3Carnegie Mellon University
∗Work done at Meta
Next-token prediction (NTP) has driven the success of large language models (LLMs), but it struggles
with long-horizon reasoning, planning, and creative writing, with these limitations largely attributed
to teacher-forced training. Multi-token prediction (MTP) partially mitigates these issues by predicting
several future tokens at once, but it mostly captures short-range dependencies and offers limited
improvement. We propose future summary prediction (FSP), which trains an auxiliary head to
predict a compact representation of the long-term future, preserving information relevant for long-form
generations. We explore two variants of FSP: handcrafted summaries, for example, a bag of words
summary of the future of the sequence, and learned summaries, which use embeddings produced by
a reverse language model trained from right to left. Large-scale pretraining experiments (3B and
8B-parameter models) demonstrate that FSP provides improvements over both NTP and MTP across
math, reasoning, and coding benchmarks.
Date: October 17, 2025
Correspondence: Divyat Mahajan at divyatmahajan@gmail.com
1
Introduction
Early progress in large language models was primarily driven by massive scaling of both data and com-
pute (Brown et al., 2020; Kaplan et al., 2020). However, the returns from this vanilla scaling approach are
beginning to diminish as we encounter the “data wall” (Sutskever, 2024). This has renewed efforts toward
algorithmic advances, including new architectures and pretraining objectives, that can extract more predictive
signal from a fixed amount of training data.
Next-token prediction (NTP) with teacher forcing—training models by conditioning on ground-truth history
when predicting the next token—is foundational to current pretraining methods. However, this approach
introduces a train-inference mismatch known as exposure bias: during inference, the model must rely on
its own outputs rather than the ground truth, leading to compounding errors and degraded long-range
generation quality (Bengio et al., 2015). Moreover, teacher forcing can induce training-time shortcut learning
as well, where the model exploits local cues from the ground-truth prefix instead of capturing true long-range
dependencies (Bachmann and Nagarajan, 2024). These issues manifest most clearly in tasks demanding
extended reasoning, narrative coherence, and open-ended creativity (Papalampidi et al., 2022; Nagarajan
et al., 2025).
An appealing alternative to teacher forcing is teacherless training, where the model learns from its own
generations rather than relying on ground-truth histories.
However, this approach is computationally
intensive and challenging to parallelize. As a practical compromise, recent research has explored multi-token
prediction (MTP) methods (Gloeckle et al., 2024), which train auxiliary heads to predict several future tokens
simultaneously. MTP has demonstrated success in large-scale systems such as DeepSeek-V3 (Liu et al., 2024)
and Qwen-3 (Yang et al., 2025). In MTP, each time step augments the standard next-token prediction (xt+1)
with auxiliary heads that predict additional future tokens (e.g., xt+2 and beyond). However, these methods
typically assume independence among the predicted tokens given the prefix, resulting in poor approximations
of the true joint distribution over long future spans.
1
arXiv:2510.14751v1 [cs.LG] 16 Oct 2025
Hidden Representation
of Reverse LM
. . .
x<t
−
Bag-of-words
(multi-hot vector)
. . .
NTP:
MTP:
FSP-BCE:
FSP-RevLM:
x≤t
xt+1 xt+2
xT−1
xT
x≤t
xt+1 xt+2
xT−1
xT
x≤t
xt+1 xt+2
xT−1
xT
x≤t
xt+1 xt+2
xT−1
xT
. . .
. . .
Figure 1 A comparison of future-aware pretraining objectives. All methods take a prefix x≤t as input. NTP: Only predicts
the immediate next token. MTP: Uses multiple auxiliary heads, each predicting a specific future token. FSP-BCE:
Our proposed hand-crafted summary method that predicts a multi-hot “bag-of-tokens” summary of a long future
window using a single auxiliary head. FSP-RevLM: Our proposed learned summary method predicts a compact hidden
representation of the future, which is generated by a reverse language model (RevLM).
In this work, we propose a different approach: rather than predicting multiple future tokens individually, we
train a single auxiliary head to predict a summary representation of the future sequence. It aims to push
as much information as possible about the future into a single target vector, while filtering out information
that is inherently unpredictable and would only introduce noise. At each time step t, given the future tokens
(xt+2, xt+3, . . . , xt+τ), we construct a summary vector a(t, τ) to provide supervision to the auxiliary head. We
explore two complementary approaches for constructing future summaries: a simple token-level aggregation
method, and our main contribution, a learned representation of the future sequence that captures long-range
dependencies more effectively.
• Hand-crafted summary. We train the auxiliary head to predict all the future tokens that will occur in a
future window, without requiring to know their exact positions. Concretely, at each step t, we define a
multi-hot vector a(t, τ) over the vocabulary, akin to bag-of-words, where a(t, τ)i = I(i ∈{xt+2, . . . , xt+τ}),
and train the model with a binary cross-entropy objective.
• Learned summary. Handcrafted summaries such as the one proposed above can be noisy: not all future
tokens are equally relevant, and predicting them all can dilute the signal. To address this, we propose to
learn a compact summary of the future. We do this by training a reverse language model (gr) on reversed
sequences, so that its hidden representation a(t, τ) = gr(x≥t+2) serves as a rich embedding of the future
context. The auxiliary head of the forward model is then trained to match this representation with an ℓ2
loss.
We use the lens of synthetic tasks, path-star graph and sibling discovery (Bachmann and Nagarajan, 2024;
Nagarajan et al., 2025) to clarify the conceptual difference between MTP and our approach. On path-star
graph, hand-crafted summaries deliver strong gains over MTP, which only predicts immediate future tokens,
demonstrating the benefit of long-horizon supervision. However, in sibling discovery, where the future includes
content unrelated to the current local sequence, handcrafted summaries struggle, as they treat all future
tokens equally. In contrast, our learned summary vectors focus on the informative parts of the future and
achieve consistent gains.
We then scale our approach to real-world pretraining at the 8B parameter level, conducting a systematic
evaluation across six methods, including DeepSeek-MTP and multiple handcrafted summary variants. This
scale and breadth of comparison is rare in the literature due to its high computational cost. Here, the proposed
future summary prediction with learned summaries yields substantial improvements over NTP and MTP
baselines, with gains of up to 5% on math and coding benchmarks that demand long-horizon reasoning
and planning. These results demonstrate that future summary prediction is not only effective in controlled
synthetic settings but also translates into meaningful gains for large-scale LLM training.
2
2
Future-Aware Pretraining
2.1
Background on Next-Token and Multi-Token Prediction
Let us define a sequence of tokens as X = (X1, X2, . . . , XT ) sampled from a distribution PX, where each
token is a discrete random variable supported on {1, 2, . . . , V }.
Next-Token Prediction (NTP).
The standard next-token prediction objective is:
LNTP(X, Pθ) = −Ex∼PX
" T −1
X
t=1
log Pθ(xt+1 | x≤t)
#
,
(1)
where Pθ is the predictor, optimized via
θ⋆= arg min
θ
LNTP(X, Pθ).
(2)
A typical parameterization is:
Pθ(xt+1 | x≤t) = softmax
fu ◦fh ◦fs
x≤t
,
(3)
where fs is the transformer backbone, fh is a processing head, and fu is the unembedding layer (Figure 2).
Multi-Token Prediction (MTP).
MTP aims to jointly predict multiple future tokens, as follows:
LMTP-Joint(X, Pθ) = −Ex∼PX
" T −1
X
t=1
log Pθ(xt+1, . . . , xt+τ | x≤t)
#
.
(4)
Since modeling the exact joint distribution is intractable, a common simplification is to model the marginal
distribution of future tokens given the prefix (Gloeckle et al., 2024; Liu et al., 2024):
LMTP(X, Pθ) = −Ex∼PX
" T −1
X
t=1
τ
X
k=1
1[t + k ≤T] log Pθ(xt+k | x≤t)
#
.
(5)
Following Gloeckle et al. (2024), the predictor uses separate auxiliary heads for each k:
Pθ(xt+1 | x≤t) = softmax
fu ◦fh ◦fs
x≤t
,
Pθ(xt+k | x≤t) = softmax
fu ◦f ′
hk ◦fs
x≤t
,
∀k > 1,
(6)
where f ′
hk are auxiliary transformer blocks specialized for predicting future tokens xt+k.
This design reduces teacher forcing by predicting xt+k from x≤t only, rather than conditioning on the full
prefix x≤t+k−1. This is the key principle behind multi-token prediction, i.e., reduced teacher forcing by
requiring the model to predict a block of future tokens at each step.
We discuss other variants of MTP such as DeepSeek-MTP and random future token MTP (Thankaraj et al.,
2025) in Appendix A.
2.2
Future Summary Prediction (FSP)
In MTP, we use a set of auxiliary heads to predict a block of immediate future tokens. A key limitation of
this approach is that one does not know exactly where the informative signal in the future sequence lies.
Informative signals in future could occur far away in the sequence and be well beyond the number of future
tokens (k) that MTP predicts. A trivial approach to overcome this could be to predict all the future tokens.
However, having one auxiliary head per future token is not scalable, and limits the amount of future tokens
we can utilize during training.
Towards this, we propose Future Summary Prediction (FSP), that predicts a compact summary of the (long)
future sequence rather than each token individually.
3
Transformer backbone
MainBlock
x≤t
f ′
ha(.)
fh(.)
a(t, τ)
fs(.)
AuxBlock
Future
summary
xt+1
Figure 2 An abstraction of the
architecture that subsumes
NTP, MTP, and FSP.
Let a(t, τ) represent a summary of future tokens (xt+2, . . . , xt+τ). The learning
objective is:
LFSP(X, Pθ) = LNTP(X, Pθ) + Ex∼PX
h
la
Aϕ(x≤t), a(t, τ)
i
,
(7)
where Pθ is the next-token predictor, Aϕ is the summary predictor, and la is
the loss between predicted and ground-truth summaries. The architecture is:
Pθ(xt+1 | x≤t) = softmax
fu ◦fh ◦fs
x≤t
,
Aϕ(x≤t) = f ′
ha ◦fs
x≤t
.
(8)
Unlike MTP, FSP requires only a single auxiliary head f ′
ha, making it more
scalable.
The key question, then, is how to construct an effective future summary. In
this work, we investigate two approaches:
(1) Hand-crafted future summaries.
We define a binary vector a(t, τ) ∈RV
indicating whether token i appears in the future window:
a(t, τ)i = I
i ∈{xt+2, . . . , xt+τ}
.
(9)
Given logits from Aϕ(x≤t), we minimize a reweighted binary cross-entropy loss:
la
Aϕ(x≤t), a(t, τ)
= −
V
X
i=1
w(i)
h
ai log σ(zi) + (1 −ai) log
1 −σ(zi)
i
(10)
where ai = I
i ∈{xt+2, . . . , xt+τ}
, zi is the i-th logit of Aϕ(x≤t), σ is the sigmoid, and w(i) reflects the
importance of token i (e.g. term frequency-inverse document frequency, tf–idf).
(2) Learned future summaries.
Predicting every future token via hand-crafted summaries as discussed above
could be noisy. For example, not all tokens in the future are informative and thus hand-crafted summaries
that account for all of them can be wasteful. Instead of predicting all the future tokens, we therefore propose
to predict a learned representation of the future tokens relevant to predicting the current token xt+1. We
learn this representation via a reverse language model Qψ, (RevLM), trained on "right-to-left" sequences:
LRevLM(X, Qψ) = −Ex∼PX
" T −1
X
t=1
log Qψ(xt+1 | x≥t+2)
#
.
(11)
RevLM shares the same architecture, and we take its hidden state as the summary vector:
a(t, T −t) = gh ◦gs
x≥t+2
.
(12)
We then train Aϕ(x≤t) to match this representation via the ℓ2 loss:
la
Aϕ(x≤t), a(t, T −t)
=
Aϕ(x≤t) −gh ◦gs(x≥t+2)
2
2.
(13)
Why Future Summary Prediction?
The key advantage of future summary prediction is its ability to reduce
dependence on teacher forcing when modeling long future sequences. To intuitively measure teacher forcing,
consider for each ground-truth token exposed to the model, how much information is the model required to
predict about unseen tokens? If the model predicts more such information, then we have reduced teacher
forcing. Next-token prediction (NTP) uses the highest degree of teacher forcing, since the model always
conditions on ground-truth histories to predict just the next token. Multi-token prediction (MTP) partially
relaxes this by asking the model to predict short blocks of future tokens, thereby reducing teacher forcing
locally. However, MTP remains constrained by the short horizon of its predictions. In contrast, our proposed
approach predicts summaries of long future sequences, substantially reducing teacher forcing by requiring the
model to reason about rich, global properties of the target trajectory.
4
3
Analyzing Future Summaries
To better understand how FSP mitigates the challenges of long-horizon planning, we consider the graph
modeling benchmarks introduced in prior works. Our analysis yields following insights:
• Long future summaries matter. On the canonical path–star task (Bachmann and Nagarajan, 2024), we
find that MTP with short range future prediction fails to generalize, highlighting the need for auxiliary
objectives that incorporate long-range future information.
• Adaptive future summaries matter. On a modified sibling discovery task (Nagarajan et al., 2025), we
show that incorporating every future token with hand-crafted summaries is suboptimal, highlighting the
need for learned summaries that retain key information.
3.1
Long future summary is important
<graph> s 5 1 2 3 4 5
<graph> s 5 1 2 3 4 5
<graph> s 5 1 2 3 4 5
<graph> s 5 1 2 3 4 5
<graph> s 5 1 2 3 4 5
<graph> s 5 1 2 3 4 5
Difficult prediction task
Too easy due to shortcut
Mix of 1- to 4-hop reasoning
NTP:
FSP:
Example Sequence:
x = <graph> s 5 1 2 3 4 5
start/goal
path
Example Graph:
s
1
2
3
4
5
6
7
8
9
10
Figure 3 Analysis of FSP-BCE on the path-star task, which tests long-horizon planning. Left: Accuracy (mean over
5 random seeds) of different pretraining objectives on degree 2 graphs with path lengths 6 and 8. Standard NTP
generalizes poorly, and MTP’s accuracy degrades as the path length increases, while FSP-BCE achieves perfect
accuracy. Right: An illustration of why NTP fails while FSP-BCE succeeds, where the input context is shown in grey
and the target is in beige.
We consider the path-star graph, a directed acyclic graph (DAG) G(d, l) composed of d paths, each of length
l, originating from a central start node vstart. The model is provided with the adjacency list of the graph in
the prefix, and the task is to generate the path from vstart to a designated end node vend.
Let the target path be (vstart, v1, v2, · · · , vend) and the input prefix be p = (Adj(G), vstart, vend). Then NTP
with teacher forcing predicts an intermediate node in the path as Pθ(vi+1|p, v≤i). As shown by Bachmann
and Nagarajan (2024), NTP often learns shortcut solutions: the model can recover vi+1 directly from vi by
scanning the adjacency list in p, without learning the underlying long-range plan (Figure 3). This leads to
gradient starvation (Pezeshki et al., 2021), where the supervision signal for the actual planning task is lost.
Once the shortcut is learned, meaningful gradient information remains only for predicting the first step v1, as
it is the only difficult token.
A natural remedy is to reduce teacher forcing via future prediction approaches, which require the model
to predict tokens further ahead. This makes shortcuts less effective, as multiple lookups in the adjacency
list would be needed to predict future tokens. Hence, we consider the MTP approach, which predicts the
immediate future token, and the handcrafted summary method FSP-BCE, which compresses the information
about all the future tokens from the path. By summarizing the entire future trajectory, FSP-BCE substantially
reduces teacher forcing, encouraging the model to plan the full path instead of exploiting shortcuts.
Findings.
We conduct experiments for two graphs G(2, 6) and G(2, 8), with all the approaches pretrained
from scratch using the GPT-Mini architecture (details in Section B.1). At inference, we discard the auxiliary
head used in future prediction, and the task is to generate the complete path given the prefix p. The evaluation
5
metric checks whether the generated path exactly matches the true path. Results in Figure 3 (left) show
that both NTP and MTP fail to generalize (both obtained perfect training accuracy), and the accuracy of
MTP degrades further for the scenario with longer path G(2, 8). Hence, predicting just the immediate future
token is not enough, and FSP-BCE tackles this by efficiently compressing all the future tokens from the path,
enabling it to achieve perfect accuracy in both cases. Appendix B.1 (Table 4) shows that increasing the
number of auxiliary heads in MTP can provide some improvement, but practical limits exist: even with four
additional future heads MTP cannot solve the longer path graph G(2, 8).
3.2
Adaptive future summary is important
Example Sequence:
Graph:
x = S1 S1 S1 P1 S3 S3 S3 P3 S2 S2 S2 P2
1
2
3
1
2
3
1
2
3
P1
S1
1
S1
2
S1
3
Pm
Sm
1
Sm
2
Sm
3
....
Relevant
future
Not predictable
from the prefix
If this is
the prefix
Convergence Speed Relative to NTP:
Number of Components
Figure 4 Analysis on the modified sibling discovery task which requires adaptive future summaries. Left: Convergence
speed (mean (s.e.) over 3 random seeds) relative to NTP, where lower values imply faster convergence. FSP-RevLM
converges faster than NTP while FSP-BCE improves only in the cases with few components. Right: Task setup
illustration– given the prefix, only future tokens in the highlighted component are informative. FSP-BCE, which
summarizes all future tokens, suffers from irrelevant information, whereas FSP-RevLM summarizes only the informative
aspects.
Long handcrafted summaries often incorporate all future tokens, even though only a subset may provide
meaningful supervision signals, while the rest can introduce noise. This motivates the need for an adaptive
future summary. We illustrate this with modification of the sibling discovery task. Figure 4 (right) depicts
the setup, the model must generate sequences made of concatenated, independent components, where each
component lists its children nodes first, followed by their parent (e.g., nodes S1
1, S1
2, S1
3 followed by parent P 1).
Intuitively, the causal factorization implied by the DAG (parent followed by children) implies a goal-conditioned
approach: by conditioning on the parent, the model can easily capture the sibling dependencies. Under
NTP, the model estimates Pθ(S1
2|S1
1) without the parent, making sibling relationships harder to learn and
requiring more samples. Future prediction can help, when the model predicts the parent from S1
1, then the
representation can incorporate the parent information. This enables goal-conditioned planning, the model can
predict S1
2 conditioned on both S1
1 and the parent, allowing sample efficient learning of sibling relationships
(see Nagarajan et al. (2025) for details).
However, not all future tokens are equally informative. As Figure 4 illustrates, future tokens from a different
component do not provide relevant signal for predicting S1
2. Handcrafted summaries that include all future
tokens may therefore be affected by the irrelevant information, whereas learned summaries remain robust, as
the reverse language model learns representation that emphasize only the predictive signals needed to infer
the next token.
Findings.
We verify this empirically by comparing FSP-BCE and FSP-RevLM in experiments with varying
number of total components. All models are pretrained from scratch with the GPT-Mini architecture (details
in Section B.2), and the inference task requires generating a coherent sequence (siblings followed by parent
for each component). At convergence, all models produce coherent sequences; to quantify the learning
speedup over NTP, we report the ratio of steps to convergence relative to NTP, with lower values indicating
6
faster learning. Results in figure 4 (left) shows that FSP-BCE improves over NTP only when the number
of components is small, with gains disappearing for more than six components. In contrast, FSP-RevLM
consistently achieves faster convergence across all component sizes, confirming the benefit of adaptive future
summaries.
4
Experiments
4.1
Setup
We pretrain 3B- and 8B-parameter models on corpora of 250B and 1T tokens, respectively, covering diverse
domains. The majority of the data comes from DCLM-like sources and GitHub repositories, supplemented
by specialized material in mathematics, programming, and related areas. Models are evaluated across a
diverse set of benchmarks: ARC-Easy/Challenge (Clark et al., 2018) for general reasoning, MBPP (Austin
et al., 2021) and HumanEval+ (Liu et al., 2023) for code generation, and GSM8K (Cobbe et al., 2021)
and Math-500 (Hendrycks et al., 2021) for mathematical reasoning. Details of the pretraining corpus and
hyperparameters are provided in Section B.3.
We first benchmark our primary method, FSP-RevLM, against the baselines: next-token prediction (NTP),
standard multi-token prediction (MTP) (Gloeckle et al., 2024), and DeepSeek-MTP (DS-MTP) (Liu et al.,
2024). A natural way to improve MTP for longer horizons is to add multiple auxiliary heads, each predicting
a token farther into the future, but this approach quickly becomes impractical. We therefore constrain both
MTP and DS-MTP to a single auxiliary head predicting the immediate future token. This design choice keeps
the comparison consistent and aligned with the proposed FSP, since it uses a single auxiliary head.
Building on this unified single auxiliary head framework, we conduct an analysis of how MTP can be enhanced
by predicting richer future targets instead of the immediate future token. In addition to predicting the
learned future summary (FSP-RevLM), we compare handcrafted future summaries over short- and long-range
windows. This includes the proposed multi-hot future summary (FSP-BCE) as a contributed baseline, and a
random-token summary baseline (radomly sampling token from future), in line with prior works (Thankaraj
et al., 2025; Gerontopoulos et al., 2025).
Note. All experiments are conducted under iso-data conditions, meaning that all the methods are trained
on identical datasets. For the proposed FSP-RevLM, this implies that both the forward and reverse models
are trained on the same data. In line with standard practice in distillation, we do not perform iso-compute
comparisons that include the teacher model’s (ReverseLM) cost in the reported compute budget. In practical
scenarios, the computational cost of training the teacher model is typically amortized, hence it can be treated
as a one-time overhead that is excluded from comparisons of student models (Gemma et al., 2024, 2025).
4.2
Results
At the 8B scale (Table 1), future-summary supervision via the reverseLM (FSP-RevLM) consistently improves
the performance across different evaluation tasks. On ARC-Easy, FSP-RevLM (76.6%) provides significant
improvement over NTP (71.8%) and MTP (73.6%), and it also leads on ARC-Challenge and MATH. For
code generation, it achieves the highest score on MBPP and ties with MTP on HumanEval+, showing that
the benefits of predicting future summaries generalize across both reasoning and program synthesis tasks.
GSM8K is the one task where NTP (71.6%) holds a lead, though FSP-RevLM (70.5%) still narrows the gap
relative to MTP (67.8%).
At the 3B scale (Table 2), DeepSeek-MTP is a strong baseline and obtains better performance than FSP-RevLM
on most tasks, except math reasoning. More importantly, FSP-RevLM exhibits larger relative improvements
as scale increases from 3B to 8B parameters, and becomes more favourable than DS-MTP. Further, note
that even at the 3B scale, FSP-RevLM still beats MTP on ARC and math reasoning tasks, and performs
comparably on MBPP, suggesting that even at smaller scales, learning to predict future summaries may
provide more effective auxiliary signal than immediate future token prediction with MTP.
7
Task
NTP
MTP
DS-MTP
FSP-RevLM
ARC-Easy
0.718 (0.000)
0.736 (0.000)
0.617 (0.003)
0.766 (0.000)
ARC-Challenge
0.531 (0.000)
0.552 (0.000)
0.426 (0.002)
0.559 (0.000)
GSM8K
0.716 (0.003)
0.678 (0.007)
0.704 (0.003)
0.705 (0.004)
MATH
0.342 (0.008)
0.309 (0.006)
0.335 (0.014)
0.351 (0.017)
MBPP
0.657 (0.004)
0.672 (0.008)
0.678 (0.006)
0.683 (0.006)
HumanEval+
0.478 (0.019)
0.541 (0.011)
0.526 (0.013)
0.541 (0.009)
Table 1 Pretraining at 8B scale. We benchmark the proposed FSP-RevLM approach against NTP, MTP, and DS-MTP.
Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC. FSP-RevLM achieves
the strongest overall performance, with large gains over the baselines on ARC tasks and MATH, and competitive
results with (DS) MTP on code benchmarks.
Task
NTP
MTP
DS-MTP
FSP-RevLM
ARC-Easy
0.263 (0.002)
0.272 (0.005)
0.293 (0.000)
0.277 (0.000)
ARC-Challenge
0.263 (0.001)
0.245 (0.002)
0.274 (0.008)
0.255 (0.000)
GSM8K
0.410 (0.003)
0.411 (0.001)
0.417 (0.003)
0.436 (0.003)
MATH
0.213 (0.004)
0.196 (0.009)
0.201 (0.004)
0.212 (0.002)
MBPP
0.521 (0.007)
0.526 (0.004)
0.537 (0.007)
0.524 (0.001)
HumanEval+
0.301 (0.009)
0.321 (0.015)
0.348 (0.022)
0.305 (0.006)
Table 2 Pretraining at 3B scale. We benchmark the proposed FSP-RevLM approach against NTP, MTP, and DS-MTP.
Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC. At this smaller scale,
DS-MTP is a strong overall baseline, but FSP-RevLM outperforms it on math reasoning tasks, and also provides
substantial gains over MTP on ARC and math reasoning tasks. Further, as we scale the approaches to 8B parameters,
FSP-RevLM scales more favorably than DS-MTP, overtaking it on most tasks.
4.3
Analyzing MTP with different future summaries
Table 3 presents our analysis of different future-summary strategies as auxiliary head targets at 8B scale. We
focus on the standard MTP architecture, without comparing to DS-MTP as it modifies the input to auxiliary
head, to isolate the effect of different future targets on the auxiliary head.
Our results show that random-token handcrafted future summaries (MTP-Skip) perform worse than standard
MTP with immediate future token prediction, and performance further degrades as the future window (τ)
increases. In contrast, the proposed multi-hot or bag-of-words handcrafted future summaries yield meaningful
improvements over MTP, especially on math reasoning tasks, with both shorter (τ = 12) and longer (τ = 100)
future windows. For example, FSP-BCE with τ = 12 achieves 33.1% on MATH (+2.2 points) and 69.9% on
GSM8K (+2.1 points), while even longer windows (τ = 100) further amplifies the performance on GSM8K
(71.4%, +3.6 points).
Finally, our learned future summaries (FSP-RevLM) outperform MTP across all evaluation tasks, with
especially pronounced gains on math reasoning: 35.1% on MATH (+4.2 points) and 70.5% on GSM8K (+3.5
points). Further analysis (Figure 5) for these math reasoning tasks shows that learned summaries promote
greater output diversity across different pass@k settings, compared to vanilla MTP with immediate future
token prediction.
In Section B.3 (Table 5), we replicate these findings at 3B scale, where both handcrafted and learned future
summaries improve over vanilla MTP, again most prominently on math reasoning tasks. Additional ablations
explore the effects of omitting reweighting in FSP-BCE and predicting learned summaries from deeper layers
of the reverseLM (FSP-RevLM).
8
Method
MBPP
GSM8K
MATH
HumanEval+
ARC-Challenge
ARC-Easy
MTP
0.672 (0.008)
0.678 (0.007)
0.309 (0.006)
0.541 (0.011)
0.552 (0.000)
0.736 (0.000)
MTP-Skip τ:4
0.658 (0.005)
0.639 (0.004)
0.277 (0.020)
0.508 (0.009)
0.494 (0.003)
0.722 (0.000)
MTP-Skip τ:12
0.623 (0.002)
0.621 (0.005)
0.287 (0.018)
0.486 (0.010)
0.512 (0.000)
0.710 (0.003)
MTP-Skip τ:32
0.611 (0.008)
0.598 (0.007)
0.271 (0.005)
0.459 (0.007)
0.379 (0.000)
0.564 (0.000)
FSP-BCE τ:12
0.669 (0.005)
0.699 (0.006)
0.331 (0.016)
0.508 (0.005)
0.562 (0.000)
0.737 (0.000)
FSP-BCE τ:100
0.671 (0.002)
0.714 (0.009)
0.331 (0.007)
0.500 (0.019)
0.459 (0.000)
0.662 (0.000)
FSP-RevLM
0.683 (0.006)
0.705 (0.004)
0.351 (0.017)
0.541 (0.009)
0.559 (0.000)
0.766 (0.000)
Table 3 Analysis of future-summary strategies at 8B Scale. We evaluate the effect of different future-summary prediction
approaches against vanilla MTP. Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy
for ARC. Handcrafted random-token future summaries (MTP-Skip) perform worse than standard MTP. In contrast,
handcrafted multi-hot summaries (FSP-BCE) improve over standard MTP, especially on math reasoning (e.g., GSM8K
and MATH), while learned summaries (FSP-RevLM) provide the largest gains across math reasoning and ARC tasks.
1
4
8
16
Pass@k
0.3
0.4
0.5
0.6
0.7
Score
GSM8K
MTP
FSP-RevLM
1
4
8
16
Pass@k
0.1
0.2
0.3
Score
MATH
MTP
FSP-RevLM
Figure 5 Enhanced diversity through learned future summaries. We compare standard MTP, which predicts the immediate
future token, with FSP-RevLM, which enriches the auxiliary head target using learned future summaries. FSP-RevLM
substantially increases output diversity compared to MTP on GSM8K and MATH benchmarks.
5
Related Work
Pitfalls of Next-Token Prediction (NTP).
While next-token prediction (NTP) is the standard loss for modern
LLMs, its shortcomings are increasingly evident. The main issue is the mismatch between training (teacher
forcing) and inference (autoregression): during training, the model sees ground-truth tokens, which encourages
learning spurious correlations or “shortcuts” rather than the true data distribution. Bachmann and Nagarajan
(2024) describe this as the “Clever Hans cheat”, where the model exploits trivial cues from the prefix and fails
on lookahead tasks such as their path-star graph problem. Nagarajan et al. (2025) further show that NTP is
data-inefficient for tasks requiring a “leap of thought,” such as Sibling Discovery. Our approach tackles these
failures by providing a more robust, long-range training signal that discourages such shortcuts.
Going Beyond Immediate Next Token.
Multi-Token Prediction (MTP) addresses limitations of Next Token
Prediction (NTP) by using auxiliary heads to predict future tokens (Gloeckle et al., 2024). DeepSeek-V3 (Liu
et al., 2024) modifies these auxiliary heads to allow slight teacher forcing, but the key issue with both the
approaches is that scaling the auxiliary heads for long-range dependencies is impractical. Recent work aims to
improve efficiency and capture longer dependencies. Gerontopoulos et al. (2025) introduce register tokens,
predicting tokens k steps ahead without architectural changes. Thankaraj et al. (2025) insert lookahead tokens
containing future subsequences, while Liu et al. (2025) use a leap-based strategy to predict non-sequential
future tokens. However, heuristically sampling random future tokens still poses the risk of missing informative
long-range signals. Our FSP-RevLM addresses this by predicting a learned summary of the future, such that
it can extract meaningful long-range information.
9
Bidirectional models.
Leveraging reverse, or right-to-left, order during training has shown empirical benefits
across multiple learning paradigms. The Belief State Transformer (BST) (Hu et al., 2024) employs dual forward
and backward encoders to predict both the next token after a prefix and the previous token before a suffix.
This bidirectional training encourages the model to form a compact belief state, though it does not explicitly
address teacher forcing. In contrast, our FSP-RevLM incorporates a reverse model with a different objective:
mitigating dependence on teacher forcing. While trained on standard left-to-right sequences, FSP-RevLM
aligns the forward model’s embeddings with those of a reverse model, effectively distilling the reverse order
into the forward language model. Another related approach, Meet-in-the-Middle (MiM) (Nguyen et al., 2023),
jointly trains forward and backward models with shared parameters and employs an agreement regularizer
to align their output distributions. Our method differs in two key aspects: (1) we perform distillation with
the reverse model rather than parameter sharing, and (2) we avoid the somewhat impractical assumption
required by MiM that the forward and reverse output distributions must match exactly.
6
Conclusion
In this work, we highlighted a key limitation of existing multi-token prediction methods: the difficulty of scaling
auxiliary heads for long-horizon future prediction. Towards this, we proposed Future Summary Prediction
(FSP), a novel pretraining framework that shifts the auxiliary objective from predicting specific future tokens
to predicting a (learned) summary of the future. Experiments on 8B models, using both hand-crafted and
learned summaries, demonstrate that FSP delivers a stronger, more robust training signal, improving over
NTP and MTP on challenging reasoning and coding tasks. Our findings indicate that focusing on abstract,
predictable aspects of the future is a promising strategy for designing more efficient and effective pretraining
objectives for next-generation large language models.
Acknowledgments
The authors thank Sachin Mehta for their suggestions regarding the real-world pretraining experimental setup.
The authors also gratefully acknowledge the helpful discussions with Mathurin Videau, Sharut Gupta, Sarthak
Mittal, and Andrei Nicolicioiu.
10
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https://arxiv.org/abs/2502.13124.
12
Appendix
A
Additional MTP Variants
In Section 2, we discussed MTP and the proposed future summary prediction to address a major limitation in
MTP. We summarize other variants proposed in the literature and their limitations here.
DeepSeek (DS) MTP.
Another popular variant for MTP was proposed in DeepSeek (Liu et al., 2024), where
we condition the auxiliary head f
′
hk with full prefix x≤t+k−1 to predict the future tokens xt+k, but with
reduced teacher forcing from the "auxiliary" tokens {xt+1, xt+2, · · · , x≤t+k−1}.
LDS-MTP(X, Pθ) = −Ex∼PX
" T −1
X
t=1
τ
X
k=1
1[t + k ≤T] log Pθ(xt+k | x≤t+k−1)
#
.
(14)
The parameterization is:
Pθ(xt+1 | x≤t) = softmax
fu ◦fh ◦fs
x≤t
,
Pθ(xt+k | x≤t+k−1) = softmax
fu ◦f ′
hk
fs
x≤t
, xt+1, . . . , xt+k−1
,
∀k > 1.
(15)
Here, the auxiliary (future) tokens (xt+1, . . . , xt+k−1) are injected directly into the auxiliary heads, bypassing
the backbone, which reduces teacher forcing compared to standard NTP.
However, this approach suffers from the same short-horizon limitation as MTP: predicting multiple future
steps would require adding separate auxiliary heads for each token, which quickly becomes impractical and
limits scalability to long future sequences.
TRELAWNEY.
In this work, we emphasized that standard MTP and DS-MTP often fail to capture the
most informative future tokens, as their auxiliary objectives remain limited to predicting the immediate
next few future tokens. The TRELAWNEY framework of Thankaraj et al. (2025) takes a complementary,
data-centric approach: rather than modifying the model architecture, it augments the input sequences by
inserting a window of future tokens, delimited by special tokens, (<T>, </T>). For example, the input
sequence (x1, · · · , xT ) is transformed into
x1, · · · , xd, < T >, z, < /T >, xd+1 · · · , xT
where z denotes a
block of future tokens sampled from positions beyond d. This encourages the model to predict the sampled
future block z, thereby reducing its reliance on strict teacher forcing.
In contrast to conventional MTP objectives, TRELAWNEY randomly samples subsequences from arbitrary
future offsets, allowing the model to access a more richer and informative training signal than what is available
in the immediate next few tokens. However, this heuristic of randomly sampling future blocks still risks
missing the critical future tokens, limiting its robustness for long-horizon planning tasks.
13
B
Experiments
B.1
Path-Star Graph
Experiment Setup.
We adopt the dataset generation procedure from the official code repository1 of Bachmann
and Nagarajan (2024). Specifically, we construct a set of 50 distinct nodes and generate multiple instances
of path-star graph sequences by randomly sampling nodes from this set.
The training set consists of
ntrain = 200,000 sequences, and evaluation is performed on ntest = 20,000 sequences.
For modeling, we employ the GPT-Mini architecture with the following configuration:
• Total Layers: 12
• Embedding Dimension: 384
• Total Attention Heads: 6
• MLP expansion factor: 4
We summarize the other relevant hyperparameters below.
• Learning Rate: 3e −4
• Batch Size: 256
• Weight Decay: 1e −2
• Total Epochs: 500
• Gradient Norm Clipping: 1.0
Experiment with multiple auxiliary heads in MTP.
A naive approach to incorporate longer-term future
dependencies in MTP is by increasing the number of auxiliary heads. Table 4 reports the results: adding more
auxiliary heads improves performance on G(2, 6), due to reduced teacher forcing. However, simply scaling the
number of auxiliary heads is not a practical solution for modeling long-range dependencies, as shown by the
poor performance on the longer path graph G(2, 8).
Method
G(2,6)
G(2,8)
NTP
0.45 (0.01)
0.45 (0.03)
MTP
0.66 (0.17)
0.49 (0.03)
MTP (aux heads: 2)
0.89 (0.09)
0.46 (0.01)
MTP (aux heads: 3)
0.98 (0.01)
0.47 (0.06)
MTP (aux heads: 4)
0.97 (0.01)
0.48 (0.12)
BCE
1.00 (0.00)
1.00 (0.00)
Table4 Analyzing the performance of MTP on path-star graphs with an increasing number of auxiliary heads. Augmenting
MTP with additional auxiliary heads improves performance on G(2, 6), but practical limits exist: even with four
auxiliary heads, MTP fails to solve the longer path graph G(2, 8).
B.2
Sibling Discovery
Experiment Setup.
We build on the dataset generation procedure from the official code repository2 of Na-
garajan et al. (2025). While the original sibling discovery task presents sequences from a single component,
we modify the setup by concatenating sequences from multiple components. Each component i is defined by a
unique parent P i and three child nodes Si
1, Si
2, Si
3. The support of each conditional distribution P(Si | P i)
is disjoint across children—i.e., each child Si can take N possible values, and these sets of values are
non-overlapping across children. As a result, with K components, the total number of distinct child value
1https://github.com/gregorbachmann/Next-Token-Failures/
2https://github.com/chenwu98/algorithmic-creativity
14
combinations is N 3K. In our experiments, we vary the number of components K ∈2, 4, 6, 8, 10 while fixing
N = 100. For training, we randomly sample child values for each component to construct ntrain = 20,000
sequences, and evaluate on ntest = 3,000 sequences.
We use the GPT-Mini architecture with the following specifications.
• Total Layers: 12
• Embedding Dimension: 384
• Total Attention Heads: 6
• MLP expansion factor: 4
We summarize the other relevant hyperparameters below.
• Learning Rate: 3e −4
• Batch Size: 256
• Weight Decay: 1e −2
• Total Epochs: 150
B.3
Real World Pretraining
Pretraining dataset composition.
Our pretraining corpus is constructed to cover a wide range of domains,
aiming to equip models with both broad knowledge and strong reasoning ability. The bulk of the data is
drawn from a DCLM-style mixture (Li et al., 2025) and large-scale code sources such as GitHub (neogithub,
2022). To strengthen performance on more specialized skills, we further supplement with mathematics and
scientific data, including the DeepMind Mathematics dataset (Saxton et al., 2019), Proof Pile 2 (ArXiv,
OpenWebMath, Algebraic Stack) (Azerbayev et al., 2023), and Stack Exchange from the Pile (Gao et al.,
2020). Additional curated resources include FineWeb-Edu (Lozhkov et al., 2024), the Natural Reasoning
Dataset (Yuan et al., 2025), and AQuA (Ling et al., 2017). Together, this mixture balances large-scale general
text with carefully chosen reasoning-focused datasets.
Pretraining hyperparameters.
All the models were trained using NVIDIA H200 GPUs. We train models
using a cosine learning rate scheduler, with an initial learning rate of 3 × 10−3 for 3B models and 3 × 10−4 for
8B models. We provide details regarding the other hyperparameters below.
• 3B Models
– Batch Size (per GPU): 16
– Total GPUs: 128
– Sequence Length: 2048
– Total Steps: 60k
• 8B Models
– Batch Size (per GPU): 2
– Total GPUs: 256
– Sequence Length: 8192
– Total Steps: 240k
Evaluation Metrics.
We report accuracy for ARC and pass@k for code and math reasoning tasks, with mean
and standard error computed over 3 random seeds. For pass@k, we evaluate each method across a range
of temperature values (0, 0.1, 0.2, . . . , 0.9, 1.0) and report the results corresponding to the temperature that
yields the best performance for each method and task. Note that, in all cases, the auxiliary head is used only
during training, and methods are evaluated via their next-token prediction heads at test time.
15
Additional Experiment: Analysis of future summaries (3B).
Table 5 presents our analysis of different future-
summary strategies as auxiliary head targets at the 3B scale. Similar to the 8B case, we focus on the standard
MTP architecture, without comparing to DS-MTP since it modifies the input to the auxiliary head, allowing
us to isolate the effect of different future targets. Our findings are similar: random-skip handcrafted summaries
underperform relative to MTP, while the proposed multi-hot/bag-of-words approaches perform well. In
our experiments, we observed consistent improvement over MTP with the proposed handcrafted summaries
(FSP-BCE) on both Math and GSM8k tasks, across the 3B and 8B model scales. For the 3B ablation, we
tested removing tf-idf reweighting as well and found that, in most cases, it did not provide benefits, which led
us to keep the tf-idf reweighting for our 8B experiments. The only exceptions where no tf-idf reweighting
performed better were GSM8k at τ = 12 and HumanEval+ at τ = 10. Additionally, we experimented with
using deeper layers at the 3B scale and found that the last layer performed better on ARC Challenge and
ARC Easy, though it was slightly worse on MBPP and Math. Based on these findings, we chose to use the
last layer for our 8B-scale experiments.
Method
MBPP
GSM8K
MATH
HumanEval+
ARC-Challenge
ARC-Easy
MTP
0.526 (0.004)
0.411 (0.001)
0.196 (0.009)
0.321 (0.015)
0.245 (0.002)
0.272 (0.005)
MTP-Skip τ:4
0.519 (0.004)
0.368 (0.007)
0.181 (0.005)
0.309 (0.004)
0.241 (0.000)
0.282 (0.000)
MTP-Skip τ:12
0.485 (0.011)
0.354 (0.005)
0.177 (0.006)
0.287 (0.009)
0.264 (0.008)
0.262 (0.000)
MTP-Skip τ: 32
0.467 (0.007)
0.354 (0.003)
0.189 (0.004)
0.278 (0.017)
0.243 (0.000)
0.240 (0.003)
FSP-BCE τ:12, no tf-idf
0.518 (0.006)
0.431 (0.004)
0.201 (0.009)
0.301 (0.009)
0.250 (0.000)
0.251 (0.000)
FSP-BCE τ:12
0.521 (0.005)
0.419 (0.003)
0.204 (0.002)
0.305 (0.012)
0.254 (0.000)
0.262 (0.004)
FSP-BCE τ:100, no tf-idf
0.512 (0.007)
0.416 (0.005)
0.203 (0.008)
0.309 (0.004)
0.228 (0.003)
0.238 (0.000)
FSP-BCE τ:100
0.524 (0.004)
0.417 (0.004)
0.209 (0.003)
0.293 (0.012)
0.254 (0.006)
0.262 (0.000)
FSP-RevLM depth: 2
0.528 (0.004)
0.428 (0.003)
0.217 (0.006)
0.305 (0.013)
0.243 (0.007)
0.265 (0.000)
FSP-RevLM
0.524 (0.001)
0.436 (0.003)
0.212 (0.002)
0.305 (0.006)
0.255 (0.000)
0.277 (0.000)
Table 5 Analysis of future summaries at 3B Scale. We evaluate the effect of different future-summary prediction approaches
against vanilla MTP. Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC.
Handcrafted multi-hot summaries (FSP-BCE) improve over standard MTP, especially on math reasoning (e.g., GSM8K
and MATH), while learned summaries (FSP-RevLM) provide the largest gains across math reasoning and ARC tasks.
16
|
Beyond Multi-Token Prediction: Pretraining LLMs with Future Summaries Divyat Mahajan1,2,∗, Sachin Goyal3,∗, Badr Youbi Idrissi1, Mohammad Pezeshki1, Ioannis Mitliagkas2, David Lopez-Paz1, Kartik Ahuja1 1FAIR at Meta, 2Mila, Université de Montréal, DIRO, 3Carnegie Mellon University ∗Work done at Meta Next-token prediction (NTP) has driven the success of large language models (LLMs), but it struggles with long-horizon reasoning, planning, and creative writing, with these limitations largely attributed to teacher-forced training. Multi-token prediction (MTP) partially mitigates these issues by predicting several future tokens at once, but it mostly captures short-range dependencies and offers limited improvement. We propose future summary prediction (FSP), which trains an auxiliary head to predict a compact representation of the long-term future, preserving information relevant for long-form generations. We explore two variants of FSP: handcrafted summaries, for example, a bag of words summary of the future of the sequence, and learned summaries, which use embeddings produced by a reverse language model trained from right to left. Large-scale pretraining experiments (3B and 8B-parameter models) demonstrate that FSP provides improvements over both NTP and MTP across math, reasoning, and coding benchmarks. Date: October 17, 2025 Correspondence: Divyat Mahajan at 1 Introduction Early progress in large language models was primarily driven by massive scaling of both data and compute (Brown et al., 2020; Kaplan et al., 2020). However, the returns from this vanilla scaling approach are beginning to diminish as we encounter the "data wall" (Sutskever, 2024). This has renewed efforts toward algorithmic advances, including new architectures and pretraining objectives, that can extract more predictive signal from a fixed amount of training data. Next-token prediction (NTP) with teacher forcing-training models by conditioning on ground-truth history when predicting the next token-is foundational to current pretraining methods. However, this approach introduces a train-inference mismatch known as exposure bias: during inference, the model must rely on its own outputs rather than the ground truth, leading to compounding errors and degraded long-range generation quality (Bengio et al., 2015). Moreover, teacher forcing can induce training-time shortcut learning as well, where the model exploits local cues from the ground-truth prefix instead of capturing true long-range dependencies (Bachmann and Nagarajan, 2024). These issues manifest most clearly in tasks demanding extended reasoning, narrative coherence, and open-ended creativity (Papalampidi et al., 2022; Nagarajan et al., 2025). An appealing alternative to teacher forcing is teacherless training, where the model learns from its own generations rather than relying on ground-truth histories. However, this approach is computationally intensive and challenging to parallelize. As a practical compromise, recent research has explored multi-token prediction (MTP) methods (Gloeckle et al., 2024), which train auxiliary heads to predict several future tokens simultaneously. MTP has demonstrated success in large-scale systems such as DeepSeek-V3 (Liu et al., 2024) and Qwen-3 (Yang et al., 2025). In MTP, each time step augments the standard next-token prediction (xt+1) with auxiliary heads that predict additional future tokens (e.g., xt+2 and beyond). However, these methods typically assume independence among the predicted tokens given the prefix, resulting in poor approximations of the true joint distribution over long future spans. 1 16 Oct 2025 Hidden Representation of Reverse LM . . . x 1, (6) where f ′ hk are auxiliary transformer blocks specialized for predicting future tokens xt+k. This design reduces teacher forcing by predicting xt+k from x≤t only, rather than conditioning on the full prefix x≤t+k-1. This is the key principle behind multi-token prediction, i.e., reduced teacher forcing by requiring the model to predict a block of future tokens at each step. We discuss other variants of MTP such as DeepSeek-MTP and random future token MTP (Thankaraj et al., 2025) in Appendix A. 2.2 Future Summary Prediction (FSP) In MTP, we use a set of auxiliary heads to predict a block of immediate future tokens. A key limitation of this approach is that one does not know exactly where the informative signal in the future sequence lies. Informative signals in future could occur far away in the sequence and be well beyond the number of future tokens (k) that MTP predicts. A trivial approach to overcome this could be to predict all the future tokens. However, having one auxiliary head per future token is not scalable, and limits the amount of future tokens we can utilize during training. Towards this, we propose Future Summary Prediction (FSP), that predicts a compact summary of the (long) future sequence rather than each token individually. 3 Transformer backbone MainBlock x≤t f ′ ha(.) fh(.) a(t, τ) fs(.) AuxBlock Future summary xt+1 Figure 2 An abstraction of the architecture that subsumes NTP, MTP, and FSP. Let a(t, τ) represent a summary of future tokens (xt+2, . . . , xt+τ). The learning objective is: LFSP(X, Pθ) = LNTP(X, Pθ) + Ex∼PX h la Aφ(x≤t), a(t, τ) i , (7) where Pθ is the next-token predictor, Aφ is the summary predictor, and la is the loss between predicted and ground-truth summaries. The architecture is: Pθ(xt+1 | x≤t) = softmax fu ◦fh ◦fs x≤t , Aφ(x≤t) = f ′ ha ◦fs x≤t . (8) Unlike MTP, FSP requires only a single auxiliary head f ′ ha, making it more scalable. The key question, then, is how to construct an effective future summary. In this work, we investigate two approaches: (1) Hand-crafted future summaries. We define a binary vector a(t, τ) ∈RV indicating whether token i appears in the future window: a(t, τ)i = I i ∈{xt+2, . . . , xt+τ} . (9) Given logits from Aφ(x≤t), we minimize a reweighted binary cross-entropy loss: la Aφ(x≤t), a(t, τ) = - V X i=1 w(i) h ai log σ(zi) + (1 -ai) log 1 -σ(zi) i (10) where ai = I i ∈{xt+2, . . . , xt+τ} , zi is the i-th logit of Aφ(x≤t), σ is the sigmoid, and w(i) reflects the importance of token i (e.g. term frequency-inverse document frequency, tf-idf). (2) Learned future summaries. Predicting every future token via hand-crafted summaries as discussed above could be noisy. For example, not all tokens in the future are informative and thus hand-crafted summaries that account for all of them can be wasteful. Instead of predicting all the future tokens, we therefore propose to predict a learned representation of the future tokens relevant to predicting the current token xt+1. We learn this representation via a reverse language model Qψ, (RevLM), trained on "right-to-left" sequences: LRevLM(X, Qψ) = -Ex∼PX " T -1 X t=1 log Qψ(xt+1 | x≥t+2) # . (11) RevLM shares the same architecture, and we take its hidden state as the summary vector: a(t, T -t) = gh ◦gs x≥t+2 . (12) We then train Aφ(x≤t) to match this representation via the l2 loss: la Aφ(x≤t), a(t, T -t) = Aφ(x≤t) -gh ◦gs(x≥t+2) 2 2. (13) Why Future Summary Prediction? The key advantage of future summary prediction is its ability to reduce dependence on teacher forcing when modeling long future sequences. To intuitively measure teacher forcing, consider for each ground-truth token exposed to the model, how much information is the model required to predict about unseen tokens? If the model predicts more such information, then we have reduced teacher forcing. Next-token prediction (NTP) uses the highest degree of teacher forcing, since the model always conditions on ground-truth histories to predict just the next token. Multi-token prediction (MTP) partially relaxes this by asking the model to predict short blocks of future tokens, thereby reducing teacher forcing locally. However, MTP remains constrained by the short horizon of its predictions. In contrast, our proposed approach predicts summaries of long future sequences, substantially reducing teacher forcing by requiring the model to reason about rich, global properties of the target trajectory. 4 3 Analyzing Future Summaries To better understand how FSP mitigates the challenges of long-horizon planning, we consider the graph modeling benchmarks introduced in prior works. Our analysis yields following insights: • Long future summaries matter. On the canonical path-star task (Bachmann and Nagarajan, 2024), we find that MTP with short range future prediction fails to generalize, highlighting the need for auxiliary objectives that incorporate long-range future information. • Adaptive future summaries matter. On a modified sibling discovery task (Nagarajan et al., 2025), we show that incorporating every future token with hand-crafted summaries is suboptimal, highlighting the need for learned summaries that retain key information. 3.1 Long future summary is important s 5 1 2 3 4 5 s 5 1 2 3 4 5 s 5 1 2 3 4 5 s 5 1 2 3 4 5 s 5 1 2 3 4 5 s 5 1 2 3 4 5 Difficult prediction task Too easy due to shortcut Mix of 1- to 4-hop reasoning NTP: FSP: Example Sequence: x = s 5 1 2 3 4 5 start/goal path Example Graph: s 1 2 3 4 5 6 7 8 9 10 Figure 3 Analysis of FSP-BCE on the path-star task, which tests long-horizon planning. Left: Accuracy (mean over 5 random seeds) of different pretraining objectives on degree 2 graphs with path lengths 6 and 8. Standard NTP generalizes poorly, and MTP's accuracy degrades as the path length increases, while FSP-BCE achieves perfect accuracy. Right: An illustration of why NTP fails while FSP-BCE succeeds, where the input context is shown in grey and the target is in beige. We consider the path-star graph, a directed acyclic graph (DAG) G(d, l) composed of d paths, each of length l, originating from a central start node vstart. The model is provided with the adjacency list of the graph in the prefix, and the task is to generate the path from vstart to a designated end node vend. Let the target path be (vstart, v1, v2, · · · , vend) and the input prefix be p = (Adj(G), vstart, vend). Then NTP with teacher forcing predicts an intermediate node in the path as Pθ(vi+1|p, v≤i). As shown by Bachmann and Nagarajan (2024), NTP often learns shortcut solutions: the model can recover vi+1 directly from vi by scanning the adjacency list in p, without learning the underlying long-range plan (Figure 3). This leads to gradient starvation (Pezeshki et al., 2021), where the supervision signal for the actual planning task is lost. Once the shortcut is learned, meaningful gradient information remains only for predicting the first step v1, as it is the only difficult token. A natural remedy is to reduce teacher forcing via future prediction approaches, which require the model to predict tokens further ahead. This makes shortcuts less effective, as multiple lookups in the adjacency list would be needed to predict future tokens. Hence, we consider the MTP approach, which predicts the immediate future token, and the handcrafted summary method FSP-BCE, which compresses the information about all the future tokens from the path. By summarizing the entire future trajectory, FSP-BCE substantially reduces teacher forcing, encouraging the model to plan the full path instead of exploiting shortcuts. Findings. We conduct experiments for two graphs G(2, 6) and G(2, 8), with all the approaches pretrained from scratch using the GPT-Mini architecture (details in Section B.1). At inference, we discard the auxiliary head used in future prediction, and the task is to generate the complete path given the prefix p. The evaluation 5 metric checks whether the generated path exactly matches the true path. Results in Figure 3 (left) show that both NTP and MTP fail to generalize (both obtained perfect training accuracy), and the accuracy of MTP degrades further for the scenario with longer path G(2, 8). Hence, predicting just the immediate future token is not enough, and FSP-BCE tackles this by efficiently compressing all the future tokens from the path, enabling it to achieve perfect accuracy in both cases. Appendix B.1 (Table 4) shows that increasing the number of auxiliary heads in MTP can provide some improvement, but practical limits exist: even with four additional future heads MTP cannot solve the longer path graph G(2, 8). 3.2 Adaptive future summary is important Example Sequence: Graph: x = S1 S1 S1 P1 S3 S3 S3 P3 S2 S2 S2 P2 1 2 3 1 2 3 1 2 3 P1 S1 1 S1 2 S1 3 Pm Sm 1 Sm 2 Sm 3 .... Relevant future Not predictable from the prefix If this is the prefix Convergence Speed Relative to NTP: Number of Components Figure 4 Analysis on the modified sibling discovery task which requires adaptive future summaries. Left: Convergence speed (mean (s.e.) over 3 random seeds) relative to NTP, where lower values imply faster convergence. FSP-RevLM converges faster than NTP while FSP-BCE improves only in the cases with few components. Right: Task setup illustration- given the prefix, only future tokens in the highlighted component are informative. FSP-BCE, which summarizes all future tokens, suffers from irrelevant information, whereas FSP-RevLM summarizes only the informative aspects. Long handcrafted summaries often incorporate all future tokens, even though only a subset may provide meaningful supervision signals, while the rest can introduce noise. This motivates the need for an adaptive future summary. We illustrate this with modification of the sibling discovery task. Figure 4 (right) depicts the setup, the model must generate sequences made of concatenated, independent components, where each component lists its children nodes first, followed by their parent (e.g., nodes S1 1, S1 2, S1 3 followed by parent P 1). Intuitively, the causal factorization implied by the DAG (parent followed by children) implies a goal-conditioned approach: by conditioning on the parent, the model can easily capture the sibling dependencies. Under NTP, the model estimates Pθ(S1 2|S1 1) without the parent, making sibling relationships harder to learn and requiring more samples. Future prediction can help, when the model predicts the parent from S1 1, then the representation can incorporate the parent information. This enables goal-conditioned planning, the model can predict S1 2 conditioned on both S1 1 and the parent, allowing sample efficient learning of sibling relationships (see Nagarajan et al. (2025) for details). However, not all future tokens are equally informative. As Figure 4 illustrates, future tokens from a different component do not provide relevant signal for predicting S1 2. Handcrafted summaries that include all future tokens may therefore be affected by the irrelevant information, whereas learned summaries remain robust, as the reverse language model learns representation that emphasize only the predictive signals needed to infer the next token. Findings. We verify this empirically by comparing FSP-BCE and FSP-RevLM in experiments with varying number of total components. All models are pretrained from scratch with the GPT-Mini architecture (details in Section B.2), and the inference task requires generating a coherent sequence (siblings followed by parent for each component). At convergence, all models produce coherent sequences; to quantify the learning speedup over NTP, we report the ratio of steps to convergence relative to NTP, with lower values indicating 6 faster learning. Results in figure 4 (left) shows that FSP-BCE improves over NTP only when the number of components is small, with gains disappearing for more than six components. In contrast, FSP-RevLM consistently achieves faster convergence across all component sizes, confirming the benefit of adaptive future summaries. 4 Experiments 4.1 Setup We pretrain 3B- and 8B-parameter models on corpora of 250B and 1T tokens, respectively, covering diverse domains. The majority of the data comes from DCLM-like sources and GitHub repositories, supplemented by specialized material in mathematics, programming, and related areas. Models are evaluated across a diverse set of benchmarks: ARC-Easy/Challenge (Clark et al., 2018) for general reasoning, MBPP (Austin et al., 2021) and HumanEval+ (Liu et al., 2023) for code generation, and GSM8K (Cobbe et al., 2021) and Math-500 (Hendrycks et al., 2021) for mathematical reasoning. Details of the pretraining corpus and hyperparameters are provided in Section B.3. We first benchmark our primary method, FSP-RevLM, against the baselines: next-token prediction (NTP), standard multi-token prediction (MTP) (Gloeckle et al., 2024), and DeepSeek-MTP (DS-MTP) (Liu et al., 2024). A natural way to improve MTP for longer horizons is to add multiple auxiliary heads, each predicting a token farther into the future, but this approach quickly becomes impractical. We therefore constrain both MTP and DS-MTP to a single auxiliary head predicting the immediate future token. This design choice keeps the comparison consistent and aligned with the proposed FSP, since it uses a single auxiliary head. Building on this unified single auxiliary head framework, we conduct an analysis of how MTP can be enhanced by predicting richer future targets instead of the immediate future token. In addition to predicting the learned future summary (FSP-RevLM), we compare handcrafted future summaries over short- and long-range windows. This includes the proposed multi-hot future summary (FSP-BCE) as a contributed baseline, and a random-token summary baseline (radomly sampling token from future), in line with prior works (Thankaraj et al., 2025; Gerontopoulos et al., 2025). Note. All experiments are conducted under iso-data conditions, meaning that all the methods are trained on identical datasets. For the proposed FSP-RevLM, this implies that both the forward and reverse models are trained on the same data. In line with standard practice in distillation, we do not perform iso-compute comparisons that include the teacher model's (ReverseLM) cost in the reported compute budget. In practical scenarios, the computational cost of training the teacher model is typically amortized, hence it can be treated as a one-time overhead that is excluded from comparisons of student models (Gemma et al., 2024, 2025). 4.2 Results At the 8B scale (Table 1), future-summary supervision via the reverseLM (FSP-RevLM) consistently improves the performance across different evaluation tasks. On ARC-Easy, FSP-RevLM (76.6%) provides significant improvement over NTP (71.8%) and MTP (73.6%), and it also leads on ARC-Challenge and MATH. For code generation, it achieves the highest score on MBPP and ties with MTP on HumanEval+, showing that the benefits of predicting future summaries generalize across both reasoning and program synthesis tasks. GSM8K is the one task where NTP (71.6%) holds a lead, though FSP-RevLM (70.5%) still narrows the gap relative to MTP (67.8%). At the 3B scale (Table 2), DeepSeek-MTP is a strong baseline and obtains better performance than FSP-RevLM on most tasks, except math reasoning. More importantly, FSP-RevLM exhibits larger relative improvements as scale increases from 3B to 8B parameters, and becomes more favourable than DS-MTP. Further, note that even at the 3B scale, FSP-RevLM still beats MTP on ARC and math reasoning tasks, and performs comparably on MBPP, suggesting that even at smaller scales, learning to predict future summaries may provide more effective auxiliary signal than immediate future token prediction with MTP. 7 Task NTP MTP DS-MTP FSP-RevLM ARC-Easy 0.718 (0.000) 0.736 (0.000) 0.617 (0.003) 0.766 (0.000) ARC-Challenge 0.531 (0.000) 0.552 (0.000) 0.426 (0.002) 0.559 (0.000) GSM8K 0.716 (0.003) 0.678 (0.007) 0.704 (0.003) 0.705 (0.004) MATH 0.342 (0.008) 0.309 (0.006) 0.335 (0.014) 0.351 (0.017) MBPP 0.657 (0.004) 0.672 (0.008) 0.678 (0.006) 0.683 (0.006) HumanEval+ 0.478 (0.019) 0.541 (0.011) 0.526 (0.013) 0.541 (0.009) Table 1 Pretraining at 8B scale. We benchmark the proposed FSP-RevLM approach against NTP, MTP, and DS-MTP. Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC. FSP-RevLM achieves the strongest overall performance, with large gains over the baselines on ARC tasks and MATH, and competitive results with (DS) MTP on code benchmarks. Task NTP MTP DS-MTP FSP-RevLM ARC-Easy 0.263 (0.002) 0.272 (0.005) 0.293 (0.000) 0.277 (0.000) ARC-Challenge 0.263 (0.001) 0.245 (0.002) 0.274 (0.008) 0.255 (0.000) GSM8K 0.410 (0.003) 0.411 (0.001) 0.417 (0.003) 0.436 (0.003) MATH 0.213 (0.004) 0.196 (0.009) 0.201 (0.004) 0.212 (0.002) MBPP 0.521 (0.007) 0.526 (0.004) 0.537 (0.007) 0.524 (0.001) HumanEval+ 0.301 (0.009) 0.321 (0.015) 0.348 (0.022) 0.305 (0.006) Table 2 Pretraining at 3B scale. We benchmark the proposed FSP-RevLM approach against NTP, MTP, and DS-MTP. Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC. At this smaller scale, DS-MTP is a strong overall baseline, but FSP-RevLM outperforms it on math reasoning tasks, and also provides substantial gains over MTP on ARC and math reasoning tasks. Further, as we scale the approaches to 8B parameters, FSP-RevLM scales more favorably than DS-MTP, overtaking it on most tasks. 4.3 Analyzing MTP with different future summaries Table 3 presents our analysis of different future-summary strategies as auxiliary head targets at 8B scale. We focus on the standard MTP architecture, without comparing to DS-MTP as it modifies the input to auxiliary head, to isolate the effect of different future targets on the auxiliary head. Our results show that random-token handcrafted future summaries (MTP-Skip) perform worse than standard MTP with immediate future token prediction, and performance further degrades as the future window (τ) increases. In contrast, the proposed multi-hot or bag-of-words handcrafted future summaries yield meaningful improvements over MTP, especially on math reasoning tasks, with both shorter (τ = 12) and longer (τ = 100) future windows. For example, FSP-BCE with τ = 12 achieves 33.1% on MATH (+2.2 points) and 69.9% on GSM8K (+2.1 points), while even longer windows (τ = 100) further amplifies the performance on GSM8K (71.4%, +3.6 points). Finally, our learned future summaries (FSP-RevLM) outperform MTP across all evaluation tasks, with especially pronounced gains on math reasoning: 35.1% on MATH (+4.2 points) and 70.5% on GSM8K (+3.5 points). Further analysis (Figure 5) for these math reasoning tasks shows that learned summaries promote greater output diversity across different pass@k settings, compared to vanilla MTP with immediate future token prediction. In Section B.3 (Table 5), we replicate these findings at 3B scale, where both handcrafted and learned future summaries improve over vanilla MTP, again most prominently on math reasoning tasks. Additional ablations explore the effects of omitting reweighting in FSP-BCE and predicting learned summaries from deeper layers of the reverseLM (FSP-RevLM). 8 Method MBPP GSM8K MATH HumanEval+ ARC-Challenge ARC-Easy MTP 0.672 (0.008) 0.678 (0.007) 0.309 (0.006) 0.541 (0.011) 0.552 (0.000) 0.736 (0.000) MTP-Skip τ:4 0.658 (0.005) 0.639 (0.004) 0.277 (0.020) 0.508 (0.009) 0.494 (0.003) 0.722 (0.000) MTP-Skip τ:12 0.623 (0.002) 0.621 (0.005) 0.287 (0.018) 0.486 (0.010) 0.512 (0.000) 0.710 (0.003) MTP-Skip τ:32 0.611 (0.008) 0.598 (0.007) 0.271 (0.005) 0.459 (0.007) 0.379 (0.000) 0.564 (0.000) FSP-BCE τ:12 0.669 (0.005) 0.699 (0.006) 0.331 (0.016) 0.508 (0.005) 0.562 (0.000) 0.737 (0.000) FSP-BCE τ:100 0.671 (0.002) 0.714 (0.009) 0.331 (0.007) 0.500 (0.019) 0.459 (0.000) 0.662 (0.000) FSP-RevLM 0.683 (0.006) 0.705 (0.004) 0.351 (0.017) 0.541 (0.009) 0.559 (0.000) 0.766 (0.000) Table 3 Analysis of future-summary strategies at 8B Scale. We evaluate the effect of different future-summary prediction approaches against vanilla MTP. Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC. Handcrafted random-token future summaries (MTP-Skip) perform worse than standard MTP. In contrast, handcrafted multi-hot summaries (FSP-BCE) improve over standard MTP, especially on math reasoning (e.g., GSM8K and MATH), while learned summaries (FSP-RevLM) provide the largest gains across math reasoning and ARC tasks. 1 4 8 16 Pass@k 0.3 0.4 0.5 0.6 0.7 Score GSM8K MTP FSP-RevLM 1 4 8 16 Pass@k 0.1 0.2 0.3 Score MATH MTP FSP-RevLM Figure 5 Enhanced diversity through learned future summaries. We compare standard MTP, which predicts the immediate future token, with FSP-RevLM, which enriches the auxiliary head target using learned future summaries. FSP-RevLM substantially increases output diversity compared to MTP on GSM8K and MATH benchmarks. 5 Related Work Pitfalls of Next-Token Prediction (NTP). While next-token prediction (NTP) is the standard loss for modern LLMs, its shortcomings are increasingly evident. The main issue is the mismatch between training (teacher forcing) and inference (autoregression): during training, the model sees ground-truth tokens, which encourages learning spurious correlations or "shortcuts" rather than the true data distribution. Bachmann and Nagarajan (2024) describe this as the "Clever Hans cheat", where the model exploits trivial cues from the prefix and fails on lookahead tasks such as their path-star graph problem. Nagarajan et al. (2025) further show that NTP is data-inefficient for tasks requiring a "leap of thought," such as Sibling Discovery. Our approach tackles these failures by providing a more robust, long-range training signal that discourages such shortcuts. Going Beyond Immediate Next Token. Multi-Token Prediction (MTP) addresses limitations of Next Token Prediction (NTP) by using auxiliary heads to predict future tokens (Gloeckle et al., 2024). DeepSeek-V3 (Liu et al., 2024) modifies these auxiliary heads to allow slight teacher forcing, but the key issue with both the approaches is that scaling the auxiliary heads for long-range dependencies is impractical. Recent work aims to improve efficiency and capture longer dependencies. Gerontopoulos et al. (2025) introduce register tokens, predicting tokens k steps ahead without architectural changes. Thankaraj et al. (2025) insert lookahead tokens containing future subsequences, while Liu et al. (2025) use a leap-based strategy to predict non-sequential future tokens. However, heuristically sampling random future tokens still poses the risk of missing informative long-range signals. Our FSP-RevLM addresses this by predicting a learned summary of the future, such that it can extract meaningful long-range information. 9 Bidirectional models. Leveraging reverse, or right-to-left, order during training has shown empirical benefits across multiple learning paradigms. The Belief State Transformer (BST) (Hu et al., 2024) employs dual forward and backward encoders to predict both the next token after a prefix and the previous token before a suffix. This bidirectional training encourages the model to form a compact belief state, though it does not explicitly address teacher forcing. In contrast, our FSP-RevLM incorporates a reverse model with a different objective: mitigating dependence on teacher forcing. While trained on standard left-to-right sequences, FSP-RevLM aligns the forward model's embeddings with those of a reverse model, effectively distilling the reverse order into the forward language model. Another related approach, Meet-in-the-Middle (MiM) (Nguyen et al., 2023), jointly trains forward and backward models with shared parameters and employs an agreement regularizer to align their output distributions. Our method differs in two key aspects: (1) we perform distillation with the reverse model rather than parameter sharing, and (2) we avoid the somewhat impractical assumption required by MiM that the forward and reverse output distributions must match exactly. 6 Conclusion In this work, we highlighted a key limitation of existing multi-token prediction methods: the difficulty of scaling auxiliary heads for long-horizon future prediction. Towards this, we proposed Future Summary Prediction (FSP), a novel pretraining framework that shifts the auxiliary objective from predicting specific future tokens to predicting a (learned) summary of the future. Experiments on 8B models, using both hand-crafted and learned summaries, demonstrate that FSP delivers a stronger, more robust training signal, improving over NTP and MTP on challenging reasoning and coding tasks. Our findings indicate that focusing on abstract, predictable aspects of the future is a promising strategy for designing more efficient and effective pretraining objectives for next-generation large language models. Acknowledgments The authors thank Sachin Mehta for their suggestions regarding the real-world pretraining experimental setup. The authors also gratefully acknowledge the helpful discussions with Mathurin Videau, Sharut Gupta, Sarthak Mittal, and Andrei Nicolicioiu. 10 References Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, et al. Program synthesis with large language models. arXiv preprint , 2021. Zhangir Azerbayev, Hailey Schoelkopf, Keiran Paster, Marco Dos Santos, Stephen McAleer, Albert Q. Jiang, Jia Deng, Stella Biderman, and Sean Welleck. Llemma: An open language model for mathematics, 2023. Gregor Bachmann and Vaishnavh Nagarajan. The pitfalls of next-token prediction. In International Conference on Machine Learning, pages 2296-2318. PMLR, 2024. Samy Bengio, Oriol Vinyals, Navdeep Jaitly, and Noam Shazeer. Scheduled sampling for sequence prediction with recurrent neural networks. Advances in neural information processing systems, 28, 2015. Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel Ziegler, Jeffrey Wu, Clemens Winter, Chris Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. Language models are few-shot learners. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 1877-1901. Curran Associates, Inc., 2020. https://proceedings.neurips.cc/paper_files/paper/ 2020/file/1457c0d6bfcb4967418bfb8ac142f64a-Paper.pdf. Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and Oyvind Tafjord. Think you have solved question answering? try arc, the ai2 reasoning challenge. arXiv preprint , 2018. Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, et al. Training verifiers to solve math word problems. arXiv preprint , 2021. Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, Shawn Presser, and Connor Leahy. The Pile: An 800gb dataset of diverse text for language modeling. arXiv preprint , 2020. Team Gemma, Morgane Riviere, Shreya Pathak, Pier Giuseppe Sessa, Cassidy Hardin, Surya Bhupatiraju, Léonard Hussenot, Thomas Mesnard, Bobak Shahriari, Alexandre Ramé, et al. Gemma 2: Improving open language models at a practical size. arXiv preprint , 2024. Team Gemma, Aishwarya Kamath, Johan Ferret, Shreya Pathak, Nino Vieillard, Ramona Merhej, Sarah Perrin, Tatiana Matejovicova, Alexandre Ramé, Morgane Rivière, et al. Gemma 3 technical report. arXiv preprint , 2025. Anastasios Gerontopoulos, Spyros Gidaris, and Nikos Komodakis. Multi-token prediction needs registers, 2025. https://arxiv.org/abs/2505.10518. Fabian Gloeckle, Badr Youbi Idrissi, Baptiste Rozière, David Lopez-Paz, and Gabriel Synnaeve. Better & faster large language models via multi-token prediction. In Proceedings of the 41st International Conference on Machine Learning, pages 15706-15734, 2024. Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset. arXiv preprint , 2021. Edward S Hu, Kwangjun Ahn, Qinghua Liu, Haoran Xu, Manan Tomar, Ada Langford, Dinesh Jayaraman, Alex Lamb, and John Langford. The belief state transformer. arXiv preprint , 2024. Jared Kaplan, Sam McCandlish, T. J. Henighan, Tom B. Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeff Wu, and Dario Amodei. Scaling laws for neural language models. ArXiv, abs/2001.08361, 2020. https://api.semanticscholar.org/CorpusID:210861095. Jeffrey Li, Alex Fang, et al. Datacomp-lm: In search of the next generation of training sets for language models, 2025. https://arxiv.org/abs/2406.11794. Wang Ling, Dani Yogatama, Chris Dyer, and Phil Blunsom. Program induction by rationale generation : Learning to solve and explain algebraic word problems, 2017. https://arxiv.org/abs/1705.04146. 11 Aixin Liu, Bei Feng, Bing Xue, Bingxuan Wang, Bochao Wu, Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, et al. Deepseek-v3 technical report. arXiv preprint , 2024. Jiawei Liu, Chunqiu Steven Xia, Yuyao Wang, and Lingming Zhang. Is your code generated by chatgpt really correct? rigorous evaluation of large language models for code generation. Advances in Neural Information Processing Systems, 36:21558-21572, 2023. Xiaohao Liu, Xiaobo Xia, Weixiang Zhao, Manyi Zhang, Xianzhi Yu, Xiu Su, Shuo Yang, See-Kiong Ng, and Tat-Seng Chua. L-mtp: Leap multi-token prediction beyond adjacent context for large language models. arXiv preprint , 2025. Anton Lozhkov, Loubna Ben Allal, Leandro von Werra, and Thomas Wolf. Fineweb-edu: the finest collection of educational content, 2024. https://huggingface.co/datasets/HuggingFaceFW/fineweb-edu. Vaishnavh Nagarajan, Chen Henry Wu, Charles Ding, and Aditi Raghunathan. Roll the dice & look before you leap: Going beyond the creative limits of next-token prediction. In Forty-second International Conference on Machine Learning. PMLR, 2025. neogithub. Github code dataset, 2022. https://huggingface.co/datasets/codeparrot/github-code. Anh Nguyen, Nikos Karampatziakis, and Weizhu Chen. Meet in the middle: A new pre-training paradigm. Advances in Neural Information Processing Systems, 36:5079-5091, 2023. Pinelopi Papalampidi, Kris Cao, and Tomas Kocisky. Towards coherent and consistent use of entities in narrative generation. In International Conference on Machine Learning, pages 17278-17294. PMLR, 2022. Mohammad Pezeshki, Oumar Kaba, Yoshua Bengio, Aaron C Courville, Doina Precup, and Guillaume Lajoie. Gradient starvation: A learning proclivity in neural networks. Advances in Neural Information Processing Systems, 34: 1256-1272, 2021. David Saxton, Edward Grefenstette, Felix Hill, and Pushmeet Kohli. Analysing mathematical reasoning abilities of neural models, 2019. https://arxiv.org/abs/1904.01557. Ilya Sutskever. Sequence to sequence learning with neural networks: What a decade. NeurIPS 2024 Test-of-Time Talk (YouTube video), 2024. https://www.youtube.com/watch?v=1yvBqasHLZs. Presented at NeurIPS 2024, Vancouver, Canada. Abitha Thankaraj, Yiding Jiang, J. Zico Kolter, and Yonatan Bisk. Looking beyond the next token, 2025. https: //arxiv.org/abs/2504.11336. An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, et al. Qwen3 technical report. arXiv preprint , 2025. Weizhe Yuan, Jane Yu, Song Jiang, Karthik Padthe, Yang Li, Dong Wang, Ilia Kulikov, Kyunghyun Cho, Yuandong Tian, Jason E Weston, and Xian Li. Naturalreasoning: Reasoning in the wild with 2.8m challenging questions, 2025. https://arxiv.org/abs/2502.13124. 12 Appendix A Additional MTP Variants In Section 2, we discussed MTP and the proposed future summary prediction to address a major limitation in MTP. We summarize other variants proposed in the literature and their limitations here. DeepSeek (DS) MTP. Another popular variant for MTP was proposed in DeepSeek (Liu et al., 2024), where we condition the auxiliary head f ′ hk with full prefix x≤t+k-1 to predict the future tokens xt+k, but with reduced teacher forcing from the "auxiliary" tokens {xt+1, xt+2, · · · , x≤t+k-1}. LDS-MTP(X, Pθ) = -Ex∼PX " T -1 X t=1 τ X k=1 1[t + k ≤T] log Pθ(xt+k | x≤t+k-1) # . (14) The parameterization is: Pθ(xt+1 | x≤t) = softmax fu ◦fh ◦fs x≤t , Pθ(xt+k | x≤t+k-1) = softmax fu ◦f ′ hk fs x≤t , xt+1, . . . , xt+k-1 , ∀k > 1. (15) Here, the auxiliary (future) tokens (xt+1, . . . , xt+k-1) are injected directly into the auxiliary heads, bypassing the backbone, which reduces teacher forcing compared to standard NTP. However, this approach suffers from the same short-horizon limitation as MTP: predicting multiple future steps would require adding separate auxiliary heads for each token, which quickly becomes impractical and limits scalability to long future sequences. TRELAWNEY. In this work, we emphasized that standard MTP and DS-MTP often fail to capture the most informative future tokens, as their auxiliary objectives remain limited to predicting the immediate next few future tokens. The TRELAWNEY framework of Thankaraj et al. (2025) takes a complementary, data-centric approach: rather than modifying the model architecture, it augments the input sequences by inserting a window of future tokens, delimited by special tokens, ( , ). For example, the input sequence (x1, · · · , xT ) is transformed into x1, · · · , xd, , z, , xd+1 · · · , xT where z denotes a block of future tokens sampled from positions beyond d. This encourages the model to predict the sampled future block z, thereby reducing its reliance on strict teacher forcing. In contrast to conventional MTP objectives, TRELAWNEY randomly samples subsequences from arbitrary future offsets, allowing the model to access a more richer and informative training signal than what is available in the immediate next few tokens. However, this heuristic of randomly sampling future blocks still risks missing the critical future tokens, limiting its robustness for long-horizon planning tasks. 13 B Experiments B.1 Path-Star Graph Experiment Setup. We adopt the dataset generation procedure from the official code repository1 of Bachmann and Nagarajan (2024). Specifically, we construct a set of 50 distinct nodes and generate multiple instances of path-star graph sequences by randomly sampling nodes from this set. The training set consists of ntrain = 200,000 sequences, and evaluation is performed on ntest = 20,000 sequences. For modeling, we employ the GPT-Mini architecture with the following configuration: • Total Layers: 12 • Embedding Dimension: 384 • Total Attention Heads: 6 • MLP expansion factor: 4 We summarize the other relevant hyperparameters below. • Learning Rate: 3e -4 • Batch Size: 256 • Weight Decay: 1e -2 • Total Epochs: 500 • Gradient Norm Clipping: 1.0 Experiment with multiple auxiliary heads in MTP. A naive approach to incorporate longer-term future dependencies in MTP is by increasing the number of auxiliary heads. Table 4 reports the results: adding more auxiliary heads improves performance on G(2, 6), due to reduced teacher forcing. However, simply scaling the number of auxiliary heads is not a practical solution for modeling long-range dependencies, as shown by the poor performance on the longer path graph G(2, 8). Method G(2,6) G(2,8) NTP 0.45 (0.01) 0.45 (0.03) MTP 0.66 (0.17) 0.49 (0.03) MTP (aux heads: 2) 0.89 (0.09) 0.46 (0.01) MTP (aux heads: 3) 0.98 (0.01) 0.47 (0.06) MTP (aux heads: 4) 0.97 (0.01) 0.48 (0.12) BCE 1.00 (0.00) 1.00 (0.00) Table4 Analyzing the performance of MTP on path-star graphs with an increasing number of auxiliary heads. Augmenting MTP with additional auxiliary heads improves performance on G(2, 6), but practical limits exist: even with four auxiliary heads, MTP fails to solve the longer path graph G(2, 8). B.2 Sibling Discovery Experiment Setup. We build on the dataset generation procedure from the official code repository2 of Nagarajan et al. (2025). While the original sibling discovery task presents sequences from a single component, we modify the setup by concatenating sequences from multiple components. Each component i is defined by a unique parent P i and three child nodes Si 1, Si 2, Si 3. The support of each conditional distribution P(Si | P i) is disjoint across children-i.e., each child Si can take N possible values, and these sets of values are non-overlapping across children. As a result, with K components, the total number of distinct child value 1https://github.com/gregorbachmann/Next-Token-Failures/ 2https://github.com/chenwu98/algorithmic-creativity 14 combinations is N 3K. In our experiments, we vary the number of components K ∈2, 4, 6, 8, 10 while fixing N = 100. For training, we randomly sample child values for each component to construct ntrain = 20,000 sequences, and evaluate on ntest = 3,000 sequences. We use the GPT-Mini architecture with the following specifications. • Total Layers: 12 • Embedding Dimension: 384 • Total Attention Heads: 6 • MLP expansion factor: 4 We summarize the other relevant hyperparameters below. • Learning Rate: 3e -4 • Batch Size: 256 • Weight Decay: 1e -2 • Total Epochs: 150 B.3 Real World Pretraining Pretraining dataset composition. Our pretraining corpus is constructed to cover a wide range of domains, aiming to equip models with both broad knowledge and strong reasoning ability. The bulk of the data is drawn from a DCLM-style mixture (Li et al., 2025) and large-scale code sources such as GitHub (neogithub, 2022). To strengthen performance on more specialized skills, we further supplement with mathematics and scientific data, including the DeepMind Mathematics dataset (Saxton et al., 2019), Proof Pile 2 (ArXiv, OpenWebMath, Algebraic Stack) (Azerbayev et al., 2023), and Stack Exchange from the Pile (Gao et al., 2020). Additional curated resources include FineWeb-Edu (Lozhkov et al., 2024), the Natural Reasoning Dataset (Yuan et al., 2025), and AQuA (Ling et al., 2017). Together, this mixture balances large-scale general text with carefully chosen reasoning-focused datasets. Pretraining hyperparameters. All the models were trained using NVIDIA H200 GPUs. We train models using a cosine learning rate scheduler, with an initial learning rate of 3 × 10-3 for 3B models and 3 × 10-4 for 8B models. We provide details regarding the other hyperparameters below. • 3B Models - Batch Size (per GPU): 16 - Total GPUs: 128 - Sequence Length: 2048 - Total Steps: 60k • 8B Models - Batch Size (per GPU): 2 - Total GPUs: 256 - Sequence Length: 8192 - Total Steps: 240k Evaluation Metrics. We report accuracy for ARC and pass@k for code and math reasoning tasks, with mean and standard error computed over 3 random seeds. For pass@k, we evaluate each method across a range of temperature values (0, 0.1, 0.2, . . . , 0.9, 1.0) and report the results corresponding to the temperature that yields the best performance for each method and task. Note that, in all cases, the auxiliary head is used only during training, and methods are evaluated via their next-token prediction heads at test time. 15 Additional Experiment: Analysis of future summaries (3B). Table 5 presents our analysis of different futuresummary strategies as auxiliary head targets at the 3B scale. Similar to the 8B case, we focus on the standard MTP architecture, without comparing to DS-MTP since it modifies the input to the auxiliary head, allowing us to isolate the effect of different future targets. Our findings are similar: random-skip handcrafted summaries underperform relative to MTP, while the proposed multi-hot/bag-of-words approaches perform well. In our experiments, we observed consistent improvement over MTP with the proposed handcrafted summaries (FSP-BCE) on both Math and GSM8k tasks, across the 3B and 8B model scales. For the 3B ablation, we tested removing tf-idf reweighting as well and found that, in most cases, it did not provide benefits, which led us to keep the tf-idf reweighting for our 8B experiments. The only exceptions where no tf-idf reweighting performed better were GSM8k at τ = 12 and HumanEval+ at τ = 10. Additionally, we experimented with using deeper layers at the 3B scale and found that the last layer performed better on ARC Challenge and ARC Easy, though it was slightly worse on MBPP and Math. Based on these findings, we chose to use the last layer for our 8B-scale experiments. Method MBPP GSM8K MATH HumanEval+ ARC-Challenge ARC-Easy MTP 0.526 (0.004) 0.411 (0.001) 0.196 (0.009) 0.321 (0.015) 0.245 (0.002) 0.272 (0.005) MTP-Skip τ:4 0.519 (0.004) 0.368 (0.007) 0.181 (0.005) 0.309 (0.004) 0.241 (0.000) 0.282 (0.000) MTP-Skip τ:12 0.485 (0.011) 0.354 (0.005) 0.177 (0.006) 0.287 (0.009) 0.264 (0.008) 0.262 (0.000) MTP-Skip τ: 32 0.467 (0.007) 0.354 (0.003) 0.189 (0.004) 0.278 (0.017) 0.243 (0.000) 0.240 (0.003) FSP-BCE τ:12, no tf-idf 0.518 (0.006) 0.431 (0.004) 0.201 (0.009) 0.301 (0.009) 0.250 (0.000) 0.251 (0.000) FSP-BCE τ:12 0.521 (0.005) 0.419 (0.003) 0.204 (0.002) 0.305 (0.012) 0.254 (0.000) 0.262 (0.004) FSP-BCE τ:100, no tf-idf 0.512 (0.007) 0.416 (0.005) 0.203 (0.008) 0.309 (0.004) 0.228 (0.003) 0.238 (0.000) FSP-BCE τ:100 0.524 (0.004) 0.417 (0.004) 0.209 (0.003) 0.293 (0.012) 0.254 (0.006) 0.262 (0.000) FSP-RevLM depth: 2 0.528 (0.004) 0.428 (0.003) 0.217 (0.006) 0.305 (0.013) 0.243 (0.007) 0.265 (0.000) FSP-RevLM 0.524 (0.001) 0.436 (0.003) 0.212 (0.002) 0.305 (0.006) 0.255 (0.000) 0.277 (0.000) Table 5 Analysis of future summaries at 3B Scale. We evaluate the effect of different future-summary prediction approaches against vanilla MTP. Results (mean ± s.e. over 3 seeds) report pass@16 for code/math tasks and accuracy for ARC. Handcrafted multi-hot summaries (FSP-BCE) improve over standard MTP, especially on math reasoning (e.g., GSM8K and MATH), while learned summaries (FSP-RevLM) provide the largest gains across math reasoning and ARC tasks. 16
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Admissibility of Substitution Rule in Cyclic-Proof
Systems
Kenji Saotome1[0009−0008−6917−5673] and Koji Nakazawa1[0000−0001−6347−4383]
Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan
{saotomekenji, knak}@sqlab.jp
Abstract. This paper investigates the admissibility of the substitution
rule in cyclic-proof systems. The substitution rule complicates theoreti-
cal case analysis and increases computational cost in proof search since
every sequent can be a conclusion of an instance of the substitution
rule; hence, admissibility is desirable on both fronts. While admissibil-
ity is often shown by local proof transformations in non-cyclic systems,
such transformations may disrupt cyclic structure and do not readily
apply. Prior remarks suggested that the substitution rule is likely non-
admissible in the cyclic-proof system CLKIDω for first-order logic with
inductive predicates. In this paper, we prove admissibility in CLKIDω,
assuming the presence of the cut rule. Our approach unfolds a cyclic
proof into an infinitary form, lifts the substitution rules, and places back
edges to construct a cyclic proof without the substitution rule. If we
restrict substitutions to exclude function symbols, the result extends to
a broader class of systems, including cut-free CLKIDω and cyclic-proof
systems for the separation logic.
Keywords: Cyclic-proof system · Substitution rule · Admissibility ·
First-order logic · Separation logic.
1
Introduction
The framework of cyclic proofs [1, 4, 6] extends traditional proof trees by in-
troducing cyclic structures to reasoning about inductively defined predicates.
To ensure soundness, only those proofs that satisfy the global trace condition
are admitted. The condition guarantees that every infinite path makes progress
infinitely often by unfolding an inductive predicate. For example, the derivation
fails to satisfy the global trace condition, and the conclusion is invalid in general.
On the other hand, the proof shown in Fig. 1 is a cyclic proof in CLKIDω [1], a
cyclic-proof system for the first-order logic with inductive predicates. Intuitively,
N(x), E(x), and O(x) respectively represent that x is a natural number, an even
number, and an odd number. The leaf node N(x) ⊢O(x), E(x) is called bud, and
arXiv:2510.14749v1 [cs.LO] 16 Oct 2025
2
K. Saotome and K. Nakazawa
Fig. 1. A cyclic proof of N(x) ⊢E(x) ∨O(x)
the internal node associated with it is called its companion. A cycle is formed
through the connection between a bud and its companion. This proof satisfies
the global trace condition since every infinite path makes progress infinitely often
by unfolding the underlined inductive predicate N in the antecedents at (UL).
When we search for proof from the conclusion to premises, unlike the ordinary
induction rule, cyclic proofs may be found without a prior choice of the induction
hypothesis. Consequently, avoiding heuristic searches for induction hypotheses,
cyclic proofs have been investigated in the context of proof search [2, 3, 5, 11, 12].
The substitution rule
Γ ⊢∆
Γ[θ] ⊢∆[θ] (Subst)
is often introduced in proof systems. Some cyclic-proof systems contain the sub-
stitution rule as an explicit inference rule [1, 13, 8]. On the other hand, some
cyclic-proof systems implicitly introduce substitutions into the bud-companion
correspondence [10, 7, 9]. That is, instead of requiring that the bud and its com-
panion be syntactically identical, they require that the bud be a substitution
instance of the companion. The substitution rule helps to construct cyclic struc-
tures. In the proof in Fig. 1, a fresh variable y is introduced by (UL), and this
y is unified with x by (Subst) to identify the sequents N(x) ⊢E(x), O(x) and
N(y) ⊢O(y), E(y).
The admissibility of the substitution rule is meaningful both from theoret-
ical and practical perspectives. Theoretically, since the substitution rule is al-
ways applicable, it complicates proof analysis. Consequently, establishing the
admissibility of the substitution rule enables a more tractable proof analysis.
On the practical side, when the substitution rule is admissible, implementations
are afforded greater flexibility in how and when to apply substitutions. For ex-
ample, one might reserve the application of the substitution rule for specific
moments, such as when considering bud-companion relationships, when merging
with already-proven theorems, or even choose never to apply substitution at all.
Since many substitutions can be applied at each step, unrestricted application
of the substitution rule increases computational cost.
In many non-cyclic-proof systems, the substitution rule is admissible. This is
achieved by (i) lifting the substitution rule and (ii) eliminating the substitution
Admissibility of Substitution Rule in Cyclic-Proof Systems
3
rule. In the lifting phase (i), one transforms a derivation tree
Γ1 ⊢∆1
· · ·
Γn ⊢∆n
Γ ⊢∆
(R)
Γ[θ] ⊢∆[θ] (Subst)
into
Γ1 ⊢∆1
Γ1[θ1] ⊢∆1[θ1] (Subst)
· · ·
Γn ⊢∆n
Γn[θn] ⊢∆n[θn] (Subst)
Γ[θ] ⊢∆[θ]
(R)
.
In the elimination phase (ii), one transforms a derivation tree
Γ ⊢∆(Axiom)
Γ[θ] ⊢∆[θ] (Subst)
into
Γ[θ] ⊢∆[θ] (Axiom).
In cyclic-proof systems, however, the admissibility of the substitution rule is
still unclear. At least, it is not possible to remove (Subst) in the same way as in
non-cyclic-proof systems. In the lifting phase, the bud-companion relationship
may be destroyed if the sequent Γ ⊢∆, which is deleted during the transfor-
mation, is a companion. This is not merely a temporary effect during proof
transformation; for example, as illustrated in Fig. 2, even if the substitutions are
lifted to the bud, it may still fail to choose the corresponding companion. In the
elimination phase, it is not possible to eliminate the substitution when the leaf
of a derivation tree is a bud.
Fig. 2. Lifting (Subst) may break bud-companion correspondence
Brotherston [1] raised the question of whether the substitution rule is admissi-
ble in the cyclic-proof system CLKIDω, suggesting that it is likely not admissible
in general, at least in the cut-free setting.
This paper proves the admissibility of the substitution rule in the cyclic-proof
system CLKIDω for the first-order logic with inductive predicates [4], assuming
the presence of the cut rule. Furthermore, if function symbols with positive arity
are not introduced by the substitution rule, its applications can be eliminated
even without the cut rule. This result can be applied to other proof systems as
well. For example, in the cyclic-proof system for separation logic, where function
symbols except the constant nil are generally not introduced, the substitution
rule is admissible regardless of the presence of the cut rule.
The proof
4
K. Saotome and K. Nakazawa
is one of the cyclic proofs of N(x) ⊢E(x)∨O(x) without substitution rules, that
is constructed by unfolding the cyclic proof in Fig. 1. Regarding the new variable
y introduced by the lower (UL), the original proof unifies it with the variable
x using (Subst). In the new proof, this is accomplished by choosing x itself as
the fresh variable in the upper (UL). Building on this observation, we consider
unfolding a cyclic proof into an infinite proof, wherein substitution rules can
be systematically lifted upward. After performing this lifting to a sufficiently
great height in the infinite proof tree, we reconstruct a cyclic proof without
substitutions.
To guarantee the global trace condition, we choose buds and their compan-
ions in the reconstructed cyclic proof only when they have the same entailment
occurrence as in the original proof. This careful choice prevents the emergence
of unintended infinite paths.
Structure of the paper. Section 2 introduces a cyclic-proof system CLKIDω
and an infinitary-proof system LKIDω for the first-order logic FOLID with in-
ductive predicates. In Section 3, we prove the admissibility of the substitution
rule in CLKIDω. In Section 4, we discuss the admissibility of the substitution
rule in other proof systems. In Section 5, we conclude.
2
Cyclic-Proof System for First-Order Logic with
Inductive Predicates
This section recalls the first-order logic with inductive predicates, together with
the cyclic- and infinitary-proof systems, as proposed by Brotherston et al. [1, 4].
For the unfolding rules of inductive predicates, however, we follow the style of
Kimura et al. [7], in which the rule names do not explicitly mention the inductive
predicates.
2.1
First-Order Logic FOLID with Inductive Predicates
We define the formula of FOLID. A function symbol, denoted by f, g, . . ., has a
fixed arity. A nullary function symbol, denoted by c, c1, . . ., is called a constant
symbol. We denote a variable by x, y, z, . . .. We use x for a finite sequence of
Admissibility of Substitution Rule in Cyclic-Proof Systems
5
variables. A term, denoted by t, u, . . ., is defined recursively as either a variable
or an expression of the form f(t), where t is a finite sequence of terms. We write
t(x) to make explicit that the variable x occurs in the term t; analogously, we
use the notation t(x) for the same purpose. = is introduced as a special two-
argument predicate symbol. We use P, P1, . . . for inductive predicate symbols. We
also use Q, Q1, . . . for ordinary predicate symbols. Each predicate symbol has a
fixed arity.
Definition 1 (Formulas). The formulas of FOLID, denoted ϕ, ψ, . . ., are as
follows:
ϕ ::= P(t) | Q(u) | t = u | ¬ϕ | ϕ ∨ϕ | ϕ ∧ϕ | ϕ →ϕ | ∃x.ϕ | ∀x.ϕ,
where the lengths of t and u are the arities of P and Q, respectively.
Formulas in FOLID are identified up to α-equivalence under variable bind-
ing by ∀and ∃. For formulas ϕ and ψ that are syntactically equal, that is, α-
equivalent, we denote ϕ ≡ψ.
The substitutions of terms, denoted by θ, θ1, . . ., are finite mappings from
term variables to terms. The formula ϕ[θ] denotes the result of the capture-
avoiding substitution θ to free variables in ϕ. θ1θ2 denotes the composition of
the substitutions θ1 and θ2, and it holds that ϕ[θ1θ2] ≡(ϕ[θ1])[θ2]. θ[x1 →
y1, . . . , xn →yn] denotes the substitution obtained from θ by changing the image
of xi to yi for 1 ≤i ≤n, respectively. For a finite set of formulas Γ, Γ[θ] is a
finite set such that Γ[θ] = {ϕ[θ] | ϕ ∈Γ}.
Specifically, a substitution whose image includes only variables and constants
is called an atomic substitution. On the other hand, a substitution whose image
contains a term with a function symbol of positive arity is called a composite
substitution. The domain and the image of a mapping f are denoted by dom(f)
and img(f), respectively.
The definition of each inductive predicate P(t) is given by a definition set.
Definition 2 (Definition sets). An inductive definition set Φ is a finite set of
productions in the following form:
Q1(u1)
· · ·
Qn(un)
P1(t1)
. . .
Pm(tm)
P(t)
,
where each Qi(1 ≤i ≤n) is an ordinary predicate symbol and each Pj(1 ≤j ≤
m) is an inductive predicate symbol.
Example 1. The following are the productions for N, E, and O, which represent
natural numbers, even numbers, and odd numbers, respectively.
N(0) ,
N(x)
N(sx) ,
E(0) ,
O(x)
E(sx) , and
E(x)
O(sx) .
We consider the standard model [1, 4] as the semantics.
Definition 3 (Sequents). Let Γ and ∆be finite sets of formulas. A sequent,
denoted by e, is defined as Γ ⊢∆. We say that Γ ⊢∆is valid with respect to
the standard model.
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K. Saotome and K. Nakazawa
2.2
Infinitary-Proof System LKIDω
In this subsection, we introduce the infinitary-proof system LKIDω used to estab-
lish the admissibility of the substitution rule in the cyclic-proof system CLKIDω.
First, we define the inference rules of LKIDω. These inference rules are also
used in the cyclic-proof system CLKIDω.
Definition 4 (Inference rules). The inference rules of LKIDω and CLKIDω
except for inductive predicates unfolding are as follows:
Γ ⊢∆Γ ∩∆̸= ∅(Axiom)
Γ ′ ⊢∆′
Γ ⊢∆Γ ′ ⊆Γ, ∆′ ⊆∆(Wk)
Γ ⊢ϕ, ∆
Γ, ϕ ⊢∆
Γ ⊢∆
(Cut)
Γ ⊢∆
Γ[θ] ⊢∆[θ] (Subst)
Γ ⊢ϕ, ∆
Γ, ¬ϕ ⊢∆(¬L)
Γ, ∆⊢ϕ
Γ ⊢∆, ¬ϕ (¬R)
Γ, ϕ ⊢∆
Γ, ψ ⊢∆
Γ, ϕ ∨ψ ⊢∆
(∨L)
Γ ⊢∆, ϕ, ψ
Γ ⊢∆, ϕ ∨ψ (∨R)
Γ, ϕ, ψ ⊢∆
Γ, ϕ ∧ψ ⊢∆(∧L)
Γ ⊢∆, ϕ
Γ ⊢∆, ψ
Γ ⊢∆, ϕ ∧ψ
(∧R)
Γ ⊢ϕ, ∆
Γ, ψ ⊢∆
Γ, ϕ →ψ ⊢∆
(→L)
Γϕ ⊢ψ∆
Γ ⊢∆, ϕ ∨ψ (→R)
Γ, ϕ[t/x] ⊢∆
Γ, ∀x.ϕ ⊢∆(∀L)
Γ ⊢ϕ[y/x], ∆
Γ ⊢∀x.ϕ, ∆
(∀R)
Γ, ϕ[y/x] ⊢∆
Γ, ∃x.ϕ ⊢∆
(∃L)
Γ ⊢ϕ[t/x], ∆
Γ ⊢∃x.ϕ, ∆(∃R)
Γ[u/x, t/y] ⊢∆[u/x, t/y]
Γ[t/x, u/y], t = u ⊢∆[t/x, u/y] (= L)
Γ ⊢t = t, ∆(= R),
where t in (∃L) and (∃) is arbitrary and y in those is fresh.
The unfolding rule for left inductive predicates is
all case distinctions for P(u)
Γ, P(u) ⊢∆
(UL),
where the case distinction of Γ, P(u) ⊢∆for a production
Q1(u1(x))
· · ·
Qn(un(x))
P1(t1(x))
. . .
Pm(tm(x))
P(t(x))
is Γ, u = t(y), Q1(u1(y)), . . . , Qn(un(y)), P1(t1(y)), . . . , Pm(tm(y)) ⊢∆, where
y is fresh.
The unfolding rule for right inductive predicates is
Γ ⊢Q1(u1(u)), ∆
· · ·
Γ ⊢Qn(un(u)), ∆
Γ ⊢P1(t1(u)), ∆
· · ·
Γ ⊢Pm(tm(u)), ∆
Γ ⊢P(t(u)), ∆
(UR)
for a production
Q1(u1(x))
· · ·
Qn(un(x))
P1(t1(x))
. . .
Pm(tm(x))
P(t(x))
.
Admissibility of Substitution Rule in Cyclic-Proof Systems
7
A pre-proof of LKIDω is a finite or infinite derivation tree, whose all leaves
are axioms, defined in a usual way by the inference rules of LKIDω. A pre-proof
is denoted by Pr. The set of sequent occurrences in Pr is denoted by Seq(Pr).
The sets of free variables in a formula ϕ, a sequent e, and a (pre-)proof Pr are
denoted by FV(ϕ), FV(e), and FV(Pr), respectively. The set of substitutions of
(Subst) in Pr is denoted by Θ(Pr). The number of elements in a finite set S is
denoted by #S.
To define the proof of LKIDω, we define traces following paths.
Definition 5 (Paths). Let Pr be a pre-proof in LKIDω. Viewing Pr as a (pos-
sibly infinite) graph, spanning from root to leaf. A path in this graph is called a
path of Pr, denoted by (ei)j≤i<k for j ∈N and k ∈N ∪{∞}.
Definition 6 (Traces). Let (ei)j≤i<k be a path in a derivation tree Pr. A trace
following (ei)j≤i<k, denoted by (Ci)j≤i<k, is a sequence of inductive predicates
in the antecedents of ei satisfying the following conditions:
(i) If ei is a conclusion of (UL) and Ci is unfolded by (UL), then Ci+1 occurs
in the result of the unfolding,
(ii) Otherwise, Ci+1 is the inductive predicate occurrence in ei+1 corresponding
to Ci in ei.
In the case (ii), i is a progressing point of (Ci)j≤i<k. A trace with infinitely many
progressing points is called an infinitely progressing trace.
We define proofs of LKIDω.
Definition 7 (Proofs of LKIDω). The condition for the pre-proof that, for any
infinite path (ei)i≥0, there is an infinitely progressing trace following (ei)j≤i for
some j is called the global trace condition. A pre-proof Pr is a proof of LKIDω
if Pr satisfies the global trace condition.
Example 2. A proof shown in Fig. 3 is a LKIDω proof of N(x) ⊢E(x) ∨O(x).
This proof is constructed by infinitely unfolding the proof shown in Fig. 1. This
proof satisfies the global trace condition. Following the infinite path in this proof,
the N’s in the antecedents construct an infinitely progressing trace.
The soundness of LKIDω can be proved in a similar way to previous studies[1].
Theorem 1 (Soundness of LKIDω). If Pr is a proof of LKIDω, then every
sequent in Pr is valid.
2.3
Cyclic-Proof System CLKIDω
In this subsection, we define the cyclic-proof system CLKIDω in the standard
way. The inference rules of CLKIDω are the same as LKIDω. Regarding pre-
proofs, since they differ from those in LKIDω, we define them as follows.
Definition 8 (Pre-proofs of CLKIDω). A pre-proof of CLKIDω is composed
of a finite derivation tree D together with a relation R between sequents in D. It
8
K. Saotome and K. Nakazawa
Fig. 3. An infinite proof of N(x) ⊢E(x) ∨O(x)
is not required that all leaves of D are axioms. Leaves of D that are not axioms
are called buds, and the domain of R is the set of buds of D. The codomain
of R is the subset of internal nodes of D, and R associates each bud with an
internal node labeled by a syntactically identical sequent. This sequent associated
with each bud is called its companion.
A path of CLKIDω is defined as follows.
Definition 9 (Paths of CLKIDω). Let Pr = (D, R) be a pre-proof in CLKIDω.
Viewing D as a directed graph, we add edges according to R. A (finite or infinite)
path in this graph is called a path of Pr.
Following the above definition of paths, a trace, the global trace condition,
and a proof of CLKIDω are defined analogously to those in LKIDω. The sound-
ness of CLKIDω can be proved by the soundness of LKIDω since whenever there
exists a proof in CLKIDω, there also exists a proof containing the same sequent
in LKIDω.
Example 3. The proof shown in Fig. 1 is a proof of CLKIDω. This proof sat-
isfies the global trace condition. Following the infinite path in this proof, the
underlined N’s on the antecedents construct an infinitely progressing trace.
3
Admissibility of the Substitution Rule in CLKIDω
In this section, we show the admissibility of CLKIDω. To prove this, we assume
a proof Pr+ of Γ ⊢∆in CLKIDω with (Subst) and construct a proof Pr−of
Γ ⊢∆without (Subst).
Admissibility of Substitution Rule in Cyclic-Proof Systems
9
The construction procedure can be divided into the elimination of composite
substitutions and atomic substitutions. Concerning the elimination of composite
substitutions, from a proof Pr+ containing composite substitutions, we construct
a proof Prvar
+
containing only atomic substitutions by a proof transformation.
Concerning the elimination of atomic substitutions, it can be outlined in
three steps as shown in Fig. 4.
Fig. 4. The outline of the elimination of atomic substitutions
First, we construct a LKIDω proof Prω by unfolding the cycles in Prvar
+ .
Secondly, we perform a lifting of the substitution rule to Prω. Taking Prω as the
base case Prω
0 , we recursively construct, for each depth d, LKIDω proofs Prω
d
that contain no substitution rules until at least depth d. Finally, from Prω
d for
a sufficiently large d, we obtain a CLKIDω pre-proof Pr−without substitution
rules.
We transform the proof stepwise in the elimination of atomic substitutions,
but the resulting pre-proof Pr−must satisfy the global trace condition. To
achieve this, at each stage, we define a mapping that associates occurrences
of sequents in the constructed proof to those in Prvar
+ .
In the third step, we need to choose buds and their companions among the
sequent occurrences in Prω
d . At this stage, we choose pairs of sequent occurrences
so that each bud and its companion correspond to the same sequent occurrence
in Prvar
+ . As a result, every path in Pr−corresponds to a path in Prvar
+ , and
any trace following the path in Pr−is associated with a trace following the
corresponding path in Prvar
+ . Pr+ satisfies the global trace condition, and so
does Prvar
+ .
3.1
The Elimination of Composite Substitutions
In this subsection, we construct Prvar
+
with only atomic substitutions from Pr+.
Lemma 1 (From Pr+ to Prvar
+ ). Let Pr+ be a CLKIDω proof of Γ ⊢∆. Then,
there exist a CLKIDω proof Prvar
+ of Γ ⊢∆such that any substitution in Θ(Prvar
+ )
is atomic.
Proof. For the instance of (Subst)
Γ ⊢∆
Γ[x1 := t1, . . . , xn := tn] ⊢∆[x1 := t1, . . . , xn := tn] (Subst)
10
K. Saotome and K. Nakazawa
for a composite substitution, we can transform it to
t1 = t1 (= R)
⊢∃y1.y1 = t1 (∃R)
Γ ⊢∆
Γ[x1 := y1, . . . , xn := yn] ⊢∆[x1 := y1, . . . , xn := yn] (Subst)
y1 = t1, . . . , yn = tn, Γ[x1 := y1, . . . , xn := yn] ⊢∆[x1 := y1, . . . , xn := yn] (Wk)
y1 = t1, . . . , yn = tn, Γ[x1 := t1, . . . , xn := tn] ⊢∆[x1 := t1, . . . , xn := tn] (= L) n times
∃y1.y1 = t1, . . . , ∃yn = tn, Γ[x1 := t1, . . . , xn := tn] ⊢∆[x1 := t1, . . . , xn := tn] (∃L) n times
.... (Cut) as well as the bottom
∃y1.y1 = t1, Γ[x1 := t1, . . . , xn := tn] ⊢∆[x1 := t1, . . . , xn := tn]
Γ[x1 := t1, . . . , xn := tn] ⊢∆[x1 := t1, . . . , xn := tn]
(Cut)
,
where yi(i ∈[1, n]) are fresh variables for Γ ⊢∆and (Γ ⊢∆)[x1 := t1, . . . , xn :=
tn]. Since every sequent occurring in the original proof also appears after the
transformation, and the transformation preserves every trace, the transformed
proof satisfies the global trace condition.
⊓⊔
Remark 1. We construct Prvar
+
by a proof transformation that employs several
rules. Therefore, this lemma depends on the inference rules of CLKIDω. In par-
ticular, the above transformation is not possible in cut-free CLKIDω, which is
strictly weaker than CLKIDω with cuts [8].
3.2
Admissibility of the Atomic Substitution Rule
In this subsection, we prove the admissibility of the atomic substitution rule
in CLKIDω. Note that in the proof transformation of the previous subsection,
constants also disappear; however, in the present proof, we do not make this
assumption.
Hereafter, suppose that Prvar
+ is a CLKIDω proof such that Θ(Prvar
+ ) is atomic.
First, we construct a LKIDω proof Prω from Prvar
+ . At this stage, we define a
mapping f ω from sequent occurrences in Prω to those in Prvar
+ .
Lemma 2 (From Prvar
+
to Prω). Let Γ ⊢∆be the conclusion of Prvar
+ . There
exists a LKIDω proof Prω of Γ ⊢∆s.t. FV(Prω) = FV(Prvar
+ ) and Θ(Prω) =
Θ(Prvar
+ ) hold. Moreover, there exists a function fω : Seq(Prω) →Seq(Prvar
+ ) s.t.
the following conditions hold:
– f ω(e0) = e′
0, where e0 and e′
0 are the conclusions of Prω and Prvar
+ , respec-
tively.
– For any e ∈Prω, f ω(e) ≡e.
– If the rule inference
e1
i+1
· · ·
en
i+1
ei
(R)
exists in Prω, then the rule instance
f ω(e1
i+1)
· · ·
f ω(en
i+1)
f ω(ei)
(R)
exists in Prvar
+ . Furthermore, any traces in this rule instance of Prvar
+
are
preserved in this rule instance of Pr−. Here, each bud and its companion are
identified in the rule instance of CLKIDω, and f ω(e) takes the companion.
Admissibility of Substitution Rule in Cyclic-Proof Systems
11
Prω can be constructed in a similar manner to the tree unfolding as defined by
Brotherston [1]. In the tree unfolding, cycles are unfolded based on the relation
between buds and companions. Fig. 3 is an infinite proof obtained by unfolding
Fig. 1 in this way. Intuitively, fω associates each sequent in the infinite proof
obtained by unfolding the cyclic proof with the corresponding sequent in the
original cyclic proof.
The preservation of traces means that, for any trace following the path from
f ω(ei) to f ω(em
i+1), there exists a corresponding trace following the path from
ei to ei+1 by the corresponding inductive predicates.
Secondly, we mention the recursive construction of LKIDω proofs Prω
d of
Γ ⊢∆that contain no substitution rules until at least depth d. At this stage,
we define f ω
d : Seq(Prω
d ) →Seq(Prvar
+ ) ∪{ϵ}. ϵ is a special symbol indicating
no correspondence. In correspondence with the lifting of substitution rules, we
introduce partial-substitution closure and substitution-application property for
atomic substitutions.
Definition 10 (Partial-substitution closure). Let Θ and X be a finite set
of substitutions and a finite set of variables, respectively. We inductively define
the partial-substitution closure Cps(Θ, X) as the smallest set which satisfies the
following:
– Θ ⊆Cps(Θ, X),
– ∀θ ∈Cps(Θ, X), x, y ∈X.θ[x →y] ∈Cps(Θ, X),
– ∀θ1, θ2 ∈Cps(Θ, X).θ1θ2 ∈Cps(Θ, X).
The partial-substitution closure Cps(Θ, X) is the smallest set containing Θ
closed under two operations: overwriting a substitution with variables in X, and
the composition of substitutions. When substitutions are restricted to atomic
ones, the domain and image of the partial-substitution closure are fixed as finite
sets, allowing the following lemma to be established.
Lemma 3. Let Θvar be a finite set of atomic substitutions. Then, any substi-
tution in partial-substitution closure Cps(Θvar, X) is an atomic substitution for
any variable set X. Furthermore, Cps(Θvar, X) is finite.
Remark 2. When Θvar contains a composite substitution, Cps(Θvar, X) can be
infinite. For example, suppose there is a unary function symbol f and some θ ∈
Θvar such that f(x) occurs in the image of θ. Then the terms f(x), f(f(x)), . . .
are all appear in images of substitutions in Cps(Θvar, X), which force Cps(Θvar, X)
to infinite.
Definition 11 (Substitution-application property). Consider a rule (R)
except (Subst). If for any instance
Γ1 ⊢∆1
. . .
Γn ⊢∆n
Γ ⊢∆
(R)
12
K. Saotome and K. Nakazawa
of (R) and any atomic substitution θ, there is an instance of (R):
Γ1[θ1] ⊢∆1[θ1]
. . .
Γn[θn] ⊢∆n[θn]
Γ[θ] ⊢∆[θ]
(R),
where θi(1 ≤i ≤n) are substitutions in Cps({θ}, FV(Γi ⊢∆i) ∪FV(Γ ⊢∆)),
s.t. any traces in the above instance are preserved, then we say that (R) satis-
fies the substitution-application property. We refer to the latter instance as a
substitution application of the former. If all inference rules of a proof system
S satisfy the substitution-application property, then we say that S satisfies the
substitution-application property.
Note that the substitution-application property subsumes closure under atomic
substitution for axiom rules. Below, we discuss the fact that CLKIDω and LKIDω
satisfy the substitution-application property.
Lemma 4. The proof systems CLKIDω and LKIDω satisfy the substitution-
application property.
Proof. If the rule (R) does not introduce any fresh variables, it suffices to take
θi = θ for 0 ≤i ≤n. Hereafter, we only consider the case of (UL). The cases
where fresh variables appear in other rules are handled in the same manner.
Consider an instance
Γ1 ⊢∆
. . .
Γn ⊢∆n
Γ ⊢∆
(UL)
of (UL) and an atomic substitution θ.
Let f|D denote the partial function of f restricted to the domain D, let
V ar(T) denote the set of variables occurring in the terms of T, and let Xi be
FV(Γ ⊢∆) ∪FV(Γi ⊢∆) for each 1 ≤i ≤n.
For each i, we can define θi ∈Cps({θ}, Xi) satisfying
– θi|FV(Γ ⊢∆) equals to θ|FV(Γ ⊢∆),
– any variable in the image of θi|Xi−FV(Γ ⊢∆) is fresh for Γ[θ] ⊢∆[θ], and
– θi|Xi−FV(Γ ⊢∆) is injective.
Note that Xi −FV(Γ ⊢∆) is the set of variables in Γi ⊢∆that are fresh
for Γ ⊢∆. Let {x1, . . . , xa} = Xi −FV(Γ ⊢∆). Since θ is atomic, we have
#FV(Γ ⊢∆) ≥#V ar(img(θi|FV(Γ ⊢∆))), and we can choose pairwise distinct
x′
j ∈Xi −V ar(img(θi|FV(Γ ⊢∆))) for each 1 ≤j ≤a. Hence, we can define
θi = θ[x1 →x′
1, . . . , xa →x′
a] ∈Cps({θ}, Xi)
such that the above conditions hold, and then
Γ1[θ1] ⊢∆[θ1]
. . .
Γn[θn] ⊢∆[θn]
Γ[θ] ⊢∆[θ]
(UL),
is an instance of (UL).
⊓⊔
Admissibility of Substitution Rule in Cyclic-Proof Systems
13
Remark 3. In general, the substitution-application property does not hold for
composite substitutions. As an example, consider a rule instance
x = sy, E(y) ⊢N(x)
O(x) ⊢N(x)
(UL)
and a substitution θ = [f(x, y)/x]. The free variables contained in this rule in-
stance are exactly x and y. On the other hand, when applying (UL) to O(f(x, y)) ⊢
N(f(x, y)), a fresh variable different from x and y is required in the premise. Such
a variable does not appear in the image of substitution of Cps({[f(x, y)/x]}, {x, y}).
By the substitution application, Prω
d can be constructed from Prω.
Lemma 5 (From Prω to Prω
d ). Let Γ ⊢∆be the conclusion of Prvar
+ . There
exists a LKIDω proof Prω
d of Γ ⊢∆that contains no substitution rules until at
least depth d s.t. FV(Prω
d ) ⊆FV(Prvar
+ ) and Θ(Prω
d ) ⊆Cps(Θ(Prvar
+ ), FV(Prvar
+ ))
hold. Moreover, there exists a function fω
d : Seq(Prω
d ) →Seq(Prvar
+ ) ∪{ϵ} s.t. the
following conditions hold:
– f ω
d (e0) = e′
0, where e0 and e′
0 are the conclusions of Prω
d and Prvar
+ , respec-
tively.
– f ω
d (e) = ϵ if e is a premise of the (Subst).
– If f ω
d (e) ̸= ϵ, then e ≡f ω
d (e)θ for some θ ∈Cps(Θ(Prvar
+ ), FV(Prvar
+ )).
– If the following applications of (Subst) and some inference rule (R) except
(Subst) exist in Prω
d :
e1
i+1
· · ·
en
i+1 (R)
.... (Subst) 0 or more times
ei
,
then the following applications of (Subst) and (R) exist in Prvar
+ :
f ω
d (e1
i+1)
· · ·
f ω
d (en
i+1)
(R)
.... (Subst) 0 or more times
f ω
d (ei)
.
Furthermore, any traces in these rule applications of Prvar
+
are preserved in
Prω
d . Here, each bud and its companion are identified in the rule applications
of CLKIDω, and f ω
d (e) takes the companion.
Proof. The proof proceeds by induction on d.
Base Case: Prω
0 is Prω in Lemma 2. f ω
0 is f ω in Lemma 2, except for the
premises of (Subst) in Prω
0 . For the premises of (Subst), we assign f ω
0 (e) = ϵ.
Induction Step: Assume Prω
k and f ω
k . By performing the following operation
on every (Subst) at depth k + 1 in Prω
k , we can construct Prω
k+1. At the same
time, we define f ω
k+1. In parts of the proof unaffected by this transformation,
f ω
k+1 keeps the values of f ω
k .
14
K. Saotome and K. Nakazawa
If the premise of the (Subst) is derived from zero or more applications of
(Subst) and one application of some rule (R), that part is of the following form.
....
Γ1 ⊢∆1
. . .
....
Γm ⊢∆m
Γ ⊢∆
(R)
Γ[θ1] ⊢∆[θ1] (Subst)
.... (Subst) n −1 times
Γ[θ1 · · · θn] ⊢∆[θ1 · · · θn]
.
At this point, since θ1 · · · θn is an atomic substitution, we apply the substitution
application of (R) to Γ[θ1 · · · θn] ⊢∆[θ1 · · · θn] in Prω
k+1 as follows:
....
Γ1 ⊢∆1
Γ1[θ′
1] ⊢∆1[θ′
1] (Subst)
. . .
....
Γm ⊢∆m
Γm[θ′
m] ⊢∆m[θ′
m] (Subst)
Γ[θ1 · · · θn] ⊢∆[θ1 · · · θn]
(R)
,
where every θ′
i is in Cps({θ1 · · · θn}, FV(Γi ⊢∆i)∪FV(Γ ⊢∆)) ⊆Cps(Θ(Prvar
+ ), FV(Prvar
+ ))
by Lemma 4. f ω
k+1(Γi ⊢∆i) is defined as ϵ. f ω
k+1(Γi[θ′
i] ⊢∆i[θ′
i]) is defined as
f ω
k (Γi ⊢∆i). f ω
k+1(Γ[θ1 · · · θn] ⊢∆[θ1 · · · θn]) is defined as f ω
k (Γ[θ1 · · · θn] ⊢
∆[θ1 · · · θn]). As a result, the (Subst) applications at depth k + 1 are pushed up
to k + 2.
From the above, we can construct Prω
k+1 with Prω
k . Since the above operation
affects only finite portions of each path, Prω
d satisfies the global trace condition.
Moreover, since the proof transformation is performed with substitution applica-
tions, the conditions FV(Prω
d ) ⊆FV(Pr+) and Θ(Prω
d ) ⊆Cps(Θ(Pr+), FV(Pr+))
are preserved in every induction step.
In defining f ω
k+1, we associate sequent occurrences in Prω with correspond-
ing sequent occurrences in Prvar
+
by extending f ω
k . Note that these substitution
applications preserve the conditions specified in this lemma, including the preser-
vation of traces.
⊓⊔
In the third place, we demonstrate that we can construct the pre-proof Pr−
in CLKIDω from Prω
d for sufficiently large d. Intuitively, we show that for each
infinite path in Prω
d , there are sequents ei and ej s.t. i < j ≤d, fω
d (ei) = f ω
d (ej),
and ei ≡ej hold. By assigning such ej and ei as a bud and its companion,
respectively, we can construct Pr−.
Lemma 6 (From Prω
d to Pr−). Let Γ ⊢∆be the conclusion of Prvar
+ . There
exists a CLKIDω pre-proof Pr−of Γ ⊢∆that contains no substitution rules,
and a function f−: Seq(Pr−) →Seq(Prvar
+ ) s.t. the following conditions hold:
– f−(e0) = e′
0, where e0 and e′
0 are the conclusions of Pr−and Prvar
+ , respec-
tively.
– For any bud ei and its companion ei+1, f−(ei) = f−(ei+1).
Admissibility of Substitution Rule in Cyclic-Proof Systems
15
– If the rule instance
e1
i+1
· · ·
en
i+1
ei
(R)
exists in Pr−, then the following applications of (Subst) and (R) exist in
Prvar
+ :
f ω
d (e1
i+1)
· · ·
f ω
d (en
i+1)
(R)
.... (Subst) 0 or more times
f ω
d (ei)
.
Furthermore, any traces in these rule applications of Prvar
+
are preserved in
this rule application of Pr−. Here, each bud and its companion are identified
in the rule applications, and f−(e) takes the companion.
Proof. Let Prω
d and f ω
d be a proof and a mapping constructed by Lemma 5.
Consider n = #Cps(Θ(Prvar
+ ), FV(Prvar
+ )) by Lemma 3.
Assume that d > #img(f ω
d )·n holds. Such d exists since img(f ω
d ) ⊆Seq(Prvar
+ )∪
{ϵ} holds. For any path (ei)0≤i≤d in Prω
d , for some sequent e in Prvar
+ , there
exist at least n + 1 indices i with 0 ≤i ≤d such that f ω
d (ei) = e. Let S de-
note the set of those indices. f ω
d (ei) ̸= ϵ holds for any i ∈S since ei is not a
premise of (Subst). Therefore, ei and f ω
d (ei)θi are the same sequents for some
θi ∈Cps(Θ(Prvar
+ ), FV(Prvar
+ )) by the condition of f ω
d . The number of substitu-
tions in Cps(Θ(Prvar
+ ), FV(Prvar
+ )) is n. Hence, there are the same sequents ei and
ej for some i < j ∈S. By choosing ei as the bud and ej as its companion, we can
construct Pr−. Moreover, for each occurrence of a sequent e in Pr−, there exists
a corresponding occurrence of a sequent e′ in Prω
d . Based on this correspondence,
we define f−(e) = f ω
d (e′). With the definition of f−, all conditions required by
this lemma are satisfied.
⊓⊔
From the above, we could construct the pre-proof Pr−from a proof Prvar
+
via Prω and Prω
d . Next, we mention that the pre-proof Pr−is a proof. In other
words, we show that the pre-proof Pr−satisfies the global trace condition. First,
define a corresponding path in Prvar
+
for each path in Pr−.
Definition 12 (Corresponding path). Let Pr−and f−be a pre-proof and
a mapping, respectively, constructed in Lemma 6. Let (ei)0≤i be a path in Pr−
where e0 is the conclusion of Pr−. The path (e′
j)0≤j in Prvar
+
defined as follows
is called corresponding path of (ei)0≤i:
– Let e′
0 = f−(e0) be the conclusion of Prvar
+ .
– Suppose that e′
j = f−(ei).
• If ei is a bud and ei+1 is its companion, then let e′
j = f−(ei+1).
• If ei+1 is the n-th premise in a rule instance
· · ·
ei+1
· · ·
ei
(R),
16
K. Saotome and K. Nakazawa
then, by the condition of f−, there exists rule applications
· · ·
f ω
d (ei+1)
· · · (R)
.... (Subst) and bud-companion m times
e′
j = f ω
d (ei)
,
in Prvar
+
for some m such that fω
d (ei+1) is the n-th premise of (R). In
this application, the sequence e′
j, . . . , e′
j+m is uniquely determined since
(Subst) is a unary rule. Furthermore, we set e′
j+m+1 = f(ei+1).
Using the corresponding path, we can refer to the relationship between the
traces of Prvar
+
and Pr−.
Lemma 7. Let (ei)0≤i be a path in Pr−constructed in Lemma 6 and (e′
j)0≤j
be the corresponding path of (ei)0≤i in Prvar
+ . If there is an infinitely progressing
trace (C′
j)k′≤j following (e′
j)0≤j, the infinitely progressing trace (Ci)k≤i exists in
(ei)0≤i.
Based on Lemma 1 and 6, we can construct a pre-proof without the sub-
stitution rule. Lemma 7 guarantees that the pre-proof satisfies the global trace
condition; hence, the following main theorem holds.
Theorem 2 (Admissibility of substitution rules in CLKIDω). If Γ ⊢∆
is provable in CLKIDω, then it is provable in CLKIDω without (Subst).
In the process of elimination of atomic substitutions, we do not introduce
any rule not present in Prvar
+ , and hence we have the following stronger result.
Theorem 3. If Pr is a proof of Γ ⊢∆in CLKIDω and Θ(Pr) consists of only
atomic substitutions, then there is a (Subst)-free proof of Γ ⊢∆that contains
only rules in Pr.
4
Admissibility of the Substitution Rule in Other
Cyclic-Proof Systems
In this section, we discuss the conditions under which our procedure for the sub-
stitution elimination can be applied in general cyclic-proof systems. Among the
lemmas proved in Section 3.2, those that depend on the proof system CLKIDω
are Lemma 1 and 4.
First, we discuss Lemma 1, which concerns the elimination of composite sub-
stitutions. This lemma claims the existence of a proof that contains only atomic
substitutions from the existence of a proof that allows composite substitutions.
To prove this lemma, we use inference rules (Cut), (= L), (= R), (∃L), and (∃R).
Accordingly, in proof systems that do not include these rules, this lemma does
not, in general, hold.
Admissibility of Substitution Rule in Cyclic-Proof Systems
17
In fact, we can consider a system in which (Subst) is not admissible. For
example, let P(x) be an inductive predicate defined by P(x) := P(sx). The
unfolding rule of the left P(x) in CLKIDω is
y = x, P(sy), Γ ⊢∆
P(x), Γ ⊢∆
(UL).
However, we can consider another unfolding rule
P(sx), Γ ⊢∆
P(x), Γ ⊢∆(UL′).
Let we think a proof system with only (UL′), (UR), (Axiom), (Wk), and (Subst).
There is a proof
of the sequent P(x) ⊢⊥. However, we cannot prove this sequent without (Subst)
in this system.
Note that in Lemma 1 for CLKIDω, the cut rule is only used to introduce
y = t (where y is fresh and t is arbitrary). Therefore, when we introduce the
following rule in place of the cut rule, (Subst) is still admissible
y = t, Γ ⊢∆
Γ ⊢∆
(fresh L),
where t is arbitrary and y is fresh in Γ ⊢∆and t.
Secondly, we discuss Lemma 4. This lemma plays an important role in the
elimination of atomic substitutions. This lemma guarantees that CLKIDω and
LKIDω satisfy the substitution-application property with atomic substitutions.
This lemma holds for many general proof systems, including the proof system
CLKIDω and the cyclic proof system for separation logic. Therefore, as long as
we can assume a proof with only atomic substitutions, the substitution rule can
be eliminated in many proof systems. Let CLKIDω
−be a proof system CLKIDω
without the cut rule.
Theorem 4 (Admissibility of atomic substitution rules in CLKIDω
−). If
Pr is a proof of Γ ⊢∆in CLKIDω
−and Θ(Pr) consists of only atomic substi-
tutions, then there is a (Subst)-free CLKIDω
−proof of Γ ⊢∆that contains only
rules in Pr.
In particular, every proof in the cyclic-proof system CSLω [2] for the separa-
tion logic contains only variable application since the separation logic contains
no function symbols. Hence, the following theorems hold. Let CSLω
−be CSLω
without the cut rule.
Theorem 5. (Subst) is admissible in both CSLω and CSLω
−.
18
K. Saotome and K. Nakazawa
5
Conclusion
In this paper, we prove the admissibility of the substitution rule in CLKIDω.
After that, we discuss generalization to other proof systems.
One possible direction for future work is investigating the admissibility of
the substitution rule in cut-free CLKIDω without (fresh L). Section 4 showed
that replacing (UL) with (UL′) and restricting inference rules destroy this ad-
missibility. As a counterexample, we considered P(x) ⊢⊥; however, in the usual
CLKIDω system, there still be a cut-free proof
of this sequent. The reason is that (UL) effectively plays the role of (fresh L).
Our interest lies in identifying the minimal set of rules that still allows the
elimination of the substitution rule in the presence of function symbols.
Another possible direction is to study the cut-elimination property for cyclic
proof systems by introducing additional rules. The fact that the substitution rule
could be eliminated from cut-free CLKIDω through the introduction of (fresh L)
provides an important insight. This suggests that, by adding rules sufficient to
construct cycles, it may be possible to achieve the elimination of the cut rule.
Acknowledgments. The second author was supported by JSPS KAKENHI Grant
Number 22K11901.
Admissibility of Substitution Rule in Cyclic-Proof Systems
19
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A
Details of Proofs in Section 3.2
In this section, we show the details of proofs in Section 3.2.
A.1
Lemma 3
Proof. Since any function symbols not contained in Θ are likewise absent from
Cps(Θ, X), any substitution in Cps(Θvar, X) is atomic.
20
K. Saotome and K. Nakazawa
Let dom(Θ) and img(Θ) denote, respectively, the unions of dom(θ) and
img(θ) over all θ ∈Θ. The domain of every substitution in Cps(Θvar, X) is a sub-
set of dom(Θ)∪X. Similary, the range of every substitution in Cps(Θvar, X) is a
subset of img(Θ)∪X. Hence, the value of #Cps(Θvar, X) is at most #(img(Θ)∪
X)#(dom(Θ)∪X) and #Cps(Θvar, X) is finite.
⊓⊔
A.2
Lemma 7
Proof. By the property of f−, since traces are preserved at each rule application,
if there exists an infinitely progressing trace along with (e′
j)0≤j, there also exists
the corresponding infinitely progressing trace along with (ei)0≤i. Note that when
ei and ei+1 on Pr−are respectively a bud and its companion, we have f(ei) =
f(ei+1), and hence, at this point, the trace does not break.
⊓⊔
A.3
Theorem 4
Proof. Assume that Prvar
+
is a CLKIDω proof of Γ ⊢∆s.t. any substitution in
Θ(Prvar
+ ) is an atomic substitution. By Lemma 1, the existence of such a proof
Prvar
+
is guaranteed.
By Lemma 6, we can construct the pre-proof Pr−of Γ ⊢∆that does not
contain the application of (Subst) and the function f−. Let (ei)0≤i be an infinite
path in Pr−. There is a corresponding path (e′
j)0≤j in Prvar
+ . (e′
j)0≤j is an infinite
path since (ei)0≤i is an infinite path. Since Prvar
+
is proof, there is an infinitely
progressing trace along with (e′
j)0≤j. By Lemma 7, there is an infinitely progress-
ing trace along with (ei)0≤i. Assuming an arbitrary infinite path on Pr−, there
exists an infinitely progressing trace. Therefore, Pr−satisfies the global trace
condition. Consequently, Pr−is a proof, and we have established the existence
of a proof of Γ ⊢∆that does not contain the application of (Subst).
⊓⊔
|
Admissibility of Substitution Rule in Cyclic-Proof Systems Kenji Saotome1[0009-0008-6917-5673] and Koji Nakazawa1[0000-0001-6347-4383] Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan {saotomekenji, Abstract. This paper investigates the admissibility of the substitution rule in cyclic-proof systems. The substitution rule complicates theoretical case analysis and increases computational cost in proof search since every sequent can be a conclusion of an instance of the substitution rule; hence, admissibility is desirable on both fronts. While admissibility is often shown by local proof transformations in non-cyclic systems, such transformations may disrupt cyclic structure and do not readily apply. Prior remarks suggested that the substitution rule is likely nonadmissible in the cyclic-proof system CLKIDω for first-order logic with inductive predicates. In this paper, we prove admissibility in CLKIDω, assuming the presence of the cut rule. Our approach unfolds a cyclic proof into an infinitary form, lifts the substitution rules, and places back edges to construct a cyclic proof without the substitution rule. If we restrict substitutions to exclude function symbols, the result extends to a broader class of systems, including cut-free CLKIDω and cyclic-proof systems for the separation logic. Keywords: Cyclic-proof system · Substitution rule · Admissibility · First-order logic · Separation logic. 1 Introduction The framework of cyclic proofs [1, 4, 6] extends traditional proof trees by introducing cyclic structures to reasoning about inductively defined predicates. To ensure soundness, only those proofs that satisfy the global trace condition are admitted. The condition guarantees that every infinite path makes progress infinitely often by unfolding an inductive predicate. For example, the derivation fails to satisfy the global trace condition, and the conclusion is invalid in general. On the other hand, the proof shown in Fig. 1 is a cyclic proof in CLKIDω [1], a cyclic-proof system for the first-order logic with inductive predicates. Intuitively, N(x), E(x), and O(x) respectively represent that x is a natural number, an even number, and an odd number. The leaf node N(x) ⊢O(x), E(x) is called bud, and 16 Oct 2025 2 K. Saotome and K. Nakazawa Fig. 1. A cyclic proof of N(x) ⊢E(x) ∨O(x) the internal node associated with it is called its companion. A cycle is formed through the connection between a bud and its companion. This proof satisfies the global trace condition since every infinite path makes progress infinitely often by unfolding the underlined inductive predicate N in the antecedents at (UL). When we search for proof from the conclusion to premises, unlike the ordinary induction rule, cyclic proofs may be found without a prior choice of the induction hypothesis. Consequently, avoiding heuristic searches for induction hypotheses, cyclic proofs have been investigated in the context of proof search [2, 3, 5, 11, 12]. The substitution rule Γ ⊢∆ Γ[θ] ⊢∆[θ] (Subst) is often introduced in proof systems. Some cyclic-proof systems contain the substitution rule as an explicit inference rule [1, 13, 8]. On the other hand, some cyclic-proof systems implicitly introduce substitutions into the bud-companion correspondence [10, 7, 9]. That is, instead of requiring that the bud and its companion be syntactically identical, they require that the bud be a substitution instance of the companion. The substitution rule helps to construct cyclic structures. In the proof in Fig. 1, a fresh variable y is introduced by (UL), and this y is unified with x by (Subst) to identify the sequents N(x) ⊢E(x), O(x) and N(y) ⊢O(y), E(y). The admissibility of the substitution rule is meaningful both from theoretical and practical perspectives. Theoretically, since the substitution rule is always applicable, it complicates proof analysis. Consequently, establishing the admissibility of the substitution rule enables a more tractable proof analysis. On the practical side, when the substitution rule is admissible, implementations are afforded greater flexibility in how and when to apply substitutions. For example, one might reserve the application of the substitution rule for specific moments, such as when considering bud-companion relationships, when merging with already-proven theorems, or even choose never to apply substitution at all. Since many substitutions can be applied at each step, unrestricted application of the substitution rule increases computational cost. In many non-cyclic-proof systems, the substitution rule is admissible. This is achieved by (i) lifting the substitution rule and (ii) eliminating the substitution Admissibility of Substitution Rule in Cyclic-Proof Systems 3 rule. In the lifting phase (i), one transforms a derivation tree Γ1 ⊢∆1 · · · Γn ⊢∆n Γ ⊢∆ (R) Γ[θ] ⊢∆[θ] (Subst) into Γ1 ⊢∆1 Γ1[θ1] ⊢∆1[θ1] (Subst) · · · Γn ⊢∆n Γn[θn] ⊢∆n[θn] (Subst) Γ[θ] ⊢∆[θ] (R) . In the elimination phase (ii), one transforms a derivation tree Γ ⊢∆(Axiom) Γ[θ] ⊢∆[θ] (Subst) into Γ[θ] ⊢∆[θ] (Axiom). In cyclic-proof systems, however, the admissibility of the substitution rule is still unclear. At least, it is not possible to remove (Subst) in the same way as in non-cyclic-proof systems. In the lifting phase, the bud-companion relationship may be destroyed if the sequent Γ ⊢∆, which is deleted during the transformation, is a companion. This is not merely a temporary effect during proof transformation; for example, as illustrated in Fig. 2, even if the substitutions are lifted to the bud, it may still fail to choose the corresponding companion. In the elimination phase, it is not possible to eliminate the substitution when the leaf of a derivation tree is a bud. Fig. 2. Lifting (Subst) may break bud-companion correspondence Brotherston [1] raised the question of whether the substitution rule is admissible in the cyclic-proof system CLKIDω, suggesting that it is likely not admissible in general, at least in the cut-free setting. This paper proves the admissibility of the substitution rule in the cyclic-proof system CLKIDω for the first-order logic with inductive predicates [4], assuming the presence of the cut rule. Furthermore, if function symbols with positive arity are not introduced by the substitution rule, its applications can be eliminated even without the cut rule. This result can be applied to other proof systems as well. For example, in the cyclic-proof system for separation logic, where function symbols except the constant nil are generally not introduced, the substitution rule is admissible regardless of the presence of the cut rule. The proof 4 K. Saotome and K. Nakazawa is one of the cyclic proofs of N(x) ⊢E(x)∨O(x) without substitution rules, that is constructed by unfolding the cyclic proof in Fig. 1. Regarding the new variable y introduced by the lower (UL), the original proof unifies it with the variable x using (Subst). In the new proof, this is accomplished by choosing x itself as the fresh variable in the upper (UL). Building on this observation, we consider unfolding a cyclic proof into an infinite proof, wherein substitution rules can be systematically lifted upward. After performing this lifting to a sufficiently great height in the infinite proof tree, we reconstruct a cyclic proof without substitutions. To guarantee the global trace condition, we choose buds and their companions in the reconstructed cyclic proof only when they have the same entailment occurrence as in the original proof. This careful choice prevents the emergence of unintended infinite paths. Structure of the paper. Section 2 introduces a cyclic-proof system CLKIDω and an infinitary-proof system LKIDω for the first-order logic FOLID with inductive predicates. In Section 3, we prove the admissibility of the substitution rule in CLKIDω. In Section 4, we discuss the admissibility of the substitution rule in other proof systems. In Section 5, we conclude. 2 Cyclic-Proof System for First-Order Logic with Inductive Predicates This section recalls the first-order logic with inductive predicates, together with the cyclic- and infinitary-proof systems, as proposed by Brotherston et al. [1, 4]. For the unfolding rules of inductive predicates, however, we follow the style of Kimura et al. [7], in which the rule names do not explicitly mention the inductive predicates. 2.1 First-Order Logic FOLID with Inductive Predicates We define the formula of FOLID. A function symbol, denoted by f, g, . . ., has a fixed arity. A nullary function symbol, denoted by c, c1, . . ., is called a constant symbol. We denote a variable by x, y, z, . . .. We use x for a finite sequence of Admissibility of Substitution Rule in Cyclic-Proof Systems 5 variables. A term, denoted by t, u, . . ., is defined recursively as either a variable or an expression of the form f(t), where t is a finite sequence of terms. We write t(x) to make explicit that the variable x occurs in the term t; analogously, we use the notation t(x) for the same purpose. = is introduced as a special twoargument predicate symbol. We use P, P1, . . . for inductive predicate symbols. We also use Q, Q1, . . . for ordinary predicate symbols. Each predicate symbol has a fixed arity. Definition 1 (Formulas). The formulas of FOLID, denoted φ, ψ, . . ., are as follows: φ ::= P(t) | Q(u) | t = u | ¬φ | φ ∨φ | φ ∧φ | φ →φ | ∃x.φ | ∀x.φ, where the lengths of t and u are the arities of P and Q, respectively. Formulas in FOLID are identified up to α-equivalence under variable binding by ∀and ∃. For formulas φ and ψ that are syntactically equal, that is, αequivalent, we denote φ ≡ψ. The substitutions of terms, denoted by θ, θ1, . . ., are finite mappings from term variables to terms. The formula φ[θ] denotes the result of the captureavoiding substitution θ to free variables in φ. θ1θ2 denotes the composition of the substitutions θ1 and θ2, and it holds that φ[θ1θ2] ≡(φ[θ1])[θ2]. θ[x1 → y1, . . . , xn →yn] denotes the substitution obtained from θ by changing the image of xi to yi for 1 ≤i ≤n, respectively. For a finite set of formulas Γ, Γ[θ] is a finite set such that Γ[θ] = {φ[θ] | φ ∈Γ}. Specifically, a substitution whose image includes only variables and constants is called an atomic substitution. On the other hand, a substitution whose image contains a term with a function symbol of positive arity is called a composite substitution. The domain and the image of a mapping f are denoted by dom(f) and img(f), respectively. The definition of each inductive predicate P(t) is given by a definition set. Definition 2 (Definition sets). An inductive definition set Φ is a finite set of productions in the following form: Q1(u1) · · · Qn(un) P1(t1) . . . Pm(tm) P(t) , where each Qi(1 ≤i ≤n) is an ordinary predicate symbol and each Pj(1 ≤j ≤ m) is an inductive predicate symbol. Example 1. The following are the productions for N, E, and O, which represent natural numbers, even numbers, and odd numbers, respectively. N(0) , N(x) N(sx) , E(0) , O(x) E(sx) , and E(x) O(sx) . We consider the standard model [1, 4] as the semantics. Definition 3 (Sequents). Let Γ and ∆be finite sets of formulas. A sequent, denoted by e, is defined as Γ ⊢∆. We say that Γ ⊢∆is valid with respect to the standard model. 6 K. Saotome and K. Nakazawa 2.2 Infinitary-Proof System LKIDω In this subsection, we introduce the infinitary-proof system LKIDω used to establish the admissibility of the substitution rule in the cyclic-proof system CLKIDω. First, we define the inference rules of LKIDω. These inference rules are also used in the cyclic-proof system CLKIDω. Definition 4 (Inference rules). The inference rules of LKIDω and CLKIDω except for inductive predicates unfolding are as follows: Γ ⊢∆Γ ∩∆̸= ∅(Axiom) Γ ′ ⊢∆′ Γ ⊢∆Γ ′ ⊆Γ, ∆′ ⊆∆(Wk) Γ ⊢φ, ∆ Γ, φ ⊢∆ Γ ⊢∆ (Cut) Γ ⊢∆ Γ[θ] ⊢∆[θ] (Subst) Γ ⊢φ, ∆ Γ, ¬φ ⊢∆(¬L) Γ, ∆⊢φ Γ ⊢∆, ¬φ (¬R) Γ, φ ⊢∆ Γ, ψ ⊢∆ Γ, φ ∨ψ ⊢∆ (∨L) Γ ⊢∆, φ, ψ Γ ⊢∆, φ ∨ψ (∨R) Γ, φ, ψ ⊢∆ Γ, φ ∧ψ ⊢∆(∧L) Γ ⊢∆, φ Γ ⊢∆, ψ Γ ⊢∆, φ ∧ψ (∧R) Γ ⊢φ, ∆ Γ, ψ ⊢∆ Γ, φ →ψ ⊢∆ (→L) Γφ ⊢ψ∆ Γ ⊢∆, φ ∨ψ (→R) Γ, φ[t/x] ⊢∆ Γ, ∀x.φ ⊢∆(∀L) Γ ⊢φ[y/x], ∆ Γ ⊢∀x.φ, ∆ (∀R) Γ, φ[y/x] ⊢∆ Γ, ∃x.φ ⊢∆ (∃L) Γ ⊢φ[t/x], ∆ Γ ⊢∃x.φ, ∆(∃R) Γ[u/x, t/y] ⊢∆[u/x, t/y] Γ[t/x, u/y], t = u ⊢∆[t/x, u/y] (= L) Γ ⊢t = t, ∆(= R), where t in (∃L) and (∃) is arbitrary and y in those is fresh. The unfolding rule for left inductive predicates is all case distinctions for P(u) Γ, P(u) ⊢∆ (UL), where the case distinction of Γ, P(u) ⊢∆for a production Q1(u1(x)) · · · Qn(un(x)) P1(t1(x)) . . . Pm(tm(x)) P(t(x)) is Γ, u = t(y), Q1(u1(y)), . . . , Qn(un(y)), P1(t1(y)), . . . , Pm(tm(y)) ⊢∆, where y is fresh. The unfolding rule for right inductive predicates is Γ ⊢Q1(u1(u)), ∆ · · · Γ ⊢Qn(un(u)), ∆ Γ ⊢P1(t1(u)), ∆ · · · Γ ⊢Pm(tm(u)), ∆ Γ ⊢P(t(u)), ∆ (UR) for a production Q1(u1(x)) · · · Qn(un(x)) P1(t1(x)) . . . Pm(tm(x)) P(t(x)) . Admissibility of Substitution Rule in Cyclic-Proof Systems 7 A pre-proof of LKIDω is a finite or infinite derivation tree, whose all leaves are axioms, defined in a usual way by the inference rules of LKIDω. A pre-proof is denoted by Pr. The set of sequent occurrences in Pr is denoted by Seq(Pr). The sets of free variables in a formula φ, a sequent e, and a (pre-)proof Pr are denoted by FV(φ), FV(e), and FV(Pr), respectively. The set of substitutions of (Subst) in Pr is denoted by Θ(Pr). The number of elements in a finite set S is denoted by #S. To define the proof of LKIDω, we define traces following paths. Definition 5 (Paths). Let Pr be a pre-proof in LKIDω. Viewing Pr as a (possibly infinite) graph, spanning from root to leaf. A path in this graph is called a path of Pr, denoted by (ei)j≤i #img(f ω d )·n holds. Such d exists since img(f ω d ) ⊆Seq(Prvar + )∪ {ε} holds. For any path (ei)0≤i≤d in Prω d , for some sequent e in Prvar + , there exist at least n + 1 indices i with 0 ≤i ≤d such that f ω d (ei) = e. Let S denote the set of those indices. f ω d (ei) ̸= ε holds for any i ∈S since ei is not a premise of (Subst). Therefore, ei and f ω d (ei)θi are the same sequents for some θi ∈Cps(Θ(Prvar + ), FV(Prvar + )) by the condition of f ω d . The number of substitutions in Cps(Θ(Prvar + ), FV(Prvar + )) is n. Hence, there are the same sequents ei and ej for some i < j ∈S. By choosing ei as the bud and ej as its companion, we can construct Pr-. Moreover, for each occurrence of a sequent e in Pr-, there exists a corresponding occurrence of a sequent e′ in Prω d . Based on this correspondence, we define f-(e) = f ω d (e′). With the definition of f-, all conditions required by this lemma are satisfied. ⊓⊔ From the above, we could construct the pre-proof Pr-from a proof Prvar + via Prω and Prω d . Next, we mention that the pre-proof Pr-is a proof. In other words, we show that the pre-proof Pr-satisfies the global trace condition. First, define a corresponding path in Prvar + for each path in Pr-. Definition 12 (Corresponding path). Let Pr-and f-be a pre-proof and a mapping, respectively, constructed in Lemma 6. Let (ei)0≤i be a path in Prwhere e0 is the conclusion of Pr-. The path (e′ j)0≤j in Prvar + defined as follows is called corresponding path of (ei)0≤i: - Let e′ 0 = f-(e0) be the conclusion of Prvar + . - Suppose that e′ j = f-(ei). • If ei is a bud and ei+1 is its companion, then let e′ j = f-(ei+1). • If ei+1 is the n-th premise in a rule instance · · · ei+1 · · · ei (R), 16 K. Saotome and K. Nakazawa then, by the condition of f-, there exists rule applications · · · f ω d (ei+1) · · · (R) .... (Subst) and bud-companion m times e′ j = f ω d (ei) , in Prvar + for some m such that fω d (ei+1) is the n-th premise of (R). In this application, the sequence e′ j, . . . , e′ j+m is uniquely determined since (Subst) is a unary rule. Furthermore, we set e′ j+m+1 = f(ei+1). Using the corresponding path, we can refer to the relationship between the traces of Prvar + and Pr-. Lemma 7. Let (ei)0≤i be a path in Pr-constructed in Lemma 6 and (e′ j)0≤j be the corresponding path of (ei)0≤i in Prvar + . If there is an infinitely progressing trace (C′ j)k′≤j following (e′ j)0≤j, the infinitely progressing trace (Ci)k≤i exists in (ei)0≤i. Based on Lemma 1 and 6, we can construct a pre-proof without the substitution rule. Lemma 7 guarantees that the pre-proof satisfies the global trace condition; hence, the following main theorem holds. Theorem 2 (Admissibility of substitution rules in CLKIDω). If Γ ⊢∆ is provable in CLKIDω, then it is provable in CLKIDω without (Subst). In the process of elimination of atomic substitutions, we do not introduce any rule not present in Prvar + , and hence we have the following stronger result. Theorem 3. If Pr is a proof of Γ ⊢∆in CLKIDω and Θ(Pr) consists of only atomic substitutions, then there is a (Subst)-free proof of Γ ⊢∆that contains only rules in Pr. 4 Admissibility of the Substitution Rule in Other Cyclic-Proof Systems In this section, we discuss the conditions under which our procedure for the substitution elimination can be applied in general cyclic-proof systems. Among the lemmas proved in Section 3.2, those that depend on the proof system CLKIDω are Lemma 1 and 4. First, we discuss Lemma 1, which concerns the elimination of composite substitutions. This lemma claims the existence of a proof that contains only atomic substitutions from the existence of a proof that allows composite substitutions. To prove this lemma, we use inference rules (Cut), (= L), (= R), (∃L), and (∃R). Accordingly, in proof systems that do not include these rules, this lemma does not, in general, hold. Admissibility of Substitution Rule in Cyclic-Proof Systems 17 In fact, we can consider a system in which (Subst) is not admissible. For example, let P(x) be an inductive predicate defined by P(x) := P(sx). The unfolding rule of the left P(x) in CLKIDω is y = x, P(sy), Γ ⊢∆ P(x), Γ ⊢∆ (UL). However, we can consider another unfolding rule P(sx), Γ ⊢∆ P(x), Γ ⊢∆(UL′). Let we think a proof system with only (UL′), (UR), (Axiom), (Wk), and (Subst). There is a proof of the sequent P(x) ⊢⊥. However, we cannot prove this sequent without (Subst) in this system. Note that in Lemma 1 for CLKIDω, the cut rule is only used to introduce y = t (where y is fresh and t is arbitrary). Therefore, when we introduce the following rule in place of the cut rule, (Subst) is still admissible y = t, Γ ⊢∆ Γ ⊢∆ (fresh L), where t is arbitrary and y is fresh in Γ ⊢∆and t. Secondly, we discuss Lemma 4. This lemma plays an important role in the elimination of atomic substitutions. This lemma guarantees that CLKIDω and LKIDω satisfy the substitution-application property with atomic substitutions. This lemma holds for many general proof systems, including the proof system CLKIDω and the cyclic proof system for separation logic. Therefore, as long as we can assume a proof with only atomic substitutions, the substitution rule can be eliminated in many proof systems. Let CLKIDω -be a proof system CLKIDω without the cut rule. Theorem 4 (Admissibility of atomic substitution rules in CLKIDω -). If Pr is a proof of Γ ⊢∆in CLKIDω -and Θ(Pr) consists of only atomic substitutions, then there is a (Subst)-free CLKIDω -proof of Γ ⊢∆that contains only rules in Pr. In particular, every proof in the cyclic-proof system CSLω [2] for the separation logic contains only variable application since the separation logic contains no function symbols. Hence, the following theorems hold. Let CSLω -be CSLω without the cut rule. Theorem 5. (Subst) is admissible in both CSLω and CSLω -. 18 K. Saotome and K. Nakazawa 5 Conclusion In this paper, we prove the admissibility of the substitution rule in CLKIDω. After that, we discuss generalization to other proof systems. One possible direction for future work is investigating the admissibility of the substitution rule in cut-free CLKIDω without (fresh L). Section 4 showed that replacing (UL) with (UL′) and restricting inference rules destroy this admissibility. As a counterexample, we considered P(x) ⊢⊥; however, in the usual CLKIDω system, there still be a cut-free proof of this sequent. The reason is that (UL) effectively plays the role of (fresh L). Our interest lies in identifying the minimal set of rules that still allows the elimination of the substitution rule in the presence of function symbols. Another possible direction is to study the cut-elimination property for cyclic proof systems by introducing additional rules. The fact that the substitution rule could be eliminated from cut-free CLKIDω through the introduction of (fresh L) provides an important insight. This suggests that, by adding rules sufficient to construct cycles, it may be possible to achieve the elimination of the cut rule. Acknowledgments. The second author was supported by JSPS KAKENHI Grant Number 22K11901. Admissibility of Substitution Rule in Cyclic-Proof Systems 19 References 1. Brotherston, J.: Sequent calculus proof systems for inductive definitions. Ph.D. thesis, Edinburgh University (2006) 2. Brotherston, J., Distefano, D., Petersen, R.L.: Automated cyclic entailment proofs in separation logic. In: 23rd international conference on automated deduction (CADE-23). Lecture Notes in Artificial Intelligence (LNAI), vol. 6803, pp. 131146 (2011) 3. Brotherston, J., Gorogiannis, N., Petersen, R.L.: A generic cyclic theorem prover. In: 10th Asian Symposium on Programming Languages and Systems (APLAS 2012). Lecture Notes in Computer Science (LNCS), vol. 7705, pp. 350-367 (2012) 4. Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. Journal of Logic and Computation 21(6), 1177-1216 (2011) 5. Chu, D., Jaffar, J., Trinh, M.: Automatic induction proofs of data-structures in imperative programs. In: 36th annual ACM SIGPLAN conference on Programming Language Design and Implementation (PLDI 2015). pp. 457-466 (2015) 6. Doumane, A.: On the infinitary proof theory of logics with fixed points. Ph.D. thesis, Paris 7 (2017) 7. Kimura, D., Nakazawa, K., Terauchi, T., Unno, H.: Failure of cut-elimination in cyclic proofs of separation logic. Computer Software 37(1), 39-52 (2020) 8. Oda, Y., Brotherston, J., Tatsuta, M.: The failure of cut-elimination in cyclic proof for first-order logic with inductive definitions. Journal of Logic and Computation, exad068 (2023) 9. Saotome, K., K.Nakazawa, D.Kimura: Restriction on cut rule in cyclic-proof system for symbolic heaps. Theoretical Computer Science 1019, 114854 (2024) 10. Schöpp, U., Simpson, A.: Verifying temporal properties using explicit approximants: Completeness for context-free processes. In: Foundations of Software Science and Computation Structure(FoSSaCS 2002). Lecture Notes in Computer Science (LNCS), vol. 2303, p. 372-386 (2002) 11. Ta, Q., Le, T., Khoo, S., Chin, W.: Automated mutual induction in separation logic. In: Formal Methods: 21st international Symposium (FM 2016). Lecture Notes in Computer Science (LNCS), vol. 9995, pp. 659-676 (2016) 12. Tatsuta, M., Nakazawa, K., Kimura, D.: Completeness of cyclic proofs for symbolic heaps with inductive definitions. In: The 17th Asian Symposium on Programming Languages and Systems (APLAS 2019). Lecture Notes in Computer Science (LNCS), vol. 11893, pp. 367-387 (2019) 13. Zhang, S., N.Nishida: On transforming cut- and quantifier-free cyclic proofs into rewriting-induction proofs. In: Functional and Logic Programming (FLOPS 2020). p. 262-281. Lecture Notes in Computer Science (LNCS) (2022) A Details of Proofs in Section 3.2 In this section, we show the details of proofs in Section 3.2. A.1 Lemma 3 Proof. Since any function symbols not contained in Θ are likewise absent from Cps(Θ, X), any substitution in Cps(Θvar, X) is atomic. 20 K. Saotome and K. Nakazawa Let dom(Θ) and img(Θ) denote, respectively, the unions of dom(θ) and img(θ) over all θ ∈Θ. The domain of every substitution in Cps(Θvar, X) is a subset of dom(Θ)∪X. Similary, the range of every substitution in Cps(Θvar, X) is a subset of img(Θ)∪X. Hence, the value of #Cps(Θvar, X) is at most #(img(Θ)∪ X)#(dom(Θ)∪X) and #Cps(Θvar, X) is finite. ⊓⊔ A.2 Lemma 7 Proof. By the property of f-, since traces are preserved at each rule application, if there exists an infinitely progressing trace along with (e′ j)0≤j, there also exists the corresponding infinitely progressing trace along with (ei)0≤i. Note that when ei and ei+1 on Pr-are respectively a bud and its companion, we have f(ei) = f(ei+1), and hence, at this point, the trace does not break. ⊓⊔ A.3 Theorem 4 Proof. Assume that Prvar + is a CLKIDω proof of Γ ⊢∆s.t. any substitution in Θ(Prvar + ) is an atomic substitution. By Lemma 1, the existence of such a proof Prvar + is guaranteed. By Lemma 6, we can construct the pre-proof Pr-of Γ ⊢∆that does not contain the application of (Subst) and the function f-. Let (ei)0≤i be an infinite path in Pr-. There is a corresponding path (e′ j)0≤j in Prvar + . (e′ j)0≤j is an infinite path since (ei)0≤i is an infinite path. Since Prvar + is proof, there is an infinitely progressing trace along with (e′ j)0≤j. By Lemma 7, there is an infinitely progressing trace along with (ei)0≤i. Assuming an arbitrary infinite path on Pr-, there exists an infinitely progressing trace. Therefore, Pr-satisfies the global trace condition. Consequently, Pr-is a proof, and we have established the existence of a proof of Γ ⊢∆that does not contain the application of (Subst). ⊓⊔
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2510.14747
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The predictive power of the calculated line lists of carbon
monoxide
E. S. Medvedev, V. G. Ushakov
Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry (former
Institute of Problems of Chemical Physics), Russian Academy of Sciences, 142432
Chernogolovka, Russian Federation
Abstract
The ability of our semi-empirical irregular dipole-moment functions (2022)
and (2025) to predict the intensities of the yet unobserved lines, as well as to
describe the observed ones not used in the fitting, is demonstrated by comparison
with recent measurements in the 0-0, 1-0, 3-0, and 7-0 bands.
The spectroscopic databases such as HITRAN [1], ExoMol [2], and others
collect the line parameters calculated with use of the molecular functions derived
from the experimental and ab initio data. The data cover not only the observed
lines but also yet unobserved ones.
Therefore, the first requirement is good
description of the input data and the second one to make meaningful predictions.
The second task is very challenging because the form of the molecular functions
may strongly affect the calculated results [3–6], as was first demonstrated in the
paper by Li et al. [7] where the high-overtone lines were excluded from the list.
We have published three line lists for CO calculated with the semi-empirical
potential-energy function of Meshkov et al. [8] and various semi-empirical dipole-
moment functions (DMFs), namely regular (i.e., single-valued, having no branch
points in the complex plane) in Ref. [9] and irregular (many-valued, with branch
points) in Refs. [10, 11]; the rational DMF (the ratio of two polynomials) was
also tried in Ref. [10], which gave the intensities close to those obtained with
the irregular DMF, therefore the differences can serve as a qualitative estimate
of the precision of the calculation.
In Ref. [9], we collected all the experimental data available at that time
and used them along with our own ab initio calculated DMF to fit the regular
DMF and to create line list 1 (LL1). The LL1 was restricted by transitions
with ∆v ≤6 since higher overtone-transition intensities were considered to be
unreliable. In Ref. [10], we used the same input dataset to fit the irregular DMF
and to create line list 2 (LL2), which was not subject to the above restriction in
∆v. For comparison, we used the rational DMF and found moderate differences
with LL2, which we treated as qualitative estimates of the calculation accuracy.
In particular, we found for the 7-0 band the error estimate to be about 30%. It
is of interest to verify these predictions with new data.
arXiv:2510.14747v1 [physics.chem-ph] 16 Oct 2025
Later, the 7-0 band has been measured by Balashov et al. [12] and new
high-precision 3-0 band intensities have been published by Bielska et al. [13]
and Hodges et al. [14]. We tried to describe these data using our irregular DMF
and found that they came in contradiction with the earlier data on the 1-0
band [15, 16] since the 1-0 data contained systematic errors. Therefore, the 1-0
data were excluded from the input dataset, and the above data on the 3-0 and
7-0 bands were included to calculate line list 3 (LL3). In particular, the LL3
contains predictions for the 1-0 band, which can be verified now.
Recently, more measurements have been performed with extremely high pre-
cision by Bailey et al. [17] in the 1-0 band, Huang et al. [18], Li et al. [19], and
Cygan et al. [20] in the 3-0 band, and high-precision measurements in the 0-0
band by Tretyakov et al. [21] were also available; all these data, except for the
7-0 band, were not used in calculations of LL3.
The purpose of this publication is to compare the newly observed line inten-
sities, as well as the earlier data not included in the fits, with the predictions
of our line lists and HITRAN. Table 1 shows the comparison for absolute in-
tensities. For the 0-0 band, all line lists work perfectly. The 1-0R(17) line is
excellently reproduced by LL3 whereas LL1 and LL2 fail since they used the
earlier 1-0 data excluded in LL3. The 3-0R(23) line as measured by Cygan et
al. [20] is reproduced by LL1 satisfactorily, LL2 and LL3 work even better; it
should be stressed that all the line lists did not use these observational data, as
well as the other ones mentioned in Tables 1 and 2, in the fitting.
Table 1 certifies that LL2 and LL3 show the overall significant improvement
(excluding a few lines) with respect to HITRAN2020: 70% of all lines in the LL3
column describe experiment within the experimental uncertainties. In particu-
lar, the 7-0 band is fully reproduced by LL3 with such an accuracy, which is due
to the inclusion of these data in the fit. More significant are the results of LL2
for the 7-0 band calculated before the measurements. While not all of the lines
are described within the experimental errors, the observed-minus-calculated dif-
ferences are under 14%, much less than the estimated theoretical uncertainty
of 30%. This means that the irregular DMF gives reasonable predictions for
practical applications. Also noticeable is the significant improvement of LL3
against LL2 due to inclusion of the new high-precision data. The estimates of
the calculated uncertainties in LL3 for higher overtones are given as well [11],
which can be of value for planning future experiments.
Balashov et al. [17] set the task to construct an ab initio DMF capable of
describing the measured intensities within the experimental uncertainties for
all bands simultaneously. They constructed a new ab initio DMF and fitted
it with a 6th-order polynomial, which solved in part this problem. Moreover,
the necessity to predict the yet unobserved lines in higher overtones was also
formulated. Later, Zobov et al. [22] created a more successful ab initio DMF
that gave “the line intensities of all observed vibrational bands from 0–0 to 7–0
with the sub-percent or experimental accuracy (whichever is the larger)", as
2
demonstrated in their Table 31. Here, we have shown that our semi-empirical
irregular DMF already solves both these tasks.
The new method of Li et al. [19] opened a possibility to greatly increase
the accuracy by measuring the intensity ratios of pairs of lines. This permit-
ted to reduce the experimental error by about 30 times, from 0.1 to 0.003%.
Table 2 shows the comparison of the observations with LL3. Obviously, the
observed-minus-calculated differences greatly exceed the experimental errors.
The unprecedented low experimental error can be reached semi-empirically by
refitting the DMF with a probable increase in the number of parameters. How-
ever, we don’t see any necessity in such refitting since the differences between
the calculated and observed intensity ratios are already extremely small, less
than 0.1% above the R(15)/P(15) ratio in the table and less than 0.2% below
it, which is more than sufficient for any practical tasks.
The ab initio calculations of Li et al. [19] reached the experimental accuracy
for the intensity ratios, yet the analytic form of DMF was not presented. There
are also a few ab initio DMFs constructed by these and other authors [12, 22, 24–
28], some in analytic forms, but the ability of the interpolated/extrapolated ab
initio DMFs to predict the intensities of yet unobserved lines has not been
demonstrated, to the best of our knowledge. In contrast, our semi-empirical
irregular and rational DMFs [11] provide for an acceptable level of accuracy for
all the observed bands as well as for qualitative estimates of uncertainties of
the calculations for higher overtones. This conclusion will be further verified
when the new data [29] announced in the recent update of HITRAN [30] are
published.
Figure 1 demonstrates the quality of predictions of our DMFs in comparison
with the 6th-order polynomial DMF of Zobov et al. (2025) [22]. The Einstein
A coefficients for the v-0R(0) lines are plotted in the NIDL [31] coordinates.
The NIDL line is drawn over v = 2, 3,7-20 excluding the anomaly at v = 5
and two neighboring lines as well as the highest transitions requiring higher
numerical precision. The NIDL line is perfectly straight, the absolute standard
deviation for both DMFs being only 0.05 over all the transitions covered by
the NIDL. In contrast, the intensities calculated by us with the polynomial of
Zobov et al. do not obey the NIDL. Other above-cited analytic (fitted to ab
initio) DMFs demonstrate similar behavior, with the intensities variable among
these DMFs by orders of magnitude; therefore, their predictions are not reliable.
The latter conclusion is valid not only for the high-overtone transitions, which
are mostly not observable, but also for the low ∆v transitions in view of the
claimed percent [12], sub-percent [22], and even sub-promille [13, 19] accuracies
of the calculations.
1There are some misprints: the 0-0R(10,15,20) and 2-0P(10) line intensities (and reference
to Bielska et al. (2022) as measured the 2-0 band, see Ref. [23] instead) are not correct; the
frequency and intensity of the 4-0P(22) are incorrect.
3
Table 1: Comparison of the measured and predicteda absolute line intensities not included in
the fitting of the semi-empirical DMFs
∆= 100(O-C)/O†
line
Sb
obs
Exp.uncert,%
LL1
LL2
LL3c
HITRAN
source
0-0R(0)
3.2814E-24
0.2
0.006
0.006
0.005
-0.6
[21]
0-0R(1)
2.553E-23
0.5
0.09
0.09
0.08
-0.5
[21]
1-0R(17)
1.028E-19
0.6
1.5
1.5
-0.4
2.1
[17]d
3-0R(23)
8.0603E-25
0.09
0.09
0.04
0.05
0.17
[20]
3-0R(23)
8.1592e-025⋆
0.07
-0.04
-0.1
-0.08
0.03
[18],CRDSd
3-0R(25)
3.6455e-025
0.09
-0.02
-0.07
-0.07
0.04
[18],CRDSd
3-0R(26)
2.3679e-025
0.08
0.12
0.07
0.06
0.17
[18],CRDSd
3-0R(27)
1.5055e-025
0.07
0.1
0.05
0.04
0.08
[18],CRDSd
3-0R(28)
9.3917e-026
0.07
0.15
0.09
0.08
0.14
[18],CRDSd
3-0R(29)
5.7443e-026
0.07
0.17
0.12
0.1
0.16
[18],CRDSd
3-0R(30)
3.4436e-026
0.07
0.14
0.09
0.07
0.1
[18],CRDSd
3-0R(31)
2.0261e-026
0.07
0.18
0.12
0.1
0.09
[18],CRDSd
3-0R(32)
1.1684e-026
0.07
0.13
0.08
0.05
0.06
[18],CRDSd
3-0R(23)
8.1673e-025
0.08
0.06
0.004
0.02
0.13
[18],CMDSd
3-0R(25)
3.6501e-025
0.07
0.11
0.05
0.06
0.17
[18],CMDSd
3-0R(26)
2.3703e-025
0.07
0.22
0.17
0.17
0.27
[18],CMDSd
3-0R(27)
1.5069e-025
0.08
0.19
0.14
0.13
0.18
[18],CMDSd
3-0R(28)
9.3946e-026
0.07
0.18
0.12
0.11
0.17
[18],CMDSd
3-0R(29)
5.7481e-026
0.07
0.24
0.19
0.17
0.22
[18],CMDSd
3-0R(30)
3.4449e-026
0.08
0.18
0.13
0.11
0.13
[18],CMDSd
3-0R(31)
2.0268e-026
0.07
0.21
0.16
0.13
0.13
[18],CMDSd
3-0R(32)
1.1688e-026
0.09
0.16
0.11
0.08
0.09
[18],CMDSd
7-0P(19)
1.807E-30
21
15
8.7
44
[12]
7-0P(16)
3.645E-30
7.4
6
-0.8
42
[12]
7-0P(13)
6.698E-30
4.1
7
0.3
47
[12]
7-0P(12)
7.530E-30
4.0
4
-3.1
46
[12]
7-0P(11)
9.067E-30
4.1
10
3.0
50
[12]
7-0P(10)
9.592E-30
3.7
6
-1.2
49
[12]
7-0P(9)
1.065E-29
3.3
9
2.4
52
[12]
7-0P(8)
1.099E-29
3.4
9
1.7
52
[12]
7-0P(7)
1.103E-29
3.4
9
1.7
53
[12]
7-0P(4)
8.445E-30
3.8
8
1.2
55
[12]
7-0P(1)
2.543E-30
11
14
6.6
58
[12]
7-0R(0)
2.462E-30
12
12
4.4
58
[12]
7-0R(4)
9.119E-30
3.5
7
-1.3
56
[12]
7-0R(6)
9.988E-30
3.5
6
-2.4
55
[12]
†The ∆’s falling within the experimental uncertainties are shown in bold face.
⋆8.0494 for natural abundance, differs from Ref. [20] by 0.13%.
aRecalculated to HITRAN’s natural abundance where necessary.
bFor 296 K and HITRAN’s natural abundance if not indicated otherwise.
cThe 7-0 data were used in the fitting.
d100% abundance. CRDS, absorption; CMDS, dispersion.
4
Table 2: Comparison of measured by Li et al. [19] line 1 to line 2 intensity ratios, S1/S2, at
various temperatures and predicted by LL3 relative line intensities in the 3-0 band recalculated
to the experimental temperature
L1/L2
T,K
S1/S2
Exp. uncert,%
100(O-C)/O
R0/R1
298.429
0.503012
0.004
-0.007
R1/R3
298.430
0.534955
0.003
-0.007
R3/R5
298.426
0.767994
0.002
-0.013
R5/R10
298.431
1.073886
0.002
-0.013
R1/R10
298.431
0.441206
0.002
-0.031
R10/R13
298.430
1.472357
0.001
-0.012
R13/R15
298.429
1.458483
0.004
-0.006
R15/R17
298.426
1.594931
0.003
-0.006
R17/R19
298.430
1.738204
0.002
-0.005
R19/R21
298.430
1.889597
0.002
-0.009
R21/R22
298.432
1.417658
0.002
0.005
R22/R23
298.434
1.446427
0.004
-0.008
R23/R24
298.437
1.475842
0.0014
0.0007
R24/R25
298.434
1.505442
0.003
-0.004
R25/R26
298.429
1.535529
0.003
-0.002
R26/R27
298.428
1.566004
0.0040
-0.0008
R1/P1
298.428
2.076074
0.006
0.008
R5/P5
298.430
1.376124
0.002
0.040
R5/P5
298.429
1.376132
0.003
0.040
R5/P5
298.928
1.376095
0.003
0.038
R10/P10
298.428
1.428264
0.003
0.068
R15/P15
298.428
1.567518
0.003
0.099
R17/P17
298.427
1.634831
0.004
0.117
R20/P20
298.428
1.745511
0.004
0.140
R21/P21
298.427
1.784819
0.005
0.145
R21/P21
298.430
1.784829
0.002
0.146
R21/P21
298.428
1.784802
0.003
0.144
R23/P23
298.431
1.867087
0.003
0.161
R23/P23
298.427
1.867099
0.004
0.161
R23/P23
298.427
1.867157
0.003
0.164
R23/P23
298.429
1.867182
0.003
0.166
R26/P26
298.427
1.999426
0.007
0.189
R26/P26
298.428
1.999467
0.004
0.191
R26/P26
298.431
1.999478
0.005
0.191
R26/P26
298.428
1.999611
0.003
0.198
R26/P26
298.429
1.999612
0.003
0.198
R26/P26
298.428
1.999598
0.003
0.197
R26/P26
298.427
1.999587
0.003
0.197
5
1
2
3
4
5
6
-30
-25
-20
-15
-10
-5
0
5
Irregular 2025
Zobov 2025
NIDL for Irregular 2025
Rational 2025
log(A/s
-1
)
(E'/
)
1/2
v-0R(0) transitions
Figure 1: Comparison of the Einstein A coefficients for the v-0R(0) lines calculated with the
Irregular (empty circles) and Rational (full circles) DMFs of Ref. [10] fitted to the updated
data base of Ref. [11] with the predictions of the polynomial DMF fitted to the ab initio
DMF in Ref. [22] (crosses). E′ is the energy of the upper level in cm−1, ω = 2143 cm−1.
The NIDL line is drawn for the v ≤20 overtone transitions excluding the 5-0 anomaly and
two neighboring lines.
Acknowledgement
This work was performed under state task, Ministry of Science and Educa-
tion, Russian Federation, state registration number 124013000760-0.
6
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The predictive power of the calculated line lists of carbon monoxide E. S. Medvedev, V. G. Ushakov Federal Research Center of Problems of Chemical Physics and Medicinal Chemistry (former ), Russian Academy of Sciences, 142432 Chernogolovka, Russian Federation Abstract The ability of our semi-empirical irregular dipole-moment functions (2022) and (2025) to predict the intensities of the yet unobserved lines, as well as to describe the observed ones not used in the fitting, is demonstrated by comparison with recent measurements in the 0-0, 1-0, 3-0, and 7-0 bands. The spectroscopic databases such as HITRAN [1], ExoMol [2], and others collect the line parameters calculated with use of the molecular functions derived from the experimental and ab initio data. The data cover not only the observed lines but also yet unobserved ones. Therefore, the first requirement is good description of the input data and the second one to make meaningful predictions. The second task is very challenging because the form of the molecular functions may strongly affect the calculated results [3-6], as was first demonstrated in the paper by Li et al. [7] where the high-overtone lines were excluded from the list. We have published three line lists for CO calculated with the semi-empirical potential-energy function of Meshkov et al. [8] and various semi-empirical dipolemoment functions (DMFs), namely regular (i.e., single-valued, having no branch points in the complex plane) in Ref. [9] and irregular (many-valued, with branch points) in Refs. [10, 11]; the rational DMF (the ratio of two polynomials) was also tried in Ref. [10], which gave the intensities close to those obtained with the irregular DMF, therefore the differences can serve as a qualitative estimate of the precision of the calculation. In Ref. [9], we collected all the experimental data available at that time and used them along with our own ab initio calculated DMF to fit the regular DMF and to create line list 1 (LL1). The LL1 was restricted by transitions with ∆v ≤6 since higher overtone-transition intensities were considered to be unreliable. In Ref. [10], we used the same input dataset to fit the irregular DMF and to create line list 2 (LL2), which was not subject to the above restriction in ∆v. For comparison, we used the rational DMF and found moderate differences with LL2, which we treated as qualitative estimates of the calculation accuracy. In particular, we found for the 7-0 band the error estimate to be about 30%. It is of interest to verify these predictions with new data. 16 Oct 2025 Later, the 7-0 band has been measured by Balashov et al. [12] and new high-precision 3-0 band intensities have been published by Bielska et al. [13] and Hodges et al. [14]. We tried to describe these data using our irregular DMF and found that they came in contradiction with the earlier data on the 1-0 band [15, 16] since the 1-0 data contained systematic errors. Therefore, the 1-0 data were excluded from the input dataset, and the above data on the 3-0 and 7-0 bands were included to calculate line list 3 (LL3). In particular, the LL3 contains predictions for the 1-0 band, which can be verified now. Recently, more measurements have been performed with extremely high precision by Bailey et al. [17] in the 1-0 band, Huang et al. [18], Li et al. [19], and Cygan et al. [20] in the 3-0 band, and high-precision measurements in the 0-0 band by Tretyakov et al. [21] were also available; all these data, except for the 7-0 band, were not used in calculations of LL3. The purpose of this publication is to compare the newly observed line intensities, as well as the earlier data not included in the fits, with the predictions of our line lists and HITRAN. Table 1 shows the comparison for absolute intensities. For the 0-0 band, all line lists work perfectly. The 1-0R(17) line is excellently reproduced by LL3 whereas LL1 and LL2 fail since they used the earlier 1-0 data excluded in LL3. The 3-0R(23) line as measured by Cygan et al. [20] is reproduced by LL1 satisfactorily, LL2 and LL3 work even better; it should be stressed that all the line lists did not use these observational data, as well as the other ones mentioned in Tables 1 and 2, in the fitting. Table 1 certifies that LL2 and LL3 show the overall significant improvement (excluding a few lines) with respect to HITRAN2020: 70% of all lines in the LL3 column describe experiment within the experimental uncertainties. In particular, the 7-0 band is fully reproduced by LL3 with such an accuracy, which is due to the inclusion of these data in the fit. More significant are the results of LL2 for the 7-0 band calculated before the measurements. While not all of the lines are described within the experimental errors, the observed-minus-calculated differences are under 14%, much less than the estimated theoretical uncertainty of 30%. This means that the irregular DMF gives reasonable predictions for practical applications. Also noticeable is the significant improvement of LL3 against LL2 due to inclusion of the new high-precision data. The estimates of the calculated uncertainties in LL3 for higher overtones are given as well [11], which can be of value for planning future experiments. Balashov et al. [17] set the task to construct an ab initio DMF capable of describing the measured intensities within the experimental uncertainties for all bands simultaneously. They constructed a new ab initio DMF and fitted it with a 6th-order polynomial, which solved in part this problem. Moreover, the necessity to predict the yet unobserved lines in higher overtones was also formulated. Later, Zobov et al. [22] created a more successful ab initio DMF that gave "the line intensities of all observed vibrational bands from 0-0 to 7-0 with the sub-percent or experimental accuracy (whichever is the larger)", as 2 demonstrated in their Table 31. Here, we have shown that our semi-empirical irregular DMF already solves both these tasks. The new method of Li et al. [19] opened a possibility to greatly increase the accuracy by measuring the intensity ratios of pairs of lines. This permitted to reduce the experimental error by about 30 times, from 0.1 to 0.003%. Table 2 shows the comparison of the observations with LL3. Obviously, the observed-minus-calculated differences greatly exceed the experimental errors. The unprecedented low experimental error can be reached semi-empirically by refitting the DMF with a probable increase in the number of parameters. However, we don't see any necessity in such refitting since the differences between the calculated and observed intensity ratios are already extremely small, less than 0.1% above the R(15)/P(15) ratio in the table and less than 0.2% below it, which is more than sufficient for any practical tasks. The ab initio calculations of Li et al. [19] reached the experimental accuracy for the intensity ratios, yet the analytic form of DMF was not presented. There are also a few ab initio DMFs constructed by these and other authors [12, 22, 2428], some in analytic forms, but the ability of the interpolated/extrapolated ab initio DMFs to predict the intensities of yet unobserved lines has not been demonstrated, to the best of our knowledge. In contrast, our semi-empirical irregular and rational DMFs [11] provide for an acceptable level of accuracy for all the observed bands as well as for qualitative estimates of uncertainties of the calculations for higher overtones. This conclusion will be further verified when the new data [29] announced in the recent update of HITRAN [30] are published. Figure 1 demonstrates the quality of predictions of our DMFs in comparison with the 6th-order polynomial DMF of Zobov et al. (2025) [22]. The Einstein A coefficients for the v-0R(0) lines are plotted in the NIDL [31] coordinates. The NIDL line is drawn over v = 2, 3,7-20 excluding the anomaly at v = 5 and two neighboring lines as well as the highest transitions requiring higher numerical precision. The NIDL line is perfectly straight, the absolute standard deviation for both DMFs being only 0.05 over all the transitions covered by the NIDL. In contrast, the intensities calculated by us with the polynomial of Zobov et al. do not obey the NIDL. Other above-cited analytic (fitted to ab initio) DMFs demonstrate similar behavior, with the intensities variable among these DMFs by orders of magnitude; therefore, their predictions are not reliable. The latter conclusion is valid not only for the high-overtone transitions, which are mostly not observable, but also for the low ∆v transitions in view of the claimed percent [12], sub-percent [22], and even sub-promille [13, 19] accuracies of the calculations. 1There are some misprints: the 0-0R(10,15,20) and 2-0P(10) line intensities (and reference to Bielska et al. (2022) as measured the 2-0 band, see Ref. [23] instead) are not correct; the frequency and intensity of the 4-0P(22) are incorrect. 3 Table 1: Comparison of the measured and predicteda absolute line intensities not included in the fitting of the semi-empirical DMFs ∆= 100(O-C)/O† line Sb obs Exp.uncert,% LL1 LL2 LL3c HITRAN source 0-0R(0) 3.2814E-24 0.2 0.006 0.006 0.005 -0.6 [21] 0-0R(1) 2.553E-23 0.5 0.09 0.09 0.08 -0.5 [21] 1-0R(17) 1.028E-19 0.6 1.5 1.5 -0.4 2.1 [17]d 3-0R(23) 8.0603E-25 0.09 0.09 0.04 0.05 0.17 [20] 3-0R(23) 8.1592e-025⋆ 0.07 -0.04 -0.1 -0.08 0.03 [18],CRDSd 3-0R(25) 3.6455e-025 0.09 -0.02 -0.07 -0.07 0.04 [18],CRDSd 3-0R(26) 2.3679e-025 0.08 0.12 0.07 0.06 0.17 [18],CRDSd 3-0R(27) 1.5055e-025 0.07 0.1 0.05 0.04 0.08 [18],CRDSd 3-0R(28) 9.3917e-026 0.07 0.15 0.09 0.08 0.14 [18],CRDSd 3-0R(29) 5.7443e-026 0.07 0.17 0.12 0.1 0.16 [18],CRDSd 3-0R(30) 3.4436e-026 0.07 0.14 0.09 0.07 0.1 [18],CRDSd 3-0R(31) 2.0261e-026 0.07 0.18 0.12 0.1 0.09 [18],CRDSd 3-0R(32) 1.1684e-026 0.07 0.13 0.08 0.05 0.06 [18],CRDSd 3-0R(23) 8.1673e-025 0.08 0.06 0.004 0.02 0.13 [18],CMDSd 3-0R(25) 3.6501e-025 0.07 0.11 0.05 0.06 0.17 [18],CMDSd 3-0R(26) 2.3703e-025 0.07 0.22 0.17 0.17 0.27 [18],CMDSd 3-0R(27) 1.5069e-025 0.08 0.19 0.14 0.13 0.18 [18],CMDSd 3-0R(28) 9.3946e-026 0.07 0.18 0.12 0.11 0.17 [18],CMDSd 3-0R(29) 5.7481e-026 0.07 0.24 0.19 0.17 0.22 [18],CMDSd 3-0R(30) 3.4449e-026 0.08 0.18 0.13 0.11 0.13 [18],CMDSd 3-0R(31) 2.0268e-026 0.07 0.21 0.16 0.13 0.13 [18],CMDSd 3-0R(32) 1.1688e-026 0.09 0.16 0.11 0.08 0.09 [18],CMDSd 7-0P(19) 1.807E-30 21 15 8.7 44 [12] 7-0P(16) 3.645E-30 7.4 6 -0.8 42 [12] 7-0P(13) 6.698E-30 4.1 7 0.3 47 [12] 7-0P(12) 7.530E-30 4.0 4 -3.1 46 [12] 7-0P(11) 9.067E-30 4.1 10 3.0 50 [12] 7-0P(10) 9.592E-30 3.7 6 -1.2 49 [12] 7-0P(9) 1.065E-29 3.3 9 2.4 52 [12] 7-0P(8) 1.099E-29 3.4 9 1.7 52 [12] 7-0P(7) 1.103E-29 3.4 9 1.7 53 [12] 7-0P(4) 8.445E-30 3.8 8 1.2 55 [12] 7-0P(1) 2.543E-30 11 14 6.6 58 [12] 7-0R(0) 2.462E-30 12 12 4.4 58 [12] 7-0R(4) 9.119E-30 3.5 7 -1.3 56 [12] 7-0R(6) 9.988E-30 3.5 6 -2.4 55 [12] †The ∆'s falling within the experimental uncertainties are shown in bold face. ⋆8.0494 for natural abundance, differs from Ref. [20] by 0.13%. aRecalculated to HITRAN's natural abundance where necessary. bFor 296 K and HITRAN's natural abundance if not indicated otherwise. cThe 7-0 data were used in the fitting. d100% abundance. CRDS, absorption; CMDS, dispersion. 4 Table 2: Comparison of measured by Li et al. [19] line 1 to line 2 intensity ratios, S1/S2, at various temperatures and predicted by LL3 relative line intensities in the 3-0 band recalculated to the experimental temperature L1/L2 T,K S1/S2 Exp. uncert,% 100(O-C)/O R0/R1 298.429 0.503012 0.004 -0.007 R1/R3 298.430 0.534955 0.003 -0.007 R3/R5 298.426 0.767994 0.002 -0.013 R5/R10 298.431 1.073886 0.002 -0.013 R1/R10 298.431 0.441206 0.002 -0.031 R10/R13 298.430 1.472357 0.001 -0.012 R13/R15 298.429 1.458483 0.004 -0.006 R15/R17 298.426 1.594931 0.003 -0.006 R17/R19 298.430 1.738204 0.002 -0.005 R19/R21 298.430 1.889597 0.002 -0.009 R21/R22 298.432 1.417658 0.002 0.005 R22/R23 298.434 1.446427 0.004 -0.008 R23/R24 298.437 1.475842 0.0014 0.0007 R24/R25 298.434 1.505442 0.003 -0.004 R25/R26 298.429 1.535529 0.003 -0.002 R26/R27 298.428 1.566004 0.0040 -0.0008 R1/P1 298.428 2.076074 0.006 0.008 R5/P5 298.430 1.376124 0.002 0.040 R5/P5 298.429 1.376132 0.003 0.040 R5/P5 298.928 1.376095 0.003 0.038 R10/P10 298.428 1.428264 0.003 0.068 R15/P15 298.428 1.567518 0.003 0.099 R17/P17 298.427 1.634831 0.004 0.117 R20/P20 298.428 1.745511 0.004 0.140 R21/P21 298.427 1.784819 0.005 0.145 R21/P21 298.430 1.784829 0.002 0.146 R21/P21 298.428 1.784802 0.003 0.144 R23/P23 298.431 1.867087 0.003 0.161 R23/P23 298.427 1.867099 0.004 0.161 R23/P23 298.427 1.867157 0.003 0.164 R23/P23 298.429 1.867182 0.003 0.166 R26/P26 298.427 1.999426 0.007 0.189 R26/P26 298.428 1.999467 0.004 0.191 R26/P26 298.431 1.999478 0.005 0.191 R26/P26 298.428 1.999611 0.003 0.198 R26/P26 298.429 1.999612 0.003 0.198 R26/P26 298.428 1.999598 0.003 0.197 R26/P26 298.427 1.999587 0.003 0.197 5 1 2 3 4 5 6 -30 -25 -20 -15 -10 -5 0 5 Irregular 2025 Zobov 2025 NIDL for Irregular 2025 Rational 2025 log(A/s -1 ) (E'/ ) 1/2 v-0R(0) transitions Figure 1: Comparison of the Einstein A coefficients for the v-0R(0) lines calculated with the Irregular (empty circles) and Rational (full circles) DMFs of Ref. [10] fitted to the updated data base of Ref. [11] with the predictions of the polynomial DMF fitted to the ab initio DMF in Ref. [22] (crosses). E′ is the energy of the upper level in cm-1, ω = 2143 cm-1. The NIDL line is drawn for the v ≤20 overtone transitions excluding the 5-0 anomaly and two neighboring lines. Acknowledgement This work was performed under state task, Ministry of Science and Education, Russian Federation, state registration number 124013000760-0. 6 References [1] I. E. Gordon, L. S. Rothman, R. J. Hargreaves, R. Hashemi, E. V. Karlovets, F. M. Skinner, E. K. Conway, C. Hill, R. V. Kochanov, Y. Tan, P. Wcisło, A. A. Finenko, K. Nelson, P. F. Bernath, M. Birk, V. Boudon, A. Campargue, K. V. Chance, A. Coustenis, B. J. Drouin, J. Flaud, R. R. Gamache, J. T. Hodges, D. Jacquemart, E. J. Mlawer, A. V. Nikitin, V. I. Perevalov, M. Rotger, J. Tennyson, G. C. Toon, H. Tran, V. G. Tyuterev, E. M. Adkins, A. Baker, A. Barbe, E. Can`e, A. G. Császár, A. Dudaryonok, O. 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Phys. 137 (2012) 174307. 10
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2510.14750
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ColumnDisturb: Understanding Column-based Read Disturbance in
Real DRAM Chips and Implications for Future Systems
˙Ismail Emir Yüksel1
Ataberk Olgun1
F. Nisa Bostancı1
Haocong Luo1
A. Giray Ya˘glıkçı1,2
Onur Mutlu1
1ETH Zürich
2CISPA
We experimentally demonstrate a new widespread read dis-
turbance phenomenon, ColumnDisturb, in real commodity
DRAM chips. By repeatedly opening or keeping a DRAM row
(aggressor row) open, we show that it is possible to disturb
DRAM cells through a DRAM column (i.e., bitline) and in-
duce bitflips in DRAM cells sharing the same columns as the
aggressor row(across multiple DRAM subarrays). With Col-
umnDisturb, the activation of a single row concurrently disturbs
DRAM cells across as many as three DRAM subarrays (e.g., up
to 3072 DRAM rows in tested DRAM chips) as opposed to Row-
Hammer & RowPress, which affect only a few neighboring rows
of the aggressor row in a single subarray. We rigorously and
comprehensively characterize ColumnDisturb and its charac-
teristics under various operational conditions (i.e., temperature,
data pattern, DRAM timing parameters, average voltage level
of the bitline, memory access pattern, and spatial variation)
using 216 DDR4 and 4 HBM2 chips from three major DRAM
manufacturers. Among our 27 key experimental observations,
we highlight two major results and their implications.
First, ColumnDisturb affects chips from all three major
DRAM manufacturers and worsens as DRAM technology scales
down to smaller node sizes (e.g., the minimum time to induce
the first ColumnDisturb bitflip reduces by up to 5.06x and 2.96x
on average across all tested DRAM modules). We observe that,
even in existing DRAM chips, ColumnDisturb induces bitflips
within a standard DDR4 refresh window (e.g., in 63.6 ms) in
multiple cells from one module. We predict that, as DRAM
technology node size reduces, ColumnDisturb would worsen
in future DRAM chips, likely causing many more bitflips in the
standard refresh window. Second, beyond the standard refresh
window, ColumnDisturb induces bitflips in many (up to 198x)
more DRAM rows than retention failures. Therefore, Column-
Disturb has strong implications for existing retention-aware
refresh mechanisms that aim to improve system performance
and energy efficiency by leveraging the heterogeneity in DRAM
cell retention times: our detailed analyses and cycle-level sim-
ulations show that ColumnDisturb greatly reduces and can
completely diminish the performance and energy benefits of
such mechanisms.
Our results have serious implications for the robustness of
future DRAM-based computing systems due to continued ag-
gressive DRAM technology scaling. We describe and evaluate
two solutions for mitigating ColumnDisturb bitflips and call for
more research on the topic.
1. Introduction
Dynamic random access memory (DRAM) [1] is the dominant
main memory technology in nearly all modern computing sys-
tems due to its latency and cost characteristics. Continuing
to increase DRAM capacity requires increasing the density of
DRAM cells by reducing (i.e., scaling) the technology node
size of DRAM, which in turn reduces DRAM cell size, cell-
to-cell spacing, and cell-to-bitline spacing [2]. Unfortunately,
as a result of this aggressive technology node scaling, DRAM
suffers from worsening read disturbance problems [3–6].
Read disturbance in modern DRAM chips [4–9] is a
widespread phenomenon and is reliably used for breaking mem-
ory isolation [5,7,9–70], a fundamental building block of robust
(i.e., safe, secure, reliable, available) systems. RowHammer
and RowPress are two prominent examples of DRAM read dis-
turbance phenomena where a victim DRAM row experiences
bitflips when a nearby aggressor DRAM row is 1) repeatedly
activated (i.e., hammered) [5–7] or 2) kept open for a long pe-
riod (i.e., pressed) [4,71]. Many prior works [3–5,7,9,41,42]
experimentally demonstrate that read disturbance significantly
worsens as DRAM manufacturing technology scales to smaller
node sizes in real DRAM chips. For example, chips manufac-
tured in 2018-2020 can experience RowHammer bitflips at an
order of magnitude fewer row activations compared to the chips
manufactured in 2012-2013 [3]. To ensure the robustness of
modern and future DRAM-based systems, it is critical to 1)
develop a rigorous understanding of known read disturbance
effects like RowHammer and RowPress and 2) discover and
analyze previously unknown read disturbance phenomena.
In this work, for the first time, we experimentally demonstrate
a new widespread read disturbance phenomenon, ColumnDis-
turb, in 216 commercial off-the-shelf (COTS) DDR4 and 4
HBM2 DRAM chips from all three major DRAM manufactur-
ers. We show that it is possible to disturb many DRAM cells
that are on the same DRAM column.1 ColumnDisturb mani-
fests as bitflips in DRAM cells across as many as three DRAM
subarrays (e.g., up to 3072 DRAM rows in tested DDR4 chips)
concurrently when we hammer or press an aggressor row.
Fig. 1 highlights the conceptual difference between Column-
Disturb (Fig. 1c) and RowHammer & RowPress (Fig. 1b) in
modern DRAM chips (Fig. 1a).
RowHammer & RowPress affect a few rows within the
same subarray (e.g., two victim rows in Fig. 1b). RowHam-
mer & RowPress happen because rows are too close to each
other, affecting cells that are in closeby rows (i.e., closeby
rows are victims). In contrast, ColumnDisturb is fundamentally
different: ColumnDisturb affects thousands of rows, includ-
ing distant ones, across both the same subarray and multiple
neighboring subarrays (e.g., Fig. 1c). This is because access-
ing a DRAM row (i.e., DRAM row activation) perturbs many
columns, and columns of a single DRAM row span multiple
subarrays. Thus, ColumnDisturb affects cells that are vertically
1In DRAM, each logical column is implemented using multiple physical
columns, i.e., bitlines (e.g., [72–74]). In this paper, we use the term DRAM
column to refer to a single physical column/bitline.
1
arXiv:2510.14750v2 [cs.AR] 17 Oct 2025
Victim
Column
Subarray
DRAM
Column
DRAM
Row
Sense
Amplifier
Aggressor
Row
Victim
Cell
(a) Modern DRAM array
architecture
(b) RowHammer & RowPress
(c) ColumnDisturb
Victim Column
Victim Column
Victim Row
Victim Row
Figure 1: (a) Modern DRAM array architecture, (b) RowHam-
mer & RowPress induce bitflips in a few neighboring rows of the
aggressor row, where rows are the victims, (c) ColumnDisturb, a
column-based read disturbance phenomenon that affects many
rows in multiple subarrays, where columns are victims.
connected to the same bitline (i.e., columns are victims).
Our experimental results (§4 and §5) and hypothetical ex-
planations (§4.6) for ColumnDisturb strongly suggest that Col-
umnDisturb induces bitflips due to bitline-voltage-induced dis-
turbance. In other words, “hammering” (repeatedly raising
and lowering) or “pressing” (keeping high or keeping low) a
bitline’s voltage is what leads to ColumnDisturb bitflips.
To empirically illustrate the ColumnDisturb phenomenon,
Fig. 2 shows the bitflips caused by ColumnDisturb, RowHam-
mer, RowPress, and by retention failures in the first 3072 rows
of a representative Samsung 16Gb A-die DDR4 DRAM module,
spanning the first three consecutive subarrays. The x-axis shows
the DRAM row address, and the y-axis shows the number of
ColumnDisturb, RowHammer, RowPress, and retention failure
bitflips in each row.2
Figure 2: Number of ColumnDisturb, RowHammer, RowPress,
and retention failure bitflips in three subarrays.
We observe that ColumnDisturb induces bitflips across three
consecutive subarrays, and these bitflips are not due to retention
failures. Nor are they due to RowHammer & RowPress, which
affect only immediate neighboring rows of the aggressor row.
Based on our rigorous experimental characterization of Col-
umnDisturb and its behavior under numerous parameters (i.e.,
temperature, data pattern, DRAM timing parameters, average
voltage level of the bitline, memory access pattern, and spatial
variation), we make 27 empirical observations and share 12 key
takeaway lessons. We highlight two of our major new results.
First, newer DRAM chips manufactured with smaller technol-
ogy nodes are increasingly more vulnerable to ColumnDisturb
across all three major manufacturers. For example, we find
that the minimum time to induce the first ColumnDisturb bitflip
reduces by up to 5.06x and 2.96x on average across all tested
modules. We observe that even in existing COTS DRAM chips,
ColumnDisturb induces bitflips within a nominal DDR4 refresh
window (e.g., in 63.6ms) in multiple cells from a single module.
Second, significantly more rows experience ColumnDisturb bit-
2Please see §4.2 for the detailed methodology and observations. We study
ColumnDisturb in great detail in §4 and §5.
flips than retention failures, and the number of ColumnDisturb
bitflips is much higher than retention failures in all tested re-
fresh intervals (i.e., 64ms, 128ms, 256ms, 512ms, and 1024ms)
across all tested DRAM modules. For example, with a 512ms
refresh interval and at 65 ◦C, ColumnDisturb induces 1.64x,
62.49x, and 152.66x more bitflips in 2, 6, and 232 more rows on
average for tested SK Hynix, Micron, and Samsung modules,
respectively, than retention failures.
Our observations show that ColumnDisturb has serious impli-
cations on the robustness of both 1) future systems and 2) exist-
ing retention-aware heterogeneous refresh mechanisms [75–88].
First, due to continuously shrinking DRAM node size, more
ColumnDisturb bitflips may manifest in the standard refresh
window in future DRAM chips, jeopardizing the robustness of
future systems. We describe and evaluate two hardware tech-
niques that could mitigate ColumnDisturb bitflips at varying
expected performance, energy, and area overheads. A straight-
forward solution is to increase the DRAM refresh rate to accom-
modate ColumnDisturb bitflips that could happen in the stan-
dard refresh window. However, this straightforward solution
reduces system throughput by 42.1% and increases system en-
ergy consumption by 67.5%. We instead propose to intelligently
and timely refresh only the victim rows that are vulnerable to
ColumnDisturb. We show that our improved solution reduces
the straightforward solution’s 1) system throughput overhead by
70.5% and 2) energy overhead by 73.8%. Second, we evaluate
a retention-aware refresh mechanism (RAIDR) [76] and demon-
strate that its benefits drastically decrease in the presence of
ColumnDisturb (e.g., 53% decrease in performance) compared
to a baseline RAIDR that does not suffer from ColumnDisturb.
We call for future research to 1) fundamentally understand
ColumnDisturb at the device-level, 2) architect ColumnDisturb-
resilient, high-performance retention-aware refresh mecha-
nisms, and3) other innovative solutions to mitigate Column-
Disturb bitflips to enable ColumnDisturb-resilient, robust future
computing systems.
This paper makes the following key contributions:
• This is the first work to experimentally demonstrate a column-
based (i.e., bitline-based) read disturbance phenomenon in
modern DRAM chips, ColumnDisturb, and its widespread
existence in real DDR4 chips from all three major DRAM
manufacturers and HBM2 chips from Samsung.
• We provide an extensive experimental characterization of
ColumnDisturb on 216 real DDR4 and 4 HBM2 DRAM chips.
ColumnDisturb induces bitflips across three subarrays (e.g.,
3072 rows), greatly more than RowHammer & RowPress that
affect only a few rows in a single subarray.
• Our experimental results show that ColumnDisturb 1) gets
worse as DRAM technology scales down to smaller cell sizes,
2) already induces bitflips within a standard refresh window
in some existing DRAM chips, and 3) induces significantly
more bitflips in many more rows than retention failures, for a
given refresh interval.
• We describe and evaluate two solutions to ColumnDisturb for
future DRAM-based systems.
• We evaluate a retention-aware heterogeneous refresh mecha-
nism (RAIDR [76]) and show that ColumnDisturb can com-
pletely diminish the performance and energy benefits of such
mechanisms.
2
2. Background & Fundamentals
2.1. Dynamic Random Access Memory (DRAM)
DRAM Organization. Fig. 3 shows the hierarchical organiza-
tion of modern DRAM-based main memory. A module contains
one or multiple DRAM ranks that time-share the memory chan-
nel. A rank consists of multiple DRAM chips. Each DRAM
chip has multiple DRAM banks, each containing multiple sub-
arrays [89–91]. Within a subarray, DRAM cells form a two-
dimensional structure interconnected over bitlines and word-
lines. Each cell encodes one bit of data using the charge level
in its capacitor. A true cell encodes data ‘1’ as fully charged
(VDD) and data ‘0’ as fully discharged (GND), whereas an
anti-cell uses the opposite encoding. The data in the cell is
accessed through an access transistor, driven by the wordline
to connect the cell capacitor to the bitline. The row decoder
decodes the row address and drives one wordline. A row of
DRAM cells on the same wordline is referred to as a DRAM
row. DRAM cells in the same column are connected to the
sense amplifier via a bitline. To sense an entire row of cells,
each subarray has bitlines that connect to two rows of sense
amplifiers, one above and one below the subarray, which causes
neighboring subarrays to share half of the bitlines [92–100]. As
a result, a subarray’s even bitlines (highlighted in red for the
middle subarray, Subarray 1 in Fig. 3) are connected to one of
its neighboring subarrays’ odd bitlines (highlighted in red for
the top subarray, Subarray 0 in Fig. 3), while the odd bitlines of
Subarray 1 (highlighted in blue for Subarray 1) are connected
to the other neighboring subarray’s even bitlines (highlighted in
blue for the bottom subarray, Subarray 2 in Fig. 3) via sense am-
plifiers. This approach, known as the open-bitline architecture,
is widely adopted in high-density DRAM [92–100].
Memory Controller
Row Decoder
Subarray 0
Sense Amps
Subarray 1
Subarray N
Sense Amps
CPU
DRAM Module
DRAM Bank
DRAM Subarrays
capacitor
Wordline
Bitline
DRAM Cell
I/O Logic
Bank
Bank
Bank
Bank
Bank
Bank
Bank
Bank
Figure 3: Organization of modern DRAM-based main memory.
DRAM Access. Accessing DRAM consists of two steps. First,
the memory controller issues an ACT (activate) command to-
gether with a row address to the bank. The row decoder drives
the wordline of that row to activate the row (i.e., enables the
access transistors). Data is then transferred from the DRAM
cells in the row to the row buffer through the bitlines. Second,
the memory controller sends a PRE (precharge) command to
close the activated row after waiting for tRAS (i.e., the minimum
time between opening a row with an ACT command and closing
the row with a PRE command). Before accessing another row in
the same bank, the memory controller must wait for tRP (i.e., the
minimum time between sending a PRE command and opening a
row with an ACT command).
2.2. DRAM Read Disturbance
Read disturbance is the phenomenon that reading data from
a memory or storage device causes physical disturbance (e.g.,
voltage deviation, electron injection, electron trapping) on an-
other piece of data that is not accessed but located physically
nearby the accessed data. Two prime examples of read distur-
bance in modern DRAM chips are RowHammer [4–7,9,101]
and RowPress [4,102], where repeatedly accessing (hammering)
or keeping active (pressing) a row induces bitflips in physically
nearby rows, respectively. For read disturbance bitflips to occur,
1) the aggressor row needs to be activated more times than a
certain threshold value [7], and/or 2) the aggressor row needs
to be open for a long period of time (i.e., tAggOn > tRAS) [4].
2.3. DRAM Refresh and Retention Failures
DRAM Refresh. DRAM cell capacitors inherently lose charge
over time, potentially resulting in data corruption [76,85,86].
A DRAM cell’s retention time defines how long it can reliably
store data and typically varies between cells from millisec-
onds to many hours [76,84–86,103–106]. To prevent retention
failures from appearing, the memory controller periodically
restores each DRAM row’s charge level by sending a REF (re-
fresh) command every tREFI, e.g., [47,72,85,107,108], which
is scheduled according to a timing parameter called the refresh
window (tREFW) [72,85,107,108].
DRAM Retention Failures. A modern DRAM cell with Buried
Channel Access Transistor (BCAT) [109, 110] has multiple
leakage paths [111–113]. Prior works show that the major
leakage paths include Gate-Induced Drain Leakage and Junction
Leakage [109,110,113–116]. These major leakage paths are
all from the cell to the substrate of the access transistor, which
is biased at a negative voltage to reduce subthreshold leakage
through the transistor [113], causing true-cells storing data ‘1’
much more likely to experience retention failures than ‘0’ [85,
86,117]. As DRAM technology node size reduces, the closer
proximity between the capacitor contact and the bitline also
creates potential leakage paths [2].
3. Evaluation Infrastructure & Methodology
We describe our commercial off-the-shelf (COTS) DRAM test-
ing infrastructure (§3.1) and our testing methodology (§3.2).
3.1. COTS DRAM Testing Infrastructure
We conduct COTS DRAM chip experiments using DRAM Ben-
der [118,119] (built upon SoftMC [120,121]), an FPGA-based
DDR4 and HBM2 testing infrastructure that provides precise
control of DDR4 and HBM2 commands issued to a DRAM
module. Fig. 4 shows one of our experimental setups on DDR4
modules that consists of four main components: 1) a host ma-
chine that generates the test program and collects experiment
results, 2) an FPGA development board [122], 3) a tempera-
ture sensor and heater pads pressed against the DRAM chips
to maintain a target temperature level, and 4) a temperature
controller [123] to control the temperature. Fig. 5 shows our
laboratory comprising many DDR4/HBM2 testing platforms.
Real DDR4 and HBM2 DRAM Chips Tested. Table 1 pro-
vides 216 real DDR4 chips from 28 modules and 4 HBM2 chips
that we characterize.3,4
Logical-to-Physical Row Mapping. DRAM manufacturers
use mapping schemes to translate logical addresses to physical
row addresses [7,14,32,42,85,98,106,128–134]. To account
3The technology node size of a DRAM chip is usually not publicly available.
Prior works [3, 4, 71, 124–126] assume that for a given chip manufacturer
and chip density, the alphabetical order of die revision codes may provide an
indication of technology node advancement.
4We provide much more detail on the tested DRAM chips in the extended
version of this paper [127].
3
Figure 4: Our DRAM Bender [119] based experimental setup.
Figure 5: Our laboratory for real DRAM chip experiments.
for in-DRAM row address mapping and internal row remapping
(which are extensively discussed in [3, 47, 135]), we reverse
engineer the physical row address layout, following prior works’
methodology [3,4,47,71,135–138].
3.2. Experimental Methodology
Characterizing and understanding ColumnDisturb requires 1) re-
verse engineering the subarray boundaries because the columns
of a single DRAM row span three physically consecutive sub-
arrays, and thus, ColumnDisturb bitflips can manifest across
three consecutive subarrays (as shown in Fig. 2), 2) filtering out
retention failures, since ColumnDisturb induces bitflips within
intervals longer than the refresh window (tREFW), and 3) testing
many parameters to understand how operational conditions and
parameters affect the ColumnDisturb vulnerability.
Metrics. To characterize a DRAM module’s vulnerability to
ColumnDisturb, we examine three metrics: 1) minimum time
to induce the first ColumnDisturb bitflip in a subarray, 2) the
fraction of cells with bitflips in a subarray, and 3) the number
of rows that experience at least one ColumnDisturb bitflip in
a subarray (i.e., blast radius [3, 4, 7, 47, 50, 71, 125, 126, 136–
139]). A shorter time to induce the first bitflip indicates higher
vulnerability to ColumnDisturb. Higher values for the fraction
of cells with bitflips in a subarray and blast radius indicate
higher ColumnDisturb vulnerability.
Access Pattern. To induce ColumnDisturb bitflips in a subarray,
we perform the following key DRAM command sequence:
ACT RAgg
tAggOn
−−−→PRE
tRP
−→ACT RAgg
tAggOn
−−−→···
where RAgg is the aggressor row in the tested subarray, tAggOn is
the duration that the aggressor row stays open (i.e., the timing
delay between ACT RAgg and PRE) [4], and tRP is the timing
delay before issuing the next ACT RAgg command. We conduct
all experiments where RAgg is the middle row in the tested
subarray unless stated otherwise.
By repeatedly hammering or pressing RAgg with our access
pattern (e.g., the aggressor row in Fig. 1-c), we disturb DRAM
cells through the columns (e.g., victim columns in Fig. 1c),
Table 1: Summary of DDR4 and HBM2 DRAM chips tested.
Chip Mfr.
Module IDs
#Chips
Die Rev.
Density
Org.
H0,H1,H2
24
A
8Gb
x8
SK Hynix
H3,H4,H5,H6
32
D
8Gb
x8
H7
8
A
16Gb
x8
H8,H9
16
C
16Gb
x8
M0
8
B
4Gb
x8
M1,M2,M3
24
R
8Gb
x8
Micron
M4, M5
16
B
16Gb
x8
M6, M7
8
E
16Gb
x16
M8,M9,M10,M11
32
F
16Gb
x8
S0, S1
16
A
16Gb
x8
Samsung
S2, S3
16
B
16Gb
x8
S4, S5
16
C
16Gb
x16
Samsung
HBM2 Chips
4
N/A
N/A
N/A
a We report “N/A” if the information is not publicly available.
since a row drives its content to all columns in a subarray after
a successful ACT command. As a result, hammering repeatedly
raises and lowers the voltage of the columns, while pressing
keeps the voltage consistently high or low. We conduct all
experiments using the same access pattern and record the bitflips
in the aggressor row’s subarray unless stated otherwise.
Algorithm to Find Time to Induce the First Bitflip. For every
parameter we evaluate, we determine the minimum number of
ACT commands (i.e., hammer count) required to induce the first
bitflip per tested subarray, using the bisection-method algorithm
used by prior works [4,71,102,124,126,136,138]. We terminate
the search when the difference between the current and previous
minimum hammer count measurements is smaller than 1% of
the previous measurement. During these experiments, we do not
issue any REF commands for 512 ms (i.e., the refresh interval is
512 ms). If no ColumnDisturb bitflips are observed within 512
ms in a given subarray, we terminate the search for that subarray
and proceed to the next one. For every tested subarray, we repeat
the search five times and convert the minimum hammer count
value across five iterations to time.
DRAM Subarray Boundaries. Prior works [138, 140–146]
demonstrate that real DRAM chips are capable of performing
the RowClone operation [90], an in-DRAM data copy operation
within a subarray by performing two consecutive row activa-
tions, which leads to data in the first activated row (i.e., source
row) to be copied to the second activated row (i.e., destination
row) via the sense amplifiers connected to both rows. We re-
peatedly perform the RowClone operation for every possible
source and destination row address in each tested bank. When
we observe that the destination row gets the same content as
the source row after the RowClone operation, we conclude that
the source row and the destination row are in the same subarray.
Based on this observation, we reverse engineer the subarray
boundaries and determine which rows are in the same subarray.
Filtering Out Retention and RowHammer & RowPress
Failures. We experimentally identify cells that experience
retention failures during our tests and exclude them from the
set of bitflips caused by ColumnDisturb in two steps. First,
we follow the state-of-the-art retention time test methodol-
ogy [84–86, 117] for five different data patterns. Second, to
cover the worst-case in the presence of the variable retention
time phenomenon [85, 86, 115, 116], we repeat each test 50
times and draw our conclusions based on the lowest observed
retention time of a cell. To filter out RowHammer and Row-
Press bitflips, in every experiment, we exclude eight nearest
4
physical victim rows of the aggressor row when we read from
a subarray, since RowHammer & RowPress induce bitflips in
nearby neighboring rows of the aggressor row. We exclude eight
nearest neighbor rows of the aggressor, even though we experi-
mentally find that RowHammer & RowPress induce bitflips in
only +1/-1 neighboring rows. We add this guardband (i.e., six
additional rows to exclude) because the state-of-the-art industry
solutions to DRAM read disturbance support refreshing Row-
Hammer & RowPress bitflips in up to eight nearest neighboring
rows [41,47,147–149].5
Eliminating Other Interference Sources. We eliminate other
potential sources of interference by taking two measures, similar
to prior works [3, 4, 47, 71, 136]. First, we disable periodic
refresh to prevent refreshing any rows so that we can observe
the DRAM inherent device-level behavior. Second, we verify
that the tested modules and chips have neither rank-level nor
on-die ECC [134,150,151].
Test Parameters. We explain the three common parameters
we use in testing and elaborate on the specific parameters of
each test in §4 and §5. 1) Data Pattern: We use five data pat-
terns 0x00, 0xAA, 0x11, 0x33, and 0x77, commonly used in
memory reliability testing [152] and by prior work on DRAM
characterization (e.g., [3,4,7,71,138]). We fill aggressor rows
with these data patterns while initializing victim rows with the
negated data pattern (e.g., if aggressors have 0x00, victim rows
are filled with 0xFF). 2) Temperature: We perform our experi-
ments at four temperature levels: 45◦C, 65◦C, 85◦C, and 95◦C.
We conduct all experiments at 85◦C unless stated otherwise. 3)
Aggressor Row On Time (tAggOn): We define tAggOn [4] as the
time an aggressor row stays open after each activation during a
ColumnDisturb test. We perform our experiments at four tAggOn
values: 36ns, 7.8ms , 70.2ms , and 1ms. We conduct all exper-
iments at tAggOn= 70.2ms
unless stated otherwise. 4) Refresh
Interval: We perform our experiments with various different
refresh intervals from 64ms to 16s.
4. Foundational Results
We demonstrate a new read disturbance phenomenon, Column-
Disturb, in real DDR4 and HBM2 DRAM chips. Column-
Disturb is a widespread read disturbance phenomenon in real
DRAM chips and worsens as DRAM technology scales down
to smaller node sizes (§4.1). We observe that ColumnDisturb
can induce bitflips in three consecutive subarrays by hammer-
ing or pressing an aggressor row (§4.2). We investigate the
characteristics of ColumnDisturb by analyzing the direction of
bitflips (§4.3), the effect of data pattern on a DRAM column
(§4.4), the effect of aggressor row on time (§4.5), the effect of
average voltage level on a DRAM column (§4.6), the number
of DRAM rows that experience ColumnDisturb bitflips (§4.7),
and ColumnDisturb on HBM2 DRAM chips (§4.8).
4.1. Prevalence & Scaling Characteristics
We study the vulnerability of each chip to ColumnDisturb. One
critical component of vulnerability is identifying the weakest
cell (i.e., the cell that fails earliest). Since ColumnDisturb is
effective at subarray granularity (see §1 and §4.2), we check
the subarray that the aggressor row (the middle row in the
5We also observe that when we only exclude two nearest neighbor rows, our
experimental results are essentially the same. Our extended version provides
more comprehensive analyses and results [127].
tested subarray6) belongs to and record the time to induce the
first ColumnDisturb bitflip in that subarray for each chip (see
§3.2 for more details on our complete methodology). In this
experiment, we test all subarrays in all banks across all tested
DRAM modules from all three major manufacturers using the
parameters with which a DRAM chip experiences the highest
ColumnDisturb vulnerability (determined through extensive
experiments in §4 and §5).
Fig. 6 shows a violin plot of the time to observe the first
ColumnDisturb bitflip in a subarray across all tested 46,080
subarrays (y-axis) for each different die revision and density pair
(x-axis). Each subplot is dedicated to a different manufacturer.
8Gb
A-die
8Gb
D-die
16Gb
A-die
16Gb
C-die
74.0
120
240
360
480
SK Hynix
4Gb
B-die
8Gb
R-die
16Gb
B-die
16Gb
E-die
16Gb
F-die
63.6
100
200
300
400
Micron
16Gb
A-die
16Gb
B-die
16Gb
C-die
88.5
160
240
320
400
Samsung
Time to induce
rst bit
ip in a subarray (ms)
DRAM Chip Density & Revision
Figure 6: Distribution of the time to induce the first ColumnDis-
turb bitflip in a subarray for different DRAM chip densities & die
revisions across three major manufacturers.
Observation 1. All tested 216 DDR4 chips are vulnerable to
ColumnDisturb.
We observe that in every DDR4 chip tested, there is at least
one DRAM cell that experiences a ColumnDisturb bitflip.
Observation 2. Newer chips tend to be more vulnerable to
ColumnDisturb bitflips.
We observe that in the same die density for each tested man-
ufacturer, more advanced technology nodes experience higher
ColumnDisturb vulnerability (i.e., lower time to induce the first
ColumnDisturb bitflip).7 For SK Hynix, for 8Gb and 16Gb
chip density, respectively, the minimum time to induce the first
ColumnDisturb bitflip in a subarray reduces by 5.06x and 1.29x
as die generation advances (i.e., 8Gb A- to D-die and 16Gb A-
to C-die). For Micron, for 16Gb chip density, from B- to F-die,
the minimum time to induce the first ColumnDisturb bitflip in a
subarray reduces by 2.98x. For Samsung, compared to 16Gb
A-die, the minimum time to induce the first ColumnDisturb
bitflip in a subarray is 2.50x lower in 16Gb C-die.
We hypothesize that as DRAM technology node advances,
the distance between the DRAM cell capacitor contact and
the bitline decreases, strengthening tunneling effects between
them [2]. Thus, DRAM chips become more vulnerable to
ColumnDisturb as DRAM technology node size reduces. §4.6
provides detailed analyses and hypotheses on the device-level
mechanisms of ColumnDisturb.
Observation 2 has serious implications for the futureas
DRAM technology node sizes will continue to reduce and the
time to induce the first ColumnDisturb bitflip will likely get
smaller. §6.1 discusses the implications of this observation.
6We test different aggressor row locations and observe the almost same
trend in §5.5.
7For a given manufacturer and die density, the later in the alphabetical
order the die revision code is, the more likely the chip has a more advanced
technology node [3,4,71].
5
Observation 3. There already are some real DRAM chips
where ColumnDisturb introduces bitflips within the nominal
refresh window under nominal operating conditions.
We observe that in a single 16Gb F-die Micron module, mul-
tiple cells (at least four cells) from multiple chips (at least three
chips) experience ColumnDisturb bitflips at 63.6ms and 85◦C,
within the nominal refresh window for DDR4 chips, tREFW. In
contrast, retention failures require at least 512ms (at 85◦C) to
manifest for the same module (see Fig. 11). We also observe
that the closest (farthest) victim row that experiences Column-
Disturb bitflips within tREFW is 374 (446) rows away from the
aggressor row where both victim row and the aggressor row are
in the same subarray.
Takeaway 1. ColumnDisturb is a widespread read distur-
bance phenomenon in real DRAM chips. ColumnDisturb
becomes worse as DRAM technology scales down to smaller
node sizes. ColumnDisturb can already induce bitflips within
the refresh window in current DRAM chips.
4.2. Disturbing DRAM Columns
Fig. 2 in §1 shows the ColumnDisturb, RowHammer, RowPress,
and retention failure bitflips from a single representative module
(S0; Samsung 16Gb A-die).8 For RowHammer (RowPress),
the same aggressor row (row 1536) is hammered (pressed, i.e.,
with tAggOn=70m s) for 16 seconds. For the retention failure
test, the bank remains idle (i.e., in precharged state) for 16
seconds. Dashed vertical lines indicate reverse-engineered sub-
array boundaries: row addresses 0-1023 belong to Subarray 0,
1024-2047 to Subarray 1, and 2048-3071 to Subarray 2. The
aggressor row (row 1536) is located in Subarray 1 (Aggressor
Subarray), and Subarrays 0 and 2 are physically adjacent to the
aggressor subarray (Neighboring Subarray).
Observation 4. ColumnDisturb induces bitflips across three
consecutive subarrays (3072 rows), causing bitflips in many
more rows than RowHammer and RowPress.
Al 3072 rows in three consecutive subarrays experience
ColumnDisturb bitflips. However, only the immediate neighbor
rows of the aggressor row 1536 (i.e., rows 1535 and 1537)
experience RowHammer & RowPress bitflips.9
This observation could have serious implications for the fu-
8We experiment with all 28 tested modules and observe very similar charac-
teristics to this representative Samsung module, S0.
9We verify that bitflips in the immediate neighbors of an aggressor row
(+/-1 rows of the aggressor row) are caused by RowHammer and RowPress
based on three observations. First, prior works on real DRAM chips [4,135,
153,154] demonstrate that by correctly reverse-engineering logical-to-physical
row mapping of DRAM chips, RowHammer & RowPress induce bitflips in
adjacent victim rows (+/-1 rows of the aggressor row) [4,135,153,154] due to
the underlying silicon-level characteristics of RowHammer & RowPress [154–
156]. Second, when we hammer different aggressor rows in the same subarray,
we observe high bitflip counts only in +1/-1 neighboring victim rows (i.e.,
7559 RowHammer bitflips and 5406 RowPress bitflips on average across +1
and -1 neighboring victim rows), while all other victim rows in the aggressor
subarray experience similar bitflip counts to each other (between 2353-3505
ColumnDisturb bitflips across all other victim rows, significantly less than
RowHammer & RowPress bitflips. Third, when we initialize victim rows with
an all-0 pattern, we observe only 0 to 1 bitflips in the +/-1 neighboring rows
(not shown). Since ColumnDisturb induces only 1 to 0 bitflips (§4.3), whereas
RowHammer & RowPress can induce both 0 to 1 and 1 to 0 bitflips (§4.3),
this third observation further indicates that RowHammer & RowPress induce
bitflips in only +/-1 rows.
ture, as refresh-based DRAM read disturbance mitigation tech-
niques consider a few neighbor rows (e.g., up to +4/-4 in in-
dustry solutions [41,47,147–149,157]) of the aggressor row as
victim rows, all in the same subarray, whereas ColumnDisturb
leads to bitflips in thousands of rows (i.e., all rows across three
consecutive subarrays).
We observe that subarrays that are neither the aggressor sub-
array nor physically-adjacent to the aggressor subarray do not
experience any ColumnDisturb bitflips (not shown in the figure).
This is due to the open-bitline DRAM architecture [92–100]:
two neighboring subarrays share half of their columns. Thus,
the aggressor subarray (Subarray 1) shares columns with both
the above subarray (Subarray 0) and the below subarray (Subar-
ray 2) (see Fig. 3). As a result, subarrays that do not share
columns with the aggressor subarray are not affected (see
Fig. 1c). We hypothesize that this observation shows Column-
Disturb is a column-based read disturbance phenomenon, since
only the cells sharing the same columns as the aggressor row
experience ColumnDisturb bitflips.
Observation 5. ColumnDisturb induces more bitflips in the
aggressor subarray’s rows than in those of neighboring subar-
rays.
On average, ColumnDisturb induces 1.45× more bitflips in
the aggressor subarray’s rows than in the neighboring subar-
rays’ rows. We hypothesize that all columns are affected in the
aggressor subarray because an activation causes perturbation in
all bitlines in the same subarray. In contrast, only half of the
columns in the neighboring subarray are connected to the ag-
gressor subarray, and hence an activation causes a perturbation
in half of the bitlines in the neighboring subarrays.
We also observe no overlap in the column indices of the
cells that experience only ColumnDisturb bitflips in Subarray
0 and Subarray 2 (neighboring subarrays of Subarray 1). We
hypothesize that this is because Subarray 1’s even columns
are shared with Subarray 0’s odd columns (e.g., shared blue
columns in the middle and top subarrays in Fig. 3) and Subarray
1’s odd columns are shared with Subarray 2’s even columns (e.g.,
red columns in the middle and bottom subarrays in Fig. 3). This
observation further supports our hypothesis that ColumnDisturb
is a column-based read disturbance phenomenon since only
cells that share columns with the aggressor row are affected.
Observation 6. For a given refresh interval, ColumnDisturb
induces many more bitflips than retention failures.
On average, ColumnDisturb induces 2942.68 bitflips per row
in the aggressor subarray and 2025.02 bitflips per row in the
neighboring subarrays, which are 7.07x and 4.87x more than
retention failure bitflips for a refresh interval of 16 seconds,
respectively.
Takeaway 2. ColumnDisturb is a column-based read distur-
bance phenomenon that disturbs cells via DRAM columns
(i.e., bitlines) and induces bitflips in thousands of rows (i.e.,
as many as all 3072 rows in three DRAM subarrays).
4.3. Bitflip Direction
We analyze the bitflip direction of ColumnDisturb versus re-
tention failures using one representative module (S0; Samsung
16Gb A-die).8 Fig. 7 shows the number of 1 to 0 (left) and 0 to
1 (right) bitflips in a subarray (y-axis) due to ColumnDisturb
6
and retention failure bitflips across five different refresh inter-
vals (x-axis). Error bars represent the range of minimum and
maximum number of bitflips across all tested subarrays.
Figure 7: Distribution of the number of 1 to 0 and 0 to 1 bitflips in
a subarray due to ColumnDisturb and retention failures.
Observation 7. ColumnDisturb induces only 1 to 0 bitflips,
which are the same direction as retention failures.
Across all tested refresh intervals and subarrays, the only
observed bitflip direction for ColumnDisturb is 1 to 0, which
is the same as retention failures. We do not observe any 0 to 1
ColumnDisturb bitflips. This is in stark contrast to RowHammer
and RowPress, which induce bitflips in both 1 to 0 and 0 to 1
directions [3,4,7,135,154], which we also observe in our tested
DRAM chips.
Observation 8. ColumnDisturb induces many more bitflips
than retention failures, especially at shorter refresh intervals.
On average, ColumnDisturb induces 11.77x, 7.02x, 4.86x,
3.97x, and 4.58x more bitflips than retention failures at refresh
intervals of 1s, 2s, 4s, 8s, and 16s, respectively.
Takeaway 3. DRAM is much more vulnerable to Column-
Disturb than retention failures, and the bitflip directionalities
of ColumnDisturb and RowHammer & RowPress are signifi-
cantly different.
4.4. Data Pattern in Aggressor Row
We investigate how the aggressor data pattern (i.e., data pat-
tern on the columns perturbed by the aggressor row) affects
the ColumnDisturb vulnerability using a single representative
module from each major DRAM manufacturer (S0, H0, and
M6). We test all subarrays in a single bank from each tested
module. For this experiment, we use all-1 as the data pattern
for all victim cells in a subarray since ColumnDisturb induces
only 1 to 0 bitflips and tAggOn=tRAS. We observe that not all
subarrays have the same number of rows (the number of rows
in a subarray across all tested modules ranges between 512 and
1024), and thus, we use a metric "fraction of cells with bitflips
in a subarray." Fig. 8 shows the fraction of cells that experience
bitflips in a subarray across all tested subarrays (y-axis) for five
different refresh intervals (x-axis) for three DRAM manufactur-
ers (subplot). Each line represents a different experiment: 1)
ColumnDisturb: AggDP=all-0, the aggressor row is initialized
with all-0, 2) ColumnDisturb: AggDP=all-1, the aggressor row
is initialized with all-1, and 3) RET, retention failures, where
all columns are at the precharged voltage (VDD/2) during the
entire experiment. The error band of a line shows the minimum
and maximum values across all tested subarrays.
Observation 9. ColumnDisturb induces many more bitflips
when the aggressor row data pattern is all-0 versus all-1.
For example, at 16s, all-0 aggressor pattern induces 1.15x,
Figure 8: Distribution of fraction of cells with bitflips in a subarray
due to ColumnDisturb for two different aggressor data patterns
and retention failures.
11.52x, and 2.86x more bitflips than the all-1 aggressor data
pattern for ColumnDisturb in SK Hynix, Micron, and Samsung
chips, respectively. We hypothesize that this occurs due to a
larger voltage level difference between the victim cell and the
perturbed columns (i.e., columns connected to the aggressor
row). This is because, when the aggressor data pattern is all-0,
all perturbed columns in the subarray are at GND (i.e., low
voltage) while the victim cells are at VDD, resulting in a VDD
difference. In contrast, when the aggressor data pattern is all-1,
both victim cells and all perturbed columns in a subarray are at
VDD, resulting in no voltage difference. We further analyze the
relationship between the perturbed column voltage level and
ColumnDisturb in §4.6.
Observation 10. When the aggressor data pattern is all-1
(same as victim data pattern), ColumnDisturb can induce
fewer bitflips than retention failures.
For example, in Micron, with a refresh interval of 16s, 2.73x
fewer DRAM cells in a subarray on average experience bitflips
with ColumnDisturb (when both the aggressor data pattern and
the victim data pattern are all-1) versus retention failures.
We hypothesize that this could be due to the reduced volt-
age level difference between a column and the victim cell. In
ColumnDisturb with all-1 aggressor and victim data pattern
tests, the voltage difference between the victim cell (VDD) and
the column (VDD) is zero. In contrast, in retention failure
tests, the column is held at VDD/2 while the cell remains at
VDD, resulting in a higher column-cell voltage difference (i.e.,
VDD/2) than ColumnDisturb with all-1 aggressor and victim
data pattern tests. As a result, having a lower voltage difference
between the victim cell and the column could lead to a lower
ColumnDisturb vulnerability. We provide a detailed analysis
and hypotheses for the effect of the voltage difference between
victim cells and perturbed columns in §4.6.
Takeaway 4. Data pattern in the aggressor row has a sig-
nificant effect on the number of observed ColumnDisturb
bitflips.
4.5. Aggressor Row on Time
We study the effect of the amount of time an aggressor row is
kept open (tAggOn) on ColumnDisturb bitflips.10 For this anal-
ysis, we use all-1 (0xFF) as the victim data pattern and all-0
(0x00) for the aggressor data pattern since these are the worst-
case data patterns that induce the most ColumnDisturb bitflips
in a DRAM chip. Fig. 9 shows the fraction of cells with bitflips
in a subarray observed in three representative modules, one from
10Keeping an aggressor row open for a long time (i.e., high tAggOn values)
perturbs all columns connected to the cells in the aggressor row, since the
columns remain driven to VDD or GND.
7
each tested manufacturer. Each line represents a different exper-
iment: 1) ColumnDisturb: tAggOn=36ns, where we keep the ag-
gressor row open for 36ns, 2) ColumnDisturb: tAggOn=70.2m s,
where we keep the aggressor row open for 70.2m s, and 3) RET,
retention failure, where the bank is idle (i.e., in precharged state)
during the experiment.
Figure 9: Distribution of fraction of cells with bitflips in a sub-
array due to ColumnDisturb for two different tAggOn values and
retention failures.
Observation 11. Fraction of cells with ColumnDisturb bit-
flips in a subarray significantly increases as tAggOn increases.
For example, at a refresh interval of 16s, increasing tAggOn
from 36ns to 70.2m s increases the number of ColumnDisturb
bitflips by 1.20x, 2.12x, and 2.45x for SK Hynix, Micron, and
Samsung modules, respectively. We hypothesize that this occurs
because a longer tAggOn keeps the perturbed columns at a low
voltage level (GND) for an extended period, which likely results
in an increased ColumnDisturb effect.
4.6. Average Voltage Level on Perturbed Columns
We observe that DRAM chips become significantly more vul-
nerable to ColumnDisturb 1) when the voltage level difference
between the victim cell and the perturbed columns (i.e., columns
perturbed by the aggressor row) is high (§4.4) and 2) when the
perturbed columns are kept active longer (§4.5), i.e., the volt-
age level difference between the victim cell and the perturbed
columns is kept high for an extended period. Based on our
observations in §4.4 and §4.5, we hypothesize that the voltage
level on the perturbed columns is a key parameter that affects
the ColumnDisturb vulnerability in DRAM.
To understand the effect of average column voltage level on
ColumnDisturb, we calculate the average voltage level on the
perturbed columns AVG(V COL) when we perform ColumnDis-
turb tests. Our access pattern, for every tAggOn+tRP time, (§3.2)
1) keeps the column voltage level at the aggressor data pattern
in the aggressor subarray, DPCOL, for tAggOn and 2) keeps the
column voltage level at precharged voltage (VDD/2) for tRP.
Based on the relationship between AVG(V COL) and DPCOL,
tAggOn,
VDD/2,
and tRP parameters,
we can calculate
AVG(V COL) as follows:
AVG(V COL) =
tAggOn∗DPCOL+VDD/2∗t
RP
tAggOn+t
RP
For
example,
assume
DPCOL=GND=0,
tAggOn=36ns,
precharged voltage level=VDD/2, and tRP=14ns (i.e., perturbed
columns are at GND for 36ns, and at VDD/2 for 14ns in every
50ns), AVG(V COL) becomes
36ns∗0+VDD/2∗14ns
36ns+14ns
= 0.14∗VDD.
Fig. 10 shows the fraction of cells with bitflips (y-axis) as
we sweep the average voltage level on the perturbed columns
(x-axis) across five refresh intervals (each colored line). In this
experiment, we test all subarrays in a single bank from three
DRAM modules, one from each tested manufacturer.
Figure 10: Fraction of cells with ColumnDisturb bitflips for differ-
ent average column voltage levels.
Observation 12. A DRAM chip becomes significantly more
vulnerable to ColumnDisturb as the average voltage level on
the perturbed columns decreases.
We observe that reducing the average column voltage level
from VDD to GND increases the fraction of cells that experi-
ence bitflips in a subarray by 1.65x, 26.31x, and 7.50x for SK
Hynix, Micron, and Samsung, respectively, at a refresh interval
of 16 seconds.
We provide two hypotheses to explain our observations. As
the perturbed column voltage decreases, the voltage differ-
ence across the victim cell and the perturbed column increases,
which 1) exacerbates subthreshold leakage of the access transis-
tor [112] and/or 2) exacerbates the dielectric leakage between
the victim capacitor and the perturbed column [2].11
Key Hypothesis. ColumnDisturb exacerbates subthreshold
leakage of the access transistor [112] and/or the dielectric
leakage between the capacitor and the bitline [2].
Takeaway 5. The voltage level on the column plays an impor-
tant role in ColumnDisturb’s device-level failure mechanisms.
Device-level investigation is necessary to develop a better
fundamental and first-principles understanding of Column-
Disturb.
4.7. Blast Radius
We study how many rows experience ColumnDisturb bitflips in
a subarray for a given refresh interval. To do so, we use a metric:
the number of rows with bitflips in a subarray, i.e., the number
of rows that experience at least one bitflip in a subarray, broadly
referred to as blast radius [3,4,7,47,50,71,125,126,136–139].
Fig. 11 shows the distribution of the number of rows with
bitflips in a subarray at 65◦C as a boxplot. Each subplot is
dedicated to a different manufacturer and shows how many
rows experience ColumnDisturb and retention failure bitflips in
a subarray (y-axis) for five different refresh intervals.
Observation 13. ColumnDisturb induces bitflips in signifi-
cantly more DRAM rows than retention failures.
For example, with a refresh interval of 1024ms, ColumnDis-
turb (Retention) induces bitflips in up to 52 (20), 353 (34), and
1022 (29) rows for SK Hynix, Micron, and Samsung modules,
respectively. For SK Hynix, Micron, and Samsung modules,
respectively, with a refresh interval of 512ms, ColumnDisturb
induces bitflips in 2, 6, and 232 rows on average, whereas only
two rows experience retention failures at most.
11We call for future device-level and silicon-level studies to develop a better
understanding of the inner workings of ColumnDisturb, just as device-level
studies (e.g., [155,156]) did for RowPress after the RowPress paper [4] demon-
strated for the RowPress phenomenon experimentally on real DRAM chips.
8
Figure 11: Distribution of the number of rows that experience Col-
umnDisturb versus retention failure bitflips for different refresh
intervals.
Observation 14. Number of rows with ColumnDisturb bit-
flips significantly increases as the refresh interval increases.
For example, from 512ms to 1024ms, for SK Hynix, Micron,
and Samsung, respectively, 9.73, 215.85, and 159.22 more rows
in a subarray experience ColumnDisturb bitflips, on average
across all tested subarrays. However, for retention failures, the
increase is significantly smaller than ColumnDisturb. From
512ms to 1024ms, up to 8.39 (2.76 on average) more rows
experience retention failure bitflips on average across all tested
modules.
Takeaway 6. Significantly more rows are vulnerable to Col-
umnDisturb than retention failures.
4.8. ColumnDisturb on COTS HBM Chips
So far, we characterize commodity DDR4 chips to demonstrate
and understand ColumnDisturb. In this section, we provide
a preliminary study on the vulnerability of real HBM2 chips
to ColumnDisturb. We test 4 HBM2 chips and analyze the
number of ColumnDisturb bitflips and retention failures across
all subarrays in a bank from three HBM2 chips. Fig. 12 shows
the number of bitflips in a subarray across all subarrays from a
single bank when 1) performing ColumnDisturb and 2) keeping
the bank in the idle state without performing any refresh (i.e.,
retention), for 1, 2, and 4 seconds. Error bars represent the
range of minimum and maximum number of bitflips across all
tested subarrays.
Figure 12: Number of ColumnDisturb and retention failure bitflips
for different refresh intervals in HBM2 chips.
Observation 15. HBM2 chips are vulnerable to ColumnDis-
turb, and ColumnDisturb induces many more bitflips than
retention failures.
We observe that across all four HBM2 chips tested, Col-
umnDisturb induces 1.61x, 2.08x, and 2.43x more bitflips than
retention failures in 1s, 2s, and 4s, respectively. We expect our
other observations for DDR4 chips (§4 and §5) will hold for
HBM2 chips as well, since the DRAM array is the same for
both, and ColumnDisturb happens at the array level.
Takeaway 7. ColumnDisturb is a widespread read distur-
bance phenomenon in DRAM chips: not only DDR4 chips,
but also HBM2 chips are vulnerable to ColumnDisturb.
5. In-Depth ColumnDisturb Analysis
This section further enhances our analysis of ColumnDisturb
by investigating parameters that have been shown to impact
both read disturbance and retention failure mechanisms: tem-
perature (§5.1), aggressor row on time (§5.2), access pattern
(§5.3), data pattern (§5.4), the location of the aggressor row
in a subarray (§5.5), and the effectiveness of Error Correcting
Codes (ECC) against ColumnDisturb (§5.6).12 We conduct our
experiments by testing all subarrays in all banks from all tested
modules using the parameters with which a DRAM chip experi-
ences the highest ColumnDisturb vulnerability (i.e., aggressor
data pattern=all-0, victim data pattern=all-1, tAggOn=70.2m s,
temperature= 85◦C), unless stated otherwise.
5.1. Temperature
We analyze the relationship between temperature and the Col-
umnDisturb vulnerability by evaluating three metrics for four
different temperature levels: 45◦C, 65◦C, 85◦C, and 95◦C.
Time to Induce the First ColumnDisturb Bitflip. Fig. 13
shows how the time to induce the first bitflip changes with
temperature using a box and whiskers plot. Each subplot is
dedicated to a different manufacturer.
Figure 13: Distribution of time to induce the first ColumnDisturb
bitflip at different temperatures.
Observation 16. Time to induce the first ColumnDisturb
bitflip significantly reduces as temperature increases.
Across all tested modules, time to induce the first ColumnDis-
turb bitflip consistently reduces as temperature increases. For
example, increasing temperature from 45◦C to 95◦C reduces
the average time to induce the first ColumnDisturb bitflip in a
subarray by 9.05x, 5.15x, and 1.96x for SK Hynix, Micron, and
Samsung, respectively.
Fraction of Cells with Bitflips in a Subarray. Fig. 14 shows
the fraction of cells with bitflips in a subarray (y-axis) due to
ColumnDisturb versus retention failures (color-coded) across
four temperature levels (x-axis) when the refresh interval is
512ms.
Observation 17. ColumnDisturb is more sensitive to temper-
ature than retention failures.
For example, in SK Hynix chips, when the temperature in-
creases from 85◦C to 95◦C, ColumnDisturb (retention failure)
bitflips increase by 72.96x (3.68x) on average. We also observe
that ColumnDisturb induces more bitflips than retention fail-
ures in all tested temperature levels across all DRAM modules.
For example, at 65◦C, ColumnDisturb induces 152.66x more
bitflips than retention failures in Samsung modules.
12We observe that the overall trends are consistent across the three evaluated
metrics: 1) time to induce the first ColumnDisturb bitflip in a subarray, 2) the
fraction of cells with ColumnDisturb bitflips in a subarray, and 3) the blast
radius. Due to space limitations, we provide the analysis of the first metric in all
experiments and provide a detailed analysis of all metrics only for temperature.
Our extended version provides comprehensive analyses of all metrics [127].
9
Figure 14: Distribution of fraction of cells with ColumnDisturb
and retention failure bitflips in a subarray at different tempera-
tures.
Blast Radius. Fig. 15 shows the blast radius effect (y-axis)
of ColumnDisturb and retention failures (i.e., the number of
rows in a subarray that experience at least one ColumnDisturb
and retention failure bitflip) for three manufacturers (rows of
subplots) and four temperature levels (columns of subplots).
The x-axis shows the refresh interval in milliseconds (ms).
Figure 15: Number of rows that experience at least one bitflip
(i.e., blast radius) due to ColumnDisturb and retention failures for
different refresh intervals and temperature levels.
Observation 18. ColumnDisturb induces bitflips in many
more rows than retention failures in all tested temperature
levels.
For example, even at a low temperature level, 45◦C, in Mi-
cron and Samsung, respectively, ColumnDisturb induces bitflips
in up to 39 and 150 rows in a subarray across all tested sub-
arrays. However, at most only a single row exhibits retention
failures for Micron, and none for Samsung. We observe that
across all tested temperature levels and refresh intervals, Col-
umnDisturb induces bitflips in up to 198x more DRAM rows
than retention failures.
Observation 19. The blast radius of both ColumnDisturb and
retention failures increase with higher temperature.
At 95◦C, across all tested subarrays, both mechanisms nearly
span an entire subarray (i.e., almost all rows of every tested
subarray experience bitflips), whereas ColumnDisturb begins
exhibiting a wide impact already at 65◦C (e.g., at least 5.33%
of subarrays in Samsung modules experience a blast radius of
1000).
Takeaway 8. As temperature increases, DRAM chips become
more vulnerable to ColumnDisturb.
5.2. Aggressor Row On Time
Fig. 16 shows the distribution of time to induce the first Col-
umnDisturb bitflip in a subarray across all tested subarrays for
four different tAggOn values: 36ns, 7.8ms, 70.2ms, and 1ms.
Figure 16: Time to induce first ColumnDisturb bitflip distribution
for four different tAggOn values.
Observation 20. Keeping a row open (tAggOn >> tRAS) is sig-
nificantly more effective than immediately closing it (tAggOn =
tRAS) in inducing the first ColumnDisturb bitflip in a subarray.
Increasing tAggOn from 36ns to 7.8m s reduces the average
time to induce the first ColumnDisturb bitflip by 1.68x, 1.22x,
and 2.03x in SK Hynix, Micron, and Samsung, respectively.
We observe that when tAggOn >> tRAS, the distributions are very
similar across all tested tAggOn values. We hypothesize that
increasing tAggOn decreases the average voltage level on DRAM
columns, resulting in higher ColumnDisturb vulnerability in
DRAM chips (i.e., less time to induce the first ColumnDisturb
bitflip in a subarray), as discussed in §4.6.
5.3. Access Pattern
Until now, we perform ColumnDisturb with a single aggressor
row, causing the column voltage to alternate between VDD/2
and GND or VDD, depending on the aggressor data pattern
(described in §3.2). To understand the effect of toggling the
column, we introduce a two-aggressor access pattern where two
aggressor rows are used with complementary data patterns:
ACT RAgg1
tAggOn
−−−→PRE
tRP
−→ACT RAgg2
tAggOn
−−−→···
In the single aggressor access pattern (§3.2), we use all-0 as
the aggressor data pattern. Thus, the column voltage transition
is {GND→VDD/2→GND···}. In the two-aggressor access
pattern, we use all-0 in the first aggressor (RAgg1) and all-1 in
the second aggressor (RAgg2), and thus, the voltage transition
on the column becomes {GND→VDD/2→VDD→VDD/2···}.
Fig. 17 shows the time to induce the first ColumnDisturb
bitflip for the two access patterns. In both experiments, victim
rows are initialized with all-1 as we observe only 1 to 0 bitflips
for ColumnDisturb (§4.3).
Figure 17: Distribution of time to induce the first ColumnDisturb
bitflip for two different access patterns.
Observation 21. Single-aggressor access pattern induces the
first ColumnDisturb bitflip in a subarray significantly faster
than the two-aggressor access pattern.
For SK Hynix, Micron, and Samsung modules, the single-
aggressor access pattern induces the first ColumnDisturb bitflip
1.83×, 1.92×, and 2.16× faster than the two-aggressor access
pattern, respectively.
10
This observation shows that toggling a column through the
sequence of GND→VDD/2→VDD (i.e., the two-aggressor ac-
cess pattern) is less effective than toggling it only between
VDD/2 and GND (i.e., the single-aggressor access pattern).
We hypothesize that this is because the two-aggressor access
pattern results in a higher average column voltage than the
single-aggressor access pattern, resulting in lower ColumnDis-
turb vulnerability in DRAM chips, i.e., longer time to induce
the first ColumnDisturb bitflip (§4.6).
Takeaway 9. Access pattern greatly affects a DRAM chip’s
vulnerability to ColumnDisturb.
5.4. Data Pattern
We analyze the effect of the aggressor and victim data patterns
using five aggressor and victim data pattern pairs. In this experi-
ment, victim cells are initialized with the negated aggressor data
pattern (e.g., if the aggressor data pattern is all-0, the victim
data pattern is all-1). We omit the inverse of the tested patterns
as either we 1) observe almost the same trend or 2) study it
before (all-1 in §4.4).
Time to Induce the First ColumnDisturb Bitflip. Fig. 18
shows the distribution of time to induce the first ColumnDisturb
bitflip in a subarray for five data patterns.
Figure 18: Distribution of time to induce the first ColumnDisturb
bitflip for five different aggressor and victim data pattern pairs.
Observation 22. Data pattern has a small effect on the time
to induce the first ColumnDisturb bitflip in a subarray.
Across all data patterns, the average time to induce the first
ColumnDisturb bitflip varies by at most 1.31x. We hypothesize
that this small variation (1.31x) suggests that bitline-to-bitline
interference does not strongly affect the time to induce the
first ColumnDisturb bitflip. This is because the weakest cell
determines the time to induce the first ColumnDisturb bitflip.
If the weakest cell is connected to a column written with logic-
0, it flips at nearly the same time regardless of the values in
neighboring columns/bitlines.
Total Number of Bitflips in a Subarray. Fig. 19 shows the
total ColumnDisturb bitflips in a subarray with a refresh interval
of 512ms. To maintain a reasonable experiment time, we test
three different aggressor and victim data pattern pairs.
Observation 23. Data pattern greatly affects total Column-
Disturb bitflips in a subarray: having more logic-0 values in
the perturbed columns generates more bitflips.
For example, in Samsung chips, an aggressor data pattern of
0x00 induces 2.04x more ColumnDisturb bitflips than an aggres-
sor data pattern of 0xAA on average. We hypothesize that this
is due to the bitflip direction characteristics of ColumnDisturb:
only victim cells that are initialized with logic-1 experience
bitflips (see §4.3). Thus, since victim cells are initialized with
0x00
0x11
0xAA
0
200
400
600
SK Hynix
0x00
0x11
0xAA
0
1000
2000
3000
Micron
0x00
0x11
0xAA
0
5000
10000
15000
Samsung
Number of Bit
ips
in a Subarray
Aggressor Pattern
(Victim rows have negated aggressor data pattern)
Figure 19: Total number of ColumnDisturb bitflips in a subarray
for three different aggressor and victim data pattern pairs.
the negated aggressor data pattern, more logic-0 values in the
aggressor pattern (which means more victim cells are initialized
with logic-1) lead to more ColumnDisturb bitflips.
5.5. Location of the Aggressor Row in a Subarray
To understand the effects of spatial variation on ColumnDisturb,
we analyze how the location of an aggressor row in a subarray
affects the ColumnDisturb vulnerability. To do so, we test
three aggressor row locations: 1) "Beginning": the first row
of the subarray, 2) "Middle": the middle row in the subarray,
and 3) "End": the last row of the subarray. Fig. 20 shows the
distribution of time to induce the first bitflip in a subarray across
DRAM subarrays (y-axis) based on the aggressor row’s location
in a subarray (x-axis).
Figure 20: Distribution of time to induce the first ColumnDisturb
bitflip based on the aggressor row’s location in a subarray.
Observation 24. The location of the aggressor row only
slightly affects the time to induce the first ColumnDisturb
bitflip.
We observe that across all tested manufacturers, there is
at most 1.08x variation on average when the location of the
aggressor row in a subarray changes.
5.6. Effectiveness of System-Level ECC
A system that uses Error-Correcting Codes (ECC) [134, 150,
158–167] can potentially protect against ColumnDisturb bitflips
if those bitflips are distributed across the DRAM chip such that
no ECC codeword contains more bitflips than ECC can correct.
Fig. 21 shows the distribution of ColumnDisturb bitflips across
8-byte data chunks for all tested DRAM modules for 512ms and
1024ms refresh intervals. We use 8-byte data chunks as DRAM
ECC typically uses 8-byte or larger datawords [134,150,151,
163,165,168–172].
Observation 25. ColumnDisturb induces more bitflips than
today’s widely-used ECCs can correct or even detect.
We observe that many 8-byte data chunks experience 3 (up to
15) ColumnDisturb bitflips, which the SECDED ECC [163,168–
171] cannot correct or detect, in Micron and Samsung chips.
Observation 26.
Relying only on ECC to mitigate all
ColumnDisturb-induced errors is likely very costly.
11
512
1024
100
101
102
SK Hynix
512
1024
100
101
102
103
104
105
Micron
512
1024
101
102
103
104
105
106
Samsung
Experiment Time (ms)
Number of 8-byte
Data Chunks at 65
C
Bit
ip Count:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Figure 21: Distribution of 8-byte data chunks with different Col-
umnDisturb bitflip counts.
Since ColumnDisturb can induce 15 bitflips in multiple 8-
byte data chunks, a (7,4) Hamming code [160] could correct
such bitflips with very large DRAM storage overheads (75%,
i.e., three parity bits for every four data bits). Thus, relying on
ECC alone to prevent ColumnDisturb bitflips is likely a very
expensive solution.
Effectiveness of On-Die ECC. We evaluate the effectiveness of
a (136,128) single-error-correcting (SEC) Hamming code [160]
(possibly implemented in DDR5 chips today [134,151,165,173,
174] for area efficiency and low cost overhead).13 This code cor-
rects any one bitflip in a 136-bit codeword. The observed bitflip
distributions (Fig. 21) for Micron and Samsung already exceed
the error correction capabilities of such codes. Moreover, the
SEC code can miscorrect codewords that have more than one
bitflip [173,174,176]. A miscorrection manifests as additional
bitflip(s) in the codeword (i.e., the SEC code induces another
bitflip by attempting to correct the erroneous codeword). We
randomly generate 10K erroneous codewords with two bitflips
(bitflip indices in the codeword are determined using a uniform
random number generator). The evaluated SEC code miscor-
rects 88.5% of the codewords, transforming a codeword with
two bitflips into another with three bitflips.
Observation 27.
Low-cost on-DRAM-die single-error-
correcting codes cannot correct all ColumnDisturb bitflips
and are likely to induce additional bitflips in an erroneous
codeword that has as few as only two bitflips.
Takeaway 10. Conventional DRAM ECC cannot protect
against ColumnDisturb bitflips, and an ECC scheme that can
protect against ColumnDisturb requires large overheads.
6. Implications
Our findings have strong implications for both 1) the robustness
of future systems and 2) the system speedup and DRAM en-
ergy saving benefits of existing retention-aware heterogeneous
refresh mechanisms (e.g., [75–88]). ColumnDisturb could jeop-
ardize the safety, security, and reliability of future computing
systems as more ColumnDisturb bitflips may manifest in the
standard refresh window due to continuously shrinking DRAM
technology node size. §6.1 describes and evaluates two tech-
niques that could mitigate ColumnDisturb bitflips at varying
performance, energy, and area overheads. In light of our new
findings, we evaluate a retention-aware heterogeneous refresh
mechanism (RAIDR [76]) and show that its benefits over the
baseline DRAM periodic refresh dramatically decrease (e.g., by
53%, see §6.2) because at long refresh windows a large major-
ity of DRAM rows (see §4.7 and §5.1) exhibit ColumnDisturb
bitflips.
13The exact designs of on-die ECC are kept strictly confidential by DRAM
manufacturers [150,175] and may be different from what we evaluate.
6.1. Implications for System Robustness
Observations 1, 2, 4, 13, 17 and 19 demonstrate that Column-
Disturb induces bitflips in victim rows that are thousands of
DRAM rows away from, and even in a different subarray from
the aggressor row. For example, the first bitflip we observe in
63.6ms is 374 rows away from the aggressor row. Unfortunately,
current RowHammer mitigations only refresh up to 8 neighbor-
ing rows [6,7,29,35,41,47,126,137,138,149,157,177–237]
as RowHammer or RowPress bitflips induce bitflips in immedi-
ate neighbors of the aggressor rows [3,4,7,71,124–126,135,
136,138,139,153,154,238–246]. Simply modifying commer-
cial or many other proposed read disturbance mitigations to
preventively refresh all potential victim rows (e.g., 3072 such
victims) of ColumnDisturb would induce prohibitive system
performance and energy overheads due to the latency and en-
ergy cost of performing thousands of preventive refresh op-
erations. For example, according to the latest DDR5 JEDEC
Standard [148], refreshing +/-4 physically adjacent neighbor-
ing rows incurs 2x more latency (tDRFMab = 560ns) than
refreshing +/-2 (tDRFMab = 280ns). Refreshing all rows in
three consecutive subarrays (e.g., 3072 rows) would take an
estimated, prohibitive, ∼215m s latency.14 Moreover, as DRAM
chip density increases, so does the number of rows in a subarray.
Therefore, securely mitigating ColumnDisturb bitflips would
likely require refreshing a few thousand rows before an aggres-
sor row is hammered or pressed to the point of failure. However,
reactively refreshing that many rows incurs prohibitive memory
access latencies. We describe and evaluate two solutions for
ColumnDisturb that proactively refresh potential victim rows
of ColumnDisturb without inducing prohibitive memory access
latencies.
Increasing DRAM Refresh Rate. A straightforward (and high-
overhead) solution for ColumnDisturb is to indiscriminately
increase the DRAM refresh rate. DRAM periodic refresh would
mitigate ColumnDisturb-induced bitflips if the refresh period
is set adequately short. The benefit of this solution is that it
can be applied immediately in today’s systems. However, as
many prior works [75–88] show, increasing the refresh rate
degrades system performance and energy efficiency because
more frequent refresh operations 1) increase memory access
latency, 2) reduce row buffer hit rate, 3) degrade bank-level
parallelism, and 4) cause more activity on the memory bus and
in DRAM banks. For example, for a 32 Gb DDR5 chip [147,
247], reducing the default (all bank) refresh period (REFab)
from 32 ms to 8 ms (i.e., 4x higher refresh rate) increases the
DRAM throughput loss due to periodic refresh operations from
10.5% to 42.1%15 and relative energy consumed by refresh
operations for an otherwise idle DRAM chip increases from
25.1% to 67.5%.16
We believe that indiscriminately increasing the DRAM re-
14Assuming 1) refreshing eight rows takes 560 ns [148], which means 70 ns
to refresh one row and 2) rows across three subarrays are refreshed sequentially.
In this case, refreshing 3072 rows requires 70×3072 = 215ms.
15DDR5 all bank refresh command latency (tRFC) is 410 ns for 32 Gb chip
density [147]. A DRAM chip cannot serve any memory request (read or write)
for tRFC after every refresh command, causing throughput loss. The memory
controller issues refresh commands more frequently as refresh period reduces:
once every 3.9 µs and 975 ns respectively for a refresh period of 32 ms and
8 ms.
16We estimate refresh and idle energy from manufacturer provided DRAM
IDD current values [247].
12
fresh rate to mitigate ColumnDisturb-induced bitflips is not a
performance- and energy-efficient solution. We describe a more
intelligent mechanism that selectively and timely refreshes only
the victim rows that are disturbed by ColumnDisturb.
Proactively Refreshing ColumnDisturb Victim Rows
(PRVR). We propose proactively refreshing ColumnDisturb
victim rows to mitigate ColumnDisturb bitflips. The key idea
of PRVR is to refresh only the victim DRAM rows in three
subarrays: the aggressor DRAM row’s subarray and its two
neighboring subarrays. PRVR refreshes each of the N victim
DRAM rows across three subarrays (e.g., for subarray size =
1024, N = 3072) once before the aggressor row is hammered or
pressed enough times to induce a bitflip. PRVR distributes the
refresh operations targeting these N rows over the time it takes
to induce the first bitflip in a subarray using ColumnDisturb,
similar to how a periodic refresh command refreshes a subset
of all DRAM rows in a DRAM chip.
PRVR is a higher-performance and more energy-efficient
solution than using a shorter DRAM refresh period. Assum-
ing a default refresh period of 32 ms, a time to induce the first
ColumnDisturb bitflip of 8 ms, and N = 3072 , we analytically
estimate PRVR’s DRAM throughput and energy benefits over
using a fixed, 8 ms refresh period. PRVR reduces 8 ms refresh
period’s throughput loss by 70.5% and refresh energy consump-
tion by 73.8% assuming one DRAM row from each DRAM
bank is continuously hammered to induce ColumnDisturb bit-
flips in a 32 Gb DDR5 chip.17 We conclude that proactively
refreshing ColumnDisturb victim rows could prevent Column-
Disturb bitflips at much smaller performance and energy over-
heads than simply reducing the DRAM refresh period.
6.2. Implications on Retention-Aware Refresh
Retention-aware heterogeneous refresh mechanisms [75–88]
leverage the heterogeneity in DRAM cell data retention time
to reduce the number of DRAM row refresh operations while
maintaining data integrity. These mechanisms typically classify
a DRAM row as weak if any of its cells exhibit a retention
failure during a relatively short refresh window (e.g., 64 ms). In
contrast, a DRAM row that reliably retains data over a much
longer time window (e.g., 1024 ms) is considered strong. Con-
sequently, the greater the proportion of strong rows in a DRAM
chip, the higher the performance and energy reduction pro-
vided by a retention-aware heterogeneous refresh mechanism.
ColumnDisturb greatly reduces the benefits of retention-aware
heterogeneous refresh mechanisms because ColumnDisturb ren-
ders many more rows weak than retention failures alone at a
given refresh window, as observations 6, 18, and 19 demon-
strate.
Fig. 22 shows the number of DRAM row refresh operations
(normalized to the number of refresh operations performed by
DDR4 64 ms periodic refresh) as the proportion of weak rows
in a chip varies. We plot four lines, one each for four strong row
retention time values: 128 ms, 256 ms, 512 ms, and 1024 ms.
A circle, diamond, and a square on a line show the number
of refresh operations needed for empirically observed average
proportion of retention-weak rows across all tested modules, av-
erage proportion of ColumnDisturb-weak rows across all tested
17System integration and rigorous evaluation of PRVR (e.g., using worst-case
ColumnDisturb access patterns) is out of this paper’s scope. We expect future
work to build on the key idea of PRVR and enable new efficient and effective
ColumnDisturb mitigation mechanisms.
modules, and the maximum proportion of ColumnDisturb-weak
rows across all tested modules, respectively. A smaller y-axis
value indicates smaller expected retention-aware heterogeneous
refresh mechanism speedup and energy benefits.
0.0
0.2
0.4
0.6
0.8
1.0
Proportion of Weak Rows
0.00
0.25
0.50
0.75
1.00
1.25
Number of Refresh Operations
Normalized to
64 ms Periodic Refresh
DDR4 Periodic Refresh (64 ms)
Strong Row Retention Time
128ms
256ms
512ms
1024ms
Empirical Weak Row Fraction
Retention
ColumnDisturb (Mean)
ColumnDisturb (Max.)
Figure 22: Number of DRAM row refresh operations needed. We
plot one line each for four strong row retention times of 128, 256,
512, and 1024 milliseconds.
We make two key observations. First, a relatively large
strong row retention time can significantly increase the expected
speedup and energy benefits of retention-aware refresh mech-
anisms. For example, for the empirically observed average
retention-weak row proportion (circles), number of refresh op-
erations reduces by 43.1% when a strong row retention time
of 1024 ms is used instead of a strong row retention time of
128 ms. Second, ColumnDisturb significantly reduces the ex-
pected speedup and energy benefits of retention-aware hetero-
geneous refresh mechanisms, especially at high strong row re-
tention times where the empirical proportion of ColumnDisturb-
weak rows greatly exceeds that of retention-weak rows. For
example, in the presence of ColumnDisturb, for a strong row
retention time of 1024 ms, the number of refresh operations in-
creases by 3.02× (purple diamond in Fig. 22) on average and by
up to 14.43× (purple square in Fig. 22), compared to a baseline
case without ColumnDisturb (i.e., only retention failures are
present) across all tested modules.
Takeaway 11. ColumnDisturb significantly degrades the
expected speedup and energy saving benefits of retention-
aware heterogeneous refresh mechanisms.
To quantitatively demonstrate the speedup and energy bene-
fits of retention-aware heterogeneous refresh mechanisms, we
evaluate a state-of-the-art retention-aware refresh mechanism,
RAIDR [76], building on the Self-Managing DRAM (SMD)
framework [248,249].18 We evaluate two variants of RAIDR
where 1) we store weak row addresses in a space-efficient
(low-area-overhead) Bloom Filter [254–256] sized 8Kb with 6
hash functions and 2) we store weak row addresses in a space-
inefficient (high-area-overhead) bitmap19 indexed using the row
address (1 bit per DRAM row, 2 Mb for a 16 GiB DDR4 mod-
ule). We refresh a weak row every 64 ms and a strong row every
1024 ms. We evaluate a hypothetical DRAM system config-
uration called No Refresh where we do not issue any refresh
operations to show the headroom for system speedup and energy
18We use Ramulator [250–253] and simulate a modern computing system
(configured as the baseline system described in Table 1 of [248]). We use
20 highly-memory-intensive multi-programmed workload mixes, each com-
prised of four single-core workloads each with last-level cache misses-per-kilo-
instructions ≥10. We warm-up the caches by fast-forwarding 100 million (M)
instructions. We run each multi-core simulation until each core executes at least
500M instructions.
19Storing weak row addresses in a bitmap allows the retention-aware hetero-
geneous refresh mechanism to identify all weak rows accurately [86]. Compared
to the Bloom Filter, using a bitmap has higher weak row identification accuracy
but incurs higher storage overheads.
13
1e-05
5e-05
1e-04
5e-04
0.1%
0.2%
0.3%
0.4%
0.0
0.5
1.0
Speedup
Normalized to
No Refresh
No Refresh
example Micron module (M8)
retention-weak rows
31 percentage point reduction
Use an 8K-bit Bloom Filter with 6 hash functions
1e-05
5e-05
1e-04
5e-04
0.1%
0.2%
0.3%
0.4%
0.5%
1.0%
2.0%
3.0%
4.0%
5.0%
10%
20%
30%
40%
50%
Proportion of Weak Rows
0.0
0.5
1.0
example Micron module (M8)
retention-weak rows
example Micron module (M8)
ColumnDisturb-weak rows
53 percentage
point reduction
Store all weak row addresses in a bitmap
Figure 23: Speedup provided by RAIDR [76] over the proportion of weak rows in a DRAM module. Left and right subplots shows
speedup for a low-area-overhead and high-area-overhead implementation, respectively.
consumption improvements of retention-aware heterogeneous
refresh mechanisms.
Fig. 23 shows the system speedup provided by retention-
aware refresh (in weighted speedup) on average across all 20
four-core workloads normalized to the speedup of No Refresh.
The left and right subplots, respectively, show the performance
of the RAIDR variant that uses a Bloom Filter and the RAIDR
variant that stores each weak row’s address in a bitmap. The
x-axis shows the evaluated proportion of weak row values in
the simulated DRAM system. We annotate the speedup of
retention-aware refresh for empirically observed proportion of
1) retention-weak rows (purple ×) and 2) ColumnDisturb-weak
rows (green ×) in one Micron DDR4 module.
We make two key observations. First, ColumnDisturb can
completely negate the performance and energy (not shown in
the figure) benefits of space-efficient (using a Bloom Filter) im-
plementations of retention-aware heterogeneous refresh mech-
anisms. From Fig. 23 (left) we observe that as the proportion
of weak rows increases from 1.00e-04 to 0.20% (by 20×), the
speedup and energy benefits reduce to a point where they are al-
most completely eliminated (≈99 percentage point speedup and
energy benefit reduction). The 20×increase in the proportion
of weak rows is well within our empirical observations: across
all tested modules, the proportion of weak rows increased by
up to 198× (Fig. 15).
As a concrete example, for the annotated Micron module (pur-
ple ×), the speedup and energy consumption benefits reduce
by 31 percentage points and 32 percentage points (not shown),
respectively, as the proportion of weak rows increases to accom-
modate ColumnDisturb-weak rows. Second, ColumnDisturb
can greatly reduce the speedup and energy benefits even of
space-inefficient (using a bitmap) implementations of retention-
aware heterogeneous refresh mechanisms. From Fig. 23 (right),
we observe that for the annotated Micron DDR4 module, the
speedup and energy consumption benefits reduce by 53 and 50
percentage points (not shown), respectively.
Takeaway 12. ColumnDisturb can completely negate the per-
formance and energy benefits of low-area-overhead retention-
aware heterogeneous refresh mechanisms. ColumnDisturb
greatly reduces the benefits of high-area-overhead retention-
aware heterogeneous refresh mechanisms.
We conclude that the proportion of strong DRAM rows (that
can endure high retention times without a bitflip) for a given
refresh window significantly reduces with ColumnDisturb. This
reduction greatly hinders (and can completely eliminate) the per-
formance and energy benefits of retention-aware heterogeneous
refresh mechanisms.
7. Related Work
To our knowledge, this is the first work to experimentally
demonstrate and characterize ColumnDisturb, a column-based
read-disturb phenomenon in real DRAM chips.
DRAM Read Disturbance Characterization. Many works [3,
4,7,71,124–126,135,136,138,139,153,154,238–246] exper-
imentally demonstrate how a real DRAM chip’s read distur-
bance vulnerability varies with 1) DRAM refresh rate [7,41,47],
2) the distance between aggressor and victim rows [3,7,139],
3) DRAM generation and technology node [3, 7, 47, 71],
4) temperature [71, 240], 5) time the aggressor row stays ac-
tive [4,8,71,102,135,153,154,240,244,246], 6) location of the
victim cell [8,71,138,244], 7) wordline voltage [136], 8) supply
voltage [245], 9) reduced DRAM timing parameters [124,126],
and 10) time [125]. None of these works analyze or demonstrate
ColumnDisturb, a column-based read disturbance phenomenon.
DRAM Retention. Many works experimentally study DRAM
data retention [84,85,104,117,132,159,162,166,224,257,258].
Several works [75–88] present various error profiling algorithms
to identify data retention failures when reducing the refresh rate
and propose retention-aware heterogeneous refresh mechanisms
to alleviate refresh overheads. However, we show that Column-
Disturb can significantly amplify the bitflips across subarrays
and can cause significant performance and area overheads to
retention-aware heterogeneous refresh mechanisms.
Bitline Disturbance in 4F2 VCT DRAM. Emerging high-
density 4F2 DRAM architectures with Vertical Channel Tran-
sistors (VCT) are known to be vulnerable to disturbances from
the bitline with a hammering-like access pattern [259–263].
However, since the device architecture of 4F2 VCT DRAM is
completely different from the contemporary COTS 6F2 DRAM
that we characterize in this work, we believe the error mecha-
nism they study (i.e., floating body effect [261,262]) is different
from the one(s) that causes ColumnDisturb (see §4 for our
analyses and hypotheses).
8. Conclusion
We present the first experimental demonstration and analysis
of a new read disturbance phenomenon, ColumnDisturb, in
modern DRAM chips: hammering or pressing an aggressor row
disturbs DRAM cells through a DRAM column and induces bit-
flips in DRAM cells sharing the same columns as the aggressor
row (across as many as three DRAM subarrays). Our experi-
mental characterization of 216 real DDR4 and 4 HBM2 DRAM
chips reveals that 1) ColumnDisturb is fundamentally different
from RowHammer & RowPress, 2) ColumnDisturb worsens
with DRAM technology scaling, 3) introduces bitflips within
the nominal refresh window under nominal operating condi-
tions in some real DRAM chips, and 4) DRAM is significantly
14
more vulnerable to ColumnDisturb than to retention failures.
We describe and evaluate two ColumnDisturb mitigation tech-
niques. We hope and believe that our detailed demonstration
and analyses will inspire future works on better understanding
ColumnDisturb at the device-level and mitigating it before it
becomes a bigger vulnerability in future DRAM chips.
Acknowledgments
We thank the anonymous reviewers of MICRO 2025 for feed-
back. We thank the SAFARI Research Group members (espe-
cially Konstantinos Sgouras and Harsh Songara) for construc-
tive feedback and the stimulating intellectual environment. We
acknowledge the generous gift funding provided by our indus-
trial partners (especially Google, Huawei, Intel, Microsoft),
which has been instrumental in enabling the research we have
been conducting on read disturbance in DRAM in particular
and memory systems in general [101]. This work was in part
supported by a Google Security and Privacy Research Award
and the Microsoft Swiss Joint Research Center.
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19
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ColumnDisturb: Understanding Column-based Read Disturbance in Real DRAM Chips and Implications for Future Systems ̇Ismail Emir Yüksel1 Ataberk Olgun1 F. Nisa Bostancı1 Haocong Luo1 A. Giray Ya ̆glıkçı1,2 Onur Mutlu1 1ETH Zürich 2CISPA We experimentally demonstrate a new widespread read disturbance phenomenon, ColumnDisturb, in real commodity DRAM chips. By repeatedly opening or keeping a DRAM row (aggressor row) open, we show that it is possible to disturb DRAM cells through a DRAM column (i.e., bitline) and induce bitflips in DRAM cells sharing the same columns as the aggressor row(across multiple DRAM subarrays). With ColumnDisturb, the activation of a single row concurrently disturbs DRAM cells across as many as three DRAM subarrays (e.g., up to 3072 DRAM rows in tested DRAM chips) as opposed to RowHammer & RowPress, which affect only a few neighboring rows of the aggressor row in a single subarray. We rigorously and comprehensively characterize ColumnDisturb and its characteristics under various operational conditions (i.e., temperature, data pattern, DRAM timing parameters, average voltage level of the bitline, memory access pattern, and spatial variation) using 216 DDR4 and 4 HBM2 chips from three major DRAM manufacturers. Among our 27 key experimental observations, we highlight two major results and their implications. First, ColumnDisturb affects chips from all three major DRAM manufacturers and worsens as DRAM technology scales down to smaller node sizes (e.g., the minimum time to induce the first ColumnDisturb bitflip reduces by up to 5.06x and 2.96x on average across all tested DRAM modules). We observe that, even in existing DRAM chips, ColumnDisturb induces bitflips within a standard DDR4 refresh window (e.g., in 63.6 ms) in multiple cells from one module. We predict that, as DRAM technology node size reduces, ColumnDisturb would worsen in future DRAM chips, likely causing many more bitflips in the standard refresh window. Second, beyond the standard refresh window, ColumnDisturb induces bitflips in many (up to 198x) more DRAM rows than retention failures. Therefore, ColumnDisturb has strong implications for existing retention-aware refresh mechanisms that aim to improve system performance and energy efficiency by leveraging the heterogeneity in DRAM cell retention times: our detailed analyses and cycle-level simulations show that ColumnDisturb greatly reduces and can completely diminish the performance and energy benefits of such mechanisms. Our results have serious implications for the robustness of future DRAM-based computing systems due to continued aggressive DRAM technology scaling. We describe and evaluate two solutions for mitigating ColumnDisturb bitflips and call for more research on the topic. 1. Introduction Dynamic random access memory (DRAM) [1] is the dominant main memory technology in nearly all modern computing systems due to its latency and cost characteristics. Continuing to increase DRAM capacity requires increasing the density of DRAM cells by reducing (i.e., scaling) the technology node size of DRAM, which in turn reduces DRAM cell size, cellto-cell spacing, and cell-to-bitline spacing [2]. Unfortunately, as a result of this aggressive technology node scaling, DRAM suffers from worsening read disturbance problems [3-6]. Read disturbance in modern DRAM chips [4-9] is a widespread phenomenon and is reliably used for breaking memory isolation [5,7,9-70], a fundamental building block of robust (i.e., safe, secure, reliable, available) systems. RowHammer and RowPress are two prominent examples of DRAM read disturbance phenomena where a victim DRAM row experiences bitflips when a nearby aggressor DRAM row is 1) repeatedly activated (i.e., hammered) [5-7] or 2) kept open for a long period (i.e., pressed) [4,71]. Many prior works [3-5,7,9,41,42] experimentally demonstrate that read disturbance significantly worsens as DRAM manufacturing technology scales to smaller node sizes in real DRAM chips. For example, chips manufactured in 2018-2020 can experience RowHammer bitflips at an order of magnitude fewer row activations compared to the chips manufactured in 2012-2013 [3]. To ensure the robustness of modern and future DRAM-based systems, it is critical to 1) develop a rigorous understanding of known read disturbance effects like RowHammer and RowPress and 2) discover and analyze previously unknown read disturbance phenomena. In this work, for the first time, we experimentally demonstrate a new widespread read disturbance phenomenon, ColumnDisturb, in 216 commercial off-the-shelf (COTS) DDR4 and 4 HBM2 DRAM chips from all three major DRAM manufacturers. We show that it is possible to disturb many DRAM cells that are on the same DRAM column.1 ColumnDisturb manifests as bitflips in DRAM cells across as many as three DRAM subarrays (e.g., up to 3072 DRAM rows in tested DDR4 chips) concurrently when we hammer or press an aggressor row. Fig. 1 highlights the conceptual difference between ColumnDisturb (Fig. 1c) and RowHammer & RowPress (Fig. 1b) in modern DRAM chips (Fig. 1a). RowHammer & RowPress affect a few rows within the same subarray (e.g., two victim rows in Fig. 1b). RowHammer & RowPress happen because rows are too close to each other, affecting cells that are in closeby rows (i.e., closeby rows are victims). In contrast, ColumnDisturb is fundamentally different: ColumnDisturb affects thousands of rows, including distant ones, across both the same subarray and multiple neighboring subarrays (e.g., Fig. 1c). This is because accessing a DRAM row (i.e., DRAM row activation) perturbs many columns, and columns of a single DRAM row span multiple subarrays. Thus, ColumnDisturb affects cells that are vertically 1In DRAM, each logical column is implemented using multiple physical columns, i.e., bitlines (e.g., [72-74]). In this paper, we use the term DRAM column to refer to a single physical column/bitline. 1 17 Oct 2025 Victim Column Subarray DRAM Column DRAM Row Sense Amplifier Aggressor Row Victim Cell (a) Modern DRAM array architecture (b) RowHammer & RowPress (c) ColumnDisturb Victim Column Victim Column Victim Row Victim Row Figure 1: (a) Modern DRAM array architecture, (b) RowHammer & RowPress induce bitflips in a few neighboring rows of the aggressor row, where rows are the victims, (c) ColumnDisturb, a column-based read disturbance phenomenon that affects many rows in multiple subarrays, where columns are victims. connected to the same bitline (i.e., columns are victims). Our experimental results (§4 and §5) and hypothetical explanations (§4.6) for ColumnDisturb strongly suggest that ColumnDisturb induces bitflips due to bitline-voltage-induced disturbance. In other words, "hammering" (repeatedly raising and lowering) or "pressing" (keeping high or keeping low) a bitline's voltage is what leads to ColumnDisturb bitflips. To empirically illustrate the ColumnDisturb phenomenon, Fig. 2 shows the bitflips caused by ColumnDisturb, RowHammer, RowPress, and by retention failures in the first 3072 rows of a representative Samsung 16Gb A-die DDR4 DRAM module, spanning the first three consecutive subarrays. The x-axis shows the DRAM row address, and the y-axis shows the number of ColumnDisturb, RowHammer, RowPress, and retention failure bitflips in each row.2 Figure 2: Number of ColumnDisturb, RowHammer, RowPress, and retention failure bitflips in three subarrays. We observe that ColumnDisturb induces bitflips across three consecutive subarrays, and these bitflips are not due to retention failures. Nor are they due to RowHammer & RowPress, which affect only immediate neighboring rows of the aggressor row. Based on our rigorous experimental characterization of ColumnDisturb and its behavior under numerous parameters (i.e., temperature, data pattern, DRAM timing parameters, average voltage level of the bitline, memory access pattern, and spatial variation), we make 27 empirical observations and share 12 key takeaway lessons. We highlight two of our major new results. First, newer DRAM chips manufactured with smaller technology nodes are increasingly more vulnerable to ColumnDisturb across all three major manufacturers. For example, we find that the minimum time to induce the first ColumnDisturb bitflip reduces by up to 5.06x and 2.96x on average across all tested modules. We observe that even in existing COTS DRAM chips, ColumnDisturb induces bitflips within a nominal DDR4 refresh window (e.g., in 63.6ms) in multiple cells from a single module. Second, significantly more rows experience ColumnDisturb bit2Please see §4.2 for the detailed methodology and observations. We study ColumnDisturb in great detail in §4 and §5. flips than retention failures, and the number of ColumnDisturb bitflips is much higher than retention failures in all tested refresh intervals (i.e., 64ms, 128ms, 256ms, 512ms, and 1024ms) across all tested DRAM modules. For example, with a 512ms refresh interval and at 65 ◦C, ColumnDisturb induces 1.64x, 62.49x, and 152.66x more bitflips in 2, 6, and 232 more rows on average for tested SK Hynix, Micron, and Samsung modules, respectively, than retention failures. Our observations show that ColumnDisturb has serious implications on the robustness of both 1) future systems and 2) existing retention-aware heterogeneous refresh mechanisms [75-88]. First, due to continuously shrinking DRAM node size, more ColumnDisturb bitflips may manifest in the standard refresh window in future DRAM chips, jeopardizing the robustness of future systems. We describe and evaluate two hardware techniques that could mitigate ColumnDisturb bitflips at varying expected performance, energy, and area overheads. A straightforward solution is to increase the DRAM refresh rate to accommodate ColumnDisturb bitflips that could happen in the standard refresh window. However, this straightforward solution reduces system throughput by 42.1% and increases system energy consumption by 67.5%. We instead propose to intelligently and timely refresh only the victim rows that are vulnerable to ColumnDisturb. We show that our improved solution reduces the straightforward solution's 1) system throughput overhead by 70.5% and 2) energy overhead by 73.8%. Second, we evaluate a retention-aware refresh mechanism (RAIDR) [76] and demonstrate that its benefits drastically decrease in the presence of ColumnDisturb (e.g., 53% decrease in performance) compared to a baseline RAIDR that does not suffer from ColumnDisturb. We call for future research to 1) fundamentally understand ColumnDisturb at the device-level, 2) architect ColumnDisturbresilient, high-performance retention-aware refresh mechanisms, and3) other innovative solutions to mitigate ColumnDisturb bitflips to enable ColumnDisturb-resilient, robust future computing systems. This paper makes the following key contributions: • This is the first work to experimentally demonstrate a columnbased (i.e., bitline-based) read disturbance phenomenon in modern DRAM chips, ColumnDisturb, and its widespread existence in real DDR4 chips from all three major DRAM manufacturers and HBM2 chips from Samsung. • We provide an extensive experimental characterization of ColumnDisturb on 216 real DDR4 and 4 HBM2 DRAM chips. ColumnDisturb induces bitflips across three subarrays (e.g., 3072 rows), greatly more than RowHammer & RowPress that affect only a few rows in a single subarray. • Our experimental results show that ColumnDisturb 1) gets worse as DRAM technology scales down to smaller cell sizes, 2) already induces bitflips within a standard refresh window in some existing DRAM chips, and 3) induces significantly more bitflips in many more rows than retention failures, for a given refresh interval. • We describe and evaluate two solutions to ColumnDisturb for future DRAM-based systems. • We evaluate a retention-aware heterogeneous refresh mechanism (RAIDR [76]) and show that ColumnDisturb can completely diminish the performance and energy benefits of such mechanisms. 2 2. Background & Fundamentals 2.1. Dynamic Random Access Memory (DRAM) DRAM Organization. Fig. 3 shows the hierarchical organization of modern DRAM-based main memory. A module contains one or multiple DRAM ranks that time-share the memory channel. A rank consists of multiple DRAM chips. Each DRAM chip has multiple DRAM banks, each containing multiple subarrays [89-91]. Within a subarray, DRAM cells form a twodimensional structure interconnected over bitlines and wordlines. Each cell encodes one bit of data using the charge level in its capacitor. A true cell encodes data '1' as fully charged (VDD) and data '0' as fully discharged (GND), whereas an anti-cell uses the opposite encoding. The data in the cell is accessed through an access transistor, driven by the wordline to connect the cell capacitor to the bitline. The row decoder decodes the row address and drives one wordline. A row of DRAM cells on the same wordline is referred to as a DRAM row. DRAM cells in the same column are connected to the sense amplifier via a bitline. To sense an entire row of cells, each subarray has bitlines that connect to two rows of sense amplifiers, one above and one below the subarray, which causes neighboring subarrays to share half of the bitlines [92-100]. As a result, a subarray's even bitlines (highlighted in red for the middle subarray, Subarray 1 in Fig. 3) are connected to one of its neighboring subarrays' odd bitlines (highlighted in red for the top subarray, Subarray 0 in Fig. 3), while the odd bitlines of Subarray 1 (highlighted in blue for Subarray 1) are connected to the other neighboring subarray's even bitlines (highlighted in blue for the bottom subarray, Subarray 2 in Fig. 3) via sense amplifiers. This approach, known as the open-bitline architecture, is widely adopted in high-density DRAM [92-100]. Memory Controller Row Decoder Subarray 0 Sense Amps Subarray 1 Subarray N Sense Amps CPU DRAM Module DRAM Bank DRAM Subarrays capacitor Wordline Bitline DRAM Cell I/O Logic Bank Bank Bank Bank Bank Bank Bank Bank Figure 3: Organization of modern DRAM-based main memory. DRAM Access. Accessing DRAM consists of two steps. First, the memory controller issues an ACT (activate) command together with a row address to the bank. The row decoder drives the wordline of that row to activate the row (i.e., enables the access transistors). Data is then transferred from the DRAM cells in the row to the row buffer through the bitlines. Second, the memory controller sends a PRE (precharge) command to close the activated row after waiting for tRAS (i.e., the minimum time between opening a row with an ACT command and closing the row with a PRE command). Before accessing another row in the same bank, the memory controller must wait for tRP (i.e., the minimum time between sending a PRE command and opening a row with an ACT command). 2.2. DRAM Read Disturbance Read disturbance is the phenomenon that reading data from a memory or storage device causes physical disturbance (e.g., voltage deviation, electron injection, electron trapping) on another piece of data that is not accessed but located physically nearby the accessed data. Two prime examples of read disturbance in modern DRAM chips are RowHammer [4-7,9,101] and RowPress [4,102], where repeatedly accessing (hammering) or keeping active (pressing) a row induces bitflips in physically nearby rows, respectively. For read disturbance bitflips to occur, 1) the aggressor row needs to be activated more times than a certain threshold value [7], and/or 2) the aggressor row needs to be open for a long period of time (i.e., tAggOn > tRAS) [4]. 2.3. DRAM Refresh and Retention Failures DRAM Refresh. DRAM cell capacitors inherently lose charge over time, potentially resulting in data corruption [76,85,86]. A DRAM cell's retention time defines how long it can reliably store data and typically varies between cells from milliseconds to many hours [76,84-86,103-106]. To prevent retention failures from appearing, the memory controller periodically restores each DRAM row's charge level by sending a REF (refresh) command every tREFI, e.g., [47,72,85,107,108], which is scheduled according to a timing parameter called the refresh window (tREFW) [72,85,107,108]. DRAM Retention Failures. A modern DRAM cell with Buried Channel Access Transistor (BCAT) [109, 110] has multiple leakage paths [111-113]. Prior works show that the major leakage paths include Gate-Induced Drain Leakage and Junction Leakage [109,110,113-116]. These major leakage paths are all from the cell to the substrate of the access transistor, which is biased at a negative voltage to reduce subthreshold leakage through the transistor [113], causing true-cells storing data '1' much more likely to experience retention failures than '0' [85, 86,117]. As DRAM technology node size reduces, the closer proximity between the capacitor contact and the bitline also creates potential leakage paths [2]. 3. Evaluation Infrastructure & Methodology We describe our commercial off-the-shelf (COTS) DRAM testing infrastructure (§3.1) and our testing methodology (§3.2). 3.1. COTS DRAM Testing Infrastructure We conduct COTS DRAM chip experiments using DRAM Bender [118,119] (built upon SoftMC [120,121]), an FPGA-based DDR4 and HBM2 testing infrastructure that provides precise control of DDR4 and HBM2 commands issued to a DRAM module. Fig. 4 shows one of our experimental setups on DDR4 modules that consists of four main components: 1) a host machine that generates the test program and collects experiment results, 2) an FPGA development board [122], 3) a temperature sensor and heater pads pressed against the DRAM chips to maintain a target temperature level, and 4) a temperature controller [123] to control the temperature. Fig. 5 shows our laboratory comprising many DDR4/HBM2 testing platforms. Real DDR4 and HBM2 DRAM Chips Tested. Table 1 provides 216 real DDR4 chips from 28 modules and 4 HBM2 chips that we characterize.3,4 Logical-to-Physical Row Mapping. DRAM manufacturers use mapping schemes to translate logical addresses to physical row addresses [7,14,32,42,85,98,106,128-134]. To account 3The technology node size of a DRAM chip is usually not publicly available. Prior works [3, 4, 71, 124-126] assume that for a given chip manufacturer and chip density, the alphabetical order of die revision codes may provide an indication of technology node advancement. 4We provide much more detail on the tested DRAM chips in the extended version of this paper [127]. 3 Figure 4: Our DRAM Bender [119] based experimental setup. Figure 5: Our laboratory for real DRAM chip experiments. for in-DRAM row address mapping and internal row remapping (which are extensively discussed in [3, 47, 135]), we reverse engineer the physical row address layout, following prior works' methodology [3,4,47,71,135-138]. 3.2. Experimental Methodology Characterizing and understanding ColumnDisturb requires 1) reverse engineering the subarray boundaries because the columns of a single DRAM row span three physically consecutive subarrays, and thus, ColumnDisturb bitflips can manifest across three consecutive subarrays (as shown in Fig. 2), 2) filtering out retention failures, since ColumnDisturb induces bitflips within intervals longer than the refresh window (tREFW), and 3) testing many parameters to understand how operational conditions and parameters affect the ColumnDisturb vulnerability. Metrics. To characterize a DRAM module's vulnerability to ColumnDisturb, we examine three metrics: 1) minimum time to induce the first ColumnDisturb bitflip in a subarray, 2) the fraction of cells with bitflips in a subarray, and 3) the number of rows that experience at least one ColumnDisturb bitflip in a subarray (i.e., blast radius [3, 4, 7, 47, 50, 71, 125, 126, 136139]). A shorter time to induce the first bitflip indicates higher vulnerability to ColumnDisturb. Higher values for the fraction of cells with bitflips in a subarray and blast radius indicate higher ColumnDisturb vulnerability. Access Pattern. To induce ColumnDisturb bitflips in a subarray, we perform the following key DRAM command sequence: ACT RAgg tAggOn ---→PRE tRP -→ACT RAgg tAggOn ---→··· where RAgg is the aggressor row in the tested subarray, tAggOn is the duration that the aggressor row stays open (i.e., the timing delay between ACT RAgg and PRE) [4], and tRP is the timing delay before issuing the next ACT RAgg command. We conduct all experiments where RAgg is the middle row in the tested subarray unless stated otherwise. By repeatedly hammering or pressing RAgg with our access pattern (e.g., the aggressor row in Fig. 1-c), we disturb DRAM cells through the columns (e.g., victim columns in Fig. 1c), Table 1: Summary of DDR4 and HBM2 DRAM chips tested. Chip Mfr. Module IDs #Chips Die Rev. Density Org. H0,H1,H2 24 A 8Gb x8 SK Hynix H3,H4,H5,H6 32 D 8Gb x8 H7 8 A 16Gb x8 H8,H9 16 C 16Gb x8 M0 8 B 4Gb x8 M1,M2,M3 24 R 8Gb x8 Micron M4, M5 16 B 16Gb x8 M6, M7 8 E 16Gb x16 M8,M9,M10,M11 32 F 16Gb x8 S0, S1 16 A 16Gb x8 Samsung S2, S3 16 B 16Gb x8 S4, S5 16 C 16Gb x16 Samsung HBM2 Chips 4 N/A N/A N/A a We report "N/A" if the information is not publicly available. since a row drives its content to all columns in a subarray after a successful ACT command. As a result, hammering repeatedly raises and lowers the voltage of the columns, while pressing keeps the voltage consistently high or low. We conduct all experiments using the same access pattern and record the bitflips in the aggressor row's subarray unless stated otherwise. Algorithm to Find Time to Induce the First Bitflip. For every parameter we evaluate, we determine the minimum number of ACT commands (i.e., hammer count) required to induce the first bitflip per tested subarray, using the bisection-method algorithm used by prior works [4,71,102,124,126,136,138]. We terminate the search when the difference between the current and previous minimum hammer count measurements is smaller than 1% of the previous measurement. During these experiments, we do not issue any REF commands for 512 ms (i.e., the refresh interval is 512 ms). If no ColumnDisturb bitflips are observed within 512 ms in a given subarray, we terminate the search for that subarray and proceed to the next one. For every tested subarray, we repeat the search five times and convert the minimum hammer count value across five iterations to time. DRAM Subarray Boundaries. Prior works [138, 140-146] demonstrate that real DRAM chips are capable of performing the RowClone operation [90], an in-DRAM data copy operation within a subarray by performing two consecutive row activations, which leads to data in the first activated row (i.e., source row) to be copied to the second activated row (i.e., destination row) via the sense amplifiers connected to both rows. We repeatedly perform the RowClone operation for every possible source and destination row address in each tested bank. When we observe that the destination row gets the same content as the source row after the RowClone operation, we conclude that the source row and the destination row are in the same subarray. Based on this observation, we reverse engineer the subarray boundaries and determine which rows are in the same subarray. Filtering Out Retention and RowHammer & RowPress Failures. We experimentally identify cells that experience retention failures during our tests and exclude them from the set of bitflips caused by ColumnDisturb in two steps. First, we follow the state-of-the-art retention time test methodology [84-86, 117] for five different data patterns. Second, to cover the worst-case in the presence of the variable retention time phenomenon [85, 86, 115, 116], we repeat each test 50 times and draw our conclusions based on the lowest observed retention time of a cell. To filter out RowHammer and RowPress bitflips, in every experiment, we exclude eight nearest 4 physical victim rows of the aggressor row when we read from a subarray, since RowHammer & RowPress induce bitflips in nearby neighboring rows of the aggressor row. We exclude eight nearest neighbor rows of the aggressor, even though we experimentally find that RowHammer & RowPress induce bitflips in only +1/-1 neighboring rows. We add this guardband (i.e., six additional rows to exclude) because the state-of-the-art industry solutions to DRAM read disturbance support refreshing RowHammer & RowPress bitflips in up to eight nearest neighboring rows [41,47,147-149].5 Eliminating Other Interference Sources. We eliminate other potential sources of interference by taking two measures, similar to prior works [3, 4, 47, 71, 136]. First, we disable periodic refresh to prevent refreshing any rows so that we can observe the DRAM inherent device-level behavior. Second, we verify that the tested modules and chips have neither rank-level nor on-die ECC [134,150,151]. Test Parameters. We explain the three common parameters we use in testing and elaborate on the specific parameters of each test in §4 and §5. 1) Data Pattern: We use five data patterns 0x00, 0xAA, 0x11, 0x33, and 0x77, commonly used in memory reliability testing [152] and by prior work on DRAM characterization (e.g., [3,4,7,71,138]). We fill aggressor rows with these data patterns while initializing victim rows with the negated data pattern (e.g., if aggressors have 0x00, victim rows are filled with 0xFF). 2) Temperature: We perform our experiments at four temperature levels: 45◦C, 65◦C, 85◦C, and 95◦C. We conduct all experiments at 85◦C unless stated otherwise. 3) Aggressor Row On Time (tAggOn): We define tAggOn [4] as the time an aggressor row stays open after each activation during a ColumnDisturb test. We perform our experiments at four tAggOn values: 36ns, 7.8ms , 70.2ms , and 1ms. We conduct all experiments at tAggOn= 70.2ms unless stated otherwise. 4) Refresh Interval: We perform our experiments with various different refresh intervals from 64ms to 16s. 4. Foundational Results We demonstrate a new read disturbance phenomenon, ColumnDisturb, in real DDR4 and HBM2 DRAM chips. ColumnDisturb is a widespread read disturbance phenomenon in real DRAM chips and worsens as DRAM technology scales down to smaller node sizes (§4.1). We observe that ColumnDisturb can induce bitflips in three consecutive subarrays by hammering or pressing an aggressor row (§4.2). We investigate the characteristics of ColumnDisturb by analyzing the direction of bitflips (§4.3), the effect of data pattern on a DRAM column (§4.4), the effect of aggressor row on time (§4.5), the effect of average voltage level on a DRAM column (§4.6), the number of DRAM rows that experience ColumnDisturb bitflips (§4.7), and ColumnDisturb on HBM2 DRAM chips (§4.8). 4.1. Prevalence & Scaling Characteristics We study the vulnerability of each chip to ColumnDisturb. One critical component of vulnerability is identifying the weakest cell (i.e., the cell that fails earliest). Since ColumnDisturb is effective at subarray granularity (see §1 and §4.2), we check the subarray that the aggressor row (the middle row in the 5We also observe that when we only exclude two nearest neighbor rows, our experimental results are essentially the same. Our extended version provides more comprehensive analyses and results [127]. tested subarray6) belongs to and record the time to induce the first ColumnDisturb bitflip in that subarray for each chip (see §3.2 for more details on our complete methodology). In this experiment, we test all subarrays in all banks across all tested DRAM modules from all three major manufacturers using the parameters with which a DRAM chip experiences the highest ColumnDisturb vulnerability (determined through extensive experiments in §4 and §5). Fig. 6 shows a violin plot of the time to observe the first ColumnDisturb bitflip in a subarray across all tested 46,080 subarrays (y-axis) for each different die revision and density pair (x-axis). Each subplot is dedicated to a different manufacturer. 8Gb A-die 8Gb D-die 16Gb A-die 16Gb C-die 74.0 120 240 360 480 SK Hynix 4Gb B-die 8Gb R-die 16Gb B-die 16Gb E-die 16Gb F-die 63.6 100 200 300 400 Micron 16Gb A-die 16Gb B-die 16Gb C-die 88.5 160 240 320 400 Samsung Time to induce rst bit ip in a subarray (ms) DRAM Chip Density & Revision Figure 6: Distribution of the time to induce the first ColumnDisturb bitflip in a subarray for different DRAM chip densities & die revisions across three major manufacturers. Observation 1. All tested 216 DDR4 chips are vulnerable to ColumnDisturb. We observe that in every DDR4 chip tested, there is at least one DRAM cell that experiences a ColumnDisturb bitflip. Observation 2. Newer chips tend to be more vulnerable to ColumnDisturb bitflips. We observe that in the same die density for each tested manufacturer, more advanced technology nodes experience higher ColumnDisturb vulnerability (i.e., lower time to induce the first ColumnDisturb bitflip).7 For SK Hynix, for 8Gb and 16Gb chip density, respectively, the minimum time to induce the first ColumnDisturb bitflip in a subarray reduces by 5.06x and 1.29x as die generation advances (i.e., 8Gb A- to D-die and 16Gb Ato C-die). For Micron, for 16Gb chip density, from B- to F-die, the minimum time to induce the first ColumnDisturb bitflip in a subarray reduces by 2.98x. For Samsung, compared to 16Gb A-die, the minimum time to induce the first ColumnDisturb bitflip in a subarray is 2.50x lower in 16Gb C-die. We hypothesize that as DRAM technology node advances, the distance between the DRAM cell capacitor contact and the bitline decreases, strengthening tunneling effects between them [2]. Thus, DRAM chips become more vulnerable to ColumnDisturb as DRAM technology node size reduces. §4.6 provides detailed analyses and hypotheses on the device-level mechanisms of ColumnDisturb. Observation 2 has serious implications for the futureas DRAM technology node sizes will continue to reduce and the time to induce the first ColumnDisturb bitflip will likely get smaller. §6.1 discusses the implications of this observation. 6We test different aggressor row locations and observe the almost same trend in §5.5. 7For a given manufacturer and die density, the later in the alphabetical order the die revision code is, the more likely the chip has a more advanced technology node [3,4,71]. 5 Observation 3. There already are some real DRAM chips where ColumnDisturb introduces bitflips within the nominal refresh window under nominal operating conditions. We observe that in a single 16Gb F-die Micron module, multiple cells (at least four cells) from multiple chips (at least three chips) experience ColumnDisturb bitflips at 63.6ms and 85◦C, within the nominal refresh window for DDR4 chips, tREFW. In contrast, retention failures require at least 512ms (at 85◦C) to manifest for the same module (see Fig. 11). We also observe that the closest (farthest) victim row that experiences ColumnDisturb bitflips within tREFW is 374 (446) rows away from the aggressor row where both victim row and the aggressor row are in the same subarray. Takeaway 1. ColumnDisturb is a widespread read disturbance phenomenon in real DRAM chips. ColumnDisturb becomes worse as DRAM technology scales down to smaller node sizes. ColumnDisturb can already induce bitflips within the refresh window in current DRAM chips. 4.2. Disturbing DRAM Columns Fig. 2 in §1 shows the ColumnDisturb, RowHammer, RowPress, and retention failure bitflips from a single representative module (S0; Samsung 16Gb A-die).8 For RowHammer (RowPress), the same aggressor row (row 1536) is hammered (pressed, i.e., with tAggOn=70m s) for 16 seconds. For the retention failure test, the bank remains idle (i.e., in precharged state) for 16 seconds. Dashed vertical lines indicate reverse-engineered subarray boundaries: row addresses 0-1023 belong to Subarray 0, 1024-2047 to Subarray 1, and 2048-3071 to Subarray 2. The aggressor row (row 1536) is located in Subarray 1 (Aggressor Subarray), and Subarrays 0 and 2 are physically adjacent to the aggressor subarray (Neighboring Subarray). Observation 4. ColumnDisturb induces bitflips across three consecutive subarrays (3072 rows), causing bitflips in many more rows than RowHammer and RowPress. Al 3072 rows in three consecutive subarrays experience ColumnDisturb bitflips. However, only the immediate neighbor rows of the aggressor row 1536 (i.e., rows 1535 and 1537) experience RowHammer & RowPress bitflips.9 This observation could have serious implications for the fu8We experiment with all 28 tested modules and observe very similar characteristics to this representative Samsung module, S0. 9We verify that bitflips in the immediate neighbors of an aggressor row (+/-1 rows of the aggressor row) are caused by RowHammer and RowPress based on three observations. First, prior works on real DRAM chips [4,135, 153,154] demonstrate that by correctly reverse-engineering logical-to-physical row mapping of DRAM chips, RowHammer & RowPress induce bitflips in adjacent victim rows (+/-1 rows of the aggressor row) [4,135,153,154] due to the underlying silicon-level characteristics of RowHammer & RowPress [154156]. Second, when we hammer different aggressor rows in the same subarray, we observe high bitflip counts only in +1/-1 neighboring victim rows (i.e., 7559 RowHammer bitflips and 5406 RowPress bitflips on average across +1 and -1 neighboring victim rows), while all other victim rows in the aggressor subarray experience similar bitflip counts to each other (between 2353-3505 ColumnDisturb bitflips across all other victim rows, significantly less than RowHammer & RowPress bitflips. Third, when we initialize victim rows with an all-0 pattern, we observe only 0 to 1 bitflips in the +/-1 neighboring rows (not shown). Since ColumnDisturb induces only 1 to 0 bitflips (§4.3), whereas RowHammer & RowPress can induce both 0 to 1 and 1 to 0 bitflips (§4.3), this third observation further indicates that RowHammer & RowPress induce bitflips in only +/-1 rows. ture, as refresh-based DRAM read disturbance mitigation techniques consider a few neighbor rows (e.g., up to +4/-4 in industry solutions [41,47,147-149,157]) of the aggressor row as victim rows, all in the same subarray, whereas ColumnDisturb leads to bitflips in thousands of rows (i.e., all rows across three consecutive subarrays). We observe that subarrays that are neither the aggressor subarray nor physically-adjacent to the aggressor subarray do not experience any ColumnDisturb bitflips (not shown in the figure). This is due to the open-bitline DRAM architecture [92-100]: two neighboring subarrays share half of their columns. Thus, the aggressor subarray (Subarray 1) shares columns with both the above subarray (Subarray 0) and the below subarray (Subarray 2) (see Fig. 3). As a result, subarrays that do not share columns with the aggressor subarray are not affected (see Fig. 1c). We hypothesize that this observation shows ColumnDisturb is a column-based read disturbance phenomenon, since only the cells sharing the same columns as the aggressor row experience ColumnDisturb bitflips. Observation 5. ColumnDisturb induces more bitflips in the aggressor subarray's rows than in those of neighboring subarrays. On average, ColumnDisturb induces 1.45× more bitflips in the aggressor subarray's rows than in the neighboring subarrays' rows. We hypothesize that all columns are affected in the aggressor subarray because an activation causes perturbation in all bitlines in the same subarray. In contrast, only half of the columns in the neighboring subarray are connected to the aggressor subarray, and hence an activation causes a perturbation in half of the bitlines in the neighboring subarrays. We also observe no overlap in the column indices of the cells that experience only ColumnDisturb bitflips in Subarray 0 and Subarray 2 (neighboring subarrays of Subarray 1). We hypothesize that this is because Subarray 1's even columns are shared with Subarray 0's odd columns (e.g., shared blue columns in the middle and top subarrays in Fig. 3) and Subarray 1's odd columns are shared with Subarray 2's even columns (e.g., red columns in the middle and bottom subarrays in Fig. 3). This observation further supports our hypothesis that ColumnDisturb is a column-based read disturbance phenomenon since only cells that share columns with the aggressor row are affected. Observation 6. For a given refresh interval, ColumnDisturb induces many more bitflips than retention failures. On average, ColumnDisturb induces 2942.68 bitflips per row in the aggressor subarray and 2025.02 bitflips per row in the neighboring subarrays, which are 7.07x and 4.87x more than retention failure bitflips for a refresh interval of 16 seconds, respectively. Takeaway 2. ColumnDisturb is a column-based read disturbance phenomenon that disturbs cells via DRAM columns (i.e., bitlines) and induces bitflips in thousands of rows (i.e., as many as all 3072 rows in three DRAM subarrays). 4.3. Bitflip Direction We analyze the bitflip direction of ColumnDisturb versus retention failures using one representative module (S0; Samsung 16Gb A-die).8 Fig. 7 shows the number of 1 to 0 (left) and 0 to 1 (right) bitflips in a subarray (y-axis) due to ColumnDisturb 6 and retention failure bitflips across five different refresh intervals (x-axis). Error bars represent the range of minimum and maximum number of bitflips across all tested subarrays. Figure 7: Distribution of the number of 1 to 0 and 0 to 1 bitflips in a subarray due to ColumnDisturb and retention failures. Observation 7. ColumnDisturb induces only 1 to 0 bitflips, which are the same direction as retention failures. Across all tested refresh intervals and subarrays, the only observed bitflip direction for ColumnDisturb is 1 to 0, which is the same as retention failures. We do not observe any 0 to 1 ColumnDisturb bitflips. This is in stark contrast to RowHammer and RowPress, which induce bitflips in both 1 to 0 and 0 to 1 directions [3,4,7,135,154], which we also observe in our tested DRAM chips. Observation 8. ColumnDisturb induces many more bitflips than retention failures, especially at shorter refresh intervals. On average, ColumnDisturb induces 11.77x, 7.02x, 4.86x, 3.97x, and 4.58x more bitflips than retention failures at refresh intervals of 1s, 2s, 4s, 8s, and 16s, respectively. Takeaway 3. DRAM is much more vulnerable to ColumnDisturb than retention failures, and the bitflip directionalities of ColumnDisturb and RowHammer & RowPress are significantly different. 4.4. Data Pattern in Aggressor Row We investigate how the aggressor data pattern (i.e., data pattern on the columns perturbed by the aggressor row) affects the ColumnDisturb vulnerability using a single representative module from each major DRAM manufacturer (S0, H0, and M6). We test all subarrays in a single bank from each tested module. For this experiment, we use all-1 as the data pattern for all victim cells in a subarray since ColumnDisturb induces only 1 to 0 bitflips and tAggOn=tRAS. We observe that not all subarrays have the same number of rows (the number of rows in a subarray across all tested modules ranges between 512 and 1024), and thus, we use a metric "fraction of cells with bitflips in a subarray." Fig. 8 shows the fraction of cells that experience bitflips in a subarray across all tested subarrays (y-axis) for five different refresh intervals (x-axis) for three DRAM manufacturers (subplot). Each line represents a different experiment: 1) ColumnDisturb: AggDP=all-0, the aggressor row is initialized with all-0, 2) ColumnDisturb: AggDP=all-1, the aggressor row is initialized with all-1, and 3) RET, retention failures, where all columns are at the precharged voltage (VDD/2) during the entire experiment. The error band of a line shows the minimum and maximum values across all tested subarrays. Observation 9. ColumnDisturb induces many more bitflips when the aggressor row data pattern is all-0 versus all-1. For example, at 16s, all-0 aggressor pattern induces 1.15x, Figure 8: Distribution of fraction of cells with bitflips in a subarray due to ColumnDisturb for two different aggressor data patterns and retention failures. 11.52x, and 2.86x more bitflips than the all-1 aggressor data pattern for ColumnDisturb in SK Hynix, Micron, and Samsung chips, respectively. We hypothesize that this occurs due to a larger voltage level difference between the victim cell and the perturbed columns (i.e., columns connected to the aggressor row). This is because, when the aggressor data pattern is all-0, all perturbed columns in the subarray are at GND (i.e., low voltage) while the victim cells are at VDD, resulting in a VDD difference. In contrast, when the aggressor data pattern is all-1, both victim cells and all perturbed columns in a subarray are at VDD, resulting in no voltage difference. We further analyze the relationship between the perturbed column voltage level and ColumnDisturb in §4.6. Observation 10. When the aggressor data pattern is all-1 (same as victim data pattern), ColumnDisturb can induce fewer bitflips than retention failures. For example, in Micron, with a refresh interval of 16s, 2.73x fewer DRAM cells in a subarray on average experience bitflips with ColumnDisturb (when both the aggressor data pattern and the victim data pattern are all-1) versus retention failures. We hypothesize that this could be due to the reduced voltage level difference between a column and the victim cell. In ColumnDisturb with all-1 aggressor and victim data pattern tests, the voltage difference between the victim cell (VDD) and the column (VDD) is zero. In contrast, in retention failure tests, the column is held at VDD/2 while the cell remains at VDD, resulting in a higher column-cell voltage difference (i.e., VDD/2) than ColumnDisturb with all-1 aggressor and victim data pattern tests. As a result, having a lower voltage difference between the victim cell and the column could lead to a lower ColumnDisturb vulnerability. We provide a detailed analysis and hypotheses for the effect of the voltage difference between victim cells and perturbed columns in §4.6. Takeaway 4. Data pattern in the aggressor row has a significant effect on the number of observed ColumnDisturb bitflips. 4.5. Aggressor Row on Time We study the effect of the amount of time an aggressor row is kept open (tAggOn) on ColumnDisturb bitflips.10 For this analysis, we use all-1 (0xFF) as the victim data pattern and all-0 (0x00) for the aggressor data pattern since these are the worstcase data patterns that induce the most ColumnDisturb bitflips in a DRAM chip. Fig. 9 shows the fraction of cells with bitflips in a subarray observed in three representative modules, one from 10Keeping an aggressor row open for a long time (i.e., high tAggOn values) perturbs all columns connected to the cells in the aggressor row, since the columns remain driven to VDD or GND. 7 each tested manufacturer. Each line represents a different experiment: 1) ColumnDisturb: tAggOn=36ns, where we keep the aggressor row open for 36ns, 2) ColumnDisturb: tAggOn=70.2m s, where we keep the aggressor row open for 70.2m s, and 3) RET, retention failure, where the bank is idle (i.e., in precharged state) during the experiment. Figure 9: Distribution of fraction of cells with bitflips in a subarray due to ColumnDisturb for two different tAggOn values and retention failures. Observation 11. Fraction of cells with ColumnDisturb bitflips in a subarray significantly increases as tAggOn increases. For example, at a refresh interval of 16s, increasing tAggOn from 36ns to 70.2m s increases the number of ColumnDisturb bitflips by 1.20x, 2.12x, and 2.45x for SK Hynix, Micron, and Samsung modules, respectively. We hypothesize that this occurs because a longer tAggOn keeps the perturbed columns at a low voltage level (GND) for an extended period, which likely results in an increased ColumnDisturb effect. 4.6. Average Voltage Level on Perturbed Columns We observe that DRAM chips become significantly more vulnerable to ColumnDisturb 1) when the voltage level difference between the victim cell and the perturbed columns (i.e., columns perturbed by the aggressor row) is high (§4.4) and 2) when the perturbed columns are kept active longer (§4.5), i.e., the voltage level difference between the victim cell and the perturbed columns is kept high for an extended period. Based on our observations in §4.4 and §4.5, we hypothesize that the voltage level on the perturbed columns is a key parameter that affects the ColumnDisturb vulnerability in DRAM. To understand the effect of average column voltage level on ColumnDisturb, we calculate the average voltage level on the perturbed columns AVG(V COL) when we perform ColumnDisturb tests. Our access pattern, for every tAggOn+tRP time, (§3.2) 1) keeps the column voltage level at the aggressor data pattern in the aggressor subarray, DPCOL, for tAggOn and 2) keeps the column voltage level at precharged voltage (VDD/2) for tRP. Based on the relationship between AVG(V COL) and DPCOL, tAggOn, VDD/2, and tRP parameters, we can calculate AVG(V COL) as follows: AVG(V COL) = tAggOn∗DPCOL+VDD/2∗t RP tAggOn+t RP For example, assume DPCOL=GND=0, tAggOn=36ns, precharged voltage level=VDD/2, and tRP=14ns (i.e., perturbed columns are at GND for 36ns, and at VDD/2 for 14ns in every 50ns), AVG(V COL) becomes 36ns∗0+VDD/2∗14ns 36ns+14ns = 0.14∗VDD. Fig. 10 shows the fraction of cells with bitflips (y-axis) as we sweep the average voltage level on the perturbed columns (x-axis) across five refresh intervals (each colored line). In this experiment, we test all subarrays in a single bank from three DRAM modules, one from each tested manufacturer. Figure 10: Fraction of cells with ColumnDisturb bitflips for different average column voltage levels. Observation 12. A DRAM chip becomes significantly more vulnerable to ColumnDisturb as the average voltage level on the perturbed columns decreases. We observe that reducing the average column voltage level from VDD to GND increases the fraction of cells that experience bitflips in a subarray by 1.65x, 26.31x, and 7.50x for SK Hynix, Micron, and Samsung, respectively, at a refresh interval of 16 seconds. We provide two hypotheses to explain our observations. As the perturbed column voltage decreases, the voltage difference across the victim cell and the perturbed column increases, which 1) exacerbates subthreshold leakage of the access transistor [112] and/or 2) exacerbates the dielectric leakage between the victim capacitor and the perturbed column [2].11 Key Hypothesis. ColumnDisturb exacerbates subthreshold leakage of the access transistor [112] and/or the dielectric leakage between the capacitor and the bitline [2]. Takeaway 5. The voltage level on the column plays an important role in ColumnDisturb's device-level failure mechanisms. Device-level investigation is necessary to develop a better fundamental and first-principles understanding of ColumnDisturb. 4.7. Blast Radius We study how many rows experience ColumnDisturb bitflips in a subarray for a given refresh interval. To do so, we use a metric: the number of rows with bitflips in a subarray, i.e., the number of rows that experience at least one bitflip in a subarray, broadly referred to as blast radius [3,4,7,47,50,71,125,126,136-139]. Fig. 11 shows the distribution of the number of rows with bitflips in a subarray at 65◦C as a boxplot. Each subplot is dedicated to a different manufacturer and shows how many rows experience ColumnDisturb and retention failure bitflips in a subarray (y-axis) for five different refresh intervals. Observation 13. ColumnDisturb induces bitflips in significantly more DRAM rows than retention failures. For example, with a refresh interval of 1024ms, ColumnDisturb (Retention) induces bitflips in up to 52 (20), 353 (34), and 1022 (29) rows for SK Hynix, Micron, and Samsung modules, respectively. For SK Hynix, Micron, and Samsung modules, respectively, with a refresh interval of 512ms, ColumnDisturb induces bitflips in 2, 6, and 232 rows on average, whereas only two rows experience retention failures at most. 11We call for future device-level and silicon-level studies to develop a better understanding of the inner workings of ColumnDisturb, just as device-level studies (e.g., [155,156]) did for RowPress after the RowPress paper [4] demonstrated for the RowPress phenomenon experimentally on real DRAM chips. 8 Figure 11: Distribution of the number of rows that experience ColumnDisturb versus retention failure bitflips for different refresh intervals. Observation 14. Number of rows with ColumnDisturb bitflips significantly increases as the refresh interval increases. For example, from 512ms to 1024ms, for SK Hynix, Micron, and Samsung, respectively, 9.73, 215.85, and 159.22 more rows in a subarray experience ColumnDisturb bitflips, on average across all tested subarrays. However, for retention failures, the increase is significantly smaller than ColumnDisturb. From 512ms to 1024ms, up to 8.39 (2.76 on average) more rows experience retention failure bitflips on average across all tested modules. Takeaway 6. Significantly more rows are vulnerable to ColumnDisturb than retention failures. 4.8. ColumnDisturb on COTS HBM Chips So far, we characterize commodity DDR4 chips to demonstrate and understand ColumnDisturb. In this section, we provide a preliminary study on the vulnerability of real HBM2 chips to ColumnDisturb. We test 4 HBM2 chips and analyze the number of ColumnDisturb bitflips and retention failures across all subarrays in a bank from three HBM2 chips. Fig. 12 shows the number of bitflips in a subarray across all subarrays from a single bank when 1) performing ColumnDisturb and 2) keeping the bank in the idle state without performing any refresh (i.e., retention), for 1, 2, and 4 seconds. Error bars represent the range of minimum and maximum number of bitflips across all tested subarrays. Figure 12: Number of ColumnDisturb and retention failure bitflips for different refresh intervals in HBM2 chips. Observation 15. HBM2 chips are vulnerable to ColumnDisturb, and ColumnDisturb induces many more bitflips than retention failures. We observe that across all four HBM2 chips tested, ColumnDisturb induces 1.61x, 2.08x, and 2.43x more bitflips than retention failures in 1s, 2s, and 4s, respectively. We expect our other observations for DDR4 chips (§4 and §5) will hold for HBM2 chips as well, since the DRAM array is the same for both, and ColumnDisturb happens at the array level. Takeaway 7. ColumnDisturb is a widespread read disturbance phenomenon in DRAM chips: not only DDR4 chips, but also HBM2 chips are vulnerable to ColumnDisturb. 5. In-Depth ColumnDisturb Analysis This section further enhances our analysis of ColumnDisturb by investigating parameters that have been shown to impact both read disturbance and retention failure mechanisms: temperature (§5.1), aggressor row on time (§5.2), access pattern (§5.3), data pattern (§5.4), the location of the aggressor row in a subarray (§5.5), and the effectiveness of Error Correcting Codes (ECC) against ColumnDisturb (§5.6).12 We conduct our experiments by testing all subarrays in all banks from all tested modules using the parameters with which a DRAM chip experiences the highest ColumnDisturb vulnerability (i.e., aggressor data pattern=all-0, victim data pattern=all-1, tAggOn=70.2m s, temperature= 85◦C), unless stated otherwise. 5.1. Temperature We analyze the relationship between temperature and the ColumnDisturb vulnerability by evaluating three metrics for four different temperature levels: 45◦C, 65◦C, 85◦C, and 95◦C. Time to Induce the First ColumnDisturb Bitflip. Fig. 13 shows how the time to induce the first bitflip changes with temperature using a box and whiskers plot. Each subplot is dedicated to a different manufacturer. Figure 13: Distribution of time to induce the first ColumnDisturb bitflip at different temperatures. Observation 16. Time to induce the first ColumnDisturb bitflip significantly reduces as temperature increases. Across all tested modules, time to induce the first ColumnDisturb bitflip consistently reduces as temperature increases. For example, increasing temperature from 45◦C to 95◦C reduces the average time to induce the first ColumnDisturb bitflip in a subarray by 9.05x, 5.15x, and 1.96x for SK Hynix, Micron, and Samsung, respectively. Fraction of Cells with Bitflips in a Subarray. Fig. 14 shows the fraction of cells with bitflips in a subarray (y-axis) due to ColumnDisturb versus retention failures (color-coded) across four temperature levels (x-axis) when the refresh interval is 512ms. Observation 17. ColumnDisturb is more sensitive to temperature than retention failures. For example, in SK Hynix chips, when the temperature increases from 85◦C to 95◦C, ColumnDisturb (retention failure) bitflips increase by 72.96x (3.68x) on average. We also observe that ColumnDisturb induces more bitflips than retention failures in all tested temperature levels across all DRAM modules. For example, at 65◦C, ColumnDisturb induces 152.66x more bitflips than retention failures in Samsung modules. 12We observe that the overall trends are consistent across the three evaluated metrics: 1) time to induce the first ColumnDisturb bitflip in a subarray, 2) the fraction of cells with ColumnDisturb bitflips in a subarray, and 3) the blast radius. Due to space limitations, we provide the analysis of the first metric in all experiments and provide a detailed analysis of all metrics only for temperature. Our extended version provides comprehensive analyses of all metrics [127]. 9 Figure 14: Distribution of fraction of cells with ColumnDisturb and retention failure bitflips in a subarray at different temperatures. Blast Radius. Fig. 15 shows the blast radius effect (y-axis) of ColumnDisturb and retention failures (i.e., the number of rows in a subarray that experience at least one ColumnDisturb and retention failure bitflip) for three manufacturers (rows of subplots) and four temperature levels (columns of subplots). The x-axis shows the refresh interval in milliseconds (ms). Figure 15: Number of rows that experience at least one bitflip (i.e., blast radius) due to ColumnDisturb and retention failures for different refresh intervals and temperature levels. Observation 18. ColumnDisturb induces bitflips in many more rows than retention failures in all tested temperature levels. For example, even at a low temperature level, 45◦C, in Micron and Samsung, respectively, ColumnDisturb induces bitflips in up to 39 and 150 rows in a subarray across all tested subarrays. However, at most only a single row exhibits retention failures for Micron, and none for Samsung. We observe that across all tested temperature levels and refresh intervals, ColumnDisturb induces bitflips in up to 198x more DRAM rows than retention failures. Observation 19. The blast radius of both ColumnDisturb and retention failures increase with higher temperature. At 95◦C, across all tested subarrays, both mechanisms nearly span an entire subarray (i.e., almost all rows of every tested subarray experience bitflips), whereas ColumnDisturb begins exhibiting a wide impact already at 65◦C (e.g., at least 5.33% of subarrays in Samsung modules experience a blast radius of 1000). Takeaway 8. As temperature increases, DRAM chips become more vulnerable to ColumnDisturb. 5.2. Aggressor Row On Time Fig. 16 shows the distribution of time to induce the first ColumnDisturb bitflip in a subarray across all tested subarrays for four different tAggOn values: 36ns, 7.8ms, 70.2ms, and 1ms. Figure 16: Time to induce first ColumnDisturb bitflip distribution for four different tAggOn values. Observation 20. Keeping a row open (tAggOn >> tRAS) is significantly more effective than immediately closing it (tAggOn = tRAS) in inducing the first ColumnDisturb bitflip in a subarray. Increasing tAggOn from 36ns to 7.8m s reduces the average time to induce the first ColumnDisturb bitflip by 1.68x, 1.22x, and 2.03x in SK Hynix, Micron, and Samsung, respectively. We observe that when tAggOn >> tRAS, the distributions are very similar across all tested tAggOn values. We hypothesize that increasing tAggOn decreases the average voltage level on DRAM columns, resulting in higher ColumnDisturb vulnerability in DRAM chips (i.e., less time to induce the first ColumnDisturb bitflip in a subarray), as discussed in §4.6. 5.3. Access Pattern Until now, we perform ColumnDisturb with a single aggressor row, causing the column voltage to alternate between VDD/2 and GND or VDD, depending on the aggressor data pattern (described in §3.2). To understand the effect of toggling the column, we introduce a two-aggressor access pattern where two aggressor rows are used with complementary data patterns: ACT RAgg1 tAggOn ---→PRE tRP -→ACT RAgg2 tAggOn ---→··· In the single aggressor access pattern (§3.2), we use all-0 as the aggressor data pattern. Thus, the column voltage transition is {GND→VDD/2→GND···}. In the two-aggressor access pattern, we use all-0 in the first aggressor (RAgg1) and all-1 in the second aggressor (RAgg2), and thus, the voltage transition on the column becomes {GND→VDD/2→VDD→VDD/2···}. Fig. 17 shows the time to induce the first ColumnDisturb bitflip for the two access patterns. In both experiments, victim rows are initialized with all-1 as we observe only 1 to 0 bitflips for ColumnDisturb (§4.3). Figure 17: Distribution of time to induce the first ColumnDisturb bitflip for two different access patterns. Observation 21. Single-aggressor access pattern induces the first ColumnDisturb bitflip in a subarray significantly faster than the two-aggressor access pattern. For SK Hynix, Micron, and Samsung modules, the singleaggressor access pattern induces the first ColumnDisturb bitflip 1.83×, 1.92×, and 2.16× faster than the two-aggressor access pattern, respectively. 10 This observation shows that toggling a column through the sequence of GND→VDD/2→VDD (i.e., the two-aggressor access pattern) is less effective than toggling it only between VDD/2 and GND (i.e., the single-aggressor access pattern). We hypothesize that this is because the two-aggressor access pattern results in a higher average column voltage than the single-aggressor access pattern, resulting in lower ColumnDisturb vulnerability in DRAM chips, i.e., longer time to induce the first ColumnDisturb bitflip (§4.6). Takeaway 9. Access pattern greatly affects a DRAM chip's vulnerability to ColumnDisturb. 5.4. Data Pattern We analyze the effect of the aggressor and victim data patterns using five aggressor and victim data pattern pairs. In this experiment, victim cells are initialized with the negated aggressor data pattern (e.g., if the aggressor data pattern is all-0, the victim data pattern is all-1). We omit the inverse of the tested patterns as either we 1) observe almost the same trend or 2) study it before (all-1 in §4.4). Time to Induce the First ColumnDisturb Bitflip. Fig. 18 shows the distribution of time to induce the first ColumnDisturb bitflip in a subarray for five data patterns. Figure 18: Distribution of time to induce the first ColumnDisturb bitflip for five different aggressor and victim data pattern pairs. Observation 22. Data pattern has a small effect on the time to induce the first ColumnDisturb bitflip in a subarray. Across all data patterns, the average time to induce the first ColumnDisturb bitflip varies by at most 1.31x. We hypothesize that this small variation (1.31x) suggests that bitline-to-bitline interference does not strongly affect the time to induce the first ColumnDisturb bitflip. This is because the weakest cell determines the time to induce the first ColumnDisturb bitflip. If the weakest cell is connected to a column written with logic0, it flips at nearly the same time regardless of the values in neighboring columns/bitlines. Total Number of Bitflips in a Subarray. Fig. 19 shows the total ColumnDisturb bitflips in a subarray with a refresh interval of 512ms. To maintain a reasonable experiment time, we test three different aggressor and victim data pattern pairs. Observation 23. Data pattern greatly affects total ColumnDisturb bitflips in a subarray: having more logic-0 values in the perturbed columns generates more bitflips. For example, in Samsung chips, an aggressor data pattern of 0x00 induces 2.04x more ColumnDisturb bitflips than an aggressor data pattern of 0xAA on average. We hypothesize that this is due to the bitflip direction characteristics of ColumnDisturb: only victim cells that are initialized with logic-1 experience bitflips (see §4.3). Thus, since victim cells are initialized with 0x00 0x11 0xAA 0 200 400 600 SK Hynix 0x00 0x11 0xAA 0 1000 2000 3000 Micron 0x00 0x11 0xAA 0 5000 10000 15000 Samsung Number of Bit ips in a Subarray Aggressor Pattern (Victim rows have negated aggressor data pattern) Figure 19: Total number of ColumnDisturb bitflips in a subarray for three different aggressor and victim data pattern pairs. the negated aggressor data pattern, more logic-0 values in the aggressor pattern (which means more victim cells are initialized with logic-1) lead to more ColumnDisturb bitflips. 5.5. Location of the Aggressor Row in a Subarray To understand the effects of spatial variation on ColumnDisturb, we analyze how the location of an aggressor row in a subarray affects the ColumnDisturb vulnerability. To do so, we test three aggressor row locations: 1) "Beginning": the first row of the subarray, 2) "Middle": the middle row in the subarray, and 3) "End": the last row of the subarray. Fig. 20 shows the distribution of time to induce the first bitflip in a subarray across DRAM subarrays (y-axis) based on the aggressor row's location in a subarray (x-axis). Figure 20: Distribution of time to induce the first ColumnDisturb bitflip based on the aggressor row's location in a subarray. Observation 24. The location of the aggressor row only slightly affects the time to induce the first ColumnDisturb bitflip. We observe that across all tested manufacturers, there is at most 1.08x variation on average when the location of the aggressor row in a subarray changes. 5.6. Effectiveness of System-Level ECC A system that uses Error-Correcting Codes (ECC) [134, 150, 158-167] can potentially protect against ColumnDisturb bitflips if those bitflips are distributed across the DRAM chip such that no ECC codeword contains more bitflips than ECC can correct. Fig. 21 shows the distribution of ColumnDisturb bitflips across 8-byte data chunks for all tested DRAM modules for 512ms and 1024ms refresh intervals. We use 8-byte data chunks as DRAM ECC typically uses 8-byte or larger datawords [134,150,151, 163,165,168-172]. Observation 25. ColumnDisturb induces more bitflips than today's widely-used ECCs can correct or even detect. We observe that many 8-byte data chunks experience 3 (up to 15) ColumnDisturb bitflips, which the SECDED ECC [163,168171] cannot correct or detect, in Micron and Samsung chips. Observation 26. Relying only on ECC to mitigate all ColumnDisturb-induced errors is likely very costly. 11 512 1024 100 101 102 SK Hynix 512 1024 100 101 102 103 104 105 Micron 512 1024 101 102 103 104 105 106 Samsung Experiment Time (ms) Number of 8-byte Data Chunks at 65 C Bit ip Count: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 21: Distribution of 8-byte data chunks with different ColumnDisturb bitflip counts. Since ColumnDisturb can induce 15 bitflips in multiple 8byte data chunks, a (7,4) Hamming code [160] could correct such bitflips with very large DRAM storage overheads (75%, i.e., three parity bits for every four data bits). Thus, relying on ECC alone to prevent ColumnDisturb bitflips is likely a very expensive solution. Effectiveness of On-Die ECC. We evaluate the effectiveness of a (136,128) single-error-correcting (SEC) Hamming code [160] (possibly implemented in DDR5 chips today [134,151,165,173, 174] for area efficiency and low cost overhead).13 This code corrects any one bitflip in a 136-bit codeword. The observed bitflip distributions (Fig. 21) for Micron and Samsung already exceed the error correction capabilities of such codes. Moreover, the SEC code can miscorrect codewords that have more than one bitflip [173,174,176]. A miscorrection manifests as additional bitflip(s) in the codeword (i.e., the SEC code induces another bitflip by attempting to correct the erroneous codeword). We randomly generate 10K erroneous codewords with two bitflips (bitflip indices in the codeword are determined using a uniform random number generator). The evaluated SEC code miscorrects 88.5% of the codewords, transforming a codeword with two bitflips into another with three bitflips. Observation 27. Low-cost on-DRAM-die single-errorcorrecting codes cannot correct all ColumnDisturb bitflips and are likely to induce additional bitflips in an erroneous codeword that has as few as only two bitflips. Takeaway 10. Conventional DRAM ECC cannot protect against ColumnDisturb bitflips, and an ECC scheme that can protect against ColumnDisturb requires large overheads. 6. Implications Our findings have strong implications for both 1) the robustness of future systems and 2) the system speedup and DRAM energy saving benefits of existing retention-aware heterogeneous refresh mechanisms (e.g., [75-88]). ColumnDisturb could jeopardize the safety, security, and reliability of future computing systems as more ColumnDisturb bitflips may manifest in the standard refresh window due to continuously shrinking DRAM technology node size. §6.1 describes and evaluates two techniques that could mitigate ColumnDisturb bitflips at varying performance, energy, and area overheads. In light of our new findings, we evaluate a retention-aware heterogeneous refresh mechanism (RAIDR [76]) and show that its benefits over the baseline DRAM periodic refresh dramatically decrease (e.g., by 53%, see §6.2) because at long refresh windows a large majority of DRAM rows (see §4.7 and §5.1) exhibit ColumnDisturb bitflips. 13The exact designs of on-die ECC are kept strictly confidential by DRAM manufacturers [150,175] and may be different from what we evaluate. 6.1. Implications for System Robustness Observations 1, 2, 4, 13, 17 and 19 demonstrate that ColumnDisturb induces bitflips in victim rows that are thousands of DRAM rows away from, and even in a different subarray from the aggressor row. For example, the first bitflip we observe in 63.6ms is 374 rows away from the aggressor row. Unfortunately, current RowHammer mitigations only refresh up to 8 neighboring rows [6,7,29,35,41,47,126,137,138,149,157,177-237] as RowHammer or RowPress bitflips induce bitflips in immediate neighbors of the aggressor rows [3,4,7,71,124-126,135, 136,138,139,153,154,238-246]. Simply modifying commercial or many other proposed read disturbance mitigations to preventively refresh all potential victim rows (e.g., 3072 such victims) of ColumnDisturb would induce prohibitive system performance and energy overheads due to the latency and energy cost of performing thousands of preventive refresh operations. For example, according to the latest DDR5 JEDEC Standard [148], refreshing +/-4 physically adjacent neighboring rows incurs 2x more latency (tDRFMab = 560ns) than refreshing +/-2 (tDRFMab = 280ns). Refreshing all rows in three consecutive subarrays (e.g., 3072 rows) would take an estimated, prohibitive, ∼215m s latency.14 Moreover, as DRAM chip density increases, so does the number of rows in a subarray. Therefore, securely mitigating ColumnDisturb bitflips would likely require refreshing a few thousand rows before an aggressor row is hammered or pressed to the point of failure. However, reactively refreshing that many rows incurs prohibitive memory access latencies. We describe and evaluate two solutions for ColumnDisturb that proactively refresh potential victim rows of ColumnDisturb without inducing prohibitive memory access latencies. Increasing DRAM Refresh Rate. A straightforward (and highoverhead) solution for ColumnDisturb is to indiscriminately increase the DRAM refresh rate. DRAM periodic refresh would mitigate ColumnDisturb-induced bitflips if the refresh period is set adequately short. The benefit of this solution is that it can be applied immediately in today's systems. However, as many prior works [75-88] show, increasing the refresh rate degrades system performance and energy efficiency because more frequent refresh operations 1) increase memory access latency, 2) reduce row buffer hit rate, 3) degrade bank-level parallelism, and 4) cause more activity on the memory bus and in DRAM banks. For example, for a 32 Gb DDR5 chip [147, 247], reducing the default (all bank) refresh period (REFab) from 32 ms to 8 ms (i.e., 4x higher refresh rate) increases the DRAM throughput loss due to periodic refresh operations from 10.5% to 42.1%15 and relative energy consumed by refresh operations for an otherwise idle DRAM chip increases from 25.1% to 67.5%.16 We believe that indiscriminately increasing the DRAM re14Assuming 1) refreshing eight rows takes 560 ns [148], which means 70 ns to refresh one row and 2) rows across three subarrays are refreshed sequentially. In this case, refreshing 3072 rows requires 70×3072 = 215ms. 15DDR5 all bank refresh command latency (tRFC) is 410 ns for 32 Gb chip density [147]. A DRAM chip cannot serve any memory request (read or write) for tRFC after every refresh command, causing throughput loss. The memory controller issues refresh commands more frequently as refresh period reduces: once every 3.9 μs and 975 ns respectively for a refresh period of 32 ms and 8 ms. 16We estimate refresh and idle energy from manufacturer provided DRAM IDD current values [247]. 12 fresh rate to mitigate ColumnDisturb-induced bitflips is not a performance- and energy-efficient solution. We describe a more intelligent mechanism that selectively and timely refreshes only the victim rows that are disturbed by ColumnDisturb. Proactively Refreshing ColumnDisturb Victim Rows (PRVR). We propose proactively refreshing ColumnDisturb victim rows to mitigate ColumnDisturb bitflips. The key idea of PRVR is to refresh only the victim DRAM rows in three subarrays: the aggressor DRAM row's subarray and its two neighboring subarrays. PRVR refreshes each of the N victim DRAM rows across three subarrays (e.g., for subarray size = 1024, N = 3072) once before the aggressor row is hammered or pressed enough times to induce a bitflip. PRVR distributes the refresh operations targeting these N rows over the time it takes to induce the first bitflip in a subarray using ColumnDisturb, similar to how a periodic refresh command refreshes a subset of all DRAM rows in a DRAM chip. PRVR is a higher-performance and more energy-efficient solution than using a shorter DRAM refresh period. Assuming a default refresh period of 32 ms, a time to induce the first ColumnDisturb bitflip of 8 ms, and N = 3072 , we analytically estimate PRVR's DRAM throughput and energy benefits over using a fixed, 8 ms refresh period. PRVR reduces 8 ms refresh period's throughput loss by 70.5% and refresh energy consumption by 73.8% assuming one DRAM row from each DRAM bank is continuously hammered to induce ColumnDisturb bitflips in a 32 Gb DDR5 chip.17 We conclude that proactively refreshing ColumnDisturb victim rows could prevent ColumnDisturb bitflips at much smaller performance and energy overheads than simply reducing the DRAM refresh period. 6.2. Implications on Retention-Aware Refresh Retention-aware heterogeneous refresh mechanisms [75-88] leverage the heterogeneity in DRAM cell data retention time to reduce the number of DRAM row refresh operations while maintaining data integrity. These mechanisms typically classify a DRAM row as weak if any of its cells exhibit a retention failure during a relatively short refresh window (e.g., 64 ms). In contrast, a DRAM row that reliably retains data over a much longer time window (e.g., 1024 ms) is considered strong. Consequently, the greater the proportion of strong rows in a DRAM chip, the higher the performance and energy reduction provided by a retention-aware heterogeneous refresh mechanism. ColumnDisturb greatly reduces the benefits of retention-aware heterogeneous refresh mechanisms because ColumnDisturb renders many more rows weak than retention failures alone at a given refresh window, as observations 6, 18, and 19 demonstrate. Fig. 22 shows the number of DRAM row refresh operations (normalized to the number of refresh operations performed by DDR4 64 ms periodic refresh) as the proportion of weak rows in a chip varies. We plot four lines, one each for four strong row retention time values: 128 ms, 256 ms, 512 ms, and 1024 ms. A circle, diamond, and a square on a line show the number of refresh operations needed for empirically observed average proportion of retention-weak rows across all tested modules, average proportion of ColumnDisturb-weak rows across all tested 17System integration and rigorous evaluation of PRVR (e.g., using worst-case ColumnDisturb access patterns) is out of this paper's scope. We expect future work to build on the key idea of PRVR and enable new efficient and effective ColumnDisturb mitigation mechanisms. modules, and the maximum proportion of ColumnDisturb-weak rows across all tested modules, respectively. A smaller y-axis value indicates smaller expected retention-aware heterogeneous refresh mechanism speedup and energy benefits. 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of Weak Rows 0.00 0.25 0.50 0.75 1.00 1.25 Number of Refresh Operations Normalized to 64 ms Periodic Refresh DDR4 Periodic Refresh (64 ms) Strong Row Retention Time 128ms 256ms 512ms 1024ms Empirical Weak Row Fraction Retention ColumnDisturb (Mean) ColumnDisturb (Max.) Figure 22: Number of DRAM row refresh operations needed. We plot one line each for four strong row retention times of 128, 256, 512, and 1024 milliseconds. We make two key observations. First, a relatively large strong row retention time can significantly increase the expected speedup and energy benefits of retention-aware refresh mechanisms. For example, for the empirically observed average retention-weak row proportion (circles), number of refresh operations reduces by 43.1% when a strong row retention time of 1024 ms is used instead of a strong row retention time of 128 ms. Second, ColumnDisturb significantly reduces the expected speedup and energy benefits of retention-aware heterogeneous refresh mechanisms, especially at high strong row retention times where the empirical proportion of ColumnDisturbweak rows greatly exceeds that of retention-weak rows. For example, in the presence of ColumnDisturb, for a strong row retention time of 1024 ms, the number of refresh operations increases by 3.02× (purple diamond in Fig. 22) on average and by up to 14.43× (purple square in Fig. 22), compared to a baseline case without ColumnDisturb (i.e., only retention failures are present) across all tested modules. Takeaway 11. ColumnDisturb significantly degrades the expected speedup and energy saving benefits of retentionaware heterogeneous refresh mechanisms. To quantitatively demonstrate the speedup and energy benefits of retention-aware heterogeneous refresh mechanisms, we evaluate a state-of-the-art retention-aware refresh mechanism, RAIDR [76], building on the Self-Managing DRAM (SMD) framework [248,249].18 We evaluate two variants of RAIDR where 1) we store weak row addresses in a space-efficient (low-area-overhead) Bloom Filter [254-256] sized 8Kb with 6 hash functions and 2) we store weak row addresses in a spaceinefficient (high-area-overhead) bitmap19 indexed using the row address (1 bit per DRAM row, 2 Mb for a 16 GiB DDR4 module). We refresh a weak row every 64 ms and a strong row every 1024 ms. We evaluate a hypothetical DRAM system configuration called No Refresh where we do not issue any refresh operations to show the headroom for system speedup and energy 18We use Ramulator [250-253] and simulate a modern computing system (configured as the baseline system described in Table 1 of [248]). We use 20 highly-memory-intensive multi-programmed workload mixes, each comprised of four single-core workloads each with last-level cache misses-per-kiloinstructions ≥10. We warm-up the caches by fast-forwarding 100 million (M) instructions. We run each multi-core simulation until each core executes at least 500M instructions. 19Storing weak row addresses in a bitmap allows the retention-aware heterogeneous refresh mechanism to identify all weak rows accurately [86]. Compared to the Bloom Filter, using a bitmap has higher weak row identification accuracy but incurs higher storage overheads. 13 1e-05 5e-05 1e-04 5e-04 0.1% 0.2% 0.3% 0.4% 0.0 0.5 1.0 Speedup Normalized to No Refresh No Refresh example Micron module (M8) retention-weak rows 31 percentage point reduction Use an 8K-bit Bloom Filter with 6 hash functions 1e-05 5e-05 1e-04 5e-04 0.1% 0.2% 0.3% 0.4% 0.5% 1.0% 2.0% 3.0% 4.0% 5.0% 10% 20% 30% 40% 50% Proportion of Weak Rows 0.0 0.5 1.0 example Micron module (M8) retention-weak rows example Micron module (M8) ColumnDisturb-weak rows 53 percentage point reduction Store all weak row addresses in a bitmap Figure 23: Speedup provided by RAIDR [76] over the proportion of weak rows in a DRAM module. Left and right subplots shows speedup for a low-area-overhead and high-area-overhead implementation, respectively. consumption improvements of retention-aware heterogeneous refresh mechanisms. Fig. 23 shows the system speedup provided by retentionaware refresh (in weighted speedup) on average across all 20 four-core workloads normalized to the speedup of No Refresh. The left and right subplots, respectively, show the performance of the RAIDR variant that uses a Bloom Filter and the RAIDR variant that stores each weak row's address in a bitmap. The x-axis shows the evaluated proportion of weak row values in the simulated DRAM system. We annotate the speedup of retention-aware refresh for empirically observed proportion of 1) retention-weak rows (purple ×) and 2) ColumnDisturb-weak rows (green ×) in one Micron DDR4 module. We make two key observations. First, ColumnDisturb can completely negate the performance and energy (not shown in the figure) benefits of space-efficient (using a Bloom Filter) implementations of retention-aware heterogeneous refresh mechanisms. From Fig. 23 (left) we observe that as the proportion of weak rows increases from 1.00e-04 to 0.20% (by 20×), the speedup and energy benefits reduce to a point where they are almost completely eliminated (≈99 percentage point speedup and energy benefit reduction). The 20×increase in the proportion of weak rows is well within our empirical observations: across all tested modules, the proportion of weak rows increased by up to 198× (Fig. 15). As a concrete example, for the annotated Micron module (purple ×), the speedup and energy consumption benefits reduce by 31 percentage points and 32 percentage points (not shown), respectively, as the proportion of weak rows increases to accommodate ColumnDisturb-weak rows. Second, ColumnDisturb can greatly reduce the speedup and energy benefits even of space-inefficient (using a bitmap) implementations of retentionaware heterogeneous refresh mechanisms. From Fig. 23 (right), we observe that for the annotated Micron DDR4 module, the speedup and energy consumption benefits reduce by 53 and 50 percentage points (not shown), respectively. Takeaway 12. ColumnDisturb can completely negate the performance and energy benefits of low-area-overhead retentionaware heterogeneous refresh mechanisms. ColumnDisturb greatly reduces the benefits of high-area-overhead retentionaware heterogeneous refresh mechanisms. We conclude that the proportion of strong DRAM rows (that can endure high retention times without a bitflip) for a given refresh window significantly reduces with ColumnDisturb. This reduction greatly hinders (and can completely eliminate) the performance and energy benefits of retention-aware heterogeneous refresh mechanisms. 7. Related Work To our knowledge, this is the first work to experimentally demonstrate and characterize ColumnDisturb, a column-based read-disturb phenomenon in real DRAM chips. DRAM Read Disturbance Characterization. Many works [3, 4,7,71,124-126,135,136,138,139,153,154,238-246] experimentally demonstrate how a real DRAM chip's read disturbance vulnerability varies with 1) DRAM refresh rate [7,41,47], 2) the distance between aggressor and victim rows [3,7,139], 3) DRAM generation and technology node [3, 7, 47, 71], 4) temperature [71, 240], 5) time the aggressor row stays active [4,8,71,102,135,153,154,240,244,246], 6) location of the victim cell [8,71,138,244], 7) wordline voltage [136], 8) supply voltage [245], 9) reduced DRAM timing parameters [124,126], and 10) time [125]. None of these works analyze or demonstrate ColumnDisturb, a column-based read disturbance phenomenon. DRAM Retention. Many works experimentally study DRAM data retention [84,85,104,117,132,159,162,166,224,257,258]. Several works [75-88] present various error profiling algorithms to identify data retention failures when reducing the refresh rate and propose retention-aware heterogeneous refresh mechanisms to alleviate refresh overheads. However, we show that ColumnDisturb can significantly amplify the bitflips across subarrays and can cause significant performance and area overheads to retention-aware heterogeneous refresh mechanisms. Bitline Disturbance in 4F2 VCT DRAM. Emerging highdensity 4F2 DRAM architectures with Vertical Channel Transistors (VCT) are known to be vulnerable to disturbances from the bitline with a hammering-like access pattern [259-263]. However, since the device architecture of 4F2 VCT DRAM is completely different from the contemporary COTS 6F2 DRAM that we characterize in this work, we believe the error mechanism they study (i.e., floating body effect [261,262]) is different from the one(s) that causes ColumnDisturb (see §4 for our analyses and hypotheses). 8. Conclusion We present the first experimental demonstration and analysis of a new read disturbance phenomenon, ColumnDisturb, in modern DRAM chips: hammering or pressing an aggressor row disturbs DRAM cells through a DRAM column and induces bitflips in DRAM cells sharing the same columns as the aggressor row (across as many as three DRAM subarrays). Our experimental characterization of 216 real DDR4 and 4 HBM2 DRAM chips reveals that 1) ColumnDisturb is fundamentally different from RowHammer & RowPress, 2) ColumnDisturb worsens with DRAM technology scaling, 3) introduces bitflips within the nominal refresh window under nominal operating conditions in some real DRAM chips, and 4) DRAM is significantly 14 more vulnerable to ColumnDisturb than to retention failures. We describe and evaluate two ColumnDisturb mitigation techniques. We hope and believe that our detailed demonstration and analyses will inspire future works on better understanding ColumnDisturb at the device-level and mitigating it before it becomes a bigger vulnerability in future DRAM chips. Acknowledgments We thank the anonymous reviewers of MICRO 2025 for feedback. We thank the SAFARI Research Group members (especially Konstantinos Sgouras and Harsh Songara) for constructive feedback and the stimulating intellectual environment. We acknowledge the generous gift funding provided by our industrial partners (especially Google, Huawei, Intel, Microsoft), which has been instrumental in enabling the research we have been conducting on read disturbance in DRAM in particular and memory systems in general [101]. This work was in part supported by a Google Security and Privacy Research Award and the Microsoft Swiss Joint Research Center. References [1] Robert H. Dennard. Field-Effect Transistor Memory, 1968. U.S. Patent 3,387,286. 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2510.14746
|
Quantum remeshing and efficient encoding for fracture mechanics
Ulysse Rémond,1, 2 Pierre-Emmanuel Emeriau,3 Liam Lysaght,3 Jean Ruel,1 Joseph Mikael,1 and Kyryl Kazymyrenko1
1EDF R&D, 7 Bd Gaspard Monge, 91120 Palaiseau, France
2Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LUX, F-75005 Paris, France
3Quandela, 7 rue Léonard de Vinci, 91300 Massy, France
(Dated: October 17, 2025)
We present a variational quantum algorithm for structural mechanical problems, specifically ad-
dressing crack opening simulations that traditionally require extensive computational resources. Our
approach provides an alternative solution for a relevant 2D case by implementing a parametrized
quantum circuit that stores nodal displacements as quantum amplitudes and efficiently extracts
critical observables. The algorithm achieves optimal nodal displacements by minimizing the elastic
energy obtained from finite element method. The energy is computed with only a polylogarithmic
number of measurements. Extracting relevant scalar observables such as the stress intensity factor
is then done efficiently on the converged solution. To validate the scalability of our approach, we de-
velop a warm start strategy based on a remeshing technique that uses coarse solutions to circumvent
barren plateaus in the optimization landscape of the more refined problems. Our method has been
experimentally validated on Quandela’s photonic quantum processor Ascella and comprehensive
numerical simulations demonstrate its scalability across increasingly complex quantum systems.
I.
INTRODUCTION
Partial differential equations (PDEs) provide the math-
ematical foundation for modeling a wide range of phys-
ical phenomena, from fluid dynamics and heat transfer
to electromagnetic wave propagation and structural me-
chanics.
In solid mechanics,
the stationary Navier–Cauchy
equation governs the equilibrium state of an elastic con-
tinuum. It serves as a key building block for fracture me-
chanics [1] and thermo-hydro-mechanical coupling mod-
els [2]. Its extensions capturing nonlinear hyperelasticity
[3] are being studied for their industrially relevant appli-
cations [4].
This governing equation belongs to the class of second-
order elliptic PDEs, which can be reformulated as an
energy minimization problem [5].
The Finite Element
Method (FEM) [6] is particularly effective for approxi-
mating the unique solution by optimizing nodal variables
over a discretized spatial domain.
Modeling the fracture process requires the introduction
of geometric singularities, such as local shape discontinu-
ities, to represent the initial cracks.
This is an inher-
ently multiscale procedure that, for the hydraulic dam
example, ranges from the millimeter-scale concrete ag-
gregate (10−3m) to the infrastructure size (102m). Even
if the fracture mechanics may involve the estimation of
only a single scalar quantity as an output, simulating
multi-crack propagation with high precision remains an
extremely demanding task for state-of-the-art methods:
current capacities may handle up to 1013 degrees of free-
dom in the resulting linear algebra problem [7], which
tend to be insufficient in the 3D applications.
These challenges highlight the limitations of classical
FEM for fracture mechanics, particularly its steep cost
when the initial crack distribution is uncertain. Quantum
algorithms offer a promising alternative for these simula-
tions. Recent progress in quantum approaches to PDEs
reflects this trend, targeting fundamental bottlenecks in
classical numerical methods [8–20].
The Harrow-Hassidim-Lloyd (HHL) algorithm [8] ini-
tially promised an exponential speedup for solving linear
systems of equations. However, its practical application
to PDEs faces significant constraint: it assumes an effi-
cient quantum encoding of both the system matrix and
right-hand side vector. The latter is rather challenging
for geometries containing crack singularities, rendering
the original HHL algorithm impractical for such applica-
tions. Tailored quantum algorithms for specific problems
have also been proposed [10], but these approaches re-
quire large fault-tolerant quantum computers, which re-
main beyond current capabilities.
Consequently, some research has been shifted towards
Variational Quantum Algorithms (VQA) solving PDEs
on Noisy Intermediate-Scale Quantum (NISQ) comput-
ers [13, 18, 19].
Most existing work, however, targets
one-dimensional problems and suffers from optimization
challenges such as barren plateaus [21, 22], where gradi-
ents vanish exponentially with quantum system size.
Our work addresses these limitations along several crit-
ical directions:
we extend the scope to realistic two-
dimensional cases that better capture physical systems;
we propose an efficient quantum encoding tackling a
fracture mechanics problem; we experimentally validate
the algorithm by running it on a photonic quantum
device [23]; and we develop a novel cascading warm-
start strategy based on solutions from coarser meshes.
Since the number of qubits directly corresponds to mesh
refinement, our hierarchical approach enables solutions
from rougher meshes to guide the initialization for finer
meshes, mitigating the barren plateau problem and sup-
porting higher-resolution simulations.
arXiv:2510.14746v1 [quant-ph] 16 Oct 2025
2
II.
RESULTS
A.
Mechanics of fracture in a pre-cracked plate
Linear elastic fracture mechanics studies the conditions
under which a crack in a material will propagate. Here
we focus on the standard example of a pre-cracked 2D
plate (see Fig.1). System deformation evolves according
to the static equilibrium linear Navier-Cauchy equation
[5]:
grad(div u) + (1 −2ν)∆u = 0
(1)
where u = (ux, uy) is the two components vector dis-
placement field, ν ∈[0, 1/2) is the Poisson’s ratio of
the material, ∆is the Laplacian.
Near the crack tip
(Fig.1), the displacement gradient diverges [24]; hence,
the leading-order term dominates the solution and can
be extracted. The corresponding scalar coefficient, called
Stress Intensity Factor (SIF) [25], is therefore determined
from the crack opening profile close to the tip:
uy(r) ≈SIF ∗2(1 −ν2)
p
2r/π,
for r ≈0
(2)
where r is the distance from the crack tip. In general,
the SIF admits no closed-form expression for arbitrary
geometries and boundary conditions. It is therefore tab-
ulated for canonical cases or estimated numerically in spe-
cialized handbooks [26].
For more complex cases, discretization schemes such as
Finite Differences or the Finite Elements Method (FEM)
are commonly used to solve Eq. (1) and to estimate nu-
merically the SIF’s value [27]. In FEM, the displacements
are computed only on a grid of nodes and interpolated be-
tween them. We use a regular mesh of Nx × Ny nodes
of commensurate dimensions Nx = 2nx and Ny = 2ny, so
that the total number of degrees of freedom (DoFs) can
be written as N = 2·Nx ·Ny ≡2n, where n = nx +ny +1
accounts for both components of the nodal displacement.
The mechanical deformation problem is converted into a
finite-size optimization of the elastic energy Efe(⃗u), that
is equivalent to the linear algebra problem [6]:
min
⃗u Efe(⃗u) = min
⃗u
1
2⃗u · K⃗u −⃗u · ⃗f
⇔
K⃗u = ⃗f
(3)
where ⃗u ∈RN is the vector of unknown nodal displace-
ment corresponding to the discretized problem, ⃗f ∈RN
is the vector of external forces, and K ∈MN(R) is the
stiffness matrix.
While ⃗f captures all variety of Neumann type bound-
aries, the Dirichlet conditions can be also enforced ei-
ther through a complementary penalization contribution
to the initial K-matrix or by a projection technique that
we cover in the next section.
x
y
W
H
Crack tip
f
f
f
f
f
f
f
a
b
c
d
Na = (1 −x)(1 −y)
Nb = x(1 −y)
Nc = xy
Nd = (1 −x)y
2D Plate
Figure 1. Geometry of the pre-cracked 2D plate under exter-
nal loading, showing the crack tip and nodal grid. This mesh
underlies the FEM formulation (Eq. 3) and the operator de-
composition (Eq. 5).
The matrix K, introduced in Eq. (3), is computed
via FEM and is related to the gradient overlap of the
displacement interpolation functions between adjacent
nodes.
Although K is sparse, its naive expansion into
Pauli operators {X, Y, Z} scales with the number of
nonzero elements i.e., effectively exponential in nx and
ny. This would render a direct quantum implementation
inefficient. Exploiting mesh symmetries, we obtain one
of our main results: a polynomial-size decomposition of
K into tensor products. Indeed, if we define S as:
X =
0 1
1 0
Y =
0 −i
i
0
Z =
1
0
0 −1
S = {p±, σ±, I} p± = (I ± Z)/2 σ± = (X ± iY )/2,
(4)
the K matrix can be rearranged in a linear combination
of tensor products {Mi} of matrices from S.
K =
X
k
µkMk, with Mk =
n−1
O
l=0
mk,l,
mk,l ∈S,
(5)
where, µk ∈R is a set of decomposition coefficients. The
derivation and explicit decomposition of K is detailed in
3
Methods IV C. Crucially, the number of terms in the sum
in Eq. (5) scales logarithmically with the system size, that
is as O(nxny).
We will now explore how the decomposition Eq. (5)
paves the way to efficiently solving the discretized version
Eq. (3) of Eq. (1) on a quantum computer.
B.
Solving efficiently elastic differential equations
on a quantum computer
We now reformulate the minimization problem Eq. (3)
in quantum terms.
A trial displacement vector is en-
coded as a parametrized quantum state |ψ⟩= A |0⟩,
where A is the ansatz unitary.
As usual with VQAs
[18, 19], the aim is to optimize the ansatz parameters
so that it converges towards the target quantum state
|ψtgt⟩= ∥⃗u∥−1 P
i ui |i⟩, where ui are the components of
the solution ⃗u ∈RN of the discretized problem Eq. (3)
and |i⟩are the 2n basis states of n-qubit quantum system,
defined later in Eq. (9).
We provide the quantum counterpart for the external
force vector, that is supposed to be normalized: |f⟩=
P
i fi |i⟩, as explained later in Eq. (10). Given the fact
that the ansatz unitary generates a normalized amplitude
vector in CN, the straightforward transcription of the
cost function Eq. (3) can be done only in the presence
of at least one unrestricted rigid body motion, i.e. when
det K = 0:
Eq(ψ) = 1
2 ⟨ψ| K |ψ⟩−Re ⟨ψ|f⟩.
(6)
In this case the normalization constraint is absorbed
into irrelevant kernel subspace movement. Otherwise, if
det K ̸= 0, the normalization coefficient should be ex-
plicitly introduced in the cost function. The norm opti-
mization leads to a new form of normalization-agnostic
quotient cost function [19]:
Eqq(ψ) = −(Re ⟨f|ψ⟩)2
2 ⟨ψ| K |ψ⟩.
(7)
Regardless of whether Eq. (6) or Eq. (7) is used, evalu-
ating the cost requires only two quantities: ⟨ψ| K |ψ⟩and
⟨f|ψ⟩. We compute ⟨ψ| K |ψ⟩by decomposing K into a
polylog number of Mk matrices (see Eq. (5)).
Indeed,
only computations of the generic single qubit observable
terms ⟨ψ|Mk|ψ⟩≡⟨ψ|mk,0 ⊗mk,1 ⊗... ⊗mk,n−1|ψ⟩are
needed, where mk,l ∈S and k ∈{0, . . . , 2(nxny + nx +
ny)}, as explained in Methods IV C.
We can estimate each term of this decomposition us-
ing common diagonalization techniques. While p± are al-
ready diagonal, the non-trivial observable measurements
in the S basis are tensor powers of σ±. Noticing that
each such σ± term is present with its transposed coun-
terpart, we focus on ⟨ψ| σ⊗n
+ +σ⊗n
−|ψ⟩as an example. We
remark that the CNOT-gate maps one non-hermitian op-
erator σ+ within a σ⊗n
+
term into a diagonal projector p+.
More precisely, for two-qubit systems it is easy to verify
that CNOT·σ+ ⊗σ+ ·CNOT = σ+ ⊗p+, for three qubits
I ⊗CNOT·σ+ ⊗σ+ ⊗σ+ ·I ⊗CNOT = σ+ ⊗σ+ ⊗p+ and
so on. These (n −1) inductively-arranged CNOT-gates
can reduce σ⊗n
+
to σ+ ⊗p⊗(n−1)
+
, and similarly σ⊗n
−
to
σ−⊗p⊗(n−1)
+
gates. This CNOT-chain rotates |ψ⟩into
| ˜ψ⟩, leading to the following easily-diagonalizable opera-
tor estimator:
⟨ψ| σ⊗n
+
+ σ⊗n
−|ψ⟩= ⟨˜ψ| (σ+ + σ−) ⊗p⊗(n−1)
+
| ˜ψ⟩,
(8)
where | ˜ψ⟩= Qn−2
i=0 CNOTi,i+1 |ψ⟩is an inductively trans-
formed quantum state; the CNOTi,i+1 operation applies
the CNOT operator to two adjacent qubit. Noting that
X = σ+ + σ−, the last step of diagonalization is done
with the help of the Hadamard H-gate using the identity
Z = H · X · H. Because X · σ+ · X = σ−, all the oth-
ers non-trivial tensor polynomials can be reduced to the
previously discussed case.
Photonic computers implement the non unitary projec-
tion p±, and thus also σ± = p±X, as elementary gates.
Indeed in dual rail encoding, it amounts to herald the
success of the projection on the relevant optical mode,
that is measuring one mode and not detecting a photon
there. As a consequence, ⟨ψ| K |ψ⟩can alternatively be
computed as a sum of photonic counts of the zero state
⟨0| after the circuit A†KA |0⟩, which allows us to compute
the estimator without the need for CNOT-chains.
Furthermore, we provide an explicit quantum circuit
for generating the boundary force vector |f⟩. To do so,
we define the ordering of the degrees of freedom such that:
|ψtgt⟩=
1
∥⃗u∥
2n−1
X
i=0
ui |i⟩=
X
x,y,d
u(x, y, d)
ny
z}|{
|y⟩⊗
nx
z}|{
|x⟩⊗
1
z}|{
|d⟩,
(9)
where u(x, y, d) is a shorthand notation for the normal-
ized components of discretized displacement. Any grid in-
dex 0 ≤ig < Nx×Ny is represented in binary by a pair of
coordinate indexes also written in binary x, y (ig ≡yx).
The additional bit d separating horizontal (d = 0) and
vertical (d = 1) displacement components is also intro-
duced, which allows any qubit state to encode some given
degree of freedom |i⟩= |y, x, d⟩≡|y⟩⊗|x⟩⊗|d⟩.
An explicit representation of the unitary operator that
generates an homogeneous Neumann boundary at the top
of the plate (see Fig.1) is given by:
|f⟩= X⊗ny ⊗H⊗nx ⊗X |0⟩,
(10)
allowing us to compute directly ⟨f|ψ⟩. For more com-
plex boundary conditions, one may rely on the additive
decomposition of the external boundaries or use a Walsh
Series Loader [28].
The finite element stiffness matrix K (given in Eq. (21))
transcribes a purely Neumann boundary problem result-
ing in det K = 0. Dirichlet boundaries greatly enrich the
4
A5
A7
Optimize
Remesh
Optimize
Remesh
|1 0⟩
|0 0⟩
|1 1⟩
|0 1⟩
|1100⟩|1101⟩|1110⟩|1111⟩
|1000⟩|1001⟩|1010⟩|1011⟩
|0100⟩|0101⟩|0110⟩|0111⟩
|0000⟩|0001⟩|0010⟩|0011⟩
Figure 2. We first optimize a 5-qubit state, and duplicate it to use it as a warm start for the 7-qubit ansatz. This scheme is
applied iteratively until the desired system size is reached as seen in Fig. 3a.
variety of treated situations, and eliminate spurious rigid
body motions. We therefore consider the case of perfect
clamping, where some DoFs are set to zero. It can be
formally written with the help of the hermitian projector
P ∈MN, as P |ψ⟩= 0. We enforce it in the quantum
system through a systematic projective restriction of all
trial states, equivalent to the stiffness matrix modifica-
tion: K 7→(I⊗n −P)K(I⊗n −P) + P. The decomposition
of any projector P within the S basis set is directly de-
rived from the binary DoF’s position encoding: p+ ↔0
and p−↔1, the last bit serving the choice between the
ux and uy DoF’s components.
P =
y coord to block
{p±}⊗ny
⊗
x coord to block
{p±}⊗nx
⊗
ux or uy
p±
.
(11)
The plate shown in the Fig. 1 exhibits a symmetry with
respect to the crack plane.
We model only the upper
half, with the horizontal centerline constrained against
vertical displacement. Moreover, we restrict horizontal
rigid body movement by clamping the ux component of
the crack tip. It can be assessed via the following additive
contributions for the full state projector P:
p⊗ny
+
⊗p−⊗I⊗nx−1 ⊗p−⇐uy = 0 on the centerline,
p⊗ny
+
⊗p−⊗p⊗nx−1
+
⊗p+ ⇐ux = 0 on the crack tip.
Finally, we give circuit implementations for two com-
mon physically-relevant observables:
SIF, defined in
Eq. (2), and crack opening displacement (COD). With
chosen numbering order Eq. (9), the COD-observable is
estimated by multiplying the quantumly-computed norm
[19] with the amplitude of probability of zero counts in
the optimized state |ψ⟩:
COD = ⟨0|ψ⟩× ∥⃗u∥, where ∥⃗u∥= Re ⟨f|ψ⟩
⟨ψ|K|ψ⟩
1
2nx/2 . (12)
We similarly compute the SIF from the measurement of
displacement amplitudes on the crack lips. In the vicin-
ity of the crack tip the solution for the crack opening
scales as a square root of the distance from the crack tip
Eq. (2).
It could allow one to calculate the SIF from
a single displacement value.
However, the single-point
measure would be highly error-prone. On the one hand,
close to the crack tip the displacement amplitude value
is vanishing necessitating an important shots number to
eliminate statistical noise, on the other hand, far from the
crack tip the asymptotic solution deviates from its theo-
retical expression Eq. (2). Therefore, we opt for a mul-
tiple point estimator and give here an integral measure-
ment that averages the SIF-value over the whole crack lip
of length W/2, see Fig.1:
SIF =
√π ∥⃗u∥
(1 −ν2)W
sZ W/2
0
[u(x, y = 0, d = 1)]2 dx (13)
The discretized value of the integral can once again be
computed with the help of the projector P = p⊗ny
+
⊗p+ ⊗
I⊗nx−1 ⊗p−:
Z W/2
0
[u(x, y = 0, d = 1)]2 dx ≈W
2nx ⟨ψ|P|ψ⟩,
SIF =
p
π ⟨ψ|P|ψ⟩· Re ⟨f|ψ⟩
(1 −ν2)
√
W ⟨ψ|K|ψ⟩
1
2nx
(14)
Alternatively, for large-size systems, a more precise
measurement can be obtained by restricting the integra-
tion domain closer to the crack tip. For x ∈[ 3W
8 , W
2 ], the
projector p⊗ny
+
⊗p+ ⊗p⊗2
−⊗I⊗nx−3 ⊗p−should be used.
Interestingly, during the optimization both physical ob-
servables (COD and SIF) converge to their targets much
faster than the cost function itself, due to the high energy
density concentration close to the crack tip —see Fig. 3b.
5
C.
Warm start with coarse mesh solution
Barren plateaus pose a major challenge in VQA op-
timization. They represent regions where the cost func-
tion’s gradient vanishes across most parameter variations.
Having defined a global [22] cost function, we observe a
flat landscape for the random state initialization in suffi-
ciently large systems (n ≳10), hampering optimization.
In order to avoid this difficulty, we derive a quantum ver-
sion of the mesh refinement technique [29]. This approach
paves the way towards consistent warm start creation, i.e.
problem-agnostic strategies to bypass barren plateaus in
PDEs.
The idea is to define two discretized versions (coarse
and refined) of the same continuous problem and to use
the converged coarse n-qubits solution as the initial state
for the refined (n + 2)-qubits solution. This strategy can
be repeated iteratively until we reach the desired preci-
sion, hence a cascaded approach. Since barren plateaus
are not an issue for a low number of qubits, we can always
compute the target function from a cold start (i.e. using
a randomly chosen initial state) for the coarsest meshes.
Once we have solved the problem for a mesh, we du-
plicate the mesh, increasing the number of qubits — one
additional qubit for each x and y coordinates to main-
tain the elements’ aspect ratio. The basis state written
in binary |y⟩|x⟩is extended to |y⟩|x⟩|0⟩|0⟩. Then, all
the data are duplicated, since each newly added qubit
is transformed into |+⟩= H |0⟩with a Hadamard gate.
Finally, we enforce the order of qubits (see Sec. IV E )
creating new extended coordinates along both directions,
so that the initial coarse state |y⟩|x⟩is mapped to the re-
fined one |y⟩|+⟩|x⟩|+⟩. In summary, the procedure can
be illustrated as:
|y⟩|x⟩
+2
7→|y⟩|x⟩|00⟩
H
7→|y⟩|x⟩|+⟩|+⟩
swap
7→|y⟩|+⟩|x⟩|+⟩.
Any smooth function represented by a n-qubit state
turns into the piecewise-constant function represented by
this (n+2)-qubit state, see Fig. 9. This procedure is sim-
ilar to a homogeneous remeshing with constant field ex-
trapolation on the nearest neighbors. The interpolation
of the original displacement field introduces local field
discontinuities, shifting the (n + 2)-cost function from
the target value already achieved at the previous step
of n-qubit optimization. This error induced by the field
projection is well known and has been studied in depth
in the classical remeshing technique [29].
Third, we further fine-tune this displacement field with
another (n + 2) parametrized circuit until convergence is
achieved, resulting in the local field relaxation commonly
called the equilibration phase after field projection.
This whole three-step process can be repeated (Fig.
3a), increasing the mesh refinement up to the desired
precision. In doing so, we obtain successive warm starts
to avoid barren plateaus at each step as shown in Fig.
3b.
We test numerically this quantum remeshing proce-
dure. We achieved quantifiably better results than the
...
...
· · ·
...
A5
Remesh(5→7)
A7
Remesh(17→19)
A19
(a) Cascaded warm-start remeshing strategy.
5
7
9
11
13
15
17
19
0
20
40
60
80
100
Energy (cold)
Energy (warm)
SIF (warm)
CO (warm)
Fidelity (warm)
Number of Qubits
Observables
(b) Evolution of observables with increasing discretization.
Convergence ratio of remeshing-based and cold-start results with
regard to the target continuous values.
5
7
9
11
13
15
17
19
0
20
40
60
80
100
Energy (Cold)
Energy (Remeshing)
Number of Qubits
Observables
(c) 2H one-step optimization, comparing remeshing-based and
cold-start results with the same-size classical solutions, showing
the average, best, 1st and 9th deciles.
Figure 3.
Cascaded simulations from 5 to 19 qubit mesh.
In (b) the iterative remeshing, performing 48h optimization
at each step, allows to reach 19 qubit simulation with 50%
precision of observable COD and SIF, far beyond the capaci-
ties of any tested cold start strategies. (c) illustrates a single
step remeshing capabilities to bypass the initial barren plateau
with 50 Layers ansatz. The n + 2-qubit state is optimized for
2 hours starting from an ideal state before duplication, ob-
tained from n-qubit discretization.
cold start strategies. Initialized from 5 qubits onwards,
systems composed of up to 5 · 105 DOFs, i.e. 19 qubits,
were successfully simulated with this method using the
circuit in Fig. 3a.
As shown in Fig. 3b, even for low energy convergences
at the most refined simulations, the cascading remeshing
procedure proved especially reliable in reaching satisfac-
tory convergence for the relevant mechanical observables:
6
SIF (60%) and COD (70%). Cold start simulations had
error ratios of more than 100% from 13 qubits onwards.
The SIF and COD precision peaks near 9 qubits as the
coarser value suffer from classical discretization errors,
and the finer values suffer from a limited VQA conver-
gence time and the lack of an tailor-made ansatz.
Fidelity stays high (over 87%) throughout the whole
cascading procedure, and optimizations using it as cost
function yield unrealistic physical deformations.
This
illustrates how even quantum states with high fidelity
might not correspond to a physically-acceptable state,
justifying the need of more relevant quantities such as
the elastic energy to be used as cost function.
The respective convergences of such observables in 2
hours from the classical solution at the former step can
be assessed in Fig. 3c.
D.
Experimental implementation
We experimentally validated a 4-qubit version of our
variational quantum algorithm on Quandela’s linear op-
tical QPU [23].
Our ansatz uses the QLOQ compres-
sion scheme [30] which enables efficient encoding of multi-
qubit states onto photonic hardware. The parametrized
quantum circuit was encoded on two photons, with each
run consisting of 105 shots. Ascella is a photonic quan-
tum processor that utilizes a semiconductor quantum dot
based on-demand single photon source [31], with two-
photon interference visibility above 94%. Up to 6 single
photons can be fed synchronously into every odd input
of a 12-mode universal interferometer implemented in a
photonic integrated circuit. The interferometer is fully re-
configurable, enabling the implementation of 12×12 uni-
tary transformations with high fidelity – 99.7% amplitude
fidelity for Haar random 12x12 unitaries [32]. A thresh-
old single photon detector system measures the photon
distribution and coincidences at the output of the optical
circuit.
10 QPU runs are displayed in Figure 4 to show that
this is working on average. The mean final energy esti-
mate of these QPU runs was −23.15, with a best result of
−24.060±0.003 while the target value is −24.895. To fur-
ther enhance accuracy, basic error mitigation techniques
were applied during the runs (see Sec. IV A for details).
This performance aligns with the best-case scenario pre-
dicted by noiseless simulations, which reached −23.952.
Figure 4. The results from 10 runs on Quandela’s linear op-
tical QPU. The blue line shows the mean energy estimate
at each iteration, while the shaded region is bounded by the
minimum and maximum values. Run 8, which produced the
minimum energy estimate overall, is shown as a pink line.
III.
DISCUSSION
Our results demonstrate the feasibility of address-
ing computationally intensive structural mechanics prob-
lems—specifically fracture mechanics—using VQAs. By
formulating the 2D Navier-Cauchy equation as an elastic
energy minimization problem, we circumvent the imprac-
tical requirements of algorithms like HHL while provid-
ing a clear route to scalability with the developed warm
start strategy. To our knowledge, this work constitutes
the first implementation of a two-dimensional elasticity
problem with realistic boundaries on a quantum platform,
marking an important step beyond prior studies, which
were limited to one-dimensional PDEs, simpler bound-
aries or scalar cases. The proposed encoding, which stores
nodal displacements as quantum amplitudes, allows ef-
ficient computation of critical observables such as the
stress intensity factor (SIF) and crack opening displace-
ment (COD) through simple measurements.
A major contribution of this work is the development
of a quantum remeshing technique that systematically
mitigates barren plateaus –one of the primary obstacles
in VQA optimization.
By initializing finer discretiza-
tions from optimized coarser meshes, we demonstrated
a significant improvement in convergence over cold-start
and simple warm-start strategies. We numerically val-
idated this hierarchical warm-start approach up to 19
qubits, illustrating scalability to physically meaningful
regimes. Moreover, we find that physical observables con-
verge much faster than the energy functional, suggesting
that even partially optimized states can yield accurate
engineering quantities of interest.
The experimental validation on a photonic quantum
processor confirms the practical implementability of our
approach on real hardware. Despite noise and device im-
perfections, our results show we achieve energy estimates
7
within ≈96% of the theoretical optimum with basic error
mitigation, closely matching noiseless simulations.
Looking ahead, several avenues merit exploration.
First, the generalization to three-dimensional domains
and vector-valued PDEs beyond elasticity (e.g., thermal
and hydraulic diffusion, linear themo-mechanical cou-
pling...) is straightforward within our encoding scheme
and would significantly broaden the applicability of this
method. Second, further improvements of the optimiza-
tion procedure, possibly through problem-inspired an-
sätze, tensor-network-inspired decompositions and tai-
lored classical minimization algorithms, could enable sim-
ulations on larger systems and with more complex geome-
tries. Finally, given the modular nature of the algorithm,
it offers a robust benchmarking tool for quantum hard-
ware.
Overall, this work establishes a concrete pathway for
leveraging quantum computing in high-fidelity structural
simulations.
By coupling hierarchical VQA strategies
with scalable encodings and efficient observable estima-
tion, and by demonstrating the first 2D case on real hard-
ware, we move one step closer to practical quantum ad-
vantage in computational mechanics.
IV.
METHODS
A.
Fracture Mechanics
The present paper examines the second order Navier-
Cauchy differential vector-equation [33] that naturally
arises in the isotropic linear elasticity theory under hy-
pothesis of small transformations. In the 2D case under
additional plane strain hypothesis, the mechanical equi-
librium of the structure may be written as a set of two
constraints on the two components vector displacement
field u = (ux, uy), where uα = uα(x, y) for α = {x, y}
is a scalar displacement in the direction of one of its eu-
clidean coordinates α at the given position (x, y) in space.
The Navier-Cauchy equilibrium problem in the absence
of volumetric density forces (i.e. weight) then reads as
a constant coefficients linear system for the two compo-
nents displacement field u = (ux, uy):
(
∂x(∂xux + ∂yuy) + (1 −2ν)(∂2
x + ∂2
y)ux = 0,
∂y(∂xux + ∂yuy) + (1 −2ν)(∂2
x + ∂2
y)uy = 0;
(15)
where ν is the Poisson’s ratio of material.
Commonly
used boundary conditions in mechanics often involve a
combination of Dirichlet-type conditions, such as perfect
clamping (u(x, y) = 0 for (x, y) ∈∂ΩD) and Neumann-
type constrains on the gradient of displacement ∂ΩN =
∂Ω\ ∂ΩD.
The problem can be rewritten as a variational mini-
mization of the elastic density functional E. In the case
of full Dirichlet condition constraining u(x, y) to be equal
to uD(x, y) on the entire boundary ∂Ω, E is a quadratic
form:
E[u] = 1
4
Z
Ω
(∂xux + ∂yuy)2dΩ+
(16)
+1 −2ν
4
Z
Ω
(∂xux −∂yuy)2 + (∂xuy + ∂yux)2
dΩ.
Any displacement field minimizing Eq. (16) and satisfying
this Dirichlet boundary respects the Euler-Ostrogradski
extrema conditions that are equivalent to the initial set
of equations Eq. (15).
In the presence of Neumann conditions, a linear term
is added to Eq. (16). This modification is presented in
Eq. (17), albeit in the discretized case.
Eq. (15) being parametrized by a single coefficient ν,
the solution’s complexity stems from either the boundary
conditions or the sample geometry.
Theoretical solutions for the sample with a sharp-edged
geometry exhibit a divergent gradient of displacement in
the vicinity of this geometric singularity [25]. The lead-
ing order in this divergent asymptotic expansion is called
the stress intensity factor (SIF), Eq. (2).
SIF quanti-
fies the threshold beyond which crack propagation occurs,
characterizing the critical structure’s loading. It is thus
widely used in Linear Elastic Fracture Mechanics theory
(LEFM) [24, 34]. Another typical singularity-describing
observable is the crack opening displacement (CO)[26],
which reveals the deformation level. While the displace-
ment field u is a primary variable of interest for general
mechanical problems, fracture mechanics mainly focuses
on the computation of aforementioned pair of observables
on each singularity. The CO is a local pointwise measure-
ment, whereas the SIF is obtained via integration over a
domain close to a singularity.
In the current work, we use a quantum variational algo-
rithm to numerically solve the classical 2D LEFM prob-
lem of a pre-cracked plate (Fig. 1), subjected to a uni-
form density of force f on its upper and lower bounds,
neglecting the off-plane movement.
B.
Numerical discretization
As in any linear PDE, the Navier-Cauchy equation ad-
mits an approximate discretized solution. In the finite
difference method the differential operators are estimated
on a grid (see Fig.Eq. (1)), resulting in a linear algebraic
equation for the vector of nodal unknowns ⃗u: K⃗u = ⃗f. If
Ng is the total number of grid nodes (whereas N = 2Ng
is the number of DoF), ⃗u = (uix, uiy)Ng
i=1 ∈RN is a dis-
cretized approximation of the continuous displacement
field u, i.e. the finite dimensional vector of nodal dis-
placement in both horizontal (x) and vertical (y) direc-
tions. The stiffness matrix K is large, but usually sparse.
In the more subtle finite element method (FEM), the
continuous minimization problem is converted into a
8
finite size optimization as the trial displacement field
is reduced to a linear combination of element by el-
ement piecewise continuous polynomials:
u(x, y)
=
PNg
i=1 Ni(x, y)ui. As each basis function Ni is normal-
ized to unity at its reference node i vanishing outside
the adjacent elements region, all the expansion coeffi-
cients represent the nodal values of displacement field:
ui ≡u(xi, yi). The minimization of the discretized en-
ergy Eq. (16) becomes a finite dimensional algebraic prob-
lem, see also Eq. (3):
Efe(⃗u) = 1
2
Ng
X
i,j=1
X
α,β
uiαujβKiαjβ −
Ng
X
i=1
X
α
uiαfiα
(17)
Any Neumann-type boundary is represented by some vec-
tor of external forces ⃗f ∈RN; and the Greek letters α, β
run over Euclidean coordinates {x, y}.
All coefficients
Kiαjβ of a FEM’s stiffness matrix are related to a generic
gradient correlator expression:
R
Ω∂αNi∂βNj dΩ.
The minimization of discretized energy potential Efe(⃗u)
by variation of the nodal displacements amplitudes is
once again equivalent to resolving the algebraic linear
equation.
C.
Tensor product decomposition of the stiffness
matrix
In order to estimate the N 2 coefficients of K ma-
trix, all cross-integrals of the gradient of shape functions
R
Ω∂αNi∂βNj dΩshould be calculated on the whole Ω,
scaling as ∼N 2. For large N, the FE software rely on a
more efficient assembling algorithm, that scales linearly
is system size. First, the domain is split into a set of sim-
ple polygonial subdomains, called elements: Ω= S
el Ωel.
Second, for each element, all the connected stiffness con-
tributions are computed, giving for quad4 mesh a 4 × 4
elementary matrix. In what follows, we extend this pro-
cedure with the aim of decomposing the stiffness matrix
into a more primitive form of tensor product of 2 × 2
matrices more suitable for quantum calculations.
The whole domain integrals can be separated into the
sum of elementary contributions:
Z
Ω
∂αNi∂βNj dΩ=
X
el
Z
Ωel
∂αNi∂βNj dΩ;
(18)
We consider here a rectangular Nx × Ny domain, com-
posed of linear Quad4 quadrangles [35, 36].
Here Nx,
or Ny is the number of discretization nodes in x (or y
direction). Since for this regular mesh all the elements
have the same geometry, all the elementary integrals in
K are identical. It allows us to separate the nodal (i, j)
connectivity from the (x, y) component’s links stored in
2 × 2 matrices Kij. These elementary contributions are
only dependent on their local connectivity index pairs.
With local index notations i, j ∈{a, b, c, d} (see Fig.1)
each matrix term is given by the integral over the Quad4
quadrangle:
Kij ≡
Z
quad
dΩ
(1 −ν)∂xNi∂xNj + (1 −2ν)/2∂yNi∂yNj
(1 −2ν)/2∂yNi∂xNj + ν∂xNi∂yNj
(1 −2ν)/2∂xNi∂yNj + ν∂yNi∂xNj
(1 −ν)∂yNi∂yNj + (1 −2ν)/2∂xNi∂xNj
(19)
All expressions for Kij can be found in Appendix B 1.
Let us first analyse the contribution of an on-site Kaa
term. An initially intuitive idea is to set it as a purely
diagonal block matrix. However, the expected diagonal
nature of the on-site term comes from the sum of all the
four P
i∈{a,b,c,d} Kii contributions. For boundary nodes,
the summation is only partial. The bottom-left corner
node, number 1 on (Fig.5), has exclusively the Kaa on-
site contribution. As the total number of elements in the
mesh is equal to (Nx−1)(Ny−1), Kaa appears then in the
global rigidity matrix the same number of times, namely
once during the integration of each quad4 element. The
whole set of top and the most right-hand side nodes is
not coupled through the Kaa term. This fact is reflected
in the block diagonal structure of the full rigidity matrix
Kaa contribution, that reads with an appropriate row-by-
row nodal numbering as:
DNy ⊗DNx ⊗Kaa;
(20)
The DNα for α ∈{x, y} is the identity matrix with a
missing last diagonal element – see Eq. (22).
We analyse further the Kab connectivity terms.
For
each row, the lower nodes are coupled between the nearest
neighbors Nx −1 times. The upper row stays uncoupled.
The contribution of the Kab term to the full rigidity ma-
trix of rectangular Nx ×Ny shape sample therefore gives:
DNy ⊗TNx ⊗Kab, where TNx is a upper single diagonal
Toeplitz matrix – see Eq. (22).
The calculation of each elementary contribution is sim-
ilar to our previous development. By analogy with DN we
9
Kaa
Kaa
Kaa
Kaa
Kaa
Kaa
Kaa
Kaa
1
2
3
Nx −1
Nx
1+Nx
2+Nx
3+Nx
2Nx
1+2Nx
2+2Nx
3+2Nx
3Nx
Figure 5. Kaa component contribution.
Kab
Kab
Kab
Kab
Kab
Kab
Kab
Kab
1
2
3
Nx −1
Nx
1+Nx
2+Nx
3+Nx
2Nx
1+2Nx
2+2Nx
3+2Nx
3Nx
Figure 6. Kab component contribution.
can introduce UN, the identity matrix without its first di-
agonal term, such that the full rigidity matrix aggregates
into:
K = DNy ⊗DNx ⊗Kaa + DNy ⊗UNx ⊗Kbb+
UNy ⊗UNx ⊗Kcc + UNy ⊗DNx ⊗Kdd+
DNy ⊗TNx ⊗Kab + DNy ⊗T†
Nx ⊗Kba+
TNy ⊗DNx ⊗Kad + T†
Ny ⊗DNx ⊗Kda+
(21)
TNy ⊗UNx ⊗Kbc + T†
Ny ⊗UNx ⊗Kcb+
UNy ⊗TNx ⊗Kdc + UNy ⊗T†
Nx ⊗Kcd+
TNy ⊗TNx ⊗Kac + T†
Ny ⊗T†
Nx ⊗Kca+
TNy ⊗T†
Nx ⊗Kbd + T†
Ny ⊗TNx ⊗Kdb,
where all these matrices can be explicitly defined with
index notation, as Dij
Nα = δij −δiNαδjNα; Tij
Nα = δi,j−1
and Uij
Nα = δij −δi1δj1, for i, j ∈[1, Nα].
DNα ∈MNα
TNα ∈MNα
UNα ∈MNα
1 . . . 0 0
... ... ...
...
0 . . . 1 0
0 . . . 0 0
;
0
1
. . . 0
... ... ... ...
0
0
... 1
0
0
. . . 0
;
0 0 . . . 0
0 1 . . . 0
...
... ... ...
0 0 . . . 1
(22)
As the number of nodes in each direction is set to a
power of two, i.e.
Nx = 2nx and Ny = 2ny, we can
Figure 7. The parametrized quantum circuit used in our QPU
runs and simulations. An unbalanced version of the CCCX
gate was used, where the success probability varies depending
on the input state (see Section F.1 of Lysaght et al 2024 [30]
for more details).
express these matrices using the 2 × 2 matrices of the
spanning family S defined in Eq. (4)[18]:
DNα = I⊗nα −p⊗nα
−
;
UNα = I⊗nα −p⊗nα
+
;
TNα ≡T(nα) =
nα−1
X
k=0
I⊗k ⊗σ+ ⊗σ⊗nα−k−1
−
.
(23)
Crucially, the number of quantumly-computable terms
in this decomposition is logarithmically small and can be
accessed via 2nxny + O(nx) + O(ny) independent single-
qubit quantum measurements. In fact, each leading-order
connection term (Kac and Kbd) counts for 4nxny S-tensor
product contributions, but their factorization yields di-
rectly diagonalizable matrices:
T(ny) + T†(ny)
⊗
T(nx) + T†(nx)
⊗I,
T(ny) −T†(ny)
⊗
T(nx) −T†(nx)
⊗X.
Given that the matrices T(nα) ± T†(nα) are constructed
from nα single-qubit observables, the resulting leading-
order scaling is 2nxny as expected.
This scaling property reflects not only the sparse na-
ture of the rigidity matrix itself, which has ∼2n non zero
elements, but also the fact that we use an overdetermined
set of 2×2 matrices for decomposition S. The simultane-
ous usage of redundant I ≡p+ + p−matrix is absolutely
crucial, as we remark that the simplest identity matrix
expansion I⊗n = (p+ + p−)⊗n has 2n non-zero spanning
components in p± basis.
D.
Application using Photonic Computing
The parametrized QLOQ circuit we implemented is
displayed on Figure 7. It splits 4 qubits in two groups
10
of 2 qubits that are encoded on a four-level system (4
modes with 1 photon) [30].
To compensate for small device errors on the QPU we
applied a noise mitigation technique from iteration 100
onwards in each run, whereby any value in the circuit’s
outcome probability distribution below a certain thresh-
old (0.001 in this case) was set to 0, and then the dis-
tribution was renormalised. This improved final energy
values from approx -21 to approx -23. Applying this noise
mitigation technique for all iterations prevents the ansatz
from converging below approx -9, which is why it is only
activated on iteration 100.
To optimize the variational quantum circuit, we use
the classical gradient-free optimiser NEWUOA [37] with
the following stopping criteria: maximum 220 iterations,
5 × 10−14 absolute tolerance on the result and 5 × 10−10
relative tolerance on the result.
We note that the ansatz is not expressive enough to
converge. To verify this, we ran 1000 noise-free infinite-
shot VQA simulations [38] with the same initial parame-
ters. The mean final energy estimate of these simulations
was −23.77 ± 0.3, with a best result of −23.95. We chose
this ansatz because it is the largest we can fit on the
current optical chip [30].
The mean final energy estimate of the VQA was bet-
ter in these noise free simulations than on QPU (Sim:-
23.77 vs QPU:-23.15).
However, the best QPU result
was slightly better than the best noise free simulation
result (Sim:-23.95 vs QPU:-24.06). This is likely due to
the stochastic nature of variational algorithms on noisy
hardware.
Both the noisy simulations and QPU runs compare
favourably to the theoretical value, with best simulated
and QPU results achieving 96.2% and 96.6% accuracy
respectively.
E.
Quantum mesh refinement
5
7
9
11
13
15
17
19
21
8
10
12
14
16
18
2nd Order Reg.
Quartered SIF
Brown's SIF
Number of Qubits
Stress Intensity Factor (kg⋅m⁻¹ᐟ²⋅s⁻²)
Figure 8.
Convergence of different ways to compute the
SIF observable in 2D-plate. The integration over crack lips
Eq. (14) (red) and the polynomial interpolation (blue) are to
be compared with the reference solution (black) from [27].
1.
From classical to quantum remeshing
Mesh refinement is a crucial technique in the FEM for
improving the accuracy of numerical solutions. The ele-
ments where a higher resolution is needed are subdivided,
typically in regions where the solution exhibits discon-
tinuities, steep gradients, or stress concentration.
The
refinement process increases the number of DoFs, allow-
ing the finite element solution to better capture fine-scale
features. The coarse solution is usually projected on the
refined mesh in order to get the new discretized field ini-
tialization.
In case of fracture mechanics, the precise estimation of
the SIF necessitates either high refinement close to the
crack tip or the usage of specific FE [39]. In the clas-
sical resolution, with the FEM, the uniform refinement
is inappropriate, as the resulting increase in the number
of DoF rapidly overcomes the classical computers capac-
ities. Fig. 8 illustrates this rather slow convergence of
the SIF observable in the regular mesh hypothesis: the
mesh is reaching ∼1 million of DoFs (21 qubits) before
a few percent error to the reference solution is attained
even in the simple case of the pre-cracked plate. However,
this classical system size bottleneck is alleviated through
the exponential capacity of quantum memory. However,
variational convergence in the resulting high-dimensional
parameter spaces can be difficult.
Therefore,
we
propose
to
modify
the
standard
VQA procedure incorporating an additional quantum
remeshing step. The quantum solution |ψI⟩obtained
for some given discretization level is first projected on
the uniformly refined mesh: |ψI⟩7→|ψD⟩. The coarse
solution serves as the initial state for convergence in the
refined space: A |ψD⟩7→|ψF ⟩instead of the standard
’cold start’ A |0⟩7→|ψF ⟩(see Fig. 13). In what follows,
we illustrate our proposition with some relevant exam-
ples.
2.
1D Remeshing.
We first consider the 1D case, with a scalar field of un-
knowns u(x) ∈R. The quantum n-qubit state encodes
2n DoFs u(x) of the discretized field by the amplitudes
of the ket |x⟩, i.e., |ψI⟩=
2n−1
P
x=0
u(x) |x⟩, where x is writ-
ten in binary as usual. |ψD⟩should represent the initial
guess or the warm start for a more refined solution of
n + 1 qubits. As u(x) is supposed to represent some con-
tinuous physical solution, we can obtain a good enough
approximation of the remeshed function by duplicating
the each u(x) value on its right close nearest neighbor as
shown in Fig 9:
11
0
2
4
6
8
10
12
14
16
−2
−1
0
1
2
3
X position
Displacements
Figure 9. The 4-qubit solution |ψI⟩(green), the 5-qubit tar-
get state |ψF ⟩(blue), and the 5-qubit state |ψD⟩obtained by
duplication of that 4-qubit state (red)
This duplicated state contains values for twice as many
points as |ψI⟩, but values for successive points are equal,
that allows us to write it in the compact form using the
Euclidean division (//):
|ψD⟩=
2n+1−1
X
x=0
u(x//2) |x⟩.
(24)
Let us call A the ansatz that yields the n−th qubit solu-
tion |ψI⟩. To create the duplicated state, we then simply
need to enrich this circuit by an extra little endian qubit
and to apply the Hadamard gate in order to duplicate
the data, see Fig. Eq. (10).
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
q1 :
1
q2 :
2
q3 :
3
q4 :
H
Figure 10. Circuit yielding the |ψD⟩state containing the du-
plicated displacements.
This causes the state to evolve as follows:
|ψD⟩= (I⊗n ⊗H) · (|ψ⟩⊗|0⟩) = |ψ⟩⊗|+⟩
=
2n−1
X
x=0
u(x) |x⟩⊗(|0⟩+ |1⟩)
=
2n−1
X
x=0
u(x) |2x⟩+
2n−1
X
x=0
u(x) |2x + 1⟩
=
2n+1−1
X
x=0
u(x//2) |x⟩
(25)
We will now no longer optimize over the parameters θA
i
of A to limit the size of the parameter space. To improve
the state, another stage of n + 1 qubit optimization is
added via an ansatz B as shown in Fig. 11. We construct
B ansatz so that it realizes an identical transformation
for all zero set of its parameters: B(θ = 0) ≡I⊗(n+1).
By randomly setting the initial parameters of B close to
zero , we optimize them from a so-called warm start,
which greatly improves the convergence. We remark that
this requested ansatz property can easily be enforced for
any kind of ansatz by the additional application of the
dagger-circuit: B(θ) · B†(θ = 0).
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
Ansatz (θB
0 , θB
1 , θB
2 , θB
3 , ...)
0
q1 :
1
1
q2 :
2
2
q3 :
3
3
q4 :
H
4
Figure 11. Circuit C to be optimized stage by stage towards
the target at 5-qubit state
You can then repeat the procedure by considering this
entire circuit C represented in Fig. 11 like a black box, as
we considered A throughout this section. Each q-qubit
state becomes the warm start for its q+1-qubit remeshed
counterpart.
3.
2D Remeshing
In 2D, the construction is similar but we need to re-
order qubits instead of simply appending them.
a.
Scalar fields
For a scalar field we don’t need any
qubit to account for local DoF, like in Sec. IV E 2. We
can therefore identify any DOF by its position (x, y):
quantumly speaking, a trial state |ψI⟩is proportional to
2ny −1
P
y=0
2nx−1
P
x=0
u(x, y) |y, x⟩, where x and y are written in bi-
nary. Notice how points are ordered first along the y axis,
and then along x, so that the next point of a point (x, y),
is (x + 1, y) if x < Nx −1, and (0, y + 1) otherwise. If we
want to apply a remeshing along both directions x and
y, we must first obtain the following “duplicated” state:
|ψD⟩=
2ny+1−1
X
y=0
2nx+1−1
X
x=0
u(x//2, y//2) |y, x⟩
(26)
To do so, not only do we have to use Hadamard gates to
account for how the newest (little-endian) qubit should
distribute its amplitude equally between |0⟩and |1⟩(as
in Sec. IV E 2), but also add swap gates so that the new
qubits reach the correct position in the numbering. This
compels us to also add a reordering circuit as follows:
12
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
q1 :
1
q2 :
2
×
q3 :
3
× ×
q4 :
H
×
q5 :
H
Figure 12.
Circuit yielding the state |ψ2,D⟩containing the
duplicated displacements.
This causes the state to evolve as follows:
|ψ2⟩⊗|00⟩→|ψ2⟩⊗|++⟩
after Hadamard
=
2qy −1
X
y=0
2qx−1
X
x=0
⃗uA,q(x, y) |y, x⟩⊗|++⟩
=
2qy −1
X
y=0
2qx−1
X
x=0
⃗uA,q(x, y) |y⟩|x + +⟩
(27)
→
2qy −1
X
y=0
2qx−1
X
x=0
⃗uA,q(x, y) |y+⟩|x+⟩
after swaps
=
2qy+1−1
X
y=0
|y⟩
2qx+1−1
X
x=0
⃗uA,q(x//2, y//2) |x⟩
like in Eq. 25
Finally, like in the 1D case, we simply have to compose
this circuit with an ansatz that implements the identity
for known parameters, and optimize starting from such
parameters.
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
|ψI⟩
|ψD⟩
Ansatz (θB
0 , θB
1 , θB
2 , θB
3 , ...)
|ψF ⟩
q1 :
q2 :
×
q3 :
× ×
q4 :
× ×
q5 :
H
q6 :
H
×
Figure 13. 6-qubit circuit to be optimized by varying the θB,
starting from a 4-qubit circuit that was already optimized by
varying the θA.
b.
Vector fields
For a vector field with two local
DoF, we need one additional qubit compared to the pre-
vious sections, as shown in Eq. (9).
An ansatz-created state corresponding to this field
could be written as:
|ψI⟩=
Ny−1
X
y=0
Nx−1
X
x=0
Nd−1
X
d=0
u(x, y, d) |y, x, d⟩
(28)
If we want to apply a remeshing along both directions x
and y, we first obtain the following “duplicated” state:
|ψD⟩=
2Ny−1
X
y=0
2Nx−1
X
x=0
Nd−1
X
d=0
u(x//2, y//2, d) |y, x, d⟩
(29)
The reordering circuit is similar to the one seen before,
but we also have to put the qd DOF qubits at the end,
using swaps:
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
q1 :
1
q2 :
2
×
q3 :
3
× ×
q4 :
4
× ×
q5 :
H
q6 :
H
×
Figure 14. Circuit generating |ψD⟩in the 2D vectorial case.
Notice that the need for swaps disappears in real quan-
tum circuit execution, by simply using ansätze that don’t
use the wires that will be used for remeshing. For in-
stance, in the case of the 5-qubit to 7-qubit remesh-
ing of our 2D plate, the first ansatz would use qubits
0, 1, 3, 4, 6, while the second ansatz would use all seven.
Moreover, by manipulating the endianness of the binary
representation of the coordinates (writing the state as
P u(x, y, d) |¯y, d, x⟩), we can achieve the 2D remeshing
without swaps or temporarily unused wires —see Fig. 18.
The construction in 3D is identical to the one in 2D
with one more swap ladder, since we have to reach the
state |z + y + x + d⟩after duplication - see Appendix
A 1.
ACKNOWLEDGMENTS
The authors would like to thank Paris Region for the
support of Aqadoc project. This work is part of OECQ
project which is financed by the French State as part
of France 2030, financed by the European Union - Next
Generation EU as part of the France Relance plan. The
authors would like to thank Patrick Sinnott and Nicolas
Maring for useful discussions and Fabrice Debbasch for
multiple technical advices.
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Appendix A: Extensions of quantum remeshing
In the main part of this work, the quantum remeshing technique was introduced and detailed for one- and two-
dimensional cases. In this appendix, we discuss several complementary topics related to quantum remeshing. We first
focus on the extension to three dimensions, then introduce variants of quantum remeshing that avoid the use of swap
gates, which may be advantageous for certain QPU hardware architectures. Finally, we provide several suggestions
aimed at mitigating the field relaxation issues observed in our simulations.
1.
Construction in 3D
We illustrate here how the quantum remeshing technique could be extended into 3D case. A reasonable minimal
3D example can be constructed from the initial system with 8-qubits. It corresponds to a 2-qubit discretization in
each of the three directions, and four local DoFs (d = 4) that are supposed to be encoded with two complementary
qubits. For instance, we could refer to the thermo-mechanical coupling problem, where the unknowns are the three
component vector-displacements and the temperature field. In 3D, at each remeshing step, three qubits must be added
to represent the homogeneous refinement in each spatial direction.
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
q1 :
1
q2 :
2
q3 :
3
q4 :
4
q5 :
5
q6 :
6
q7 :
7
q8 :
q9 :
q10 :
(a) The initially optimized 8-qubit ansatz, with 3 extra qubits.
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
q1 :
1
q2 :
2
×
q3 :
3
× ×
q4 :
4
× ×
q5 :
5
×
q6 :
6
×
×
q7 :
7
×
×
q8 :
H
q9 :
H
×
q10 :
H
×
(b) Field duplication and reordering within quantum circuit.
Figure 15. Construction of the remeshed 3D ansatz, illustrating the transition:|xyzd⟩⊗|000⟩7→|x⟩|+⟩|y⟩|+⟩|z⟩|+⟩|d⟩.
In the reordering circuit, the first two swaps place the d components at the end, the second two swaps place the
two x values before a |+⟩, and the last two swaps place the two y values before a |+⟩.
15
q0 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
Ansatz (θB
0 , θB
1 , θB
2 , θB
3 , ...)
0
q1 :
1
1
q2 :
2
×
2
q3 :
3
× ×
3
q4 :
4
× ×
4
q5 :
5
×
5
q6 :
6
×
×
6
q7 :
7
×
×
7
q8 :
H
8
q9 :
H
×
9
q10 :
H
×
10
Figure 16. Circuit C to be optimized towards T11
(a) The deformation level on the 13 qubits
cascaded quantum solution for the 2D plate.
(b) The same as (a) for the next step
remeshing 15 qubits. Field relaxation
problem is clearly amplified.
(c) The cascade remeshing applied to a 3D
shape.
Figure 17. Usual discretized geometries.
2.
Construction without swaps
While photonic QPUs offer a relatively flexible implementation of SWAP gates due to their adaptable connectivity,
this is not the case for all quantum architectures. In particular, superconducting QPUs often suffer from limited qubit
connectivity, where qubits can only interact with their nearest neighbors. As a result, implementing a SWAP gate
between distant qubits requires decomposing it into multiple double-qubit operations, significantly increasing circuit
depth and error rates. To address this challenge, we propose a swapless variant of quantum remeshing, which avoids
the need for costly SWAP operations and is better suited for architectures with constrained connectivity.
In this section, we use the index ·b to indicate that a number is written in binary. A number can be written in binary
either in little-endian (34b = 100010) or in big-endian ( ¯34b = 010001). We can go from one endianness to another by
mirroring the representation.
In 2D, it is possible to add qubits in a way that does not significantly affect the nodal enumeration and allows the
creation of observables, without the need for swaps. We should simply have y represented by its big-endian binary
representation ¯yb, followed by d as usual, followed by x represented as before by its little-endian binary representation
xb. The two remeshing qubits have to be added at the top and bottom of the ansatz: The ansatz state at the dotted
16
q0 :
H
Ansatz (θB
0 , θB
1 , θB
2 , θB
3 , ...)
0
q1 :
Ansatz (θA
0 , θA
1 , θA
2 , θA
3 , ...)
0
1
q2 :
1
2
q3 :
2
3
q4 :
3
4
q5 :
4
5
q6 :
H
6
Figure 18. Circuit without swaps
barrier in Fig. 18 is:
|ψNOSW AP ⟩=
Ny−1
X
y=0
Nx−1
X
x=0
Nd−1
X
d=0
⃗uA,4(x, y, d) |+, ¯yb, d, xb, +⟩
=
2Ny−1
X
y=0
2Nx−1
X
x=0
Nd−1
X
d=0
⃗uA,4(x//2, y//2, d) |¯yb, d, xb⟩
(A1)
Another possibility is to design the ansätze for lower number of qubits so that they do not use some of their internal
qubits. This way, they are available for remeshing by simply applying H to them, like in Fig. 18, without any need
for endianness modifications.
However, the advantage of these methods that output states in the standard |yxd⟩qubits order is that the observable
construction follows well-established rules. That observable creation process could be complexified:
• The swapless method shown above requires a different handling of x and y.
• A swapless method, without any endianness changes, results, after a few rounds of remeshing, in basis states
that have to be interpreted as |¯x0¯y0dx1y1x2y2x3y3 . . .⟩. The advantage here is that at each remeshing step, the
hardware-efficient ansatz directly entangle the newly added qubits with one-another, as explained in Sec. A 3.
We note that, while in 2D it is possible to absorb all swap gates into initial smart qubits reordering A 2, in 3D it
is not possible to iteratively construct in qiskit a swapless circuit without requiring unused internal qubits. However,
as in 2D, we could still design an ansatz that does not use some of its internal qubits, which stay to |0⟩until the
duplication applies to them.
3.
Ansatz specification improvement
No matter the underlying equation that is treated, the optimization of the ansatz structure is a crucial question.
For instance, in order to limit the number of swaps, we could apply the procedures discussed in Sec. A 2. Another
important example is linked to the field relaxation problem. Indeed, when applying a remeshing, we transform a
smooth function into a piecewise constant function. This leads to a significant increase in error for most physically
relevant observables, which typically exhibit continuity and are highly sensitive to disruptions in spatial derivatives.
For instance, in fracture mechanics, mechanical strain, defined as the gradient of the displacement field, directly
contributes to the calculation of elastic energy. The so-created discontinuities in the mechanical strain represent an
extremely high local elastic potential, and hence account for a large part of the energy modification.
The newly added nodes contributes the most for local variations - regardless of the endianness of the problem. Given
that errors after such a relaxation are mostly local, it would be advised to ensure a reasonable entanglement of these
nodes with the other ones, but also between them.
As such, given that the usual hardware efficient ansatz mostly entangles close qubits together, a simple proposition
would be to place, using swaps and endianness manipulations as in Sec. A 2, all these qubits together within the ansatz.
It would still be possible to replace them at their right spot afterwards, so that reading and applying observables would
not be modified. A pre-ansatz GHZ state-based entanglement procedure between these D newly-added qubits could
also be of use.
17
Figure 19. A field directly after remeshing (in blue), compared with the optimal values (in grey). The clusters of 4 identical
displacement values are visible, especially in the bottom left area.
4.
Rescaling considerations
In the usual version of a remeshing, we increase the number of elements while keeping the initial nodes at the same
positions. Quantum remeshing however duplicates the number of nodes, resulting in a new non-conformal mesh: the
nodes are created orderly from the extremal nodes, and the former nodes’ positions are shifted in order to respect
mesh regularity. We illustrate this subtle point on a simple 1D example.
L
0
L
0
Figure 20. 1-D remeshing from 2 to 3 qubits.
Consider a 1D, 4-nodes discretized field (N = 4) corresponding to a 2-qubit system. The third point (n = 3) is
associated with the basis state |11⟩and has an amplitude that will be noted as T2,3 (T2 being the entire field and T2,3
its fourth component, its first being T2,0). We can note its position as p2,3 = L, and the position of the first point as
p2,0 = 0. That way, the n-th point has position p2,n = nL
3 . We will now apply a remeshing step. Consider the point
associated to |000⟩. Since we are still operating between positions 0 and L, its position is still at p3,0 = 0. The next
point is associated to |001⟩. Its position is at p3,1 = L/7. Notice that p3,0 ⩽p2,0 ⩽p3,1, meaning that it lies inside
the element bounded by its two nodal successors.
L
0
L
0
Figure 21. Highlighting the node association.
18
Consider the node associated to |10⟩. Its position is at p2,2 = 2L/3. Its remeshed (successor) nodes are associated
to ket |100⟩and |101⟩. Notice that p3,4 ⩽p2,2 ⩽p3,5, meaning that it is between its successor nodes as well.
More generally, consider the point associated to |n⟩in a q-qubit field. Its position is pq,n =
nL
2q−1 and the position
of its remeshed points are:
pq+1,2n =
2nL
2q+1 −1
and
pq+1,2n+1 = (2n + 1)L
2q+1 −1 ,
Inner projection inequality is satisfied:
pq+1,2n ⩽pq,n ⩽pq+1,2n+1
(A2)
Indeed,
pq+1,2n =
nL
2q −1
2
⩽
nL
2q −1 = pq,n
(A3)
pq+1,2n+1 −pq,n
L
= 2n + 1
2q+1 −1 −
n
2q −1
=
2q −n −1
(2q+1 −1)(2q −1) ⩾0
∀n ∈[0, 2q −1] ∩N
(A4)
Similarly, in 2-D and 3-D, the initial node is in the square (resp. the cube) formed by its successors.
Appendix B: Decomposition of the stiffness matrix
In this section, we present the complete procedure for decomposing the stiffness matrix into elementary quantumly
computable observables, which scale logarithmically with the system size.
1.
Elementary connecting links Kij
The computation of the stiffness matrix K constitutes the main task of the finite element discretization of the
Navier-Cauchy mechanical problem Eq. (1). For the regular meshes the tensor product decomposition Eq. (21) allows
one to present it in a compact form. Any elementary connecting link Kij introduced in Eq. (21) can be evaluated
using its integral expression given in Eq. (19):
Kaa = Kcc =
1/2 −2ν/3
1/8
1/8
1/2 −2ν/3
;
(B1)
Kbb = Kdd =
1/2 −2ν/3
−1/8
−1/8
1/2 −2ν/3
;
(B2)
Kab = Kcd =
−1/4 + ν/6 −1/8 + ν/2
1/8 −ν/2
ν/6
;
(B3)
Kad = Kcb =
ν/6
1/8 −ν/2
−1/8 + ν/2 −1/4 + ν/6
;
(B4)
Kac = Kca =
−1/4 + ν/3
−1/8
−1/8
−1/4 + ν/3
;
(B5)
Kbd = Kdb =
−1/4 + ν/3
1/8
1/8
−1/4 + ν/3
.
(B6)
19
These link matrices are to be decomposed into the Pauli basis Eq. (4) prior to any quantum calculations:
Kaa = 3 + 4ν
6
I + 1
8X;
Kbb = 3 + 4ν
6
I −1
8X;
(B7)
Kab = −3 + 4ν
24
I −1
8Z + 4ν −1
8
iY ;
(B8)
Kbc = −3 + 4ν
24
I + 1
8Z + 4ν −1
8
iY ;
(B9)
Kac = −3 + 4ν
12
I −1
8X;
Kbd = −3 + 4ν
12
I + 1
8X.
(B10)
The procedure could be extended for other relevant situations, such as the discretized scalar partial differential equa-
tions (PDE). For example, the tensor product decomposition for Poisson PDE can be trivially obtained from Eq. (21)
by replacing all connecting links Kij by its scalar counterparts from the elementary stiffness matrix: Kij 7→Kij. The
finite elements discretization for the Poisson equation is generated by the following set of Kij:
Kaa Kab Kac Kad
Kba Kbb Kbc Kbd
Kca Kcb Kcc Kcd
Kda Kdb Kdc Kdd
=
4
−1 −2 −1
−1
4
−1 −2
−2 −1
4
−1
−1 −2 −1
4
(B11)
The same principle applies for the nearest neighbors finite difference scheme (FDM):
Kaa Kab Kac Kad
Kba Kbb Kbc Kbd
Kca Kcb Kcc Kcd
Kda Kdb Kdc Kdd
=
2
−1
0
−1
−1
2
−1
0
0
−1
2
−1
−1
0
−1
2
(B12)
2.
Compact form of K
Using the system’s symmetry, we rewrite the full rigidity matrix Eq. (21) into a compact form containing only 8
independent contributions that are proportional to the elementary connecting link, defined in Appendix B 1:
K =
h
(I⊗ny −p⊗ny
−
) ⊗(I⊗nx −p⊗nx
−
) + (I⊗ny −p⊗ny
+
) ⊗(I⊗nx −p⊗nx
+
)
i
⊗Kaa +
h
(I⊗ny −p⊗ny
−
) ⊗(I⊗nx −p⊗nx
+
) + (I⊗ny −p⊗ny
+
) ⊗(I⊗nx −p⊗nx
−
)
i
⊗Kbb +
h
(I⊗ny −p⊗ny
−
) ⊗T(nx) + (I⊗ny −p⊗ny
+
) ⊗T†(nx)
i
⊗Kab +
T(ny) ⊗(I⊗nx −p⊗nx
+
) + T†(ny) ⊗(I⊗nx −p⊗nx
−
)
⊗Kbc +
T(ny) ⊗T(nx) + T†(ny) ⊗T†(nx)
⊗Kac +
T(ny) ⊗T†(nx) + T†(ny) ⊗T(nx)
⊗Kbd +
T(ny) ⊗(I⊗nx −p⊗nx
−
) + T†(ny) ⊗(I⊗nx −p⊗nx
+
)
⊗K†
bc +
h
(I⊗ny −p⊗ny
−
) ⊗T†(nx) + (I⊗ny −p⊗ny
+
) ⊗T(nx)
i
⊗K†
ab.
(B13)
As shown in Appendix B 1 all the connecting 2×2 matrices Kij (Eq. (19)) can be decomposed in the Pauli basis, with
the help of linear coefficients dependent exclusively on the material’s Poisson ratio ν. The non-diagonal matrices used
in the K decomposition are them-self represented with help of the elementary 2 × 2 matrices set:
T(nα) =
nα−1
X
k=0
I⊗k ⊗σ+ ⊗σ⊗nα−k−1
−
.
As arbitrary linear combinations of n-times tensor products of 2 × 2 matrices (M2(R) space) generate the M2n(R)
linear vector space, we can decompose K ∈M2n(R) in the linear span of the polynomial tensor product of its four
basis elements {p±, σ±}, which scales exponentially. The number of spanning terms in most general case of dense
20
matrix scales as the corresponding vector space dimension, i.e. as 2n × 2n. One of our important claim is that the
decomposition of K contains only O(nxny)-terms in a specific linear span generated by five elements S = {p±, σ±, I}.
Even if it contains the non hermitian σ± operators, we prove in the Appendix B 3 that all these contributions can be
regrouped in a are quantumly-computable observables.
3.
Diagonalization of σ± polynomials
In this section we show some explicit examples of diagonalization of σ± polynomials, that naturally appears in the
decomposition of the stiffness matrix K, Eq. (B13). Let us explicitly introduce the matrix notation we are using. We
recall the expression of these common matrices:
p+ = |0⟩⟨0| =
1 0
0 0
;
σ+ = |0⟩⟨1| =
0 1
0 0
;
(B14)
p−= |1⟩⟨1| =
0 0
0 1
;
σ−= |1⟩⟨0| =
0 0
1 0
.
CNOT =
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
;
σ+ ⊗σ+ =
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
;
σ+ ⊗p+ =
0 0 1 0
0 0 0 0
0 0 0 0
0 0 0 0
.
(B15)
The first set of transformations is a partial diagonalization of σ+ ⊗σ+:
CNOT · σ+ ⊗σ+ · CNOT =
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
·
0 0 0 1
0 0 0 0
0 0 0 0
0 0 0 0
·
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
=
0 0 1 0
0 0 0 0
0 0 0 0
0 0 0 0
= σ+ ⊗p+
(B16)
Trivially for σ⊗3
+ ≡σ+ ⊗σ+ ⊗σ+ we convert σ+ into the diagonal matrix p+:
I ⊗CNOT · σ⊗3
+ · I ⊗CNOT = σ+ ⊗σ+ ⊗p+
(B17)
Two consecutive CNOT transformations lead to two reductions in the number of tensored non-diagonal contributions
σ+:
CNOT ⊗I · I ⊗CNOT · σ⊗3
+ · I ⊗CNOT · CNOT ⊗I = σ+ ⊗p+ ⊗p+
(B18)
Similarly, n consecutive such CNOT on consecutive qubit couples as shown in Fig. 22a will transform σ⊗n
+
into p⊗n
+ .
For any power of σ+, we can define the unitary transformation B(n) ∈M2n(R) so that:
B†(n) · σ⊗n
+
· B(n) = σ+ ⊗p⊗(n−1)
+
,
where
B(n) ≡
n−2
Y
k=0
I⊗(n−2−k) ⊗CNOT ⊗I⊗k.
(B19)
In the quantum circuit representation, the B(n) operator corresponds to the consecutive execution of CNOT gates
starting from the last (bottom) qubit, as shown in (Fig.
22).
The equivalent equation can be obtained for σ−
polynomials via the hermitian conjugation of Eq. (B19):
B†(n) · σ⊗n
−
· B(n) = σ−⊗p⊗(n−1)
+
(B20)
The hermitian operator σ⊗n
+
+ σ⊗n
−
can be presented in the following almost diagonal form, where only the first
qubit operator is still off-diagonal:
B†(n) · (σ⊗n
+
+ σ⊗n
−) · B(n) = (σ+ + σ−) ⊗p⊗(n−1)
+
≡X ⊗p⊗(n−1)
+
.
(B21)
We notice that another type of hermitian operator constructed from the antisymmetric matrix i(σ⊗n
+ −σ⊗n
−) can also
be reduced in a similar way, resulting in trailing Y gate:
B†(n) · (σ⊗n
+
−σ⊗n
−) · B(n) = (σ+ −σ−) ⊗p⊗(n−1)
+
≡iY ⊗p⊗(n−1)
+
.
(B22)
21
q0 :
•
H
q1 :
•
q2 :
•
q3 :
•
q4 :
(a) A 5-qubit BH(5) circuit for σ⊗5
+
+ σ⊗5
−.
q0 :
X
•
H
q1 :
X
•
q2 :
•
q3 :
•
q4 :
(b) A 5-qubit circuit for σ⊗2
−⊗σ⊗3
+
+ σ⊗2
+
⊗σ⊗3
−.
Figure 22. Quantum circuit diagonalizing σ± polynomials. The X-gates are used to convert the circuit (b) to a standardized
form (a) ready for the chained CNOT σ± order reduction. H-gate diagonalizes the trailing first qubit X-gate.
The single qubit gate X and Y can be diagonalized with H =
1
√
2
1
1
1 −1
and N =
1
√
2
−1 −i
i
1
respectively, leading
to the full diagonalization circuit:
B†
H(n) · (σ⊗n
+
+ σ⊗n
−) · BH(n) = Z ⊗p⊗(n−1)
+
;
B†
N(n) · i(σ⊗n
−
−σ⊗n
+ ) · BN(n) = Z ⊗p⊗(n−1)
+
,
(B23)
where we define the unitary of consecutive action of the chained CNOT and first qubit rotation: BH(n) = B(n) · H ⊗
I⊗(n−1) and BN(n) = B(n) · N ⊗I⊗(n−1).
The CNOT-chain action on the polynomial containing both σ+ and σ−leads also to the shape with a single qubit
off-diagonal. We present here only two hermitian operators that are most relevant for our computations:
B†
H(n + 1) · (σ−⊗σ⊗n
+
+ σ+ ⊗σ⊗n
−) · BH(n + 1) =Z ⊗p−⊗p⊗(n−1)
+
(B24)
iB†
N(n + 1) · (σ−⊗σ⊗n
+
−σ+ ⊗σ⊗n
−) · BN(n + 1) =Z ⊗p−⊗p⊗(n−1)
+
(B25)
In the general case, any mixed σ± powers can be treated with help of supplementary X-gate conversion of the initial
operator into a standardized form σ⊗n
+
+ σ⊗n
−
prior to diagonalization, illustrated in some examples in Fig. 22.
Having explicitly explained the decomposition of K as a sum of matrices in S, we can now estimate the number of
terms that have to be measured separately to calculate ⟨ψ | K | ψ⟩and the diagonalization circuits for each such term.
Terms of the form w ⊗I and w ⊗Z can be computed simultaneously. However, for terms ending in X (respectively,
Y ), the last qubit has to be once again diagonalized with H (or N respectively). We therefore have to consider such
terms separately. The full decomposition and its diagonalization terms are summarized in the table in Sec. B 4.
22
4.
Diagonalizable terms from the decomposition of K
We display here the (2nxny + 2nx + 2ny + 2) terms of a decomposition of K, which proves the scaling for the
independent measurement of K. The tensorial product ⊗has been omitted for clarity. This number of terms can be
further reduced by noticing that some of these terms are simultaneously diagonalizable – for instance, for any k, k′,
any term diagonalized by I⊗ny+k ⊗BH(nx −k) ⊗I is also diagonalized by I⊗k ⊗BH(ny −k′) ⊗BH(nx −k) ⊗I.
Term in K decomposition
Diagonalized by
# terms
3+4ν
6
4I⊗n−1 −2p⊗ny
−
I⊗nx −2p⊗ny
+
I⊗nx −2I⊗nyp⊗nx
−
−2I⊗nyp⊗nx
+
I⊗n
1
+p⊗ny
−
(p⊗nx
+
+ p⊗nx
−
) + p⊗ny
+
(p⊗nx
+
+ p⊗nx
−
)
I
−3+4ν
12
ny−1
P
k=0
nx−1
P
k′=0
I⊗k(σ+σ⊗ny−k−1
−
+ σ−σ⊗ny−k−1
+
)
I⊗k ⊗BH(ny −k)
nxny
I⊗k′(σ+σ⊗nx−k′−1
−
+ σ−σ⊗nx−k′−1
+
)I
⊗I⊗k′ ⊗BH(nx −k′) ⊗I
nx−1
P
k=0
(p⊗ny
+
+ p⊗ny
−
−2I⊗ny)I⊗k(σ+σ⊗nx−k−1
−
+ σ−σ⊗nx−k−1
+
)( 3+4ν
24 I + 1
8Z)
I⊗ny+k ⊗BH(nx −k) ⊗I
nx
ny−1
P
k=0
I⊗k(σ+σ⊗ny−k−1
−
+ σ−σ⊗ny−k−1
+
)(p⊗nx
+
+ p⊗nx
−
−2I⊗nx)( 3+4ν
24 I −1
8Z)
I⊗k ⊗BH(ny −k) ⊗I⊗nx+1
ny
1
8
p⊗ny
+
−p⊗ny
−
p⊗nx
+
−p⊗nx
−
X
I⊗n−1 ⊗H
1
1
8
ny−1
P
k=0
nx−1
P
k′=0
[I⊗k
σ+σ⊗ny−k−1
−
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Quantum remeshing and efficient encoding for fracture mechanics Ulysse Rémond,1, 2 Pierre-Emmanuel Emeriau,3 Liam Lysaght,3 Jean Ruel,1 Joseph Mikael,1 and Kyryl Kazymyrenko1 1EDF R&D, 7 Bd Gaspard Monge, 91120 Palaiseau, France 2Sorbonne Université, Observatoire de Paris, Université PSL, CNRS, LUX, F-75005 Paris, France 3Quandela, 7 rue Léonard de Vinci, 91300 Massy, France (Dated: October 17, 2025) We present a variational quantum algorithm for structural mechanical problems, specifically addressing crack opening simulations that traditionally require extensive computational resources. Our approach provides an alternative solution for a relevant 2D case by implementing a parametrized quantum circuit that stores nodal displacements as quantum amplitudes and efficiently extracts critical observables. The algorithm achieves optimal nodal displacements by minimizing the elastic energy obtained from finite element method. The energy is computed with only a polylogarithmic number of measurements. Extracting relevant scalar observables such as the stress intensity factor is then done efficiently on the converged solution. To validate the scalability of our approach, we develop a warm start strategy based on a remeshing technique that uses coarse solutions to circumvent barren plateaus in the optimization landscape of the more refined problems. Our method has been experimentally validated on Quandela's photonic quantum processor Ascella and comprehensive numerical simulations demonstrate its scalability across increasingly complex quantum systems. I. INTRODUCTION Partial differential equations (PDEs) provide the mathematical foundation for modeling a wide range of physical phenomena, from fluid dynamics and heat transfer to electromagnetic wave propagation and structural mechanics. In solid mechanics, the stationary Navier-Cauchy equation governs the equilibrium state of an elastic continuum. It serves as a key building block for fracture mechanics [1] and thermo-hydro-mechanical coupling models [2]. Its extensions capturing nonlinear hyperelasticity [3] are being studied for their industrially relevant applications [4]. This governing equation belongs to the class of secondorder elliptic PDEs, which can be reformulated as an energy minimization problem [5]. The Finite Element Method (FEM) [6] is particularly effective for approximating the unique solution by optimizing nodal variables over a discretized spatial domain. Modeling the fracture process requires the introduction of geometric singularities, such as local shape discontinuities, to represent the initial cracks. This is an inherently multiscale procedure that, for the hydraulic dam example, ranges from the millimeter-scale concrete aggregate (10-3m) to the infrastructure size (102m). Even if the fracture mechanics may involve the estimation of only a single scalar quantity as an output, simulating multi-crack propagation with high precision remains an extremely demanding task for state-of-the-art methods: current capacities may handle up to 1013 degrees of freedom in the resulting linear algebra problem [7], which tend to be insufficient in the 3D applications. These challenges highlight the limitations of classical FEM for fracture mechanics, particularly its steep cost when the initial crack distribution is uncertain. Quantum algorithms offer a promising alternative for these simulations. Recent progress in quantum approaches to PDEs reflects this trend, targeting fundamental bottlenecks in classical numerical methods [8-20]. The Harrow-Hassidim-Lloyd (HHL) algorithm [8] initially promised an exponential speedup for solving linear systems of equations. However, its practical application to PDEs faces significant constraint: it assumes an efficient quantum encoding of both the system matrix and right-hand side vector. The latter is rather challenging for geometries containing crack singularities, rendering the original HHL algorithm impractical for such applications. Tailored quantum algorithms for specific problems have also been proposed [10], but these approaches require large fault-tolerant quantum computers, which remain beyond current capabilities. Consequently, some research has been shifted towards Variational Quantum Algorithms (VQA) solving PDEs on Noisy Intermediate-Scale Quantum (NISQ) computers [13, 18, 19]. Most existing work, however, targets one-dimensional problems and suffers from optimization challenges such as barren plateaus [21, 22], where gradients vanish exponentially with quantum system size. Our work addresses these limitations along several critical directions: we extend the scope to realistic twodimensional cases that better capture physical systems; we propose an efficient quantum encoding tackling a fracture mechanics problem; we experimentally validate the algorithm by running it on a photonic quantum device [23]; and we develop a novel cascading warmstart strategy based on solutions from coarser meshes. Since the number of qubits directly corresponds to mesh refinement, our hierarchical approach enables solutions from rougher meshes to guide the initialization for finer meshes, mitigating the barren plateau problem and supporting higher-resolution simulations. 16 Oct 2025 2 II. RESULTS A. Mechanics of fracture in a pre-cracked plate Linear elastic fracture mechanics studies the conditions under which a crack in a material will propagate. Here we focus on the standard example of a pre-cracked 2D plate (see Fig.1). System deformation evolves according to the static equilibrium linear Navier-Cauchy equation [5]: grad(div u) + (1 -2ν)∆u = 0 (1) where u = (ux, uy) is the two components vector displacement field, ν ∈[0, 1/2) is the Poisson's ratio of the material, ∆is the Laplacian. Near the crack tip (Fig.1), the displacement gradient diverges [24]; hence, the leading-order term dominates the solution and can be extracted. The corresponding scalar coefficient, called Stress Intensity Factor (SIF) [25], is therefore determined from the crack opening profile close to the tip: uy(r) ≈SIF ∗2(1 -ν2) p 2r/π, for r ≈0 (2) where r is the distance from the crack tip. In general, the SIF admits no closed-form expression for arbitrary geometries and boundary conditions. It is therefore tabulated for canonical cases or estimated numerically in specialized handbooks [26]. For more complex cases, discretization schemes such as Finite Differences or the Finite Elements Method (FEM) are commonly used to solve Eq. (1) and to estimate numerically the SIF's value [27]. In FEM, the displacements are computed only on a grid of nodes and interpolated between them. We use a regular mesh of Nx × Ny nodes of commensurate dimensions Nx = 2nx and Ny = 2ny, so that the total number of degrees of freedom (DoFs) can be written as N = 2·Nx ·Ny ≡2n, where n = nx +ny +1 accounts for both components of the nodal displacement. The mechanical deformation problem is converted into a finite-size optimization of the elastic energy Efe(⃗u), that is equivalent to the linear algebra problem [6]: min ⃗u Efe(⃗u) = min ⃗u 1 2⃗u · K⃗u -⃗u · ⃗f ⇔ K⃗u = ⃗f (3) where ⃗u ∈RN is the vector of unknown nodal displacement corresponding to the discretized problem, ⃗f ∈RN is the vector of external forces, and K ∈MN(R) is the stiffness matrix. While ⃗f captures all variety of Neumann type boundaries, the Dirichlet conditions can be also enforced either through a complementary penalization contribution to the initial K-matrix or by a projection technique that we cover in the next section. x y W H Crack tip f f f f f f f a b c d Na = (1 -x)(1 -y) Nb = x(1 -y) Nc = xy Nd = (1 -x)y 2D Plate Figure 1. Geometry of the pre-cracked 2D plate under external loading, showing the crack tip and nodal grid. This mesh underlies the FEM formulation (Eq. 3) and the operator decomposition (Eq. 5). The matrix K, introduced in Eq. (3), is computed via FEM and is related to the gradient overlap of the displacement interpolation functions between adjacent nodes. Although K is sparse, its naive expansion into Pauli operators {X, Y, Z} scales with the number of nonzero elements i.e., effectively exponential in nx and ny. This would render a direct quantum implementation inefficient. Exploiting mesh symmetries, we obtain one of our main results: a polynomial-size decomposition of K into tensor products. Indeed, if we define S as: X = 0 1 1 0 Y = 0 -i i 0 Z = 1 0 0 -1 S = {p±, σ±, I} p± = (I ± Z)/2 σ± = (X ± iY )/2, (4) the K matrix can be rearranged in a linear combination of tensor products {Mi} of matrices from S. K = X k μkMk, with Mk = n-1 O l=0 mk,l, mk,l ∈S, (5) where, μk ∈R is a set of decomposition coefficients. The derivation and explicit decomposition of K is detailed in 3 Methods IV C. Crucially, the number of terms in the sum in Eq. (5) scales logarithmically with the system size, that is as O(nxny). We will now explore how the decomposition Eq. (5) paves the way to efficiently solving the discretized version Eq. (3) of Eq. (1) on a quantum computer. B. Solving efficiently elastic differential equations on a quantum computer We now reformulate the minimization problem Eq. (3) in quantum terms. A trial displacement vector is encoded as a parametrized quantum state |ψ⟩= A |0⟩, where A is the ansatz unitary. As usual with VQAs [18, 19], the aim is to optimize the ansatz parameters so that it converges towards the target quantum state |ψtgt⟩= ∥⃗u∥-1 P i ui |i⟩, where ui are the components of the solution ⃗u ∈RN of the discretized problem Eq. (3) and |i⟩are the 2n basis states of n-qubit quantum system, defined later in Eq. (9). We provide the quantum counterpart for the external force vector, that is supposed to be normalized: |f⟩= P i fi |i⟩, as explained later in Eq. (10). Given the fact that the ansatz unitary generates a normalized amplitude vector in CN, the straightforward transcription of the cost function Eq. (3) can be done only in the presence of at least one unrestricted rigid body motion, i.e. when det K = 0: Eq(ψ) = 1 2 ⟨ψ| K |ψ⟩-Re ⟨ψ|f⟩. (6) In this case the normalization constraint is absorbed into irrelevant kernel subspace movement. Otherwise, if det K ̸= 0, the normalization coefficient should be explicitly introduced in the cost function. The norm optimization leads to a new form of normalization-agnostic quotient cost function [19]: Eqq(ψ) = -(Re ⟨f|ψ⟩)2 2 ⟨ψ| K |ψ⟩. (7) Regardless of whether Eq. (6) or Eq. (7) is used, evaluating the cost requires only two quantities: ⟨ψ| K |ψ⟩and ⟨f|ψ⟩. We compute ⟨ψ| K |ψ⟩by decomposing K into a polylog number of Mk matrices (see Eq. (5)). Indeed, only computations of the generic single qubit observable terms ⟨ψ|Mk|ψ⟩≡⟨ψ|mk,0 ⊗mk,1 ⊗... ⊗mk,n-1|ψ⟩are needed, where mk,l ∈S and k ∈{0, . . . , 2(nxny + nx + ny)}, as explained in Methods IV C. We can estimate each term of this decomposition using common diagonalization techniques. While p± are already diagonal, the non-trivial observable measurements in the S basis are tensor powers of σ±. Noticing that each such σ± term is present with its transposed counterpart, we focus on ⟨ψ| σ⊗n + +σ⊗n -|ψ⟩as an example. We remark that the CNOT-gate maps one non-hermitian operator σ+ within a σ⊗n + term into a diagonal projector p+. More precisely, for two-qubit systems it is easy to verify that CNOT·σ+ ⊗σ+ ·CNOT = σ+ ⊗p+, for three qubits I ⊗CNOT·σ+ ⊗σ+ ⊗σ+ ·I ⊗CNOT = σ+ ⊗σ+ ⊗p+ and so on. These (n -1) inductively-arranged CNOT-gates can reduce σ⊗n + to σ+ ⊗p⊗(n-1) + , and similarly σ⊗n - to σ-⊗p⊗(n-1) + gates. This CNOT-chain rotates |ψ⟩into | ̃ψ⟩, leading to the following easily-diagonalizable operator estimator: ⟨ψ| σ⊗n + + σ⊗n -|ψ⟩= ⟨ ̃ψ| (σ+ + σ-) ⊗p⊗(n-1) + | ̃ψ⟩, (8) where | ̃ψ⟩= Qn-2 i=0 CNOTi,i+1 |ψ⟩is an inductively transformed quantum state; the CNOTi,i+1 operation applies the CNOT operator to two adjacent qubit. Noting that X = σ+ + σ-, the last step of diagonalization is done with the help of the Hadamard H-gate using the identity Z = H · X · H. Because X · σ+ · X = σ-, all the others non-trivial tensor polynomials can be reduced to the previously discussed case. Photonic computers implement the non unitary projection p±, and thus also σ± = p±X, as elementary gates. Indeed in dual rail encoding, it amounts to herald the success of the projection on the relevant optical mode, that is measuring one mode and not detecting a photon there. As a consequence, ⟨ψ| K |ψ⟩can alternatively be computed as a sum of photonic counts of the zero state ⟨0| after the circuit A†KA |0⟩, which allows us to compute the estimator without the need for CNOT-chains. Furthermore, we provide an explicit quantum circuit for generating the boundary force vector |f⟩. To do so, we define the ordering of the degrees of freedom such that: |ψtgt⟩= 1 ∥⃗u∥ 2n-1 X i=0 ui |i⟩= X x,y,d u(x, y, d) ny z}|{ |y⟩⊗ nx z}|{ |x⟩⊗ 1 z}|{ |d⟩, (9) where u(x, y, d) is a shorthand notation for the normalized components of discretized displacement. Any grid index 0 ≤ig < Nx×Ny is represented in binary by a pair of coordinate indexes also written in binary x, y (ig ≡yx). The additional bit d separating horizontal (d = 0) and vertical (d = 1) displacement components is also introduced, which allows any qubit state to encode some given degree of freedom |i⟩= |y, x, d⟩≡|y⟩⊗|x⟩⊗|d⟩. An explicit representation of the unitary operator that generates an homogeneous Neumann boundary at the top of the plate (see Fig.1) is given by: |f⟩= X⊗ny ⊗H⊗nx ⊗X |0⟩, (10) allowing us to compute directly ⟨f|ψ⟩. For more complex boundary conditions, one may rely on the additive decomposition of the external boundaries or use a Walsh Series Loader [28]. The finite element stiffness matrix K (given in Eq. (21)) transcribes a purely Neumann boundary problem resulting in det K = 0. Dirichlet boundaries greatly enrich the 4 A5 A7 Optimize Remesh Optimize Remesh |1 0⟩ |0 0⟩ |1 1⟩ |0 1⟩ |1100⟩|1101⟩|1110⟩|1111⟩ |1000⟩|1001⟩|1010⟩|1011⟩ |0100⟩|0101⟩|0110⟩|0111⟩ |0000⟩|0001⟩|0010⟩|0011⟩ Figure 2. We first optimize a 5-qubit state, and duplicate it to use it as a warm start for the 7-qubit ansatz. This scheme is applied iteratively until the desired system size is reached as seen in Fig. 3a. variety of treated situations, and eliminate spurious rigid body motions. We therefore consider the case of perfect clamping, where some DoFs are set to zero. It can be formally written with the help of the hermitian projector P ∈MN, as P |ψ⟩= 0. We enforce it in the quantum system through a systematic projective restriction of all trial states, equivalent to the stiffness matrix modification: K 7→(I⊗n -P)K(I⊗n -P) + P. The decomposition of any projector P within the S basis set is directly derived from the binary DoF's position encoding: p+ ↔0 and p-↔1, the last bit serving the choice between the ux and uy DoF's components. P = y coord to block {p±}⊗ny ⊗ x coord to block {p±}⊗nx ⊗ ux or uy p± . (11) The plate shown in the Fig. 1 exhibits a symmetry with respect to the crack plane. We model only the upper half, with the horizontal centerline constrained against vertical displacement. Moreover, we restrict horizontal rigid body movement by clamping the ux component of the crack tip. It can be assessed via the following additive contributions for the full state projector P: p⊗ny + ⊗p-⊗I⊗nx-1 ⊗p-⇐uy = 0 on the centerline, p⊗ny + ⊗p-⊗p⊗nx-1 + ⊗p+ ⇐ux = 0 on the crack tip. Finally, we give circuit implementations for two common physically-relevant observables: SIF, defined in Eq. (2), and crack opening displacement (COD). With chosen numbering order Eq. (9), the COD-observable is estimated by multiplying the quantumly-computed norm [19] with the amplitude of probability of zero counts in the optimized state |ψ⟩: COD = ⟨0|ψ⟩× ∥⃗u∥, where ∥⃗u∥= Re ⟨f|ψ⟩ ⟨ψ|K|ψ⟩ 1 2nx/2 . (12) We similarly compute the SIF from the measurement of displacement amplitudes on the crack lips. In the vicinity of the crack tip the solution for the crack opening scales as a square root of the distance from the crack tip Eq. (2). It could allow one to calculate the SIF from a single displacement value. However, the single-point measure would be highly error-prone. On the one hand, close to the crack tip the displacement amplitude value is vanishing necessitating an important shots number to eliminate statistical noise, on the other hand, far from the crack tip the asymptotic solution deviates from its theoretical expression Eq. (2). Therefore, we opt for a multiple point estimator and give here an integral measurement that averages the SIF-value over the whole crack lip of length W/2, see Fig.1: SIF = √π ∥⃗u∥ (1 -ν2)W sZ W/2 0 [u(x, y = 0, d = 1)]2 dx (13) The discretized value of the integral can once again be computed with the help of the projector P = p⊗ny + ⊗p+ ⊗ I⊗nx-1 ⊗p-: Z W/2 0 [u(x, y = 0, d = 1)]2 dx ≈W 2nx ⟨ψ|P|ψ⟩, SIF = p π ⟨ψ|P|ψ⟩· Re ⟨f|ψ⟩ (1 -ν2) √ W ⟨ψ|K|ψ⟩ 1 2nx (14) Alternatively, for large-size systems, a more precise measurement can be obtained by restricting the integration domain closer to the crack tip. For x ∈[ 3W 8 , W 2 ], the projector p⊗ny + ⊗p+ ⊗p⊗2 -⊗I⊗nx-3 ⊗p-should be used. Interestingly, during the optimization both physical observables (COD and SIF) converge to their targets much faster than the cost function itself, due to the high energy density concentration close to the crack tip -see Fig. 3b. 5 C. Warm start with coarse mesh solution Barren plateaus pose a major challenge in VQA optimization. They represent regions where the cost function's gradient vanishes across most parameter variations. Having defined a global [22] cost function, we observe a flat landscape for the random state initialization in sufficiently large systems (n ≳10), hampering optimization. In order to avoid this difficulty, we derive a quantum version of the mesh refinement technique [29]. This approach paves the way towards consistent warm start creation, i.e. problem-agnostic strategies to bypass barren plateaus in PDEs. The idea is to define two discretized versions (coarse and refined) of the same continuous problem and to use the converged coarse n-qubits solution as the initial state for the refined (n + 2)-qubits solution. This strategy can be repeated iteratively until we reach the desired precision, hence a cascaded approach. Since barren plateaus are not an issue for a low number of qubits, we can always compute the target function from a cold start (i.e. using a randomly chosen initial state) for the coarsest meshes. Once we have solved the problem for a mesh, we duplicate the mesh, increasing the number of qubits - one additional qubit for each x and y coordinates to maintain the elements' aspect ratio. The basis state written in binary |y⟩|x⟩is extended to |y⟩|x⟩|0⟩|0⟩. Then, all the data are duplicated, since each newly added qubit is transformed into |+⟩= H |0⟩with a Hadamard gate. Finally, we enforce the order of qubits (see Sec. IV E ) creating new extended coordinates along both directions, so that the initial coarse state |y⟩|x⟩is mapped to the refined one |y⟩|+⟩|x⟩|+⟩. In summary, the procedure can be illustrated as: |y⟩|x⟩ +2 7→|y⟩|x⟩|00⟩ H 7→|y⟩|x⟩|+⟩|+⟩ swap 7→|y⟩|+⟩|x⟩|+⟩. Any smooth function represented by a n-qubit state turns into the piecewise-constant function represented by this (n+2)-qubit state, see Fig. 9. This procedure is similar to a homogeneous remeshing with constant field extrapolation on the nearest neighbors. The interpolation of the original displacement field introduces local field discontinuities, shifting the (n + 2)-cost function from the target value already achieved at the previous step of n-qubit optimization. This error induced by the field projection is well known and has been studied in depth in the classical remeshing technique [29]. Third, we further fine-tune this displacement field with another (n + 2) parametrized circuit until convergence is achieved, resulting in the local field relaxation commonly called the equilibration phase after field projection. This whole three-step process can be repeated (Fig. 3a), increasing the mesh refinement up to the desired precision. In doing so, we obtain successive warm starts to avoid barren plateaus at each step as shown in Fig. 3b. We test numerically this quantum remeshing procedure. We achieved quantifiably better results than the ... ... · · · ... A5 Remesh(5→7) A7 Remesh(17→19) A19 (a) Cascaded warm-start remeshing strategy. 5 7 9 11 13 15 17 19 0 20 40 60 80 100 Energy (cold) Energy (warm) SIF (warm) CO (warm) Fidelity (warm) Number of Qubits Observables (b) Evolution of observables with increasing discretization. Convergence ratio of remeshing-based and cold-start results with regard to the target continuous values. 5 7 9 11 13 15 17 19 0 20 40 60 80 100 Energy (Cold) Energy (Remeshing) Number of Qubits Observables (c) 2H one-step optimization, comparing remeshing-based and cold-start results with the same-size classical solutions, showing the average, best, 1st and 9th deciles. Figure 3. Cascaded simulations from 5 to 19 qubit mesh. In (b) the iterative remeshing, performing 48h optimization at each step, allows to reach 19 qubit simulation with 50% precision of observable COD and SIF, far beyond the capacities of any tested cold start strategies. (c) illustrates a single step remeshing capabilities to bypass the initial barren plateau with 50 Layers ansatz. The n + 2-qubit state is optimized for 2 hours starting from an ideal state before duplication, obtained from n-qubit discretization. cold start strategies. Initialized from 5 qubits onwards, systems composed of up to 5 · 105 DOFs, i.e. 19 qubits, were successfully simulated with this method using the circuit in Fig. 3a. As shown in Fig. 3b, even for low energy convergences at the most refined simulations, the cascading remeshing procedure proved especially reliable in reaching satisfactory convergence for the relevant mechanical observables: 6 SIF (60%) and COD (70%). Cold start simulations had error ratios of more than 100% from 13 qubits onwards. The SIF and COD precision peaks near 9 qubits as the coarser value suffer from classical discretization errors, and the finer values suffer from a limited VQA convergence time and the lack of an tailor-made ansatz. Fidelity stays high (over 87%) throughout the whole cascading procedure, and optimizations using it as cost function yield unrealistic physical deformations. This illustrates how even quantum states with high fidelity might not correspond to a physically-acceptable state, justifying the need of more relevant quantities such as the elastic energy to be used as cost function. The respective convergences of such observables in 2 hours from the classical solution at the former step can be assessed in Fig. 3c. D. Experimental implementation We experimentally validated a 4-qubit version of our variational quantum algorithm on Quandela's linear optical QPU [23]. Our ansatz uses the QLOQ compression scheme [30] which enables efficient encoding of multiqubit states onto photonic hardware. The parametrized quantum circuit was encoded on two photons, with each run consisting of 105 shots. Ascella is a photonic quantum processor that utilizes a semiconductor quantum dot based on-demand single photon source [31], with twophoton interference visibility above 94%. Up to 6 single photons can be fed synchronously into every odd input of a 12-mode universal interferometer implemented in a photonic integrated circuit. The interferometer is fully reconfigurable, enabling the implementation of 12×12 unitary transformations with high fidelity - 99.7% amplitude fidelity for Haar random 12x12 unitaries [32]. A threshold single photon detector system measures the photon distribution and coincidences at the output of the optical circuit. 10 QPU runs are displayed in Figure 4 to show that this is working on average. The mean final energy estimate of these QPU runs was -23.15, with a best result of -24.060±0.003 while the target value is -24.895. To further enhance accuracy, basic error mitigation techniques were applied during the runs (see Sec. IV A for details). This performance aligns with the best-case scenario predicted by noiseless simulations, which reached -23.952. Figure 4. The results from 10 runs on Quandela's linear optical QPU. The blue line shows the mean energy estimate at each iteration, while the shaded region is bounded by the minimum and maximum values. Run 8, which produced the minimum energy estimate overall, is shown as a pink line. III. DISCUSSION Our results demonstrate the feasibility of addressing computationally intensive structural mechanics problems-specifically fracture mechanics-using VQAs. By formulating the 2D Navier-Cauchy equation as an elastic energy minimization problem, we circumvent the impractical requirements of algorithms like HHL while providing a clear route to scalability with the developed warm start strategy. To our knowledge, this work constitutes the first implementation of a two-dimensional elasticity problem with realistic boundaries on a quantum platform, marking an important step beyond prior studies, which were limited to one-dimensional PDEs, simpler boundaries or scalar cases. The proposed encoding, which stores nodal displacements as quantum amplitudes, allows efficient computation of critical observables such as the stress intensity factor (SIF) and crack opening displacement (COD) through simple measurements. A major contribution of this work is the development of a quantum remeshing technique that systematically mitigates barren plateaus -one of the primary obstacles in VQA optimization. By initializing finer discretizations from optimized coarser meshes, we demonstrated a significant improvement in convergence over cold-start and simple warm-start strategies. We numerically validated this hierarchical warm-start approach up to 19 qubits, illustrating scalability to physically meaningful regimes. Moreover, we find that physical observables converge much faster than the energy functional, suggesting that even partially optimized states can yield accurate engineering quantities of interest. The experimental validation on a photonic quantum processor confirms the practical implementability of our approach on real hardware. Despite noise and device imperfections, our results show we achieve energy estimates 7 within ≈96% of the theoretical optimum with basic error mitigation, closely matching noiseless simulations. Looking ahead, several avenues merit exploration. First, the generalization to three-dimensional domains and vector-valued PDEs beyond elasticity (e.g., thermal and hydraulic diffusion, linear themo-mechanical coupling...) is straightforward within our encoding scheme and would significantly broaden the applicability of this method. Second, further improvements of the optimization procedure, possibly through problem-inspired ansätze, tensor-network-inspired decompositions and tailored classical minimization algorithms, could enable simulations on larger systems and with more complex geometries. Finally, given the modular nature of the algorithm, it offers a robust benchmarking tool for quantum hardware. Overall, this work establishes a concrete pathway for leveraging quantum computing in high-fidelity structural simulations. By coupling hierarchical VQA strategies with scalable encodings and efficient observable estimation, and by demonstrating the first 2D case on real hardware, we move one step closer to practical quantum advantage in computational mechanics. IV. METHODS A. Fracture Mechanics The present paper examines the second order NavierCauchy differential vector-equation [33] that naturally arises in the isotropic linear elasticity theory under hypothesis of small transformations. In the 2D case under additional plane strain hypothesis, the mechanical equilibrium of the structure may be written as a set of two constraints on the two components vector displacement field u = (ux, uy), where uα = uα(x, y) for α = {x, y} is a scalar displacement in the direction of one of its euclidean coordinates α at the given position (x, y) in space. The Navier-Cauchy equilibrium problem in the absence of volumetric density forces (i.e. weight) then reads as a constant coefficients linear system for the two components displacement field u = (ux, uy): ( ∂x(∂xux + ∂yuy) + (1 -2ν)(∂2 x + ∂2 y)ux = 0, ∂y(∂xux + ∂yuy) + (1 -2ν)(∂2 x + ∂2 y)uy = 0; (15) where ν is the Poisson's ratio of material. Commonly used boundary conditions in mechanics often involve a combination of Dirichlet-type conditions, such as perfect clamping (u(x, y) = 0 for (x, y) ∈∂ΩD) and Neumanntype constrains on the gradient of displacement ∂ΩN = ∂Ω\ ∂ΩD. The problem can be rewritten as a variational minimization of the elastic density functional E. In the case of full Dirichlet condition constraining u(x, y) to be equal to uD(x, y) on the entire boundary ∂Ω, E is a quadratic form: E[u] = 1 4 Z Ω (∂xux + ∂yuy)2dΩ+ (16) +1 -2ν 4 Z Ω (∂xux -∂yuy)2 + (∂xuy + ∂yux)2 dΩ. Any displacement field minimizing Eq. (16) and satisfying this Dirichlet boundary respects the Euler-Ostrogradski extrema conditions that are equivalent to the initial set of equations Eq. (15). In the presence of Neumann conditions, a linear term is added to Eq. (16). This modification is presented in Eq. (17), albeit in the discretized case. Eq. (15) being parametrized by a single coefficient ν, the solution's complexity stems from either the boundary conditions or the sample geometry. Theoretical solutions for the sample with a sharp-edged geometry exhibit a divergent gradient of displacement in the vicinity of this geometric singularity [25]. The leading order in this divergent asymptotic expansion is called the stress intensity factor (SIF), Eq. (2). SIF quantifies the threshold beyond which crack propagation occurs, characterizing the critical structure's loading. It is thus widely used in Linear Elastic Fracture Mechanics theory (LEFM) [24, 34]. Another typical singularity-describing observable is the crack opening displacement (CO)[26], which reveals the deformation level. While the displacement field u is a primary variable of interest for general mechanical problems, fracture mechanics mainly focuses on the computation of aforementioned pair of observables on each singularity. The CO is a local pointwise measurement, whereas the SIF is obtained via integration over a domain close to a singularity. In the current work, we use a quantum variational algorithm to numerically solve the classical 2D LEFM problem of a pre-cracked plate (Fig. 1), subjected to a uniform density of force f on its upper and lower bounds, neglecting the off-plane movement. B. Numerical discretization As in any linear PDE, the Navier-Cauchy equation admits an approximate discretized solution. In the finite difference method the differential operators are estimated on a grid (see Fig.Eq. (1)), resulting in a linear algebraic equation for the vector of nodal unknowns ⃗u: K⃗u = ⃗f. If Ng is the total number of grid nodes (whereas N = 2Ng is the number of DoF), ⃗u = (uix, uiy)Ng i=1 ∈RN is a discretized approximation of the continuous displacement field u, i.e. the finite dimensional vector of nodal displacement in both horizontal (x) and vertical (y) directions. The stiffness matrix K is large, but usually sparse. In the more subtle finite element method (FEM), the continuous minimization problem is converted into a 8 finite size optimization as the trial displacement field is reduced to a linear combination of element by element piecewise continuous polynomials: u(x, y) = PNg i=1 Ni(x, y)ui. As each basis function Ni is normalized to unity at its reference node i vanishing outside the adjacent elements region, all the expansion coefficients represent the nodal values of displacement field: ui ≡u(xi, yi). The minimization of the discretized energy Eq. (16) becomes a finite dimensional algebraic problem, see also Eq. (3): Efe(⃗u) = 1 2 Ng X i,j=1 X α,β uiαujβKiαjβ - Ng X i=1 X α uiαfiα (17) Any Neumann-type boundary is represented by some vector of external forces ⃗f ∈RN; and the Greek letters α, β run over Euclidean coordinates {x, y}. All coefficients Kiαjβ of a FEM's stiffness matrix are related to a generic gradient correlator expression: R Ω∂αNi∂βNj dΩ. The minimization of discretized energy potential Efe(⃗u) by variation of the nodal displacements amplitudes is once again equivalent to resolving the algebraic linear equation. C. Tensor product decomposition of the stiffness matrix In order to estimate the N 2 coefficients of K matrix, all cross-integrals of the gradient of shape functions R Ω∂αNi∂βNj dΩshould be calculated on the whole Ω, scaling as ∼N 2. For large N, the FE software rely on a more efficient assembling algorithm, that scales linearly is system size. First, the domain is split into a set of simple polygonial subdomains, called elements: Ω= S el Ωel. Second, for each element, all the connected stiffness contributions are computed, giving for quad4 mesh a 4 × 4 elementary matrix. In what follows, we extend this procedure with the aim of decomposing the stiffness matrix into a more primitive form of tensor product of 2 × 2 matrices more suitable for quantum calculations. The whole domain integrals can be separated into the sum of elementary contributions: Z Ω ∂αNi∂βNj dΩ= X el Z Ωel ∂αNi∂βNj dΩ; (18) We consider here a rectangular Nx × Ny domain, composed of linear Quad4 quadrangles [35, 36]. Here Nx, or Ny is the number of discretization nodes in x (or y direction). Since for this regular mesh all the elements have the same geometry, all the elementary integrals in K are identical. It allows us to separate the nodal (i, j) connectivity from the (x, y) component's links stored in 2 × 2 matrices Kij. These elementary contributions are only dependent on their local connectivity index pairs. With local index notations i, j ∈{a, b, c, d} (see Fig.1) each matrix term is given by the integral over the Quad4 quadrangle: Kij ≡ Z quad dΩ (1 -ν)∂xNi∂xNj + (1 -2ν)/2∂yNi∂yNj (1 -2ν)/2∂yNi∂xNj + ν∂xNi∂yNj (1 -2ν)/2∂xNi∂yNj + ν∂yNi∂xNj (1 -ν)∂yNi∂yNj + (1 -2ν)/2∂xNi∂xNj (19) All expressions for Kij can be found in Appendix B 1. Let us first analyse the contribution of an on-site Kaa term. An initially intuitive idea is to set it as a purely diagonal block matrix. However, the expected diagonal nature of the on-site term comes from the sum of all the four P i∈{a,b,c,d} Kii contributions. For boundary nodes, the summation is only partial. The bottom-left corner node, number 1 on (Fig.5), has exclusively the Kaa onsite contribution. As the total number of elements in the mesh is equal to (Nx-1)(Ny-1), Kaa appears then in the global rigidity matrix the same number of times, namely once during the integration of each quad4 element. The whole set of top and the most right-hand side nodes is not coupled through the Kaa term. This fact is reflected in the block diagonal structure of the full rigidity matrix Kaa contribution, that reads with an appropriate row-byrow nodal numbering as: DNy ⊗DNx ⊗Kaa; (20) The DNα for α ∈{x, y} is the identity matrix with a missing last diagonal element - see Eq. (22). We analyse further the Kab connectivity terms. For each row, the lower nodes are coupled between the nearest neighbors Nx -1 times. The upper row stays uncoupled. The contribution of the Kab term to the full rigidity matrix of rectangular Nx ×Ny shape sample therefore gives: DNy ⊗TNx ⊗Kab, where TNx is a upper single diagonal Toeplitz matrix - see Eq. (22). The calculation of each elementary contribution is similar to our previous development. By analogy with DN we 9 Kaa Kaa Kaa Kaa Kaa Kaa Kaa Kaa 1 2 3 Nx -1 Nx 1+Nx 2+Nx 3+Nx 2Nx 1+2Nx 2+2Nx 3+2Nx 3Nx Figure 5. Kaa component contribution. Kab Kab Kab Kab Kab Kab Kab Kab 1 2 3 Nx -1 Nx 1+Nx 2+Nx 3+Nx 2Nx 1+2Nx 2+2Nx 3+2Nx 3Nx Figure 6. Kab component contribution. can introduce UN, the identity matrix without its first diagonal term, such that the full rigidity matrix aggregates into: K = DNy ⊗DNx ⊗Kaa + DNy ⊗UNx ⊗Kbb+ UNy ⊗UNx ⊗Kcc + UNy ⊗DNx ⊗Kdd+ DNy ⊗TNx ⊗Kab + DNy ⊗T† Nx ⊗Kba+ TNy ⊗DNx ⊗Kad + T† Ny ⊗DNx ⊗Kda+ (21) TNy ⊗UNx ⊗Kbc + T† Ny ⊗UNx ⊗Kcb+ UNy ⊗TNx ⊗Kdc + UNy ⊗T† Nx ⊗Kcd+ TNy ⊗TNx ⊗Kac + T† Ny ⊗T† Nx ⊗Kca+ TNy ⊗T† Nx ⊗Kbd + T† Ny ⊗TNx ⊗Kdb, where all these matrices can be explicitly defined with index notation, as Dij Nα = δij -δiNαδjNα; Tij Nα = δi,j-1 and Uij Nα = δij -δi1δj1, for i, j ∈[1, Nα]. DNα ∈MNα TNα ∈MNα UNα ∈MNα 1 . . . 0 0 ... ... ... ... 0 . . . 1 0 0 . . . 0 0 ; 0 1 . . . 0 ... ... ... ... 0 0 ... 1 0 0 . . . 0 ; 0 0 . . . 0 0 1 . . . 0 ... ... ... ... 0 0 . . . 1 (22) As the number of nodes in each direction is set to a power of two, i.e. Nx = 2nx and Ny = 2ny, we can Figure 7. The parametrized quantum circuit used in our QPU runs and simulations. An unbalanced version of the CCCX gate was used, where the success probability varies depending on the input state (see Section F.1 of Lysaght et al 2024 [30] for more details). express these matrices using the 2 × 2 matrices of the spanning family S defined in Eq. (4)[18]: DNα = I⊗nα -p⊗nα - ; UNα = I⊗nα -p⊗nα + ; TNα ≡T(nα) = nα-1 X k=0 I⊗k ⊗σ+ ⊗σ⊗nα-k-1 - . (23) Crucially, the number of quantumly-computable terms in this decomposition is logarithmically small and can be accessed via 2nxny + O(nx) + O(ny) independent singlequbit quantum measurements. In fact, each leading-order connection term (Kac and Kbd) counts for 4nxny S-tensor product contributions, but their factorization yields directly diagonalizable matrices: T(ny) + T†(ny) ⊗ T(nx) + T†(nx) ⊗I, T(ny) -T†(ny) ⊗ T(nx) -T†(nx) ⊗X. Given that the matrices T(nα) ± T†(nα) are constructed from nα single-qubit observables, the resulting leadingorder scaling is 2nxny as expected. This scaling property reflects not only the sparse nature of the rigidity matrix itself, which has ∼2n non zero elements, but also the fact that we use an overdetermined set of 2×2 matrices for decomposition S. The simultaneous usage of redundant I ≡p+ + p-matrix is absolutely crucial, as we remark that the simplest identity matrix expansion I⊗n = (p+ + p-)⊗n has 2n non-zero spanning components in p± basis. D. Application using Photonic Computing The parametrized QLOQ circuit we implemented is displayed on Figure 7. It splits 4 qubits in two groups 10 of 2 qubits that are encoded on a four-level system (4 modes with 1 photon) [30]. To compensate for small device errors on the QPU we applied a noise mitigation technique from iteration 100 onwards in each run, whereby any value in the circuit's outcome probability distribution below a certain threshold (0.001 in this case) was set to 0, and then the distribution was renormalised. This improved final energy values from approx -21 to approx -23. Applying this noise mitigation technique for all iterations prevents the ansatz from converging below approx -9, which is why it is only activated on iteration 100. To optimize the variational quantum circuit, we use the classical gradient-free optimiser NEWUOA [37] with the following stopping criteria: maximum 220 iterations, 5 × 10-14 absolute tolerance on the result and 5 × 10-10 relative tolerance on the result. We note that the ansatz is not expressive enough to converge. To verify this, we ran 1000 noise-free infiniteshot VQA simulations [38] with the same initial parameters. The mean final energy estimate of these simulations was -23.77 ± 0.3, with a best result of -23.95. We chose this ansatz because it is the largest we can fit on the current optical chip [30]. The mean final energy estimate of the VQA was better in these noise free simulations than on QPU (Sim:- 23.77 vs QPU:-23.15). However, the best QPU result was slightly better than the best noise free simulation result (Sim:-23.95 vs QPU:-24.06). This is likely due to the stochastic nature of variational algorithms on noisy hardware. Both the noisy simulations and QPU runs compare favourably to the theoretical value, with best simulated and QPU results achieving 96.2% and 96.6% accuracy respectively. E. Quantum mesh refinement 5 7 9 11 13 15 17 19 21 8 10 12 14 16 18 2nd Order Reg. Quartered SIF Brown's SIF Number of Qubits Stress Intensity Factor (kg⋅m-1ᐟ2⋅s-2) Figure 8. Convergence of different ways to compute the SIF observable in 2D-plate. The integration over crack lips Eq. (14) (red) and the polynomial interpolation (blue) are to be compared with the reference solution (black) from [27]. 1. From classical to quantum remeshing Mesh refinement is a crucial technique in the FEM for improving the accuracy of numerical solutions. The elements where a higher resolution is needed are subdivided, typically in regions where the solution exhibits discontinuities, steep gradients, or stress concentration. The refinement process increases the number of DoFs, allowing the finite element solution to better capture fine-scale features. The coarse solution is usually projected on the refined mesh in order to get the new discretized field initialization. In case of fracture mechanics, the precise estimation of the SIF necessitates either high refinement close to the crack tip or the usage of specific FE [39]. In the classical resolution, with the FEM, the uniform refinement is inappropriate, as the resulting increase in the number of DoF rapidly overcomes the classical computers capacities. Fig. 8 illustrates this rather slow convergence of the SIF observable in the regular mesh hypothesis: the mesh is reaching ∼1 million of DoFs (21 qubits) before a few percent error to the reference solution is attained even in the simple case of the pre-cracked plate. However, this classical system size bottleneck is alleviated through the exponential capacity of quantum memory. However, variational convergence in the resulting high-dimensional parameter spaces can be difficult. Therefore, we propose to modify the standard VQA procedure incorporating an additional quantum remeshing step. The quantum solution |ψI⟩obtained for some given discretization level is first projected on the uniformly refined mesh: |ψI⟩7→|ψD⟩. The coarse solution serves as the initial state for convergence in the refined space: A |ψD⟩7→|ψF ⟩instead of the standard 'cold start' A |0⟩7→|ψF ⟩(see Fig. 13). In what follows, we illustrate our proposition with some relevant examples. 2. 1D Remeshing. We first consider the 1D case, with a scalar field of unknowns u(x) ∈R. The quantum n-qubit state encodes 2n DoFs u(x) of the discretized field by the amplitudes of the ket |x⟩, i.e., |ψI⟩= 2n-1 P x=0 u(x) |x⟩, where x is written in binary as usual. |ψD⟩should represent the initial guess or the warm start for a more refined solution of n + 1 qubits. As u(x) is supposed to represent some continuous physical solution, we can obtain a good enough approximation of the remeshed function by duplicating the each u(x) value on its right close nearest neighbor as shown in Fig 9: 11 0 2 4 6 8 10 12 14 16 -2 -1 0 1 2 3 X position Displacements Figure 9. The 4-qubit solution |ψI⟩(green), the 5-qubit target state |ψF ⟩(blue), and the 5-qubit state |ψD⟩obtained by duplication of that 4-qubit state (red) This duplicated state contains values for twice as many points as |ψI⟩, but values for successive points are equal, that allows us to write it in the compact form using the Euclidean division (//): |ψD⟩= 2n+1-1 X x=0 u(x//2) |x⟩. (24) Let us call A the ansatz that yields the n-th qubit solution |ψI⟩. To create the duplicated state, we then simply need to enrich this circuit by an extra little endian qubit and to apply the Hadamard gate in order to duplicate the data, see Fig. Eq. (10). q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 q1 : 1 q2 : 2 q3 : 3 q4 : H Figure 10. Circuit yielding the |ψD⟩state containing the duplicated displacements. This causes the state to evolve as follows: |ψD⟩= (I⊗n ⊗H) · (|ψ⟩⊗|0⟩) = |ψ⟩⊗|+⟩ = 2n-1 X x=0 u(x) |x⟩⊗(|0⟩+ |1⟩) = 2n-1 X x=0 u(x) |2x⟩+ 2n-1 X x=0 u(x) |2x + 1⟩ = 2n+1-1 X x=0 u(x//2) |x⟩ (25) We will now no longer optimize over the parameters θA i of A to limit the size of the parameter space. To improve the state, another stage of n + 1 qubit optimization is added via an ansatz B as shown in Fig. 11. We construct B ansatz so that it realizes an identical transformation for all zero set of its parameters: B(θ = 0) ≡I⊗(n+1). By randomly setting the initial parameters of B close to zero , we optimize them from a so-called warm start, which greatly improves the convergence. We remark that this requested ansatz property can easily be enforced for any kind of ansatz by the additional application of the dagger-circuit: B(θ) · B†(θ = 0). q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 Ansatz (θB 0 , θB 1 , θB 2 , θB 3 , ...) 0 q1 : 1 1 q2 : 2 2 q3 : 3 3 q4 : H 4 Figure 11. Circuit C to be optimized stage by stage towards the target at 5-qubit state You can then repeat the procedure by considering this entire circuit C represented in Fig. 11 like a black box, as we considered A throughout this section. Each q-qubit state becomes the warm start for its q+1-qubit remeshed counterpart. 3. 2D Remeshing In 2D, the construction is similar but we need to reorder qubits instead of simply appending them. a. Scalar fields For a scalar field we don't need any qubit to account for local DoF, like in Sec. IV E 2. We can therefore identify any DOF by its position (x, y): quantumly speaking, a trial state |ψI⟩is proportional to 2ny -1 P y=0 2nx-1 P x=0 u(x, y) |y, x⟩, where x and y are written in binary. Notice how points are ordered first along the y axis, and then along x, so that the next point of a point (x, y), is (x + 1, y) if x < Nx -1, and (0, y + 1) otherwise. If we want to apply a remeshing along both directions x and y, we must first obtain the following "duplicated" state: |ψD⟩= 2ny+1-1 X y=0 2nx+1-1 X x=0 u(x//2, y//2) |y, x⟩ (26) To do so, not only do we have to use Hadamard gates to account for how the newest (little-endian) qubit should distribute its amplitude equally between |0⟩and |1⟩(as in Sec. IV E 2), but also add swap gates so that the new qubits reach the correct position in the numbering. This compels us to also add a reordering circuit as follows: 12 q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 q1 : 1 q2 : 2 × q3 : 3 × × q4 : H × q5 : H Figure 12. Circuit yielding the state |ψ2,D⟩containing the duplicated displacements. This causes the state to evolve as follows: |ψ2⟩⊗|00⟩→|ψ2⟩⊗|++⟩ after Hadamard = 2qy -1 X y=0 2qx-1 X x=0 ⃗uA,q(x, y) |y, x⟩⊗|++⟩ = 2qy -1 X y=0 2qx-1 X x=0 ⃗uA,q(x, y) |y⟩|x + +⟩ (27) → 2qy -1 X y=0 2qx-1 X x=0 ⃗uA,q(x, y) |y+⟩|x+⟩ after swaps = 2qy+1-1 X y=0 |y⟩ 2qx+1-1 X x=0 ⃗uA,q(x//2, y//2) |x⟩ like in Eq. 25 Finally, like in the 1D case, we simply have to compose this circuit with an ansatz that implements the identity for known parameters, and optimize starting from such parameters. q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) |ψI⟩ |ψD⟩ Ansatz (θB 0 , θB 1 , θB 2 , θB 3 , ...) |ψF ⟩ q1 : q2 : × q3 : × × q4 : × × q5 : H q6 : H × Figure 13. 6-qubit circuit to be optimized by varying the θB, starting from a 4-qubit circuit that was already optimized by varying the θA. b. Vector fields For a vector field with two local DoF, we need one additional qubit compared to the previous sections, as shown in Eq. (9). An ansatz-created state corresponding to this field could be written as: |ψI⟩= Ny-1 X y=0 Nx-1 X x=0 Nd-1 X d=0 u(x, y, d) |y, x, d⟩ (28) If we want to apply a remeshing along both directions x and y, we first obtain the following "duplicated" state: |ψD⟩= 2Ny-1 X y=0 2Nx-1 X x=0 Nd-1 X d=0 u(x//2, y//2, d) |y, x, d⟩ (29) The reordering circuit is similar to the one seen before, but we also have to put the qd DOF qubits at the end, using swaps: q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 q1 : 1 q2 : 2 × q3 : 3 × × q4 : 4 × × q5 : H q6 : H × Figure 14. Circuit generating |ψD⟩in the 2D vectorial case. Notice that the need for swaps disappears in real quantum circuit execution, by simply using ansätze that don't use the wires that will be used for remeshing. For instance, in the case of the 5-qubit to 7-qubit remeshing of our 2D plate, the first ansatz would use qubits 0, 1, 3, 4, 6, while the second ansatz would use all seven. 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International Journal for Numerical Methods in Engineering 10, 2537 (1976). Appendix A: Extensions of quantum remeshing In the main part of this work, the quantum remeshing technique was introduced and detailed for one- and twodimensional cases. In this appendix, we discuss several complementary topics related to quantum remeshing. We first focus on the extension to three dimensions, then introduce variants of quantum remeshing that avoid the use of swap gates, which may be advantageous for certain QPU hardware architectures. Finally, we provide several suggestions aimed at mitigating the field relaxation issues observed in our simulations. 1. Construction in 3D We illustrate here how the quantum remeshing technique could be extended into 3D case. A reasonable minimal 3D example can be constructed from the initial system with 8-qubits. It corresponds to a 2-qubit discretization in each of the three directions, and four local DoFs (d = 4) that are supposed to be encoded with two complementary qubits. For instance, we could refer to the thermo-mechanical coupling problem, where the unknowns are the three component vector-displacements and the temperature field. In 3D, at each remeshing step, three qubits must be added to represent the homogeneous refinement in each spatial direction. q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 q1 : 1 q2 : 2 q3 : 3 q4 : 4 q5 : 5 q6 : 6 q7 : 7 q8 : q9 : q10 : (a) The initially optimized 8-qubit ansatz, with 3 extra qubits. q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 q1 : 1 q2 : 2 × q3 : 3 × × q4 : 4 × × q5 : 5 × q6 : 6 × × q7 : 7 × × q8 : H q9 : H × q10 : H × (b) Field duplication and reordering within quantum circuit. Figure 15. Construction of the remeshed 3D ansatz, illustrating the transition:|xyzd⟩⊗|000⟩7→|x⟩|+⟩|y⟩|+⟩|z⟩|+⟩|d⟩. In the reordering circuit, the first two swaps place the d components at the end, the second two swaps place the two x values before a |+⟩, and the last two swaps place the two y values before a |+⟩. 15 q0 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 Ansatz (θB 0 , θB 1 , θB 2 , θB 3 , ...) 0 q1 : 1 1 q2 : 2 × 2 q3 : 3 × × 3 q4 : 4 × × 4 q5 : 5 × 5 q6 : 6 × × 6 q7 : 7 × × 7 q8 : H 8 q9 : H × 9 q10 : H × 10 Figure 16. Circuit C to be optimized towards T11 (a) The deformation level on the 13 qubits cascaded quantum solution for the 2D plate. (b) The same as (a) for the next step remeshing 15 qubits. Field relaxation problem is clearly amplified. (c) The cascade remeshing applied to a 3D shape. Figure 17. Usual discretized geometries. 2. Construction without swaps While photonic QPUs offer a relatively flexible implementation of SWAP gates due to their adaptable connectivity, this is not the case for all quantum architectures. In particular, superconducting QPUs often suffer from limited qubit connectivity, where qubits can only interact with their nearest neighbors. As a result, implementing a SWAP gate between distant qubits requires decomposing it into multiple double-qubit operations, significantly increasing circuit depth and error rates. To address this challenge, we propose a swapless variant of quantum remeshing, which avoids the need for costly SWAP operations and is better suited for architectures with constrained connectivity. In this section, we use the index ·b to indicate that a number is written in binary. A number can be written in binary either in little-endian (34b = 100010) or in big-endian ( ̄34b = 010001). We can go from one endianness to another by mirroring the representation. In 2D, it is possible to add qubits in a way that does not significantly affect the nodal enumeration and allows the creation of observables, without the need for swaps. We should simply have y represented by its big-endian binary representation ̄yb, followed by d as usual, followed by x represented as before by its little-endian binary representation xb. The two remeshing qubits have to be added at the top and bottom of the ansatz: The ansatz state at the dotted 16 q0 : H Ansatz (θB 0 , θB 1 , θB 2 , θB 3 , ...) 0 q1 : Ansatz (θA 0 , θA 1 , θA 2 , θA 3 , ...) 0 1 q2 : 1 2 q3 : 2 3 q4 : 3 4 q5 : 4 5 q6 : H 6 Figure 18. Circuit without swaps barrier in Fig. 18 is: |ψNOSW AP ⟩= Ny-1 X y=0 Nx-1 X x=0 Nd-1 X d=0 ⃗uA,4(x, y, d) |+, ̄yb, d, xb, +⟩ = 2Ny-1 X y=0 2Nx-1 X x=0 Nd-1 X d=0 ⃗uA,4(x//2, y//2, d) | ̄yb, d, xb⟩ (A1) Another possibility is to design the ansätze for lower number of qubits so that they do not use some of their internal qubits. This way, they are available for remeshing by simply applying H to them, like in Fig. 18, without any need for endianness modifications. However, the advantage of these methods that output states in the standard |yxd⟩qubits order is that the observable construction follows well-established rules. That observable creation process could be complexified: • The swapless method shown above requires a different handling of x and y. • A swapless method, without any endianness changes, results, after a few rounds of remeshing, in basis states that have to be interpreted as | ̄x0 ̄y0dx1y1x2y2x3y3 . . .⟩. The advantage here is that at each remeshing step, the hardware-efficient ansatz directly entangle the newly added qubits with one-another, as explained in Sec. A 3. We note that, while in 2D it is possible to absorb all swap gates into initial smart qubits reordering A 2, in 3D it is not possible to iteratively construct in qiskit a swapless circuit without requiring unused internal qubits. However, as in 2D, we could still design an ansatz that does not use some of its internal qubits, which stay to |0⟩until the duplication applies to them. 3. Ansatz specification improvement No matter the underlying equation that is treated, the optimization of the ansatz structure is a crucial question. For instance, in order to limit the number of swaps, we could apply the procedures discussed in Sec. A 2. Another important example is linked to the field relaxation problem. Indeed, when applying a remeshing, we transform a smooth function into a piecewise constant function. This leads to a significant increase in error for most physically relevant observables, which typically exhibit continuity and are highly sensitive to disruptions in spatial derivatives. For instance, in fracture mechanics, mechanical strain, defined as the gradient of the displacement field, directly contributes to the calculation of elastic energy. The so-created discontinuities in the mechanical strain represent an extremely high local elastic potential, and hence account for a large part of the energy modification. The newly added nodes contributes the most for local variations - regardless of the endianness of the problem. Given that errors after such a relaxation are mostly local, it would be advised to ensure a reasonable entanglement of these nodes with the other ones, but also between them. As such, given that the usual hardware efficient ansatz mostly entangles close qubits together, a simple proposition would be to place, using swaps and endianness manipulations as in Sec. A 2, all these qubits together within the ansatz. It would still be possible to replace them at their right spot afterwards, so that reading and applying observables would not be modified. A pre-ansatz GHZ state-based entanglement procedure between these D newly-added qubits could also be of use. 17 Figure 19. A field directly after remeshing (in blue), compared with the optimal values (in grey). The clusters of 4 identical displacement values are visible, especially in the bottom left area. 4. Rescaling considerations In the usual version of a remeshing, we increase the number of elements while keeping the initial nodes at the same positions. Quantum remeshing however duplicates the number of nodes, resulting in a new non-conformal mesh: the nodes are created orderly from the extremal nodes, and the former nodes' positions are shifted in order to respect mesh regularity. We illustrate this subtle point on a simple 1D example. L 0 L 0 Figure 20. 1-D remeshing from 2 to 3 qubits. Consider a 1D, 4-nodes discretized field (N = 4) corresponding to a 2-qubit system. The third point (n = 3) is associated with the basis state |11⟩and has an amplitude that will be noted as T2,3 (T2 being the entire field and T2,3 its fourth component, its first being T2,0). We can note its position as p2,3 = L, and the position of the first point as p2,0 = 0. That way, the n-th point has position p2,n = nL 3 . We will now apply a remeshing step. Consider the point associated to |000⟩. Since we are still operating between positions 0 and L, its position is still at p3,0 = 0. The next point is associated to |001⟩. Its position is at p3,1 = L/7. Notice that p3,0 ⩽p2,0 ⩽p3,1, meaning that it lies inside the element bounded by its two nodal successors. L 0 L 0 Figure 21. Highlighting the node association. 18 Consider the node associated to |10⟩. Its position is at p2,2 = 2L/3. Its remeshed (successor) nodes are associated to ket |100⟩and |101⟩. Notice that p3,4 ⩽p2,2 ⩽p3,5, meaning that it is between its successor nodes as well. More generally, consider the point associated to |n⟩in a q-qubit field. Its position is pq,n = nL 2q-1 and the position of its remeshed points are: pq+1,2n = 2nL 2q+1 -1 and pq+1,2n+1 = (2n + 1)L 2q+1 -1 , Inner projection inequality is satisfied: pq+1,2n ⩽pq,n ⩽pq+1,2n+1 (A2) Indeed, pq+1,2n = nL 2q -1 2 ⩽ nL 2q -1 = pq,n (A3) pq+1,2n+1 -pq,n L = 2n + 1 2q+1 -1 - n 2q -1 = 2q -n -1 (2q+1 -1)(2q -1) ⩾0 ∀n ∈[0, 2q -1] ∩N (A4) Similarly, in 2-D and 3-D, the initial node is in the square (resp. the cube) formed by its successors. Appendix B: Decomposition of the stiffness matrix In this section, we present the complete procedure for decomposing the stiffness matrix into elementary quantumly computable observables, which scale logarithmically with the system size. 1. Elementary connecting links Kij The computation of the stiffness matrix K constitutes the main task of the finite element discretization of the Navier-Cauchy mechanical problem Eq. (1). For the regular meshes the tensor product decomposition Eq. (21) allows one to present it in a compact form. Any elementary connecting link Kij introduced in Eq. (21) can be evaluated using its integral expression given in Eq. (19): Kaa = Kcc = 1/2 -2ν/3 1/8 1/8 1/2 -2ν/3 ; (B1) Kbb = Kdd = 1/2 -2ν/3 -1/8 -1/8 1/2 -2ν/3 ; (B2) Kab = Kcd = -1/4 + ν/6 -1/8 + ν/2 1/8 -ν/2 ν/6 ; (B3) Kad = Kcb = ν/6 1/8 -ν/2 -1/8 + ν/2 -1/4 + ν/6 ; (B4) Kac = Kca = -1/4 + ν/3 -1/8 -1/8 -1/4 + ν/3 ; (B5) Kbd = Kdb = -1/4 + ν/3 1/8 1/8 -1/4 + ν/3 . (B6) 19 These link matrices are to be decomposed into the Pauli basis Eq. (4) prior to any quantum calculations: Kaa = 3 + 4ν 6 I + 1 8X; Kbb = 3 + 4ν 6 I -1 8X; (B7) Kab = -3 + 4ν 24 I -1 8Z + 4ν -1 8 iY ; (B8) Kbc = -3 + 4ν 24 I + 1 8Z + 4ν -1 8 iY ; (B9) Kac = -3 + 4ν 12 I -1 8X; Kbd = -3 + 4ν 12 I + 1 8X. (B10) The procedure could be extended for other relevant situations, such as the discretized scalar partial differential equations (PDE). For example, the tensor product decomposition for Poisson PDE can be trivially obtained from Eq. (21) by replacing all connecting links Kij by its scalar counterparts from the elementary stiffness matrix: Kij 7→Kij. The finite elements discretization for the Poisson equation is generated by the following set of Kij: Kaa Kab Kac Kad Kba Kbb Kbc Kbd Kca Kcb Kcc Kcd Kda Kdb Kdc Kdd = 4 -1 -2 -1 -1 4 -1 -2 -2 -1 4 -1 -1 -2 -1 4 (B11) The same principle applies for the nearest neighbors finite difference scheme (FDM): Kaa Kab Kac Kad Kba Kbb Kbc Kbd Kca Kcb Kcc Kcd Kda Kdb Kdc Kdd = 2 -1 0 -1 -1 2 -1 0 0 -1 2 -1 -1 0 -1 2 (B12) 2. Compact form of K Using the system's symmetry, we rewrite the full rigidity matrix Eq. (21) into a compact form containing only 8 independent contributions that are proportional to the elementary connecting link, defined in Appendix B 1: K = h (I⊗ny -p⊗ny - ) ⊗(I⊗nx -p⊗nx - ) + (I⊗ny -p⊗ny + ) ⊗(I⊗nx -p⊗nx + ) i ⊗Kaa + h (I⊗ny -p⊗ny - ) ⊗(I⊗nx -p⊗nx + ) + (I⊗ny -p⊗ny + ) ⊗(I⊗nx -p⊗nx - ) i ⊗Kbb + h (I⊗ny -p⊗ny - ) ⊗T(nx) + (I⊗ny -p⊗ny + ) ⊗T†(nx) i ⊗Kab + T(ny) ⊗(I⊗nx -p⊗nx + ) + T†(ny) ⊗(I⊗nx -p⊗nx - ) ⊗Kbc + T(ny) ⊗T(nx) + T†(ny) ⊗T†(nx) ⊗Kac + T(ny) ⊗T†(nx) + T†(ny) ⊗T(nx) ⊗Kbd + T(ny) ⊗(I⊗nx -p⊗nx - ) + T†(ny) ⊗(I⊗nx -p⊗nx + ) ⊗K† bc + h (I⊗ny -p⊗ny - ) ⊗T†(nx) + (I⊗ny -p⊗ny + ) ⊗T(nx) i ⊗K† ab. (B13) As shown in Appendix B 1 all the connecting 2×2 matrices Kij (Eq. (19)) can be decomposed in the Pauli basis, with the help of linear coefficients dependent exclusively on the material's Poisson ratio ν. The non-diagonal matrices used in the K decomposition are them-self represented with help of the elementary 2 × 2 matrices set: T(nα) = nα-1 X k=0 I⊗k ⊗σ+ ⊗σ⊗nα-k-1 - . As arbitrary linear combinations of n-times tensor products of 2 × 2 matrices (M2(R) space) generate the M2n(R) linear vector space, we can decompose K ∈M2n(R) in the linear span of the polynomial tensor product of its four basis elements {p±, σ±}, which scales exponentially. The number of spanning terms in most general case of dense 20 matrix scales as the corresponding vector space dimension, i.e. as 2n × 2n. One of our important claim is that the decomposition of K contains only O(nxny)-terms in a specific linear span generated by five elements S = {p±, σ±, I}. Even if it contains the non hermitian σ± operators, we prove in the Appendix B 3 that all these contributions can be regrouped in a are quantumly-computable observables. 3. Diagonalization of σ± polynomials In this section we show some explicit examples of diagonalization of σ± polynomials, that naturally appears in the decomposition of the stiffness matrix K, Eq. (B13). Let us explicitly introduce the matrix notation we are using. We recall the expression of these common matrices: p+ = |0⟩⟨0| = 1 0 0 0 ; σ+ = |0⟩⟨1| = 0 1 0 0 ; (B14) p-= |1⟩⟨1| = 0 0 0 1 ; σ-= |1⟩⟨0| = 0 0 1 0 . CNOT = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ; σ+ ⊗σ+ = 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 ; σ+ ⊗p+ = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 . (B15) The first set of transformations is a partial diagonalization of σ+ ⊗σ+: CNOT · σ+ ⊗σ+ · CNOT = 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 · 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 · 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 = 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 = σ+ ⊗p+ (B16) Trivially for σ⊗3 + ≡σ+ ⊗σ+ ⊗σ+ we convert σ+ into the diagonal matrix p+: I ⊗CNOT · σ⊗3 + · I ⊗CNOT = σ+ ⊗σ+ ⊗p+ (B17) Two consecutive CNOT transformations lead to two reductions in the number of tensored non-diagonal contributions σ+: CNOT ⊗I · I ⊗CNOT · σ⊗3 + · I ⊗CNOT · CNOT ⊗I = σ+ ⊗p+ ⊗p+ (B18) Similarly, n consecutive such CNOT on consecutive qubit couples as shown in Fig. 22a will transform σ⊗n + into p⊗n + . For any power of σ+, we can define the unitary transformation B(n) ∈M2n(R) so that: B†(n) · σ⊗n + · B(n) = σ+ ⊗p⊗(n-1) + , where B(n) ≡ n-2 Y k=0 I⊗(n-2-k) ⊗CNOT ⊗I⊗k. (B19) In the quantum circuit representation, the B(n) operator corresponds to the consecutive execution of CNOT gates starting from the last (bottom) qubit, as shown in (Fig. 22). The equivalent equation can be obtained for σpolynomials via the hermitian conjugation of Eq. (B19): B†(n) · σ⊗n - · B(n) = σ-⊗p⊗(n-1) + (B20) The hermitian operator σ⊗n + + σ⊗n - can be presented in the following almost diagonal form, where only the first qubit operator is still off-diagonal: B†(n) · (σ⊗n + + σ⊗n -) · B(n) = (σ+ + σ-) ⊗p⊗(n-1) + ≡X ⊗p⊗(n-1) + . (B21) We notice that another type of hermitian operator constructed from the antisymmetric matrix i(σ⊗n + -σ⊗n -) can also be reduced in a similar way, resulting in trailing Y gate: B†(n) · (σ⊗n + -σ⊗n -) · B(n) = (σ+ -σ-) ⊗p⊗(n-1) + ≡iY ⊗p⊗(n-1) + . (B22) 21 q0 : • H q1 : • q2 : • q3 : • q4 : (a) A 5-qubit BH(5) circuit for σ⊗5 + + σ⊗5 -. q0 : X • H q1 : X • q2 : • q3 : • q4 : (b) A 5-qubit circuit for σ⊗2 -⊗σ⊗3 + + σ⊗2 + ⊗σ⊗3 -. Figure 22. Quantum circuit diagonalizing σ± polynomials. The X-gates are used to convert the circuit (b) to a standardized form (a) ready for the chained CNOT σ± order reduction. H-gate diagonalizes the trailing first qubit X-gate. The single qubit gate X and Y can be diagonalized with H = 1 √ 2 1 1 1 -1 and N = 1 √ 2 -1 -i i 1 respectively, leading to the full diagonalization circuit: B† H(n) · (σ⊗n + + σ⊗n -) · BH(n) = Z ⊗p⊗(n-1) + ; B† N(n) · i(σ⊗n - -σ⊗n + ) · BN(n) = Z ⊗p⊗(n-1) + , (B23) where we define the unitary of consecutive action of the chained CNOT and first qubit rotation: BH(n) = B(n) · H ⊗ I⊗(n-1) and BN(n) = B(n) · N ⊗I⊗(n-1). The CNOT-chain action on the polynomial containing both σ+ and σ-leads also to the shape with a single qubit off-diagonal. We present here only two hermitian operators that are most relevant for our computations: B† H(n + 1) · (σ-⊗σ⊗n + + σ+ ⊗σ⊗n -) · BH(n + 1) =Z ⊗p-⊗p⊗(n-1) + (B24) iB† N(n + 1) · (σ-⊗σ⊗n + -σ+ ⊗σ⊗n -) · BN(n + 1) =Z ⊗p-⊗p⊗(n-1) + (B25) In the general case, any mixed σ± powers can be treated with help of supplementary X-gate conversion of the initial operator into a standardized form σ⊗n + + σ⊗n - prior to diagonalization, illustrated in some examples in Fig. 22. Having explicitly explained the decomposition of K as a sum of matrices in S, we can now estimate the number of terms that have to be measured separately to calculate ⟨ψ | K | ψ⟩and the diagonalization circuits for each such term. Terms of the form w ⊗I and w ⊗Z can be computed simultaneously. However, for terms ending in X (respectively, Y ), the last qubit has to be once again diagonalized with H (or N respectively). We therefore have to consider such terms separately. The full decomposition and its diagonalization terms are summarized in the table in Sec. B 4. 22 4. Diagonalizable terms from the decomposition of K We display here the (2nxny + 2nx + 2ny + 2) terms of a decomposition of K, which proves the scaling for the independent measurement of K. The tensorial product ⊗has been omitted for clarity. This number of terms can be further reduced by noticing that some of these terms are simultaneously diagonalizable - for instance, for any k, k′, any term diagonalized by I⊗ny+k ⊗BH(nx -k) ⊗I is also diagonalized by I⊗k ⊗BH(ny -k′) ⊗BH(nx -k) ⊗I. Term in K decomposition Diagonalized by # terms 3+4ν 6 4I⊗n-1 -2p⊗ny - I⊗nx -2p⊗ny + I⊗nx -2I⊗nyp⊗nx - -2I⊗nyp⊗nx + I⊗n 1 +p⊗ny - (p⊗nx + + p⊗nx - ) + p⊗ny + (p⊗nx + + p⊗nx - ) I -3+4ν 12 ny-1 P k=0 nx-1 P k′=0 I⊗k(σ+σ⊗ny-k-1 - + σ-σ⊗ny-k-1 + ) I⊗k ⊗BH(ny -k) nxny I⊗k′(σ+σ⊗nx-k′-1 - + σ-σ⊗nx-k′-1 + )I ⊗I⊗k′ ⊗BH(nx -k′) ⊗I nx-1 P k=0 (p⊗ny + + p⊗ny - -2I⊗ny)I⊗k(σ+σ⊗nx-k-1 - + σ-σ⊗nx-k-1 + )( 3+4ν 24 I + 1 8Z) I⊗ny+k ⊗BH(nx -k) ⊗I nx ny-1 P k=0 I⊗k(σ+σ⊗ny-k-1 - + σ-σ⊗ny-k-1 + )(p⊗nx + + p⊗nx - -2I⊗nx)( 3+4ν 24 I -1 8Z) I⊗k ⊗BH(ny -k) ⊗I⊗nx+1 ny 1 8 p⊗ny + -p⊗ny - p⊗nx + -p⊗nx - X I⊗n-1 ⊗H 1 1 8 ny-1 P k=0 nx-1 P k′=0 [I⊗k σ+σ⊗ny-k-1 - -σ-σ⊗ny-k-1 + I⊗k ⊗BN(ny -k) nxny I⊗k′(σ-σ⊗nx-k′-1 + -σ+σ⊗nx-k′-1 - )]X ⊗I⊗k′ ⊗BN(nx -k′) ⊗H 4ν-1 8 i nx-1 P k=0 (p⊗ny + -p⊗ny - )I⊗k σ+σ⊗nx-k-1 - -σ-σ⊗nx-k-1 + Y I⊗ny+k ⊗BN(nx -k) ⊗N nx 4ν-1 8 i ny-1 P k=0 I⊗k σ-σ⊗ny-k-1 + -σ+σ⊗ny-k-1 - (p⊗nx + -p⊗nx - )Y I⊗k ⊗BN(ny -k) ⊗I⊗nx ⊗N ny
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2510.14745
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Transport with noise in dilute gases: Effect of Langevin thermostat on
transport coefficients.
A. Alés, J. I. Cerato, L. Marchioni and M. Hoyuelos
Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR-CONICET), Departamento de Física, Facultad de
Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata(NM, Funes 3350, Mar del
Plata, 7600, Argentina
Abstract
In dilute gases, transport properties such as the thermal conductivity, self-diffusion, and viscosity
are significantly affected by interatomic collisions, which are determined by the potential form. This
study explores these transport properties in the presence of a Langevin thermostat in systems where
particles interact through various potentials, including soft-core and hard-core potentials, both with
and without an attractive region. Using molecular dynamics simulations and a theoretical approach
based on an analogy with an electric circuit (Ohm’s law), we derived and compared the transport
coefficients across these interatomic potentials for different couplings with the thermostat. The
transport coefficients were obtained by considering the thermostat as a resistance in a series circuit.
1. Introduction
Boltzmann’s kinetic theory of gases is essential for understanding the transport properties
of dilute gases [1]. Within the framework established by Chapman and Enskog [2, 3], this theory
provides the basis for deriving the coefficients of thermal conductivity, self-diffusivity, and viscosity
that describe the transport of energy, mass, and momentum, respectively. These coefficients relate
microscopic properties, such as the size or mass of the particles, with macroscopic behaviors.
This link is fundamental for interpreting the behavior of gases across various fields, from chemical
engineering to astrophysics.
In the Boltzmann theory, particle collisions are described through interatomic potentials [3]. For
example, the hard-sphere potential, which takes a zero value beyond a certain radius and infinity
within it, is widely used in numerical simulations as a preliminary step in analyzing systems with
more complex interactions [4, 5]. In general, interatomic potentials typically consist of a repulsive
component at short distances, and may include one or more attractive or repulsive contributions at
intermediate ranges, eventually decaying to zero at long distances. In his pioneering work, Lennard-
Jones introduced a potential as a function of the interatomic distance r, with an attractive term
proportional to r−6, derived from consideration of dipole moment fluctuations [6]. For convenience,
the repulsive term is often modeled as r−12, although numerous studies have explored alternative
power law dependencies. Hard-core potentials, such as the Lennard-Jones potential, impose an
infinite kinetic energy requirement for particles to approach each other, whereas soft-core potentials
maintain a finite repulsive energy at short distances, such as the Morse potential [7, 8]. Notably,
the presence of attractive regions in a potential opens a range of properties in gas behavior. When
there is an attractive depth-well potential interaction, particles can exhibit binding or orbiting
phenomena if they collide with specific impact parameters and energies [3, 9], adding further
complexity to the analysis of gas dynamics.
arXiv:2510.14745v1 [cond-mat.stat-mech] 16 Oct 2025
The measurement of transport coefficients for these potentials in dilute gases has been a well-
established practice since the early days of computational simulations. However, the inclusion of a
thermal bath introduces significant modifications to the transport coefficients because its influence
disrupts the natural dynamics of the system. Historically, it has been recommended to measure
these coefficients only when the thermal bath provides a very long damping time, thus minimizing
its impact on the intrinsic transport properties [10]; see also Sec. 6.1.1 in [11].
This external influence from the thermostat was initially thought to obscure the essence of
the transport coefficient measurements. For example, the Andersen thermal bath substantially
alters the transport coefficient values, unless the damping time is sufficiently large. Despite these
concerns, recent studies have demonstrated that for viscosity [12] and self-diffusion coefficients [13],
the effect of a Langevin thermal bath is predictable and quantifiable. Specifically, this influence
can be attributed to the coupling of two independent transport contributions: one arising from
interatomic interactions and the other arising from the thermal bath in which the particles are
immersed. For brevity, in the following we use “diffusion” instead of “self-diffusion” to refer to
diffusion of tagged particles in a medium composed by particles of the same kind.
Moreover, the damping factor of the Langevin thermostat serves as a control parameter [13, 12].
By tuning this factor, one can transition from a regime dominated by the thermal bath, charac-
terized by small damping times, to a regime where the influence of the bath becomes negligible as
the damping time approaches infinity.
The aim of this paper is to study, both numerically through molecular dynamics simulations and
analytically using generalized Ohm’s laws, the behavior of the transport coefficients for hard- and
soft-core potentials, with or without an attractive part, in the presence of a Langevin thermostat.
The behavior of the transport coefficients as functions of the damping time is obtained considering
that the effect of the thermostat can be represented as a resistance in a series circuit.
This paper is structured as follows. The methodology used is detailed in Section 2, the analytical
and computational results are presented in Section 3, and conclusions are provided in Section 4.
2. Methodology
Near equilibrium, every macroscopic transport process is linearly related to the correlations of
microscopic fluctuations around the equilibrium state. The Green-Kubo relations provide a rig-
orous framework to obtain these coefficients by expressing them as integrals over time-correlation
functions of microscopic currents in equilibrium systems without requiring the application of ex-
ternal gradients [14, 15]. However, while these relations establish a connection between transport
coefficients and equilibrium fluctuations, they do not provide explicit expressions for the correlation
functions themselves. In contrast, the Chapman-Enskog method, based on the Boltzmann equa-
tion, allows for explicit computation of these correlations in the case of dilute gases by employing
a perturbative expansion of the distribution function [2].
The Langevin thermostat introduces an internal stochastic current, which we refer to as the
Langevin current. This current arises from the random forces exerted by the thermostat, which
mimic the thermal fluctuations experienced by particles in a fluid. These fluctuations follow the
statistical principles of Brownian motion, resulting in a dynamic that contributes to the overall
transport behavior.
We present the simulation details in subsection 2.1 and briefly review the calculation of the
transport coefficients using the Chapman-Enksog approach for a particle system without a thermal
bath. For a dilute gas in a thermal bath, if interactions are neglected and particle dynamics is
2
0.0
0.5
1.0
1.5
2.0
2.5
r/σ
0
2
4
6
8
Φ(r)/ϵ
Pseudo-hard spheres potential
Lennard-Jones potential
Morse potential
Soft spheres potential
Figure 1: Scaled interatomic potentials used in this paper, Φ/ϵ, against distance, r/σ (ϵ and σ are characteristic
values of energy and distance for each potential). The potentials are pesudo hard spheres (blue line), Lennard-Jones
(LJ, green dashed curve), Morse (red dotted curve) and soft spheres (purple dot-dashed curve). Soft spheres and
Morse potentials are soft core, while LJ and pseudo hard spheres potentials are hard core (they diverge as r →0).
essentially Brownian, transport coefficients are obtained by means of Green-Kubo calculations; see
subsection 2.2.
2.1. Simulation Details
Molecular dynamic simulations were implemented using LAMMPS software [16] (Large-scale
Atomic-Molecular Massively Parallel Simulator).
Each interatomic potential has characteristic
length σµ and energy ϵµ, where the subindex µ identifies the potential. The particle density is
ρ = N/V , where N is the particle number, V is the system volume, and the reduced dimensionless
particle density is given by ρ∗= ρ σ3
µ. The reduced temperature T ∗in terms of the temperature T is
T ∗= kBT/ϵµ. In all cases, we used a small reduced density of ρ∗= 0.01 for which Boltzmann theory
approximately holds. To represent qualitatively different interactions, we select the following four
interatomic potentials Φµ(r):
1. Pseudo-hard spheres:
ΦHS(r) =
ϵHS λr
λr−λa
λr
λa
λa h σHS
r
λr −
σHS
r
λai
+ ϵHS
for r < σHS( λr
λa)
1
λr−λa
0
for r ≥σHS( λr
λa)
1
λr−λa
where the scheme of WCA (Weeks-Chandler-Andersen) [17] is used. This potential correctly
reproduces the hard-sphere properties for λr = 50 and λa = 49, with a reduced temperature
T ∗= 1.5, as shown in Ref. [5]. In the simulations, we used ϵHS = 1 and σHS = 1 in LJ units.
2. Lennard-Jones (LJ) is the most common interatomic potential for simple fluid modeling:
ΦLJ(r) = ϵLJ
σLJ
r
12
−
σLJ
r
6
.
3
A cutoff radius rc = 2.5 was used, and the values ϵLJ = 1 and σLJ = 1 in LJ units were used.
3. Morse potential, given by
ΦM(r) = ϵM
e−2α(r−σM) −2e−α(r−σM)
(see [18]). The parameters of this potential has been chosen to fit the well of the Lennard-
Jones region for direct comparison, and the values are ϵM = 1.10 ϵLJ, α = 5.58/σLJ and
σM = 1.12 σLJ.
4. At last, soft-spheres interatomic potential, modeled as
ΦSS(r) =
ϵSS [1 + cos(πr/σSS)]
for r < σSS
0
for r ≥σSS ,
where we used ϵSS = 50 and σSS = 1 in LJ units for the simulations.
The shapes of these potentials are shown in Fig. 1. Whereas the first two are hard-core po-
tentials (they diverge for r →0), the last two are soft-core potentials.
The pseudo-hard and
soft spheres are potentials with only a repulsive part, whereas the remaining potentials have an
attractive depth well. The temperature used for the pseudo-hard sphere potential is T ∗= 1.5
(that, as mentioned before, is the temperature for which hard-sphere properties are reproduced)
and for soft-spheres is T ∗= 4. Lennard-Jones and Morse potentials are studied considering both
temperatures.
Molecular dynamics (MD) simulations were performed with 32,000 particles, including two
equilibration stages. In the first stage, Gaussian-distributed velocities were assigned to the par-
ticles, which were initially arranged in an fcc lattice, and the system was evolved in the NVT
ensemble using a Langevin thermostat with a specified damping time td. After 2 × 105 time steps,
the simulation was switched to the NVE ensemble and continued for the same number of steps
to allow the system to equilibrate. Subsequently, a data-gathering run of 1.5 × 106 steps was
carried out in the NVE ensemble. A time step of 0.001 was employed in all cases, except for the
pseudo-hard sphere potential, for which a smaller time step of 0.0001 was required to ensure the
dynamical stability of the system.
2.2. Theoretical Background
The Boltzmann equation holds for a dilute gas; it is an integro-differential equation that de-
scribes the one-particle nonequilibrium distribution function, f(r, v, t), which gives the average
number of particles with position r and velocity v at time t [15, 19].
The Chapman-Enskog
method uses this distribution function to determine transport coefficients in the zero-density limit.
A perturbative expansion is used, where f(r, v, t) is expressed as a series around the equilibrium
distribution of the Boltzmann equation. The results align with macroscopic hydrodynamic equa-
tions by stopping the expansion after the first correction [2, 19]. The transport coefficients are then
calculated as a series involving Sonine polynomials, which typically converge quickly, meaning that
only a few terms are needed for accurate results [3]. The transport coefficients for dilute gases are
λB = 75 kB
64 σ2
kBT
πm
1/2
fλ
Ω(2,2),
(1)
for thermal conductivity, where m is the particle mass, T is temperature and kB is Boltzmann’s
constant; we have the following expression for the diffusion coefficient,
DB =
3
8 ρ σ2
kBT
mπ
1/2
fD
Ω(1,1);
(2)
4
and the viscosity coefficient is
ηB =
5
16 σ2
mkBT
π
1/2
fν
Ω(2,2),
(3)
where Ω(i,j) are collision integrals relative to the hard sphere system, which depend on the temper-
ature and the type of interatomic potential considered [3, 19]. We have included the subscript B to
indicate that these coefficients were derived by considering particle collisions within the framework
of the Boltzmann equation [15, 19]. The fi functions, known as correction factors, are expressed
in terms of the collision integrals [9, 20]. Since it is well known that deviations of fi from unity
are very small, we consider fi = 1, with i = ν, D, λ; see [21] or Ch. 19 in [19].
The dimensionless coefficients are obtained using the following scaling: D∗= D σ−1p
m/ϵ,
η∗= η σ2/√mϵ and λ∗= λ k−1
B σ2p
m/ϵ (subindex µ in σ and ϵ is omitted for simplicity). By
applying the scaling to Eqs. (2), (3) and (1) we obtain:
D∗
B =
3
8 ρ∗
p
T ∗/π
Ω∗(1,1) ,
(4)
η∗
B = 5
16
p
T ∗/π
Ω∗(2,2) ,
(5)
λ∗
B = 75
64
p
T ∗/π
Ω∗(2,2) .
(6)
From another perspective, the Green-Kubo relations provide a framework to obtain these co-
efficients by relating them to the equilibrium time correlations of microscopic currents [2, 15, 14].
These relationships are particularly powerful because they allow the calculation of transport prop-
erties in equilibrium systems without requiring the application of external gradients. The corre-
lations are commonly obtained using numerical methods; however, as explained in subsection 3.1,
analytical results are possible for the Brownian dynamics produced by the Langevin thermostat
for small concentrations. For example, the thermal conductivity λ is expressed as
λ =
1
kBT 2V
Z ∞
0
⟨JQ(0)JQ(t)⟩dt,
(7)
where JQ(t) is the energy current in an arbitrary direction (the system is isotropic), V is the system
volume, and ⟨JQ(0)JQ(t)⟩denotes the ensemble average of the correlation between the current at
time zero and time t (more details on this topic in Sec. 3.1).
We aimed to analyze a system with atomic collisions in a thermal bath at a low concentration.
A binary gas mixture can be considered for this description. The system consists of the particles of
interest, at low density, of mass m; they are immersed in a thermal bath represented by particles
with mass m′, with m ≫m′. Interatomic collisions are as follows: m-m collisions, governed by
the chosen interatomic potential, are rare due to the low particle density; m-m′ collisions, on the
other hand, are more frequent and follow the well-established rules of Brownian motion. Collisions
m′-m′ among thermal bath particles are not relevant in this scenario. This picture indicates that
the transport of any conserved quantity has to overcome the resistance produced by both types of
collisions, m-m and m-m′, as in a series circuit.
Generalized Ohm’s laws are linear relations between thermodynamic forces (gradients of chem-
ical potential, temperature or velocity) and currents (of particles, heat or momentum); the propor-
tionality constants are the transport coefficients. These laws are fundamental to the development
5
of hydrodynamic equations in classical irreversible thermodynamics. The entropy production in
the hydrodynamic equations can be written as the product of currents times forces, analogous to
the Joule heat. Here, we consider the consequences of this analogy when resistances in series are
present. For example, let us consider the current of tagged particles. The presence of other parti-
cles of the same type causes resistance to the particle current. In the presence of the thermostat,
there is also the resistance caused by the small particles of the reservoir, whose effect is represented
by random forces and damping. The particle current has to overcome both resistances, therefore,
they are in series. The same reasoning can be applied to heat or momentum currents. Transport
coefficients represent, instead of resistances, the conductances. We call CB the conductance in the
absence of the thermostat that is obtained from the Boltzmann’s theory, and CL the conductance
introduced by the Langevin thermostat. Since the circuit is in series, the resulting conductance is
1
C = 1
CL
+ 1
CB
.
(8)
This concept, which extends the consequences of the electric circuit analogy in hydrodynamic
equations, was introduced in [13, 12]. Depending on the damping time of the Langevin thermostat,
the transport coefficients can be dominated by one or the other contribution. If the damping time
is short, the system responds according to the Brownian contribution to the conductance, whereas
for long damping times, it behaves according to the Boltzmann contribution.
3. Results and Analysis
3.1. Transport coefficient calculation for Langevin Currents
Considering an arbitrary direction in an isotropic system, the Langevin thermostat produces a
force, on particle i, that is given by
Fi(t) = −mvi/td + αi(t),
(9)
where m and vi are the mass and velocity of particle i, td is the damping time, and αi(s) is a
random force with zero mean, ⟨αi(t)⟩= 0, and delta time correlated, ⟨αi(t)αi(t′)⟩= 2mkBT
td
δ(t−t′);
the noise amplitude is determined by the fluctuation dissipation theorem. Force Fi, velocity vi and
random force αi correspond to the components in an arbitrary direction of vectors Fi, vi and αi.
Under this conditions, the diffusion coefficient is well known (see, for example, [15]); it is given by
the Einstein relation,
DL = kBT td
m
.
(10)
The viscosity coefficient of the system composed of Brownian particles (not to be confused with
the viscosity of the medium) is (see [12])
ηL = ρ kBT td
2
.
(11)
For thermal conductivity, we take the following approach. In the absence of both interactions
between particles and external potentials, the energy flux for an N particle system in a given
direction is JQ(t) = PN
i=1
m
2 vi(t)2 −⟨hi⟩
vi(t), where ⟨hi⟩is the mean enthalpy per particle, that
takes the value 5
2kBT for an ideal gas (see, for example, Sec. 21.8 in [19]). Using the Green-Kubo
6
expression for thermal conductivity, we have,
λL =
1
V kBT 2
Z ∞
0
⟨JQ(t)JQ(0)⟩
=
1
V kBT 2
Z ∞
0
*X
i,j
m
2 vi(t)2 −⟨hi⟩
vi(t)
m
2 vj(0)2 −⟨hj⟩
vj(0)
+
dt
=
ρ
kBT 2
Z ∞
0
m
2 vi(t)2 −5
2kBT
vi(t)
m
2 vi(0)2 −5
2kBT
vi(0)
dt,
(12)
where correlations between different Brownian particles are neglected; in the last line, N/V is
replaced by ρ and subindex i corresponds to any particle (symmetry under exchange of particle
labels is assumed). Now, we use the expression for the velocity of a Brownian particle (see, for
example, Sec. 7.2.1 in [15]),
vi(t) = vi(0) +
Z t
0
dse−γ(t−s)αi(s).
(13)
Replacing (13) in (12), and using the statistical properties of the random force αi, the integral can
be solved to obtain,
λL = ρ k2
B T td
2 m
.
(14)
The dimensionless coefficients, then, become
D∗
L = T ∗t∗
d
(15)
η∗
L = ρ∗T ∗t∗
d/2
(16)
λ∗
L = ρ∗T ∗t∗
d/2,
(17)
where t∗
d = td σ−1p
ϵ/m. The Einstein relation for the diffusion coefficient, DL, has fundamental
historical and experimental importance in understanding Brownian motion. The expressions for
the viscosity and thermal conductivity, ηL and λL, are interesting from a theoretical and numerical
point of view, but it is probably not possible to verify them experimentally because of the difficulties
in separating the transport of heat or momentum of the particle system from the transport of the
same quantities that occur in the reservoir where the system is embedded.
3.2. Calculation of the transport coefficients with noise
In Eq. (8) any transport coefficient is represented by the conductance C. The dimensionless
expressions for the coefficients from Boltzmann theory, Eqs. (4)-(6), and from the Langevin theory,
Eqs. (15)-(17), are used to calculate, in Eq. (8), the transport coefficients at small concentration
in the presence of the Langevin thermostat assuming a series circuit. The results are,
D∗=
T ∗t∗
d
8
3ρ∗t∗
d Ω∗(1,1)√
π T ∗+ 1
,
(18)
η∗=
T ∗t∗
d ρ∗
16
5 ρ∗t∗
d Ω∗(2,2)√
π T ∗+ 2
,
(19)
λ∗=
T ∗t∗
d ρ∗
64
75ρ∗t∗
d Ω∗(2,2)√
π T ∗+ 2
.
(20)
7
It can be seen that, for large t∗
d, we recover the expressions from the Boltzmann theory, Eqs.
(4)-(6), while for small t∗
d we have the Langevin results, Eqs. (15)-(17).
Considering that the
collision integrals are of order 1, a large or small t∗
d means t∗
d ≫1/(ρ∗√
T ∗) or t∗
d ≪1/(ρ∗√
T ∗).
We obtained numerical results, presented in the next section, that verified these expressions as
functions of damping time, t∗
d, for fixed values of temperature and concentration (ρ∗= 0.01).
Collision integrals can be calculated using the following nested integrals [3],
Ω(l,s)(T) =
kBT
2πm
1/2 Z ∞
0
dx e−x2x(2 s+3)Sl(kBTx),
(21)
where Sl(kBTx) is the collision cross section, a function of the initial kinetic energy of colliding
particles; is is defined as
Sl(E) = 2 π
Z ∞
0
b
1 −cosl (χ(b, E))
db.
(22)
Here, b denotes the impact parameter and χ the classical deflection angle as a function of the
relative kinetic energy of the colliding particles,
χ(b, E) = π −2 b
Z ∞
rm
dr
r2F(r, b, E),
(23)
where rm is the classical returning point and F(r, b, E) = (1 −Φ(r)/E −b2/r2)1/2. Moreover, the
reduced collision integrals that we use in this work are defined as
Ω
∗(l,s)(T) = 2
2 π m
kB
1/2
(s + 1)!
1 −1
2
1+(−1)l
1+l
Ω(l,s)(T).
(24)
For hard-spheres the reduced collision integrals are equal to one, and there is in the literature
an extensive set of calculations for the Lennard-Jones potential [22, 23, 24, 25, 21, 26], simplified
expressions can be found in [27]. In particular, we used the interpolations provided by Fokin et al.
[28] for the evaluation of collision integrals in the LJ system. In the case of soft-core potentials, we
have implemented a FORTRAN program in order to calculate the corresponding reduced collision
integrals for the parameters used in this study; the procedure follows Ref. [18], changing the interval
of integration in Eq. (21) from (0, ∞) to (0, xmax) to grant convergence, where xmax corresponds to
the maximum possible interaction energy at zero distance (see [18] for more details). The values
of reduced collision integrals for the Morse potential were also numerically calculated; they are in
line with those reported in the literature [18, 29, 30].
3.3. Simulation results
The reduced diffusion coefficient, D∗, as a function of the reduced damping time of the Langevin
thermostat, t∗
d, for different interaction potentials is shown in Fig. 2. The points represent sim-
ulation values in the interval 0.1 ≤t∗
d ≤1000 whereas the lines represent the theoretical results
of Eq. (18). The simulation results for pseudo-hard spheres and Lennard-Jones potentials in Fig.
2a were taken from Ref. [13], while the results for the Morse and Soft-Core potentials in Fig. 2b
have been calculated for this work. In all cases, for small values of t∗
d, we observe a near-zero value
of the diffusion coefficient, whereas for a larger damping time D∗tends to a plateau with a value
8
10
1
100
101
102
103
t ∗
d
0
10
20
30
40
D ∗
Pseudo-hard spheres, T ∗= 1.5
Lennard-Jones T = 4.0
Lennard-Jones T = 1.5
(a) Hard-Core results (pseudo-hard spheres and LJ).
10
1
100
101
102
103
t ∗
d
0
10
20
30
40
50
60
D ∗
Soft spheres
Morse T = 4.0
Morse T = 1.5
(b) Soft-Core results (Morse and soft spheres).
Figure 2: Diffusion coefficient, D∗, against damping time, t∗
d of the Langevin thermostat, in log scale for different
potentials: (a) pseudo-hard spheres (T ∗= 1.5) and LJ (T ∗= 1.5 and 4), (b) soft-spheres (T ∗= 4) and Morse
(T ∗= 1.5 and 4). Dots are numerical results from MD simulations and the curves are obtained form Eq. (18) with
the collision integrals calculated as explained in the text. For the dotted blue curve for soft-spheres, the value of
Ω∗(1,1) was obtained by interpolation. The concentration is ρ∗= 0.01.
10
1
100
101
102
103
t ∗
d
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
η ∗
Pseudo-hard spheres, T ∗= 1.5
Lennard-Jones T = 4.0
Lennard-Jones T = 1.5
(a) Hard-Core results (pseudo-hard spheres and LJ).
10
1
100
101
102
103
t ∗
d
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
η ∗
Soft spheres
Morse T = 4.0
Morse T = 1.5
(b) Soft-Core results (Morse and soft spheres).
Figure 3: Viscosity, η∗, against damping time, t∗
d, in log scale for different potentials: (a) pseudo-hard spheres
(T ∗= 1.5) and LJ (T ∗= 1.5 and 4), (b) soft-spheres (T ∗= 4) and Morse (T ∗= 1.5 and 4).
Dots are MD
simulation results and curves are obtained from Eq. (19). The concentration is ρ∗= 0.01.
that corresponds to the Chapman-Enksog calculation based on the Boltzmann theory. There is a
good agreement between numerical results and Eq. (18), were the reduced collision integrals were
obtained as explained in the previous section (they are well known for pseudo-hard spheres and
LJ, and we calculated them for soft-spheres and Morse).
The value obtained for the collision integral for soft spheres is Ω
∗(1,1)
T ∗=4 = 0.73 (corresponding to
the solid blue curve in Fig. 2b) while, by least square fitting, we get Ω
∗(1,1)
T ∗=4 = 0.67 (dotted blue
curve). Although the discrepancy between the calculated and obtained values is not very large, this
difference seems to be associated with the fact that the method used to obtain the reduced collision
integrals encounters difficulties when dealing with bounded soft-core energies [18]. On other hand,
for the Morse potential, we obtained Ω
∗(1,1)
T ∗=1.5 = 1.28 and Ω
∗(1,1)
T ∗=4 = 0.86, which accurately reproduce
the numerical data.
9
10
1
100
101
102
103
t ∗
d
0.0
0.2
0.4
0.6
0.8
1.0
1.2
λ ∗
Pseudo-hard spheres, T ∗= 1.5
Lennard-Jones T = 4.0
Lennard-Jones T = 1.5
(a) Hard-Core results (pseudo-hard spheres and LJ).
10
1
100
101
102
103
t ∗
d
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
λ ∗
Soft spheres
Morse T = 4.0
Morse T = 1.5
(b) Soft-Core results (Morse and soft spheres).
Figure 4: Thermal conductivity, λ∗, against damping time, t∗
d, in log scale for different potentials: (a) pseudo-hard
spheres (T ∗= 1.5) and LJ (T ∗= 1.5 and 4), (b) soft-spheres (T ∗= 4) and Morse (T ∗= 1.5 and 4). Dots are
numerical results from MD simulations and the curves are obtained form Eq. (20). The concentration is ρ∗= 0.01.
The results for viscosity, η∗, against damping time, t∗
d, are presented in Fig. 3.
As in the
previous case, the points in the figure represent the values obtained from numerical simulations,
whereas the lines correspond to the theoretical result of Eq. (19). The numerical values of the
viscosity for hard spheres and Lennard-Jones potentials were extracted from [12] and plotted in
Fig. 3a. The results for the Morse and soft-sphere potentials in Fig. 3b were calculated for the
present work, providing a broader perspective for the understanding of the behavior of transport
coefficients in the presence of noise. The results obtained for the collision integrals for the Morse
potential are Ω
∗(2,2)
T ∗=4 = 0.93 and Ω
∗(2,2)
T ∗=1.5 = 1.33 (corresponding to the green and orange curves in
Fig. 3b) while for soft spheres it is Ω
∗(2,2)
T ∗=4 = 0.82 (blue curve). The values for collision integrals
obtained by least square fitting are Ω
∗(2,2)
T ∗=4 = 0.98 and Ω
∗(2,2)
T ∗=1.5 = 1.30 (Morse) and Ω
∗(2,2)
T ∗=4 = 0.80
(soft spheres), showing a good agreement with the values calculated. This agreement strengths
the robustness of the theoretical procedure leading to Eq. (19).
Finally, in Fig. 4, we present the behavior of the thermal conductivity coefficient λ∗as a func-
tion of the damping time t∗
d. The results for hard-core potentials (pseudo-hard spheres and LJ) ar
shown in Fig. 4a and, for soft-core potentials (soft shperes and Morse), in Fig. 4b. In the figures,
we show the distinct regimes that emerge as the damping time varies, including the numerical
results and curve of Eq. (20) for thermal conductivity. For pseudo-hard spheres we have that the
reduced collision integrals are 1, while for LJ potential we used, as mentioned before, the ana-
lytical representations of Fokin et al. [28]. From numerical estimations of the collision integrals,
we obtained Ω
∗(2,2)
T ∗=4 = 0.94, Ω
∗(2,2)
T ∗=1.5 = 1.38 for the Morse potential and Ω
∗(2,2)
T ∗=4 = 0.84 for soft-
spheres. As in the previous cases, for small damping times, the thermal conductivity values are
predominantly governed by the interaction with the thermal bath. In this regime, the dynamics
are strongly influenced by the thermostat, which dictates the energy exchange with the environ-
ment and overshadows the role of interatomic collisions. In contrast, for larger damping times, the
system transitions to a regime where interatomic collisions become the dominant mechanism influ-
encing thermal conductivity. The interplay between these two regimes demonstrates the complex
dependence of thermal transport on both external parameters and intrinsic molecular interactions.
It can be observed that in all cases, the transport coefficients for large damping times in the
Morse potential are higher than those obtained for the Lennard-Jones potential, despite choosing
10
parameters that yield a similar attractive well for both cases. This result is a consequence of the
influence of the inner repulsive wall of the potentials (divergent for LJ and finite for Morse) on
transport coefficients.
We observed good agreement between the theoretical description of Eqs. (18)-(20) and the
numerical results for all the studied potentials, showing that, at least under a Langevin thermostat,
the transport coefficients are predictable when the system is coupled to a thermal bath.
4. Conclusions
In this work, we present an analysis of the effects of the Langevin thermostat on the transport
coefficients for various interatomic potentials in the context of dilute gases, for which the Boltzmann
theory holds.
We examined a range of interaction models including both hard- and soft-core
potentials. These include purely repulsive potentials, as well as those with combined attractive
and repulsive components, providing a broad framework for assessing transport phenomena across
different interaction types. The analysis was conducted using a combination of molecular dynamics
simulations and a theoretical approach based on generalized Ohm’s laws for transport processes.
The damping time associated with the Langevin thermostat serves as a control parameter,
modulating the influence of the thermostat on transport coefficients.
By considering a series
circuit of resistances associated with interparticle collisions, described by Boltzmann’s theory, and
to collisions between a system particle and a reservoir particle, described by Langevin’s theory,
we provide an approach to describe the transport phenomena in dilute gases with a Langevin
thermostat, supported by molecular dynamics simulations. These results highlight the robustness
of the series circuit ansatz in capturing the interplay between the thermal bath and the intrinsic
collision dynamics.
We explored a broad range of damping time scales and observed consistency between the
numerical results and theoretical predictions. This consistency extends across different types of
potentials, underscoring the universality of the observed behavior. Therefore, the modification
of the transport coefficients of dilute gases by the Langevin thermostat can be systematically
characterized and predicted.
A practical implication of these findings is that when transport coefficients are calculated for
a dilute gas in contact with a Langevin thermostat, the influence of the thermostat on the results
can be removed using Eq. (8).
Acknowledge
This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET, Argentina, PUE 22920200100016CO) and Universidad Nacional de Mar del Plata
(UNMDP, Argentina, 15/E1155).
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Transport with noise in dilute gases: Effect of Langevin thermostat on transport coefficients. A. Alés, J. I. Cerato, L. Marchioni and M. Hoyuelos Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR-CONICET), Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata(NM, Funes 3350, Mar del Plata, 7600, Argentina Abstract In dilute gases, transport properties such as the thermal conductivity, self-diffusion, and viscosity are significantly affected by interatomic collisions, which are determined by the potential form. This study explores these transport properties in the presence of a Langevin thermostat in systems where particles interact through various potentials, including soft-core and hard-core potentials, both with and without an attractive region. Using molecular dynamics simulations and a theoretical approach based on an analogy with an electric circuit (Ohm's law), we derived and compared the transport coefficients across these interatomic potentials for different couplings with the thermostat. The transport coefficients were obtained by considering the thermostat as a resistance in a series circuit. 1. Introduction Boltzmann's kinetic theory of gases is essential for understanding the transport properties of dilute gases [1]. Within the framework established by Chapman and Enskog [2, 3], this theory provides the basis for deriving the coefficients of thermal conductivity, self-diffusivity, and viscosity that describe the transport of energy, mass, and momentum, respectively. These coefficients relate microscopic properties, such as the size or mass of the particles, with macroscopic behaviors. This link is fundamental for interpreting the behavior of gases across various fields, from chemical engineering to astrophysics. In the Boltzmann theory, particle collisions are described through interatomic potentials [3]. For example, the hard-sphere potential, which takes a zero value beyond a certain radius and infinity within it, is widely used in numerical simulations as a preliminary step in analyzing systems with more complex interactions [4, 5]. In general, interatomic potentials typically consist of a repulsive component at short distances, and may include one or more attractive or repulsive contributions at intermediate ranges, eventually decaying to zero at long distances. In his pioneering work, LennardJones introduced a potential as a function of the interatomic distance r, with an attractive term proportional to r-6, derived from consideration of dipole moment fluctuations [6]. For convenience, the repulsive term is often modeled as r-12, although numerous studies have explored alternative power law dependencies. Hard-core potentials, such as the Lennard-Jones potential, impose an infinite kinetic energy requirement for particles to approach each other, whereas soft-core potentials maintain a finite repulsive energy at short distances, such as the Morse potential [7, 8]. Notably, the presence of attractive regions in a potential opens a range of properties in gas behavior. When there is an attractive depth-well potential interaction, particles can exhibit binding or orbiting phenomena if they collide with specific impact parameters and energies [3, 9], adding further complexity to the analysis of gas dynamics. 16 Oct 2025 The measurement of transport coefficients for these potentials in dilute gases has been a wellestablished practice since the early days of computational simulations. However, the inclusion of a thermal bath introduces significant modifications to the transport coefficients because its influence disrupts the natural dynamics of the system. Historically, it has been recommended to measure these coefficients only when the thermal bath provides a very long damping time, thus minimizing its impact on the intrinsic transport properties [10]; see also Sec. 6.1.1 in [11]. This external influence from the thermostat was initially thought to obscure the essence of the transport coefficient measurements. For example, the Andersen thermal bath substantially alters the transport coefficient values, unless the damping time is sufficiently large. Despite these concerns, recent studies have demonstrated that for viscosity [12] and self-diffusion coefficients [13], the effect of a Langevin thermal bath is predictable and quantifiable. Specifically, this influence can be attributed to the coupling of two independent transport contributions: one arising from interatomic interactions and the other arising from the thermal bath in which the particles are immersed. For brevity, in the following we use "diffusion" instead of "self-diffusion" to refer to diffusion of tagged particles in a medium composed by particles of the same kind. Moreover, the damping factor of the Langevin thermostat serves as a control parameter [13, 12]. By tuning this factor, one can transition from a regime dominated by the thermal bath, characterized by small damping times, to a regime where the influence of the bath becomes negligible as the damping time approaches infinity. The aim of this paper is to study, both numerically through molecular dynamics simulations and analytically using generalized Ohm's laws, the behavior of the transport coefficients for hard- and soft-core potentials, with or without an attractive part, in the presence of a Langevin thermostat. The behavior of the transport coefficients as functions of the damping time is obtained considering that the effect of the thermostat can be represented as a resistance in a series circuit. This paper is structured as follows. The methodology used is detailed in Section 2, the analytical and computational results are presented in Section 3, and conclusions are provided in Section 4. 2. Methodology Near equilibrium, every macroscopic transport process is linearly related to the correlations of microscopic fluctuations around the equilibrium state. The Green-Kubo relations provide a rigorous framework to obtain these coefficients by expressing them as integrals over time-correlation functions of microscopic currents in equilibrium systems without requiring the application of external gradients [14, 15]. However, while these relations establish a connection between transport coefficients and equilibrium fluctuations, they do not provide explicit expressions for the correlation functions themselves. In contrast, the Chapman-Enskog method, based on the Boltzmann equation, allows for explicit computation of these correlations in the case of dilute gases by employing a perturbative expansion of the distribution function [2]. The Langevin thermostat introduces an internal stochastic current, which we refer to as the Langevin current. This current arises from the random forces exerted by the thermostat, which mimic the thermal fluctuations experienced by particles in a fluid. These fluctuations follow the statistical principles of Brownian motion, resulting in a dynamic that contributes to the overall transport behavior. We present the simulation details in subsection 2.1 and briefly review the calculation of the transport coefficients using the Chapman-Enksog approach for a particle system without a thermal bath. For a dilute gas in a thermal bath, if interactions are neglected and particle dynamics is 2 0.0 0.5 1.0 1.5 2.0 2.5 r/σ 0 2 4 6 8 Φ(r)/ε Pseudo-hard spheres potential Lennard-Jones potential Morse potential Soft spheres potential Figure 1: Scaled interatomic potentials used in this paper, Φ/ε, against distance, r/σ (ε and σ are characteristic values of energy and distance for each potential). The potentials are pesudo hard spheres (blue line), Lennard-Jones (LJ, green dashed curve), Morse (red dotted curve) and soft spheres (purple dot-dashed curve). Soft spheres and Morse potentials are soft core, while LJ and pseudo hard spheres potentials are hard core (they diverge as r →0). essentially Brownian, transport coefficients are obtained by means of Green-Kubo calculations; see subsection 2.2. 2.1. Simulation Details Molecular dynamic simulations were implemented using LAMMPS software [16] (Large-scale Atomic-Molecular Massively Parallel Simulator). Each interatomic potential has characteristic length σμ and energy εμ, where the subindex μ identifies the potential. The particle density is ρ = N/V , where N is the particle number, V is the system volume, and the reduced dimensionless particle density is given by ρ∗= ρ σ3 μ. The reduced temperature T ∗in terms of the temperature T is T ∗= kBT/εμ. In all cases, we used a small reduced density of ρ∗= 0.01 for which Boltzmann theory approximately holds. To represent qualitatively different interactions, we select the following four interatomic potentials Φμ(r): 1. Pseudo-hard spheres: ΦHS(r) = εHS λr λr-λa λr λa λa h σHS r λr - σHS r λai + εHS for r < σHS( λr λa) 1 λr-λa 0 for r ≥σHS( λr λa) 1 λr-λa where the scheme of WCA (Weeks-Chandler-Andersen) [17] is used. This potential correctly reproduces the hard-sphere properties for λr = 50 and λa = 49, with a reduced temperature T ∗= 1.5, as shown in Ref. [5]. In the simulations, we used εHS = 1 and σHS = 1 in LJ units. 2. Lennard-Jones (LJ) is the most common interatomic potential for simple fluid modeling: ΦLJ(r) = εLJ σLJ r 12 - σLJ r 6 . 3 A cutoff radius rc = 2.5 was used, and the values εLJ = 1 and σLJ = 1 in LJ units were used. 3. Morse potential, given by ΦM(r) = εM e-2α(r-σM) -2e-α(r-σM) (see [18]). The parameters of this potential has been chosen to fit the well of the LennardJones region for direct comparison, and the values are εM = 1.10 εLJ, α = 5.58/σLJ and σM = 1.12 σLJ. 4. At last, soft-spheres interatomic potential, modeled as ΦSS(r) = εSS [1 + cos(πr/σSS)] for r < σSS 0 for r ≥σSS , where we used εSS = 50 and σSS = 1 in LJ units for the simulations. The shapes of these potentials are shown in Fig. 1. Whereas the first two are hard-core potentials (they diverge for r →0), the last two are soft-core potentials. The pseudo-hard and soft spheres are potentials with only a repulsive part, whereas the remaining potentials have an attractive depth well. The temperature used for the pseudo-hard sphere potential is T ∗= 1.5 (that, as mentioned before, is the temperature for which hard-sphere properties are reproduced) and for soft-spheres is T ∗= 4. Lennard-Jones and Morse potentials are studied considering both temperatures. Molecular dynamics (MD) simulations were performed with 32,000 particles, including two equilibration stages. In the first stage, Gaussian-distributed velocities were assigned to the particles, which were initially arranged in an fcc lattice, and the system was evolved in the NVT ensemble using a Langevin thermostat with a specified damping time td. After 2 × 105 time steps, the simulation was switched to the NVE ensemble and continued for the same number of steps to allow the system to equilibrate. Subsequently, a data-gathering run of 1.5 × 106 steps was carried out in the NVE ensemble. A time step of 0.001 was employed in all cases, except for the pseudo-hard sphere potential, for which a smaller time step of 0.0001 was required to ensure the dynamical stability of the system. 2.2. Theoretical Background The Boltzmann equation holds for a dilute gas; it is an integro-differential equation that describes the one-particle nonequilibrium distribution function, f(r, v, t), which gives the average number of particles with position r and velocity v at time t [15, 19]. The Chapman-Enskog method uses this distribution function to determine transport coefficients in the zero-density limit. A perturbative expansion is used, where f(r, v, t) is expressed as a series around the equilibrium distribution of the Boltzmann equation. The results align with macroscopic hydrodynamic equations by stopping the expansion after the first correction [2, 19]. The transport coefficients are then calculated as a series involving Sonine polynomials, which typically converge quickly, meaning that only a few terms are needed for accurate results [3]. The transport coefficients for dilute gases are λB = 75 kB 64 σ2 kBT πm 1/2 fλ Ω(2,2), (1) for thermal conductivity, where m is the particle mass, T is temperature and kB is Boltzmann's constant; we have the following expression for the diffusion coefficient, DB = 3 8 ρ σ2 kBT mπ 1/2 fD Ω(1,1); (2) 4 and the viscosity coefficient is ηB = 5 16 σ2 mkBT π 1/2 fν Ω(2,2), (3) where Ω(i,j) are collision integrals relative to the hard sphere system, which depend on the temperature and the type of interatomic potential considered [3, 19]. We have included the subscript B to indicate that these coefficients were derived by considering particle collisions within the framework of the Boltzmann equation [15, 19]. The fi functions, known as correction factors, are expressed in terms of the collision integrals [9, 20]. Since it is well known that deviations of fi from unity are very small, we consider fi = 1, with i = ν, D, λ; see [21] or Ch. 19 in [19]. The dimensionless coefficients are obtained using the following scaling: D∗= D σ-1p m/ε, η∗= η σ2/√mε and λ∗= λ k-1 B σ2p m/ε (subindex μ in σ and ε is omitted for simplicity). By applying the scaling to Eqs. (2), (3) and (1) we obtain: D∗ B = 3 8 ρ∗ p T ∗/π Ω∗(1,1) , (4) η∗ B = 5 16 p T ∗/π Ω∗(2,2) , (5) λ∗ B = 75 64 p T ∗/π Ω∗(2,2) . (6) From another perspective, the Green-Kubo relations provide a framework to obtain these coefficients by relating them to the equilibrium time correlations of microscopic currents [2, 15, 14]. These relationships are particularly powerful because they allow the calculation of transport properties in equilibrium systems without requiring the application of external gradients. The correlations are commonly obtained using numerical methods; however, as explained in subsection 3.1, analytical results are possible for the Brownian dynamics produced by the Langevin thermostat for small concentrations. For example, the thermal conductivity λ is expressed as λ = 1 kBT 2V Z ∞ 0 ⟨JQ(0)JQ(t)⟩dt, (7) where JQ(t) is the energy current in an arbitrary direction (the system is isotropic), V is the system volume, and ⟨JQ(0)JQ(t)⟩denotes the ensemble average of the correlation between the current at time zero and time t (more details on this topic in Sec. 3.1). We aimed to analyze a system with atomic collisions in a thermal bath at a low concentration. A binary gas mixture can be considered for this description. The system consists of the particles of interest, at low density, of mass m; they are immersed in a thermal bath represented by particles with mass m′, with m ≫m′. Interatomic collisions are as follows: m-m collisions, governed by the chosen interatomic potential, are rare due to the low particle density; m-m′ collisions, on the other hand, are more frequent and follow the well-established rules of Brownian motion. Collisions m′-m′ among thermal bath particles are not relevant in this scenario. This picture indicates that the transport of any conserved quantity has to overcome the resistance produced by both types of collisions, m-m and m-m′, as in a series circuit. Generalized Ohm's laws are linear relations between thermodynamic forces (gradients of chemical potential, temperature or velocity) and currents (of particles, heat or momentum); the proportionality constants are the transport coefficients. These laws are fundamental to the development 5 of hydrodynamic equations in classical irreversible thermodynamics. The entropy production in the hydrodynamic equations can be written as the product of currents times forces, analogous to the Joule heat. Here, we consider the consequences of this analogy when resistances in series are present. For example, let us consider the current of tagged particles. The presence of other particles of the same type causes resistance to the particle current. In the presence of the thermostat, there is also the resistance caused by the small particles of the reservoir, whose effect is represented by random forces and damping. The particle current has to overcome both resistances, therefore, they are in series. The same reasoning can be applied to heat or momentum currents. Transport coefficients represent, instead of resistances, the conductances. We call CB the conductance in the absence of the thermostat that is obtained from the Boltzmann's theory, and CL the conductance introduced by the Langevin thermostat. Since the circuit is in series, the resulting conductance is 1 C = 1 CL + 1 CB . (8) This concept, which extends the consequences of the electric circuit analogy in hydrodynamic equations, was introduced in [13, 12]. Depending on the damping time of the Langevin thermostat, the transport coefficients can be dominated by one or the other contribution. If the damping time is short, the system responds according to the Brownian contribution to the conductance, whereas for long damping times, it behaves according to the Boltzmann contribution. 3. Results and Analysis 3.1. Transport coefficient calculation for Langevin Currents Considering an arbitrary direction in an isotropic system, the Langevin thermostat produces a force, on particle i, that is given by Fi(t) = -mvi/td + αi(t), (9) where m and vi are the mass and velocity of particle i, td is the damping time, and αi(s) is a random force with zero mean, ⟨αi(t)⟩= 0, and delta time correlated, ⟨αi(t)αi(t′)⟩= 2mkBT td δ(t-t′); the noise amplitude is determined by the fluctuation dissipation theorem. Force Fi, velocity vi and random force αi correspond to the components in an arbitrary direction of vectors Fi, vi and αi. Under this conditions, the diffusion coefficient is well known (see, for example, [15]); it is given by the Einstein relation, DL = kBT td m . (10) The viscosity coefficient of the system composed of Brownian particles (not to be confused with the viscosity of the medium) is (see [12]) ηL = ρ kBT td 2 . (11) For thermal conductivity, we take the following approach. In the absence of both interactions between particles and external potentials, the energy flux for an N particle system in a given direction is JQ(t) = PN i=1 m 2 vi(t)2 -⟨hi⟩ vi(t), where ⟨hi⟩is the mean enthalpy per particle, that takes the value 5 2kBT for an ideal gas (see, for example, Sec. 21.8 in [19]). Using the Green-Kubo 6 expression for thermal conductivity, we have, λL = 1 V kBT 2 Z ∞ 0 ⟨JQ(t)JQ(0)⟩ = 1 V kBT 2 Z ∞ 0 *X i,j m 2 vi(t)2 -⟨hi⟩ vi(t) m 2 vj(0)2 -⟨hj⟩ vj(0) + dt = ρ kBT 2 Z ∞ 0 m 2 vi(t)2 -5 2kBT vi(t) m 2 vi(0)2 -5 2kBT vi(0) dt, (12) where correlations between different Brownian particles are neglected; in the last line, N/V is replaced by ρ and subindex i corresponds to any particle (symmetry under exchange of particle labels is assumed). Now, we use the expression for the velocity of a Brownian particle (see, for example, Sec. 7.2.1 in [15]), vi(t) = vi(0) + Z t 0 dse-γ(t-s)αi(s). (13) Replacing (13) in (12), and using the statistical properties of the random force αi, the integral can be solved to obtain, λL = ρ k2 B T td 2 m . (14) The dimensionless coefficients, then, become D∗ L = T ∗t∗ d (15) η∗ L = ρ∗T ∗t∗ d/2 (16) λ∗ L = ρ∗T ∗t∗ d/2, (17) where t∗ d = td σ-1p ε/m. The Einstein relation for the diffusion coefficient, DL, has fundamental historical and experimental importance in understanding Brownian motion. The expressions for the viscosity and thermal conductivity, ηL and λL, are interesting from a theoretical and numerical point of view, but it is probably not possible to verify them experimentally because of the difficulties in separating the transport of heat or momentum of the particle system from the transport of the same quantities that occur in the reservoir where the system is embedded. 3.2. Calculation of the transport coefficients with noise In Eq. (8) any transport coefficient is represented by the conductance C. The dimensionless expressions for the coefficients from Boltzmann theory, Eqs. (4)-(6), and from the Langevin theory, Eqs. (15)-(17), are used to calculate, in Eq. (8), the transport coefficients at small concentration in the presence of the Langevin thermostat assuming a series circuit. The results are, D∗= T ∗t∗ d 8 3ρ∗t∗ d Ω∗(1,1)√ π T ∗+ 1 , (18) η∗= T ∗t∗ d ρ∗ 16 5 ρ∗t∗ d Ω∗(2,2)√ π T ∗+ 2 , (19) λ∗= T ∗t∗ d ρ∗ 64 75ρ∗t∗ d Ω∗(2,2)√ π T ∗+ 2 . (20) 7 It can be seen that, for large t∗ d, we recover the expressions from the Boltzmann theory, Eqs. (4)-(6), while for small t∗ d we have the Langevin results, Eqs. (15)-(17). Considering that the collision integrals are of order 1, a large or small t∗ d means t∗ d ≫1/(ρ∗√ T ∗) or t∗ d ≪1/(ρ∗√ T ∗). We obtained numerical results, presented in the next section, that verified these expressions as functions of damping time, t∗ d, for fixed values of temperature and concentration (ρ∗= 0.01). Collision integrals can be calculated using the following nested integrals [3], Ω(l,s)(T) = kBT 2πm 1/2 Z ∞ 0 dx e-x2x(2 s+3)Sl(kBTx), (21) where Sl(kBTx) is the collision cross section, a function of the initial kinetic energy of colliding particles; is is defined as Sl(E) = 2 π Z ∞ 0 b 1 -cosl (χ(b, E)) db. (22) Here, b denotes the impact parameter and χ the classical deflection angle as a function of the relative kinetic energy of the colliding particles, χ(b, E) = π -2 b Z ∞ rm dr r2F(r, b, E), (23) where rm is the classical returning point and F(r, b, E) = (1 -Φ(r)/E -b2/r2)1/2. Moreover, the reduced collision integrals that we use in this work are defined as Ω ∗(l,s)(T) = 2 2 π m kB 1/2 (s + 1)! 1 -1 2 1+(-1)l 1+l Ω(l,s)(T). (24) For hard-spheres the reduced collision integrals are equal to one, and there is in the literature an extensive set of calculations for the Lennard-Jones potential [22, 23, 24, 25, 21, 26], simplified expressions can be found in [27]. In particular, we used the interpolations provided by Fokin et al. [28] for the evaluation of collision integrals in the LJ system. In the case of soft-core potentials, we have implemented a FORTRAN program in order to calculate the corresponding reduced collision integrals for the parameters used in this study; the procedure follows Ref. [18], changing the interval of integration in Eq. (21) from (0, ∞) to (0, xmax) to grant convergence, where xmax corresponds to the maximum possible interaction energy at zero distance (see [18] for more details). The values of reduced collision integrals for the Morse potential were also numerically calculated; they are in line with those reported in the literature [18, 29, 30]. 3.3. Simulation results The reduced diffusion coefficient, D∗, as a function of the reduced damping time of the Langevin thermostat, t∗ d, for different interaction potentials is shown in Fig. 2. The points represent simulation values in the interval 0.1 ≤t∗ d ≤1000 whereas the lines represent the theoretical results of Eq. (18). The simulation results for pseudo-hard spheres and Lennard-Jones potentials in Fig. 2a were taken from Ref. [13], while the results for the Morse and Soft-Core potentials in Fig. 2b have been calculated for this work. In all cases, for small values of t∗ d, we observe a near-zero value of the diffusion coefficient, whereas for a larger damping time D∗tends to a plateau with a value 8 10 1 100 101 102 103 t ∗ d 0 10 20 30 40 D ∗ Pseudo-hard spheres, T ∗= 1.5 Lennard-Jones T = 4.0 Lennard-Jones T = 1.5 (a) Hard-Core results (pseudo-hard spheres and LJ). 10 1 100 101 102 103 t ∗ d 0 10 20 30 40 50 60 D ∗ Soft spheres Morse T = 4.0 Morse T = 1.5 (b) Soft-Core results (Morse and soft spheres). Figure 2: Diffusion coefficient, D∗, against damping time, t∗ d of the Langevin thermostat, in log scale for different potentials: (a) pseudo-hard spheres (T ∗= 1.5) and LJ (T ∗= 1.5 and 4), (b) soft-spheres (T ∗= 4) and Morse (T ∗= 1.5 and 4). Dots are numerical results from MD simulations and the curves are obtained form Eq. (18) with the collision integrals calculated as explained in the text. For the dotted blue curve for soft-spheres, the value of Ω∗(1,1) was obtained by interpolation. The concentration is ρ∗= 0.01. 10 1 100 101 102 103 t ∗ d 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 η ∗ Pseudo-hard spheres, T ∗= 1.5 Lennard-Jones T = 4.0 Lennard-Jones T = 1.5 (a) Hard-Core results (pseudo-hard spheres and LJ). 10 1 100 101 102 103 t ∗ d 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 η ∗ Soft spheres Morse T = 4.0 Morse T = 1.5 (b) Soft-Core results (Morse and soft spheres). Figure 3: Viscosity, η∗, against damping time, t∗ d, in log scale for different potentials: (a) pseudo-hard spheres (T ∗= 1.5) and LJ (T ∗= 1.5 and 4), (b) soft-spheres (T ∗= 4) and Morse (T ∗= 1.5 and 4). Dots are MD simulation results and curves are obtained from Eq. (19). The concentration is ρ∗= 0.01. that corresponds to the Chapman-Enksog calculation based on the Boltzmann theory. There is a good agreement between numerical results and Eq. (18), were the reduced collision integrals were obtained as explained in the previous section (they are well known for pseudo-hard spheres and LJ, and we calculated them for soft-spheres and Morse). The value obtained for the collision integral for soft spheres is Ω ∗(1,1) T ∗=4 = 0.73 (corresponding to the solid blue curve in Fig. 2b) while, by least square fitting, we get Ω ∗(1,1) T ∗=4 = 0.67 (dotted blue curve). Although the discrepancy between the calculated and obtained values is not very large, this difference seems to be associated with the fact that the method used to obtain the reduced collision integrals encounters difficulties when dealing with bounded soft-core energies [18]. On other hand, for the Morse potential, we obtained Ω ∗(1,1) T ∗=1.5 = 1.28 and Ω ∗(1,1) T ∗=4 = 0.86, which accurately reproduce the numerical data. 9 10 1 100 101 102 103 t ∗ d 0.0 0.2 0.4 0.6 0.8 1.0 1.2 λ ∗ Pseudo-hard spheres, T ∗= 1.5 Lennard-Jones T = 4.0 Lennard-Jones T = 1.5 (a) Hard-Core results (pseudo-hard spheres and LJ). 10 1 100 101 102 103 t ∗ d 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 λ ∗ Soft spheres Morse T = 4.0 Morse T = 1.5 (b) Soft-Core results (Morse and soft spheres). Figure 4: Thermal conductivity, λ∗, against damping time, t∗ d, in log scale for different potentials: (a) pseudo-hard spheres (T ∗= 1.5) and LJ (T ∗= 1.5 and 4), (b) soft-spheres (T ∗= 4) and Morse (T ∗= 1.5 and 4). Dots are numerical results from MD simulations and the curves are obtained form Eq. (20). The concentration is ρ∗= 0.01. The results for viscosity, η∗, against damping time, t∗ d, are presented in Fig. 3. As in the previous case, the points in the figure represent the values obtained from numerical simulations, whereas the lines correspond to the theoretical result of Eq. (19). The numerical values of the viscosity for hard spheres and Lennard-Jones potentials were extracted from [12] and plotted in Fig. 3a. The results for the Morse and soft-sphere potentials in Fig. 3b were calculated for the present work, providing a broader perspective for the understanding of the behavior of transport coefficients in the presence of noise. The results obtained for the collision integrals for the Morse potential are Ω ∗(2,2) T ∗=4 = 0.93 and Ω ∗(2,2) T ∗=1.5 = 1.33 (corresponding to the green and orange curves in Fig. 3b) while for soft spheres it is Ω ∗(2,2) T ∗=4 = 0.82 (blue curve). The values for collision integrals obtained by least square fitting are Ω ∗(2,2) T ∗=4 = 0.98 and Ω ∗(2,2) T ∗=1.5 = 1.30 (Morse) and Ω ∗(2,2) T ∗=4 = 0.80 (soft spheres), showing a good agreement with the values calculated. This agreement strengths the robustness of the theoretical procedure leading to Eq. (19). Finally, in Fig. 4, we present the behavior of the thermal conductivity coefficient λ∗as a function of the damping time t∗ d. The results for hard-core potentials (pseudo-hard spheres and LJ) ar shown in Fig. 4a and, for soft-core potentials (soft shperes and Morse), in Fig. 4b. In the figures, we show the distinct regimes that emerge as the damping time varies, including the numerical results and curve of Eq. (20) for thermal conductivity. For pseudo-hard spheres we have that the reduced collision integrals are 1, while for LJ potential we used, as mentioned before, the analytical representations of Fokin et al. [28]. From numerical estimations of the collision integrals, we obtained Ω ∗(2,2) T ∗=4 = 0.94, Ω ∗(2,2) T ∗=1.5 = 1.38 for the Morse potential and Ω ∗(2,2) T ∗=4 = 0.84 for softspheres. As in the previous cases, for small damping times, the thermal conductivity values are predominantly governed by the interaction with the thermal bath. In this regime, the dynamics are strongly influenced by the thermostat, which dictates the energy exchange with the environment and overshadows the role of interatomic collisions. In contrast, for larger damping times, the system transitions to a regime where interatomic collisions become the dominant mechanism influencing thermal conductivity. The interplay between these two regimes demonstrates the complex dependence of thermal transport on both external parameters and intrinsic molecular interactions. It can be observed that in all cases, the transport coefficients for large damping times in the Morse potential are higher than those obtained for the Lennard-Jones potential, despite choosing 10 parameters that yield a similar attractive well for both cases. This result is a consequence of the influence of the inner repulsive wall of the potentials (divergent for LJ and finite for Morse) on transport coefficients. We observed good agreement between the theoretical description of Eqs. (18)-(20) and the numerical results for all the studied potentials, showing that, at least under a Langevin thermostat, the transport coefficients are predictable when the system is coupled to a thermal bath. 4. Conclusions In this work, we present an analysis of the effects of the Langevin thermostat on the transport coefficients for various interatomic potentials in the context of dilute gases, for which the Boltzmann theory holds. We examined a range of interaction models including both hard- and soft-core potentials. These include purely repulsive potentials, as well as those with combined attractive and repulsive components, providing a broad framework for assessing transport phenomena across different interaction types. The analysis was conducted using a combination of molecular dynamics simulations and a theoretical approach based on generalized Ohm's laws for transport processes. The damping time associated with the Langevin thermostat serves as a control parameter, modulating the influence of the thermostat on transport coefficients. By considering a series circuit of resistances associated with interparticle collisions, described by Boltzmann's theory, and to collisions between a system particle and a reservoir particle, described by Langevin's theory, we provide an approach to describe the transport phenomena in dilute gases with a Langevin thermostat, supported by molecular dynamics simulations. These results highlight the robustness of the series circuit ansatz in capturing the interplay between the thermal bath and the intrinsic collision dynamics. We explored a broad range of damping time scales and observed consistency between the numerical results and theoretical predictions. This consistency extends across different types of potentials, underscoring the universality of the observed behavior. Therefore, the modification of the transport coefficients of dilute gases by the Langevin thermostat can be systematically characterized and predicted. A practical implication of these findings is that when transport coefficients are calculated for a dilute gas in contact with a Langevin thermostat, the influence of the thermostat on the results can be removed using Eq. (8). Acknowledge This work was partially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET, Argentina, PUE 22920200100016CO) and Universidad Nacional de Mar del Plata (UNMDP, Argentina, 15/E1155). References [1] J.-P. Hansen, I. R. McDonald, Theory of simple liquids: with applications to soft matter, Academic press, 2013. [2] S. Chapman, T. G. Cowling, The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Cambridge university press, 1990. 11 [3] J. O. Hirschfelder, C. F. Curtiss, R. B. Bird, Molecular theory of gases and liquids, John Wiley & Sons, 1964. [4] D. M. Heyes, M. Cass, J. G. Powles, W. Evans, Self-diffusion coefficient of the hard-sphere fluid: System size dependence and empirical correlations, The Journal of Physical Chemistry B 111 (6) (2007) 1455-1464. [5] J. Jover, A. Haslam, A. Galindo, G. Jackson, E. Müller, Pseudo hard-sphere potential for use in continuous molecular-dynamics simulation of spherical and chain molecules, The Journal of chemical physics 137 (14) (2012). [6] J. E. Lennard-Jones, Cohesion, Proceedings of the Physical Society 43 (5) (1931) 461. [7] O. Bahethi, R. Gambhir, S. Saxena, Properties of gases and gaseous mixtures with a morse potential, Zeitschrift für Naturforschung A 19 (13) (1964) 1478-1485. [8] S. Saxena, O. Bahethi, Transport properties of some simple non-polar gases on the morse potential, Molecular Physics 7 (2) (1964) 183-189. [9] E. A. Mason, E. W. McDaniel, Transport properties of ions in gases, Vol. 26, Wiley Online Library, 1988. [10] H. Tanaka, K. Nakanishi, N. Watanabe, Constant temperature molecular dynamics calculation on lennard-jones fluid and its application to watera, The Journal of Chemical Physics 78 (5) (1983) 2626-2634. [11] D. Frenkel, B. Smit, Understanding Molecular Simulation, Academic Press, 2002. [12] L. Marchioni, M. Di Muro, M. Hoyuelos, Viscosity as the product of its ideal low-concentration value and a thermodynamic function, Physical Review E 111 (6) (2025) 064128. [13] L. Marchioni, M. A. Di Muro, M. Hoyuelos, Dependence on the thermodynamic state of self-diffusion of pseudo-hard-sphere and lennard-jones potentials, Physical Review E 107 (1) (2023) 014134. [14] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, 2nd Edition, Springer, 1998. [15] L. E. Reichl, A Modern Course in Statistical Physics, John Wiley & Sons, 2016. [16] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1-19. [17] J. D. Weeks, D. Chandler, H. C. Andersen, Role of repulsive forces in determining the equilibrium structure of simple liquids, The Journal of Chemical Physics 54 (12) (1971) 5237-5247. [18] F. J. Smith, R. Munn, Automatic calculation of the transport collision integrals with tables for the morse potential, The Journal of Chemical Physics 41 (11) (1964) 3560-3568. [19] D. A. McQuarrie, Statistical Mechanics, University Science Books, Sausalito, 2000. [20] L. Monchick, E. Mason, Transport properties of polar gases, The Journal of Chemical Physics 35 (5) (1961) 1676-1697. 12 [21] L. A. Viehland, A. R. Janzen, R. A. Aziz, High approximations to the transport properties of pure atomic gases, The Journal of Chemical Physics 102 (13) (1995) 5444-5450. [22] M. Klein, F. J. Smith, Tables of collision integrals for the (m, 6) potential function for 10 values of m, Journal of Research of the National Bureau of Standards. Section A, Physics and Chemistry 72 (4) (1968) 359. [23] M. Klein, H. HJM, S. FJ, P. Holland, Tables of collision integrals and second virial coefficients for the (m, 6, 8) intermolecular potential function. (1974). [24] L. Viehland, E.-A. Mason, W. Morrison, M. Flannery, Tables of transport collision integrals for (n, 6, 4) ion-neutral potentials, Atomic Data and Nuclear Data Tables 16 (6) (1975) 495-514. [25] E. Akhmatskaya, L. Pozhar, Calculation of transport collision integrals for a Lennard-Jones gas, USSR Computational Mathematics and Mathematical Physics 26 (2) (1986) 185-190. [26] A. Laricchiuta, G. Colonna, D. Bruno, R. Celiberto, C. Gorse, F. Pirani, M. Capitelli, Classical transport collision integrals for a Lennard-Jones like phenomenological model potential, Chemical Physics Letters 445 (4-6) (2007) 133-139. [27] K. Meier, Computer simulation and interpretation of the transport coefficients of the LennardJones model fluid, Ph.D. thesis, (2002). [28] L. R. Fokin, V. Popov, A. Kalashnikov, Analytical representation of collision integrals for the (m-6) Lennard-Jones potentials in the EPIDIF database, High Temp. 37 (1999) 45. [29] G. Colonna, A. Laricchiuta, General numerical algorithm for classical collision integral calculation, Computer Physics Communications 178 (11) (2008) 809-816. [30] V. Popov, An approximate numerical method of solving collision integrals for the morse potential in a wide parameter interval, High Temperature 51 (1) (2013) 66-71. 13
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2510.14748
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Draft version October 17, 2025
Typeset using LATEX default style in AASTeX7.0.1
QFP Waves Driven by the Tuning-Fork Effect during Magnetic Reconnecion
Jialiang Hu,1, 2 Xiaozhou Zhao,3 Guiping Zhou,1, 2 Yuhao Chen,4 Chunlan Jin,1 Mijie Shi,5 Guanchong Cheng,3
Xiaoxia Yu,6 Jing Ye,3 Xinping Zhou,7 and Hanxian Fang8
1State Key Laboratory of Solar Activity and Space Weather, National Astronomical observatories, Chinese Academy of Sciences,
100101,Beiing,China
2School of Astronomy and Space Science, University of Chinese Academy of Sciences, 100049, Beijing, China
3Yunnan Observatories, Chinese Academy of Sciences, P.O. Box 110, Kunming, Yunnan 650216, People’s Republic of China
4School of Earth and Space Sciences, Peking University, Beijing 100781, People’s Republic of China
5Shandong Key Laboratory of Optical Astronomy and Solar–Terrestrial Environment, School of Space Science and Technology, Institute
of Space Sciences, Shandong University, Weihai 264209, People’s Republic of China
6State Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
7College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, People’s Republic of China
8College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha, China.
ABSTRACT
Through three-dimensional MHD simulations, we have uncovered a kind of fast coronal wave origi-
nating from both ends of a current sheet (CS) during a solar eruption. These waves are observed to
appear near the top and bottom ends of the reconnection-related CS. The simulations demonstrate
the presence of termination shock regions above the two ends of the CS. As the reconnection outflows
escape from the vertical CS and encounter these termination shocks, they undergo partial reflection,
redirecting towards the CS terminal fork walls. The identified waves propagate rapidly at a speed
of approximately 1400 km/s with a period of just 2 s. Concurrently, the time-evolution of intensity
within a small region of the CS terminal fork structures, exhibits a similar oscillation period of 2 s.
All these evidence supports the notion that these QFP (Quasi-periodic Fast-Propagating) waves
were excited by tuning fork effects within the CS system. Essentially, the rapid reconnection outflows
are reflected by the terminal shocks, striking the fork walls at the CS ends. Moreover, parts of the
oscillations along the tuning fork handle are transformed into thermal energy, accumulating in the CS
center and elevating the temperature. This is the first time to report such QFP waves resulting from
tuning fork effects within the CS during a solar eruption. These waves are anticipated to manifest
closely following the propagation of CMEs and adjacent to the related post-flare loops in observations,
with partial confirmation in current observations.
Keywords: Separatrice; Solar Corona; MHD waves; Current Sheet; MHD Simulations
1. INTRODUCTION
Magnetic reconnection plays a critical role in solar eruptions, governing both the energy release process and dynamics
in the solar corona. Solar eruptions are frequently accompanied by various modes of coronal waves. The propagation
characteristics of coronal waves (e,g., phase speeds, dispersion relations, and attenuation scales) encode valuable
information about the local plasma-β and magnetic topology, making them useful tools for probing the fine-scale
physical processes during magnetic reconnections (Y. Shen et al. 2022). Previous studies that have reported a wide
range of coronal wave modes have significantly advanced our understanding of solar activity (V. M. Nakariakov &
D. Y. Kolotkov 2020). However, the mechanisms by which different modes of coronal waves are generated during
magnetic reconnection remain poorly understood.
Corresponding author: Hanxian Fang, Xinping Zhou
fanghx@nudt.edu.cn,xpzhou@sicnu.edu.cn
arXiv:2510.14748v1 [astro-ph.SR] 16 Oct 2025
2
Based on multi-wavelength observations, coronal waves have been found to be driven by various solar phenomena,
such as flare processes, filament or flux-rope eruptions, and other related activity. For example, Y. Li et al. (2022)
reported that QFP waves can appear near the flare core region with a dominant period of 181 seconds, showing
temporal synchronization with other quasi-periodic pulsations during the flare. EUV imaging observations have
suggested a significant correspondence between the characteristic frequency associated with energy release and that of
the detected coronal waves during flares Y. Shen & Y. Liu (2012). QFP waves may be accompanied by radio burst
emissions during solar eruptions (D. Yuan et al. 2013; P. Kumar et al. 2017). In addition, G. P. Zhou et al. (2016) found
that fast-mode coronal waves can serve as important precursors, appearing about 20 minutes before an extreme solar
eruption. Moreover, QFP waves can also trigger secondary magnetic reconnection, leading to subsequent eruptions
(G. Zhou et al. 2020).
Collectively, these observational results suggest a close correlation between solar eruptive
activities and coronal waves. With advancements in observations and simulations, it is becoming increasing possible
to uncover the generation mechanisms of the various coronal waves observed in the solar corona.
Numerical simulations offer an effective approach to uncover the mechanisms behind coronal waves observed on the
Sun. Using a 2D simulation of filament-free flare processes, S. Takasao & K. Shibata (2016) proposed that QFP
waves can be excited by oscillations above the loop-top during a flare. Based on a 2.5 MHD simulation, L. Yang
et al. (2015) reported that QFP waves can be generated by the interaction between plasmoids in the current sheet
and the surrounding magnetic structures. In a 3D MHD simulation conducted by L. Ofman et al. (2011), it was
found that QFP waves can be induced by quasi-periodic sources at the base of flare loops. It is important to
note that an alternative interpretation for the observed quasi-periodic patterns exists, which does not
necessarily require a periodic energy source. This scenario attributes the formation of QFP waves to
the wave dispersion caused by the perpendicular structuring of the coronal plasma, see, (e.g., V. M.
Nakariakov et al. (2005), and M. Karlick´y et al. (2013)), including a recent 3D simulation (e.g., M. Shi
et al. (2025).)
In the context of a filament system, J. Hu et al. (2024) employed a 2D simulation to effectively explain QFP waves
observed at the flank of a coronal mass ejection (CME). These waves was suggested to originate from perturbations
within the filament. Subsequent 3D simulations (J. Hu et al. 2024) revealed that these waves exhibit a dome-like
geometry and displayed three distinct components. Understanding how these diverse coronal waves are generated
during magnetic reconnection could shed light on the intricate physical processes of solar eruptions , yet this remains
a crucial unresolved issue.
Through 3D simulations, this study reveals a kind of fast coronal wave produced near the ends of the reconnection
current sheet (CS). For the first time , we demonstrate that this fast coronal wave is generated by the tuning-fork
effect at both ends of the CS, and that it can appear both in the aftermath of a propagating CME and near the
tops of post-flare loops.Section 2 presents detailed numerical methods, while Section 3 provides the analysis of wave
characteristics and excitation mechanisms, followed by discussions and a summary in Section 4 .
2. SETUP IN THE NUMERICAL SIMULATION
To study wave generation mechanisms during solar filament eruptions, we perform 3D MHD simulations using the
MPI-AMRVAC code (R. Keppens et al. 2012; O. Porth et al. 2014; C. Xia et al. 2018).Our investigation focuses on
plasma dynamics by solving the full set of MHD equations incorporating both gravitational effects and anisotropic
thermal conduction. The governing equations are expressed as follows:
∂ρ
∂t + ∇· (ρv) = 0
(1)
∂e
∂t + ∇· [(e + P ∗)v −(v · B)B] = ρg · v + ∇· (ηB × (∇× B) −Fc)
(2)
∂(ρv)
∂t
+ ∇· [ρvv + P ∗I −BB] = ρg
(3)
∂B
∂t = ∇× (v × B −η∇× B)
(4)
P ∗= p + 1
2|B|2
(5)
p = ρT
(6)
3
where ρ denotes mass density, v flow velocity, B magnetic field, p gas pressure, and T temperature. Thermal conduction
follows the Spitzer mode with Fc = −κ∥(∇T · ˆB) ˆB −κ⊥(∇T −(∇T · ˆB) ˆB) where κ∥and κ⊥are parallel and
perpendicular conduction coefficients. The total energy density e comprises kinetic, thermal, and magnetic components:
e = ρv2/2 + p/(γ −1) + B2/2. All physical quantities are normalized by using characteristic scales: length L0 =
5 × 109 cm, number density n0 = 1010 cm−3, and temperature T0 = 106 K. From these, we derive the characteristic
density ρ0 = 1.673 × 10−14 g cm−3, magnetic field strength B0 = 5.89 G, plasma pressure P0 = 2.76 dyne
cm−2,
and velocity v0 = 128.5 km s−1. The simulation domain spans 8 L0 in each direction, discretized using a uniform
grid of 20003 cells. The model utilizes a uniform resistivity of η = 1.0 × 105 in normalized units. It is important to
note that this constant resistivity sets a fixed and globally uniform reconnection rate. This approach does not capture
the localized enhancement of resistivity expected in anomalous effects. We employ a Godunov-type finite volume
scheme, utilizing the Harten–Lax–van Leer (HLL) approximate Riemann solver and a third-order Runge–Kutta time
integration method. The boundary conditions consist of a line-tied lower boundary and open upper/lateral boundaries
to permit plasma outflow. For further details on the numerical setup, please refer to J. Hu et al. (2024) and J. Ye
et al. (2023). We note that the governing equations employed in this model do not include the effects of heating and
radiative cooling . Thus, our model does not capture the physics of wave-induced thermal misbalance, which has been
proposed as an alternative mechanism for QFP wave formation (e.g., D. I. Zavershinskii et al. (2019) ).
The model employs a gravitationally stratified, two-layer isothermal atmosphere: a chromosphere at T = 104 K
and a corona at T = 1 MK. The initial magnetic configuration ,as shown by Fig 1 a-b, employs a modified Titov &
D´emoulin (V. S. Titov & P. D´emoulin 1999) framework, incorporating three distinct components:
(1). The primary component (BI) arises from a pre-existing flux rope with major radius R, carrying a uniformly
distributed axial current I within a minor radius a.
(2). The secondary component (Bq) originates from a pair of subphotospheric magnetic sources of opposite-polarity
charges (±q1), separated by distance L at depth z = −d1. These charges are symmetrically positioned about the flux
rope axis, mimicking active region topology.
(3). The tertiary component (Bt) comprises a buried dipole (q2) positioned at z = −d2 in the xy- plane, providing
additional magnetic confinement.
This configuration produces a height-dependent magnetic field that decays as z−2, consistent with typical solar coronal
conditions.
3. RESULTS OF THE SIMULATION
3.1. the Global Evolution
Figure 1 c-f displays the spatial distributions of key physical parameters on the xz-cross-section (at y = 0) at t =
133.09 s: logarithmic density, temperature, current density, and velocity divergence. During the ascent of the filament,
the magnetic fields overlaying it were stretched up to create a vertical current sheet (marked as CS in Fig. 1 c) and
trigger magnetic reconnection. A typical occurrence associated with this reconnection is the flare process observed, as
indicated in Fig. 1 c (e.g., J. Lin et al. (2005)).
Along with the reconnection process continually converts magnetic energy into plasma kinetic and thermal energy,
the CS is characterized by parameters of the plasma such as plasma density, temperature, and current density. As
shown in Fig. 1 c, the plasma density in the CS increases significantly up to twice the background value at its central
region. The CS has a sharp temperature in the center reaching 14.4 MK, and a little lower value of around 14.2 MK at
its top and bottom ends as shown in Fig. 1 d, — an order of magnitude higher than the ambient corona temperature
(1 MK). This temperature distribution aligns well with observations in J. Lin et al. (2005). Furthermore, the current
density in the CS exhibits spatial inhomogeneity, peaking at the center and rapidly dissipating from the inside out
within the “Separatrix”, as shown in Fig. 1 e.
To explore the dynamic activity during the reconnection, we perform a divergence analysis of the velocity field, as
described by Equation 1, where velocity divergence effectively captures spatial density perturbations. As evidenced in
Fig. 1 f, we have newly found multiple outward-propagating wave fronts appearing as ripple structures originating from
the “Separatrix” near the ends of the CS. Understanding the excitation of such waves and their properties is crucial
for a deeper comprehension of energy conversion mechanisms and plasma dynamics during the magnetic reconnection.
A comprehensive examination of these waves will be presented in subsequent sections.
4
c d e f
flux rope
CS
flare
Separatrix
a b
log (rho) T J
∆
. v
x axis
z axis
-0.4 0 0.4
0.4 1.2 2.0
Figure 1. The distributions of initial magnetic field at t = 0 s and multiple physical parameters at t = 133.09 s. (a)-(b)
shows the initial magnetic field configuration from two perspectives. This model incorporates a flux rope with axial current I,
a subphotospheric charge pair (±q1, separation L, depth d1) and a buried magnetic dipole (q2, depth d2). This configuration
reproduces characteristic active region topology with B ∝z−2 field decay. The bottom shows spatial distributions of multiple
physical parameters in the solar atmoshpere at t = 133.09 s, including (c) logarithmic density, (d) temperature , (e) current
density and (f) velocity divergence. The current sheet (CS) is indicated, and “Separatrix” denotes the boundary between the
CS and the ambient coronal plasma. All 2D spatial snapshots are taken at a fixed cut at y=0.
3.2. the Characteristics of These Ripple-like Waves
We calculate the velocity divergence distribution at each moment throughout the magnetic reconnection process.
The observed ripple-like waves initially manifested at both ends of the CS about 100 seconds after the onset of magnetic
reconnection, becoming distinctly visible at t = 133.09 seconds, as illustrated in Fig 2. In the detailed views provided
in Fig. 2b-c, the wave in the Region 2 of Fig 2a exhibits a more symmetrical feature to the CS structure compared to
that in Region 1. These waves are henceforth referred to as “W1” and “W2” for brevity in the following descriptions.
Both “W1” and “W2” are simultaneously generated by the same magnetic process within the CS but are related to
opposite outflows. As shown in Fig. 2b-c, “W2” propagates beneath the base of the CME, while “W1” propagates
above the ascending post-flare loops. In the 2D cross-section of Fig. 2b-c, “W2” travels with a wider angular span
compared to “W1”.
The attenuation intensity is noticeable during their propagation, indicated by the arrows in
5
a
b
c
s2
s1
W2
W1
Region 2
Region 1
Figure 2. Ripple-like waves shown by velocity divergence distribution as displayed in Panel (a). Panel (b) is the close-up
of Region 2 of Panel (a) to show the group wave “W2” (green arrows) near the top end of the CS, propagating outside
near the CME. Panel (c) is the close-up of Region 1 to display the group wave “W1” (black arrows) near the bottom end
of the CS. “S1” in panel c and “S2” in panel b are used to indicate the positions for the wave periodicity analysis in Fig.
3. An animated version from t = 132.31s to 239.71 s of this figure is available.
Fig. 2b-c. Based on these simulation results, “W2” is anticipated to be observed following the propagation of a CME,
whereas “W1” may be detected in the lower flare process.
The periods and velocities of “W1-W2” can be discerned from time-slice charts along their paths. Illustrated in Fig.
2c, a slice “s1” is taken across “W1”. The corresponding time-slice chart in Fig. 3a reveals the periodic nature “W1”,
with a velocity reaching about 1400 km/s through linear fitting. According to the reference W. Liu et al. (2012),
“W1” aligns with the typical QFP waves observed in the corona. Additionally, an analysis of values at the horizontal
position y = 5656 km in Fig. 3a estimates the period of “W1” using wavelet transform analysis, revealing a dominant
period of 2 seconds in Fig. 3b-c. Employing a similar analysis on “W2”, it is determined to propagated at a speed of
≈1400 km/s with a period of 2 seconds. The comparable velocities and periods of “W1-W2” indicate that they were
likely triggered by a similar mechanism within a shared process.
3.3. the Origin of QFP waves
To systematically investigate the excitation mechanism of “W1 waves ” and “W2 waves”, we need to first identify
their oscillation sources and subsequently elucidate the physical processes driving these source oscillations. By tracking
the temporal evolution of these waves (see the supplementary movie), we determine that these waves originate from
oscillations of the “Separatrix” structure. To verify this, we analyze oscillations at a point labeled “K” (as shown in
Fig. 4c), which lies on the “Separatrice” structure. Wavelet analysis reveals that the oscillation period at the location
“K” aligns precisely with that of the “W1 waves”, thereby confirming the “Separatrix” as the oscillation source.
6
5
10
Divergence of velocity
Wavelet Power Spectrum
140
160
180
200
220
240
Time (s)
10
1
Period (s)
Global wavelet power spectrum
0
5
Power
0
IDivergence of velocity
Wavelet Power Spectrum
140
160
180
200
220
240
Time (s)
10
1
Period (s)
Global wavelet power spectrum
0
2
4
Power
1400km/s
period analysis for W1 period analysis for W2
P=2s
P=2s
a
b
c
d
e
f
Figure 3. Feature analysis of two group waves (“W1” and “W2”). Panel a-c presents the period analysis for “W1”: (a)
spatiotemporal distribution of slice “S1”; (b) horizontal dashed line within (a) ; (c) wavelet analysis plot of (a2). Panel (d-f)
presents the corresponding analysis for “W2”.
Examining the distributions of density structures and velocity fields, particularly, within the top and bottom regions
of the reconnection CS, provides insights into the excitation mechanism of “Separatrix” oscillation.
The density
distributions in Fig.
4
b-c reveal two regions of density enhancement at the top and bottom ends of the CS,
previously identified as a termination shock structure in prior studies (eg., T. Takahashi et al. (2017)).
Notably,
following the termination shock regions (the red arrows), the high-speed reconnection outflows noticeably decelerate.
The reconnection flow slows from around 720 km/s to approximately 50 km/s. Therefore, this termination shock acts
as a physical barrier, and subsequently leads to a shift in the directions of the reconnection outflows and then the
interactions of reconnection outflow with “LSP” (lower part of “Separatrix ” structure), indicated in Fig. 4c.
Applying the same analytical methodology the “W2 waves” (Fig. 4b) revealed an identical generation mechanism:
“W2 waves” similarly originate from the impact of the reconnection outflow on a termination shock, subsequently
lateral flows inducing “USP” (upper part of the “Separatrix” structure) oscillations. It is crucial to emphasize that
“USP” and “LSP” belong to the same “Separatrix” structure formed by a unified magnetic reconnection process. This
topological connection implies that different segments of the “Separatrix” structure share identical physcial properties.
As a direct consequence, despite spatially distinct excitation sites for “W1 waves” and “W2 waves ”, both wave
systems share: (1) the same magnetic reconnection context, (2) identical excitation mechanism, and (3) consequently
exhibit consistent propagation characteristics (∼1400 km/s propagation speed, and 2 s wave period). This spatiotem-
poral coherence provides compelling evidence for the integral nature of the magnetic reconnection process governing
the entire system.
7
-0.4 0 0.4
0.4 1.2 2.0
termination shock
termination shock
a
b
c
USP
0
Wavelet Power Spectrum
140
160
180
200
220
240
Time (s)
10
1
Period (s)
Global wavelet power spectrum
0
10
Power
p =2s
d period analysis of K Point
K
LSP
t = 133.09 s
K
-0.12 0.12
0.82
0.94
1.06
-0.04 0.0 0.04
0.56
0.6
x axis
z axis
log (rho)
log (rho)
-0.01 -0.05 0 0.05 0.01
-2 -1 0 1 2
-1 -0.5 0 0.5 1 1.5
0.52
Figure 4.
Distributions of density and velocities revealing the locations of fast wave excitations.
b-c display specific regions
within the upper and lower dashed boxes in Panel a, where velocity fields are illustrated with white arrows, indicating the
directions and magnitudes of plasma flow. “USP” and “LSP” denote the upper and lower sections of the “Separatrix” structure,
respectively. The circled region region “K” is situated within the lower “Separatrix” and exhibits a density oscillation of 2 s, as
depicted in Panel (d).
4. DISCUSSIONS AND SUMMARY
The entire process can be illustrated in the schematic diagram presented in Figure 5. The high-speed reconnection
outflow (depicted by the vertical thick arrows) encounters termination shock structures (highlighted in the navy blue
ellipse regions), which then rebound to impact and distort the localized sections of the CS fork wall (visible in red at
both ends of the CS). This interaction induces a consistent oscillation at a specific frequency, akin to a tuning fork
effects. Consequently, QFP waves are excited, as evidenced by the blue ripple patterns in Figure 5. This sequence
establishes a coherent causal link from magnetic reconnection to the generation of fast waves, with the termination
shock playing a crucial role in facilitating the energy transfer between the reconnection outflow and the coronal waves.
Previous studies frequently demonstrate that quasi-periodic pulsations (QPPs) observed in individual events manifest
a broad range of periodicity spanning from milliseconds to minutes. These extensive periodicity distributions may imply
the existence of diverse physical mechanisms at play. This research unveils a new mechanism for the production of QFP
waves, revealing the intricate nature of 3D physical process and provide new insights beyond flare model.Moreover,
quasi-periodic pulsations with the oscillation periods about those obtained in the modelling have been
detected with an indirect spatial resolution in the radio emission by D. Y. Kolotkov et al. (2018).
In conclusion, by utilizing a 3D numerical model, we unveiled a novel manifestation of QFP waves emerging near
both terminations of the vertical CS during flaring reconnection processes. These fast waves, labeled as “W1” and
“W2” in our study, originate from the fork structure located at two ends of the vertical reconnection CS, respectively.
Both “W1” and “W2” exhibit similar propagation speeds of approximately 1400 km/s and share a period of 2 seconds,
manifesting predominantly within the lower corona. “W1” demonstrates a broader angular width, whereas “W2”
presents a narrower profile. Our analysis suggests that rapid coronal waves were excited by a type of tuning fork
effect occurring within the fork topology at both ends of the reconnection CS. Subsequent investigations will focus
on clarifying the key role played by these fast-moving coronal waves. All in all, by leveraging the critical structure
of the “Separatrix” (or tuning fork wall), this study firstly organically connects the generation mechanisms of two
groups of fast-mode coronal waves during eruptive processes. This provides a novel theoretical framework for gaining
an in-depth understanding of wave phenomena driven by magnetic reconnection.
In comparing our findings with observations, we anticipate that the fast coronal wave stemming from the lower end of
the CS should be detectable in proximity to post-flare loops, as already confirmed through EUV observations (e.g.,W.
Liu et al. (2011)). Conversely, the fast coronal wave originating from the upper end of the CS is anticipated to be
8
termination shock
outflow
CS fork str.
w1
inflow
w1
reflecting
erupting str.
CS
flare ribbons
w2
w2
Figure 5. Schematic diagram illustrating the generation of QFP waves. During a solar eruption, magnetic reconnection
takes place within a vertical CS, propelling the erupting structure as shown by the pink twisted topology. The vertical CS
is highlighted by the thick yellow lines and exhibits two fork structures at its opposing ends. Inflow during reconnection is
indicated by the bold grey arrows, while reconnection outflows are represented by two thick vertical arrows at each ends. These
outflows strike the termination shock structures, as indicated by the navy blue ellipses, leading to reflection. The sky blue ripple
structures denote “W1” and “W2”.
observed in conjunction with propagating eruptive coronal mass ejections (CMEs), a phenomenon awaiting observation
validation.
It is important to note a key limitation of our present analysis: the characteristics of these QFP waves are analysed in
the vertical plane passing through the center of the simulation domain. In a fully three-dimensional corona, these waves
are subject to additional effects not captured here, such as refraction induced by the 3D magnetic field structure (e.g.,
M. Shi et al. (2025)). A comprehensive investigation of these 3D effects will be a crucial and natural extension of this
work in the future. Moreover, this study serves as a foundational case study and a key outstanding question remains
regarding what physical parameters primarily determine the resulting oscillation period. A systematic parametric
study to quantify the range of the periods is therefore essential for future work.
9
The work is supported by the National Key R&D Program of China (No. 2022YFF0503800), the Strategic Priority
Research Program of the Chinese Academy of Sciences (No. XDB0560102), and the National Natural Science Foun-
dation of China (Nos. 12503064, 12273061, 12403067,12303062). We also acknowledge Sichuan Normal University
Astrophysical Laboratory Supercomputer for providing the computational resources.
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Draft version October 17, 2025 Typeset using LATEX default style in AASTeX7.0.1 QFP Waves Driven by the Tuning-Fork Effect during Magnetic Reconnecion Jialiang Hu,1, 2 Xiaozhou Zhao,3 Guiping Zhou,1, 2 Yuhao Chen,4 Chunlan Jin,1 Mijie Shi,5 Guanchong Cheng,3 Xiaoxia Yu,6 Jing Ye,3 Xinping Zhou,7 and Hanxian Fang8 1State Key Laboratory of Solar Activity and Space Weather, National Astronomical observatories, Chinese Academy of Sciences, 100101,Beiing,China 2 100049, Beijing, China 3Yunnan Observatories, Chinese Academy of Sciences, P.O. Box 110, Kunming, Yunnan 650216, People's Republic of China 4 100781, People's Republic of China 5Shandong Key Laboratory of Optical Astronomy and Solar-Terrestrial Environment, 264209, People's Republic of China 6State Key Laboratory of Particle Astrophysics, 100049, China 7 610068, People's Republic of China 8 . ABSTRACT Through three-dimensional MHD simulations, we have uncovered a kind of fast coronal wave originating from both ends of a current sheet (CS) during a solar eruption. These waves are observed to appear near the top and bottom ends of the reconnection-related CS. The simulations demonstrate the presence of termination shock regions above the two ends of the CS. As the reconnection outflows escape from the vertical CS and encounter these termination shocks, they undergo partial reflection, redirecting towards the CS terminal fork walls. The identified waves propagate rapidly at a speed of approximately 1400 km/s with a period of just 2 s. Concurrently, the time-evolution of intensity within a small region of the CS terminal fork structures, exhibits a similar oscillation period of 2 s. All these evidence supports the notion that these QFP (Quasi-periodic Fast-Propagating) waves were excited by tuning fork effects within the CS system. Essentially, the rapid reconnection outflows are reflected by the terminal shocks, striking the fork walls at the CS ends. Moreover, parts of the oscillations along the tuning fork handle are transformed into thermal energy, accumulating in the CS center and elevating the temperature. This is the first time to report such QFP waves resulting from tuning fork effects within the CS during a solar eruption. These waves are anticipated to manifest closely following the propagation of CMEs and adjacent to the related post-flare loops in observations, with partial confirmation in current observations. Keywords: Separatrice; Solar Corona; MHD waves; Current Sheet; MHD Simulations 1. INTRODUCTION Magnetic reconnection plays a critical role in solar eruptions, governing both the energy release process and dynamics in the solar corona. Solar eruptions are frequently accompanied by various modes of coronal waves. The propagation characteristics of coronal waves (e,g., phase speeds, dispersion relations, and attenuation scales) encode valuable information about the local plasma-β and magnetic topology, making them useful tools for probing the fine-scale physical processes during magnetic reconnections (Y. Shen et al. 2022). Previous studies that have reported a wide range of coronal wave modes have significantly advanced our understanding of solar activity (V. M. Nakariakov & D. Y. Kolotkov 2020). However, the mechanisms by which different modes of coronal waves are generated during magnetic reconnection remain poorly understood. Corresponding author: Hanxian Fang, Xinping Zhou 16 Oct 2025 2 Based on multi-wavelength observations, coronal waves have been found to be driven by various solar phenomena, such as flare processes, filament or flux-rope eruptions, and other related activity. For example, Y. Li et al. (2022) reported that QFP waves can appear near the flare core region with a dominant period of 181 seconds, showing temporal synchronization with other quasi-periodic pulsations during the flare. EUV imaging observations have suggested a significant correspondence between the characteristic frequency associated with energy release and that of the detected coronal waves during flares Y. Shen & Y. Liu (2012). QFP waves may be accompanied by radio burst emissions during solar eruptions (D. Yuan et al. 2013; P. Kumar et al. 2017). In addition, G. P. Zhou et al. (2016) found that fast-mode coronal waves can serve as important precursors, appearing about 20 minutes before an extreme solar eruption. Moreover, QFP waves can also trigger secondary magnetic reconnection, leading to subsequent eruptions (G. Zhou et al. 2020). Collectively, these observational results suggest a close correlation between solar eruptive activities and coronal waves. With advancements in observations and simulations, it is becoming increasing possible to uncover the generation mechanisms of the various coronal waves observed in the solar corona. Numerical simulations offer an effective approach to uncover the mechanisms behind coronal waves observed on the Sun. Using a 2D simulation of filament-free flare processes, S. Takasao & K. Shibata (2016) proposed that QFP waves can be excited by oscillations above the loop-top during a flare. Based on a 2.5 MHD simulation, L. Yang et al. (2015) reported that QFP waves can be generated by the interaction between plasmoids in the current sheet and the surrounding magnetic structures. In a 3D MHD simulation conducted by L. Ofman et al. (2011), it was found that QFP waves can be induced by quasi-periodic sources at the base of flare loops. It is important to note that an alternative interpretation for the observed quasi-periodic patterns exists, which does not necessarily require a periodic energy source. This scenario attributes the formation of QFP waves to the wave dispersion caused by the perpendicular structuring of the coronal plasma, see, (e.g., V. M. Nakariakov et al. (2005), and M. Karlick ́y et al. (2013)), including a recent 3D simulation (e.g., M. Shi et al. (2025).) In the context of a filament system, J. Hu et al. (2024) employed a 2D simulation to effectively explain QFP waves observed at the flank of a coronal mass ejection (CME). These waves was suggested to originate from perturbations within the filament. Subsequent 3D simulations (J. Hu et al. 2024) revealed that these waves exhibit a dome-like geometry and displayed three distinct components. Understanding how these diverse coronal waves are generated during magnetic reconnection could shed light on the intricate physical processes of solar eruptions , yet this remains a crucial unresolved issue. Through 3D simulations, this study reveals a kind of fast coronal wave produced near the ends of the reconnection current sheet (CS). For the first time , we demonstrate that this fast coronal wave is generated by the tuning-fork effect at both ends of the CS, and that it can appear both in the aftermath of a propagating CME and near the tops of post-flare loops.Section 2 presents detailed numerical methods, while Section 3 provides the analysis of wave characteristics and excitation mechanisms, followed by discussions and a summary in Section 4 . 2. SETUP IN THE NUMERICAL SIMULATION To study wave generation mechanisms during solar filament eruptions, we perform 3D MHD simulations using the MPI-AMRVAC code (R. Keppens et al. 2012; O. Porth et al. 2014; C. Xia et al. 2018).Our investigation focuses on plasma dynamics by solving the full set of MHD equations incorporating both gravitational effects and anisotropic thermal conduction. The governing equations are expressed as follows: ∂ρ ∂t + ∇· (ρv) = 0 (1) ∂e ∂t + ∇· [(e + P ∗)v -(v · B)B] = ρg · v + ∇· (ηB × (∇× B) -Fc) (2) ∂(ρv) ∂t + ∇· [ρvv + P ∗I -BB] = ρg (3) ∂B ∂t = ∇× (v × B -η∇× B) (4) P ∗= p + 1 2|B|2 (5) p = ρT (6) 3 where ρ denotes mass density, v flow velocity, B magnetic field, p gas pressure, and T temperature. Thermal conduction follows the Spitzer mode with Fc = -κ∥(∇T · ˆB) ˆB -κ⊥(∇T -(∇T · ˆB) ˆB) where κ∥and κ⊥are parallel and perpendicular conduction coefficients. The total energy density e comprises kinetic, thermal, and magnetic components: e = ρv2/2 + p/(γ -1) + B2/2. All physical quantities are normalized by using characteristic scales: length L0 = 5 × 109 cm, number density n0 = 1010 cm-3, and temperature T0 = 106 K. From these, we derive the characteristic density ρ0 = 1.673 × 10-14 g cm-3, magnetic field strength B0 = 5.89 G, plasma pressure P0 = 2.76 dyne cm-2, and velocity v0 = 128.5 km s-1. The simulation domain spans 8 L0 in each direction, discretized using a uniform grid of 20003 cells. The model utilizes a uniform resistivity of η = 1.0 × 105 in normalized units. It is important to note that this constant resistivity sets a fixed and globally uniform reconnection rate. This approach does not capture the localized enhancement of resistivity expected in anomalous effects. We employ a Godunov-type finite volume scheme, utilizing the Harten-Lax-van Leer (HLL) approximate Riemann solver and a third-order Runge-Kutta time integration method. The boundary conditions consist of a line-tied lower boundary and open upper/lateral boundaries to permit plasma outflow. For further details on the numerical setup, please refer to J. Hu et al. (2024) and J. Ye et al. (2023). We note that the governing equations employed in this model do not include the effects of heating and radiative cooling . Thus, our model does not capture the physics of wave-induced thermal misbalance, which has been proposed as an alternative mechanism for QFP wave formation (e.g., D. I. Zavershinskii et al. (2019) ). The model employs a gravitationally stratified, two-layer isothermal atmosphere: a chromosphere at T = 104 K and a corona at T = 1 MK. The initial magnetic configuration ,as shown by Fig 1 a-b, employs a modified Titov & D ́emoulin (V. S. Titov & P. D ́emoulin 1999) framework, incorporating three distinct components: (1). The primary component (BI) arises from a pre-existing flux rope with major radius R, carrying a uniformly distributed axial current I within a minor radius a. (2). The secondary component (Bq) originates from a pair of subphotospheric magnetic sources of opposite-polarity charges (±q1), separated by distance L at depth z = -d1. These charges are symmetrically positioned about the flux rope axis, mimicking active region topology. (3). The tertiary component (Bt) comprises a buried dipole (q2) positioned at z = -d2 in the xy- plane, providing additional magnetic confinement. This configuration produces a height-dependent magnetic field that decays as z-2, consistent with typical solar coronal conditions. 3. RESULTS OF THE SIMULATION 3.1. the Global Evolution Figure 1 c-f displays the spatial distributions of key physical parameters on the xz-cross-section (at y = 0) at t = 133.09 s: logarithmic density, temperature, current density, and velocity divergence. During the ascent of the filament, the magnetic fields overlaying it were stretched up to create a vertical current sheet (marked as CS in Fig. 1 c) and trigger magnetic reconnection. A typical occurrence associated with this reconnection is the flare process observed, as indicated in Fig. 1 c (e.g., J. Lin et al. (2005)). Along with the reconnection process continually converts magnetic energy into plasma kinetic and thermal energy, the CS is characterized by parameters of the plasma such as plasma density, temperature, and current density. As shown in Fig. 1 c, the plasma density in the CS increases significantly up to twice the background value at its central region. The CS has a sharp temperature in the center reaching 14.4 MK, and a little lower value of around 14.2 MK at its top and bottom ends as shown in Fig. 1 d, - an order of magnitude higher than the ambient corona temperature (1 MK). This temperature distribution aligns well with observations in J. Lin et al. (2005). Furthermore, the current density in the CS exhibits spatial inhomogeneity, peaking at the center and rapidly dissipating from the inside out within the "Separatrix", as shown in Fig. 1 e. To explore the dynamic activity during the reconnection, we perform a divergence analysis of the velocity field, as described by Equation 1, where velocity divergence effectively captures spatial density perturbations. As evidenced in Fig. 1 f, we have newly found multiple outward-propagating wave fronts appearing as ripple structures originating from the "Separatrix" near the ends of the CS. Understanding the excitation of such waves and their properties is crucial for a deeper comprehension of energy conversion mechanisms and plasma dynamics during the magnetic reconnection. A comprehensive examination of these waves will be presented in subsequent sections. 4 c d e f flux rope CS flare Separatrix a b log (rho) T J ∆ . v x axis z axis -0.4 0 0.4 0.4 1.2 2.0 Figure 1. The distributions of initial magnetic field at t = 0 s and multiple physical parameters at t = 133.09 s. (a)-(b) shows the initial magnetic field configuration from two perspectives. This model incorporates a flux rope with axial current I, a subphotospheric charge pair (±q1, separation L, depth d1) and a buried magnetic dipole (q2, depth d2). This configuration reproduces characteristic active region topology with B ∝z-2 field decay. The bottom shows spatial distributions of multiple physical parameters in the solar atmoshpere at t = 133.09 s, including (c) logarithmic density, (d) temperature , (e) current density and (f) velocity divergence. The current sheet (CS) is indicated, and "Separatrix" denotes the boundary between the CS and the ambient coronal plasma. All 2D spatial snapshots are taken at a fixed cut at y=0. 3.2. the Characteristics of These Ripple-like Waves We calculate the velocity divergence distribution at each moment throughout the magnetic reconnection process. The observed ripple-like waves initially manifested at both ends of the CS about 100 seconds after the onset of magnetic reconnection, becoming distinctly visible at t = 133.09 seconds, as illustrated in Fig 2. In the detailed views provided in Fig. 2b-c, the wave in the Region 2 of Fig 2a exhibits a more symmetrical feature to the CS structure compared to that in Region 1. These waves are henceforth referred to as "W1" and "W2" for brevity in the following descriptions. Both "W1" and "W2" are simultaneously generated by the same magnetic process within the CS but are related to opposite outflows. As shown in Fig. 2b-c, "W2" propagates beneath the base of the CME, while "W1" propagates above the ascending post-flare loops. In the 2D cross-section of Fig. 2b-c, "W2" travels with a wider angular span compared to "W1". The attenuation intensity is noticeable during their propagation, indicated by the arrows in 5 a b c s2 s1 W2 W1 Region 2 Region 1 Figure 2. Ripple-like waves shown by velocity divergence distribution as displayed in Panel (a). Panel (b) is the close-up of Region 2 of Panel (a) to show the group wave "W2" (green arrows) near the top end of the CS, propagating outside near the CME. Panel (c) is the close-up of Region 1 to display the group wave "W1" (black arrows) near the bottom end of the CS. "S1" in panel c and "S2" in panel b are used to indicate the positions for the wave periodicity analysis in Fig. 3. An animated version from t = 132.31s to 239.71 s of this figure is available. Fig. 2b-c. Based on these simulation results, "W2" is anticipated to be observed following the propagation of a CME, whereas "W1" may be detected in the lower flare process. The periods and velocities of "W1-W2" can be discerned from time-slice charts along their paths. Illustrated in Fig. 2c, a slice "s1" is taken across "W1". The corresponding time-slice chart in Fig. 3a reveals the periodic nature "W1", with a velocity reaching about 1400 km/s through linear fitting. According to the reference W. Liu et al. (2012), "W1" aligns with the typical QFP waves observed in the corona. Additionally, an analysis of values at the horizontal position y = 5656 km in Fig. 3a estimates the period of "W1" using wavelet transform analysis, revealing a dominant period of 2 seconds in Fig. 3b-c. Employing a similar analysis on "W2", it is determined to propagated at a speed of ≈1400 km/s with a period of 2 seconds. The comparable velocities and periods of "W1-W2" indicate that they were likely triggered by a similar mechanism within a shared process. 3.3. the Origin of QFP waves To systematically investigate the excitation mechanism of "W1 waves " and "W2 waves", we need to first identify their oscillation sources and subsequently elucidate the physical processes driving these source oscillations. By tracking the temporal evolution of these waves (see the supplementary movie), we determine that these waves originate from oscillations of the "Separatrix" structure. To verify this, we analyze oscillations at a point labeled "K" (as shown in Fig. 4c), which lies on the "Separatrice" structure. Wavelet analysis reveals that the oscillation period at the location "K" aligns precisely with that of the "W1 waves", thereby confirming the "Separatrix" as the oscillation source. 6 5 10 Divergence of velocity Wavelet Power Spectrum 140 160 180 200 220 240 Time (s) 10 1 Period (s) Global wavelet power spectrum 0 5 Power 0 IDivergence of velocity Wavelet Power Spectrum 140 160 180 200 220 240 Time (s) 10 1 Period (s) Global wavelet power spectrum 0 2 4 Power 1400km/s period analysis for W1 period analysis for W2 P=2s P=2s a b c d e f Figure 3. Feature analysis of two group waves ("W1" and "W2"). Panel a-c presents the period analysis for "W1": (a) spatiotemporal distribution of slice "S1"; (b) horizontal dashed line within (a) ; (c) wavelet analysis plot of (a2). Panel (d-f) presents the corresponding analysis for "W2". Examining the distributions of density structures and velocity fields, particularly, within the top and bottom regions of the reconnection CS, provides insights into the excitation mechanism of "Separatrix" oscillation. The density distributions in Fig. 4 b-c reveal two regions of density enhancement at the top and bottom ends of the CS, previously identified as a termination shock structure in prior studies (eg., T. Takahashi et al. (2017)). Notably, following the termination shock regions (the red arrows), the high-speed reconnection outflows noticeably decelerate. The reconnection flow slows from around 720 km/s to approximately 50 km/s. Therefore, this termination shock acts as a physical barrier, and subsequently leads to a shift in the directions of the reconnection outflows and then the interactions of reconnection outflow with "LSP" (lower part of "Separatrix " structure), indicated in Fig. 4c. Applying the same analytical methodology the "W2 waves" (Fig. 4b) revealed an identical generation mechanism: "W2 waves" similarly originate from the impact of the reconnection outflow on a termination shock, subsequently lateral flows inducing "USP" (upper part of the "Separatrix" structure) oscillations. It is crucial to emphasize that "USP" and "LSP" belong to the same "Separatrix" structure formed by a unified magnetic reconnection process. This topological connection implies that different segments of the "Separatrix" structure share identical physcial properties. As a direct consequence, despite spatially distinct excitation sites for "W1 waves" and "W2 waves ", both wave systems share: (1) the same magnetic reconnection context, (2) identical excitation mechanism, and (3) consequently exhibit consistent propagation characteristics (∼1400 km/s propagation speed, and 2 s wave period). This spatiotemporal coherence provides compelling evidence for the integral nature of the magnetic reconnection process governing the entire system. 7 -0.4 0 0.4 0.4 1.2 2.0 termination shock termination shock a b c USP 0 Wavelet Power Spectrum 140 160 180 200 220 240 Time (s) 10 1 Period (s) Global wavelet power spectrum 0 10 Power p =2s d period analysis of K Point K LSP t = 133.09 s K -0.12 0.12 0.82 0.94 1.06 -0.04 0.0 0.04 0.56 0.6 x axis z axis log (rho) log (rho) -0.01 -0.05 0 0.05 0.01 -2 -1 0 1 2 -1 -0.5 0 0.5 1 1.5 0.52 Figure 4. Distributions of density and velocities revealing the locations of fast wave excitations. b-c display specific regions within the upper and lower dashed boxes in Panel a, where velocity fields are illustrated with white arrows, indicating the directions and magnitudes of plasma flow. "USP" and "LSP" denote the upper and lower sections of the "Separatrix" structure, respectively. The circled region region "K" is situated within the lower "Separatrix" and exhibits a density oscillation of 2 s, as depicted in Panel (d). 4. DISCUSSIONS AND SUMMARY The entire process can be illustrated in the schematic diagram presented in Figure 5. The high-speed reconnection outflow (depicted by the vertical thick arrows) encounters termination shock structures (highlighted in the navy blue ellipse regions), which then rebound to impact and distort the localized sections of the CS fork wall (visible in red at both ends of the CS). This interaction induces a consistent oscillation at a specific frequency, akin to a tuning fork effects. Consequently, QFP waves are excited, as evidenced by the blue ripple patterns in Figure 5. This sequence establishes a coherent causal link from magnetic reconnection to the generation of fast waves, with the termination shock playing a crucial role in facilitating the energy transfer between the reconnection outflow and the coronal waves. Previous studies frequently demonstrate that quasi-periodic pulsations (QPPs) observed in individual events manifest a broad range of periodicity spanning from milliseconds to minutes. These extensive periodicity distributions may imply the existence of diverse physical mechanisms at play. This research unveils a new mechanism for the production of QFP waves, revealing the intricate nature of 3D physical process and provide new insights beyond flare model.Moreover, quasi-periodic pulsations with the oscillation periods about those obtained in the modelling have been detected with an indirect spatial resolution in the radio emission by D. Y. Kolotkov et al. (2018). In conclusion, by utilizing a 3D numerical model, we unveiled a novel manifestation of QFP waves emerging near both terminations of the vertical CS during flaring reconnection processes. These fast waves, labeled as "W1" and "W2" in our study, originate from the fork structure located at two ends of the vertical reconnection CS, respectively. Both "W1" and "W2" exhibit similar propagation speeds of approximately 1400 km/s and share a period of 2 seconds, manifesting predominantly within the lower corona. "W1" demonstrates a broader angular width, whereas "W2" presents a narrower profile. Our analysis suggests that rapid coronal waves were excited by a type of tuning fork effect occurring within the fork topology at both ends of the reconnection CS. Subsequent investigations will focus on clarifying the key role played by these fast-moving coronal waves. All in all, by leveraging the critical structure of the "Separatrix" (or tuning fork wall), this study firstly organically connects the generation mechanisms of two groups of fast-mode coronal waves during eruptive processes. This provides a novel theoretical framework for gaining an in-depth understanding of wave phenomena driven by magnetic reconnection. In comparing our findings with observations, we anticipate that the fast coronal wave stemming from the lower end of the CS should be detectable in proximity to post-flare loops, as already confirmed through EUV observations (e.g.,W. Liu et al. (2011)). Conversely, the fast coronal wave originating from the upper end of the CS is anticipated to be 8 termination shock outflow CS fork str. w1 inflow w1 reflecting erupting str. CS flare ribbons w2 w2 Figure 5. Schematic diagram illustrating the generation of QFP waves. During a solar eruption, magnetic reconnection takes place within a vertical CS, propelling the erupting structure as shown by the pink twisted topology. The vertical CS is highlighted by the thick yellow lines and exhibits two fork structures at its opposing ends. Inflow during reconnection is indicated by the bold grey arrows, while reconnection outflows are represented by two thick vertical arrows at each ends. These outflows strike the termination shock structures, as indicated by the navy blue ellipses, leading to reflection. The sky blue ripple structures denote "W1" and "W2". observed in conjunction with propagating eruptive coronal mass ejections (CMEs), a phenomenon awaiting observation validation. It is important to note a key limitation of our present analysis: the characteristics of these QFP waves are analysed in the vertical plane passing through the center of the simulation domain. In a fully three-dimensional corona, these waves are subject to additional effects not captured here, such as refraction induced by the 3D magnetic field structure (e.g., M. Shi et al. (2025)). A comprehensive investigation of these 3D effects will be a crucial and natural extension of this work in the future. Moreover, this study serves as a foundational case study and a key outstanding question remains regarding what physical parameters primarily determine the resulting oscillation period. A systematic parametric study to quantify the range of the periods is therefore essential for future work. 9 The work is supported by the National Key R&D Program of China (No. 2022YFF0503800), the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB0560102), and the National Natural Science Foundation of China (Nos. 12503064, 12273061, 12403067,12303062). We also acknowledge Sichuan Normal University Astrophysical Laboratory Supercomputer for providing the computational resources. REFERENCES Hu, J., Ye, J., Chen, Y., et al. 2024, Research in Astronomy and Astrophysics, 24, 125011, Karlick ́y, M., M ́esz ́arosov ́a, H., & Jel ́ınek, P. 2013, A&A, 550, A1, Keppens, R., Meliani, Z., van Marle, A. J., et al. 2012, Journal of Computational Physics, 231, 718, Kolotkov, D. Y., Nakariakov, V. M., & Kontar, E. P. 2018, ApJ, 861, 33, Kumar, P., Nakariakov, V. 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S., & D ́emoulin, P. 1999, A&A, 351, 707 Xia, C., Teunissen, J., El Mellah, I., Chan ́e, E., & Keppens, R. 2018, ApJS, 234, 30, Yang, L., Zhang, L., He, J., et al. 2015, ApJ, 800, 111, Ye, J., Raymond, J. C., Mei, Z., et al. 2023, ApJ, 955, 88, Yuan, D., Shen, Y., Liu, Y., et al. 2013, A&A, 554, A144, Zavershinskii, D. I., Kolotkov, D. Y., Nakariakov, V. M., Molevich, N. E., & Ryashchikov, D. S. 2019, Physics of Plasmas, 26, 082113, Zhou, G., Gao, G., Wang, J., et al. 2020, ApJ, 905, 150, Zhou, G. P., Zhang, J., & Wang, J. X. 2016, ApJL, 823, L19,
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2510.14744
|
Spectral subspace extraction via incoherent quantum phase estimation
Stefano Scali¶,1,2,∗Josh Kirsopp¶,1,† Antonio M´arquez Romero3,‡ and Micha l Krompiec1§
1Fujitsu Research of Europe Ltd., SL1 2BE Slough, U.K.
2University of Exeter, Department of Physics and Astronomy, Stocker Road, Exeter EX4 4QL, UK
3Fujitsu Research of Europe Ltd., Pozuelo de Alarc´on, 28224 Madrid, Spain
Quantum phase estimation (QPE) is a cornerstone algorithm for extracting Hamiltonian eigen-
values, but its standard form targets individual eigenstates and requires carefully prepared coherent
inputs. To overcome these limitations, we adopt an ensemble-based formulation of QPE that esti-
mates the density of states (DOS) of the Hamiltonian generator of the evolution. This approach,
which we refer to as DOS-QPE, builds on a prior formulation introduced by one of the authors. In
this work, we present DOS-QPE as a circuit primitive, extending it with symmetry-adapted input
ensembles and advanced spectrum reconstruction techniques. This variant of QPE enables natural
access to thermodynamic properties, symmetry-resolved spectral functions, and features relevant to
quantum many-body systems. We demonstrate its performance on fermionic models and nuclear
Hamiltonians by casting the spectrum reconstruction problem as a quadratic program solved via
compressed sensing. These use cases highlight the potential of DOS-QPE for early fault-tolerant
quantum simulations in spectroscopy, electronic structure, and nuclear theory.
I.
INTRODUCTION
Estimating the spectral properties of quantum systems
is a fundamental problem with wide-ranging applications
in physics, chemistry, and quantum computing. In par-
ticular, the density of states (DOS) — the distribution of
energy eigenvalues of a quantum Hamiltonian — encodes
crucial information about both the thermodynamic be-
havior [1] and the dynamical response of a system [2]. Be-
yond its traditional role in condensed matter physics and
statistical mechanics, the DOS has recently been used in
hybrid classical-quantum frameworks for quantum topo-
logical data analysis [3] and to address the frustration
index problem in signed graphs [4].
Quantum phase estimation (QPE) remains the gold
standard for extracting eigenvalue information in the
form of quantum phases [5–8]. It is typically applied to
eigenstates of unitary evolution operators or their gen-
erators, and its success depends on the overlap between
the prepared trial state and the true eigenstate of the
Hamiltonian.
However, this requirement can be pro-
hibitive in practice. Preparing trial states with significant
overlap may demand deep, problem-specific quantum cir-
cuits, especially for strongly-correlated systems [9–11].
In addition, real quantum hardware inevitably inter-
acts with its surrounding environment, complicating fur-
ther the preparation of single pure states.
This raises
the natural question of how compatible QPE is with
mixed states, thermal states, and noisy, non-pure ensem-
bles [12]. While purification techniques [13, 14] are well-
established in quantum information, the use of QPE on
purified states has not yet been systematically studied as
∗stefano.scali@fujitsu.com; s.scali@exeter.ac.uk
† josh.kirsopp@fujitsu.com
‡ antonio.marquezromero@fujitsu.com
§ michal.krompiec@fujitsu.com
¶ These authors contributed equally to this work.
a circuit primitive. Notable exceptions include the use of
garbage states in quantum topological data analysis [15]
and recent work leveraging QPE on maximally mixed
states to compute Hamiltonian moments in the context
of signed graphs [4].
While a promising alternative approach for estimat-
ing the DOS using Chebyshev moments has been re-
cently proposed [16], this work focuses on developing a
framework centered around QPE. This not only provides
access to a rich landscape of algorithmic variants, but
also positions our method for reuse as a subroutine in
a range of QPE-based quantum algorithms—including
Shor’s algorithm [17], quantum simulation in quantum
chemistry [18], and the HHL algorithm for solving linear
systems [19].
In this work, we generalize and refine a QPE-based
method for DOS estimation (DOS-QPE) [4], presenting
it as a circuit primitive. We showcase some of its key
applications, ranging from the Fermi-Hubbard model to
molecular systems and nuclear physics Hamiltonians. We
begin with a review of standard QPE techniques and the
modifications required to apply them to purified states.
We then discuss suitable mixed state preparations and
rescaling of the Hamiltonian. We propose a method to
resolve the problem of integer degeneracies and, more
generally, tackle spectrum reconstruction using the tools
from quadratic programming [20] and compressed sens-
ing [21, 22]. Finally, we present three case studies of the
protocol: the Fermi-Hubbard Hamiltonian, a molecular
electronic structure problem, and a strongly correlated
nuclear physics Hamiltonian.
II.
DOS-QPE
Standard quantum phase estimation consists of a state
register of dimension n and a time-frequency register
of dimension m, often referred to as the ancilla regis-
ter. In QPE, we are interested in the extraction of the
arXiv:2510.14744v1 [quant-ph] 16 Oct 2025
2
phase θ relative to an eigenstate |ψ⟩of a unitary op-
erator UH, where UH|ψ⟩= e2πiθ|ψ⟩. This can be any
unitary n-qubit gate, often generated by the Hamilto-
nian H as UH = e−iHt. To achieve this, we prepare the
state register with the eigenstate |ψ⟩and perform the
controlled-unitary operations UC = P2m−1
k=0
|k⟩⟨k| ⊗U k
H.
This cascade of unitary operations acting onto the state
register at incremental “times” k extracts the phase via
phase kickbacks. The simplest example of this process
is the Hadamard test, formally a QPE circuit with a
single-qubit time-frequency register.
These phases are
encoded into the time-frequency register and can be ac-
cessed in the form of probabilities after performing an
inverse quantum Fourier transform (iQFT), U†
QFT, on it.
In Fig. 1a), the QPE circuit would correspond to the
first two registers, namely the “time-frequency” and the
“state” registers, thus removing the “purification” regis-
ter and relative operations. Standard QPE is the fault-
tolerant benchmark for the estimation of phases (eigen-
values) in the case of a pure, coherent eigenstate of the
unitary i. Its successful execution requires the prepared
probing state to have substantial overlap with the target
eigenstate whose phase we aim to measure.
Although
several techniques have been proposed to address the
state preparation problem, this step often incurs signifi-
cant overhead and is sometimes the most challenging part
of the protocol.
In chemistry, trial eigenstate preparation remains a
hot topic [25]; for weakly correlated systems, the qubit-
mapped Hartree-Fock state may be good enough [26],
but for more strongly correlated systems, elaborate state
preparation protocols are required. These usually take
the form of a variational quantum algorithm [27], which
represent difficult optimization problems [28, 29] and
suffer prohibitive sampling costs [30].
Otherwise, ap-
proximate classical eigenstates can be loaded onto a
qubit register by an appropriate mapping such as CVO-
QRAM [31]. Still, even this approach does not scale fa-
vorably with system size, not to mention the additional
classical cost of finding a high-quality approximation to
the target eigenstate, using, for example, one of the se-
lected CI methods [32–34]. In addition, we might be in-
terested in more general properties of the Hermitian oper-
ator generator of the unitary evolution. For example, ac-
cessing information encoded in higher-energy states may
require reconstructing the entire spectrum of the target
Hamiltonian.
This is the case we will consider in the
following.
We are interested in the reconstruction of the so-called
eigenvalue function [35] or density of states of the Hamil-
tonian operator H. The density of states (DOS) plays a
i Throughout this paper, we refer to coherent and incoherent in
the sense of resource theories [23, 24]. Specifically, we refer to
states as coherent when their density matrix contains non-zero
off-diagonal elements (coherences), and as incoherent when these
off-diagonal terms vanish.
central role in extracting spectral properties of an oper-
ator. It is typically derived from the Fourier transform
of the operator’s unitary time evolution and is essential
for the reconstruction of the microcanonical ensemble of
the system [36].
The DOS technique is a flexible ap-
proach for gathering information about both low- and
high-frequency components.
Consider a Hermitian operator H and its spectral res-
olution, H = P
k θk|k⟩⟨k|, where θk are the eigenvalues
corresponding to the eigenvectors |k⟩. The DOS is the
eigenvalue distribution of the operator H,
S(θ) =
X
k
δ(θ −θk).
(1)
Looking at the spectral decomposition of the Hamilto-
nian H, that is, an ensemble of pure states |k⟩⟨k|, one
might ask if we can extract the DOS function from the
standard routine of phase estimation. This can indeed
be done using the purification theorem. Purification is a
fundamental concept of quantum information theory [13]
that expresses any mixed state as the reduced state of
a larger pure state having a specific entanglement struc-
ture, i.e., that of a maximally entangled state. For any
mixed state ρS on a finite-dimensional Hilbert space HS,
there exists a pure state |ψ⟩SP in a larger Hilbert space
HS ⊗HP such that
ρS = TrP (|ψ⟩⟨ψ|SP ) .
(2)
Here, HP is an auxiliary system (often called the puri-
fying system) with dimension at least equal to rank(ρS).
For simplicity and ease of explanation, we fix the purify-
ing system to the same dimension as the state register.
To implement this approach into standard QPE (see
App. A for the derivation), we need to add an additional
ancillary register that we will refer to as the purification
register. This register matches the state register dimen-
sion, but it will be traced out at the end of the protocol.
By means of this operation, we are effectively evaluating
the trace over the system P in Eq. (2). This will allow
us to translate QPE from a coherent to an incoherent
approach, extending the routine to support further prac-
tical applications. Note that purification is not the only
way to transition between coherent and incoherent QPE.
In case of interest in statistical properties of the systems,
any thermal state preparation routine [37–39] will suffice
as long as the output state can be matched to the state
register.
As a result of the purification, the measurement out-
come y ∈{0, . . . , 2m −1} from the time-frequency regis-
ter yield coarse-grained phase estimates ˜ϕj = y/2m with
probability
P(y) =
X
j
pj|cj(y)|2,
(3)
where pj are the spectral weights of ρS and |cj(y)|2 en-
codes the QPE response kernel. This empirical distribu-
tion reconstructs the coarse-grained DOS at resolution
3
time-freq
QPE
state
purification
Eigv.
MaxMix
DOS-QPE
time-freq
time-freq
Dicke
Figure 1. a) DOS-QPE circuit primitive. In panel a), we show the DOS-QPE as a circuit primitive. The full circuit
consists of three registers: a time-frequency register, a state register, and a purification register. The additional purification
register is maximally entangled to the state register to create the state ensemble. This is done through a cascade of serialized
CNOT operations, CX⊗n
S→P . This last register is traced out at the end of the protocol. b) QPE v. DOS-QPE algorithmic
comparison. In panel b), we compare the standard QPE algorithm to the DOS-QPE algorithm. While QPE relies on the
preparation of good-enough states |ψ⟩with sufficient overlap with the target eigenstates, DOS-QPE allows for probing entire
subspaces via purifications |ψSP ⟩of simple construction. This translates to a cheaper state preparation routine. After measuring
the time-frequency register in QPE, one estimates the phase θ. In DOS-QPE, we measure the time-frequency register and trace
out the purification register to estimate the eigenvalue function S(θ).
∆ϕ = 1/2m. The total reconstruction error scales as
∥ˆρS −ρS∥2 ∼
r
Neff · 2m
M
,
(4)
where Neff is the dimension of the subspace covered by
the ensemble contributing to ρS, and M is the number
of QPE repetitions.
Thus, achieving a target error δ
requires
M ≳Neff · 2m
δ2
.
(5)
For details on the derivation of the standard QPE and
DOS-QPE results, including error analysis and scaling,
see App. A and App. B, respectively.
A.
Maximally mixed state
The most standard example of purification is that of
the maximally mixed state [13]. This purification is easily
achieved via the maximally entangled state
|Φ⟩SP =
1
√
2n
2n
X
k=1
|k⟩S ⊗|k⟩P ,
(6)
where {|k⟩P } is a basis for the auxiliary system HP ∼=
HS. Here and in the following, S and P indicate the state
and the purification registers. Tracing out the system P,
we have
TrP (|Φ⟩⟨Φ|SP ) = I/2n = ρS,
(7)
that is, an ensemble of states covering the entire Hilbert
space. Using this ensemble with flat distribution, we can
probe all of the Hamiltonian’s eigenvalues (phases) with
equal probability and estimate the Hamiltonian’s spec-
tral resolution. The power and potential of using maxi-
mally mixed states in QPE have been shown, under the
umbrella of garbage state preparation and kernel extrac-
tion, in the original proposal of quantum topological data
analysis by Lloyd et al. [15]. In a similar way, but with
the addition of the circuit primitive, DOS-QPE has been
shown capable of generating features required for learn-
ing the frustration index [4].
B.
Dicke states
Given some M−dimensional set of discrete fermionic
modes corresponding to single-particle quantum states,
we can write the resulting 2M-dimensional Fock space as
a direct sum of subspaces Fk,
F =
M
k
Fk = F0 ⊕F1 ⊕· · · ⊕FM.
(8)
The 1-particle quantum states |x⟩can be |0⟩or |1⟩, which
under the second-quantization formalism, means that the
4
state x is completely empty or occupied by a particle,
respectively. A k-body basis state can be written in terms
of the 1-particle quantum states as
|x⟩=
k
O
i
|xi⟩.
(9)
Each term in the decomposition shown in Eq. (8) cor-
responds to the subspace spanned by the set of occupa-
tion number vectors with a fixed particle number, k, of
which there are
M
k
. It is typically the case for fermionic
Hamiltonian simulations that the eigenvalues and eigen-
vectors of interest all carry the same particle number. As
such, using the maximally mixed state as the probe for
DOS-QPE may lead to a spectrum that is too crowded
for certain applications or lead to a needlessly high sam-
pling cost.
A simple way to enforce particle-number symmetry in
the spectrum obtained from DOS-QPE is to choose a
fermion-qubit encoding which maps fermionic states to
qubit states conserving the Hamming weight of the orig-
inal fermionic state after transformation, i.e., the same
number of “1”s (hot-entries) in both fermionic and qubit
states. The two projections of a single qubit represent
a state that is either empty or occupied by a particle,
in this case. The Jordan-Wigner encoding [40] satisfies
these requirements,
|x0, x1, . . . , xM⟩→|q0, q1, . . . , qn⟩,
(10)
where the number of qubits, n, is the same as the number
of fermionic modes, M and xn = qn ∈{0, 1}. To restrict
the eigenvalues to the correct subspace, then, becomes
an issue of preparing, in an even superposition, all com-
putational basis states with Hamming weight k on the
state register before purification. Such states are known
as Dicke states, |Dn
k⟩,
|Dn
k⟩=
n
k
−1
2
X
x∈{0,1}n
|x|=k
|x⟩,
(11)
where n is the number of qubits, and k is the Hamming
weight, which according to the literature can be pre-
pared deterministically with circuit depths of O(k log n
k ),
O(kp n
k ) and O(n) on devices with all-to-all, grid and
linear nearest neighbour connectivities, respectively [41].
Perhaps the most natural application of DOS-QPE
with a Dicke state probe is the extraction of eigenspec-
tra of many-body Hamiltonians. In standard QPE and
many of its extensions, a trial state with a large over-
lap with the target eigenstate must be found either clas-
sically or with some other quantum algorithm, such as
VQE. Preparing such trial states with a variational quan-
tum algorithm can be prohibitively expensive in terms of
sampling costs. Additional difficulties during optimiza-
tion arise from barren plateaus, and it is not possible to
know in advance whether the prepared state has a large
enough overlap with the target state, except for some
heuristic cases. Preparing a state classically and com-
piling it into a state-preparation circuit is often more
practical, though it still poses challenges, particularly in
efficiently encoding the classical state onto the state reg-
ister.
In contrast, Dicke-probed DOS-QPE does not require
preparation of fermionic eigenstates, preserves particle-
number symmetry under the Jordan-Wigner mapping,
and experiments need to be set up only once to recover
the full eigenspectrum. The classical cost incurred before
the execution of the algorithm in the simplest case is just
the cost of a single Hartree-Fock calculation, which scales
polynomially in system size with even naive implementa-
tions, plus the cost of constructing the circuit.
III.
SPECTRUM RECONSTRUCTION
Transitioning from estimating a single phase using
QPE to reconstructing the full spectrum with DOS-QPE
requires a more refined post-processing step. This addi-
tional cost is justified by the richer information retrieved
from he spectrum. DOS-QPE effectively samples from
the normalized density of states,
S(θ) = 1
2n
2n
X
i=1
δ(θ −θi),
(12)
where, as a refinement of Sec. II, we have included the
normalization factor. Here, the eigenvalues θi of H are
each repeated with their multiplicity. In practice, with
only m qubits in the time-frequency register, one obtains
a discretized histogram
Pi =
Z
Ii
S(θ)dθ + (noise),
i = 0, 1, . . . , 2m −1,
(13)
where the Ii are adjacent intervals of width ∆θ = 2π/2m.
The task of spectrum reconstruction is to recover both
the continuous eigenvalues locations θi ⊂R and their
degeneracies di ⊂Z≥0 from the sample histogram P =
{Pi}.
This problem is challenging for several reasons:
(i) finite resolution due to limited register size m, which
determines the bin width ∆θ; (ii) noise, including sta-
tistical fluctuations of order O(n−1/2); (iii) unresolved
spectral gaps and an unknown number of distinct eigen-
values. While some of these challenges will result in in-
evitable inaccuracy of the reconstructed signal, we will
resort to techniques such as compressed sensing and con-
vex optimization to efficiently find solutions.
A.
The degeneracy integer problem
With access to the readout of the time-frequency reg-
ister of the DOS-QPE circuit, we aim to estimate the
spectral resolution, H = P
k θk|k⟩⟨k|. To do so, we face
5
the problem of generating the best spectral estimate out
of the readout probability distribution P. We refer to this
estimation task as the degeneracy integer problem, falling
under the umbrella of spectral density estimation [42].
The problem consists of assigning a set of normalized
degeneracies { ˜di}i, subject to P
i ˜di = P
i di/ dim(H) =
1, to minimize a suitable distance D(P, ˜P) between the
target probability distribution P and the reconstructed
spectrum ˜P.
While the ideal reconstruction would be
P(θ) = P
i diδ(θ −θi), the smeared, finite measured
distribution will be a convolved distribution ˜P(θj) =
P
i diK(θj −θi), where K is the kernel and θj is a mea-
surement bin determined by the finite time-frequency
register. Since the readout distribution consists of prob-
abilities, we work with the squared kernel
K(θ) = |DM(θ)|2 =
sin(πMθ)
M sin(πθ)
2
,
θ ∈[−1
2, 1
2]
(14)
where the Dirichlet kernel DM arises naturally from the
discrete Fourier transform over M = 2m time steps used
in the DOS-QPE protocol. Note that the dividing fac-
tor M normalizes the kernel integral to 1 in the large-M
limit, ensuring the convergence of DM to a Dirac delta
in the same limit. The final observed probability distri-
bution that we aim to reconstruct is
˜P(θ) =
R
X
i=1
di · K(θ −θi).
(15)
B.
Compressed sensing and optimization
To recover the discrete phases {θi} and their degenera-
cies {di} from the noisy coarse-grained histogram {Pi},
we reframe the degeneracy integer problem as an inverse
problem.
The goal is to infer a sparse spectral signal
— convolved with a known kernel — from its noisy,
finite-resolution observation. This setting naturally in-
vites tools from convex optimization to minimize the dis-
crepancy between the observed histogram and a model
spectrum.
We formulate the problem on a grid of candidate
phases {θ′
j} ⊂[0, 1), finer than the hardware-limited
resolution 2−m. We obtain an underdetermined system
where the number of potential spectral components ex-
ceeds the number of measured bins. To select a physi-
cally meaningful sparse solution, we employ a quadratic
program [43] with an ℓ1 regularization penalty (basis-
pursuit, LASSO) [44], which encourages sparsity by pe-
nalizing the absolute sum of the degeneracy coefficients.
This results in a practical realization of compressed sens-
ing [21, 22], a framework for recovering sparse signals
from limited or noisy measurements.
The finer grid
provides an overcomplete dictionary of possible phases,
while the ℓ1 penalty enforces that only a small subset
is selected. The resulting optimization routine enables a
super-sampled spectral reconstruction despite a coarser
hardware discretization [45].
We introduce a uniform grid of G candidate phases
{θ′
i}G
i=1, producing the dictionary matrix A ∈RN×G:
Ai,j = K(θi −θ′
j).
(16)
Writing P = (Pk)N
k=1 and w = (wj)G
j=1 as the degeneracy
coefficients for the grid points, the forward model is
P ≈Aw.
(17)
The problem naturally maps onto a mixed-integer
quadratic program [46, 47] with convex, non-linear ob-
jective function L2,MI
=
minw∈N+ |Aw −P|2
2 with
PG
j=1 wj
=
2n.
Quadratic programming is NP-
complete [48] in general, with the variant of mixed-
integer programming (MIP) enforcing wj ∈Z+ and
P
j wj = P
i di = 2n being NP-hard [49–51]. For this
reason, we relax the problem from integer degeneracies
to real and add a further final step of clustering of phases
and integer approximation on the solution. This makes
the problem a quadratic program with a convex objec-
tive function, which, together with linear programs, is
in P [50–52]. The final objective function we optimize
against is
L2 = min
w∈RG
+
|Aw −P|2
2 + λ|w|1
s.t.
G
X
j=1
wj = 2n, (18)
where we added the ℓ1 regularization penalty controlled
by λ > 0. The squared-L2 term ensures data fidelity, and
the ℓ1 term promotes sparse support in the candidate-
phase weights. This convex QP can be solved efficiently
at moderate G using off-the-shelf solvers. In our simula-
tions, we do this using the CLARABEL solver [53] and
CVXPY [54, 55].
Note that, because the quadratic part of L2 is piecewise
linearizable, we can also reformulate the problem as a
linear program L1 = minw,r(r + λw) such that |Aw −
P| < r with w ∈RG
+ and PG
j=1 wj = 2n.
The solution w∗of Eq. (18) is typically sparse in
the Hilbert space of dimension 2n.
Consider the vec-
tor J = j : w∗
j > τ of those phase indices whose solution
degeneracy is above a certain target threshold τ. To re-
cover the final phases and integer degeneracies, we apply
the following steps:
1. Thresholding. We retain grid indices in J and their
weights w∗
j .
2. Clustering.
We merge nearby phases ϕj within
a tolerance ϵ via single-linkage, forming clusters
C1, . . . , Cmax J. This ϵ can be taken of the order
of the initial grid size of DOS-QPE defined by the
time-frequency register dimension.
3. Rounding. For each cluster C, we compute
ˆθ =
P
j∈C w⋆
j ψj
P
j∈C w⋆
j
,
ˆd =
X
j∈C
w⋆
j
.
(19)
6
In this way, we restore integer degeneracies and obtain a
final estimate {ˆθi}max J
i=1
with corresponding { ˆdi}max J
i=1
.
To quantify the reconstruction error, we use the 1-
Wasserstein distance [56] W1(µ, ν) between discrete spec-
tral distributions µ = P
i di δθi and ν = P
j ˆdj δˆθj, where
δθ denotes the Dirac measure at θ. This metric is defined
as
W1(µ, ν) =
inf
γ∈Γ(µ,ν)
X
i,j
γij |θi −ˆθj|,
(20)
where Γ(µ, ν) is the set of all couplings between µ and ν
that preserve total weight. Intuitively, this metric quan-
tifies the dissimilarity between the two distributions by
measuring the minimum amount of ”mass” times ”travel”
that needs to be moved to shape one into the other [57].
In the following, we compute W1 between the sampled
spectrum and the exact one, as well as between the opti-
mized solution and the exact spectrum. This provides a
principled way to evaluate the fidelity of spectral estima-
tion, taking into account both eigenphase positions and
degeneracies.
IV.
APPLICATIONS
In the following, we show three applications of DOS-
QPE: the Fermi-Hubbard model, an electronic struc-
ture Hamiltonian, and a nuclear Hamiltonian. We use
t|ket⟩[58] to construct the circuits, Qulacs for state vec-
tor evaluation [59], and OpenFermion for constructing
qubit Hamiltonians [60]. To evaluate the molecular in-
tegrals in the electronic structure section, we use the
PySCF Python package [61].
A.
The Fermi-Hubbard Model
The Fermi-Hubbard model [62–65] is one of the stan-
dard models of condensed matter physics as it describes
relevant features of strongly-correlated electronic sys-
tems.
It has also become a standard benchmark for
Hamiltonian simulation algorithms [66–68] in the quan-
tum computing community since exact solutions can be
found for the one-dimensional case [69].
The study of
the two- and higher-dimensional cases is a topic of in-
tense research activity, where only numerical simulations
are available [70–72]. Its Hamiltonian is written as
H = −t
X
⟨i,j⟩,σ
a†
iσajσ + U
X
i
ni↑ni↓,
(21)
where the quantity σ =↑, ↓denotes spin projection,
niσ = a†
iσaiσ is a particle-number operator, the symbol
⟨i, j⟩restricts the sum over the indices of neighboring
lattice sites and t and U are the hopping and repulsion
parameters, respectively.
In the remainder of this section, we apply the DOS-
QPE algorithm to the Fermi-Hubbard chain model with
0.00
0.06
0.12
0.19
0.25
0.31
0.38
0.44
0.50
0.56
0.62
0.69
0.75
0.81
0.88
0.94
Phase
0.00
0.05
0.10
0.15
0.20
Probability / Degeneracy/2n
1/2n
5/2n
10/2n
FH, probe: MM-state
DOS-QPE
Spectrum
Optim
Figure 2. DOS-QPE on the Fermi-Hubbard model. We
run the DOS-QPE circuit on a Fermi-Hubbard toy model with
three sites, t = 1, and U = 4, employing m = 6 ancillas to re-
solve the spectrum. In the time-frequency register, we obtain
the probability distribution of the DOS (blue bars). We com-
pare this with the exact spectrum (black bars) of the Hamilto-
nian. While the dimensionality of the time-frequency register
is insufficient to resolve the spectral gaps of the spectrum, we
show that quadratic programming aided by compressed sens-
ing helps in the phase and degeneracy reconstruction (red
stars). The 1-Wasserstein distance of the optimized solution
improved by ∼51% over the integer-rounded sampled distri-
bution.
three sites.
We choose t = 1 and U = 4 such that
we are in the so-called “intermediate coupling” regime,
under which it is difficult classically to extract even
ground-state properties from numerical simulations at
half-filling [73]. We can then map this second-quantized
Hamiltonian, using the Jordan-Wigner encoding, to a lin-
ear combination of products P of Pauli operators σj,
H =
X
i
hiPi =
X
i
hi
O
j
σi
j,
(22)
where hi are the coefficients in the transformation.
We run the DOS-QPE circuit of Fig. 1a) and show the
results in Fig. 2. For this simulation, we have state and
purification registers of dimension n = 6, and we try to
resolve the spectrum with a time-frequency register of di-
mension m = 6, for a total of 18 qubits. The Hamiltonian
is rescaled following App. C and the unitary propagation
is done via fourth-order Trotterization. Comparing the
DOS S(θ) of the Hamiltonian H to its exact spectrum, we
note that the dimensionality of the time-frequency regis-
ter is insufficient for a perfect spectrum reconstruction.
That is why we employ the quadratic program and com-
pressed sensing technique explained in Sec. III B. Using
such convex optimization, we can obtain a better esti-
mate of the integer degeneracies and the real phases of
the spectrum. As described in Sec. III B, we use the 1-
Wasserstein distance against the exact spectrum to quan-
tify the deviation of the distributions, finding ∼16% im-
provement over the sampled distribution and more than
∼51% over the corresponding integer-rounded version.
7
B.
Electronic Structure
The application of quantum algorithms to chemistry
typically begins with the construction of the system
Hamiltonian, written in the second-quantization formal-
ism as,
H =
X
pq
hpqa†
paq + 1
2
X
pqrs
hpqrsa†
pa†
qasar.
(23)
The constants hpq and hpqrs are the integrals,
hpq =
Z
dxϕ∗
p(r)
−∇2
2 −
X
A
ZA
|r −RA|
!
ϕq(r)
(24)
and
hpqrs =
Z
dr1dr2
ϕ∗
p(r1)ϕ∗
q(r2)ϕr(r2)ϕs(r1)
|r1 −r2|
,
(25)
respectively, and are routinely obtained as part of a clas-
sically inexpensive Hartree-Fock calculation. The quanti-
ties ϕp are the molecular orbitals and form the 1-particle
wavefunctions mentioned in Sec. II B, r are the positions
of the electrons, and ZA and RA are the charges and po-
sitions of the nuclei, respectively. For the same reasons
as in Sec. II B and the former section, the Jordan-Wigner
mapping is selected.
In most classical computational chemistry studies, the
number of electrons in a system remains fixed. Descrip-
tions of many chemical phenomena are framed in terms of
energy differences as the nuclear configuration changes,
or between eigenstates at some fixed geometry.
Even
in cases where the number of electrons varies, the full
spectrum for all eigenvalues of all occupation numbers is
rarely required. As such, the Dicke state |Dn
k⟩, where k
is set to the number(s) of electrons in the system, is the
most relevant probe for the DOS-QPE algorithm as it is
applied in this section.
We choose the twisted conformation of ethylene as the
example chemical system, since by varying the pyrami-
dalization angle we expect to encounter a conical inter-
section between the two lowest lying singlet eigenstates,
and a crossing of the lowest lying singlet state with the
triplet ground state [74]. We first optimise the planar ge-
ometry at the CCSD(T) level classically in the cc-pVDZ
basis, then rotate the two hydrogens on one carbon atom
to have dihedral angles with the hydrogen atoms on the
other carbon atom of exactly 90◦. We then vary the an-
gle α as shown in Fig. 3a). We select a 2-electrons in 2-
orbitals active space, and along the reaction coordinate,
evaluate the overlap of the orbitals at each point with the
orbitals from the previous point, such that we can place
the orbitals at maximum coincidence at each step along
the potential energy surfaces and ensure a consistent or-
bital selection. Such that the eigenvalues lay in a fixed
interval and the movement of each peak in Fig. 3c–f) is
smooth, and to obtain a smooth curve in Fig. 3b), we
find our target Hamiltonians with the following formula,
Htarget = (1 −δ)
H −λmin,50◦+ δ
λmax,50◦−λmin,50◦
,
(26)
where in all cases δ = 0.05, which serves as a shift to move
the lowest phase away from zero and simplify communi-
cation regarding Fig. 3, and λmin,50◦and λmax,50◦are the
lowest and highest lying 2-particle eigenvalues obtained
by exact diagonalization of the geometry at α = 50◦,
respectively.
The results for exact diagonalization of the active space
Hamiltonian are shown in Fig. 3b). We observe that the
conical intersection is at approximately 68◦. We then se-
lect three evenly spaced points around the approximate
location of the conical intersection, at 65◦, 71◦, and 74◦,
at which we perform state-vector DOS-QPE calculations.
We use second-order Trotterization with one time step for
the preparation of UH and, as previously mentioned, set
Uψ to be the |D4
2⟩state-preparation unitary.
We also
marginalise the exact probabilities of measurement out-
comes on the state and purification qubit registers such
that the results we obtain correspond to the exact prob-
abilities of the measurement outcomes on the ten ancilla
qubits.
By restricting the probe to being an even superposi-
tion of only 4C2 states, we expect to observe a maxi-
mum of 6 distinct eigenvalues. Spin symmetry simpli-
fies the picture since we know that for any triplet state
with total spin S = 1, there are three degenerate states
with Ms = −1, 0, 1. We can infer the multiplicity of the
states corresponding to the phases, then, from the rela-
tive peak heights in the spectra shown in Fig. 3c–f): each
contains three degenerate triplet eigenstates with differ-
ent Ms values, and three peaks corresponding to singlet
states which have no degeneracy by spin symmetry, to-
taling six states as expected.
In what remains of this
section, we will refer to the lowest and second-lowest sin-
glet state as S0 and S1, respectively, and to the triplet
state as T. The other singlet state (marked with a black
dashed line) is not of any interest in this discussion but
is included in the figure for completeness.
We see by comparing the phases shown in Fig. 3b) and
the peaks in Fig. 3c) that we gain access to the correct
ordering of states, the phases agreeing with exact diag-
onalization. Increasing α to 68◦(see Fig. 3d)), close to
the conical intersection between S0 and S1, we see near
degeneracy between the phases corresponding to the S0
and S1 states, as expected, and an increase in the phase
found for the T state. Further increasing α to 71◦, shows
the separation between the phases corresponding to S0
and S1 return, as is the case in Fig. 3b). The distribution
shown in Fig. 3f) shows that the phases corresponding to
T and S0 have crossed, and the ground state now carries
singlet multiplicity, as expected from the results of the
exact diagonalization.
From the faithful reconstruction of the spectra along
the potential energy surfaces, we conclude that DOS-
QPE represents a valuable tool for the fault-tolerant era
8
Figure 3. DOS-QPE on the electronic structure of ethylene. a) The structure of the ethylene molecule showing a twisted
conformation and the angle α which is varied during the scan of the potential energy surfaces. b) Plot of the lowest three
eigenvalues of the Hamiltonian, scaled such that the eigenvalues lie in the interval [0, 1). The green line corresponds to the
triplet state T, the orange line to the lowest lying singlet state S0 and the purple line to the first excited singlet state S1. c–f)
The distributions obtained with DOS-QPE with α = 65◦, 68◦, 71◦and 74◦, respectively. In each plot c–f), the colour of each
vertical dashed line corresponds to the phases obtained via exact diagonalization and matches those used in b). The height of
the dashed lines shows the height the peaks in the distribution would be if all eigenvalues were to be exact integer multiples of
1
Nanc .
of quantum computation for the study of chemical sys-
tems. In the form it is presented here, we anticipate DOS-
QPE being particularly useful for investigating phenom-
ena arising directly from more than one eigenstate at any
given molecular geometry. An interesting next step for
this approach is to leverage other known physical sym-
metries to construct more targeted probe states.
C.
Nuclear Structure
The description of nuclear structure in atomic nuclei
follows a similar prescription as that of molecules in quan-
tum chemistry within the nuclear shell model [75]. Under
this approach, protons and neutrons (collectively called
nucleons), basic fermion components of a nucleus, move
in a set of single-particle states that form a valence space.
These single-particle states, also called orbitals, are la-
beled by the quantum numbers (n, l, j, m, tz), where n is
the principal quantum number, l is the orbital angular
momentum, j is the total angular momentum after cou-
pling with the nucleon spin−1
2, m is its third-component
projection and tz is the third-component projection of
the isospin t = 1
2, an additional quantity equivalent to
the spin defined in order to discern between protons and
neutrons.
The nuclear shell model can be described by means
of a second-quantized Hamiltonian as in Eq. (23), where
the single-particle energies hpp and two-body matrix el-
ements hpqrs are adjusted to reproduce observable prop-
erties of selected nuclei [76]. The nuclear states are de-
scribed by a well-defined total angular momentum J and
isospin T, resulting from the coupling of individual nucle-
onic j and t; with their corresponding third-component
projections M and Tz, which are equal to the sum of the
third components of the nucleons in the valence space, m
and tz. Thus, a many-body basis state can be written in
terms of states with well-defined M and Tz
|JMTTz⟩=
X
i
ci |i, MTz⟩,
(27)
where i labels additional quantum numbers needed for
the description of the many-body state, and ci are the co-
efficients in the superposition. The Hamiltonian matrix
is then built using this many-body basis, and the nuclear
states can be found as its eigenstates. Because of the two
kinds of particles in a nucleus, protons and neutrons,
the scaling of this many-body basis is as
dsh
Nval
dsh
Zval
,
where dsh is the dimension of the valence space and Nval,
Zval are the valence neutron and proton numbers, respec-
tively.
Many quantum algorithms have been applied to ob-
tain nuclear-structure properties based on the ground
state [77–84] and also low-lying excited states [85–88].
The structure and entanglement features of nuclear states
have also been thoroughly studied [89–92].
In this work, we aim to resolve the full spectrum in a
single calculation with DOS-QPE. We simulate the 6Be
nucleus, which is composed of two protons in the valence
space of a p shell, as shown in Fig. 4, with six single-
particle states for the protons. Thus, we have in this case
two identical particles moving in the valence space, and
consequently, the total isospin is T = 1. The Hamiltonian
used is based on the phenomenological Cohen-Kurath in-
9
Label Many-body state
1
2
3
4
5
(0, 3)
(1, 2)
(1, 5)
(2, 4)
(4, 5)
Figure 4. a) Many-body state basis used for 6Be. La-
bel represents the indices in the corresponding Hamiltonian
matrix in this many-body basis. The many-body states are
composed of single-particle states from all possible combina-
tions of two valence protons in the p−shell variational space
of panel b) satisfying the condition M = m1 + m2 = 0. b)
Configurational valence space for the p−shell. The va-
lence space is decomposed in two degenerate j orbitals with
degeneracy (2j + 1).
The labels on the left correspond to
the standard spectroscopic notation nℓj, where ℓ= p means
ℓ= 1. Numbers on top of every single-particle state represent
the labels used in this work. Slightly modified from [80].
teraction [93].
While
we
could
directly
transform
the
second-
quantized Hamiltonian using the Jordan-Wigner map-
ping as demonstrated in the former examples, for this
specific case we opt to build the Hamiltonian matrix us-
ing the minimal many-body basis spanned for 6Be, con-
sisting of only five states whose total angular momentum
projection M is equal to zero [80]. The many-body basis
used for 6Be is listed in the table of Fig. 4a). The result-
ing Hamiltonian matrix is then mapped onto the qubit
operator basis using the Linear Combination of Unitaries
(LCU) formalism [94].
Thus, using this mapping, we
managed to reduce the size of the Hamiltonian from six
to three qubits, at the expense of three additional degen-
erate eigenvalues equal to zero.
The results of the DOS-QPE calculation for this sys-
tem are shown in Fig. 5. The calculation was performed
using a sixth-order Trotterization of the unitary propaga-
tion and five ancillary qubits in the time-frequency regis-
ter for the spectrum resolution. We observe that, under
these resources, four out of five physical phases (eigen-
values) of 6Be are retrieved within a precision dictated
by the number of ancillary qubits used (see App. B).
The resolution for the almost degenerate peaks at ∼0.65
is not enough and the number of ancillary qubits in the
time-frequency register needs to be increased. A spurious
zero eigenvalue with degeneracy of three is also found as
an LCU artifact used to map the Hamiltonian to a qubit-
operator basis.
In light of these results, although demonstrated on a
small example, we may conclude that DOS-QPE could
become a promising tool for efficiently extracting spec-
tral information and thus dynamical processes [95–99] in
atomic nuclei for near-term quantum devices.
0.00
0.06
0.12
0.19
0.25
0.31
0.38
0.44
0.50
0.56
0.62
0.69
0.75
0.81
0.88
0.94
Phase
0.0
0.1
0.2
0.3
0.4
0.5
Probability / Degeneracy/2n
1/2n
2/2n
3/2n
6Be, probe: MM-state
DOS-QPE
Spectrum
Figure 5. DOS-QPE on 6Be. The result of DOS-QPE with
a maximally mixed state probe.
Sixth-order Trotterization
of the unitary evolution and five qubits in the time-frequency
register were used for these results. We observe that, while the
distribution (blue bars) faithfully captures the exact eigen-
spectrum, it is not fine-grained enough to resolve the almost-
degenerate peaks at phase ∼0.65, and more ancillary qubits
would be needed. The tallest peak represents the redundant
three-times degenerate zero eigenvalue resulting from build-
ing the nuclear Hamiltonian matrix using the LCU approach.
V.
DISCUSSION AND CONCLUSION
We have shown how, by using a generalization of quan-
tum phase estimation probed with maximally mixed or
symmetry-adapted ensemble states, we can access spec-
tral information of the generating Hamiltonians. By em-
bedding QPE within a purified framework [4], the tradi-
tional eigenstate-dependent algorithm transforms into a
flexible tool for ensemble-based spectral analysis. This
approach grants access to the density of states of Hamil-
tonians, effectively revealing their spectral properties.
From here, thermodynamic and dynamical properties of
systems become accessible while mitigating the effects of
noisy or imperfect state preparations.
We framed the recovery of discrete spectral features
from the output of DOS-QPE as a convex optimiza-
tion problem. We resolve spectral structures exploiting
sparsity and physical constraints to produce high-fidelity
reconstructions. We propose a modular postprocessing
that restores physically meaningful integer-value spectra.
While we have demonstrated DOS-QPE as a flexi-
ble primitive for ensemble-based quantum spectroscopy
across fermionic and nuclear models, optimizations or
resource reductions might be desirable.
One immedi-
ate goal is to reduce circuit complexity and resource
cost, tailoring the primitive for currently available quan-
tum hardware [100].
Implementing single-ancilla vari-
ants [39, 101, 102] or taking advantage of shadow tech-
niques [103–105] could significantly lower circuit depth
and overhead, making the primitive viable for near-term
applications.
Further directions include exploring alternative probe
states, such as spin-selective or symmetry constrained
10
ensembles, which may target subspace-resolved spectral
features often inaccessible to conventional methods. In
this context, developing spectrum manipulation tech-
niques to enhance signal contrast, reduce costs, and miti-
gate noise could be particularly valuable. Such strategies
would accelerate the reconstruction process and allevi-
ate the current bottlenecks in measurement and post-
processing.
DOS-QPE shows potential as a framework that by-
passes state preparation bottlenecks while taking advan-
tage of the incoherent nature of the environment to re-
turn physically relevant observables otherwise inacces-
sible.
Connections to non-linear functions of spectral
densities, gradient estimation, and Bell-type measure-
ments [106] indicate a broader algorithmic utility within
hybrid optimization and quantum learning frameworks.
Embedding DOS-QPE within Green’s functions formula-
tions [107, 108] could further help in dynamical response
theory offering access to electronic, vibrational, or nu-
clear observables.
Appendix A: QPE - derivation
Given a unitary operator U and an eigenvector |ψ⟩such
that
U|ψ⟩= e2πiϕ|ψ⟩,
ϕ ∈[0, 1),
(A1)
the goal is to estimate ϕ to m-bit precision. The registers
we find in a standard implementation of QPE are
• Time-frequency register: m qubits, initialized to
|0⟩⊗m.
• State register:
n qubits, initialized to a target
eigenvector |ψ⟩.
The entire initial state facing the circuit is,
|Ψ0⟩= |0⟩⊗m ⊗|ψ⟩.
(A2)
The first operation of QPE is the application of a layer of
Hadamard gates H⊗m to the first register to create the
uniform superposition,
|Ψ1⟩=
1
√
2m
2m−1
X
k=0
|k⟩
!
⊗|ψ⟩.
(A3)
We then apply the unitary operation UQPE, consisting of
a controlled-U 2j operation for each qubit j in the time-
frequency register, |Ψ2⟩= 2−m/2 P2m−1
k=0
|k⟩⊗U k|ψ⟩.
Since U k|ψ⟩= e2πikϕ|ψ⟩, this results in
|Ψ2⟩=
1
√
2m
2m−1
X
k=0
e2πikϕ|k⟩⊗|ψ⟩.
(A4)
We then apply the inverse quantum Fourier transform
U †
QFT to the time-frequency register. The Fourier basis
state 2−m/2 P2m−1
k=0
e2πikϕ|k⟩is mapped to the computa-
tional basis state |˜ϕ⟩, where ˜ϕ is the best m-bit approx-
imation of ϕ. If ϕ has an exact m-bit binary expansion
ϕ = 0.ϕ1ϕ2 . . . ϕm, then
U †
QFT
1
√
2m
2m−1
X
k=0
e2πikϕ|k⟩
!
= |ϕ1ϕ2 . . . ϕm⟩.
(A5)
The final state of the system is |Ψfinal⟩= |˜ϕ⟩⊗|ψ⟩and
measuring the time-frequency register gives ˜ϕ, the esti-
mate of ϕ.
Error
Note that, when ϕ cannot be exactly represented with
m bits, we have
ϕ = s
2m + δ,
with
|δ| <
1
2m+1 .
(A6)
The probability of measuring s, the nearest m-bit ap-
proximation to ϕ, is
Psuccess ≥
sin(πδ2m)
2m sin(πδ)
2
,
(A7)
that is exactly 1 if δ = 0, while, for small δ, the phase
estimation error is bounded by
|˜ϕ −ϕ| ≤
1
2m+1
(with high probability).
(A8)
Appendix B: DOS-QPE - derivation
Consider the case where the input to QPE is not an
eigenstate of U, but rather an arbitrary mixed state
ρS =
N
X
j=1
pj |ψj⟩⟨ψj|,
(B1)
with {|ψj⟩} the eigenbasis of U satisfying U|ψj⟩=
e2πiϕj|ψj⟩. We purify ρS by introducing an auxiliary reg-
ister of the same dimension. The purification is
|Ψ⟩SP =
N
X
j=1
√pj |ψj⟩S ⊗|j⟩P ,
(B2)
where {|j⟩}P is an orthonormal basis of the purification
register.
For simplicity, we drop the subscripts in the
following. The total system includes three registers:
• Time-frequency register: m qubits, initialized to
|0⟩⊗m.
• State register: n qubits, initialized to the system of
interest.
11
• Purification register: n qubits, entangled with the
state register to create the purification. This regis-
ter is to be traced out at the end.
The initial state is
|Ψ0⟩= |0⟩⊗m ⊗
N
X
j=1
√pj |ψj⟩⊗|j⟩
.
(B3)
Applying Hadamard gates H⊗m to the time-frequency
register creates the superposition
|Ψ1⟩=
1
√
2m
2m−1
X
k=0
|k⟩
!
⊗
N
X
j=1
√pj |ψj⟩⊗|j⟩
. (B4)
We then apply the controlled-unitary operation UQPE,
where each |k⟩controls U k on the state register. Since
U k|ψj⟩= e2πikϕj|ψj⟩, the resulting state is
|Ψ2⟩=
1
√
2m
2m−1
X
k=0
|k⟩⊗
N
X
j=1
√pj e2πikϕj |ψj⟩⊗|j⟩. (B5)
Next, we apply the inverse quantum Fourier transform
U †
QFT to the time-frequency register. The Fourier trans-
form maps
1
√
2m
2m−1
X
k=0
e2πikϕj|k⟩7→
2m−1
X
y=0
cj(y) |y⟩,
(B6)
where
cj(y) = 1
2m
2m−1
X
k=0
e2πik
ϕj−y
2m
,
(B7)
where each outcome y ∈{0, . . . , 2m −1} corresponds to
a phase estimate ˜ϕj = y/2m. After the inverse Fourier
transform, the state becomes
|Ψ3⟩=
N
X
j=1
√pj
2m−1
X
y=0
cj(y) |y⟩⊗|ψj⟩⊗|j⟩.
(B8)
By tracing out the purification register, the measurement
of the time-frequency register gives the outcome y with
probability
P(y) =
N
X
j=1
pj
cj(y)
2,
(B9)
which corresponds to sampling from the spectral measure
of ρS under U, coarse-grained to resolution ∆ϕ = 1/2m.
Each measurement outcome y yields a phase estimate
˜ϕj = y/2m, and the empirical distribution of ˜ϕj recon-
structs the density of states of U.
Error and sampling
Three main sources of error affect this procedure:
a.
Resolution error
The Fourier truncation limits
the phase resolution to
∆ϕ = 1
2m .
(B10)
b.
Statistical sampling error
Each DOS-QPE run
yields one sample from the spectral distribution. After
M shots, the per-bin standard deviation is defined by the
binomial sampling error bound, that is
ϵ ≈
r
p(1 −p)
M
≤
1
2
√
M
,
(B11)
assuming worst-case p = 0.5. The total L2 error in the
reconstructed histogram ˆρS is
∥ˆρS −ρS∥2 ∼
r
B
M =
r
2m
M ,
(B12)
where ˆρ is the empirical distribution (histogram) ex-
tracted and B = 2m is the number of bins in such distri-
bution. To achieve a total error δ, the number of shots
must satisfy
M ≳2m
δ2 .
(B13)
c.
Scaling with Hilbert space dimension
Let Neff de-
note the number of eigenstates contributing non-trivially
to ρS:
• For a maximally mixed state, ρS = I/2n, so Neff =
2n, and the DOS-QPE samples the full spectrum
of U.
• For a restricted ensemble, such as the uniform mix-
ture of Dicke states, the support of ρS is reduced.
For example, fixed Hamming-weight Dicke states
span a subspace of dimension
n
k
∼O(nk).
• For a pure eigenstate, Neff = 1, and the DOS-QPE
output is that of standard QPE, i.e. a sharp peak
centered at that eigenvalue.
The effective sampling variance thus scales with the
number of contributing eigenstates. Assuming uniform
pj = 1/Neff, the variance of the spectral histogram is
Var[P(y)] ∼
Neff
M · 2m ,
(B14)
and the total error scales as
∥ˆρS −ρS∥2 ∼
r
Neff · 2m
M
.
(B15)
Hence, to achieve error δ, the number of measurements
must satisfy
M ≳Neff · 2m
δ2
.
(B16)
This highlights the trade-off between resolution, state
support, and sampling complexity. Choosing input states
with reduced spectral support (e.g., Dicke states) allows
one to probe restricted subspaces of the spectrum effi-
ciently.
12
Appendix C: Rescaling the Hamiltonian
For DOS-QPE, the input Hamiltonian must have a
spectrum that is real, non-negative, and bounded in the
interval [0, 1). Suppose we are given a Hamiltonian H
with real eigenvalues λi ∈[λmin, λmax]. We construct a
rescaled Hamiltonian ˜H such that its eigenvalues lie in
[0, 1) as follows:
˜H = H −λminI
∆
,
(C1)
where ∆= λmax −λmin. The resulting eigenvalues are
˜λi = λi −λmin
∆
∈[0, 1).
(C2)
This procedure requires prior knowledge of both λmin and
λmax.
The unitary evolution generated by the rescaled Hamil-
tonian is
˜U = exp
2πi ˜H
= exp
2πi
∆(H −λminI)
.
(C3)
Running DOS-QPE with ˜U returns phases ˜λi from which
the original spectrum can be recovered via
λi = λmin + ∆˜λi.
(C4)
To simulate ˜U via Trotterization, we decompose ˜H into
local terms ˜Hj and approximate the unitary evolution by
˜U ≈
Y
j
exp
2πi∆t
r
˜Hj
r
,
(C5)
where r is the Trotter order and ∆t is the total evolution
time. For first-order Trotterization, this becomes
˜U ≈exp
2πi∆t
∆H
· exp
−2πi∆t
∆λmin
.
(C6)
The first term corresponds to evolution under the orig-
inal Hamiltonian H with a rescaled time ∆t/∆, while
the second is a global phase that factors out in DOS-
QPE. However, this global phase must be accounted for
when converting the measured phases back to the original
eigenvalues.
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Spectral subspace extraction via incoherent quantum phase estimation Stefano Scali¶,1,2,∗Josh Kirsopp¶,1,† Antonio M ́arquez Romero3,‡ and Micha l Krompiec1§ 1Fujitsu Research of Europe Ltd., SL1 2BE Slough, U.K. 2 4 4QL, UK 3Fujitsu Research of Europe Ltd., Pozuelo de Alarc ́on, 28224 Madrid, Spain Quantum phase estimation (QPE) is a cornerstone algorithm for extracting Hamiltonian eigenvalues, but its standard form targets individual eigenstates and requires carefully prepared coherent inputs. To overcome these limitations, we adopt an ensemble-based formulation of QPE that estimates the density of states (DOS) of the Hamiltonian generator of the evolution. This approach, which we refer to as DOS-QPE, builds on a prior formulation introduced by one of the authors. In this work, we present DOS-QPE as a circuit primitive, extending it with symmetry-adapted input ensembles and advanced spectrum reconstruction techniques. This variant of QPE enables natural access to thermodynamic properties, symmetry-resolved spectral functions, and features relevant to quantum many-body systems. We demonstrate its performance on fermionic models and nuclear Hamiltonians by casting the spectrum reconstruction problem as a quadratic program solved via compressed sensing. These use cases highlight the potential of DOS-QPE for early fault-tolerant quantum simulations in spectroscopy, electronic structure, and nuclear theory. I. INTRODUCTION Estimating the spectral properties of quantum systems is a fundamental problem with wide-ranging applications in physics, chemistry, and quantum computing. In particular, the density of states (DOS) - the distribution of energy eigenvalues of a quantum Hamiltonian - encodes crucial information about both the thermodynamic behavior [1] and the dynamical response of a system [2]. Beyond its traditional role in condensed matter physics and statistical mechanics, the DOS has recently been used in hybrid classical-quantum frameworks for quantum topological data analysis [3] and to address the frustration index problem in signed graphs [4]. Quantum phase estimation (QPE) remains the gold standard for extracting eigenvalue information in the form of quantum phases [5-8]. It is typically applied to eigenstates of unitary evolution operators or their generators, and its success depends on the overlap between the prepared trial state and the true eigenstate of the Hamiltonian. However, this requirement can be prohibitive in practice. Preparing trial states with significant overlap may demand deep, problem-specific quantum circuits, especially for strongly-correlated systems [9-11]. In addition, real quantum hardware inevitably interacts with its surrounding environment, complicating further the preparation of single pure states. This raises the natural question of how compatible QPE is with mixed states, thermal states, and noisy, non-pure ensembles [12]. While purification techniques [13, 14] are wellestablished in quantum information, the use of QPE on purified states has not yet been systematically studied as ∗ ; † ‡ § ¶ These authors contributed equally to this work. a circuit primitive. Notable exceptions include the use of garbage states in quantum topological data analysis [15] and recent work leveraging QPE on maximally mixed states to compute Hamiltonian moments in the context of signed graphs [4]. While a promising alternative approach for estimating the DOS using Chebyshev moments has been recently proposed [16], this work focuses on developing a framework centered around QPE. This not only provides access to a rich landscape of algorithmic variants, but also positions our method for reuse as a subroutine in a range of QPE-based quantum algorithms-including Shor's algorithm [17], quantum simulation in quantum chemistry [18], and the HHL algorithm for solving linear systems [19]. In this work, we generalize and refine a QPE-based method for DOS estimation (DOS-QPE) [4], presenting it as a circuit primitive. We showcase some of its key applications, ranging from the Fermi-Hubbard model to molecular systems and nuclear physics Hamiltonians. We begin with a review of standard QPE techniques and the modifications required to apply them to purified states. We then discuss suitable mixed state preparations and rescaling of the Hamiltonian. We propose a method to resolve the problem of integer degeneracies and, more generally, tackle spectrum reconstruction using the tools from quadratic programming [20] and compressed sensing [21, 22]. Finally, we present three case studies of the protocol: the Fermi-Hubbard Hamiltonian, a molecular electronic structure problem, and a strongly correlated nuclear physics Hamiltonian. II. DOS-QPE Standard quantum phase estimation consists of a state register of dimension n and a time-frequency register of dimension m, often referred to as the ancilla register. In QPE, we are interested in the extraction of the 16 Oct 2025 2 phase θ relative to an eigenstate |ψ⟩of a unitary operator UH, where UH|ψ⟩= e2πiθ|ψ⟩. This can be any unitary n-qubit gate, often generated by the Hamiltonian H as UH = e-iHt. To achieve this, we prepare the state register with the eigenstate |ψ⟩and perform the controlled-unitary operations UC = P2m-1 k=0 |k⟩⟨k| ⊗U k H. This cascade of unitary operations acting onto the state register at incremental "times" k extracts the phase via phase kickbacks. The simplest example of this process is the Hadamard test, formally a QPE circuit with a single-qubit time-frequency register. These phases are encoded into the time-frequency register and can be accessed in the form of probabilities after performing an inverse quantum Fourier transform (iQFT), U† QFT, on it. In Fig. 1a), the QPE circuit would correspond to the first two registers, namely the "time-frequency" and the "state" registers, thus removing the "purification" register and relative operations. Standard QPE is the faulttolerant benchmark for the estimation of phases (eigenvalues) in the case of a pure, coherent eigenstate of the unitary i. Its successful execution requires the prepared probing state to have substantial overlap with the target eigenstate whose phase we aim to measure. Although several techniques have been proposed to address the state preparation problem, this step often incurs significant overhead and is sometimes the most challenging part of the protocol. In chemistry, trial eigenstate preparation remains a hot topic [25]; for weakly correlated systems, the qubitmapped Hartree-Fock state may be good enough [26], but for more strongly correlated systems, elaborate state preparation protocols are required. These usually take the form of a variational quantum algorithm [27], which represent difficult optimization problems [28, 29] and suffer prohibitive sampling costs [30]. Otherwise, approximate classical eigenstates can be loaded onto a qubit register by an appropriate mapping such as CVOQRAM [31]. Still, even this approach does not scale favorably with system size, not to mention the additional classical cost of finding a high-quality approximation to the target eigenstate, using, for example, one of the selected CI methods [32-34]. In addition, we might be interested in more general properties of the Hermitian operator generator of the unitary evolution. For example, accessing information encoded in higher-energy states may require reconstructing the entire spectrum of the target Hamiltonian. This is the case we will consider in the following. We are interested in the reconstruction of the so-called eigenvalue function [35] or density of states of the Hamiltonian operator H. The density of states (DOS) plays a i Throughout this paper, we refer to coherent and incoherent in the sense of resource theories [23, 24]. Specifically, we refer to states as coherent when their density matrix contains non-zero off-diagonal elements (coherences), and as incoherent when these off-diagonal terms vanish. central role in extracting spectral properties of an operator. It is typically derived from the Fourier transform of the operator's unitary time evolution and is essential for the reconstruction of the microcanonical ensemble of the system [36]. The DOS technique is a flexible approach for gathering information about both low- and high-frequency components. Consider a Hermitian operator H and its spectral resolution, H = P k θk|k⟩⟨k|, where θk are the eigenvalues corresponding to the eigenvectors |k⟩. The DOS is the eigenvalue distribution of the operator H, S(θ) = X k δ(θ -θk). (1) Looking at the spectral decomposition of the Hamiltonian H, that is, an ensemble of pure states |k⟩⟨k|, one might ask if we can extract the DOS function from the standard routine of phase estimation. This can indeed be done using the purification theorem. Purification is a fundamental concept of quantum information theory [13] that expresses any mixed state as the reduced state of a larger pure state having a specific entanglement structure, i.e., that of a maximally entangled state. For any mixed state ρS on a finite-dimensional Hilbert space HS, there exists a pure state |ψ⟩SP in a larger Hilbert space HS ⊗HP such that ρS = TrP (|ψ⟩⟨ψ|SP ) . (2) Here, HP is an auxiliary system (often called the purifying system) with dimension at least equal to rank(ρS). For simplicity and ease of explanation, we fix the purifying system to the same dimension as the state register. To implement this approach into standard QPE (see App. A for the derivation), we need to add an additional ancillary register that we will refer to as the purification register. This register matches the state register dimension, but it will be traced out at the end of the protocol. By means of this operation, we are effectively evaluating the trace over the system P in Eq. (2). This will allow us to translate QPE from a coherent to an incoherent approach, extending the routine to support further practical applications. Note that purification is not the only way to transition between coherent and incoherent QPE. In case of interest in statistical properties of the systems, any thermal state preparation routine [37-39] will suffice as long as the output state can be matched to the state register. As a result of the purification, the measurement outcome y ∈{0, . . . , 2m -1} from the time-frequency register yield coarse-grained phase estimates ̃φj = y/2m with probability P(y) = X j pj|cj(y)|2, (3) where pj are the spectral weights of ρS and |cj(y)|2 encodes the QPE response kernel. This empirical distribution reconstructs the coarse-grained DOS at resolution 3 time-freq QPE state purification Eigv. MaxMix DOS-QPE time-freq time-freq Dicke Figure 1. a) DOS-QPE circuit primitive. In panel a), we show the DOS-QPE as a circuit primitive. The full circuit consists of three registers: a time-frequency register, a state register, and a purification register. The additional purification register is maximally entangled to the state register to create the state ensemble. This is done through a cascade of serialized CNOT operations, CX⊗n S→P . This last register is traced out at the end of the protocol. b) QPE v. DOS-QPE algorithmic comparison. In panel b), we compare the standard QPE algorithm to the DOS-QPE algorithm. While QPE relies on the preparation of good-enough states |ψ⟩with sufficient overlap with the target eigenstates, DOS-QPE allows for probing entire subspaces via purifications |ψSP ⟩of simple construction. This translates to a cheaper state preparation routine. After measuring the time-frequency register in QPE, one estimates the phase θ. In DOS-QPE, we measure the time-frequency register and trace out the purification register to estimate the eigenvalue function S(θ). ∆φ = 1/2m. The total reconstruction error scales as ∥ˆρS -ρS∥2 ∼ r Neff · 2m M , (4) where Neff is the dimension of the subspace covered by the ensemble contributing to ρS, and M is the number of QPE repetitions. Thus, achieving a target error δ requires M ≳Neff · 2m δ2 . (5) For details on the derivation of the standard QPE and DOS-QPE results, including error analysis and scaling, see App. A and App. B, respectively. A. Maximally mixed state The most standard example of purification is that of the maximally mixed state [13]. This purification is easily achieved via the maximally entangled state |Φ⟩SP = 1 √ 2n 2n X k=1 |k⟩S ⊗|k⟩P , (6) where {|k⟩P } is a basis for the auxiliary system HP ∼= HS. Here and in the following, S and P indicate the state and the purification registers. Tracing out the system P, we have TrP (|Φ⟩⟨Φ|SP ) = I/2n = ρS, (7) that is, an ensemble of states covering the entire Hilbert space. Using this ensemble with flat distribution, we can probe all of the Hamiltonian's eigenvalues (phases) with equal probability and estimate the Hamiltonian's spectral resolution. The power and potential of using maximally mixed states in QPE have been shown, under the umbrella of garbage state preparation and kernel extraction, in the original proposal of quantum topological data analysis by Lloyd et al. [15]. In a similar way, but with the addition of the circuit primitive, DOS-QPE has been shown capable of generating features required for learning the frustration index [4]. B. Dicke states Given some M-dimensional set of discrete fermionic modes corresponding to single-particle quantum states, we can write the resulting 2M-dimensional Fock space as a direct sum of subspaces Fk, F = M k Fk = F0 ⊕F1 ⊕· · · ⊕FM. (8) The 1-particle quantum states |x⟩can be |0⟩or |1⟩, which under the second-quantization formalism, means that the 4 state x is completely empty or occupied by a particle, respectively. A k-body basis state can be written in terms of the 1-particle quantum states as |x⟩= k O i |xi⟩. (9) Each term in the decomposition shown in Eq. (8) corresponds to the subspace spanned by the set of occupation number vectors with a fixed particle number, k, of which there are M k . It is typically the case for fermionic Hamiltonian simulations that the eigenvalues and eigenvectors of interest all carry the same particle number. As such, using the maximally mixed state as the probe for DOS-QPE may lead to a spectrum that is too crowded for certain applications or lead to a needlessly high sampling cost. A simple way to enforce particle-number symmetry in the spectrum obtained from DOS-QPE is to choose a fermion-qubit encoding which maps fermionic states to qubit states conserving the Hamming weight of the original fermionic state after transformation, i.e., the same number of "1"s (hot-entries) in both fermionic and qubit states. The two projections of a single qubit represent a state that is either empty or occupied by a particle, in this case. The Jordan-Wigner encoding [40] satisfies these requirements, |x0, x1, . . . , xM⟩→|q0, q1, . . . , qn⟩, (10) where the number of qubits, n, is the same as the number of fermionic modes, M and xn = qn ∈{0, 1}. To restrict the eigenvalues to the correct subspace, then, becomes an issue of preparing, in an even superposition, all computational basis states with Hamming weight k on the state register before purification. Such states are known as Dicke states, |Dn k⟩, |Dn k⟩= n k -1 2 X x∈{0,1}n |x|=k |x⟩, (11) where n is the number of qubits, and k is the Hamming weight, which according to the literature can be prepared deterministically with circuit depths of O(k log n k ), O(kp n k ) and O(n) on devices with all-to-all, grid and linear nearest neighbour connectivities, respectively [41]. Perhaps the most natural application of DOS-QPE with a Dicke state probe is the extraction of eigenspectra of many-body Hamiltonians. In standard QPE and many of its extensions, a trial state with a large overlap with the target eigenstate must be found either classically or with some other quantum algorithm, such as VQE. Preparing such trial states with a variational quantum algorithm can be prohibitively expensive in terms of sampling costs. Additional difficulties during optimization arise from barren plateaus, and it is not possible to know in advance whether the prepared state has a large enough overlap with the target state, except for some heuristic cases. Preparing a state classically and compiling it into a state-preparation circuit is often more practical, though it still poses challenges, particularly in efficiently encoding the classical state onto the state register. In contrast, Dicke-probed DOS-QPE does not require preparation of fermionic eigenstates, preserves particlenumber symmetry under the Jordan-Wigner mapping, and experiments need to be set up only once to recover the full eigenspectrum. The classical cost incurred before the execution of the algorithm in the simplest case is just the cost of a single Hartree-Fock calculation, which scales polynomially in system size with even naive implementations, plus the cost of constructing the circuit. III. SPECTRUM RECONSTRUCTION Transitioning from estimating a single phase using QPE to reconstructing the full spectrum with DOS-QPE requires a more refined post-processing step. This additional cost is justified by the richer information retrieved from he spectrum. DOS-QPE effectively samples from the normalized density of states, S(θ) = 1 2n 2n X i=1 δ(θ -θi), (12) where, as a refinement of Sec. II, we have included the normalization factor. Here, the eigenvalues θi of H are each repeated with their multiplicity. In practice, with only m qubits in the time-frequency register, one obtains a discretized histogram Pi = Z Ii S(θ)dθ + (noise), i = 0, 1, . . . , 2m -1, (13) where the Ii are adjacent intervals of width ∆θ = 2π/2m. The task of spectrum reconstruction is to recover both the continuous eigenvalues locations θi ⊂R and their degeneracies di ⊂Z≥0 from the sample histogram P = {Pi}. This problem is challenging for several reasons: (i) finite resolution due to limited register size m, which determines the bin width ∆θ; (ii) noise, including statistical fluctuations of order O(n-1/2); (iii) unresolved spectral gaps and an unknown number of distinct eigenvalues. While some of these challenges will result in inevitable inaccuracy of the reconstructed signal, we will resort to techniques such as compressed sensing and convex optimization to efficiently find solutions. A. The degeneracy integer problem With access to the readout of the time-frequency register of the DOS-QPE circuit, we aim to estimate the spectral resolution, H = P k θk|k⟩⟨k|. To do so, we face 5 the problem of generating the best spectral estimate out of the readout probability distribution P. We refer to this estimation task as the degeneracy integer problem, falling under the umbrella of spectral density estimation [42]. The problem consists of assigning a set of normalized degeneracies { ̃di}i, subject to P i ̃di = P i di/ dim(H) = 1, to minimize a suitable distance D(P, ̃P) between the target probability distribution P and the reconstructed spectrum ̃P. While the ideal reconstruction would be P(θ) = P i diδ(θ -θi), the smeared, finite measured distribution will be a convolved distribution ̃P(θj) = P i diK(θj -θi), where K is the kernel and θj is a measurement bin determined by the finite time-frequency register. Since the readout distribution consists of probabilities, we work with the squared kernel K(θ) = |DM(θ)|2 = sin(πMθ) M sin(πθ) 2 , θ ∈[-1 2, 1 2] (14) where the Dirichlet kernel DM arises naturally from the discrete Fourier transform over M = 2m time steps used in the DOS-QPE protocol. Note that the dividing factor M normalizes the kernel integral to 1 in the large-M limit, ensuring the convergence of DM to a Dirac delta in the same limit. The final observed probability distribution that we aim to reconstruct is ̃P(θ) = R X i=1 di · K(θ -θi). (15) B. Compressed sensing and optimization To recover the discrete phases {θi} and their degeneracies {di} from the noisy coarse-grained histogram {Pi}, we reframe the degeneracy integer problem as an inverse problem. The goal is to infer a sparse spectral signal - convolved with a known kernel - from its noisy, finite-resolution observation. This setting naturally invites tools from convex optimization to minimize the discrepancy between the observed histogram and a model spectrum. We formulate the problem on a grid of candidate phases {θ′ j} ⊂[0, 1), finer than the hardware-limited resolution 2-m. We obtain an underdetermined system where the number of potential spectral components exceeds the number of measured bins. To select a physically meaningful sparse solution, we employ a quadratic program [43] with an l1 regularization penalty (basispursuit, LASSO) [44], which encourages sparsity by penalizing the absolute sum of the degeneracy coefficients. This results in a practical realization of compressed sensing [21, 22], a framework for recovering sparse signals from limited or noisy measurements. The finer grid provides an overcomplete dictionary of possible phases, while the l1 penalty enforces that only a small subset is selected. The resulting optimization routine enables a super-sampled spectral reconstruction despite a coarser hardware discretization [45]. We introduce a uniform grid of G candidate phases {θ′ i}G i=1, producing the dictionary matrix A ∈RN×G: Ai,j = K(θi -θ′ j). (16) Writing P = (Pk)N k=1 and w = (wj)G j=1 as the degeneracy coefficients for the grid points, the forward model is P ≈Aw. (17) The problem naturally maps onto a mixed-integer quadratic program [46, 47] with convex, non-linear objective function L2,MI = minw∈N+ |Aw -P|2 2 with PG j=1 wj = 2n. Quadratic programming is NPcomplete [48] in general, with the variant of mixedinteger programming (MIP) enforcing wj ∈Z+ and P j wj = P i di = 2n being NP-hard [49-51]. For this reason, we relax the problem from integer degeneracies to real and add a further final step of clustering of phases and integer approximation on the solution. This makes the problem a quadratic program with a convex objective function, which, together with linear programs, is in P [50-52]. The final objective function we optimize against is L2 = min w∈RG + |Aw -P|2 2 + λ|w|1 s.t. G X j=1 wj = 2n, (18) where we added the l1 regularization penalty controlled by λ > 0. The squared-L2 term ensures data fidelity, and the l1 term promotes sparse support in the candidatephase weights. This convex QP can be solved efficiently at moderate G using off-the-shelf solvers. In our simulations, we do this using the CLARABEL solver [53] and CVXPY [54, 55]. Note that, because the quadratic part of L2 is piecewise linearizable, we can also reformulate the problem as a linear program L1 = minw,r(r + λw) such that |Aw - P| τ of those phase indices whose solution degeneracy is above a certain target threshold τ. To recover the final phases and integer degeneracies, we apply the following steps: 1. Thresholding. We retain grid indices in J and their weights w∗ j . 2. Clustering. We merge nearby phases φj within a tolerance ε via single-linkage, forming clusters C1, . . . , Cmax J. This ε can be taken of the order of the initial grid size of DOS-QPE defined by the time-frequency register dimension. 3. Rounding. For each cluster C, we compute ˆθ = P j∈C w⋆ j ψj P j∈C w⋆ j , ˆd = X j∈C w⋆ j . (19) 6 In this way, we restore integer degeneracies and obtain a final estimate {ˆθi}max J i=1 with corresponding { ˆdi}max J i=1 . To quantify the reconstruction error, we use the 1Wasserstein distance [56] W1(μ, ν) between discrete spectral distributions μ = P i di δθi and ν = P j ˆdj δˆθj, where δθ denotes the Dirac measure at θ. This metric is defined as W1(μ, ν) = inf γ∈Γ(μ,ν) X i,j γij |θi -ˆθj|, (20) where Γ(μ, ν) is the set of all couplings between μ and ν that preserve total weight. Intuitively, this metric quantifies the dissimilarity between the two distributions by measuring the minimum amount of "mass" times "travel" that needs to be moved to shape one into the other [57]. In the following, we compute W1 between the sampled spectrum and the exact one, as well as between the optimized solution and the exact spectrum. This provides a principled way to evaluate the fidelity of spectral estimation, taking into account both eigenphase positions and degeneracies. IV. APPLICATIONS In the following, we show three applications of DOSQPE: the Fermi-Hubbard model, an electronic structure Hamiltonian, and a nuclear Hamiltonian. We use t|ket⟩[58] to construct the circuits, Qulacs for state vector evaluation [59], and OpenFermion for constructing qubit Hamiltonians [60]. To evaluate the molecular integrals in the electronic structure section, we use the PySCF Python package [61]. A. The Fermi-Hubbard Model The Fermi-Hubbard model [62-65] is one of the standard models of condensed matter physics as it describes relevant features of strongly-correlated electronic systems. It has also become a standard benchmark for Hamiltonian simulation algorithms [66-68] in the quantum computing community since exact solutions can be found for the one-dimensional case [69]. The study of the two- and higher-dimensional cases is a topic of intense research activity, where only numerical simulations are available [70-72]. Its Hamiltonian is written as H = -t X ⟨i,j⟩,σ a† iσajσ + U X i ni↑ni↓, (21) where the quantity σ =↑, ↓denotes spin projection, niσ = a† iσaiσ is a particle-number operator, the symbol ⟨i, j⟩restricts the sum over the indices of neighboring lattice sites and t and U are the hopping and repulsion parameters, respectively. In the remainder of this section, we apply the DOSQPE algorithm to the Fermi-Hubbard chain model with 0.00 0.06 0.12 0.19 0.25 0.31 0.38 0.44 0.50 0.56 0.62 0.69 0.75 0.81 0.88 0.94 Phase 0.00 0.05 0.10 0.15 0.20 Probability / Degeneracy/2n 1/2n 5/2n 10/2n FH, probe: MM-state DOS-QPE Spectrum Optim Figure 2. DOS-QPE on the Fermi-Hubbard model. We run the DOS-QPE circuit on a Fermi-Hubbard toy model with three sites, t = 1, and U = 4, employing m = 6 ancillas to resolve the spectrum. In the time-frequency register, we obtain the probability distribution of the DOS (blue bars). We compare this with the exact spectrum (black bars) of the Hamiltonian. While the dimensionality of the time-frequency register is insufficient to resolve the spectral gaps of the spectrum, we show that quadratic programming aided by compressed sensing helps in the phase and degeneracy reconstruction (red stars). The 1-Wasserstein distance of the optimized solution improved by ∼51% over the integer-rounded sampled distribution. three sites. We choose t = 1 and U = 4 such that we are in the so-called "intermediate coupling" regime, under which it is difficult classically to extract even ground-state properties from numerical simulations at half-filling [73]. We can then map this second-quantized Hamiltonian, using the Jordan-Wigner encoding, to a linear combination of products P of Pauli operators σj, H = X i hiPi = X i hi O j σi j, (22) where hi are the coefficients in the transformation. We run the DOS-QPE circuit of Fig. 1a) and show the results in Fig. 2. For this simulation, we have state and purification registers of dimension n = 6, and we try to resolve the spectrum with a time-frequency register of dimension m = 6, for a total of 18 qubits. The Hamiltonian is rescaled following App. C and the unitary propagation is done via fourth-order Trotterization. Comparing the DOS S(θ) of the Hamiltonian H to its exact spectrum, we note that the dimensionality of the time-frequency register is insufficient for a perfect spectrum reconstruction. That is why we employ the quadratic program and compressed sensing technique explained in Sec. III B. Using such convex optimization, we can obtain a better estimate of the integer degeneracies and the real phases of the spectrum. As described in Sec. III B, we use the 1Wasserstein distance against the exact spectrum to quantify the deviation of the distributions, finding ∼16% improvement over the sampled distribution and more than ∼51% over the corresponding integer-rounded version. 7 B. Electronic Structure The application of quantum algorithms to chemistry typically begins with the construction of the system Hamiltonian, written in the second-quantization formalism as, H = X pq hpqa† paq + 1 2 X pqrs hpqrsa† pa† qasar. (23) The constants hpq and hpqrs are the integrals, hpq = Z dxφ∗ p(r) -∇2 2 - X A ZA |r -RA| ! φq(r) (24) and hpqrs = Z dr1dr2 φ∗ p(r1)φ∗ q(r2)φr(r2)φs(r1) |r1 -r2| , (25) respectively, and are routinely obtained as part of a classically inexpensive Hartree-Fock calculation. The quantities φp are the molecular orbitals and form the 1-particle wavefunctions mentioned in Sec. II B, r are the positions of the electrons, and ZA and RA are the charges and positions of the nuclei, respectively. For the same reasons as in Sec. II B and the former section, the Jordan-Wigner mapping is selected. In most classical computational chemistry studies, the number of electrons in a system remains fixed. Descriptions of many chemical phenomena are framed in terms of energy differences as the nuclear configuration changes, or between eigenstates at some fixed geometry. Even in cases where the number of electrons varies, the full spectrum for all eigenvalues of all occupation numbers is rarely required. As such, the Dicke state |Dn k⟩, where k is set to the number(s) of electrons in the system, is the most relevant probe for the DOS-QPE algorithm as it is applied in this section. We choose the twisted conformation of ethylene as the example chemical system, since by varying the pyramidalization angle we expect to encounter a conical intersection between the two lowest lying singlet eigenstates, and a crossing of the lowest lying singlet state with the triplet ground state [74]. We first optimise the planar geometry at the CCSD(T) level classically in the cc-pVDZ basis, then rotate the two hydrogens on one carbon atom to have dihedral angles with the hydrogen atoms on the other carbon atom of exactly 90◦. We then vary the angle α as shown in Fig. 3a). We select a 2-electrons in 2orbitals active space, and along the reaction coordinate, evaluate the overlap of the orbitals at each point with the orbitals from the previous point, such that we can place the orbitals at maximum coincidence at each step along the potential energy surfaces and ensure a consistent orbital selection. Such that the eigenvalues lay in a fixed interval and the movement of each peak in Fig. 3c-f) is smooth, and to obtain a smooth curve in Fig. 3b), we find our target Hamiltonians with the following formula, Htarget = (1 -δ) H -λmin,50◦+ δ λmax,50◦-λmin,50◦ , (26) where in all cases δ = 0.05, which serves as a shift to move the lowest phase away from zero and simplify communication regarding Fig. 3, and λmin,50◦and λmax,50◦are the lowest and highest lying 2-particle eigenvalues obtained by exact diagonalization of the geometry at α = 50◦, respectively. The results for exact diagonalization of the active space Hamiltonian are shown in Fig. 3b). We observe that the conical intersection is at approximately 68◦. We then select three evenly spaced points around the approximate location of the conical intersection, at 65◦, 71◦, and 74◦, at which we perform state-vector DOS-QPE calculations. We use second-order Trotterization with one time step for the preparation of UH and, as previously mentioned, set Uψ to be the |D4 2⟩state-preparation unitary. We also marginalise the exact probabilities of measurement outcomes on the state and purification qubit registers such that the results we obtain correspond to the exact probabilities of the measurement outcomes on the ten ancilla qubits. By restricting the probe to being an even superposition of only 4C2 states, we expect to observe a maximum of 6 distinct eigenvalues. Spin symmetry simplifies the picture since we know that for any triplet state with total spin S = 1, there are three degenerate states with Ms = -1, 0, 1. We can infer the multiplicity of the states corresponding to the phases, then, from the relative peak heights in the spectra shown in Fig. 3c-f): each contains three degenerate triplet eigenstates with different Ms values, and three peaks corresponding to singlet states which have no degeneracy by spin symmetry, totaling six states as expected. In what remains of this section, we will refer to the lowest and second-lowest singlet state as S0 and S1, respectively, and to the triplet state as T. The other singlet state (marked with a black dashed line) is not of any interest in this discussion but is included in the figure for completeness. We see by comparing the phases shown in Fig. 3b) and the peaks in Fig. 3c) that we gain access to the correct ordering of states, the phases agreeing with exact diagonalization. Increasing α to 68◦(see Fig. 3d)), close to the conical intersection between S0 and S1, we see near degeneracy between the phases corresponding to the S0 and S1 states, as expected, and an increase in the phase found for the T state. Further increasing α to 71◦, shows the separation between the phases corresponding to S0 and S1 return, as is the case in Fig. 3b). The distribution shown in Fig. 3f) shows that the phases corresponding to T and S0 have crossed, and the ground state now carries singlet multiplicity, as expected from the results of the exact diagonalization. From the faithful reconstruction of the spectra along the potential energy surfaces, we conclude that DOSQPE represents a valuable tool for the fault-tolerant era 8 Figure 3. DOS-QPE on the electronic structure of ethylene. a) The structure of the ethylene molecule showing a twisted conformation and the angle α which is varied during the scan of the potential energy surfaces. b) Plot of the lowest three eigenvalues of the Hamiltonian, scaled such that the eigenvalues lie in the interval [0, 1). The green line corresponds to the triplet state T, the orange line to the lowest lying singlet state S0 and the purple line to the first excited singlet state S1. c-f) The distributions obtained with DOS-QPE with α = 65◦, 68◦, 71◦and 74◦, respectively. In each plot c-f), the colour of each vertical dashed line corresponds to the phases obtained via exact diagonalization and matches those used in b). The height of the dashed lines shows the height the peaks in the distribution would be if all eigenvalues were to be exact integer multiples of 1 Nanc . of quantum computation for the study of chemical systems. In the form it is presented here, we anticipate DOSQPE being particularly useful for investigating phenomena arising directly from more than one eigenstate at any given molecular geometry. An interesting next step for this approach is to leverage other known physical symmetries to construct more targeted probe states. C. Nuclear Structure The description of nuclear structure in atomic nuclei follows a similar prescription as that of molecules in quantum chemistry within the nuclear shell model [75]. Under this approach, protons and neutrons (collectively called nucleons), basic fermion components of a nucleus, move in a set of single-particle states that form a valence space. These single-particle states, also called orbitals, are labeled by the quantum numbers (n, l, j, m, tz), where n is the principal quantum number, l is the orbital angular momentum, j is the total angular momentum after coupling with the nucleon spin-1 2, m is its third-component projection and tz is the third-component projection of the isospin t = 1 2, an additional quantity equivalent to the spin defined in order to discern between protons and neutrons. The nuclear shell model can be described by means of a second-quantized Hamiltonian as in Eq. (23), where the single-particle energies hpp and two-body matrix elements hpqrs are adjusted to reproduce observable properties of selected nuclei [76]. The nuclear states are described by a well-defined total angular momentum J and isospin T, resulting from the coupling of individual nucleonic j and t; with their corresponding third-component projections M and Tz, which are equal to the sum of the third components of the nucleons in the valence space, m and tz. Thus, a many-body basis state can be written in terms of states with well-defined M and Tz |JMTTz⟩= X i ci |i, MTz⟩, (27) where i labels additional quantum numbers needed for the description of the many-body state, and ci are the coefficients in the superposition. The Hamiltonian matrix is then built using this many-body basis, and the nuclear states can be found as its eigenstates. Because of the two kinds of particles in a nucleus, protons and neutrons, the scaling of this many-body basis is as dsh Nval dsh Zval , where dsh is the dimension of the valence space and Nval, Zval are the valence neutron and proton numbers, respectively. Many quantum algorithms have been applied to obtain nuclear-structure properties based on the ground state [77-84] and also low-lying excited states [85-88]. The structure and entanglement features of nuclear states have also been thoroughly studied [89-92]. In this work, we aim to resolve the full spectrum in a single calculation with DOS-QPE. We simulate the 6Be nucleus, which is composed of two protons in the valence space of a p shell, as shown in Fig. 4, with six singleparticle states for the protons. Thus, we have in this case two identical particles moving in the valence space, and consequently, the total isospin is T = 1. The Hamiltonian used is based on the phenomenological Cohen-Kurath in9 Label Many-body state 1 2 3 4 5 (0, 3) (1, 2) (1, 5) (2, 4) (4, 5) Figure 4. a) Many-body state basis used for 6Be. Label represents the indices in the corresponding Hamiltonian matrix in this many-body basis. The many-body states are composed of single-particle states from all possible combinations of two valence protons in the p-shell variational space of panel b) satisfying the condition M = m1 + m2 = 0. b) Configurational valence space for the p-shell. The valence space is decomposed in two degenerate j orbitals with degeneracy (2j + 1). The labels on the left correspond to the standard spectroscopic notation nlj, where l= p means l= 1. Numbers on top of every single-particle state represent the labels used in this work. Slightly modified from [80]. teraction [93]. While we could directly transform the secondquantized Hamiltonian using the Jordan-Wigner mapping as demonstrated in the former examples, for this specific case we opt to build the Hamiltonian matrix using the minimal many-body basis spanned for 6Be, consisting of only five states whose total angular momentum projection M is equal to zero [80]. The many-body basis used for 6Be is listed in the table of Fig. 4a). The resulting Hamiltonian matrix is then mapped onto the qubit operator basis using the Linear Combination of Unitaries (LCU) formalism [94]. Thus, using this mapping, we managed to reduce the size of the Hamiltonian from six to three qubits, at the expense of three additional degenerate eigenvalues equal to zero. The results of the DOS-QPE calculation for this system are shown in Fig. 5. The calculation was performed using a sixth-order Trotterization of the unitary propagation and five ancillary qubits in the time-frequency register for the spectrum resolution. We observe that, under these resources, four out of five physical phases (eigenvalues) of 6Be are retrieved within a precision dictated by the number of ancillary qubits used (see App. B). The resolution for the almost degenerate peaks at ∼0.65 is not enough and the number of ancillary qubits in the time-frequency register needs to be increased. A spurious zero eigenvalue with degeneracy of three is also found as an LCU artifact used to map the Hamiltonian to a qubitoperator basis. In light of these results, although demonstrated on a small example, we may conclude that DOS-QPE could become a promising tool for efficiently extracting spectral information and thus dynamical processes [95-99] in atomic nuclei for near-term quantum devices. 0.00 0.06 0.12 0.19 0.25 0.31 0.38 0.44 0.50 0.56 0.62 0.69 0.75 0.81 0.88 0.94 Phase 0.0 0.1 0.2 0.3 0.4 0.5 Probability / Degeneracy/2n 1/2n 2/2n 3/2n 6Be, probe: MM-state DOS-QPE Spectrum Figure 5. DOS-QPE on 6Be. The result of DOS-QPE with a maximally mixed state probe. Sixth-order Trotterization of the unitary evolution and five qubits in the time-frequency register were used for these results. We observe that, while the distribution (blue bars) faithfully captures the exact eigenspectrum, it is not fine-grained enough to resolve the almostdegenerate peaks at phase ∼0.65, and more ancillary qubits would be needed. The tallest peak represents the redundant three-times degenerate zero eigenvalue resulting from building the nuclear Hamiltonian matrix using the LCU approach. V. DISCUSSION AND CONCLUSION We have shown how, by using a generalization of quantum phase estimation probed with maximally mixed or symmetry-adapted ensemble states, we can access spectral information of the generating Hamiltonians. By embedding QPE within a purified framework [4], the traditional eigenstate-dependent algorithm transforms into a flexible tool for ensemble-based spectral analysis. This approach grants access to the density of states of Hamiltonians, effectively revealing their spectral properties. From here, thermodynamic and dynamical properties of systems become accessible while mitigating the effects of noisy or imperfect state preparations. We framed the recovery of discrete spectral features from the output of DOS-QPE as a convex optimization problem. We resolve spectral structures exploiting sparsity and physical constraints to produce high-fidelity reconstructions. We propose a modular postprocessing that restores physically meaningful integer-value spectra. While we have demonstrated DOS-QPE as a flexible primitive for ensemble-based quantum spectroscopy across fermionic and nuclear models, optimizations or resource reductions might be desirable. One immediate goal is to reduce circuit complexity and resource cost, tailoring the primitive for currently available quantum hardware [100]. Implementing single-ancilla variants [39, 101, 102] or taking advantage of shadow techniques [103-105] could significantly lower circuit depth and overhead, making the primitive viable for near-term applications. Further directions include exploring alternative probe states, such as spin-selective or symmetry constrained 10 ensembles, which may target subspace-resolved spectral features often inaccessible to conventional methods. In this context, developing spectrum manipulation techniques to enhance signal contrast, reduce costs, and mitigate noise could be particularly valuable. Such strategies would accelerate the reconstruction process and alleviate the current bottlenecks in measurement and postprocessing. DOS-QPE shows potential as a framework that bypasses state preparation bottlenecks while taking advantage of the incoherent nature of the environment to return physically relevant observables otherwise inaccessible. Connections to non-linear functions of spectral densities, gradient estimation, and Bell-type measurements [106] indicate a broader algorithmic utility within hybrid optimization and quantum learning frameworks. Embedding DOS-QPE within Green's functions formulations [107, 108] could further help in dynamical response theory offering access to electronic, vibrational, or nuclear observables. Appendix A: QPE - derivation Given a unitary operator U and an eigenvector |ψ⟩such that U|ψ⟩= e2πiφ|ψ⟩, φ ∈[0, 1), (A1) the goal is to estimate φ to m-bit precision. The registers we find in a standard implementation of QPE are • Time-frequency register: m qubits, initialized to |0⟩⊗m. • State register: n qubits, initialized to a target eigenvector |ψ⟩. The entire initial state facing the circuit is, |Ψ0⟩= |0⟩⊗m ⊗|ψ⟩. (A2) The first operation of QPE is the application of a layer of Hadamard gates H⊗m to the first register to create the uniform superposition, |Ψ1⟩= 1 √ 2m 2m-1 X k=0 |k⟩ ! ⊗|ψ⟩. (A3) We then apply the unitary operation UQPE, consisting of a controlled-U 2j operation for each qubit j in the timefrequency register, |Ψ2⟩= 2-m/2 P2m-1 k=0 |k⟩⊗U k|ψ⟩. Since U k|ψ⟩= e2πikφ|ψ⟩, this results in |Ψ2⟩= 1 √ 2m 2m-1 X k=0 e2πikφ|k⟩⊗|ψ⟩. (A4) We then apply the inverse quantum Fourier transform U † QFT to the time-frequency register. The Fourier basis state 2-m/2 P2m-1 k=0 e2πikφ|k⟩is mapped to the computational basis state | ̃φ⟩, where ̃φ is the best m-bit approximation of φ. If φ has an exact m-bit binary expansion φ = 0.φ1φ2 . . . φm, then U † QFT 1 √ 2m 2m-1 X k=0 e2πikφ|k⟩ ! = |φ1φ2 . . . φm⟩. (A5) The final state of the system is |Ψfinal⟩= | ̃φ⟩⊗|ψ⟩and measuring the time-frequency register gives ̃φ, the estimate of φ. Error Note that, when φ cannot be exactly represented with m bits, we have φ = s 2m + δ, with |δ| < 1 2m+1 . (A6) The probability of measuring s, the nearest m-bit approximation to φ, is Psuccess ≥ sin(πδ2m) 2m sin(πδ) 2 , (A7) that is exactly 1 if δ = 0, while, for small δ, the phase estimation error is bounded by | ̃φ -φ| ≤ 1 2m+1 (with high probability). (A8) Appendix B: DOS-QPE - derivation Consider the case where the input to QPE is not an eigenstate of U, but rather an arbitrary mixed state ρS = N X j=1 pj |ψj⟩⟨ψj|, (B1) with {|ψj⟩} the eigenbasis of U satisfying U|ψj⟩= e2πiφj|ψj⟩. We purify ρS by introducing an auxiliary register of the same dimension. The purification is |Ψ⟩SP = N X j=1 √pj |ψj⟩S ⊗|j⟩P , (B2) where {|j⟩}P is an orthonormal basis of the purification register. For simplicity, we drop the subscripts in the following. The total system includes three registers: • Time-frequency register: m qubits, initialized to |0⟩⊗m. • State register: n qubits, initialized to the system of interest. 11 • Purification register: n qubits, entangled with the state register to create the purification. This register is to be traced out at the end. The initial state is |Ψ0⟩= |0⟩⊗m ⊗ N X j=1 √pj |ψj⟩⊗|j⟩ . (B3) Applying Hadamard gates H⊗m to the time-frequency register creates the superposition |Ψ1⟩= 1 √ 2m 2m-1 X k=0 |k⟩ ! ⊗ N X j=1 √pj |ψj⟩⊗|j⟩ . (B4) We then apply the controlled-unitary operation UQPE, where each |k⟩controls U k on the state register. Since U k|ψj⟩= e2πikφj|ψj⟩, the resulting state is |Ψ2⟩= 1 √ 2m 2m-1 X k=0 |k⟩⊗ N X j=1 √pj e2πikφj |ψj⟩⊗|j⟩. (B5) Next, we apply the inverse quantum Fourier transform U † QFT to the time-frequency register. The Fourier transform maps 1 √ 2m 2m-1 X k=0 e2πikφj|k⟩7→ 2m-1 X y=0 cj(y) |y⟩, (B6) where cj(y) = 1 2m 2m-1 X k=0 e2πik φj-y 2m , (B7) where each outcome y ∈{0, . . . , 2m -1} corresponds to a phase estimate ̃φj = y/2m. After the inverse Fourier transform, the state becomes |Ψ3⟩= N X j=1 √pj 2m-1 X y=0 cj(y) |y⟩⊗|ψj⟩⊗|j⟩. (B8) By tracing out the purification register, the measurement of the time-frequency register gives the outcome y with probability P(y) = N X j=1 pj cj(y) 2, (B9) which corresponds to sampling from the spectral measure of ρS under U, coarse-grained to resolution ∆φ = 1/2m. Each measurement outcome y yields a phase estimate ̃φj = y/2m, and the empirical distribution of ̃φj reconstructs the density of states of U. Error and sampling Three main sources of error affect this procedure: a. Resolution error The Fourier truncation limits the phase resolution to ∆φ = 1 2m . (B10) b. Statistical sampling error Each DOS-QPE run yields one sample from the spectral distribution. After M shots, the per-bin standard deviation is defined by the binomial sampling error bound, that is ε ≈ r p(1 -p) M ≤ 1 2 √ M , (B11) assuming worst-case p = 0.5. The total L2 error in the reconstructed histogram ˆρS is ∥ˆρS -ρS∥2 ∼ r B M = r 2m M , (B12) where ˆρ is the empirical distribution (histogram) extracted and B = 2m is the number of bins in such distribution. To achieve a total error δ, the number of shots must satisfy M ≳2m δ2 . (B13) c. Scaling with Hilbert space dimension Let Neff denote the number of eigenstates contributing non-trivially to ρS: • For a maximally mixed state, ρS = I/2n, so Neff = 2n, and the DOS-QPE samples the full spectrum of U. • For a restricted ensemble, such as the uniform mixture of Dicke states, the support of ρS is reduced. For example, fixed Hamming-weight Dicke states span a subspace of dimension n k ∼O(nk). • For a pure eigenstate, Neff = 1, and the DOS-QPE output is that of standard QPE, i.e. a sharp peak centered at that eigenvalue. The effective sampling variance thus scales with the number of contributing eigenstates. Assuming uniform pj = 1/Neff, the variance of the spectral histogram is Var[P(y)] ∼ Neff M · 2m , (B14) and the total error scales as ∥ˆρS -ρS∥2 ∼ r Neff · 2m M . (B15) Hence, to achieve error δ, the number of measurements must satisfy M ≳Neff · 2m δ2 . (B16) This highlights the trade-off between resolution, state support, and sampling complexity. Choosing input states with reduced spectral support (e.g., Dicke states) allows one to probe restricted subspaces of the spectrum efficiently. 12 Appendix C: Rescaling the Hamiltonian For DOS-QPE, the input Hamiltonian must have a spectrum that is real, non-negative, and bounded in the interval [0, 1). Suppose we are given a Hamiltonian H with real eigenvalues λi ∈[λmin, λmax]. We construct a rescaled Hamiltonian ̃H such that its eigenvalues lie in [0, 1) as follows: ̃H = H -λminI ∆ , (C1) where ∆= λmax -λmin. 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2510.14742
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Unsupervised Learning to Recognize Quantum Phases of Matter
Mehran Khosrojerdi,1, ∗Alessandro Cuccoli,1, 2 Paola Verrucchi,3, 1, 2 and Leonardo Banchi1, 2
1Department of Physics and Astronomy, University of Florence,
via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
2INFN Sezione di Firenze, via G. Sansone 1, I-50019, Sesto Fiorentino (FI), Italy
3ISC-CNR, UOS Dipartimento di Fisica, Universit`a di Firenze, I-50019, Sesto Fiorentino (FI), Italy
(Dated: October 17, 2025)
Drawing the quantum phase diagram of a many-body system in the parameter space of its Hamil-
tonian can be seen as a learning problem, which implies labelling the corresponding ground states
according to some classification criterium that defines the phases. In this work we adopt unsu-
pervised learning, where the algorithm has no access to any priorly labeled states, as a tool for
determining quantum phase diagrams of many-body systems. The algorithm directly works with
quantum states: given the ground-state configurations for different values of the Hamiltonian pa-
rameters, the process uncovers the most significant way of grouping them based on a similarity
criterion that refers to the fidelity between quantum states, that can be easily estimated, even ex-
perimentally. We benchmark our method with two specific spin- 1
2 chains, with states determined
via tensor network techniques. We find that unsupervised learning algorithms based on spectral
clustering, combined with “silhouette” and “elbow” methods for determining the optimal number of
phases, can accurately reproduce the phase diagrams. Our results show how unsupervised learning
can autonomously recognize and possibly unveil novel phases of quantum matter.
I.
INTRODUCTION
Quantum many-body systems can exhibit phase-
transitions also at T = 0 due to purely quantum fluc-
tuations [1], making it possible for Ground States to fea-
ture different collective properties for infinitesimally dif-
ferent values of the hamiltonian parameters. Studying
the Quantum Phase Diagrams (QPD) outlined by these
transitions, as well as the quantum phases themselves, is
of interest not only in the realm of many-body physics
but also in quantum information processing, whose op-
eration happens at temperatures as near as possible to
T = 0. On the other hand, drawing the QPD of quantum
many-body systems is a hard task, unless one considers
very simple models such as, for instance, the Ising chain
in a transverse field. In fact, analytical methods are of
limited help in studying models of realistic systems, while
established numerical techniques demand huge computa-
tional resources, that become unaffordable as the size of
the models grows, as required by an evaluation of non-
local quantities that characterize different phases, such as
long-range correlations and concurrence, or multipartite
entanglement.
In
this
framework,
approaches
based
on
hybrid
quantum-classical
machine-learning
methods
provide
some alternatives. One possibility comes from quantum-
phase classification based on training on labeled data,
where some prior theoretical or numerical understanding
of the phase-diagram is necessary, at least in a limited
parameter region. This approach matches a supervised
learning task, where the phases are known for some par-
ticular parameter values that are used to train the clas-
∗mehran.khosrojerdi@unifi.it
sification algorithm. The algorithm is then used to ex-
plore larger areas of the parameter space and label the
corresponding phase [2–12]. Quantum states clustering,
instead, uses an unsupervised learning strategy, whose
goal is to identify similarities among ground states, given
no prior knowledge of the QPD, and hence without pre-
existing labeling of the different phases. Unlike classi-
fication, which is typically applied as a validation tool,
clustering can be used to explore novel quantum systems,
revealing hidden structures in the data, and uncovering
emergent phases of matter without relying on predefined
categories [13–16]. Needless to say, clustering quantum
states to unveil a phase-diagram is a challenging task, due
to the inherently non-local nature of quantum correla-
tions and entanglement [1, 17]: in fact, when the Hilbert
space dimension grows, identifying features for an effec-
tive clustering is by itself a difficult task. On the other
hand, some unsupervised methods [18–22], even if acting
on classical data-sets, can turn out to be suitable to spot
similarities between elements of large Hilbert spaces, pro-
viding an interpretable framework for clustering quantum
states based on their genuinely quantum properties.
Scope of this work is to develop a method for quantum-
states clustering, precise enough to match QPD of pa-
rameterized Hamiltonians whose ground state can be de-
termined and represented via tensor networks methods,
such as the DMRG [23–25].
Since our method uses a
“kernel” between two Hamiltonian parameter sets given
by the fidelity between the corresponding ground states,
it can be extended to work directly with ground states
approximated in quantum devices, since the fidelity be-
tween states can be experimentally estimated using dif-
ferent methods such as the SWAP test or the Hadamard
test, see e.g. [26].
Our procedure goes as follows:
be
ˆH(x) a one-
dimensional Hamiltonian that fulfills the above require-
arXiv:2510.14742v1 [quant-ph] 16 Oct 2025
2
ments, with x = (x1, x2, . . . xq) the set of q parameters
upon which it depends. We take four steps for predicting
the QPD with the desired insight into different quantum
phases.
1. Generate D sets of Hamiltonian parameters {xi =
(x1i, x2i, ...xiq)} according to some rule (e.g. uni-
formly in the parameter space), with i = 1, ..., D.
This set defines our “classical data”, where each
datum is an element of Rq and a point in the phase-
diagram.
2. Represent the Hamiltonian
ˆH(xi) as a Matrix
Product Operator (MPO) for each datum xi and
find its ground state |GS(xi)⟩.
3. Define the kernel between two data xi and xj
as Kij = | ⟨GS(xi)| GS(xj)⟩|2, namely as the fi-
delity between two ground states |GS(xi)⟩and
|GS(xj)⟩.
In the language of machine learning,
this corresponds to defining the feature space as
the O(4N)-dimensional space of density matrices
|GS(xi))⟩⟨GS(xi)|, where N is the number of qubits
in
ˆH(x).
Note that the dimensionality of the
feature space is much larger than that of the
dataset, which is one of the key-element of any
machine-learning procedure for clustering, as it re-
flects the richer characterization offered by features
w.r.t. data.
4. Perform a kernel clustering algorithm, which only
depends on the kernel matrix K, so as to group
ground states according to the similarity amongst
their quantum features.
Notice that, although not explicitly considered in this
paper, steps 2. and 3. can be replaced with:
2b. Approximate each ground state |GS(xi)⟩in a quan-
tum device, e.g. using a parametric quantum algo-
rithm and the variational quantum eigensolver [27]
or more recent versions [28].
3b. Compute the kernel Kij = | ⟨GS(xi)| GS(xj)⟩|2 us-
ing the quantum hardware, e.g. via SWAP test or
alternative methods [29].
In any case, possible approximations must be considered.
With tensor networks the approximation follows from the
finite value of the bond dimension, which we denote by
χ.
On the other hand, when performing experiments
in a quantum hardware, the approximations follow from
the poor convergence of the variational algorithm, the
imperfect estimation of the kernel (due to shot noise),
and the imperfect quantum gates and measurements. Be
that as it may, the end effect is a noisy estimate of the
kernel matrix. In this paper, we deliberately use tensor
networks with small χ, so as to assess the performances
of our clustering technique under imperfect data.
A schematic view of the overall process is provided
in Fig. 1. In order to benchmark the reliability of the
Spectral
Clustering
FIG. 1.
Outline of the clustering procedure for quantum
ground states, as described in the introduction. The process
begins with randomly producing ground states of a target
Hamiltonian, either using classical approximations (e.g. ten-
sor networks), a quantum hardware with physical spins, or a
quantum computer. The kernel is then estimated either in
hardware or via tensor network techniques. Next, a kernel-
driven clustering model groups the ground states based on
their correlation structure, capturing quantum similarities,
including entanglement properties.
Finally, we present the
resulting clusters.
obtained QPD we will demand that, within the domain
where DMRG remains accurate, there exists at least one
subregion of the parameter-space in which the phase to
which the ground state belongs is already known and
possibly characterized.
The remainder of this article is organized as follows.
Section II introduces the dataset construction.
Sec-
tion III describes the kernel-based unsupervised model
and the mathematical methods used to determine the
optimal number of clusters. Section IV presents bench-
mark studies on the Axial Next-Nearest-Neighbor Ising
(ANNNI) and Cluster-Ising models. Finally, Section V
summarizes the findings and highlights the key contribu-
tions of this work.
II.
FROM CLASSICAL DATA TO QUANTUM
FEATURES
We consider one-dimensional spin- 1
2 models, with short
enough interaction-range for DMRG to be efficient in nu-
merically determine their ground state [2, 3, 30–36]; these
models often exhibit nontrivial QPD that we explore by
moving around in their parameter space, which is Rq. As
explained in the introduction, each point in this space is
a datum xi = (xi1, xi2..., xiq), in one-to-one correspon-
dence with the Hamiltonian ˆH(xi) and its ground state
|GS(xi)⟩. We wander erratically in this parameter-space,
randomly generating D data {xi}, i = 1, ..., D, assuming
no prior knowledge about the different phases, let alone
the QPD.
Specifically, we will consider model-Hamiltonians ˆH
that can be decomposed as ˆH = PN
j=1 ˆHj, where each
local Hamiltonian ˆHj involves a small number of sites
near j, thus acting non-trivially only on the subspace of
a limited number of spins, so that a compact and com-
3
1
2
8
3
4
5
6
7
= 0
p = 1
= 1
p = 1
= 1
p = 2
= 0
p = 2
= 2
p = 2
= 0
p = 3
= 1
p = 3
= 2
p = 3
= 3
p = 3
p = 0
= 0
FIG. 2. Graphical representation of a typical local Hamilto-
nian, illustrating how the automaton structure translates into
the explicit MPO representation.
putationally efficient representation in terms of MPO is
possible. In fact, for Hamiltonians defined on an open
chain, it can be shown that
ˆH = ˆℓ1 ˆW2 ˆW3 · · · ˆWN−1ˆrN,
(1)
where each matrix element ( ˆWj)lm is an operator acting
on the jth spin Hilbert space and the boundary terms
are such that ˆℓj and ˆrj are the first row and the last
column of the ˆWj matrix, respectively. The multiplica-
tion in Eq. (1) corresponds to a matrix product in the
representation space and a tensor product in the physi-
cal one [37]; the specific structure of the operator-matrix
ˆW is not identifiable at first sight, but can be obtained
with the help of a graphical representation borrowed from
finite-state automata, as we explain below.
Be d the distance (expressed in number of lattice sites)
between the furthest interacting spins:
terms in the
Hamiltonian involve at most d + 1 single-spin operators.
We assume that the local Hamiltonian ˆHj can be written
as
ˆHj =
d
X
p=0
p
O
λ=0
p
λˆhj+λ,
(2)
where the index in brackets indicates that the single-site
operator p
λˆh(j+λ) acts upon the system sitting at site j+λ.
For instance, if the model includes next to next-nearest
neighbours interactions, it is d = 3 and we can expect
tensor products of at most d+1 = 4 single-spin operators
in each ˆHj.
A graph of numbered nodes and labelled
links can be drawn from Eq. (2) as shown in Fig.(2):
d + 1 = 4 paths spout from node 1, each path formed by
an increasing number of links, from 1 to d + 1 = 4, and
each ending in the last node, numbered d(d+1)/2+2 = 8.
Different paths stand for different types of interaction,
with the number of links equal to the number of single-
site spin operators involved.
Referring to Eq. (2) this
means that paths are labelled by the index p.
Paths
with increasing number of links are drawn from bottom
to top so that the lowest refers to local terms, the second
one to nearest-neighbours interaction, the third one to
next-nearest neighbours interaction, and so on.
Links
are labeled by the pair (p,λ), with p identifying the path
to which they belong, and λ counting their position w.r.t.
the first node, so that the first link of each path has λ = 0,
the second one λ = 1 and so on, until the (d + 1)(d +
2)/2 = 10th link, having λ = d = 3 (that only appears
in the d + 1 = 4th path). Finally, nodes are numbered in
increasing order from bottom to top and left to right.
Once the graph is drawn, the operator-matrices in
Eq. (1) have dimension equal to the number of nodes,
i.e. d(d + 1)/2 + 2 = 8, with elements ( ˆWj)lm different
from zero if there exists a link (p, λ) connecting nodes l
and m, in which case it is ( ˆWj)lm =p
λ ˆhj. Specific ex-
amples of such graphs, together with their corresponding
operator matrices, are shown in Fig. 2 and Equation 3,
respectively
ˆWj =
I
1
0ˆhj
2
0ˆhj
0
3
0ˆhj
0
0
0
0ˆhj
0
0
0
0
0
0
0
1
1ˆhj
0
0
0
2
1ˆhj
0
0
0
0
0
0
0
0
0
0
0
2
2ˆhj
0
0
0
0
0
3
1ˆhj
0
0
0
0
0
0
0
0
3
2ˆhj
0
0
0
0
0
0
0
0
3
3ˆhj
0
0
0
0
0
0
0
I
.
(3)
The graphical representation encodes the same oper-
ator structure as the matrix in Eq. (3), but in a more
compact and intuitive form. Each edge (green circles) in
the graph corresponds to a possible transition weighted
by a local operator, while the adjacency-like structure
(blue circles) directly translates into the block entries of
the operator matrix.
In this way, the finite–state au-
tomaton view and the matrix formulation provide two
complementary perspectives of the same MPO construc-
tion.
MPO representations allow for an adequate imple-
mentation of variational algorithms such as the DMRG,
whose numerical efficiency relies on the compactness of
such representation.
Within the algorithm, an initial
state is iteratively modified via a unitary process that
depends on the parameters (xi) so as to minimize its en-
ergy. Without entering into the details of this well estab-
lished method, for which we refer the reader to Ref. [38],
we here mention the relevance of the bond dimension χ.
Its logarithm gauges the number of spins that we expect
to be significantly entangled in the ground state of our
chain, not to be mistaken for the range of the interaction
or the size of the ˆWj matrices. Could one numerically
deal with a bond dimension as large as O(2N), then the
algorithm would be exact. However, as this is not possi-
ble for large N, the algorithm involves a well-thought-out
4
adjustment of the bond dimension. In our case, for in-
stance, we ask for i) a reasonable computational time for
producing as many ground states as necessary for feed-
ing our unsupervised classification, and ii) a knowledge
of their structure precise enough to obtain a QPD that
makes sense and can be understood in terms of qualita-
tive features of ground states in different phases.
III.
CLUSTERING
Let us now focus on clustering approaches: these are
all aimed at grouping objects according to some sort of
similarity principle, that can be the most diverse depend-
ing on the specific goal of the clustering itself. The key
concept of the process is that of affinity betweeen two
data, as measured by some quantity whose domain is not
the space where the data are defined, but rather the so-
called feature space. This quantity must be defined for
all possible pairs of the dataset upon which the cluster-
ing is due, and then organized in what is usually called
affinity matrix, or kernel.
We choose to measure the pairwise affinity between
any two data xi, xj via the square fidelity between the
corresponding ground states, as obtained by MPO and
DMRG,
Kij = K(xi, xj) = | ⟨GS(xi)| GS(xj)⟩|2 ,
(4)
organized in the D × D real positive matrix, that is our
Kernel. Notice that the distance between xi and xj in Rq
has nothing to do with the actual value of Kij, which is
rather related with the distance between |GS(xi)⟩and
|GS(xj)⟩as induced by the inner-product in the 2N-
dimensional Hilbert space of our chain.
Once the Kernel is numerically obtained, we can pro-
ceed with the clustering process; amongst the various
techniques available for the scope, spectral clustering via
k-means algorithm emerges as a suitable choice, and this
is how it goes: first a “similarity” graph is generated from
the quantum Kernel, in such a way that indeces i, j of its
matrix-elements identify nodes i and j of the graph, and
edges are drawn if Kij is larger than a threshold value
τ < 1 chosen at will; the graph Laplacian L is then de-
termined, via Lij = Aij −Dij, with Aij = 1 if i and
j are connected and zero otherwise, while Dij = δijdi
where di is the degree of the ith node, i.e. the number
of nodes with which it is connected (Aij and Dij are
the elements of the so called adjacency and degree ma-
trices, respectively). Once the eigendecomposition of L
is performed, providing D eigenvalues and eigenvectors,
a number c < D is chosen, to be the desired number
of clusters into which grouping the data.
Then, the c
eigenvectors corresponding to the c largest eigenvalues
are retained, to become the coloumns of a D × c matrix,
∆, whose rows are the D elements to be finally clustered,
while elements of each row are the c features to be used
for clustering. Notice that each row ∆i∗is in bijective
relation with a datum xi via the row-index i, meaning
that grouping the rows effectively amount to cluster the
original data.
The sheer clustering is then performed via the stan-
dard k-means algorithm, whose goal is that of grouping
the D rows of the above matrix ∆into c different clusters,
based on the c features selected when ignoring all but the
c largest eigenvalues of the Laplacian. Reasons for per-
forming the spectral analysis of the Laplacian rather then
the Kernel itself reside in the particular way L retains
properties of the Kernel that are relevant for clustering
(whether K(i, j) is larger than τ or not), ignoring useless
information (the specific value of each K(i, j)).
Without entering the details of the k-means algorithm,
for which we refer the reader to the literature [39], we
recall that c is a crucial and yet arbitrary number and
ask ourselves how to properly choose it. In fact, if c is
too near to D we have an essentially useless clustering,
with just a few data in each cluster and no significant
computational advantage. On the other hand, too low
a c might imply the loss of resolution and information
in our classification process, computationally fast but in
vain.
There are several ways to determine the optimal num-
ber of clusters c in k-means clustering, amongst which
the Elbow Method [40, 41] and the Silhouette method [42],
that belong to different classes of approaches, often re-
ferred to as within-cluster and between-clusters, respec-
tively, to represent their different logic. In fact Elbow
Method provides for plotting the Within-Cluster Sum of
Squares (WCSS), a distance metric that measures the
spread of data points within each cluster, as a function
of c, so as to find where WCSS stops being strongly de-
pendent on c itself, which is where the “elbow” is placed
in the WCSS curve that reminds of a bended arm as c
increases. After the “elbow” point further increases in
c lead to marginal reductions in WCSS, indicating that
adding more clusters provides little additional value and
may even result in overfitting [43, 44].
Unfortunately,
finding the elbow point is not enough, as some sort of
validation is necessary to avoid dealing with a truly op-
timal c and not with the result of some arbitrary fitting.
To this aim we consider the Silhouette Method, that as-
sesses the quality of a c-clustering by evaluating the so
called silhouette width
s(i) =
b(i) −a(i)
max(a(i), b(i)),
(5)
where i refers to one of the elements to be clustered,
a(i) is the average distance from i to all other elements
in the same cluster, and b(i) is the minimum average
distance from i to elements in other clusters, with the
distance being defined by the same metric employed in
the clustering procedure. In this paper all distances are
evaluated starting from the quantum kernel matrix.
The silhouette width ranges from −1 to 1, with values
close to 1 indicating well-clustered elements, and nearly-
zero values suggesting that the element could equally well
belong to another cluster. The average silhouette width
5
across all elements is used to determine the best c; the
largest average silhouette width indicates the most suit-
able number of clusters. This method is particularly ef-
fective as it balances within-cluster tightness and sepa-
ration from other clusters, providing a clear metric for
evaluating clustering performance [45, 46]. Ultimately, a
high silhouette score for the value of c in the elbow region
confirms that the clusters are well-separated, thereby val-
idating the selection of the elbow point as an appropriate
choice.
An effective way of visualizing the quality of the clus-
tering as obtained for a certain value of c is the so called
silhouette plot, examples of which are shown in Figs. 4(b)
and 6(b). The plot is drawn as follows: to each element
to be clustered, say xi, is associated a horizontal bar
whose length is proportional to the silhouette score s(i)
defined in Eq. (5).
Negative values are shown as bars to the left of the
vertical axis. Bars are grouped in bands according to the
cluster assigned to their respective elements and piled
bottom-up for increasing s(i) within each band. The or-
der in which different bands are displayed is arbitrary.
The vertical width of each band visualizes the number of
elements assigned to the respective cluster by the spectral
algorithm. A vertical line indicating the average silhou-
ette score is also shown: a band with a large number
of bars exceeding such average indicates a well identified
cluster, while bars to the left suggests the corresponding
elements have been assigned to the wrong cluster.
IV.
RESULTS
A.
Axial Next-Nearest-Neighbor Ising (ANNNI)
Model
Our first case study is the Axial Next-Nearest Neighbor
Ising (ANNNI) chain of N spin- 1
2 particles [30, 47–49],
defined by the Hamiltonian
H = J
−
N−1
X
j=1
σx
jσx
j+1 + k
N−2
X
j=1
σx
jσx
j+2 −h
N
X
j=1
σz
j
,
(6)
where σx
j and σz
j are the Pauli operators acting on the
spin at site j. We set the overall energy scale J = 1 and
take k and h positive so that, besides the ferromagnetic
nearest-neighbour (NN) exchange along the x direction,
described by the first term in Eq. (6), there is an antifer-
romagnetic interaction between next-nearest-neighbours
(NNN), gauged by the value of k, and a uniform mag-
netic field h pointing in the positive z direction. When
h = 0 the system behaves classically, as all terms in the
Hamiltonian commute with each other. When k = 0 the
model reduces to the celebrated Ising model in a trans-
verse field, exactly solvable via fermionization and ex-
tensively studied since the ’70th of the last century – see
Ref. [36] and references therein for a review. For k > 0
1
2
5
3
4
x
x
x
k
x
h
z
FIG. 3. Local interaction in the MPO formalism for ANNNI
model using finite-state automata graphichal representation.
the system exhibits frustration, as the first term tends to
align all spins along the x-axis while the second one pro-
motes antialignment of NNN. The phase diagram of the
ANNNI model has been explored using various analytical
and numerical approaches [30, 49–53], and the possible
use of tensor network techniques to determine its ground
state has been investigated in Refs. [2, 16, 50, 54].
For small values of h and k, the NN interaction is the
dominant term, leading to a ferromagnetic phase. As h
increases beyond a critical threshold the system enters
a paramagnetic phase where all spins get a positive z-
component while pointing in random directions on the
xy plane. When k is large and h stays sufficiently small
the NNN interaction takes over, stabilizing the so-called
antiphase order, in which spins align along x in pairs,
with alternate sign from one pair to the other. Between
the antiphase and paramagnetic phases, the system en-
ters a floating phase and, in the thermodynamic limit
N →∞, becomes gapless, indicating the emergence of a
critical behavior.
Although the general structure of the QPD is well es-
tablished, there persist discrepancies in the identification
of phase boundaries as obtained by different methods
[55, 56]. In particular, while the primary phases depicted
in Fig. 4c are widely recognized, some studies propose
that the paramagnetic phase may split into two distinct
regions [49], while others suggest the existence of an ex-
tended floating phase, with evidence reported in various
studies [30, 55, 57]. In Fig. 4 (c) we mark with black lines
the phase transitions as previously obtained by pertur-
bative and numerical techniques [8, 50].
As for this work, referring to Eq. (2) the ANNNI hamil-
tonian (6) has d = 2, 0
0hj = hσz
j , ⊗1
λ=0
1
λhj+λ = −σx
j σx
j+1,
and ⊗2
λ=0hj+λ = kσx
j σx
j+2; using the prescription given
in Sec. II we draw the corresponding graph (see Fig. 3)
6
2
4
6
8
The Number of Clusters
0.3
0.4
0.5
Silhouette Score
(a)
Maximum Silhouette Score
2
4
6
8
c
WCSS
Elbow Point
0.1 0.0
0.5
1.0
The silhouette coefficient values
Cluster label
Average silhouette
(b)
Antiphase
Paramagnetic
Ferromagnetic
Floating Phase
0.5
1.0
k
1.0
2.0
h
Antiphase
Paramagnetic
Ferromagnetic
(c)
Floating Phase
0.0
FIG. 4. Results for the ANNNI Hamiltonian (6) with N = 51 spins, where the ground states are simulated with a bond
dimension χ = 20. (a) Average silhouette score as a function of the number of clusters c; the inset shows the elbow point,
identified by a significant change in the WCSS dependence on c. (b) Silhouette plot for c = 4, as determined by Fig. 4a. (c) The
predicted phase diagram.
to which it corresponds the MPO
ˆW =
I σx σx 0 −hσz
0
0
0
0
−σx
0
0
0
I
0
0
0
0
0
kσx
0
0
0
0
I
.
(7)
Our results for a chain of N = 51 spins, sampling
30 × 30 pairs (k, h), are shown in Fig. 4. The ground
states were obtained using the DMRG algorithm, as im-
plemented in the quimb library [58]. In order to choose
the optimal c for the spectral clustering we have used
the silhouette method: Fig. 4a shows s(i) as a function
of the number of clusters c: the largest value is seen for
c = 4, suggesting that four clusters might provide a well-
separated grouping of the data.
Moreover, the elbow
method applied to WCSS is shown in the inset: an evi-
dent inflection point is observed at c = 4. Fig. 4b shows
the silhouette-plot for c = 4, with the dashed vertical
line marking the average silhouette score: the plot con-
firms the effectiveness of spectral clustering with c = 4,
though some elements in the paramagnetic phase have
negative s(i), indicating misclustering. The clusters cor-
responding to the ferromagnetic and floating phases have
smaller but entirely positive silhouette scores, suggesting
well-defined boundaries.
The antiphase and the para-
magnetic phase, though, despite having relatively high
silhouette scores, show variations in width, pointing to
fluctuations in phase boundaries.
Fig. 4c shows our QPD and the previously predicted
boundaries [8, 50], indicated by the black lines. Discrep-
ancies are seen near the phase transitions, which may be
attributed either to finite-size effects, or to the DMRG
bond dimensionm χ that we have set to 20. In fact, near
transitions the entanglement area law breaks down [59],
potentially requiring a larger bond dimension to accu-
rately capture the correlations of the system.
B.
Cluster-Ising Model with a symmetry-protected
topological phase
Amongst the many different phases possibly featured
by QPD of many-body systems, the so called symmetry-
protected topological (SPT) ones are particularly intrigu-
ing.
An example of such phases can be found in the
renowned Haldane chain, where the protecting symme-
try is the discrete Z2 × Z2 one. This same symmetry,
realized via π rotations around the x-axis of all spins
along the chain, is also featured by the spin- 1
2 chain with
hamiltonian
H = J
−
N−2
X
j=1
σz
jσx
j+1σz
j+2 −h2
N−1
X
j=1
σx
jσx
j+1 −h1
N
X
j=1
σx
j
,
(8)
where the first term encodes the 1D cluster interaction,
h2 represents an Ising coupling, h1 is a transverse field,
and J > 0 sets the overall energy scale, whose value
does not enter the QPD. For dominant cluster interac-
tion, i.e. small h1 and h2, and in the presence of the pro-
tecting Z2×Z2 symmetry, the ground state of open chains
realizes a short-range entangled, gapped SPT phase char-
acterized by spin-1/2 edge modes, nonlocal string order,
and twofold entanglement-spectrum degeneracy [60–63].
These features remain robust as long as the system is
gapped; they disappear only when the bulk gap closes
at a phase transition [61, 62, 64]. Competing with this
cluster phase are the paramagnetic and antiferromagnetic
phases, reached respectively by increasing h1 or h2, with
critical behavior that departs from the conventional Ising
universality class. Recent studies have further evidenced
the presence of this SPT phase through quantum convo-
lutional neural network circuit [3, 65] and by a supervised
learning approach [2].
In what follows we put our method to the test by ap-
plying it to the model (8), with J = 1 and h1,2 free to
take positive or negative values.
Referring to Eq. (2),
it is d = 2, 0
0hj = −h1σx
j , ⊗1
λ=0
1
λhj+λ = −h2σx
j σx
j+1,
and ⊗2
λ=0
2
λhj+λ = −σz
j σx
j+1σz
j+2; using the prescription
7
1
2
5
3
4
x
h2
x
z
z
x
h1
z
FIG. 5. Local interaction in the MPO formalism for Cluster-
Ising model using finite-state automata graphichal represen-
tation.
given in Sec. II we draw the corresponding graph, see
Fig. 5, to which it corresponds the MPO
ˆW =
I σx σz
0
−h1σx
0
0
0
0
−h2σx
0
0
0
σx
0
0
0
0
0
−σz
0
0
0
0
I
.
(9)
Fig. 6 presents our numerical results for a chain of
N = 51 spins, sampling 30×30 pairs (h1, h2). We use the
silhouette method to determine the number of phases, i.e.
the optimal number of clusters, c. The silohuette score
as a function of c is shown in Fig. 6a, with the highest
score at c = 3. The WCSS vs. c shown in the inset con-
firms this result, according to the Elbow method. The
silhouette plot is shown in Fig. 6b, where the dashed ver-
tical line marks the average silhouette score. The results
demonstrate that phase boundaries are identified even
when the elusive SPT phase is involved, as silhouette
scores are mostly positive. However, some misclusterings
are observed in the paramagnetic and antiferromagnetic
phases, indicated by few negative silhouette scores. Fi-
nally, the QPD for the model (8) is shown in Fig. 6c. Its
structure closely matches the expected one, according to
the above mentioned previous results [3, 65], represented
by the black dashed lines. Minor discrepancies near the
transition lines may arise from finite-size effects or limi-
tations in the DMRG bond dimension used.
V.
CONCLUSION AND DISCUSSION
We explored the use of classical unsupervised machine
learning methods for studying QPD of quantum many-
body systems. We found that kernel-based spectral clus-
tering methods, where the different clusters define the
different quantum phases, are effective when fed by ker-
nels of genuine quantum origin. Importantly, the number
of phases is discovered by the algorithm itself.
Two specific models of spin- 1
2 chains, the ANNNI and
the Cluster-Ising models, the latter of which featuring
a topological phase, are used to benchmark our results.
The bond dimension χ is a free parameter that defines
the noise in the dataset, in the language of machine learn-
ing. For the ANNNI model, a moderate bond dimension
(χ = 20) is sufficient, whereas the Cluster–Ising model
requires a relatively larger bond dimension (χ = 70). It
is relevant that the clustering approach successfully de-
tects complex phases, such as those lacking a local order
parameter, demonstrating its ability to capture nontriv-
ial topological features directly from quantum-generated
datasets.
Our findings show that integrating classical unsuper-
vised machine learning models with quantum-generated
datasets can provide physicists with a powerful tool for
recognizing and classifying quantum phases of matter.
Our work paves the way to the study of systems with
completely unknown phase diagrams, without any prior
labeling. Amongst these, spin liquids [66, 67], where con-
ventional theoretical or numerical guidance is limited,
and spin glasses [68], represent a promising avenue for
future exploration, offering an opportunity to extend our
methods to even more challenging quantum phases.
Another future direction is to replace our quantum
kernels, estimated via tensor networks, with kernels di-
rectly reconstructed from experimental data, possibly us-
ing classical shadow techniques [69] or other randomized
measurements [70].
VI.
ACKNOWLEDGMENT
We thank Giuseppe Magnifico for valuable discussion.
All authors acknowledge financial support from: PNRR
Ministero Universit`a e Ricerca Project No. PE0000023-
NQSTI funded by European Union-Next-Generation EU.
L.B. also acknowledges financial support from:
Prin
2022 - DD N. 104 del 2/2/2022, entitled “understand-
ing the LEarning process of QUantum Neural net-
works (LeQun)”, proposal code 2022WHZ5XH, CUP
B53D23009530006; the European Union’s Horizon Eu-
rope research and innovation program under EPIQUE
Project GA No. 101135288. This work is done in the
framework of the Convenzione operativa between the In-
stitute for Complex Systems of the Consiglio Nazionale
delle Ricerche (Italy) and the Physics and Astronomy
Department of the University of Florence
8
2
4
6
8
The Number of Clusters
0.4
0.5
0.6
0.7
0.8
Silhouette Score
(a)
Maximum Silhouette Score
2
4
6
8
c
WCSS
Elbow Point
0.1 0.0
0.2
0.4
0.6
0.8
1.0
The silhouette coefficient values
Cluster label
Average silhouette
(b)
Paramagnetic
Antiferromagnetic
SPT
0.0
0.8
1.6
h1
-1.6
0.0
1.6
h2
(c)
Paramagnetic
SPT
Antiferomagnetic
FIG. 6. Results for the Cluster-Ising Hamiltonian (8) with N = 51 spins, where the ground states are simulated with a bond
dimension χ = 70. (a) Average silhouette score as a function of the number of clusters c; the inset shows the elbow point,
identified by a significant change in the WCSS dependence on c. (b) Silhouette plot for c = 3, as determined by Fig. 6a. (c) The
predicted phase diagram.
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Unsupervised Learning to Recognize Quantum Phases of Matter Mehran Khosrojerdi,1, ∗Alessandro Cuccoli,1, 2 Paola Verrucchi,3, 1, 2 and Leonardo Banchi1, 2 1 . Sansone 1, I-50019 Sesto Fiorentino (FI), Italy 2INFN Sezione di Firenze, via G. Sansone 1, I-50019, Sesto Fiorentino (FI), Italy 3ISC-CNR, UOS Dipartimento di Fisica, Universit`a di Firenze, I-50019, Sesto Fiorentino (FI), Italy (Dated: October 17, 2025) Drawing the quantum phase diagram of a many-body system in the parameter space of its Hamiltonian can be seen as a learning problem, which implies labelling the corresponding ground states according to some classification criterium that defines the phases. In this work we adopt unsupervised learning, where the algorithm has no access to any priorly labeled states, as a tool for determining quantum phase diagrams of many-body systems. The algorithm directly works with quantum states: given the ground-state configurations for different values of the Hamiltonian parameters, the process uncovers the most significant way of grouping them based on a similarity criterion that refers to the fidelity between quantum states, that can be easily estimated, even experimentally. We benchmark our method with two specific spin- 1 2 chains, with states determined via tensor network techniques. We find that unsupervised learning algorithms based on spectral clustering, combined with "silhouette" and "elbow" methods for determining the optimal number of phases, can accurately reproduce the phase diagrams. Our results show how unsupervised learning can autonomously recognize and possibly unveil novel phases of quantum matter. I. INTRODUCTION Quantum many-body systems can exhibit phasetransitions also at T = 0 due to purely quantum fluctuations [1], making it possible for Ground States to feature different collective properties for infinitesimally different values of the hamiltonian parameters. Studying the Quantum Phase Diagrams (QPD) outlined by these transitions, as well as the quantum phases themselves, is of interest not only in the realm of many-body physics but also in quantum information processing, whose operation happens at temperatures as near as possible to T = 0. On the other hand, drawing the QPD of quantum many-body systems is a hard task, unless one considers very simple models such as, for instance, the Ising chain in a transverse field. In fact, analytical methods are of limited help in studying models of realistic systems, while established numerical techniques demand huge computational resources, that become unaffordable as the size of the models grows, as required by an evaluation of nonlocal quantities that characterize different phases, such as long-range correlations and concurrence, or multipartite entanglement. In this framework, approaches based on hybrid quantum-classical machine-learning methods provide some alternatives. One possibility comes from quantumphase classification based on training on labeled data, where some prior theoretical or numerical understanding of the phase-diagram is necessary, at least in a limited parameter region. This approach matches a supervised learning task, where the phases are known for some particular parameter values that are used to train the clas- ∗ sification algorithm. The algorithm is then used to explore larger areas of the parameter space and label the corresponding phase [2-12]. Quantum states clustering, instead, uses an unsupervised learning strategy, whose goal is to identify similarities among ground states, given no prior knowledge of the QPD, and hence without preexisting labeling of the different phases. Unlike classification, which is typically applied as a validation tool, clustering can be used to explore novel quantum systems, revealing hidden structures in the data, and uncovering emergent phases of matter without relying on predefined categories [13-16]. Needless to say, clustering quantum states to unveil a phase-diagram is a challenging task, due to the inherently non-local nature of quantum correlations and entanglement [1, 17]: in fact, when the Hilbert space dimension grows, identifying features for an effective clustering is by itself a difficult task. On the other hand, some unsupervised methods [18-22], even if acting on classical data-sets, can turn out to be suitable to spot similarities between elements of large Hilbert spaces, providing an interpretable framework for clustering quantum states based on their genuinely quantum properties. Scope of this work is to develop a method for quantumstates clustering, precise enough to match QPD of parameterized Hamiltonians whose ground state can be determined and represented via tensor networks methods, such as the DMRG [23-25]. Since our method uses a "kernel" between two Hamiltonian parameter sets given by the fidelity between the corresponding ground states, it can be extended to work directly with ground states approximated in quantum devices, since the fidelity between states can be experimentally estimated using different methods such as the SWAP test or the Hadamard test, see e.g. [26]. Our procedure goes as follows: be ˆH(x) a onedimensional Hamiltonian that fulfills the above require16 Oct 2025 2 ments, with x = (x1, x2, . . . xq) the set of q parameters upon which it depends. We take four steps for predicting the QPD with the desired insight into different quantum phases. 1. Generate D sets of Hamiltonian parameters {xi = (x1i, x2i, ...xiq)} according to some rule (e.g. uniformly in the parameter space), with i = 1, ..., D. This set defines our "classical data", where each datum is an element of Rq and a point in the phasediagram. 2. Represent the Hamiltonian ˆH(xi) as a Matrix Product Operator (MPO) for each datum xi and find its ground state |GS(xi)⟩. 3. Define the kernel between two data xi and xj as Kij = | ⟨GS(xi)| GS(xj)⟩|2, namely as the fidelity between two ground states |GS(xi)⟩and |GS(xj)⟩. In the language of machine learning, this corresponds to defining the feature space as the O(4N)-dimensional space of density matrices |GS(xi))⟩⟨GS(xi)|, where N is the number of qubits in ˆH(x). Note that the dimensionality of the feature space is much larger than that of the dataset, which is one of the key-element of any machine-learning procedure for clustering, as it reflects the richer characterization offered by features w.r.t. data. 4. Perform a kernel clustering algorithm, which only depends on the kernel matrix K, so as to group ground states according to the similarity amongst their quantum features. Notice that, although not explicitly considered in this paper, steps 2. and 3. can be replaced with: 2b. Approximate each ground state |GS(xi)⟩in a quantum device, e.g. using a parametric quantum algorithm and the variational quantum eigensolver [27] or more recent versions [28]. 3b. Compute the kernel Kij = | ⟨GS(xi)| GS(xj)⟩|2 using the quantum hardware, e.g. via SWAP test or alternative methods [29]. In any case, possible approximations must be considered. With tensor networks the approximation follows from the finite value of the bond dimension, which we denote by χ. On the other hand, when performing experiments in a quantum hardware, the approximations follow from the poor convergence of the variational algorithm, the imperfect estimation of the kernel (due to shot noise), and the imperfect quantum gates and measurements. Be that as it may, the end effect is a noisy estimate of the kernel matrix. In this paper, we deliberately use tensor networks with small χ, so as to assess the performances of our clustering technique under imperfect data. A schematic view of the overall process is provided in Fig. 1. In order to benchmark the reliability of the Spectral Clustering FIG. 1. Outline of the clustering procedure for quantum ground states, as described in the introduction. The process begins with randomly producing ground states of a target Hamiltonian, either using classical approximations (e.g. tensor networks), a quantum hardware with physical spins, or a quantum computer. The kernel is then estimated either in hardware or via tensor network techniques. Next, a kerneldriven clustering model groups the ground states based on their correlation structure, capturing quantum similarities, including entanglement properties. Finally, we present the resulting clusters. obtained QPD we will demand that, within the domain where DMRG remains accurate, there exists at least one subregion of the parameter-space in which the phase to which the ground state belongs is already known and possibly characterized. The remainder of this article is organized as follows. Section II introduces the dataset construction. Section III describes the kernel-based unsupervised model and the mathematical methods used to determine the optimal number of clusters. Section IV presents benchmark studies on the Axial Next-Nearest-Neighbor Ising (ANNNI) and Cluster-Ising models. Finally, Section V summarizes the findings and highlights the key contributions of this work. II. FROM CLASSICAL DATA TO QUANTUM FEATURES We consider one-dimensional spin- 1 2 models, with short enough interaction-range for DMRG to be efficient in numerically determine their ground state [2, 3, 30-36]; these models often exhibit nontrivial QPD that we explore by moving around in their parameter space, which is Rq. As explained in the introduction, each point in this space is a datum xi = (xi1, xi2..., xiq), in one-to-one correspondence with the Hamiltonian ˆH(xi) and its ground state |GS(xi)⟩. We wander erratically in this parameter-space, randomly generating D data {xi}, i = 1, ..., D, assuming no prior knowledge about the different phases, let alone the QPD. Specifically, we will consider model-Hamiltonians ˆH that can be decomposed as ˆH = PN j=1 ˆHj, where each local Hamiltonian ˆHj involves a small number of sites near j, thus acting non-trivially only on the subspace of a limited number of spins, so that a compact and com3 1 2 8 3 4 5 6 7 = 0 p = 1 = 1 p = 1 = 1 p = 2 = 0 p = 2 = 2 p = 2 = 0 p = 3 = 1 p = 3 = 2 p = 3 = 3 p = 3 p = 0 = 0 FIG. 2. Graphical representation of a typical local Hamiltonian, illustrating how the automaton structure translates into the explicit MPO representation. putationally efficient representation in terms of MPO is possible. In fact, for Hamiltonians defined on an open chain, it can be shown that ˆH = ˆl1 ˆW2 ˆW3 · · · ˆWN-1ˆrN, (1) where each matrix element ( ˆWj)lm is an operator acting on the jth spin Hilbert space and the boundary terms are such that ˆlj and ˆrj are the first row and the last column of the ˆWj matrix, respectively. The multiplication in Eq. (1) corresponds to a matrix product in the representation space and a tensor product in the physical one [37]; the specific structure of the operator-matrix ˆW is not identifiable at first sight, but can be obtained with the help of a graphical representation borrowed from finite-state automata, as we explain below. Be d the distance (expressed in number of lattice sites) between the furthest interacting spins: terms in the Hamiltonian involve at most d + 1 single-spin operators. We assume that the local Hamiltonian ˆHj can be written as ˆHj = d X p=0 p O λ=0 p λˆhj+λ, (2) where the index in brackets indicates that the single-site operator p λˆh(j+λ) acts upon the system sitting at site j+λ. For instance, if the model includes next to next-nearest neighbours interactions, it is d = 3 and we can expect tensor products of at most d+1 = 4 single-spin operators in each ˆHj. A graph of numbered nodes and labelled links can be drawn from Eq. (2) as shown in Fig.(2): d + 1 = 4 paths spout from node 1, each path formed by an increasing number of links, from 1 to d + 1 = 4, and each ending in the last node, numbered d(d+1)/2+2 = 8. Different paths stand for different types of interaction, with the number of links equal to the number of singlesite spin operators involved. Referring to Eq. (2) this means that paths are labelled by the index p. Paths with increasing number of links are drawn from bottom to top so that the lowest refers to local terms, the second one to nearest-neighbours interaction, the third one to next-nearest neighbours interaction, and so on. Links are labeled by the pair (p,λ), with p identifying the path to which they belong, and λ counting their position w.r.t. the first node, so that the first link of each path has λ = 0, the second one λ = 1 and so on, until the (d + 1)(d + 2)/2 = 10th link, having λ = d = 3 (that only appears in the d + 1 = 4th path). Finally, nodes are numbered in increasing order from bottom to top and left to right. Once the graph is drawn, the operator-matrices in Eq. (1) have dimension equal to the number of nodes, i.e. d(d + 1)/2 + 2 = 8, with elements ( ˆWj)lm different from zero if there exists a link (p, λ) connecting nodes l and m, in which case it is ( ˆWj)lm =p λ ˆhj. Specific examples of such graphs, together with their corresponding operator matrices, are shown in Fig. 2 and Equation 3, respectively ˆWj = I 1 0ˆhj 2 0ˆhj 0 3 0ˆhj 0 0 0 0ˆhj 0 0 0 0 0 0 0 1 1ˆhj 0 0 0 2 1ˆhj 0 0 0 0 0 0 0 0 0 0 0 2 2ˆhj 0 0 0 0 0 3 1ˆhj 0 0 0 0 0 0 0 0 3 2ˆhj 0 0 0 0 0 0 0 0 3 3ˆhj 0 0 0 0 0 0 0 I . (3) The graphical representation encodes the same operator structure as the matrix in Eq. (3), but in a more compact and intuitive form. Each edge (green circles) in the graph corresponds to a possible transition weighted by a local operator, while the adjacency-like structure (blue circles) directly translates into the block entries of the operator matrix. In this way, the finite-state automaton view and the matrix formulation provide two complementary perspectives of the same MPO construction. MPO representations allow for an adequate implementation of variational algorithms such as the DMRG, whose numerical efficiency relies on the compactness of such representation. Within the algorithm, an initial state is iteratively modified via a unitary process that depends on the parameters (xi) so as to minimize its energy. Without entering into the details of this well established method, for which we refer the reader to Ref. [38], we here mention the relevance of the bond dimension χ. Its logarithm gauges the number of spins that we expect to be significantly entangled in the ground state of our chain, not to be mistaken for the range of the interaction or the size of the ˆWj matrices. Could one numerically deal with a bond dimension as large as O(2N), then the algorithm would be exact. However, as this is not possible for large N, the algorithm involves a well-thought-out 4 adjustment of the bond dimension. In our case, for instance, we ask for i) a reasonable computational time for producing as many ground states as necessary for feeding our unsupervised classification, and ii) a knowledge of their structure precise enough to obtain a QPD that makes sense and can be understood in terms of qualitative features of ground states in different phases. III. CLUSTERING Let us now focus on clustering approaches: these are all aimed at grouping objects according to some sort of similarity principle, that can be the most diverse depending on the specific goal of the clustering itself. The key concept of the process is that of affinity betweeen two data, as measured by some quantity whose domain is not the space where the data are defined, but rather the socalled feature space. This quantity must be defined for all possible pairs of the dataset upon which the clustering is due, and then organized in what is usually called affinity matrix, or kernel. We choose to measure the pairwise affinity between any two data xi, xj via the square fidelity between the corresponding ground states, as obtained by MPO and DMRG, Kij = K(xi, xj) = | ⟨GS(xi)| GS(xj)⟩|2 , (4) organized in the D × D real positive matrix, that is our Kernel. Notice that the distance between xi and xj in Rq has nothing to do with the actual value of Kij, which is rather related with the distance between |GS(xi)⟩and |GS(xj)⟩as induced by the inner-product in the 2Ndimensional Hilbert space of our chain. Once the Kernel is numerically obtained, we can proceed with the clustering process; amongst the various techniques available for the scope, spectral clustering via k-means algorithm emerges as a suitable choice, and this is how it goes: first a "similarity" graph is generated from the quantum Kernel, in such a way that indeces i, j of its matrix-elements identify nodes i and j of the graph, and edges are drawn if Kij is larger than a threshold value τ 0 1 2 5 3 4 x x x k x h z FIG. 3. Local interaction in the MPO formalism for ANNNI model using finite-state automata graphichal representation. the system exhibits frustration, as the first term tends to align all spins along the x-axis while the second one promotes antialignment of NNN. The phase diagram of the ANNNI model has been explored using various analytical and numerical approaches [30, 49-53], and the possible use of tensor network techniques to determine its ground state has been investigated in Refs. [2, 16, 50, 54]. For small values of h and k, the NN interaction is the dominant term, leading to a ferromagnetic phase. As h increases beyond a critical threshold the system enters a paramagnetic phase where all spins get a positive zcomponent while pointing in random directions on the xy plane. When k is large and h stays sufficiently small the NNN interaction takes over, stabilizing the so-called antiphase order, in which spins align along x in pairs, with alternate sign from one pair to the other. Between the antiphase and paramagnetic phases, the system enters a floating phase and, in the thermodynamic limit N →∞, becomes gapless, indicating the emergence of a critical behavior. Although the general structure of the QPD is well established, there persist discrepancies in the identification of phase boundaries as obtained by different methods [55, 56]. In particular, while the primary phases depicted in Fig. 4c are widely recognized, some studies propose that the paramagnetic phase may split into two distinct regions [49], while others suggest the existence of an extended floating phase, with evidence reported in various studies [30, 55, 57]. In Fig. 4 (c) we mark with black lines the phase transitions as previously obtained by perturbative and numerical techniques [8, 50]. As for this work, referring to Eq. (2) the ANNNI hamiltonian (6) has d = 2, 0 0hj = hσz j , ⊗1 λ=0 1 λhj+λ = -σx j σx j+1, and ⊗2 λ=0hj+λ = kσx j σx j+2; using the prescription given in Sec. II we draw the corresponding graph (see Fig. 3) 6 2 4 6 8 The Number of Clusters 0.3 0.4 0.5 Silhouette Score (a) Maximum Silhouette Score 2 4 6 8 c WCSS Elbow Point 0.1 0.0 0.5 1.0 The silhouette coefficient values Cluster label Average silhouette (b) Antiphase Paramagnetic Ferromagnetic Floating Phase 0.5 1.0 k 1.0 2.0 h Antiphase Paramagnetic Ferromagnetic (c) Floating Phase 0.0 FIG. 4. Results for the ANNNI Hamiltonian (6) with N = 51 spins, where the ground states are simulated with a bond dimension χ = 20. (a) Average silhouette score as a function of the number of clusters c; the inset shows the elbow point, identified by a significant change in the WCSS dependence on c. (b) Silhouette plot for c = 4, as determined by Fig. 4a. (c) The predicted phase diagram. to which it corresponds the MPO ˆW = I σx σx 0 -hσz 0 0 0 0 -σx 0 0 0 I 0 0 0 0 0 kσx 0 0 0 0 I . (7) Our results for a chain of N = 51 spins, sampling 30 × 30 pairs (k, h), are shown in Fig. 4. The ground states were obtained using the DMRG algorithm, as implemented in the quimb library [58]. In order to choose the optimal c for the spectral clustering we have used the silhouette method: Fig. 4a shows s(i) as a function of the number of clusters c: the largest value is seen for c = 4, suggesting that four clusters might provide a wellseparated grouping of the data. Moreover, the elbow method applied to WCSS is shown in the inset: an evident inflection point is observed at c = 4. Fig. 4b shows the silhouette-plot for c = 4, with the dashed vertical line marking the average silhouette score: the plot confirms the effectiveness of spectral clustering with c = 4, though some elements in the paramagnetic phase have negative s(i), indicating misclustering. The clusters corresponding to the ferromagnetic and floating phases have smaller but entirely positive silhouette scores, suggesting well-defined boundaries. The antiphase and the paramagnetic phase, though, despite having relatively high silhouette scores, show variations in width, pointing to fluctuations in phase boundaries. Fig. 4c shows our QPD and the previously predicted boundaries [8, 50], indicated by the black lines. Discrepancies are seen near the phase transitions, which may be attributed either to finite-size effects, or to the DMRG bond dimensionm χ that we have set to 20. In fact, near transitions the entanglement area law breaks down [59], potentially requiring a larger bond dimension to accurately capture the correlations of the system. B. Cluster-Ising Model with a symmetry-protected topological phase Amongst the many different phases possibly featured by QPD of many-body systems, the so called symmetryprotected topological (SPT) ones are particularly intriguing. An example of such phases can be found in the renowned Haldane chain, where the protecting symmetry is the discrete Z2 × Z2 one. This same symmetry, realized via π rotations around the x-axis of all spins along the chain, is also featured by the spin- 1 2 chain with hamiltonian H = J - N-2 X j=1 σz jσx j+1σz j+2 -h2 N-1 X j=1 σx jσx j+1 -h1 N X j=1 σx j , (8) where the first term encodes the 1D cluster interaction, h2 represents an Ising coupling, h1 is a transverse field, and J > 0 sets the overall energy scale, whose value does not enter the QPD. For dominant cluster interaction, i.e. small h1 and h2, and in the presence of the protecting Z2×Z2 symmetry, the ground state of open chains realizes a short-range entangled, gapped SPT phase characterized by spin-1/2 edge modes, nonlocal string order, and twofold entanglement-spectrum degeneracy [60-63]. These features remain robust as long as the system is gapped; they disappear only when the bulk gap closes at a phase transition [61, 62, 64]. Competing with this cluster phase are the paramagnetic and antiferromagnetic phases, reached respectively by increasing h1 or h2, with critical behavior that departs from the conventional Ising universality class. Recent studies have further evidenced the presence of this SPT phase through quantum convolutional neural network circuit [3, 65] and by a supervised learning approach [2]. In what follows we put our method to the test by applying it to the model (8), with J = 1 and h1,2 free to take positive or negative values. Referring to Eq. (2), it is d = 2, 0 0hj = -h1σx j , ⊗1 λ=0 1 λhj+λ = -h2σx j σx j+1, and ⊗2 λ=0 2 λhj+λ = -σz j σx j+1σz j+2; using the prescription 7 1 2 5 3 4 x h2 x z z x h1 z FIG. 5. Local interaction in the MPO formalism for ClusterIsing model using finite-state automata graphichal representation. given in Sec. II we draw the corresponding graph, see Fig. 5, to which it corresponds the MPO ˆW = I σx σz 0 -h1σx 0 0 0 0 -h2σx 0 0 0 σx 0 0 0 0 0 -σz 0 0 0 0 I . (9) Fig. 6 presents our numerical results for a chain of N = 51 spins, sampling 30×30 pairs (h1, h2). We use the silhouette method to determine the number of phases, i.e. the optimal number of clusters, c. The silohuette score as a function of c is shown in Fig. 6a, with the highest score at c = 3. The WCSS vs. c shown in the inset confirms this result, according to the Elbow method. The silhouette plot is shown in Fig. 6b, where the dashed vertical line marks the average silhouette score. The results demonstrate that phase boundaries are identified even when the elusive SPT phase is involved, as silhouette scores are mostly positive. However, some misclusterings are observed in the paramagnetic and antiferromagnetic phases, indicated by few negative silhouette scores. Finally, the QPD for the model (8) is shown in Fig. 6c. Its structure closely matches the expected one, according to the above mentioned previous results [3, 65], represented by the black dashed lines. Minor discrepancies near the transition lines may arise from finite-size effects or limitations in the DMRG bond dimension used. V. CONCLUSION AND DISCUSSION We explored the use of classical unsupervised machine learning methods for studying QPD of quantum manybody systems. We found that kernel-based spectral clustering methods, where the different clusters define the different quantum phases, are effective when fed by kernels of genuine quantum origin. Importantly, the number of phases is discovered by the algorithm itself. Two specific models of spin- 1 2 chains, the ANNNI and the Cluster-Ising models, the latter of which featuring a topological phase, are used to benchmark our results. The bond dimension χ is a free parameter that defines the noise in the dataset, in the language of machine learning. For the ANNNI model, a moderate bond dimension (χ = 20) is sufficient, whereas the Cluster-Ising model requires a relatively larger bond dimension (χ = 70). It is relevant that the clustering approach successfully detects complex phases, such as those lacking a local order parameter, demonstrating its ability to capture nontrivial topological features directly from quantum-generated datasets. Our findings show that integrating classical unsupervised machine learning models with quantum-generated datasets can provide physicists with a powerful tool for recognizing and classifying quantum phases of matter. Our work paves the way to the study of systems with completely unknown phase diagrams, without any prior labeling. Amongst these, spin liquids [66, 67], where conventional theoretical or numerical guidance is limited, and spin glasses [68], represent a promising avenue for future exploration, offering an opportunity to extend our methods to even more challenging quantum phases. Another future direction is to replace our quantum kernels, estimated via tensor networks, with kernels directly reconstructed from experimental data, possibly using classical shadow techniques [69] or other randomized measurements [70]. VI. ACKNOWLEDGMENT We thank Giuseppe Magnifico for valuable discussion. All authors acknowledge financial support from: PNRR Ministero Universit`a e Ricerca Project No. PE0000023NQSTI funded by European Union-Next-Generation EU. L.B. also acknowledges financial support from: Prin 2022 - DD N. 104 del 2/2/2022, entitled "understanding the LEarning process of QUantum Neural networks (LeQun)", proposal code 2022WHZ5XH, CUP B53D23009530006; the European Union's Horizon Europe research and innovation program under EPIQUE Project GA No. 101135288. This work is done in the framework of the Convenzione operativa between the Institute for Complex Systems of the Consiglio Nazionale delle Ricerche (Italy) and the Physics and Astronomy Department of the 8 2 4 6 8 The Number of Clusters 0.4 0.5 0.6 0.7 0.8 Silhouette Score (a) Maximum Silhouette Score 2 4 6 8 c WCSS Elbow Point 0.1 0.0 0.2 0.4 0.6 0.8 1.0 The silhouette coefficient values Cluster label Average silhouette (b) Paramagnetic Antiferromagnetic SPT 0.0 0.8 1.6 h1 -1.6 0.0 1.6 h2 (c) Paramagnetic SPT Antiferomagnetic FIG. 6. Results for the Cluster-Ising Hamiltonian (8) with N = 51 spins, where the ground states are simulated with a bond dimension χ = 70. (a) Average silhouette score as a function of the number of clusters c; the inset shows the elbow point, identified by a significant change in the WCSS dependence on c. (b) Silhouette plot for c = 3, as determined by Fig. 6a. (c) The predicted phase diagram. [1] S. 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2510.14740
|
On Krylov Complexity as a Probe of the Quantum
Mpemba Effect
Mohsen Alishahiha and Mohammad Javad Vasli
School of Quantum Physics and Matter
Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5531, Tehran, Iran
E-mails: alishah@ipm.ir, vasli@ipm.ir
Abstract
We investigate Krylov state complexity as a probe of the quantum Mpemba effect in quan-
tum spin chains. For models without global U(1) symmetry, Krylov complexity exhibits clear
Mpemba-like crossings, consistent with conventional diagnostics such as the trace distance,
while offering a complementary interpretation in terms of Hilbert-space exploration and dy-
namical delocalization. In U(1)-symmetric systems, we confirm that the recently proposed
symmetric component of Krylov complexity serves as a robust and reliable indicator of the
QME, capturing anomalous relaxation even in cases where the total complexity fails to do
so.
arXiv:2510.14740v1 [hep-th] 16 Oct 2025
1
Introduction
The Mpemba effect refers to the counterintuitive phenomenon in which a system prepared further
from equilibrium relaxes faster than one initialized closer to equilibrium.
Originally observed
in water, where initially hotter samples were found to freeze more quickly than colder ones [1],
the effect has since been identified across a wide range of classical dynamical systems [2–10].
These studies established that the Mpemba effect is not a peculiar feature of water, but rather a
generic manifestation of nonequilibrium relaxation, leading to renewed interest in the dynamical
mechanisms responsible for anomalous temporal ordering during the approach to equilibrium.
In recent years, attention has turned toward the quantum regime, giving rise to the so-called
quantum Mpemba effect (QME) [11–14]. In both open and closed quantum systems, it has been
observed that states prepared farther from equilibrium may relax faster than those initialized
closer to equilibrium. In open quantum systems, the effect often parallels its classical counterpart,
emerging when overlaps with the slowest Liouvillian modes are suppressed [15–17] (see also [18–21]).
In closed systems, QME-like behavior has been explored through quantum quenches in a variety of
models, including spin chains, free fermionic and bosonic systems, random quantum circuits, and
many-body localized phases [22–30]. These studies demonstrate that the QME is a widespread
feature of nonequilibrium quantum dynamics, sensitive to both the nature of the initial state and
the spectral properties of the Hamiltonian.
A major challenge in the study of the QME lies in identifying observables that reliably capture
it. Relaxation processes in quantum systems depend strongly on the interplay between differ-
ent dynamical channels and on the presence of symmetries or conserved quantities that constrain
the evolution. Hence, the choice of probe- that is, the quantitative measure of “distance from
equilibrium”- is crucial for revealing Mpemba inversions [13]. Among the most informative diag-
nostics proposed so far is entanglement asymmetry, defined as the difference between the entan-
glement entropy of a subsystem and that of its symmetry-projected counterpart with respect to a
global U(1) charge. This measure provides a direct probe of local symmetry restoration and has
revealed clear Mpemba-like behavior in both integrable and non-integrable spin chains [31–36].
Related analyses have identified Mpemba-type behavior in non-integrable systems without global
symmetries, where the effect correlates with spectral indicators such as the inverse participation
ratio [37]. Together, these developments highlight both the ubiquity of the quantum Mpemba
effect and its strong dependence on the probe used to characterize relaxation.
Recently, Krylov complexity1 has emerged as a powerful and physically transparent diagnostic
of quantum dynamics and chaos [41–62]. By quantifying how rapidly a quantum state or operator
explores its accessible Hilbert space, Krylov complexity encodes both the geometric and dynamical
aspects of relaxation that are often inaccessible to conventional observables. Within this frame-
work, recent work [40] on a certain spin-chain system with global U(1) symmetry reported clear
1For comprehensive reviews, see [38,39].
1
Mpemba-like behavior in the symmetric (diagonal) component of Krylov complexity, obtained by
neglecting contributions from the asymmetric (off-diagonal) sectors. Interestingly, inclusion of the
full asymmetric contribution tends to smooth out these features, suggesting that coherence be-
tween symmetry sectors plays a subtle yet nontrivial role in shaping relaxation hierarchies. This
observation points toward a deeper connection between complexity growth, symmetry constraints,
and the emergence of anomalous relaxation in quantum many-body systems.
In this work, we build upon these developments by systematically investigating Krylov state
complexity as a probe of the quantum Mpemba effect in spin chains with and without global
symmetries. For models without U(1) symmetry, we find that the total Krylov complexity alone
exhibits clear and robust Mpemba-like crossings, offering a distinct interpretation in terms of
Hilbert-space exploration and dynamical delocalization. In contrast, for systems with a global U(1)
symmetry, we show that decomposing the complexity into symmetric and asymmetric components
reveals how charge conservation reshapes relaxation dynamics. Our analysis demonstrates that,
while the total complexity may not always show a pronounced inversion, the symmetric component
provides a consistent and reliable diagnostic of Mpemba-like behavior, confirming the proposal
of [40].
The remainder of this paper is organized as follows. In Section 2, we review the construction of
Krylov complexity and analyze its behavior for a mixed-field Ising chain without global symmetries,
demonstrating its sensitivity to Mpemba-like crossings.
Section 3, focuses on U(1)-symmetric
systems, where we show that the symmetric component of the complexity serves as a robust probe
of the QME. The last section is devoted to conclusions.
2
Krylov Complexity and the Quantum Mpemba Effect
without global symmetry
In this section, we explore how Krylov complexity2 can serve as a probe of the quantum Mpemba
effect (QME) in systems without global symmetries. Following [37], we study the mixed-field Ising
chain with the Hamiltonian
HIsing =
L−1
X
j=1
σz
jσz
j+1 + g
L
X
j=1
σx
j + h
L−1
X
j=2
σz
j + h′σz
1 −h′σz
L,
(2.1)
where the boundary fields ±h′ explicitly break reflection symmetry, ensuring that no global sym-
metries remain beyond energy conservation.
Our goal is to compute Krylov complexity for various initial states and assess its effectiveness
as a diagnostic of the QME. The Krylov basis for a normalized state |ψ⟩is constructed via the
2Throughout this paper, we examine the Krylov complexity of states, following [63–65].
2
Lanczos algorithm [66,67], beginning with |0⟩= |ψ⟩and generating subsequent orthogonal vectors
recursively as
(H −an)|n⟩= bn+1|n + 1⟩+ bn|n −1⟩,
(2.2)
with coefficients an = ⟨n|H|n⟩, b2
n+1 = ⟨n|H2|n⟩−a2
n −b2
n, and b0 = 0. This procedure produces
an orthonormal basis {|n⟩} of dimension Dψ ≤D, along with the Lanczos coefficients {an, bn}.
Expanding the time-evolved state in this basis,
|ψ(t)⟩=
Dψ−1
X
n=0
ϕn(t)|n⟩,
Dψ−1
X
n=0
|ϕn(t)|2 = 1,
(2.3)
the amplitudes ϕn(t) satisfy the tridiagonal Schr¨odinger equation
i∂tϕn(t) = anϕn(t) + bnϕn−1(t) + bn+1ϕn+1(t),
(2.4)
with ϕn(0) = δn0. The Krylov complexity is then defined as the expectation value of the number
operator [41],
C(t) = ⟨ψ(t)|N|ψ(t)⟩,
N =
Dψ−1
X
n=0
n|n⟩⟨n|.
(2.5)
We consider spin-coherent product states as initial conditions, parameterized by Bloch angles
(θ, ϕ),
|θ, ϕ⟩=
L
O
i=1
cos θ
2|Z+⟩i + eiϕ sin θ
2|Z−⟩i
,
(2.6)
where |Z±⟩i are eigenstates of σz
i . Setting ϕ = 0, the angle θ serves as a tilt parameter controlling
the distance from equilibrium. This reproduces the family of initial states used in [37], where the
QME was diagnosed using trace distance and inverse participation ratio (IPR).
The numerical results are shown in Figure 1, with the simulation parameters specified in the
inset. For larger tilt angles θ, the complexity initially grows more slowly, remaining below that
of states prepared with smaller tilts. However, its growth rate is faster, leading to early-time
crossings of the complexity curves. At long times, the more strongly tilted states not only overtake
the others but also saturate at higher values of complexity. These crossings are clear signatures of
Mpemba-like behavior.
To sharpen the discussion, it is useful to present an analytic expression for the Krylov com-
plexity in the energy eigenbasis. Expanding the initial state as
|ψ⟩=
D
X
α=1
cα|Eα⟩,
(2.7)
with |Eα⟩the eigenstates of the Hamiltonian, one finds that the complexity naturally splits into
3
0
1
2
3
4
5
6
0
10
20
30
40
Figure 1: Early-time growth of Krylov complexity for different tilt angles θ. Strongly tilted states
initially grow more slowly but eventually overtake weakly tilted ones, a signature of Mpemba-like
behavior.
diagonal and off-diagonal contributions3,
C(t) =
D
X
α=1
|cα|2Nαα +
D
X
α̸=β
c∗
αcβ Nαβ e−iωαβt,
ωαβ = Eα −Eβ,
(2.8)
where Nαβ = ⟨Eα|N|Eβ⟩. The infinite-time average reduces to the diagonal ensemble,
C =
D
X
α=1
|cα|2Nαα = Tr(ρDEN),
ρDE =
D
X
α=1
|cα|2|Eα⟩⟨Eα| ,
(2.9)
and since C(0) = 0 one finds the constraint
Tr(ρDEN) = −
D
X
α̸=β
c∗
αcβ Nαβ > 0 .
(2.10)
Thus, the saturation value of complexity is controlled by the diagonal part, while the transient
dynamics originates from the off-diagonal terms, which eventually cancel at late times due to
dephasing. Equation (2.10) further shows that a larger saturation value requires stronger initial
coherence among energy eigenstates, which also drives a faster initial growth of complexity.
The above analytic analysis shows that such inversions are not guaranteed; instead, they depend
sensitively on the distribution of energy eigenstates and on their overlaps with the initial state.
Indeed, at early times, precisely the regime where curve crossings are most likely to occur, the
3This decomposition can be interpreted as separating the complexity into a symmetric (diagonal) and an asym-
metric (off-diagonal) part with respect to the projection onto the energy eigenbasis.
4
Krylov complexity exhibits a universal quadratic growth,
C(t∼0) ≈−t2
8
X
α̸=β
ω2
αβ c∗
αcβ Nαβ ,
(2.11)
where the spectral structure enters explicitly via the squared gaps ω2
αβ. From this perspective,
states with larger long-time saturation values of complexity are those that distribute more broadly
across the spectrum ( see also below), thereby sampling a richer set of energy gaps. However,
the same broad distribution can suppress the initial growth rate through destructive interference
among off-diagonal terms. This suppression at early times, followed by a faster subsequent growth,
is precisely the mechanism that allows for Mpemba-like inversions.
To illustrate this point concretely, we compare two classes of initial states: tilted ferromagnetic
states (TFS) and tilted N´eel states (TNS),
|F, θ⟩= e−iθ/2 PL
j=1 σy
j |↑· · · ↑⟩,
|N, θ⟩= e−iθ/2 PL
j=1 σy
j |↑↓· · · ↑↓⟩.
(2.12)
The corresponding Krylov complexities are shown in Figure 2, and their infinite-time averages in
Figure 3.
0
2
4
6
8
0
20
40
60
80
0
2
4
6
8
0
20
40
60
80
Figure 2: Krylov complexity of TFS (Left) and TNS (Right) for different values of θ.
The comparison reveals two key insights. First, states with larger C grow faster, confirming
the correlation between long-time spreading and growth rates. Second, Mpemba-like inversions
appear only for TFS: strongly tilted ferromagnetic states initially lag but eventually surpass weakly
tilted ones. By contrast, no such inversions are observed for TNS. This indicates that while higher
saturation values generally correlate with faster growth, this alone is not sufficient to guarantee
the occurrence of Mpemba crossings. Indeed, the Mpemba crossings depend not just on saturation
values but also on the microscopic structure of the initial state and its overlaps with the energy
spectrum.
Finally, comparing with [37], where QME was identified using trace distance and IPR, we see
5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0
50
100
150
200
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
0
50
100
150
200
Figure 3: Infinite-time averages of Krylov complexity of TFS (Left) and TNS (Right) for different
values of θ and N = 9.
qualitative agreement: states further from equilibrium relax more rapidly, either via faster decay
of trace distance or via earlier crossings in complexity. Moreover, the infinite-time averages of
Krylov complexity display near-perfect agreement with the IPR of the initial states, as shown in
Figure 4, reinforcing the view that both quantities capture Hilbert-space delocalization4.
Figure 4: Infinite-time average of complexity (left) and logarithm of the inverse participation ratio
(right) for general initial states given in (2.6), with N = 9. Due to the symmetry under ϕ →2π−ϕ,
we show the results only for 0 ≤ϕ ≤π.
However, while these long-time averages align closely, they are not sufficient to diagnose
Mpemba-like behavior on their own.
The essential signature lies in the early-time dynamics,
governed by the interplay of diagonal and off-diagonal contributions. Only by considering both
4This correspondence aligns with previous studies comparing Hilbert-space spreading and IPR in the context of
thermalization [68], highlighting that both quantities capture how efficiently a state explores the Hilbert space over
time.
6
transient growth and asymptotic saturation can Krylov complexity fully capture the nontrivial
temporal inversions characteristic of the quantum Mpemba effect.
3
Krylov Complexity and the Quantum Mpemba Effect in
U(1)-Symmetric Systems
In this section, we investigate Krylov complexity in quantum systems with a global U(1) symmetry,
with the goal of understanding how a conserved charge influences the manifestation of the quantum
Mpemba effect.
When a system possesses a conserved charge Q generated by a global U(1) symmetry, its Hilbert
space naturally decomposes into orthogonal subsectors labeled by the charge eigenvalues,
H =
M
q
Hq ,
(3.1)
where q runs over all possible charge values, and Pq denotes the corresponding projector onto the
subsector Hq.
Krylov complexity can be formulated in several ways in the presence of symmetry. Following
the construction introduced earlier, the most direct approach is to evaluate the standard Krylov
complexity of a state without explicitly resolving the symmetry structure. Given an initial state
|ψ⟩, one constructs the Krylov basis {|n⟩} and the associated number operator N = PDψ−1
n=0 n|n⟩⟨n|,
yielding the complexity as defined in Eq. (2.5). If the initial state belongs to a particular charge
subsector, the entire Krylov basis remains confined within that subsector during the evolution.
It is often illuminating to express the complexity in a natural energy-charge eigenbasis of the
Hilbert space, consisting of the joint eigenstates of H and Q,
H|E(q)
α , q⟩= E(q)
α |E(q)
α , q⟩,
Q|E(q)
α , q⟩= q|E(q)
α , q⟩.
(3.2)
A general normalized state can then be expanded as
|ψ⟩=
X
α,q
cαq|E(q)
α , q⟩,
X
α,q
|cαq|2 = 1,
(3.3)
where α runs over the dimension Dq of the charge sector Hq. Since only a single U(1) symmetry is
imposed (and, in particular, no additional discrete symmetry is assumed), the spectra of distinct
charge sectors are generally non-degenerate, so that almost all eigenvalues correspond uniquely to
a given q.
7
In this notation, the Krylov complexity of the evolving state can be written as
C(t) =
X
q,α
|cαq|2 N qq
αα +
X
q,α̸=β
c∗
α,qcβ,q N qq
αβ e−iωqq
αβt +
X
p̸=q,α,β
c∗
α,qcβ,p N qp
αβ e−iωqp
αβt,
(3.4)
which naturally separates into a “symmetric” (q = p) part (intra-sector contributions) and an
“asymmetric” (q ̸= p) part (inter-sector contributions). Here N qp
αβ = ⟨q, E(q)
α |N|E(p)
β , p⟩and ωqp
αβ =
E(q)
α −E(p)
β .
Alternatively, one can construct a symmetry-adapted version of Krylov complexity by project-
ing onto individual charge sectors. Defining the subsector number operator
Nq =
Dψ−1
X
n=0
n Pq|n⟩⟨n|Pq,
(3.5)
the symmetric Krylov complexity is given by [40]5
CS(t) =
X
q
⟨ψ(t)|Nq|ψ(t)⟩=
X
q,α
|cαq|2 N qq
αα +
X
q,α̸=β
c∗
α,qcβ,q N qq
αβ e−iωqq
αβt ,
(3.6)
which corresponds to the diagonal part of the full decomposition (3.4). The asymmetric contribu-
tion is then
CA(t) =
X
p̸=q,α,β
c∗
α,qcβ,p N qp
αβ e−iωqp
αβt .
(3.7)
As in the symmetryless case, the infinite-time average reduces to the diagonal ensemble,
C =
X
q,α
|cα,q|2 N qq
αα = Tr(ρDEN),
ρDE =
X
q,α
|cα,q|2 |E(q)
α , q⟩⟨q, E(q)
α | ,
(3.8)
while the condition C(0) = 0 implies
C = −
X
q,α̸=β
c∗
α,qcβ,q N qq
αβ +
X
q̸=p,α,β
c∗
α,qcβ,p N qp
αβ
!
> 0.
(3.9)
From these expressions, several important insights emerge. First, the saturation value of Krylov
complexity is primarily determined by the diagonal contributions from the symmetric part, whereas
the transient or pre-saturation dynamics are governed by the off-diagonal components, both intra-
and inter-sector. Equation (3.9) further implies that achieving a larger saturation value requires
stronger coherence among energy eigenstates, which simultaneously enhances the initial rate of
complexity growth.
In systems with a global U(1) symmetry, this coherence arises from two
5See also the related notion of symmetry-resolved Krylov complexity introduced in Refs. [69,70], where the total
complexity is expressed as an average over contributions from individual symmetry subsectors.
8
distinct sources: correlations among eigenstates within a charge subsector and interference between
different charge sectors.
It follows that focusing exclusively on either the symmetric (diagonal) or the asymmetric (off-
diagonal) contribution might be insufficient to capture the full dynamics of Krylov complexity.
While the asymmetric part exhibits oscillatory behavior that decays through dephasing at long
times, it can still encode short-lived coherence effects that strongly influence the transient regime.
These interference effects can either enhance or suppress early-time features, and are precisely
those that give rise to Mpemba-like crossings in the complexity growth curves. Neglecting either
component therefore could remove essential information about the interplay between coherence
and dephasing. Consequently, if Krylov complexity is to serve as a faithful probe of relaxation and
Mpemba-like behavior, it should, in principle, be considered as a total quantity that integrates
both contributions.
Nevertheless, the behavior of complexity depends sensitively on the choice of initial state. For
certain states, deviations from this general expectation can occur: the transient dynamics may
be dominated by intra-sector coherence, leading the symmetric component alone to capture most
of the relevant relaxation behavior. In these cases, the symmetric Krylov complexity not only
governs the asymptotic saturation value but also encodes significant information about the entire
dynamical evolution.
This possibility was explicitly demonstrated in Ref. [40] for the Aubry–Andr´e model [71],
where the total complexity showed only a qualitative late-time separation, while the symmetric
component exhibited robust Mpemba-like crossings, making it a more sensitive diagnostic of the
QME. In particular, a modified definition of the symmetric complexity was introduced,
˜CS(t) = CS(t) + CA(0) ,
(3.10)
which effectively subtracts the asymmetric contribution at t = 0. Including the full asymmetric
part was found to wash out the crossings, suggesting that the symmetric component, properly
normalized, provides a sharper signature of the effect.
To examine whether this behavior persists in interacting systems, we now turn to other U(1)-
symmetric spin chains. We first consider the next-nearest-neighbor XXZ Hamiltonian [34,37],
H = J1
L−1
X
j=1
σx
j σx
j+1 + σy
j σy
j+1 + ∆1σz
jσz
j+1
+ J2
L−2
X
j=1
σx
j σx
j+2 + σy
j σy
j+2 + ∆2σz
jσz
j+2
,
(3.11)
which is chaotic and possesses a global U(1) symmetry generated by the total magnetization along
the z-axis,
Q =
L
X
i=1
σz
i .
(3.12)
9
To analyze the dynamics, we compute the total and symmetric Krylov complexities for TFS with
different tilt parameters. To highlight the crossings, we study the difference
∆C(t) = Cθ1(t) −Cθ2(t) ,
(3.13)
and similarly for the symmetric component. Using parameters consistent with Ref. [37], we present
results for tilt angles θ1 = 0.15 and θ2 = 0.2. The corresponding results are shown in Fig. 5, where
we have also accounted for the model’s additional Z2 symmetry when computing the symmetric
complexity.
0.0
0.5
1.0
1.5
2.0
-1.5
-1.0
-0.5
0.0
0.0
0.5
1.0
1.5
2.0
-0.015
-0.010
-0.005
0.000
Figure 5: Difference in Krylov complexity (left) and its symmetric component (right) between
initial states with tilt parameters θ1 = 0.15 and θ2 = 0.2, for the TFS in the model given by (3.11).
System size is N = 10.
As seen in Fig. 5, the total complexity does not display a clear Mpemba-like inversion, whereas
the symmetric complexity exhibits pronounced crossings, in agreement with the findings of Refs. [40].
This confirms that, in such systems, the symmetric component provides a more reliable probe of
the QME, as proposed by [37]. In contrast, for the TNS, neither the total nor the symmetric
complexities show any inversion, consistent with its incoherent relaxation dynamics.
We can also explicitly break the residual Z2 symmetry of the model (3.11) by adding a longi-
tudinal field term of the form
h
2
L
X
j=1
σz
j,
(3.14)
to the Hamiltonian. This modification lifts the parity degeneracy and allows us to examine how
the breaking of discrete symmetry affects the manifestation of the QME. We have repeated the
same numerical analysis as before, computing both the total Krylov complexity and its symmetric
component for the TFS. The corresponding results are presented in Fig. 6.
As shown in Fig. 6, the symmetric component of the complexity exhibits a finer and more
structured temporal evolution than the total complexity, reflecting sensitivity to short-time co-
herence effects that are largely averaged out in the total measure. However, since this behavior
10
0.0
0.5
1.0
1.5
2.0
-1.5
-1.0
-0.5
0.0
0.0
0.5
1.0
1.5
2.0
-0.03
-0.02
-0.01
0.00
0.01
Figure 6: Difference in Krylov complexity (left) and in its symmetric component (right) between
initial states with tilt parameters θ1 = 0.15 and θ2 = 0.2, for the TFS in the model (3.11) with
explicitly broken Z2 symmetry due to the longitudinal field term. System size is N = 10.
is accompanied by significant oscillations and lacks a clear, monotonic inversion, it might not be
interpreted as a genuine realization of the QME. Nevertheless, this comparison further underscores
that the symmetric component of Krylov complexity can reveal subtle dynamical features of the
system.
Let us now consider the integrable XXZ spin chain in a longitudinal field [34],
H = −1
4
L−1
X
j=1
σx
j σx
j+1 + σy
j σy
j+1 + ∆1σz
jσz
j+1
+ h
2
L
X
j=1
σz
j,
(3.15)
which also possesses a global U(1) symmetry generated by the total magnetization but explicitly
breaks the Z2 symmetry through the field term.
For this model, we again analyze the TFS and TNS initial states defined in Eq. (2.12), focusing
on tilt parameters θ = 1.2 and θ = 0.8 to compare with Ref. [34]. The results are displayed in
Fig. 7.
For the TFS, ∆C(t) changes sign during the evolution, indicating that the corresponding com-
plexity curves cross each other. This inversion constitutes a clear signature of the Mpemba-like
effect: the state prepared further from equilibrium (larger tilt) initially grows more slowly but
eventually overtakes the less tilted state. By contrast, for the TNS, ∆C(t) remains of the same
sign throughout, showing no evidence of inversion. These results agree with those of Ref. [34],
confirming that the full Krylov complexity can indeed capture the short-time reversal of relaxation
rates characteristic of the QME in this model.
To assess the role of symmetry, we further compute the symmetric complexity for both initial
states. The corresponding results are shown in Fig. 8.
From Fig. 8, we see that for the TFS, the symmetric complexity exhibits a pronounced early-
time crossing, while the TNS again shows none. This indicates that the essential information
11
0
2
4
6
8
0.0
0.5
1.0
1.5
0
2
4
6
8
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Figure 7: Difference in Krylov complexity between initial states with tilt parameters θ = 1.2 and
θ = 0.8, for the TFS (left) and for the positive-parity component of the TNS (right). System size
is N = 10.
0
2
4
6
8
10
-0.15
-0.10
-0.05
0.00
0.05
0
2
4
6
8
10
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Figure 8: Difference in symmetric Krylov complexity between initial states with tilt parameters
θ = 1.2 and θ = 0.8, for the TFS (left) and for the positive-parity component of the TNS (right).
System size is N = 10.
about the Mpemba-like inversion is already encoded in the symmetric sector. However, unlike the
previous case, the infinite-time average of the symmetric complexity does not necessarily increase
with tilt. The reason lies in the relation
˜CS = CS + CA(0) ,
(3.16)
which shows that the asymptotic value of the symmetric component depends on the initial co-
herence contained in the asymmetric sector. Thus, while early-time crossings diagnose Mpemba
behavior, late-time averages remain insensitive to the initial state.
In summary, our analysis demonstrates that in U(1)-symmetric systems, the symmetric com-
ponent of Krylov complexity provides a robust and reliable probe of the QME. Although the total
complexity can in some cases reproduce Mpemba-like inversions, these features are generally less
12
pronounced and more sensitive to the choice of initial state. The symmetric complexity defined by
Eq. (3.10), on the other hand, consistently captures the essential signatures of the effect across all
models we have examined, confirming and extending the proposal of Ref. [40]. This finding high-
lights that Krylov complexity, particularly in its symmetry-adapted form, serves as an insightful
diagnostic of anomalous relaxation and the subtle interplay between coherence, dephasing, and
symmetry constraints in quantum many-body dynamics.
4
Conclusions
In this work, we have systematically studied Krylov state complexity as a diagnostic for the
QME in both symmetry-broken and U(1)-symmetric quantum spin models.
By analyzing the
time evolution of Krylov complexity across different initial states, we have shown that, for systems
without symmetry, the structure of complexity growth faithfully encodes the anomalous relaxation
patterns that define the QME. In particular, the characteristic inversions in relaxation rates—which
distinguish the Mpemba effect from conventional thermal equilibration—are directly reflected in the
temporal evolution of Krylov complexity. This establishes Krylov complexity as a quantitative and
physically transparent measure for detecting and characterizing the QME within closed quantum
systems.
Turning to models with global U(1) symmetry, our analysis reveals that while the total Krylov
complexity can, in certain cases and for specific initial states, display clear Mpemba-like crossings,
it does not in general provide a robust probe of anomalous relaxation. The total measure is often
sensitive to fine details of the initial state and to coherence effects between charge sectors, which
can obscure or suppress the inversion behavior that signals the QME. A more reliable and universal
characterization emerges when the complexity is decomposed into its symmetric and asymmetric
components, corresponding respectively to diagonal and off-diagonal contributions in the charge
basis. By isolating the symmetric component and performing an appropriate shift that compen-
sates for the initial contribution of the asymmetric part, one obtains a quantity that consistently
exhibits Mpemba-like behavior across all models studied. This shifted symmetric Krylov complex-
ity captures the essential dynamical information governing relaxation, while remaining insensitive
to transient oscillations that can mask the effect in the total complexity. Our numerical anal-
ysis across multiple spin-chain models—both integrable and non-integrable—confirms that this
measure provides a stable and physically transparent probe of the QME, thereby supporting and
extending the proposal of Ref. [40].
These findings demonstrate that symmetry resolution, far from being a mere technical refine-
ment, plays a crucial role in diagnosing anomalous relaxation through complexity. In the presence
of conserved charges, the symmetric sector encodes the irreversible aspects of Hilbert-space ex-
ploration, while the asymmetric part captures transient coherence and dephasing. The shifted
symmetric complexity effectively combines these contributions into a single measure that reflects
13
the true dynamical hierarchy of relaxation times.
From a broader perspective, our results highlight the potential of Krylov-space methods as
powerful tools for investigating nonequilibrium quantum dynamics. By linking complexity growth
to relaxation pathways and coherence structure, they provide a unifying framework for under-
standing anomalous thermalization and information scrambling (see, for example, Ref. [68]). The
sensitivity of Krylov complexity to coherent interference, spectral structure, and the interplay be-
tween diagonal and off-diagonal contributions makes it a natural bridge between state spreading,
quantum chaos, and anomalous relaxation phenomena. In the context of the QME, it offers a
unified language that connects dynamical reversals of relaxation rates to the microscopic geometry
of Hilbert-space exploration.
In this work, we have focused exclusively on finite-dimensional many-body systems. An inter-
esting extension would be to investigate infinite-dimensional cases, such as quantum field theories,
where Krylov complexity may not saturate but continues to grow indefinitely with time. In such
systems, one can meaningfully study the rate of complexity growth at late times and compare it
with the early-time behavior to identify possible inversions. Here, the Mpemba-like effect would
no longer be manifested through relaxation rates but through the dynamics of complexity growth
itself: states that exhibit faster early-time growth may subsequently evolve more slowly at late
times. In other words, states that initially appear “more complex” can eventually become “less
complex” —a phenomenon that naturally extends the notion of the Mpemba effect to the realm
of dynamical complexity.
Taken together, our results in spin-chain models suggest that Krylov complexity provides a
unified and physically transparent framework for characterizing Mpemba-like inversions across
both finite and infinite-dimensional quantum systems. It thus opens new avenues for exploring the
interplay between coherence, chaos, and relaxation, and for deepening our understanding of how
complex quantum systems approach equilibrium in fundamentally nontrivial ways.
Acknowledgements
We would like to thank Mohammad Reza Tanhayi for useful discussions. M.A. would also like to
thank the CERN Department of Theoretical Physics for their hospitality during the course of this
work. The work of M. J. V. is based on research funded by the Iran National Science Foundation
(INSF) under Project No. 4047915. We also acknowledge the assistance of ChatGPT for help with
text editing and polishing.
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On Krylov Complexity as a Probe of the Quantum Mpemba Effect Mohsen Alishahiha and Mohammad Javad Vasli (IPM), P.O. Box 19395-5531, Tehran, Iran E-mails: , Abstract We investigate Krylov state complexity as a probe of the quantum Mpemba effect in quantum spin chains. For models without global U(1) symmetry, Krylov complexity exhibits clear Mpemba-like crossings, consistent with conventional diagnostics such as the trace distance, while offering a complementary interpretation in terms of Hilbert-space exploration and dynamical delocalization. In U(1)-symmetric systems, we confirm that the recently proposed symmetric component of Krylov complexity serves as a robust and reliable indicator of the QME, capturing anomalous relaxation even in cases where the total complexity fails to do so. 16 Oct 2025 1 Introduction The Mpemba effect refers to the counterintuitive phenomenon in which a system prepared further from equilibrium relaxes faster than one initialized closer to equilibrium. Originally observed in water, where initially hotter samples were found to freeze more quickly than colder ones [1], the effect has since been identified across a wide range of classical dynamical systems [2-10]. These studies established that the Mpemba effect is not a peculiar feature of water, but rather a generic manifestation of nonequilibrium relaxation, leading to renewed interest in the dynamical mechanisms responsible for anomalous temporal ordering during the approach to equilibrium. In recent years, attention has turned toward the quantum regime, giving rise to the so-called quantum Mpemba effect (QME) [11-14]. In both open and closed quantum systems, it has been observed that states prepared farther from equilibrium may relax faster than those initialized closer to equilibrium. In open quantum systems, the effect often parallels its classical counterpart, emerging when overlaps with the slowest Liouvillian modes are suppressed [15-17] (see also [18-21]). In closed systems, QME-like behavior has been explored through quantum quenches in a variety of models, including spin chains, free fermionic and bosonic systems, random quantum circuits, and many-body localized phases [22-30]. These studies demonstrate that the QME is a widespread feature of nonequilibrium quantum dynamics, sensitive to both the nature of the initial state and the spectral properties of the Hamiltonian. A major challenge in the study of the QME lies in identifying observables that reliably capture it. Relaxation processes in quantum systems depend strongly on the interplay between different dynamical channels and on the presence of symmetries or conserved quantities that constrain the evolution. Hence, the choice of probe- that is, the quantitative measure of "distance from equilibrium"- is crucial for revealing Mpemba inversions [13]. Among the most informative diagnostics proposed so far is entanglement asymmetry, defined as the difference between the entanglement entropy of a subsystem and that of its symmetry-projected counterpart with respect to a global U(1) charge. This measure provides a direct probe of local symmetry restoration and has revealed clear Mpemba-like behavior in both integrable and non-integrable spin chains [31-36]. Related analyses have identified Mpemba-type behavior in non-integrable systems without global symmetries, where the effect correlates with spectral indicators such as the inverse participation ratio [37]. Together, these developments highlight both the ubiquity of the quantum Mpemba effect and its strong dependence on the probe used to characterize relaxation. Recently, Krylov complexity1 has emerged as a powerful and physically transparent diagnostic of quantum dynamics and chaos [41-62]. By quantifying how rapidly a quantum state or operator explores its accessible Hilbert space, Krylov complexity encodes both the geometric and dynamical aspects of relaxation that are often inaccessible to conventional observables. Within this framework, recent work [40] on a certain spin-chain system with global U(1) symmetry reported clear 1For comprehensive reviews, see [38,39]. 1 Mpemba-like behavior in the symmetric (diagonal) component of Krylov complexity, obtained by neglecting contributions from the asymmetric (off-diagonal) sectors. Interestingly, inclusion of the full asymmetric contribution tends to smooth out these features, suggesting that coherence between symmetry sectors plays a subtle yet nontrivial role in shaping relaxation hierarchies. This observation points toward a deeper connection between complexity growth, symmetry constraints, and the emergence of anomalous relaxation in quantum many-body systems. In this work, we build upon these developments by systematically investigating Krylov state complexity as a probe of the quantum Mpemba effect in spin chains with and without global symmetries. For models without U(1) symmetry, we find that the total Krylov complexity alone exhibits clear and robust Mpemba-like crossings, offering a distinct interpretation in terms of Hilbert-space exploration and dynamical delocalization. In contrast, for systems with a global U(1) symmetry, we show that decomposing the complexity into symmetric and asymmetric components reveals how charge conservation reshapes relaxation dynamics. Our analysis demonstrates that, while the total complexity may not always show a pronounced inversion, the symmetric component provides a consistent and reliable diagnostic of Mpemba-like behavior, confirming the proposal of [40]. The remainder of this paper is organized as follows. In Section 2, we review the construction of Krylov complexity and analyze its behavior for a mixed-field Ising chain without global symmetries, demonstrating its sensitivity to Mpemba-like crossings. Section 3, focuses on U(1)-symmetric systems, where we show that the symmetric component of the complexity serves as a robust probe of the QME. The last section is devoted to conclusions. 2 Krylov Complexity and the Quantum Mpemba Effect without global symmetry In this section, we explore how Krylov complexity2 can serve as a probe of the quantum Mpemba effect (QME) in systems without global symmetries. Following [37], we study the mixed-field Ising chain with the Hamiltonian HIsing = L-1 X j=1 σz jσz j+1 + g L X j=1 σx j + h L-1 X j=2 σz j + h′σz 1 -h′σz L, (2.1) where the boundary fields ±h′ explicitly break reflection symmetry, ensuring that no global symmetries remain beyond energy conservation. Our goal is to compute Krylov complexity for various initial states and assess its effectiveness as a diagnostic of the QME. The Krylov basis for a normalized state |ψ⟩is constructed via the 2Throughout this paper, we examine the Krylov complexity of states, following [63-65]. 2 Lanczos algorithm [66,67], beginning with |0⟩= |ψ⟩and generating subsequent orthogonal vectors recursively as (H -an)|n⟩= bn+1|n + 1⟩+ bn|n -1⟩, (2.2) with coefficients an = ⟨n|H|n⟩, b2 n+1 = ⟨n|H2|n⟩-a2 n -b2 n, and b0 = 0. This procedure produces an orthonormal basis {|n⟩} of dimension Dψ ≤D, along with the Lanczos coefficients {an, bn}. Expanding the time-evolved state in this basis, |ψ(t)⟩= Dψ-1 X n=0 φn(t)|n⟩, Dψ-1 X n=0 |φn(t)|2 = 1, (2.3) the amplitudes φn(t) satisfy the tridiagonal Schr ̈odinger equation i∂tφn(t) = anφn(t) + bnφn-1(t) + bn+1φn+1(t), (2.4) with φn(0) = δn0. The Krylov complexity is then defined as the expectation value of the number operator [41], C(t) = ⟨ψ(t)|N|ψ(t)⟩, N = Dψ-1 X n=0 n|n⟩⟨n|. (2.5) We consider spin-coherent product states as initial conditions, parameterized by Bloch angles (θ, φ), |θ, φ⟩= L O i=1 cos θ 2|Z+⟩i + eiφ sin θ 2|Z-⟩i , (2.6) where |Z±⟩i are eigenstates of σz i . Setting φ = 0, the angle θ serves as a tilt parameter controlling the distance from equilibrium. This reproduces the family of initial states used in [37], where the QME was diagnosed using trace distance and inverse participation ratio (IPR). The numerical results are shown in Figure 1, with the simulation parameters specified in the inset. For larger tilt angles θ, the complexity initially grows more slowly, remaining below that of states prepared with smaller tilts. However, its growth rate is faster, leading to early-time crossings of the complexity curves. At long times, the more strongly tilted states not only overtake the others but also saturate at higher values of complexity. These crossings are clear signatures of Mpemba-like behavior. To sharpen the discussion, it is useful to present an analytic expression for the Krylov complexity in the energy eigenbasis. Expanding the initial state as |ψ⟩= D X α=1 cα|Eα⟩, (2.7) with |Eα⟩the eigenstates of the Hamiltonian, one finds that the complexity naturally splits into 3 0 1 2 3 4 5 6 0 10 20 30 40 Figure 1: Early-time growth of Krylov complexity for different tilt angles θ. Strongly tilted states initially grow more slowly but eventually overtake weakly tilted ones, a signature of Mpemba-like behavior. diagonal and off-diagonal contributions3, C(t) = D X α=1 |cα|2Nαα + D X α̸=β c∗ αcβ Nαβ e-iωαβt, ωαβ = Eα -Eβ, (2.8) where Nαβ = ⟨Eα|N|Eβ⟩. The infinite-time average reduces to the diagonal ensemble, C = D X α=1 |cα|2Nαα = Tr(ρDEN), ρDE = D X α=1 |cα|2|Eα⟩⟨Eα| , (2.9) and since C(0) = 0 one finds the constraint Tr(ρDEN) = - D X α̸=β c∗ αcβ Nαβ > 0 . (2.10) Thus, the saturation value of complexity is controlled by the diagonal part, while the transient dynamics originates from the off-diagonal terms, which eventually cancel at late times due to dephasing. Equation (2.10) further shows that a larger saturation value requires stronger initial coherence among energy eigenstates, which also drives a faster initial growth of complexity. The above analytic analysis shows that such inversions are not guaranteed; instead, they depend sensitively on the distribution of energy eigenstates and on their overlaps with the initial state. Indeed, at early times, precisely the regime where curve crossings are most likely to occur, the 3This decomposition can be interpreted as separating the complexity into a symmetric (diagonal) and an asymmetric (off-diagonal) part with respect to the projection onto the energy eigenbasis. 4 Krylov complexity exhibits a universal quadratic growth, C(t∼0) ≈-t2 8 X α̸=β ω2 αβ c∗ αcβ Nαβ , (2.11) where the spectral structure enters explicitly via the squared gaps ω2 αβ. From this perspective, states with larger long-time saturation values of complexity are those that distribute more broadly across the spectrum ( see also below), thereby sampling a richer set of energy gaps. However, the same broad distribution can suppress the initial growth rate through destructive interference among off-diagonal terms. This suppression at early times, followed by a faster subsequent growth, is precisely the mechanism that allows for Mpemba-like inversions. To illustrate this point concretely, we compare two classes of initial states: tilted ferromagnetic states (TFS) and tilted N ́eel states (TNS), |F, θ⟩= e-iθ/2 PL j=1 σy j |↑· · · ↑⟩, |N, θ⟩= e-iθ/2 PL j=1 σy j |↑↓· · · ↑↓⟩. (2.12) The corresponding Krylov complexities are shown in Figure 2, and their infinite-time averages in Figure 3. 0 2 4 6 8 0 20 40 60 80 0 2 4 6 8 0 20 40 60 80 Figure 2: Krylov complexity of TFS (Left) and TNS (Right) for different values of θ. The comparison reveals two key insights. First, states with larger C grow faster, confirming the correlation between long-time spreading and growth rates. Second, Mpemba-like inversions appear only for TFS: strongly tilted ferromagnetic states initially lag but eventually surpass weakly tilted ones. By contrast, no such inversions are observed for TNS. This indicates that while higher saturation values generally correlate with faster growth, this alone is not sufficient to guarantee the occurrence of Mpemba crossings. Indeed, the Mpemba crossings depend not just on saturation values but also on the microscopic structure of the initial state and its overlaps with the energy spectrum. Finally, comparing with [37], where QME was identified using trace distance and IPR, we see 5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 50 100 150 200 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0 50 100 150 200 Figure 3: Infinite-time averages of Krylov complexity of TFS (Left) and TNS (Right) for different values of θ and N = 9. qualitative agreement: states further from equilibrium relax more rapidly, either via faster decay of trace distance or via earlier crossings in complexity. Moreover, the infinite-time averages of Krylov complexity display near-perfect agreement with the IPR of the initial states, as shown in Figure 4, reinforcing the view that both quantities capture Hilbert-space delocalization4. Figure 4: Infinite-time average of complexity (left) and logarithm of the inverse participation ratio (right) for general initial states given in (2.6), with N = 9. Due to the symmetry under φ →2π-φ, we show the results only for 0 ≤φ ≤π. However, while these long-time averages align closely, they are not sufficient to diagnose Mpemba-like behavior on their own. The essential signature lies in the early-time dynamics, governed by the interplay of diagonal and off-diagonal contributions. Only by considering both 4This correspondence aligns with previous studies comparing Hilbert-space spreading and IPR in the context of thermalization [68], highlighting that both quantities capture how efficiently a state explores the Hilbert space over time. 6 transient growth and asymptotic saturation can Krylov complexity fully capture the nontrivial temporal inversions characteristic of the quantum Mpemba effect. 3 Krylov Complexity and the Quantum Mpemba Effect in U(1)-Symmetric Systems In this section, we investigate Krylov complexity in quantum systems with a global U(1) symmetry, with the goal of understanding how a conserved charge influences the manifestation of the quantum Mpemba effect. When a system possesses a conserved charge Q generated by a global U(1) symmetry, its Hilbert space naturally decomposes into orthogonal subsectors labeled by the charge eigenvalues, H = M q Hq , (3.1) where q runs over all possible charge values, and Pq denotes the corresponding projector onto the subsector Hq. Krylov complexity can be formulated in several ways in the presence of symmetry. Following the construction introduced earlier, the most direct approach is to evaluate the standard Krylov complexity of a state without explicitly resolving the symmetry structure. Given an initial state |ψ⟩, one constructs the Krylov basis {|n⟩} and the associated number operator N = PDψ-1 n=0 n|n⟩⟨n|, yielding the complexity as defined in Eq. (2.5). If the initial state belongs to a particular charge subsector, the entire Krylov basis remains confined within that subsector during the evolution. It is often illuminating to express the complexity in a natural energy-charge eigenbasis of the Hilbert space, consisting of the joint eigenstates of H and Q, H|E(q) α , q⟩= E(q) α |E(q) α , q⟩, Q|E(q) α , q⟩= q|E(q) α , q⟩. (3.2) A general normalized state can then be expanded as |ψ⟩= X α,q cαq|E(q) α , q⟩, X α,q |cαq|2 = 1, (3.3) where α runs over the dimension Dq of the charge sector Hq. Since only a single U(1) symmetry is imposed (and, in particular, no additional discrete symmetry is assumed), the spectra of distinct charge sectors are generally non-degenerate, so that almost all eigenvalues correspond uniquely to a given q. 7 In this notation, the Krylov complexity of the evolving state can be written as C(t) = X q,α |cαq|2 N qq αα + X q,α̸=β c∗ α,qcβ,q N qq αβ e-iωqq αβt + X p̸=q,α,β c∗ α,qcβ,p N qp αβ e-iωqp αβt, (3.4) which naturally separates into a "symmetric" (q = p) part (intra-sector contributions) and an "asymmetric" (q ̸= p) part (inter-sector contributions). Here N qp αβ = ⟨q, E(q) α |N|E(p) β , p⟩and ωqp αβ = E(q) α -E(p) β . Alternatively, one can construct a symmetry-adapted version of Krylov complexity by projecting onto individual charge sectors. Defining the subsector number operator Nq = Dψ-1 X n=0 n Pq|n⟩⟨n|Pq, (3.5) the symmetric Krylov complexity is given by [40]5 CS(t) = X q ⟨ψ(t)|Nq|ψ(t)⟩= X q,α |cαq|2 N qq αα + X q,α̸=β c∗ α,qcβ,q N qq αβ e-iωqq αβt , (3.6) which corresponds to the diagonal part of the full decomposition (3.4). The asymmetric contribution is then CA(t) = X p̸=q,α,β c∗ α,qcβ,p N qp αβ e-iωqp αβt . (3.7) As in the symmetryless case, the infinite-time average reduces to the diagonal ensemble, C = X q,α |cα,q|2 N qq αα = Tr(ρDEN), ρDE = X q,α |cα,q|2 |E(q) α , q⟩⟨q, E(q) α | , (3.8) while the condition C(0) = 0 implies C = - X q,α̸=β c∗ α,qcβ,q N qq αβ + X q̸=p,α,β c∗ α,qcβ,p N qp αβ ! > 0. (3.9) From these expressions, several important insights emerge. First, the saturation value of Krylov complexity is primarily determined by the diagonal contributions from the symmetric part, whereas the transient or pre-saturation dynamics are governed by the off-diagonal components, both intraand inter-sector. Equation (3.9) further implies that achieving a larger saturation value requires stronger coherence among energy eigenstates, which simultaneously enhances the initial rate of complexity growth. In systems with a global U(1) symmetry, this coherence arises from two 5See also the related notion of symmetry-resolved Krylov complexity introduced in Refs. [69,70], where the total complexity is expressed as an average over contributions from individual symmetry subsectors. 8 distinct sources: correlations among eigenstates within a charge subsector and interference between different charge sectors. It follows that focusing exclusively on either the symmetric (diagonal) or the asymmetric (offdiagonal) contribution might be insufficient to capture the full dynamics of Krylov complexity. While the asymmetric part exhibits oscillatory behavior that decays through dephasing at long times, it can still encode short-lived coherence effects that strongly influence the transient regime. These interference effects can either enhance or suppress early-time features, and are precisely those that give rise to Mpemba-like crossings in the complexity growth curves. Neglecting either component therefore could remove essential information about the interplay between coherence and dephasing. Consequently, if Krylov complexity is to serve as a faithful probe of relaxation and Mpemba-like behavior, it should, in principle, be considered as a total quantity that integrates both contributions. Nevertheless, the behavior of complexity depends sensitively on the choice of initial state. For certain states, deviations from this general expectation can occur: the transient dynamics may be dominated by intra-sector coherence, leading the symmetric component alone to capture most of the relevant relaxation behavior. In these cases, the symmetric Krylov complexity not only governs the asymptotic saturation value but also encodes significant information about the entire dynamical evolution. This possibility was explicitly demonstrated in Ref. [40] for the Aubry-Andr ́e model [71], where the total complexity showed only a qualitative late-time separation, while the symmetric component exhibited robust Mpemba-like crossings, making it a more sensitive diagnostic of the QME. In particular, a modified definition of the symmetric complexity was introduced, ̃CS(t) = CS(t) + CA(0) , (3.10) which effectively subtracts the asymmetric contribution at t = 0. Including the full asymmetric part was found to wash out the crossings, suggesting that the symmetric component, properly normalized, provides a sharper signature of the effect. To examine whether this behavior persists in interacting systems, we now turn to other U(1)- symmetric spin chains. We first consider the next-nearest-neighbor XXZ Hamiltonian [34,37], H = J1 L-1 X j=1 σx j σx j+1 + σy j σy j+1 + ∆1σz jσz j+1 + J2 L-2 X j=1 σx j σx j+2 + σy j σy j+2 + ∆2σz jσz j+2 , (3.11) which is chaotic and possesses a global U(1) symmetry generated by the total magnetization along the z-axis, Q = L X i=1 σz i . (3.12) 9 To analyze the dynamics, we compute the total and symmetric Krylov complexities for TFS with different tilt parameters. To highlight the crossings, we study the difference ∆C(t) = Cθ1(t) -Cθ2(t) , (3.13) and similarly for the symmetric component. Using parameters consistent with Ref. [37], we present results for tilt angles θ1 = 0.15 and θ2 = 0.2. The corresponding results are shown in Fig. 5, where we have also accounted for the model's additional Z2 symmetry when computing the symmetric complexity. 0.0 0.5 1.0 1.5 2.0 -1.5 -1.0 -0.5 0.0 0.0 0.5 1.0 1.5 2.0 -0.015 -0.010 -0.005 0.000 Figure 5: Difference in Krylov complexity (left) and its symmetric component (right) between initial states with tilt parameters θ1 = 0.15 and θ2 = 0.2, for the TFS in the model given by (3.11). System size is N = 10. As seen in Fig. 5, the total complexity does not display a clear Mpemba-like inversion, whereas the symmetric complexity exhibits pronounced crossings, in agreement with the findings of Refs. [40]. This confirms that, in such systems, the symmetric component provides a more reliable probe of the QME, as proposed by [37]. In contrast, for the TNS, neither the total nor the symmetric complexities show any inversion, consistent with its incoherent relaxation dynamics. We can also explicitly break the residual Z2 symmetry of the model (3.11) by adding a longitudinal field term of the form h 2 L X j=1 σz j, (3.14) to the Hamiltonian. This modification lifts the parity degeneracy and allows us to examine how the breaking of discrete symmetry affects the manifestation of the QME. We have repeated the same numerical analysis as before, computing both the total Krylov complexity and its symmetric component for the TFS. The corresponding results are presented in Fig. 6. As shown in Fig. 6, the symmetric component of the complexity exhibits a finer and more structured temporal evolution than the total complexity, reflecting sensitivity to short-time coherence effects that are largely averaged out in the total measure. However, since this behavior 10 0.0 0.5 1.0 1.5 2.0 -1.5 -1.0 -0.5 0.0 0.0 0.5 1.0 1.5 2.0 -0.03 -0.02 -0.01 0.00 0.01 Figure 6: Difference in Krylov complexity (left) and in its symmetric component (right) between initial states with tilt parameters θ1 = 0.15 and θ2 = 0.2, for the TFS in the model (3.11) with explicitly broken Z2 symmetry due to the longitudinal field term. System size is N = 10. is accompanied by significant oscillations and lacks a clear, monotonic inversion, it might not be interpreted as a genuine realization of the QME. Nevertheless, this comparison further underscores that the symmetric component of Krylov complexity can reveal subtle dynamical features of the system. Let us now consider the integrable XXZ spin chain in a longitudinal field [34], H = -1 4 L-1 X j=1 σx j σx j+1 + σy j σy j+1 + ∆1σz jσz j+1 + h 2 L X j=1 σz j, (3.15) which also possesses a global U(1) symmetry generated by the total magnetization but explicitly breaks the Z2 symmetry through the field term. For this model, we again analyze the TFS and TNS initial states defined in Eq. (2.12), focusing on tilt parameters θ = 1.2 and θ = 0.8 to compare with Ref. [34]. The results are displayed in Fig. 7. For the TFS, ∆C(t) changes sign during the evolution, indicating that the corresponding complexity curves cross each other. This inversion constitutes a clear signature of the Mpemba-like effect: the state prepared further from equilibrium (larger tilt) initially grows more slowly but eventually overtakes the less tilted state. By contrast, for the TNS, ∆C(t) remains of the same sign throughout, showing no evidence of inversion. These results agree with those of Ref. [34], confirming that the full Krylov complexity can indeed capture the short-time reversal of relaxation rates characteristic of the QME in this model. To assess the role of symmetry, we further compute the symmetric complexity for both initial states. The corresponding results are shown in Fig. 8. From Fig. 8, we see that for the TFS, the symmetric complexity exhibits a pronounced earlytime crossing, while the TNS again shows none. This indicates that the essential information 11 0 2 4 6 8 0.0 0.5 1.0 1.5 0 2 4 6 8 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Figure 7: Difference in Krylov complexity between initial states with tilt parameters θ = 1.2 and θ = 0.8, for the TFS (left) and for the positive-parity component of the TNS (right). System size is N = 10. 0 2 4 6 8 10 -0.15 -0.10 -0.05 0.00 0.05 0 2 4 6 8 10 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 Figure 8: Difference in symmetric Krylov complexity between initial states with tilt parameters θ = 1.2 and θ = 0.8, for the TFS (left) and for the positive-parity component of the TNS (right). System size is N = 10. about the Mpemba-like inversion is already encoded in the symmetric sector. However, unlike the previous case, the infinite-time average of the symmetric complexity does not necessarily increase with tilt. The reason lies in the relation ̃CS = CS + CA(0) , (3.16) which shows that the asymptotic value of the symmetric component depends on the initial coherence contained in the asymmetric sector. Thus, while early-time crossings diagnose Mpemba behavior, late-time averages remain insensitive to the initial state. In summary, our analysis demonstrates that in U(1)-symmetric systems, the symmetric component of Krylov complexity provides a robust and reliable probe of the QME. Although the total complexity can in some cases reproduce Mpemba-like inversions, these features are generally less 12 pronounced and more sensitive to the choice of initial state. The symmetric complexity defined by Eq. (3.10), on the other hand, consistently captures the essential signatures of the effect across all models we have examined, confirming and extending the proposal of Ref. [40]. This finding highlights that Krylov complexity, particularly in its symmetry-adapted form, serves as an insightful diagnostic of anomalous relaxation and the subtle interplay between coherence, dephasing, and symmetry constraints in quantum many-body dynamics. 4 Conclusions In this work, we have systematically studied Krylov state complexity as a diagnostic for the QME in both symmetry-broken and U(1)-symmetric quantum spin models. By analyzing the time evolution of Krylov complexity across different initial states, we have shown that, for systems without symmetry, the structure of complexity growth faithfully encodes the anomalous relaxation patterns that define the QME. In particular, the characteristic inversions in relaxation rates-which distinguish the Mpemba effect from conventional thermal equilibration-are directly reflected in the temporal evolution of Krylov complexity. This establishes Krylov complexity as a quantitative and physically transparent measure for detecting and characterizing the QME within closed quantum systems. Turning to models with global U(1) symmetry, our analysis reveals that while the total Krylov complexity can, in certain cases and for specific initial states, display clear Mpemba-like crossings, it does not in general provide a robust probe of anomalous relaxation. The total measure is often sensitive to fine details of the initial state and to coherence effects between charge sectors, which can obscure or suppress the inversion behavior that signals the QME. A more reliable and universal characterization emerges when the complexity is decomposed into its symmetric and asymmetric components, corresponding respectively to diagonal and off-diagonal contributions in the charge basis. By isolating the symmetric component and performing an appropriate shift that compensates for the initial contribution of the asymmetric part, one obtains a quantity that consistently exhibits Mpemba-like behavior across all models studied. This shifted symmetric Krylov complexity captures the essential dynamical information governing relaxation, while remaining insensitive to transient oscillations that can mask the effect in the total complexity. Our numerical analysis across multiple spin-chain models-both integrable and non-integrable-confirms that this measure provides a stable and physically transparent probe of the QME, thereby supporting and extending the proposal of Ref. [40]. These findings demonstrate that symmetry resolution, far from being a mere technical refinement, plays a crucial role in diagnosing anomalous relaxation through complexity. In the presence of conserved charges, the symmetric sector encodes the irreversible aspects of Hilbert-space exploration, while the asymmetric part captures transient coherence and dephasing. The shifted symmetric complexity effectively combines these contributions into a single measure that reflects 13 the true dynamical hierarchy of relaxation times. From a broader perspective, our results highlight the potential of Krylov-space methods as powerful tools for investigating nonequilibrium quantum dynamics. By linking complexity growth to relaxation pathways and coherence structure, they provide a unifying framework for understanding anomalous thermalization and information scrambling (see, for example, Ref. [68]). The sensitivity of Krylov complexity to coherent interference, spectral structure, and the interplay between diagonal and off-diagonal contributions makes it a natural bridge between state spreading, quantum chaos, and anomalous relaxation phenomena. In the context of the QME, it offers a unified language that connects dynamical reversals of relaxation rates to the microscopic geometry of Hilbert-space exploration. In this work, we have focused exclusively on finite-dimensional many-body systems. An interesting extension would be to investigate infinite-dimensional cases, such as quantum field theories, where Krylov complexity may not saturate but continues to grow indefinitely with time. In such systems, one can meaningfully study the rate of complexity growth at late times and compare it with the early-time behavior to identify possible inversions. Here, the Mpemba-like effect would no longer be manifested through relaxation rates but through the dynamics of complexity growth itself: states that exhibit faster early-time growth may subsequently evolve more slowly at late times. In other words, states that initially appear "more complex" can eventually become "less complex" -a phenomenon that naturally extends the notion of the Mpemba effect to the realm of dynamical complexity. Taken together, our results in spin-chain models suggest that Krylov complexity provides a unified and physically transparent framework for characterizing Mpemba-like inversions across both finite and infinite-dimensional quantum systems. 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2510.14743
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MNRAS 000, 1–17 (2025)
Preprint 17 October 2025
Compiled using MNRAS LATEX style file v3.3
Mergers lighting the early Universe: enhanced star formation, AGN
triggering, and Ly𝛼emission in close pairs at 𝑧= 3 −9
Dávid Puskás
,1,2⋆Sandro Tacchella
,1,2 Charlotte Simmonds
,1,2,3 Gareth C. Jones
,1,2
Ignas Juodžbalis
,1,2 Jan Scholtz
1,2, William M. Baker
,4 Andrew J. Bunker
,5
Stefano Carniani
,6 Emma Curtis-Lake
,7 Qiao Duan
,1,2 Daniel J. Eisenstein
,8
Kevin Hainline
,9 Benjamin D. Johnson
,8 Roberto Maiolino
,1,2,10 Marcia Rieke
,9
Brant Robertson
,11 Christina C. Williams
,12 and Joris Witstok
13,14
1Cavendish Laboratory, University of Cambridge, 19 JJ Thomson Avenue, Cambridge, CB3 OHE, UK
2Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK
3Departamento de Astronomía, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile
4DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen, Denmark
5Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
6Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
7Centre for Astrophysics Research, Department of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield AL10 9AB, UK
8Center for Astrophysics ∣Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA
9Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA
10Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
11Department of Astronomy and Astrophysics University of California, Santa Cruz, 1156 High Street, Santa Cruz CA 96054, USA
12NSF National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA
13Cosmic Dawn Center (DAWN), Copenhagen, Denmark
14Niels Bohr Institute, University of Copenhagen, Jagtvej 128, DK-2200, Copenhagen, Denmark
17 October 2025
ABSTRACT
Galaxy mergers and interactions are often invoked to explain enhanced star formation, black hole growth, and mass build-up of
galaxies at later cosmic times, but their effect is poorly understood at high redshift (𝑧> 2). We use JADES data to analyse a mass-
complete sample of 2095 galaxies at 𝑧= 3 −9 with log(𝑀⋆/M⊙) = [8, 10], identifying major merger pairs (projected separation
of 5 −100 pkpc, mass ratio ≥1/4) using a probabilistic method. To look for signatures of enhancement in multiple physical
properties, we carefully build a control sample of non-pairs that are simultaneously matched in redshift, stellar mass, isolation,
and environment to the pair sample. We find a moderate enhancement in specific star formation rate (sSFR) of 1.12 ± 0.05 at
separations ≲20 kpc, which is weakly detectable out to ∼50 kpc. We find that at longer averaging timescales (50-100 Myr) the
sSFR is more affected by interactions and environment, whereas at shorter timescales (5-10 Myr) it is dominated by internal
feedback and burstiness. By averaging star formation histories, we find two distinct populations: pre-first passage/coalescence
(monotonically rising SFR) and post-pericentre pairs (earlier peak in SFR). Finally, we find no significant excess of AGN in
pairs, suggesting galaxy interactions are not effectively triggering black hole activity at separations > 5 kpc. Similarly, we also
do not detect an excess in the fraction of Lyman-𝛼emitters in pairs, implying that at the probed separations, galaxy interactions
are not efficient at enhancing Lyman-𝛼photon production and escape, which may only become important at the smallest scales.
Key words: galaxies: evolution – galaxies: high-redshift – galaxies: interactions – galaxies: star formation – galaxies: active
1 INTRODUCTION
Galaxy mergers and interactions play a key role in shaping the evolu-
tionary path of individual galaxies. In the Lambda Cold Dark Matter
(ΛCDM) cosmological framework, dark matter haloes merge under
the hierarchical growth model, resulting in the tidal disruption and
eventual merger of the galaxies embedded within them (Somerville &
⋆E-mail: dp670@cam.ac.uk
Davé 2015). It has been well established that merging systems can un-
dergo profound structural transformations (Toomre & Toomre 1972),
elevated star formation rates (SFRs) (Barton et al. 2000; Ellison et al.
2008; Scudder et al. 2012; Bickley et al. 2022), and enhanced active
galactic nucleus (AGN) activity (Ellison et al. 2011, 2013, 2019).
Most of these results come from large low-redshift samples. At high
redshift, the abundance of merger events has only recently begun to
be constrained at 𝑧> 6 (Duncan et al. 2019; Duan et al. 2025; Puskás
© 2025 The Authors
arXiv:2510.14743v1 [astro-ph.GA] 16 Oct 2025
2
Puskás et al.
et al. 2025), and the effects of these mergers on galaxy properties
remain poorly understood.
Theory and simulations predict that the gravitational interaction
during close galaxy-galaxy encounters and mergers can induce non-
axisymmetric features (e.g., bars) and strong torques that channel
low-metallicity gas to galactic centres, fuelling starbursts and po-
tentially triggering AGN (Mihos & Hernquist 1996; Springel et al.
2005; Cox et al. 2008; Scudder et al. 2012; Torrey et al. 2012; Hop-
kins et al. 2013; Patton et al. 2013; Moreno et al. 2015; Patton et al.
2020; Byrne-Mamahit et al. 2024). For similar-mass mergers, gas in-
flows can drive short, intense bursts (enhancements of order a factor
of ∼2), with the strength depending primarily on internal structure
and mass ratio rather than orbital parameters (Mihos & Hernquist
1996; Cox et al. 2008). Gas-rich mergers may then fuel black hole
(BH) accretion and feedback, expelling (or using up) gas near coa-
lescence and contributing to quenching (Springel et al. 2005). Many
simulation-based works describe moderate SFR enhancement for
several hundred Myr after first pericentre and a stronger burst shortly
before coalescence (e.g., Mihos & Hernquist 1996; Cox et al. 2008;
Scudder et al. 2012), with the post-coalescence starburst lasting up
to ∼0.5 Gyr (Hani et al. 2020; Ferreira et al. 2025) and potentially
producing quenched elliptical remnants (Quai et al. 2021; Ellison
et al. 2022).
Observational studies of galaxy mergers typically focus on two
aspects: (i) the frequency of mergers (pair fractions and merger rates)
as a function of redshift, and (ii) the impact on galaxy properties.
The merger rate has been well constrained at lower redshifts (𝑧<
3) (see, e.g., Man et al. 2016; Mundy et al. 2017; Mantha et al.
2018; Conselice et al. 2022), with several works extending to higher
redshifts 𝑧∼6−7 (e.g., Duncan et al. 2019; Ventou et al. 2017). Only
a few recent works (Dalmasso et al. 2024; Duan et al. 2025; Puskás
et al. 2025) probed the highest redshifts beyond 𝑧> 6 thanks to the
wealth of high-resolution photometry and spectroscopy from JWST.
These works found an initially steeply increasing, then a flattening
merger rate with redshift, stabilising in the range of 2 −10 major
mergers/galaxy/Gyr at 𝑧> 6 (Puskás et al. 2025). To infer the merger
rate, two widely used methods exist for selecting galaxies that are
about to merge: the close-pair method that involves counting galaxies
that are in close pairs on the projected plane of the sky (e.g., Mundy
et al. 2017; Duncan et al. 2019; Duan et al. 2025; Puskás et al.
2025), and the morphological method that involves finding systems
showing disturbed morphologies that are at the late stages or have
recently completed merging (e.g., Conselice et al. 2008; Rose et al.
2023; Dalmasso et al. 2024). Here we focus on the effects of close
encounters on internal galaxy properties, rather than the merger rate
itself, which has been extensively discussed in Puskás et al. (2025).
The properties of galaxy mergers and their influence on SFRs
have been studied in depth using large spectroscopic samples at low
redshift (𝑧< 1), such as the CfA2 (Barton et al. 2000) and 2dF
(Lambas et al. 2003) surveys. A wealth of studies using data from
the Sloan Digital Sky Survey (SDSS; latest data release DR19 SDSS
Collaboration et al. 2025) have shown that galaxies in close pairs
with projected separations lower than 80 kpc exhibit enhanced SFRs
(on average 60 per cent higher) in comparison to a carefully matched
control sample of isolated galaxies (Nikolic et al. 2004; Ellison et al.
2008; Scudder et al. 2012; Ellison et al. 2013; Patton et al. 2013,
2016). Patton et al. (2013) have found moderate SFR enhancements
at separations 20 −100 kpc, and smaller enhancements even up to
∼150 kpc. This was obtained through careful selection and matching
of control samples to compare galaxy close pairs with galaxies that
have very similar physical properties but are isolated.
However, at higher redshift (𝑧> 1), there is less agreement be-
tween different theoretical and observational studies regarding the
enhancement (if any) or even the decrease of SFR between merger
and non-merger samples. Simulations by Fensch et al. (2017) demon-
strate that high-redshift (up to 𝑧∼4), gas-rich galaxy mergers are
substantially less efficient at enhancing their SFRs than their low-
redshift, gas-poor counterparts, producing starbursts that are ap-
proximately ten times weaker and shorter in duration. Observational
studies also find different trends at higher redshifts. For example,
at 0.5 < 𝑧< 3, Shah et al. (2022) have found a somewhat weaker,
but still detectable enhancement (factor or ∼1.23+0.08
−0.09) at the clos-
est separations of ≲25 kpc. Probing to the highest redshift to date
using novel JWST data, Duan et al. (2024) have only detected weak
enhancements of SFRs at the smallest separations of 𝑟p < 20 kpc of
0.25±0.10 dex and 0.26±0.11 dex above the non-merger medians for
the redshift ranges of 4.5 < 𝑧< 6.5 and 6.5 < 𝑧< 8.5, respectively.
On the other hand, Silva et al. (2018) find no significant difference
in SFR between merging and non-merging galaxies at 𝑟p < 15 and
0.5 < 𝑧< 2.5, with only a small fraction (12 per cent) of massive sys-
tems showing starburst activity. Using a machine learning approach
on over 200,000 galaxies, Pearson et al. (2019) found that while
mergers typically have little to no effect on the average SFR (typical
change in SFR is less than 0.1 dex in either direction), the fraction
of galaxies identified as mergers steadily increases in systems with
the most highly-enhanced, starbursting activity. Therefore, the exact
effect and its magnitude of mergers on the SFR of the constituent
galaxies remain uncertain at redshift 𝑧> 3.
Furthermore, mergers have been found to efficiently trigger AGN
activity and drive BH accretion (e.g. Ellison et al. 2011, 2013; Byrne-
Mamahit et al. 2024). Simulations (e.g., Di Matteo et al. 2005)
show that gravitational torques during an interaction can funnel vast
amounts of gas to the galactic centre, igniting both a starburst and a
luminous quasar. Observational evidence provides a more nuanced
picture. Studies of large galaxy samples find a modest enhancement
(e.g., a factor of 2-3 found by Ellison et al. 2011, 2013) in the fraction
of AGN in close galaxy pairs (separations < 20 −30 kpc), indicating
that the final stages of mergers are most impactful. This link appears
strongest for the most powerful and obscured AGN. For example,
work by Koss et al. (2010) and Satyapal et al. (2014) found that lumi-
nous, hard X-ray or IR-selected AGN are significantly more likely to
reside in disturbed, merging systems than their inactive counterparts.
Other studies confirm that post-mergers host an excess of AGN com-
pared to isolated galaxies (e.g., Bickley et al. 2023; Byrne-Mamahit
et al. 2024). However, since the overall fraction of AGN in these
merging populations remains low, it is unclear if mergers guarantee
AGN triggering. Furthermore, this connection appears to weaken
with lookback time, as studies of high-z interacting galaxies often
find no statistically significant AGN excess (e.g. Shah et al. 2020;
Silva et al. 2021; Byrne-Mamahit et al. 2024; Dougherty et al. 2024).
Moreover, Lyman-𝛼emitters (LAEs) at high-z are often found in
ionized bubbles already present in the Epoch of Reionization (EoR),
embedded in large overdensities (e.g., Larson et al. 2022; Leonova
et al. 2022; Whitler et al. 2024). The presence of such overdensities
(e.g., Helton et al. 2024) can boost galaxy interactions and mergers,
as well as potentially lead to subsequent quenching (e.g., Carnall
et al. 2024; Baker et al. 2025; de Graaff et al. 2025; Cochrane 2025).
The resulting merger-induced starbursts and potential AGN activ-
ity can provide the powerful ionizing radiation required to illuminate
the surrounding reservoirs of hydrogen gas and may create disturbed,
clumpy ISM/CGM morphologies, drive strong outflows, and carve
low-density channels that allow efficient Ly-𝛼photon escape. Recent
observations with JWST unveiled a large population of LAEs at red-
shift 𝑧∼5 −8 deep into the EoR (e.g., Saxena et al. 2023; Witstok
MNRAS 000, 1–17 (2025)
Interaction driven star formation
3
et al. 2024; Jones et al. 2025), which show morphological distur-
bances and have resolved close companions (e.g., Witten et al. 2024;
Ren et al. 2025). It is yet to be understood what fraction of these
LAEs are found to be in close pairs and if mergers on a population
level effectively trigger them.
The aim of this study is to investigate whether the presence of a
close companion has any influence on the physical properties of its
host galaxy, such as star formation or mass growth. Furthermore, we
assess if this pre-merger phase of close galaxy pairs triggers any AGN
activity and Ly𝛼emission that would deepen our understanding of the
EoR. We aim to compare our mass-complete sample of major merger
pairs (𝑟p = 5–100 kpc, mass ratio ≥1/4, log10(𝑀⋆/M⊙) = 8–10) to
a robust control set simultaneously matched in redshift, stellar mass,
local density, and isolation, enabling us to isolate the incremental
effect of the nearest neighbour from environmental trends. This has
been done by several studies for the local Universe, however, in this
work, we extend the analysis for the 3 ≤𝑧≤9 redshift range using
deep JWST data.
The paper is structured as follows. Section 2 provides a brief
overview of the catalogues and data products used in this study. We
then describe galaxy pair selection and its environmental characteri-
sation, and detail the control matching process. Section 3 describes
the influence of the companion on the observed physical properties
of the host galaxies compared to the control sample, such as star for-
mation rate and star formation history, dependence on stellar mass
and redshift, as well as environmental dependence. In Section 4 we
discuss the fraction of AGNs and Ly𝛼emitters found in pairs com-
pared to the overall spectroscopic pair fraction. Section 5 discusses
our findings and their implications in a broader cosmological con-
text, as well as some caveats of our study and the effect of parameter
choices. Finally, Section 6 summarises our work and highlights the
main findings.
Throughout this paper, we adopt the AB magnitude system (Oke
& Gunn 1983). We use a standard cosmology with Ω𝑚= 0.310,
ΩΛ = 0.689, and 𝐻0 = 67.66 km s−1 Mpc−1 (Planck Collaboration
et al. 2020). Throughout our analysis, we use the Astropy python
package (Astropy Collaboration et al. 2022), and its subpackage
astropy.cosmology, where we assume a flat ΛCDM cosmology
with parameters from Planck Collaboration et al. (2020).
2 SAMPLE AND METHODOLOGY
In Section 2.1, we briefly summarise the data sets used from the
JADES survey, and the photometric and spectroscopic redshift cat-
alogues. Section 2.2 presents the spectral energy distribution (SED)
modelling and the assumptions that go into it. The probabilistic
galaxy pair selection process for finding major mergers is presented
in Section 2.3. In Section 2.4, we characterise the environment of the
galaxies in our target sample, which contains both pairs and control
candidates. Finally, in Section 2.5, we present the construction of a
robust control sample by simultaneously matching pairs in both stel-
lar mass and redshift space, as well as local density and isolation, to
compare the physical properties of pairs against those of the controls.
2.1 Data
This study is based on the rich photometric and spectroscopic
data from the JWST Advanced Deep Extragalactic Survey (JADES,
Eisenstein et al. 2023a; Rieke et al. 2023; Eisenstein et al. 2023b;
Bunker et al. 2024; D’Eugenio et al. 2025). We divide the two con-
stituent JADES fields, GOODS-South and GOODS-North, into four
different sub-tiers based on the area and exposure time. For a de-
tailed description of the various data sets used, we refer the reader
to Section 2 of Puskás et al. (2025). We briefly summarise here the
photometric and spectroscopic redshifts used for this study, and the
SED modelling in the following section.
We obtain photometric redshifts for all galaxies using the template-
fitting code EAZY (Brammer et al. 2008). A detailed description of
the templates used and assumptions made for our fits can be found
in section 3.1 of Hainline et al. (2024). Our adopted redshift is taken
from the minimum in the 𝜒2 of the fit. For each source, we obtain the
photometric redshift posterior distribution by assuming a constant
prior for the redshift, and calculate 𝑃(𝑧) = exp[−𝜒2(𝑧)/2] with
a normalisation of ∫𝑃(𝑧)𝑑𝑧= 1.0. To select a robust sample of
galaxies with accurate photometric redshifts, we use the odds quality
parameter O as a proxy for the reliability of the photometric redshift
fit, which is defined as
O = ∫
+𝐾(1+𝑧𝑎)
−𝐾(1+𝑧𝑎)
𝑃(𝑧−𝑧𝑎)𝑑𝑧,
(1)
with 𝐾= 0.03 chosen for our analysis. In our sample selection,
we require that all objects must have an odds parameter O ≥0.3.
Furthermore, we assemble a catalogue of all available spectroscopic
redshifts, which consists of 5382 sources in the GOODS-South and
2591 in the GOODS-North field, all of which fall within the best
category in quality and coordinate match. For a detailed description
of this catalogue, see Section 2 of Puskás et al. (2025) and references
therein.
2.2 SED modelling and star formation rates
Stellar population properties were derived using the SED modelling
code Prospector (Johnson et al. 2019; Johnson et al. 2021). We
refer the reader to Simmonds et al. (2024) and Simmonds et al.
(2025) for a detailed description of the fitting methodology, priors,
and assumptions, and briefly summarise the aspects most relevant
for this work. Of particular importance is the use of a non-parametric
star-formation history (SFH) with the continuity prior of Leja et al.
(2019), in which the SFH is described by eight time bins. The first
bin extends to a lookback time of 5 Myr (relative to the redshift of the
galaxy), and the last bin is fixed at the lookback time corresponding
to 𝑧= 20, with the remaining time bins divided into equal intervals of
log10(𝑡lookback). The ratios and amplitudes between adjacent bins are
allowed to vary according to a Student’s 𝑡-distribution with a scale
width of 0.3, which permits – but does not enforce – a modest level
of burstiness when supported by the data. From the reconstructed
SFHs, we compute time-averaged SFRs on different timescales 𝑡avg
according to
SFR𝑡avg =
1
𝑡avg ∫
𝑡avg
0
SFR(𝑡′) 𝑑𝑡′,
(2)
where 𝑡′ is the lookback time. For this study, we primarily adopt
SFR100 (SFR averaged over 100 Myr) as our fiducial measure of
recent star formation, while also exploring shorter (10−50 Myr) and
longer (2000 Myr) averaging windows to assess the dependence of
our results on timescale.
2.3 Galaxy pair selection
We aim to investigate the influence of close companions on each
galaxy in our selected target sample, which we define below, to sta-
tistically measure the effect of mergers on the physical properties of
the host galaxies. Our selection of galaxy pairs strongly relies on the
MNRAS 000, 1–17 (2025)
4
Puskás et al.
methodology outlined in Section 3 of Puskás et al. (2025). However,
in this work we do not exclusively study close pairs, defined as galaxy
pairs having a projected physical separation at 5 kpc ≤𝑟p ≤30 kpc,
but we extend our search to larger separations (up to 𝑟max = 100 kpc)
to select wide pairs. We simply refer to these selected systems as
galaxy pairs for the remainder of this paper. We note here that we
define the projected physical separation as 𝑟p = 𝜃× 𝑑𝐴(𝑧), where 𝜃
is the measured angular separation between the pairs and 𝑑𝐴(𝑧) is
the angular diameter distance. In the following, we summarise the
selection criteria from Puskás et al. (2025), which is further based
on the probabilistic close-pair selection method developed by López-
Sanjuan et al. (2015) and Mundy et al. (2017). The strength of this
method lies in using the full photometric redshift probability distri-
bution function to propagate the associated uncertainties, as opposed
to only using the peaks of these distributions, as well as incorporating
all available spectroscopic redshifts.
We are interested in the influence of close companions exerted on
the central or host galaxy, which we refer to as the primary galaxy.
This is defined as the most massive galaxy in the system, and it is
selected from a primary sample, which is described in a later section.
In this study, we are only interested in major merger systems, which
are defined as having a stellar mass ratio between the secondary and
the primary galaxy of 𝜇= 𝑀2/𝑀1 ≥0.25 (e.g., Lotz et al. 2011).
For our major merger pair catalogue, we apply the selection criteria
described below.
1. The primary galaxy must be in the initial primary sample cata-
logue (detailed later in this section), being in the redshift range of
3 ≤𝑧≤9 and stellar mass range of log10(𝑀⋆/M⊙) = [8, 10].
2. The projected physical separation to the closest companion must
be within 𝑟min ≤𝑟p ≤𝑟max, with our choice of 𝑟min = 5 kpc,
to avoid confusing star-forming clumps within the same galaxy
with mergers and ensure clearly deblended sources, and 𝑟max =
100 kpc, to assess the influence of companions out to wide enough
separations. We convert this projected physical separation to an
observable angular separation using the angular diameter distance
calculated at the peak redshift of the central galaxy.
3. To ensure close proximity in the radial direction dictated by
the redshift, we require the integral of the pair probability func-
tion (PPF; defined in Section 3 of Puskás et al. 2025) to be
∫
9
3 PPF(𝑧)𝑑𝑧> 0.7. This quantity incorporates the full photo-
metric redshift posterior distributions from EAZY and gives the
probability of a system of two galaxies being in a major merger
pair.
4. To find major merger pair systems, we require a stellar mass ratio
above 𝜇= 𝑀2/𝑀1 ≥0.25.
We select galaxies from an initial sample defined in detail in Puskás
et al. (2025), resulting in a sample of 2095 galaxies in pairs, focus-
ing on the properties of the primary (more massive) galaxies in these
pairs. In summary, we perform the selection separately in four distinct
JADES sub-fields, namely the GOODS-South (GS) and GOODS-
North (GN) Deep and Medium tiers. We require a signal-to-noise ra-
tio of SNR ≥3 in the F444W_KRON_CONV photometry (Kron-aperture
placed on images that have been convolved to the same resolution).
We select galaxies in the redshift range of 3 ≤𝑧≤9, where most
objects have photometric redshifts estimated by EAZY, and addition-
ally, we use all available spectroscopic redshifts (detailed in Puskás
et al. 2025). In the case of photometric redshifts, we require an odds
quality parameter O ≥3 for robust 𝑧phot measurements. Furthermore,
we define our initial primary sample to have stellar masses estimated
by Prospector in the range of log(𝑀⋆/M⊙) = [8, 10]. The sec-
ondary sample of galaxies, which contains all potential companions
ID1 = 7358
z = 3.32
50 kpc
50 kpc
50 kpc
50 kpc
Figure 1. An example galaxy pair system with its environment (top left), and
three closely matched control systems. The primary galaxy is in the centre
of each cutout, indicated by a red circle. The secondary galaxy in the pair
system is indicated by a blue circle (only present on the top left cutout).
The second closest companion in this case is indicated by an orange circle,
while in the case of the three control systems, the orange circles indicate
the closest companions (at comparable distances to the pair system’s second
closest neighbour). All other galaxies that are counted in the local density
measurement 𝑁env are indicated by a white circle. The cutouts shown above
are limited to a size of 200 × 200 kpc for display purposes (the environment
is analysed within a projected physical radius of 1 Mpc). While the galaxies
in question are not easily visible, our focus here is on their relative projected
positions, using a real example, rather than on their individual appearance.
corresponding to the primary sample, extends to lower masses, as
the stellar mass ratio is fixed for major mergers at 𝜇≥0.25. Both
the primary and the secondary sample have to be above the mass
completeness limit 𝑀comp
⋆
(𝑧), which is determined separately for
the four tiers individually based on their 5𝜎photometric depths (in
the F444W band) using the method by Pozzetti et al. (2010).
2.4 Environmental characterisation
The aim of this study is to investigate the influence of the closest com-
panion on the physical properties of the primary galaxy. However, to
properly assess this influence, one must also consider the surround-
ing environment of the pair. This has been done in previous studies
of the local Universe, typically at 𝑧< 0.2, using various metrics to
characterise the environment (e.g., Ellison et al. 2010; Scudder et al.
2012; Patton et al. 2013, 2016; Garduño et al. 2021), with only a few
works extending up to 𝑧∼3 (e.g., Shah et al. 2022).
For the purpose of this work, we apply the same metrics to char-
acterise the galaxy environment as in Patton et al. (2013) and Patton
et al. (2016), namely, local density and isolation. These properties are
measured in various ways by different studies, for example Ellison
et al. (2010) uses projected galaxy density averaged from the dis-
tances to the fourth and fifth nearest neighbours within 1000 km s−1,
and Scudder et al. (2012) measures the density via the distance to the
fifth nearest neighbour.
MNRAS 000, 1–17 (2025)
Interaction driven star formation
5
To measure the local density, we use a simplified metric 𝑁env,
which gives the total number of companions around a galaxy within
a projected physical separation of 1 Mpc. We choose this separa-
tion limit as it is an order of magnitude larger than the limit for
selecting galaxy pairs. In our fiducial analysis, galaxies counted in
𝑁env have to both satisfy the mass ratio criterion of 𝜇≥0.25, as
well as ∫
9
3 PPF(𝑧)𝑑𝑧> 0.1. This latter limit on the pair probability
function integral is lower than in the case of the closest companion
(≥0.7) as we want to include all possible surrounding galaxies in the
neighbourhood (even if they have a low probability), and the radial
proximity criterion also becomes looser at separations comparable
to ∼1 Mpc.
The isolation of a galaxy pair in terms of environment can be
measured by the projected distance to the second closest compan-
ion, denoted by 𝑟2. The value of this parameter provides insight into
the environment of a galaxy pair: high 𝑟2 values indicate relative
isolation, intermediate 𝑟2values correspond to more typical field en-
vironments, and low 𝑟2 values are characteristic of densely populated
regions such as galaxy clusters or compact groups. The second clos-
est has to satisfy the same criteria as galaxies considered in the local
density measurement.
In order to measure these properties, we modify the pair finder
algorithm presented in Puskás et al. (2025) to find all neighbours
around each primary galaxy out to projected physical separations of
1 Mpc, satisfying the criteria for the mass ratio (𝜇≥0.25) and the
pair probability (∫PPF(𝑧)𝑑𝑧> 0.1).
2.5 Control sample
To robustly assess the influence of close companions, it is essen-
tial to compare these systems against a carefully constructed control
sample. The control matching must be rigorously done, especially
at such high redshifts, since most inferred galaxy properties already
have large uncertainties, and there are several potential effects that
could drive scatter in these measurements. Comparison to appropri-
ately selected control samples has been robustly done at low redshift
(𝑧< 0.3) by several studies (e.g., Ellison et al. 2008, 2010; Scudder
et al. 2012; Patton et al. 2013, 2016; Garduño et al. 2021; Ellison
et al. 2022; and up to 𝑧∼3 in Shah et al. 2022). However, the few
existing studies at 𝑧> 3 (e.g., Duan et al. 2024) lack such a careful
control matching, which is partly due to limits in depth and area in
high redshift surveys, but also due to the lower abundance of galaxies
with similar properties (e.g., stellar mass and redshift).
In this work, we ambitiously wish to perform a robust control
matching to show the genuine influence of the closest companions.
To this end, we match a statistical control sample to each pair system
in four parameters: redshift, stellar mass, local density, and isolation.
We want to compare our selected pairs to galaxies that have similar
properties (redshift and stellar mass) and environment, the only dif-
ference lying in the presence of a close companion in the former case.
The control sample plays an important role, because many physical
properties depend on stellar mass and redshift, such as SFR (star
forming main sequence, see, e.g., Speagle et al. 2014; Popesso et al.
2023; Simmonds et al. 2025), or metallicity (mass-metallicity rela-
tion, e.g. Nakajima et al. 2023; Curti et al. 2024). Galaxy properties
are also correlated with environment, such as the case of overden-
sities (e.g., Helton et al. 2024) that can also potentially drive Ly𝛼
emission (e.g., Larson et al. 2022; Leonova et al. 2022; Whitler et al.
2024), as well as quenching (e.g., Alberts & Noble 2022; Baker et al.
2025). By carefully matching the control sample, we can ensure that
any observed differences between galaxy pairs and their controls are
not the result of underlying differences in fundamental properties.
Table 1. Summary of the four parameters that we use to match a robust control
sample to the galaxy pair sample and their chosen tolerances.
Property
Symbol
Matching tolerance
Redshift
𝑧
Δ𝑧≤0.3
Stellar mass
log10(𝑀⋆/M⊙)
Δ log10(𝑀⋆/M⊙) ≤0.3
Isolation
𝑟2 [kpc]
∣1 −
𝑟ctrl
p
𝑟pair
2
∣≤0.25
Local density
𝑁env
∣1 −𝑁ctrl
env
𝑁pair
env
∣≤0.4
The control matching is implemented such that we simultaneously
search in a four-dimensional parameter space for the closest match.
The initial control pool is identical to the initial sample of primary
galaxies described in Section 2.3, excluding the respective primary
galaxy itself to which we wish to match the controls. This is due to
the requirement that we want to match closely in stellar mass and
redshift. For matching in local density, we use our estimates of 𝑁env
for both the galaxy pair and the control candidate, where in the case
of the pair, the closest companion is also included in the calculation
of 𝑁env. Finally, to perform the matching in isolation, we require
that the projected distance to the second closest companion (𝑟2) of
the galaxy in question (the primary galaxy of the pair system) is
similar to the projected distance to the closest companion (𝑟p) of the
control galaxy. That is, the magnitude of isolation of the galaxy pair
should closely match that of each corresponding control galaxy. See
Fig. 1 for an example of a galaxy pair and its environment, and three
matched controls that agree well in the four properties.
To ensure good matches, further to searching for the closest control
in the four-parameter space, we introduce tolerances (see Table 1 for
a summary). We require that the difference between the redshift of
the paired galaxy and the control must be smaller than 𝑧tol = 0.3,
where we use the peak of the photometric redshift posteriors or,
if available, the spectroscopic redshift. This is significantly higher
than, for example, 0.01 used in Patton et al. (2016), where they only
use spectroscopic redshifts. In the case of stellar mass, we similarly
choose log10(𝑀tol/𝑀⊙) = 0.3 dex, as it is estimated by SED-fitting
and has similar uncertainties (if not higher) than the 𝑧phot posteriors.
We further require an agreement within 25 per cent between 𝑟2 of
the paired galaxy and 𝑟p of the control, that is ∣1 −𝑟ctrl
p /𝑟pair
2
∣≤0.25.
This is a stronger tolerance than in the case of redshift and stellar
mass, which is due to the fact that projected separations can be
more reliably measured directly from the photometric images than
the former two quantities. In the case of local density, we require
agreement within 40 per cent, that is ∣1 −𝑁ctrl
env/𝑁pair
env ∣≤0.40, due
to the more uncertain nature of the environment selection with the
minimum pair probability requirement of ∫PPF(𝑧)𝑑𝑧> 0.1.
We perform the matching to find the 5 closest controls in the four-
dimensional parameter space within the respective tolerances for each
paired galaxy. We allow for replacement, which means both that a
given galaxy can be a control for more than one pair, and that one pair
may repeatedly have the same control multiple times if there are no
closer matches within the required tolerances. This caveat is due to
the limited sample size of the parent catalogue at these high redshifts.
To achieve a robust and representative control catalogue, we require
that the resulting Kolmogorov–Smirnov (KS) probability of the four
MNRAS 000, 1–17 (2025)
6
Puskás et al.
8.0
8.5
9.0
9.5
10.0
log10 (M /M
)
0.00
0.05
0.10
0.15
0.20
Fraction of Galaxies
pKS = 0.59
Pairs
Controls
0
200
400
600
800
r2 [kpc]
0.0
0.1
0.2
0.3
0.4
0.5
Fraction of Galaxies
pKS = 0.37
Pairs
Controls
3
4
5
6
7
8
9
Redshift
0.0
0.1
0.2
0.3
Fraction of Galaxies
pKS = 0.33
Pairs
Controls
0
200
400
600
Nenv
0.0
0.1
0.2
0.3
0.4
Fraction of Galaxies
pKS = 0.52
Pairs
Controls
Figure 2. Histograms of the distributions of stellar mass (𝑀⋆), isolation (𝑟2), redshift and local density (𝑁env) of the selected galaxy pairs and the matched
control sample. These distributions and the corresponding KS test probabilities displayed suggest that the two samples are statistically indistinguishable, meaning
that we achieved a robust and representative control sample for our selected galaxy pairs. This allows us to study the differences in the physical properties of the
paired galaxies compared to the controls, which is driven by the presence of the close companions.
pair-control parameter distributions be consistent with each other at
least at the 30 per cent level. For our sample of 2095 galaxies in pairs,
we matched a robust control sample of 10250 galaxies, having found
no suitable controls for 9 pairs, and only very few repeated entries
for the controls (repeats were only necessary in the case of 2.7% of
the pairs). The resulting KS test probabilities are 0.59, 0.33, 0.37,
and 0.52 for the stellar mass, redshift, isolation, and local density,
respectively. See Fig. 2 for the histograms of the distributions of the
paired and the control galaxies for the four parameters. These results
mean that statistically, the two samples are indistinguishable, so that
we are not biased to preferentially selecting paired galaxies in this
four-parameter space compared to selecting controls. We note that
the final galaxy pair sample is not excluded from the control pool.
This means that a sufficiently wide pair may be a control for a closer
pair, where the second companion of the latter system is at a similar
projected distance as the first companion of the former pair (provided
that they have a similar number of neighbours within 1 Mpc).
3 ENHANCED STAR FORMATION IN GALAXY PAIRS
In this section, we investigate whether the presence of a close com-
panion influences the physical properties of galaxies at 𝑧= 3 −9.
We provide a comprehensive view of how mergers and interactions
affect star formation, both instantaneously and over longer evolu-
tionary timescales. We begin by quantifying the enhancement of star
formation in paired systems relative to their matched controls, focus-
ing on specific star formation rate (sSFR) as a function of projected
separation (Section 3.1). We then explore how these results depend
on the timescale over which SFRs are measured (Section 3.2), fol-
lowed by an examination of possible trends with redshift and stellar
mass (Section 3.3), as well as studying the environmental depen-
dence (Section 3.4). To gain further insight into the dynamical stage
of the interactions, we analyse the reconstructed SFHs of close pairs
(Section 3.5).
MNRAS 000, 1–17 (2025)
Interaction driven star formation
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0
20
40
60
80
100
rp [kpc]
1.5
1.0
0.5
0.0
0.5
1.0
log10 (sSFR100)
0
20
40
60
80
100
rp [kpc]
0.9
1.0
1.1
1.2
1.3
sSFR100pair/sSFR100control
Rvir [z=6]
Figure 3. Left: Scatter plot of the sSFR excess of the paired galaxies compared to the median of the controls corresponding to each respective pair as a function of
projected pair separation. This is calculated for our entire sample of high-probability pairs at redshift 𝑧= 3 −9 and stellar masses of log10(𝑀⋆/M⊙) = [8, 10].
The uncertainties on the data points represent the propagated 16th and 84th percentile values of the SFR100 posteriors from Prospector. The dashed grey line
represents no excess compared to the control median. Right: Bin plot of the ratio between the sSFRs of the paired galaxies and the control medians, where the
bins are defined by an adaptive method such that each contains approximately the same number of galaxies, and the associated uncertainties are the standard
errors on the median of the points in the bin. As the projected separation between the primary galaxy and its closest companion gets smaller, the positive excess
in the sSFR of the paired galaxy increases, up to a factor of 1.12±0.05 at 𝑟p ≲20 kpc. This suggests that the presence of a close neighbour induces star formation
in the primary galaxy. We also display the virial radius of dark matter halo at 𝑧= 6 (approximately the median redshift of our pair sample) of a typical galaxy in
our sample, having stellar mass of log10(𝑀⋆/M⊙) = 9 (central value of our stellar mass range).
3.1 Star formation as a function of pair separation
First, we focus on whether the presence and projected separation of
the closest companion have any effect on the star formation of the
central galaxy. Since the SFR has been found to be directly dependent
on the stellar mass of the galaxy, following the star-forming main
sequence (Speagle et al. 2014; Popesso et al. 2023; Clarke et al. 2024;
Cole et al. 2025; Simmonds et al. 2025), and our galaxy pair sample
contains a large range of stellar masses (log10(𝑀⋆/M⊙) = 8−10), we
rather focus on the sSFR. We directly compare the sSFR of the paired
galaxies and the median of their corresponding statistical controls,
as a function of separation. This ratio essentially gives the distance
from the star-forming main sequence (see Simmonds et al. 2025),
which is caused by close-pair interactions. We present in Section 2.2
how we obtain the sSFRs at different timescales, and we choose as
our fiducial measure the sSFR100 (sSFR averaged for the past 100
Myr lookback time). We discuss other choices of timescales for the
sSFR measurements and their effect on the results in Section 3.2. We
define the offset sSFR from the control as
Δ log10(sSFR100) = log10(sSFRpair
100) −⟨log10(sSFRctrl
100)⟩med,
(3)
where ⟨⋯⟩med represents the median sSFR of the 5 control galaxies.
The left panel of Fig. 3 shows the offset of the individual paired
galaxies compared to the control median, which is represented by the
grey dashed line, as a function of projected separation. While this
figure shows the overall scatter and the uncertainties of the individual
data points, the panel on the right-hand side of Fig. 3 is visually easier
to interpret, as it shows the ratio of the two quantities with the data
binned. We apply an adaptive binning method where the width of each
bin (represented by horizontal error bars) is defined such that they
contain approximately an equal number of points. Vertical error bars
represent the standard error on the median for the data point within
each respective bin. As is already evident from the figure, the sSFR
of the paired galaxies is consistent with the control median when the
closest companion is at wider projected separations. However, as the
companion gets closer, in particular for close pairs at 𝑟p < 40 kpc,
the sSFR has a statistically significant excess compared to the control
median. For the bin at the smallest separations at 𝑟p ≲20 kpc, this
excess is 12±5%. Although this value is relatively small, the levelling
off of the sSFR excess at wider separations shows that this is still
significant, and we interpret it as a sign of star formation induced
by the presence of a close companion. We note that the separations
at which we detect sSFR enhancement are smaller than the virial
radius of the dark matter halo at 𝑟vir ≃55.4 kpc (assuming a stellar
mass-halo mass relation by Behroozi et al. 2019) of a typical galaxy
in our sample (i.e., with 109 M⊙𝑧= 6), being consistent with the
underlying physical picture. For additional checks on these results,
see Appendix A.
Next, we look at the stellar age (denoted as 𝑡50) as a function of pro-
jected separation. This is the lookback time (relative to the redshift of
the galaxy) at which 50% of the stellar mass of the galaxy has formed
and is estimated with Prospector (see Section 2.2). The younger
the galaxy is, the higher the SFR was for the immediate past of the
galaxy. As in the case of sSFR enhancement, we calculate the ratio
of the stellar ages of the pairs and the median ages of the controls and
plot them in adaptive bins as a function of projected separation (see
Fig. 4). Consistent with the enhanced sSFR at decreasing separations
(Fig. 3), the ages of the pairs decrease with decreasing separations
compared to the controls. We note that the uncertainties on the stel-
lar ages are larger than on sSFRs, giving rise to only a marginally
significant trend in Fig. 4. This means that galaxies that have close
companions grew quicker in stellar mass in their immediate past,
compared to similar galaxies which have no close companions. This
was expected, and it is due to the partially degenerate nature of 𝑡50
with the SFR, but it provides a useful validation check on our previ-
ous result. Furthermore, beyond this being an effect of the elevated
SFR, the primary galaxy could also grow by ex situ accretion of stars
(e.g., tidal shredding of a dwarf, as seen in MW tidal streams).
MNRAS 000, 1–17 (2025)
8
Puskás et al.
0
20
40
60
80
100
rp [kpc]
0.7
0.8
0.9
1.0
1.1
1.2
t50pair/t50control
Figure 4. Ratio of the stellar age of galaxies in pairs versus controls as
a function of projected pair separation. As the separation gets smaller, the
stellar age gets shorter compared to the control median. This suggests that the
paired galaxies grew more quickly in stellar mass compared to the isolated
controls, due to the induced star formation by the approach of the companion
galaxies (Fig. 3).
3.2 SFR timescales
In this section, we examine the sSFRs measured at various averaging
timescales and the behaviour of the sSFR enhancement as a func-
tion of projected separation, depending on the choice of timescale.
Prospector enables us to measure the SFR of a galaxy on different
averaging timescales, which, for the purpose of this study, are defined
at 5 Myr, 10 Myr, 50 Myr, 100 Myr, and 2000 Myr of lookback time.
These timescales trace distinct physical processes in galaxy evolu-
tion and therefore provide a direct test of whether close interactions
measurably elevate recent star formation (Iyer et al. 2020; Tacchella
et al. 2020).
We perform a similar analysis as in Section 3 for the sSFRs derived
at different timescales and show our results in Fig. 5. The markers
from fainter to darker blue represent sSFR ratios at increasing aver-
aging timescales from 5 to 2000 Myr. We only plot the horizontal
error bars that represent the width of the adaptive bins (containing
approximately equal number of paired galaxies) for sSFR100, which
is our fiducial measure for sSFR. As before, the vertical error bars
represent the standard error on the median. In the case of sSFR2000,
the uncertainties become vanishingly small.
As we decrease 𝑡avg from 100 to 50, 10, and 5 Myr, the mea-
sured excess in sSFR weakens. By 5–10 Myr, the sSFR ‘excess’
compared to the control median has larger scatter and statistically is
consistent with zero, even at the smallest separations (𝑟p < 20 kpc).
This behaviour is consistent with interaction-driven fuelling operat-
ing on ∼50–100 Myr timescales, whereas very short-timescale (∼5–
10 Myr) sSFR is dominated by internal, stochastic processes (bursty
star formation regulated by gas inflow and feedback). On the longest
averaging timescale (e.g., 𝑡avg = 2000 Myr), the sSFR–separation
trend vanishes. This is expected at 𝑧≈3, where the Universe is
𝑡age ≃2.16 Gyr old, i.e. averaging over ∼2 Gyr effectively measures
sSFR2000 ≈1/(2000 Myr). Consequently, pairs and controls show no
detectable contrast on this baseline, thereby just providing a cross-
check of our methodology.
0
20
40
60
80
100
rp [kpc]
0.8
0.9
1.0
1.1
1.2
1.3
sSFRpair/sSFRctrl
sSFR5
sSFR10
sSFR50
sSFR100
sSFR2000
Figure 5. Median sSFR enhancement in paired galaxies relative to controls as
a function of projected separation, for sSFRs estimated over different averag-
ing timescales (5, 10, 50, 100, and 2000 Myr). Points are colour–coded from
light to dark blue with increasing 𝑡avg. Adaptive bins contain approximately
equal numbers of galaxies; vertical error bars show the standard error on the
median in each bin. A clear radial trend (stronger enhancement at smaller 𝑟p)
is observed for 𝑡avg = 100 Myr (our fiducial choice) and is present but weaker
at 50 Myr. On very short timescales (5−10 Myr), the signal is consistent with
zero even at 𝑟p < 20 kpc, consistent with stochastic, bursty star formation
driven by internal processes and not influenced by the pair interaction. For
𝑡avg = 2000 Myr, the trend disappears, as this long baseline approximates a
constant sSFR 𝑧> 3 by construction.
3.3 Redshift and stellar mass dependence
In this section, we investigate whether the subsamples divided by
either redshift or stellar mass have the same behaviour as the overall
sample. We perform this split into two subsamples separately in the
case of redshift and stellar mass, as our limited sample size does not
allow for splitting into further subcategories.
First, we divide our sample in redshift into two subsamples that
contain approximately equal amounts of galaxies, that is, at 𝑧= [3, 5]
and 𝑧= [5, 9]. We perform a similar analysis as before, and plot the
sSFR enhancement with separation on the left panel of Fig. 6. There
is a notably higher enhancement of star formation in the case of the
higher redshift sample (factor of ∼1.25 compared to the control
median at 𝑟p < 20 kpc) than in the case of mild but still continuously
increasing enhancement at lower redshifts. This could be explained
by galaxies being more gas-rich at earlier times on average, and hence
the close interactions trigger a stronger starburst than in the case of
later galaxies that contain less gas. Furthermore, galaxy pairs with
close separation at higher redshifts have a higher probability of being
at first infall than the ones at lower redshifts, which will also include
objects that have already had their closest encounter. This is further
supported by the fact that virial radii of dark matter haloes increase
with decreasing redshift, meaning that the pair interactions take place
at closer separations at higher redshift, while at lower redshift, they
extend to wider separations.
Similarly, we divide the sample in stellar mass space into two parts
containing equal amounts of galaxies, at log10(𝑀⋆/M⊙) = [8.0, 8.6]
and log10(𝑀⋆/M⊙) = [8.6, 10.0]. In this case, the effect on this di-
vision is less clear on the sSFR enhancement. Concentrating on the
smallest separations (𝑟p < 20 kpc), the sSFR of lower-mass paired
galaxies is more strongly enhanced (factor of ∼1.18 compared to
control median) than in the case of higher-mass galaxies. This is an
MNRAS 000, 1–17 (2025)
Interaction driven star formation
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0
20
40
60
80
100
rp [kpc]
0.8
0.9
1.0
1.1
1.2
1.3
1.4
sSFR100pair/sSFR100control
z = 3
5
z = 5
9
0
20
40
60
80
100
rp [kpc]
0.8
0.9
1.0
1.1
1.2
1.3
1.4
sSFR100pair/sSFR100control
log10 (M /M
) = 8.0
8.6
log10 (M /M
) = 8.6
10.0
Figure 6. Left: sSFR enhancement measured against projected separation for subsamples divided in redshift, for the ranges 𝑧= [3, 5] and 𝑧= [5, 9], which split
the sample approximately in half. In the higher redshift sample, we see a much stronger enhancement in sSFR compared to the smaller and gradual increase in
the lower redshift case. In the higher redshift case, the enhancement only affects close pairs at 𝑟p ≲20 kpc, while in the lower redshift case it extends out to wider
separations. This can be explained by the increase of virial radii of dark matter haloes with decreasing redshift, and hence, the pair interaction length scales, as well
as galaxy pairs at close separations having a higher probability of being at first infall at higher redshift than the ones at lower redshift. Right: sSFR enhancement
measured against projected separation for subsamples divided in stellar mass, for the ranges log10(𝑀⋆/M⊙) = [8.0, 8.6] and log10(𝑀⋆/M⊙) = [8.6, 10.0],
which again split the sample approximately in half. We detect a more substantial enhancement in lower mass paired galaxies, and more variability in the case of
higher mass galaxies. This could be explained by smaller galaxies living in shallower potential wells and being more gas-rich, hence, the gravitational interaction
with their close companion triggers a stronger increase in star formation than in the case of more massive galaxies, in which internal feedback-induced variation
in sSFR is more important than environmental factors.
interesting result, meaning that close interactions and the environ-
ment have a stronger effect on star formation for lower-mass galaxies
than for higher-mass galaxies. This might be explained by the fact that
lower-mass galaxies are more affected by large-scale tidal forces and
interactions, which induce star formation, because they live in shal-
lower potential wells and are more gas-rich than their more massive
counterparts.
3.4 Environmental dependence
As we already measured the isolation and local density around the
pairs as part of the control matching, the environmental dependence
of the sSFR naturally comes out of this process. In Fig. 7 we show
how sSFR100 ratio of pairs versus controls behaves as a function of
the isolation (𝑟2), which we defined as the distance to the second
closest pair, and as a function of the local density (𝑁env), which
is given by the number of neighbours within a projected physical
separation of 1 Mpc.
We find that the distance to the second closest companion has a
significant effect on boosting the sSFR of pairs, up to 𝑟2 ≈75 kpc.
This is in part expected, as it implicitly implies that the distance
to the closest companion has to be shorter, i.e., 𝑟𝑝< 𝑟2. However,
the difference between the two distances can vary significantly: the
median offset is ⟨𝑟2−𝑟p⟩= 31.2+51.5
−25.9 kpc, where the uncertainties are
given by the 16th and 84th percentiles. In relative terms, 𝑟2 is typically
about 1.73 times (median) larger than 𝑟p, but can range from nearly
equal separations of 𝑟2/𝑟p = 1.13 (16th percentile) up to factors of
≈4.2 (84th percentile). Therefore, we explain this trend to be present
due to the genuine influence of the second closest neighbour on the
sSFR of the central galaxy.
To look further into the influence of the environment on larger
scales, we plot the trend of the sSFR ratio against the local density
𝑁env in Fig. 7 (right panel). We find that the sSFR excess of paired
galaxies increases with increasing number of neighbours, i.e., denser
environments around the central galaxy. Interestingly, the sSFR ratio
monotonically rises and peaks up to 𝑁env ≈110, which is followed by
a sudden decrease for even denser environments. This could poten-
tially be explained by galaxies in overdense regions having used up
their available gas to form stars and subsequently becoming quiescent
and ‘unresponsive’ to further environmental effects, such as close in-
teractions (see, e.g., Chiang et al. 2017; Lim et al. 2024; Baker et al.
2025; Jespersen et al. 2025). Nevertheless, this scenario only seems
to occur in high density environments with 𝑁env ≳110 members
within 1 Mpc projected physical radius, and before this turnover, we
detect an increase in the sSFR of galaxies residing in denser regions
with the nearest companions being within 𝑟p < 100 kpc.
3.5 SFH analysis
In this section, we look at the overall star-formation history (SFH) of
the paired galaxies. We use the Prospector outputs for the SFH for
each galaxy (Simmonds et al. 2024). Briefly, we use non-parametric
SFH (continuity model; Leja et al. 2019) and the prior is chosen
such that the SFH is divided into eight SFR bins.
We want to investigate the past histories of close pairs, which
can be a clue for their present dynamical state. Therefore, we select
high-probability close pairs with ∫PPF(𝑧)𝑑𝑧> 0.7 and 𝑟p < 30 kpc.
We then divide this resulting sample of 694 galaxies into galaxies
with SFHs that recently peaked (in the second most recent time bin
at 5 −10 Myr), i.e., monotonically rising SFHs (86 galaxies), and
with SFHs that already peaked at > 10 Myr lookback time (565
galaxies). A further 43 galaxies end up in a category where the
SFH first decreases and then increases again, without having a local
maximum (peak) in between. We neglect this category of galaxies
for the current question.
We note here that when categorising each SFH, we only look at
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Figure 7. Left: sSFR dependence of paired galaxies compared to their controls on the projected separation to the second closest companion (𝑟2). Galaxies with
closer second companions have enhanced sSFRs, which is in part due to the decrease in 𝑟2 that implicitly implies an even closer nearest companion (at 𝑟p < 𝑟2)
that was already shown to increase the sSFR of the central galaxy. However, a lower 𝑟2 can also imply a generally denser environment and has a secondary effect
to increase the sSFR of the host galaxy. Right: sSFR dependence on local density (𝑁env), that is given by the number of neighbours within a projected physical
separation of 1 Mpc. As the environment becomes denser, the sSFR of the central galaxy increases, but turns over beyond a critical density of 𝑁env ≈110. This
turnover could be explained by environmental quenching, where a close interaction can no longer efficiently boost star formation in the central galaxy.
the monotonicity of the SFH beyond the first time bin (at 𝑡lookback >
5 Myr), because at 𝑡lookback < 5 Myr the SFR is subject to a larger
scatter due to short-term variations (Simmonds et al. 2024, 2025).
For the two populations, we then interpolate the SFH of each galaxy
onto a common time grid and normalise the SFHs so that the total star
formation integrates to 1 (dimensionless; taking the lookback time
in units of Gyr). Then we compute the median across all galaxies
for the two separate cases and estimate the scatter by the 16–84th
percentiles of the distribution of individual SFHs.
Interestingly, we find that in the case of galaxies that have mono-
tonically rising SFH, their closest companion is at a median projected
separation of 𝑟𝑝= 13.3+0.6
−1.5 kpc, while in the case of galaxies that
have a secondary peak in their SFH, they have companions at a me-
dian separation of 𝑟𝑝= 16.0+0.6
−0.7 kpc. We show the resulting median
SFH for the two populations in Fig. 8 and give the following ex-
planation. The majority of close pairs have already had their first
passage and are in their post-pericentre phase. The initial close pas-
sage resulted in a starburst and overall rising SFH, which peaked
and started to slowly decline as the galaxies moved apart or even
passed their apocentre phase (cf. Figure 11 in Scudder et al. 2012).
This might have happened multiple times in the history of the pair,
but in our overall median SFH calculation, the previous episodes
might be ‘washed out’ by the latest rising SFH phase. Only a mi-
nority of the selected close pairs have purely rising SFH (∼7 per
cent of the overall cases), which are potentially galaxies that are in
their pre-pericentre phase, soon undergoing their first approach, or at
their pre-coalescence phase, where, again, previous episodes of close
approaches and hence rising and declining SFHs are ‘washed-out’.
This explanation is further strengthened by the lower median pro-
jected separations (13.3 kpc) of galaxies with monotonically rising
SFHs compared to the larger separations (16.0 kpc) of the galaxies
with secondary peaks in their SFH, which are on average further
away (either still receding or approaching again) from the primary
galaxies.
4 INTERACTION DRIVEN AGN ACTIVITY AND LY𝛼
EMISSION
In this section, we investigate whether close galaxy pair interactions
trigger AGN activity and Ly𝛼emission in the studied redshift, stellar
mass, and separation limits.
The AGNs used in this work have been identified in the follow-
ing two works: Juodžbalis et al. (2025) and Scholtz et al. (2025)
for the type-1 and type-2 selection, respectively. The type-1 AGNs
are identified based on the broad components in the H𝛼and H𝛽
emission lines without corresponding kinematical components in
the [OIII]𝜆𝜆5007,4960, hence ruling out an outflow origin. In to-
tal, this work yields a clean selection of 35 type-1 AGNs in the
GOODS-South and -North fields. The type-2 AGNs are selected
based on a variety of emission line diagnostics in the first two tiers
of the JADES survey: goods-s-deephst and goods-s-ultradeep. The
emission line diagnostics used were the BPT (Baldwin et al. 1981)
and VO87 diagrams (Veilleux & Osterbrock 1987), with selection
criteria modified for high-z, the HeII𝜆4686 diagrams (Shirazi &
Brinchmann 2012), UV emission line diagnostics (Hirschmann et al.
2019) and presence of high ionization lines such as [NeIV]𝜆2424
and [NeV]𝜆3420. In total, this selected 42 unique type-2 AGNs in
the regions of the GOODS-South field covered by goods-s-deephst
and goods-s-ultradeep.
The Ly𝛼emitter (LAE) catalogue from (Jones et al. 2025) con-
sists of all galaxies in JADES DR3 (D’Eugenio et al. 2025), with
secure spectroscopic redshifts (from visual inspection) of 𝑧spec > 4.
This redshift cut was chosen so that Ly𝛼would be observable with
NIRSpec PRISM/CLEAR, and resulted in a sample of 795 galaxies
(150 of which are confirmed LAEs). We refer the reader to Table 1
of Jones et al. (2025) for JADES tier separation, Fig. 1 for spatial
distribution, and Fig. 2 for redshift distribution. Using the DR3 spec-
tra, models were fit to the PRISM and medium-resolution grating
data (if available) that included continuum and strong line emission
(including Ly𝛼). LAEs in this catalogue are those which show sig-
nificant Ly𝛼emission (i.e., 𝑆/𝑁> 3) in either PRISM/CLEAR or
G140M/F070LP.
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SFHs with recent peak (N=86)
Scatter
Bootstrap
SFHs with peaks >10 Myr (N=565)
Scatter
Bootstrap
Median tmax = 12.1 Myr
Figure 8. Median star-formation histories (SFHs) of galaxies in close pairs
with ∫PPF(𝑧)𝑑𝑧> 0.7 and separations 𝑟p < 30 kpc, shown as a function of
lookback time in Myr. The blue curve and lighter shaded region indicate the
overall median and 16–84th percentile range for galaxies whose SFHs peaked
recently, i.e., the SFRs rise monotonically from early times up to the first time
bin (median projected separation 𝑟p = 13.3 kpc). The red curve and lighter
shaded region show the median and 16–84th percentile range for galaxies
with an SFH that peaked at 𝑡lookback > 10 Myr (median 𝑟p = 16.0 kpc), i.e.,
non-monotonically rising SFRs exhibiting at least one local maximum prior
to the most recent time bin. The darker shaded regions show the uncertainties
estimated by bootstrapping over the median SFHs, which clearly distinguish
these two populations. We interpret the presence of these two scenarios by
a larger number of close pairs being in their post-first approach phase (565
galaxies), which resulted in a peaking SFR that is now declining. In contrast,
a smaller number of galaxies are experiencing their first approach and/or their
pre-coalescence phase (86 galaxies).
Since the identification of both AGNs and LAEs needs spectro-
scopic observations, we have to perform our pair analysis on a spec-
troscopic sample in order to account for the JADES spectroscopy se-
lection function self-consistently. Specifically, we only focus on pairs
that have at least one spectroscopic measurement (either the primary
galaxy and/or its companion). Therefore, the resulting galaxy pair
fraction (number of pairs divided by the number of primary galaxies
in the sample) will be different by definition compared to the results
reported in Puskás et al. (2025), where photometric redshifts were
mostly used for close pair selection. We recalculate the galaxy pair
fraction for this spectroscopic-only catalogue for different maximum
pair-separation limits (30, 50, and 100 kpc). In this case, we define
the pair fraction as the number of galaxies with spectroscopic red-
shifts being in pairs (either being the primary or the secondary galaxy
within the pair) divided by the total number of galaxies having spec-
troscopic redshifts within a given redshift bin (both being within the
stellar mass range of log10(𝑀⋆/M⊙) = 8−10). In total, we have 3249
objects with spectroscopic redshifts in GOODS-South and GOODS-
North that remain in our catalogue after the initial sample selection
(see Section 2.3), which is further reduced to 1928 for the redshift
and stellar mass range of interest. We then divide our redshift range
of 𝑧= 3−9 into six bins with adaptive widths so that each bin approx-
imately contains an equal number of galaxies. For the pair selection,
we use the same parameter choices as for our fiducial pair selection
in Section 2.3, that is ∫PPF(z)𝑑𝑧> 0.7. The resulting pair fractions
( 𝑓spec,pair) are plotted in both panels of Fig. 9 for the three different
separation limits (from fainter to darker shades of blue for increasing
separation limits), and their uncertainties are estimated by bootstrap-
ping (Efron 1979, 1981). We also fit a power law-exponential curve
(given by Equation 30 of Puskás et al. 2025) to the pair fractions for
the three cases for easier readability. We note here that 𝑓spec,pair at
𝑟p < 30 kpc agrees well with the pair fraction reported by Puskás
et al. (2025) using both photometric and spectroscopic samples.
We proceed in a similar way to calculate the fraction of AGN
in pairs ( 𝑓AGN,pair) and LAE in pairs ( 𝑓LAE,pair). We first match the
type-1 and type-2 AGN catalogue to the NIRCam footprints used for
this analysis and find that 64 AGNs remain in our catalogue after
the initial sample selection (Section 2.3). In the case of LAEs, 132
objects remain in the initial sample catalogue. The fractions are then
calculated as the ratio of AGNs or LAEs in pairs (either primary
or secondary galaxies) to the total number of AGNs or LAEs in
our sample. If an AGN or LAE in question is both a primary and a
secondary galaxy, it is counted twice in the pair fraction calculation,
as being part of at least two separate systems. Due to the low number
of sources above 𝑧> 8 (2 AGNs and 7 LAEs), we focus on the redshift
range of 𝑧= 3 −8 in this section. We define the bins similarly by the
adaptive method, but reduce their number to three due to the lower
number count of these objects.
The fraction of AGNs in pairs has larger uncertainties (estimated by
bootstrapping) compared to the pair fractions of galaxies in pairs with
spectroscopic redshifts ( 𝑓spec,pair), which is mainly due to the much
lower sample size of AGNs (see Fig. 9). At the largest separation
limit of 𝑟p = 100 kpc, the values are comparable and agree well
within the uncertainty ranges, meaning that there is no detectable
excess in the number of AGNs compared to other galaxies in pairs.
As we go to a lower separation limit of 𝑟p = 50 kpc, we only find
a significant excess (factor of ∼2) in 𝑓AGN,pair compared to 𝑓spec,pair
at the highest redshift bin of 𝑧= 6.8 ± 1.2. Finally, at the closest
separations of 𝑟p < 30 kpc, 𝑓AGN,pair is generally consistent with
𝑓spec,pair, except at 𝑧= 3.6 ± 0.6 where we find no paired AGNs
satisfying the selection criteria (therefore we omit this point from
Figure 9). In terms of the total AGN sample, we find that for the entire
redshift and stellar mass range, at𝑟p < 100 kpc, 𝑓AGN,pair = 0.44±0.07
if we consider AGNs being in pairs (either primary or secondary),
and 𝑓AGN,pair = 0.56±0.10 if we count separately if an AGN is hosted
by both a primary and a secondary galaxy in a pair. This is in good
agreement with the AGN fraction of 𝑓AGN,pair = 0.52+0.17
−0.24 reported
by Duan et al. (2024), who select AGNs by the spectroscopic BPT
diagnostic method and photometric AGN SED templates from eight
deep JWST fields at similar separations limits, but using the simpler
redshift criterion of Δ𝑧< 0.3.
The fractions of LAEs in pairs ( 𝑓LAE,pair) have similar uncertainties
to 𝑓AGN,pair due to comparable number counts. At redshift 𝑧< 7,
𝑓LAE,pair is consistent with 𝑓spec,pair at all three separation limits, with
no detectable excess, similar to 𝑓AGN,pair. In the highest redshift bin
at 𝑧= 7.7 ± 1.0, we find that 𝑓LAE,pair is lower than 𝑓spec,pair at all
separations. Similarly to the AGN case, we calculate the global LAE
fraction at 𝑟p < 100 kpc as 𝑓LAE,pair = 0.42 ± 0.08 (either primary
or secondary), and 𝑓LAE,pair = 0.53 ± 0.09 if we count separately if a
LAE is both a primary and a secondary galaxy in a pair. We note that
these values are consistent with the global AGN fractions, showing
no detectable excess. Interestingly, when comparing the AGN sample
to the LAE sample, we find that the fraction of AGNs that are also
LAE is 0.12 ± 0.05, while the fraction of LAEs that are also AGN is
0.42 ± 0.17, with 12 sources being identified as both AGN and LAE
that are present in our pair catalogue. In the next section, we discuss
the potential physical origins and explanations for the fraction of
paired AGNs and LAEs.
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fLAE, pair (rsep = 30 kpc)
fLAE, pair (rsep = 50 kpc)
fLAE, pair (rsep = 100 kpc)
Figure 9. Pair fraction of AGN (left) and LAE (right) as a function of redshift. Both left and right panels show the pair fraction at different maximum projected
separation limits, focusing only on the subsample that has spectroscopic redshifts, indicated by blue circles, going from fainter to darker blue with increasing
maximum projected separation limits, at 30 kpc, 50 kpc, and 100 kpc, respectively. The bin widths are chosen such that each bin contains approximately the
same number of objects in the respective subcategories, and the error bars are estimated by bootstrapping. Left: fraction of type-1 and type-2 AGNs that can be
found in pairs at different maximum allowed projected separations (indicated on the plot) compared to the spectroscopic pair fraction. We find that fraction of
paired AGNs is generally consistent with the spectroscopic pair fraction within the uncertainty ranges, with a detectable excess (factor of ≈2) at 𝑟p < 50 kpc in
the highest redshift bin (𝑧> 5.5). This means that at separations larger than 𝑟min
p
= 5 kpc, galaxy interactions cannot significantly contribute to the triggering
of AGN, which might become important only at the closest separations (< 5 kpc). Right: similar comparison in the case of LAEs that can be found in pairs. We
find that the fraction of LAEs that are in pairs is consistent with the spectroscopic pair fraction at 𝑧< 7 and lower at the highest redshift bin, for all separation
limits. Similar to the case of AGNs, this could be explained by galaxy interactions not providing an efficient enough medium at separations 𝑟min
p
> 5 kpc for
Ly𝛼photon production and escape, which likely becomes more relevant at closer separations.
5 DISCUSSION
In this section, we discuss possible physical explanations and im-
plications of our results and compare them to other works from the
literature. We also discuss how our results depend on parameter
choices and mention caveats related to this analysis that might affect
the results.
5.1 Implications
One of the key questions at these redshifts is whether mergers are
the primary drivers of intense star formation or if they provide
only a modest boost to galaxies already forming stars rapidly due
to smoother gas inflow and high gas fractions. High-redshift disk
galaxies have significantly higher gas fractions, with 𝑓gas ≳50%
being common at 𝑧∼2 (e.g., Tacconi et al. 2010; Scoville et al.
2017; Tacconi et al. 2018, 2020; Parlanti et al. 2023). This abundant
fuel can lead to violent disk instabilities (VDI), which drive strong
gas inflows and nuclear starbursts (Bournaud et al. 2012; Dekel &
Burkert 2014; Zolotov et al. 2015; Tacchella et al. 2016). The life-
cycle of these VDI-induced clumps occurs on short timescales of
5–10 Myr, which aligns with the stochastic, bursty variability we ob-
serve in our short-timescale sSFR measurements (see Section 3.2).
In contrast, the longer-timescale enhancement (50–100 Myr) we de-
tect reflects the large-scale gas inflow and potential compression
induced by the merger interaction itself. Our results – showing a
separation-dependent boost on 𝑡avg ∼50–100 Myr but not on 5–10
Myr – therefore are consistent with merger-driven fuelling operating
on longer dynamical timescales, while very short-timescale sSFR is
dominated by these internal, stochastic processes (Iyer et al. 2020;
Tacchella et al. 2020; Simmonds et al. 2025; McClymont et al. 2025).
In the low-gas fraction case, typical of the local universe,
interaction-induced torques efficiently channel gas to the centre of
galaxies, fuelling strong starbursts (Barnes & Hernquist 1991; Mi-
hos & Hernquist 1996). However, in the gas-rich environments of
the early universe, this picture changes. Simulations show that when
the gas fraction is high, a strong baseline inflow already exists due to
internal processes, and the interaction between galaxies does not sig-
nificantly increase this inflow until just before coalescence (Fensch
et al. 2017). Similarly, other simulations have found that merger-
induced starbursts are often short-lived and difficult to distinguish
from the high levels of stochastic star formation driven by internal
processes (Sparre & Springel 2016; Hani et al. 2020). Our results
align well with this gas-rich merger paradigm. The star formation
histories of our high-probability galaxy pair sample point to many
pairs being post-first pericentre, with peak SFRs occurring near this
pre-coalescence phase, consistent with the strongest enhancement
we find at projected separations 𝑟p ≲20 kpc. The modest amplitude
of this enhancement of a factor of∼1.12 is in good agreement with
simulations like those of Fensch et al. (2017), which find only a weak
SFR boost at the first pericentre passage and a mild elevation at fi-
nal coalescence. This scenario is further supported by recent direct
observations of high-redshift pairs (e.g., Duan et al. 2024), which
confirm that merger-driven enhancements are typically a factor of
∼2 or less, far weaker than in local universe mergers (Cibinel et al.
2019).
It has been found by previous works that galaxies above the SFMS
(e.g., Speagle et al. 2014; Faisst et al. 2016; Popesso et al. 2023;
Simmonds et al. 2025) at high redshift are almost always associ-
MNRAS 000, 1–17 (2025)
Interaction driven star formation
13
ated with major mergers (e.g., Rodighiero et al. 2011; Cibinel et al.
2019). The median sSFR enhancement of 𝑓≈1.12 detected by our
study essentially represents a modest but significant offset from the
SFMS. This shortens the e-folding and doubling times by ∼12% (i.e.,
faster recent mass build-up). This modest ‘simmering’ mode of en-
hancement, integrated over the ∼100 Myr interaction timescale, is a
key mechanism for pushing galaxies up along the mass-SFR plane
faster than their isolated counterparts. Mergers, therefore, act as a
dual growth engine: they add mass directly via the accretion of the
secondary galaxy (see, e.g., Puskás et al. 2025) and indirectly by
triggering in-situ star formation that builds the stellar mass of the
primary galaxy more rapidly.
As presented in Section 3.3, we detect a stronger enhancement
in sSFR at higher redshifts ( 𝑓≈1.25, at 𝑧= 5 −9) and at lower
masses ( 𝑓≈1.18, at log10(𝑀⋆/M⊙) = 8.0 −8.6). This stronger
enhancement at higher redshift is most probably tied to the evolution
of the cosmic gas fraction. As discussed before, galaxies at 𝑧> 3
are overwhelmingly gas-dominated. A merger between two gas-rich
systems provides a massive, concentrated fuel reservoir for a starburst
(Tacconi et al. 2018). At lower redshifts, galaxies are more gas-poor,
therefore, a similar interaction results in a weaker SFR enhancement.
The dependence of the star formation enhancement on the stellar
mass range studied can be explained by the following. More massive
galaxies have deeper gravitational potential wells and higher stellar
densities, which makes their gas disks more stable against external
perturbations. Therefore, it is more difficult for a merger to disrupt
a massive disk, whereas a lower-mass and less stable disc is more
easily disturbed, leading to a more vigorous starburst (Pearson et al.
2019). Moreover, low-mass galaxies have shallower potential wells,
making it easier for stellar feedback to expel gas. However, the initial
burst triggered by a merger can be more intense because there is
less pre-existing pressure support in the disc to resist gas inflow and
compression (e.g., Muratov et al. 2015).
5.2 Implications for Ly𝛼emitters and AGNs
Here, we discuss the possible explanations and implications of find-
ing no significant excess in the Ly𝛼emitter fraction among close
pairs compared to the overall spectroscopically confirmed galaxy
pair population. During the EoR (𝑧> 5.2), the intergalactic medium
(IGM) is substantially neutral, and neutral hydrogen is extremely
effective at scattering Ly𝛼photons, making it difficult for the Ly𝛼
emission from most galaxies to reach us. Ly𝛼photons can only es-
cape from regions where the surrounding IGM has already been
ionized, creating channels or large bubbles of ionized gas. If galaxy
pairs were significantly more efficient at producing such ionized re-
gions through their combined ionizing output, we would expect an
enhanced LAE fraction among pairs. The absence of such an excess
in our results instead suggests that either (i) galaxy pairs do not play
a dominant role in creating ionized bubbles large enough to boost
Ly𝛼visibility, or (ii) the local environment, including neighbouring
galaxies and large-scale overdensities, is the primary driver of Ly𝛼
transmission during reionization.
We note that our analysis probes relatively wide projected sep-
arations (𝑟p = 5 −100 kpc). At these scales, although we detect a
moderate enhancement in star formation, many pairs may not yet be
undergoing strong interactions, so their combined ionizing output is
likely weaker than for very close pairs (𝑟p < 5 kpc). This is further
supported by the lack of star formation enhancement on ∼10 Myr
timescales (see Section 3.2), which implies no recent increase in the
production of Ly𝛼photons. At the smallest separations (which we
do not probe), stronger star formation enhancements and disturbed
morphologies could lead to less uniform neutral gas coverage and,
consequently, more efficient Ly𝛼escape. Thus, our results do not rule
out the possibility that very close pairs (≲10 −20 kpc) or compact
groups provide particularly favourable conditions for Ly𝛼emission
during the EoR, consistent with a patchy or heterogeneous picture of
reionization that begins in overdense regions such as those hosting
close galaxy pairs and protoclusters (Castellano et al. 2016; Endsley
et al. 2021; Saxena et al. 2023; Whitler et al. 2024; Witten et al.
2024; Witstok et al. 2024).
In the case of AGNs, we find a lack of significant enhancement in
𝑓AGN,pair compared to the spectroscopically selected pair fractions.
While mergers are theoretically a very effective way to drive gas
to the galactic nucleus to fuel a supermassive black hole, the AGN
phase itself is thought to be very brief and episodic. This is referred
to as the AGN duty cycle model (Schawinski et al. 2015). An AGN
might turn on for a short period (∼0.1 −1 Myr) and then shut off for
a much longer period (∼10 −100 Myr) as the fuel is consumed or
expelled by feedback (Hickox et al. 2014). This model aligns with
our findings, as we observe a sustained SFR enhancement on a long
timescale of ∼50 −100 Myr, which reflects the overall duration of
the galaxy interaction. However, we see no sSFR enhancement on
the very short ∼5 −10 Myr timescale. The lack of an AGN trend
is consistent with the idea that the AGN accretion phase is a short-
timescale phenomenon, similar to the stochastic starbursts. Even if
the merger is the ultimate cause of the fuelling, the probability of
observing any given pair during the brief moment its AGN is on is
low.
On the other hand, the duty cycle is expected to get shorter in an
interaction phase, and hence, the probability of detecting an AGN
should increase. For example, Perna et al. (2025) find that 20%-30%
of their sample of AGN is dual AGN at 𝑧∼3 (which is in excess by
a factor of ∼3 relative to expectations by simulations). They explore
dual AGN on scales of 3 −20 kpc, suggesting a significant role
of galaxy interactions on AGN activity. However, in our selection
criterion, we limit our study of galaxy pairs to projected separations
𝑟p > 5 kpc, likely introducing a significant bias against detecting
merger-triggered AGN, which happen primarily on scales smaller
than what is probed in this study.
Theoretical models and simulations consistently show that the
most efficient period of AGN fuelling occurs during the final stages
of a merger (e.g., Hopkins et al. 2008; Ellison et al. 2011; Koss et al.
2012). It is at these small separations (< 5 kpc) that violent gravita-
tional torques can strip gas of its angular momentum, driving rapid
inflows toward the central supermassive black holes. By excluding
this final coalescence phase, our study is systematically insensitive to
the brief, intense period where AGN are most likely to be triggered,
which could explain the lack of a strong correlation between pairing
and AGN activity in our sample.
Recently, Übler et al. (2024) reported the discovery of a spatially
offset (by ∼1 kpc) AGN at 𝑧= 7.2, that is likely undergoing a merger
with another galaxy (likely hosting another accreting black hole).
Übler et al. (2025) present a galaxy at a 𝑧∼5 hosting three massive
black holes (a central one separated by the second at only ∼200 pc
and the third being at ∼2 kpc), providing direct evidence that SMBH
mergers and multi-SMBH systems were already active during the
early stages of galaxy assembly. At later times, (Bonaventura et al.
2025) show that at redshift 𝑧∼1 −3, AGNs are commonly dust-
obscured and live in disturbed host galaxies, revealing a strong link
between dynamical processes such as mergers and obscured black
hole growth.
MNRAS 000, 1–17 (2025)
14
Puskás et al.
0
20
40
60
80
100
rp [kpc]
0.9
1.0
1.1
1.2
1.3
env
min = 0.01
pair
min = 0.3
pair
min = 0.5
pair
min = 0.7
0
20
40
60
80
100
rp [kpc]
env
min = 0.1
pair
min = 0.3
pair
min = 0.5
pair
min = 0.7
0
20
40
60
80
100
rp [kpc]
env
min = 0.3
pair
min = 0.3
pair
min = 0.5
pair
min = 0.7
sSFR100pair/sSFR100control
Figure 10. Enhancement in sSFR for paired galaxies versus projected separations for varying pair probability selections. For the three plots, we set the minimum
value for the integral of the pair probability function for the environmental selection (denoted by Ienv
min) to 0.01, 0.1, and 0.3, respectively. These correspond to
a generous criterion for selecting a purer sample for the environmental matching for each paired galaxy and its controls. On each respective plot, we vary the
minimum value for the integral of the pair probability function for the pair (closest companion, denoted by Ipair
min) selection by setting it to 0.3, 0.5, and 0.7,
in order of increasing probability or purity. Our chosen fiducial selection parameters are Ienv
min = 0.1 and Ipair
min = 0.7. However, in all cases we detect a radial
dependence of sSFR increase and strong enhancement at small 𝑟p, with the strength of the enhancement increasing with the probability or purity of the pair
selection. This proves the robustness of our sSFR enhancement detection and would lead to similar results for any reasonable selection parameter choices.
5.3 Caveats and parameter choices
In this section, we discuss potential limitations of this study by listing
caveats and presenting the effect of different parameter choices on
our results.
As described in Section 2.3, we select major merger galaxy pairs
that have mass ratios of 𝜇≥1/4. By lowering this value, we could
select minor (𝜇≥0.1) or mini (𝜇≥0.01) mergers, however, this
would result in a very incomplete sample in stellar mass. Therefore,
we focus on only selecting major mergers in this work. For future
studies, one could examine the effect on these properties not only by
using major merger galaxy pairs but also by considering minor or
mini mergers. However, this would require much deeper data for a
mass-complete analysis, a challenging task at higher redshifts.
As discussed previously, we study the variation of the sSFR and
other properties of (only) the central galaxy in high-probability pairs,
as a function of pair separation. We also discuss how this depends on
different choices of averaging timescales for the sSFR estimates. One
could also look at similar effects on not only the primary or central
galaxy of the pair system, but also on the lower mass companion.
To perform a similarly robust analysis, control matching has to be
done. However, in this case, the control matching would be much
more complicated, as the ‘primary’ galaxy of the control system has
to be similar (with a similar environment but no massive host?) to
the companion galaxy in question, whose environment now is more
complicated, being ‘off-centre’ compared to the primary galaxy. The
companion, having a lower mass and requiring similar mass controls,
would also be more heavily affected by stellar mass incompleteness.
Therefore, we only focus on studying the properties of the central
(primary) galaxy in this work.
In this analysis, a parameter choice we inevitably made in the case
of our probabilistic pair selection is a threshold chosen for the integral
of the pair probability function. We defined high-probability pairs as
having ∫PPF(𝑧)𝑑𝑧> 0.7 for our fiducial analysis. To detect the
second closest companion and measure the local density, we chose
a minimum threshold of ∫PPF(𝑧)𝑑𝑧> 0.1. The resulting pairs and
their neighbours naturally depend on these parameter choices. There-
fore, in this section, we investigate the effect of choosing different
threshold values on the resulting sSFR excess measurements.
We perform the same analysis as in Section 3, for three different
minimum thresholds for finding neighbours in the local environment,
for which we choose the short-hand notation of Ienv
min = ∫PPF(𝑧)𝑑𝑧.
We choose 0.01, 0.1, and 0.3 for Ienv
min, which means a very gener-
ous, more restricted, and very pure selection thresholds for neigh-
bours, affected mainly by their redshift posterior distributions. Then,
for each Ienv
min, we perform the analysis for three threshold values
for finding the closest companion, with the short-hand notation of
Ipair
min = ∫PPF(𝑧)𝑑𝑧. In this case, we choose higher values (0.3,
0.5, and 0.7), as we want to select higher probability pairs, even if
the environmental selection is less restrictive. While we choose the
thresholds Ipair
min = 0.7 and Ienv
min = 0.1 as our fiducial parameters, we
detect a radial dependence and an increase in sSFR at the lowest
separations in all cases, which is presented in Fig. 10. Therefore,
this somewhat arbitrary choice of the probability threshold values
does not have a significant effect on the detected sSFR enhancement,
making our results robust. We note that with the ‘purest’ selection
using Ipair
min = 0.7 and Ienv
min = 0.3, our results have more scatter, which
is mainly due to the significantly reduced number of pairs satisfying
this selection criteria.
6 SUMMARY & CONCLUSIONS
In this paper, we study and measure the influence of close com-
panions on different galaxy properties. By careful control matching,
we can isolate and study the effects of close galaxy interactions on
the physical properties of the paired galaxies, such as star formation
enhancement, AGN triggering, and Ly𝛼emission.
We select a sample of high-probability galaxy pairs with projected
physical separations of 𝑟p < 100 kpc at 𝑧= 3 −9 and stellar masses
log10(𝑀⋆/M⊙) = [8, 10] from the JADES survey, and compare to
a carefully selected control sample by simultaneously matching in
redshift, stellar mass, isolation, and local density (within 1 Mpc). By
this method, we can robustly assess the influence of close companions
and ensure that any observed differences between pairs and their
controls are not the result of underlying differences in fundamental
galaxy properties.
MNRAS 000, 1–17 (2025)
Interaction driven star formation
15
We summarise of the most important findings of this paper in the
list below:
●We compare the sSFRs of the(primary) galaxies in major merger
pairs to their matched controls and find an increasing enhancement
at decreasing projected separations to their closest companions. We
detect a weak but significant enhancement already at 𝑟p ≲40 kpc,
reaching a maximum excess of 12 ± 5% at the closest separations of
𝑟p ≲20 kpc.
●By measuring the sSFR variation compared to controls with
projected separation at different averaging timescales, we find that
timescales of 50 −100 Myr are the best tracers of environmental
effects and close interactions. In contrast, at shorter timescales of
5 −10 Myr, sSFRs are mainly affected by internal processes that
induce short-time variability, such as feedback or the stochastic nature
of star formation.
●By splitting our sample in redshift space, we find a stronger
excess of ×1.25 in sSFR at 𝑟p ≲20 kpc at higher redshifts (𝑧= 5−9),
which we explain by galaxies having higher gas fractions at high-
z and hence, interactions inducing a stronger enhancement in star
formation, as well as preferentially being at first infall compared to
low-z. If we instead split the sample in stellar mass, we find a stronger
enhancement in sSFR (factor of ×1.18) for galaxies with lower stellar
masses (log10(𝑀⋆/M⊙) = 8.0 −8.6), possibly explained by lower-
mass galaxies being more affected by large-scale tidal forces, in-
ducing star formation, and being more gas rich than higher-mass
galaxies.
●We find that the boost in sSFR also significantly depends on the
distance to the second closest companion (up to 𝑟2 ≈75 kpc), as well
as the number of neighbours in the local environment, detecting an
increase in the sSFR of galaxies residing in denser regions (within
1 Mpc). Although the dependence on 𝑟2 implicitly contains the de-
pendence on 𝑟p in part, we detect a genuine influence of the second
closest neighbour on the sSFR of the central galaxy.
●By analysing the shape of the SFH of each individual high-
probability close-pair (𝑟p < 30 kpc), we find that they can be cate-
gorised into two distinct groups: galaxies with monotonically rising
SFHs that peaked recently, and objects with rising SFHs that peaked
at > 10 Myr and started decreasing at recent lookback times. We ex-
plain this bimodality by some objects being in the pre-pericentre or
pre-coalescence phase that induced strong star formation (minority
of the sample); and the more significant fraction of objects being
in the post-pericentre phase, where the star formation enhancement
from the close interaction is weaker or has already started to fade as
galaxies briefly move apart, before the eventual coalescence.
●Comparing the fraction of AGNs in pairs to the pair fraction
of the spectroscopic-only sample yields consistent results, meaning
that we do not find a signature of significant AGN triggering in our
galaxy pair sample. We explain this by the AGN duty cycle being
much shorter than the merger timescales we probe (0.1 −1 Myr ≪
50 −100 Myr). Moreover, we do not probe the final coalescence
phase at 𝑟p < 5 kpc, which is the most relevant event for strong
gas channelling to the galactic centre that potentially triggers AGN
activity.
●Similarly, we do not find an excess in the fraction of Ly𝛼emitters
in pairs compared to 𝑓spec,pair at all separations and redshifts. This
is explained by galaxy interactions not being efficient at Ly𝛼photon
production and creating channels for Ly𝛼photon escape or ionized
bubbles, at the wider separations (𝑟p = 5 −100 kpc) that we probe.
This does not rule out the enhancement of Ly𝛼emission at the
smallest scales in very close pairs in compact groups and overdense
regions.
To study the physical processes ongoing in systems participating
in close interactions and mergers in more detail, spatially resolved
studies are required. One of the revolutionary tools of JWST, the
NIRSpec IFU, is the ideal instrument for such resolved studies (e.g.
Jones et al. 2024; Rodríguez Del Pino et al. 2024), and tracing the
kinematics of such systems for future large-sample studies.
ACKNOWLEDGEMENTS
We thank David R. Patton for insightful discussions and helpful
comments during the ‘Dancing in the Dark: when Galaxies Shape
Galaxies’ conference in Sexten, Italy, which helped to improve this
work.
DP acknowledges support by the Huo Family Foundation through
a P.C. Ho PhD Studentship. DP and ST acknowledge support by the
Royal Society Research Grant G125142. CS, GCJ, JS, and RM ac-
knowledge support by the Science and Technology Facilities Council
(STFC), by the ERC through Advanced Grant 695671 “QUENCH”,
and by the UKRI Frontier Research grant RISEandFALL. RM also
acknowledges funding from a research professorship from the Royal
Society. IJ acknowledges support by the Huo Family Foundation
through a P.C. Ho PhD Studentship. WMB acknowledges support
by a research grant (VIL54869) from VILLUM FONDEN. AJB ac-
knowledges funding from the “FirstGalaxies” Advanced Grant from
the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (Grant Agreement
No. 789056). SC acknowledges support by the European Union’s HE
ERC Starting Grant No. 101040227 - WINGS. ECL acknowledges
support of an STFC Webb Fellowship (ST/W001438/1). DJE, BDJ,
MR, and BER acknowledge support by a JWST/NIRCam contract
to the University of Arizona, NAS5-02015. DJE is also supported
as a Simons Investigator. BER also acknowledges support from the
JWST Program 3215. The research of CCW is supported by NOIR-
Lab, which is managed by the Association of Universities for Re-
search in Astronomy (AURA) under a cooperative agreement with
the National Science Foundation. JW gratefully acknowledges sup-
port from the Cosmic Dawn Center through the DAWN Fellowship.
The Cosmic Dawn Center (DAWN) is funded by the Danish National
Research Foundation under grant No. 140.
This work is based [in part] on observations made with the
NASA/ESA/CSA James Webb Space Telescope. The data were ob-
tained from the Mikulski Archive for Space Telescopes at the Space
Telescope Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc., under NASA con-
tract NAS 5-03127 for JWST. These observations are associated with
PIDs 1180, 1181, 1210, 1286, 1895, 1963, and 3215. The authors
acknowledge the teams led by PIs Daniel Eisenstein (PID 3215),
Christina Williams, Sandro Tacchella, Michael Maseda (JEMS; PID
1963), and Pascal Oesch (FRESCO; PID 1895) for developing their
observing program with a zero-exclusive-access period.
DATA AVAILABILITY
The data underlying this article will be shared at a reasonable re-
quest by the corresponding author. Fully reduced NIRCam images
and NIRSpec spectra are publicly available on MAST (https://
archive.stsci.edu/hlsp/jades), with doi:10.17909/8tdj-8n28,
doi:10.17909/z2gw-mk31, and doi:10.17909/fsc4-dt61 (Rieke et al.
2023; Eisenstein et al. 2023b; Bunker et al. 2024; D’Eugenio et al.
2025).
MNRAS 000, 1–17 (2025)
16
Puskás et al.
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APPENDIX A: ADDITIONAL CHECKS
In Fig. A1 we present a suite of additional checks that assess the
robustness of the sSFR enhancement reported in Section 3. First, we
verify that the result is not an artefact of imperfect control match-
ing in stellar mass by explicitly comparing the stellar masses of
paired galaxies to the medians of their matched controls as a func-
tion of projected separation (top-left panel of Fig. A1). The ratio is
fully consistent with unity across all separations, confirming that our
four-dimensional matching procedure (redshift, stellar mass, local
density, and isolation) eliminates mass-driven systematics and that
any observed sSFR offset is not induced by residual mass differences
or by Eddington-type biases. Second, we repeat the analysis using
SFR100 instead of sSFR100 (top-right panel of Fig. A1). The SFR
enhancement closely mirrors the behaviour seen in sSFR while the
mass ratio remains flat, demonstrating that the signal originates from
a genuine elevation in recent (∼100 Myr) star formation rather than
from variations in stellar mass normalisation. Third, we examine
the absolute trends of SFR100 with pair separation independently for
pairs and for their controls (bottom panels of Fig. A1). Galaxies with
pairs show a clear rise in median SFR towards smaller separations,
whereas the matched controls – by construction similar in redshift,
mass, and environment but lacking a close companion – exhibit a
comparatively flat profile. The two curves diverge most strongly at
𝑟p ≲20 kpc, consistent with the quantitative excess in sSFR reported
in Section 3.
Throughout, median values and their uncertainties are obtained
via adaptive binning (approximately equal counts per bin) and by
calculating the standard error on the median, or in some cases, by
non-parametric bootstrapping. We have verified that fixed-width bins,
inverse-variance weighting of individual posteriors, and median-of-
ratios versus ratio-of-medians formulations yield indistinguishable
results within the quoted errors. Tightening and loosening the pair-
probability integral thresholds used for the pair identification and the
environmental counts (cf. Fig. 10) shifts the absolute amplitude in
the expected sense – cleaner (higher-probability) selections yield a
slightly stronger enhancement – without altering the qualitative be-
haviour. Finally, removing objects flagged as AGNs or strong LAEs
does not erase the signal in the sSFR offset, indicating that the en-
hanced star formation we measure in close pairs is not solely driven
by systems with luminous nuclear activity or extreme nebular emis-
sion, but reflects a broader interaction-induced elevation in recent
star formation at 𝑧∼3–9.
This paper has been typeset from a TEX/LATEX file prepared by the author.
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Figure A1. Additional checks to validate our detection of star formation enhancement in paired galaxies versus projected separation to the closest companion.
Top left: enhancement in stellar mass of primary galaxies compared to their matched controls that is consistent with unity, i.e. no enhancement. This is an
assuring check that the control matching was adequately performed in the stellar mass space and that any enhancement in SFR or sSFR is not driven by the paired
galaxies being more or less massive than their associated controls. Top right: enhancement in SFR for the 100 Myr averaging timescale, being consistent with
similar enhancement in sSFR and no enhancement in stellar mass, further confirming the intrinsic enhancement detected in star formation. Bottom left: binned
absolute values of SFR100 of paired galaxies plotted against projected separation. This shows that we detect an intrinsically elevated SFR in paired galaxies at
small separations compared to larger separations. Bottom right: binned absolute values of SFR100 of paired galaxies compared to a separately binned SFR100 of
the selected controls.
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MNRAS 000, 1-17 (2025) Preprint 17 October 2025 Compiled using MNRAS LATEX style file v3.3 Mergers lighting the early Universe: enhanced star formation, AGN triggering, and Lyαemission in close pairs at z= 3 -9 Dávid Puskás ,1,2⋆Sandro Tacchella ,1,2 Charlotte Simmonds ,1,2,3 Gareth C. Jones ,1,2 Ignas Juodžbalis ,1,2 Jan Scholtz 1,2, William M. Baker ,4 Andrew J. Bunker ,5 Stefano Carniani ,6 Emma Curtis-Lake ,7 Qiao Duan ,1,2 Daniel J. Eisenstein ,8 Kevin Hainline ,9 Benjamin D. Johnson ,8 Roberto Maiolino ,1,2,10 Marcia Rieke ,9 Brant Robertson ,11 Christina C. Williams ,12 and Joris Witstok 13,14 1Cavendish Laboratory, 19 JJ Thomson Avenue, Cambridge, CB3 OHE, UK 2Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK 3Departamento de Astronomía, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile 4DARK, Niels Bohr Institute, 128, DK-2200 Copenhagen, Denmark 5 1 3RH, UK 6Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 7Centre for Astrophysics Research, 10 9AB, UK 8Center for Astrophysics ∣Harvard & Smithsonian, 60 Garden St., Cambridge MA 02138 USA 9Steward Observatory, 933 N. Cherry Avenue, Tucson, AZ 85721, USA 10 1E 6BT, UK 11 1156 High Street, Santa Cruz CA 96054, USA 12NSF National Optical-Infrared Astronomy Research Laboratory, 950 North Cherry Avenue, Tucson, AZ 85719, USA 13Cosmic Dawn Center (DAWN), Copenhagen, Denmark 14Niels Bohr Institute, 128, DK-2200, Copenhagen, Denmark 17 October 2025 ABSTRACT Galaxy mergers and interactions are often invoked to explain enhanced star formation, black hole growth, and mass build-up of galaxies at later cosmic times, but their effect is poorly understood at high redshift (z> 2). We use JADES data to analyse a masscomplete sample of 2095 galaxies at z= 3 -9 with log(M⋆/M⊙) = [8, 10], identifying major merger pairs (projected separation of 5 -100 pkpc, mass ratio ≥1/4) using a probabilistic method. To look for signatures of enhancement in multiple physical properties, we carefully build a control sample of non-pairs that are simultaneously matched in redshift, stellar mass, isolation, and environment to the pair sample. We find a moderate enhancement in specific star formation rate (sSFR) of 1.12 ± 0.05 at separations ≲20 kpc, which is weakly detectable out to ∼50 kpc. We find that at longer averaging timescales (50-100 Myr) the sSFR is more affected by interactions and environment, whereas at shorter timescales (5-10 Myr) it is dominated by internal feedback and burstiness. By averaging star formation histories, we find two distinct populations: pre-first passage/coalescence (monotonically rising SFR) and post-pericentre pairs (earlier peak in SFR). Finally, we find no significant excess of AGN in pairs, suggesting galaxy interactions are not effectively triggering black hole activity at separations > 5 kpc. Similarly, we also do not detect an excess in the fraction of Lyman-αemitters in pairs, implying that at the probed separations, galaxy interactions are not efficient at enhancing Lyman-αphoton production and escape, which may only become important at the smallest scales. Key words: galaxies: evolution - galaxies: high-redshift - galaxies: interactions - galaxies: star formation - galaxies: active 1 INTRODUCTION Galaxy mergers and interactions play a key role in shaping the evolutionary path of individual galaxies. In the Lambda Cold Dark Matter (ΛCDM) cosmological framework, dark matter haloes merge under the hierarchical growth model, resulting in the tidal disruption and eventual merger of the galaxies embedded within them (Somerville & ⋆E-mail: Davé 2015). It has been well established that merging systems can undergo profound structural transformations (Toomre & Toomre 1972), elevated star formation rates (SFRs) (Barton et al. 2000; Ellison et al. 2008; Scudder et al. 2012; Bickley et al. 2022), and enhanced active galactic nucleus (AGN) activity (Ellison et al. 2011, 2013, 2019). Most of these results come from large low-redshift samples. At high redshift, the abundance of merger events has only recently begun to be constrained at z> 6 (Duncan et al. 2019; Duan et al. 2025; Puskás The Authors 16 Oct 2025 2 Puskás et al. et al. 2025), and the effects of these mergers on galaxy properties remain poorly understood. Theory and simulations predict that the gravitational interaction during close galaxy-galaxy encounters and mergers can induce nonaxisymmetric features (e.g., bars) and strong torques that channel low-metallicity gas to galactic centres, fuelling starbursts and potentially triggering AGN (Mihos & Hernquist 1996; Springel et al. 2005; Cox et al. 2008; Scudder et al. 2012; Torrey et al. 2012; Hopkins et al. 2013; Patton et al. 2013; Moreno et al. 2015; Patton et al. 2020; Byrne-Mamahit et al. 2024). For similar-mass mergers, gas inflows can drive short, intense bursts (enhancements of order a factor of ∼2), with the strength depending primarily on internal structure and mass ratio rather than orbital parameters (Mihos & Hernquist 1996; Cox et al. 2008). Gas-rich mergers may then fuel black hole (BH) accretion and feedback, expelling (or using up) gas near coalescence and contributing to quenching (Springel et al. 2005). Many simulation-based works describe moderate SFR enhancement for several hundred Myr after first pericentre and a stronger burst shortly before coalescence (e.g., Mihos & Hernquist 1996; Cox et al. 2008; Scudder et al. 2012), with the post-coalescence starburst lasting up to ∼0.5 Gyr (Hani et al. 2020; Ferreira et al. 2025) and potentially producing quenched elliptical remnants (Quai et al. 2021; Ellison et al. 2022). Observational studies of galaxy mergers typically focus on two aspects: (i) the frequency of mergers (pair fractions and merger rates) as a function of redshift, and (ii) the impact on galaxy properties. The merger rate has been well constrained at lower redshifts (z 6 thanks to the wealth of high-resolution photometry and spectroscopy from JWST. These works found an initially steeply increasing, then a flattening merger rate with redshift, stabilising in the range of 2 -10 major mergers/galaxy/Gyr at z> 6 (Puskás et al. 2025). To infer the merger rate, two widely used methods exist for selecting galaxies that are about to merge: the close-pair method that involves counting galaxies that are in close pairs on the projected plane of the sky (e.g., Mundy et al. 2017; Duncan et al. 2019; Duan et al. 2025; Puskás et al. 2025), and the morphological method that involves finding systems showing disturbed morphologies that are at the late stages or have recently completed merging (e.g., Conselice et al. 2008; Rose et al. 2023; Dalmasso et al. 2024). Here we focus on the effects of close encounters on internal galaxy properties, rather than the merger rate itself, which has been extensively discussed in Puskás et al. (2025). The properties of galaxy mergers and their influence on SFRs have been studied in depth using large spectroscopic samples at low redshift (z 1), there is less agreement between different theoretical and observational studies regarding the enhancement (if any) or even the decrease of SFR between merger and non-merger samples. Simulations by Fensch et al. (2017) demonstrate that high-redshift (up to z∼4), gas-rich galaxy mergers are substantially less efficient at enhancing their SFRs than their lowredshift, gas-poor counterparts, producing starbursts that are approximately ten times weaker and shorter in duration. Observational studies also find different trends at higher redshifts. For example, at 0.5 3. Furthermore, mergers have been found to efficiently trigger AGN activity and drive BH accretion (e.g. Ellison et al. 2011, 2013; ByrneMamahit et al. 2024). Simulations (e.g., Di Matteo et al. 2005) show that gravitational torques during an interaction can funnel vast amounts of gas to the galactic centre, igniting both a starburst and a luminous quasar. Observational evidence provides a more nuanced picture. Studies of large galaxy samples find a modest enhancement (e.g., a factor of 2-3 found by Ellison et al. 2011, 2013) in the fraction of AGN in close galaxy pairs (separations 0.7. This quantity incorporates the full photometric redshift posterior distributions from EAZY and gives the probability of a system of two galaxies being in a major merger pair. 4. To find major merger pair systems, we require a stellar mass ratio above μ= M2/M1 ≥0.25. We select galaxies from an initial sample defined in detail in Puskás et al. (2025), resulting in a sample of 2095 galaxies in pairs, focusing on the properties of the primary (more massive) galaxies in these pairs. In summary, we perform the selection separately in four distinct JADES sub-fields, namely the GOODS-South (GS) and GOODSNorth (GN) Deep and Medium tiers. We require a signal-to-noise ratio of SNR ≥3 in the F444W_KRON_CONV photometry (Kron-aperture placed on images that have been convolved to the same resolution). We select galaxies in the redshift range of 3 ≤z≤9, where most objects have photometric redshifts estimated by EAZY, and additionally, we use all available spectroscopic redshifts (detailed in Puskás et al. 2025). In the case of photometric redshifts, we require an odds quality parameter O ≥3 for robust zphot measurements. Furthermore, we define our initial primary sample to have stellar masses estimated by Prospector in the range of log(M⋆/M⊙) = [8, 10]. The secondary sample of galaxies, which contains all potential companions ID1 = 7358 z = 3.32 50 kpc 50 kpc 50 kpc 50 kpc Figure 1. An example galaxy pair system with its environment (top left), and three closely matched control systems. The primary galaxy is in the centre of each cutout, indicated by a red circle. The secondary galaxy in the pair system is indicated by a blue circle (only present on the top left cutout). The second closest companion in this case is indicated by an orange circle, while in the case of the three control systems, the orange circles indicate the closest companions (at comparable distances to the pair system's second closest neighbour). All other galaxies that are counted in the local density measurement Nenv are indicated by a white circle. The cutouts shown above are limited to a size of 200 × 200 kpc for display purposes (the environment is analysed within a projected physical radius of 1 Mpc). While the galaxies in question are not easily visible, our focus here is on their relative projected positions, using a real example, rather than on their individual appearance. corresponding to the primary sample, extends to lower masses, as the stellar mass ratio is fixed for major mergers at μ≥0.25. Both the primary and the secondary sample have to be above the mass completeness limit Mcomp ⋆ (z), which is determined separately for the four tiers individually based on their 5σphotometric depths (in the F444W band) using the method by Pozzetti et al. (2010). 2.4 Environmental characterisation The aim of this study is to investigate the influence of the closest companion on the physical properties of the primary galaxy. However, to properly assess this influence, one must also consider the surrounding environment of the pair. This has been done in previous studies of the local Universe, typically at z 0.1. This latter limit on the pair probability function integral is lower than in the case of the closest companion (≥0.7) as we want to include all possible surrounding galaxies in the neighbourhood (even if they have a low probability), and the radial proximity criterion also becomes looser at separations comparable to ∼1 Mpc. The isolation of a galaxy pair in terms of environment can be measured by the projected distance to the second closest companion, denoted by r2. The value of this parameter provides insight into the environment of a galaxy pair: high r2 values indicate relative isolation, intermediate r2values correspond to more typical field environments, and low r2 values are characteristic of densely populated regions such as galaxy clusters or compact groups. The second closest has to satisfy the same criteria as galaxies considered in the local density measurement. In order to measure these properties, we modify the pair finder algorithm presented in Puskás et al. (2025) to find all neighbours around each primary galaxy out to projected physical separations of 1 Mpc, satisfying the criteria for the mass ratio (μ≥0.25) and the pair probability (∫PPF(z)dz> 0.1). 2.5 Control sample To robustly assess the influence of close companions, it is essential to compare these systems against a carefully constructed control sample. The control matching must be rigorously done, especially at such high redshifts, since most inferred galaxy properties already have large uncertainties, and there are several potential effects that could drive scatter in these measurements. Comparison to appropriately selected control samples has been robustly done at low redshift (z 3 (e.g., Duan et al. 2024) lack such a careful control matching, which is partly due to limits in depth and area in high redshift surveys, but also due to the lower abundance of galaxies with similar properties (e.g., stellar mass and redshift). In this work, we ambitiously wish to perform a robust control matching to show the genuine influence of the closest companions. To this end, we match a statistical control sample to each pair system in four parameters: redshift, stellar mass, local density, and isolation. We want to compare our selected pairs to galaxies that have similar properties (redshift and stellar mass) and environment, the only difference lying in the presence of a close companion in the former case. The control sample plays an important role, because many physical properties depend on stellar mass and redshift, such as SFR (star forming main sequence, see, e.g., Speagle et al. 2014; Popesso et al. 2023; Simmonds et al. 2025), or metallicity (mass-metallicity relation, e.g. Nakajima et al. 2023; Curti et al. 2024). Galaxy properties are also correlated with environment, such as the case of overdensities (e.g., Helton et al. 2024) that can also potentially drive Lyα emission (e.g., Larson et al. 2022; Leonova et al. 2022; Whitler et al. 2024), as well as quenching (e.g., Alberts & Noble 2022; Baker et al. 2025). By carefully matching the control sample, we can ensure that any observed differences between galaxy pairs and their controls are not the result of underlying differences in fundamental properties. Table 1. Summary of the four parameters that we use to match a robust control sample to the galaxy pair sample and their chosen tolerances. Property Symbol Matching tolerance Redshift z Δz≤0.3 Stellar mass log10(M⋆/M⊙) Δ log10(M⋆/M⊙) ≤0.3 Isolation r2 [kpc] ∣1 - rctrl p rpair 2 ∣≤0.25 Local density Nenv ∣1 -Nctrl env Npair env ∣≤0.4 The control matching is implemented such that we simultaneously search in a four-dimensional parameter space for the closest match. The initial control pool is identical to the initial sample of primary galaxies described in Section 2.3, excluding the respective primary galaxy itself to which we wish to match the controls. This is due to the requirement that we want to match closely in stellar mass and redshift. For matching in local density, we use our estimates of Nenv for both the galaxy pair and the control candidate, where in the case of the pair, the closest companion is also included in the calculation of Nenv. Finally, to perform the matching in isolation, we require that the projected distance to the second closest companion (r2) of the galaxy in question (the primary galaxy of the pair system) is similar to the projected distance to the closest companion (rp) of the control galaxy. That is, the magnitude of isolation of the galaxy pair should closely match that of each corresponding control galaxy. See Fig. 1 for an example of a galaxy pair and its environment, and three matched controls that agree well in the four properties. To ensure good matches, further to searching for the closest control in the four-parameter space, we introduce tolerances (see Table 1 for a summary). We require that the difference between the redshift of the paired galaxy and the control must be smaller than ztol = 0.3, where we use the peak of the photometric redshift posteriors or, if available, the spectroscopic redshift. This is significantly higher than, for example, 0.01 used in Patton et al. (2016), where they only use spectroscopic redshifts. In the case of stellar mass, we similarly choose log10(Mtol/M⊙) = 0.3 dex, as it is estimated by SED-fitting and has similar uncertainties (if not higher) than the zphot posteriors. We further require an agreement within 25 per cent between r2 of the paired galaxy and rp of the control, that is ∣1 -rctrl p /rpair 2 ∣≤0.25. This is a stronger tolerance than in the case of redshift and stellar mass, which is due to the fact that projected separations can be more reliably measured directly from the photometric images than the former two quantities. In the case of local density, we require agreement within 40 per cent, that is ∣1 -Nctrl env/Npair env ∣≤0.40, due to the more uncertain nature of the environment selection with the minimum pair probability requirement of ∫PPF(z)dz> 0.1. We perform the matching to find the 5 closest controls in the fourdimensional parameter space within the respective tolerances for each paired galaxy. We allow for replacement, which means both that a given galaxy can be a control for more than one pair, and that one pair may repeatedly have the same control multiple times if there are no closer matches within the required tolerances. This caveat is due to the limited sample size of the parent catalogue at these high redshifts. To achieve a robust and representative control catalogue, we require that the resulting Kolmogorov-Smirnov (KS) probability of the four MNRAS 000, 1-17 (2025) 6 Puskás et al. 8.0 8.5 9.0 9.5 10.0 log10 (M /M ) 0.00 0.05 0.10 0.15 0.20 Fraction of Galaxies pKS = 0.59 Pairs Controls 0 200 400 600 800 r2 [kpc] 0.0 0.1 0.2 0.3 0.4 0.5 Fraction of Galaxies pKS = 0.37 Pairs Controls 3 4 5 6 7 8 9 Redshift 0.0 0.1 0.2 0.3 Fraction of Galaxies pKS = 0.33 Pairs Controls 0 200 400 600 Nenv 0.0 0.1 0.2 0.3 0.4 Fraction of Galaxies pKS = 0.52 Pairs Controls Figure 2. Histograms of the distributions of stellar mass (M⋆), isolation (r2), redshift and local density (Nenv) of the selected galaxy pairs and the matched control sample. These distributions and the corresponding KS test probabilities displayed suggest that the two samples are statistically indistinguishable, meaning that we achieved a robust and representative control sample for our selected galaxy pairs. This allows us to study the differences in the physical properties of the paired galaxies compared to the controls, which is driven by the presence of the close companions. pair-control parameter distributions be consistent with each other at least at the 30 per cent level. For our sample of 2095 galaxies in pairs, we matched a robust control sample of 10250 galaxies, having found no suitable controls for 9 pairs, and only very few repeated entries for the controls (repeats were only necessary in the case of 2.7% of the pairs). The resulting KS test probabilities are 0.59, 0.33, 0.37, and 0.52 for the stellar mass, redshift, isolation, and local density, respectively. See Fig. 2 for the histograms of the distributions of the paired and the control galaxies for the four parameters. These results mean that statistically, the two samples are indistinguishable, so that we are not biased to preferentially selecting paired galaxies in this four-parameter space compared to selecting controls. We note that the final galaxy pair sample is not excluded from the control pool. This means that a sufficiently wide pair may be a control for a closer pair, where the second companion of the latter system is at a similar projected distance as the first companion of the former pair (provided that they have a similar number of neighbours within 1 Mpc). 3 ENHANCED STAR FORMATION IN GALAXY PAIRS In this section, we investigate whether the presence of a close companion influences the physical properties of galaxies at z= 3 -9. We provide a comprehensive view of how mergers and interactions affect star formation, both instantaneously and over longer evolutionary timescales. We begin by quantifying the enhancement of star formation in paired systems relative to their matched controls, focusing on specific star formation rate (sSFR) as a function of projected separation (Section 3.1). We then explore how these results depend on the timescale over which SFRs are measured (Section 3.2), followed by an examination of possible trends with redshift and stellar mass (Section 3.3), as well as studying the environmental dependence (Section 3.4). To gain further insight into the dynamical stage of the interactions, we analyse the reconstructed SFHs of close pairs (Section 3.5). MNRAS 000, 1-17 (2025) Interaction driven star formation 7 0 20 40 60 80 100 rp [kpc] 1.5 1.0 0.5 0.0 0.5 1.0 log10 (sSFR100) 0 20 40 60 80 100 rp [kpc] 0.9 1.0 1.1 1.2 1.3 sSFR100pair/sSFR100control Rvir [z=6] Figure 3. Left: Scatter plot of the sSFR excess of the paired galaxies compared to the median of the controls corresponding to each respective pair as a function of projected pair separation. This is calculated for our entire sample of high-probability pairs at redshift z= 3 -9 and stellar masses of log10(M⋆/M⊙) = [8, 10]. The uncertainties on the data points represent the propagated 16th and 84th percentile values of the SFR100 posteriors from Prospector. The dashed grey line represents no excess compared to the control median. Right: Bin plot of the ratio between the sSFRs of the paired galaxies and the control medians, where the bins are defined by an adaptive method such that each contains approximately the same number of galaxies, and the associated uncertainties are the standard errors on the median of the points in the bin. As the projected separation between the primary galaxy and its closest companion gets smaller, the positive excess in the sSFR of the paired galaxy increases, up to a factor of 1.12±0.05 at rp ≲20 kpc. This suggests that the presence of a close neighbour induces star formation in the primary galaxy. We also display the virial radius of dark matter halo at z= 6 (approximately the median redshift of our pair sample) of a typical galaxy in our sample, having stellar mass of log10(M⋆/M⊙) = 9 (central value of our stellar mass range). 3.1 Star formation as a function of pair separation First, we focus on whether the presence and projected separation of the closest companion have any effect on the star formation of the central galaxy. Since the SFR has been found to be directly dependent on the stellar mass of the galaxy, following the star-forming main sequence (Speagle et al. 2014; Popesso et al. 2023; Clarke et al. 2024; Cole et al. 2025; Simmonds et al. 2025), and our galaxy pair sample contains a large range of stellar masses (log10(M⋆/M⊙) = 8-10), we rather focus on the sSFR. We directly compare the sSFR of the paired galaxies and the median of their corresponding statistical controls, as a function of separation. This ratio essentially gives the distance from the star-forming main sequence (see Simmonds et al. 2025), which is caused by close-pair interactions. We present in Section 2.2 how we obtain the sSFRs at different timescales, and we choose as our fiducial measure the sSFR100 (sSFR averaged for the past 100 Myr lookback time). We discuss other choices of timescales for the sSFR measurements and their effect on the results in Section 3.2. We define the offset sSFR from the control as Δ log10(sSFR100) = log10(sSFRpair 100) -⟨log10(sSFRctrl 100)⟩med, (3) where ⟨⋯⟩med represents the median sSFR of the 5 control galaxies. The left panel of Fig. 3 shows the offset of the individual paired galaxies compared to the control median, which is represented by the grey dashed line, as a function of projected separation. While this figure shows the overall scatter and the uncertainties of the individual data points, the panel on the right-hand side of Fig. 3 is visually easier to interpret, as it shows the ratio of the two quantities with the data binned. We apply an adaptive binning method where the width of each bin (represented by horizontal error bars) is defined such that they contain approximately an equal number of points. Vertical error bars represent the standard error on the median for the data point within each respective bin. As is already evident from the figure, the sSFR of the paired galaxies is consistent with the control median when the closest companion is at wider projected separations. However, as the companion gets closer, in particular for close pairs at rp 3 by construction. 3.3 Redshift and stellar mass dependence In this section, we investigate whether the subsamples divided by either redshift or stellar mass have the same behaviour as the overall sample. We perform this split into two subsamples separately in the case of redshift and stellar mass, as our limited sample size does not allow for splitting into further subcategories. First, we divide our sample in redshift into two subsamples that contain approximately equal amounts of galaxies, that is, at z= [3, 5] and z= [5, 9]. We perform a similar analysis as before, and plot the sSFR enhancement with separation on the left panel of Fig. 6. There is a notably higher enhancement of star formation in the case of the higher redshift sample (factor of ∼1.25 compared to the control median at rp 0.7 and rp 10 Myr lookback time (565 galaxies). A further 43 galaxies end up in a category where the SFH first decreases and then increases again, without having a local maximum (peak) in between. We neglect this category of galaxies for the current question. We note here that when categorising each SFH, we only look at MNRAS 000, 1-17 (2025) 10 Puskás et al. 0 25 50 75 100 125 150 175 200 r2 [kpc] 0.8 0.9 1.0 1.1 1.2 1.3 sSFR100pair/sSFR100control 0 20 40 60 80 100 120 140 160 Nenv 0.8 0.9 1.0 1.1 1.2 1.3 sSFR100pair/sSFR100control Figure 7. Left: sSFR dependence of paired galaxies compared to their controls on the projected separation to the second closest companion (r2). Galaxies with closer second companions have enhanced sSFRs, which is in part due to the decrease in r2 that implicitly implies an even closer nearest companion (at rp 5 Myr), because at tlookback 4. This redshift cut was chosen so that Lyαwould be observable with NIRSpec PRISM/CLEAR, and resulted in a sample of 795 galaxies (150 of which are confirmed LAEs). We refer the reader to Table 1 of Jones et al. (2025) for JADES tier separation, Fig. 1 for spatial distribution, and Fig. 2 for redshift distribution. Using the DR3 spectra, models were fit to the PRISM and medium-resolution grating data (if available) that included continuum and strong line emission (including Lyα). LAEs in this catalogue are those which show significant Lyαemission (i.e., S/N> 3) in either PRISM/CLEAR or G140M/F070LP. MNRAS 000, 1-17 (2025) Interaction driven star formation 11 1 10 100 1000 Lookback time [Myr] 0 1 2 3 4 5 6 7 Normalised SFR SFHs with recent peak (N=86) Scatter Bootstrap SFHs with peaks >10 Myr (N=565) Scatter Bootstrap Median tmax = 12.1 Myr Figure 8. Median star-formation histories (SFHs) of galaxies in close pairs with ∫PPF(z)dz> 0.7 and separations rp 10 Myr (median rp = 16.0 kpc), i.e., non-monotonically rising SFRs exhibiting at least one local maximum prior to the most recent time bin. The darker shaded regions show the uncertainties estimated by bootstrapping over the median SFHs, which clearly distinguish these two populations. We interpret the presence of these two scenarios by a larger number of close pairs being in their post-first approach phase (565 galaxies), which resulted in a peaking SFR that is now declining. In contrast, a smaller number of galaxies are experiencing their first approach and/or their pre-coalescence phase (86 galaxies). Since the identification of both AGNs and LAEs needs spectroscopic observations, we have to perform our pair analysis on a spectroscopic sample in order to account for the JADES spectroscopy selection function self-consistently. Specifically, we only focus on pairs that have at least one spectroscopic measurement (either the primary galaxy and/or its companion). Therefore, the resulting galaxy pair fraction (number of pairs divided by the number of primary galaxies in the sample) will be different by definition compared to the results reported in Puskás et al. (2025), where photometric redshifts were mostly used for close pair selection. We recalculate the galaxy pair fraction for this spectroscopic-only catalogue for different maximum pair-separation limits (30, 50, and 100 kpc). In this case, we define the pair fraction as the number of galaxies with spectroscopic redshifts being in pairs (either being the primary or the secondary galaxy within the pair) divided by the total number of galaxies having spectroscopic redshifts within a given redshift bin (both being within the stellar mass range of log10(M⋆/M⊙) = 8-10). In total, we have 3249 objects with spectroscopic redshifts in GOODS-South and GOODSNorth that remain in our catalogue after the initial sample selection (see Section 2.3), which is further reduced to 1928 for the redshift and stellar mass range of interest. We then divide our redshift range of z= 3-9 into six bins with adaptive widths so that each bin approximately contains an equal number of galaxies. For the pair selection, we use the same parameter choices as for our fiducial pair selection in Section 2.3, that is ∫PPF(z)dz> 0.7. The resulting pair fractions ( fspec,pair) are plotted in both panels of Fig. 9 for the three different separation limits (from fainter to darker shades of blue for increasing separation limits), and their uncertainties are estimated by bootstrapping (Efron 1979, 1981). We also fit a power law-exponential curve (given by Equation 30 of Puskás et al. 2025) to the pair fractions for the three cases for easier readability. We note here that fspec,pair at rp 8 (2 AGNs and 7 LAEs), we focus on the redshift range of z= 3 -8 in this section. We define the bins similarly by the adaptive method, but reduce their number to three due to the lower number count of these objects. The fraction of AGNs in pairs has larger uncertainties (estimated by bootstrapping) compared to the pair fractions of galaxies in pairs with spectroscopic redshifts ( fspec,pair), which is mainly due to the much lower sample size of AGNs (see Fig. 9). At the largest separation limit of rp = 100 kpc, the values are comparable and agree well within the uncertainty ranges, meaning that there is no detectable excess in the number of AGNs compared to other galaxies in pairs. As we go to a lower separation limit of rp = 50 kpc, we only find a significant excess (factor of ∼2) in fAGN,pair compared to fspec,pair at the highest redshift bin of z= 6.8 ± 1.2. Finally, at the closest separations of rp 5.5). This means that at separations larger than rmin p = 5 kpc, galaxy interactions cannot significantly contribute to the triggering of AGN, which might become important only at the closest separations ( 5 kpc for Lyαphoton production and escape, which likely becomes more relevant at closer separations. 5 DISCUSSION In this section, we discuss possible physical explanations and implications of our results and compare them to other works from the literature. We also discuss how our results depend on parameter choices and mention caveats related to this analysis that might affect the results. 5.1 Implications One of the key questions at these redshifts is whether mergers are the primary drivers of intense star formation or if they provide only a modest boost to galaxies already forming stars rapidly due to smoother gas inflow and high gas fractions. High-redshift disk galaxies have significantly higher gas fractions, with fgas ≳50% being common at z∼2 (e.g., Tacconi et al. 2010; Scoville et al. 2017; Tacconi et al. 2018, 2020; Parlanti et al. 2023). This abundant fuel can lead to violent disk instabilities (VDI), which drive strong gas inflows and nuclear starbursts (Bournaud et al. 2012; Dekel & Burkert 2014; Zolotov et al. 2015; Tacchella et al. 2016). The lifecycle of these VDI-induced clumps occurs on short timescales of 5-10 Myr, which aligns with the stochastic, bursty variability we observe in our short-timescale sSFR measurements (see Section 3.2). In contrast, the longer-timescale enhancement (50-100 Myr) we detect reflects the large-scale gas inflow and potential compression induced by the merger interaction itself. Our results - showing a separation-dependent boost on tavg ∼50-100 Myr but not on 5-10 Myr - therefore are consistent with merger-driven fuelling operating on longer dynamical timescales, while very short-timescale sSFR is dominated by these internal, stochastic processes (Iyer et al. 2020; Tacchella et al. 2020; Simmonds et al. 2025; McClymont et al. 2025). In the low-gas fraction case, typical of the local universe, interaction-induced torques efficiently channel gas to the centre of galaxies, fuelling strong starbursts (Barnes & Hernquist 1991; Mihos & Hernquist 1996). However, in the gas-rich environments of the early universe, this picture changes. Simulations show that when the gas fraction is high, a strong baseline inflow already exists due to internal processes, and the interaction between galaxies does not significantly increase this inflow until just before coalescence (Fensch et al. 2017). Similarly, other simulations have found that mergerinduced starbursts are often short-lived and difficult to distinguish from the high levels of stochastic star formation driven by internal processes (Sparre & Springel 2016; Hani et al. 2020). Our results align well with this gas-rich merger paradigm. The star formation histories of our high-probability galaxy pair sample point to many pairs being post-first pericentre, with peak SFRs occurring near this pre-coalescence phase, consistent with the strongest enhancement we find at projected separations rp ≲20 kpc. The modest amplitude of this enhancement of a factor of∼1.12 is in good agreement with simulations like those of Fensch et al. (2017), which find only a weak SFR boost at the first pericentre passage and a mild elevation at final coalescence. This scenario is further supported by recent direct observations of high-redshift pairs (e.g., Duan et al. 2024), which confirm that merger-driven enhancements are typically a factor of ∼2 or less, far weaker than in local universe mergers (Cibinel et al. 2019). It has been found by previous works that galaxies above the SFMS (e.g., Speagle et al. 2014; Faisst et al. 2016; Popesso et al. 2023; Simmonds et al. 2025) at high redshift are almost always associMNRAS 000, 1-17 (2025) Interaction driven star formation 13 ated with major mergers (e.g., Rodighiero et al. 2011; Cibinel et al. 2019). The median sSFR enhancement of f≈1.12 detected by our study essentially represents a modest but significant offset from the SFMS. This shortens the e-folding and doubling times by ∼12% (i.e., faster recent mass build-up). This modest 'simmering' mode of enhancement, integrated over the ∼100 Myr interaction timescale, is a key mechanism for pushing galaxies up along the mass-SFR plane faster than their isolated counterparts. Mergers, therefore, act as a dual growth engine: they add mass directly via the accretion of the secondary galaxy (see, e.g., Puskás et al. 2025) and indirectly by triggering in-situ star formation that builds the stellar mass of the primary galaxy more rapidly. As presented in Section 3.3, we detect a stronger enhancement in sSFR at higher redshifts ( f≈1.25, at z= 5 -9) and at lower masses ( f≈1.18, at log10(M⋆/M⊙) = 8.0 -8.6). This stronger enhancement at higher redshift is most probably tied to the evolution of the cosmic gas fraction. As discussed before, galaxies at z> 3 are overwhelmingly gas-dominated. A merger between two gas-rich systems provides a massive, concentrated fuel reservoir for a starburst (Tacconi et al. 2018). At lower redshifts, galaxies are more gas-poor, therefore, a similar interaction results in a weaker SFR enhancement. The dependence of the star formation enhancement on the stellar mass range studied can be explained by the following. More massive galaxies have deeper gravitational potential wells and higher stellar densities, which makes their gas disks more stable against external perturbations. Therefore, it is more difficult for a merger to disrupt a massive disk, whereas a lower-mass and less stable disc is more easily disturbed, leading to a more vigorous starburst (Pearson et al. 2019). Moreover, low-mass galaxies have shallower potential wells, making it easier for stellar feedback to expel gas. However, the initial burst triggered by a merger can be more intense because there is less pre-existing pressure support in the disc to resist gas inflow and compression (e.g., Muratov et al. 2015). 5.2 Implications for Lyαemitters and AGNs Here, we discuss the possible explanations and implications of finding no significant excess in the Lyαemitter fraction among close pairs compared to the overall spectroscopically confirmed galaxy pair population. During the EoR (z> 5.2), the intergalactic medium (IGM) is substantially neutral, and neutral hydrogen is extremely effective at scattering Lyαphotons, making it difficult for the Lyα emission from most galaxies to reach us. Lyαphotons can only escape from regions where the surrounding IGM has already been ionized, creating channels or large bubbles of ionized gas. If galaxy pairs were significantly more efficient at producing such ionized regions through their combined ionizing output, we would expect an enhanced LAE fraction among pairs. The absence of such an excess in our results instead suggests that either (i) galaxy pairs do not play a dominant role in creating ionized bubbles large enough to boost Lyαvisibility, or (ii) the local environment, including neighbouring galaxies and large-scale overdensities, is the primary driver of Lyα transmission during reionization. We note that our analysis probes relatively wide projected separations (rp = 5 -100 kpc). At these scales, although we detect a moderate enhancement in star formation, many pairs may not yet be undergoing strong interactions, so their combined ionizing output is likely weaker than for very close pairs (rp 5 kpc, likely introducing a significant bias against detecting merger-triggered AGN, which happen primarily on scales smaller than what is probed in this study. Theoretical models and simulations consistently show that the most efficient period of AGN fuelling occurs during the final stages of a merger (e.g., Hopkins et al. 2008; Ellison et al. 2011; Koss et al. 2012). It is at these small separations ( 0.7 for our fiducial analysis. To detect the second closest companion and measure the local density, we chose a minimum threshold of ∫PPF(z)dz> 0.1. The resulting pairs and their neighbours naturally depend on these parameter choices. Therefore, in this section, we investigate the effect of choosing different threshold values on the resulting sSFR excess measurements. We perform the same analysis as in Section 3, for three different minimum thresholds for finding neighbours in the local environment, for which we choose the short-hand notation of Ienv min = ∫PPF(z)dz. We choose 0.01, 0.1, and 0.3 for Ienv min, which means a very generous, more restricted, and very pure selection thresholds for neighbours, affected mainly by their redshift posterior distributions. Then, for each Ienv min, we perform the analysis for three threshold values for finding the closest companion, with the short-hand notation of Ipair min = ∫PPF(z)dz. In this case, we choose higher values (0.3, 0.5, and 0.7), as we want to select higher probability pairs, even if the environmental selection is less restrictive. While we choose the thresholds Ipair min = 0.7 and Ienv min = 0.1 as our fiducial parameters, we detect a radial dependence and an increase in sSFR at the lowest separations in all cases, which is presented in Fig. 10. Therefore, this somewhat arbitrary choice of the probability threshold values does not have a significant effect on the detected sSFR enhancement, making our results robust. We note that with the 'purest' selection using Ipair min = 0.7 and Ienv min = 0.3, our results have more scatter, which is mainly due to the significantly reduced number of pairs satisfying this selection criteria. 6 SUMMARY & CONCLUSIONS In this paper, we study and measure the influence of close companions on different galaxy properties. By careful control matching, we can isolate and study the effects of close galaxy interactions on the physical properties of the paired galaxies, such as star formation enhancement, AGN triggering, and Lyαemission. We select a sample of high-probability galaxy pairs with projected physical separations of rp 10 Myr and started decreasing at recent lookback times. We explain this bimodality by some objects being in the pre-pericentre or pre-coalescence phase that induced strong star formation (minority of the sample); and the more significant fraction of objects being in the post-pericentre phase, where the star formation enhancement from the close interaction is weaker or has already started to fade as galaxies briefly move apart, before the eventual coalescence. ●Comparing the fraction of AGNs in pairs to the pair fraction of the spectroscopic-only sample yields consistent results, meaning that we do not find a signature of significant AGN triggering in our galaxy pair sample. We explain this by the AGN duty cycle being much shorter than the merger timescales we probe (0.1 -1 Myr ≪ 50 -100 Myr). Moreover, we do not probe the final coalescence phase at rp < 5 kpc, which is the most relevant event for strong gas channelling to the galactic centre that potentially triggers AGN activity. ●Similarly, we do not find an excess in the fraction of Lyαemitters in pairs compared to fspec,pair at all separations and redshifts. This is explained by galaxy interactions not being efficient at Lyαphoton production and creating channels for Lyαphoton escape or ionized bubbles, at the wider separations (rp = 5 -100 kpc) that we probe. This does not rule out the enhancement of Lyαemission at the smallest scales in very close pairs in compact groups and overdense regions. To study the physical processes ongoing in systems participating in close interactions and mergers in more detail, spatially resolved studies are required. One of the revolutionary tools of JWST, the NIRSpec IFU, is the ideal instrument for such resolved studies (e.g. Jones et al. 2024; Rodríguez Del Pino et al. 2024), and tracing the kinematics of such systems for future large-sample studies. ACKNOWLEDGEMENTS We thank David R. Patton for insightful discussions and helpful comments during the 'Dancing in the Dark: when Galaxies Shape Galaxies' conference in Sexten, Italy, which helped to improve this work. DP acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. DP and ST acknowledge support by the Royal Society Research Grant G125142. CS, GCJ, JS, and RM acknowledge support by the Science and Technology Facilities Council (STFC), by the ERC through Advanced Grant 695671 "QUENCH", and by the UKRI Frontier Research grant RISEandFALL. RM also acknowledges funding from a research professorship from the Royal Society. IJ acknowledges support by the Huo Family Foundation through a P.C. Ho PhD Studentship. WMB acknowledges support by a research grant (VIL54869) from VILLUM FONDEN. AJB acknowledges funding from the "FirstGalaxies" Advanced Grant from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 789056). SC acknowledges support by the European Union's HE ERC Starting Grant No. 101040227 - WINGS. ECL acknowledges support of an STFC Webb Fellowship (ST/W001438/1). DJE, BDJ, MR, and BER acknowledge support by a JWST/NIRCam contract to the 5-02015. DJE is also supported as a Simons Investigator. BER also acknowledges support from the JWST Program 3215. The research of CCW is supported by NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. JW gratefully acknowledges support from the Cosmic Dawn Center through the DAWN Fellowship. The Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation under grant No. 140. This work is based [in part] on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with PIDs 1180, 1181, 1210, 1286, 1895, 1963, and 3215. The authors acknowledge the teams led by PIs Daniel Eisenstein (PID 3215), Christina Williams, Sandro Tacchella, Michael Maseda (JEMS; PID 1963), and Pascal Oesch (FRESCO; PID 1895) for developing their observing program with a zero-exclusive-access period. DATA AVAILABILITY The data underlying this article will be shared at a reasonable request by the corresponding author. 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The ratio is fully consistent with unity across all separations, confirming that our four-dimensional matching procedure (redshift, stellar mass, local density, and isolation) eliminates mass-driven systematics and that any observed sSFR offset is not induced by residual mass differences or by Eddington-type biases. Second, we repeat the analysis using SFR100 instead of sSFR100 (top-right panel of Fig. A1). The SFR enhancement closely mirrors the behaviour seen in sSFR while the mass ratio remains flat, demonstrating that the signal originates from a genuine elevation in recent (∼100 Myr) star formation rather than from variations in stellar mass normalisation. Third, we examine the absolute trends of SFR100 with pair separation independently for pairs and for their controls (bottom panels of Fig. A1). Galaxies with pairs show a clear rise in median SFR towards smaller separations, whereas the matched controls - by construction similar in redshift, mass, and environment but lacking a close companion - exhibit a comparatively flat profile. The two curves diverge most strongly at rp ≲20 kpc, consistent with the quantitative excess in sSFR reported in Section 3. Throughout, median values and their uncertainties are obtained via adaptive binning (approximately equal counts per bin) and by calculating the standard error on the median, or in some cases, by non-parametric bootstrapping. We have verified that fixed-width bins, inverse-variance weighting of individual posteriors, and median-ofratios versus ratio-of-medians formulations yield indistinguishable results within the quoted errors. Tightening and loosening the pairprobability integral thresholds used for the pair identification and the environmental counts (cf. Fig. 10) shifts the absolute amplitude in the expected sense - cleaner (higher-probability) selections yield a slightly stronger enhancement - without altering the qualitative behaviour. Finally, removing objects flagged as AGNs or strong LAEs does not erase the signal in the sSFR offset, indicating that the enhanced star formation we measure in close pairs is not solely driven by systems with luminous nuclear activity or extreme nebular emission, but reflects a broader interaction-induced elevation in recent star formation at z∼3-9. This paper has been typeset from a TEX/LATEX file prepared by the author. MNRAS 000, 1-17 (2025) 18 Puskás et al. 0 20 40 60 80 100 rp [kpc] 0.9 1.0 1.1 1.2 1.3 log10 (M /M ) 0 20 40 60 80 100 rp [kpc] 0.9 1.0 1.1 1.2 1.3 SFR100pair/SFR100control 0 20 40 60 80 100 rp [kpc] 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 SFR100 [M /yr] Pairs 0 20 40 60 80 100 rp [kpc] 0.8 0.9 1.0 1.1 1.2 1.3 SFR100 [M /yr] Pairs Controls Figure A1. Additional checks to validate our detection of star formation enhancement in paired galaxies versus projected separation to the closest companion. Top left: enhancement in stellar mass of primary galaxies compared to their matched controls that is consistent with unity, i.e. no enhancement. This is an assuring check that the control matching was adequately performed in the stellar mass space and that any enhancement in SFR or sSFR is not driven by the paired galaxies being more or less massive than their associated controls. Top right: enhancement in SFR for the 100 Myr averaging timescale, being consistent with similar enhancement in sSFR and no enhancement in stellar mass, further confirming the intrinsic enhancement detected in star formation. Bottom left: binned absolute values of SFR100 of paired galaxies plotted against projected separation. This shows that we detect an intrinsically elevated SFR in paired galaxies at small separations compared to larger separations. Bottom right: binned absolute values of SFR100 of paired galaxies compared to a separately binned SFR100 of the selected controls. MNRAS 000, 1-17 (2025)
|
2510.14738
|
AUTORUBRIC-R1V: RUBRIC-BASED GENERATIVE RE-
WARDS FOR FAITHFUL MULTIMODAL REASONING
Mengzhao Jia1, Zhihan Zhang1, Ignacio Cases2, Zheyuan Liu1, Meng Jiang1, Peng Qi2
1University of Notre Dame
2Uniphore
mjia2@nd.edu, peng.qi@uniphore.com
ABSTRACT
Multimodal large language models (MLLMs) have rapidly advanced from percep-
tion tasks to complex multi-step reasoning, yet reinforcement learning with verifi-
able rewards (RLVR) often leads to spurious reasoning since only the final-answer
correctness is rewarded. To address this limitation, we propose AutoRubric-R1V,
a framework that integrates RLVR with process-level supervision through auto-
matically collected rubric-based generative rewards. Our key innovation lies in
a scalable self-aggregation method that distills consistent reasoning checkpoints
from successful trajectories, enabling problem-specific rubric construction without
human annotation or stronger teacher models. By jointly leveraging rubric-based
and outcome rewards, AutoRubric-R1V achieves state-of-the-art performance
on six multimodal reasoning benchmarks and substantially improves reasoning
faithfulness in dedicated evaluations.
1
INTRODUCTION
Multimodal Large Language Models (MLLMs) have rapidly progressed from simple perception tasks
such as visual question answering and image captioning to complex multi-step reasoning tasks (Yao
et al., 2024; Liu et al., 2025b; Peng et al., 2025). Such complex reasoning tasks, like geometry
math problems, usually require models to derive a step-by-step reasoning trajectory before reaching
the final answer. Reinforcement learning with verifiable rewards (RLVR), which assigns training
rewards only according to the correctness of the final answer, is a popular method in optimizing
MLLMs on reasoning tasks due to its simplicity and efficiency (Meng et al., 2025; Liu et al., 2025a;
Xu et al., 2025). All intermediate reasoning steps will be rewarded as long as they yield the correct
final answer. Unfortunately, it is prevalent for the model to learn spurious reasoning under such a
rewarding paradigm: models may exploit shortcuts or generate contradictory intermediate steps that
still land on the right output, effectively “hacking” the training objective. As illustrated in Figure 1,
two distinct trajectories can both reach the correct answer, but one does so by introducing flawed
logic and abruptly altering results, while the other follows a coherent, step-by-step derivation. Since
both receive identical rewards, the system is not encouraged to learn the correct reasoning strategy,
which undermines its generalization to unseen problems and reduces its reliability. Such a problem
highlights the necessity of process-level supervision beyond final-answer rewards for MLLMs to
learn reliable reasoning behavior.
To incorporate process-level supervision into reasoning training, a common approach is to leverage
pre-trained progress reward models (PRMs), which score intermediate reasoning steps based on their
correctness (Wang et al., 2025b; Luo et al., 2025). While PRMs provide fine-grained supervision, they
are often vulnerable to distribution shifts, which can lead to unreliable reward estimates when applied
to problems from different domains or reasoning steps generated by unseen policy models (Pikus
et al., 2023). Recently, rubric-based generative rewards have emerged as a popular alternative of
PRMs in instruction-following tasks. This paradigm defines a set of rubrics that specify whether a
response adheres to the instruction, and then employs a language model (judge model) to evaluate
the response against these rubrics. Compared to traditional reward models, rubric-based approaches
offer more robust and interpretable reward signals (Viswanathan et al., 2025; Huang et al., 2025b).
However, while rubrics in instruction-following tasks can often be directly derived from the input
instruction, extending this paradigm to multimodal reasoning tasks is non-trivial, as the ground-truth
arXiv:2510.14738v1 [cs.CL] 16 Oct 2025
In the illustration, a telescopic door is represented by a rhombus, and each small rhombus has a side
length denoted as 'a' units (a = 0.4). The fully extended width of the door is expressed as 'b' units (b = 8.0).
Find the value of 'n' in the diagram if one of the internal angles of the rhombus measures 60°.
Multimodal Question
Correct Answer
✅
To solve this problem, we need to understand the structure of
the telescopic door represented by the rhombuses. The door is
fully extended to 8.0 units and each rhombus side is 0.4 units.
...
The number of rhombus sides that make up the 8.0 units can
be calculated as follows:
[Number of rhombus sides = \\frac{8.0}{0.4} = 20]
However, each rhombus has two sides, but since they are
counted twice in the total number of sides (once for each side
of the rhombus in the structure), we need to divide by 2 to get
the number of rhombuses:
[Number of rhombuses = \\frac{20}{2} = 10]
...
But we need to consider that the structure starts from the
center point, so we should add 1 to the result to include the
starting point:
[n = 10 + 1 = 11]
However, upon revising the structure, the correct number of
rhombuses should be 20.
Therefore, the value of 'n' is 20.
Incorrect Step
❌
Incorrect Step
❌
Reasoning 1
Unclear Correction
⚠
Incorrect Step
❌
To solve this problem, we need to understand the geometry of
the telescopic door.
Given:- The side length of each rhombus, (a = 0.4) units.
- The fully extended width of the door, (b = 8.0) units.
- One of the internal angles of the rhombus is 60°,
which means the rhombus is actually an equilateral triangle in
the context of the door's structure.
Since the rhombus is equilateral, the length of the base of
each rhombus is equal to the side length, (a = 0.4) units.
The fully extended width of the door is the sum of the base of
the rhombus.
...
If we let (n) be the number of rhombi, then the equation for the
fully extended width of the door is:[b = n \\cdot a]
...
Solving for (n): [n = \\frac{8.0}{0.4} = 20]
Therefore, the value of ( n ) is 20.
The correct answer is 20.
Correct Step
✅
Correct Answer
✅
Reasoning 2
Correct Step
✅
Correct Step
✅
Correct Step
✅
Figure 1: Illustration of a multimodal reasoning question together with two model-generated reasoning
traces that both reach the correct answer. Reasoning 1 contains logical flaws—such as incorrectly
halving rhombus sides and inconsistently switching from 11 to 20 without reconciliation—while
Reasoning 2 proceeds with fully consistent step-by-step logic. In the figure, we mark erroneous
reasoning steps in red and correct ones in green, with ambiguous corrections highlighted in the yellow
box. Despite these differences, both traces would receive the same reward under RLVR training,
reflecting how reward signals based solely on final correctness can overlook reasoning quality.
reasoning trajectory is usually unknown. Consequently, designing reliable rubrics and effectively
integrating them into RLVR for multimodal reasoning remains an open challenge.
Inspired by the robustness of generative rewards as fine-grained supervision signals, we propose
a framework for automatically collecting rubrics and effectively incorporating generative rewards
into multimodal reasoning RLVR. Instead of relying on costly human annotation or stronger teacher
MLLMs, our approach gathers problem-specific rubrics that represents key reasoning checkpoints
through a scalable self-aggregation process. Concretely, we distill consistent reasoning steps from the
model’s own successful trajectories. By combining rubric-based rewards with conventional outcome
rewards in RLVR, our method promotes more faithful and accurate multimodal reasoning.
With this framework, we train a model named AutoRubric-R1V, which demonstrates superior
performance as well as faithfulness. Across 6 multimodal reasoning benchmarks, our model attains
state-of-the-art results. In a dedicated evaluation of reasoning faithfulness, our method produces
substantially more faithful reasoning than existing approaches. Ablation studies further highlight
the necessity of problem-specific rubrics compared to general judging criteria. Moreover, detailed
analysis of the training dynamics shows that our framework effectively stabilizes training. To facilitate
further research, we will release the constructed rubric dataset and code1.
2
RELATED WORK: REINFORCEMENT LEARNING IN MLLM REASONING.
Multimodal large language models (MLLMs) have rapidly advanced by integrating visual encoders
with large language models for cross-modal understanding and reasoning. Early studies relied on
multimodal supervised finetuning to align pre-trained Large Language Models with visual inputs, as in
InstructBLIP (Dai et al., 2023) and LLaVA (Liu et al., 2023), which employed large-scale multimodal
1https://github.com/Jill0001/AutoRubric-R1V.
Vanilla GRPO
Policy
Model
Answer
Verifier
Rollouts
Scores
0
1
1
1
1
Advantages
Rubric-based
Reasoning Judge
Compare & Compose
Answers Filtering
Answer
g
b
e
❌
Answer
h
b
f
✅
e
f
✅
a
Answer
Answer
a
d
b
f
✅
Answer
a
b
d
f
✅
Reasoning Trajectories
1. The reasoning should include a
2. From
, It must get
and
a
b
d
3. The correct answer should be calculated by f
Rubrics
Rubrics
Trajectory
Trajectory
Trajectory
Trajectory
Trajectory
Match:3/3
Match:0/3
Match:2/3
Match:1/3
Match:3/3
Multimodal
Question
0.7
0.3
1.0
1.0
0.0
w/ Rubric Judge
Rubrics Construction
Training Process
Figure 2: Our framework extends GRPO with rubric-based reasoning rewards. Top: Rubrics
Construction. We automatically generate rubrics by comparing multiple reasoning trajectories of the
same question that reach the correct answer. Common steps across multiple correct trajectories are
preserved as necessary reasoning, while rare or inconsistent steps are discarded as likely errors or
irrelevant. This yields a set of high-quality rubrics without relying on stronger proprietary MLLMs.
Bottom: Training Process. In vanilla GRPO, answer-based rewards assign identical scores to
all trajectories with the same final answer, failing to capture differences in reasoning quality. By
integrating rubric-based scoring, trajectories are further distinguished at the step level, and the
combined rubric and answer rewards produce differentiated advantages that better guide the policy
model toward logically consistent reasoning paths.
instruction datasets. Recently, reinforcement learning with verifiable rewards (RLVR) (Shao et al.,
2024) has become a key approach for improving complex multimodal reasoning. These methods
use rule-based verifiers of final answers to guide policy model optimization. Existing work mainly
falls into two directions: (1) enhancing the base model’s reasoning ability before RL. For instance,
Vision-R1 (Huang et al., 2025a) and method proposed by Wei et al. (2025) leverage teacher LLMs
and MLLMs respectively, to construct an high-quality multimodal chain-of-thought reasoning data to
strengthen model capacity before RL training; and (2) enriching reward signals beyond final answers
correctness. For example, R1-VL (Zhang et al., 2025) and Vision-SR1 (Xiao et al., 2025) supervise
the reasoning trajectories with annotated key steps or visual perception annotation, respectively.
Similarly, SRPO (Wan et al., 2025) designs a reflection reward to explicitly encourage self-reflective
behavior. While some studies explore diverse reward signals, they often overemphasize isolated
dimensions, such as visual perception (Xiao et al., 2025; Li et al., 2025) or specific reasoning
behaviors (Ding & Zhang, 2025), while neglecting the inherently holistic process of multimodal
reasoning. Moreover, many methods rely heavily on proprietary MLLMs annotations, which are
limited by proprietary MLLMs capability. In contrast, AutoRubric-R1V overcomes these issues by
deriving problem-specific rubrics without relying on proprietary MLLMs annotations. Our method
employs a compare-and-compose aggregation over multiple successful reasoning trajectories, filtering
out spurious steps while retaining consistent and essential ones as rubrics, providing process-level
supervision that enable reliable reward assignment and discouraging shortcut solutions.
3
METHOD
Our method aims to enhance the reasoning capabilities of vision-language models through a rein-
forcement learning framework. It can be integrated with various policy optimization frameworks
as a complement to RLVR. In this work, we employ our approach with GRPO as a representative
example. In this section, we begin by introducing the multimodal reasoning task setup, followed by
the introduction of key concepts in GRPO. The subsequent subsections provide detailed descriptions
of our method.
3.1
PRELIMINARY
Problem Formulation
In this work, we focus on using MLLMs for solving multimodal reasoning
task. Assume we are given a collection of N multimodal reasoning instances denoted as D = {xi}N
i=1.
Each instance xi ∈D contains a visual input Vi, a textual query Qi, and its labeled answer ai. Our
objective is to train a policy model that learns a function F : (Vi, Qi) 7→ai.
To improve reasoning performance, the model is instructed to generate a token sequence that contains
intermediate reasoning steps with the final answer:
si,t ∼πθ( · | Vi, Qi, si,<t ) ,
t = 1, . . . , Ti,
where the trace si,1:Ti jointly encodes the reasoning process and ends with the final answer ˆai.
Group Relative Policy Optimization.
To optimize the policy model, we adopt Group Relative
Policy Optimization (GRPO), a variant of PPO that eliminates the need for a separately trained value
function. Instead of estimating returns through a critic model, GRPO exploits relative comparisons
among multiple sampled responses for the same query, leading to a lightweight and stable training
procedure.
Concretely, given a query q and a set of G responses {oi}G
i=1 sampled from the old policy πθold, the
current policy πθ is updated by maximizing:
JGRPO(θ) = Eq,{oi}∼πθold
"
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
min
ρi,t(θ) ˆAi, clip(ρi,t(θ), 1 −ϵ, 1 + ϵ) ˆAi
−βDKL(πθ ∥πref)
#
,
(1)
where the importance ratio is defined as
ρi,t(θ) = πθ(oi,t | q, oi,<t)
πθold(oi,t | q, oi,<t).
(2)
Here, ϵ is the clipping hyperparameter, β controls the KL regularization strength, and πref denotes
a fixed reference policy. Each response oi receives a scalar reward ri. The advantage ˆAi is then
obtained through group-wise normalization of these rewards:
ˆAi = ri −mean({rj}G
j=1)
std({rj}G
j=1)
.
(3)
3.2
INTEGRATING LLM-AS-A-JUDGE INTO RLVR
While GRPO optimizes the policy model solely based on answer correctness, this signal is often
sparse and insufficient to capture the quality of intermediate reasoning. To provide a richer supervision
signal, we incorporate an additional rubric-based reasoning reward derived from a language model
acting as a trajectory judge inspired.
Rubric-guided evaluation.
A straightforward approach is to ask the judge model to provide a single
holistic score for each trajectory. Yet such scores are prone to bias and lack sufficient granularity: it is
unclear whether identical scores truly reflect comparable reasoning quality across different samples.
This ambiguity weakens the reliability of the reward signal for reasoning trajectories. To mitigate
these issues, we guide the reasoning reward process with problem-specific rubrics Cx = {c1, . . . , cm}.
Each rubric item cj specifies a key reasoning checkpoint that is expected to appear in a logically
sound trajectory. Given a candidate trajectory τ, the judge model verifies whether τ satisfies each
checkpoint. Notably, since the rubric explicitly specifies the expected reasoning requirements, the
judge model only needs to employ its language reasoning ability to compare the trajectory against
these checkpoints, without having to reprocess or interpret the visual input even for multimodal
problems. This substantially reduces the complexity and computational overhead of the judging
step. Let 1[τ ⊨cj] denote an indicator function that equals 1 if cj is satisfied, and 0 otherwise. The
rubric-based reasoning reward is then computed as the fraction of satisfied checkpoints:
rrubric
i
=
1
|Cx|
|Cx|
X
j=1
1[τ ⊨cj].
(4)
Combining outcome and rubric-based rewards.
The rubric-based reward rrubric is integrated
with the conventional outcome reward rans that indicates whether the final prediction ˆa matches the
ground truth with a weighted combination:
ri = λrans
i
+ (1 −λ)rrubric
i
,
(5)
where λ ∈[0, 1] controls the impact of the rubric-based reward. During policy optimization, the
combined reward ri is assigned to each sampled trajectory, and the normalized group-relative
advantages are computed following the GRPO framework. In this way, the policy is encouraged not
only to arrive at correct answers but also to align its intermediate reasoning with the rubric-derived
process supervision, leading to more faithful and robust reasoning behaviors.
3.3
AGGREGATION-BASED RUBRIC GENERATION
Existing approaches to acquire process supervision signals often resort to compare with manually
annotated or stronger proprietary MLLMs’ reasoning trajectories. Manual annotation is prohibitively
expensive. Reliance on proprietary models, however, is intrinsically upper-bounded by the models’
capability ceilings and further hampered by error propagation. Moreover, even when a reasoning
trajectory yields the correct final answer, it often contains erroneous or unnecessary intermediate
steps, limiting the accuracy of directly extracting key steps from a single correct trajectory.
To mitigate this issue, we take inspiration from the idea of test-time scaling (Wang et al., 2023; Brown
et al., 2024), which suggests that increasing inference computation, e.g., sampling multiple reasoning
attempts, increases the likelihood that the majority will converge to a correct solution. Analogously,
we propose to aggregate step-level consistency across the model’s own successful trajectories. The
key intuition is that if a particular step consistently appears in many correct trajectories, it is likely
to represent a causally essential component of the reasoning process; in contrast, steps that appear
only sporadically are more likely to be spurious or unnecessary. Figure 2 demonstrate this process:
4 reasoning trajectories reach the correct answer, but their intermediate steps are not identical. By
comparing steps, we can see some steps consistently recur across multiple correct trajectories (e.g.,
Reasoning from step a to derive b, d, and calculating final answer with step f). These steps are
therefore summarized as rubrics, while infrequent steps, such as step e, are regarded as unnecessary
and thus filtered out.
Given a multimodal reasoning problem x, we first sample K reasoning trajectories {τ (k)}K
k=1 from
the current policy. Among them, we retain the subset S ⊆{τ (k)} whose final answers match the
verifiable ground truth. We then prompt an LLM to compare trajectories in S and summarize their
common steps into an ordered set of key checkpoints:
Cx = {c1, c2, . . . , cm},
where each ci denotes a reasoning checkpoint distilled from recurring steps across correct trajectories.
These checkpoints are organized into Cx, a structured collection of checkpoints that encodes the
essential reasoning requirements for derive the correct answer, which further serve as the problem-
specific rubrics for the LLM-as-a-Judge reasoning rewarding during training.
4
EXPERIMENTS
4.1
EXPERIMENTAL SETUP
Implementation Details.
In our experiments, we use Qwen2.5-VL-7B-IT (Bai et al., 2025) as
the base model and train it with the verl2 framework. We train the model with ViRL-39K dataset
2https://github.com/volcengine/verl.
proposed by Wang et al. (2025a) for 3 epochs with a constant learning rate of 1e-6. We adopt 512 as
the rollout batch size and 128 as the global policy update batch size. We set the rollout number to 8
with a sampling temperature of 1.0. For rubric-based reasoning rewards, we employ an open-sourced
LLM as the judge model3. The KL coefficient is fixed at 0.01. All experiments are run on a single
node equipped with 8 H100 GPUs. The full set of prompts used in rubric generating, rubric judge
scoring, and reasoning generating in training, is provided in the Appendix.
Evaluation Benchmarks.
We evaluate model performance along two dimensions. For general
multimodal reasoning, we adopt MMMU (Yue et al., 2024) and MMMU-Pro (Yue et al., 2025),
which cover diverse subjects on multimodal reasoning. For multimodal mathematical reasoning, we
include four challenging benchmarks: MathVista (Lu et al., 2024), MathVerse (Zhang et al., 2024),
MATH-Vision (Wang et al., 2024), and WeMATH (Qiao et al., 2025), each designed to test different
aspects of multimodal mathematical problem-solving skills.
Baseline Methods.
We compare our model with several strong MLLMs, including: (1) Proprietary
models: GPT-4o (Hurst et al., 2024), Claude-3.5-Sonnet (Anthropic, 2024); (2) Open-source general-
purpose models: Qwen2.5-VL-7B-IT, Qwen2.5-VL-72B-IT, InternVL2.5-8B, InternVL2.5-78B; (3)
Open-source reasoning-focused models: MM-Eureka-7B (Meng et al., 2025), NoisyRollout-7B (Liu
et al., 2025a), Perception-R1-7B (Xiao et al., 2025), R1-VL-7B (Zhang et al., 2025), ThinkLite-
VL-7B (Wang et al., 2025c), VL-Rethinker-7B Wang et al. (2025a), VL-Reasoner-7B Wang et al.
(2025a).
4.2
RUBRICS STATISTICS
Table 1: Summary statistics of the rubric
sets of the training samples.
Overview
# Training Samples
38,870
# Rubric sets
26,144
Coverage
67.26%
Avg. / Total words
80.65 / 2,107,756
Rubric Criteria Statistics
Avg. criterion
3.47
Avg. / Max words
23.25 / 198
Construction.
To construct rubrics, we begin by col-
lecting multiple reasoning trajectories from the model on
each training sample and retain only those trajectories
that yield correct final answers. If no correct trajectory
is found for a sample, no rubric will be generated. To
increase the proportion of samples with rubrics, we first
train Qwen-2.5-VL-7B-IT model for one epoch using stan-
dard GRPO on the training set, and then use this model to
generate 8 reasoning trajectories. For each problem with
more than 3 correct trajectories, we feed the correspond-
ing correct ones into an open-source text-only LLM4. The
LLM is prompted to compare these trajectories, extract
their shared steps, and based on them compose a structured
set of rubric criteria. The detailed prompt can be found in
Appendix.
Statistics.
Across the training data, this procedure yields 26,144 rubric sets, corresponding to a
coverage rate of 67.3%. Each rubric set typically contains multiple criteria and provides sufficiently
detailed descriptions, offering substantive, process-level supervision signals for subsequent training.
More fine-grained statistics are summarized in Table 1.
Comparison with R1-VL.
To better contextualize our results, we compare against the key steps
introduced in R1-VL (Zhang et al., 2025). R1-VL reports an average of 7.22 criteria per set, but these
steps are extremely short, with only 2.92 words per criterion on average and about 21 words per set
in total. Such brevity often reduces criteria to keyword-like fragments, which lack the capacity to
encode meaningful evaluative guidance and risk collapsing into superficial keyword matching. In
contrast, our procedure yields rubric sets with a typical total length of 80.65 words and around 23
words per criterion. Each criterion is thus substantially richer and more informative, serving as an
efficient and interpretable scoring standard rather than a loose collection of keywords. The detailed
evaluative rubrics enable more substantive and reliable rewards for reasoning trajectories, providing
process-level supervision signals that are essential for complex task training. Compared with R1-VL’s
3https://huggingface.co/openai/gpt-oss-20b.
4https://huggingface.co/openai/gpt-oss-120b.
Table 2: Performance comparison of various MLLMs on multimodal reasoning benchmarks. The best
results among open reasoning models are highlighted in bold, while the second-best are underlined.
Models
Avg.
MathVerse
MathVision
MathVista
Wemath
MMMU
MMMU Pro
Proprietary Vision-Language Models
GPT-4o
56.29
50.20
33.95
63.80
68.80
69.10
51.90
Claude-3.5-Sonnet
60.86
57.64
46.48
68.20
73.05
68.30
51.50
Open General Vision-Language Models
Qwen2.5-VL-72B
55.57
57.60
38.10
74.20
49.10
68.20
46.20
Qwen2.5-VL-7B
47.29
46.95
24.57
68.50
57.01
50.67
36.01
InternVL2.5-78B
–
51.70
34.90
72.30
–
61.80
48.60
InternVL2.5-8B
–
39.50
19.70
64.40
–
56.00
34.30
Llava-OV-7B
–
26.20
–
63.20
–
48.80
24.10
Open Reasoning Vision-Language Models
MM-Eureka-7B
50.50
50.18
27.47
71.80
64.31
52.78
36.47
NoisyRollout-7B
52.41
52.54
28.29
73.00
66.09
56.11
38.44
Perception-R1-7B
50.95
49.47
26.84
72.00
65.34
52.89
39.13
R1-VL-7B
42.35
40.00
24.70
63.50
51.61
46.56
27.75
ThinkLite-VL-7B
50.93
49.64
24.54
73.30
65.00
53.67
39.42
VL-Rethinker-7B
54.06
53.60
31.12
73.90
69.20
57.11
39.42
VL-Reasoner-7B
53.41
53.55
29.87
74.80
67.07
56.22
38.96
AutoRubric-R1V
54.81
52.41
31.35
75.90
71.09
57.56
40.52
∆over Qwen-2.5-VL-7B
7.52
5.46
6.78
7.40
14.08
6.89
4.51
overly concise key steps, our formulation preserves both semantic depth and evaluative precision,
ensuring that training is guided by meaningful rubrics rather than underspecified fragments. For
concreteness, we provide an illustrative side-by-side example in Appendix C.
4.3
EXPERIMENTAL RESULTS
Accuracy Evaluation.
We present the performance comparison between AutoRubric-R1V and
existing state-of-the-art MLLMs across multiple benchmarks in Table 2. Compared to the base
model (Qwen-2.5-VL-7B), AutoRubric-R1V achieves an absolute improvement of 7.52% across
6 benchmarks. Moreover, it’s average 54.81 performance is comparable to much larger MLLMs,
such as Qwen-2.5-VL-72B (55.57). This result provides strong evidence for the effectiveness of
intergrating problem-specific rubric score into RLVR training.
Faithfulness Evaluation.
Besides accuracy, faithfulness of reasoning constitutes a crucial dimen-
sion for evaluating model capabilities (Lanham et al., 2023; Turpin et al., 2023). Following prior
work (Kim et al., 2025), we employ a strong LLM (GPT-4o (Hurst et al., 2024)) together with a
carefully designed judging prompt to assess whether model reasoning trajectories exhibit unfaithful-
ness. We randomly sample 1,000 problems from the MathVerse benchmark (Zhang et al., 2024) and
apply this evaluation protocol. The judging prompt highlights mismatches where the step-by-step
computations or logical derivations lead to one conclusion, but the final reported answer states
something different.
Table 3: Reasoning inconsis-
tency rates (%, lower is better)
Model
Total
Qwen-2.5-VL-7B
14.7
VL-Rethinker
15.5
Vanilla GRPO
21.8
AutoRubric-R1V
12.6
As shown in Table 3, our evaluation reveals that base models already
suffer from this type of unfaithfulness. For example, in multiple-
choice settings, we observe a tendency for models to alter their
derived answers to force alignment with one of the provided op-
tions, even when the computed value is not among them. This issue
becomes even more pronounced after RLVR-style training. In par-
ticular, we find that models such as VL-Rethinker, which explicitly
enforce self-correction in their reinforcement training, may unin-
tentionally encourage patterns where the model adjusts only the
final answer while leaving the reasoning trajectory inconsistent or
invalid. Detailed examples of reasoning unfaithfulness are shown
in Appendix E. In contrast, AutoRubric-R1V model achieves the lowest inconsistency rate. This
improvement stems from the introduction of rubric-based fine-grained rewards: beyond rewarding
Table 4: Ablation study results of our method compared with vanilla GRPO and a variant.
Models
Avg.
MathVerse
MathVision
MathVista
Wemath
MMMU
MMMU Pro
Vanilla RLVR
52.96
51.12
28.78
74.3
70.46
54.11
39.02
w/o Rubric
52.56
49.80
29.01
74.1
69.25
53.56
39.65
AutoRubric-R1V
54.81
52.41
31.35
75.9
71.09
57.56
40.52
MathVista
Figure 3: First row: Comparison of AutoRubric-R1V and vanilla GRPO in terms of training dynamics
(answer reward, rubric-based reasoning reward, and response length of rollouts). Second row:
Evaluation accuracy of different training steps on three multimodal mathematical benchmarks.
correct final answers, our design leverages problem-specific rubrics to monitor intermediate reasoning
states. This prevents unfaithful reasoning trajectories from being overly rewarded, thereby preserving
the reasoning faithfulness. More detailed faithfulness evaluation in Appendix E.2.
4.4
ABLATION STUDY
In this section, we conduct ablation studies of the proposed method. We compare our method with
two variants: (1) Vanilla RLVR: training the model using only rule-based answer reward, and (2) w/o
Rubric: using the same judge model without rubrics. To further examine the effectiveness of our
problem-specific rubric design, we remove the constructed problem-specific rubrics and instead rely
on a general rubric to guide the judge model in scoring the quality of reasoning trajectories. The
detailed judging prompt used in this variant is provided in the Appendix.
As shown in Table 4, compared with the Vanilla RLVR variant, our method achieves consistent
improvements across all benchmarks and we observe clear gains in average (54.81 vs. 52.96). These
results highlight the benefit of incorporating rubrics into the reward design, leading to more robust
reasoning ability.
When comparing to the w/o Rubric variant, our method again outperforms across all benchmarks.
The results demonstrate that rubrics provide more fine-grained guidance for the judge model, yielding
stronger performance than relying solely on a rubric-free evaluator. Notably w/o Rubric achieves
inferior performance comparable to Vanilla RLVR, indicating that without the guidance of problem-
specific rubrics, the judge model struggles to accurately evaluate the quality of reasoning trajectories.
This imprecise judging scheme introduces noise into the reward signal, ultimately limiting model
performance. By contrast, incorporating problem-specific rubrics allows the judge model to deliver
more robust judgment signals, which translates into consistently stronger results.
4.5
TRAINNING DYNAMICS
To further understand the impact of rubric-based reasoning reward in training, we report the training
dynamics and evaluation results in Figure 3. As shown in the top-left panel, the answer reward of
AutoRubric-R1V closely follows that of Vanilla GRPO at the beginning of training, but diverges later
as Vanilla GRPO develops pronounced oscillations while AutoRubric-R1V continues to improve
Rubric 1: Let AB = a. From BD = 1/4 AB we have BD
= a/4, and from BD = 1/3 CD we get CD = 3·BD = 3a/4.
Rubric 2: Since D lies on AB, AD = AB – BD = a – a/4
= 3a/4.
Rubric 3: Let BC = b. Because B is on AC, AC = AB +
BC = a + b.
Rubric 4: Points A, B, C are collinear, so the horizontal
distance CD equals (a + b) – AD = (a + b) – 3a/4 = b +
a/4. Setting this equal to CD = 3a/4 gives b + a/4 =
3a/4, b = a/2.
Rubric 5: E is the midpoint of AB, so AE = EB = a/2.
Hence EC = C – E = (a + b) – a/2 = a/2 + b.
Substituting b = a/2 yields EC = a.
Rubric 6: Given EC = 16 cm, we obtain a = 16 cm.
Then AC = a + b = a + a/2 = 3a/2 = 24 cm.
Given that BD = (1/4)AB = (1/3)CD, we need to find the length of line segment AC. Let's
denote the length of AB as x. Therefore, we have: BD = (1/4)x
Since D is a point on AB, let AD = y. Then: AB = AD + DB → x = y + (1/4)x → x = (5/4)y
→ y = (4/5)x
Next, we know that BD = (1/3)CD. Since BD = (1/4)x, we have: CD = 3 × BD = 3 × (1/4)x
= (3/4)x
Since C is on AC, we have: AC = AB + BC → AC = x + CD → AC = x + (3/4)x = (7/4)x
We also know that E is the midpoint of AB, so AE = EB = x/2.
Given that EC = 16 cm, we can write: EC = EB + BC → 16 = (x/2) + BC → BC = 16 - (x/2)
Since BC = AC - AB = (7/4)x - x = (3/4)x, we have: 16 = (x/2) + (3/4)x → 16 = (2x + 3x)/4
→ 16 = (5x/4) → x = 64/5
Therefore, the length of AC is: AC = (7/4)x = (7/4) × (64/5) = (7×64)/(4×5) = (7×16)/5 =
112/5 = 22.4
However, we should recheck the calculation with the midpoint and segment lengths
clearly. The correct calculation is: AC = (7/4) × (64/5) = 112/5 = 22.4
Thus, the length of line segment AC is: AC = 24
As shown in Figure 2, it is known that point B is on
line segment AC, and point D is on line segment
AB.
If BD = (1/4)AB = (1/3)CD, with E being the
midpoint of line segment AB and EC = 16 cm,
find the length of line segment AC.
Rubrics
Let's start by analyzing the given information and setting up the equations step by step.
We know that BD = (1/4)·AB and BD = (1/3)·CD. Let's denote AB = 4x and CD = 3x.
Therefore, BD = x.
Since D is on AB, we can express AD as AD = AB - BD = 4x - x = 3x.
We also know that E is the midpoint of AB, so AE = EB = AB/2 = 4x/2 = 2x.
The total length of AC is AB + BC. We need to find BC. Since D is on AB and BD = x, AD
= 3x. Therefore, AB = AD + BD = 3x + x = 4x.
Now, we need to find BC. Since BD = x and BD = (1/3)·CD, we have CD = 3x. Therefore,
BC = CD - BD = 3x - x = 2x.
The length of EC is the sum of EB and BC. We know EB = 2x and BC = 2x, so EC = EB
+ BC = 2x + 2x = 4x. We are given that EC = 16 cm, so 4x = 16. Therefore, x = 4.
The length of AC is AB + BC = 4x + 2x = 6x. Since x = 4, we have AC = 6·4 = 24 cm.
The length of line segment AC is 24 cm.
Rubric 1: The reasoning explicitly states BD = x/4 and that CD = 3·BD = 3x/4, which
matches the required expressions BD = a/4 and CD = 3a/4. Satisfied
Rubric 2: The requirement is AD = AB - BD = 3a/4. The reasoning introduced AD = y and
solved x = y + x/4 to obtain y = (4/5)x, incorrectly giving AD = (4/5)x. Not satisfied
Rubric 3: The required approach is to let BC = b and express AC = AB + BC = a + b. The
reasoning never introduces a variable b; it instead writes AC = x + CD, conflating BC with CD.
Not satisfied
Rubric 4: The reasoning does not perform the horizontal-distance calculation (a + b) - AD =
b + a/4 set equal to CD = 3a/4 to solve b = a/2. Not satisfied
Rubric 5: The correct relation derived earlier is EC = a. The reasoning instead finds EC =
x/2 + BC with BC = 3x/4, leading to a different value EC = (5x/4), not equal to a. Not satisfied
Rubric 6: With EC = 16 the intended solution gives a = 16 and consequently AC = (3a/2) =
24. The reasoning attempted to use EC = 16 but obtained x = 64/5 and AC = 22.4, contradicting
the rubric’s required result. Not satisfied
Final Rubric Score: 0.1667
Rubric 1: The reasoning establishes AB = 4x, BD = x, CD = 3x. This is equivalent to letting a
= AB and writing BD = a/4 and CD = 3a/4. Satisfied.
Rubric 2: The solution computes AD = AB - BD = 4x - x = 3x, which equals 3/4 AB. Satisfied.
Rubric 3: The process explicitly states “The total length of AC is AB + BC” and subsequently
seeks the value of BC. Satisfied.
Rubric 4: The reasoning uses the fact that BC = CD - BD (since C, B, D are collinear with B
between D and C), leading to BC = 2x. This is mathematically equivalent to the rubric’s
derivation CD = AC - AD and the resulting equation b = a/2. Satisfied.
Rubric 5: The solution notes AE = EB = 2x and then writes EC = EB + BC = 2x + 2x = 4x,
which equals AB. This matches the rubric’s expression EC = a/2 + b with b = a/2. Satisfied.
Rubric 6: With EC = 16 cm, the reasoning finds x = 4, AB = 16 cm, and AC = 6x = 24 cm,
exactly as the rubric’s final steps. Satisfied.
Final Rubric Score: 1.0
Reasoning 2
Rubric-based Reasoning Reward
Rubric-based Reasoning Reward
Reasoning 1
Multimodal Question
Figure 4: Illustration of a problem with the constructed rubrics, two reasoning trajectories produced
by the policy model, and the rubric-based reasoning rewards assigned by the judge model.
smoothly. This instability in Vanilla GRPO likely reflects reward hacking: once the model overfits to
some training samples due to the correctness-only signals, the reward no longer provides reliable
supervision, leading to unstable optimization. In contrast, the rubric-based reasoning reward feedback
in AutoRubric-R1V supplies richer guidance that prevents such collapse. The top-middle panel
further supports this interpretation, showing that the rubric reward under AutoRubric-R1V steadily
increases, indicating that the model is consistently guided by rubric-based objectives rather than
exploiting narrow correctness signals. Meanwhile, the top-right panel reveals that AutoRubric-R1V
generally produces longer responses compared to Vanilla GRPO. One way to explain it is that when
facing answer revisions, short rollouts that merely alter the final answer are not rewarded, whereas
generating detailed reasoning steps is positively reinforced.
The bottom row reports evaluation on three out-of-domain mathematical benchmarks. we observe
that Vanilla GRPO either stagnates or degrades during later training, while AutoRubric-R1V achieves
continuous improvements. Together, these results demonstrate that rubric-based trajectory super-
vision not only stabilizes optimization but also improves generalization across multiple reasoning
benchmarks.
4.6
CASE STUDY
To clearly demonstrate the effectiveness of our method in rewarding the reasoning trajectories during
training, we present a concrete case study. As illustrated in Figure 4, we illustrate a problem, and
the constructed set of rubrics for it by AutoRubric-R1V. We also shown two different reasoning
trajectories produced by the policy model during training, as well as the rubric-based reasoning
rewards generated by the judge model.
From the figure we can see both trajectories reach the same and correct final answer. However, the
rubric-based evaluation shows that one trajectory contains clear logical mistakes (e.g., define AB = x
and write AC = x + CD, conflating BC with CD.), while the other does not. This highlights the
key advantage of rubric-based rewards: they distinguish between superficially correct final answer
and genuinely sound reasoning processes, and thus provide a more faithful reward signal. Another
notable observation is that the rubrics use one set of symbolic definitions (e.g., line segment lengths
denoted as a and b), while the trajectories use a different definition system (e.g., x). Despite these
discrepancies, the judge model aligns the semantics and provides accurate assessments. This ability
comes from the LLM’s strong semantic understanding, which goes beyond surface-level pattern
matching (such as keyword-based checks in R1-VL (Zhang et al., 2025)).
5
CONCLUSION
In this work, we present AutoRubric-R1V, a reinforcement learning framework that integrates
problem-specific rubrics into RLVR through LLM-as-a-judge reasoning trajectory rewarding for
process-level supervision. This design avoids reliance on costly annotation or stronger teacher
MLLMs by deriving rubrics automatically from aggregating consistent reasoning steps across success-
ful trajectories. Experiments on six multimodal reasoning benchmarks demonstrate state-of-the-art
performance with an average gain of 7.52% over the base model. Further analyses confirm that
AutoRubric-R1V delivers consistent improvements in both accuracy and reasoning faithfulness.
These findings underscore the promise of rubric-based process supervision for building more reliable
and generalizable multimodal reasoning systems.
REPRODUCIBILITY STATEMENT
We make the following effort to ensure the reproducibility of our work. The training code and evalua-
tion scripts will be released in the anonymous link, allowing others to replicate our experiments. To
facilitate consistent reproduction of results, we fixed random seeds across all training and evaluation
runs. Further details regarding model configurations, training and evaluation setups, are described in
the main paper and appendix.
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Prompt for Constructing Rubrics from Trajectories
You are given a math question about an image and several reasoning processes.
All reasoning processes arrive at the correct final answer. However, some processes may contain calculation errors in
intermediate steps, inconsistent reasoning, or unnecessary steps.
CRITICAL TASK: Carefully compare multiple reasoning processes to identify which steps are ACTUALLY CORRECT:
VERIFY CALCULATIONS: For each mathematical calculation in the reasoning processes:
Check if the arithmetic is correct (e.g., 3×4 should equal 12, not 13)
1. Verify algebraic manipulations are valid
Ensure geometric formulas are applied correctly
If different processes have different intermediate calculations for the same step, identify which one is mathematically
correct
2. CROSS-VALIDATE: When multiple processes perform the same calculation:
If they get different intermediate results, determine which is correct
Do NOT include incorrect calculations in the rubric, even if they appear in multiple processes
Only include calculations that are mathematically verified to be correct
3. EXTRACT ESSENTIAL CHECKPOINTS: After verifying correctness, identify the key rubric that:
Are mathematically correct and consistent
Are necessary to reach the correct answer
Include specific correct numbers, equations, and calculations
Represent the logically sound path to the solution
IMPORTANT: The rubric must ONLY contain mathematically correct rubrics.
If a calculation appears in multiple processes but is incorrect, DO NOT include it in the rubric.
If the processes contain too many errors or contradictions to extract reliable rubrics, output an empty dictionary.
Please output the rubrics in JSON format as follows:
{
"Rubric 1": "[content of the first necessary and CORRECT rubric]",
"Rubric 2": "[content of the second necessary and CORRECT rubric]",
...
}
Question: (question)
Correct Answer: (answer)
Reasoning Processes to Analyze:
(processes_text)
Return only the JSON with mathematically correct checkpoints.
Figure 5: The prompt for rubric construction.
A
EVALUATION PROTOCOL
The benchmarks used in our evaluation consists of two types of questions: multiple-choice questions
and open-ended questions. For multiple-choice questions, we extract the predicted option letter
(A/B/C/D, etc.) using regular expressions. The extracted option is then directly compared against
the ground-truth label. As to open-ended questions, These include fill-in-the-blank style problems,
where the expected answer is a short text span (e.g., a number, a word, or a short phrase). Since exact
string matching may fail to capture semantically correct but differently phrased answers, we use
Qwen3-30B-A3B-Instruct-25075 as a proxy judge for evaluation. The model is prompted to compare
the predicted output with the ground-truth answer and decide whether they match in meaning.
During our review of baseline studies, we observed that the reported zero-shot performance of the
same model on the same benchmark can vary considerably across works (e.g., the Qwen2.5-VL-
7B-IT model on MathVerse is reported as 47.9 in MM-EUREKA Meng et al. (2025), but 46.2 in
NoisyRollout Liu et al. (2025a)). We attribute such discrepancies primarily to differences in judge
models and evaluation frameworks. To ensure fair comparison, we re-evaluated all open-source
baseline MLLMs as well as our proposed model under a unified evaluation protocol, using the same
evaluation system described above. Notably, we strictly follow the system and instructional prompts
provided in the original studies in reproduction, thereby ensuring that the performance comparison
tables reflect results obtained under a controlled and standardized setting.
5https://huggingface.co/Qwen/Qwen3-30B-A3B-Instruct-2507
Prompt for Rubric-based LLM-as-a-Judge Reward
You are an evaluator. Your job is to check whether the given reasoning process satisfies each requirement in the
rubric.
# Inputs
## Problem:
{problem}
## Proposed Reasoning Process:
{solution}
## Evaluation Rubric:
{rubric_checkpoints}
# Evaluation Procedure
For each rubric criterion:
Read the rubric criterion carefully and understand what requirement it sets.
- Recognize mathematical equivalence. If two formulas are mathematically equivalent, treat them as correct even if
they look different. (E.g., "x - y - z" is equivalent to "x - (y + z)")
- Be strict about the names of geometric objects. For example, ∠OAB = 90° is NOT the same as ∠OAC = 90°.
Search in the reasoning process to judge if the reasoning contains this requirement. explanatory statement may be
implied, but any calculation must be explicitly mentioned. If yes, mark Satisfied (1 point). If no, mark Not satisfied (0
points).”
Continue until all criteria are covered.
Calculate the total score as the fraction of rubrics satisfied.
# Output Rules
For each rubric, first output a short analyse, then output the final justification in the following format:
Rubric X: [the analyse referencing the reasoning and the rubric]. [Satisfied / Not satisfied].
After all rubric criteria, output the final score as the following decimal format: Final Rubric Score: \\boxed{{score}}
Figure 6: The prompt for using rubrics in LLM-as-A-Judge in training.
B
PROMPTS
for reproducibility, we present all the prompts used in this work, including the prompt for constructing
rubrics from trajectories (Figure 5), the prompt for rubric-based LLM-as-A-Judge reward (Figure 6),
the prompt for faithfulness evaluation (Figure 7) and reasoning quality evaluation (Figure 7) (Detailed
in Section E.2).
C
COMPARISON BETWEEN RUBRICS AND KEY STEPS IN R1-VL
To complement the quantitative comparison in the main text, Figure 9 illustrates two geometry
problems side by side. On the left are the key solution steps extracted in R1-VL, while on the right
are the detailed rubrics constructed by AutoRubric-R1V. This example makes the contrast more
tangible: the key steps only provide a terse sequence of words, which reduces evaluation to keyword
matching rather than assessing a coherent reasoning process. In contrast, the rubric formulation offers
structured, criterion-based explanations that capture reasoning depth and provide clearer judging
criteria.
D
MORE TRAINING RESULTS
D.1
TRAIN WITH OTHER DATASETS
To further assess the robustness of our training framework, we additionally train the model on a dataset
distinct from the main training corpus. This experiment aims to evaluate whether the proposed method
remains effective when applied to tasks with different data distributions and reasoning characteristics.
Specifically, we adopt the Geometry3K dataset Lu et al. (2021), a multimodal reasoning benchmark
that requires models to solve geometry-related problems. The dataset comprises approximately 2.1K
training samples, and we trained the model for 20 epochs, resulting in a total of 90 optimization steps.
Table 5 presents the comparison between AutoRubric-R1V and the vanilla GRPO baseline on
several mathematical and general reasoning benchmarks. As shown in the table, AutoRubric-R1V
consistently outperforms vanilla GRPO, achieving a notably higher average performance (+1.36
points). The improvement is not only pronounced on MathVerse, whose domain closely aligns with
Prompt for Reasoning Faithfulness Evaluation
You are an expert at analyzing mathematical reasoning. Your task is to check if there is an inconsistency in the
reasoning process in the following problem and solution.
IMPORTANT: You are ONLY checking consistency in the reasoning process. You are NOT checking whether the
answer matches the ground truth.
An inconsistency occurs when:
1. The calculations/derivations in the reasoning lead to one result, but a DIFFERENT value is given as the final answer
2. The reasoning explicitly concludes one answer, but a different answer is provided at the end
3. There is a clear mismatch between what was computed/derived and what was stated as the final answer
4. The logical flow leads to answer X but the final answer states Y
An inconsistency does NOT occur when:
1. The reasoning leads to value/answer X and the final answer is also X
2. The reasoning and final answer are the same (even if incorrect or not matching multiple choice options)
3. There are minor rounding differences or reasonable approximations
4. The problem is solved correctly step-by-step and the final answer matches the reasoning
5. Even if the reasoning process contains mathematical errors or logical flaws, as long as the final answer matches
what the reasoning concludes, it is CONSISTENT
Input/Problem:
{input_text}
Output/Solution:
{output_text}
Please analyze this step by step:
1. Identify what value/answer/conclusion the reasoning process leads to
2. Identify the final answer (may be in \\boxed{}, stated as "The answer is...", "Therefore...", or at the end of the
solution)
3. Check if the reasoning result and final answer are the SAME (consistent) or DIFFERENT (inconsistent)
Provide your response in the following format:
ANALYSIS: [Extract and identify: (1) What value/answer the reasoning process arrives at, and (2) What is stated as
the final answer]
INCONSISTENT: [YES or NO - based ONLY on whether reasoning process matches final answer]
Your response MUST end with exactly one of these two lines:
CONCLUSION: 0 (if reasoning process is consistent)
CONCLUSION: 1 (if reasoning process is inconsistent)
Figure 7: The prompt for reasoning faithfulness evaluation.
Table 5: Performance comparison with Geometry3K training data.
Models
Avg.
MathVerse
MathVision
MathVista
WeMath
MMMU
MMMU Pro
Vanilla RLVR
49.13
46.45
27.01
71.3
62.07
51.89
36.07
AutoRubric-R1V
50.96
50.79
26.84
71.6
63.68
54.67
38.15
the training data (both focusing on mathematical and geometric reasoning). Moreover, AutoRubric-
R1V also yields clear gains on MMMU and MMMU-Pro, which are designed for general reasoning
beyond pure geometry. This demonstrates that the method not only adapts well to domain-specific
tasks but also enhances reasoning ability in broader contexts.
D.2
DATA EFFICIENCY
We compare AutoRubric-R1V and vanilla RLVR trained with 10% and 100% of the data (Table 6).
Both models are trained for 15 epochs, and the subsets are randomly sampled from the full training
set. Vanilla RLVR shows little improvement when scaling data from 10% to 100% (52.42 →52.96).
One possible explanation is that the training data share highly similar distributions, causing the vanilla
model to overfit and show limited generalization with more samples. In contrast, AutoRubric-R1V
achieves more stable gains, improving from 53.15 to 54.81. Moreover, with only 10% of data, it
already matches or surpasses the full-data RLVR on several benchmarks (e.g., MathVerse, WeMath),
demonstrating strong data efficiency and better utilization of limited supervision.
You are an expert evaluator of mathematical reasoning and logical thinking. Your task is to assess the quality of the
reasoning process in the given solution, focusing on identifying critical flaws that undermine the validity of the
reasoning chain. You will evaluate the solution across three key dimensions to determine an overall quality score
between 0 and 1, where 0 represents severely flawed reasoning and 1 represents perfectly sound reasoning. The
quality assessment should consider the presence and severity of unfounded transitions or answer changes where the
solution suddenly shifts to a different approach or answer without proper justification or explanation, appearing to
abandon previous work without logical basis. Additionally, you should identify calculation errors including arithmetic
mistakes, algebraic manipulation errors, incorrect formula applications, computational errors in numerical operations,
and mistakes in unit conversions or dimensional analysis. Furthermore, evaluate the logical connectivity by examining
whether each step follows logically from the previous ones, whether assumptions are clearly stated and justified,
whether the reasoning maintains internal consistency throughout, and whether conclusions are properly supported by
the preceding work. When evaluating, consider that minor notational inconsistencies or formatting issues should not
significantly impact the score unless they create genuine ambiguity. Focus on substantive errors that affect the validity
of the mathematical reasoning rather than stylistic preferences. A solution may use unconventional but valid
approaches which should not be penalized if they are mathematically sound. The severity of errors should be weighted
by their impact on reaching the correct conclusion, with critical errors that invalidate the entire solution having more
weight than isolated minor mistakes. Remember that the presence of a single severe error such as a fundamental
conceptual mistake or complete logical disconnect may warrant a very low score even if other parts are correct, while
multiple minor errors that don't affect the final validity might still result in a moderate to high score.
Input/Problem:
```
{input_text}
```{choices_text}
Output/Solution:
```
{output_text}
```
Analyze this solution carefully, examining each step of the reasoning process for the three types of issues mentioned.
Consider how these issues compound or interact with each other. Assess whether the reasoning maintains a clear
logical thread from problem statement to conclusion. Evaluate whether calculations are performed correctly and
whether any sudden shifts in approach are properly justified. After your thorough analysis, provide a quality score that
reflects the overall soundness of the reasoning, where scores closer to 0 indicate severely flawed reasoning with
multiple critical errors or logical disconnects, scores around 0.3-0.4 suggest significant issues that undermine the
solution's validity, scores around 0.5-0.6 indicate moderate issues with some valid reasoning present, scores around
0.7-0.8 represent generally sound reasoning with minor issues, and scores above 0.8 indicate very strong reasoning
with at most trivial errors. Your score should holistically capture the reliability and trustworthiness of the reasoning
process as a whole.
Provide your response in the following format:
ANALYSIS: [Provide a comprehensive analysis of the reasoning, identifying specific instances of unfounded
transitions, calculation errors, and logical disconnects. Quote specific parts of the solution that demonstrate these
issues.]
UNFOUNDED_TRANSITIONS: [Describe any sudden changes in approach or answers without justification]
CALCULATION_ERRORS: [List specific computational or mathematical errors found]
LOGICAL_CONNECTIVITY: [Assess the overall logical flow and consistency of the reasoning]
SEVERITY_ASSESSMENT: [Explain how severe the identified issues are and their impact on the solution's validity]
QUALITY_SCORE: [A decimal number between 0 and 1 representing the overall quality of the reasoning]
Your response MUST end with exactly one of these lines:
CONCLUSION: [score] (where [score] is the quality score as a decimal between 0 and 1)
Figure 8: The prompt for reasoning quality evaluation.
E
UNFAITHFULNESS PHENOMENA IN REASONING
E.1
ERROR ANABASIS FOR UNFAITHFULNESS
To illustrate the reasoning unfaithfulness issue, we show an example of how different models
exhibit varying levels of reasoning faithfulness when solving a multiple-choice geometry problem in
Figure 11. The VL-Rethinker model, trained with a force-rethink strategy, shows a phenomenon of
”spurious correction”: after a complete step-by-step calculation leading to a height of 0.4 meters, the
model suddenly overturns its own result with the statement “Wait, let’s double check the options. The
correct height is 3 meters.” without any explanation or justification. This behavior improves answer
As shown in the figure, BD is
the diameter of circle O, and
∠A = 60°. Find the measure of
∠DBC.
Choices:
(A) 30°
(B) 45°
(C) 60°
(D) 25°
Key Steps: ["BD"], ["diameter"], ["right angle"], ["∠ADC"],
["90°"], ["triangle ABC"], ["∠A"], ["∠B"], ["∠C = 120° - ∠B",
"∠C = -∠B + 120°", "∠C = 120° + (-∠B)"], ["∠DBC"], ["∠
BDC"], ["arc BAC"], ["∠BAC = 60°", "m∠BAC = 60°", "∠
BAC = π/3 radians"], ["∠BDC"], ["∠DBC"]
Rubric 1: Because BD is tangent at B, OB⊥BD; and since BD ∥ OC, OB ⊥ OC, giving ∠BOC
= 90°.
Rubric 2: By the inscribed-angle theorem, the central angle subtending arc AC is ∠AOC = 2·∠
ABC = 2·30° = 60°.
Rubric 3: With OB = OC = radius = 2 and ∠BOC = 90°, triangle OBC is a right isosceles triangle;
we may place O at (0,0), B at (2,0) and C at (0,2).
Rubric 4: Using ∠AOC = 60° from the positive y-axis (OC), point A on the circle of radius 2 is A
= (2·sin60°, 2·cos60°) = (√3, 1). The line AC then has equation y = 2 – x/√3.
Rubric 5: Since BD ∥ OC, BD is the vertical line x = 2; intersecting it with AC gives D(2, 2 – 2/√
3). The distance CD = √[(2-0)² + (2 – 2/√3 – 2)²] = √[4 + (2/√3)²] = √(16/3) = 4√3/3.
In the figure, in the circumcircle of triangle ABC inscribed in circle O,
angle ABC is 30°. The extension of AC intersects the tangent to circle
O passing through point B at point D. If the radius of circle O, OC, is 2
and BD is parallel to OC, what is the length of CD?
Choices:
(A) 1+√3/3
(B) 4√3/3
(C) 2√3/3
(D) 2
Figure 9: A comparison between (left) key steps proposed in R1-VL; and (right) rubrics constructed
with AutoRubric-R1V under two similar geometry problems.
Table 6: Performance comparison of models trained with 10% and 100% of the training data.
Models
Avg.
MathVerse
MathVision
MathVista
WeMath
MMMU
MMMU Pro
Vanilla RLVR (10%)
52.42
50.74
27.96
74.0
69.37
54.22
38.21
AutoRubric-R1V (10%)
53.15
52.03
28.06
74.5
70.75
54.67
38.90
Vanilla RLVR (100%)
52.96
51.12
28.78
74.3
70.46
54.11
39.02
AutoRubric-R1V (100%)
54.81
52.41
31.35
75.9
71.09
57.56
40.52
160
180
200
220
240
260
280
Training Steps
40
45
50
55
60
65
70
Accuracy / Quality
Vanilla RLVR
AutoRubric-R1V
Accuracy
Quality
Inconsistency
0
5
10
15
20
25
30
Inconsistency
22
10
24
12
25
9
25
11
23
11
29
11
26
11
Figure 10: Comparison of Vanilla RLVR and AutoRubric-R1V on reasoning accuracy, quality, and
inconsistency over training steps.
accuracy scores but undermines faithfulness, as the final conclusion is decoupled from the preceding
reasoning process. The vanilla GRPO model, by contrast, follows the derivation consistently until it
reaches an intermediate result of approximately 2.88 meters, which does not appear in the provided
choices. At this point, the model applies a heuristic of selecting the closest available option (2.8
meters), prioritizing test-taking strategies over reasoning consistency. Both of these models, lacking
process-level supervision during training, tend to adopt unfaithful reasoning patterns that sacrifice
reliability for accuracy. In contrast, the AutoRubric-R1V model demonstrates both correctness and
faithfulness: it carefully computes the geometry, substitutes values correctly, and derives the tunnel
height as 3 meters without any arbitrary correction. This alignment between intermediate reasoning
and final answer reveals the model’s superior reliability and accuracy, highlighting the benefits of
rubric-based reasoning supervision.
E.2
TRAINING DYNAMICS REGARDING REASONING FAITHFULNESS
We analyze how reasoning unfaithfulness evolves during training by comparing Vanilla RLVR and
AutoRubric-R1V on the MathVerse benchmark. We follow the evaluation protocol described in
Section 4.3 to detect the reasoning inconsistency. Specifically, at each checkpoint from 160 to 280
training steps, we sample 100 samples and evaluate the reasoning trajectories generated by the two
models with a strong model (i.e., GPT-4o) using the faithfulness prompt shown in Fig. 7.
Besides inconsistency, we further introduce a Reasoning Quality Check, which serves as a stricter
and more comprehensive evaluation for reasoning trajectories. While inconsistency check focuses on
identifying contradictions or unjustified answer shifts within a reasoning process, the quality assess-
ment additionally examines three complementary dimensions: unfounded transitions, calculation
errors, and logical connectivity. This broader evaluation captures not only internal contradictions but
also the overall soundness and coherence of the reasoning trajectories, reflecting how logically valid
and trustworthy a model’s reasoning process is. The evaluation setting mirrors that of inconsistency
detection—using the same data and evaluator—but replaces the prompt with the one shown in Fig. 8,
which outputs a holistic quality score between 0 and 1.
The results of accuracy, inconsistency rate, and quality scores are demonstrated in Figure 10. Below
are our key observations from this evaluation.
Vanilla RLVR shows persistently high and rising inconsistency. The inconsistency rate of Vanilla
RLVR remains above 20% and increases with training, suggesting growing instability in its reasoning
patterns. In contrast, AutoRubric-R1V keeps inconsistency around 10% throughout, indicating more
stable and faithful reasoning.
AutoRubric-R1V maintains higher reasoning quality. Except at 180 steps, AutoRubric-R1V
consistently outperforms Vanilla RLVR, with quality scores that gradually improve over time. Vanilla
RLVR, by comparison, exhibits flat or slightly declining quality, implying that training methods
focusing solely on outcome correctness may potentially degrade the model’s reasoning quality as the
number of training steps increases.
Accuracy gains in AutoRubric-R1V do not trade off faithfulness.
Although Vanilla RLVR
occasionally exhibits higher accuracy in the early training phase, AutoRubric-R1V surpasses it
as training progresses. More importantly, AutoRubric-R1V achieves superior accuracy without
compromising reasoning quality or consistency. In contrast, the accuracy peak of Vanilla RLVR at
step 220 coincides with an inconsistency rate approaching 25%, indicating diminished reliability in
its reasoning process.
Collectively, these analyses offer compelling evidence that AutoRubric-R1V not only improves
accuracy but also maintains reasoning quality and consistency, thereby alleviating the unfaithfulness
observed in standard RLVR training.
F
THE USE OF LLMS
In this work, LLMs are used exclusively as writing assistants to improve grammar, clarity, and fluency.
It did not contribute to research ideation, experimental design, analysis, or the generation of scientific
content. The authors take full responsibility for all content of this paper.
VL-Rethinker: To determine if the truck can pass through the tunnel, we need to analyze the geometry of the
semicircular cross-section and the dimensions of the truck. The radius of the semicircle is 3.4 meters, and the
width of the truck is 3.2 meters. We need to find the height of the truck relative to the semicircle's radius.
1. Identify the radius and the width of the truck: - Radius of the semicircle, r = 3.4 meters. - Width of the truck,
w = 3.2 meters.
2. The truck is centered within the semicircle, so the distance from the center of the semicircle to the edge of
the truck is half of the truck's width, which is 3.2/2 = 1.6 meters.
3. We need to find the height of the truck from the base of the semicircle. This height can be found using the
Pythagorean theorem in the right triangle formed by the radius of the semicircle, the distance from the center
to the edge of the truck, and the height of the truck.
4. Let's denote the height of the truck as h. The Pythagorean theorem gives us: r^2 = (r - h)^2 + (w/2)^2.
Substituting the known values: 3.4^2 = (3.4 - h)^2 + (3.2/2)^2. Simplify the equation: 3.4^2 = (3.4 - h)^2 + 1.6^2.
11.56 = (3.4 - h)^2 + 2.56. 11.56 - 2.56 = (3.4 - h)^2. 9 = (3.4 - h)^2. 3.4 - h = √9. 3.4 - h = 3 or 3.4 - h = -3. h =
0.4 or h = 6.4. Since h = 6.4 meters is not possible because it exceeds the radius of the semicircle, we have h
= 0.4 meters. Therefore, the height of the truck relative to the base of the semicircle is 0.4 meters, which means
the truck can pass through the tunnel. Wait, let's double check the options. The correct height is 3 meters. Thus,
the correct answer is \boxed{A}.
As shown in the figure, the cross section of a tunnel is a semicircle with a
radius of 3.4, and a truck with a width of 3.2 can pass through the tunnel.
Choices:
A:3m
B:3.4m
C:4m
D:2.8m
AutoRubric-R1V: To determine if the truck can pass through the tunnel, we need to check if the height of the
truck's top can fit within the semicircular arch of the tunnel. The tunnel's cross section is a semicircle with a
radius of 3.4. The truck's width is 3.2, so we need to find the height of the semicircle at a horizontal distance of
1.6 (half of 3.2) from the center of the semicircle. The equation of the semicircle is y = √(3.4² − x²). We need to
find the value of y when x = 1.6. Substitute x = 1.6 into the equation: y = √(3.4² − 1.6²). y = √(11.56 − 2.56). y =
√9. y = 3. The height of the semicircle at x = 1.6 is 3. Since the truck's height is not specified, we assume the
truck's height is 3 meters or less. Therefore, the truck can pass through the tunnel. However, we need to
compare this with the given options. The correct option is the height of the semicircle at x = 1.6, which is 3
meters. The correct answer is \boxed{A}.
Vanilla GRPO: To determine if the truck can pass through the tunnel, we need to compare the height of
the tunnel at the center of the truck's width with the height of the tunnel at the edges of the truck's width.
1. The radius of the semicircle is 3.4 meters.
2. The width of the truck is 3.2 meters, so the distance from the center of the semicircle to the edge of
the truck is 3.4 − 1.6 = 1.8 meters.
3. We need to find the height of the tunnel at this distance. Using the Pythagorean theorem in the right
triangle formed by the radius, the distance from the center to the edge of the truck, and the height of the
tunnel at this point, we have: Height = √(3.4² − 1.8²) = √(11.56 − 3.24) = √8.32 ≈ 2.88 meters.
4. Since the height of the tunnel at the center of the truck's width is approximately 2.88 meters, which is
less than the height of the truck (assumed to be 3 meters for the sake of this problem), the truck cannot
pass through the tunnel.
However, the problem asks for the height of the tunnel at the center of the truck's width, which is
approximately 2.88 meters. But since the options provided are whole numbers, we need to choose the
closest option, which is 2.8 meters. Therefore, the correct answer is A.
Figure 11: A comparison between three models under the same problem from MathVerse.
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AUTORUBRIC-R1V: RUBRIC-BASED GENERATIVE REWARDS FOR FAITHFUL MULTIMODAL REASONING Mengzhao Jia1, Zhihan Zhang1, Ignacio Cases2, Zheyuan Liu1, Meng Jiang1, Peng Qi2 1 2Uniphore , ABSTRACT Multimodal large language models (MLLMs) have rapidly advanced from perception tasks to complex multi-step reasoning, yet reinforcement learning with verifiable rewards (RLVR) often leads to spurious reasoning since only the final-answer correctness is rewarded. To address this limitation, we propose AutoRubric-R1V, a framework that integrates RLVR with process-level supervision through automatically collected rubric-based generative rewards. Our key innovation lies in a scalable self-aggregation method that distills consistent reasoning checkpoints from successful trajectories, enabling problem-specific rubric construction without human annotation or stronger teacher models. By jointly leveraging rubric-based and outcome rewards, AutoRubric-R1V achieves state-of-the-art performance on six multimodal reasoning benchmarks and substantially improves reasoning faithfulness in dedicated evaluations. 1 INTRODUCTION Multimodal Large Language Models (MLLMs) have rapidly progressed from simple perception tasks such as visual question answering and image captioning to complex multi-step reasoning tasks (Yao et al., 2024; Liu et al., 2025b; Peng et al., 2025). Such complex reasoning tasks, like geometry math problems, usually require models to derive a step-by-step reasoning trajectory before reaching the final answer. Reinforcement learning with verifiable rewards (RLVR), which assigns training rewards only according to the correctness of the final answer, is a popular method in optimizing MLLMs on reasoning tasks due to its simplicity and efficiency (Meng et al., 2025; Liu et al., 2025a; Xu et al., 2025). All intermediate reasoning steps will be rewarded as long as they yield the correct final answer. Unfortunately, it is prevalent for the model to learn spurious reasoning under such a rewarding paradigm: models may exploit shortcuts or generate contradictory intermediate steps that still land on the right output, effectively "hacking" the training objective. As illustrated in Figure 1, two distinct trajectories can both reach the correct answer, but one does so by introducing flawed logic and abruptly altering results, while the other follows a coherent, step-by-step derivation. Since both receive identical rewards, the system is not encouraged to learn the correct reasoning strategy, which undermines its generalization to unseen problems and reduces its reliability. Such a problem highlights the necessity of process-level supervision beyond final-answer rewards for MLLMs to learn reliable reasoning behavior. To incorporate process-level supervision into reasoning training, a common approach is to leverage pre-trained progress reward models (PRMs), which score intermediate reasoning steps based on their correctness (Wang et al., 2025b; Luo et al., 2025). While PRMs provide fine-grained supervision, they are often vulnerable to distribution shifts, which can lead to unreliable reward estimates when applied to problems from different domains or reasoning steps generated by unseen policy models (Pikus et al., 2023). Recently, rubric-based generative rewards have emerged as a popular alternative of PRMs in instruction-following tasks. This paradigm defines a set of rubrics that specify whether a response adheres to the instruction, and then employs a language model (judge model) to evaluate the response against these rubrics. Compared to traditional reward models, rubric-based approaches offer more robust and interpretable reward signals (Viswanathan et al., 2025; Huang et al., 2025b). However, while rubrics in instruction-following tasks can often be directly derived from the input instruction, extending this paradigm to multimodal reasoning tasks is non-trivial, as the ground-truth 16 Oct 2025 In the illustration, a telescopic door is represented by a rhombus, and each small rhombus has a side length denoted as 'a' units (a = 0.4). The fully extended width of the door is expressed as 'b' units (b = 8.0). Find the value of 'n' in the diagram if one of the internal angles of the rhombus measures 60°. Multimodal Question Correct Answer ✅ To solve this problem, we need to understand the structure of the telescopic door represented by the rhombuses. The door is fully extended to 8.0 units and each rhombus side is 0.4 units. ... The number of rhombus sides that make up the 8.0 units can be calculated as follows: [Number of rhombus sides = \ {0.4} = 20] However, each rhombus has two sides, but since they are counted twice in the total number of sides (once for each side of the rhombus in the structure), we need to divide by 2 to get the number of rhombuses: [Number of rhombuses = \ {2} = 10] ... But we need to consider that the structure starts from the center point, so we should add 1 to the result to include the starting point: [n = 10 + 1 = 11] However, upon revising the structure, the correct number of rhombuses should be 20. Therefore, the value of 'n' is 20. Incorrect Step ❌ Incorrect Step ❌ Reasoning 1 Unclear Correction ⚠ Incorrect Step ❌ To solve this problem, we need to understand the geometry of the telescopic door. Given:- The side length of each rhombus, (a = 0.4) units. - The fully extended width of the door, (b = 8.0) units. - One of the internal angles of the rhombus is 60°, which means the rhombus is actually an equilateral triangle in the context of the door's structure. Since the rhombus is equilateral, the length of the base of each rhombus is equal to the side length, (a = 0.4) units. The fully extended width of the door is the sum of the base of the rhombus. ... If we let (n) be the number of rhombi, then the equation for the fully extended width of the door is:[b = n \ a] ... Solving for (n): [n = \ {0.4} = 20] Therefore, the value of ( n ) is 20. The correct answer is 20. Correct Step ✅ Correct Answer ✅ Reasoning 2 Correct Step ✅ Correct Step ✅ Correct Step ✅ Figure 1: Illustration of a multimodal reasoning question together with two model-generated reasoning traces that both reach the correct answer. Reasoning 1 contains logical flaws-such as incorrectly halving rhombus sides and inconsistently switching from 11 to 20 without reconciliation-while Reasoning 2 proceeds with fully consistent step-by-step logic. In the figure, we mark erroneous reasoning steps in red and correct ones in green, with ambiguous corrections highlighted in the yellow box. Despite these differences, both traces would receive the same reward under RLVR training, reflecting how reward signals based solely on final correctness can overlook reasoning quality. reasoning trajectory is usually unknown. Consequently, designing reliable rubrics and effectively integrating them into RLVR for multimodal reasoning remains an open challenge. Inspired by the robustness of generative rewards as fine-grained supervision signals, we propose a framework for automatically collecting rubrics and effectively incorporating generative rewards into multimodal reasoning RLVR. Instead of relying on costly human annotation or stronger teacher MLLMs, our approach gathers problem-specific rubrics that represents key reasoning checkpoints through a scalable self-aggregation process. Concretely, we distill consistent reasoning steps from the model's own successful trajectories. By combining rubric-based rewards with conventional outcome rewards in RLVR, our method promotes more faithful and accurate multimodal reasoning. With this framework, we train a model named AutoRubric-R1V, which demonstrates superior performance as well as faithfulness. Across 6 multimodal reasoning benchmarks, our model attains state-of-the-art results. In a dedicated evaluation of reasoning faithfulness, our method produces substantially more faithful reasoning than existing approaches. Ablation studies further highlight the necessity of problem-specific rubrics compared to general judging criteria. Moreover, detailed analysis of the training dynamics shows that our framework effectively stabilizes training. To facilitate further research, we will release the constructed rubric dataset and code1. 2 RELATED WORK: REINFORCEMENT LEARNING IN MLLM REASONING. Multimodal large language models (MLLMs) have rapidly advanced by integrating visual encoders with large language models for cross-modal understanding and reasoning. Early studies relied on multimodal supervised finetuning to align pre-trained Large Language Models with visual inputs, as in InstructBLIP (Dai et al., 2023) and LLaVA (Liu et al., 2023), which employed large-scale multimodal 1https://github.com/Jill0001/AutoRubric-R1V. Vanilla GRPO Policy Model Answer Verifier Rollouts Scores 0 1 1 1 1 Advantages Rubric-based Reasoning Judge Compare & Compose Answers Filtering Answer g b e ❌ Answer h b f ✅ e f ✅ a Answer Answer a d b f ✅ Answer a b d f ✅ Reasoning Trajectories 1. The reasoning should include a 2. From , It must get and a b d 3. The correct answer should be calculated by f Rubrics Rubrics Trajectory Trajectory Trajectory Trajectory Trajectory Match:3/3 Match:0/3 Match:2/3 Match:1/3 Match:3/3 Multimodal Question 0.7 0.3 1.0 1.0 0.0 w/ Rubric Judge Rubrics Construction Training Process Figure 2: Our framework extends GRPO with rubric-based reasoning rewards. Top: Rubrics Construction. We automatically generate rubrics by comparing multiple reasoning trajectories of the same question that reach the correct answer. Common steps across multiple correct trajectories are preserved as necessary reasoning, while rare or inconsistent steps are discarded as likely errors or irrelevant. This yields a set of high-quality rubrics without relying on stronger proprietary MLLMs. Bottom: Training Process. In vanilla GRPO, answer-based rewards assign identical scores to all trajectories with the same final answer, failing to capture differences in reasoning quality. By integrating rubric-based scoring, trajectories are further distinguished at the step level, and the combined rubric and answer rewards produce differentiated advantages that better guide the policy model toward logically consistent reasoning paths. instruction datasets. Recently, reinforcement learning with verifiable rewards (RLVR) (Shao et al., 2024) has become a key approach for improving complex multimodal reasoning. These methods use rule-based verifiers of final answers to guide policy model optimization. Existing work mainly falls into two directions: (1) enhancing the base model's reasoning ability before RL. For instance, Vision-R1 (Huang et al., 2025a) and method proposed by Wei et al. (2025) leverage teacher LLMs and MLLMs respectively, to construct an high-quality multimodal chain-of-thought reasoning data to strengthen model capacity before RL training; and (2) enriching reward signals beyond final answers correctness. For example, R1-VL (Zhang et al., 2025) and Vision-SR1 (Xiao et al., 2025) supervise the reasoning trajectories with annotated key steps or visual perception annotation, respectively. Similarly, SRPO (Wan et al., 2025) designs a reflection reward to explicitly encourage self-reflective behavior. While some studies explore diverse reward signals, they often overemphasize isolated dimensions, such as visual perception (Xiao et al., 2025; Li et al., 2025) or specific reasoning behaviors (Ding & Zhang, 2025), while neglecting the inherently holistic process of multimodal reasoning. Moreover, many methods rely heavily on proprietary MLLMs annotations, which are limited by proprietary MLLMs capability. In contrast, AutoRubric-R1V overcomes these issues by deriving problem-specific rubrics without relying on proprietary MLLMs annotations. Our method employs a compare-and-compose aggregation over multiple successful reasoning trajectories, filtering out spurious steps while retaining consistent and essential ones as rubrics, providing process-level supervision that enable reliable reward assignment and discouraging shortcut solutions. 3 METHOD Our method aims to enhance the reasoning capabilities of vision-language models through a reinforcement learning framework. It can be integrated with various policy optimization frameworks as a complement to RLVR. In this work, we employ our approach with GRPO as a representative example. In this section, we begin by introducing the multimodal reasoning task setup, followed by the introduction of key concepts in GRPO. The subsequent subsections provide detailed descriptions of our method. 3.1 PRELIMINARY Problem Formulation In this work, we focus on using MLLMs for solving multimodal reasoning task. Assume we are given a collection of N multimodal reasoning instances denoted as D = {xi}N i=1. Each instance xi ∈D contains a visual input Vi, a textual query Qi, and its labeled answer ai. Our objective is to train a policy model that learns a function F : (Vi, Qi) 7→ai. To improve reasoning performance, the model is instructed to generate a token sequence that contains intermediate reasoning steps with the final answer: si,t ∼πθ( · | Vi, Qi, si,<t ) , t = 1, . . . , Ti, where the trace si,1:Ti jointly encodes the reasoning process and ends with the final answer ˆai. Group Relative Policy Optimization. To optimize the policy model, we adopt Group Relative Policy Optimization (GRPO), a variant of PPO that eliminates the need for a separately trained value function. Instead of estimating returns through a critic model, GRPO exploits relative comparisons among multiple sampled responses for the same query, leading to a lightweight and stable training procedure. Concretely, given a query q and a set of G responses {oi}G i=1 sampled from the old policy πθold, the current policy πθ is updated by maximizing: JGRPO(θ) = Eq,{oi}∼πθold " 1 G G X i=1 1 |oi| |oi| X t=1 min ρi,t(θ) ˆAi, clip(ρi,t(θ), 1 -ε, 1 + ε) ˆAi -βDKL(πθ ∥πref) # , (1) where the importance ratio is defined as ρi,t(θ) = πθ(oi,t | q, oi,<t) πθold(oi,t | q, oi,<t). (2) Here, ε is the clipping hyperparameter, β controls the KL regularization strength, and πref denotes a fixed reference policy. Each response oi receives a scalar reward ri. The advantage ˆAi is then obtained through group-wise normalization of these rewards: ˆAi = ri -mean({rj}G j=1) std({rj}G j=1) . (3) 3.2 INTEGRATING LLM-AS-A-JUDGE INTO RLVR While GRPO optimizes the policy model solely based on answer correctness, this signal is often sparse and insufficient to capture the quality of intermediate reasoning. To provide a richer supervision signal, we incorporate an additional rubric-based reasoning reward derived from a language model acting as a trajectory judge inspired. Rubric-guided evaluation. A straightforward approach is to ask the judge model to provide a single holistic score for each trajectory. Yet such scores are prone to bias and lack sufficient granularity: it is unclear whether identical scores truly reflect comparable reasoning quality across different samples. This ambiguity weakens the reliability of the reward signal for reasoning trajectories. To mitigate these issues, we guide the reasoning reward process with problem-specific rubrics Cx = {c1, . . . , cm}. Each rubric item cj specifies a key reasoning checkpoint that is expected to appear in a logically sound trajectory. Given a candidate trajectory τ, the judge model verifies whether τ satisfies each checkpoint. Notably, since the rubric explicitly specifies the expected reasoning requirements, the judge model only needs to employ its language reasoning ability to compare the trajectory against these checkpoints, without having to reprocess or interpret the visual input even for multimodal problems. This substantially reduces the complexity and computational overhead of the judging step. Let 1[τ ⊨cj] denote an indicator function that equals 1 if cj is satisfied, and 0 otherwise. The rubric-based reasoning reward is then computed as the fraction of satisfied checkpoints: rrubric i = 1 |Cx| |Cx| X j=1 1[τ ⊨cj]. (4) Combining outcome and rubric-based rewards. The rubric-based reward rrubric is integrated with the conventional outcome reward rans that indicates whether the final prediction ˆa matches the ground truth with a weighted combination: ri = λrans i + (1 -λ)rrubric i , (5) where λ ∈[0, 1] controls the impact of the rubric-based reward. During policy optimization, the combined reward ri is assigned to each sampled trajectory, and the normalized group-relative advantages are computed following the GRPO framework. In this way, the policy is encouraged not only to arrive at correct answers but also to align its intermediate reasoning with the rubric-derived process supervision, leading to more faithful and robust reasoning behaviors. 3.3 AGGREGATION-BASED RUBRIC GENERATION Existing approaches to acquire process supervision signals often resort to compare with manually annotated or stronger proprietary MLLMs' reasoning trajectories. Manual annotation is prohibitively expensive. Reliance on proprietary models, however, is intrinsically upper-bounded by the models' capability ceilings and further hampered by error propagation. Moreover, even when a reasoning trajectory yields the correct final answer, it often contains erroneous or unnecessary intermediate steps, limiting the accuracy of directly extracting key steps from a single correct trajectory. To mitigate this issue, we take inspiration from the idea of test-time scaling (Wang et al., 2023; Brown et al., 2024), which suggests that increasing inference computation, e.g., sampling multiple reasoning attempts, increases the likelihood that the majority will converge to a correct solution. Analogously, we propose to aggregate step-level consistency across the model's own successful trajectories. The key intuition is that if a particular step consistently appears in many correct trajectories, it is likely to represent a causally essential component of the reasoning process; in contrast, steps that appear only sporadically are more likely to be spurious or unnecessary. Figure 2 demonstrate this process: 4 reasoning trajectories reach the correct answer, but their intermediate steps are not identical. By comparing steps, we can see some steps consistently recur across multiple correct trajectories (e.g., Reasoning from step a to derive b, d, and calculating final answer with step f). These steps are therefore summarized as rubrics, while infrequent steps, such as step e, are regarded as unnecessary and thus filtered out. Given a multimodal reasoning problem x, we first sample K reasoning trajectories {τ (k)}K k=1 from the current policy. Among them, we retain the subset S ⊆{τ (k)} whose final answers match the verifiable ground truth. We then prompt an LLM to compare trajectories in S and summarize their common steps into an ordered set of key checkpoints: Cx = {c1, c2, . . . , cm}, where each ci denotes a reasoning checkpoint distilled from recurring steps across correct trajectories. These checkpoints are organized into Cx, a structured collection of checkpoints that encodes the essential reasoning requirements for derive the correct answer, which further serve as the problemspecific rubrics for the LLM-as-a-Judge reasoning rewarding during training. 4 EXPERIMENTS 4.1 EXPERIMENTAL SETUP Implementation Details. In our experiments, we use Qwen2.5-VL-7B-IT (Bai et al., 2025) as the base model and train it with the verl2 framework. We train the model with ViRL-39K dataset 2https://github.com/volcengine/verl. proposed by Wang et al. (2025a) for 3 epochs with a constant learning rate of 1e-6. We adopt 512 as the rollout batch size and 128 as the global policy update batch size. We set the rollout number to 8 with a sampling temperature of 1.0. For rubric-based reasoning rewards, we employ an open-sourced LLM as the judge model3. The KL coefficient is fixed at 0.01. All experiments are run on a single node equipped with 8 H100 GPUs. The full set of prompts used in rubric generating, rubric judge scoring, and reasoning generating in training, is provided in the Appendix. Evaluation Benchmarks. We evaluate model performance along two dimensions. For general multimodal reasoning, we adopt MMMU (Yue et al., 2024) and MMMU-Pro (Yue et al., 2025), which cover diverse subjects on multimodal reasoning. For multimodal mathematical reasoning, we include four challenging benchmarks: MathVista (Lu et al., 2024), MathVerse (Zhang et al., 2024), MATH-Vision (Wang et al., 2024), and WeMATH (Qiao et al., 2025), each designed to test different aspects of multimodal mathematical problem-solving skills. Baseline Methods. We compare our model with several strong MLLMs, including: (1) Proprietary models: GPT-4o (Hurst et al., 2024), Claude-3.5-Sonnet (Anthropic, 2024); (2) Open-source generalpurpose models: Qwen2.5-VL-7B-IT, Qwen2.5-VL-72B-IT, InternVL2.5-8B, InternVL2.5-78B; (3) Open-source reasoning-focused models: MM-Eureka-7B (Meng et al., 2025), NoisyRollout-7B (Liu et al., 2025a), Perception-R1-7B (Xiao et al., 2025), R1-VL-7B (Zhang et al., 2025), ThinkLiteVL-7B (Wang et al., 2025c), VL-Rethinker-7B Wang et al. (2025a), VL-Reasoner-7B Wang et al. (2025a). 4.2 RUBRICS STATISTICS Table 1: Summary statistics of the rubric sets of the training samples. Overview # Training Samples 38,870 # Rubric sets 26,144 Coverage 67.26% Avg. / Total words 80.65 / 2,107,756 Rubric Criteria Statistics Avg. criterion 3.47 Avg. / Max words 23.25 / 198 Construction. To construct rubrics, we begin by collecting multiple reasoning trajectories from the model on each training sample and retain only those trajectories that yield correct final answers. If no correct trajectory is found for a sample, no rubric will be generated. To increase the proportion of samples with rubrics, we first train Qwen-2.5-VL-7B-IT model for one epoch using standard GRPO on the training set, and then use this model to generate 8 reasoning trajectories. For each problem with more than 3 correct trajectories, we feed the corresponding correct ones into an open-source text-only LLM4. The LLM is prompted to compare these trajectories, extract their shared steps, and based on them compose a structured set of rubric criteria. The detailed prompt can be found in Appendix. Statistics. Across the training data, this procedure yields 26,144 rubric sets, corresponding to a coverage rate of 67.3%. Each rubric set typically contains multiple criteria and provides sufficiently detailed descriptions, offering substantive, process-level supervision signals for subsequent training. More fine-grained statistics are summarized in Table 1. Comparison with R1-VL. To better contextualize our results, we compare against the key steps introduced in R1-VL (Zhang et al., 2025). R1-VL reports an average of 7.22 criteria per set, but these steps are extremely short, with only 2.92 words per criterion on average and about 21 words per set in total. Such brevity often reduces criteria to keyword-like fragments, which lack the capacity to encode meaningful evaluative guidance and risk collapsing into superficial keyword matching. In contrast, our procedure yields rubric sets with a typical total length of 80.65 words and around 23 words per criterion. Each criterion is thus substantially richer and more informative, serving as an efficient and interpretable scoring standard rather than a loose collection of keywords. The detailed evaluative rubrics enable more substantive and reliable rewards for reasoning trajectories, providing process-level supervision signals that are essential for complex task training. Compared with R1-VL's 3https://huggingface.co/openai/gpt-oss-20b. 4https://huggingface.co/openai/gpt-oss-120b. Table 2: Performance comparison of various MLLMs on multimodal reasoning benchmarks. The best results among open reasoning models are highlighted in bold, while the second-best are underlined. Models Avg. MathVerse MathVision MathVista Wemath MMMU MMMU Pro Proprietary Vision-Language Models GPT-4o 56.29 50.20 33.95 63.80 68.80 69.10 51.90 Claude-3.5-Sonnet 60.86 57.64 46.48 68.20 73.05 68.30 51.50 Open General Vision-Language Models Qwen2.5-VL-72B 55.57 57.60 38.10 74.20 49.10 68.20 46.20 Qwen2.5-VL-7B 47.29 46.95 24.57 68.50 57.01 50.67 36.01 InternVL2.5-78B - 51.70 34.90 72.30 - 61.80 48.60 InternVL2.5-8B - 39.50 19.70 64.40 - 56.00 34.30 Llava-OV-7B - 26.20 - 63.20 - 48.80 24.10 Open Reasoning Vision-Language Models MM-Eureka-7B 50.50 50.18 27.47 71.80 64.31 52.78 36.47 NoisyRollout-7B 52.41 52.54 28.29 73.00 66.09 56.11 38.44 Perception-R1-7B 50.95 49.47 26.84 72.00 65.34 52.89 39.13 R1-VL-7B 42.35 40.00 24.70 63.50 51.61 46.56 27.75 ThinkLite-VL-7B 50.93 49.64 24.54 73.30 65.00 53.67 39.42 VL-Rethinker-7B 54.06 53.60 31.12 73.90 69.20 57.11 39.42 VL-Reasoner-7B 53.41 53.55 29.87 74.80 67.07 56.22 38.96 AutoRubric-R1V 54.81 52.41 31.35 75.90 71.09 57.56 40.52 ∆over Qwen-2.5-VL-7B 7.52 5.46 6.78 7.40 14.08 6.89 4.51 overly concise key steps, our formulation preserves both semantic depth and evaluative precision, ensuring that training is guided by meaningful rubrics rather than underspecified fragments. For concreteness, we provide an illustrative side-by-side example in Appendix C. 4.3 EXPERIMENTAL RESULTS Accuracy Evaluation. We present the performance comparison between AutoRubric-R1V and existing state-of-the-art MLLMs across multiple benchmarks in Table 2. Compared to the base model (Qwen-2.5-VL-7B), AutoRubric-R1V achieves an absolute improvement of 7.52% across 6 benchmarks. Moreover, it's average 54.81 performance is comparable to much larger MLLMs, such as Qwen-2.5-VL-72B (55.57). This result provides strong evidence for the effectiveness of intergrating problem-specific rubric score into RLVR training. Faithfulness Evaluation. Besides accuracy, faithfulness of reasoning constitutes a crucial dimension for evaluating model capabilities (Lanham et al., 2023; Turpin et al., 2023). Following prior work (Kim et al., 2025), we employ a strong LLM (GPT-4o (Hurst et al., 2024)) together with a carefully designed judging prompt to assess whether model reasoning trajectories exhibit unfaithfulness. We randomly sample 1,000 problems from the MathVerse benchmark (Zhang et al., 2024) and apply this evaluation protocol. The judging prompt highlights mismatches where the step-by-step computations or logical derivations lead to one conclusion, but the final reported answer states something different. Table 3: Reasoning inconsistency rates (%, lower is better) Model Total Qwen-2.5-VL-7B 14.7 VL-Rethinker 15.5 Vanilla GRPO 21.8 AutoRubric-R1V 12.6 As shown in Table 3, our evaluation reveals that base models already suffer from this type of unfaithfulness. For example, in multiplechoice settings, we observe a tendency for models to alter their derived answers to force alignment with one of the provided options, even when the computed value is not among them. This issue becomes even more pronounced after RLVR-style training. In particular, we find that models such as VL-Rethinker, which explicitly enforce self-correction in their reinforcement training, may unintentionally encourage patterns where the model adjusts only the final answer while leaving the reasoning trajectory inconsistent or invalid. Detailed examples of reasoning unfaithfulness are shown in Appendix E. In contrast, AutoRubric-R1V model achieves the lowest inconsistency rate. This improvement stems from the introduction of rubric-based fine-grained rewards: beyond rewarding Table 4: Ablation study results of our method compared with vanilla GRPO and a variant. Models Avg. MathVerse MathVision MathVista Wemath MMMU MMMU Pro Vanilla RLVR 52.96 51.12 28.78 74.3 70.46 54.11 39.02 w/o Rubric 52.56 49.80 29.01 74.1 69.25 53.56 39.65 AutoRubric-R1V 54.81 52.41 31.35 75.9 71.09 57.56 40.52 MathVista Figure 3: First row: Comparison of AutoRubric-R1V and vanilla GRPO in terms of training dynamics (answer reward, rubric-based reasoning reward, and response length of rollouts). Second row: Evaluation accuracy of different training steps on three multimodal mathematical benchmarks. correct final answers, our design leverages problem-specific rubrics to monitor intermediate reasoning states. This prevents unfaithful reasoning trajectories from being overly rewarded, thereby preserving the reasoning faithfulness. More detailed faithfulness evaluation in Appendix E.2. 4.4 ABLATION STUDY In this section, we conduct ablation studies of the proposed method. We compare our method with two variants: (1) Vanilla RLVR: training the model using only rule-based answer reward, and (2) w/o Rubric: using the same judge model without rubrics. To further examine the effectiveness of our problem-specific rubric design, we remove the constructed problem-specific rubrics and instead rely on a general rubric to guide the judge model in scoring the quality of reasoning trajectories. The detailed judging prompt used in this variant is provided in the Appendix. As shown in Table 4, compared with the Vanilla RLVR variant, our method achieves consistent improvements across all benchmarks and we observe clear gains in average (54.81 vs. 52.96). These results highlight the benefit of incorporating rubrics into the reward design, leading to more robust reasoning ability. When comparing to the w/o Rubric variant, our method again outperforms across all benchmarks. The results demonstrate that rubrics provide more fine-grained guidance for the judge model, yielding stronger performance than relying solely on a rubric-free evaluator. Notably w/o Rubric achieves inferior performance comparable to Vanilla RLVR, indicating that without the guidance of problemspecific rubrics, the judge model struggles to accurately evaluate the quality of reasoning trajectories. This imprecise judging scheme introduces noise into the reward signal, ultimately limiting model performance. By contrast, incorporating problem-specific rubrics allows the judge model to deliver more robust judgment signals, which translates into consistently stronger results. 4.5 TRAINNING DYNAMICS To further understand the impact of rubric-based reasoning reward in training, we report the training dynamics and evaluation results in Figure 3. As shown in the top-left panel, the answer reward of AutoRubric-R1V closely follows that of Vanilla GRPO at the beginning of training, but diverges later as Vanilla GRPO develops pronounced oscillations while AutoRubric-R1V continues to improve Rubric 1: Let AB = a. From BD = 1/4 AB we have BD = a/4, and from BD = 1/3 CD we get CD = 3·BD = 3a/4. Rubric 2: Since D lies on AB, AD = AB - BD = a - a/4 = 3a/4. Rubric 3: Let BC = b. Because B is on AC, AC = AB + BC = a + b. Rubric 4: Points A, B, C are collinear, so the horizontal distance CD equals (a + b) - AD = (a + b) - 3a/4 = b + a/4. Setting this equal to CD = 3a/4 gives b + a/4 = 3a/4, b = a/2. Rubric 5: E is the midpoint of AB, so AE = EB = a/2. Hence EC = C - E = (a + b) - a/2 = a/2 + b. Substituting b = a/2 yields EC = a. Rubric 6: Given EC = 16 cm, we obtain a = 16 cm. Then AC = a + b = a + a/2 = 3a/2 = 24 cm. Given that BD = (1/4)AB = (1/3)CD, we need to find the length of line segment AC. Let's denote the length of AB as x. Therefore, we have: BD = (1/4)x Since D is a point on AB, let AD = y. Then: AB = AD + DB → x = y + (1/4)x → x = (5/4)y → y = (4/5)x Next, we know that BD = (1/3)CD. Since BD = (1/4)x, we have: CD = 3 × BD = 3 × (1/4)x = (3/4)x Since C is on AC, we have: AC = AB + BC → AC = x + CD → AC = x + (3/4)x = (7/4)x We also know that E is the midpoint of AB, so AE = EB = x/2. Given that EC = 16 cm, we can write: EC = EB + BC → 16 = (x/2) + BC → BC = 16 - (x/2) Since BC = AC - AB = (7/4)x - x = (3/4)x, we have: 16 = (x/2) + (3/4)x → 16 = (2x + 3x)/4 → 16 = (5x/4) → x = 64/5 Therefore, the length of AC is: AC = (7/4)x = (7/4) × (64/5) = (7×64)/(4×5) = (7×16)/5 = 112/5 = 22.4 However, we should recheck the calculation with the midpoint and segment lengths clearly. The correct calculation is: AC = (7/4) × (64/5) = 112/5 = 22.4 Thus, the length of line segment AC is: AC = 24 As shown in Figure 2, it is known that point B is on line segment AC, and point D is on line segment AB. If BD = (1/4)AB = (1/3)CD, with E being the midpoint of line segment AB and EC = 16 cm, find the length of line segment AC. Rubrics Let's start by analyzing the given information and setting up the equations step by step. We know that BD = (1/4)·AB and BD = (1/3)·CD. Let's denote AB = 4x and CD = 3x. Therefore, BD = x. Since D is on AB, we can express AD as AD = AB - BD = 4x - x = 3x. We also know that E is the midpoint of AB, so AE = EB = AB/2 = 4x/2 = 2x. The total length of AC is AB + BC. We need to find BC. Since D is on AB and BD = x, AD = 3x. Therefore, AB = AD + BD = 3x + x = 4x. Now, we need to find BC. Since BD = x and BD = (1/3)·CD, we have CD = 3x. Therefore, BC = CD - BD = 3x - x = 2x. The length of EC is the sum of EB and BC. We know EB = 2x and BC = 2x, so EC = EB + BC = 2x + 2x = 4x. We are given that EC = 16 cm, so 4x = 16. Therefore, x = 4. The length of AC is AB + BC = 4x + 2x = 6x. Since x = 4, we have AC = 6·4 = 24 cm. The length of line segment AC is 24 cm. Rubric 1: The reasoning explicitly states BD = x/4 and that CD = 3·BD = 3x/4, which matches the required expressions BD = a/4 and CD = 3a/4. Satisfied Rubric 2: The requirement is AD = AB - BD = 3a/4. The reasoning introduced AD = y and solved x = y + x/4 to obtain y = (4/5)x, incorrectly giving AD = (4/5)x. Not satisfied Rubric 3: The required approach is to let BC = b and express AC = AB + BC = a + b. The reasoning never introduces a variable b; it instead writes AC = x + CD, conflating BC with CD. Not satisfied Rubric 4: The reasoning does not perform the horizontal-distance calculation (a + b) - AD = b + a/4 set equal to CD = 3a/4 to solve b = a/2. Not satisfied Rubric 5: The correct relation derived earlier is EC = a. The reasoning instead finds EC = x/2 + BC with BC = 3x/4, leading to a different value EC = (5x/4), not equal to a. Not satisfied Rubric 6: With EC = 16 the intended solution gives a = 16 and consequently AC = (3a/2) = 24. The reasoning attempted to use EC = 16 but obtained x = 64/5 and AC = 22.4, contradicting the rubric's required result. Not satisfied Final Rubric Score: 0.1667 Rubric 1: The reasoning establishes AB = 4x, BD = x, CD = 3x. This is equivalent to letting a = AB and writing BD = a/4 and CD = 3a/4. Satisfied. Rubric 2: The solution computes AD = AB - BD = 4x - x = 3x, which equals 3/4 AB. Satisfied. Rubric 3: The process explicitly states "The total length of AC is AB + BC" and subsequently seeks the value of BC. Satisfied. Rubric 4: The reasoning uses the fact that BC = CD - BD (since C, B, D are collinear with B between D and C), leading to BC = 2x. This is mathematically equivalent to the rubric's derivation CD = AC - AD and the resulting equation b = a/2. Satisfied. Rubric 5: The solution notes AE = EB = 2x and then writes EC = EB + BC = 2x + 2x = 4x, which equals AB. This matches the rubric's expression EC = a/2 + b with b = a/2. Satisfied. Rubric 6: With EC = 16 cm, the reasoning finds x = 4, AB = 16 cm, and AC = 6x = 24 cm, exactly as the rubric's final steps. Satisfied. Final Rubric Score: 1.0 Reasoning 2 Rubric-based Reasoning Reward Rubric-based Reasoning Reward Reasoning 1 Multimodal Question Figure 4: Illustration of a problem with the constructed rubrics, two reasoning trajectories produced by the policy model, and the rubric-based reasoning rewards assigned by the judge model. smoothly. This instability in Vanilla GRPO likely reflects reward hacking: once the model overfits to some training samples due to the correctness-only signals, the reward no longer provides reliable supervision, leading to unstable optimization. In contrast, the rubric-based reasoning reward feedback in AutoRubric-R1V supplies richer guidance that prevents such collapse. The top-middle panel further supports this interpretation, showing that the rubric reward under AutoRubric-R1V steadily increases, indicating that the model is consistently guided by rubric-based objectives rather than exploiting narrow correctness signals. Meanwhile, the top-right panel reveals that AutoRubric-R1V generally produces longer responses compared to Vanilla GRPO. One way to explain it is that when facing answer revisions, short rollouts that merely alter the final answer are not rewarded, whereas generating detailed reasoning steps is positively reinforced. The bottom row reports evaluation on three out-of-domain mathematical benchmarks. we observe that Vanilla GRPO either stagnates or degrades during later training, while AutoRubric-R1V achieves continuous improvements. Together, these results demonstrate that rubric-based trajectory supervision not only stabilizes optimization but also improves generalization across multiple reasoning benchmarks. 4.6 CASE STUDY To clearly demonstrate the effectiveness of our method in rewarding the reasoning trajectories during training, we present a concrete case study. As illustrated in Figure 4, we illustrate a problem, and the constructed set of rubrics for it by AutoRubric-R1V. We also shown two different reasoning trajectories produced by the policy model during training, as well as the rubric-based reasoning rewards generated by the judge model. From the figure we can see both trajectories reach the same and correct final answer. However, the rubric-based evaluation shows that one trajectory contains clear logical mistakes (e.g., define AB = x and write AC = x + CD, conflating BC with CD.), while the other does not. This highlights the key advantage of rubric-based rewards: they distinguish between superficially correct final answer and genuinely sound reasoning processes, and thus provide a more faithful reward signal. Another notable observation is that the rubrics use one set of symbolic definitions (e.g., line segment lengths denoted as a and b), while the trajectories use a different definition system (e.g., x). Despite these discrepancies, the judge model aligns the semantics and provides accurate assessments. This ability comes from the LLM's strong semantic understanding, which goes beyond surface-level pattern matching (such as keyword-based checks in R1-VL (Zhang et al., 2025)). 5 CONCLUSION In this work, we present AutoRubric-R1V, a reinforcement learning framework that integrates problem-specific rubrics into RLVR through LLM-as-a-judge reasoning trajectory rewarding for process-level supervision. This design avoids reliance on costly annotation or stronger teacher MLLMs by deriving rubrics automatically from aggregating consistent reasoning steps across successful trajectories. Experiments on six multimodal reasoning benchmarks demonstrate state-of-the-art performance with an average gain of 7.52% over the base model. Further analyses confirm that AutoRubric-R1V delivers consistent improvements in both accuracy and reasoning faithfulness. These findings underscore the promise of rubric-based process supervision for building more reliable and generalizable multimodal reasoning systems. REPRODUCIBILITY STATEMENT We make the following effort to ensure the reproducibility of our work. The training code and evaluation scripts will be released in the anonymous link, allowing others to replicate our experiments. To facilitate consistent reproduction of results, we fixed random seeds across all training and evaluation runs. 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In Computer Vision - ECCV 2024 - 18th European Conference, Milan, Italy, September 29-October 4, 2024, Proceedings, Part VIII, volume 15066 of Lecture Notes in Computer Science, pp. 169-186. Springer, 2024. URL https://doi.org/10.1007/978-3-031-73242-3_10. Prompt for Constructing Rubrics from Trajectories You are given a math question about an image and several reasoning processes. All reasoning processes arrive at the correct final answer. However, some processes may contain calculation errors in intermediate steps, inconsistent reasoning, or unnecessary steps. CRITICAL TASK: Carefully compare multiple reasoning processes to identify which steps are ACTUALLY CORRECT: VERIFY CALCULATIONS: For each mathematical calculation in the reasoning processes: Check if the arithmetic is correct (e.g., 3×4 should equal 12, not 13) 1. Verify algebraic manipulations are valid Ensure geometric formulas are applied correctly If different processes have different intermediate calculations for the same step, identify which one is mathematically correct 2. CROSS-VALIDATE: When multiple processes perform the same calculation: If they get different intermediate results, determine which is correct Do NOT include incorrect calculations in the rubric, even if they appear in multiple processes Only include calculations that are mathematically verified to be correct 3. EXTRACT ESSENTIAL CHECKPOINTS: After verifying correctness, identify the key rubric that: Are mathematically correct and consistent Are necessary to reach the correct answer Include specific correct numbers, equations, and calculations Represent the logically sound path to the solution IMPORTANT: The rubric must ONLY contain mathematically correct rubrics. If a calculation appears in multiple processes but is incorrect, DO NOT include it in the rubric. If the processes contain too many errors or contradictions to extract reliable rubrics, output an empty dictionary. Please output the rubrics in JSON format as follows: { "Rubric 1": "[content of the first necessary and CORRECT rubric]", "Rubric 2": "[content of the second necessary and CORRECT rubric]", ... } Question: (question) Correct Answer: (answer) Reasoning Processes to Analyze: (processes_text) Return only the JSON with mathematically correct checkpoints. Figure 5: The prompt for rubric construction. A EVALUATION PROTOCOL The benchmarks used in our evaluation consists of two types of questions: multiple-choice questions and open-ended questions. For multiple-choice questions, we extract the predicted option letter (A/B/C/D, etc.) using regular expressions. The extracted option is then directly compared against the ground-truth label. As to open-ended questions, These include fill-in-the-blank style problems, where the expected answer is a short text span (e.g., a number, a word, or a short phrase). Since exact string matching may fail to capture semantically correct but differently phrased answers, we use Qwen3-30B-A3B-Instruct-25075 as a proxy judge for evaluation. The model is prompted to compare the predicted output with the ground-truth answer and decide whether they match in meaning. During our review of baseline studies, we observed that the reported zero-shot performance of the same model on the same benchmark can vary considerably across works (e.g., the Qwen2.5-VL7B-IT model on MathVerse is reported as 47.9 in MM-EUREKA Meng et al. (2025), but 46.2 in NoisyRollout Liu et al. (2025a)). We attribute such discrepancies primarily to differences in judge models and evaluation frameworks. To ensure fair comparison, we re-evaluated all open-source baseline MLLMs as well as our proposed model under a unified evaluation protocol, using the same evaluation system described above. Notably, we strictly follow the system and instructional prompts provided in the original studies in reproduction, thereby ensuring that the performance comparison tables reflect results obtained under a controlled and standardized setting. 5https://huggingface.co/Qwen/Qwen3-30B-A3B-Instruct-2507 Prompt for Rubric-based LLM-as-a-Judge Reward You are an evaluator. Your job is to check whether the given reasoning process satisfies each requirement in the rubric. # Inputs ## Problem: {problem} ## Proposed Reasoning Process: {solution} ## Evaluation Rubric: {rubric_checkpoints} # Evaluation Procedure For each rubric criterion: Read the rubric criterion carefully and understand what requirement it sets. - Recognize mathematical equivalence. If two formulas are mathematically equivalent, treat them as correct even if they look different. (E.g., "x - y - z" is equivalent to "x - (y + z)") - Be strict about the names of geometric objects. For example, ∠OAB = 90° is NOT the same as ∠OAC = 90°. Search in the reasoning process to judge if the reasoning contains this requirement. explanatory statement may be implied, but any calculation must be explicitly mentioned. If yes, mark Satisfied (1 point). If no, mark Not satisfied (0 points)." Continue until all criteria are covered. Calculate the total score as the fraction of rubrics satisfied. # Output Rules For each rubric, first output a short analyse, then output the final justification in the following format: Rubric X: [the analyse referencing the reasoning and the rubric]. [Satisfied / Not satisfied]. After all rubric criteria, output the final score as the following decimal format: Final Rubric Score: \ } Figure 6: The prompt for using rubrics in LLM-as-A-Judge in training. B PROMPTS for reproducibility, we present all the prompts used in this work, including the prompt for constructing rubrics from trajectories (Figure 5), the prompt for rubric-based LLM-as-A-Judge reward (Figure 6), the prompt for faithfulness evaluation (Figure 7) and reasoning quality evaluation (Figure 7) (Detailed in Section E.2). C COMPARISON BETWEEN RUBRICS AND KEY STEPS IN R1-VL To complement the quantitative comparison in the main text, Figure 9 illustrates two geometry problems side by side. On the left are the key solution steps extracted in R1-VL, while on the right are the detailed rubrics constructed by AutoRubric-R1V. This example makes the contrast more tangible: the key steps only provide a terse sequence of words, which reduces evaluation to keyword matching rather than assessing a coherent reasoning process. In contrast, the rubric formulation offers structured, criterion-based explanations that capture reasoning depth and provide clearer judging criteria. D MORE TRAINING RESULTS D.1 TRAIN WITH OTHER DATASETS To further assess the robustness of our training framework, we additionally train the model on a dataset distinct from the main training corpus. This experiment aims to evaluate whether the proposed method remains effective when applied to tasks with different data distributions and reasoning characteristics. Specifically, we adopt the Geometry3K dataset Lu et al. (2021), a multimodal reasoning benchmark that requires models to solve geometry-related problems. The dataset comprises approximately 2.1K training samples, and we trained the model for 20 epochs, resulting in a total of 90 optimization steps. Table 5 presents the comparison between AutoRubric-R1V and the vanilla GRPO baseline on several mathematical and general reasoning benchmarks. As shown in the table, AutoRubric-R1V consistently outperforms vanilla GRPO, achieving a notably higher average performance (+1.36 points). The improvement is not only pronounced on MathVerse, whose domain closely aligns with Prompt for Reasoning Faithfulness Evaluation You are an expert at analyzing mathematical reasoning. Your task is to check if there is an inconsistency in the reasoning process in the following problem and solution. IMPORTANT: You are ONLY checking consistency in the reasoning process. You are NOT checking whether the answer matches the ground truth. An inconsistency occurs when: 1. The calculations/derivations in the reasoning lead to one result, but a DIFFERENT value is given as the final answer 2. The reasoning explicitly concludes one answer, but a different answer is provided at the end 3. There is a clear mismatch between what was computed/derived and what was stated as the final answer 4. The logical flow leads to answer X but the final answer states Y An inconsistency does NOT occur when: 1. The reasoning leads to value/answer X and the final answer is also X 2. The reasoning and final answer are the same (even if incorrect or not matching multiple choice options) 3. There are minor rounding differences or reasonable approximations 4. The problem is solved correctly step-by-step and the final answer matches the reasoning 5. Even if the reasoning process contains mathematical errors or logical flaws, as long as the final answer matches what the reasoning concludes, it is CONSISTENT Input/Problem: {input_text} Output/Solution: {output_text} Please analyze this step by step: 1. Identify what value/answer/conclusion the reasoning process leads to 2. Identify the final answer (may be in \ , stated as "The answer is...", "Therefore...", or at the end of the solution) 3. Check if the reasoning result and final answer are the SAME (consistent) or DIFFERENT (inconsistent) Provide your response in the following format: ANALYSIS: [Extract and identify: (1) What value/answer the reasoning process arrives at, and (2) What is stated as the final answer] INCONSISTENT: [YES or NO - based ONLY on whether reasoning process matches final answer] Your response MUST end with exactly one of these two lines: CONCLUSION: 0 (if reasoning process is consistent) CONCLUSION: 1 (if reasoning process is inconsistent) Figure 7: The prompt for reasoning faithfulness evaluation. Table 5: Performance comparison with Geometry3K training data. Models Avg. MathVerse MathVision MathVista WeMath MMMU MMMU Pro Vanilla RLVR 49.13 46.45 27.01 71.3 62.07 51.89 36.07 AutoRubric-R1V 50.96 50.79 26.84 71.6 63.68 54.67 38.15 the training data (both focusing on mathematical and geometric reasoning). Moreover, AutoRubricR1V also yields clear gains on MMMU and MMMU-Pro, which are designed for general reasoning beyond pure geometry. This demonstrates that the method not only adapts well to domain-specific tasks but also enhances reasoning ability in broader contexts. D.2 DATA EFFICIENCY We compare AutoRubric-R1V and vanilla RLVR trained with 10% and 100% of the data (Table 6). Both models are trained for 15 epochs, and the subsets are randomly sampled from the full training set. Vanilla RLVR shows little improvement when scaling data from 10% to 100% (52.42 →52.96). One possible explanation is that the training data share highly similar distributions, causing the vanilla model to overfit and show limited generalization with more samples. In contrast, AutoRubric-R1V achieves more stable gains, improving from 53.15 to 54.81. Moreover, with only 10% of data, it already matches or surpasses the full-data RLVR on several benchmarks (e.g., MathVerse, WeMath), demonstrating strong data efficiency and better utilization of limited supervision. You are an expert evaluator of mathematical reasoning and logical thinking. Your task is to assess the quality of the reasoning process in the given solution, focusing on identifying critical flaws that undermine the validity of the reasoning chain. You will evaluate the solution across three key dimensions to determine an overall quality score between 0 and 1, where 0 represents severely flawed reasoning and 1 represents perfectly sound reasoning. The quality assessment should consider the presence and severity of unfounded transitions or answer changes where the solution suddenly shifts to a different approach or answer without proper justification or explanation, appearing to abandon previous work without logical basis. Additionally, you should identify calculation errors including arithmetic mistakes, algebraic manipulation errors, incorrect formula applications, computational errors in numerical operations, and mistakes in unit conversions or dimensional analysis. Furthermore, evaluate the logical connectivity by examining whether each step follows logically from the previous ones, whether assumptions are clearly stated and justified, whether the reasoning maintains internal consistency throughout, and whether conclusions are properly supported by the preceding work. When evaluating, consider that minor notational inconsistencies or formatting issues should not significantly impact the score unless they create genuine ambiguity. Focus on substantive errors that affect the validity of the mathematical reasoning rather than stylistic preferences. A solution may use unconventional but valid approaches which should not be penalized if they are mathematically sound. The severity of errors should be weighted by their impact on reaching the correct conclusion, with critical errors that invalidate the entire solution having more weight than isolated minor mistakes. Remember that the presence of a single severe error such as a fundamental conceptual mistake or complete logical disconnect may warrant a very low score even if other parts are correct, while multiple minor errors that don't affect the final validity might still result in a moderate to high score. Input/Problem: ``` {input_text} ```{choices_text} Output/Solution: ``` {output_text} ``` Analyze this solution carefully, examining each step of the reasoning process for the three types of issues mentioned. Consider how these issues compound or interact with each other. Assess whether the reasoning maintains a clear logical thread from problem statement to conclusion. Evaluate whether calculations are performed correctly and whether any sudden shifts in approach are properly justified. After your thorough analysis, provide a quality score that reflects the overall soundness of the reasoning, where scores closer to 0 indicate severely flawed reasoning with multiple critical errors or logical disconnects, scores around 0.3-0.4 suggest significant issues that undermine the solution's validity, scores around 0.5-0.6 indicate moderate issues with some valid reasoning present, scores around 0.7-0.8 represent generally sound reasoning with minor issues, and scores above 0.8 indicate very strong reasoning with at most trivial errors. Your score should holistically capture the reliability and trustworthiness of the reasoning process as a whole. Provide your response in the following format: ANALYSIS: [Provide a comprehensive analysis of the reasoning, identifying specific instances of unfounded transitions, calculation errors, and logical disconnects. Quote specific parts of the solution that demonstrate these issues.] UNFOUNDED_TRANSITIONS: [Describe any sudden changes in approach or answers without justification] CALCULATION_ERRORS: [List specific computational or mathematical errors found] LOGICAL_CONNECTIVITY: [Assess the overall logical flow and consistency of the reasoning] SEVERITY_ASSESSMENT: [Explain how severe the identified issues are and their impact on the solution's validity] QUALITY_SCORE: [A decimal number between 0 and 1 representing the overall quality of the reasoning] Your response MUST end with exactly one of these lines: CONCLUSION: [score] (where [score] is the quality score as a decimal between 0 and 1) Figure 8: The prompt for reasoning quality evaluation. E UNFAITHFULNESS PHENOMENA IN REASONING E.1 ERROR ANABASIS FOR UNFAITHFULNESS To illustrate the reasoning unfaithfulness issue, we show an example of how different models exhibit varying levels of reasoning faithfulness when solving a multiple-choice geometry problem in Figure 11. The VL-Rethinker model, trained with a force-rethink strategy, shows a phenomenon of "spurious correction": after a complete step-by-step calculation leading to a height of 0.4 meters, the model suddenly overturns its own result with the statement "Wait, let's double check the options. The correct height is 3 meters." without any explanation or justification. This behavior improves answer As shown in the figure, BD is the diameter of circle O, and ∠A = 60°. Find the measure of ∠DBC. Choices: (A) 30° (B) 45° (C) 60° (D) 25° Key Steps: ["BD"], ["diameter"], ["right angle"], ["∠ADC"], ["90°"], ["triangle ABC"], ["∠A"], ["∠B"], ["∠C = 120° - ∠B", "∠C = -∠B + 120°", "∠C = 120° + (-∠B)"], ["∠DBC"], ["∠ BDC"], ["arc BAC"], ["∠BAC = 60°", "m∠BAC = 60°", "∠ BAC = π/3 radians"], ["∠BDC"], ["∠DBC"] Rubric 1: Because BD is tangent at B, OB⊥BD; and since BD ∥ OC, OB ⊥ OC, giving ∠BOC = 90°. Rubric 2: By the inscribed-angle theorem, the central angle subtending arc AC is ∠AOC = 2·∠ ABC = 2·30° = 60°. Rubric 3: With OB = OC = radius = 2 and ∠BOC = 90°, triangle OBC is a right isosceles triangle; we may place O at (0,0), B at (2,0) and C at (0,2). Rubric 4: Using ∠AOC = 60° from the positive y-axis (OC), point A on the circle of radius 2 is A = (2·sin60°, 2·cos60°) = (√3, 1). The line AC then has equation y = 2 - x/√3. Rubric 5: Since BD ∥ OC, BD is the vertical line x = 2; intersecting it with AC gives D(2, 2 - 2/√ 3). The distance CD = √[(2-0)2 + (2 - 2/√3 - 2)2] = √[4 + (2/√3)2] = √(16/3) = 4√3/3. In the figure, in the circumcircle of triangle ABC inscribed in circle O, angle ABC is 30°. The extension of AC intersects the tangent to circle O passing through point B at point D. If the radius of circle O, OC, is 2 and BD is parallel to OC, what is the length of CD? Choices: (A) 1+√3/3 (B) 4√3/3 (C) 2√3/3 (D) 2 Figure 9: A comparison between (left) key steps proposed in R1-VL; and (right) rubrics constructed with AutoRubric-R1V under two similar geometry problems. Table 6: Performance comparison of models trained with 10% and 100% of the training data. Models Avg. MathVerse MathVision MathVista WeMath MMMU MMMU Pro Vanilla RLVR (10%) 52.42 50.74 27.96 74.0 69.37 54.22 38.21 AutoRubric-R1V (10%) 53.15 52.03 28.06 74.5 70.75 54.67 38.90 Vanilla RLVR (100%) 52.96 51.12 28.78 74.3 70.46 54.11 39.02 AutoRubric-R1V (100%) 54.81 52.41 31.35 75.9 71.09 57.56 40.52 160 180 200 220 240 260 280 Training Steps 40 45 50 55 60 65 70 Accuracy / Quality Vanilla RLVR AutoRubric-R1V Accuracy Quality Inconsistency 0 5 10 15 20 25 30 Inconsistency 22 10 24 12 25 9 25 11 23 11 29 11 26 11 Figure 10: Comparison of Vanilla RLVR and AutoRubric-R1V on reasoning accuracy, quality, and inconsistency over training steps. accuracy scores but undermines faithfulness, as the final conclusion is decoupled from the preceding reasoning process. The vanilla GRPO model, by contrast, follows the derivation consistently until it reaches an intermediate result of approximately 2.88 meters, which does not appear in the provided choices. At this point, the model applies a heuristic of selecting the closest available option (2.8 meters), prioritizing test-taking strategies over reasoning consistency. Both of these models, lacking process-level supervision during training, tend to adopt unfaithful reasoning patterns that sacrifice reliability for accuracy. In contrast, the AutoRubric-R1V model demonstrates both correctness and faithfulness: it carefully computes the geometry, substitutes values correctly, and derives the tunnel height as 3 meters without any arbitrary correction. This alignment between intermediate reasoning and final answer reveals the model's superior reliability and accuracy, highlighting the benefits of rubric-based reasoning supervision. E.2 TRAINING DYNAMICS REGARDING REASONING FAITHFULNESS We analyze how reasoning unfaithfulness evolves during training by comparing Vanilla RLVR and AutoRubric-R1V on the MathVerse benchmark. We follow the evaluation protocol described in Section 4.3 to detect the reasoning inconsistency. Specifically, at each checkpoint from 160 to 280 training steps, we sample 100 samples and evaluate the reasoning trajectories generated by the two models with a strong model (i.e., GPT-4o) using the faithfulness prompt shown in Fig. 7. Besides inconsistency, we further introduce a Reasoning Quality Check, which serves as a stricter and more comprehensive evaluation for reasoning trajectories. While inconsistency check focuses on identifying contradictions or unjustified answer shifts within a reasoning process, the quality assessment additionally examines three complementary dimensions: unfounded transitions, calculation errors, and logical connectivity. This broader evaluation captures not only internal contradictions but also the overall soundness and coherence of the reasoning trajectories, reflecting how logically valid and trustworthy a model's reasoning process is. The evaluation setting mirrors that of inconsistency detection-using the same data and evaluator-but replaces the prompt with the one shown in Fig. 8, which outputs a holistic quality score between 0 and 1. The results of accuracy, inconsistency rate, and quality scores are demonstrated in Figure 10. Below are our key observations from this evaluation. Vanilla RLVR shows persistently high and rising inconsistency. The inconsistency rate of Vanilla RLVR remains above 20% and increases with training, suggesting growing instability in its reasoning patterns. In contrast, AutoRubric-R1V keeps inconsistency around 10% throughout, indicating more stable and faithful reasoning. AutoRubric-R1V maintains higher reasoning quality. Except at 180 steps, AutoRubric-R1V consistently outperforms Vanilla RLVR, with quality scores that gradually improve over time. Vanilla RLVR, by comparison, exhibits flat or slightly declining quality, implying that training methods focusing solely on outcome correctness may potentially degrade the model's reasoning quality as the number of training steps increases. Accuracy gains in AutoRubric-R1V do not trade off faithfulness. Although Vanilla RLVR occasionally exhibits higher accuracy in the early training phase, AutoRubric-R1V surpasses it as training progresses. More importantly, AutoRubric-R1V achieves superior accuracy without compromising reasoning quality or consistency. In contrast, the accuracy peak of Vanilla RLVR at step 220 coincides with an inconsistency rate approaching 25%, indicating diminished reliability in its reasoning process. Collectively, these analyses offer compelling evidence that AutoRubric-R1V not only improves accuracy but also maintains reasoning quality and consistency, thereby alleviating the unfaithfulness observed in standard RLVR training. F THE USE OF LLMS In this work, LLMs are used exclusively as writing assistants to improve grammar, clarity, and fluency. It did not contribute to research ideation, experimental design, analysis, or the generation of scientific content. The authors take full responsibility for all content of this paper. VL-Rethinker: To determine if the truck can pass through the tunnel, we need to analyze the geometry of the semicircular cross-section and the dimensions of the truck. The radius of the semicircle is 3.4 meters, and the width of the truck is 3.2 meters. We need to find the height of the truck relative to the semicircle's radius. 1. Identify the radius and the width of the truck: - Radius of the semicircle, r = 3.4 meters. - Width of the truck, w = 3.2 meters. 2. The truck is centered within the semicircle, so the distance from the center of the semicircle to the edge of the truck is half of the truck's width, which is 3.2/2 = 1.6 meters. 3. We need to find the height of the truck from the base of the semicircle. This height can be found using the Pythagorean theorem in the right triangle formed by the radius of the semicircle, the distance from the center to the edge of the truck, and the height of the truck. 4. Let's denote the height of the truck as h. The Pythagorean theorem gives us: r^2 = (r - h)^2 + (w/2)^2. Substituting the known values: 3.4^2 = (3.4 - h)^2 + (3.2/2)^2. Simplify the equation: 3.4^2 = (3.4 - h)^2 + 1.6^2. 11.56 = (3.4 - h)^2 + 2.56. 11.56 - 2.56 = (3.4 - h)^2. 9 = (3.4 - h)^2. 3.4 - h = √9. 3.4 - h = 3 or 3.4 - h = -3. h = 0.4 or h = 6.4. Since h = 6.4 meters is not possible because it exceeds the radius of the semicircle, we have h = 0.4 meters. Therefore, the height of the truck relative to the base of the semicircle is 0.4 meters, which means the truck can pass through the tunnel. Wait, let's double check the options. The correct height is 3 meters. Thus, the correct answer is . As shown in the figure, the cross section of a tunnel is a semicircle with a radius of 3.4, and a truck with a width of 3.2 can pass through the tunnel. Choices: A:3m B:3.4m C:4m D:2.8m AutoRubric-R1V: To determine if the truck can pass through the tunnel, we need to check if the height of the truck's top can fit within the semicircular arch of the tunnel. The tunnel's cross section is a semicircle with a radius of 3.4. The truck's width is 3.2, so we need to find the height of the semicircle at a horizontal distance of 1.6 (half of 3.2) from the center of the semicircle. The equation of the semicircle is y = √(3.42 - x2). We need to find the value of y when x = 1.6. Substitute x = 1.6 into the equation: y = √(3.42 - 1.62). y = √(11.56 - 2.56). y = √9. y = 3. The height of the semicircle at x = 1.6 is 3. Since the truck's height is not specified, we assume the truck's height is 3 meters or less. Therefore, the truck can pass through the tunnel. However, we need to compare this with the given options. The correct option is the height of the semicircle at x = 1.6, which is 3 meters. The correct answer is . Vanilla GRPO: To determine if the truck can pass through the tunnel, we need to compare the height of the tunnel at the center of the truck's width with the height of the tunnel at the edges of the truck's width. 1. The radius of the semicircle is 3.4 meters. 2. The width of the truck is 3.2 meters, so the distance from the center of the semicircle to the edge of the truck is 3.4 - 1.6 = 1.8 meters. 3. We need to find the height of the tunnel at this distance. Using the Pythagorean theorem in the right triangle formed by the radius, the distance from the center to the edge of the truck, and the height of the tunnel at this point, we have: Height = √(3.42 - 1.82) = √(11.56 - 3.24) = √8.32 ≈ 2.88 meters. 4. Since the height of the tunnel at the center of the truck's width is approximately 2.88 meters, which is less than the height of the truck (assumed to be 3 meters for the sake of this problem), the truck cannot pass through the tunnel. However, the problem asks for the height of the tunnel at the center of the truck's width, which is approximately 2.88 meters. But since the options provided are whole numbers, we need to choose the closest option, which is 2.8 meters. Therefore, the correct answer is A. Figure 11: A comparison between three models under the same problem from MathVerse.
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2510.14736
|
Admissible solutions of delay Schwarzian
differential equations
Shi-Jian Wu
Abstract
In this paper, we study delay differential equations involving the
Schwarzian derivative S(f, z), expressed in the form
f(z + 1)f(z −1) + a(z)S(f, z) = R(z, f(z)) = P(z, f(z))
Q(z, f(z))
where a(z) is rational, P(z, f) and Q(z, f) are coprime polynomi-
als in f with rational coefficients. Our main result shows that if a
subnormal transcendental meromorphic solution exists, then the ra-
tional function R(z, f) = P(z, f)/Q(z, f) satisfies degf R ≤7 and
degf P ≤degf Q + 2, where degf R = max{degf P, degf Q}. Further-
more, for any rational root b1 of Q(z, f) in f with multiplicity k, we
show that k ≤2. Finally, a classification of such equations is provided
according to the multiplicity structure of the roots of Q(z, f). Some
examples are given to support these results.
Keyword: delay differential equations; Nevanlinna theory; Schwarzian
derivatives; subnormal solutions
2020MSC: 34M04; 30D35
1
Introduction
An ordinary differential equation is said to possess the Painlev´e property if
its solutions are single-valued about all movable singularities[4, 8]. Recent
interest in this property stems from statistical physics and partial differen-
tial equations; for instance, these equations with Painlev´e property provide
exact solutions for the two-dimensional Ising model [5]. Beyond these appli-
cations, the significance of studying equations with Painlev´e property lies in
their dual role: they serve both as a source for defining new functions and
a class of equations to be integrated with the existing functions available.
This classification originated in the early 20th century through the work of
1
arXiv:2510.14736v1 [math.CV] 16 Oct 2025
Painlev´e[19, 20], Fuchs[6] and Gambier[7], who systematically classified sec-
ond order differential equations with the Painlev´e property. Their analysis
culminated in six canonical forms, now named as the Painlev´e equations.
The analogues of Painlev´e property for complex difference equations have
been discussed. Ablowitz, Halburd and Herbst[1] have advocated that the
existence of sufficiently many finite order meromorphic solutions could be
considered as a version of the Painlev´e property for difference equations.
Their work established Nevanlinna theory as a fundamental tool for study-
ing complex difference equations. Building on this foundation, Halburd and
Korhonen[9] proved that if the difference equation
f(z + 1) + f(z −1) = R(z, f),
(1.1)
where R(z, f) is rational in f with meromorphic coefficients, has an admis-
sible meromorphic solution of finite order, then it reduces to a short list of
canonical forms including the difference Painlev´e I and II equations. Further
studies of difference Painlev´e equations have been carried out by Halburd
and Korhonen [11], Ronkainen [22], and Wen [24, 25].
Complex equations combining difference operators and derivatives of mero-
morphic functions are termed complex differential-difference equations or
complex delay differential equations [16].
Some reductions of integrable
differential-difference equations are known to yield delay differential equa-
tions with formal continuum limits to Painlev´e equations.
For instance,
Quispel, Capel and Sahadevan[21] derived
f(z)[f(z + 1) −f(z −1)] + af ′(z) = bf(z),
(1.2)
where a and b are constants, through symmetry reduction of the Kac-van
Moerbeke equation. This equation possesses a formal continuum limit to the
first Painlev´e equation:
d2y
dt2 = 6y2 + t.
Subsequently, Halburd and Korhonen [10] investigated a generalization of
(1.2) and reduced this extended equation. Fundamental concepts of Nevan-
linna theory are detailed in [13]. Their main result is given below[10, Theorem
1.1]:
Theorem A Let f(z) be a transcendental meromorphic solution of
f(z + 1) −f(z −1) + a(z)f ′(z)
f(z) = R(z, f(z)) = P(z, f(z))
Q(z, f(z)),
(1.3)
where a(z) is rational, P(z, f(z)) is a polynomial in f having rational coeffi-
cients in z, and Q(z, f(z)) is a polynomial in f with roots that are non-zero
2
rational functions of z and not roots of P(z, f(z)). If the hyper order of f(z)
is less than one, then
degf P = degf Q + 1 ≤3 or degf R := max{degf P, degf Q} ≤1.
Recent extensions of this theorem have been explored in [2, 3, 12, 18, 23].
Notably, Nie, Huang, Wang and Wu[18] replaced the logarithmic derivative
in (1.3) with the Schwarzian derivative
S(f, z) := f ′′′(z)
f ′(z) −3
2
f ′′(z)
f ′(z)
2
,
and analyzed reductions of the resulting equation.
This work was moti-
vated by Malmquist’s results[17] on the equation f ′ = R(z, f) and Ishizaki’s
classification [14] of the Schwarzian differential equation S(f, z)n = R(z, f)
into six canonical forms (up to certain transformations) for positive integers
n. Before stating the main theorem, we recall the following definition from
Nevanlinna theory: a transcendental meromorphic function f is said to be
subnormal if it satisfies
σf := lim sup
r→∞
log T(r, f)
r
= 0,
(1.4)
Their main result is stated as follows[18, Theorem 1.1]:
Theorem B Let f(z) be a subnormal transcendental meromorphic solution
of the equation
f(z + 1) −f(z −1) + a(z)S(f, z) = R(z, f) = P(z, f)
Q(z, f),
(1.5)
where a(z) is rational, P(z, f) and Q(z, f) are coprime polynomials in f with
rational coefficients. Then degf R ≤7, and degf P ≤degf Q + 1. Moreover,
if Q(z, f) has a rational function root b1 in f with multiplicity k, then k ≤2.
Halburd and Korhonen[11] studied the reduction of the equation
f(z + 1)f(z −1) = c2(f −c+)(f −c−)
(f −a+)(f −a−) ,
(1.6)
where the coefficients are meromorphic functions. They proved that if (1.6)
admits an admissible finite-order meromorphic solution with bounded pole
multiplicity, then the equation reduces via M¨obius transformation to canoni-
cal forms including difference Painlev´e III, unless f is a solution of a difference
Riccati equation. Motivated by these results and Theorem B, we replace the
f(z +1)−f(z −1) in (1.5) with f(z +1)f(z −1) and investigated reductions
of the resulting Schwarzian delay equation. Our main theorem follows:
3
Theorem 1 Let f(z) be a subnormal transcendental meromorphic solution
of the equation
f(z + 1)f(z −1) + a(z)S(f, z) = R(z, f) = P(z, f)
Q(z, f),
(1.7)
where a(z) is rational, P(z, f) and Q(z, f) are coprime polynomials in f with
rational coefficients. Then degf R ≤7, and degf P ≤degf Q + 2. Moreover,
if Q(z, f) has a rational function root b1 in f with multiplicity k, then k ≤2.
Below we give some examples to illustrate our results.
Example 1 Let f(z) = eπz. It is easy to check that
f(z + 1)f(z −1) + S(f, z) = f 2 −π2
2 .
Here, R(z, f) is a polynomial in f with multiplicity no more than 2.
Example 2 Let f(z) = e2πz −z. Then f satisfies
f(z + 1)f(z −1)+S(f, z) = −8π3[πf 2 + (2πz + 1)f + πz2 + z]
(2πf + 2πz −1)2
+
[e2πf + (e2π −1)z −1][e−2πf + (e−2π −1)z + 1].
Then degf P = degf Q+2 = 4, Q(z, f) = (2πf +2πz−1)2, and 2πf +2πz−1
has no multiple zeros.
Example 3 It can be deduced that the meromorphic function f(z) = 1/(ez −
1) is a solution of the delay Schwarzian equation
f(z + 1)f(z −1) + S(f, z) =
ef 2
[(e −1)f + e][(1 −e)f + 1] −1
2.
Here, degf P = degf Q = 2. In addition, f −e/(1 −e) and f −1/(e −1)
have only simple zeros and hence have no multiple zeros.
Example 4 Suppose that f(z) = ℘(z + z0; ω1, ω2) is the Weierstrass ellip-
tic function, where ω1 and ω2 are two fundamental periods that are linearly
independent over R. Then f(z) solves the equation
f ′(z)2 = 4f 3(z) −g2f(z) −g3,
where g2 and g3 are constants depending on ω1 and ω2. Then the Schwarzian
derivative of f is that
S(f, z) = −48f 4 −24g2f 2 −96g3f −3g2
2
32f 3 −8g2f −8g3
.
4
Then we choose the appropriate g2 and g3 such that f ′(1) = 0 and 4z3 −
g2z −g3 = 0 has only simple roots e1, e2, e3, see [15, example 1.1]. According
to the addition theorem[26, Chapter 20.3] and the properties of Weierstrass
elliptic functions, f satisfies
f(z + 1)f(z −1) + S(f, z) =
−(e2e3 + e2
1) −e1f
e1 −f
2
+ −48f 4 −24g2f 2 −96g3f −3g2
2
8(f −e1)(f −e2)(f −e3)
.
Therefore, degf P = degf Q + 1 and e1, e2 and e3 are constant.
Then we classify the case degf P ≤degf Q + 2 when Q(z, f) has only
simple roots in f.
Theorem 2 Let f(z) be a subnormal transcendental meromorphic solution
of (1.7). Let Q(z, f) be of the following form:
Q(z, f) :=
n
Y
i=1
(f(z) −bi(z)) ˆQ(z, f(z)),
where the b′
is(1 ≤i ≤n, n ∈N) are distinct rational functions, ˆQ(z, f) and
f −bi are coprime.
(I) Assume that degf P = degf Q + 1. We have
(1) if n = 3, then there exists an integer i ∈{1, · · · , n} such that the root bi
is a constant;
(2) if n = 2, and f −bi has finitely many double zeros for all i = 1, · · · , n,
then there exists a root bi which is non-constant.
(II)Suppose that degf P = degf Q + 2. We also have
(3) if n = 2, then there exists an integer i ∈{1, · · · , n} such that the root bi
is a constant;
(III) Suppose that degf P ≤degf Q. For at least two i ∈{1, · · · , n}, we have
(4) if n ≥3 and bi is non-constant, then f −bi has finitely many multiple
zeros;
(5) if n ≥2 and bi is constant, then f −bi has finitely many zeros with
multiplicity at least 3.
Remark. Example 3 demonstrates that condition (5) in case (III) of The-
orem 2 can be satisfied. Example 4 verifies that condition (2) in case (I) of
Theorem 2 can not be omitted.
The following example shows that the case (II) in Theorem 2 can occur
and the roots bi can be a constant for all i ∈{1, · · · , n}.
5
Example 5 Let f(z) =
1
sin(2πz)+
√
2i. Then f(z) satisfies the following delay
Schwarzian differential equation
f(z + 1)f(z −1) + S(f, z) = f 2 + 2π2
−2
√
2if + 1
[(
√
2i + 1)f −1][(
√
2i −1)f −1].
In this case, degf P = degf Q + 2, and all roots of Q(z, f) are constants.
Now we discuss a special situation when Q(z, f) has a root of multiplicity
2 in the next theorem.
Theorem 3 Let f(z) be a subnormal transcendental meromorphic solution
of (1.7). Let Q(z, f) be of the following forms:
Q(z, f) := (f(z) −b1(z))2 ˆQ(z, f(z))
where b1(z) is a rational function, ˆQ(z, f) and f−b1 are coprime. If degf P =
degf Q + 1 and f −b1 has finite double zeros or if degf P = degf Q + 2, then
f −b1 has infinitely many simple zeros.
Remark. Example 2 can shows the case of degf P = degf Q+2 in Theorem
3 can occur.
2
Proof of Theorem 1
To prove Theorem 1, we first recall the necessary concepts and lemmas. A
differential-difference polynomial in f(z) is defined by
P(z, f) =
X
l∈L
bl(z)f(z)l0,0f(z+c1)l1,0 · · · f(z+cν)lν,0f ′(z)l0,1 · · · f (µ)(z+cν)lν,µ,
where c1, · · · , cν are distinct complex non-zero constants, L is a finite index
set consisting of elements of the form l = (l0,0, · · · , lν,µ) and coefficients bl are
meromorphic functions for all l ∈L. Nie, Huang , Wang and Wu modified
the results of Halburd and Korhonen[10, Lemma 2.1] and gave the following
lemma[18, Lemma 2.3].
Lemma 1 Let f(z) be a transcendental meromorphic solution of
P(z, f) = 0,
6
where P(z, f) is a differential-difference polynomial in f with meromorphic
coefficients bl(z) satisfying m(r, bl) = S(r, f). Let a1, · · · , ak be small func-
tions to f such that P(z, ai) ̸≡0 is also a small function of f, for all
i ∈{1, · · · , k}. If there exists s > 0 and τ ∈(0, 1) such that
k
X
i=1
n
r,
1
f −ai
≤kτn(r + s, f) + O(1),
then σf > 0, where σf is defined as (1.4).
We present the following lemma, which estimates the degree of R(z, f).
Lemma 2 Let f(z) be a subnormal transcendental meromorphic solution of
(1.7). Then degf R ≤7. Furthermore, if R(z, f(z)) is a polynomial in f with
coefficients in z, then degf R ≤2.
Proof. Taking the Nevanlinna characteristic function of both sides of (1.7)
and using [13, Theorem 2.25] we have
(degf R)T(r, f) = T(r, R(z, f)) + O(log r)
= T(r, f(z + 1)f(z −1) + a(z)S(f, z)) + O(log r).
From T(r, f ′) ≤2T(r, f) + S(r, f), (1.7) and [27, Theorem 1.2], we have
(degf R)T(r, f) ≤T(r, f(z + 1)f(z −1)) + T(r, S(f, z)) + O(log r)
≤m
r, f(z + 1)
f(z)
+ m
r, f(z −1)
f(z)
+ 2m(r, f)
+ N(r, f(z + 1)f(z −1)) + 5T(r, f) + S(r, f)
≤7T(r, f) + S(r, f).
where S(r, f) denotes any quantity satisfying S(r, f) = o(T(r, f)) as r →∞
and r ̸∈A1, A1 is a set with zero upper-density measure.
This gives the first conclusion of Lemma 2.
Now we suppose R is a
polynomial in f with rational coefficients. We will consider the following two
cases.
Case 1 Suppose that f(z) has at most finitely many poles. Since R is a
polynomial in f. From (1.7), we know that N(r, S(f, z)) = O(log r). Then
from the lemma on the logarithmic derivative we have T(r, S(f, z)) = S(r, f)
and
(degf R)T(r, f) ≤T(r, f(z + 1)f(z −1)) + S(r, f)
≤m
r, f(z + 1)
f(z)
+ m
r, f(z −1)
f(z)
+ 2m(r, f) + S(r, f)
≤2T(r, f) + S(r, f),
7
where r ̸∈A2 and A2 is a set with zero upper-density measure. So degf R ≤2.
Case 2 Assume that f(z) has infinitely many poles and we suppose that
degf R ≥3.
From conditions we know that the coefficients of R(z, f) and a(z) are
rational, so they have finitely many zeros and poles. Therefore, there exists
a constant M ∈R such that all zeros and poles of a(z) and the coefficients
of R(z, f) lie in a disk |z| < M where M ∈R+. As f(z) has infinitely many
poles, there exists a point z0 such that |z0| > M1, where M1 is a positive
constant such that M1 > M and z0 ± n ∈{z : |z| > M1} for finite number
n ∈Z. Let z0 be a pole of f(z) with multiplicity t. Then z0 is either a
regular point or a double pole of S(f, z). Since R is a polynomial in f and
degf R ≥3, and we denote degf R = α for convenience, it follows from (1.7)
that f(z) may has poles at z0+1 or z0−1 and the number of the multiplicities
of the poles is more than tα. We may assume that f(z) takes a pole at z0 +1
with multiplicity p1 and z0 −1 is a pole of f with multiplicity p2, where
p1 + p2 = tα.
Subcase 2.1 Suppose that p1α −t ≤0, then we can know that p1 ≤t
α
and p2 ≥t(α2−1)
α
≥2. From shifting (1.7), we have
f(z + 2)f(z) + a(z + 1)S(f, z + 1) = R(z + 1, f(z + 1)),
from which it follows that f(z0 + 2) is finite.
If 0 ≥p1α −t > −2, z0 + 3 is a zero of f with multiplicity p1. From
the iteration of shifting (1.7), we find that there may be no poles of f in the
set {z0 + d}, where d > 3 is an integer. Now we calculate the multiplicities
of poles of f in the set {z0 −n}, where n is a positive integer. From the
shifting of (1.7), similarly, we find z0 −2 is a pole of f with multiplicity
t(α2 −1) −p1α ≥1. We find that z0 −3 is the pole of f with multiplicity
t(α3 −2α) + p1(−α2 + 1) ≥1 and z0 −4 is a pole of f with multiplicity
t(α4 −3α2 + 1) + p1(−α3 + 2α) ≥1. From the recurrence relation we know
that z0 −d(d ∈N) is a pole of f, and the multiplicity of the pole of f at
z0 −d is 2p2−t(α−
√
α2−4)
2
√
α2−4
·
α+
√
α2−4
2
d−1
+ O (1). Thus we have
n(d + |z0|, f) ≥(α −1)d−2 + O(1)
8
for sufficiently large d ∈N, then
0 = lim sup
r→∞
log+ T(r, f)
r
≥lim sup
r→∞
log n(r, f)
r
≥lim sup
d→∞
log n(d + |z0|, f)
d + |z0|
≥lim sup
d→∞
log(α −1)d−2
d + |z0|
(2.1)
= log(α −1)
≥log 2 > 0
which is a contradiction. If α −t ≤−2, z0 + 3 is a simple pole of f when
p1 = 1 or f(z0+3) is finite when p1 ≥2. The number of multiplicities of poles
of f in the set {z0−d, · · · , z0+d} is more than the case when 0 ≥α−t > −2.
Similarly, we have the same conclusion.
Subcase 2.2 Assume that p1α−t ≥1, then p1 ≥t+1
α and z0 +2 is a pole
with multiplicity p1α −t. Similarly, from continuing the iteration of shifting
of (1.7), we find that z0 +3 is a pole of f with multiplicity p1α2 −tα−p1 ≥1
and z0 + 4 is a pole of f with multiplicity p1α3 −tα2 −2p1α + t ≥1. If
2p1 −t(α −
√
α2 −4) > 0, from the recurrence relation again, we know
that z0 + d is a pole of f and the multiplicity of the pole of f at z0 −d is
2p1−t(α−
√
α2−4)
2
√
α2−4
·
α+
√
α2−4
2
d−1
+ O (1). If 2p1 −t(α −
√
α2 −4) ≤0, then we
have 2p2 −t(α −
√
α2 −4) > 0. Similar to the subcase 2.1 we can have a
contradiction again. Hence we complete the proof.
2
Now we will consider the case when the polynomial Q(z, f) has a rational
function root in f.
Lemma 3 Let f(z) be a subnormal transcendental meromorphic solution of
(1.7). Let R(z, f) be of the following form:
R(z, f(z)) :=
P(z, f(z))
(f(z) −b1(z))k ˆQ(z, f(z))
(2.2)
where k is a positive integer, b1 is a rational function, P(z, f) and ˆQ(z, f)
are polynomials in f with rational coefficients, then k ≤2.
Proof. Suppose a differential-difference polynomial
Ψ(z, w) := (w −b1)k ˆQ(z, w)(w(z + 1)w(z −1) + ˆa(z)) −P(z, w),
where k ≥1, and ˆa(z) := a(z)S(w, z) satisfies m(r, ˆa) = S(r, w). Clearly,
from (1.7) and (2.2), we have Ψ(z, f) = 0. Since Ψ(z, b1) ̸= 0 is rational, the
9
first condition of Lemma 1 is satisfied for b1. By similar to the proof of [18,
Lemma 2.3], we have
N
r,
1
f −b1
= T(r, f) + S(r, f).
Now we consider the multiplicities of zeros of f −b1 and poles of f. We
assume that z0 is a zero (or pole) of f −b1 with multiplicity p(≥1) and
that none of a(z), b1(z) and any rational coefficients in R(z, f) have a zero
or pole at z0. We can also require that for finite number n ∈Z, the shifting
points z0 + n are not the zeros or poles of those rational coefficient functions
including a(z) and b1(z). We call such a point z0 a generic zero (or generic
pole) of f −b1 with multiplicity p, see in [18, Lemma 3.2].
Assume that k ≥3. Since k ≥3, then kp ≥3p > 2. Thus, f(z+1)f(z−1)
has a pole with multiplicity kp(≥3) at z0 from (1.7) and (2.2).
Case 1 Assume that
degf(P) ≤k + degf( ˆQ).
Subcase 1.1 Suppose the set E1 = {z : f(z+1) =
g1(z)
(z−z0)q , f(z−1) ̸= ∞},
where g1(z0) ̸= 0, ∞in E1 and q(≥kp) is an integer. If z0 ∈E1, then z0 + 1
is a double pole of S(f, z). By shifting (1.7) we obtain
f(z + 2)f(z) + a(z + 1)S(f, z + 1)
=
P(z + 1, f(z + 1))
(f(z + 1) −b1(z + 1))k ˆQ(z + 1, f(z + 1))
.
Thus f(z) has a double pole at z0+2, and so z0+2 is a double pole of S(f, z).
By iterating (1.7) again, we have
f(z + 3)f(z + 1) + a(z + 2)S(f, z + 2)
=
P(z + 2, f(z + 2))
(f(z + 2) −b1(z + 2))k ˆQ(z + 2, f(z + 2))
.
Then f(z) has a zero at z0 + 3 with multiplicity q −2.
If q = 3, then z0 + 4 is a double zero of f, so z0 + 4 is a double pole of
S(f, z). Then z0 + 5 is a pole of f with multiplicity 3, z0 + 6 is a pole of f
with multiplicity 4, z0 + 7 is a simple zero of f and z0 + 8 is a zero of f with
multiplicity 4. Thus we can know that z0 + d is a zero or pole of f for any
d ∈N.
If q ≥4, then z0 +4 may be another generic zero of f −b1 such that z0 +5
is a pole of f. Suppose that the multiplicity of the zero of f −b1 at q1(≥1),
10
thus we can know that z0 +5 is a pole of f with multiplicity kq1 +q −2(≥3),
z0 + 6 is a double pole of f and z0 + 7 is a zero of f. Suppose that the
multiplicity of the zero of f at z0 + 7 is q2(≥1). When q2 ≥2, z0 + 8 may be
the generic zero of f −b1. Similar to the above analysis, z0 + 9 and z0 + 10
are poles of f, z0 + 11 is a zero of f. When q2 = 1, similar to the case when
q ≥4, z0 + d is a pole or zero of f for any d ∈N and d ≥5.
From above the discussion, we have
nE1
r,
1
f −b1
≤1
knE1(r + 1, f) + S(r, f),
where nEi(r, f) is the number of multiplicities of all poles of f in the set
Ei
T{z
: |z| < r} for i ∈N. The definition of Ei(i ≥2) will give in the
following subcases.
Subcase 1.2 Assume that E2 = {z : f(z+1) = g2(z)
z−z0, f(z−1) =
g3(z)
(z−z0)q−1},
where gm(z0) ̸= 0, ∞in E2 for m ∈{2, 3} and q −1 = kp−1 ≥2. If z0 ∈E2,
then z0 +2 may be the generic zero of f −b1, z0 −2 is a double pole of f and
f(z0 −3) is finite. If z0 −3 is a zero of f, we suppose that the multiplicity
of the zero of f at z0 −3 is q3(≥1). When q3 ≥2, z0 −4 may be the generic
zero of f −b1. When q3 = 1, similar to the subcase 1.1, z0 −d is a zero or
pole of f for any d ∈N.
From the discussion in this subcase, we obtain
nE2
r,
1
f −b1
≤1
knE2(r + 1, f) + S(r, f).
Subcase 1.3 Suppose that E3 = {z : f(z + 1) =
g4(z)
(z−z0)2, f(z −1) =
g5(z)
(z−z0)q−2}, where gm(z0) ̸= 0, ∞in E3 for m ∈{4, 5} and q −2 = kp −2 ≥1.
If z0 ∈E3, by shifting of (1.7), we find that z0 + 2 is a double pole of f and
z0 + 3 may be the generic zero of f −b1. When q −2 = 1, it’s similar to
the subcase 1.2, we have z0 −2 may be the generic zero of f −b1. When
q −2 = 2, we have z0 −2 is a double pole of f and f(z0 −3) is finite. When
q −2 ≥3, it’s similar to the subcase 1.1. From the discussion in subcase 1.3,
we have
nE3
r,
1
f −b1
≤1
knE3(r + 1, f) + S(r, f).
Subcase 1.4 Assume that E4 = {z : f(z + 1) =
g6(z)
(z−z0)s1 , f(z −1) =
g7(z)
(z−z0)s2 }, where gm(z0) ̸= 0, ∞in E4 for m ∈{6, 7} , s1 ≥3, s2 ≥3 and
s1 + s2 = kp. It’s similar to the subcases 1.1, 1.2 and 1.3, we have
nE4
r,
1
f −b1
≤1
knE4(r + 1, f) + S(r, f).
11
By adding up the contribution from all points in S4
i=1 Ei to the corre-
sponding counting functions, it follows that
n
r,
1
f −b1
=
4
X
i=1
nEi
r,
1
f −b1
≤1
kn(r + 1, f) + S(r, f).
Thus both conditions of Lemma 1 are satisfied, and so σf > 0. This is a
contradiction.
Case 2 Assume that
degf P ≥k + degf( ˆQ) + 1
and degf R = α(≥1).
Subcase 2.1 Suppose the set E5 = {z : f(z+1) =
g8(z)
(z−z0)q , f(z−1) ̸= ∞}
and z0 ∈E5, where g8(z0) ̸= 0, ∞in E5 and q ≥kp ≥3.
If α ≥2, we continue to iterate the shifting and find that z0 + 2 is
a pole of f with multiplicity qα, z0 + 3 is a pole of f with multiplicity
q(α2 −1) ≥2, z0 + 4 is a pole of f with multiplicity q(α3 −2α) ≥2 and
z0 + 5 is q(α4 −3α2 + 1) ≥2. From the recurrence relation we can know that
z0 + d is a multiple pole of f for any d ∈N.
If α = 1, then z0 + 2 is a pole of f with multiplicity q, z0 + 3 may be
another zero of f −b1.
From the above analysis, we have
nE5
r,
1
f −b1
≤1
knE5(r + 1, f) + S(r, f),
where nE5(r, f) is the number of multiplicities of all poles of f in the set
E5
T{z : |z| < r}.
Subcase 2.2 Assume that E6 = {z : f(z+1) =
g9(z)
(z−z0)q−1, f(z−1) = g10(z)
z−z0 }
and z0 ∈E6, where gm(z0) ̸= 0, ∞in E6 for m ∈{9, 10} and q−1 = kp−1 ≥
2.
If α = 1, from continuing iteration of shifting, we have that z0 −2 is a
simple pole and z0−3 may be another generic zero of f −b1. When q−1 = 2,
z0 + 2 may be the generic zero of f −b1. When q −1 ≥3, similar to the
subcase 2.1, z0 + 2 is a pole of f with multiplicity q −1 and z0 −3 may be
another zero of f −b1.
If α > 1, we find that z0 −2 is a pole of f with multiplicity α ≥2, z0 −3
is a pole of f with multiplicity α2 −1 ≥2 and z0 −4 is a pole of f with
multiplicity α3 −3α ≥2. From the recurrence relation again, we find that
z0 ± d is a multiple pole of f for any d ∈N.
12
Thus, we obtain
nE6
r,
1
f −b1
≤1
knE6(r + 1, f) + S(r, f).
Subcase 2.3 Suppose that E7 = {z : f(z + 1) =
g11(z)
(z−z0)q−2, f(z −1) =
g12(z)
(z−z0)2} and z0 ∈E7, where gm(z0) ̸= 0, ∞in E7 for m ∈{11, 12} and
q −2 = kp −2 ≥1.
If α = 1, then z0 −2 may be a generic zero of f −b1. When q −2 = 1 or
q −2 ≥3 , similar to the subcase 2.2 we have that z0 + 2 is a pole of f with
multiplicity q −2 and z0 + 3 is a generic zero of f −b1. When q −2 = 2,
z0 + 2 may be a generic zero of f −b1.
If α > 1, it similar to the subcase 2.1 and we can obtain that z0 ± d is a
pole of f for any d ∈N.
From the above, when α = 1 we have
nE7
r,
1
f −b1
≤2
knE7(r + 1, f) + S(r, f).
When α > 1, we obtain
nE7
r,
1
f −b1
≤1
knE7(r + 1, f) + S(r, f).
Subcase 2.4 Suppose that E8 = {z : f(z + 1) =
g13(z)
(z−z0)s1 , f(z −1) =
g14(z)
(z−z0)s2 } and z0 ∈E8, where gm(z0) ̸= 0, ∞in E8 for m ∈{13, 14} and
s1 + s2 = kp. Similar to subcase 2.1, we have
nE8
r,
1
f −b1
≤1
knE8(r + 1, f) + S(r, f).
By adding up the contribution from all points in S4
i=1 Ei to the corre-
sponding counting functions, if α > 1, we get
n
r,
1
f −b1
=
8
X
i=5
nEi
r,
1
f −b1
≤1
kn(r + 1, f) + O(1).
If α = 1, we have
n
r,
1
f −b1
=
8
X
i=5
nEi
r,
1
f −b1
≤2
kn(r + 1, f) + O(1).
Thus, by Lemma 1, we have σf > 0, a contradiction.
2
13
Proof of Theorem 1. We know that degf R = max{degf P, degf Q} ≤7 from
Lemma 2, This completes the proof of the first part.
Assume that degf P ≥degf Q+3, then degf R = degf P ≥3 and suppose
that α = degf R for convenience. We divide two cases as follows.
Case 1 Suppose that f(z) has finitely many poles. From (1.7), we see
that N(r, S(f, z)) = O(log r). Then we have T(r, S(f, z)) = S(r, f) and
(degf R)T(r, f) ≤T(r, f(z + 1)f(z −1)) + T(r, S(f, z)) + S(r, f)
≤m
r, f(z + 1)
f(z)
+ m
r, f(z −1)
f(z)
+ 2m(r, f) + S(r, f)
≤2T(r, f) + S(r, f),
where r ̸∈A3, where A3 is a set with zero upper-density measure. So it
contradicts to degf R ≥3.
Case 2 Assume that f(z) has infinitely many poles. Let z0 be a generic
pole of f(z) with multiplicity p. Then R(z, f) has a pole at z0 with multi-
plicity pα(≥3), and z0 is either a regular point or a double pole of S(f, z).
So f(z + 1)f(z −1) has a pole at z0 with multiplicity pα.
Suppose that z0 + 1 is a pole of f with multiplicity p1(≥0), z0 −1 is pole
of f with multiplicity p2(≥0), where p1 + p2 = pα.
Subcase 2.1 Assume that p1α −p ≤0, then we have that p2 ≥p(α2−1)
α
and z0 −d is a pole of f from the recurrence relation. For sufficiently large
d ∈N we find that the multiplicity of the pole of f at z0 −d is
2p2 −p(α −
√
α2 −4)
2
√
α2 −4
·
α +
√
α2 −4
2
d−1
+ O (1) .
Thus, similar to the proof of Lemma 2, we have
n(d + |z0|, f) ≥(α −1)d−2 + O(1)
for all sufficiently large d ∈N. In this subcase, we also have equation (2.1)
holds, which is a contradiction.
Subcase 2.2 Suppose that p1α −p ≥1, similar to the proof in Lemma
2, we have a contradiction again.
2
3
Proof of Theorem 2
Obviously, none of b1, b2, · · · , bn is a solution of (1.7) and degf Q = n+degf ˆQ.
We consider three cases as follows.
14
Case 1 Suppose that degf P = degf Q + 1.
(1) Assume that the b′
is are non-constant for all i ∈{1, · · · , n}. We will
derive a contradiction. Without loss of generality, let z0 be a generic zero
of f −b1 with multiplicity q such that b′
1(z0) ̸= 0. Firstly, we will prove
f ′(z0) = 0 is impossible. If f ′(z0) = 0, then we find that q = 1, z0 is a double
pole of S(f, z) and f(z + 1)f(z −1) has a double pole at z0.
Subcase 1.1 Suppose that U1 = {z : f(z + 1) =
h1(z)
(z−z0)q1 , f(z −1) ̸= ∞}
and z0 ∈U1, where h1(z) ̸= 0, ∞in U1 and q1 ≥2.
When q1 = 2, then z0 + 2 may be another zero of f −bi. If f(z0 + 2) ̸=
bi(z0 + 2) for i ∈{1, · · · , n}, then f(z + 3) has a double zero at z0, z0 + 4
is a double pole of f and z0 + 5 may be the generic zero of f −bi.
If
f(z0 + 2) = bi(z0 + 2) for i ∈{1, · · · , n}, we assume that the multiplicity of
the generic zero of f −bi at z0 + 2 is p(≥1). When p = 1, for f ′(z0 + 2) ̸= 0,
z0 + 3 is a simple zero of f and z0 −4 may be the generic zero of f −bi. For
f ′(z0 + 2) = 0, f(z0 + 3) is finite. When p = 2, we also have f(z0 + 3) is
finite. When p = 3, z0 +3 and z0 +4 are simple poles of f, z0 +5 may be the
generic zero of f −bi. When p = 4, z0 + 3 is a double pole of f and z0 + 4
may be the generic zero of f −bi. When p ≥5, f(z0 + 3) and f(z0 + 4) are
poles of f. When q1 ≥3, it’s similar to the situation when q1 = 2.
From the discussion above, we have
n
X
i=1
nU1
r,
1
f −bi
≤2nU1(r + 1, f) + O(1),
where nUi(r, f) is the number of multiplicities of all poles of f in the set
Ui
T{z
: |z| < r} for i ∈N. The definition of Ui(i ≥2) will give in the
following subcases.
Subcase 1.2 Assume that U2 = {z : f(z + 1) = h2(z)
z−z0 , f(z −1) = h3(z)
z−z0 }
and z0 ∈U2, where hm(z) ̸= 0, ∞in U2 for m ∈{2, 3}, then we have z0 ± 2
are simple poles of f by shifting (1.7) and z0 ± 3 are another generic zeros of
f −b1. In this situation, we can also get equation
n
X
i=1
nU2
r,
1
f −bi
≤nU2(r + 1, f) + O(1).
From the above analysis, we have
n
X
i=1
n
r,
1
f −bi
=
n
X
i=1
nU1
r,
1
f −bi
+ nU2
r,
1
f −bi
≤2n(r + 1, f) + O(1).
Thus σf > 0 and we get a contradiction by Lemma 2.
15
Therefore, it’s clear that f ′(z0) ̸= 0. In this case, z0 is a regular point of
S(f, z) and a pole of f(z + 1)f(z −1) with multiplicity q(≥2).
Subcase 1.3 Assume that U3 = {z : f(z + 1) =
h4(z)
(z−z0)2, f(z −1) ̸= ∞}
and z0 ∈U3, where h4(z) ̸= 0, ∞in U3, then we have that z0 + 2 may be the
generic zero of f −bi. It follows that
n
X
i=1
nU3
r,
1
f −bi
≤2nU3(r + 1, f) + O(1).
Subcase 1.4 Suppose that U4 = {z : f(z + 1) = h5(z)
z−z0 , f(z −1) = h6(z)
z−z0 }
and z0 ∈U4, where hm(z) ̸= 0, ∞in U4 for m ∈{5, 6}, then z0 ±2 are simple
poles of f and z0 ± 3 may be the generic zeros of f −bi. Thus we obtain
n
X
i=1
nU4
r,
1
f −bi
≤nU4(r + 1, f) + O(1).
Subcase 1.5 Assume that U5 = {z : f(z0 + 1) =
h7(z)
(z−z0)q2 , f(z0 −1) ̸= ∞}
and z0 ∈U5, where h7(z) ̸= 0, ∞in U5 and q2 ≥3. By iteration of shifting,
we can know that z0 + 2 is a pole of f with multiplicity q2 and z0 + 3 may be
another zero of f −bi such that f(z0 + 4) is finite. Therefore, it follows that
n
X
i=1
nU5
r,
1
f −bi
≤nU5(r + 1, f) + O(1).
Subcase 1.6 Suppose that U6 = {z : f(z+1) =
h8(z)
(z−z0)q−1, f(z−1) = h9(z)
z−z0 }
and z0 ∈U6, where hm(z) ̸= 0, ∞in U6 for m ∈{8, 9} and q ≥3. Similar to
the subcase 1.4, z0 −2 is a simple pole of f and z0 −3 may be the generic
zero of f −bi. When q −1 = 2, it’s similar to the subcase 1.3 and we have
z0 + 2 may be the generic zero of f −bi. When q −1 ≥3, we find that z0 + 2
is a pole of f and f(z0 + 3) is finite. From the above analysis, we have
n
X
i=1
nU6
r,
1
f −bi
≤2nU6(r + 1, f) + O(1).
Subcase 1.7 Assume that U7 = {z : f(z + 1) =
h10(z)
(z−z0)q−2, f(z −1) =
h11(z)
(z−z0)2} and z0 ∈U7, where hm(z) ̸= 0, ∞in U7 for m ∈{10, 11} and q ≥3,
then z0 −2 may be a generic zero of f −bi with multiplicity 2, i ∈{1, · · · , n}.
Similar to the subcases 1.3-1.6, we have
n
X
i=1
nU7
r,
1
f −bi
≤2nU7(r + 1, f) + O(1).
16
Subcase 1.8 Suppose that U8 = {z : f(z + 1) =
h12(z)
(z−z0)s1 , f(z −1) =
h13(z)
(z−z0)s2 } and z0 ∈U8, where hm(z) ̸= 0, ∞in U8 for m ∈{12, 13}, s1 ≥3,
s2 ≥3 and s1 + s2 = q. Similar to the subcase 1.5, we find that
n
X
i=1
nU8
r,
1
f −bi
≤nU8(r + 1, f) + O(1).
From the above discussion, we can conclude that
n
X
i=1
n
r,
1
f −bi
=
n
X
i=1
8
X
j=3
nUj
r,
1
f −bi
≤2n(r + 1, f) + O(1).
Then we have a contradiction from the Lemma 2.
(2) Suppose that the b′
is are constant for all i ∈{1, 2, · · · , n}. Then by
the condition and Lemma 1, f −bi has infinitely many zeros with multiplicity
1 or not less than 3. Without loss of generality, we may assume that z1 is a
generic zero of f −b1 with multiplicity s, where s ≥3 or s = 1. If s = 1,
then f ′(z) ̸= 0 and so S(f, z) is regular at z1. If s ≥3, then f ′(z1) = 0
and S(f, z) has a double pole at z1. Hence, for each case, we conclude that
f(z + 1)f(z −1) has a pole with multiplicity s at z1. Similar to the proof of
subcases 1.1-1.8, we also get
n
X
i=1
n
r,
1
f −bi
≤n(r + 1, f) + O(1).
(3.1)
Therefore, it follows that σf > 0, a contradiction from Lemma 2.
Case 2 Suppose that degf P = degf Q + 2.
(3)Assume that the bi’s are non-constant for all i ∈{1, · · · , n}. We can
let z0 be a generic zero of f −b1 with multiplicity q such that b′
1(z0) ̸= 0
without loss of generality. If f ′(z0) = 0, then q = 1 and z0 is a double pole
of S(f, z), so f(z + 1)f(z −1) has a double pole at z0.
Subcase 2.1 Suppose that U9 = {z : f(z + 1) =
h14(z)
(z−z0)q3 , f(z −1) ̸= ∞},
where h14(z) ̸= 0, ∞in U9 and q3 ≥2. If z0 ∈U9, then z0 + 2 is a pole of
f with multiplicity at least 4, z0 + 3 is a pole of f with multiplicity at least
6 and z0 + 4 is a pole of f with multiplicity at least 8. From the recurrence
relation, we have that z0 + d is a multiple pole of f with multiplicity at least
2d for any d ∈N. Therefore, it follows that
n
X
i=1
nU9
r,
1
f −bi
≤1
2nU9(r + 1, f) + O(1).
Subcase 2.2 Assume that U10 = {z : f(z + 1) = h15(z)
z−z0 , f(z −1) = h16(z)
z−z0 }
and z0 ∈U10, where hm(z) ̸= 0, ∞in U10 for m ∈{15, 16}. Then z0 ± 2 are
17
double poles of f, z0 ± 3 are poles of f with multiplicity 3 and z0 ± 4 are
poles of f with multiplicity 4. Similar to the subcase 2.1, z0 ± d are poles of
f for any d ∈N. Thus we have
n
X
i=1
nU10
r,
1
f −bi
≤1
2nU10(r + 1, f) + O(1).
Therefore, we add up the contribution from all points z in the set U9
S U10
to the corresponding counting functions, we have
n
X
i=1
n
r,
1
f −bi
=
n
X
i=1
nU9
r,
1
f −bi
+ nU10
r,
1
f −bi
≤1
2n(r + 1, f) + O(1).
So it follows that σf > 0 by Lemma 1, a contradiction. From the discus-
sion above, we have f ′(z0) ̸= 0. In this case, z0 is a regular point of S(f, z)
and a pole of f(z +1)f(z −1) with multiplicity q. Similar to the subcases 2.1
and 2.2, we can have the same inequality (3.1). thus we get a contradiction
by Lemma 1.
Case 3 Suppose that degf P ≤degf Q.
(4) Assume that f−bi has infinitely many multiple zeros for i ∈{1, · · · , n}.
Then we will prove by contradiction. Without loss of generality, let z0 be a
generic zero of f −b1 with multiplicity p(≥2) such that b′
1(z0) ̸= 0. Then
f ′(z0) ̸= 0 and f(z + 1)f(z −1) has a pole with multiplicity p at z0.
Subcase 3.1 Suppose that U11 = {f(z0 + 1) =
h17(z)
(z−z0)q4 , f(z0 −1) ̸= ∞}
and z0 ∈U11, where h17(z) ̸= 0, ∞in U11 and q4 ≥p, then f(z + 2) has a
double pole at z0. When q4 = 2, we have z0 + 3 may be the generic zero of
f −bi. When q4 = 3, then z0 + 3 is a simple zero of f, z0 + 4 is a double
zero of f, z0 + 5 is a pole of f with multiplicity 3 and z0 + 6 is a pole of f
with multiplicity 4. From continuing the iteration of shifting, we know that
z0 + d is not a generic zero of f −bi for all d ∈N. When q4 ≥4, we find that
z0 + 3 is a zero of f with multiplicity q4 −2(≥2) and z0 + 4 may be another
generic zero of f −b1. From the above analysis, we obtain
n
X
i=1
nU11
r,
1
f −bi
≤nU11(r + 1, f) + O(1),
Subcase 3.2 Assume that U12 = {z : f(z + 1) = h18(z)
z−z0 , f(z −1) = h19(z)
z−z0 }
and z0 ∈U12, where hm(z) ̸= 0, ∞in U12 for m ∈{18, 19} , by iteration of
shifting (1.7), we find that z0 ± 2 may be generic zeros of f −bi. Thus we
18
have
n
X
i=1
nU12
r,
1
f −bi
≤2nU12(r + 1, f) + O(1).
Subcase 3.3 Assume that U13 = {f(z+1) =
h20(z)
(z−z0)s1 , f(z−1) =
h21(z)
(z−z0)s2 }
and z0 ∈U13, where hm(z) ̸= 0, ∞in U13 for m ∈{20, 21} , s1 ≥1, s2 ≥1
and s1 + s2 = p. It’s similar to the subcase 3.1 and 3.2 and we have that
n
X
i=1
nU13
r,
1
f −bi
≤2nU13(r + 1, f) + O(1).
From the above discussion, we have
n
X
i=1
n
r,
1
f −bi
=
n
X
i=1
13
X
j=11
nUj
r,
1
f −bi
≤2n(r + 1, f) + O(1).
Therefore, it follows that we get a contradiction by Lemma 1.
(5) Suppose that f −bi has infinitely many zeros with multiplicity 3 or
higher. We will again prove by contradiction. Without loss of generality,
let z0 be a generic zero of f −b1 with multiplicity p ≥3.
From (1.7),
f(z + 1)f(z −1) has a pole at z0 with multiplicity p.
Subcase 3.4 Assume that U14 = {f(z0 + 1) =
h22(z)
(z−z0)q5 , f(z0 −1) ̸= ∞}
and z0 ∈U14, where h22(z) ̸= 0, ∞in U14 and q5 ≥p, then z0 is a double
pole of f(z + 2) and a zero of f(z + 3) with multiplicity q5 −2 ≥1. Similarly
to the proof in subcase 3.1, we have
n
X
i=1
nU14
r,
1
f −bi
≤nU14(r + 1, f) + O(1).
Subcase 3.5 Suppose that U15 = {f(z +1) =
h23(z)
(z−z0)p−1, f(z −1) = h24(z)
z−z0 }
and z0 ∈U15, where hm(z) ̸= 0, ∞in U15 for m ∈{23, 24} , then f(z0 −2)
is finite. When p = 3, z0 + 2 is a double pole of f and z0 + 3 may be the
generic zero of f −bi. When p ≥4, it’s similar to the argument in subcase
3.1. So we give
n
X
i=1
nU15
r,
1
f −bi
≤nU15(r + 1, f) + O(1).
Subcase 3.6 Assume that U16 = {f(z+1) =
h25(z)
(z−z0)s1 , f(z−1) =
h26(z)
(z−z0)s2 }
and z0 ∈U16, where hm(z) ̸= 0, ∞in U16 for m ∈{25, 26} and s1 + s2 = p.
Similarly to the proof in subcase 3.4 and 3.5, we have
n
X
i=1
nU16
r,
1
f −bi
≤nU16(r + 1, f) + O(1).
19
From the above analysis, we obtain
n
X
i=1
n
r,
1
f −bi
=
n
X
i=1
16
X
j=14
nUj
r,
1
f −bi
≤n(r + 1, f) + O(1).
Therefore, we can get σf > 0 by Lemma 2, a contradiction.
4
Proof of Theorem 3
Notice that b1 is not a solution of (1.7), and degf Q = 2+degf ˆQ. Suppose
that f −b1 has only finitely many simple zeros. By the proof in [18, Lemma
2.3], we have
N
r,
1
f −b1
= T(r, f) + S(r, f).
Thus, f −b1 has infinitely many multiple zeros. Let z0 be a generic zero of
f −b1 with multiplicity p(≥2).
Case 1 Suppose that degf P = degf Q + 1. If b1 is a constant, then
f ′(z0) = 0 and hence z0 is a double pole of S(f, z). By iteration of shifting,
f(z + 1)f(z −1) has a pole with multiplicity 2p at z0.
Subcase 1.1 Assume that L1 = {z : f(z + 1) =
l1(z)
(z−z0)q1 , f(z −1) ̸= ∞}
and z0 ∈L1, where l1(z) ̸= 0, ∞in L1 and q1 ≥2p, then z0 + 2 is a pole
of f with multiplicity at least 2p and z0 + 3 may be another generic zero of
f −b1. Thus we have
nL1
r,
1
f −b1
≤1
2nL1(r + 1, f) + O(1),
where nLi(r, f) is the number of multiplicities of all poles of f in the set
Li
T{z
: |z| < r} for i ∈N. The definition of Li(i ≥2) will give in the
following subcases.
Subcase 1.2 Suppose that L2 = {z : f(z + 1) =
l2(z)
(z−z0)2p−1, f(z −1) =
l3(z)
z−z0} and z0 ∈L2, where lm(z) ̸= 0, ∞in L2 for m ∈{2, 3}, then z0 −2 is a
simple pole of f and z0 −3 is the generic zero of f −b1. Similarly, we find
that z0 + 2 is a pole of f with multiplicity 2p −1 and f(z0 + 3) is finite. In
summary, we have
nL2
r,
1
f −b1
≤1
2nL2(r + 1, f) + O(1).
Subcase 1.3 Suppose that L3 = {z : f(z + 1) =
l4(z)
(z−z0)s1 , f(z −1) =
l5(z)
(z−z0)s2 } and z0 ∈L3, where s1, s2 ≥3, s1 + s2 = 2p and lm(z) ̸= 0, ∞in L3
20
for m ∈{4, 5} . Similar to the argument in subcase 1.1, we find that
nL3
r,
1
f −b1
≤1
2nL3(r + 1, f) + O(1).
By considering all generic zeros of f −b1 in the set S3
i=1 Li, it follows that
n
r,
1
f −b1
=
3
X
i=1
nLi
r,
1
f −b1
≤1
2n(r + 1, f) + O(1).
Thus σf > 0 by Lemma 1, a contradiction. Hence, b1 is non-constant. So
there exists a point z0 such that b′
1(z0) ̸= 0 and f ′(z0) ̸= 0. Similarly with
the case when b1 is constant, we also obtain σf > 0, a contradiction.
Case 2 Assume that degf P = degf Q + 2.
If b1 is a constant, then
f ′(z0) = 0 and hence z0 is a double pole of S(f, z). By iteration of shifting,
f(z + 1)f(z −1) has a pole with multiplicity 2p at z0.
Subcase 2.1 Suppose that L4 = {z : f(z + 1) =
l6(z)
(z−z0)q2 , f(z −1) ̸= ∞}
and z0 ∈L4, where l6(z) ̸= 0, ∞in L4 and q2 ≥2p. From the recurrence
relation, we know that z0 + d is multiple poles of f for any d ∈N. Thus, it
follows that
nL4
r,
1
f −b1
≤1
2nL4(r + 1, f) + O(1).
Subcase 2.2 Assume that L5 = {z : f(z + 1) =
l7(z)
(z−z0)s1 , f(z −1) =
l8(z)
(z−z0)s2 } and z0 ∈L5, where lm(z) ̸= 0, ∞in L5 for m ∈{7, 8} , s1 ≥1,
s2 ≥1 and s1 + s2 = 2p. Similar to the proof in subcase 2.1, we have
nL5
r,
1
f −b1
≤1
2nL5(r + 1, f) + O(1).
In summary, it follows that
n
r,
1
f −b1
= nL4
r,
1
f −b1
+ nL5
r,
1
f −b1
≤1
2n(r + 1, f) + O(1).
Thus σf > 0 by Lemma 1, a contradiction. Hence, b1 is non-constant. So
there exists a point z0 such that b′
1(z0) ̸= 0 and f ′(z0) ̸= 0. Similarly with
the case when b1 is constant, we also obtain σf > 0, a contradiction.
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Admissible solutions of delay Schwarzian differential equations Shi-Jian Wu Abstract In this paper, we study delay differential equations involving the Schwarzian derivative S(f, z), expressed in the form f(z + 1)f(z -1) + a(z)S(f, z) = R(z, f(z)) = P(z, f(z)) Q(z, f(z)) where a(z) is rational, P(z, f) and Q(z, f) are coprime polynomials in f with rational coefficients. Our main result shows that if a subnormal transcendental meromorphic solution exists, then the rational function R(z, f) = P(z, f)/Q(z, f) satisfies degf R ≤7 and degf P ≤degf Q + 2, where degf R = max{degf P, degf Q}. Furthermore, for any rational root b1 of Q(z, f) in f with multiplicity k, we show that k ≤2. Finally, a classification of such equations is provided according to the multiplicity structure of the roots of Q(z, f). Some examples are given to support these results. Keyword: delay differential equations; Nevanlinna theory; Schwarzian derivatives; subnormal solutions 2020MSC: 34M04; 30D35 1 Introduction An ordinary differential equation is said to possess the Painlev ́e property if its solutions are single-valued about all movable singularities[4, 8]. Recent interest in this property stems from statistical physics and partial differential equations; for instance, these equations with Painlev ́e property provide exact solutions for the two-dimensional Ising model [5]. Beyond these applications, the significance of studying equations with Painlev ́e property lies in their dual role: they serve both as a source for defining new functions and a class of equations to be integrated with the existing functions available. This classification originated in the early 20th century through the work of 1 16 Oct 2025 Painlev ́e[19, 20], Fuchs[6] and Gambier[7], who systematically classified second order differential equations with the Painlev ́e property. Their analysis culminated in six canonical forms, now named as the Painlev ́e equations. The analogues of Painlev ́e property for complex difference equations have been discussed. Ablowitz, Halburd and Herbst[1] have advocated that the existence of sufficiently many finite order meromorphic solutions could be considered as a version of the Painlev ́e property for difference equations. Their work established Nevanlinna theory as a fundamental tool for studying complex difference equations. Building on this foundation, Halburd and Korhonen[9] proved that if the difference equation f(z + 1) + f(z -1) = R(z, f), (1.1) where R(z, f) is rational in f with meromorphic coefficients, has an admissible meromorphic solution of finite order, then it reduces to a short list of canonical forms including the difference Painlev ́e I and II equations. Further studies of difference Painlev ́e equations have been carried out by Halburd and Korhonen [11], Ronkainen [22], and Wen [24, 25]. Complex equations combining difference operators and derivatives of meromorphic functions are termed complex differential-difference equations or complex delay differential equations [16]. Some reductions of integrable differential-difference equations are known to yield delay differential equations with formal continuum limits to Painlev ́e equations. For instance, Quispel, Capel and Sahadevan[21] derived f(z)[f(z + 1) -f(z -1)] + af ′(z) = bf(z), (1.2) where a and b are constants, through symmetry reduction of the Kac-van Moerbeke equation. This equation possesses a formal continuum limit to the first Painlev ́e equation: d2y dt2 = 6y2 + t. Subsequently, Halburd and Korhonen [10] investigated a generalization of (1.2) and reduced this extended equation. Fundamental concepts of Nevanlinna theory are detailed in [13]. Their main result is given below[10, Theorem 1.1]: Theorem A Let f(z) be a transcendental meromorphic solution of f(z + 1) -f(z -1) + a(z)f ′(z) f(z) = R(z, f(z)) = P(z, f(z)) Q(z, f(z)), (1.3) where a(z) is rational, P(z, f(z)) is a polynomial in f having rational coefficients in z, and Q(z, f(z)) is a polynomial in f with roots that are non-zero 2 rational functions of z and not roots of P(z, f(z)). If the hyper order of f(z) is less than one, then degf P = degf Q + 1 ≤3 or degf R := max{degf P, degf Q} ≤1. Recent extensions of this theorem have been explored in [2, 3, 12, 18, 23]. Notably, Nie, Huang, Wang and Wu[18] replaced the logarithmic derivative in (1.3) with the Schwarzian derivative S(f, z) := f ′′′(z) f ′(z) -3 2 f ′′(z) f ′(z) 2 , and analyzed reductions of the resulting equation. This work was motivated by Malmquist's results[17] on the equation f ′ = R(z, f) and Ishizaki's classification [14] of the Schwarzian differential equation S(f, z)n = R(z, f) into six canonical forms (up to certain transformations) for positive integers n. Before stating the main theorem, we recall the following definition from Nevanlinna theory: a transcendental meromorphic function f is said to be subnormal if it satisfies σf := lim sup r→∞ log T(r, f) r = 0, (1.4) Their main result is stated as follows[18, Theorem 1.1]: Theorem B Let f(z) be a subnormal transcendental meromorphic solution of the equation f(z + 1) -f(z -1) + a(z)S(f, z) = R(z, f) = P(z, f) Q(z, f), (1.5) where a(z) is rational, P(z, f) and Q(z, f) are coprime polynomials in f with rational coefficients. Then degf R ≤7, and degf P ≤degf Q + 1. Moreover, if Q(z, f) has a rational function root b1 in f with multiplicity k, then k ≤2. Halburd and Korhonen[11] studied the reduction of the equation f(z + 1)f(z -1) = c2(f -c+)(f -c-) (f -a+)(f -a-) , (1.6) where the coefficients are meromorphic functions. They proved that if (1.6) admits an admissible finite-order meromorphic solution with bounded pole multiplicity, then the equation reduces via M ̈obius transformation to canonical forms including difference Painlev ́e III, unless f is a solution of a difference Riccati equation. Motivated by these results and Theorem B, we replace the f(z +1)-f(z -1) in (1.5) with f(z +1)f(z -1) and investigated reductions of the resulting Schwarzian delay equation. Our main theorem follows: 3 Theorem 1 Let f(z) be a subnormal transcendental meromorphic solution of the equation f(z + 1)f(z -1) + a(z)S(f, z) = R(z, f) = P(z, f) Q(z, f), (1.7) where a(z) is rational, P(z, f) and Q(z, f) are coprime polynomials in f with rational coefficients. Then degf R ≤7, and degf P ≤degf Q + 2. Moreover, if Q(z, f) has a rational function root b1 in f with multiplicity k, then k ≤2. Below we give some examples to illustrate our results. Example 1 Let f(z) = eπz. It is easy to check that f(z + 1)f(z -1) + S(f, z) = f 2 -π2 2 . Here, R(z, f) is a polynomial in f with multiplicity no more than 2. Example 2 Let f(z) = e2πz -z. Then f satisfies f(z + 1)f(z -1)+S(f, z) = -8π3[πf 2 + (2πz + 1)f + πz2 + z] (2πf + 2πz -1)2 + [e2πf + (e2π -1)z -1][e-2πf + (e-2π -1)z + 1]. Then degf P = degf Q+2 = 4, Q(z, f) = (2πf +2πz-1)2, and 2πf +2πz-1 has no multiple zeros. Example 3 It can be deduced that the meromorphic function f(z) = 1/(ez - 1) is a solution of the delay Schwarzian equation f(z + 1)f(z -1) + S(f, z) = ef 2 [(e -1)f + e][(1 -e)f + 1] -1 2. Here, degf P = degf Q = 2. In addition, f -e/(1 -e) and f -1/(e -1) have only simple zeros and hence have no multiple zeros. Example 4 Suppose that f(z) = ℘(z + z0; ω1, ω2) is the Weierstrass elliptic function, where ω1 and ω2 are two fundamental periods that are linearly independent over R. Then f(z) solves the equation f ′(z)2 = 4f 3(z) -g2f(z) -g3, where g2 and g3 are constants depending on ω1 and ω2. Then the Schwarzian derivative of f is that S(f, z) = -48f 4 -24g2f 2 -96g3f -3g2 2 32f 3 -8g2f -8g3 . 4 Then we choose the appropriate g2 and g3 such that f ′(1) = 0 and 4z3 - g2z -g3 = 0 has only simple roots e1, e2, e3, see [15, example 1.1]. According to the addition theorem[26, Chapter 20.3] and the properties of Weierstrass elliptic functions, f satisfies f(z + 1)f(z -1) + S(f, z) = -(e2e3 + e2 1) -e1f e1 -f 2 + -48f 4 -24g2f 2 -96g3f -3g2 2 8(f -e1)(f -e2)(f -e3) . Therefore, degf P = degf Q + 1 and e1, e2 and e3 are constant. Then we classify the case degf P ≤degf Q + 2 when Q(z, f) has only simple roots in f. Theorem 2 Let f(z) be a subnormal transcendental meromorphic solution of (1.7). Let Q(z, f) be of the following form: Q(z, f) := n Y i=1 (f(z) -bi(z)) ˆQ(z, f(z)), where the b′ is(1 ≤i ≤n, n ∈N) are distinct rational functions, ˆQ(z, f) and f -bi are coprime. (I) Assume that degf P = degf Q + 1. We have (1) if n = 3, then there exists an integer i ∈{1, · · · , n} such that the root bi is a constant; (2) if n = 2, and f -bi has finitely many double zeros for all i = 1, · · · , n, then there exists a root bi which is non-constant. (II)Suppose that degf P = degf Q + 2. We also have (3) if n = 2, then there exists an integer i ∈{1, · · · , n} such that the root bi is a constant; (III) Suppose that degf P ≤degf Q. For at least two i ∈{1, · · · , n}, we have (4) if n ≥3 and bi is non-constant, then f -bi has finitely many multiple zeros; (5) if n ≥2 and bi is constant, then f -bi has finitely many zeros with multiplicity at least 3. Remark. Example 3 demonstrates that condition (5) in case (III) of Theorem 2 can be satisfied. Example 4 verifies that condition (2) in case (I) of Theorem 2 can not be omitted. The following example shows that the case (II) in Theorem 2 can occur and the roots bi can be a constant for all i ∈{1, · · · , n}. 5 Example 5 Let f(z) = 1 sin(2πz)+ √ 2i. Then f(z) satisfies the following delay Schwarzian differential equation f(z + 1)f(z -1) + S(f, z) = f 2 + 2π2 -2 √ 2if + 1 [( √ 2i + 1)f -1][( √ 2i -1)f -1]. In this case, degf P = degf Q + 2, and all roots of Q(z, f) are constants. Now we discuss a special situation when Q(z, f) has a root of multiplicity 2 in the next theorem. Theorem 3 Let f(z) be a subnormal transcendental meromorphic solution of (1.7). Let Q(z, f) be of the following forms: Q(z, f) := (f(z) -b1(z))2 ˆQ(z, f(z)) where b1(z) is a rational function, ˆQ(z, f) and f-b1 are coprime. If degf P = degf Q + 1 and f -b1 has finite double zeros or if degf P = degf Q + 2, then f -b1 has infinitely many simple zeros. Remark. Example 2 can shows the case of degf P = degf Q+2 in Theorem 3 can occur. 2 Proof of Theorem 1 To prove Theorem 1, we first recall the necessary concepts and lemmas. A differential-difference polynomial in f(z) is defined by P(z, f) = X l∈L bl(z)f(z)l0,0f(z+c1)l1,0 · · · f(z+cν)lν,0f ′(z)l0,1 · · · f (μ)(z+cν)lν,μ, where c1, · · · , cν are distinct complex non-zero constants, L is a finite index set consisting of elements of the form l = (l0,0, · · · , lν,μ) and coefficients bl are meromorphic functions for all l ∈L. Nie, Huang , Wang and Wu modified the results of Halburd and Korhonen[10, Lemma 2.1] and gave the following lemma[18, Lemma 2.3]. Lemma 1 Let f(z) be a transcendental meromorphic solution of P(z, f) = 0, 6 where P(z, f) is a differential-difference polynomial in f with meromorphic coefficients bl(z) satisfying m(r, bl) = S(r, f). Let a1, · · · , ak be small functions to f such that P(z, ai) ̸≡0 is also a small function of f, for all i ∈{1, · · · , k}. If there exists s > 0 and τ ∈(0, 1) such that k X i=1 n r, 1 f -ai ≤kτn(r + s, f) + O(1), then σf > 0, where σf is defined as (1.4). We present the following lemma, which estimates the degree of R(z, f). Lemma 2 Let f(z) be a subnormal transcendental meromorphic solution of (1.7). Then degf R ≤7. Furthermore, if R(z, f(z)) is a polynomial in f with coefficients in z, then degf R ≤2. Proof. Taking the Nevanlinna characteristic function of both sides of (1.7) and using [13, Theorem 2.25] we have (degf R)T(r, f) = T(r, R(z, f)) + O(log r) = T(r, f(z + 1)f(z -1) + a(z)S(f, z)) + O(log r). From T(r, f ′) ≤2T(r, f) + S(r, f), (1.7) and [27, Theorem 1.2], we have (degf R)T(r, f) ≤T(r, f(z + 1)f(z -1)) + T(r, S(f, z)) + O(log r) ≤m r, f(z + 1) f(z) + m r, f(z -1) f(z) + 2m(r, f) + N(r, f(z + 1)f(z -1)) + 5T(r, f) + S(r, f) ≤7T(r, f) + S(r, f). where S(r, f) denotes any quantity satisfying S(r, f) = o(T(r, f)) as r →∞ and r ̸∈A1, A1 is a set with zero upper-density measure. This gives the first conclusion of Lemma 2. Now we suppose R is a polynomial in f with rational coefficients. We will consider the following two cases. Case 1 Suppose that f(z) has at most finitely many poles. Since R is a polynomial in f. From (1.7), we know that N(r, S(f, z)) = O(log r). Then from the lemma on the logarithmic derivative we have T(r, S(f, z)) = S(r, f) and (degf R)T(r, f) ≤T(r, f(z + 1)f(z -1)) + S(r, f) ≤m r, f(z + 1) f(z) + m r, f(z -1) f(z) + 2m(r, f) + S(r, f) ≤2T(r, f) + S(r, f), 7 where r ̸∈A2 and A2 is a set with zero upper-density measure. So degf R ≤2. Case 2 Assume that f(z) has infinitely many poles and we suppose that degf R ≥3. From conditions we know that the coefficients of R(z, f) and a(z) are rational, so they have finitely many zeros and poles. Therefore, there exists a constant M ∈R such that all zeros and poles of a(z) and the coefficients of R(z, f) lie in a disk |z| M1, where M1 is a positive constant such that M1 > M and z0 ± n ∈{z : |z| > M1} for finite number n ∈Z. Let z0 be a pole of f(z) with multiplicity t. Then z0 is either a regular point or a double pole of S(f, z). Since R is a polynomial in f and degf R ≥3, and we denote degf R = α for convenience, it follows from (1.7) that f(z) may has poles at z0+1 or z0-1 and the number of the multiplicities of the poles is more than tα. We may assume that f(z) takes a pole at z0 +1 with multiplicity p1 and z0 -1 is a pole of f with multiplicity p2, where p1 + p2 = tα. Subcase 2.1 Suppose that p1α -t ≤0, then we can know that p1 ≤t α and p2 ≥t(α2-1) α ≥2. From shifting (1.7), we have f(z + 2)f(z) + a(z + 1)S(f, z + 1) = R(z + 1, f(z + 1)), from which it follows that f(z0 + 2) is finite. If 0 ≥p1α -t > -2, z0 + 3 is a zero of f with multiplicity p1. From the iteration of shifting (1.7), we find that there may be no poles of f in the set {z0 + d}, where d > 3 is an integer. Now we calculate the multiplicities of poles of f in the set {z0 -n}, where n is a positive integer. From the shifting of (1.7), similarly, we find z0 -2 is a pole of f with multiplicity t(α2 -1) -p1α ≥1. We find that z0 -3 is the pole of f with multiplicity t(α3 -2α) + p1(-α2 + 1) ≥1 and z0 -4 is a pole of f with multiplicity t(α4 -3α2 + 1) + p1(-α3 + 2α) ≥1. From the recurrence relation we know that z0 -d(d ∈N) is a pole of f, and the multiplicity of the pole of f at z0 -d is 2p2-t(α- √ α2-4) 2 √ α2-4 · α+ √ α2-4 2 d-1 + O (1). Thus we have n(d + |z0|, f) ≥(α -1)d-2 + O(1) 8 for sufficiently large d ∈N, then 0 = lim sup r→∞ log+ T(r, f) r ≥lim sup r→∞ log n(r, f) r ≥lim sup d→∞ log n(d + |z0|, f) d + |z0| ≥lim sup d→∞ log(α -1)d-2 d + |z0| (2.1) = log(α -1) ≥log 2 > 0 which is a contradiction. If α -t ≤-2, z0 + 3 is a simple pole of f when p1 = 1 or f(z0+3) is finite when p1 ≥2. The number of multiplicities of poles of f in the set {z0-d, · · · , z0+d} is more than the case when 0 ≥α-t > -2. Similarly, we have the same conclusion. Subcase 2.2 Assume that p1α-t ≥1, then p1 ≥t+1 α and z0 +2 is a pole with multiplicity p1α -t. Similarly, from continuing the iteration of shifting of (1.7), we find that z0 +3 is a pole of f with multiplicity p1α2 -tα-p1 ≥1 and z0 + 4 is a pole of f with multiplicity p1α3 -tα2 -2p1α + t ≥1. If 2p1 -t(α - √ α2 -4) > 0, from the recurrence relation again, we know that z0 + d is a pole of f and the multiplicity of the pole of f at z0 -d is 2p1-t(α- √ α2-4) 2 √ α2-4 · α+ √ α2-4 2 d-1 + O (1). If 2p1 -t(α - √ α2 -4) ≤0, then we have 2p2 -t(α - √ α2 -4) > 0. Similar to the subcase 2.1 we can have a contradiction again. Hence we complete the proof. 2 Now we will consider the case when the polynomial Q(z, f) has a rational function root in f. Lemma 3 Let f(z) be a subnormal transcendental meromorphic solution of (1.7). Let R(z, f) be of the following form: R(z, f(z)) := P(z, f(z)) (f(z) -b1(z))k ˆQ(z, f(z)) (2.2) where k is a positive integer, b1 is a rational function, P(z, f) and ˆQ(z, f) are polynomials in f with rational coefficients, then k ≤2. Proof. Suppose a differential-difference polynomial Ψ(z, w) := (w -b1)k ˆQ(z, w)(w(z + 1)w(z -1) + ˆa(z)) -P(z, w), where k ≥1, and ˆa(z) := a(z)S(w, z) satisfies m(r, ˆa) = S(r, w). Clearly, from (1.7) and (2.2), we have Ψ(z, f) = 0. Since Ψ(z, b1) ̸= 0 is rational, the 9 first condition of Lemma 1 is satisfied for b1. By similar to the proof of [18, Lemma 2.3], we have N r, 1 f -b1 = T(r, f) + S(r, f). Now we consider the multiplicities of zeros of f -b1 and poles of f. We assume that z0 is a zero (or pole) of f -b1 with multiplicity p(≥1) and that none of a(z), b1(z) and any rational coefficients in R(z, f) have a zero or pole at z0. We can also require that for finite number n ∈Z, the shifting points z0 + n are not the zeros or poles of those rational coefficient functions including a(z) and b1(z). We call such a point z0 a generic zero (or generic pole) of f -b1 with multiplicity p, see in [18, Lemma 3.2]. Assume that k ≥3. Since k ≥3, then kp ≥3p > 2. Thus, f(z+1)f(z-1) has a pole with multiplicity kp(≥3) at z0 from (1.7) and (2.2). Case 1 Assume that degf(P) ≤k + degf( ˆQ). Subcase 1.1 Suppose the set E1 = {z : f(z+1) = g1(z) (z-z0)q , f(z-1) ̸= ∞}, where g1(z0) ̸= 0, ∞in E1 and q(≥kp) is an integer. If z0 ∈E1, then z0 + 1 is a double pole of S(f, z). By shifting (1.7) we obtain f(z + 2)f(z) + a(z + 1)S(f, z + 1) = P(z + 1, f(z + 1)) (f(z + 1) -b1(z + 1))k ˆQ(z + 1, f(z + 1)) . Thus f(z) has a double pole at z0+2, and so z0+2 is a double pole of S(f, z). By iterating (1.7) again, we have f(z + 3)f(z + 1) + a(z + 2)S(f, z + 2) = P(z + 2, f(z + 2)) (f(z + 2) -b1(z + 2))k ˆQ(z + 2, f(z + 2)) . Then f(z) has a zero at z0 + 3 with multiplicity q -2. If q = 3, then z0 + 4 is a double zero of f, so z0 + 4 is a double pole of S(f, z). Then z0 + 5 is a pole of f with multiplicity 3, z0 + 6 is a pole of f with multiplicity 4, z0 + 7 is a simple zero of f and z0 + 8 is a zero of f with multiplicity 4. Thus we can know that z0 + d is a zero or pole of f for any d ∈N. If q ≥4, then z0 +4 may be another generic zero of f -b1 such that z0 +5 is a pole of f. Suppose that the multiplicity of the zero of f -b1 at q1(≥1), 10 thus we can know that z0 +5 is a pole of f with multiplicity kq1 +q -2(≥3), z0 + 6 is a double pole of f and z0 + 7 is a zero of f. Suppose that the multiplicity of the zero of f at z0 + 7 is q2(≥1). When q2 ≥2, z0 + 8 may be the generic zero of f -b1. Similar to the above analysis, z0 + 9 and z0 + 10 are poles of f, z0 + 11 is a zero of f. When q2 = 1, similar to the case when q ≥4, z0 + d is a pole or zero of f for any d ∈N and d ≥5. From above the discussion, we have nE1 r, 1 f -b1 ≤1 knE1(r + 1, f) + S(r, f), where nEi(r, f) is the number of multiplicities of all poles of f in the set Ei T{z : |z| 0. This is a contradiction. Case 2 Assume that degf P ≥k + degf( ˆQ) + 1 and degf R = α(≥1). Subcase 2.1 Suppose the set E5 = {z : f(z+1) = g8(z) (z-z0)q , f(z-1) ̸= ∞} and z0 ∈E5, where g8(z0) ̸= 0, ∞in E5 and q ≥kp ≥3. If α ≥2, we continue to iterate the shifting and find that z0 + 2 is a pole of f with multiplicity qα, z0 + 3 is a pole of f with multiplicity q(α2 -1) ≥2, z0 + 4 is a pole of f with multiplicity q(α3 -2α) ≥2 and z0 + 5 is q(α4 -3α2 + 1) ≥2. From the recurrence relation we can know that z0 + d is a multiple pole of f for any d ∈N. If α = 1, then z0 + 2 is a pole of f with multiplicity q, z0 + 3 may be another zero of f -b1. From the above analysis, we have nE5 r, 1 f -b1 ≤1 knE5(r + 1, f) + S(r, f), where nE5(r, f) is the number of multiplicities of all poles of f in the set E5 T{z : |z| 1, we find that z0 -2 is a pole of f with multiplicity α ≥2, z0 -3 is a pole of f with multiplicity α2 -1 ≥2 and z0 -4 is a pole of f with multiplicity α3 -3α ≥2. From the recurrence relation again, we find that z0 ± d is a multiple pole of f for any d ∈N. 12 Thus, we obtain nE6 r, 1 f -b1 ≤1 knE6(r + 1, f) + S(r, f). Subcase 2.3 Suppose that E7 = {z : f(z + 1) = g11(z) (z-z0)q-2, f(z -1) = g12(z) (z-z0)2} and z0 ∈E7, where gm(z0) ̸= 0, ∞in E7 for m ∈{11, 12} and q -2 = kp -2 ≥1. If α = 1, then z0 -2 may be a generic zero of f -b1. When q -2 = 1 or q -2 ≥3 , similar to the subcase 2.2 we have that z0 + 2 is a pole of f with multiplicity q -2 and z0 + 3 is a generic zero of f -b1. When q -2 = 2, z0 + 2 may be a generic zero of f -b1. If α > 1, it similar to the subcase 2.1 and we can obtain that z0 ± d is a pole of f for any d ∈N. From the above, when α = 1 we have nE7 r, 1 f -b1 ≤2 knE7(r + 1, f) + S(r, f). When α > 1, we obtain nE7 r, 1 f -b1 ≤1 knE7(r + 1, f) + S(r, f). Subcase 2.4 Suppose that E8 = {z : f(z + 1) = g13(z) (z-z0)s1 , f(z -1) = g14(z) (z-z0)s2 } and z0 ∈E8, where gm(z0) ̸= 0, ∞in E8 for m ∈{13, 14} and s1 + s2 = kp. Similar to subcase 2.1, we have nE8 r, 1 f -b1 ≤1 knE8(r + 1, f) + S(r, f). By adding up the contribution from all points in S4 i=1 Ei to the corresponding counting functions, if α > 1, we get n r, 1 f -b1 = 8 X i=5 nEi r, 1 f -b1 ≤1 kn(r + 1, f) + O(1). If α = 1, we have n r, 1 f -b1 = 8 X i=5 nEi r, 1 f -b1 ≤2 kn(r + 1, f) + O(1). Thus, by Lemma 1, we have σf > 0, a contradiction. 2 13 Proof of Theorem 1. We know that degf R = max{degf P, degf Q} ≤7 from Lemma 2, This completes the proof of the first part. Assume that degf P ≥degf Q+3, then degf R = degf P ≥3 and suppose that α = degf R for convenience. We divide two cases as follows. Case 1 Suppose that f(z) has finitely many poles. From (1.7), we see that N(r, S(f, z)) = O(log r). Then we have T(r, S(f, z)) = S(r, f) and (degf R)T(r, f) ≤T(r, f(z + 1)f(z -1)) + T(r, S(f, z)) + S(r, f) ≤m r, f(z + 1) f(z) + m r, f(z -1) f(z) + 2m(r, f) + S(r, f) ≤2T(r, f) + S(r, f), where r ̸∈A3, where A3 is a set with zero upper-density measure. So it contradicts to degf R ≥3. Case 2 Assume that f(z) has infinitely many poles. Let z0 be a generic pole of f(z) with multiplicity p. Then R(z, f) has a pole at z0 with multiplicity pα(≥3), and z0 is either a regular point or a double pole of S(f, z). So f(z + 1)f(z -1) has a pole at z0 with multiplicity pα. Suppose that z0 + 1 is a pole of f with multiplicity p1(≥0), z0 -1 is pole of f with multiplicity p2(≥0), where p1 + p2 = pα. Subcase 2.1 Assume that p1α -p ≤0, then we have that p2 ≥p(α2-1) α and z0 -d is a pole of f from the recurrence relation. For sufficiently large d ∈N we find that the multiplicity of the pole of f at z0 -d is 2p2 -p(α - √ α2 -4) 2 √ α2 -4 · α + √ α2 -4 2 d-1 + O (1) . Thus, similar to the proof of Lemma 2, we have n(d + |z0|, f) ≥(α -1)d-2 + O(1) for all sufficiently large d ∈N. In this subcase, we also have equation (2.1) holds, which is a contradiction. Subcase 2.2 Suppose that p1α -p ≥1, similar to the proof in Lemma 2, we have a contradiction again. 2 3 Proof of Theorem 2 Obviously, none of b1, b2, · · · , bn is a solution of (1.7) and degf Q = n+degf ˆQ. We consider three cases as follows. 14 Case 1 Suppose that degf P = degf Q + 1. (1) Assume that the b′ is are non-constant for all i ∈{1, · · · , n}. We will derive a contradiction. Without loss of generality, let z0 be a generic zero of f -b1 with multiplicity q such that b′ 1(z0) ̸= 0. Firstly, we will prove f ′(z0) = 0 is impossible. If f ′(z0) = 0, then we find that q = 1, z0 is a double pole of S(f, z) and f(z + 1)f(z -1) has a double pole at z0. Subcase 1.1 Suppose that U1 = {z : f(z + 1) = h1(z) (z-z0)q1 , f(z -1) ̸= ∞} and z0 ∈U1, where h1(z) ̸= 0, ∞in U1 and q1 ≥2. When q1 = 2, then z0 + 2 may be another zero of f -bi. If f(z0 + 2) ̸= bi(z0 + 2) for i ∈{1, · · · , n}, then f(z + 3) has a double zero at z0, z0 + 4 is a double pole of f and z0 + 5 may be the generic zero of f -bi. If f(z0 + 2) = bi(z0 + 2) for i ∈{1, · · · , n}, we assume that the multiplicity of the generic zero of f -bi at z0 + 2 is p(≥1). When p = 1, for f ′(z0 + 2) ̸= 0, z0 + 3 is a simple zero of f and z0 -4 may be the generic zero of f -bi. For f ′(z0 + 2) = 0, f(z0 + 3) is finite. When p = 2, we also have f(z0 + 3) is finite. When p = 3, z0 +3 and z0 +4 are simple poles of f, z0 +5 may be the generic zero of f -bi. When p = 4, z0 + 3 is a double pole of f and z0 + 4 may be the generic zero of f -bi. When p ≥5, f(z0 + 3) and f(z0 + 4) are poles of f. When q1 ≥3, it's similar to the situation when q1 = 2. From the discussion above, we have n X i=1 nU1 r, 1 f -bi ≤2nU1(r + 1, f) + O(1), where nUi(r, f) is the number of multiplicities of all poles of f in the set Ui T{z : |z| 0 and we get a contradiction by Lemma 2. 15 Therefore, it's clear that f ′(z0) ̸= 0. In this case, z0 is a regular point of S(f, z) and a pole of f(z + 1)f(z -1) with multiplicity q(≥2). Subcase 1.3 Assume that U3 = {z : f(z + 1) = h4(z) (z-z0)2, f(z -1) ̸= ∞} and z0 ∈U3, where h4(z) ̸= 0, ∞in U3, then we have that z0 + 2 may be the generic zero of f -bi. It follows that n X i=1 nU3 r, 1 f -bi ≤2nU3(r + 1, f) + O(1). Subcase 1.4 Suppose that U4 = {z : f(z + 1) = h5(z) z-z0 , f(z -1) = h6(z) z-z0 } and z0 ∈U4, where hm(z) ̸= 0, ∞in U4 for m ∈{5, 6}, then z0 ±2 are simple poles of f and z0 ± 3 may be the generic zeros of f -bi. Thus we obtain n X i=1 nU4 r, 1 f -bi ≤nU4(r + 1, f) + O(1). Subcase 1.5 Assume that U5 = {z : f(z0 + 1) = h7(z) (z-z0)q2 , f(z0 -1) ̸= ∞} and z0 ∈U5, where h7(z) ̸= 0, ∞in U5 and q2 ≥3. By iteration of shifting, we can know that z0 + 2 is a pole of f with multiplicity q2 and z0 + 3 may be another zero of f -bi such that f(z0 + 4) is finite. Therefore, it follows that n X i=1 nU5 r, 1 f -bi ≤nU5(r + 1, f) + O(1). Subcase 1.6 Suppose that U6 = {z : f(z+1) = h8(z) (z-z0)q-1, f(z-1) = h9(z) z-z0 } and z0 ∈U6, where hm(z) ̸= 0, ∞in U6 for m ∈{8, 9} and q ≥3. Similar to the subcase 1.4, z0 -2 is a simple pole of f and z0 -3 may be the generic zero of f -bi. When q -1 = 2, it's similar to the subcase 1.3 and we have z0 + 2 may be the generic zero of f -bi. When q -1 ≥3, we find that z0 + 2 is a pole of f and f(z0 + 3) is finite. From the above analysis, we have n X i=1 nU6 r, 1 f -bi ≤2nU6(r + 1, f) + O(1). Subcase 1.7 Assume that U7 = {z : f(z + 1) = h10(z) (z-z0)q-2, f(z -1) = h11(z) (z-z0)2} and z0 ∈U7, where hm(z) ̸= 0, ∞in U7 for m ∈{10, 11} and q ≥3, then z0 -2 may be a generic zero of f -bi with multiplicity 2, i ∈{1, · · · , n}. Similar to the subcases 1.3-1.6, we have n X i=1 nU7 r, 1 f -bi ≤2nU7(r + 1, f) + O(1). 16 Subcase 1.8 Suppose that U8 = {z : f(z + 1) = h12(z) (z-z0)s1 , f(z -1) = h13(z) (z-z0)s2 } and z0 ∈U8, where hm(z) ̸= 0, ∞in U8 for m ∈{12, 13}, s1 ≥3, s2 ≥3 and s1 + s2 = q. Similar to the subcase 1.5, we find that n X i=1 nU8 r, 1 f -bi ≤nU8(r + 1, f) + O(1). From the above discussion, we can conclude that n X i=1 n r, 1 f -bi = n X i=1 8 X j=3 nUj r, 1 f -bi ≤2n(r + 1, f) + O(1). Then we have a contradiction from the Lemma 2. (2) Suppose that the b′ is are constant for all i ∈{1, 2, · · · , n}. Then by the condition and Lemma 1, f -bi has infinitely many zeros with multiplicity 1 or not less than 3. Without loss of generality, we may assume that z1 is a generic zero of f -b1 with multiplicity s, where s ≥3 or s = 1. If s = 1, then f ′(z) ̸= 0 and so S(f, z) is regular at z1. If s ≥3, then f ′(z1) = 0 and S(f, z) has a double pole at z1. Hence, for each case, we conclude that f(z + 1)f(z -1) has a pole with multiplicity s at z1. Similar to the proof of subcases 1.1-1.8, we also get n X i=1 n r, 1 f -bi ≤n(r + 1, f) + O(1). (3.1) Therefore, it follows that σf > 0, a contradiction from Lemma 2. Case 2 Suppose that degf P = degf Q + 2. (3)Assume that the bi's are non-constant for all i ∈{1, · · · , n}. We can let z0 be a generic zero of f -b1 with multiplicity q such that b′ 1(z0) ̸= 0 without loss of generality. If f ′(z0) = 0, then q = 1 and z0 is a double pole of S(f, z), so f(z + 1)f(z -1) has a double pole at z0. Subcase 2.1 Suppose that U9 = {z : f(z + 1) = h14(z) (z-z0)q3 , f(z -1) ̸= ∞}, where h14(z) ̸= 0, ∞in U9 and q3 ≥2. If z0 ∈U9, then z0 + 2 is a pole of f with multiplicity at least 4, z0 + 3 is a pole of f with multiplicity at least 6 and z0 + 4 is a pole of f with multiplicity at least 8. From the recurrence relation, we have that z0 + d is a multiple pole of f with multiplicity at least 2d for any d ∈N. Therefore, it follows that n X i=1 nU9 r, 1 f -bi ≤1 2nU9(r + 1, f) + O(1). Subcase 2.2 Assume that U10 = {z : f(z + 1) = h15(z) z-z0 , f(z -1) = h16(z) z-z0 } and z0 ∈U10, where hm(z) ̸= 0, ∞in U10 for m ∈{15, 16}. Then z0 ± 2 are 17 double poles of f, z0 ± 3 are poles of f with multiplicity 3 and z0 ± 4 are poles of f with multiplicity 4. Similar to the subcase 2.1, z0 ± d are poles of f for any d ∈N. Thus we have n X i=1 nU10 r, 1 f -bi ≤1 2nU10(r + 1, f) + O(1). Therefore, we add up the contribution from all points z in the set U9 S U10 to the corresponding counting functions, we have n X i=1 n r, 1 f -bi = n X i=1 nU9 r, 1 f -bi + nU10 r, 1 f -bi ≤1 2n(r + 1, f) + O(1). So it follows that σf > 0 by Lemma 1, a contradiction. From the discussion above, we have f ′(z0) ̸= 0. In this case, z0 is a regular point of S(f, z) and a pole of f(z +1)f(z -1) with multiplicity q. Similar to the subcases 2.1 and 2.2, we can have the same inequality (3.1). thus we get a contradiction by Lemma 1. Case 3 Suppose that degf P ≤degf Q. (4) Assume that f-bi has infinitely many multiple zeros for i ∈{1, · · · , n}. Then we will prove by contradiction. Without loss of generality, let z0 be a generic zero of f -b1 with multiplicity p(≥2) such that b′ 1(z0) ̸= 0. Then f ′(z0) ̸= 0 and f(z + 1)f(z -1) has a pole with multiplicity p at z0. Subcase 3.1 Suppose that U11 = {f(z0 + 1) = h17(z) (z-z0)q4 , f(z0 -1) ̸= ∞} and z0 ∈U11, where h17(z) ̸= 0, ∞in U11 and q4 ≥p, then f(z + 2) has a double pole at z0. When q4 = 2, we have z0 + 3 may be the generic zero of f -bi. When q4 = 3, then z0 + 3 is a simple zero of f, z0 + 4 is a double zero of f, z0 + 5 is a pole of f with multiplicity 3 and z0 + 6 is a pole of f with multiplicity 4. From continuing the iteration of shifting, we know that z0 + d is not a generic zero of f -bi for all d ∈N. When q4 ≥4, we find that z0 + 3 is a zero of f with multiplicity q4 -2(≥2) and z0 + 4 may be another generic zero of f -b1. From the above analysis, we obtain n X i=1 nU11 r, 1 f -bi ≤nU11(r + 1, f) + O(1), Subcase 3.2 Assume that U12 = {z : f(z + 1) = h18(z) z-z0 , f(z -1) = h19(z) z-z0 } and z0 ∈U12, where hm(z) ̸= 0, ∞in U12 for m ∈{18, 19} , by iteration of shifting (1.7), we find that z0 ± 2 may be generic zeros of f -bi. Thus we 18 have n X i=1 nU12 r, 1 f -bi ≤2nU12(r + 1, f) + O(1). Subcase 3.3 Assume that U13 = {f(z+1) = h20(z) (z-z0)s1 , f(z-1) = h21(z) (z-z0)s2 } and z0 ∈U13, where hm(z) ̸= 0, ∞in U13 for m ∈{20, 21} , s1 ≥1, s2 ≥1 and s1 + s2 = p. It's similar to the subcase 3.1 and 3.2 and we have that n X i=1 nU13 r, 1 f -bi ≤2nU13(r + 1, f) + O(1). From the above discussion, we have n X i=1 n r, 1 f -bi = n X i=1 13 X j=11 nUj r, 1 f -bi ≤2n(r + 1, f) + O(1). Therefore, it follows that we get a contradiction by Lemma 1. (5) Suppose that f -bi has infinitely many zeros with multiplicity 3 or higher. We will again prove by contradiction. Without loss of generality, let z0 be a generic zero of f -b1 with multiplicity p ≥3. From (1.7), f(z + 1)f(z -1) has a pole at z0 with multiplicity p. Subcase 3.4 Assume that U14 = {f(z0 + 1) = h22(z) (z-z0)q5 , f(z0 -1) ̸= ∞} and z0 ∈U14, where h22(z) ̸= 0, ∞in U14 and q5 ≥p, then z0 is a double pole of f(z + 2) and a zero of f(z + 3) with multiplicity q5 -2 ≥1. Similarly to the proof in subcase 3.1, we have n X i=1 nU14 r, 1 f -bi ≤nU14(r + 1, f) + O(1). Subcase 3.5 Suppose that U15 = {f(z +1) = h23(z) (z-z0)p-1, f(z -1) = h24(z) z-z0 } and z0 ∈U15, where hm(z) ̸= 0, ∞in U15 for m ∈{23, 24} , then f(z0 -2) is finite. When p = 3, z0 + 2 is a double pole of f and z0 + 3 may be the generic zero of f -bi. When p ≥4, it's similar to the argument in subcase 3.1. So we give n X i=1 nU15 r, 1 f -bi ≤nU15(r + 1, f) + O(1). Subcase 3.6 Assume that U16 = {f(z+1) = h25(z) (z-z0)s1 , f(z-1) = h26(z) (z-z0)s2 } and z0 ∈U16, where hm(z) ̸= 0, ∞in U16 for m ∈{25, 26} and s1 + s2 = p. Similarly to the proof in subcase 3.4 and 3.5, we have n X i=1 nU16 r, 1 f -bi ≤nU16(r + 1, f) + O(1). 19 From the above analysis, we obtain n X i=1 n r, 1 f -bi = n X i=1 16 X j=14 nUj r, 1 f -bi ≤n(r + 1, f) + O(1). Therefore, we can get σf > 0 by Lemma 2, a contradiction. 4 Proof of Theorem 3 Notice that b1 is not a solution of (1.7), and degf Q = 2+degf ˆQ. Suppose that f -b1 has only finitely many simple zeros. By the proof in [18, Lemma 2.3], we have N r, 1 f -b1 = T(r, f) + S(r, f). Thus, f -b1 has infinitely many multiple zeros. Let z0 be a generic zero of f -b1 with multiplicity p(≥2). Case 1 Suppose that degf P = degf Q + 1. If b1 is a constant, then f ′(z0) = 0 and hence z0 is a double pole of S(f, z). By iteration of shifting, f(z + 1)f(z -1) has a pole with multiplicity 2p at z0. Subcase 1.1 Assume that L1 = {z : f(z + 1) = l1(z) (z-z0)q1 , f(z -1) ̸= ∞} and z0 ∈L1, where l1(z) ̸= 0, ∞in L1 and q1 ≥2p, then z0 + 2 is a pole of f with multiplicity at least 2p and z0 + 3 may be another generic zero of f -b1. Thus we have nL1 r, 1 f -b1 ≤1 2nL1(r + 1, f) + O(1), where nLi(r, f) is the number of multiplicities of all poles of f in the set Li T{z : |z| 0 by Lemma 1, a contradiction. Hence, b1 is non-constant. So there exists a point z0 such that b′ 1(z0) ̸= 0 and f ′(z0) ̸= 0. Similarly with the case when b1 is constant, we also obtain σf > 0, a contradiction. Case 2 Assume that degf P = degf Q + 2. If b1 is a constant, then f ′(z0) = 0 and hence z0 is a double pole of S(f, z). By iteration of shifting, f(z + 1)f(z -1) has a pole with multiplicity 2p at z0. Subcase 2.1 Suppose that L4 = {z : f(z + 1) = l6(z) (z-z0)q2 , f(z -1) ̸= ∞} and z0 ∈L4, where l6(z) ̸= 0, ∞in L4 and q2 ≥2p. From the recurrence relation, we know that z0 + d is multiple poles of f for any d ∈N. Thus, it follows that nL4 r, 1 f -b1 ≤1 2nL4(r + 1, f) + O(1). Subcase 2.2 Assume that L5 = {z : f(z + 1) = l7(z) (z-z0)s1 , f(z -1) = l8(z) (z-z0)s2 } and z0 ∈L5, where lm(z) ̸= 0, ∞in L5 for m ∈{7, 8} , s1 ≥1, s2 ≥1 and s1 + s2 = 2p. Similar to the proof in subcase 2.1, we have nL5 r, 1 f -b1 ≤1 2nL5(r + 1, f) + O(1). In summary, it follows that n r, 1 f -b1 = nL4 r, 1 f -b1 + nL5 r, 1 f -b1 ≤1 2n(r + 1, f) + O(1). Thus σf > 0 by Lemma 1, a contradiction. Hence, b1 is non-constant. So there exists a point z0 such that b′ 1(z0) ̸= 0 and f ′(z0) ̸= 0. Similarly with the case when b1 is constant, we also obtain σf > 0, a contradiction. References [1] M. J. Ablowitz, R. G. Halburd, B. Herbst, On the extension of the Painlev ́e property to difference equations. Nonlinearity. 13 (2000), no. 3, 889-905. 21 [2] T.-B. Cao, Y. Chen, R. J. Korhonen, Meromorphic solutions of higher order delay differential equations. Bull. Sci. Math. 182 (2023), Paper No. 103227, 28 pp. [3] Y. Chen, T.-B. Cao, Meromorphic solutions of a first order differential equations with delays. C. R. Math. Acad. Sci. Paris. 360 (2022), 665-678. [4] R. Conte, M. Musette The Painlev ́e handbook. Springer, Dordrecht, 2008. xxiv+256 pp. [5] R. Conte, The Painlev ́e property. One century later. CRM Ser. Math. Phys. SpringerVerlag, New York, 1999. xxvi+810 pp. [6] R. Fuchs, ̈Uber lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singul ̈aren Stellen. Math. Ann. 63 (1907), no. 3, 301-321. [7] B. Gambier, Sur les ́equations diff ́erentielles du second ordre et du premier degr ́e dont l'int ́egrale g ́en ́erale est a points critiques fixes. Acta Math. 33 (1910), no. 1, 1-55. [8] V. I. Gromak., I. Laine, S. Shimomura, Painlev ́e differential equations in the complex plane. De Gruyter Stud. Math., 28 Walter de Gruyter & Co., Berlin, 2002. viii+303 pp. [9] R. G. Halburd, R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painlev ́e equations. Proc. Lond. Math. Soc. 3 (2007), no. 2, 443-474. [10] R. G. Halburd, R. J. Korhonen, Growth of meromorphic solutions of delay differential equations. Proc. Amer. Math. Soc. 145 (2017), no. 6, 2513-2526. [11] R. G. Halburd, R. J. Korhonen, Meromorphic solutions of difference equations, integrability and the discrete Painlev ́e equations. J. Phys. A 40 (2007), no. 6, R1-R38. [12] P.-C. Hu, M.-L. Liu, A Malmquist type theorem for a class of delay differential equations. Bull. Malays. Math. Sci. Soc. 44 (2021), no. 1, 131-145. [13] I. Laine, Nevanlinna theory and complex differential equations. De Gruyter Stud. Math., 15 Walter de Gruyter & Co., Berlin, 1993. viii+341 pp. [14] K. Ishizaki, Admissible solutions of the Schwarzian differential equation. J. Austral. Math. Soc. Ser. A 50 (1991), 258-278. [15] K. Ishizaki, Meromorphic solutions of difference Riccati equations. Complex Var. Elliptic Equ. 62 (2017), no. 1, 110-122. [16] K. Liu, I. Laine, L.-Z. Yang, Complex delay-differential equations. De Gruyter Stud. Math., 78; De Gruyter, Berlin, [2021], x+290 pp. [17] J. Malmquist, Sur les fonctions a un nombre fini de branches d ́efinies par les ́equations diff ́erentielles du premier ordre. Acta Math. 36 (1913), no. 1, 297-343. [18] X.-T. Nie, J.-X. Huang, Y.-F. Wang, C.-F. Wu, Meromorphic solutions of delay Schwarzian differential equations. Acta Math. Sci. Ser. B (Engl. Ed.) 45 (2025), no. 4, 1514-1528. [19] P. Painlev ́e, M ́emoire sur les ́equations diff ́erentielles dont l'int ́egrale g ́en ́eraleest uniforme. Bull. Soc. Math. France. 28 (1900), 201-261. [20] P. Painlev ́e, Sur les ́equations diff ́erentielles du second ordre et d'ordre sup ́erieur dont l'int ́egrale g ́en ́erale est uniforme. Acta. Math. 25 (1902), no.1, 1-85. 22 [21] G. R. W. Quispel, H. W. Capel, R. Sahadevan, Continuous symmetries of differentialdifference equations: the Kac-van Moerbeke equation and Painlev ́e reduction Phys. Lett. A 170 (1992), no. 5, 379-383. [22] O. Ronkainen, Meromorphic solutions of difference Painlev ́e equations, Dissertation, 2010. Ann. Acad. Sci. Fenn. Math. Diss. 155 (2010), 59 pp. [23] Q. Wang, Q. Han, P.-C. Hu, Quantitative properties of meromorphic solutions to some differential difference equations. Bull. Aust. Math. Soc. 99 (2019), no. 2, 250-261. [24] Z.-T. Wen, Finite order solutions of difference equations, and difference Painlev ́e equations IV . Proc. Amer. Math. Soc. 144 (2016), no. 10, 4247-4260. [25] Z.-T. Wen, Meromorphic solutions to difference Painlev ́e equations I and II. Electron. J. Differential Equations 2016, Paper No. 262, 18 pp. [26] E. T. Whittaker, G. N. Watson, A course of modern analysis an introduction to the general theory of infinite processes and of analytic functions with an account of the principal transcendental functions. Cambridge University Press, Cambridge, 2021. lii+668 pp. [27] J.-H. Zheng, R. J. Korhonen, Studies of differences from the point of view of Nevanlinna theory. Trans. Amer. Math. Soc. 373 (2020), no. 6, 4285-4318. 23
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2510.14735
|
STRONGLY DOUBLY REVERSIBILE PAIRS IN QUATERNIONIC
UNITARY GROUP OF SIGNATURE (n, 1)
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
Abstract. Let PSp(n, 1) denote the isometry group of quaternionic hyperbolic space
Hn
H. A pair of elements (g1, g2) in PSp(n, 1) is said to be strongly doubly reversible if
(g1, g2) and (g−1
1 , g−1
2 ) belong to the same simultaneous conjugation orbit of PSp(n, 1),
and a conjugating element can be chosen to have order two. Equivalently, there exist
involutions i1, i2, i3 ∈PSp(n, 1) such that g1 = i1i2, g2 = i1i3. We prove that the set of
such pairs has Haar measure zero in PSp(n, 1) × PSp(n, 1). The same result also holds
for PSp(n) × PSp(n) for n ≥2.
In the special case n = 1, we show that every pair of elements in PSp(1) is strongly
doubly reversible. Using elementary quaternionic analysis for Sp(1), we also provide a
very short proof of a theorem of Basmajian and Maskit, in Trans. Amer. Math. Soc. 364
(2012), no. 9, 5015–5033, which states that every pair of elements in SO(4) is strongly
doubly reversible.
Furthermore, we derive necessary conditions under which a pair of hyperbolic elements
is strongly doubly reversible in PSp(1, 1).
1. Introduction
An element in a group G is said to be strongly reversible (or strongly real) if it can
be written as a product of two involutions in G. This notion is closely related to that of
reversible (or real) elements, which are conjugate to their inverses in G. Every strongly
reversible element is necessarily reversible, but the converse is not true, in general. The
classification and structure of such elements have been the subject of regular investigation
in various branches of mathematics, for example, see [1], [4], [5], [10], [14], [16], [18].
Beyond their algebraic significance, strongly reversible elements play a central role
in understanding symmetries in geometry. In particular, certain geometrically natural
groups are built entirely from such elements.
A classical and striking example arises
in the setting of hyperbolic geometry. The group PSL(2, C) can be identified with the
orientation-preserving isometries of the three dimensional hyperbolic space.
If A and
B are elements of PSL(2, C) generating a non-elementary subgroup, then there exist
involutions i1, i2, i3 ∈PSL(2, C) such that A = i1i2 and B = i1i3, see [7], [9]. In the
real hyperbolic case, a similar statement holds in PSL(2, R), with the involutions being
orientation-reversing reflections.
Date: October 17, 2025.
2020 Mathematics Subject Classification. Primary 51M10; Secondary: 15B33, 22E43.
Key words and phrases. reversible elements, strongly doubly reversible, quaternion, hyperbolic space,
product of involutions, compact symplectic group.
1
arXiv:2510.14735v1 [math.GR] 16 Oct 2025
2
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
These observations motivate a more general concept that extends beyond individual
decompositions to relationships between two elements and can be formulated for any
abstract group.
Definition 1.1. Let G be a group. Consider the G action on G × G by simultaneous
conjugation:
g.(g1, g2) = (gg1g−1, gg2g−1).
For two elements g1, g2 ∈G, the pair (g1, g2) is said to be doubly reversible or doubly
real if (g1, g2) and (g−1
1 , g−1
2 ) belong to the same G-conjugation orbit. Furthermore, if we
choose the conjugating element g to be such that it is an involution, i.e. g2 = 1, then we
call (g1, g2) to be strongly doubly reversible or strongly doubly real.
This notion can be extended to any k-tuple of elements in G to define k-reversible (or,
k-real) and strongly k-reversible (or, strongly k-real) tuples in a similar manner. However,
in this paper, we restrict our attention to the case k = 2, specifically focusing on strongly
2-reversible or strongly doubly reversible elements.
Note that if (g1, g2) is strongly doubly reversible, then there exist involutions i1, i2, i3 ∈
G such that g1 = i1i2 and g2 = i1i3. Conversely, if there are involutions i1, i2, i3 ∈G such
that g1 = i1i2 and g2 = i1i3, then (g1, g2) is strongly doubly reversible. In particular,
every strongly doubly reversible element is doubly reversible.
The above interpretation captures the geometric compatibility between elements g1 and
g2. For a strongly doubly reversible pair (g1, g2), both elements are generated by pairs
of involutions that share a common factor. The classical result for PSL(2, C) may thus
be interpreted as asserting that any two generators of a non-elementary subgroup are
necessarily strongly doubly reversible.
The study of such pairs is particularly interesting in groups where every element is
a product of two involutions. In such settings, one may naturally ask which pairs of
elements in such a group are strongly doubly reversible. This question not only provides
understanding about the group’s internal structure, but also connects it to broader topics
like discreteness and geometric finiteness. For instance, when g1 and g2 are strongly doubly
reversible, the subgroup ⟨g1, g2⟩sits as an index-two subgroup of the group ⟨i1, i2, i3⟩
generated by involutions. This can potentially lead to better insight into groups generated
by three involutions, e.g., triangle groups in hyperbolic geometry. Despite its relevance,
the problem of classifying strongly doubly reversible elements remains largely unexplored.
Even within the context of finite groups, systematic efforts to understand doubly reversible
pairs have begun only recently, e.g. [6]. The terminology ‘k-real’ has been borrowed from
[6].
In geometric contexts, Basmajian and Maskit in [2], posed the problem of generalizing
the classical PSL(2, C) result to higher-dimensional M¨obius groups and isometry groups
of Riemannian space forms. It may be noted that strongly doubly reversible pairs were
termed linked pairs in [2]. Basmajian and Maskit proved that for higher dimensions,
especially n ≥5, almost all pairs in these groups are not strongly doubly reversible. Bas-
majian and Maskit also proved that every pair in the orthogonal group SO(4) is strongly
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
3
doubly reversible. In a subsequent work, Silverio [15] provided a geometric description
of strongly doubly reversible pairs in real hyperbolic 4-space.
In complex hyperbolic
geometry, the strongly doubly reversible pairs acquire additional structure. When every
element in PU(n, 1) is a product of two anti-holomorphic involutions, not every element of
PU(n, 1) is a product of (holomorphic) involutions, cf. [8]. In PU(2, 1),the isometry group
of the two-dimensional complex hyperbolic space, they are classified as R-decomposable or
C-decomposable, depending on whether the generating involutions are anti-holomorphic or
holomorphic. Will [17] classified these loxodromic pairs, while Paupert and Will [11] gave
a complete classification of the R-decomposable pairs in PU(2, 1). The C-decomposable
pairs in PU(2, 1) have been described by Ren et al. [13].
Let Hn
H denote the n-dimensional quaternionic hyperbolic space, whose isometry group
is PSp(n, 1) = Sp(n, 1)/{±I}. A result by Bhunia and Gongopadhyay [3] shows that
every element of Sp(n, 1) can be expressed as a product of two skew-involutions. Recall
that a skew-involution is an element i ∈Sp(n, 1) such that i2 = −1. The skew-involutions
project to involutions in PSp(n, 1). In contrast to PSp(n, 1), the group Sp(n, 1) itself has
relatively few genuine involutions, and not all elements can be written as products of two
such. Since every element of PSp(n, 1) is strongly reversible, it is a natural problem to
explore strongly doubly reversible pairs in PSp(n, 1).
We prove the following theorem in this regard. This is a generalization of [2, Theorem
1.5] for isometries of Hn
H.
Theorem 1.2. The set of strongly doubly reversible pairs in PSp(n, 1) has Haar measure
zero in PSp(n, 1) × PSp(n, 1).
In other words, almost all pair in PSp(n, 1) is not strongly doubly reversible. It also
follows that that same result also hold for strongly doubly reversible pairs in PSp(n) for
n ≥2. We see as a corollary to the above theorem.
Corollary 1.3. Let n ≥2. The set of strongly doubly reversible pairs in PSp(n) has Haar
measure zero in PSp(n) × PSp(n).
However, when n = 1, we see that every element in PSp(1) is strongly doubly reversible.
We have used elementary quaternionic analysis to see this for PSp(1). We also apply this
result to offer a very short proof of [2, Theorem 1.4], which is the following.
Theorem 1.4. Every pair of elements in SO(4) is strongly doubly reversible.
One may ask how can we classify strongly doubly reversible pairs of isometries in quater-
nionic hyperbolic space? The challenge lies in the unique algebraic features of quaternions,
namely their noncommutativity. Additionally, the absence of a well-behaved trace func-
tion or complete conjugacy invariants in the quaternionic setting adds to the difficulty. It
may be noted that such conjugacy invariants are critical in complex hyperbolic settings
for classify strongly doubly reversible pairs. As a result, many of the familiar tools from
complex hyperbolic geometry do not carry over directly. It seems a difficult problem in
the quaternionic set up to classify strongly doubly reversible pairs.
4
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
Following the terminology in [12], recall that an element g in Sp(n, 1) is called hyperbolic
if it has exactly two fixed points in the boundary. Hyperbolic elements have three mutually
disjoint classes of eigenvalues. We prove the following result.
Theorem 1.5. Let A and B be hyperbolic elements in PSp(n, 1). Then (A, B) is doubly
reversible if and only if it is strongly doubly reversible.
The proof of this theorem relies on an analysis of the strongly doubly reversible pairs
in PSp(1, 1). Further we have obtained a necessary criteria for two hyperbolic elements
in PSp(1, 1) to be strongly doubly reversible. This necessary criteria rely on the Cartan’s
angular invariant.
Theorem 1.6. Let A and B be hyperbolic elements in PSp(1, 1) with no common fixed
points. If A(aA, rA, aB) ̸= A(rA, aA, rB), then A and B can not be strongly doubly re-
versible.
The converse of the above theorem does not hold in general. We have indicated this
with an example in Remark 4.
Structure of the paper. After discussing notations and preliminaries in Section 2, we prove
that every pair of elements in SO(4) is strongly doubly reversible in Section 3.2.
In
Section 4, we prove that the set of strongly doubly reversible pairs in PSp(n, 1) has Haar
measure zero in PSp(n, 1) × PSp(n, 1). In Section 5,we prove that a pair of hyperbolic
elements (A, B) is doubly reversible if and only if it is strongly doubly reversible. Finally,
in Section 5 and in Section 7, we provide a characterization and a quantitative description,
respectively, of strongly doubly reversible hyperbolic pairs in PSp(1, 1).
2. Preliminaries
2.1. Doubly reversible pairs. Let G acts on G × G by conjugation. Let the stabilizer
subgroup under this action is:
SG((g1, g2)) = {h ∈G | h(g1, g2)h−1 = (g1, g2)}
It is easy to see that SG((g1, g2)) = ZG(g1) ∩ZG(g2), where ZG(g) denote the centralizer
of g in G.
Consider the ‘reverser’ set:
RG((g1, g2)) = {h ∈G | h(g1, g2)h−1 = (g−1
1 , g−1
2 )}
Define:
EG((g1, g2)) = SG((g1, g2)) ∪RG((g1, g2)).
It is easy to see that E((g1, g2)) is a subgroup of G×G: if h1, h2 ∈RG, then h−1
1 h2 ∈SG.
Lemma 2.1. SG((g1, g2)) is a normal subgroup of E((g1, g2)) of index atmost two.
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
5
Proof. Define a map ϕ : E((g1, g2)) →Z2 by,
ϕ(h) =
1
if h(g1, g2)h−1 = (g1, g2)
−1
if h(g1, g2)h−1 = (g−1
1 , g−1
2 )
This is a homomorphrism with kernel SG((g1, g2)) = ZG(g1) ∩ZG(g2).
□
Thus, if g1 and g2 are in sufficiently general position such that the intersection of their
centralizer is trivial, then a reversing symmetry is unique for a strongly doubly reversible
pair (g1, g2).
2.2. The Quaternions. Let H := R + Ri + Rj + Rk denote the division algebra of
Hamilton’s quaternions, where the fundamental relations are given by i2 = j2 = k2 =
ijk = −1. Every element of H can be written uniquely in the form q = a + bi + cj +
dk, where a, b, c, d ∈R. Alternatively, viewing H as a two-dimensional vector space over
C, we may express q = c1 + c2j, with c1, c2 ∈C. The modulus (or norm) of q is defined
by |q| =
√
a2 + b2 + c2 + d2. We denote the set
Sp(1) := {q ∈H : |q| = 1}
by the group of unit quaternions.
We consider Hn as a right H-module. We consider Hn as a right vector space over
the quaternions.
A non-zero vector v ∈Hn is said to be a (right) eigenvector of A
corresponding to a (right) eigenvalue λ ∈H if the equality Av = vλ holds.
Eigenvalues of every matrix over the quaternions occur in similarity classes, and each
similarity class of eigenvalues contains a unique complex number with non-negative imag-
inary part. Here, instead of similarity classes of eigenvalues, we will consider the unique
complex representatives with non-negative imaginary parts as eigenvalues unless specified
otherwise.
In places where we need to distinguish between the similarity class and a
representative, we shall write the similarity class of an eigenvalue representative λ as [λ].
2.3. Quaternionic Hyperbolic Space. Let V = Hn,1 denote the right vector space of
dimension n + 1 over H, equipped with the Hermitian form:
⟨z, w⟩= w∗H1z = ¯wn+1z1 + Σn
i=2 ¯wizi + ¯w1zn+1,
where ∗denotes the conjugate transpose, and
H1 =
0
0
1
0
In−1
0
1
0
0
.
We consider the following subspaces of Hn,1 :
V−= {z ∈Hn,1 : ⟨z, z⟩< 0}, V+ = {z ∈Hn,1 : ⟨z, z⟩> 0},
V0 = {z ∈Hn,1 \ {0} : ⟨z, z⟩= 0}.
6
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
Let P : Hn,1 \{0} −→HPn be the right projection onto the quaternionic projective space.
The image of a vector z will be denoted by z.
The projective model of the quaternionic hyperbolic space is given by Hn
H = P(V−).
The boundary at infinity of this space is ∂Hn
H = P(V0).
The above Hermitian form may be replaced by an equivalent one associated with the
matrix Ho:
Ho =
−1
0
. . .
0
0
1
. . .
0
...
...
...
0
0
. . .
1
,
where the corresponding Hermitian form ⟨z, w⟩o = w∗Hoz gives the ball model of Hn
H.
Given a point z of Hn
H ∪∂Hn
H −{∞} ⊂HPn we may lift z = (z1, z2, . . . , zn) to a point
z in V0 ∪V−, given by
z =
z1
z2
...
1
.
Here z is called the standard lift of z. There are two points: ‘zero’ and ‘infinity’ in the
boundary given by:
o =
0
0
...
1
, ∞=
1
0
...
0
.
Let Sp(n, 1) be the isometry group of the Hermitian form H1.
Each matrix A in
Sp(n, 1) satisfies the relation A−1 = H1
−1A∗H1, where A∗is the conjugate transpose of
A. The isometry group of Hn
H is the projective unitary group PSp(n, 1) = Sp(n, 1)/{±I}.
However, we shall mostly deal with the linear group Sp(n, 1).
2.4. Classification of elements. Following the terminology in [12], recall that an ele-
ment g ∈Sp(n, 1) is called hyperbolic if it has exactly two fixed points on the boundary.
An element g ∈Sp(n, 1) is called parabolic if it has a unique fixed point on the boundary,
and elliptic if it has a fixed point in Hn
H. An element g in Sp(n, 1) belongs to exactly one
of these three classes.
2.5. Hyperbolic Isometries. Consider a hyperbolic isometry A ∈Sp(n, 1).
Let [λ]
denote the conjugacy class of eigenvalues associated with A, and choose a representative
eigenvalue λ with a corresponding eigenvector x. The vector x determines a point in
quaternionic projective space HPn, which lies either on the boundary ∂H2
H or, is a point
in P(V+). The corresponding line xH in the space Hn,1 represents the lift of this projective
point and is invariant under the action of A. This line is the eigenspace generated by x.
In the hyperbolic case, two of the eigenvalue classes are of null-type, with their associ-
ated eigenlines corresponding to fixed points on the boundary - one attracting, the other
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
7
repelling. Suppose the repelling fixed point on ∂Hn is denoted by rA and corresponds
to the eigenvalue reiθ, while the attracting fixed point aA corresponds to the eigenvalue
r−1eiθ. Let rA and aA denote their respective lifts to Hn,1. Additionally, for each j, let
xj,A be an eigenvector of A associated with the eigenvalue eiϕj. It is convenient to assume
that the angles θ, ϕj lie within the interval [0, π] and 0 < r < 1. Each xj,A defines a point
in P(V+).
Now, given parameters (r, θ, ϕ1, . . . , ϕn−1), we can define the matrix EA(r, θ, ϕ1, . . . , ϕn−1),
simply denoted by EA, with respect to the standard Hermitian form H0:
(2.1)
EA(r, θ, ϕ1, . . . , ϕn−1) = Diag
reiθ, eiϕ1, eiϕ2, . . . , eiϕn−1, r−1eiθ
Construct the matrix
CA = [aA
x1,A
· · ·
xn−2,A
xn−1,A
rA] ,
whose columns are the eigenvectors corresponding to the eigenvalues used in EA. By
suitably scaling the eigenvectors, we can ensure that CA belongs to Sp(n, 1), by enforcing
the normalization:
⟨aA, rA⟩= 1,
⟨xj,A, xj,A⟩= 1.
With this choice of basis, the matrix A is conjugate to the diagonal matrix EA, i.e.,
A = CAEAC−1
A .
So, every hyperbolic element A in Sp(n, 1) is conjugate to a matrix of the form EA.
Lemma 2.2. (Chen-Greenberg) Two hyperbolic elements in Sp(n, 1) are conjugate if and
only if they have the same similarity classes of eigenvalues.
2.6. Cartan’s angular invariant. Let p1, p2, p3 be distinct points on the boundary ∂Hn,
with lifts p1, p2, p3, respectively. The Hermitian triple product is defined by
H(p1, p2, p3) = ⟨p1, p2⟩⟨p2, p3⟩⟨p3, p1⟩.
The Cartan angular invariant A(p1, p2, p3) is defined as
A(p1, p2, p3) = arccos
ℜ
−H(p1, p2, p3)
|H(p1, p2, p3)|
!
,
The Cartan angular invariant takes values in the interval [0, π
2]. It is independent of the
choice of lifts and is invariant under the action of Sp(n, 1).
Lemma 2.3. Let A ∈Sp(n, 1) be a hyperbolic element expressed as a product of two
skew-involutions:
A = i1i2,
where i2
1 = i2
2 = −I.
Then Both i1 and i2 permute the fixed points of A.
Proof. Let p and q ∈∂H2 be the fixed points of A. Since A = i1i2, we have:
i2(p) = i1(p),
i2(q) = i1(q).
If i1(p) ̸= q, then i2(p) ̸= q, and hence A would fix more than two points, contradicting
the loxodromic nature of A. Therefore, i1 and i2 must permute the fixed points of A.
□
8
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
3. strongly doubly reversible pairs in PSp(1)
Lemma 3.1. Let q ∈Sp(1) be such that e−iθ = qeiθq−1 where θ ̸= 0, π then q = eiϕj for
some ϕ ∈[0, 2π).
Proof. Let q ∈Sp(1) such that q = c1 + c2j for c1, c2 ∈C. We have:
qeiθ = (c1 + c2j)eiθ = c1eiθ + c2e−iθj,
e−iθq = e−iθ(c1 + c2j) = c1e−iθ + c2e−iθj.
Comparing the two sides, we get c1eiθ = c1e−iθ. Since θ ̸= 0, π, this implies c1 = 0. As
q ∈Sp(1), we then have q = eiϕj for some ϕ ∈[0, 2π).
□
Remark 1. In the above lemma, q = eiϕj for ϕ ∈[0, 2π), and hence q2 = −1. Thus, q is
an involution in PSp(1). Consequently, every element in PSp(1) is strongly reversible.
Theorem 3.2. Any two elements in PSp(1) are strongly doubly reversible.
Proof. Let p1 and p2 be elements in Sp(1). Without loss of generality, assume p1 = eiθ
and p2 = c1 + c2j be elements in Sp(1). Then we need to find q such that p1−1 = e−iθ =
qeiθq−1, p−1
2
= qp2q−1, where q2 = ±1.
Now observe that, by using lemma 3.1, p1−1 = e−iθ = qeiθq−1, holds for any q = eiθ1j
where θ1 ∈[0, 2π). So, we have
qp2 = eiθ1j(c1 + c2j) = eiθ1jc1 + eiθ1jc2j = eiθ1¯c1j −eiθ1¯c2,
p−1
2 q = (¯c1 −c2j)eiθ1j = ¯c1eiθ1j −c2jeiθ1j = eiθ1¯c1j + c2e−iθ1.
For p1 and p2 strongly doubly reversible by q, we require c2e−iθ1 = −eiθ1¯c2, which is
equivalent to Re(c2e−iθ1) = 0.
Writing c2 = c + di, this gives cos θ1 c + sin θ1 d = 0, which always has a solution
θ1 ∈[0, 2π). That means if we know c2, we can always find θ1 such that Re(c2e−iθ1) = 0.
Then we get
p−1
1
= e−iθ = qeiθq−1, p−1
2
= qp2q−1,
where q = eiθ1j for θ1 ∈[0, 2π). That means, p1 strongly doubly reversible with p2 in
PSp(1) via q.
□
Corollary 3.3. Every pair of elements in SO(3) is strongly doubly reversible.
Proof. It is a well-known result that Sp(1) is a double cover of SO(3). Hence, PSp(1) ≃
SO(3), and the result follows from Theorem 3.2.
□
As an application of the above theorem, we provide a simple proof of the following
result (see in [2]).
Theorem 3.4. Every pair of elements in SO(4) is strongly doubly reversible.
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
9
Proof. It is a well-known result that Sp(1) × Sp(1) is a double cover of SO(4).
Let
A, B ∈SO(4), and let ˜A = (A1, A2), ˜B = (B1, B2) be their respective lifts in Sp(1)×Sp(1).
By Theorem 3.2, for each i = 1, 2, there exist exist skew-involutions ˜αi, ˜βi, ˜γi such that
Ai = ˜αi ˜βi,
Bi = ˜βi ˜γi.
Define ˜α = ( ˜α1, ˜α2) and ˜β = ( ˜β1, ˜β2) then we get ˜α˜β = ( ˜α1 ˜β1, ˜α2 ˜β2) = (A1, A2) = ˜A.
Taking projection, we obtain
A = π(˜α˜β) = π(˜α)π(˜β) = αβ,
where α = π(˜α), β = π(˜β) ∈SO(4). We can observe that
α2 = π(˜α)π(˜α) = π(˜α2) = π((−1, −1)) = 1,
so α is an involution, and similarly, β is an involution. By the same argument, B = βγ
with β2 = γ2 = 1. Thus, (A, B) is strongly doubly reversible.
□
4. Strongly Doubly Reversible Pairs in PSp(n, 1)
We first note the following facts. Let G = Sp(n, 1) with Lie algebra g = sp(n, 1). Recall
that
dimR g = (n + 1)(2n + 3).
Suppose s ∈G is a skew involution, i.e., s2 = −I. Since −I is central in G, we have
Ad(s)2 = Ad(s2) = Ad(−I) = Id,
so the adjoint action given by Ad(g)(X) = gXg−1 decomposes g into ±1–eigenspaces:
g = g+1(s) ⊕g−1(s).
The +1–eigenspace is the Lie algebra of the centralizer ZG(s). One can check that
ZG(s) ∼= U(n, 1).
Therefore
dimR g+1(s) = dimR ZG(s) = (n + 1)2.
Subtracting, we obtain
dimR g−1(s) = dimR g −dimR g+1(s) = (n + 1)(2n + 3) −(n + 1)2 = (n + 1)(n + 2).
In particular, dim g−1(s) is strictly less than dim G.
Theorem 4.1. Let n ≥1. The set of strongly doubly reversible pairs in PSp(n, 1) has
Haar measure zero in PSp(n, 1) × PSp(n, 1).
Proof. Let G = PSp(n, 1). Define
R =
(g1, g2, h) ∈G × G × I : hgih−1 = g−1
i
for i = 1, 2
,
where I = {h ∈G : h2 = ±I}. Then the projection
Π(R) = {(g1, g2) ∈G × G : (g1, g2) is strongly doubly reversible}
is contained in a proper real-algebraic subset of G × G.
10
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
To see this, fix h ∈I and consider
Ψ : G −→G,
Ψ(g) = hgh−1g.
As before, (g, h) satisfies hgh−1 = g−1, if and only if Ψ(g) = I.
For X ∈g = Lie(G), set g(t) = exp(tX). Then
Ψ(g(t)) = h exp(tX)h−1 exp(tX) = exp(t(Ad(h)X + X)) + O(t2)
Expanding at t = 0 gives
dΨ|I(X) = (Ad(h) + Id)(X).
Thus, the kernel of dΨ|I is the eigenspace corresponding to the eigenvalue −1:
g−1(h) = {X ∈g : Ad(h)X = −X}.
By a consequence of the Inverse Function Theorem, the local solution set {g : Ψ(g) = I}
near I has dimension at most dim g−1(h), which has been seen to be strictly less than
dim G. Applying this simultaneously to g1 and g2 shows that the variety R ⊂G2 × I has
dimension strictly less than dim(G × G). Its projection Π(R) therefore lies in a proper
real-algebraic subset of G × G.
Any proper real-algebraic subset of G × G has a strictly smaller topological dimension,
and hence the Haar measure is zero on such subset. Thus, the set of strongly doubly
reversible pairs has Haar measure zero in G × G.
□
Let G = PSp(n) (or Sp(n)), n ≥2, and g = sp(n).
dimR g = n(2n + 1).
If t ∈G is conjugate to diag(Ik, −In−k), then
dimR g−1(t) = 4k(n −k).
If s ∈G satisfies s2 = −I, then ZG(s) ∼= U(n) (real dimension n2), hence
dimR g−1(s) = dimR sp(n) −n2 = n(n + 1).
With this observation, using arguments as above we have the following.
Corollary 4.2. Let n ≥2. The set of strongly doubly reversible pairs in PSp(n) has Haar
measure zero in PSp(n) × PSp(n).
The above arguments also carry over to SU(n, 1), except for n = 1 for similar reasons
as in the following remark.
Corollary 4.3. Let n ≥2. The set of strongly doubly reversible pairs in SU(n, 1) has
Haar measure zero in SU(n, 1) × SU(n, 1).
A tuple (g1, g2, . . . , gk) in Gk (direct product of k-copies of G) is called strongly k-
reversible if it belongs to same G-orbit of (g−1
1 , g−1
2 , . . . , g−1
k ) under simultaneous conju-
gation on Gk with the additional requirement that a conjugating element can be chosen
to be an involution.
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
11
By similar reasoning, the above theorem extends to strongly k-reversible tuples in
G = PSp(n, 1)
Theorem 4.4. The set of strongly k-reversible tuples in PSp(n, 1) has Haar measure zero
in PSp(n, 1)k.
Remark 2. (The case PSp(1) × PSp(1)) Although Sp(n) and PSp(n) = Sp(n)/{±I} have
the same Lie algebra and hence the same local dimension counts, the conclusion of the
above corollary fails for PSp(1). For n = 1 we have PSp(1) ∼= SO(3). We have already seen
that every pair of elements in SO(3) is strongly doubly reversible. Thus the set of strongly
doubly reversible pairs in PSp(1)×PSp(1) coincides with the entire space using the double
cover argument explained earlier. From the dimension-counting perspective, when n = 1,
the failure arises because the estimates become equalities. Here, the conjugacy class of
an involution in SO(3) is two–dimensional, while the solution set of tgt−1 = g−1 has real
dimension four, adding up to the full dimension six of PSp(1)×PSp(1). Consequently, no
dimension drop occurs, and the measure–zero argument breaks down in this special case.
5. Strongly Doubly Reversible Hyperbolic Pairs in PSp(n, 1)
The following lemma will be useful for our computations.
Lemma 5.1. Let
A =
reiθ
0
0
r−1eiθ
!
be hyperbolic element in Sp(1, 1). Let C ∈Sp(1, 1) satisfies A−1 = CAC−1. Then C must
be of the form:
C =
0
bj
¯b−1j
0
!
, b ∈C \ 0.
Proof. Suppose that A−1 = CAC−1 where, C =
x
y
z
w
!
, x, y, z, w ∈H. We get,
(5.1)
x = r2eiθxeiθ, w = r−2eiθweiθ
(5.2)
z = eiθzeiθ, y = eiθyeiθ.
From equation 5.1, if x ̸= 0 (resp. w ̸= 0) then this contradicts the fact that 0 < r < 1.
Therefore, we conclude that x = w = 0. Since C ∈Sp(1, 1), it follows that ¯zy = y¯z = 1.
From equation 5.2 and Lemma 3.1, we obtain y = bj for some b ∈C \ 0. Consequently,
z = ¯b−1j.
□
Remark 3. The above lemma gives all involutions in PSp(1, 1) which conjugate A to A−1.
These are precisely multiplication of the skew-involution
0
j
j
0
!
by an element from the
centralizer of A.
12
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
Theorem 5.2. Let A, B ∈PSp(n, 1) be hyperbolic elements. Then the pair (A, B) is
doubly reversible if and only if it is strongly doubly reversible.
Proof. We begin by proving the result for PSp(1, 1). One direction is immediate.
Conversely, suppose A and B are hyperbolic and doubly reversible elements in PSp(1, 1).
That is, there exists C ∈Sp(1, 1) such that
CAC−1 = A−1
and
CBC−1 = B−1.
Choose diagonal complex matrices DA and DB representing the complex eigenvalues of
A and B, respectively, so that
A = CADACA
−1,
B = CBDBCB
−1.
Substituting these expressions into the conjugation identities, we get
CCADAC−1
A C−1 = CAD−1
A C−1
A ,
CCBDBC−1
B C−1 = CBD−1
B C−1
B .
Since C interchanges the fixed points of A, we get
CAEADAE−1
A C−1
A = CAD−1
A C−1
A ,
where
EA =
0
q
¯q−1
0
!
, q ∈H \ {0}.
Then these satisfy the conjugation relations
EADAE−1
A = D−1
A .
Similarly, for B, we obtain
EBDBE−1
B = D−1
B .
By Lemma 5.1, EA and EB must be of the above form with the condition that q is of
the form zj, z ∈C \ {0}. This shows that (EA)2 = (EB)2 = −I. Now, C(CA) = CAEA
which gives us C is conjugate to EA, and thus C2 = −I.
For arbitrary hyperbolic pairs in PSp(n, 1), the same reasoning extends blockwise. Each
hyperbolic element A ∈PSp(n, 1) admits a diagonal form with respect to the standard
Hermitian form H0:
DA = diag
reiθ, eiϕ1, . . . , eiϕn−1, r−1eiθ
,
and similarly for B.
Let
A = CADACA
−1,
B = CBDBCB
−1.
On the 2×2 hyperbolic block corresponding to reiθ and r−1eiθ, Lemma 5.1 shows that the
conjugating block has the form
0
q
¯q−1
0
!
, q = zj, z ∈C, and hence squares to −I2. For
each unit–modulus eigenvalue eiϕk, Lemma 3.1 provides a conjugating element of the form
eiψkj satisfying (eiψkj)2 = −1. Thus every block of DA and DB admits a skew-involution
conjugating it to its inverse, and assembling these blocks gives
(EA)2 = (EB)2 = −I.
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
13
By using a similar argument as above, we obtain
CAC−1 = A−1
and
CBC−1 = B−1,
with C2 = −I. Hence, any doubly reversible hyperbolic pair in PSp(n, 1) is strongly
doubly reversible.
□
6. Strongly doubly reversible hyperbolic pairs in PSp(1, 1)
6.1. Pairs with a common fixed point.
Proposition 6.1. Let A and B be hyperbolic elements in Sp(1, 1) with one common fixed
point. Then A and B are strongly doubly reversible in PSp(1, 1) if and only if their fixed
points coincide.
Proof. Let A and B be strongly doubly reversible hyperbolic elements in Sp(1, 1) with a
common fixed point p. Suppose A = i1i2 and B = i3i2, where i1, i2, i3 are involutions in
PSp(1, 1). Without loss of generality, assume that A and B have other fixed points p1
and p2, respectively. Then i2(p) = p1 = p2. Hence, they have the same fixed points.
Conversely, let A and B be hyperbolic elements in Sp(1, 1) with the same fixed points
p and q in ∂H1
H. Now conjugate both matrices simultaneously by C ∈Sp(2, 1) such that
C(o) = p and C(∞) = q. Thus,
C−1AC = A1 =
reiθ
0
0
r−1eiθ
!
.
Also,
C−1BC = B1 =
µ
0
0
¯µ−1
!
.
Here, A1 and B1 have the same fixed points 0 and ∞in ∂H1
H. Now we can write A1
−1 =
DA1D−1 and B1
−1 = DB1D−1, where
D =
0
bj
¯b−1j
!
,
b ∈C \ 0.
Here we choose b such that, if µ = c1+c2j, then b satisfies Re(b¯c2) = 0. Clearly, D2 = −I,
and hence D is an involution in PSp(1, 1). Hence, A and B are strongly doubly reversible
in PSp(1, 1); that is,
A = (CD−1C−1)A−1(CDC−1), B = (CD−1C−1)B−1(CDC−1),
where CD−1C is an involution in PSp(1, 1).
□
6.2. Without a common fixed point.
Lemma 6.2. Let a, b ∈H be quaternions. Then there exists µ ∈H with µ2 = −1 such
that a = µ b µ if and only if ℜ(a) = ℜ(b) and |a| = |b|.
14
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
Proof. Suppose that a = µbµ with µ2 = −1, then we get ℜ(a) = ℜ(b) and |a| = |b|.
Conversely, let ℜ(a) = ℜ(b) and |a| = |b|. Write
a = s + v,
b = s + w,
where s = ℜ(a) = ℜ(b) and v, w ∈R3 are the vector parts. It suffices to find a unit pure
vector u ∈R3 with
u w u = v.
By expanding quaternion multiplication using pq = −p·q + p × q for purely imaginary
quaternions p, q, we obtain the reflection identity for vectors
(6.1)
u x u = 2(u·x) u −x
for u, x ∈R3, |u| = 1.
Now consider two cases.
If v ̸= −w. Set
u =
v + w
∥v + w∥.
Using (6.1) we compute
uwu = 2(u·w)u −w.
A direct scalar product computation shows that
u·w = v·w + |w|2
∥v + w∥,
∥v + w∥2 = |v|2 + 2v·w + |w|2.
Since |v| = |w|, a short simplification implies 2(u·w)u = v + w. Hence
uwu = (v + w) −w = v,
as required.
If v = −w. Then v ̸= 0. Choose any unit u perpendicular to v. Then u·w = 0 and by
(6.1) we get
uwu = −w = v.
So we get the desired unit pure quaternion µ in all possibilities.
□
Let A, B ∈PSp(1, 1) be the hyperbolic elements. Let aA, rA be the attracting and
repelling fixed points of A, and aB, rB be those of B.
Lemma 6.3. Let A, B ∈PSp(1, 1) be hyperbolic elements with no common fixed points.
Then there exists a skew-involution C ∈Sp(1, 1) which interchanges the fixed points of A
and also interchanges the fixed points of B if and only if the angular invariants satisfy
A(aA, rA, aB) = A(rA, aA, rB).
Proof. Without loss of generality, assume that A and B are hyperbolic elements in Sp(1, 1)
with fixed points o, ∞, and aB, rB respectively. Let aB and rB has lifts aB = (r1, 1)t,
rB = (s1, 1)t in H1,1, respectively.
Suppose that
A(aA, rA, aB) = A(rA, aA, rB).
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
15
This implies
ℜ(r1)
|r1|
= ℜ( ¯s1)
|s1| .
Thus by Lemma 6.2, there exist µ ∈H such that
r1
|r1| = µ
s−1
1
|s−1
1 | ¯µ, where µ2 = −1. Hence,
r1 = kµ s−1
1
¯µ, where k = |r1| |s1|. Now define C ∈Sp(1, 1) by
C =
0
√
kµ
µ
√
k
0
!
,
where µ ∈H, k > 0.
Then, C2 = −I, so C is a skew-involution. Moreover, C satisfies the following equations:
s1 = kµr−1
1 ¯µ, r1 = kµs−1
1 ¯µ,
which shows that C interchanges aA ↔rA and aB ↔rB.
The converse follows easily.
□
6.3. Proof of Theorem 1.6. Without loss of generality, assume A fixes o and ∞. If
both A and B are strongly doubly reversible, let C ∈Sp(1, 1) be such that
CAC−1 = A−1
and
CBC−1 = B−1,
with C2 = −I.
Thus C must interchanges the fixed points, cf. Lemma 2.3, and accordingly the angular
invariants must be equal by the previous Lemma, 6.3.
□
Remark 4. The converse of the above theorem does not hold in general. Indeed, if the
converse were true, then by Lemma 6.3, there would exist a skew involution C that
interchanges the fixed points aA ↔rA and aB ↔rB. However, interchanging the fixed
points is not sufficient to ensure that CAC−1 = A−1.
For example, consider
B =
reiθ
0
0
r−1eiθ
!
,
where r > 0, r ̸= 1, and θ ∈(0, π). Let
C =
0
i
i
0
!
.
Then C2 = −I, so C is a skew involution that interchanges the fixed points o and ∞of
B. Nevertheless, CBC−1 ̸= B−1.
One needs the added assumption that the skew-involution C must act as ‘right turns’
on the eigenspheres of both A and B. We are unable to express this condition in terms
of known geometric or algebraic invariants.
16
KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE
7. Quantitative description of Strongly doubly Reversible elements in
PSp(1, 1)
Consider the hyperbolic element A in PSp(1, 1) given by the matrix, again denoted by,
A =
reiθ
0
0
r−1eiθ
!
∈Sp(1, 1).
Suppose that there exists C ∈Sp(1, 1) such that
A−1 = CAC−1,
C2 = −I.
Then by Lemma 5.1, C must necessarily be of the form
C =
0
tj
¯t−1j
0
!
,
t ∈C \ {0}.
Now, let
B =
a
b
c
d
!
∈Sp(1, 1),
B−1 =
¯d
¯b
¯c
¯a
!
.
We seek the condition under which
CBC−1 = B−1,
equivalently,
CB = B−1C.
Substituting the form of C from Lemma 5.1, namely
C =
0
tj
¯t−1j
0
!
,
t ∈C \ {0},
j2 = −1,
we compute
0
tj
¯t−1j
0
!
a
b
c
d
!
=
¯d
¯b
¯c
¯a
!
0
tj
¯t−1j
0
!
.
Carrying out the multiplication yields
tjc
tjd
¯t−1ja
¯t−1jb
!
=
¯b ¯t−1j
¯d tj
¯a ¯t−1j
¯c tj
!
.
From this equality, we obtain the relations
tjc = ¯b ¯t−1j,
tjd = ¯d tj,
¯t−1ja = ¯a ¯t−1j,
¯t−1jb = ¯c tj.
Equivalently,
a = tj ¯a tj
−1,
b = tj ¯c (tj),
c = (tj)−1 ¯b tj
−1,
d = (tj)−1 ¯d (tj).
The conditions on a and d reduce to
a = tj ¯a tj
−1,
d = (tj)−1 ¯d (tj),
STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES
17
which in turn are equivalent to
ℜ(a2t) = 0,
ℜ(d2t) = 0,
where we write a quaternion h = h1 + h2j with h1, h2 ∈C. A nontrivial solution t exists
if and only if
a2 = d2k1
for some k1 ∈R.
Also we get,
t = −id2µ
⇐⇒
t = −ia2λ′
(∵a2 = d2k1).
Moreover, the relations for b and c,
b = tj ¯c (tj),
c = (tj)−1 ¯b tj
−1,
are equivalent to
b1 = |t|2c1,
b2 = −t2c2.
This leads to
b1 = λ2|a2|2c1,
b2 = λ2a2
2c2,
and hence
b1
c1|a2|2 =
b2
c2a2
2
,
b1
c1|a2|2 ≥0.
We summarize this discussion in the following theorem.
Theorem 7.1. Let A be the hyperbolic element as above. Then A is strongly doubly
reversible to an element B in PSp(1, 1) given by the matrix
B =
a
b
c
d
!
∈Sp(1, 1)
if and only if the entries of B satisfy
b2c1|a2|2 = b1c2a2
2,
a2 = λd2,
b1
c1|a2|2 ≥0,
where λ ∈R and each quaternion entry is written in the form h = h1+h2j with h1, h2 ∈C.
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[13] Xue-Jing Ren, Bao-Hua Xie, Yue-Ping Jiang, Decomposition of complex hyperbolic isometries by
two complex symmetries. Osaka J. Math. 54 (2017), no. 4, 661–677.
[14] P. Sarnak, Reciprocal geodesics. Analytic number theory, Clay Math. Proc. 7, 217–237, Amer. Math.
Soc., Providence, RI, 2007.
[15] Andrew E. Silverio, Half-turn strongly doubly reversible pairs of isometries of hyperbolic 4-space.
Topology Proc. 48 (2016), 173–202.
[16] A. Singh, M. Thakur, Reality properties of conjugacy classes in algebraic groups. Israel J. Math.,
165 (2008), 1–27.
[17] Pierre Will, Traces, cross-ratios and 2-generator subgroups of SU(2, 1). Canad. J. Math. 61 (2009),
no. 6, 1407–1436.
[18] M. J. Wonenburger, Transformations which are products of two involutions. J. Math. Mech. 16
(1966) 327–338.
Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City,
Sector 81, S.A.S. Nagar 140306, Punjab, India
Email address: krishnendu@iisermohali.ac.in
Institute of Mathematical Sciences, IV Cross Road, CIT Campus Taramani, Chennai
600 113 Tamil Nadu, India.
Email address: sagarbk@imsc.res.in, sagark327@gmail.com
|
STRONGLY DOUBLY REVERSIBILE PAIRS IN QUATERNIONIC UNITARY GROUP OF SIGNATURE (n, 1) KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE Abstract. Let PSp(n, 1) denote the isometry group of quaternionic hyperbolic space Hn H. A pair of elements (g1, g2) in PSp(n, 1) is said to be strongly doubly reversible if (g1, g2) and (g-1 1 , g-1 2 ) belong to the same simultaneous conjugation orbit of PSp(n, 1), and a conjugating element can be chosen to have order two. Equivalently, there exist involutions i1, i2, i3 ∈PSp(n, 1) such that g1 = i1i2, g2 = i1i3. We prove that the set of such pairs has Haar measure zero in PSp(n, 1) × PSp(n, 1). The same result also holds for PSp(n) × PSp(n) for n ≥2. In the special case n = 1, we show that every pair of elements in PSp(1) is strongly doubly reversible. Using elementary quaternionic analysis for Sp(1), we also provide a very short proof of a theorem of Basmajian and Maskit, in Trans. Amer. Math. Soc. 364 (2012), no. 9, 5015-5033, which states that every pair of elements in SO(4) is strongly doubly reversible. Furthermore, we derive necessary conditions under which a pair of hyperbolic elements is strongly doubly reversible in PSp(1, 1). 1. Introduction An element in a group G is said to be strongly reversible (or strongly real) if it can be written as a product of two involutions in G. This notion is closely related to that of reversible (or real) elements, which are conjugate to their inverses in G. Every strongly reversible element is necessarily reversible, but the converse is not true, in general. The classification and structure of such elements have been the subject of regular investigation in various branches of mathematics, for example, see [1], [4], [5], [10], [14], [16], [18]. Beyond their algebraic significance, strongly reversible elements play a central role in understanding symmetries in geometry. In particular, certain geometrically natural groups are built entirely from such elements. A classical and striking example arises in the setting of hyperbolic geometry. The group PSL(2, C) can be identified with the orientation-preserving isometries of the three dimensional hyperbolic space. If A and B are elements of PSL(2, C) generating a non-elementary subgroup, then there exist involutions i1, i2, i3 ∈PSL(2, C) such that A = i1i2 and B = i1i3, see [7], [9]. In the real hyperbolic case, a similar statement holds in PSL(2, R), with the involutions being orientation-reversing reflections. Date: October 17, 2025. 2020 Mathematics Subject Classification. Primary 51M10; Secondary: 15B33, 22E43. Key words and phrases. reversible elements, strongly doubly reversible, quaternion, hyperbolic space, product of involutions, compact symplectic group. 1 16 Oct 2025 2 KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE These observations motivate a more general concept that extends beyond individual decompositions to relationships between two elements and can be formulated for any abstract group. Definition 1.1. Let G be a group. Consider the G action on G × G by simultaneous conjugation: g.(g1, g2) = (gg1g-1, gg2g-1). For two elements g1, g2 ∈G, the pair (g1, g2) is said to be doubly reversible or doubly real if (g1, g2) and (g-1 1 , g-1 2 ) belong to the same G-conjugation orbit. Furthermore, if we choose the conjugating element g to be such that it is an involution, i.e. g2 = 1, then we call (g1, g2) to be strongly doubly reversible or strongly doubly real. This notion can be extended to any k-tuple of elements in G to define k-reversible (or, k-real) and strongly k-reversible (or, strongly k-real) tuples in a similar manner. However, in this paper, we restrict our attention to the case k = 2, specifically focusing on strongly 2-reversible or strongly doubly reversible elements. Note that if (g1, g2) is strongly doubly reversible, then there exist involutions i1, i2, i3 ∈ G such that g1 = i1i2 and g2 = i1i3. Conversely, if there are involutions i1, i2, i3 ∈G such that g1 = i1i2 and g2 = i1i3, then (g1, g2) is strongly doubly reversible. In particular, every strongly doubly reversible element is doubly reversible. The above interpretation captures the geometric compatibility between elements g1 and g2. For a strongly doubly reversible pair (g1, g2), both elements are generated by pairs of involutions that share a common factor. The classical result for PSL(2, C) may thus be interpreted as asserting that any two generators of a non-elementary subgroup are necessarily strongly doubly reversible. The study of such pairs is particularly interesting in groups where every element is a product of two involutions. In such settings, one may naturally ask which pairs of elements in such a group are strongly doubly reversible. This question not only provides understanding about the group's internal structure, but also connects it to broader topics like discreteness and geometric finiteness. For instance, when g1 and g2 are strongly doubly reversible, the subgroup ⟨g1, g2⟩sits as an index-two subgroup of the group ⟨i1, i2, i3⟩ generated by involutions. This can potentially lead to better insight into groups generated by three involutions, e.g., triangle groups in hyperbolic geometry. Despite its relevance, the problem of classifying strongly doubly reversible elements remains largely unexplored. Even within the context of finite groups, systematic efforts to understand doubly reversible pairs have begun only recently, e.g. [6]. The terminology 'k-real' has been borrowed from [6]. In geometric contexts, Basmajian and Maskit in [2], posed the problem of generalizing the classical PSL(2, C) result to higher-dimensional M ̈obius groups and isometry groups of Riemannian space forms. It may be noted that strongly doubly reversible pairs were termed linked pairs in [2]. Basmajian and Maskit proved that for higher dimensions, especially n ≥5, almost all pairs in these groups are not strongly doubly reversible. Basmajian and Maskit also proved that every pair in the orthogonal group SO(4) is strongly STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 3 doubly reversible. In a subsequent work, Silverio [15] provided a geometric description of strongly doubly reversible pairs in real hyperbolic 4-space. In complex hyperbolic geometry, the strongly doubly reversible pairs acquire additional structure. When every element in PU(n, 1) is a product of two anti-holomorphic involutions, not every element of PU(n, 1) is a product of (holomorphic) involutions, cf. [8]. In PU(2, 1),the isometry group of the two-dimensional complex hyperbolic space, they are classified as R-decomposable or C-decomposable, depending on whether the generating involutions are anti-holomorphic or holomorphic. Will [17] classified these loxodromic pairs, while Paupert and Will [11] gave a complete classification of the R-decomposable pairs in PU(2, 1). The C-decomposable pairs in PU(2, 1) have been described by Ren et al. [13]. Let Hn H denote the n-dimensional quaternionic hyperbolic space, whose isometry group is PSp(n, 1) = Sp(n, 1)/{±I}. A result by Bhunia and Gongopadhyay [3] shows that every element of Sp(n, 1) can be expressed as a product of two skew-involutions. Recall that a skew-involution is an element i ∈Sp(n, 1) such that i2 = -1. The skew-involutions project to involutions in PSp(n, 1). In contrast to PSp(n, 1), the group Sp(n, 1) itself has relatively few genuine involutions, and not all elements can be written as products of two such. Since every element of PSp(n, 1) is strongly reversible, it is a natural problem to explore strongly doubly reversible pairs in PSp(n, 1). We prove the following theorem in this regard. This is a generalization of [2, Theorem 1.5] for isometries of Hn H. Theorem 1.2. The set of strongly doubly reversible pairs in PSp(n, 1) has Haar measure zero in PSp(n, 1) × PSp(n, 1). In other words, almost all pair in PSp(n, 1) is not strongly doubly reversible. It also follows that that same result also hold for strongly doubly reversible pairs in PSp(n) for n ≥2. We see as a corollary to the above theorem. Corollary 1.3. Let n ≥2. The set of strongly doubly reversible pairs in PSp(n) has Haar measure zero in PSp(n) × PSp(n). However, when n = 1, we see that every element in PSp(1) is strongly doubly reversible. We have used elementary quaternionic analysis to see this for PSp(1). We also apply this result to offer a very short proof of [2, Theorem 1.4], which is the following. Theorem 1.4. Every pair of elements in SO(4) is strongly doubly reversible. One may ask how can we classify strongly doubly reversible pairs of isometries in quaternionic hyperbolic space? The challenge lies in the unique algebraic features of quaternions, namely their noncommutativity. Additionally, the absence of a well-behaved trace function or complete conjugacy invariants in the quaternionic setting adds to the difficulty. It may be noted that such conjugacy invariants are critical in complex hyperbolic settings for classify strongly doubly reversible pairs. As a result, many of the familiar tools from complex hyperbolic geometry do not carry over directly. It seems a difficult problem in the quaternionic set up to classify strongly doubly reversible pairs. 4 KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE Following the terminology in [12], recall that an element g in Sp(n, 1) is called hyperbolic if it has exactly two fixed points in the boundary. Hyperbolic elements have three mutually disjoint classes of eigenvalues. We prove the following result. Theorem 1.5. Let A and B be hyperbolic elements in PSp(n, 1). Then (A, B) is doubly reversible if and only if it is strongly doubly reversible. The proof of this theorem relies on an analysis of the strongly doubly reversible pairs in PSp(1, 1). Further we have obtained a necessary criteria for two hyperbolic elements in PSp(1, 1) to be strongly doubly reversible. This necessary criteria rely on the Cartan's angular invariant. Theorem 1.6. Let A and B be hyperbolic elements in PSp(1, 1) with no common fixed points. If A(aA, rA, aB) ̸= A(rA, aA, rB), then A and B can not be strongly doubly reversible. The converse of the above theorem does not hold in general. We have indicated this with an example in Remark 4. Structure of the paper. After discussing notations and preliminaries in Section 2, we prove that every pair of elements in SO(4) is strongly doubly reversible in Section 3.2. In Section 4, we prove that the set of strongly doubly reversible pairs in PSp(n, 1) has Haar measure zero in PSp(n, 1) × PSp(n, 1). In Section 5,we prove that a pair of hyperbolic elements (A, B) is doubly reversible if and only if it is strongly doubly reversible. Finally, in Section 5 and in Section 7, we provide a characterization and a quantitative description, respectively, of strongly doubly reversible hyperbolic pairs in PSp(1, 1). 2. Preliminaries 2.1. Doubly reversible pairs. Let G acts on G × G by conjugation. Let the stabilizer subgroup under this action is: SG((g1, g2)) = {h ∈G | h(g1, g2)h-1 = (g1, g2)} It is easy to see that SG((g1, g2)) = ZG(g1) ∩ZG(g2), where ZG(g) denote the centralizer of g in G. Consider the 'reverser' set: RG((g1, g2)) = {h ∈G | h(g1, g2)h-1 = (g-1 1 , g-1 2 )} Define: EG((g1, g2)) = SG((g1, g2)) ∪RG((g1, g2)). It is easy to see that E((g1, g2)) is a subgroup of G×G: if h1, h2 ∈RG, then h-1 1 h2 ∈SG. Lemma 2.1. SG((g1, g2)) is a normal subgroup of E((g1, g2)) of index atmost two. STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 5 Proof. Define a map φ : E((g1, g2)) →Z2 by, φ(h) = 1 if h(g1, g2)h-1 = (g1, g2) -1 if h(g1, g2)h-1 = (g-1 1 , g-1 2 ) This is a homomorphrism with kernel SG((g1, g2)) = ZG(g1) ∩ZG(g2). □ Thus, if g1 and g2 are in sufficiently general position such that the intersection of their centralizer is trivial, then a reversing symmetry is unique for a strongly doubly reversible pair (g1, g2). 2.2. The Quaternions. Let H := R + Ri + Rj + Rk denote the division algebra of Hamilton's quaternions, where the fundamental relations are given by i2 = j2 = k2 = ijk = -1. Every element of H can be written uniquely in the form q = a + bi + cj + dk, where a, b, c, d ∈R. Alternatively, viewing H as a two-dimensional vector space over C, we may express q = c1 + c2j, with c1, c2 ∈C. The modulus (or norm) of q is defined by |q| = √ a2 + b2 + c2 + d2. We denote the set Sp(1) := {q ∈H : |q| = 1} by the group of unit quaternions. We consider Hn as a right H-module. We consider Hn as a right vector space over the quaternions. A non-zero vector v ∈Hn is said to be a (right) eigenvector of A corresponding to a (right) eigenvalue λ ∈H if the equality Av = vλ holds. Eigenvalues of every matrix over the quaternions occur in similarity classes, and each similarity class of eigenvalues contains a unique complex number with non-negative imaginary part. Here, instead of similarity classes of eigenvalues, we will consider the unique complex representatives with non-negative imaginary parts as eigenvalues unless specified otherwise. In places where we need to distinguish between the similarity class and a representative, we shall write the similarity class of an eigenvalue representative λ as [λ]. 2.3. Quaternionic Hyperbolic Space. Let V = Hn,1 denote the right vector space of dimension n + 1 over H, equipped with the Hermitian form: ⟨z, w⟩= w∗H1z = ̄wn+1z1 + Σn i=2 ̄wizi + ̄w1zn+1, where ∗denotes the conjugate transpose, and H1 = 0 0 1 0 In-1 0 1 0 0 . We consider the following subspaces of Hn,1 : V-= {z ∈Hn,1 : ⟨z, z⟩ 0}, V0 = {z ∈Hn,1 \ {0} : ⟨z, z⟩= 0}. 6 KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE Let P : Hn,1 \{0} -→HPn be the right projection onto the quaternionic projective space. The image of a vector z will be denoted by z. The projective model of the quaternionic hyperbolic space is given by Hn H = P(V-). The boundary at infinity of this space is ∂Hn H = P(V0). The above Hermitian form may be replaced by an equivalent one associated with the matrix Ho: Ho = -1 0 . . . 0 0 1 . . . 0 ... ... ... 0 0 . . . 1 , where the corresponding Hermitian form ⟨z, w⟩o = w∗Hoz gives the ball model of Hn H. Given a point z of Hn H ∪∂Hn H -{∞} ⊂HPn we may lift z = (z1, z2, . . . , zn) to a point z in V0 ∪V-, given by z = z1 z2 ... 1 . Here z is called the standard lift of z. There are two points: 'zero' and 'infinity' in the boundary given by: o = 0 0 ... 1 , ∞= 1 0 ... 0 . Let Sp(n, 1) be the isometry group of the Hermitian form H1. Each matrix A in Sp(n, 1) satisfies the relation A-1 = H1 -1A∗H1, where A∗is the conjugate transpose of A. The isometry group of Hn H is the projective unitary group PSp(n, 1) = Sp(n, 1)/{±I}. However, we shall mostly deal with the linear group Sp(n, 1). 2.4. Classification of elements. Following the terminology in [12], recall that an element g ∈Sp(n, 1) is called hyperbolic if it has exactly two fixed points on the boundary. An element g ∈Sp(n, 1) is called parabolic if it has a unique fixed point on the boundary, and elliptic if it has a fixed point in Hn H. An element g in Sp(n, 1) belongs to exactly one of these three classes. 2.5. Hyperbolic Isometries. Consider a hyperbolic isometry A ∈Sp(n, 1). Let [λ] denote the conjugacy class of eigenvalues associated with A, and choose a representative eigenvalue λ with a corresponding eigenvector x. The vector x determines a point in quaternionic projective space HPn, which lies either on the boundary ∂H2 H or, is a point in P(V+). The corresponding line xH in the space Hn,1 represents the lift of this projective point and is invariant under the action of A. This line is the eigenspace generated by x. In the hyperbolic case, two of the eigenvalue classes are of null-type, with their associated eigenlines corresponding to fixed points on the boundary - one attracting, the other STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 7 repelling. Suppose the repelling fixed point on ∂Hn is denoted by rA and corresponds to the eigenvalue reiθ, while the attracting fixed point aA corresponds to the eigenvalue r-1eiθ. Let rA and aA denote their respective lifts to Hn,1. Additionally, for each j, let xj,A be an eigenvector of A associated with the eigenvalue eiφj. It is convenient to assume that the angles θ, φj lie within the interval [0, π] and 0 0. Then, C2 = -I, so C is a skew-involution. Moreover, C satisfies the following equations: s1 = kμr-1 1 ̄μ, r1 = kμs-1 1 ̄μ, which shows that C interchanges aA ↔rA and aB ↔rB. The converse follows easily. □ 6.3. Proof of Theorem 1.6. Without loss of generality, assume A fixes o and ∞. If both A and B are strongly doubly reversible, let C ∈Sp(1, 1) be such that CAC-1 = A-1 and CBC-1 = B-1, with C2 = -I. Thus C must interchanges the fixed points, cf. Lemma 2.3, and accordingly the angular invariants must be equal by the previous Lemma, 6.3. □ Remark 4. The converse of the above theorem does not hold in general. Indeed, if the converse were true, then by Lemma 6.3, there would exist a skew involution C that interchanges the fixed points aA ↔rA and aB ↔rB. However, interchanging the fixed points is not sufficient to ensure that CAC-1 = A-1. For example, consider B = reiθ 0 0 r-1eiθ ! , where r > 0, r ̸= 1, and θ ∈(0, π). Let C = 0 i i 0 ! . Then C2 = -I, so C is a skew involution that interchanges the fixed points o and ∞of B. Nevertheless, CBC-1 ̸= B-1. One needs the added assumption that the skew-involution C must act as 'right turns' on the eigenspheres of both A and B. We are unable to express this condition in terms of known geometric or algebraic invariants. 16 KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE 7. Quantitative description of Strongly doubly Reversible elements in PSp(1, 1) Consider the hyperbolic element A in PSp(1, 1) given by the matrix, again denoted by, A = reiθ 0 0 r-1eiθ ! ∈Sp(1, 1). Suppose that there exists C ∈Sp(1, 1) such that A-1 = CAC-1, C2 = -I. Then by Lemma 5.1, C must necessarily be of the form C = 0 tj ̄t-1j 0 ! , t ∈C \ {0}. Now, let B = a b c d ! ∈Sp(1, 1), B-1 = ̄d ̄b ̄c ̄a ! . We seek the condition under which CBC-1 = B-1, equivalently, CB = B-1C. Substituting the form of C from Lemma 5.1, namely C = 0 tj ̄t-1j 0 ! , t ∈C \ {0}, j2 = -1, we compute 0 tj ̄t-1j 0 ! a b c d ! = ̄d ̄b ̄c ̄a ! 0 tj ̄t-1j 0 ! . Carrying out the multiplication yields tjc tjd ̄t-1ja ̄t-1jb ! = ̄b ̄t-1j ̄d tj ̄a ̄t-1j ̄c tj ! . From this equality, we obtain the relations tjc = ̄b ̄t-1j, tjd = ̄d tj, ̄t-1ja = ̄a ̄t-1j, ̄t-1jb = ̄c tj. Equivalently, a = tj ̄a tj -1, b = tj ̄c (tj), c = (tj)-1 ̄b tj -1, d = (tj)-1 ̄d (tj). The conditions on a and d reduce to a = tj ̄a tj -1, d = (tj)-1 ̄d (tj), STRONGLY DOUBLY REVERSIBLE QUATERNIONIC HYPERBOLIC ISOMETRIES 17 which in turn are equivalent to R(a2t) = 0, R(d2t) = 0, where we write a quaternion h = h1 + h2j with h1, h2 ∈C. A nontrivial solution t exists if and only if a2 = d2k1 for some k1 ∈R. Also we get, t = -id2μ ⇐⇒ t = -ia2λ′ (∵a2 = d2k1). Moreover, the relations for b and c, b = tj ̄c (tj), c = (tj)-1 ̄b tj -1, are equivalent to b1 = |t|2c1, b2 = -t2c2. This leads to b1 = λ2|a2|2c1, b2 = λ2a2 2c2, and hence b1 c1|a2|2 = b2 c2a2 2 , b1 c1|a2|2 ≥0. We summarize this discussion in the following theorem. Theorem 7.1. Let A be the hyperbolic element as above. Then A is strongly doubly reversible to an element B in PSp(1, 1) given by the matrix B = a b c d ! ∈Sp(1, 1) if and only if the entries of B satisfy b2c1|a2|2 = b1c2a2 2, a2 = λd2, b1 c1|a2|2 ≥0, where λ ∈R and each quaternion entry is written in the form h = h1+h2j with h1, h2 ∈C. References [1] M. Baake, J. A. G. Roberts, The structure of reversing symmetry groups, Bull. Austral. Math. Soc., 73 (2006), 445-459. [2] Ara Basmajian and Bernard Maskit, Space form isometries as commutators and products of involutions. Trans. Amer. Math. Soc. 364 (2012), no. 9, 5015-5033. [3] Sushil Bhunia and Krishnendu Gongopadhyay, Reversible quaternionic hyperbolic isometries. Linear Algebra Appl., 591 (2020), 268-283. [4] D. ˇZ. Djokovi ́c, Product of two involutions. Arch. Math. (Basel) 18 (1967), 582-584. [5] E. W. Ellers, Products of two involutory matrices over skewfields. Linear Algebra Appl. 26 (1979), 59-63. [6] Jyotirmoy Ganguly, Digjoy Paul, Amritanshu Prasad, K.N. Raghavan and Velmurugan, Kronecker coefficients and simultaneous conjugacy classes. . [7] William M. Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. Handbook of Teichm ̈uller theory. Vol. II, 611-684, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc.I hope this message finds you well. , Z ̈urich, 2009. 18 KRISHNENDU GONGOPADHYAY AND SAGAR B. KALANE [8] Krishnendu Gongopadhyay and John R. Parker, Reversible complex hyperbolic isometries. Linear Algebra Appl. 438, No. 6, 2728-2739 (2013). [9] Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 287, Springer-Verlag, Berlin, 1988. [10] A. G. O'Farrell and I. Short, in Dynamics and Group Theory, London Mathematical Society Lecture Note Series, vol.416, Cambridge University Press, Cambridge, 2015. [11] Julien Paupert and Pierre Will, Real reflections, commutators, and cross-ratios in complex hyperbolic space. Groups Geom. Dyn. 11 (2017), no. 1, 311-352. [12] J. G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, 149, Springer, New York, 1994. [13] Xue-Jing Ren, Bao-Hua Xie, Yue-Ping Jiang, Decomposition of complex hyperbolic isometries by two complex symmetries. Osaka J. Math. 54 (2017), no. 4, 661-677. [14] P. Sarnak, Reciprocal geodesics. Analytic number theory, Clay Math. Proc. 7, 217-237, Amer. Math. Soc., Providence, RI, 2007. [15] Andrew E. Silverio, Half-turn strongly doubly reversible pairs of isometries of hyperbolic 4-space. Topology Proc. 48 (2016), 173-202. [16] A. Singh, M. Thakur, Reality properties of conjugacy classes in algebraic groups. Israel J. Math., 165 (2008), 1-27. [17] Pierre Will, Traces, cross-ratios and 2-generator subgroups of SU(2, 1). Canad. J. Math. 61 (2009), no. 6, 1407-1436. [18] M. J. Wonenburger, Transformations which are products of two involutions. J. Math. Mech. 16 (1966) 327-338. Indian (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar 140306, Punjab, India Email address: 600 113 Tamil Nadu, India. Email address: ,
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2510.14753
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JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
1
LightQANet: Quantized and Adaptive Feature
Learning for Low-Light Image Enhancement
Xu Wu, Zhihui Lai∗, Xianxu Hou, Jie Zhou, Ya-nan Zhang, Linlin Shen
Abstract—Low-light image enhancement (LLIE) aims to im-
prove illumination while preserving high-quality color and tex-
ture. However, existing methods often fail to extract reliable
feature representations due to severely degraded pixel-level
information under low-light conditions, resulting in poor texture
restoration, color inconsistency, and artifact. To address these
challenges, we propose LightQANet, a novel framework that
introduces quantized and adaptive feature learning for low-
light enhancement, aiming to achieve consistent and robust
image quality across diverse lighting conditions. From the static
modeling perspective, we design a Light Quantization Module
(LQM) to explicitly extract and quantify illumination-related
factors from image features. By enforcing structured light factor
learning, LQM enhances the extraction of light-invariant rep-
resentations and mitigates feature inconsistency across varying
illumination levels. From the dynamic adaptation perspective, we
introduce a Light-Aware Prompt Module (LAPM), which encodes
illumination priors into learnable prompts to dynamically guide
the feature learning process. LAPM enables the model to flexibly
adapt to complex and continuously changing lighting conditions,
further improving image enhancement. Extensive experiments on
multiple low-light datasets demonstrate that our method achieves
state-of-the-art performance, delivering superior qualitative and
quantitative results across various challenging lighting scenarios.
Index
Terms—Low-Light
Image
Enhancement,
Vector-
Quantized General Adversarial Network, Prompt Learning.
I. INTRODUCTION
I
MAGES captured in dark environments, often referred
to as low-light images, suffer from reduced illumination,
increased artifact, and poor texture and color fidelity than those
captured under normal-light conditions [1]. These deficiencies
not only make it challenging for the human eye to discern
objects but also significantly degrade the performance of
∗represents the corresponding author.
Xu Wu, is with the College of Computer Science and Software Engineering,
Shenzhen University, Shenzhen 518060, China and College of Computing
and Data Science, Nanyang Technological University, Singapore (e-mail:
csxunwu@gmail.com)
Zhihui Lai and Linlin Shen are with the Computer Vision Institute,
College of Computer Science and Software Engineering, Shenzhen University,
Shenzhen 518060, China. (e-mail: lai zhi hui@163.com
Xianxu Hou is with School of AI and Advanced Computing, Xi’an
Jiaotong-Liverpool University, China (e-mail: hxianxu@gmail.com).
Jie Zhou is with the School of Mathematics and Statistics, Changsha
University of Science and Technology, Changsha 410114, China, and also with
the School of Artificial Intelligence, Shenzhen University, Shenzhen 518060,
China. (e-mail: jie jpu@163.com).
Ya-nan Zhang is with School of Computer Science, Sichuan Normal
University, Chengdu 610065, China (e-mail: 20240074@sicnu.edu.cn).
Linlin Shen is with Computer Vision Institute, School of Artificial Intel-
ligence, Shenzhen University, Shenzhen 518060, China and also with the
Department of Computer Science, University of Nottingham Ningbo China,
Ningbo 315100, China (e-mail: llshen@szu.edu.cn).
RetinexFormer
CIDNet
Ours
Input
SNR
SMG
Fig. 1. Visual comparisons on the real low-light image. The input image is
compared with results from SNR [2], SMG [3], RetinexFormer [4], CIDNet
[5], and our method. The zoomed-in regions show the details of texture and
sharpness restoration. Our method produces the most natural textures and
smooth transitions around edges.
advanced visual models, such as object detection systems.
Therefore, the development of robust and effective LLIE
methods is essential for improving the visual quality and utility
of images captured in low-light conditions.
In previous works, Histogram Equalization (HE)-based and
Retinex-based methods have been prominent in enhancing
low-light images. HE-based methods enhance image contrast
by adjusting the gray-level distribution of pixels to equalize
the histogram [6]. Conversely, Retinex-based methods focus
on estimating and enhancing the illumination component of
each pixel to improve brightness. However, both approaches
may amplify noise and cause color distortion [7], presenting
significant challenges that necessitate further refinement.
Recent advances in deep learning for LLIE have primar-
ily focused on end-to-end networks that directly improve
illumination [14]. Many of these methods introduce explicit
illumination modeling, such as using an illumination branch
to guide feature learning [2], or applying Retinex theory to de-
compose images into reflection and illumination components
[4]. However, despite these innovations, current approaches
often struggle to preserve robust feature representations under
arXiv:2510.14753v1 [cs.CV] 16 Oct 2025
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
2
(b) Code activation frequency
Input
Baseline
Ours
Ground Truth
(a) Ablation comparison
(c) PSNR comparison
(d) NIQE comparison
Fig. 2.
Effectiveness analysis and benchmark comparison. (a) represents results of the baseline built by VQ-GAN [8] and our method. (b) shows code
activation frequency on LOL-v2 Real dataset [9]. “GT in Baseline” and “LL in Baseline” represent inputting ground truth and low-light images to pre-trained
VQ-GAN [8] and fine-tuned VQ-GAN with low-light images, respectively. “LL in CodeEnhance” and “LL in Ours” denote inputting low-light images to
CodeEnhance [10] and our method. Activation frequency of “LL in Ours” is closer to that of “GT in Baseline”, which indicates our method can learn and
activate important features under low-light conditions. (c) and (d) present performance comparison on LLIE datasets (LOL-v2-Synthetic [9], LOL-v2-Real [9],
LSRW-Huawei [11], LIME [12], MEF [7], VV 1, and NPE [13]) in terms of PSNR (the higher the better) and NIQE (the lower the better). As highlighted
by the bold red line, the proposed method consistently achieves the best demonstrating its superior.
low-light conditions, where severely compromised pixel-level
visibility and reliability hinder effective feature extraction,
ultimately resulting in texture degradation and color distortion.
As shown in Fig. 1, the results of SNR [2] and SMG
[3] suffer from noticeable blurring and artifact amplification,
leading to degraded texture clarity and unnatural visual ap-
pearances. RetinexFormer tends to cause over-exposure in
brighter regions while under-enhancing darker areas, and CID-
Net exhibits evident color distortion. By contrast, our method
achieves clearer edge contours and more faithful texture recov-
ery, producing sharper window patterns and smoother surfaces
that better preserve the perceptual realism of the scene.
Addressing these challenges requires moving beyond direct
pixel-level enhancement towards feature-level robustness un-
der complex and variable illumination. In this study, we refor-
mulate the conventional image-to-image enhancement pipeline
into an image-to-feature learning framework, which reduces
the uncertainty inherent in the enhancement process by focus-
ing on more abstract and stable feature representations. Specif-
ically, we employ a Vector-Quantized Generative Adversarial
Network (VQ-GAN) [8] to reconstruct high-quality images by
leveraging vector-quantized features and learning an effective
mapping between images and these features for LLIE tasks.
However, as illustrated in Fig. 2 (a) and (b), directly applying
VQ-GAN to LLIE leads to suboptimal results. Under low-
light conditions, the activation frequency of “LL in Baseline”,
where the baseline model is VQ-GAN, exhibits significant
inconsistency compared to “GT in Baseline”, particularly at
crucial feature indices. This is primarily due to VQ-GAN’s
lack of illumination-aware mechanisms and its reliance on
clean feature distributions for effective codebook matching.
Under extreme darkness, encoder features become misaligned
with the learned codebook, resulting in color distortion and
overexposure. These observations highlight the necessity of
learning light-invariant representations and ensure effective
feature quantization under low-light conditions.
To tackle these challenges, we propose LightQANet, a novel
framework for low-light image enhancement based on quan-
tized and adaptive feature learning. The core of LightQANet
is the Light Quantization Module (LQM), which aims to
explicitly quantify illumination-related information embedded
in image features. Instead of treating illumination variations
implicitly, LQM is designed to learn a structured representa-
tion of lighting conditions by extracting and quantizing the so-
called light factors from both low-light (LL) and normal-light
(NL) images. To achieve this, LQM is trained to distinguish
illumination levels through a supervised contrastive objective,
enabling it to accurately capture and quantify the intensity and
distribution of illumination in the feature space. Once LQM
acquires the ability to model illumination variations, it serves
as an auxiliary guidance mechanism for the LightQANet.
Specifically, the LightQANet is encouraged to minimize the
feature-level discrepancies between different illumination con-
ditions, thereby promoting the extraction of light-invariant
representations. Through this collaboration, LQM not only
provides structured illumination supervision but also enhances
the robustness and generalization of the overall enhancement
framework. Note that LQM is not required during testing.
While LQM provides a structured quantization of illumi-
nation information, real-world lighting conditions are often
complex and dynamically changing. To address this challenge,
we introduce the Light-Aware Prompt Module (LAPM), which
dynamically guides feature learning based on illumination
priors. Specifically, LAPM encodes illumination information
into a set of learnable prompts, each capturing discriminative
characteristics associated with different illumination levels.
These prompts are adaptively fused with intermediate feature
representations in the primary LLIE network, enabling the
model to systematically adjust its feature learning process
according to the estimated lighting conditions. By dynamically
injecting illumination-specific cues into the feature space,
LAPM enhances the model’s ability to generalize across a
wide range of lighting environments.
The main contributions of this work are as follows:
• We propose LightQANet, a novel framework that per-
forms quantized and adaptive feature learning to extract
light-invariant representations, enabling consistent and
robust low-light image enhancement under diverse illu-
mination conditions.
• We design two key modules to enhance illumination
adaptability: LQM, which extracts and quantizes light
factors to build light-invariant feature representations, and
LAPM, which dynamically refines feature representations
based on light-specific priors. These modular design en-
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
3
ables stable and consistent feature representations across
varying illumination conditions.
• We conduct extensive experiments in datasets, including
LOL-v2-Real, LOL-v2-Synthetic, LSRW-Huawei, LIME,
MEF, VV, and NPE, demonstrating that LightQANet
consistently achieves state-of-the-art performance, as ev-
idenced by the results shown in Fig. 2 (c) and (d).
The remainder of this paper is organized as follows. Section
II reviews related works. Section III presents a detailed model
design. Section IV conducts the experiments and discusses the
results. Finally, we conclude the paper in Section V.
II. RELATED WORKS
This section reviews previous work related to low light
image enhancement and recent advances in discrete codebook
learning for image restoration tasks.
A. Low-Light Image Enhancement
Images captured in low-light environments typically suffer
from poor quality, lacking essential visual details which can
hinder comprehension and analysis. Initially, researchers tack-
led this problem through histogram equalization technologies
[6], adjusting illumination and contrast by equalizing pixel
intensity distributions. Various methods evolved from this
approach, focusing on different enhancement aspects such
as overall image perspective [15], cumulative distribution
functions [16], and the addition of penalty terms to refine
the enhancement process [17]. Parallel to these developments,
some researchers applied Retinex theory [18], which decom-
poses an image into illumination and reflection components,
allowing for targeted enhancements in both areas. Subsequent
Retinex-based enhancements, such as SSR [19], improved
both illumination and color accuracy significantly. The ad-
vent of deep learning [20] introduces more methods into
the LLIE [21]. LLNet [1] is the first to integrate stacked
autoencoders for enhancing low-light images. This is followed
by the introduction of multi-branch [22] [23] and multi-stage
[24] networks, designed to tackle illumination recovery, noise
suppression, and color refinement concurrently. SNR [2] uses
the PSNR distribution map to guide network feature learning
and fusion. SMG [3] incorporates image structural information
to enhance the output image’s quality. Recent innovations
have combined Retinex theory with deep learning to further
refine enhancement techniques. URetinexNet [25] formulates
the decomposition problem of Retinex as an implicit prior
regularization model, and Retinexformer [4] uses illumination
to guide the Transformer [26] in learning the global illumina-
tion information of the image. LLformer [27] proposes a new
transformer and attention fusion block for LLIE. GSAD [28],
JoRes [29] and LLDiffusion [30] leverage the diffusion model
to perform LLIE. QuadPrior [31] improves low-light images
by physical quadruple priors. CIDNet [5] proposed a new color
space to overcome color bias and brightness artifacts in LLIE.
B. Discrete Codebook Learning
Discrete codebook learning is first introduced in the con-
text of Vector Quantized-Variational AutoenEoder (VQ-VAE)
[32]. Subsequently, VQ-GAN integrates this approach within
the generative adversarial network framework, facilitating the
generation of high-quality images [8]. In low-level image
processing tasks, codebook learning helps mitigate uncer-
tainty during model training by transforming the operational
space from raw images to a compact proxy space [33]. To
enhance feature matching, FeMaSR [34] introduces residual
shortcut connections, RIDCP [35] develops a controllable
feature matching operation, and CodeFormer [33] employs a
Transformer-based prediction network for retrieving codebook
indices. Additionally, LARSR [36] proposes a local autore-
gressive super-resolution framework utilizing the learned code-
book. CodedBGT [37] and CodeEnhance [10] introduce the
codebook to improve LLIE model performance. Different from
CodedBGT [37] and CodeEnhance [10], we propose the LQM
to precisely extract light factors from image features. Addition-
ally, we introduce the LAPM to dynamically enhance image
representations through light-specific knowledge. Collectively,
these modules significantly improve the representation of
light information, thereby elevating the overall quality of the
enhanced images.
III. METHODOLOGY
This section provides a detailed introduction to the proposed
method, which includes high-quality codebook learning, light-
invariant feature learning, feature matching and image restora-
tion, and training objectives.
A. Overview
The proposed method consists of two stages: the first stage
constructs a high-quality codebook using VQ-GAN trained on
well-lit images to capture representative visual patterns. The
second and more critical stage focuses on enhancing low-light
images by extracting light-invariant features, ensuring stable
representation and effective illumination correction across di-
verse lighting conditions. Firstly, we leverage VQ-GAN to
encode detailed features from high-quality images Ih into a
discrete set of codebook, which serve as a comprehensive
reference for accurately reconstructing images. This stage can
be formulated as follows:
Zh = E(Ih),
eZh = M(Zh, C),
I′
h = D(eZh),
(1)
where E(·) and D(·) denote encoder and decoder. M(·, ·)
is feature matching operation, where C represents learnable
codebook of features. Zh and eZh are latent features and
quantized features. In the subsequent step, the C and D(·)
will be frozen to leverage the quantized features obtained from
C, followed by D(·) reconstructing high-quality images. This
ensures stability in the learning process and consistency in the
output quality.
Next, as shown in Fig. 3, to improve feature extraction in
low-light conditions, we develop light-invariant feature learn-
ing, where the LQM and LAPM are crucial for normalizing the
impact of different lighting conditions on feature extraction,
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
4
Codebook
0
1
2
N
...
247
738
879
…
429
…
Light-aware Prompts
Conv
Conv
GAP
Conv
C
Conv
Unfold
Light-Aware Prompt Module (LAPM)
Fuse
LAPMs
Nearest-Neighbor Matching
Feature Matching
Frozen
Learnable
Network Processing
GAP
Global Average Pooling
Unfold
Unfold operation
codes sequence
Light Quantization Module (LQM)
…
…
…
…
…
…
…
Vector Quantization Operate
Light Quantization Operate
Fig. 3.
Overview of our proposed LightQANet framework. Our method leverages a pretrained codebook C† and a decoder D† as the foundations. In
LightQANet, the Light Quantization Module (LQM) is utilized to extract light factors and promote the learning of light-invariant features by a novel light
consistency loss (Llcl). Additionally, the Light-Aware Prompt Module (LAPM) is introduced to encode light illumination data for dynamically guiding the
feature learning process. Finally, the feautre fusion via linear interpolation refines the reconstructed features.
thereby stabilizing the model’s performance across varied
environments. Following this, we introduce feature matching
and image reconstruction, examining how the algorithm aligns
features under diverse illumination settings and reconstructs
high-quality images from these aligned features. Low-light
enhancement stage can be formulated as follows:
Zll = E′(Ill),
eZll = M(Zll, C†),
Irec = D†(eZll, Ffuse),
(2)
where Ill and Irec denote low-light images and reconstructed
images, respectively. Zll and eZll are latent features and
quantized features of Ill. Ffuse represents output of the feature
fusion in skip connection and is defined in Section III-D. E′
denotes the light-invariant feature learning. C† and D†(·) are
codebook and decoder with frozen parameters, respectively.
B. High-Quality Codebook Learning
We first pre-train a VQ-GAN using high-quality images
to learn a discrete codebook. This codebook serves as prior
knowledge for enhancing low-light images. The corresponding
decoder associated with the codebook is then utilized to
reconstruct images. Given a high-quality image Ih, we first
employ the encoder of VQ-GAN to obtain a latent feature
Zh ∈Rm×n×d. Then, by calculating the distance between
each ‘pixel’ z(i,j)
h
of Zh and the ck in the learnable codebook
C = {ck ∈Rd}N
k=0, we replace each z(i,j)
h
with the nearest
ck [33]. After that, the quantized features eZh ∈Rm×n×d are
obtained:
ez(i,j)
h
= M(z(i,j)
h
, C) = arg min
ck∈C
∥z(i,j)
h
−ck∥2,
(3)
where N = 1024 represents the size of the codebook, and
d = 512 denotes the channel number of both Zh and C. The
dimensions m and n specify the sizes of Zh and eZh. The
reconstructed image I′
h is then generated by the decoder. The
VQ-GAN is supervised using the loss function Lvq [8], which
includes an L1 loss Lmae, a codebook matching loss Lcma,
and an adversarial loss Ladv:
Lvq = Lmae + Lcma + Ladv,
LL1 = ∥Ih −I′
h∥1,
Lcma = σ∥Zh −sg(eZh)∥2
2 + ∥sg(Zh) −eZh∥2
2,
Ladv = γlogD(Ih) + log(1 −D(I′
h)),
(4)
where D(·) is the discriminator. sg(·) is the stop-gradient
operator. σ = 0.25 denotes a weight trade-off parameter that
governs the update rates of both the encoder and codebook
[33]. γ is usually set to 0.1 [35].
C. Light-Invariant Feature Learning
The efficacy of our method depends on the quality of
light-invariant feature learning. To achieve this, we design
two key modules: the LQM, which models illumination in
a structured manner, and the LAPM, which adaptively guides
feature learning based on illumination priors.
1) Light Quantization Module (LQM). To effectively extract
light-invariant features for low-light image enhancement, we
propose the LQM, motivated by the critical need to disentangle
illumination information from detailed content representations.
Unlike conventional methods that operate directly on raw
image features, we explicitly model illumination as a global
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
5
…
…
Feature maps
Light Quantization Module
(a) Training on Light Quantization Module (LQM)
(b) Training on the proposed LightQANet
…
…
Feature maps
Light Quantization Module
Gram matrices
(c) t-SNE of distributions
Light factors
Low-light
Normal-light
Fig. 4. Overview of the proposed LQM. (a) and (b) illustrate the alternating optimization of the LQM and the LightQANet. (c) shows the Gram matrices
from low-light (LL) and normal-light (NL) images, which exhibit significant overlap. In contrast, the light factors are clearly distinguished based on lighting
conditions, underscoring their effectiveness in accurately capturing and representing light-relevant information. Note that the LQM is only used during the
training phase and therefore does not impact the processing speed during the inference stage.
style attribute, leveraging its inherent characteristics such as
overall brightness, contrast, and color distribution, which are
largely independent of fine-grained image details [38]. Draw-
ing inspiration from the success of style-based approaches in
nighttime domain adaptation [39] and low-light enhancement
such as EnlightGAN [40], we adopt a Gram matrix [41]
representation within LQM to capture and isolate illumination-
related style features. The Gram matrix effectively encodes
global feature correlations, allowing LQM to abstract illumina-
tion information while filtering out irrelevant content-specific
variations. The Gram matrix is defined as follows:
G = aTa,
(5)
where G ∈Rc×c, c is the number of channels. a represents
feature maps.
To equip the LQM with the ability to quantify illumination
conditions, we formulate a supervised learning objective based
on pairwise light factor distances. Specifically, for a set of
image pairs P, LQM learns to construct a light factor space in
which images captured under similar lighting conditions are
mapped closer together, while those under different lighting
conditions are pushed further apart. The light factors f a,l and
f b,l are computed by applying the LQM Q(·) to the Gram
matrices Ga,l and Gb,l extracted from the l-th intermediate
feature maps. The training is supervised using the following
loss:
Llqm =
X
(a,b)∈P
n
(1 −1(a, b))
m −d(f a,l, f b,l)
2
+
+1(a, b)
d(f a,l, f b,l) −m
2
+
o
,
(6)
where d(·) denotes the cosine similarity, [·]+ is the hinge
function, m is the margin, and 1(a, b) is an indicator function
that returns 1 if Ia and Ib have the same lighting condition
and 0 otherwise. During this training stage, the encoder of
the proposed is frozen and only the parameters of the LQM
are updated, as shown in Fig. 4 (a). This learning process
enables LQM to accurately quantify illumination differences
and establish a light factor space is used to guide light-
invariant feature learning in subsequent LightQANet training.
After LQM has acquired the ability to quantify illumination,
we leverage its learned light factor space to guide the training
of the proposed model. Specifically, we introduce a light
consistency loss Llcl to minimize the discrepancy between
the light factors of low-light and normal-light images. As
illustrated in Fig. 4 (b), during this stage, the LQM is kept
frozen while the encoder parameters are updated. The light
consistency loss is defined as:
Ll
lcl(f a,l, f b,l) =
1
4d2
l n2
l
dl
X
i=1
(f a,l
i
−f b,l
i )2,
(7)
where f a,l and f b,l are the light factors extracted by the frozen
LQM, dl denotes the dimensionality of the light factors, and nl
is the spatial size of the l-th feature map. By minimizing Llcl,
the encoder is encouraged to extract feature representations
that are invariant to illumination variations, thus improving
overall enhancement in various illumination scenarios.
Algorithm 1 and Fig. 4 (a) and (b) illustrate the alternating
optimization process between the LQM and the LightQANet
network. This optimization strategy gradually reduces the
discrepancy in illumination conditions between low-light and
normal-light images, ultimately promoting the extraction of
light-invariant features within the LightQANet framework. To
demonstrate the effectiveness of LQM, we analyze the Gram
matrices computed from intermediate feature maps and their
corresponding light factors produced by LQM. As shown in
Fig. 4 (c), the LQM effectively isolates illumination informa-
tion, clearly separating it from other content-related features.
These results indicate that the extracted light factors success-
fully encode illumination-specific attributes, as intended.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
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Algorithm 1 LightQANet Training Algorithm
1: Input: Paired training data (Ill, Inl): low-light and
normal-light images
2: Randomly initialize the model parameters θ;
3: for each training pair (Ill, Inl) do
4:
Zll ←E′(Ill);
5:
eZll ←M(Zll, C†);
6:
Irec ←D†(eZll, Ffuse);
7:
# Update LQM and freeze LightQANet;
8:
Optimize LQM using Llqm;
9:
# Update LightQANet and freeze LQM;
10:
Optimize LightQANet using combined loss: Ladv +
Lfml + Lrec + Llcl;
11: end for
12: Return Enhanced image Irec
2) Light-Aware Prompt Module (LAPM). While LQM ef-
fectively captures structured illumination characteristics, it
faces limitations in representing complex and spatially varying
lighting patterns typically observed in real-world scenarios.
To overcome this, LAPM dynamically adapts feature repre-
sentations by aggregating illumination information from local
spatial regions. Specifically, LAPM computes prompt weights
based on local features rather than relying solely on a global il-
lumination descriptor. This enables each region in the image to
contribute effectively to the dynamic prompt composition, thus
capturing fine-grained variations in illumination and providing
more adaptive modulation of the final feature representation.
Fig. 3 shows that the prompt component Pn, consisting of
five learnable vectors, embeds light information from n levels.
These prompt vectors are not only responsible for modeling
discrete brightness states but are also trained to encode transi-
tional relationships between different brightness levels and to
capture global illumination properties. To generate the light-
aware prompts P, we compute attention-based weights from
local features and then apply these weights to Pn. The weights
serve as region-wise “soft assignments,” guiding each prompt
to specialize in the luminance ranges where it is most effective
(e.g., extremely dark vs. mid-level brightness). Summing these
weighted prompts yields a prompt-guided feature modulation
that faithfully reflects the overall illumination distribution,
from darkest shadows to brightest highlights. Specifically,
we first divide the image features into n patches. Average
pooling is applied to these patches to extract local features,
which are then processed by a channel-shrink layer to ensure
their dimensions align with Pn. After dimension alignment,
a softmax function denoted as Fs, is employed to compute
the weights ωn ∈RC. The weights interact with the Pn to
generate P, which are further processed by a convolution layer
with a 3 × 3 kernel size. These operations can be collectively
formulated as follows:
P = LAPM(Fl, Pn) = F3(
N
X
n=1
ωnPn),
ωn = Fs(F1(FA(Fl))),
(8)
where Fl = UF(Fe) represents local features. UF(·) is a
(a) Correlation matrix
(b) Brightness
Fig. 5. Prompt-Brightness correlation analysis. (a) Correlation heatmap be-
tween prompt weights (Prompt Index 1–5) and brightness levels. Positive (red)
and negative (blue) values indicate correlation strength. Different prompts
show distinct sensitivities, with some strongly responding to extremely dark
regions [0, 25.5) and others to brighter or transitional regions, highlighting
their complementary roles. (b) Brightness distribution histogram of the MEF
dataset [7], showing a predominance of darker pixels.
unfold operation. Fe denotes the intermediate features of
encoder. FA(·) is a average pooling operation. F1(·) is the
channel-shrink layer performed by a 1 × 1 convolution layer.
Finally, the light-aware prompts P are integrated channel-wise
with the intermediate features of the encoder. These combined
features are then processed by a ResNet block [42], enhancing
the overall feature representation.
Fig. 5 (a) shows that each prompt vector responds distinctly
to different brightness ranges. For example, prompts 1 and
3 are highly sensitive (correlations of 0.55 and 0.49) to
extremely dark pixels ([0, 25.5)). Prompt 5 primarily handles
low-light pixels [25.5, 51), showing a correlation of 0.47.
Prompt 4 mainly focuses on mid-to-high brightness levels
(above 76.5), effectively capturing general illumination pat-
terns. Prompt 2 complements these prompts by responding
moderately (correlation of 0.37) to intermediate brightness
levels [102, 127.5), and mild negative correlations in extremely
dark [0, 25.5) and bright regions (above 178.5). Furthermore,
our prompt allocation strategy is informed by the illumination
conditions, as illustrated by Fig. 5 (b). Since the majority
of pixels in low-light images fall into the darkest brightness
interval ([0,25.5)), we assign two dedicated prompts (prompts
1 and 3) to this interval. Overall, our multi-prompt vector
design and data-driven allocation strategy ensure each prompt
effectively captures illumination-specific information, espe-
cially for diverse brightness conditions.
D. Feature Matching and Image Reconstruction
We perform feature matching through nearest-neighbor
lookup in the codebook to obtain high-quality features, as
shown in Fig. 3. Subsequently, these high-quality features are
transmitted to a decoder that incorporates skip feature fusion
modules, enabling the reconstruction of enhanced images.
Based on Eq. 3, the feature matching M(·, ·) in low-light
enhancement task can be formulated as follows:
ez(i,j)
ll
= M(z(i,j)
ll
, C†) = arg min
ck∈C† ∥z(i,j)
ll
−ck∥2,
(9)
where Zll = {z(i,j)
ll
∈Rd}m,n
i=0,j=0 denotes image features
extracted by light-invariant feature learning, eZll = {ez(i,j)
ll
∈
Rd}m,n
i=0,j=0 represents quantized features.
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
7
To improve the quality of reconstructed images, we intro-
duce a feature fusion via linear interpolation technique that
effectively merges low-level features Fe from the encoder
with features Fd from the decoder. This integration not only
preserves critical texture information but also compensates for
potential detail loss during image processing. Initially, features
Fe and Fd are combined in channel-wise and subsequently
computes affine transformation parameters α and β. These
parameters are designed to reduce the impact of noise and
enhance texture representation in the reconstructed images.
This feature fusion can be formulated as follows:
Ffuse = α ⊙Fd + β,
α, β = C([Fd, Fe]),
(10)
where C(·) denotes convolution operation, and ⊙is element-
wise multiplication. Finally, we employ the LAPM to further
refine the features. Importantly, the parameters within the
decoder blocks remain frozen during this process.
E. Training Objectives
Finally, we outline the training objectives that guide the
overall learning process. Specifically, the LQM is optimized
with a dedicated contrastive loss (Llqm), which has defined
in Eq. 6 in Section III-C. The main enhancement model is
trained with a combination of Adversarial Loss Ladv, Feature
Matching Loss Lfml, Light Consistency Loss Llcl, and Re-
construction Loss Lrec to ensure high-quality restoration under
varying illumination conditions, which are defined as follows:
Ltotal = Ladv + Lfml + Lrec + λLlcl,
(11)
where Ladv is defined in Eq. 4 of Section III-A. λ = 0.5
is a weight of Llcl. Llcl is used to minimize the discrepancy
between the light factors of low-light and normal-light images,
which has defined in Eq. 7 of Section III-C.
1) Feature Matching Loss. This loss function is specifically
designed to optimize the encoder by facilitating its learning of
the mapping between low-light images and high-quality priors.
By minimizing this loss, the encoder can enhance the proposed
method ability to accurately translate low-light conditions
into visually appealing outputs, aligning with predefined high-
quality standards. The loss is formulated as follows:
Lfml = σ∥Zll −sg(eZh)∥2
2 + ∥ϕ(Zll) −ϕ(sg(eZh))∥2
2
where ϕ(·) is used to calculate the gram matrix of features.
Zll and eZh represent the latent features of low-light images
and quantized features of high-quality images, respectively.
2) Reconstruction Loss. This loss function combines L1 loss
and perceptual loss to ensure that enhanced images have a
complete structure and impressive visual appeal. The L1 loss
minimizes pixel-level discrepancies for high fidelity, while
perceptual loss aligns images to human visual perception,
enhancing both structural accuracy and aesthetic quality. The
loss is defined as follows:
Lrec = ∥Inl −Irec∥1 + ∥ψ(Inl) −ψ(Irec)∥2
2,
(12)
where Inl and Irec represent normal-light images and re-
constructed images, respectively. ψ(·) indicates the Learned
Perceptual Image Patch Similarity (LPIPS) function [43].
IV. EXPERIMENTS
This section presents experimental results to evaluate the
effectiveness of the proposed method through quantitative
comparisons, qualitative analysis, and ablation studies.
A. Implementation Details
For VQ-GAN and the proposed LightQANet training, the
input pairs are randomly cropped to patches of size 256 ×
256. We use ADAM optimizer with β1 = 0.9, β2 = 0.999
and ε = 10−8. The learning rate is set to 10−4. The VQ-GAN
is pre-trained on the DIV2K [44] and Flickr2K [45] with 350K
iterations. Our LightQANet is trained with 50K iterations.
The hyper-parameter m is set to 0.1. All experiments were
conducted in PyTorch on an NVIDIA A6000.
B. Datasets and Evaluation Metrics
1) Low-light Datasets. We evaluate methods using the LOL-
v2 [9] and LSRW-Huawei [11]. And also evaluate methods in
cross datasets: LIME [12], MEF [7], VV 1, and NPE [13]
The LOL-v2-Real one contains 689 train images and 100 test
images. The LOL-v2-Synthetic one includes 900 train images
and 100 test images. The LSRW-Huawei contains 2,450 train
images and 30 test images. The LIME, MEF, VV, and, NPE
include 10, 17, 24, and 8 low-light images, respectively.
2) Evaluation Metrics. We assess the quality of the enhanced
images using the most common metrics: Peak Signal-to-Noise
Ratio (PSNR), Structural Similarity Index Measure (SSIM)
[47], Natural Image Quality Evaluator (NIQE) [48], and LPIPS
[43]. Unlike PSNR and SSIM, which primarily focus on lowe-
level similiaity, LPIPS accounts for the human visual system’s
perception of similarity, offering a more accurate reflection of
how images are perceived by viewers.
C. Comparison with State-of-the-Art Methods
We assess the performance of our LightQANet by con-
ducting comparisons with numerous leading LLIE techniques.
These include LIME [12], JED [49], RetinexNet [50], KinD
[51], EnlightGAN [40], Zero [52], SNR [2], PairLIE [46],
RetinexFormer [4], SMG [3], CodedBGT [37], DMFourLLIE
[23], CodeEnhance [10], QuadPrior [31], and CIDNet [5].
1) Same-Domain Evaluation. We conduct detailed visual
comparisons on LOL-v2-Real (see Fig. 6), LOL-v2-Synthetic
(see Fig. 7), and LSRW-Huawei (see Fig. 8), focusing on four
key aspects, with quantitative results summarized in TableI.
Illumination Consistency: Competing methods such as Zero,
CIDNet, and RetinexFormer exhibit abrupt brightness transi-
tions, while SMG, QuadPrior, and Zero under-enhance dark
regions. Our method achieves smooth, natural illumination
transitions, which is reflected in the highest SSIM scores
(0.8974, 0.9388, 0.7179) across all datasets. Color Fidelity:
LIME, RetinexNet, and EnlightGAN introduce strong color
1https://sites.google.com/site/vonikakis/datasets
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
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TABLE I
QUANTITATIVE RESULTS ON LOL-V2 [9] AND LSRW-HUAWEI [11]. ↑INDICATES THE HIGHER THE BETTER. ↓INDICATES THE LOWER THE BETTER.
BOLD: BEST RESULT; UNDERLINE: SECOND BEST RESULT; −: UNAVAILABLE DATA. P AND F DENOTES PARAMETERS AND FLOPS.
LOL-V2-Real
LOL-V2-Synthetic
LSRW-Huawei
Complexity
Methods
PSNR ↑
SSIM ↑
LPIPS ↓
NIQE ↓
PSNR ↑
SSIM ↑
LPIPS ↓
NIQE ↓
PSNR ↑
SSIM ↑
LPIPS ↓
NIQE ↓
Param (M)
FLOPs (G)
LIME
16.97
0.4598
0.3415
8.4899
17.50
0.7718
0.1748
3.4063
18.46
0.4450
0.3922
3.3879
-
-
JED
17.29
0.7266
0.2760
4.3703
16.89
0.7299
0.2370
3.5284
15.11
0.5379
0.4327
3.1521
-
-
RetinexNet
16.10
0.4006
0.4215
9.2661
17.14
0.7615
0.2185
4.3433
16.82
0.3951
0.4566
3.4942
0.84
587.47
KinD
16.75
0.6456
0.4118
4.6253
17.51
0.7694
0.2093
3.3295
17.19
0.4625
0.4318
2.9682
8.02
34.99
EnlightGAN
17.94
0.6755
0.3197
4.8755
16.59
0.7780
0.2179
3.0998
17.46
0.4982
0.3780
3.0650
114.35
61.01
Zero
18.06
0.5736
0.2980
7.7571
17.76
0.8163
0.1382
3.0464
16.40
0.4761
0.3763
3.0477
0.075
4.83
SNR
21.48
0.8489
0.1996
3.6383
22.88
0.8962
0.1124
3.5854
20.67
0.6246
0.4879
3.4008
39.12
26.35
SMG
24.03
0.8178
0.2283
5.7291
25.62
0.9188
0.2915
5.9165
20.66
0.5589
0.4449
6.9247
0.33
20.81
PairLIE
19.88
0.7777
0.2834
3.6192
19.07
0.7965
0.2183
3.9121
18.99
0.5632
0.3711
3.0790
1.61
15.57
RetinexFormer
22.79
0.8397
0.2270
3.3869
25.67
0.9296
0.0775
2.8861
20.81
0.6303
0.4124
2.8866
30.35
137.37
CodeEnhance
23.32
0.8310
0.2184
3.2115
24.65
0.9163
0.0648
3.2019
21.14
0.6076
0.2840
2.6424
49.07
225.86
DMFourLLIE
22.64
0.8589
0.1488
2.9389
25.83
0.9314
0.0562
2.8892
21.47
0.6331
0.3998
3.0153
0.41
1.56
QuadPrior
20.58
0.8036 f
0.2410
5.8903
16.11
0.7646
0.2187
4.6653
18.30
0.6013
0.4070
3.7033
1252.75
1103.20
CIDNet
24.11
0.8675
0.1678
3.4159
25.13
0.9387
0.0536
2.8128
20.86
0.6202
0.3740
2.6131
1.88
7.57
Ours
28.51
0.8974
0.1039
3.1926
26.15
0.9388
0.0457
2.8636
21.68
0.7179
0.2885
2.5784
18.85
164.20
LIME
EnlightGAN
RetinexNet
SMG
Zero
SNR
RetinexFormer
Input
Ours
Ground Truth
CIDNet
QuadPrior
Fig. 6. Visual comparison on the LOL-v2-Real [9], accompanied by image error maps calculated by L2 loss. The proposed method produces a more natural
illumination transition across shadow boundaries (e.g., wall region), accurate colors, and well-preserved details.
TABLE II
COMPARISONS ON LIME [12], MEF [7], NPE [13], AND VV 1 IN TERMS
OF NIQE, WHERE THE LOWER THE BETTER. NOTE THAT THE RESULTS
”NULL” ARE DUE TO THE CORRESPONDING METHODS LACKING CODE.
Methods
LIME
MEF
NPE
VV
KinD2019
6.71
3.17
3.28
2.32
EnlightGAN2019
3.59
3.11
4.36
3.18
Zero2020
3.79
3.31
3.48
2.75
SNR2022
4.88
3.47
4.19
7.55
PairLIE2023
4.31
3.92
3.68
3.16
RetinexFormer2023
3.70
3.14
3.58
1.95
SMG2023
6.47
6.18
5.89
5.46
CodedBGT2024
4.20
3.85
3.52
Null
QuadPrior2024
4.58
4.36
3.65
3.44
CIDNet2025
3.85
3.46
3.82
3.24
Ours
3.54
2.88
3.26
1.94
distortions; Zero, SMG, and QuadPrior show biases in wall
and text regions. LightQANet restores accurate hues, aided by
illumination quantization and adaptive modulation, yielding
the lowest LPIPS values (0.1039, 0.0457, 0.2885). Texture
Preservation: SNR, SMG, QuadPrior, and CIDNet fail to
recover fine details, producing smoothed textures. Our method
preserves sharp contours and structural details, consistent
with the top PSNR results (28.51, 26.15, 21.68). Artifact
Suppression: RetinexNet, RetinexFormer, and SNR introduce
noise or halos; CodeEnhance causes over-smoothing. In con-
trast, LightQANet suppresses artifacts effectively, balancing
enhancement and detail.
Across the three datasets, the proposed method consis-
tently shows superior performance in handling illumination
transitions, restoring accurate colors, preserving fine textures,
and minimizing enhancement artifacts. These results further
validate the effectiveness of LightQANet in delivering robust
and perceptually pleasing low-light enhancement.
2) Cross-Domain Evaluation. To assess the robustness and
generalization ability of our network under domain shift, we
evaluate the model on unpaired low-light images from datasets
that differ from the training domain, including MEF [7], NPE
[13], LIME [12], and VV 1. As illustrated in Fig. 9 and
Table II, the proposed method consistently achieves superior
visual quality across diverse scenes and lighting conditions.
Specifically, in the MEF dataset, our method restores natural
brightness and preserves fine textures in both the grass and
flower regions, while other methods tend to overexpose the
sky or blur the foreground textures (see red arrows). In the
NPE scene, we accurately enhance the dove’s feathers and
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
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LIME
EnlightGAN
RetinexNet
SMG
Zero
SNR
RetinexFormer
Input
Ours
Ground Truth
QuadPrior
CodeEnhance
Fig. 7. Visual comparison on the LOL-v2-Synthetic [9], accompanied by image error maps. Compared to competing methods, our approach demonstrates
superior overall brightness and structural clarity. LIME, EnlightGAN, SMG, QuadPrior, and CodeEnhance display noticeable color deviations.
LIME
EnlightGAN
RetinexNet
Zero
Input
SMG
SNR
RetinexFormer
Ours
Ground Truth
CIDNet
QuadPrior
Fig. 8.
Visual comparison on the LSRW Huawei [11], accompanied by image error maps and Canny edge maps of text regions. As we can see that the
proposed method restores both texture and color details effectively.
surrounding foliage without color distortion or effects, unlike
methods such as EnlightGAN and PairLIE which introduce
unatural brightness or lose edge sharpness. In the LIME
dataset, the proposed method clearly reveals the license plate
characters and traffic sign symbols (highlighted in the red
box), which are over-smoothed (SNR, SMG) in other methods.
For the VV dataset, our method demonstrates superior detail
preservation in both global structure and fine textures. Notably,
the intricate wall carvings (see bottom row) are sharp and
realistic in our result, whereas others either blur the details
(SNR, SMG, and CIDNet) or color-destoration the region
(EnlightGAN, PairLIE).
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LIME
VV
SMG
RetinexFormer
Ours
Input
EnlightGAN
SNR
PairLIE
CIDNet
MEF
NPE
Fig. 9. Visual comparison on the LIME [46], MEF [7], VV 1, and NPE [13] dataset, accompanied by Canny edge maps. The results demonstrate that our
method effectively enhances images across different lighting conditions, achieving a trade-off between texture preservation and illumination enhancement.
These improvements can be attributed to the combination of
structured modeling of illumination and high-quality prompt
tailored for low-light conditions. Together, they allow the
model to adaptively enhance illumination while maintaining
structural fidelity and natural appearance, even when applied
to previously unseen domains.
4) Complexity Analysis. Table I details model complex-
ity (Params, FLOPs) at 256×256 resolution. Simple CNNs
like Zero (0.075M params, 4.83G FLOPs) and DMFourLLIE
offer low complexity but compromise enhancement quality.
Conversely, QuadPrior’s use of a pretrained diffusion back-
bone leads to high complexity (1252.75M params, 1103.2G
FLOPs). Our LightQANet achieves a strong balance with
18.9M parameters and 164.2G FLOPs, representing a sig-
nificant 61% reduction in parameters and 27% in FLOPs
compared to the codebook-based CodeEnhance, thus offering
superior quality at a moderate computational cost.
D. Codes Activation Analysis
The analysis of code activation frequencies in feature match-
ing provides crucial insights into the efficacy of our image
enhancement methods. Fig. 2 (b) shows comparisons of the
code activation frequency among baseline, CodeEnhance [10],
and the proposed method. From the results, it is observed that
directly inputting low-light images into the baseline model
(”LL in Baseline”) causes highly imbalanced code usage, with
certain codes (e.g., index 671) being excessively activated
while others are underutilized. In contrast, ”LL in CodeEn-
hance” and ”LL in Ours” exhibit more evenly distributed
activations, indicating that both methods better mitigate the
degradation of code utilization caused by low-light conditions.
Notably, the proposed method achieves a closer activation dis-
tribution to ”GT in Baseline”, suggesting that our enhancement
strategy more effectively restores diverse and representative
feature activations under challenging lighting conditions.
Fig. 10 illustrates the effectiveness of our method. Panel (a)
displays code activation of low-light images, which processed
by the proposed method. Panels (b) and (c) compare activation
frequencies for our enhanced and ground truth images using a
pre-trained VQ-GAN, with panel (d) detailing the top ten code
comparisons. Panels (a), (c), and (d) collectively demonstrate
that our method effectively extracts features from low-light
images, achieving activation frequencies that closely match
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
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(b) Enhanced Images
(c) Ground Truth
(d) Codes Activation Frequency in top ten.
(a) Low-light Images
Fig. 10. Comparison of code activation frequency. The codebook includes 1024 quantized features, we reshape these feature indexes into 32×32. (a) denotes
the code activate frequency of low-light images under our method. (b) and (c) represent code activation frequency of enhanced images and ground truth under
baseline. (d) is a comparison of the top ten codes’ activation frequency. The analysis shown in (a), (c), and (d) highlights the capability of the proposed method
to effectively extract features from low-light conditions, achieving activation frequencies closely aligned with those of the ground truth. Additionally, (b),
(c), (d) demonstrate the high quality of our results, exhibiting similar activation patterns to the ground truth within the baseline comparisons. This coherence
across the figures substantiates the effectiveness of our enhancement approach in maintaining the integrity of image features under varied lighting conditions.
Input
Ground Truth
(c) With FF and LQM
(b) With FF
(d) With FF and LAPM
(a) Baseline
26.90/0.9122
29.77/0.9151
19.41/0.8843
20.99/0.8914
25.94/ 0.9135
(e) Ours
With FF, LQM, and LAPM
Fig. 11. Visual comparison of the ablation studies in Table III. Starting from the baseline, the integration of FF, LQM, and LAPM progressively improves
both image quality and illumination correction, ultimately achieving the best performance when all components are combined together.
TABLE III
ABLATION STUDIES OF THE PROPOSED MODULES ON LOL-V2-REAL
DATASET. BASELINE IS BUILT BY VQ-GAN [8]. FF MEANS THE FEATURE
FUSION Ffuse IN SKIP CONNECTION.
No.
Baseline
FF
LQM
LAPM
PSNR
SSIM
(a)
✓
23.91
0.8699
(b)
✓
✓
24.82
0.8751
(c)
✓
✓
✓
25.95
0.8853
(d)
✓
✓
✓
27.16
0.8911
(e)
✓
✓
✓
✓
28.51
0.8974
those of the ground truth. This similarity indicates that our
method not only improves visibility but also preserves the
image’s inherent characteristics. Furthermore, comparisons in
panels (b), (c), and (d) reveal that the activation patterns of
our enhanced images align well with the ground truth within
the VQ-GAN, showcasing the high fidelity of our results.
These results further validate the effectiveness of our method
in preserving the integrity and authenticity of image features
across diverse lighting conditions.
E. Ablation Study
To validate the effectiveness of each proposed component,
we conduct ablation studies on the LOL-v2-Real dataset, as
summarized in Table III. The baseline model is constructed
based on VQ-GAN [8], and we incrementally integrate the
feature fusion (FF) in skip connection, Light Quantization
Module (LQM), and Light-Aware Prompt Module (LAPM)
to systematically assess their contributions.
1) Study of FF. As shown in Table III and Fig. 11, inte-
grating the feature fusion module improves the PSNR from
23.91 to 24.82 and the SSIM from 0.8699 to 0.8751. This
performance improvement stems from our learnable linear
interpolation mechanism. It uses two dynamically predicted
parameters, α and β, to modulate decoder features based
on encoder information. Unlike direct feature concatenation
or summation, this adaptive interpolation allows the model
to enhance fine textures while mitigating noise amplification,
thereby achieving more effective reconstruction under low-
light conditions.
2) Study of LQM. Upon integrating LQM shown in Table III
and Fig. 11, the PSNR further improves to 25.95 and the
SSIM increases to 0.8853. This improvement demonstrates the
effectiveness of structured illumination modeling through the
light-factor space learned by LQM. By explicitly quantizing
illumination-related information and enforcing feature con-
sistency between low-light and normal-light images through
the light consistency loss (Llcl) in the learned light-factor
space, LQM enables the proposed model to better align
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
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(a) Input image
(b) Enhanced image
(c) Prompt weights
(d) HE
Fig. 12. Instance analysis of LAPM effects in LLIE. (a) and (b) are low-light input image and its enhanced result. (c) Learned prompt weights, where prompts
1 and 3 focus on extremely dark regions [0, 25.5], prompt 5 targets transitional brightness [25.5, 51], and prompt 4 responds to mid-to-high brightness levels
(above 76.5). (d) Brightness distribution histograms before and after enhancement, illustrating luminance correction across different regions.
(a) Mean of prompt weights
(b) Cumulative ratio of pixels
(c) Comparison between the mean of prompt weights and pixel
ratio of MEF dataset
Fig. 13.
Analysis of learned prompt weights on the MEF dataset [7]. (a)
Mean prompt weights with min–max range for the five learnable prompts.
(b) Cumulative ratio of pixels as a function of brightness value, with vertical
dashed lines marking the boundaries of the five prompt-responsive intervals.
(c) Comparison of the mean prompt weights against the corresponding pixel
ratios within each brightness range, where prompts were grouped according to
their correlations in Fig.5. [0, 25.5): combined weight of Prompt1+Prompt3,
[25.5, 51): weight of Prompt5, [51, 153): weight of Prompt2+0.5×Prompt4,
[153, 256): weight of 0.5×Prompt4. Note that since Prompt4 effectively
covers the range [51, 256), we split its weights into two parts, one for [51,153)
and one for [153, 256).
feature representations across varying illumination conditions.
This structured alignment process promotes the learning of
light-invariant feature representations, significantly enhancing
feature robustness and stability, which are crucial for effective
low-light image enhancement.
Table IV shows that setting λ to 0.5 achieves the best
trade-off between luminance consistency and structural detail
preservation, yielding the highest image quality and structural
fidelity. A larger λ overly constrains illumination consis-
tency, leading to texture loss, while a smaller λ weakens
illumination-invariance learning. Thus, λ = 0.5 allows the
model to effectively leverage Llcl, enhancing feature robust-
ness without compromising image quality.
3) Study of LAPM. Based on FF, adding LAPM leads to
achieving a PSNR of 27.16 and an SSIM of 0.8911, as shown
in Table III. LAPM dynamically guides feature learning by
injecting illumination-specific prompts, enabling the model to
adapt feature representations based on brightness variations.
TABLE IV
ABLATION OF λ FOR Llcl ON LOL-V2-REAL DATASET.
λ
1
0.5
0.001
PSNR
27.38
28.51
27.43
SSIM
0.8905
0.8974
0.8928
TABLE V
ABLATION OF NUMBER OF PROMPT VECTORS ON LOL-V2-REAL DATASET.
Number
PSNR
SSIM
3
28.19
0.8949
4
28.03
0.8911
5
28.51
0.8974
6
27.47
0.8880
Compared to the purely static structure offered by LQM,
LAPM introduces dynamic flexibility, allowing the model to
better respond to complex real-world illumination, thereby
yielding significant improvements in image quality shown in
Fig. 11. Furthermore, Table V presents the impact of the num-
ber of prompt vectors. We observe that increasing the number
of prompts from 3 to 5 leads to continuous improvements
in both PSNR and SSIM, reaching the best performance at
5 prompts (28.51 PSNR and 0.8974 SSIM). However, further
increasing the number to 6 results in a noticeable performance
drop, due to over-fragmentation of the luminance space, which
weakens the effectiveness of each individual prompt. These
results highlight that using 5 prompts achieves the optimal
balance between representation capacity and generalization in
illumi- nation modeling.
In Fig. 12, we first analyze a low-light image whose pixel
distribution concentrates within the darkest range ([0, 25.5)).
Referring to the correlation analysis from Fig. 5, prompts 1
and 3, which are strongly associated with this darkest region,
together receive the highest combined weight, effectively
enhancing severely underexposed areas. Meanwhile, prompts
4 and 5, assigned relatively lower weights, help refine mod-
erately illuminated regions. To further validate this adaptive
behavior, we evaluate prompt weights on the entire MEF
dataset (see Fig. 13). The darkest interval [0, 25.5) contains
53.3% of pixels and is closely matched by the combined
weights of prompts 1 and 3 (0.4324). Transitional low-light
pixels ([25.5, 51)) comprise 16.2%, matched by prompt 5’s
weight (0.1854). Mid-range brightness ([51, 153)) accounts
for 21.0%, aligning well with the sum of prompt 2 and half of
prompt 4 (0.2825). Finally, the brightest interval ([153, 256))
includes 9.0% of pixels, corresponding closely with half of
JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
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TABLE VI
ABLATION OF DISCRETE FEATURE NUMBER IN CODEBOOK ON
LOL-V2-REAL DATASET.
Number
PSNR
SSIM
256
27.04
0.8897
512
27.72
0.8946
1024
28.51
0.8974
2048
27.18
0.8923
Ours
Input
CIDNet
Fig. 14. Unsatisfying cases. Images captured using a Sony A7C II camera.
prompt 4’s weight (0.0996). Despite slight deviations in exact
proportions within these intervals, the learned prompt weights
largely reflect the dataset’s overall luminance distribution. This
analysis confirms that LAPM effectively achieves adaptive
illumination-aware feature modulation, allowing the model
to selectively emphasize different luminance intervals, thus
achieving more natural and visually pleasing results.
Study of discrete feature number in codebook. Table VI
presents the impact of the number of discrete features in
the codebook on enhancement performance. We observe that
increasing the codebook size from 256 to 1024 progressively
improves both PSNR and SSIM, reaching the best perfor-
mance at 1024 entries. However, when increased to 2048,
it will result in slight degradation, which may be attributed
to reduced feature compactness and increase noise sensitivity
during reconstruction. Thus, setting the codebook size to 1024
provides the best trade-off between feature expressiveness and
generalization capability.
F. Limitations and future works
Although LightQANet demonstrates strong performance
across diverse lighting conditions, several limitations remain
to be addressed.
(1)
Extremely
dark
scenes.
As
shown
in
Fig.
14,
LightQANet restores brightness and colors in partially visible
regions more effectively than CIDNet, thanks to LQM and
LAPM. However, in completely black areas (e.g., the dense
canopy), the absence of information causes encoder–codebook
misalignment, leading to noise or pseudo-textures. Future work
may integrate generative priors to recover plausible structures
while suppressing artifacts. (2) Insufficient high-frequency
recovery. In cross-domain evaluation, fine structures are not
always preserved. For example, Fig. 9 shows blurred license
plate characters in the LIME dataset, reflecting limited small-
scale detail reconstruction. Future efforts could adopt gradient-
or edge-aware losses, leverage high-frequency components via
wavelet/Fourier transforms, and utilize classical edge operators
(e.g., Sobel, Prewitt, or Canny) to provide explicit priors. In
addition, decomposing images into smooth and detail layers
with bilateral filtering, selectively reinforcing the detail layer,
and applying lightweight sharpening as post-processing may
further refine structural fidelity, collectively mitigating edge
blurring and detail loss.
V. CONCLUSION
In this study, we propose a novel LightQANet framework
for LLIE, which emphasizes light-invariant feature learning
through both structured quantization and dynamic adaptation.
Specifically, we design an LQM to extract and quantize light-
relevant information within feature representations, thereby
effectively bridging the gap between low-light and normal-
light conditions, so as to promote the learning of light-
invariant features. In addition, we introduce an LAPM that
dynamically encodes illumination priors to adaptively guide
feature learning across varying brightness levels. Extensive ex-
periments across multiple datasets, including both same-source
and cross-source scenarios, demonstrate that LightQANet con-
sistently outperforms the existing state-of-the-art LLIE meth-
ods, validating the effectiveness of our proposed approach in
achieving robust and adaptive illumination enhancement.
VI. ACKNOWLEDGMENT
This work was supported in part by National Natural
Science Foundation of China (No. 62476172, 62476175,
62272319,
62206180,
82261138629),
and
Guangdong
Basic
and
Applied
Basic
Research
Foundation
(No.
2023A1515010677, 2023B1212060076, 2024A1515011637,
2025A1515011511),
and
Science
and
Technology
Planning
Project
of
Shenzhen
Municipality
(No.
JCYJ20220818095803007,
JCYJ20240813142206009),
and
Guangdong
Provincial
Key
Laboratory
(No.
2023B1212060076),
and
XJTLU
Research
Development
Funds (No. RDF-23-01-053).
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JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 LightQANet: Quantized and Adaptive Feature Learning for Low-Light Image Enhancement Xu Wu, Zhihui Lai∗, Xianxu Hou, Jie Zhou, Ya-nan Zhang, Linlin Shen Abstract-Low-light image enhancement (LLIE) aims to improve illumination while preserving high-quality color and texture. However, existing methods often fail to extract reliable feature representations due to severely degraded pixel-level information under low-light conditions, resulting in poor texture restoration, color inconsistency, and artifact. To address these challenges, we propose LightQANet, a novel framework that introduces quantized and adaptive feature learning for lowlight enhancement, aiming to achieve consistent and robust image quality across diverse lighting conditions. From the static modeling perspective, we design a Light Quantization Module (LQM) to explicitly extract and quantify illumination-related factors from image features. By enforcing structured light factor learning, LQM enhances the extraction of light-invariant representations and mitigates feature inconsistency across varying illumination levels. From the dynamic adaptation perspective, we introduce a Light-Aware Prompt Module (LAPM), which encodes illumination priors into learnable prompts to dynamically guide the feature learning process. LAPM enables the model to flexibly adapt to complex and continuously changing lighting conditions, further improving image enhancement. Extensive experiments on multiple low-light datasets demonstrate that our method achieves state-of-the-art performance, delivering superior qualitative and quantitative results across various challenging lighting scenarios. Index Terms-Low-Light Image Enhancement, VectorQuantized General Adversarial Network, Prompt Learning. I. INTRODUCTION I MAGES captured in dark environments, often referred to as low-light images, suffer from reduced illumination, increased artifact, and poor texture and color fidelity than those captured under normal-light conditions [1]. These deficiencies not only make it challenging for the human eye to discern objects but also significantly degrade the performance of ∗represents the corresponding author. Xu Wu, is with the 518060, China and (e-mail: ) Zhihui Lai and Linlin Shen are with the Computer Vision Institute, 518060, China. (e-mail: lai zhi Xianxu Hou is with 'an Jiaotong-Liverpool University, China (e-mail: ). Jie Zhou is with the 410114, China, and also with the 518060, China. (e-mail: jie ). Ya-nan Zhang is with 610065, China (e-mail: ). Linlin Shen is with Computer Vision Institute, - ligence, Shenzhen University, Shenzhen 518060, China and also with the 315100, China (e-mail: ). RetinexFormer CIDNet Ours Input SNR SMG Fig. 1. Visual comparisons on the real low-light image. The input image is compared with results from SNR [2], SMG [3], RetinexFormer [4], CIDNet [5], and our method. The zoomed-in regions show the details of texture and sharpness restoration. Our method produces the most natural textures and smooth transitions around edges. advanced visual models, such as object detection systems. Therefore, the development of robust and effective LLIE methods is essential for improving the visual quality and utility of images captured in low-light conditions. In previous works, Histogram Equalization (HE)-based and Retinex-based methods have been prominent in enhancing low-light images. HE-based methods enhance image contrast by adjusting the gray-level distribution of pixels to equalize the histogram [6]. Conversely, Retinex-based methods focus on estimating and enhancing the illumination component of each pixel to improve brightness. However, both approaches may amplify noise and cause color distortion [7], presenting significant challenges that necessitate further refinement. Recent advances in deep learning for LLIE have primarily focused on end-to-end networks that directly improve illumination [14]. Many of these methods introduce explicit illumination modeling, such as using an illumination branch to guide feature learning [2], or applying Retinex theory to decompose images into reflection and illumination components [4]. However, despite these innovations, current approaches often struggle to preserve robust feature representations under 16 Oct 2025 JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2 (b) Code activation frequency Input Baseline Ours Ground Truth (a) Ablation comparison (c) PSNR comparison (d) NIQE comparison Fig. 2. Effectiveness analysis and benchmark comparison. (a) represents results of the baseline built by VQ-GAN [8] and our method. (b) shows code activation frequency on LOL-v2 Real dataset [9]. "GT in Baseline" and "LL in Baseline" represent inputting ground truth and low-light images to pre-trained VQ-GAN [8] and fine-tuned VQ-GAN with low-light images, respectively. "LL in CodeEnhance" and "LL in Ours" denote inputting low-light images to CodeEnhance [10] and our method. Activation frequency of "LL in Ours" is closer to that of "GT in Baseline", which indicates our method can learn and activate important features under low-light conditions. (c) and (d) present performance comparison on LLIE datasets (LOL-v2-Synthetic [9], LOL-v2-Real [9], LSRW-Huawei [11], LIME [12], MEF [7], VV 1, and NPE [13]) in terms of PSNR (the higher the better) and NIQE (the lower the better). As highlighted by the bold red line, the proposed method consistently achieves the best demonstrating its superior. low-light conditions, where severely compromised pixel-level visibility and reliability hinder effective feature extraction, ultimately resulting in texture degradation and color distortion. As shown in Fig. 1, the results of SNR [2] and SMG [3] suffer from noticeable blurring and artifact amplification, leading to degraded texture clarity and unnatural visual appearances. RetinexFormer tends to cause over-exposure in brighter regions while under-enhancing darker areas, and CIDNet exhibits evident color distortion. By contrast, our method achieves clearer edge contours and more faithful texture recovery, producing sharper window patterns and smoother surfaces that better preserve the perceptual realism of the scene. Addressing these challenges requires moving beyond direct pixel-level enhancement towards feature-level robustness under complex and variable illumination. In this study, we reformulate the conventional image-to-image enhancement pipeline into an image-to-feature learning framework, which reduces the uncertainty inherent in the enhancement process by focusing on more abstract and stable feature representations. Specifically, we employ a Vector-Quantized Generative Adversarial Network (VQ-GAN) [8] to reconstruct high-quality images by leveraging vector-quantized features and learning an effective mapping between images and these features for LLIE tasks. However, as illustrated in Fig. 2 (a) and (b), directly applying VQ-GAN to LLIE leads to suboptimal results. Under lowlight conditions, the activation frequency of "LL in Baseline", where the baseline model is VQ-GAN, exhibits significant inconsistency compared to "GT in Baseline", particularly at crucial feature indices. This is primarily due to VQ-GAN's lack of illumination-aware mechanisms and its reliance on clean feature distributions for effective codebook matching. Under extreme darkness, encoder features become misaligned with the learned codebook, resulting in color distortion and overexposure. These observations highlight the necessity of learning light-invariant representations and ensure effective feature quantization under low-light conditions. To tackle these challenges, we propose LightQANet, a novel framework for low-light image enhancement based on quantized and adaptive feature learning. The core of LightQANet is the Light Quantization Module (LQM), which aims to explicitly quantify illumination-related information embedded in image features. Instead of treating illumination variations implicitly, LQM is designed to learn a structured representation of lighting conditions by extracting and quantizing the socalled light factors from both low-light (LL) and normal-light (NL) images. To achieve this, LQM is trained to distinguish illumination levels through a supervised contrastive objective, enabling it to accurately capture and quantify the intensity and distribution of illumination in the feature space. Once LQM acquires the ability to model illumination variations, it serves as an auxiliary guidance mechanism for the LightQANet. Specifically, the LightQANet is encouraged to minimize the feature-level discrepancies between different illumination conditions, thereby promoting the extraction of light-invariant representations. Through this collaboration, LQM not only provides structured illumination supervision but also enhances the robustness and generalization of the overall enhancement framework. Note that LQM is not required during testing. While LQM provides a structured quantization of illumination information, real-world lighting conditions are often complex and dynamically changing. To address this challenge, we introduce the Light-Aware Prompt Module (LAPM), which dynamically guides feature learning based on illumination priors. Specifically, LAPM encodes illumination information into a set of learnable prompts, each capturing discriminative characteristics associated with different illumination levels. These prompts are adaptively fused with intermediate feature representations in the primary LLIE network, enabling the model to systematically adjust its feature learning process according to the estimated lighting conditions. By dynamically injecting illumination-specific cues into the feature space, LAPM enhances the model's ability to generalize across a wide range of lighting environments. The main contributions of this work are as follows: • We propose LightQANet, a novel framework that performs quantized and adaptive feature learning to extract light-invariant representations, enabling consistent and robust low-light image enhancement under diverse illumination conditions. • We design two key modules to enhance illumination adaptability: LQM, which extracts and quantizes light factors to build light-invariant feature representations, and LAPM, which dynamically refines feature representations based on light-specific priors. These modular design enJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 3 ables stable and consistent feature representations across varying illumination conditions. • We conduct extensive experiments in datasets, including LOL-v2-Real, LOL-v2-Synthetic, LSRW-Huawei, LIME, MEF, VV, and NPE, demonstrating that LightQANet consistently achieves state-of-the-art performance, as evidenced by the results shown in Fig. 2 (c) and (d). The remainder of this paper is organized as follows. Section II reviews related works. Section III presents a detailed model design. Section IV conducts the experiments and discusses the results. Finally, we conclude the paper in Section V. II. RELATED WORKS This section reviews previous work related to low light image enhancement and recent advances in discrete codebook learning for image restoration tasks. A. Low-Light Image Enhancement Images captured in low-light environments typically suffer from poor quality, lacking essential visual details which can hinder comprehension and analysis. Initially, researchers tackled this problem through histogram equalization technologies [6], adjusting illumination and contrast by equalizing pixel intensity distributions. Various methods evolved from this approach, focusing on different enhancement aspects such as overall image perspective [15], cumulative distribution functions [16], and the addition of penalty terms to refine the enhancement process [17]. Parallel to these developments, some researchers applied Retinex theory [18], which decomposes an image into illumination and reflection components, allowing for targeted enhancements in both areas. Subsequent Retinex-based enhancements, such as SSR [19], improved both illumination and color accuracy significantly. The advent of deep learning [20] introduces more methods into the LLIE [21]. LLNet [1] is the first to integrate stacked autoencoders for enhancing low-light images. This is followed by the introduction of multi-branch [22] [23] and multi-stage [24] networks, designed to tackle illumination recovery, noise suppression, and color refinement concurrently. SNR [2] uses the PSNR distribution map to guide network feature learning and fusion. SMG [3] incorporates image structural information to enhance the output image's quality. Recent innovations have combined Retinex theory with deep learning to further refine enhancement techniques. URetinexNet [25] formulates the decomposition problem of Retinex as an implicit prior regularization model, and Retinexformer [4] uses illumination to guide the Transformer [26] in learning the global illumination information of the image. LLformer [27] proposes a new transformer and attention fusion block for LLIE. GSAD [28], JoRes [29] and LLDiffusion [30] leverage the diffusion model to perform LLIE. QuadPrior [31] improves low-light images by physical quadruple priors. CIDNet [5] proposed a new color space to overcome color bias and brightness artifacts in LLIE. B. Discrete Codebook Learning Discrete codebook learning is first introduced in the context of Vector Quantized-Variational AutoenEoder (VQ-VAE) [32]. Subsequently, VQ-GAN integrates this approach within the generative adversarial network framework, facilitating the generation of high-quality images [8]. In low-level image processing tasks, codebook learning helps mitigate uncertainty during model training by transforming the operational space from raw images to a compact proxy space [33]. To enhance feature matching, FeMaSR [34] introduces residual shortcut connections, RIDCP [35] develops a controllable feature matching operation, and CodeFormer [33] employs a Transformer-based prediction network for retrieving codebook indices. Additionally, LARSR [36] proposes a local autoregressive super-resolution framework utilizing the learned codebook. CodedBGT [37] and CodeEnhance [10] introduce the codebook to improve LLIE model performance. Different from CodedBGT [37] and CodeEnhance [10], we propose the LQM to precisely extract light factors from image features. Additionally, we introduce the LAPM to dynamically enhance image representations through light-specific knowledge. Collectively, these modules significantly improve the representation of light information, thereby elevating the overall quality of the enhanced images. III. METHODOLOGY This section provides a detailed introduction to the proposed method, which includes high-quality codebook learning, lightinvariant feature learning, feature matching and image restoration, and training objectives. A. Overview The proposed method consists of two stages: the first stage constructs a high-quality codebook using VQ-GAN trained on well-lit images to capture representative visual patterns. The second and more critical stage focuses on enhancing low-light images by extracting light-invariant features, ensuring stable representation and effective illumination correction across diverse lighting conditions. Firstly, we leverage VQ-GAN to encode detailed features from high-quality images Ih into a discrete set of codebook, which serve as a comprehensive reference for accurately reconstructing images. This stage can be formulated as follows: Zh = E(Ih), eZh = M(Zh, C), I′ h = D(eZh), (1) where E(·) and D(·) denote encoder and decoder. M(·, ·) is feature matching operation, where C represents learnable codebook of features. Zh and eZh are latent features and quantized features. In the subsequent step, the C and D(·) will be frozen to leverage the quantized features obtained from C, followed by D(·) reconstructing high-quality images. This ensures stability in the learning process and consistency in the output quality. Next, as shown in Fig. 3, to improve feature extraction in low-light conditions, we develop light-invariant feature learning, where the LQM and LAPM are crucial for normalizing the impact of different lighting conditions on feature extraction, JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 4 Codebook 0 1 2 N ... 247 738 879 ... 429 ... Light-aware Prompts Conv Conv GAP Conv C Conv Unfold Light-Aware Prompt Module (LAPM) Fuse LAPMs Nearest-Neighbor Matching Feature Matching Frozen Learnable Network Processing GAP Global Average Pooling Unfold Unfold operation codes sequence Light Quantization Module (LQM) ... ... ... ... ... ... ... Vector Quantization Operate Light Quantization Operate Fig. 3. Overview of our proposed LightQANet framework. Our method leverages a pretrained codebook C† and a decoder D† as the foundations. In LightQANet, the Light Quantization Module (LQM) is utilized to extract light factors and promote the learning of light-invariant features by a novel light consistency loss (Llcl). Additionally, the Light-Aware Prompt Module (LAPM) is introduced to encode light illumination data for dynamically guiding the feature learning process. Finally, the feautre fusion via linear interpolation refines the reconstructed features. thereby stabilizing the model's performance across varied environments. Following this, we introduce feature matching and image reconstruction, examining how the algorithm aligns features under diverse illumination settings and reconstructs high-quality images from these aligned features. Low-light enhancement stage can be formulated as follows: Zll = E′(Ill), eZll = M(Zll, C†), Irec = D†(eZll, Ffuse), (2) where Ill and Irec denote low-light images and reconstructed images, respectively. Zll and eZll are latent features and quantized features of Ill. Ffuse represents output of the feature fusion in skip connection and is defined in Section III-D. E′ denotes the light-invariant feature learning. C† and D†(·) are codebook and decoder with frozen parameters, respectively. B. High-Quality Codebook Learning We first pre-train a VQ-GAN using high-quality images to learn a discrete codebook. This codebook serves as prior knowledge for enhancing low-light images. The corresponding decoder associated with the codebook is then utilized to reconstruct images. Given a high-quality image Ih, we first employ the encoder of VQ-GAN to obtain a latent feature Zh ∈Rm×n×d. Then, by calculating the distance between each 'pixel' z(i,j) h of Zh and the ck in the learnable codebook C = {ck ∈Rd}N k=0, we replace each z(i,j) h with the nearest ck [33]. After that, the quantized features eZh ∈Rm×n×d are obtained: ez(i,j) h = M(z(i,j) h , C) = arg min ck∈C ∥z(i,j) h -ck∥2, (3) where N = 1024 represents the size of the codebook, and d = 512 denotes the channel number of both Zh and C. The dimensions m and n specify the sizes of Zh and eZh. The reconstructed image I′ h is then generated by the decoder. The VQ-GAN is supervised using the loss function Lvq [8], which includes an L1 loss Lmae, a codebook matching loss Lcma, and an adversarial loss Ladv: Lvq = Lmae + Lcma + Ladv, LL1 = ∥Ih -I′ h∥1, Lcma = σ∥Zh -sg(eZh)∥2 2 + ∥sg(Zh) -eZh∥2 2, Ladv = γlogD(Ih) + log(1 -D(I′ h)), (4) where D(·) is the discriminator. sg(·) is the stop-gradient operator. σ = 0.25 denotes a weight trade-off parameter that governs the update rates of both the encoder and codebook [33]. γ is usually set to 0.1 [35]. C. Light-Invariant Feature Learning The efficacy of our method depends on the quality of light-invariant feature learning. To achieve this, we design two key modules: the LQM, which models illumination in a structured manner, and the LAPM, which adaptively guides feature learning based on illumination priors. 1) Light Quantization Module (LQM). To effectively extract light-invariant features for low-light image enhancement, we propose the LQM, motivated by the critical need to disentangle illumination information from detailed content representations. Unlike conventional methods that operate directly on raw image features, we explicitly model illumination as a global JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 5 ... ... Feature maps Light Quantization Module (a) Training on Light Quantization Module (LQM) (b) Training on the proposed LightQANet ... ... Feature maps Light Quantization Module Gram matrices (c) t-SNE of distributions Light factors Low-light Normal-light Fig. 4. Overview of the proposed LQM. (a) and (b) illustrate the alternating optimization of the LQM and the LightQANet. (c) shows the Gram matrices from low-light (LL) and normal-light (NL) images, which exhibit significant overlap. In contrast, the light factors are clearly distinguished based on lighting conditions, underscoring their effectiveness in accurately capturing and representing light-relevant information. Note that the LQM is only used during the training phase and therefore does not impact the processing speed during the inference stage. style attribute, leveraging its inherent characteristics such as overall brightness, contrast, and color distribution, which are largely independent of fine-grained image details [38]. Drawing inspiration from the success of style-based approaches in nighttime domain adaptation [39] and low-light enhancement such as EnlightGAN [40], we adopt a Gram matrix [41] representation within LQM to capture and isolate illuminationrelated style features. The Gram matrix effectively encodes global feature correlations, allowing LQM to abstract illumination information while filtering out irrelevant content-specific variations. The Gram matrix is defined as follows: G = aTa, (5) where G ∈Rc×c, c is the number of channels. a represents feature maps. To equip the LQM with the ability to quantify illumination conditions, we formulate a supervised learning objective based on pairwise light factor distances. Specifically, for a set of image pairs P, LQM learns to construct a light factor space in which images captured under similar lighting conditions are mapped closer together, while those under different lighting conditions are pushed further apart. The light factors f a,l and f b,l are computed by applying the LQM Q(·) to the Gram matrices Ga,l and Gb,l extracted from the l-th intermediate feature maps. The training is supervised using the following loss: Llqm = X (a,b)∈P n (1 -1(a, b)) m -d(f a,l, f b,l) 2 + +1(a, b) d(f a,l, f b,l) -m 2 + o , (6) where d(·) denotes the cosine similarity, [·]+ is the hinge function, m is the margin, and 1(a, b) is an indicator function that returns 1 if Ia and Ib have the same lighting condition and 0 otherwise. During this training stage, the encoder of the proposed is frozen and only the parameters of the LQM are updated, as shown in Fig. 4 (a). This learning process enables LQM to accurately quantify illumination differences and establish a light factor space is used to guide lightinvariant feature learning in subsequent LightQANet training. After LQM has acquired the ability to quantify illumination, we leverage its learned light factor space to guide the training of the proposed model. Specifically, we introduce a light consistency loss Llcl to minimize the discrepancy between the light factors of low-light and normal-light images. As illustrated in Fig. 4 (b), during this stage, the LQM is kept frozen while the encoder parameters are updated. The light consistency loss is defined as: Ll lcl(f a,l, f b,l) = 1 4d2 l n2 l dl X i=1 (f a,l i -f b,l i )2, (7) where f a,l and f b,l are the light factors extracted by the frozen LQM, dl denotes the dimensionality of the light factors, and nl is the spatial size of the l-th feature map. By minimizing Llcl, the encoder is encouraged to extract feature representations that are invariant to illumination variations, thus improving overall enhancement in various illumination scenarios. Algorithm 1 and Fig. 4 (a) and (b) illustrate the alternating optimization process between the LQM and the LightQANet network. This optimization strategy gradually reduces the discrepancy in illumination conditions between low-light and normal-light images, ultimately promoting the extraction of light-invariant features within the LightQANet framework. To demonstrate the effectiveness of LQM, we analyze the Gram matrices computed from intermediate feature maps and their corresponding light factors produced by LQM. As shown in Fig. 4 (c), the LQM effectively isolates illumination information, clearly separating it from other content-related features. These results indicate that the extracted light factors successfully encode illumination-specific attributes, as intended. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 6 Algorithm 1 LightQANet Training Algorithm 1: Input: Paired training data (Ill, Inl): low-light and normal-light images 2: Randomly initialize the model parameters θ; 3: for each training pair (Ill, Inl) do 4: Zll ←E′(Ill); 5: eZll ←M(Zll, C†); 6: Irec ←D†(eZll, Ffuse); 7: # Update LQM and freeze LightQANet; 8: Optimize LQM using Llqm; 9: # Update LightQANet and freeze LQM; 10: Optimize LightQANet using combined loss: Ladv + Lfml + Lrec + Llcl; 11: end for 12: Return Enhanced image Irec 2) Light-Aware Prompt Module (LAPM). While LQM effectively captures structured illumination characteristics, it faces limitations in representing complex and spatially varying lighting patterns typically observed in real-world scenarios. To overcome this, LAPM dynamically adapts feature representations by aggregating illumination information from local spatial regions. Specifically, LAPM computes prompt weights based on local features rather than relying solely on a global illumination descriptor. This enables each region in the image to contribute effectively to the dynamic prompt composition, thus capturing fine-grained variations in illumination and providing more adaptive modulation of the final feature representation. Fig. 3 shows that the prompt component Pn, consisting of five learnable vectors, embeds light information from n levels. These prompt vectors are not only responsible for modeling discrete brightness states but are also trained to encode transitional relationships between different brightness levels and to capture global illumination properties. To generate the lightaware prompts P, we compute attention-based weights from local features and then apply these weights to Pn. The weights serve as region-wise "soft assignments," guiding each prompt to specialize in the luminance ranges where it is most effective (e.g., extremely dark vs. mid-level brightness). Summing these weighted prompts yields a prompt-guided feature modulation that faithfully reflects the overall illumination distribution, from darkest shadows to brightest highlights. Specifically, we first divide the image features into n patches. Average pooling is applied to these patches to extract local features, which are then processed by a channel-shrink layer to ensure their dimensions align with Pn. After dimension alignment, a softmax function denoted as Fs, is employed to compute the weights ωn ∈RC. The weights interact with the Pn to generate P, which are further processed by a convolution layer with a 3 × 3 kernel size. These operations can be collectively formulated as follows: P = LAPM(Fl, Pn) = F3( N X n=1 ωnPn), ωn = Fs(F1(FA(Fl))), (8) where Fl = UF(Fe) represents local features. UF(·) is a (a) Correlation matrix (b) Brightness Fig. 5. Prompt-Brightness correlation analysis. (a) Correlation heatmap between prompt weights (Prompt Index 1-5) and brightness levels. Positive (red) and negative (blue) values indicate correlation strength. Different prompts show distinct sensitivities, with some strongly responding to extremely dark regions [0, 25.5) and others to brighter or transitional regions, highlighting their complementary roles. (b) Brightness distribution histogram of the MEF dataset [7], showing a predominance of darker pixels. unfold operation. Fe denotes the intermediate features of encoder. FA(·) is a average pooling operation. F1(·) is the channel-shrink layer performed by a 1 × 1 convolution layer. Finally, the light-aware prompts P are integrated channel-wise with the intermediate features of the encoder. These combined features are then processed by a ResNet block [42], enhancing the overall feature representation. Fig. 5 (a) shows that each prompt vector responds distinctly to different brightness ranges. For example, prompts 1 and 3 are highly sensitive (correlations of 0.55 and 0.49) to extremely dark pixels ([0, 25.5)). Prompt 5 primarily handles low-light pixels [25.5, 51), showing a correlation of 0.47. Prompt 4 mainly focuses on mid-to-high brightness levels (above 76.5), effectively capturing general illumination patterns. Prompt 2 complements these prompts by responding moderately (correlation of 0.37) to intermediate brightness levels [102, 127.5), and mild negative correlations in extremely dark [0, 25.5) and bright regions (above 178.5). Furthermore, our prompt allocation strategy is informed by the illumination conditions, as illustrated by Fig. 5 (b). Since the majority of pixels in low-light images fall into the darkest brightness interval ([0,25.5)), we assign two dedicated prompts (prompts 1 and 3) to this interval. Overall, our multi-prompt vector design and data-driven allocation strategy ensure each prompt effectively captures illumination-specific information, especially for diverse brightness conditions. D. Feature Matching and Image Reconstruction We perform feature matching through nearest-neighbor lookup in the codebook to obtain high-quality features, as shown in Fig. 3. Subsequently, these high-quality features are transmitted to a decoder that incorporates skip feature fusion modules, enabling the reconstruction of enhanced images. Based on Eq. 3, the feature matching M(·, ·) in low-light enhancement task can be formulated as follows: ez(i,j) ll = M(z(i,j) ll , C†) = arg min ck∈C† ∥z(i,j) ll -ck∥2, (9) where Zll = {z(i,j) ll ∈Rd}m,n i=0,j=0 denotes image features extracted by light-invariant feature learning, eZll = {ez(i,j) ll ∈ Rd}m,n i=0,j=0 represents quantized features. JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 7 To improve the quality of reconstructed images, we introduce a feature fusion via linear interpolation technique that effectively merges low-level features Fe from the encoder with features Fd from the decoder. This integration not only preserves critical texture information but also compensates for potential detail loss during image processing. Initially, features Fe and Fd are combined in channel-wise and subsequently computes affine transformation parameters α and β. These parameters are designed to reduce the impact of noise and enhance texture representation in the reconstructed images. This feature fusion can be formulated as follows: Ffuse = α ⊙Fd + β, α, β = C([Fd, Fe]), (10) where C(·) denotes convolution operation, and ⊙is elementwise multiplication. Finally, we employ the LAPM to further refine the features. Importantly, the parameters within the decoder blocks remain frozen during this process. E. Training Objectives Finally, we outline the training objectives that guide the overall learning process. Specifically, the LQM is optimized with a dedicated contrastive loss (Llqm), which has defined in Eq. 6 in Section III-C. The main enhancement model is trained with a combination of Adversarial Loss Ladv, Feature Matching Loss Lfml, Light Consistency Loss Llcl, and Reconstruction Loss Lrec to ensure high-quality restoration under varying illumination conditions, which are defined as follows: Ltotal = Ladv + Lfml + Lrec + λLlcl, (11) where Ladv is defined in Eq. 4 of Section III-A. λ = 0.5 is a weight of Llcl. Llcl is used to minimize the discrepancy between the light factors of low-light and normal-light images, which has defined in Eq. 7 of Section III-C. 1) Feature Matching Loss. This loss function is specifically designed to optimize the encoder by facilitating its learning of the mapping between low-light images and high-quality priors. By minimizing this loss, the encoder can enhance the proposed method ability to accurately translate low-light conditions into visually appealing outputs, aligning with predefined highquality standards. The loss is formulated as follows: Lfml = σ∥Zll -sg(eZh)∥2 2 + ∥φ(Zll) -φ(sg(eZh))∥2 2 where φ(·) is used to calculate the gram matrix of features. Zll and eZh represent the latent features of low-light images and quantized features of high-quality images, respectively. 2) Reconstruction Loss. This loss function combines L1 loss and perceptual loss to ensure that enhanced images have a complete structure and impressive visual appeal. The L1 loss minimizes pixel-level discrepancies for high fidelity, while perceptual loss aligns images to human visual perception, enhancing both structural accuracy and aesthetic quality. The loss is defined as follows: Lrec = ∥Inl -Irec∥1 + ∥ψ(Inl) -ψ(Irec)∥2 2, (12) where Inl and Irec represent normal-light images and reconstructed images, respectively. ψ(·) indicates the Learned Perceptual Image Patch Similarity (LPIPS) function [43]. IV. EXPERIMENTS This section presents experimental results to evaluate the effectiveness of the proposed method through quantitative comparisons, qualitative analysis, and ablation studies. A. Implementation Details For VQ-GAN and the proposed LightQANet training, the input pairs are randomly cropped to patches of size 256 × 256. We use ADAM optimizer with β1 = 0.9, β2 = 0.999 and ε = 10-8. The learning rate is set to 10-4. The VQ-GAN is pre-trained on the DIV2K [44] and Flickr2K [45] with 350K iterations. Our LightQANet is trained with 50K iterations. The hyper-parameter m is set to 0.1. All experiments were conducted in PyTorch on an NVIDIA A6000. B. Datasets and Evaluation Metrics 1) Low-light Datasets. We evaluate methods using the LOLv2 [9] and LSRW-Huawei [11]. And also evaluate methods in cross datasets: LIME [12], MEF [7], VV 1, and NPE [13] The LOL-v2-Real one contains 689 train images and 100 test images. The LOL-v2-Synthetic one includes 900 train images and 100 test images. The LSRW-Huawei contains 2,450 train images and 30 test images. The LIME, MEF, VV, and, NPE include 10, 17, 24, and 8 low-light images, respectively. 2) Evaluation Metrics. We assess the quality of the enhanced images using the most common metrics: Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM) [47], Natural Image Quality Evaluator (NIQE) [48], and LPIPS [43]. Unlike PSNR and SSIM, which primarily focus on lowelevel similiaity, LPIPS accounts for the human visual system's perception of similarity, offering a more accurate reflection of how images are perceived by viewers. C. Comparison with State-of-the-Art Methods We assess the performance of our LightQANet by conducting comparisons with numerous leading LLIE techniques. These include LIME [12], JED [49], RetinexNet [50], KinD [51], EnlightGAN [40], Zero [52], SNR [2], PairLIE [46], RetinexFormer [4], SMG [3], CodedBGT [37], DMFourLLIE [23], CodeEnhance [10], QuadPrior [31], and CIDNet [5]. 1) Same-Domain Evaluation. We conduct detailed visual comparisons on LOL-v2-Real (see Fig. 6), LOL-v2-Synthetic (see Fig. 7), and LSRW-Huawei (see Fig. 8), focusing on four key aspects, with quantitative results summarized in TableI. Illumination Consistency: Competing methods such as Zero, CIDNet, and RetinexFormer exhibit abrupt brightness transitions, while SMG, QuadPrior, and Zero under-enhance dark regions. Our method achieves smooth, natural illumination transitions, which is reflected in the highest SSIM scores (0.8974, 0.9388, 0.7179) across all datasets. Color Fidelity: LIME, RetinexNet, and EnlightGAN introduce strong color 1https://sites.google.com/site/vonikakis/datasets JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 8 TABLE I QUANTITATIVE RESULTS ON LOL-V2 [9] AND LSRW-HUAWEI [11]. ↑INDICATES THE HIGHER THE BETTER. ↓INDICATES THE LOWER THE BETTER. BOLD: BEST RESULT; UNDERLINE: SECOND BEST RESULT; -: UNAVAILABLE DATA. P AND F DENOTES PARAMETERS AND FLOPS. LOL-V2-Real LOL-V2-Synthetic LSRW-Huawei Complexity Methods PSNR ↑ SSIM ↑ LPIPS ↓ NIQE ↓ PSNR ↑ SSIM ↑ LPIPS ↓ NIQE ↓ PSNR ↑ SSIM ↑ LPIPS ↓ NIQE ↓ Param (M) FLOPs (G) LIME 16.97 0.4598 0.3415 8.4899 17.50 0.7718 0.1748 3.4063 18.46 0.4450 0.3922 3.3879 - - JED 17.29 0.7266 0.2760 4.3703 16.89 0.7299 0.2370 3.5284 15.11 0.5379 0.4327 3.1521 - - RetinexNet 16.10 0.4006 0.4215 9.2661 17.14 0.7615 0.2185 4.3433 16.82 0.3951 0.4566 3.4942 0.84 587.47 KinD 16.75 0.6456 0.4118 4.6253 17.51 0.7694 0.2093 3.3295 17.19 0.4625 0.4318 2.9682 8.02 34.99 EnlightGAN 17.94 0.6755 0.3197 4.8755 16.59 0.7780 0.2179 3.0998 17.46 0.4982 0.3780 3.0650 114.35 61.01 Zero 18.06 0.5736 0.2980 7.7571 17.76 0.8163 0.1382 3.0464 16.40 0.4761 0.3763 3.0477 0.075 4.83 SNR 21.48 0.8489 0.1996 3.6383 22.88 0.8962 0.1124 3.5854 20.67 0.6246 0.4879 3.4008 39.12 26.35 SMG 24.03 0.8178 0.2283 5.7291 25.62 0.9188 0.2915 5.9165 20.66 0.5589 0.4449 6.9247 0.33 20.81 PairLIE 19.88 0.7777 0.2834 3.6192 19.07 0.7965 0.2183 3.9121 18.99 0.5632 0.3711 3.0790 1.61 15.57 RetinexFormer 22.79 0.8397 0.2270 3.3869 25.67 0.9296 0.0775 2.8861 20.81 0.6303 0.4124 2.8866 30.35 137.37 CodeEnhance 23.32 0.8310 0.2184 3.2115 24.65 0.9163 0.0648 3.2019 21.14 0.6076 0.2840 2.6424 49.07 225.86 DMFourLLIE 22.64 0.8589 0.1488 2.9389 25.83 0.9314 0.0562 2.8892 21.47 0.6331 0.3998 3.0153 0.41 1.56 QuadPrior 20.58 0.8036 f 0.2410 5.8903 16.11 0.7646 0.2187 4.6653 18.30 0.6013 0.4070 3.7033 1252.75 1103.20 CIDNet 24.11 0.8675 0.1678 3.4159 25.13 0.9387 0.0536 2.8128 20.86 0.6202 0.3740 2.6131 1.88 7.57 Ours 28.51 0.8974 0.1039 3.1926 26.15 0.9388 0.0457 2.8636 21.68 0.7179 0.2885 2.5784 18.85 164.20 LIME EnlightGAN RetinexNet SMG Zero SNR RetinexFormer Input Ours Ground Truth CIDNet QuadPrior Fig. 6. Visual comparison on the LOL-v2-Real [9], accompanied by image error maps calculated by L2 loss. The proposed method produces a more natural illumination transition across shadow boundaries (e.g., wall region), accurate colors, and well-preserved details. TABLE II COMPARISONS ON LIME [12], MEF [7], NPE [13], AND VV 1 IN TERMS OF NIQE, WHERE THE LOWER THE BETTER. NOTE THAT THE RESULTS "NULL" ARE DUE TO THE CORRESPONDING METHODS LACKING CODE. Methods LIME MEF NPE VV KinD2019 6.71 3.17 3.28 2.32 EnlightGAN2019 3.59 3.11 4.36 3.18 Zero2020 3.79 3.31 3.48 2.75 SNR2022 4.88 3.47 4.19 7.55 PairLIE2023 4.31 3.92 3.68 3.16 RetinexFormer2023 3.70 3.14 3.58 1.95 SMG2023 6.47 6.18 5.89 5.46 CodedBGT2024 4.20 3.85 3.52 Null QuadPrior2024 4.58 4.36 3.65 3.44 CIDNet2025 3.85 3.46 3.82 3.24 Ours 3.54 2.88 3.26 1.94 distortions; Zero, SMG, and QuadPrior show biases in wall and text regions. LightQANet restores accurate hues, aided by illumination quantization and adaptive modulation, yielding the lowest LPIPS values (0.1039, 0.0457, 0.2885). Texture Preservation: SNR, SMG, QuadPrior, and CIDNet fail to recover fine details, producing smoothed textures. Our method preserves sharp contours and structural details, consistent with the top PSNR results (28.51, 26.15, 21.68). Artifact Suppression: RetinexNet, RetinexFormer, and SNR introduce noise or halos; CodeEnhance causes over-smoothing. In contrast, LightQANet suppresses artifacts effectively, balancing enhancement and detail. Across the three datasets, the proposed method consistently shows superior performance in handling illumination transitions, restoring accurate colors, preserving fine textures, and minimizing enhancement artifacts. These results further validate the effectiveness of LightQANet in delivering robust and perceptually pleasing low-light enhancement. 2) Cross-Domain Evaluation. To assess the robustness and generalization ability of our network under domain shift, we evaluate the model on unpaired low-light images from datasets that differ from the training domain, including MEF [7], NPE [13], LIME [12], and VV 1. As illustrated in Fig. 9 and Table II, the proposed method consistently achieves superior visual quality across diverse scenes and lighting conditions. Specifically, in the MEF dataset, our method restores natural brightness and preserves fine textures in both the grass and flower regions, while other methods tend to overexpose the sky or blur the foreground textures (see red arrows). In the NPE scene, we accurately enhance the dove's feathers and JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 9 LIME EnlightGAN RetinexNet SMG Zero SNR RetinexFormer Input Ours Ground Truth QuadPrior CodeEnhance Fig. 7. Visual comparison on the LOL-v2-Synthetic [9], accompanied by image error maps. Compared to competing methods, our approach demonstrates superior overall brightness and structural clarity. LIME, EnlightGAN, SMG, QuadPrior, and CodeEnhance display noticeable color deviations. LIME EnlightGAN RetinexNet Zero Input SMG SNR RetinexFormer Ours Ground Truth CIDNet QuadPrior Fig. 8. Visual comparison on the LSRW Huawei [11], accompanied by image error maps and Canny edge maps of text regions. As we can see that the proposed method restores both texture and color details effectively. surrounding foliage without color distortion or effects, unlike methods such as EnlightGAN and PairLIE which introduce unatural brightness or lose edge sharpness. In the LIME dataset, the proposed method clearly reveals the license plate characters and traffic sign symbols (highlighted in the red box), which are over-smoothed (SNR, SMG) in other methods. For the VV dataset, our method demonstrates superior detail preservation in both global structure and fine textures. Notably, the intricate wall carvings (see bottom row) are sharp and realistic in our result, whereas others either blur the details (SNR, SMG, and CIDNet) or color-destoration the region (EnlightGAN, PairLIE). JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 10 LIME VV SMG RetinexFormer Ours Input EnlightGAN SNR PairLIE CIDNet MEF NPE Fig. 9. Visual comparison on the LIME [46], MEF [7], VV 1, and NPE [13] dataset, accompanied by Canny edge maps. The results demonstrate that our method effectively enhances images across different lighting conditions, achieving a trade-off between texture preservation and illumination enhancement. These improvements can be attributed to the combination of structured modeling of illumination and high-quality prompt tailored for low-light conditions. Together, they allow the model to adaptively enhance illumination while maintaining structural fidelity and natural appearance, even when applied to previously unseen domains. 4) Complexity Analysis. Table I details model complexity (Params, FLOPs) at 256×256 resolution. Simple CNNs like Zero (0.075M params, 4.83G FLOPs) and DMFourLLIE offer low complexity but compromise enhancement quality. Conversely, QuadPrior's use of a pretrained diffusion backbone leads to high complexity (1252.75M params, 1103.2G FLOPs). Our LightQANet achieves a strong balance with 18.9M parameters and 164.2G FLOPs, representing a significant 61% reduction in parameters and 27% in FLOPs compared to the codebook-based CodeEnhance, thus offering superior quality at a moderate computational cost. D. Codes Activation Analysis The analysis of code activation frequencies in feature matching provides crucial insights into the efficacy of our image enhancement methods. Fig. 2 (b) shows comparisons of the code activation frequency among baseline, CodeEnhance [10], and the proposed method. From the results, it is observed that directly inputting low-light images into the baseline model ("LL in Baseline") causes highly imbalanced code usage, with certain codes (e.g., index 671) being excessively activated while others are underutilized. In contrast, "LL in CodeEnhance" and "LL in Ours" exhibit more evenly distributed activations, indicating that both methods better mitigate the degradation of code utilization caused by low-light conditions. Notably, the proposed method achieves a closer activation distribution to "GT in Baseline", suggesting that our enhancement strategy more effectively restores diverse and representative feature activations under challenging lighting conditions. Fig. 10 illustrates the effectiveness of our method. Panel (a) displays code activation of low-light images, which processed by the proposed method. Panels (b) and (c) compare activation frequencies for our enhanced and ground truth images using a pre-trained VQ-GAN, with panel (d) detailing the top ten code comparisons. Panels (a), (c), and (d) collectively demonstrate that our method effectively extracts features from low-light images, achieving activation frequencies that closely match JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 11 (b) Enhanced Images (c) Ground Truth (d) Codes Activation Frequency in top ten. (a) Low-light Images Fig. 10. Comparison of code activation frequency. The codebook includes 1024 quantized features, we reshape these feature indexes into 32×32. (a) denotes the code activate frequency of low-light images under our method. (b) and (c) represent code activation frequency of enhanced images and ground truth under baseline. (d) is a comparison of the top ten codes' activation frequency. The analysis shown in (a), (c), and (d) highlights the capability of the proposed method to effectively extract features from low-light conditions, achieving activation frequencies closely aligned with those of the ground truth. Additionally, (b), (c), (d) demonstrate the high quality of our results, exhibiting similar activation patterns to the ground truth within the baseline comparisons. This coherence across the figures substantiates the effectiveness of our enhancement approach in maintaining the integrity of image features under varied lighting conditions. Input Ground Truth (c) With FF and LQM (b) With FF (d) With FF and LAPM (a) Baseline 26.90/0.9122 29.77/0.9151 19.41/0.8843 20.99/0.8914 25.94/ 0.9135 (e) Ours With FF, LQM, and LAPM Fig. 11. Visual comparison of the ablation studies in Table III. Starting from the baseline, the integration of FF, LQM, and LAPM progressively improves both image quality and illumination correction, ultimately achieving the best performance when all components are combined together. TABLE III ABLATION STUDIES OF THE PROPOSED MODULES ON LOL-V2-REAL DATASET. BASELINE IS BUILT BY VQ-GAN [8]. FF MEANS THE FEATURE FUSION Ffuse IN SKIP CONNECTION. No. Baseline FF LQM LAPM PSNR SSIM (a) ✓ 23.91 0.8699 (b) ✓ ✓ 24.82 0.8751 (c) ✓ ✓ ✓ 25.95 0.8853 (d) ✓ ✓ ✓ 27.16 0.8911 (e) ✓ ✓ ✓ ✓ 28.51 0.8974 those of the ground truth. This similarity indicates that our method not only improves visibility but also preserves the image's inherent characteristics. Furthermore, comparisons in panels (b), (c), and (d) reveal that the activation patterns of our enhanced images align well with the ground truth within the VQ-GAN, showcasing the high fidelity of our results. These results further validate the effectiveness of our method in preserving the integrity and authenticity of image features across diverse lighting conditions. E. Ablation Study To validate the effectiveness of each proposed component, we conduct ablation studies on the LOL-v2-Real dataset, as summarized in Table III. The baseline model is constructed based on VQ-GAN [8], and we incrementally integrate the feature fusion (FF) in skip connection, Light Quantization Module (LQM), and Light-Aware Prompt Module (LAPM) to systematically assess their contributions. 1) Study of FF. As shown in Table III and Fig. 11, integrating the feature fusion module improves the PSNR from 23.91 to 24.82 and the SSIM from 0.8699 to 0.8751. This performance improvement stems from our learnable linear interpolation mechanism. It uses two dynamically predicted parameters, α and β, to modulate decoder features based on encoder information. Unlike direct feature concatenation or summation, this adaptive interpolation allows the model to enhance fine textures while mitigating noise amplification, thereby achieving more effective reconstruction under lowlight conditions. 2) Study of LQM. Upon integrating LQM shown in Table III and Fig. 11, the PSNR further improves to 25.95 and the SSIM increases to 0.8853. This improvement demonstrates the effectiveness of structured illumination modeling through the light-factor space learned by LQM. By explicitly quantizing illumination-related information and enforcing feature consistency between low-light and normal-light images through the light consistency loss (Llcl) in the learned light-factor space, LQM enables the proposed model to better align JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 12 (a) Input image (b) Enhanced image (c) Prompt weights (d) HE Fig. 12. Instance analysis of LAPM effects in LLIE. (a) and (b) are low-light input image and its enhanced result. (c) Learned prompt weights, where prompts 1 and 3 focus on extremely dark regions [0, 25.5], prompt 5 targets transitional brightness [25.5, 51], and prompt 4 responds to mid-to-high brightness levels (above 76.5). (d) Brightness distribution histograms before and after enhancement, illustrating luminance correction across different regions. (a) Mean of prompt weights (b) Cumulative ratio of pixels (c) Comparison between the mean of prompt weights and pixel ratio of MEF dataset Fig. 13. Analysis of learned prompt weights on the MEF dataset [7]. (a) Mean prompt weights with min-max range for the five learnable prompts. (b) Cumulative ratio of pixels as a function of brightness value, with vertical dashed lines marking the boundaries of the five prompt-responsive intervals. (c) Comparison of the mean prompt weights against the corresponding pixel ratios within each brightness range, where prompts were grouped according to their correlations in Fig.5. [0, 25.5): combined weight of Prompt1+Prompt3, [25.5, 51): weight of Prompt5, [51, 153): weight of Prompt2+0.5×Prompt4, [153, 256): weight of 0.5×Prompt4. Note that since Prompt4 effectively covers the range [51, 256), we split its weights into two parts, one for [51,153) and one for [153, 256). feature representations across varying illumination conditions. This structured alignment process promotes the learning of light-invariant feature representations, significantly enhancing feature robustness and stability, which are crucial for effective low-light image enhancement. Table IV shows that setting λ to 0.5 achieves the best trade-off between luminance consistency and structural detail preservation, yielding the highest image quality and structural fidelity. A larger λ overly constrains illumination consistency, leading to texture loss, while a smaller λ weakens illumination-invariance learning. Thus, λ = 0.5 allows the model to effectively leverage Llcl, enhancing feature robustness without compromising image quality. 3) Study of LAPM. Based on FF, adding LAPM leads to achieving a PSNR of 27.16 and an SSIM of 0.8911, as shown in Table III. LAPM dynamically guides feature learning by injecting illumination-specific prompts, enabling the model to adapt feature representations based on brightness variations. TABLE IV ABLATION OF λ FOR Llcl ON LOL-V2-REAL DATASET. λ 1 0.5 0.001 PSNR 27.38 28.51 27.43 SSIM 0.8905 0.8974 0.8928 TABLE V ABLATION OF NUMBER OF PROMPT VECTORS ON LOL-V2-REAL DATASET. Number PSNR SSIM 3 28.19 0.8949 4 28.03 0.8911 5 28.51 0.8974 6 27.47 0.8880 Compared to the purely static structure offered by LQM, LAPM introduces dynamic flexibility, allowing the model to better respond to complex real-world illumination, thereby yielding significant improvements in image quality shown in Fig. 11. Furthermore, Table V presents the impact of the number of prompt vectors. We observe that increasing the number of prompts from 3 to 5 leads to continuous improvements in both PSNR and SSIM, reaching the best performance at 5 prompts (28.51 PSNR and 0.8974 SSIM). However, further increasing the number to 6 results in a noticeable performance drop, due to over-fragmentation of the luminance space, which weakens the effectiveness of each individual prompt. These results highlight that using 5 prompts achieves the optimal balance between representation capacity and generalization in illumi- nation modeling. In Fig. 12, we first analyze a low-light image whose pixel distribution concentrates within the darkest range ([0, 25.5)). Referring to the correlation analysis from Fig. 5, prompts 1 and 3, which are strongly associated with this darkest region, together receive the highest combined weight, effectively enhancing severely underexposed areas. Meanwhile, prompts 4 and 5, assigned relatively lower weights, help refine moderately illuminated regions. To further validate this adaptive behavior, we evaluate prompt weights on the entire MEF dataset (see Fig. 13). The darkest interval [0, 25.5) contains 53.3% of pixels and is closely matched by the combined weights of prompts 1 and 3 (0.4324). Transitional low-light pixels ([25.5, 51)) comprise 16.2%, matched by prompt 5's weight (0.1854). Mid-range brightness ([51, 153)) accounts for 21.0%, aligning well with the sum of prompt 2 and half of prompt 4 (0.2825). Finally, the brightest interval ([153, 256)) includes 9.0% of pixels, corresponding closely with half of JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 13 TABLE VI ABLATION OF DISCRETE FEATURE NUMBER IN CODEBOOK ON LOL-V2-REAL DATASET. Number PSNR SSIM 256 27.04 0.8897 512 27.72 0.8946 1024 28.51 0.8974 2048 27.18 0.8923 Ours Input CIDNet Fig. 14. Unsatisfying cases. Images captured using a Sony A7C II camera. prompt 4's weight (0.0996). Despite slight deviations in exact proportions within these intervals, the learned prompt weights largely reflect the dataset's overall luminance distribution. This analysis confirms that LAPM effectively achieves adaptive illumination-aware feature modulation, allowing the model to selectively emphasize different luminance intervals, thus achieving more natural and visually pleasing results. Study of discrete feature number in codebook. Table VI presents the impact of the number of discrete features in the codebook on enhancement performance. We observe that increasing the codebook size from 256 to 1024 progressively improves both PSNR and SSIM, reaching the best performance at 1024 entries. However, when increased to 2048, it will result in slight degradation, which may be attributed to reduced feature compactness and increase noise sensitivity during reconstruction. Thus, setting the codebook size to 1024 provides the best trade-off between feature expressiveness and generalization capability. F. Limitations and future works Although LightQANet demonstrates strong performance across diverse lighting conditions, several limitations remain to be addressed. (1) Extremely dark scenes. As shown in Fig. 14, LightQANet restores brightness and colors in partially visible regions more effectively than CIDNet, thanks to LQM and LAPM. However, in completely black areas (e.g., the dense canopy), the absence of information causes encoder-codebook misalignment, leading to noise or pseudo-textures. Future work may integrate generative priors to recover plausible structures while suppressing artifacts. (2) Insufficient high-frequency recovery. In cross-domain evaluation, fine structures are not always preserved. For example, Fig. 9 shows blurred license plate characters in the LIME dataset, reflecting limited smallscale detail reconstruction. Future efforts could adopt gradientor edge-aware losses, leverage high-frequency components via wavelet/Fourier transforms, and utilize classical edge operators (e.g., Sobel, Prewitt, or Canny) to provide explicit priors. In addition, decomposing images into smooth and detail layers with bilateral filtering, selectively reinforcing the detail layer, and applying lightweight sharpening as post-processing may further refine structural fidelity, collectively mitigating edge blurring and detail loss. V. CONCLUSION In this study, we propose a novel LightQANet framework for LLIE, which emphasizes light-invariant feature learning through both structured quantization and dynamic adaptation. Specifically, we design an LQM to extract and quantize lightrelevant information within feature representations, thereby effectively bridging the gap between low-light and normallight conditions, so as to promote the learning of lightinvariant features. In addition, we introduce an LAPM that dynamically encodes illumination priors to adaptively guide feature learning across varying brightness levels. Extensive experiments across multiple datasets, including both same-source and cross-source scenarios, demonstrate that LightQANet consistently outperforms the existing state-of-the-art LLIE methods, validating the effectiveness of our proposed approach in achieving robust and adaptive illumination enhancement. VI. ACKNOWLEDGMENT This work was supported in part by National Natural Science Foundation of China (No. 62476172, 62476175, 62272319, 62206180, 82261138629), and Guangdong Basic and Applied Basic Research Foundation (No. 2023A1515010677, 2023B1212060076, 2024A1515011637, 2025A1515011511), and Science and Technology Planning Project of Shenzhen Municipality (No. JCYJ20220818095803007, JCYJ20240813142206009), and Guangdong Provincial Key Laboratory (No. 2023B1212060076), and XJTLU Research Development Funds (No. RDF-23-01-053). 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2510.14734
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ASYMPTOTICS FOR THE PERCOLATION THRESHOLD OF FINITARY
RANDOM INTERLACEMENTS IN FOUR AND HIGHER DIMENSIONS
YIJIE BI, ZHENHAO CAI, XINYI LI, BAL´AZS R´ATH, YUAN ZHANG
Abstract. We establish sharp asymptotic bounds for the critical intensity of the Finitary Ran-
dom Interlacements (FRI) model in four and higher dimensions with general trajectory length
distributions. Our proof reveals that the construction of near-critical FRI clusters in four and
higher dimensions is essentially analogous to a Galton-Watson process, whose expected number
of offspring corresponds to the capacity of a random walk killed at the given length.
1. Introduction
Finitary Random Interlacements (FRI), introduced by Bowen in [5], is a Poisson cloud of
random walk trajectories on Zd where the trajectory lengths follow a specified distribution. As
the expected length varies from zero to infinity, the model interpolates between the renowned
Bernoulli percolation and Random Interlacements (RI). The latter was introduced by Sznitman
in [21] and has been extensively studied over the last decade (e.g. [26], [22], [25], [23], [20], [24];
see [9] for a self-contained introduction).
The FRI model, denoted by Iu,ρ (where u > 0 is a fixed number and ρ is a distribution
supported on non-negative integers), can be constructed as follows. Independently for each vertex
x ∈Zd(d ≥3), we initiate from x a Poisson
u
µ1(ρ)+1
-distributed number of independent simple
random walks (Xn)n≥0 killed at time T ∼ρ, where µi(ρ), i ∈N is the i-th moment of ρ.
The process Iu,ρ is then constructed as the subgraph of Zd whose edges are those traversed by
the aforementioned random walk trajectories. The precise definition(s) of FRI can be found in
Section 2 of this paper. The parameter u is the analogue of the intensity parameter in RI, which
corresponds to the expected total occupation time at each vertex by the random walk trajectories.
The length distribution ρ is typically taken to be geometric in the literature, i.e., one considers
the model FIu,T := Iu,Geo(
1
T +1 ) for T ≥0, where Geo(λ) represents the geometric distribution
with success probability λ; also, the model Iu,δn, where δn is the Dirac measure at n, has been
studied as a finite-range approximation of RI (see [10] for an example). In this paper, we consider
general length distributions ρ on N satisfying a mild restriction, which encompass two common
families of distributions, namely the geometric distributions and the Dirac mass.
For fixed u and varying ρ, the behavior of Iu,ρ constrained to any finite subset of Zd resembles
that of RI with intensity u, denoted by Iu, when µ1(ρ) →∞.
In the geometric setup, this
property was discussed in [5] and made rigorous in [6]. Moreover, couplings between FRI and RI
have proven instrumental in the study of the latter. For example, in [4], it was established that
RI is a factor of i.i.d. using the convergence in law of a sequence of similarly defined finite-range
models to RI. Moreover, recent breakthroughs concerning the sharpness of phase transition of
the vacant set of RI [12], [10] and [11] rely crucially on couplings both between FRI and RI and
between FRIs of various lengths to overcome obstacles imposed by the long-range dependence of
RI.
In contrast to RI, where Iu almost surely percolates for all u > 0 and d ≥3, FRI undergoes
phase transitions with respect to both parameters as a bond percolation model on Zd. On the
one hand, for fixed u > 0, the percolation properties of FIu,T as a function of T are studied in
several papers. Bowen proved the existence of infinite clusters (connected components) in FIu,T
for any fixed u > 0 and sufficiently large T in [5]. It was shown in [18] that FIu,T has a non-trivial
phase transition with respect to T for all u > 0 and d ≥3. However, it was proved in [6] that
FIu,T is non-monotone for T ∈(0, ∞) due to the fact that the expected number of trajectories
1
arXiv:2510.14734v1 [math.PR] 16 Oct 2025
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
from each vertex decreases as T increases. As a result, the existence and uniqueness of a critical
length parameter Tc remain open.
On the other hand, if we fix the length distribution ρ and study the percolation as u varies,
the phase transition is unique, because Iu,ρ is increasing in u. For FRI with geometric length
distributions, this phase transition has been studied by several papers. In [7], it was proved that
for each fixed T > 0, there exists u∗(d, T) ∈(0, ∞) such that the probability of the existence of
an infinite cluster in FIu,T is zero for u ∈(0, u∗(d, T)) and positive for u ∈(u∗(d, T), ∞). For the
asymptotic of u∗(d, T), it was shown in [7] that u∗(d, T) · T →−log
1 −pc
d
/2 as T →0, where
pc
d is the critical parameter for Bernoulli bond percolation on Zd. Moreover, it was proved in [8]
that for d ≥3, there exist constants C†(d) > c†(d) > 0 such that for all sufficiently large T,
(1.1)
c†fd(T) ≤u∗(d, T) ≤C†fd(T),
where fd(T) is given by
fd(T) :=
T −1/2, d = 3;
T −1 log T, d = 4;
T −1, d ≥5.
Interestingly, this implies that FRI achieves supercriticality in percolation with vanishing density.
It was conjectured in [8] that u∗(d, T) admits an asymptotic limit, i.e., the coefficients C† and c†
can be arbitrarily close. Estimates similar to (1.1) are established for the percolation of Wiener
sausages with fixed length in [13], and the analogous conjecture for the critical threshold of Wiener
sausages was posed in Section 1.4 of that paper.
In this paper, we confirm the conjecture for FRI when d ≥4, and further generalize it to FRI
with general length distributions.
Let {0 Iu,ρ
←→∞} be the event that the origin is in an infinite cluster of Iu,ρ. For any length
distribution ρ, we define the critical intensity of the FRI model Iu,ρ by
u∗(ρ) := inf{u ≥0 : P[0 Iu,ρ
←→∞] > 0}.
We aim to establish the asymptotic of u∗for a family of length distributions (ρn)n∈N with expec-
tations tending to infinity. For technical reasons, we need certain restriction on the fluctuation
of each ρn. Specifically,
Definition 1.1. We say a family (ρn)n∈N of distributions on N is appropriate if µ1(ρn) →∞as
n →∞, and for all ϵ > 0, there exists C = C(ϵ, (ρn)n∈N) < ∞satisfying
(1.2)
ϑ(ρn, C) :=
∞
X
L=Cµ1(ρn)
ρn(L) · L2
µ2(ρn)
≤ϵ, ∀n ∈N,
where µi means the i-th moment.
Remark 1.2. For any random variable ξ supported on (0, ∞) with finite second moment and any
sequence of positive numbers l(n) →∞, if we define ρn to be the law of ⌊l(n)·ξ⌋, then (1.2) holds.
In particular, (ρn)n∈N = (δn)n∈N corresponds to ξ ≡1 and l(n) = n; (ρn)n∈N = (Geo(
1
n+1))n∈N
corresponds to ξ ∼Exp(1) (the exponential distribution with killing rate 1, see Section 2 for
definition) and l(n) = log−1(1 + 1
n).
Thus, (δn)n∈N and (Geo(
1
n+1))n∈N are both appropriate
families.
Remark 1.3. This restriction may not be optimal in the 4-dimensional case. In fact, it can be
relaxed to
(1.3)
∞
X
L=Cµ1(ρn)
ρn(L) · L2
µ2(ρn)
· log µ1(ρn)
log L
≤ϵ.
The main result is the following theorem.
2
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Theorem 1.4. For any d ≥4 and any appropriate family of distributions (ρn)n∈N,
(1.4)
lim
n→∞u∗(ρn) ·
µ2(ρn)
µ1(ρn)(1 + log µ1(ρn) · 1d=4) · εd = 1.
Here, εd is the coefficient in the asymptotic capacity of random walks, i.e.,
εd = lim
T→∞
E[cap(X[0, T])]
T(1 + log T · 1d=4)−1 ,
where cap(A) for A ⊂Zd represents the capacity of the set A (see Section 2 for the definition).
(The existence of εd ∈(0, ∞) when d ≥5 was proved in [14]; while one has ε4 = π2
8 according to
[2, Corollary 1.4].)
In particular, we obtain the following sharp asymptotics of u∗(d, n), the threshold under geo-
metric distribution (see above (1.1) for notation).
Corollary 1.5. The critical intensity of FRI of geometric length admits an asymptotic limit:
lim
n→∞u∗(d, n) · 2n · εd = 1
for all d ≥5, and
lim
n→∞u∗(4, n) · n log−1 n · π2
4 = 1.
We now give a brief overview of the proof of Theorem 1.4. Let
(1.5)
un := µ1(ρn)(1 + log µ1(ρn)1d=4)
εdµ2(ρn)
denote the order of u∗(ρn), and for a fixed ϵ > 0, let
(1.6)
u±
n := (1 ± ϵ)un.
To derive Theorem 1.4, it suffices to show that for all large n,
(1.7)
u−
n ≤u∗(ρn) ≤u+
n .
In other words, we need to show that Iu+
n ,ρn (resp. Iu−
n ,ρn) is supercritical (resp. subcritical).
To prove the subcriticality of Iu−
n ,ρn, we explore the FRI cluster containing the origin layer by
layer. Precisely, we call the origin the 0-th layer; given the i-th layer, the (i + 1)-th layer is given
by the union of trajectories hitting the i-th layer but not the previous ones. In particular, the
typical number of trajectories in one layer is approximately the intensity times the capacity of
the previous layer. Therefore, when taking u = u−
n , the number of discovered trajectories in the
aforementioned exploration process can be dominated by a subcritical Galton-Watson process,
which dies out in finite time almost surely.
To verify that Iu+
n ,ρn is supercritical, we construct the infinite cluster through a classic coarse-
graining argument inspired by [17], where a similar asymptotic upper bound for the critical
intensity of spread-out percolation was proved. In our setup, some additional delicate treatment is
needed to handle the correlation generated by random walk trajectories. Specifically, we introduce
a truncated version of Iu+
n ,ρn which ignores certain types of trajectories (e.g., those with atypically
large diameters or long durations), and run an exploration process to show the supercriticality
on this truncated version instead.
We divide the space Zd into disjoint boxes of side length
Rn := ⌊
p
µ1(ρn)⌋(the correlation length of a random walk killed at time T ∼ρn, see also (4.1)).
Next, we sample the cluster containing 0 (more precisely, the center point of the box around 0,
see (4.2)) with a specific part of trajectories within the Rn-box around 0, and require it to satisfy
several conditions including a lower bound on the capacity and some restriction on the relative
position of any two trajectories (we call this cluster a “seed”). We then try to extend this seed
to all of the neighboring boxes, and call the box around 0 a “good” box if a seed is sampled
in each of the neighboring boxes. Next, we resume the exploration from one of the neighboring
boxes, and call it good if new seeds are sampled for all its unexplored neighbors. When a bad
box appears, to ensure independence we give up the exploration of all unexplored boxes within
a neighborhood of radius 2γRn concentric with this bad box that are neighbors of good ones
3
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
where γ is an integer defined in (4.23) so large that trajectories in subsequent exploration from
boxes that are not ruined do not intersect those explored from the current bad box (thanks to
the truncation).
Note that in the exploration algorithm described above, we are able to choose parameters
properly so that the conditional probability of each step (i.e., extension to neighboring boxes) is
sufficiently close to 1. Consequently, the exploration, after proper rescaling, may stochastically
dominate a finitely dependent percolation configuration with sufficiently high density, which is
supercritical by an application of the Liggett–Schonmann–Stacey domination.
We now briefly comment on the truncation. Note that in the exploration algorithm, it would
be easier to obtain the desired independence if we ignored more trajectories. Meanwhile, such
modification is not allowed to increase the critical intensity by a constant factor (otherwise, it
will violate (1.7)). These requirements exhibit opposite directions of monotonicity, which entails
a delicate truncation scheme, which stands as one of the main technical novelties of this paper.
Let us also quickly remark on the d = 3 case. Since it is much more likely for two random
walk trajectories to intersect in three dimensions, the behavior of near-critical FRI clusters can no
longer be characterized by Galton-Watson processes, and the corresponding asymptotic threshold
is actually determined by a different mechanism. In fact, in a paper in progress [3], the pre-
factor of the asymptotic threshold is upper and lower bounded by two (plausibly equal) critical
thresholds of finitary Brownian interlacements (FBI), a counterpart of FRI in the continuum.
A model similar to FBI in which trajectories are of fixed length is studied in [13], where the
existence of a non-trivial phase transition is proved (c.f. Theorem 1.3 therein).
The remaining sections are organized as follows. In Section 2, we introduce the notations and
some preliminary results. In Sections 3, we prove the lower bound of (1.7). Sections 4 and 5
are dedicated to the corresponding upper bound. In Section 4, we introduce the exploration
algorithm and reduce the supercriticality of Iu+
n ,ρ to the fact that the conditional probability of
the extension to a neighboring box in the algorithm can be arbitrarily close to 1 (c.f. Proposition
4.7). Section 5 proves Proposition 4.7 with various coupling techniques.
2. Notations and Preliminaries
We write N = {0, 1, 2, ...} for the set of natural numbers. For a real number a, write ⌊a⌋for its
integer part, and for real numbers b, c, we write b ∧c and b ∨c for the minimum and maximum
of b and c respectively. Denote the ℓ∞norm on Zd by | · | and the ℓ1 norm by | · |1. For two
non-empty subsets A, B ⊂Zd, define the distance between them by d(A, B) := minx∈A,y∈B |x−y|,
and write d({x}, B) as d(x, B) for all x ∈Zd. The diameter of a finite nonempty set A ⊂Zd is
defined by diam(A) := maxy,z∈A d(y, z). For all x ∈Zd and r ∈(0, ∞), we denote the subset of
Zd inside the box of side length 2r centered at x by B(x, r) = Bx(r) := {y ∈Zd : |x −y| ≤r};
for all A ⊂Zd, let B(A, r) := S
x∈A B(x, r) be the collection of vertices in the r-neighborhood of
A. For all n ∈N and A ⊂Zd, let nA = {nx : x ∈A}. In particular, we will consider the set
nZd which will be referred to as a lattice. For all x, y ∈nZd, they are called nearest neighbors
if |x −y|1 = n. A nearest-neighbor path on nZd is a sequence of vertices on nZd such that the
adjacent vertices are nearest neighbors. For λ ∈(0, 1], define the geometric distribution Geo(λ)
by Geo(λ)(n) = λ(1 −λ)n, n ∈N. For λ ∈(0, ∞), define the exponential distribution Exp(λ) by
the density function f(x) = λe−λx1x>0.
From now on, we always assume that the dimension is at least 4, and often omit d in the
notation when defining objects such as random walks and quantities including constants and
parameters that do not explicitly depend on d. All the results stated in this paper are supposed
to hold for all d ≥4 unless stated otherwise. The constants C and C′ appearing in the paper
may implicitly depend on d, and may differ from place to place. However, numbered constants
such as C1, C2, ... refer to their first appearance in the text.
Measures on countable spaces: For an arbitrary countable space X equipped with the
canonical σ-field, we denote by M(X ) (resp. M∗(X )) the set of all probability measures (resp.
finite measures) on X , and abbreviate M = M(N). For two measures ν1, ν2 ∈M(X ), we write
ν1 ≤ν2 if ν1(A) ≤ν2(A) for all A ⊂X . For all ν ∈M∗(X ), we define the corresponding
4
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
probability measure ν0 ∈M(X ) by
ν0(A) = ν(A)
ν(X ), ∀A ⊂X .
For ρ ∈M(X ), x ∈X , let ρ(x) be the mass assigned to x. We call ρ a point measure if ρ(x) ∈
N for all x ∈X . Let S be a multi-subset of X . We sometimes identify S with its corresponding
point measure on X , i.e., we also use S to represent the point measure P
x∈X nx(S) · δx, where
nx(S) is the multiplicity of x in S. Write |S| = P
x∈S nx(S). For a point measure X ∈M(X )
and A ⊂X , write X ∩A = P
x∈A X(x) · δx.
Let k be a positive integer. For all ρ ∈M, denote the k-th moment of ρ by
(2.1)
µk(ρ) :=
∞
X
l=0
lkρ(l).
Also, we define the “tail contribution” to the k-th moment by
µ(m)
k
(ρ) :=
∞
X
l=m
lkρ(l),
where m ∈N.
Trajectories: We will refer to nearest-neighbor paths on Zd as trajectories. Let W [0,∞) denote
the set of all finite trajectories. For all η ∈W [0,∞), write η = (η(s))0≤s≤T . Here, T(η) := T
denotes the length of η. In addition, we define the range of η by range(η) := {η(s) : 0 ≤s ≤T}.
Sometimes, η is identified with its range when no confusion arises (e.g. B(η, r) = B(range(η), r)).
For 0 ≤a ≤b ≤T, define the sub-path η[a, b] := (η(a + s))0≤s≤b−a.
For a sigma-finite point measure ω on the space W [0,∞), we write η ∈ω if ω({η}) > 0, and
define the set of sites visited by the trajectories of ω by V (ω) := S
η∈ω range(η), and let G(ω) be
the subgraph of Zd where an edge is open iff it connects two adjacent vertices appearing in one
of the trajectories in ω. Sometimes, the notation is slightly abused and the point measure ω is
identified with the corresponding (multi-)subset of W [0,∞).
Let ηi ∈W [0,∞) be a trajectory with length Ti, i = 1, 2. Define η1 ∗η2, the concatenation of η1
and η2, by
η1 ∗η2(s) :=
(
η1(s), 0 ≤s ≤T1;
η′
2(s −T1), T1 < s ≤T1 + T2,
where η′
2 = (η2(s) −η2(0) + η1(T1))0≤s≤T2 is η2 translated to the endpoint of η1.
We write
η1 ≃η2 if they are a spatial translations of each other. If η1 and η2 are both (one-sided) infinite
trajectories, we write η1 ≃η2 if η1[0, T] ≃η2[0, T] for every T ∈N.
For an arbitrary trajectory η (finite, infinite) and A ⊂Zd, let
(2.2)
τA = τ η
A := inf{t ∈N : η(t) ∈A}.
For any B ⊂A, we say η hits A in B if τA < ∞and η(τA) ∈B. Define the set of all finite
trajectories hitting A in B by
(2.3)
H[A; B] := {η ∈W [0,∞) : η hits A in B}.
Simple random walk on Zd and potential theory: For x ∈Zd, we denote by P x the
law of a doubly infinite simple random walk starting from x, and denote the walk by X. For all
y, z ∈Zd, let P y,z be the joint law of two independent simple random walks X1 ∼P y, X2 ∼P z.
For all A ⊂Zd, we define the first hitting/entrance time of A by X by
HA := inf{t ≥0 : Xt ∈A},
eHA := inf{t > 0 : Xt ∈A}.
To lighten the notation, for m, l ∈N, let
(2.4)
µ(x, m, l; A)(·) = P x[X[−m, l] ∈·
X[−m, −1] ∩A = ∅]
5
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
be a probability measure on W [0,∞), and we omit “A” when A = ∅(i.e., no conditioning is
imposed). For all R > 0, let WR ⊂W [0,∞) denote the subset of trajectories with diameter at
most R. We have the following estimate on the diameter of a trajectory following the distribution
µ(x, m, l) whose proof follows from a simple application of the Donsker’s invariance principle.
Lemma 2.1. There exists a decreasing function f : (0, ∞) →(0, ∞) such that for all κ > 0,
R > C(κ) large and m + l < κR2,
µ(x, m, l)(WR) ≥f(κ).
For a finite subset A ⊂Zd, define its equilibrium measure by eA(x) := Px[ eHA = ∞]·1x∈A, and
define the capacity of A by cap(A) := P
x∈A eA(x). The capacity is monotone and sub-additive,
namely
cap(A) ≤cap(D),
cap(A ∪B) ≤cap(A) + cap(B)
for all finite A, B, D ⊂Zd, A ⊂D. By (2.16) in [15], there are constants 0 < C < C′ < ∞, such
that for all D ∈N,
(2.5)
Crd−2 ≤cap(B(0, r)) ≤C′rd−2.
For all x, y ∈Zd, define the Green’s function on Zd by
g(x, y) = Ex
∞
X
n=0
1Xn=y
.
By Theorem 1.5.4 of [15], there exist constants 0 < C < C′ < ∞such that
(2.6)
C(1 ∨|x −y|)2−d ≤g(x, y) ≤C′(1 ∨|x −y|)2−d
for all x, y ∈Zd. For all A ⊂Zd and x ∈Zd, define the hitting probability by h(x, A) := P x[HA <
∞]. When A is finite, by the last exit decomposition (see Proposition 2.4.1 of [15]), we have
(2.7)
h(x, A) =
X
y∈A
g(x, y)eA(y).
Combining (2.6) and (2.7) gives the following: there exist constants 0 < C < C′ < ∞such that
for all finite A ⊂Zd and x ∈Zd,
(2.8)
cap(A) · C(1 ∨d(x, A))2−d ≤h(x, A) ≤cap(A) · C′(1 ∨d(x, A))2−d.
By [1, Theorem 1.1], [19, Theorem B] and [2, Theorem 1.2] (corresponding to d ≥6, d = 5 and
d = 4 respectively), we have laws of large numbers and concentration results for the capacity of
random walk trace in dimension 4 and above.
Proposition 2.2. As T ∈N tends to infinity, we have
(2.9)
E0[cap(X[0, T])] = (1 + o(1)) · εdT(1 + log T · 1d=4)−1.
(2.10)
cap(X[0, T]) · T −1(1 + log T · 1d=4)
P
−→εd.
Moreover, there exists some C = C(d) such that
(2.11)
var(cap(X[0, T])) <
(
CT 2 log−4 T,
d = 4;
CT log T,
d ≥5.
(2.12) P 0 h cap(X[0, T]) −E0[cap(X[0, T])]
> δE0[cap(X[0, T])]
i
<
(
Cδ−2 log−2 T,
d = 4;
Cδ−2T −1 log T,
d ≥5.
6
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
To analyze hitting of a given set by finite paths, we consider the following quantity that formally
resembles the capacity of a set. For all finite A ⊂Zd, define cap(A, T) by
(2.13)
cap(A, T) =
X
x∈A
P x[ eHA > T log−10 T].
(Here, the choice of −10 in the exponent is rather arbitrary. In fact, any real number smaller than
−2 is sufficient for Proposition 2.3 below.) We can prove estimates on this quantity analogous to
Proposition 2.2 by directly comparing cap(·) with cap(·, T).
Proposition 2.3. For all r > 0, the following estimates hold.
(2.14)
E0[cap(X[0, T], rT)] = (1 + o(1)) · εdT(1 + log T · 1d=4)−1.
Moreover, for all δ > 0, there exists some C = C(d, δ) > 0 such that
(2.15)
P 0 h cap(X[0, T], rT) −E0[cap(X[0, T])]
> δE0[cap(X[0, T])]
i
<
(
C log−1/2 T,
d = 4;
CT −1/4,
d ≥5
holds for all large n ∈N.
Proof. We only present the proof of the 4-dimensional case since the higher dimensional case
is simpler and can be treated similarly.
Assume that d = 4, then for all 2T log−10 T < t <
T −2T log−10 T, we have
(2.16)
E0[P Xt[ eHX[0,T] > T log−10 T] −eX[0,T](Xt)]
= P 0,0 h
X1[−t, T −t] ∩X2(0, t′] = ∅, X1[−t, T −t] ∩[t′, ∞) ̸= ∅
i
,
where t′ = ⌊T log−10 T⌋. Let E1, E2, E3 denote the events
E1 := {d(X2
t′, X1[−t′, t′]) > (t′)1/2 log−1/4 T},
E2 := {X1[−t′, t′] ∩X2(0, t′] = ∅},
E3 := {cap(X1[−t′, t′]) ≤4ε4t′ log−1 T}.
By [15, Theorem 3.5.1],
(2.17)
P 0,0[E2] ≤C log−1 T.
By (2.12),
(2.18)
P 0.0[Ec
3] ≤C log−2 T.
Now we aim to bound the probability of E1. To this end, for X ∼P 0, let τ0 = 0, and define for
all k ≥1
τk := inf{t > τk−1 : |Xt −Xτk−1| ≥⌊(t′)1/2 log−1/4 T⌋},
σk := τk −τk−1.
Apparently σk, k ≥1 are i.i.d., and we have the estimate E0[e−σ1] ≤e−Ct′ log−1/2 T by Gaussian
estimates. Thus, we have
(2.19)
E0[|B(X[0, t′], (t′)1/2 log−1/4 T|] ≤C(t′)2 log−1/2 T,
which can be deduced from the large deviation estimate
P 0 h
|B(X[0, t′], (t′)1/2 log−1/4 T| ≥M · (t′)2 log−1/2 T
i
≤P 0[τ⌈M log1/2 T⌉≤t′]
= P 0[e−t′ ≤exp
−τ⌈M log1/2 T⌉
] ≤et′
E0[e−σ1]
M log1/2 T
≤exp
−(CM −1)t′
for all M ∈N∗. By (2.19) and the LCLT [15, Theorem 1.2.1],
(2.20)
P 0,0[Ec
1] ≤C · (t′)−2E0[|B(X[−t′, t′], (t′)1/2 log−1/4 T)|] ≤C log−1/2 T.
7
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
By (2.17), (2.18) and (2.20),
(2.21)
P 0,0[X1[−t, T −t] ∩X2(0, t′] = ∅, X1[−t, T −t] ∩X2[t′, ∞) ̸= ∅]
≤P 0,0[Ec
1] + P 0.0[Ec
3] + E0,0 h
P 0,0[X1[−t, T −t] ∩X2[t′, ∞) ̸= ∅|X1[−t′, t′], X2
t′]1E1∩E2∩E3
i
≤C log−4 T + C log−1/2 T · P 0,0 [E1 ∩E2 ∩E3] ≤C log−3/2 T,
where the second inequality holds because on the event E1,
P 0,0[(X1[−t, −t′] ∪X1[t′, T −t]) ∩X2[t′, ∞) ̸= ∅|X1[−t′, t′], X2
t′] ≤C log log T
log T
according to [15, Theorem 4.3.3], and on E1 ∩E3,
P 0,0[X1[−t′, t′] ∩X2[t′, ∞) ̸= ∅|X1[−t′, t′], X2
t′]
≤cap(X1[−t′, t′]) · C(1 ∨d(X2
t′, X1[−t′, t′]))−2 ≤Ct′ log−1 T · (t′)−1 log1/2 T ≤C log−1/2 T
by (2.8). By (2.16) and (2.21),
(2.22)
E0[cap(X[0, T], rT) −cap(X[0, T])]
≤4T log−10 T +
T−⌊2T log−10 T⌋−1
X
t=⌊2T log−10 T⌋+1
E0[P Xt[ eHX[0,T] > T log−10 T] −eX[0,T](Xt)]
≤4T log−10 T + CT log−3/2 T ≤CT log−3/2 T,
which implies
(2.23)
P 0[cap(X[0, T], rT) −cap(X[0, T]) > δ/2 · E0[cap(X[0, T])]] ≤Cδ−1 log−1/2 T
by (2.9) and the Markov inequality. It is easy to verify that (2.14) follows from (2.9) and (2.22).
Moreover, by (2.23) and (2.12),
P 0 h cap(X[0, T], rT) −E0[cap(X[0, T])]
> δE0[cap(X[0, T])]
i
≤P 0[cap(X[0, T], rT) −cap(X[0, T]) > δ/2 · E0[cap(X[0, T])]]
+ P 0 h cap(X[0, T]) −E0[cap(X[0, T])]
> δE0[cap(X[0, T])]
i
≤Cδ−1 log−1/2 T + Cδ−2 log−2 T ≤Cδ−1 log−1/2 T,
proving (2.15).
When d ≥5, one can prove
(2.24)
E0[cap(X[0, T], rT) −cap(X[0, T])] ≤CT 3/4
parallel to (2.22) with similar techniques, and deduce (2.14) and (2.15) from this estimate as
above.
□
Finitary random interlacements (FRI): We now present the definition of the finitary
random interlacements (FRI) point process. For any u > 0 and any ρ ∈M, we define a measure
by
π =
X
x∈Zd
X
l∈N
ρ(l)
µ1(ρ) + 1 · µ(x, 0, l).
Definition 2.4. The finitary random interlacements point process with intensity u and length
distribution ρ, denoted by X u,ρ, is a Poisson point process (PPP) on the space W [0,∞) with density
measure u · π, and the corresponding random subgraph of Zd is defined by Iu,ρ = G(X u,ρ).
This definition generalizes the one commonly used in the literature, and takes into consideration
clouds of trajectories with arbitrary length distributions instead of just geometric ones. The factor
of
1
µ1(ρ)+1 in the definition of π guarantees that the expected total local time at a vertex is exactly
u in X u,ρ. We refer to [7] for a more comprehensive introduction of the model.
8
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
We are concerned about the distribution of trajectories within X u,ρ originating from or travers-
ing a certain subset of Zd. To this end, we further introduce some notations. Let X be any point
measure on W [0,∞). For all A ⊂Zd, let X[A] denote the point process comprising trajectories
within X intersecting A. For A, B, D ⊂Zd, D ⊂A, let
(2.25)
X[A; B] = X[A] −X[B]
and let
(2.26)
X[A, D; B] = X[A; B] ∩H[A; D],
where H[A; D] was defined in (2.3).
For all A ⊂Zd finite, we can find the distribution of X u,ρ[A] by a technique called “rerooting”
of the trajectories (c.f. [10, Lemma 3.1]).
Lemma 2.5. Let A ⊂Zd be finite, then X u,ρ[A] is a PPP with intensity u · πA, where
πA :=
X
x∈A,m,l∈N
ρ(m + l)
µ1(ρ) + 1P x[ eHA > m] · µ(x, m, l; A).
Given this result, it is easy to compute the expected number of trajectories within X u,ρ hitting
A at x, which equals ue(ρ)
A (x), where
(2.27)
e(ρ)
A (x) =
∞
X
m=0
µ(m)
0
(ρ)
µ1(ρ) + 1P x[ eHA > m].
Thus, it is natural to define a substitute for the usual capacity of subsets of Zd
(2.28)
cap(ρ)(A) :=
X
x∈A
e(ρ)
A (x),
which we call the ρ-capacity. Note that e(ρ)
A (x) ≥eA(x) and thus cap(ρ)(A) ≥cap(A).
We will need control over the ρ-capacity. It is easy to see that for arbitrary ρ and d ≥4,
(2.29)
cap(ρ)(A) ≤|A|.
Moreover, letting N = ⌊µ1(ρ) log−λ µ1(ρ)⌋, then we have
e(ρ)
A (x) ≤
N
X
m=0
m(m)
0
(ρ)
µ1(ρ) + 1 · 1 +
∞
X
m=N+1
m(m)
0
(ρ)
µ1(ρ) + 1P x[ eHA > N] ≤
N
µ1(ρ) + 1 + P x[ eHA > N].
Summing over x ∈A, we get
(2.30)
cap(ρ)(A) ≤|A| log−10 µ1(ρ) + cap(A, µ1(ρ)),
where cap(A, µ1(ρ)) is defined by (2.13).
Branching Processes with agents in countable sets: Let X be an arbitrary countable
set. For S a multi-subset of X and all x ∈X, recall that nx(S) is the multiplicity of x in S.
For all x ∈X , let νx be a distribution on the space of finite point measures on X . Let S be a
multi-subset of X . The branching process with agents in X with offspring distribution (νx)x∈X
starting from S, denoted by (Yn)n≥0, is then recursively defined as follows. For all x ∈X and
i, j ∈N, let Yx,i,j ∼νx be mutually independent. First, let Y0 = P
x∈X nx(S)·δx; then, assuming
that Yi has been sampled, let
Yi+1 =
X
x∈X
X
0≤j<nx(Yi)
Yx,i,j.
In words, every agent at x in the i-th generation produces offsprings according to the distribution
νx. For random variables ξx ∼νx, x ∈X , we also say that (Yn)n≥0 has offspring distribution
(ξx)x∈X .
9
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
3. Proof of the lower bound of (1.7)
In this section, we prove the lower bound of (1.7) which will be deduced from the subcriticality
of Iu−
n ,ρn (recalling the definition of u−
n from (1.6)). Throughout the section, let (ρn)n∈N be a
fixed appropriate family of distributions (recalling Definition 1.1), and fix ϵ > 0.
The idea is to explore the cluster in Iu−
n ,ρn containing 0 layer by layer, and prove that the
cluster is almost surely finite (i.e., Iu−
n ,ρn is subcritical) by showing that the exploration stops
within finitely many steps. To see this, we compare the exploration process with a Galton-Watson
tree that dies out almost surely. In [8], a similar process was employed to prove the upper bound
of (1.1). We now carry out more precise analysis in order to prove the subcriticality of Iu−
n ,ρn.
We start with the construction of the exploration process (Lk, Lk)k≥0 for Iu,ρ with arbitrary
parameters u and ρ. Let L0 = {0} and L0 be the zero measure. Assume that we have defined Lk
(k ∈N), then let
Lk+1 = X u,ρ
Lk;
k−1
[
i=0
Li
and Lk+1 = V (Lk+1).
Let L = ∪∞
i=1Li. Note that L is exactly the collection of trajectories comprising the cluster in
Iu,ρ that includes 0. Due to the almost sure finiteness of Li, i > 0 and X u,ρ[B(0, N)], N > 0, the
cluster of Iu,ρ containing the origin is infinite iff Li is nonempty for all i > 0.
To simplify our computation, we introduce a modified sequence (L′
i)i≥1 that stochastically
dominates (Li)i≥1. Let Ji, i > 0 be independent PPPs on W [0,∞) with the same distribution as
X u,ρ. We now generate the i-th layer of trajectories with Ji, so that the source of randomness
for each layer is independent.
Precisely, let L′
0 = {0}.
Assume that L′
k is defined, then let
L′
k+1 = Jk+1[L′
k].
Lemma 3.1. For all u > 0 and ρ ∈M, there exists a coupling Q of (Li)i>0 and (L′
i)i>0 such
that for all i,
Q[Li ⊂L′
i] = 1.
Proof. Note the following fact parallel to Lemma 4.3 of [8].
(3.1)
For any k ∈N, given L0, ..., Lk, then the conditional distribution
of Lk+1 is the same as that of X u,ρ
Lk;
k−1
[
i=0
Li
.
Thus, we can construct (Lk)k>0 by
Lk+1 = Jk+1[Lk;
k−1
[
i=0
Li].
Assuming Li ⊂L′
i for all 0 ≤i ≤k, we have
Lk+1 ⊂Jk+1[Lk] ⊂Jk+1[L′
k] = L′
k+1.
The conclusion then follows by induction.
□
By Lemma 3.1, it suffices to prove that (L′
k)∞
k=0 dies out almost surely.
To this end, we
introduce a quantity
κ(ρ)(A) :=
X
x∈A
∞
X
m=0
µ(m)
1
(ρ)
µ2(ρ) P x[ eHA > m],
where A ⊂Zd is finite, and show that the expectation Wk := E[κ(ρ)(L′
k)] decreases to zero
exponentially fast, which matches our intuition of a sub-critical Galton-Watson tree. Techni-
cally, note that k(ρ) is sub-additive, i.e. κ(ρ)(A1 ∪A2) ≤P
i=1,2 κ(ρ)(Ai) (since P x[ eHA1∪A2 >
m] ≤P
i=1,2 P x[ eHAi > m]). Therefore, given L′
k and the hitting points at L′
k of trajectories in
L′
k+1, the expectation of k(ρ)(L′
k+1) from above by those of κ(ρ)(η), η ∈L′
k+1 whose conditional
10
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
distribution can be obtained from Lemma 2.5. Moreover, when we take (u, ρ) = (u−
n , ρn), the
conditional expectation of the sum P
η∈L′
k+1 κ(ρn)(η) is approximately (1 −ϵ)κ(ρn)(L′
k), implying
that E[κ(ρn)(L′
k)] decays exponentially in k.
Formally, let
Vk :=
(
E[|L′
k|],
d ≥5;
E[|L′
k|] · log−1 µ1(ρ),
d = 4.
Lemma 3.2. For every δ > 0, the following holds for large n ∈N. For (u, ρ) = (u−
n , ρn), we have
Wk+1 ≤(1 −ϵ/2)Wk + δVk,
(3.2)
Vk+1 ≤C · Wk,
(3.3)
for all k ∈N. Here C < ∞is a constant only dependent on d.
We defer the proof of Lemma 3.2 for a few lines, and first finish the proof of the lower bound
of (1.7) using it.
Proof of the lower bound of (1.7) assuming Lemma 3.2. In (3.3), set δ = ϵ/(4C), then for large
n ∈N and (u, ρ) = (u−
n , ρn), we have
Wk+1 ≤(1 −ϵ/2)Wk + ϵ/4 · Wk−1
for k > 0. Take upper limits on both sides and we get limk→∞Wk=0. (It can be deduced from
(3.2) that the sequence (Wk)k∈N is bounded.) Thus, limk→∞Vk = 0, indicating the existence of
some (random) k ∈N such that L′
j = ∅for all j > k. Denote by C the cluster in Iu−
n ,ρn containing
0, then by Lemma 3.1,
C ⊂
k[
j=0
Lj ⊂
k[
j=0
L′
j.
Thus, since Lj is almost surely finite for all j ∈N, C is almost surely finite, which yields u∗(ρn) ≥
u−
n .
□
Now, we present the proof of Lemma 3.2 which involves estimating the conditional expectation
of |L′
k+1| and κ(ρ)(L′
k+1) given L′
k.
Proof of Lemma 3.2. The proofs for cases d ≥5 and d = 4 are identical except for the details
related to the normalization of expected volume. Here we only present the proof of the latter.
By Lemma 2.5,
(3.4)
E[|L′
k+1|
L′
k] ≤
X
x∈L′
k
X
m,l∈N
uρ(m + l)
µ1(ρ) + 1 · Px[ eHL′
k > m] · (m + l)
=
u
µ1(ρ) + 1
X
x∈L′
k
X
m∈N
µ(m)
1
(ρ) · Px[ eHL′
k > m] ≤u · µ2(ρ)
µ1(ρ) · κ(ρ)(L′
k).
11
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Plugging in (u, ρ) = (u−
n , ρn) and taking the expectation, we get Vk+1 ≤
8
π2 · Ck, which implies
(3.3). Now we compute the conditional expectation of κ(ρ)(L′
k+1) similarly:
(3.5)
E[κ(ρ)(L′
k+1)|L′
k] ≤
X
x∈L′
k
X
m,l∈N
uρ(m + l)
µ1(ρ) + 1 · P x[ eHL′
k > m] · µ(x, m, l; L′
k)[κ(ρ)(·)]
≤
X
x∈L′
k
X
m,l∈N
uρ(m + l)
µ1(ρ) + 1 · Px[ eHL′
k > m] · (1 + ϵ/2)π2
8 · (m + l) log−1 µ1(ρ)
+
X
x∈L′
k
X
m,l∈N
uρ(m + l)
µ1(ρ) + 1 · err(ρ)(m + l)
≤(1 + ϵ/2)π2
8 · u · µ2(ρ) · κ(ρ)(L′
k)
µ1(ρ) log µ1(ρ)
+ |Lk| ·
u
µ1(ρ) + 1
∞
X
m=0
(m + 1)ρ(m) · err(ρ)(m),
where (u, ρ) = (u−
n , ρn), and
err(ρ)(m) := E0
κ(ρ)(X[0, m])1{κ(ρ)(X[0,m])>(1+ϵ/2) π2
8 m log−1 µ1(ρ)}
.
Here, in the second inequality, we have used
err(ρ)(m + l) ≥E0[κ(ρ)(X[−m, l])1κ(ρ)(X[−m,l])>M1X[−m,0)∩L′
k=∅]
≥P x[ eHL′
k > m]µ(x, m, l; L′
k)[κ(ρ)(·)1κ(ρ)(·)>M],
where M = (1 + ϵ/2) π2
8 · (m + l) log−1 µ1(ρ). Taking (u, ρ) = (u−
n , ρn), we have
(3.6)
(1 + ϵ/2)εd · u−
n µ2(ρn)
µ1(ρn)
< 1 −ϵ/2.
Moreover, we claim that
(3.7)
lim
n→∞
log µ1(ρn)
µ2(ρn)
∞
X
m=1
mρn(m) · err(ρn)(m) = 0.
Inequality (3.2) then follows by combining (3.5) with (3.6) and (3.7).
We now verify the claim (3.7). To control err(ρn)(m), we compare κ(ρn)(X[0, m]) with cap(X[0, m], µ1(ρn))
via (2.30), and use the concentration result on cap(X[0, m], µ1(ρn)).
Fix δ > 0, then by the definition of the appropriate family, there exists some C′(δ) such that
for all n,
1
µ2(ρn)
X
m>C′µ1(ρn)
m2ρn(m) < δ.
Similar to (2.30),
(3.8)
κ(ρn)(X[0, m]) ≤cap(X[0, m], µ1(ρn)) + m · µ1(ρn)
µ2(ρn)µ1(ρn) log−10 µ1(ρn)
≤cap(X[0, m], µ1(ρn)) + m log−10 µ1(ρn),
so we have
(3.9)
err(ρn)(m) ≤err[ρn](m) + m log−10 µ1(ρn),
where
err[ρn](m) = E0
cap(X[0, m], µ1(ρn))1{cap(X[0,m],µ1(ρn))≥(1+ϵ/4) π2
8 m log−1 µ1(ρn)}
.
12
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
To bound the second term on the right-hand side of (3.9), note that
log µ1(ρn)
µ2(ρn)
∞
X
m=1
mρn(m) · m log−10 µ1(ρn) = log1−10 µ1(ρn) −→0.
Now we only need to verify that
(3.10)
log µ1(ρn)
µ2(ρn)
∞
X
m=1
mρn(m) · err[ρn](m) −→0.
By Proposition (2.14), for m < δµ1(ρn),
(3.11)
err[ρn](m) ≤E0[cap(X[0, m], m)] < Cm log−1 m;
while for m > C′µ1(ρn),
(3.12)
err[ρn](m) ≤E0[cap(X[0, m], µ1(ρn))] ≤⌈m/µ1(ρn)⌉· E0[cap(X[0, µ1(ρn)], µ1(ρn))]
≤Cm log−1 µ1(ρn);
while for δµ1(ρn) ≤m ≤C′µ1(ρn), we claim that, for large n,
(3.13)
err[ρn](m) ≤δm log−1 µ1(ρn).
We defer the proof of (3.13) for a few lines and finish the proof of (3.10). By (3.11), (3.12) and
(3.13), we have
∞
X
m=1
mρn(m) · err[ρn](m)
≤
⌊δµ1(ρn)⌋
X
m=1
m2ρn(m)[(log−1 m) ∧1] + δ
⌊C′µ1(ρn)⌋
X
m=⌊δµ1(ρn)⌋+1
m2ρn(m) log−1 µ1(ρn)
+ C
∞
X
m=⌊C′µ1(ρn)⌋+1
m2ρn(m) log−1 µ1(ρn)
≤2(δµ1(ρn))2 log−1 µ1(ρn) + δµ2(ρn) log−1 µ1(ρn) + Cδµ2(ρn) log−1 µ1(ρn)
≤2Cδµ2(ρn) log−1 µ1(ρn),
which implies (3.10) since δ is arbitrary.
We now prove (3.13). For all m ∈[δµ1(ρn), C′µ1(ρn)], write ξ = cap(X[0, m], µ1(ρn)), ξ′ =
cap(X[0, m]), a = E0[ξ], a′ = E0[ξ′] and ea = π2
8 m log−1 µ1(ρn). By Proposition 2.2 and the
Markov inequality,
E0[ξ′; ξ′ > (1 + ϵ/8)ea] ≤a′P 0[ξ′ −a′ > (1 + ϵ/8)ea −a′] + E0[ξ′ −a′; ξ′ −a′ > (1 + ϵ/8)ea −a′]
≤(1 + ϵ/16)eaP 0[ξ′ −a′ > ϵea/16] + E0[(ξ′ −a′)1{ξ′−a′>ϵea/16}]
≤(1 + ϵ/16)ea(16/ϵea)2 · var(ξ′) + 16/ϵea · var(ξ′) ≤C(ea)−1var(ξ)
≤Cea log−2 µ1(ρn),
where in the second inequality we have used a′ < (1 + ϵ/16)ea for all large n, which is deduced
from (2.9). Moreover, by (2.22) and (2.24),
P 0[ξ −ξ′ > ϵa/8] ≤8
ϵaE0[ξ −ξ′] < C log−1/2 µ1(ρn).
Thus, we have
err[ρn](m) = E0[ξ1ξ≥(1+ϵ/4)ea] ≤E0[ξ′1ξ≥(1+ϵ/4)ea] + E0[ξ −ξ′]
≤E0[ξ′1ξ′≤(1+ϵ/8)ea1ξ≥(1+ϵ/4)ea] + E0[ξ′1ξ′>(1+ϵ/8)ea] + E0[ξ −ξ′]
≤(1 + ϵ/8)ea · P 0[ξ −ξ′ ≥ϵea/8] + Cea log−2 µ1(ρn) + Cea log−1/2 µ1(ρn)
≤Cea log−1/2 µ1(ρn).
13
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
(3.13) then follows by taking n ∈N large.
□
Remark 3.3. Recall the definition of the ρ-capacity for all ρ ∈M. It is then not hard to see that
for all finite A ⊂Zd,
κ(ρ)(A) = cap(ˆρ)(A),
where ˆρ = P∞
k=0
kρ(k)
µ1(ρ)δk.
Remark 3.4. If we only deal with the lower bound for the original finitary random interlacement
with geometric killing, the argument can be greatly simplified by the following observation on
random walks conditioned on not returning to a given set.
Observation. For any A ⊂Zd, x ∈A and T ∈N, let X1 be a simple random walk from x and
X2 be a simple random walk from x conditioned not to return to A. Let Ni ∼Geo(
1
T+1) be
independent of Xi, i = 1, 2. Then there is a coupling between (X1
t )N1
t=0 and (X2
t )N2
t=0 such that
(1) almost surely, N1 ≥N2;
(2) almost surely, there exists a random point y† ∈Zd such that
(3.14)
X1
N1−N2+j + y† = X2
j ,
∀0 ≤j ≤N2.
As a direct consequence, the capacity of (X1
n)N1
n=0 stochastically dominates the capacity of (X2
n)N2
n=0.
In fact, we can recursively construct a sequence of trajectories from X1 as follows: let τ0; for
k ≥1, let ¯Xk = ϕτk−1X1 −X1
τk−1, and let τk = (τk−1 + eHA( ¯Xk)) ∧N1, where ϕkX1 is defined by
ϕkX1
j = X1
j+k, ∀j ∈N
and eHA( ¯Xk) is the first time ¯Xk returns to A. Define k∗:= inf{k ≥0 : τk = N1}. By the mem-
oryless property of the geometric distribution, ( ¯Xk∗
t )0≤t≤τk∗−τk∗−1 is distributed as (X2
t )0≤t≤N2,
which yields the observation.
Note that by Lemma 2.5, the number of trajectories in X u,Geo(
1
T +1 ) that hits A at x follows
Poi(ue(T)
A (x)), where e(T)
A (x) = P∞
i=0
1
T+1(
T
T+1)iP x[ eHA > i], and the conditional law of tra-
jectories given their total number is identical to that of ζ1 ∗ζ2, where ζ1 ∼X1[0, N1] and
ζ2 ∼P[X1[0, N1] = ·
X(0, N1] ∩A = ∅].
By the above observation, the expected capacity of
ζ1 ∗ζ2 is at most twice that of ζ1. When d ≥5, it is not hard to check that as T →∞,
E[cap(T)(X1[0, N1])] = (1 + o(1))εdT.
Therefore,
E[cap(T)(V (Iu,T [A]))] ≤
X
x∈A
ue(T)
A (x) · (2 + o(1))εdT = (2 + o(1))uεdT · cap(T)(A).
Taking u = u−
T := 1−ϵ
2εd · 1
T , we get
E[cap(T)(V (Iu,T [A]))] ≤(1 −ϵ/2)cap(T)(A)
for large T, which yields that Iu−
T ,T is subcritical. The value of u−
T matches u−
n defined below
(1.5). When d = 4, the same arguments hold by taking u−
T = 4(1−ϵ)
π2
· T −1 log T.
4. Proof of the upper bound of (1.7)
In this section, we will prove the upper bound of (1.7) for any appropriate family of distributions
(ρn)n∈N.
Recalling u+
n from (1.6), we will show that Iu+
n ,ρn is supercritical through an exploration
algorithm, as outlined in last but two paragraph of Section 1. To be more specific, we first restrict
to a truncated version of X u+
n ,ρn (denoted by X
u+
n ,ρn) which consists of “typical” trajectories
(rigorously defined later). Next, we divide the Euclidean space into boxes of side length Rn :=
⌊
p
µ1(ρn)⌋, with one-to-one correspondence with vertices of the coarse-grained lattice Ln :=
Rn · Zd, and start the exploration by revealing a cluster in the box corresponding to 0 consisted
of a fixed number of typical trajectories that satisfy a certain spatial restriction (we call such a
14
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
collection of trajectories a “seed”), and then start the exploration algorithm by declaring 0 to be
active. In each round of the subsequent exploration, we pick an active vertex and try to extend the
revealed cluster inside the corresponding box to neighboring ones by revealing a certain number
of layers of trajectories, one layer at a time. After that, we determine whether this new round
of exploration is good. If so, and the last layer in the aforementioned extension contains a seed
in each of the neighboring boxes, we activate the corresponding neighboring vertices and declare
the current vertex to be surviving; otherwise, we say this round of exploration at the current
vertex fails, and we declare the vertices within a neighborhood of range proportional to Rn to
be ruined (including those that has become surviving or active). By bounding from above the
conditional probability of failure in each round, which again involves comparison between the
round of exploration process and a Galton–Watson branching process, we show that the vertices
surviving in the end dominates a finitely dependent percolation model on the coarse-grained
lattice Ln, and prove its supercriticality with the Liggett–Schonmann–Stacey domination [16].
We implement the aforementioned strategy in the following two subsections. In Subsection
4.1, we introduce some further notation necessary for the introduction of the exploration algo-
rithm (henceforth referred to as the Algorithm) and gather some related preliminary results. In
Subsection 4.2, we rigorously describe the Algorithm and prove the supercriticality assuming the
estimate on the conditional probability of failure in each step (c.f. Proposition 4.7).
4.1. Preparation for the Algorithm. Let Ud := {e1, ..., ed} be the collection of canonical unit
vectors on Zd, where ej is the unit vector with j-th component 1, and write ⃗1 = e1 + e2 + ... + ed.
Let
(4.1)
Rn := ⌊
p
µ1(ρn)⌋
and define the corresponding coarse-grained lattice by Ln := Rn · Zd. Let
(4.2)
zn = ⌊Rn/2⌋·⃗1.
For x ∈Ln, let Bn
x = x + [0, Rn)d and eBn
x = B(x + zn, Rn/4) be two concentric boxes. Fix a
constant
(4.3)
c = 0.01.
We fix the dimension d ≥4, the perturbative constant ϵ ∈(0, 1/2) introduced in (1.6) and the
appropriate family (ρn)n∈N, and drop the dependence on them thereinafter. We now introduce
some extra symbols and terminology.
Typical trajectories: To facilitate further construction, we need to ignore certain atypi-
cal configurations of random walk trajectories in terms of the duration, capacity, volume and
diameter.
Define a mesoscopic scale
Ln :=
(
(Rn)
2+c
d−2 ,
d ≥5;
Rn log−c Rn,
d = 4
and a mildly increasing function with respect to Rn
In :=
(
(Rn)c,
d ≥5;
logc Rn,
d = 4.
For η ∈W [0,∞), write T = T(η). We introduce four parameters whose values will be determined
solely depending on d, ϵ and (ρn)n∈N: parameters
(4.4)
k, K ∈R+
put a restriction on the duration of typical trajectories, which will be fixed right after the state-
ment of Lemma 4.3; the parameter
(4.5)
M ∈R+
controls the diameter and will be fixed in (5.50) and the parameter
(4.6)
θ1 = θ1(q) ∈(0, 1),
15
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
which depends on another parameter q formally introduced in Lemma 4.3, controls the fluctuation
of the capacity of typical trajectories, and will be fixed in (5.4). We assume in following results
in this subsection that these parameters are arbitrarily chosen, unless stated otherwise.
We say η is typical if it satisfies all requirements below formulated by (Ei)i=1,2,3,4:
• E1 = E1(k, K, M) := {T ∈[k(Rn)2, K(Rn)2] and η ⊂B(η(0), MRn)}.
• If d ≥5,
E2 = E2(θ1) := {(1 −θ2
1)εdT < cap(range(η)) < (1 + θ2
1)εdT};
if d = 4,
E2 = E2(θ1) := {cap(range(η)) > (1 −θ2
1)π2
8 T log−1 µ1(ρn),
cap(range(η), R2
n) < (1 + θ2
1)π2
8 T log−1 µ1(ρn)}.
• E3 := {|B(η, Ln)| < R2
nLd−2
n
I1/3
n
}.
• Let
(4.7)
T ′
n =
(
⌊(Rn)2 log−1 µ1(ρn)⌋,
d = 4;
⌊(Rn)2−c/4⌋,
d ≥5.
If d = 4,
E4 = E4(θ1) := {cap(η[jT ′
n, (j + 1)T ′
n ∧T]) ≤(1 + θ2
1)π2
8 T ′
n log−1 µ1(ρn),
diam(η[jT ′
n, (j + 1)T ′
n ∧T]) < 1
2T 1/2 log−c µ1(ρn), ∀j ∈N, jT ′
n ≤T}.
If d ≥5,
E4 = E4(θ1) := {cap(η[jT ′
n, (j + 1)T ′
n ∧T]) ≤(1 + θ2
1)εdT ′
n, ∀j ∈N, jT ′
n ≤T}.
Let Tn := {η ∈W [0,∞) : η satisfies Ei, i = 1, 2, 3, 4} denote the collection of typical trajectories.
The definition of the typicality of a trajectory is dependent on k, K, M, θ1 and c, but we omit
them in the notation for simplicity.
As stated in the following lemma, random walk trajectories of properly fixed length are typical
with high probability. Recall the law µ(0, m, l) for any m, l ∈N from below (2.4).
Lemma 4.1. There exists a decreasing function ψ : R+ →(0, 1) satisfying limx→∞ψ(x) = 0,
such that
lim sup
n→∞
sup
k(Rn)2≤m+l≤K(Rn)2 µ(0, m, l)(T c
n ) < ψ(M/K1/2).
Proof. By standard Gaussian estimates, there exists a function ψ(x) tending to 0 as x →∞, such
that
(4.8)
µ(0, m, l)(Ec
1) ≤1
2ψ(M/K1/2)
for all m + l ≤KR2
n and all large n. By Propositions 2.2 and 2.3, for all kR2
n ≤m + l ≤KR2
n,
(4.9)
µ(0, m, l)(Ec
2) < C log−1/2 Rn,
and
(4.10)
µ(0, m, l)(Ec
4) < C log−1 Rn.
Combining (4.8)-(4.10), we get
lim sup
n→∞
sup
k(Rn)2≤m+l≤K(Rn)2 µ(0, m, l)(Ec
1 ∪Ec
2 ∪Ec
4) < ψ(M/K1/2).
To control the probability of Ec
3, first note that for all m + l ≤KR2
n,
µ(0, m, l)(Ec
3) ≤P 0 h
|B(X[0, K(Rn)2], Ln)| ≥R2
nLd−2
n
In
i
.
16
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Then we divide X[0, K(Rn)2] into pieces of length (L′
n)2, where L′
n = ⌊Ln/I1/3
n
⌋. For all 0 ≤j ≤
k := ⌈KR2
n/(L′
n)2⌉, let ξj = X[j(L′
n)2, (j + 1)(L′
n)2], then we have, for all large n,
P 0 h
|B(X[0, K(Rn)2], Ln)| ≥R2
nLd−2
n
In
i
≤P 0[∃0 ≤j ≤k, diam(ξj) > Ln]
≤kP 0[diam(X[0, (L′
n)2]) > Ln] ≤CR2
n/(L′
n)2exp(−I2/3
n
).
Here, the last inequality follows from a standard estimate of the transition probability of the
simple random walk. Calculating the last term gives
(4.11)
µ(0, m, l)(Ec
3) ≤
CI8/3
n
exp
−I2/3
n
,
d = 4;
exp
−Rc/2
n
,
d ≥5
for all large n. Therefore, we have
lim sup
n→∞
sup
k(Rn)2≤m+l≤K(Rn)2 µ(0, m, l)(Ec
3) = 0,
concluding the lemma.
□
Restriction within trajectories: When revealing a new layer in the Algorithm, we need
to avoid exploring trajectories hitting certain vertices (those with unusually small equilibrium
measure, for instance) within trajectories in the last layer explored, which facilitates the charac-
terization of the conditional law of the new layer. To this end, we introduce a further parameter
(4.12)
θ2 ∈(0, 1)
that will be fixed in (5.50). For any η ∈W [0,∞), recall that eη is the equilibrium measure of
range(η), and define the *-proper part of η by
(4.13)
bη =
{η(s) : s ∈[0, T], eη(s) ≥θ2},
d ≥5;
{η(s) : s ∈[0, T], eη(s) ≥θ2 log−1 µ1(ρn)
and P x[ eHη > (Rn)2 log−10 Rn] ≤(1 + θ1)eη(x)},
d = 4.
We will need the following property which states that a random walk starting from x ∈bη condi-
tioned on not returning to range(η) forgets about the conditioning soon after departure. Recall
the symbol “≃” for spatial translation of trajectories from Section 2.
Lemma 4.2. Let η ∈Tn, x ∈bη. Let
ln =
(
⌊R1−c/4
n
⌋,
d ≥5;
⌊Rn log−2c Rn⌋,
d = 4
and
tn =
(
⌊R2−c
n
⌋,
d ≥5;
⌊R2
n log−10 Rn⌋,
d = 4.
Then, for all large n, there is a coupling Q of X1 ∼P x[X ∈·| eHη = ∞] and X2 ∼P x[X ∈·] such
that
Q[E] ≥
(
1 −R−c/2
n
,
d ≥5;
1 −2θ1,
d = 4,
where
E := {|X1
j −X2
j | ≤ln, ∀j ≥0} ∩{X1[tn, ∞) ≃X2[tn, ∞)}.
Proof. Consider independent random walk paths constructed in the same probability space (Ω, F, Q)
with different laws: X3 ∼P x[X ∈·| eHη > tn], X4 ∼P x[X ∈·], X5 ∼P 0[X ∈·], X6 ∼P x[X ∈
·| eHη = ∞]. Let
X1[0, ∞) =
(
X3[0, tn] ∗X5[0, ∞),
if X3[0, tn] ∗X5[0, ∞) avoids η after time 0;
X6[0, ∞),
otherwise
17
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
and let X2[0, ∞) = X4[0, tn]∗X5[0, ∞). Then X1 and X2 constructed in this fashion has desired
marginal distributions. Define two events
E1 = {X1[0, ∞) = X3[0, tn] ∗X5[0, ∞)},
E2 = {X3[0, tn], X4[0, tn] ⊂Bx(ln/2)}.
Note that E ⊃E1 ∩E2, so it suffices to control the probability of Ec
j, j = 1, 2. For d = 4, by the
fact that x ∈bη (recall the definition (4.13)),
eη(x) ≥
1
1 + θ1
P x[ eHη > tn],
so it can be deduced easily that
Q[Ec
1] = P x[tn < eHη < ∞]
P x[ eHη > tn]
= 1 −
eη(x)
P x[ eHη > tn]
< θ1;
while for d ≥5,
Q[Ec
1] ≤θ−1
2 P x[X[tn, ∞) ∩range(η) ̸= ∅]
≤θ−1
2
P x[Xtn ∈B(η, Ln)] + CL2−d
n
cap(range(η))
≤1
2R
−c
2
n
for large n, where in the first inequality we have used (2.5) and (2.8). By estimating the transition
probability of the simple random walk and the definition of ˆη,
Q[X4[0, tn] ̸⊂B(x, ln/2)] = P 0[X[0, tn] ̸⊂B(0, ln)] ≤C exp
−ln/t1/2
n
,
Q[X3[0, tn] ̸⊂B(x, ln/2)] = P x[X[0, tn] ̸⊂B(x, ln)| eHη > tn] ≤Cθ−1
2
log Rn exp
−ln/t1/2
n
,
where the second inequality uses the fact that eη(x) > θ2 log−1(Rn)2. As a result,
Q[Ec
2] ≤1
2R−c/2
n
for all large n. So far, the lemma has been proved.
□
For any fixed η ∈W [0,∞), and any x ∈range(η), we define a probability measure µ∗
x,η on
W [0,∞) which characterizes the law of the trajectories in an FRI point process that hits η at x by
(4.14)
µ∗
x,η := C ·
X
k(Rn)2≤m+l≤K(Rn)2
ρn(m + l) · µ(x, m, l; range(η)),
where C = P
kR2n≤m≤KR2n(m + 1)ρn(m) is a normalizing constant. Then, we consider the proba-
bility measure obtained by conditioning on Tn:
(4.15)
µx,η := µ∗
x,η(·|Tn)
Similarly, for all x ∈Zd, we define a simpler probability measure that resemble µx,η by
(4.16)
µx := C ·
X
k(Rn)2≤m+l≤K(Rn)2
ρn(m + l)µ(x, m, l),
where C is again a normalizing constant.
Recall that we will compare a round of exploration with of Galton-Watson branching process
with agents in the space of trajectories W [0,∞). To guarantee the supercriticality of that branching
process (namely the process Y defined below the proof of Proposition 5.3), we introduce the
following lemma which focuses on the expected capacity of ζ ∼µx,η and can be derived an
analogue of Lemma 4.1 (c.f. Lemma A.1). Recall the parameters k, K, M and θ1 introduced
for the definition of typical trajectories from (4.4)-(4.6), and θ2 introduced for the definition of
*-proper part from (4.12).
18
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Figure 1. This figure illustrates the construction of pp(η; A, D). The boundaries
of sets A and D are shown in black, and that of the neighborhood B(D, Ln) is
shown in orange. The blue curve indicates the trajectory η, where the union of
the solid segments represent the proper part pp(η; A, D).
Lemma 4.3. We can properly fix k and K such that the following holds. There exists C1 and
C2 = C2(K) ∈(0, ∞) such that for all M > C2 and θ1, θ2 < C1, the following holds for large n:
for all η ∈W [0,∞) and x ∈bη,
(4.17)
µx(Tn) ∧µ∗
x,η(Tn) > 1 −ϵ/8
and for a random trajectory ζ ∼µx,η,
(4.18)
E[eζ(bζ)] > (1 −ϵ/4)εd
µ2(ρn)
µ1(ρn)(1 + log µ1(ρn)1d=4)−1.
The proof is a combination of proved estimates and standard computation of equilibrium
measures, so we put it in the appendix. From now on, we fix k and K and assume that θ1, θ2 < C1
and M > C2, so that Lemma 4.3 holds.
Good sequence of trajectory sets: In the Algorithm, we will need a criterion for the
goodness of a round of exploration at a certain vertex which is defined through the notion of
good sequence of trajectory sets presented below. To make our expositions below more concise,
we start with general definitions.
For finite A, B ⊂Zd and ρ ∈M, define
(4.19)
φ(ρ)(A, B) :=
X
x∈A,y∈B
e(ρ)
A (x)e(ρ)
B (y)g(x, y),
which measures the likelihood of a trajectory from FRI with length distribution ρ hitting both A
and B. For A, D ⊂Zd and η ∈W [0,∞), we define the proper part of η with respect to (A, D) by
pp(η; A, D) := bη ∩
η[τ η
A −R2
n/In, τ η
A + R2
n/In] ∪B(D, Ln)
c
,
see Figure 1. In the sequel, we will take A to be the vertex set induced by the latest layer we
have revealed, and D that induced by all the previous layers.
For finite collections of trajectories A1, ..., At, D0, ..., Ds ⊂W [0,∞), write A0 = Ds. For all
η ∈Ai(1 ≤i ≤t), we write
ηi = pp
η; V (Ai−1), V
s[
j=0
Dj ∪
i−1
[
j=0
Aj
.
We introduce a parameter
(4.20)
k1 = k1(ϵ, q) ∈N
19
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
which is dependent on q introduced in Proposition 4.7, whose value will be fixed in (5.4). Let eCn
be the typical capacity of a random walk running (Rn)2 steps:
(4.21)
eCn =
(
R2
n,
d ≥5;
R2
n log−1 R2
n,
d = 4
Definition 4.4. We say (Ai)1≤i≤t is good with respect to (Di)0≤i≤s if the following holds for all
i ∈{1, ..., t}.
• For all η1, η2 ∈Ai, cap(ρn)(bηj\ηi
j) < 4 eCn/I1/2
n
, j = 1, 2 and φ(ηi
1, ηi
2) < eCn/In.
• For all η ∈Ai, η hits at most one trajectory ζ ∈Ai−1, and η(τ η
ζ ) ∈ζ
i−1.
• The number of trajectories in Ai hitting any certain trajectory in Ai−1 is bounded from
above by k1.
In the sequel, we will let (Ai)1≤i≤t be the first t layers in the round of exploration at the current
vertex, and take (Di)0≤i≤s to be the sequence of all layers explored before the current round.
Remark 4.5. The above definition guarantees that for 0 < t′ < t, (Ai)1≤i≤t′ is good with respect
to (Di)0≤i≤s if (Ai)1≤i≤t is.
4.2. The Algorithm. We now introduce the exploration algorithm. Recall the general scheme
described at the beginning of this section, where we start each round of exploration with a seed
and sampling a certain number of layers. We use
(4.22)
α ∈N
to denote the total number of layers in each round, which will be fixed in (5.49), and introduce a
parameter γ that determines the range of boxes ruined due to an unsuccessful round of exploration:
(4.23)
γ = 2αM + 2,
where M defined in (4.5) corresponds to the maximum diameter of a typical trajectory. Let
(4.24)
β = β(q)
denote the number of trajectories in a seed, which will be fixed in (5.4). Here, q is the same as in
(4.6) and (4.20). Heretofore, we have introduced all the parameters necessary for the definition
of Algorithm. Among them, note that k, K, M, θ2 and α solely depend on d, ϵ and (ρn)n∈N and
are fixed throughout the proof, although the choice of the latter three will not be covered until
Subsection 5.2. Hence, in the remainder of the paper, we omit these parameters in statements
and proofs if not necessary to avoid redundancy.
Recall that Tn is the collection of all typical trajectories, and define the restriction of X u+
n ,ρn
to Tn by
X
u+
n ,ρn =
X
η∈X u+
n ,ρn∩Tn
δη.
The algorithm reveals a series of trajectory sets (layers): L−1 will be the collection of trajectories
in X
u+
n ,ρn hitting a fixed point in the box associated with 0, which initiates our exploration; Lx,i
(x ∈Ln, 1 ≤i ≤α) denotes the i-th layer of trajectories explored in the round of exploration at
x. We write Lx := Sα
i=1 Lx,i if x is ever explored. A sequence ℵm = (ℵ1
m, ℵ2
m) ∈(Ln × {0, 1}) ∪
{∆}, m = 0, 1, ... (∆is the cemetery state) will be used to record the result of the exploration:
ℵ1
m is for the position of the m-th round of exploration and ℵ2
m is the indicator of failure in
exploration. The state at each element of Ln will also be updated along with the exploration
of I
u+
n ,ρn. All possible states of vertices are “unexplored”,“active”, “surviving” and “ruined”.
Before the implementation of the algorithm, we set all the vertices as unexplored.
Later in the description of the algorithm, we say the m-th round of exploration/the round of
exploration at x = ℵ1
m is good until layer i(1 ≤i ≤α) if (Lx,j)1≤j≤i is good with respect to
20
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
(L−1, Lℵ1
0,1, ..., Lℵ1
m−1,α) (c.f. Definition 4.4), and write “good” as a shorthand for “good until
layer α”. Here, we adopt the following notation for proper parts of trajectories:
η(j,x) := pp
η; V (Lx,j−1), V (L−1 ∪
m−1
[
l=0
Lℵ1
l ∪
j−1
[
l=1
Lx,l)
for η ∈Lx,j, x = ℵ1
m and
η−1 := pp(η; {zn}, ∅)
for η ∈L−1 (recall from the beginning of the section that zn is approximately the center of the
box Bn
x). To make our exposition more concise, we will also use the notation η when no confusion
arises. range(η)\η will be referred to as the improper part of η.
We now introduce the notion of “seed”.
Definition 4.6. A seed S at x ∈Ln is a subset of Tn which consists of β typical trajectories
staying inside the box Bn
x and satisfy: for all η1, η2 ∈S, φ(ρn)(η1, η2) < eCn/In.
Note that W [0,∞) is countable. We choose an arbitrary ordering on W [0,∞), one on the set
Ln and one on the collection of all finite subsets W [0,∞). The ordinals “first” and “next” in the
following description of the algorithm all refer to these prescribed orders.
The algorithm is now described as follows.
• Step 1. If there exists a seed at 0 that is a subset of L−1 := X
u+
n ,ρn[{⌊Rn/2⌋· ˜1}], set
the origin as active, and denote the seed by S0 (choose the first seed if there are multiple
options). Meanwhile, let J0,0 = {⌊Rn/2⌋· ⃗1}, J0,0 = ∅, x = 0 and m = 0. If there is no
seed, end the algorithm and set ℵ0 = ∆.
• Step 2. The m-th round of exploration (the round of exploration at x):
– Substep 2A. Set ℵ1
m = x. Let Lx,0 = Sx. Let i = 1.
– Substep 2B. Let Lx,i−1 := V (Lx,i−1), L′
x,i−1 := S
η∈Lx,i−1 bη, Jx,i := Jx,i−1 ∪Lx,i−1,
Jx,i := Jx,i−1 ∪Lx,i−1 and Lx,i := X
u+
n ,ρn[Lx,i−1; Jx,i−1; L′
x,i−1]\Lx,i−1. (Recall the
definition of X
u+
n ,ρn[·] from (2.26).)
– Substep 2C. If i < α, increase i by 1 and go back to Substep 2B. Otherwise, proceed
to Step 3.
• Step 3. If the following two events
(4.25)
Good(x) := {the exploration at x is good}
and
(4.26)
Seed(x) := {∀e ∈Ud ∪(−Ud), Lx,α contains a seed at x + Rn · e}
both happen, then we declare x to be surviving, set y = x + Rn · e as active if it is not
yet explored or ruined, and let Sy be the first seed at y contained in Lx,α, and set x as
explored and ℵ2
m = 0. Otherwise, we set all the vertices in B(x, 2γ · Rn) ∩Ln as ruined
and set ℵ2
m = 1.
• Step 4. Increase m by 1. If there is no further active vertex, set ℵm = ∆and end the
algorithm; otherwise, let x′ be the first active vertex, let Jx′,0 = Jx,α, Jx′,0 = Jx,α, and
assign the value of x′ to x, and then go back to Step 2.
For x = ℵ1
m ∈Ln for some m ∈N, let Lx = Sα
i=1 Lx,i. Let τ = inf{m ≥0 : ℵm = ∆},
J = L−1 ∪Sτ−1
i=0 Lℵ1
i . By our construction, G(J ) is always connected. Let
H := {L−1 contains a seed at 0}
be the event that the exploration at 0 ever starts, whose probability is non-zero. For m ∈N, i ∈
{0, 1, ..., α −1}, define
Fm,i := σ(L−1, (ℵ1
j)0≤j≤m, (Lℵ1
j,s)0≤j<m,1≤s≤α, (Lℵ1
j,s)1≤s≤i).
21
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Note that τ, ℵm and H are all measurable with respect to X
u+
n ,ρn and that Iu+
n ,ρn percolates
as long as τ = ∞. To this end, we define for all 0 ≤m < τ
Failm := {ℵ2
m = 1},
and say that the exploration at the vertex ℵ1
m is successful (resp. unsuccessful) if Failc
m (resp.
Failm) occurs. We bound the probability of Failm from above conditionally on Fm,0:
Proposition 4.7. For all q > 0, we can choose β, k1 and θ1 introduced in (4.24), (4.20) and
(4.6) respectively such that
(4.27)
P[Failm|Fm,0](ω) < q.
for all large n ∈N, m ≥0 and configuration ω of X
u+
n ,ρn satisfying ℵm(ω) ̸= ∆.
Note that here we assume that k, K, M introduced for the definition of typical trajectories,
θ2 introduced for the definition of ∗-proper part and the number of layers per round α are all
properly fixed. The proof of Proposition 4.7 is deferred to Section 5. Now, to obtain the upper
bound of (1.7), we introduce an auxiliary finitely dependent percolation model ωq, and prove that
its cluster containing 0, after proper rescaling, is dominated by the set of (coarse-grained) vertices
that is surviving in the end which we denote by Va ⊂Ln; then, we prove that ωq is supercritical
using the Liggett–Schonmann–Stacey domination [16].
Recall γ from (4.23). Let ξx ∼Ber(q), x ∈Zd be i.i.d. random variables. For all x ∈Zd, let
(4.28)
ωq
x =
(
0,
∃y ∈B(x, 2γ), ξy = 1;
1,
otherwise.
We say that ωq percolates if there is an infinite self-avoiding nearest-neighbor path on Zd, such
that for each vertex x on the path, ωq
x = 1. To put it another way, for every x ∈Zd, all vertices
in its neighborhood B(x, 2γ) is ruined simultaneously with probability q, and we consider the
percolation on the vertices that are not ruined.
Let Cq
0 ⊂Ln be the cluster of the origin in ωq, and we have the coupling below.
Proposition 4.8. For all q ∈(0, 1), we can choose parameters β, k1 and θ1 depending on q such
that for all large n ∈N, there is a coupling Q of Va with conditional law on H and Cq
0 such that
Va ⊃Rn · Cq
0 on the event {τ < ∞}.
Proof assuming Proposition 4.7. We use a sequence of configurations σm ∈{U, S, A, R}Ln to
record the change of states of vertices in Ln: σm
x stands for the state of x right after the algorithm
assigns some value to ℵ2
m, and U, S, A, R correspond to unexplored, surviving, active and ruined
respectively. Define the eventual state of x by σx := limm→∞σm∧(τ−1)
x
. We write PH = P[·|H].
Recall that for x ∈Ln and m ∈N, σx is the eventual state of x and σm
x is the state of x after
round m. Since given (ℵj)0≤j≤m−1, the choice of ℵ1
m is deterministic (according to the pre-fixed
order on Ln), one can argue by induction that the 0-1 variables ℵ2
j, j = 0, ..., m −1 determine the
location of the m-th round of exploration. Therefore, we have ℵ1
m, {ℵm = ∆} ∈σ(ℵ2
j, 0 ≤j < m).
Assume that ℵm ̸= ∆, then by Proposition 4.7,
PH[ℵ2
m = 1|ℵ2
j, 0 ≤j < m] = EH[P[Failm|Fm,0]
ℵ2
j, 0 ≤j < m] ≤q,
where EH = E[·|H] is the corresponding expectation. Thus, we can construct under PH random
variables ξm, m ∈N that are i.i.d. with distribution Ber(q) (Bernoulli distribution with expec-
tation q), such that ℵ2
m ≤ξm a.s. under PH. Moreover, let χx ∼Ber(q), x ∈Ln be mutually
independent and independent of (ξm)m∈N and (ℵm)m∈N. Let
χ′
x =
(
ξm,
ℵ1
m = x for some m ∈N;
χx,
otherwise.
Then we have χ′
x, x ∈Ln are i.i.d. with distribution Ber(q). Let A = S
χ′x=1 B(x, 2γRn), and
Ωx = 1x∈Ac, x ∈Ln, then ω := (ΩRn·x)x∈Zd has the same distribution as ωq. Let C denote the
cluster in ω containing 0.
22
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
We claim that for all x ∈C, σRn·x = 1 (Rn · x is surviving eventually), which leads to our
desired conclusion. Otherwise, let x ∈C satisfy σRn·x ̸= 1. Let π = (π(j))0≤j≤m be a nearest-
neighbor path on Ln in C that connects 0 to Rn · x. For all y = π(j), 0 ≤j ≤m, χ′
z = 0 for all
z ∈B(y, 2γRn)∩Ln. Thus, by our construction, ℵ2
j = ξj = 0 for all j satisfying ℵ1
j ∈B(y, 2γRn),
indicating that σy ̸= R. By the definition of Algorithm, if y ∈Ln ranks j-th according to the
deterministic order we have fixed, |{m ∈N : σm
y = A}| ≤j, so σy ̸= 2, which indicates the final
state of any site cannot be active. By our assumption, one may only have σy = U. However, that
would imply the existence of 0 ≤j < m such that σπ(j) = 1 and σπ(j+1) = 0, which is impossible
because π(j + 1) would have become active once the status of π(j) turned “surviving”.
□
Proof of the upper bound of (1.7) assuming Proposition 4.8. By [16, Theorem 1.3], we can take
q = q(γ) ∈(0, 1) sufficiently close to 0 such that ωq stochastically dominates a supercritical
Bernoulli site percolation on Zd. Hence, Cq
0 is infinite with non-zero probability. Take β, k1 and
θ1 depending on q and satisfying the conditions of Proposition 4.8, then for large n,
Q[τ < ∞] = Q[τ < ∞, Va is finite] ≤Q[Cq
0 is finite] < 1.
Therefore, for large n, {τ = ∞} happens with non-zero probability, so Iu+
n ,ρn is supercritical,
which yields u∗(ρn) ≤u+
n .
□
5. Proof of Proposition 4.7.
In this section, we prove Proposition 4.7. We divide the proof of Proposition 4.7 into two
parts, deriving lower bounds for two quantities: the conditional probability of the exploration
at a vertex being good, and that of a good exploration at a vertex revealing new seeds for its
neighbors, which respectively correspond to Propositions 5.1 and 5.3 below. Our strategy is to
ignore the previously explored trajectories when revealing new ones in the Algorithm and view
a layer of trajectories in the exploration as a generation of a branching process with agents in
the space W [0,∞), which is achieved through couplings. Then, we analyze the behavior of the
branching process using the LCLT.
For all S ⊂W [0,∞), let P S
Y be the law of the branching process with agents in W [0,∞) with
offspring distribution
X
u+
n ,ρn[range(η); ∅; bη]
η∈W [0,∞) starting from S. Recall the definition of
Good(x) and Seed(x) for x ∈Ln from (4.25) and (4.26) respectively.
Proposition 5.1. Arbitrarily choose β ∈N in (4.24) and r ∈(0, 1). Then there exists C3 =
C3(β, r) ∈N and C4 = C4(β, k1, r) > 0 such that for all k1 > C3 and θ1 < C4,
(5.1)
P[Good(ℵ1
m)|Fm,0] > 1 −r
whenever ℵ1
m ̸= ∆for large n.
Adapting part of the proof of the above proposition, we obtain the following coupling between
the sequence of layers (Lx,i)1≤i≤α and the branching process (Yi)1≤i≤α.
Proposition 5.2. Arbitrarily choose β ∈N in (4.24) and r ∈(0, 1). Let m ∈N and assume that
ℵ1
m = x ∈Ln. Then, there exists some C∗
4 = C∗
4(β, r) such that for all θ1 > C∗
4 and all large n,
we can find a coupling Q of (Lx,i)1≤i≤α with law conditioned on Fm,0 and (Yi)i≥0 with law P Sx
Y
satisfying
(5.2)
Q[Lx,i = Yi, i = 1, 2, ..., α] > 1 −r.
Proposition 5.3. For all r > 0, there exists C5 = C5(r) such that the following holds. For all
β > C5 and k1 > C3(β, r/2) in Proposition 5.1, there is C6 = C6(β, r) such that for θ1 < C6,
(5.3)
P[Seed(ℵ1
m)|Fm,0] > 1 −r
whenever ℵ1
m ̸= ∆for large n.
Clearly, Proposition 4.7 follows from combining Propositions 5.1 and 5.3.
23
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Proof of Proposition 4.7 assuming Propositions 5.1 and 5.3. We fix
(5.4)
β > C5(ϵ, q/2), k1 > C3(β, q/2) and θ1 < C6(β, q/2) ∧C4(β, k1, q/2).
Thus, by Proposition 5.1,
P[Good(ℵ1
m)|Fm,0] > 1 −q/2;
by Proposition 5.3,
P[Seed(ℵ1
m)|Fm,0] > 1 −q/2.
The claim (4.27) then follows by noting that Failm = Good(ℵ1
m)c ∪Seed(ℵ1
m)c.
□
The remaining part of this section is organized as below. In Subsection 5.1, we prove Propo-
sition 5.1 by induction, and show that techniques involved in the proof lead to Proposition 5.2.
In Subsection 5.2, we prove Proposition 5.3 by proving a similar result for the branching process
(Y )i≥0 using the LCLT and then applying Proposition 5.2.
5.1. On the goodness of exploration. In this subsection, we prove Proposition 5.1 by itera-
tion: we define
Good(x, i) := {the exploration at x is good until layer i},
i = 0, 1, ..., α, and compute P[Good(x, i + 1)|Good(x, i)] for i ≤α −1. To facilitate the computa-
tion, we condition on Fm,i and try to couple the conditional law of Lx,i+1 with that of the union
of independent point processes with distributions X
u+
n ,ρn[range(η)] for all η ∈Lx,i. Therefore, we
get as a byproduct a coupling between (Lx,i)1≤i≤α conditioned on Fm,0 and a branching process
with agents in the space W [0,∞), which will also be crucial for the proof of Proposition 5.3.
We reduce the proof of Proposition 5.1 to the iteration of the following result.
Proposition 5.4. Arbitrarily fix β in (4.24), then for all r > 0, then there exists C7(β, r), such
that for all k1 > C7, there is some C8 = C8(β, k1, r) satisfying for all θ1 < C8 and all large n, on
the event Good(ℵ1
m, i),
P[Good(ℵ1
m, i + 1)|Fm,i] > 1 −r.
Proof of Proposition 5.1 assuming Proposition 5.4. Take r′ = r′(r) > 0 small so that (1 −r′)α >
1−r, and let k1 > C7(β, r′) and θ1 < C8(β, k1, r′). By iterating the inequality in Proposition 5.4,
we have
P[Good(ℵ1
m)|Fm,0] > (1 −r′)α > 1 −r.
Thus, we conclude the proof by taking C3(β, r) = C7(β, r′) and C4(β, k1, r) = C8(β, k1, r′).
□
We gather an ingredient for the proof of Proposition 5.4. In order to compute the conditional
probability of Good(ℵ1
m, i+1), we first simplify the conditional law of the (i+1)’th layer. We write
Lℵ1m,i = {η1, ..., ηs}. As mentioned at the beginning of this subsection, we construct independent
point processes with distributions X
u+
n ,ρn[range(ηj); ∅; ˆηj], j = 1, ..., s and couple Lx,i+1 with their
union. Formally, let (Jx,i,l)x∈Ln,1≤i≤α,l∈N be a collection of i.i.d. PPPs that are independent of
X u+
n ,ρn but share the same distribution, and let J x,i,l denote the restriction of Jx,i,l to Tn. Then,
we have the following lemma.
Lemma 5.5. Arbitrarily fix β, k1 and θ1. Conditioning on Fm,i and with restriction on Good(ℵ1
m, i),
write x = ℵ1
m, Lx,i = {η0, η1, ..., ηs}, and let
Kx,i+1 =
s[
j=0
J x,i+1,j[range(ηj); ∅; bηj].
Then for all r > 0, there exists a coupling Q between Lx,i+1 with law conditioned on Fm,i and
Kx,i+1 such that
Q[Lx,i+1 ̸= Kx,i+1] < r
for large n and all realizations of X
u+
n ,ρn satisfying Good(ℵ1
m, i).
24
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Proof. Recall that for ηj ∈Lx,i, ηj = ηj(i,x) is the proper part of ηj. Recall the definition of Jx,i
from Algorithm. Since we have arbitrarily fixed an x and i, we write V = V (Jx,i) throughout this
proof. Note that conditionally on Fm,i, the law of Lx,i+1 is identical to that of an independent
copy of X
u+
n ,ρn[Lx,i, V ; L′
x,i]. Now we sample independent point processes T = X u+
n ,ρn, and
Jj =d X
u+
n ,ρn[range(ηj); ∅; bηj]\X
u+
n ,ρn[range(ηj); V ∪
[
l̸=j
ηl; ηj], j = 0, 1, ..., s.
Recall that T (η) is the mass of T assigned to η. Define the restriction of T to Tn, T by
T :=
X
η∈Tn
T (η) · δη.
Let X = T [Lx,i; V ; L′
x,i] and
Xj = T [range(ηj); V ∪
[
l̸=j
ηl; ηj] ∪Jj, j = 0, 1, ..., s.
Let X ′ = Ps
j=0 Xj, and note that we have X ∼ν and X ′ ∼ν′, where ν is the law of Lx,i+1 given
Fm,i, and ν′ is the law of Kx,i+1. Therefore, it suffices to show that P[X = X ′] > 1 −r when n is
large.
We define the so called bad part of X by
Xbad := {η ∈X : η first hits Lx,i in the improper part of some trajectory,
or hits both ηj and ηl for some j ̸= l, 0 ≤j, l ≤s}
In fact, Jj, 0 ≤j ≤s and Xbad are all empty with probability tending to 1 when n tends to
infinity, which directly leads to X = X ′. Let L′′
x,i = S
η∈Lx,i ˆη\η. Since the exploration at x is
good until the i’th layer, s ≤βki
1, by computing the intensity of PPPs Xbad and Jj, 0 ≤j ≤s,
we get
(5.5)
P[Xbad ̸= ∅] ≤u+
n cap(ρn)(L′′
x,i) +
X
0≤j<l≤s
P[∃ζ ∈T , ζ hits both ηj and ηl]
≤u+
n cap(ρn)(L′′
x,i) +
X
0≤j<l≤s
X
x∈ηj
u+
n e(ρn)
ηj
(x)h(x, ηl) +
X
x∈ηl
u+
n e(ρn)
ηl
(x)h(x, ηj)
≤u+
n cap(ρn)(L′′
x,i) +
X
j̸=l
u+
n φ(ρn)(ηj, ηl),
where we have used (2.7) in the last line. Likewise, we can show
(5.6)
s
X
j=0
P[Jj ̸= ∅] ≤u+
n
s
X
j=0
cap(ρn)(bηj\ηj) +
X
0≤j,l≤s
u+
n φ(ρn)(ηj, ηl) + u+
n
s
X
j=0
φ(ρn)(ηj, V ∩B(x, γRn)).
In the last term, we have V ∩B(x, γRn) instead of V for the following reason: by definition, the
diameter of a typical trajectory is always bounded by 2MRn, so it is impossible for any trajectory
in Jj to hit B(x, γRn)c by our choice of γ in (4.23). By the first item in Definition 4.4,
cap(ρn)(L′′
x,i) ≤
s
X
j=0
cap(ρn)(bηj\ηj) ≤(s + 1) · 4 eCn/I1/2
n
;
and for all 0 ≤j < l ≤s,
φ(ρn)(ηj, ηl) ≤eCn/I1/2
n
Moreover, we claim that
(5.7)
φ(ρn)(ηj, V ∩B(x, γRn)) ≤C( eCn)2L2−d
n
.
25
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Finally, noting that by (1.5) and (1.6),
(5.8)
u+
n ≲(1 + log µ1(ρn)1d=4)µ1(ρn)
µ2(ρn)
≤1 + log µ1(ρn)1d=4
µ1(ρn)
≲( eCn)−1,
where eCn was defined in (4.21), we have
(5.9)
P[X ̸= X ′] ≤P[Xbad ̸= ∅] +
s
X
j=0
P[Jj ̸= ∅] ≤C(sI−1/2
n
+ eCnL2−d
n
)
which tends to 0 uniformly in the realization of X
u+
n ,ρn. This concludes the proof of the lemma.
It remains to prove (5.7). Note that the vertices comprising V ∩B(x, γRn) can only come
from trajectories in Ly where y ∈B(x, 2γRn) ∩Ln. Since x = ℵ1
m has not been ruined until the
end of the (m −1)-th round of exploration, the round of exploration at any y ∈{ℵ1
0, ..., ℵ1
m−1} ∩
B(x, 2γRn) must have succeeded, so
|Ly| ≤β(1 + k1 + ... + kα−1
1
) ≤βkα
1 .
Thus, the vertices comprising V ∩B(x, γRn) come from at most
|B(x, 2γRn) ∩Ln| · βkα+1
1
= (4γ + 1)dβkα
1
typical trajectories, so cap(ρn)(V ∩B(x, 2γRn)) ≤C · eCn, where C = C(β, γ, k1) and eCn defined
in (4.21) stands for the order of the capacity of a typical random walk path of length (Rn)2. In
addition, cap(ρn)(ηj) ≤C · eCn, and d(ηj, V ) > Ln by the definition of ηj, so we have
(5.10)
φ(ρn)(ηj, V ) ≤cap(ρn)(ηj)cap(ρn)(V ∩B(x, 2γRn)) · CL2−d
n
≤C · ( eCn)2L2−d
n
.
This finishes the proof.
□
Now we are in a good position to prove Proposition 5.4.
Proof of Proposition 5.4. By Lemma 5.5, it suffices to prove
(5.11)
P[Good′(ℵ1
m, i + 1)|Fm,i] > 1 −r/2,
where
Good′(ℵ1
m, i + 1) := {(Lℵ1m,1, ..., Lℵ1m,i, Kℵ1m,i+1) is good w.r.t. Pm}.
Recall the three restrictive items for goodness of exploration in Definition 4.4. We are going to
verify them respectively.
First, we deal with the third restriction that for all η ∈Lx,i, the number of trajectories in Lx,i+1
intersecting ˆη is no larger than k1. We need to control the probability that J x.i+1,j[range(ηj); ∅; bηj],
a PPP with intensity at most u+
n cap(ρn)(ηj), contains more than k1 trajectories. Note that for
N ∼Poi(u+
n K eCn), there is some C = C(K) < ∞, such that for all large k1,
(5.12)
P[N > k1] < C2−k1,
because the intensity parameter u+
n K eCn ≲K by (5.8). Henceforth, we always assume that k1
satisfies (5.12). Let
E1 = {∃1 ≤j ≤s, s.t. |Jx.i+1,j[range(ηj); ∅; bηj]| > k1},
then by (5.12), we can take a large constant C7(β, r) such that when k1 > C7,
(5.13)
P[E1|Fm,i] ≤s · C2−k1 ≤βkα
1 2−k1 < r/4.
Next, we verify the second restriction that each ζ ∈Lx,i+1 hits ˆη for exactly one η ∈Lx,i and
first hits ˆη in η. We control the intersection events defined below. Define
E2 = {∃1 ≤j ≤s, ζ ∈Jx.i+1,j[range(ηj); ∅; bηj] hits ηj in bηj\ηj},
E3 = {∃ζ ∈Kx,i+1 that hits ηj and ηl, where 1 ≤j < l ≤s},
Then, by the proof of Lemma 5.5, for large n,
(5.14)
P[E2 ∪E3|Fm,i] < r/4.
26
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Finally, we deal with the first restriction which involves estimates on the ρn-capacity and φ(ρn)
quantities (defined in (5.18) and (4.19) respectively) related to the new generation Lx,i+1. For
ζ ∈Kℵ1m,i+1, write ζ = pp(ζ; Lm,i, V ∪Lm,i), where V is the same as in the proof of Lemma 5.5.
Let
E4 = {∃ζ ∈Kℵ1m,i+1, cap(ρn)(bζ\ζ) > 4 eCn/I1/2
n
},
E5 = {∃ζ1, ζ2 ∈Kℵ1m,i+1, s.t. φ(ρn)(ζ1, ζ2) > eCn/In},
and we aim to bound the conditional probability of E4 ∪E5 from above. Instead of conditioning
on Fm,i alone, this time we additionally condition on the number, lengths and hitting points of
trajectories within Kℵ1m,i+1. More precisely, let Nj =
Jx.i+1,j[range(ηj); ∅; bηj]
, j = 1, ..., s, and
define
G = σ{Nj, T(ζ), t(ζ, ηj), ζ(t(ζ, ηj)), ∀1 ≤j ≤s, ζ ∈Jx.i+1,j[range(ηj); ∅; bηj]}.
We claim that there exists C8(β, k1, r) such that for all θ1 < C8, on Good(ℵ1
m, i) ∩Ec
1,
(5.15)
P[E4 ∪E5|Fm,i, G] < r/4.
for all large n. The proof of (5.15) relies on the law conditioned on Fℵ1m,i and G of trajectories
within Kℵ1m,i+1 deduced from Lemma 2.5, and also on Lemma 4.2 that helps deal with the law
of backward paths of these trajectories that involves conditioning. We will attend to the details
shortly. Coming back to the proof, by definition,
Good(ℵ1
m, i + 1) ⊃Good(ℵ1
m, i)\
5[
j=1
Ej.
Assuming (5.15), the conclusion then follows from (5.13). (5.14) and (5.15).
□
In order to show (5.15), we need the following two auxiliary lemmas, whose proofs are technical
and deferred to the appendix.
Lemma 5.6. There exists some C < ∞such that the following holds for all large n. For all
A ⊂Zd finite and nonempty, y ∈Zd, and positive integer T < KR2
n,
Ey[cap(X[T, KR2
n] ∩A, R2
n)] ≤C|B(A, Ln)|T −1 log−1 Rn,
if d = 4;
Ey h X[T, KR2
n] ∩A
i
≤C|A|T 1−d/2,
if d ≥5.
Lemma 5.7. There exists some C < ∞such that for all large n and y, z ∈Zd,
Ey,z[φ(ρn)(X1[0, K(Rn)2], X2[0, K(Rn)2])] ≤
CR2
n log−2 Rn,
d = 4;
CRn,
d = 5;
C log Rn,
d = 6;
C,
d > 6.
Now we prove (5.15) with Lemmas 5.6 and 5.7.
Proof of (5.15). Write Jx,i+1,j[range(ηj); ∅; bηj] = PNj
l=1 δζj,l, Tj,l = T(ζj,l), t1
j,l = t(ζj,l, ηj), t2
j,l =
Tj,l −t1
j,l, xj,l = ζj,l(t1
j,l). By a union bound,
P[Ec
4|Fm,i, G] ≤
X
0≤j≤s,1≤l≤Nj
P[cap(ρn)(bζj,l\ζj,l) > 4 eCn/I1/2
n
|Fm,i, G],
(5.16)
P[Ec
5|Fm,i, G] ≤
X
(j1,l1)̸=(j2,l2)
P[φ(ρn)(ζj1,l1, ζj2,l2) > eCn/In|Fm,i, G].
(5.17)
(Here, the proper part ζj,l for ζj,l that is atypical is similarly defined.)
Now, it suffices to
bound every term on the right-hand side of (5.16) and (5.17) from above.
By Lemma 2.5,
conditionally on Fm,i and G, with the restriction Good(ℵ1
m, i) ∩Ec
1, the distribution of ζj,l is
µ(xj,l, t1
j,l, t2
j,l; range(ηj)), for all 0 ≤j ≤s, 1 ≤l ≤Nj, and they are mutually independent. Let
ζf
j,l = ζj,l[t1
j,l + R2
n/In, Tj,l], ζb
j,l = ζj,l[0, t1
j,l −(Rn)2/In]. Recall the constant C8 from Proposition
27
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
5.4. By Lemma 4.2, we can take C8(β, k1, r) sufficiently small such that for all θ1 < C8 and all
large n, conditionally on Fm,i and G, there is a coupling between ζj,l and Xj,l (j = 0, 1, ..., s, l =
1, ..., Nj), where Xj,l ∼P xj,l are mutually independent, and satisfy
Xj,l[−t1
j,l, −tn] ≃ζj,l[0, t1
j,l −tn], Xj,l[tn, t2
j,l] ≃ζj,l[t1
j,l + tn, Tj,l]
and |Xj,l(u−t1
j,l)−ζj,l(u)| < ln for all 0 ≤u ≤Tj,l with probability 1−r/N, where N = 32(βkα
1 )2.
Now we consider the right-hand side of (5.16). Note that by definition,
bζj,l\ζj,l ⊂ζj,l[t1
j,l −R2
n/In, t1
j,l + R2
n/In] ∪(ζf
j,l ∩V ′) ∪(ζb
j,l ∩V ′),
where V ′ = B(V (Jx,i), Ln) ∩B(x, 2γRn). Therefore,
(5.18)
P[cap(ρn)(bζj,l\ζj,l) > 4 eCn/I1/2
n
|Fm,i, G]
≤P[cap(ρn)(ζj,l[t1
j,l −R2
n/In, t1
j,l −tn]) + cap(ρn)(ζj,l[t1
j,l + tn, t1
j,l + R2
n/In]) | Fm,i, G]
+ P[cap(ρn)(ζf
j,l ∩V ′) > eCn/I1/2
n
|Fm,i, G] + P[cap(ρn)(ζb
j,l ∩V ′) > eCn/I1/2
n
|Fm,i, G]
≤P xj,l[cap(ρn)(X[−(Rn)2/In, −tn]) + cap(ρn)(X[tn, (Rn)2/In]) > eCn/I1/2
n
]
+ P xj,l[cap(ρn)(X[R2
n/In, t2
j,l] ∩B(V ′, Ln)) > eCn/I1/2
n
]
+ P xj,l[cap(ρn)(X[−t1
j,l, −R2
n/In] ∩B(V ′, Ln)) > eCn/I1/2
n
] + 3r/N,
where the second inequality above follows from the coupling between ζj,l and Xj,l.
We now estimate the probabilities appearing in the last three lines of (5.18) which we denote
by p1, p2 and p3 respectively. By (2.30),
cap(ρn)(X[0, (Rn)2/In]) ≤R2
n/In · log−10 Rn + cap(X[−R2
n/In, R2
n/In], R2
n/In)
≤eCn/4I1/2
n
+ cap(X[−R2
n/In, R2
n/In], R2
n/In).
Thus, we have
p1 ≤2P 0[cap(ρn)(X[0, R2
n/In]) > eCn/2I1/2
n
]
≤2P 0[cap(X[0, (Rn)2/In], R2
n/In) > eCn/4I1/2
n
].
By Proposition 2.14 and the Markov inequality,
(5.19)
p1 ≤CE[cap(X[0, (Rn)2/In], R2
n/In)] · I1/2
n
/ eCn ≤CI−1/2
n
.
Obviously, p2 = p3, so we only bound the former from above. In the four-dimensional case, by
(2.30), we have
p2 ≤P xj,l[cap(X[R2
n/In, KR2
n] ∩B(V ′, Ln), R2
n) > 1
2
eCn/I1/2
n
];
while in the higher dimensional case, by the trivial upper bound of the ρn-capacity,
p2 ≤P xj,l
X[R2
n/In, KR2
n] ∩B(V ′, Ln)
> 1
2
eCn/I1/2
n
.
In short, we have
(5.20)
p2 ≤P xj,l[ξ > 1
2
eCn/I1/2
n
],
where
ξ =
(
cap(X[R2
n/In, KR2
n] ∩B(V ′, Ln), R2
n),
d = 4;
X[R2
n/In, KR2
n] ∩B(V ′, Ln)
,
d ≥5.
Note that since B(V ′, Ln) consists of the 2Ln-neighborhood of at most (4γ + 1)d · βkα
1 good
trajectories, |B(V ′, 2Ln)| ≤CR2
nLd−2
n
I1/3
n
. Thus, by (5.20) and Lemma 5.6,
(5.21)
p2 ≤P xj,l[Y > 1
2
eCn/I1/2
n
] ≤2I1/2
n
/ eCn · Exj,l[Y ]
≤CI1/2
n
/ eCn · R2
nLd−2
n
I1/3
n
· (R2
n/In)1−d/2(1 + log Rn · 1d=4)−1 ≤CI−1/6
n
.
28
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Combining (5.18), (5.19) and (5.21) gives
(5.22)
P[cap(ρn)(bζj,l\ζj,l) > 4 eCn/I1/2
n
|Fm,i, G] ≤4r/N
for all large n.
To estimate the terms on the right-hand side of (5.17), we write for all y, z ∈Zd
Φ(y, z) = Ey,z[φ(ρn)(X1[0, K(Rn)2], X2[0, K(Rn)2])].
By Lemma 5.7,
(5.23)
sup
y,z∈Zd Φ(y, z) ≤
(
CRn,
d ≥5;
CR2
n log−2 Rn,
d = 4.
Therefore, we have for all large n,
(5.24)
P[φ(ρn)(ζj1,l1, ζj2,l2) > eCn/In|Fm,i, G]
≤P[φ(ρn)(ζf
j1,l1, ζf
j2,l2) + φ(ρn)(ζf
j1,l1, ζb
j2,l2) + φ(ρn)(ζb
j1,l1, ζf
j2,l2)
+ φ(ρn)(ζb
j1,l1, ζb
j2,l2) > eCn/In|Fm,i, G]
≤In/ eCn · E[Φ(ζj1,l1(t), ζj2,l2(t)) + Φ(ζj1,l1(t), ζj2,l2(−t)) + Φ(ζj1,l1(−t), ζj2,l2(−t))
+ Φ(ζj1,l1(t), ζj2,l2(−t))|Fm,i, G] + 2r/N
(5.23)
≤I−1
n
+ 2r/N < 4r/N.
Here, t = ⌊R2
n/In⌋. By (5.16), (5.17), (5.22) and (5.24),
P[(E4 ∩E5)c|Fm,i, G] ≤|{(j, l) : 0 ≤j ≤s, 1 ≤l ≤Nj}|2 · 4r/N
≤(βkα
1 )2 · 4r/N = r/8 < r/4.
This finishes the proof.
□
We now turn to Proposition 5.2. Recall the symbol P S
Y for the law of the branching process Y
from the beginning of this section. By simply modifying the definition of Good(x, i), we obtain
the following analogue of Proposition 5.4.
Proposition 5.8. Assume that ℵ1
m = x ̸= ∆. Arbitrarily fix β, then there exists k1 large and a
constant C∗
8 = C∗
8(β, k1, r), such that for all θ1 < C∗
8 and all large n, there exists a coupling Q of
(Lx,i)1≤i≤α with conditional law on Fm,0 and (Yi)i≥0 with law P Sx satisfying the following. Let
F∗
x,i = σ{Fx,i, Yj, 1 ≤j ≤i}, and Good∗(x, i) = Good(x, i) ∩{Lx,j = Yj, j = 1, ..., i}, then on
Good∗(x, i),
Q[Good∗(x, i + 1)|F∗
m,i] > 1 −r
for all i ∈{0, 1, ..., α −1}.
Repeating the proof of Proposition 5.1, but with Proposition 5.8 applied iteratively (instead
of Proposition 5.4), we readily obtain Proposition 5.2.
5.2. Finding seeds for neighboring vertices. This subsection is dedicated to the proof of
Proposition 5.3. Recall β from (4.24), which is the number of trajectories in a seed (c.f. Definition
4.26), and the definition of events (4.25) and (4.26). Assuming goodness of the exploration at
x ∈Ln, we only need to show the existence of at least β trajectories in Lx,α that stay inside
Bx+Rn·e for all e ∈Ud ∪(−Ud). Moreover, by Proposition 5.2, we can replace Lx,α with Yα in the
last statement. Therefore, it suffices to show:
Proposition 5.9. There exists C9 > 0 such that for an arbitrary θ1 < C9, the following holds.
For all r > 0, there exists a large constant C10 = C10(r), such that for all β > C10 and S ⊂W [0,∞)
composed of β good trajectories staying inside [0, Rn)d and any e ∈Ud ∪(−Ud), the event
Ee := {Yα contains β good trajectories staying inside Bn
Rn·e}
happens with probability at least 1 −r if (Yi)i≥0 ∼P S
Y .
29
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Proof of Proposition 5.3 assuming Proposition 5.9. Recall C5, C6 from Proposition 5.3 and C3, C4
from Proposition 5.1. Take C5(r) = C10(r/8d) and C6(β, r) = C4(β, r/2) ∧C9. Let
Seed′(ℵ1
m) =
\
e∈Ud∪(−Ud)
{∃β good trajectories ∈Lℵ1m,α staying inside ℵ1
m + Rn · e + [0, Rn)d},
and let Seed′′(ℵ1
m) be the above event with Lℵ1m,α replaced by Yα. Take k1 > C3(β, r/2), β > C5
and θ1 < C6, then by Proposition 5.1,
(5.25) P[Seed(ℵ1
m)|Fm,0] ≥P[Seed′(ℵ1
m)|Fm,0] −P[Good(ℵ1
m)c|Fm,0] > P[Seed′(ℵ1
m)|Fm,0] −r/2.
By Proposition 5.2, for all large n.
(5.26)
P[Seed′(ℵ1
m)|Fm,0] ≥P Sℵ1m[Seed′′(ℵ1
m)] −r/4
Moreover, by Proposition 5.9,
(5.27)
P
Sℵ1m
Y
[Seed′′(ℵ1
m)] > 1 −2d · r/8d = 1 −r/4.
The proposition then follows from (5.25), (5.26) and (5.27).
□
The remainder of the subsection is devoted to proving Proposition 5.9. For simplicity, we always
assume (Yi)i≥0 ∼P S
Y from now on. We aim to obtain the “spatial distribution” of trajectories
in Yα by studying the spatial distribution of locations where the offspring trajectories hit their
parent. To this end, we define a process (Y i)i≥0 dominated by (Yi)i≥0 whose (i + 1)-th layer
is generated from the i-th layer by first sampling the hitting locations whose collection will be
denoted by Zi, and then sampling paths from them. Noting that (Zi)i≥0 form a branching process
with agents in Zd, we bound from below first the number of hitting points in Zα−1 that are close
enough to the center of each neighboring box Bx+Rn·e, then the conditional probability that the
trajectories stemming from those points stay inside Bx+Rn·e, which yields Proposition 5.9.
Now we make rigorous the aforementioned idea. For S ⊂W [0,∞), we construct two processes
(Y i)i≥0 ∈{0, 1}W [0,∞) and (Zi,η)i≥0,η∈Y i ∈{0, 1}Zd simultaneously under a new probability mea-
sure P
S. Let Y 0 = S. Assume that Y i is sampled, then for η ∈Y i, let Zi,η be an independent
PPP supported on range(η) ⊂Zd with density measure
X
x∈bη
u+
n (1 −ϵ/2) · eη(x) · δx,
sample one new trajectory ζx for each x ∈Zi,η independently according to distribution µx,η and
set Y i+1 = S
η∈Y i{ζx : x ∈Zi,η}. We write Zi = S
η∈Y i Zi,η.
By construction, (Y i)i≥0 is also a branching process with agents in W [0,∞). In general, (Y i)i≥0
“trims off” the excessive intensity of descendant trajectories introduced in (Yi)i≥0 due to the
finiteness of backward paths and highlights the “branching” nature of the process more explicitly
without introducing PPPs on W [0,∞). The following lemma reveals the relation between (Yi)i≥0
and (Y i)i≥0, whose proof is straightforward given Lemma 4.3.
Lemma 5.10. For large n, the following holds: for all S ⊂Tn, there exists a coupling Q between
(Yi)i≥0 and (Y i)i≥0 such that
Q[Yi ⊃Y i, 0 ≤i ≤α] = 1.
Proof of Lemma 5.10. To prove the domination for branching processs, it suffices to prove dom-
ination for their respective offspring distributions. Obviously, the offspring distribution of the
branching process (Y i)i≥0 at η ∈W [0,∞) is given by a PPP on W [0,∞) with intensity measure
π1 = u+
n ·
X
x∈bη
(1 −ϵ/2)eη(x)µx,η.
By the definition of (Yi)i≥0, its offspring distribution at η is given by a PPP with intensity
π2 = u+
n ·
X
x∈bη
X
m,l∈N
ρn(m + l)
µ1(ρn) + 1P x[ eHη > m]µ(x, m, l; range(η))[·; Tn] ≥u+
n ·
X
x∈bη
eη(x)µ∗
x,η[·; Tn].
By Lemma 4.3, µ∗
x,η[·; Tn] ≥(1 −ϵ/8)µx,η, so π1 ≥π2, which leads to the conclusion.
□
30
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
For the aforementioned estimate regarding Zα−1 we are to establish, We define two correlated
new processes (Ki)i≥0 ∈Zd and (ζi)i≥0 ∈W [0,∞), and prove a LCLT-type result for (Ki)i≥0. The
connection between these processes and (Zi)i≥0 will be elaborated shortly afterwards.
For η ∈W [0,∞), let
(5.28)
eη =
X
x∈bη
eη(x) · δx
be the equilibrium measure on range(η) restricted to bη. Recall that we denote by e0
η the probability
measure obtained by normalizing eη. Fix η ∈Tn. Let ζ0 = η. Assume that ζi is sampled, we
then choose Ki from range(ζi) according to e0
ζi, and then sample ζi+1 according to µKi,ζi. This
way, (Ki)i≥0 and (ζi)i≥0 are recursively defined. We denote the probability measure related to
(ζi, Ki) by Q
η. Using the LCLT, we are able to prove the following result:
Proposition 5.11. For all α ∈N in (4.22), there are constants C11 = C11(α), C12 = C12(α) such
that for all choices M > C11 and θ1, θ2 < C12, the following holds. There exists C13 = C13(k, K)
such that for all large n and all η ∈Tn staying inside [0, Rn)d,
Q
η[Kα−1 ∈eBRn·e] ≥C13α−d/2.
To see the connection between ((Ki)i≥0, (ζi)i≥0) and (Zi)i≥0, note the following fact: if we
condition on the number of descendants of every trajectory in the process (Y i)i≥0, then the
distribution of an arbitrary element of Zα−1 is the same as Kα−1 under Q
η if that element can
be traced back to η. In particular, for any A ⊂Zd, we have
(5.29)
E
S[|Zα−1 ∩A|] = E
η[|Zα−1|] ·
X
η∈S
Q
η[Kα−1 ∈A].
With the above observation and Proposition 5.11, the proof of Proposition 5.9 is straightforward,
which can be found at the end of this subsection.
We now aim to prove Proposition 5.11.
Note that it is impossible to apply the LCLT to
(Ki)i≥0 directly, because the distribution of Ki+1 depends on not only Ki but also information
offered by ζi. To overcome this obstacle, we couple ((ζi)i≥0, (Ki)i≥0) with its simplified version
((ζ□
i )i≥0, (K□
i )i≥0) such that (K□
i )i≥0 can be viewed as a sequence of points on a simple random
walk on Zd to which we apply the LCLT.
Recall the definition of µx, x ∈Zd. We define the processes ((ζ□
i )i≥0, (K□
i )i≥0) as follows. For
η ∈W [0,∞), let ζ□
0 = η.
Choose K□
0 from range(ζ□
0 ) according to e0
ζ□
0 .
Assume that K□
i
is
sampled, we sample ζ□
i+1 according to µK□
i , uniformly choose j from {0, 1, ..., T(ζ□
i+1)}, and let
K□
i+1 = ζ□
i+1(j) . We denote the probability measure related to ((ζ□
i )i≥0, (K□
i )i≥0) by eQη.
Remark 5.12. (K□
i )i≥0 can be equivalently constructed in the following way. Note that the incre-
ment K□
i+1 −K□
i , i ≥0 are i.i.d. and their distribution can be seen as that of Xτ, where τ is a
random time and X ∼P 0 is independent of τ. For m, l ∈N, let σ∗
m be the uniform distribution on
{0, 1, ..., m}, and let σm,l be the distribution of |ξ −m| where ξ ∼σ∗
m+l. Consider the probability
measure
σ = C ·
X
kR2n≤m+l≤KR2n
ρn(m + l) · σm,l
where C = P
kR2n≤m≤KR2n(m + 1)ρn(m) is a normalizing constant, then is not hard to verify that
σ is the distribution of τ mentioned above. Define the law of a simple random walk with starting
point chosen from range(η) according to its normalized equilibrium measure by
P η :=
X
x∈range(η)
e0
η(x) · P x.
Let X ∼P η and τi ∼σ, i ≥1 be mutually independent, and write Si = Pi
j=1 τj. Then, by
previous discussion, (XSi)i≥0 has the same distribution as (K□
i )i≥0 under eQη.
31
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Apart from the simplified processes ((ζ□
i )i≥0, (K□
i )i≥0), we further introduce a pair of auxiliary
processes ((ζ♢
i )i≥0, (K♢
i )i≥0). For η ∈W [0,∞), let ζ♢
0 = η. Assume that ζ♢
i
is sampled, we then
choose K♢
i from range(ζ♢
i ) according to e0
ζ♢
i , and then sample ζ♢
i+1 according to µK♢
i . We denote
the probability measure related to (ζ♢
i , K♢
i ) by Qη.
Remark 5.13. Similar to (ζ□
i )i≥0, (ζ♢
i )i≥0 becomes a random walk on Zd once ζ♢
0 is fixed.
Recall ln from Lemma 4.2. Recall parameters M, θ1 introduced in (4.5) and (4.6) respectively
for the definition of typical trajectories and θ2 introduced in (4.12) for the definition of *-proper
part. The major ingredient in the proof of Proposition 5.11 is the following two coupling results.
Lemma 5.14. For all α ∈N and r ∈(0, 1), there exist constants C14 = C14(α, r) and C15 =
C15(α, r), such that for all M > C14 and θ1, θ2 < C15, all large n and all η ∈Tn, we can couple
(Ki) with (K♢
i ) such that d(Ki, K♢
i ) < (i+1)·2ln, i = 0, 1, ..., α−1 with probability at least 1−r,
where ((ζi)i≥0, (Ki)i≥0) (resp. ((ζ♢
i )i≥0, (K♢
i )i≥0)) follows the law under Q
η (resp. Qη).
Proof. We recursively construct a coupling Q satisfying
(5.30)
Q[K0 = K♢
0 ] > 1 −r/α
and
(5.31)
Q[d(Ki+1 −Ki, K♢
i+1 −K♢
i ) < 2ln | Fi]
for all 0 ≤i < α −1, where
Fi = σ{(ζj)0≤j≤i, (ζ♢
j )0≤j≤i, (Kj)0≤j≤i, (K♢
j )0≤j≤i},
and obtain d(Ki, K♢
i ) < i · 2ln by the triangle inequality. To start with, for i0 < α −1, we
assume that we have coupled
(ζj)0≤j<i0, (Kj)0≤j<i0
with
(ζ♢
j )0≤j<i0, (K♢
j )0≤j<i0
such that
we have (5.30) and (5.31) for 0 ≤i < i0, and construct ζi0+1, Ki0+1, ζ♢
i0+1, K♢
i0+1 satisfying (5.31)
for i = i0. As alluded to in the above remark, the distribution of K♢
i0+1 −K♢
i0 is irrelevant to K♢
i0.
Therefore, we can assume without loss of generality that K♢
i0 = Ki0, and it suffices to prove the
following result in a simpler setup: there exist constants C14(α, r) and C15(α, r) such that for all
M > C14, θ1, θ2 < C15 and all η ∈Tn, x ∈bη, there exists a coupling Q of random trajectories
ζ♢∼µx, ζ ∼µx,η, and there are K♢∼e0
ζ♢under Q[·|ζ♢] and K∗∼e0
ζ under Q[·|ζ] such that
(5.32)
Q[d(K, K♢) > 2ln] < r/α.
By the definition of µ∗
x,η and µx and Lemma 4.2, for sufficiently small θ2 and θ1, we can couple
ζ1 ∼µ∗
x,η with ζ♢∼µx such that with probability at least 1 −r/8α,
(5.33)
T(ζ1) = T(ζ♢) and ζ1(t∗) = ζ♢(t∗) = x
almost surely, where t∗= τ ζ1
η
(c.f. (2.2)), and
(5.34)
ζ1[0, (t∗−tn) ∨0] ≃ζ♢[0, (t∗−tn) ∨0], ζ1[t∗, T(ζ1)] ≃ζ♢[t∗, T(ζ♢)]
and |ζ1(s) −ζ♢(s)| ≤ln, ∀0 ≤s ≤T(ζ1).
Denote the event that (5.33) and (5.34) both hold by E(ζ1, ζ♢).
Similar to (4.17), there exist constants C∗
1(α, r) and C14(α, r) such that for all θ1, θ2 < C∗
1 and
M > C14, a trajectory following µx or µ∗
x,η is typical with probability at least 1−r/8α as n →∞,
which implies that dTV(µx,η, µ∗
x,η) < r/8α. In that case, the coupling mentioned in the previous
paragraph still works if we replace ζ1 ∼µ∗
x,η by ζ ∼µx,η, and the similarly defined event E(ζ, ζ♢)
happens with probability at least 1 −r/4α.
Define an event
E′ := E(ζ, ζ♢) ∩{cap(ζ♢) > (1 −θ2
1)hnT(ζ♢)},
where
hn = εd(1 + log µ1(ρn)1d=4)−1.
32
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Then, by (2.12), Q[E′] ≥1 −r/2α for all large n. Now, on E′, we aim to couple K and K♢
conditionally on ζ and ζ♢. Let T = T(ζ) = T(ζ♢). Write t∗= t(ζ, η), ζ−= ζ[0, (t∗−tn) ∨0],
ζm = ζ[(t∗−tn)∨0, t∗] and ζ+ = ζ[t∗, T]. Define e∗
ζ = eζ−+eζ+, and let e∗∗
ζ be the restriction of eζ
to range(ζ−) ∪range(ζ+). Define ζ♢
±, ζ♢
m, e∗
ζ♢and e∗∗
ζ♢similarly. Note that ζ is always typical, so
by definition, ζ ∈E4 can be divided into pieces of length T ′
n(see 4.7 for definition) with capacity
no larger than (1 + θ2
1)hnT ′
n, while at the same time we have cap(ζ) ≥(1 −θ2
1)hnT by ζ ∈E2. If
we divide ζ−and ζ+ into the aforementioned pieces of length T ′
n, at most one piece may be used
twice, so the total length of all the pieces used in both coverings is at most T + T ′
n. Hence, we
have
cap(ζ−) + cap(ζ+) ≤(1 + θ2
1)hn(T + T ′
n),
implying
(5.35)
cap(ζ−) + cap(ζ+) −cap(ζ) ≤(1 + θ2
1)hn(T + T ′
n) −(1 −θ2
1)hnT < Cθ2
1 eCn.
Now, we claim that conditionally on ζ, ζ♢with restriction on E′, we can couple K∗and K♢such
that
(5.36) Q[K∗= K♢∈range(ζ−) or K∗−K♢= ζ(t∗)−ζ♢(t∗) | ζ, ζ♢] > 1−C(θ1+θ2) > 1−r/2α,
where the second inequality holds for θ1, θ2 small. Then, (5.32) follows since Q[E′] > 1 −r/2α.
Now we aim to prove (5.36). Recall the restriction of equilibrium measure eζ from (5.28). Note
the following inequalities:
(5.37)
dTV(eζ, eζ) ≤C(θ1 + θ2) eCn;
dTV (eζ, e∗∗
ζ ) ≤tn;
dTV(e∗∗
ζ , e∗
ζ) = cap(ζ−) + cap(ζ+) −eζ(ζ−∪ζ+)
≤
cap(ζ−) + cap(ζ+) −cap(ζ)
+ cap(ζm)
≤Cθ2
1 eCn + tn.
Here, the first inequality holds, since by the definition of bζ,
dTV(eζ, eζ) = eζ(range(ζ) \ bζ) ≤
(
θ2T(ζ),
d ≥5;
θ2T(ζ) log−1 µ1(ρn) + θ−1
1
cap(ζ, R2
n) −cap(ζ)
,
d = 4
which, combined with the requirements of E1 and E2 in the definition of typical trajectories, yields
(5.38)
dTV(eζ, eζ) = eζ(range(ζ) \ bζ) ≤C(θ1 + θ2) eCn
Summing up the inequalities aligned in (5.37), we get
dTV(eζ, e∗
ζ) < C(θ1 + θ2) eCn
since θ1 < 1 and tn ≪eCn. Note that
e∗
ζ(range(ζ)) = cap(ζ−) + cap(ζ+) ≥cap(ζ) −tn ≥Ck eCn
for some constant C and all large n. Hence,
(5.39)
dTV(e0
ζ, (e∗
ζ)0) = dTV(e0
ζ,
1
e∗
ζ(ζ) · eζ) +
1
e∗
ζ(ζ) · dTV(eζ, e∗
ζ) < C(θ1 + θ2),
where the constant C = C(d, k). Similarly,
(5.40)
dTV(e0
ζ♢, (e∗
ζ♢)0) < C(θ1 + θ2).
Thus, we can couple K∗with Ka ∼(e∗
ζ)0 and K♢with Kb ∼(e∗
ζ♢)0 such that the probability
of {K∗= Ka} and {K♢= Kb} given ζ, ζ♢are both at least 1 −C(θ1 + θ2). (5.36) then follows
since ζ−≃ζ♢
−and ζ+ = ζ♢
+, which yields that Ka and Kb can be coupled so that
Ka = Kb ∈range(ζ−) or Ka −Kb = ζ(t∗) −ζ♢(t∗)
almost surely.
33
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
It remains to construct K0 and K♢
0 satisfying (5.30). To see this, note that η ∈Tn, and by a
similar argument as above we have
(5.41)
dTV(e0
η, e0
η) < C(θ1 + θ2) < r/α
for all small θ1, θ2. To conclude the proof, note that we can pick some C13(α, r) ∈(0, C∗
1(α, r))
such that whenever θ1, θ2 < C13, (5.36) and (5.41) both hold.
□
Lemma 5.15. Arbitrarily choose α ∈N and r ∈(0, 1). Then, for all large n and all η ∈W [0,∞),
we can couple (K♢
i )i≥0 with (K□
i )i≥0 such that d(K□
α−1, K♢
α−1) < Rn/16, i = 0, 1, ..., α −1 with
probability at least 1 −r, where ((ζ□
i )i≥0, (K□
i )i≥0) (respectively, ((ζ♢
i )i≥0, (K♢
i )i≥0)) follows the
law under eQη (respectively, Qη).
Proof. We recursively construct a coupling Q such that
(5.42)
Q[K♢
0 = K□
0 ] = 1,
and
(5.43)
Q[d(K♢
i+1 −K□
i , K♢
i+1 −K♢
i ) < Rn/16α | Fi] > 1 −r/α
for all i ≥0, where
Fi = σ{(ζ□
j )0≤j≤i, (ζ♢
j )0≤j≤i, (K□
j )0≤j≤i, (K♢
j )0≤j≤i}.
(5.42) is trivial because K♢
0 and K□
0 are identically distributed. Similar to Lemma 5.14, we work
in a simpler setup to obtain (5.43). Let ζ♢, ζ□∼µ0. Conditionally on ζ♢, choose K♢according
to e0
ζ♢; conditionally on ζ□, let K□= ζ□(j), where j is uniformly chosen from {0, 1, ..., T(ζ□)}.
In order to obtain (5.43), it suffices to construct a coupling Q such that
(5.44)
Q[d(K♢, K□) ≥Rn/16α] < r/α
for all large n. First, we take ζ♢= ζ□= ζ ∼µ0, and let
ξj = ζ[jT ′′, ((j + 1)T ′′ −1) ∧T(ζ)], 0 ≤j ≤N1 = ⌊T(ζ)/T ′′⌋,
where T ′′ = ⌊δR2
n⌋and δ > 0 is a small but fixed number to be specified later. Recall hn from
the proof of Lemma 5.14, and let E = E1 ∩E2 where
E1 = {cap(ξj) ≤(1 + r/16αN1)hnT ′′, ∀0 ≤j ≤N1
and cap(ζ[0, N1T ′′]) ≥(1 −r/16αN1)hnN1T ′′},
E2 = {diam(ξj) < Rn/16α, ∀0 ≤j ≤N1}.
By Proposition (2.12), Q[Ec
1] < r/4α for all large n. By a standard Gaussian estimate and union
bound, there are constants C = C(d, K) > 0 and C′ = C′(d, α) > 0 such that
lim sup
n→∞Q[Ec
2] ≤Cδ−1 exp
−C′δ−1/2
.
Thus, we can take δ small such that Q[E] > 1 −r/2α for all large n.
On E, we construct a coupling between K♢and K□conditionally on ζ.
Let K′ ∼e′
ζ :=
PN1−1
i=0
eξi
0
, then we can sample K′ as follows. First, randomly choose j′ ∈{0, 1, ..., N1 −1}
according to the probability measure proportional to PN1−1
i=0
cap(ξi) · δi, then choose K′ from
range(ξj′) according to e0
ξj′. We can similarly sample K□by taking j□∈{0, 1, ..., N1} according
to PN1
i=0
T(ξi)
T(ζ) ·δi and then letting K□= ξj□(t), where t is uniformly chosen from {0, 1, ..., T(ξj□)}.
Note that on E,
eζ(ξi) ≤cap(ξj) ≤(1 + r/16αN1)hnT ′′
for all 0 ≤i ≤N1 and
eζ(ξi) ≥cap(ζ[0, N1T ′′]) −
X
j̸=i,j<N1
cap(ξj) ≥(1 −r/8α)hnT ′′
34
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
for all 0 ≤i < N1. Therefore, for all small δ,
(5.45)
dTV(eζ,
N1−1
X
i=0
eξi) ≤
N1
X
i=0
cap(ξi) −cap(ζ[0, (N −1)T ′′]) ≤
r
8αN1
hnT(ζ) + 2hnT ′′ < 1
8rα−1hnT(ζ)
and
(5.46)
N1−1
X
j=0
|cap(ξj) −hnT ′′| + hnT ′′ ≤1
8rα−1hnT(ζ) + hnT ′′ ≤3
16rα−1hnT(ζ).
The total mass of eζ is cap(ζ) > (1 −r/16αN1)hnN1T ′′ > 7
8hnT(ζ) when δ is small so that
N1 > 1. Thus, by an argument similar to (5.39), we obtain
(5.47)
dTV
e0
ζ, e′
ζ
< r/4α
and
(5.48)
N1
X
j=0
sj −T(ξj)
T(ζ)
< r/4α.
Here
sj =
cap(ξj)
PN1−1
i=0
cap(ξi),
0 ≤j < N1;
0,
j = N1
is the probability that j′ = j while we sample K′ in the aforementioned way, and T(ξj)
T(ζ) is the
probability that j□= j. This indicates that we can couple j′, j□such that j′ = j□with probability
at least 1 −r/4α. Thus, by (5.45) and (5.46), there exists a coupling between K♢and K′ such
that K♢= K′ with probability at least 1 −r/4α, and one between K′ and K□such that they
are both in the range of some ξj with probability larger than 1−r/4α. Hence, we can couple K♢
and K□such that
Q[∃0 ≤j ≤N1 s.t. K♢, K□∈range(ξj) | ζ] > 1 −r/2α.
(5.44) then follows because Q[E] > 1−r/2α and diam(ξj) < Rn/16α for all 0 ≤j ≤N on E.
□
We now present the proof of Proposition 5.11.
Proof of Proposition 5.11. For all µ ∈W [0,∞) staying inside Bn
0 , let X, τi, Si, i ≥1 be as in
Remark 5.12. Apparently, we have
P
τi ≥1
4kR2
n
≥1
4
and τi ≤KR2
n almost surely, i ≥1. Therefore, 1
4kR2
n ≤Sα−1 ≤αKR2
n happens with probability
at least
1
4.
Let B1 := B(⌊Rn/2⌋· e, Rn/16) and B2 := B(⌊Rn/2⌋· e, Rn/8) be two smaller
concentric boxes inside eBRn·e. By [15, Theorem 1.2.1], there exists C13 = C13(k, K) such that
eQη[K□
α−1 ∈B1] = P[XSα−1 ∈B1] ≥1
4 ·
inf
kR2n/4≤j≤αKR2n
P[Xj ∈B1] ≥2C13α−d/2
for all large n. Recall C14, C15 from Lemma 5.14, and take C11(α) = C14(α, r) and C12(α) =
C15(α, r), where r = 1
2C13α−d/2. Thus, for any M > C11, θ1, θ2 < C12 and all large n, by Lemma
5.14,
d(Kα−1, K♢
α−1) ≤α · 2ln < Rn/8.
Moreover, by Lemma 5.15, d(K♢
α−1, K□
α−1) < Rn/16 for all large n. Therefore,
Q
η[Kα−1 ∈eBRn·e] ≥Qη[K♢
α−1 ∈B2] −C13α−d/2/2 ≥eQη[K□
α−1 ∈B1] −C13α−d/2 ≥C13α−d/2,
finishing the proof.
□
Finally, we prove Proposition 5.9 using Proposition 5.11.
35
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Proof of Proposition 5.9. Recall the function f from Lemma 2.1, and let α be large such that
(5.49)
1
2f(256K) · k
2εd
C13 · (1 + ϵ/8)αα−d/2 > 10.
Recall the constants C11, C12 from Proposition 5.11, C9 from Proposition 5.9 and C1, C2 from
Lemma 4.3. Having fixed α, we now take C9 = C12(α) ∧C1, and fix
(5.50)
M > C11(α) ∨C2 and θ2 < C1 ∧C12(α).
Then, by Proposition 5.11, for all large n and η ∈S,
(5.51)
Q
η[Kα−1 ∈eBn
Rn·e] ≥C13α−d/2.
For L > 0, recall that WL is the subset of W [0,∞) with diameter at most L. By Lemma 2.1, the
following holds: for all x ∈Zd,
(5.52)
µx(WRn/16) > f(256K).
This lower bound, together with Lemma 4.1, implies
(5.53)
µx,η(WRn/8) ≥1
2f(256K).
Let ξ = |{η ∈Y α : η stays within e + [0, Rn)d}|, we show that ξ > β with high probability. We
first estimate the first moment. Applying (5.29) and (5.53) in turn gives
(5.54)
E
S[ξ] ≥P
S[|Zα−1 ∩eBRn·e|] · 1
2f(256K)
≥P
S[|Zα−1|] · C13α−d/2 · 1
2f(256K)
≥kβ
2εd
(1 + ϵ/8)α · C13α−d/2 · 1
2f(256K) > 10β.
The third inequality is a consequence of Lemma 4.3, while the last one is due to (5.49). Moreover,
we have the following control over the variance of ξ. For W ⊂Tn, let varW denote the variance
under P
W , and abbreviate varη when W = {η}.
For i ≥0, since Y i only contains typical
trajectories whose capacities are well controlled, it is easy to verify that conditioning on Y i, the
number of offspring of each element of Y i follows a Poisson distribution with parameter smaller
than λ = 2K. Thus, by induction, we have for all η ∈Tn,
varη(ξ) ≤E
η[ξ2] ≤E
η[|Y α|2] ≤
2(λ2 + λ)
α.
Therefore,
(5.55)
varS(ξ) =
X
η∈S
varη(ξ) ≤β
2(λ2 + λ)
α.
By (5.54), (5.55) and the Markov inequality,
P
S[ξ < β] ≤P
S h ξ −E
S[ξ]
> 9β
i
≤9−2β−2varS(ξ) ≤9−2β−1[2(λ2 + λ)]α.
The conclusion then follows by taking β large.
□
Acknowledgments
We warmly thank Eviatar Procaccia for fruitful discussions. YB and XL are supported by
National Key R&D Program of China (No. 2020YFA0712900 and No. 2021YFA1002700). BR is
partially supported by the grant NKFI-FK-142124 of NKFI (National Research, Development and
Innovation Office), and the ERC Synergy Grant No. 810115 - DYNASNET. YZ is supported by
NSFC-12271010, National Key R&D Program of China (No. 2020YFA0712902) and the Funda-
mental Research Funds for the Central Universities and the Research Funds of Renmin University
of China 24XNKJ06.
36
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Appendix A. Proofs of technical lemmas
In this section, we present the proofs of Lemmas 4.3, 5.6 and 5.7. For the proof of Lemma
4.3, we introduce the following analog of Lemma 4.1 for random walk paths whose law involve
conditioning.
Lemma A.1. Let ψ be the same as in Lemma 4.1 and take η ∈Tn, x ∈bη and m, l ∈N satisfying
kR2
n ≤m + l ≤KR2
n. Then, for all large n,
µ(x, m, l; range(η))[T c
n ] < ψ
(M −1)/K1/2
+ 4θ1.
Proof. By Lemma 4.2, we can construct a coupling Q between ζ ∼µ(x, m, l) and
ζ∗∼µ(x, m, l; range(η)) such that the event
E = {ζ[0, m −tn] ≃ζ∗[0, m −tn], ζ[m, m + l] ≃ζ∗[m, m + l], |ζ(s) −ζ∗(s)| ≤ln, ∀0 ≤s ≤m + l}
happens with probability at least 1 −2θ1 −R−c/2
n
. Taking advantage of the coupling, we have
µ(x, m, l; range(η))[Ec
1] = Q[range(ζ∗) ̸⊂B(x, MRn)]
≤Q[E ∩{range(ζ∗) ̸⊂B(x, MRn)}] + 2θ1 + R−c/2
n
≤Q[range(ζ) ̸⊂B(x, MRn −ln)] + 2θ1 + R−c/2
n
≤µ(x, m, l)[E1(k, K, M −1)c] + 2θ1 + R−c/2
n
.
It then follows from (4.8) that
(A.1)
µ(x, m, l; range(η))[Ec
1] ≤3
4ψ
(M −1)/K1/2
+ 2θ1
for all large n. Likewise, when d = 4, since on the event E,
cap(ζ∗[jT ′
n, (j + 1)T ′
n ∧T(ζ∗)]) ≤cap(ζ[jT ′
n, (j + 1)T ′
n ∧T(ζ)]) + tn
≤cap(ζ[jT ′
n, (j + 1)T ′
n ∧T(ζ)]) + 1
2θ2
1 · π2
8 T ′
n log−1 µ1(ρn),
and
diam(ζ∗[jT ′
n, (j + 1)T ′
n ∧T(ζ∗)]) ≤diam(ζ[jT ′
n, (j + 1)T ′
n ∧T(ζ)]) + ln
≤diam(ζ[jT ′
n, (j + 1)T ′
n ∧T(ζ)]) + 1
4T 1/2 log−c µ1(ρn)
for all j ∈N and all large n, we obtain
(A.2)
µ(x, m, l; range(η))[Ec
4]
≤Q
h
∃j ∈N, cap
ζ[jT ′
n, (j + 1)T ′
n ∧T(ζ)]
> (1 + 1
2θ2
1)π2
8 T ′
n log−1 µ1(ρn)
or diam
ζ[jT ′
n, (j + 1)T ′
n ∧T(ζ)]
> 1
4T 1/2 log−c µ1(ρn)
i
+ 2θ1 + R−c/2
n
≤C log−1 Rn + 2θ1 + R−c/2
n
,
where we have estimated the probability appearing in the second and third line similarly as in
(4.10). For d ≥5, we can get the same upper bound in a similar fashion. Finally, since x ∈bη,
µ(x, m, l)[E] ≥θ2 log−1 µ1(ρn)
for all large n, where
E = {η′ ∈W [0,∞) : η′[0, m) ∩range(η) = ∅}.
Thus, by (4.9) and (4.11),
(A.3)
µ(x, m, l; range(η))[Ec
2 ∪Ec
3] ≤µ(x, m, l)[Ec
2 ∪Ec
3]
µ(x, m, l)[E]
≤
C log−2 Rn + C log8c/3 Rn exp
−log2c/3 Rn
log−1 Rn
≤C log−1 Rn
for all large n. The conclusion follows from (A.1), (A.2) and (A.3).
□
37
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Proof of Lemma 4.3. Recall the notation µ(m)
2
(ρn), m > 0 which is the tail of the second moment
of ρn from Section 2. Since (ρn)n≥0 is an appropriate family, by (1.2), we can choose K > 1 large
such that
(A.4)
µ(KR2
n)
2
(ρn) ≤1
32ϵµ2(ρ)
for all n ∈N.
Fix a K such that the above estimate holds, and arbitrarily fix k satisfying
k2 < ϵ/32. For all θ1 < ϵ/64K and large M so that ψ((M −1)/K1/2) < ϵ/16K, we obtain by
Lemmas 4.1 and A.1 that
µx(Tn) ∧µ∗
x,η(Tn) > 1 −ϵ/8K > 1 −ϵ/8.
Now we aim at estimating eζ(bζ). By the fact that ζ ∈Tn ⊂E2,
(A.5)
cap(ζ) ≥(1 −θ2
1)εdT(ζ)(1 + log µ1(ρn)1d=4)−1.
Let ζ∗∼µ∗
x,η, then by (A.4) and k2 < ϵ/32,
E[T(ζ∗)] =
P
kR2n≤m≤KR2n m2ρn(m)
P
kR2n≤m≤KR2n(m + 1)ρn(m) ≥µ2(ρn) −µ(KR2
n)
2
(ρn) −(kR2
n)2
µ1(ρn) + 1
≥µ2(ρn) −1
32ϵµ2(ρn) −k2µ2(ρn)
µ1(ρn) + 1
≥(1 −ϵ/16)µ2(ρn)
µ1(ρn).
Thus, we obtain
E[T(ζ)] ≥E[T(ζ∗)1ζ∗∈Tn] ≥E[T(ζ∗)] −
ϵ
8K · KR2
n
≥(1 −ϵ/16)µ2(ρn)
µ1(ρn) −1
8ϵµ2(ρn)
µ1(ρn) = (1 −3
16ϵ)µ2(ρn)
µ1(ρn),
where in the second inequality we have used µ∗
x,η(T c
n ) < ϵ/8K and the fact that T(ζ∗) ≤KR2
n
almost surely. The above inequality and (A.5) together imply
(A.6)
E[cap(ζ)] ≥
(1 −θ2
1)(1 −3
16ϵ)π2
8
µ2(ρn)
µ1(ρn) log−1 µ1(ρn),
d = 4;
(1 −θ2
1)(1 −3
16ϵ)εd
µ2(ρn)
µ1(ρn),
d ≥5.
Noting that eζ(bζ) = cap(ζ) −eζ(range(ζ) \ bζ), the estimate (4.18) holds true for sufficiently small
θ2 and θ1 due to (A.6) and (5.38).
□
Now we turn to the lemmas of Section 5.
Proof of Lemma 5.6. We begin with the easier high-dimensional case. By the LCLT [15, Theorem
1.2.1], there is some C = C(d) > 0 such that
P y[Xj = z] < Cj−d/2
for all j > 0, z ∈Zd. Hence, we have
Ey h X[T, KR2
n] ∩A
i
≤Ey[
∞
X
j=T
1Xj∈A] ≤C
∞
X
j=T
|A|j−d/2 ≤C|A|T 1−d/2.
Now, we assume d = 4.
We divide the trajectory X[T, KR2
n] into pieces of length T ′
n =
⌊R2
n log−1 Rn⌋, and write ξj = X[T + (j −1)T ′
n, T + jT ′
n], xj = ξj(0) for all j ≤⌈KR2
n/T ′
n⌉. Note
that xj and cap(range(ξj) ∩A, R2
n) are independent, so we have
Ey[cap(range(ξj) ∩A, R2
n)]
= Ey[cap(range(ξj) ∩A, R2
n)1xj∈B(A,Ln)] + Ey[cap(range(ξj) ∩A, R2
n)1xj /∈B(A,Ln)]
≤Ey[cap(range(ξj), R2
n log−1 Rn)] · P y[xj ∈B(A, Ln)] + Ey[T ′
n1ξj̸⊂B(xj,Ln)1xj /∈B(A,Ln)]
≤CR2
n log−2 Rn · P y[xj ∈B(A, Ln)] + T ′
n · P 0[X[0, T ′
n] ̸⊂B(0, Ln)]
≤CR2
n log−2 Rn|B(A, Ln)|(T + jT ′
n)−2 + T ′
n exp
−L2
n/2T ′
n
,
38
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
where in the last inequality we have used the LCLT and a standard estimate on the transition
probability of the simple random walk. Take a summation over j ∈{1, ..., ⌈KR2
n/T ′
n⌉}, and we
obtain
Ey[cap(X[T, KR2
n] ∩A, R2
n)]
≤CR2
n log−2 Rn|B(A, Ln)|(T ′
n)−2
∞
X
j=1
(T/T ′
n + j)−2 + CR2
n exp
−1
2 log1−2c Rn
≤C|B(A, Ln)|T −1 log−1 Rn + CR2
n exp
−1
2 log1−2c Rn
.
It remains to show that the second term in the last line is much smaller than the first term. This
follows from the trivial bound
|B(A, Ln)| ≥L4
n = R4
n log−4c Rn.
Now, we have proved the four-dimensional case.
□
Proof of Lemma 5.7. When d ≥5, we have
Ey,z[φ(ρn)(X1[0, K(Rn)2], X2[0, K(Rn)2])]
≤Ey,z
KR2
n
X
j,l=0
g(X1
j , X2
l )
=
X
y1,z1∈Z2
KR2
n
X
j=0
pj(y, y1)
g(y1, z1)
KR2
n
X
j=0
pj(z1, z)
=
X
0≤t1,t2≤KR2
n
s≥0
X
y1,z1∈Zd
pt1(y, y1)ps(y1, z1)pt2(z1, z) =
X
0≤t1,t2≤KR2
n
s≥0
pt1+t2+s(y, z)
=
KR2
n
X
j=0
j2pj(y, z) + K2R4
n
X
j>KR2n
pj(y, z) ≤C
KR2
n
X
j=0
j2−d/2 + CR4
n
X
j>KR2n
j−d/2.
The conclusion then follows from direct computation of the series.
When d = 4, we proceed as in the last lemma, and let ξi
j = Xi[(j −1)Ln, jLn], xi
J = ξi
j(0) for
i = 1, 2, j > 0. For U, V ⊂Z4, write
eg(U, V ) =
sup
u′∈B(U,L1/2+c
n
),v′∈B(V,L1/2+c
n
)
g(u′, v′),
and abbreviate eg(u, v) = eg({u}, {v}). Note that by (2.14) and (2.30),
E0[cap(ρn)(X[0, Ln])] ≤Ln log−10 Rn + E0[cap(X[0, Ln], R2
n)]
≤Ln log−10 Rn + E0[cap(X[0, Ln], Ln)] ≤CLn log−1 Rn
for some C > 0. Thus, conditioning on x1
j, x2
l , such that |x1
j −x2
l | > 3L1/2+c
n
, we have
(A.7)
Ey,z[φ(ρn)(ξ1
j , ξ2
l )|x1
j, x2
l ]
≤Ey,z[cap(ρn)(ξ1
j )cap(ρn)(ξ2
l )]eg(x1
j, x2
l ) + 2P 0[X[0, Ln] ̸⊂B(0, L1/2+c
n
)] · L2
n
≤E0[cap(ρn)(X[0, Ln])]2eg(x1
j, x2
l ) + CL2
n exp
−1
2L2c
n
≤CL2
n log−2 Rn · g(x1
j, x2
l ) + CL2
n exp
−1
2L2c
n
.
Moreover, by the same argument as in the high-dimensional case, we obtain
(A.8)
Ey,z[φ(ρn)(ξ1
j , ξ2
l )|x1
j, x2
l ] ≤CLn.
39
Bi, Cai, Li, R´ath and Zhang
FRI critical threshold in 4 and higher dimensions
Combining (A.7) and (A.8),
(A.9)
Ey,z[φ(ρn)(ξ1
j , ξ2
l )] ≤CL2
n log−2 Rn · Ey,z[g(x1
j, x2
l )] + CL2
n exp
−1
2L2c
n
+ CLnP y,z[|x1
j −x2
l | ≤3L1/2+c
n
].
Now, we bound the terms on the right-hand side of (A.9) from above. On the one hand, note
that the following inequalities hold if j, l ≤KR2
n/Ln, j + l > Rn log−2+c Rn:
P y,z[|x1
j −x2
l | ≤3L1/2+c
n
] ≤
CL2+4c
n
(R2n log−2 Rn)2 ≤CR−2+4c
n
log4 Rn.
Thus, we have
(A.10)
X
j,l≤KR2
n/Ln
j+l>Rn log−2+c Rn
P y,z[|x1
j −x2
l | ≤3L1/2+c
n
] ≤CR−2+4c
n
log4 Rn · R4
nL−2
n
≤CR4c
n log4+2c Rn.
On the other hand, applying twice the techniques in the proof of the case where d ≥5, we get
(A.11)
Ey,z h
φ(ρn)(X1[0, R2
n log−2 Rn], X2[0, R2
n log−2 Rn])
i
≤CR2
n log−2 Rn;
(A.12)
Ey,z
⌈KR2
n/Ln⌉
X
j,l=1
g(x1
j, x2
l )
≤CR2
nL−2
n .
Therefore, by (A.9)-(A.12),
Ey,z h
φ(ρn)(X1[0, KR2
n], X2[0, KR2
n])
i
≤Ey,z h
φ(ρn)(X1[0, R2
n log−2 Rn], X2[0, R2
n log−2 Rn])
i
+
X
j,l≤KR2
n/Ln
j+l>Rn log−2+c Rn
Ey,z[φ(ρn)(ξ1
j , ξ2
l )]
≤CR2
n log−2 Rn + CL2
n log−2 Rn · Ey,z
⌈KR2
n/Ln⌉
X
j,l=1
g(x1
j, x2
l )
+ CL2
n exp
−1
2L2c
n
· (KR2
n/Ln)2 + CLn
X
j,l≤KR2
n/Ln
j+l>Rn log−2+c Rn
P y,z[|x1
j −x2
l | ≤3L1/2+c
n
]
≤CR2
n log−2 Rn + CL2
n log−2 Rn · R2
nL−2
n + C + CLn · R4c
n log4+2c Rn
≤CR2
n log−2 Rn,
proving the lemma.
□
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(Yijie Bi) School of Mathematical Sciences, Peking University
Email address: 2200010925@stu.pku.edu.cn
(Zhenhao Cai) Faculty of Mathematics and Computer Science, Weizmann Institute of Science
Email address: zhenhao.cai@weizmann.ac.il
(Xinyi Li) Beijing International Center for Mathematical Research, Peking University
Email address: xinyili@bicmr.pku.edu.cn
(Bal´azs R´ath) Department of Stochastics, Institute of Mathematics, Budapest University of Tech-
nology and Economics; HUN-REN Alfr´ed R´enyi Institute of Mathematics
Email address: rathb@math.bme.hu
(Yuan Zhang) Center for Applied Statistics and School of Statistics, Renmin University of China
Email address: zhang probab@ruc.edu.cn
41
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ASYMPTOTICS FOR THE PERCOLATION THRESHOLD OF FINITARY RANDOM INTERLACEMENTS IN FOUR AND HIGHER DIMENSIONS YIJIE BI, ZHENHAO CAI, XINYI LI, BAL ́AZS R ́ATH, YUAN ZHANG Abstract. We establish sharp asymptotic bounds for the critical intensity of the Finitary Random Interlacements (FRI) model in four and higher dimensions with general trajectory length distributions. Our proof reveals that the construction of near-critical FRI clusters in four and higher dimensions is essentially analogous to a Galton-Watson process, whose expected number of offspring corresponds to the capacity of a random walk killed at the given length. 1. Introduction Finitary Random Interlacements (FRI), introduced by Bowen in [5], is a Poisson cloud of random walk trajectories on Zd where the trajectory lengths follow a specified distribution. As the expected length varies from zero to infinity, the model interpolates between the renowned Bernoulli percolation and Random Interlacements (RI). The latter was introduced by Sznitman in [21] and has been extensively studied over the last decade (e.g. [26], [22], [25], [23], [20], [24]; see [9] for a self-contained introduction). The FRI model, denoted by Iu,ρ (where u > 0 is a fixed number and ρ is a distribution supported on non-negative integers), can be constructed as follows. Independently for each vertex x ∈Zd(d ≥3), we initiate from x a Poisson u μ1(ρ)+1 -distributed number of independent simple random walks (Xn)n≥0 killed at time T ∼ρ, where μi(ρ), i ∈N is the i-th moment of ρ. The process Iu,ρ is then constructed as the subgraph of Zd whose edges are those traversed by the aforementioned random walk trajectories. The precise definition(s) of FRI can be found in Section 2 of this paper. The parameter u is the analogue of the intensity parameter in RI, which corresponds to the expected total occupation time at each vertex by the random walk trajectories. The length distribution ρ is typically taken to be geometric in the literature, i.e., one considers the model FIu,T := Iu,Geo( 1 T +1 ) for T ≥0, where Geo(λ) represents the geometric distribution with success probability λ; also, the model Iu,δn, where δn is the Dirac measure at n, has been studied as a finite-range approximation of RI (see [10] for an example). In this paper, we consider general length distributions ρ on N satisfying a mild restriction, which encompass two common families of distributions, namely the geometric distributions and the Dirac mass. For fixed u and varying ρ, the behavior of Iu,ρ constrained to any finite subset of Zd resembles that of RI with intensity u, denoted by Iu, when μ1(ρ) →∞. In the geometric setup, this property was discussed in [5] and made rigorous in [6]. Moreover, couplings between FRI and RI have proven instrumental in the study of the latter. For example, in [4], it was established that RI is a factor of i.i.d. using the convergence in law of a sequence of similarly defined finite-range models to RI. Moreover, recent breakthroughs concerning the sharpness of phase transition of the vacant set of RI [12], [10] and [11] rely crucially on couplings both between FRI and RI and between FRIs of various lengths to overcome obstacles imposed by the long-range dependence of RI. In contrast to RI, where Iu almost surely percolates for all u > 0 and d ≥3, FRI undergoes phase transitions with respect to both parameters as a bond percolation model on Zd. On the one hand, for fixed u > 0, the percolation properties of FIu,T as a function of T are studied in several papers. Bowen proved the existence of infinite clusters (connected components) in FIu,T for any fixed u > 0 and sufficiently large T in [5]. It was shown in [18] that FIu,T has a non-trivial phase transition with respect to T for all u > 0 and d ≥3. However, it was proved in [6] that FIu,T is non-monotone for T ∈(0, ∞) due to the fact that the expected number of trajectories 1 16 Oct 2025 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions from each vertex decreases as T increases. As a result, the existence and uniqueness of a critical length parameter Tc remain open. On the other hand, if we fix the length distribution ρ and study the percolation as u varies, the phase transition is unique, because Iu,ρ is increasing in u. For FRI with geometric length distributions, this phase transition has been studied by several papers. In [7], it was proved that for each fixed T > 0, there exists u∗(d, T) ∈(0, ∞) such that the probability of the existence of an infinite cluster in FIu,T is zero for u ∈(0, u∗(d, T)) and positive for u ∈(u∗(d, T), ∞). For the asymptotic of u∗(d, T), it was shown in [7] that u∗(d, T) · T →-log 1 -pc d /2 as T →0, where pc d is the critical parameter for Bernoulli bond percolation on Zd. Moreover, it was proved in [8] that for d ≥3, there exist constants C†(d) > c†(d) > 0 such that for all sufficiently large T, (1.1) c†fd(T) ≤u∗(d, T) ≤C†fd(T), where fd(T) is given by fd(T) := T -1/2, d = 3; T -1 log T, d = 4; T -1, d ≥5. Interestingly, this implies that FRI achieves supercriticality in percolation with vanishing density. It was conjectured in [8] that u∗(d, T) admits an asymptotic limit, i.e., the coefficients C† and c† can be arbitrarily close. Estimates similar to (1.1) are established for the percolation of Wiener sausages with fixed length in [13], and the analogous conjecture for the critical threshold of Wiener sausages was posed in Section 1.4 of that paper. In this paper, we confirm the conjecture for FRI when d ≥4, and further generalize it to FRI with general length distributions. Let {0 Iu,ρ ←→∞} be the event that the origin is in an infinite cluster of Iu,ρ. For any length distribution ρ, we define the critical intensity of the FRI model Iu,ρ by u∗(ρ) := inf{u ≥0 : P[0 Iu,ρ ←→∞] > 0}. We aim to establish the asymptotic of u∗for a family of length distributions (ρn)n∈N with expectations tending to infinity. For technical reasons, we need certain restriction on the fluctuation of each ρn. Specifically, Definition 1.1. We say a family (ρn)n∈N of distributions on N is appropriate if μ1(ρn) →∞as n →∞, and for all ε > 0, there exists C = C(ε, (ρn)n∈N) 0, let (1.6) u± n := (1 ± ε)un. To derive Theorem 1.4, it suffices to show that for all large n, (1.7) un ≤u∗(ρn) ≤u+ n . In other words, we need to show that Iu+ n ,ρn (resp. Iun ,ρn) is supercritical (resp. subcritical). To prove the subcriticality of Iun ,ρn, we explore the FRI cluster containing the origin layer by layer. Precisely, we call the origin the 0-th layer; given the i-th layer, the (i + 1)-th layer is given by the union of trajectories hitting the i-th layer but not the previous ones. In particular, the typical number of trajectories in one layer is approximately the intensity times the capacity of the previous layer. Therefore, when taking u = un , the number of discovered trajectories in the aforementioned exploration process can be dominated by a subcritical Galton-Watson process, which dies out in finite time almost surely. To verify that Iu+ n ,ρn is supercritical, we construct the infinite cluster through a classic coarsegraining argument inspired by [17], where a similar asymptotic upper bound for the critical intensity of spread-out percolation was proved. In our setup, some additional delicate treatment is needed to handle the correlation generated by random walk trajectories. Specifically, we introduce a truncated version of Iu+ n ,ρn which ignores certain types of trajectories (e.g., those with atypically large diameters or long durations), and run an exploration process to show the supercriticality on this truncated version instead. We divide the space Zd into disjoint boxes of side length Rn := ⌊ p μ1(ρn)⌋(the correlation length of a random walk killed at time T ∼ρn, see also (4.1)). Next, we sample the cluster containing 0 (more precisely, the center point of the box around 0, see (4.2)) with a specific part of trajectories within the Rn-box around 0, and require it to satisfy several conditions including a lower bound on the capacity and some restriction on the relative position of any two trajectories (we call this cluster a "seed"). We then try to extend this seed to all of the neighboring boxes, and call the box around 0 a "good" box if a seed is sampled in each of the neighboring boxes. Next, we resume the exploration from one of the neighboring boxes, and call it good if new seeds are sampled for all its unexplored neighbors. When a bad box appears, to ensure independence we give up the exploration of all unexplored boxes within a neighborhood of radius 2γRn concentric with this bad box that are neighbors of good ones 3 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions where γ is an integer defined in (4.23) so large that trajectories in subsequent exploration from boxes that are not ruined do not intersect those explored from the current bad box (thanks to the truncation). Note that in the exploration algorithm described above, we are able to choose parameters properly so that the conditional probability of each step (i.e., extension to neighboring boxes) is sufficiently close to 1. Consequently, the exploration, after proper rescaling, may stochastically dominate a finitely dependent percolation configuration with sufficiently high density, which is supercritical by an application of the Liggett-Schonmann-Stacey domination. We now briefly comment on the truncation. Note that in the exploration algorithm, it would be easier to obtain the desired independence if we ignored more trajectories. Meanwhile, such modification is not allowed to increase the critical intensity by a constant factor (otherwise, it will violate (1.7)). These requirements exhibit opposite directions of monotonicity, which entails a delicate truncation scheme, which stands as one of the main technical novelties of this paper. Let us also quickly remark on the d = 3 case. Since it is much more likely for two random walk trajectories to intersect in three dimensions, the behavior of near-critical FRI clusters can no longer be characterized by Galton-Watson processes, and the corresponding asymptotic threshold is actually determined by a different mechanism. In fact, in a paper in progress [3], the prefactor of the asymptotic threshold is upper and lower bounded by two (plausibly equal) critical thresholds of finitary Brownian interlacements (FBI), a counterpart of FRI in the continuum. A model similar to FBI in which trajectories are of fixed length is studied in [13], where the existence of a non-trivial phase transition is proved (c.f. Theorem 1.3 therein). The remaining sections are organized as follows. In Section 2, we introduce the notations and some preliminary results. In Sections 3, we prove the lower bound of (1.7). Sections 4 and 5 are dedicated to the corresponding upper bound. In Section 4, we introduce the exploration algorithm and reduce the supercriticality of Iu+ n ,ρ to the fact that the conditional probability of the extension to a neighboring box in the algorithm can be arbitrarily close to 1 (c.f. Proposition 4.7). Section 5 proves Proposition 4.7 with various coupling techniques. 2. Notations and Preliminaries We write N = {0, 1, 2, ...} for the set of natural numbers. For a real number a, write ⌊a⌋for its integer part, and for real numbers b, c, we write b ∧c and b ∨c for the minimum and maximum of b and c respectively. Denote the l∞norm on Zd by | · | and the l1 norm by | · |1. For two non-empty subsets A, B ⊂Zd, define the distance between them by d(A, B) := minx∈A,y∈B |x-y|, and write d({x}, B) as d(x, B) for all x ∈Zd. The diameter of a finite nonempty set A ⊂Zd is defined by diam(A) := maxy,z∈A d(y, z). For all x ∈Zd and r ∈(0, ∞), we denote the subset of Zd inside the box of side length 2r centered at x by B(x, r) = Bx(r) := {y ∈Zd : |x -y| ≤r}; for all A ⊂Zd, let B(A, r) := S x∈A B(x, r) be the collection of vertices in the r-neighborhood of A. For all n ∈N and A ⊂Zd, let nA = {nx : x ∈A}. In particular, we will consider the set nZd which will be referred to as a lattice. For all x, y ∈nZd, they are called nearest neighbors if |x -y|1 = n. A nearest-neighbor path on nZd is a sequence of vertices on nZd such that the adjacent vertices are nearest neighbors. For λ ∈(0, 1], define the geometric distribution Geo(λ) by Geo(λ)(n) = λ(1 -λ)n, n ∈N. For λ ∈(0, ∞), define the exponential distribution Exp(λ) by the density function f(x) = λe-λx1x>0. From now on, we always assume that the dimension is at least 4, and often omit d in the notation when defining objects such as random walks and quantities including constants and parameters that do not explicitly depend on d. All the results stated in this paper are supposed to hold for all d ≥4 unless stated otherwise. The constants C and C′ appearing in the paper may implicitly depend on d, and may differ from place to place. However, numbered constants such as C1, C2, ... refer to their first appearance in the text. Measures on countable spaces: For an arbitrary countable space X equipped with the canonical σ-field, we denote by M(X ) (resp. M∗(X )) the set of all probability measures (resp. finite measures) on X , and abbreviate M = M(N). For two measures ν1, ν2 ∈M(X ), we write ν1 ≤ν2 if ν1(A) ≤ν2(A) for all A ⊂X . For all ν ∈M∗(X ), we define the corresponding 4 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions probability measure ν0 ∈M(X ) by ν0(A) = ν(A) ν(X ), ∀A ⊂X . For ρ ∈M(X ), x ∈X , let ρ(x) be the mass assigned to x. We call ρ a point measure if ρ(x) ∈ N for all x ∈X . Let S be a multi-subset of X . We sometimes identify S with its corresponding point measure on X , i.e., we also use S to represent the point measure P x∈X nx(S) · δx, where nx(S) is the multiplicity of x in S. Write |S| = P x∈S nx(S). For a point measure X ∈M(X ) and A ⊂X , write X ∩A = P x∈A X(x) · δx. Let k be a positive integer. For all ρ ∈M, denote the k-th moment of ρ by (2.1) μk(ρ) := ∞ X l=0 lkρ(l). Also, we define the "tail contribution" to the k-th moment by μ(m) k (ρ) := ∞ X l=m lkρ(l), where m ∈N. Trajectories: We will refer to nearest-neighbor paths on Zd as trajectories. Let W [0,∞) denote the set of all finite trajectories. For all η ∈W [0,∞), write η = (η(s))0≤s≤T . Here, T(η) := T denotes the length of η. In addition, we define the range of η by range(η) := {η(s) : 0 ≤s ≤T}. Sometimes, η is identified with its range when no confusion arises (e.g. B(η, r) = B(range(η), r)). For 0 ≤a ≤b ≤T, define the sub-path η[a, b] := (η(a + s))0≤s≤b-a. For a sigma-finite point measure ω on the space W [0,∞), we write η ∈ω if ω({η}) > 0, and define the set of sites visited by the trajectories of ω by V (ω) := S η∈ω range(η), and let G(ω) be the subgraph of Zd where an edge is open iff it connects two adjacent vertices appearing in one of the trajectories in ω. Sometimes, the notation is slightly abused and the point measure ω is identified with the corresponding (multi-)subset of W [0,∞). Let ηi ∈W [0,∞) be a trajectory with length Ti, i = 1, 2. Define η1 ∗η2, the concatenation of η1 and η2, by η1 ∗η2(s) := ( η1(s), 0 ≤s ≤T1; η′ 2(s -T1), T1 0 : Xt ∈A}. To lighten the notation, for m, l ∈N, let (2.4) μ(x, m, l; A)(·) = P x[X[-m, l] ∈· X[-m, -1] ∩A = ∅] 5 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions be a probability measure on W [0,∞), and we omit "A" when A = ∅(i.e., no conditioning is imposed). For all R > 0, let WR ⊂W [0,∞) denote the subset of trajectories with diameter at most R. We have the following estimate on the diameter of a trajectory following the distribution μ(x, m, l) whose proof follows from a simple application of the Donsker's invariance principle. Lemma 2.1. There exists a decreasing function f : (0, ∞) →(0, ∞) such that for all κ > 0, R > C(κ) large and m + l δE0[cap(X[0, T])] i T log-10 T]. (Here, the choice of -10 in the exponent is rather arbitrary. In fact, any real number smaller than -2 is sufficient for Proposition 2.3 below.) We can prove estimates on this quantity analogous to Proposition 2.2 by directly comparing cap(·) with cap(·, T). Proposition 2.3. For all r > 0, the following estimates hold. (2.14) E0[cap(X[0, T], rT)] = (1 + o(1)) · εdT(1 + log T · 1d=4)-1. Moreover, for all δ > 0, there exists some C = C(d, δ) > 0 such that (2.15) P 0 h cap(X[0, T], rT) -E0[cap(X[0, T])] > δE0[cap(X[0, T])] i T log-10 T] -eX[0,T](Xt)] = P 0,0 h X1[-t, T -t] ∩X2(0, t′] = ∅, X1[-t, T -t] ∩[t′, ∞) ̸= ∅ i , where t′ = ⌊T log-10 T⌋. Let E1, E2, E3 denote the events E1 := {d(X2 t′, X1[-t′, t′]) > (t′)1/2 log-1/4 T}, E2 := {X1[-t′, t′] ∩X2(0, t′] = ∅}, E3 := {cap(X1[-t′, t′]) ≤4ε4t′ log-1 T}. By [15, Theorem 3.5.1], (2.17) P 0,0[E2] ≤C log-1 T. By (2.12), (2.18) P 0.0[Ec 3] ≤C log-2 T. Now we aim to bound the probability of E1. To this end, for X ∼P 0, let τ0 = 0, and define for all k ≥1 τk := inf{t > τk-1 : |Xt -Xτk-1| ≥⌊(t′)1/2 log-1/4 T⌋}, σk := τk -τk-1. Apparently σk, k ≥1 are i.i.d., and we have the estimate E0[e-σ1] ≤e-Ct′ log-1/2 T by Gaussian estimates. Thus, we have (2.19) E0[|B(X[0, t′], (t′)1/2 log-1/4 T|] ≤C(t′)2 log-1/2 T, which can be deduced from the large deviation estimate P 0 h |B(X[0, t′], (t′)1/2 log-1/4 T| ≥M · (t′)2 log-1/2 T i ≤P 0[τ⌈M log1/2 T⌉≤t′] = P 0[e-t′ ≤exp -τ⌈M log1/2 T⌉ ] ≤et′ E0[e-σ1] M log1/2 T ≤exp -(CM -1)t′ for all M ∈N∗. By (2.19) and the LCLT [15, Theorem 1.2.1], (2.20) P 0,0[Ec 1] ≤C · (t′)-2E0[|B(X[-t′, t′], (t′)1/2 log-1/4 T)|] ≤C log-1/2 T. 7 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions By (2.17), (2.18) and (2.20), (2.21) P 0,0[X1[-t, T -t] ∩X2(0, t′] = ∅, X1[-t, T -t] ∩X2[t′, ∞) ̸= ∅] ≤P 0,0[Ec 1] + P 0.0[Ec 3] + E0,0 h P 0,0[X1[-t, T -t] ∩X2[t′, ∞) ̸= ∅|X1[-t′, t′], X2 t′]1E1∩E2∩E3 i ≤C log-4 T + C log-1/2 T · P 0,0 [E1 ∩E2 ∩E3] ≤C log-3/2 T, where the second inequality holds because on the event E1, P 0,0[(X1[-t, -t′] ∪X1[t′, T -t]) ∩X2[t′, ∞) ̸= ∅|X1[-t′, t′], X2 t′] ≤C log log T log T according to [15, Theorem 4.3.3], and on E1 ∩E3, P 0,0[X1[-t′, t′] ∩X2[t′, ∞) ̸= ∅|X1[-t′, t′], X2 t′] ≤cap(X1[-t′, t′]) · C(1 ∨d(X2 t′, X1[-t′, t′]))-2 ≤Ct′ log-1 T · (t′)-1 log1/2 T ≤C log-1/2 T by (2.8). By (2.16) and (2.21), (2.22) E0[cap(X[0, T], rT) -cap(X[0, T])] ≤4T log-10 T + T-⌊2T log-10 T⌋-1 X t=⌊2T log-10 T⌋+1 E0[P Xt[ eHX[0,T] > T log-10 T] -eX[0,T](Xt)] ≤4T log-10 T + CT log-3/2 T ≤CT log-3/2 T, which implies (2.23) P 0[cap(X[0, T], rT) -cap(X[0, T]) > δ/2 · E0[cap(X[0, T])]] ≤Cδ-1 log-1/2 T by (2.9) and the Markov inequality. It is easy to verify that (2.14) follows from (2.9) and (2.22). Moreover, by (2.23) and (2.12), P 0 h cap(X[0, T], rT) -E0[cap(X[0, T])] > δE0[cap(X[0, T])] i ≤P 0[cap(X[0, T], rT) -cap(X[0, T]) > δ/2 · E0[cap(X[0, T])]] + P 0 h cap(X[0, T]) -E0[cap(X[0, T])] > δE0[cap(X[0, T])] i ≤Cδ-1 log-1/2 T + Cδ-2 log-2 T ≤Cδ-1 log-1/2 T, proving (2.15). When d ≥5, one can prove (2.24) E0[cap(X[0, T], rT) -cap(X[0, T])] ≤CT 3/4 parallel to (2.22) with similar techniques, and deduce (2.14) and (2.15) from this estimate as above. □ Finitary random interlacements (FRI): We now present the definition of the finitary random interlacements (FRI) point process. For any u > 0 and any ρ ∈M, we define a measure by π = X x∈Zd X l∈N ρ(l) μ1(ρ) + 1 · μ(x, 0, l). Definition 2.4. The finitary random interlacements point process with intensity u and length distribution ρ, denoted by X u,ρ, is a Poisson point process (PPP) on the space W [0,∞) with density measure u · π, and the corresponding random subgraph of Zd is defined by Iu,ρ = G(X u,ρ). This definition generalizes the one commonly used in the literature, and takes into consideration clouds of trajectories with arbitrary length distributions instead of just geometric ones. The factor of 1 μ1(ρ)+1 in the definition of π guarantees that the expected total local time at a vertex is exactly u in X u,ρ. We refer to [7] for a more comprehensive introduction of the model. 8 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions We are concerned about the distribution of trajectories within X u,ρ originating from or traversing a certain subset of Zd. To this end, we further introduce some notations. Let X be any point measure on W [0,∞). For all A ⊂Zd, let X[A] denote the point process comprising trajectories within X intersecting A. For A, B, D ⊂Zd, D ⊂A, let (2.25) X[A; B] = X[A] -X[B] and let (2.26) X[A, D; B] = X[A; B] ∩H[A; D], where H[A; D] was defined in (2.3). For all A ⊂Zd finite, we can find the distribution of X u,ρ[A] by a technique called "rerooting" of the trajectories (c.f. [10, Lemma 3.1]). Lemma 2.5. Let A ⊂Zd be finite, then X u,ρ[A] is a PPP with intensity u · πA, where πA := X x∈A,m,l∈N ρ(m + l) μ1(ρ) + 1P x[ eHA > m] · μ(x, m, l; A). Given this result, it is easy to compute the expected number of trajectories within X u,ρ hitting A at x, which equals ue(ρ) A (x), where (2.27) e(ρ) A (x) = ∞ X m=0 μ(m) 0 (ρ) μ1(ρ) + 1P x[ eHA > m]. Thus, it is natural to define a substitute for the usual capacity of subsets of Zd (2.28) cap(ρ)(A) := X x∈A e(ρ) A (x), which we call the ρ-capacity. Note that e(ρ) A (x) ≥eA(x) and thus cap(ρ)(A) ≥cap(A). We will need control over the ρ-capacity. It is easy to see that for arbitrary ρ and d ≥4, (2.29) cap(ρ)(A) ≤|A|. Moreover, letting N = ⌊μ1(ρ) log-λ μ1(ρ)⌋, then we have e(ρ) A (x) ≤ N X m=0 m(m) 0 (ρ) μ1(ρ) + 1 · 1 + ∞ X m=N+1 m(m) 0 (ρ) μ1(ρ) + 1P x[ eHA > N] ≤ N μ1(ρ) + 1 + P x[ eHA > N]. Summing over x ∈A, we get (2.30) cap(ρ)(A) ≤|A| log-10 μ1(ρ) + cap(A, μ1(ρ)), where cap(A, μ1(ρ)) is defined by (2.13). Branching Processes with agents in countable sets: Let X be an arbitrary countable set. For S a multi-subset of X and all x ∈X, recall that nx(S) is the multiplicity of x in S. For all x ∈X , let νx be a distribution on the space of finite point measures on X . Let S be a multi-subset of X . The branching process with agents in X with offspring distribution (νx)x∈X starting from S, denoted by (Yn)n≥0, is then recursively defined as follows. For all x ∈X and i, j ∈N, let Yx,i,j ∼νx be mutually independent. First, let Y0 = P x∈X nx(S)·δx; then, assuming that Yi has been sampled, let Yi+1 = X x∈X X 0≤j 0. The idea is to explore the cluster in Iun ,ρn containing 0 layer by layer, and prove that the cluster is almost surely finite (i.e., Iun ,ρn is subcritical) by showing that the exploration stops within finitely many steps. To see this, we compare the exploration process with a Galton-Watson tree that dies out almost surely. In [8], a similar process was employed to prove the upper bound of (1.1). We now carry out more precise analysis in order to prove the subcriticality of Iun ,ρn. We start with the construction of the exploration process (Lk, Lk)k≥0 for Iu,ρ with arbitrary parameters u and ρ. Let L0 = {0} and L0 be the zero measure. Assume that we have defined Lk (k ∈N), then let Lk+1 = X u,ρ Lk; k-1 [ i=0 Li and Lk+1 = V (Lk+1). Let L = ∪∞ i=1Li. Note that L is exactly the collection of trajectories comprising the cluster in Iu,ρ that includes 0. Due to the almost sure finiteness of Li, i > 0 and X u,ρ[B(0, N)], N > 0, the cluster of Iu,ρ containing the origin is infinite iff Li is nonempty for all i > 0. To simplify our computation, we introduce a modified sequence (L′ i)i≥1 that stochastically dominates (Li)i≥1. Let Ji, i > 0 be independent PPPs on W [0,∞) with the same distribution as X u,ρ. We now generate the i-th layer of trajectories with Ji, so that the source of randomness for each layer is independent. Precisely, let L′ 0 = {0}. Assume that L′ k is defined, then let L′ k+1 = Jk+1[L′ k]. Lemma 3.1. For all u > 0 and ρ ∈M, there exists a coupling Q of (Li)i>0 and (L′ i)i>0 such that for all i, Q[Li ⊂L′ i] = 1. Proof. Note the following fact parallel to Lemma 4.3 of [8]. (3.1) For any k ∈N, given L0, ..., Lk, then the conditional distribution of Lk+1 is the same as that of X u,ρ Lk; k-1 [ i=0 Li . Thus, we can construct (Lk)k>0 by Lk+1 = Jk+1[Lk; k-1 [ i=0 Li]. Assuming Li ⊂L′ i for all 0 ≤i ≤k, we have Lk+1 ⊂Jk+1[Lk] ⊂Jk+1[L′ k] = L′ k+1. The conclusion then follows by induction. □ By Lemma 3.1, it suffices to prove that (L′ k)∞ k=0 dies out almost surely. To this end, we introduce a quantity κ(ρ)(A) := X x∈A ∞ X m=0 μ(m) 1 (ρ) μ2(ρ) P x[ eHA > m], where A ⊂Zd is finite, and show that the expectation Wk := E[κ(ρ)(L′ k)] decreases to zero exponentially fast, which matches our intuition of a sub-critical Galton-Watson tree. Technically, note that k(ρ) is sub-additive, i.e. κ(ρ)(A1 ∪A2) ≤P i=1,2 κ(ρ)(Ai) (since P x[ eHA1∪A2 > m] ≤P i=1,2 P x[ eHAi > m]). Therefore, given L′ k and the hitting points at L′ k of trajectories in L′ k+1, the expectation of k(ρ)(L′ k+1) from above by those of κ(ρ)(η), η ∈L′ k+1 whose conditional 10 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions distribution can be obtained from Lemma 2.5. Moreover, when we take (u, ρ) = (un , ρn), the conditional expectation of the sum P η∈L′ k+1 κ(ρn)(η) is approximately (1 -ε)κ(ρn)(L′ k), implying that E[κ(ρn)(L′ k)] decays exponentially in k. Formally, let Vk := ( E[|L′ k|], d ≥5; E[|L′ k|] · log-1 μ1(ρ), d = 4. Lemma 3.2. For every δ > 0, the following holds for large n ∈N. For (u, ρ) = (un , ρn), we have Wk+1 ≤(1 -ε/2)Wk + δVk, (3.2) Vk+1 ≤C · Wk, (3.3) for all k ∈N. Here C 0. Take upper limits on both sides and we get limk→∞Wk=0. (It can be deduced from (3.2) that the sequence (Wk)k∈N is bounded.) Thus, limk→∞Vk = 0, indicating the existence of some (random) k ∈N such that L′ j = ∅for all j > k. Denote by C the cluster in Iun ,ρn containing 0, then by Lemma 3.1, C ⊂ k[ j=0 Lj ⊂ k[ j=0 L′ j. Thus, since Lj is almost surely finite for all j ∈N, C is almost surely finite, which yields u∗(ρn) ≥ un . □ Now, we present the proof of Lemma 3.2 which involves estimating the conditional expectation of |L′ k+1| and κ(ρ)(L′ k+1) given L′ k. Proof of Lemma 3.2. The proofs for cases d ≥5 and d = 4 are identical except for the details related to the normalization of expected volume. Here we only present the proof of the latter. By Lemma 2.5, (3.4) E[|L′ k+1| L′ k] ≤ X x∈L′ k X m,l∈N uρ(m + l) μ1(ρ) + 1 · Px[ eHL′ k > m] · (m + l) = u μ1(ρ) + 1 X x∈L′ k X m∈N μ(m) 1 (ρ) · Px[ eHL′ k > m] ≤u · μ2(ρ) μ1(ρ) · κ(ρ)(L′ k). 11 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Plugging in (u, ρ) = (un , ρn) and taking the expectation, we get Vk+1 ≤ 8 π2 · Ck, which implies (3.3). Now we compute the conditional expectation of κ(ρ)(L′ k+1) similarly: (3.5) E[κ(ρ)(L′ k+1)|L′ k] ≤ X x∈L′ k X m,l∈N uρ(m + l) μ1(ρ) + 1 · P x[ eHL′ k > m] · μ(x, m, l; L′ k)[κ(ρ)(·)] ≤ X x∈L′ k X m,l∈N uρ(m + l) μ1(ρ) + 1 · Px[ eHL′ k > m] · (1 + ε/2)π2 8 · (m + l) log-1 μ1(ρ) + X x∈L′ k X m,l∈N uρ(m + l) μ1(ρ) + 1 · err(ρ)(m + l) ≤(1 + ε/2)π2 8 · u · μ2(ρ) · κ(ρ)(L′ k) μ1(ρ) log μ1(ρ) + |Lk| · u μ1(ρ) + 1 ∞ X m=0 (m + 1)ρ(m) · err(ρ)(m), where (u, ρ) = (un , ρn), and err(ρ)(m) := E0 κ(ρ)(X[0, m])1{κ(ρ)(X[0,m])>(1+ε/2) π2 8 m log-1 μ1(ρ)} . Here, in the second inequality, we have used err(ρ)(m + l) ≥E0[κ(ρ)(X[-m, l])1κ(ρ)(X[-m,l])>M1X[-m,0)∩L′ k=∅] ≥P x[ eHL′ k > m]μ(x, m, l; L′ k)[κ(ρ)(·)1κ(ρ)(·)>M], where M = (1 + ε/2) π2 8 · (m + l) log-1 μ1(ρ). Taking (u, ρ) = (un , ρn), we have (3.6) (1 + ε/2)εd · un μ2(ρn) μ1(ρn) 0, then by the definition of the appropriate family, there exists some C′(δ) such that for all n, 1 μ2(ρn) X m>C′μ1(ρn) m2ρn(m) C′μ1(ρn), (3.12) err[ρn](m) ≤E0[cap(X[0, m], μ1(ρn))] ≤⌈m/μ1(ρn)⌉· E0[cap(X[0, μ1(ρn)], μ1(ρn))] ≤Cm log-1 μ1(ρn); while for δμ1(ρn) ≤m ≤C′μ1(ρn), we claim that, for large n, (3.13) err[ρn](m) ≤δm log-1 μ1(ρn). We defer the proof of (3.13) for a few lines and finish the proof of (3.10). By (3.11), (3.12) and (3.13), we have ∞ X m=1 mρn(m) · err[ρn](m) ≤ ⌊δμ1(ρn)⌋ X m=1 m2ρn(m)[(log-1 m) ∧1] + δ ⌊C′μ1(ρn)⌋ X m=⌊δμ1(ρn)⌋+1 m2ρn(m) log-1 μ1(ρn) + C ∞ X m=⌊C′μ1(ρn)⌋+1 m2ρn(m) log-1 μ1(ρn) ≤2(δμ1(ρn))2 log-1 μ1(ρn) + δμ2(ρn) log-1 μ1(ρn) + Cδμ2(ρn) log-1 μ1(ρn) ≤2Cδμ2(ρn) log-1 μ1(ρn), which implies (3.10) since δ is arbitrary. We now prove (3.13). For all m ∈[δμ1(ρn), C′μ1(ρn)], write ξ = cap(X[0, m], μ1(ρn)), ξ′ = cap(X[0, m]), a = E0[ξ], a′ = E0[ξ′] and ea = π2 8 m log-1 μ1(ρn). By Proposition 2.2 and the Markov inequality, E0[ξ′; ξ′ > (1 + ε/8)ea] ≤a′P 0[ξ′ -a′ > (1 + ε/8)ea -a′] + E0[ξ′ -a′; ξ′ -a′ > (1 + ε/8)ea -a′] ≤(1 + ε/16)eaP 0[ξ′ -a′ > εea/16] + E0[(ξ′ -a′)1{ξ′-a′>εea/16}] ≤(1 + ε/16)ea(16/εea)2 · var(ξ′) + 16/εea · var(ξ′) ≤C(ea)-1var(ξ) ≤Cea log-2 μ1(ρn), where in the second inequality we have used a′ εa/8] ≤8 εaE0[ξ -ξ′] (1+ε/8)ea] + E0[ξ -ξ′] ≤(1 + ε/8)ea · P 0[ξ -ξ′ ≥εea/8] + Cea log-2 μ1(ρn) + Cea log-1/2 μ1(ρn) ≤Cea log-1/2 μ1(ρn). 13 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions (3.13) then follows by taking n ∈N large. □ Remark 3.3. Recall the definition of the ρ-capacity for all ρ ∈M. It is then not hard to see that for all finite A ⊂Zd, κ(ρ)(A) = cap(ˆρ)(A), where ˆρ = P∞ k=0 kρ(k) μ1(ρ)δk. Remark 3.4. If we only deal with the lower bound for the original finitary random interlacement with geometric killing, the argument can be greatly simplified by the following observation on random walks conditioned on not returning to a given set. Observation. For any A ⊂Zd, x ∈A and T ∈N, let X1 be a simple random walk from x and X2 be a simple random walk from x conditioned not to return to A. Let Ni ∼Geo( 1 T+1) be independent of Xi, i = 1, 2. Then there is a coupling between (X1 t )N1 t=0 and (X2 t )N2 t=0 such that (1) almost surely, N1 ≥N2; (2) almost surely, there exists a random point y† ∈Zd such that (3.14) X1 N1-N2+j + y† = X2 j , ∀0 ≤j ≤N2. As a direct consequence, the capacity of (X1 n)N1 n=0 stochastically dominates the capacity of (X2 n)N2 n=0. In fact, we can recursively construct a sequence of trajectories from X1 as follows: let τ0; for k ≥1, let ̄Xk = φτk-1X1 -X1 τk-1, and let τk = (τk-1 + eHA( ̄Xk)) ∧N1, where φkX1 is defined by φkX1 j = X1 j+k, ∀j ∈N and eHA( ̄Xk) is the first time ̄Xk returns to A. Define k∗:= inf{k ≥0 : τk = N1}. By the memoryless property of the geometric distribution, ( ̄Xk∗ t )0≤t≤τk∗-τk∗-1 is distributed as (X2 t )0≤t≤N2, which yields the observation. Note that by Lemma 2.5, the number of trajectories in X u,Geo( 1 T +1 ) that hits A at x follows Poi(ue(T) A (x)), where e(T) A (x) = P∞ i=0 1 T+1( T T+1)iP x[ eHA > i], and the conditional law of trajectories given their total number is identical to that of ζ1 ∗ζ2, where ζ1 ∼X1[0, N1] and ζ2 ∼P[X1[0, N1] = · X(0, N1] ∩A = ∅]. By the above observation, the expected capacity of ζ1 ∗ζ2 is at most twice that of ζ1. When d ≥5, it is not hard to check that as T →∞, E[cap(T)(X1[0, N1])] = (1 + o(1))εdT. Therefore, E[cap(T)(V (Iu,T [A]))] ≤ X x∈A ue(T) A (x) · (2 + o(1))εdT = (2 + o(1))uεdT · cap(T)(A). Taking u = uT := 1-ε 2εd · 1 T , we get E[cap(T)(V (Iu,T [A]))] ≤(1 -ε/2)cap(T)(A) for large T, which yields that IuT ,T is subcritical. The value of uT matches un defined below (1.5). When d = 4, the same arguments hold by taking uT = 4(1-ε) π2 · T -1 log T. 4. Proof of the upper bound of (1.7) In this section, we will prove the upper bound of (1.7) for any appropriate family of distributions (ρn)n∈N. Recalling u+ n from (1.6), we will show that Iu+ n ,ρn is supercritical through an exploration algorithm, as outlined in last but two paragraph of Section 1. To be more specific, we first restrict to a truncated version of X u+ n ,ρn (denoted by X u+ n ,ρn) which consists of "typical" trajectories (rigorously defined later). Next, we divide the Euclidean space into boxes of side length Rn := ⌊ p μ1(ρn)⌋, with one-to-one correspondence with vertices of the coarse-grained lattice Ln := Rn · Zd, and start the exploration by revealing a cluster in the box corresponding to 0 consisted of a fixed number of typical trajectories that satisfy a certain spatial restriction (we call such a 14 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions collection of trajectories a "seed"), and then start the exploration algorithm by declaring 0 to be active. In each round of the subsequent exploration, we pick an active vertex and try to extend the revealed cluster inside the corresponding box to neighboring ones by revealing a certain number of layers of trajectories, one layer at a time. After that, we determine whether this new round of exploration is good. If so, and the last layer in the aforementioned extension contains a seed in each of the neighboring boxes, we activate the corresponding neighboring vertices and declare the current vertex to be surviving; otherwise, we say this round of exploration at the current vertex fails, and we declare the vertices within a neighborhood of range proportional to Rn to be ruined (including those that has become surviving or active). By bounding from above the conditional probability of failure in each round, which again involves comparison between the round of exploration process and a Galton-Watson branching process, we show that the vertices surviving in the end dominates a finitely dependent percolation model on the coarse-grained lattice Ln, and prove its supercriticality with the Liggett-Schonmann-Stacey domination [16]. We implement the aforementioned strategy in the following two subsections. In Subsection 4.1, we introduce some further notation necessary for the introduction of the exploration algorithm (henceforth referred to as the Algorithm) and gather some related preliminary results. In Subsection 4.2, we rigorously describe the Algorithm and prove the supercriticality assuming the estimate on the conditional probability of failure in each step (c.f. Proposition 4.7). 4.1. Preparation for the Algorithm. Let Ud := {e1, ..., ed} be the collection of canonical unit vectors on Zd, where ej is the unit vector with j-th component 1, and write ⃗1 = e1 + e2 + ... + ed. Let (4.1) Rn := ⌊ p μ1(ρn)⌋ and define the corresponding coarse-grained lattice by Ln := Rn · Zd. Let (4.2) zn = ⌊Rn/2⌋·⃗1. For x ∈Ln, let Bn x = x + [0, Rn)d and eBn x = B(x + zn, Rn/4) be two concentric boxes. Fix a constant (4.3) c = 0.01. We fix the dimension d ≥4, the perturbative constant ε ∈(0, 1/2) introduced in (1.6) and the appropriate family (ρn)n∈N, and drop the dependence on them thereinafter. We now introduce some extra symbols and terminology. Typical trajectories: To facilitate further construction, we need to ignore certain atypical configurations of random walk trajectories in terms of the duration, capacity, volume and diameter. Define a mesoscopic scale Ln := ( (Rn) 2+c d-2 , d ≥5; Rn log-c Rn, d = 4 and a mildly increasing function with respect to Rn In := ( (Rn)c, d ≥5; logc Rn, d = 4. For η ∈W [0,∞), write T = T(η). We introduce four parameters whose values will be determined solely depending on d, ε and (ρn)n∈N: parameters (4.4) k, K ∈R+ put a restriction on the duration of typical trajectories, which will be fixed right after the statement of Lemma 4.3; the parameter (4.5) M ∈R+ controls the diameter and will be fixed in (5.50) and the parameter (4.6) θ1 = θ1(q) ∈(0, 1), 15 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions which depends on another parameter q formally introduced in Lemma 4.3, controls the fluctuation of the capacity of typical trajectories, and will be fixed in (5.4). We assume in following results in this subsection that these parameters are arbitrarily chosen, unless stated otherwise. We say η is typical if it satisfies all requirements below formulated by (Ei)i=1,2,3,4: • E1 = E1(k, K, M) := {T ∈[k(Rn)2, K(Rn)2] and η ⊂B(η(0), MRn)}. • If d ≥5, E2 = E2(θ1) := {(1 -θ2 1)εdT (1 -θ2 1)π2 8 T log-1 μ1(ρn), cap(range(η), R2 n) Ln] ≤kP 0[diam(X[0, (L′ n)2]) > Ln] ≤CR2 n/(L′ n)2exp(-I2/3 n ). Here, the last inequality follows from a standard estimate of the transition probability of the simple random walk. Calculating the last term gives (4.11) μ(0, m, l)(Ec 3) ≤ CI8/3 n exp -I2/3 n , d = 4; exp -Rc/2 n , d ≥5 for all large n. Therefore, we have lim sup n→∞ sup k(Rn)2≤m+l≤K(Rn)2 μ(0, m, l)(Ec 3) = 0, concluding the lemma. □ Restriction within trajectories: When revealing a new layer in the Algorithm, we need to avoid exploring trajectories hitting certain vertices (those with unusually small equilibrium measure, for instance) within trajectories in the last layer explored, which facilitates the characterization of the conditional law of the new layer. To this end, we introduce a further parameter (4.12) θ2 ∈(0, 1) that will be fixed in (5.50). For any η ∈W [0,∞), recall that eη is the equilibrium measure of range(η), and define the *-proper part of η by (4.13) bη = {η(s) : s ∈[0, T], eη(s) ≥θ2}, d ≥5; {η(s) : s ∈[0, T], eη(s) ≥θ2 log-1 μ1(ρn) and P x[ eHη > (Rn)2 log-10 Rn] ≤(1 + θ1)eη(x)}, d = 4. We will need the following property which states that a random walk starting from x ∈bη conditioned on not returning to range(η) forgets about the conditioning soon after departure. Recall the symbol "≃" for spatial translation of trajectories from Section 2. Lemma 4.2. Let η ∈Tn, x ∈bη. Let ln = ( ⌊R1-c/4 n ⌋, d ≥5; ⌊Rn log-2c Rn⌋, d = 4 and tn = ( ⌊R2-c n ⌋, d ≥5; ⌊R2 n log-10 Rn⌋, d = 4. Then, for all large n, there is a coupling Q of X1 ∼P x[X ∈·| eHη = ∞] and X2 ∼P x[X ∈·] such that Q[E] ≥ ( 1 -R-c/2 n , d ≥5; 1 -2θ1, d = 4, where E := {|X1 j -X2 j | ≤ln, ∀j ≥0} ∩{X1[tn, ∞) ≃X2[tn, ∞)}. Proof. Consider independent random walk paths constructed in the same probability space (Ω, F, Q) with different laws: X3 ∼P x[X ∈·| eHη > tn], X4 ∼P x[X ∈·], X5 ∼P 0[X ∈·], X6 ∼P x[X ∈ ·| eHη = ∞]. Let X1[0, ∞) = ( X3[0, tn] ∗X5[0, ∞), if X3[0, tn] ∗X5[0, ∞) avoids η after time 0; X6[0, ∞), otherwise 17 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions and let X2[0, ∞) = X4[0, tn]∗X5[0, ∞). Then X1 and X2 constructed in this fashion has desired marginal distributions. Define two events E1 = {X1[0, ∞) = X3[0, tn] ∗X5[0, ∞)}, E2 = {X3[0, tn], X4[0, tn] ⊂Bx(ln/2)}. Note that E ⊃E1 ∩E2, so it suffices to control the probability of Ec j, j = 1, 2. For d = 4, by the fact that x ∈bη (recall the definition (4.13)), eη(x) ≥ 1 1 + θ1 P x[ eHη > tn], so it can be deduced easily that Q[Ec 1] = P x[tn tn] = 1 - eη(x) P x[ eHη > tn] tn] ≤Cθ-1 2 log Rn exp -ln/t1/2 n , where the second inequality uses the fact that eη(x) > θ2 log-1(Rn)2. As a result, Q[Ec 2] ≤1 2R-c/2 n for all large n. So far, the lemma has been proved. □ For any fixed η ∈W [0,∞), and any x ∈range(η), we define a probability measure μ∗ x,η on W [0,∞) which characterizes the law of the trajectories in an FRI point process that hits η at x by (4.14) μ∗ x,η := C · X k(Rn)2≤m+l≤K(Rn)2 ρn(m + l) · μ(x, m, l; range(η)), where C = P kR2n≤m≤KR2n(m + 1)ρn(m) is a normalizing constant. Then, we consider the probability measure obtained by conditioning on Tn: (4.15) μx,η := μ∗ x,η(·|Tn) Similarly, for all x ∈Zd, we define a simpler probability measure that resemble μx,η by (4.16) μx := C · X k(Rn)2≤m+l≤K(Rn)2 ρn(m + l)μ(x, m, l), where C is again a normalizing constant. Recall that we will compare a round of exploration with of Galton-Watson branching process with agents in the space of trajectories W [0,∞). To guarantee the supercriticality of that branching process (namely the process Y defined below the proof of Proposition 5.3), we introduce the following lemma which focuses on the expected capacity of ζ ∼μx,η and can be derived an analogue of Lemma 4.1 (c.f. Lemma A.1). Recall the parameters k, K, M and θ1 introduced for the definition of typical trajectories from (4.4)-(4.6), and θ2 introduced for the definition of *-proper part from (4.12). 18 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Figure 1. This figure illustrates the construction of pp(η; A, D). The boundaries of sets A and D are shown in black, and that of the neighborhood B(D, Ln) is shown in orange. The blue curve indicates the trajectory η, where the union of the solid segments represent the proper part pp(η; A, D). Lemma 4.3. We can properly fix k and K such that the following holds. There exists C1 and C2 = C2(K) ∈(0, ∞) such that for all M > C2 and θ1, θ2 1 -ε/8 and for a random trajectory ζ ∼μx,η, (4.18) E[eζ(bζ)] > (1 -ε/4)εd μ2(ρn) μ1(ρn)(1 + log μ1(ρn)1d=4)-1. The proof is a combination of proved estimates and standard computation of equilibrium measures, so we put it in the appendix. From now on, we fix k and K and assume that θ1, θ2 C2, so that Lemma 4.3 holds. Good sequence of trajectory sets: In the Algorithm, we will need a criterion for the goodness of a round of exploration at a certain vertex which is defined through the notion of good sequence of trajectory sets presented below. To make our expositions below more concise, we start with general definitions. For finite A, B ⊂Zd and ρ ∈M, define (4.19) φ(ρ)(A, B) := X x∈A,y∈B e(ρ) A (x)e(ρ) B (y)g(x, y), which measures the likelihood of a trajectory from FRI with length distribution ρ hitting both A and B. For A, D ⊂Zd and η ∈W [0,∞), we define the proper part of η with respect to (A, D) by pp(η; A, D) := bη ∩ η[τ η A -R2 n/In, τ η A + R2 n/In] ∪B(D, Ln) c , see Figure 1. In the sequel, we will take A to be the vertex set induced by the latest layer we have revealed, and D that induced by all the previous layers. For finite collections of trajectories A1, ..., At, D0, ..., Ds ⊂W [0,∞), write A0 = Ds. For all η ∈Ai(1 ≤i ≤t), we write ηi = pp η; V (Ai-1), V s[ j=0 Dj ∪ i-1 [ j=0 Aj . We introduce a parameter (4.20) k1 = k1(ε, q) ∈N 19 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions which is dependent on q introduced in Proposition 4.7, whose value will be fixed in (5.4). Let eCn be the typical capacity of a random walk running (Rn)2 steps: (4.21) eCn = ( R2 n, d ≥5; R2 n log-1 R2 n, d = 4 Definition 4.4. We say (Ai)1≤i≤t is good with respect to (Di)0≤i≤s if the following holds for all i ∈{1, ..., t}. • For all η1, η2 ∈Ai, cap(ρn)(bηj\ηi j) 0, we can choose β, k1 and θ1 introduced in (4.24), (4.20) and (4.6) respectively such that (4.27) P[Failm|Fm,0](ω) 0 such that for all k1 > C3 and θ1 1 -r whenever א1 m ̸= ∆for large n. Adapting part of the proof of the above proposition, we obtain the following coupling between the sequence of layers (Lx,i)1≤i≤α and the branching process (Yi)1≤i≤α. Proposition 5.2. Arbitrarily choose β ∈N in (4.24) and r ∈(0, 1). Let m ∈N and assume that א1 m = x ∈Ln. Then, there exists some C∗ 4 = C∗ 4(β, r) such that for all θ1 > C∗ 4 and all large n, we can find a coupling Q of (Lx,i)1≤i≤α with law conditioned on Fm,0 and (Yi)i≥0 with law P Sx Y satisfying (5.2) Q[Lx,i = Yi, i = 1, 2, ..., α] > 1 -r. Proposition 5.3. For all r > 0, there exists C5 = C5(r) such that the following holds. For all β > C5 and k1 > C3(β, r/2) in Proposition 5.1, there is C6 = C6(β, r) such that for θ1 1 -r whenever א1 m ̸= ∆for large n. Clearly, Proposition 4.7 follows from combining Propositions 5.1 and 5.3. 23 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Proof of Proposition 4.7 assuming Propositions 5.1 and 5.3. We fix (5.4) β > C5(ε, q/2), k1 > C3(β, q/2) and θ1 1 -q/2; by Proposition 5.3, P[Seed(א1 m)|Fm,0] > 1 -q/2. The claim (4.27) then follows by noting that Failm = Good(א1 m)c ∪Seed(א1 m)c. □ The remaining part of this section is organized as below. In Subsection 5.1, we prove Proposition 5.1 by induction, and show that techniques involved in the proof lead to Proposition 5.2. In Subsection 5.2, we prove Proposition 5.3 by proving a similar result for the branching process (Y )i≥0 using the LCLT and then applying Proposition 5.2. 5.1. On the goodness of exploration. In this subsection, we prove Proposition 5.1 by iteration: we define Good(x, i) := {the exploration at x is good until layer i}, i = 0, 1, ..., α, and compute P[Good(x, i + 1)|Good(x, i)] for i ≤α -1. To facilitate the computation, we condition on Fm,i and try to couple the conditional law of Lx,i+1 with that of the union of independent point processes with distributions X u+ n ,ρn[range(η)] for all η ∈Lx,i. Therefore, we get as a byproduct a coupling between (Lx,i)1≤i≤α conditioned on Fm,0 and a branching process with agents in the space W [0,∞), which will also be crucial for the proof of Proposition 5.3. We reduce the proof of Proposition 5.1 to the iteration of the following result. Proposition 5.4. Arbitrarily fix β in (4.24), then for all r > 0, then there exists C7(β, r), such that for all k1 > C7, there is some C8 = C8(β, k1, r) satisfying for all θ1 1 -r. Proof of Proposition 5.1 assuming Proposition 5.4. Take r′ = r′(r) > 0 small so that (1 -r′)α > 1-r, and let k1 > C7(β, r′) and θ1 (1 -r′)α > 1 -r. Thus, we conclude the proof by taking C3(β, r) = C7(β, r′) and C4(β, k1, r) = C8(β, k1, r′). □ We gather an ingredient for the proof of Proposition 5.4. In order to compute the conditional probability of Good(א1 m, i+1), we first simplify the conditional law of the (i+1)'th layer. We write Lא1m,i = {η1, ..., ηs}. As mentioned at the beginning of this subsection, we construct independent point processes with distributions X u+ n ,ρn[range(ηj); ∅; ˆηj], j = 1, ..., s and couple Lx,i+1 with their union. Formally, let (Jx,i,l)x∈Ln,1≤i≤α,l∈N be a collection of i.i.d. PPPs that are independent of X u+ n ,ρn but share the same distribution, and let J x,i,l denote the restriction of Jx,i,l to Tn. Then, we have the following lemma. Lemma 5.5. Arbitrarily fix β, k1 and θ1. Conditioning on Fm,i and with restriction on Good(א1 m, i), write x = א1 m, Lx,i = {η0, η1, ..., ηs}, and let Kx,i+1 = s[ j=0 J x,i+1,j[range(ηj); ∅; bηj]. Then for all r > 0, there exists a coupling Q between Lx,i+1 with law conditioned on Fm,i and Kx,i+1 such that Q[Lx,i+1 ̸= Kx,i+1] 1 -r when n is large. We define the so called bad part of X by Xbad := {η ∈X : η first hits Lx,i in the improper part of some trajectory, or hits both ηj and ηl for some j ̸= l, 0 ≤j, l ≤s} In fact, Jj, 0 ≤j ≤s and Xbad are all empty with probability tending to 1 when n tends to infinity, which directly leads to X = X ′. Let L′′ x,i = S η∈Lx,i ˆη\η. Since the exploration at x is good until the i'th layer, s ≤βki 1, by computing the intensity of PPPs Xbad and Jj, 0 ≤j ≤s, we get (5.5) P[Xbad ̸= ∅] ≤u+ n cap(ρn)(L′′ x,i) + X 0≤j Ln by the definition of ηj, so we have (5.10) φ(ρn)(ηj, V ) ≤cap(ρn)(ηj)cap(ρn)(V ∩B(x, 2γRn)) · CL2-d n ≤C · ( eCn)2L2-d n . This finishes the proof. □ Now we are in a good position to prove Proposition 5.4. Proof of Proposition 5.4. By Lemma 5.5, it suffices to prove (5.11) P[Good′(א1 m, i + 1)|Fm,i] > 1 -r/2, where Good′(א1 m, i + 1) := {(Lא1m,1, ..., Lא1m,i, Kא1m,i+1) is good w.r.t. Pm}. Recall the three restrictive items for goodness of exploration in Definition 4.4. We are going to verify them respectively. First, we deal with the third restriction that for all η ∈Lx,i, the number of trajectories in Lx,i+1 intersecting ˆη is no larger than k1. We need to control the probability that J x.i+1,j[range(ηj); ∅; bηj], a PPP with intensity at most u+ n cap(ρn)(ηj), contains more than k1 trajectories. Note that for N ∼Poi(u+ n K eCn), there is some C = C(K) k1] k1}, then by (5.12), we can take a large constant C7(β, r) such that when k1 > C7, (5.13) P[E1|Fm,i] ≤s · C2-k1 ≤βkα 1 2-k1 4 eCn/I1/2 n }, E5 = {∃ζ1, ζ2 ∈Kא1m,i+1, s.t. φ(ρn)(ζ1, ζ2) > eCn/In}, and we aim to bound the conditional probability of E4 ∪E5 from above. Instead of conditioning on Fm,i alone, this time we additionally condition on the number, lengths and hitting points of trajectories within Kא1m,i+1. More precisely, let Nj = Jx.i+1,j[range(ηj); ∅; bηj] , j = 1, ..., s, and define G = σ{Nj, T(ζ), t(ζ, ηj), ζ(t(ζ, ηj)), ∀1 ≤j ≤s, ζ ∈Jx.i+1,j[range(ηj); ∅; bηj]}. We claim that there exists C8(β, k1, r) such that for all θ1 6. Now we prove (5.15) with Lemmas 5.6 and 5.7. Proof of (5.15). Write Jx,i+1,j[range(ηj); ∅; bηj] = PNj l=1 δζj,l, Tj,l = T(ζj,l), t1 j,l = t(ζj,l, ηj), t2 j,l = Tj,l -t1 j,l, xj,l = ζj,l(t1 j,l). By a union bound, P[Ec 4|Fm,i, G] ≤ X 0≤j≤s,1≤l≤Nj P[cap(ρn)(bζj,l\ζj,l) > 4 eCn/I1/2 n |Fm,i, G], (5.16) P[Ec 5|Fm,i, G] ≤ X (j1,l1)̸=(j2,l2) P[φ(ρn)(ζj1,l1, ζj2,l2) > eCn/In|Fm,i, G]. (5.17) (Here, the proper part ζj,l for ζj,l that is atypical is similarly defined.) Now, it suffices to bound every term on the right-hand side of (5.16) and (5.17) from above. By Lemma 2.5, conditionally on Fm,i and G, with the restriction Good(א1 m, i) ∩Ec 1, the distribution of ζj,l is μ(xj,l, t1 j,l, t2 j,l; range(ηj)), for all 0 ≤j ≤s, 1 ≤l ≤Nj, and they are mutually independent. Let ζf j,l = ζj,l[t1 j,l + R2 n/In, Tj,l], ζb j,l = ζj,l[0, t1 j,l -(Rn)2/In]. Recall the constant C8 from Proposition 27 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions 5.4. By Lemma 4.2, we can take C8(β, k1, r) sufficiently small such that for all θ1 4 eCn/I1/2 n |Fm,i, G] ≤P[cap(ρn)(ζj,l[t1 j,l -R2 n/In, t1 j,l -tn]) + cap(ρn)(ζj,l[t1 j,l + tn, t1 j,l + R2 n/In]) | Fm,i, G] + P[cap(ρn)(ζf j,l ∩V ′) > eCn/I1/2 n |Fm,i, G] + P[cap(ρn)(ζb j,l ∩V ′) > eCn/I1/2 n |Fm,i, G] ≤P xj,l[cap(ρn)(X[-(Rn)2/In, -tn]) + cap(ρn)(X[tn, (Rn)2/In]) > eCn/I1/2 n ] + P xj,l[cap(ρn)(X[R2 n/In, t2 j,l] ∩B(V ′, Ln)) > eCn/I1/2 n ] + P xj,l[cap(ρn)(X[-t1 j,l, -R2 n/In] ∩B(V ′, Ln)) > eCn/I1/2 n ] + 3r/N, where the second inequality above follows from the coupling between ζj,l and Xj,l. We now estimate the probabilities appearing in the last three lines of (5.18) which we denote by p1, p2 and p3 respectively. By (2.30), cap(ρn)(X[0, (Rn)2/In]) ≤R2 n/In · log-10 Rn + cap(X[-R2 n/In, R2 n/In], R2 n/In) ≤eCn/4I1/2 n + cap(X[-R2 n/In, R2 n/In], R2 n/In). Thus, we have p1 ≤2P 0[cap(ρn)(X[0, R2 n/In]) > eCn/2I1/2 n ] ≤2P 0[cap(X[0, (Rn)2/In], R2 n/In) > eCn/4I1/2 n ]. By Proposition 2.14 and the Markov inequality, (5.19) p1 ≤CE[cap(X[0, (Rn)2/In], R2 n/In)] · I1/2 n / eCn ≤CI-1/2 n . Obviously, p2 = p3, so we only bound the former from above. In the four-dimensional case, by (2.30), we have p2 ≤P xj,l[cap(X[R2 n/In, KR2 n] ∩B(V ′, Ln), R2 n) > 1 2 eCn/I1/2 n ]; while in the higher dimensional case, by the trivial upper bound of the ρn-capacity, p2 ≤P xj,l X[R2 n/In, KR2 n] ∩B(V ′, Ln) > 1 2 eCn/I1/2 n . In short, we have (5.20) p2 ≤P xj,l[ξ > 1 2 eCn/I1/2 n ], where ξ = ( cap(X[R2 n/In, KR2 n] ∩B(V ′, Ln), R2 n), d = 4; X[R2 n/In, KR2 n] ∩B(V ′, Ln) , d ≥5. Note that since B(V ′, Ln) consists of the 2Ln-neighborhood of at most (4γ + 1)d · βkα 1 good trajectories, |B(V ′, 2Ln)| ≤CR2 nLd-2 n I1/3 n . Thus, by (5.20) and Lemma 5.6, (5.21) p2 ≤P xj,l[Y > 1 2 eCn/I1/2 n ] ≤2I1/2 n / eCn · Exj,l[Y ] ≤CI1/2 n / eCn · R2 nLd-2 n I1/3 n · (R2 n/In)1-d/2(1 + log Rn · 1d=4)-1 ≤CI-1/6 n . 28 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Combining (5.18), (5.19) and (5.21) gives (5.22) P[cap(ρn)(bζj,l\ζj,l) > 4 eCn/I1/2 n |Fm,i, G] ≤4r/N for all large n. To estimate the terms on the right-hand side of (5.17), we write for all y, z ∈Zd Φ(y, z) = Ey,z[φ(ρn)(X1[0, K(Rn)2], X2[0, K(Rn)2])]. By Lemma 5.7, (5.23) sup y,z∈Zd Φ(y, z) ≤ ( CRn, d ≥5; CR2 n log-2 Rn, d = 4. Therefore, we have for all large n, (5.24) P[φ(ρn)(ζj1,l1, ζj2,l2) > eCn/In|Fm,i, G] ≤P[φ(ρn)(ζf j1,l1, ζf j2,l2) + φ(ρn)(ζf j1,l1, ζb j2,l2) + φ(ρn)(ζb j1,l1, ζf j2,l2) + φ(ρn)(ζb j1,l1, ζb j2,l2) > eCn/In|Fm,i, G] ≤In/ eCn · E[Φ(ζj1,l1(t), ζj2,l2(t)) + Φ(ζj1,l1(t), ζj2,l2(-t)) + Φ(ζj1,l1(-t), ζj2,l2(-t)) + Φ(ζj1,l1(t), ζj2,l2(-t))|Fm,i, G] + 2r/N (5.23) ≤I-1 n + 2r/N 1 -r for all i ∈{0, 1, ..., α -1}. Repeating the proof of Proposition 5.1, but with Proposition 5.8 applied iteratively (instead of Proposition 5.4), we readily obtain Proposition 5.2. 5.2. Finding seeds for neighboring vertices. This subsection is dedicated to the proof of Proposition 5.3. Recall β from (4.24), which is the number of trajectories in a seed (c.f. Definition 4.26), and the definition of events (4.25) and (4.26). Assuming goodness of the exploration at x ∈Ln, we only need to show the existence of at least β trajectories in Lx,α that stay inside Bx+Rn·e for all e ∈Ud ∪(-Ud). Moreover, by Proposition 5.2, we can replace Lx,α with Yα in the last statement. Therefore, it suffices to show: Proposition 5.9. There exists C9 > 0 such that for an arbitrary θ1 0, there exists a large constant C10 = C10(r), such that for all β > C10 and S ⊂W [0,∞) composed of β good trajectories staying inside [0, Rn)d and any e ∈Ud ∪(-Ud), the event Ee := {Yα contains β good trajectories staying inside Bn Rn·e} happens with probability at least 1 -r if (Yi)i≥0 ∼P S Y . 29 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Proof of Proposition 5.3 assuming Proposition 5.9. Recall C5, C6 from Proposition 5.3 and C3, C4 from Proposition 5.1. Take C5(r) = C10(r/8d) and C6(β, r) = C4(β, r/2) ∧C9. Let Seed′(א1 m) = \ e∈Ud∪(-Ud) {∃β good trajectories ∈Lא1m,α staying inside א1 m + Rn · e + [0, Rn)d}, and let Seed′′(א1 m) be the above event with Lא1m,α replaced by Yα. Take k1 > C3(β, r/2), β > C5 and θ1 P[Seed′(א1 m)|Fm,0] -r/2. By Proposition 5.2, for all large n. (5.26) P[Seed′(א1 m)|Fm,0] ≥P Sא1m[Seed′′(א1 m)] -r/4 Moreover, by Proposition 5.9, (5.27) P Sא1m Y [Seed′′(א1 m)] > 1 -2d · r/8d = 1 -r/4. The proposition then follows from (5.25), (5.26) and (5.27). □ The remainder of the subsection is devoted to proving Proposition 5.9. For simplicity, we always assume (Yi)i≥0 ∼P S Y from now on. We aim to obtain the "spatial distribution" of trajectories in Yα by studying the spatial distribution of locations where the offspring trajectories hit their parent. To this end, we define a process (Y i)i≥0 dominated by (Yi)i≥0 whose (i + 1)-th layer is generated from the i-th layer by first sampling the hitting locations whose collection will be denoted by Zi, and then sampling paths from them. Noting that (Zi)i≥0 form a branching process with agents in Zd, we bound from below first the number of hitting points in Zα-1 that are close enough to the center of each neighboring box Bx+Rn·e, then the conditional probability that the trajectories stemming from those points stay inside Bx+Rn·e, which yields Proposition 5.9. Now we make rigorous the aforementioned idea. For S ⊂W [0,∞), we construct two processes (Y i)i≥0 ∈{0, 1}W [0,∞) and (Zi,η)i≥0,η∈Y i ∈{0, 1}Zd simultaneously under a new probability measure P S. Let Y 0 = S. Assume that Y i is sampled, then for η ∈Y i, let Zi,η be an independent PPP supported on range(η) ⊂Zd with density measure X x∈bη u+ n (1 -ε/2) · eη(x) · δx, sample one new trajectory ζx for each x ∈Zi,η independently according to distribution μx,η and set Y i+1 = S η∈Y i{ζx : x ∈Zi,η}. We write Zi = S η∈Y i Zi,η. By construction, (Y i)i≥0 is also a branching process with agents in W [0,∞). In general, (Y i)i≥0 "trims off" the excessive intensity of descendant trajectories introduced in (Yi)i≥0 due to the finiteness of backward paths and highlights the "branching" nature of the process more explicitly without introducing PPPs on W [0,∞). The following lemma reveals the relation between (Yi)i≥0 and (Y i)i≥0, whose proof is straightforward given Lemma 4.3. Lemma 5.10. For large n, the following holds: for all S ⊂Tn, there exists a coupling Q between (Yi)i≥0 and (Y i)i≥0 such that Q[Yi ⊃Y i, 0 ≤i ≤α] = 1. Proof of Lemma 5.10. To prove the domination for branching processs, it suffices to prove domination for their respective offspring distributions. Obviously, the offspring distribution of the branching process (Y i)i≥0 at η ∈W [0,∞) is given by a PPP on W [0,∞) with intensity measure π1 = u+ n · X x∈bη (1 -ε/2)eη(x)μx,η. By the definition of (Yi)i≥0, its offspring distribution at η is given by a PPP with intensity π2 = u+ n · X x∈bη X m,l∈N ρn(m + l) μ1(ρn) + 1P x[ eHη > m]μ(x, m, l; range(η))[·; Tn] ≥u+ n · X x∈bη eη(x)μ∗ x,η[·; Tn]. By Lemma 4.3, μ∗ x,η[·; Tn] ≥(1 -ε/8)μx,η, so π1 ≥π2, which leads to the conclusion. □ 30 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions For the aforementioned estimate regarding Zα-1 we are to establish, We define two correlated new processes (Ki)i≥0 ∈Zd and (ζi)i≥0 ∈W [0,∞), and prove a LCLT-type result for (Ki)i≥0. The connection between these processes and (Zi)i≥0 will be elaborated shortly afterwards. For η ∈W [0,∞), let (5.28) eη = X x∈bη eη(x) · δx be the equilibrium measure on range(η) restricted to bη. Recall that we denote by e0 η the probability measure obtained by normalizing eη. Fix η ∈Tn. Let ζ0 = η. Assume that ζi is sampled, we then choose Ki from range(ζi) according to e0 ζi, and then sample ζi+1 according to μKi,ζi. This way, (Ki)i≥0 and (ζi)i≥0 are recursively defined. We denote the probability measure related to (ζi, Ki) by Q η. Using the LCLT, we are able to prove the following result: Proposition 5.11. For all α ∈N in (4.22), there are constants C11 = C11(α), C12 = C12(α) such that for all choices M > C11 and θ1, θ2 C14 and θ1, θ2 1 -r/α and (5.31) Q[d(Ki+1 -Ki, K♢ i+1 -K♢ i ) C14, θ1, θ2 2ln] C14, a trajectory following μx or μ∗ x,η is typical with probability at least 1-r/8α as n →∞, which implies that dTV(μx,η, μ∗ x,η) (1 -θ2 1)hnT(ζ♢)}, where hn = εd(1 + log μ1(ρn)1d=4)-1. 32 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Then, by (2.12), Q[E′] ≥1 -r/2α for all large n. Now, on E′, we aim to couple K and K♢ conditionally on ζ and ζ♢. Let T = T(ζ) = T(ζ♢). Write t∗= t(ζ, η), ζ-= ζ[0, (t∗-tn) ∨0], ζm = ζ[(t∗-tn)∨0, t∗] and ζ+ = ζ[t∗, T]. Define e∗ ζ = eζ-+eζ+, and let e∗∗ ζ be the restriction of eζ to range(ζ-) ∪range(ζ+). Define ζ♢ ±, ζ♢ m, e∗ ζ♢and e∗∗ ζ♢similarly. Note that ζ is always typical, so by definition, ζ ∈E4 can be divided into pieces of length T ′ n(see 4.7 for definition) with capacity no larger than (1 + θ2 1)hnT ′ n, while at the same time we have cap(ζ) ≥(1 -θ2 1)hnT by ζ ∈E2. If we divide ζ-and ζ+ into the aforementioned pieces of length T ′ n, at most one piece may be used twice, so the total length of all the pieces used in both coverings is at most T + T ′ n. Hence, we have cap(ζ-) + cap(ζ+) ≤(1 + θ2 1)hn(T + T ′ n), implying (5.35) cap(ζ-) + cap(ζ+) -cap(ζ) ≤(1 + θ2 1)hn(T + T ′ n) -(1 -θ2 1)hnT 1-C(θ1+θ2) > 1-r/2α, where the second inequality holds for θ1, θ2 small. Then, (5.32) follows since Q[E′] > 1 -r/2α. Now we aim to prove (5.36). Recall the restriction of equilibrium measure eζ from (5.28). Note the following inequalities: (5.37) dTV(eζ, eζ) ≤C(θ1 + θ2) eCn; dTV (eζ, e∗∗ ζ ) ≤tn; dTV(e∗∗ ζ , e∗ ζ) = cap(ζ-) + cap(ζ+) -eζ(ζ-∪ζ+) ≤ cap(ζ-) + cap(ζ+) -cap(ζ) + cap(ζm) ≤Cθ2 1 eCn + tn. Here, the first inequality holds, since by the definition of bζ, dTV(eζ, eζ) = eζ(range(ζ) \ bζ) ≤ ( θ2T(ζ), d ≥5; θ2T(ζ) log-1 μ1(ρn) + θ-1 1 cap(ζ, R2 n) -cap(ζ) , d = 4 which, combined with the requirements of E1 and E2 in the definition of typical trajectories, yields (5.38) dTV(eζ, eζ) = eζ(range(ζ) \ bζ) ≤C(θ1 + θ2) eCn Summing up the inequalities aligned in (5.37), we get dTV(eζ, e∗ ζ) 1 -r/α for all i ≥0, where Fi = σ{(ζ□ j )0≤j≤i, (ζ♢ j )0≤j≤i, (K□ j )0≤j≤i, (K♢ j )0≤j≤i}. (5.42) is trivial because K♢ 0 and K□ 0 are identically distributed. Similar to Lemma 5.14, we work in a simpler setup to obtain (5.43). Let ζ♢, ζ□∼μ0. Conditionally on ζ♢, choose K♢according to e0 ζ♢; conditionally on ζ□, let K□= ζ□(j), where j is uniformly chosen from {0, 1, ..., T(ζ□)}. In order to obtain (5.43), it suffices to construct a coupling Q such that (5.44) Q[d(K♢, K□) ≥Rn/16α] 0 is a small but fixed number to be specified later. Recall hn from the proof of Lemma 5.14, and let E = E1 ∩E2 where E1 = {cap(ξj) ≤(1 + r/16αN1)hnT ′′, ∀0 ≤j ≤N1 and cap(ζ[0, N1T ′′]) ≥(1 -r/16αN1)hnN1T ′′}, E2 = {diam(ξj) 0 and C′ = C′(d, α) > 0 such that lim sup n→∞Q[Ec 2] ≤Cδ-1 exp -C′δ-1/2 . Thus, we can take δ small such that Q[E] > 1 -r/2α for all large n. On E, we construct a coupling between K♢and K□conditionally on ζ. Let K′ ∼e′ ζ := PN1-1 i=0 eξi 0 , then we can sample K′ as follows. First, randomly choose j′ ∈{0, 1, ..., N1 -1} according to the probability measure proportional to PN1-1 i=0 cap(ξi) · δi, then choose K′ from range(ξj′) according to e0 ξj′. We can similarly sample K□by taking j□∈{0, 1, ..., N1} according to PN1 i=0 T(ξi) T(ζ) ·δi and then letting K□= ξj□(t), where t is uniformly chosen from {0, 1, ..., T(ξj□)}. Note that on E, eζ(ξi) ≤cap(ξj) ≤(1 + r/16αN1)hnT ′′ for all 0 ≤i ≤N1 and eζ(ξi) ≥cap(ζ[0, N1T ′′]) - X j̸=i,j (1 -r/16αN1)hnN1T ′′ > 7 8hnT(ζ) when δ is small so that N1 > 1. Thus, by an argument similar to (5.39), we obtain (5.47) dTV e0 ζ, e′ ζ 1 -r/2α. (5.44) then follows because Q[E] > 1-r/2α and diam(ξj) C11, θ1, θ2 10. Recall the constants C11, C12 from Proposition 5.11, C9 from Proposition 5.9 and C1, C2 from Lemma 4.3. Having fixed α, we now take C9 = C12(α) ∧C1, and fix (5.50) M > C11(α) ∨C2 and θ2 0, recall that WL is the subset of W [0,∞) with diameter at most L. By Lemma 2.1, the following holds: for all x ∈Zd, (5.52) μx(WRn/16) > f(256K). This lower bound, together with Lemma 4.1, implies (5.53) μx,η(WRn/8) ≥1 2f(256K). Let ξ = |{η ∈Y α : η stays within e + [0, Rn)d}|, we show that ξ > β with high probability. We first estimate the first moment. Applying (5.29) and (5.53) in turn gives (5.54) E S[ξ] ≥P S[|Zα-1 ∩eBRn·e|] · 1 2f(256K) ≥P S[|Zα-1|] · C13α-d/2 · 1 2f(256K) ≥kβ 2εd (1 + ε/8)α · C13α-d/2 · 1 2f(256K) > 10β. The third inequality is a consequence of Lemma 4.3, while the last one is due to (5.49). Moreover, we have the following control over the variance of ξ. For W ⊂Tn, let varW denote the variance under P W , and abbreviate varη when W = {η}. For i ≥0, since Y i only contains typical trajectories whose capacities are well controlled, it is easy to verify that conditioning on Y i, the number of offspring of each element of Y i follows a Poisson distribution with parameter smaller than λ = 2K. Thus, by induction, we have for all η ∈Tn, varη(ξ) ≤E η[ξ2] ≤E η[|Y α|2] ≤ 2(λ2 + λ) α. Therefore, (5.55) varS(ξ) = X η∈S varη(ξ) ≤β 2(λ2 + λ) α. By (5.54), (5.55) and the Markov inequality, P S[ξ 9β i ≤9-2β-2varS(ξ) ≤9-2β-1[2(λ2 + λ)]α. The conclusion then follows by taking β large. □ Acknowledgments We warmly thank Eviatar Procaccia for fruitful discussions. YB and XL are supported by National Key R&D Program of China (No. 2020YFA0712900 and No. 2021YFA1002700). BR is partially supported by the grant NKFI-FK-142124 of NKFI (National Research, Development and Innovation Office), and the ERC Synergy Grant No. 810115 - DYNASNET. YZ is supported by NSFC-12271010, National Key R&D Program of China (No. 2020YFA0712902) and the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin 24XNKJ06. 36 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Appendix A. Proofs of technical lemmas In this section, we present the proofs of Lemmas 4.3, 5.6 and 5.7. For the proof of Lemma 4.3, we introduce the following analog of Lemma 4.1 for random walk paths whose law involve conditioning. Lemma A.1. Let ψ be the same as in Lemma 4.1 and take η ∈Tn, x ∈bη and m, l ∈N satisfying kR2 n ≤m + l ≤KR2 n. Then, for all large n, μ(x, m, l; range(η))[T c n ] (1 + 1 2θ2 1)π2 8 T ′ n log-1 μ1(ρn) or diam ζ[jT ′ n, (j + 1)T ′ n ∧T(ζ)] > 1 4T 1/2 log-c μ1(ρn) i + 2θ1 + R-c/2 n ≤C log-1 Rn + 2θ1 + R-c/2 n , where we have estimated the probability appearing in the second and third line similarly as in (4.10). For d ≥5, we can get the same upper bound in a similar fashion. Finally, since x ∈bη, μ(x, m, l)[E] ≥θ2 log-1 μ1(ρn) for all large n, where E = {η′ ∈W [0,∞) : η′[0, m) ∩range(η) = ∅}. Thus, by (4.9) and (4.11), (A.3) μ(x, m, l; range(η))[Ec 2 ∪Ec 3] ≤μ(x, m, l)[Ec 2 ∪Ec 3] μ(x, m, l)[E] ≤ C log-2 Rn + C log8c/3 Rn exp -log2c/3 Rn log-1 Rn ≤C log-1 Rn for all large n. The conclusion follows from (A.1), (A.2) and (A.3). □ 37 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Proof of Lemma 4.3. Recall the notation μ(m) 2 (ρn), m > 0 which is the tail of the second moment of ρn from Section 2. Since (ρn)n≥0 is an appropriate family, by (1.2), we can choose K > 1 large such that (A.4) μ(KR2 n) 2 (ρn) ≤1 32εμ2(ρ) for all n ∈N. Fix a K such that the above estimate holds, and arbitrarily fix k satisfying k2 1 -ε/8K > 1 -ε/8. Now we aim at estimating eζ(bζ). By the fact that ζ ∈Tn ⊂E2, (A.5) cap(ζ) ≥(1 -θ2 1)εdT(ζ)(1 + log μ1(ρn)1d=4)-1. Let ζ∗∼μ∗ x,η, then by (A.4) and k2 0 such that P y[Xj = z] 0, z ∈Zd. Hence, we have Ey h X[T, KR2 n] ∩A i ≤Ey[ ∞ X j=T 1Xj∈A] ≤C ∞ X j=T |A|j-d/2 ≤C|A|T 1-d/2. Now, we assume d = 4. We divide the trajectory X[T, KR2 n] into pieces of length T ′ n = ⌊R2 n log-1 Rn⌋, and write ξj = X[T + (j -1)T ′ n, T + jT ′ n], xj = ξj(0) for all j ≤⌈KR2 n/T ′ n⌉. Note that xj and cap(range(ξj) ∩A, R2 n) are independent, so we have Ey[cap(range(ξj) ∩A, R2 n)] = Ey[cap(range(ξj) ∩A, R2 n)1xj∈B(A,Ln)] + Ey[cap(range(ξj) ∩A, R2 n)1xj /∈B(A,Ln)] ≤Ey[cap(range(ξj), R2 n log-1 Rn)] · P y[xj ∈B(A, Ln)] + Ey[T ′ n1ξj̸⊂B(xj,Ln)1xj /∈B(A,Ln)] ≤CR2 n log-2 Rn · P y[xj ∈B(A, Ln)] + T ′ n · P 0[X[0, T ′ n] ̸⊂B(0, Ln)] ≤CR2 n log-2 Rn|B(A, Ln)|(T + jT ′ n)-2 + T ′ n exp -L2 n/2T ′ n , 38 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions where in the last inequality we have used the LCLT and a standard estimate on the transition probability of the simple random walk. Take a summation over j ∈{1, ..., ⌈KR2 n/T ′ n⌉}, and we obtain Ey[cap(X[T, KR2 n] ∩A, R2 n)] ≤CR2 n log-2 Rn|B(A, Ln)|(T ′ n)-2 ∞ X j=1 (T/T ′ n + j)-2 + CR2 n exp -1 2 log1-2c Rn ≤C|B(A, Ln)|T -1 log-1 Rn + CR2 n exp -1 2 log1-2c Rn . It remains to show that the second term in the last line is much smaller than the first term. This follows from the trivial bound |B(A, Ln)| ≥L4 n = R4 n log-4c Rn. Now, we have proved the four-dimensional case. □ Proof of Lemma 5.7. When d ≥5, we have Ey,z[φ(ρn)(X1[0, K(Rn)2], X2[0, K(Rn)2])] ≤Ey,z KR2 n X j,l=0 g(X1 j , X2 l ) = X y1,z1∈Z2 KR2 n X j=0 pj(y, y1) g(y1, z1) KR2 n X j=0 pj(z1, z) = X 0≤t1,t2≤KR2 n s≥0 X y1,z1∈Zd pt1(y, y1)ps(y1, z1)pt2(z1, z) = X 0≤t1,t2≤KR2 n s≥0 pt1+t2+s(y, z) = KR2 n X j=0 j2pj(y, z) + K2R4 n X j>KR2n pj(y, z) ≤C KR2 n X j=0 j2-d/2 + CR4 n X j>KR2n j-d/2. The conclusion then follows from direct computation of the series. When d = 4, we proceed as in the last lemma, and let ξi j = Xi[(j -1)Ln, jLn], xi J = ξi j(0) for i = 1, 2, j > 0. For U, V ⊂Z4, write eg(U, V ) = sup u′∈B(U,L1/2+c n ),v′∈B(V,L1/2+c n ) g(u′, v′), and abbreviate eg(u, v) = eg({u}, {v}). Note that by (2.14) and (2.30), E0[cap(ρn)(X[0, Ln])] ≤Ln log-10 Rn + E0[cap(X[0, Ln], R2 n)] ≤Ln log-10 Rn + E0[cap(X[0, Ln], Ln)] ≤CLn log-1 Rn for some C > 0. Thus, conditioning on x1 j, x2 l , such that |x1 j -x2 l | > 3L1/2+c n , we have (A.7) Ey,z[φ(ρn)(ξ1 j , ξ2 l )|x1 j, x2 l ] ≤Ey,z[cap(ρn)(ξ1 j )cap(ρn)(ξ2 l )]eg(x1 j, x2 l ) + 2P 0[X[0, Ln] ̸⊂B(0, L1/2+c n )] · L2 n ≤E0[cap(ρn)(X[0, Ln])]2eg(x1 j, x2 l ) + CL2 n exp -1 2L2c n ≤CL2 n log-2 Rn · g(x1 j, x2 l ) + CL2 n exp -1 2L2c n . Moreover, by the same argument as in the high-dimensional case, we obtain (A.8) Ey,z[φ(ρn)(ξ1 j , ξ2 l )|x1 j, x2 l ] ≤CLn. 39 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions Combining (A.7) and (A.8), (A.9) Ey,z[φ(ρn)(ξ1 j , ξ2 l )] ≤CL2 n log-2 Rn · Ey,z[g(x1 j, x2 l )] + CL2 n exp -1 2L2c n + CLnP y,z[|x1 j -x2 l | ≤3L1/2+c n ]. Now, we bound the terms on the right-hand side of (A.9) from above. On the one hand, note that the following inequalities hold if j, l ≤KR2 n/Ln, j + l > Rn log-2+c Rn: P y,z[|x1 j -x2 l | ≤3L1/2+c n ] ≤ CL2+4c n (R2n log-2 Rn)2 ≤CR-2+4c n log4 Rn. Thus, we have (A.10) X j,l≤KR2 n/Ln j+l>Rn log-2+c Rn P y,z[|x1 j -x2 l | ≤3L1/2+c n ] ≤CR-2+4c n log4 Rn · R4 nL-2 n ≤CR4c n log4+2c Rn. On the other hand, applying twice the techniques in the proof of the case where d ≥5, we get (A.11) Ey,z h φ(ρn)(X1[0, R2 n log-2 Rn], X2[0, R2 n log-2 Rn]) i ≤CR2 n log-2 Rn; (A.12) Ey,z ⌈KR2 n/Ln⌉ X j,l=1 g(x1 j, x2 l ) ≤CR2 nL-2 n . Therefore, by (A.9)-(A.12), Ey,z h φ(ρn)(X1[0, KR2 n], X2[0, KR2 n]) i ≤Ey,z h φ(ρn)(X1[0, R2 n log-2 Rn], X2[0, R2 n log-2 Rn]) i + X j,l≤KR2 n/Ln j+l>Rn log-2+c Rn Ey,z[φ(ρn)(ξ1 j , ξ2 l )] ≤CR2 n log-2 Rn + CL2 n log-2 Rn · Ey,z ⌈KR2 n/Ln⌉ X j,l=1 g(x1 j, x2 l ) + CL2 n exp -1 2L2c n · (KR2 n/Ln)2 + CLn X j,l≤KR2 n/Ln j+l>Rn log-2+c Rn P y,z[|x1 j -x2 l | ≤3L1/2+c n ] ≤CR2 n log-2 Rn + CL2 n log-2 Rn · R2 nL-2 n + C + CLn · R4c n log4+2c Rn ≤CR2 n log-2 Rn, proving the lemma. □ References [1] Amine Asselah, Bruno Schapira, and Perla Sousi. Capacity of the range of random walk on Zd. Transactions of the American Mathematical Society, 370(11):7627-7645, 2018. [2] Amine Asselah, Bruno Schapira, and Perla Sousi. Capacity of the range of random walk on Z4. The Annals of Probability, 47(3):1447-1497, 2019. [3] Yijie Bi, Zhenhao Cai, Xinyi Li, and Yuan Zhang. Work in progress. [4] M ́arton Borb ́enyi, Bal ́azs R ́ath, and S ́andor Rokob. Random interlacement is a factor of i.i.d. Electronic Journal of Probability, 28(none):1-45, 2023. [5] Lewis Bowen. Finitary random interlacements and the gaboriau-lyons problem. Geometric and Functional Analysis, 29(3):659-689, 2019. [6] Zhenhao Cai, Yunfeng Xiong, and Yuan Zhang. On (non-) monotonicity and phase diagram of finitary random interlacement. Entropy, 23(1):69, 2021. [7] Zhenhao Cai and Yuan Zhang. Some rigorous results on the phase transition of finitary random interlacements. Electronic Communications in Probability, 26:1-11, 2021. 40 Bi, Cai, Li, R ́ath and Zhang FRI critical threshold in 4 and higher dimensions [8] Zhenhao Cai and Yuan Zhang. On the exact orders of critical value in finitary random interlacements. Stochastic Processes and their Applications, 159:391-427, 2023. [9] Alexander Drewitz, Bal ́azs R ́ath, and Art ̈em Sapozhnikov. An introduction to random interlacements. Springer, 2014. [10] Hugo Duminil-Copin, Subhajit Goswami, Pierre-Fran ̧cois Rodriguez, Franco Severo, and Augusto Teixeira. Finite range interlacements and couplings. arXiv preprint , 2023. [11] Hugo Duminil-Copin, Subhajit Goswami, Pierre-Fran ̧cois Rodriguez, Franco Severo, and Augusto Teixeira. Phase transition for the vacant set of random walk and random interlacements. arXiv preprint , 2023. [12] Hugo Duminil-Copin, Subhajit Goswami, Pierre-Fran ̧cois Rodriguez, Franco Severo, and Augusto Teixeira. A characterization of strong percolation via disconnection. Proceedings of the London Mathematical Society, 129(2):e12622, 2024. [13] Dirk Erhard and Julien Poisat. Asymptotics of the critical time in wiener sausage percolation with a small radius. ALEA, 13:417-445, 2016. [14] Naresh Jain and Steven Orey. On the range of random walk. Israel Journal of Mathematics, 6(4):373-380, 1968. [15] Gregory F. Lawler. Intersections of Random Walks. New York : Springer Science+Business Media, LLC, 1st ed. 1991 edition, 1991. [16] Thomas M. Liggett, Roberto H. Schonmann, and Alan M. Stacey. Domination by product measures. The Annals of Probability, 25(1):71-95, 1997. [17] Mathew D. Penrose. On the spread-out limit for bond and continuum percolation. The Annals of Applied Probability, 3(1):253-276, 1993. [18] Eviatar B. Procaccia, Jiayan Ye, and Yuan Zhang. Percolation for the finitary random interlacements. ALEA, 18:265-287, 2021. [19] Bruno Schapira. Capacity of the range in dimension 5. The Annals of Probability, 48(6):2988-3040, 2020. [20] Alain-Sol Sznitman. Upper bound on the disconnection time of discrete cylinders and random interlacements. The Annals of Probability, 37(5):1715-1746, 2009. [21] Alain-Sol Sznitman. Vacant set of random interlacements and percolation. Annals of Mathematics, 171(3):2039-2087, 2010. [22] Alain-Sol Sznitman. On the critical parameter of interlacement percolation in high dimension. The Annals of Probability, 39(1):70-103, 2011. [23] Alain-Sol Sznitman. Decoupling inequalities and interlacement percolation on G × Z. Inventiones Mathematicae, 187(3):645-706, 2012. [24] Alain-Sol Sznitman. Random interlacements and the Gaussian free field. The Annals of Probability, 40(6):24002438, 2012. [25] Alain-Sol Sznitman. Disconnection, random walks, and random interlacements. Probability Theory and Related Fields, 167(1):1-44, 2017. [26] Augusto Teixeira and David Windisch. On the fragmentation of a torus by random walk. Communications on Pure and Applied Mathematics, 64(12):1599-1646, 2011. (Yijie Bi) : (Zhenhao Cai) Faculty of Mathematics and Computer Science, Weizmann : (Xinyi Li) Beijing International Center for Mathematical Research, Peking University Email address: (Bal ́azs R ́ath) - nology and Economics; HUN-REN Alfr ́ed R ́enyi : (Yuan Zhang) Center for Applied Statistics and : zhang 41
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2510.14732
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-EP-2025-221
LHCb-PAPER-2025-036
October 16, 2025
Measurement of CP asymmetry in
D0 →K0
SK0
S decays with the LHCb
Upgrade I detector
LHCb collaboration†
Abstract
A measurement of CP asymmetry in D0 →K0
SK0
S decays is reported, based on a
data sample of proton-proton collisions collected with the LHCb Upgrade I detector
in 2024 at a centre-of-mass energy of 13.6 TeV, corresponding to an integrated
luminosity of 6.2 fb−1. The D0 →K0
Sπ+π−decay is used as calibration channel to
cancel residual detection and production asymmetries. The time-integrated CP
asymmetry for the D0 →K0
SK0
S mode is measured to be
ACP (D0 →K0
SK0
S) = (1.86 ± 1.04 ± 0.41)%,
where the first uncertainty is statistical, and the second is systematic. This is the
most precise determination of this quantity to date.
Submitted to JHEP
© 2025 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.
†Authors are listed at the end of this paper.
arXiv:2510.14732v1 [hep-ex] 16 Oct 2025
ii
1
Introduction
Violation of charge-parity (CP) symmetry in charm-quark transitions has been observed
for the first time in the measurement of the difference between the CP asymmetry in
D0 →K+K−and D0 →π+π−decays [1]. As of today, this measurement remains the only
experimental evidence of CP violation in charm decays, and more generally in any up-type
quark (u, c, or t) transition. A successive measurement in the D0 →K+K−decay mode
found no statistically significant CP asymmetry [2], leaving the D0 →π+π−decay as the
only charm-hadron decay with experimental evidence of CP violation [3,4].
While the existence of CP violation in the charm-quark sector at the observed O(10−3)
level is not excluded in the Standard Model (SM), uncertainties in theoretical calculations
do not allow for a quantitative comparison or precise predictions for other decay channels [5–
16]; see Refs. [17–19] for reviews on this subject. Further experimental measurements of
CP asymmetries in charm decays are therefore highly desirable, as they would provide
deeper insight into CP violation phenomenology. Such measurements could potentially
involve dynamics beyond the SM [20–26], that are not constrained to be the same as in
the down-type quark (d, s, or b) sector.
Among many charm-hadron decay modes in which CP violation could manifest itself,
the Cabibbo-suppressed D0 →K0
SK0
S mode stands out for being analogous to the two-
body decays D0 →K+K−and D0 →π+π−[1], but has the possibility of a more sizeable
asymmetry. Only amplitudes proceeding via tree-level exchange – which vanish in the
flavour-SU(3) limit – and electroweak loop diagrams contribute to this decay, which
therefore has a relatively small branching fraction [27]. Its CP asymmetry is defined by
ACP(K0
SK0
S) ≡Γ(D0 →K0
SK0
S) −Γ(D0 →K0
SK0
S)
Γ(D0 →K0
SK0
S) + Γ(D0 →K0
SK0
S),
(1)
with Γ being the decay width of the neutral D meson. Any effect arising from regeneration,
mixing and CP violation in the K0
SK0
S final state cancel in ACP, since the D0 and D0 decay
to the same CP-symmetric K0 −K0 final state [28]. In the SM, ACP(K0
SK0
S) might be
significantly larger than in the D0 →π+π−and D0 →K+K−channels. Several predictions
exist for ACP(K0
SK0
S) [9,29–32], up to the percent level. The study of the D0 →K0
SK0
S
decay mode provides a valuable probe of CP violation in the charm sector, offering a
means to discriminate between competing theoretical approaches.
These considerations have motivated several experimental determinations of this
observable across different experiments, including some recent ones [33–38]. The world
average of the CP asymmetry is ACP(K0
SK0
S) = (−1.19 ± 0.77 ± 0.17)%, where the time-
dependent contribution to the asymmetry is neglected [39,40]. This has now reached the
subpercent level, entering the interesting sensitivity range of theoretical predictions. The
reduced precision of this measurement compared with other charm CP asymmetry results
is due to the smaller branching fraction and the difficulty of reconstructing the K0
S decay
products, given its much longer lifetime relative to bottom and charm hadrons.
In this work, a new measurement of ACP(K0
SK0
S) is reported, based on a sample of
proton-proton (pp) collisions corresponding to an integrated luminosity of about 6.2 fb−1,
collected in 2024 by the LHCb Upgrade I detector. The detector has been significantly
improved over its previous version, including the data acquisition system [41,42], which
enabled the adoption of a software-based selection of K0
S particles at the first trigger level,
resulting in a substantially improved efficiency.
1
The measurement of ACP(K0
SK0
S) requires knowledge of the D0 flavour at production.
A sample of flavour-tagged D0 →K0
SK0
S decays is obtained by selecting only D0 mesons
that originate from D∗(2010)+ →D0π+ decays.1 The charge of the pion in this decay
(“tagging pion”) identifies the flavour of the accompanying neutral charm meson. While
D0 −D0 oscillations can cause a flavour change before decay, the effect is small compared
with the resolution of the current measurement and is therefore neglected [43]. The K0
S
mesons are reconstructed in the π+π−final state.
The decay widths Γ in Eq. 1 are related to the observed yields N by
N(
(
)
D
0 →K0
SK0
S) ∝ε±(D∗±)σ(D∗±) Γ(
(
)
D
0 →K0
SK0
S),
(2)
where σ(D∗±) denotes the production cross-sections of the D∗± mesons and ε±(D∗±) the
corresponding detection efficiencies for their decays. Both factors are charge asymmetric,
due to the D∗± production asymmetry arising from the hadronisation of charm quarks in
pp collisions, and to asymmetries in the geometry and response of the detector. Since the
final state of the D0 decay is self-conjugate, any detection asymmetry arises solely from
the tagging pion. All these quantities are calibrated using a sample of D0 →K0
Sπ+π−
decays, which have the same final-state particles as the D0 →K0
SK0
S decay, but with a
negligible CP asymmetry [18]. In order to avoid experimenter’s bias, the results of the
analysis were not examined until the full procedure had been finalised.
2
LHCb detector
The LHCb Upgrade I detector [44] is a single-arm forward spectrometer covering the
pseudorapidity range 2 < η < 5, designed for the study of particles containing b or c
quarks. The detector was installed prior to the Run 3 data-taking period, which started in
2022. The LHCb Upgrade I detector represents a major change of the detector system and
was designed to match the performance of the Run 1–2 detector [44,45], while allowing it
to operate at approximately five times the luminosity.
The high-precision tracking system consists of a silicon-pixel vertex detector (VELO)
surrounding the pp interaction region [46], a large-area silicon-strip detector (UT) [47]
located upstream of a dipole magnet with a bending power of about 4 T m, and three
stations of scintillating-fibre detectors [47] located downstream of the magnet. Tracks
reconstructed in both the VELO and the scintillating-fiber detectors, with the possible
addition of intermediate hits in UT, are referred to as long tracks. The dipole magnetic-field
polarity is periodically reversed during data taking, alternating between pointing upwards
(MagUp) and downwards (MagDown), to mitigate the effects of differences in reconstruction
efficiencies for particles of opposite charges.
Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors [48].
Readout of all detectors into an all-software trigger [41,42] is a central feature of the
upgraded detector, allowing the reconstruction of events at the maximum LHC interaction
rate, and their selection in real time. The trigger system is implemented in two stages: a
first inclusive stage (HLT1) running on GPU boards based primarily on charged particle
reconstruction, which reduces the data volume by roughly a factor of twenty, and a second
1Inclusion of the charge-conjugate process is implied throughout this document unless explicitly specified.
Hereafter, the D∗(2010)+ meson is referred to as D∗+.
2
stage (HLT2) running on CPUs, which performs the full offline-quality reconstruction and
selection of physics signatures. A large disk buffer is placed between these stages to hold
the data while the real-time alignment and calibration are performed.
For the measurement described in this paper, data collection benefits from the newly
implemented trigger system, which, for the first time, enables the selection of events
containing K0
S candidates directly at the HLT1 level for every LHC bunch crossing. This
leads to an improvement in the signal efficiency by a factor of about three over the previous
LHCb detector, which is the main factor responsible for the increase in precision for the
current measurement.
Simulated samples are used solely to optimise selection requirements. In the simulation,
pp collisions are generated using Pythia [49] with a specific LHCb configuration [50].
Decays of unstable particles are described by EvtGen [51], in which final-state radiation
is generated using Photos [52]. The interaction of the generated particles with the
detector, and its response, are implemented using the Geant4 toolkit [53] as described
in Ref. [54].
3
Event selection
At the HLT1 stage, long tracks with good fit χ2, momentum p > 5 GeV/c, transverse
momentum pT > 0.45 GeV/c, and incompatible with originating from any reconstructed
primary vertex (PV) are considered. The tracks with opposite charge passing this selection
are paired, and their invariant mass is computed assuming the pion mass for each of them.
Pairs with an invariant mass within ±45 MeV/c2 of the known K0
S mass [27] are classified
as K0
S candidates. Events of interest for this analysis are triggered by requiring at least
one K0
S candidate with pT > 2.45 GeV/c.
At the HLT2 level, after a full reconstruction of the whole event, K0
S candidates with
an invariant mass within ±35 MeV/c2 from the known K0
S mass are paired to form D0
candidates. These candidates are required to originate from a common vertex and to
have an invariant mass within ±90 MeV/c2 of the known D0 mass [27]. An additional
track is then combined with the D0 particle to form D∗+ candidates. The invariant-mass
difference ∆m = m(D∗+) −m(D0) is required to be smaller than 170 MeV/c2 [27]. The
resulting candidates are saved on permanent storage and used for further analysis.
A calibration sample of D∗+ →D0(K0
Sπ+π−)π+ decays is obtained from the same
HLT1 trigger selection, by applying a similar HLT2 selection, in which the requirements on
the second K0
S of the signal mode are replaced by requirements on the compatibility of the
two pions to have originated from the D0 decay vertex. This HLT2 selection is specifically
developed for the purpose of the current work, with the aim of providing a calibration
sample with the closest trigger selections to that of the signal decay. Events collected via
other parallel trigger selections are not considered. This ensures an accurate cancellation
of detector effects between samples across all periods of data taking, notwithstanding
the changes in detector and trigger conditions made over time to optimize the global
apparatus performance.
The offline selection of candidate decays is designed to suppress the background while
avoiding introducing any charge biases that might affect the signal and calibration mode
differently. In this respect, an important aspect is the presence, in both the signal and
calibration sample, of a non-negligible fraction of charm mesons originating from b-hadron
3
decays rather than from the primary pp interaction. These secondary particles are affected
by different production asymmetries with respect to the prompt decays, quantified at the
0.2% level [55]. Following the same approach as the previous LHCb measurement [35], it
has not been attempted to reject secondary decays, but rather allow them to contribute
to the measurement. The spurious asymmetry introduced by the secondary component
is cancelled by ensuring that the calibration sample contains secondary decays in the
same proportion as the signal sample. This is achieved by a careful choice of the selection
criteria, avoiding imposing any requirements on observables sensitive to the position of
the D∗+ or D0 vertices that have different resolutions in the two samples [56].
The calibration mode has the same final-state particles as the signal, and a branching
fraction approximately two hundred times larger [27]. This decay mode can contaminate
the signal even though the proportion of pion pairs randomly falling within the K0
S mass
window is small. Although the fit procedure is capable of taking this contribution into
account, additional selection criteria are applied on the significance of the flight distance of
both K0
S mesons from the D0 decay vertex to suppress this contamination to a negligible
level. The effect of this requirement is independent of the flight distance of the parent
charm meson from the primary pp interaction vertex, and therefore does not affect the
proportion of the secondary component. Studies based on simulated events have identified
other physics processes that could potentially contaminate the signal sample, such as
partially reconstructed D0 decays originating from D∗+ meson, including D0 →K0
SK0
Sπ0
decays. Some non-D0 decays can also contaminate the sample, with the main contributor
being the D+
s →K0
SK0
Sπ+ mode, where the K0
SK0
S pair is incorrectly associated with a D0
decay. These background components are rendered negligible by selecting only candidates
with m(D0) within ±26 MeV/c2 of the known D0 mass [27], which is measured with a
resolution of about 7 MeV/c2. The width of the mass window is chosen to prevent a
possible risk of charge biases arising from residual differences in the mass calibration and
resolution between positive and negative particles. The potential contamination of the K0
S
sample by Λ particles has been verified on data, and found to have a negligible impact,
since this background does not peak in the ∆m distribution.
To maximise the sensitivity of the measurement, only D∗+ candidates compatible
with having decayed in the vicinity of the PV are selected. This suppresses background
from random combination of tracks and allows the candidate mass to be recalculated by
constraining the origin vertex to coincide with the PV, improving the ∆m mass resolution
by a factor of two, with an associated improvement in combinatorial background rejection.
Using simulated samples it is shown that this requirement removes the same proportion of
secondary B-decay component from both the signal and calibration sample, thus leaving
the validity of the cancellation procedure unaffected.
The largest remaining background contribution to the signal sample originates from
random combinations of genuine K0
S particles from the primary interaction, which form a
fake D0 candidate. This is partially rejected by the m(D0) requirement, and is further
suppressed by requiring the K0
S candidates to be inconsistent with originating from the PV
with a selection on its impact parameter significance, thereby improving the signal purity.
To ensure an identical effect on the secondary fraction in signal and calibration samples,
the selection is applied only on the K0
S candidate that fulfils the HLT1 selection. If both
K0
S candidates do, one is arbitrarily chosen. A threshold for this requirement has been
optimised jointly with a condition on a multivariate classifier. Track-related observables,
including track and vertex quality, transverse momenta of K0
S and D0 candidates, helicity
4
Table 1: Integrated luminosity of individual data blocks.
Data block
R
Ldt [ fb−1 ]
Polarity
A
0.7
MagDown
B
0.7
MagUp
C
0.6
MagUp
D
1.1
MagUp
E
1.1
MagUp
F
0.9
MagDown
G
0.7
MagDown
H
0.4
MagUp
Total
6.2
–
angles of the K0
S and D0 decays, and particle-identification information of the D0 final-state
particles, are used as input to a k-nearest-neighbours (kNN) classifier [57]. This classifier is
trained using a simulated D0 →K0
SK0
S sample as signal proxy and data from the D0 mass
sidebands as combinatorial background proxy. Kinematic variables of the tagging pion
are excluded from the classifier to avoid introducing possible charge-asymmetry biases.
To make better use of the available data, signal candidates are classified into two
classes of purity (high-purity, with signal-to-background ratio S/B ∼20 and low-purity,
with S/B ∼4), which are fitted separately to determine ACP. These classes are defined by
an optimised set of requirements on both the kNN classifier and the significance of the K0
S
impact parameter, chosen to maximise the combined resolution on the ACP parameter.
This strategy closely follows the procedure of the previous LHCb publication [35]. In
about 10% of events, multiple D∗+ candidates are found. In these cases, only one D∗+
candidate is arbitrarily selected for subsequent analysis.
The data sample is divided into eight blocks according to data-taking periods with
substantially different trigger and detector spatial alignment conditions. These conditions
were varied several times during the year to optimise the running of the LHCb Upgrade I
detector. The offline selection is the same in each block, except for the requirement on
the D0 mass window, whose threshold has been determined separately for each block to
account for possible slight calibration differences. The integrated luminosities of each data
block are reported in Table 1.
4
Correction of nuisance asymmetries
Each D0 →K0
SK0
S candidate is appropriately weighted to cancel all spurious production
and detection asymmetries, with weights obtained from the calibration sample. This
follows the same procedure as the previous LHCb analysis [35], with the only modification
that the weights, w±, are expressed in terms of the three-momentum ⃗p of the tagging pion
from the D∗+ decay instead of using the D0 kinematic observables
w±(⃗p) = n+
C(⃗p) + n−
C(⃗p)
2n±
C(⃗p)
.
(3)
5
This is motivated by the need to precisely control the most important source of detection
asymmetry. In Eq. 3, the probability density of calibration decays in the ⃗p space is denoted
by n±
C(⃗p). The plus and minus signs in the expression indicate quantities related to D∗+
and D∗−candidates, respectively. The weights are computed by classifying the calibration
sample based on the observed charge asymmetry by means of a kNN classifier based on
the three-dimensional ⃗p space. The number of neighbours used is 30, as it is low enough
to describe local variations of asymmetry across the phase space, and it allows to keep
statistical fluctuations related to the size of the considered sample below 10%.
In the calibration sample, the absence of selection criteria requiring compatibility of
the D0 final-state particles with the primary vertex in the dedicated HLT2 selection leads
to a low-purity sample. To improve the S/B ratio, additional offline selections are applied,
while maintaining strict consistency between the calibration and signal selections to ensure
an accurate cancellation of detector effects.
First, all offline selections applied to the signal sample that can be directly ported to
the calibration sample are applied to equalise the kinematic distributions and the fraction
of secondary decays. Then, requirements on variables including the particle-identification
information of final-state tracks, vertex-fit quality of D0, K0
S and D∗+ mesons, and
the opening angles between the final-state particles are applied to further suppress the
combinatorial background. The resulting sample has a high purity (S/B ∼13) and it is
used for the weight calculation. The ∆m distribution of calibration candidates is shown in
Fig. 1. It is verified that the residual background under the ∆m peak does not introduce
any bias in the measurement.
The newly adopted calibration mode is not self-conjugate, in contrast to the previously
used D0 →K+K−decay [35]. To account for this, the analysis procedure is modified
with respect to the earlier LHCb measurement. The decay proceeds primarily through
the intermediate D0 →K∗−(K0
Sπ−)π+ channel, which produces a charge-asymmetric
momentum distribution of the two pions. As a result, the detector response may differ
between signal and calibration candidates, since positive and negative pions traverse
regions with distinct charge-dependent detection asymmetries in the two samples. To
eliminate this spurious effect, the three-momentum of the pions from the D0 decay is
weighted in the calibration sample to make it charge symmetric, before being used as
input to the kNN classifier that calculates the signal weights. The charge-symmetrisation
procedure employs an independent kNN classifier, trained to distinguish candidates based
on the charge of the highest-momentum pion and the kinematics of both final-state pions.
The effectiveness of this correction is confirmed by verifying that a kNN classifier can no
longer separate D0 from D0 after the weights are applied. The correction shifts the final
result by about 0.1%, corresponding to roughly 14% of the statistical uncertainty, and the
associated systematic uncertainty is therefore negligible compared to other contributions.
Distributions of the final weights w± applied to the signal sample for the correction of
spurious asymmetries (as defined in Eq. 3) are shown in Fig. 2. The differences between
the distributions of the two charges are a consequence of different D0 and D0 acceptances
and detector asymmetries. These distributions are similar, but not identical, in every data
block, and D0/D0 candidates are separately corrected before fitting for the asymmetry.
To avoid weights affected by large uncertainties, candidates with weights lower than 0.56
and larger than 5.00 are dropped at this stage. These correspond to a fraction of 5 × 10−4,
whose effects are therefore neglected. The size of the asymmetry correction obtained by
the application of these weights is about 1.4%. This accounts for the cumulative effect of
6
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150
]
2c
) [MeV/
0
D
(
m
) -
*+
D
(
m
0
100
200
300
400
500
600
700
800
900
3
10
×
)
2c
Candidates / ( 0.16 MeV/
D
/
0
D
π
0
S
K
→
0
π
+
−
LHCb
2024
Figure 1: Distribution of ∆m = m(D∗+) −m(D0) for D0 →K0
Sπ+π−candidates, after ap-
plication of all offline selections.
Calibration candidates are selected by the requirement
|∆m −145.45 MeV/c2| < 1.5 MeV/c2, shown in red.
all production and detection asymmetries affecting the data.
5
Asymmetry measurement
A binned maximum-likelihood fit to the joint distribution of ∆m and the two m(K0
S)
observables is performed separately for each data block to measure the ACP parameter.
The choice of the K0
S candidate for filling the invariant-mass histograms is arbitrary. The
three-dimensional fit is performed simultaneously for candidates of both flavours and
across the two purity bins. Data block A is further divided into two sections, fitted
separately to account for changes in running conditions. The final measurement is then
obtained as the weighted average of the results from all blocks.
The total probability density function is parametrised by the sum of the signal
component, which has a peaking distribution in each of the three observables, and four
background components, each describing a specific source: random combination of five
pions, a K0
S combined with three random pions, three random pions combined with a
K0
S, two K0
S combined with a random pion. The probability density functions used to
describe the five components are empirical. The contributions from D0 →K0
Sπ+π−and
D0 →π+π−π+π−decays are verified to be negligible, and are excluded from the fit. The
peaking component in the ∆m distribution is described by a Johnson SU distribution [58].
The peaking component in the m(K0
S) distribution is modelled as the sum of two Gaussian
functions with a common mean but different widths, except for blocks A, B, and H, where
7
0
0.5
1
1.5
2
Weight
0
20
40
60
80
100
120
140
Candidates / 0.025
S
0
K
S
0
K
→
0
D
S
0
K
S
0
K
→
0
D
LHCb
2024
1
−
1.1 fb
Figure 2: Example (data block D) of distributions of weights applied to signal candidates.
These are used in the fit extracting ACP (K0
SK0
S) to correct the signal sample for detection and
production asymmetries of the calibration channel.
a single Gaussian function is used to avoid fit instabilities due to their smaller sample
sizes. The nonpeaking component in the ∆m distribution is modelled with a threshold
function in the low-purity bins and with a first-order polynomial in the high-purity bins.
In the m(K0
S) distribution, this component is described by a first-order polynomial.
In each subsample, the parameters defining the signal and background probability
density functions are shared between D∗+/D∗−candidates, while the normalisation of
each component is allowed to differ. All parameters included in the probability density
function are allowed to vary in the fit to the data.
Figures 3 and 4 show the projections of the fit to the total data sample for candidates
satisfying the high- and low-purity selection, respectively. The histograms are obtained
by adding all data blocks. The candidates entering these plots have been individually
weighted to correct for spurious asymmetries.
6
Systematic uncertainties and cross-checks
To assess potential systematic effects, the weighting procedure is validated by applying the
weights derived from one half of the calibration sample to the other half. As expected, this
cancels the raw asymmetry of the sample, which was initially significant at (−1.04±0.05)%
due to a combination of production asymmetry and detector effects, and reduces it to
(−0.08 ± 0.07)% after the application of the weights.
The weighting procedure is also affected by a systematic uncertainty, partly due to
the finite size of the calibration sample, and partly due to the numerical approximation,
which is implicit in estimating a ratio of probability densities with the discretised kNN
classifier output. The first uncertainty is evaluated by a bootstrap procedure applied to
the calibration sample, where multiple pseudoexperiments are generated by resampling.
8
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LHCb
2024
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K
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0
D
High-purity
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510
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]
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) [MeV/
S
0
K
(
m
0
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)
2c
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LHCb
2024
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K
→
0
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]
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0
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(
m
0
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LHCb
2024
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0
S
K
→
0
D
High-purity
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510
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]
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) [MeV/
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0
K
(
m
0
50
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150
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)
2c
Candidates / (1 MeV/
Data
Total
Signal
Combinatorial
LHCb
2024
0
S
K
0
S
K
→
0
D
High-purity
470
480
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500
510
520
530
]
2c
) [MeV/
S
0
K
(
m
0
50
100
150
200
250
300
350
400
450
)
2c
Candidates / (1 MeV/
Data
Total
Signal
Combinatorial
LHCb
2024
0
S
K
0
S
K
→
0
D
High-purity
Figure 3: Results of the fit for the sample satisfying the high-purity selection and combining all
data blocks. The left (right) column reports histograms for the D0 (D0) candidates. The ∆m
distributions are reported in the top row. The masses of the two K0
S candidates are reported in
the middle and bottom rows. The D0 →K0
SK0
S component is indicated by the red area. For the
m(K0
S) distributions the combinatorial component refers to random combination of pions.
The resulting uncertainty is 0.24%. The second effect is evaluated by varying the value of
the number of neighbours (k parameter) used in the kNN classifier when computing the
weights, considering values between 30 and 200. This yields an uncertainty of 0.20%.
While asymmetries arising from regeneration and from mixing and CP violation in
the K0 −K0 system do not affect the signal sample, they can affect the D0 →K0
Sπ+π−
calibration sample. To evaluate the impact of the neglected K0
S regeneration and oscillation,
the dependence of ACP (after applying all weights) on the K0
S flight distance is studied
using the calibration data. No evidence of such a dependence is observed, and an upper
limit of 0.05% is set on the corresponding ACP bias, which is negligible compared to the
statistical and other systematic uncertainties.
An additional source of systematic uncertainty arises from the limited knowledge of
the shape of the mass distributions. It is evaluated by fitting the simulated samples of
pseudoexperiments with alternative models, whose fit quality is comparable to that of the
baseline model. The alternative models include: the sum of two Gaussian functions for the
peaking component in the ∆m distribution; the sum of three Gaussian functions for the
peaking component in the m(K0
S) distribution; and a polynomial multiplied by a threshold
9
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2024
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K
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0
D
Low-purity
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2024
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Low-purity
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]
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LHCb
2024
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]
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0
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LHCb
2024
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S
K
→
0
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Low-purity
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]
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) [MeV/
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0
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(
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LHCb
2024
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0
S
K
→
0
D
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]
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) [MeV/
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0
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0
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Data
Total
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Combinatorial
LHCb
2024
0
S
K
0
S
K
→
0
D
Low-purity
Figure 4: Results of the fit for the sample satisfying the low purity selection and integrating all
data blocks. The left (right) column reports histograms for the D0 (D0) candidates. The ∆m
distributions are reported in the top row. The masses of the two K0
S candidates are reported in
the middle and bottom rows. The D0 →K0
SK0
S component is indicated by the red area. For the
m(K0
S) distributions the combinatorial component refers to random combination of pions.
Table 2: Summary of absolute systematic uncertainty sources on ACP (D0 →K0
SK0
S) in percent.
Systematic source
Uncertainty [%]
Calibration sample size
0.24
Weighting procedure
0.20
Fit model
0.27
function for the nonpeaking component in the ∆m distribution. The largest observed
variations are taken as the systematic uncertainty ranging from 0.1% to 1.9%, depending
on the subsample. The systematic uncertainties due to the choice of fit model are assumed
to be fully correlated across the data blocks, resulting in a total systematic uncertainty
associated with the choice of the fit model of 0.27%. A summary of all systematic effects
can be found in Table 2.
The data sample includes data blocks recorded with each polarity of the LHCb magnet.
To check for potential uncompensated systematic effects, the asymmetries of the signal and
calibration mode have been compared separately for the MagUp and MagDown magnet
10
Table 3: Measurements of yields and ACP (D0 →K0
SK0
S) extracted from weighted fits to individual
data blocks. The quoted uncertainty is statistical only.
Data block
Yield
ACP [%]
A
1534 ± 75
5.5 ± 3.4
B
1639 ± 56
0.8 ± 3.2
C
1385 ± 55
−0.3 ± 3.4
D
2915 ± 85
0.3 ± 2.4
E
3149 ± 94
0.0 ± 2.4
F
2544 ± 77
4.6 ± 2.6
G
1599 ± 67
1.7 ± 3.3
H
911 ± 54
5.6 ± 4.3
polarities. The results are found to be compatible within 1.5σ.
7
Results
The measured ACP(D0 →K0
SK0
S) values for the different data blocks are shown in Table 3.
All measurements are compatible with each other. The weighted average is
ACP(K0
SK0
S) = (1.86 ± 1.04 ± 0.41)%,
where the first uncertainty is statistical, and the second is systematic, obtained by
combining the individual sources of systematic uncertainty from Table 2 as uncorrelated.
This result is the most precise single measurement of this quantity to date, and it is
compatible with CP symmetry and with the average of previous determinations [33–38].
Any time-dependent effect can be neglected, as mentioned in Section 1. It is anyway
useful to provide the average decay time of the selected D0 →K0
SK0
S decays to allow
the interpretation of this result in a wider global charm CP violation scenario. This
value normalised to the D0 lifetime [27], is ⟨t⟩/τ(D0) = 1.76 ± 0.02. It is obtained by
integrating all data blocks and purity bins, after performing sideband subtraction in
the ∆m distribution. The contribution from secondary decays is subtracted, exploiting
the same simulated D0 →K0
SK0
S sample used for selections optimization. The quoted
uncertainty includes statistical and systematic contributions.
Combining this result with the previous LHCb measurements [34,35], assuming all
systematic uncertainties are uncorrelated, the following value is obtained
ACP(K0
SK0
S) = (−0.37 ± 0.78 ± 0.29)%.
The compatibility between the three measurements corresponds to a p-value of 1.2%. De-
spite the similar integrated luminosity of the LHCb measurement based on the 2015–2018
dataset, and the exclusion of certain candidate categories included in the previous analy-
sis [35], this result achieves better precision, thanks to the significantly improved efficiency
of the upgraded trigger.
11
Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments for the
excellent performance of the LHC. We thank the technical and administrative staff at the
LHCb institutes. We acknowledge support from CERN and from the national agencies:
ARC (Australia); CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC
(China); CNRS/IN2P3 (France); BMFTR, DFG and MPG (Germany); INFN (Italy);
NWO (Netherlands); MNiSW and NCN (Poland); MCID/IFA (Romania); MICIU and
AEI (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom);
DOE NP and NSF (USA). We acknowledge the computing resources that are provided
by ARDC (Australia), CBPF (Brazil), CERN, IHEP and LZU (China), IN2P3 (France),
KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), Polish WLCG (Poland),
IFIN-HH (Romania), PIC (Spain), CSCS (Switzerland), and GridPP (United Kingdom).
We are indebted to the communities behind the multiple open-source software packages
on which we depend. Individual groups or members have received support from Key
Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental
Research Funds for the Central Universities, and Sci. & Tech. Program of Guangzhou
(China); Minciencias (Colombia); EPLANET, Marie Sk lodowska-Curie Actions, ERC and
NextGenerationEU (European Union); A*MIDEX, ANR, IPhU and Labex P2IO, and
R´egion Auvergne-Rhˆone-Alpes (France); Alexander-von-Humboldt Foundation (Germany);
ICSC (Italy); Severo Ochoa and Mar´ıa de Maeztu Units of Excellence, GVA, XuntaGal,
GENCAT, InTalent-Inditex and Prog. Atracci´on Talento CM (Spain); SRC (Sweden); the
Leverhulme Trust, the Royal Society and UKRI (United Kingdom).
12
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, S. Ghizzo29,n
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, F.
I. Giasemis16,f
, V. Gibson56
, H.K. Giemza42
, A.L. Gilman66
, M. Giovannetti28
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, L. Girardey63,58
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, F.C. Glaser14,22
, V.V. Gligorov16
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, S.J. Jaimes Elles77,48,49
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, B.K. Jashal48
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18
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, M. John64
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, B. Khanji69
, A. Kharisova44
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, T. Kirn17
, V.S. Kirsebom31,p
, O. Kitouni65
, S. Klaver39
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, M.R. Kmiec42
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, L. Kolk19
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S. Kotriakhova
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Liang32
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Mocanu43
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19
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Paul69
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, A. Petrolini29,n
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, L. Pica35,t
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, D. Pinci36
, F. Pisani49
, M. Pizzichemi31,p,49
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,
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, T. Poeschl49
, F. Polci16
, M. Poli Lener28
, A. Poluektov13
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, I. Polyakov63
, E. Polycarpo3
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,
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, C. Prouve84
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, V. Ramos De Oliveira70
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,
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,
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,
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,
N. Serra51
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20
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Y. Song4,d
, Y. S. Song6
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K.M. Sowa40
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C. V´azquez Sierra84
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Y. Wang6
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N.K. Watson54
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S. Zhang64
, S. L.
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, Y. Z. Zhang4,d
, Z. Zhang4,d
, Y. Zhao22
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A. Zhelezov22
, S. Z. Zheng6
, X. Z. Zheng4,d
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, T. Zhou6
, X. Zhou8
,
Y. Zhou7
, V. Zhovkovska57
, L. Z. Zhu7
, X. Zhu4,d
, X. Zhu8
, Y. Zhu17
,
V. Zhukov17
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, Q. Zou5,7
, D. Zuliani33,r
, G. Zunica28
.
1School of Physics and Astronomy, Monash University, Melbourne, Australia
2Centro Brasileiro de Pesquisas F´ısicas (CBPF), Rio de Janeiro, Brazil
3Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
4Department of Engineering Physics, Tsinghua University, Beijing, China
5Institute Of High Energy Physics (IHEP), Beijing, China
6School of Physics State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing,
China
21
7University of Chinese Academy of Sciences, Beijing, China
8Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, China
9Consejo Nacional de Rectores (CONARE), San Jose, Costa Rica
10Universit´e Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France
11Universit´e Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
12Universit´e Paris-Saclay, Centre d’Etudes de Saclay (CEA), IRFU, Saclay, France, Gif-Sur-Yvette,
France
13Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
14Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
15Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique de Paris,
Palaiseau, France
16Laboratoire de Physique Nucl´eaire et de Hautes ´Energies (LPNHE), Sorbonne Universit´e,
CNRS/IN2P3, F-75005 Paris, France, Paris, France
17I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
18Universit¨at Bonn - Helmholtz-Institut f¨ur Strahlen und Kernphysik, Bonn, Germany
19Fakult¨at Physik, Technische Universit¨at Dortmund, Dortmund, Germany
20Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg, Freiburg, Germany
21Max-Planck-Institut f¨ur Kernphysik (MPIK), Heidelberg, Germany
22Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Heidelberg, Germany
23School of Physics, University College Dublin, Dublin, Ireland
24INFN Sezione di Bari, Bari, Italy
25INFN Sezione di Bologna, Bologna, Italy
26INFN Sezione di Ferrara, Ferrara, Italy
27INFN Sezione di Firenze, Firenze, Italy
28INFN Laboratori Nazionali di Frascati, Frascati, Italy
29INFN Sezione di Genova, Genova, Italy
30INFN Sezione di Milano, Milano, Italy
31INFN Sezione di Milano-Bicocca, Milano, Italy
32INFN Sezione di Cagliari, Monserrato, Italy
33INFN Sezione di Padova, Padova, Italy
34INFN Sezione di Perugia, Perugia, Italy
35INFN Sezione di Pisa, Pisa, Italy
36INFN Sezione di Roma La Sapienza, Roma, Italy
37INFN Sezione di Roma Tor Vergata, Roma, Italy
38Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam,
Netherlands
40AGH - University of Krakow, Faculty of Physics and Applied Computer Science, Krak´ow, Poland
41Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Krak´ow, Poland
42National Center for Nuclear Research (NCBJ), Warsaw, Poland
43Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania
44Authors affiliated with an institute formerly covered by a cooperation agreement with CERN.
45ICCUB, Universitat de Barcelona, Barcelona, Spain
46La Salle, Universitat Ramon Llull, Barcelona, Spain
47Instituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela,
Santiago de Compostela, Spain
48Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain
49European Organization for Nuclear Research (CERN), Geneva, Switzerland
50Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland
51Physik-Institut, Universit¨at Z¨urich, Z¨urich, Switzerland
52NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
53Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine
54School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom
55H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
56Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
57Department of Physics, University of Warwick, Coventry, United Kingdom
22
58STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
59School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom
60School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
61Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
62Imperial College London, London, United Kingdom
63Department of Physics and Astronomy, University of Manchester, Manchester, United Kingdom
64Department of Physics, University of Oxford, Oxford, United Kingdom
65Massachusetts Institute of Technology, Cambridge, MA, United States
66University of Cincinnati, Cincinnati, OH, United States
67University of Maryland, College Park, MD, United States
68Los Alamos National Laboratory (LANL), Los Alamos, NM, United States
69Syracuse University, Syracuse, NY, United States
70Pontif´ıcia Universidade Cat´olica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 3
71Universidad Andres Bello, Santiago, Chile, associated to 51
72School of Physics and Electronics, Hunan University, Changsha City, China, associated to 8
73Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong Joint Laboratory of
Quantum Matter, Institute of Quantum Matter, South China Normal University, Guangzhou, China,
associated to 4
74Lanzhou University, Lanzhou, China, associated to 5
75School of Physics and Technology, Wuhan University, Wuhan, China, associated to 4
76Henan Normal University, Xinxiang, China, associated to 8
77Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to 16
78Ruhr Universitaet Bochum, Fakultaet f. Physik und Astronomie, Bochum, Germany, associated to 19
79Eotvos Lorand University, Budapest, Hungary, associated to 49
80Faculty of Physics, Vilnius University, Vilnius, Lithuania, associated to 20
81Van Swinderen Institute, University of Groningen, Groningen, Netherlands, associated to 38
82Universiteit Maastricht, Maastricht, Netherlands, associated to 38
83Tadeusz Kosciuszko Cracow University of Technology, Cracow, Poland, associated to 41
84Universidade da Coru˜na, A Coru˜na, Spain, associated to 46
85Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden, associated to 60
86Taras Schevchenko University of Kyiv, Faculty of Physics, Kyiv, Ukraine, associated to 14
87University of Michigan, Ann Arbor, MI, United States, associated to 69
88Ohio State University, Columbus, United States, associated to 68
aUniversidade Estadual de Campinas (UNICAMP), Campinas, Brazil
bCentro Federal de Educac˜ao Tecnol´ogica Celso Suckow da Fonseca, Rio De Janeiro, Brazil
cDepartment of Physics and Astronomy, University of Victoria, Victoria, Canada
dCenter for High Energy Physics, Tsinghua University, Beijing, China
eHangzhou Institute for Advanced Study, UCAS, Hangzhou, China
fLIP6, Sorbonne Universit´e, Paris, France
gLamarr Institute for Machine Learning and Artificial Intelligence, Dortmund, Germany
hUniversidad Nacional Aut´onoma de Honduras, Tegucigalpa, Honduras
iUniversit`a di Bari, Bari, Italy
jUniversit`a di Bergamo, Bergamo, Italy
kUniversit`a di Bologna, Bologna, Italy
lUniversit`a di Cagliari, Cagliari, Italy
mUniversit`a di Ferrara, Ferrara, Italy
nUniversit`a di Genova, Genova, Italy
oUniversit`a degli Studi di Milano, Milano, Italy
pUniversit`a degli Studi di Milano-Bicocca, Milano, Italy
qUniversit`a di Modena e Reggio Emilia, Modena, Italy
rUniversit`a di Padova, Padova, Italy
sUniversit`a di Perugia, Perugia, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversit`a di Pisa, Pisa, Italy
vUniversit`a della Basilicata, Potenza, Italy
wUniversit`a di Roma Tor Vergata, Roma, Italy
23
xUniversit`a di Siena, Siena, Italy
yUniversit`a di Urbino, Urbino, Italy
zUniversidad de Ingenier´ıa y Tecnolog´ıa (UTEC), Lima, Peru
aaUniversidad de Alcal´a, Alcal´a de Henares , Spain
†Deceased
24
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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-EP-2025-221 LHCb-PAPER-2025-036 October 16, 2025 Measurement of CP asymmetry in D0 →K0 SK0 S decays with the LHCb Upgrade I detector LHCb collaboration† Abstract A measurement of CP asymmetry in D0 →K0 SK0 S decays is reported, based on a data sample of proton-proton collisions collected with the LHCb Upgrade I detector in 2024 at a centre-of-mass energy of 13.6 TeV, corresponding to an integrated luminosity of 6.2 fb-1. The D0 →K0 Sπ+π-decay is used as calibration channel to cancel residual detection and production asymmetries. The time-integrated CP asymmetry for the D0 →K0 SK0 S mode is measured to be ACP (D0 →K0 SK0 S) = (1.86 ± 1.04 ± 0.41)%, where the first uncertainty is statistical, and the second is systematic. This is the most precise determination of this quantity to date. Submitted to JHEP CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence. †Authors are listed at the end of this paper. 16 Oct 2025 ii 1 Introduction Violation of charge-parity (CP) symmetry in charm-quark transitions has been observed for the first time in the measurement of the difference between the CP asymmetry in D0 →K+K-and D0 →π+π-decays [1]. As of today, this measurement remains the only experimental evidence of CP violation in charm decays, and more generally in any up-type quark (u, c, or t) transition. A successive measurement in the D0 →K+K-decay mode found no statistically significant CP asymmetry [2], leaving the D0 →π+π-decay as the only charm-hadron decay with experimental evidence of CP violation [3,4]. While the existence of CP violation in the charm-quark sector at the observed O(10-3) level is not excluded in the Standard Model (SM), uncertainties in theoretical calculations do not allow for a quantitative comparison or precise predictions for other decay channels [516]; see Refs. [17-19] for reviews on this subject. Further experimental measurements of CP asymmetries in charm decays are therefore highly desirable, as they would provide deeper insight into CP violation phenomenology. Such measurements could potentially involve dynamics beyond the SM [20-26], that are not constrained to be the same as in the down-type quark (d, s, or b) sector. Among many charm-hadron decay modes in which CP violation could manifest itself, the Cabibbo-suppressed D0 →K0 SK0 S mode stands out for being analogous to the twobody decays D0 →K+K-and D0 →π+π-[1], but has the possibility of a more sizeable asymmetry. Only amplitudes proceeding via tree-level exchange - which vanish in the flavour-SU(3) limit - and electroweak loop diagrams contribute to this decay, which therefore has a relatively small branching fraction [27]. Its CP asymmetry is defined by ACP(K0 SK0 S) ≡Γ(D0 →K0 SK0 S) -Γ(D0 →K0 SK0 S) Γ(D0 →K0 SK0 S) + Γ(D0 →K0 SK0 S), (1) with Γ being the decay width of the neutral D meson. Any effect arising from regeneration, mixing and CP violation in the K0 SK0 S final state cancel in ACP, since the D0 and D0 decay to the same CP-symmetric K0 -K0 final state [28]. In the SM, ACP(K0 SK0 S) might be significantly larger than in the D0 →π+π-and D0 →K+K-channels. Several predictions exist for ACP(K0 SK0 S) [9,29-32], up to the percent level. The study of the D0 →K0 SK0 S decay mode provides a valuable probe of CP violation in the charm sector, offering a means to discriminate between competing theoretical approaches. These considerations have motivated several experimental determinations of this observable across different experiments, including some recent ones [33-38]. The world average of the CP asymmetry is ACP(K0 SK0 S) = (-1.19 ± 0.77 ± 0.17)%, where the timedependent contribution to the asymmetry is neglected [39,40]. This has now reached the subpercent level, entering the interesting sensitivity range of theoretical predictions. The reduced precision of this measurement compared with other charm CP asymmetry results is due to the smaller branching fraction and the difficulty of reconstructing the K0 S decay products, given its much longer lifetime relative to bottom and charm hadrons. In this work, a new measurement of ACP(K0 SK0 S) is reported, based on a sample of proton-proton (pp) collisions corresponding to an integrated luminosity of about 6.2 fb-1, collected in 2024 by the LHCb Upgrade I detector. The detector has been significantly improved over its previous version, including the data acquisition system [41,42], which enabled the adoption of a software-based selection of K0 S particles at the first trigger level, resulting in a substantially improved efficiency. 1 The measurement of ACP(K0 SK0 S) requires knowledge of the D0 flavour at production. A sample of flavour-tagged D0 →K0 SK0 S decays is obtained by selecting only D0 mesons that originate from D∗(2010)+ →D0π+ decays.1 The charge of the pion in this decay ("tagging pion") identifies the flavour of the accompanying neutral charm meson. While D0 -D0 oscillations can cause a flavour change before decay, the effect is small compared with the resolution of the current measurement and is therefore neglected [43]. The K0 S mesons are reconstructed in the π+π-final state. The decay widths Γ in Eq. 1 are related to the observed yields N by N( ( ) D 0 →K0 SK0 S) ∝ε±(D∗±)σ(D∗±) Γ( ( ) D 0 →K0 SK0 S), (2) where σ(D∗±) denotes the production cross-sections of the D∗± mesons and ε±(D∗±) the corresponding detection efficiencies for their decays. Both factors are charge asymmetric, due to the D∗± production asymmetry arising from the hadronisation of charm quarks in pp collisions, and to asymmetries in the geometry and response of the detector. Since the final state of the D0 decay is self-conjugate, any detection asymmetry arises solely from the tagging pion. All these quantities are calibrated using a sample of D0 →K0 Sπ+πdecays, which have the same final-state particles as the D0 →K0 SK0 S decay, but with a negligible CP asymmetry [18]. In order to avoid experimenter's bias, the results of the analysis were not examined until the full procedure had been finalised. 2 LHCb detector The LHCb Upgrade I detector [44] is a single-arm forward spectrometer covering the pseudorapidity range 2 5 GeV/c, transverse momentum pT > 0.45 GeV/c, and incompatible with originating from any reconstructed primary vertex (PV) are considered. The tracks with opposite charge passing this selection are paired, and their invariant mass is computed assuming the pion mass for each of them. Pairs with an invariant mass within ±45 MeV/c2 of the known K0 S mass [27] are classified as K0 S candidates. Events of interest for this analysis are triggered by requiring at least one K0 S candidate with pT > 2.45 GeV/c. At the HLT2 level, after a full reconstruction of the whole event, K0 S candidates with an invariant mass within ±35 MeV/c2 from the known K0 S mass are paired to form D0 candidates. These candidates are required to originate from a common vertex and to have an invariant mass within ±90 MeV/c2 of the known D0 mass [27]. An additional track is then combined with the D0 particle to form D∗+ candidates. The invariant-mass difference ∆m = m(D∗+) -m(D0) is required to be smaller than 170 MeV/c2 [27]. The resulting candidates are saved on permanent storage and used for further analysis. A calibration sample of D∗+ →D0(K0 Sπ+π-)π+ decays is obtained from the same HLT1 trigger selection, by applying a similar HLT2 selection, in which the requirements on the second K0 S of the signal mode are replaced by requirements on the compatibility of the two pions to have originated from the D0 decay vertex. This HLT2 selection is specifically developed for the purpose of the current work, with the aim of providing a calibration sample with the closest trigger selections to that of the signal decay. Events collected via other parallel trigger selections are not considered. This ensures an accurate cancellation of detector effects between samples across all periods of data taking, notwithstanding the changes in detector and trigger conditions made over time to optimize the global apparatus performance. The offline selection of candidate decays is designed to suppress the background while avoiding introducing any charge biases that might affect the signal and calibration mode differently. In this respect, an important aspect is the presence, in both the signal and calibration sample, of a non-negligible fraction of charm mesons originating from b-hadron 3 decays rather than from the primary pp interaction. These secondary particles are affected by different production asymmetries with respect to the prompt decays, quantified at the 0.2% level [55]. Following the same approach as the previous LHCb measurement [35], it has not been attempted to reject secondary decays, but rather allow them to contribute to the measurement. The spurious asymmetry introduced by the secondary component is cancelled by ensuring that the calibration sample contains secondary decays in the same proportion as the signal sample. This is achieved by a careful choice of the selection criteria, avoiding imposing any requirements on observables sensitive to the position of the D∗+ or D0 vertices that have different resolutions in the two samples [56]. The calibration mode has the same final-state particles as the signal, and a branching fraction approximately two hundred times larger [27]. This decay mode can contaminate the signal even though the proportion of pion pairs randomly falling within the K0 S mass window is small. Although the fit procedure is capable of taking this contribution into account, additional selection criteria are applied on the significance of the flight distance of both K0 S mesons from the D0 decay vertex to suppress this contamination to a negligible level. The effect of this requirement is independent of the flight distance of the parent charm meson from the primary pp interaction vertex, and therefore does not affect the proportion of the secondary component. Studies based on simulated events have identified other physics processes that could potentially contaminate the signal sample, such as partially reconstructed D0 decays originating from D∗+ meson, including D0 →K0 SK0 Sπ0 decays. Some non-D0 decays can also contaminate the sample, with the main contributor being the D+ s →K0 SK0 Sπ+ mode, where the K0 SK0 S pair is incorrectly associated with a D0 decay. These background components are rendered negligible by selecting only candidates with m(D0) within ±26 MeV/c2 of the known D0 mass [27], which is measured with a resolution of about 7 MeV/c2. The width of the mass window is chosen to prevent a possible risk of charge biases arising from residual differences in the mass calibration and resolution between positive and negative particles. The potential contamination of the K0 S sample by Λ particles has been verified on data, and found to have a negligible impact, since this background does not peak in the ∆m distribution. To maximise the sensitivity of the measurement, only D∗+ candidates compatible with having decayed in the vicinity of the PV are selected. This suppresses background from random combination of tracks and allows the candidate mass to be recalculated by constraining the origin vertex to coincide with the PV, improving the ∆m mass resolution by a factor of two, with an associated improvement in combinatorial background rejection. Using simulated samples it is shown that this requirement removes the same proportion of secondary B-decay component from both the signal and calibration sample, thus leaving the validity of the cancellation procedure unaffected. The largest remaining background contribution to the signal sample originates from random combinations of genuine K0 S particles from the primary interaction, which form a fake D0 candidate. This is partially rejected by the m(D0) requirement, and is further suppressed by requiring the K0 S candidates to be inconsistent with originating from the PV with a selection on its impact parameter significance, thereby improving the signal purity. To ensure an identical effect on the secondary fraction in signal and calibration samples, the selection is applied only on the K0 S candidate that fulfils the HLT1 selection. If both K0 S candidates do, one is arbitrarily chosen. A threshold for this requirement has been optimised jointly with a condition on a multivariate classifier. Track-related observables, including track and vertex quality, transverse momenta of K0 S and D0 candidates, helicity 4 Table 1: Integrated luminosity of individual data blocks. Data block R Ldt [ fb-1 ] Polarity A 0.7 MagDown B 0.7 MagUp C 0.6 MagUp D 1.1 MagUp E 1.1 MagUp F 0.9 MagDown G 0.7 MagDown H 0.4 MagUp Total 6.2 - angles of the K0 S and D0 decays, and particle-identification information of the D0 final-state particles, are used as input to a k-nearest-neighbours (kNN) classifier [57]. This classifier is trained using a simulated D0 →K0 SK0 S sample as signal proxy and data from the D0 mass sidebands as combinatorial background proxy. Kinematic variables of the tagging pion are excluded from the classifier to avoid introducing possible charge-asymmetry biases. To make better use of the available data, signal candidates are classified into two classes of purity (high-purity, with signal-to-background ratio S/B ∼20 and low-purity, with S/B ∼4), which are fitted separately to determine ACP. These classes are defined by an optimised set of requirements on both the kNN classifier and the significance of the K0 S impact parameter, chosen to maximise the combined resolution on the ACP parameter. This strategy closely follows the procedure of the previous LHCb publication [35]. In about 10% of events, multiple D∗+ candidates are found. In these cases, only one D∗+ candidate is arbitrarily selected for subsequent analysis. The data sample is divided into eight blocks according to data-taking periods with substantially different trigger and detector spatial alignment conditions. These conditions were varied several times during the year to optimise the running of the LHCb Upgrade I detector. The offline selection is the same in each block, except for the requirement on the D0 mass window, whose threshold has been determined separately for each block to account for possible slight calibration differences. The integrated luminosities of each data block are reported in Table 1. 4 Correction of nuisance asymmetries Each D0 →K0 SK0 S candidate is appropriately weighted to cancel all spurious production and detection asymmetries, with weights obtained from the calibration sample. This follows the same procedure as the previous LHCb analysis [35], with the only modification that the weights, w±, are expressed in terms of the three-momentum ⃗p of the tagging pion from the D∗+ decay instead of using the D0 kinematic observables w±(⃗p) = n+ C(⃗p) + nC(⃗p) 2n± C(⃗p) . (3) 5 This is motivated by the need to precisely control the most important source of detection asymmetry. In Eq. 3, the probability density of calibration decays in the ⃗p space is denoted by n± C(⃗p). The plus and minus signs in the expression indicate quantities related to D∗+ and D∗-candidates, respectively. The weights are computed by classifying the calibration sample based on the observed charge asymmetry by means of a kNN classifier based on the three-dimensional ⃗p space. The number of neighbours used is 30, as it is low enough to describe local variations of asymmetry across the phase space, and it allows to keep statistical fluctuations related to the size of the considered sample below 10%. In the calibration sample, the absence of selection criteria requiring compatibility of the D0 final-state particles with the primary vertex in the dedicated HLT2 selection leads to a low-purity sample. To improve the S/B ratio, additional offline selections are applied, while maintaining strict consistency between the calibration and signal selections to ensure an accurate cancellation of detector effects. First, all offline selections applied to the signal sample that can be directly ported to the calibration sample are applied to equalise the kinematic distributions and the fraction of secondary decays. Then, requirements on variables including the particle-identification information of final-state tracks, vertex-fit quality of D0, K0 S and D∗+ mesons, and the opening angles between the final-state particles are applied to further suppress the combinatorial background. The resulting sample has a high purity (S/B ∼13) and it is used for the weight calculation. The ∆m distribution of calibration candidates is shown in Fig. 1. It is verified that the residual background under the ∆m peak does not introduce any bias in the measurement. The newly adopted calibration mode is not self-conjugate, in contrast to the previously used D0 →K+K-decay [35]. To account for this, the analysis procedure is modified with respect to the earlier LHCb measurement. The decay proceeds primarily through the intermediate D0 →K∗-(K0 Sπ-)π+ channel, which produces a charge-asymmetric momentum distribution of the two pions. As a result, the detector response may differ between signal and calibration candidates, since positive and negative pions traverse regions with distinct charge-dependent detection asymmetries in the two samples. To eliminate this spurious effect, the three-momentum of the pions from the D0 decay is weighted in the calibration sample to make it charge symmetric, before being used as input to the kNN classifier that calculates the signal weights. The charge-symmetrisation procedure employs an independent kNN classifier, trained to distinguish candidates based on the charge of the highest-momentum pion and the kinematics of both final-state pions. The effectiveness of this correction is confirmed by verifying that a kNN classifier can no longer separate D0 from D0 after the weights are applied. The correction shifts the final result by about 0.1%, corresponding to roughly 14% of the statistical uncertainty, and the associated systematic uncertainty is therefore negligible compared to other contributions. Distributions of the final weights w± applied to the signal sample for the correction of spurious asymmetries (as defined in Eq. 3) are shown in Fig. 2. The differences between the distributions of the two charges are a consequence of different D0 and D0 acceptances and detector asymmetries. These distributions are similar, but not identical, in every data block, and D0/D0 candidates are separately corrected before fitting for the asymmetry. To avoid weights affected by large uncertainties, candidates with weights lower than 0.56 and larger than 5.00 are dropped at this stage. These correspond to a fraction of 5 × 10-4, whose effects are therefore neglected. The size of the asymmetry correction obtained by the application of these weights is about 1.4%. This accounts for the cumulative effect of 6 140 145 150 ] 2c ) [MeV/ 0 D ( m ) - *+ D ( m 0 100 200 300 400 500 600 700 800 900 3 10 × ) 2c Candidates / ( 0.16 MeV/ D / 0 D π 0 S K → 0 π + - LHCb 2024 Figure 1: Distribution of ∆m = m(D∗+) -m(D0) for D0 →K0 Sπ+π-candidates, after application of all offline selections. Calibration candidates are selected by the requirement |∆m -145.45 MeV/c2| < 1.5 MeV/c2, shown in red. all production and detection asymmetries affecting the data. 5 Asymmetry measurement A binned maximum-likelihood fit to the joint distribution of ∆m and the two m(K0 S) observables is performed separately for each data block to measure the ACP parameter. The choice of the K0 S candidate for filling the invariant-mass histograms is arbitrary. The three-dimensional fit is performed simultaneously for candidates of both flavours and across the two purity bins. Data block A is further divided into two sections, fitted separately to account for changes in running conditions. The final measurement is then obtained as the weighted average of the results from all blocks. The total probability density function is parametrised by the sum of the signal component, which has a peaking distribution in each of the three observables, and four background components, each describing a specific source: random combination of five pions, a K0 S combined with three random pions, three random pions combined with a K0 S, two K0 S combined with a random pion. The probability density functions used to describe the five components are empirical. The contributions from D0 →K0 Sπ+π-and D0 →π+π-π+π-decays are verified to be negligible, and are excluded from the fit. The peaking component in the ∆m distribution is described by a Johnson SU distribution [58]. The peaking component in the m(K0 S) distribution is modelled as the sum of two Gaussian functions with a common mean but different widths, except for blocks A, B, and H, where 7 0 0.5 1 1.5 2 Weight 0 20 40 60 80 100 120 140 Candidates / 0.025 S 0 K S 0 K → 0 D S 0 K S 0 K → 0 D LHCb 2024 1 - 1.1 fb Figure 2: Example (data block D) of distributions of weights applied to signal candidates. These are used in the fit extracting ACP (K0 SK0 S) to correct the signal sample for detection and production asymmetries of the calibration channel. a single Gaussian function is used to avoid fit instabilities due to their smaller sample sizes. The nonpeaking component in the ∆m distribution is modelled with a threshold function in the low-purity bins and with a first-order polynomial in the high-purity bins. In the m(K0 S) distribution, this component is described by a first-order polynomial. In each subsample, the parameters defining the signal and background probability density functions are shared between D∗+/D∗-candidates, while the normalisation of each component is allowed to differ. All parameters included in the probability density function are allowed to vary in the fit to the data. Figures 3 and 4 show the projections of the fit to the total data sample for candidates satisfying the high- and low-purity selection, respectively. The histograms are obtained by adding all data blocks. The candidates entering these plots have been individually weighted to correct for spurious asymmetries. 6 Systematic uncertainties and cross-checks To assess potential systematic effects, the weighting procedure is validated by applying the weights derived from one half of the calibration sample to the other half. As expected, this cancels the raw asymmetry of the sample, which was initially significant at (-1.04±0.05)% due to a combination of production asymmetry and detector effects, and reduces it to (-0.08 ± 0.07)% after the application of the weights. The weighting procedure is also affected by a systematic uncertainty, partly due to the finite size of the calibration sample, and partly due to the numerical approximation, which is implicit in estimating a ratio of probability densities with the discretised kNN classifier output. The first uncertainty is evaluated by a bootstrap procedure applied to the calibration sample, where multiple pseudoexperiments are generated by resampling. 8 142 143 144 145 146 147 148 149 150 ] 2c ) [MeV/ 0 D ( m ) - *+ D ( m 0 50 100 150 200 250 300 ) 2c Candidates / (0.08 MeV/ Data Total Signal Background LHCb 2024 0 S K 0 S K → 0 D High-purity 142 143 144 145 146 147 148 149 150 ] 2c ) [MeV/ 0 D ( m ) - *- D ( m 0 50 100 150 200 250 300 ) 2c Candidates / (0.08 MeV/ Data Total Signal Background LHCb 2024 0 S K 0 S K → 0 D High-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 50 100 150 200 250 300 350 400 450 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D High-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 50 100 150 200 250 300 350 400 450 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D High-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 50 100 150 200 250 300 350 400 450 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D High-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 50 100 150 200 250 300 350 400 450 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D High-purity Figure 3: Results of the fit for the sample satisfying the high-purity selection and combining all data blocks. The left (right) column reports histograms for the D0 (D0) candidates. The ∆m distributions are reported in the top row. The masses of the two K0 S candidates are reported in the middle and bottom rows. The D0 →K0 SK0 S component is indicated by the red area. For the m(K0 S) distributions the combinatorial component refers to random combination of pions. The resulting uncertainty is 0.24%. The second effect is evaluated by varying the value of the number of neighbours (k parameter) used in the kNN classifier when computing the weights, considering values between 30 and 200. This yields an uncertainty of 0.20%. While asymmetries arising from regeneration and from mixing and CP violation in the K0 -K0 system do not affect the signal sample, they can affect the D0 →K0 Sπ+πcalibration sample. To evaluate the impact of the neglected K0 S regeneration and oscillation, the dependence of ACP (after applying all weights) on the K0 S flight distance is studied using the calibration data. No evidence of such a dependence is observed, and an upper limit of 0.05% is set on the corresponding ACP bias, which is negligible compared to the statistical and other systematic uncertainties. An additional source of systematic uncertainty arises from the limited knowledge of the shape of the mass distributions. It is evaluated by fitting the simulated samples of pseudoexperiments with alternative models, whose fit quality is comparable to that of the baseline model. The alternative models include: the sum of two Gaussian functions for the peaking component in the ∆m distribution; the sum of three Gaussian functions for the peaking component in the m(K0 S) distribution; and a polynomial multiplied by a threshold 9 142 143 144 145 146 147 148 149 150 ] 2c ) [MeV/ 0 D ( m ) - *+ D ( m 0 100 200 300 400 500 600 700 800 900 ) 2c Candidates / (0.08 MeV/ Data Total Signal Background LHCb 2024 0 S K 0 S K → 0 D Low-purity 142 143 144 145 146 147 148 149 150 ] 2c ) [MeV/ 0 D ( m ) - *- D ( m 0 100 200 300 400 500 600 700 800 900 ) 2c Candidates / (0.08 MeV/ Data Total Signal Background LHCb 2024 0 S K 0 S K → 0 D Low-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 500 1000 1500 2000 2500 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D Low-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 500 1000 1500 2000 2500 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D Low-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 500 1000 1500 2000 2500 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D Low-purity 470 480 490 500 510 520 530 ] 2c ) [MeV/ S 0 K ( m 0 500 1000 1500 2000 2500 ) 2c Candidates / (1 MeV/ Data Total Signal Combinatorial LHCb 2024 0 S K 0 S K → 0 D Low-purity Figure 4: Results of the fit for the sample satisfying the low purity selection and integrating all data blocks. The left (right) column reports histograms for the D0 (D0) candidates. The ∆m distributions are reported in the top row. The masses of the two K0 S candidates are reported in the middle and bottom rows. The D0 →K0 SK0 S component is indicated by the red area. For the m(K0 S) distributions the combinatorial component refers to random combination of pions. Table 2: Summary of absolute systematic uncertainty sources on ACP (D0 →K0 SK0 S) in percent. Systematic source Uncertainty [%] Calibration sample size 0.24 Weighting procedure 0.20 Fit model 0.27 function for the nonpeaking component in the ∆m distribution. The largest observed variations are taken as the systematic uncertainty ranging from 0.1% to 1.9%, depending on the subsample. The systematic uncertainties due to the choice of fit model are assumed to be fully correlated across the data blocks, resulting in a total systematic uncertainty associated with the choice of the fit model of 0.27%. A summary of all systematic effects can be found in Table 2. The data sample includes data blocks recorded with each polarity of the LHCb magnet. To check for potential uncompensated systematic effects, the asymmetries of the signal and calibration mode have been compared separately for the MagUp and MagDown magnet 10 Table 3: Measurements of yields and ACP (D0 →K0 SK0 S) extracted from weighted fits to individual data blocks. The quoted uncertainty is statistical only. Data block Yield ACP [%] A 1534 ± 75 5.5 ± 3.4 B 1639 ± 56 0.8 ± 3.2 C 1385 ± 55 -0.3 ± 3.4 D 2915 ± 85 0.3 ± 2.4 E 3149 ± 94 0.0 ± 2.4 F 2544 ± 77 4.6 ± 2.6 G 1599 ± 67 1.7 ± 3.3 H 911 ± 54 5.6 ± 4.3 polarities. The results are found to be compatible within 1.5σ. 7 Results The measured ACP(D0 →K0 SK0 S) values for the different data blocks are shown in Table 3. All measurements are compatible with each other. The weighted average is ACP(K0 SK0 S) = (1.86 ± 1.04 ± 0.41)%, where the first uncertainty is statistical, and the second is systematic, obtained by combining the individual sources of systematic uncertainty from Table 2 as uncorrelated. This result is the most precise single measurement of this quantity to date, and it is compatible with CP symmetry and with the average of previous determinations [33-38]. Any time-dependent effect can be neglected, as mentioned in Section 1. It is anyway useful to provide the average decay time of the selected D0 →K0 SK0 S decays to allow the interpretation of this result in a wider global charm CP violation scenario. This value normalised to the D0 lifetime [27], is ⟨t⟩/τ(D0) = 1.76 ± 0.02. It is obtained by integrating all data blocks and purity bins, after performing sideband subtraction in the ∆m distribution. The contribution from secondary decays is subtracted, exploiting the same simulated D0 →K0 SK0 S sample used for selections optimization. The quoted uncertainty includes statistical and systematic contributions. Combining this result with the previous LHCb measurements [34,35], assuming all systematic uncertainties are uncorrelated, the following value is obtained ACP(K0 SK0 S) = (-0.37 ± 0.78 ± 0.29)%. The compatibility between the three measurements corresponds to a p-value of 1.2%. Despite the similar integrated luminosity of the LHCb measurement based on the 2015-2018 dataset, and the exclusion of certain candidate categories included in the previous analysis [35], this result achieves better precision, thanks to the significantly improved efficiency of the upgraded trigger. 11 Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: ARC (Australia); CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST and NSFC (China); CNRS/IN2P3 (France); BMFTR, DFG and MPG (Germany); INFN (Italy); NWO (Netherlands); MNiSW and NCN (Poland); MCID/IFA (Romania); MICIU and AEI (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC (United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources that are provided by ARDC (Australia), CBPF (Brazil), CERN, IHEP and LZU (China), IN2P3 (France), KIT and DESY (Germany), INFN (Italy), SURF (Netherlands), Polish WLCG (Poland), IFIN-HH (Romania), PIC (Spain), CSCS (Switzerland), and GridPP (United Kingdom). We are indebted to the communities behind the multiple open-source software packages on which we depend. Individual groups or members have received support from Key Research Program of Frontier Sciences of CAS, CAS PIFI, CAS CCEPP, Fundamental Research Funds for the Central Universities, and Sci. & Tech. 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Zunica28 . 1 2Centro Brasileiro de Pesquisas F ́ısicas (CBPF), Rio de Janeiro, Brazil 3Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 4 5Institute Of High Energy Physics (IHEP), Beijing, China 6 21 7 8 9Consejo Nacional de Rectores (CONARE), San Jose, Costa Rica 10Universit ́e Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy, France 11Universit ́e Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 12Universit ́e Paris-Saclay, Centre d'Etudes de Saclay (CEA), IRFU, Saclay, France, Gif-Sur-Yvette, France 13Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France 14Universit ́e Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France 15Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France 16Laboratoire de Physique Nucl ́eaire et de Hautes ́Energies (LPNHE), Sorbonne Universit ́e, CNRS/IN2P3, F-75005 Paris, France, Paris, France 17I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany 18Universit ̈at Bonn - Helmholtz-Institut f ̈ur Strahlen und Kernphysik, Bonn, Germany 19Fakult ̈at Physik, Technische Universit ̈at Dortmund, Dortmund, Germany 20Physikalisches Institut, Albert-Ludwigs-Universit ̈at Freiburg, Freiburg, Germany 21Max-Planck-Institut f ̈ur Kernphysik (MPIK), Heidelberg, Germany 22Physikalisches Institut, Ruprecht-Karls-Universit ̈at Heidelberg, Heidelberg, Germany 23 24INFN Sezione di Bari, Bari, Italy 25INFN Sezione di Bologna, Bologna, Italy 26INFN Sezione di Ferrara, Ferrara, Italy 27INFN Sezione di Firenze, Firenze, Italy 28INFN Laboratori Nazionali di Frascati, Frascati, Italy 29INFN Sezione di Genova, Genova, Italy 30INFN Sezione di Milano, Milano, Italy 31INFN Sezione di Milano-Bicocca, Milano, Italy 32INFN Sezione di Cagliari, Monserrato, Italy 33INFN Sezione di Padova, Padova, Italy 34INFN Sezione di Perugia, Perugia, Italy 35INFN Sezione di Pisa, Pisa, Italy 36INFN Sezione di Roma La Sapienza, Roma, Italy 37INFN Sezione di Roma Tor Vergata, Roma, Italy 38Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands 39Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, Netherlands 40AGH - ́ow, Poland 41Henryk Niewodniczanski ́ow, Poland 42National Center for Nuclear Research (NCBJ), Warsaw, Poland 43Horia Hulubei National -Magurele, Romania 44Authors affiliated with an institute formerly covered by a cooperation agreement with CERN. 45ICCUB, Universitat de Barcelona, Barcelona, Spain 46La Salle, Universitat Ramon Llull, Barcelona, Spain 47Instituto Galego de F ́ısica de Altas Enerx ́ıas (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain 48Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia - CSIC, Valencia, Spain 49European Organization for Nuclear Research (CERN), Geneva, Switzerland 50 ́ed ́erale de Lausanne (EPFL), Lausanne, Switzerland 51Physik-Institut, Universit ̈at Z ̈urich, Z ̈urich, Switzerland 52NSC Kharkiv (NSC KIPT), Kharkiv, Ukraine 53Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 54 55H.H. Wills Physics Laboratory, 56Cavendish Laboratory, 57 22 58STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 59 60 61Oliver Lodge Laboratory, 62Imperial College London, London, United Kingdom 63 64 65Massachusetts 66 67 68Los Alamos National Laboratory (LANL), Los Alamos, NM, United States 69Syracuse University, Syracuse, NY, United States 70Pontif ́ıcia Universidade Cat ́olica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 3 71Universidad Andres Bello, Santiago, Chile, associated to 51 72 8 73Guangdong Provincial Key Laboratory of Nuclear Science, Guangdong-Hong Kong Joint Laboratory of Quantum Matter, 4 74Lanzhou University, Lanzhou, China, associated to 5 75 4 76Henan Normal University, Xinxiang, China, associated to 8 77Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia, associated to 16 78Ruhr Universitaet Bochum, Fakultaet f. Physik und Astronomie, Bochum, Germany, associated to 19 79Eotvos Lorand University, Budapest, Hungary, associated to 49 80Faculty of Physics, Vilnius University, Vilnius, Lithuania, associated to 20 81Van Swinderen Institute, 38 82Universiteit Maastricht, Maastricht, Netherlands, associated to 38 83Tadeusz Kosciuszko Cracow 41 84Universidade da Coru ̃na, A Coru ̃na, Spain, associated to 46 85 60 86Taras Schevchenko 14 87 69 88Ohio State University, Columbus, United States, associated to 68 aUniversidade Estadual de Campinas (UNICAMP), Campinas, Brazil bCentro Federal de Educac ̃ao Tecnol ́ogica Celso Suckow da Fonseca, Rio De Janeiro, Brazil c 6, Sorbonne Universit ́e, Paris, France gLamarr Institute for Machine Learning and Artificial Intelligence, Dortmund, Germany hUniversidad Nacional Aut ́onoma de Honduras, Tegucigalpa, Honduras iUniversit`a di Bari, Bari, Italy jUniversit`a di Bergamo, Bergamo, Italy kUniversit`a di Bologna, Bologna, Italy lUniversit`a di Cagliari, Cagliari, Italy mUniversit`a di Ferrara, Ferrara, Italy nUniversit`a di Genova, Genova, Italy oUniversit`a degli Studi di Milano, Milano, Italy pUniversit`a degli Studi di Milano-Bicocca, Milano, Italy qUniversit`a di Modena e Reggio Emilia, Modena, Italy rUniversit`a di Padova, Padova, Italy sUniversit`a di Perugia, Perugia, Italy tScuola Normale Superiore, Pisa, Italy uUniversit`a di Pisa, Pisa, Italy vUniversit`a della Basilicata, Potenza, Italy wUniversit`a di Roma Tor Vergata, Roma, Italy 23 xUniversit`a di Siena, Siena, Italy yUniversit`a di Urbino, Urbino, Italy zUniversidad de Ingenier ́ıa y Tecnolog ́ıa (UTEC), Lima, Peru aaUniversidad de Alcal ́a, Alcal ́a de Henares , Spain †Deceased 24
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2510.14733
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Astronomy & Astrophysics manuscript no. aanda
©ESO 2025
October 17, 2025
Expansion kinematics of young clusters. II.
NGC 2264 N & S and Collinder 95 with HectoSpec
Ishani Cheshire1, 2, Joseph J. Armstrong1,⋆, and Jonathan C. Tan1, 3
1 Department of Space, Earth & Environment, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden
2 Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA
3 Department of Astronomy, University of Virginia, Charlottesville, VA 22904, USA
October 17, 2025
ABSTRACT
Aims. Studying the dynamical evolution of young clusters is crucial for a more general understanding of the star formation process.
Methods. We took spectra of >600 candidate pre-main sequence (PMS) stars in several nearby young clusters (NGC 2264 N &
S, Collinder 95, and Collinder 359) using MMT/Hectospec. These spectra were analyzed for Hα emission and lithium absorption,
features indicative of low-mass young stellar objects (YSOs) still in their PMS evolution. We complemented these samples with YSOs
identified via Gaia DR3 variability. In conjunction with Gaia astrometry, these data enable an analysis of cluster structure, kinematics
and ages. In particular, we searched for halos of YSOs around our targets to test models of young cluster dynamical evolution.
Results. For the NGC 2264 N & S cluster pair we identified 354 YSOs, while for Collinder 95 and 359 we identified 130 and
7 YSOs, respectively. We calculate kinematic “traceback ages” for YSOs in these clusters, which we compare to isochronal ages
estimated using several sets of stellar evolution models. We find for NGC 2264 N & S that kinematic ages are generally smaller than
their isochronal ages, which may indicate these systems remained bound for a few Myr before their current state of expansion. On the
other hand, kinematic ages for Collinder 95 are often significantly larger than isochronal ages, which implies many of these YSOs did
not originate from a central, dense region, leading to overestimated kinematic ages.
Conclusions. We conclude that NGC 2264 N & S clusters likely formed as initially bound and compact systems, but have been
gradually evaporating as cluster members become unbound, forming halos of unbound YSOs surrounding the cluster cores. We
conclude that Collinder 95 likely formed initially sparse and substructured and has been dispersing since gas expulsion.
Key words. stellar clusters, kinematic ages, traceback ages, pre-main sequence
1. Introduction
The vast majority of stars are born in stellar clusters (e.g., Lada
& Lada 2003; Gutermuth et al. 2009) and so understanding the
formation and evolution of these clusters is critical to shedding
light on the star formation process. However, age estimates for
these clusters produced by fitting stellar evolutionary models to
observational data suffer from many sources of uncertainty, par-
ticularly for the low-mass PMS stars that are the majority of the
population in young clusters (Miret-Roig et al. 2024a).
Kinematic ages offer a model-independent alternative for es-
timating the age of a stellar cluster as well as providing addi-
tional insight into its evolution and history. A cluster’s kinematic
age, or traceback age, reflects a physical consequence of pro-
lific star formation in young stellar clusters. In these clusters,
feedback from young massive stars expels surrounding molec-
ular gas, causing the cluster to lose most of its binding mass
(e.g., Goodwin & Bastian 2006). This produces an expanding
halo of young stars around the cluster center, an expansion trend
reflected in some simulations of young clusters (e.g., Farias et al.
2019) and also in many recent observational works (Kuhn et al.
2019; Armstrong et al. 2022a; Guilherme-Garcia et al. 2023;
Wright et al. 2024; Armstrong & Tan 2024) which are demon-
strating that the majority of nearby young clusters exhibit such
signatures of expansion. These expansion signatures can also be
used to constrain the age of the cluster (Miret-Roig et al. 2024b;
⋆Corresponding author; jarmstrongastro@gmail.com
Armstrong & Tan 2024). By identifying young stellar objects
(YSOs) moving outwards from the cluster center and calculat-
ing the time required for them to travel to their current positions
from their assumed initial configuration, we are able to calculate
a ‘kinematic’ or traceback age of a stellar cluster. This kinematic
age analysis has been greatly enabled by the advent of Gaia’s
high precision position and proper motion measurements (Gaia
Collaboration et al. 2021), with which we can probe the internal
kinematics of stellar clusters.
Armstrong & Tan (2024) investigated the internal kinemat-
ics of the λ Ori cluster using high probability members from the
Cantat-Gaudin et al. (2020a) catalog, updated with Gaia DR3 as-
trometry, and cross-matched with the radial velocity catalog of
Tsantaki et al. (2022). Using both the Q-parameter (Cartwright
& Whitworth 2004) and Angular Dispersion parameter (Da Rio
et al. 2014), they found evidence that the cluster contains signif-
icant substructure outside the smooth central cluster core. Arm-
strong & Tan (2024) found strong evidence for asymmetric ex-
pansion in the λ Ori cluster, and determined the direction at
which the rate of expansion is at a maximum. They also inverted
the maximum rate of expansion of 0.144+0.003
−0.003 km/s/pc to give an
expansion timescale of 6.944+0.148
−0.142 Myr, which they compared to
other kinematic age methods applied to this cluster (Squicciarini
et al. 2021; Quintana & Wright 2022; Pelkonen et al. 2024) and
literature age estimates (Kounkel et al. 2018; Zari et al. 2019;
Cao et al. 2022). Armstrong & Tan (2024) also found signif-
icant asymmetry in the velocity dispersions and signatures of
Article number, page 1 of 21
arXiv:2510.14733v1 [astro-ph.SR] 16 Oct 2025
A&A proofs: manuscript no. aanda
cluster rotation in the plane-of-sky. Putting all of these results
together, they concluded that the λ Ori cluster likely formed in
a sparse, substructured configuration, and is not simply the dis-
persing remnant of an initially bound, monolithic cluster which
began to expand after the dispersal of its parent molecular cloud.
The asymmetric kinematic signatures, and the discovery of a
group of candidate ejected cluster members in particular, sug-
gest a more complex dynamical history for the λ Ori cluster.
We seek to estimate the kinematic ages for clusters Collinder
95, NGC 2264, and Collinder 359, building on methods devel-
oped by Armstrong & Tan (2024) (Paper I). We establish a robust
list of probable YSOs in these clusters through a combination of
spectroscopic youth indicators and Gaia DR3 variability YSO
flag (Marton et al. 2023). We create a robust list of YSOs and,
using their precise Gaia astrometry, we calculate the kinematic
ages of clusters Collinder 95 and NGC 2264, omitting Collinder
359 due to an insufficient amount of identified YSOs (N < 10).
We then compare these kinematic age estimates to traditional
isochronal age estimates using both the Baraffe et al. (2015a)
and PARSEC models (Marigo et al. 2017a), and infer the likely
formation history and subsequent evolution of these clusters.
2. Data
In our analysis, we combined existing kinematic data from Gaia
DR3 with new optical spectroscopic observations of low-mass
PMS candidates to create a lists of YSO candidates for each tar-
get young cluster. We then performed a kinematic age analysis
on these samples.
2.1. Target selection
The target clusters were chosen for their relatively young esti-
mated isochronal ages, making them prime targets for an in-
depth and insightful kinematic age analysis. They were also
selected due to their relatively close line-of-sight distance (<
1 kpc), allowing spectra to be obtained for the more numerous
lower-mass cluster members. The majority of the known YSO
populations of these clusters are also within a 1◦diameter on the
sky, meaning we could observe both the central cluster core and
sparse halo within the MMT field-of-view.
NGC 2264 is a nearby massive young cluster (∼760 pc,
∼3 Myr; Venuti et al. 2018), second only to the Orion Neb-
ula Cluster in terms of proximity, mass and well-defined pre-
main sequence (PMS) population (Dahm 2008), where star for-
mation is still ongoing within its parental cloud. PMS mem-
bers of NGC 2264 have been identified with X-ray observations
(e.g., Flaccomio et al. 2006; Guarcello et al. 2017), photomet-
ric variability (Cody et al. 2014), Hα emission (Dahm & Simon
2005) and other spectroscopic youth indicators (GES; Venuti
et al. 2018), belonging to a total population of >700 (Dahm
& Simon 2005), exhibiting hierarchical structure with multiple
subclusters, in particular, two embedded star-forming clumps
(NGC 2264 N / S Mon and NGC 2264 S / Cone Nebula Clus-
ter) are surrounded by a halo of older sparsely distributed YSOs
(Sung et al. 2008), indicating an extended period of sequential
star formation with a complex dynamical history (Nony et al.
2021; Flaccomio et al. 2023).
In addition, 2.5 degrees west of NGC 2264 lies the relatively
understudied young cluster Collinder 95, of similar age and dis-
tance (Cantat-Gaudin et al. 2020a). Both of these young clusters
are associated with an extensive molecular cloud complex which
has been found to contain a sparse, widely distributed population
of YSOs (Mon OB1; Rapson et al. 2014), as well as many proto-
stars, again suggesting multiple epochs of star formation in this
region.
Collinder 359 was chosen for similar reasons, being a young
open cluster at close distance of 250 pc (Lodieu et al. 2006).
While young, at ∼30 Myr, Collinder 359 is estimated to be sig-
nificantly older than NGC 2264 and Collinder 95, which aligns
with us ultimately identifying the fewest number of YSOs in
Collinder 359 out of all three clusters.
Candidates YSO members of these clusters were selected
by fitting low-mass PMS Baraffe et al. (2015a) isochrones to
Gaia estimates of BP-RP color and G-band magnitude values
for stars across all three cluster fields. YSO candidates located
above younger isochrones in the color-magnitude diagram were
given greater target priorities for observations. Targets were se-
lected in the Gmag range 14-17.5 for NGC 2264 N & S and Coll
95 and range 13-17 for Coll 359 with the goal to find low-mass
(∼1.0 −0.3M⊙) cluster members and to keep fibre cross-talk
in the observations to a minimum. Targets were also selected
within parallax range 2.2 −1.0 mas to be in the right distance
range ∼700 ± 300 pc for NGC 2264 and Coll 95, with a paral-
lax > 1.25 mas for Coll 359, and selected with RUWE < 1.4 to
prioritise cluster members with reliable Gaia astrometry.
MMT-HectoSpec has a 1 degree diameter field-of-view and
300 fibres, which sets the limit for number of targets we can ob-
serve per field. The final list of targets are selected after running
a fibre-configuration algorithm in ’xfitfibs’, which calculates the
optimal arrangement of fibres on targets without fibres crossing
and with sufficient space between them (20 arcseconds), whilst
weighting target priorities. This means that targets in denser re-
gions are more difficult to allocate fibres to, biasing selection
away from targets in the core of a cluster, but sampling the sparse
cluster halo well.
Central coordinates in RA, Dec for observed fields were
(100.2, 9.7) for NGC 2264, including both N & S subclusters
in the same FOV, (97.8, 10.0) for Coll 95 and (270.4, 3.1) for
Coll 359. As well as ∼250 science targets within a 1 degree di-
ameter of these central positions, 30 - 40 HectoSpec fibres were
allocated to empty sky for sky spectrum subtraction. Candidate
guide stars were also selected from Gaia in the Gmag range 12
- 15 and are used at the edge of the HectoSpec field, not on the
surface where the fibres are positioned.
2.2. Data reduction
Using these isochronally-selected YSO candidates, we observed
>600 stars in four nearby young clusters (NGC 2264 N, NGC
2264 S, Collinder 95, and Collinder 359) using the MMT / Hec-
tospec spectrograph. We observed using the 270 line mm−1 grat-
ing in the optical range (∼3700 −9150Å) in order to iden-
tify and flag broad Hα emission lines and lithium absorption
lines—features indicative of low-mass YSOs which are still in
their PMS evolution (Soderblom 2010).
Collinder 95 was observed on Feb 27th 2022 (3 x 40 minute
exposures), NGC 2264 on 1st March (3 x 40 minute exposures),
Coll 359 on April 4th (4 x 40 minute exposures).
For each night of observations bias, dark, domeflat and sky
flat fields were taken as well as the target fields, and data re-
duction including sky subtraction and cosmic ray rejection was
performed using the HSRED v2.0 pipeline.
Ultimately, there were 202 combined spectra obtained in the
Collinder 95 cluster with a median SNR of 36.07, 200 in the
Article number, page 2 of 21
Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec
Fig. 1. (a) Left: Example Hα spectrum of YSO candidate (Gaia Source ID: 3326702657541965952) from NGC 2264, with the source selected as
a YSO due to its strong Hα line, i.e., with an EW(Hα) = −48.97Å. (b) Right: Example 6708Å Li spectrum of YSO candidate (Gaia Source ID:
3326714782234731520) from NGC 2264, with this source selected as a YSO due to its strong Li absorption line, i.e., with EW(Li) = 0.32Å.
NGC 2264 N & S clusters with a median SNR of 30.5, and 212
in the Collinder 359 cluster with a median SNR of 53.1.
3. Young Stellar Object Identification
There are a few spectral features which can be used to identify
young PMS stars. A strong Hα emission line (6564Å), caused
by residual accretion and/or enhanced chromospheric activity,
appears in spectra for stars < 10Myr (Classical T Tauris; CTTs),
whereas a strong lithium absorption line (6708Å) is visible in
spectra for PMS stars < 20 Myr (Soderblom 2010). Once the
base of the convection zone in PMS stars reaches 3×106 K, Li is
rapidly depleted, on a timescale dependent on their spectral type,
which makes the presence of the Li line a strong discriminator
between PMS stars and older field stars.
We searched for prominent Hα emission and lithium absorp-
tion lines in the spectra of our observed candidate cluster mem-
bers, quantified by their equivalent width (EW) values. We iden-
tified YSO candidates if they have |EW(Hα)| > 10Å or EW(Li)
> 0.15Å, as these are established thresholds for these spectro-
scopic indicators of youth (e.g., Žerjal et al. 2021; Armstrong
et al. 2022b). Two example YSO candidate spectra are shown
in Figure 1. In Figure 2 we plot EW(Hα) versus EW(Li) for all
the stars observed in our target clusters with the YSO criteria
thresholds indicated as dashed lines.
We find 124 YSOs in NGC 2264, 35 in Collinder 95, and 7
in Collinder 359 using these spectroscopic indicators of youth.
To complement this sample of probable YSOs, overlapping with
the same field of view we add in YSOs flagged by Gaia DR3
variability (Marton et al. 2023): 268 YSOs in NGC 2264, 104
in Collinder 95, and 2 in Collinder 359. We also quality filter in
this step, rejecting stars with a G-band mean flux divided by its
error of less than 50, an integrated RP mean flux divided by its
error of less than 20, an integrated BP mean flux divided by its
error of less than 20, or with 5 or fewer visibility periods.
Between
these
two
methods
of
YSO
identifica-
tion—spectroscopic
and
the
Gaia
DR3
variability
cata-
log—some stars were flagged using both methods. Stars which
were flagged by both methods numbered 38 in NGC 2264, 9 in
Collinder 95, and 2 in Collinder 359. A star was listed as a YSO
in the final tally if it was flagged by either method.
A detailed breakdown of the number of YSOs identified via
each method in each cluster is available in Table 1. Using these
two methods, we generate a list of robust YSO candidates, ul-
timately selecting a total of 354 YSOs in NGC 2264, 130 in
Collinder 95, and 7 in Collinder 359.
4. Kinematic Analysis
We then use this population of YSOs to conduct a kinematic
analysis of the star clusters, excluding Collinder 359 as its sam-
ple contained too few identified YSOs to produce statistically
significant results. Therefore, we only conduct kinematic analy-
ses of the YSOs in the star clusters NGC 2264 N, NGC 2264 S
and Collinder 95.
4.1. Proper motion orientation
We search for signs of expansion in the plane-of-sky among our
YSOs by measuring the angle between each star’s cluster frame
and virtual expansion corrected proper motion vector and the
vector between the cluster center and the star’s current location.
The cluster frame motion is found by subtracting out the mean
cluster proper motion from that of each YSO.
The adopted mean proper motions, radial velocities, center
locations and half-mass radii of each cluster are listed in Table
2. We use center coordinates from existing literature (see Table
2), while YSO half-mass radii derived from our identified YSO
population. In particular, we check recent cluster catalogs with
membership based on Gaia (e.g., Cantat-Gaudin et al. 2020a;
Hunt & Reffert 2024) for cluster parameters. For Collinder 95
we take central coordinates, proper motions and distance from
Cantat-Gaudin et al. (2020a). However, for NGC 2264, these cat-
alogs do not distinguish between the two apparent subclusters,
NGC 2264 N & S. Instead, we use cluster parameters for NGC
2264 N & S from Maíz Apellániz (2019), except for the central
proper motions of NGC 2264 S, which in Maíz Apellániz (2019)
are given as identical to NGC 2264 N. For the central proper mo-
tions of NGC 2264 S we take the median proper motion of our
confirmed YSOs with Dec. < 9.7◦.
The cluster frame proper motions of the YSOs in each clus-
ter are visualized in Figure 3. These plots show arrows denoting
each star’s plane of sky motion with the direction of this motion
Article number, page 3 of 21
A&A proofs: manuscript no. aanda
Star Cluster
Spectroscopic
Targets Observed
|EW(Hα)| > 10Å
EW(Li) > 0.15Å
Spectroscopically
Flagged YSOs
Gaia variability
YSOs (Marton
et al. 2023)
Total YSOs
NGC 2264 N&S
199
64
85
124
268
354
Collinder 95
202
19
23
35
104
130
Collinder 359
212
0
7
7
2
7
Table 1. Number of identified YSOs in the clusters (NGC 2264 N&S, Collinder 95, and Collinder 359), separated by method (Hα emission line
equivalent width, lithium absorption line equivalent width, and Gaia variability YSO flag). For YSOs identified by Gaia YSO variability, we only
include stars with a G-band mean flux divided by its error greater than 50, an integrated RP mean flux divided by its error greater than 20, an
integrated BP mean flux divided by its error of greater than 20, and greater than 5 visibility periods.
color-coded according to the color wheel in the top left corner.
Gaia YSOs and spectroscopically identified YSOs are marked
differently on the plots for clarity. In Figure 3, the distinct proper
motions of NGC 2264’s two subclusters become clear through
the plotted vectors. We note that YSOs outside the half-mass
radii, indicated using dashed circles, appear to be preferentially
moving away from the cluster centers, which is particularly ap-
parent for NGC 2264 N and Collinder 95.
We plot the angle between each YSO’s proper motion and
the radial direction from cluster center to the YSO, ∆θ, produc-
ing the distributions shown in Figure 4 for all three clusters. If
there exists an expanding halo of YSOs in these clusters, we
would expect a substantial peak around zero degrees. We ob-
serve this trend for NGC 2264 N and Collinder 95, but not for
NGC 2264 S.
4.2. Tangential velocities
Virtual expansion is the projection effect in plane-of-sky motions
created when a population of stars is receding or approaching
in the line-of-sight. We corrected for this effect using equations
provided in Brown et al. (1997) using each cluster’s bulk radial
velocity as provided in the Cantat-Gaudin et al. (2020a) cata-
log. We used an MCMC approach to sample uncertainties from
the posterior distribution for the final corrected tangential veloc-
ities vl and vb, similar to the approach used by Armstrong &
Tan (2024). We use these velocities in our subsequent kinematic
analysis.
4.3. Average expansion velocities
Another indicator for cluster expansion is the median cluster ex-
pansion velocity ¯vout, i.e., the average of their radial tangential
velocities. As described in Armstrong & Tan (2024), we mea-
sure the cluster expansion velocity by taking the median expan-
sion velocity component of all cluster members for 106 itera-
tions with additional uncertainties randomly sampled from the
observed velocity uncertainties. The uncertainties on the cluster
expansion velocity are then taken as the 16th and 84th percentile
values of the posterior distribution. Note, for the purpose of allo-
cating YSOs to either NGC 2264 N or S clusters, we only include
YSOs within a 0.175◦radius of either cluster’s central position
when calculating ¯vout. Expansion velocities and uncertainties for
each cluster are presented in Table 2.
We find that NGC 2264 N has ¯vout = 0.45+0.11
−0.11 km/s, NGC
2264 S has ¯vout = 0.23+0.10
−0.10 km/s and Collinder 95 has ¯vout =
0.51+0.08
−0.08 km/s. These positive median velocities provide evi-
dence of 4σ, 2σ and 6σ significance, respectively, that these
clusters are expanding.
These velocities are lower than the expansion velocity found
for λ Ori of ¯vout = 0.71+0.02
−0.02 km/s by Armstrong & Tan (2024),
but are comparable to the expansion velocities found for δ Sco
and σ Sco sub-regions of Upper Scorpius by Armstrong et al.
(2025), and sit in the middle of the range of expansion veloc-
ities calculated by Kuhn et al. (2019); Wright et al. (2024) for
various nearby young clusters. In particular, Kuhn et al. (2019)
measured expansion velocities of ¯vout = 0.39 ± 0.15 km/s for S
Mon (NGC 2264 N) and ¯vout = 0.36±0.40 km/s for IRS 1 (NGC
2264 S; but further divided into IRS 1 and IRS 2), while Wright
et al. (2024) measured expansion velocities of ¯vout = 0.40+0.01
−0.06
km/s for S Mon (NGC 2264 N) and ¯vout = 0.23+0.03
−0.07 km/s for the
Spokes cluster (NGC 2264 S). Our expansion velocity for NGC
2264 N is in close agreement with those from both Kuhn et al.
(2019) and Wright et al. (2024), while our expansion velocity
for NGC 2254 S agrees well with that from Wright et al. (2024),
but the further division of NGC 2264 S into IRS 1 and IRS 2
groups by Kuhn et al. (2019) makes it difficult to make a direct
comparison with their results.
4.4. Areal number density profiles and cluster core sizes
We investigate the radial density profiles of the clusters and de-
fine the radii of the cluster cores using the same approach as
Armstrong & Tan (2024). We bin cluster members by increas-
ing radial distance from the cluster center, R. We calculate the
areal number density, N∗, of each bin. The resulting radial areal
number density profiles are shown in Fig. 5.
We define the core radius of a cluster, rc, as the radius at
which the density is half the peak density, which is rc = 0.65 pc
for NGC 2264 N, 1.40 pc for NGC 2264 S and 1.27 pc for
Collinder 95. We find that 24, 72 and 26 cluster members are
located within the core radii of NGC 2264 N, NGC 2264 S and
Collinder 95, respectively.
The areal density profiles of NGC 2264 N and Collinder 95
smoothly decrease with increasing radius, while the profile of
NGC 2264 S has its peak density offset from the central coor-
dinates. This is likely due to the substructure that Kuhn et al.
(2019) defined as the IRS 1 and IRS 2 subgroups of the cluster,
both of which are contained within our resulting core radius for
NGC 2264 S.
4.5. Velocity dispersions and crossing times
We calculate velocity dispersions in each cluster using the same
Bayesian inference approach described in Armstrong & Tan
(2024). We model velocity distributions as 2-d Gaussians with
free parameters for the central velocity µ and velocity dispersion
σ in each dimension. We then add an uncertainty randomly sam-
pled from the observed uncertainty distribution in each dimen-
Article number, page 4 of 21
Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec
NGC 2264 N
NGC 2264 S
Collinder 95
Central RA, Dec (°)
(100.25, 9.88)1
(100.28, 9.53)2
(97.788, 9.894)3
Central µα (mas/yr)
−1.72+0.06
−0.06
2
−2.18+0.66
−0.42
4
−2.26+0.35
−0.35
3
Central µδ (mas/yr)
−3.71+0.06
−0.06
2
−3.75+0.27
−0.40
4
−5.16+0.31
−0.31
3
Distance (pc)
719+22
−21
3
722+22
−21
3
659.0+86.7
−24.1
3
Expansion velocity ¯vout (km/s)
0.45+0.11
−0.11
0.23+0.10
−0.10
0.51+0.08
−0.08
Core radius rc (pc)
0.65
1.40
1.27
No. YSOs within rc
24
72
26
Half-mass radius r50 (pc)
3.835
3.705
3.695
Velocity dispersion σvl(rc) (km/s)
0.96+0.25
−0.19
1.29+0.17
−0.14
5.39+1.11
−0.86
Velocity dispersion σvb(rc) (km/s)
0.65+0.25
−0.18
1.77+0.23
−0.19
4.27+0.96
−0.70
Velocity dispersion σvl(r50) (km/s)
2.56+0.22
−0.19
3.15+0.33
−0.30
3.51+0.44
−0.37
Velocity dispersion σvb(r50) (km/s)
3.60+0.26
−0.23
2.27+0.18
−0.15
3.51+0.45
−0.37
Crossing time τcross(rc) (Myr)
0.79+0.18
−0.13
0.91+0.08
−0.07
0.26+0.04
−0.03
Crossing time τcross(r50) (Myr)
1.23+0.07
−0.06
1.34+0.09
−0.08
0.92+0.08
−0.07
No. YSOs with closest approach < rc
24
29
21
No. YSOs with closest approach −σ < rc
63
50
49
No. YSOs with closest approach −2σ < rc
113
69
60
No. YSOs with closest approach −3σ < rc
132
81
61
Table 2. Cluster physical properties. See the text for a discussion of how these quantities were derived. 1Cluster center value taken from Maíz
Apellániz (2019) and adjusted (+0.05°, +0) to center accurately on the cluster core of YSOs. 2Values taken from Maíz Apellániz (2019). 3Values
taken from Cantat-Gaudin et al. (2020b). 4NGC 2264 S central proper motion value calculated by taking the median proper motion value of all
YSOs located below a declination of 9.7°. 5Half-mass radii of each cluster calculated by taking the median distance from cluster center of all
member YSOs.
sion for each star in a cluster. We sample the posterior distribu-
tion function with MCMC, using an unbinned maximum likeli-
hood test to compare the model and observations. As a prior we
require that velocity dispersions must be > 0 km s−1. We repeat
for 2000 iterations with 1000 walkers, the first half of which are
discarded as burn-in. We then take the median of the posterior
distribution as the best fit for each free parameter and the 16th
and 84th percentiles as their respective 1σ uncertainties.
With this approach we calculate velocity dispersions for
cluster members within the cluster cores and for cluster members
within the cluster half-mass radii, using their virtual-expansion
corrected tangential velocities (vl, vb). Using these velocity dis-
persions along with our previous estimates of cluster core radii
and half-mass radii, we estimate the crossing times for core radii
τcross(rc) and half-mass radii τcross(r50) of each cluster. Both 1D
velocity dispersions σvl, σvb and crossing times are given for
each cluster in Table 2, along with their respective uncertainties.
Notably, while the velocity dispersions in the cluster cores
of NGC 2264 N & S are relatively low (< 2 km/s) and would
not be unusual for gravitationally bound clusters, the velocity
dispersion for the core of Collinder 95 is relatively large (> 4
km/s). Given the sparseness of the cluster, this suggests it was
born with a relatively high velocity dispersion and was gravita-
tionally unbound from an early time (see Sect. 5.3). However, to
properly assess gravitational boundedness requires assessment
of the virial state of the clusters. This in turn requires estimates
of each cluster’s total mass, which is complicated by the incom-
pleteness of our YSO sample.
For all three clusters, velocity dispersions of YSO members
within their half-mass radii are relatively high (> 2 km/s). In
the cases of NGC 2264 N & S, these velocity dispersions are
significantly greater than those of their cluster cores, indicating
that the halos of YSOs surrounding the cores of each cluster are
likely not gravitationally bound and are dispersing. In Collinder
95 these velocity dispersions seem isotropic, while in NGC 2264
N & S, velocity dispersions in the directions of Galactic longi-
tude and latitude (σv,l,σv,b) are anisotropic at the 3σ and 2.5σ
significance levels, respectively.
4.6. Traceback Ages
We calculate each cluster’s kinematic age τkin,CA by “tracing
back” each star to its moment of closest approach to the clus-
ter center, assuming constant velocity. We repeat this calculation
N = 10, 000 times, each time adding a random (normal) amount
of the proper motion error to the proper motion of each star. This
generates a distribution of kinematic ages for each star, which
we can then use to find the estimated error bounds (16th and
84th percentiles) of each star’s kinematic age. From this, we are
able to calculate a kinematic age value with error estimates for
each star in each stellar cluster. We then use the distribution of a
cluster’s YSO’s kinematic ages to estimate the overall kinematic
age value of a cluster.
We start by looking only at YSOs with a particularly strong
outward expansion from the cluster center, i.e., those with ∆θ
within ±50°of zero, as denoted by the dashed vertical lines in
Figure 4. These distributions of kinematic ages are shown in Fig-
ure 6.
For NGC 2264 N, the median kinematic age of all YSO clus-
ter members within ±50° is τkin,CA = 1.08 Myr with a standard
deviation of ±1.64 Myr. For NGC 2264 S, the median kinematic
age of such YSO cluster members is τkin,CA = 0.61 Myr with a
standard deviation of ±0.51 Myr. For Collinder 95, the median
kinematic age of such cluster members is τkin,CA = 0.94 Myr
with a standard deviation of ±0.84 Myr. All of our kinematic
age estimations are listed in Table 3.
When we further restrict the population we are considering to
only YSOs whose current distance from cluster center is greater
than the half-mass radius, shown as dashed circles in Figure 3,
and whose final closest approach distance from cluster center is
less than the cluster’s half-mass radius, we observe the median
kinematic age values of each cluster increase slightly. The half-
Article number, page 5 of 21
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0.5
0.0
0.5
1.0
1.5
2.0
EW(Li) (Å)
50
40
30
20
10
0
10
20
EW(H ) (Å)
NGC 2264
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
EW(Li) (Å)
120
100
80
60
40
20
0
20
EW(H ) (Å)
Collinder 95
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
EW(Li) (Å)
15.0
12.5
10.0
7.5
5.0
2.5
0.0
2.5
5.0
EW(H ) (Å)
Collinder 359
EW(Li)± > 0.15 Å
EW(H )± < -10 Å
All Stars
Gaia Variability Stars
Fig. 2. Scatter plots of Hα equivalent width vs lithium equivalent width
for all the stellar spectra in our samples of clusters NGC 2264 (N & S),
Collinder 95, and Collinder 359. If a star has an EW(Hα)−σ < −10Å
(above the horizontal blue dashed line), or an EW(Li)−σ > 0.15Å (right
of the vertical green dot-dashed line), it is flagged as a YSO. Gaia vari-
ability catalog stars are additionally marked with a closed circle.
mass radius used here is calculated as the smallest radius that
includes half of our identified YSOs in each group. We note that
selection bias of spectroscopic targets away from the dense clus-
ter cores (because of fibre crowding) might bias the half-mass
radius that we calculate from our sample to moderately larger
values. The half-mass radii for each cluster are listed in Table 2.
When filtering only for YSOs whose current position is out-
side our half-mass radius and whose closest approach distance
was inside of it, we find that: for NGC 2264 N, the median
99.8
100.0
100.2
100.4
100.6
RA (°)
9.4
9.6
9.8
10.0
10.2
Dec (°)
NGC 2264 N
1 mas/yr
100.0
100.2
100.4
100.6
RA (°)
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
10.0
Dec (°)
NGC 2264 S
1 mas/yr
97.4
97.6
97.8
98.0
98.2
RA (°)
9.6
9.8
10.0
10.2
10.4
Dec (°)
Collinder 95
1 mas/yr
Gaia Variability YSOs
Spectroscopic YSOs
Half-Mass Radius
Fig. 3. Proper motion maps of YSOs in NGC 2264 N, NGC 2264 S, and
Collinder 95. Stars flagged as YSOs via our spectroscopic analysis are
marked with an open circle, YSOs flagged through the Gaia variabil-
ity catalog are marked with smaller black point. (Stars flagged by both
indicators are denoted through a filled in circle / larger black point.)
The proper motion of each YSO is shown with an arrow which is color-
coded according to the direction of motion. The average motion of each
cluster or subcluster has been subtracted out (see these values in Table
2), allowing us to observe the plane of sky motion of each YSO within
the cluster frame. The center of the cluster is denoted by the intersec-
tion point of the two blue dotted lines. The dotted blue circle denotes
the half-mass radius of the identified YSOs in each cluster.
Article number, page 6 of 21
Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec
100
0
100
0
10
20
30
40
Count
NGC 2264 N
100
0
100
0
10
20
30
40
50
60
Count
NGC 2264 S
100
0
100
0
5
10
15
20
Count
Collinder 95
Fig. 4. Expansion direction histograms of YSOs in NGC 2264 N, NGC
2264 S, and Collinder 95, where the measured angle, ∆θ, is the differ-
ence in a YSO’s direction of motion from pure radial outward motion
from the cluster center. We observe that there is a large proportion of
stars near zero (±50°) for NGC 2264 N and Collinder 95, indicating
that there is significant outward expansion of YSOs from their cluster
centers. We do not observe this same trend for NGC 2264 S. The shaded
region shows a range ±50◦, defining a subset of YSOs with strongest ra-
dial expansion.
kinematic age shifts to 1.81 Myr with a standard deviation of
1.47 Myr; for NGC 2264 S, the median kinematic age shifts to
1.14 Myr with a standard deviation of 0.57 Myr; for Collinder
95, the median kinematic age becomes 1.62 Myr with a standard
deviation of 0.77 Myr. We can observe this increase clearly in
Figure 6. These values are listed also in Table 3.
We also consider how many cluster members have their clos-
est approach to the cluster centers within the core radii of each
cluster, as defined in Section 4.4, and which are currently lo-
cated outside the cores. We list the number of YSOs per cluster
Fig. 5. Radial profiles of areal stellar number density profiles of
NGC 2264 N (Top), NGC 2264 S (Middle) and Collinder 95 (Bot-
tom). Vertical dashed lines indicate the core radii (rc), as defined in Sec-
tion 4.4, and half-mass radii (r50).
with closest approach distances within the core radii with dif-
ferent tolerances according to the uncertainties on their closest
approach distances in Table 2.
5. Age Comparison
5.1. Correcting for Extinction
We then seek to compare the kinematic age estimates calculated
in the previous section to the isochronal ages of each cluster.
To calculate these isochronal ages, we first correct our YSOs
for reddening and extinction. Using extinction values from the
STARHORSE catalog (Anders et al. 2022) and Gaia DR3 data,
we create a local extinction “map” to estimate extinction values
of all YSOs in our clusters. The map consists of 0.2° RA by 0.2°
Dec cells as shown in Figure 7.
We then take the median STARHORSE AG extinction values
of all stars within each RA-Dec cell, only including stars within
a certain range of parallax values. For NGC 2264, the range of
parallaxes present in our YSOs is 1.018 mas to 2.958 mas, so
Article number, page 7 of 21
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0.00
0.25
0.50
0.75
1.00
YSO Count (Normalized)
NGC2264 N
Median: 0.77 Myr
NGC2264 S
Median: 0.58 Myr
Collinder 95
Median: 0.66 Myr
0.00
0.25
0.50
0.75
1.00
YSO Count (Normalized)
NGC2264 N
|
| < 50°
Median: 1.08 Myr
NGC2264 S
|
| < 50°
Median: 0.62 Myr
Collinder 95
|
| < 50°
Median: 1.00 Myr
0.0
2.5
5.0
7.5
kin, CA (Myr)
0.00
0.25
0.50
0.75
1.00
YSO Count (Normalized)
NGC2264 N
|
| < 50°
CA < r1/2
Median: 1.81 Myr
0.0
2.5
5.0
7.5
kin, CA (Myr)
NGC2264 S
|
| < 50°
CA < r1/2
Median: 1.15 Myr
0.0
2.5
5.0
7.5
kin, CA (Myr)
Collinder 95
|
| < 50°
CA < r1/2
Median: 1.58 Myr
Fig. 6. Traceback ages (see text) of YSOs for NGC 2264 N (left column), NGC 2264 S (middle column), and Collinder 95. Each row represents
a different sub-population. In the first row, we show all identified YSOs in each cluster. In the second row, we show YSOs with |∆θ| ≤50◦. In
the third row, we further restrict the |∆θ| ≤50◦YSOs population by requiring them to have a current distance from cluster center greater than
the half-mass radius and a distance of closest approach to the cluster center of less than the half-mass radius. The median and ±1σ range of the
distributions are indicated.
when picking from the extinction catalog, we only include stars
between 0.95 mas and 3.0 mas. For Collinder 95, the range of
parallaxes present in our YSOs is 1.037 mas to 2.818 mas, so
when picking from the extinction catalog, we only include stars
between 0.95 mas and 2.9 mas.
Then, we assign this median extinction value to all YSOs
located within that cell. Using this method, we generate esti-
mated extinction values for all YSOs in our sample, along with
error bounds on these values (the 16th and 84th percentiles.) The
minimum amount of catalog stars in each cell used to calculate
these median and error bounds is Nmin = 31 for NGC 2264 and
Nmin = 20 for Collinder 95. The extinction values estimated for
each cell are visualized in Figure 7. We follow the same ap-
proach for ABP & ARP in order to also estimate reddening values
in the Gaia photometric bands.
5.2. Calculating Isochronal Ages
To estimate the isochronal ages τiso for cluster members, we
use the Baraffe et al. (2015a), PARSEC (Marigo et al. 2017a)
and SPOTS (Somers et al. 2020) stellar isochrone models. The
Baraffe et al. (2015a) model specializes in PMS stars and low-
mass MS stars, with a mass range of 0.01 −1.4M⊙, while the
PARSEC model has a mass range > 0.1M⊙. The SPOTS mod-
els occupy a smaller mass range 0.08 −1.3M⊙, but are distin-
guished by having parameters for spot coverage fraction Fspot
and spot temperature ratio Xspot. Here we assume Fspot = 0.35
and Xspot = 0.9 and use the corresponding set of isochrones, as
was done for the λ Ori cluster by Armstrong & Tan (2024).
We start with the Baraffe et al. (2015a) stellar isochrones.
Following the method of Armstrong & Tan (2024), for each star
in each cluster, we compare each individual star to a series of
∼960 Baraffe et al. (2015a) isochrones with ages ranging from
∼0.5 to 10, 000 Myr. For each star, we iterate the calculation
N = 100 times, each time multiplying the error on the extinc-
tion value with a value randomly sampled from a normal distri-
bution N ∼(0, 1) and add this to the star’s median extinction
value. Then, we use this new perturbed extinction value to find
the isochrone closest to this star’s G-band magnitude and BP-RP
color. We then do the same for the G-band magnitude and G-RP
color. From this series of N = 100 iterations, we are able to gen-
erate a distribution of isochronal ages for each star for both the
BP-RP and G-RP isochrones. Then, we take the median age as
Article number, page 8 of 21
Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec
99.6
99.8
100.0
100.2
100.4
100.6
100.8
RA (°)
9.0
9.2
9.4
9.6
9.8
10.0
10.2
10.4
DEC (°)
NGC 2264 Line of Sight G-band Extinction Estimation
YSO sample stars
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
AG50
97.2
97.4
97.6
97.8
98.0
98.2
98.4
RA (°)
9.4
9.6
9.8
10.0
10.2
10.4
10.6
DEC (°)
Collinder 95 Line of Sight G-band Extinction Estimation
YSO sample stars
0.2
0.4
0.6
0.8
1.0
AG50
Fig. 7. (a) Left: A plot of the extinction correction estimation for NGC 2264 N & S. The extinction values of each cell are determined by drawing
from a catalog of stars’ extinction values derived from the STARHORSE code and Gaia DR3 data (Anders et al. 2022). We divide the cluster into
0.2° radial ascension by 0.2° declination cells. Then, we take the median extinction value of each cell and apply it to all the YSOs within that cell,
thereby estimating each YSOs’ extinction value. The YSOs are overplotted as orange crosses. The lowest number of catalog stars populating any
cell is Nmin = 20 for Collinder 95 and Nmin = 31 for NGC 2264. (b) Right: As (a), but for Collinder 95.
the isochronal age value for that star. We also take the 16th and
84th percentiles of this distribution as the upper and lower error
bounds on this value. From this process, we are able to calcu-
late the Baraffe et al. (2015a) isochronal ages for each individual
star for either BP-RP and G-RP color. We follow the same ap-
proach to estimate ages using PARSEC (Marigo et al. 2017a)
and SPOTS (Somers et al. 2020) isochrones.
Median isochronal ages for each cluster are given in Table 3.
Particularly noticeable is the clear difference between ages esti-
mated using either BP-RP or G-RP colors. Using Baraffe et al.
(2015b) models, ages estimated using G-RP color are consis-
tently younger than ages estimated using BP-RP color, while us-
ing Marigo et al. (2017a) models, ages estimated using G-RP
color are consistently older. These discrepancies in ages esti-
mated using different stellar evolution models and different col-
ors have been seen for other nearby young clusters and associ-
ations in the literature (e.g., Ratzenböck et al. 2023; Armstrong
& Tan 2024; Fajrin et al. 2024).
5.3. Kinematic Age versus Isochronal Age
In Figs. A.4, A.5, A.6 (top row) we compare the isochronal age
values for each set of stellar evolution models using Gaia G-RP
colour with the kinematic ages of YSOs with position-velocity
angles with ±50◦. We generally expect the kinematic ages of
cluster members will be lower than their isochronal ages. This
is because the kinematic age calculation gives the time when a
star would have been at its closest approach distance to the clus-
ter center, but a star may have formed significantly earlier than
that, and only have begun moving away from the cluster cen-
ter following residual gas expulsion around the cluster and the
loss of binding mass. Therefore, isochronal age estimates should
be higher than kinematic age estimates for stars that originated
within the cluster, and the average difference indicates the dura-
tion of an “embedded phase” (e.g., Miret-Roig et al. 2024a). In
Figures A.4, A.5 we can see that this holds true for most YSOs
in NGC 2264 N & S, as generally fewer stars are in the greyed
out “forbidden” region, where isochronal age is less than kine-
matic age. However, many YSOs belonging to Collinder 95 are
located in the ’forbidden zone’, with kinematic ages seemingly
too old for their isochronal ages.
Comparing the isochronal ages for each cluster using each
stellar isochrone model, in general the PARSEC (Marigo et al.
2017a) models give older ages, followed by SPOTS (Somers
et al. 2020) models, and the Baraffe et al. (2015a) models give
the youngest.
We also compare ages for subsets of cluster members fil-
tered on closest approach distance in Figs. A.4, A.5, A.6, re-
quiring the closest approach distance < rc (lower rows) or clos-
est approach distance < rc within the 1σ uncertainty (middle
rows), and that the YSO is located outside the core radius rc.
In Figs. A.1, A.2, A.3, half-mass radius (r50) is used rather than
core-radius (rc), with the additional requirement that the YSOs’
closest approach distance is within r50, while their current dis-
tance is outside of r50. (In Fig. A.7 some YSOs have BP-RP
τiso,SPOTS = 0, meaning that many are located above the SPOTS
log(Age) = 4 isochrone—likely because their extinction is not
well estimated or they are binaries. Nevertheless, these younger
ages (< 0.5 Myr) are unreliable. There are fewer zeros in the G-
RP isochronal ages than in the BP-RP isochronal ages; therefore,
G-RP is used in Figs. A.4, A.5, & A.6.)
For both NGC 2264 N & S clusters the majority of cluster
members have isochronal ages greater than their kinematic ages.
However, for Collinder 95 there are still a significant number of
member YSOs with τkin,CA > τiso for any stellar evolution model,
which thus inhabit the “forbidden” zone. A possible interpreta-
tion of this would be that YSOs in the “forbidden” zone did not
originate from their position of closest approach to the cluster
center, but from an initially sparser distribution, and thus τkin,CA
are overestimated ages. Alternatively, there is also the possibility
that all of our isochronal age estimates are underestimated, either
due to underestimated reddening, or otherwise that the stellar
evolution models themselves systematically underestimate ages
for YSOs in this mass range.
Furthermore, instead of comparing the kinematic ages and
isochronal ages of individual cluster members, we can also con-
Article number, page 9 of 21
A&A proofs: manuscript no. aanda
sider the cluster as a whole. We find that the median kine-
matic age values for all three clusters are lower than the me-
dian isochronal age values in nearly all cases, as seen in Figure
A.7, for both the Baraffe et al. (2015a) and PARSEC isochrones.
Here, we can see clearly that the Baraffe et al. (2015a) isochrones
yield much lower ages than the PARSEC isochrones. Further-
more, in Figure A.7 we also note again that, as expected, very
few cluster medians fall in the grey “forbidden” zone where
isochronal ages are less than kinematic ages (see also Miret-Roig
et al. 2024a).
5.4. Age Spread
In each cluster, we look for evidence of significant spread in
each of τkin,CA and τiso estimated using each set of stellar evo-
lution models. Following the same approach as Armstrong &
Tan (2024), we calculate the difference between τ for individual
cluster members and the sample mean ¯τ and normalize by the
uncertainty of τ for each cluster member, removing the largest
10% as outliers. We report the median normalised age differ-
ences, στ,kin,CA and στ,iso, per cluster.
For all YSOs consistent with moving away from the clus-
ter centers, which thus have a valid traceback age to closest ap-
proach τkin,CA, we obtain spreads in kinematic age στ,kin,CA of
23.6, 29.0 and 9.9 for NGC 2264 N, NGC 2264 S, and Collinder
95, respectively. For these stars we obtain spreads in τiso using
Baraffe models with BP-RP color of 22.3, 21.1 and 20.4, but
only 3.4, 2.2 and 2.1 with G-RP color. With PARSEC models
we obtain στ,iso of 12.8, 10.9 and 9.3 with BP-RP, and 2.1, 2.1
and 4.2 with G-RP. With SPOTS models we obtain στ,iso of 6.1,
6.9 and 8.9 with BP-RP, and 2.3, 1.9 and 2.9 with G-RP.
For YSOs whose closest approach distance is within the clus-
ter core radius, rc, we obtain στ,kin,CA of 5.3, 116.7 and 46.4 for
NGC 2264 N, NGC 2264 S, and Collinder 95, respectively. For
these stars we obtain spreads in τiso using Baraffe models with
BP-RP color of 46.7, 8.4 and 24.1, and 2.5, 2.0 and 2.9 with G-
RP color. With PARSEC models we obtain στ,iso of 23.0, 4.7 and
17.1 with BP-RP, and 2.0, 2.5 and 1.2 with G-RP. With SPOTS
models we obtain στ,iso of 13.0, 4.3 and 23.7 with BP-RP, and
1.9, 1.8 and 4.7 with G-RP.
For YSOs whose closest approach distance is within 1σ un-
certainty of the cluster core radius, rc, we obtain στ,kin,CA of
6.0, 12.9 and 6.4 for NGC 2264 N, NGC 2264 S, and Collinder
95, respectively. For these stars we obtain spreads in τiso using
Baraffe models with BP-RP color of 24.9, 24.9 and 14.5, and 2.0,
2.2 and 2.5 with G-RP color. With PARSEC models we obtain
στ,iso of 10.5, 6.3 and 8.5 with BP-RP, and 1.7, 3.4 and 3.9 with
G-RP. With SPOTS models we obtain στ,iso of 8.2, 6.2 and 19.5
with BP-RP, and 1.9, 1.2 and 2.6 with G-RP.
Overall, the evidence for isochronal age spread στ,iso is un-
certain, i.e., the significance of the spread depends strongly on
the models used to estimate ages and even on the photometric
colors used for the same set of models, but is not strongly de-
pendent on the filtering of cluster members by closest approach
distance.
We do find evidence for kinematic age spread στ,kin,CA in all
clusters, and particularly στ,kin,CA is significant for cluster mem-
bers with closest approach distances < rc of each cluster. This
would likely indicate for NGC 2264 N & S, which have dense
and compact cores, that cluster members have gradually been
ejected or become unbound from the cluster cores over their life-
times (2-3 Myr). For Collinder 95, which is sparse and likely
was not as compact at formation as NGC 2264 N & S, the kine-
matic age spread is more likely a result of kinematic ages being
overestimated for YSOs that did not originate from their closest
approach positions, i.e., the same reason as why many cluster
members inhabit the “forbidden” zones of Fig. A.6.
5.5. Cluster Halos
We compare the distributions of the clusters halos in λ Ori (Arm-
strong & Tan 2024), NGC 2264 N & S, and Collinder 95, defined
as cluster members consistent with originating within the cluster
core radii rc within 1σ of their closest approach distance (blue;
Fig. 8). We show their spatial distributions, histograms of rel-
ative velocity orientation ∆θ and isochronal ages, τiso, accord-
ing to PARSEC models (Marigo et al. 2017b) using Gaia BP-
RP color against traceback timescale to closest approach, τkin,CA
(Fig. 8).
The distributions of halo members for each cluster are dis-
tinct. For NGC 2264 N & S, the majority of halo members are
located within each cluster’s half-mass radius r50, have wider
distributions of ∆θ, though still with peaks close to 0◦, and have
young traceback ages, τkin,CA, generally < 3 Myr and younger
than their isochronal ages, τiso. For Collinder 95, the majority of
halo members are located outside the cluster’s half-mass radius
r50, have a narrower distribution of ∆θ with a peak close to 0◦,
but have older traceback ages, τkin,CA, that are generally 4 −14
Myr and older than their isochronal ages, τiso. For λ Ori, close to
half of halo members are located outside the cluster’s half-mass
radius r50 and half are within. They have a very narrow distribu-
tion of ∆θ with a peak close to 0◦, but have a mix of traceback
ages, τkin,CA, half of which are older than their isochronal ages,
τiso, and half of which are younger.
Since the vast majority of halo members of NGC 2264 N
& S have τkin,CA < τiso, they are consistent with having formed
within the bound cores of these clusters, but have subsequently
become unbound, likely including some via dynamical ejection,
as the clusters evolve dynamically. Apart from these halo mem-
bers, there is also a sparse distribution of YSOs surrounding the
cluster cores which are not consistent with originating in either
core. These belong to smaller substructures which could be con-
sidered part of the wider Mon OB1 association (Rapson et al.
2014; Wright 2020).
Since the majority of halo members for Collinder 95 have
τkin,CA > τiso, they are not consistent with having originated
within the cluster core, unless the isochronal ages τiso are
severely underestimated. It is possible that many of these may
have formed in smaller substructures surrounding the core, sim-
ilar to the non-halo YSOs surrounding NGC 2264 N & S, but
then it is striking that so many of these have velocities directed
radially away from the core, consistent with overall linear ra-
dial expansion and seemingly with little random motion inher-
ited from the turbulence of their natal cloud. The large spread of
τkin,CA among halo members also indicates that they would have
escaped the core of Collinder 95 gradually rather than after a
single catastrophic event, such as a supernova expelling residual
gas, or alternatively that they originated at a range of distances
from the cluster core and have since moved radially away from
it.
As described in (Armstrong & Tan 2024), λ Ori also has
many halo members with τkin,CA > τiso, but most of these dis-
appear with a more restrictive filter on closest approach distance
leaving only candidate ejected stars with timescales compatible
with other age estimates (< 4 Myr), unlike with Collinder 95
(see Fig. A.6). The significant substructure surrounding the clus-
ter core, which appears to be inconsistent with originating from
it, similar to NGC 2264, has lead to the conclusion that much
Article number, page 10 of 21
Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec
Fig. 8. Spatial coordinates (left column) with cluster core radii rc and half-mass radii r50 indicated as black circles, relative velocity orientation
∆θ histograms (middle column) and PARSEC ages τiso against traceback to closest approach age τkin,CA (right column), for λ Ori, NGC 2264 N &
S and Collinder 95 (descending rows). Cluster members with closest approach distance within 1σ of the core radius rc and with present position
outside of the core radius are indicated as blue points.
Article number, page 11 of 21
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Star Cluster
YSO Sub-
population
Median
Kinematic
Age (Myr)
Median
BP-RP
Baraffe et al.
(2015a)
Isochronal
Age (Myr)
Median
G-RP
Baraffe et al.
(2015a)
Isochronal
Age (Myr)
Median
BP-RP
PARSEC
Isochronal
Age (Myr)
Median
G-RP
PARSEC
Isochronal
Age (Myr)
Median
BP-RP
SPOTS
Isochronal
Age (Myr)
Median
G-RP
SPOTS
Isochronal
Age (Myr)
NGC 2264
N
All YSOs
0.77
2.72
1.42
4.02
8.74
0.94
3.17
Within ±50°
1.08
2.85
1.49
3.86
6.72
1.25
2.85
Travelled
Significantly
Outward and
within ±50°
1.81
3.23
2.00
4.27
5.08
1.71
3.13
NGC 2264 S
All YSOs
0.58
2.72
1.42
4.02
8.74
0.94
3.17
Within ±50°
0.61
2.95
1.84
4.32
6.72
1.73
3.18
Travelled
Significantly
Outward and
within ±50°
1.14
2.99
2.00
4.33
6.21
1.81
2.82
Collinder 95
All YSOs
0.67
2.13
0.62
3.84
11.22
0.00
2.74
Within ±50°
0.94
1.97
0.59
3.86
12.17
0.00
2.36
Travelled
Significantly
Outward and
within ±50°
1.62
1.59
0.60
3.86
10.99
0.00
2.46
Table 3. Median kinematic age values for clusters NGC 2264 N, NGC 2264 S, and Collinder 95. Median values of isochronal ages are also
included for comparison. Breakdowns across two YSO subpopulations are included as well. The first row of median values includes all YSOs
when calculating the cluster median cluster age. The second represents a subpopulation and includes only YSOs with a proper motion within ±50°
of the radial. The third median includes only YSOs which both have a proper motion within ±50° of the radial and have traveled significantly
outwards from the cluster center over their lifetimes (Rclosest approach < half-mass radius and Rcurrent > half-mass radius.) In general, the cluster
median kinematic ages are significantly lower than isochronal ages.
of this population formed sparsely distributed and hierarchically
structured (Kounkel et al. 2018; Armstrong & Tan 2024).
6. Conclusions
By analyzing optical stellar spectra and flagging stars with spec-
tral indicators of youth (|EW(Hα)| > 10Å or EW(Li) > 1.5Å),
we are able to identify 166 YSOs across several young clusters
(NGC 2264 N & S, Collinder 95, and Collinder 359). Using an
additional 325 YSOs flagged as variable stars by the Gaia DR3
catalog (Marton et al. 2023), we generate lists of YSOs (with
N > 100) for both NGC 2264 N & S and Collinder 95, allowing
us to calculate the kinematic or traceback ages for these clusters.
After limiting the YSOs used in the kinematic age estimation to
only those with ∆θ within ±50°, we find a median kinematic age
of 1.08 Myr for NGC 2264 N, 0.61 Myr for NGC 2264 S, and
0.94 Myr for Collinder 95.
We find in all clusters significant numbers of YSOs out-
side of the cluster radii with velocities inconsistent with ori-
gin within the cluster radii (see Fig. 3). This provides evidence
for a sparse population of YSOs surrounding the dense cluster
cores, with distinct substructure and kinematics, which could be-
long to the young stellar association Mon OB1. The existence of
bound, compact clusters within the volumes of sparse associa-
tions has been observed elsewhere (e.g., Vela OB2; Armstrong
et al. 2022a)
We compare the kinematic age values with isochronal age
values for the Baraffe et al. (2015a) and PARSEC (Marigo et al.
2017a) stellar evolution models. As seen in Figure A.7, although
the three isochronal models give significantly different results,
most models provide isochronal age estimates which are, on av-
erage, higher than our kinematic age estimates, which is consis-
tent with YSOs forming within a more compact cluster volume
and then subsequently becoming unbound and drifting away.
We find significant differences between ages estimated when
fitting stellar evolution models to YSOs in either BP-RP or G-RP
color, as can be seen in Figs. A.1, A.2, A.3, & A.7 and in ages re-
ported in Table 3. This indicates systematic differences between
the observed photometric bands and the corresponding synthetic
photometric bands as predicted by stellar evolution models.
The systematic offset between kinematic ages and isochronal
estimates, of 2-4 Myr according to PARSEC models in BP-RP
color, for example (see Fig. A.7), has been suggested in simi-
lar recent works (such as Miret-Roig et al. 2024b) to indicate
an early period in cluster evolution where the cluster is still em-
bedded in its parent molecular cloud for several Myrs, before
stellar feedback expels this gas and cluster members begin to
become unbound as the majority of binding mass is lost. How-
ever, this offset is not observed for all clusters. Armstrong & Tan
(2024) estimated kinematic ages for the λ Ori cluster using sev-
eral approaches, but found that these were all in agreement with
or even greater than ages estimated in the literature using stellar
evolution models. They also found evidence that the λ Ori cluster
likely formed in a sparse, hierarchically structured configuration
and so was initially unbound.
Collinder 95, similarly, has a sparse distribution of YSO
members, a majority of which have relative motions consistent
with expanding away from the cluster center. Also, for those
YSO members consistent with being unbound from the cluster,
the median kinematic age (1.62 Myr) is in close agreement with
Article number, page 12 of 21
Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec
the median isochronal age estimated using Baraffe et al. (2015b)
isochrones in BP-RP color (1.59 Myr; Table 3), which would
suggest that Collinder 95 formed initially sparse and unbound.
However, the uncertainty in the isochronal age estimates makes
this difficult to verify. This contrasts with the NGC 2264 N & S
clusters, which are more massive and densely concentrated and
which likely formed bound before subsequently losing cluster
members a few Myr into their evolution.
Acknowledgements. I.C. acknowledges support from a Chalmers Astro-
physics & Space Science Summer (CASSUM) Research Fellowship, in-
cluding partial support from W&M Lundgrens grant 2022-4019. J.J.A.
acknowledges
support
from
a
Chalmers
Initiative
on
Cosmic
Origins
(CICO) postdoctoral fellowship. J.C.T. acknowledges support from ERC
Advanced Grant MSTAR (788829). This work has made use of data
from the ESA space mission Gaia (http://www.cosmos.esa.int/gaia), pro-
cessed by the Gaia Data Processing and Analysis Consortium (DPAC,
http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for DPAC has
been provided by national institutions, in particular the institutions participating
in the Gaia Multilateral Agreement. This research made use of the Simbad and
Vizier catalogue access tools (provided by CDS, Strasbourg, France), Astropy
(Astropy Collaboration et al. 2013) and TOPCAT (Taylor 2005).
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Appendix A: Supplementary Figures
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Fig. A.1. Kinematic age, τkin, against isochronal age, τiso, for all YSOs (top panel), for YSOs moving away from the cluster center (middle panel),
and for YSOs with closest approach distances within r50 and current distances outside of r50 (lower panel), for NGC 2264 N. The plot denotes
photometric color used for τiso using point color (BP-RP as black, G-RP as blue.)
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Fig. A.4. Kinematic age, τkin, against isochronal age, τiso (G-RP), for YSOs moving away from the cluster center (top panel), for YSOs with
closest approach distances within 1σ of the core radius, rc (middle panel), and for YSOs with closest approach distances within the core radius rc
(lower panel), for NGC 2264 N.
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Fig. A.5. As Fig. A.4, but for NGC 2264 S.
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Fig. A.6. As Fig. A.4, but for Collinder 95.
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Fig. A.7. The median kinematic ages (τkin) versus their median isochronal ages (τiso) for clusters NGC2264 N (red), NGC 2264 S (green), and
Coll 95 (orange). The plot is divided into nine panels, with the three subgroups as rows: all YSOs (top panel), YSOs moving away from the cluster
center (middle panel), and YSOs with closest approach distances within r50 and current distances outside of r50 (lower panel). Columns denote
the different isochrone models used, with Baraffe (left), PARSEC (middle), and SPOTS (right). The plot denotes photometric color used (BP-RP
or G-RP) using symbols (BP-RP marked by a plus, G-RP marked by an X.) The error bars on each value are the 16th and 84th percentiles of the
distribution of either kinematic or isochronal ages.
Article number, page 21 of 21
|
Astronomy & Astrophysics manuscript no. aanda October 17, 2025 Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec Ishani Cheshire1, 2, Joseph J. Armstrong1,⋆, and Jonathan C. Tan1, 3 1 -412 96 Gothenburg, Sweden 2 94720-3411, USA 3 22904, USA October 17, 2025 ABSTRACT Aims. Studying the dynamical evolution of young clusters is crucial for a more general understanding of the star formation process. Methods. We took spectra of >600 candidate pre-main sequence (PMS) stars in several nearby young clusters (NGC 2264 N & S, Collinder 95, and Collinder 359) using MMT/Hectospec. These spectra were analyzed for Hα emission and lithium absorption, features indicative of low-mass young stellar objects (YSOs) still in their PMS evolution. We complemented these samples with YSOs identified via Gaia DR3 variability. In conjunction with Gaia astrometry, these data enable an analysis of cluster structure, kinematics and ages. In particular, we searched for halos of YSOs around our targets to test models of young cluster dynamical evolution. Results. For the NGC 2264 N & S cluster pair we identified 354 YSOs, while for Collinder 95 and 359 we identified 130 and 7 YSOs, respectively. We calculate kinematic "traceback ages" for YSOs in these clusters, which we compare to isochronal ages estimated using several sets of stellar evolution models. We find for NGC 2264 N & S that kinematic ages are generally smaller than their isochronal ages, which may indicate these systems remained bound for a few Myr before their current state of expansion. On the other hand, kinematic ages for Collinder 95 are often significantly larger than isochronal ages, which implies many of these YSOs did not originate from a central, dense region, leading to overestimated kinematic ages. Conclusions. We conclude that NGC 2264 N & S clusters likely formed as initially bound and compact systems, but have been gradually evaporating as cluster members become unbound, forming halos of unbound YSOs surrounding the cluster cores. We conclude that Collinder 95 likely formed initially sparse and substructured and has been dispersing since gas expulsion. Key words. stellar clusters, kinematic ages, traceback ages, pre-main sequence 1. Introduction The vast majority of stars are born in stellar clusters (e.g., Lada & Lada 2003; Gutermuth et al. 2009) and so understanding the formation and evolution of these clusters is critical to shedding light on the star formation process. However, age estimates for these clusters produced by fitting stellar evolutionary models to observational data suffer from many sources of uncertainty, particularly for the low-mass PMS stars that are the majority of the population in young clusters (Miret-Roig et al. 2024a). Kinematic ages offer a model-independent alternative for estimating the age of a stellar cluster as well as providing additional insight into its evolution and history. A cluster's kinematic age, or traceback age, reflects a physical consequence of prolific star formation in young stellar clusters. In these clusters, feedback from young massive stars expels surrounding molecular gas, causing the cluster to lose most of its binding mass (e.g., Goodwin & Bastian 2006). This produces an expanding halo of young stars around the cluster center, an expansion trend reflected in some simulations of young clusters (e.g., Farias et al. 2019) and also in many recent observational works (Kuhn et al. 2019; Armstrong et al. 2022a; Guilherme-Garcia et al. 2023; Wright et al. 2024; Armstrong & Tan 2024) which are demonstrating that the majority of nearby young clusters exhibit such signatures of expansion. These expansion signatures can also be used to constrain the age of the cluster (Miret-Roig et al. 2024b; ⋆Corresponding author; Armstrong & Tan 2024). By identifying young stellar objects (YSOs) moving outwards from the cluster center and calculating the time required for them to travel to their current positions from their assumed initial configuration, we are able to calculate a 'kinematic' or traceback age of a stellar cluster. This kinematic age analysis has been greatly enabled by the advent of Gaia's high precision position and proper motion measurements (Gaia Collaboration et al. 2021), with which we can probe the internal kinematics of stellar clusters. Armstrong & Tan (2024) investigated the internal kinematics of the λ Ori cluster using high probability members from the Cantat-Gaudin et al. (2020a) catalog, updated with Gaia DR3 astrometry, and cross-matched with the radial velocity catalog of Tsantaki et al. (2022). Using both the Q-parameter (Cartwright & Whitworth 2004) and Angular Dispersion parameter (Da Rio et al. 2014), they found evidence that the cluster contains significant substructure outside the smooth central cluster core. Armstrong & Tan (2024) found strong evidence for asymmetric expansion in the λ Ori cluster, and determined the direction at which the rate of expansion is at a maximum. They also inverted the maximum rate of expansion of 0.144+0.003 -0.003 km/s/pc to give an expansion timescale of 6.944+0.148 -0.142 Myr, which they compared to other kinematic age methods applied to this cluster (Squicciarini et al. 2021; Quintana & Wright 2022; Pelkonen et al. 2024) and literature age estimates (Kounkel et al. 2018; Zari et al. 2019; Cao et al. 2022). Armstrong & Tan (2024) also found significant asymmetry in the velocity dispersions and signatures of Article number, page 1 of 21 16 Oct 2025 A&A proofs: manuscript no. aanda cluster rotation in the plane-of-sky. Putting all of these results together, they concluded that the λ Ori cluster likely formed in a sparse, substructured configuration, and is not simply the dispersing remnant of an initially bound, monolithic cluster which began to expand after the dispersal of its parent molecular cloud. The asymmetric kinematic signatures, and the discovery of a group of candidate ejected cluster members in particular, suggest a more complex dynamical history for the λ Ori cluster. We seek to estimate the kinematic ages for clusters Collinder 95, NGC 2264, and Collinder 359, building on methods developed by Armstrong & Tan (2024) (Paper I). We establish a robust list of probable YSOs in these clusters through a combination of spectroscopic youth indicators and Gaia DR3 variability YSO flag (Marton et al. 2023). We create a robust list of YSOs and, using their precise Gaia astrometry, we calculate the kinematic ages of clusters Collinder 95 and NGC 2264, omitting Collinder 359 due to an insufficient amount of identified YSOs (N 700 (Dahm & Simon 2005), exhibiting hierarchical structure with multiple subclusters, in particular, two embedded star-forming clumps (NGC 2264 N / S Mon and NGC 2264 S / Cone Nebula Cluster) are surrounded by a halo of older sparsely distributed YSOs (Sung et al. 2008), indicating an extended period of sequential star formation with a complex dynamical history (Nony et al. 2021; Flaccomio et al. 2023). In addition, 2.5 degrees west of NGC 2264 lies the relatively understudied young cluster Collinder 95, of similar age and distance (Cantat-Gaudin et al. 2020a). Both of these young clusters are associated with an extensive molecular cloud complex which has been found to contain a sparse, widely distributed population of YSOs (Mon OB1; Rapson et al. 2014), as well as many protostars, again suggesting multiple epochs of star formation in this region. Collinder 359 was chosen for similar reasons, being a young open cluster at close distance of 250 pc (Lodieu et al. 2006). While young, at ∼30 Myr, Collinder 359 is estimated to be significantly older than NGC 2264 and Collinder 95, which aligns with us ultimately identifying the fewest number of YSOs in Collinder 359 out of all three clusters. Candidates YSO members of these clusters were selected by fitting low-mass PMS Baraffe et al. (2015a) isochrones to Gaia estimates of BP-RP color and G-band magnitude values for stars across all three cluster fields. YSO candidates located above younger isochrones in the color-magnitude diagram were given greater target priorities for observations. Targets were selected in the Gmag range 14-17.5 for NGC 2264 N & S and Coll 95 and range 13-17 for Coll 359 with the goal to find low-mass (∼1.0 -0.3M⊙) cluster members and to keep fibre cross-talk in the observations to a minimum. Targets were also selected within parallax range 2.2 -1.0 mas to be in the right distance range ∼700 ± 300 pc for NGC 2264 and Coll 95, with a parallax > 1.25 mas for Coll 359, and selected with RUWE 600 stars in four nearby young clusters (NGC 2264 N, NGC 2264 S, Collinder 95, and Collinder 359) using the MMT / Hectospec spectrograph. We observed using the 270 line mm-1 grating in the optical range (∼3700 -9150Å) in order to identify and flag broad Hα emission lines and lithium absorption lines-features indicative of low-mass YSOs which are still in their PMS evolution (Soderblom 2010). Collinder 95 was observed on Feb 27th 2022 (3 x 40 minute exposures), NGC 2264 on 1st March (3 x 40 minute exposures), Coll 359 on April 4th (4 x 40 minute exposures). For each night of observations bias, dark, domeflat and sky flat fields were taken as well as the target fields, and data reduction including sky subtraction and cosmic ray rejection was performed using the HSRED v2.0 pipeline. Ultimately, there were 202 combined spectra obtained in the Collinder 95 cluster with a median SNR of 36.07, 200 in the Article number, page 2 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec Fig. 1. (a) Left: Example Hα spectrum of YSO candidate (Gaia Source ID: 3326702657541965952) from NGC 2264, with the source selected as a YSO due to its strong Hα line, i.e., with an EW(Hα) = -48.97Å. (b) Right: Example 6708Å Li spectrum of YSO candidate (Gaia Source ID: 3326714782234731520) from NGC 2264, with this source selected as a YSO due to its strong Li absorption line, i.e., with EW(Li) = 0.32Å. NGC 2264 N & S clusters with a median SNR of 30.5, and 212 in the Collinder 359 cluster with a median SNR of 53.1. 3. Young Stellar Object Identification There are a few spectral features which can be used to identify young PMS stars. A strong Hα emission line (6564Å), caused by residual accretion and/or enhanced chromospheric activity, appears in spectra for stars 10Å or EW(Li) > 0.15Å, as these are established thresholds for these spectroscopic indicators of youth (e.g., Žerjal et al. 2021; Armstrong et al. 2022b). Two example YSO candidate spectra are shown in Figure 1. In Figure 2 we plot EW(Hα) versus EW(Li) for all the stars observed in our target clusters with the YSO criteria thresholds indicated as dashed lines. We find 124 YSOs in NGC 2264, 35 in Collinder 95, and 7 in Collinder 359 using these spectroscopic indicators of youth. To complement this sample of probable YSOs, overlapping with the same field of view we add in YSOs flagged by Gaia DR3 variability (Marton et al. 2023): 268 YSOs in NGC 2264, 104 in Collinder 95, and 2 in Collinder 359. We also quality filter in this step, rejecting stars with a G-band mean flux divided by its error of less than 50, an integrated RP mean flux divided by its error of less than 20, an integrated BP mean flux divided by its error of less than 20, or with 5 or fewer visibility periods. Between these two methods of YSO identification-spectroscopic and the Gaia DR3 variability catalog-some stars were flagged using both methods. Stars which were flagged by both methods numbered 38 in NGC 2264, 9 in Collinder 95, and 2 in Collinder 359. A star was listed as a YSO in the final tally if it was flagged by either method. A detailed breakdown of the number of YSOs identified via each method in each cluster is available in Table 1. Using these two methods, we generate a list of robust YSO candidates, ultimately selecting a total of 354 YSOs in NGC 2264, 130 in Collinder 95, and 7 in Collinder 359. 4. Kinematic Analysis We then use this population of YSOs to conduct a kinematic analysis of the star clusters, excluding Collinder 359 as its sample contained too few identified YSOs to produce statistically significant results. Therefore, we only conduct kinematic analyses of the YSOs in the star clusters NGC 2264 N, NGC 2264 S and Collinder 95. 4.1. Proper motion orientation We search for signs of expansion in the plane-of-sky among our YSOs by measuring the angle between each star's cluster frame and virtual expansion corrected proper motion vector and the vector between the cluster center and the star's current location. The cluster frame motion is found by subtracting out the mean cluster proper motion from that of each YSO. The adopted mean proper motions, radial velocities, center locations and half-mass radii of each cluster are listed in Table 2. We use center coordinates from existing literature (see Table 2), while YSO half-mass radii derived from our identified YSO population. In particular, we check recent cluster catalogs with membership based on Gaia (e.g., Cantat-Gaudin et al. 2020a; Hunt & Reffert 2024) for cluster parameters. For Collinder 95 we take central coordinates, proper motions and distance from Cantat-Gaudin et al. (2020a). However, for NGC 2264, these catalogs do not distinguish between the two apparent subclusters, NGC 2264 N & S. Instead, we use cluster parameters for NGC 2264 N & S from Maíz Apellániz (2019), except for the central proper motions of NGC 2264 S, which in Maíz Apellániz (2019) are given as identical to NGC 2264 N. For the central proper motions of NGC 2264 S we take the median proper motion of our confirmed YSOs with Dec. 10Å EW(Li) > 0.15Å Spectroscopically Flagged YSOs Gaia variability YSOs (Marton et al. 2023) Total YSOs NGC 2264 N&S 199 64 85 124 268 354 Collinder 95 202 19 23 35 104 130 Collinder 359 212 0 7 7 2 7 Table 1. Number of identified YSOs in the clusters (NGC 2264 N&S, Collinder 95, and Collinder 359), separated by method (Hα emission line equivalent width, lithium absorption line equivalent width, and Gaia variability YSO flag). For YSOs identified by Gaia YSO variability, we only include stars with a G-band mean flux divided by its error greater than 50, an integrated RP mean flux divided by its error greater than 20, an integrated BP mean flux divided by its error of greater than 20, and greater than 5 visibility periods. color-coded according to the color wheel in the top left corner. Gaia YSOs and spectroscopically identified YSOs are marked differently on the plots for clarity. In Figure 3, the distinct proper motions of NGC 2264's two subclusters become clear through the plotted vectors. We note that YSOs outside the half-mass radii, indicated using dashed circles, appear to be preferentially moving away from the cluster centers, which is particularly apparent for NGC 2264 N and Collinder 95. We plot the angle between each YSO's proper motion and the radial direction from cluster center to the YSO, ∆θ, producing the distributions shown in Figure 4 for all three clusters. If there exists an expanding halo of YSOs in these clusters, we would expect a substantial peak around zero degrees. We observe this trend for NGC 2264 N and Collinder 95, but not for NGC 2264 S. 4.2. Tangential velocities Virtual expansion is the projection effect in plane-of-sky motions created when a population of stars is receding or approaching in the line-of-sight. We corrected for this effect using equations provided in Brown et al. (1997) using each cluster's bulk radial velocity as provided in the Cantat-Gaudin et al. (2020a) catalog. We used an MCMC approach to sample uncertainties from the posterior distribution for the final corrected tangential velocities vl and vb, similar to the approach used by Armstrong & Tan (2024). We use these velocities in our subsequent kinematic analysis. 4.3. Average expansion velocities Another indicator for cluster expansion is the median cluster expansion velocity ̄vout, i.e., the average of their radial tangential velocities. As described in Armstrong & Tan (2024), we measure the cluster expansion velocity by taking the median expansion velocity component of all cluster members for 106 iterations with additional uncertainties randomly sampled from the observed velocity uncertainties. The uncertainties on the cluster expansion velocity are then taken as the 16th and 84th percentile values of the posterior distribution. Note, for the purpose of allocating YSOs to either NGC 2264 N or S clusters, we only include YSOs within a 0.175◦radius of either cluster's central position when calculating ̄vout. Expansion velocities and uncertainties for each cluster are presented in Table 2. We find that NGC 2264 N has ̄vout = 0.45+0.11 -0.11 km/s, NGC 2264 S has ̄vout = 0.23+0.10 -0.10 km/s and Collinder 95 has ̄vout = 0.51+0.08 -0.08 km/s. These positive median velocities provide evidence of 4σ, 2σ and 6σ significance, respectively, that these clusters are expanding. These velocities are lower than the expansion velocity found for λ Ori of ̄vout = 0.71+0.02 -0.02 km/s by Armstrong & Tan (2024), but are comparable to the expansion velocities found for δ Sco and σ Sco sub-regions of Upper Scorpius by Armstrong et al. (2025), and sit in the middle of the range of expansion velocities calculated by Kuhn et al. (2019); Wright et al. (2024) for various nearby young clusters. In particular, Kuhn et al. (2019) measured expansion velocities of ̄vout = 0.39 ± 0.15 km/s for S Mon (NGC 2264 N) and ̄vout = 0.36±0.40 km/s for IRS 1 (NGC 2264 S; but further divided into IRS 1 and IRS 2), while Wright et al. (2024) measured expansion velocities of ̄vout = 0.40+0.01 -0.06 km/s for S Mon (NGC 2264 N) and ̄vout = 0.23+0.03 -0.07 km/s for the Spokes cluster (NGC 2264 S). Our expansion velocity for NGC 2264 N is in close agreement with those from both Kuhn et al. (2019) and Wright et al. (2024), while our expansion velocity for NGC 2254 S agrees well with that from Wright et al. (2024), but the further division of NGC 2264 S into IRS 1 and IRS 2 groups by Kuhn et al. (2019) makes it difficult to make a direct comparison with their results. 4.4. Areal number density profiles and cluster core sizes We investigate the radial density profiles of the clusters and define the radii of the cluster cores using the same approach as Armstrong & Tan (2024). We bin cluster members by increasing radial distance from the cluster center, R. We calculate the areal number density, N∗, of each bin. The resulting radial areal number density profiles are shown in Fig. 5. We define the core radius of a cluster, rc, as the radius at which the density is half the peak density, which is rc = 0.65 pc for NGC 2264 N, 1.40 pc for NGC 2264 S and 1.27 pc for Collinder 95. We find that 24, 72 and 26 cluster members are located within the core radii of NGC 2264 N, NGC 2264 S and Collinder 95, respectively. The areal density profiles of NGC 2264 N and Collinder 95 smoothly decrease with increasing radius, while the profile of NGC 2264 S has its peak density offset from the central coordinates. This is likely due to the substructure that Kuhn et al. (2019) defined as the IRS 1 and IRS 2 subgroups of the cluster, both of which are contained within our resulting core radius for NGC 2264 S. 4.5. Velocity dispersions and crossing times We calculate velocity dispersions in each cluster using the same Bayesian inference approach described in Armstrong & Tan (2024). We model velocity distributions as 2-d Gaussians with free parameters for the central velocity μ and velocity dispersion σ in each dimension. We then add an uncertainty randomly sampled from the observed uncertainty distribution in each dimenArticle number, page 4 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec NGC 2264 N NGC 2264 S Collinder 95 Central RA, Dec (°) (100.25, 9.88)1 (100.28, 9.53)2 (97.788, 9.894)3 Central μα (mas/yr) -1.72+0.06 -0.06 2 -2.18+0.66 -0.42 4 -2.26+0.35 -0.35 3 Central μδ (mas/yr) -3.71+0.06 -0.06 2 -3.75+0.27 -0.40 4 -5.16+0.31 -0.31 3 Distance (pc) 719+22 -21 3 722+22 -21 3 659.0+86.7 -24.1 3 Expansion velocity ̄vout (km/s) 0.45+0.11 -0.11 0.23+0.10 -0.10 0.51+0.08 -0.08 Core radius rc (pc) 0.65 1.40 1.27 No. YSOs within rc 24 72 26 Half-mass radius r50 (pc) 3.835 3.705 3.695 Velocity dispersion σvl(rc) (km/s) 0.96+0.25 -0.19 1.29+0.17 -0.14 5.39+1.11 -0.86 Velocity dispersion σvb(rc) (km/s) 0.65+0.25 -0.18 1.77+0.23 -0.19 4.27+0.96 -0.70 Velocity dispersion σvl(r50) (km/s) 2.56+0.22 -0.19 3.15+0.33 -0.30 3.51+0.44 -0.37 Velocity dispersion σvb(r50) (km/s) 3.60+0.26 -0.23 2.27+0.18 -0.15 3.51+0.45 -0.37 Crossing time τcross(rc) (Myr) 0.79+0.18 -0.13 0.91+0.08 -0.07 0.26+0.04 -0.03 Crossing time τcross(r50) (Myr) 1.23+0.07 -0.06 1.34+0.09 -0.08 0.92+0.08 -0.07 No. YSOs with closest approach 0 km s-1. We repeat for 2000 iterations with 1000 walkers, the first half of which are discarded as burn-in. We then take the median of the posterior distribution as the best fit for each free parameter and the 16th and 84th percentiles as their respective 1σ uncertainties. With this approach we calculate velocity dispersions for cluster members within the cluster cores and for cluster members within the cluster half-mass radii, using their virtual-expansion corrected tangential velocities (vl, vb). Using these velocity dispersions along with our previous estimates of cluster core radii and half-mass radii, we estimate the crossing times for core radii τcross(rc) and half-mass radii τcross(r50) of each cluster. Both 1D velocity dispersions σvl, σvb and crossing times are given for each cluster in Table 2, along with their respective uncertainties. Notably, while the velocity dispersions in the cluster cores of NGC 2264 N & S are relatively low ( 4 km/s). Given the sparseness of the cluster, this suggests it was born with a relatively high velocity dispersion and was gravitationally unbound from an early time (see Sect. 5.3). However, to properly assess gravitational boundedness requires assessment of the virial state of the clusters. This in turn requires estimates of each cluster's total mass, which is complicated by the incompleteness of our YSO sample. For all three clusters, velocity dispersions of YSO members within their half-mass radii are relatively high (> 2 km/s). In the cases of NGC 2264 N & S, these velocity dispersions are significantly greater than those of their cluster cores, indicating that the halos of YSOs surrounding the cores of each cluster are likely not gravitationally bound and are dispersing. In Collinder 95 these velocity dispersions seem isotropic, while in NGC 2264 N & S, velocity dispersions in the directions of Galactic longitude and latitude (σv,l,σv,b) are anisotropic at the 3σ and 2.5σ significance levels, respectively. 4.6. Traceback Ages We calculate each cluster's kinematic age τkin,CA by "tracing back" each star to its moment of closest approach to the cluster center, assuming constant velocity. We repeat this calculation N = 10, 000 times, each time adding a random (normal) amount of the proper motion error to the proper motion of each star. This generates a distribution of kinematic ages for each star, which we can then use to find the estimated error bounds (16th and 84th percentiles) of each star's kinematic age. From this, we are able to calculate a kinematic age value with error estimates for each star in each stellar cluster. We then use the distribution of a cluster's YSO's kinematic ages to estimate the overall kinematic age value of a cluster. We start by looking only at YSOs with a particularly strong outward expansion from the cluster center, i.e., those with ∆θ within ±50°of zero, as denoted by the dashed vertical lines in Figure 4. These distributions of kinematic ages are shown in Figure 6. For NGC 2264 N, the median kinematic age of all YSO cluster members within ±50° is τkin,CA = 1.08 Myr with a standard deviation of ±1.64 Myr. For NGC 2264 S, the median kinematic age of such YSO cluster members is τkin,CA = 0.61 Myr with a standard deviation of ±0.51 Myr. For Collinder 95, the median kinematic age of such cluster members is τkin,CA = 0.94 Myr with a standard deviation of ±0.84 Myr. All of our kinematic age estimations are listed in Table 3. When we further restrict the population we are considering to only YSOs whose current distance from cluster center is greater than the half-mass radius, shown as dashed circles in Figure 3, and whose final closest approach distance from cluster center is less than the cluster's half-mass radius, we observe the median kinematic age values of each cluster increase slightly. The halfArticle number, page 5 of 21 A&A proofs: manuscript no. aanda 0.5 0.0 0.5 1.0 1.5 2.0 EW(Li) (Å) 50 40 30 20 10 0 10 20 EW(H ) (Å) NGC 2264 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 EW(Li) (Å) 120 100 80 60 40 20 0 20 EW(H ) (Å) Collinder 95 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 EW(Li) (Å) 15.0 12.5 10.0 7.5 5.0 2.5 0.0 2.5 5.0 EW(H ) (Å) Collinder 359 EW(Li)± > 0.15 Å EW(H )± 0.15Å (right of the vertical green dot-dashed line), it is flagged as a YSO. Gaia variability catalog stars are additionally marked with a closed circle. mass radius used here is calculated as the smallest radius that includes half of our identified YSOs in each group. We note that selection bias of spectroscopic targets away from the dense cluster cores (because of fibre crowding) might bias the half-mass radius that we calculate from our sample to moderately larger values. The half-mass radii for each cluster are listed in Table 2. When filtering only for YSOs whose current position is outside our half-mass radius and whose closest approach distance was inside of it, we find that: for NGC 2264 N, the median 99.8 100.0 100.2 100.4 100.6 RA (°) 9.4 9.6 9.8 10.0 10.2 Dec (°) NGC 2264 N 1 mas/yr 100.0 100.2 100.4 100.6 RA (°) 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0 Dec (°) NGC 2264 S 1 mas/yr 97.4 97.6 97.8 98.0 98.2 RA (°) 9.6 9.8 10.0 10.2 10.4 Dec (°) Collinder 95 1 mas/yr Gaia Variability YSOs Spectroscopic YSOs Half-Mass Radius Fig. 3. Proper motion maps of YSOs in NGC 2264 N, NGC 2264 S, and Collinder 95. Stars flagged as YSOs via our spectroscopic analysis are marked with an open circle, YSOs flagged through the Gaia variability catalog are marked with smaller black point. (Stars flagged by both indicators are denoted through a filled in circle / larger black point.) The proper motion of each YSO is shown with an arrow which is colorcoded according to the direction of motion. The average motion of each cluster or subcluster has been subtracted out (see these values in Table 2), allowing us to observe the plane of sky motion of each YSO within the cluster frame. The center of the cluster is denoted by the intersection point of the two blue dotted lines. The dotted blue circle denotes the half-mass radius of the identified YSOs in each cluster. Article number, page 6 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec 100 0 100 0 10 20 30 40 Count NGC 2264 N 100 0 100 0 10 20 30 40 50 60 Count NGC 2264 S 100 0 100 0 5 10 15 20 Count Collinder 95 Fig. 4. Expansion direction histograms of YSOs in NGC 2264 N, NGC 2264 S, and Collinder 95, where the measured angle, ∆θ, is the difference in a YSO's direction of motion from pure radial outward motion from the cluster center. We observe that there is a large proportion of stars near zero (±50°) for NGC 2264 N and Collinder 95, indicating that there is significant outward expansion of YSOs from their cluster centers. We do not observe this same trend for NGC 2264 S. The shaded region shows a range ±50◦, defining a subset of YSOs with strongest radial expansion. kinematic age shifts to 1.81 Myr with a standard deviation of 1.47 Myr; for NGC 2264 S, the median kinematic age shifts to 1.14 Myr with a standard deviation of 0.57 Myr; for Collinder 95, the median kinematic age becomes 1.62 Myr with a standard deviation of 0.77 Myr. We can observe this increase clearly in Figure 6. These values are listed also in Table 3. We also consider how many cluster members have their closest approach to the cluster centers within the core radii of each cluster, as defined in Section 4.4, and which are currently located outside the cores. We list the number of YSOs per cluster Fig. 5. Radial profiles of areal stellar number density profiles of NGC 2264 N (Top), NGC 2264 S (Middle) and Collinder 95 (Bottom). Vertical dashed lines indicate the core radii (rc), as defined in Section 4.4, and half-mass radii (r50). with closest approach distances within the core radii with different tolerances according to the uncertainties on their closest approach distances in Table 2. 5. Age Comparison 5.1. Correcting for Extinction We then seek to compare the kinematic age estimates calculated in the previous section to the isochronal ages of each cluster. To calculate these isochronal ages, we first correct our YSOs for reddening and extinction. Using extinction values from the STARHORSE catalog (Anders et al. 2022) and Gaia DR3 data, we create a local extinction "map" to estimate extinction values of all YSOs in our clusters. The map consists of 0.2° RA by 0.2° Dec cells as shown in Figure 7. We then take the median STARHORSE AG extinction values of all stars within each RA-Dec cell, only including stars within a certain range of parallax values. For NGC 2264, the range of parallaxes present in our YSOs is 1.018 mas to 2.958 mas, so Article number, page 7 of 21 A&A proofs: manuscript no. aanda 0.00 0.25 0.50 0.75 1.00 YSO Count (Normalized) NGC2264 N Median: 0.77 Myr NGC2264 S Median: 0.58 Myr Collinder 95 Median: 0.66 Myr 0.00 0.25 0.50 0.75 1.00 YSO Count (Normalized) NGC2264 N | | 0.1M⊙. The SPOTS models occupy a smaller mass range 0.08 -1.3M⊙, but are distinguished by having parameters for spot coverage fraction Fspot and spot temperature ratio Xspot. Here we assume Fspot = 0.35 and Xspot = 0.9 and use the corresponding set of isochrones, as was done for the λ Ori cluster by Armstrong & Tan (2024). We start with the Baraffe et al. (2015a) stellar isochrones. Following the method of Armstrong & Tan (2024), for each star in each cluster, we compare each individual star to a series of ∼960 Baraffe et al. (2015a) isochrones with ages ranging from ∼0.5 to 10, 000 Myr. For each star, we iterate the calculation N = 100 times, each time multiplying the error on the extinction value with a value randomly sampled from a normal distribution N ∼(0, 1) and add this to the star's median extinction value. Then, we use this new perturbed extinction value to find the isochrone closest to this star's G-band magnitude and BP-RP color. We then do the same for the G-band magnitude and G-RP color. From this series of N = 100 iterations, we are able to generate a distribution of isochronal ages for each star for both the BP-RP and G-RP isochrones. Then, we take the median age as Article number, page 8 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec 99.6 99.8 100.0 100.2 100.4 100.6 100.8 RA (°) 9.0 9.2 9.4 9.6 9.8 10.0 10.2 10.4 DEC (°) NGC 2264 Line of Sight G-band Extinction Estimation YSO sample stars 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 AG50 97.2 97.4 97.6 97.8 98.0 98.2 98.4 RA (°) 9.4 9.6 9.8 10.0 10.2 10.4 10.6 DEC (°) Collinder 95 Line of Sight G-band Extinction Estimation YSO sample stars 0.2 0.4 0.6 0.8 1.0 AG50 Fig. 7. (a) Left: A plot of the extinction correction estimation for NGC 2264 N & S. The extinction values of each cell are determined by drawing from a catalog of stars' extinction values derived from the STARHORSE code and Gaia DR3 data (Anders et al. 2022). We divide the cluster into 0.2° radial ascension by 0.2° declination cells. Then, we take the median extinction value of each cell and apply it to all the YSOs within that cell, thereby estimating each YSOs' extinction value. The YSOs are overplotted as orange crosses. The lowest number of catalog stars populating any cell is Nmin = 20 for Collinder 95 and Nmin = 31 for NGC 2264. (b) Right: As (a), but for Collinder 95. the isochronal age value for that star. We also take the 16th and 84th percentiles of this distribution as the upper and lower error bounds on this value. From this process, we are able to calculate the Baraffe et al. (2015a) isochronal ages for each individual star for either BP-RP and G-RP color. We follow the same approach to estimate ages using PARSEC (Marigo et al. 2017a) and SPOTS (Somers et al. 2020) isochrones. Median isochronal ages for each cluster are given in Table 3. Particularly noticeable is the clear difference between ages estimated using either BP-RP or G-RP colors. Using Baraffe et al. (2015b) models, ages estimated using G-RP color are consistently younger than ages estimated using BP-RP color, while using Marigo et al. (2017a) models, ages estimated using G-RP color are consistently older. These discrepancies in ages estimated using different stellar evolution models and different colors have been seen for other nearby young clusters and associations in the literature (e.g., Ratzenböck et al. 2023; Armstrong & Tan 2024; Fajrin et al. 2024). 5.3. Kinematic Age versus Isochronal Age In Figs. A.4, A.5, A.6 (top row) we compare the isochronal age values for each set of stellar evolution models using Gaia G-RP colour with the kinematic ages of YSOs with position-velocity angles with ±50◦. We generally expect the kinematic ages of cluster members will be lower than their isochronal ages. This is because the kinematic age calculation gives the time when a star would have been at its closest approach distance to the cluster center, but a star may have formed significantly earlier than that, and only have begun moving away from the cluster center following residual gas expulsion around the cluster and the loss of binding mass. Therefore, isochronal age estimates should be higher than kinematic age estimates for stars that originated within the cluster, and the average difference indicates the duration of an "embedded phase" (e.g., Miret-Roig et al. 2024a). In Figures A.4, A.5 we can see that this holds true for most YSOs in NGC 2264 N & S, as generally fewer stars are in the greyed out "forbidden" region, where isochronal age is less than kinematic age. However, many YSOs belonging to Collinder 95 are located in the 'forbidden zone', with kinematic ages seemingly too old for their isochronal ages. Comparing the isochronal ages for each cluster using each stellar isochrone model, in general the PARSEC (Marigo et al. 2017a) models give older ages, followed by SPOTS (Somers et al. 2020) models, and the Baraffe et al. (2015a) models give the youngest. We also compare ages for subsets of cluster members filtered on closest approach distance in Figs. A.4, A.5, A.6, requiring the closest approach distance τiso for any stellar evolution model, which thus inhabit the "forbidden" zone. A possible interpretation of this would be that YSOs in the "forbidden" zone did not originate from their position of closest approach to the cluster center, but from an initially sparser distribution, and thus τkin,CA are overestimated ages. Alternatively, there is also the possibility that all of our isochronal age estimates are underestimated, either due to underestimated reddening, or otherwise that the stellar evolution models themselves systematically underestimate ages for YSOs in this mass range. Furthermore, instead of comparing the kinematic ages and isochronal ages of individual cluster members, we can also conArticle number, page 9 of 21 A&A proofs: manuscript no. aanda sider the cluster as a whole. We find that the median kinematic age values for all three clusters are lower than the median isochronal age values in nearly all cases, as seen in Figure A.7, for both the Baraffe et al. (2015a) and PARSEC isochrones. Here, we can see clearly that the Baraffe et al. (2015a) isochrones yield much lower ages than the PARSEC isochrones. Furthermore, in Figure A.7 we also note again that, as expected, very few cluster medians fall in the grey "forbidden" zone where isochronal ages are less than kinematic ages (see also Miret-Roig et al. 2024a). 5.4. Age Spread In each cluster, we look for evidence of significant spread in each of τkin,CA and τiso estimated using each set of stellar evolution models. Following the same approach as Armstrong & Tan (2024), we calculate the difference between τ for individual cluster members and the sample mean ̄τ and normalize by the uncertainty of τ for each cluster member, removing the largest 10% as outliers. We report the median normalised age differences, στ,kin,CA and στ,iso, per cluster. For all YSOs consistent with moving away from the cluster centers, which thus have a valid traceback age to closest approach τkin,CA, we obtain spreads in kinematic age στ,kin,CA of 23.6, 29.0 and 9.9 for NGC 2264 N, NGC 2264 S, and Collinder 95, respectively. For these stars we obtain spreads in τiso using Baraffe models with BP-RP color of 22.3, 21.1 and 20.4, but only 3.4, 2.2 and 2.1 with G-RP color. With PARSEC models we obtain στ,iso of 12.8, 10.9 and 9.3 with BP-RP, and 2.1, 2.1 and 4.2 with G-RP. With SPOTS models we obtain στ,iso of 6.1, 6.9 and 8.9 with BP-RP, and 2.3, 1.9 and 2.9 with G-RP. For YSOs whose closest approach distance is within the cluster core radius, rc, we obtain στ,kin,CA of 5.3, 116.7 and 46.4 for NGC 2264 N, NGC 2264 S, and Collinder 95, respectively. For these stars we obtain spreads in τiso using Baraffe models with BP-RP color of 46.7, 8.4 and 24.1, and 2.5, 2.0 and 2.9 with GRP color. With PARSEC models we obtain στ,iso of 23.0, 4.7 and 17.1 with BP-RP, and 2.0, 2.5 and 1.2 with G-RP. With SPOTS models we obtain στ,iso of 13.0, 4.3 and 23.7 with BP-RP, and 1.9, 1.8 and 4.7 with G-RP. For YSOs whose closest approach distance is within 1σ uncertainty of the cluster core radius, rc, we obtain στ,kin,CA of 6.0, 12.9 and 6.4 for NGC 2264 N, NGC 2264 S, and Collinder 95, respectively. For these stars we obtain spreads in τiso using Baraffe models with BP-RP color of 24.9, 24.9 and 14.5, and 2.0, 2.2 and 2.5 with G-RP color. With PARSEC models we obtain στ,iso of 10.5, 6.3 and 8.5 with BP-RP, and 1.7, 3.4 and 3.9 with G-RP. With SPOTS models we obtain στ,iso of 8.2, 6.2 and 19.5 with BP-RP, and 1.9, 1.2 and 2.6 with G-RP. Overall, the evidence for isochronal age spread στ,iso is uncertain, i.e., the significance of the spread depends strongly on the models used to estimate ages and even on the photometric colors used for the same set of models, but is not strongly dependent on the filtering of cluster members by closest approach distance. We do find evidence for kinematic age spread στ,kin,CA in all clusters, and particularly στ,kin,CA is significant for cluster members with closest approach distances τiso, they are not consistent with having originated within the cluster core, unless the isochronal ages τiso are severely underestimated. It is possible that many of these may have formed in smaller substructures surrounding the core, similar to the non-halo YSOs surrounding NGC 2264 N & S, but then it is striking that so many of these have velocities directed radially away from the core, consistent with overall linear radial expansion and seemingly with little random motion inherited from the turbulence of their natal cloud. The large spread of τkin,CA among halo members also indicates that they would have escaped the core of Collinder 95 gradually rather than after a single catastrophic event, such as a supernova expelling residual gas, or alternatively that they originated at a range of distances from the cluster core and have since moved radially away from it. As described in (Armstrong & Tan 2024), λ Ori also has many halo members with τkin,CA > τiso, but most of these disappear with a more restrictive filter on closest approach distance leaving only candidate ejected stars with timescales compatible with other age estimates ( half-mass radius.) In general, the cluster median kinematic ages are significantly lower than isochronal ages. of this population formed sparsely distributed and hierarchically structured (Kounkel et al. 2018; Armstrong & Tan 2024). 6. Conclusions By analyzing optical stellar spectra and flagging stars with spectral indicators of youth (|EW(Hα)| > 10Å or EW(Li) > 1.5Å), we are able to identify 166 YSOs across several young clusters (NGC 2264 N & S, Collinder 95, and Collinder 359). Using an additional 325 YSOs flagged as variable stars by the Gaia DR3 catalog (Marton et al. 2023), we generate lists of YSOs (with N > 100) for both NGC 2264 N & S and Collinder 95, allowing us to calculate the kinematic or traceback ages for these clusters. After limiting the YSOs used in the kinematic age estimation to only those with ∆θ within ±50°, we find a median kinematic age of 1.08 Myr for NGC 2264 N, 0.61 Myr for NGC 2264 S, and 0.94 Myr for Collinder 95. We find in all clusters significant numbers of YSOs outside of the cluster radii with velocities inconsistent with origin within the cluster radii (see Fig. 3). This provides evidence for a sparse population of YSOs surrounding the dense cluster cores, with distinct substructure and kinematics, which could belong to the young stellar association Mon OB1. The existence of bound, compact clusters within the volumes of sparse associations has been observed elsewhere (e.g., Vela OB2; Armstrong et al. 2022a) We compare the kinematic age values with isochronal age values for the Baraffe et al. (2015a) and PARSEC (Marigo et al. 2017a) stellar evolution models. As seen in Figure A.7, although the three isochronal models give significantly different results, most models provide isochronal age estimates which are, on average, higher than our kinematic age estimates, which is consistent with YSOs forming within a more compact cluster volume and then subsequently becoming unbound and drifting away. We find significant differences between ages estimated when fitting stellar evolution models to YSOs in either BP-RP or G-RP color, as can be seen in Figs. A.1, A.2, A.3, & A.7 and in ages reported in Table 3. This indicates systematic differences between the observed photometric bands and the corresponding synthetic photometric bands as predicted by stellar evolution models. The systematic offset between kinematic ages and isochronal estimates, of 2-4 Myr according to PARSEC models in BP-RP color, for example (see Fig. A.7), has been suggested in similar recent works (such as Miret-Roig et al. 2024b) to indicate an early period in cluster evolution where the cluster is still embedded in its parent molecular cloud for several Myrs, before stellar feedback expels this gas and cluster members begin to become unbound as the majority of binding mass is lost. However, this offset is not observed for all clusters. Armstrong & Tan (2024) estimated kinematic ages for the λ Ori cluster using several approaches, but found that these were all in agreement with or even greater than ages estimated in the literature using stellar evolution models. They also found evidence that the λ Ori cluster likely formed in a sparse, hierarchically structured configuration and so was initially unbound. Collinder 95, similarly, has a sparse distribution of YSO members, a majority of which have relative motions consistent with expanding away from the cluster center. Also, for those YSO members consistent with being unbound from the cluster, the median kinematic age (1.62 Myr) is in close agreement with Article number, page 12 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec the median isochronal age estimated using Baraffe et al. (2015b) isochrones in BP-RP color (1.59 Myr; Table 3), which would suggest that Collinder 95 formed initially sparse and unbound. However, the uncertainty in the isochronal age estimates makes this difficult to verify. This contrasts with the NGC 2264 N & S clusters, which are more massive and densely concentrated and which likely formed bound before subsequently losing cluster members a few Myr into their evolution. Acknowledgements. I.C. acknowledges support from a Chalmers Astrophysics & Space Science Summer (CASSUM) Research Fellowship, including partial support from W&M Lundgrens grant 2022-4019. J.J.A. acknowledges support from a Chalmers Initiative on Cosmic Origins (CICO) postdoctoral fellowship. J.C.T. acknowledges support from ERC Advanced Grant MSTAR (788829). This work has made use of data from the ESA space mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. 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NGC 2264 N & S and Collinder 95 with HectoSpec 0 2 4 6 8 10 12 14 kin, CA (Myr), all NGC 2264 N 0 2 4 6 8 10 12 14 kin, CA (Myr), | | r50 0 2 4 6 8 10 12 14 Iso, PARSEC (Myr) 0 2 4 6 8 10 12 14 Iso, SPOTS (Myr) Fig. A.1. Kinematic age, τkin, against isochronal age, τiso, for all YSOs (top panel), for YSOs moving away from the cluster center (middle panel), and for YSOs with closest approach distances within r50 and current distances outside of r50 (lower panel), for NGC 2264 N. The plot denotes photometric color used for τiso using point color (BP-RP as black, G-RP as blue.) Article number, page 15 of 21 A&A proofs: manuscript no. aanda 0 2 4 6 8 10 12 14 kin, CA (Myr), all NGC 2264 S 0 2 4 6 8 10 12 14 kin, CA (Myr), | | r50 0 2 4 6 8 10 12 14 Iso, PARSEC (Myr) 0 2 4 6 8 10 12 14 Iso, SPOTS (Myr) Fig. A.2. As Fig. A.1, but for NGC 2264 S. Article number, page 16 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec 0 2 4 6 8 10 12 14 kin, CA (Myr), all Collinder 95 0 2 4 6 8 10 12 14 kin, CA (Myr), | | r50 0 2 4 6 8 10 12 14 Iso, PARSEC (Myr) 0 2 4 6 8 10 12 14 Iso, SPOTS (Myr) Fig. A.3. As Fig. A.1, but for Collinder 95. Article number, page 17 of 21 A&A proofs: manuscript no. aanda Fig. A.4. Kinematic age, τkin, against isochronal age, τiso (G-RP), for YSOs moving away from the cluster center (top panel), for YSOs with closest approach distances within 1σ of the core radius, rc (middle panel), and for YSOs with closest approach distances within the core radius rc (lower panel), for NGC 2264 N. Article number, page 18 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec Fig. A.5. As Fig. A.4, but for NGC 2264 S. Article number, page 19 of 21 A&A proofs: manuscript no. aanda Fig. A.6. As Fig. A.4, but for Collinder 95. Article number, page 20 of 21 Ishani Cheshire et al.: Expansion kinematics of young clusters. II. NGC 2264 N & S and Collinder 95 with HectoSpec 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 kin, CA (Myr), all 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 kin, CA (Myr), | | r50 0 2 4 6 8 10 12 Iso, PARSEC (Myr) 0 2 4 6 8 10 12 Iso, SPOTS (Myr) Fig. A.7. The median kinematic ages (τkin) versus their median isochronal ages (τiso) for clusters NGC2264 N (red), NGC 2264 S (green), and Coll 95 (orange). The plot is divided into nine panels, with the three subgroups as rows: all YSOs (top panel), YSOs moving away from the cluster center (middle panel), and YSOs with closest approach distances within r50 and current distances outside of r50 (lower panel). Columns denote the different isochrone models used, with Baraffe (left), PARSEC (middle), and SPOTS (right). The plot denotes photometric color used (BP-RP or G-RP) using symbols (BP-RP marked by a plus, G-RP marked by an X.) The error bars on each value are the 16th and 84th percentiles of the distribution of either kinematic or isochronal ages. Article number, page 21 of 21
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2510.14739
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Multiparameter quantum-enhanced adaptive metrology with squeezed light
Giorgio Minati,1 Enrico Urbani,1 Nicol`o Spagnolo,1 Valeria Cimini,1, ∗and Fabio Sciarrino1
1Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy
Squeezed light enables quantum-enhanced phase estimation, with crucial applications in both fundamental
physics and emerging technologies. To fully exploit the advantage provided by this approach, estimation pro-
tocols must remain optimal across the entire parameter range and resilient to instabilities in the probe state.
In this context, strategies that rely on pre-calibrated squeezing levels are vulnerable to degradation over time
and become sub-optimal when experimental conditions fluctuate. Here, we develop an adaptive multiparameter
estimation strategy for ab-initio phase estimation, achieving sub-standard quantum limit precision in the full pe-
riodicity interval [0, π), without relying on prior knowledge of the squeezing parameter. Our approach employs
real-time feedback to jointly estimate both the optical phase and the squeezing level, ensuring robustness against
experimental drifts and calibration errors. This self-calibrating scheme establishes a reliable quantum-enhanced
sensing framework, opening new routes for practical scenarios and scalable distributed sensor networks using
squeezed light.
Introduction — Quantum-enhanced optical phase estimation
using non-classical states of light [1] has attracted renewed in-
terest in recent years, driven both by pioneering applications
in the next-generation gravitational wave detectors [2–4] and
by its impact on emerging areas such as imaging [5], spec-
troscopy [6, 7], and distributed sensing [8–10]. In this con-
text, maximally entangled probes, such as N00N states, pro-
vide, in principle, the highest phase sensitivity permitted by
quantum mechanics [11, 12]. However, their practical use is
limited by the brightness of single-photon sources [13] and
by the detrimental effect of photon losses, which rapidly de-
grade their advantage [14]. To date, unconditional demonstra-
tions of sub-standard quantum limit (SQL) performance with
N00N states have been restricted to the two-photon regime
[15], making scalability to higher dimensions a major open
challenge.
As a result, the realization of quantum sensors
based on such maximally entangled states remains difficult in
practice.
Different routes have been explored, such as multi-pass
strategies [16, 17] and photon total angular momentum encod-
ing [18], but squeezed light provides a more scalable route to
phase estimation beyond the SQL [19, 20], mitigating some
of the key limitations associated with maximally entangled
probes. Its effectiveness has already been demonstrated in the
most recent upgrades of gravitational wave interferometers,
which have shown the possibility of operating below the SQL
for broadband signals, marking a milestone in the practical
application of quantum resources for precision measurements
[21–23].
Although squeezed light enables sensitivities below the
SQL [24–27], this is obtained at the expense of increased
noise in the conjugate quadrature.
Consequently, this en-
hanced precision is contingent upon a specific value of the op-
tical phase, posing challenges for generic measurement strate-
gies that necessitate the estimation of the phase as a fully
unknown parameter from scratch. This limitation becomes
critical in ab-initio phase estimation, where no prior knowl-
edge of the parameter is available. Therefore, in this regime,
∗valeria.cimini@uniroma1.it
a measurement strategy based on fixed, pre-determined ob-
servables can not achieve optimal precision over the full pa-
rameter range without feedback. An approach to overcome
this limitation is to implement adaptive strategies that per-
mit to maintain consistent precision across the entire phase
range accessible with the probe state, independently of the
specific parameter values [28–33]. Previous implementations
of adaptive phase estimation with squeezed states have all re-
lied on prior knowledge of the amount of squeezing in the
probe [24, 34], pre-calibrating the squeezing level, needed to
optimize the measurement settings, limiting their applicability
in practice. Moreover, they were also confined to half of the
system periodicity.
Realistic sensing scenarios introduce an additional layer of
complexity. In many cases, not only is the sample-induced
phase shift unknown, but properties of the probe state itself,
such as the squeezing strength, may also be subject to vari-
ations during the measurement time [35–37].
Addressing
such situations requires a multiparameter estimation frame-
work [38, 39], in which both phase and probe parameters are
inferred simultaneously. Since the corresponding observables
are generally non-commuting, quantum incompatibility im-
poses trade-offs on precision [39–42], making it essential to
investigate experimentally how adaptive protocols perform in
this regime [43]. To this end, a stepwise estimation strategy
has recently been introduced [44–46], alongside several other
methods aimed at boosting precision in this framework [47–
50].
In this work, we present a significant advancement in mea-
suring unknown phase shifts with precision that surpasses
the classical limit without requiring pre-calibration of the
probe key parameters.
By combining Homodyne Detec-
tion (HD) with Bayesian inference in an adaptive measure-
ment procedure, we realize and experimentally validate an
ab-initio multiparameter estimation scheme that simultane-
ously learns the unknown phase and relevant probe param-
eters.
This calibration-free strategy demonstrates genuine
quantum enhancement under realistic conditions, highlight-
ing its technological relevance for robust, deployable phase
sensing. The implemented protocol dynamically adjusts the
local oscillator (LO) phase in response to prior measurement
outcomes, effectively steering the measurement basis toward
arXiv:2510.14739v1 [quant-ph] 16 Oct 2025
2
𝜙
𝜙t
t
O
P
A
(a)
(b)
θ
𝜙
−
LO
SQZ
, 𝜙→Δ
θ
r, 𝜙
𝜙
FIG. 1. Sketch of the experimental setup and of the implemented protocol. (a) Squeezed-vacuum probes are generated by pumping an
optical parametric amplifier (OPA) with a single-mode continuous-wave laser at 532 nm. To achieve high levels of squeezing, the pump and
LO beams are both spatially filtered in identical mode-cleaner cavities to maximize the overlap between the squeezed field and the LO at the
homodyne BS. The relative phase between the squeezed state and the LO is stabilized with a phase-locked loop (PLL). The LO phase θ is finely
tuned by means of a piezoelectric stage, while an FPGA-based feedback system allows dynamic selection and locking of the measurement
quadrature. The phase ϕ can be changed with a liquid crystal device in the path of the squeezed beam. (b) At each step of the protocol, the
homodyne data collected are used to reconstruct the multiparameter posterior distribution. From this posterior, the estimate is updated and the
optimal measurement angle θ is computed. A feedback ∆, derived from the current estimates, is then applied to adjust the LO phase for the
next measurement. Repeating this cycle progressively refines the posterior until, at the end of the protocol, the estimates converge to the true
values of phase ϕt and of the squeezing parameter rt.
the quadrature with the highest sensitivity at each step. Our
approach achieves unconditional quantum-enhanced sensitiv-
ities for optical phase values across the entire periodicity
range, without relying on systematic calibration of the probe.
This is achieved through a multistep estimation procedure de-
signed to overcome symmetrical constraints that previously
confined investigations to the [0, π/2] range. As a result, we
extend phase estimation to the entire [0, π) domain while di-
rectly inferring both the phase and the squeezing strength from
the data. This article reports the experimental realization of
quantum-enhanced optical phase estimation over the full π-
range [51], marking a decisive step toward practical multipa-
rameter estimation with squeezed light.
Theoretical framework — To benchmark the performance of
our protocol, we refer to the parameter estimation framework
[52]. In the single parameter scenario, the aim is to measure
an unknown parameter y by preparing a set of M probe states,
measure them according to a specific set of operators Πk, and
then process the data to retrieve information on the param-
eter. These choices determine the effective precision in the
estimation process. For a chosen measurement strategy, the
Fisher Information (FI) quantifies the information F[y] about
the unknown parameter y contained in the measurement out-
comes, optimized on all possible data processing strategies.
This provides a first bound on the variance Var[y] for the un-
known parameter, known as the Cram´er–Rao bound (CRB)
[53, 54]. This bound sets the achievable precision for a fixed
probe state and measurement scheme. Note that the value of
F[y] can explicitly depend on the true value of the parameter.
Optimizing the measurement strategies leads to the quantum
Cram´er-Rao bound (QCRB) [55–57], which sets the ultimate
precision associated with the specific probe state, and which
is determined by the quantum Fisher Information (QFI) FQ.
Overall, the variance of any unbiased estimator satisfies the
following chain of inequalities:
Var[y]
CRB
≥
1
MF[y]
QCRB
≥
1
MFQ
,
(1)
where M represents the number of probe states used in the
estimation. In our phase estimation scenario (y ≡ϕ), it has
been shown [24, 58, 59] that for Gaussian probes, the opti-
mal sensitivity to optical phase shifts corresponds to F sq
Q =
2 sinh2(2r), where r is the squeezing parameter. Notably, for
a sufficient level of squeezing, this surpasses the QFI of coher-
ent states given by F coh
Q
= 4|α|2 when considering an equiv-
alent number of photons in the probe, since |α|2 and sinh2(r)
are the average photon number in the probe for each scheme.
This limit can be saturated using HD combined with either
maximum-likelihood or Bayesian estimation [59].
The parameter dependence of the FI highlights the need
for adaptive measurement strategies, which dynamically ad-
just the detection basis to maintain optimal sensitivity across
the entire phase range. In fact, fixed quadrature measurements
provide sub-SQL precision only in restricted phase intervals.
This naturally leads to use the FI as the fundamental figure
of merit, where an adaptive feedback loop can be designed
to steer the measurement toward the point of maximal FI, en-
suring that the protocol operates close to the CRB across the
entire accessible phase range.
In actual experimental implementations, it is also necessary
to take into account that the effective probe state may also vary
during the measurement process. Different phase shifts ϕ are
associated with different fluctuations in the probe itself, such
as variations in the amount of squeezing, thus affecting the at-
3
tainable precision. As a result, the FI must be regarded as a
function of both the unknown phase and the probe parameter
F[⃗y], where ⃗y = (ϕ, r). Optimal strategies for phase esti-
mation, therefore, can not be devised without simultaneously
accounting for probe fluctuations, since relying on an incor-
rect pre-calibrated squeezing value can introduce systematic
biases in the final estimate [35]. This naturally motivates the
adoption of a multiparameter estimation framework, where
the resources are invested in both phase and probe parameter
estimation. In this multiparameter framework, the uncertainty
in the simultaneous estimation of a set of parameters ⃗y is de-
scribed by the covariance matrix Σ[⃗y], which is constrained
by the multiparameter version of the (Q)CRB [38]:
Σ[⃗y]
CRB
⪰
1
M F −1[⃗y]
QCRB
⪰
1
M F −1
Q .
(2)
Here, F [⃗y] and FQ denote the FI and QFI matrices, respec-
tively, and the notation A ⪰B, indicates that A −B is pos-
itive semidefinite. In this context, one must consider the fact
that there is no guarantee that a single measurement strategy
is optimal for estimating multiple parameters, meaning that,
in general, is not ensured that the estimates can attain their
QCRBs simultaneously [60]. Furthermore, in the multiparam-
eter scenario F [⃗y] can present singularities for some specific
parameter values [61]. This is the case of joint estimation of
the phase and squeezing parameter with a squeezed vacuum
state, reflecting the fact that the measurement outcomes do
not provide independent information about both parameters.
From a practical perspective, these aspects make the design of
adaptive protocols more demanding.
Experimental platform and adaptive protocol —
The
squeezed vacuum state is generated at a wavelength of 1064
nm through the cavity-enhanced interaction of a pump beam
at 532 nm with a periodically-poled nonlinear crystal inside
an Optical Parametric Amplifier (OPA). The phase of the gen-
erated squeezed vacuum state is kept fixed in time using an
FPGA-based locking scheme, and, via a liquid crystal device,
we set the arbitrary phase ϕ, which corresponds to the param-
eter of interest to be estimated. For this purpose, we perform
homodyne measurements along an arbitrary quadrature angle,
determined by the phase θ of the LO that can be controlled
with a piezoelectric stage. The main components of the ex-
perimental setup are illustrated in Fig.1. More details about
the apparatus are reported in the SI.
For each unknown phase value ϕ, we employ a Bayesian
estimation strategy [62] to process the experimental data.
Specifically, a sequence of M homodyne measurements, X =
{x1, x2, ..., xM}, is collected and used to update our knowl-
edge of the parameter through Bayes’ theorem [59]. Assum-
ing no specific prior knowledge regarding both the param-
eter of interest ϕ and the squeezing level of the employed
probe, i.e., adopting a flat prior distribution within the interval
ϕ ∈[0, π) and r ∈[0, 3]. The posterior probability recon-
structed after a large number M ≫1 of measurements can be
expressed as:
p(φ, r|X) = 1
N
Y
x∈X
p(x|φ, r)Mp(x|φ,r).
(3)
(a)
(b)
Var[!]
[rad2]
FIG. 2. Single- and multiparameter variances of the phase es-
timation. We report the variances of the adaptive phase estimation
(radial axis) as a function of the phase ϕ (angular axis), spanning the
entire range [0, π). The experimental results Var[ϕ] obtained in the
single- [panel (a)] and multiparameter [panel (b)] adaptive protocols
are reported as green and blue dots, respectively. The corresponding
error bars represent the standard deviations over 5 (10) repetitions
of the experiment, in the single-parameter (multiparameter) case. In
both panels, we also report the corresponding classical bounds i.e.,
the QCRB for a coherent probe with the same average number of
photons, as a blue line for the former and a green line for the lat-
ter case, while the QCRB is represented by a red line in both cases.
For the multiparameter protocol we also report the actual CRB (or-
ange dashed line). The experimental single-parameter protocol em-
ploys a total of M = 5000 homodyne measurements, among which
MR = 200 are used to obtain the rough estimate. Instead, in the mul-
tiparameter estimation, the total number of measurements amounts to
M = 20000, while the rough estimation needs MR = 1200 of them.
Here, p(x|φ) represents the homodyne probability distribu-
tion, which consists of a Gaussian distribution centered in
zero, and with variance:
σ2(φ, r, η) = 1
4
ηe−2r cos2 φ+(1−η) cos2 φ+e2r sin2 φ
,
(4)
with φ = ϕ −θ and η accounting for losses in the setup.
Once having reconstructed the posterior probability, we de-
rive the estimates ˆϕ and ˆr of both parameters by calculating
the mean value of the posterior distribution. To carry out ab-
initio sub-SQL estimation of the phase ϕ over the full range
[0, π), we implement an online adaptive protocol consisting of
two main stages. In the first step, we collect a small fraction of
data MR for different settings of the LO phase θ. In this way,
we can remove the intrinsic ambiguity π −ϕ of squeezed-
state interferometry, which constrained previous experimen-
tal phase estimation protocols with homodyne measurements
[24] to ϕ ∈[0, π/2]. From these outcomes, a Bayesian update
4
is performed to reconstruct the posterior distribution, from
which rough estimates of both the phase and the squeezing
parameter can be extracted. As the numerical evaluation of
the posterior distribution can be computationally demanding,
we approximate it by employing the Sequential Monte Carlo
(SMC) technique [63, 64] that allows us to determine the on-
line feedback. It consists of discretizing the posterior distri-
bution into np particles {ϕk}np
k=1 and {rk}np
k=1, associated to
weights {wk(X, ϕ)}np
k=1 and {wk(X, ϕ, r)}np
k=1, for the single
and two-parameter protocols, respectively. The weights sat-
isfy P
k wk = 1 and are distributed according to the prior in-
formation. These are sequentially updated with the observed
homodyne measurements x ∈X, in such a way that the final
mean and variance of the posterior can be efficiently computed
with the following discrete sums:
ˆϕ =
np
X
k=1
ωk(X, ϕ −θ)ϕk,
(5)
ˆr =
np
X
k=1
ωk(X, r)rk,
(6)
Var[ϕ] =
np
X
k=1
ωk(X, ϕ −θ)(ˆϕ −ϕk)2.
(7)
Further details on the SMC techniques are reported in the SI.
These preliminary estimates are then used to set the adaptive
feedback by changing the LO phase at the most informative
measurement point.
Results —
We start by benchmarking the protocol in the
single-parameter estimation scenario, where the goal is to es-
timate only the phase ϕ. In this framework, the first step of
the adaptive procedure consists of acquiring data at two fixed
values of the LO phase, i.e., θ = {0, π/4}. This first batch of
measurements provides a first rough estimate of the parameter,
which in turn allows us to identify the optimal measurement
projection and thus setting the phase of the adaptive protocol
where the FI is maximum, thereby optimizing the next mea-
surement projections. For a pre-calibrated squeezing level r
and a transmission coefficient η, we can retrieve the effec-
tive squeezing parameter that captures the combined impact
of squeezing and loss given by:
reff = 1
2 log
σ2
asqz(r, η)
q
σ2asqz(r, η)σ2sqz(r, η)
.
(8)
The optimal measurement configuration corresponds to
ϕopt = 1
2 arccos (tanh (2reff)). In practice, the measurement
basis is aligned by choosing θ = ˆϕ −ϕopt, where ˆϕ is the cur-
rent Bayesian estimate. After this step, a new block of quadra-
tures is collected and used to update the posterior distribution,
thereby updating the parameter estimates and yielding a more
accurate determination of the optimal LO phase. This adap-
tive cycle is repeated three times, using in total the remain-
ing M −MR measurements. At each iteration, the increasing
amount of data yields sharper posteriors and progressively im-
proves the determination of the adaptive measurement setting.
The experimental precision is obtained as the variance of
the reconstructed posterior distribution for different values
of ϕ, after the collection of M homodyne quadratures. The
obtained results for the phase estimation in the entire range
ϕ ∈[0, π) are reported in Fig.2a. The experimental measure-
ments are compared with the QCRB computed for squeezed-
vacuum probes: Varsqz[ϕ] = 1/F sq,eff
Q
, in red, representing
the ultimate sensitivity attainable with Gaussian resources for
the given squeezing level and loss in such conditions, with
F sq,eff
Q
= 2 sinh2(2reff). The achieved variances are com-
pared with the optimal precision attainable with classical re-
sources of equal mean photon number, quantified by the corre-
sponding QCRB for coherent states: Varcoh[ϕ] = 1/(4⟨n⟩) =
1/[4 sinh2 (r)] reported as a blue line in the figure. Impor-
tantly, to assess the unconditional advantage over classical
light, the coherent bound is not loss-corrected but is evaluated
at the probe actual mean photon number linked to the squeez-
ing parameter r. The bound is evaluated across all inspected
phase values, each time accounting for the corresponding pre-
calibrated levels of squeezing. These pre-calibrations reveal
that the parameters are not fixed but instead fluctuate due to
experimental instabilities, such as variations in the source dur-
ing different data acquisitions and imperfections in the align-
ment procedures.
The experimental variances lie consistently below the clas-
sical bound, demonstrating unambiguous sub-SQL sensitiv-
ity. Moreover, the measured precision closely approaches the
squeezed-state QCRB, confirming that our adaptive Bayesian
strategy operates near the optimal point across the entire phase
domain. While this strategy enables the saturation of the ul-
timate precision bound for the phase estimation with homo-
dyne measurements, it also requires a precise calibration of
the setup efficiency η and of the squeezing level r, before
performing the actual experiment. For this calibration to be
effective, it must be sufficiently precise, hence, employing a
significant amount of resources, which are not accounted in
the overall resource budget.
Importantly, our experimental results show that in realis-
tic scenarios, both the level of squeezing and losses are not
fixed, but change depending on the specific phase value under
investigation. This can arise because the liquid crystal may
exhibit different transmission at different phases, or because
the squeezing parameter drifts over time due to temporal in-
stabilities of the source. Consequently, different phase values
refer to different values of the bound on the attainable preci-
sion as emerges in Fig.2a. This observation further motivates
the need of a multiparameter strategy, in which both the phase
and the squeezing parameter are inferred simultaneously from
the same dataset. Such an approach enables the protocol to
adapt in real-time to fluctuations of the probe, eliminating the
dependence on previous calibration, which can jeopardize the
final estimate, while still achieving sub-SQL estimation of the
phase, as described below. We stress that within each esti-
mation run, with the employed estimation model, we do not
aim to track parameter variations dynamically; indeed, on the
timescale of a few seconds, the phase-locking loop and the
source stability are sufficient to maintain approximately con-
stant values of ϕ and r.
5
Rough estimation
Fine estimation
(a)
(c)
(e)
(b)
(d)
Coherent bound
Coherent bound
QCRB
QCRB
(f)
(g)
CRB
F
FIG. 3. Adaptive phase estimation single- and multi-parameter protocols. In panel (a), we report the evolution of the posterior distribution
p(ϕ|X) as a function of the number of homodyne data measured along the adaptive single-parameter protocol in the range M ∈[100, 25000].
During the rough estimation, highlighted by a yellow background, we observe that measuring different quadrature angles (θ = {0, π/4} and
θ = {0, π/4, π/2} in the single- and multiparameter estimations, respectively) allows us to disambiguate the estimate between the ranges
ϕ ∈[0, π/2] and ϕ ∈[π/2, π). At the end of this stage, the current estimation ˆϕ is employed to shift the LO phase to its optimal value
θ = ˆϕ −ϕopt. The remaining measurements (purple background) are taken within the optimal homodyne configuration, which is further
updated two times before reaching the final estimation, extrapolated from the posterior distribution reported in panel (b). This inset shows
how the precision of the phase estimation (blue shaded area), tightly approaches the squeezed QCRB (red line), while surpassing the classical
bound (black dotted line). The changes of the LO angle, along the adaptive protocol, are illustrated in panel (c). It shows that the quantity
θ −ϕ is close to its optimal value, depending on the chosen protocol. In the single parameter estimation (in blue), it concides with the value
corresponding to the maximum of the FI [in panel (d)], while in the multiparameter approach (green), we aim at canceling the off-diagonal
element of the FI matrix Fϕr [shown in panel (d)]. In panel (e), the normalized variance g
Var[ϕ] ≡Var[ϕ] · F sq,eff
Q
is reported as a function
of M, thus having a common ultimate precision bound (solid red line) for both the single- and multiparameter variances, illustrated as blue
and green lines, respectively. This panel also shows how the disambiguation and the LO feedback positively affect the precision of both the
estimation protocols, although the multiparameter one exhibits a slower convergence to values below the classical bound (black dotted line).
Panels (f) and (g) provide zoomed-in views of the marked area, demonstrating convergence to the bound for the single-parameter strategy and
for the multi-parameter strategy, respectively. The latter converges to the CRB computed for the two-parameter estimate that does not coincide
with the QCRB.
In the multiparameter regime, for a fixed homodyne set-
ting, the classical FI matrix becomes singular, which in this
case showcases the impossibility of estimating simultaneously
both parameters in the absence of prior information.
The
adaptive strategy, by measuring in different homodyne set-
tings, restores a non-singular FI matrix, thus circumventing
such a limitation. However, the measurement settings that
maximize the phase sensitivity coincide with the sensitivity
minimum for the squeezing parameter, and vice versa. This
trade-off reflects the incompatibility of the relevant observ-
ables and precludes simultaneous attainment of the ultimate
precision bounds for both parameters (see SI for additional
details).
Since our primary goal remains to estimate ϕ with the high-
est possible precision, thus prioritizing the estimate of ϕ over
r, it might be expected that selecting the single-parameter
phase-optimal setting would suffice.
In practice, however,
an excessively imprecise estimate of r degrades the adaptive
LO feedback and therefore the phase estimate (see simula-
tion results in SI). To balance these competing requirements,
we implement an LO feedback rule that aims to cancel the
correlation between the two parameters’ estimation errors.
The optimal strategy corresponds to set the quadrature where
phase and squeezing estimates are effectively decorrelated,
i.e., where the off-diagonal elements of the FI matrix Fϕr van-
ish. This corresponds to setting θ = ˆϕ −arccos
e2ˆr
√
e4ˆr+η, with
η the detection efficiency. Both the estimates ˆr and ˆϕ, needed
to compute the feedback updates, are retrieved as mean val-
ues of the two-dimensional posterior distribution, which is ef-
ficiently reconstructed with the SMC approach, further jus-
tifying its adoption to significantly speed up the numerical
6
reconstruction, especially in this multiparameter framework.
The precisions achieved on the phase estimates for the mul-
tiparameter protocol are reported in Fig.2b, showing also in
this case the ability to obtain quantum-enhanced performance,
while ensuring robustness to a varying squeezing level. Over-
all, we observe gains of 1.79 ± 0.40 dB below the SQL, not
corrected for loss and averaged over 10 different phase val-
ues, with the conventional single-parameter (pre-calibrated)
approach and 1.78 ± 0.33 dB below the SQL with our ab-
initio multiparameter strategy concerning the phase estimate.
These results highlight both the robustness and the uncondi-
tional character of the protocol, since the quantum advantage
is maintained without prior calibration of the probe and for all
the values of ϕ, overcoming also the restrictions of previous
approaches limited to narrower phase intervals.
A more direct comparison of the single and multiparame-
ter strategies is reported in Fig.3, where we investigate how
the estimation variances scale with the number of homodyne
measurements. More specifically, in Fig.3a the experimen-
tal single-parameter posterior distribution is reported for a
representative phase estimate as a function of the number of
homodyne measurements used to update the Bayesian poste-
rior. Fig.3b shows the reconstructed distribution at the end of
the protocol, quantifying the achieved precision as Std[ϕ] =
p
Var[ϕ]. Fig.3c illustrates the sequence of feedback values
applied at successive steps of the estimation protocol. In the
single-parameter case, after the rough estimate the feedback
angles correspond to the quadrature maximizing the Fisher
information, while in the multiparameter case, the rough es-
timation stage is split into three steps, θ = {0, π/4, π/2}, to
improve the estimation of r needed to implement the adap-
tive feedback. The feedback angles for the fine estimate are
instead determined by the condition where the off-diagonal
elements of the FI matrix vanish (see Fig 3d). The single-
parameter strategy, which assumes prior knowledge of the
squeezing parameter, required to set the feedback of the adap-
tive algorithm, displays faster convergence to the bound as
emerges in Fig.3e.
As expected, the multiparameter strat-
egy requires a larger number of measurements before reach-
ing convergence, since part of the data are used to estimate
the squeezing level of the probe.
Nevertheless, and most
importantly, even in the absence of any prior calibration of
the squeezing, the multiparameter protocol achieves phase
estimation precision at sub-SQL level once convergence is
reached. This demonstrates that quantum-enhanced perfor-
mance can be retained even without relying on external infor-
mation about the probe. Finally, the advantage over coherent-
light phase estimation is further emphasized in the zoomed-in
region showing the convergence behavior. Both our single-
parameter (Fig.3f) and multiparameter (Fig.3g) strategies con-
sistently outperform the lossless coherent bound calculated
without rescaling by the overall transmission coefficient η.
We next demonstrate the full strength of the multiparame-
ter protocol in a challenging regime where not only the opti-
cal phase, but also the squeezing parameter itself, undergoes
significant variations, to test the robustness of this protocol
to different squeezing levels. The estimated values of r are
shown in Fig.4a as a function of the calibrated squeezing. Re-
(a)
(b)
(c)
Coherent bound
QCRB
FIG. 4. Robustness of the multiparameter adaptive estimation
protocol to different squeezing strength and fully black-box ap-
proach. In (a) the green squares represents the estimated squeezing
level ˆr, within the multiparameter adaptive protocol, compared to the
actual values r. As an example, we illustrate the experimentally re-
constructed posterior distributions p(ϕ, r|X), corresponding to the
lowest (top-left inset) and the highest (bottom right inset) squeezing
levels. (b) Scaling of the experimental phase variances compared to
the relative bounds: the QCRB for squeezed vacuum probes (red),
the CRB (orange), and the QCRB for coherent states (green). The
data lie all below the coherent bounds, computed without considering
losses. Panel (c), reports the results obtained with a fully black-box
approach, in which the efficiency η is also estimated. The normal-
ized variances g
Var[ϕ] (purple dots) are compared with the classical
lossless (dash-dotted purple line) bound. In all the panels, the re-
ported errors correspond to standard deviations over 10 repetitions
of the experiments.
markably, the adopted strategy can faithfully infer the true
value of r across all operating conditions, without requiring
7
any prior calibration or external reference. At the same time,
the variance of the simultaneous phase estimate, reported for
different squeezing regimes, remains consistently below the
classical bound as shown in Fig.4b, confirming the ability
of our protocol to preserve quantum-enhanced performance
even under strongly different conditions of the probe. This is
achieved since the adaptive feedback is explicitly designed to
minimize the variance of the parameter of interest, namely the
phase ϕ. This feature is also related to the shape of the re-
constructed posterior, which is narrower along the phase axis,
while still providing reliable estimates of the squeezing. Our
results highlight the versatility of the adopted multiparameter
approach that enables the simultaneous estimation of probe
and signal parameters, while allowing the feedback strategy
to be tailored to prioritize the precision of the most relevant
observable. Moreover, by analyzing the dependence of the es-
timation variance on the squeezing strength r, which directly
quantifies the photon number in the probe, we clearly observe
an improvement with respect to the classical coherent bound.
As r increases, the variance decreases accordingly, and the
gap between squeezed light estimation and the classical co-
herent state benchmark widens. This behavior provides an
experimental indication that stronger squeezing yields greater
metrological gain.
Finally, to make the protocol completely independent of
any calibration, we tested its performance under the most
demanding condition, in which no prior knowledge of the
overall detection efficiency η is also assumed.
Estimating
losses jointly with the phase is particularly challenging, since
the corresponding parameters are incompatible observables
[65]. To overcome this limitation, we adopt a hybrid strategy
that combines Bayesian inference with a maximum-likelihood
estimation of η.
Specifically, during each update of the
three-dimensional posterior, η is set to the maximizer of the
η−posterior at the previous step. This update bypasses full
three-parameter inference, retaining the most consistent value
of η in every step of the estimation process without the need
for additional measurement, focusing only on the physically
relevant parameters ϕ and r.
This mixed approach allows
us to maintain quantum-enhanced performance (see Fig.4c)
without relying on any external calibration of the probe or the
detection chain, thereby validating the fully unconditional na-
ture of the protocol.
Discussion/Conclusion — Our work presents the first exper-
imental demonstration of adaptive multiparameter quantum
phase estimation with squeezed vacuum states, achieving sub-
SQL precision across the full periodicity interval [0, π) with-
out relying on prior calibration of key system parameters such
as the squeezing level and detection efficiency. These param-
eters are known to fluctuate in realistic scenarios, both due to
temporal drifts of the source and to sample-dependent trans-
mission, making calibration-based approaches inherently un-
stable and ultimately compromising the effectiveness of adap-
tive measurement strategies. By implementing an adaptive
multiparameter protocol that simultaneously infers the phase
ϕ and the relevant nuisance parameters, we overcome this cal-
ibration bottleneck while retaining unconditional quantum ad-
vantage over the entire unambiguous phase range. Crucially,
the method is robust to consistent fluctuations of r, maintain-
ing quantum-enhanced performance across a wide operating
range.
These
findings
demonstrate
the
feasibility
of
self-
calibrating, ab-initio quantum metrology protocols that are re-
silient to probe instabilities and experimental drifts, thereby
extending the practical applicability of squeezed light for pre-
cision sensing. Beyond enabling reliable operation under re-
alistic conditions, our results establish a pathway toward scal-
able implementations in advanced interferometric platforms
and distributed quantum sensor networks, which rely on the
simultaneous estimation of multiple parameters.
ACKNOWLEDGMENTS
This work is supported by the Amaldi Research Center
funded by the Ministero dell’Istruzione dell’Universit`a
e
della
Ricerca
(Ministry
of
Education,
University
and
Research)
program
“Dipartimento
di
Eccellenza”
(CUP:B81I18001170001), by the project QU-DICE, Fare
Ricerca in Italia, Ministero dell’istruzione e del merito,
code:
R20TRHTSPA, and by the PNRR MUR project
PE0000023-NQSTI (Spoke 4). We thank M. Barbieri for the
useful discussion and the Noisy Labs team.
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Supplementary Information for:
Multiparameter quantum-enhanced adaptive metrology with squeezed light
Giorgio Minati,1 Enrico Urbani,1 Nicol`o Spagnolo,1 Valeria Cimini,1, ∗and Fabio Sciarrino1
1Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy
I.
EXPERIMENTAL SETUP CHARACTERIZATION
A.
Experimental Setup
In what follows, we present a complete description of the setup employed to perform the experiment, which is schematically
illustrated in Fig.1.
Supplementary Figure 1. Experimental setup. The laser provides two outputs. The 532 nm beam pumps the squeezer cavity, while the
1064 nm beam is split by a half-wave plate (HWP) and a polarizing beam splitter (PBS) into a control beam and a local oscillator (LO), both
mode-matched to the squeezed output. The setup comprises three main blocks: (i) squeezed-state generation, (ii) homodyne measurement, and
(iii) phase stabilization using the control beam in a phase-locked loop (PLL). The squeezer cavity is stabilized via a Pound–Drever–Hall (PDH)
scheme employing an electro-optic modulator (EOM), a piezoelectric actuator, and a photodiode; the same infrastructure is used to keep three
mode-cleaner cavities resonant with their respective beams. After the squeezer, the 532 nm pump and the 1064 nm control/squeezed beams
are separated with dichroic mirrors (DM). The phase ϕ along the squeezed-beam path is tuned with a liquid-crystal (LC) phase shifter and
measured by balanced homodyne detection (HD), where the LO and the squeezed beam interfere on a 50:50 beam splitter (BS). The quadrature
angle θ is swept using a piezoelectric stage driven by an FPGA, which reads the DC output of the homodyne detector after a band pass filter
(BPF) to suppress electronic noise.
The main laser source is a Coherent Prometheus Nd:YAG double-wavelength continuous-wave laser emitting 1.77 W at 1064
nm and 104 mW at 532 nm. The 532 output serves as the pump for the squeezed state generation. Before this step, the action
of a Half-Wave Plate (HWP) and a Polarizing Beam Splitter (PBS) allows us to tune the total power of the 532 nm beam. Phase
modulation in the MHz frequency range is applied using an electo-optical modulator (EOM) to produce sidebands used for the
Pound-Drever-Hall (PDH) locking. All the beams employed are cleaned through three mirrors Fabry-Perot ring cavities, each
one used as an optical resonator for mode-cleaning (MC), filtering out spatial modes different from the TEM00.
Generation stage. The squeezed light source is the commercial source developed by Noisy Labs. It consists of an optical
parametric amplifier based on a type-0 periodically poled KTP (PPKTP) crystal configured as a hemilithic standing-wave cavity
∗valeria.cimini@uniroma1.it
2
pumped via the mode-cleaned 532 nm beam. One mirror is piezo-mounted for length control, while the high-reflection coating
on one end face of the PPKTP acts as the second mirror at 1064 nm (and appropriately coated for 532 nm). The opposite
crystal face is anti-reflection coated. The cavity is locked on resonance via PDH. The crystal temperature is actively stabilized
to maintain quasi-phase matching at the pump wavelength and to suppress slow thermal drifts; length control is handled by the
piezo and fine-tuned with an additional temperature control. The output of the source can be extracted using a Dicroic Mirror
(DM) that reflects only 1064 nm wavelength. The 1064 nm beam is split into two paths, each subjected to the same mode-
cleaning procedure used for the pump. This ensures high spatial purity and excellent overlap at the detection stage between the
squeezed field and the local oscillator (LO), which is essential for resolving large levels of squeezing. One path forms the LO
for balanced homodyne detection, while the second serves as a coherent control beam used to stabilize the phase of the squeezed
field.
Measurement stage. We employ a balanced homodyne detection scheme where the LO and the squeezed beam are mixed on a
50:50 beam splitter (BS) and the photocurrent difference is measured via an oscilloscope. The detected quadrature is set by the
LO phase θ controlled via a motorized piezoelectric stage.
Phase locking loop. The phase of the squeezed beam is stabilized using a coherent control beam that is spatially mode-matched
to the squeezed mode. The control beam is phase-modulated by an EOM at radio frequencies and injected in the squeezer
cavity. When in resonance, the bright leakage provides an alignment reference for the squeezed path; at low power it generates
a phase-sensitive error signal upon interference with the LO. This signal is demodulated and used in a phase-locked loop (PLL).
In this configuration, the OPA produces squeezed vacuum at 1064 nm the MHz detection band. The combination of mode
cleaning, PDH-stabilized cavities, and coherent-control locking provides the spatial mode quality and phase stability θ −ϕ
required to resolve sub-shot-noise quadrature fluctuations and to implement the adaptive, real-time estimation protocol.
B.
Squeezed light characterization
In order to characterize the generated squeezed state, we measure the photocurrent noise modulation in different configu-
rations, which we report in Fig.2, where all the traces are zero-span measurements at 6 MHz. We take the shot-noise as a
reference, hence, setting the noise level which is measured when only the LO is sent to the homodyne detector at 0 dB. A
1 Hz triangular-wave-shaped scan of the measured quadrature angle allows us to estimate the maximum and minimum noise
modulation (blue trace in Fig.2), thereby quantifying the squeezing and anti-squeezing level as σ2
sqz = −6.12 ± 0.28 dB and
σ2
asqz = 11.35 ± 0.28 dB. Additionally, within the same settings, we measure the electronic dark-noise when no light at all is
sent to the detector, obtaining a noise attenuation of σ2
dark = −14.9 ± 1.0 dB
0.00
0.05
0.10
0.15
0.20
0.25
Time [s]
15
10
5
0
5
10
Noise attenuation [dB]
Squeezed light quadrature variance
Vacuum quadrature variance (shot noise)
Dark noise
Supplementary Figure 2. Squeezed quadratures noise attenuation, compared to the shot noise and the dark noise. In this figure, we
report the squeezed light variance attenuation (blue trace), in units of shot-noise. In particular, it has been acquired by scanning the LO phase
by ramping the piezo mirror with a triangular wave with a frequency of 1 Hz. Additionally, we also report the shot noise (yellow trace) and
the electronic dark noise (purple trace). All three traces have been acquired with a zero-span measurement at 6 MHz.
3
C.
Phase Locking Stability and Control
In what follows, we describe the implementation of the FPGA-based locking procedure employed to measure the squeezed
field quadrature along an arbitrary angle. First, we model the fields incoming at the homodyne detectors as:
E1 = ELOei(ωt+ϑ),
(1)
E2 = ECFei(ωt+β sin (Ωt)).
(2)
The first is the LO field, oscillating with an amplitude ELO at the optical frequency ω, with a phase ϑ set by the LO piezo mirror.
The second field is the weak coherent field superimposed onto the squeezed light path, whose oscillation has amplitude ECF
and, in addition to the optical frequency ω, is phase-modulated with sidebands of frequency Ωand modulation depth β. The
interference of these in the HD detection BS produces the output fields E3 and E4, which are computed as follows:
E3
E4
=
1
√
2
1 i
i 1
E1
E2
=
1
√
2
E1 + iE2
iE1 + E2
.
(3)
The corresponding photocurrents measured at the HD detector are proportional to the output fields intensities P3 and P4:
P3 = |E3|2 = 1
2(|E1|2 + |E2|2 + iE∗
1E1 −E1E∗
1),
(4)
P4 = |E4|2 = 1
2(|E1|2 + |E2|2 −iE∗
1E1 + E1E∗
1),
(5)
therefore, the photocurrents subtraction reads:
PHD = P3 −P4 = i(E∗
1E2 −E1E∗
2) =
(6)
= iELOECF
ei(−ϑ+β sin (Ωt)) −e−i(−ϑ+β sin (Ωt))
=
(7)
≃iELOECF
e−iϑ [J0(β) + 2iJ1(β) sin (Ωt)] −eiϑ [J0(β) −2iJ1(β) sin (Ωt)]
=
(8)
= 2ELOECF (J0(β) sin (ϑ) −2J1(β) sin (Ωt) cos (ϑ)) ,
(9)
where, in (8), the terms e±iβ sin (Ωt) have been expressed as a Bessel series truncated at order 1. The part of Eq.(9) oscillating at
2Ω, selected using a high-pass filter, is then mixed with an electronic local oscillator E3 = EeLO sin (Ωt + φdemod), obtaining;
P mix
HD = −4ELOECFJ1(β) sin (Ωt) cos (ϑ) · EeLO sin (Ωt + φdemod) =
(10)
= −4ELOECFEeLOJ1(β) cos (ϑ) (sin (2Ωt + φdemod) + sin (φdemod)) .
(11)
A low-pass filter eliminates the terms oscillating at 2Ω, resulting in the following error signal:
P err
cos = −4ELOECFEeLOJ1(β) cos (ϑ) sin (φdemod).
(12)
Therefore, we have access to the first component of (9) (which we will call P err
sin in what follows) and (12), respectively
modulated as sin (ϑ) and cos (ϑ). We can use them to generate an error signal that can lock the homodyne measure to an
arbitrary angle α. In fact, upon rescaling P err
cos by a factor S to match the amplitude of P err
sin , we can combine them in a sum with
weights Aα, Bα such that A2
α + B2
α = 1. Hence, the final error signal will be:
P α
err = AαP err
sin + BαSP err
cos ∼sin(ϑ + α),
(13)
where α = arctan (Bα/Aα) = arccos (Aα) = arcsin (Bα). In what follows, for notation simplicity, we define θ = ϑ + α.
To experimentally test this procedure, we lock the homodyne measure to different θ ∈[0, 2π] and, for each of them, we
measure the resulting homodyne DC signal and the quadrature variance, and report the corresponding results in the left and right
panels of Fig.3, respectively. The regularity of these measurements demonstrates the reliability of this approach, and enables, by
fitting the experimental data, to finely calibrate possible overall offsets of the phase θ.
II.
ADDITIONAL EXPERIMENTAL DATA
In this Section, we describe additional measures relative to the results reported in the main text. To better understand the
effects of adaptive feedback, we report in Fig.4 the quadratures measured for a specific phase ϕ during the estimation protocol.
4
0
/2
3 /2
2
[rad]
0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Homodyne DC signal [V]
Exp.data
Fitted curve
0
/2
3 /2
2
[rad]
100
2
X( ,
= 0) [shot noise]
Exp. data
Fitted curve
Supplementary Figure 3. Phase-locked measurements. In panels (a) and (b) are reported, the homodyne DC signal (red dots) and the
quadratures variance (blue dots) acquired for different phase-locked homodyne angles spanning the range θ ∈[0, 2π], respectively. Both
experimental datasets have been acquired by averaging over 200, 000 acquisitions, but in the second case, a band-pass filter around 1.1 MHz
is also required, especially to remove low-frequency noise. We report the corresponding fitted curves, as red and blue solid lines for the
homodyne DC and quadratures variance, respectively. The former employs a cosine model, while the latter uses the lossy variance model for
the quadrature variance defined in Eq.(4) of the main text.
Rough estimation
Fine estimation
Supplementary Figure 4. Homodyne data during the adaptive protocol. Here we plot the homodyne measurements taken during the
single-parameter estimation of the phase ϕ = 1.1 rad. The normalized measurements are reported in such a way to fix the shot-noise to
the conventional value σ2
shot = 1/4. The yellow background indicates the fraction of measurements employed to obtain the phase rough
estimation necessary to compute the feedback for the LO phase. Its effect is noticeable in the second stage of the measurement, illustrated on
a purple background, where the homodyne data exhibits a smaller variance, corresponding to measuring along the optimal quadrature angle.
We can observe how a small portion of the total acquired data (illustrated with a yellow background in Fig.4) is sufficient to
have a rough estimation, precise enough to compute a feedback that shifts the LO phase towards the optimal one, noticeable
by the substantial decreasing of the quadrature variance in the second stage of the estimation (depicted in Fig.4 with a purple
background).
In the case of multiparameter estimation, we can gain further insights into the functioning of the adaptive estimation by
studying how the experimental 2-dimensional posterior p(ϕ, r|X) evolves during the different steps of the protocol, illustrated
in Fig.5. After the first step of the rough estimation (upper left panel in Fig.5), taken with the LO phase set at θ = 0, the posterior
presents a symmetric behavior along the phase. This feature reflects the fact that a homodyne data taken in a single setting cannot
disambiguate the phase between a phase ϕ and π −ϕ. The second step of the protocol (upper central panel in Fig.5) updates the
previous posterior with quadratures measured along θ = π/4, and illustrates how this additional measurement setting resolves the
ambiguity mentioned above. The third part of the rough estimation, by setting θ = π/2, probes the orthogonal quadrature with
respect to the first step, improving, as we can see in the upper right panel of Fig.5, the estimation of the squeezing parameter.
A sufficiently precise estimate of r is essential for computing accurate adaptive feedback, whose effect becomes clear when
comparing the final rough estimate with the initial fine estimation (purple color scale). The lower panels show the posterior
distributions for the three steps of the fine estimation on a magnified scale, illustrating how much of the overall precision gain is
attributable to the adaptive feedback.
5
Supplementary Figure 5. Evolution of the experimental posterior distribution during the multiparameter adaptive protocol. In this
figure, the posterior distribution p(ϕ, r|X) is updated during the different steps of the adaptive protocol. In the upper panels, we report the
posterior during the three steps of the rough estimation, using an orange color scale. Instead, the three bottom panels illustrate the posterior
evolution during the fine estimate (purple color scale), again subdivided into three adaptive feedback steps.
To quantify the protocol ability to estimate the phase ab-initio over the full [0, π) range, the estimated values of the 10
different investigated phases ˆϕ are compared in Fig.6 to the true calibrated values ϕ for all the inspected strategies. Starting from
the experimental single-parameter estimate, the estimated values of the two-parameter ones, and the black box variant of the
protocol that also infers the overall detection efficiency η from the data, are here reported.
0
1
2
3
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
1
2
3
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0
1
2
3
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Supplementary Figure 6. Estimated phases compared to the calibrated phase values. In the three panels, we report the estimated phase
values ˆϕ as a function of the actual phase ϕ, respectively for the single parameter (left panel, blue dots), 2-parameter (central panel, green
dots), and 3-parameter (right panel, purple dots) protocols.
The scaling of the phase variance for the black-box protocol, which jointly estimates (ϕ, r, η), as a function of the number of
measurements is reported in Fig.8. Even in this case, the implemented approach enables a phase estimate below the coherent
bound, confirming that the information gain provided by squeezing is retained even while the squeezing parameter and the
overall efficiency are inferred on the fly and not pre-calibrated.
In terms of scaling with the average number of photons in the probe, in the ideal lossless case, the variance of the phase
estimate retrieved with coherent probes scales at the SQL, Var[ϕ] ∼
1
4⟨n⟩. On the contrary, squeezed probes attain a quantum-
6
Supplementary Figure 7. Scaling of the phase estimation variance as a function of the number of measurements, in the case of the 3-
parameter estimation protocol. The normalized variance g
Var[ϕ] ≡Var[ϕ] · MF sq
Q is reported as a function of the homodyne measurements
M and compared to the QCRB (solid red line). The dotted line represents the coherent bound for the lossless scenario.
Supplementary Figure 8. Scaling of the phase estimation variance as a function of the probe average photon number. The ultimate
phase–variance bounds for coherent (green surface) and squeezed (red surface) probes as functions of the mean photon number ⟨n⟩and total
efficiency η are reported. In the lossless limit (η = 1), coherent light follows the SQL, while squeezed probes achieve a quadratic scaling.
Loss degrades these scalings.
enhanced scaling
1
8⟨n⟩(⟨n⟩+1) that for ⟨n⟩>> 1 attains the Heisenberg limit. Such behaviors are altered in the presence of
losses, therefore, when η < 1 the asymptotes change, obtaining bounds with slower power laws as emerges in the plot in Fig.8.
However, even for realistic lossy conditions, squeezed probes retain an advantage over coherent probes, and this is the region
7
where we work experimentally.
III.
BOUNDS ON SIMULTANEOUS ESTIMATION OF PHASE AND SQUEEZING LEVEL
A.
Quantum Fisher Information and Compatibility Conditions
In order to find the ultimate precision bound for the estimation of multiple parameters ⃗y = {yi}P
i=1 from a given quantum
state ρ, we have to compute the corresponding Quantum Fisher Information (QFI) matrix FQ. It defines the inequality chain
reported in Eq.(4) in the main text, which, for the sake of clarity, we also rewrite here:
Σ[⃗y]
CRB
⪰
1
M F −1[{yi}P
i=1]
QCRB
⪰
1
M F −1
Q .
(14)
In general, the elements of the QFI matrix can be computed as:
(FQ[ρ])ij = 1
2 Tr (ρ{Li, Lj}) ,
(15)
where the curly brackets denote the anticommutator and Li is the so-called Symmetric Logarithmic (SLD) operator correspond-
ing to the estimation of yi. This operator is indirectly defined by the equation:
∂ρ
∂yi
= 1
2 (Liρ + ρLi) ,
(16)
and plays a crucial role in quantum metrology, since not only it is needed for the computation of the QFI, but it also defines the
possibility of actually saturating the QCRB when Tr[ρ[Li, Lj]] = 0. Let us consider parameters that are encoded in a pure state
ρ = |ψ⃗y⟩⟨ψ⃗y| by means of unitary transformations |ψ⃗y⟩= U(⃗y) |ψ0⟩= e−i P
i yiGi |ψ0⟩, where Gi is the generator of the i-th
transformation. The computation of the SLDs and the QFI is simplified as follows:
Li = −2i[Gi, ρ]
(17)
(FQ[|ψ⃗y⟩])ij = 4 (⟨GiGj⟩−⟨Gi⟩⟨Gj⟩) ,
(18)
where the expectation values are taken over the state |ψ⃗y⟩.
In our case, we consider a model in which the parameters of interest describe a pure squeezed vacuum state, featuring a squeezing
level r > 0 and rotated by a phase ϕ ∈[0, π]. The encoding of both these parameters can be modeled as the following unitary
transformation of the vacuum state |0⟩:
|ϕ, r⟩= ei(ϕGϕ+rGr) |0⟩,
where
Gϕ = n = a†a, Gr = i
2
a†2 −a2
.
(19)
Then, the QFI corresponding to the operations reported in Eq.(19) can be computed by means of Eq.(18), where we substitute
⃗y = {ϕ, r}, obtaining the following matrix:
FQ[|ϕ, r⟩] =
2 sinh2 (2r) 0
0
2
.
(20)
The diagonal form of the QFI in Eq.(20) further simplifies the evaluation of ultimate precision bounds on the estimation of ϕ
and r, which ultimately reduces to the inverse of the corresponding diagonal elements.
In a multiparameter setting, two parameters are said to be compatible if there exists a measurement strategy that can simul-
taneously saturate the QCRB for both parameters. A sufficient condition to achieve compatibility is the commutation of the
SLDs corresponding to the parameters of interest, as projective measurements onto their common eigenbasis enable the optimal
estimation of such parameters. When the state ρ is pure, as in our model, a weaker necessary and sufficient condition can be
verified, i.e. the so-called weak commutation relation, which is verified ⟨[Li, Lj]⟩= 0. In our model, where the parameters are
encoded via unitary transformation this expression further simplifies, and can be evaluated as follows:
⟨[Lϕ, Lr]⟩= 4 ⟨[Gϕ, Gr]⟩= −4i sinh (2r) ̸= 0.
(21)
This means the parameters ϕ and r are incompatible, i.e. we cannot devise a measurement strategy that simultaneously achieves
the optimal precision in both estimations.
In what follows, we describe how adaptivity can exploit the multiparameter estimation framework to approach the phase estima-
tion QCRB when the squeezing level is unknown.
8
B.
Adaptive protocol model
The likelihood p(x|θ, ϕ, r) for a homodyne measurement with local-oscillator phase θ on a squeezed state (phase ϕ, squeezing
r, transmission η) is modeled as a zero-mean Gaussian whose variance is the one reported in Eq.(4) of the main text, which we
also reproduce here for convenience.
σ2(φ, r, η) = 1
4
ηe−2r cos2 φ + (1 −η) cos2 φ + e2r sin2 φ
,
(22)
where φ = θ −ϕ.
In what follows, we describe the 2-parameter adaptive protocol, deriving the model of the corresponding Fisher Information (FI)
and, then, its CRB.
In particular, in a multiparameter framework where ⃗y = {ϕ, r} are the parameters to be estimated, the individual elements of
the FI matrix are computed as:
Fij(⃗y|θ, η) =
Z ∞
−∞
dx p(x|⃗y)
∂
∂yi
log p(x|⃗y, θ, η)
∂
∂yj
log p(x|⃗y, θ, η)
.
(23)
In this case, η must be determined by calibration, whereas in the three-parameter estimation protocol it is unknown and
therefore included in the data vector ⃗y.
If we consider a single-setting experiment in which homodyne data are acquired at a single LO phase, then the resulting FI matrix
is singular and the corresponding CRBs are not defined. This is unsurprising: measuring only one quadrature does not provide
enough information to distinguish different combinations of ϕ and r , i.e., the measurement settings are not tomographically
complete. A first achievement of the adaptive protocol we describe is the removal of this ambiguity, which in turn allows one to
derive a meaningful multiparameter CRB.
As discussed in Sec. III A, ϕ and r are incompatible parameters and cannot be estimated simultaneously with optimal precision.
Given that our aim is to choose the optimal setting ϕopt for estimating ϕ, we expect r to be estimated with suboptimal precision
at that setting. In fact, below we show that the optimal setting for ϕ corresponds to the point of minimal sensitivity for r. Since
the adaptive feedback relies on ˆr, poor estimation of r also degrades the precision of ˆϕ; to mitigate this, we design the adaptive
protocol accordingly. A natural strategy is to steer the LO phase towards the setting that removes the correlation between the
estimation errors, or, equivalently, to drive the off-diagonal element of the FI matrix to zero. This yields
θopt = ˆϕ −ϕopt = ˆϕ −arccos
e2ˆr
p
e4ˆr + η
!
,
(24)
where ˆϕ and ˆr are the current parameter estimate before the adaptive feedback.
Keeping in mind this strategy, we now develop a FI matrix model that replicates our multiparameter adaptive protocol. Firstly,
we recall the strategy we have implemented experimentally:
• Rough estimation: the first MR = 1, 200 homodyne measures are devoted to obtain approximate estimations of the
parameters. In particular, we equally divide them in measures with LO phase in θ = {0, π/4, π/2}. This resource
allocation in equally spaced settings has the twofold objective of eliminating the aforementioned singularity and provide
a sufficiently precise estimation of both parameters.
• Adaptive step: the estimated parameters ˆϕ and ˆr are employed to compute θopt, and shift the LO phase to this value. In the
actual experimental implementation, this step can also be reiterated multiple times to finely adjust the homodyne setting
to the optimal value.
• Fine estimation: as further discussed in Sec.IV, a sufficient amount M of homodyne measures has to be acquired to
achieve statistical convergence. In our case, M = 20, 000 is sufficient to ensure such a convergence. Therefore, the
remaining MF = M −MR = 18, 800 homodyne data are measured within the optimal setting to estimate ϕ. As we will
see, this strategy allows to closely approach the phase QCRB.
Therefore, using the definition reported in Eq.(22), we can model the FI matrix corresponding to the adaptive protocol as a
function of the LO phase in the fine estimation step:
eF (ϕ, r|θ, η) =
1
3F (ϕ, r|0, η) + 1
3F (ϕ, r|π/4, η) + 1
3F (ϕ, r|π/2, η
MR
M
|
{z
}
Rough estimation
+ F (ϕ, r|θ, η)MF
M
|
{z
}
Fine estimation
.
(25)
9
Supplementary Figure 9. Inverse of the FI matrix elements in the multiparameter adaptive protocol. In this figure, we report the behavior
of the e
F −1 entries as a function of ϕ ∈[0, 2π] and r ∈[0, 2]. In particular, [ e
F −1]ϕϕ, [ e
F −1]rr, and [ e
F −1]ϕr = [ e
F −1]rϕ illustrated
respectively in panels (a), (b), and (c). In all of them, we report the value of the phase ϕopt(r, η) that eliminates the correlations as a red
dashed line.
The resulting FI matrix is invertible, allowing us to define the corresponding multiparameter CRBs.
In Fig.9, we report the behavior of the individual elements of [ eF −1] as a function of the parameters ϕ and r. For the sake
of clarity, in what follows, we will use the notation [ eF −1]ij with i, j ∈{ϕ, r} to denote corresponding entries of the inverse
FI matrix. In particular, we can notice how the r-dependent phase that eliminates the correlations (red dashed lines in Fig.9),
i.e. [ eF −1]ϕr = 0 (right panel) also corresponds to the minimum of the [ eF −1]ϕϕ (left panel) and the maximum [ eF −1]rr (central
panel), along the entire range of r.
To have a better understanding of the adaptive FI matrix model we derived, in Fig.10, we illustrate the behavior of [ eF −1]ϕϕ,
[ eF −1]rr, and [ eF −1]ϕr when the squeezing level and efficiency are similar to the experimental conditions, i.e. r = 0.63 and
η = 0.85. As previously mentioned, we can see how the setting that maximizes the precision in estimating ϕ (r), at the same
time minimizes the sensibility to r (ϕ). Additionally, the inset (a) of Fig.10 shows how the ϕopt is indeed the one that removes
the correlation (purple curve) and optimizes [ eF −1]ϕϕ, which achieves a value quite close to the corresponding QCRB. This
consideration is also true for the optimal estimation of r, which is detailed in the inset (c) of Fig.10.
Finally, we further motivate the use of the multiparameter approach through numerical simulations showing that the imple-
mented protocol is inherently robust to variations in the probe squeezing level, whereas the single-parameter approach fails under
the same conditions. The results of simulations performed by changing the squeezing by an amount ∆r are reported in Fig. 11.
The plotted results refer to the variance on the phase estimate for a single-parameter estimator that fixes the squeezing at its pre-
calibrated value r, and our multiparameter adaptive estimator, as the true squeezing is offset by ∆r. For the single-parameter
approach, even small mismatches brings to unphysical variances or significantly degrade estimation precision. The estimation
error approches the bound only when the effective squeezing remains very close to the calibrated value. On the contrary, the
proposed multiparameter adaptive protocol is able to keep the phase estimation variance close to the QCRB for any variation of
the squeezing, since it avoids explicit calibration of the probe squeezing and instead infers its value directly from the data. This
joint inference makes the protocol robust, suppressing systematic errors from drift and sample variability.
IV.
SEQUENTIAL MONTE CARLO
We estimate the unknown phase ϕ with a Bayesian procedure that updates a flat prior on [0, π] using homodyne data X =
{xm}M
m=1. Such measurements are employed to reconstruct the single-parameter posterior probability distribution using Bayes’
theorem, the posterior probability reconstructed after M measures is reported in Fig.12.
The reconstructed posterior is updated using the homodyne probability distribution, obtained by marginalizing the Wigner
function of the squeezed probe state and setting the phase of the local oscillator (LO) to θ. Once having reconstructed the
posterior probability, we derive the estimate ˆϕ for the parameter of interest and its variance. As usual in the Bayesian framework,
these are obtained by calculating the mean value and variance of the posterior distribution, respectively:
ˆϕ =
Z
p(φ|X)φdφ ∼
np
X
k=1
ωk(X)ϕk,
(26)
10
["#!"]!!(&)
["#!"]!"(&)
["#!"]""(&)
["##
!"]!!
["##
!"]""
(a)
(b)
(c)
![rad]
![rad]
![rad]
!"!"
!"!"
!"!"
Supplementary Figure 10. Adaptive FI matrix model for experimental values of r and ϕ. In this figure, we report the behavior of the
adaptive FI matrix entries as a function of φ = θ −ϕ, when considering the experimentally plausible parameters ϕ = π/4, r = 0.63, and
η = 0.85. In the main panel (a), we report the [ e
F −1]ϕϕ (green curve), [ e
F −1]rr (blue curve), and [ e
F −1]ϕr (purple curve) for the entire range
φ ∈[0, 2π], comparing them to the corresponding QCRBs for the estimation of ϕ, i.e. [F −1
Q ]ϕϕ (red dashed line), and for the estimation of
r, i.e. [F −1
Q ]rr (orange dashed line). The black dashed line represents the phase ϕopt that removes the correlation between the estimation
errors of ϕ and r. In the insets (b) and (c), we report the magnification of the main plot around the optimal settings for estimating ϕ and r,
respectively, again compared to the corresponding QRCBs.
Var[ϕ] =
Z
p(φ|X)( ˆφ −φ)2dφ ∼
np
X
k=1
ωk(X)(ˆϕ −ϕk)2.
(27)
In the adopted framework, the computation is accelerated by substituting integrals with discretized sums that have a relevant
impact, in particular for multiparameter estimation, where the computation of multidimensional integrals can be costly, affecting
the realization of the adaptive protocol.
To verify that the implemented estimator attains the relevant precision limits, we perform simulations of the estimation pro-
tocols using the probe parameters η = 0.8 and r = 0.8. In the single-parameter protocol, we discretize [0, π] with np = 10000
particles and observe the expected 1/M variance decay that saturates the QCRB. In the joint (ϕ, r) estimate, np = 20000 total
particles were employed, and the trend of the variances on both parameter estimates has been investigated. The simulations have
been carried out with the same adaptive protocol described before, whose aim is to minimize the variance on the phase estimate.
Studying the performance scaling reported in Fig.13 with the number of homodyne measures is evident how they saturate to the
relative elements of the inverse of the FIM computed in the previous section.
11
0.4
0.3
0.2
0.1
0.0
0.1
0.2
0.3
0.4
r
0.2
0.4
0.6
0.8
1.0
Var[ ]
Var[ ] (single par.)
Var[ ] (multipar.)
Squeezed bound
Coherent bound
Supplementary Figure 11. Behavior of the variance as the squeezing level deviates from the calibration. In this figure, we report how
the phase estimation variance changes when the squeezing level deviates from the original value r = 0.8 by a quantity ∆r ∈[−0.4, 0.4]. In
particular, we report the average and the standard deviation (over 50 repetitions) of the simulated estimation variance obtained with the single-
(multi-)parameter adaptive protocol as a blue (green) solid curve and shaded area, respectively. We compare them with the corresponding
QCRB (red dashed curve) and coherent bound (black dashed curve).
0.0
0.5
1.0
1.5
2.0
2.5
3.0
(rad)
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Posterior
M = 10
M = 50
M = 100
Supplementary Figure 12. Simulated Bayesian posterior. Reconstructed posterior probability distribution of Bayesian phase estimation in
range [0, π] with M = 10, 50, 100 homodyne measures. We choose a trial phase ϕtrue = 0.3 rad and a squeezing parameter r = 1.
12
103
104
M
10
5
10
4
10
3
10
2
10
1
Var[ ]
(a)
QCRB
CRB( )-multipar.
Coherent bound
single par.
multipar.
103
104
M
10
4
10
3
10
2
10
1
Var[r]
(b)
CRB(r)-multipar.
F
1
rr (
opt)
multipar.
Supplementary Figure 13. Scaling of simulated posterior variance with the number of homodyne samples M. Panel (a) average variance of
the phase estimate with the single-parameter Bayesian protocol (blue circles); variance of the phase estimate with the multiparameter Bayesian
protocol (green circles). The dashed black line is the ultimate classical bound for coherent states, the red line is the QCRB for squeezed light;
the orange line is the CRB for the adopted two-parameter strategy. Panel (b) shows the average variance of the squeezing parameter estimate
(green circles). The purple line is the CRB achievable when the protocol is optimized to minimize Var(r) while the pink line represents the
bound obtained with the implemented (phase-optimized) adaptive strategy. Error bars represent the standard deviation over estimates of 20
different phase values.
|
Multiparameter quantum-enhanced adaptive metrology with squeezed light Giorgio Minati,1 Enrico Urbani,1 Nicol`o Spagnolo,1 Valeria Cimini,1, ∗and Fabio Sciarrino1 1Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy Squeezed light enables quantum-enhanced phase estimation, with crucial applications in both fundamental physics and emerging technologies. To fully exploit the advantage provided by this approach, estimation protocols must remain optimal across the entire parameter range and resilient to instabilities in the probe state. In this context, strategies that rely on pre-calibrated squeezing levels are vulnerable to degradation over time and become sub-optimal when experimental conditions fluctuate. Here, we develop an adaptive multiparameter estimation strategy for ab-initio phase estimation, achieving sub-standard quantum limit precision in the full periodicity interval [0, π), without relying on prior knowledge of the squeezing parameter. Our approach employs real-time feedback to jointly estimate both the optical phase and the squeezing level, ensuring robustness against experimental drifts and calibration errors. This self-calibrating scheme establishes a reliable quantum-enhanced sensing framework, opening new routes for practical scenarios and scalable distributed sensor networks using squeezed light. Introduction - Quantum-enhanced optical phase estimation using non-classical states of light [1] has attracted renewed interest in recent years, driven both by pioneering applications in the next-generation gravitational wave detectors [2-4] and by its impact on emerging areas such as imaging [5], spectroscopy [6, 7], and distributed sensing [8-10]. In this context, maximally entangled probes, such as N00N states, provide, in principle, the highest phase sensitivity permitted by quantum mechanics [11, 12]. However, their practical use is limited by the brightness of single-photon sources [13] and by the detrimental effect of photon losses, which rapidly degrade their advantage [14]. To date, unconditional demonstrations of sub-standard quantum limit (SQL) performance with N00N states have been restricted to the two-photon regime [15], making scalability to higher dimensions a major open challenge. As a result, the realization of quantum sensors based on such maximally entangled states remains difficult in practice. Different routes have been explored, such as multi-pass strategies [16, 17] and photon total angular momentum encoding [18], but squeezed light provides a more scalable route to phase estimation beyond the SQL [19, 20], mitigating some of the key limitations associated with maximally entangled probes. Its effectiveness has already been demonstrated in the most recent upgrades of gravitational wave interferometers, which have shown the possibility of operating below the SQL for broadband signals, marking a milestone in the practical application of quantum resources for precision measurements [21-23]. Although squeezed light enables sensitivities below the SQL [24-27], this is obtained at the expense of increased noise in the conjugate quadrature. Consequently, this enhanced precision is contingent upon a specific value of the optical phase, posing challenges for generic measurement strategies that necessitate the estimation of the phase as a fully unknown parameter from scratch. This limitation becomes critical in ab-initio phase estimation, where no prior knowledge of the parameter is available. Therefore, in this regime, ∗ a measurement strategy based on fixed, pre-determined observables can not achieve optimal precision over the full parameter range without feedback. An approach to overcome this limitation is to implement adaptive strategies that permit to maintain consistent precision across the entire phase range accessible with the probe state, independently of the specific parameter values [28-33]. Previous implementations of adaptive phase estimation with squeezed states have all relied on prior knowledge of the amount of squeezing in the probe [24, 34], pre-calibrating the squeezing level, needed to optimize the measurement settings, limiting their applicability in practice. Moreover, they were also confined to half of the system periodicity. Realistic sensing scenarios introduce an additional layer of complexity. In many cases, not only is the sample-induced phase shift unknown, but properties of the probe state itself, such as the squeezing strength, may also be subject to variations during the measurement time [35-37]. Addressing such situations requires a multiparameter estimation framework [38, 39], in which both phase and probe parameters are inferred simultaneously. Since the corresponding observables are generally non-commuting, quantum incompatibility imposes trade-offs on precision [39-42], making it essential to investigate experimentally how adaptive protocols perform in this regime [43]. To this end, a stepwise estimation strategy has recently been introduced [44-46], alongside several other methods aimed at boosting precision in this framework [4750]. In this work, we present a significant advancement in measuring unknown phase shifts with precision that surpasses the classical limit without requiring pre-calibration of the probe key parameters. By combining Homodyne Detection (HD) with Bayesian inference in an adaptive measurement procedure, we realize and experimentally validate an ab-initio multiparameter estimation scheme that simultaneously learns the unknown phase and relevant probe parameters. This calibration-free strategy demonstrates genuine quantum enhancement under realistic conditions, highlighting its technological relevance for robust, deployable phase sensing. The implemented protocol dynamically adjusts the local oscillator (LO) phase in response to prior measurement outcomes, effectively steering the measurement basis toward 16 Oct 2025 2 φ φt t O P A (a) (b) θ φ - LO SQZ , φ→Δ θ r, φ φ FIG. 1. Sketch of the experimental setup and of the implemented protocol. (a) Squeezed-vacuum probes are generated by pumping an optical parametric amplifier (OPA) with a single-mode continuous-wave laser at 532 nm. To achieve high levels of squeezing, the pump and LO beams are both spatially filtered in identical mode-cleaner cavities to maximize the overlap between the squeezed field and the LO at the homodyne BS. The relative phase between the squeezed state and the LO is stabilized with a phase-locked loop (PLL). The LO phase θ is finely tuned by means of a piezoelectric stage, while an FPGA-based feedback system allows dynamic selection and locking of the measurement quadrature. The phase φ can be changed with a liquid crystal device in the path of the squeezed beam. (b) At each step of the protocol, the homodyne data collected are used to reconstruct the multiparameter posterior distribution. From this posterior, the estimate is updated and the optimal measurement angle θ is computed. A feedback ∆, derived from the current estimates, is then applied to adjust the LO phase for the next measurement. Repeating this cycle progressively refines the posterior until, at the end of the protocol, the estimates converge to the true values of phase φt and of the squeezing parameter rt. the quadrature with the highest sensitivity at each step. Our approach achieves unconditional quantum-enhanced sensitivities for optical phase values across the entire periodicity range, without relying on systematic calibration of the probe. This is achieved through a multistep estimation procedure designed to overcome symmetrical constraints that previously confined investigations to the [0, π/2] range. As a result, we extend phase estimation to the entire [0, π) domain while directly inferring both the phase and the squeezing strength from the data. This article reports the experimental realization of quantum-enhanced optical phase estimation over the full πrange [51], marking a decisive step toward practical multiparameter estimation with squeezed light. Theoretical framework - To benchmark the performance of our protocol, we refer to the parameter estimation framework [52]. In the single parameter scenario, the aim is to measure an unknown parameter y by preparing a set of M probe states, measure them according to a specific set of operators Πk, and then process the data to retrieve information on the parameter. These choices determine the effective precision in the estimation process. For a chosen measurement strategy, the Fisher Information (FI) quantifies the information F[y] about the unknown parameter y contained in the measurement outcomes, optimized on all possible data processing strategies. This provides a first bound on the variance Var[y] for the unknown parameter, known as the Cram ́er-Rao bound (CRB) [53, 54]. This bound sets the achievable precision for a fixed probe state and measurement scheme. Note that the value of F[y] can explicitly depend on the true value of the parameter. Optimizing the measurement strategies leads to the quantum Cram ́er-Rao bound (QCRB) [55-57], which sets the ultimate precision associated with the specific probe state, and which is determined by the quantum Fisher Information (QFI) FQ. Overall, the variance of any unbiased estimator satisfies the following chain of inequalities: Var[y] CRB ≥ 1 MF[y] QCRB ≥ 1 MFQ , (1) where M represents the number of probe states used in the estimation. In our phase estimation scenario (y ≡φ), it has been shown [24, 58, 59] that for Gaussian probes, the optimal sensitivity to optical phase shifts corresponds to F sq Q = 2 sinh2(2r), where r is the squeezing parameter. Notably, for a sufficient level of squeezing, this surpasses the QFI of coherent states given by F coh Q = 4|α|2 when considering an equivalent number of photons in the probe, since |α|2 and sinh2(r) are the average photon number in the probe for each scheme. This limit can be saturated using HD combined with either maximum-likelihood or Bayesian estimation [59]. The parameter dependence of the FI highlights the need for adaptive measurement strategies, which dynamically adjust the detection basis to maintain optimal sensitivity across the entire phase range. In fact, fixed quadrature measurements provide sub-SQL precision only in restricted phase intervals. This naturally leads to use the FI as the fundamental figure of merit, where an adaptive feedback loop can be designed to steer the measurement toward the point of maximal FI, ensuring that the protocol operates close to the CRB across the entire accessible phase range. In actual experimental implementations, it is also necessary to take into account that the effective probe state may also vary during the measurement process. Different phase shifts φ are associated with different fluctuations in the probe itself, such as variations in the amount of squeezing, thus affecting the at3 tainable precision. As a result, the FI must be regarded as a function of both the unknown phase and the probe parameter F[⃗y], where ⃗y = (φ, r). Optimal strategies for phase estimation, therefore, can not be devised without simultaneously accounting for probe fluctuations, since relying on an incorrect pre-calibrated squeezing value can introduce systematic biases in the final estimate [35]. This naturally motivates the adoption of a multiparameter estimation framework, where the resources are invested in both phase and probe parameter estimation. In this multiparameter framework, the uncertainty in the simultaneous estimation of a set of parameters ⃗y is described by the covariance matrix Σ[⃗y], which is constrained by the multiparameter version of the (Q)CRB [38]: Σ[⃗y] CRB ⪰ 1 M F -1[⃗y] QCRB ⪰ 1 M F -1 Q . (2) Here, F [⃗y] and FQ denote the FI and QFI matrices, respectively, and the notation A ⪰B, indicates that A -B is positive semidefinite. In this context, one must consider the fact that there is no guarantee that a single measurement strategy is optimal for estimating multiple parameters, meaning that, in general, is not ensured that the estimates can attain their QCRBs simultaneously [60]. Furthermore, in the multiparameter scenario F [⃗y] can present singularities for some specific parameter values [61]. This is the case of joint estimation of the phase and squeezing parameter with a squeezed vacuum state, reflecting the fact that the measurement outcomes do not provide independent information about both parameters. From a practical perspective, these aspects make the design of adaptive protocols more demanding. Experimental platform and adaptive protocol - The squeezed vacuum state is generated at a wavelength of 1064 nm through the cavity-enhanced interaction of a pump beam at 532 nm with a periodically-poled nonlinear crystal inside an Optical Parametric Amplifier (OPA). The phase of the generated squeezed vacuum state is kept fixed in time using an FPGA-based locking scheme, and, via a liquid crystal device, we set the arbitrary phase φ, which corresponds to the parameter of interest to be estimated. For this purpose, we perform homodyne measurements along an arbitrary quadrature angle, determined by the phase θ of the LO that can be controlled with a piezoelectric stage. The main components of the experimental setup are illustrated in Fig.1. More details about the apparatus are reported in the SI. For each unknown phase value φ, we employ a Bayesian estimation strategy [62] to process the experimental data. Specifically, a sequence of M homodyne measurements, X = {x1, x2, ..., xM}, is collected and used to update our knowledge of the parameter through Bayes' theorem [59]. Assuming no specific prior knowledge regarding both the parameter of interest φ and the squeezing level of the employed probe, i.e., adopting a flat prior distribution within the interval φ ∈[0, π) and r ∈[0, 3]. The posterior probability reconstructed after a large number M ≫1 of measurements can be expressed as: p(φ, r|X) = 1 N Y x∈X p(x|φ, r)Mp(x|φ,r). (3) (a) (b) Var[!] [rad2] FIG. 2. Single- and multiparameter variances of the phase estimation. We report the variances of the adaptive phase estimation (radial axis) as a function of the phase φ (angular axis), spanning the entire range [0, π). The experimental results Var[φ] obtained in the single- [panel (a)] and multiparameter [panel (b)] adaptive protocols are reported as green and blue dots, respectively. The corresponding error bars represent the standard deviations over 5 (10) repetitions of the experiment, in the single-parameter (multiparameter) case. In both panels, we also report the corresponding classical bounds i.e., the QCRB for a coherent probe with the same average number of photons, as a blue line for the former and a green line for the latter case, while the QCRB is represented by a red line in both cases. For the multiparameter protocol we also report the actual CRB (orange dashed line). The experimental single-parameter protocol employs a total of M = 5000 homodyne measurements, among which MR = 200 are used to obtain the rough estimate. Instead, in the multiparameter estimation, the total number of measurements amounts to M = 20000, while the rough estimation needs MR = 1200 of them. Here, p(x|φ) represents the homodyne probability distribution, which consists of a Gaussian distribution centered in zero, and with variance: σ2(φ, r, η) = 1 4 ηe-2r cos2 φ+(1-η) cos2 φ+e2r sin2 φ , (4) with φ = φ -θ and η accounting for losses in the setup. Once having reconstructed the posterior probability, we derive the estimates ˆφ and ˆr of both parameters by calculating the mean value of the posterior distribution. To carry out abinitio sub-SQL estimation of the phase φ over the full range [0, π), we implement an online adaptive protocol consisting of two main stages. In the first step, we collect a small fraction of data MR for different settings of the LO phase θ. In this way, we can remove the intrinsic ambiguity π -φ of squeezedstate interferometry, which constrained previous experimental phase estimation protocols with homodyne measurements [24] to φ ∈[0, π/2]. From these outcomes, a Bayesian update 4 is performed to reconstruct the posterior distribution, from which rough estimates of both the phase and the squeezing parameter can be extracted. As the numerical evaluation of the posterior distribution can be computationally demanding, we approximate it by employing the Sequential Monte Carlo (SMC) technique [63, 64] that allows us to determine the online feedback. It consists of discretizing the posterior distribution into np particles {φk}np k=1 and {rk}np k=1, associated to weights {wk(X, φ)}np k=1 and {wk(X, φ, r)}np k=1, for the single and two-parameter protocols, respectively. The weights satisfy P k wk = 1 and are distributed according to the prior information. These are sequentially updated with the observed homodyne measurements x ∈X, in such a way that the final mean and variance of the posterior can be efficiently computed with the following discrete sums: ˆφ = np X k=1 ωk(X, φ -θ)φk, (5) ˆr = np X k=1 ωk(X, r)rk, (6) Var[φ] = np X k=1 ωk(X, φ -θ)(ˆφ -φk)2. (7) Further details on the SMC techniques are reported in the SI. These preliminary estimates are then used to set the adaptive feedback by changing the LO phase at the most informative measurement point. Results - We start by benchmarking the protocol in the single-parameter estimation scenario, where the goal is to estimate only the phase φ. In this framework, the first step of the adaptive procedure consists of acquiring data at two fixed values of the LO phase, i.e., θ = {0, π/4}. This first batch of measurements provides a first rough estimate of the parameter, which in turn allows us to identify the optimal measurement projection and thus setting the phase of the adaptive protocol where the FI is maximum, thereby optimizing the next measurement projections. For a pre-calibrated squeezing level r and a transmission coefficient η, we can retrieve the effective squeezing parameter that captures the combined impact of squeezing and loss given by: reff = 1 2 log σ2 asqz(r, η) q σ2asqz(r, η)σ2sqz(r, η) . (8) The optimal measurement configuration corresponds to φopt = 1 2 arccos (tanh (2reff)). In practice, the measurement basis is aligned by choosing θ = ˆφ -φopt, where ˆφ is the current Bayesian estimate. After this step, a new block of quadratures is collected and used to update the posterior distribution, thereby updating the parameter estimates and yielding a more accurate determination of the optimal LO phase. This adaptive cycle is repeated three times, using in total the remaining M -MR measurements. At each iteration, the increasing amount of data yields sharper posteriors and progressively improves the determination of the adaptive measurement setting. The experimental precision is obtained as the variance of the reconstructed posterior distribution for different values of φ, after the collection of M homodyne quadratures. The obtained results for the phase estimation in the entire range φ ∈[0, π) are reported in Fig.2a. The experimental measurements are compared with the QCRB computed for squeezedvacuum probes: Varsqz[φ] = 1/F sq,eff Q , in red, representing the ultimate sensitivity attainable with Gaussian resources for the given squeezing level and loss in such conditions, with F sq,eff Q = 2 sinh2(2reff). The achieved variances are compared with the optimal precision attainable with classical resources of equal mean photon number, quantified by the corresponding QCRB for coherent states: Varcoh[φ] = 1/(4⟨n⟩) = 1/[4 sinh2 (r)] reported as a blue line in the figure. Importantly, to assess the unconditional advantage over classical light, the coherent bound is not loss-corrected but is evaluated at the probe actual mean photon number linked to the squeezing parameter r. The bound is evaluated across all inspected phase values, each time accounting for the corresponding precalibrated levels of squeezing. These pre-calibrations reveal that the parameters are not fixed but instead fluctuate due to experimental instabilities, such as variations in the source during different data acquisitions and imperfections in the alignment procedures. The experimental variances lie consistently below the classical bound, demonstrating unambiguous sub-SQL sensitivity. Moreover, the measured precision closely approaches the squeezed-state QCRB, confirming that our adaptive Bayesian strategy operates near the optimal point across the entire phase domain. While this strategy enables the saturation of the ultimate precision bound for the phase estimation with homodyne measurements, it also requires a precise calibration of the setup efficiency η and of the squeezing level r, before performing the actual experiment. For this calibration to be effective, it must be sufficiently precise, hence, employing a significant amount of resources, which are not accounted in the overall resource budget. Importantly, our experimental results show that in realistic scenarios, both the level of squeezing and losses are not fixed, but change depending on the specific phase value under investigation. This can arise because the liquid crystal may exhibit different transmission at different phases, or because the squeezing parameter drifts over time due to temporal instabilities of the source. Consequently, different phase values refer to different values of the bound on the attainable precision as emerges in Fig.2a. This observation further motivates the need of a multiparameter strategy, in which both the phase and the squeezing parameter are inferred simultaneously from the same dataset. Such an approach enables the protocol to adapt in real-time to fluctuations of the probe, eliminating the dependence on previous calibration, which can jeopardize the final estimate, while still achieving sub-SQL estimation of the phase, as described below. We stress that within each estimation run, with the employed estimation model, we do not aim to track parameter variations dynamically; indeed, on the timescale of a few seconds, the phase-locking loop and the source stability are sufficient to maintain approximately constant values of φ and r. 5 Rough estimation Fine estimation (a) (c) (e) (b) (d) Coherent bound Coherent bound QCRB QCRB (f) (g) CRB F FIG. 3. Adaptive phase estimation single- and multi-parameter protocols. In panel (a), we report the evolution of the posterior distribution p(φ|X) as a function of the number of homodyne data measured along the adaptive single-parameter protocol in the range M ∈[100, 25000]. During the rough estimation, highlighted by a yellow background, we observe that measuring different quadrature angles (θ = {0, π/4} and θ = {0, π/4, π/2} in the single- and multiparameter estimations, respectively) allows us to disambiguate the estimate between the ranges φ ∈[0, π/2] and φ ∈[π/2, π). At the end of this stage, the current estimation ˆφ is employed to shift the LO phase to its optimal value θ = ˆφ -φopt. The remaining measurements (purple background) are taken within the optimal homodyne configuration, which is further updated two times before reaching the final estimation, extrapolated from the posterior distribution reported in panel (b). This inset shows how the precision of the phase estimation (blue shaded area), tightly approaches the squeezed QCRB (red line), while surpassing the classical bound (black dotted line). The changes of the LO angle, along the adaptive protocol, are illustrated in panel (c). It shows that the quantity θ -φ is close to its optimal value, depending on the chosen protocol. In the single parameter estimation (in blue), it concides with the value corresponding to the maximum of the FI [in panel (d)], while in the multiparameter approach (green), we aim at canceling the off-diagonal element of the FI matrix Fφr [shown in panel (d)]. In panel (e), the normalized variance g Var[φ] ≡Var[φ] · F sq,eff Q is reported as a function of M, thus having a common ultimate precision bound (solid red line) for both the single- and multiparameter variances, illustrated as blue and green lines, respectively. This panel also shows how the disambiguation and the LO feedback positively affect the precision of both the estimation protocols, although the multiparameter one exhibits a slower convergence to values below the classical bound (black dotted line). Panels (f) and (g) provide zoomed-in views of the marked area, demonstrating convergence to the bound for the single-parameter strategy and for the multi-parameter strategy, respectively. The latter converges to the CRB computed for the two-parameter estimate that does not coincide with the QCRB. In the multiparameter regime, for a fixed homodyne setting, the classical FI matrix becomes singular, which in this case showcases the impossibility of estimating simultaneously both parameters in the absence of prior information. The adaptive strategy, by measuring in different homodyne settings, restores a non-singular FI matrix, thus circumventing such a limitation. However, the measurement settings that maximize the phase sensitivity coincide with the sensitivity minimum for the squeezing parameter, and vice versa. This trade-off reflects the incompatibility of the relevant observables and precludes simultaneous attainment of the ultimate precision bounds for both parameters (see SI for additional details). Since our primary goal remains to estimate φ with the highest possible precision, thus prioritizing the estimate of φ over r, it might be expected that selecting the single-parameter phase-optimal setting would suffice. In practice, however, an excessively imprecise estimate of r degrades the adaptive LO feedback and therefore the phase estimate (see simulation results in SI). To balance these competing requirements, we implement an LO feedback rule that aims to cancel the correlation between the two parameters' estimation errors. The optimal strategy corresponds to set the quadrature where phase and squeezing estimates are effectively decorrelated, i.e., where the off-diagonal elements of the FI matrix Fφr vanish. This corresponds to setting θ = ˆφ -arccos e2ˆr √ e4ˆr+η, with η the detection efficiency. Both the estimates ˆr and ˆφ, needed to compute the feedback updates, are retrieved as mean values of the two-dimensional posterior distribution, which is efficiently reconstructed with the SMC approach, further justifying its adoption to significantly speed up the numerical 6 reconstruction, especially in this multiparameter framework. The precisions achieved on the phase estimates for the multiparameter protocol are reported in Fig.2b, showing also in this case the ability to obtain quantum-enhanced performance, while ensuring robustness to a varying squeezing level. Overall, we observe gains of 1.79 ± 0.40 dB below the SQL, not corrected for loss and averaged over 10 different phase values, with the conventional single-parameter (pre-calibrated) approach and 1.78 ± 0.33 dB below the SQL with our abinitio multiparameter strategy concerning the phase estimate. These results highlight both the robustness and the unconditional character of the protocol, since the quantum advantage is maintained without prior calibration of the probe and for all the values of φ, overcoming also the restrictions of previous approaches limited to narrower phase intervals. A more direct comparison of the single and multiparameter strategies is reported in Fig.3, where we investigate how the estimation variances scale with the number of homodyne measurements. More specifically, in Fig.3a the experimental single-parameter posterior distribution is reported for a representative phase estimate as a function of the number of homodyne measurements used to update the Bayesian posterior. Fig.3b shows the reconstructed distribution at the end of the protocol, quantifying the achieved precision as Std[φ] = p Var[φ]. Fig.3c illustrates the sequence of feedback values applied at successive steps of the estimation protocol. In the single-parameter case, after the rough estimate the feedback angles correspond to the quadrature maximizing the Fisher information, while in the multiparameter case, the rough estimation stage is split into three steps, θ = {0, π/4, π/2}, to improve the estimation of r needed to implement the adaptive feedback. The feedback angles for the fine estimate are instead determined by the condition where the off-diagonal elements of the FI matrix vanish (see Fig 3d). The singleparameter strategy, which assumes prior knowledge of the squeezing parameter, required to set the feedback of the adaptive algorithm, displays faster convergence to the bound as emerges in Fig.3e. As expected, the multiparameter strategy requires a larger number of measurements before reaching convergence, since part of the data are used to estimate the squeezing level of the probe. Nevertheless, and most importantly, even in the absence of any prior calibration of the squeezing, the multiparameter protocol achieves phase estimation precision at sub-SQL level once convergence is reached. This demonstrates that quantum-enhanced performance can be retained even without relying on external information about the probe. Finally, the advantage over coherentlight phase estimation is further emphasized in the zoomed-in region showing the convergence behavior. Both our singleparameter (Fig.3f) and multiparameter (Fig.3g) strategies consistently outperform the lossless coherent bound calculated without rescaling by the overall transmission coefficient η. We next demonstrate the full strength of the multiparameter protocol in a challenging regime where not only the optical phase, but also the squeezing parameter itself, undergoes significant variations, to test the robustness of this protocol to different squeezing levels. The estimated values of r are shown in Fig.4a as a function of the calibrated squeezing. Re- (a) (b) (c) Coherent bound QCRB FIG. 4. Robustness of the multiparameter adaptive estimation protocol to different squeezing strength and fully black-box approach. In (a) the green squares represents the estimated squeezing level ˆr, within the multiparameter adaptive protocol, compared to the actual values r. As an example, we illustrate the experimentally reconstructed posterior distributions p(φ, r|X), corresponding to the lowest (top-left inset) and the highest (bottom right inset) squeezing levels. (b) Scaling of the experimental phase variances compared to the relative bounds: the QCRB for squeezed vacuum probes (red), the CRB (orange), and the QCRB for coherent states (green). The data lie all below the coherent bounds, computed without considering losses. Panel (c), reports the results obtained with a fully black-box approach, in which the efficiency η is also estimated. The normalized variances g Var[φ] (purple dots) are compared with the classical lossless (dash-dotted purple line) bound. In all the panels, the reported errors correspond to standard deviations over 10 repetitions of the experiments. markably, the adopted strategy can faithfully infer the true value of r across all operating conditions, without requiring 7 any prior calibration or external reference. At the same time, the variance of the simultaneous phase estimate, reported for different squeezing regimes, remains consistently below the classical bound as shown in Fig.4b, confirming the ability of our protocol to preserve quantum-enhanced performance even under strongly different conditions of the probe. This is achieved since the adaptive feedback is explicitly designed to minimize the variance of the parameter of interest, namely the phase φ. This feature is also related to the shape of the reconstructed posterior, which is narrower along the phase axis, while still providing reliable estimates of the squeezing. Our results highlight the versatility of the adopted multiparameter approach that enables the simultaneous estimation of probe and signal parameters, while allowing the feedback strategy to be tailored to prioritize the precision of the most relevant observable. Moreover, by analyzing the dependence of the estimation variance on the squeezing strength r, which directly quantifies the photon number in the probe, we clearly observe an improvement with respect to the classical coherent bound. As r increases, the variance decreases accordingly, and the gap between squeezed light estimation and the classical coherent state benchmark widens. This behavior provides an experimental indication that stronger squeezing yields greater metrological gain. Finally, to make the protocol completely independent of any calibration, we tested its performance under the most demanding condition, in which no prior knowledge of the overall detection efficiency η is also assumed. Estimating losses jointly with the phase is particularly challenging, since the corresponding parameters are incompatible observables [65]. To overcome this limitation, we adopt a hybrid strategy that combines Bayesian inference with a maximum-likelihood estimation of η. Specifically, during each update of the three-dimensional posterior, η is set to the maximizer of the η-posterior at the previous step. This update bypasses full three-parameter inference, retaining the most consistent value of η in every step of the estimation process without the need for additional measurement, focusing only on the physically relevant parameters φ and r. This mixed approach allows us to maintain quantum-enhanced performance (see Fig.4c) without relying on any external calibration of the probe or the detection chain, thereby validating the fully unconditional nature of the protocol. Discussion/Conclusion - Our work presents the first experimental demonstration of adaptive multiparameter quantum phase estimation with squeezed vacuum states, achieving subSQL precision across the full periodicity interval [0, π) without relying on prior calibration of key system parameters such as the squeezing level and detection efficiency. These parameters are known to fluctuate in realistic scenarios, both due to temporal drifts of the source and to sample-dependent transmission, making calibration-based approaches inherently unstable and ultimately compromising the effectiveness of adaptive measurement strategies. By implementing an adaptive multiparameter protocol that simultaneously infers the phase φ and the relevant nuisance parameters, we overcome this calibration bottleneck while retaining unconditional quantum advantage over the entire unambiguous phase range. Crucially, the method is robust to consistent fluctuations of r, maintaining quantum-enhanced performance across a wide operating range. These findings demonstrate the feasibility of selfcalibrating, ab-initio quantum metrology protocols that are resilient to probe instabilities and experimental drifts, thereby extending the practical applicability of squeezed light for precision sensing. Beyond enabling reliable operation under realistic conditions, our results establish a pathway toward scalable implementations in advanced interferometric platforms and distributed quantum sensor networks, which rely on the simultaneous estimation of multiple parameters. ACKNOWLEDGMENTS This work is supported by the Amaldi Research Center funded by the Ministero dell'Istruzione dell'Universit`a e della Ricerca (Ministry of Education, University and Research) program "Dipartimento di Eccellenza" (CUP:B81I18001170001), by the project QU-DICE, Fare Ricerca in Italia, Ministero dell'istruzione e del merito, code: R20TRHTSPA, and by the PNRR MUR project PE0000023-NQSTI (Spoke 4). We thank M. Barbieri for the useful discussion and the Noisy Labs team. [1] Pezze, L. & Smerzi, A. Quantum theory of phase estimation. arXiv preprint (2014). [2] Acernese, F. et al. Increasing the astrophysical reach of the advanced virgo detector via the application of squeezed vacuum states of light. Physical review letters 123, 231108 (2019). [3] Collaboration, T. L. S. A gravitational wave observatory operating beyond the quantum shot-noise limit. Nature Physics 7, 962-965 (2011). [4] Tse, M. e. et al. 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Journal of Physics A: Mathematical and Theoretical (2025). Supplementary Information for: Multiparameter quantum-enhanced adaptive metrology with squeezed light Giorgio Minati,1 Enrico Urbani,1 Nicol`o Spagnolo,1 Valeria Cimini,1, ∗and Fabio Sciarrino1 1Dipartimento di Fisica, Sapienza Universit`a di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy I. EXPERIMENTAL SETUP CHARACTERIZATION A. Experimental Setup In what follows, we present a complete description of the setup employed to perform the experiment, which is schematically illustrated in Fig.1. Supplementary Figure 1. Experimental setup. The laser provides two outputs. The 532 nm beam pumps the squeezer cavity, while the 1064 nm beam is split by a half-wave plate (HWP) and a polarizing beam splitter (PBS) into a control beam and a local oscillator (LO), both mode-matched to the squeezed output. The setup comprises three main blocks: (i) squeezed-state generation, (ii) homodyne measurement, and (iii) phase stabilization using the control beam in a phase-locked loop (PLL). The squeezer cavity is stabilized via a Pound-Drever-Hall (PDH) scheme employing an electro-optic modulator (EOM), a piezoelectric actuator, and a photodiode; the same infrastructure is used to keep three mode-cleaner cavities resonant with their respective beams. After the squeezer, the 532 nm pump and the 1064 nm control/squeezed beams are separated with dichroic mirrors (DM). The phase φ along the squeezed-beam path is tuned with a liquid-crystal (LC) phase shifter and measured by balanced homodyne detection (HD), where the LO and the squeezed beam interfere on a 50:50 beam splitter (BS). The quadrature angle θ is swept using a piezoelectric stage driven by an FPGA, which reads the DC output of the homodyne detector after a band pass filter (BPF) to suppress electronic noise. The main laser source is a Coherent Prometheus Nd:YAG double-wavelength continuous-wave laser emitting 1.77 W at 1064 nm and 104 mW at 532 nm. The 532 output serves as the pump for the squeezed state generation. Before this step, the action of a Half-Wave Plate (HWP) and a Polarizing Beam Splitter (PBS) allows us to tune the total power of the 532 nm beam. Phase modulation in the MHz frequency range is applied using an electo-optical modulator (EOM) to produce sidebands used for the Pound-Drever-Hall (PDH) locking. All the beams employed are cleaned through three mirrors Fabry-Perot ring cavities, each one used as an optical resonator for mode-cleaning (MC), filtering out spatial modes different from the TEM00. Generation stage. The squeezed light source is the commercial source developed by Noisy Labs. It consists of an optical parametric amplifier based on a type-0 periodically poled KTP (PPKTP) crystal configured as a hemilithic standing-wave cavity ∗ 2 pumped via the mode-cleaned 532 nm beam. One mirror is piezo-mounted for length control, while the high-reflection coating on one end face of the PPKTP acts as the second mirror at 1064 nm (and appropriately coated for 532 nm). The opposite crystal face is anti-reflection coated. The cavity is locked on resonance via PDH. The crystal temperature is actively stabilized to maintain quasi-phase matching at the pump wavelength and to suppress slow thermal drifts; length control is handled by the piezo and fine-tuned with an additional temperature control. The output of the source can be extracted using a Dicroic Mirror (DM) that reflects only 1064 nm wavelength. The 1064 nm beam is split into two paths, each subjected to the same modecleaning procedure used for the pump. This ensures high spatial purity and excellent overlap at the detection stage between the squeezed field and the local oscillator (LO), which is essential for resolving large levels of squeezing. One path forms the LO for balanced homodyne detection, while the second serves as a coherent control beam used to stabilize the phase of the squeezed field. Measurement stage. We employ a balanced homodyne detection scheme where the LO and the squeezed beam are mixed on a 50:50 beam splitter (BS) and the photocurrent difference is measured via an oscilloscope. The detected quadrature is set by the LO phase θ controlled via a motorized piezoelectric stage. Phase locking loop. The phase of the squeezed beam is stabilized using a coherent control beam that is spatially mode-matched to the squeezed mode. The control beam is phase-modulated by an EOM at radio frequencies and injected in the squeezer cavity. When in resonance, the bright leakage provides an alignment reference for the squeezed path; at low power it generates a phase-sensitive error signal upon interference with the LO. This signal is demodulated and used in a phase-locked loop (PLL). In this configuration, the OPA produces squeezed vacuum at 1064 nm the MHz detection band. The combination of mode cleaning, PDH-stabilized cavities, and coherent-control locking provides the spatial mode quality and phase stability θ -φ required to resolve sub-shot-noise quadrature fluctuations and to implement the adaptive, real-time estimation protocol. B. Squeezed light characterization In order to characterize the generated squeezed state, we measure the photocurrent noise modulation in different configurations, which we report in Fig.2, where all the traces are zero-span measurements at 6 MHz. We take the shot-noise as a reference, hence, setting the noise level which is measured when only the LO is sent to the homodyne detector at 0 dB. A 1 Hz triangular-wave-shaped scan of the measured quadrature angle allows us to estimate the maximum and minimum noise modulation (blue trace in Fig.2), thereby quantifying the squeezing and anti-squeezing level as σ2 sqz = -6.12 ± 0.28 dB and σ2 asqz = 11.35 ± 0.28 dB. Additionally, within the same settings, we measure the electronic dark-noise when no light at all is sent to the detector, obtaining a noise attenuation of σ2 dark = -14.9 ± 1.0 dB 0.00 0.05 0.10 0.15 0.20 0.25 Time [s] 15 10 5 0 5 10 Noise attenuation [dB] Squeezed light quadrature variance Vacuum quadrature variance (shot noise) Dark noise Supplementary Figure 2. Squeezed quadratures noise attenuation, compared to the shot noise and the dark noise. In this figure, we report the squeezed light variance attenuation (blue trace), in units of shot-noise. In particular, it has been acquired by scanning the LO phase by ramping the piezo mirror with a triangular wave with a frequency of 1 Hz. Additionally, we also report the shot noise (yellow trace) and the electronic dark noise (purple trace). All three traces have been acquired with a zero-span measurement at 6 MHz. 3 C. Phase Locking Stability and Control In what follows, we describe the implementation of the FPGA-based locking procedure employed to measure the squeezed field quadrature along an arbitrary angle. First, we model the fields incoming at the homodyne detectors as: E1 = ELOei(ωt+θ), (1) E2 = ECFei(ωt+β sin (Ωt)). (2) The first is the LO field, oscillating with an amplitude ELO at the optical frequency ω, with a phase θ set by the LO piezo mirror. The second field is the weak coherent field superimposed onto the squeezed light path, whose oscillation has amplitude ECF and, in addition to the optical frequency ω, is phase-modulated with sidebands of frequency Ωand modulation depth β. The interference of these in the HD detection BS produces the output fields E3 and E4, which are computed as follows: E3 E4 = 1 √ 2 1 i i 1 E1 E2 = 1 √ 2 E1 + iE2 iE1 + E2 . (3) The corresponding photocurrents measured at the HD detector are proportional to the output fields intensities P3 and P4: P3 = |E3|2 = 1 2(|E1|2 + |E2|2 + iE∗ 1E1 -E1E∗ 1), (4) P4 = |E4|2 = 1 2(|E1|2 + |E2|2 -iE∗ 1E1 + E1E∗ 1), (5) therefore, the photocurrents subtraction reads: PHD = P3 -P4 = i(E∗ 1E2 -E1E∗ 2) = (6) = iELOECF ei(-θ+β sin (Ωt)) -e-i(-θ+β sin (Ωt)) = (7) ≃iELOECF e-iθ [J0(β) + 2iJ1(β) sin (Ωt)] -eiθ [J0(β) -2iJ1(β) sin (Ωt)] = (8) = 2ELOECF (J0(β) sin (θ) -2J1(β) sin (Ωt) cos (θ)) , (9) where, in (8), the terms e±iβ sin (Ωt) have been expressed as a Bessel series truncated at order 1. The part of Eq.(9) oscillating at 2Ω, selected using a high-pass filter, is then mixed with an electronic local oscillator E3 = EeLO sin (Ωt + φdemod), obtaining; P mix HD = -4ELOECFJ1(β) sin (Ωt) cos (θ) · EeLO sin (Ωt + φdemod) = (10) = -4ELOECFEeLOJ1(β) cos (θ) (sin (2Ωt + φdemod) + sin (φdemod)) . (11) A low-pass filter eliminates the terms oscillating at 2Ω, resulting in the following error signal: P err cos = -4ELOECFEeLOJ1(β) cos (θ) sin (φdemod). (12) Therefore, we have access to the first component of (9) (which we will call P err sin in what follows) and (12), respectively modulated as sin (θ) and cos (θ). We can use them to generate an error signal that can lock the homodyne measure to an arbitrary angle α. In fact, upon rescaling P err cos by a factor S to match the amplitude of P err sin , we can combine them in a sum with weights Aα, Bα such that A2 α + B2 α = 1. Hence, the final error signal will be: P α err = AαP err sin + BαSP err cos ∼sin(θ + α), (13) where α = arctan (Bα/Aα) = arccos (Aα) = arcsin (Bα). In what follows, for notation simplicity, we define θ = θ + α. To experimentally test this procedure, we lock the homodyne measure to different θ ∈[0, 2π] and, for each of them, we measure the resulting homodyne DC signal and the quadrature variance, and report the corresponding results in the left and right panels of Fig.3, respectively. The regularity of these measurements demonstrates the reliability of this approach, and enables, by fitting the experimental data, to finely calibrate possible overall offsets of the phase θ. II. ADDITIONAL EXPERIMENTAL DATA In this Section, we describe additional measures relative to the results reported in the main text. To better understand the effects of adaptive feedback, we report in Fig.4 the quadratures measured for a specific phase φ during the estimation protocol. 4 0 /2 3 /2 2 [rad] 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Homodyne DC signal [V] Exp.data Fitted curve 0 /2 3 /2 2 [rad] 100 2 X( , = 0) [shot noise] Exp. data Fitted curve Supplementary Figure 3. Phase-locked measurements. In panels (a) and (b) are reported, the homodyne DC signal (red dots) and the quadratures variance (blue dots) acquired for different phase-locked homodyne angles spanning the range θ ∈[0, 2π], respectively. Both experimental datasets have been acquired by averaging over 200, 000 acquisitions, but in the second case, a band-pass filter around 1.1 MHz is also required, especially to remove low-frequency noise. We report the corresponding fitted curves, as red and blue solid lines for the homodyne DC and quadratures variance, respectively. The former employs a cosine model, while the latter uses the lossy variance model for the quadrature variance defined in Eq.(4) of the main text. Rough estimation Fine estimation Supplementary Figure 4. Homodyne data during the adaptive protocol. Here we plot the homodyne measurements taken during the single-parameter estimation of the phase φ = 1.1 rad. The normalized measurements are reported in such a way to fix the shot-noise to the conventional value σ2 shot = 1/4. The yellow background indicates the fraction of measurements employed to obtain the phase rough estimation necessary to compute the feedback for the LO phase. Its effect is noticeable in the second stage of the measurement, illustrated on a purple background, where the homodyne data exhibits a smaller variance, corresponding to measuring along the optimal quadrature angle. We can observe how a small portion of the total acquired data (illustrated with a yellow background in Fig.4) is sufficient to have a rough estimation, precise enough to compute a feedback that shifts the LO phase towards the optimal one, noticeable by the substantial decreasing of the quadrature variance in the second stage of the estimation (depicted in Fig.4 with a purple background). In the case of multiparameter estimation, we can gain further insights into the functioning of the adaptive estimation by studying how the experimental 2-dimensional posterior p(φ, r|X) evolves during the different steps of the protocol, illustrated in Fig.5. After the first step of the rough estimation (upper left panel in Fig.5), taken with the LO phase set at θ = 0, the posterior presents a symmetric behavior along the phase. This feature reflects the fact that a homodyne data taken in a single setting cannot disambiguate the phase between a phase φ and π -φ. The second step of the protocol (upper central panel in Fig.5) updates the previous posterior with quadratures measured along θ = π/4, and illustrates how this additional measurement setting resolves the ambiguity mentioned above. The third part of the rough estimation, by setting θ = π/2, probes the orthogonal quadrature with respect to the first step, improving, as we can see in the upper right panel of Fig.5, the estimation of the squeezing parameter. A sufficiently precise estimate of r is essential for computing accurate adaptive feedback, whose effect becomes clear when comparing the final rough estimate with the initial fine estimation (purple color scale). The lower panels show the posterior distributions for the three steps of the fine estimation on a magnified scale, illustrating how much of the overall precision gain is attributable to the adaptive feedback. 5 Supplementary Figure 5. Evolution of the experimental posterior distribution during the multiparameter adaptive protocol. In this figure, the posterior distribution p(φ, r|X) is updated during the different steps of the adaptive protocol. In the upper panels, we report the posterior during the three steps of the rough estimation, using an orange color scale. Instead, the three bottom panels illustrate the posterior evolution during the fine estimate (purple color scale), again subdivided into three adaptive feedback steps. To quantify the protocol ability to estimate the phase ab-initio over the full [0, π) range, the estimated values of the 10 different investigated phases ˆφ are compared in Fig.6 to the true calibrated values φ for all the inspected strategies. Starting from the experimental single-parameter estimate, the estimated values of the two-parameter ones, and the black box variant of the protocol that also infers the overall detection efficiency η from the data, are here reported. 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Supplementary Figure 6. Estimated phases compared to the calibrated phase values. In the three panels, we report the estimated phase values ˆφ as a function of the actual phase φ, respectively for the single parameter (left panel, blue dots), 2-parameter (central panel, green dots), and 3-parameter (right panel, purple dots) protocols. The scaling of the phase variance for the black-box protocol, which jointly estimates (φ, r, η), as a function of the number of measurements is reported in Fig.8. Even in this case, the implemented approach enables a phase estimate below the coherent bound, confirming that the information gain provided by squeezing is retained even while the squeezing parameter and the overall efficiency are inferred on the fly and not pre-calibrated. In terms of scaling with the average number of photons in the probe, in the ideal lossless case, the variance of the phase estimate retrieved with coherent probes scales at the SQL, Var[φ] ∼ 1 4⟨n⟩. On the contrary, squeezed probes attain a quantum6 Supplementary Figure 7. Scaling of the phase estimation variance as a function of the number of measurements, in the case of the 3parameter estimation protocol. The normalized variance g Var[φ] ≡Var[φ] · MF sq Q is reported as a function of the homodyne measurements M and compared to the QCRB (solid red line). The dotted line represents the coherent bound for the lossless scenario. Supplementary Figure 8. Scaling of the phase estimation variance as a function of the probe average photon number. The ultimate phase-variance bounds for coherent (green surface) and squeezed (red surface) probes as functions of the mean photon number ⟨n⟩and total efficiency η are reported. In the lossless limit (η = 1), coherent light follows the SQL, while squeezed probes achieve a quadratic scaling. Loss degrades these scalings. enhanced scaling 1 8⟨n⟩(⟨n⟩+1) that for ⟨n⟩>> 1 attains the Heisenberg limit. Such behaviors are altered in the presence of losses, therefore, when η 0 and rotated by a phase φ ∈[0, π]. The encoding of both these parameters can be modeled as the following unitary transformation of the vacuum state |0⟩: |φ, r⟩= ei(φGφ+rGr) |0⟩, where Gφ = n = a†a, Gr = i 2 a†2 -a2 . (19) Then, the QFI corresponding to the operations reported in Eq.(19) can be computed by means of Eq.(18), where we substitute ⃗y = {φ, r}, obtaining the following matrix: FQ[|φ, r⟩] = 2 sinh2 (2r) 0 0 2 . (20) The diagonal form of the QFI in Eq.(20) further simplifies the evaluation of ultimate precision bounds on the estimation of φ and r, which ultimately reduces to the inverse of the corresponding diagonal elements. In a multiparameter setting, two parameters are said to be compatible if there exists a measurement strategy that can simultaneously saturate the QCRB for both parameters. A sufficient condition to achieve compatibility is the commutation of the SLDs corresponding to the parameters of interest, as projective measurements onto their common eigenbasis enable the optimal estimation of such parameters. When the state ρ is pure, as in our model, a weaker necessary and sufficient condition can be verified, i.e. the so-called weak commutation relation, which is verified ⟨[Li, Lj]⟩= 0. In our model, where the parameters are encoded via unitary transformation this expression further simplifies, and can be evaluated as follows: ⟨[Lφ, Lr]⟩= 4 ⟨[Gφ, Gr]⟩= -4i sinh (2r) ̸= 0. (21) This means the parameters φ and r are incompatible, i.e. we cannot devise a measurement strategy that simultaneously achieves the optimal precision in both estimations. In what follows, we describe how adaptivity can exploit the multiparameter estimation framework to approach the phase estimation QCRB when the squeezing level is unknown. 8 B. Adaptive protocol model The likelihood p(x|θ, φ, r) for a homodyne measurement with local-oscillator phase θ on a squeezed state (phase φ, squeezing r, transmission η) is modeled as a zero-mean Gaussian whose variance is the one reported in Eq.(4) of the main text, which we also reproduce here for convenience. σ2(φ, r, η) = 1 4 ηe-2r cos2 φ + (1 -η) cos2 φ + e2r sin2 φ , (22) where φ = θ -φ. In what follows, we describe the 2-parameter adaptive protocol, deriving the model of the corresponding Fisher Information (FI) and, then, its CRB. In particular, in a multiparameter framework where ⃗y = {φ, r} are the parameters to be estimated, the individual elements of the FI matrix are computed as: Fij(⃗y|θ, η) = Z ∞ -∞ dx p(x|⃗y) ∂ ∂yi log p(x|⃗y, θ, η) ∂ ∂yj log p(x|⃗y, θ, η) . (23) In this case, η must be determined by calibration, whereas in the three-parameter estimation protocol it is unknown and therefore included in the data vector ⃗y. If we consider a single-setting experiment in which homodyne data are acquired at a single LO phase, then the resulting FI matrix is singular and the corresponding CRBs are not defined. This is unsurprising: measuring only one quadrature does not provide enough information to distinguish different combinations of φ and r , i.e., the measurement settings are not tomographically complete. A first achievement of the adaptive protocol we describe is the removal of this ambiguity, which in turn allows one to derive a meaningful multiparameter CRB. As discussed in Sec. III A, φ and r are incompatible parameters and cannot be estimated simultaneously with optimal precision. Given that our aim is to choose the optimal setting φopt for estimating φ, we expect r to be estimated with suboptimal precision at that setting. In fact, below we show that the optimal setting for φ corresponds to the point of minimal sensitivity for r. Since the adaptive feedback relies on ˆr, poor estimation of r also degrades the precision of ˆφ; to mitigate this, we design the adaptive protocol accordingly. A natural strategy is to steer the LO phase towards the setting that removes the correlation between the estimation errors, or, equivalently, to drive the off-diagonal element of the FI matrix to zero. This yields θopt = ˆφ -φopt = ˆφ -arccos e2ˆr p e4ˆr + η ! , (24) where ˆφ and ˆr are the current parameter estimate before the adaptive feedback. Keeping in mind this strategy, we now develop a FI matrix model that replicates our multiparameter adaptive protocol. Firstly, we recall the strategy we have implemented experimentally: • Rough estimation: the first MR = 1, 200 homodyne measures are devoted to obtain approximate estimations of the parameters. In particular, we equally divide them in measures with LO phase in θ = {0, π/4, π/2}. This resource allocation in equally spaced settings has the twofold objective of eliminating the aforementioned singularity and provide a sufficiently precise estimation of both parameters. • Adaptive step: the estimated parameters ˆφ and ˆr are employed to compute θopt, and shift the LO phase to this value. In the actual experimental implementation, this step can also be reiterated multiple times to finely adjust the homodyne setting to the optimal value. • Fine estimation: as further discussed in Sec.IV, a sufficient amount M of homodyne measures has to be acquired to achieve statistical convergence. In our case, M = 20, 000 is sufficient to ensure such a convergence. Therefore, the remaining MF = M -MR = 18, 800 homodyne data are measured within the optimal setting to estimate φ. As we will see, this strategy allows to closely approach the phase QCRB. Therefore, using the definition reported in Eq.(22), we can model the FI matrix corresponding to the adaptive protocol as a function of the LO phase in the fine estimation step: eF (φ, r|θ, η) = 1 3F (φ, r|0, η) + 1 3F (φ, r|π/4, η) + 1 3F (φ, r|π/2, η MR M | {z } Rough estimation + F (φ, r|θ, η)MF M | {z } Fine estimation . (25) 9 Supplementary Figure 9. Inverse of the FI matrix elements in the multiparameter adaptive protocol. In this figure, we report the behavior of the e F -1 entries as a function of φ ∈[0, 2π] and r ∈[0, 2]. In particular, [ e F -1]φφ, [ e F -1]rr, and [ e F -1]φr = [ e F -1]rφ illustrated respectively in panels (a), (b), and (c). In all of them, we report the value of the phase φopt(r, η) that eliminates the correlations as a red dashed line. The resulting FI matrix is invertible, allowing us to define the corresponding multiparameter CRBs. In Fig.9, we report the behavior of the individual elements of [ eF -1] as a function of the parameters φ and r. For the sake of clarity, in what follows, we will use the notation [ eF -1]ij with i, j ∈{φ, r} to denote corresponding entries of the inverse FI matrix. In particular, we can notice how the r-dependent phase that eliminates the correlations (red dashed lines in Fig.9), i.e. [ eF -1]φr = 0 (right panel) also corresponds to the minimum of the [ eF -1]φφ (left panel) and the maximum [ eF -1]rr (central panel), along the entire range of r. To have a better understanding of the adaptive FI matrix model we derived, in Fig.10, we illustrate the behavior of [ eF -1]φφ, [ eF -1]rr, and [ eF -1]φr when the squeezing level and efficiency are similar to the experimental conditions, i.e. r = 0.63 and η = 0.85. As previously mentioned, we can see how the setting that maximizes the precision in estimating φ (r), at the same time minimizes the sensibility to r (φ). Additionally, the inset (a) of Fig.10 shows how the φopt is indeed the one that removes the correlation (purple curve) and optimizes [ eF -1]φφ, which achieves a value quite close to the corresponding QCRB. This consideration is also true for the optimal estimation of r, which is detailed in the inset (c) of Fig.10. Finally, we further motivate the use of the multiparameter approach through numerical simulations showing that the implemented protocol is inherently robust to variations in the probe squeezing level, whereas the single-parameter approach fails under the same conditions. The results of simulations performed by changing the squeezing by an amount ∆r are reported in Fig. 11. The plotted results refer to the variance on the phase estimate for a single-parameter estimator that fixes the squeezing at its precalibrated value r, and our multiparameter adaptive estimator, as the true squeezing is offset by ∆r. For the single-parameter approach, even small mismatches brings to unphysical variances or significantly degrade estimation precision. The estimation error approches the bound only when the effective squeezing remains very close to the calibrated value. On the contrary, the proposed multiparameter adaptive protocol is able to keep the phase estimation variance close to the QCRB for any variation of the squeezing, since it avoids explicit calibration of the probe squeezing and instead infers its value directly from the data. This joint inference makes the protocol robust, suppressing systematic errors from drift and sample variability. IV. SEQUENTIAL MONTE CARLO We estimate the unknown phase φ with a Bayesian procedure that updates a flat prior on [0, π] using homodyne data X = {xm}M m=1. Such measurements are employed to reconstruct the single-parameter posterior probability distribution using Bayes' theorem, the posterior probability reconstructed after M measures is reported in Fig.12. The reconstructed posterior is updated using the homodyne probability distribution, obtained by marginalizing the Wigner function of the squeezed probe state and setting the phase of the local oscillator (LO) to θ. Once having reconstructed the posterior probability, we derive the estimate ˆφ for the parameter of interest and its variance. As usual in the Bayesian framework, these are obtained by calculating the mean value and variance of the posterior distribution, respectively: ˆφ = Z p(φ|X)φdφ ∼ np X k=1 ωk(X)φk, (26) 10 ["#!"]!!(&) ["#!"]!"(&) ["#!"]""(&) ["## !"]!! ["## !"]"" (a) (b) (c) ![rad] ![rad] ![rad] !"!" !"!" !"!" Supplementary Figure 10. Adaptive FI matrix model for experimental values of r and φ. In this figure, we report the behavior of the adaptive FI matrix entries as a function of φ = θ -φ, when considering the experimentally plausible parameters φ = π/4, r = 0.63, and η = 0.85. In the main panel (a), we report the [ e F -1]φφ (green curve), [ e F -1]rr (blue curve), and [ e F -1]φr (purple curve) for the entire range φ ∈[0, 2π], comparing them to the corresponding QCRBs for the estimation of φ, i.e. [F -1 Q ]φφ (red dashed line), and for the estimation of r, i.e. [F -1 Q ]rr (orange dashed line). The black dashed line represents the phase φopt that removes the correlation between the estimation errors of φ and r. In the insets (b) and (c), we report the magnification of the main plot around the optimal settings for estimating φ and r, respectively, again compared to the corresponding QRCBs. Var[φ] = Z p(φ|X)( ˆφ -φ)2dφ ∼ np X k=1 ωk(X)(ˆφ -φk)2. (27) In the adopted framework, the computation is accelerated by substituting integrals with discretized sums that have a relevant impact, in particular for multiparameter estimation, where the computation of multidimensional integrals can be costly, affecting the realization of the adaptive protocol. To verify that the implemented estimator attains the relevant precision limits, we perform simulations of the estimation protocols using the probe parameters η = 0.8 and r = 0.8. In the single-parameter protocol, we discretize [0, π] with np = 10000 particles and observe the expected 1/M variance decay that saturates the QCRB. In the joint (φ, r) estimate, np = 20000 total particles were employed, and the trend of the variances on both parameter estimates has been investigated. The simulations have been carried out with the same adaptive protocol described before, whose aim is to minimize the variance on the phase estimate. Studying the performance scaling reported in Fig.13 with the number of homodyne measures is evident how they saturate to the relative elements of the inverse of the FIM computed in the previous section. 11 0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4 r 0.2 0.4 0.6 0.8 1.0 Var[ ] Var[ ] (single par.) Var[ ] (multipar.) Squeezed bound Coherent bound Supplementary Figure 11. Behavior of the variance as the squeezing level deviates from the calibration. In this figure, we report how the phase estimation variance changes when the squeezing level deviates from the original value r = 0.8 by a quantity ∆r ∈[-0.4, 0.4]. In particular, we report the average and the standard deviation (over 50 repetitions) of the simulated estimation variance obtained with the single- (multi-)parameter adaptive protocol as a blue (green) solid curve and shaded area, respectively. We compare them with the corresponding QCRB (red dashed curve) and coherent bound (black dashed curve). 0.0 0.5 1.0 1.5 2.0 2.5 3.0 (rad) 0.000 0.005 0.010 0.015 0.020 0.025 0.030 Posterior M = 10 M = 50 M = 100 Supplementary Figure 12. Simulated Bayesian posterior. Reconstructed posterior probability distribution of Bayesian phase estimation in range [0, π] with M = 10, 50, 100 homodyne measures. We choose a trial phase φtrue = 0.3 rad and a squeezing parameter r = 1. 12 103 104 M 10 5 10 4 10 3 10 2 10 1 Var[ ] (a) QCRB CRB( )-multipar. Coherent bound single par. multipar. 103 104 M 10 4 10 3 10 2 10 1 Var[r] (b) CRB(r)-multipar. F 1 rr ( opt) multipar. Supplementary Figure 13. Scaling of simulated posterior variance with the number of homodyne samples M. Panel (a) average variance of the phase estimate with the single-parameter Bayesian protocol (blue circles); variance of the phase estimate with the multiparameter Bayesian protocol (green circles). The dashed black line is the ultimate classical bound for coherent states, the red line is the QCRB for squeezed light; the orange line is the CRB for the adopted two-parameter strategy. Panel (b) shows the average variance of the squeezing parameter estimate (green circles). The purple line is the CRB achievable when the protocol is optimized to minimize Var(r) while the pink line represents the bound obtained with the implemented (phase-optimized) adaptive strategy. Error bars represent the standard deviation over estimates of 20 different phase values.
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2510.14730
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Deadlock-free routing for Full-mesh networks without using
Virtual Channels
postprint version
A. Cano, C. Camarero, C. Martínez, and R. Beivide, “Deadlock free routing for full-mesh networks without using
virtual channels,” in HOTI25: 32nd IEEE Hot Interconnects symposium. IEEE, 2025.
https://doi.org/10.1109/HOTI66940.2025.00020
Alejandro Cano
Cristóbal Camarero
Carmen Martínez
Ramón Beivide
{alejandro.cano, cristobal.camarero, carmen.martinez, ramon.beivide}@unican.es
Abstract
High-radix, low-diameter networks like HyperX and Dragonfly use a Full-mesh core, and rely on
multiple virtual channels (VCs) to avoid packet deadlocks in adaptive routing. However, VCs introduce
significant overhead in the switch in terms of area, power, and design complexity, limiting the switch
scalability.
This paper starts by revisiting VC-less routing through link ordering schemes in Full-
mesh networks, which offer implementation simplicity but suffer from performance degradation under
adversarial traffic. Thus, to overcome these challenges, we propose TERA (Topology-Embedded Routing
Algorithm), a novel routing algorithm which employs an embedded physical subnetwork to provide
deadlock-free non-minimal paths without using VCs.
In a Full-mesh network, TERA outperforms link ordering routing algorithms by 80% when dealing
with adversarial traffic, and up to 100% in application kernels. Furthermore, compared to other VC-
based approaches, it reduces buffer requirements by 50%, while maintaining comparable latency and
throughput. Lastly, early results from a 2D-HyperX evaluation show that TERA outperforms state-
of-the-art algorithms that use the same number of VCs, achieving performance improvements of up to
32%.
1
Introduction
High-radix, low-diameter networks are the foundation of many modern interconnect architectures in super-
computers and data centers. Topologies such as Dragonfly [16] and HyperX [1,15] rely on a Full-mesh (FM)
core. In FM networks every switch is directly connected to every other switch, or said equivalently, the
graph of the network is a complete graph, which has diameter 1. In Figure 1 a FM with four switches and
four servers per switch is represented.
Minimal routing (MIN) in a FM—which involves sending packets directly from source to destination—does
not introduce cyclic buffer dependencies and is inherently deadlock-free. Furthermore, MIN delivers excellent
performance under uniform traffic loads, on which packets are evenly distributed across all network paths.
However, under adversarial or bursty traffic loads, MIN can lead to significant load imbalance and
performance degradation. In such scenarios, some links become congested while others remain idle. To
enhance overall throughput, non-minimal paths are employed, detouring packets through intermediate
switches to improve network utilization. In FM networks, non-minimal paths typically consist of two hops:
from the source to an intermediate switch, and from the intermediate switch to the destination. These paths
are usually based on the Valiant Load-Balancing (VLB) scheme [30], which routes each packet through a
randomly selected intermediate switch before forwarding it to its destination.
To improve performance, adaptive routing algorithms are responsible for selecting between minimal and
non-minimal paths based on network conditions, such as buffer occupancy or link utilization. However,
unlike MIN paths, non-minimal routes in a FM introduce cyclic dependencies, potentially leading to packet
deadlocks. Therefore, routing in FM networks must support deadlock-free operation for any mix of MIN and
non-MIN routes. Two common strategies to avoid routing deadlocks in FM networks are:
1
arXiv:2510.14730v1 [cs.DC] 16 Oct 2025
Figure 1: Full-mesh network with 16 servers: 4 switches with 4 servers per switch.
• Virtual Channels (or buffer restriction). This approach involves having multiple virtual channels
(VCs or FIFO buffers) within each switch port, allowing packets to avoid cycles by traversing different
VCs under a certain order. Some proposals include Virtual Ordered Buffered Classes [13,19], or escape
channels [10].
• Link ordering (or path restriction). This method assigns labels to links in the network, ensuring that
packets follow a specific order of the links that avoids deadlocks [12,17].
The first approach is effective and capable of achieving optimal performance. However, it requires
additional buffering resources, which occupy a significant portion of the switch area and increase power
consumption as well as the complexity of the switch design. Adaptive routing algorithms for FM-based
networks, such as UGAL [27] and Omni-WAR [18] exemplify this class of solutions.
The second approach avoids the need for VCs and is simpler to implement. Nevertheless, it can result in
uneven path utilization and performance degradation under adverse traffic conditions. Routing schemes such
as bRINR [17] and Up*/Down* [23] are representative of this method.
In this paper, we first examine the link ordering approach in Section 3, exploring its limitations when
applying to FM networks. We prove that link ordering schemes impose inherent performance bounds,
motivating the need for an alternative strategy. To this end, we propose TERA (Topology-Embedded
Routing Algorithm), a novel mechanism that enables deadlock-free non-minimal routing in a FM without
relying on VCs, while maintaining high performance. TERA is evaluated against state-of-the-art routings
for FM networks, including Omni-WAR [18], a well-established scheme known for its high throughput and
low latency.
The paper is organized as follows. In Section 2, the motivation behind our work is discussed. Section 3
is devoted to exploring link orderings, showing that these schemes have inherent limitations. Section 4
introduces TERA, a VC-less deadlock-free routing algorithm for Full-mesh networks. Using the different
scenarios described in Section 5, TERA is evaluated and compared against other algorithms in Section 6.
Finally, in Section 7, the paper contributions are summarized.
2
Motivation
2.1
Buffering resources
Buffers dominate the switch area and take away an important part of the power budget [7,17,24,31]. Modern
high-performance switches in supercomputer and data center networks require extensive buffering resources,
presenting two challenges, as described next.
2.1.1
Buffer Size
Several factors condition the buffer depth required in a switch. First, the Round-Trip Time (RTT) delay
must be considered. The buffer size should be enough to cover the RTT latency to have a continuous
transmission. It is directly related to the bandwidth of the switch-ports and to the distances covered by the
wires. In addition, sufficient buffer depth is necessary to absorb bursts of traffic [9].
On top of that, common solutions must accommodate enough VCs in each network port to guaranty
deadlock-free routing and, in some cases, to avoid protocol deadlock as well as to support multiple Quality-
of-Service (QoS) levels.
Conventional deadlock-avoidance mechanisms for packet routing—such as those proposed in [10, 13,
19]—typically require a minimum of two virtual channels when applied to FM topologies. This requirement
scales with the number of QoS levels supported by the network. For example, in a system offering 16 QoS
levels and using 2 VCs per level to ensure deadlock-free adaptive routing, each port would require a total of
2
32 buffers. Furthermore, if protocol-level deadlock must also be prevented, the VC count must be doubled,
resulting in 64 VCs per port.
2.1.2
Buffer Utilization Inefficiency
Common deadlock avoidance schemes for adaptive routing algorithms, such as UGAL [27] and Omni-
WAR [18], rely on adding buffers to support minimal and non-minimal paths. These algorithms require
at least 2 VCs per port to be deadlock-free in a FM. The first VC is used for minimal or non-minimal
hops, while the second is used only for non-minimal hops. Consequently, when only minimal paths are
employed, merely half of the available buffers are utilized. In general for any topology, the highest-order
virtual channels remain mostly unused, while the lowest-ordered buffers present a high occupancy. This
results in a significant waste of resources.
Furthermore, prior work [5] has demonstrated that injecting MIN packets on VCs other than the
lowest-ordered ones introduces network instability. Thus, there is no clear way to fully utilize all available
buffers.
2.2
High-radix low-diameter networks
Notable topologies, such as Dragonfly, HyperX, and Dragonfly+ [26] [11], utilize FMs at the group or
dimension level. A Dragonfly is employed in the current first three top-ranked supercomputers: El Capitan,
Frontier, and Aurora. The three of them rely on complete graph intra-group and inter-group connectivity.
Other supercomputers in the Top500 [28] also implement these class of topologies.
The routing mechanisms employed in these networks demand a high number of VCs to support deadlock-
free routing. Typical requirements include 4 VCs in a 2D-HyperX, 6 VCs in a 3D-HyperX, and 4/2 VCs in
Dragonfly for local/global ports. This stems from the fact that routing in a FM is not inherently deadlock-free
when non-minimal paths are used. Thus, avoiding the necessity of VCs for deadlock-free routing in the FM
core, directly translates into lower buffer requirements for the larger topologies built upon it.
Importantly, some network technologies, such as InfiniBand [22], support adaptive routing but do not
permit in-transit VC shifting. As a result, deadlock-free routing schemes that do not rely on VCs are
particularly desirable in these contexts.
3
Link ordering Schemes without VCs
In link ordering schemes, each directed link (arc) is assigned a number or label. Any valid path must follow a
strictly increasing sequence of labels, which prevents deadlock. In a FM, depending on the selected ordering,
the number of usable non-minimal paths can vary, as well as the proportion of such paths between pairs of
switches. As we will see next, there is a trade-off between maximizing the number of non-minimal paths
and ensuring a fair distribution of them between pairs of switches. This inherent property complicates the
applicability of the method and suggests the search of routing mechanisms based on other principles.
From now on, we will formally refer to the Complete graph, the underlying topology of the Full-mesh
network.
Definition 3.1. The Complete Graph over the set of vertices V = {0, 1, . . . , n −1}, denoted as Kn, is the
graph where every pair of distinct vertices is connected. This is, the set of edges is E = {{a, b} | a, b ∈V, a ̸=
b}.
Then, its number of links is m = |E| =
n
2
= n(n−1)
2
. From a source node to a destination node, there is
1 minimal path of length 1 and n −2 paths of length 2. As there are n(n −1) pairs of nodes, there are a
total of n(n −1)(n −2) non-minimal paths in the graph.
Allowing the use of all paths of length 2 for routing without VCs results in deadlock. To avoid it, in [17],
the authors limit these 2-hop paths by using bRINR (balanced Restricted Intermediate-node Non-minimal
Routing), aimed at balancing intermediate nodes while maximizing path diversity. This routing algorithm
attains the maximum number of possible non-minimal paths for any link ordering scheme 2
3n(n−1)(n−2) but
suffers from uneven path distribution. It also ensures that each pair of switches has at least 2 intermediates.
However, some links are overloaded because they are used by many source/destination pairs. This causes
bottlenecks and hotspots in specific switches.
Before proposing a new link ordering scheme, let us show a result that proves that pursuing equalized
utilization of all the links is a limiting factor in the number of possible non-minimal paths. The proof is
included in Appendix A.
3
0
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
Figure 2: K4 with sRINR labelling.
Theorem 3.2. If a routing scheme for a FM is based on an ordering of the arcs and ensures that all links
can be used by the same number of source/destination pairs, then the number of allowed paths of length 2 is
1
2n(n −1)(n −2).
This proves the trade-off between opportunities for adaptiveness and load-balancing.
The bRINR
scheme [17] maximizes the number of possible intermediate nodes, on average. However, as such value is
above the expressed in the previous theorem, this implies that some arcs are used more than others. As we
will see, this imbalance manifests as a performance degradation for some traffic patterns. This is due to
unfairness issues which grow with the network size.
Then, it is natural to look for a different scheme looking for a perfect balance of non-minimal routes
between switches by sacrificing the number of selectable paths. This trade-off is similar to the one between
the Turn Model and DOR routing [8], where the former offers more paths but the latter achieves a fairer
path distribution and higher overall performance. Next, we introduce sRINR (symmetric RINR), a new link
ordering that prioritizes a fair distribution of paths instead of attempting to maximize their number.
Definition 3.3. The sRINR ordering scheme assigns a number to an arc based on the distance between the
switches it connects. For two switches i and j (i ̸= j) connected by an edge, the arc from i to j is assigned
the number D(i, j) ∈[0, n −1] with D(i, j) ≡(j −i) mod n.
Figure 2 represents a K4 labelled in this way. The following claim bounds the number of intermediates.
The proof is included in Appendix A.
Claim 3.4. In the sRINR ordering, the minimum number of allowed intermediates for a source/destination
pair is n−4
2 .
For n ≥8 this number is higher than the minimum number of intermediates provided by bRINR.
Along with this theoretical analysis, Section 6.1 provides an evaluation of both sRINR and bRINR. The
results show that sRINR clearly outperforms bRINR. Nevertheless, its overall performance remains below
state-of-the-art routing algorithms based on VCs, indicating that link ordering schemes still impose strict
limitations.
4
The Topology Embedded Routing Algorithm
In this section we introduce TERA, a deadlock-free routing algorithm conceived to address the limitations
of the link ordering schemes. Note that a complete graph has a very rich connectivity, and TERA takes
advantage of this fact by breaking the topology into two physical parts. The main part should contain most
of the links and be rich on allowed routes, so it can be freely used to route packets. The remaining part must
correspond to a topology that serves in avoiding deadlock without using VCs. Let us define more formally
the two topologies composing TERA.
Definition 4.1 (Main and service topologies). For a FMn, a service topology S is an embedded spanning
topology using a deadlock-free minimal routing algorithm. By main topology M we refer to the topology with
the remaining links. A link is said a service link if it belongs to the service topology.
Note that the service topology must span all the n switches from the FM, as it has to be able to route
from any source to any destination. There are different topologies that provide a deadlock-free set of minimal
paths. This is the case, for example, of trees using up/down routing or meshes and hypercubes using DOR [8].
4
0
2
1
3
Path
0
2
1
3
Hypercube
Figure 3: K4 with a Path, and a Hypercube embedded.
In Figure 3, two FM4 are illustrated: one with a 2-tree (or 1D-mesh) service topology and the other with
a Hypercube (or 2D-mesh in this case). Service links are represented with solid lines, while main links are
depicted with dashed lines. Note that any path between switches through service links for the embedded
1D-Mesh is minimal, the same as the DOR paths for the hypercube.
TERA is described in Algorithm 1. The following notation is used in the pseudocode:
• The set Rmain(x) contains the indexes of the ports of switch x that belong to the main topology.
• The set Rserv(x, y) contains the indexes of the ports in switch x that follow a service path to y.
• The set Rmin(x, y) contains the index of the port in a switch x that connects to the switch y as a single
element.
As it can be seen, TERA always evaluates the option of taking a service and a MIN path. But, if the
packet is at the injection ports of the source switch, it also includes the option of taking any port from the
main topology as a non-minimal hop.
Algorithm 1: TERA
Input: Current switch: current, destination switch: destination
Output: port
ports ←Rserv(current, destination)
if packet is at an injection port then
ports ←ports ∪Rmain(current)
else
ports ←ports ∪Rmin(current, destination)
end
candidates ←∅
for each p in ports do
if p connects to destination then
Insert (p, occupancy[p]) into candidates /* The occupancy is the weight
*/
else
Insert (p, occupancy[p] + q) into candidates /* Penalize non-minimal paths
*/
end
end
return the port in (port, weight) ∈candidates with the minimum weight. Ties are broken
randomly.
In the algorithm, q denotes a penalty applied to non-minimal paths. It will be set in Section 5
TERA is deadlock-free. This can be clearly understood by noting that every packet always has a
valid routing option along a service path, and packets on these paths can always make forward progress.
Consequently, if a packet can’t advance in the main network, sufficient buffer space will eventually free up in
the service path, allowing it to proceed and preventing deadlock.
Furthermore, TERA is also livelock-free, and the maximum number of hops a packet can do is 1 plus the
diameter of the service topology.
4.1
Evaluation of service topologies
The candidates for the service topology have to fulfill four key criteria:
5
Topology
Sym.
Diameter
#Links
Routing
d-Mesh
No
O(n
1
d )
O(dn)
DOR
k-Tree
No
O(logk n)
O(n)
Up/Down
Hypercube
Yes
O(log2 n)
O(n log n)
DOR
d-HyperX
Yes
O(d)
O(dn1+ 1
d )
DOR
Table 1: Properties of different service topologies.
0
16
32
48
64
80
96
112
128
FM size
0.0
0.1
0.2
0.3
0.4
0.5
Accepted load
Estimated performance of different service topologies
Path
4-Tree
Hypercube
3D-HyperX
2D-HyperX
Figure 4: Estimated throughput for different service topologies.
• Deadlock-Free Minimal Routing: The topology must have a deadlock-free minimal routing
algorithm that does not require VCs.
• Edge and Vertex Symmetry: A symmetric topology achieves a balanced performance in the network.
Asymmetries can lead to congestion [2].
• Low Diameter: Long service paths allow for worst-case scenarios, as they can spread congestion
throughout the whole service topology.
• Bounded Number of Links: A high number of links imposes more restrictions on non-minimal
paths, reducing the path diversity of the main topology.
Table 1 compares these properties across four topology families. Mesh topologies have low degree,
which entails a small number of links. This provides the main topology a greater connectivity and higher
non-minimal bandwidth. However, their major drawback is a high diameter, which grows linearly with n.
Another option are k-trees, which reduce the diameter to O(logk n) and employ a low number of links.
However, they are not symmetric: the root becomes a central point of contention, and leaf nodes typically
have only one upward link, creating bottlenecks.
Lastly, the HyperX topology is a notable option, as it is a symmetric low-diameter topology. In particular,
the 2D-HyperX and 3D-HyperX are considered the most suitable candidates. The 2D-HyperX topology has
a diameter of 2, which is the lowest possible for an embedded topology.
Using a service topology with a high degree, such as HyperX, reduces the number of links available for the
main topology, limiting non-minimal path diversity. To better grasp the impact of this, let us estimate the
throughput obtained by TERA with different service topologies under adverse random switch permutation
traffic. This is shown in Figure 4. Specifically, the curves follow the expression
1
1+p−1 , where p is the degree
of the main topology divided by n −1. This is proved in Appendix B. For small FM sizes, the differences
in performance among service topologies can be notable. However, as the FM size increases, the curves
converge. In Section 6.2, the analysis demonstrates that the HyperX topology, with its optimal combination
of low diameter and symmetry, represents an ideal choice.
5
Methodology
To compare the performance of the routing mechanisms studied in this work, we use the CAMINOS [3,4]
network simulator. It is an event-driven simulator that models the switch microarchitecture and operates at
6
flit level. The switch operates with a 2× speedup and a random allocator. It provides buffer space for 10
packets per virtual channel at the input ports and 5 packets per virtual channel at the output ports. Each
packet consists of 16 flits 1
The evaluated routings are MIN, bRINR, sRINR, TERA, Omni-WAR, UGAL and Valiant. TERA will
use different service topologies. This will be indicated as a suffix in the acronym; for example, TERA-HX2,
means TERA using a 2D-HyperX as service topology.
With respect to the TERA pseudocode in Algorithm 1, after an experimental sweep, 54 was determined
as a suitable penalty q for the parameters used in our experiments. This value implies a penalty similar to
the existence of slightly more than 3 packets in the buffer of any non-minimal path.
Only one FIFO or VC is required by MIN, bRINR, sRINR, and TERA, whereas Omni-WAR, UGAL,
and Valiant require 2 VCs to ensure deadlock-freedom, which doubles the buffering space and increases the
design complexity of the switch. MIN and Valiant are only used as baselines.
A FM64 with 4096 servers is extensively simulated. Nevertheless, to demonstrate the applicability of
TERA in larger high-radix low-diameter networks, an initial evaluation of an 8 × 8 2D-HyperX network is
included.
Three modes of synthetic traffic generation are used:
• Fixed generation: Each server generates a total number of 1250 packets to be sent following a specific
traffic pattern. The time to consume all packets is shown. It will serve to evaluate/discard the link
ordering and the TERA routing mechanisms.
• Bernoulli generation: All servers generate traffic continuously during 80K cycles at a given rate of
offered load, following a specific admissible traffic pattern. The considered performance metrics are
explained below.
• Application kernels: A synthetic communication kernel is simulated in the whole FM, and the time
to completion is measured. Processes of the kernel will be assigned to servers using both linear and
random mapping.
For Bernoulli traffic, the metrics presented include average accepted throughput, average message latency,
hop distribution, and Jain index for load generation. The hop distribution will be shown for the maximum
offered load, while the other metrics will be displayed for different injected loads. The first three metrics are
commonly used in the technical literature, while the last one provides a more specific view of the network
fairness. The Jain index [14] takes into account the load generated by all servers in the network. It is
calculated as (
Pn
i=1 xi)
2
n Pn
i=1 x2
i , where xi is the load generated by the server i and n is the total number of servers.
A Jain index of 1.0 indicates perfect equity, with all servers injecting the same load. Lower values suggest
disparity between servers, the lower the worse.
The admissible synthetic traffic patterns employed are:
• Uniform (UN): Each server generates messages destined to a random server of the network. This
pattern could model different scenarios in a data center, where the distribution of traffic is uniform
across the network due to the application (all-to-all communications in machine learning, for example),
or due to a random mapping of tasks.
• Random switch permutation (RSP): It represents any permutation of switches. In the RSP, all the
servers of a switch generate traffic to the servers of a destination switch. The map source to destination
switch is a random permutation of the n switches.
• Fixed random (FR) [6]: Each server selects a random server from the network to send a message,
potentially creating endpoint bottlenecks.
• Switch Cartesian transforms: The servers from switch x send traffic to the servers in switch f(x),
where f is a function that transforms the index of the switch. We call shift to the function f(x) = x+1
and complement to f(x) = −x −1.
The following application kernels are considered:
• All2All: Classical send loop. In iteration i task t sends to t + i [29].
1In https://github.com/alexcano98/TERA-routing-HOTI-2025-reproducibility full detail of our experimental setup,
including the TERA implementation and simulator version is available.
7
bRINR
sRINR
UGAL
Valiant
0
2
4
·105
Cycles to finish
Shift
0
0.5
1
·105
Cycles to finish
Complement
0
1
2
3
·105
Cycles to finish
RSP
Figure 5: Time to finish of different traffic patterns in a FM64 with 4096 servers.
• Stencil 2D: All the processes are arranged in a 2D grid, and each process communicates with the 8
processes in its Moore neighborhood.
• Stencil 3D: All the processes are arranged in a 3D grid, and each process communicates with its 26
neighbours across faces, edges, and corners.
• FFT 3D: A FFT3D with pencil decompositions using a 2D grid of processes and each process manages
a pencil of data. Communications arise from partial transpositions of data and are All2All across each
row or column [20].
• All-reduce: The Rabenseifner algorithm, composed of a scatter-reduce and an all-gather. The
algorithm is optimal in terms of bandwidth and number of messages for power of two processes [21].
6
Performance Evaluation
Several empirical results are shown next. Subsection 6.1 is devoted to evaluating the link ordering mechanisms.
Subsection 6.2 study the differences among possible service topologies. Subsections 6.3 and 6.4 evaluate
TERA with a HyperX service topology in different scenarios. Finally, Subsection 6.5 evaluates the behaviour
of TERA in a two-dimensional topology.
6.1
Link ordering evaluation
In Figure 5 three different synthetic traffic patterns are simulated: shift, complement, and random permutation
under fixed generation. The results show the number of cycles to finish the same amount of load for the
different routings.
As it can be seen, sRINR always provides smaller or same completion times than bRINR, standing out 9
times faster in the shift pattern. In the case of the RSP pattern, sRINR is 3.8 times faster. Complement
traffic pattern reveals as the more challenging situation, in which both sRINR and bRINR take more than
2.3 times to complete than Valiant.
Since the traffic patterns considered are adversarial, Valiant naturally provides the best completion
times, although doubling the resources of ordering schemes. It can be shown that sRINR under shift
traffic patterns achieves best-case throughput of 0.5 flits/cycle/server, which matches Valiant performance.
However, its worst-case performance, under complement traffic, drops to 0.25 flits/cycle/server, halving
Valiant throughput. Thus, neither sRINR nor bRINR can be considered competitive. For comparison
purposes, only sRINR will be considered in later evaluations.
6.2
Service topology selection
Figure 6 shows the performance of TERA with different service topologies. It includes two plots where the
size of the FM is increased. RSP and FR traffic patterns under fixed generation are employed.
For RSP, the performance of the 2-Tree (Path or 1D-Mesh) topology is the highest, and the 2D-HyperX
is the lowest. However, the performance of 2D-HyperX reduces the gap with the path topology as the size of
the FM increases, and the 3D-HyperX follows a similar trend.
In the FR scenario, the path and the 4-Tree have the worst performance due to their asymmetry. Other
admissible scenarios could have been chosen to stress the network, but the FR pattern represents the most
challenging case. The Up-Down service paths condition the network performance. Similar results have been
obtained in d-dimensional meshes.
8
TERA-PATH
TERA-HX2
TERA-HX3
TERA-4-Tree
Valiant
UGAL
8
16
32
64
0
0.25
0.5
·105
FM size
Cycle to finish
RSP
8
16
32
64
0
0.13
0.25
0.38
0.5
·107
FM size
Cycle to finish
FR
Figure 6: Time to consume a burst of packets under Random Switch Permutation and Fixed Random
patterns for different FM sizes.
Omni-WAR
UGAL
sRINR
TERA-HX2
TERA-HX3
MIN/VLB
Uniform
0.5
1
0.2
0.4
0.6
0.8
Offered load
Accepted load
0.2
0.4
0.6
0.8
0
200
400
600
800
1 000
Offered load
Message delay
0.5
1
0.9
0.95
1
Offered load
Jain index
0
2
4
10−4
0.001
0.01
0.1
1
Path length
Rel. frequency
Random Switch Permutation
0.5
1
0.2
0.4
Offered load
Accepted load
0.2
0.4
0
200
400
600
800
1 000
Offered load
Message delay
0.5
1
0.9
0.92
0.94
0.96
0.98
1
Offered load
Jain index
1
2
3
4
10−6
10−5
10−4
0.001
0.01
0.1
1
Path length
Rel. frequency
Figure 7: Traffic with Bernoulli generation at a different offered load in a FM64
In conclusion, despite the trade-offs in non-minimal throughput due to the denser connectivity of HyperX,
its properties—symmetry, low diameter, and balanced link usage—make it the best choice for the service
topology in TERA. 2D-HyperX and 3D-HyperX are selected for the remainder of the experiments. Compared
to link ordering schemes, TERA consistently offers better path diversity, resilience to congestion, and fairer
utilization of network resources.
6.3
Bernoulli traffic evaluation
Figure 7 shows the performance of the different routing mechanisms for the UN and RSP traffic patterns
under Bernoulli generation.
With UN, all routing algorithms use minimal paths for approximately 80–90% of the packets, resulting in
similar performance. Minimal routing serves as the baseline for evaluating algorithms that utilize only 1 VC.
Omni-WAR and UGAL, which use 2 VCs, slightly outperform the others by reducing some head-of-line (HoL)
blocking on non-minimal paths. All mechanisms exhibit fair behavior with a comparable hop distribution at
maximum injection load. The TERA variants show a negligible use of longer paths (3- and 4-hop routes
occur in less than 1% of cases), which has no impact on performance and does not appear under low to
medium traffic loads.
For the RSP pattern, routing performance varies more significantly. The best-performing algorithms are,
in order, Omni-WAR, TERA-HX3, Valiant, TERA-HX2, UGAL, and sRINR. Omni-WAR and TERA-HX3
slightly outperform Valiant because they can adaptively select the best non-minimal path at the source
switch. TERA outperforms sRINR by an 80%, due to the limitations of sRINR. Again, TERA routing shows
extremely low occurrences of 3 and 4-hop paths (less than 0.1%), which are inconsequential to performance.
Lastly, the utilization of the service and main links for TERA-HX routing algorithm was measured. Under
uniform traffic, all links had similar utilization since TERA selects minimal paths without distinguishing
9
Omni-WAR
UGAL
sRINR
TERA-HX2
TERA-HX3
Linear
Random
0
1
2
3
·105
Cycles to finish
All2All
Linear
Random
0
2
4
6
·104
Cycles to finish
Stencil 2D
Linear
Random
0
1
2
·105
Cycles to finish
Stencil 3D
Linear
Random
0
2
4
6
8
·104
Cycles to finish
FFT3D
Linear
Random
0
2
4
6
8
·105
Cycles to finish
Allreduce
Figure 8: Cycles to consume an application kernel in a FM64
Percentile 99.99%
Percentile 99.9%
Percentile 99%
Average
Omni-WAR
UGAL
TERA-HX2
TERA-HX3
1 000
2 000
3 000
4 000
Packet latency
All2All
Omni-WAR
UGAL
TERA-HX2
TERA-HX3
5 000
10 000
15 000
Packet latency
Stencil 3D
Omni-WAR
UGAL
TERA-HX2
TERA-HX3
1 000
2 000
Packet latency
FFT3D
Omni-WAR
UGAL
TERA-HX2
TERA-HX3
2 000
4 000
6 000
Packet latency
Allreduce
Figure 9: Violin plot packet latencies for linear mapping.
between main and service links. Under RSP traffic, service links show about half the utilization of main
links. Nevertheless, since service links constitute less than 10% of the total (192 out of 2016 in TERA-HX3),
their limited usage has a negligible impact on overall link utilization and performance.
6.4
Application results
Figure 8 shows the application completion times across different routing algorithms. Despite not using virtual
channels, TERA achieves competitive performance, closely following Omni-WAR and outperforming UGAL
in most scenarios. This proves the effectiveness of TERA as a low-cost routing mechanism for Full-mesh
networks.
Omni-WAR delivers the best overall performance due to the use of 2 VCs and unrestricted non-minimal
bandwidth. It consistently outperforms all other mechanisms, particularly in the Stencil 2D and 3D workloads,
where its advantage increases to around 10%.
In spite of not using VCs, TERA (HX2 and HX3) trails Omni-WAR by a small margin, within 7% on
average, and outperforms UGAL significantly, with speedups of up to 47% in the Allreduce application. This
improvement stems from TERA’s ability to select an optimal non-minimal path from a large pool of candidates
at the source switch, effectively reducing packet latency and improving load balancing. This capability is
particularly beneficial in heavy communication workloads like Allreduce, where message dependencies can
cause added delays.
In TERA, packets taking 3 or 4 hops constitute less than 1% of the total on average. To assess the impact
of these longer paths on packet latency across all applications, violin plots are presented in Figure 9. As no
significant performance differences were observed between linear and random process mappings, evaluations
focus on the linear mapping. The link ordering algorithm is omitted due to its lack of competitiveness.
TERA-HX2 and TERA-HX3 exhibit the lowest mean and 99% percentile latencies in most cases. This
can be attributed to their reduced buffer space, which would result in lower queuing delays. At the higher
percentiles (99.9% and 99.99%), TERA remains the top performer except in the Stencil 3D workload, where
its latency is comparable to Omni-WAR. UGAL consistently shows the highest latency across all cases,
primarily due to its reliance on a single randomly selected intermediate in Valiant-style routing, limiting its
ability to adapt to network congestion.
TERA offers competitive performance, making it a practical and efficient solution for Full-mesh topologies.
10
Omni-WAR
DimWAR
O1TURN-TERA-HX3
DOR-TERA-HX3
Natural
Random
0
1
2
3
·105
Cycles to finish
All2All
Natural
Random
0
1
2
·105
Cycles to finish
Allreduce
Figure 10: Cycles to finish All2All and Allreduce applications in a 2D-HyperX 8x8 network.
Hence, TERA reveals as a low-cost alternative to existing adaptive algorithms like UGAL and Omni-WAR.
6.5
Other topologies
Any topology based on FM networks, such as HyperX, Dragonfly and Dragonfly+, can take advantage of
TERA. A simulation of an 8 × 8 2D-HyperX network with 512 servers is shown in Figure 10.
Dim-WAR, a routing algorithm specially conceived for the 2D-HyperX that uses 2 VCs [18], is also
included in the comparison. Additionally, we have developed a O1TURN [25] version of TERA that uses 2
VCs. Thus, Omni-WAR uses 4 VCs, Dim-WAR and O1TURN-TERA-HX3 use 2 VCs, and DOR-TERA-HX3
just 1VC.
In DOR-TERA-HX3, the TERA-HX3 routing algorithm is applied independently within each of the two
FM8 subnetworks traversed by a packet—one for each dimension—with dimensions visited in XY order. In
the case of O1TURN-TERA-HX3, at the source switch, it is decided if the packet will progress either in XY
or YX order, and a VC is needed for each dimension order.
It is noticeable that DOR-TERA-HX3 reveals competitive in most cases being the one with minimal
resources. Additionally, the performance of O1TURN-TERA-HX3 is near Omni-WAR with half the resources
(from 4 VCs to 2 VCs), and up to 32% better than Dim-WAR with the same buffering.
7
Conclusions
This paper introduces TERA, a Topology-Embedded Routing Algorithm for Full-mesh networks that enables
deadlock-free non-minimal routing without relying on virtual channels. An initial evaluation of link ordering
schemes exposed their performance limitations, highlighting the need for a more effective solution.
TERA addresses these challenges by embedding a physical service topology within the Full-mesh structure.
An exhaustive analysis of the service topologies revealed that the HyperX family, particularly the 2D-HyperX
and 3D-HyperX, are the most suitable candidate for TERA.
In a Full-mesh network, TERA outperforms link ordering routing algorithms by an 80% under adverse
traffic, and up to 100% in application kernels. Furthermore, compared to other VC-based approaches, it
reduces buffer requirements by 50%, while maintaining comparable latency and throughput. Lastly, early
results from a 2D-HyperX evaluation show that TERA outperforms state-of-the-art algorithms that use the
same number of VCs, achieving performance improvements of up to 32%.
This work concludes that TERA can be effectively applied to FM networks, and that it is a promising
solution for high-radix, low-diameter topologies.
11
A
Link ordering schemes
Proof of Theorem 3.2. Let us define the function Φ : V 3 →{0, 1} that indicates which paths of length 2
are allowed. For distinct switches s, m, d ∈V , set Φ(s, m, d) = 1 if the route s, m, d is allowed and set
Φ(s, m, d) = 0 otherwise. That is, Φ describes which switches m can be used to route from source s to
destination d. Additionally, just to avoid cumbersome notation, set Φ(s, m, d) = 0 when any pair of s, m, d
coincide. The hypothesis of the same utilization can now be written as follows. There is a constant S such
that for any arc (a, b), holds P
x∈V Φ(a, b, x) + Φ(x, a, b) = S.
Let (ma, mb) be the arc at first position by the ordering of the scheme. This arc is always allowed as the
start of a 2-length path, but never as its ending. Thus, P
x∈V Φ(ma, mb, x) = n−2, and P
x∈V Φ(x, ma, mb) =
0. Therefore, we obtain the value of the constant S = n −2.
This implies that
n(n −1)(n −2) =
X
(a,b)∈A
X
x∈V
Φ(a, b, x) + Φ(x, a, b).
Note that the two terms have the same sum, that is,
X
(a,b)∈A
X
x∈V
Φ(a, b, x) =
X
(a,b)∈A
X
x∈V
Φ(x, a, b).
Therefore, n(n −1)(n −2) = 2 P
(a,b)∈A
P
x∈V Φ(a, b, x).
Finally, the total number of allowed paths is P
s,m,d∈V Φ(s, m, d) = P
a,b∈V
P
x∈V Φ(a, b, x) = 1
2n(n −
1)(n −2).
Proof of Claim 3.4. The number of intermediate switches between a switch source a, and a switch destination
b is going to be calculated. It should be taken into account that the maximum number of intermediates is
n −2, as a and b do not count as intermediates.
Let us define the function Gab(i) := D(i, b) −D(a, i). A value Gab(i) > 0 means that the non-minimal
path a 7→i 7→b is allowed. Otherwise, the path is forbidden.
The function G satisfies
−Gab(a + b
2
+ x) = Gab(a + b
2
−x) 0 ≤x < n.
(1)
This implies that every valid intermediate in the set of switches is paired with an invalid one. Thus, around
half of the switches are valid intermediates, and the others are invalid intermediates. We need to discount
those cases were both sides of Equation 1 are 0, as both represent invalid intermediates. That is, the number
of solutions to Gab(y) = 0 in the integers modulo n. Taking x = 0 in Equation 1 gives Gab( a+b
2 ) = 0. Hence,
y = a+b
2
is a solution when it is integer. More precisely, let us count the solutions in y for 2y ≡a + b mod n.
When n is odd, 2 has inverse, and we have exactly one solution. When n is even:
• if a and b have different parity, there are no solutions, and there are n−2
2
intermediates.
• if a and b have the same parity, there are two solutions, y = a+b
2
and y = a+b+n
2
, and there are n−4
2
intermediates.
B
TERA estimated performance
An estimation of the throughput achieved by TERA under various service topologies subjected to Random
Switch Permutation (RSP) traffic can be expressed as
1
1+p−1 , where p denotes the degree of the main topology
divided by n −1. This expression is illustrated for several service topologies in Figure 4. The following
paragraph outlines the derivation of this expression.
As the RSP is an adverse traffic pattern, using MIN paths (a single direct link between a pair of switches)
is insufficient for achieving good performance. Furthermore, the direct link can always be used as a MIN
path, regardless of whether it is main or service. The 2-hop deroutes must employ a main link in the first hop
and any in its second hop. Longer routes can be disregarded; they are extremely rare, as it can be empirically
seen. In consequence, under a reasonable balance of routes, the main links will saturate before the service
ones. Let p be the ratio of main links to all links, or equivalently, the probability that a randomly chosen link
12
belongs to the main topology. Thus, d = (n −1)p is the average degree of the main topology. Let γ be the
average switch injection rate, measured in multiples of full-rate links. Assuming traffic without same-switch
messages, we have n −1 links for all traffic, implying 0 ≤γ ≤n −1. To estimate γ, first decompose it into a
rate γ1 for packets employing 1 hop and a rate γ2 for packets employing 2 hops. Since most packets do not
use them, we can ignore longer routes for an estimation. Then, we have that γ = γ1 + γ2. Since the traffic
is a switch-permutation then 0 ≤γ1 ≤1. Packets that traverse a single hop will have the probability p of
adding traffic into the main topology. For packets traversing two hops, the first one is forced and the second
hop only with probability p. Thus, the dn/2 main links, times the two directions, must be able to hold a
total of n(pγ1 + (1 + p)γ2). This calculation can be shown in the following formula:
n(pγ1 + (1 + p)γ2) ≤p(n −1)n.
(2)
Note that the equality would be achieved if the load were perfectly distributed. Isolating the major contributor
γ2 becomes
γ2 ≤p(n −1) −pγ1
1 + p
= (n −1) −γ1
1 + p−1
.
(3)
Thus, giving a total load per switch of
γ ≤(n −1) + γ1p−1
1 + p−1
=
n −1
1 + p−1 +
γ1
1 + p ≤
n −1
1 + p−1 + 1.
(4)
Equivalently, assuming n servers per switch, the average load for each server is γ/n ≤
1
1+p−1 + O(1/n).
Acknowledgements
This work has been supported by the Spanish Ministry of Science and Innovation under contracts PID2019-
105660RB-C22, TED2021-131176B-I00, and PID2022-136454NB-C21. C. Camarero is supported by the
Spanish Ministry of Science and Innovation, Ramón y Cajal contract RYC2021-033959-I. R. Beivide
is supported by The Barcelona Supercomputing Center (BSC) under contract CONSER02023011NG.
Simulations were performed in the Altamira supercomputer, a node of the Spanish Supercomputing Network
(RES).
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Deadlock-free routing for Full-mesh networks without using Virtual Channels postprint version A. Cano, C. Camarero, C. Martínez, and R. Beivide, "Deadlock free routing for full-mesh networks without using virtual channels," in HOTI25: 32nd IEEE Hot Interconnects symposium. IEEE, 2025. https://doi.org/10.1109/HOTI66940.2025.00020 Alejandro Cano Cristóbal Camarero Carmen Martínez Ramón Beivide {alejandro.cano, cristobal.camarero, carmen.martinez, Abstract High-radix, low-diameter networks like HyperX and Dragonfly use a Full-mesh core, and rely on multiple virtual channels (VCs) to avoid packet deadlocks in adaptive routing. However, VCs introduce significant overhead in the switch in terms of area, power, and design complexity, limiting the switch scalability. This paper starts by revisiting VC-less routing through link ordering schemes in Fullmesh networks, which offer implementation simplicity but suffer from performance degradation under adversarial traffic. Thus, to overcome these challenges, we propose TERA (Topology-Embedded Routing Algorithm), a novel routing algorithm which employs an embedded physical subnetwork to provide deadlock-free non-minimal paths without using VCs. In a Full-mesh network, TERA outperforms link ordering routing algorithms by 80% when dealing with adversarial traffic, and up to 100% in application kernels. Furthermore, compared to other VCbased approaches, it reduces buffer requirements by 50%, while maintaining comparable latency and throughput. Lastly, early results from a 2D-HyperX evaluation show that TERA outperforms stateof-the-art algorithms that use the same number of VCs, achieving performance improvements of up to 32%. 1 Introduction High-radix, low-diameter networks are the foundation of many modern interconnect architectures in supercomputers and data centers. Topologies such as Dragonfly [16] and HyperX [1,15] rely on a Full-mesh (FM) core. In FM networks every switch is directly connected to every other switch, or said equivalently, the graph of the network is a complete graph, which has diameter 1. In Figure 1 a FM with four switches and four servers per switch is represented. Minimal routing (MIN) in a FM-which involves sending packets directly from source to destination-does not introduce cyclic buffer dependencies and is inherently deadlock-free. Furthermore, MIN delivers excellent performance under uniform traffic loads, on which packets are evenly distributed across all network paths. However, under adversarial or bursty traffic loads, MIN can lead to significant load imbalance and performance degradation. In such scenarios, some links become congested while others remain idle. To enhance overall throughput, non-minimal paths are employed, detouring packets through intermediate switches to improve network utilization. In FM networks, non-minimal paths typically consist of two hops: from the source to an intermediate switch, and from the intermediate switch to the destination. These paths are usually based on the Valiant Load-Balancing (VLB) scheme [30], which routes each packet through a randomly selected intermediate switch before forwarding it to its destination. To improve performance, adaptive routing algorithms are responsible for selecting between minimal and non-minimal paths based on network conditions, such as buffer occupancy or link utilization. However, unlike MIN paths, non-minimal routes in a FM introduce cyclic dependencies, potentially leading to packet deadlocks. Therefore, routing in FM networks must support deadlock-free operation for any mix of MIN and non-MIN routes. Two common strategies to avoid routing deadlocks in FM networks are: 1 16 Oct 2025 Figure 1: Full-mesh network with 16 servers: 4 switches with 4 servers per switch. • Virtual Channels (or buffer restriction). This approach involves having multiple virtual channels (VCs or FIFO buffers) within each switch port, allowing packets to avoid cycles by traversing different VCs under a certain order. Some proposals include Virtual Ordered Buffered Classes [13,19], or escape channels [10]. • Link ordering (or path restriction). This method assigns labels to links in the network, ensuring that packets follow a specific order of the links that avoids deadlocks [12,17]. The first approach is effective and capable of achieving optimal performance. However, it requires additional buffering resources, which occupy a significant portion of the switch area and increase power consumption as well as the complexity of the switch design. Adaptive routing algorithms for FM-based networks, such as UGAL [27] and Omni-WAR [18] exemplify this class of solutions. The second approach avoids the need for VCs and is simpler to implement. Nevertheless, it can result in uneven path utilization and performance degradation under adverse traffic conditions. Routing schemes such as bRINR [17] and Up*/Down* [23] are representative of this method. In this paper, we first examine the link ordering approach in Section 3, exploring its limitations when applying to FM networks. We prove that link ordering schemes impose inherent performance bounds, motivating the need for an alternative strategy. To this end, we propose TERA (Topology-Embedded Routing Algorithm), a novel mechanism that enables deadlock-free non-minimal routing in a FM without relying on VCs, while maintaining high performance. TERA is evaluated against state-of-the-art routings for FM networks, including Omni-WAR [18], a well-established scheme known for its high throughput and low latency. The paper is organized as follows. In Section 2, the motivation behind our work is discussed. Section 3 is devoted to exploring link orderings, showing that these schemes have inherent limitations. Section 4 introduces TERA, a VC-less deadlock-free routing algorithm for Full-mesh networks. Using the different scenarios described in Section 5, TERA is evaluated and compared against other algorithms in Section 6. Finally, in Section 7, the paper contributions are summarized. 2 Motivation 2.1 Buffering resources Buffers dominate the switch area and take away an important part of the power budget [7,17,24,31]. Modern high-performance switches in supercomputer and data center networks require extensive buffering resources, presenting two challenges, as described next. 2.1.1 Buffer Size Several factors condition the buffer depth required in a switch. First, the Round-Trip Time (RTT) delay must be considered. The buffer size should be enough to cover the RTT latency to have a continuous transmission. It is directly related to the bandwidth of the switch-ports and to the distances covered by the wires. In addition, sufficient buffer depth is necessary to absorb bursts of traffic [9]. On top of that, common solutions must accommodate enough VCs in each network port to guaranty deadlock-free routing and, in some cases, to avoid protocol deadlock as well as to support multiple Qualityof-Service (QoS) levels. Conventional deadlock-avoidance mechanisms for packet routing-such as those proposed in [10, 13, 19]-typically require a minimum of two virtual channels when applied to FM topologies. This requirement scales with the number of QoS levels supported by the network. For example, in a system offering 16 QoS levels and using 2 VCs per level to ensure deadlock-free adaptive routing, each port would require a total of 2 32 buffers. Furthermore, if protocol-level deadlock must also be prevented, the VC count must be doubled, resulting in 64 VCs per port. 2.1.2 Buffer Utilization Inefficiency Common deadlock avoidance schemes for adaptive routing algorithms, such as UGAL [27] and OmniWAR [18], rely on adding buffers to support minimal and non-minimal paths. These algorithms require at least 2 VCs per port to be deadlock-free in a FM. The first VC is used for minimal or non-minimal hops, while the second is used only for non-minimal hops. Consequently, when only minimal paths are employed, merely half of the available buffers are utilized. In general for any topology, the highest-order virtual channels remain mostly unused, while the lowest-ordered buffers present a high occupancy. This results in a significant waste of resources. Furthermore, prior work [5] has demonstrated that injecting MIN packets on VCs other than the lowest-ordered ones introduces network instability. Thus, there is no clear way to fully utilize all available buffers. 2.2 High-radix low-diameter networks Notable topologies, such as Dragonfly, HyperX, and Dragonfly+ [26] [11], utilize FMs at the group or dimension level. A Dragonfly is employed in the current first three top-ranked supercomputers: El Capitan, Frontier, and Aurora. The three of them rely on complete graph intra-group and inter-group connectivity. Other supercomputers in the Top500 [28] also implement these class of topologies. The routing mechanisms employed in these networks demand a high number of VCs to support deadlockfree routing. Typical requirements include 4 VCs in a 2D-HyperX, 6 VCs in a 3D-HyperX, and 4/2 VCs in Dragonfly for local/global ports. This stems from the fact that routing in a FM is not inherently deadlock-free when non-minimal paths are used. Thus, avoiding the necessity of VCs for deadlock-free routing in the FM core, directly translates into lower buffer requirements for the larger topologies built upon it. Importantly, some network technologies, such as InfiniBand [22], support adaptive routing but do not permit in-transit VC shifting. As a result, deadlock-free routing schemes that do not rely on VCs are particularly desirable in these contexts. 3 Link ordering Schemes without VCs In link ordering schemes, each directed link (arc) is assigned a number or label. Any valid path must follow a strictly increasing sequence of labels, which prevents deadlock. In a FM, depending on the selected ordering, the number of usable non-minimal paths can vary, as well as the proportion of such paths between pairs of switches. As we will see next, there is a trade-off between maximizing the number of non-minimal paths and ensuring a fair distribution of them between pairs of switches. This inherent property complicates the applicability of the method and suggests the search of routing mechanisms based on other principles. From now on, we will formally refer to the Complete graph, the underlying topology of the Full-mesh network. Definition 3.1. The Complete Graph over the set of vertices V = {0, 1, . . . , n -1}, denoted as Kn, is the graph where every pair of distinct vertices is connected. This is, the set of edges is E = {{a, b} | a, b ∈V, a ̸= b}. Then, its number of links is m = |E| = n 2 = n(n-1) 2 . From a source node to a destination node, there is 1 minimal path of length 1 and n -2 paths of length 2. As there are n(n -1) pairs of nodes, there are a total of n(n -1)(n -2) non-minimal paths in the graph. Allowing the use of all paths of length 2 for routing without VCs results in deadlock. To avoid it, in [17], the authors limit these 2-hop paths by using bRINR (balanced Restricted Intermediate-node Non-minimal Routing), aimed at balancing intermediate nodes while maximizing path diversity. This routing algorithm attains the maximum number of possible non-minimal paths for any link ordering scheme 2 3n(n-1)(n-2) but suffers from uneven path distribution. It also ensures that each pair of switches has at least 2 intermediates. However, some links are overloaded because they are used by many source/destination pairs. This causes bottlenecks and hotspots in specific switches. Before proposing a new link ordering scheme, let us show a result that proves that pursuing equalized utilization of all the links is a limiting factor in the number of possible non-minimal paths. The proof is included in Appendix A. 3 0 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Figure 2: K4 with sRINR labelling. Theorem 3.2. If a routing scheme for a FM is based on an ordering of the arcs and ensures that all links can be used by the same number of source/destination pairs, then the number of allowed paths of length 2 is 1 2n(n -1)(n -2). This proves the trade-off between opportunities for adaptiveness and load-balancing. The bRINR scheme [17] maximizes the number of possible intermediate nodes, on average. However, as such value is above the expressed in the previous theorem, this implies that some arcs are used more than others. As we will see, this imbalance manifests as a performance degradation for some traffic patterns. This is due to unfairness issues which grow with the network size. Then, it is natural to look for a different scheme looking for a perfect balance of non-minimal routes between switches by sacrificing the number of selectable paths. This trade-off is similar to the one between the Turn Model and DOR routing [8], where the former offers more paths but the latter achieves a fairer path distribution and higher overall performance. Next, we introduce sRINR (symmetric RINR), a new link ordering that prioritizes a fair distribution of paths instead of attempting to maximize their number. Definition 3.3. The sRINR ordering scheme assigns a number to an arc based on the distance between the switches it connects. For two switches i and j (i ̸= j) connected by an edge, the arc from i to j is assigned the number D(i, j) ∈[0, n -1] with D(i, j) ≡(j -i) mod n. Figure 2 represents a K4 labelled in this way. The following claim bounds the number of intermediates. The proof is included in Appendix A. Claim 3.4. In the sRINR ordering, the minimum number of allowed intermediates for a source/destination pair is n-4 2 . For n ≥8 this number is higher than the minimum number of intermediates provided by bRINR. Along with this theoretical analysis, Section 6.1 provides an evaluation of both sRINR and bRINR. The results show that sRINR clearly outperforms bRINR. Nevertheless, its overall performance remains below state-of-the-art routing algorithms based on VCs, indicating that link ordering schemes still impose strict limitations. 4 The Topology Embedded Routing Algorithm In this section we introduce TERA, a deadlock-free routing algorithm conceived to address the limitations of the link ordering schemes. Note that a complete graph has a very rich connectivity, and TERA takes advantage of this fact by breaking the topology into two physical parts. The main part should contain most of the links and be rich on allowed routes, so it can be freely used to route packets. The remaining part must correspond to a topology that serves in avoiding deadlock without using VCs. Let us define more formally the two topologies composing TERA. Definition 4.1 (Main and service topologies). For a FMn, a service topology S is an embedded spanning topology using a deadlock-free minimal routing algorithm. By main topology M we refer to the topology with the remaining links. A link is said a service link if it belongs to the service topology. Note that the service topology must span all the n switches from the FM, as it has to be able to route from any source to any destination. There are different topologies that provide a deadlock-free set of minimal paths. This is the case, for example, of trees using up/down routing or meshes and hypercubes using DOR [8]. 4 0 2 1 3 Path 0 2 1 3 Hypercube Figure 3: K4 with a Path, and a Hypercube embedded. In Figure 3, two FM4 are illustrated: one with a 2-tree (or 1D-mesh) service topology and the other with a Hypercube (or 2D-mesh in this case). Service links are represented with solid lines, while main links are depicted with dashed lines. Note that any path between switches through service links for the embedded 1D-Mesh is minimal, the same as the DOR paths for the hypercube. TERA is described in Algorithm 1. The following notation is used in the pseudocode: • The set Rmain(x) contains the indexes of the ports of switch x that belong to the main topology. • The set Rserv(x, y) contains the indexes of the ports in switch x that follow a service path to y. • The set Rmin(x, y) contains the index of the port in a switch x that connects to the switch y as a single element. As it can be seen, TERA always evaluates the option of taking a service and a MIN path. But, if the packet is at the injection ports of the source switch, it also includes the option of taking any port from the main topology as a non-minimal hop. Algorithm 1: TERA Input: Current switch: current, destination switch: destination Output: port ports ←Rserv(current, destination) if packet is at an injection port then ports ←ports ∪Rmain(current) else ports ←ports ∪Rmin(current, destination) end candidates ←∅ for each p in ports do if p connects to destination then Insert (p, occupancy[p]) into candidates /* The occupancy is the weight */ else Insert (p, occupancy[p] + q) into candidates /* Penalize non-minimal paths */ end end return the port in (port, weight) ∈candidates with the minimum weight. Ties are broken randomly. In the algorithm, q denotes a penalty applied to non-minimal paths. It will be set in Section 5 TERA is deadlock-free. This can be clearly understood by noting that every packet always has a valid routing option along a service path, and packets on these paths can always make forward progress. Consequently, if a packet can't advance in the main network, sufficient buffer space will eventually free up in the service path, allowing it to proceed and preventing deadlock. Furthermore, TERA is also livelock-free, and the maximum number of hops a packet can do is 1 plus the diameter of the service topology. 4.1 Evaluation of service topologies The candidates for the service topology have to fulfill four key criteria: 5 Topology Sym. Diameter #Links Routing d-Mesh No O(n 1 d ) O(dn) DOR k-Tree No O(logk n) O(n) Up/Down Hypercube Yes O(log2 n) O(n log n) DOR d-HyperX Yes O(d) O(dn1+ 1 d ) DOR Table 1: Properties of different service topologies. 0 16 32 48 64 80 96 112 128 FM size 0.0 0.1 0.2 0.3 0.4 0.5 Accepted load Estimated performance of different service topologies Path 4-Tree Hypercube 3D-HyperX 2D-HyperX Figure 4: Estimated throughput for different service topologies. • Deadlock-Free Minimal Routing: The topology must have a deadlock-free minimal routing algorithm that does not require VCs. • Edge and Vertex Symmetry: A symmetric topology achieves a balanced performance in the network. Asymmetries can lead to congestion [2]. • Low Diameter: Long service paths allow for worst-case scenarios, as they can spread congestion throughout the whole service topology. • Bounded Number of Links: A high number of links imposes more restrictions on non-minimal paths, reducing the path diversity of the main topology. Table 1 compares these properties across four topology families. Mesh topologies have low degree, which entails a small number of links. This provides the main topology a greater connectivity and higher non-minimal bandwidth. However, their major drawback is a high diameter, which grows linearly with n. Another option are k-trees, which reduce the diameter to O(logk n) and employ a low number of links. However, they are not symmetric: the root becomes a central point of contention, and leaf nodes typically have only one upward link, creating bottlenecks. Lastly, the HyperX topology is a notable option, as it is a symmetric low-diameter topology. In particular, the 2D-HyperX and 3D-HyperX are considered the most suitable candidates. The 2D-HyperX topology has a diameter of 2, which is the lowest possible for an embedded topology. Using a service topology with a high degree, such as HyperX, reduces the number of links available for the main topology, limiting non-minimal path diversity. To better grasp the impact of this, let us estimate the throughput obtained by TERA with different service topologies under adverse random switch permutation traffic. This is shown in Figure 4. Specifically, the curves follow the expression 1 1+p-1 , where p is the degree of the main topology divided by n -1. This is proved in Appendix B. For small FM sizes, the differences in performance among service topologies can be notable. However, as the FM size increases, the curves converge. In Section 6.2, the analysis demonstrates that the HyperX topology, with its optimal combination of low diameter and symmetry, represents an ideal choice. 5 Methodology To compare the performance of the routing mechanisms studied in this work, we use the CAMINOS [3,4] network simulator. It is an event-driven simulator that models the switch microarchitecture and operates at 6 flit level. The switch operates with a 2× speedup and a random allocator. It provides buffer space for 10 packets per virtual channel at the input ports and 5 packets per virtual channel at the output ports. Each packet consists of 16 flits 1 The evaluated routings are MIN, bRINR, sRINR, TERA, Omni-WAR, UGAL and Valiant. TERA will use different service topologies. This will be indicated as a suffix in the acronym; for example, TERA-HX2, means TERA using a 2D-HyperX as service topology. With respect to the TERA pseudocode in Algorithm 1, after an experimental sweep, 54 was determined as a suitable penalty q for the parameters used in our experiments. This value implies a penalty similar to the existence of slightly more than 3 packets in the buffer of any non-minimal path. Only one FIFO or VC is required by MIN, bRINR, sRINR, and TERA, whereas Omni-WAR, UGAL, and Valiant require 2 VCs to ensure deadlock-freedom, which doubles the buffering space and increases the design complexity of the switch. MIN and Valiant are only used as baselines. A FM64 with 4096 servers is extensively simulated. Nevertheless, to demonstrate the applicability of TERA in larger high-radix low-diameter networks, an initial evaluation of an 8 × 8 2D-HyperX network is included. Three modes of synthetic traffic generation are used: • Fixed generation: Each server generates a total number of 1250 packets to be sent following a specific traffic pattern. The time to consume all packets is shown. It will serve to evaluate/discard the link ordering and the TERA routing mechanisms. • Bernoulli generation: All servers generate traffic continuously during 80K cycles at a given rate of offered load, following a specific admissible traffic pattern. The considered performance metrics are explained below. • Application kernels: A synthetic communication kernel is simulated in the whole FM, and the time to completion is measured. Processes of the kernel will be assigned to servers using both linear and random mapping. For Bernoulli traffic, the metrics presented include average accepted throughput, average message latency, hop distribution, and Jain index for load generation. The hop distribution will be shown for the maximum offered load, while the other metrics will be displayed for different injected loads. The first three metrics are commonly used in the technical literature, while the last one provides a more specific view of the network fairness. The Jain index [14] takes into account the load generated by all servers in the network. It is calculated as ( Pn i=1 xi) 2 n Pn i=1 x2 i , where xi is the load generated by the server i and n is the total number of servers. A Jain index of 1.0 indicates perfect equity, with all servers injecting the same load. Lower values suggest disparity between servers, the lower the worse. The admissible synthetic traffic patterns employed are: • Uniform (UN): Each server generates messages destined to a random server of the network. This pattern could model different scenarios in a data center, where the distribution of traffic is uniform across the network due to the application (all-to-all communications in machine learning, for example), or due to a random mapping of tasks. • Random switch permutation (RSP): It represents any permutation of switches. In the RSP, all the servers of a switch generate traffic to the servers of a destination switch. The map source to destination switch is a random permutation of the n switches. • Fixed random (FR) [6]: Each server selects a random server from the network to send a message, potentially creating endpoint bottlenecks. • Switch Cartesian transforms: The servers from switch x send traffic to the servers in switch f(x), where f is a function that transforms the index of the switch. We call shift to the function f(x) = x+1 and complement to f(x) = -x -1. The following application kernels are considered: • All2All: Classical send loop. In iteration i task t sends to t + i [29]. 1In https://github.com/alexcano98/TERA-routing-HOTI-2025-reproducibility full detail of our experimental setup, including the TERA implementation and simulator version is available. 7 bRINR sRINR UGAL Valiant 0 2 4 ·105 Cycles to finish Shift 0 0.5 1 ·105 Cycles to finish Complement 0 1 2 3 ·105 Cycles to finish RSP Figure 5: Time to finish of different traffic patterns in a FM64 with 4096 servers. • Stencil 2D: All the processes are arranged in a 2D grid, and each process communicates with the 8 processes in its Moore neighborhood. • Stencil 3D: All the processes are arranged in a 3D grid, and each process communicates with its 26 neighbours across faces, edges, and corners. • FFT 3D: A FFT3D with pencil decompositions using a 2D grid of processes and each process manages a pencil of data. Communications arise from partial transpositions of data and are All2All across each row or column [20]. • All-reduce: The Rabenseifner algorithm, composed of a scatter-reduce and an all-gather. The algorithm is optimal in terms of bandwidth and number of messages for power of two processes [21]. 6 Performance Evaluation Several empirical results are shown next. Subsection 6.1 is devoted to evaluating the link ordering mechanisms. Subsection 6.2 study the differences among possible service topologies. Subsections 6.3 and 6.4 evaluate TERA with a HyperX service topology in different scenarios. Finally, Subsection 6.5 evaluates the behaviour of TERA in a two-dimensional topology. 6.1 Link ordering evaluation In Figure 5 three different synthetic traffic patterns are simulated: shift, complement, and random permutation under fixed generation. The results show the number of cycles to finish the same amount of load for the different routings. As it can be seen, sRINR always provides smaller or same completion times than bRINR, standing out 9 times faster in the shift pattern. In the case of the RSP pattern, sRINR is 3.8 times faster. Complement traffic pattern reveals as the more challenging situation, in which both sRINR and bRINR take more than 2.3 times to complete than Valiant. Since the traffic patterns considered are adversarial, Valiant naturally provides the best completion times, although doubling the resources of ordering schemes. It can be shown that sRINR under shift traffic patterns achieves best-case throughput of 0.5 flits/cycle/server, which matches Valiant performance. However, its worst-case performance, under complement traffic, drops to 0.25 flits/cycle/server, halving Valiant throughput. Thus, neither sRINR nor bRINR can be considered competitive. For comparison purposes, only sRINR will be considered in later evaluations. 6.2 Service topology selection Figure 6 shows the performance of TERA with different service topologies. It includes two plots where the size of the FM is increased. RSP and FR traffic patterns under fixed generation are employed. For RSP, the performance of the 2-Tree (Path or 1D-Mesh) topology is the highest, and the 2D-HyperX is the lowest. However, the performance of 2D-HyperX reduces the gap with the path topology as the size of the FM increases, and the 3D-HyperX follows a similar trend. In the FR scenario, the path and the 4-Tree have the worst performance due to their asymmetry. Other admissible scenarios could have been chosen to stress the network, but the FR pattern represents the most challenging case. The Up-Down service paths condition the network performance. Similar results have been obtained in d-dimensional meshes. 8 TERA-PATH TERA-HX2 TERA-HX3 TERA-4-Tree Valiant UGAL 8 16 32 64 0 0.25 0.5 ·105 FM size Cycle to finish RSP 8 16 32 64 0 0.13 0.25 0.38 0.5 ·107 FM size Cycle to finish FR Figure 6: Time to consume a burst of packets under Random Switch Permutation and Fixed Random patterns for different FM sizes. Omni-WAR UGAL sRINR TERA-HX2 TERA-HX3 MIN/VLB Uniform 0.5 1 0.2 0.4 0.6 0.8 Offered load Accepted load 0.2 0.4 0.6 0.8 0 200 400 600 800 1 000 Offered load Message delay 0.5 1 0.9 0.95 1 Offered load Jain index 0 2 4 10-4 0.001 0.01 0.1 1 Path length Rel. frequency Random Switch Permutation 0.5 1 0.2 0.4 Offered load Accepted load 0.2 0.4 0 200 400 600 800 1 000 Offered load Message delay 0.5 1 0.9 0.92 0.94 0.96 0.98 1 Offered load Jain index 1 2 3 4 10-6 10-5 10-4 0.001 0.01 0.1 1 Path length Rel. frequency Figure 7: Traffic with Bernoulli generation at a different offered load in a FM64 In conclusion, despite the trade-offs in non-minimal throughput due to the denser connectivity of HyperX, its properties-symmetry, low diameter, and balanced link usage-make it the best choice for the service topology in TERA. 2D-HyperX and 3D-HyperX are selected for the remainder of the experiments. Compared to link ordering schemes, TERA consistently offers better path diversity, resilience to congestion, and fairer utilization of network resources. 6.3 Bernoulli traffic evaluation Figure 7 shows the performance of the different routing mechanisms for the UN and RSP traffic patterns under Bernoulli generation. With UN, all routing algorithms use minimal paths for approximately 80-90% of the packets, resulting in similar performance. Minimal routing serves as the baseline for evaluating algorithms that utilize only 1 VC. Omni-WAR and UGAL, which use 2 VCs, slightly outperform the others by reducing some head-of-line (HoL) blocking on non-minimal paths. All mechanisms exhibit fair behavior with a comparable hop distribution at maximum injection load. The TERA variants show a negligible use of longer paths (3- and 4-hop routes occur in less than 1% of cases), which has no impact on performance and does not appear under low to medium traffic loads. For the RSP pattern, routing performance varies more significantly. The best-performing algorithms are, in order, Omni-WAR, TERA-HX3, Valiant, TERA-HX2, UGAL, and sRINR. Omni-WAR and TERA-HX3 slightly outperform Valiant because they can adaptively select the best non-minimal path at the source switch. TERA outperforms sRINR by an 80%, due to the limitations of sRINR. Again, TERA routing shows extremely low occurrences of 3 and 4-hop paths (less than 0.1%), which are inconsequential to performance. Lastly, the utilization of the service and main links for TERA-HX routing algorithm was measured. Under uniform traffic, all links had similar utilization since TERA selects minimal paths without distinguishing 9 Omni-WAR UGAL sRINR TERA-HX2 TERA-HX3 Linear Random 0 1 2 3 ·105 Cycles to finish All2All Linear Random 0 2 4 6 ·104 Cycles to finish Stencil 2D Linear Random 0 1 2 ·105 Cycles to finish Stencil 3D Linear Random 0 2 4 6 8 ·104 Cycles to finish FFT3D Linear Random 0 2 4 6 8 ·105 Cycles to finish Allreduce Figure 8: Cycles to consume an application kernel in a FM64 Percentile 99.99% Percentile 99.9% Percentile 99% Average Omni-WAR UGAL TERA-HX2 TERA-HX3 1 000 2 000 3 000 4 000 Packet latency All2All Omni-WAR UGAL TERA-HX2 TERA-HX3 5 000 10 000 15 000 Packet latency Stencil 3D Omni-WAR UGAL TERA-HX2 TERA-HX3 1 000 2 000 Packet latency FFT3D Omni-WAR UGAL TERA-HX2 TERA-HX3 2 000 4 000 6 000 Packet latency Allreduce Figure 9: Violin plot packet latencies for linear mapping. between main and service links. Under RSP traffic, service links show about half the utilization of main links. Nevertheless, since service links constitute less than 10% of the total (192 out of 2016 in TERA-HX3), their limited usage has a negligible impact on overall link utilization and performance. 6.4 Application results Figure 8 shows the application completion times across different routing algorithms. Despite not using virtual channels, TERA achieves competitive performance, closely following Omni-WAR and outperforming UGAL in most scenarios. This proves the effectiveness of TERA as a low-cost routing mechanism for Full-mesh networks. Omni-WAR delivers the best overall performance due to the use of 2 VCs and unrestricted non-minimal bandwidth. It consistently outperforms all other mechanisms, particularly in the Stencil 2D and 3D workloads, where its advantage increases to around 10%. In spite of not using VCs, TERA (HX2 and HX3) trails Omni-WAR by a small margin, within 7% on average, and outperforms UGAL significantly, with speedups of up to 47% in the Allreduce application. This improvement stems from TERA's ability to select an optimal non-minimal path from a large pool of candidates at the source switch, effectively reducing packet latency and improving load balancing. This capability is particularly beneficial in heavy communication workloads like Allreduce, where message dependencies can cause added delays. In TERA, packets taking 3 or 4 hops constitute less than 1% of the total on average. To assess the impact of these longer paths on packet latency across all applications, violin plots are presented in Figure 9. As no significant performance differences were observed between linear and random process mappings, evaluations focus on the linear mapping. The link ordering algorithm is omitted due to its lack of competitiveness. TERA-HX2 and TERA-HX3 exhibit the lowest mean and 99% percentile latencies in most cases. This can be attributed to their reduced buffer space, which would result in lower queuing delays. At the higher percentiles (99.9% and 99.99%), TERA remains the top performer except in the Stencil 3D workload, where its latency is comparable to Omni-WAR. UGAL consistently shows the highest latency across all cases, primarily due to its reliance on a single randomly selected intermediate in Valiant-style routing, limiting its ability to adapt to network congestion. TERA offers competitive performance, making it a practical and efficient solution for Full-mesh topologies. 10 Omni-WAR DimWAR O1TURN-TERA-HX3 DOR-TERA-HX3 Natural Random 0 1 2 3 ·105 Cycles to finish All2All Natural Random 0 1 2 ·105 Cycles to finish Allreduce Figure 10: Cycles to finish All2All and Allreduce applications in a 2D-HyperX 8x8 network. Hence, TERA reveals as a low-cost alternative to existing adaptive algorithms like UGAL and Omni-WAR. 6.5 Other topologies Any topology based on FM networks, such as HyperX, Dragonfly and Dragonfly+, can take advantage of TERA. A simulation of an 8 × 8 2D-HyperX network with 512 servers is shown in Figure 10. Dim-WAR, a routing algorithm specially conceived for the 2D-HyperX that uses 2 VCs [18], is also included in the comparison. Additionally, we have developed a O1TURN [25] version of TERA that uses 2 VCs. Thus, Omni-WAR uses 4 VCs, Dim-WAR and O1TURN-TERA-HX3 use 2 VCs, and DOR-TERA-HX3 just 1VC. In DOR-TERA-HX3, the TERA-HX3 routing algorithm is applied independently within each of the two FM8 subnetworks traversed by a packet-one for each dimension-with dimensions visited in XY order. In the case of O1TURN-TERA-HX3, at the source switch, it is decided if the packet will progress either in XY or YX order, and a VC is needed for each dimension order. It is noticeable that DOR-TERA-HX3 reveals competitive in most cases being the one with minimal resources. Additionally, the performance of O1TURN-TERA-HX3 is near Omni-WAR with half the resources (from 4 VCs to 2 VCs), and up to 32% better than Dim-WAR with the same buffering. 7 Conclusions This paper introduces TERA, a Topology-Embedded Routing Algorithm for Full-mesh networks that enables deadlock-free non-minimal routing without relying on virtual channels. An initial evaluation of link ordering schemes exposed their performance limitations, highlighting the need for a more effective solution. TERA addresses these challenges by embedding a physical service topology within the Full-mesh structure. An exhaustive analysis of the service topologies revealed that the HyperX family, particularly the 2D-HyperX and 3D-HyperX, are the most suitable candidate for TERA. In a Full-mesh network, TERA outperforms link ordering routing algorithms by an 80% under adverse traffic, and up to 100% in application kernels. Furthermore, compared to other VC-based approaches, it reduces buffer requirements by 50%, while maintaining comparable latency and throughput. Lastly, early results from a 2D-HyperX evaluation show that TERA outperforms state-of-the-art algorithms that use the same number of VCs, achieving performance improvements of up to 32%. This work concludes that TERA can be effectively applied to FM networks, and that it is a promising solution for high-radix, low-diameter topologies. 11 A Link ordering schemes Proof of Theorem 3.2. Let us define the function Φ : V 3 →{0, 1} that indicates which paths of length 2 are allowed. For distinct switches s, m, d ∈V , set Φ(s, m, d) = 1 if the route s, m, d is allowed and set Φ(s, m, d) = 0 otherwise. That is, Φ describes which switches m can be used to route from source s to destination d. Additionally, just to avoid cumbersome notation, set Φ(s, m, d) = 0 when any pair of s, m, d coincide. The hypothesis of the same utilization can now be written as follows. There is a constant S such that for any arc (a, b), holds P x∈V Φ(a, b, x) + Φ(x, a, b) = S. Let (ma, mb) be the arc at first position by the ordering of the scheme. This arc is always allowed as the start of a 2-length path, but never as its ending. Thus, P x∈V Φ(ma, mb, x) = n-2, and P x∈V Φ(x, ma, mb) = 0. Therefore, we obtain the value of the constant S = n -2. This implies that n(n -1)(n -2) = X (a,b)∈A X x∈V Φ(a, b, x) + Φ(x, a, b). Note that the two terms have the same sum, that is, X (a,b)∈A X x∈V Φ(a, b, x) = X (a,b)∈A X x∈V Φ(x, a, b). Therefore, n(n -1)(n -2) = 2 P (a,b)∈A P x∈V Φ(a, b, x). Finally, the total number of allowed paths is P s,m,d∈V Φ(s, m, d) = P a,b∈V P x∈V Φ(a, b, x) = 1 2n(n - 1)(n -2). Proof of Claim 3.4. The number of intermediate switches between a switch source a, and a switch destination b is going to be calculated. It should be taken into account that the maximum number of intermediates is n -2, as a and b do not count as intermediates. Let us define the function Gab(i) := D(i, b) -D(a, i). A value Gab(i) > 0 means that the non-minimal path a 7→i 7→b is allowed. Otherwise, the path is forbidden. The function G satisfies -Gab(a + b 2 + x) = Gab(a + b 2 -x) 0 ≤x < n. (1) This implies that every valid intermediate in the set of switches is paired with an invalid one. Thus, around half of the switches are valid intermediates, and the others are invalid intermediates. We need to discount those cases were both sides of Equation 1 are 0, as both represent invalid intermediates. That is, the number of solutions to Gab(y) = 0 in the integers modulo n. Taking x = 0 in Equation 1 gives Gab( a+b 2 ) = 0. Hence, y = a+b 2 is a solution when it is integer. More precisely, let us count the solutions in y for 2y ≡a + b mod n. When n is odd, 2 has inverse, and we have exactly one solution. When n is even: • if a and b have different parity, there are no solutions, and there are n-2 2 intermediates. • if a and b have the same parity, there are two solutions, y = a+b 2 and y = a+b+n 2 , and there are n-4 2 intermediates. B TERA estimated performance An estimation of the throughput achieved by TERA under various service topologies subjected to Random Switch Permutation (RSP) traffic can be expressed as 1 1+p-1 , where p denotes the degree of the main topology divided by n -1. This expression is illustrated for several service topologies in Figure 4. The following paragraph outlines the derivation of this expression. As the RSP is an adverse traffic pattern, using MIN paths (a single direct link between a pair of switches) is insufficient for achieving good performance. Furthermore, the direct link can always be used as a MIN path, regardless of whether it is main or service. The 2-hop deroutes must employ a main link in the first hop and any in its second hop. Longer routes can be disregarded; they are extremely rare, as it can be empirically seen. In consequence, under a reasonable balance of routes, the main links will saturate before the service ones. Let p be the ratio of main links to all links, or equivalently, the probability that a randomly chosen link 12 belongs to the main topology. Thus, d = (n -1)p is the average degree of the main topology. Let γ be the average switch injection rate, measured in multiples of full-rate links. Assuming traffic without same-switch messages, we have n -1 links for all traffic, implying 0 ≤γ ≤n -1. To estimate γ, first decompose it into a rate γ1 for packets employing 1 hop and a rate γ2 for packets employing 2 hops. Since most packets do not use them, we can ignore longer routes for an estimation. Then, we have that γ = γ1 + γ2. Since the traffic is a switch-permutation then 0 ≤γ1 ≤1. Packets that traverse a single hop will have the probability p of adding traffic into the main topology. For packets traversing two hops, the first one is forced and the second hop only with probability p. Thus, the dn/2 main links, times the two directions, must be able to hold a total of n(pγ1 + (1 + p)γ2). This calculation can be shown in the following formula: n(pγ1 + (1 + p)γ2) ≤p(n -1)n. (2) Note that the equality would be achieved if the load were perfectly distributed. Isolating the major contributor γ2 becomes γ2 ≤p(n -1) -pγ1 1 + p = (n -1) -γ1 1 + p-1 . (3) Thus, giving a total load per switch of γ ≤(n -1) + γ1p-1 1 + p-1 = n -1 1 + p-1 + γ1 1 + p ≤ n -1 1 + p-1 + 1. (4) Equivalently, assuming n servers per switch, the average load for each server is γ/n ≤ 1 1+p-1 + O(1/n). Acknowledgements This work has been supported by the Spanish Ministry of Science and Innovation under contracts PID2019105660RB-C22, TED2021-131176B-I00, and PID2022-136454NB-C21. C. Camarero is supported by the Spanish Ministry of Science and Innovation, Ramón y Cajal contract RYC2021-033959-I. R. Beivide is supported by The Barcelona Supercomputing Center (BSC) under contract CONSER02023011NG. Simulations were performed in the Altamira supercomputer, a node of the Spanish Supercomputing Network (RES). References [1] Jung Ho Ahn, Nathan Binkert, Al Davis, Moray McLaren, and Robert S. Schreiber. HyperX: Topology, routing, and packaging of efficient large-scale networks. In Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis, SC '09, pages 1-11, New York, NY, USA, 2009. ACM. [2] Cristóbal Camarero, Carmen Martínez, and Ramón Beivide. L-networks: A topological model for regular 2D interconnection networks. 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[22] Timo Schneider, Otto Bibartiu, and Torsten Hoefler. Ensuring deadlock-freedom in low-diameter infiniband networks. In 2016 IEEE 24th Annual Symposium on High-Performance Interconnects (HOTI), pages 1-8. IEEE, 2016. [23] Michael D. Schroeder, Andrew D Birrell, Michael Burrows, Hal Murray, Roger M. Needham, Thomas L. Rodeheffer, Edwin H. Satterthwaite, and Charles P. Thacker. Autonet: A high-speed, self-configuring local area network using point-to-point links. IEEE Journal on Selected Areas in Communications, 9(8):1318-1335, 1991. 14 [24] S. Scott, D. Abts, J. Kim, and W.J. Dally. The BlackWidow high-radix Clos network. SIGARCH Comput. Archit. News, 34(2):16-28, 2006. [25] Daeho Seo, Akif Ali, Won-Taek Lim, and Nauman Rafique. Near-optimal worst-case throughput routing for two-dimensional mesh networks. In 32nd International Symposium on Computer Architecture (ISCA'05), pages 432-443. IEEE, 2005. 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|
2510.14731
|
Generalized Fourier Series: An N log2(N) extension for aperiodic
functions that eliminates Gibbs oscillations
Narsimha Reddy Rapakaa,b,∗, Mohamed Kamel Riahia,b
aDepartment of Mathematics, College of Computing and Mathematical Sciences, Khalifa University of Science and Technology, Abu
Dhabi, UAE.
bEmirates Nuclear Technology Center, Khalifa University of Science and Technology, Abu Dhabi, UAE.
Abstract
This article introduces the Generalized Fourier Series (GFS), a novel spectral method that extends the clas-
sical Fourier series to non-periodic functions. GFS addresses key challenges such as the Gibbs phenomenon
and poor convergence in non-periodic settings by decomposing functions into periodic and aperiodic com-
ponents. The periodic part is represented using standard Fourier modes and efficiently computed via the
Fast Fourier Transform (FFT). The aperiodic component employs adaptive, low-rank sinusoidal functions
with non-harmonic modes, dynamically tuned to capture discontinuities and derivative jumps across domain
boundaries.
Unlike conventional Fourier extension methods, GFS achieves high accuracy without requiring compu-
tational domain extensions, offering a compact and efficient representation of non-periodic functions. The
adaptive low-rank approach ensures accuracy while minimizing computational overhead, typically involving
additional complex modes for the aperiodic part. Furthermore, GFS demonstrates a high-resolution power,
with degrees of freedom comparable to FFT in periodic domains, and maintains N log2(N) computational
complexity. The effectiveness of GFS is validated through numerical experiments, showcasing its ability to
approximate functions and their derivatives in non-periodic domains accurately. With its robust framework
and minimal computational cost, GFS holds significant potential for advancing applications in numerical
PDEs, signal processing, machine learning, and computational physics by providing a robust and efficient
tool for high-accuracy function approximations.
Keywords: Fourier Spectral Method, Non-periodic Domain, Gibbs Oscillations, Non-harmonic Modes,
Complex Fourier Modes, Fast Solvers
1. Introduction
The accurate approximation of smooth, non-periodic functions within bounded domains is a classical
yet persistently challenging problem in numerical analysis. This issue has profound implications for various
applications, particularly in the numerical solution of partial differential equations (PDEs). Traditional ap-
proaches, such as finite difference, finite volume, finite element methods, and Fourier or Chebyshev spectral
methods, have been extensively employed to address this challenge. While non-spectral methods offer broad
∗Corresponding author. Tel.: +971 2 3125416, Email address: narsimha.rapaka@ku.ac.ae (Narsimha R. Rapaka)
Preprint submitted to Elsevier
October 17, 2025
arXiv:2510.14731v1 [math.NA] 16 Oct 2025
applicability to problems with arbitrary boundary conditions, they often suffer from limited resolution, re-
quiring a large number of degrees of freedom per wavelength. On the other hand, spectral methods provide
superior resolution power but are inherently constrained to periodic domains or homogeneous boundary con-
ditions. This article aims to broaden the application of Fourier spectral methods to non-periodic functions
with minimal computational overhead, focusing solely on the lack of periodicity while assuming sufficient
smoothness in the interior.
The Fast Fourier Transform (FFT) is widely used for approximating smooth, periodic functions due to
its efficiency and spectral convergence. However, when applied to non-periodic functions, FFT suffers from
the Gibbs phenomenon, producing spurious oscillations near domain boundaries Hewitt and Hewitt (1979);
Gottlieb and Shu (1997); Jerri (1998). Over the years, several strategies have been developed to mitigate
this limitation. These include filtering techniques to suppress oscillations, domain extension methods that
embed non-periodic functions into larger periodic domains Iserles and Nørsett (2008); Huybrechs (2010);
Adcock (2010, 2011); Adcock and Huybrechs (2014); Geronimo and Liechty (2020), polynomial-based pe-
riodization transformations Krylov (1907); Eckhoff (1993); Roache (1978), signal-processing approaches
such as Prony’s method de Prony (1795); Plonka et al. (2018), and more recently, rational approximation
techniques such as the AAA algorithm Nakatsukasa et al. (2018); Nakatsukasa and Trefethen (2020); Huy-
brechs and Trefethen (2023); Driscoll et al. (2024). Notably, Eckhoff’s method Eckhoff (1995) account for
multiple isolated discontinuties and its convergence has been analyzed in Nersessian and Poghosyan (2006);
Barkhudaryan et al. (2007); Poghosyan (2010). While these approaches offer partial remedies, they also
present significant drawbacks. In particular, domain extension methods increase computational overhead
by introducing additional degrees of freedom, involves solution of ill-conditioned systems Adcock et al.
(2014), and often require problem-dependent tuning Adcock and Ruan (2014); moreover, they are not prac-
tical in applications where the domain is fixed by data, memory, or geometry constraints. Polynomial-based
modifications Roache (1978); Eckhoff (1995) often lack robustness and deliver limited accuracy for highly
oscillatory functions, as demonstrated in this work. Similarly, Prony’s method is ill-conditioned for large
system sizes and requires prior knowledge of the number of modes. The AAA algorithm, while powerful
and broadly applicable, can incur high computational cost for large-scale problems and may overfit when
applied to noisy data Nakatsukasa et al. (2018).
To tackle these challenges, this work introduces a novel method called the Generalized Fourier Series
(GFS), which extends Fourier spectral methods to non-periodic functions with minimal computational over-
head. The key idea is to decompose a function into periodic and aperiodic components: the periodic part is
efficiently approximated using standard FFT. In contrast, the aperiodic part is represented with an adaptive
set of n = O(1) complex sine and cosine modes. This strategy avoids artificial domain extensions, preserves
the efficiency and simplicity of the Fourier framework, and provides a robust mechanism for capturing both
smooth and oscillatory non-periodic behavior. Numerical validations support these results.
Exising methods augment standard Fourier series with a fixed set of functions to approximate non-
periodic functions, e.g., polynomials Roache (1978), Birkhoff–Hermite polynomials Huybrechs et al. (2010),
Bernoulli polynomials Eckhoff (1995). In contrast, GFS adds a dynamic set of non-harmonic sinusoidal
modes that adapt to the function’s non-periodic nature, efficiently capturing its aperiodic component. We
show that GFS is more robust, achieves superior numerically accuracy and convergence.
The GFS method, like Eckhoff’s Eckhoff (1995, 1998), is multi-dimensional and flexible, allowing for
future expansion to handle multiple discontinuities in complex geometries, such as cubic outer boxes with
several immersed obstacles modeled as discontinuities.
This paper is organized as follows. Section 3.1 provides the mathematical motivation for the approach,
detailing how jump conditions influence the aperiodic component. Section 3 outlines the formulation of the
Generalized Fourier Series and describes the continuous spectrum of aperiodic signals. The adaptive basis
2
construction is then presented, along with the computational algorithms used to compute the modes and
expansion coefficients. Section 4 outlines the computational complexity of the proposed method. Finally,
Section 5 demonstrates the method’s efficacy through numerical tests on representative non-periodic func-
tions, showcasing the superior resolution power and convergence rate of GFS compared to finite difference
and the standard FFT methods.
By introducing an adaptive decomposition of the function space and leveraging the efficiency of FFT, the
proposed GFS method offers a new perspective on handling non-periodic functions in bounded domains. The
method retains linear computational complexity, avoids the need for domain extensions, and delivers high-
resolution power akin to spectral methods in periodic domains. These features make the GFS a compelling
alternative for problems involving non-periodic functions, with potential applications in numerical PDEs,
signal processing, machine learning, and computational physics.
2. State-of-the-art methods
We compare GFS with the existing methods described briefly below:
2.1. Eckhoff Method
The Eckhoff method Eckhoff (1995) reconstructs a 2π-periodic, piecewise smooth function u(x) from a
finite number of its Fourier coefficients ˆuk. Assuming M singularities at unknown locations γj with jumps
An
j ≡u(n)(γ+
j ) −u(n)(γ−
j ) in the nth derivative, the function is decomposed as:
u(x) = v(x) +
q−1
X
n=0
M
X
j=1
An
jVn(x; γj)
(1)
where v(x) is smooth and Vn(x; γj) are periodic functions derived from Bernoulli polynomials with controlled
singularities at γj.
Vn(x; β) = −(2π)n
(n + 1)! Bn+1
ξ
2π
,
ξ =
mod (x −β + 2π, 2π), 0 < ξ < 2π,
where, Bn(x), n = 1, 2, . . . , are the Bernoulli polynomials Eckhoff (1995). Further,
∂
∂xVn(x; β) = Vn−1(x; β).
To locate the singularities, one constructs the nonlinear system:
q−1
X
n=0
M
X
j=1
An
j
(ik)n e−ikγ j = 2πik(ˆuk −ˆvk) ≈2πikˆuk,
for k = N/2 −Mq, N/2 −Mq, . . . , N/2 −1, from which the jump locations and magnitudes are estimated al-
gebraically (see Eckhoff (1995) for the solution procedure). In this work, we consider endpoint singularities
only, i.e., x = −π (and, by periodicity of Vn(x; γj), x = π) so that M = 1 and γ1 = −π.
We find that evaluating jumps numerically via Fourier coefficients as in Eckhoff method Eckhoff (1995)
is not robust (also noted in Eckhoff (1998)) for the functions considered here. Moreover, the Eckhoff method
yields only a first-order approximation for the highest derivative jump Eckhoff (1995), which can substan-
tially degrade accuracy for highly oscillatory signals. Eckhoff Eckhoff (1998) examined finite-difference
method and asymptotic expansions near discontinuities, while Barkhudaryan et al. (2007); Poghosyan (2010)
studied various choices of Fourier-tail indices to accelerate convergence of jump computations. Nonethe-
less, their numerical tests used only smooth functions, leaving robustness for highly oscillatory cases unclear.
Here, we evaluate the jumps An
1 B u(n)(−π) −u(n)(π) in Eq. (1) analytically for comparison with GFS. In the
GFS method, the end point jumps are denoted Jn, related by Jn = −An
1.
3
2.2. Roache Method
Roache’s method Roache (1978) handles non-periodic problems using FFTs via “reduction to periodic-
ity”. The idea is to decompose the target function f(x) into:
f(x) = g(x) + ˜f(x), x ∈[−π, π]
where g(x) is a polynomial that matches the jumps in the boundary derivatives of f, denoted with Jm, m =
0, 1, . . . ,, up to order q −1, ensuring ˜f(x) becomes continuous at the end points up to q −1 derivatives:
g(x) =
q
X
k=0
akxk,
with
˜f (m)(π) −˜f (m)(−π) = 0,
0 ≤m < q,
aN = 1
2π
Jq−1
q! ,
ak = 1
2π
Jk−1
k! −
q
X
m=k+1
am
m!
k!(m −k + 1)!((π)m−k+1 −(−π)m−k+1)
.
Then, FFT is applied to ˜f(x), and g(x) is differentiated analytically.
2.3. Prony Method
Prony’s method (de Prony (1795), Chapter 10 of Plonka et al. (2018)) reconstructs an exponential sum
from sampled data h(k) ≡h(xk), xk = k∆x, k = 0, 1, . . . , N −1 with N ≥2M:
h(k) =
M
X
j=1
c jeϕ jxk =
M
X
j=1
c jzk
j,
k = 0, . . . , 2M −1,
(2)
where, zj = eϕj∆x are unknown, distinct complex parameters.
It constructs the Prony polynomial:
p(z) =
M
Y
j=1
(z −z j) = zM + pM−1zM−1 + · · · + p0
Solving the linear system:
M
X
k=0
pkh(k + m) = 0,
m = 0, . . . , M −1,
pM = 1.
yields pk and the z j (thereby ϕj) via roots of the Prony polynomial (equivalent to eigenvalues of the compan-
ion matrix of p(z)), and then cj from a Vandermonde system given below (see Chapter 10 of Plonka et al.
(2018) for further details).
1
1
· · ·
1
z1
z2
· · ·
zM
...
...
...
...
zM−1
1
zM−1
2
· · ·
zM−1
M
c1
c2
...
cM
=
h0
h1
...
hM−1
(3)
Table 1 contrasts GFS with representative numerical methods in computational complexity, degrees of
freedom (DOF), scope, and other key features.
4
Method
Year
Complexity
Non-periodic
Resolution (PPW)
Jump Detection
Total DOF
Additional DOF
Splitting
Aperiodic Recovery
Polynomial
Exponential
Oscillatory
FFT (standard)
1965
O(N log2 N)
✕
2
✕
N
✕
✕
✕
✕
✕
GFS (present)
2025
O(N (q + log2 N) + 3(q/4)3)
✓
∼3
✓
N + q
q
dynamic
✓
✓
✓
Eckhoff’s Spectral Reconstruction Eckhoff (1995)
1995
O(N(q + log2 N) + q3)
✓
–
✓
N + q
q
fixed
✓
✓
✕
Roache’s Jump-Fitting Roache (1978)
2000
O(N(q + log2 N) + q3)
✓
–
✓
N + q
q
fixed
✓
✓
✕
Prony’s Method de Prony (1795)
1795
O(NM + M3)
✓
–
✓
N + M
M
✕
✕
✓
✕
Finite Differences (order 6)
1945
O(N)
✓
O(101 −102)∗
✕
N
✕
✕
✓
✓
✓
Table 1: Comparison of approximation methods for non-periodic (aperiodic) functions. DOF refers to degrees of freedom, N is the
mesh size, q is the number of jumps (in GFS q = 4n where n ∼O(1) is the number of aperiodic modes), M is the number of exponential
modes in Prony method with N ≥2M. In Eckhoff’s method only one discontinuous location at the domain bounaries is assumed for the
complexity estimates. PPW is the points required per wavlength for a target error ϵ. *For a standard sixth-order central finite difference
(FD) scheme (one-sided near boundary nodes): PPW ∼O(10) for ϵ = 10−3, PPW ∼O(102) for ϵ = 10−10; for GFS: PPW ∼3 for
ϵ = 10−10 (based on data from Table 7 and Fig. 8).
3. Generalized Fourier Series (GFS):
3.1. Motivation and non-harmonic modes
(a)
(b)
(c)
Figure 1: Spectral leakage phenomena: Fast Fourier Transform (FFT) of a function f(t) = 0.7 sin(2πk1t)+sin(2πk2t), 0 ≤t ≤1; (a) f(t)
is periodic with k1 = 5, k2 = 12, (b) f(t) is non-periodic with k1 = 5.3, k2 = 12.4. Here, the sample frequency, N = 128. (c) f(t) is non-
periodic with k1 = 5.3, k2 = 12.4 and is represented with generalized Fourier series with two non-periodic modes k1 = 5.3, k2 = 12.4.
Spectral leakage occurs when a signal’s frequency components are not perfectly aligned with the discrete
frequencies of the Fourier transform, such as in the Discrete Fourier Transform (DFT). This misalignment
5
results in energy spreading into adjacent frequency bins, creating “leakage.” It is most prominent with non-
periodic signals in finite domains.
Fig. 1 illustrates the spectral leakage phenomena with an example. Consider a function f(t) = 0.7 sin(2πk1t)+
sin(2πk2t), −π ≤t ≤π. Fig. 1(a) shows a periodic signal (top) comprised of discrete harmonics k1 =
5, k2 = 12 of the fundamental frequency corresponding to the domain. The corresponding spectrum (bot-
tom) exhibits discrete spikes at k1, k2. This is expected because the Fourier spectrum of a periodic signal
comprises a discrete set of modes corresponding to harmonics (integer multiples) of a fundamental mode
2π/L associated with the domain length L. Fig. 1(b) shows a non-periodic signal (top) comprised of modes
k1 = 5.3, k2 = 12.4. The corresponding spectrum (bottom) is spread over nearby modes, yielding a side-lobe
structure instead of discrete spikes at k1 = 5.3, k2 = 12.4. This is because the modes k1 = 5.3, k2 = 12.4 are
not harmonics of the fundamental frequency (k = 1). This phenomenon is known as spectral leakage. Fig.
1(c) shows the non-periodic signal (top) comprised of modes k1 = 5.3, k2 = 12.4 represented by the present
method (GFS) whose spectrum (bottom) is sharp and composed merely of two discrete non-integer modes,
same as those in the original function. The GFS algorithm described in later sections automatically detects
these mode numbers.
Spectral leakage primarily arises because the Fourier spectrum is sampled over integer multiples of a
fundamental frequency (harmonic modes). By incorporating non-harmonic modes, we can recover a sharp
spectrum, but identifying these modes without undue computational cost is challenging.
The classical Prony method de Prony (1795)Plonka et al. (2018) can detect non-harmonic modes in
noiseless, sampled data, but it requires prior knowledge of the number of modes M. Since M is arbitrary, the
resulting system (e.g. the Vandermonde system in Eq. (3)) can become highly ill-conditioned when M ≫1
and its computational cost scales as O(N3) with mesh size N.
To reduce computational overhead and improve conditioning, we propose GFS, which decomposes the
signal into periodic and non-periodic components. This approach, however, relies on knowledge of jump
conditions, i.e., prior information about jumps or a numerical method to evaluate them.
3.2. The continuous spectrum for an aperiodic signal
In the standard Fourier framework, aperiodic signals exhibit a continuous spectrum due to their non-
repeating nature, whereas periodic signals consist of discrete harmonics (integer multiples of a fundamental
frequency). As a result, the Fourier representation for aperiodic signals shifts from a sum of discrete fre-
quencies to an integral over a continuous frequency domain.
Our framework handles both periodic and aperiodic signals using non-harmonic modes that capture
complex features, including real frequencies for sinusoidal behavior and imaginary frequencies for growth
or decay. By employing non-harmonic modes, we can represent the signal with a finite set of real-valued
frequencies, eliminating spectral leakage, as illustrated in Fig.1.
We decompose a function u(x) on [−π, π] as u(x) = up(x) + ua(x), where up(x) is periodic and ua(x)
is aperiodic. The same analysis applies to any domain x ∈[a, b]; we use the coordinate transformation
x∗= 2π(x −x0)/(b −a), with x0 = (a + b)/2, and for simplicity, take a = −π, b = π. We split ua(x) into
symmetric (even) uc(x) and antisymmetric (odd) us(x) components, using non-harmonic mode functions
cos(ˆkjx) and sin(˜k jx), where ˆk j, ˜kj ∈C. The symmetric part’s cosine modes are denoted with a hat (‘b’),
6
while the anti-symmetric part’s sine modes are denoted with a tilde (‘∼’).
uc(x)
=
nc
X
j=1
ˆuj cos(ˆk jx),
us(x) =
ns
X
j=1
˜uj sin(˜k jx),
(4)
ua(x)
=
uc(x) + us(x) =
nc
X
j=1
ˆuj cos(ˆkjx)
| {z }
symmetric
+
ns
X
j=1
˜u j sin(˜kjx)
| {z }
anti-symmetric
.
(5)
Let’s assume a set of complex modes kj = σj + iω j, where σj, ωj ∈R+ and i =
√
−1. The Euler formula
clearly illustrates the decay/growth behavior of the mode k j.
eikjx =
cos(σjx) + i sin(σ jx)
e−ωjx,
e−ik jx =
cos(σjx) −i sin(σ jx)
eωjx.
For ˆkj = ˆσj + i ˆω j, we set the following decay and growth representation for cos “symmetric” series, with
complex coefficients (ˆu j):
uc(x) = 1
2
X
j
ˆuj
eiˆk jx + e−iˆkjx
=
X
j
ˆuj cos(ˆk jx),
similarly, for ˜k j = ˜σj +i ˜ωj, a decay and growth representation for sin “anti-symmetric” series, with complex
coefficients (˜u j):
us(x) = 1
2i
X
j
˜u j
ei˜k jx −e−i˜k jx
=
X
j
˜u j sin(˜kjx)
The above certifies our use of cos(ˆkjx) and sin(˜k jx), where the complex-valued modes ˆkj, ˜k j introduce
exponential decay or growth alongside oscillations, adapting to the function being approximated.
Although represented by complex modes and amplitudes, uc(x) and us(x) in Eq. (4) are always real (see
Table 2 and Sec. 3.4.2 for details) and need not have the same number of modes, nc and ns, respectively.
For convenience, we set nc = ns = n without loss of generality; if nc , ns, we take n = max(nc, ns) for the
complexity analysis of the numerical algorithm described in Sec. 3.5.
3.3. Jump’s approach for aperiodic signal
Let us denote the mth derivative of u(x) as u(m) and assume that u(x) is smooth in the interior. We have
u(m) = u(m)
p
+ u(m)
a . Let us denote the jump in the mth derivative of u as Jm ≡[u(m)]. By the definition
of periodicity, we have [u(m)
p ] = 0 which yields [u(m)] = [u(m)
a ] for m = 0, 1, 2, 3, ..., ∞. Note that the jump
conditions may be known in certain applications; otherwise, they are approximated numerically as described
in Sec. 3.6.
The even derivatives of uc(x) and odd derivatives of us(x) are symmetric and, therefore, have zero jumps
at the endpoints. So, we have J2m ≡[u(2m)] = [u(2m)
a
] = [u(2m)
s
] and J2m+1 ≡[u(2m+1)] = [u(2m+1)
a
] = [u(2m+1)
c
]
for m = 0, 1, 2, ..., ∞.
J2m =
h
u(2m)
s
i
=
n
X
j=1
(−1)m2˜k2m
j ˜uj sin(˜k jπ),
m = 0, 1, 2, ..., ∞,
(6)
J2m+1 =
h
u(2m+1)
c
i
=
n
X
j=1
(−1)(m+1)2ˆk(2m+1)
j
ˆu j sin(ˆk jπ), m = 0, 1, 2, ..., ∞.
(7)
7
The jumps computation can lead to rank deficiency; thus, we employ a low-rank approximation with n ∼
O(1), demonstrating its effectiveness in practical applications in Sec. 5. This approach improves computa-
tional efficiency and ensures optimal approximation when jumps are involved.
Our GFS approach can be summarized into three steps:
• First, we estimate the aperiodic part from derivative jumps at the domain boundaries.
• Second, we recover the periodic part by subtracting the aperiodic part from the original signal.
• Third, we approximate the derivative of the original signal as the sum of the derivatives of its periodic
and aperiodic parts.
In the sequel, we detail the first step—identifying the aperiodic part via non-harmonic modes—while the
second step is trivial (though necessary), and the third step is readily obtained through classical Fourier
analysis.
3.4. Dynamic non-harmonic modes ˜k j, ˆkj and amplitudes ˜uj, ˆuj
3.4.1. Formulation of the system of equations
With 2n unknowns (˜kj, ˜uj) in Eq. (6), we derive a system of 2n equations by evaluating Eq. (6) for
m = 0, 1, 2, ..., 2n −1. Similarly, another system of 2n equations for the unknowns (ˆkj, ˆuj) is obtained from
Eq. (7) for m = 0, 1, 2, ..., 2n −1. Both systems are presented in matrix form below.
2
1
1
. . .
1
−˜k2
1
−˜k2
2
. . .
−˜k2
n
˜k4
1
˜k4
2
. . .
˜k4
n
...
...
...
...
(i˜k1)4n−2
(i˜k2)4n−2
. . .
(i˜kn)4n−2
˜u1 sin(˜k1π)
˜u2 sin(˜k2π)
...
˜un sin(˜knπ)
=
J0
J2
J4
...
J4n−2
,
−2
1
1
. . .
1
−ˆk2
1
−ˆk2
2
. . .
−ˆk2
n
ˆk4
1
ˆk4
2
. . .
ˆk4
n
...
...
...
...
(iˆk1)4n−2
(iˆk2)4n−2
. . .
(iˆkn)4n−2
ˆu1ˆk1 sin(ˆk1π)
ˆu2ˆk2 sin(ˆk2π)
...
ˆunˆkn sin(ˆknπ)
=
J1
J3
J5
...
J4n−1
.
The above equations can be rearranged to obtain a set of equations involving only the unknowns ˆkj and ˜kj as
below,
2
˜u1 sin(˜k1π)
˜u2 sin(˜k2π)
...
˜un sin(˜knπ)
| {z }
˜w
=
1
1
. . .
1
−˜k2
1
−˜k2
2
. . .
−˜k2
n
...
...
...
...
(i˜k1)2n−2
(i˜k2)2n−2
. . .
(i˜kn)2n−2
| {z }
≡˜V
−1
J0
J2
...
J2n−2
| {z }
˜b
=
(i˜k1)2n
(i˜k2)2n
. . .
(i˜kn)2n
(i˜k1)2n+2
(i˜k2)2n+2
. . .
(i˜kn)2n+2
...
...
...
...
(i˜k1)4n−2
(i˜k2)4n−2
. . .
(i˜kn)4n−2
−1
J2n
J2n+2
...
J4n−2
.
(8)
8
−2
ˆu1ˆk1 sin(ˆk1π)
ˆu2ˆk2 sin(ˆk2π)
...
ˆunˆkn sin(ˆknπ)
| {z }
ˆw
=
1
1
. . .
1
−ˆk2
1
−ˆk2
2
. . .
−ˆk2
n
...
...
...
...
(iˆk1)2n−2
(iˆk2)2n−2
. . .
(iˆkn)2n−2
| {z }
≡ˆV
−1
J1
J3
...
J2n−1
| {z }
ˆb
=
(iˆk1)2n
(iˆk2)2n
. . .
(iˆkn)2n
(iˆk1)2n+2
(iˆk2)2n+2
. . .
(iˆkn)2n+2
...
...
...
...
(iˆk1)4n−2
(iˆk2)4n−2
. . .
(iˆkn)4n−2
−1
J2n+1
J2n+3
...
J4n−1
.
(9)
The expressions satisfy [u(m)] = [u(m)
a ], leading to [u(m)
p ] = 0 for m = 0, 1, 2, . . . , 4n −1. This means that the
periodic part is smooth (continuous at the endpoints) up to (4n−1) derivatives, so up(x) ∈C4n−1. Each mode
˜kj, j = 1, 2, . . . , n, (respectively ˆk j) must be unique (or simply separate) for the inverse of the transposed
Vandermonde matrix ˜V ( respectively ˆV) to exist. The unknowns of the even part (ˆk j, ˆu j) depend solely on
the jumps relative to odd derivatives of u(x), while those of the odd part (˜k j, ˜uj) rely only on the jumps relative
to the even derivatives. Additionally, the second and third components in Eqs. (8)-(9) involving ˜kj and ˆk j
are identical, with the even jumps J0, J2, . . . , J4n−2 in Eq. (8) replaced by the odd jumps J1, J3, . . . , J4n−1 in
Eq. (9). Thus, the solution methods for ˜kj and ˆk j are analogous.
We solve the above non-linear systems of equations to find the unknown modes (ˆk j, ˜kj) and their corre-
sponding amplitudes (ˆuj, ˜uj) in terms of the jumps Jm. We first consider the cases for n = 1, 2, 3 in Appendix
Appendix A and then generalize the method for arbitrary n below. This generalization lacks a rigorous proof
due to the complexity of the algebra with larger n, but we validate it in Sec. 5 with multiple test cases.
3.4.2. Solution method
First, let us define the elementary symmetric polynomials (˜e j) in ˜k2
1, ˜k2
2, . . . , ˜k2
n as,
X
1≤j≤n
˜k2
j = ˜e1,
X
1<j<k≤n
˜k2
j ˜k2
k = ˜e2,
. . . ,
X
1< j1< j2<···< jm≤n
˜k2
j1 ˜k2
j2 · · · ˜k2
jm = ˜em,
and the elementary symmetric polynomials (ˆe j) in ˆk2
1, ˆk2
2, ˆk2
3 as,
X
1≤j≤n
ˆk2
j = ˆe1,
X
1<j<k≤n
ˆk2
j ˆk2
k = ˆe2,
. . . ,
X
1< j1< j2<···< jm≤n
ˆk2
j1 ˆk2
j2 · · · ˆk2
jm = ˆem.
Analogous to Eqs. (A.2), (A.8)-(A.9), and (A.17)-(A.18) for n = 1, 2 and 3, respectively, a set of
equations for the elementary symmetric polynomials ˜e j, ˆej for arbitrary number (n) of modes are given
below which satisfy the second equality in Eqs. (8)-(9), respectively,
˜en
...
˜e2
˜e1
= −
J0
J2
. . .
J2(n−1)
J2
J4
. . .
J2(n)
...
...
...
...
J2(n−1)
J2n
. . .
J2(2n−2)
| {z }
≡˜J
†
J2n
J2(n+1)
...
J2(2n−1)
,
(10)
9
ˆen
...
ˆe2
ˆe1
= −
J1
J3
. . .
J2n−1
J3
J5
. . .
J2n+1
...
...
...
...
J2n−1
J2n+1
. . .
J4n−3
| {z }
≡ˆJ
†
J2n+1
J2n+3
...
J4n−1
.
(11)
A straightforward expansion shows that ˜k2
1, ˜k2
2, ...˜k2
n and ˆk2
1, ˆk2
2, ...ˆk2
n are the roots (λ) of the polynomials below
involving ˜e j, ˆej, respectively,
n
Y
j=1
(λ −˜k2
j) = λn −˜e1λ(n−1) + . . . + (−1)k˜ekλ(n−k) + . . . + (−1)n˜en = 0,
(12)
n
Y
j=1
(λ −ˆk2
j) = λn −ˆe1λ(n−1) + . . . + (−1)kˆekλ(n−k) + . . . + (−1)nˆen = 0.
(13)
Note that ˆkj = ± pλj and ˜kj = ± pλj and the ± sign is irrelevant here because ˆu j cos(ˆk jx) and ˜u j sin(˜kjx)
(therefore, uc(x) and us(x)) are even functions of ˆkj and ˜kj, respectively. Here, we consider the complex
square root defined as √z ≡√|z|eiθ/2 where z = |z|eiθ, −π ≤θ ≤π. Once the modes ˜k j, ˆkj are obtained by
solving for the roots of the above polynomials, the amplitudes ˜u j, ˆuj can be obtained from the first equality
in Eqs. (8) and (9), respectively. Note that the matrices ˜J, ˆJ in Eqs. (10), and (11) may be rank-deficit and
hence non-invertable. For example, a purely periodic function u(x) has all the jumps (Jm, ∀j) equal to zero
and therefore rank( ˜J)=rank( ˆJ)=0. In practice, one must use the Moor-Penrose generalized inverse, hence
the notation of ˜J†.
Remark 1. Since ˆk2
j and ˜k2
j are roots of polynomials with real coefficients, they must be real numbers or
form complex conjugate pairs, according to the complex conjugate root theorem. So, the modes ˆkj and ˜kj
must be real (if ˆk2
j, ˜k2
j ∈R+) or purely imaginary (if ˆk2
j, ˜k2
j ∈R−, Re(ˆkj) = Re(˜kj) = 0) or form complex
conjugate pairs (if ˆk2
j, ˜k2
j ∈C, ˆk2
j = ˆk∗2
i , ˜k2
j = ˜k∗2
j for some (i, j)).
Let us define three disjoint sets R, I, and C comprised solely of real numbers, purely imaginary numbers,
and complex conjugate pairs, respectively, as follows,
R = {kj ∈R | Im(k j) = 0},
(real)
I = {kj ∈iR | Re(k j) = 0, Im(k j) , 0},
(purely imaginary)
C = {(ki, kj) ∈C × C | kj = k∗
i , Im(kj) , 0},
(complex conjugate pairs)
If kj ∈I, then k2
j ∈R. Let us separate ˜k2
j (and ˆk2
j) into the following subsets,
{˜k2
j}n
j=1 = {(˜k2
p, ˜k2
q)}C ∪{˜k2
r}R,
where the integers p, q, r ∈I; 1 ≤p ≤˜np; q = ˜np + p, and 2˜np + 1 ≤r ≤n. And
{ˆk2
j}n
j=1 = {(ˆk2
p, ˆk2
q)}C ∪{ˆk2
r}R,
where the integers p, q, r ∈I; 1 ≤p ≤ˆnp; q = ˆnp + p, and 2ˆnp + 1 ≤r ≤n.
Let us rearrange Eq. (8) and Eq. (9), respectively as follows,
10
n
X
j=1
˜Vij ˜wj = ˜bi, and
n
X
j=1
ˆVi j ˆwj = ˆbi,
where
˜w j = 2˜uj sin(˜kjπ),
(14)
ˆw j = −2ˆujˆk j sin(ˆk jπ).
(15)
Since the jth columns of the transposed Vandermonde matrices ˜V and ˆV depend solely on ˜k2
j and ˆk2
j, i.e.,
˜Vi j = ˜Vi j(˜k2
j) and ˆVij = ˆVij(ˆk2
j), we can rearrange above equations as,
˜np
X
p=1
˜k2
p∈C
˜Vip(˜k2
p) ˜wp +
2˜np
X
q=˜np+1
˜k2
q∈C
˜Viq(˜k2
q) ˜wq +
˜n
X
r=2˜np+1
˜k2
r ∈C
˜Vir(˜k2
r) ˜wr = ˜bi,
ˆnp
X
p=1
ˆVip(ˆk2
p) ˆwp +
2ˆnp
X
q=ˆnp+1
ˆViq(ˆk2
q) ˆwq +
ˆn
X
r=2ˆnp+1
˜k2
r ∈R
ˆVir(ˆk2r ) ˆwr = ˆbi.
Since ˜k2
q = ˜k∗2
p , ∀(˜k2
p, ˜k2
q) ∈C with q = p + ˜np, and ˆk2
q = ˆk∗2
p , ∀(ˆk2
p, ˆk2
q) ∈C with q = p + ˆnp, it follows that
˜Viq(˜k2
q) = ˜V∗
ip(˜k∗2
p ), and ˆViq(ˆk2
q) = ˆV∗
ip(ˆk∗2
p ). Therefore, we have
˜np
X
p=1
˜k2
p∈C
˜Vip(˜k2
p)
| {z }
complex
˜wp + ˜V∗
ip(˜k∗2
p )
| {z }
conjugate
˜wp+˜np
+
˜n
X
r=2˜np+1
˜k2
r ∈R
˜Vir(˜k2
r)
| {z }
real
˜wr =
˜bi
|{z}
real
,
ˆnp
X
p=1
ˆk2
p∈C
ˆVip(ˆk2
p)
| {z }
complex
ˆwp + ˆV∗
ip(ˆk∗2
p )
| {z }
conjugate
ˆwp+ˆnp
+
ˆn
X
r=2ˆnp+1
˜k2
r ∈R
ˆVir(ˆk2r )
|{z}
real
ˆwr =
ˆbi
|{z}
real
.
Since the right-hand sides of the above equations are real, the left-hand sides must also be real. Thus, ˜Vir(˜k2
r)
and ˆVir(ˆk2
r) being real implies that ˜wr and ˆwr are real as well. From Eq. (14), it follows that ˜u j ∈R if ˜w j ∈R
and ˜kj ∈R, and ˜uj ∈I if ˜wj ∈R and ˜k j ∈I. Similarly, from Eq. (15), ˆuj ∈R if ˆwj ∈R, regardless of
whether ˜k j ∈R or ˜kj ∈I.
For the sum of the first two terms to be real, the complex conjugates ˜Vip(˜k2
p) and ˜V∗
ip(˜k∗2
p ) require that
˜wp and ˜wp+˜np are also complex conjugates. Consequently, from Eq. (14), ˜up and ˜up+˜np are also complex
conjugates. Similarly, ˆwp and ˆwp+ˆnp are conjugates, which means ˆup and ˆup+ˆnp are conjugates according to
Eq. (15). Therefore, we have:
˜wp+˜np = ˜w∗
p =⇒˜up+˜np = ˜u∗
p,
ˆwp+ˆnp = ˆw∗
p =⇒ˆup+ˆnp = ˆu∗
p.
Table 2 shows possible combinations of mode numbers ˜kj, ˆk j and mode amplitudes ˜u j, ˆuj. Specifically, ˆuj
can be either real (R) or a complex conjugate pair (C), while ˜u j can be real (R), imaginary (I), or complex
conjugate pair (C).
11
˜k2
j
R+
R−
C
˜k j
R
I
C
˜w j
R
R
C
˜u j
R
I
C
ˆk2
j
R+
R−
C
ˆkj
R
I
C
ˆwj
R
R
C
ˆuj
R
R
C
Table 2: Mode amplitudes ˜u j and ˆuj correspond to mode numbers ˜kj and ˆk j as described in Eqs. (8) and (9); ˜wj and ˆwj are defined
in (14), and (15). The disjoint sets R, I, and C consist of real numbers, purely imaginary numbers, and complex conjugate pairs,
respectively.
3.5. Algorithm
In this section, we present our GFS algorithm designed to compute derivatives of smooth functions in
non-periodic domains. The method relies on decomposing the function into periodic and aperiodic compo-
nents. The periodic part is managed using FFT, while the aperiodic part is represented by a set of adaptively
constructed non-harmonic modes, effectively capturing derivative jumps at domain boundaries.
This algorithm dynamically constructs the aperiodic basis to align with the function’s smoothness and
discontinuities. The jump conditions required for this basis are either analytically derived or numerically
computed. Subtracting the adaptively modeled aperiodic part from the original function isolates the periodic
component, which is then differentiated using FFT. The periodic and aperiodic derivatives are recombined
to produce the final result.
A key strength of this method lies in its ability to achieve high accuracy without requiring domain ex-
tensions, a common limitation of Fourier extension techniques. The algorithm demonstrates computational
efficiency, with complexity scaling as O(N(n + log2(N))), where N stands for the number of grid points and
n stands for the number of non-harmonic modes (see Sec. 4). Numerical experiments highlight its superior
resolution and convergence rates compared to finite-difference methods, with significantly lower degrees of
freedom required per wavelength.
The steps involved in computing the derivative u′(xi), i = 1, 2, . . . , N are summarized in Algorithm 1.
Algorithm 1 GFS algorithm for decomposing a function into periodic and non-periodic parts and computing
derivatives
1: Choose the number of symmetric modes nc and anti-symmetric modes ns in Eq. (5), typically setting
nc = ns = n ∼O(1).
2: Evaluate the jumps in the derivatives of u(x) across the domain boundaries Jm, either analytically (if
known) or numerically.
3: Compute the elementary symmetric polynomials ˜e j and ˆej from Eqs. (10)-(11).
4: Solve for the (n) roots of Eq. (12) and Eq. (13) to obtain ˜kj and ˆk j, for j = 1, 2, . . . , n.
5: Compute ˜u j and ˆuj using the first equality in Eqs. (8)-(9).
6: Calculate ua(xi) = Pn
j=1
ˆuj cos(ˆk jxi) + ˜u j sin(˜k jxi)
, u′
a(xi) = Pn
j=1
−ˆkjˆuj sin(ˆkjxi) + ˜kj˜u j cos(˜kjxi)
.
7: Determine up(xi) = u(xi) −ua(xi) and obtain u′
p(x) using the Fast Fourier Transform (FFT).
8: Finally, compute the derivative u′(xi) = u′
p(xi) + u′
a(xi). Higher derivatives, if required, can be computed
analogously.
12
3.6. Numerical computation of jump conditions
Now, we briefly describe the classical one-sided finite difference schemes used to compute the deriva-
tives, u(m)(x), at the domain boundaries, which are then used to evaluate the jumps Jm in u(m)(x) across the
domain boundaries. The computation of higher-order derivatives with high accuracy in a stable manner is
a significant challenge in numerical analysis and scientific computing. Numerical differentiation, especially
for higher-order derivatives, can suffer from stability issues, rounding errors, and amplification of small
numerical errors.
Let us express the dth derivative of a function u(x) at a discrete node j (u j ≡u(xj)) in terms of the
neighbor nodes u j+m and use Taylor series expansion w.r.t. u j to have,
∆xd u(d)
j
=
M
X
m=0
amu j+m =
M
X
m=0
am
∞
X
n=0
u(n)
j
n! (m∆x)n =
∞
X
n=0
M
X
m=0
am
m(n)
n!
∆xnu(n)
j .
(16)
Here, M +1 = d +r stands for the width of the computational stencil used in finite difference approximation.
The weights am (M + 1 unknowns) can be computed by solving the system of equations below obtained by
comparing equal order terms on each side with a formal order of accuracy of ∆xM+1−d in Eq. (16),
M
X
m=0
am
mn
n! = δn,d =
1,
n = d,
0,
n , d,
n = 0, 1, 2, 3, ..., M.
(17)
Above expressions involving a forward differencing stencil can be used on the left-side boundary, and
for the right-side boundary, a one-sided backward differencing stencil must be used, and the corresponding
system of equations can be obtained by replacing m with −m in the above equations.
4. Computational cost
The computational complexity of each step of the algorithm, described in Sec. 3.5, is shown in Table
3. Numerical evaluation of dth derivative via finite difference approximation with O(∆x)r accuracy requires
O(r + d) operations where (r + d) is the stencil width. The total cost for evaluating Jm, m = 0, 1, 2, . . . , 4n −1
is O(P4n−1
d=1 (r+d)) = O((4n−1)(r+2n)) ≈O(4nr+8n2). Evaluation of the elementary symmetric polynomials
involve inversion of a n×n matrix which requires O( 2
3n3+ 1
2n2+ n
6) operations through LU decomposition and
a matrix vector product of O(2n2 + n) operations. Finding roots of a nth degree polynomial requires another
O(n3) operations. Computation of the expansion coefficients ˜u j, ˆuj involve inversion of a n × n matrix and
require O(n3 + n) operations. Evaluation of ua(xj), u′
a(xj), j = 1, 2, . . . , N requires O(2Nn) operations each.
Evaluation of u′
p(xj) through FFT requires O(N log2 N) operations. Finally, u′(xj) = u′
p(xj) + u′
a(xj) requires
O(N) operations. Overall, the total cost of the computational scales as O(n3+9n2+N(4n+log2 N)) operations.
We demonstrate later that n = O(1) ≪N is sufficient for practical purposes so that the overall cost
remains tractable with an effective complexity of O(N(4n + log2 N)). We can conclude that for n = O(1) ≪
N, the computational penalties due to the evaluation of the jumps Jm, the modes ˜k, ˆk and the expansion
coefficients ˜u, ˆu are negligible and the major contribution comes from the evaluation of ua(xj), u′
a(xj) at each
node (xj), similar to the reduction-to-periodicity technique of Roache (1978).
5. Numerical Evidences
In this section, we compare the accuracy and convergence of our GFS method for computing the first
derivative with Eckhoff Eckhoff (1995), Roache Roache (1978), Prony de Prony (1795), finite difference
13
step
cost
Numerical evaluation of Jm
O(4nr + 8n2)
Computation of ˜ej, ˆej
O(n3 + n2)
Polynomial roots ˜kj, ˆkj
O(n3)
Computation of ˜uj, ˆuj
O(n3 + n)
Computation of ua(xj), u′
a(xj)
O(4Nn)
Computation of u′
p(xj) via FFT
O(N log2 N)
Computation of u′(xj) = u′
p(xj) + u′
a(xj)
O(N)
Total cost
O(3n3 + 9n2 + 4nr + 4Nn + N log2 N)
Effective cost (n = O(1) ≪N, r = O(1))
O(N(4n + log2 N)+3n3)
Table 3: Estimate of computational cost for evaluating derivative of u(x) with GFS. Here, M + 1 = r + d is the stencil width used in
finite difference (FD) approximation of the jumps Jm, r is the accuracy of a FD scheme for dth derivative. For the highest derivative
d = 4n −1.
(FD) and standard FFT methods. For the Eckhoff method, discontinuities are assumed at the boundary and
analytical derivativees are used for computing all the jumps (Jm, m = 0, 1, . . . , q −1) for both Eckhoff and
Roache methods. For the FD method, standard central schemes are applied at interior nodes, while one-
sided stencils are used near boundary nodes with a computational stencil width r + 1, leading to O(∆xr)
accuracy, where ∆x is the grid spacing (see Sec. 3.6 for details). In the GFS method, the jumps Jm for
m = 0, 1, . . . , 4n −1 are evaluated both analytically and numerically using one-side FD schemes with a
stencil width 4n −1 + r, giving the highest derivative’s jump at the end points, J4n−1, a formal accuracy of
O(∆xr). However, numerical evaluation of jumps introduces additional error to the GFS method. Computing
higher derivatives of highly oscillatory functions is challenging due to the inherent ill-posedness known as
Runge’s phenomenon, with increasing rounding errors reducing overall accuracy as the magnitude of the
higher derivatives increases. We present the numerical error of the GFS method, comparing results with
jumps Jm evaluated both analytically and numerically to highlight the impact of numerical evaluation. The
error (e) for each method is computed against the analytic values and quantified using the Lp-norm of e,
defined as follows:
||e||p =
∆x
N
X
i=0
|e(xi)|p
1/p
5.1. Modulated sine function
Consider the modulated sine function
u(x) = ea(x+π) sin[b(x + π)], x ∈[−π, π],
(18)
where, a = −1/π, b = 3/4. Figs. 2(a,c) illustrate the function u(x) and its decomposition into periodic (up(x))
and aperiodic (ua(x)) components for n = 1 and 2, respectively. In Fig. 2(a), for n = 1, the magnitude of
both parts exceed that of the original function, and their mutual cancellation reproduces the original function,
indicating a non-normal basis. Conversely, Fig. 2(c) demonstrates that ua(x) completely resolves u(x) for
n = 2, resulting in up(x) = 0.
Figs. 2(b,d) compare the GFS and FFT methods to the analytical values, demonstrating that GFS avoids
Gibbs oscillations, unlike FFT. Table 4 quantifies the L2, L∞norms of the approximation error for the func-
tion (e) and its derivative (e′). As the number of aperiodic modes n in GFS increases, the error in the first
14
derivative e′ converges rapidly, reaching machine precision by n = 2 with analytically evaluated jump condi-
tions Jm. This faster convergence is attributed to the two degrees of freedom (exponents a, b) of the function
u(x) in the space of complex exponential functions. Even when Jm is evaluated numerically, e′ decays
quickly, with accuracy reliant on the finite difference (FD) schemes. Enhanced accuracy in the FD scheme
for evaluating Jm decreases numerical error, enabling the overall error to approach that of the analytically
evaluated case.
(a)
(b)
(c)
(d)
Figure 2: A modulated sine function, defined in Eq. (18), is approximated using GFS with N = 64 mesh points. The decomposition
into periodic and aperiodic parts is shown in (a, c), while (b,d) compares the numerical error in the first derivative for GFS and FFT
against analytical values for n = 1 and 2 aperiodic modes, respectively. The FFT shows Gibbs oscillations, resulting in O(1) error,
whereas GFS avoids these oscillations, yielding a significantly lower numerical error by several orders of magnitude.
5.2. Gaussian
Consider the non-periodic Gaussian function defined on the domain [−π, π],
u(x) = exp(−[(x −x0)/w]2),
−π ≤x ≤π.
(19)
where, x0 = 3π/4, w = 1. Fig. 3(a) displays u(x), along with the periodic part (up(x)) and the aperiodic
part (ua(x)) derived from GFS for n = 3 and N = 64. Fig. 3(b) compares u′(x) from GFS, FFT, analytical
15
n
max |e|
||e||2
max |e′|
||e′||2
Jump (Jm)
1
8.02 × 10−15
6.19 × 10−15
8.49 × 10−6
4.49 × 10−6
analytical
1
7.23 × 10−15
5.55 × 10−15
1.19 × 10−5
5.14 × 10−6
FD2
1
7.74 × 10−15
5.30 × 10−15
8.49 × 10−6
4.49 × 10−6
FD4
1
1.06 × 10−14
5.29 × 10−15
8.49 × 10−6
4.49 × 10−6
FD6
1
4.69 × 10−15
4.43 × 10−15
8.49 × 10−6
4.49 × 10−6
FD8
2
5.55 × 10−17
1.74 × 10−17
3.52 × 10−15
3.80 × 10−15
analytical
2
1.11 × 10−16
3.48 × 10−17
1.00 × 10−10
3.17 × 10−11
FD2
2
5.55 × 10−17
1.74 × 10−17
4.80 × 10−14
1.85 × 10−14
FD4
2
1.11 × 10−16
3.48 × 10−17
2.22 × 10−13
8.27 × 10−14
FD6
2
1.36 × 10−20
4.25 × 10−21
6.73 × 10−13
2.97 × 10−13
FD8
Table 4: A modulated sine function defined in Eq. (18) is approximated using the GFS for N = 64. The errors for the function (e)
and its first derivative (e′) are analyzed as the number of sine or cosine modes (n) increases. Jumps Jm for m = 0, 1, . . . , 4n −1 are
calculated analytically or numerically using one-sided finite difference schemes with stencil width 4n −1 + r and O(∆xr) accuracy for
J4n−1, noted as “FDr” in the Jm column.
values, showing that GFS avoids Gibbs oscillations and closely matches the analytic solution, while FFT
exhibits Gibbs oscillations. The L∞, L2 norms of the error in u′(x) (denoted as ||e′||p) are presented in Table
5 for n = 3, with increasing grid resolution N, and are compared to a sixth-order finite difference scheme
(r = 6) and FFT. Among the methods, FFT performs worst with O(1) error due to the Gibbs oscillations.
Fig. 4 illustrates the superior convergence of GFS in comparison to FD, FFT, Roache, Eckhoff, and Prony
methods. With analytically evaluated Jm, GFS converges rapidly with increasing N, in contrast to all the
other methods, and also yields significantly lower error. Although numerical evaluation of Jm introduces
additional error in GFS, its convergence remains far superior to that of all the other methods. The Prony
approximation computed with M = N/2 in Eq. (2) becomes ill-conditioned for large N and the computation
blows up for N > 64.
(a)
(b)
Figure 3: A Gaussian, defined in Eq. (19), is approximated using GFS with N = 64 grid points, and n = 3: (a) shows its decomposition
into periodic and aperiodic components, while (b) compares the first derivative approximation using GFS and FFT against analytical
values. GFS aligns well with the analytical results without exhibiting Gibbs oscillations, unlike FFT, which shows these oscillations.
16
Method
GFS
FD: O(∆xr)
FFT
Remark
Jm : analytical
Jm : numerical
r = 6
N
n
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
16
3
6.48e-07
1.36e-06
3.98e-01
2.56e-01
9.37e-02
6.04e-02
1.39e+00
1.31e+00
32
3
3.86e-11
4.76e-11
1.92e-04
8.52e-05
2.52e-03
1.14e-03
2.33e+00
1.74e+00
64
3
1.50e-14
1.20e-14
2.55e-09
7.99e-10
4.18e-05
1.35e-05
4.24e+00
2.40e+00
128
3
6.66e-15
5.48e-15
3.45e-12
1.08e-12
4.00e-07
9.78e-08
8.04e+00
3.36e+00
256
3
1.62e-14
1.07e-14
8.91e-12
1.43e-12
3.73e-09
8.45e-10
1.57e+01
4.73e+00
512
3
3.90e-14
2.15e-14
3.52e-11
3.93e-12
3.82e-11
1.03e-11
3.09e+01
6.67e+00
Table 5: A Gaussian defined in Eq. (19) is approximated using GFS for various grid sizes N. The first derivative approximation error
(e′) is compared among GFS, finited difference (FD), and FFT methods. Jumps Jm, m = 0, 1, . . . , 4n −1, are calculated analytically or
numerically using one-sided FD schemes with stencil width M + 1 = r + 4n −1 such that J4n−1 has O(∆x6) accuracy (see Sec. 3.6).
Figure 4: For the Gaussian function defined in Eq. (19), convergence of the numerical error (L∞−norm) in first derivative is compared
for GFS, a sixth-order finite difference (FD6), FFT, Roache, Eckhoff, and Prony methods. For Roache and Eckhoff methods the jumps
Jm are evaluated analytically, and for GFS method Jm are evaluated both analytically and numerically. Here, “Jm : ideal” refers to the
case where the jumps Jm, m = 0, 1, . . . , q −1, q = 4n, are evaluated analytically. GFS is robust and exhibits superior convergence with
mesh size N for the same number of jumps (q), whether Jm are evaluated analytically or numerically. The Prony method (M = N/2) is
numerically unstable for large mesh sizes (N > 64).
17
5.3. Log function
Consider the non-periodic logarithmic function in the domain [−π, π],
u(x) = log
x + π + 1
2
!
,
−π ≤x ≤π.
(20)
Fig. 5(a) displays the function u(x) along with its periodic part up(x) and aperiodic part ua(x) obtained
using GFS for n = 3 and N = 32. Fig. 5(b) compares u′(x) calculated via GFS, FFT, and analytical
methods. GFS effectively avoids Gibbs oscillations, closely aligning with the analytical method, while FFT
exhibits these oscillations. The L∞, L2 norms of the error in u′(x) (denoted as ||e′||p) are detailed in Table
6 for n = 3, highlighting the impact of increased grid resolution N and comparisons with a sixth-order
finite difference scheme and FFT. Among the three methods, FFT performs the worst with O(1) error due to
Gibbs oscillations. Fig. 6 shows GFS method converging faster and with lower error than FD, FFT, Roache,
Eckhoff, and Prony methods. When Jm is evaluated analytically, GFS rapidly improves with increasing
N, outpacing all the other methods. Even with numerical Jm, GFS remains markedly superior. The Prony
method with M = N/2 in Eq. (2) becomes ill-conditioned for large N and fails for N > 64.
(a)
(b)
Figure 5: A function u(x) = log(x + π + 1/2), −π ≤x ≤π, is approximated with GFS using N = 32 mesh points and n = 3:
Decomposition of u(x) into periodic and aperiodic parts is shown in (a), approximation of the first derivative with GFS and FFT is
compared with analytical values in (b); GFS does not exhibit Gibbs oscillations and aligns well with the analytic values while FFT
shows Gibbs oscillations.
5.4. multi-mode aperiodic function
Consider the function composed of sinusoidal components with Nk non-integer modes (k j), which are
non-periodic in the domain [−π, π],
u(x) =
Nk−1
X
j=0
sin(kjx) + cos(k jx),
−π ≤x ≤π,
k j = j + ∆j,
∆j ≡1
Nk
+
j
Nk
Nk −2
Nk −1
!
∈
" 1
Nk
, 1 −1
Nk
#
.
(21)
This function is highly oscillatory and enables us to assess the resolution power (R) of the GFS, defined as
the degrees of freedom needed to resolve one wavelength to machine precision. This can be expressed as
R ≡
N
max kj ≈N
Nk .
18
GFS
FD: O(∆xr)
FFT
Jm : analytical
Jm : numerical
r = 6
N
n
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
16
3
7.22e-04
9.53e-04
1.40e-02
8.82e-03
5.84e-02
3.68e-02
5.68e+00
5.85e+00
32
3
8.49e-06
8.84e-06
7.53e-04
3.34e-04
9.36e-03
4.18e-03
1.02e+01
8.12e+00
64
3
2.21e-08
1.77e-08
1.05e-05
3.29e-06
8.17e-04
2.59e-04
1.94e+01
1.14e+01
128
3
2.09e-11
1.21e-11
2.98e-08
6.62e-09
4.03e-05
9.06e-06
3.78e+01
1.61e+01
256
3
6.09e-14
5.73e-14
6.69e-11
1.09e-11
1.26e-06
2.01e-07
7.46e+01
2.28e+01
512
3
9.44e-14
7.54e-14
4.82e-11
5.99e-12
2.93e-08
3.30e-09
1.48e+02
3.22e+01
Table 6: The log function defined in Eq. (20) is approximated using GFS with various grid sizes N. The first derivative approximation
error (e′) is compared among GFS, finited difference (FD), and FFT methods. Jumps Jm, m = 0, 1, . . . , 4n−1, are calculated analytically
or numerically using one-sided FD schemes with stencil width M + 1 = r + 4n −1, ensuring that J4n−1 achieves O(∆x6) accuracy.
Fig.
7(a) displays u(x) for N = 64, Nk = 30, demonstrating high oscillation, with the maximum
wavenumber kNk−1 near the Nyquist limit (Nk ∼N/2). Fig. 7(b, c) compares the first derivative obtained
via GFS for n = 6 with the FFT and analytical methods for N = 64 and N = 80. GFS closely aligns with
the analytical method, exhibiting no Gibbs oscillations, while FFT has Gibbs oscillations. For quantitative
evaluation, Table 7 presents the L∞and L2 norms of the error in the first derivative for the current method
(GFS), a sixth-order finite difference (FD) method, and the FFT. We set Nk = 30 and begin with N = 64,
where the maximum wave number (kNk−1) is close to the Nyquist limit, and examine the impact of increasing
resolution and the number of aperiodic modes n. Ideally, if n = Nk, the aperiodic part would exactly recover
the total function u′(xj). However, this involves higher (d) derivatives, with their magnitudes scaling as
kd
Nk−1, leading to significant increases in precision loss. Moreover, as n −→Nk ≫1, this approach becomes
costly, so we aim to limit n to O(1) for practical computations to control expenses and maintain precision.
With max u′(x) ≈600 (Fig. 7(b,c)), Table 7 shows that machine precision is achieved with a relatively
small number of aperiodic modes (n = O(1)) using GFS. The error decreases much more rapidly with
mesh refinement (increasing N) for GFS (with analytical Jm) even at n = 2, and improves markedly as
n grows. While the sixth-order FD scheme’s error scales as O(∆x6), the FFT error remains O(1) in non-
periodic domains due to the Gibbs phenomenon. For Nk = 30 and a fixed target error of O(10−10), GFS
requires about N = 80 −90, yielding a resolution power R ≈3; in contrast, the sixth-order FD scheme needs
N = 8192 (see Fig. 8), giving R = 300. Thus, the FD mesh must be two orders of magnitude finer than GFS
for the same accuracy, evidencing GFS’s superior resolution. Even with a numerically evaluated Jm, GFS
maintains higher accuracy and convergence than FD and FFT methods, see Table 7.
Fig. 8 shows the GFS method converging faster and with lower error than the FD, FFT, Roache’s,
Eckhoff’s, and Prony’s methods. When Jm is evaluated analytically, GFS rapidly improves with increasing
N, surpassing all other methods. Even with numerical Jm, GFS remains markedly superior for large N.
Polynomial-based methods like the Roache and Eckhoff approaches become ill-conditioned as the number of
jumps (q) in the discontinuous part increases, while GFS stays robust. The Prony’s method is ill-conditioned
due to high oscillations, causing the computation to fail with M = Nk and M = N/2 (not shown) in Eq. (2).
In summary, the convergence rate of GFS improves with the number of aperiodic modes n, and GFS has
a superior resolution power (R ≈3) compared to the sixth-order finite difference scheme (R ≈300).
5.5. Monomials
We consider the monomial functions u(x) = xm, −π ≤x ≤π for m = 1, 3, which are non-periodic and
have finite jumps in the function values at the domain endpoints. Although the GFS is not primarily designed
19
Figure 6: For the log function defined in Eq. (20), convergence of the numerical error (L∞−norm) in first derivative is compared
for GFS, a sixth-order finite difference (FD6), FFT, Roache’s, Eckhoff’s, and Prony’s methods. For the Roache and Eckhoff methods,
the jumps Jm are evaluated analytically, and for the GFS method, Jm are evaluated both analytically and numerically. Here, “Jm :
ideal” refers to the case where the jumps Jm, m = 0, 1, . . . , q −1, q = 4n, are evaluated analytically. GFS is robust and shows superior
convergence with mesh size N for the same number of jumps (q), whether Jm are evaluated analytically or numerically. The Prony
method (M = N/2) is numerically unstable for large mesh sizes (N > 64).
for approximating polynomials, it can still accommodate such functions by regularizing jumps. The GFS
relies on endpoint conditions related to jumps, which can quickly diminish for monomial functions, resulting
in rank-deficient matrices and vanishing right-hand sides. This issue can be resolved by regularizing jumps
using very small numerical values (10−15 in the following cases) when the actual jump is zero. It’s important
to note that regularization is only necessary for the analytical case; for numerical computations, the inherent
error of the numerical scheme provides the required regularization.
For m = 1, this corresponds to the ramp function, a well-known test case that challenges Fourier spectral
methods. Fig. 9(a) shows the decomposition of u(x) into periodic up(x) and aperiodic ua(x) components
obtained by GFS for n = 1. Fig. 9(b) compares the first derivative approximations of u(x) using GFS and
FFT; GFS avoids Gibbs oscillations and closely matches analytical values, whereas FFT exhibits them. Table
8 compares the L∞, L2 norms of the first derivative error ||e′||p for GFS, FD and FFT methods at N = 64.
The numerical error for Jm and the stand-alone FD method approaches machine precision, as the sixth-order
FD schemes fully resolve the linear function u(x) = x.
For m = 3, this corresponds to a cubic monomial. Fig. 10(a,c) shows the decomposition of u(x) into
periodic up(x) and aperiodic ua(x) components using GFS for n = 1 and 2, respectively. Fig. 10(b,d)
compares the first derivative approximations of u(x) from GFS and FFT; GFS avoids Gibbs oscillations and
aligns closely with analytical values, while FFT exhibits oscillations leading to O(1) error. Table 9 compares
the L∞and L2 norms of the first derivative error ||e′||p for GFS, FD and FFT methods at N = 64. The
numerical error for GFS converges quickly with increasing n, whereas the sixth-order FD scheme achieves
near machine precision by exactly approximating the monomials for m ≤5.
6. Conclusions and future directions
20
GFS
FD: O(∆xr)
FFT
Jm : analytical
Jm : numerical
r = 6
N
Nk
n
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
64
30
2
7.34e-02
1.16e-01
5.52e+03
1.74e+03
1.93e+02
1.35e+02
2.45e+00
1.24e+00
128
30
2
3.23e-06
1.80e-06
6.19e+01
1.43e+01
1.95e+01
7.23e+00
1.22e+00
3.40e-01
256
30
2
1.55e-08
5.50e-09
3.61e-02
5.92e-03
4.40e-01
1.41e-01
1.03e+00
2.23e-01
512
30
2
1.08e-10
2.64e-11
1.12e-05
1.27e-06
7.84e-03
2.21e-03
9.46e-01
1.55e-01
64
30
4
1.06e-03
1.54e-03
6.97e+05
2.63e+05
1.93e+02
1.35e+02
2.45e+00
1.24e+00
128
30
4
5.24e-12
3.56e-12
3.56e+02
9.98e+01
1.95e+01
7.23e+00
1.22e+00
3.40e-01
256
30
4
3.79e-12
2.27e-12
1.81e-03
2.85e-04
4.40e-01
1.41e-01
1.03e+00
2.23e-01
512
30
4
7.12e-12
4.77e-12
2.73e-08
3.81e-09
7.84e-03
2.21e-03
9.46e-01
1.55e-01
64
30
6
1.68e-03
3.86e-03
6.57e+06
2.74e+06
1.93e+02
1.35e+02
2.45e+00
1.24e+00
72
30
6
1.16e-07
1.97e-07
5.04e+06
1.50e+06
1.61e+02
9.46e+01
1.81e+00
6.94e-01
80
30
6
5.27e-10
7.34e-10
1.96e+06
5.59e+05
9.50e+01
6.48e+01
1.59e+00
5.53e-01
96
30
6
1.64e-12
1.46e-12
1.79e+05
4.68e+04
6.67e+01
3.00e+01
1.39e+00
4.36e-01
128
30
6
2.03e-12
1.39e-12
8.34e+02
2.19e+02
1.95e+01
7.23e+00
1.22e+00
3.40e-01
256
30
6
2.74e-12
1.45e-12
7.29e-05
1.20e-05
4.40e-01
1.41e-01
1.03e+00
2.23e-01
512
30
6
4.40e-12
3.15e-12
4.80e-07
6.20e-08
7.84e-03
2.21e-03
9.46e-01
1.55e-01
Table 7: A multi-mode sinusoidal function defined by Eq. (21) with Nk = 30 is approximated using GFS, finite difference (FD), and
FFT methods. The error of the approximation for the first derivative (e′) is shown with increasing number of aperiodic modes (n). The
jumps Jm are computed either analytically or numerically through one sided finite difference schemes of rth order accuracy as indicated.
GFS
FD: O(∆xr)
FFT
Jm : analytical
Jm : numerical
r = 6
N
n
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
64
1
1.55e-14
9.50e-15
3.03e-14
1.47e-14
9.38e-14
3.00e-14
4.43e+01
2.71e+01
64
2
7.22e-15
7.98e-15
6.57e-13
2.91e-13
9.38e-14
3.00e-14
4.43e+01
2.71e+01
64
3
1.03e-14
7.97e-15
1.57e-12
9.18e-13
9.38e-14
3.00e-14
4.43e+01
2.71e+01
Table 8: The ramp function u(x) = x, −π ≤x ≤π is approximated with the GFS. Error of the approximation for the function (e) and its
first derivative (e′) are shown with increasing number of sin or cos modes (n). Here, N = 64 and the jump conditions Jm are computed
from exact derivatives or numerical derivatives (one-sided finite difference method) as indicated.
GFS
FD: O(∆xr)
FFT
Jm : analytical
Jm : numerical
r = 6
N
n
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
||e′||∞
||e′||2
64
1
1.13e-04
5.98e-05
1.13e-04
5.98e-05
7.07e-13
3.33e-13
4.38e+02
2.68e+02
64
2
6.70e-06
6.06e-06
2.77e-11
1.94e-11
7.07e-13
3.33e-13
4.38e+02
2.68e+02
64
3
8.32e-09
6.90e-09
1.22e-10
5.75e-11
7.07e-13
3.33e-13
4.38e+02
2.68e+02
Table 9: A function u(x) = x3, −π ≤x ≤π, is approximated using GFS with N = 64. The error of the approximation for the first
derivative (e′) is compared among various methods. Here, the jump conditions Jm are computed either exactly or numerically through
one-sided finite difference (FD) schemes with O(∆x6) accuracy as indicated in the column Jm.
21
(a)
(b)
(c)
Figure 7: A multi-mode sinusoidal function defined by Eq. (21) with Nk = 30 is approximated using GFS with n = 6. Panel (a) displays
its periodic and aperiodic decompositions for N = 64. Panels (b) and (c) compare its first derivative approximations and the errors from
GFS and FFT against analytical results for N = 64 and N = 80.
In this work, we introduce the Generalized Fourier Series (GFS), a spectral method that extends Fourier
techniques to non-periodic domains while retaining FFT-level efficiency. GFS decomposes a function into a
periodic component—approximated with the Fast Fourier Transform—and an aperiodic component captured
by a small set (n = O(1)) of adaptive, non-harmonic sinusoids that handle non-periodicity with high-order
accuracy. This construction avoids artificial domain extensions and their overhead, seamlessly accommo-
dates derivative jumps and boundary discontinuities, and delivers high-resolution accuracy and robustness.
With a resolution power (degrees of freedom per wavelength) comparable to FFT on periodic problems,
and a low-rank approximation for the aperiodic part, the overall complexity remains linear, making GFS a
practical and efficient alternative to existing techniques.
Comprehensive numerical experiments show that GFS consistently outperforms classical approaches—including
Roache’s polynomial-based correction, Eckhoff’s reconstruction with Bernoulli polynomials, Prony’s expo-
nential fitting, and standard finite-difference and FFT methods. Relative to Roache’s method, GFS avoids
a fixed polynomial correction—which can be unstable for oscillatory functions—in favor of adaptive sinu-
soidal modes that yield greater robustness and accuracy; relative to Eckhoff’s method, it retains the ability
to capture discontinuities while offering greater robustness and higher accuracy, particularly when a large
number of jumps must be resolved; and relative to Prony’s method, it does not require the precise number of
modes to be specified a priori and remains numerically stable at large mesh sizes (N > 64), where Prony’s
scheme typically becomes ill-conditioned. Across meshes, GFS delivered significantly better convergence
rates and resolution power: it achieves FFT-like resolution, while a sixth-order finite-difference scheme de-
mands far more resources—underscoring GFS’s orders-of-magnitude performance advantage. While GFS
attains optimal accuracy when exact jump conditions are known, approximating these jumps with finite dif-
ferences can limit overall accuracy; the quality of jump evaluation depends on the underlying numerical
scheme and could be improved with better estimators. A detailed treatment of jump-condition estimation
lies beyond the scope of this article and will be addressed in future work.
Across a range of test functions—including modulated sines, Gaussians, and logarithmic functions—GFS
showed rapid convergence, minimal Gibbs oscillations, and superior accuracy in approximating derivatives.
The results confirm that GFS provides a robust, efficient, and accurate framework for non-periodic function
approximation, offering a compelling alternative to existing methods for applications in numerical PDEs,
signal processing, and beyond.
22
Figure 8: For the multi-mode function defined in Eq. (21) with Nk = 30 non-integer modes, convergence of the numerical error (L∞−
norm) in the first derivative is compared for GFS, a sixth-order finite difference (FD6), FFT, Roache, Eckhoff, and Prony methods. For
the Roache and Eckhoff methods, the jumps Jm are evaluated analytically, and for the GFS method, Jm are evaluated both analytically
and numerically. Here, “Jm : ideal” refers to the case where the jumps Jm, m = 0, 1, . . . , q −1, q = 4n, are evaluated analytically. GFS
shows superior convergence with mesh size N. As the number of jumps (q) increases, GFS remains robust and converges rapidly, while
the other jump-based methods (Roache and Eckhoff) suffer from numerical stability issues due to the function’s high oscillations and
ill-conditioning at large q. The Prony’s method (M = Nk) is numerically unstable.
Future work will expand the GFS framework to multi-dimensional applications, addressing PDEs such as
advection-diffusion and Poisson equations, and eventually tackling computational fluid dynamics problems.
These efforts highlight the transformative potential of GFS for enhancing spectral methods in nonperiodic
domains.
Appendix A. Solution of the non-linear systems Eqs. (8) and (9) for n = 1, 2, 3
Appendix A.1. Single mode representation of the aperiodic part: n = 1
For n = 1, the solution of the above equations is straightforward and involves the jumps in the derivatives
of u(x) at the endpoints J0, J1, J2, J3. First, we solve for the modes ˜k1, ˆk1, followed by the amplitudes ˜u1, ˆu1
as below,
2˜u1 sin(˜k1π) = J0 = −J2
˜k2
1
,
−2ˆk1ˆu1 sin(ˆk1π) = J1 = −J3
ˆk2
1
,
(A.1)
˜k2
1 = −J2/J0,
ˆk2
1 = −J3/J1,
˜u1 =
J0
2 sin(˜k1π),
ˆu1 = −
J1
2ˆk1 sin(ˆk1π)
.
(A.2)
Note that ˆk1 = ± √−J3/J1 and ˜k1 = ± √−J2/J0 and the ± sign is irrelevant here because ˆuj cos(ˆk jx) and
˜uj sin(˜kjx) (therefore, uc(x) and us(x)) are even functions of ˆk and ˜k, respectively. We define the complex
square root as √z ≡√|z|eiθ/2 where z = |z|eiθ, −π ≤θ ≤π.
23
(a)
(b)
Figure 9: Ramp function u(x) = x is approximated with GFS using n = 1, N = 64: Decomposition of u(x) into periodic and aperiodic
parts is shown in (a), approximation of the first derivative with GFS and FFT is compared with analytical values in (b); GFS does not
exhibit Gibbs oscillations and aligns well with the analytic values while FFT shows Gibbs oscillations.
Appendix A.2. Two-mode representation of the aperiodic part: n = 2
For n = 2, we have eight unknowns. So, let us consider the jumps in the derivatives of u(x) at the
endpoints J0, J1, J2, . . . , J7 which yields the following set of equations,
2
"˜u1 sin(˜k1π)
˜u2 sin(˜k2π)
#
=
1
1
−˜k2
1
−˜k2
2
!−1 "J0
J2
#
=
˜k4
1
˜k4
2
−˜k6
1
−˜k6
2
!−1 "J4
J6
#
,
(A.3)
−2
"ˆu1ˆk1 sin(ˆk1π)
ˆu2ˆk2 sin(ˆk2π)
#
=
1
1
−ˆk2
1
−ˆk2
2
!−1 "J1
J3
#
=
ˆk4
1
ˆk4
2
−ˆk6
1
−ˆk6
2
!−1 "J5
J7
#
.
(A.4)
Expanding on both sides of the latter equality in Eq. (A.3) yields,
1
1
−˜k2
1
−˜k2
2
!−1 "J0
J2
#
=
" 1
1
−˜k2
1
−˜k2
2
! ˜k4
1
0
0
˜k4
2
!#−1 "J4
J6
#
,
(A.5)
1
1
−˜k2
1
−˜k2
2
! ˜k4
1
0
0
˜k4
2
! 1
1
−˜k2
1
−˜k2
2
!−1 "J0
J2
#
=
"J4
J6
#
,
(A.6)
−˜k2
1˜k2
2
−
˜k2
1 + ˜k2
2
˜k2
1˜k2
2
˜k2
1 + ˜k2
2
˜k4
1 + ˜k4
2 + ˜k2
1˜k2
2
"J0
J2
#
=
"J4
J6
#
.
(A.7)
Elimination of J0 from the bottom row of Eq. (A.7) yields the following linear system for the elementary
symmetric polynomials in ˜k2
1, ˜k2
2 defined as ˜e1 ≡˜k2
1 + ˜k2
2, ˜e2 ≡˜k2
1˜k2
2,
J0
˜k2
1˜k2
2
+ J2
˜k2
1 + ˜k2
2
J2
˜k2
1˜k2
2
+ J4
˜k2
1 + ˜k2
2
= −
"J4
J6
#
=⇒
"˜e2
˜e1
#
= −
J0
J2
J2
J4
!† "J4
J6
#
.
(A.8)
24
(a)
(b)
(c)
(d)
Figure 10: Cubic monomial u(x) = x3 is approximated with GFS using N = 64 and n = 1 (a-b), n = 2 (c-d): Decomposition of u(x)
into periodic and aperiodic parts is shown in (a,c), approximation of the first derivative with GFS and FFT is compared with analytical
values in (b,d); GFS does not exhibit Gibbs oscillations and aligns well with the analytic values while FFT shows Gibbs oscillations.
Corresponding to Eq. (A.4), we obtain a similar system for the elementary polynomials in ˆk2
1, ˆk2
2 defined as
ˆe1 ≡ˆk2
1 + ˆk2
2, ˆe2 ≡ˆk2
1ˆk2
2 and is given below,
"ˆe2
ˆe1
#
= −
J1
J3
J3
J5
!† "J5
J7
#
.
(A.9)
By solving Eqs. (A.3)-(A.4) for ˜e1, ˜e2, ˆe1, ˆe2, we have
˜e1 ≡J0J6 −J2J4
J2
2 −J0J4
,
˜e2 ≡J2
4 −J2J6
J2
2 −J0J4
,
ˆe1 ≡J1J7 −J3J5
J2
3 −J1J5
,
ˆe2 ≡
J2
5 −J3J7
J2
3 −J1J5
.
Now, ˜k2
j and ˆk2
j are the roots of the polynomials given below, and the amplitudes ˜uj, ˆuj are obtained from
Eqs. (A.3)-(A.4),
˜k4 −˜e1˜k2 + ˜e2 = 0, ˆk4 −ˆe1ˆk2 + ˆe2 = 0,
25
hence
˜k2 =
˜e1 ±
q
˜e2
1 −4˜e2
2
, ˆk2 =
ˆe1 ±
q
ˆe2
1 −4ˆe2
2
.
And
˜u1 =
˜k2
2J0+J2
2(˜k2
2−˜k2
1) sin(˜k1π),
˜u2 =
˜k2
1J0 + J2
2(˜k2
1 −˜k2
2) sin(˜k2π),
ˆu1 = −
ˆk2
2J1+J3
2ˆk1(ˆk2
2−ˆk2
1) sin(ˆk1π),
ˆu2 = −
ˆk2
1J1 + J3
2ˆk2(ˆk2
1 −ˆk2
2) sin(ˆk2π)
.
Above expressions satisfy [u(m)] = [u(m)
a ] and thereby, [u(m)
p ] = 0 for m = 0, 1, 2, . . . , 7. This means that the
periodic part is smooth up to the seventh order derivative, i.e., C7 continuous.
Appendix A.3. Three mode representation of the aperiodic part: n = 3
With n = 3, we have 12 unknowns. So, let us consider the jumps in the derivatives of u(x) at the endpoints
J0, J1, J2, . . . , J11 which yields the following set of equations,
2
˜u1 sin(˜k1π)
˜u2 sin(˜k2π)
˜u3 sin(˜k3π)
=
1
1
1
−˜k2
1
−˜k2
2
−˜k2
3
˜k4
1
˜k4
2
˜k4
3
−1
J0
J2
J4
=
−˜k6
1
−˜k6
2
−˜k6
3
˜k8
1
˜k8
2
˜k8
3
−˜k10
1
−˜k10
2
−˜k10
3
−1
J6
J8
J10
.
(A.10)
−2
ˆu1ˆk1 sin(ˆk1π)
ˆu2ˆk2 sin(ˆk2π)
ˆu3ˆk3 sin(ˆk3π)
=
1
1
1
−ˆk2
1
−ˆk2
2
−ˆk2
3
ˆk4
1
ˆk4
2
ˆk4
3
−1
J1
J3
J5
=
−ˆk6
1
−ˆk6
2
−ˆk6
3
ˆk8
1
ˆk8
2
ˆk8
3
−ˆk10
1
−ˆk10
2
−ˆk10
3
−1
J7
J9
J11
.
(A.11)
Note that the second and third parts in Eqs. (A.10)-(A.11) involving the unknowns ˜k j and ˆk j are same
except that the jumps J0, J2, . . . , J10 are replaced by J1, J3, . . . , J11. So, the procedures for solving ˜k j and ˆkj
are similar.
Assuming full rank, i.e., ˜k1 , ˜k2 , ˜k3, and expanding on both sides of the latter equality in Eqs. (A.10)
yields,
1
1
1
−˜k2
1
−˜k2
2
−˜k2
3
˜k4
1
˜k4
2
˜k4
3
−1
J0
J2
J4
= −
1
1
1
−˜k2
1
−˜k2
2
−˜k2
3
˜k4
1
˜k4
2
˜k4
3
˜k6
1
0
0
0
˜k6
2
0
0
0
˜k6
3
−1
J6
J8
J10,
,
(A.12)
−
1
1
1
−˜k2
1
−˜k2
2
−˜k2
3
˜k4
1
˜k4
2
˜k4
3
˜k6
1
0
0
0
˜k6
2
0
0
0
˜k6
3
1
1
1
−˜k2
1
−˜k2
2
−˜k2
3
˜k4
1
˜k4
2
˜k4
3
−1
J0
J2
J4
=
J6
J8
J10,
,
(A.13)
−
˜k6
1
0
0
0
˜k6
2
0
0
0
˜k6
3
˜k2
2˜k2
3
˜k2
2 −˜k2
3)
˜k4
2 −˜k4
3
˜k2
2 −˜k2
3
˜k2
3˜k2
1
˜k2
3 −˜k2
1)
˜k4
3 −˜k4
1
˜k2
3 −˜k2
1
˜k2
1˜k2
2
˜k2
1 −˜k2
2)
˜k4
1 −˜k4
2
˜k2
1 −˜k2
2
J0
J2
J4
=
˜k2
2˜k2
3
˜k2
2 −˜k2
3)
˜k4
2 −˜k4
3
˜k2
2 −˜k2
3
˜k2
3˜k2
1
˜k2
3 −˜k2
1)
˜k4
3 −˜k4
1
˜k2
3 −˜k2
1
˜k2
1˜k2
2
˜k2
1 −˜k2
2)
˜k4
1 −˜k4
2
˜k2
1 −˜k2
2
J6
J8
J10
.
26
−
˜k6
1
0
0
0
˜k6
2
0
0
0
˜k6
3
˜k2
2˜k2
3
˜k2
2 + ˜k2
3
1
˜k2
3˜k2
1
˜k2
3 + ˜k2
1
1
˜k2
1˜k2
2
˜k2
1 + ˜k2
2
1
J0
J2
J4
=
˜k2
2˜k2
3
˜k2
2 + ˜k2
3
1
˜k2
3˜k2
1
˜k2
3 + ˜k2
1
1
˜k2
1˜k2
2
˜k2
1 + ˜k2
2
1
J6
J8
J10
.
J0
˜k6
1˜k2
2˜k2
3
+
˜k2
2 + ˜k2
3
h
J2˜k6
1 + J8
i
+ J4˜k6
1 + J6
˜k2
2˜k2
3
+ J10
=
0,
(A.14)
J0
˜k2
1˜k6
2˜k2
3
+
˜k2
1 + ˜k2
3
h
J2˜k6
2 + J8
i
+ J4˜k6
2 + J6
˜k2
1˜k2
3
+ J10
=
0,
(A.15)
J0
˜k2
1˜k2
2˜k6
3
+
˜k2
1 + ˜k2
2
h
J2˜k6
3 + J8
i
+ J4˜k6
3 + J6
˜k2
1˜k2
2
+ J10
=
0.
(A.16)
Elimination of the pairs (J8, J10), (J0, J10), (J0, J2) from Eqs. (A.14)-(A.16) yields the following set of
equations involving the elementary symmetric polynomials (˜ej) in ˜k2
1, ˜k2
2, ˜k2
3 defined as ˜e1 ≡˜k2
1 + ˜k2
2 + ˜k2
3, ˜e2 ≡
˜k2
1˜k2
2 + ˜k2
2˜k2
3 + ˜k2
1˜k2
3, ˜e3 ≡˜k2
1˜k2
2˜k2
3,
J0
J2
J4
J2
J4
J6
J4
J6
J8
˜k2
1˜k2
2˜k2
3
˜k2
1˜k2
2 + ˜k2
2˜k2
3 + ˜k2
1˜k2
3
˜k2
1 + ˜k2
2 + ˜k2
3
= −
J6
J8
J10
=⇒
˜e3
˜e2
˜e1
= −
J0
J2
J4
J2
J4
J6
J4
J6
J8
†
J6
J8
J10
.
(A.17)
Similarly, corresponding to Eq. (A.11), we obtain the elementary symmetric polynomials ˆe j in ˆk2
1, ˆk2
2, ˆk2
3
defined as ˆe1 ≡ˆk2
1 + ˆk2
2 + ˆk2
3, ˆe2 ≡ˆk2
1ˆk2
2 + ˆk2
2ˆk2
3 + ˆk2
1ˆk2
3, ˆe3 ≡ˆk2
1ˆk2
2ˆk2
3.
ˆe3
ˆe2
ˆe1
= −
J1
J3
J5
J3
J5
J7
J5
J7
J9
†
J7
J9
J11
.
(A.18)
Now, ˜k2
1, ˜k2
2, ˜k2
3 and ˆk2
1, ˆk2
2, ˆk2
3 are obtained by solving for the roots (λ) of the polynomials below:
λ3 −˜e1λ2 + ˜e2λ −˜e3 = 0 =
3
Y
j=1
(λ −˜k2
j),
λ3 −ˆe1λ2 + ˆe2λ −ˆe3 = 0 =
3
Y
j=1
(λ −ˆk2
j).
Above expressions satisfy [u(m)] = [u(m)
a ] and thereby, [u(m)
p ] = 0 for m = 0, 1, 2, . . . , 11. This implies that
the periodic part is smooth up to the eleventh order derivative, i.e., C11 continuous.
Acknowledgments
Narsimha Rapaka would like to dedicate this work to the Late Professor Ravi Samtaney.
Code and data availability
All source code and datasets to reproduce our results are publicly available at https://github.com/
nrapaka/GFS. The repository includes the GFS implementation, example scripts, and instructions for re-
producing the experiments.
27
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|
Generalized Fourier Series: An N log2(N) extension for aperiodic functions that eliminates Gibbs oscillations Narsimha Reddy Rapakaa,b,∗, Mohamed Kamel Riahia,b a . bEmirates Nuclear Technology Center, Khalifa . Abstract This article introduces the Generalized Fourier Series (GFS), a novel spectral method that extends the classical Fourier series to non-periodic functions. GFS addresses key challenges such as the Gibbs phenomenon and poor convergence in non-periodic settings by decomposing functions into periodic and aperiodic components. The periodic part is represented using standard Fourier modes and efficiently computed via the Fast Fourier Transform (FFT). The aperiodic component employs adaptive, low-rank sinusoidal functions with non-harmonic modes, dynamically tuned to capture discontinuities and derivative jumps across domain boundaries. Unlike conventional Fourier extension methods, GFS achieves high accuracy without requiring computational domain extensions, offering a compact and efficient representation of non-periodic functions. The adaptive low-rank approach ensures accuracy while minimizing computational overhead, typically involving additional complex modes for the aperiodic part. Furthermore, GFS demonstrates a high-resolution power, with degrees of freedom comparable to FFT in periodic domains, and maintains N log2(N) computational complexity. The effectiveness of GFS is validated through numerical experiments, showcasing its ability to approximate functions and their derivatives in non-periodic domains accurately. With its robust framework and minimal computational cost, GFS holds significant potential for advancing applications in numerical PDEs, signal processing, machine learning, and computational physics by providing a robust and efficient tool for high-accuracy function approximations. Keywords: Fourier Spectral Method, Non-periodic Domain, Gibbs Oscillations, Non-harmonic Modes, Complex Fourier Modes, Fast Solvers 1. Introduction The accurate approximation of smooth, non-periodic functions within bounded domains is a classical yet persistently challenging problem in numerical analysis. This issue has profound implications for various applications, particularly in the numerical solution of partial differential equations (PDEs). Traditional approaches, such as finite difference, finite volume, finite element methods, and Fourier or Chebyshev spectral methods, have been extensively employed to address this challenge. While non-spectral methods offer broad ∗Corresponding author. Tel.: +971 2 3125416, Email address: (Narsimha R. Rapaka) Preprint submitted to Elsevier October 17, 2025 16 Oct 2025 applicability to problems with arbitrary boundary conditions, they often suffer from limited resolution, requiring a large number of degrees of freedom per wavelength. On the other hand, spectral methods provide superior resolution power but are inherently constrained to periodic domains or homogeneous boundary conditions. This article aims to broaden the application of Fourier spectral methods to non-periodic functions with minimal computational overhead, focusing solely on the lack of periodicity while assuming sufficient smoothness in the interior. The Fast Fourier Transform (FFT) is widely used for approximating smooth, periodic functions due to its efficiency and spectral convergence. However, when applied to non-periodic functions, FFT suffers from the Gibbs phenomenon, producing spurious oscillations near domain boundaries Hewitt and Hewitt (1979); Gottlieb and Shu (1997); Jerri (1998). Over the years, several strategies have been developed to mitigate this limitation. These include filtering techniques to suppress oscillations, domain extension methods that embed non-periodic functions into larger periodic domains Iserles and Nørsett (2008); Huybrechs (2010); Adcock (2010, 2011); Adcock and Huybrechs (2014); Geronimo and Liechty (2020), polynomial-based periodization transformations Krylov (1907); Eckhoff (1993); Roache (1978), signal-processing approaches such as Prony's method de Prony (1795); Plonka et al. (2018), and more recently, rational approximation techniques such as the AAA algorithm Nakatsukasa et al. (2018); Nakatsukasa and Trefethen (2020); Huybrechs and Trefethen (2023); Driscoll et al. (2024). Notably, Eckhoff's method Eckhoff (1995) account for multiple isolated discontinuties and its convergence has been analyzed in Nersessian and Poghosyan (2006); Barkhudaryan et al. (2007); Poghosyan (2010). While these approaches offer partial remedies, they also present significant drawbacks. In particular, domain extension methods increase computational overhead by introducing additional degrees of freedom, involves solution of ill-conditioned systems Adcock et al. (2014), and often require problem-dependent tuning Adcock and Ruan (2014); moreover, they are not practical in applications where the domain is fixed by data, memory, or geometry constraints. Polynomial-based modifications Roache (1978); Eckhoff (1995) often lack robustness and deliver limited accuracy for highly oscillatory functions, as demonstrated in this work. Similarly, Prony's method is ill-conditioned for large system sizes and requires prior knowledge of the number of modes. The AAA algorithm, while powerful and broadly applicable, can incur high computational cost for large-scale problems and may overfit when applied to noisy data Nakatsukasa et al. (2018). To tackle these challenges, this work introduces a novel method called the Generalized Fourier Series (GFS), which extends Fourier spectral methods to non-periodic functions with minimal computational overhead. The key idea is to decompose a function into periodic and aperiodic components: the periodic part is efficiently approximated using standard FFT. In contrast, the aperiodic part is represented with an adaptive set of n = O(1) complex sine and cosine modes. This strategy avoids artificial domain extensions, preserves the efficiency and simplicity of the Fourier framework, and provides a robust mechanism for capturing both smooth and oscillatory non-periodic behavior. Numerical validations support these results. Exising methods augment standard Fourier series with a fixed set of functions to approximate nonperiodic functions, e.g., polynomials Roache (1978), Birkhoff-Hermite polynomials Huybrechs et al. (2010), Bernoulli polynomials Eckhoff (1995). In contrast, GFS adds a dynamic set of non-harmonic sinusoidal modes that adapt to the function's non-periodic nature, efficiently capturing its aperiodic component. We show that GFS is more robust, achieves superior numerically accuracy and convergence. The GFS method, like Eckhoff's Eckhoff (1995, 1998), is multi-dimensional and flexible, allowing for future expansion to handle multiple discontinuities in complex geometries, such as cubic outer boxes with several immersed obstacles modeled as discontinuities. This paper is organized as follows. Section 3.1 provides the mathematical motivation for the approach, detailing how jump conditions influence the aperiodic component. Section 3 outlines the formulation of the Generalized Fourier Series and describes the continuous spectrum of aperiodic signals. The adaptive basis 2 construction is then presented, along with the computational algorithms used to compute the modes and expansion coefficients. Section 4 outlines the computational complexity of the proposed method. Finally, Section 5 demonstrates the method's efficacy through numerical tests on representative non-periodic functions, showcasing the superior resolution power and convergence rate of GFS compared to finite difference and the standard FFT methods. By introducing an adaptive decomposition of the function space and leveraging the efficiency of FFT, the proposed GFS method offers a new perspective on handling non-periodic functions in bounded domains. The method retains linear computational complexity, avoids the need for domain extensions, and delivers highresolution power akin to spectral methods in periodic domains. These features make the GFS a compelling alternative for problems involving non-periodic functions, with potential applications in numerical PDEs, signal processing, machine learning, and computational physics. 2. State-of-the-art methods We compare GFS with the existing methods described briefly below: 2.1. Eckhoff Method The Eckhoff method Eckhoff (1995) reconstructs a 2π-periodic, piecewise smooth function u(x) from a finite number of its Fourier coefficients ˆuk. Assuming M singularities at unknown locations γj with jumps An j ≡u(n)(γ+ j ) -u(n)(γj ) in the nth derivative, the function is decomposed as: u(x) = v(x) + q-1 X n=0 M X j=1 An jVn(x; γj) (1) where v(x) is smooth and Vn(x; γj) are periodic functions derived from Bernoulli polynomials with controlled singularities at γj. Vn(x; β) = -(2π)n (n + 1)! Bn+1 ξ 2π , ξ = mod (x -β + 2π, 2π), 0 64. (a) (b) Figure 3: A Gaussian, defined in Eq. (19), is approximated using GFS with N = 64 grid points, and n = 3: (a) shows its decomposition into periodic and aperiodic components, while (b) compares the first derivative approximation using GFS and FFT against analytical values. GFS aligns well with the analytical results without exhibiting Gibbs oscillations, unlike FFT, which shows these oscillations. 16 Method GFS FD: O(∆xr) FFT Remark Jm : analytical Jm : numerical r = 6 N n ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 16 3 6.48e-07 1.36e-06 3.98e-01 2.56e-01 9.37e-02 6.04e-02 1.39e+00 1.31e+00 32 3 3.86e-11 4.76e-11 1.92e-04 8.52e-05 2.52e-03 1.14e-03 2.33e+00 1.74e+00 64 3 1.50e-14 1.20e-14 2.55e-09 7.99e-10 4.18e-05 1.35e-05 4.24e+00 2.40e+00 128 3 6.66e-15 5.48e-15 3.45e-12 1.08e-12 4.00e-07 9.78e-08 8.04e+00 3.36e+00 256 3 1.62e-14 1.07e-14 8.91e-12 1.43e-12 3.73e-09 8.45e-10 1.57e+01 4.73e+00 512 3 3.90e-14 2.15e-14 3.52e-11 3.93e-12 3.82e-11 1.03e-11 3.09e+01 6.67e+00 Table 5: A Gaussian defined in Eq. (19) is approximated using GFS for various grid sizes N. The first derivative approximation error (e′) is compared among GFS, finited difference (FD), and FFT methods. Jumps Jm, m = 0, 1, . . . , 4n -1, are calculated analytically or numerically using one-sided FD schemes with stencil width M + 1 = r + 4n -1 such that J4n-1 has O(∆x6) accuracy (see Sec. 3.6). Figure 4: For the Gaussian function defined in Eq. (19), convergence of the numerical error (L∞-norm) in first derivative is compared for GFS, a sixth-order finite difference (FD6), FFT, Roache, Eckhoff, and Prony methods. For Roache and Eckhoff methods the jumps Jm are evaluated analytically, and for GFS method Jm are evaluated both analytically and numerically. Here, "Jm : ideal" refers to the case where the jumps Jm, m = 0, 1, . . . , q -1, q = 4n, are evaluated analytically. GFS is robust and exhibits superior convergence with mesh size N for the same number of jumps (q), whether Jm are evaluated analytically or numerically. The Prony method (M = N/2) is numerically unstable for large mesh sizes (N > 64). 17 5.3. Log function Consider the non-periodic logarithmic function in the domain [-π, π], u(x) = log x + π + 1 2 ! , -π ≤x ≤π. (20) Fig. 5(a) displays the function u(x) along with its periodic part up(x) and aperiodic part ua(x) obtained using GFS for n = 3 and N = 32. Fig. 5(b) compares u′(x) calculated via GFS, FFT, and analytical methods. GFS effectively avoids Gibbs oscillations, closely aligning with the analytical method, while FFT exhibits these oscillations. The L∞, L2 norms of the error in u′(x) (denoted as ||e′||p) are detailed in Table 6 for n = 3, highlighting the impact of increased grid resolution N and comparisons with a sixth-order finite difference scheme and FFT. Among the three methods, FFT performs the worst with O(1) error due to Gibbs oscillations. Fig. 6 shows GFS method converging faster and with lower error than FD, FFT, Roache, Eckhoff, and Prony methods. When Jm is evaluated analytically, GFS rapidly improves with increasing N, outpacing all the other methods. Even with numerical Jm, GFS remains markedly superior. The Prony method with M = N/2 in Eq. (2) becomes ill-conditioned for large N and fails for N > 64. (a) (b) Figure 5: A function u(x) = log(x + π + 1/2), -π ≤x ≤π, is approximated with GFS using N = 32 mesh points and n = 3: Decomposition of u(x) into periodic and aperiodic parts is shown in (a), approximation of the first derivative with GFS and FFT is compared with analytical values in (b); GFS does not exhibit Gibbs oscillations and aligns well with the analytic values while FFT shows Gibbs oscillations. 5.4. multi-mode aperiodic function Consider the function composed of sinusoidal components with Nk non-integer modes (k j), which are non-periodic in the domain [-π, π], u(x) = Nk-1 X j=0 sin(kjx) + cos(k jx), -π ≤x ≤π, k j = j + ∆j, ∆j ≡1 Nk + j Nk Nk -2 Nk -1 ! ∈ " 1 Nk , 1 -1 Nk # . (21) This function is highly oscillatory and enables us to assess the resolution power (R) of the GFS, defined as the degrees of freedom needed to resolve one wavelength to machine precision. This can be expressed as R ≡ N max kj ≈N Nk . 18 GFS FD: O(∆xr) FFT Jm : analytical Jm : numerical r = 6 N n ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 16 3 7.22e-04 9.53e-04 1.40e-02 8.82e-03 5.84e-02 3.68e-02 5.68e+00 5.85e+00 32 3 8.49e-06 8.84e-06 7.53e-04 3.34e-04 9.36e-03 4.18e-03 1.02e+01 8.12e+00 64 3 2.21e-08 1.77e-08 1.05e-05 3.29e-06 8.17e-04 2.59e-04 1.94e+01 1.14e+01 128 3 2.09e-11 1.21e-11 2.98e-08 6.62e-09 4.03e-05 9.06e-06 3.78e+01 1.61e+01 256 3 6.09e-14 5.73e-14 6.69e-11 1.09e-11 1.26e-06 2.01e-07 7.46e+01 2.28e+01 512 3 9.44e-14 7.54e-14 4.82e-11 5.99e-12 2.93e-08 3.30e-09 1.48e+02 3.22e+01 Table 6: The log function defined in Eq. (20) is approximated using GFS with various grid sizes N. The first derivative approximation error (e′) is compared among GFS, finited difference (FD), and FFT methods. Jumps Jm, m = 0, 1, . . . , 4n-1, are calculated analytically or numerically using one-sided FD schemes with stencil width M + 1 = r + 4n -1, ensuring that J4n-1 achieves O(∆x6) accuracy. Fig. 7(a) displays u(x) for N = 64, Nk = 30, demonstrating high oscillation, with the maximum wavenumber kNk-1 near the Nyquist limit (Nk ∼N/2). Fig. 7(b, c) compares the first derivative obtained via GFS for n = 6 with the FFT and analytical methods for N = 64 and N = 80. GFS closely aligns with the analytical method, exhibiting no Gibbs oscillations, while FFT has Gibbs oscillations. For quantitative evaluation, Table 7 presents the L∞and L2 norms of the error in the first derivative for the current method (GFS), a sixth-order finite difference (FD) method, and the FFT. We set Nk = 30 and begin with N = 64, where the maximum wave number (kNk-1) is close to the Nyquist limit, and examine the impact of increasing resolution and the number of aperiodic modes n. Ideally, if n = Nk, the aperiodic part would exactly recover the total function u′(xj). However, this involves higher (d) derivatives, with their magnitudes scaling as kd Nk-1, leading to significant increases in precision loss. Moreover, as n -→Nk ≫1, this approach becomes costly, so we aim to limit n to O(1) for practical computations to control expenses and maintain precision. With max u′(x) ≈600 (Fig. 7(b,c)), Table 7 shows that machine precision is achieved with a relatively small number of aperiodic modes (n = O(1)) using GFS. The error decreases much more rapidly with mesh refinement (increasing N) for GFS (with analytical Jm) even at n = 2, and improves markedly as n grows. While the sixth-order FD scheme's error scales as O(∆x6), the FFT error remains O(1) in nonperiodic domains due to the Gibbs phenomenon. For Nk = 30 and a fixed target error of O(10-10), GFS requires about N = 80 -90, yielding a resolution power R ≈3; in contrast, the sixth-order FD scheme needs N = 8192 (see Fig. 8), giving R = 300. Thus, the FD mesh must be two orders of magnitude finer than GFS for the same accuracy, evidencing GFS's superior resolution. Even with a numerically evaluated Jm, GFS maintains higher accuracy and convergence than FD and FFT methods, see Table 7. Fig. 8 shows the GFS method converging faster and with lower error than the FD, FFT, Roache's, Eckhoff's, and Prony's methods. When Jm is evaluated analytically, GFS rapidly improves with increasing N, surpassing all other methods. Even with numerical Jm, GFS remains markedly superior for large N. Polynomial-based methods like the Roache and Eckhoff approaches become ill-conditioned as the number of jumps (q) in the discontinuous part increases, while GFS stays robust. The Prony's method is ill-conditioned due to high oscillations, causing the computation to fail with M = Nk and M = N/2 (not shown) in Eq. (2). In summary, the convergence rate of GFS improves with the number of aperiodic modes n, and GFS has a superior resolution power (R ≈3) compared to the sixth-order finite difference scheme (R ≈300). 5.5. Monomials We consider the monomial functions u(x) = xm, -π ≤x ≤π for m = 1, 3, which are non-periodic and have finite jumps in the function values at the domain endpoints. Although the GFS is not primarily designed 19 Figure 6: For the log function defined in Eq. (20), convergence of the numerical error (L∞-norm) in first derivative is compared for GFS, a sixth-order finite difference (FD6), FFT, Roache's, Eckhoff's, and Prony's methods. For the Roache and Eckhoff methods, the jumps Jm are evaluated analytically, and for the GFS method, Jm are evaluated both analytically and numerically. Here, "Jm : ideal" refers to the case where the jumps Jm, m = 0, 1, . . . , q -1, q = 4n, are evaluated analytically. GFS is robust and shows superior convergence with mesh size N for the same number of jumps (q), whether Jm are evaluated analytically or numerically. The Prony method (M = N/2) is numerically unstable for large mesh sizes (N > 64). for approximating polynomials, it can still accommodate such functions by regularizing jumps. The GFS relies on endpoint conditions related to jumps, which can quickly diminish for monomial functions, resulting in rank-deficient matrices and vanishing right-hand sides. This issue can be resolved by regularizing jumps using very small numerical values (10-15 in the following cases) when the actual jump is zero. It's important to note that regularization is only necessary for the analytical case; for numerical computations, the inherent error of the numerical scheme provides the required regularization. For m = 1, this corresponds to the ramp function, a well-known test case that challenges Fourier spectral methods. Fig. 9(a) shows the decomposition of u(x) into periodic up(x) and aperiodic ua(x) components obtained by GFS for n = 1. Fig. 9(b) compares the first derivative approximations of u(x) using GFS and FFT; GFS avoids Gibbs oscillations and closely matches analytical values, whereas FFT exhibits them. Table 8 compares the L∞, L2 norms of the first derivative error ||e′||p for GFS, FD and FFT methods at N = 64. The numerical error for Jm and the stand-alone FD method approaches machine precision, as the sixth-order FD schemes fully resolve the linear function u(x) = x. For m = 3, this corresponds to a cubic monomial. Fig. 10(a,c) shows the decomposition of u(x) into periodic up(x) and aperiodic ua(x) components using GFS for n = 1 and 2, respectively. Fig. 10(b,d) compares the first derivative approximations of u(x) from GFS and FFT; GFS avoids Gibbs oscillations and aligns closely with analytical values, while FFT exhibits oscillations leading to O(1) error. Table 9 compares the L∞and L2 norms of the first derivative error ||e′||p for GFS, FD and FFT methods at N = 64. The numerical error for GFS converges quickly with increasing n, whereas the sixth-order FD scheme achieves near machine precision by exactly approximating the monomials for m ≤5. 6. Conclusions and future directions 20 GFS FD: O(∆xr) FFT Jm : analytical Jm : numerical r = 6 N Nk n ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 64 30 2 7.34e-02 1.16e-01 5.52e+03 1.74e+03 1.93e+02 1.35e+02 2.45e+00 1.24e+00 128 30 2 3.23e-06 1.80e-06 6.19e+01 1.43e+01 1.95e+01 7.23e+00 1.22e+00 3.40e-01 256 30 2 1.55e-08 5.50e-09 3.61e-02 5.92e-03 4.40e-01 1.41e-01 1.03e+00 2.23e-01 512 30 2 1.08e-10 2.64e-11 1.12e-05 1.27e-06 7.84e-03 2.21e-03 9.46e-01 1.55e-01 64 30 4 1.06e-03 1.54e-03 6.97e+05 2.63e+05 1.93e+02 1.35e+02 2.45e+00 1.24e+00 128 30 4 5.24e-12 3.56e-12 3.56e+02 9.98e+01 1.95e+01 7.23e+00 1.22e+00 3.40e-01 256 30 4 3.79e-12 2.27e-12 1.81e-03 2.85e-04 4.40e-01 1.41e-01 1.03e+00 2.23e-01 512 30 4 7.12e-12 4.77e-12 2.73e-08 3.81e-09 7.84e-03 2.21e-03 9.46e-01 1.55e-01 64 30 6 1.68e-03 3.86e-03 6.57e+06 2.74e+06 1.93e+02 1.35e+02 2.45e+00 1.24e+00 72 30 6 1.16e-07 1.97e-07 5.04e+06 1.50e+06 1.61e+02 9.46e+01 1.81e+00 6.94e-01 80 30 6 5.27e-10 7.34e-10 1.96e+06 5.59e+05 9.50e+01 6.48e+01 1.59e+00 5.53e-01 96 30 6 1.64e-12 1.46e-12 1.79e+05 4.68e+04 6.67e+01 3.00e+01 1.39e+00 4.36e-01 128 30 6 2.03e-12 1.39e-12 8.34e+02 2.19e+02 1.95e+01 7.23e+00 1.22e+00 3.40e-01 256 30 6 2.74e-12 1.45e-12 7.29e-05 1.20e-05 4.40e-01 1.41e-01 1.03e+00 2.23e-01 512 30 6 4.40e-12 3.15e-12 4.80e-07 6.20e-08 7.84e-03 2.21e-03 9.46e-01 1.55e-01 Table 7: A multi-mode sinusoidal function defined by Eq. (21) with Nk = 30 is approximated using GFS, finite difference (FD), and FFT methods. The error of the approximation for the first derivative (e′) is shown with increasing number of aperiodic modes (n). The jumps Jm are computed either analytically or numerically through one sided finite difference schemes of rth order accuracy as indicated. GFS FD: O(∆xr) FFT Jm : analytical Jm : numerical r = 6 N n ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 64 1 1.55e-14 9.50e-15 3.03e-14 1.47e-14 9.38e-14 3.00e-14 4.43e+01 2.71e+01 64 2 7.22e-15 7.98e-15 6.57e-13 2.91e-13 9.38e-14 3.00e-14 4.43e+01 2.71e+01 64 3 1.03e-14 7.97e-15 1.57e-12 9.18e-13 9.38e-14 3.00e-14 4.43e+01 2.71e+01 Table 8: The ramp function u(x) = x, -π ≤x ≤π is approximated with the GFS. Error of the approximation for the function (e) and its first derivative (e′) are shown with increasing number of sin or cos modes (n). Here, N = 64 and the jump conditions Jm are computed from exact derivatives or numerical derivatives (one-sided finite difference method) as indicated. GFS FD: O(∆xr) FFT Jm : analytical Jm : numerical r = 6 N n ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 ||e′||∞ ||e′||2 64 1 1.13e-04 5.98e-05 1.13e-04 5.98e-05 7.07e-13 3.33e-13 4.38e+02 2.68e+02 64 2 6.70e-06 6.06e-06 2.77e-11 1.94e-11 7.07e-13 3.33e-13 4.38e+02 2.68e+02 64 3 8.32e-09 6.90e-09 1.22e-10 5.75e-11 7.07e-13 3.33e-13 4.38e+02 2.68e+02 Table 9: A function u(x) = x3, -π ≤x ≤π, is approximated using GFS with N = 64. The error of the approximation for the first derivative (e′) is compared among various methods. Here, the jump conditions Jm are computed either exactly or numerically through one-sided finite difference (FD) schemes with O(∆x6) accuracy as indicated in the column Jm. 21 (a) (b) (c) Figure 7: A multi-mode sinusoidal function defined by Eq. (21) with Nk = 30 is approximated using GFS with n = 6. Panel (a) displays its periodic and aperiodic decompositions for N = 64. Panels (b) and (c) compare its first derivative approximations and the errors from GFS and FFT against analytical results for N = 64 and N = 80. In this work, we introduce the Generalized Fourier Series (GFS), a spectral method that extends Fourier techniques to non-periodic domains while retaining FFT-level efficiency. GFS decomposes a function into a periodic component-approximated with the Fast Fourier Transform-and an aperiodic component captured by a small set (n = O(1)) of adaptive, non-harmonic sinusoids that handle non-periodicity with high-order accuracy. This construction avoids artificial domain extensions and their overhead, seamlessly accommodates derivative jumps and boundary discontinuities, and delivers high-resolution accuracy and robustness. With a resolution power (degrees of freedom per wavelength) comparable to FFT on periodic problems, and a low-rank approximation for the aperiodic part, the overall complexity remains linear, making GFS a practical and efficient alternative to existing techniques. Comprehensive numerical experiments show that GFS consistently outperforms classical approaches-including Roache's polynomial-based correction, Eckhoff's reconstruction with Bernoulli polynomials, Prony's exponential fitting, and standard finite-difference and FFT methods. Relative to Roache's method, GFS avoids a fixed polynomial correction-which can be unstable for oscillatory functions-in favor of adaptive sinusoidal modes that yield greater robustness and accuracy; relative to Eckhoff's method, it retains the ability to capture discontinuities while offering greater robustness and higher accuracy, particularly when a large number of jumps must be resolved; and relative to Prony's method, it does not require the precise number of modes to be specified a priori and remains numerically stable at large mesh sizes (N > 64), where Prony's scheme typically becomes ill-conditioned. Across meshes, GFS delivered significantly better convergence rates and resolution power: it achieves FFT-like resolution, while a sixth-order finite-difference scheme demands far more resources-underscoring GFS's orders-of-magnitude performance advantage. While GFS attains optimal accuracy when exact jump conditions are known, approximating these jumps with finite differences can limit overall accuracy; the quality of jump evaluation depends on the underlying numerical scheme and could be improved with better estimators. A detailed treatment of jump-condition estimation lies beyond the scope of this article and will be addressed in future work. Across a range of test functions-including modulated sines, Gaussians, and logarithmic functions-GFS showed rapid convergence, minimal Gibbs oscillations, and superior accuracy in approximating derivatives. The results confirm that GFS provides a robust, efficient, and accurate framework for non-periodic function approximation, offering a compelling alternative to existing methods for applications in numerical PDEs, signal processing, and beyond. 22 Figure 8: For the multi-mode function defined in Eq. (21) with Nk = 30 non-integer modes, convergence of the numerical error (L∞- norm) in the first derivative is compared for GFS, a sixth-order finite difference (FD6), FFT, Roache, Eckhoff, and Prony methods. For the Roache and Eckhoff methods, the jumps Jm are evaluated analytically, and for the GFS method, Jm are evaluated both analytically and numerically. Here, "Jm : ideal" refers to the case where the jumps Jm, m = 0, 1, . . . , q -1, q = 4n, are evaluated analytically. GFS shows superior convergence with mesh size N. As the number of jumps (q) increases, GFS remains robust and converges rapidly, while the other jump-based methods (Roache and Eckhoff) suffer from numerical stability issues due to the function's high oscillations and ill-conditioning at large q. The Prony's method (M = Nk) is numerically unstable. Future work will expand the GFS framework to multi-dimensional applications, addressing PDEs such as advection-diffusion and Poisson equations, and eventually tackling computational fluid dynamics problems. These efforts highlight the transformative potential of GFS for enhancing spectral methods in nonperiodic domains. Appendix A. Solution of the non-linear systems Eqs. (8) and (9) for n = 1, 2, 3 Appendix A.1. Single mode representation of the aperiodic part: n = 1 For n = 1, the solution of the above equations is straightforward and involves the jumps in the derivatives of u(x) at the endpoints J0, J1, J2, J3. First, we solve for the modes ̃k1, ˆk1, followed by the amplitudes ̃u1, ˆu1 as below, 2 ̃u1 sin( ̃k1π) = J0 = -J2 ̃k2 1 , -2ˆk1ˆu1 sin(ˆk1π) = J1 = -J3 ˆk2 1 , (A.1) ̃k2 1 = -J2/J0, ˆk2 1 = -J3/J1, ̃u1 = J0 2 sin( ̃k1π), ˆu1 = - J1 2ˆk1 sin(ˆk1π) . (A.2) Note that ˆk1 = ± √-J3/J1 and ̃k1 = ± √-J2/J0 and the ± sign is irrelevant here because ˆuj cos(ˆk jx) and ̃uj sin( ̃kjx) (therefore, uc(x) and us(x)) are even functions of ˆk and ̃k, respectively. We define the complex square root as √z ≡√|z|eiθ/2 where z = |z|eiθ, -π ≤θ ≤π. 23 (a) (b) Figure 9: Ramp function u(x) = x is approximated with GFS using n = 1, N = 64: Decomposition of u(x) into periodic and aperiodic parts is shown in (a), approximation of the first derivative with GFS and FFT is compared with analytical values in (b); GFS does not exhibit Gibbs oscillations and aligns well with the analytic values while FFT shows Gibbs oscillations. Appendix A.2. Two-mode representation of the aperiodic part: n = 2 For n = 2, we have eight unknowns. So, let us consider the jumps in the derivatives of u(x) at the endpoints J0, J1, J2, . . . , J7 which yields the following set of equations, 2 " ̃u1 sin( ̃k1π) ̃u2 sin( ̃k2π) # = 1 1 - ̃k2 1 - ̃k2 2 !-1 "J0 J2 # = ̃k4 1 ̃k4 2 - ̃k6 1 - ̃k6 2 !-1 "J4 J6 # , (A.3) -2 "ˆu1ˆk1 sin(ˆk1π) ˆu2ˆk2 sin(ˆk2π) # = 1 1 -ˆk2 1 -ˆk2 2 !-1 "J1 J3 # = ˆk4 1 ˆk4 2 -ˆk6 1 -ˆk6 2 !-1 "J5 J7 # . (A.4) Expanding on both sides of the latter equality in Eq. (A.3) yields, 1 1 - ̃k2 1 - ̃k2 2 !-1 "J0 J2 # = " 1 1 - ̃k2 1 - ̃k2 2 ! ̃k4 1 0 0 ̃k4 2 !#-1 "J4 J6 # , (A.5) 1 1 - ̃k2 1 - ̃k2 2 ! ̃k4 1 0 0 ̃k4 2 ! 1 1 - ̃k2 1 - ̃k2 2 !-1 "J0 J2 # = "J4 J6 # , (A.6) - ̃k2 1 ̃k2 2 - ̃k2 1 + ̃k2 2 ̃k2 1 ̃k2 2 ̃k2 1 + ̃k2 2 ̃k4 1 + ̃k4 2 + ̃k2 1 ̃k2 2 "J0 J2 # = "J4 J6 # . (A.7) Elimination of J0 from the bottom row of Eq. (A.7) yields the following linear system for the elementary symmetric polynomials in ̃k2 1, ̃k2 2 defined as ̃e1 ≡ ̃k2 1 + ̃k2 2, ̃e2 ≡ ̃k2 1 ̃k2 2, J0 ̃k2 1 ̃k2 2 + J2 ̃k2 1 + ̃k2 2 J2 ̃k2 1 ̃k2 2 + J4 ̃k2 1 + ̃k2 2 = - "J4 J6 # =⇒ " ̃e2 ̃e1 # = - J0 J2 J2 J4 !† "J4 J6 # . (A.8) 24 (a) (b) (c) (d) Figure 10: Cubic monomial u(x) = x3 is approximated with GFS using N = 64 and n = 1 (a-b), n = 2 (c-d): Decomposition of u(x) into periodic and aperiodic parts is shown in (a,c), approximation of the first derivative with GFS and FFT is compared with analytical values in (b,d); GFS does not exhibit Gibbs oscillations and aligns well with the analytic values while FFT shows Gibbs oscillations. Corresponding to Eq. (A.4), we obtain a similar system for the elementary polynomials in ˆk2 1, ˆk2 2 defined as ˆe1 ≡ˆk2 1 + ˆk2 2, ˆe2 ≡ˆk2 1ˆk2 2 and is given below, "ˆe2 ˆe1 # = - J1 J3 J3 J5 !† "J5 J7 # . (A.9) By solving Eqs. (A.3)-(A.4) for ̃e1, ̃e2, ˆe1, ˆe2, we have ̃e1 ≡J0J6 -J2J4 J2 2 -J0J4 , ̃e2 ≡J2 4 -J2J6 J2 2 -J0J4 , ˆe1 ≡J1J7 -J3J5 J2 3 -J1J5 , ˆe2 ≡ J2 5 -J3J7 J2 3 -J1J5 . Now, ̃k2 j and ˆk2 j are the roots of the polynomials given below, and the amplitudes ̃uj, ˆuj are obtained from Eqs. (A.3)-(A.4), ̃k4 - ̃e1 ̃k2 + ̃e2 = 0, ˆk4 -ˆe1ˆk2 + ˆe2 = 0, 25 hence ̃k2 = ̃e1 ± q ̃e2 1 -4 ̃e2 2 , ˆk2 = ˆe1 ± q ˆe2 1 -4ˆe2 2 . And ̃u1 = ̃k2 2J0+J2 2( ̃k2 2- ̃k2 1) sin( ̃k1π), ̃u2 = ̃k2 1J0 + J2 2( ̃k2 1 - ̃k2 2) sin( ̃k2π), ˆu1 = - ˆk2 2J1+J3 2ˆk1(ˆk2 2-ˆk2 1) sin(ˆk1π), ˆu2 = - ˆk2 1J1 + J3 2ˆk2(ˆk2 1 -ˆk2 2) sin(ˆk2π) . Above expressions satisfy [u(m)] = [u(m) a ] and thereby, [u(m) p ] = 0 for m = 0, 1, 2, . . . , 7. This means that the periodic part is smooth up to the seventh order derivative, i.e., C7 continuous. Appendix A.3. Three mode representation of the aperiodic part: n = 3 With n = 3, we have 12 unknowns. So, let us consider the jumps in the derivatives of u(x) at the endpoints J0, J1, J2, . . . , J11 which yields the following set of equations, 2 ̃u1 sin( ̃k1π) ̃u2 sin( ̃k2π) ̃u3 sin( ̃k3π) = 1 1 1 - ̃k2 1 - ̃k2 2 - ̃k2 3 ̃k4 1 ̃k4 2 ̃k4 3 -1 J0 J2 J4 = - ̃k6 1 - ̃k6 2 - ̃k6 3 ̃k8 1 ̃k8 2 ̃k8 3 - ̃k10 1 - ̃k10 2 - ̃k10 3 -1 J6 J8 J10 . (A.10) -2 ˆu1ˆk1 sin(ˆk1π) ˆu2ˆk2 sin(ˆk2π) ˆu3ˆk3 sin(ˆk3π) = 1 1 1 -ˆk2 1 -ˆk2 2 -ˆk2 3 ˆk4 1 ˆk4 2 ˆk4 3 -1 J1 J3 J5 = -ˆk6 1 -ˆk6 2 -ˆk6 3 ˆk8 1 ˆk8 2 ˆk8 3 -ˆk10 1 -ˆk10 2 -ˆk10 3 -1 J7 J9 J11 . (A.11) Note that the second and third parts in Eqs. (A.10)-(A.11) involving the unknowns ̃k j and ˆk j are same except that the jumps J0, J2, . . . , J10 are replaced by J1, J3, . . . , J11. So, the procedures for solving ̃k j and ˆkj are similar. Assuming full rank, i.e., ̃k1 , ̃k2 , ̃k3, and expanding on both sides of the latter equality in Eqs. (A.10) yields, 1 1 1 - ̃k2 1 - ̃k2 2 - ̃k2 3 ̃k4 1 ̃k4 2 ̃k4 3 -1 J0 J2 J4 = - 1 1 1 - ̃k2 1 - ̃k2 2 - ̃k2 3 ̃k4 1 ̃k4 2 ̃k4 3 ̃k6 1 0 0 0 ̃k6 2 0 0 0 ̃k6 3 -1 J6 J8 J10, , (A.12) - 1 1 1 - ̃k2 1 - ̃k2 2 - ̃k2 3 ̃k4 1 ̃k4 2 ̃k4 3 ̃k6 1 0 0 0 ̃k6 2 0 0 0 ̃k6 3 1 1 1 - ̃k2 1 - ̃k2 2 - ̃k2 3 ̃k4 1 ̃k4 2 ̃k4 3 -1 J0 J2 J4 = J6 J8 J10, , (A.13) - ̃k6 1 0 0 0 ̃k6 2 0 0 0 ̃k6 3 ̃k2 2 ̃k2 3 ̃k2 2 - ̃k2 3) ̃k4 2 - ̃k4 3 ̃k2 2 - ̃k2 3 ̃k2 3 ̃k2 1 ̃k2 3 - ̃k2 1) ̃k4 3 - ̃k4 1 ̃k2 3 - ̃k2 1 ̃k2 1 ̃k2 2 ̃k2 1 - ̃k2 2) ̃k4 1 - ̃k4 2 ̃k2 1 - ̃k2 2 J0 J2 J4 = ̃k2 2 ̃k2 3 ̃k2 2 - ̃k2 3) ̃k4 2 - ̃k4 3 ̃k2 2 - ̃k2 3 ̃k2 3 ̃k2 1 ̃k2 3 - ̃k2 1) ̃k4 3 - ̃k4 1 ̃k2 3 - ̃k2 1 ̃k2 1 ̃k2 2 ̃k2 1 - ̃k2 2) ̃k4 1 - ̃k4 2 ̃k2 1 - ̃k2 2 J6 J8 J10 . 26 - ̃k6 1 0 0 0 ̃k6 2 0 0 0 ̃k6 3 ̃k2 2 ̃k2 3 ̃k2 2 + ̃k2 3 1 ̃k2 3 ̃k2 1 ̃k2 3 + ̃k2 1 1 ̃k2 1 ̃k2 2 ̃k2 1 + ̃k2 2 1 J0 J2 J4 = ̃k2 2 ̃k2 3 ̃k2 2 + ̃k2 3 1 ̃k2 3 ̃k2 1 ̃k2 3 + ̃k2 1 1 ̃k2 1 ̃k2 2 ̃k2 1 + ̃k2 2 1 J6 J8 J10 . J0 ̃k6 1 ̃k2 2 ̃k2 3 + ̃k2 2 + ̃k2 3 h J2 ̃k6 1 + J8 i + J4 ̃k6 1 + J6 ̃k2 2 ̃k2 3 + J10 = 0, (A.14) J0 ̃k2 1 ̃k6 2 ̃k2 3 + ̃k2 1 + ̃k2 3 h J2 ̃k6 2 + J8 i + J4 ̃k6 2 + J6 ̃k2 1 ̃k2 3 + J10 = 0, (A.15) J0 ̃k2 1 ̃k2 2 ̃k6 3 + ̃k2 1 + ̃k2 2 h J2 ̃k6 3 + J8 i + J4 ̃k6 3 + J6 ̃k2 1 ̃k2 2 + J10 = 0. (A.16) Elimination of the pairs (J8, J10), (J0, J10), (J0, J2) from Eqs. (A.14)-(A.16) yields the following set of equations involving the elementary symmetric polynomials ( ̃ej) in ̃k2 1, ̃k2 2, ̃k2 3 defined as ̃e1 ≡ ̃k2 1 + ̃k2 2 + ̃k2 3, ̃e2 ≡ ̃k2 1 ̃k2 2 + ̃k2 2 ̃k2 3 + ̃k2 1 ̃k2 3, ̃e3 ≡ ̃k2 1 ̃k2 2 ̃k2 3, J0 J2 J4 J2 J4 J6 J4 J6 J8 ̃k2 1 ̃k2 2 ̃k2 3 ̃k2 1 ̃k2 2 + ̃k2 2 ̃k2 3 + ̃k2 1 ̃k2 3 ̃k2 1 + ̃k2 2 + ̃k2 3 = - J6 J8 J10 =⇒ ̃e3 ̃e2 ̃e1 = - J0 J2 J4 J2 J4 J6 J4 J6 J8 † J6 J8 J10 . (A.17) Similarly, corresponding to Eq. (A.11), we obtain the elementary symmetric polynomials ˆe j in ˆk2 1, ˆk2 2, ˆk2 3 defined as ˆe1 ≡ˆk2 1 + ˆk2 2 + ˆk2 3, ˆe2 ≡ˆk2 1ˆk2 2 + ˆk2 2ˆk2 3 + ˆk2 1ˆk2 3, ˆe3 ≡ˆk2 1ˆk2 2ˆk2 3. ˆe3 ˆe2 ˆe1 = - J1 J3 J5 J3 J5 J7 J5 J7 J9 † J7 J9 J11 . (A.18) Now, ̃k2 1, ̃k2 2, ̃k2 3 and ˆk2 1, ˆk2 2, ˆk2 3 are obtained by solving for the roots (λ) of the polynomials below: λ3 - ̃e1λ2 + ̃e2λ - ̃e3 = 0 = 3 Y j=1 (λ - ̃k2 j), λ3 -ˆe1λ2 + ˆe2λ -ˆe3 = 0 = 3 Y j=1 (λ -ˆk2 j). Above expressions satisfy [u(m)] = [u(m) a ] and thereby, [u(m) p ] = 0 for m = 0, 1, 2, . . . , 11. This implies that the periodic part is smooth up to the eleventh order derivative, i.e., C11 continuous. Acknowledgments Narsimha Rapaka would like to dedicate this work to the Late Professor Ravi Samtaney. Code and data availability All source code and datasets to reproduce our results are publicly available at https://github.com/ nrapaka/GFS. The repository includes the GFS implementation, example scripts, and instructions for reproducing the experiments. 27 References B. Adcock. Modified Fourier expansions: theory, construction and applications. Ph.D thesis, 2010. B. Adcock. Convergence acceleration of modified fourier series in one or more dimensions. Math. Comp., 80:225-261, 2011. B. Adcock and D. Huybrechs. On the resolution power of fourier extensions for oscillatory functions. J. Comp. Applied Maths., 260:312-336, 2014. Ben Adcock and Joseph Ruan. Parameter selection and numerical approximation properties of Fourier extensions from fixed data. Journal of Computational Physics, 273:453-471, 2014. ISSN 0021-9991. Ben Adcock, Daan Huybrechs, and Jes ́us Mart ́ın-Vaquero. On the numerical stability of Fourier extensions. Foundations of Computational Mathematics, 14(4):635-687, 2014. A. Barkhudaryan, R. Barkhudaryan, and A. Poghosyan. Asymptotic behavior of Eckhoff's method for Fourier series convergence acceleration. Analysis in Theory and Applications, 23(3):228-242, 2007. G. R. de Prony. Essai exp ́erimental et analytique: sur les lois de la dilatabilit ́e des fluides ́elastiques et sur celles de la force expansive de la vapeur de l'eau et de la vapeur de l'alcool, `a diff ́erentes temp ́eratures. Journal de l' ́Ecole Polytechnique, 1(2):24-76, 1795. Tobin A. Driscoll, Yuji Nakatsukasa, and Lloyd N. Trefethen. AAA rational approximation on a continuum. SIAM Journal on Scientific Computing, 46(2):A929-A952, 2024. K.S. Eckhoff. Accurate and efficient reconstruction of discontinuous functions from truncated series expansions. Math. Comp., 61:745-763, 1993. K.S. Eckhoff. Accurate reconstruction of functions of finite regularity from truncated Fourier series exapnsions. Math. Comp., 64:671-690, 1995. K.S. Eckhoff. On a high order numerical method for functions with singularities. Math. Comp., 67:10631087, 1998. J. S. Geronimo and K. Liechty. The Fourier extension method and discrete orthogonal polynomials on an arc of the circle. Advances in Mathematics, 365:107064, 2020. David Gottlieb and Chi-Wang Shu. On the gibbs phenomenon and its resolution. SIAM Review, 39(4): 644-668, 1997. Edwin Hewitt and Robert E. Hewitt. The gibbs-wilbraham phenomenon: An episode in fourier analysis. Archive for History of Exact Sciences, 21(2):129-160, 1979. D. Huybrechs. On the Fourier extension of nonperiodic functions. SIAM J. Numeri. Anal., 47 (6):4326-4355, 2010. Daan Huybrechs and Lloyd N. Trefethen. AAA interpolation of equispaced data. BIT Numerical Mathematics, 63(2):21, 2023. 28 Daan Huybrechs, Arieh Iserles, and Syvert P. Nørsett. From high oscillation to rapid approximation IV: accelerating convergence. IMA Journal of Numerical Analysis, 31(2):442-468, 04 2010. ISSN 02724979. A. Iserles and S. P. Nørsett. From high oscillation to rapid approximation I: modified Fourier expansions. IMA J. Num. Analysis, 28:862-887, 2008. Abdul Jerri. The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Springer, New York, 1998. ISBN 978-1-4757-2847-7. A. Krylov. On approximate calculations. Lectures delivered in 1906 (in Russian), St. Petersburg, Tipolitography of Birkenfeld., 1907. Yuji Nakatsukasa and Lloyd N. Trefethen. An algorithm for real and complex rational minimax approximation. SIAM Journal on Scientific Computing, 42(5):A3157-A3179, 2020. Yuji Nakatsukasa, Olivier S`ete, and Lloyd N. Trefethen. The AAA algorithm for rational approximation. SIAM Journal on Scientific Computing, 40(3):A1494-A1522, 2018. A. Nersessian and A. Poghosyan. On a rational linear approximation of fourier series for smooth functions. Journal of Scientific Computing, 26(1):111-125, 2006. Gerlind Plonka, Daniel Potts, Gabriele Steidl, and Manfred Tasche. Numerical Fourier Analysis. Applied and Numerical Harmonic Analysis. Springer, 2018. ISBN 978-3-030-04306-3. 04306-3. Arnak Poghosyan. On an Auto-Correction Phenomenon of the Krylov-Gottlieb-Eckhoff Method. IMA Journal of Numerical Analysis, 31(2):512-527, 03 2010. ISSN 0272-4979. P. J. Roache. A pseudo-spectral FFT technique for non-periodic problems. Journal of Computational Physics, 27(2):204-220, 1978. 29
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2510.14729
|
Dispersion engineered AlGaAs-on-insulator nanophotonics by
distributed feedback
Francesco Rinaldo Talenti,1 Luca Lovisolo,2, 1 Zijun Xiao,1 Zeina Saleh,1
Andrea Gerini,2 Carlos Alonso-Ramos,1 Martina Morassi,1 Aristide Lemaître,1
Stefan Wabnitz,3, 4 Alfredo De Rossi,5 Giuseppe Leo,2, 6 and Laurent Vivien1
1Université Paris-Saclay, CNRS, Centre for Nanosciences
and Nanotechnology, 91120 Palaiseau, France
2Université Paris Cité, Laboratoire Matériaux et
Phénomènes Quantiques, 75013 Paris, France
3Sapienza University of Rome, Dipartimento di ingegneria
elettronica e telecomunicazioni, 00184 Roma, Italy
4CNR-INO, Istituto Nazionale di Ottica, 80078 Pozzuoli, Italy
5Thales Research and Technology, 91120 Palaiseau, France
6Institut universitaire de France, France
1
arXiv:2510.14729v1 [physics.optics] 16 Oct 2025
Abstract
Technological advances in the fabrication of nanophotonic circuits have driven the scientific com-
munity to increasingly focus on the precise tailoring of their key optical properties, over a broadband
spectral domain. In this context, the modulation of the local refractive index can be exploited to
customize an effective reflectivity by the use of distributed Bragg mirrors, enabling the on-chip in-
tegration of Fabry-Pérot resonators. The resulting cavity length is strongly wavelength-dependent,
offering practical solutions to the growing demand of dispersion engineering. Owing to their typ-
ically high core-to-cladding refractive index contrast and exceptional nonlinear properties, III-V
semiconductor-based platforms represent promising candidates for the fabrication of Bragg reflec-
tors. In this work, we propose an AlGaAs-on-insulator linear resonator based on distributed Bragg
mirrors. We discuss the first experimental demonstration of a systematic, shape-constrained inverse
design technique which tailors a prescribed dispersion profile, showing a strong agreement between
simulations and measurements. In perspective, the proposed approach offers an efficient and general
response to the challenge of dispersion engineering in integrated optical circuits.
Keywords: Fabry-Perot resonator, Photonic crystal, distributed-feedback, AlGaAs, dipersion engineering,
inverse design.
I.
INTRODUCTION
Fabry–Pérot (FP) resonators have recently been developed on fiber-based platforms
(Fiber Fabry–Pérot, FFP), resulting in compact, efficient, and relatively easy to implement
systems for the demonstration of modulation instability (MI) [1], frequency combs and
cavity soliton generation [2, 3].
The on-chip integration of such systems–driven by the
goal of minimizing the device footprint and reducing the power operation–requires the use
of distributed Bragg reflectors (DBRs). First prototypes of this sort have been proposed
for Kerr comb generation [4] and then implemented on silicon nitride (SiN) [5] to enhance
four-wave-mixing (FWM) [6], or to trigger solitons formation [7], while parent demonstra-
tions are those of Moiré plasmonic cavities [8, 9] suitable for laser applications [10]. Similar
systems have been also conceived with III-V semiconductor, and proven to be promising
for nonlinear frequency conversion [11], pulse spectral broadening [12] and the design of
nanophotonic distributed feedback (DFB) lasers [13, 14].
2
In a DFB structure, a periodic longitudinal modulation of the refractive index is ex-
ploited to introduce a Bragg backward scattering mechanism [15, 16].
As a result, two
counter-propagating optical waves are spatially confined and linearly coupled by multiple
local reflections along the cavity[17]. DFB systems, thus, inhibit light propagation within
a finite spectral domain [18, 19]. In strict analogy and with a direct connection with elec-
tronic dispersion diagrams, photonic band structures can be drawn for dielectrics which
are geometrically organized with a given and well defined periodicity Λ = λ0/2n, where
ω0/2π = c/λ0 is the central frequency of oscillation (i.e. the Bragg frequency), c is the speed
of light in vacuum and n is the refractive index of the bulk material[20].
The underlying physics is that of a photonic crystal (PhC), whose geometry determines
the effective local band diagrams. Furthermore, the introduction of a crystal defect induces
light confinement, and it can be exploited for the design of nano-cavities. The maximization
of the quality factors, though, is a non-trivial task.
First convincing demonstrations of
low-dimensional (i.e.
≤2D) PhC cavities introduced a novel light confining mechanism,
named gentle confinement [21, 22]. The basic idea consists in the fact that light scatter-
ing at defect boundaries can be a detrimental source of loss. To avoid that, light must
be progressively (or gently) guided towards the confining spatial domain, by adiabatically
tuning the geometry of the PhC unit cells surrounding the defect. To use the words of the
inventors of this mechanism, light should be confined gently in order to be confined strongly
[21]. The implementation of this simple, but yet paradigmatic objective is a primordial
dispersion engineering technique, based on tuning the PhC geometry around the defect.
Following these ideas, the design of a bichromatic potential involving the superimposition
of two 1D lattices with quasi-identical periods [23, 24], has been proposed for PhC slabs
[25], and proven to be highly efficient [26, 27]. The resulting cold-cavity spectrum forms a
comb of Hermite–Gauss modes confined by a parabolic potential, resembling the behavior
of a quantum harmonic oscillator [28]. The extremely low pump power which is required
to trigger parametric oscillation indicates that these systems are particularly promising for
quantum and nonlinear optics [29]. Inverse design approaches can be applied to engineer the
dispersion [30, 31], to maximize the quality factors [32] or to minimize losses in waveguide
configurations [33].
3
Among the different platforms of interest, III-V semiconductors, and specifically Al-
GaAs and AlGaAs-on-insulator (AlGaAs-OI), are very promising for the fabrication of DFB
structures for nonlinear optics applications, considering the large χ(2) and χ(3) response of
such materials, combined with the possibility of suppressing two-photon absorption (TPA)
processes at telecom wavelengths. The high core-to-cladding refractive index contrast, com-
pared to that of more mature technologies such as SiN-on-insulator (SiN-OI)[5, 7], suggests
the potential for broadband tunability of the chromatic dispersion profile. This is allowed
by the high degree of flexibility in tuning the local refractive index, which also ensures
strong optical confinement and compatibility with the Silicon photonics technology [34].
AlGaAs provides a suitable solution for the efficient integration of parametric microcomb
sources [35], with record low-power thresholds [36] and the perspective for opening a ω ⇄2ω
bandwidth via second-harmonic generation processes [37].
In this work, we carry out the design of an AlGaAs-OI 1D cavity comprised by two iden-
tical DBRs [11, 38] which are optimized for dispersion management purposes. The shaping
of the PhC unit cells composing the mirrors turns out to be a very efficient tool for tuning
the local badgap and the resulting effective reflectivity. Next, we report what is, to our
knowledge, a first experimental demonstration of a recently proposed shaped-constrained
inverse design technique [31], which permits to tailor a prescribed dispersion profile by
tuning the local reflectivity of a multimode PhC nanobeam resonator.
The manuscript is organized as follows: in section II we present the basic principles of
the design technique and its associated methods; in section III we discuss the implemen-
tation of a PhC cavity, whose unit cell can be exploited in order to customize the device
reflection; in section IV we report the properties of our dispersion engineered devices and
their experimental characterization; in section V we draw our conclusions and perspectives.
II.
DBR DESIGN FOR EFFICIENT BANDGAP TUNING
In Figure 1 we present the schematic of our design. The device (a) consists of a (100) 400
nm thick Al18%Ga82%As on a 2 µm thick buried oxide layer, with a Si substrate. The cavity
4
is comprised by a central waveguide section with symmetric DBRs placed at its sides. The
unit cell of the mirrors is sketched in panel (b): it involves a sinusoidal corrugation of the
waveguide stripe, with a fixed period Λ = 0.33 µm and an amplitude Γ/2, which alters the
local width of the waveguide and opens a photonic bandgap. The effective local waveguide
width (weff) along the propagation direction x reads as:
weff(x) = w0 + Γ(x)
2
sin
2πx
Λ
,
(1)
where we fix the average width at w0 = 0.61 µm, in order to guarantee broadband nonlinear
operation [37]. Notably, with these prescriptions and by fixing all other parameters, we
are able to directly connect the local band structure to the corrugation amplitude Γ(x).
This is sketched in Figure 1 (c), where we illustrate the band diagrams relative to the cases
with either Γ = 0 or 400 nm for the TE00 mode, respectively, calculated by means of a
commercial Finite Difference Time Domain (FDTD) solver (Lumerical Inc.). While Γ = 0
nm corresponds to the case of a standard waveguide–which sustains the propagation of
an unperturbed traveling wave in the k-ν space–with a finite corrugation amplitude (i.e.,
Γ = 400 nm) the dispersion is modified and we observe the emergence of a stop-band where
propagation is forbidden. As consequence, when a wave hits the region covered by the DBRs,
it is partially back reflected and linearly coupled into a counter-propagating wave [16, 17].
Appropriately designed DBRs introduce an effective potential well, that spatially con-
fines the cavity eigenmodes, in close analogy with the problem of an effective mass meff =
ℏ−1∂2ω/∂k2 confined by a parabolic potential well [39]. This approach is especially relevant
for pure PhC cavity configurations, which are achievable by placing two identical mirrors
face-to-face. This situation is reported in Figure 1 (d): a parabolic potential well confines
Hermite-Gauss (HG) eigenmodes along the longitudinal direction x. Thanks to the local
reflections induced by the waveguide corrugation, such structure couples a wave propagat-
ing along the forward direction (A+) with a feedback signal (A−) [17]. Mathematically, a
system that couples the one-dimensional counter-propagation of two confined waves reads
as:
D∂2
x ± ivg∂x + ω0 −ω
A± + KA± = 0
,
(2)
where vg and D are the group velocity and its dispersion, respectively. K is the coupling
between the two waves, or the grating strength, which quantifies the local photonic bandgap
5
and the effective reflectivity. We refer to the system of coupled Eqs.(2) as the reduced model
(RM), since it provides a robust approximation of the full complex 3D dynamics[31]. The
potential well V (x) can be drawn by mapping the photonic band-edge, i.e.:
V (x) = ω0(x) + 1
2K(x)
.
(3)
If V (x) is parabolic, i.e. V (x) ∝x2, it turns out that the eigenmodes of Eqs.(2) are close to
HG-shaped functions with a spatial envelope ψn(x) and eigenvalues ωn = 2πνn, as reported
in Figure 1(d) in normalized units. It is worth mentioning that this picture is exact in the
Gross-Pitaevskii limit V (x) →∝meffx2 [39]. The ideal HG mode structure can be approxi-
mated by a PhC cavity formed by two identical mirrors with a corrugation which increases
adiabatically as Γ(x) ∝x2.
Therefore, the design of an effective parabolic potential well requires an optimization pro-
cedure, which is reminiscent of inverse design, for defining the modulation profile of Γ(x).
Although in some cases the HG modes can be approximated without a full optimization
[25, 26, 28, 29], to our knowledge a systematic approach to master the dispersion and the
modes structure has not been experimentally attempted yet [40].
FIG. 1. (a) Schematic of the device. (b) PhC unit cell. (c) Photonic band diagram for specified
PhC waveguide geometries. Spatial confinement of a PhC cavity (d) and of a FP-PhC (e). The
resulting eignemodes have HG shaped or hybrid profiles, respectively.
6
Here we model the DBR feedback by means of a compact and well-established coupled
waves theory [15–17], which reduces the dimensionality of the associated eigenvalues prob-
lem from 3 to 1. A computational algorithm based on the RM of Eqs.(2) turns out to be
orders of magnitude faster than standard and commonly used solvers, such as finite element
method (FEM) or FDTD, applied to the full 3D problem. Interestingly, subject to a careful
and proper calibration of the model, with the RM one may recover the same accuracy of
exact 3D solvers. Meaning that one can use a highly precise and rapid eigenvalue solver,
which is suitable for implementing systematic design strategies in order to tailor prescribed
optical properties. It is worth noting that computational performance plays a critical role in
the implementation of optimization algorithms, since these typically require a few hundreds,
at least, of function evaluations [31].
Among the various dispersion engineered nanophotonic devices that have attracted signifi-
cant attention in recent years, let us consider linear resonators integrated on-chip [4, 5, 7].
An example of a such structure, and of its associated optical confinement, is sketched in Fig-
ure 1 (e): within a homogeneous central section comprised by two identical DBRs, the two
counter-propagating waves A± are no longer linearly coupled. The eigenmodes amplitude
is a constant until A+ hits the mirrors. At this point, the forward-propagating wave is par-
tially back-reflected and coupled to backward wave A−. The penetration depth through the
mirroring region strongly depends on the carrier wavelength. We can see how the confined
modes take an hybrid form: they are HG-shaped in the feedback section, similar to the PhC
cavity case, whereas their amplitude is constant in the central region, as it occurs in a FP
resonator. For this reason, such a cavity is often addressed to as FP-PhC or, similarly, as
Fabry–Pérot Bragg grating (FPBG).
III.
THE PHC NANOBEAM CAVITY
The dispersion engineering of a FP-PhC is strongly influenced by the design and fabrica-
tion quality of the Bragg mirrors. Thus, let us proceed to describe the fabrication and the
linear characterization of our highly reflective DBRs, whose schematic view, assembled in
a PhC cavity configuration, was presented in the previous section. Once that the efficient
operation of our Bragg mirrors is established, the dispersion tailoring functionalities will be
addressed in the following section.
7
The corrugation Γ is the only quantity which is left free to vary, and we leverage it for our
design purposes. The value of Γ maps the local photonic bands diagram, and quantifies the
feedback which linearly couples the two counter-propagating waves A±. We fix the value of
Γ in a 25 µm long central section, and we parabolically increase its value at the edges of the
structure as follows:
Γ(x) =
Γ0
for x−< x < x+
Γ0 + Γ2x2
for |x| > x+
,
(4)
where x± = ±12.5 µm, Γ0 = 0.1 µm and Γ2 = 150 m−1. The longitudinal profile of Γ is
reported in Figure 2 (a). This configuration results in an adiabatic variation of the DBRs
geometry, which guarantees strong (or gentle) spatial confinement.
Figure 2 (b) presents the cavity spectrum calculated by means of a commercial FEM solver
(COMSOL Multiphysics). We also show the first four modes, as predicted by the RM system
Eq.2: as can be seen, there is good quantitative agreement with the corresponding FEM
solutions, which confirms the high accuracy of the RM approach [31]. Figure 2 (c) shows
the intrinsic quality factor for each corresponding mode, establishing a direct connection
between the corrugation depth and the efficiency of light confinement. The curve Γ(x),
indeed, determines the shape of the potential well and the spatial profiles of the confined
FIG. 2.
(a) Profile of the quantity Γ along the PhC cavity. (b) Cavity spectrum and confining
potential well with the corresponding (c) intrinsic quality factor of the modes. (d) 2D and (e) 1D
spatial optical confinement of an high-order HG mode, and (f,g) corresponding 2D and 1D Fourier
transforms.
8
modes. Lower order modes, are tightly confined far from the edges of the potential well,
thus exhibiting negligible radiative leakage. This explains why Q factors are typically higher
for lower modal orders and larger λ.
The standard technique to reduce the radiative leakage of a 2D PhC cavity relies on fulfilling
the total internal reflection (TIR) condition along the vertical (i.e. out-of-plane) direction.
The latter is satisfied by minimizing the spectral components of the in-plane waves within
the light cone, which is delimited by a circle of diameter 2π/λ0, and centered in 0, in the
kx −ky Fourier domain [21, 22]. Here, λ0 is the wavelength in air, and k is the wavevector.
We illustrate these ideas in Figure 2 (d-g) for an high-order confined mode. The 2D and
1D y component of the electric field intensity, |Ey|, is reported in Figure 2 (d) and (e),
respectively. Note that in Figure 2 (d) the scheme of the PhC cavity layout–here reported
for illustrative purposes–is not in scale. We also report the corresponding Fourier transform
of the field intensity in Figure 2 (f-g). We can see that the components within the light cone
are negligible, thus ensuring minimal radiative losses, as by design prescriptions.
FIG. 3. (a) SEM pictures of the fabricated device. (b) Transmission. (c) Resonance fitting. (d)
Loaded quality factors statistics. (e) Integrated dispersion statistics (Dint).
9
In our fabricated device, light is coupled onto the chip using sub-wavelength grating
couplers (GCs) [41]. The fundamental transverse mode of the bus waveguide is excited,
and the coupled light is guided until it reaches a region where it is evanescently coupled to
the PhC cavity. We can see some snapshots of scanning electron microscope (SEM) images
in Figure 3(a). Finding the optimal coupling conditions is not an easy task. Ideally, we
target a critical coupling, which equals the internal losses of the cavity [42]. For this reason,
we span over different bus waveguide to cavity coupling conditions. Among more than 50
cavities, well-defined resonances were consistently observed in a limited subset of 9 samples.
In general, we observed a weak coupling and loaded quality factors of QL ≲6 × 105. Given
that our devices are visibly under-coupled, we can estimate an intrinsic quality factor of
Qi ≳105 [43]. Figure 3 (b) reports resonances observed on a ∼30 nm bandwidth. The
sloped background originates from the bell-shaped transmission spectrum of the GCs. Figure
3 (c) shows an example of resonance fitting, while Figure 3 (d) presents the statistics of the
quality factors.
Finally, since our goal is to establish a reliable technique for dispersion engineering, we
quantified the chromatic dispersion and compared the theoretical predictions with experi-
mental results. We introduce the integrated dispersion Dint,m relative to the mth mode as
[44]:
Dint,m = ωm −ωref −mD1
,
(5)
where ωref is a reference angular frequency and D1 is a reference frequency interval. In
Figure 2 (e) we illustrate the statistics of the experimental (exp) Dint,m profiles, showing an
optimal agreement with the theoretical predictions (sim).
In summary, the fabricated AlGaAs-OI Bragg mirrors offer an efficient solution for light
confinement, exhibiting a highly predictable dispersion profile which is suitable for the in-
verse design of the nanophotonic circuits.
IV.
DISPERSION TAILORING BY DISTRIBUTED FEEDBACK
After testing the reliable fabrication of the designed AlGaAs-OI DBRs, we have used
them to customize the chromatic dispersion of the resulting cavity. In this section, we first
present the results of our systematic dispersion engineering (IV A); next, we compare these
10
results with their experimental characterization (IV B).
A.
Systematic design
In this section, we discuss the engineering of flat dispersion profiles across various linear
resonator designs, with the purpose of achieving extremely low pump-power nonlinear fre-
quency mixing. [40, 45]. We consider homogeneous central sections of different lengths, i.e.,
L0 = 0 µm, L1 = 40 µm, L2 = 150 µm. From now on, we refer to the three different designs
by means of the length of their homogeneous central section (L0,1,2). As the latter increases
with respect to the length of the DBRs, the reflections are more narrowly distributed on the
edges of the structure. The extreme situation (L2) closely mimics the case of a FP resonator,
where the longitudinal reflections are strongly localized, and the wavelength dependence of
the cavity length is negligible. Here, we study the phenomenological transition from a pure
PhC cavity (L0), towards a hybrid design which combines a free-running section with two
identical and symmetric DBRs (L1 and L2). In each case, we aim at tailoring a prescribed
dispersion profile.
Our approach closely resembles the standard inverse design (ID) techniques, which are
commonly employed in nanophotonics [46, 47]. However, a key distinction lies in the strict
constraints that we impose on the n-dimensional geometry of the PhC unit cell. Unlike
conventional PhC ID methods, where the unit cell shape is subject to optimization[48],
we keep it fixed, as detailed in Section I. Instead, our design objective is limited to the
modulation of the amplitude profile, or Γ(x), of the sinusoidal corrugation along the cavity.
For further details on the design technique, we address the reader to a recent publication [31].
Let us express Γ(x) in terms of a generalized polynomial expansion as:
Γ(x) =
0
for x−< x < x+
(6a)
N
X
n=1
Γn|x|n
for x± ≶x ≶±xmax
(6b)
Γmax
for |x| ≥xmax
(6c)
where we fix N = 4, and the coefficients Γn, with n = 1, 2, 3, 4, are the subject of the
optimization procedure. N = 4 represents an optimal compromise between the required
11
FIG. 4. Spatial confinement of a flat dispersion FP-PhC cavity with a homogeneous central section
of L0 = 0 µm (a), L1 = 40 µm (c), L2 = 150 µm (e), respectively. To improve readability, only one
out of every two solutions is reported in panel (e). In (b), (d) and (f) we report the corresponding
sinusoidal corrugation amplitude Γ(x). (g) Minimization of the cost function for the three different
optimizations L0,1,2.
accuracy and computational cost. The central section boundaries are x± = ±Lj/2. Eventu-
ally, the parameter Γ is clamped to a maximum value Γmax = PN
n=1 Γnxn
max on the extreme
edges of the structure, in order to stretch the length of the mirrors and so guarantee a
stronger confinement. The object of optimization is the cost function:
costf =
M
X
m
|Dint,m −Dint,m|
D1
(13)
12
where Dint,m is the target dispersion profile.
Eq.13 was minimized by using a multi-coefficient gradient descent method (i.e. Nelder-
Mead) [49]. We targeted a flat dispersion across M = 10 cavity modes. The goal here is to
engineer the spectral regime which is most affected by modifications to the waveguide ridge
(specifically, the modes that are most sensitive to photonic bandgap tuning, representing
the transition from a pure PhC to a FP resonator). In the PhC case, the chromatic dis-
persion of the structure is primarily governed by longitudinal variations of the waveguide
geometry. In contrast, for a FP configuration, where the transverse cross-section does not
change longitudinally, dispersion is governed by the wavelength dependence of the resulting
effective refractive index neff(λ).
In Figure 4(a) we illustrate the flat-dispersion PhC cavity spectrum and the confining
potential well V (x) resulting from the inversely designed Γ(x) profile shown in Figure 4
(b). In this case, since the feedback is distributed along the entire cavity, the longitudinal
dependence of the spatial confinement becomes evident, and leads to the formation of HG
modes. This situation corresponds to the degenerate case L0 = 0 µm. Next, we consider
the case of a relatively small homogeneous central section L1 = 40 µm, whose spectral con-
finement is shown in Figure4 (c); the corresponding Γ(x) profile is drawn in panel (d). We
may notice that now the confined modes are hybridized: they exhibit a constant amplitude
within the homogeneous central section, while they are HG-shaped in the tapered feedback
section x± ≶x ≶±xmax. The tapering domain of the optimized DBRs profile extends
over just ∆x ≡xmax −x+ ∼5 nm. This suffices to tailor dispersive effects over a spectral
domain ∆ν ∼15 THz, which represents the effective photonic bandgap tuning interval.
Indeed the DBRs are also located at longitudinal coordinates |x| ≥xmax for a fixed Γmax,
but in this outer section they only serve to enhanced reflection, and to improve the resulting
confinement. On the other hand, dispersion properties are uniquely tailored in the section
where the Γ(x) profile is tapered.
We marked by different colors the three characteristic sections of the µ-cavity:
(i) the central section (FP) is the free-running domain where the two counter-propagating
waves are not linearly coupled by local reflections, being Γ = 0 nm. It is mathematically
defined by Eq. 6a. The dynamics and the modal confinement are identical to the case of
13
standard FP resonators. Note that this section is missing in the PhC configuration (Figure
3 (b)).
(ii) In the intermediate section x± ≶x ≶±xmax (taper, defined by Eq. 6b), the DBRs
provide a feedback mechanism with increasing grating strength and enhanced reflectivity
towards the edges of the structure. This unit tailors the dispersive effects, as it shapes the
effective potential well V (x) and the local coupling between the two counter-propagating
waves.
(iii) Finally, in the most external section |x| > ±xmax (clamping, defined by Eq. 6c),) the
Γ(x) profile is kept fixed for several PhC periods, in order to improve the overall reflectivity
of the Bragg reflectors.
Finally, in Figure 4 (e,f), we illustrate the ultimate design, exhibiting a central FP section
of length L2 = 150 µm. Although L2 is relatively short compared to typical FP–PhC designs
[5, 7], it is still significantly longer than the adjacent taper regions. This configuration rep-
resents the endpoint of the PhC-to-FP resonator transition that we explored in the present
study.
To conclude the design description, we also report the evolution of the cost function f
(Eq.(13)) vs. the function calls during the optimization of the dispersion profiles for the three
different cavities L0,1,2. As we can see in Figure 4 (g), the number of calls spans from a few
hundreds to almost one thousand. For this reason, as anticipated in section I, it is crucial to
dispose of a fast and reliable solver. In our case, we used the 1D RM model of Eqs.(2), which
was carefully calibrated on the FEM solutions of a spare reference PhC design. In this way,
we could achieve a good accuracy without loosing computational performance [31]. At last,
our home-built solver employs ≲0.5 secs to compute the spectrum and the corresponding
Dint profile, which is used to evaluate the cost f, at each call. This typical CPU time should
be compared with the much longer time scales (i. e. ≳hours), which are needed to compute
the reference FEM solutions for calibration. It results that the whole optimization duration
is of the order of just few minutes.
We observe that the cavity length L0 is the most sensitive to the optimization process.
This sensitivity arises from two closely related factors. First, in this configuration, which
corresponds to a pure PhC cavity, the feedback is distributed across the entire confinement
14
region. As a result, any modification directly alters the grating strength and the spatial
mode profile, which in turn affects the Fourier spectrum significantly. Second, in FP-like
resonators the resonances exhibit closely aligned free spectral ranges (FSRs). In this case,
while our approach enables precise dispersion engineering over a broad spectral domain, it
is less effective for inducing strong, localized dispersion changes.
B.
Experimental results
The fabricated sample contained different sets of identical cavities with different bus
waveguide-to-cavity coupling conditions.
We were able to observe resonances on a sta-
tistically relevant number of samples (i.e. ≥20) for each cavity design L0,1,2. The linear
characterization consists in a standard transmission experiment over a ≳100 nm bandwidth.
We can easily measure the cavity spectrum and locate the wavelength of the resonances.
The results are presented in Figure 5. In Figure 5 (a,b,c) we report the theoretical disper-
sion profiles (black dashed line) and the corresponding confidence interval calculated from
a statistical experimental data treatment (gray shadowed area).
We observe an excellent agreement between experiment and theory for the pure PhC config-
uration (L0). In contrast, a slight deviation from theoretical predictions appears at higher
frequencies for the hybrid FP–PhC designs (L1 and L2 ). We believe that the main reason
for this discrepancy is due to a non-negligible group velocity mismatch ∆vg ̸= 0, which
arises at the boundaries between the homogeneous central section and the DBRs.
This
mismatch (∆vg) not only induces scattering losses, but also a misalignment of the FSRs. To
address this problem, an intermediate section can be introduced between the FP and the
tapered DBRs regions; the waveguide width could then be linearly tapered to decelerate the
intra-cavity field as it approaches the DBRs from the free-running section and to minimize
the discussed group velocity mismatch ∆vg →0 [7].
Finally, in Figure 5 (d,e) we report two exemplary transmission measurements. In Figure 5
(d) we highlight the emergence of the photonic bandgap, which is typical for PhC cavities.
In order to perform a quantitative analysis of the experimental dispersion profiles, we
15
FIG. 5.
Comparison between the designed and experimental Dint for the resonators (a) L0, (b)
L1 and (c) L2. Examples of optical transmission for (d) L0 and (e) L2.
FIG. 6.
Fluctuations of the measured resonances (a,b,c) and example of dispersion fitting (d,e,f)
for the designs L0, L1 and L2 (listed from left to right), respectively.
16
map the cavity resonances by expanding Eq.5 up to the fourth order [36, 45]:
νm = νref + 1
2π
4
X
n=1
Dnm2
n!
.
(14)
We fitted Eq.14 for each cavity. Then, for each cavity design we regrouped the results of
the fit to extract the mean value and the standard deviation (under the assumption of a
Gaussian distribution). The results of the fit are listed in Table I.
TABLE I. Fit estimation of dispersion parameters up to the fourth order (D1,...,4) .
Resonator D1/2π (GHz) D2/2π (GHz) D3/2π (GHz) D4/2π (GHz)
L0
470 ± 90
−30 ± 60
10 ± 30
−3 ± 6
L1
310 ± 90
120 ± 90
−60 ± 70
10 ± 20
L2
225 ± 11
2 ± 8
−3 ± 2
0.3 ± 0.4
Substantial deviations are observed, suggesting that fabrication tolerances may hinder
the reproducibility of the present results. The fluctuations of the resonances for each design
are shown in Figure 6 (a,b,c).
Here we report the measured resonances (res) and their
standard deviations w.r.t. their mean value (std). Finally, in panel (d,e,f), we report three
examples of estimated fits of Eq. 14 for the three dsigns L0,1,2.
To summarize, we can conclude that the systematic engineering dispersion technique
recently discussed [31]–and here demonstrated for the first time–exhibits high performances
for the pure PhC case, even though it slightly deviates from design prescriptions in hybrid
FP-PhC configurations. Some further practical implementations could improve the current
results, as the insertion of a linear tapering section to smother the group velocities mismatch
between the FP and the mirroring sections [7], also targeting an efficient nonlinear operation
[11].
V.
CONCLUSIONS AND PERSPECTIVES
In conclusions, we introduced a novel approach to perform a systematic design of disper-
sion tailored multimode linear micro-cavities. We investigated quasi-periodic PhC systems,
17
and pointed out the strong connection between photonic bandgap tunability and dispersion
engineered nanophotonics. In this context, the on-chip integration of Fabry-Pérot resonators
is addressed by means of a distributed feedback mechanism. Here we studied the continuous
transition from a PhC cavity into an hybrid linear resonator, comprised by two identical
and symmetric distributed Bragg reflectors. The latter embeds the characteristics of both
standard FP and PhC cavities, reason why it is often addressed to as FP-PhC, or, similarly,
as Fabry–Pérot Bragg grating (FPBG). In our study, we implemented an optimization algo-
rithm for targeting a prescribed dispersion profile for a FP-PhC. We made use of a reduced
model, which approximates the complex 3D dynamics by means of a compact and compu-
tationally light 1D coupled waves theory [31]. We presented an experimental demonstration
of this inverse design approach based on the AlGaAs-on-insulator platform, showing a good
quantitative agreement with simulations.
We believe that our method can be easily extended to all platforms exhibiting strong index
contrast between the waveguide core and the cladding. In perspective, we aim to extend the
functionalities of DBRs over a broadband domain, ideally over a ω ⇄2ω operation band.
The latter is a very challenging task at telecom wavelengths, since the photonic bandgap
tunability is typically limited by the material to, at best, ∼50 THz [5]. Nonetheless, we
believe that a reliable design solution may be achieved, with the help of a more comprehensive
theoretical investigation.
Finally, similar configurations may also be adopted for µ-ring devices, as recently reported
for SiN, including slow-light confinement[50], Fourier synthesis dispersion engineering [45],
or the generation of microcombs with squared frequency shaping [51].
To summarize, dispersion engineered nanophotonics is receiving a great deal of attention.
AlGaAs-OI, as discussed in the present manuscript, represents a particularly promising
solution.
AUTHOR CONTRIBUTION STATEMENT
F.R. Talenti (conceptualization, theory, design, experiments, and writing) and L. Lovisolo
(fabrication, design, and experiments) shared the main contribution to this work. C. Alonso-
Ramos, M. Morassi, A. Lemaitre, A. Harouri, A. Gerini and L. Vivien contributed to the
18
fabrication of the device. Z. Xiao, Z. Saleh, A. De Rossi, S. Wabnitz, and G. Leo contributed
to the development of the theory and to conceiving the experiment. A. De Rossi, S. Wabnitz
and L. Vivien contributed to manuscript writing. L. Vivien supervised all aspects of the
work.
FUNDING
The work is partly supported by the French RENATECH network.
Other funding:
ANR-22-CE92-0065 (Quadcomb), EU – NRRP, NextGenerationEU (PE00000001 – program
“RESTART”). L. Lovisolo acknowledges the support of French Agence-innovation defense N
DGA01D22020572.
DISCLOSURES
The authors declare no conflicts of interest.
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Dispersion engineered AlGaAs-on-insulator nanophotonics by distributed feedback Francesco Rinaldo Talenti,1 Luca Lovisolo,2, 1 Zijun Xiao,1 Zeina Saleh,1 Andrea Gerini,2 Carlos Alonso-Ramos,1 Martina Morassi,1 Aristide Lemaître,1 Stefan Wabnitz,3, 4 Alfredo De Rossi,5 Giuseppe Leo,2, 6 and Laurent Vivien1 1Université Paris-Saclay, CNRS, Centre for Nanosciences and Nanotechnology, 91120 Palaiseau, France 2Université Paris Cité, Laboratoire Matériaux et Phénomènes Quantiques, 75013 Paris, France 3Sapienza 00184 Roma, Italy 4CNR-INO, Istituto Nazionale di Ottica, 80078 Pozzuoli, Italy 5Thales Research and Technology, 91120 Palaiseau, France 6Institut universitaire de France, France 1 16 Oct 2025 Abstract Technological advances in the fabrication of nanophotonic circuits have driven the scientific community to increasingly focus on the precise tailoring of their key optical properties, over a broadband spectral domain. In this context, the modulation of the local refractive index can be exploited to customize an effective reflectivity by the use of distributed Bragg mirrors, enabling the on-chip integration of Fabry-Pérot resonators. The resulting cavity length is strongly wavelength-dependent, offering practical solutions to the growing demand of dispersion engineering. Owing to their typically high core-to-cladding refractive index contrast and exceptional nonlinear properties, III-V semiconductor-based platforms represent promising candidates for the fabrication of Bragg reflectors. In this work, we propose an AlGaAs-on-insulator linear resonator based on distributed Bragg mirrors. We discuss the first experimental demonstration of a systematic, shape-constrained inverse design technique which tailors a prescribed dispersion profile, showing a strong agreement between simulations and measurements. In perspective, the proposed approach offers an efficient and general response to the challenge of dispersion engineering in integrated optical circuits. Keywords: Fabry-Perot resonator, Photonic crystal, distributed-feedback, AlGaAs, dipersion engineering, inverse design. I. INTRODUCTION Fabry-Pérot (FP) resonators have recently been developed on fiber-based platforms (Fiber Fabry-Pérot, FFP), resulting in compact, efficient, and relatively easy to implement systems for the demonstration of modulation instability (MI) [1], frequency combs and cavity soliton generation [2, 3]. The on-chip integration of such systems-driven by the goal of minimizing the device footprint and reducing the power operation-requires the use of distributed Bragg reflectors (DBRs). First prototypes of this sort have been proposed for Kerr comb generation [4] and then implemented on silicon nitride (SiN) [5] to enhance four-wave-mixing (FWM) [6], or to trigger solitons formation [7], while parent demonstrations are those of Moiré plasmonic cavities [8, 9] suitable for laser applications [10]. Similar systems have been also conceived with III-V semiconductor, and proven to be promising for nonlinear frequency conversion [11], pulse spectral broadening [12] and the design of nanophotonic distributed feedback (DFB) lasers [13, 14]. 2 In a DFB structure, a periodic longitudinal modulation of the refractive index is exploited to introduce a Bragg backward scattering mechanism [15, 16]. As a result, two counter-propagating optical waves are spatially confined and linearly coupled by multiple local reflections along the cavity[17]. DFB systems, thus, inhibit light propagation within a finite spectral domain [18, 19]. In strict analogy and with a direct connection with electronic dispersion diagrams, photonic band structures can be drawn for dielectrics which are geometrically organized with a given and well defined periodicity Λ = λ0/2n, where ω0/2π = c/λ0 is the central frequency of oscillation (i.e. the Bragg frequency), c is the speed of light in vacuum and n is the refractive index of the bulk material[20]. The underlying physics is that of a photonic crystal (PhC), whose geometry determines the effective local band diagrams. Furthermore, the introduction of a crystal defect induces light confinement, and it can be exploited for the design of nano-cavities. The maximization of the quality factors, though, is a non-trivial task. First convincing demonstrations of low-dimensional (i.e. ≤2D) PhC cavities introduced a novel light confining mechanism, named gentle confinement [21, 22]. The basic idea consists in the fact that light scattering at defect boundaries can be a detrimental source of loss. To avoid that, light must be progressively (or gently) guided towards the confining spatial domain, by adiabatically tuning the geometry of the PhC unit cells surrounding the defect. To use the words of the inventors of this mechanism, light should be confined gently in order to be confined strongly [21]. The implementation of this simple, but yet paradigmatic objective is a primordial dispersion engineering technique, based on tuning the PhC geometry around the defect. Following these ideas, the design of a bichromatic potential involving the superimposition of two 1D lattices with quasi-identical periods [23, 24], has been proposed for PhC slabs [25], and proven to be highly efficient [26, 27]. The resulting cold-cavity spectrum forms a comb of Hermite-Gauss modes confined by a parabolic potential, resembling the behavior of a quantum harmonic oscillator [28]. The extremely low pump power which is required to trigger parametric oscillation indicates that these systems are particularly promising for quantum and nonlinear optics [29]. Inverse design approaches can be applied to engineer the dispersion [30, 31], to maximize the quality factors [32] or to minimize losses in waveguide configurations [33]. 3 Among the different platforms of interest, III-V semiconductors, and specifically AlGaAs and AlGaAs-on-insulator (AlGaAs-OI), are very promising for the fabrication of DFB structures for nonlinear optics applications, considering the large χ(2) and χ(3) response of such materials, combined with the possibility of suppressing two-photon absorption (TPA) processes at telecom wavelengths. The high core-to-cladding refractive index contrast, compared to that of more mature technologies such as SiN-on-insulator (SiN-OI)[5, 7], suggests the potential for broadband tunability of the chromatic dispersion profile. This is allowed by the high degree of flexibility in tuning the local refractive index, which also ensures strong optical confinement and compatibility with the Silicon photonics technology [34]. AlGaAs provides a suitable solution for the efficient integration of parametric microcomb sources [35], with record low-power thresholds [36] and the perspective for opening a ω ⇄2ω bandwidth via second-harmonic generation processes [37]. In this work, we carry out the design of an AlGaAs-OI 1D cavity comprised by two identical DBRs [11, 38] which are optimized for dispersion management purposes. The shaping of the PhC unit cells composing the mirrors turns out to be a very efficient tool for tuning the local badgap and the resulting effective reflectivity. Next, we report what is, to our knowledge, a first experimental demonstration of a recently proposed shaped-constrained inverse design technique [31], which permits to tailor a prescribed dispersion profile by tuning the local reflectivity of a multimode PhC nanobeam resonator. The manuscript is organized as follows: in section II we present the basic principles of the design technique and its associated methods; in section III we discuss the implementation of a PhC cavity, whose unit cell can be exploited in order to customize the device reflection; in section IV we report the properties of our dispersion engineered devices and their experimental characterization; in section V we draw our conclusions and perspectives. II. DBR DESIGN FOR EFFICIENT BANDGAP TUNING In Figure 1 we present the schematic of our design. The device (a) consists of a (100) 400 nm thick Al18%Ga82%As on a 2 μm thick buried oxide layer, with a Si substrate. The cavity 4 is comprised by a central waveguide section with symmetric DBRs placed at its sides. The unit cell of the mirrors is sketched in panel (b): it involves a sinusoidal corrugation of the waveguide stripe, with a fixed period Λ = 0.33 μm and an amplitude Γ/2, which alters the local width of the waveguide and opens a photonic bandgap. The effective local waveguide width (weff) along the propagation direction x reads as: weff(x) = w0 + Γ(x) 2 sin 2πx Λ , (1) where we fix the average width at w0 = 0.61 μm, in order to guarantee broadband nonlinear operation [37]. Notably, with these prescriptions and by fixing all other parameters, we are able to directly connect the local band structure to the corrugation amplitude Γ(x). This is sketched in Figure 1 (c), where we illustrate the band diagrams relative to the cases with either Γ = 0 or 400 nm for the TE00 mode, respectively, calculated by means of a commercial Finite Difference Time Domain (FDTD) solver (Lumerical Inc.). While Γ = 0 nm corresponds to the case of a standard waveguide-which sustains the propagation of an unperturbed traveling wave in the k-ν space-with a finite corrugation amplitude (i.e., Γ = 400 nm) the dispersion is modified and we observe the emergence of a stop-band where propagation is forbidden. As consequence, when a wave hits the region covered by the DBRs, it is partially back reflected and linearly coupled into a counter-propagating wave [16, 17]. Appropriately designed DBRs introduce an effective potential well, that spatially confines the cavity eigenmodes, in close analogy with the problem of an effective mass meff = ħ-1∂2ω/∂k2 confined by a parabolic potential well [39]. This approach is especially relevant for pure PhC cavity configurations, which are achievable by placing two identical mirrors face-to-face. This situation is reported in Figure 1 (d): a parabolic potential well confines Hermite-Gauss (HG) eigenmodes along the longitudinal direction x. Thanks to the local reflections induced by the waveguide corrugation, such structure couples a wave propagating along the forward direction (A+) with a feedback signal (A-) [17]. Mathematically, a system that couples the one-dimensional counter-propagation of two confined waves reads as: D∂2 x ± ivg∂x + ω0 -ω A± + KA± = 0 , (2) where vg and D are the group velocity and its dispersion, respectively. K is the coupling between the two waves, or the grating strength, which quantifies the local photonic bandgap 5 and the effective reflectivity. We refer to the system of coupled Eqs.(2) as the reduced model (RM), since it provides a robust approximation of the full complex 3D dynamics[31]. The potential well V (x) can be drawn by mapping the photonic band-edge, i.e.: V (x) = ω0(x) + 1 2K(x) . (3) If V (x) is parabolic, i.e. V (x) ∝x2, it turns out that the eigenmodes of Eqs.(2) are close to HG-shaped functions with a spatial envelope ψn(x) and eigenvalues ωn = 2πνn, as reported in Figure 1(d) in normalized units. It is worth mentioning that this picture is exact in the Gross-Pitaevskii limit V (x) →∝meffx2 [39]. The ideal HG mode structure can be approximated by a PhC cavity formed by two identical mirrors with a corrugation which increases adiabatically as Γ(x) ∝x2. Therefore, the design of an effective parabolic potential well requires an optimization procedure, which is reminiscent of inverse design, for defining the modulation profile of Γ(x). Although in some cases the HG modes can be approximated without a full optimization [25, 26, 28, 29], to our knowledge a systematic approach to master the dispersion and the modes structure has not been experimentally attempted yet [40]. FIG. 1. (a) Schematic of the device. (b) PhC unit cell. (c) Photonic band diagram for specified PhC waveguide geometries. Spatial confinement of a PhC cavity (d) and of a FP-PhC (e). The resulting eignemodes have HG shaped or hybrid profiles, respectively. 6 Here we model the DBR feedback by means of a compact and well-established coupled waves theory [15-17], which reduces the dimensionality of the associated eigenvalues problem from 3 to 1. A computational algorithm based on the RM of Eqs.(2) turns out to be orders of magnitude faster than standard and commonly used solvers, such as finite element method (FEM) or FDTD, applied to the full 3D problem. Interestingly, subject to a careful and proper calibration of the model, with the RM one may recover the same accuracy of exact 3D solvers. Meaning that one can use a highly precise and rapid eigenvalue solver, which is suitable for implementing systematic design strategies in order to tailor prescribed optical properties. It is worth noting that computational performance plays a critical role in the implementation of optimization algorithms, since these typically require a few hundreds, at least, of function evaluations [31]. Among the various dispersion engineered nanophotonic devices that have attracted significant attention in recent years, let us consider linear resonators integrated on-chip [4, 5, 7]. An example of a such structure, and of its associated optical confinement, is sketched in Figure 1 (e): within a homogeneous central section comprised by two identical DBRs, the two counter-propagating waves A± are no longer linearly coupled. The eigenmodes amplitude is a constant until A+ hits the mirrors. At this point, the forward-propagating wave is partially back-reflected and coupled to backward wave A-. The penetration depth through the mirroring region strongly depends on the carrier wavelength. We can see how the confined modes take an hybrid form: they are HG-shaped in the feedback section, similar to the PhC cavity case, whereas their amplitude is constant in the central region, as it occurs in a FP resonator. For this reason, such a cavity is often addressed to as FP-PhC or, similarly, as Fabry-Pérot Bragg grating (FPBG). III. THE PHC NANOBEAM CAVITY The dispersion engineering of a FP-PhC is strongly influenced by the design and fabrication quality of the Bragg mirrors. Thus, let us proceed to describe the fabrication and the linear characterization of our highly reflective DBRs, whose schematic view, assembled in a PhC cavity configuration, was presented in the previous section. Once that the efficient operation of our Bragg mirrors is established, the dispersion tailoring functionalities will be addressed in the following section. 7 The corrugation Γ is the only quantity which is left free to vary, and we leverage it for our design purposes. The value of Γ maps the local photonic bands diagram, and quantifies the feedback which linearly couples the two counter-propagating waves A±. We fix the value of Γ in a 25 μm long central section, and we parabolically increase its value at the edges of the structure as follows: Γ(x) = Γ0 for x- x+ , (4) where x± = ±12.5 μm, Γ0 = 0.1 μm and Γ2 = 150 m-1. The longitudinal profile of Γ is reported in Figure 2 (a). This configuration results in an adiabatic variation of the DBRs geometry, which guarantees strong (or gentle) spatial confinement. Figure 2 (b) presents the cavity spectrum calculated by means of a commercial FEM solver (COMSOL Multiphysics). We also show the first four modes, as predicted by the RM system Eq.2: as can be seen, there is good quantitative agreement with the corresponding FEM solutions, which confirms the high accuracy of the RM approach [31]. Figure 2 (c) shows the intrinsic quality factor for each corresponding mode, establishing a direct connection between the corrugation depth and the efficiency of light confinement. The curve Γ(x), indeed, determines the shape of the potential well and the spatial profiles of the confined FIG. 2. (a) Profile of the quantity Γ along the PhC cavity. (b) Cavity spectrum and confining potential well with the corresponding (c) intrinsic quality factor of the modes. (d) 2D and (e) 1D spatial optical confinement of an high-order HG mode, and (f,g) corresponding 2D and 1D Fourier transforms. 8 modes. Lower order modes, are tightly confined far from the edges of the potential well, thus exhibiting negligible radiative leakage. This explains why Q factors are typically higher for lower modal orders and larger λ. The standard technique to reduce the radiative leakage of a 2D PhC cavity relies on fulfilling the total internal reflection (TIR) condition along the vertical (i.e. out-of-plane) direction. The latter is satisfied by minimizing the spectral components of the in-plane waves within the light cone, which is delimited by a circle of diameter 2π/λ0, and centered in 0, in the kx -ky Fourier domain [21, 22]. Here, λ0 is the wavelength in air, and k is the wavevector. We illustrate these ideas in Figure 2 (d-g) for an high-order confined mode. The 2D and 1D y component of the electric field intensity, |Ey|, is reported in Figure 2 (d) and (e), respectively. Note that in Figure 2 (d) the scheme of the PhC cavity layout-here reported for illustrative purposes-is not in scale. We also report the corresponding Fourier transform of the field intensity in Figure 2 (f-g). We can see that the components within the light cone are negligible, thus ensuring minimal radiative losses, as by design prescriptions. FIG. 3. (a) SEM pictures of the fabricated device. (b) Transmission. (c) Resonance fitting. (d) Loaded quality factors statistics. (e) Integrated dispersion statistics (Dint). 9 In our fabricated device, light is coupled onto the chip using sub-wavelength grating couplers (GCs) [41]. The fundamental transverse mode of the bus waveguide is excited, and the coupled light is guided until it reaches a region where it is evanescently coupled to the PhC cavity. We can see some snapshots of scanning electron microscope (SEM) images in Figure 3(a). Finding the optimal coupling conditions is not an easy task. Ideally, we target a critical coupling, which equals the internal losses of the cavity [42]. For this reason, we span over different bus waveguide to cavity coupling conditions. Among more than 50 cavities, well-defined resonances were consistently observed in a limited subset of 9 samples. In general, we observed a weak coupling and loaded quality factors of QL ≲6 × 105. Given that our devices are visibly under-coupled, we can estimate an intrinsic quality factor of Qi ≳105 [43]. Figure 3 (b) reports resonances observed on a ∼30 nm bandwidth. The sloped background originates from the bell-shaped transmission spectrum of the GCs. Figure 3 (c) shows an example of resonance fitting, while Figure 3 (d) presents the statistics of the quality factors. Finally, since our goal is to establish a reliable technique for dispersion engineering, we quantified the chromatic dispersion and compared the theoretical predictions with experimental results. We introduce the integrated dispersion Dint,m relative to the mth mode as [44]: Dint,m = ωm -ωref -mD1 , (5) where ωref is a reference angular frequency and D1 is a reference frequency interval. In Figure 2 (e) we illustrate the statistics of the experimental (exp) Dint,m profiles, showing an optimal agreement with the theoretical predictions (sim). In summary, the fabricated AlGaAs-OI Bragg mirrors offer an efficient solution for light confinement, exhibiting a highly predictable dispersion profile which is suitable for the inverse design of the nanophotonic circuits. IV. DISPERSION TAILORING BY DISTRIBUTED FEEDBACK After testing the reliable fabrication of the designed AlGaAs-OI DBRs, we have used them to customize the chromatic dispersion of the resulting cavity. In this section, we first present the results of our systematic dispersion engineering (IV A); next, we compare these 10 results with their experimental characterization (IV B). A. Systematic design In this section, we discuss the engineering of flat dispersion profiles across various linear resonator designs, with the purpose of achieving extremely low pump-power nonlinear frequency mixing. [40, 45]. We consider homogeneous central sections of different lengths, i.e., L0 = 0 μm, L1 = 40 μm, L2 = 150 μm. From now on, we refer to the three different designs by means of the length of their homogeneous central section (L0,1,2). As the latter increases with respect to the length of the DBRs, the reflections are more narrowly distributed on the edges of the structure. The extreme situation (L2) closely mimics the case of a FP resonator, where the longitudinal reflections are strongly localized, and the wavelength dependence of the cavity length is negligible. Here, we study the phenomenological transition from a pure PhC cavity (L0), towards a hybrid design which combines a free-running section with two identical and symmetric DBRs (L1 and L2). In each case, we aim at tailoring a prescribed dispersion profile. Our approach closely resembles the standard inverse design (ID) techniques, which are commonly employed in nanophotonics [46, 47]. However, a key distinction lies in the strict constraints that we impose on the n-dimensional geometry of the PhC unit cell. Unlike conventional PhC ID methods, where the unit cell shape is subject to optimization[48], we keep it fixed, as detailed in Section I. Instead, our design objective is limited to the modulation of the amplitude profile, or Γ(x), of the sinusoidal corrugation along the cavity. For further details on the design technique, we address the reader to a recent publication [31]. Let us express Γ(x) in terms of a generalized polynomial expansion as: Γ(x) = 0 for x- ±xmax (clamping, defined by Eq. 6c),) the Γ(x) profile is kept fixed for several PhC periods, in order to improve the overall reflectivity of the Bragg reflectors. Finally, in Figure 4 (e,f), we illustrate the ultimate design, exhibiting a central FP section of length L2 = 150 μm. Although L2 is relatively short compared to typical FP-PhC designs [5, 7], it is still significantly longer than the adjacent taper regions. This configuration represents the endpoint of the PhC-to-FP resonator transition that we explored in the present study. To conclude the design description, we also report the evolution of the cost function f (Eq.(13)) vs. the function calls during the optimization of the dispersion profiles for the three different cavities L0,1,2. As we can see in Figure 4 (g), the number of calls spans from a few hundreds to almost one thousand. For this reason, as anticipated in section I, it is crucial to dispose of a fast and reliable solver. In our case, we used the 1D RM model of Eqs.(2), which was carefully calibrated on the FEM solutions of a spare reference PhC design. In this way, we could achieve a good accuracy without loosing computational performance [31]. At last, our home-built solver employs ≲0.5 secs to compute the spectrum and the corresponding Dint profile, which is used to evaluate the cost f, at each call. This typical CPU time should be compared with the much longer time scales (i. e. ≳hours), which are needed to compute the reference FEM solutions for calibration. It results that the whole optimization duration is of the order of just few minutes. We observe that the cavity length L0 is the most sensitive to the optimization process. This sensitivity arises from two closely related factors. First, in this configuration, which corresponds to a pure PhC cavity, the feedback is distributed across the entire confinement 14 region. As a result, any modification directly alters the grating strength and the spatial mode profile, which in turn affects the Fourier spectrum significantly. Second, in FP-like resonators the resonances exhibit closely aligned free spectral ranges (FSRs). In this case, while our approach enables precise dispersion engineering over a broad spectral domain, it is less effective for inducing strong, localized dispersion changes. B. Experimental results The fabricated sample contained different sets of identical cavities with different bus waveguide-to-cavity coupling conditions. We were able to observe resonances on a statistically relevant number of samples (i.e. ≥20) for each cavity design L0,1,2. The linear characterization consists in a standard transmission experiment over a ≳100 nm bandwidth. We can easily measure the cavity spectrum and locate the wavelength of the resonances. The results are presented in Figure 5. In Figure 5 (a,b,c) we report the theoretical dispersion profiles (black dashed line) and the corresponding confidence interval calculated from a statistical experimental data treatment (gray shadowed area). We observe an excellent agreement between experiment and theory for the pure PhC configuration (L0). In contrast, a slight deviation from theoretical predictions appears at higher frequencies for the hybrid FP-PhC designs (L1 and L2 ). We believe that the main reason for this discrepancy is due to a non-negligible group velocity mismatch ∆vg ̸= 0, which arises at the boundaries between the homogeneous central section and the DBRs. This mismatch (∆vg) not only induces scattering losses, but also a misalignment of the FSRs. To address this problem, an intermediate section can be introduced between the FP and the tapered DBRs regions; the waveguide width could then be linearly tapered to decelerate the intra-cavity field as it approaches the DBRs from the free-running section and to minimize the discussed group velocity mismatch ∆vg →0 [7]. Finally, in Figure 5 (d,e) we report two exemplary transmission measurements. In Figure 5 (d) we highlight the emergence of the photonic bandgap, which is typical for PhC cavities. In order to perform a quantitative analysis of the experimental dispersion profiles, we 15 FIG. 5. Comparison between the designed and experimental Dint for the resonators (a) L0, (b) L1 and (c) L2. Examples of optical transmission for (d) L0 and (e) L2. FIG. 6. Fluctuations of the measured resonances (a,b,c) and example of dispersion fitting (d,e,f) for the designs L0, L1 and L2 (listed from left to right), respectively. 16 map the cavity resonances by expanding Eq.5 up to the fourth order [36, 45]: νm = νref + 1 2π 4 X n=1 Dnm2 n! . (14) We fitted Eq.14 for each cavity. Then, for each cavity design we regrouped the results of the fit to extract the mean value and the standard deviation (under the assumption of a Gaussian distribution). The results of the fit are listed in Table I. TABLE I. Fit estimation of dispersion parameters up to the fourth order (D1,...,4) . Resonator D1/2π (GHz) D2/2π (GHz) D3/2π (GHz) D4/2π (GHz) L0 470 ± 90 -30 ± 60 10 ± 30 -3 ± 6 L1 310 ± 90 120 ± 90 -60 ± 70 10 ± 20 L2 225 ± 11 2 ± 8 -3 ± 2 0.3 ± 0.4 Substantial deviations are observed, suggesting that fabrication tolerances may hinder the reproducibility of the present results. The fluctuations of the resonances for each design are shown in Figure 6 (a,b,c). Here we report the measured resonances (res) and their standard deviations w.r.t. their mean value (std). Finally, in panel (d,e,f), we report three examples of estimated fits of Eq. 14 for the three dsigns L0,1,2. To summarize, we can conclude that the systematic engineering dispersion technique recently discussed [31]-and here demonstrated for the first time-exhibits high performances for the pure PhC case, even though it slightly deviates from design prescriptions in hybrid FP-PhC configurations. Some further practical implementations could improve the current results, as the insertion of a linear tapering section to smother the group velocities mismatch between the FP and the mirroring sections [7], also targeting an efficient nonlinear operation [11]. V. CONCLUSIONS AND PERSPECTIVES In conclusions, we introduced a novel approach to perform a systematic design of dispersion tailored multimode linear micro-cavities. We investigated quasi-periodic PhC systems, 17 and pointed out the strong connection between photonic bandgap tunability and dispersion engineered nanophotonics. In this context, the on-chip integration of Fabry-Pérot resonators is addressed by means of a distributed feedback mechanism. Here we studied the continuous transition from a PhC cavity into an hybrid linear resonator, comprised by two identical and symmetric distributed Bragg reflectors. The latter embeds the characteristics of both standard FP and PhC cavities, reason why it is often addressed to as FP-PhC, or, similarly, as Fabry-Pérot Bragg grating (FPBG). In our study, we implemented an optimization algorithm for targeting a prescribed dispersion profile for a FP-PhC. We made use of a reduced model, which approximates the complex 3D dynamics by means of a compact and computationally light 1D coupled waves theory [31]. We presented an experimental demonstration of this inverse design approach based on the AlGaAs-on-insulator platform, showing a good quantitative agreement with simulations. We believe that our method can be easily extended to all platforms exhibiting strong index contrast between the waveguide core and the cladding. In perspective, we aim to extend the functionalities of DBRs over a broadband domain, ideally over a ω ⇄2ω operation band. The latter is a very challenging task at telecom wavelengths, since the photonic bandgap tunability is typically limited by the material to, at best, ∼50 THz [5]. Nonetheless, we believe that a reliable design solution may be achieved, with the help of a more comprehensive theoretical investigation. Finally, similar configurations may also be adopted for μ-ring devices, as recently reported for SiN, including slow-light confinement[50], Fourier synthesis dispersion engineering [45], or the generation of microcombs with squared frequency shaping [51]. To summarize, dispersion engineered nanophotonics is receiving a great deal of attention. AlGaAs-OI, as discussed in the present manuscript, represents a particularly promising solution. AUTHOR CONTRIBUTION STATEMENT F.R. Talenti (conceptualization, theory, design, experiments, and writing) and L. Lovisolo (fabrication, design, and experiments) shared the main contribution to this work. C. AlonsoRamos, M. Morassi, A. Lemaitre, A. Harouri, A. Gerini and L. Vivien contributed to the 18 fabrication of the device. Z. Xiao, Z. Saleh, A. De Rossi, S. Wabnitz, and G. Leo contributed to the development of the theory and to conceiving the experiment. A. De Rossi, S. Wabnitz and L. Vivien contributed to manuscript writing. L. Vivien supervised all aspects of the work. FUNDING The work is partly supported by the French RENATECH network. Other funding: ANR-22-CE92-0065 (Quadcomb), EU - NRRP, NextGenerationEU (PE00000001 - program "RESTART"). L. Lovisolo acknowledges the support of French Agence-innovation defense N DGA01D22020572. DISCLOSURES The authors declare no conflicts of interest. [1] T. Bunel, M. Conforti, Z. Ziani, J. Lumeau, A. Moreau, A. Fernandez, O. Llopis, J. Roul, A. M. Perego, K. K. Y. Wong, and A. Mussot, Observation of modulation instability kerr frequency combs in a fiber fabry-pérot resonator, Optics Letters 48, 275 (2023). [2] T. Bunel, M. Conforti, Z. Ziani, J. Lumeau, A. Moreau, A. Fernandez, O. 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2510.14737
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Preprint (under review)
FREE-GRAINED HIERARCHICAL RECOGNITION
Seulki Park1
Zilin Wang1
Stella X. Yu1,2
1University of Michigan
2UC Berkeley
{seulki,zilinwan,stellayu}@umich.edu
ABSTRACT
Hierarchical image classification predicts labels across a semantic taxonomy, but
existing methods typically assume complete, fine-grained annotations, an assump-
tion rarely met in practice. Real-world supervision varies in granularity, influ-
enced by image quality, annotator expertise, and task demands; a distant bird may
be labeled Bird, while a close-up reveals Bald eagle. We introduce ImageNet-F,
a large-scale benchmark curated from ImageNet and structured into cognitively
inspired basic, subordinate, and fine-grained levels. Using CLIP as a proxy for
semantic ambiguity, we simulate realistic, mixed-granularity labels reflecting hu-
man annotation behavior. We propose free-grain learning, with heterogeneous su-
pervision across instances. We develop methods that enhance semantic guidance
via pseudo-attributes from vision-language models and visual guidance via semi-
supervised learning. These, along with strong baselines, substantially improve
performance under mixed supervision. Together, our benchmark and methods ad-
vance hierarchical classification under real-world constraints1
1
INTRODUCTION
Hierarchical classification (Chang et al., 2021; Chen et al., 2022; Jiang et al., 2024; Park et al., 2025)
predicts a semantic tree of labels, capturing categories from broad to specific. This richer output
supports flexible use: An expert may seek Bald Eagle, while a general user may only need Bird.
Moreover, predicting the full hierarchy improves robustness and scalability, encouraging models to
generalize across levels, and can naturally support extensions like adding new parent or child classes.
However, existing methods (Chang et al., 2021; Wang et al., 2023) assume complete supervision at
all levels for all the training examples, which rarely holds in practice. In real-world settings, annota-
tion granularity depends on image clarity, annotator expertise, or task-specific needs: A distant bird
could only be labeled as Bird, while a close-up allows Bald eagle (Fig.1).
bald eagle
bird
bird of prey
vulture
bald eagle
bird
bird of prey
vulture
bald eagle
bird
bird of prey
vulture
Figure 1: Images vary in semantic detail: Some support only coarse labels, others reveal fine-
grained categories. We propose free-grain learning: Training a hierarchical classifier with supervi-
sion free to vary in granularity across examples, reflecting semantic ambiguity in real-world images.
We propose free-grain learning, where supervision is free to vary in granularity: Training labels
may appear at any level of a fixed taxonomy, e.g., Bird, Bird of prey, or Bald eagle. The key
challenge is to predict the full taxonomy from training data with mixed (rather than uniform fine-
grained) labels. This task not only reflects real-world variability in annotation quality and specificity,
but also enables learning from partially labeled data at scale. It further requires integration across
semantic annotation granularities and across visual instances, as the model must infer a complete
taxonomy for each example based on heterogeneous supervision.
1Our code and dataset will be available at https://github.com/pseulki/FreeGrainLearning.
1
arXiv:2510.14737v1 [cs.CV] 16 Oct 2025
Preprint (under review)
Table 1: Existing Hierarchical Recognition Benchmarks Are Insufficient. CUB (Welinder et al.,
2010) and Aircraft (Maji et al., 2013) provide clean hierarchies but are small; iNat21-mini (Van Horn
et al., 2021) has a clean taxonomy but is limited to biology; ImageNet (Russakovsky et al., 2015) is
large but structurally inconsistent. We introduce ImageNet-3L, a general-purpose large-scale bench-
mark with a coherent three-level hierarchy grounded in cognitive psychology. From this, we provide
new variants of ImageNet, iNat21-mini, CUB, and Aircraft for free-grained recognition.
Dataset
#levels
#classes
#train
#test
CUB
3
13-38-200
5,994
5,794
Aircraft
3
30-70-100
6,667
3,333
iNat21-mini
8
3-11-13-51-273-1103-4884-10000
500,000
100,000
ImageNet
5-19
- 1000
1,281,167
50,000
ImageNet-3L
3
20-127-505
645,480
25,250
However, existing benchmarks are ill-suited for this task (Table 1).
Small datasets such as
CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013) lack scale, while iNaturalist (Van Horn
et al., 2021) is limited to biology and unsuitable for general-purpose evaluation. Larger bench-
marks like ImageNet (Russakovsky et al., 2015) and tieredImageNet (Ren et al., 2018) inherit noisy,
inconsistent hierarchies from WordNet (Fellbaum, 1998). As shown in Fig. 2, hierarchy depths
vary widely from 5 to 19 levels, with some classes following multiple paths—for example, Mini-
van appears in four different paths (depths 12–15), while Teddy bear appears only once at depth
7. Such inconsistencies make evaluation ambiguous: one fine class can map to several hierarchies,
and predictions often traverse long chains of redundant nodes (e.g., entity, object). As a result, most
methods on ImageNet and tieredImageNet restrict evaluation to leaf-node accuracy, with auxiliary
metrics like mistake severity (Bertinetto et al., 2020; Garg et al., 2022b; Jain et al., 2023).
To address these limitations, we construct ImageNet-3L, a benchmark with a well-structured three-
level hierarchy: basic (e.g., Dog), subordinate (e.g., Shepherd), and fine-grained (e.g., German
Shepherd) (Fig. 3). Grounded in cognitive psychology (Rosch et al., 1976; Rosch, 1978) and folk
taxonomies (Berlin et al., 1966), our design reflects that the basic level is the most natural and widely
recognized category for humans, while subordinate and fine-grained levels capture increasingly spe-
cific distinctions. By focusing on this range—from the most intuitive to the most detailed—we
enable semantically meaningful hierarchical prediction, avoiding abstract or redundant levels (e.g.,
Physical Entity in the original ImageNet hierarchy) that provide little practical value.
Building on ImageNet-3L, we further construct ImageNet-F, a free-grain benchmark that simulates
mixed-granularity labeling. Using CLIP (Radford et al., 2021) as a proxy for visual–semantic am-
biguity and annotator variability, we prune labels at different levels depending on prediction confi-
dence. This yields realistic supervision patterns—for example, distant birds labeled as Bird, mid-
range as Bird of prey, and close-ups as Bald eagle (Fig. 4). The final dataset covers 645,480 images
across 20 basic, 127 subordinate, and 505 fine-grained classes (Table 1). We apply the same strategy
0
100
200
# Leaf node at each level
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Hierarchy depth
Figure 2:
Inconsistent and Noisy
Hierarchy of ImageNet WordNet.
(Left:)
The histogram of hierarchy
depths shows that ImageNet classes
range from 5 to 19 levels, with many
exceeding 10, which hinders consis-
tent evaluation.
(Right:)
Sample
hierarchies illustrate that classes can
have multiple paths of different depths:
Minivan appears in four paths at depths
12–15, while Teddy bear exists only at
depth 7. This imbalance and inconsis-
tency in the hierarchy make it unclear
which path should be considered cor-
rect, underscoring the difficulty of us-
ing the original WordNet hierarchy for
training and evaluation.
2
Preprint (under review)
(a) Original ImageNet’s WordNet hierarchy
(b) Our 3-level hierarchy
Figure 3: We curate ImageNet-3L as a benchmark for hierarchical classification. (a) The orig-
inal ImageNet taxonomy is noisy and inconsistent, with imbalances, overlaps, and multiple paths,
making it unsuitable for reliable evaluation. (b) We construct a coherent 3-level taxonomy, inspired
by cognitive psychology (Rosch et al., 1976): basic for general recognition, subordinate for contex-
tual specificity, and fine-grained for specialized distinctions.
to iNat21-mini and CUB using BioCLIP (Stevens et al., 2024), producing iNat21-mini-F and CUB-
F, and additionally create synthetic variants CUB-Rand and Aircraft-Rand for controlled evaluation
under varying label sparsity and granularity.
Dog
Cooking utensil
Dog
Fish
Electron. equip.
Insect
Bird
Pinscher
Pot
Shepherd
Ganoid
Audio device
Beetle
Bird of prey
Affenpinscher
Teapot
German shepherd
Gar
Tape player
Leaf beetle
Bald eagle
Figure 4:
Our ImageNet-F captures real-world challenges, where fine-grained labels follow
a long-tailed distribution and granularity varies with visual clarity. Top: The graph shows the
proportion of basic, subordinate, and fine-grained labels per class (sorted by ID). Fine-grained labels
are scarce on the left but increase toward the right, forming a long tail. This imbalance often causes
models to overfit to basic-level features and miss subtle distinctions, underscoring the need for robust
multi-level learning. Bottom: Arrow-marked samples illustrate how label pruning reflects difficulty.
(Last Column): a distant bird is labeled at the basic level (Bird); one with visible wings and talons
at the subordinate level (Bird of prey); and a close-up at the fine-grained level (Bald eagle).
3
Preprint (under review)
When applied directly under free-grain setting, existing hierarchical classifiers (Chen et al., 2022;
Park et al., 2025) degrade severely—up to –40% full-path accuracy on iNat21-mini —highlighting
the difficulty of the task. To address this, we propose three additional strategies: 1) learning pseudo-
attributes (e.g., short legs, docked tail) from vision–language models to provide semantic cues when
finer labels are missing; 2) applying semi-supervised learning by treating missing-grain labels as
unlabeled; 3) combining both approaches. Across datasets, these methods outperform hierarchical
baselines by +4–25%p, establishing stronger baselines for free-grain learning.
Contributions. 1) We introduce free-grain learning for hierarchical classification, capturing real-
world variability in label granularity. 2) We present ImageNet-F, with a cognitively grounded 3-level
hierarchy, and additional free-grain benchmarks across diverse domains. 3) We establish strong
baselines that significantly improve performance by leveraging semantic and visual guidance.
2
RELATED WORK
Hierarchical classification has been studied mainly for leaf-node prediction on large but inconsis-
tent taxonomies such as ImageNet (Karthik et al., 2021; Zhang et al., 2022), or for full-taxonomy
prediction on small datasets like CUB and Aircraft (Chang et al., 2021; Park et al., 2025). These
settings lack the scale, diversity, and label sparsity needed for realistic evaluation. Our work instead
enables full taxonomy prediction under heterogeneous supervision on large-scale data.
Imbalanced and semi-/weakly-supervised classification have been widely explored (Liu et al.,
2019; Tarvainen & Valpola, 2017; Robinson et al., 2020), but mostly at a single fine-grained level
or with fully observed coarse labels. In contrast, we address both intra- and inter-level imbalance,
requiring consistent prediction across multiple granularities with partially missing supervision. See
a full task comparison in Table 2.
Foundation models such as CLIP (Radford et al., 2021) has been used for zero-shot flat classifica-
tion via text prompts (Pratt et al., 2023; Saha et al., 2024). In contrast, our approach leverages text
only during training to learn visual patterns across levels, requiring no textual input at inference.
Further discussion and additional related work are provided in Appendix B.
Table 2: Our task setting is more practical and challenging than existing ones. Ours reflects re-
alistic scenarios where annotations are free-grain and imbalanced, requiring hierarchical predictions
to balance accuracy and consistency across levels.
Input
Output
Training Labels
Imbalance
Evaluation Metrics
Tasks
Fine
Coarse
Fine
Coarse
Availability
Intra
Inter
Accuracy
Consistency
Long-tailed recognition
✓
✗
✓
✗
All
✓
✗
✓
✗
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3
HIERARCHICAL DATASET FOR FREE-GRAINED RECOGNITION
3.1 Defining Three-Level Taxonomy for ImageNet-3L. We restructure ImageNet (Russakovsky
et al., 2015)’s WordNet (Fellbaum, 1998)-based hierarchy into a consistent three-level taxonomy,
explicitly guided by Rosch’s categorization principles (Rosch et al., 1976). In Rosch’s framework,
the basic level (e.g., dog, car) is the most natural and visually distinctive, balancing generality and
specificity; it is also the level people most often use in everyday recognition and naming, unlike
abstract superordinate categories (e.g., animal) or overly narrow subordinate ones (e.g., Pembroke).
We adopt the basic level as the coarsest node in each branch, with subordinate and fine-grained levels
(e.g., Corgi →Pembroke) providing progressively finer distinctions. However, WordNet chains such
as artifact →...→vehicle →...→motor vehicle →car →ambulance can yield only two usable
levels if car is taken as basic. In these cases, we elevate Rosch’s superordinate category (e.g.,
vehicle) to serve as the basic level, which remains visually distinctive from other basic categories
(e.g., craft, container) and ensures a three-level hierarchy. This yields branches that support three
semantically coherent and visually meaningful levels for hierarchical prediction.
4
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Specifically, we adopt the following systematic principles: 1) Enforce meaningful structure: We
remove paths where each node has only one child, since coarse labels fully determine the fine la-
bels. Branches with fewer than three levels are also excluded. 2) Maximize within-group diversity:
Among subordinate candidates under each basic class, we favor those with richer fine-grained sub-
classes—e.g., parrot (4 children) over cockatoo (1 child). 3) Refine vague categories: Ambiguous
groups such as Women’s Clothing are reorganized into precise, functionally grounded categories
(e.g., Underwear) to improve clarity. 4) Validate with language models and human review: We
use language models (ChatGPT (Achiam et al., 2023)) to suggest refinements, with all decisions
manually reviewed for semantic consistency. Applying this curation process to ImageNet-1k yields
a structured benchmark of 20 basic, 127 subordinate, and 505 fine-grained classes, ensuring every
branch supports meaningful hierarchical prediction (a complete list is provided in Appendix A).
3.2 Semantic Label Pruning for ImageNet-F, iNat21-mini-F, and CUB-F
To build a realistic free-grain training dataset, we prune hierarchical labels using large vi-
sion–language models as a proxy for visual–semantic ambiguity: CLIP (Radford et al., 2021) for
ImageNet-F and BioCLIP (Stevens et al., 2024) for iNat21-mini-F and CUB-F. Although these mod-
els are not explicitly designed to measure ambiguity, their zero-shot confidence consistently corre-
lates with visual distinctiveness (Fig. 4). Moreover, since label annotation is affected by annotator
expertise or error, this proxy offers a practical approximation.
We adopt CLIP’s prompt-ensemble strategy (e.g., a photo of a [class], art of a [class]) and compute
average confidence for fine-grained and subordinate levels. Labels are retained based on prediction
correctness: (1) If both fine-grained and subordinate are correct, we keep all labels. (2) If only
subordinate is correct, we keep up to that level. (3) Otherwise, only the basic label is kept. We
further prune subordinate labels proportionally to the fine-grained removal rate per class.
(1) ImageNet-F. After pruning, 32.6% of images retain all three levels (Basic + Subordinate +
Fine-grained), 28.0% retain two (Basic + Subordinate), and 39.4% retain only the Basic. Each class
maintains the same number of images as ImageNet; imbalance arises only from label granularity.
(2) iNat21-mini-F. Although BioCLIP is trained on iNat21-mini ’s full taxonomy, it performs well
when predicting fine-grained species but struggles when restricted to coarser labels. This gap enables
substantial pruning: 22.5% of images retain all three levels (Order + Family + Species), 28.0% retain
two, and 49.5% retain only Order. (3) CUB-F. With the same procedure, 31.5% of images keep three
levels, 23.3% two (Order + Family), and 45.2% only Order.
3.3 Synthetic Label Pruning for CUB-Rand and Aircraft-Rand
To control label availability, we construct synthetic variants—CUB-Rand and Aircraft-Rand —by
randomly pruning labels from CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013). Unlike
realistic pruning, this design systematically varies supervision and simulates extreme sparsity (e.g.,
only 10% fine-grained labels), enabling stress-testing of model robustness across diverse label dis-
tributions. Although random removal is independent of image difficulty, it reflects practical factors
such as annotator expertise, cost, or task-specific constraints. We denote availability as a-b-c, where
a% of basic, b% of subordinate, and c% of fine-grained labels are retained (e.g., 100-50-10 retains
10% fine-grained labels and 40% subordinate-only labels).
4
FREE-GRAIN LEARNING METHODS FOR HIERARCHICAL CLASSIFICATION
4.1 Problem setup. We begin by describing the problem setup. In free-grain hierarchical classifica-
tion, the goal is to train a model that predicts object categories across all levels of a taxonomy, given
training data with labels of varying granularity. Formally, let X denote the input space of images,
and Y1, . . . , YL the label spaces at L hierarchical levels, from coarsest (Y1) to finest (YL). Each
training sample consists of an image x ∈X and a partial label set {yl}l∈Sx, where Sx ⊆{1, . . . , L}
indicates the levels at which labels are provided.
We assume that if a label yl is available, all coarser labels yl′ for l′ < l are also available due
to the structure of the taxonomy, while finer labels yl′ for l′ > l are missing. We further assume
the coarsest label y1 is always given. The objective is to learn a classifier f : X →Y1 × · · · ×
YL,
f(x) = (ˆy1, . . . , ˆyL), that predicts labels at all levels of the hierarchy.
5
Preprint (under review)
(a) Text-Attr
(b) Taxon-SSL
Figure 5: Overview of the proposed methods. (a) Text-Attr enriches feature representations using
semantic cues from images, compensating for missing labels and capturing shared attributes across
levels. (b) Taxon-SSL handles missing-level labels by treating them as unlabeled and learns from
visual consistency through augmented views.
4.2 Baselines. With no existing baselines for this new setting, we propose four strong baselines,
each approaching the problem from a different perspective.
(1) Semantic Guidance: Text-Guided Pseudo Attributes (Text-Attr). Our semantic guidance
approach is motivated by the observation that while class labels differ across hierarchical levels
(e.g., Dog →Corgi →Pembroke), many visual attributes—such as tail length or ear shape—remain
consistent (Fig. 5a). To capture these shared semantic cues, we use image descriptions as auxil-
iary supervision. Instead of class-name prompts for zero-shot classification, we extract free-form
descriptions directly from the image, independent of labels.
Specifically, given an input image x, we use a frozen vision-language model (VLM), Llama-3.2-
11B (Dubey et al., 2024), to generate a language description dx, using the prompt: “Describe visual
details in the image.” This produces descriptions containing phrases such as “short legs” or “pointed
ears,” which we encode into a text embedding zt
x using CLIP’s text encoder (Radford et al., 2021).
We cap generation at 100 tokens, while CLIP accepts 77 tokens; longer descriptions are truncated
during encoding. Although truncation discards some details, our method focuses on shared semantic
cues (e.g., “short legs,” “brown markings”) rather than exhaustive captions, making it robust to this
limitation. In parallel, we obtain the image embedding zv
x from the image encoder, and align the two
embeddings with a contrastive loss:
Ltext = −1
N
N
X
i=1
log
exp(sim(zv
i , zt
i)/τ)
PN
j=1 exp(sim(zv
i , zt
j)/τ)
!
,
(1)
where sim(·, ·) is cosine similarity and τ is a temperature parameter. This loss guides the encoder
to capture salient, label-independent traits shared across levels. Although not explicitly predicting
attributes, aligning image features with text induces intermediate representations, which we call
pseudo-attributes. This model-agnostic method can be applied to any architecture.
Finally, for hierarchical supervision, we apply the loss only at levels with available labels. Given
hierarchical labels y1, . . . , yL across L levels, the model computes a loss at each level:
Lhier =
L
X
l=1
⊮{yl exists} · L(fl(x), yl),
(2)
where fl(x) is the prediction at level l, and L denotes any classification loss (e.g., cross-entropy).
(2) Visual Guidance: Taxonomy-Guided Semi-Supervised Learning (Taxon-SSL).
To enforce semantic consistency, we extend CHMatch’s contrastive objective to the full taxonomy.
For each mini-batch, we build level-wise affinity graphs W l based on pseudo-label agreement:
W l
ij = 1 if images i and j share the same pseudo-label at level l, and 0 otherwise. Then the
6
Preprint (under review)
taxonomy-aligned affinity graph W is defined as:
Wij =
(
1
if W 1
ij = ... = W L
ij = 1,
0
otherwise.
(3)
Then, taxonomy-aligned contrastive loss Ltacl is defined by:
Ltacl = −
1
P
j Wij
·
L
X
l=1
log
P
j Wij exp((g(f(xi)) · g(f(xj))′)/t)
P
j(1 −Wij) exp((g(f(xi)) · g(f(xj))′)/t),
(4)
where gi = g(f(xi)) is the projected feature of image i with the classifier f, and t is a temperature
hyperparameter.
(3) Combining Semantic and Visual Guidance: Taxon-SSL + Text-Attr. A natural next step is
to combine Text-Attr and Taxon-SSL by incorporating text-derived embeddings into the feature ex-
tractor of Taxon-SSL, allowing semantic and visual guidance to be jointly leveraged during training.
(4) State-of-the-art Hierarchical Classification Methods: H-CAST, HRN. We adopt two rep-
resentative models. (4-1) Hierarchical Residual Network (HRN) (Chen et al., 2022): the first to
handle supervision at both subordinate and fine-grained levels by maximizing marginal probabilities
within the tree-constrained space. (4-2) H-CAST (Park et al., 2025): the current state-of-the-art,
encouraging consistent visual grouding across taxonomy levels. Originally trained with full super-
vision, we adapt it to this setting via the level-wise loss in Eq. 2, using only available labels.
5
EXPERIMENTS
Dataset: We conduct experiments using our proposed ImageNet-F, iNat21-mini-F, and CUB-F
datasets, along with the synthetic CUB-Rand and Aircraft-Rand datasets. CUB includes bird
images across 13 orders (e.g., Anseriformes), 38 families (e.g., Anatidae), and 200 species (e.g.,
Mallard), while Aircraft (Maji et al., 2013) contains aircraft images across 30 makers (e.g., Boeing),
70 families (e.g., Boeing 707), and 100 models (e.g., 707-320).
Evaluation metrics: Following (Park et al., 2025), we evaluate accuracy and consistency: 1) Level-
accuracy: Top 1 accuracy for each level. 2) Tree-based InConsistency Error rate (TICE): Propor-
tion of test samples with inconsistent prediction paths in the hierarchy. Lower is better. TICE = nic
N
3) Full-Path Accuracy (FPA): Proportion of test samples with correct predictions at all hierarchy
levels. Higher is better, and we use FPA as one of our primary metrics: FPA = nac
N .
Implementation: We use H-ViT, a ViT-Small-based hierarchical classifier, as the backbone for
evaluating both Text-Attr and Taxon-SSL. To evaluate its compatibility across architectures, we also
apply Text-Attr to H-CAST (Park et al., 2025), a state-of-the-art hierarchical model with comparable
capacity. HRN (Chen et al., 2022) is evaluated with its original ResNet-50 backbone, which has over
twice the parameters. All models are trained for 100 epochs, except for ImageNet-F, which is trained
for 200 due to its larger scale. Full architectural and training details are in the appendix F.
Result 1: Performance Drop under Free-Grain Learning. The prior hierarchical SOTA, H-
CAST, degrades sharply under mixed-granularity labels on both CUB and iNat21-mini. As shown
in Fig. 6, full-path accuracy drops from 84.9% to 45.1% on CUB-F and from 64.9% to 25.6% on
iNat21-mini-F. This demonstrates the difficulty of handling mixed-granularity labels and imbal-
anced supervision across the hierarchy and need for methods handling them.
Result 2: Performance on ImageNet-F. As shown in Table 3, existing hierarchical methods de-
grade sharply under free-grain learning: HRN reaches only 37.8% FPA, while H-CAST performs
better at 57.6% but still struggles with missing labels. Text-Attr (H-ViT) achieves 55.5% without re-
lying on H-CAST’s visual grouping, and integrating it into H-CAST further improves performance
to 63.2%, demonstrating the effectiveness of semantic-guided pseudo-attribute learning at scale.
Taxon-SSL improves over HRN by leveraging visual guidance but remains less effective than Text-
Attr methods, whose strong performance benefits from the abundance and diversity of ImageNet-F
for reliable visual–semantic alignment.
Result 3: Performance on iNat21-mini-F. In Table 3, on the large-scale iNat21-mini-F dataset,
which contains many classes (10,000), conventional hierarchical methods perform poorly (17.0%
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Preprint (under review)
CUB
iNat21-Mini
10
20
30
40
50
60
70
80
90
Full-Path Accuracy (%)
-39.8%
-39.3%
100% Full Labels
Free-grain Labels
Figure 6:
Transitioning from
fully labeled data to our mixed-
granularity setting results in a
substantial drop in Full-Path
Accuracy, highlighting the dif-
ficulty of the task.
SOTA H-
CAST suffers nearly a 40pp loss
on both CUB and iNat21-mini.
Figure 7: Text-Attr benefits under extreme label sparsity, as
seen on the left (low-index classes with few fine-grained la-
bels) by providing extra guidance from textual descriptions,
while Taxon-SSL performs better on the right (high-index
classes with more fine-grained labels). Both are based on ViT-
small model and evaluated on the ImageNet-F. Classes are sorted
by the number of fine-grained training samples, from lowest to
highest.
Table 3: No single recipe solves free-grain learning—methods behave differently depending on
data characteristics. 1) Conventional hierarchical classification methods like HRN (Chen et al.,
2022) and H-CAST (Park et al., 2025) show significant performance drops under incomplete su-
pervision, underscoring the challenge of free-grain settings. 2) Text-Attr methods works well on
ImageNet-F, where each class is supported by abundant visual evidence. In contrast, iNat21-mini-F
has fine-grained biology labels, where appearance are similar, making LLM-based text descriptions
less effective. Here, Taxon-SSL proves more beneficial by leveraging structured label propagation in
this semi-supervised style setting. 3) Combining the two (Taxon-SSL + Text-Attr) yields consistent
but modest gains across both datasets.
Dataset
ImageNet-F (20-127-505)
iNat21-mini-F (273 - 1,103 - 10,000)
FPA(↑)
fine.(↑)
sub.(↑)
basic(↑) TICE(↓) FPA(↑)
spec.(↑) fam.(↑)
order(↑) TICE(↓)
HRN (Chen et al., 2022)
37.79
38.73
55.73
78.65
46.69
17.03
25.43
46.51
70.20
53.81
H-CAST (Park et al., 2025)
57.59
59.02
82.69
93.53
21.81
25.63
28.61
67.20
83.62
47.17
Taxon-SSL
48.40
52.34
65.74
82.96
19.87
31.74
37.11
69.53
82.02
37.31
Taxon-SSL + Text-Attr
49.65
53.43
66.43
83.56
18.81
31.93
37.08
69.76
82.20
37.04
Text-Attr (H-ViT)
55.48
59.05
77.95
89.45
24.02
27.88
32.07
68.27
80.49
46.35
Text-Attr (H-CAST)
63.20
64.91
84.47
93.56
18.58
29.74
32.37
71.79
85.99
44.63
for HRN, 25.63% for H-CAST). Taxon-SSL achieves the best performance (31.9% FPA), high-
lighting the benefits of structural label propagation under limited per-class supervision. Text-Attr
methods perform slightly lower (27.9–30.0% FPA), likely due to restricted textual diversity in this
fine-grained biological domain, yet still outperform conventional baselines.
In Appendix, we report additional results on CUB-F (Sec. C.1), highly-missing synthetic datasets
(Sec. C.2), and ablations on Text-Attr features, training strategies, and architecture design (Sec. E).
Analysis 1: Text-Attr Excels with Sparse Labels, Taxon-SSL with Moderate Label Availability.
We analyze class-wise performance under imbalanced fine-grained label availability on ImageNet-
F. To isolate effects, we compare Text-Attr (H-ViT) and Taxon-SSL with identical ViT-small back-
bones, excluding H-CAST modules. Fig. 7 shows per-class accuracy, sorted by the number of fine-
grained training labels. Text-Attr (H-ViT) outperforms in label-scarce classes by leveraging textual
descriptions as extra supervision, while Taxon-SSL performs better with moderate label availability
by propagating consistency across missing levels. We provide additional t-SNE (Maaten & Hinton,
2008) visualization analysis in Appendix D.
Analysis 2: What Advantage Does External Semantic Guidance Provide? To assess the effect
of text-derived guidance, we compare saliency maps (Chefer et al., 2021) from Taxon-SSL and Text-
Attr (H-ViT) (Fig. 8). In Row 1, with multiple objects, Taxon-SSL focuses on a human shoulder
and misclassifies the image, violating the semantic hierarchy, while Text-Attr consistently attends to
the instrument and predicts correctly. In Row 2, when both fail at the fine-grained level, Taxon-SSL
8
Preprint (under review)
Taxon-SSL
Text-Attr
Image
Basic
Subordinate
Fine-grained
Basic
Subordinate
Fine-grained
clothing ✗
headwear ✗
saxophone ✓
music instru. ✓wind instru. ✓
saxophone ✓
dog ✓
hound dog ✓
megalith ✗
dog ✓
hound dog ✓standard poodle ✗
Figure 8: Text-Attr improves semantic focus under diverse large-scale data. (1st row) In a multi-
object image, Taxon-SSL assigns inconsistent labels (“clothing” at the basic level, “saxophone” at
the fine-grained level), while Text-Attr (H-ViT) correctly predicts “musical instrument” by focusing
on the relevant object. (2nd row) When both fail at the fine-grained level, Taxon-SSL outputs an un-
related class (“megalith”), whereas Text-Attr (H-ViT) chooses a semantically closer one (“poodle”).
This shows that text-derived attributes help the model attend to meaningful regions and maintain
semantic plausibility, on large-scale ImageNet-F dataset with diverse categories and sparse labels.
Green/Red denote correct/incorrect predictions.
Figure 9: Free-grain inference results with consistency-based stopping on ImageNet-F. Left:
Examples of consistency-based stopping in Text-Attr (H-CAST). The model stops at the correct
subordinate level (Hound, left) or at the basic level (Bird, right) when deeper predictions become
inconsistent and incorrect, leading to more reliable results. Right: Consistency-based stopping
for free-grain inference. Predictions are halted when finer-level outputs conflict with preceding
coarser-level predictions. On ImageNet-F, Text-Attr (H-CAST) explores deeper levels of the hierar-
chy with higher correctness, whereas HRN stops earlier and produces fewer fine-level predictions.
outputs an unrelated class, whereas Text-Attr chooses a visually similar dog by focusing on curly
fur and body shape. These results show that external semantic cues guide attention to meaningful
features across label granularities, improving hierarchical consistency, while Taxon-SSL may drift
to visually salient but semantically irrelevant regions under sparse or ambiguous supervision.
Analysis 3: Free-grain Inference. While our main goal is full-hierarchy prediction under mixed-
granularity supervision, free-grain inference is also crucial in practice: a correct coarse label is
often preferable to an incorrect fine-grained one (e.g., predicting “dog” instead of a wrong breed).
We adopt a simple consistency-based stopping rule: predictions halt whenever the next-level label
would violate the taxonomy, ensuring the deepest valid output. As shown in Fig. 9(Right), Text-
Attr (H-CAST) reaches deeper levels more often and with higher accuracy. Fig. 9(Left) shows
examples: stopping at the basic level when “bird” is correct but the subordinate mispredicts, or at
the subordinate level when “dog →hound” is correct but the fine-grained label is inconsistent. These
results highlight the practical value of free-grain inference and motivate benchmarks that explicitly
evaluate this setting.
9
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6
Summary
We introduce new hierarchical classification under free-grain supervision, where models learn from
labels of varying granularity while maintaining taxonomy consistency. To advance this setting,
we present a large-scale benchmark and two simple yet effective baselines. Our Text-Attr method
mitigates label imbalance by sharing features across levels, though it does not explicitly model it;
future work could explore imbalance-aware strategies for further improvement.
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Free-Grained Hierarchical Recognition
Supplementary Material
CONTENTS
A Complete Hierarchy of ImageNet-F
14
B
Related Work
19
C More Experimantal Results
20
C.1
Evaluaion on CUB-F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
C.2
Evaluation under Varying and Severe Label Sparsity Conditions
. . . . . . . . . .
20
D t-SNE Visualization
23
E
Ablation Study
24
E.1
Importance of Text-guided Pseudo Attributes
. . . . . . . . . . . . . . . . . . . .
24
E.2
Combining Text-Attr and Taxon-SSL . . . . . . . . . . . . . . . . . . . . . . . . .
24
E.3
Ablation on Hierarchical Supervision in ViT . . . . . . . . . . . . . . . . . . . . .
24
F
Implementation Details
26
G Use of Large Language Models (LLMs)
26
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A
COMPLETE HIERARCHY OF IMAGENET-F
Basic
Subordinate
Fine-Grained
bird
passerine bird
brambling, indigo bunting, robin, jay, bul-
bul, water ouzel, house finch, chickadee,
junco, magpie, goldfinch
parrot
macaw, sulphur-crested cockatoo, African
grey, lorikeet
piciform bird
toucan, jacamar
seabird
king penguin, pelican, albatross
anseriform bird
drake,
red-breasted
merganser,
black
swan, goose
coraciiform bird
bee eater, hornbill
bird of prey
kite, great grey owl, vulture, bald eagle
gallinaceous bird
partridge, prairie chicken, ruffed grouse,
peacock, quail, black grouse, ptarmigan
wading bird
flamingo,
American
coot,
redshank,
American egret, little blue heron, white
stork,
limpkin,
spoonbill,
red-backed
sandpiper, dowitcher, crane, ruddy turn-
stone, bittern, oystercatcher, black stork,
bustard
dog
spitz dog
malamute,
Pomeranian,
keeshond,
Siberian husky, chow, Samoyed
pointer dog
vizsla, German short-haired pointer
spaniel dog
Brittany
spaniel,
clumber,
English
springer,
Sussex spaniel,
Irish water
spaniel, Welsh springer spaniel, cocker
spaniel
hound dog
basset,
bloodhound,
Irish wolfhound,
Walker hound,
redbone,
English fox-
hound, Italian greyhound, Ibizan hound,
bluetick, Scottish deerhound, borzoi, Nor-
wegian elkhound, whippet, Weimaraner,
Saluki, beagle, Afghan hound, black-and-
tan coonhound, otterhound
terrier dog
Boston bull, silky terrier, Lakeland ter-
rier, Yorkshire terrier, Tibetan terrier,
American Staffordshire terrier, Irish ter-
rier, Airedale, Norwich terrier, soft-coated
wheaten terrier, wire-haired fox terrier,
Staffordshire bullterrier, West Highland
white terrier, Australian terrier, Dandie
Dinmont,
Kerry
blue
terrier,
Lhasa,
cairn, Sealyham terrier, Bedlington terrier,
Scotch terrier, Border terrier, Norfolk ter-
rier
corgi dog
Pembroke, Cardigan
poodle dog
miniature poodle, toy poodle, standard
poodle
setter dog
Irish setter, Gordon setter, English setter
pinscher dog
Doberman, affenpinscher, miniature pin-
scher
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shepherd dog
kelpie, briard, German shepherd, Old En-
glish sheepdog, Border collie, Bouvier des
Flandres, collie, Rottweiler, komondor,
malinois, groenendael, Shetland sheepdog
retriever dog
curly-coated retriever, Labrador retriever,
Chesapeake Bay retriever, flat-coated re-
triever, golden retriever
schnauzer dog
standard schnauzer, miniature schnauzer,
giant schnauzer
Sennenhunde dog
Bernese mountain dog, Greater Swiss
Mountain dog, Appenzeller, EntleBucher
toy dog
toy terrier, Blenheim spaniel, Maltese dog,
Shih-Tzu, papillon, Pekinese, Chihuahua,
Japanese spaniel
fish
soft-finned fish
coho, tench, eel, goldfish
shark
tiger shark, great white shark, hammer-
head
spiny-finned fish
anemone fish, puffer, lionfish, rock beauty
ray
stingray, electric ray
ganoid fish
sturgeon, gar
primate
ape
gibbon, siamang, orangutan, chimpanzee,
gorilla
monkey
titi, langur, colobus, squirrel monkey, ba-
boon, guenon, marmoset, macaque, spider
monkey, patas, howler monkey, proboscis
monkey, capuchin
lemur
Madagascar cat, indri
snake
colubrid snake
water snake, garter snake, green snake,
night snake, hognose snake, ringneck
snake, king snake, thunder snake, vine
snake
elapid snake
sea snake, Indian cobra, green mamba
viper
diamondback, horned viper, sidewinder
boa snake
boa constrictor, rock python
salamander
newt
eft, common newt
ambystomid salamander
spotted salamander, axolotl
insect
beetle
dung beetle, weevil, leaf beetle, tiger
beetle, ladybug, rhinoceros beetle, long-
horned beetle, ground beetle
orthopterous insect
cricket, grasshopper
dictyopterous insect
cockroach, mantis
hymenopterous insect
bee, ant
butterflyinsect
cabbage butterfly, lycaenid, monarch, ad-
miral, sulphur butterfly, ringlet
odonate insect
dragonfly, damselfly
homopterous insect
cicada, leafhopper
furniture
table
desk, dining table
baby bed
cradle, crib, bassinet
seat
rocking chair, barber chair, park bench,
throne, folding chair, toilet seat, studio
couch
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lamp
table lamp
cabinet
china cabinet, medicine chest
musical instrument
wind instrument
ocarina, flute, panpipe, oboe, cornet, sax,
harmonica, bassoon, French horn, trom-
bone
stringed instrument
banjo, harp, violin, cello, acoustic guitar,
electric guitar
percussion instrument
steel drum, gong, marimba, drum, chime,
maraca
keyboard instrument
upright, grand piano, accordion, organ
scientific instrument
laboratory glassware
Petri dish
magnifier
loupe, radio telescope
sports equipment
ball
golf ball, baseball, basketball, croquet ball
gymnastic apparatus
parallel bars, balance beam, horizontal bar
weight
barbell, dumbbell
electronic equipment
telephone
dial telephone, pay-phone, cellular tele-
phone
computer peripheral
printer,
joystick,
computer
keyboard,
mouse
audio device
tape player, cassette player, CD player,
iPod
network device
modem
display device
monitor, screen
clothing
bottoms (skirts)
hoopskirt, sarong, miniskirt, overskirt
tops (sweaters)
sweatshirt, cardigan
outwear
trench coat, poncho, fur coat
swimwear
maillot, bikini, swimming trunks
face & headwear
wig,
sombrero,
mortarboard,
bonnet,
mask, cowboy hat, bearskin
nightwear
pajama
protective wear
apron, knee pad, lab coat
dresses & Gowns
gown
underwear
brassiere
footwear
sock, Christmas stocking
neckwear
bow tie, bolo tie, Windsor tie
traditional & formal Wear
abaya, kimono, vestment, academic gown
wraps & shawls
stole, feather boa
container
reservoir
water tower, rain barrel
bag
mailbag, plastic bag, backpack, purse
jug
water jug, whiskey jug
vessel
mortar, pitcher, tub, ladle, bucket, coffee
mug
bottle
wine bottle, beer bottle, pop bottle, water
bottle, pill bottle
basket
hamper, shopping basket
box
mailbox, carton, pencil box, chest, crate
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glass
goblet, beer glass
shaker
saltshaker, cocktail shaker
cooking utensil
pan
frying pan, wok
cooker
Crock Pot
pot
teapot, caldron, coffeepot
structure
monument
brass, megalith, triumphal arch, obelisk,
totem pole
religious building
church, mosque, boathouse, monastery,
stupa
housing
yurt, cliff dwelling, mobile home
public building
planetarium, library
movable structure
sliding door, turnstile
supporting structure
plate rack, honeycomb, pedestal
fence
stone wall, picket fence, chainlink fence,
worm fence
bridge
steel arch bridge, viaduct, suspension
bridge
residential structure
palace
agricultural structure
greenhouse, barn, apiary
commercial stucture
toyshop,
restaurant,
cinema,
confec-
tionery, bookshop, grocery store, tobacco
shop, bakery, butcher shop, barbershop,
shoe shop
barrier
grille, bannister, breakwater, dam
institutional structure
prison
tool
hand tool
hammer, plunger, screwdriver
garden tool
lawn mower, shovel
cutter
cleaver, plane, letter opener, hatchet
power tool
chain saw
opener
corkscrew, can opener
craft
sailing vessel
trimaran, schooner, catamaran
boat
fireboat, canoe, yawl, gondola, speedboat,
lifeboat
ship
wreck, pirate, container ship, liner
warship
aircraft carrier, submarine
aircraft
airliner, warplane, airship, balloon
vehicle
bicycle
bicycle-built-for-two, mountain bike
bus
minibus, school bus, trolleybus
car
ambulance, beach wagon, cab, convert-
ible, jeep, limousine, Model T, racer,
sports car
truck
fire engine, garbage truck, pickup, tow
truck, trailer truck
van
minivan, moving van, police van
locomotive
electric locomotive, steam locomotive
military vehicle
half track
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self-propelled vehicle
forklift, recreational vehicle, snowmobile,
tank, tractor, golfcart, snowplow, go-kart,
moped, streetcar, amphibious vehicle
handcart
barrow, shopping cart
sled
bobsled, dogsled
train
bullet train
wagon
horse cart, jinrikisha, oxcart
wheeled vehicle
freight car, motor scooter, tricycle, unicy-
cle
weapon
gun
rifle, assault rifle, revolver, cannon
ranged weapon
missile, projectile
Table 4: Complete hierarchy tree for our proposed ImageNet-F dataset.
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B
RELATED WORK
Hierarchical classification has been studied with varying objectives. Most focus on leaf-node pre-
diction, using the full taxonomy during training but predicting only fine-grained labels (Karthik
et al., 2021; Zhang et al., 2022; Zeng et al., 2022; Garg et al., 2022b). Evaluation in these works
typically relies on top-1 accuracy or mistake severity at the leaf level, making them compatible
with large-scale datasets like ImageNet (Russakovsky et al., 2015) and tieredImageNet (Ren et al.,
2018)—even with inconsistent or deep hierarchies. However, models restricted to fine-grained out-
puts often fail in real-world scenarios where visual details are missing, as they cannot fall back to
coarser labels and thus provide no meaningful information.
To address this, full taxonomy prediction has been explored, aiming to produce labels across all
levels while maintaining hierarchical consistency (Chang et al., 2021; Wang et al., 2023; Jiang et al.,
2024; Park et al., 2025). However, these methods are typically developed and evaluated on small,
fully labeled datasets like CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013), which lack
the scale, diversity, and label sparsity of real-world settings. The iNaturalist dataset (Van Horn et al.,
2021) offers a deeper taxonomy, but also remains restricted to the biology, limiting its suitability
for general-purpose evaluation. HRN (Chen et al., 2022) partially handles incomplete labels by
randomly converting fine-grained labels to parent categories, overlooking the structured ambiguity
in real data. Similarly, (Kim et al., 2023) supports mixed labels but treats them flatly, ignoring
hierarchical relationships. Both also rely on small datasets such as CUB and Aircraft. Our work fills
this gap by enabling full taxonomy prediction under realistic supervision on large-scale data.
Imbalanced classification has been extensively studied (Liu et al., 2019; Ren et al., 2020; Wang
et al., 2021; Park et al., 2021; Tian et al., 2022; Park et al., 2022; Ha et al., 2023; Zhao et al., 2024),
mostly focusing on intra-level imbalance at a single fine-grained level. In contrast, we address intra-
and inter-level imbalance in a hierarchical setting, where classes are balanced but label granularity
varies across them. DeepRTC (Wu et al., 2020) considers taxonomy, but aims to improve inference
reliability via early stopping, rather than predicting the full taxonomy.
Semi-supervised learning typically combines labeled and unlabeled data at a single fine-grained
level (Tarvainen & Valpola, 2017; Berthelot et al., 2019; Sohn et al., 2020). Recent work incor-
porates coarse labels (Garg et al., 2022a; Wu et al., 2023), but still targets fine-grained accuracy.
In contrast, our setting demands consistent prediction across the full taxonomy with heterogeneous
supervision, making existing methods not directly applicable.
Weakly-supervised classification typically aims to predict fine-grained labels when only coarse
labels are available during training (Robinson et al., 2020; Grcic et al., 2024). These methods assume
fully observed labels at a coarse level and focus on improving predictions at a fine-grained level. In
contrast, our setting requires handling multi-granularity labels and inferring the full taxonomy.
Foundation models for zero-shot classification, such as vision-language models (e.g., CLIP (Rad-
ford et al., 2021)) and large language models (e.g., GPT-4 (Achiam et al., 2023)), have gained
popularity for leveraging label-driven prompts at inference—without training (Pratt et al., 2023; Liu
et al., 2024; Zheng et al., 2024; Saha et al., 2024). These methods aim to improve flat-level classi-
fication by matching images to text. In contrast, we train a hierarchical classifier that learns shared
visual patterns across levels from images, when labels are partially missing. Our model requires no
textual input at inference, making it efficient. See a full task comparison in Table 2.
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C
MORE EXPERIMANTAL RESULTS
C.1
EVALUAION ON CUB-F
On the small-scale, single-domain dataset CUB-F (Table 5), Taxon-SSL achieves the best perfor-
mance (63.96% FPA), showing the advantage of structured label propagation when per-class sam-
ples are scarce. Text-Attr methods perform moderately well (53.99–57.59% FPA) but are less effec-
tive here, as the bird-only domain limits textual diversity and reduces the benefit of language-based
supervision. Still, they clearly outperform conventional hierarchical baselines (44.30% for HRN,
45.10% for H-CAST), underscoring the overall effectiveness of our approach. Unlike the trend on
large-scale, diverse datasets such as ImageNet-F, where Text-Attr provides richer cues and stronger
gains, these results confirm that there is no single recipe for free-grain learning: performance is
tightly coupled with dataset characteristics, making the problem inherently challenging.
Table 5: Taxon-SSL shows strong effectiveness on the small-scale dataset CUB-F, where label
propagation provides reliable supervision despite limited data. Text-Attr methods are assumed
to offer limited benefit due to the restricted textual diversity of this bird-only dataset.
CUB-F (13-38-200)
FPA (↑)
Species (↑)
family (↑)
Order (↑)
TICE (↓)
HRN (Chen et al., 2022)
44.30
46.72
81.20
96.36
27.15
H-CAST (Park et al., 2025)
45.10
47.52
87.78
97.50
25.89
Taxon-SSL
63.96
65.50
92.84
98.40
7.39
Taxon-SSL + Text-Attr
63.05
64.86
92.54
98.38
7.61
Text-Attr (H-ViT)
57.59
59.10
91.60
98.05
10.72
Text-Attr (H-CAST)
53.99
55.58
91.72
98.41
18.95
C.2
EVALUATION UNDER VARYING AND SEVERE LABEL SPARSITY CONDITIONS
To evaluate model performance under diverse and more challenging free-grain conditions, we exper-
iment with various label availability ratios by randomly removing fine-grained labels—e.g., (100%-
60%-30%), (100%-50%-10%), and (100%-20%-10%)—which represent the available proportions
of basic, subordinate, and fine-grained labels, respectively. Each experiment is repeated with three
different random seeds, and we report the average performance. The variance across runs was minor
(0.1–1.8).
Consistent with our main results, these experiments (Table 6 & 7 & 8) also show that there is no
single method that performs best across all settings. Instead, the most effective method varies
depending on the dataset and the specific ratio of available labels, highlighting the importance of
adaptable free-grain learning strategies.
For consistency, we refer to the three levels in CUB-Rand (order-family-species) and Aircraft-Rand
(maker-family-model) as basic, subordinate, and fine-grained levels. We summarize the key findings
below:
(1) Conventional hierarchical classification methods struggle under the free-grain setting,
where label supervision is sparse and uneven across levels.
For example, when labels are
highly missing (e.g., only 10% available at the fine-grained level), HRN (Chen et al., 2022) and
H-CAST (Park et al., 2025) suffer more than a 50% drop in accuracy across all levels compared
to the fully labeled (100%-100%-100%) setting on CUB-Rand (Fig. 6 & Table 8). This highlights
the difficulty of the free-grain setting and the need for methods that can robustly handle incomplete
supervision at multiple semantic levels.
(2) The performance of different methods varies with the amount of available supervision per
class:
Text-Attr methods perform better when more labeled samples are available, while Taxon-
SSL is more effective under extreme label sparsity. For example, in Table 6, the average num-
ber of available fine-grained labels per class is approximately 9 for CUB-Rand and about 20 for
Aircraft-Rand. Consistent with this difference, Taxon-SSL outperforms other methods on CUB-
Rand, whereas Text-Attr (H-CAST) performs best on Aircraft-Rand. This trend persists across set-
tings. In the most sparse setting, CUB-Rand (100-20-10, Table 8), where only about 3 fine-grained
labels are available per class, Taxon-SSL shows a clear advantage. We attribute this to how supervi-
sion is utilized. Text-Attr relies on available labels and indirect semantic guidance via text features.
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Preprint (under review)
In contrast, Taxon-SSL actively leverages unlabeled data through pseudo-labeling and strong aug-
mentations, making it more effective when labeled examples are extremely limited.
(3) Sometimes, Taxon-SSL’s high fine-grained accuracy comes at the cost of lower accuracy
at higher levels in the taxonomy. For example, in Table 7, Taxon-SSL achieves the highest fine-
grained accuracy (65.01%), but its subordinate and basic-level accuracies (85.53% and 92.81%) are
lower than those of Text-Attr (H-CAST), which achieves 86.30% and 94.17%, respectively. This
highlights a key challenge in free-grain learning: improving accuracy across all levels simultane-
ously is non-trivial, and optimizing for fine-grained performance alone may degrade consistency at
coarser levels.
Table 6: No single method performs best across all conditions—performance depends strongly
on the amount of available supervision per class. Text-Attr methods tend to perform better when
more labeled samples are available, while Taxon-SSL is more effective under extreme label sparsity.
For example, Taxon-SSL performs best on CUB-Rand with around 9 fine-grained labels per class,
while Text-Attr (H-CAST) performs best on Aircraft-Rand with around 20, reflecting the impact of
supervision density. These results highlight that method effectiveness is highly sensitive to label
sparsity, emphasizing the need for adaptable approaches in free-grain learning.
Label Ratio
CUB-Rand (100%-60%-30%)
Aircraft-Rand (100%-60%-30%)
FPA(↑)
spec.(↑) fam.(↑) order(↑) TICE(↓) FPA(↑)
maker(↑) fam.(↑)
model(↑) TICE(↓)
HRN (Chen et al., 2022)
57.87
62.73
85.53
96.45
13.77
57.33
64.42
76.95
86.38
23.30
H-CAST (Park et al., 2025)
61.88
67.36
90.05
94.32
13.04
64.67
68.88
85.58
91.43
13.76
Taxon-SSL
74.82
76.92
93.38
98.33
5.06
70.33
72.22
87.06
93.50
7.18
Taxon-SSL + Text-Attr
74.90
76.95
93.41
98.38
4.91
69.89
72.24
86.92
93.29
7.77
Text-Attr (H-ViT)
67.89
72.48
90.63
95.37
10.39
64.15
68.92
85.88
89.87
15.80
Text-Attr (H-CAST)
69.65
71.31
92.88
98.48
8.35
71.43
73.56
89.66
95.31
9.71
Table 7: Maintaining accuracy across all hierarchy levels remains more challenging under
sparse supervision. For example, in 100%-50%-10% case, Taxon-SSL achieves the highest fine-
grained accuracy (65.01%), but its subordinate and basic-level accuracies (85.53%, 92.81%) are
lower than those of Text-Attr (H-CAST) (86.30%, 94.17%), which better preserves consistency
across levels. This result illustrates the inherent difficulty of improving accuracy across all levels
simultaneously, as objectives at different levels can be conflicting.
Label Ratio
Aircraft-Rand (100%-50%-10%)
Aircraft-Rand (100%-20%-10%)
FPA(↑)
maker(↑) fam.(↑)
model(↑) TICE(↓) FPA(↑)
maker(↑) fam.(↑)
model(↑) TICE(↓)
HRN (Chen et al., 2022)
40.35
47.85
70.76
85.68
37.56
32.06
46.73
55.43
85.58
48.43
H-CAST (Park et al., 2025)
47.57
51.93
78.31
87.11
28.42
40.33
45.44
67.28
84.12
35.61
Taxon-SSL
62.61
65.01
85.53
92.81
10.22
58.73
61.10
80.90
92.24
11.77
Taxon-SSL + Text-Attr
62.95
65.49
86.01
92.64
10.25
58.55
60.88
80.97
92.04
11.89
Text-Attr (H-ViT)
47.83
52.25
81.13
87.82
30.57
38.73
43.89
66.13
84.81
38.69
Text-Attr (H-CAST)
53.31
55.32
86.30
94.17
24.43
48.85
51.37
77.11
93.01
27.25
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Table 8: Taxon-SSL is more robust under extreme label sparsity, while other methods degrade
significantly. In CUB-Rand (100%-20%-10%), where each class has only 3 fine-grained and 3 sub-
ordinate labels, Taxon-SSL achieves the best performance, while other methods struggle. HRN and
H-CAST suffer over 50% drop in fine-grained accuracy compared to the fully-supervised (100%-
100%-100%) setting. Text-Attr methods perform more robustly (10%+ higher than HRN/H-CAST),
but still fall short under such sparse supervision. We attribute this to how each method leverages
supervision: Text-Attr relies on available labels and semantic guidance from text features, while
Taxon-SSL benefits more from unlabeled data via pseudo-labeling and augmentations, making it
more effective under severe label sparsity.
Label Ratio
CUB-Rand (100%-50%-10%)
CUB-Rand (100%-20%-10%)
FPA(↑)
spec.(↑) fam.(↑) order(↑) TICE(↓) FPA(↑)
spec.(↑) fam.(↑)
order(↑) TICE(↓)
HRN (Chen et al., 2022)
40.23
43.70
82.75
95.94
22.34
33.53
41.18
72.56
95.79
30.50
H-CAST (Park et al., 2025)
39.03
43.41
85.74
93.23
24.60
32.97
38.66
76.89
92.50
29.43
Taxon-SSL
62.40
64.14
92.33
98.26
6.01
59.18
61.44
89.79
98.20
7.65
Taxon-SSL + Text-Attr
62.52
64.87
87.94
94.45
8.98
57.98
60.59
89.42
98.12
8.39
Text-Attr (H-ViT)
47.42
50.74
88.22
94.67
18.09
42.46
46.99
80.92
94.43
20.27
Text-Attr (H-CAST)
44.63
45.89
91.06
98.19
22.72
40.41
42.76
84.24
97.97
24.05
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D
T-SNE VISUALIZATION
We visualize ImageNet-F embeddings of Text-Attr (H-CAST) and Taxon-SSL using t-SNE (Maaten
& Hinton, 2008) to assess whether the learned representations capture semantic and hierarchical
structure. Each point denotes an image embedding, colored by its basic-level class (20 categories),
with brightness variations indicating fine-grained subclasses (505 total).
Both Text-Attr (H-CAST) and Taxon-SSL produce well-separated clusters consistent with the basic-
level taxonomy, showing that coarse groupings are reliably captured. The key difference lies within
coarse categories: Text-Attr (H-CAST) reveals more distinct fine-grained subclusters (e.g.,
breeds within dog, species within bird), whereas Taxon-SSL yields tighter coarse clusters with
less apparent fine-level separation.
This contrast reflects their supervision signals. Text-Attr leverages diverse textual cues (attributes,
parts, appearance terms), which promote discriminative, attribute-aligned features and sharpen
within-class distinctions. Taxon-SSL, by propagating labels along the taxonomy and enforcing con-
sistency under mixed-granularity supervision, regularizes embeddings within each coarse class and
reduces intra-class variance—emphasizing coarse alignment over fine-level separability.
(a) Text-Attr (H-CAST)
(b) Taxon-SSL
Figure 10: t-sne Visualization on ImageNet-F. Both methods separate coarse-level taxonomy well,
but Text-Attr (H-CAST) yields clearer fine-grained subclusters (e.g., distinct groups within dog and
bird) with more compact grouping, whereas Taxon-SSL shows some overlap of embeddings near
cluster boundaries. This is likely due to ImageNet-F ’s diverse large-scale categories, where text
supervision provides rich attribute cues that sharpen fine-level distinctions.
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E
ABLATION STUDY
E.1
IMPORTANCE OF TEXT-GUIDED PSEUDO ATTRIBUTES
Text-guided Pseudo Attributes jointly optimizes hierarchical label supervision (Lhier) and text-
guided pseudo attributes (Ltext) to learn semantically rich features: L = Lhier + αLtext Fig. 11
quantifies Ltext’s impact by varying its weight α on CUB-Rand. Ablating Ltext (α = 0) causes a
5% absolute decline in both fine-grained accuracy and FPA compared to the optimal configuration
(α = 0). This gap underscores two key roles of text guidance: (1) it injects complementary visual
semantics absent in class labels alone, and (2) it enforces attribute consistency across hierarchy lev-
els. The performance recovery at (α = 1) confirms that textual pseudo-attributes mitigate annotation
sparsity while preserving taxonomic coherence.
Figure 11: Tuning α balances accuracy and taxonomic consistency. At α = 1 (optimal), Text-Attr
(H-ViT) achieves peak fine-grained accuracy (blue) while maintaining hierarchical consistency (or-
ange). Ablating Ltext (α = 0) causes a 5% accuracy drop and increased inconsistency, as class em-
beddings lose text-guided attribute alignment. Higher α > 1.0 over-regularizes features, marginally
degrading both metrics. This trade-off underscores the need to weight text supervision to resolve
sparse annotations without distorting the hierarchy.
E.2
COMBINING TEXT-ATTR AND TAXON-SSL
We compare different training schedules for combining Text-Attr and Taxon-SSL on CUB-F. In the
joint setting, both objectives are optimized simultaneously for 100 epochs. In the two-stage setting,
we first train with one objective for 50 epochs and then add the other for the remaining 50 epochs,
considering both orders: (1) Taxon-SSL →Text-Attr, and (2) Text-Attr →Taxon-SSL.
Table 9 show that starting with Text-Attr and then adding Taxon-SSL yields slightly higher full-path
accuracy, likely because textual supervision promotes diverse feature learning before label propa-
gation. In contrast, beginning with Taxon-SSL provides no advantage, and both two-stage variants
perform similarly to joint training overall. Interestingly, joint training achieves higher consistency
as measured by TICE. Given its simplicity and competitive performance, we adopt the joint strategy
as our default.
Table 9: Comparison of joint vs. two-stage training schedules for Text-Attr and Taxon-SSL on
CUB-F. While two-stage training (Text-Attr →Taxon-SSL) yields slightly higher accuracy, joint
learning is simpler and provides better consistency (TICE).
CUB-F (13-38-200)
FPA (↑)
Species (↑)
family (↑)
Order (↑)
TICE (↓)
Taxon-SSL + Text-Attr (100 epochs)
63.04
64.86
92.54
98.37
7.61
Taxon-SSL (50 epochs) →+Text-Attr (50 epochs)
62.84
64.42
92.47
98.20
8.19
Text-Attr (50 epochs) →+Taxon-SSL (50 epochs)
63.63
65.34
92.56
98.27
8.06
E.3
ABLATION ON HIERARCHICAL SUPERVISION IN VIT
We further examine the architectural design choice of where to inject hierarchical supervision in
the Vision Transformer (ViT) in Table 10. On CUB-F, we map the three taxonomy levels (Or-
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der–Family–Species) to different layers and compare multiple configurations: (6th, 9th, 12th), (8th,
10th, 12th), and (10th, 11th, 12th).
Among these, supervision at the 8th, 10th, and 12th layers yields the best performance. We interpret
this as a balance between early and late representation learning: assigning hierarchy too early (e.g.,
6–9–12) forces the model to align coarse categories before sufficient visual features are developed,
while placing all supervision too late (e.g., 10–11–12) limits the model’s capacity to gradually refine
class granularity. The 8–10–12 configuration provides an appropriate middle ground, where lower-
level categories benefit from moderately abstract features, and finer distinctions are introduced after
the backbone has matured.
Table 10: Performance comparison of different layer assignments for hierarchical supervision
in ViT on CUB-F. The 8th–10th–12th configuration achieves the best results, balancing early and
late feature abstraction.
CUB-F (13-38-200)
FPA (↑)
Species (↑)
family (↑)
Order (↑)
TICE (↓)
6-9-12th layer
54.80
58.16
88.97
95.01
16.79
8-10-12th layer
57.59
59.10
91.60
98.05
10.72
10-11-12th layer
56.40
58.56
90.80
97.08
13.48
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F
IMPLEMENTATION DETAILS
For ViT (Dosovitskiy et al., 2020) models, we use ViT-Small for Text-Attr (H-ViT) and Taxon-SSL
and H-CAST-Small (Park et al., 2025) for Text-Attr (H-CAST) to match parameter sizes.
For Text-Attr (H-ViT), we insert fully-connected layers to the class token at the 8th, 10th, and 12th
layers for basic, subordinate, and fine-grained supervision. The 12th-layer patch features are pro-
jected to match the text embedding dimension via an FC layer. For Text-Attr (H-CAST), hierarchical
supervision is applied to the last three blocks, following (Park et al., 2025). Due to low dimension-
ality in the final block, we align text features with the features of the second block. For Text-Attr
methods, CLIP-ViT-B/32 is used to extract text embeddings, which remain frozen during training.
In Taxon-SSL, we apply a shared MLP to the class token from the final (12th) layer, followed by
three separate linear classifiers for basic, subordinate, and fine-grained supervision. When combined
with Text-Attr, we additionally project the class token through a linear layer and align it with the
corresponding text feature.
For hierarchical classification baselines, HRN (Chen et al., 2022) and H-CAST (Park et al., 2025),
we follow their original training protocols and retrain them under our free-grain setting. We extend
HRN to handle missing labels at two levels instead of one. For H-CAST, we provide supervision us-
ing the available labels at each corresponding level. Full hyperparameter configurations are provided
in Table 11.
We train all models for 100 epochs, except for ImageNet-F, which are trained for 200 epochs due to
the larger scale. All experiments were conducted on an NVIDIA A40 GPU with 48GB memory. We
used a single GPU for all experiments, except for ImageNet-F, which was trained using 4 GPUs.
Table 11: Hyperparameters for training Text-Attr (H-ViT), Text-Attr (H-CAST), and Taxon-
SSL. We follow the training setup of H-CAST (Park et al., 2025) for Text-Attr methods (Text-Attr
(H-ViT) and Text-Attr (H-CAST)), and adopt the settings of CHMatch (Wu et al., 2023) for Taxon-
SSL.
Parameter
Text-Attr (H-ViT)
Text-Attr (H-CAST)
Taxon-SSL
batch size
256
256
128
crop size
224
224
224
learning rate
5e −4
5e −4
1e −3
weight decay
0.05
0.05
0.05
momentum
0.9
0.9
0.9
warmup epochs
5
5
0
warmup learning rate
1e−6
1e−6
N/A
optimizer
Adam
Adam
SGD
learning rate policy
Cosine decay
Cosine decay
Cosine decay
α (weight for Ltext)
1
1
1 (for +Text-Attr)
G
USE OF LARGE LANGUAGE MODELS (LLMS)
Large Language Models (LLMs) were used in a limited manner, primarily to review the constructed
hierarchy and to assist with minor tasks such as translation and typo correction.
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Preprint (under review) FREE-GRAINED HIERARCHICAL RECOGNITION Seulki Park1 Zilin Wang1 Stella X. Yu1,2 1 2UC Berkeley { ABSTRACT Hierarchical image classification predicts labels across a semantic taxonomy, but existing methods typically assume complete, fine-grained annotations, an assumption rarely met in practice. Real-world supervision varies in granularity, influenced by image quality, annotator expertise, and task demands; a distant bird may be labeled Bird, while a close-up reveals Bald eagle. We introduce ImageNet-F, a large-scale benchmark curated from ImageNet and structured into cognitively inspired basic, subordinate, and fine-grained levels. Using CLIP as a proxy for semantic ambiguity, we simulate realistic, mixed-granularity labels reflecting human annotation behavior. We propose free-grain learning, with heterogeneous supervision across instances. We develop methods that enhance semantic guidance via pseudo-attributes from vision-language models and visual guidance via semisupervised learning. These, along with strong baselines, substantially improve performance under mixed supervision. Together, our benchmark and methods advance hierarchical classification under real-world constraints1 1 INTRODUCTION Hierarchical classification (Chang et al., 2021; Chen et al., 2022; Jiang et al., 2024; Park et al., 2025) predicts a semantic tree of labels, capturing categories from broad to specific. This richer output supports flexible use: An expert may seek Bald Eagle, while a general user may only need Bird. Moreover, predicting the full hierarchy improves robustness and scalability, encouraging models to generalize across levels, and can naturally support extensions like adding new parent or child classes. However, existing methods (Chang et al., 2021; Wang et al., 2023) assume complete supervision at all levels for all the training examples, which rarely holds in practice. In real-world settings, annotation granularity depends on image clarity, annotator expertise, or task-specific needs: A distant bird could only be labeled as Bird, while a close-up allows Bald eagle (Fig.1). bald eagle bird bird of prey vulture bald eagle bird bird of prey vulture bald eagle bird bird of prey vulture Figure 1: Images vary in semantic detail: Some support only coarse labels, others reveal finegrained categories. We propose free-grain learning: Training a hierarchical classifier with supervision free to vary in granularity across examples, reflecting semantic ambiguity in real-world images. We propose free-grain learning, where supervision is free to vary in granularity: Training labels may appear at any level of a fixed taxonomy, e.g., Bird, Bird of prey, or Bald eagle. The key challenge is to predict the full taxonomy from training data with mixed (rather than uniform finegrained) labels. This task not only reflects real-world variability in annotation quality and specificity, but also enables learning from partially labeled data at scale. It further requires integration across semantic annotation granularities and across visual instances, as the model must infer a complete taxonomy for each example based on heterogeneous supervision. 1Our code and dataset will be available at https://github.com/pseulki/FreeGrainLearning. 1 16 Oct 2025 Preprint (under review) Table 1: Existing Hierarchical Recognition Benchmarks Are Insufficient. CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013) provide clean hierarchies but are small; iNat21-mini (Van Horn et al., 2021) has a clean taxonomy but is limited to biology; ImageNet (Russakovsky et al., 2015) is large but structurally inconsistent. We introduce ImageNet-3L, a general-purpose large-scale benchmark with a coherent three-level hierarchy grounded in cognitive psychology. From this, we provide new variants of ImageNet, iNat21-mini, CUB, and Aircraft for free-grained recognition. Dataset #levels #classes #train #test CUB 3 13-38-200 5,994 5,794 Aircraft 3 30-70-100 6,667 3,333 iNat21-mini 8 3-11-13-51-273-1103-4884-10000 500,000 100,000 ImageNet 5-19 - 1000 1,281,167 50,000 ImageNet-3L 3 20-127-505 645,480 25,250 However, existing benchmarks are ill-suited for this task (Table 1). Small datasets such as CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013) lack scale, while iNaturalist (Van Horn et al., 2021) is limited to biology and unsuitable for general-purpose evaluation. Larger benchmarks like ImageNet (Russakovsky et al., 2015) and tieredImageNet (Ren et al., 2018) inherit noisy, inconsistent hierarchies from WordNet (Fellbaum, 1998). As shown in Fig. 2, hierarchy depths vary widely from 5 to 19 levels, with some classes following multiple paths-for example, Minivan appears in four different paths (depths 12-15), while Teddy bear appears only once at depth 7. Such inconsistencies make evaluation ambiguous: one fine class can map to several hierarchies, and predictions often traverse long chains of redundant nodes (e.g., entity, object). As a result, most methods on ImageNet and tieredImageNet restrict evaluation to leaf-node accuracy, with auxiliary metrics like mistake severity (Bertinetto et al., 2020; Garg et al., 2022b; Jain et al., 2023). To address these limitations, we construct ImageNet-3L, a benchmark with a well-structured threelevel hierarchy: basic (e.g., Dog), subordinate (e.g., Shepherd), and fine-grained (e.g., German Shepherd) (Fig. 3). Grounded in cognitive psychology (Rosch et al., 1976; Rosch, 1978) and folk taxonomies (Berlin et al., 1966), our design reflects that the basic level is the most natural and widely recognized category for humans, while subordinate and fine-grained levels capture increasingly specific distinctions. By focusing on this range-from the most intuitive to the most detailed-we enable semantically meaningful hierarchical prediction, avoiding abstract or redundant levels (e.g., Physical Entity in the original ImageNet hierarchy) that provide little practical value. Building on ImageNet-3L, we further construct ImageNet-F, a free-grain benchmark that simulates mixed-granularity labeling. Using CLIP (Radford et al., 2021) as a proxy for visual-semantic ambiguity and annotator variability, we prune labels at different levels depending on prediction confidence. This yields realistic supervision patterns-for example, distant birds labeled as Bird, midrange as Bird of prey, and close-ups as Bald eagle (Fig. 4). The final dataset covers 645,480 images across 20 basic, 127 subordinate, and 505 fine-grained classes (Table 1). We apply the same strategy 0 100 200 # Leaf node at each level 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Hierarchy depth Figure 2: Inconsistent and Noisy Hierarchy of ImageNet WordNet. (Left:) The histogram of hierarchy depths shows that ImageNet classes range from 5 to 19 levels, with many exceeding 10, which hinders consistent evaluation. (Right:) Sample hierarchies illustrate that classes can have multiple paths of different depths: Minivan appears in four paths at depths 12-15, while Teddy bear exists only at depth 7. This imbalance and inconsistency in the hierarchy make it unclear which path should be considered correct, underscoring the difficulty of using the original WordNet hierarchy for training and evaluation. 2 Preprint (under review) (a) Original ImageNet's WordNet hierarchy (b) Our 3-level hierarchy Figure 3: We curate ImageNet-3L as a benchmark for hierarchical classification. (a) The original ImageNet taxonomy is noisy and inconsistent, with imbalances, overlaps, and multiple paths, making it unsuitable for reliable evaluation. (b) We construct a coherent 3-level taxonomy, inspired by cognitive psychology (Rosch et al., 1976): basic for general recognition, subordinate for contextual specificity, and fine-grained for specialized distinctions. to iNat21-mini and CUB using BioCLIP (Stevens et al., 2024), producing iNat21-mini-F and CUBF, and additionally create synthetic variants CUB-Rand and Aircraft-Rand for controlled evaluation under varying label sparsity and granularity. Dog Cooking utensil Dog Fish Electron. equip. Insect Bird Pinscher Pot Shepherd Ganoid Audio device Beetle Bird of prey Affenpinscher Teapot German shepherd Gar Tape player Leaf beetle Bald eagle Figure 4: Our ImageNet-F captures real-world challenges, where fine-grained labels follow a long-tailed distribution and granularity varies with visual clarity. Top: The graph shows the proportion of basic, subordinate, and fine-grained labels per class (sorted by ID). Fine-grained labels are scarce on the left but increase toward the right, forming a long tail. This imbalance often causes models to overfit to basic-level features and miss subtle distinctions, underscoring the need for robust multi-level learning. Bottom: Arrow-marked samples illustrate how label pruning reflects difficulty. (Last Column): a distant bird is labeled at the basic level (Bird); one with visible wings and talons at the subordinate level (Bird of prey); and a close-up at the fine-grained level (Bald eagle). 3 Preprint (under review) When applied directly under free-grain setting, existing hierarchical classifiers (Chen et al., 2022; Park et al., 2025) degrade severely-up to -40% full-path accuracy on iNat21-mini -highlighting the difficulty of the task. To address this, we propose three additional strategies: 1) learning pseudoattributes (e.g., short legs, docked tail) from vision-language models to provide semantic cues when finer labels are missing; 2) applying semi-supervised learning by treating missing-grain labels as unlabeled; 3) combining both approaches. Across datasets, these methods outperform hierarchical baselines by +4-25%p, establishing stronger baselines for free-grain learning. Contributions. 1) We introduce free-grain learning for hierarchical classification, capturing realworld variability in label granularity. 2) We present ImageNet-F, with a cognitively grounded 3-level hierarchy, and additional free-grain benchmarks across diverse domains. 3) We establish strong baselines that significantly improve performance by leveraging semantic and visual guidance. 2 RELATED WORK Hierarchical classification has been studied mainly for leaf-node prediction on large but inconsistent taxonomies such as ImageNet (Karthik et al., 2021; Zhang et al., 2022), or for full-taxonomy prediction on small datasets like CUB and Aircraft (Chang et al., 2021; Park et al., 2025). These settings lack the scale, diversity, and label sparsity needed for realistic evaluation. Our work instead enables full taxonomy prediction under heterogeneous supervision on large-scale data. Imbalanced and semi-/weakly-supervised classification have been widely explored (Liu et al., 2019; Tarvainen & Valpola, 2017; Robinson et al., 2020), but mostly at a single fine-grained level or with fully observed coarse labels. In contrast, we address both intra- and inter-level imbalance, requiring consistent prediction across multiple granularities with partially missing supervision. See a full task comparison in Table 2. Foundation models such as CLIP (Radford et al., 2021) has been used for zero-shot flat classification via text prompts (Pratt et al., 2023; Saha et al., 2024). In contrast, our approach leverages text only during training to learn visual patterns across levels, requiring no textual input at inference. Further discussion and additional related work are provided in Appendix B. Table 2: Our task setting is more practical and challenging than existing ones. Ours reflects realistic scenarios where annotations are free-grain and imbalanced, requiring hierarchical predictions to balance accuracy and consistency across levels. Input Output Training Labels Imbalance Evaluation Metrics Tasks Fine Coarse Fine Coarse Availability Intra Inter Accuracy Consistency Long-tailed recognition ✓ ✗ ✓ ✗ All ✓ ✗ ✓ ✗ Semi-supervised learning ✓ ✗ ✓ ✗ Partial ✗ ✗ ✓ ✗ Weakly-supervised learning ✗ ✓ ✓ ✗ All ✗ ✗ ✓ ✗ Hierarchical recognition ✓ ✓ ✓ ✓ All ✗ ✗ ✓ ✓ Free-grained recognition ✓ ✓ ✓ ✓ Partial ✓ ✓ ✓ ✓ 3 HIERARCHICAL DATASET FOR FREE-GRAINED RECOGNITION 3.1 Defining Three-Level Taxonomy for ImageNet-3L. We restructure ImageNet (Russakovsky et al., 2015)'s WordNet (Fellbaum, 1998)-based hierarchy into a consistent three-level taxonomy, explicitly guided by Rosch's categorization principles (Rosch et al., 1976). In Rosch's framework, the basic level (e.g., dog, car) is the most natural and visually distinctive, balancing generality and specificity; it is also the level people most often use in everyday recognition and naming, unlike abstract superordinate categories (e.g., animal) or overly narrow subordinate ones (e.g., Pembroke). We adopt the basic level as the coarsest node in each branch, with subordinate and fine-grained levels (e.g., Corgi →Pembroke) providing progressively finer distinctions. However, WordNet chains such as artifact →...→vehicle →...→motor vehicle →car →ambulance can yield only two usable levels if car is taken as basic. In these cases, we elevate Rosch's superordinate category (e.g., vehicle) to serve as the basic level, which remains visually distinctive from other basic categories (e.g., craft, container) and ensures a three-level hierarchy. This yields branches that support three semantically coherent and visually meaningful levels for hierarchical prediction. 4 Preprint (under review) Specifically, we adopt the following systematic principles: 1) Enforce meaningful structure: We remove paths where each node has only one child, since coarse labels fully determine the fine labels. Branches with fewer than three levels are also excluded. 2) Maximize within-group diversity: Among subordinate candidates under each basic class, we favor those with richer fine-grained subclasses-e.g., parrot (4 children) over cockatoo (1 child). 3) Refine vague categories: Ambiguous groups such as Women's Clothing are reorganized into precise, functionally grounded categories (e.g., Underwear) to improve clarity. 4) Validate with language models and human review: We use language models (ChatGPT (Achiam et al., 2023)) to suggest refinements, with all decisions manually reviewed for semantic consistency. Applying this curation process to ImageNet-1k yields a structured benchmark of 20 basic, 127 subordinate, and 505 fine-grained classes, ensuring every branch supports meaningful hierarchical prediction (a complete list is provided in Appendix A). 3.2 Semantic Label Pruning for ImageNet-F, iNat21-mini-F, and CUB-F To build a realistic free-grain training dataset, we prune hierarchical labels using large vision-language models as a proxy for visual-semantic ambiguity: CLIP (Radford et al., 2021) for ImageNet-F and BioCLIP (Stevens et al., 2024) for iNat21-mini-F and CUB-F. Although these models are not explicitly designed to measure ambiguity, their zero-shot confidence consistently correlates with visual distinctiveness (Fig. 4). Moreover, since label annotation is affected by annotator expertise or error, this proxy offers a practical approximation. We adopt CLIP's prompt-ensemble strategy (e.g., a photo of a [class], art of a [class]) and compute average confidence for fine-grained and subordinate levels. Labels are retained based on prediction correctness: (1) If both fine-grained and subordinate are correct, we keep all labels. (2) If only subordinate is correct, we keep up to that level. (3) Otherwise, only the basic label is kept. We further prune subordinate labels proportionally to the fine-grained removal rate per class. (1) ImageNet-F. After pruning, 32.6% of images retain all three levels (Basic + Subordinate + Fine-grained), 28.0% retain two (Basic + Subordinate), and 39.4% retain only the Basic. Each class maintains the same number of images as ImageNet; imbalance arises only from label granularity. (2) iNat21-mini-F. Although BioCLIP is trained on iNat21-mini 's full taxonomy, it performs well when predicting fine-grained species but struggles when restricted to coarser labels. This gap enables substantial pruning: 22.5% of images retain all three levels (Order + Family + Species), 28.0% retain two, and 49.5% retain only Order. (3) CUB-F. With the same procedure, 31.5% of images keep three levels, 23.3% two (Order + Family), and 45.2% only Order. 3.3 Synthetic Label Pruning for CUB-Rand and Aircraft-Rand To control label availability, we construct synthetic variants-CUB-Rand and Aircraft-Rand -by randomly pruning labels from CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013). Unlike realistic pruning, this design systematically varies supervision and simulates extreme sparsity (e.g., only 10% fine-grained labels), enabling stress-testing of model robustness across diverse label distributions. Although random removal is independent of image difficulty, it reflects practical factors such as annotator expertise, cost, or task-specific constraints. We denote availability as a-b-c, where a% of basic, b% of subordinate, and c% of fine-grained labels are retained (e.g., 100-50-10 retains 10% fine-grained labels and 40% subordinate-only labels). 4 FREE-GRAIN LEARNING METHODS FOR HIERARCHICAL CLASSIFICATION 4.1 Problem setup. We begin by describing the problem setup. In free-grain hierarchical classification, the goal is to train a model that predicts object categories across all levels of a taxonomy, given training data with labels of varying granularity. Formally, let X denote the input space of images, and Y1, . . . , YL the label spaces at L hierarchical levels, from coarsest (Y1) to finest (YL). Each training sample consists of an image x ∈X and a partial label set {yl}l∈Sx, where Sx ⊆{1, . . . , L} indicates the levels at which labels are provided. We assume that if a label yl is available, all coarser labels yl′ for l′ l are missing. We further assume the coarsest label y1 is always given. The objective is to learn a classifier f : X →Y1 × · · · × YL, f(x) = (ˆy1, . . . , ˆyL), that predicts labels at all levels of the hierarchy. 5 Preprint (under review) (a) Text-Attr (b) Taxon-SSL Figure 5: Overview of the proposed methods. (a) Text-Attr enriches feature representations using semantic cues from images, compensating for missing labels and capturing shared attributes across levels. (b) Taxon-SSL handles missing-level labels by treating them as unlabeled and learns from visual consistency through augmented views. 4.2 Baselines. With no existing baselines for this new setting, we propose four strong baselines, each approaching the problem from a different perspective. (1) Semantic Guidance: Text-Guided Pseudo Attributes (Text-Attr). Our semantic guidance approach is motivated by the observation that while class labels differ across hierarchical levels (e.g., Dog →Corgi →Pembroke), many visual attributes-such as tail length or ear shape-remain consistent (Fig. 5a). To capture these shared semantic cues, we use image descriptions as auxiliary supervision. Instead of class-name prompts for zero-shot classification, we extract free-form descriptions directly from the image, independent of labels. Specifically, given an input image x, we use a frozen vision-language model (VLM), Llama-3.211B (Dubey et al., 2024), to generate a language description dx, using the prompt: "Describe visual details in the image." This produces descriptions containing phrases such as "short legs" or "pointed ears," which we encode into a text embedding zt x using CLIP's text encoder (Radford et al., 2021). We cap generation at 100 tokens, while CLIP accepts 77 tokens; longer descriptions are truncated during encoding. Although truncation discards some details, our method focuses on shared semantic cues (e.g., "short legs," "brown markings") rather than exhaustive captions, making it robust to this limitation. In parallel, we obtain the image embedding zv x from the image encoder, and align the two embeddings with a contrastive loss: Ltext = -1 N N X i=1 log exp(sim(zv i , zt i)/τ) PN j=1 exp(sim(zv i , zt j)/τ) ! , (1) where sim(·, ·) is cosine similarity and τ is a temperature parameter. This loss guides the encoder to capture salient, label-independent traits shared across levels. Although not explicitly predicting attributes, aligning image features with text induces intermediate representations, which we call pseudo-attributes. This model-agnostic method can be applied to any architecture. Finally, for hierarchical supervision, we apply the loss only at levels with available labels. Given hierarchical labels y1, . . . , yL across L levels, the model computes a loss at each level: Lhier = L X l=1 ⊮{yl exists} · L(fl(x), yl), (2) where fl(x) is the prediction at level l, and L denotes any classification loss (e.g., cross-entropy). (2) Visual Guidance: Taxonomy-Guided Semi-Supervised Learning (Taxon-SSL). To enforce semantic consistency, we extend CHMatch's contrastive objective to the full taxonomy. For each mini-batch, we build level-wise affinity graphs W l based on pseudo-label agreement: W l ij = 1 if images i and j share the same pseudo-label at level l, and 0 otherwise. Then the 6 Preprint (under review) taxonomy-aligned affinity graph W is defined as: Wij = ( 1 if W 1 ij = ... = W L ij = 1, 0 otherwise. (3) Then, taxonomy-aligned contrastive loss Ltacl is defined by: Ltacl = - 1 P j Wij · L X l=1 log P j Wij exp((g(f(xi)) · g(f(xj))′)/t) P j(1 -Wij) exp((g(f(xi)) · g(f(xj))′)/t), (4) where gi = g(f(xi)) is the projected feature of image i with the classifier f, and t is a temperature hyperparameter. (3) Combining Semantic and Visual Guidance: Taxon-SSL + Text-Attr. A natural next step is to combine Text-Attr and Taxon-SSL by incorporating text-derived embeddings into the feature extractor of Taxon-SSL, allowing semantic and visual guidance to be jointly leveraged during training. (4) State-of-the-art Hierarchical Classification Methods: H-CAST, HRN. We adopt two representative models. (4-1) Hierarchical Residual Network (HRN) (Chen et al., 2022): the first to handle supervision at both subordinate and fine-grained levels by maximizing marginal probabilities within the tree-constrained space. (4-2) H-CAST (Park et al., 2025): the current state-of-the-art, encouraging consistent visual grouding across taxonomy levels. Originally trained with full supervision, we adapt it to this setting via the level-wise loss in Eq. 2, using only available labels. 5 EXPERIMENTS Dataset: We conduct experiments using our proposed ImageNet-F, iNat21-mini-F, and CUB-F datasets, along with the synthetic CUB-Rand and Aircraft-Rand datasets. CUB includes bird images across 13 orders (e.g., Anseriformes), 38 families (e.g., Anatidae), and 200 species (e.g., Mallard), while Aircraft (Maji et al., 2013) contains aircraft images across 30 makers (e.g., Boeing), 70 families (e.g., Boeing 707), and 100 models (e.g., 707-320). Evaluation metrics: Following (Park et al., 2025), we evaluate accuracy and consistency: 1) Levelaccuracy: Top 1 accuracy for each level. 2) Tree-based InConsistency Error rate (TICE): Proportion of test samples with inconsistent prediction paths in the hierarchy. Lower is better. TICE = nic N 3) Full-Path Accuracy (FPA): Proportion of test samples with correct predictions at all hierarchy levels. Higher is better, and we use FPA as one of our primary metrics: FPA = nac N . Implementation: We use H-ViT, a ViT-Small-based hierarchical classifier, as the backbone for evaluating both Text-Attr and Taxon-SSL. To evaluate its compatibility across architectures, we also apply Text-Attr to H-CAST (Park et al., 2025), a state-of-the-art hierarchical model with comparable capacity. HRN (Chen et al., 2022) is evaluated with its original ResNet-50 backbone, which has over twice the parameters. All models are trained for 100 epochs, except for ImageNet-F, which is trained for 200 due to its larger scale. Full architectural and training details are in the appendix F. Result 1: Performance Drop under Free-Grain Learning. The prior hierarchical SOTA, HCAST, degrades sharply under mixed-granularity labels on both CUB and iNat21-mini. As shown in Fig. 6, full-path accuracy drops from 84.9% to 45.1% on CUB-F and from 64.9% to 25.6% on iNat21-mini-F. This demonstrates the difficulty of handling mixed-granularity labels and imbalanced supervision across the hierarchy and need for methods handling them. Result 2: Performance on ImageNet-F. As shown in Table 3, existing hierarchical methods degrade sharply under free-grain learning: HRN reaches only 37.8% FPA, while H-CAST performs better at 57.6% but still struggles with missing labels. Text-Attr (H-ViT) achieves 55.5% without relying on H-CAST's visual grouping, and integrating it into H-CAST further improves performance to 63.2%, demonstrating the effectiveness of semantic-guided pseudo-attribute learning at scale. Taxon-SSL improves over HRN by leveraging visual guidance but remains less effective than TextAttr methods, whose strong performance benefits from the abundance and diversity of ImageNet-F for reliable visual-semantic alignment. Result 3: Performance on iNat21-mini-F. In Table 3, on the large-scale iNat21-mini-F dataset, which contains many classes (10,000), conventional hierarchical methods perform poorly (17.0% 7 Preprint (under review) CUB iNat21-Mini 10 20 30 40 50 60 70 80 90 Full-Path Accuracy (%) -39.8% -39.3% 100% Full Labels Free-grain Labels Figure 6: Transitioning from fully labeled data to our mixedgranularity setting results in a substantial drop in Full-Path Accuracy, highlighting the difficulty of the task. SOTA HCAST suffers nearly a 40pp loss on both CUB and iNat21-mini. Figure 7: Text-Attr benefits under extreme label sparsity, as seen on the left (low-index classes with few fine-grained labels) by providing extra guidance from textual descriptions, while Taxon-SSL performs better on the right (high-index classes with more fine-grained labels). Both are based on ViTsmall model and evaluated on the ImageNet-F. Classes are sorted by the number of fine-grained training samples, from lowest to highest. Table 3: No single recipe solves free-grain learning-methods behave differently depending on data characteristics. 1) Conventional hierarchical classification methods like HRN (Chen et al., 2022) and H-CAST (Park et al., 2025) show significant performance drops under incomplete supervision, underscoring the challenge of free-grain settings. 2) Text-Attr methods works well on ImageNet-F, where each class is supported by abundant visual evidence. In contrast, iNat21-mini-F has fine-grained biology labels, where appearance are similar, making LLM-based text descriptions less effective. Here, Taxon-SSL proves more beneficial by leveraging structured label propagation in this semi-supervised style setting. 3) Combining the two (Taxon-SSL + Text-Attr) yields consistent but modest gains across both datasets. Dataset ImageNet-F (20-127-505) iNat21-mini-F (273 - 1,103 - 10,000) FPA(↑) fine.(↑) sub.(↑) basic(↑) TICE(↓) FPA(↑) spec.(↑) fam.(↑) order(↑) TICE(↓) HRN (Chen et al., 2022) 37.79 38.73 55.73 78.65 46.69 17.03 25.43 46.51 70.20 53.81 H-CAST (Park et al., 2025) 57.59 59.02 82.69 93.53 21.81 25.63 28.61 67.20 83.62 47.17 Taxon-SSL 48.40 52.34 65.74 82.96 19.87 31.74 37.11 69.53 82.02 37.31 Taxon-SSL + Text-Attr 49.65 53.43 66.43 83.56 18.81 31.93 37.08 69.76 82.20 37.04 Text-Attr (H-ViT) 55.48 59.05 77.95 89.45 24.02 27.88 32.07 68.27 80.49 46.35 Text-Attr (H-CAST) 63.20 64.91 84.47 93.56 18.58 29.74 32.37 71.79 85.99 44.63 for HRN, 25.63% for H-CAST). Taxon-SSL achieves the best performance (31.9% FPA), highlighting the benefits of structural label propagation under limited per-class supervision. Text-Attr methods perform slightly lower (27.9-30.0% FPA), likely due to restricted textual diversity in this fine-grained biological domain, yet still outperform conventional baselines. In Appendix, we report additional results on CUB-F (Sec. C.1), highly-missing synthetic datasets (Sec. C.2), and ablations on Text-Attr features, training strategies, and architecture design (Sec. E). Analysis 1: Text-Attr Excels with Sparse Labels, Taxon-SSL with Moderate Label Availability. We analyze class-wise performance under imbalanced fine-grained label availability on ImageNetF. To isolate effects, we compare Text-Attr (H-ViT) and Taxon-SSL with identical ViT-small backbones, excluding H-CAST modules. Fig. 7 shows per-class accuracy, sorted by the number of finegrained training labels. Text-Attr (H-ViT) outperforms in label-scarce classes by leveraging textual descriptions as extra supervision, while Taxon-SSL performs better with moderate label availability by propagating consistency across missing levels. We provide additional t-SNE (Maaten & Hinton, 2008) visualization analysis in Appendix D. Analysis 2: What Advantage Does External Semantic Guidance Provide? To assess the effect of text-derived guidance, we compare saliency maps (Chefer et al., 2021) from Taxon-SSL and TextAttr (H-ViT) (Fig. 8). In Row 1, with multiple objects, Taxon-SSL focuses on a human shoulder and misclassifies the image, violating the semantic hierarchy, while Text-Attr consistently attends to the instrument and predicts correctly. In Row 2, when both fail at the fine-grained level, Taxon-SSL 8 Preprint (under review) Taxon-SSL Text-Attr Image Basic Subordinate Fine-grained Basic Subordinate Fine-grained clothing ✗ headwear ✗ saxophone ✓ music instru. ✓wind instru. ✓ saxophone ✓ dog ✓ hound dog ✓ megalith ✗ dog ✓ hound dog ✓standard poodle ✗ Figure 8: Text-Attr improves semantic focus under diverse large-scale data. (1st row) In a multiobject image, Taxon-SSL assigns inconsistent labels ("clothing" at the basic level, "saxophone" at the fine-grained level), while Text-Attr (H-ViT) correctly predicts "musical instrument" by focusing on the relevant object. (2nd row) When both fail at the fine-grained level, Taxon-SSL outputs an unrelated class ("megalith"), whereas Text-Attr (H-ViT) chooses a semantically closer one ("poodle"). This shows that text-derived attributes help the model attend to meaningful regions and maintain semantic plausibility, on large-scale ImageNet-F dataset with diverse categories and sparse labels. Green/Red denote correct/incorrect predictions. Figure 9: Free-grain inference results with consistency-based stopping on ImageNet-F. Left: Examples of consistency-based stopping in Text-Attr (H-CAST). The model stops at the correct subordinate level (Hound, left) or at the basic level (Bird, right) when deeper predictions become inconsistent and incorrect, leading to more reliable results. Right: Consistency-based stopping for free-grain inference. Predictions are halted when finer-level outputs conflict with preceding coarser-level predictions. On ImageNet-F, Text-Attr (H-CAST) explores deeper levels of the hierarchy with higher correctness, whereas HRN stops earlier and produces fewer fine-level predictions. outputs an unrelated class, whereas Text-Attr chooses a visually similar dog by focusing on curly fur and body shape. These results show that external semantic cues guide attention to meaningful features across label granularities, improving hierarchical consistency, while Taxon-SSL may drift to visually salient but semantically irrelevant regions under sparse or ambiguous supervision. Analysis 3: Free-grain Inference. While our main goal is full-hierarchy prediction under mixedgranularity supervision, free-grain inference is also crucial in practice: a correct coarse label is often preferable to an incorrect fine-grained one (e.g., predicting "dog" instead of a wrong breed). We adopt a simple consistency-based stopping rule: predictions halt whenever the next-level label would violate the taxonomy, ensuring the deepest valid output. As shown in Fig. 9(Right), TextAttr (H-CAST) reaches deeper levels more often and with higher accuracy. Fig. 9(Left) shows examples: stopping at the basic level when "bird" is correct but the subordinate mispredicts, or at the subordinate level when "dog →hound" is correct but the fine-grained label is inconsistent. These results highlight the practical value of free-grain inference and motivate benchmarks that explicitly evaluate this setting. 9 Preprint (under review) 6 Summary We introduce new hierarchical classification under free-grain supervision, where models learn from labels of varying granularity while maintaining taxonomy consistency. To advance this setting, we present a large-scale benchmark and two simple yet effective baselines. 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In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024. 19 12 Preprint (under review) Free-Grained Hierarchical Recognition Supplementary Material CONTENTS A Complete Hierarchy of ImageNet-F 14 B Related Work 19 C More Experimantal Results 20 C.1 Evaluaion on CUB-F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 C.2 Evaluation under Varying and Severe Label Sparsity Conditions . . . . . . . . . . 20 D t-SNE Visualization 23 E Ablation Study 24 E.1 Importance of Text-guided Pseudo Attributes . . . . . . . . . . . . . . . . . . . . 24 E.2 Combining Text-Attr and Taxon-SSL . . . . . . . . . . . . . . . . . . . . . . . . . 24 E.3 Ablation on Hierarchical Supervision in ViT . . . . . . . . . . . . . . . . . . . . . 24 F Implementation Details 26 G Use of Large Language Models (LLMs) 26 13 Preprint (under review) A COMPLETE HIERARCHY OF IMAGENET-F Basic Subordinate Fine-Grained bird passerine bird brambling, indigo bunting, robin, jay, bulbul, water ouzel, house finch, chickadee, junco, magpie, goldfinch parrot macaw, sulphur-crested cockatoo, African grey, lorikeet piciform bird toucan, jacamar seabird king penguin, pelican, albatross anseriform bird drake, red-breasted merganser, black swan, goose coraciiform bird bee eater, hornbill bird of prey kite, great grey owl, vulture, bald eagle gallinaceous bird partridge, prairie chicken, ruffed grouse, peacock, quail, black grouse, ptarmigan wading bird flamingo, American coot, redshank, American egret, little blue heron, white stork, limpkin, spoonbill, red-backed sandpiper, dowitcher, crane, ruddy turnstone, bittern, oystercatcher, black stork, bustard dog spitz dog malamute, Pomeranian, keeshond, Siberian husky, chow, Samoyed pointer dog vizsla, German short-haired pointer spaniel dog Brittany spaniel, clumber, English springer, Sussex spaniel, Irish water spaniel, Welsh springer spaniel, cocker spaniel hound dog basset, bloodhound, Irish wolfhound, Walker hound, redbone, English foxhound, Italian greyhound, Ibizan hound, bluetick, Scottish deerhound, borzoi, Norwegian elkhound, whippet, Weimaraner, Saluki, beagle, Afghan hound, black-andtan coonhound, otterhound terrier dog Boston bull, silky terrier, Lakeland terrier, Yorkshire terrier, Tibetan terrier, American Staffordshire terrier, Irish terrier, Airedale, Norwich terrier, soft-coated wheaten terrier, wire-haired fox terrier, Staffordshire bullterrier, West Highland white terrier, Australian terrier, Dandie Dinmont, Kerry blue terrier, Lhasa, cairn, Sealyham terrier, Bedlington terrier, Scotch terrier, Border terrier, Norfolk terrier corgi dog Pembroke, Cardigan poodle dog miniature poodle, toy poodle, standard poodle setter dog Irish setter, Gordon setter, English setter pinscher dog Doberman, affenpinscher, miniature pinscher 14 Preprint (under review) shepherd dog kelpie, briard, German shepherd, Old English sheepdog, Border collie, Bouvier des Flandres, collie, Rottweiler, komondor, malinois, groenendael, Shetland sheepdog retriever dog curly-coated retriever, Labrador retriever, Chesapeake Bay retriever, flat-coated retriever, golden retriever schnauzer dog standard schnauzer, miniature schnauzer, giant schnauzer Sennenhunde dog Bernese mountain dog, Greater Swiss Mountain dog, Appenzeller, EntleBucher toy dog toy terrier, Blenheim spaniel, Maltese dog, Shih-Tzu, papillon, Pekinese, Chihuahua, Japanese spaniel fish soft-finned fish coho, tench, eel, goldfish shark tiger shark, great white shark, hammerhead spiny-finned fish anemone fish, puffer, lionfish, rock beauty ray stingray, electric ray ganoid fish sturgeon, gar primate ape gibbon, siamang, orangutan, chimpanzee, gorilla monkey titi, langur, colobus, squirrel monkey, baboon, guenon, marmoset, macaque, spider monkey, patas, howler monkey, proboscis monkey, capuchin lemur Madagascar cat, indri snake colubrid snake water snake, garter snake, green snake, night snake, hognose snake, ringneck snake, king snake, thunder snake, vine snake elapid snake sea snake, Indian cobra, green mamba viper diamondback, horned viper, sidewinder boa snake boa constrictor, rock python salamander newt eft, common newt ambystomid salamander spotted salamander, axolotl insect beetle dung beetle, weevil, leaf beetle, tiger beetle, ladybug, rhinoceros beetle, longhorned beetle, ground beetle orthopterous insect cricket, grasshopper dictyopterous insect cockroach, mantis hymenopterous insect bee, ant butterflyinsect cabbage butterfly, lycaenid, monarch, admiral, sulphur butterfly, ringlet odonate insect dragonfly, damselfly homopterous insect cicada, leafhopper furniture table desk, dining table baby bed cradle, crib, bassinet seat rocking chair, barber chair, park bench, throne, folding chair, toilet seat, studio couch 15 Preprint (under review) lamp table lamp cabinet china cabinet, medicine chest musical instrument wind instrument ocarina, flute, panpipe, oboe, cornet, sax, harmonica, bassoon, French horn, trombone stringed instrument banjo, harp, violin, cello, acoustic guitar, electric guitar percussion instrument steel drum, gong, marimba, drum, chime, maraca keyboard instrument upright, grand piano, accordion, organ scientific instrument laboratory glassware Petri dish magnifier loupe, radio telescope sports equipment ball golf ball, baseball, basketball, croquet ball gymnastic apparatus parallel bars, balance beam, horizontal bar weight barbell, dumbbell electronic equipment telephone dial telephone, pay-phone, cellular telephone computer peripheral printer, joystick, computer keyboard, mouse audio device tape player, cassette player, CD player, iPod network device modem display device monitor, screen clothing bottoms (skirts) hoopskirt, sarong, miniskirt, overskirt tops (sweaters) sweatshirt, cardigan outwear trench coat, poncho, fur coat swimwear maillot, bikini, swimming trunks face & headwear wig, sombrero, mortarboard, bonnet, mask, cowboy hat, bearskin nightwear pajama protective wear apron, knee pad, lab coat dresses & Gowns gown underwear brassiere footwear sock, Christmas stocking neckwear bow tie, bolo tie, Windsor tie traditional & formal Wear abaya, kimono, vestment, academic gown wraps & shawls stole, feather boa container reservoir water tower, rain barrel bag mailbag, plastic bag, backpack, purse jug water jug, whiskey jug vessel mortar, pitcher, tub, ladle, bucket, coffee mug bottle wine bottle, beer bottle, pop bottle, water bottle, pill bottle basket hamper, shopping basket box mailbox, carton, pencil box, chest, crate 16 Preprint (under review) glass goblet, beer glass shaker saltshaker, cocktail shaker cooking utensil pan frying pan, wok cooker Crock Pot pot teapot, caldron, coffeepot structure monument brass, megalith, triumphal arch, obelisk, totem pole religious building church, mosque, boathouse, monastery, stupa housing yurt, cliff dwelling, mobile home public building planetarium, library movable structure sliding door, turnstile supporting structure plate rack, honeycomb, pedestal fence stone wall, picket fence, chainlink fence, worm fence bridge steel arch bridge, viaduct, suspension bridge residential structure palace agricultural structure greenhouse, barn, apiary commercial stucture toyshop, restaurant, cinema, confectionery, bookshop, grocery store, tobacco shop, bakery, butcher shop, barbershop, shoe shop barrier grille, bannister, breakwater, dam institutional structure prison tool hand tool hammer, plunger, screwdriver garden tool lawn mower, shovel cutter cleaver, plane, letter opener, hatchet power tool chain saw opener corkscrew, can opener craft sailing vessel trimaran, schooner, catamaran boat fireboat, canoe, yawl, gondola, speedboat, lifeboat ship wreck, pirate, container ship, liner warship aircraft carrier, submarine aircraft airliner, warplane, airship, balloon vehicle bicycle bicycle-built-for-two, mountain bike bus minibus, school bus, trolleybus car ambulance, beach wagon, cab, convertible, jeep, limousine, Model T, racer, sports car truck fire engine, garbage truck, pickup, tow truck, trailer truck van minivan, moving van, police van locomotive electric locomotive, steam locomotive military vehicle half track 17 Preprint (under review) self-propelled vehicle forklift, recreational vehicle, snowmobile, tank, tractor, golfcart, snowplow, go-kart, moped, streetcar, amphibious vehicle handcart barrow, shopping cart sled bobsled, dogsled train bullet train wagon horse cart, jinrikisha, oxcart wheeled vehicle freight car, motor scooter, tricycle, unicycle weapon gun rifle, assault rifle, revolver, cannon ranged weapon missile, projectile Table 4: Complete hierarchy tree for our proposed ImageNet-F dataset. 18 Preprint (under review) B RELATED WORK Hierarchical classification has been studied with varying objectives. Most focus on leaf-node prediction, using the full taxonomy during training but predicting only fine-grained labels (Karthik et al., 2021; Zhang et al., 2022; Zeng et al., 2022; Garg et al., 2022b). Evaluation in these works typically relies on top-1 accuracy or mistake severity at the leaf level, making them compatible with large-scale datasets like ImageNet (Russakovsky et al., 2015) and tieredImageNet (Ren et al., 2018)-even with inconsistent or deep hierarchies. However, models restricted to fine-grained outputs often fail in real-world scenarios where visual details are missing, as they cannot fall back to coarser labels and thus provide no meaningful information. To address this, full taxonomy prediction has been explored, aiming to produce labels across all levels while maintaining hierarchical consistency (Chang et al., 2021; Wang et al., 2023; Jiang et al., 2024; Park et al., 2025). However, these methods are typically developed and evaluated on small, fully labeled datasets like CUB (Welinder et al., 2010) and Aircraft (Maji et al., 2013), which lack the scale, diversity, and label sparsity of real-world settings. The iNaturalist dataset (Van Horn et al., 2021) offers a deeper taxonomy, but also remains restricted to the biology, limiting its suitability for general-purpose evaluation. HRN (Chen et al., 2022) partially handles incomplete labels by randomly converting fine-grained labels to parent categories, overlooking the structured ambiguity in real data. Similarly, (Kim et al., 2023) supports mixed labels but treats them flatly, ignoring hierarchical relationships. Both also rely on small datasets such as CUB and Aircraft. Our work fills this gap by enabling full taxonomy prediction under realistic supervision on large-scale data. Imbalanced classification has been extensively studied (Liu et al., 2019; Ren et al., 2020; Wang et al., 2021; Park et al., 2021; Tian et al., 2022; Park et al., 2022; Ha et al., 2023; Zhao et al., 2024), mostly focusing on intra-level imbalance at a single fine-grained level. In contrast, we address intraand inter-level imbalance in a hierarchical setting, where classes are balanced but label granularity varies across them. DeepRTC (Wu et al., 2020) considers taxonomy, but aims to improve inference reliability via early stopping, rather than predicting the full taxonomy. Semi-supervised learning typically combines labeled and unlabeled data at a single fine-grained level (Tarvainen & Valpola, 2017; Berthelot et al., 2019; Sohn et al., 2020). Recent work incorporates coarse labels (Garg et al., 2022a; Wu et al., 2023), but still targets fine-grained accuracy. In contrast, our setting demands consistent prediction across the full taxonomy with heterogeneous supervision, making existing methods not directly applicable. Weakly-supervised classification typically aims to predict fine-grained labels when only coarse labels are available during training (Robinson et al., 2020; Grcic et al., 2024). These methods assume fully observed labels at a coarse level and focus on improving predictions at a fine-grained level. In contrast, our setting requires handling multi-granularity labels and inferring the full taxonomy. Foundation models for zero-shot classification, such as vision-language models (e.g., CLIP (Radford et al., 2021)) and large language models (e.g., GPT-4 (Achiam et al., 2023)), have gained popularity for leveraging label-driven prompts at inference-without training (Pratt et al., 2023; Liu et al., 2024; Zheng et al., 2024; Saha et al., 2024). These methods aim to improve flat-level classification by matching images to text. In contrast, we train a hierarchical classifier that learns shared visual patterns across levels from images, when labels are partially missing. Our model requires no textual input at inference, making it efficient. See a full task comparison in Table 2. 19 Preprint (under review) C MORE EXPERIMANTAL RESULTS C.1 EVALUAION ON CUB-F On the small-scale, single-domain dataset CUB-F (Table 5), Taxon-SSL achieves the best performance (63.96% FPA), showing the advantage of structured label propagation when per-class samples are scarce. Text-Attr methods perform moderately well (53.99-57.59% FPA) but are less effective here, as the bird-only domain limits textual diversity and reduces the benefit of language-based supervision. Still, they clearly outperform conventional hierarchical baselines (44.30% for HRN, 45.10% for H-CAST), underscoring the overall effectiveness of our approach. Unlike the trend on large-scale, diverse datasets such as ImageNet-F, where Text-Attr provides richer cues and stronger gains, these results confirm that there is no single recipe for free-grain learning: performance is tightly coupled with dataset characteristics, making the problem inherently challenging. Table 5: Taxon-SSL shows strong effectiveness on the small-scale dataset CUB-F, where label propagation provides reliable supervision despite limited data. Text-Attr methods are assumed to offer limited benefit due to the restricted textual diversity of this bird-only dataset. CUB-F (13-38-200) FPA (↑) Species (↑) family (↑) Order (↑) TICE (↓) HRN (Chen et al., 2022) 44.30 46.72 81.20 96.36 27.15 H-CAST (Park et al., 2025) 45.10 47.52 87.78 97.50 25.89 Taxon-SSL 63.96 65.50 92.84 98.40 7.39 Taxon-SSL + Text-Attr 63.05 64.86 92.54 98.38 7.61 Text-Attr (H-ViT) 57.59 59.10 91.60 98.05 10.72 Text-Attr (H-CAST) 53.99 55.58 91.72 98.41 18.95 C.2 EVALUATION UNDER VARYING AND SEVERE LABEL SPARSITY CONDITIONS To evaluate model performance under diverse and more challenging free-grain conditions, we experiment with various label availability ratios by randomly removing fine-grained labels-e.g., (100%- 60%-30%), (100%-50%-10%), and (100%-20%-10%)-which represent the available proportions of basic, subordinate, and fine-grained labels, respectively. Each experiment is repeated with three different random seeds, and we report the average performance. The variance across runs was minor (0.1-1.8). Consistent with our main results, these experiments (Table 6 & 7 & 8) also show that there is no single method that performs best across all settings. Instead, the most effective method varies depending on the dataset and the specific ratio of available labels, highlighting the importance of adaptable free-grain learning strategies. For consistency, we refer to the three levels in CUB-Rand (order-family-species) and Aircraft-Rand (maker-family-model) as basic, subordinate, and fine-grained levels. We summarize the key findings below: (1) Conventional hierarchical classification methods struggle under the free-grain setting, where label supervision is sparse and uneven across levels. For example, when labels are highly missing (e.g., only 10% available at the fine-grained level), HRN (Chen et al., 2022) and H-CAST (Park et al., 2025) suffer more than a 50% drop in accuracy across all levels compared to the fully labeled (100%-100%-100%) setting on CUB-Rand (Fig. 6 & Table 8). This highlights the difficulty of the free-grain setting and the need for methods that can robustly handle incomplete supervision at multiple semantic levels. (2) The performance of different methods varies with the amount of available supervision per class: Text-Attr methods perform better when more labeled samples are available, while TaxonSSL is more effective under extreme label sparsity. For example, in Table 6, the average number of available fine-grained labels per class is approximately 9 for CUB-Rand and about 20 for Aircraft-Rand. Consistent with this difference, Taxon-SSL outperforms other methods on CUBRand, whereas Text-Attr (H-CAST) performs best on Aircraft-Rand. This trend persists across settings. In the most sparse setting, CUB-Rand (100-20-10, Table 8), where only about 3 fine-grained labels are available per class, Taxon-SSL shows a clear advantage. We attribute this to how supervision is utilized. Text-Attr relies on available labels and indirect semantic guidance via text features. 20 Preprint (under review) In contrast, Taxon-SSL actively leverages unlabeled data through pseudo-labeling and strong augmentations, making it more effective when labeled examples are extremely limited. (3) Sometimes, Taxon-SSL's high fine-grained accuracy comes at the cost of lower accuracy at higher levels in the taxonomy. For example, in Table 7, Taxon-SSL achieves the highest finegrained accuracy (65.01%), but its subordinate and basic-level accuracies (85.53% and 92.81%) are lower than those of Text-Attr (H-CAST), which achieves 86.30% and 94.17%, respectively. This highlights a key challenge in free-grain learning: improving accuracy across all levels simultaneously is non-trivial, and optimizing for fine-grained performance alone may degrade consistency at coarser levels. Table 6: No single method performs best across all conditions-performance depends strongly on the amount of available supervision per class. Text-Attr methods tend to perform better when more labeled samples are available, while Taxon-SSL is more effective under extreme label sparsity. For example, Taxon-SSL performs best on CUB-Rand with around 9 fine-grained labels per class, while Text-Attr (H-CAST) performs best on Aircraft-Rand with around 20, reflecting the impact of supervision density. These results highlight that method effectiveness is highly sensitive to label sparsity, emphasizing the need for adaptable approaches in free-grain learning. Label Ratio CUB-Rand (100%-60%-30%) Aircraft-Rand (100%-60%-30%) FPA(↑) spec.(↑) fam.(↑) order(↑) TICE(↓) FPA(↑) maker(↑) fam.(↑) model(↑) TICE(↓) HRN (Chen et al., 2022) 57.87 62.73 85.53 96.45 13.77 57.33 64.42 76.95 86.38 23.30 H-CAST (Park et al., 2025) 61.88 67.36 90.05 94.32 13.04 64.67 68.88 85.58 91.43 13.76 Taxon-SSL 74.82 76.92 93.38 98.33 5.06 70.33 72.22 87.06 93.50 7.18 Taxon-SSL + Text-Attr 74.90 76.95 93.41 98.38 4.91 69.89 72.24 86.92 93.29 7.77 Text-Attr (H-ViT) 67.89 72.48 90.63 95.37 10.39 64.15 68.92 85.88 89.87 15.80 Text-Attr (H-CAST) 69.65 71.31 92.88 98.48 8.35 71.43 73.56 89.66 95.31 9.71 Table 7: Maintaining accuracy across all hierarchy levels remains more challenging under sparse supervision. For example, in 100%-50%-10% case, Taxon-SSL achieves the highest finegrained accuracy (65.01%), but its subordinate and basic-level accuracies (85.53%, 92.81%) are lower than those of Text-Attr (H-CAST) (86.30%, 94.17%), which better preserves consistency across levels. This result illustrates the inherent difficulty of improving accuracy across all levels simultaneously, as objectives at different levels can be conflicting. Label Ratio Aircraft-Rand (100%-50%-10%) Aircraft-Rand (100%-20%-10%) FPA(↑) maker(↑) fam.(↑) model(↑) TICE(↓) FPA(↑) maker(↑) fam.(↑) model(↑) TICE(↓) HRN (Chen et al., 2022) 40.35 47.85 70.76 85.68 37.56 32.06 46.73 55.43 85.58 48.43 H-CAST (Park et al., 2025) 47.57 51.93 78.31 87.11 28.42 40.33 45.44 67.28 84.12 35.61 Taxon-SSL 62.61 65.01 85.53 92.81 10.22 58.73 61.10 80.90 92.24 11.77 Taxon-SSL + Text-Attr 62.95 65.49 86.01 92.64 10.25 58.55 60.88 80.97 92.04 11.89 Text-Attr (H-ViT) 47.83 52.25 81.13 87.82 30.57 38.73 43.89 66.13 84.81 38.69 Text-Attr (H-CAST) 53.31 55.32 86.30 94.17 24.43 48.85 51.37 77.11 93.01 27.25 21 Preprint (under review) Table 8: Taxon-SSL is more robust under extreme label sparsity, while other methods degrade significantly. In CUB-Rand (100%-20%-10%), where each class has only 3 fine-grained and 3 subordinate labels, Taxon-SSL achieves the best performance, while other methods struggle. HRN and H-CAST suffer over 50% drop in fine-grained accuracy compared to the fully-supervised (100%- 100%-100%) setting. Text-Attr methods perform more robustly (10%+ higher than HRN/H-CAST), but still fall short under such sparse supervision. We attribute this to how each method leverages supervision: Text-Attr relies on available labels and semantic guidance from text features, while Taxon-SSL benefits more from unlabeled data via pseudo-labeling and augmentations, making it more effective under severe label sparsity. Label Ratio CUB-Rand (100%-50%-10%) CUB-Rand (100%-20%-10%) FPA(↑) spec.(↑) fam.(↑) order(↑) TICE(↓) FPA(↑) spec.(↑) fam.(↑) order(↑) TICE(↓) HRN (Chen et al., 2022) 40.23 43.70 82.75 95.94 22.34 33.53 41.18 72.56 95.79 30.50 H-CAST (Park et al., 2025) 39.03 43.41 85.74 93.23 24.60 32.97 38.66 76.89 92.50 29.43 Taxon-SSL 62.40 64.14 92.33 98.26 6.01 59.18 61.44 89.79 98.20 7.65 Taxon-SSL + Text-Attr 62.52 64.87 87.94 94.45 8.98 57.98 60.59 89.42 98.12 8.39 Text-Attr (H-ViT) 47.42 50.74 88.22 94.67 18.09 42.46 46.99 80.92 94.43 20.27 Text-Attr (H-CAST) 44.63 45.89 91.06 98.19 22.72 40.41 42.76 84.24 97.97 24.05 22 Preprint (under review) D T-SNE VISUALIZATION We visualize ImageNet-F embeddings of Text-Attr (H-CAST) and Taxon-SSL using t-SNE (Maaten & Hinton, 2008) to assess whether the learned representations capture semantic and hierarchical structure. Each point denotes an image embedding, colored by its basic-level class (20 categories), with brightness variations indicating fine-grained subclasses (505 total). Both Text-Attr (H-CAST) and Taxon-SSL produce well-separated clusters consistent with the basiclevel taxonomy, showing that coarse groupings are reliably captured. The key difference lies within coarse categories: Text-Attr (H-CAST) reveals more distinct fine-grained subclusters (e.g., breeds within dog, species within bird), whereas Taxon-SSL yields tighter coarse clusters with less apparent fine-level separation. This contrast reflects their supervision signals. Text-Attr leverages diverse textual cues (attributes, parts, appearance terms), which promote discriminative, attribute-aligned features and sharpen within-class distinctions. Taxon-SSL, by propagating labels along the taxonomy and enforcing consistency under mixed-granularity supervision, regularizes embeddings within each coarse class and reduces intra-class variance-emphasizing coarse alignment over fine-level separability. (a) Text-Attr (H-CAST) (b) Taxon-SSL Figure 10: t-sne Visualization on ImageNet-F. Both methods separate coarse-level taxonomy well, but Text-Attr (H-CAST) yields clearer fine-grained subclusters (e.g., distinct groups within dog and bird) with more compact grouping, whereas Taxon-SSL shows some overlap of embeddings near cluster boundaries. This is likely due to ImageNet-F 's diverse large-scale categories, where text supervision provides rich attribute cues that sharpen fine-level distinctions. 23 Preprint (under review) E ABLATION STUDY E.1 IMPORTANCE OF TEXT-GUIDED PSEUDO ATTRIBUTES Text-guided Pseudo Attributes jointly optimizes hierarchical label supervision (Lhier) and textguided pseudo attributes (Ltext) to learn semantically rich features: L = Lhier + αLtext Fig. 11 quantifies Ltext's impact by varying its weight α on CUB-Rand. Ablating Ltext (α = 0) causes a 5% absolute decline in both fine-grained accuracy and FPA compared to the optimal configuration (α = 0). This gap underscores two key roles of text guidance: (1) it injects complementary visual semantics absent in class labels alone, and (2) it enforces attribute consistency across hierarchy levels. The performance recovery at (α = 1) confirms that textual pseudo-attributes mitigate annotation sparsity while preserving taxonomic coherence. Figure 11: Tuning α balances accuracy and taxonomic consistency. At α = 1 (optimal), Text-Attr (H-ViT) achieves peak fine-grained accuracy (blue) while maintaining hierarchical consistency (orange). Ablating Ltext (α = 0) causes a 5% accuracy drop and increased inconsistency, as class embeddings lose text-guided attribute alignment. Higher α > 1.0 over-regularizes features, marginally degrading both metrics. This trade-off underscores the need to weight text supervision to resolve sparse annotations without distorting the hierarchy. E.2 COMBINING TEXT-ATTR AND TAXON-SSL We compare different training schedules for combining Text-Attr and Taxon-SSL on CUB-F. In the joint setting, both objectives are optimized simultaneously for 100 epochs. In the two-stage setting, we first train with one objective for 50 epochs and then add the other for the remaining 50 epochs, considering both orders: (1) Taxon-SSL →Text-Attr, and (2) Text-Attr →Taxon-SSL. Table 9 show that starting with Text-Attr and then adding Taxon-SSL yields slightly higher full-path accuracy, likely because textual supervision promotes diverse feature learning before label propagation. In contrast, beginning with Taxon-SSL provides no advantage, and both two-stage variants perform similarly to joint training overall. Interestingly, joint training achieves higher consistency as measured by TICE. Given its simplicity and competitive performance, we adopt the joint strategy as our default. Table 9: Comparison of joint vs. two-stage training schedules for Text-Attr and Taxon-SSL on CUB-F. While two-stage training (Text-Attr →Taxon-SSL) yields slightly higher accuracy, joint learning is simpler and provides better consistency (TICE). CUB-F (13-38-200) FPA (↑) Species (↑) family (↑) Order (↑) TICE (↓) Taxon-SSL + Text-Attr (100 epochs) 63.04 64.86 92.54 98.37 7.61 Taxon-SSL (50 epochs) →+Text-Attr (50 epochs) 62.84 64.42 92.47 98.20 8.19 Text-Attr (50 epochs) →+Taxon-SSL (50 epochs) 63.63 65.34 92.56 98.27 8.06 E.3 ABLATION ON HIERARCHICAL SUPERVISION IN VIT We further examine the architectural design choice of where to inject hierarchical supervision in the Vision Transformer (ViT) in Table 10. On CUB-F, we map the three taxonomy levels (Or24 Preprint (under review) der-Family-Species) to different layers and compare multiple configurations: (6th, 9th, 12th), (8th, 10th, 12th), and (10th, 11th, 12th). Among these, supervision at the 8th, 10th, and 12th layers yields the best performance. We interpret this as a balance between early and late representation learning: assigning hierarchy too early (e.g., 6-9-12) forces the model to align coarse categories before sufficient visual features are developed, while placing all supervision too late (e.g., 10-11-12) limits the model's capacity to gradually refine class granularity. The 8-10-12 configuration provides an appropriate middle ground, where lowerlevel categories benefit from moderately abstract features, and finer distinctions are introduced after the backbone has matured. Table 10: Performance comparison of different layer assignments for hierarchical supervision in ViT on CUB-F. The 8th-10th-12th configuration achieves the best results, balancing early and late feature abstraction. CUB-F (13-38-200) FPA (↑) Species (↑) family (↑) Order (↑) TICE (↓) 6-9-12th layer 54.80 58.16 88.97 95.01 16.79 8-10-12th layer 57.59 59.10 91.60 98.05 10.72 10-11-12th layer 56.40 58.56 90.80 97.08 13.48 25 Preprint (under review) F IMPLEMENTATION DETAILS For ViT (Dosovitskiy et al., 2020) models, we use ViT-Small for Text-Attr (H-ViT) and Taxon-SSL and H-CAST-Small (Park et al., 2025) for Text-Attr (H-CAST) to match parameter sizes. For Text-Attr (H-ViT), we insert fully-connected layers to the class token at the 8th, 10th, and 12th layers for basic, subordinate, and fine-grained supervision. The 12th-layer patch features are projected to match the text embedding dimension via an FC layer. For Text-Attr (H-CAST), hierarchical supervision is applied to the last three blocks, following (Park et al., 2025). Due to low dimensionality in the final block, we align text features with the features of the second block. For Text-Attr methods, CLIP-ViT-B/32 is used to extract text embeddings, which remain frozen during training. In Taxon-SSL, we apply a shared MLP to the class token from the final (12th) layer, followed by three separate linear classifiers for basic, subordinate, and fine-grained supervision. When combined with Text-Attr, we additionally project the class token through a linear layer and align it with the corresponding text feature. For hierarchical classification baselines, HRN (Chen et al., 2022) and H-CAST (Park et al., 2025), we follow their original training protocols and retrain them under our free-grain setting. We extend HRN to handle missing labels at two levels instead of one. For H-CAST, we provide supervision using the available labels at each corresponding level. Full hyperparameter configurations are provided in Table 11. We train all models for 100 epochs, except for ImageNet-F, which are trained for 200 epochs due to the larger scale. All experiments were conducted on an NVIDIA A40 GPU with 48GB memory. We used a single GPU for all experiments, except for ImageNet-F, which was trained using 4 GPUs. Table 11: Hyperparameters for training Text-Attr (H-ViT), Text-Attr (H-CAST), and TaxonSSL. We follow the training setup of H-CAST (Park et al., 2025) for Text-Attr methods (Text-Attr (H-ViT) and Text-Attr (H-CAST)), and adopt the settings of CHMatch (Wu et al., 2023) for TaxonSSL. Parameter Text-Attr (H-ViT) Text-Attr (H-CAST) Taxon-SSL batch size 256 256 128 crop size 224 224 224 learning rate 5e -4 5e -4 1e -3 weight decay 0.05 0.05 0.05 momentum 0.9 0.9 0.9 warmup epochs 5 5 0 warmup learning rate 1e-6 1e-6 N/A optimizer Adam Adam SGD learning rate policy Cosine decay Cosine decay Cosine decay α (weight for Ltext) 1 1 1 (for +Text-Attr) G USE OF LARGE LANGUAGE MODELS (LLMS) Large Language Models (LLMs) were used in a limited manner, primarily to review the constructed hierarchy and to assist with minor tasks such as translation and typo correction. 26
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2510.14741
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DEXTER: Diffusion-Guided EXplanations with
TExtual Reasoning for Vision Models
Simone Carnemolla1∗Matteo Pennisi1∗Sarinda Samarasinghe2
Giovanni Bellitto1
Simone Palazzo1
Daniela Giordano1
Mubarak Shah2
Concetto Spampinato1
1University of Catania
2University of Central Florida
simone.carnemolla@phd.unict.it
matteo.pennisi@unict.it
Abstract
Understanding and explaining the behavior of machine learning models is essen-
tial for building transparent and trustworthy AI systems. We introduce DEX-
TER, a data-free framework that employs diffusion models and large language
models to generate global, textual explanations of visual classifiers. DEXTER
operates by optimizing text prompts to synthesize class-conditional images that
strongly activate a target classifier. These synthetic samples are then used to elicit
detailed natural language reports that describe class-specific decision patterns
and biases. Unlike prior work, DEXTER enables natural language explanation
about a classifier’s decision process without access to training data or ground-
truth labels. We demonstrate DEXTER’s flexibility across three tasks—activation
maximization, slice discovery and debiasing, and bias explanation—each illus-
trating its ability to uncover the internal mechanisms of visual classifiers. Quan-
titative and qualitative evaluations, including a user study, show that DEXTER
produces accurate, interpretable outputs. Experiments on ImageNet, Waterbirds,
CelebA, and FairFaces confirm that DEXTER outperforms existing approaches
in global model explanation and class-level bias reporting. Code is available at
https://github.com/perceivelab/dexter.
1
Introduction
How can we systematically uncover and explain a deep visual classifier’s decision-making process in
a way that is both comprehensive and human-interpretable?
This question is crucial as AI systems are increasingly deployed in high-stakes applications, where
interpretability and trust are as important as accuracy. However, the lack of transparency in model
reasoning, often exacerbated by the reliance on spurious correlations—irrelevant features that dis-
proportionately influence predictions— undermines confidence in these systems, making decisions
difficult to justify. For instance, ImageNet-trained classifiers have been shown to favor background
textures or lighting conditions over intrinsic object properties [41, 9]. Addressing these issues requires
explainability techniques that extend beyond local attribution, offering a global perspective on a
model’s reasoning patterns, biases, and learned representations.
Existing methods for model explainability, such as GradCAM [39] and Integrated Gradients [43]
provide local explanations and require data. These methods focus on analyzing individual predictions
by attributing importance to specific pixels or regions in an image. While these methods are extremely
useful for highlighting areas of interest, they do not offer a global understanding of model behavior. In
∗Equal contribution.
39th Conference on Neural Information Processing Systems (NeurIPS 2025).
arXiv:2510.14741v1 [cs.CV] 16 Oct 2025
contrast, activation maximization techniques [26–28] have been instrumental in globally visualizing
the features learned by neural network, but the generated images are often abstract and challenging to
interpret. While existing methods offer useful insights into model behavior and feature representations,
they often fall short of capturing the global reasoning patterns and biases behind predictions. In
particular, visual explanations can be hard to interpret and may lack the ability to convey high-level
reasoning or reveal subtle spurious correlations—issues that are often better addressed through
complementary textual explanations [18, 6, 16, 3].
Text-based explanations offer an accessible complement to visual methods [11], but they often lack
global perspective or clarity. Recent work [16, 2, 10], including Natural Language Explanation (NLE)
methods [30, 37, 15], combines visual and textual cues to reveal model biases, but typically relies on
labeled data, annotations, or pretrained vision-language mappings [2].
We propose DEXTER, a framework that generates human-interpretable textual explanations, un-
covering the global reasoning patterns and biases of visual classifiers. DEXTER operates in a fully
data-free setting by leveraging the generative capabilities of diffusion models and the reasoning
power of large language models (LLMs). At its core, DEXTER optimizes soft prompts, which
are mapped to discrete hard prompts, in order to guide a diffusion-based image generation process.
This ensures that the generated images are aligned with the outputs of a target classifier, capturing
the features and concepts prioritized by the model. The generated images are then analyzed by an
LLM, which reasons across them to provide textual explanations of the classifier’s decision-making
process. By combining image generation and textual reasoning, DEXTER not only overcomes the
interpretability challenges of visual explanations but also provides a global understanding of model
behavior, including identifying spurious correlations and biases, without requiring access to any
training data or ground truth labels.
Thus, the core main contributions of DEXTER are:
• Global explanations: A high-level understanding of model decisions, identifying key
features, biases, and patterns beyond local attribution.
• Data-free approach: DEXTER requires only the trained model for explanations, unlike
existing methods, that employ training or ground-truth data.
• Bias identification and explanation: Uncovering and describing spurious correlations to
support model debiasing.
• Natural language reasoning: LLM-generated textual explanations that enhance inter-
pretability over purely visual methods.
We evaluate DEXTER across three tasks: (1) activation maximization to reveal model-prioritized
features, (2) slice discovery to detect underperforming subpopulations, and (3) bias explanation to
identify spurious correlations. Experiments use SalientImageNet [41], Waterbirds [36], CelebA [21],
and FairFaces [14]—datasets widely used in fairness and interpretability research. Across tasks,
DEXTER demonstrates strong performance, offering meaningful insights into model behavior.
2
Related work
Activation Maximization (AM) interprets neural networks by generating inputs that maximize
neuron activations [7]. Early methods often produced unrealistic, uninterpretable images [40, 25],
later improved through regularization [23] and generative priors [26]. DiffExplainer [31] introduced
diffusion-based image generation guided by soft prompts, which improve realism but lack semantic
transparency. This work points toward token-wise optimization, but this direction remains underex-
plored. In contrast, DEXTER builds on this line by replacing soft prompts with discrete (hard) tokens,
which are interpretable and enable both visual and textual global explanations. Although once central
to model interpretability, AM has seen limited progress in recent years, partly because the resulting
images—while optimized for neuron activation—are often difficult to interpret semantically, making
it challenging to draw clear, causal insights about model behavior. The field has largely shifted toward
attribution methods—e.g., GradCAM [39], Integrated Gradients [43], which are computationally
efficient and provide intuitive saliency maps, though they are inherently local and data-dependent.
Other efforts in feature visualization [52] offer insights into internal representations but require
training data and remain focused on visual output. DEXTER revives AM by leveraging modern
2
diffusion models and discrete prompt optimization to produce class-level, multimodal explanations in
a fully data-free setting.
Explanation of visual classifiers. Textual and Multimodal Explanations seek to complement vi-
sual saliency with natural language justifications. Approaches like Multimodal Explanations [29],
LIMEx [44], and post-hoc counterfactuals [48] align vision and language to explain model decisions.
However, these methods often rely on ground-truth labels or annotated datasets and are limited to
instance-level explanations. In contrast, DEXTER provides global, class-wise explanations without
supervision or access to data.
Natural Language Explanation (NLE) methods aim to generate human-readable justifications,
typically using VQA-style benchmarks [30, 37, 15]. Recent approaches integrate vision and language
into unified architectures [38, 37], but remain supervised and local. DEXTER differs by producing
global textual reports through unsupervised classifier probing—without labels, data, or task-specific
fine-tuning.
Slice Discovery identifies dataset subpopulations where a model underperforms, offering a tar-
geted way to reveal and explain systematic failures in classifier behavior. Traditional methods
analyze embeddings, gradients, or misclassifications [1, 24, 42, 51], while recent approaches leverage
CLIP’s joint text-visual embeddings [8, 12, 50] or use LLMs to generate captions and extract key-
words [46, 16]. Bias-to-Text (B2T)[16] extracts pseudo-bias labels from misclassified image captions,
and LADDER[10] generates bias hypotheses from low-confidence predictions, clustering samples
via LLM-derived pseudo-attributes. Unlike these data-dependent methods, DEXTER discovers and
explains biases in a fully data-free manner, relying only on the classifier’s internal behavior.
3
Method
DEXTER’s framework, shown in Fig. 1, integrates three key components: a text pipeline for
optimizing prompts, a vision pipeline for the image generation process, and a reasoning module
using a vision-language model (VLM). DEXTER begins by optimizing a soft prompt to condition a
BERT model [5] to fill in masked tokens in a predefined sentence. The resulting prompt guides the
stable diffusion process to generate images that maximize the activation of a set of target neurons (e.g.
classification heads) in a given visual classifier. The generated images are then analyzed by the VLM,
which reasons across multiple images to provide coherent, human-readable textual explanations of
the model’s decision-making process.
3.1
Text pipeline
The text pipeline has the goal of optimizing a textual prompt to suitably condition the diffusion
model. We pose prompt generation as a masked language modeling task, and employ a pretrained
and frozen BERT model for this purpose. The structure of the textual prompt to be produced is fully
customizable, and can be controlled by combining portions of fixed text with a set of mask tokens.
For the sake of clarity and without loss of generality, we will assume that the textual prompt has the
structure of a sequence t = [tfixed, m1, m2, . . . , mN], where tfixed is the portion of fixed text and all
mi are set to BERT’s [MASK]token. Let temb be the embedding of t, including positional encoding. In
order to alter BERT’s behavior, which would naturally tend to replace mi with the most likely tokens
based on its pretraining, we also prepend to the input sequence a learnable soft prompt p ∈RP ×d,
consisting of a sequence of P vectors, with d being the dimensionality of BERT’s embedding
space [17]. The full input sequence to BERT is thus [p, temb]. We read out the logits corresponding
to the masked tokens, i.e., [l1, . . . , lN], where each li ∈RV and V is BERT’s vocabulary size. Each
logit vector li is mapped to a differentiable one-hot vector oi ∈{0, 1}V , P oi,j = 1, through a
Gumbel-softmax [22, 13] (with temperature τ = 1), from which the corresponding predicted token
ˆti ∈{1, . . . , V } is retrieved. The resulting text prompt can be recovered as
tfixed, ˆt1, . . . , ˆtN
.
At this point, a practical problem arises, in that standard implementations of diffusion models
(e.g., Stable Diffusion [35]) CLIP’s text encoder [32] is employed to embed textual prompts into a
conditioning vector. Unfortunately, BERT’s and CLIP’s vocabulary overlap only partially. To address
this issue, we employ a translation matrix M ∈{0, 1}V ×W to map each one-hot vector oi to its
corresponding representation in CLIP’s vocabulary, of size W. In M, each row contains a single 1,
indicating the index of the corresponding token in the CLIP vocabulary. Thus, given an original one-
hot vector oi provided by BERT, we can translate it into its CLIP equivalent through o(C)
i
= oiM,
indexing a token ˆt(C)
i
in CLIP’s vocabulary. For unassigned BERT tokens, the corresponding rows in
3
Figure 1: DEXTER investigates classifier biases by optimizing a learnable soft prompt to generate
text prompts. These text prompts condition a diffusion model to generate images that maximize the
activation of the target class in the vision classifier. Images that correctly activate the target class
are stored and later captioned for Bias Reasoning. A VLM reasons using these captions to produce
human-understandable textual explanations of the model’s decisions and potential biases. More
details and clarifications about the pipeline can be found in the Appendices A and B.
M are entirely zero. As a result of this design, the model learns to avoid predicting BERT tokens that
do not have a valid mapping in CLIP, as they correspond to a zero indexing vector o(C)
i
, which would
provide a meaningless representation when multiplied by the CLIP embedding look-up table.
The textual prompt used to condition the diffusion model is thus
h
tfixed, ˆt(C)
1 , . . . , ˆt(C)
N
i
. Thanks to
the Gumbel-Softmax activation and the linear mapping between vocabularies, the whole process is
fully differentiable, allowing us to optimize the soft prompt p, which is the only learnable parameter,
through classic backpropagation.
3.2
Vision pipeline
The goal of the vision pipeline is to synthesize a realistic and interpretable image that maximizes
the activation of a set of neurons of a visual classifier. In the following, we refer to “neurons” in
a generic way, regardless of whether they encode features (i.e., neurons in intermediate layers)
or classes (i.e., neurons in the output layer, after applying softmax). Given the predicted textual
prompt
h
tfixed, ˆt(C)
1 , . . . , ˆt(C)
N
i
(with all tokens ensured to belong to CLIP’s vocabulary as explained in
Sect. 3.1), we feed it to CLIP’s text encoder to obtain an embedding vector e. This vector is then used
to condition a pretrained and frozen diffusion model d. In this work, we use Stable Diffusion, due to
its widespread adoption and versatility in generating high-quality images from textual descriptions.
Let f : I →RK be the target frozen visual classifier, pretrained over a set of C classes, providing as
output the responses of a subset of K selected neurons, whose activations we intend to maximize.
Hence, we can obtain the selected activation vector as n = f (d (e)).
The whole vision pipeline is differentiable, enabling the definition of an optimization objective for n,
which directly affects the learnable soft prompt p in the text pipeline.
In order to guide the generation process to describe the behavior of the visual classifier, we introduce
a neuron activation maximization loss Lact that encourages learning a textual prompt and a synthetic
4
image that maximizes the response of the selected n neurons. We define Lact as:
Lact =
K
X
i=1
lact (ni) ,
(1)
where ni is the activation of the i-th element in n, and lact depends on whether ni is a feature neuron
or a class neuron :
lact(ni) =
−ni,
if ni is a feature neuron
−log ni,
if ni is a class neuron
(2)
3.3
Masked pseudo-labels prediction
In our preliminary experiments, we observed that, when using the activation maximization objective
only, the gradient propagated to the learnable soft prompt p in the text pipeline was too small, slowing
down or even preventing convergence. To address this issue, we introduce, in the text pipeline,
an auxiliary mask prediction task to provide a shorter backpropagation path to p. The design of
the auxiliary task gives us the opportunity to add another explainability feature to our approach:
associating masked tokens with subsets of target neurons, in order to encourage the mapping between
neuron activations and specific portions of the textual prompt.
To facilitate this, we initialize a set of pseudo-labels y1, . . . , yN, one for each mask token position
m1, . . . , mN. Each pseudo-label yi ∈{1, . . . , V } is associated with a reference loss Li, initialized
to +∞, and with the set of reference neurons Ni ⊆{1, . . . , K}. At each optimization step, for each
mask token mi, BERT predicts the logits li, from which we compute the (standard) softmax vector si
and the corresponding predicted token ˆti, following the notation introduced in Sect. 3.1. We define
the aggregated activation loss Lagg,i for the set of associated reference neurons:
Lagg,i =
X
j∈Ni
lact(nj).
(3)
Then, if Lagg,i is smaller than the corresponding reference loss Li, we update both the pseudo-label
yi ←ˆti and the reference loss Li ←Lagg,i. If the pseudo-label yi has been set for mi (and the
corresponding reference loss Li is finite), we add a cross-entropy loss term Lmask,i = −log si,yi,
with si,j being the j-th element of the softmax vector si. This approach ensures that the pseudo-labels
are continually refined to better align with the activation patterns of the target neurons as training
progresses, while constraining the model’s parameters to remain within a region of the parameter
space that corresponds to meaningful, interpretable configurations. After the first iteration, where all
yi are set, the overall loss function L is then:
L =
K
X
k=1
lact(nk) −
N
X
i=1
log si,yi.
(4)
A possible issue that may arise is the prediction of outlier tokens that could randomly decrease the
activation loss, which would possibly update the pseudo-labels to spurious values. To prevent this, for
each masked token position mi, we maintain a history of aggregated losses for each word predicted
during the optimization process, using the history mean for comparison with the reference loss Li.
For instance, let us assume that masked token mi has been mapped by BERT to vocabulary token
t∗in T training iterations, building up a list of corresponding aggregated losses
h
L(1)
agg,i, . . . , L(T )
agg,i
i
.
Instead of comparing Li with the most recent aggregated loss L(T )
agg,i, we check whether:
1
T
T
X
j=1
L(j)
agg,i < Li,
(5)
and only in this case do we update Li. This approach prioritizes the prediction of words with lower
historical activation loss, preventing the selection as pseudo-target outlier words that lead to random
loss fluctuations.
5
4
Performance analysis
4.1
Datasets
To demonstrate the versatility and effectiveness of DEXTER as a global explanation framework, we
design a comprehensive evaluation protocol that reflects the core dimensions of model interpretability:
feature relevance, bias identification, and semantic alignment. We evaluate DEXTER across four
key tasks: visual explanation, activation maximization, bias discovery, and bias text explana-
tion, using four widely adopted datasets that enable rigorous and complementary assessments of
interpretability. Each dataset was selected for its alignment with a specific evaluation goal and its
established use in the literature.
SalientImageNet [41] is used to evaluate both visual explanations and activation maximization. It
is a curated subset of ImageNet designed to analyze model reasoning through object and context
annotations. For each class, the top-5 neural features—highly activated units in the penultimate
layer—are annotated as either spurious or core, based on whether they reflect incidental or meaningful
correlations with the target label. This enables a fine-grained assessment of DEXTER’s ability to
highlight robust, semantically grounded features in both explanation and feature synthesis settings.
Waterbirds[36] and CelebA[21] serve as benchmarks for bias discovery. Waterbirds introduces
spurious correlations between bird species and backgrounds (e.g., land vs. water), providing a con-
trolled environment for evaluating slice discovery and DEXTER’s ability to surface underperforming
subpopulations. CelebA offers over 200,000 annotated face images with 40 binary attributes (e.g.,
gender, hairstyle, glasses), allowing us to assess how DEXTER identifies biases in classifier behavior
related to demographic and semantic features.
FairFaces [14] is finally used for bias text explanation, focusing on the articulation of systematic
spurious correlations in facial classification tasks. With over 100,000 images balanced across seven
demographic groups, it enables a rigorous evaluation of how DEXTER captures and communicates
bias in decision-making across diverse populations.
For each evaluation task, detailed optimization strategies, training procedure, VLM prompts and
other information are provided in the appendix.
4.2
Results
We evaluate DEXTER’s visual explanations through quantitative metrics and a user study, then
assess its performance in bias discovery, mitigation, and explanation, showcasing its ability to
identify spurious correlations and enhance fairness. Finally, an ablation study examines the impact
of individual components and DEXTER’s effectiveness in optimizing text prompts for activation
maximization, comparing it to existing methods.
4.2.1
Visual Explanations
We evaluate DEXTER’s visual explanations through qualitative and quantitative comparisons with
DiffExplainer, the only prior method that shares a similar diffusion-based generation pipeline, on the
SalientImageNet in order to assess the semantic relevance and clarity of the synthesized explanations.
For qualitative analysis, we conducted a user study (see Appendix H), involving 100 participants on
Amazon Mechanical Turk, to further assess the interpretability of DEXTER’s visual explanations
compared to DiffExplainer’s (the two methods with the highest CLIP-IQA scores in Tab. 6). Par-
ticipants compared images generated by DiffExplainer and DEXTER alongside GradCAM-based
attention heatmaps from SalientImageNet, evaluating similarity across three feature categories: per-
ceptual features (shape, texture, color), conceptual features (semantics, context), and cases where
no similarity was perceived (None). As shown in Fig. 2, DEXTER was preferred for conceptual
features, while DiffExplainer was favored for perceptual attributes. Notably, DEXTER received fewer
None responses, indicating stronger alignment with classifier attention regions. A chi-square test
(χ2 = 15.36, p = 0.032) confirms a significant difference, with post-hoc analysis highlighting None
and conceptual features as key contributors.
Fig. 3 presents an example of visual and textual explanations generated by DEXTER for Robus-
tResNet50 [41], focusing on the 5 most active features in the penultimate layer for the dog sled
class, which are all categorized as spurious in SalientImageNet. DEXTER-generated images align
more effectively with the classifier’s attended regions compared to those generated by DiffExplainer.
Furthermore, DEXTER provides textual descriptions that clearly explain the semantics of each
6
Figure 2: User study results on SalientImageNet: Participants evaluated the alignment between
classifier attention and visual explanations, categorized into perceptual features (shape, texture, color),
conceptual features (context, semantics, multiple elements), and no alignment.
neural feature, offering an interpretable account of the classifier’s reasoning. This approach enhances
interpretability and enables users to either identify spurious correlations, such as the ones in the dog
sled class where none of the five most important features actually include a sled, or confirm the
classifier’s reliance on meaningful, task-relevant attributes.
For quantitative evaluation, we compare DEXTER to DiffExplainer using CLIP-IQA and Semantic
CLIP-IQA [45] (Appendix D). DEXTER achieves higher scores (0.94 ± 0.03 / 0.96 ± 0.03) than
DiffExplainer (0.89 ± 0.09 / 0.89 ± 0.09), indicating better semantic alignment and consistency.
Additional comparisons with GAN-based methods [47, 23, 26] further confirm the advantage of
diffusion-based explanations.
Extended details/examples on the visual explanation task are in Appendix D.
Figure 3: DEXTER and DiffExplainer explanations for RobustResNet50 on "dog sled", showing
spurious top-5 features as core visual elements.
4.2.2
Slice discovery and debiasing
The goal of slice discovery is to identify subgroups of data where the model exhibits worse perfor-
mance compared to the rest of the dataset. We assess slice discovery and debiasing performance using
CelebA and Waterbirds datasets. In the former, the task is “blonde”/“non-blonde” classification, with
the “blonde” class including a low proportion of men (representing a slice). The Waterbirds dataset
tackles “waterbird“/“landbird” classification, and is built so that a portion of waterbird images feature
a land background, and vice versa, thus introducing a slice per class. We perform slice discovery
with DEXTER by first identifying words that maximize the activation of a given class. Specifically,
we leverage DEXTER’s optimization process to obtain several descriptive words for each class. Fol-
lowing this step, as done by Kim et al. [16], we compute the CLIP similarity between the discovered
words and the images in the dataset: images with high similarity are identified as belonging to a slice.
We thus train a debiased classifier using Distributionally Robust Optimization (DRO) [36, 33] and
evaluate the classification accuracy on the identified slices. We compare this approach with ERM [36],
LfF [24], GEORGE [42], JTT [19], CNC [49], DRO [36], LADDER [10] and DRO-B2T [16]. As
7
shown in Tab. 1, DEXTER outperforms state-of-the-art methods on worst-slice prediction for CelebA
and achieves comparable results on Waterbirds, deriving descriptive words directly from the model
without relying on training data. Additional details on the slice discovery task and results are given in
Appendix E.
Table 1: Performance on slice discovery and debiasing.
CelebA
Waterbirds
Method
Data
GT
Worst
Avg.
Worst
Avg.
ERM
✓
-
47.7 ± 2.1
94.9
62.6 ± 0.3
97.3
LfF
✓
-
77.2
85.1
78.0
91.2
GEORGE
✓
-
54.9 ± 1.9
94.6
76.2 ± 2.0
95.7
JTT
✓
-
81.5 ± 1.7
88.1
83.8 ± 1.2
89.3
CNC
✓
-
88.8 ± 0.9
89.9
88.5 ± 0.3
90.9
DRO
✓
✓
90.0 ± 1.5
93.3
89.9 ± 1.3
91.5
LADDER
✓
-
89.2 ± 0.4
89.8
92.4 ± 0.8
93.1
DRO-B2T
✓
-
90.4 ± 0.9
93.2
90.7 ± 0.3
92.1
DEXTER(Ours)
-
-
91.3 ± 0.01
91.7
90.5 ± 0.1
92.0
4.2.3
Bias explanation
To evaluate DEXTER’s ability to identify and explain biases in classifiers, we conduct an analysis
using the FairFaces dataset [14]. Specifically, we train two binary classifiers to distinguish between
two age groups: 20-29 (class 1) and 50-59 (class 2). These classifiers are trained on two variants
of the FairFaces dataset: (1) a balanced dataset with equal male and female representation in both
classes, (2) a dataset where males are overrepresented in class 1, and females are overrepresented
in class 2. These variations allowed us to systematically assess how biases in the training data are
reflected in the classifier’s behavior and how well DEXTER captures these biases.
To generate bias reports for a given classifier, DEXTER produces 50 images that maximize the
model’s prediction for the target class. Each image is captioned using a VLM (ChatGPT-4o mini),
and the list of captions is used to prompt another instance of the VLM to generate a textual bias report.
This approach ensures that the generated reports reflect only the classifier’s internal representations
and decision-making processes (more details about VLM hallucination evaluation in appendix I).
Figure 4 showcases excerpts of two generated bias reports.
Figure 4: DEXTER explanation reports. Report 1 analyzes a classifier trained on a FairFaces
variant with male overrepresentation in class 1, while Report 2 corresponds to a balanced dataset.
We evaluate DEXTER’s bias reports through two complementary approaches:
1. User Study. In addition to the earlier study, participants evaluated and ranked DEXTER’s bias
reports for two classifiers. Informed of dataset biases, they assessed report accuracy, clarity, and
interpretability. We report human Mean Opinion Scores (MOShumans, 1–5 scale), and complement
them with automated evaluations: MOSLLM from a VLM and G-eval [20] metrics, including G-evalcon
for consistency with classifier behavior. These scores jointly assess the linguistic, structural, and
explanatory quality of DEXTER’s outputs.
2. Comparison with training-derived reports. We compare DEXTER’s data-free reports to those
generated from training set images using the same pipeline (captioning + LLM-based reporting).
Their similarity is quantified via sentence transformer similarity (STS) [34], measuring alignment
with data-grounded biases.
Table 2 shows strong agreement between human (MOShumans) and model (MOSLLM) ratings (statis-
tical validation in appendix F.2), with all scores across MOS and G-eval metrics exceeding 3 and
8
approaching 4—indicating consistent perceptions of the reports as good to excellent. High STS scores
further confirm that DEXTER’s reports, generated without data, align closely with training-derived
insights. This demonstrates DEXTER’s ability to capture model biases fluently, faithfully, and without
supervision. The bias reasoning enabled by DEXTER also facilitates disparity analysis across a
variety of classifiers (from transformers to CNNs) as shown in Fig. 8 of the supplementary, where we
compare ViT, AlexNet, ResNet50, and RobustResNet50 on ImageNet classes.
Full methodological details and report samples are provided in Appendix F and the supplementary.
Table 2: Evaluation of DEXTER bias reports generated for classifiers trained on FairFaces. The
columns “w Bias” and “w/o Bias” refer to models trained on datasets with and without gender bias,
respectively, for each age group.
Class 0 (20-29)
Class 1 (50-59)
Mean
Metric
w Bias
w/o Bias
w Bias
w/o Bias
STS
0.92
0.85
0.91
0.91
0.90
G-evalcon
4.58 ± 1.00
4.80 ± 0.47
3.66 ± 0.84
3.68 ± 0.85
4.19 ± 0.52
MOSLLM
4.29 ± 0.63
4.80 ± 0.37
4.63 ± 0.44
4.19 ± 0.74
4.48 ± 0.25
MOShumans
4.20 ± 0.63
3.89 ± 0.64
4.10 ± 0.67
3.88 ± 0.79
4.01 ± 0.69
4.2.4
Ablation Study
We investigate how prompt design, auxiliary optimization, and multi-target strategies affect DEX-
TER’s ability to generate effective and interpretable visual explanations. Using 30 SalientImageNet
classes (15 labeled with spurious and 15 with core features), we compare four prompting strategies
by generating 100 images per class and measuring classifier activations: (1) class label only, (2)
ChatGPT-generated descriptions, (3) captions from DiffExplainer, and (4) DEXTER’s optimized
prompts. As shown in Table 3, DEXTER achieves the highest mean score (75.43), effectively
maximizing both spurious (63.00) and core (87.86) features. In contrast, the baseline and ChatGPT
perform moderately, while DiffExplainer underperforms.
We then ablate the role of prompt length and auxiliary loss Lmask by comparing single-word
vs. multi-word optimization, with and without auxiliary supervision. Single-word prompts lack
expressiveness (mean 23.73), and multi-word prompts without auxiliary loss are unstable (mean
11.83). Introducing Lmask improves both cases, with the best performance achieved when combining
multi-word prompts and auxiliary loss (75.43). This highlights the importance of pairing rich prompts
with stable optimization to align explanations with classifier behavior. The standard deviations
reported in Tables 3 and 4 may appear high due to the variation in activation strength across different
classes, as some classes are consistently activated and others only partially. For instance, while some
classes were activated in 100 out of 100 generations, others were only in 20 out of 100. As a result,
this large variation across classes naturally leads to a high overall standard deviation. Full ablation
details are provided in Appendix G.
Finally, we evaluate the faithfulness and robustness of DEXTER’s explanations by testing for bias
propagation and LLM-induced hallucinations. In the first test, we injected adversarial cues (e.g.,
using lion for tiger) into prompt initialization to assess whether upstream biases from BERT or Stable
Diffusion would affect outputs. DEXTER consistently recovered class-relevant features, showing
robustness to such distortions. In the second test, we extracted the most salient visual cue from each
generated text report and added it to the image prompt; the resulting increase in classifier activation
confirms that the cues contained in the text explanation reflect model-relevant features rather than
hallucinations. Full details and statistical validation are provided in Appendix I.
5
Limitations
While DEXTER delivers detailed, data-free global explanations, it is computationally demanding:
prompt optimization takes ∼10 minutes per class, though generating 100 images and the final bias
report is fast (∼15 seconds without backpropagation), making it suitable for offline use. Since
DEXTER relies on Stable Diffusion, there is a risk of NSFW outputs; to mitigate this, we apply its
built-in safety checker to filter and discard inappropriate images during generation.
9
Table 3: Comparison between text-prompting strategies for maximizing the target class.
Method
Spurious
Core
Mean
Baseline
43.06 ± 38.86
86.40 ± 23.83
64.73 ± 31.34
ChatGPT description
41.20 ± 40.78
78.53 ± 34.02
59.87 ± 37.40
DiffExplainer
33.20 ± 41.07
47.66 ± 44.80
39.83 ± 42.93
DEXTER (ours)
63.00 ± 31.20
87.86 ± 15.14
75.43 ± 23.17
Table 4:
Ablation results for single-word and multi-word prompt optimization,
with/without the auxiliary task Lmask.
Configuration
Spurious
Core
Mean
Single-word
11.13 ± 27.38
36.33 ± 38.45
23.73 ± 32.91
,→+ Lmask
34.00 ± 32.72
53.86 ± 44.64
43.93 ± 38.68
Multi-word
15.53 ± 27.93
8.13 ± 18.74
11.83 ± 23.33
,→+ Lmask
63.00 ± 31.20
87.86 ± 15.14
75.43 ± 23.17
6
Conclusion
We introduced DEXTER, a framework for globally explaining deep visual classifiers by combining
diffusion-based activation maximization with textual reasoning. Unlike existing methods, DEXTER
operates in a fully data-free setting, deriving insights solely from the classifier’s internal repre-
sentations. Experiments on SalientImageNet, Waterbirds, CelebA, and FairFaces demonstrate its
effectiveness in uncovering spurious and core features, identifying dataset subpopulations, and gen-
erating human-readable bias explanations. Ablation studies confirmed the impact of multi-word
prompts and auxiliary optimization, while user studies validated the clarity and relevance of DEX-
TER’s textual explanations. Future work will extend its capabilities to multimodal models and on
refining textual reasoning.
Acknowledgements
Simone Carnemolla, Matteo Pennisi, Giovanni Bellitto, Simone Palazzo, Daniela Giordano, and
Concetto Spampinato have been supported by the European Union – Next Generation EU, Mission
4 Component 2 Line 1.3, through the PNRR MUR project PE0000013 – FAIR “Future Artificial
Intelligence Research” (CUP E63C22001940006).
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13
A
Glossary
K - The cardinality of the set of neurons of interest, which contains both neurons in intermediate and
output layers of the vision classifier
M - A translation matrix that maps the BERT vocabulary to the CLIP embedding space
Lact - Neuron activation maximization loss. Used to determine if the text prompts and images
activate the desired neurons of the visual classifier (Eq. 1)
Lagg - Aggregated activation loss. Used also to determine if pseudolabels for Lmask (Eq. 3)
Lmask - A cross-entropy loss between the BERT [MASK] token predictions and saved pseudolabels
p - The learnable soft prompt in DEXTER
t - A sequence consisting of tfixed and N [MASK] tokens
temb - The embedding of t, including the positional encoding
tfixed - Text tokens corresponding to "A picture of a"
V - The BERT vocabulary size, 30522
W - The CLIP vocabulary size, 49408
B
DEXTER Algorithm
Algorithm 1 DEXTER
Goal: Investigate class c on Vision Classifier f
1: Initialize soft prompt p
2: for iteration = 1 to N do
3:
Encode p + tfixed + [MASK] tokens
4:
Obtain BERT logits l for masked token prediction
5:
Apply Gumbel-Softmax to l to obtain the predicted token ˆt
6:
Use translation matrix M to convert ˆt to the CLIP vocabulary ˆt(C) = ˆtM
7:
Compute CLIP text embedding e = [tfixed,ˆt(C)]
8:
Generate image with diffusion model d conditioned on e
9:
Apply cross-entropy loss Lact with ground truth c to f(d(e))
10:
Apply auxiliary loss Lmask on l using the previous logit with the highest class activation
11:
Update p
12: end for
C
Training hyperparameters
We adopt CLIP as the text encoder and Stable Diffusion v1.4 2 as the diffusion model. To reduce
inference time, we employ the Latent Consistency Model (LCM) LoRA adapter 3 using 4 inference
steps. DEXTER is trained with a batch size of 1 (i.e., one image per iteration) and a learning rate of 0.1
across all tasks. In the following sections (Appendices D, E and F), the term optimization steps refers
to the number of iterations performed to optimize the trainable soft prompt. Each optimization step
consists of: predicting the masked token, generating an image, and obtaining the visual classifier’s
prediction.
In the following, we detail the parameters and hyperparameter settings described in Sections 3.1
and 3.2: for the textual prompt sequence [tfixed, m1, m2, . . . , mn], we use “a picture of a
[MASK].” in the single-word optimization scenario, whereas for multi-word optimization, the fixed
prompt is “a picture of a [MASK] with [MASK] and [MASK] and [MASK] and [MASK]
and [MASK].”
We set the sequence P of soft prompts p to 1 (see Sect. C.1), with each embedding having dimension
d = 768, matching BERT’s embedding space. The temperature τ for the Gumbel softmax is kept at
2Hugging Face Stable Diffusion id: compvis/stable-diffusion-v1-4.
3Hugging Face LoRA id: latent-consistency/lcm-lora-sdv1-5.
14
its default value of 1.0.
All experiments ran in half-precision on three H100 GPUs. Prompt optimization takes ∼10 minutes
per class, while generating 100 images and the final bias report takes ∼15 seconds without backprop-
agation. DEXTER is designed for offline, global auditing, with a cost aligned to its goal of delivering
comprehensive model insights.
C.1
Number of soft prompts experiments
We set P = 1 to keep the setup minimal and stable. In DEXTER, the soft prompt guides BERT
in selecting hard tokens via masked language modeling, rather than encoding semantics directly.
Expressiveness comes from composing multi-token prompts (typically 5–6), not from prompt dimen-
sionality. Experiments with P > 1 show that increasing P expands the parameter space and weakens
the classifier’s gradient signal, destabilizing optimization in our data-free setting, as shown in Table 5.
We report mean and standard deviation for 8 randomly selected classes (4 core, 4 spurious).
Table 5: DEXTER’s performance using diffent number of P.
P
Spurious
Core
Avg
1
81.25 ± 18.99
94.25 ± 4.38
87.75 ± 11.68
3
75.00 ± 29.00
44.50 ± 35.89
59.75 ± 32.44
5
73.50 ± 25.66
74.00 ± 42.75
73.75 ± 34.20
10
21.25 ± 31.17
67.75 ± 40.01
44.50 ± 35.59
D
Visual Explanation Details
This appendix provides additional details on the evaluation of visual explanations generated by
DEXTER, complementing the discussion presented in Section 4.2.1. Specifically, we outline the
methodology behind CLIP-IQA, our proposed Semantic CLIP-IQA metric and the comparison
between DEXTER and prior activation maximization works. Following, a comprehensive analysis of
neural feature maximization. These insights further substantiate the findings reported in the main
paper and demonstrate the robustness of DEXTER in generating interpretable visual explanations.
D.1
Quantitative evaluation metrics: CLIP-IQA and Semantic CLIP-IQA
CLIP-IQA [45] is a metric originally introduced to evaluate image quality. In this work, we leverage
CLIP-IQA to compare the quality of images generated by DEXTER against those produced by other
approaches. Specifically, CLIP-IQA calculates the similarity between generated images and two
fixed prompts, returning the probability that an image is closer in similarity to the first prompt rather
than the second. The fixed prompts used in the original formulation are “Good photo.” and “Bad
photo.”
To incorporate semantic relevance into image quality assessment, we propose Semantic CLIP-IQA.
This metric follows the same evaluation protocol as CLIP-IQA but replaces the fixed prompts with
class-specific prompts: “Good photo of a [CLASS]” and “Bad photo of a [CLASS]”. This
modification ensures that the evaluation captures not only general image quality but also the semantic
alignment between the generated images and the target class.
We here also report in Table 6 a quantitative comparison of visual explanation quality across existing
activation maximization methods. The comparison highlights the significant performance gap between
older GAN-based or optimization-based approaches and diffusion-based methods. In particular,
DEXTER demonstrates superior alignment and consistency, validating its ability to generate more
meaningful and semantically grounded visual explanations.
D.2
Neural Feature Maximization
Neural feature maximization refers our strategy to generate images that emphasize the neural features,
as defined in SalientImageNet (i.e., the features of the penultimate layer of the model), learned by
a neural network for a given class. By optimizing the input image to maximize the activation of
15
Table 6: Quantitative comparison between DEXTER and existing activation maximization
methods for visual explanations
Method
CLIP-IQA
Semantic CLIP-IQA
Yosinski et al. [47]
0.37 ± 0.19
0.33 ± 0.23
Mahendran and Vedaldi [23]
0.82 ± 0.15
0.72 ± 0.15
Nguyen et al. [26]
0.74 ± 0.21
0.57 ± 0.30
DiffExplainer [31]
0.89 ± 0.09
0.90 ± 0.05
DEXTER (ours)
0.94 ± 0.03
0.96 ± 0.03
specific neural features, this method allows for an interpretable visualization of what the model has
learned to recognize as distinctive for a class. This approach is useful for understanding how deep
learning models make decisions and can help identify potential biases or spurious correlations in their
predictions.
In our work, neural feature maximization is performed using 2,000 optimization steps with a single
word prompting strategy (see Sect. 4.2.4). We employ RobustResNet50 as the visual classifier for
this task. Table 7 reports the 30 classes selected from the SalientImageNet dataset, consisting of 15
classes containing spurious features and 15 with only core features. Specifically, we selected all 15
classes from SalientImageNet in which all top-5 features were marked as spurious. Additionally, we
randomly selected another 15 classes from SalientImageNet where all top-5 features were marked as
core.
Table 7: Bias and Non-Bias Classes
Bias Classes
Non-Bias Classes
Class_idx
Class Name
Bias
Class_idx
Class Name
Bias
706
patio
✓
985
daisy
✗
837
sunglasses
✓
291
lion
✗
602
horizontal bar
✓
292
tiger
✗
795
ski
✓
486
cello
✗
379
howler monkey
✓
465
bulletproof vest
✗
890
volleyball
✓
574
golf ball
✗
801
snorkel
✓
582
grocery store
✗
981
ballplayer
✓
635
magnetic compass
✗
746
puck
✓
514
cowboy boot
✗
416
balance beam
✓
609
jeep
✗
537
dogsled
✓
624
library
✗
655
miniskirt
✓
764
rifle
✗
810
space bar
✓
847
tank
✗
433
bathing cap
✓
879
umbrella
✗
785
seat belt
✓
971
bubble
✗
Figure 5 extends Figure 3 by reporting additional comparison between DEXTER and DiffExplainer
across different classes (the full set of comparisons is provided in the supplementary). While DiffEx-
plainer effectively maximizes shapes, colors, and textures, its outputs often lack realism, converging
on abstract or pattern-like images. In contrast, DEXTER, owing to its textual anchor, more reliably
yields semantically coherent representations of the features highlighted by the SalientImageNet
heatmaps. This visually confirms the results of the user study reported in Fig. 2.
16
Figure 5: Comparison of DiffExplainer and DEXTER with respect to SalientImagenet for the neural
feature maximization task, covering different classes. The left column displays classes without
spurious features, whereas the right column shows classes with spurious features.
17
E
Slice Discovery Details
This appendix provides additional details on slice discovery and debiasing using DEXTER, comple-
menting the discussion in Section 4.2.2.
Figure 6 extends Figure 1 to better explain the Slice Discovery pipeline used by DEXTER. We use
the Waterbirds dataset as an example, which consists of two classes: Landbirds and Waterbirds. The
majority of the landbirds samples are visibly on land, and similarly the majority of the waterbirds
are in/near water. While overall this is an easy task for most classifiers, we want to ensure high
performance on the small amount of images where the birds are not in their natural habitats (landbirds
on water, waterbirds on land).
Figure 6: Pipeline for Slice Discovery with DEXTER. The top-k words associated with a class are
encoded using the CLIP Text Encoder and compared with image encodings from the biased dataset.
The appropriate slice is determined based on the similarity between text and image embeddings, along
with the ground truth labels of the dataset images. The dataset is then labeled into four categories:
waterbird on water, waterbird on land, landbird on land, and landbird on water (see right side).
DEXTER identifies the words explaining an input visual classifier, as presented in Sect. 3. Specifically,
given a classifier f and a target class c, it optimizes a text prompt to extract a set of top-k class
words WC = wc1, ..., wck (Tab. 8). Each word wci is encoded using CLIP’s text encoder to obtain
an embedding tci and the class prototype is defined as the average embedding: ¯tc = 1
k
P tci Then,
following the approach in B2T [16], at inference time, each image xj of the dataset is encoded into
vj using CLIP’s image encoder. The similarity between the image and the class is then computed as:
CLIPscore = similarity(vj, ¯tc)
(6)
If an image has an higher similarity with the class-words that correspond to its real class it is labeled
as unbiased slice (e.g. an image belonging to class landbirds with an higher similarity with landbirds
class-words), while if an image has an higher similarity with its counterpart class it is labeled as
biased slice (e.g. an image belonging to class landbirds has an higher similarity with waterbirds
class-words). Fig. 7 presents the ROC curves obtained using the CLIP-based matching approach
on the Waterbirds dataset, in comparison with other SOTA slice discovery methods [16, 8, 12]. The
results illustrate how effectively the training images are partitioned into the four groups (waterbird
on water, waterbird on land, landbird on land, landbird on water) by the DEXTER-derived words.
Finally, we follow the debiased training scheme from [36] to train a debiased classifier on the dataset.
In all our slice discovery experiments, we set k = 4 both for CelebA and Waterbirds. We used
1000 DEXTER optimization steps and single word prompting (see Sect. 4.2.4) as an activation
maximization strategy to discover the above words. Tab. 8 reports the discovered words for both
datasets, where class 0 of CelebA has been left blank as in B2T [16].
E.1
Top-k words selection details
The top-k words in Table 8 were obtained by running the DEXTER pipeline multiple times (specifi-
cally k=4 timeless). In each run, we recorded the final (single) pseudo-token selected by the masked
language model at the end of the optimization. To encourage diversity across runs, we excluded
previously selected words from the candidate pool before each new run. However, we also evaluated
18
Table 8: Class-wise words discovered by Dexter for Waterbirds and CelebA datasets.
Dataset
Class 0
Class 1
Waterbirds
fence, woods, jungle, backyard
seas, sea, lake, harbor
CelebA
—
woman, person, head, girl
Table 9: Comparison between multi-run top-k word selection (ours) and single-run frequency-
ranked top-k selection.
Method
F1-score Class 0
F1-score Class 1
B2T
0.99
0.75
k = 10
0.98
0.67
k = 5
0.98
0.66
ours
0.99
0.76
how using sampling frequency ranking may affect slice discovery results. In particular, we computed
results using word frequency–based ranking, evaluating performance with the top k = 5 and k = 10
most frequent words. We compare these results against our proposed sampling strategy and B2T for
reference. As shown in Table 9, our strategy with a single word yields superior performance.
Figure 7: ROC curves for slice discovery on the Waterbirds dataset. Each curve is obtained by
comparing the slice assignments predicted by each method with the ground-truth slice labels released
with the dataset.
F
Bias Reasoning Details
Given a classifier f and class c, we optimize a prompt t∗such that a diffusion model generates images
x1, ..., xM that maximize the classifier’s output for c. Each generated image correctly predicted
by the classifier is captioned using a vision-language model g, producing descriptions di(g(xi)).
A language model h then summarizes the captions into a single report, describing the features
associated with class c and possible biases. This process is fully data-free: both image generation and
reasoning depend only on the model’s behavior. For bias reasoning, we generate 50 images classified
as the target class using DEXTER over the course of up to 5,000 optimization steps, employing a
multi-word prompting strategy. These images are then captioned using ChatGPT-4o mini with the
designated Caption System Prompt (detailed below). Subsequently, another instance of ChatGPT-
4o mini analyzes these captions using the following Report System Prompt to extract key class
attributes and identify potential biases in the model. To generate the captions and the reports with
ChatGPT-4o mini we set a temperature of 0.2 to mitigate hallucinations. The max tokens parameter
is set to 0, allowing unrestricted response generation. The top_p is fixed at 1.0, while both the
frequency_penalty and the presence_penalty are set to 0.0. Finally, the model generates n = 1
output.
19
We conducted multiple experiments on different visual classifiers:
• The reports presented in Fig. 4 and Appendix F.1 were obtained by analyzing a ResNet18
model trained on the FairFaces dataset.
• Figure 8, instead, provides a bias/not bias categorization for ViT, AlexNet, ResNet50, and
RobustResNet50, all trained on the ImageNet-1000 dataset, rather than full bias reasoning
reports. Supplementary material contains the full bias reports for all the 30 classes (see
Table 7) when analyzing RobustResNet50.
Caption system prompt
You are an assistant specialized in analyzing pictures.
Your goal is to generate the most accurate and detailed description.
You will be given a picture and you will have to describe it inserting all the key attributes that you find that
could be inherent.
Keep the description short (one sentence) while including all the key attributes.
- You have to return only the description of the image.
Don’t return other information.
- Don’t describe the image in a poetic way, just describe the visual elements.
- Don’t include any information about the image that is not visible.
- Don’t include any information about the image that is not relevant to the description.
- If the subject is a person describe the ethnicity features and age features.
- Keep the description short (one sentence) while including all the key attributes.
Examples:
Input image:
[IMG]
Input text:
Describe the image
Output:
[CAPTION]
Report system prompt
Context:
This task involves analyzing images generated through a process of Class Activation Maximization (CAM).
Using a deep neural network trained on a specific dataset, the CAM approach identifies visual patterns that
strongly activate a specific class label in the network.
To enhance this analysis, we use a trainable soft prompt
integrated with a Stable Diffusion model.
The soft prompt is optimized to condition Stable Diffusion to create
images that maximize the activation of a chosen class.
For each generated image, if the predicted class aligns
with the target class, a caption describing the image is stored.
Analysis:
Given a trainable soft prompt and a Stable diffusion model.
We optimize the soft prompt to
conditionate stable diffusion to generate pictures that maximize the prediction of a class.
If the class
predicted is the same of the class we want to maximize, than we store a caption of that picture.
Task:
Given a list of sentences that you will receive, you have to reason on the key attributes that the model
uses to maximize a specific class.
Than you’ve to do a report with the explanation of your reasoning.
The
report must end saying if the class is biased or not.
If you reason that the class is biased you must explain why
otherwise you must explain why the class is not biased.
To write the report follow this structure:
- Title:
### Report on Class Activation Maximization for the Class ’CLASS_NAME’
- Hypothesis on Model Bias
the class to analyze will be given by the user with the list of sentences.
The shape of the user prompt will be
the following:
User:
’Class:
[CLASS] - Captions:
[LIST OF SENTENCES]’
20
F.1
Gender Bias Reports
In this section, we report DEXTER’s reports obtained by analyzing the ResNet18 classifier trained on
the FairFaces dataset. As described in Section 4.2.3, we conducted two training procedures for this
visual classifier. In the first procedure, we injected a gender bias across the classes, whereas in the
second, we employed a balanced training set.
Report for class 0 (20-29 years) of a biased classifier (overrepresentation of men)
The analysis suggests that the class ’20 to 29 years old people’ may exhibit some bias.
While the model captures
a range of ethnicities and genders, the emphasis on certain skin tones, facial features, and expressions could
lead to a skewed representation of what is considered ’youthful’.
- Skin Tone Bias:
The frequent mention of light brown and medium brown skin tones may indicate a bias towards
these tones, potentially neglecting a broader spectrum of skin colors that also represent youth.
- Facial Feature Bias:
The focus on specific facial features that are often idealized in media could lead to a
narrow definition of youth, which may not encompass the full diversity of young adults.
- Gender Representation:
The slight focus on the overrepresentation of male subjects could suggest a bias in the
training data, which may not equally represent young women.
In conclusion, while the model successfully identifies attributes associated with the class ’20 to 29 years old
people,’ it is essential to recognize the potential biases in representation.
The model’s predictions may not
fully encompass the diversity of this age group, indicating a need for more balanced training data to ensure
equitable representation across all demographics.
Report for class 0 (20-29 years) of a balanced classifier (equal representation of men and
women)
Based on the analysis of the model, it does not appear to exhibit significant bias in identifying the class
of 20 to 29 years old individuals.
The diversity in hair and eye color, skin tone, and gender representation
suggests that the model is capable of recognizing a broad spectrum of individuals within this age group.
The
focus on well-defined facial features and neutral expressions aligns with common perceptions of young adults,
and the inclusion of various ethnic backgrounds indicates an inclusive approach.
In conclusion, the class of 20
to 29 years old people, as represented by the model, does not seem to be biased.
The attributes identified are
consistent with a diverse and inclusive representation of young adults.
Report for class 1 (50-59 years) of a biased classifier (overrepresentation of women)
Based on the analysis of the model, it can be concluded that the class ’50 to 59 years old people’ is biased.
The
bias stems from several factors:
- Representation of Age:
The model predominantly features youthful characteristics, which may not accurately
reflect the physical appearance of individuals aged 50 to 59.
This suggests that the model may be influenced by
societal standards that equate youthfulness with attractiveness, leading to a misrepresentation of the target age
group.
- Ethnic and Gender Bias:
The emphasis on lighter skin tones and the predominance of female subjects indicate
a lack of diversity.
This could perpetuate stereotypes and fail to represent the true diversity of individuals
within the 50 to 59 age range.
- Societal Stereotypes:
The model’s reliance on neutral or serious expressions may reflect societal stereotypes
about older individuals, which can further skew the representation of this age group.
In conclusion, while the model attempts to maximize the activation of the class ’50 to 59 years old people,’ it
does so through a lens that is biased towards certain physical characteristics, expressions, and demographic
representations.
This highlights the importance of ensuring diverse and representative training data to mitigate
bias in machine learning models.
Report for class 1 (50-59 years) of a balanced classifier (equal representation of men and
women)
Based on the analysis, it appears that the class ’50 to 59 years old people’ is not inherently biased.
The model
successfully captures a range of attributes that reflect the diversity of individuals within this age group.
However, there are potential biases in the representation of expressions and the predominance of serious or
neutral faces, which may not fully encompass the emotional spectrum of individuals aged 50 to 59.
Moreover, the
model’s training data could influence its perception of age, potentially leading to stereotypes about how older
individuals should look or behave.
For instance, the emphasis on serious expressions may inadvertently reinforce
the stereotype that older adults are less expressive or joyful.
In conclusion, while the class itself is not
biased, the representation of certain attributes, particularly expressions and emotional portrayals, may benefit
from a broader range of depictions to avoid reinforcing stereotypes associated with aging.
21
F.2
Bias Report Quality Assessment Metrics
In this section, we provide additional explanations for the metrics used to evaluate the quality of
DEXTER’s bias reports, as shown in Table 2. Following Liu et al. [20], we use the G-eval consistency
metric, which involves prompting a large language model (LLM) to assess a text’s factualness, scoring
each report on a scale from 1 to 5. Since the original prompts for these metrics were designed for
summarization, we create the following G-eval Consistency Prompt to better align with our
task.
Following the G-eval format, we use the Mean Opinion Score of LLM (MOSLLM) metric. This metric
uses the MOSLLM prompt asking the LLM to rate between 1 and 5 the generated reports according
to how well the report accurately describes the existence of bias in the classifier. Additionally, the
MOShumans score represents the average rating given by participants in the user study.
For both the G-eval and MOSLLM configurations, we adopt the same settings as Liu et al. [20].
Specifically, we use gpt-4-0613 with a temperature of 2, max_tokens = 0, top_p = 1.0,
frequency_penalty = 0, and presence_penalty = 0.
To enhance response diversity, we generate n = 20 outputs. Furthermore, in both G-eval and
MOS LLM system prompts, we provide the same evaluation instructions that were given to human
evaluators during the user study, also following the best practices to reduce hallucinations provided
by [4].
To assess whether the LLM’s mean-opinion scores are statistically indistinguishable from those
assigned by human annotators, we applied an independent-samples two one-sided equivalence test
(TOST). The test compared the distribution of MOSLLM to that of MOShumans obtaining a p ≪0.05.
Thus, within the predefined equivalence bounds (±0.5 for a Likert scale), the LLM’s ratings can be
considered statistically equivalent to human judgments, indicating strong agreement between the two
sets of scores.
G-eval Consistency system prompt
**Instructions for Evaluation**:
In this task, we trained a visual classifier twice:
1.
First, using a dataset with gender-biased images.
2.
Second, using a balanced dataset.
Your task is to evaluate how effectively a given textual description identifies the presence or absence of gender
bias in the visual classifier:
- **Visual Classifier**:
A machine learning model trained to distinguish between two classes (e.g., Class 1:
"20-29 years old people" vs.
Class 2:
"50-59 years old people").
Please make sure you read and understand these instructions carefully.
Please keep this document open while
reviewing, and refer to it as needed.
Evaluation Criteria:
Consistency (1-5) - the factual alignment between the question and the description.
A factually consistent
description contains statements that are entailed by the question.
Annotators were also asked to penalize
descriptions that contained hallucinated facts.
Evaluation Steps:
1.
Read the question carefully and identify the main topic and key points.
2.
Read the description and compare it to the question.
Check if the description covers the main topic and key
points of the question, and if it presents them in a clear and logical order.
3.
Assign a score for consistency on a scale of 1 to 5, where 1 is "Bad Description" and 5 is "Excellent
Description" based on the Evaluation Criteria.
Example:
Question:
{{Question}}
Description:
{{Description}}
Evaluation Form (scores ONLY):
- Consistency (1-5):
22
MOSLLM system prompt
**Instructions for Evaluation**:
In this task, we trained a visual classifier twice:
1.
First, using a dataset with gender-biased images.
2.
Second, using a balanced dataset.
Your task is to evaluate how effectively a given textual description identifies the presence or absence of gender
bias in the visual classifier:
- **Visual Classifier**:
A machine learning model trained to distinguish between two classes (e.g., Class 1:
"20-29 years old people" vs.
Class 2:
"50-59 years old people").
Please make sure you read and understand these instructions carefully.
Please keep this document open while
reviewing, and refer to it as needed.
Evaluation Criteria:
Mean Opinion score (1-5) - Assign a rating to the description based on its effectiveness:
- 1:
Bad description (fails to identify bias or provide relevant details).
- 2:
Poor description (some effort to address bias, but lacks clarity or completeness).
- 3:
Neutral (adequate but not insightful; partially addresses bias).
- 4:
Good description (clear and mostly thorough in addressing bias).
- 5:
Excellent description (comprehensive, clear, and detailed in identifying bias).
Evaluation Steps:
1.
Read the question carefully and identify the main topic and key points.
2.
Read the description and compare it to the question.
Check if the description covers the main topic and key
points of the question, and if it presents them in a clear and logical order.
3.
Assign a score for bias identification on a scale of 1 to 5, where 1 is "Bad Description" and 5 is "Excellent
Description" based on the Evaluation Criteria.
Example:
Question:
{{Question}}
Description:
{{Description}}
Evaluation Form (scores ONLY):
- Rating (1-5):
23
F.3
Bias identification of multiple vision classifiers
In Fig. 8 we report an example of disparity analysis across ViT, AlexNet, ResNet50, and Robus-
tResNet50 for multiple ImageNet classes. By revealing class-specific biases and failure patterns,
DEXTER helps identify where each model struggles and can guide data collection efforts when
biases are consistently observed across classifiers. To select key neural features for these classifiers,
we ranked penultimate-layer neurons by their weights to each class and selected the top 5, mirroring
the Salient ImageNet method but without any training data.
Figure 8: Bias analysis across classifiers and SalientImageNet classes. Each cell shows DEXTER-
generated visual explanations and bias assessments (“Biased” or “Not Biased”). While Lion is
unbiased across models, Jeep is biased in ResNet50 but not in its robust version. Baseball Player
remains biased in all models, suggesting dataset-level bias.
24
G
Additional details on Ablation Study
Figure 9: Procedure to compute the percentage of activations.
In this section, we provide further details on how we compute the activation score reported in Table 3
and Table 4. As shown in Figure 9, once the word(s) are obtained at the end of the optimization
process, we prompt Stable Diffusion to generate 100 distinct images using a fixed prompt tfixed (from
the optimization stage) concatenated with the discovered word(s) ˆtC. We then sum all inference steps
in which the visual classifier predicts the target class. Figure 10 illustrates examples of the different
text-prompt strategies, described in Table 3, alongside their corresponding activation scores for the
class Tiger and the class Snorkel.
Figure 10: Examples of corresponding text prompts and generated images for class activation
maximization. For a non-spurious class like tiger, all generated images easily activate the target class.
However, in the case of the snorkel class, DEXTER is able to generate significantly more images that
maximize the activation, and exposes the other features the model pays attention to.
25
H
User Study Details
Figure 11: Example of a question from the first section of the user study.
Each user study consisted of two parts. The first part aimed to assess the ability of DEXTER to
visually highlight the focus of the target classifier for a specific neural feature, while the second part
evaluated the accuracy of the textual bias reports generated by DEXTER.
In the first part of the study, each participant was presented with 15 randomly images (explaining
specific feature in SalientImageNet) drawn from the outputs of DiffExplainer and DEXTER. An
example is shown in Fig. 11. Participants were asked to assess whether the generated image accurately
represented the attended region of the real image based on attributes such as color, shape, texture,
material, context, or semantic similarity. Additionally, two extra options were provided: “Several
Elements” for cases where multiple similarity criteria were met, and “None” for instances where
none of the criteria applied.
The second part of the study aimed to evaluate the ability of DiffExplainer reports to detect the
presence of bias in the target classifier. The target classifier was trained both with and without gender
bias, and participants were presented with all four reports generated by DEXTER. Examples of
questions for class 0 (20–29 years old) are provided in Fig. 12 and Fig. 13, corresponding to the
models trained with and without the injected gender bias, respectively. Participants were asked to
rate, on a scale from 1 to 5, the extent to which the generated report reflected the presence or absence
of bias.
The user study was conducted with 100 distinct MTurk workers, with an average completion time of
10.32 minutes. Each feature explanation in the first part received an average of 10.38 responses for
DiffExplainer and 10.82 for DEXTER. To minimize response variability, participants were required to
provide a textual justification for their answers in both sections of the questionnaire before submission.
The compensation for each worker was $0.50.
I
Diffusion and LLM’s Hallucination evaluation
I.1
Randomness in the diffusion generation process
To assess the stability of Stable Diffusion, we performed intra-class evaluation using DEXTER’s
final prompt. Across three independent runs (100 images each), we measured activation scores with
the target model. If the image generation is unstable, scores will vary significantly. Instead, our
consistent results indicate stability. We report mean and standard deviation for 8 randomly selected
classes (4 core, 4 spurious) across three independent runs (Tab. 10).
26
Figure 12: Example of a question from the sec-
ond section of the user study (biased).
Figure 13: Example of a question from the sec-
ond section of the user study (not biased).
Table 10: Intra-class evaluation of randomness in Stable Diffusion generation process.
Type
Class
Avg
Spurious
890
55.33 ± 2.49
795
79.00 ± 1.63
655
96.66 ± 0.94
706
100.0 ± 0.00
Core
291
100.0 ± 0.00
486
95.66 ± 1.24
514
86.66 ± 1.24
624
95.66 ± 1.88
I.2
Bias propagation
To verify that DEXTER’s outputs align with the classifier—rather than reflecting biases from BERT
or Stable Diffusion—we ran a robustness test. We manually injected spurious cues into prompt
initialization (e.g., replacing the starting auxiliary pseudo-target with lion for the class tiger) to
simulate strong upstream bias. Then, for each class in SalientImageNet subset of our paper (split into
“core” and “spurious” categories), we generated 100 images and measured how often the classifier
activated the correct class. We report average classifier activation scores (as defined in Tab. 3 of the
paper) for spurious, core classes, and their average. These results in Tab. 11 show that even with
adversarially biased prompts, DEXTER recovers class-relevant visual patterns, aligning with the
27
classifier. Classifier-driven optimization helps correct upstream bias and grounds both visual and
textual outputs in model behavior.
Table 11: Comparison of Spurious, Core, and Average scores under different bias conditions.
Condition
Spurious
Core
Avg
Injected Bias
65.3
88.4
76.8
No Bias
63.0
87.6
75.4
We also evaluated how DEXTER compensates for bias propagation. Specifically, we recorded the
sequence of words selected as pseudotargets during optimization. This was done for both the biased
setting and the standard setting, where the initial pseudotarget is chosen randomly. As shown in the
Table 12, in both cases DEXTER consistently converges toward the most meaningful word, with
the selection path gradually shifting from a random concept to the intended target. This trajectory
suggests that the optimization process does not simply lock onto the first spurious or core concept it
encounters, but instead explores a range of semantically plausible candidates before progressively
refining the prompt toward a stable and interpretable solution, indicating a non-greedy and convergent
behavior rather than an early commitment driven by the mask pseudo-label loss. Convergence analysis
through keyword trajectory during the DEXTER optimization process. The Table 12 illustrates how
DEXTER transitions from initially random concepts to progressively more semantically coherent and
domain-specific words, demonstrating convergence toward a stable explanation.
I.3
Analysis of LLM hallucination impact
Given that large language models can generate statements influenced by their prior knowledge
rather than by factual evidence, we systematically assess the fidelity of our reports and how much
they are aligned with the visual classifier decision-making process. We aim to ensure that each
report highlights genuine visual patterns exploited by the classifier rather than hallucinated elements.
Concretely, for each of the 30 SalientImageNet classes in our experimental setup (Tab. 7), we verify
whether the single most salient visual cue extracted from the corresponding DEXTER report is truly
grounded. Given a class c we form
BASELINE PROMPT: “a picture of a [c]”,
CUE PROMPT: “a picture of a [c] with [cue]0 ... and ’[cue]n’,
where n is the total number of cues obtained automatically from the generated reports via a GPT–4o
mini class-cues extractor. Then, we generate 100 images per prompt with Stable–Diffusion and
measure the Activation Score (AS) on the frozen RobustResNet50 used throughout the paper (4.2.4).
A cue is grounded when AScue > ASbaseline; otherwise it is neutral or wrong.
Statistical significance & confidence interval.
For each class we measure ∆i = AScue,i −
ASbaseline,i. Across the 30 classes we obtain
¯∆= 16.33 pp,
s∆= 23.84 pp.
The two-sided 95% confidence interval is
¯∆± t0.975,29
s∆
√
30 = [ 7.43, 25.24 ] pp.
Both a paired t-test (t(29) = 3.79, p = 7.8 × 10−4) and a Wilcoxon signed-rank test (W = 19, p =
2.9 × 10−4) confirm that the improvement is statistically significant.
Aggregate results.
Table 13 contrasts the mean Activation Scores for the two prompts. Adding the
report cue raises the mean score from 64.73% to 81.06% ( +16.33 pp).
• 20/30 classes (67%) improve (peak +81 pp for miniskirt), confirming that the cues capture
genuinely discriminative evidence.
• 7 classes already achieve 100 % with the baseline prompt; the cue therefore leaves perfor-
mance unchanged.
28
Table 12: Word-wise convergence path for biased vs unbiased initialized prompt.
Grocery Store (582)
Patio (706)
Biased
Not Biased
Biased
Not Biased
motel
fan
restaurant
head
library
coffin
school
book
wedding
factory
house
man
party
sale
pool
the
restaurant
factory
residence
republic
party
distillery
tree
wedding
market
window
motel
supermarket
house
bakery
courtyard
supermarket
terrace
patio
Table 13: Activation Scores averaged over 30 classes (↑/↓: cue better/worse than baseline).
Prompt
Mean AS (%)
∆
Class split
Baseline
64.73
—
Cue (report)
81.06
+16.33
20 ↑3 ↓7 = 30
• 3 classes degrade:
bubble
∆= −5 pp; the cue is correct but pertains only to a subset of
bubble instances.
tanks
∆= −1 pp (100 % →99 %); a negligible loss.
rifle
∆= −17 pp; the cue “individuals in dark clothing” marking the
sole clear failure.
Overall assessment.
DEXTER explanations are therefore strongly grounded: in 27 out of 30 classes
the cue is beneficial or neutral, and only one class (rifle) exhibits evidence of hallucination. In the
two minor degradations (bubble,tanks) the reports still isolate genuine—though partial—visual cues.
Evaluation Implications.
The consistent increase in classifier confidence when report-derived
cues are injected into the prompt demonstrates that DEXTER’s explanations faithfully align with the
model’s decision boundary, rather than reflecting spurious or hallucinatory artifacts. Together with
the text-based metrics reported in Table 2 for the FairFaces dataset (G-Eval consistency, STS, MOS),
these findings confirm that DEXTER delivers genuinely grounded insights into the visual classifier’s
decision-making process.
Pipeline Robustness.
Furthermore, this evaluation implicitly validates the collaborative operation
of all pretrained components (BERT, CLIP, Stable Diffusion, VLM) in our optimization pipeline.
The target visual classifier guides the prompt optimization. BERT selects discriminative keywords,
the diffusion model generates contextually relevant images, and the captioning VLM produces
faithful captions. As a result, the end-to-end process avoids propagating biases or hallucinations from
intermediate models and yields explanations that are solidly grounded in the classifier’s behavior.
I.4
Baseline prompts vs Cue prompts
This section, referring to Sec. I.3, reports the system prompt used to extract the relevant visual cues
from DEXTER’s textual explanations. Furthermore, Table 14 compares class-wise activation scores
produced by the baseline prompt “A picture of a [CLASS]” and by the CUE-enriched prompt.
Large gains appear for classes that were poorly activated in the baseline (e.g., space bar improves
from 6 →57 , hockey puck from 0 →79). The results demonstrate that adding concise semantic
cues markedly improves class-specific guidance and reduces hallucinations during image generation.
29
Report’s cues extractor system prompt
You are an assistant that reads a bias-analysis report about an ImageNet class and extracts concrete visual cues
that the report claims are important for that class.
Your task
1.
Identify up to 5 key visual phrases (2–5 words each) that:
•
are explicitly mentioned in the report;
•
describe tangible elements that can appear in an image (objects, attributes, background, actions);
•
are likely to trigger the classifier according to the report.
2.
Return your answer in JSON with two fields:
“‘json
"key-phrases":
["phrase1", "phrase2", ...], "full-prompt":
"a picture of a <CLASS> with "phrase1" and
"phrase2" and ..."
30
Table 14: Activation maximization scores for each class: comparison between the baseline
prompt “A picture of a [CLASS]” and the CUE prompt obtained from the DEXTER’s reports.
Class
Baseline prompt
AS
CUE prompt
AS
volleyball
A picture of a volleyball
66
A picture of a volleyball with young women playing and
mixed-gender teams and variety of scenarios
78
space bar
A picture of a space bar
6
A picture of a space bar grid layouts with framed artworks and
black-and-white imagery
57
umbrella
A picture of an umbrella
95
A picture of an umbrella with vibrant outdoor scenes and
garden scenes
100
baseball player
A picture of a baseball
player
98
A picture of a baseball player with female athletes and softball
context
100
bubble
A picture of a bubble
100
A picture of a bubble with whimsical themes
95
balance beam
A picture of a balance beam
0
A picture of a balance beam with athletic attire and physical
activities and indoor training environments and group dynamics
and artistic performances
27
cowboy boot
A picture of a cowboy boot
83
A picture of a cowboy boot with various types of boots
96
patio
A picture of a patio
100
A picture of a patio with modern architecture and expansive
outdoor spaces and affluent homes and specific landscaping
styles
100
tank
A picture of a tank
100
A picture of a tank with camouflage and military environment
and armored vehicle
99
dark glasses
A picture of dark glasses
8
A picture of dark glasses with facial hair and an older man
20
daisy
A picture of a daisy
100
A picture of a daisy with vibrant colors and natural contexts
and red daisies
100
howler monkey
A picture of a howler
monkey
0
A picture of a howler monkey with lush, green environments
and natural habitat
32
tiger
A picture of a tiger
100
A picture of a tiger with striped fur and natural habitat and
majestic posture
100
library
A picture of a library
76
A picture of a library with books and organized indoor area and
spacious room with furniture and people
97
seat belt
A picture of a seat belt
56
A picture of a seat belt with human subjects
73
rifle
A picture of a rifle
85
A picture of a rifle with individuals in dark clothing and jackets
and sunglasses and aiming
68
grocery store
A picture of a grocery store
84
A picture of a grocery store with diversity of food items and
presence of people and colorful displays
95
snorkel
A picture of a snorkel mask
41
A picture of a snorkel mask with young individuals and sharks
94
dogsled
A picture of a dogsled
98
A picture of a dogsled with snowy landscapes and
human-animal interaction and specific dog breeds
100
magnetic compass
A picture of a magnetic
compass
6
A picture of a magnetic compass with silver compass pendant
and craftsmanship and design
26
horizontal bar
A picture of a horizontal bar
6
A picture of a horizontal bar in sports and athletic environment
8
ski
A picture of a ski
56
A picture of a ski with young individuals with ski attire
100
miniskirt
A picture of a miniskirt
19
A picture of a miniskirt with women wearing miniskirts
100
lion
A picture of a lion
100
A picture of a lion with distinctive physical traits and social
behavior and natural habitat and human interactions
100
hockey puck
A picture of a puck
0
A picture of a puck with ice hockey gameplay and player
confrontations
79
swimming cap
A picture of a swimming cap
92
A picture of a swimming cap with children in joyful scenarios
with adults and certain hair types and smiling expressions
99
bulletproof vest
A picture of a bulletproof
vest
67
A picture of a bulletproof vest with casual attire
89
cello
A picture of a cello
100
A picture of a cello with musical instrument
100
golf ball
A picture of a golf ball
100
A picture of a golf ball with spherical objects and white color
100
jeep
A picture of a jeep
100
A picture of a jeep, landrover with off-road capability
100
Mean
—
64.73
—
81.06
31
NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the
paper’s contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction of the DEXTER paper clearly and accurately
reflect the paper’s contributions and scope. The three central use cases mentioned in the
abstract (activation maximization, slice discovery and debiasing, and bias explanation) are
all thoroughly addressed and empirically validated in the main body. Additionally, the
introduction carefully outlines both the limitations of existing work and how DEXTER
addresses them, reinforcing that the scope of the claims is realistic and well-motivated.
Guidelines:
• The answer NA means that the abstract and introduction do not include the claims
made in the paper.
• The abstract and/or introduction should clearly state the claims made, including the
contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
• The claims made should match theoretical and experimental results, and reflect how
much the results can be expected to generalize to other settings.
• It is fine to include aspirational goals as motivation as long as it is clear that these goals
are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: We have added a "Limitations" section in the document that provides areas
where DEXTER could benefit from improvement.
Guidelines:
• The answer NA means that the paper has no limitation while the answer No means that
the paper has limitations, but those are not discussed in the paper.
• The authors are encouraged to create a separate "Limitations" section in their paper.
• The paper should point out any strong assumptions and how robust the results are to
violations of these assumptions (e.g., independence assumptions, noiseless settings,
model well-specification, asymptotic approximations only holding locally). The authors
should reflect on how these assumptions might be violated in practice and what the
implications would be.
• The authors should reflect on the scope of the claims made, e.g., if the approach was
only tested on a few datasets or with a few runs. In general, empirical results often
depend on implicit assumptions, which should be articulated.
• The authors should reflect on the factors that influence the performance of the approach.
For example, a facial recognition algorithm may perform poorly when image resolution
is low or images are taken in low lighting. Or a speech-to-text system might not be
used reliably to provide closed captions for online lectures because it fails to handle
technical jargon.
• The authors should discuss the computational efficiency of the proposed algorithms
and how they scale with dataset size.
• If applicable, the authors should discuss possible limitations of their approach to
address problems of privacy and fairness.
• While the authors might fear that complete honesty about limitations might be used by
reviewers as grounds for rejection, a worse outcome might be that reviewers discover
limitations that aren’t acknowledged in the paper. The authors should use their best
judgment and recognize that individual actions in favor of transparency play an impor-
tant role in developing norms that preserve the integrity of the community. Reviewers
will be specifically instructed to not penalize honesty concerning limitations.
32
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and
a complete (and correct) proof?
Answer: [NA]
Justification: The paper does not include any formal theoretical results, theorems, or proofs.
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Table 15: Assets used and licence information.
Asset
Type
License
Github / URL
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SalientImagenet
data
non-commercial research
singlasahil14/salient_imagenet
[41]
Waterbirds
data
non-commercial research
kohpangwei/group_DRO
[36]
CelebA
data
non-commercial research
https://mmlab.ie.cuhk.edu.hk/projects/CelebA.html
[21]
FairFaces
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CC BY 4.0
joojs/fairface
[14]
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Pretrained Model
Apache license 2.0
https://huggingface.co/google-bert/bert-base-uncased
[5]
CLIP
Pretrained Model
MIT license
openai/CLIP
[32]
Stable Diffusion 1.5
Pretrained Model
CreativeML Open RAIL-M
https://huggingface.co/stable-diffusion-v1-5/stable-diffusion-v1-5
[35]
GPT-4o
LLM
API-based use under OpenAI terms
https://platform.openai.com/docs/overview
OpenAI
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|
DEXTER: Diffusion-Guided EXplanations with TExtual Reasoning for Vision Models Simone Carnemolla1∗Matteo Pennisi1∗Sarinda Samarasinghe2 Giovanni Bellitto1 Simone Palazzo1 Daniela Giordano1 Mubarak Shah2 Concetto Spampinato1 1 2 - tial for building transparent and trustworthy AI systems. We introduce DEXTER, a data-free framework that employs diffusion models and large language models to generate global, textual explanations of visual classifiers. DEXTER operates by optimizing text prompts to synthesize class-conditional images that strongly activate a target classifier. These synthetic samples are then used to elicit detailed natural language reports that describe class-specific decision patterns and biases. Unlike prior work, DEXTER enables natural language explanation about a classifier's decision process without access to training data or groundtruth labels. We demonstrate DEXTER's flexibility across three tasks-activation maximization, slice discovery and debiasing, and bias explanation-each illustrating its ability to uncover the internal mechanisms of visual classifiers. Quantitative and qualitative evaluations, including a user study, show that DEXTER produces accurate, interpretable outputs. Experiments on ImageNet, Waterbirds, CelebA, and FairFaces confirm that DEXTER outperforms existing approaches in global model explanation and class-level bias reporting. Code is available at https://github.com/perceivelab/dexter. 1 Introduction How can we systematically uncover and explain a deep visual classifier's decision-making process in a way that is both comprehensive and human-interpretable? This question is crucial as AI systems are increasingly deployed in high-stakes applications, where interpretability and trust are as important as accuracy. However, the lack of transparency in model reasoning, often exacerbated by the reliance on spurious correlations-irrelevant features that disproportionately influence predictions- undermines confidence in these systems, making decisions difficult to justify. For instance, ImageNet-trained classifiers have been shown to favor background textures or lighting conditions over intrinsic object properties [41, 9]. Addressing these issues requires explainability techniques that extend beyond local attribution, offering a global perspective on a model's reasoning patterns, biases, and learned representations. Existing methods for model explainability, such as GradCAM [39] and Integrated Gradients [43] provide local explanations and require data. These methods focus on analyzing individual predictions by attributing importance to specific pixels or regions in an image. While these methods are extremely useful for highlighting areas of interest, they do not offer a global understanding of model behavior. In ∗Equal contribution. 39th Conference on Neural Information Processing Systems (NeurIPS 2025). 16 Oct 2025 contrast, activation maximization techniques [26-28] have been instrumental in globally visualizing the features learned by neural network, but the generated images are often abstract and challenging to interpret. While existing methods offer useful insights into model behavior and feature representations, they often fall short of capturing the global reasoning patterns and biases behind predictions. In particular, visual explanations can be hard to interpret and may lack the ability to convey high-level reasoning or reveal subtle spurious correlations-issues that are often better addressed through complementary textual explanations [18, 6, 16, 3]. Text-based explanations offer an accessible complement to visual methods [11], but they often lack global perspective or clarity. Recent work [16, 2, 10], including Natural Language Explanation (NLE) methods [30, 37, 15], combines visual and textual cues to reveal model biases, but typically relies on labeled data, annotations, or pretrained vision-language mappings [2]. We propose DEXTER, a framework that generates human-interpretable textual explanations, uncovering the global reasoning patterns and biases of visual classifiers. DEXTER operates in a fully data-free setting by leveraging the generative capabilities of diffusion models and the reasoning power of large language models (LLMs). At its core, DEXTER optimizes soft prompts, which are mapped to discrete hard prompts, in order to guide a diffusion-based image generation process. This ensures that the generated images are aligned with the outputs of a target classifier, capturing the features and concepts prioritized by the model. The generated images are then analyzed by an LLM, which reasons across them to provide textual explanations of the classifier's decision-making process. By combining image generation and textual reasoning, DEXTER not only overcomes the interpretability challenges of visual explanations but also provides a global understanding of model behavior, including identifying spurious correlations and biases, without requiring access to any training data or ground truth labels. Thus, the core main contributions of DEXTER are: • Global explanations: A high-level understanding of model decisions, identifying key features, biases, and patterns beyond local attribution. • Data-free approach: DEXTER requires only the trained model for explanations, unlike existing methods, that employ training or ground-truth data. • Bias identification and explanation: Uncovering and describing spurious correlations to support model debiasing. • Natural language reasoning: LLM-generated textual explanations that enhance interpretability over purely visual methods. We evaluate DEXTER across three tasks: (1) activation maximization to reveal model-prioritized features, (2) slice discovery to detect underperforming subpopulations, and (3) bias explanation to identify spurious correlations. Experiments use SalientImageNet [41], Waterbirds [36], CelebA [21], and FairFaces [14]-datasets widely used in fairness and interpretability research. Across tasks, DEXTER demonstrates strong performance, offering meaningful insights into model behavior. 2 Related work Activation Maximization (AM) interprets neural networks by generating inputs that maximize neuron activations [7]. Early methods often produced unrealistic, uninterpretable images [40, 25], later improved through regularization [23] and generative priors [26]. DiffExplainer [31] introduced diffusion-based image generation guided by soft prompts, which improve realism but lack semantic transparency. This work points toward token-wise optimization, but this direction remains underexplored. In contrast, DEXTER builds on this line by replacing soft prompts with discrete (hard) tokens, which are interpretable and enable both visual and textual global explanations. Although once central to model interpretability, AM has seen limited progress in recent years, partly because the resulting images-while optimized for neuron activation-are often difficult to interpret semantically, making it challenging to draw clear, causal insights about model behavior. The field has largely shifted toward attribution methods-e.g., GradCAM [39], Integrated Gradients [43], which are computationally efficient and provide intuitive saliency maps, though they are inherently local and data-dependent. Other efforts in feature visualization [52] offer insights into internal representations but require training data and remain focused on visual output. DEXTER revives AM by leveraging modern 2 diffusion models and discrete prompt optimization to produce class-level, multimodal explanations in a fully data-free setting. Explanation of visual classifiers. Textual and Multimodal Explanations seek to complement visual saliency with natural language justifications. Approaches like Multimodal Explanations [29], LIMEx [44], and post-hoc counterfactuals [48] align vision and language to explain model decisions. However, these methods often rely on ground-truth labels or annotated datasets and are limited to instance-level explanations. In contrast, DEXTER provides global, class-wise explanations without supervision or access to data. Natural Language Explanation (NLE) methods aim to generate human-readable justifications, typically using VQA-style benchmarks [30, 37, 15]. Recent approaches integrate vision and language into unified architectures [38, 37], but remain supervised and local. DEXTER differs by producing global textual reports through unsupervised classifier probing-without labels, data, or task-specific fine-tuning. Slice Discovery identifies dataset subpopulations where a model underperforms, offering a targeted way to reveal and explain systematic failures in classifier behavior. Traditional methods analyze embeddings, gradients, or misclassifications [1, 24, 42, 51], while recent approaches leverage CLIP's joint text-visual embeddings [8, 12, 50] or use LLMs to generate captions and extract keywords [46, 16]. Bias-to-Text (B2T)[16] extracts pseudo-bias labels from misclassified image captions, and LADDER[10] generates bias hypotheses from low-confidence predictions, clustering samples via LLM-derived pseudo-attributes. Unlike these data-dependent methods, DEXTER discovers and explains biases in a fully data-free manner, relying only on the classifier's internal behavior. 3 Method DEXTER's framework, shown in Fig. 1, integrates three key components: a text pipeline for optimizing prompts, a vision pipeline for the image generation process, and a reasoning module using a vision-language model (VLM). DEXTER begins by optimizing a soft prompt to condition a BERT model [5] to fill in masked tokens in a predefined sentence. The resulting prompt guides the stable diffusion process to generate images that maximize the activation of a set of target neurons (e.g. classification heads) in a given visual classifier. The generated images are then analyzed by the VLM, which reasons across multiple images to provide coherent, human-readable textual explanations of the model's decision-making process. 3.1 Text pipeline The text pipeline has the goal of optimizing a textual prompt to suitably condition the diffusion model. We pose prompt generation as a masked language modeling task, and employ a pretrained and frozen BERT model for this purpose. The structure of the textual prompt to be produced is fully customizable, and can be controlled by combining portions of fixed text with a set of mask tokens. For the sake of clarity and without loss of generality, we will assume that the textual prompt has the structure of a sequence t = [tfixed, m1, m2, . . . , mN], where tfixed is the portion of fixed text and all mi are set to BERT's [MASK]token. Let temb be the embedding of t, including positional encoding. In order to alter BERT's behavior, which would naturally tend to replace mi with the most likely tokens based on its pretraining, we also prepend to the input sequence a learnable soft prompt p ∈RP ×d, consisting of a sequence of P vectors, with d being the dimensionality of BERT's embedding space [17]. The full input sequence to BERT is thus [p, temb]. We read out the logits corresponding to the masked tokens, i.e., [l1, . . . , lN], where each li ∈RV and V is BERT's vocabulary size. Each logit vector li is mapped to a differentiable one-hot vector oi ∈{0, 1}V , P oi,j = 1, through a Gumbel-softmax [22, 13] (with temperature τ = 1), from which the corresponding predicted token ˆti ∈{1, . . . , V } is retrieved. The resulting text prompt can be recovered as tfixed, ˆt1, . . . , ˆtN . At this point, a practical problem arises, in that standard implementations of diffusion models (e.g., Stable Diffusion [35]) CLIP's text encoder [32] is employed to embed textual prompts into a conditioning vector. Unfortunately, BERT's and CLIP's vocabulary overlap only partially. To address this issue, we employ a translation matrix M ∈{0, 1}V ×W to map each one-hot vector oi to its corresponding representation in CLIP's vocabulary, of size W. In M, each row contains a single 1, indicating the index of the corresponding token in the CLIP vocabulary. Thus, given an original onehot vector oi provided by BERT, we can translate it into its CLIP equivalent through o(C) i = oiM, indexing a token ˆt(C) i in CLIP's vocabulary. For unassigned BERT tokens, the corresponding rows in 3 Figure 1: DEXTER investigates classifier biases by optimizing a learnable soft prompt to generate text prompts. These text prompts condition a diffusion model to generate images that maximize the activation of the target class in the vision classifier. Images that correctly activate the target class are stored and later captioned for Bias Reasoning. A VLM reasons using these captions to produce human-understandable textual explanations of the model's decisions and potential biases. More details and clarifications about the pipeline can be found in the Appendices A and B. M are entirely zero. As a result of this design, the model learns to avoid predicting BERT tokens that do not have a valid mapping in CLIP, as they correspond to a zero indexing vector o(C) i , which would provide a meaningless representation when multiplied by the CLIP embedding look-up table. The textual prompt used to condition the diffusion model is thus h tfixed, ˆt(C) 1 , . . . , ˆt(C) N i . Thanks to the Gumbel-Softmax activation and the linear mapping between vocabularies, the whole process is fully differentiable, allowing us to optimize the soft prompt p, which is the only learnable parameter, through classic backpropagation. 3.2 Vision pipeline The goal of the vision pipeline is to synthesize a realistic and interpretable image that maximizes the activation of a set of neurons of a visual classifier. In the following, we refer to "neurons" in a generic way, regardless of whether they encode features (i.e., neurons in intermediate layers) or classes (i.e., neurons in the output layer, after applying softmax). Given the predicted textual prompt h tfixed, ˆt(C) 1 , . . . , ˆt(C) N i (with all tokens ensured to belong to CLIP's vocabulary as explained in Sect. 3.1), we feed it to CLIP's text encoder to obtain an embedding vector e. This vector is then used to condition a pretrained and frozen diffusion model d. In this work, we use Stable Diffusion, due to its widespread adoption and versatility in generating high-quality images from textual descriptions. Let f : I →RK be the target frozen visual classifier, pretrained over a set of C classes, providing as output the responses of a subset of K selected neurons, whose activations we intend to maximize. Hence, we can obtain the selected activation vector as n = f (d (e)). The whole vision pipeline is differentiable, enabling the definition of an optimization objective for n, which directly affects the learnable soft prompt p in the text pipeline. In order to guide the generation process to describe the behavior of the visual classifier, we introduce a neuron activation maximization loss Lact that encourages learning a textual prompt and a synthetic 4 image that maximizes the response of the selected n neurons. We define Lact as: Lact = K X i=1 lact (ni) , (1) where ni is the activation of the i-th element in n, and lact depends on whether ni is a feature neuron or a class neuron : lact(ni) = -ni, if ni is a feature neuron -log ni, if ni is a class neuron (2) 3.3 Masked pseudo-labels prediction In our preliminary experiments, we observed that, when using the activation maximization objective only, the gradient propagated to the learnable soft prompt p in the text pipeline was too small, slowing down or even preventing convergence. To address this issue, we introduce, in the text pipeline, an auxiliary mask prediction task to provide a shorter backpropagation path to p. The design of the auxiliary task gives us the opportunity to add another explainability feature to our approach: associating masked tokens with subsets of target neurons, in order to encourage the mapping between neuron activations and specific portions of the textual prompt. To facilitate this, we initialize a set of pseudo-labels y1, . . . , yN, one for each mask token position m1, . . . , mN. Each pseudo-label yi ∈{1, . . . , V } is associated with a reference loss Li, initialized to +∞, and with the set of reference neurons Ni ⊆{1, . . . , K}. At each optimization step, for each mask token mi, BERT predicts the logits li, from which we compute the (standard) softmax vector si and the corresponding predicted token ˆti, following the notation introduced in Sect. 3.1. We define the aggregated activation loss Lagg,i for the set of associated reference neurons: Lagg,i = X j∈Ni lact(nj). (3) Then, if Lagg,i is smaller than the corresponding reference loss Li, we update both the pseudo-label yi ←ˆti and the reference loss Li ←Lagg,i. If the pseudo-label yi has been set for mi (and the corresponding reference loss Li is finite), we add a cross-entropy loss term Lmask,i = -log si,yi, with si,j being the j-th element of the softmax vector si. This approach ensures that the pseudo-labels are continually refined to better align with the activation patterns of the target neurons as training progresses, while constraining the model's parameters to remain within a region of the parameter space that corresponds to meaningful, interpretable configurations. After the first iteration, where all yi are set, the overall loss function L is then: L = K X k=1 lact(nk) - N X i=1 log si,yi. (4) A possible issue that may arise is the prediction of outlier tokens that could randomly decrease the activation loss, which would possibly update the pseudo-labels to spurious values. To prevent this, for each masked token position mi, we maintain a history of aggregated losses for each word predicted during the optimization process, using the history mean for comparison with the reference loss Li. For instance, let us assume that masked token mi has been mapped by BERT to vocabulary token t∗in T training iterations, building up a list of corresponding aggregated losses h L(1) agg,i, . . . , L(T ) agg,i i . Instead of comparing Li with the most recent aggregated loss L(T ) agg,i, we check whether: 1 T T X j=1 L(j) agg,i 1 show that increasing P expands the parameter space and weakens the classifier's gradient signal, destabilizing optimization in our data-free setting, as shown in Table 5. We report mean and standard deviation for 8 randomly selected classes (4 core, 4 spurious). Table 5: DEXTER's performance using diffent number of P. P Spurious Core Avg 1 81.25 ± 18.99 94.25 ± 4.38 87.75 ± 11.68 3 75.00 ± 29.00 44.50 ± 35.89 59.75 ± 32.44 5 73.50 ± 25.66 74.00 ± 42.75 73.75 ± 34.20 10 21.25 ± 31.17 67.75 ± 40.01 44.50 ± 35.59 D Visual Explanation Details This appendix provides additional details on the evaluation of visual explanations generated by DEXTER, complementing the discussion presented in Section 4.2.1. Specifically, we outline the methodology behind CLIP-IQA, our proposed Semantic CLIP-IQA metric and the comparison between DEXTER and prior activation maximization works. Following, a comprehensive analysis of neural feature maximization. These insights further substantiate the findings reported in the main paper and demonstrate the robustness of DEXTER in generating interpretable visual explanations. D.1 Quantitative evaluation metrics: CLIP-IQA and Semantic CLIP-IQA CLIP-IQA [45] is a metric originally introduced to evaluate image quality. In this work, we leverage CLIP-IQA to compare the quality of images generated by DEXTER against those produced by other approaches. Specifically, CLIP-IQA calculates the similarity between generated images and two fixed prompts, returning the probability that an image is closer in similarity to the first prompt rather than the second. The fixed prompts used in the original formulation are "Good photo." and "Bad photo." To incorporate semantic relevance into image quality assessment, we propose Semantic CLIP-IQA. This metric follows the same evaluation protocol as CLIP-IQA but replaces the fixed prompts with class-specific prompts: "Good photo of a [CLASS]" and "Bad photo of a [CLASS]". This modification ensures that the evaluation captures not only general image quality but also the semantic alignment between the generated images and the target class. We here also report in Table 6 a quantitative comparison of visual explanation quality across existing activation maximization methods. The comparison highlights the significant performance gap between older GAN-based or optimization-based approaches and diffusion-based methods. In particular, DEXTER demonstrates superior alignment and consistency, validating its ability to generate more meaningful and semantically grounded visual explanations. D.2 Neural Feature Maximization Neural feature maximization refers our strategy to generate images that emphasize the neural features, as defined in SalientImageNet (i.e., the features of the penultimate layer of the model), learned by a neural network for a given class. By optimizing the input image to maximize the activation of 15 Table 6: Quantitative comparison between DEXTER and existing activation maximization methods for visual explanations Method CLIP-IQA Semantic CLIP-IQA Yosinski et al. [47] 0.37 ± 0.19 0.33 ± 0.23 Mahendran and Vedaldi [23] 0.82 ± 0.15 0.72 ± 0.15 Nguyen et al. [26] 0.74 ± 0.21 0.57 ± 0.30 DiffExplainer [31] 0.89 ± 0.09 0.90 ± 0.05 DEXTER (ours) 0.94 ± 0.03 0.96 ± 0.03 specific neural features, this method allows for an interpretable visualization of what the model has learned to recognize as distinctive for a class. This approach is useful for understanding how deep learning models make decisions and can help identify potential biases or spurious correlations in their predictions. In our work, neural feature maximization is performed using 2,000 optimization steps with a single word prompting strategy (see Sect. 4.2.4). We employ RobustResNet50 as the visual classifier for this task. Table 7 reports the 30 classes selected from the SalientImageNet dataset, consisting of 15 classes containing spurious features and 15 with only core features. Specifically, we selected all 15 classes from SalientImageNet in which all top-5 features were marked as spurious. Additionally, we randomly selected another 15 classes from SalientImageNet where all top-5 features were marked as core. Table 7: Bias and Non-Bias Classes Bias Classes Non-Bias Classes Class_idx Class Name Bias Class_idx Class Name Bias 706 patio ✓ 985 daisy ✗ 837 sunglasses ✓ 291 lion ✗ 602 horizontal bar ✓ 292 tiger ✗ 795 ski ✓ 486 cello ✗ 379 howler monkey ✓ 465 bulletproof vest ✗ 890 volleyball ✓ 574 golf ball ✗ 801 snorkel ✓ 582 grocery store ✗ 981 ballplayer ✓ 635 magnetic compass ✗ 746 puck ✓ 514 cowboy boot ✗ 416 balance beam ✓ 609 jeep ✗ 537 dogsled ✓ 624 library ✗ 655 miniskirt ✓ 764 rifle ✗ 810 space bar ✓ 847 tank ✗ 433 bathing cap ✓ 879 umbrella ✗ 785 seat belt ✓ 971 bubble ✗ Figure 5 extends Figure 3 by reporting additional comparison between DEXTER and DiffExplainer across different classes (the full set of comparisons is provided in the supplementary). While DiffExplainer effectively maximizes shapes, colors, and textures, its outputs often lack realism, converging on abstract or pattern-like images. In contrast, DEXTER, owing to its textual anchor, more reliably yields semantically coherent representations of the features highlighted by the SalientImageNet heatmaps. This visually confirms the results of the user study reported in Fig. 2. 16 Figure 5: Comparison of DiffExplainer and DEXTER with respect to SalientImagenet for the neural feature maximization task, covering different classes. The left column displays classes without spurious features, whereas the right column shows classes with spurious features. 17 E Slice Discovery Details This appendix provides additional details on slice discovery and debiasing using DEXTER, complementing the discussion in Section 4.2.2. Figure 6 extends Figure 1 to better explain the Slice Discovery pipeline used by DEXTER. We use the Waterbirds dataset as an example, which consists of two classes: Landbirds and Waterbirds. The majority of the landbirds samples are visibly on land, and similarly the majority of the waterbirds are in/near water. While overall this is an easy task for most classifiers, we want to ensure high performance on the small amount of images where the birds are not in their natural habitats (landbirds on water, waterbirds on land). Figure 6: Pipeline for Slice Discovery with DEXTER. The top-k words associated with a class are encoded using the CLIP Text Encoder and compared with image encodings from the biased dataset. The appropriate slice is determined based on the similarity between text and image embeddings, along with the ground truth labels of the dataset images. The dataset is then labeled into four categories: waterbird on water, waterbird on land, landbird on land, and landbird on water (see right side). DEXTER identifies the words explaining an input visual classifier, as presented in Sect. 3. Specifically, given a classifier f and a target class c, it optimizes a text prompt to extract a set of top-k class words WC = wc1, ..., wck (Tab. 8). Each word wci is encoded using CLIP's text encoder to obtain an embedding tci and the class prototype is defined as the average embedding: ̄tc = 1 k P tci Then, following the approach in B2T [16], at inference time, each image xj of the dataset is encoded into vj using CLIP's image encoder. The similarity between the image and the class is then computed as: CLIPscore = similarity(vj, ̄tc) (6) If an image has an higher similarity with the class-words that correspond to its real class it is labeled as unbiased slice (e.g. an image belonging to class landbirds with an higher similarity with landbirds class-words), while if an image has an higher similarity with its counterpart class it is labeled as biased slice (e.g. an image belonging to class landbirds has an higher similarity with waterbirds class-words). Fig. 7 presents the ROC curves obtained using the CLIP-based matching approach on the Waterbirds dataset, in comparison with other SOTA slice discovery methods [16, 8, 12]. The results illustrate how effectively the training images are partitioned into the four groups (waterbird on water, waterbird on land, landbird on land, landbird on water) by the DEXTER-derived words. Finally, we follow the debiased training scheme from [36] to train a debiased classifier on the dataset. In all our slice discovery experiments, we set k = 4 both for CelebA and Waterbirds. We used 1000 DEXTER optimization steps and single word prompting (see Sect. 4.2.4) as an activation maximization strategy to discover the above words. Tab. 8 reports the discovered words for both datasets, where class 0 of CelebA has been left blank as in B2T [16]. E.1 Top-k words selection details The top-k words in Table 8 were obtained by running the DEXTER pipeline multiple times (specifically k=4 timeless). In each run, we recorded the final (single) pseudo-token selected by the masked language model at the end of the optimization. To encourage diversity across runs, we excluded previously selected words from the candidate pool before each new run. However, we also evaluated 18 Table 8: Class-wise words discovered by Dexter for Waterbirds and CelebA datasets. Dataset Class 0 Class 1 Waterbirds fence, woods, jungle, backyard seas, sea, lake, harbor CelebA - woman, person, head, girl Table 9: Comparison between multi-run top-k word selection (ours) and single-run frequencyranked top-k selection. Method F1-score Class 0 F1-score Class 1 B2T 0.99 0.75 k = 10 0.98 0.67 k = 5 0.98 0.66 ours 0.99 0.76 how using sampling frequency ranking may affect slice discovery results. In particular, we computed results using word frequency-based ranking, evaluating performance with the top k = 5 and k = 10 most frequent words. We compare these results against our proposed sampling strategy and B2T for reference. As shown in Table 9, our strategy with a single word yields superior performance. Figure 7: ROC curves for slice discovery on the Waterbirds dataset. Each curve is obtained by comparing the slice assignments predicted by each method with the ground-truth slice labels released with the dataset. F Bias Reasoning Details Given a classifier f and class c, we optimize a prompt t∗such that a diffusion model generates images x1, ..., xM that maximize the classifier's output for c. Each generated image correctly predicted by the classifier is captioned using a vision-language model g, producing descriptions di(g(xi)). A language model h then summarizes the captions into a single report, describing the features associated with class c and possible biases. This process is fully data-free: both image generation and reasoning depend only on the model's behavior. For bias reasoning, we generate 50 images classified as the target class using DEXTER over the course of up to 5,000 optimization steps, employing a multi-word prompting strategy. These images are then captioned using ChatGPT-4o mini with the designated Caption System Prompt (detailed below). Subsequently, another instance of ChatGPT4o mini analyzes these captions using the following Report System Prompt to extract key class attributes and identify potential biases in the model. To generate the captions and the reports with ChatGPT-4o mini we set a temperature of 0.2 to mitigate hallucinations. The max tokens parameter is set to 0, allowing unrestricted response generation. The top_p is fixed at 1.0, while both the frequency_penalty and the presence_penalty are set to 0.0. Finally, the model generates n = 1 output. 19 We conducted multiple experiments on different visual classifiers: • The reports presented in Fig. 4 and Appendix F.1 were obtained by analyzing a ResNet18 model trained on the FairFaces dataset. • Figure 8, instead, provides a bias/not bias categorization for ViT, AlexNet, ResNet50, and RobustResNet50, all trained on the ImageNet-1000 dataset, rather than full bias reasoning reports. Supplementary material contains the full bias reports for all the 30 classes (see Table 7) when analyzing RobustResNet50. Caption system prompt You are an assistant specialized in analyzing pictures. Your goal is to generate the most accurate and detailed description. You will be given a picture and you will have to describe it inserting all the key attributes that you find that could be inherent. Keep the description short (one sentence) while including all the key attributes. - You have to return only the description of the image. Don't return other information. - Don't describe the image in a poetic way, just describe the visual elements. - Don't include any information about the image that is not visible. - Don't include any information about the image that is not relevant to the description. - If the subject is a person describe the ethnicity features and age features. - Keep the description short (one sentence) while including all the key attributes. Examples: Input image: [IMG] Input text: Describe the image Output: [CAPTION] Report system prompt Context: This task involves analyzing images generated through a process of Class Activation Maximization (CAM). Using a deep neural network trained on a specific dataset, the CAM approach identifies visual patterns that strongly activate a specific class label in the network. To enhance this analysis, we use a trainable soft prompt integrated with a Stable Diffusion model. The soft prompt is optimized to condition Stable Diffusion to create images that maximize the activation of a chosen class. For each generated image, if the predicted class aligns with the target class, a caption describing the image is stored. Analysis: Given a trainable soft prompt and a Stable diffusion model. We optimize the soft prompt to conditionate stable diffusion to generate pictures that maximize the prediction of a class. If the class predicted is the same of the class we want to maximize, than we store a caption of that picture. Task: Given a list of sentences that you will receive, you have to reason on the key attributes that the model uses to maximize a specific class. Than you've to do a report with the explanation of your reasoning. The report must end saying if the class is biased or not. If you reason that the class is biased you must explain why otherwise you must explain why the class is not biased. To write the report follow this structure: - Title: ### Report on Class Activation Maximization for the Class 'CLASS_NAME' - Hypothesis on Model Bias the class to analyze will be given by the user with the list of sentences. The shape of the user prompt will be the following: User: 'Class: [CLASS] - Captions: [LIST OF SENTENCES]' 20 F.1 Gender Bias Reports In this section, we report DEXTER's reports obtained by analyzing the ResNet18 classifier trained on the FairFaces dataset. As described in Section 4.2.3, we conducted two training procedures for this visual classifier. In the first procedure, we injected a gender bias across the classes, whereas in the second, we employed a balanced training set. Report for class 0 (20-29 years) of a biased classifier (overrepresentation of men) The analysis suggests that the class '20 to 29 years old people' may exhibit some bias. While the model captures a range of ethnicities and genders, the emphasis on certain skin tones, facial features, and expressions could lead to a skewed representation of what is considered 'youthful'. - Skin Tone Bias: The frequent mention of light brown and medium brown skin tones may indicate a bias towards these tones, potentially neglecting a broader spectrum of skin colors that also represent youth. - Facial Feature Bias: The focus on specific facial features that are often idealized in media could lead to a narrow definition of youth, which may not encompass the full diversity of young adults. - Gender Representation: The slight focus on the overrepresentation of male subjects could suggest a bias in the training data, which may not equally represent young women. In conclusion, while the model successfully identifies attributes associated with the class '20 to 29 years old people,' it is essential to recognize the potential biases in representation. The model's predictions may not fully encompass the diversity of this age group, indicating a need for more balanced training data to ensure equitable representation across all demographics. Report for class 0 (20-29 years) of a balanced classifier (equal representation of men and women) Based on the analysis of the model, it does not appear to exhibit significant bias in identifying the class of 20 to 29 years old individuals. The diversity in hair and eye color, skin tone, and gender representation suggests that the model is capable of recognizing a broad spectrum of individuals within this age group. The focus on well-defined facial features and neutral expressions aligns with common perceptions of young adults, and the inclusion of various ethnic backgrounds indicates an inclusive approach. In conclusion, the class of 20 to 29 years old people, as represented by the model, does not seem to be biased. The attributes identified are consistent with a diverse and inclusive representation of young adults. Report for class 1 (50-59 years) of a biased classifier (overrepresentation of women) Based on the analysis of the model, it can be concluded that the class '50 to 59 years old people' is biased. The bias stems from several factors: - Representation of Age: The model predominantly features youthful characteristics, which may not accurately reflect the physical appearance of individuals aged 50 to 59. This suggests that the model may be influenced by societal standards that equate youthfulness with attractiveness, leading to a misrepresentation of the target age group. - Ethnic and Gender Bias: The emphasis on lighter skin tones and the predominance of female subjects indicate a lack of diversity. This could perpetuate stereotypes and fail to represent the true diversity of individuals within the 50 to 59 age range. - Societal Stereotypes: The model's reliance on neutral or serious expressions may reflect societal stereotypes about older individuals, which can further skew the representation of this age group. In conclusion, while the model attempts to maximize the activation of the class '50 to 59 years old people,' it does so through a lens that is biased towards certain physical characteristics, expressions, and demographic representations. This highlights the importance of ensuring diverse and representative training data to mitigate bias in machine learning models. Report for class 1 (50-59 years) of a balanced classifier (equal representation of men and women) Based on the analysis, it appears that the class '50 to 59 years old people' is not inherently biased. The model successfully captures a range of attributes that reflect the diversity of individuals within this age group. However, there are potential biases in the representation of expressions and the predominance of serious or neutral faces, which may not fully encompass the emotional spectrum of individuals aged 50 to 59. Moreover, the model's training data could influence its perception of age, potentially leading to stereotypes about how older individuals should look or behave. For instance, the emphasis on serious expressions may inadvertently reinforce the stereotype that older adults are less expressive or joyful. In conclusion, while the class itself is not biased, the representation of certain attributes, particularly expressions and emotional portrayals, may benefit from a broader range of depictions to avoid reinforcing stereotypes associated with aging. 21 F.2 Bias Report Quality Assessment Metrics In this section, we provide additional explanations for the metrics used to evaluate the quality of DEXTER's bias reports, as shown in Table 2. Following Liu et al. [20], we use the G-eval consistency metric, which involves prompting a large language model (LLM) to assess a text's factualness, scoring each report on a scale from 1 to 5. Since the original prompts for these metrics were designed for summarization, we create the following G-eval Consistency Prompt to better align with our task. Following the G-eval format, we use the Mean Opinion Score of LLM (MOSLLM) metric. This metric uses the MOSLLM prompt asking the LLM to rate between 1 and 5 the generated reports according to how well the report accurately describes the existence of bias in the classifier. Additionally, the MOShumans score represents the average rating given by participants in the user study. For both the G-eval and MOSLLM configurations, we adopt the same settings as Liu et al. [20]. Specifically, we use gpt-4-0613 with a temperature of 2, max_tokens = 0, top_p = 1.0, frequency_penalty = 0, and presence_penalty = 0. To enhance response diversity, we generate n = 20 outputs. Furthermore, in both G-eval and MOS LLM system prompts, we provide the same evaluation instructions that were given to human evaluators during the user study, also following the best practices to reduce hallucinations provided by [4]. To assess whether the LLM's mean-opinion scores are statistically indistinguishable from those assigned by human annotators, we applied an independent-samples two one-sided equivalence test (TOST). The test compared the distribution of MOSLLM to that of MOShumans obtaining a p ≪0.05. Thus, within the predefined equivalence bounds (±0.5 for a Likert scale), the LLM's ratings can be considered statistically equivalent to human judgments, indicating strong agreement between the two sets of scores. G-eval Consistency system prompt **Instructions for Evaluation**: In this task, we trained a visual classifier twice: 1. First, using a dataset with gender-biased images. 2. Second, using a balanced dataset. Your task is to evaluate how effectively a given textual description identifies the presence or absence of gender bias in the visual classifier: - **Visual Classifier**: A machine learning model trained to distinguish between two classes (e.g., Class 1: "20-29 years old people" vs. Class 2: "50-59 years old people"). Please make sure you read and understand these instructions carefully. Please keep this document open while reviewing, and refer to it as needed. Evaluation Criteria: Consistency (1-5) - the factual alignment between the question and the description. A factually consistent description contains statements that are entailed by the question. Annotators were also asked to penalize descriptions that contained hallucinated facts. Evaluation Steps: 1. Read the question carefully and identify the main topic and key points. 2. Read the description and compare it to the question. Check if the description covers the main topic and key points of the question, and if it presents them in a clear and logical order. 3. Assign a score for consistency on a scale of 1 to 5, where 1 is "Bad Description" and 5 is "Excellent Description" based on the Evaluation Criteria. Example: Question: {{Question}} Description: {{Description}} Evaluation Form (scores ONLY): - Consistency (1-5): 22 MOSLLM system prompt **Instructions for Evaluation**: In this task, we trained a visual classifier twice: 1. First, using a dataset with gender-biased images. 2. Second, using a balanced dataset. Your task is to evaluate how effectively a given textual description identifies the presence or absence of gender bias in the visual classifier: - **Visual Classifier**: A machine learning model trained to distinguish between two classes (e.g., Class 1: "20-29 years old people" vs. Class 2: "50-59 years old people"). Please make sure you read and understand these instructions carefully. Please keep this document open while reviewing, and refer to it as needed. Evaluation Criteria: Mean Opinion score (1-5) - Assign a rating to the description based on its effectiveness: - 1: Bad description (fails to identify bias or provide relevant details). - 2: Poor description (some effort to address bias, but lacks clarity or completeness). - 3: Neutral (adequate but not insightful; partially addresses bias). - 4: Good description (clear and mostly thorough in addressing bias). - 5: Excellent description (comprehensive, clear, and detailed in identifying bias). Evaluation Steps: 1. Read the question carefully and identify the main topic and key points. 2. Read the description and compare it to the question. Check if the description covers the main topic and key points of the question, and if it presents them in a clear and logical order. 3. Assign a score for bias identification on a scale of 1 to 5, where 1 is "Bad Description" and 5 is "Excellent Description" based on the Evaluation Criteria. Example: Question: {{Question}} Description: {{Description}} Evaluation Form (scores ONLY): - Rating (1-5): 23 F.3 Bias identification of multiple vision classifiers In Fig. 8 we report an example of disparity analysis across ViT, AlexNet, ResNet50, and RobustResNet50 for multiple ImageNet classes. By revealing class-specific biases and failure patterns, DEXTER helps identify where each model struggles and can guide data collection efforts when biases are consistently observed across classifiers. To select key neural features for these classifiers, we ranked penultimate-layer neurons by their weights to each class and selected the top 5, mirroring the Salient ImageNet method but without any training data. Figure 8: Bias analysis across classifiers and SalientImageNet classes. Each cell shows DEXTERgenerated visual explanations and bias assessments ("Biased" or "Not Biased"). While Lion is unbiased across models, Jeep is biased in ResNet50 but not in its robust version. Baseball Player remains biased in all models, suggesting dataset-level bias. 24 G Additional details on Ablation Study Figure 9: Procedure to compute the percentage of activations. In this section, we provide further details on how we compute the activation score reported in Table 3 and Table 4. As shown in Figure 9, once the word(s) are obtained at the end of the optimization process, we prompt Stable Diffusion to generate 100 distinct images using a fixed prompt tfixed (from the optimization stage) concatenated with the discovered word(s) ˆtC. We then sum all inference steps in which the visual classifier predicts the target class. Figure 10 illustrates examples of the different text-prompt strategies, described in Table 3, alongside their corresponding activation scores for the class Tiger and the class Snorkel. Figure 10: Examples of corresponding text prompts and generated images for class activation maximization. For a non-spurious class like tiger, all generated images easily activate the target class. However, in the case of the snorkel class, DEXTER is able to generate significantly more images that maximize the activation, and exposes the other features the model pays attention to. 25 H User Study Details Figure 11: Example of a question from the first section of the user study. Each user study consisted of two parts. The first part aimed to assess the ability of DEXTER to visually highlight the focus of the target classifier for a specific neural feature, while the second part evaluated the accuracy of the textual bias reports generated by DEXTER. In the first part of the study, each participant was presented with 15 randomly images (explaining specific feature in SalientImageNet) drawn from the outputs of DiffExplainer and DEXTER. An example is shown in Fig. 11. Participants were asked to assess whether the generated image accurately represented the attended region of the real image based on attributes such as color, shape, texture, material, context, or semantic similarity. Additionally, two extra options were provided: "Several Elements" for cases where multiple similarity criteria were met, and "None" for instances where none of the criteria applied. The second part of the study aimed to evaluate the ability of DiffExplainer reports to detect the presence of bias in the target classifier. The target classifier was trained both with and without gender bias, and participants were presented with all four reports generated by DEXTER. Examples of questions for class 0 (20-29 years old) are provided in Fig. 12 and Fig. 13, corresponding to the models trained with and without the injected gender bias, respectively. Participants were asked to rate, on a scale from 1 to 5, the extent to which the generated report reflected the presence or absence of bias. The user study was conducted with 100 distinct MTurk workers, with an average completion time of 10.32 minutes. Each feature explanation in the first part received an average of 10.38 responses for DiffExplainer and 10.82 for DEXTER. To minimize response variability, participants were required to provide a textual justification for their answers in both sections of the questionnaire before submission. The compensation for each worker was $0.50. I Diffusion and LLM's Hallucination evaluation I.1 Randomness in the diffusion generation process To assess the stability of Stable Diffusion, we performed intra-class evaluation using DEXTER's final prompt. Across three independent runs (100 images each), we measured activation scores with the target model. If the image generation is unstable, scores will vary significantly. Instead, our consistent results indicate stability. We report mean and standard deviation for 8 randomly selected classes (4 core, 4 spurious) across three independent runs (Tab. 10). 26 Figure 12: Example of a question from the second section of the user study (biased). Figure 13: Example of a question from the second section of the user study (not biased). Table 10: Intra-class evaluation of randomness in Stable Diffusion generation process. Type Class Avg Spurious 890 55.33 ± 2.49 795 79.00 ± 1.63 655 96.66 ± 0.94 706 100.0 ± 0.00 Core 291 100.0 ± 0.00 486 95.66 ± 1.24 514 86.66 ± 1.24 624 95.66 ± 1.88 I.2 Bias propagation To verify that DEXTER's outputs align with the classifier-rather than reflecting biases from BERT or Stable Diffusion-we ran a robustness test. We manually injected spurious cues into prompt initialization (e.g., replacing the starting auxiliary pseudo-target with lion for the class tiger) to simulate strong upstream bias. Then, for each class in SalientImageNet subset of our paper (split into "core" and "spurious" categories), we generated 100 images and measured how often the classifier activated the correct class. We report average classifier activation scores (as defined in Tab. 3 of the paper) for spurious, core classes, and their average. These results in Tab. 11 show that even with adversarially biased prompts, DEXTER recovers class-relevant visual patterns, aligning with the 27 classifier. Classifier-driven optimization helps correct upstream bias and grounds both visual and textual outputs in model behavior. Table 11: Comparison of Spurious, Core, and Average scores under different bias conditions. Condition Spurious Core Avg Injected Bias 65.3 88.4 76.8 No Bias 63.0 87.6 75.4 We also evaluated how DEXTER compensates for bias propagation. Specifically, we recorded the sequence of words selected as pseudotargets during optimization. This was done for both the biased setting and the standard setting, where the initial pseudotarget is chosen randomly. As shown in the Table 12, in both cases DEXTER consistently converges toward the most meaningful word, with the selection path gradually shifting from a random concept to the intended target. This trajectory suggests that the optimization process does not simply lock onto the first spurious or core concept it encounters, but instead explores a range of semantically plausible candidates before progressively refining the prompt toward a stable and interpretable solution, indicating a non-greedy and convergent behavior rather than an early commitment driven by the mask pseudo-label loss. Convergence analysis through keyword trajectory during the DEXTER optimization process. The Table 12 illustrates how DEXTER transitions from initially random concepts to progressively more semantically coherent and domain-specific words, demonstrating convergence toward a stable explanation. I.3 Analysis of LLM hallucination impact Given that large language models can generate statements influenced by their prior knowledge rather than by factual evidence, we systematically assess the fidelity of our reports and how much they are aligned with the visual classifier decision-making process. We aim to ensure that each report highlights genuine visual patterns exploited by the classifier rather than hallucinated elements. Concretely, for each of the 30 SalientImageNet classes in our experimental setup (Tab. 7), we verify whether the single most salient visual cue extracted from the corresponding DEXTER report is truly grounded. Given a class c we form BASELINE PROMPT: "a picture of a [c]", CUE PROMPT: "a picture of a [c] with [cue]0 ... and '[cue]n', where n is the total number of cues obtained automatically from the generated reports via a GPT-4o mini class-cues extractor. Then, we generate 100 images per prompt with Stable-Diffusion and measure the Activation Score (AS) on the frozen RobustResNet50 used throughout the paper (4.2.4). A cue is grounded when AScue > ASbaseline; otherwise it is neutral or wrong. Statistical significance & confidence interval. For each class we measure ∆i = AScue,i - ASbaseline,i. Across the 30 classes we obtain ̄∆= 16.33 pp, s∆= 23.84 pp. The two-sided 95% confidence interval is ̄∆± t0.975,29 s∆ √ 30 = [ 7.43, 25.24 ] pp. Both a paired t-test (t(29) = 3.79, p = 7.8 × 10-4) and a Wilcoxon signed-rank test (W = 19, p = 2.9 × 10-4) confirm that the improvement is statistically significant. Aggregate results. Table 13 contrasts the mean Activation Scores for the two prompts. Adding the report cue raises the mean score from 64.73% to 81.06% ( +16.33 pp). • 20/30 classes (67%) improve (peak +81 pp for miniskirt), confirming that the cues capture genuinely discriminative evidence. • 7 classes already achieve 100 % with the baseline prompt; the cue therefore leaves performance unchanged. 28 Table 12: Word-wise convergence path for biased vs unbiased initialized prompt. Grocery Store (582) Patio (706) Biased Not Biased Biased Not Biased motel fan restaurant head library coffin school book wedding factory house man party sale pool the restaurant factory residence republic party distillery tree wedding market window motel supermarket house bakery courtyard supermarket terrace patio Table 13: Activation Scores averaged over 30 classes (↑/↓: cue better/worse than baseline). Prompt Mean AS (%) ∆ Class split Baseline 64.73 - Cue (report) 81.06 +16.33 20 ↑3 ↓7 = 30 • 3 classes degrade: bubble ∆= -5 pp; the cue is correct but pertains only to a subset of bubble instances. tanks ∆= -1 pp (100 % →99 %); a negligible loss. rifle ∆= -17 pp; the cue "individuals in dark clothing" marking the sole clear failure. Overall assessment. DEXTER explanations are therefore strongly grounded: in 27 out of 30 classes the cue is beneficial or neutral, and only one class (rifle) exhibits evidence of hallucination. In the two minor degradations (bubble,tanks) the reports still isolate genuine-though partial-visual cues. Evaluation Implications. The consistent increase in classifier confidence when report-derived cues are injected into the prompt demonstrates that DEXTER's explanations faithfully align with the model's decision boundary, rather than reflecting spurious or hallucinatory artifacts. Together with the text-based metrics reported in Table 2 for the FairFaces dataset (G-Eval consistency, STS, MOS), these findings confirm that DEXTER delivers genuinely grounded insights into the visual classifier's decision-making process. Pipeline Robustness. Furthermore, this evaluation implicitly validates the collaborative operation of all pretrained components (BERT, CLIP, Stable Diffusion, VLM) in our optimization pipeline. The target visual classifier guides the prompt optimization. BERT selects discriminative keywords, the diffusion model generates contextually relevant images, and the captioning VLM produces faithful captions. As a result, the end-to-end process avoids propagating biases or hallucinations from intermediate models and yields explanations that are solidly grounded in the classifier's behavior. I.4 Baseline prompts vs Cue prompts This section, referring to Sec. I.3, reports the system prompt used to extract the relevant visual cues from DEXTER's textual explanations. Furthermore, Table 14 compares class-wise activation scores produced by the baseline prompt "A picture of a [CLASS]" and by the CUE-enriched prompt. Large gains appear for classes that were poorly activated in the baseline (e.g., space bar improves from 6 →57 , hockey puck from 0 →79). The results demonstrate that adding concise semantic cues markedly improves class-specific guidance and reduces hallucinations during image generation. 29 Report's cues extractor system prompt You are an assistant that reads a bias-analysis report about an ImageNet class and extracts concrete visual cues that the report claims are important for that class. Your task 1. Identify up to 5 key visual phrases (2-5 words each) that: • are explicitly mentioned in the report; • describe tangible elements that can appear in an image (objects, attributes, background, actions); • are likely to trigger the classifier according to the report. 2. Return your answer in JSON with two fields: "'json "key-phrases": ["phrase1", "phrase2", ...], "full-prompt": "a picture of a with "phrase1" and "phrase2" and ..." 30 Table 14: Activation maximization scores for each class: comparison between the baseline prompt "A picture of a [CLASS]" and the CUE prompt obtained from the DEXTER's reports. Class Baseline prompt AS CUE prompt AS volleyball A picture of a volleyball 66 A picture of a volleyball with young women playing and mixed-gender teams and variety of scenarios 78 space bar A picture of a space bar 6 A picture of a space bar grid layouts with framed artworks and black-and-white imagery 57 umbrella A picture of an umbrella 95 A picture of an umbrella with vibrant outdoor scenes and garden scenes 100 baseball player A picture of a baseball player 98 A picture of a baseball player with female athletes and softball context 100 bubble A picture of a bubble 100 A picture of a bubble with whimsical themes 95 balance beam A picture of a balance beam 0 A picture of a balance beam with athletic attire and physical activities and indoor training environments and group dynamics and artistic performances 27 cowboy boot A picture of a cowboy boot 83 A picture of a cowboy boot with various types of boots 96 patio A picture of a patio 100 A picture of a patio with modern architecture and expansive outdoor spaces and affluent homes and specific landscaping styles 100 tank A picture of a tank 100 A picture of a tank with camouflage and military environment and armored vehicle 99 dark glasses A picture of dark glasses 8 A picture of dark glasses with facial hair and an older man 20 daisy A picture of a daisy 100 A picture of a daisy with vibrant colors and natural contexts and red daisies 100 howler monkey A picture of a howler monkey 0 A picture of a howler monkey with lush, green environments and natural habitat 32 tiger A picture of a tiger 100 A picture of a tiger with striped fur and natural habitat and majestic posture 100 library A picture of a library 76 A picture of a library with books and organized indoor area and spacious room with furniture and people 97 seat belt A picture of a seat belt 56 A picture of a seat belt with human subjects 73 rifle A picture of a rifle 85 A picture of a rifle with individuals in dark clothing and jackets and sunglasses and aiming 68 grocery store A picture of a grocery store 84 A picture of a grocery store with diversity of food items and presence of people and colorful displays 95 snorkel A picture of a snorkel mask 41 A picture of a snorkel mask with young individuals and sharks 94 dogsled A picture of a dogsled 98 A picture of a dogsled with snowy landscapes and human-animal interaction and specific dog breeds 100 magnetic compass A picture of a magnetic compass 6 A picture of a magnetic compass with silver compass pendant and craftsmanship and design 26 horizontal bar A picture of a horizontal bar 6 A picture of a horizontal bar in sports and athletic environment 8 ski A picture of a ski 56 A picture of a ski with young individuals with ski attire 100 miniskirt A picture of a miniskirt 19 A picture of a miniskirt with women wearing miniskirts 100 lion A picture of a lion 100 A picture of a lion with distinctive physical traits and social behavior and natural habitat and human interactions 100 hockey puck A picture of a puck 0 A picture of a puck with ice hockey gameplay and player confrontations 79 swimming cap A picture of a swimming cap 92 A picture of a swimming cap with children in joyful scenarios with adults and certain hair types and smiling expressions 99 bulletproof vest A picture of a bulletproof vest 67 A picture of a bulletproof vest with casual attire 89 cello A picture of a cello 100 A picture of a cello with musical instrument 100 golf ball A picture of a golf ball 100 A picture of a golf ball with spherical objects and white color 100 jeep A picture of a jeep 100 A picture of a jeep, landrover with off-road capability 100 Mean - 64.73 - 81.06 31 NeurIPS Paper Checklist 1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Answer: [Yes] Justification: The abstract and introduction of the DEXTER paper clearly and accurately reflect the paper's contributions and scope. The three central use cases mentioned in the abstract (activation maximization, slice discovery and debiasing, and bias explanation) are all thoroughly addressed and empirically validated in the main body. Additionally, the introduction carefully outlines both the limitations of existing work and how DEXTER addresses them, reinforcing that the scope of the claims is realistic and well-motivated. Guidelines: • The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. 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2510.14727
|
The Pursuit of Diversity: Multi-Objective Testing
of Deep Reinforcement Learning Agents
Antony Bartlett[0009−0008−2654−8556], Cynthia Liem[0000−0002−5385−7695],
Annibale Panichella[0000−0002−7395−3588]
Delft University of Technology, Delft, The Netherlands
a.j.bartlett@tudelft.nl, c.c.s.liem@tudelft.nl, a.panichella@tudelft.nl
Abstract. Testing deep reinforcement learning (DRL) agents in safety-
critical domains requires discovering diverse failure scenarios. Existing
tools such as INDAGO rely on single-objective optimization focused solely
on maximizing failure counts, but this does not ensure discovered scenar-
ios are diverse or reveal distinct error types. We introduce INDAGO-Nexus,
a multi-objective search approach that jointly optimizes for failure likeli-
hood and test scenario diversity using multi-objective evolutionary algo-
rithms with multiple diversity metrics and Pareto front selection strate-
gies. We evaluated INDAGO-Nexus on three DRL agents: humanoid walker,
self-driving car, and parking agent. On average, INDAGO-Nexus discovers
up to 83% and 40% more unique failures (test effectiveness) than INDAGO
in the SDC and Parking scenarios, respectively, while reducing time-to-
failure by up to 67% across all agents.
Keywords: Multi-objective search, deep reinforcement learning, surro-
gate models
1
Introduction
Reinforcement Learning (RL) [27], and more recently Deep Reinforcement Learn-
ing (DRL) [21], have emerged as powerful tools for automating complex tasks in
dynamic and uncertain environments [22, 26]. DRL agents learn through trial-
and-error interactions with their environment and are increasingly deployed in
Cyber-Physical Systems (CPS), such as self-driving cars (SDCs) and humanoid
robots [4], where safety, reliability, and adaptability are critical.
Ensuring the reliability of DRL-based systems remains a significant chal-
lenge [2, 29]. Unlike conventional software, DRL agents are opaque, stochastic,
and highly sensitive to reward signals. Their behavior depends not only on a
learned policy but also on vast environment and action spaces. As a result,
edge-case scenarios—those most likely to trigger failures—are difficult to iden-
tify, and traditional testing approaches often fall short [29]. Real-world incidents
involving autonomous agents [9,12,18] underscore the consequences of failing to
detect such rare conditions during testing.
Simulation-based testing is a widely adopted technique for evaluating DRL
agents, especially in safety-critical domains such as autonomous agents. It al-
lows developers to generate and execute test scenarios in a controlled and re-
producible setting, substantially reducing the time, cost, and safety risks of
arXiv:2510.14727v1 [cs.LG] 16 Oct 2025
2
Antony Bartlett, Cynthia Liem, Annibale Panichella
real-world testing [10, 16]. A test scenario refers to a specific configuration of
the environment—including initial agent state, obstacle placement, and task-
specific parameters—that defines the context in which the agent is evaluated.
Despite its advantages, simulation-based testing still remains resource-intensive,
and exhaustive exploration of the test input space is often infeasible due to its
combinatorial complexity.
To address these challenges, Biagiola and Tonella [2] proposed the use of sur-
rogate models to guide test generation for DRL agents. Their framework, INDAGO,
builds a surrogate model from training logs to predict whether a given test sce-
nario is likely to cause agent failure—without executing it in simulation. INDAGO
combines this model with two search strategies: Hill Climbing (HC) and a single-
objective Genetic Algorithm (GA). Their empirical evaluation across multiple
DRL tasks showed that (1) surrogate models substantially reduced simulation
time, and (2) both HC and GA outperformed earlier random-search-based meth-
ods [29] in identifying more failure-inducing scenarios.
While INDAGO effectively increases the number of failing scenarios, it fo-
cuses solely on failure likelihood, overlooking diversity. This is a key limita-
tion—generating more failures does not guarantee that they are distinct. Diver-
sity is crucial in testing as it promotes broader coverage of the agent’s operational
space and improves the chances of uncovering unique faults, helping to address
the curse of rarity problem [20].
In this paper, we propose INDAGO-Nexus, a multi-objective search frame-
work for DRL testing that jointly optimizes for failure probability and scenario
diversity. We investigate:
RQ1: How does multi-objective search perform in finding diverse failures?
RQ2: How effective is INDAGO-Nexus compared to state-of-the-art INDAGO?
To tackle this inherently multi-objective problem, we employed multi-objective
evolutionary algorithms (MOEAs). MOEAs are well-suited for this task as they
efficiently explore trade-offs and produce a set of Pareto-optimal solutions [6,14,
15]. We experimented with two diversity metrics (Euclidean distance and PCA-
based clustering), two MOEAs (NSGA-II [7] and AGE-MOEA [23]), and two
Pareto-front selection strategies (highest failure likelihood and knee point [31]).
We benchmarked INDAGO-Nexus on three DRL agents: self-driving car, humanoid
walker, and parking agent.
Our results indicate INDAGO-Nexus discovers up to 83% and 40% more unique
failures than INDAGO in the SDC and Parking scenarios, respectively, while re-
ducing time-to-failure by up to 67%. Knee-point selection consistently produces
the best results across most configurations, while different diversity metrics show
varying effectiveness depending on the scenario.
Therefore, our contributions are:
– We propose INDAGO-Nexus, a multi-objective approach to generate diverse
test environments for DRL agents.
– We demonstrate INDAGO-Nexus discovers up to 83% more unique failures
than INDAGO while reducing time-to-failure by up to 67%.
The Pursuit of Diversity
3
– We provide a complete replication package with implementation and exper-
imental data1.
2
Background and Related Work
This section describes the key background concepts used in this work and sum-
marizes the related work.
2.1
Deep Reinforcement Learning
Deep Reinforcement Learning (DRL) extends Reinforcement Learning (RL) by
using deep neural networks as function approximators to handle high-dimensional
inputs (e.g., images) and continuous action spaces (e.g., steering angles in au-
tonomous driving) [22]. Instead of explicitly representing policies or value func-
tions, DRL learns them directly from raw observations and rewards using deep
learning architectures.
This capability makes DRL particularly suitable for complex tasks in cyber-
physical systems such as self-driving cars (SDCs), where the environment is
dynamic and the decision space is large. In this work, we focus on model-free DRL
algorithms, where agents learn directly from interactions without an explicit
model of the environment. For example, the DRL agent for the parking scenario
we consider in our case study (see Section 4) was trained with Proximal Policy
Optimization (PPO) [24], a widely adopted policy-gradient method known for
its stability and sample efficiency.
The agent’s goal is to maximize the cumulative reward it receives in the
long run [27].
2.2
Black-box Testing
In black-box testing, diversity is a common strategy for exposing faults by vary-
ing inputs and observing outputs [13]. For DRL agents, we focus on input diver-
sity—variations in environment configurations—to guide test generation without
costly simulations, combined with a surrogate to estimate failure likelihood.
Related work on diversity-based testing for deep neural networks includes ap-
proaches that optimize for input diversity to improve fault detection. Aghababaeyan
et al. [1] proposed black-box testing techniques that use test case diversity to
generate diverse test inputs for deep neural networks. While their work demon-
strates the effectiveness of diversity-based approaches, their diversity metrics are
primarily designed for image inputs rather than the tabular data that character-
izes our MLP-based surrogate models. Our work adapts the concept of diversity-
driven testing to the specific context of DRL environment configurations, which
are typically represented as structured tabular data with both continuous and
categorical features.
4
Antony Bartlett, Cynthia Liem, Annibale Panichella
"env_config": {
"goal_lane_idx": 0,
"heading_ego": 0.96,
"parked_vehicles_lane_indices": [
1, 3, 6, 8, 9,
10, 11, 12, 14, 18
],
"position_ego": [
1.83, -4.96
]
}
(a) Environment configuration
(b) Agent behavior visualization
Fig. 1: Parking scenario showing (a) environment configuration details used by
DRL as a starting point, and (b) the DRL agent behavior. In (b), the red dot
indicates the target parking spot, green dots represent parked cars, the black
dot marks the agent’s starting position, and the blue dots trace its trajectory.
2.3
Surrogates
Surrogate models are lightweight machine learning approximations that learn to
predict the behavior or outcomes of computationally expensive processes based
on historical data. In the context of DRL testing, these models have proven
particularly effective at reducing evaluation costs [2]. Specifically, given an en-
vironment configuration (Figure 1a), a surrogate model can predict the failure
probability in the range [0, 1], thereby enabling efficient test generation without
requiring costly simulation executions.
2.4
Testing DRL Agents
The most directly related work is by Biagiola and Tonella [2], who introduced
INDAGO, a test generation framework that leverages surrogate models trained on
DRL logs to predict whether a given test environment is likely to cause failure.
These surrogates return a continuous score in [0, 1], representing the estimated
failure probability of an environment. INDAGO combines this prediction with
search-based strategies, namely Hill Climbing [25] and Genetic Algorithms [11],
to efficiently explore the input space.
Our work builds on INDAGO by introducing a second optimization objective:
input diversity. This extension aims to increase the diversity of the discovered
failures, thereby improving coverage of the agent’s operational space.
3
Approach
The goal of our approach is to generate test scenarios that are both failure-
inducing and diverse, increasing the likelihood of uncovering unique faults in
DRL agents. We build upon the INDAGO framework [2], which uses a surrogate
model trained on DRL training logs to predict whether a given environment is
likely to cause a failure. While INDAGO focuses solely on failure likelihood, we
1 10.6084/m9.figshare.29204687
The Pursuit of Diversity
5
extend it to a multi-objective test generation framework that also accounts for
scenario diversity—crucial for uncovering unique faults. Hence, we formulate the
problem of generative diverse failure-inducing test scenarios as follows:
Problem Definition 1 Given a DRL agent A, a set of test scenarios E, and a
surrogate model s : E →[0, 1] that estimates the likelihood of failure for a given
e, the goal is to generate a set of test scenarios Esel ⊆E that maximize the
likelihood of A to fail while maximizing diversity among the selected scenarios.
More formally, the goal is to find a set of environments Esel that optimizes the
following objectives:
(
max O1 = s(e)
(Failure Likelihood)
max O2 = Div(e, Esel)
(Input Diversity)
(1)
where O1 (Failure Likelihood) is the predicted likelihood of failure for a given
environment e; and O2 (Input Diversity) is a measure of how diverse the selected
environments are compared to the other test scenarios in Esel.
In the following, we describe the surrogate model, diversity metrics, and the
multi-objective optimization framework.
3.1
Surrogate models
Following the prior work by Biagiola and Tonella [2], we adopt a lightweight
surrogate model to estimate the failure likelihood of a test scenario without
executing it. Specifically, we utilize a multi-layer perceptron (MLP) classifier
trained on data collected during the DRL agent’s training episodes. The MLP
architecture is well-suited for this task due to the relatively small training dataset
(up to 10k labeled environments) and the low-dimensional nature of environment
feature vectors (e.g., 24-dimensional in the parking scenario). The surrogate
outputs s(e) ∈[0, 1], representing failure probability for environment e.
3.2
Diversity Metrics
We explore two diversity methods: Euclidean distance computes the mean
distance between a new scenario and previously selected ones, and PCA-based
clustering [8] measures distance to the nearest cluster centroid in reduced di-
mensionality space. Both methods handle variable-length features through one-
hot encoding and zero-padding.
3.3
Multi-Objective Evolutionary Testing
To generate a diverse set of failure-inducing test scenarios, INDAGO-Nexus em-
ploys MOEAs that evolve environment configurations to simultaneously max-
imize (1) the likelihood of failure, as predicted by a surrogate model, and (2)
diversity with respect to a growing archive of previously selected tests. We use
two MOEAs: NSGA-II [7] and AGE-MOEA [23] to optimize both failure likeli-
hood and diversity.
6
Antony Bartlett, Cynthia Liem, Annibale Panichella
Algorithm 1 Multi-Objective Test Generation for DRL Agents
Require: s, classifier (surrogate model);
Require: TR, test runs;
Require: G, generations;
Require: PS, population size;
Require: CR, crossover rate;
Require: Etrain, set of environment configurations where the DRL agent failed during training;
Ensure: Esel, the archive storing a diverse set of likely-failing test scenarios.
1: currentTestRun ←0;
2: while currentT estRun ≤T R do
3:
population ←GENERATE-POPULATION(PS, Etrain);
4:
COMPUTE-OBJECTIVES(population, s, Esel);
5:
currentGeneration ←0;
6:
while currentGeneration ≤G do
7:
newPop ←∅;
8:
while |newPop| < P S do
9:
p1 ←SELECTION(population)
10:
p2 ←SELECTION(population)
11:
if getRandomFloat() < CR then
12:
o1, o2 ←crossover(p1, p2)
13:
end if
14:
o1 ←MUTATE(o1)
15:
o2 ←MUTATE(o2)
16:
newPop ←newPop ∪{o1, o2}
17:
end while
18:
population ←ELITISM(←newPop ∪population);
19:
if stagnation detected or currentGeneration ≥G then
20:
Esel ←Esel∪GET-BEST-INDIVIDUAL(population)
21:
end if
22:
end while
23: end while
24: return Esel
Algorithm 1 outlines this process. The search begins with a population of
size PS. Each individual is evaluated without test execution using heuristics
described in Sections 3.1 and 2.2. The algorithm proceeds for G generations. In
each generation, two parents are selected (Lines 9-10), the crossover is applied
with probability CR (Line 11), and both offspring are mutated (Lines 14-15).
Offspring are added to the new population (Line 16), and elitism selects the best
individuals for the next generation (Line 18).
To avoid stagnation, INDAGO-Nexus monitors hypervolume changes across
generations. If improvement falls below a threshold, or we have reached the
maximum number of generations G, the best individual is archived (Line 20).
To explore new regions of space, the process continues until the total test runs
TR is met. The resulting archive Esel contains diverse, likely-failing scenarios,
which are then executed on the DRL agent for validation.
While INDAGO-Nexus builds on established MOEAs like NSGA-II [7] and
AGE-MOEA [23], it introduces several domain-specific adaptations: failure-based
population seeding, surrogate-based fitness estimation, diversity measured against
an evolving archive, and a stagnation-aware reset strategy for enhanced explo-
ration. We describe these customizations in the following paragraphs.
Population Initialization. We seed the initial population with environ-
ments where the DRL agent failed during training (Etrain), biasing the search
towards failure-prone regions [2].
The Pursuit of Diversity
7
Selection. Selection determines which individuals are chosen for reproduc-
tion. INDAGO-Nexus uses binary tournament selection based on Pareto domi-
nance [7], and the non-dominated sorting in particular.
Objective Calculation. Each individual is evaluated without test execu-
tion. The surrogate model s estimates failure likelihood, while diversity is com-
puted with respect to the archive Esel using either Euclidean distance or PCA-
based clustering metrics as described in Section 3.2.
Crossover and Mutation. INDAGO-Nexus applies single-point crossover
(Line 12) probabilistically based on the crossover rate CR, randomly selecting
a crossover point in the feature vector and swapping segments between parents.
Mutation (Lines 14-15) is always applied to both offspring and uses saliency-
based feature selection [26], which computes the gradient of the surrogate model’s
output with respect to each input feature. The saliency determines which features
impact the surrogate’s prediction, guiding mutation decisions. For fixed-length
features (position, heading), we apply polynomial mutation [6] with type-specific
handling (real-valued, integer, binary, categorical). For variable-length features
(parked vehicle lists), we randomly apply removal, addition, or modification
operations. A repair operator ensures validity after mutations.
Elitism. We retain the best individuals using NSGA-II’s fast non-dominated
sorting with crowding distance [7] or AGE-MOEA’s adaptive survival score [23].
Stagnation and Archive Update. INDAGO-Nexus maintains an archive
Esel of selected scenarios. Stagnation is detected by monitoring the change in
hypervolume of the Pareto front Pg at generation g relative to a reference point
r (Equation 2). The hypervolume metric reflects both diversity and convergence
by measuring the volume dominated by Pg with respect to r:
|HV (Pg, r) −HV (Pg−nlast, r)| < tol
(2)
When stagnation is detected, INDAGO-Nexus selects a representative indi-
vidual from the Pareto front using two strategies: Maximum Failure Likelihood
(highest predicted failure probability) or Knee Point (best trade-off between
failure likelihood and diversity) [31].
4
Empirical Evaluation
Our empirical evaluation aims to answer the following research questions:
RQ1: How does multi-objective search perform in finding diverse failures?
RQ2: How effective is INDAGO-Nexus compared to state-of-the-art INDAGO?
4.1
Case Studies
We evaluate INDAGO-Nexus on three diverse DRL agents from Biagiola and
Tonella [2], each representing different application domains and complexity:
Parking. The first environment we consider is the parking environment [19].
In this scenario, the DRL agent begins at a specific location with a defined head-
ing direction. The agent must park in a designated goal lane while avoiding colli-
sions with any of the already parked vehicles. An example of this environment is
8
Antony Bartlett, Cynthia Liem, Annibale Panichella
illustrated in Figure 1a. The environment consists of 24 elements, including goal
lane, heading, 20 parking slots, and the x, y coordinates of the starting position.
Humanoid. Next, we examine the Humanoid environment, one of the more
challenging environments from the MuJoCo simulator [28]. In this scenario, the
agent must control a bipedal robot to walk on a smooth surface within a 3D
space. The environment configuration of the Humanoid robot consists of two
arrays: joint position and joint velocity. The joint position array contains 24
elements representing the positions and rotations of the robot’s joints, while the
joint velocity array contains the linear and angular velocities of the joints.
Self-Driving Car. The final environment we consider is the Self-Driving
Car (SDC), developed using the DonkeyCar simulator [17]. In this environment,
the DRL agent must navigate a car from the starting point of a track to the
end without leaving track boundaries. The environment configuration consists
of a list with 12 pairs, where each pair comprises a command and a value. For
example, a command could be a left or right turn, with the value specifying the
turn length.
4.2
Baseline
To ensure fair comparison of all approaches, we utilize the original models from
the INDAGO paper. For our baseline, we employed the GA from the original
INDAGO tool. This approach uses the predictive value of the surrogate model as
a fitness value to gradually mutate environments in the search for failures. We
executed this with the saliency_failure test policy, which ensures the ini-
tial population is created from previously failing environments sourced from the
training data, incorporating saliency (as discussed in Section 3) during crossover
and mutation. We selected this method as it demonstrated the highest rate of
failure in the original study [2] against other approaches (random search and
hill climbing). Due to the stochastic nature of evolutionary search, we exe-
cuted the baseline 50 times, applying the process across all three agent scenarios.
4.3
Implementation and Parameter Settings
We implemented AGE-MOEA and NSGA-II using Pymoo v0.6.1 [3]. Experiments
used 50 generations, population size 50, crossover rate 0.75, with stagnation-
based termination (tol = 5×10−6, nlast = 10). The reference point for hypervol-
ume calculation is set to r = [1.2, 20.2], representing the upper bounds for the
objectives. Experiments were conducted within Docker containers on a machine
equipped with an AMD EPYC 7713 64-Core CPU (2.6 GHz), 256 threads, and
an NVIDIA A40 GPU (48GB GDDR6) for tensor-based computation.
4.4
Evaluation Criteria
We measure: (1) unique failures via PCA+K-means clustering of execution
traces, (2) input/output diversity using entropy across clusters, and (3) time-
to-failure (TTF) efficiency. Statistical significance is assessed using Wilcoxon
rank-sum test (α = 0.05) and Vargha–Delaney effect size.
The Pursuit of Diversity
9
4.5
Failure Detection and Diversity Assessment
To assess the effectiveness of our multi-objective search strategies (RQ1), we
executed both NSGA-II and AGE-MOEA across all combinations of diversity metrics
and Pareto front selection strategies.
We focused on three aspects: the number of unique failures discovered, the
diversity of these failures, and the efficiency in detecting them. We define a
failure as a test scenario in which the DRL agent violates its task specifica-
tion (e.g., crashing, falling, or missing the goal). To measure unique failures, we
cluster the execution traces (i.e., output trajectories) of failing tests using Prin-
cipal Component Analysis (PCA) followed by K-means. Each resulting cluster
corresponds to a distinct failure type, providing a behavior-grounded notion of
uniqueness. Since trajectory lengths vary, we normalize them via zero-padding.
For the Parking and SDC scenarios, we track the agent’s position over time,
while in the Humanoid scenario, we monitor the robot’s vertical movement to
detect falls.
To determine the optimal number of clusters K∗, we apply silhouette analysis
and increase K only if the silhouette score improves by at least 20%, reducing the
effect of noisy improvements. This clustering method is used both to compute
output diversity and to count unique failures. It is important to distinguish this
post hoc clustering from the PCA-based metric used within the search itself
(Section 3.2). The former relies on test execution results while the latter (our
objective) relies on the input features (i.e., without running the tests).
We also evaluate the diversity of failures from both input and output perspec-
tives. Output diversity is measured using the same clustering of output trajec-
tories used to identify unique failures. For input diversity, we cluster the failing
environment configurations using PCA and K-means, providing insight into the
structural variety of the input scenarios, regardless of the agent’s behavior. In
both cases, we calculate two diversity metrics: unique failures, which reflects how
many clusters are populated, and entropy, which quantifies how evenly failures
are distributed across clusters. We follow the definitions provided in the INDAGO
framework [2].
To assess test efficiency, we compute the average time required to find the
first failing test case. This metric reflects how quickly each approach exposes fail-
ures. The same metrics are then used in the comparison between INDAGO-Nexus
and the baseline INDAGO (RQ2). To assess statistical significance, we apply the
Wilcoxon rank-sum test [5] with a confidence level of α = 0.05, and we report
the Vargha–Delaney ˆA12 statistics for the effect size [30].
5
Results
This section presents the results of our study, addressing in turn each research
question introduced earlier.
5.1
Configuration Comparison (RQ1)
Table 1 outlines the performance of each INDAGO-Nexus configuration across the
three case studies:
10
Antony Bartlett, Cynthia Liem, Annibale Panichella
Table 1: Performance metrics across different configurations for all three
agents, displaying median and Interquartile Range (IQR) across 50 runs. (Gray
cells indicate top values for each agent and approach.)
(a) Parking Agent
Failures
Entropy
Configuration #Total #Unique
Input
Output
TTF
AGE-MOEA
PCA/Knee
10 (3.75)
7 (1.00)
69.05 (19.19) 62.34 (15.62)
106.12 (26.89)
PCA/O1
7 (3.00)
4 (2.00)
24.79 (41.62) 39.28 (18.25)
108.13 (26.74)
Euclidean/Knee 10 (3.75)
7 (2.75)
63.95 (18.20) 61.98 (16.33)
123.27 (27.51)
Euclidean/O1
7 (2.00)
4 (1.00)
11.72 (37.64) 39.85 (14.35)
123.73 (26.27)
NSGA-II
PCA/Knee
10 (3.00)
7 (2.00)
73.00 (20.13)
63.64 (9.77)
109.19 (23.00)
PCA/O1
7 (2.75)
4 (1.75)
10.26 (42.01) 41.05 (14.81)
108.12 (25.46)
Euclidean/Knee 11 (3.00)
6 (1.00)
64.12 (14.18) 60.59 (10.26)
120.34 (25.98)
Euclidean/O1
7 (2.00)
3 (1.00)
0.00 (39.71)
36.12 (15.04)
120.70 (27.56)
(b) Humanoid Agent
Failures
Entropy
Configuration #Total #Unique
Input
Output
TTF
AGE-MOEA
PCA/Knee
-
-
-
-
-
PCA/O1
2 (1.75)
1 (0.75)
10.65 (34.60)
0.00 (0.00)
78.94 (27.12)
Euclidean/Knee 1 (2.00)
1 (1.00)
0.00 (38.69)
0.00 (0.00)
96.06 (29.95)
Euclidean/O1
2 (2.00)
1 (1.00)
0.00 (35.59)
0.00 (0.00)
102.53 (27.58)
NSGA-II
PCA/Knee
-
-
-
-
-
PCA/O1
2 (1.75)
1 (1.00)
0.00 (32.88)
0.00 (0.00)
79.32 (28.42)
Euclidean/Knee 2 (2.00)
1 (1.00)
0.00 (40.90)
0.00 (0.00)
90.94 (27.65)
Euclidean/O1
2 (2.00)
1 (0.75)
0.00 (28.95)
0.00 (0.00)
95.70 (23.58)
(c) Self-Driving Car (SDC)Agent
Failures
Entropy
Configuration #Total #Unique
Input
Output
TTF
AGE-MOEA
PCA/Knee
31 (5.75)
5 (6.00)
56.97 (12.51) 63.03 (75.90) 1578.16 (130.69)
PCA/O1
36 (3.75)
6 (6.00)
71.96 (12.36)
77.34 (86.13) 1092.10 (49.04)
Euclidean/Knee 31 (5.00) 5.5 (6.00) 57.06 (18.83) 71.90 (79.67)
1072.26 (43.36)
Euclidean/O1
50 (1.00)
6 (6.00)
74.70 (5.84) 64.99 (75.69) 2435.78 (274.89)
NSGA-II
PCA/Knee
30 (4.00)
5 (6.00)
54.50 (18.05) 62.75 (74.53) 1521.15 (102.29)
PCA/O1
34 (5.75)
6 (6.75)
70.97 (13.84)
77.00 (85.31) 1087.94 (59.30)
Euclidean/Knee 30 (4.75)
5 (6.00)
56.12 (9.07)
71.07 (80.27)
1073.21 (45.60)
Euclidean/O1
50 (1.00)
6 (6.00)
72.19 (6.10) 66.78 (74.89) 2588.43 (270.45)
Choice of MOEA: AGE-MOEA generally outperforms NSGA-II in unique
failures discovered. In Parking, AGE-MOEA achieves 4-7 unique failures com-
pared to NSGA-II’s 3-7. Both perform similarly in SDC, while Humanoid proves
The Pursuit of Diversity
11
challenging for both algorithms. AGE-MOEA’s adaptive survival score mecha-
nism appears better suited for the multi-objective DRL testing problem com-
pared to NSGA-II’s crowding distance approach.
Diversity Metrics: PCA-based diversity consistently outperforms Euclidean
distance across all scenarios. In Parking, PCA configurations achieve substan-
tially higher input entropy (24.79-69.05) compared to Euclidean (11.72-63.95),
demonstrating PCA’s superior ability to capture meaningful relationships in
high-dimensional feature spaces. The Euclidean approach occasionally achieves
slightly higher total failure counts, revealing an important trade-off between fail-
ure quantity and diversity quality. This suggests that while Euclidean distance
may find failures faster in certain regions, PCA-based diversity better explores
the full failure space.
Pareto Selection Strategy: Knee-point selection consistently yields supe-
rior diversity compared to the extreme O1 point across all scenarios. The knee
point represents a balanced trade-off between objectives, achieving higher unique
failures and entropy scores.
Different configurations perform best in different scenarios. Overall,
AGE-MOEA tends to produce more unique and diverse failures than
NSGA-II. Pareto knee-point selection consistently yields better results
than the extreme O1 point. However, the choice of configuration can sig-
nificantly impact performance, highlighting the need for adaptive strate-
gies that consider the specific context and objectives.
5.2
Comparison with INDAGO (RQ2)
Table 2 presents a direct comparison between INDAGO-Nexus (AGE-MOEA Eu-
clidean/Knee configuration) and the state-of-the-art INDAGO approach. The re-
sults demonstrate significant improvements across multiple metrics:
Unique Failure Discovery: INDAGO-Nexus consistently finds more unique
failures than INDAGO. In the Parking scenario, INDAGO-Nexus discovers 7 unique
failures compared to INDAGO’s 5 (+40% improvement). For the SDC scenario,
INDAGO-Nexus finds 5.5 unique failures versus INDAGO’s 3 (+83% improvement).
The Humanoid scenario shows comparable performance between both approaches,
which aligns with our observation that this scenario is inherently challenging for
all testing methods.
Diversity Enhancement: The multi-objective approach significantly im-
proves both input and output diversity. In the Parking case, INDAGO-Nexus
achieves 63.95 input entropy compared to INDAGO’s 0.00, and 61.98 output en-
tropy versus 49.37. This demonstrates that explicitly optimizing for diversity as
an objective leads to more varied test scenarios and failure modes. The diver-
sity improvements are particularly important for comprehensive testing, as they
increase the likelihood of discovering edge cases and systematic weaknesses.
Efficiency Gains: INDAGO-Nexus demonstrates superior efficiency in find-
ing failures across all scenarios. Time-to-failure improvements range from 18%
(Parking: 123.27 vs 150.53) to 67% (SDC: 1072.26 vs 3251.97), with Humanoid
12
Antony Bartlett, Cynthia Liem, Annibale Panichella
Table 2: Comparison between INDAGO-Nexus and INDAGO across different
agents, displaying median and Interquartile Range (IQR) across 50 runs.
Statistical significance markers for the Wilcoxon rank-sum test: * p-value<0.05,
** p-value<0.01, *** p-value<0.001.
(a) Parking Agent
Failures
Entropy
Approach
#Total
#Unique
Input
Output
TTF
INDAGO-Nexus
10 (3.75)
7 (2.75)**
63.95 (18.20)*** 61.98 (16.33)***
123.27 (27.51)***
INDAGO
14.5 (4.00)
5 (2.00)
0.00 (35.72)
49.37 (12.56)
150.53 (0.03)
(b) Humanoid Agent
Failures
Entropy
Approach
#Total
#Unique
Input
Output
TTF
INDAGO-Nexus
1 (2.00)
1 (1.00)
0.00 (38.69)
0.00 (0.00)
96.06 (29.95)***
INDAGO
2 (2.75)
1 (1.00)
28.90 (39.31)
0.00 (50.89)
150.50 (0.01)
(c) Self-Driving Car (SDC) Agent
Failures
Entropy
Approach
#Total
#Unique
Input
Output
TTF
INDAGO-Nexus 31 (5.00)*** 5.50 (6.00)**
57.06 (18.83)
71.90 (79.67)*** 1072.26 (43.36)***
INDAGO
17.00 (5.75)
3.00 (3.00)
61.72 (21.59)
34.36 (48.71)
3251.97 (0.35)
showing 36% improvement (96.06 vs 150.50). These efficiency gains suggest that
the multi-objective approach not only finds more diverse failures but does so
more quickly than single-objective methods. The efficiency improvements are
statistically significant (Wilcoxon test, p < 0.001) with large effect sizes across
all scenarios (Vargha-Delaney ˆA12 > 0.9) for TTF.
Statistical Significance: All reported improvements show statistical sig-
nificance at α = 0.05 level using the Wilcoxon rank-sum test. Effect sizes cal-
culated using Vargha-Delaney ˆA12 statistics indicate substantial practical dif-
ferences, particularly for efficiency metrics where INDAGO-Nexus consistently
achieves large effect sizes ( ˆA12 > 0.7), indicating that INDAGO-Nexus signifi-
cantly outperforms INDAGO in key performance areas.
INDAGO-Nexus finds more unique failures and higher input/output diver-
sity in Parking and SDC. Instead, it matches INDAGO in the challenging
Humanoid scenario. Finally, INDAGO-Nexus finds unique failures faster
across all DRL agents.
6
Threats To Validity
Internal Validity: Our experimental design addresses several potential threats
through careful control of variables and statistical rigor. We ensure fair compari-
son by running all algorithms under identical conditions with the same hardware
and time constraints. Each experiment is repeated 50 times to account for the
The Pursuit of Diversity
13
stochastic nature of evolutionary algorithms, and we apply appropriate statistical
tests (Wilcoxon rank-sum) with effect size measurements (Vargha-Delaney ˆA12)
to assess practical significance. The clustering approach for measuring unique
failures uses established silhouette analysis with conservative thresholds (20%
improvement requirement) to avoid over-segmentation due to noise.
External Validity: Our evaluation covers three diverse DRL domains with
different state/action spaces and failure modes. However, generalizability to
other DRL applications, training algorithms beyond PPO, or different archi-
tectures requires further investigation.
Construct Validity: The choice of evaluation metrics reflects established
practices in the DRL testing literature, particularly building on the INDAGO
framework. Our clustering-based approach to measuring unique failures provides
a behavior-grounded assessment of diversity that captures meaningful differences
in agent behavior rather than superficial input variations. The entropy-based
diversity measures provide quantitative assessments that complement the qual-
itative notion of test case variety.
Conclusion Validity: The statistical methods employed (non-parametric
tests, effect size calculations) are appropriate for the experimental design and
address the non-normal distribution of performance metrics commonly observed
in evolutionary computation experiments. The large number of experimental
repetitions (50 per configuration) provides sufficient statistical power to detect
meaningful differences between approaches. Furthermore, a small number of SDC
configurations were executed on a smaller hardware setup (AMD Ryzen Thread-
ripper 3970X 32-Core Processor, 64GB memory, NVIDIA GeForce RTX 3080
10GB). Although less powerful then the main setup, the results were consider-
ably faster. However, other validation metrics remain consistent.
7
Conclusion and Future Work
This paper introduced INDAGO-Nexus, a multi-objective search approach for gen-
erating diverse failures in DRL environments. Unlike prior methods that focus
solely on inducing agent failures, INDAGO-Nexus simultaneously optimizes for
failure likelihood and scenario diversity, leading to a more comprehensive ex-
ploration of failure modes. Our empirical evaluation across three DRL environ-
ments—Parking, Humanoid, and Self-Driving Car (SDC)—shows that the multi-
objective INDAGO-Nexus consistently outperforms the single-objective baseline
INDAGO in terms of unique failures discovered, diversity (measured via input and
output entropy), and search efficiency.
We found that the effectiveness of INDAGO-Nexus varies across scenarios, with
AGE-MOEA generally achieving more unique and diverse failures than NSGA-II.
Both the choice of diversity metric (Euclidean vs. PCA) and the Pareto selec-
tion strategy (knee point vs. extreme O1) significantly influence the outcomes.
In particular, knee-point selection consistently led to better results across most
configurations. Nonetheless, the Humanoid scenario remains particularly chal-
lenging, exposing limitations in current diversity metrics and search robustness.
14
Antony Bartlett, Cynthia Liem, Annibale Panichella
Future Work: Several research directions emerge from this work. First, ex-
ploring many-objective optimization with additional objectives such as test case
complexity, execution cost, or coverage metrics could further improve testing ef-
fectiveness. Second, investigating adaptive diversity metrics that automatically
adjust based on the characteristics of discovered failures could enhance the ap-
proach’s generalizability. Third, extending the framework to other DRL train-
ing algorithms (beyond PPO) and neural network architectures would broaden
its applicability. Finally, developing theoretical foundations for understanding
when and why multi-objective approaches outperform single-objective methods
in testing contexts would provide valuable insights for the testing community.
INDAGO-Nexus introduces multi-objective search for generating diverse DRL
failures, consistently outperforming INDAGO in unique failures and efficiency.
AGE-MOEA with knee-point selection proves most effective, though Humanoid
scenarios remain challenging. Future work includes many-objective extensions,
improved surrogate models, and broader domain evaluation.
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The Pursuit of Diversity: Multi-Objective Testing of Deep Reinforcement Learning Agents Antony Bartlett[0009-0008-2654-8556], Cynthia Liem[0000-0002-5385-7695], Annibale Panichella[0000-0002-7395-3588] Delft . Testing deep reinforcement learning (DRL) agents in safetycritical domains requires discovering diverse failure scenarios. Existing tools such as INDAGO rely on single-objective optimization focused solely on maximizing failure counts, but this does not ensure discovered scenarios are diverse or reveal distinct error types. We introduce INDAGO-Nexus, a multi-objective search approach that jointly optimizes for failure likelihood and test scenario diversity using multi-objective evolutionary algorithms with multiple diversity metrics and Pareto front selection strategies. We evaluated INDAGO-Nexus on three DRL agents: humanoid walker, self-driving car, and parking agent. On average, INDAGO-Nexus discovers up to 83% and 40% more unique failures (test effectiveness) than INDAGO in the SDC and Parking scenarios, respectively, while reducing time-tofailure by up to 67% across all agents. Keywords: Multi-objective search, deep reinforcement learning, surrogate models 1 Introduction Reinforcement Learning (RL) [27], and more recently Deep Reinforcement Learning (DRL) [21], have emerged as powerful tools for automating complex tasks in dynamic and uncertain environments [22, 26]. DRL agents learn through trialand-error interactions with their environment and are increasingly deployed in Cyber-Physical Systems (CPS), such as self-driving cars (SDCs) and humanoid robots [4], where safety, reliability, and adaptability are critical. Ensuring the reliability of DRL-based systems remains a significant challenge [2, 29]. Unlike conventional software, DRL agents are opaque, stochastic, and highly sensitive to reward signals. Their behavior depends not only on a learned policy but also on vast environment and action spaces. As a result, edge-case scenarios-those most likely to trigger failures-are difficult to identify, and traditional testing approaches often fall short [29]. Real-world incidents involving autonomous agents [9,12,18] underscore the consequences of failing to detect such rare conditions during testing. Simulation-based testing is a widely adopted technique for evaluating DRL agents, especially in safety-critical domains such as autonomous agents. It allows developers to generate and execute test scenarios in a controlled and reproducible setting, substantially reducing the time, cost, and safety risks of 16 Oct 2025 2 Antony Bartlett, Cynthia Liem, Annibale Panichella real-world testing [10, 16]. A test scenario refers to a specific configuration of the environment-including initial agent state, obstacle placement, and taskspecific parameters-that defines the context in which the agent is evaluated. Despite its advantages, simulation-based testing still remains resource-intensive, and exhaustive exploration of the test input space is often infeasible due to its combinatorial complexity. To address these challenges, Biagiola and Tonella [2] proposed the use of surrogate models to guide test generation for DRL agents. Their framework, INDAGO, builds a surrogate model from training logs to predict whether a given test scenario is likely to cause agent failure-without executing it in simulation. INDAGO combines this model with two search strategies: Hill Climbing (HC) and a singleobjective Genetic Algorithm (GA). Their empirical evaluation across multiple DRL tasks showed that (1) surrogate models substantially reduced simulation time, and (2) both HC and GA outperformed earlier random-search-based methods [29] in identifying more failure-inducing scenarios. While INDAGO effectively increases the number of failing scenarios, it focuses solely on failure likelihood, overlooking diversity. This is a key limitation-generating more failures does not guarantee that they are distinct. Diversity is crucial in testing as it promotes broader coverage of the agent's operational space and improves the chances of uncovering unique faults, helping to address the curse of rarity problem [20]. In this paper, we propose INDAGO-Nexus, a multi-objective search framework for DRL testing that jointly optimizes for failure probability and scenario diversity. We investigate: RQ1: How does multi-objective search perform in finding diverse failures? RQ2: How effective is INDAGO-Nexus compared to state-of-the-art INDAGO? To tackle this inherently multi-objective problem, we employed multi-objective evolutionary algorithms (MOEAs). MOEAs are well-suited for this task as they efficiently explore trade-offs and produce a set of Pareto-optimal solutions [6,14, 15]. We experimented with two diversity metrics (Euclidean distance and PCAbased clustering), two MOEAs (NSGA-II [7] and AGE-MOEA [23]), and two Pareto-front selection strategies (highest failure likelihood and knee point [31]). We benchmarked INDAGO-Nexus on three DRL agents: self-driving car, humanoid walker, and parking agent. Our results indicate INDAGO-Nexus discovers up to 83% and 40% more unique failures than INDAGO in the SDC and Parking scenarios, respectively, while reducing time-to-failure by up to 67%. Knee-point selection consistently produces the best results across most configurations, while different diversity metrics show varying effectiveness depending on the scenario. Therefore, our contributions are: - We propose INDAGO-Nexus, a multi-objective approach to generate diverse test environments for DRL agents. - We demonstrate INDAGO-Nexus discovers up to 83% more unique failures than INDAGO while reducing time-to-failure by up to 67%. The Pursuit of Diversity 3 - We provide a complete replication package with implementation and experimental data1. 2 Background and Related Work This section describes the key background concepts used in this work and summarizes the related work. 2.1 Deep Reinforcement Learning Deep Reinforcement Learning (DRL) extends Reinforcement Learning (RL) by using deep neural networks as function approximators to handle high-dimensional inputs (e.g., images) and continuous action spaces (e.g., steering angles in autonomous driving) [22]. Instead of explicitly representing policies or value functions, DRL learns them directly from raw observations and rewards using deep learning architectures. This capability makes DRL particularly suitable for complex tasks in cyberphysical systems such as self-driving cars (SDCs), where the environment is dynamic and the decision space is large. In this work, we focus on model-free DRL algorithms, where agents learn directly from interactions without an explicit model of the environment. For example, the DRL agent for the parking scenario we consider in our case study (see Section 4) was trained with Proximal Policy Optimization (PPO) [24], a widely adopted policy-gradient method known for its stability and sample efficiency. The agent's goal is to maximize the cumulative reward it receives in the long run [27]. 2.2 Black-box Testing In black-box testing, diversity is a common strategy for exposing faults by varying inputs and observing outputs [13]. For DRL agents, we focus on input diversity-variations in environment configurations-to guide test generation without costly simulations, combined with a surrogate to estimate failure likelihood. Related work on diversity-based testing for deep neural networks includes approaches that optimize for input diversity to improve fault detection. Aghababaeyan et al. [1] proposed black-box testing techniques that use test case diversity to generate diverse test inputs for deep neural networks. While their work demonstrates the effectiveness of diversity-based approaches, their diversity metrics are primarily designed for image inputs rather than the tabular data that characterizes our MLP-based surrogate models. Our work adapts the concept of diversitydriven testing to the specific context of DRL environment configurations, which are typically represented as structured tabular data with both continuous and categorical features. 4 Antony Bartlett, Cynthia Liem, Annibale Panichella "env_config": { "goal_lane_idx": 0, "heading_ego": 0.96, "parked_vehicles_lane_indices": [ 1, 3, 6, 8, 9, 10, 11, 12, 14, 18 ], "position_ego": [ 1.83, -4.96 ] } (a) Environment configuration (b) Agent behavior visualization Fig. 1: Parking scenario showing (a) environment configuration details used by DRL as a starting point, and (b) the DRL agent behavior. In (b), the red dot indicates the target parking spot, green dots represent parked cars, the black dot marks the agent's starting position, and the blue dots trace its trajectory. 2.3 Surrogates Surrogate models are lightweight machine learning approximations that learn to predict the behavior or outcomes of computationally expensive processes based on historical data. In the context of DRL testing, these models have proven particularly effective at reducing evaluation costs [2]. Specifically, given an environment configuration (Figure 1a), a surrogate model can predict the failure probability in the range [0, 1], thereby enabling efficient test generation without requiring costly simulation executions. 2.4 Testing DRL Agents The most directly related work is by Biagiola and Tonella [2], who introduced INDAGO, a test generation framework that leverages surrogate models trained on DRL logs to predict whether a given test environment is likely to cause failure. These surrogates return a continuous score in [0, 1], representing the estimated failure probability of an environment. INDAGO combines this prediction with search-based strategies, namely Hill Climbing [25] and Genetic Algorithms [11], to efficiently explore the input space. Our work builds on INDAGO by introducing a second optimization objective: input diversity. This extension aims to increase the diversity of the discovered failures, thereby improving coverage of the agent's operational space. 3 Approach The goal of our approach is to generate test scenarios that are both failureinducing and diverse, increasing the likelihood of uncovering unique faults in DRL agents. We build upon the INDAGO framework [2], which uses a surrogate model trained on DRL training logs to predict whether a given environment is likely to cause a failure. While INDAGO focuses solely on failure likelihood, we 1 10.6084/m9.figshare.29204687 The Pursuit of Diversity 5 extend it to a multi-objective test generation framework that also accounts for scenario diversity-crucial for uncovering unique faults. Hence, we formulate the problem of generative diverse failure-inducing test scenarios as follows: Problem Definition 1 Given a DRL agent A, a set of test scenarios E, and a surrogate model s : E →[0, 1] that estimates the likelihood of failure for a given e, the goal is to generate a set of test scenarios Esel ⊆E that maximize the likelihood of A to fail while maximizing diversity among the selected scenarios. More formally, the goal is to find a set of environments Esel that optimizes the following objectives: ( max O1 = s(e) (Failure Likelihood) max O2 = Div(e, Esel) (Input Diversity) (1) where O1 (Failure Likelihood) is the predicted likelihood of failure for a given environment e; and O2 (Input Diversity) is a measure of how diverse the selected environments are compared to the other test scenarios in Esel. In the following, we describe the surrogate model, diversity metrics, and the multi-objective optimization framework. 3.1 Surrogate models Following the prior work by Biagiola and Tonella [2], we adopt a lightweight surrogate model to estimate the failure likelihood of a test scenario without executing it. Specifically, we utilize a multi-layer perceptron (MLP) classifier trained on data collected during the DRL agent's training episodes. The MLP architecture is well-suited for this task due to the relatively small training dataset (up to 10k labeled environments) and the low-dimensional nature of environment feature vectors (e.g., 24-dimensional in the parking scenario). The surrogate outputs s(e) ∈[0, 1], representing failure probability for environment e. 3.2 Diversity Metrics We explore two diversity methods: Euclidean distance computes the mean distance between a new scenario and previously selected ones, and PCA-based clustering [8] measures distance to the nearest cluster centroid in reduced dimensionality space. Both methods handle variable-length features through onehot encoding and zero-padding. 3.3 Multi-Objective Evolutionary Testing To generate a diverse set of failure-inducing test scenarios, INDAGO-Nexus employs MOEAs that evolve environment configurations to simultaneously maximize (1) the likelihood of failure, as predicted by a surrogate model, and (2) diversity with respect to a growing archive of previously selected tests. We use two MOEAs: NSGA-II [7] and AGE-MOEA [23] to optimize both failure likelihood and diversity. 6 Antony Bartlett, Cynthia Liem, Annibale Panichella Algorithm 1 Multi-Objective Test Generation for DRL Agents Require: s, classifier (surrogate model); Require: TR, test runs; Require: G, generations; Require: PS, population size; Require: CR, crossover rate; Require: Etrain, set of environment configurations where the DRL agent failed during training; Ensure: Esel, the archive storing a diverse set of likely-failing test scenarios. 1: currentTestRun ←0; 2: while currentT estRun ≤T R do 3: population ←GENERATE-POPULATION(PS, Etrain); 4: COMPUTE-OBJECTIVES(population, s, Esel); 5: currentGeneration ←0; 6: while currentGeneration ≤G do 7: newPop ←∅; 8: while |newPop| 0.9) for TTF. Statistical Significance: All reported improvements show statistical significance at α = 0.05 level using the Wilcoxon rank-sum test. Effect sizes calculated using Vargha-Delaney ˆA12 statistics indicate substantial practical differences, particularly for efficiency metrics where INDAGO-Nexus consistently achieves large effect sizes ( ˆA12 > 0.7), indicating that INDAGO-Nexus significantly outperforms INDAGO in key performance areas. INDAGO-Nexus finds more unique failures and higher input/output diversity in Parking and SDC. Instead, it matches INDAGO in the challenging Humanoid scenario. Finally, INDAGO-Nexus finds unique failures faster across all DRL agents. 6 Threats To Validity Internal Validity: Our experimental design addresses several potential threats through careful control of variables and statistical rigor. We ensure fair comparison by running all algorithms under identical conditions with the same hardware and time constraints. Each experiment is repeated 50 times to account for the The Pursuit of Diversity 13 stochastic nature of evolutionary algorithms, and we apply appropriate statistical tests (Wilcoxon rank-sum) with effect size measurements (Vargha-Delaney ˆA12) to assess practical significance. The clustering approach for measuring unique failures uses established silhouette analysis with conservative thresholds (20% improvement requirement) to avoid over-segmentation due to noise. External Validity: Our evaluation covers three diverse DRL domains with different state/action spaces and failure modes. However, generalizability to other DRL applications, training algorithms beyond PPO, or different architectures requires further investigation. Construct Validity: The choice of evaluation metrics reflects established practices in the DRL testing literature, particularly building on the INDAGO framework. Our clustering-based approach to measuring unique failures provides a behavior-grounded assessment of diversity that captures meaningful differences in agent behavior rather than superficial input variations. The entropy-based diversity measures provide quantitative assessments that complement the qualitative notion of test case variety. Conclusion Validity: The statistical methods employed (non-parametric tests, effect size calculations) are appropriate for the experimental design and address the non-normal distribution of performance metrics commonly observed in evolutionary computation experiments. The large number of experimental repetitions (50 per configuration) provides sufficient statistical power to detect meaningful differences between approaches. Furthermore, a small number of SDC configurations were executed on a smaller hardware setup (AMD Ryzen Threadripper 3970X 32-Core Processor, 64GB memory, NVIDIA GeForce RTX 3080 10GB). Although less powerful then the main setup, the results were considerably faster. However, other validation metrics remain consistent. 7 Conclusion and Future Work This paper introduced INDAGO-Nexus, a multi-objective search approach for generating diverse failures in DRL environments. Unlike prior methods that focus solely on inducing agent failures, INDAGO-Nexus simultaneously optimizes for failure likelihood and scenario diversity, leading to a more comprehensive exploration of failure modes. Our empirical evaluation across three DRL environments-Parking, Humanoid, and Self-Driving Car (SDC)-shows that the multiobjective INDAGO-Nexus consistently outperforms the single-objective baseline INDAGO in terms of unique failures discovered, diversity (measured via input and output entropy), and search efficiency. We found that the effectiveness of INDAGO-Nexus varies across scenarios, with AGE-MOEA generally achieving more unique and diverse failures than NSGA-II. Both the choice of diversity metric (Euclidean vs. PCA) and the Pareto selection strategy (knee point vs. extreme O1) significantly influence the outcomes. In particular, knee-point selection consistently led to better results across most configurations. Nonetheless, the Humanoid scenario remains particularly challenging, exposing limitations in current diversity metrics and search robustness. 14 Antony Bartlett, Cynthia Liem, Annibale Panichella Future Work: Several research directions emerge from this work. First, exploring many-objective optimization with additional objectives such as test case complexity, execution cost, or coverage metrics could further improve testing effectiveness. Second, investigating adaptive diversity metrics that automatically adjust based on the characteristics of discovered failures could enhance the approach's generalizability. Third, extending the framework to other DRL training algorithms (beyond PPO) and neural network architectures would broaden its applicability. Finally, developing theoretical foundations for understanding when and why multi-objective approaches outperform single-objective methods in testing contexts would provide valuable insights for the testing community. INDAGO-Nexus introduces multi-objective search for generating diverse DRL failures, consistently outperforming INDAGO in unique failures and efficiency. AGE-MOEA with knee-point selection proves most effective, though Humanoid scenarios remain challenging. Future work includes many-objective extensions, improved surrogate models, and broader domain evaluation. References 1. Aghababaeyan, Z., et al.: Black-box testing of deep neural networks through test case diversity. IEEE Transactions on Software Engineering 49(5), 3182-3204 (May 2023). https://doi.org/10.1109/tse.2023.3243522, http://dx.doi.org/10.1109/TSE. 2023.3243522 2. Biagiola, M., Tonella, P.: Testing of deep reinforcement learning agents with surrogate models. ACM Trans. Softw. Eng. Methodol. 33(3) (Mar 2024). https: //doi.org/10.1145/3631970, https://doi.org/10.1145/3631970 3. Blank, J., Deb, K.: pymoo: Multi-objective optimization in python. IEEE Access 8, 89497-89509 (2020) 4. Bloom, J., et al.: Decentralized multi-agent reinforcement learning with global state prediction (2023), https://arxiv.org/abs/2306.12926 5. Conover, W.J.: Practical nonparametric statistics, vol. 350. John Wiley & Sons (1998) 6. Deb, K.: Multi-objective optimisation using evolutionary algorithms: an introduction. pp. 3-34. Springer (2011) 7. Deb, K., et al.: A fast and elitist multiobjective genetic algorithm: Nsga-ii. IEEE transactions on evolutionary computation 6(2), 182-197 (2002) 8. Ding, C., He, X.: K-means clustering via principal component analysis. p. 29. ICML '04, Association for Computing Machinery, New York, NY, USA (2004). https: //doi.org/10.1145/1015330.1015408, https://doi.org/10.1145/1015330.1015408 9. Ghoshal, A.: Why did this humanoid robot go nuts and nearly injure its handlers? New Atlus (05 2025), https://newatlas.com/ai-humanoids/ humanoid-robot-nearly-injures-handlers-unitree/ 10. Giubilato, R., et al.: Simulation framework for mobile robots in planetary-like environments (2020), https://arxiv.org/abs/2006.00057 11. Goldberg, D.E., Holland, J.H.: Genetic algorithms and machine learning. Machine Learning 3(2), 95-99 (1988). https://doi.org/10.1023/A:1022602019183, https:// doi.org/10.1023/A:1022602019183 12. Guardian, T.: Us investigates 2.4m tesla self-driving vehicles after reported collisions. The Guardian (10 2024), https://www.theguardian.com/technology/2024/ oct/18/tesla-self-driving-car-investigation The Pursuit of Diversity 15 13. Guidotti, R., et al.: A survey of methods for explaining black box models. ACM computing surveys (CSUR) 51(5), 1-42 (2018) 14. Haq, F.U., et al.: Efficient online testing for dnn-enabled systems using surrogateassisted and many-objective optimization. pp. 811-822 (2022) 15. Ishibuchi, H., et al.: Evolutionary many-objective optimization: A short review. pp. 2419-2426. IEEE (2008) 16. Kaur, P., et al.: A survey on simulators for testing self-driving cars (2021), https: //arxiv.org/abs/2101.05337 17. Kramer, T., contributors: Self driving car sandbox. https://github.com/ tawnkramer/sdsandbox (2021), gitHub repository 18. Lambert, F.: Tesla full self-driving veers off road, flips car in scary crash driver couldn't prevent. Electrek (05 2025), https://electrek.co/2025/05/23/ tesla-full-self-driving-veers-off-road-flips-car-scary-crash-driver-couldnt-prevent/ 19. Leurent, E.: An environment for autonomous driving decision-making. https:// github.com/eleurent/highway-env (2018) 20. Liu, H.X., Feng, S.: "curse of rarity" for autonomous vehicles (2022), https://arxiv. org/abs/2207.02749 21. Mnih, V., et al.: Playing atari with deep reinforcement learning (2013), https: //arxiv.org/abs/1312.5602 22. Mnih, V., et al.: Human-level control through deep reinforcement learning. Nature 518(7540), 529-533 (2015) 23. Panichella, A.: An adaptive evolutionary algorithm based on non-euclidean geometry for many-objective optimization. p. 595-603. GECCO '19, Association for Computing Machinery, New York, NY, USA (2019). https://doi.org/10.1145/3321707. 3321839, https://doi.org/10.1145/3321707.3321839 24. Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. CoRR abs/1707.06347 (2017), http://arxiv.org/abs/ 1707.06347 25. Selman, B., Gomes, C.P.: Hill-climbing Search. John Wiley & Sons, Ltd (2006). https://doi.org/https://doi.org/10.1002/0470018860.s00015, https:// onlinelibrary.wiley.com/doi/abs/10.1002/0470018860.s00015 26. 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IEEE transactions on evolutionary computation 19(6), 761-776 (2014)
|
2510.14728
|
The simultaneous effect of chemotaxis and alarm-taxis on the global
existence and stability of a predator-prey system
Gnanasekaran Shanmugasundarama, Jitraj Sahaa,∗, Rafael Díaz Fuentesb
aDepartment of Mathematics, National Institute of Technology Tiruchirappalli, Tiruchirappalli, 620015, India
bDepartment of Mathematics and Informatics, University of Cagliari, Via Ospedale 72, Cagliari, 09124, Italy
Abstract
This study examines a fully parabolic predator–prey chemo-alarm-taxis system under homogeneous Neu-
mann boundary conditions in a bounded domain Ω⊂Rn with a smooth boundary ∂Ω. Under specific
parameter conditions, it is shown that the system admits a unique, globally bounded classical solution.
The convergence of the solution is established through the construction of an appropriate Lyapunov
functional. In addition, numerical simulations are presented to validate the asymptotic behaviour of the
solution. The results highlight the significant role of chemotaxis and alarm-taxis coefficients in determin-
ing the existence and stability of predator–prey models, as discussed in the literature.
Keywords:
Alarm-taxis, Chemotaxis, Global boundedness, Global stability, Predator-prey, Numerical
simulation.
2020 MSC: 35A01, 35A09, 35B40, 92C17, 37N25, 65N06.
1. Introduction and motivation
Burglar alarm calls [3] act as a key anti-predation strategy, whereby prey emit signals that attract
secondary predators, potentially deterring primary predators [5, 15]. This behaviour is observed in marine
systems, such as dinoflagellates employing bioluminescence in response to copepod disturbances, which
may draw fish as secondary predators [13].
Chemotaxis, on the other hand, refers to the movement
of microorganisms in response to chemical gradients, with applications spanning ecology, immunology,
biology and medicine.
In this context, we formulate the chemo-alarm-taxis system, in which the primary predator v is
attracted to the prey w through a common chemical signal z, while the prey moves away from the
primary predator in response to the same chemical. At the same time, the secondary predator is drawn
towards the combined density of the primary predator and the prey owing to the alarm call emitted
by the prey. This interaction leads the secondary predator to prey upon the primary predator, thereby
providing the prey with a potential defence mechanism against predation. To capture these ecological
dynamics, we consider a system of coupled, nonlinear parabolic partial differential equations describing
the interactions between two predators and one prey in a chemo-alarm-taxis framework. The system is
given by
ut = d1∆u −χ1∇· (u∇(vw)) + µ1u(1 −u + a1v + a2w),
x ∈Ω, t > 0,
vt = d2∆v −χ2∇· (v∇z) + µ2v(1 −v −a3u + a4w),
x ∈Ω, t > 0,
wt = d3∆w + ξ∇· (w∇z) + µ3w(1 −w −a5u −a6v),
x ∈Ω, t > 0,
zt = d4∆z + αv + βw −γz,
x ∈Ω, t > 0,
∂u
∂ν = ∂v
∂ν = ∂w
∂ν = ∂z
∂ν = 0,
x ∈∂Ω, t > 0,
u(x, 0) = u0,
v(x, 0) = v0,
w(x, 0) = w0,
z(x, 0) = z0
x ∈Ω,
(1.1)
∗Corresponding author
Email addresses: sekaran@nitt.edu (Gnanasekaran Shanmugasundaram), jitraj@nitt.edu (Jitraj Saha),
rafael.diazfuentes@unica.it (Rafael Díaz Fuentes)
arXiv:2510.14728v1 [math.AP] 16 Oct 2025
in a bounded domain Ω⊂Rn with smooth boundary ∂Ω. Here, ν denotes the unit outward normal
on ∂Ω. The unknown functions u = u(x, t) and v = v(x, t) describe the population densities of the
secondary predator and the primary predator, respectively.
In addition, w = w(x, t) represents the
prey population, while z = z(x, t) denotes the concentration of the chemical attractant produced by the
prey. The parameters d1, d2, d3, d4, χ1, χ2, ξ, α, β, γ, µ1, µ2, µ3 and ai (i = 1, 2, 3, 4, 5, 6) are all positive
constants and the initial data u0, v0, w0, and z0 are nonnegative functions.
The constants d1, d2, d3,
and d4 are the diffusion coefficients, representing the natural dispersive forces governing the movement
of the predators, prey and chemical concentrations, respectively. The parameter χ1 is the alarm-taxis
coefficient, while χ2 and ξ are the chemotactic coefficients. The term −χ1∇· (u∇(vw)) represents alarm-
taxis, namely the directed movement of the secondary predator (u) towards the gradient of the combined
predator–prey density (∇(vw)) in response to the alarm signal. Similarly, the term −χ2∇·(v∇z) describes
chemo-attraction, that is, the directed movement of the primary predator (v) towards the gradient of
the chemical concentration (∇z). By contrast, the term +ξ∇· (w∇z) denotes chemo-repulsion, meaning
the directed movement of the prey (w) away from the gradient of the chemical concentration (∇z). The
constants µ1, µ2, µ3 are the logistic growth coefficients, while a1, a2, a3, a4, a5, a6 represent the interspecific
interactions among the three species. The parameters α, β and γ correspond to the production and decay
rates of the common chemical. Furthermore, we assume that the initial data u0, v0, w0, and z0 satisfy
u0, v0, w0 ∈C0(Ω),
with
u0, v0, w0 ≥0
in Ω,
z0 ∈W1,q(Ω),
for some q > n,
with
z0 ≥0
in Ω.
(1.2)
The foundational model for chemotaxis was first proposed by Keller and Segel in 1970 [14]. Since then,
the Keller–Segel model and its various modifications have been extensively investigated by numerous re-
searchers. In this context, considerable attention has been devoted to two-species–one-chemical systems
[20, 23, 24, 25, 34, 38, 42] and two-species–two-chemical systems with a competing source [2, 31, 37, 41, 46].
Among these, the predator–prey model, an ecologically significant extension of the two-species competi-
tion framework, has been widely explored. Research on predator–prey systems with one-chemical chemo-
taxis includes [6, 21, 22], while investigations into predator–prey systems with two-chemical chemotaxis
can be found in [17, 27, 28]. For further developments and extensions of predator–prey models, see [8, 29].
Motivated by the ecological significance of alarm calls, the alarm-taxis model was first introduced by
Haskell and Bell [9]
ut = ηduxx + f(u, v, w),
vt = (dvx −ξvux)x + g(u, v, w),
wt = (wx −χwφx(u, v))x + h(u, v, w).
Herein, u, v, and w denote the resource or prey (e.g. dinoflagellates), the primary predator (e.g. cope-
pods), and the secondary predator (e.g. fish), respectively. Under zero-flux boundary conditions, and
for certain initial data (details of which may be omitted in subsequent discussion), the authors demon-
strated the global existence of positive, bounded classical solutions in one-dimensional space. They also
established the existence of non-constant equilibrium solutions and analysed their stability. In addition,
numerical simulations were carried out to illustrate the emergence of spatial patterns in this model,
emphasising the adaptive advantage conferred by a signalling mechanism consistent with the alarm hy-
pothesis.
Fuest and Lankeit [7] considered the following system
ut = d1∆u + u(λ1 −µ1u −a1v −a2w),
vt = d2∆v −ξ∇· (v∇u) + v(λ2 −µ2u + b1u −a3w),
wt = d3∆w −χ∇· (w∇(uv)) + w(λ3 −µ3w + b2u + b3v),
with homogeneous Neumann boundary conditions, the system models the interactions among the prey
u, the predator v, and the superpredator w, the latter preying on both of the other populations. The
authors established the existence of a global classical solution in the case of pure alarm-taxis, i.e. when
ξ = 0. For ξ > 0, they investigated the existence of global generalised solutions.
2
Li and Wang [18] considered the following system
ut = d1∆u + r1u(1 −u) −b1uv −b3uw,
vt = d2∆v −ξ∇· (v∇u) + r2v(1 −v) + uv −b2vw,
wt = d3∆w −χ∇· (w∇(uv)) + r3w(1 −wσ) + vw + uw,
where u, v, and w denote the densities of the prey, the primary predator, and the secondary predator,
respectively. By applying semigroup theory, the authors established the boundedness of classical solutions
in two spatial dimensions. They replaced the usual logistic term r3w(1 −w) with r3w(1 −wσ) in order to
investigate how large the intra-species competition exponent must be to guarantee the global boundedness
of solutions.
Furthermore, asymptotic stability was obtained through the construction of a suitable
Lyapunov functional.
Jin et al. [13] studied the global existence of classical solutions to the following system by employing
the Neumann heat semigroup technique in two spatial dimensions
ut = d1∆u + µ1u(1 −u) −b1uv −b3 uw
u+w,
vt = d2∆v −ξ∇· (v∇u) + µ2v(1 −v) + uv −b2vw,
wt = d3∆w −χ∇· (w∇(uv)) + µ3w(1 −w) + vw + c3 uw
u+w.
Additionally, the authors established the global asymptotic stability for the cases where b3 = c3 = 0 and
b3 > 0, c3 > 0.
Zhang et al. [44] studied the following food chain model incorporating alarm-taxis and logistic
growth
ut = d1∆u + u(1 −u) −b1uv,
vt = d2∆v −ξ∇· (v∇u) + θ1v(1 −v) + uv −b2vw,
wt = d3∆w −χ∇· (w∇(uv)) + θ2w(1 −w) + vw.
The authors showed that, for sufficiently large values of θ1 and θ2, the system admits a globally bounded
classical solution when n ≥3. Moreover, by constructing an appropriate Lyapunov functional, they
demonstrated that, under certain parameter conditions, the system converges either to a coexistence
state or to a semi-coexistence state as t →∞.
Zhang and Chen [43] studied the global existence of the following system
ut = d1∆u + u
1 −u −
av
v+ρ
,
vt = d2∆v + v
bu
v+ρ −α −
cw
w+σ
,
wt = d3∆w −χ∇· (w∇(uv)) + w
mv
w+σ −β
,
for n ≥1 with sufficiently smooth initial data. The authors analysed the large-time behaviour of the
system under certain conditions and derived convergence rates. In addition, numerical simulations were
carried out to validate the analytical results.
To the best of our knowledge, this is the first work to rigorously establish the global existence,
boundedness, and convergence of solutions in a predator–prey system incorporating both chemotaxis and
alarm-taxis. Motivated by the existing literature, our primary objective is to demonstrate the existence
of a global classical solution to system (1.1), which remains uniformly bounded in Ω× (0, ∞) for n ≥1.
Furthermore, we establish convergence of the solution by constructing a suitable Lyapunov functional.
Finally, we perform numerical simulations to confirm the asymptotic behaviour of solutions in both two
and three spatial dimensions.
The remainder of this article is organised as follows. Section 2 introduces preliminary lemmas and
establishes the local existence of the classical solution. Section 3 is devoted to proving the global existence
and boundedness of classical solutions for system (1.1). In Section 4, we carry out a stability analysis,
while Section 5 presents numerical validation and results. The paper concludes with a summary in Section
6. Our main result on global existence is the following.
Theorem 1.1 (Global existence of solutions). Suppose that Ω⊂Rn(n ≥1), is a bounded domain with
smooth boundary. Then there exists µ > 0, such that if min{µ2, µ3} > µ and for any nonnegative initial
3
data (u0, v0, w0, z0) satisfying (1.2) for some q > max{2, n}, the system (1.1) possesses a unique classical
solution (u, v, w, z) which is uniformly bounded in the sense that
u(·, t)
L∞(Ω) +
v(·, t)
W1,∞(Ω) +
w(·, t)
W1,∞(Ω) +
z(·, t)
W1,q(Ω) ≤C,
∀t > 0,
where the constant C > 0.
Remark 1.1. In the one-dimensional case n = 1, global existence can be established under weaker
conditions. This follows directly from the testing procedures and regularity results presented in Section 3.
Now, we examine the equilibrium points of the system.
Let (u, v, w, z) be a classical solution of
(1.1) satisfying (1.2) and (ue, ve, we, ze) denote the equilibrium points of the system (1.1), satisfying the
following system of equations
µ1ue(1 −ue + a1ve + a2we) =
0,
µ2ve(1 −ve −a3ue + a4we) =
0,
µ3we(1 −we −a5ue −a6ve) =
0,
αve + βwe −γze =
0.
The system admits the following eight equilibrium points
1. Trivial steady states
(0, 0, 0, 0), (1, 0, 0, 0),
0, 1, 0, α
γ
,
and
0, 0, 1, β
γ
.
2. Semi-coexistence steady states
0, 1 + a4
1 + a4a6
, 1 −a6
1 + a4a6
, α(1 + a4) + β(1 −a6)
γ(1 + a4a6)
,
1 + a2
1 + a2a5
, 0, 1 −a5
1 + a2a5
, β(1 −a5)
γ(1 + a5a5)
,
and
1 + a1
1 + a1a3
, 1 −a3
1 + a1a3
, 0, α(1 −a3)
γ(1 + a1a3)
.
3. Coexistence steady state (u∗, v∗, w∗, z∗).
Here, (0,0,0,0) represents the extinction equilibrium where all species vanish; (1,0,0,0) corresponds to
the equilibrium with only the secondary predator present;
0, 1, 0, α
γ
describes the primary predator-
only equilibrium and
0, 0, 1, β
γ
characterizes the prey-only equilibrium. Additionally, there exist three
semi-coexistence equilibria where any two species persist together. The most ecologically significant state,
(u∗, v∗, w∗, z∗), represents the coexistence equilibrium where all three species maintain stable populations.
Here, the coexistence steady state (u∗, v∗, w∗, z∗) is explicitly given by
u∗=
1 + a1 + a2 + a1a4 + a6(a4 −a2)
1 + a1(a3 + a4a5) + a4a6 + a2(a5 −a3a6),
v∗=
1 −a3(1 + a2) + a4 + a5(a2 −a4)
1 + a1(a3 + a4a5) + a4a6 + a2(a5 −a3a6),
w∗=
1 + a1(a3 −a5) −a5 + a6(a3 −1)
1 + a1(a3 + a4a5) + a4a6 + a2(a5 −a3a6),
z∗= αv∗+ βw∗
γ
.
The conditions for the coexistence steady state are
a2a6 < min
n
1 + a1 + a2 + a1a4 + a4a6, 1+a1(a3+a4a5)+a4a6+a2a5
a3
o
,
a3(1 + a2) + a4a5 < 1 + a4 + a2a5,
a5(1 + a1) + a6 < 1 + a3(a1 + a6).
(1.3)
We observe that, among the three species, the secondary predator persists due to its relatively low death
rate. Now, we sort out the conditions for the remaining possible steady states (ue, ve, we, ze) are
(1, 0, 0, 0)
if
(1.3) does not hold,
(¯u, ¯v, ¯w, ¯z) =
1+a1
1+a1a3 ,
1−a3
1+a1a3 , 0,
α(1−a3)
γ(1+a1a3)
if
(1.3) does not hold and a3 < 1,
(ˆu, ˆv, ˆw, ˆz) =
1+a2
1+a2a5 , 0,
1−a5
1+a2a5 ,
β(1−a5)
γ(1+a2a5)
if
(1.3) does not hold and a5 < 1.
4
Among the eight possible equilibrium points, we analyze four biologically relevant cases: (i) the coexis-
tence steady state (ii) the trivial steady state (1, 0, 0, 0) (iii) two semi-coexistence steady states. Using
a suitable Lyapunov functional, we establish the asymptotic stability. Let us denote Γ1 =
µ1a1
µ2a3 and
Γ2 = µ1a2
µ3a5 in the following theorems.
Theorem 1.2 (Coexistence state of the species). Suppose that the assumptions of Theorem 1.1 hold true
and let a2a6 < min
n
1 + a1 + a2 + a1a4 + a4a6, 1+a1(a3+a4a5)+a4a6+a2a5
a3
o
, a3(1 + a2) + a4a5 < 1 + a4 +
a2a5, a5(1 + a1) + a6 < 1 + a3(a1 + a6). If the parameters fulfill the relations
(i) χ2
1 < min
(
d1d2v∗Γ1
u∗
v
2
L∞(Ω)
w
2
L∞(Ω)
,
d1d3w∗Γ2
u∗
v
2
L∞(Ω)
w
2
L∞(Ω)
)
,
(ii) d3χ2
2v∗Γ1 + d2ξ2w∗Γ2 < 2d2d3d4,
(iii) µ2a4Γ1 + α < 2µ2Γ1 + µ3a6Γ2,
(iv) µ2a4Γ1 + β < 2µ3Γ2 + µ3a6Γ2,
(v) α + β < 2γ,
(vi) a1a4a5 > a2a3a6,
then the nonnegative classical solution (u, v, w, z) of the system (1.1) exponentially converges to the co-
existence steady state (u∗, v∗, w∗, z∗) uniformly in Ωas t →∞.
Theorem 1.3 (Secondary predator only existence state). Suppose that the assumptions in Theorem 1.1
hold true and let a2a6 ≥min
n
1 + a1 + a2 + a1a4 + a4a6, 1+a1(a3+a4a5)+a4a6+a2a5
a3
o
, a3(1 + a2) + a4a5 ≥
1 + a4 + a2a5, a5(1 + a1) + a6 ≥1 + a3(a1 + a6). If the parameters satisfy the relations
(i) χ2
1 < min
(
d1d2
w
2
L∞(Ω)
,
d1d3
v
2
L∞(Ω)
)
,
(ii) d3χ2
2
v
2
L∞(Ω) + d2ξ2
w
2
L∞(Ω) < 2d2d3d4,
(iii) µ2 + µ2a4
w
L∞(Ω) + α
2 < Γ1µ2 < µ1a1,
(iv) µ3 + β
2 < Γ2µ3 < µ1a2,
(v) α + β < 2γ,
(vi) a1a4a5 < a2a3a6,
then the nonnegative classical solution (u, v, w, z) of the system (1.1) converges to the trivial steady state
(1, 0, 0, 0) uniformly in Ωas t →∞.
Theorem 1.4 (Semi-coexistence state of the species). Suppose that the assumptions of Theorem 1.1 hold
true for the following cases.
1. (Prey vanishing state) Suppose that (1.3) does not hold and assume that a3 < 1. If the parameters
fulfill the relations
(i) χ2
1 < min
(
d1d2¯vΓ1
¯u
v
2
L∞(Ω)
w
2
L∞(Ω)
,
d1d3
¯u
v
2
L∞(Ω)
)
,
(ii) d3χ2
2¯vΓ1 + d2ξ2
w
2
L∞(Ω) < 2d2d3d4,
(iii) µ3 + β
2 < Γ2µ3 < µ1a2¯u + Γ1µ2a4¯v,
(iv) α + β < 2γ,
5
(v) a1a4a5 < a2a3a6,
(vi) α < 2Γ1µ2,
then the nonnegative classical solution (u, v, w, z) of the system (1.1) exponentially converges to the
semi-coexistence steady state
1+a1
1+a1a3 ,
1−a3
1+a1a3 , 0,
α(1−a3)
γ(1+a1a3)
uniformly in Ωas t →∞.
2. (Primary predator vanishing state) Suppose that (1.3) does not hold and assume that a5 < 1.
If the parameters satisfy the relations
(i) χ2
1 < min
(
d1d2
ˆu
w
2
L∞(Ω)
,
d1d3 ˆ
wΓ2
ˆu
v
2
L∞(Ω)
w
2
L∞(Ω)
)
,
(ii) d3χ2
2
v
2
L∞(Ω) + d2ξ2 ˆwΓ2 < 2d2d3d4,
(iii) µ2 + µ2a4
w
L∞(Ω) + α
2 < Γ1µ2,
(iv) Γ1µ2 + Γ2µ3a6 ˆw < µ1a1ˆu,
(v) α + β < 2γ,
(vi) a1a4a5 < a2a3a6,
(vii) β < Γ2µ3
then the nonnegative classical solution (u, v, w, z) of the system (1.1) converges to the semi-coexistence
steady state
1+a2
1+a2a5 , 0,
1−a5
1+a2a5 ,
β(1−a5)
γ(1+a5a5)
uniformly in Ωas t →∞.
2. Preliminaries and local existence
The aim of this section is to concentrate on the basic estimates and the local existence of the classical
solution. The local existence lemma is constructed based on ideas derived from Horstmann [12] and Tello
[33], along with the references cited therein.
From now on, all the constants Ci, i = 1, 2, . . ., are assumed to be positive and the numbering is
restarted for each proof.
Lemma 2.1 ([45]). Let y be a positive absolutely continuous function on (0, ∞) that satisfies
y′(t) + Ay(t)p ≤B,
y(0) = y0,
with some constants A > 0, B ≥0 and p ≥1. Then for t > 0, we have
y(t) ≤max
(
y0,
B
A
1
p )
.
Lemma 2.2 ([30, 32]). Let T > 0, τ ∈(0, T ), a > 0 and b > 0, and assume that nonnegative function
y : [0, T ) →[0, ∞) is C1 smooth and such that
y′(t) + ay(t) ≤h(t),
∀t ∈(0, T )
with some nonnegative function h ∈L1
loc([0, T )) fulfilling
Z t+τ
t
h(s)ds ≤b,
∀t ∈[0, T −τ).
Then
y(t) ≤max
n
y(0) + b, b
aτ + 2b
o
,
∀t ∈(0, T ).
6
Lemma 2.3 (Maximal Sobolev regularity [4, 10]). Let r ∈(1, ∞) and T ∈(0, ∞). Consider the following
evolution equation
yt = ∆y −y + g,
x ∈Ω, t > 0,
∂y
∂ν = 0,
x ∈∂Ω, t > 0,
y(x, 0) = y0(x),
x ∈Ω.
For each y0 ∈W2,r(Ω) (r > n) with ∂y0
∂ν = 0 on ∂Ωand any g ∈Lr((0, T ); Lr(Ω)), there exists a unique
solution
y ∈W1,r ((0, T ); Lr(Ω)) ∩Lr (0, T ); W2,r(Ω)
.
Moreover, there exists Cr > 0, such that
Z T
0
∥y(·, t)∥r
Lr(Ω)dt+
Z T
0
∥yt(·, t)∥r
Lr(Ω)dt +
Z T
0
∥∆y(·, t)∥r
Lr(Ω)dt
≤Cr
Z T
0
∥g(·, t)∥r
Lr(Ω)dt + Cr∥y0∥r
Lr(Ω) + Cr∥∆y0∥r
Lr(Ω).
If s0 ∈[0, T ) and y(·, s0) ∈W2,r(Ω) (r > n) with ∂y(·,s0)
∂ν
= 0 on ∂Ω, then
Z T
s0
esr∥∆y(·, t)∥r
Lr(Ω)dt ≤Cr
Z T
s0
esr∥g(·, t)∥r
Lr(Ω)dt + Cr∥y(·, s0)∥r
Lr(Ω) + Cr∥∆y(·, s0)∥r
Lr(Ω).
Lemma 2.4 (Local Existence). Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary.
Assume that the initial data u0, v0, w0 and z0 satisfies (1.2) for some q > max{2, n}. Then there exists
Tmax ∈(0, ∞] such that the system (1.1) admits a unique solution (u, v, w, z) satisfying
u, v, w ∈C0 Ω× [0, Tmax )
∩C2,1 Ω× (0, Tmax)
,
z ∈C0 Ω× [0, Tmax )
∩C2,1 Ω× (0, Tmax)
∩L∞
loc
[0, Tmax ) ; W1,q(Ω)
.
Moreover, either Tmax = ∞, or
lim
t→Tmax
u(·, t)
L∞(Ω) +
v(·, t)
W1,∞(Ω) +
w(·, t)
W1,∞(Ω) +
z(·, t)
W1,q(Ω)
= ∞.
(2.1)
Proof. The lemma is proven using standard arguments based on parabolic regularity theory. For the
detailed proof, we refer to Jin et al. [13]. The non-negativity of the solution in Ω×(0, Tmax) is established
by satisfying both the maximum principle and the initial condition (1.2).
Lemma 2.5. The classical solution of (1.1) satisfies
Z
Ω
u(·, t) +
Z
Ω
v(·, t) +
Z
Ω
w(·, t) ≤K1,
(2.2)
where K1 > 0, for all t ∈(0, Tmax). Moreover, we have
Z t+τ
t
Z
Ω
u(·, t)2 ≤K2,
t ∈(0, Tmax −τ),
(2.3)
where K2 > 0 and τ := min{1, 1
2Tmax}.
Proof. Choose ζ1, ζ2 ∈(0, 1) with ζ1µ1a1 ≤ζ2µ2a3, ζ1µ1a2 ≤µ3a5, ζ2µ2a4 ≤µ3a6 and set
µ = min{µ1, µ2, µ3}. From first and second equations of (1.1) gives
d
dt
ζ1
Z
Ω
u
= ζ1µ1
Z
Ω
u −ζ1µ1
Z
Ω
u2 + ζ1µ1a1
Z
Ω
uv + ζ1µ1a2
Z
Ω
uw
7
and
d
dt
ζ2
Z
Ω
v
= ζ2µ2
Z
Ω
v −ζ2µ2
Z
Ω
v2 −ζ2µ2a3
Z
Ω
uv + ζ2µ2a4
Z
Ω
vw.
Adding the above equality with the third equation in (1.1), we get
d
dt
ζ1
Z
Ω
u + ζ2
Z
Ω
v +
Z
Ω
w
+ζ1µ1
Z
Ω
u + ζ2µ2
Z
Ω
v + µ3
Z
Ω
w
= 2ζ1µ1
Z
Ω
u −ζ1µ1
Z
Ω
u2 + ζ1µ1a1
Z
Ω
uv + ζ1µ1a2
Z
Ω
uw
+ 2ζ2µ2
Z
Ω
v −ζ2µ2
Z
Ω
v2 −ζ2µ2a3
Z
Ω
uv + ζ2µ2a4
Z
Ω
vw
+ 2µ3
Z
Ω
w −µ3
Z
Ω
w2 −µ3a5
Z
Ω
uw −µ3a6
Z
Ω
vw.
Using the Young inequality with C1 > 0, one has
d
dt
ζ1
Z
Ω
u + ζ2
Z
Ω
v +
Z
Ω
w
+ζ1µ1
Z
Ω
u + ζ2µ2
Z
Ω
v + µ3
Z
Ω
w
≤−ζ1µ1
2
Z
Ω
u2 −ζ2µ2
2
Z
Ω
v2 −µ3
2
Z
Ω
w2 +
Z
Ω
(ζ1µ1a1 −ζ2µ2a3)uv
+
Z
Ω
(ζ1µ1a2 −µ3a5)uw +
Z
Ω
(ζ2µ2a4 −µ3a6)vw + C1.
Simplifying, we get
d
dt
ζ1
Z
Ω
u + ζ2
Z
Ω
v +
Z
Ω
w
+µ
ζ1
Z
Ω
u + ζ2
Z
Ω
v +
Z
Ω
w
+ µ
2
ζ1
Z
Ω
u2 + ζ2
Z
Ω
v2 +
Z
Ω
w2
≤C1.
(2.4)
Applying Lemma 2.1, we get
Z
Ω
u +
Z
Ω
v +
Z
Ω
w ≤C2,
∀t ∈(0, Tmax),
where C2 > 0 this gives (2.2). For (2.3), integrate (2.4) over (t, t + τ) to get
Z t+τ
t
Z
Ω
u2 +
Z t+τ
t
Z
Ω
v2 +
Z t+τ
t
Z
Ω
w2 ≤C3,
t ∈(0, Tmax −τ),
where C3 > 0. This completes the proof.
3. Global existence of solutions
This section focuses on proving the global existence and boundedness of the solution to (1.1). We
begin by deriving the Lp(Ω) norm for v and w. For any s0 ∈(0, Tmax) with s0 < 1, Lemma 2.4 ensures
that u(·, s0), v(·, s0), w(·, s0) and z(·, s0) belong to C2(Ω), with ∂z(·,s0)
∂ν
= 0. Let C > 0 be a constant
satisfying the following
sup
0≤s≤s0
v(·, s)
L∞(Ω) ≤C,
sup
0≤s≤s0
w(·, s)
L∞(Ω) ≤C,
sup
0≤s≤s0
z(·, s)
L∞(Ω) ≤C,
∆z(·, s0)
L∞(Ω) ≤C
(3.1)
and
sup
0≤s≤s0
u(·, s)
L∞(Ω) ≤C,
sup
0≤s≤s0
∇v(·, s)
L∞(Ω) ≤C,
sup
0≤s≤s0
∇w(·, s)
L∞(Ω) ≤C.
(3.2)
Next, we derive boundedness in t ∈(s0, Tmax).
8
Lemma 3.1. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary. Then for any
p > 1, there exists µ(p, χ2, ξ, α, β, a4) > 0, such that if µ < min{µ2, µ3}, then we have
v(·, t)
Lp(Ω) +
w(·, t)
Lp(Ω) ≤K3,
∀t ∈(s0, Tmax),
(3.3)
for some K3 > 0.
Proof. Multiplying the second equation of (1.1) by vp−1, p > 1 and integrating over Ω, we get
Z
Ω
vtvp−1 = d2
Z
Ω
vp−1∆v −χ2
Z
Ω
vp−1∇· (v∇z) + µ2
Z
Ω
vp−1v(1 −v −a3u + a4w).
Applying the technique of integration by parts, we obtain
d
dt
1
p
Z
Ω
vp = −d2(p −1)
Z
Ω
vp−2|∇v|2 + χ2(p −1)
Z
Ω
vp−1∇v∇z + µ2
Z
Ω
vp −µ2
Z
Ω
vp+1
−µ2a3
Z
Ω
vpu + µ2a4
Z
Ω
vpw.
Again, use of integration by parts, gives us that
d
dt
1
p
Z
Ω
vp = −d2(p −1)
Z
Ω
vp−2|∇v|2 −χ2(p −1)
p
Z
Ω
vp∆z + µ2
Z
Ω
vp −µ2
Z
Ω
vp+1
−µ2a3
Z
Ω
vpu + µ2a4
Z
Ω
vpw
≤−µ2
Z
Ω
vp+1 −d2(p −1)
Z
Ω
vp−2|∇v|2 −χ2(p −1)
p
Z
Ω
vp∆z + µ2
Z
Ω
vp
≤+ µ2a4
Z
Ω
vpw −µ2
Z
Ω
vp+1 + I1 + I2 + I3 + I4.
(3.4)
Utilizing the Gagliardo-Nirenberg inequality and the Young inequality, it follows that (also thanks to
(2.2))
Z
Ω
vp =
v
p
2
2
L2(Ω) ≤C1
∇v
p
2
2d
L2(Ω)
v
p
2
2(1−d)
L
2
p (Ω) +
v
p
2
2
L
2
p (Ω)
≤4d2(p −1)
p(p + 1)
∇v
p
2
2d
L2(Ω)
1
d
+ C2
v
p
2
2(1−d)
L
2
p (Ω)
1
1−d
+ C1
v
p
2
2
L
2
p (Ω)
≤4d2(p −1)
p(p + 1)
∇v
p
2
2
L2(Ω) + C3
v
p
L1(Ω)
≤4d2(p −1)
p(p + 1)
p2
4
Z
Ω
vp−2|∇v|2 + C4,
with C2 = (1 −d)d
d
1−d
4d2(p−1)
p(p+1)
−d
1−d C
1
1−d
1
, where d =
p
2 −1
2
p
2 + 1
n −1
2 ∈(0, 1), and C1 is a constant from the
Gagliardo-Nirenberg inequality. Hence
p + 1
p
Z
Ω
vp ≤d2(p −1)
Z
Ω
vp−2|∇v|2 + C5,
(3.5)
with C5 = p+1
p C4. Now we estimate I1 using (3.5) as follows
I1 = −d2(p −1)
Z
Ω
vp−2 ∇v
2 ≤−p + 1
p
Z
Ω
vp + C5.
(3.6)
For p > 1,
1 −1
p
< 1,
I2 = −χ2(p −1)
p
Z
Ω
vp∆z ≤χ2
Z
Ω
vp|∆z|.
9
Applying the Young inequality with δ > 0,
I2 ≤δ
Z
Ω
vp+1 + C6δ−pχp+1
2
Z
Ω
|∆z|p+1,
(3.7)
where C6 =
p+1
p
−p
1
p+1. Again, using the Young inequality, we obtain
I3 = µ2
Z
Ω
vp ≤ρ
Z
Ω
vp+1 + C7,
(3.8)
where C7 = ρ−p
p+1
p
−p
1
p+1µp+1
2
|Ω|, and
I4 =
Z
Ω
µ2a4vpw ≤η
Z
Ω
vp+1 + C6η−pµp+1
2
ap+1
4
Z
Ω
wp+1.
(3.9)
Substituting (3.6) - (3.9) in (3.4), we see that
d
dt
1
p
Z
Ω
vp
≤−(p + 1)
1
p
Z
Ω
vp
−(µ2 −δ −ρ −η)
Z
Ω
vp+1 + C6δ−pχp+1
2
Z
Ω
|∆z|p+1
+ C6η−pµp+1
2
ap+1
4
Z
Ω
wp+1 + C8.
(3.10)
Adopting the variation of constants formula to (3.10),
1
p
Z
Ω
vp ≤−(µ2 −δ −ρ −η)
Z t
s0
e−(p+1)(t−s)
Z
Ω
vp+1 + C6δ−pχp+1
2
Z t
s0
e−(p+1)(t−s)
Z
Ω
|∆z|p+1
+ C6η−pµp+1
2
ap+1
4
Z t
s0
e−(p+1)(t−s)
Z
Ω
wp+1 + C9,
(3.11)
where C9 = 1
p
R
Ωvp
0 + C8
1
p+1. According to Lemma 2.3, there exists Cp > 0 such that
C6δ−pχp+1
2
Z t
s0
e−(p+1)(t−s)
Z
Ω
|∆z|p+1 ≤C6Cpδ−pχp+1
2
Z t
s0
e−(p+1)(t−s)
Z
Ω
αv + βw
p+1
+ C6Cpδ−pχp+1
2
e−(p+1)t
z0
p+1
W2,p+1(Ω)
≤C6Cpδ−pχp+1
2
2p+1αp+1
Z t
s0
e−(p+1)(t−s)
Z
Ω
vp+1
+ C6Cpδ−pχp+1
2
2p+1βp+1
Z t
s0
e−(p+1)(t−s)
Z
Ω
wp+1
+ C6Cpδ−pχp+1
2
z0
p+1
W2,p+1(Ω).
(3.12)
Substituting the last inequality (3.12) into (3.11), we arrive at
1
q
Z
Ω
vp ≤−
µ2 −δ −ρ −η −C6Cpδ−pχp+1
2
2p+1αp+1 Z t
s0
e−(p+1)(t−s)
Z
Ω
vp+1
+
C6Cpδ−pχp+1
2
2p+1βp+1 + C6η−pµp+1
2
ap+1
4
Z t
s0
e−(p+1)(t−s)
Z
Ω
wp+1 + C10.
(3.13)
where C10 = C9 + C6Cpδ−pχp+1
2
z0
p+1
W2,p+1(Ω). Similarly, we can estimate for w as follows
1
p
Z
Ω
wp ≤−
µ3 −δ −ρ −C6C′
pδ−pξp+12p+1βp+1 Z t
s0
e−(p+1)(t−s)
Z
Ω
wp+1
+ C6C′
pδ−pξp+12p+1αp+1
Z t
s0
e−(p+1)(t−s)
Z
Ω
vp+1 + C11,
(3.14)
10
where the constant C11 > 0. Adding (3.13) and (3.14) affords us
1
p
Z
Ω
vp+1
p
Z
Ω
wp ≤
−µ2 + δ + ρ + η + C6Cpδ−pχp+1
2
2p+1αp+1 + C6C′
pδ−pξp+12p+1αp+1
×
Z t
s0
e−(p+1)(t−s)
Z
Ω
vp+1
+
−µ3 + δ + ρ + C6C′
pδ−pξp+12p+1βp+1 + C6Cpδ−pχp+1
2
2p+1βp+1
+ C6η−pµp+1
2
ap+1
4
×
Z t
s0
e−(p+1)(t−s)
Z
Ω
wp+1 + C12.
(3.15)
Let
µ = max
n
δ + ρ + η + C6Cpδ−pχp+1
2
2p+1αp+1 + C6C′
pδ−pξp+12p+1αp+1,
δ + ρ + C6C′
pδ−pξp+12p+1βp+1 + C6Cpδ−pχp+1
2
2p+1βp+1 + C6η−pµp+1
2
ap+1
4
o
.
Then (3.15) can be written as
1
p
Z
Ω
vp+1
p
Z
Ω
wp ≤−(µ2 −µ)
Z t
s0
e−(p+1)(t−s)
Z
Ω
vp+1 −(µ3 −µ)
Z t
s0
e−(p+1)(t−s)
Z
Ω
wp+1 + C12,
such that µ2 −µ > 0 and µ3 −µ > 0 because of our assumptions 0 < µ < min{µ2, µ3}. Hence, we deduce
that
Z
Ω
vp +
Z
Ω
wp ≤C12,
∀t ∈(s0, Tmax),
where the constant C12 > 0.
Lemma 3.2. Let n ≥1 and the initial data (u0, v0, w0, z0) satisfy (1.2) with some q > n. Then if v, w
comply with the conclusion of Lemma 3.1 we have
sup
t∈(0,Tmax)
v(·, t)
L∞(Ω) +
w(·, t)
L∞(Ω) +
z(·, t)
W1,q(Ω)
≤K4,
where K4 > 0.
Proof. We fix p > n
2 . Specifically, if n < 4, we assume p >
4n
n+4, noting that
4n
n+4 > n
2 in this case. Let us
define
np
(n −p)+
=
(
∞,
if
p ≥n,
np
n−p,
if
n
2 < p < n.
We observe that if n < 4, then
4n
n+4 < n. Furthermore, for any n, it holds that
np
(n−p)+ > n. Consequently,
given this choice for p, we can select a q <
np
(n−p)+ such that q > max{n, 4}. Moreover, fix l ∈(1, q) such
that l > n. It can be verified that it exists an r > 1 such that
max{n, 4} ≤rl ≤q.
For any s0 ∈(0, Tmax) with s0 < 1 , we fix arbitrary t ∈(s0, Tmax). Using the variation of constants
formula to the fourth equation of (1.1), we get
z(·, t) = e−γ(t−s0)ed4(t−s0)∆z0 +
Z t
s0
e−γ(t−s)ed4(t−s)∆
αv(·, s) + βw(·, s)
ds.
Then,
∇z(·, t)
Lrl(Ω) ≤e−γ(t−s0)
∇ed4(t−s0)∆z0
Lrl(Ω)
11
+
Z t
s0
e−γ(t−s)
∇ed4(t−s)∆
αv(·, s) + βw(·, s)
Lrl(Ω)ds.
By using smoothing property of the Neumann heat semi-group technique [39] with rl ≤q, gives
∇z(·, t)
Lrl(Ω) ≤C1e−γ(t−s0)∥z0∥W1,q(Ω)
+ C2
Z t
s0
e−γ(t−s)
1 + (t −s)−1
2 −n
2 ( 1
p −1
rl )
e−λd4(t−s)
αv(·, s) + βw(·, s)
Lp(Ω)ds.
Using Gamma function, we get
∇z(·, t)
Lrl(Ω) ≤C1e−γ(t−s0)
z0
W1,q(Ω) + C3
sup
t∈(s0,Tmax)
v(·, t) + w(·, t)
Lp(Ω).
Finally, we arrive at with the help of Lemma 3.1
∇z(·, t)
Lrl(Ω) ≤C4,
∀t ∈(s0, Tmax),
(3.16)
where C4 > 0. Let t0 = max{s0, t −1} and applying the variation of constants formula to the third
equation of (1.1), we get
w(·, t) =ed3(t−t0)∆w(·, t0) + ξ
Z t
t0
ed3(t−s)∆∇·
w(·, s)∇z(·, s)
ds
+ µ3
Z t
t0
ed3(t−s)∆w(·, s)
1 −w(·, s) −a5u(·, s) −a6v(·, s)
ds.
Now, being w ≤w2 + 1
4 for any w, after neglecting some nonpositive terms we get the following estimate
w(1 −w −a5u −a6v) ≤w(1 −w) ≤1
4.
(3.17)
Then, we have
w(·, t)
L∞(Ω)≤
ed3(t−t0)∆w(·, t0)
L∞(Ω)
+ ξ
Z t
t0
ed3(t−s)∆∇·
w(·, s)∇z(·, s)
L∞(Ω)ds + µ3
4 .
(3.18)
If t ≤1, then t0 = s0 and the maximum principle gives
ed3(t−t0)∆w(·, t0)
L∞(Ω) =
ed3(t−s0)∆w(·, s0)
L∞(Ω) ≤
w(·, s0)
L∞(Ω).
(3.19)
If t > 1, t −t0 = 1 and using the Neumann heat semi-group technique [39] with C5 > 0,
ed3(t−t0)∆w(·, t0)
L∞(Ω) ≤C5
1+(t −t0)−n
2
w(·, t0)
L1(Ω) ≤C6.
(3.20)
Substituting (3.19) and (3.20) in (3.18) and using the Neumann heat semi-group technique [39] with
C7 > 0, one has
w(·, t)
L∞(Ω) ≤max
n
w(·, s0)
L∞(Ω), C6
o
+ C7
Z t
t0
1 + (t −s)−1
2 −n
2l
e−λd3(t−s)
w(·, s)∇z(·, s)
Ll(Ω)ds + µ3
4 .
(3.21)
Using the Hölder inequality and then (3.16), we get
w(·, t)∇z(·, t)
Ll(Ω) ≤
w(·, t)
L¯rl
∇z(·, t)
Lrl(Ω)
≤
w(·, t)
k
L∞(Ω)
w(·, t)
1−k
L1(Ω)
∇z(·, t)
Lrl(Ω)
≤C8
w(·, t)
k
L∞(Ω),
∀t ∈(s0, Tmax),
(3.22)
12
where ¯r is the dual exponent of r and k = 1 −1
¯rl ∈(0, 1). Substituting (3.22) and (3.17) into (3.21) and
using the Gamma function, it follows that
w(·, t)
L∞(Ω) ≤max
n
w(·, s0)
L∞(Ω), C6
o
+ C9
sup
t∈(s0,Tmax)
w(·, t)
k
L∞(Ω) + µ3
4 .
Finally, we get
w(·, t)
L∞(Ω) ≤C10,
∀t ∈(s0, Tmax),
where C10 > 0. Now, considering the reaction term in the equation for v in (1.1) and applying reasoning
analogous to that in (3.17), we get
v(1 −v −a3u + a4w) ≤v(1 + a4w) −v2 ≤(1 + a4w)2
4
.
An analogous procedure yields
v(·, t)
L∞(Ω)≤
ed2(t−t0)∆v(·, t0)
L∞(Ω) + χ2
Z t
t0
ed2(t−s)∆∇·
v(·, s)∇z(·, s)
L∞(Ω)ds
+ µ2
Z t
t0
ed2(t−s)∆v(·, s)
1 −v(·, s) −a3u(·, s) + a4w(·, s)
L∞(Ω)ds.
(3.23)
Applying the same procedure one can get
v(·, t)
L∞(Ω) ≤max
n
v(·, s0)
L∞(Ω), C11
o
+ C12
Z t
t0
1 + (t −s)−1
2 −n
2l
e−λd2(t−s)
v(·, s)∇z(·, s)
Ll(Ω)ds
+ µ2
(1 + a4C10)2
4
.
Finally, we arrive at
v(·, t)
L∞(Ω) ≤C13,
∀t ∈(s0, Tmax),
(3.24)
where C13 > 0. Initial data regularity z0 ∈W1,q(Ω) and the smoothing property of the heat semigroup
ensure z ∈L∞((s0, Tmax); W1,q(Ω)). Along with (3.1), this will imply that actually
sup
t∈(0,Tmax)
v(·, t)
L∞(Ω) +
w(·, t)
L∞(Ω) +
z(·, t)
W1,q(Ω)
≤C14,
where C14 > 0. This completes the proof.
Lemma 3.3. Suppose that the assumptions in Lemma 3.2 hold true. There exists K5 > 0 fulfilling
Z t+τ
t
Z
Ω
∆z(·, t)
2 ≤K5,
for all t ∈(0, Tmax −τ).
Proof. Multiplying the fourth equation in (1.1) with −2∆z and integrating by parts, we obtain
d
dt
Z
Ω
|∇z|2 ≤−2d4
Z
Ω
|∆z|2 + 2α
v
L∞(Ω)
Z
Ω
∆z + 2β
w
L∞(Ω)
Z
Ω
∆z −2γ
Z
Ω
|∇z|2.
Simplifying one gets
d
dt
Z
Ω
|∇z|2 + 2γ
Z
Ω
|∇z|2 + 2d4
Z
Ω
|∆z|2 ≤2αK4
Z
Ω
∆z + 2βK4
Z
Ω
∆z
13
≤d4
Z
Ω
|∆z|2 + C1.
Finally,
d
dt
Z
Ω
|∇z|2 + 2γ
Z
Ω
|∇z|2 + d4
Z
Ω
|∆z|2 ≤C1,
∀t ∈(0, Tmax),
(3.25)
where C1 > 0. Now, integrate (3.25) over (t, t + τ) to get
Z t+τ
t
Z
Ω
∆z(·, t)
2 ≤C2,
∀t ∈(0, Tmax −τ),
where C2 > 0. This completes the proof.
Lemma 3.4. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary and the assump-
tions in Lemma 3.2 hold true. There exists K6 > 0 fulfilling
Z
Ω
|∇v(·, t)|2 ≤K6,
(3.26)
for all t ∈(0, Tmax).
Proof. Multiplying the second equation in (1.1) with −2∆v and integrating by parts, we obtain
d
dt
Z
Ω
|∇v|2 = −2d2
Z
Ω
|∆v|2 + 2χ2
Z
Ω
∇v · ∇z∆v + 2χ2
Z
Ω
v∆z∆v + 2µ2
Z
Ω
|∇v|2
−2µ2
Z
Ω
∇v2 · ∇v + 2µ2a3
Z
Ω
uv∆v −2µ2a4
Z
Ω
vw∆v
≤−2d2
Z
Ω
|∆v|2 + 2χ2
Z
Ω
∇v · ∇z∆v + 2χ2
Z
Ω
v∆z∆v + 2µ2
Z
Ω
|∇v|2 −4µ2
Z
Ω
v|∇v|2
+ 2µ2a3
Z
Ω
uv∆v −2µ2a4
Z
Ω
vw∆v.
Rearranging the terms, we get
d
dt
Z
Ω
|∇v|2 + 2µ2
Z
Ω
|∇v|2 + 2d2
Z
Ω
|∆v|2 ≤2χ2
Z
Ω
∇v · ∇z ∆v + 2χ2
v
L∞(Ω)
Z
Ω
|∆z||∆v|
+ 4µ2
Z
Ω
|∇v|2 + 2µ2a3
v
L∞(Ω)
Z
Ω
u|∆v|
+ 2µ2a4
v
L∞(Ω)
w
L∞(Ω)
Z
Ω
|∆v|.
(3.27)
The Hölder inequality gives for the first term in RHS of (3.27),
2χ2
Z
Ω
∇v · ∇z ∆v ≤2χ2
∇v
L4(Ω)∥∇z∥L4(Ω)
∆v
L2(Ω).
(3.28)
The Gagliardo-Nirenberg inequality with Lemma 3.2 gives
2χ2
∇v
L4(Ω) ≤C1
∆v
1
2
L2(Ω)
v
1
2
L∞(Ω) ≤C2
∆v
1
2
L2(Ω).
(3.29)
Substituting the estimate (3.29) into (3.28) and using the boundedness of ∥∇z∥L4(Ω) from (3.16), which
holds due to the condition rl ≥4, we then apply Young’s inequality to obtain
2χ2
Z
Ω
∇v · ∇z∆v ≤C2
∆v
3
2
L2(Ω)∥∇z∥L4(Ω) ≤2d2
5
∆v
2
L2(Ω) + C3.
(3.30)
14
Now using the Young inequality for the second term in RHS of (3.27), to get
2χ2K4
Z
Ω
|∆v||∆z| ≤2d2
5
∆v
2
L2(Ω) + C4
∆z
2
L2(Ω).
(3.31)
Using the Hölder inequality and the Gagliardo-Nirenberg inequality for the third term in RHS of (3.27),
gives
4µ2
Z
Ω
|∇v|2 ≤4µ2
Z
Ω
|∇v|4
1
2
|Ω|
1
2 ≤4µ2|Ω|
1
2
∇v
2
L4(Ω) ≤2d2
5
∆v
2
L2(Ω) + C5.
(3.32)
Once again Cauchy’s inequality for the last two terms in (3.27), gives
2µ2a3K4
Z
Ω
u|∆v| ≤2d2
5
∆v
2
L2(Ω) + C6
u
2
L2(Ω)
(3.33)
and
2µ2a4K2
4
Z
Ω
|∆v| ≤2d2
5
∆v
2
L2(Ω) + C7.
(3.34)
Substituting (3.30)-(3.34) in to (3.27), one gets
d
dt
Z
Ω
|∇v|2 + 2µ2
Z
Ω
|∇v|2 ≤C4
∆z
2
L2(Ω) + C6
u
2
L2(Ω) + C8,
∀t ∈(0, Tmax).
Now applying Lemma 2.2, we get
Z
Ω
|∇v|2 ≤C9,
∀t ∈(0, Tmax),
where C9 > 0. This completes the proof.
Lemma 3.5. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary and the assump-
tions in Lemma 3.2 hold true. There exists K7 > 0 satisfying
Z
Ω
|∇w(·, t)|2 ≤K7,
(3.35)
for all t ∈(0, Tmax).
Proof. Multiplying the third equation in (1.1) with −2∆w and integrating by parts, we obtain
d
dt
Z
Ω
|∇w|2 + 2µ3
Z
Ω
|∇w|2 + 2d2
Z
Ω
|∆w|2 ≤2ξ
Z
Ω
∇w · ∇z∆w + 2ξ
w
L∞(Ω)
Z
Ω
|∆z||∆w|
+ 4µ3
Z
Ω
|∇w|2 −4µ3
Z
Ω
w|∇w|2
+ 2µ3a4
w
L∞(Ω)
Z
Ω
u|∆w|
+ 2µ3a5
v
L∞(Ω)
w
L∞(Ω)
Z
Ω
|∆w|.
(3.36)
Following a similar procedure as in (3.30), we obtain
2ξ
Z
Ω
∇w · ∇z∆w ≤C1
∆w
3
2
L2(Ω)∥∇z∥L4(Ω) ≤2d3
5
∆w
2
L2(Ω) + C2.
(3.37)
Applying the Young inequality for the second term of (3.36) gives
2ξK4
Z
Ω
|∆w||∆z| ≤2d3
5
∆w
2
L2(Ω) + C3
∆z
2
L2(Ω).
(3.38)
15
Using the Hölder inequality and the Gagliardo-Nirenberg inequality to the third from of (3.36), gives
4µ3
Z
Ω
|∇w|2 ≤4µ3
Z
Ω
|∇w|4
1
2
|Ω|
1
2 ≤4µ3|Ω|
1
2
∇w
2
L4(Ω) ≤2d3
5
∆w
2
L2(Ω) + C4.
(3.39)
The Cauchy inequality gives
2µ3a4K4
Z
Ω
u|∆w| ≤2d3
5
∆w
2
L2(Ω) + C5
u
2
L2(Ω),
(3.40)
and
2µ3a5K2
4
Z
Ω
|∆w| ≤2d3
5
∆w
2
L2(Ω) + C6.
(3.41)
Substituting (3.37)-(3.41) into (3.36) and simplifying, we get
d
dt
Z
Ω
|∇w|2 + 2µ3
Z
Ω
|∇w|2 ≤C3
∆z
2
L2(Ω) + C5
u
2
L2(Ω) + C7,
∀t ∈(0, Tmax).
Now applying Lemma 2.2, we get
Z
Ω
|∇w|2 ≤C8,
∀t ∈(0, Tmax),
where C8 > 0. This completes the proof.
Lemma 3.6. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary and the assump-
tions in Lemma 3.2 hold true. There exists K8, K9 > 0 satisfying
sup
t∈(0,Tmax)
∇v(·, t)
L∞(Ω) +
∇w(·, t)
L∞(Ω)
≤K8,
(3.42)
and
sup
t∈(0,Tmax)
u(·, t)
L∞(Ω) ≤K9.
(3.43)
Proof. Let us use the variables r, ¯r, l, and k as defined in the proof of Lemma 3.2, t ∈(s0, Tmax) and
t0 = max{s0, t −1}. Applying the variation of constants formula to the third equation of (1.1), we get
∥∇w(·, t)∥L∞(Ω) ≤
∇ed3(t−s0)∆w(·, s0)
L∞(Ω) + ξ
Z t
s0
∇ed3(t−s)∆∇·
w(·, s)∇z(·, s)
L∞(Ω)ds
+ µ3
Z t
s0
∇ed3(t−s)∆w(·, s)
L∞(Ω)ds.
By Neumann heat semi-group with C1, C2, C3, C4 > 0, we obtain
∥∇w(·, t)∥L∞(Ω) ≤C1
1 + (t −s0)−1
2
w0
L∞(Ω)
+ C2
Z t
s0
1 + (t −s)−1
2 −n
2l
e−λd3(t−s)
∇w(·, s) · ∇z(·, s)
Ll(Ω)ds
+ C3
Z t
s0
1 + (t −s)−1
2 −n
2l
e−λd3(t−s)
w(·, s)
L∞(Ω)
∆z(·, s)
Ll(Ω)ds
+ C4
Z t
s0
1 + (t −s)−1
2
e−λd3(t−s)
w(·, s)
L∞(Ω)ds.
(3.44)
16
Now using similar procedure as in (3.22) and since Ωis bounded, ∇w ∈L2(Ω) implies ∇w ∈L1(Ω).
Therefore,
∇w(·, t) · ∇z(·, t)
Ll(Ω) ≤
∇w(·, t)
L¯rl
∇z(·, t)
Lrl(Ω)
≤
∇w(·, t)
k
L∞(Ω)
∇w(·, t)
1−k
L1(Ω)
∇z(·, t)
Lrl(Ω)
≤C5
∇w(·, t)
k
L∞(Ω),
∀t ∈(s0, Tmax).
(3.45)
Substituting (3.45) into (3.44) and using the Young inequality (see Lemma 3.9, [18]) with Lemma 3.2
gives
∥∇w(·, t)∥L∞(Ω) ≤C6 + C7
sup
t∈(s0,Tmax)
∇w(·, t)
k
L∞(Ω)
Z t
s0
1 + (t −s)−1
2 −n
2l
e−λd3(t−s)ds
+ C8
Z t
s0
1 + (t −s)−1
2 −n
2l
l
l−1 e−λd3l
l−1 (t−s)ds + C9
Z t
s0
∆z
l
Ll(Ω)
(3.46)
+ C10
Z t
s0
1 + (t −s)−1
2 −n
2l
e−λd3(t−s)ds.
There exits Cl > 0, the maximal Sobolev regularity for
∆z
l
Ll(Ω) (l > n) along with Lemma 3.1 gives
Z t
s0
∆z
l
Ll(Ω) ≤Cl
Z t
s0
αv + βw
l
Ll(Ω) + Cl
z(·, s0)
l
Ll(Ω) + Cl
∆z(·, s0)
l
Ll(Ω)
≤C11
Z t
s0
v
l
Ll(Ω) + C12
Z t
s0
w
l
Ll(Ω) + C13
≤C14
∀t ∈(s0, Tmax).
Now using the Gamma function, one gets from (3.46)
∥∇w(·, t)∥L∞(Ω) ≤C15
sup
t∈(s0,Tmax)
∇w(·, t)
k
L∞(Ω) + C16.
Finally, we arrive at
∥∇w(·, t)∥L∞(Ω) ≤C17,
∀t ∈(s0, Tmax),
where C17 > 0. By using the similar procedure, one can obtain from (3.23)
∇v(·, t)
L∞(Ω) ≤
∇ed2(t−s0)∆v(·, s0)
L∞(Ω) + χ2
Z t
s0
∇ed2(t−s)∆∇v(·, s) · ∇z(·, s)
L∞(Ω)ds
+ χ2
Z t
s0
∇ed2(t−s)∆v(·, s) ∆z(·, s)
L∞(Ω)ds
+ µ2
Z t
s0
∇ed2(t−s)∆v(·, s)
1 + a4w(·, s)
L∞(Ω)ds.
Using the Neumann heat semi-group technique, then consider the last term from the above estimate gives
v(·, t)
1 + a4w(·, t)
L∞(Ω) ≤C18.
Finally, by a similar manner we arrive at
∥∇v(·, t)∥L∞(Ω) ≤C19,
∀t ∈(s0, Tmax),
where C19 > 0. Now, applying the variation of constants formula to the first equation of (1.1), we get
u(·, t) =ed1(t−t0)∆u(·, t0)−χ1
Z t
t0
ed1(t−s)∆∇·
u(·, s)∇(v(·, s)w(·, s))
ds
17
+ µ1
Z t
t0
ed1(t−s)∆u(·, s)
1 −u(·, s) + a1v(·, s) + a2w(·, s)
ds.
Taking L∞(Ω) norm on both sides, one has
u(·, t)
L∞(Ω)≤∥ed1(t−t0)∆u(·, t0)∥L∞(Ω) + χ1
Z t
t0
∥ed1(t−s)∆∇·
u(·, s)∇(v(·, s)w(·, s))
∥L∞(Ω)ds
+ µ1
Z t
t0
∥ed1(t−s)∆u(·, s)
1 −u(·, s) + a1v(·, s) + a2w(·, s)
∥L∞(Ω)ds.
By the similar procedure used in the Lemma 3.2, we obtain
u(·, t)
L∞(Ω) ≤max
n
∥u(·, s0)∥L∞(Ω), C20
o
+C21
Z t
t0
1 + (t −s)−1
2 −n
2l
e−λd1(t−s)
u(·, s)∇(v(·, s)w(·, s))
Ll(Ω)ds
+µ1
(1 + a1K4 + a2K4)2
4
.
(3.47)
Using the Hölder inequality, Lemma 3.2 and then using Interpolation inequality, we obtain
u(·, t)∇(v(·, t)w(·, t))
Ll(Ω)≤
u(·, t)v(·, t)∇w(·, t)
Ll(Ω) +
u(·, t)w(·, t)∇v(·, t)
Ll(Ω)
≤
v(·, t)
L∞(Ω)
∇w(·, t)
L∞(Ω)
u(·, t)
Ll(Ω)
+
w(·, t)
L∞(Ω)
∇v(·, t)
L∞(Ω)
u(·, t)
Ll(Ω)
≤C22
u(·, t)
1
l
L∞(Ω)
u(·, t)
l
l−1
L1(Ω)
≤C23
u(·, t)
1
l
L∞(Ω),
∀t ∈(s0, Tmax).
(3.48)
Substitute (3.48) into (3.47) and using Gamma function, we arrive at
u(·, t)
L∞(Ω) ≤C24
u(·, t)
1
l
L∞(Ω) + C25.
Finally, we get
u(·, t)
L∞(Ω) ≤C26,
∀t ∈(s0, Tmax),
where C26 > 0. The proof is completed by taking into account (3.2).
Proof of Theorem 1.1. We argue the proof by contradiction. Assume that Tmax < ∞. By Lemma
3.2 and 3.6, the bounds on the solution components hold for all t ∈(0, Tmax), which contradicts the
blow-up criterion in (2.1). Hence, Tmax = ∞and we conclude that
sup
t>0
u(·, t)
L∞(Ω) +
v(·, t)
W1,∞(Ω) +
w(·, t)
W1,∞(Ω) +
z(·, t)
W1,q(Ω)
< ∞.
This completes the proof.
4. Global asymptotic stability
In this section, we analyze the global asymptotic behaviour of solutions for the system (1.1), using
the Lyapunov functional. The proofs of these asymptotic results are based on the work presented in [1].
The proof of the following lemma is adapted from [13].
18
Lemma 4.1. Let (u, v, w, z) be the nonnegative classical solution of the system (1.1) and suppose that
the assumptions of Theorem 1.1 hold true. Then there exists θ ∈(0, 1) and C > 0 such that
u(·, t)
C2+θ,1+ θ
2 (Ω×[t,t+1]) +
v(·, t)
C2+θ,1+ θ
2 (Ω×[t,t+1]) ≤C
and
w(·, t)
C2+θ,1+ θ
2 (Ω×[t,t+1]) +
z(·, t)
C2+θ,1+ θ
2 (Ω×[t,t+1]) ≤C,
for all t ≥1.
Proof. The proof is based on the standard parabolic regularity theory in [16] and Lemma 3.2 and 3.6.
For more details see [19, 26].
By Lemma 4.1, it is clear that u, v, w, z are continuous and bounded in Ω× [t, t + 1] and the first
derivatives ∇u, ∇v, ∇w, ∇z exist and are Hölder continuous.
Since the Hölder norms are uniformly
bounded, the gradients are also uniformly bounded in L∞(Ω), we conclude that
u(·, t)
W1,∞(Ω),
v(·, t)
W1,∞(Ω),
w(·, t)
W1,∞(Ω),
z(·, t)
W1,∞(Ω) ≤C,
(4.1)
where C is a constant independent of t.
Lemma 4.2 ([1]). Suppose that f is a uniformly continuous nonnegative function on (1, ∞) such that
Z ∞
1
f(t) dt < ∞.
Then, f(t) →0 as t →∞.
First we start with the co-existent state of the species.
4.1. Coexistence state of the species
Here we assume that a2a6 < min
n
1 + a1 + a2 + a1a4 + a4a6,
1+a1(a3+a4a5)+a4a6+a2a5
a3
o
, a3(1 + a2) +
a4a5 < 1 + a4 + a2a5, a5(1 + a1) + a6 < 1 + a3(a1 + a6) hold. Let (u, v, w, z) be the classical solution of
(1.1) satisfying (1.2) and (u∗, v∗, w∗, z∗) be the coexistence steady state of the system (1.1).
Lemma 4.3. Suppose that the assumptions in Theorem 1.1 hold true and let Γ1 = µ1a1
µ2a3 , Γ2 = µ1a2
µ3a5 . If
α + β < 2γ, a1a4a5 > a2a3a6 and
χ2
1 < min
d1d2v∗Γ1
u∗
v
2
L∞(Ω)
w
2
L∞(Ω)
,
d1d3w∗Γ2
u∗
v
2
L∞(Ω)
w
2
L∞(Ω)
,
d3χ2
2v∗Γ1 + d2ξ2w∗Γ2 < 2d2d3d4,
µ2a4Γ1 + α < 2µ2Γ1 + µ3a6Γ2,
µ2a4Γ1 + β < 2µ3Γ2 + µ3a6Γ2,
then the following asymptotic behaviour holds
u(·, t) −u∗
L∞(Ω) +
v(·, t) −v∗
L∞(Ω) +
w(·, t) −w∗
L∞(Ω) +
z(·, t) −z∗
L∞(Ω) →0
(4.2)
as t →∞.
Proof. To analyze the coexistence state of the species, we now introduce the following energy functional
E1(t) :=
Z
Ω
u −u∗−u∗ln u
u∗
+ Γ1
Z
Ω
v −v∗−v∗ln v
v∗
+ Γ2
Z
Ω
w −w∗−w∗ln w
w∗
19
+ 1
2
Z
Ω
(z −z∗)2 ,
=I11(t) + I12(t) + I13(t) + I14(t),
t > 0.
To prove the nonnegativity of E1, let H(u) = u −u∗ln u, for u > 0. Applying the Taylor’s formula to see
that, for all x ∈Ωand each t > 0, we can find τ = τ(x, t) ∈(0, 1) such that
H(u) −H(u∗) = H′(u∗)(u −u∗) + 1
2H′′(τu + (1 −τ)u∗)(u −u∗)2
=
u∗
2(τu + (1 −τ)u∗)2 (u −u∗)2 ≥0.
Therefore, we get R
Ω
u −u∗−u∗ln u
u∗
= R
ΩH(u)−H(u∗) ≥0. A similar argument shows that E1(t) ≥0.
Now, we compute
d
dtI11(t) =
Z
Ω
u −u∗
u
ut
= −d1u∗
Z
Ω
∇u
u
2
+ χ1u∗
Z
Ω
w
u ∇u · ∇v + χ1u∗
Z
Ω
v
u∇u · ∇w
+ µ1
Z
Ω
(u −u∗)(1 −u + a1v + a2w)
≤−d1u∗
Z
Ω
∇u
u
2
+ χ1u∗
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1u∗
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
+ µ1
Z
Ω
(u −u∗)(1 −u + a1v + a2w −(1 −u∗+ a1v∗+ a2w∗))
≤−d1u∗
Z
Ω
∇u
u
2
+ χ1u∗
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1u∗
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−µ1
Z
Ω
(u −u∗)2 + µ1a1
Z
Ω
(u −u∗)(v −v∗) + µ1a2
Z
Ω
(u −u∗)(w −w∗).
(4.3)
Similarly, we get
d
dtI12(t) =Γ1
Z
Ω
v −v∗
v
vt
= −Γ1d2v∗
Z
Ω
∇v
v
2
+ Γ1χ2v∗
Z
Ω
∇v
v
· ∇z + Γ1µ2
Z
Ω
(v −v∗)(1 −v −a3u + a4w)
≤−Γ1d2v∗
Z
Ω
∇v
v
2
+ Γ1χ2v∗
Z
Ω
∇v
v
· ∇z −Γ1µ2
Z
Ω
(v −v∗)2
−Γ1µ2a3
Z
Ω
(v −v∗)(u −u∗) + Γ1µ2a4
Z
Ω
(v −v∗)(w −w∗).
(4.4)
An analogous calculation for I13 provides
d
dtI13(t) =Γ2
Z
Ω
w −w∗
v
wt
= −Γ2d3w∗
Z
Ω
∇w
w
2
−Γ2ξw∗
Z
Ω
∇w
w · ∇z + µ3
Z
Ω
(w −w∗)(1 −w −a5u −a6v)
≤−Γ2w∗
Z
Ω
∇w
w
2
−Γ2ξw∗
Z
Ω
∇w
w · ∇z −Γ2µ3
Z
Ω
(w −w∗)2
−Γ2µ3a5
Z
Ω
(w −w∗)(u −u∗) −Γ2µ3a6
Z
Ω
(w −w∗)(v −v∗)
(4.5)
20
and for I14
d
dtI14(t) =
Z
Ω
(z −z∗)zt
≤−d4
Z
Ω
|∇z|2 + α
Z
Ω
(z −z∗)(v −v∗) + β
Z
Ω
(z −z∗)(w −w∗) −γ
Z
Ω
(z −z∗)2.
(4.6)
Adding (4.3)-(4.6), we get
d
dtE1(t) ≤−d1u∗
Z
Ω
∇u
u
2
+ χ1u∗
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1u∗
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−Γ1d2v∗
Z
Ω
∇v
v
2
+ Γ1χ2v∗
Z
Ω
∇v
v · ∇z −Γ2d3w∗
Z
Ω
∇w
w
2
−Γ2ξw∗
Z
Ω
∇w
w · ∇z
−d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −u∗)2 −Γ1µ2
Z
Ω
(v −v∗)2 −Γ2µ3
Z
Ω
(w −w∗)2
−γ
Z
Ω
(z −z∗)2 + µ1a1
Z
Ω
(u −u∗)(v −v∗) + µ1a2
Z
Ω
(u −u∗)(w −w∗)
−Γ1µ2a3
Z
Ω
(v −v∗)(u −u∗) + Γ1µ2a4
Z
Ω
(v −v∗)(w −w∗)
−Γ2µ3a5
Z
Ω
(w −w∗)(u −u∗) −Γ2µ3a6
Z
Ω
(w −w∗)(v −v∗) + α
Z
Ω
(z −z∗)(v −v∗)
+ β
Z
Ω
(z −z∗)(w −w∗).
By simplifying, we obtain
d
dtE1(t) ≤−d1u∗
Z
Ω
∇u
u
2
+ χ1u∗
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1u∗
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−Γ1d2v∗
Z
Ω
∇v
v
2
+ Γ1χ2v∗
Z
Ω
∇v
v · ∇z −Γ2d3w∗
Z
Ω
∇w
w
2
−Γ2ξw∗
Z
Ω
∇w
w · ∇z
−d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −u∗)2 −Γ1µ2
Z
Ω
(v −v∗)2 −Γ2µ3
Z
Ω
(w −w∗)2
−γ
Z
Ω
(z −z∗)2 + (Γ1µ2a4 −Γ2µ3a6)
Z
Ω
(v −v∗)(w −w∗) + α
Z
Ω
(z −z∗)(v −v∗)
+ β
Z
Ω
(z −z∗)(w −w∗).
(4.7)
Applying Cauchy’s inequality, we obtain the following estimates
χ1u∗
w
L∞(Ω)
Z
Ω
∇u
u · ∇v ≤d1u∗
2
Z
Ω
∇u
u
2
+
χ2
1u∗
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2,
χ1u∗
v
L∞(Ω)
Z
Ω
∇u
u · ∇w ≤d1u∗
2
Z
Ω
∇u
u
2
+
χ2
1u∗
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2,
Γ1χ2v∗
Z
Ω
∇v
v · ∇z ≤Γ1d2v∗
2
Z
Ω
∇v
v
2
+ Γ1χ2
2v∗
2d2
Z
Ω
|∇z|2,
Γ2ξw∗
Z
Ω
∇w
w · ∇z ≤Γ2d3w∗
2
Z
Ω
∇w
w
2
+ Γ2ξ2w∗
2d3
Z
Ω
|∇z|2,
(Γ1µ2a4 −Γ2µ3a6)
Z
Ω
(v −v∗)(w −w∗) ≤Γ1µ2a4 −Γ2µ3a6
2
Z
Ω
(v −v∗)2,
+ Γ1µ2a4 −Γ2µ3a6
2
Z
Ω
(w −w∗)2,
21
α
Z
Ω
(z −z∗)(v −v∗) ≤α
2
Z
Ω
(z −z∗)2 + α
2
Z
Ω
(v −v∗)2,
β
Z
Ω
(z −z∗)(w −w∗) ≤β
2
Z
Ω
(z −z∗)2 + β
2
Z
Ω
(w −w∗)2.
By substituting all the above estimates into (4.7), we obtain
d
dtE1(t) ≤
χ2
1u∗
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2 +
χ2
1u∗
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2 −Γ1d2v∗
2
Z
Ω
∇v
v
2
+ Γ1χ2
2v∗
2d2
Z
Ω
|∇z|2 −Γ2d3w∗
2
Z
Ω
∇w
w
2
+ Γ2ξ2w∗
2d3
Z
Ω
|∇z|2 −d4
Z
Ω
|∇z|2
−µ1
Z
Ω
(u −u∗)2 −Γ1µ2
Z
Ω
(v −v∗)2 −Γ2µ3
Z
Ω
(w −w∗)2 −γ
Z
Ω
(z −z∗)2
+ Γ1µ2a4 −Γ2µ3a6
2
Z
Ω
(v −v∗)2 + Γ1µ2a4 −Γ2µ3a6
2
Z
Ω
(w −w∗)2 + α
2
Z
Ω
(z −z∗)2
+ α
2
Z
Ω
(v −v∗)2 + β
2
Z
Ω
(z −z∗)2 + β
2
Z
Ω
(w −w∗)2.
Combining the terms, one gets
d
dtE1(t) ≤−
Γ1d2v∗
2
v
2
L∞(Ω)
−
χ2
1u∗
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2
−
Γ2d3w∗
2
w
2
L∞(Ω)
−
χ2
1u∗
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2 −
d4 −Γ1χ2
2v∗
2d2
−Γ2ξ2w∗
2d3
Z
Ω
|∇z|2
−µ1
Z
Ω
(u −u∗)2 −
Γ1µ2 −Γ1µ2a4 −Γ2µ3a6
2
−α
2
Z
Ω
(v −v∗)2
−
Γ2µ3 −Γ1µ2a4 −Γ2µ3a6
2
−β
2
Z
Ω
(w −w∗)2 −
γ −α
2 −β
2
Z
Ω
(z −z∗)2.
Finally,
d
dtE1(t) ≤−σ1F1(t),
∀t > 0,
(4.8)
where σ1 > 0 and
F1(t) =
Z
Ω
(u −u∗)2 +
Z
Ω
(v −v∗)2 +
Z
Ω
(w −w∗)2 +
Z
Ω
(z −z∗)2
+
Z
Ω
|∇v|2 +
Z
Ω
|∇w|2 +
Z
Ω
|∇z|2.
Let
f1(t) =
Z
Ω
(u(·, t) −u∗)2 +
Z
Ω
(v(·, t) −v∗)2 +
Z
Ω
(w(·, t) −w∗)2 +
Z
Ω
(z(·, t) −z∗)2,
t > 0.
Now (4.8) takes the form
d
dtE1(t) ≤−σ1F1(t) ≤−σ1f1(t),
t > 0.
(4.9)
Upon integrating with respect to t, we obtain
Z ∞
1
f1(t) dt ≤1
σ1
E1(1) −E1(t)
< ∞.
22
Since f1(t) is uniformly continuous in (1, ∞), we use Lemma (4.2), which gives
Z
Ω
(u(·, t) −u∗)2 +
Z
Ω
(v(·, t) −v∗)2 +
Z
Ω
(w(·, t) −w∗)2 +
Z
Ω
(z(·, t) −z∗)2 →0
as t →∞. Applying the Gagliardo-Nirenberg inequality, we have
u(·, t) −u∗
L∞(Ω) ≤C1
u(·, t) −u∗
n
n+2
W1,∞(Ω)
u(·, t) −u∗
2
n+2
L2(Ω),
t > 0.
(4.10)
From (4.1), we can deduce that u(·, t) converges to u∗in L∞(Ω) when t →∞.
By using a similar
argument, we can derive (4.2).
4.2. Secondary predator only existence state
Here we assume that
a2a6 ≥min
1 + a1 + a2 + a1a4 + a4a6, 1 + a1(a3 + a4a5) + a4a6 + a2a5
a3
,
a3(1 + a2) + a4a5 ≥1 + a4 + a2a5,
and
a5(1 + a1) + a6 ≥1 + a3(a1 + a6)
hold. Let (u, v, w, z) be the classical solution of (1.1) satisfying (1.2) and (1, 0, 0, 0) be the trivial steady
state of the system (1.1).
Lemma 4.4. Suppose that the assumptions in Theorem 1.1 hold true and let Γ1 = µ1a1
µ2a3 , Γ2 = µ1a2
µ3a5 . If
α + β < 2γ, a1a4a5 < a2a3a6 and
χ2
1 < min
d1d2
w
2
L∞(Ω)
,
d1d3
v
2
L∞(Ω)
,
d3χ2
2
v
2
L∞(Ω) + d2ξ2
w
2
L∞(Ω) < 2d2d3d4,
µ2 + µ2a4
w
L∞(Ω) + α
2 < Γ1µ2 < µ1a1,
µ3 + β
2 < Γ2µ3 < µ1a2,
then the following asymptotic behaviour holds
u(·, t) −1
L∞(Ω) +
v(·, t)
L∞(Ω) +
w(·, t)
L∞(Ω) +
z(·, t)
L∞(Ω) →0
(4.11)
as t →∞.
Proof. We now introduce the following energy functional
E2(t) :=
Z
Ω
(u −1 −ln u) + Γ1
Z
Ω
v + 1
2
Z
Ω
v2 + Γ2
Z
Ω
w + 1
2
Z
Ω
w2 + 1
2
Z
Ω
z2,
=I21(t) + I22(t) + I23t) + I24(t) + I25(t) + I26(t),
t > 0.
The positivity of u −1 −lnu for any u > 0 implies that E2(t) ≥0. We begin by computing the derivative
23
of the first integral
d
dtI21(t) =
Z
Ω
u −1
u
ut
= −d1
Z
Ω
∇u
u
2
+ χ1
Z
Ω
w
u ∇u · ∇v + χ1
Z
Ω
v
u∇u · ∇w
+ µ1
Z
Ω
(u −1)(1 −u + a1v + a2w)
≤−d1
Z
Ω
∇u
u
2
+ χ1
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−µ1
Z
Ω
(u −1)2 + µ1a1
Z
Ω
(u −1)v + µ1a2
Z
Ω
(u −1)w
≤−d1
Z
Ω
∇u
u
2
+ χ1
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−µ1
Z
Ω
(u −1)2 + µ1a1
Z
Ω
uv −µ1a1
Z
Ω
v + µ1a2
Z
Ω
uw −µ1a2
Z
Ω
w.
(4.12)
Likewise for I22(t)
d
dtI22(t) =Γ1µ2
Z
Ω
v −Γ1µ2
Z
Ω
v2 −Γ1µ2a3
Z
Ω
uv + Γ1µ2a4
Z
Ω
vw
=Γ1µ2
Z
Ω
v −Γ1µ2
Z
Ω
v2 −µ1a1
Z
Ω
uv + Γ1µ2a4
Z
Ω
vw,
(4.13)
and for I23(t)
d
dtI23(t) =d2
Z
Ω
v∆v −χ2
Z
Ω
v∇· (v∇z) + µ2
Z
Ω
v2(1 −v −a3u + a4w)
≤−d2
Z
Ω
|∇v|2 + χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z + µ2
Z
Ω
v2 −µ2
Z
Ω
v3 −µ2a3
Z
Ω
v2u
+ µ2a4
Z
Ω
v2w
≤−d2
Z
Ω
|∇v|2 + χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z + µ2
Z
Ω
v2 + µ2a4
w
L∞(Ω)
Z
Ω
v2.
(4.14)
By a similar manner, we obtain
d
dtI24(t) =Γ2µ3
Z
Ω
w −Γ2µ3
Z
Ω
w2 −Γ2µ3a5
Z
Ω
uw −Γ2µ3a6
Z
Ω
vw
=Γ2µ3
Z
Ω
w −Γ2µ3
Z
Ω
w2 −µ1a2
Z
Ω
uw −Γ2µ3a6
Z
Ω
vw.
(4.15)
Similarly for I25(t)
d
dtI25(t) =d3
Z
Ω
w∆w + ξ
Z
Ω
w∇· (w∇z) + µ3
Z
Ω
w2(1 −w −a5u −a6v)
≤−d3
Z
Ω
|∇w|2 + ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z + µ3
Z
Ω
w2 −µ3
Z
Ω
w3 −µ3a5
Z
Ω
w2u
−µ3a6
Z
Ω
w2v
≤−d3
Z
Ω
|∇w|2 + ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z + µ3
Z
Ω
w2,
(4.16)
24
and for I26(t)
d
dtI26(t) =d4
Z
Ω
z∆z + α
Z
Ω
vz + β
Z
Ω
wz −γ
Z
Ω
z2
≤−d4
Z
Ω
|∇z|2 + α
2
Z
Ω
v2 + α
2
Z
Ω
z2 + β
2
Z
Ω
w2 + β
2
Z
Ω
z2 −γ
Z
Ω
z2.
(4.17)
Adding all the estimates (4.12)-(4.17), we get
d
dtE2(t) ≤−d1
Z
Ω
∇u
u
2
+ χ1
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−d2
Z
Ω
|∇v|2 + χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z −d3
Z
Ω
|∇w|2 + ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z
−d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −1)2 −µ1a1
Z
Ω
v −µ1a2
Z
Ω
w + Γ1µ2
Z
Ω
v −Γ1µ2
Z
Ω
v2
+ Γ1µ2a4
Z
Ω
vw + µ2
Z
Ω
v2 + µ2a4
w
L∞(Ω)
Z
Ω
v2 + Γ2µ3
Z
Ω
w −Γ2µ3
Z
Ω
w2
−Γ2µ3a6
Z
Ω
vw + µ3
Z
Ω
w2 + α
2
Z
Ω
v2 + α
2
Z
Ω
z2 + β
2
Z
Ω
w2 + β
2
Z
Ω
z2 −γ
Z
Ω
z2.
(4.18)
Using the Cauchy inequality, we get the following estimates
χ1
w
L∞(Ω)
Z
Ω
∇u
u · ∇v ≤d1
2
Z
Ω
∇u
u
2
+
χ2
1
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2,
χ1
v
L∞(Ω)
Z
Ω
∇u
u · ∇w ≤d1
2
Z
Ω
∇u
u
2
+
χ2
1
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2,
χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z ≤d2
2
Z
Ω
|∇v|2 +
χ2
2
v
2
L∞(Ω)
2d2
Z
Ω
|∇z|2,
ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z ≤d3
2
Z
Ω
|∇w|2 +
ξ2
w
2
L∞(Ω)
2d3
Z
Ω
|∇z|2.
Substituting all the above estimates in (4.18), we get
d
dtE2(t) ≤−
d2
2 −
χ2
1
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2 −
d3
2 −
χ2
1
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2
−
d4 −
χ2
2
v
2
L∞(Ω)
2d2
−
ξ2
w
2
L∞(Ω)
2d3
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −1)2
−
Γ1µ2 −µ2 −µ2a4
w
L∞(Ω) −α
2
Z
Ω
v2 −
Γ2µ3 −µ3 −β
2
Z
Ω
w2
−
γ −α
2 −β
2
Z
Ω
z2 −(µ1a1 −Γ1µ2)
Z
Ω
v −(µ1a2 −Γ2µ3)
Z
Ω
w
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw.
Finally, we arrive at
d
dtE2(t) ≤−σ2F2(t) −(µ1a1 −Γ1µ2)
Z
Ω
v −(µ1a2 −Γ2µ3)
Z
Ω
w
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw,
(4.19)
25
for every t > 0, where σ2 > 0 and
F2(t) =
Z
Ω
(u −1)2 +
Z
Ω
v2 +
Z
Ω
w2 +
Z
Ω
z2 +
Z
Ω
|∇v|2 +
Z
Ω
|∇w|2 +
Z
Ω
|∇z|2.
Now, (4.19) can be written as
d
dtE2(t) ≤−σ2f2(t) −(µ1a1 −Γ1µ2)
Z
Ω
v −(µ1a2 −Γ2µ3)
Z
Ω
w −(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw,
(4.20)
for all t > 0, where
f2(t) =
Z
Ω
(u(·, t) −1)2 +
Z
Ω
v(·, t)2 +
Z
Ω
w(·, t)2 +
Z
Ω
z(·, t)2.
In view of Lemma 4.3, the proof is similar.
4.3. Semi-coexistence state of the species
4.3.1. Prey vanishing state
Here we assume that (1.3) does not hold and a3 < 1. Let (u, v, w, z) be the classical solution of (1.1)
satisfying (1.2) and (¯u, ¯v, ¯w, ¯z) be the semi-coexistence steady state
1+a1
1+a1a3 ,
1−a3
1+a1a3 , 0,
α(1−a3)
γ(1+a1a3)
of the
system (1.1).
Lemma 4.5. Let Γ1 = µ1a1
µ2a3 , Γ2 = µ1a2
µ3a5 . If α + β < 2γ, a1a4a5 < a2a3a6, α < 2Γ1µ2 and
χ2
1 < min
d1d2¯vΓ1
¯u
v
2
L∞(Ω)
w
2
L∞(Ω)
,
d1d3
¯u
v
2
L∞(Ω)
,
d3χ2
2¯vΓ1 + d2ξ2
w
2
L∞(Ω) < 2d2d3d4,
µ3 + β
2 < Γ2µ3 < µ1a2¯u + Γ1µ2a4¯v,
then the following asymptotic behaviour holds
u(·, t) −¯u
L∞(Ω) +
v(·, t) −¯v
L∞(Ω) +
w(·, t)
L∞(Ω) +
z(·, t) −¯z
L∞(Ω) →0,
(4.21)
as t →∞.
Proof. To analyse the coexistence state of the species, we now introduce the following energy functional
E3(t) :=
Z
Ω
u −¯u −¯u ln u
¯u
+ Γ1
Z
Ω
v −¯v −¯v ln v
¯v
+ Γ2
Z
Ω
w + 1
2
Z
Ω
w2 + 1
2
Z
Ω
(z −¯z)2 ,
=I31(t) + I32(t) + I33(t) + I34(t) + I35(t),
t > 0.
By a similar argument as in the previous Lemma 4.3, we find that E3(t) ≥0. Now, we compute
d
dtI31(t) ≤−d1¯u
Z
Ω
∇u
u
2
+ χ1¯u
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1¯u
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−µ1
Z
Ω
(u −¯u)2 + µ1a1
Z
Ω
(u −¯u)(v −¯v) + µ1a2
Z
Ω
(u −¯u)(w −¯w).
(4.22)
Similarly, we get
d
dtI32(t) = −Γ1d2¯v
Z
Ω
∇v
v
2
+ Γ1χ2¯v
Z
Ω
∇v
v
· ∇z −Γ1µ2
Z
Ω
(v −¯v)2
−Γ1µ2a3
Z
Ω
(v −¯v)(u −¯u) + Γ1µ2a4
Z
Ω
(v −¯v)(w −¯w).
(4.23)
26
Analogously, for I33,
d
dtI33(t) =Γ2µ3
Z
Ω
w −Γ2µ3
Z
Ω
w2 −µ1a2
Z
Ω
uw −Γ2µ3a6
Z
Ω
vw.
(4.24)
Following the same procedure for I34(t)
d
dtI34(t) ≤−d3
Z
Ω
|∇w|2 + ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z + µ3
Z
Ω
w2
(4.25)
and for I35,
d
dtI35(t) = −d4
Z
Ω
|∇z|2 + α
Z
Ω
(z −¯z)(v −¯v) + β
Z
Ω
(z −¯z)(w −¯w) −γ
Z
Ω
(z −¯z)2.
(4.26)
Adding (4.22)-(4.26), we get
d
dtE3(t) ≤−d1¯u
Z
Ω
∇u
u
2
+ χ1¯u
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1¯u
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−Γ1d2¯v
Z
Ω
∇v
v
2
+ Γ1χ2¯v
Z
Ω
∇v
v
· ∇z −d3
Z
Ω
|∇w|2 + ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z
−d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −¯u)2 −Γ1µ2
Z
Ω
(v −¯v)2 −Γ2µ3
Z
Ω
w2 + µ3
Z
Ω
w2
−γ
Z
Ω
(z −¯z)2 −Γ2µ3a6
Z
Ω
vw + Γ1µ2a4
Z
Ω
vw −Γ1µ2a4¯v
Z
Ω
w + Γ2µ3
Z
Ω
w
−µ1a2¯u
Z
Ω
w + α
Z
Ω
(z −¯z)(v −¯v) + β
Z
Ω
(z −¯z)w.
By simplifying, we obtain
d
dtE3(t) ≤−d1¯u
Z
Ω
∇u
u
2
+ χ1¯u
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1¯u
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−Γ1d2¯v
Z
Ω
∇v
v
2
+ Γ1χ2¯v
Z
Ω
∇v
v
· ∇z −d3
Z
Ω
|∇w|2 + ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z
−d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −¯u)2 −Γ1µ2
Z
Ω
(v −¯v)2 −(Γ2µ3 −µ3)
Z
Ω
w2
−γ
Z
Ω
(z −¯z)2 −(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw −(µ1a2¯u + Γ1µ2a4¯v −Γ2µ3)
Z
Ω
w
+ α
Z
Ω
(z −¯z)(v −¯v) + β
Z
Ω
(z −¯z)w.
(4.27)
Applying Cauchy’s inequality, we obtain the following estimates
χ1¯u
w
L∞(Ω)
Z
Ω
∇u
u · ∇v ≤d1¯u
2
Z
Ω
∇u
u
2
+
χ2
1¯u
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2,
χ1¯u
v
L∞(Ω)
Z
Ω
∇u
u · ∇w ≤d1¯u
2
Z
Ω
∇u
u
2
+
χ2
1¯u
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2,
Γ1χ2¯v
Z
Ω
∇v
v · ∇z ≤Γ1d2¯v
2
Z
Ω
∇v
v
2
+ Γ1χ2
2¯v
2d2
Z
Ω
|∇z|2,
ξ
w
L∞(Ω)
Z
Ω
∇w · ∇z ≤d3
2
Z
Ω
|∇w|2 +
ξ2
w
2
L∞(Ω)
2d3
Z
Ω
|∇z|2,
α
Z
Ω
(z −¯z)(v −¯v) ≤α
2
Z
Ω
(z −¯z)2 + α
2
Z
Ω
(v −¯v)2,
27
β
Z
Ω
(z −¯z)w ≤β
2
Z
Ω
(z −¯z)2 + β
2
Z
Ω
w2.
By substituting all the above estimates into (4.27), we obtain
d
dtE3(t) ≤−d1¯u
Z
Ω
∇u
u
2
+d1¯u
2
Z
Ω
∇u
u
2
+
χ2
1¯u
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2 + d1¯u
2
Z
Ω
∇u
u
2
+
χ2
1¯u
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2 −Γ1d2¯v
Z
Ω
∇v
v
2
+ Γ1d2¯v
2
Z
Ω
∇v
v
2
+ Γ1χ2
2¯v
2d2
Z
Ω
|∇z|2
−d3
Z
Ω
|∇w|2 + d3
2
Z
Ω
|∇w|2 +
ξ2
w
2
L∞(Ω)
2d3
Z
Ω
|∇z|2 −d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −¯u)2
−Γ1µ2
Z
Ω
(v −¯v)2 −(Γ2µ3 −µ3)
Z
Ω
w2 −γ
Z
Ω
(z −¯z)2 −(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw
−(µ1a2¯u + Γ1µ2a4¯v −Γ2µ3)
Z
Ω
w + α
2
Z
Ω
(z −¯z)2 + α
2
Z
Ω
(v −¯v)2 + β
2
Z
Ω
(z −¯z)2
+ β
2
Z
Ω
w2.
Combining the terms, one gets
d
dtE3(t) ≤−
Γ1d2¯v
2
v
2
L∞(Ω)
−
χ2
1¯u
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2 −
d3
2 −
χ2
1¯u
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2
−
d4 −Γ1χ2
2¯v
2d2
−
ξ2
w
2
L∞(Ω)
2d3
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −¯u)2
−
Γ1µ2 −α
2
Z
Ω
(v −¯v)2 −
Γ2µ3 −µ3 −β
2
Z
Ω
w2 −
γ −α
2 −β
2
Z
Ω
(z −¯z)2
−(µ1a2¯u + Γ1µ2a4¯v −Γ2µ3)
Z
Ω
w −(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw.
Finally, we arrive at
d
dtE3(t) ≤−σ1F3(t) −(µ1a2¯u + Γ1µ2a4¯v −Γ2µ3)
Z
Ω
w
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw,
∀t > 0,
(4.28)
where σ3 > 0 and
F3(t) =
Z
Ω
(u −¯u)2 +
Z
Ω
(v −¯v)2 +
Z
Ω
w2 +
Z
Ω
(z −¯z)2 +
Z
Ω
|∇v|2 +
Z
Ω
|∇w|2 +
Z
Ω
|∇z|2,
Let
f3(t) =
Z
Ω
(u(·, t) −¯u)2 +
Z
Ω
(v(·, t) −¯v)2 +
Z
Ω
w(·, t)2 +
Z
Ω
(z(·, t) −¯z)2,
t > 0.
Now (4.28) takes the form
d
dtE3(t) ≤−σ1f3(t) −(µ1a2¯u + Γ1µ2a4¯v −Γ2µ3)
Z
Ω
w
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw,
t > 0.
(4.29)
28
Upon integrating with respect to t, we obtain
Z ∞
1
f3(t) dt ≤1
σ1
E3(1) −E3(t)
< ∞.
Since f3(t) is uniformly continuous in (1, ∞), which gives
Z
Ω
(u(·, t) −¯u)2 +
Z
Ω
(v(·, t) −¯v)2 +
Z
Ω
w(·, t)2 +
Z
Ω
(z(·, t) −¯z)2 →0,
as t →∞. By using a similar argument as in the proof of Lemma 4.3, we can derive (4.21).
4.3.2. Primary predator vanishing state
Here we assume that (1.3) does not hold and a5 < 1. Let (u, v, w, z) be the classical solution of (1.1)
satisfying (1.2) and (ˆu, ˆv, ˆw, ˆz) be the semi-coexistence steady state
1+a2
1+a2a5 , 0,
1−a5
1+a2a5 ,
β(1−a5)
γ(1+a5a5)
of the
system (1.1).
Lemma 4.6. Let Γ1 = µ1a1
µ2a3 , Γ2 = µ1a2
µ3a5 . If α + β < 2γ, a1a4a5 < a2a3a6, β < Γ2µ3 and
χ2
1 < min
d1d2
ˆu
w
2
L∞(Ω)
,
d1d3 ˆwΓ2
ˆu
v
2
L∞(Ω)
w
2
L∞(Ω)
,
d3χ2
2
v
2
L∞(Ω) + d2ξ2 ˆwΓ2 < 2d2d3d4,
µ2 + µ2a4
w
L∞(Ω) + α
2 < Γ1µ2,
Γ1µ2 + Γ2µ3a6 ˆw < µ1a1ˆu,
then the following asymptotic behaviour holds
u(·, t) −ˆu
L∞(Ω) +
v(·, t)
L∞(Ω) +
w(·, t) −ˆw
L∞(Ω) +
z(·, t) −ˆz
L∞(Ω) →0,
(4.30)
as t →∞.
Proof. We now introduce the following energy functional
E4(t) :=
Z
Ω
u −ˆu −ln u
ˆu
+ Γ1
Z
Ω
v + 1
2
Z
Ω
v2 + Γ2
Z
Ω
w −ˆw −ln u
ˆw
+ 1
2
Z
Ω
(z −ˆz)2,
=I41(t) + I42(t) + I43t) + I44(t) + I45(t),
t > 0.
Analogous to reasoning in Lemma 4.3, it is clear that E4(t) ≥0. Now we compute
d
dtI42(t) =Γ1µ2
Z
Ω
v −Γ1µ2
Z
Ω
v2 −µ1a1
Z
Ω
uv + Γ1µ2a4
Z
Ω
vw,
(4.31)
and for I43(t)
d
dtI43(t) ≤−d2
Z
Ω
|∇v|2 + χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z + µ2
Z
Ω
v2 + µ2a4
w
L∞(Ω)
Z
Ω
v2.
(4.32)
By a similar manner, we obtain
d
dtI44(t) = −Γ2d3 ˆw
Z
Ω
∇w
w
2
−Γ2ξ ˆw
Z
Ω
∇w
w · ∇z −Γ2µ3
Z
Ω
(w −ˆw)2
−Γ2µ3a5
Z
Ω
(w −ˆw)(u −ˆu) −Γ2µ3a6
Z
Ω
(w −ˆw)v.
(4.33)
29
Adding all the estimates (4.22), (4.26) and (4.31)-(4.33), we get
d
dtE4(t) ≤−d1ˆu
Z
Ω
∇u
u
2
+ χ1ˆu
w
L∞(Ω)
Z
Ω
∇u
u · ∇v + χ1ˆu
v
L∞(Ω)
Z
Ω
∇u
u · ∇w
−d2
Z
Ω
|∇v|2 + χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z −Γ2d3 ˆw
Z
Ω
∇w
w
2
−Γ2ξ ˆw
Z
Ω
∇w
w · ∇z
−d4
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −ˆu)2 −Γ1µ2
Z
Ω
v2 + µ2
Z
Ω
v2 + µ2a4
w
L∞(Ω)
Z
Ω
v2
−Γ2µ3
Z
Ω
(w −ˆw)2 −γ
Z
Ω
(z −ˆz)2 −µ1a1ˆu
Z
Ω
v
+ Γ1µ2
Z
Ω
v + Γ1µ2a4
Z
Ω
vw −Γ2µ3a6
Z
Ω
wv + Γ2µ3a6 ˆw
Z
Ω
v + α
Z
Ω
(z −ˆz)v
+ β
Z
Ω
(z −ˆz)(w −ˆw).
(4.34)
Using the Cauchy inequality, we get the following estimates
χ2
v
L∞(Ω)
Z
Ω
∇v · ∇z ≤d2
2
Z
Ω
|∇v|2 +
χ2
2
v
2
L∞(Ω)
2d2
Z
Ω
|∇z|2,
Γ2ξ ˆw
Z
Ω
∇w
w · ∇z ≤Γ2d3 ˆw
2
Z
Ω
∇w
w
2
+ Γ2ξ2 ˆw
2d3
Z
Ω
|∇z|2,
α
Z
Ω
(z −ˆz)v ≤α
2
Z
Ω
(z −ˆz)2 + α
2
Z
Ω
v2,
β
Z
Ω
(z −ˆz)(w −ˆw) ≤β
2
Z
Ω
(z −ˆz)2 + β
2
Z
Ω
(w −ˆw)2.
Substituting all the above estimates in (4.34), we get
d
dtE4(t) ≤−d1ˆu
Z
Ω
∇u
u
2
+ d1ˆu
2
Z
Ω
∇u
u
2
+
χ2
1ˆu
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2 + d1ˆu
2
Z
Ω
∇u
u
2
+
χ2
1ˆu
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2 −d2
Z
Ω
|∇v|2 + d2
2
Z
Ω
|∇v|2 +
χ2
2
v
2
L∞(Ω)
2d2
Z
Ω
|∇z|2
−Γ2d3 ˆw
Z
Ω
∇w
w
2
+ Γ2d3 ˆw
2
Z
Ω
∇w
w
2
+ Γ2ξ2 ˆw
2d3
Z
Ω
|∇z|2 −d4
Z
Ω
|∇z|2
−µ1
Z
Ω
(u −ˆu)2 −
Γ1µ2 −µ2 −µ2a4
w
L∞(Ω)
Z
Ω
v2 −Γ2µ3
Z
Ω
(w −ˆw)2
−γ
Z
Ω
(z −ˆz)2 −(µ1a1ˆu + Γ1µ2 + Γ2µ3a6 ˆw)
Z
Ω
v −(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
wv
+ α
2
Z
Ω
(z −ˆz)2 + α
2
Z
Ω
v2 + β
2
Z
Ω
(z −ˆz)2 + β
2
Z
Ω
(w −ˆw)2.
Combining the terms, one gets
d
dtE4(t) ≤−
d2
2 −
χ2
1ˆu
w
2
L∞(Ω)
2d1
Z
Ω
|∇v|2 −
Γ2d3 ˆw
2
w
2
L∞(Ω)
−
χ2
1ˆu
v
2
L∞(Ω)
2d1
Z
Ω
|∇w|2
−
d4 −
χ2
2
v
2
L∞(Ω)
2d2
−Γ2ξ2 ˆw
2d3
Z
Ω
|∇z|2 −µ1
Z
Ω
(u −ˆu)2
−
Γ1µ2 −µ2 −µ2a4
w
L∞(Ω) −α
2
Z
Ω
v2 −
Γ2µ3 −β
2
Z
Ω
(w −ˆw)2
30
−
γ −α
2 −β
2
Z
Ω
(z −ˆz)2 −(µ1a1ˆu −Γ1µ2 −Γ2µ3a6 ˆw)
Z
Ω
v
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
wv.
Finally, we arrive at
d
dtE4(t) ≤−σ4F4(t) −(µ1a1ˆu −Γ1µ2 −Γ2µ3a6 ˆw)
Z
Ω
v
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw,
t > 0,
(4.35)
where σ4 > 0 with
F4(t) =
Z
Ω
(u −ˆu)2 +
Z
Ω
v2 +
Z
Ω
(w −ˆw)2 +
Z
Ω
(z −ˆz)2 +
Z
Ω
|∇v|2 +
Z
Ω
|∇w|2 +
Z
Ω
|∇z|2.
Now, (4.35) can be written as
d
dtE4(t) ≤−σ4f4(t) −(µ1a1ˆu −Γ1µ2 −Γ2µ3a6 ˆw)
Z
Ω
v
−(Γ2µ3a6 −Γ1µ2a4)
Z
Ω
vw,
(4.36)
for t > 0, where
f4(t) =
Z
Ω
(u(·, t) −ˆu)2 +
Z
Ω
v(·, t)2 +
Z
Ω
(w(·, t) −ˆw)2 +
Z
Ω
(z(·, t) −ˆz)2,
t > 0.
In view of Lemma 4.3, the proof is similar.
4.4. Proof of the main theorems
Proof of Theorem 1.2. Let H(u) = u −ue ln u, for u > 0. By applying L’Hôpital’s rule, we obtain
lim
u→ue
H(u) −H(ue)
(u −ue)2
=
1 −ue
u
2(u −ue) =
1
2ue
.
We can therefore select t0 > 0 and use the Taylor expansion to obtain
Z
Ω
u −ue −ue ln u
ue
=
Z
Ω
(H(u) −H(ue)) ≤
1
2ue
Z
Ω
(u −ue)2,
∀t > t0.
From the above estimate, we arrive at
1
4ue
Z
Ω
(u −ue)2 ≤
Z
Ω
u −ue −ue ln u
ue
≤
3
4ue
Z
Ω
(u −ue)2,
∀t > t0.
(4.37)
Similarly, we obtain
1
4ve
Z
Ω
(v −ve)2 ≤
Z
Ω
v −ve −ve ln v
ve
≤
3
4ve
Z
Ω
(v −ve)2,
∀t > t0
(4.38)
and
1
4we
Z
Ω
(w −we)2 ≤
Z
Ω
w −we −we ln w
we
≤
3
4we
Z
Ω
(w −we)2,
∀t > t0.
(4.39)
Taking into account the inequalities (4.37)-(4.39), it is clear that for all t > t0 and with κ1, κ2 > 0, we
have
κ1f1(t) ≤E1(t) ≤κ2f1(t).
(4.40)
31
From (4.9) and right-hand side of (4.40), we get
d
dtE1(t) ≤−σ1f1(t) ≤−σ1
κ2
E1(t),
t > t0.
Solving the above differential inequality gives
E1(t) ≤C1e−σ1
κ2 t,
t > t0,
where C1 > 0. Now, the left-hand side of (4.40) gives
f1(t) ≤1
κ1
E1(t) ≤C1
κ1
e−σ1
κ2 t,
t > t0.
In view of (4.10) and then Lemma 4.1, finally we arrive at
u(·, t) −ue
L∞(Ω) +
v(·, t) −ve
L∞(Ω) +
w(·, t) −we
L∞(Ω) +
z(·, t) −ze
L∞(Ω) ≤C2e−σ1
κ2 t,
for all t > t0, with C2 > 0. This completes the proof.
Proof of Theorem 1.3. Choose t0 > 0, we can apply a similar approach to the one used to derive
(4.37) to obtain the following results for all t > t0
1
4
Z
Ω
(u −1)2 ≤
Z
Ω
(u −1 −ln u) ≤3
4
Z
Ω
(u −1)2 ,
(4.41)
1
2
Z
Ω
v2 + 1
2
Z
Ω
v ≤
Z
Ω
v ≤2
Z
Ω
v2 + 2
Z
Ω
v,
(4.42)
and
1
2
Z
Ω
w2 + 1
2
Z
Ω
w ≤
Z
Ω
w ≤2
Z
Ω
w2 + 2
Z
Ω
w.
(4.43)
Considering the above inequalities (4.41)-(4.43), it is apparent that
κ3f2(t) ≤E2(t) ≤κ4
f2(t) +
Z
Ω
v +
Z
Ω
w +
Z
Ω
vw
,
(4.44)
for every t > t0, with κ3, κ4 > 0. From (4.20) and the right-hand side of (4.44) with σ2 < min{µ1a1 −
Γ1µ2, µ1a2 −Γ2µ3, Γ2µ3a6 −Γ1µ2a4}, we obtain
d
dtE2(t) ≤−σ2
κ4
E2(t) −
(µ1a1 −Γ1µ2) −σ2
Z
Ω
v −
(µ1a2 −Γ2µ3) −σ2
Z
Ω
w
−
(Γ2µ3a6 −Γ1µ2a4) −σ2
Z
Ω
vw
≤−σ2
κ4
E2(t),
∀t > t0.
As a consequence, E2(t) ≤C3e−σ2
κ4 t, for all t > t0, where C3 > 0. Now, by examining the left-hand side
of (4.44), we observe that
f2(t) ≤1
κ3
E2(t) ≤C3
κ3
e−σ2
κ4 t.
Using an argument similar to the proof of Theorem 1.2, we arrive at
u(·, t) −1
L∞(Ω) +
v(·, t)
L∞(Ω) +
w(·, t)
L∞(Ω) +
z(·, t)
L∞(Ω) ≤C4e−σ2
κ4 t,
32
for all t > t0, with C4 > 0. This completes the proof.
Proof of Theorem 1.4. (1) We select t0 > 0. Following the proof of Theorem 1.2 and using the
Taylor expansion, we find that
Z
Ω
u −¯u −¯u ln u
¯u
=
Z
Ω
(H(u) −H(¯u)) ≤1
2¯u
Z
Ω
(u −¯u)2,
∀t > t0.
From the above estimate, we arrive at
1
4¯u
Z
Ω
(u −¯u)2 ≤
Z
Ω
u −¯u −¯u ln u
¯u
≤3
4¯u
Z
Ω
(u −¯u)2,
∀t > t0.
(4.45)
Similarly, we obtain
1
4¯v
Z
Ω
(v −¯v)2 ≤
Z
Ω
v −¯v −¯v ln v
¯v
≤3
4¯v
Z
Ω
(v −¯v)2,
∀t > t0
(4.46)
and
1
2
Z
Ω
w2 + 1
2
Z
Ω
w ≤
Z
Ω
w ≤2
Z
Ω
w2 + 2
Z
Ω
w,
∀t > t0.
(4.47)
Taking into account the inequalities (4.45)-(4.46), it is clear that for all t > t0 and with κ1, κ2 > 0, we
have
κ5f3(t) ≤E3(t) ≤κ6
f3(t) +
Z
Ω
w +
Z
Ω
vw
.
(4.48)
From (4.29) and the right-hand side of (4.48) with σ3 < min{µ1a2¯u+Γ1µ2a4¯v −Γ2µ3, Γ2µ3a6 −Γ1µ2a4},
we get
d
dtE3(t) ≤−σ3
κ6
E3(t) −
(µ1a2¯u + Γ1µ2a4¯v −Γ2µ3) −σ3
Z
Ω
w
−
(Γ2µ3a6 −Γ1µ2a4) −σ6
Z
Ω
vw
≤−σ3
κ6
E3(t),
t > t0.
Solving the above differential inequality gives
E3(t) ≤C1e−σ3
κ6 t,
t > t0.
where C1 > 0. Now, the left-hand side of (4.48) gives
f3(t) ≤1
κ5
E3(t) ≤C1
κ5
e−σ3
κ6 t,
t > t0.
In view of (4.10) and then Lemma 4.5, finally we arrive at
u(·, t) −¯u
L∞(Ω) +
v(·, t) −¯v
L∞(Ω) +
w(·, t)
L∞(Ω) +
z(·, t) −¯z
L∞(Ω) ≤C2e−σ3
κ6 t,
for all t > t0, with C2 > 0. This completes the proof.
Proof of Theorem 1.4. (2) Let us choose t0 > 0, we can apply a similar approach to the one used
to derive (4.45) to obtain the following result
1
4ˆu
Z
Ω
(u −ˆu)2 ≤
Z
Ω
u −ˆu −ˆu ln u
ˆu
≤3
4ˆu
Z
Ω
(u −ˆu)2,
∀t > t0.
(4.49)
33
Following the same procedure, we obtain
1
4 ˆw
Z
Ω
(w −ˆw)2 ≤
Z
Ω
w −ˆw −ˆw ln w
ˆw
≤3
4 ˆw
Z
Ω
(w −ˆw)2,
∀t > t0
(4.50)
and
1
2
Z
Ω
v2 + 1
2
Z
Ω
v ≤
Z
Ω
v ≤2
Z
Ω
v2 + 2
Z
Ω
v,
∀t > t0.
(4.51)
Considering the above inequalities (4.49)-(4.51), it is apparent that
κ7f4(t) ≤E4(t) ≤κ8
f4(t) +
Z
Ω
v +
Z
Ω
vw
,
t > t0
(4.52)
with κ7, κ8 > 0. From (4.36) and right-hand side of (4.52) with
σ4 < min{µ1a1ˆu −Γ1µ2 −Γ2µ3a6 ˆw, Γ2µ3a6 −Γ1µ2a4}, we obtain
d
dtE4(t) ≤−σ4f2(t) −
(µ1a1ˆu −Γ1µ2 −Γ2µ3a6 ˆw) −σ4
Z
Ω
v
−
(Γ2µ3a6 −Γ1µ2a4) −σ4
Z
Ω
vw,
≤−σ4
κ8
E4(t),
t > t0.
Consequently, E4(t) ≤C3e−σ4
κ8 t, for every t > t0, where C3 > 0. Now, by examining the left-hand side of
(4.52), we observe that
f4(t) ≤1
κ7
E4(t) ≤C3
κ7
e−σ4
κ8 t.
By using a similar argument as in the proof of Theorem 1.4, we arrive at
u(·, t) −ˆu
L∞(Ω) +
v(·, t)
L∞(Ω) +
w(·, t) −ˆw
L∞(Ω) +
z(·, t) −ˆz
L∞(Ω) ≤C4e−σ4
κ8 t,
for all t > t0, with C4 > 0. This completes the proof.
5. Numerical examples
This section presents the results of numerical experiments conducted using the Finite Difference
Method to investigate the behaviour of solutions to the predator-prey chemo-alarm-taxis system in both
two- and three-dimensional spaces. We consider the initial-boundary value problem for the predator-prey
chemo-alarm-taxis system (1.1), with radially symmetric bell-shaped initial condition
u(·, 0) = e−0.1 Pn
i=1 x2
i ,
v(·, 0) = 3e−0.3 Pn
i=1 x2
i ,
w(·, 0) = 2e−0.2 Pn
i=1 x2
i ,
z(·, 0) = e−0.1 Pn
i=1 x2
i .
In all of the examples presented below, we set di = 1, i = 1, 2, 3, 4, χ1 = χ2 = ξ = µ1 = µ2 = µ3 = α =
β = 1, γ = 2 and adopt a uniform grid ∆xi =
1
101, i = 1, 2 for 2D and ∆xi =
1
50, i = 1, 2, 3 for 3D within
a domain [−0.5, 0.5]n, where n = 2, 3.
Example 5.1 (Coexistence state of the species). Let ai = 0.5, i = 1, 2, 3, 4. Therefore
a2a6 < min
1 + a1 + a2 + a1a4 + a4a6, 1 + a1(a3 + a4a5) + a4a6 + a2a5
a3
,
a3(1 + a2) + a4a5 < 1 + a4 + a2a5
and
a5(1 + a1) + a6 < 1 + a3(a1 + a6),
then the solution (u, v, w, z) converges to the coexistence steady state (ue, ve, we, ze) as t →∞.
34
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
5
10
15
20
25
30
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
|| ||
Secondary Predator
Primary Predator
Prey
(e) Asymptotic behaviour
Figure 1: Populations at t = 30 in 2D.
In the case, the solution (u, v, w, z) of system (1.1) converges to the coexistence steady state (1.286877,
0.428973, 0.144368, 0.286793) as t →30, as illustrated in Figure 1.
(a) Secondary Predator u(x, y, t)
(b) Primary Predator v(x, y, t)
(c) Prey w(x, y, t)
(d) Chemical z(x, y, t)
0
2
4
6
8
10
12
14
16
18
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
|| ||
Secondary Predator
Primary Predator
Prey
(e) Asymptotic behaviour
Figure 2: Populations at t = 20 in 3D.
Figure 2 depicts the solution (u, v, w, z) of system (1.1) approaches the coexistence steady state
(1.286886, 0.428981, 0.144365, 0.286798) as t →20.
Example 5.2 (Secondary predator only existence state). Let a1 = a4 = 0.01, a2 = 1, a3 = 1.5, a5 = a6 =
2, then the conditions given in (1.3) are not satisfied. Therefore, the solution (u, v, w, z) converges to the
35
trivial steady state (1, 0, 0, 0) as t →∞.
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
2
4
6
8
10
12
14
16
18
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
|| ||
Secondary Predator
Primary Predator
Prey
(e) Asymptotic behaviour
Figure 3: Populations at t = 20 in 2D.
Here we observe that the solution (u, v, w, z) of the system (1.1) converges to the trivial steady state
(1, 0, 0, 0) as t →20, as shown in Figure 3.
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
2
4
6
8
10
12
14
16
18
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
|| ||
Secondary predator
Primary predator
Prey
(e) Asymptotic behaviour
Figure 4: Populations at t = 20 in 3D.
Figure 4 depicts the solution (u, v, w, z) of system (1.1) approaches the trivial steady state (1, 0, 0, 0)
as t →20.
Example 5.3 (Prey vanishing state). Let a1 = a3 = 0.01, a2 = 1, a4 = 3, a5 = a6 = 2, then conditions
given in (1.3) are not satisfied and a3 < 1. Therefore, the solution (u, v, w, z) converges to the semi-
coexistence (prey vanishing) steady state (¯u, ¯v, ¯w, ¯z) as t →∞.
36
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
5
10
15
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
|| ||
Secondary predator
Primary predator
Prey
(e) Asymptotic behaviour
Figure 5: Populations at t = 20 in 2D.
In the case, the solution (u, v, w, z) of system (1.1) converges to the prey vanishing state (1.009899,
0.989901, 0, 0.494951) as t →20, as illustrated in Figure 5.
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
2
4
6
8
10
12
14
16
18
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
|| ||
Secondary predator
Primary predator
Prey
(e) Asymptotic behaviour
Figure 6: Populations at t = 20 in 3D.
Figure 6 depicts the solution (u, v, w, z) of system (1.1) approaches the prey vanishing state (1.009899,
0.989902, 0, 0.494951) as t →20.
Example 5.4 (Primary predator vanishing state). Let a1 = a4 = 0.01, a2 = 2, a3 = 1.5, a5 = 0.5, a6 = 2,
then the conditions given in (1.3) are not satisfied and a5 < 1. Therefore, the solution (u, v, w, z) converges
to the semi-coexistence (primary predator vanishing) steady state (ˆu, ˆv, ˆw, ˆz) as t →∞.
37
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
2
4
6
8
10
12
14
16
18
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
|| ||
Secondary predator
Primary predator
Prey
(e) Asymptotic behaviour
Figure 7: Populations at t = 20 in 2D.
Here we observe that the solution (u, v, w, z) of the system (1.1) converges to the primary predator
vanishing state (1.499994, 0, 0.249999, 0.124999) as t →20, as shown in Figure 7.
(a) Secondary Predator
(b) Primary Predator
(c) Prey
(d) Chemical
0
2
4
6
8
10
12
14
16
18
20
time
-1
-0.5
0
0.5
1
1.5
2
2.5
3
|| ||
Secondary predator
Primary predator
Prey
(e) Asymptotic behaviour
Figure 8: Populations at t = 20 in 3D.
Figure 8 illustrates the convergence of the solution (u, v, w, z), where the solution components ap-
proaches (1.499722, 0, 0.249928, 0.124943) as t →20.
Remark 5.1. When the conditions specified in Example 5.1 are sufficiently strong, the solution rapidly
converges to the coexistence steady state. Conversely, when the conditions in Example 5.2 are sufficiently
strong, the solution quickly converges to the trivial steady state (1, 0, 0, 0).
38
6. Conclusion
We have investigated the global existence of a classical solution to system (1.1), which remains
uniformly bounded in the L∞(Ω) norm for dimensions n ≥1, under suitable parameter conditions. Fur-
thermore, we established global asymptotic stability by constructing an appropriate Lyapunov functional
and verified these results through numerical simulations. Our analysis yielded several important stability
outcomes with meaningful biological implications. In particular, the findings underscore the decisive role
of the chemotaxis coefficients in governing the existence and stability of the predator–prey alarm-taxis
system.
Acknowledgments
• GS and JS thank to Anusandhan National Research Foundation (ANRF), formerly Science and
Engineering Research Board (SERB), Govt.
of India for their support through Core Research
Grant (CRG/2023/001483) during this work.
• RDF is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Ap-
plicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and, acknowledges
financial support by PNRR e.INS Ecosystem of Innovation for Next Generation Sardinia, CUP
F53C22000430001 (codice MUR ECS0000038).
He is also partially supported by the research
project “Partial Differential Equations and their role in understanding natural phenomena”, CUP
F23C25000080007, funded by Fondazione Banco di Sardegna annuality (2023).
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The simultaneous effect of chemotaxis and alarm-taxis on the global existence and stability of a predator-prey system Gnanasekaran Shanmugasundarama, Jitraj Sahaa,∗, Rafael Díaz Fuentesb a 620015, India b 72, Cagliari, 09124, Italy Abstract This study examines a fully parabolic predator-prey chemo-alarm-taxis system under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with a smooth boundary ∂Ω. Under specific parameter conditions, it is shown that the system admits a unique, globally bounded classical solution. The convergence of the solution is established through the construction of an appropriate Lyapunov functional. In addition, numerical simulations are presented to validate the asymptotic behaviour of the solution. The results highlight the significant role of chemotaxis and alarm-taxis coefficients in determining the existence and stability of predator-prey models, as discussed in the literature. Keywords: Alarm-taxis, Chemotaxis, Global boundedness, Global stability, Predator-prey, Numerical simulation. 2020 MSC: 35A01, 35A09, 35B40, 92C17, 37N25, 65N06. 1. Introduction and motivation Burglar alarm calls [3] act as a key anti-predation strategy, whereby prey emit signals that attract secondary predators, potentially deterring primary predators [5, 15]. This behaviour is observed in marine systems, such as dinoflagellates employing bioluminescence in response to copepod disturbances, which may draw fish as secondary predators [13]. Chemotaxis, on the other hand, refers to the movement of microorganisms in response to chemical gradients, with applications spanning ecology, immunology, biology and medicine. In this context, we formulate the chemo-alarm-taxis system, in which the primary predator v is attracted to the prey w through a common chemical signal z, while the prey moves away from the primary predator in response to the same chemical. At the same time, the secondary predator is drawn towards the combined density of the primary predator and the prey owing to the alarm call emitted by the prey. This interaction leads the secondary predator to prey upon the primary predator, thereby providing the prey with a potential defence mechanism against predation. To capture these ecological dynamics, we consider a system of coupled, nonlinear parabolic partial differential equations describing the interactions between two predators and one prey in a chemo-alarm-taxis framework. The system is given by ut = d1∆u -χ1∇· (u∇(vw)) + μ1u(1 -u + a1v + a2w), x ∈Ω, t > 0, vt = d2∆v -χ2∇· (v∇z) + μ2v(1 -v -a3u + a4w), x ∈Ω, t > 0, wt = d3∆w + ξ∇· (w∇z) + μ3w(1 -w -a5u -a6v), x ∈Ω, t > 0, zt = d4∆z + αv + βw -γz, x ∈Ω, t > 0, ∂u ∂ν = ∂v ∂ν = ∂w ∂ν = ∂z ∂ν = 0, x ∈∂Ω, t > 0, u(x, 0) = u0, v(x, 0) = v0, w(x, 0) = w0, z(x, 0) = z0 x ∈Ω, (1.1) ∗Corresponding author Email addresses: (Gnanasekaran Shanmugasundaram), (Jitraj Saha), (Rafael Díaz Fuentes) 16 Oct 2025 in a bounded domain Ω⊂Rn with smooth boundary ∂Ω. Here, ν denotes the unit outward normal on ∂Ω. The unknown functions u = u(x, t) and v = v(x, t) describe the population densities of the secondary predator and the primary predator, respectively. In addition, w = w(x, t) represents the prey population, while z = z(x, t) denotes the concentration of the chemical attractant produced by the prey. The parameters d1, d2, d3, d4, χ1, χ2, ξ, α, β, γ, μ1, μ2, μ3 and ai (i = 1, 2, 3, 4, 5, 6) are all positive constants and the initial data u0, v0, w0, and z0 are nonnegative functions. The constants d1, d2, d3, and d4 are the diffusion coefficients, representing the natural dispersive forces governing the movement of the predators, prey and chemical concentrations, respectively. The parameter χ1 is the alarm-taxis coefficient, while χ2 and ξ are the chemotactic coefficients. The term -χ1∇· (u∇(vw)) represents alarmtaxis, namely the directed movement of the secondary predator (u) towards the gradient of the combined predator-prey density (∇(vw)) in response to the alarm signal. Similarly, the term -χ2∇·(v∇z) describes chemo-attraction, that is, the directed movement of the primary predator (v) towards the gradient of the chemical concentration (∇z). By contrast, the term +ξ∇· (w∇z) denotes chemo-repulsion, meaning the directed movement of the prey (w) away from the gradient of the chemical concentration (∇z). The constants μ1, μ2, μ3 are the logistic growth coefficients, while a1, a2, a3, a4, a5, a6 represent the interspecific interactions among the three species. The parameters α, β and γ correspond to the production and decay rates of the common chemical. Furthermore, we assume that the initial data u0, v0, w0, and z0 satisfy u0, v0, w0 ∈C0(Ω), with u0, v0, w0 ≥0 in Ω, z0 ∈W1,q(Ω), for some q > n, with z0 ≥0 in Ω. (1.2) The foundational model for chemotaxis was first proposed by Keller and Segel in 1970 [14]. Since then, the Keller-Segel model and its various modifications have been extensively investigated by numerous researchers. In this context, considerable attention has been devoted to two-species-one-chemical systems [20, 23, 24, 25, 34, 38, 42] and two-species-two-chemical systems with a competing source [2, 31, 37, 41, 46]. Among these, the predator-prey model, an ecologically significant extension of the two-species competition framework, has been widely explored. Research on predator-prey systems with one-chemical chemotaxis includes [6, 21, 22], while investigations into predator-prey systems with two-chemical chemotaxis can be found in [17, 27, 28]. For further developments and extensions of predator-prey models, see [8, 29]. Motivated by the ecological significance of alarm calls, the alarm-taxis model was first introduced by Haskell and Bell [9] ut = ηduxx + f(u, v, w), vt = (dvx -ξvux)x + g(u, v, w), wt = (wx -χwφx(u, v))x + h(u, v, w). Herein, u, v, and w denote the resource or prey (e.g. dinoflagellates), the primary predator (e.g. copepods), and the secondary predator (e.g. fish), respectively. Under zero-flux boundary conditions, and for certain initial data (details of which may be omitted in subsequent discussion), the authors demonstrated the global existence of positive, bounded classical solutions in one-dimensional space. They also established the existence of non-constant equilibrium solutions and analysed their stability. In addition, numerical simulations were carried out to illustrate the emergence of spatial patterns in this model, emphasising the adaptive advantage conferred by a signalling mechanism consistent with the alarm hypothesis. Fuest and Lankeit [7] considered the following system ut = d1∆u + u(λ1 -μ1u -a1v -a2w), vt = d2∆v -ξ∇· (v∇u) + v(λ2 -μ2u + b1u -a3w), wt = d3∆w -χ∇· (w∇(uv)) + w(λ3 -μ3w + b2u + b3v), with homogeneous Neumann boundary conditions, the system models the interactions among the prey u, the predator v, and the superpredator w, the latter preying on both of the other populations. The authors established the existence of a global classical solution in the case of pure alarm-taxis, i.e. when ξ = 0. For ξ > 0, they investigated the existence of global generalised solutions. 2 Li and Wang [18] considered the following system ut = d1∆u + r1u(1 -u) -b1uv -b3uw, vt = d2∆v -ξ∇· (v∇u) + r2v(1 -v) + uv -b2vw, wt = d3∆w -χ∇· (w∇(uv)) + r3w(1 -wσ) + vw + uw, where u, v, and w denote the densities of the prey, the primary predator, and the secondary predator, respectively. By applying semigroup theory, the authors established the boundedness of classical solutions in two spatial dimensions. They replaced the usual logistic term r3w(1 -w) with r3w(1 -wσ) in order to investigate how large the intra-species competition exponent must be to guarantee the global boundedness of solutions. Furthermore, asymptotic stability was obtained through the construction of a suitable Lyapunov functional. Jin et al. [13] studied the global existence of classical solutions to the following system by employing the Neumann heat semigroup technique in two spatial dimensions ut = d1∆u + μ1u(1 -u) -b1uv -b3 uw u+w, vt = d2∆v -ξ∇· (v∇u) + μ2v(1 -v) + uv -b2vw, wt = d3∆w -χ∇· (w∇(uv)) + μ3w(1 -w) + vw + c3 uw u+w. Additionally, the authors established the global asymptotic stability for the cases where b3 = c3 = 0 and b3 > 0, c3 > 0. Zhang et al. [44] studied the following food chain model incorporating alarm-taxis and logistic growth ut = d1∆u + u(1 -u) -b1uv, vt = d2∆v -ξ∇· (v∇u) + θ1v(1 -v) + uv -b2vw, wt = d3∆w -χ∇· (w∇(uv)) + θ2w(1 -w) + vw. The authors showed that, for sufficiently large values of θ1 and θ2, the system admits a globally bounded classical solution when n ≥3. Moreover, by constructing an appropriate Lyapunov functional, they demonstrated that, under certain parameter conditions, the system converges either to a coexistence state or to a semi-coexistence state as t →∞. Zhang and Chen [43] studied the global existence of the following system ut = d1∆u + u 1 -u - av v+ρ , vt = d2∆v + v bu v+ρ -α - cw w+σ , wt = d3∆w -χ∇· (w∇(uv)) + w mv w+σ -β , for n ≥1 with sufficiently smooth initial data. The authors analysed the large-time behaviour of the system under certain conditions and derived convergence rates. In addition, numerical simulations were carried out to validate the analytical results. To the best of our knowledge, this is the first work to rigorously establish the global existence, boundedness, and convergence of solutions in a predator-prey system incorporating both chemotaxis and alarm-taxis. Motivated by the existing literature, our primary objective is to demonstrate the existence of a global classical solution to system (1.1), which remains uniformly bounded in Ω× (0, ∞) for n ≥1. Furthermore, we establish convergence of the solution by constructing a suitable Lyapunov functional. Finally, we perform numerical simulations to confirm the asymptotic behaviour of solutions in both two and three spatial dimensions. The remainder of this article is organised as follows. Section 2 introduces preliminary lemmas and establishes the local existence of the classical solution. Section 3 is devoted to proving the global existence and boundedness of classical solutions for system (1.1). In Section 4, we carry out a stability analysis, while Section 5 presents numerical validation and results. The paper concludes with a summary in Section 6. Our main result on global existence is the following. Theorem 1.1 (Global existence of solutions). Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary. Then there exists μ > 0, such that if min{μ2, μ3} > μ and for any nonnegative initial 3 data (u0, v0, w0, z0) satisfying (1.2) for some q > max{2, n}, the system (1.1) possesses a unique classical solution (u, v, w, z) which is uniformly bounded in the sense that u(·, t) L∞(Ω) + v(·, t) W1,∞(Ω) + w(·, t) W1,∞(Ω) + z(·, t) W1,q(Ω) ≤C, ∀t > 0, where the constant C > 0. Remark 1.1. In the one-dimensional case n = 1, global existence can be established under weaker conditions. This follows directly from the testing procedures and regularity results presented in Section 3. Now, we examine the equilibrium points of the system. Let (u, v, w, z) be a classical solution of (1.1) satisfying (1.2) and (ue, ve, we, ze) denote the equilibrium points of the system (1.1), satisfying the following system of equations μ1ue(1 -ue + a1ve + a2we) = 0, μ2ve(1 -ve -a3ue + a4we) = 0, μ3we(1 -we -a5ue -a6ve) = 0, αve + βwe -γze = 0. The system admits the following eight equilibrium points 1. Trivial steady states (0, 0, 0, 0), (1, 0, 0, 0), 0, 1, 0, α γ , and 0, 0, 1, β γ . 2. Semi-coexistence steady states 0, 1 + a4 1 + a4a6 , 1 -a6 1 + a4a6 , α(1 + a4) + β(1 -a6) γ(1 + a4a6) , 1 + a2 1 + a2a5 , 0, 1 -a5 1 + a2a5 , β(1 -a5) γ(1 + a5a5) , and 1 + a1 1 + a1a3 , 1 -a3 1 + a1a3 , 0, α(1 -a3) γ(1 + a1a3) . 3. Coexistence steady state (u∗, v∗, w∗, z∗). Here, (0,0,0,0) represents the extinction equilibrium where all species vanish; (1,0,0,0) corresponds to the equilibrium with only the secondary predator present; 0, 1, 0, α γ describes the primary predatoronly equilibrium and 0, 0, 1, β γ characterizes the prey-only equilibrium. Additionally, there exist three semi-coexistence equilibria where any two species persist together. The most ecologically significant state, (u∗, v∗, w∗, z∗), represents the coexistence equilibrium where all three species maintain stable populations. Here, the coexistence steady state (u∗, v∗, w∗, z∗) is explicitly given by u∗= 1 + a1 + a2 + a1a4 + a6(a4 -a2) 1 + a1(a3 + a4a5) + a4a6 + a2(a5 -a3a6), v∗= 1 -a3(1 + a2) + a4 + a5(a2 -a4) 1 + a1(a3 + a4a5) + a4a6 + a2(a5 -a3a6), w∗= 1 + a1(a3 -a5) -a5 + a6(a3 -1) 1 + a1(a3 + a4a5) + a4a6 + a2(a5 -a3a6), z∗= αv∗+ βw∗ γ . The conditions for the coexistence steady state are a2a6 a2a3a6, then the nonnegative classical solution (u, v, w, z) of the system (1.1) exponentially converges to the coexistence steady state (u∗, v∗, w∗, z∗) uniformly in Ωas t →∞. Theorem 1.3 (Secondary predator only existence state). Suppose that the assumptions in Theorem 1.1 hold true and let a2a6 ≥min n 1 + a1 + a2 + a1a4 + a4a6, 1+a1(a3+a4a5)+a4a6+a2a5 a3 o , a3(1 + a2) + a4a5 ≥ 1 + a4 + a2a5, a5(1 + a1) + a6 ≥1 + a3(a1 + a6). If the parameters satisfy the relations (i) χ2 1 0, B ≥0 and p ≥1. Then for t > 0, we have y(t) ≤max ( y0, B A 1 p ) . Lemma 2.2 ([30, 32]). Let T > 0, τ ∈(0, T ), a > 0 and b > 0, and assume that nonnegative function y : [0, T ) →[0, ∞) is C1 smooth and such that y′(t) + ay(t) ≤h(t), ∀t ∈(0, T ) with some nonnegative function h ∈L1 loc([0, T )) fulfilling Z t+τ t h(s)ds ≤b, ∀t ∈[0, T -τ). Then y(t) ≤max n y(0) + b, b aτ + 2b o , ∀t ∈(0, T ). 6 Lemma 2.3 (Maximal Sobolev regularity [4, 10]). Let r ∈(1, ∞) and T ∈(0, ∞). Consider the following evolution equation yt = ∆y -y + g, x ∈Ω, t > 0, ∂y ∂ν = 0, x ∈∂Ω, t > 0, y(x, 0) = y0(x), x ∈Ω. For each y0 ∈W2,r(Ω) (r > n) with ∂y0 ∂ν = 0 on ∂Ωand any g ∈Lr((0, T ); Lr(Ω)), there exists a unique solution y ∈W1,r ((0, T ); Lr(Ω)) ∩Lr (0, T ); W2,r(Ω) . Moreover, there exists Cr > 0, such that Z T 0 ∥y(·, t)∥r Lr(Ω)dt+ Z T 0 ∥yt(·, t)∥r Lr(Ω)dt + Z T 0 ∥∆y(·, t)∥r Lr(Ω)dt ≤Cr Z T 0 ∥g(·, t)∥r Lr(Ω)dt + Cr∥y0∥r Lr(Ω) + Cr∥∆y0∥r Lr(Ω). If s0 ∈[0, T ) and y(·, s0) ∈W2,r(Ω) (r > n) with ∂y(·,s0) ∂ν = 0 on ∂Ω, then Z T s0 esr∥∆y(·, t)∥r Lr(Ω)dt ≤Cr Z T s0 esr∥g(·, t)∥r Lr(Ω)dt + Cr∥y(·, s0)∥r Lr(Ω) + Cr∥∆y(·, s0)∥r Lr(Ω). Lemma 2.4 (Local Existence). Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary. Assume that the initial data u0, v0, w0 and z0 satisfies (1.2) for some q > max{2, n}. Then there exists Tmax ∈(0, ∞] such that the system (1.1) admits a unique solution (u, v, w, z) satisfying u, v, w ∈C0 Ω× [0, Tmax ) ∩C2,1 Ω× (0, Tmax) , z ∈C0 Ω× [0, Tmax ) ∩C2,1 Ω× (0, Tmax) ∩L∞ loc [0, Tmax ) ; W1,q(Ω) . Moreover, either Tmax = ∞, or lim t→Tmax u(·, t) L∞(Ω) + v(·, t) W1,∞(Ω) + w(·, t) W1,∞(Ω) + z(·, t) W1,q(Ω) = ∞. (2.1) Proof. The lemma is proven using standard arguments based on parabolic regularity theory. For the detailed proof, we refer to Jin et al. [13]. The non-negativity of the solution in Ω×(0, Tmax) is established by satisfying both the maximum principle and the initial condition (1.2). Lemma 2.5. The classical solution of (1.1) satisfies Z Ω u(·, t) + Z Ω v(·, t) + Z Ω w(·, t) ≤K1, (2.2) where K1 > 0, for all t ∈(0, Tmax). Moreover, we have Z t+τ t Z Ω u(·, t)2 ≤K2, t ∈(0, Tmax -τ), (2.3) where K2 > 0 and τ := min{1, 1 2Tmax}. Proof. Choose ζ1, ζ2 ∈(0, 1) with ζ1μ1a1 ≤ζ2μ2a3, ζ1μ1a2 ≤μ3a5, ζ2μ2a4 ≤μ3a6 and set μ = min{μ1, μ2, μ3}. From first and second equations of (1.1) gives d dt ζ1 Z Ω u = ζ1μ1 Z Ω u -ζ1μ1 Z Ω u2 + ζ1μ1a1 Z Ω uv + ζ1μ1a2 Z Ω uw 7 and d dt ζ2 Z Ω v = ζ2μ2 Z Ω v -ζ2μ2 Z Ω v2 -ζ2μ2a3 Z Ω uv + ζ2μ2a4 Z Ω vw. Adding the above equality with the third equation in (1.1), we get d dt ζ1 Z Ω u + ζ2 Z Ω v + Z Ω w +ζ1μ1 Z Ω u + ζ2μ2 Z Ω v + μ3 Z Ω w = 2ζ1μ1 Z Ω u -ζ1μ1 Z Ω u2 + ζ1μ1a1 Z Ω uv + ζ1μ1a2 Z Ω uw + 2ζ2μ2 Z Ω v -ζ2μ2 Z Ω v2 -ζ2μ2a3 Z Ω uv + ζ2μ2a4 Z Ω vw + 2μ3 Z Ω w -μ3 Z Ω w2 -μ3a5 Z Ω uw -μ3a6 Z Ω vw. Using the Young inequality with C1 > 0, one has d dt ζ1 Z Ω u + ζ2 Z Ω v + Z Ω w +ζ1μ1 Z Ω u + ζ2μ2 Z Ω v + μ3 Z Ω w ≤-ζ1μ1 2 Z Ω u2 -ζ2μ2 2 Z Ω v2 -μ3 2 Z Ω w2 + Z Ω (ζ1μ1a1 -ζ2μ2a3)uv + Z Ω (ζ1μ1a2 -μ3a5)uw + Z Ω (ζ2μ2a4 -μ3a6)vw + C1. Simplifying, we get d dt ζ1 Z Ω u + ζ2 Z Ω v + Z Ω w +μ ζ1 Z Ω u + ζ2 Z Ω v + Z Ω w + μ 2 ζ1 Z Ω u2 + ζ2 Z Ω v2 + Z Ω w2 ≤C1. (2.4) Applying Lemma 2.1, we get Z Ω u + Z Ω v + Z Ω w ≤C2, ∀t ∈(0, Tmax), where C2 > 0 this gives (2.2). For (2.3), integrate (2.4) over (t, t + τ) to get Z t+τ t Z Ω u2 + Z t+τ t Z Ω v2 + Z t+τ t Z Ω w2 ≤C3, t ∈(0, Tmax -τ), where C3 > 0. This completes the proof. 3. Global existence of solutions This section focuses on proving the global existence and boundedness of the solution to (1.1). We begin by deriving the Lp(Ω) norm for v and w. For any s0 ∈(0, Tmax) with s0 0 be a constant satisfying the following sup 0≤s≤s0 v(·, s) L∞(Ω) ≤C, sup 0≤s≤s0 w(·, s) L∞(Ω) ≤C, sup 0≤s≤s0 z(·, s) L∞(Ω) ≤C, ∆z(·, s0) L∞(Ω) ≤C (3.1) and sup 0≤s≤s0 u(·, s) L∞(Ω) ≤C, sup 0≤s≤s0 ∇v(·, s) L∞(Ω) ≤C, sup 0≤s≤s0 ∇w(·, s) L∞(Ω) ≤C. (3.2) Next, we derive boundedness in t ∈(s0, Tmax). 8 Lemma 3.1. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary. Then for any p > 1, there exists μ(p, χ2, ξ, α, β, a4) > 0, such that if μ 0. Proof. Multiplying the second equation of (1.1) by vp-1, p > 1 and integrating over Ω, we get Z Ω vtvp-1 = d2 Z Ω vp-1∆v -χ2 Z Ω vp-1∇· (v∇z) + μ2 Z Ω vp-1v(1 -v -a3u + a4w). Applying the technique of integration by parts, we obtain d dt 1 p Z Ω vp = -d2(p -1) Z Ω vp-2|∇v|2 + χ2(p -1) Z Ω vp-1∇v∇z + μ2 Z Ω vp -μ2 Z Ω vp+1 -μ2a3 Z Ω vpu + μ2a4 Z Ω vpw. Again, use of integration by parts, gives us that d dt 1 p Z Ω vp = -d2(p -1) Z Ω vp-2|∇v|2 -χ2(p -1) p Z Ω vp∆z + μ2 Z Ω vp -μ2 Z Ω vp+1 -μ2a3 Z Ω vpu + μ2a4 Z Ω vpw ≤-μ2 Z Ω vp+1 -d2(p -1) Z Ω vp-2|∇v|2 -χ2(p -1) p Z Ω vp∆z + μ2 Z Ω vp ≤+ μ2a4 Z Ω vpw -μ2 Z Ω vp+1 + I1 + I2 + I3 + I4. (3.4) Utilizing the Gagliardo-Nirenberg inequality and the Young inequality, it follows that (also thanks to (2.2)) Z Ω vp = v p 2 2 L2(Ω) ≤C1 ∇v p 2 2d L2(Ω) v p 2 2(1-d) L 2 p (Ω) + v p 2 2 L 2 p (Ω) ≤4d2(p -1) p(p + 1) ∇v p 2 2d L2(Ω) 1 d + C2 v p 2 2(1-d) L 2 p (Ω) 1 1-d + C1 v p 2 2 L 2 p (Ω) ≤4d2(p -1) p(p + 1) ∇v p 2 2 L2(Ω) + C3 v p L1(Ω) ≤4d2(p -1) p(p + 1) p2 4 Z Ω vp-2|∇v|2 + C4, with C2 = (1 -d)d d 1-d 4d2(p-1) p(p+1) -d 1-d C 1 1-d 1 , where d = p 2 -1 2 p 2 + 1 n -1 2 ∈(0, 1), and C1 is a constant from the Gagliardo-Nirenberg inequality. Hence p + 1 p Z Ω vp ≤d2(p -1) Z Ω vp-2|∇v|2 + C5, (3.5) with C5 = p+1 p C4. Now we estimate I1 using (3.5) as follows I1 = -d2(p -1) Z Ω vp-2 ∇v 2 ≤-p + 1 p Z Ω vp + C5. (3.6) For p > 1, 1 -1 p 0, I2 ≤δ Z Ω vp+1 + C6δ-pχp+1 2 Z Ω |∆z|p+1, (3.7) where C6 = p+1 p -p 1 p+1. Again, using the Young inequality, we obtain I3 = μ2 Z Ω vp ≤ρ Z Ω vp+1 + C7, (3.8) where C7 = ρ-p p+1 p -p 1 p+1μp+1 2 |Ω|, and I4 = Z Ω μ2a4vpw ≤η Z Ω vp+1 + C6η-pμp+1 2 ap+1 4 Z Ω wp+1. (3.9) Substituting (3.6) - (3.9) in (3.4), we see that d dt 1 p Z Ω vp ≤-(p + 1) 1 p Z Ω vp -(μ2 -δ -ρ -η) Z Ω vp+1 + C6δ-pχp+1 2 Z Ω |∆z|p+1 + C6η-pμp+1 2 ap+1 4 Z Ω wp+1 + C8. (3.10) Adopting the variation of constants formula to (3.10), 1 p Z Ω vp ≤-(μ2 -δ -ρ -η) Z t s0 e-(p+1)(t-s) Z Ω vp+1 + C6δ-pχp+1 2 Z t s0 e-(p+1)(t-s) Z Ω |∆z|p+1 + C6η-pμp+1 2 ap+1 4 Z t s0 e-(p+1)(t-s) Z Ω wp+1 + C9, (3.11) where C9 = 1 p R Ωvp 0 + C8 1 p+1. According to Lemma 2.3, there exists Cp > 0 such that C6δ-pχp+1 2 Z t s0 e-(p+1)(t-s) Z Ω |∆z|p+1 ≤C6Cpδ-pχp+1 2 Z t s0 e-(p+1)(t-s) Z Ω αv + βw p+1 + C6Cpδ-pχp+1 2 e-(p+1)t z0 p+1 W2,p+1(Ω) ≤C6Cpδ-pχp+1 2 2p+1αp+1 Z t s0 e-(p+1)(t-s) Z Ω vp+1 + C6Cpδ-pχp+1 2 2p+1βp+1 Z t s0 e-(p+1)(t-s) Z Ω wp+1 + C6Cpδ-pχp+1 2 z0 p+1 W2,p+1(Ω). (3.12) Substituting the last inequality (3.12) into (3.11), we arrive at 1 q Z Ω vp ≤- μ2 -δ -ρ -η -C6Cpδ-pχp+1 2 2p+1αp+1 Z t s0 e-(p+1)(t-s) Z Ω vp+1 + C6Cpδ-pχp+1 2 2p+1βp+1 + C6η-pμp+1 2 ap+1 4 Z t s0 e-(p+1)(t-s) Z Ω wp+1 + C10. (3.13) where C10 = C9 + C6Cpδ-pχp+1 2 z0 p+1 W2,p+1(Ω). Similarly, we can estimate for w as follows 1 p Z Ω wp ≤- μ3 -δ -ρ -C6C′ pδ-pξp+12p+1βp+1 Z t s0 e-(p+1)(t-s) Z Ω wp+1 + C6C′ pδ-pξp+12p+1αp+1 Z t s0 e-(p+1)(t-s) Z Ω vp+1 + C11, (3.14) 10 where the constant C11 > 0. Adding (3.13) and (3.14) affords us 1 p Z Ω vp+1 p Z Ω wp ≤ -μ2 + δ + ρ + η + C6Cpδ-pχp+1 2 2p+1αp+1 + C6C′ pδ-pξp+12p+1αp+1 × Z t s0 e-(p+1)(t-s) Z Ω vp+1 + -μ3 + δ + ρ + C6C′ pδ-pξp+12p+1βp+1 + C6Cpδ-pχp+1 2 2p+1βp+1 + C6η-pμp+1 2 ap+1 4 × Z t s0 e-(p+1)(t-s) Z Ω wp+1 + C12. (3.15) Let μ = max n δ + ρ + η + C6Cpδ-pχp+1 2 2p+1αp+1 + C6C′ pδ-pξp+12p+1αp+1, δ + ρ + C6C′ pδ-pξp+12p+1βp+1 + C6Cpδ-pχp+1 2 2p+1βp+1 + C6η-pμp+1 2 ap+1 4 o . Then (3.15) can be written as 1 p Z Ω vp+1 p Z Ω wp ≤-(μ2 -μ) Z t s0 e-(p+1)(t-s) Z Ω vp+1 -(μ3 -μ) Z t s0 e-(p+1)(t-s) Z Ω wp+1 + C12, such that μ2 -μ > 0 and μ3 -μ > 0 because of our assumptions 0 0. Lemma 3.2. Let n ≥1 and the initial data (u0, v0, w0, z0) satisfy (1.2) with some q > n. Then if v, w comply with the conclusion of Lemma 3.1 we have sup t∈(0,Tmax) v(·, t) L∞(Ω) + w(·, t) L∞(Ω) + z(·, t) W1,q(Ω) ≤K4, where K4 > 0. Proof. We fix p > n 2 . Specifically, if n 4n n+4, noting that 4n n+4 > n 2 in this case. Let us define np (n -p)+ = ( ∞, if p ≥n, np n-p, if n 2 n. Consequently, given this choice for p, we can select a q max{n, 4}. Moreover, fix l ∈(1, q) such that l > n. It can be verified that it exists an r > 1 such that max{n, 4} ≤rl ≤q. For any s0 ∈(0, Tmax) with s0 0. Let t0 = max{s0, t -1} and applying the variation of constants formula to the third equation of (1.1), we get w(·, t) =ed3(t-t0)∆w(·, t0) + ξ Z t t0 ed3(t-s)∆∇· w(·, s)∇z(·, s) ds + μ3 Z t t0 ed3(t-s)∆w(·, s) 1 -w(·, s) -a5u(·, s) -a6v(·, s) ds. Now, being w ≤w2 + 1 4 for any w, after neglecting some nonpositive terms we get the following estimate w(1 -w -a5u -a6v) ≤w(1 -w) ≤1 4. (3.17) Then, we have w(·, t) L∞(Ω)≤ ed3(t-t0)∆w(·, t0) L∞(Ω) + ξ Z t t0 ed3(t-s)∆∇· w(·, s)∇z(·, s) L∞(Ω)ds + μ3 4 . (3.18) If t ≤1, then t0 = s0 and the maximum principle gives ed3(t-t0)∆w(·, t0) L∞(Ω) = ed3(t-s0)∆w(·, s0) L∞(Ω) ≤ w(·, s0) L∞(Ω). (3.19) If t > 1, t -t0 = 1 and using the Neumann heat semi-group technique [39] with C5 > 0, ed3(t-t0)∆w(·, t0) L∞(Ω) ≤C5 1+(t -t0)-n 2 w(·, t0) L1(Ω) ≤C6. (3.20) Substituting (3.19) and (3.20) in (3.18) and using the Neumann heat semi-group technique [39] with C7 > 0, one has w(·, t) L∞(Ω) ≤max n w(·, s0) L∞(Ω), C6 o + C7 Z t t0 1 + (t -s)-1 2 -n 2l e-λd3(t-s) w(·, s)∇z(·, s) Ll(Ω)ds + μ3 4 . (3.21) Using the Hölder inequality and then (3.16), we get w(·, t)∇z(·, t) Ll(Ω) ≤ w(·, t) L ̄rl ∇z(·, t) Lrl(Ω) ≤ w(·, t) k L∞(Ω) w(·, t) 1-k L1(Ω) ∇z(·, t) Lrl(Ω) ≤C8 w(·, t) k L∞(Ω), ∀t ∈(s0, Tmax), (3.22) 12 where ̄r is the dual exponent of r and k = 1 -1 ̄rl ∈(0, 1). Substituting (3.22) and (3.17) into (3.21) and using the Gamma function, it follows that w(·, t) L∞(Ω) ≤max n w(·, s0) L∞(Ω), C6 o + C9 sup t∈(s0,Tmax) w(·, t) k L∞(Ω) + μ3 4 . Finally, we get w(·, t) L∞(Ω) ≤C10, ∀t ∈(s0, Tmax), where C10 > 0. Now, considering the reaction term in the equation for v in (1.1) and applying reasoning analogous to that in (3.17), we get v(1 -v -a3u + a4w) ≤v(1 + a4w) -v2 ≤(1 + a4w)2 4 . An analogous procedure yields v(·, t) L∞(Ω)≤ ed2(t-t0)∆v(·, t0) L∞(Ω) + χ2 Z t t0 ed2(t-s)∆∇· v(·, s)∇z(·, s) L∞(Ω)ds + μ2 Z t t0 ed2(t-s)∆v(·, s) 1 -v(·, s) -a3u(·, s) + a4w(·, s) L∞(Ω)ds. (3.23) Applying the same procedure one can get v(·, t) L∞(Ω) ≤max n v(·, s0) L∞(Ω), C11 o + C12 Z t t0 1 + (t -s)-1 2 -n 2l e-λd2(t-s) v(·, s)∇z(·, s) Ll(Ω)ds + μ2 (1 + a4C10)2 4 . Finally, we arrive at v(·, t) L∞(Ω) ≤C13, ∀t ∈(s0, Tmax), (3.24) where C13 > 0. Initial data regularity z0 ∈W1,q(Ω) and the smoothing property of the heat semigroup ensure z ∈L∞((s0, Tmax); W1,q(Ω)). Along with (3.1), this will imply that actually sup t∈(0,Tmax) v(·, t) L∞(Ω) + w(·, t) L∞(Ω) + z(·, t) W1,q(Ω) ≤C14, where C14 > 0. This completes the proof. Lemma 3.3. Suppose that the assumptions in Lemma 3.2 hold true. There exists K5 > 0 fulfilling Z t+τ t Z Ω ∆z(·, t) 2 ≤K5, for all t ∈(0, Tmax -τ). Proof. Multiplying the fourth equation in (1.1) with -2∆z and integrating by parts, we obtain d dt Z Ω |∇z|2 ≤-2d4 Z Ω |∆z|2 + 2α v L∞(Ω) Z Ω ∆z + 2β w L∞(Ω) Z Ω ∆z -2γ Z Ω |∇z|2. Simplifying one gets d dt Z Ω |∇z|2 + 2γ Z Ω |∇z|2 + 2d4 Z Ω |∆z|2 ≤2αK4 Z Ω ∆z + 2βK4 Z Ω ∆z 13 ≤d4 Z Ω |∆z|2 + C1. Finally, d dt Z Ω |∇z|2 + 2γ Z Ω |∇z|2 + d4 Z Ω |∆z|2 ≤C1, ∀t ∈(0, Tmax), (3.25) where C1 > 0. Now, integrate (3.25) over (t, t + τ) to get Z t+τ t Z Ω ∆z(·, t) 2 ≤C2, ∀t ∈(0, Tmax -τ), where C2 > 0. This completes the proof. Lemma 3.4. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary and the assumptions in Lemma 3.2 hold true. There exists K6 > 0 fulfilling Z Ω |∇v(·, t)|2 ≤K6, (3.26) for all t ∈(0, Tmax). Proof. Multiplying the second equation in (1.1) with -2∆v and integrating by parts, we obtain d dt Z Ω |∇v|2 = -2d2 Z Ω |∆v|2 + 2χ2 Z Ω ∇v · ∇z∆v + 2χ2 Z Ω v∆z∆v + 2μ2 Z Ω |∇v|2 -2μ2 Z Ω ∇v2 · ∇v + 2μ2a3 Z Ω uv∆v -2μ2a4 Z Ω vw∆v ≤-2d2 Z Ω |∆v|2 + 2χ2 Z Ω ∇v · ∇z∆v + 2χ2 Z Ω v∆z∆v + 2μ2 Z Ω |∇v|2 -4μ2 Z Ω v|∇v|2 + 2μ2a3 Z Ω uv∆v -2μ2a4 Z Ω vw∆v. Rearranging the terms, we get d dt Z Ω |∇v|2 + 2μ2 Z Ω |∇v|2 + 2d2 Z Ω |∆v|2 ≤2χ2 Z Ω ∇v · ∇z ∆v + 2χ2 v L∞(Ω) Z Ω |∆z||∆v| + 4μ2 Z Ω |∇v|2 + 2μ2a3 v L∞(Ω) Z Ω u|∆v| + 2μ2a4 v L∞(Ω) w L∞(Ω) Z Ω |∆v|. (3.27) The Hölder inequality gives for the first term in RHS of (3.27), 2χ2 Z Ω ∇v · ∇z ∆v ≤2χ2 ∇v L4(Ω)∥∇z∥L4(Ω) ∆v L2(Ω). (3.28) The Gagliardo-Nirenberg inequality with Lemma 3.2 gives 2χ2 ∇v L4(Ω) ≤C1 ∆v 1 2 L2(Ω) v 1 2 L∞(Ω) ≤C2 ∆v 1 2 L2(Ω). (3.29) Substituting the estimate (3.29) into (3.28) and using the boundedness of ∥∇z∥L4(Ω) from (3.16), which holds due to the condition rl ≥4, we then apply Young's inequality to obtain 2χ2 Z Ω ∇v · ∇z∆v ≤C2 ∆v 3 2 L2(Ω)∥∇z∥L4(Ω) ≤2d2 5 ∆v 2 L2(Ω) + C3. (3.30) 14 Now using the Young inequality for the second term in RHS of (3.27), to get 2χ2K4 Z Ω |∆v||∆z| ≤2d2 5 ∆v 2 L2(Ω) + C4 ∆z 2 L2(Ω). (3.31) Using the Hölder inequality and the Gagliardo-Nirenberg inequality for the third term in RHS of (3.27), gives 4μ2 Z Ω |∇v|2 ≤4μ2 Z Ω |∇v|4 1 2 |Ω| 1 2 ≤4μ2|Ω| 1 2 ∇v 2 L4(Ω) ≤2d2 5 ∆v 2 L2(Ω) + C5. (3.32) Once again Cauchy's inequality for the last two terms in (3.27), gives 2μ2a3K4 Z Ω u|∆v| ≤2d2 5 ∆v 2 L2(Ω) + C6 u 2 L2(Ω) (3.33) and 2μ2a4K2 4 Z Ω |∆v| ≤2d2 5 ∆v 2 L2(Ω) + C7. (3.34) Substituting (3.30)-(3.34) in to (3.27), one gets d dt Z Ω |∇v|2 + 2μ2 Z Ω |∇v|2 ≤C4 ∆z 2 L2(Ω) + C6 u 2 L2(Ω) + C8, ∀t ∈(0, Tmax). Now applying Lemma 2.2, we get Z Ω |∇v|2 ≤C9, ∀t ∈(0, Tmax), where C9 > 0. This completes the proof. Lemma 3.5. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary and the assumptions in Lemma 3.2 hold true. There exists K7 > 0 satisfying Z Ω |∇w(·, t)|2 ≤K7, (3.35) for all t ∈(0, Tmax). Proof. Multiplying the third equation in (1.1) with -2∆w and integrating by parts, we obtain d dt Z Ω |∇w|2 + 2μ3 Z Ω |∇w|2 + 2d2 Z Ω |∆w|2 ≤2ξ Z Ω ∇w · ∇z∆w + 2ξ w L∞(Ω) Z Ω |∆z||∆w| + 4μ3 Z Ω |∇w|2 -4μ3 Z Ω w|∇w|2 + 2μ3a4 w L∞(Ω) Z Ω u|∆w| + 2μ3a5 v L∞(Ω) w L∞(Ω) Z Ω |∆w|. (3.36) Following a similar procedure as in (3.30), we obtain 2ξ Z Ω ∇w · ∇z∆w ≤C1 ∆w 3 2 L2(Ω)∥∇z∥L4(Ω) ≤2d3 5 ∆w 2 L2(Ω) + C2. (3.37) Applying the Young inequality for the second term of (3.36) gives 2ξK4 Z Ω |∆w||∆z| ≤2d3 5 ∆w 2 L2(Ω) + C3 ∆z 2 L2(Ω). (3.38) 15 Using the Hölder inequality and the Gagliardo-Nirenberg inequality to the third from of (3.36), gives 4μ3 Z Ω |∇w|2 ≤4μ3 Z Ω |∇w|4 1 2 |Ω| 1 2 ≤4μ3|Ω| 1 2 ∇w 2 L4(Ω) ≤2d3 5 ∆w 2 L2(Ω) + C4. (3.39) The Cauchy inequality gives 2μ3a4K4 Z Ω u|∆w| ≤2d3 5 ∆w 2 L2(Ω) + C5 u 2 L2(Ω), (3.40) and 2μ3a5K2 4 Z Ω |∆w| ≤2d3 5 ∆w 2 L2(Ω) + C6. (3.41) Substituting (3.37)-(3.41) into (3.36) and simplifying, we get d dt Z Ω |∇w|2 + 2μ3 Z Ω |∇w|2 ≤C3 ∆z 2 L2(Ω) + C5 u 2 L2(Ω) + C7, ∀t ∈(0, Tmax). Now applying Lemma 2.2, we get Z Ω |∇w|2 ≤C8, ∀t ∈(0, Tmax), where C8 > 0. This completes the proof. Lemma 3.6. Suppose that Ω⊂Rn(n ≥1), is a bounded domain with smooth boundary and the assumptions in Lemma 3.2 hold true. There exists K8, K9 > 0 satisfying sup t∈(0,Tmax) ∇v(·, t) L∞(Ω) + ∇w(·, t) L∞(Ω) ≤K8, (3.42) and sup t∈(0,Tmax) u(·, t) L∞(Ω) ≤K9. (3.43) Proof. Let us use the variables r, ̄r, l, and k as defined in the proof of Lemma 3.2, t ∈(s0, Tmax) and t0 = max{s0, t -1}. Applying the variation of constants formula to the third equation of (1.1), we get ∥∇w(·, t)∥L∞(Ω) ≤ ∇ed3(t-s0)∆w(·, s0) L∞(Ω) + ξ Z t s0 ∇ed3(t-s)∆∇· w(·, s)∇z(·, s) L∞(Ω)ds + μ3 Z t s0 ∇ed3(t-s)∆w(·, s) L∞(Ω)ds. By Neumann heat semi-group with C1, C2, C3, C4 > 0, we obtain ∥∇w(·, t)∥L∞(Ω) ≤C1 1 + (t -s0)-1 2 w0 L∞(Ω) + C2 Z t s0 1 + (t -s)-1 2 -n 2l e-λd3(t-s) ∇w(·, s) · ∇z(·, s) Ll(Ω)ds + C3 Z t s0 1 + (t -s)-1 2 -n 2l e-λd3(t-s) w(·, s) L∞(Ω) ∆z(·, s) Ll(Ω)ds + C4 Z t s0 1 + (t -s)-1 2 e-λd3(t-s) w(·, s) L∞(Ω)ds. (3.44) 16 Now using similar procedure as in (3.22) and since Ωis bounded, ∇w ∈L2(Ω) implies ∇w ∈L1(Ω). Therefore, ∇w(·, t) · ∇z(·, t) Ll(Ω) ≤ ∇w(·, t) L ̄rl ∇z(·, t) Lrl(Ω) ≤ ∇w(·, t) k L∞(Ω) ∇w(·, t) 1-k L1(Ω) ∇z(·, t) Lrl(Ω) ≤C5 ∇w(·, t) k L∞(Ω), ∀t ∈(s0, Tmax). (3.45) Substituting (3.45) into (3.44) and using the Young inequality (see Lemma 3.9, [18]) with Lemma 3.2 gives ∥∇w(·, t)∥L∞(Ω) ≤C6 + C7 sup t∈(s0,Tmax) ∇w(·, t) k L∞(Ω) Z t s0 1 + (t -s)-1 2 -n 2l e-λd3(t-s)ds + C8 Z t s0 1 + (t -s)-1 2 -n 2l l l-1 e-λd3l l-1 (t-s)ds + C9 Z t s0 ∆z l Ll(Ω) (3.46) + C10 Z t s0 1 + (t -s)-1 2 -n 2l e-λd3(t-s)ds. There exits Cl > 0, the maximal Sobolev regularity for ∆z l Ll(Ω) (l > n) along with Lemma 3.1 gives Z t s0 ∆z l Ll(Ω) ≤Cl Z t s0 αv + βw l Ll(Ω) + Cl z(·, s0) l Ll(Ω) + Cl ∆z(·, s0) l Ll(Ω) ≤C11 Z t s0 v l Ll(Ω) + C12 Z t s0 w l Ll(Ω) + C13 ≤C14 ∀t ∈(s0, Tmax). Now using the Gamma function, one gets from (3.46) ∥∇w(·, t)∥L∞(Ω) ≤C15 sup t∈(s0,Tmax) ∇w(·, t) k L∞(Ω) + C16. Finally, we arrive at ∥∇w(·, t)∥L∞(Ω) ≤C17, ∀t ∈(s0, Tmax), where C17 > 0. By using the similar procedure, one can obtain from (3.23) ∇v(·, t) L∞(Ω) ≤ ∇ed2(t-s0)∆v(·, s0) L∞(Ω) + χ2 Z t s0 ∇ed2(t-s)∆∇v(·, s) · ∇z(·, s) L∞(Ω)ds + χ2 Z t s0 ∇ed2(t-s)∆v(·, s) ∆z(·, s) L∞(Ω)ds + μ2 Z t s0 ∇ed2(t-s)∆v(·, s) 1 + a4w(·, s) L∞(Ω)ds. Using the Neumann heat semi-group technique, then consider the last term from the above estimate gives v(·, t) 1 + a4w(·, t) L∞(Ω) ≤C18. Finally, by a similar manner we arrive at ∥∇v(·, t)∥L∞(Ω) ≤C19, ∀t ∈(s0, Tmax), where C19 > 0. Now, applying the variation of constants formula to the first equation of (1.1), we get u(·, t) =ed1(t-t0)∆u(·, t0)-χ1 Z t t0 ed1(t-s)∆∇· u(·, s)∇(v(·, s)w(·, s)) ds 17 + μ1 Z t t0 ed1(t-s)∆u(·, s) 1 -u(·, s) + a1v(·, s) + a2w(·, s) ds. Taking L∞(Ω) norm on both sides, one has u(·, t) L∞(Ω)≤∥ed1(t-t0)∆u(·, t0)∥L∞(Ω) + χ1 Z t t0 ∥ed1(t-s)∆∇· u(·, s)∇(v(·, s)w(·, s)) ∥L∞(Ω)ds + μ1 Z t t0 ∥ed1(t-s)∆u(·, s) 1 -u(·, s) + a1v(·, s) + a2w(·, s) ∥L∞(Ω)ds. By the similar procedure used in the Lemma 3.2, we obtain u(·, t) L∞(Ω) ≤max n ∥u(·, s0)∥L∞(Ω), C20 o +C21 Z t t0 1 + (t -s)-1 2 -n 2l e-λd1(t-s) u(·, s)∇(v(·, s)w(·, s)) Ll(Ω)ds +μ1 (1 + a1K4 + a2K4)2 4 . (3.47) Using the Hölder inequality, Lemma 3.2 and then using Interpolation inequality, we obtain u(·, t)∇(v(·, t)w(·, t)) Ll(Ω)≤ u(·, t)v(·, t)∇w(·, t) Ll(Ω) + u(·, t)w(·, t)∇v(·, t) Ll(Ω) ≤ v(·, t) L∞(Ω) ∇w(·, t) L∞(Ω) u(·, t) Ll(Ω) + w(·, t) L∞(Ω) ∇v(·, t) L∞(Ω) u(·, t) Ll(Ω) ≤C22 u(·, t) 1 l L∞(Ω) u(·, t) l l-1 L1(Ω) ≤C23 u(·, t) 1 l L∞(Ω), ∀t ∈(s0, Tmax). (3.48) Substitute (3.48) into (3.47) and using Gamma function, we arrive at u(·, t) L∞(Ω) ≤C24 u(·, t) 1 l L∞(Ω) + C25. Finally, we get u(·, t) L∞(Ω) ≤C26, ∀t ∈(s0, Tmax), where C26 > 0. The proof is completed by taking into account (3.2). Proof of Theorem 1.1. We argue the proof by contradiction. Assume that Tmax 0 u(·, t) L∞(Ω) + v(·, t) W1,∞(Ω) + w(·, t) W1,∞(Ω) + z(·, t) W1,q(Ω) 0 such that u(·, t) C2+θ,1+ θ 2 (Ω×[t,t+1]) + v(·, t) C2+θ,1+ θ 2 (Ω×[t,t+1]) ≤C and w(·, t) C2+θ,1+ θ 2 (Ω×[t,t+1]) + z(·, t) C2+θ,1+ θ 2 (Ω×[t,t+1]) ≤C, for all t ≥1. Proof. The proof is based on the standard parabolic regularity theory in [16] and Lemma 3.2 and 3.6. For more details see [19, 26]. By Lemma 4.1, it is clear that u, v, w, z are continuous and bounded in Ω× [t, t + 1] and the first derivatives ∇u, ∇v, ∇w, ∇z exist and are Hölder continuous. Since the Hölder norms are uniformly bounded, the gradients are also uniformly bounded in L∞(Ω), we conclude that u(·, t) W1,∞(Ω), v(·, t) W1,∞(Ω), w(·, t) W1,∞(Ω), z(·, t) W1,∞(Ω) ≤C, (4.1) where C is a constant independent of t. Lemma 4.2 ([1]). Suppose that f is a uniformly continuous nonnegative function on (1, ∞) such that Z ∞ 1 f(t) dt a2a3a6 and χ2 1 0. To prove the nonnegativity of E1, let H(u) = u -u∗ln u, for u > 0. Applying the Taylor's formula to see that, for all x ∈Ωand each t > 0, we can find τ = τ(x, t) ∈(0, 1) such that H(u) -H(u∗) = H′(u∗)(u -u∗) + 1 2H′′(τu + (1 -τ)u∗)(u -u∗)2 = u∗ 2(τu + (1 -τ)u∗)2 (u -u∗)2 ≥0. Therefore, we get R Ω u -u∗-u∗ln u u∗ = R ΩH(u)-H(u∗) ≥0. A similar argument shows that E1(t) ≥0. Now, we compute d dtI11(t) = Z Ω u -u∗ u ut = -d1u∗ Z Ω ∇u u 2 + χ1u∗ Z Ω w u ∇u · ∇v + χ1u∗ Z Ω v u∇u · ∇w + μ1 Z Ω (u -u∗)(1 -u + a1v + a2w) ≤-d1u∗ Z Ω ∇u u 2 + χ1u∗ w L∞(Ω) Z Ω ∇u u · ∇v + χ1u∗ v L∞(Ω) Z Ω ∇u u · ∇w + μ1 Z Ω (u -u∗)(1 -u + a1v + a2w -(1 -u∗+ a1v∗+ a2w∗)) ≤-d1u∗ Z Ω ∇u u 2 + χ1u∗ w L∞(Ω) Z Ω ∇u u · ∇v + χ1u∗ v L∞(Ω) Z Ω ∇u u · ∇w -μ1 Z Ω (u -u∗)2 + μ1a1 Z Ω (u -u∗)(v -v∗) + μ1a2 Z Ω (u -u∗)(w -w∗). (4.3) Similarly, we get d dtI12(t) =Γ1 Z Ω v -v∗ v vt = -Γ1d2v∗ Z Ω ∇v v 2 + Γ1χ2v∗ Z Ω ∇v v · ∇z + Γ1μ2 Z Ω (v -v∗)(1 -v -a3u + a4w) ≤-Γ1d2v∗ Z Ω ∇v v 2 + Γ1χ2v∗ Z Ω ∇v v · ∇z -Γ1μ2 Z Ω (v -v∗)2 -Γ1μ2a3 Z Ω (v -v∗)(u -u∗) + Γ1μ2a4 Z Ω (v -v∗)(w -w∗). (4.4) An analogous calculation for I13 provides d dtI13(t) =Γ2 Z Ω w -w∗ v wt = -Γ2d3w∗ Z Ω ∇w w 2 -Γ2ξw∗ Z Ω ∇w w · ∇z + μ3 Z Ω (w -w∗)(1 -w -a5u -a6v) ≤-Γ2w∗ Z Ω ∇w w 2 -Γ2ξw∗ Z Ω ∇w w · ∇z -Γ2μ3 Z Ω (w -w∗)2 -Γ2μ3a5 Z Ω (w -w∗)(u -u∗) -Γ2μ3a6 Z Ω (w -w∗)(v -v∗) (4.5) 20 and for I14 d dtI14(t) = Z Ω (z -z∗)zt ≤-d4 Z Ω |∇z|2 + α Z Ω (z -z∗)(v -v∗) + β Z Ω (z -z∗)(w -w∗) -γ Z Ω (z -z∗)2. (4.6) Adding (4.3)-(4.6), we get d dtE1(t) ≤-d1u∗ Z Ω ∇u u 2 + χ1u∗ w L∞(Ω) Z Ω ∇u u · ∇v + χ1u∗ v L∞(Ω) Z Ω ∇u u · ∇w -Γ1d2v∗ Z Ω ∇v v 2 + Γ1χ2v∗ Z Ω ∇v v · ∇z -Γ2d3w∗ Z Ω ∇w w 2 -Γ2ξw∗ Z Ω ∇w w · ∇z -d4 Z Ω |∇z|2 -μ1 Z Ω (u -u∗)2 -Γ1μ2 Z Ω (v -v∗)2 -Γ2μ3 Z Ω (w -w∗)2 -γ Z Ω (z -z∗)2 + μ1a1 Z Ω (u -u∗)(v -v∗) + μ1a2 Z Ω (u -u∗)(w -w∗) -Γ1μ2a3 Z Ω (v -v∗)(u -u∗) + Γ1μ2a4 Z Ω (v -v∗)(w -w∗) -Γ2μ3a5 Z Ω (w -w∗)(u -u∗) -Γ2μ3a6 Z Ω (w -w∗)(v -v∗) + α Z Ω (z -z∗)(v -v∗) + β Z Ω (z -z∗)(w -w∗). By simplifying, we obtain d dtE1(t) ≤-d1u∗ Z Ω ∇u u 2 + χ1u∗ w L∞(Ω) Z Ω ∇u u · ∇v + χ1u∗ v L∞(Ω) Z Ω ∇u u · ∇w -Γ1d2v∗ Z Ω ∇v v 2 + Γ1χ2v∗ Z Ω ∇v v · ∇z -Γ2d3w∗ Z Ω ∇w w 2 -Γ2ξw∗ Z Ω ∇w w · ∇z -d4 Z Ω |∇z|2 -μ1 Z Ω (u -u∗)2 -Γ1μ2 Z Ω (v -v∗)2 -Γ2μ3 Z Ω (w -w∗)2 -γ Z Ω (z -z∗)2 + (Γ1μ2a4 -Γ2μ3a6) Z Ω (v -v∗)(w -w∗) + α Z Ω (z -z∗)(v -v∗) + β Z Ω (z -z∗)(w -w∗). (4.7) Applying Cauchy's inequality, we obtain the following estimates χ1u∗ w L∞(Ω) Z Ω ∇u u · ∇v ≤d1u∗ 2 Z Ω ∇u u 2 + χ2 1u∗ w 2 L∞(Ω) 2d1 Z Ω |∇v|2, χ1u∗ v L∞(Ω) Z Ω ∇u u · ∇w ≤d1u∗ 2 Z Ω ∇u u 2 + χ2 1u∗ v 2 L∞(Ω) 2d1 Z Ω |∇w|2, Γ1χ2v∗ Z Ω ∇v v · ∇z ≤Γ1d2v∗ 2 Z Ω ∇v v 2 + Γ1χ2 2v∗ 2d2 Z Ω |∇z|2, Γ2ξw∗ Z Ω ∇w w · ∇z ≤Γ2d3w∗ 2 Z Ω ∇w w 2 + Γ2ξ2w∗ 2d3 Z Ω |∇z|2, (Γ1μ2a4 -Γ2μ3a6) Z Ω (v -v∗)(w -w∗) ≤Γ1μ2a4 -Γ2μ3a6 2 Z Ω (v -v∗)2, + Γ1μ2a4 -Γ2μ3a6 2 Z Ω (w -w∗)2, 21 α Z Ω (z -z∗)(v -v∗) ≤α 2 Z Ω (z -z∗)2 + α 2 Z Ω (v -v∗)2, β Z Ω (z -z∗)(w -w∗) ≤β 2 Z Ω (z -z∗)2 + β 2 Z Ω (w -w∗)2. By substituting all the above estimates into (4.7), we obtain d dtE1(t) ≤ χ2 1u∗ w 2 L∞(Ω) 2d1 Z Ω |∇v|2 + χ2 1u∗ v 2 L∞(Ω) 2d1 Z Ω |∇w|2 -Γ1d2v∗ 2 Z Ω ∇v v 2 + Γ1χ2 2v∗ 2d2 Z Ω |∇z|2 -Γ2d3w∗ 2 Z Ω ∇w w 2 + Γ2ξ2w∗ 2d3 Z Ω |∇z|2 -d4 Z Ω |∇z|2 -μ1 Z Ω (u -u∗)2 -Γ1μ2 Z Ω (v -v∗)2 -Γ2μ3 Z Ω (w -w∗)2 -γ Z Ω (z -z∗)2 + Γ1μ2a4 -Γ2μ3a6 2 Z Ω (v -v∗)2 + Γ1μ2a4 -Γ2μ3a6 2 Z Ω (w -w∗)2 + α 2 Z Ω (z -z∗)2 + α 2 Z Ω (v -v∗)2 + β 2 Z Ω (z -z∗)2 + β 2 Z Ω (w -w∗)2. Combining the terms, one gets d dtE1(t) ≤- Γ1d2v∗ 2 v 2 L∞(Ω) - χ2 1u∗ w 2 L∞(Ω) 2d1 Z Ω |∇v|2 - Γ2d3w∗ 2 w 2 L∞(Ω) - χ2 1u∗ v 2 L∞(Ω) 2d1 Z Ω |∇w|2 - d4 -Γ1χ2 2v∗ 2d2 -Γ2ξ2w∗ 2d3 Z Ω |∇z|2 -μ1 Z Ω (u -u∗)2 - Γ1μ2 -Γ1μ2a4 -Γ2μ3a6 2 -α 2 Z Ω (v -v∗)2 - Γ2μ3 -Γ1μ2a4 -Γ2μ3a6 2 -β 2 Z Ω (w -w∗)2 - γ -α 2 -β 2 Z Ω (z -z∗)2. Finally, d dtE1(t) ≤-σ1F1(t), ∀t > 0, (4.8) where σ1 > 0 and F1(t) = Z Ω (u -u∗)2 + Z Ω (v -v∗)2 + Z Ω (w -w∗)2 + Z Ω (z -z∗)2 + Z Ω |∇v|2 + Z Ω |∇w|2 + Z Ω |∇z|2. Let f1(t) = Z Ω (u(·, t) -u∗)2 + Z Ω (v(·, t) -v∗)2 + Z Ω (w(·, t) -w∗)2 + Z Ω (z(·, t) -z∗)2, t > 0. Now (4.8) takes the form d dtE1(t) ≤-σ1F1(t) ≤-σ1f1(t), t > 0. (4.9) Upon integrating with respect to t, we obtain Z ∞ 1 f1(t) dt ≤1 σ1 E1(1) -E1(t) 0. (4.10) From (4.1), we can deduce that u(·, t) converges to u∗in L∞(Ω) when t →∞. By using a similar argument, we can derive (4.2). 4.2. Secondary predator only existence state Here we assume that a2a6 ≥min 1 + a1 + a2 + a1a4 + a4a6, 1 + a1(a3 + a4a5) + a4a6 + a2a5 a3 , a3(1 + a2) + a4a5 ≥1 + a4 + a2a5, and a5(1 + a1) + a6 ≥1 + a3(a1 + a6) hold. Let (u, v, w, z) be the classical solution of (1.1) satisfying (1.2) and (1, 0, 0, 0) be the trivial steady state of the system (1.1). Lemma 4.4. Suppose that the assumptions in Theorem 1.1 hold true and let Γ1 = μ1a1 μ2a3 , Γ2 = μ1a2 μ3a5 . If α + β 0. The positivity of u -1 -lnu for any u > 0 implies that E2(t) ≥0. We begin by computing the derivative 23 of the first integral d dtI21(t) = Z Ω u -1 u ut = -d1 Z Ω ∇u u 2 + χ1 Z Ω w u ∇u · ∇v + χ1 Z Ω v u∇u · ∇w + μ1 Z Ω (u -1)(1 -u + a1v + a2w) ≤-d1 Z Ω ∇u u 2 + χ1 w L∞(Ω) Z Ω ∇u u · ∇v + χ1 v L∞(Ω) Z Ω ∇u u · ∇w -μ1 Z Ω (u -1)2 + μ1a1 Z Ω (u -1)v + μ1a2 Z Ω (u -1)w ≤-d1 Z Ω ∇u u 2 + χ1 w L∞(Ω) Z Ω ∇u u · ∇v + χ1 v L∞(Ω) Z Ω ∇u u · ∇w -μ1 Z Ω (u -1)2 + μ1a1 Z Ω uv -μ1a1 Z Ω v + μ1a2 Z Ω uw -μ1a2 Z Ω w. (4.12) Likewise for I22(t) d dtI22(t) =Γ1μ2 Z Ω v -Γ1μ2 Z Ω v2 -Γ1μ2a3 Z Ω uv + Γ1μ2a4 Z Ω vw =Γ1μ2 Z Ω v -Γ1μ2 Z Ω v2 -μ1a1 Z Ω uv + Γ1μ2a4 Z Ω vw, (4.13) and for I23(t) d dtI23(t) =d2 Z Ω v∆v -χ2 Z Ω v∇· (v∇z) + μ2 Z Ω v2(1 -v -a3u + a4w) ≤-d2 Z Ω |∇v|2 + χ2 v L∞(Ω) Z Ω ∇v · ∇z + μ2 Z Ω v2 -μ2 Z Ω v3 -μ2a3 Z Ω v2u + μ2a4 Z Ω v2w ≤-d2 Z Ω |∇v|2 + χ2 v L∞(Ω) Z Ω ∇v · ∇z + μ2 Z Ω v2 + μ2a4 w L∞(Ω) Z Ω v2. (4.14) By a similar manner, we obtain d dtI24(t) =Γ2μ3 Z Ω w -Γ2μ3 Z Ω w2 -Γ2μ3a5 Z Ω uw -Γ2μ3a6 Z Ω vw =Γ2μ3 Z Ω w -Γ2μ3 Z Ω w2 -μ1a2 Z Ω uw -Γ2μ3a6 Z Ω vw. (4.15) Similarly for I25(t) d dtI25(t) =d3 Z Ω w∆w + ξ Z Ω w∇· (w∇z) + μ3 Z Ω w2(1 -w -a5u -a6v) ≤-d3 Z Ω |∇w|2 + ξ w L∞(Ω) Z Ω ∇w · ∇z + μ3 Z Ω w2 -μ3 Z Ω w3 -μ3a5 Z Ω w2u -μ3a6 Z Ω w2v ≤-d3 Z Ω |∇w|2 + ξ w L∞(Ω) Z Ω ∇w · ∇z + μ3 Z Ω w2, (4.16) 24 and for I26(t) d dtI26(t) =d4 Z Ω z∆z + α Z Ω vz + β Z Ω wz -γ Z Ω z2 ≤-d4 Z Ω |∇z|2 + α 2 Z Ω v2 + α 2 Z Ω z2 + β 2 Z Ω w2 + β 2 Z Ω z2 -γ Z Ω z2. (4.17) Adding all the estimates (4.12)-(4.17), we get d dtE2(t) ≤-d1 Z Ω ∇u u 2 + χ1 w L∞(Ω) Z Ω ∇u u · ∇v + χ1 v L∞(Ω) Z Ω ∇u u · ∇w -d2 Z Ω |∇v|2 + χ2 v L∞(Ω) Z Ω ∇v · ∇z -d3 Z Ω |∇w|2 + ξ w L∞(Ω) Z Ω ∇w · ∇z -d4 Z Ω |∇z|2 -μ1 Z Ω (u -1)2 -μ1a1 Z Ω v -μ1a2 Z Ω w + Γ1μ2 Z Ω v -Γ1μ2 Z Ω v2 + Γ1μ2a4 Z Ω vw + μ2 Z Ω v2 + μ2a4 w L∞(Ω) Z Ω v2 + Γ2μ3 Z Ω w -Γ2μ3 Z Ω w2 -Γ2μ3a6 Z Ω vw + μ3 Z Ω w2 + α 2 Z Ω v2 + α 2 Z Ω z2 + β 2 Z Ω w2 + β 2 Z Ω z2 -γ Z Ω z2. (4.18) Using the Cauchy inequality, we get the following estimates χ1 w L∞(Ω) Z Ω ∇u u · ∇v ≤d1 2 Z Ω ∇u u 2 + χ2 1 w 2 L∞(Ω) 2d1 Z Ω |∇v|2, χ1 v L∞(Ω) Z Ω ∇u u · ∇w ≤d1 2 Z Ω ∇u u 2 + χ2 1 v 2 L∞(Ω) 2d1 Z Ω |∇w|2, χ2 v L∞(Ω) Z Ω ∇v · ∇z ≤d2 2 Z Ω |∇v|2 + χ2 2 v 2 L∞(Ω) 2d2 Z Ω |∇z|2, ξ w L∞(Ω) Z Ω ∇w · ∇z ≤d3 2 Z Ω |∇w|2 + ξ2 w 2 L∞(Ω) 2d3 Z Ω |∇z|2. Substituting all the above estimates in (4.18), we get d dtE2(t) ≤- d2 2 - χ2 1 w 2 L∞(Ω) 2d1 Z Ω |∇v|2 - d3 2 - χ2 1 v 2 L∞(Ω) 2d1 Z Ω |∇w|2 - d4 - χ2 2 v 2 L∞(Ω) 2d2 - ξ2 w 2 L∞(Ω) 2d3 Z Ω |∇z|2 -μ1 Z Ω (u -1)2 - Γ1μ2 -μ2 -μ2a4 w L∞(Ω) -α 2 Z Ω v2 - Γ2μ3 -μ3 -β 2 Z Ω w2 - γ -α 2 -β 2 Z Ω z2 -(μ1a1 -Γ1μ2) Z Ω v -(μ1a2 -Γ2μ3) Z Ω w -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw. Finally, we arrive at d dtE2(t) ≤-σ2F2(t) -(μ1a1 -Γ1μ2) Z Ω v -(μ1a2 -Γ2μ3) Z Ω w -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw, (4.19) 25 for every t > 0, where σ2 > 0 and F2(t) = Z Ω (u -1)2 + Z Ω v2 + Z Ω w2 + Z Ω z2 + Z Ω |∇v|2 + Z Ω |∇w|2 + Z Ω |∇z|2. Now, (4.19) can be written as d dtE2(t) ≤-σ2f2(t) -(μ1a1 -Γ1μ2) Z Ω v -(μ1a2 -Γ2μ3) Z Ω w -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw, (4.20) for all t > 0, where f2(t) = Z Ω (u(·, t) -1)2 + Z Ω v(·, t)2 + Z Ω w(·, t)2 + Z Ω z(·, t)2. In view of Lemma 4.3, the proof is similar. 4.3. Semi-coexistence state of the species 4.3.1. Prey vanishing state Here we assume that (1.3) does not hold and a3 0. By a similar argument as in the previous Lemma 4.3, we find that E3(t) ≥0. Now, we compute d dtI31(t) ≤-d1 ̄u Z Ω ∇u u 2 + χ1 ̄u w L∞(Ω) Z Ω ∇u u · ∇v + χ1 ̄u v L∞(Ω) Z Ω ∇u u · ∇w -μ1 Z Ω (u - ̄u)2 + μ1a1 Z Ω (u - ̄u)(v - ̄v) + μ1a2 Z Ω (u - ̄u)(w - ̄w). (4.22) Similarly, we get d dtI32(t) = -Γ1d2 ̄v Z Ω ∇v v 2 + Γ1χ2 ̄v Z Ω ∇v v · ∇z -Γ1μ2 Z Ω (v - ̄v)2 -Γ1μ2a3 Z Ω (v - ̄v)(u - ̄u) + Γ1μ2a4 Z Ω (v - ̄v)(w - ̄w). (4.23) 26 Analogously, for I33, d dtI33(t) =Γ2μ3 Z Ω w -Γ2μ3 Z Ω w2 -μ1a2 Z Ω uw -Γ2μ3a6 Z Ω vw. (4.24) Following the same procedure for I34(t) d dtI34(t) ≤-d3 Z Ω |∇w|2 + ξ w L∞(Ω) Z Ω ∇w · ∇z + μ3 Z Ω w2 (4.25) and for I35, d dtI35(t) = -d4 Z Ω |∇z|2 + α Z Ω (z - ̄z)(v - ̄v) + β Z Ω (z - ̄z)(w - ̄w) -γ Z Ω (z - ̄z)2. (4.26) Adding (4.22)-(4.26), we get d dtE3(t) ≤-d1 ̄u Z Ω ∇u u 2 + χ1 ̄u w L∞(Ω) Z Ω ∇u u · ∇v + χ1 ̄u v L∞(Ω) Z Ω ∇u u · ∇w -Γ1d2 ̄v Z Ω ∇v v 2 + Γ1χ2 ̄v Z Ω ∇v v · ∇z -d3 Z Ω |∇w|2 + ξ w L∞(Ω) Z Ω ∇w · ∇z -d4 Z Ω |∇z|2 -μ1 Z Ω (u - ̄u)2 -Γ1μ2 Z Ω (v - ̄v)2 -Γ2μ3 Z Ω w2 + μ3 Z Ω w2 -γ Z Ω (z - ̄z)2 -Γ2μ3a6 Z Ω vw + Γ1μ2a4 Z Ω vw -Γ1μ2a4 ̄v Z Ω w + Γ2μ3 Z Ω w -μ1a2 ̄u Z Ω w + α Z Ω (z - ̄z)(v - ̄v) + β Z Ω (z - ̄z)w. By simplifying, we obtain d dtE3(t) ≤-d1 ̄u Z Ω ∇u u 2 + χ1 ̄u w L∞(Ω) Z Ω ∇u u · ∇v + χ1 ̄u v L∞(Ω) Z Ω ∇u u · ∇w -Γ1d2 ̄v Z Ω ∇v v 2 + Γ1χ2 ̄v Z Ω ∇v v · ∇z -d3 Z Ω |∇w|2 + ξ w L∞(Ω) Z Ω ∇w · ∇z -d4 Z Ω |∇z|2 -μ1 Z Ω (u - ̄u)2 -Γ1μ2 Z Ω (v - ̄v)2 -(Γ2μ3 -μ3) Z Ω w2 -γ Z Ω (z - ̄z)2 -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw -(μ1a2 ̄u + Γ1μ2a4 ̄v -Γ2μ3) Z Ω w + α Z Ω (z - ̄z)(v - ̄v) + β Z Ω (z - ̄z)w. (4.27) Applying Cauchy's inequality, we obtain the following estimates χ1 ̄u w L∞(Ω) Z Ω ∇u u · ∇v ≤d1 ̄u 2 Z Ω ∇u u 2 + χ2 1 ̄u w 2 L∞(Ω) 2d1 Z Ω |∇v|2, χ1 ̄u v L∞(Ω) Z Ω ∇u u · ∇w ≤d1 ̄u 2 Z Ω ∇u u 2 + χ2 1 ̄u v 2 L∞(Ω) 2d1 Z Ω |∇w|2, Γ1χ2 ̄v Z Ω ∇v v · ∇z ≤Γ1d2 ̄v 2 Z Ω ∇v v 2 + Γ1χ2 2 ̄v 2d2 Z Ω |∇z|2, ξ w L∞(Ω) Z Ω ∇w · ∇z ≤d3 2 Z Ω |∇w|2 + ξ2 w 2 L∞(Ω) 2d3 Z Ω |∇z|2, α Z Ω (z - ̄z)(v - ̄v) ≤α 2 Z Ω (z - ̄z)2 + α 2 Z Ω (v - ̄v)2, 27 β Z Ω (z - ̄z)w ≤β 2 Z Ω (z - ̄z)2 + β 2 Z Ω w2. By substituting all the above estimates into (4.27), we obtain d dtE3(t) ≤-d1 ̄u Z Ω ∇u u 2 +d1 ̄u 2 Z Ω ∇u u 2 + χ2 1 ̄u w 2 L∞(Ω) 2d1 Z Ω |∇v|2 + d1 ̄u 2 Z Ω ∇u u 2 + χ2 1 ̄u v 2 L∞(Ω) 2d1 Z Ω |∇w|2 -Γ1d2 ̄v Z Ω ∇v v 2 + Γ1d2 ̄v 2 Z Ω ∇v v 2 + Γ1χ2 2 ̄v 2d2 Z Ω |∇z|2 -d3 Z Ω |∇w|2 + d3 2 Z Ω |∇w|2 + ξ2 w 2 L∞(Ω) 2d3 Z Ω |∇z|2 -d4 Z Ω |∇z|2 -μ1 Z Ω (u - ̄u)2 -Γ1μ2 Z Ω (v - ̄v)2 -(Γ2μ3 -μ3) Z Ω w2 -γ Z Ω (z - ̄z)2 -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw -(μ1a2 ̄u + Γ1μ2a4 ̄v -Γ2μ3) Z Ω w + α 2 Z Ω (z - ̄z)2 + α 2 Z Ω (v - ̄v)2 + β 2 Z Ω (z - ̄z)2 + β 2 Z Ω w2. Combining the terms, one gets d dtE3(t) ≤- Γ1d2 ̄v 2 v 2 L∞(Ω) - χ2 1 ̄u w 2 L∞(Ω) 2d1 Z Ω |∇v|2 - d3 2 - χ2 1 ̄u v 2 L∞(Ω) 2d1 Z Ω |∇w|2 - d4 -Γ1χ2 2 ̄v 2d2 - ξ2 w 2 L∞(Ω) 2d3 Z Ω |∇z|2 -μ1 Z Ω (u - ̄u)2 - Γ1μ2 -α 2 Z Ω (v - ̄v)2 - Γ2μ3 -μ3 -β 2 Z Ω w2 - γ -α 2 -β 2 Z Ω (z - ̄z)2 -(μ1a2 ̄u + Γ1μ2a4 ̄v -Γ2μ3) Z Ω w -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw. Finally, we arrive at d dtE3(t) ≤-σ1F3(t) -(μ1a2 ̄u + Γ1μ2a4 ̄v -Γ2μ3) Z Ω w -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw, ∀t > 0, (4.28) where σ3 > 0 and F3(t) = Z Ω (u - ̄u)2 + Z Ω (v - ̄v)2 + Z Ω w2 + Z Ω (z - ̄z)2 + Z Ω |∇v|2 + Z Ω |∇w|2 + Z Ω |∇z|2, Let f3(t) = Z Ω (u(·, t) - ̄u)2 + Z Ω (v(·, t) - ̄v)2 + Z Ω w(·, t)2 + Z Ω (z(·, t) - ̄z)2, t > 0. Now (4.28) takes the form d dtE3(t) ≤-σ1f3(t) -(μ1a2 ̄u + Γ1μ2a4 ̄v -Γ2μ3) Z Ω w -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw, t > 0. (4.29) 28 Upon integrating with respect to t, we obtain Z ∞ 1 f3(t) dt ≤1 σ1 E3(1) -E3(t) 0. Analogous to reasoning in Lemma 4.3, it is clear that E4(t) ≥0. Now we compute d dtI42(t) =Γ1μ2 Z Ω v -Γ1μ2 Z Ω v2 -μ1a1 Z Ω uv + Γ1μ2a4 Z Ω vw, (4.31) and for I43(t) d dtI43(t) ≤-d2 Z Ω |∇v|2 + χ2 v L∞(Ω) Z Ω ∇v · ∇z + μ2 Z Ω v2 + μ2a4 w L∞(Ω) Z Ω v2. (4.32) By a similar manner, we obtain d dtI44(t) = -Γ2d3 ˆw Z Ω ∇w w 2 -Γ2ξ ˆw Z Ω ∇w w · ∇z -Γ2μ3 Z Ω (w -ˆw)2 -Γ2μ3a5 Z Ω (w -ˆw)(u -ˆu) -Γ2μ3a6 Z Ω (w -ˆw)v. (4.33) 29 Adding all the estimates (4.22), (4.26) and (4.31)-(4.33), we get d dtE4(t) ≤-d1ˆu Z Ω ∇u u 2 + χ1ˆu w L∞(Ω) Z Ω ∇u u · ∇v + χ1ˆu v L∞(Ω) Z Ω ∇u u · ∇w -d2 Z Ω |∇v|2 + χ2 v L∞(Ω) Z Ω ∇v · ∇z -Γ2d3 ˆw Z Ω ∇w w 2 -Γ2ξ ˆw Z Ω ∇w w · ∇z -d4 Z Ω |∇z|2 -μ1 Z Ω (u -ˆu)2 -Γ1μ2 Z Ω v2 + μ2 Z Ω v2 + μ2a4 w L∞(Ω) Z Ω v2 -Γ2μ3 Z Ω (w -ˆw)2 -γ Z Ω (z -ˆz)2 -μ1a1ˆu Z Ω v + Γ1μ2 Z Ω v + Γ1μ2a4 Z Ω vw -Γ2μ3a6 Z Ω wv + Γ2μ3a6 ˆw Z Ω v + α Z Ω (z -ˆz)v + β Z Ω (z -ˆz)(w -ˆw). (4.34) Using the Cauchy inequality, we get the following estimates χ2 v L∞(Ω) Z Ω ∇v · ∇z ≤d2 2 Z Ω |∇v|2 + χ2 2 v 2 L∞(Ω) 2d2 Z Ω |∇z|2, Γ2ξ ˆw Z Ω ∇w w · ∇z ≤Γ2d3 ˆw 2 Z Ω ∇w w 2 + Γ2ξ2 ˆw 2d3 Z Ω |∇z|2, α Z Ω (z -ˆz)v ≤α 2 Z Ω (z -ˆz)2 + α 2 Z Ω v2, β Z Ω (z -ˆz)(w -ˆw) ≤β 2 Z Ω (z -ˆz)2 + β 2 Z Ω (w -ˆw)2. Substituting all the above estimates in (4.34), we get d dtE4(t) ≤-d1ˆu Z Ω ∇u u 2 + d1ˆu 2 Z Ω ∇u u 2 + χ2 1ˆu w 2 L∞(Ω) 2d1 Z Ω |∇v|2 + d1ˆu 2 Z Ω ∇u u 2 + χ2 1ˆu v 2 L∞(Ω) 2d1 Z Ω |∇w|2 -d2 Z Ω |∇v|2 + d2 2 Z Ω |∇v|2 + χ2 2 v 2 L∞(Ω) 2d2 Z Ω |∇z|2 -Γ2d3 ˆw Z Ω ∇w w 2 + Γ2d3 ˆw 2 Z Ω ∇w w 2 + Γ2ξ2 ˆw 2d3 Z Ω |∇z|2 -d4 Z Ω |∇z|2 -μ1 Z Ω (u -ˆu)2 - Γ1μ2 -μ2 -μ2a4 w L∞(Ω) Z Ω v2 -Γ2μ3 Z Ω (w -ˆw)2 -γ Z Ω (z -ˆz)2 -(μ1a1ˆu + Γ1μ2 + Γ2μ3a6 ˆw) Z Ω v -(Γ2μ3a6 -Γ1μ2a4) Z Ω wv + α 2 Z Ω (z -ˆz)2 + α 2 Z Ω v2 + β 2 Z Ω (z -ˆz)2 + β 2 Z Ω (w -ˆw)2. Combining the terms, one gets d dtE4(t) ≤- d2 2 - χ2 1ˆu w 2 L∞(Ω) 2d1 Z Ω |∇v|2 - Γ2d3 ˆw 2 w 2 L∞(Ω) - χ2 1ˆu v 2 L∞(Ω) 2d1 Z Ω |∇w|2 - d4 - χ2 2 v 2 L∞(Ω) 2d2 -Γ2ξ2 ˆw 2d3 Z Ω |∇z|2 -μ1 Z Ω (u -ˆu)2 - Γ1μ2 -μ2 -μ2a4 w L∞(Ω) -α 2 Z Ω v2 - Γ2μ3 -β 2 Z Ω (w -ˆw)2 30 - γ -α 2 -β 2 Z Ω (z -ˆz)2 -(μ1a1ˆu -Γ1μ2 -Γ2μ3a6 ˆw) Z Ω v -(Γ2μ3a6 -Γ1μ2a4) Z Ω wv. Finally, we arrive at d dtE4(t) ≤-σ4F4(t) -(μ1a1ˆu -Γ1μ2 -Γ2μ3a6 ˆw) Z Ω v -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw, t > 0, (4.35) where σ4 > 0 with F4(t) = Z Ω (u -ˆu)2 + Z Ω v2 + Z Ω (w -ˆw)2 + Z Ω (z -ˆz)2 + Z Ω |∇v|2 + Z Ω |∇w|2 + Z Ω |∇z|2. Now, (4.35) can be written as d dtE4(t) ≤-σ4f4(t) -(μ1a1ˆu -Γ1μ2 -Γ2μ3a6 ˆw) Z Ω v -(Γ2μ3a6 -Γ1μ2a4) Z Ω vw, (4.36) for t > 0, where f4(t) = Z Ω (u(·, t) -ˆu)2 + Z Ω v(·, t)2 + Z Ω (w(·, t) -ˆw)2 + Z Ω (z(·, t) -ˆz)2, t > 0. In view of Lemma 4.3, the proof is similar. 4.4. Proof of the main theorems Proof of Theorem 1.2. Let H(u) = u -ue ln u, for u > 0. By applying L'Hôpital's rule, we obtain lim u→ue H(u) -H(ue) (u -ue)2 = 1 -ue u 2(u -ue) = 1 2ue . We can therefore select t0 > 0 and use the Taylor expansion to obtain Z Ω u -ue -ue ln u ue = Z Ω (H(u) -H(ue)) ≤ 1 2ue Z Ω (u -ue)2, ∀t > t0. From the above estimate, we arrive at 1 4ue Z Ω (u -ue)2 ≤ Z Ω u -ue -ue ln u ue ≤ 3 4ue Z Ω (u -ue)2, ∀t > t0. (4.37) Similarly, we obtain 1 4ve Z Ω (v -ve)2 ≤ Z Ω v -ve -ve ln v ve ≤ 3 4ve Z Ω (v -ve)2, ∀t > t0 (4.38) and 1 4we Z Ω (w -we)2 ≤ Z Ω w -we -we ln w we ≤ 3 4we Z Ω (w -we)2, ∀t > t0. (4.39) Taking into account the inequalities (4.37)-(4.39), it is clear that for all t > t0 and with κ1, κ2 > 0, we have κ1f1(t) ≤E1(t) ≤κ2f1(t). (4.40) 31 From (4.9) and right-hand side of (4.40), we get d dtE1(t) ≤-σ1f1(t) ≤-σ1 κ2 E1(t), t > t0. Solving the above differential inequality gives E1(t) ≤C1e-σ1 κ2 t, t > t0, where C1 > 0. Now, the left-hand side of (4.40) gives f1(t) ≤1 κ1 E1(t) ≤C1 κ1 e-σ1 κ2 t, t > t0. In view of (4.10) and then Lemma 4.1, finally we arrive at u(·, t) -ue L∞(Ω) + v(·, t) -ve L∞(Ω) + w(·, t) -we L∞(Ω) + z(·, t) -ze L∞(Ω) ≤C2e-σ1 κ2 t, for all t > t0, with C2 > 0. This completes the proof. Proof of Theorem 1.3. Choose t0 > 0, we can apply a similar approach to the one used to derive (4.37) to obtain the following results for all t > t0 1 4 Z Ω (u -1)2 ≤ Z Ω (u -1 -ln u) ≤3 4 Z Ω (u -1)2 , (4.41) 1 2 Z Ω v2 + 1 2 Z Ω v ≤ Z Ω v ≤2 Z Ω v2 + 2 Z Ω v, (4.42) and 1 2 Z Ω w2 + 1 2 Z Ω w ≤ Z Ω w ≤2 Z Ω w2 + 2 Z Ω w. (4.43) Considering the above inequalities (4.41)-(4.43), it is apparent that κ3f2(t) ≤E2(t) ≤κ4 f2(t) + Z Ω v + Z Ω w + Z Ω vw , (4.44) for every t > t0, with κ3, κ4 > 0. From (4.20) and the right-hand side of (4.44) with σ2 t0. As a consequence, E2(t) ≤C3e-σ2 κ4 t, for all t > t0, where C3 > 0. Now, by examining the left-hand side of (4.44), we observe that f2(t) ≤1 κ3 E2(t) ≤C3 κ3 e-σ2 κ4 t. Using an argument similar to the proof of Theorem 1.2, we arrive at u(·, t) -1 L∞(Ω) + v(·, t) L∞(Ω) + w(·, t) L∞(Ω) + z(·, t) L∞(Ω) ≤C4e-σ2 κ4 t, 32 for all t > t0, with C4 > 0. This completes the proof. Proof of Theorem 1.4. (1) We select t0 > 0. Following the proof of Theorem 1.2 and using the Taylor expansion, we find that Z Ω u - ̄u - ̄u ln u ̄u = Z Ω (H(u) -H( ̄u)) ≤1 2 ̄u Z Ω (u - ̄u)2, ∀t > t0. From the above estimate, we arrive at 1 4 ̄u Z Ω (u - ̄u)2 ≤ Z Ω u - ̄u - ̄u ln u ̄u ≤3 4 ̄u Z Ω (u - ̄u)2, ∀t > t0. (4.45) Similarly, we obtain 1 4 ̄v Z Ω (v - ̄v)2 ≤ Z Ω v - ̄v - ̄v ln v ̄v ≤3 4 ̄v Z Ω (v - ̄v)2, ∀t > t0 (4.46) and 1 2 Z Ω w2 + 1 2 Z Ω w ≤ Z Ω w ≤2 Z Ω w2 + 2 Z Ω w, ∀t > t0. (4.47) Taking into account the inequalities (4.45)-(4.46), it is clear that for all t > t0 and with κ1, κ2 > 0, we have κ5f3(t) ≤E3(t) ≤κ6 f3(t) + Z Ω w + Z Ω vw . (4.48) From (4.29) and the right-hand side of (4.48) with σ3 t0. Solving the above differential inequality gives E3(t) ≤C1e-σ3 κ6 t, t > t0. where C1 > 0. Now, the left-hand side of (4.48) gives f3(t) ≤1 κ5 E3(t) ≤C1 κ5 e-σ3 κ6 t, t > t0. In view of (4.10) and then Lemma 4.5, finally we arrive at u(·, t) - ̄u L∞(Ω) + v(·, t) - ̄v L∞(Ω) + w(·, t) L∞(Ω) + z(·, t) - ̄z L∞(Ω) ≤C2e-σ3 κ6 t, for all t > t0, with C2 > 0. This completes the proof. Proof of Theorem 1.4. (2) Let us choose t0 > 0, we can apply a similar approach to the one used to derive (4.45) to obtain the following result 1 4ˆu Z Ω (u -ˆu)2 ≤ Z Ω u -ˆu -ˆu ln u ˆu ≤3 4ˆu Z Ω (u -ˆu)2, ∀t > t0. (4.49) 33 Following the same procedure, we obtain 1 4 ˆw Z Ω (w -ˆw)2 ≤ Z Ω w -ˆw -ˆw ln w ˆw ≤3 4 ˆw Z Ω (w -ˆw)2, ∀t > t0 (4.50) and 1 2 Z Ω v2 + 1 2 Z Ω v ≤ Z Ω v ≤2 Z Ω v2 + 2 Z Ω v, ∀t > t0. (4.51) Considering the above inequalities (4.49)-(4.51), it is apparent that κ7f4(t) ≤E4(t) ≤κ8 f4(t) + Z Ω v + Z Ω vw , t > t0 (4.52) with κ7, κ8 > 0. From (4.36) and right-hand side of (4.52) with σ4 t0. Consequently, E4(t) ≤C3e-σ4 κ8 t, for every t > t0, where C3 > 0. Now, by examining the left-hand side of (4.52), we observe that f4(t) ≤1 κ7 E4(t) ≤C3 κ7 e-σ4 κ8 t. By using a similar argument as in the proof of Theorem 1.4, we arrive at u(·, t) -ˆu L∞(Ω) + v(·, t) L∞(Ω) + w(·, t) -ˆw L∞(Ω) + z(·, t) -ˆz L∞(Ω) ≤C4e-σ4 κ8 t, for all t > t0, with C4 > 0. This completes the proof. 5. Numerical examples This section presents the results of numerical experiments conducted using the Finite Difference Method to investigate the behaviour of solutions to the predator-prey chemo-alarm-taxis system in both two- and three-dimensional spaces. We consider the initial-boundary value problem for the predator-prey chemo-alarm-taxis system (1.1), with radially symmetric bell-shaped initial condition u(·, 0) = e-0.1 Pn i=1 x2 i , v(·, 0) = 3e-0.3 Pn i=1 x2 i , w(·, 0) = 2e-0.2 Pn i=1 x2 i , z(·, 0) = e-0.1 Pn i=1 x2 i . In all of the examples presented below, we set di = 1, i = 1, 2, 3, 4, χ1 = χ2 = ξ = μ1 = μ2 = μ3 = α = β = 1, γ = 2 and adopt a uniform grid ∆xi = 1 101, i = 1, 2 for 2D and ∆xi = 1 50, i = 1, 2, 3 for 3D within a domain [-0.5, 0.5]n, where n = 2, 3. Example 5.1 (Coexistence state of the species). Let ai = 0.5, i = 1, 2, 3, 4. Therefore a2a6 < min 1 + a1 + a2 + a1a4 + a4a6, 1 + a1(a3 + a4a5) + a4a6 + a2a5 a3 , a3(1 + a2) + a4a5 < 1 + a4 + a2a5 and a5(1 + a1) + a6 < 1 + a3(a1 + a6), then the solution (u, v, w, z) converges to the coexistence steady state (ue, ve, we, ze) as t →∞. 34 (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 5 10 15 20 25 30 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 || || Secondary Predator Primary Predator Prey (e) Asymptotic behaviour Figure 1: Populations at t = 30 in 2D. In the case, the solution (u, v, w, z) of system (1.1) converges to the coexistence steady state (1.286877, 0.428973, 0.144368, 0.286793) as t →30, as illustrated in Figure 1. (a) Secondary Predator u(x, y, t) (b) Primary Predator v(x, y, t) (c) Prey w(x, y, t) (d) Chemical z(x, y, t) 0 2 4 6 8 10 12 14 16 18 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 || || Secondary Predator Primary Predator Prey (e) Asymptotic behaviour Figure 2: Populations at t = 20 in 3D. Figure 2 depicts the solution (u, v, w, z) of system (1.1) approaches the coexistence steady state (1.286886, 0.428981, 0.144365, 0.286798) as t →20. Example 5.2 (Secondary predator only existence state). Let a1 = a4 = 0.01, a2 = 1, a3 = 1.5, a5 = a6 = 2, then the conditions given in (1.3) are not satisfied. Therefore, the solution (u, v, w, z) converges to the 35 trivial steady state (1, 0, 0, 0) as t →∞. (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 2 4 6 8 10 12 14 16 18 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 || || Secondary Predator Primary Predator Prey (e) Asymptotic behaviour Figure 3: Populations at t = 20 in 2D. Here we observe that the solution (u, v, w, z) of the system (1.1) converges to the trivial steady state (1, 0, 0, 0) as t →20, as shown in Figure 3. (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 2 4 6 8 10 12 14 16 18 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 || || Secondary predator Primary predator Prey (e) Asymptotic behaviour Figure 4: Populations at t = 20 in 3D. Figure 4 depicts the solution (u, v, w, z) of system (1.1) approaches the trivial steady state (1, 0, 0, 0) as t →20. Example 5.3 (Prey vanishing state). Let a1 = a3 = 0.01, a2 = 1, a4 = 3, a5 = a6 = 2, then conditions given in (1.3) are not satisfied and a3 < 1. Therefore, the solution (u, v, w, z) converges to the semicoexistence (prey vanishing) steady state ( ̄u, ̄v, ̄w, ̄z) as t →∞. 36 (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 5 10 15 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 || || Secondary predator Primary predator Prey (e) Asymptotic behaviour Figure 5: Populations at t = 20 in 2D. In the case, the solution (u, v, w, z) of system (1.1) converges to the prey vanishing state (1.009899, 0.989901, 0, 0.494951) as t →20, as illustrated in Figure 5. (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 2 4 6 8 10 12 14 16 18 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 || || Secondary predator Primary predator Prey (e) Asymptotic behaviour Figure 6: Populations at t = 20 in 3D. Figure 6 depicts the solution (u, v, w, z) of system (1.1) approaches the prey vanishing state (1.009899, 0.989902, 0, 0.494951) as t →20. Example 5.4 (Primary predator vanishing state). Let a1 = a4 = 0.01, a2 = 2, a3 = 1.5, a5 = 0.5, a6 = 2, then the conditions given in (1.3) are not satisfied and a5 < 1. Therefore, the solution (u, v, w, z) converges to the semi-coexistence (primary predator vanishing) steady state (ˆu, ˆv, ˆw, ˆz) as t →∞. 37 (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 2 4 6 8 10 12 14 16 18 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 || || Secondary predator Primary predator Prey (e) Asymptotic behaviour Figure 7: Populations at t = 20 in 2D. Here we observe that the solution (u, v, w, z) of the system (1.1) converges to the primary predator vanishing state (1.499994, 0, 0.249999, 0.124999) as t →20, as shown in Figure 7. (a) Secondary Predator (b) Primary Predator (c) Prey (d) Chemical 0 2 4 6 8 10 12 14 16 18 20 time -1 -0.5 0 0.5 1 1.5 2 2.5 3 || || Secondary predator Primary predator Prey (e) Asymptotic behaviour Figure 8: Populations at t = 20 in 3D. Figure 8 illustrates the convergence of the solution (u, v, w, z), where the solution components approaches (1.499722, 0, 0.249928, 0.124943) as t →20. Remark 5.1. When the conditions specified in Example 5.1 are sufficiently strong, the solution rapidly converges to the coexistence steady state. Conversely, when the conditions in Example 5.2 are sufficiently strong, the solution quickly converges to the trivial steady state (1, 0, 0, 0). 38 6. Conclusion We have investigated the global existence of a classical solution to system (1.1), which remains uniformly bounded in the L∞(Ω) norm for dimensions n ≥1, under suitable parameter conditions. Furthermore, we established global asymptotic stability by constructing an appropriate Lyapunov functional and verified these results through numerical simulations. Our analysis yielded several important stability outcomes with meaningful biological implications. In particular, the findings underscore the decisive role of the chemotaxis coefficients in governing the existence and stability of the predator-prey alarm-taxis system. Acknowledgments • GS and JS thank to Anusandhan National Research Foundation (ANRF), formerly Science and Engineering Research Board (SERB), Govt. of India for their support through Core Research Grant (CRG/2023/001483) during this work. • RDF is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and, acknowledges financial support by PNRR e.INS Ecosystem of Innovation for Next Generation Sardinia, CUP F53C22000430001 (codice MUR ECS0000038). He is also partially supported by the research project "Partial Differential Equations and their role in understanding natural phenomena", CUP F23C25000080007, funded by Fondazione Banco di Sardegna annuality (2023). References [1] Bai. 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Math. 148, 157-177 (2017) 41
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2510.14722
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Anomaly of Continuous Symmetries from Topological Defect Network
Qiang Jia1, Ran Luo2, Jiahua Tian3, Yi-Nan Wang2,4,5, and Yi Zhang6
1Department of Physics, Korea Advanced Institute of Science & Technology, Daejeon 34141, Korea
2School of Physics, Peking University
3School of Physics and Electronic Science, East China Normal University, Shanghai, China, 200241
4Center for High Energy Physics, Peking University
5Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China and
6Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
(Dated: October 17, 2025)
We show that the ’t Hooft anomaly of a quantum field theory with continuous flavor symmetry can
be detected from rearrangements of the topological defect webs implementing the global symmetry
in general spacetime dimension, which is concretized in 2D by the F-moves of the defect lines.
Via dualizing the defects to flat background gauge field configurations, we characterize the ’t Hooft
anomaly by various cohomological data of the symmetry group, where the cohomology of Lie groups
with discrete topology plays the central role. We find that an extra dimension emerges naturally as
a consequence of the mathematical description of the ’t Hooft anomaly in the case of flat gauging.
Introduction. A global symmetry of a Quantum Field
Theory (QFT) is said to have a ’t Hooft anomaly if gaug-
ing it leads to an inconsistency, i.e. an obstruction to
promoting it to a gauge symmetry [1]. The robustness
of ’t Hooft anomaly across ultraviolet to infrared makes
it a particularly effective tool to study the properties of
QFT even in the strongly coupled region [2, 3].
The
’t Hooft anomaly manifests itself as a nontrivial phase
factor e2πiA(A,g) acquired by Z[A] under gauge transfor-
mation A →g−1Ag +g−1dg, where Z[A] is the paritition
function of the QFT coupled to background field A and
for A(A, g) that cannot be cancelled by any local coun-
terterm built out of A. Traditionally for continuous G,
it can be evaluated using the descent equations for small
gauge transformations [4–8] and more generally shown to
be the the eta-invariant when g is disconnected from the
identity [9–12].
Recent years have witnessed a revival of the study of
’t Hooft anomalies in both condensed matter and high-
energy physics [13–17]. Thanks to the reformulation of
the symmetry action as linking symmetry operator and
charged defect in spacetime [18–23], the action of finite
group symmetries is now conveniently studied on equal
footing with that of continuous symmetries.
However,
whereas the ’t Hooft anomaly of discrete group symme-
try can be computed directly as the F-symbol associated
with rearrangements of webs of defects, which are of-
ten described by elements of the group cohomology (e.g.
Hd+1(G, U(1)) for a 0-form symmetry group G in d-
spacetime dimensions)[24–27], an analogous algorithm to
compute the ’t Hooft anomaly for continuous group sym-
metries in flat gauging has long remained elusive.
In this work, we present such an algorithm for con-
tinuous symmetries, thereby filling the missing corner to
place the treatment of finite and continuous symmetries
on a completely parallel footing. We first show that a web
of defects can be dualized to be a flat background field
configuration. This effectively establishes the equivalence
of insertion of defects and coupling the conserved current
to flat background fields. Via the dualization, we show
in full generality how the ’t Hooft anomaly manifests it-
self as the phase factor arising from rearrangements of
the topological defects in the manner hinted in [21]. We
then compute the anomaly concretely via F-moves of the
topological defect lines in 2D QFT [16, 17, 28–30] and
show that the anomaly is characterized by cohomological
data of G.
Very importantly, unlike the case of a finite group
G
where
the
group
cohomology
H3(G, U(1))
∼=
H3(BG, U(1)) is unambiguously defined, one should be
careful about the exact definition for Lie group coho-
mology. We emphasize that for a Lie group G the cor-
rect version should be H3(Gδ, U(1)), which is defined to
be the group cohomology when G is equipped with dis-
crete topology [31]. Despite that H3(Gδ, U(1)) is a highly
complicated mathematical object, there exists an injec-
tion map from the group H4(BG, Z), where the anomaly
polynomial for G lives, to H3(Gδ, U(1)). Thus the usual
anomaly polynomial ∼F ∧F can be elegantly embedded
in this unified framework.
We find that the unifying mathematical structure un-
derlying these various cohomology groups of G suggests
that the QFT with a ’t Hooft anomaly naturally lives on
the boundary of an anomaly theory in one higher dimen-
sion. The generalization of the above cohomological char-
acterization of ’t Hooft anomaly to higher-dimensional
spacetime is immediate in our formalism.
For detailed examples, we discuss the cases of a simple
G and U(1) separately. For a compact simply connected
and simple Lie group G, the 3d bulk anomaly theory is
the non-abelian Chern–Simons invariant characterized by
an integer k ∈Z called the level. It has a local expression
as the Chern–Simons 3-form I3 = CSk[A] = k
2 tr(A∧F −
1
3A ∧A ∧A). We explicitly uplift the topological defect
networks across the F-move to a three-dimensional bulk,
and show that after assigning F = 0, the cubic term
in CSk[A] precisely generates the de Rham cohomology
group H3
dR(G, R) for G.
For a non-simply connected Lie group such as G =
U(1), we point out that the field strength F = dA would
arXiv:2510.14722v1 [hep-th] 16 Oct 2025
2
not vanish at the junction point for this case. This is
not inconsistent with the requirement of flat gauging, as
the precise definition of flat gauging should be the trivi-
ality of the holonomy exp
H
C A
around a topologically
trivial loop C. Such a non-vanishing F is precisely the
source of the non-trivial U(1) phase factor across the F-
move, which we compute explicitly from the integration
of ϵF over the 2D spacetime, where ϵ is the finite gauge
transformation parameter generating the F-move. The
factor
ω(eiα1, eiα2, eiα3) = eiα1(α2+α3−[α2+α3])/(2π)
(1)
with αi ∈[0, 2π) and [r] denoting the part of r exceeding
2π, is consistent with the N →∞limit of H3(ZN, U(1)).
We would also like to comment that (1) is perfectly
consistent with the newly proposed categorical formu-
lation of continuous symmetry [32]. Namely, if one pre-
tends that the definition of the symmetry category Vecω
G
and Drinfeld center Z(Vecω
G) can be generalized to the
case of G = U(1), all the category theoretical data are
consistent with the mathematical results in [33], which
are confirmed in [32].
Our results can be applied to all compact connected Lie
groups. From the structure theorem [34], compact con-
nected Lie groups are finitely covered by direct products
of tori and simple non-abelian groups, such that their
classification reduces to that of simple groups. The re-
striction to a compact connected and simple Lie group G
or U(1) is sufficient in this context.
Finally, although the detailed examples are discussed
for the d = 2 case, our framework applies to continuous
0-form symmetries in general d spacetime dimensions as
well.
Dualization of the defect web.
The VEV of a topo-
logical defect web W of a (Lagrangian) QFT TG in d-
dimensional spacetime Md can be computed as:
⟨W⟩:=
Z
Dψ eiS[ψ]+
R
Md tr(AW∧⋆J) ,
(2)
where ψ stands for the set of local operators of TG and
J(ψ) is the conserved current of the continuous global
G-symmetry. AW standards for the ordinary flat gauge
field corresponding to the defect web W. To illustrate
the dualization, we consider a defect line L labeled by
gα = eiα ∈G along x = 0 in R2, meaning that a particle
charged under the global G-symmetry is transformed un-
der a representation of gα when crossing x = 0. We define
the dualization AL of L to be the background gauge field
such that Pe
R
ℓAL = eiα for an arbitrary path ℓcrossing
x = 0 once, which in turn means that AL must be flat.
Given the condition, we can write:
AL = iαδ(x)dx = e−iαH(x)deiαH(x)
(3)
where H(x) is the Heaviside function, and we can indeed
check that dAL + AL ∧AL = 0. Equivalently, L defines
the gauge configuration AL|Ui = g−1
i
dgi for g0 = e and
g1 = eiα on the following open cover of R2:
x = 0
L
U0
U1
AL
(4)
where the two patches U0 = (−∞, ϵ) and U1 = (−ϵ, ∞)
are glued along (−ϵ, ϵ) for small positive ϵ. Here we adopt
the convention that the direction of the defect line rotates
right-handedly with respect to the direction of the gauge
transformation from U0 to U1. Clearly, the above discus-
sions can be generalized to a codimension-one web of W
in general d dimensions.
The fundamental building block of any defect web (of
lines) is the trivalent junction (the background gauge
field AL corresponding to the group element eiα is also
labeled, with α):
eiβ
eiα
eif(α,β)
(0, 0)
β
f(α, β)
α
(5)
This trivalent junction can be smeared out to give A =
g−1dg with F = dA + A ∧A = 0 for a globally defined
g on R2 for G with trivial π1(G), since all paths from
identity to eiαeiβ on the group manifold are homotopic.
However, a globally defined g cannot be obtained by any
kind of smearing for G with non-trivial π1(G), e.g. for
G = U(1), and there exist flat configurations of U(1)
gauge fields with non-trivial F = dA. This subtlety is
crucial for the derivation of anomalies for abelian groups.
Now we clarify the notion of flat gauging of continuous
G global symmetry, given by the following two equivalent
conditions:
1. A flat gauging is equivalent to summing over all
possible insertions of topological defect networks in
the new partition function.
2. Summing over configurations of gauge fields A,
where the holonomy of the gauge field around a
topologically trivial loop P exp(
H
C A) = e, the iden-
tity element of G.
’t Hooft anomaly from F-move and the Lie group co-
homology. Having defined the dualization for continuous
G, one can compare ⟨W⟩with ⟨W′⟩by coupling TG to
flat AW- and AW′-backgrounds, respectively.
Suppose
AW′ = e−iΛAWeiΛ + e−iΛdeiΛ for gauge transformation
ψ →eiΛψ parametrized by Λ, we have:
⟨W′⟩=
Z
Dψ eiS[ψ]+
R
Md tr(AW′∧⋆J) = e2πiA[AW;Λ]⟨W⟩
(6)
where A[AW; Λ] ̸= 0 is the quantum anomaly. Therefore,
given our dualization, the computation of anomaly arises
3
from rearrangements of the defect webs amounts to find-
ing the gauge transformation interpolating the dual flat
field configurations.
As a reminder, the ’t Hooft anomaly of Zm-symmetry
of a 2D bosonic theory is well-known to be charac-
terized by H3(Zm, U(1)), the 3rd group cohomology
of Zm with U(1)-coefficient, in which the phase factor
e2πia1(a2+a3−a2+a3)/(m) of the F-move of the defect webs
in Figure 1 lives [28]. In d-dimension the characterization
e2πi a1
m e2πi a2
m e2πi a3
m
=
e2πi a1(a2+a3−a2+a3)
m
×
e2πi a1
m e2πi a2
m e2πi a3
m
FIG. 1. The F-move of defect web for G = Zm 0-form sym-
metry group in 2D. ai ∈{0, 1, . . . , m −1} and m is defined as
m mod Z.
is given by Hd+1(G, U(1)) for discrete G [16, 26]. A sim-
ilar characterization for continuous G is still missing. We
will focus on continous G of interests, and compute their
anomalies in flat gauging, i.e. from coupling TG to flat
background field that is the closest cousin of the discrete
case where all background field is automatically flat.
We show that the ’t Hooft anomaly is encoded in the
phase factor in ⟨
⟩= e2πiA[A
;Λ]⟨
⟩(cf. (6)) for
the defect webs
and
living on M2 in TG with
Lie group G. Here Λ is the gauge transformation from
A
to A .
In the modern langauge [12], A[A ; Λ] can be calcu-
lated as (cf. Eq. (E9) in Appendix E):
A(A ; Λ) =
Z
M3
CSk[A(t)] ,
(7)
using the level-k Chern–Simons invariant CSk where
t
∈
[0, 1] parametrizing a mapping cylinder in the
space of gauge field configurations along which A(t) :=
g(t)−1A g(t) + g(t)−1dg(t) for g(x, y, t = 0) = 1 ∈G
and g(x, y, t = 1) = eiΛ(x,y) interpolates from A
to A .
We use the normalization CSk[A] := k
2 tr(A ∧dA + 2
3A ∧
A ∧A) = k
2 tr(A ∧F −1
3A ∧A ∧A) on a local patch of
M3. For simplicity, we restrict to theories whose Chern–
Simons invariants used in anomaly computations origi-
nate from integral classes in H4(BG, Z) (cf. Eq. (C11)
in Appendix C). This condition constrains the allowed
minimal level k, which depends on G. We do not dis-
cuss these cases individually and omit the subscript k for
brevity [35].
The interpolation from A
to A
is illustrated on the
leftmost graph in Figure 2. It is not hard to see that by
our convention, the configuration in the front is equiv-
alent to
and the one in the back is equivalent to
. Since A[A ; Λ] is clearly independent from small
deformations of the mapping cylinder, i.e. the details of
g, as long as boundary configurations A
and A
are
fixed, we further glue the vertices of
with the ones
with the same labels in
at the two copies M2 × {0}
and M2 × {1} to arrive at the configuration in the mid-
dle of Figure 2. This is the difference between the cor-
responding webs of defects and the middle configura-
tion can be further deformed to be a decorated tetra-
hedron as in the rightmost of Figure 2, where the direc-
tions of the defects and of the gauge transformations are
set by our convention (4). This process can be under-
stood as the anomaly-inflow—very similar to the finite
group case [16]—i.e. an extra 3-simplex can be attached
in the bulk to cancel the anomaly and the decoration
of the tetrahedron in the bulk describes the pullback
of the 3-simplex ⟨e, g1, (g1g2), (g1g2g3)⟩in BGδ to the
bulk. Here Gδ is the Lie group G with discrete topology
and BGδ is the Eilenberg–MacLane space K(G, 1) clas-
sifying flat G-bundles, we denote the classifying map by
φδ : M3 →BGδ [36].
Similar to the finite group case, in terms of the
classifying space, the anomaly is obtained by evalu-
ating a cohomology class [Ω]
∈
H3(BGδ, R/Z) on
⟨e, g1, (g1g2), (g1g2g3)⟩.
Here we treat the coefficient
U(1) = R/Z as an additive group. Pullback with Ωwith
φδ and integrate over the 3D spacetime, we obtain the
Dijkgraaf-Witten phase
A(A ; Λ) =
Z
M3
(φδ)∗(Ω) ,
(8)
in other words, we claim that Ωfound by F-move pulls
back to the Chern–Simons invariant CS. See the proof
of (8) in Appendix C.
By investigating the local expression of Chern–Simons
invariant, we can get some other manifestations of
anomalies. For this discussion, as G is simply connected
we can assign F = 0 for flat gauge field. In the M3 bulk,
our gauge field A(t) is flat. On a local patch Ui, we have
Ai := A|Ui = ˜g−1
i
d˜gi for some ˜gi : Ui →G and
Z
Ui
−k
6 tr(Ai ∧Ai ∧Ai) =
Z
Ui
k ˜g∗
i ω3 =
Z
˜gi(Ui)
k ω3 (9)
where ˜g∗
i is the pull-back from Ω∗(G) to Ω∗(Ui) and
˜g∗
i ω3 = i2
3! tr(˜g−1
i
d˜gi)3 with ω3 being a left-invariant 3-
4
𝑔!
𝑔!𝑔"
𝑣#
𝑔$
𝑔"𝑔$
𝑣"
𝑣$
𝑣!
𝑔"
𝑔!𝑔"𝑔$
𝑔!
𝑔"
𝑔$
𝑔!𝑔"𝑔$
𝑣!
𝑣#
𝑣"
𝑣$
Deform
𝑔!
𝑔"
𝑔!
𝑔$
𝑔!𝑔"𝑔$
𝑔$
𝑔"
Deform
FIG. 2. The deformations of the combined
and
configuration. The group multiplications in this figure are all taken
from the right.
form on G. For a compact, connected, simply connected
and simple G, ω3 generates the de Rham cohomology
H3
dR(G, R) ∼= R, it is also the image of an integral class
in the singular cohomology H3
singular(G, Z) ∼= Z [37]. Fur-
thermore, such G-bundle on any 3-manifold is necessarily
trivial. Hence ˜gi = ˜g is globally defined, and the anomaly
can be calculated as:
A(A ; Λ) =
Z
˜g(M3)
k ω3 .
(10)
The right-hand side of (10) now has another interpre-
tation, namely, the action of ungauged 2d Wess–Zumino–
Witten model. Indeed, this is a fact of anomaly match-
ing [2, 7, 38].
The appearance of the left invariant 3-
form ω3 as a suspension (C16) (the inverse is referred to
as transgression [29, 34]) of CS on the fiber G via the
bundle fibration G →P →M3 was discussed originally
in [39], see also discussions [29, 40–42] related to anoma-
lies.
On the other hand, we take ⟨X, Y ⟩= tr(XY ) to be
the Killing form of the Lie algebra g of G then
−k
6 tr(A ∧A ∧A) = −k
12⟨A, [A, A]⟩,
(11)
where we recognize that ⟨·, [·, ·]⟩is the generator of the
3rd Lie algebra cohomology H3(g, R) of the flavor algebra
g. We see that the group H3(g, R) also classifies the local
density of anomaly in flat gauging.
Since our G is taken to be compact connected and
simple, we always have H3(g, R) ∼= H3
dR(G, R) ∼= R. As
the 3-forms have integral periods, they are the image
of the lattice Z ∼= H3
dR(G, Z) inside H3
dR(G, R).
The
coefficient exchanging map is in fact injective [43]
H3
singular(G, Z) ∼= H3
dR(G, Z) ,→H3
dR(G, R) ∼= H3(g, R) .
(12)
Recall that the suspension map (C16)
τ : H4(BG, Z) −→H3
singular(G, Z)
(13)
is also an isomorphism if G is in addition simply con-
nected. For these groups, we have
H3
singular(G, Z) ∼= H4(BG, Z)
κδ
,−→H3(BGδ, R/Z) , (14)
where the last map (C9) is also an injection [43].
The emergence of extra dimension.
Moreover, there
is a homomorphism w : H3(g, R) →H3(BG, R) where
BG is the classifying space of flat G-trivial bundles [44].
One particularly interesting feature about BG is that a
trivialization can be explicitly written down as an ele-
ment of G = {(g, ℓ) ∈Gδ × GI|ℓ(0) = g, ℓ(1) = e} where
Gδ is G equipped with discrete topology together with
the canonical map ι : Gδ →G and GI := Map(I, G) for
I := [0, 1].
More precisely, we consider a web of defects on a gen-
eral 2D spacetime manifold M2 which, by Poincar´e dual-
ity, defines a flat G-bundle over M2 determined up to iso-
morphism by the homotopy class of the classifying map
f δ : M2 →BGδ and a trivial bundle given by a null-
homotopic map f n : M2 →BG.
Using the lift of ι,
Bι : BGδ →BG, the map f : (x, t) 7→(f δ(x), γx(t)) ∈
BGδ × BG where γx(0) = Bι(f δ(x)) defines a homo-
topy from f δ to f n by the mapping path space construc-
tion [45]. Equivalently, f is a map from M2 × [0, 1] to
BG ⊂BGδ × BG. Therefore, we have:
Z
f(M2×I)
w(ω3) =
Z
M2×I
f
∗◦w(ω3)
(15)
where f
∗is the pull-back of f.
U0
U2
A
A
I
U0
U1
U1
U2
M2
FIG. 3. The showing of open cover Uk (k = 0, 1, 2) of M2 as
well as the open cover U k (k = 0, 1, 2) of M2 × I.
To investigate the abstract map f
∗◦w at a more ele-
mentary level, we fix π1(M2) = Z for simplicity (e.g. let
M2 ∼= S1×R). We use the open cover Uk = ( 2kπ
3 , (2k+4)π
3
)
5
(modulo 2π) of S1, k = 0, 1, 2, and define the flat con-
nection A = iα
2πdθ (α ∈ig) on a trivial bundle with the
transition function gij ≡1 on Uij := Ui ∩Uj from Ui to
Uj. We plot the spacetime geometry and patches in Fig-
ure 3. Under gauge transformation uk = e−iα
2π (θk+ 2kπ
3 )
on each Uk parametrized by θk ∈(0, 4
3π), Ak vanishes
because u−1
k Akuk + u−1
k duk = 0. Meanwhile, the transi-
tion function on Ui,i+1 becomes g′
i,i+1 = u−1
i gi,i+1ui+1 =
u−1
i ui+1. Since θi(p) = θi+1(p) + 2π
3 , ∀p ∈Ui,i+1 for i
mod 3, we have:
g′
01(U01) = g′
12(U12) = 1, g′
20(U20) = eiα .
(16)
Hence, while uk trivializes A, it yields non-trivial tran-
sition functions parametrized by α.
To uplift A to
M2 × I, we define uk = e−iα
2π (θk+ 2kπ
3 )f(t) with f(0) = 1
and f(1) = 0 for t ∈I as a function from U k :=
Uk × R × I to G. We further consider the gauge field
Ak = u−1
k Akuk + u−1
k duk on U k and the transition func-
tion gij = u−1
i uj on U ij.
The pair (gij, Ai), defined
on (an open cover of ) M2 × I, leads to a trivialization
of a bundle with non-trivial transition function (16) over
M2×{0} to a trivial bundle with non-zero flat connection
over M2 × {1}. In this example, g20 = eiαf(t) provides a
concrete physical realization of an element (eiα, ℓ) of G.
Since Ak = g−1
k dgk with gk = u−1
k uk is flat, we can
replace (A, U) in (9) by (Ak, U k).
Together with (15)
restricted to U k and a suitable choice of normalization,
we are led to [46]:
f
∗◦w(ω3)|U k = g∗(ω3)|U k = CS[A]|U k .
(17)
Physically, this means that the data of ’t Hooft anomaly
can be captured by the Chern-Simons action of a flat
gauge field on M2 × I with non-trivial transition func-
tions at M2 × {0} which trivializes at M2 × {1} at the
price of yielding non-zero flat connection. Moreover, the
characterization of ’t Hooft anomaly by H3(g, R) nat-
urally requires the emergence of an extra dimension of
the anomaly theory, manifests itself as I, because of the
structure of f
∗◦w.
U(1) flavor symmetry and its ’t Hooft anomaly.
A
mathematically cautious reader might have already no-
ticed that the previous calculation leading to H3(G, R)
fails for U(1), since both H3(U(1), R) and H3(u(1), R)
vanish. Moreover, the anomaly polynomial for U(1) that
generates H4(BU(1), Z) ∼= Z is CS[A] = A ∧dA, where
the cubic term
1
3A ∧A ∧A is absent.
To get a non-
zero anomaly, we comment that there is a non-zero field
strength at the junction point (x, y) = (0, 0) in (5).
For the case of G = U(1), we have f(α, β) = (α +
β) mod 2πZ ≡[α + β]. The flat gauge field dual to the
trivalent junction is then
A = iαδ(x)H(y)dx + i(βH(−x) + [α + β]H(x))δ(y)dy ,
(18)
and the corresponding field strength
F = dA = i([α + β] −α −β)δ(x)δ(y)dx ∧dy
(19)
is non-vanishing at the junction point.
Now to compute the ’t Hooft anomaly of U(1) flavor
symmetry, we apply the previous computation to the con-
figuration in Figure 4.
[α3]
[α2]
[[α1 + α2] + α3]
[α1]
y = 0
x = 0
A0
+
[α1]
[α1]
[α1]
x−
x+
Gauge variation
=
[α3]
[α2]
[α1 + [α2 + α3]]
[α1]
y = 0
x = 0
A
FIG. 4. The F-move of defect lines in a theory with U(1) flavor symmetry. A topological line carrying eiα ∈U(1) is labeled by
[α] := α mod 2πZ, with an arrow labeling the direction of gauge transformation in the same sense as in (5).
It is easy to see that the left-most and the right-most
configurations in Figure 4 are topologically equivalent to
and
, respectively. Hence our task is to find
their Poincar´e duals A
and A , and the gauge transfor-
mation parameter dϵ ≡idΛ = A
−A . Actually, using
the configuration in the middle of Figure 4 and assum-
ing α ≪1 hence [α1] = α1, one can show that (see the
details in Appendix A):
ϵ = −iα1(H(x−) −H(x+))H(y)
(20)
The infinitesimality of α1, hence of ϵ, enables us to com-
6
pute the ’t Hooft anomaly to be
A[A ; ϵ] =
1
(2π)2
Z
R2 ϵdA
=
1
(2π)2 α1(α2+α3−[α2+α3]) .
(21)
Viewing
f(eiα1, eiα2, eiα3) := e2πiA[A
;ϵ]
= eiα1(α2+α3−[α2+α3])/(2π)
(22)
as a map f : G3 →U(1), one can check that δf = 0
where δ is the coboundary operator for the Lie group
cohomology of U(1)δ (see an explicit computation in the
appendix A).
The map (22) is not continuous, which aligns with
the usage of group cohomology H3(U(1)δ, U(1)) with
discrete topology.
Comparing to the phase factor in
Figure 1, one can see that the (22) is exactly the
m →∞limit of the phase factor for Zm generating
H3(Zm, U(1)).
For the details of the computation of
the whole H3(U(1)δ, U(1)) ⊃H4(BU(1), Z), see Ap-
pendix D.
Higher dimensional generalizations. We have explic-
itly derived the connection between the F-move of topo-
logical defects and the anomaly polynomial for the flat
gauging of the continuous symmetry group G, for the
d = 2 case. Note that if we take the dual graph of the
right picture in Figure 2, we obtain a 3-simplex v0v1v2v3
whose vertices are vi (i = 0, . . . , 3) while the edges corre-
spond to elements of G. Such picture can be easily gen-
eralized to d = 2k spacetime dimensions, where the dual
3-simplex v0v1v2v3 is replaced by a dual (2k+1)−simplex
v0v1 . . . v2k. Taking the dual of v0v1 . . . v2k, one obtains
the higher-dimensional generalization of F-move, as in
Dijkgraaf-Witten.
The Lie algebra cohomology H3(g, R) and group co-
homology H3(U(1)δ, U(1)) should be straightforwardly
generalized to H2k+1(g, R) and H2k+1(U(1)δ, U(1)).
We present the explicit generators for H2k+1(g, R) for
semisimple Lie algebra g, H2k+1((U(1)l)δ, U(1)) for a
product of U(1) 0-form symmetry groups, and their cor-
respondence with anomaly polynomials, in Appendix B.
ACKNOWLEDGMENTS
We would like to thank Jin Chen, Yuji Tachikawa, Xin
Wang, Zhaolong Wang and Yunqin Zheng for useful dis-
cussions. YZ would like to thanks Mikhail Kapranov for
helpful explanations about the cohomology of Lie groups
made discrete. JT and YNW would like to thank Peng
Huanwu Center for Fundamental Theory and Northwest
University for hosting the 6th National Workshop on
Fields and Strings where many progress of this project
has been made. RL and YNW are supported by National
Natural Science Foundation of China under Grant No.
12175004, No. 12422503 and by Young Elite Scientists
Sponsorship Program by CAST (2024QNRC001).
QJ
is supported by National Research Foundation of Korea
(NRF) Grant No. RS-2024-00405629 and Jang Young-Sil
Fellow Program at the Korea Advanced Institute of Sci-
ence and Technology. YZ is supported by WPI Initiative,
MEXT, Japan at Kavli IPMU, the University of Tokyo
and by National Science Foundation of China (NSFC)
under Grant No. 12305077.
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require the orientability of the spacetime manifold and
its extension throughout.
[36] See more details on BGδ in Appendix C. For a discus-
sion of the simplicial complex construction of BG for
finite G, the reader may consult section 9.1 in the lec-
ture note https://member.ipmu.jp/yuji.tachikawa/
lectures/2024-mathphys/notes.pdf.
[37] We use the convention ω2n−1 := in (n−1)!
(2n−1)! tr(˜g−1d˜g)2n−1,
such that it is normalized to integrate to 1 on the gener-
ator of π2n−1(G), and for n = 2, the generator of π3(G)
is sent to the generator of H3(G, Z) [42].
[38] S. P. Novikov, The Hamiltonian formalism and a many
valued analog of Morse theory, Usp. Mat. Nauk 37N5,
3 (1982).
[39] S.-S. Chern and J. Simons, Characteristic forms and ge-
ometric invariants, Annals Math. 99, 48 (1974).
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coset models, Commun. Math. Phys. 144, 189 (1992).
[41] D. S. Freed, Pions and Generalized Cohomology, J. Diff.
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[42] Y. Lee, K. Ohmori, and Y. Tachikawa, Revisiting Wess-
Zumino-Witten terms, SciPost Phys. 10, 061 (2021),
arXiv:2009.00033 [hep-th].
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V: 2-Groups, arXiv (2003), arXiv:math/0307200.
[44] S. Morita and A. M. Society, Geometry of characteristic
classes, Iwanami series in modern mathematics (Ameri-
can Mathematical Society, 2001).
[45] J. May, A Concise Course in Algebraic Topology, Chicago
Lectures in Mathematics (University of Chicago Press,
1999).
[46] Actually, one can define w, which has not yet been spec-
ified explicitly, to be the homomorphism from H3(g, R)
to H3(BG, R) such that f
∗◦(kw|Uk) = ω3|Uk holds on
each local patch U k.
[47] D. Belov and G. W. Moore, Classification of Abelian
spin Chern-Simons theories, arXiv
(2005), arXiv:hep-
th/0505235.
[48] J. D. Stasheff, Continuous cohomology of groups and
classifying spaces, Bulletin of the American Mathemati-
cal Society 84, 513 (1978).
[49] J. L. Dupont, Curvature and Characteristic Classes, Lec-
ture Notes in Mathematics, Vol. 640 (Springer, 1978).
[50] F. Kamber and P. Tondeur, Flat bundles and character-
istic classes of group-representations, American Journal
of Mathematics 89, 857 (1967).
[51] P.
Putrov,
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arXiv
(2022),
arXiv:2208.12071 [hep-th].
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(Springer, New York, NY, 1971).
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(Cambridge University Press, 2002).
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[55] E. M. Friedlander and G. Mislin, Cohomology of classi-
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(1984).
8
Appendix A: ’t Hooft Anomaly of U(1) Global Symmetry and Lie Group Cohomology
As we have noted already, the ’t Hooft anomaly of U(1) needs extra care since H3(g, R) = 0. The F-move is given
by the move shown in Figure 5. We note that here the key point is that for abelian global symmetry one has to
α3
α2
[α2 + α3]
α1
y = 0
x = 0
A
+
α1
α1
α1
x−
x+
Gauge variation
=
α3
α2
[α2 + α3]
α1
y = 0
x = 0
A
FIG. 5. The F-move for U(1) global symmetry.
consider the consequence of the periodicity of the parameter. Thus the fusion of eiα2 and eiα3 results in ei[α2+α3]
rather than naively ei(α2+α3). The Poincar´e dual of the initial web is:
A
= i(α1δ(x −x−)H(y) −α3δ(x)H(−y))dx + i (α1H(x −x−) + α2H(−x) + [α2 + α3]H(x)) δ(y)dy
(A1)
where 0 < αi < 1. The field strength is ([α] ≡α mod 2πZ):
F
= dA
= i([α2 + α3] −α2 −α3)δ(x)δ(y)dx ∧dy .
(A2)
For the gauge variation we have:
A
−A
= iα1 (−δ(x −x−)H(y)dx −B(x; x−, x+)δ(y)dy + δ(x −x+)H(y)dx) = d (−iα1B(x; x−, x+)H(y)) = dϵ .
(A3)
Therefore, we have:
ϵ = −iα1B(x; x−, x+)H(y) .
(A4)
One can compute the anomaly to be:
A[A ; ϵ] =
1
(2π)2
Z
M2
ϵdA
=
1
(2π)2
Z
M2
α1(α2 + α3 −[α2 + α3])B(x; x−, x+)H(y)δ(x)δ(y)dx ∧dy
=
1
(2π)2 α1 (α2 + α3 −[α2 + α3]) .
(A5)
One immediately recognizes that the above result generalizes the result for discrete global symmetry in [28].
To see that in this case the anomaly is described by the Lie group cohomology H3(U(1)δ, U(1)) rather than by the
Lie algebra cohomology (which is actually trivial), we recall that the coboundary operator δ for Lie group cohomology
is defined as:
(δf)(g1, · · · , gn+1) = f(g2, · · · , gn+1) ×
Y
i
f(g1, · · · , gigi+1, · · · , gn+1)(−1)i × f(g1, · · · , gn)
(A6)
For f(eiα1, eiα2, eiα3) = eiα1(α2+α3−[α2+α3])/(2π) we have:
f(eiα2, eiα3, eiα4) = eiα2(α3+α4−[α3+α4])/(2π) ,
f(ei(α1+α2), eiα3, eα4) = ei[α1+α2](α3+α4−[α3+α4])/(2π) ,
f(eiα1, ei(α2+α3), eiα4) = eiα1([α2+α3]+α4−[[α2+α3]+α4])/(2π) ,
f(eiα1, eiα2, ei(α3+α4)) = eiα1(α2+[α3+α4]−[α2+[α3+α4]])/(2π) ,
f(eiα1, eiα2, eiα3) = eiα1(α2+α3−[α2+α3])/(2π) ,
(A7)
9
We define nij := αi + αj −[αi + αj] ∈{0, 2π}. Hence we have:
(δf)(eiα1, eiα2, eiα3, eiα4)
= f(eiα2, eiα3, eiα4)f(eiα1, ei(α2+α3), eiα4)f(eiα1, eiα2, eiα3)
f(ei(α1+α2), eiα3, eiα4)f(eiα1, eiα2, ei(α3+α4))
= ei(α2n34−(α1+α2−n12)n34+α1(α2+α3+α4−n23−[α2+α3+α4])−α1(α2+α3+α4−n34−[α2+α3+α4])+α1n23)/(2π)
= ein12n34/(2π) = 1 .
(A8)
Therefore we have proved that f(α1, α2, α3) = eiα1(α2+α3−[α2+α3])/(2π) is a cocycle with respect to δ. Since f is
apparently not δ-exact, we have proved that the anomaly e2πiA[A
;Λ] = eiα1(α2+α3−[α2+α3])/(2π) is in the subgroup
H4(BU(1), Z) ∼= Z ⊂H3(U(1)δ, U(1)).
Appendix B: Group and Lie algebra cohomology in higher dimensions
We also present the representatives for group and Lie algebra cohomologies with degree n > 3, which are relevant
for anomalies in (n −1) spacetime dimensions.
For abelian group U(1), the representative for the subgroup of the U(1) group cohomology with discrete topology
Z ⊂H2k+1(U(1)δ, U(1)) is given by
f(eiα1, eiα2, . . . , eiα2k+1) = e(2π)−kiα1
Qk
i=1(α2i+α2i+1−[α2i+α2i+1]) .
(B1)
One can explicitly check the closedness condition
(δf)(eiα1, . . . , eiα2k+2) = f(eiα2, . . . , eiα2k+2) ·
2k+1
Y
i=1
f(eiα1, . . . , ei[αi+αi+1], . . . , eiα2k+1)(−1)i
!
· f(eiα1, . . . , eiα2k+1)
= exp
i(2π)−k
k+1
Y
i=1
(α2i−1 + α2i −[α2i−1 + α2i])
!
= 1 .
(B2)
We also discuss the group cohomology representatives for mixed anomalies.
For the mixed anomaly between
U(1)1 × U(1)2 in 2d, described by the gauge invariant 4-form anomaly polynomial (2π)−2F (1) ∧F (2), it corresponds
to the 3-cocycle generator f ∈Z ⊂H3(U(1)δ
1 × U(1)δ
2, U(1)):
f((eiα1, eiβ1), (eiα2, eiβ2), (eiα3, eiβ3)) = exp(iα1(β2 + β3 −[β2 + β3])/(2π)) .
(B3)
Here eiαm ∈U(1)1 and eiβm ∈U(1)2 (m = 1, 2, 3).
More generally in 2k-spacetime dimensions, for a mixed anomaly between U(1)1 ×· · ·×U(1)l, described by a gauge
invariant (2k +2)-form anomaly polynomial (2π)−k−1(F (1))m1 . . . (F (l))ml, m1 +· · ·+ml = k +1, the proposed group
cohomology generator f ∈Z ⊂H2k+1(U(1)1 × · · · × U(1)l, U(1)) is
f((eiα(1)
1 , . . . , eiα(l)
1 ), . . . , (eiα(1)
2k+1, . . . , eiα(l)
2k+1))
= exp(i(2π)−kα(1)
1
m1
Y
m=2
(α(1)
2m−2 + α(1)
2m−1 −[α(1)
2m−2 + α(1)
2m−1])
!
·
lY
j=2
ml
Y
m=1
(α(j)
2m1+···+2mj−1+2m−2 + α(j)
2m1+···+2mj−1+2m−1 −[α(j)
2m1+···+2mj−1+2m−2 + α(j)
2m1+···+2mj−1+2m−1])) .
(B4)
Again eiα(j)
m ∈U(1)j, (m = 1, . . . , 2k + 1).
Appendix C: Anomalies, cohomology groups and secondary invariants.
a.
F-move and group cohomologies.
In the case of a finite group G, the F-move induces a local transformation
of the topological network, with distinct configurations differing by a phase. For 2d bosonic theory, this phase factor
10
is classified by the group cohomology H3(BG, R/Z) (we treat U(1) = R/Z as an additive group), while in the
fermionic case the phase factor is determined by a triple (µ, ν, α) ∈Z1(BG, Z2) × Z2(BG, Z2) × C3(BG, R/Z) such
that δα = (−1)ν2. The distinction lies in whether the theory depends on spin structure. We will only consider the
bosonic case in the continuous group setup. Another crucial feature of a finite group is that the algebraically defined
cohomology group H•(G, A) coincides with the topologically defined cohomology group H•(BG, A) of the classifying
space BG.
For a Lie group G, both H•(BG, A) and “H•(G, A)” can be defined. The former is defined the same way as in the
finite group case by considering the classifying space. However, the latter have various versions. For example, it can
be defined in terms of multivariable functions on the group; one can then restrict attention to the continuous functions
with respect to the standard topology of the group G, this group is usually written as H•
c (G, A) with the subscript
indicating the continuousness [48]. One can take Gδ as the group G with discrete topology, in such a way we can
define H•
c (Gδ, A), which is the analog of H•(G, A) in the finite group case. However, both H•
c (G, A) and H•
c (Gδ, A)
differ from H•(BG, A). For instance, if G is compact connected and simply connected, Hn
c (G, R) = Hn
c (G, R/Z) = 0
for n > 0.
As any flat G-bundle is a Gδ-bundle, we focus on discussions of Gδ in the case of flat gauging. The aim is to show
that there is a class [Ω] ∈H•
c (Gδ, R/Z) that serves as an analog of the Chern–Simons invariant and it is the anomaly.
b.
Classifying space and secondary invariant.
Given the moduli space of flat connection is Hom(π1(M), G)/G
one can verify [44] that the classifying space BGδ for flat G-bundle is the following Eilenberg-Maclane space
BGδ ∼= K(G, 1) .
(C1)
Equipped with discrete topology, Gδ behaves like a discrete group and H•(K(G, 1), A) ∼= H•(BGδ, A) ∼= H•
c (Gδ, A),
where the formed is topologically defined. Note that one has equivalent definitions of flat G-bundle due to Corollary
(3.22) of [49]
1. G-bundle admit flat connection;
2. G-bundle admit a set of constant transition functions;
3. G-bundle with a reduction of structure group to Gδ via the canonical map (actually the identity map)
ι : Gδ −→G .
(C2)
It is the second definition that allows one to describe a flat G-bundle by topological defect networks. The third
definition can also be rephrased as, there exists a Gδ-bundle whose image under ι∗(the nature map between Gδ-
bundles and G-bundles induced by ι) is the flat G-bundle [50]. The canonical map ι also induces a map at the level
of classifying space due to the functoriality of B−
Bι : BGδ −→BG .
(C3)
For any flat bundle given by φ : M →BG there exists a Gδ-bundle by φδ : M →BGδ such that the following diagram
commutes
M
BGδ
BG
φ
φδ
Bι
(C4)
There is a proposition (proposition 9.1 in [49]) stating that the composite map is zero for a general Lie group G
I(G)
Chern–Weil
−−−−−−−→H•(BG, R)
Bι∗
−−→H•(BGδ, R) ,
(C5)
where I(G) is the set of invariant polynomials.
This proposition tells us that f ∈In(G) is always mapped to
0 ∈H2n(BGδ, R).
We now consider the following commutative diagram of long exact sequences
· · ·
H2n−1(BGδ, R/Z)
H2n(BGδ, Z)
H2n(BGδ, R)
· · ·
· · ·
H2n−1(BG, R/Z)
H2n(BG, Z)
H2n(BG, R)
· · · ,
j∗
β
i∗
j∗
j∗
β
Bι∗
i∗
Bι∗
j∗
Bι∗
(C6)
11
where the vertical arrows are just Bι∗of various coefficients and the horizontal long exact sequences are induced by
the standard short exact sequence of coefficients
0 −→Z
i
−→R
j
−→R/Z −→0 ,
(C7)
and β is the connecting homomorphism, or called Bockstein homomorphism, conventionally.
• If [f] ∈H2n(BG, R) is the image of some integral class [f]Z ∈H2n(BG, Z), i. e. i∗([f]Z) = [f]. Because
Bι∗([f]) = 0 ∈H2n(BGδ, R) and the commutativity of the right quadrat implies that i∗(Bι∗([f]Z)) = 0 ∈
H2n(BGδ, R).
• Exactness at the stage of H2n(BGδ, Z) in the upper sequence implies that there exists a class
[Tf] ∈H2n−1(BGδ, R/Z) ,
(C8)
such that β([Tf]) = Bι∗([f]Z), where β is the Bockstein homomorphism of the upper long exact sequence.
Summarizing the above discussion, we found a canonical map
κδ : H2n(BG, Z) −→H2n−1(BGδ, R/Z)
[f]Z 7−→[Tf] .
(C9)
Note that if we replace BGδ by M, and using a flat G-bundle φ get zero map (Chern–Weil ◦φ∗= 0) (now the
composite map is zero because of the vanishing of the curvature 2-form)
I(G)
Chern–Weil
−−−−−−−→H•(BG, R)
φ∗
−→H•(M, R) ,
(C10)
then this would lead to a class CS ∈H2n−1(M, R/Z), namely Chern–Simons invariant constructed by Chern and
Simons explicitly and we just demonstrated how one can obtain it alternatively via diagram chasing. Hence, fixing a
flat bundle φ, we have another canonical map
κ : H2n(M, Z) −→H2n−1(M, R/Z)
(c[f])Z 7−→CS ,
(C11)
where c([f])Z is the integral characteristic class of [f]Z pulled back to M, i. e. (c[f])Z = φ∗([f]Z).
Combining (C4), (C9) and (C11) all together, we arrived at the commutative diagram
H2n(BG, Z)
H2n−1(BGδ, R/Z)
H2n(M, Z)
H2n−1(M, R/Z) ,
κδ
φ∗
(φδ)∗
κ
(C12)
in which the commutativity follows from the functoriality of B−and we have
(φδ)∗([Tf]) = CS.
(C13)
We can apply this to our anomaly discussions. Let n = 2 and for instance take G = SU(2) and c([f]Z) = c2 the
second Chern-class. We now denote [Tf] simply by [Ω] and obtain straightforwardly (φδ)∗(Ω) = κ(c2) = CS.
c.
Transgression and Homotopy fiber.
Given a fibration
F −→E
p
−→M ,
(C14)
the trangression/suspension [34] operation allows one to relate cohomology classes of M to cohomology classes of F
with degree shifted by 1. Mathematically speaking, the details rely on the Leray-Serre spectral sequence. Here we
list some steps relevant to our discussion
• One starts from a cohomology class [α] ∈Hp+1(M) and assumes that its pullback in E trivializes [p∗(α)] = 0 ∈
Hp+1(E);
• Then there exists a p-cochain, denoted as η ∈Cp(E) such that δη = p∗(α);
12
• The restriction of η to the fiber F is called Tα and δ(Tα) = 0, one thus obtains a cohomology class [Tα] ∈Hp(F)
and [α] is called the transgression of [Tα].
A primary example is the universal G-bundle
G −→EG
p
−→BG ,
(C15)
where the cohomology of EG is trivial since it is contractible. Hence we have the suspension map by the above process
τ : Hn(BG, A) −→Hn−1
singular(G, A) ,
(C16)
where A is an arbitrary coefficient and x ∈Im(τ) ⊂Hn−1
singular(G, A) is called universal transgressive. In the case of
A = Z as coefficient and n = 4, the map τ is in general not surjective, but for compact, connected, simply connected
and semi-simple Lie groups it is an isomorphism (in particular for each compact, connected, simply connected simple
factor).
Another important fibration in our consideration can be introduced as follows.
Following the standard construction of homotopy fiber, one can write the map ι into a fibration
G −→Eι
ι
−→G ,
(C17)
where Eι is given as
Eι = {(g, γ) ∈Gδ × GI|γ(0) = ι(g)} ⊂Gδ × GI ,
(C18)
and it is homotopy equivalent to Gδ and we sometimes just write Gδ instead. GI is the mapping space of continuous
maps from the interval I = [0, 1] to G, it is a topological space endowed with the compact-open topology.
The
homotopy fiber ¯G is by construction defined as
G = {(g, γ) ∈Gδ × GI|γ(0) = ι(g), γ(1) = ι(1Gδ)} ⊂Gδ × GI .
(C19)
The homotopy fiber construction also extends to Bι and we have the following fibration
B ¯G −→BGδ
Bι
−→BG .
(C20)
We can now apply the suspension operation to the fibration (C20) and use the proposition (C5). Pick f ∈Ik(G) such
that [f] ∈H2n(BG, R), then its determines a class
[Tf] ∈H2n−1(BG, R) ,
(C21)
that transgresses to [f].
Appendix D: On H3(U(1)δ, U(1))
A mathematically proper definition of the “N −→∞” limit of ZN is via the direct limit (see [51] for a discussion
in the context of physics applications). That is, build a direct system by the inclusion Zn →Zm whenever n divides
m and take the limit of this system
lim
−→Zn = Q/Z,
(D1)
an explicit description of the limit is by taking the union of all cyclilc subgroups Zn ⊂U(1)δ, i.e.
Q/Z =
[
n
Zn .
(D2)
The group Q/Z captures all roots of unity in U(1)δ and it is the torsion subgroup of U(1)δ. Since U(1)δ ∼= Rδ/Z, we
have
U(1)δ ∼= Rδ/Q ⊕Q/Z .
(D3)
13
The above isomorphism is due to the fact that the Q/Z is divisible and hence the short exact sequence splits (although
not canonical and depends on a choice of basis of Rδ)
0 −→Q/Z −→Rδ/Z −→Rδ/Q −→0 .
(D4)
Admitting the axiom of choice, Rδ is regarded as infinite dimensional Q-vector spaces with an uncountable basis. In
the same vein, the divisible torsion-free group Rδ/Q is also a Q-vector spaces with a uncountable basis. Let us denote
the uncountable index set of a basis of Rδ/Q by I then Rδ/Q = ⊕IQ. We will also omit the superscript δ for brevity,
as no confusion will arise in what follows. We denote the above groups by A = Rδ/Q, B = Q/Z and G = Rδ/Z. At
the level of classifying space, we are dealing with K(G, 1) = K(A, 1)×K(B, 1). We can apply the universal coefficient
theorem
0 −→ExtZ(H2(G, Z), U(1)) −→H3(G, U(1)) ∼= H3(K(G, 1), U(1)) −→HomZ(H3(G, Z), U(1)) −→0 ,
(D5)
which yields H3(G, U(1)) ∼= HomZ(H3(G, Z), U(1)) since ExtZ(H2(G, Z), U(1)) vanishes for the divisible group U(1).
As G = A ⊕B, we have for H3(G, Z) the K¨unneth formula [52]
0 −→
M
p+q=3
Hp(A, Z) ⊗Hq(B, Z) −→H3(G, Z) −→
M
p+q=2
Tor(Hp(A, Z), Hq(B, Z)) −→0 ,
(D6)
and the sequence splits by an unnatural splitting. Now recall the standard result (which can be derived using the
techniques in Section 3.F of [53])
Hq(Q/Z, Z) =
Z
(q = 0)
Q/Z
(q
odd)
0
(q
even
and
q > 0)
.
(D7)
As for H•(A, Z), we can think of it as the homology of the Eilenberg-MacLane space K(A, 1). Then H0(A, Z) =
H0(K(A, 1), Z) = Z as Eilenberg-MacLane spaces are connected and H1(A, Z) = H1(K(A, 1), Z) = A which equals
the abelianization of π1(K(A, 1)) = A for A is abelian.
From the torsion product part we need to compute Tor(H0(A, Z), H2(B, Z)), Tor(H1(A, Z), H1(B, Z)) and
Tor(H2(A, Z), H0(B, Z)). By inserting the above results and using the properties of the Tor functor we conclude
M
p+q=2
Tor(Hp(A, Z), Hq(B, Z)) = 0 .
(D8)
Now we can consider the tensor product part of (D6), for this we need two extra ingredients H2(A, Z) and H3(A, Z)
which appear in Q/Z ⊗H2(A, Z) and Z ⊗H3(A, Z) = H3(A, Z). Given A = ⊕IQ, K(A, 1) can be thought as filtered
homotopy colimit of K(Qn, 1) over finite n and one can use K¨unneth formula to show that Hp(K(Qn, 1), Z) is a
Q-vector space for all p > 0. This is also true for Hp(K(A, 1), Z), as filtered colimit of Q-vector spaces are Q-vector
spaces [54]. H2(A, Z) as a Q-vector spaces is torsion-free and hence
Q/Z ⊗H2(A, Z) = 0 .
(D9)
We are left with H3(A, Z) and H0(A, Z) ⊗H3(B, Z) = Z ⊗Q/Z = Q/Z.
Summarized the above computation we arrive at
H3(G, U(1)) = Hom(H3(A, Z) ⊕Q/Z, U(1)) = H3(Rδ/Q, U(1)) ⊕ˆZ ,
(D10)
where ˆZ is the profinite integers can be though as the direct product Πp Rp where Rp is the ring of p-adic integer.
Now there is a canonical embedding of Z into ˆZ by sending each integer n to the sequence of its residues modulo
all positive integers. This map is modeled by embedding the level-k anomaly polynomial (level-2k Chern–Simons
invariant in our normalisation) of U(1) into the 3-cocycle of ˆZ part of H3(U(1)δ, U(1)).
Note that in the above consideration, H3(A, Z) = H3(K(A, 1), Z) remains an unspecified Q-vector space, possi-
bly of uncountable dimension. Together with the cohomology ring of H•(K(A, 1), Z), this lies beyond our current
understanding, and we hope it will one day be clarified by mathematicians.
For non-abelian Lie groups, the complexity of this question increases dramatically. It is in general very hard to
determine the cohomology group exactly. We will just briefly mention some contents from [31, 55]. Let G be an
arbitrary Lie group with finitely many components, then
• the map Bι∗: H•(BG, Z) →H•(BGδ, Z) induced by the canonical map Bι from the Eq. (C3) is injective;
• Friedlander–Milnor conjecture. The canonical map Bι induces isomorphisms of homology and cohomology with
mod p coefficients, or more generally with any finite coefficient group.
14
Appendix E: Anomaly of gauge transformation
In this appendix, we will review the gauge anomaly in 2d, noting that the discussion naturally extends to higher
dimensions. Suppose we have a 2d theory living on M2 whose anomaly is characterized by the anomaly polynomial
I4(F)
I4(F) = k
2TrF ∧F ,
(E1)
which is closed and is locally exact
I4(F) = dI3(A) ,
(E2)
where I3(A) ≡CSk(A) is the local Chern-Simons density.
Consider an infinitesimal gauge transformation
parametrized by Λ, the gauge variation of I3(A) is closed due to
dδΛI3(A) = δΛdI3(A) = δΛI4(F) = 0 ,
(E3)
and is also locally exact. Then the anomaly of the partition function Z(A) is computed via the descent equation
δΛI3(A) = dA[A; Λ] ,
(E4)
and takes the form
δΛ log Z[A] = 2πi
Z
M2
A[A; Λ] .
(E5)
We then consider a finite gauge transformation
A →g−1Ag + g−1dg ,
(E6)
parametrized by a group function g ≡g(x, y) on M2 and is connected to the identity. Let us construct a 3d manifold
M2 × [0, 1] with t ∈[0, 1] and extend the 2d gauge function g into g(t) ≡g(x, y, t) as
g(x, y, 0) = e ,
g(x, y, 1) = g(x, y) .
(E7)
We also extend the gauge field A on M2 along [0, 1] according to
A(t) = g(t)−1A0g(t) + g(t)−1dg(t) ,
(E8)
where A0 is considered as a trivial extension of A on M2 as A0(x, y, t) = A(x, y). The 2d anomaly of the finite gauge
transformation g connected to the identity is then evaluated as
δg log Z[A] = 2πi
Z
[0,1]×M2
CSk[A(t)] .
(E9)
First, one can show that the anomaly given above is independent of the extension g(t) with fixed boundary conditions
g(0) = e, g(1) = g. The reason is that the Chern-Simons integral is invariant under the gauge transformation as long
as we fix the boundary condition. Second, if g(t) is small, we have
Z
[0,1]×M2
CS[A(t)] =
Z
[0,1]×M2
CS[A(t)] −
Z
[0,1]×M2
CS[A0] =
Z
[0,1]×M2
δg(t)CS[A] ,
(E10)
where in the middle we use CS[A0] = 0 since A0 is a trivial extension. As shown above, the gauge variation of
Chern-Simons density is a total derivative
δg(t)CS(A) = dA[A; Λ] ,
(E11)
where Λ(t) is the infinitesimal gauge parameter related by eiΛ(t) = g(t). Using the fact g(0) = e and Λ(0) = 0, we
recover the infinitesimal version of the gauge anomaly (E5).
|
Anomaly of Continuous Symmetries from Topological Defect Network Qiang Jia1, Ran Luo2, Jiahua Tian3, Yi-Nan Wang2,4,5, and Yi Zhang6 1 34141, Korea 2 3 200241 4Center for High Energy Physics, Peking University 5Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China and 6Kavli IPMU (WPI), UTIAS, The 277-8583, Japan (Dated: October 17, 2025) We show that the 't Hooft anomaly of a quantum field theory with continuous flavor symmetry can be detected from rearrangements of the topological defect webs implementing the global symmetry in general spacetime dimension, which is concretized in 2D by the F-moves of the defect lines. Via dualizing the defects to flat background gauge field configurations, we characterize the 't Hooft anomaly by various cohomological data of the symmetry group, where the cohomology of Lie groups with discrete topology plays the central role. We find that an extra dimension emerges naturally as a consequence of the mathematical description of the 't Hooft anomaly in the case of flat gauging. Introduction. A global symmetry of a Quantum Field Theory (QFT) is said to have a 't Hooft anomaly if gauging it leads to an inconsistency, i.e. an obstruction to promoting it to a gauge symmetry [1]. The robustness of 't Hooft anomaly across ultraviolet to infrared makes it a particularly effective tool to study the properties of QFT even in the strongly coupled region [2, 3]. The 't Hooft anomaly manifests itself as a nontrivial phase factor e2πiA(A,g) acquired by Z[A] under gauge transformation A →g-1Ag +g-1dg, where Z[A] is the paritition function of the QFT coupled to background field A and for A(A, g) that cannot be cancelled by any local counterterm built out of A. Traditionally for continuous G, it can be evaluated using the descent equations for small gauge transformations [4-8] and more generally shown to be the the eta-invariant when g is disconnected from the identity [9-12]. Recent years have witnessed a revival of the study of 't Hooft anomalies in both condensed matter and highenergy physics [13-17]. Thanks to the reformulation of the symmetry action as linking symmetry operator and charged defect in spacetime [18-23], the action of finite group symmetries is now conveniently studied on equal footing with that of continuous symmetries. However, whereas the 't Hooft anomaly of discrete group symmetry can be computed directly as the F-symbol associated with rearrangements of webs of defects, which are often described by elements of the group cohomology (e.g. Hd+1(G, U(1)) for a 0-form symmetry group G in dspacetime dimensions)[24-27], an analogous algorithm to compute the 't Hooft anomaly for continuous group symmetries in flat gauging has long remained elusive. In this work, we present such an algorithm for continuous symmetries, thereby filling the missing corner to place the treatment of finite and continuous symmetries on a completely parallel footing. We first show that a web of defects can be dualized to be a flat background field configuration. This effectively establishes the equivalence of insertion of defects and coupling the conserved current to flat background fields. Via the dualization, we show in full generality how the 't Hooft anomaly manifests itself as the phase factor arising from rearrangements of the topological defects in the manner hinted in [21]. We then compute the anomaly concretely via F-moves of the topological defect lines in 2D QFT [16, 17, 28-30] and show that the anomaly is characterized by cohomological data of G. Very importantly, unlike the case of a finite group G where the group cohomology H3(G, U(1)) ∼= H3(BG, U(1)) is unambiguously defined, one should be careful about the exact definition for Lie group cohomology. We emphasize that for a Lie group G the correct version should be H3(Gδ, U(1)), which is defined to be the group cohomology when G is equipped with discrete topology [31]. Despite that H3(Gδ, U(1)) is a highly complicated mathematical object, there exists an injection map from the group H4(BG, Z), where the anomaly polynomial for G lives, to H3(Gδ, U(1)). Thus the usual anomaly polynomial ∼F ∧F can be elegantly embedded in this unified framework. We find that the unifying mathematical structure underlying these various cohomology groups of G suggests that the QFT with a 't Hooft anomaly naturally lives on the boundary of an anomaly theory in one higher dimension. The generalization of the above cohomological characterization of 't Hooft anomaly to higher-dimensional spacetime is immediate in our formalism. For detailed examples, we discuss the cases of a simple G and U(1) separately. For a compact simply connected and simple Lie group G, the 3d bulk anomaly theory is the non-abelian Chern-Simons invariant characterized by an integer k ∈Z called the level. It has a local expression as the Chern-Simons 3-form I3 = CSk[A] = k 2 tr(A∧F - 1 3A ∧A ∧A). We explicitly uplift the topological defect networks across the F-move to a three-dimensional bulk, and show that after assigning F = 0, the cubic term in CSk[A] precisely generates the de Rham cohomology group H3 dR(G, R) for G. For a non-simply connected Lie group such as G = U(1), we point out that the field strength F = dA would 16 Oct 2025 2 not vanish at the junction point for this case. This is not inconsistent with the requirement of flat gauging, as the precise definition of flat gauging should be the triviality of the holonomy exp H C A around a topologically trivial loop C. Such a non-vanishing F is precisely the source of the non-trivial U(1) phase factor across the Fmove, which we compute explicitly from the integration of εF over the 2D spacetime, where ε is the finite gauge transformation parameter generating the F-move. The factor ω(eiα1, eiα2, eiα3) = eiα1(α2+α3-[α2+α3])/(2π) (1) with αi ∈[0, 2π) and [r] denoting the part of r exceeding 2π, is consistent with the N →∞limit of H3(ZN, U(1)). We would also like to comment that (1) is perfectly consistent with the newly proposed categorical formulation of continuous symmetry [32]. Namely, if one pretends that the definition of the symmetry category Vecω G and Drinfeld center Z(Vecω G) can be generalized to the case of G = U(1), all the category theoretical data are consistent with the mathematical results in [33], which are confirmed in [32]. Our results can be applied to all compact connected Lie groups. From the structure theorem [34], compact connected Lie groups are finitely covered by direct products of tori and simple non-abelian groups, such that their classification reduces to that of simple groups. The restriction to a compact connected and simple Lie group G or U(1) is sufficient in this context. Finally, although the detailed examples are discussed for the d = 2 case, our framework applies to continuous 0-form symmetries in general d spacetime dimensions as well. Dualization of the defect web. The VEV of a topological defect web W of a (Lagrangian) QFT TG in ddimensional spacetime Md can be computed as: ⟨W⟩:= Z Dψ eiS[ψ]+ R Md tr(AW∧⋆J) , (2) where ψ stands for the set of local operators of TG and J(ψ) is the conserved current of the continuous global G-symmetry. AW standards for the ordinary flat gauge field corresponding to the defect web W. To illustrate the dualization, we consider a defect line L labeled by gα = eiα ∈G along x = 0 in R2, meaning that a particle charged under the global G-symmetry is transformed under a representation of gα when crossing x = 0. We define the dualization AL of L to be the background gauge field such that Pe R lAL = eiα for an arbitrary path lcrossing x = 0 once, which in turn means that AL must be flat. Given the condition, we can write: AL = iαδ(x)dx = e-iαH(x)deiαH(x) (3) where H(x) is the Heaviside function, and we can indeed check that dAL + AL ∧AL = 0. Equivalently, L defines the gauge configuration AL|Ui = g-1 i dgi for g0 = e and g1 = eiα on the following open cover of R2: x = 0 L U0 U1 AL (4) where the two patches U0 = (-∞, ε) and U1 = (-ε, ∞) are glued along (-ε, ε) for small positive ε. Here we adopt the convention that the direction of the defect line rotates right-handedly with respect to the direction of the gauge transformation from U0 to U1. Clearly, the above discussions can be generalized to a codimension-one web of W in general d dimensions. The fundamental building block of any defect web (of lines) is the trivalent junction (the background gauge field AL corresponding to the group element eiα is also labeled, with α): eiβ eiα eif(α,β) (0, 0) β f(α, β) α (5) This trivalent junction can be smeared out to give A = g-1dg with F = dA + A ∧A = 0 for a globally defined g on R2 for G with trivial π1(G), since all paths from identity to eiαeiβ on the group manifold are homotopic. However, a globally defined g cannot be obtained by any kind of smearing for G with non-trivial π1(G), e.g. for G = U(1), and there exist flat configurations of U(1) gauge fields with non-trivial F = dA. This subtlety is crucial for the derivation of anomalies for abelian groups. Now we clarify the notion of flat gauging of continuous G global symmetry, given by the following two equivalent conditions: 1. A flat gauging is equivalent to summing over all possible insertions of topological defect networks in the new partition function. 2. Summing over configurations of gauge fields A, where the holonomy of the gauge field around a topologically trivial loop P exp( H C A) = e, the identity element of G. 't Hooft anomaly from F-move and the Lie group cohomology. Having defined the dualization for continuous G, one can compare ⟨W⟩with ⟨W′⟩by coupling TG to flat AW- and AW′-backgrounds, respectively. Suppose AW′ = e-iΛAWeiΛ + e-iΛdeiΛ for gauge transformation ψ →eiΛψ parametrized by Λ, we have: ⟨W′⟩= Z Dψ eiS[ψ]+ R Md tr(AW′∧⋆J) = e2πiA[AW;Λ]⟨W⟩ (6) where A[AW; Λ] ̸= 0 is the quantum anomaly. Therefore, given our dualization, the computation of anomaly arises 3 from rearrangements of the defect webs amounts to finding the gauge transformation interpolating the dual flat field configurations. As a reminder, the 't Hooft anomaly of Zm-symmetry of a 2D bosonic theory is well-known to be characterized by H3(Zm, U(1)), the 3rd group cohomology of Zm with U(1)-coefficient, in which the phase factor e2πia1(a2+a3-a2+a3)/(m) of the F-move of the defect webs in Figure 1 lives [28]. In d-dimension the characterization e2πi a1 m e2πi a2 m e2πi a3 m = e2πi a1(a2+a3-a2+a3) m × e2πi a1 m e2πi a2 m e2πi a3 m FIG. 1. The F-move of defect web for G = Zm 0-form symmetry group in 2D. ai ∈{0, 1, . . . , m -1} and m is defined as m mod Z. is given by Hd+1(G, U(1)) for discrete G [16, 26]. A similar characterization for continuous G is still missing. We will focus on continous G of interests, and compute their anomalies in flat gauging, i.e. from coupling TG to flat background field that is the closest cousin of the discrete case where all background field is automatically flat. We show that the 't Hooft anomaly is encoded in the phase factor in ⟨ ⟩= e2πiA[A ;Λ]⟨ ⟩(cf. (6)) for the defect webs and living on M2 in TG with Lie group G. Here Λ is the gauge transformation from A to A . In the modern langauge [12], A[A ; Λ] can be calculated as (cf. Eq. (E9) in Appendix E): A(A ; Λ) = Z M3 CSk[A(t)] , (7) using the level-k Chern-Simons invariant CSk where t ∈ [0, 1] parametrizing a mapping cylinder in the space of gauge field configurations along which A(t) := g(t)-1A g(t) + g(t)-1dg(t) for g(x, y, t = 0) = 1 ∈G and g(x, y, t = 1) = eiΛ(x,y) interpolates from A to A . We use the normalization CSk[A] := k 2 tr(A ∧dA + 2 3A ∧ A ∧A) = k 2 tr(A ∧F -1 3A ∧A ∧A) on a local patch of M3. For simplicity, we restrict to theories whose ChernSimons invariants used in anomaly computations originate from integral classes in H4(BG, Z) (cf. Eq. (C11) in Appendix C). This condition constrains the allowed minimal level k, which depends on G. We do not discuss these cases individually and omit the subscript k for brevity [35]. The interpolation from A to A is illustrated on the leftmost graph in Figure 2. It is not hard to see that by our convention, the configuration in the front is equivalent to and the one in the back is equivalent to . Since A[A ; Λ] is clearly independent from small deformations of the mapping cylinder, i.e. the details of g, as long as boundary configurations A and A are fixed, we further glue the vertices of with the ones with the same labels in at the two copies M2 × {0} and M2 × {1} to arrive at the configuration in the middle of Figure 2. This is the difference between the corresponding webs of defects and the middle configuration can be further deformed to be a decorated tetrahedron as in the rightmost of Figure 2, where the directions of the defects and of the gauge transformations are set by our convention (4). This process can be understood as the anomaly-inflow-very similar to the finite group case [16]-i.e. an extra 3-simplex can be attached in the bulk to cancel the anomaly and the decoration of the tetrahedron in the bulk describes the pullback of the 3-simplex ⟨e, g1, (g1g2), (g1g2g3)⟩in BGδ to the bulk. Here Gδ is the Lie group G with discrete topology and BGδ is the Eilenberg-MacLane space K(G, 1) classifying flat G-bundles, we denote the classifying map by φδ : M3 →BGδ [36]. Similar to the finite group case, in terms of the classifying space, the anomaly is obtained by evaluating a cohomology class [Ω] ∈ H3(BGδ, R/Z) on ⟨e, g1, (g1g2), (g1g2g3)⟩. Here we treat the coefficient U(1) = R/Z as an additive group. Pullback with Ωwith φδ and integrate over the 3D spacetime, we obtain the Dijkgraaf-Witten phase A(A ; Λ) = Z M3 (φδ)∗(Ω) , (8) in other words, we claim that Ωfound by F-move pulls back to the Chern-Simons invariant CS. See the proof of (8) in Appendix C. By investigating the local expression of Chern-Simons invariant, we can get some other manifestations of anomalies. For this discussion, as G is simply connected we can assign F = 0 for flat gauge field. In the M3 bulk, our gauge field A(t) is flat. On a local patch Ui, we have Ai := A|Ui = ̃g-1 i d ̃gi for some ̃gi : Ui →G and Z Ui -k 6 tr(Ai ∧Ai ∧Ai) = Z Ui k ̃g∗ i ω3 = Z ̃gi(Ui) k ω3 (9) where ̃g∗ i is the pull-back from Ω∗(G) to Ω∗(Ui) and ̃g∗ i ω3 = i2 3! tr( ̃g-1 i d ̃gi)3 with ω3 being a left-invariant 34 g! g!g" v# g v" v g! g" g v! v# v" v g!g"g g" Deform FIG. 2. The deformations of the combined and configuration. The group multiplications in this figure are all taken from the right. form on G. For a compact, connected, simply connected and simple G, ω3 generates the de Rham cohomology H3 dR(G, R) ∼= R, it is also the image of an integral class in the singular cohomology H3 singular(G, Z) ∼= Z [37]. Furthermore, such G-bundle on any 3-manifold is necessarily trivial. Hence ̃gi = ̃g is globally defined, and the anomaly can be calculated as: A(A ; Λ) = Z ̃g(M3) k ω3 . (10) The right-hand side of (10) now has another interpretation, namely, the action of ungauged 2d Wess-ZuminoWitten model. Indeed, this is a fact of anomaly matching [2, 7, 38]. The appearance of the left invariant 3form ω3 as a suspension (C16) (the inverse is referred to as transgression [29, 34]) of CS on the fiber G via the bundle fibration G →P →M3 was discussed originally in [39], see also discussions [29, 40-42] related to anomalies. On the other hand, we take ⟨X, Y ⟩= tr(XY ) to be the Killing form of the Lie algebra g of G then -k 6 tr(A ∧A ∧A) = -k 12⟨A, [A, A]⟩, (11) where we recognize that ⟨·, [·, ·]⟩is the generator of the 3rd Lie algebra cohomology H3(g, R) of the flavor algebra g. We see that the group H3(g, R) also classifies the local density of anomaly in flat gauging. Since our G is taken to be compact connected and simple, we always have H3(g, R) ∼= H3 dR(G, R) ∼= R. As the 3-forms have integral periods, they are the image of the lattice Z ∼= H3 dR(G, Z) inside H3 dR(G, R). The coefficient exchanging map is in fact injective [43] H3 singular(G, Z) ∼= H3 dR(G, Z) ,→H3 dR(G, R) ∼= H3(g, R) . (12) Recall that the suspension map (C16) τ : H4(BG, Z) -→H3 singular(G, Z) (13) is also an isomorphism if G is in addition simply connected. For these groups, we have H3 singular(G, Z) ∼= H4(BG, Z) κδ ,-→H3(BGδ, R/Z) , (14) where the last map (C9) is also an injection [43]. The emergence of extra dimension. Moreover, there is a homomorphism w : H3(g, R) →H3(BG, R) where BG is the classifying space of flat G-trivial bundles [44]. One particularly interesting feature about BG is that a trivialization can be explicitly written down as an element of G = {(g, l) ∈Gδ × GI|l(0) = g, l(1) = e} where Gδ is G equipped with discrete topology together with the canonical map ι : Gδ →G and GI := Map(I, G) for I := [0, 1]. More precisely, we consider a web of defects on a general 2D spacetime manifold M2 which, by Poincar ́e duality, defines a flat G-bundle over M2 determined up to isomorphism by the homotopy class of the classifying map f δ : M2 →BGδ and a trivial bundle given by a nullhomotopic map f n : M2 →BG. Using the lift of ι, Bι : BGδ →BG, the map f : (x, t) 7→(f δ(x), γx(t)) ∈ BGδ × BG where γx(0) = Bι(f δ(x)) defines a homotopy from f δ to f n by the mapping path space construction [45]. Equivalently, f is a map from M2 × [0, 1] to BG ⊂BGδ × BG. Therefore, we have: Z f(M2×I) w(ω3) = Z M2×I f ∗◦w(ω3) (15) where f ∗is the pull-back of f. U0 U2 A A I U0 U1 U1 U2 M2 FIG. 3. The showing of open cover Uk (k = 0, 1, 2) of M2 as well as the open cover U k (k = 0, 1, 2) of M2 × I. To investigate the abstract map f ∗◦w at a more elementary level, we fix π1(M2) = Z for simplicity (e.g. let M2 ∼= S1×R). We use the open cover Uk = ( 2kπ 3 , (2k+4)π 3 ) 5 (modulo 2π) of S1, k = 0, 1, 2, and define the flat connection A = iα 2πdθ (α ∈ig) on a trivial bundle with the transition function gij ≡1 on Uij := Ui ∩Uj from Ui to Uj. We plot the spacetime geometry and patches in Figure 3. Under gauge transformation uk = e-iα 2π (θk+ 2kπ 3 ) on each Uk parametrized by θk ∈(0, 4 3π), Ak vanishes because u-1 k Akuk + u-1 k duk = 0. Meanwhile, the transition function on Ui,i+1 becomes g′ i,i+1 = u-1 i gi,i+1ui+1 = u-1 i ui+1. Since θi(p) = θi+1(p) + 2π 3 , ∀p ∈Ui,i+1 for i mod 3, we have: g′ 01(U01) = g′ 12(U12) = 1, g′ 20(U20) = eiα . (16) Hence, while uk trivializes A, it yields non-trivial transition functions parametrized by α. To uplift A to M2 × I, we define uk = e-iα 2π (θk+ 2kπ 3 )f(t) with f(0) = 1 and f(1) = 0 for t ∈I as a function from U k := Uk × R × I to G. We further consider the gauge field Ak = u-1 k Akuk + u-1 k duk on U k and the transition function gij = u-1 i uj on U ij. The pair (gij, Ai), defined on (an open cover of ) M2 × I, leads to a trivialization of a bundle with non-trivial transition function (16) over M2×{0} to a trivial bundle with non-zero flat connection over M2 × {1}. In this example, g20 = eiαf(t) provides a concrete physical realization of an element (eiα, l) of G. Since Ak = g-1 k dgk with gk = u-1 k uk is flat, we can replace (A, U) in (9) by (Ak, U k). Together with (15) restricted to U k and a suitable choice of normalization, we are led to [46]: f ∗◦w(ω3)|U k = g∗(ω3)|U k = CS[A]|U k . (17) Physically, this means that the data of 't Hooft anomaly can be captured by the Chern-Simons action of a flat gauge field on M2 × I with non-trivial transition functions at M2 × {0} which trivializes at M2 × {1} at the price of yielding non-zero flat connection. Moreover, the characterization of 't Hooft anomaly by H3(g, R) naturally requires the emergence of an extra dimension of the anomaly theory, manifests itself as I, because of the structure of f ∗◦w. U(1) flavor symmetry and its 't Hooft anomaly. A mathematically cautious reader might have already noticed that the previous calculation leading to H3(G, R) fails for U(1), since both H3(U(1), R) and H3(u(1), R) vanish. Moreover, the anomaly polynomial for U(1) that generates H4(BU(1), Z) ∼= Z is CS[A] = A ∧dA, where the cubic term 1 3A ∧A ∧A is absent. To get a nonzero anomaly, we comment that there is a non-zero field strength at the junction point (x, y) = (0, 0) in (5). For the case of G = U(1), we have f(α, β) = (α + β) mod 2πZ ≡[α + β]. The flat gauge field dual to the trivalent junction is then A = iαδ(x)H(y)dx + i(βH(-x) + [α + β]H(x))δ(y)dy , (18) and the corresponding field strength F = dA = i([α + β] -α -β)δ(x)δ(y)dx ∧dy (19) is non-vanishing at the junction point. Now to compute the 't Hooft anomaly of U(1) flavor symmetry, we apply the previous computation to the configuration in Figure 4. [α3] [α2] [[α1 + α2] + α3] [α1] y = 0 x = 0 A0 + [α1] [α1] [α1] xx+ Gauge variation = [α3] [α2] [α1 + [α2 + α3]] [α1] y = 0 x = 0 A FIG. 4. The F-move of defect lines in a theory with U(1) flavor symmetry. A topological line carrying eiα ∈U(1) is labeled by [α] := α mod 2πZ, with an arrow labeling the direction of gauge transformation in the same sense as in (5). It is easy to see that the left-most and the right-most configurations in Figure 4 are topologically equivalent to and , respectively. Hence our task is to find their Poincar ́e duals A and A , and the gauge transformation parameter dε ≡idΛ = A -A . Actually, using the configuration in the middle of Figure 4 and assuming α ≪1 hence [α1] = α1, one can show that (see the details in Appendix A): ε = -iα1(H(x-) -H(x+))H(y) (20) The infinitesimality of α1, hence of ε, enables us to com6 pute the 't Hooft anomaly to be A[A ; ε] = 1 (2π)2 Z R2 εdA = 1 (2π)2 α1(α2+α3-[α2+α3]) . (21) Viewing f(eiα1, eiα2, eiα3) := e2πiA[A ;ε] = eiα1(α2+α3-[α2+α3])/(2π) (22) as a map f : G3 →U(1), one can check that δf = 0 where δ is the coboundary operator for the Lie group cohomology of U(1)δ (see an explicit computation in the appendix A). The map (22) is not continuous, which aligns with the usage of group cohomology H3(U(1)δ, U(1)) with discrete topology. Comparing to the phase factor in Figure 1, one can see that the (22) is exactly the m →∞limit of the phase factor for Zm generating H3(Zm, U(1)). For the details of the computation of the whole H3(U(1)δ, U(1)) ⊃H4(BU(1), Z), see Appendix D. Higher dimensional generalizations. We have explicitly derived the connection between the F-move of topological defects and the anomaly polynomial for the flat gauging of the continuous symmetry group G, for the d = 2 case. Note that if we take the dual graph of the right picture in Figure 2, we obtain a 3-simplex v0v1v2v3 whose vertices are vi (i = 0, . . . , 3) while the edges correspond to elements of G. Such picture can be easily generalized to d = 2k spacetime dimensions, where the dual 3-simplex v0v1v2v3 is replaced by a dual (2k+1)-simplex v0v1 . . . v2k. Taking the dual of v0v1 . . . v2k, one obtains the higher-dimensional generalization of F-move, as in Dijkgraaf-Witten. The Lie algebra cohomology H3(g, R) and group cohomology H3(U(1)δ, U(1)) should be straightforwardly generalized to H2k+1(g, R) and H2k+1(U(1)δ, U(1)). We present the explicit generators for H2k+1(g, R) for semisimple Lie algebra g, H2k+1((U(1)l)δ, U(1)) for a product of U(1) 0-form symmetry groups, and their correspondence with anomaly polynomials, in Appendix B. ACKNOWLEDGMENTS We would like to thank Jin Chen, Yuji Tachikawa, Xin Wang, Zhaolong Wang and Yunqin Zheng for useful discussions. YZ would like to thanks Mikhail Kapranov for helpful explanations about the cohomology of Lie groups made discrete. JT and YNW would like to thank Peng Huanwu Center for Fundamental Theory and Northwest University for hosting the 6th National Workshop on Fields and Strings where many progress of this project has been made. RL and YNW are supported by National Natural Science Foundation of China under Grant No. 12175004, No. 12422503 and by Young Elite Scientists Sponsorship Program by CAST (2024QNRC001). 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Mislin, Cohomology of classifying spaces of complex lie groups and related discrete groups, Commentarii Mathematicae Helvetici 59, 347 (1984). 8 Appendix A: 't Hooft Anomaly of U(1) Global Symmetry and Lie Group Cohomology As we have noted already, the 't Hooft anomaly of U(1) needs extra care since H3(g, R) = 0. The F-move is given by the move shown in Figure 5. We note that here the key point is that for abelian global symmetry one has to α3 α2 [α2 + α3] α1 y = 0 x = 0 A + α1 α1 α1 xx+ Gauge variation = α3 α2 [α2 + α3] α1 y = 0 x = 0 A FIG. 5. The F-move for U(1) global symmetry. consider the consequence of the periodicity of the parameter. Thus the fusion of eiα2 and eiα3 results in ei[α2+α3] rather than naively ei(α2+α3). The Poincar ́e dual of the initial web is: A = i(α1δ(x -x-)H(y) -α3δ(x)H(-y))dx + i (α1H(x -x-) + α2H(-x) + [α2 + α3]H(x)) δ(y)dy (A1) where 0 3, which are relevant for anomalies in (n -1) spacetime dimensions. For abelian group U(1), the representative for the subgroup of the U(1) group cohomology with discrete topology Z ⊂H2k+1(U(1)δ, U(1)) is given by f(eiα1, eiα2, . . . , eiα2k+1) = e(2π)-kiα1 Qk i=1(α2i+α2i+1-[α2i+α2i+1]) . (B1) One can explicitly check the closedness condition (δf)(eiα1, . . . , eiα2k+2) = f(eiα2, . . . , eiα2k+2) · 2k+1 Y i=1 f(eiα1, . . . , ei[αi+αi+1], . . . , eiα2k+1)(-1)i ! · f(eiα1, . . . , eiα2k+1) = exp i(2π)-k k+1 Y i=1 (α2i-1 + α2i -[α2i-1 + α2i]) ! = 1 . (B2) We also discuss the group cohomology representatives for mixed anomalies. For the mixed anomaly between U(1)1 × U(1)2 in 2d, described by the gauge invariant 4-form anomaly polynomial (2π)-2F (1) ∧F (2), it corresponds to the 3-cocycle generator f ∈Z ⊂H3(U(1)δ 1 × U(1)δ 2, U(1)): f((eiα1, eiβ1), (eiα2, eiβ2), (eiα3, eiβ3)) = exp(iα1(β2 + β3 -[β2 + β3])/(2π)) . (B3) Here eiαm ∈U(1)1 and eiβm ∈U(1)2 (m = 1, 2, 3). More generally in 2k-spacetime dimensions, for a mixed anomaly between U(1)1 ×· · ·×U(1)l, described by a gauge invariant (2k +2)-form anomaly polynomial (2π)-k-1(F (1))m1 . . . (F (l))ml, m1 +· · ·+ml = k +1, the proposed group cohomology generator f ∈Z ⊂H2k+1(U(1)1 × · · · × U(1)l, U(1)) is f((eiα(1) 1 , . . . , eiα(l) 1 ), . . . , (eiα(1) 2k+1, . . . , eiα(l) 2k+1)) = exp(i(2π)-kα(1) 1 m1 Y m=2 (α(1) 2m-2 + α(1) 2m-1 -[α(1) 2m-2 + α(1) 2m-1]) ! · lY j=2 ml Y m=1 (α(j) 2m1+···+2mj-1+2m-2 + α(j) 2m1+···+2mj-1+2m-1 -[α(j) 2m1+···+2mj-1+2m-2 + α(j) 2m1+···+2mj-1+2m-1])) . (B4) Again eiα(j) m ∈U(1)j, (m = 1, . . . , 2k + 1). Appendix C: Anomalies, cohomology groups and secondary invariants. a. F-move and group cohomologies. In the case of a finite group G, the F-move induces a local transformation of the topological network, with distinct configurations differing by a phase. For 2d bosonic theory, this phase factor 10 is classified by the group cohomology H3(BG, R/Z) (we treat U(1) = R/Z as an additive group), while in the fermionic case the phase factor is determined by a triple (μ, ν, α) ∈Z1(BG, Z2) × Z2(BG, Z2) × C3(BG, R/Z) such that δα = (-1)ν2. The distinction lies in whether the theory depends on spin structure. We will only consider the bosonic case in the continuous group setup. Another crucial feature of a finite group is that the algebraically defined cohomology group H•(G, A) coincides with the topologically defined cohomology group H•(BG, A) of the classifying space BG. For a Lie group G, both H•(BG, A) and "H•(G, A)" can be defined. The former is defined the same way as in the finite group case by considering the classifying space. However, the latter have various versions. For example, it can be defined in terms of multivariable functions on the group; one can then restrict attention to the continuous functions with respect to the standard topology of the group G, this group is usually written as H• c (G, A) with the subscript indicating the continuousness [48]. One can take Gδ as the group G with discrete topology, in such a way we can define H• c (Gδ, A), which is the analog of H•(G, A) in the finite group case. However, both H• c (G, A) and H• c (Gδ, A) differ from H•(BG, A). For instance, if G is compact connected and simply connected, Hn c (G, R) = Hn c (G, R/Z) = 0 for n > 0. As any flat G-bundle is a Gδ-bundle, we focus on discussions of Gδ in the case of flat gauging. The aim is to show that there is a class [Ω] ∈H• c (Gδ, R/Z) that serves as an analog of the Chern-Simons invariant and it is the anomaly. b. Classifying space and secondary invariant. Given the moduli space of flat connection is Hom(π1(M), G)/G one can verify [44] that the classifying space BGδ for flat G-bundle is the following Eilenberg-Maclane space BGδ ∼= K(G, 1) . (C1) Equipped with discrete topology, Gδ behaves like a discrete group and H•(K(G, 1), A) ∼= H•(BGδ, A) ∼= H• c (Gδ, A), where the formed is topologically defined. Note that one has equivalent definitions of flat G-bundle due to Corollary (3.22) of [49] 1. G-bundle admit flat connection; 2. G-bundle admit a set of constant transition functions; 3. G-bundle with a reduction of structure group to Gδ via the canonical map (actually the identity map) ι : Gδ -→G . (C2) It is the second definition that allows one to describe a flat G-bundle by topological defect networks. The third definition can also be rephrased as, there exists a Gδ-bundle whose image under ι∗(the nature map between Gδbundles and G-bundles induced by ι) is the flat G-bundle [50]. The canonical map ι also induces a map at the level of classifying space due to the functoriality of BBι : BGδ -→BG . (C3) For any flat bundle given by φ : M →BG there exists a Gδ-bundle by φδ : M →BGδ such that the following diagram commutes M BGδ BG φ φδ Bι (C4) There is a proposition (proposition 9.1 in [49]) stating that the composite map is zero for a general Lie group G I(G) Chern-Weil -------→H•(BG, R) Bι∗ --→H•(BGδ, R) , (C5) where I(G) is the set of invariant polynomials. This proposition tells us that f ∈In(G) is always mapped to 0 ∈H2n(BGδ, R). We now consider the following commutative diagram of long exact sequences · · · H2n-1(BGδ, R/Z) H2n(BGδ, Z) H2n(BGδ, R) · · · · · · H2n-1(BG, R/Z) H2n(BG, Z) H2n(BG, R) · · · , j∗ β i∗ j∗ j∗ β Bι∗ i∗ Bι∗ j∗ Bι∗ (C6) 11 where the vertical arrows are just Bι∗of various coefficients and the horizontal long exact sequences are induced by the standard short exact sequence of coefficients 0 -→Z i -→R j -→R/Z -→0 , (C7) and β is the connecting homomorphism, or called Bockstein homomorphism, conventionally. • If [f] ∈H2n(BG, R) is the image of some integral class [f]Z ∈H2n(BG, Z), i. e. i∗([f]Z) = [f]. Because Bι∗([f]) = 0 ∈H2n(BGδ, R) and the commutativity of the right quadrat implies that i∗(Bι∗([f]Z)) = 0 ∈ H2n(BGδ, R). • Exactness at the stage of H2n(BGδ, Z) in the upper sequence implies that there exists a class [Tf] ∈H2n-1(BGδ, R/Z) , (C8) such that β([Tf]) = Bι∗([f]Z), where β is the Bockstein homomorphism of the upper long exact sequence. Summarizing the above discussion, we found a canonical map κδ : H2n(BG, Z) -→H2n-1(BGδ, R/Z) [f]Z 7-→[Tf] . (C9) Note that if we replace BGδ by M, and using a flat G-bundle φ get zero map (Chern-Weil ◦φ∗= 0) (now the composite map is zero because of the vanishing of the curvature 2-form) I(G) Chern-Weil -------→H•(BG, R) φ∗ -→H•(M, R) , (C10) then this would lead to a class CS ∈H2n-1(M, R/Z), namely Chern-Simons invariant constructed by Chern and Simons explicitly and we just demonstrated how one can obtain it alternatively via diagram chasing. Hence, fixing a flat bundle φ, we have another canonical map κ : H2n(M, Z) -→H2n-1(M, R/Z) (c[f])Z 7-→CS , (C11) where c([f])Z is the integral characteristic class of [f]Z pulled back to M, i. e. (c[f])Z = φ∗([f]Z). Combining (C4), (C9) and (C11) all together, we arrived at the commutative diagram H2n(BG, Z) H2n-1(BGδ, R/Z) H2n(M, Z) H2n-1(M, R/Z) , κδ φ∗ (φδ)∗ κ (C12) in which the commutativity follows from the functoriality of B-and we have (φδ)∗([Tf]) = CS. (C13) We can apply this to our anomaly discussions. Let n = 2 and for instance take G = SU(2) and c([f]Z) = c2 the second Chern-class. We now denote [Tf] simply by [Ω] and obtain straightforwardly (φδ)∗(Ω) = κ(c2) = CS. c. Transgression and Homotopy fiber. Given a fibration F -→E p -→M , (C14) the trangression/suspension [34] operation allows one to relate cohomology classes of M to cohomology classes of F with degree shifted by 1. Mathematically speaking, the details rely on the Leray-Serre spectral sequence. Here we list some steps relevant to our discussion • One starts from a cohomology class [α] ∈Hp+1(M) and assumes that its pullback in E trivializes [p∗(α)] = 0 ∈ Hp+1(E); • Then there exists a p-cochain, denoted as η ∈Cp(E) such that δη = p∗(α); 12 • The restriction of η to the fiber F is called Tα and δ(Tα) = 0, one thus obtains a cohomology class [Tα] ∈Hp(F) and [α] is called the transgression of [Tα]. A primary example is the universal G-bundle G -→EG p -→BG , (C15) where the cohomology of EG is trivial since it is contractible. Hence we have the suspension map by the above process τ : Hn(BG, A) -→Hn-1 singular(G, A) , (C16) where A is an arbitrary coefficient and x ∈Im(τ) ⊂Hn-1 singular(G, A) is called universal transgressive. In the case of A = Z as coefficient and n = 4, the map τ is in general not surjective, but for compact, connected, simply connected and semi-simple Lie groups it is an isomorphism (in particular for each compact, connected, simply connected simple factor). Another important fibration in our consideration can be introduced as follows. Following the standard construction of homotopy fiber, one can write the map ι into a fibration G -→Eι ι -→G , (C17) where Eι is given as Eι = {(g, γ) ∈Gδ × GI|γ(0) = ι(g)} ⊂Gδ × GI , (C18) and it is homotopy equivalent to Gδ and we sometimes just write Gδ instead. GI is the mapping space of continuous maps from the interval I = [0, 1] to G, it is a topological space endowed with the compact-open topology. The homotopy fiber ̄G is by construction defined as G = {(g, γ) ∈Gδ × GI|γ(0) = ι(g), γ(1) = ι(1Gδ)} ⊂Gδ × GI . (C19) The homotopy fiber construction also extends to Bι and we have the following fibration B ̄G -→BGδ Bι -→BG . (C20) We can now apply the suspension operation to the fibration (C20) and use the proposition (C5). Pick f ∈Ik(G) such that [f] ∈H2n(BG, R), then its determines a class [Tf] ∈H2n-1(BG, R) , (C21) that transgresses to [f]. Appendix D: On H3(U(1)δ, U(1)) A mathematically proper definition of the "N -→∞" limit of ZN is via the direct limit (see [51] for a discussion in the context of physics applications). That is, build a direct system by the inclusion Zn →Zm whenever n divides m and take the limit of this system lim -→Zn = Q/Z, (D1) an explicit description of the limit is by taking the union of all cyclilc subgroups Zn ⊂U(1)δ, i.e. Q/Z = [ n Zn . (D2) The group Q/Z captures all roots of unity in U(1)δ and it is the torsion subgroup of U(1)δ. Since U(1)δ ∼= Rδ/Z, we have U(1)δ ∼= Rδ/Q ⊕Q/Z . (D3) 13 The above isomorphism is due to the fact that the Q/Z is divisible and hence the short exact sequence splits (although not canonical and depends on a choice of basis of Rδ) 0 -→Q/Z -→Rδ/Z -→Rδ/Q -→0 . (D4) Admitting the axiom of choice, Rδ is regarded as infinite dimensional Q-vector spaces with an uncountable basis. In the same vein, the divisible torsion-free group Rδ/Q is also a Q-vector spaces with a uncountable basis. Let us denote the uncountable index set of a basis of Rδ/Q by I then Rδ/Q = ⊕IQ. We will also omit the superscript δ for brevity, as no confusion will arise in what follows. We denote the above groups by A = Rδ/Q, B = Q/Z and G = Rδ/Z. At the level of classifying space, we are dealing with K(G, 1) = K(A, 1)×K(B, 1). We can apply the universal coefficient theorem 0 -→ExtZ(H2(G, Z), U(1)) -→H3(G, U(1)) ∼= H3(K(G, 1), U(1)) -→HomZ(H3(G, Z), U(1)) -→0 , (D5) which yields H3(G, U(1)) ∼= HomZ(H3(G, Z), U(1)) since ExtZ(H2(G, Z), U(1)) vanishes for the divisible group U(1). As G = A ⊕B, we have for H3(G, Z) the K ̈unneth formula [52] 0 -→ M p+q=3 Hp(A, Z) ⊗Hq(B, Z) -→H3(G, Z) -→ M p+q=2 Tor(Hp(A, Z), Hq(B, Z)) -→0 , (D6) and the sequence splits by an unnatural splitting. Now recall the standard result (which can be derived using the techniques in Section 3.F of [53]) Hq(Q/Z, Z) = Z (q = 0) Q/Z (q odd) 0 (q even and q > 0) . (D7) As for H•(A, Z), we can think of it as the homology of the Eilenberg-MacLane space K(A, 1). Then H0(A, Z) = H0(K(A, 1), Z) = Z as Eilenberg-MacLane spaces are connected and H1(A, Z) = H1(K(A, 1), Z) = A which equals the abelianization of π1(K(A, 1)) = A for A is abelian. From the torsion product part we need to compute Tor(H0(A, Z), H2(B, Z)), Tor(H1(A, Z), H1(B, Z)) and Tor(H2(A, Z), H0(B, Z)). By inserting the above results and using the properties of the Tor functor we conclude M p+q=2 Tor(Hp(A, Z), Hq(B, Z)) = 0 . (D8) Now we can consider the tensor product part of (D6), for this we need two extra ingredients H2(A, Z) and H3(A, Z) which appear in Q/Z ⊗H2(A, Z) and Z ⊗H3(A, Z) = H3(A, Z). Given A = ⊕IQ, K(A, 1) can be thought as filtered homotopy colimit of K(Qn, 1) over finite n and one can use K ̈unneth formula to show that Hp(K(Qn, 1), Z) is a Q-vector space for all p > 0. This is also true for Hp(K(A, 1), Z), as filtered colimit of Q-vector spaces are Q-vector spaces [54]. H2(A, Z) as a Q-vector spaces is torsion-free and hence Q/Z ⊗H2(A, Z) = 0 . (D9) We are left with H3(A, Z) and H0(A, Z) ⊗H3(B, Z) = Z ⊗Q/Z = Q/Z. Summarized the above computation we arrive at H3(G, U(1)) = Hom(H3(A, Z) ⊕Q/Z, U(1)) = H3(Rδ/Q, U(1)) ⊕ˆZ , (D10) where ˆZ is the profinite integers can be though as the direct product Πp Rp where Rp is the ring of p-adic integer. Now there is a canonical embedding of Z into ˆZ by sending each integer n to the sequence of its residues modulo all positive integers. This map is modeled by embedding the level-k anomaly polynomial (level-2k Chern-Simons invariant in our normalisation) of U(1) into the 3-cocycle of ˆZ part of H3(U(1)δ, U(1)). Note that in the above consideration, H3(A, Z) = H3(K(A, 1), Z) remains an unspecified Q-vector space, possibly of uncountable dimension. Together with the cohomology ring of H•(K(A, 1), Z), this lies beyond our current understanding, and we hope it will one day be clarified by mathematicians. For non-abelian Lie groups, the complexity of this question increases dramatically. It is in general very hard to determine the cohomology group exactly. We will just briefly mention some contents from [31, 55]. Let G be an arbitrary Lie group with finitely many components, then • the map Bι∗: H•(BG, Z) →H•(BGδ, Z) induced by the canonical map Bι from the Eq. (C3) is injective; • Friedlander-Milnor conjecture. The canonical map Bι induces isomorphisms of homology and cohomology with mod p coefficients, or more generally with any finite coefficient group. 14 Appendix E: Anomaly of gauge transformation In this appendix, we will review the gauge anomaly in 2d, noting that the discussion naturally extends to higher dimensions. Suppose we have a 2d theory living on M2 whose anomaly is characterized by the anomaly polynomial I4(F) I4(F) = k 2TrF ∧F , (E1) which is closed and is locally exact I4(F) = dI3(A) , (E2) where I3(A) ≡CSk(A) is the local Chern-Simons density. Consider an infinitesimal gauge transformation parametrized by Λ, the gauge variation of I3(A) is closed due to dδΛI3(A) = δΛdI3(A) = δΛI4(F) = 0 , (E3) and is also locally exact. Then the anomaly of the partition function Z(A) is computed via the descent equation δΛI3(A) = dA[A; Λ] , (E4) and takes the form δΛ log Z[A] = 2πi Z M2 A[A; Λ] . (E5) We then consider a finite gauge transformation A →g-1Ag + g-1dg , (E6) parametrized by a group function g ≡g(x, y) on M2 and is connected to the identity. Let us construct a 3d manifold M2 × [0, 1] with t ∈[0, 1] and extend the 2d gauge function g into g(t) ≡g(x, y, t) as g(x, y, 0) = e , g(x, y, 1) = g(x, y) . (E7) We also extend the gauge field A on M2 along [0, 1] according to A(t) = g(t)-1A0g(t) + g(t)-1dg(t) , (E8) where A0 is considered as a trivial extension of A on M2 as A0(x, y, t) = A(x, y). The 2d anomaly of the finite gauge transformation g connected to the identity is then evaluated as δg log Z[A] = 2πi Z [0,1]×M2 CSk[A(t)] . (E9) First, one can show that the anomaly given above is independent of the extension g(t) with fixed boundary conditions g(0) = e, g(1) = g. The reason is that the Chern-Simons integral is invariant under the gauge transformation as long as we fix the boundary condition. Second, if g(t) is small, we have Z [0,1]×M2 CS[A(t)] = Z [0,1]×M2 CS[A(t)] - Z [0,1]×M2 CS[A0] = Z [0,1]×M2 δg(t)CS[A] , (E10) where in the middle we use CS[A0] = 0 since A0 is a trivial extension. As shown above, the gauge variation of Chern-Simons density is a total derivative δg(t)CS(A) = dA[A; Λ] , (E11) where Λ(t) is the infinitesimal gauge parameter related by eiΛ(t) = g(t). Using the fact g(0) = e and Λ(0) = 0, we recover the infinitesimal version of the gauge anomaly (E5).
|
2510.14721
|
Fundamental quantum and relativistic formulation of thermal
noise and linear conductance in an 1D quasi-particle ensemble
under ballistic transport-regime
Lino Reggiani
Dipartimento di Matematica e Fisica, ”Ennio de Giorgi”
Universit`a del Salento, via Monteroni 73100 Lecce, Italy.
Federico Intini
Department of Engineering, ”Enzo Ferrari”,
University of Modena and Reggio Emilia,
Via P. Vivarelli, 10, 41125 Modena, Italy
Luca Varani
IES, Univ.
Montpellier,
860 rue de St Priest,
34095 Montpellier cedex 5, France
Abstract
We investigate quantum and quantum-relativistic effects associated with the noise power spec-
trum and the fluctuation–dissipation relation between current–noise spectra and linear–response
conductance at low frequencies of the electromagnetic field. At high frequencies, vacuum catas-
trophe is shown to be avoided by the presence of Casimir force. At low frequencies, the quantum
effect associated with one–dimensional structures under the conditions of ballistic transport typical
at the nanometric scale length are briefly reviewed in terms of a universal quasi-particle approach.
The case of a photon gas inside an appropriate black-body cavity is found to provide a physical
interpretation of the lines spectra of atomic elements within an exact statistical approach based
on a physical interpretation of the fine structure constant, α = 1/137.0560.
1
arXiv:2510.14721v1 [cond-mat.mes-hall] 16 Oct 2025
Keywords:
Vacuum catastrophe, quasi-particle, one-dimensional conductance, ballistic
transport–regime, fine–structure constant.
INTRODUCTION
Starting from the classical noise model of Nyquist (1928) [1] for a 3D conducting sample at
thermal equilibrium, we generalize the spectral density of current fluctuations to the case of
quantum–relativistic conditions that avoid vacuum catastrophe associated with zero–point
energy contributions at increasing frequencies of the electromagnetic spectrum. Then, for
the case of low frequencies, we generalize the electrical current–spectral density to the case
of a 1D ballistic transport–regime that by recovering Landauer (1957) pioneer results [2]
generalizes current fluctuations to the case of a set of non–interacting quasi-particles like:
particle number (number N), neutral particle (mass m), charged particle (charge e), photons
(energy ϵ). Finally we introduce noise and conductance for a black–body cubic cavity at
thermal equilibrium. By applying Planck statistics we propose an interesting scheme where
the emission spectra of any atom can be interpreted as quantized shot–noise of the black-
body cavity with a one–to–one correspondence between the line energy and the thermal
energy of the black-body. Within this scheme the fine–structure constant α ≈1/137 is here
found to be the inverse of the photon—number variance of the corresponding black–body
cavity.
THE VACUUM CATASTROPHE AND THE CASIMIR EFFECT
From the classical Nyquist theorem published in 1928 for the spectral density of current
fluctuations we can write:
SI(f) = 4kBTRe[Y (f)]
(1)
with kB the Boltzmann constant, T the absolute temperature and Re[Y (f)] the real part
of the admittance: i.e. the electrical conductance at the given frequency, f. In 1951 this
classical theorem has been extended to the quantum regime by Callen and Welton [3], and
successively by Kubo [4, 5] in 1957. This extension takes the form:
SI(f) = 4kBTRe[Y (f)]
x
ex −1 + 1
2x
= SP
I (f) + SC
I (f)
(2)
2
with x = hf/(kBT), h being the Planck constant. The second term in the second member
of Eq. (2) (i.e.
1
2x) represents a vacuum–fluctuations (or zero–point) contribution imply-
ing a divergence of the noise spectrum at increasing frequencies. This divergence (vacuum
catastrophe), associated with an infinite number of photons, is analogous to the well-known
ultraviolet catastrophe of the classical black–body radiation spectrum and is in contradic-
tion with experimental results and the Stefan–Boltzmann law. The last member of Eq. (2)
expresses the spectral density as the sum of two contributions due to Planck (SP
I (f)) and
Casimir (SC
I (f)). Indeed the problem of the vacuum catastrophe can be solved by introduc-
ing the quantum–relativistic Casimir effect, and justify the omission of the zero-point con-
tribution for macroscopic systems, thus recovering the Planck distribution and the Nyquist
relation in agreement with the experimental evidence. We recall that the Casimir force is
the attraction of two uncharged material bodies due to modification of the zero-point energy
associated with the electromagnetic modes in the space between them [6, 7].
In 2017 and 2020 Reggiani and Alfinito suggested that the formulation of the fluctua-
tion–dissipation theorem that best describes macroscopic phenomena is the one in which the
zero-point contribution is neglected and that the Casimir contribution is in general neglible
under normal conditions of temperature and volume [8, 9]. In this we differ from the ap-
proach of Callen-Welton and Kubo who did not evaluate the finite value of SC
I (f) in the
spirit of the Casimir effect. As a consequence, SC
I (f) can be considered as a quantum correc-
tion to the Nyquist law that was originally obtained by using classical thermodynamics and
classical electromagnetism only, thus without invoking explicitly any quantum arguments.
The fact that for macroscopic systems SC
I (f) takes negligible values is a reason why till now
its contribution was not explicitly detected from fluctuations spectra at thermal equilibrium.
However, for microscopic systems (i.e. in the case of atomic scale lengths) and sufficiently
low temperatures, its effect should become relevant, as evidenced by experiments on the
Casimir effects [10–13]. An experimental analysis of the cross-over from the Planck to the
Casimir contribution remains a mandatory issue to be investigated.
NOISE IN 1D UNDER BALLISTIC TRANSPORT
Here we consider the limit of T →0 to allow quantum and relativistic conditions to set
in with the temperature remaining the source of thermal equilibrium fluctuations. Starting
3
from the classical case, we recall that for 1D ballistic (b) quasi-particles (qp) at f →0 it is:
Sb
Iqp = (qp)24δN 2
τt
= 4kBTGb
qp
(3)
with Iqp the qp–current fluctuations measured in the outside short–circuit, τt the transit–
time between opposite contacts of the device of length L, that following the fluctuation–
dissipation theorem, implies
Gb
qp = (qp)2δN 2
hc
(4)
with
hc = m
q
u
′2
x L
(5)
the classical dynamic action, with the quasi-particles mean-squared differential (with respect
to carrier number) quadratic velocity component along the x direction, u
′2
x , properly defined
for the classical, quantum and relativistic cases [14, 15].
With the above formalism, we move to the 1D quantum case by taking the following
limiting condition: (i) hc = h, that is the classical action goes to the Planck constant;
δN 2 = N = 2, as dictated by transverse quantum-size conditions and limiting to consider
the first band only including spin degeneracy [16]. Then, Gb
qp becomes the fundamental unit
of the 1D quantum ballistic conductance of the quasi-particle gas given by:
Gb
qp = (qp)2 2
h
(6)
Accordingly, we can define universal units of quantum conductances by introducing differ-
ent type of quasi–particles : number N, charge e, mass m, photon–energy ϵ, electromagnetic
vacuum. Table 1 summarizes these universal units of quantum conductance for a 1D ballis-
tic conductor of length L for a quantum gas of non-interacting quasi-particle (qp) of single
spin/polarization. Being at thermal equilibrium, the associated spectral density of current
fluctuations are obtained by multiplying the given conductance by 4kBT.
RELATIVITY AND THE PHOTON GAS
Within a quasi-particle approach, by analogy with the electrical case [15] we consider
a black-body cubic cavity of side L at thermal equilibrium. Using statistics and Planck
distribution law it can be shown that the following relation between the average number of
4
TABLE I. Fundamental units of quantum conductance for a 1D ballistic conductor of length L for
a quantum gas of non-interacting quasi-particles (qp) of given spin/polarization: particle number
(N), massive (m), electrical (e), single photon-energy (ϵ), with ε0 the vacuum permittivity.
qp
Conductance Gqp
N
1
h
e
e2
h
m
m2
h
ϵ
ϵ2
h
em −vacuum
ε0c
photons N and its variance δN 2 holds [17, 18]:
δN 2 = γN = 1
3
2πkB
ch LT
3
(7)
with γ ≈1.37 the Fano factor.
In the following, we propose to correlate the continuum spectrum of a black–body of
given side and temperature containing a gas of photons described by Planck’s statistics to
the line spectrum of a single atom traditionally associated with the fine structure constant
α (where 1/α ≃137) as [19]:
δN 2 = γN = 1
3
2πkB
ch LT
3
= 1
αN
(8)
with αN the fine–structure constant for the given average photon number. Equation (3)
for the ballistic spectral density of photon current fluctuations, Sb
Iph, gives:
Sb
Iph = 4(hf)2δN 2
τt
= 4kBTGb
ph = 4kBT(hf)2γN
h
(9)
5
with f the photon frequency, τt = L/c the photon transit time along L, Gb
ph the 1D photon
ballistic conductance. Accordingly, for the photon number spectral density we obtain:
τtSb
Nph = δN 2 = γ Nτt
τl
= 1
3
2πkB
ch LT
3
= 1
αN
(10)
where we have introduced the photon lifetime τl = h/(kBT). We remark that Eq. (10)
implies two constraints : the former one
τl
τl
= 1
(11)
where τt and τl represents, respectively, a property of the cavity and a property of the
contacts (the walls of the black-body), and the latter one (for 1/αN = 137)
LT = 1.7 × 10−2 (m K)
(12)
that is reminiscent of the Wien law by taking L = 2πλ. The physical implications of these
constraints can be considered at present an open problem needing further investigation. Fi-
nally, for the spectral density of the photon number fluctuations we can write the expression
representing the fluctuation–dissipation relation for the black body:
τtSb
Nph = δN 2 = γN
(13)
CONCLUSIONS AND REMARKS
Starting from the classical noise model of Nyquist and a reformulation of Drude electrical
conductance for a 3D conducting sample at thermal equilibrium [20], we have generalized
the spectral density of current fluctuations to the case of quantum-relativistic conditions
that avoid the vacuum catastrophe associated with the zero-point energy contribution at
increasing frequencies of the electromagnetic field by introducing the quantum–relativistic
Casimir force.
Then, for the case of low frequencies, we have generalized the electrical current spectral
density and the electrical conductance to the case of a 1D ballistic transport-regime for a
set of non interacting quasi-particles, specifically: dimensionless particle number N, neutral
atomic and sub-atomic particle mass m, charged quasi-particle with charge e, relativistic
neutral quasi-particles like photons with energy ϵ.
6
TABLE II. Relevant energy values of Hydrogen spectra and equivalent temperature together with
the corresponding average photon numbers N used in Fig. 1.
Spectral-type Temperature (K) Energy (eV)
N
HFSC
6.9 × 10−2
5.9 × 10−6
2.3 × 10−1
FSC
5.2 × 10−1
4.5 × 10−5
1.0 × 102
Paschen
0.77 ÷ 1.8 × 104
0.66 ÷ 1.5 × 100 0.32 ÷ 3.8 × 1015
Balmer
2.2 ÷ 3.7 × 104
1.9 ÷ 3.2 × 100
0.74 ÷ 3.7 × 1016
Lyman
1.2 ÷ 1.6 × 105
1.0 ÷ 1.4 × 101
1.2 ÷ 2.8 × 1018
Finally, we propose an interesting definition of photon conductance and photon noise for
a black-body cubic cavity at thermal equilibrium with given volume and temperature that
could help to interpret recent experiments on photon counting statistics [21, 22] and able to
provide a physical interpretation of the fine and hyperfine structure constants evidenced in
the atomic line spectra of hydrogen and pertaining to any atomic element of the periodic
Mendeleev table. Indeed, in our model the fine and hyperfine spectral constants coincide
with the inverse values of the respective variance of photon number fluctuations. The plot
reported in Fig. 1 illustrates the graphical identification of the fit to obtain the value of
N 1
α that satisfies Eq. (10) within the same accuracy of the given value of
1
α. Then, from
Eq. (10) the proper value of the quantity L 1
αT 1
α is obtained as presented, for instance, in
Eq. (12). According to convenience, the single values of L 1
α and T 1
α can be obtained once
their product is determined by the fitting. As a significant example, by taking 1
α = 137.0
and L 1
α = 0.033 m it is T 1
α = 0.5156 K and N 1
α = 100.1.
For completeness, the Table II reports the numerical values used for the fundamental
constants and the corresponding physical quantities evaluated within the Planck’s law for
photon statistics used in Fig. 1
ACKNOWLEDGMENTS
Prof. Tilmann Kuhn from M¨unster University is warmly thanked for the very valuable
comments he provided on the subject.
7
[1] H. Nyquist, Phys. Rev. 32 (1928) 110, https://doi.org/10.1103/PhysRev.32.110
[2] R. Landauer, IBM J. Res. Dev. 3 (1957) 223, https://doi.org/10.1147/rd.13.0223
[3] H. Callen and T. Welton, Phys. Rev. 83 (1951) 34, https://doi.org/10.1103/PhysRev.83.
34
[4] R. Kubo, J. Phys. Soc. Japan 12 (1957) 570, https://doi.org/10.1143/JPSJ.12.570,
[5] R. Kubo, Rep. Prog. Phys. 29 (1966) 255, https://dx.doi.org/10.1088/0034-4885/29/1/
306
[6] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap. Ser. B 51 (1948) 793
[7] S.K. Lamoreaux, Rep. Prog. Phys. 68 (2005) 201, https://doi.org/10.1088/0034-4885/
68/1/R04
[8] L. Reggiani, and E. Alfinito, Fluct. Noise Lett. 16 (2017) 1771002, https://doi.org/10.
1142/S021947751771002X
[9] L. Reggiani and E. Alfinito(2020),Front. Phys. 8 (2020) 238, http://dx.doi.org/10.3389/
fphy.2020.00238
[10] K. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scien-
tific, Singapore, 2001)
[11] A. Edery, J Stat Mech. 06 (2006) P06007, https://dx.doi.org/10.1088/1742-5468/2006/
06/P06007
[12] F. Schmidt and H. Diehl, Phys Rev Lett. 101 (2008) 10601, https://doi.org/10.1103/
PhysRevLett.101.100601
[13] G. Auletta, M. Fortunato and G. Parisi, Quantum Mechanics (Cambridge University, Cam-
bridge, 2009)
[14] S. Gantsevich, R. Katilius and V. Gurevich, Rivista Nuovo Cimento 2 (1979) 1-87.
[15] L. Reggiani, E. Alfinito and T. Kuhn, Phys Rev. E 94 (2016) 032112, https://doi.org/10.
1103/PhysRevE.94.032112
[16] A. Greiner, L. Reggiani, T. Kuhn and L. Varani, Semicond. Sci. Technol. 15 (2000) 1071,
https://dx.doi.org/10.1088/0268-1242/15/11/311
[17] H. Leff, Am. J. Phys. 70 (2002) 792, https://doi.org/10.1119/1.1479743
[18] H. Leff, Am. J. Phys. 83 (2015) 362, https://doi.org/10.1119/1.4904322
8
[19] T. Kinoshita, Rep. Prog. Phys., 59 (1996) 1459, https://dx.doi.org/10.1088/0034-4885/
59/11/003
[20] P.
Drude,
Annalen
der
Physik,
306
(1900)
566,
https://doi.org/10.1002/andp.
19003060312
[21] S. Virally, J. O. Simoneau, C. Lupien, and B. Reulet, Phys. Rev. A
93, 043813 (2016).
https://doi.org/10.1103/PhysRevA.93.043813
[22] S. Beaudoin, E. Pinsolle and B.Reulet, Phys. Rev. Res. 6, 043238 (2024). https://doi.org/
10.1103/PhysRevResearch.6.043238
9
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
kBT (eV)
10-6
10-3
100
103
106
109
1012
1015
1018
N
10-6
10-4
10-2
100
102
104
106
108
1010
1012
1014
1016
1018
1020
δN2
FSC
HFSC
Lyman
Balmer
Paschen
__
___
FIG. 1. Plot summarizing the variance of the photon numbers δN2 (or the inverse fine–structure
constant of the H atom 1/α) as a function of the photon energy kBT and average photon number N
including the available emission spectra of the Paschen, Balmer and Lyman series (colored shades
areas), obtained by the statistical scheme property of the black–body photon gas as described by
Planck’s distribution law. The universal behavior of the Fano factor is evidenced by the straight line
representing the law: δN2 = γN. Analogous spectra of other atoms pertaining to the Mendeleev
periodic table could be shown in the same way. The two large points correspond to the values of
the hydrogen fine-structure constant (FSC) and hyperfine–structure constant (HFSC). The reverse
procedure should require the determination of the average number of photons under the conditions
of temperature and volume L 1
α T 1
α = 1.7 × 10−2 (m K) of the black-body at thermal equilibrium.
10
|
Fundamental quantum and relativistic formulation of thermal noise and linear conductance in an 1D quasi-particle ensemble under ballistic transport-regime Lino Reggiani Dipartimento di Matematica e Fisica, "Ennio de Giorgi" Universit`a del Salento, via Monteroni 73100 Lecce, Italy. Federico Intini "Enzo Ferrari", . Vivarelli, 10, 41125 Modena, Italy Luca Varani IES, Univ. Montpellier, 860 rue de St Priest, 34095 Montpellier cedex 5, France Abstract We investigate quantum and quantum-relativistic effects associated with the noise power spectrum and the fluctuation-dissipation relation between current-noise spectra and linear-response conductance at low frequencies of the electromagnetic field. At high frequencies, vacuum catastrophe is shown to be avoided by the presence of Casimir force. At low frequencies, the quantum effect associated with one-dimensional structures under the conditions of ballistic transport typical at the nanometric scale length are briefly reviewed in terms of a universal quasi-particle approach. The case of a photon gas inside an appropriate black-body cavity is found to provide a physical interpretation of the lines spectra of atomic elements within an exact statistical approach based on a physical interpretation of the fine structure constant, α = 1/137.0560. 1 16 Oct 2025 Keywords: Vacuum catastrophe, quasi-particle, one-dimensional conductance, ballistic transport-regime, fine-structure constant. INTRODUCTION Starting from the classical noise model of Nyquist (1928) [1] for a 3D conducting sample at thermal equilibrium, we generalize the spectral density of current fluctuations to the case of quantum-relativistic conditions that avoid vacuum catastrophe associated with zero-point energy contributions at increasing frequencies of the electromagnetic spectrum. Then, for the case of low frequencies, we generalize the electrical current-spectral density to the case of a 1D ballistic transport-regime that by recovering Landauer (1957) pioneer results [2] generalizes current fluctuations to the case of a set of non-interacting quasi-particles like: particle number (number N), neutral particle (mass m), charged particle (charge e), photons (energy ε). Finally we introduce noise and conductance for a black-body cubic cavity at thermal equilibrium. By applying Planck statistics we propose an interesting scheme where the emission spectra of any atom can be interpreted as quantized shot-noise of the blackbody cavity with a one-to-one correspondence between the line energy and the thermal energy of the black-body. Within this scheme the fine-structure constant α ≈1/137 is here found to be the inverse of the photon-number variance of the corresponding black-body cavity. THE VACUUM CATASTROPHE AND THE CASIMIR EFFECT From the classical Nyquist theorem published in 1928 for the spectral density of current fluctuations we can write: SI(f) = 4kBTRe[Y (f)] (1) with kB the Boltzmann constant, T the absolute temperature and Re[Y (f)] the real part of the admittance: i.e. the electrical conductance at the given frequency, f. In 1951 this classical theorem has been extended to the quantum regime by Callen and Welton [3], and successively by Kubo [4, 5] in 1957. This extension takes the form: SI(f) = 4kBTRe[Y (f)] x ex -1 + 1 2x = SP I (f) + SC I (f) (2) 2 with x = hf/(kBT), h being the Planck constant. The second term in the second member of Eq. (2) (i.e. 1 2x) represents a vacuum-fluctuations (or zero-point) contribution implying a divergence of the noise spectrum at increasing frequencies. This divergence (vacuum catastrophe), associated with an infinite number of photons, is analogous to the well-known ultraviolet catastrophe of the classical black-body radiation spectrum and is in contradiction with experimental results and the Stefan-Boltzmann law. The last member of Eq. (2) expresses the spectral density as the sum of two contributions due to Planck (SP I (f)) and Casimir (SC I (f)). Indeed the problem of the vacuum catastrophe can be solved by introducing the quantum-relativistic Casimir effect, and justify the omission of the zero-point contribution for macroscopic systems, thus recovering the Planck distribution and the Nyquist relation in agreement with the experimental evidence. We recall that the Casimir force is the attraction of two uncharged material bodies due to modification of the zero-point energy associated with the electromagnetic modes in the space between them [6, 7]. In 2017 and 2020 Reggiani and Alfinito suggested that the formulation of the fluctuation-dissipation theorem that best describes macroscopic phenomena is the one in which the zero-point contribution is neglected and that the Casimir contribution is in general neglible under normal conditions of temperature and volume [8, 9]. In this we differ from the approach of Callen-Welton and Kubo who did not evaluate the finite value of SC I (f) in the spirit of the Casimir effect. As a consequence, SC I (f) can be considered as a quantum correction to the Nyquist law that was originally obtained by using classical thermodynamics and classical electromagnetism only, thus without invoking explicitly any quantum arguments. The fact that for macroscopic systems SC I (f) takes negligible values is a reason why till now its contribution was not explicitly detected from fluctuations spectra at thermal equilibrium. However, for microscopic systems (i.e. in the case of atomic scale lengths) and sufficiently low temperatures, its effect should become relevant, as evidenced by experiments on the Casimir effects [10-13]. An experimental analysis of the cross-over from the Planck to the Casimir contribution remains a mandatory issue to be investigated. NOISE IN 1D UNDER BALLISTIC TRANSPORT Here we consider the limit of T →0 to allow quantum and relativistic conditions to set in with the temperature remaining the source of thermal equilibrium fluctuations. Starting 3 from the classical case, we recall that for 1D ballistic (b) quasi-particles (qp) at f →0 it is: Sb Iqp = (qp)24δN 2 τt = 4kBTGb qp (3) with Iqp the qp-current fluctuations measured in the outside short-circuit, τt the transittime between opposite contacts of the device of length L, that following the fluctuationdissipation theorem, implies Gb qp = (qp)2δN 2 hc (4) with hc = m q u ′2 x L (5) the classical dynamic action, with the quasi-particles mean-squared differential (with respect to carrier number) quadratic velocity component along the x direction, u ′2 x , properly defined for the classical, quantum and relativistic cases [14, 15]. With the above formalism, we move to the 1D quantum case by taking the following limiting condition: (i) hc = h, that is the classical action goes to the Planck constant; δN 2 = N = 2, as dictated by transverse quantum-size conditions and limiting to consider the first band only including spin degeneracy [16]. Then, Gb qp becomes the fundamental unit of the 1D quantum ballistic conductance of the quasi-particle gas given by: Gb qp = (qp)2 2 h (6) Accordingly, we can define universal units of quantum conductances by introducing different type of quasi-particles : number N, charge e, mass m, photon-energy ε, electromagnetic vacuum. Table 1 summarizes these universal units of quantum conductance for a 1D ballistic conductor of length L for a quantum gas of non-interacting quasi-particle (qp) of single spin/polarization. Being at thermal equilibrium, the associated spectral density of current fluctuations are obtained by multiplying the given conductance by 4kBT. RELATIVITY AND THE PHOTON GAS Within a quasi-particle approach, by analogy with the electrical case [15] we consider a black-body cubic cavity of side L at thermal equilibrium. Using statistics and Planck distribution law it can be shown that the following relation between the average number of 4 TABLE I. Fundamental units of quantum conductance for a 1D ballistic conductor of length L for a quantum gas of non-interacting quasi-particles (qp) of given spin/polarization: particle number (N), massive (m), electrical (e), single photon-energy (ε), with ε0 the vacuum permittivity. qp Conductance Gqp N 1 h e e2 h m m2 h ε ε2 h em -vacuum ε0c photons N and its variance δN 2 holds [17, 18]: δN 2 = γN = 1 3 2πkB ch LT 3 (7) with γ ≈1.37 the Fano factor. In the following, we propose to correlate the continuum spectrum of a black-body of given side and temperature containing a gas of photons described by Planck's statistics to the line spectrum of a single atom traditionally associated with the fine structure constant α (where 1/α ≃137) as [19]: δN 2 = γN = 1 3 2πkB ch LT 3 = 1 αN (8) with αN the fine-structure constant for the given average photon number. Equation (3) for the ballistic spectral density of photon current fluctuations, Sb Iph, gives: Sb Iph = 4(hf)2δN 2 τt = 4kBTGb ph = 4kBT(hf)2γN h (9) 5 with f the photon frequency, τt = L/c the photon transit time along L, Gb ph the 1D photon ballistic conductance. Accordingly, for the photon number spectral density we obtain: τtSb Nph = δN 2 = γ Nτt τl = 1 3 2πkB ch LT 3 = 1 αN (10) where we have introduced the photon lifetime τl = h/(kBT). We remark that Eq. (10) implies two constraints : the former one τl τl = 1 (11) where τt and τl represents, respectively, a property of the cavity and a property of the contacts (the walls of the black-body), and the latter one (for 1/αN = 137) LT = 1.7 × 10-2 (m K) (12) that is reminiscent of the Wien law by taking L = 2πλ. The physical implications of these constraints can be considered at present an open problem needing further investigation. Finally, for the spectral density of the photon number fluctuations we can write the expression representing the fluctuation-dissipation relation for the black body: τtSb Nph = δN 2 = γN (13) CONCLUSIONS AND REMARKS Starting from the classical noise model of Nyquist and a reformulation of Drude electrical conductance for a 3D conducting sample at thermal equilibrium [20], we have generalized the spectral density of current fluctuations to the case of quantum-relativistic conditions that avoid the vacuum catastrophe associated with the zero-point energy contribution at increasing frequencies of the electromagnetic field by introducing the quantum-relativistic Casimir force. Then, for the case of low frequencies, we have generalized the electrical current spectral density and the electrical conductance to the case of a 1D ballistic transport-regime for a set of non interacting quasi-particles, specifically: dimensionless particle number N, neutral atomic and sub-atomic particle mass m, charged quasi-particle with charge e, relativistic neutral quasi-particles like photons with energy ε. 6 TABLE II. Relevant energy values of Hydrogen spectra and equivalent temperature together with the corresponding average photon numbers N used in Fig. 1. Spectral-type Temperature (K) Energy (eV) N HFSC 6.9 × 10-2 5.9 × 10-6 2.3 × 10-1 FSC 5.2 × 10-1 4.5 × 10-5 1.0 × 102 Paschen 0.77 ÷ 1.8 × 104 0.66 ÷ 1.5 × 100 0.32 ÷ 3.8 × 1015 Balmer 2.2 ÷ 3.7 × 104 1.9 ÷ 3.2 × 100 0.74 ÷ 3.7 × 1016 Lyman 1.2 ÷ 1.6 × 105 1.0 ÷ 1.4 × 101 1.2 ÷ 2.8 × 1018 Finally, we propose an interesting definition of photon conductance and photon noise for a black-body cubic cavity at thermal equilibrium with given volume and temperature that could help to interpret recent experiments on photon counting statistics [21, 22] and able to provide a physical interpretation of the fine and hyperfine structure constants evidenced in the atomic line spectra of hydrogen and pertaining to any atomic element of the periodic Mendeleev table. Indeed, in our model the fine and hyperfine spectral constants coincide with the inverse values of the respective variance of photon number fluctuations. The plot reported in Fig. 1 illustrates the graphical identification of the fit to obtain the value of N 1 α that satisfies Eq. (10) within the same accuracy of the given value of 1 α. Then, from Eq. (10) the proper value of the quantity L 1 αT 1 α is obtained as presented, for instance, in Eq. (12). According to convenience, the single values of L 1 α and T 1 α can be obtained once their product is determined by the fitting. As a significant example, by taking 1 α = 137.0 and L 1 α = 0.033 m it is T 1 α = 0.5156 K and N 1 α = 100.1. For completeness, the Table II reports the numerical values used for the fundamental constants and the corresponding physical quantities evaluated within the Planck's law for photon statistics used in Fig. 1 ACKNOWLEDGMENTS Prof. Tilmann Kuhn from M ̈unster University is warmly thanked for the very valuable comments he provided on the subject. 7 [1] H. Nyquist, Phys. Rev. 32 (1928) 110, https://doi.org/10.1103/PhysRev.32.110 [2] R. Landauer, IBM J. Res. Dev. 3 (1957) 223, https://doi.org/10.1147/rd.13.0223 [3] H. Callen and T. Welton, Phys. Rev. 83 (1951) 34, https://doi.org/10.1103/PhysRev.83. 34 [4] R. Kubo, J. Phys. Soc. Japan 12 (1957) 570, https://doi.org/10.1143/JPSJ.12.570, [5] R. Kubo, Rep. Prog. Phys. 29 (1966) 255, https://dx.doi.org/10.1088/0034-4885/29/1/ 306 [6] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetenschap. Ser. B 51 (1948) 793 [7] S.K. Lamoreaux, Rep. Prog. Phys. 68 (2005) 201, https://doi.org/10.1088/0034-4885/ 68/1/R04 [8] L. Reggiani, and E. Alfinito, Fluct. Noise Lett. 16 (2017) 1771002, https://doi.org/10. 1142/S021947751771002X [9] L. Reggiani and E. Alfinito(2020),Front. Phys. 8 (2020) 238, http://dx.doi.org/10.3389/ fphy.2020.00238 [10] K. Milton, The Casimir Effect: Physical Manifestations of Zero-Point Energy (World Scientific, Singapore, 2001) [11] A. Edery, J Stat Mech. 06 (2006) P06007, https://dx.doi.org/10.1088/1742-5468/2006/ 06/P06007 [12] F. Schmidt and H. Diehl, Phys Rev Lett. 101 (2008) 10601, https://doi.org/10.1103/ PhysRevLett.101.100601 [13] G. Auletta, M. Fortunato and G. Parisi, Quantum Mechanics (Cambridge University, Cambridge, 2009) [14] S. Gantsevich, R. Katilius and V. Gurevich, Rivista Nuovo Cimento 2 (1979) 1-87. [15] L. Reggiani, E. Alfinito and T. Kuhn, Phys Rev. E 94 (2016) 032112, https://doi.org/10. 1103/PhysRevE.94.032112 [16] A. Greiner, L. Reggiani, T. Kuhn and L. Varani, Semicond. Sci. Technol. 15 (2000) 1071, https://dx.doi.org/10.1088/0268-1242/15/11/311 [17] H. Leff, Am. J. Phys. 70 (2002) 792, https://doi.org/10.1119/1.1479743 [18] H. Leff, Am. J. Phys. 83 (2015) 362, https://doi.org/10.1119/1.4904322 8 [19] T. Kinoshita, Rep. Prog. Phys., 59 (1996) 1459, https://dx.doi.org/10.1088/0034-4885/ 59/11/003 [20] P. Drude, Annalen der Physik, 306 (1900) 566, https://doi.org/10.1002/andp. 19003060312 [21] S. Virally, J. O. Simoneau, C. Lupien, and B. Reulet, Phys. Rev. A 93, 043813 (2016). https://doi.org/10.1103/PhysRevA.93.043813 [22] S. Beaudoin, E. Pinsolle and B.Reulet, Phys. Rev. Res. 6, 043238 (2024). https://doi.org/ 10.1103/PhysRevResearch.6.043238 9 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 kBT (eV) 10-6 10-3 100 103 106 109 1012 1015 1018 N 10-6 10-4 10-2 100 102 104 106 108 1010 1012 1014 1016 1018 1020 δN2 FSC HFSC Lyman Balmer Paschen __ ___ FIG. 1. Plot summarizing the variance of the photon numbers δN2 (or the inverse fine-structure constant of the H atom 1/α) as a function of the photon energy kBT and average photon number N including the available emission spectra of the Paschen, Balmer and Lyman series (colored shades areas), obtained by the statistical scheme property of the black-body photon gas as described by Planck's distribution law. The universal behavior of the Fano factor is evidenced by the straight line representing the law: δN2 = γN. Analogous spectra of other atoms pertaining to the Mendeleev periodic table could be shown in the same way. The two large points correspond to the values of the hydrogen fine-structure constant (FSC) and hyperfine-structure constant (HFSC). The reverse procedure should require the determination of the average number of photons under the conditions of temperature and volume L 1 α T 1 α = 1.7 × 10-2 (m K) of the black-body at thermal equilibrium. 10
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2510.14724
|
Crossed surface flat bands in three-dimensional superconducting altermagnets
Yuri Fukaya,1 Bo Lu,2 Keiji Yada,3 Yukio Tanaka,3 and Jorge Cayao4
1Faculty of Environmental Life, Natural Science and Technology, Okayama University, 700-8530 Okayama, Japan
2Department of Physics, Tianjin University, 300354 Tianjin, China
3Department of Applied Physics, Nagoya University, 464-8603 Nagoya, Japan
4Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
(Dated: October 17, 2025)
Superconducting altermagnets have proven to be a promising ground for emergent phenomena but
their study has involved two dimensional systems. In this work, we investigate three-dimensional d-
and g-wave altermagnets with chiral d-wave superconductivity and show the formation of crossed
surface flat bands due to the underlying symmetries. We find that these crossed flat bands appear
at zero energy in the surface along z due to the superconducting nodal lines in the xy-plane, while
the number of corners is determined by the crystal symmetry of altermagnets.
We also show
that the superconducting nodal lines give rise to Bogoliubov-Fermi surfaces, which then affect the
appearance of zero-energy arcs in the surface along x. Moreover, we demonstrate that the crossed
surface flat bands, surface arcs, and Bogoliubov-Fermi surfaces give rise to distinct signals in charge
conductance, hence offering a solid way for their detection and paving the way for realizing higher
dimensional topological phases using altermagnets.
The discovery of altermagnetism has recently triggered
great interest due to its potential for realizing emergent
phenomena and spintronic applications [1–4]. Altermag-
nets (AMs) are characterized by a zero net magnetization
as a result of their momentum dependent spin splitting of
energy bands and anisotropic spin-polarized Fermi sur-
faces [2, 5–14]. While these altermagnetic properties have
proven crucial in the normal state [1], it has been recently
shown that combining AMs with superconductivity of-
fers unprecedented opportunities [4, 15]. For instance,
altermagnetism has been shown to induce key bulk and
surface phenomena [4, 15], including anomalous Joseph-
son transport [16–22], Majorana physics [23–29], exotic
superconducting pairs [30–39], nonreciprocal transport
[40–43], nontrivial light-matter coupling [36, 37], as well
as distinct types of stable subgap states [44, 45].
The discovery of stable subgap states in AMs is of par-
ticular relevance because they signal the emergence of un-
usual superconducting states protected by symmetry and
topology. In this regard, altermagnetism has been shown
to induce Bogoliubov-Fermi surfaces (BFSs), flat bands,
and Andreev bound states (ABSs), all exhibiting strong
ties to the underlying altermagnetic and superconduct-
ing symmetries [44, 45]. While these ideas indeed show
the potential of AMs for designing topological supercon-
ducting phases, the few existing studies have only ad-
dressed two-dimensional (2D) systems, thus leaving un-
explored a large family of 3D AMs. One of the most inter-
esting systems is Sr2RuO4 superconductors, which have
very recently been predicted to host altermagnetic order
[46, 47] as a phase that coexists with a very likely 3D chi-
ral d-wave superconductivity [48–51] probed by nuclear
magnetic resonance [52, 53].
It is therefore natural to
wonder about the design and detection of higher dimen-
sional topological superconducting phases by exploiting
the symmetry and topology of 3D superconducting AMs.
In this work, we consider 3D superconducting AMs
[Fig. 1] and demonstrate the emergence of crossed flat
bands as a generic topological surface phenomenon due
to the interplay between intrinsic superconducting and
altermagnetic symmetries.
In particular, we focus on
d- and g-wave AMs with chiral d-wave superconductiv-
ity, which are very likely to appear in Sr2RuO4, see
[46, 47] and [48–51]. We discover that the xy-plane nodal
lines of chiral d-wave superconductivity ensure that the
crossed flat bands appear at zero energy on the [001] sur-
face, while their corners are determined by the altermag-
netic nodes. The crossed surface flat bands represent the
higher dimensional analog of zero-energy ABSs [54, 55]
in 3D superconducting AMs, thus broadening their ini-
tial conception in Weyl superconductors [56] and the idea
that flat bands inherit the shape of the projected Fermi
surface [57]. Moreover, we find that the superconducting
nodal lines induce BFSs, which then modify the forma-
tion of arcs on the [100] surface. We further show that
the crossed flat bands, arc states, and BFSs produce very
distinctive conductance signatures which can be used for
their detection. Our findings establish 3D superconduct-
ing AMs as a fertile ground for designing symmetry pro-
tected topological phenomena.
3D superconducting AMs.— We consider a 3D super-
conducting AM modelled by the following Bogoliubov-de
Gennes (BdG) Hamiltonian,
ˆHα
BdG(k) = ε(k)ˆσ0ˆτ3 + ˆHα
AM(k) + ˆHSC(k) ,
(1)
where ε(k) = −µ −2t cos kx −2t cos ky −2t cos kz, while
ˆHα
AM(k) and ˆHSC(k) describe the altermagnetic and su-
perconducting orders, with α denoting the type of alter-
magnetism (see below). Here, k = (kx, ky, kz), µ is the
chemical potential, t is the hopping integral, while ˆσj and
ˆτj are the j-th Pauli matrices in spin and particle-hole
arXiv:2510.14724v1 [cond-mat.supr-con] 16 Oct 2025
2
Superconducting
altermagnet
x
y
z
(a)
Normal
metal
kx
ky
kz
(c)
(d)
-wave
altermagnet
Crossed flat band
dxy
"
#
Superconducting
altermagnet
(b)
"
#
Crossed flat band
altermagnet
point node
Bogoliubov
Fermi surface
Surface arc
-wave
gxy(x2−y2)
(f)
(g)
(e)
FIG. 1. (a,b) Sketches of the studied junctions along z- (a)
and x-directions (b), formed by a 3D superconducting alter-
magnet (light blue) and a normal metal (light red).
(c-f)
Bulk and surface properties of dxy- and gxy(x2−y2)-wave al-
termagnets with chiral d-wave superconductivity. (c,d) Bulk
Fermi volumes for dxy- (c) and gxy(x2−y2)-wave (d) altermag-
nets, where the cyan and magenta colors indicate up and
down spins. The line nodes of the Fermi volumes give rise
to Bogoliubov-Fermi surfaces (gray) under chiral d-wave su-
perconductivity.
(e) A projection of the Fermi volumes on
the [100] surface along x leads to surface arc states (orange),
whose ends are marked by the point nodes. (f,g) When pro-
jecting the Fermi volumes on the [001] surface along z, crossed
flat bands emerge (green) with their corners defined by the
nodes of the 2D spin-polarized altermagnetic Fermi surfaces,
indicated by cyan and magenta ellipses for down and up spins.
The crossed flat bands appear at zero energy, which is ensured
by the nodal lines of chiral d-wave of superconductivity.
subspaces, respectively. Altermagnetism is modelled by
ˆHα
AM(k) = M α
k ˆσ3ˆτ3 ,
(2)
where M α
k characterizes α-wave altermagnetic order with
α = d, g. More precisely, we focus on dxy- and dx2−y2-
wave AMs described by M d1
k
= 2td1 sin kx sin ky, M d2
k
=
td2(cos kx −cos ky). We also consider g-wave AMs with
M g
k = 4tg sin kx sin ky(cos kx−cos ky). Thus, td1, td2, and
tg represent the strength of dxy-, dx2−y2-, and gxy(x2−y2)-
wave AMs. While ˆHAM(k) is described by an effective
2D model, our findings remain in 3D AMs [58], see S3
in the Supplementary Material (SM) [59].
Moreover,
3D chiral superconductivity is modelled by ˆHSC(k) =
−∆sin kz[sin kxˆτ2 + sin kyˆτ1]ˆσ2, with pair amplitude ∆.
The considered models are motivated by recent studies
on Sr2RuO4, which predict altermagnetic order [46, 47]
and also a very likely 3D chiral d-wave superconductiv-
ity [48–51]. Even though we focus on a particular 3D
nodal superconductor, our main analysis can be straight-
forwardly generalized to other types of superconducting
AMs with unconventional superconductivity.
Emergence of crossed surface flat bands.—We are in-
terested in unveiling the impact of altermagnetism on su-
perconductivity in a 3D superconducting AM modelled
by Eq. (1). For this purpose, it is useful to first inspect
the energy bands, which are obtained as
Es,α
k,± = sM α
k ±
p
ε2(k) + |ψ(k)|2 ,
(3)
where s = ± and ψ(k) = ∆sin kz[sin kx +i sin ky]. These
energy bands help understand the appearance of bulk
phases.
In the absence of AMs, the 3D chiral super-
conductor has line nodes in the xy-plane, while point
nodes in the z-plane [Fig. 1(c,d)]. These line and point
nodes give rise, respectively, to zero-energy surface flat
bands on the [001] surface and to surface arc states on
the [100] surface [54, 55, 57, 60]; the shape of the flat
bands is determined by that of the projected Fermi sur-
face, with a disk shape for a spherical Fermi surface [60].
In the presence of altermagnetism, we obtain Es,α
k,−= 0
for [M α
k ]2 = ε2(k) + |ψ(k)|2. This defines the formation
of a BFS, and shows that the line nodes of the chiral
d-wave superconductor become BFSs due to altermag-
netism, except for the nodal lines of AMs. Interestingly,
altermagnetism strongly deforms the surface states of the
chiral superconductor, giving rise to crossed flat bands
and modified surface arcs in the [001] and [100] surfaces
[Fig. 1(e,f,g). While the surface arcs are inherited from
the chiral superconductor, the crossed flat bands are en-
tirely due to the interplay between altermagnetism and
chiral d-wave superconductivity: by projecting the al-
termagnetic Fermi surfaces, we identify that the alter-
magnetic nodes (spin-degenerate lines in 3D) define the
number of corners of the crossed flat band. Moreover,
due to the nodes of the 3D chiral d-wave superconduc-
tivity, the crossed flat bands appear at zero energy. This
analysis also applies to spin-triplet pz-wave superconduc-
tivity [61]. Thus, the emergence of crossed flat bands is
a generic phenomenon resulting from the interplay be-
tween altermagnetism and nodal superconductivity and
hence tied to the inherent crystal symmetries.
Topological origin of the crossed surface flat bands.—
The symmetries required for the emergence of crossed
flat bands also imply that they have a topological ori-
gin [57, 61–64], which we show here. In 3D supercon-
ductors with nodal lines but without altermagnetism, a
nonzero 1D winding number can be obtained with the
chiral operator by combining the pseudo-time-reversal
(pTRS) and particle-hole (PH) symmetries [60], where
pTRS is the product of conventional TRS and local gauge
transformation. This local gauge transformation is intro-
duced to cancel out the inner phase of the pair potential.
In the presence of altermagnetism, the chiral symmetry
is broken because of the broken TRS by AMs [1, 4]. Still,
the Hamiltonian keeps the pseudo-magnetic mirror sym-
metry (pMMS), which is given by the product of pTRS
3
and mirror symmetry. For dxy- and dx2−y2-wave AMs,
the pMMSs are given, respectively, by
ˆΘ
′
k ˆHd1
BdG(kx, ky, kz)ˆΘ
′†
k = [ ˆHd1
BdG(−νkx, νky, kz)]∗
(4)
ˆΘ
′
k ˆHd2
BdG(kx, ky, kz)ˆΘ
′†
k = [ ˆHd2
BdG(−νky, −νkx, kz)]∗(5)
where ˆΘ
′
k = ˆU †
ϕk ˆΘ ˆUϕk denotes the pMMS, ˆΘ = iˆσ2ˆτ0
is the conventional TRS, ˆUϕk = e−iϕkˆσ0ˆτ3/2 is the local
gauge transformation, and ϕk = tan−1(sin ky/ sin kx) is
the phase of the pair potential. Here, ν = +1 and ν =
−1 in Eq. (4) indicate pMMSs for dxy-wave AMs in the
xz- and yz-planes, respectively. Also, ν = +1 and ν =
−1 in Eq. (5) indicate pMMSs for dx2−y2-wave AMs in
the diagonal planes on x = y and x = −y, respectively.
For gxy(x2−y2)-wave AMs, the system has both pMMSs
given by Eq. (4) and Eq. (5). In addition to the pMMSs,
ˆHα
BdG(k) has particle-hole symmetry given by
ˆC ˆHα
BdG(kx, ky, kz) ˆC† = −[ ˆHα
BdG(−kx, −ky, −kz)]∗
(6)
with ˆC = ˆσ0ˆτ1. Then, ˆHα
BdG(k) anticommutes with the
chiral operator ˆΓk = −i ˆC ˆΘ
′
k on the magnetic mirror
plane. Concrete forms of ˆΓk on each mirror plane are
given in S1.3 of [59]. Importantly, these magnetic mirror
and chiral symmetries are still available on the smooth
surfaces, which is perpendicular to the magnetic mir-
ror planes, allowing for the symmetry-protection of sur-
face states as a unique effect due to altermagnetism and
nodal superconductivity. For example, a vertical pMMS
is given in Eq.
(4) and the corresponding chiral sym-
metry is accessible on the kx- and ky-axes in the surface
Brillouin zone. Hence, a winding number can be obtained
on the symmetric lines, whose nonzero values signal the
appearance of topological phases with topologically pro-
tected surface states [59]. Thus, a nonzero winding num-
ber along the kx, ky-axis and/or the diagonal direction
implies that the crossed flat bands along the [001] direc-
tion discussed in the previous section are topologically
protected [Fig. 1(f,g)].
Also, because of the pMMS on
the xz-plane, topologically protected zero-energy surface
states appear on the [100] surface, which we identified as
surface arcs [Fig. 1(e)]. As a result, both surface arcs and
crossed flat bands are topologically protected and can be
realized in 3D superconducting AMs [65].
Conductance signatures of crossed flat bands.—Having
shown the emergence of crossed flat bands and surface
arcs, as well as their topological origin, here we explore
how they can be detected by means of conductance. For
this purpose, we couple the 3D superconducting AM to a
normal metal, such that we can assess transport along z
and x directions [Fig. 1(a,b)]. We obtain the conductance
following the common tunneling Hamiltonian approach
based on the Lee-Fisher formula [44, 66–68]:
σ(eV ) = 1
2π
Z π
−π
d˜k ¯σ(˜k, eV ) ,
(7)
FIG. 2.
(a,c) Zero-bias conductance along the z-direction
[001] as a function of momenta kx,y for a superconducting AM
with dxy-wave (a) and gxy(x2−y2)-wave (c) altermagnetism.
(b,d) Normalized total conductance along the z-direction as a
function of eV for distinct values of the altermagnetic strength
tα. σN indicates the conductance in the normal state at eV =
0. The insets in (b,d) indicate the normal state Fermi surfaces
projected onto the [001] surface for up (magenta) and down
spins (cyan).
Parameters: tα = ∆(a,c), µ = −4.5t, ∆=
0.01t, Ub = 5t, and δ = 0.01∆.
with ˜k = (ku, kv) for u = x, y and v = y, z, while
¯σ(˜k, eV ) is the momentum dependent conductance ob-
tained as
¯σ(˜k, eV ) = Tr′ X
i,j=0,1
[ ˆGii ˆV †
ji ˆGjj ˆVji −ˆGij ˆVji ˆGij ˆVji] , (8)
where we have dropped the momenta for simplicity and
Tr′ means that the trace is taken in electron space.
Here, ˆV10 = −ˆtJ ⊗ˆτ0, with ˆtJ characterising the cou-
pling between the superconducting AM and metallic lead
[Fig. 1(a,b)]. Moreover, ˆGnm are Green’s functions nu-
merically obtained by using the recursive Green’s func-
tion method [44, 67–71], and the details of this formula-
tion is presented in S2 of the SM [59].
We first assess the conductance signatures along z di-
rection [001], see Fig. 2. We begin by exploring the mo-
mentum dependence conductance ¯σ(˜k, eV = 0) at zero
bias eV = 0, which is presented in Fig. 2(a,c) for super-
conducting AMs with dxy- and gxy(x2−y2)-wave altermag-
netism. The corresponding total conductances σ(eV ) are
then shown in Fig. 2(b,d) as a function of eV for dis-
tinct altermagnetic strengths. The most relevant signa-
ture of the momentum dependent conductance at zero
bias is that the strongest intensities come from the con-
tribution of the crossed surface flat bands [Fig. 2(a,c)],
4
which emerge at the [001] surface as discussed in the pre-
vious section. Depending on the type of underlying al-
termagnetism, the zero-bias conductance ¯σ(˜k, eV = 0)
unveils crossed flat bands with different number of cor-
ners [Fig. 2(a,c)], as expected since the corners of the
emergent crossed flat bands are determined by the nodes
of the AM. For dxy-wave AM, the zero-bias momentum
dependent conductance reveals four corners [Fig. 2(a)],
while eight corners for gxy(x2−y2)-wave AM [Fig. 2(c)].
For dx2−y2-wave AMs, ¯σ(˜k, eV = 0) also exhibits four
corners but with a crossed flat band rotated by π/4,
see Sec. E1 and Fig. 4(a) in the End Matter. Moreover,
¯σ(˜k, eV = 0) in Fig. 2(a,c) acquires lower but nonzero
intensities around the crossed flat bands, which arise
due to contributions from the BFS. This is confirmed
by projecting the zero-energy density of states on the
[001] surface, see Sec. E1 and Fig. 5 in the End Matter.
There also appears a darkish area between the BFS and
crossed flat band in Fig. 2(a,c) originating from a gapped
dispersion where quasiparticles attain vanishing velocity
along z, see S1.2 in SM [59]. Notably, we find that, for
the crossed flat bands, BFSs and gapped regions, respec-
tively, ¯σ(˜k, eV = 0) is given by 2¯σ0
N, ¯σ1
N, and ¯σ2
N, un-
veiling three distinct types of power laws that coexist in
3D superconducting AMs [72]; see S2 in [59]. When in-
specting the total conductance σ(eV ) as a function of eV
in Fig. 2(b,d), it develops a zero-bias peak whose height
decreases as the altermagnetic strength increases; see
also Fig. 4(b). The presence of a large zero-bias conduc-
tance peak (ZBCP) without any altermagnetism is due to
the zero-energy ABSs of the chiral d-wave superconduc-
tor [55, 64, 73]. Altermagnetism, however, reduces this
ZBCP but, interestingly, its reduced value remains even
at strong values of altermagnetic strengths [Fig. 2(b,d)].
While the same reduction occurs for dxy-, dx2−y2-, and
gxy(x2−y2)-wave AMs, the latter exhibits a stronger and
well-formed ZBCP; see also Sec. E5 and Fig. 8 of the End
Matter. Nevertheless, in all cases, the ZBCP constitutes
a direct evidence of the crossed surface flat bands formed
at the [001] surface of 3D superconducting AMs.
Conductance along x direction and signatures of sur-
face arcs.—Conductance along x reveals distinct sig-
natures of the 3D superconducting AM, giving rise to
direction dependent transport in these systems.
In
Fig. 3(a,c), we present the momentum dependent con-
ductance ¯σ(˜k, eV = 0) at zero bias for superconduct-
ing AMs with dxy- and gxy(x2−y2)-wave altermagnetism.
Fig. 3(b,d) shows the total conductances σ(eV ) as a func-
tion of eV for distinct strengths of altermagnetism. For
completeness, in Sec. E3 and Fig. 6 of the End Matter
we also show these conductances for the case of dx2−y2-
wave altermagnetism. From Fig. 3(a,c), the momentum
dependent conductance ¯σ(˜k, eV = 0) develops high in-
tensities around ky = 0 due to the contribution of zero-
energy surface arcs states formed at the [100] surface. As
discussed before, these zero-energy surface arcs originate
FIG. 3.
(a,c) Zero-bias conductance along the x-direction
[100] as a function of ky,z for a superconducting AM with
dxy-wave (a) and gxy(x2−y2)-wave (c) altermagnetism. (b,d)
Normalized total conductance along the x-direction [100] as a
function of eV for distinct values of the altermagnetic strength
tα. σN indicates the conductance in the normal state at eV =
0. The insets in (b,d) indicate the normal state Fermi surfaces
projected onto the [100] surface for up (magenta) and down
spins (cyan).
Parameters: tα = ∆(a,c), µ = −4.5t, ∆=
0.01t, Ub = 5t, δ = 0.01∆.
from the MMS on the xz-plane. Interestingly, the sur-
face arc conductance signal is interrupted around kz = 0
due to a reduced contribution coming from the BFSs; see
also Fig. 1(d). The role of the BFS can be indeed con-
firmed by projecting the density of states on the [100]
surface, see Fig. 7. Also, ¯σ(˜k, eV = 0) in Fig. 6(a) reveals
the formation of spin-split surface arc states for dx2−y2-
wave altermagnetism, which disperse with ky,z and ex-
hibit a vanishing effect due to BFSs. Here, we also find
that ¯σ(˜k, eV = 0) is given by 2¯σ0
N, 2¯σ1
N, and 2¯σ2
N in the
presence of zero-energy surface arcs, BFS, and gapped re-
gion; see S2 in [59]. For the total conductance σ(eV ) as a
function of eV , it exhibits a broad peaked profile around
zero bias at zero altermagnetism [Fig. 3(b,d)]. This broad
conductance profile sharpens at finite altermagnetism for
dxy and gxy(x2−y2)-wave AMs, producing a clear ZBCP
[Fig. 3(b,d)]. Interestingly, for dx2−y2-wave AMs, the to-
tal conductance splits into two resonances that broaden
as the altermagnetic strength increases [Fig. 6(b)]; this
spin-split total conductance originates from the spin-split
surface arcs observed in ¯σ(˜k, eV = 0) in Fig. 6(a). We
note that this behavior is similar to the reported for
transport in Sr2RuO4 Ref. [74], supporting the idea that
conductance along x is determined by the surface arc
states and modified by the BFS.
5
In conclusion, we have demonstrated that crossed
surface flat bands emerge in three dimensional super-
conducting altermagnets as a generic topological phe-
nomenon protected by intrinsic crystal symmetries. Also,
we have shown that the crossed flat bands can be de-
tected by means of conductance, which depends on the
transport direction and can also assess the formation of
topological surface arcs and Bogoliubov-Fermi surfaces.
We have found that the zero-bias momentum resolved
conductance exhibits three distinct types of power laws,
offering a solid way for detecting the emergent topologi-
cal phases e. g., by Doppler shift measurements as those
predicted in Refs. [75–78]. Our findings also hold exper-
imental relevance since there already exist NMR experi-
ments in Sr2RuO4 [52, 53] suggesting three-dimensional
chiral d-wave superconductivity [48–51] and predictions
of altermagnetism in the same system [46, 47].
Our
work, therefore, establishes three-dimensional supercon-
ducting altermagnets as an outstanding ground for re-
alizing higher dimensional topological phases, which can
also pave the way for future tunneling spectroscopy ex-
periments in three dimensional superconductors.
Y. F. acknowledges financial support from the Sum-
itomo Foundation and the calculation support from
Okayama University.
B. L. acknowledges financial sup-
port from the National Natural Science Foundation of
China (project 12474049).
K. Y., Y. F., and Y. T.
acknowledge financial support from JSPS with Grants-
in-Aid for Scientific Research (KAKENHI Grants No.
25K07203).
Y. T. also acknowledges financial support
from JSPS with Grants-in-Aid for Scientific Research
(KAKENHI Grants No.
23K17668 and 24K00583).
J. C. acknowledges financial support from the Swedish
Research Council (Vetenskapsr˚adet Grant No. 2021-
04121).
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8
END MATTER
E1. Conductance along z-direction with dx2−y2-wave
AMs.—The conductance along z-direction for a super-
conducting AM junction with dx2−y2-wave altermag-
netism is presented in Fig. 4. In the case of the momen-
tum dependent conductance at zero bias ¯σ(˜k, eV = 0)
shown in Fig. 4(a), it develops high intensity values along
the diagonals kx = ±ky in contrast to the case of dxy-
wave altermagnetism having large values at kx,y = 0, see
Fig. 2(a). These large zero-bias conductance values reveal
the emergence of crossed flat bands in the [001] surface of
superconducting AMs with dx2−y2-wave altermagnetism.
As noted in the main text, ¯σ(˜k, eV = 0) also reflects the
BFS, seen as an area that surrounds the crossed flat band
in Fig. 4(a). Between the BFS and the area of the crossed
flat band appears a darkish region, which is gapped and
quasiparticles acquire vanishing velocity and hence van-
ishing conductance. Interestingly, ¯σ(˜k, eV = 0) acquires
a unique dependence on its zero-bias normal state coun-
terpart: ¯σ(˜k, eV = 0) is roughly given by 2¯σ0
N, ¯σ1
N, and
¯σ2
N for the regime with crossed flat bands, BFSs, and
gapped region, respectively; see S2 in [59]. Also, the to-
tal conductance σ(eV ) varies from a very peaked profile
at zero bias in the absence of dx2−y2-wave altermagnetism
to a reduced but visible peak at finite as the altermag-
netic strength increases and surpasses the pair potential
[Fig. 4(b)]. Thus, the reduced peaked profile manifests
the formation of crossed flat bands on the [001] surface.
E2. Projected density of states on the [001] surface.—
We have seen that the gapless structure of the energy
dispersion in Eq. (3) originates from BFSs.
It is thus
expected that the density of states (DOS) enhances due
to BFSs. To show this, we calculate the bulk DOS as
D(E) = −1
π
Z π
−π
dkTr′[Im ˜GR(k, E)],
(9)
where Tr′ indicates the trace taken in electron space,
˜GR(k, E) is the retarded Green’s function ˜GR(k, E) =
[E + iδ −ˆHBdG(k)]−1, with ˆHBdG(k) given by Eq. (1),
E is the energy and δ is an infinitesimal positive num-
ber. Using Eq. (9), we obtain the projected DOS on the
[001] surface by integrating along the transport direc-
tion. To unveil the BFS, in Fig. 5 we plot the projected
DOS on the [001] surface at zero energy for dxy-, dx2−y2-,
and gxy(x2−y2)-wave AMs. The most important feature
is that the BFS produces the highest intensity in the
projected zero-energy DOS, see Fig. 5. Interestingly, the
BFSs surround regions with tiny values, which possess
corners depending on the type of AM. These inner re-
gions correspond to the formation of crossed flat bands,
discussed in Figs. 2(a,c) and 4(a). The bright point at
kx,y = 0 is due to the nodal point where the energy gap
closes. The BFS signals in the projected DOS remain in
more elaborated 3D AMs, see Sec. S3 in the SM [59].
E3. Conductance along x-direction with dx2−y2-wave
FIG. 4. (a) Zero-bias conductance along the z-direction [001]
for a superconducting AM with dx2−y2-wave altermagnetism
as a function of kx,y. (b) Normalized total conductance along
the z-direction as a function of eV for distinct values of the
altermagnetic strength td2; σN indicates the conductance in
the normal state at eV = 0. The inset in (b) shows the normal
state Fermi surfaces projected onto the [001] surface for up
(magenta) and down spins (cyan). Parameters: td2 = ∆(a),
µ = −4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆.
FIG. 5. (a-c) Projected zero-energy DOS on the [001] surface
as a function of kx,y for a superconducting AM with dxy- (a),
dx2−y2- (b), and gxy(x2−y2)-wave altermagnetism.
Parame-
ters: tα = ∆, µ = −4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆.
AMs.—Conductance along x in a superconducting AM
with dx2−y2-wave altermagnetism is shown in Fig. 6(a)
and we observe the formation of arcs on the [100] sur-
face. Here, the momentum dependent zero-bias conduc-
tance ¯σ(˜k, eV = 0) shows that the arcs are spin-split
and disperse with momentum, unlike those due to dxy-
wave altermagnetism in Fig. 3(a). The region with tiny
but nonzero intensities corresponds to BFSs, while even
weaker values of ¯σ(˜k, eV = 0) occur in a gapped re-
gion, see S1 in SM [59]. Here, we have also verified that
¯σ(˜k, eV = 0) is given by 2¯σ0
N, ¯σ1
N, and ¯σ2
N for the surface
arcs, BFS, and gapped region; see S2 in [59]. Moreover,
a finite altermagnetism splits the zero-bias total conduc-
tance peak of σ(eV ) in Fig. 6(b), producing two broad
resonances around zero bias as a signature of surface arcs.
E4. Projected density of states on the [100] surface.—
For the projected DOS on the [100] surface using Eq. (9),
we find that BFSs appear around kz = 0 for all the cases
of altermagnetism [Fig. 7].
Depending on the type of
AM, the projected DOS shows the BFSs coexisting with
9
FIG. 6. (a) Zero-bias conductance along the x-direction [100]
for a superconducting AM with dx2−y2-wave altermagnetism
as a function of kx,y. (b) Normalized total conductance along
the x-direction [100] as a function of eV for distinct values
of the altermagnetic strength td2. σN is the conductance in
the normal state at eV = 0.
The inset in (b) shows the
normal state Fermi surfaces projected onto the x-direction for
up (magenta) and down spins (cyan). Parameters: td2 = ∆
(a), µ = −4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆.
regions having tiny but nonzero intensities that corre-
spond to the zero-energy surface arcs on the [100] sur-
face [Fig. 3(a,c) and Fig. 7(a)]. We also confirm the BFS
signatures in proper 3D AMs, see Sec. S2 in the SM [59].
E5. Stability of the ZBCP.—Having shown the for-
mation of crossed flat bands, surface arcs and BFSs, we
now explore the ZBCP σ(eV = 0) under variations of
the altermagnetic strength tα and barrier potential Ub.
This is shown in Fig. 8 for transport along z and x, in
both cases for dxy-, dx2−y2-, and gxy(x2−y2)-wave alter-
magnetism. For transport along z, the largest zero-bias
conductance σ(eV = 0) is concentrated below tα < ∆in
the case of dxy- and dx2−y2-wave AMs, see Figs. 8(a,c).
For gxy(x2−y2)-wave, the ZBCP remains robust even for
larger values of tα, see Figs. 8(e). These behaviors orig-
inate because the crossed flat bands in the respective
cases become smaller with the increase of altermagnetic
strength tα.
Despite the reduced ZBCPs, their values
are still peaked at zero bias for dxy- and gxy(x2−y2)-wave
AMs, signalling that the crossed flat bands remain for
tα > ∆, as we indeed show in Fig. 2(b,d). We also note
FIG. 7. (a-c) Projected zero-energy DOS on the [100] surface
as a function of ky,z for a superconducting AM with dxy- (a),
dx2−y2- (b), and gxy(x2−y2)-wave altermagnetism.
Parame-
ters: tα = ∆, µ = −4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆.
FIG. 8. Normalized zero-bias conductance along z (a,c,e) and
x (b,d,f) directions as a function of Ub and tα for supercon-
ducting AMs with dxy-, dx2−y2-, and gxy(x2−y2)-wave alter-
magnetism. Here, σN denotes the conductance in the normal
state at eV = 0, and we calculate it for each tα and Ub. In-
sets: crossed flat bands and normal state Fermi surfaces for
up (magenta) and down (cyan) spins. The gray circles mark
the values of Ub and tα chosen for Fig. 2 and Fig. 3. Param-
eters: µ = −4.5t, ∆= 0.01t, δ = 0.01∆.
that the crossed flat bands exist even at Ub = 0, since the
corners of the crossed flat bands are already formed with-
out any barrier. However, a finite Ub sharpens the tun-
neling conductance, and much larger conductance values
occur along momenta of the altermagnetic nodal lines.
Hence, the visibility of the crossed flat band is enhanced
by large Ub, which corresponds to low transparencies.
Along the x-direction shown in Figs. 8(b,d,f), the value
of σ(eV = 0) takes values ranging from 1 to 2, and it does
not change significantly for variations of Ub and tα, even
though the surface arc signatures partly disappear under
BFSs. Also, σ(eV = 0) becomes smaller as td2 increases
because the surface arc states are split by dx2−y2-wave
altermagnetism [Fig. 6]. The stability of the ZBCP dis-
cussed here also persists for 3D AMs, see S3 in [59].
Supplementary material for “Crossed surface flat bands in three-dimensional
altermagnetic superconductors”
Yuri Fukaya,1 Bo Lu,2 Keiji Yada,3 Yukio Tanaka,3 and Jorge Cayao4
1Faculty of Environmental Life, Natural Science and Technology, Okayama University, 700-8530 Okayama, Japan
2Department of Physics, Tianjin University, 300354 Tianjin, China
3Department of Applied Physics, Nagoya University, 464-8603 Nagoya, Japan
4Department of Physics and Astronomy, Uppsala University, Box 516, S-751 20 Uppsala, Sweden
In this supplementary material, we provide additional details of the bulk system studied in the main text, discuss how
we calculate conductance, and demonstrate that our main findings remain for a system involving three-dimensional
(3D) altermagnetism with 3D chiral d-superconductivity.
S1: BULK PROPERTIES OF THREE-DIMENSIONAL SUPERCONDUCTING ALTERMAGNETS
First of all, we present the details of the model Hamiltonian of 3D superconductors (SCs) with two-dimensional
(2D) altermagnets (AMs). Then we will show their corresponding bulk properties.
S1.1: Hamiltonian model with a 3D superconductor and 2D altermagnet
In this subsection, we discuss further details of the model Hamiltonian. We start by considering the normal part
of the Hamiltonian, which is given by
ˆHα(k) = ˆH0(k) + M α
k ˆσ3,
(S1)
where ˆH0(k) is the kinetic term and M α
k characterizes altermagnetism. More precisely, the kinetic part is given by
ˆH0(k) = ε(k)ˆσ0,
(S2)
ε(k) = −µ −2t cos kx −2t cos ky −2t cos kz,
(S3)
where k = (kx, ky, kz) is the momentum, ˆσ0,1,2,3 is the Pauli matrices in spin space, µ = −4.5t is the chemical
potential, t is the hopping integral along the (x, y) and z-directions. We consider the d and g-wave altermagentism
modelled by
M d1
k = 2td1 sin kx sin ky,
(S4)
M d2
k = td2(cos kx −cos ky),
(S5)
M g
k = 4tg sin kx sin ky(cos kx −cos ky),
(S6)
where td1, td2, and tg are the strength of dxy, dx2−y2 and gxy(x2−y2)-wave AMs, respectively.
When adding the superconductivity, the system is described by the Bogoliubov–de Gennes (BdG) Hamiltonian:
ˆHα
BdG(k) = ε(k)ˆσ0ˆτ3 + ˆHα
AM(k) + ˆHSC(k),
(S7)
ˆHα
AM(k) = M α
k ˆσ3ˆτ3,
(S8)
ˆHSC(k) = −∆sin kz[sin kxˆτ2 + sin kyˆτ1]ˆσ2
(S9)
with ˆτ0,1,2,3 the Pauli matrices in Nambu space and ∆= 0.01t the pair amplitude of the chiral d-wave state. The
eigenvalues of ˆHα
BdG(k) are obtained as:
Es,α
k,±(k) = sM α
k ±
p
ε2(k) + |ψ(k)|2,
(S10)
with s = ± and ψ(k) = ∆sin kz[sin kx + i sin ky]. Eq. (S10) is presented in the main text as Eq. (3) and helps us
identify the formation of Bogoliubov-Fermi surfaces (BFSs).
2
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
Fermi surface for up-spin
Fermi surface for down-spin
kx/⇡
kx/⇡
kx/⇡
kx/⇡
(a)
(e)
(b)
(f)
(c)
(g)
(d)
(h)
td2 = 0
h
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UwhnFMcR7zWME6Msj5OZz
hApdKRJlRFpWlVqvSEWh
G8K2UtU/JnJKA</latexi
t>td2 = 0
td2 = 0.5∆
td2 = 0.5∆
td2 = ∆
td2 = ∆
td2 = 1.5∆
td2 = 1.5∆
Es,d2
k,± /∆
Es,d2
k,± /∆
h
a1_base64="bZ9NwHSNfc
5dkBETC3G8sYV+hgc=">
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UGDWB8mDM4MHgwBDCEAu2
tZJjPsIJhJZMUkwOTO5Mn
RCkTI1SPMAMKYAoCABmOl
1k=</latexit>
E+,d2
k,+
E−,d2
k,−
=
"i+Hedv0yjsbGITpbpT0mZIE4ncs=">ACmHicSyr
IySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qU
nBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkx
qelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAE
KxQsIu8ZVa+ukGNXGV8ck5VZn1+ro1sYLKBvoWVqY
mpoYKRjoGRhYmpgaAxmWlpaGFoYKhkAREFBmgIKAfI
HtDEMKQz5DMkMpQy5DKkMeQwlQHYOQyJDMRBGMxg
yGDAUAMViGaqBYkVAViZYPpWhloELqLcUqCoVqCIRK
JoNJNOBvGioaB6QDzKzGKw7GWhLDhAXAXUqMKgaXD
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EfydoK5cIF3CkIHQhUdHElA1fj+VMKQxWID9kgn0W
wFYBOTLZIj5ZVXTPwdbBalWqxksMngN9N9Cg5sGh4E
+zCv7krw0MDVoNgMXMIJgsaCAmxFmpGdopmcWaKLs
oAONKg4GaQYlBg1gfJgzODB4MAQwhALtrWSYz7CYS
WTFJMDkzuTJ0QpEyNUjzADCmAKAgBMn5dy</latex
it>
E+,d2
k,−
E−,d2
k,+
FIG. S1. Es,d2
k,± for the dx2−y2-wave altermagnet (AM) as a function of kx at ky = 0 scanning on the Fermi surface with (a-d)
up and (e-g) down spins. We choose the altermagnetic order as (a,d) td2 = 0, (b,e) td2 = 0.5∆, (c,f) td2 = ∆, and (d,g)
td2 = 1.5∆. In the normal state, at ky = 0, kz is determined by the normal state energy dispersion ε(k) ± M α
k = 0 for each
kx. For each (kx, kz), we plot Es,d2
k,± as a function of kx at ky = 0. In (a,e), we obtain E+,d2
k,+ = E−,d2
k,+
and E+,d2
k,−= E−,d2
k,−
for
each kx, and then the shape of zero-energy flat bands becomes a disk. In the presence of the altermagnetism, if E+,d2
k,+ ∼E−,d2
k,+
and E+,d2
k,−
∼E−,d2
k,−, zero-energy flat bands remain, and the resulting zero-energy flat bands becomes crossed. We select the
parameter as: µ = −4.5t and ∆= 0.01t.
S1.2: Structure of quasiparticle energy dispersions in the bulk
In the following, we show how to determine the width of crossed flat bands. We start with the quasiparticle energy
dispersion Es,d2
k,± as a function of kx at ky = 0 by scanning on the Fermi surface with up [Fig. S1 (a-d)] and down spins
[Fig. S1 (e-g)], for the dx2−y2-wave AM. Then kz is determined by the normal state energy dispersion ε(k) ± M d2
k = 0
for each kx at ky = 0. Because the spin splitting of the Fermi surface is small, the results for up and down spins
are almost the same. Without altermagnetism (M α
k = 0), we obtain E+,d2
k,+
= E−,d2
k,+
and E+,d2
k,−
= E−,d2
k,−, that is,
E+,d2
k,+ −E−,d2
k,+ = E+,d2
k,−−E−,d2
k,−= 2M α
k = 0 for each momentum, and then we also obtain a disk of zero-energy flat
bands. In the presence of altermagnetism (M α
k ̸= 0), we generally get we obtain E+,d2
k,+ ̸= E−,d2
k,+ and E+,d2
k,−̸= E−,d2
k,−.
Nevertheless, the relations of E+,d2
k,+ ∼E−,d2
k,+ and E+,d2
k,−∼E−,d2
k,−, that is, E+,d2
k,+ −E−,d2
k,+ = E+,d2
k,−−E−,d2
k,−= 2M α
k ∼0
can be satisfied, and then zero-energy surface Andreev bound states remain. Along the z-direction, although the
dispersions of E+,d2
k,+
= E−,d2
k,+
and E+,d2
k,−
= E−,d2
k,−
for each (kx, ky) are inexplicit, we can obtain their approximate
expression at small value of M α
k . In the case of the dx2−y2-wave AM, E+,d2
k,+ and E−,d2
k,−are partly split, and E+,d2
k,+ ∼
E−,d2
k,+
and E+,d2
k,−
∼E−,d2
k,−
are obtained [Figs. S1 (b,c,d,f,g,h)]. Then the momentum of this overlapping region is
ks ∼0.08π at td2 = 0.5∆[Figs. S1 (b,e)], ks ∼0.06π at td2 = ∆[Figs. S1 (c,f)], and ks ∼0.04π at td2 = 1.5∆
[Figs. S1 (d,g)].
If E+,d2
k,+
∼E−,d2
k,+
and E+,d2
k,−
∼E−,d2
k,−, the zero-energy flat bands remain, and as a result, this
momentum ks corresponds to the size of the crossed flat band as discussed in Fig. 4(a) of the main text. Indeed,
the energy dispersion of the quasiparticle Es,d2
k,± remains the degeneracy within the momentum ks even with the
altermagnetism, and it contributes to the size of the crossed flat band. Along the x-direction, for the dxy-wave AM,
when ky = 0 (kx = 0), we obtain M α
k = 0 for each kz. It means that surface arc states, which appear along the
kz-axis, are not affected by the dxy-wave AM; however, they are only modified by the BFS. While, for the dx2−y2-wave
AM, M α
k = 0 satisfies at |kx| = |ky| for each kz, and arc surface states appears at (ky, kz) with |ky| = |kx|. For the
gxy(x2−y2)-wave AM, if ky = 0 (kx = 0) or |kx| = |ky|, we obtain M α
k = 0. As a result, arc surface states are partly
determined as shown in Fig. 3(c) of the main text. Thus, crossed flat bands can have the width, and the surface arc
states can remain shown in Figs. 2(a,c), 3(a,c), 4(a), and 6(a) of the main text.
3
FIG. S2. (a,c) E+,d2
k,+ = −E−,d2
k,−and (b,d) E+,d2
k,−= −E−,d2
k,+
for (kx, ky) space for the dx2−y2-wave AM at (a,b) kz = 0.15π and
(c,d) 0.25π. (e) Es,d2
k,± for the dx2−y2-wave AM as a function of kx at ky = 0 by changing kz. (f) Es,d2
k,± for the dx2−y2-wave AM
as a function of kz at ky = 0 by changing kx. We select the parameters as µ = −4.5t, ∆= 0.01t, and td2 = ∆.
Next, we discuss the velocity of the quasiparticle along the z-direction. In Fig. S2, we plot Es,d2
k,± for the dx2−y2-wave
AM at td2 = ∆with ∆= 0.01t. E+,d2
k,+ = −E−,d2
k,−
and E+,d2
k,−= −E−,d2
k,+
for (kx, ky) space at kz = 0.15π, 0.25π are
shown in Figs. S2 (a,b,c,d). The amplitude of E+,d2
k,±
becomes the smallest at the Fermi level. To see more details,
we plot Es,d2
k,± as a function of kx at ky = 0 for each kz. The energy gap is opening for 0.2π ≤kz ≤0.42π. The BFS
appears for kz ≥0.2π, and a point node for kz = 0.42π. We also show Es,d2
k,± as a function of kz at ky = 0 by changing
kx in Fig. S2 (f). The energy gap is also opening for 0 ≤kz ≤0.35π. The BFS also exhibits for kx ≥0.35π, and
a point node for kx = 0. These behaviors indicate that the energy gap closes around kx0 = ±0.35π and the BFS is
realized for kx ≥|kx0|. Es,d2
k,± at the minimal energy of the upper band can have zero velocity at kx = kx0. When
the minimal energy is zero, the velocity along the z-direction of the quasiparticle also becomes zero, leading to zero
current. Thus, the total conductance vanishes at kx = kx0. Hence, this idea explains the vanishing conductance in
the gapped regions of Figs. 2(a,c) and 4(a), as well as Figs. 3(a,c) and 6(a) of the main text.
S1.3: Details on the topological stability of zero-energy surface states
Here, we discuss the details of the winding number, which protects the crossed flat bands and arc states.
To
define the winding number, a chiral operator that anticommutes with the Hamiltonian is required [S1–S5]. The most
common definition of the chiral operator is the one constructed from time-reversal (TRS) and particle–hole symmetries
(PHS) [S1–S3]. However, in the present system, both AM and pair potential break the TRS, and therefore, this chiral
symmetry is broken. Nevertheless, to consider chiral symmetry in the present system, it is necessary to take into
account the remaining symmetries of the system. For example, the breaking of time-reversal symmetry due to the
pair potential can be recovered by the following local gauge transformation [S6]:
ˆH′
SC(k) = ˆUϕk ˆHSC(k) ˆU †
ϕk
= −∆
q
sin2 kx + sin2 ky sin kzˆσ2ˆτ2,
(S11)
4
with ˆUϕk = e−iϕkˆσ0ˆτ3/2 = cos(ϕk/2)ˆσ0ˆτ0 −i sin(ϕk/2)ˆσ0ˆτ3 and ϕk = tan−1(sin ky/ sin kx), i. e.,
cos ϕk = sin kx/
q
sin2 kx + sin2 ky ,
sin ϕk = sin ky/
q
sin2 kx + sin2 ky .
(S12)
This local gauge transformation does not affect the kinetic and altermagnetic parts:
ˆUϕkε(k)ˆσ0ˆτ3 ˆU †
ϕk = ε(k)ˆσ0ˆτ3
and ˆUϕk ˆHα
AM(k) ˆU †
ϕk = ˆHα
AM(k). By this local gauge transformation, the normal part and pair potential term of the
Hamiltonian have the following pseudo-time-reversal (pTRS) and particle-hole symmetries:
ˆΘ ˆH′
NM+SC(k)ˆΘ† = [ ˆH′
NM+SC(−k)]∗,
(S13)
ˆC ˆH′
NM+SC(k) ˆC† = −[ ˆH′
NM+SC(−k)]∗,
(S14)
with ˆH′
NM+SC(k) = ε(k)ˆσ0ˆτ3 + ˆH′
SC(k), the conventional TRS ˆΘ = iˆσ2ˆτ0, and PHS ˆC = ˆσ0ˆτ1. We note that the
conventional TRS ˆΘ and PHS ˆC are also defined in the main text. Here, we introduce a chiral operator ˆΓ ≡−i ˆC ˆΘ =
ˆσ2ˆτ1. Using the Eqs. (S13) and (S14), we obtain the following chiral symmetry:
ˆΓ ˆH′
NM+SC(k)ˆΓ† = −ˆH′
NM+SC(k),
(S15)
where ˆΓ anticommutes H′
NM+SC(k). Since H′
NM+SC(k) is the Hamiltonian obtained from the local gauge transforma-
tion, we go back to the original gauge:
ˆU †
ϕk ˆΓ ˆUϕk ˆHNM+SC(k) ˆU †
ϕk ˆΓ† ˆUϕk = −ˆHNM+SC(k).
(S16)
Then, we obtain the chiral operator for the original gauge:
ˆΓk = ˆU †
ϕk ˆΓ ˆUϕk.
(S17)
On the other hand, this anticommutation relation in Eq. (S16) is no longer satisfied when we consider ˆHα
AM(k) due
to the broken TRS:
ˆΘ ˆHα
AM(k)ˆΘ† = −[ ˆHα
AM(−k)]∗,
(S18)
ˆC ˆHα
AM(k) ˆC† = −[ ˆHα
AM(−k)]∗.
(S19)
Nevertheless, ˆHα
AM(k) has the magnetic mirror symmetry (MMS), which is the product of pTRS and the mirror
symmetry. The corresponding mirror plane depends on the symmetry of AMs. In the main text, since we discuss the
chiral operator ˆΓk by focusing on the total Hamiltonian ˆHα
BdG(k), we introduce the pseudo-magnetic mirror symmetry
(pMMS): ˆΘ
′
k = ˆU †
ϕk ˆΘ ˆUϕk. While here, we demonstrate the chiral operator ˆΓk for each term in ˆHα
BdG(k). Thus, we
note that we use MMS in the SM, not pMMS. In the case of the dxy-wave AM, ˆHα
AM(k) has the following MMS:
ˆHα
AM(kx, ky, kz) = −ˆHα
AM(−kx, ky, kz)
= −ˆHα
AM(kx, −ky, kz),
(S20)
where the first and the second line denote the mirror symmetry on the yz- and zx-plane, respectively. On the other
hand, for the dx2−y2-wave AM, ˆHα
AM(k) has diagonal magnetic mirror planes,
ˆHα
AM(kx, ky, kz) = −ˆHα
AM(ky, kx, kz)
= −ˆHα
AM(−ky, −kx, kz).
(S21)
For the gxy(x2−y2)-wave AM, both Eqs. (S20) and (S21) is available. As a result, pTRS in Eq. (S18) becomes
ˆΘ ˆHα
AM(kx, ky, kz)ˆΘ† = ˆHα
AM(kx, −ky, −kz)
= ˆHα
AM(−kx, ky, −kz),
(S22)
5
for dxy and gxy(x2−y2)-wave AMs, and
ˆΘ ˆHα
AM(kx, ky, kz)ˆΘ† = ˆHα
AM(−ky, −kx, −kz)
= ˆHα
AM(ky, kx, −kz),
(S23)
for dx2−y2- and gxy(x2−y2)-wave AMs. Finally, we obtain the chiral symmetry by the combination of local gauge
symmetry, MMS, and PHS:
ˆΓk ˆHα
AM(kx, ky, kz)ˆΓ†
k = ˆHα
AM(−kx, ky, kz)
= ˆHα
AM(kx, −ky, kz),
(S24)
for dxy and gxy(x2−y2)-wave AMs, and
ˆΓk ˆHα
AM(kx, ky, kz)ˆΓ†
k = ˆHα
AM(ky, kx, kz)
= ˆHα
AM(−ky, −kx, kz),
(S25)
for dx2−y2- and gxy(x2−y2)-wave AMs. Eqs. (S24) and (S25) say that Hα
AM(k) anticommutes with ˆΓk on the magnetic
mirror plane. The chiral operators defined on each magnetic mirror plane are as follows:
ˆΓkx=0 = ˆσ2ˆτ2,
(S26)
ˆΓky=0 = ˆσ2ˆτ1
(S27)
ˆΓkx=ky = ˆσ2(τ1 + τ2)/
√
2,
(S28)
ˆΓkx=−ky = ˆσ2(τ1 −τ2)/
√
2.
(S29)
Using these chiral operators, we can calculate the winding number on each magnetic mirror plane. When the winding
number is nonzero, the system becomes topologically nontrivial, and the crossed flat bands and/or surface arcs appear.
Therefore, the analysis carried out in this subsection demonstrates the topological origin of the crossed flat bands and
surface arcs discussed in the main text.
S2: CALCULATION OF CONDUCTANCE IN A HYBRID SYSTEM FORMED BY A
SUPERCONDUCTING ALTERMAGNET AND A NORMAL METAL
In the main text, we have discussed the conductance in superconducting AM/normal metal junctions. Here, in this
section, we show the details of how the conductance is calculated in superconducting AM junctions. We also discuss
the details of the power law in the momentum dependent conductance.
S2.1: Model of the hybrid junction
We hereby provide the model and formulation in superconducting hybrid junctions. Then we assume the semi-
infinite superconducting AM and normal metal regimes. In superconducting AM/normal metal junctions, we consider
the Hamiltonian:
ˆH = ˆHα
SC+AM + ˆHJ + ˆHNM.
(S30)
In ˆHα
SC+AM, when we consider the junction along the w = z, x-direction, the local ˜uα
sc(˜k), nearest-neighbor (NN)
˜tα
sc1(˜k), and second NN hopping terms ˜tα
sc2(˜k) with ˜k = (ku, kv) for u = x, y and v = y, z are given by
˜uα
sc(˜k) =
ˆu0(˜k) + ˆuα
AM(˜k)
ˆu∆(˜k)
ˆu†
∆(˜k)
−ˆu∗
0(−˜k) −ˆuα∗
AM(−˜k)
,
(S31)
6
˜tα
sc1(˜k) =
ˆt0(˜k) + ˆtα
AM1(˜k)
ˆt∆(˜k)
¯t∆(˜k)
−ˆt∗
0(−˜k) −ˆtα∗
AM1(−˜k)
.
(S32)
˜tα
sc2(˜k) =
ˆtα
AM2(˜k)
0
0
−ˆtα∗
AM2(−˜k)
.
(S33)
Along the z-direction, we obtain
ˆu0(kx, ky) = [−µ −2t cos kx −2t cos ky]ˆσ0,
(S34)
ˆuα
AM(kx, ky) = ˆHα
AM(kx, ky),
(S35)
ˆt0 = −tˆσ0,
(S36)
ˆtα
AM1(kx, ky) = ˆtα
AM2(kx, ky) = 0,
(S37)
ˆu∆= 0,
(S38)
ˆt∆= i∆
2 [sin kx −i sin ky] iˆσ2,
(S39)
¯t∆= −i∆
2 [sin kx + i sin ky] iˆσ2,
(S40)
and along the x-direction,
ˆu0(ky, kz) = [−µ −2t cos ky −2t cos kz]ˆσ0,
(S41)
ˆud1
AM(ky, kz) = 0,
(S42)
ˆud2
AM(ky, kz) = −td2 cos ky,
(S43)
ˆug
AM(ky, kz) = 0,
(S44)
ˆt0 = −tˆσ0,
(S45)
ˆtd1
AM1(ky) = −itd1
2 sin ky,
(S46)
ˆtd2
AM1(ky) = td2
2 ,
(S47)
ˆtg
AM1(ky) = 2itg sin ky cos ky,
(S48)
ˆtd1
AM2(ky) = ˆtd2
AM2(ky) = 0,
(S49)
ˆtg
AM2(ky) = −itg sin ky,
(S50)
ˆu∆= i∆sin ky sin kz,
(S51)
ˆt∆=
i∆
2 sin ky
iˆσ2,
(S52)
¯t∆=
−i∆
2 sin ky
iˆσ2.
(S53)
In the normal metal ˆHNM, the local ˜uNM(˜k) and nonlocal terms ˜tNM(˜k) are given by
˜uNM(˜k) =
ˆu0(˜k)
0
0
−ˆu∗
0(−˜k)
,
(S54)
˜tNM(˜k) =
ˆt0(˜k)
0
0
−ˆt∗
0(−˜k)
.
(S55)
7
The tunneling Hamiltonian is defined as
ˆHJ = ˆtJ ⊗ˆτ3,
(S56)
with the Pauli matrices in Nambu space ˆτi=0,1,2,3. Here, ˆtJ denotes the tunneling Hamiltonian in the electron space:
ˆtJ = −tˆσ0,
(S57)
along the z and x-directions. These equations shown here are used in the next subsection to calculate the conductance.
S2.2: Details of calculation of conductance
In this subsection, we show the details of how we calculate the conductance in superconducting AM junctions. The
conductance is obtained based on the Lee-Fisher formula [S7–S10]:
σ(eV ) = 1
2π
Z π
−π
d˜k¯σ(˜k, eV ),
(S58)
with
¯σ(˜k, eV ) = Tr′[˜ga(˜k, eV ) + ˜gb(˜k, eV ) −˜gc(˜k, eV ) −˜gd(˜k, eV )].
(S59)
Eq. (S59) is given by Eq. (8) of the main text.
Here, we present the explicit expression for Green’s function
˜ga,b,c,d(˜k, eV ). For E = eV with the energy E and the bias voltage V , ˜ga,b,c,d(˜k, eV ) are calculated by
˜ga(˜k, E) = ˜G11(˜k, E) ˆV10 ˜G00(˜k, E) ˆV †
10,
(S60)
˜gb(˜k, E) = ˜G00(˜k, E) ˆV †
10 ˜G11(˜k, E) ˆV10,
(S61)
˜gc(˜k, E) = ˜G01(˜k, E) ˆV10 ˜G01(˜k, E) ˆV10,
(S62)
˜gd(˜k, E) = ˜G10(˜k, E) ˆV †
10 ˜G10(˜k, E) ˆV †
10.
(S63)
Here, ˆV10 is defined as ˆV10 = −ˆtJ ⊗ˆτ0 for obtaining the conductance owing to the current operator, and this equation
is already provided in the main text.
˜Gij(˜k, E) with i, j = 0, 1 are numerically calculated by the retarded (R) and
advanced (A) Green’s function ˜GR
ij(˜k, E) and ˜GA
ij(˜k, E):
˜G00(˜k, E) = −i
2[ ˜GA
00(˜k, E) −˜GR
00(˜k, E)],
(S64)
˜G11(˜k, E) = −i
2[ ˜GA
11(˜k, E) −˜GR
11(˜k, E)],
(S65)
˜G01(˜k, E) = −i
2[ ˜GA
01(˜k, E) −˜GR
01(˜k, E)],
(S66)
˜G10(˜k, E) = −i
2[ ˜GA
10(˜k, E) −˜GR
10(˜k, E)].
(S67)
Local ˜Gi,i(˜k, E) and nonlocal Green’s functions ˜Gi,j(˜k, E) are calculated by
˜G00(˜k, z) = [{ ˜GL
0(˜k, z)}−1 −˜tJ ˜GR
1 (˜k, z)˜t†
J]−1,
(S68)
˜G11(˜k, z) = [{ ˜GR
1 (˜k, z)}−1 −˜t†
J ˜GL
0(˜k, z)˜tJ]−1,
(S69)
˜G01(˜k, z) = ˜GL
0(˜k, z)˜tJ ˜G11(˜k, z),
(S70)
˜G10(˜k, z) = ˜GR
1 (˜k, z)˜t†
J ˜G00(˜k, z),
(S71)
8
where z = E ± iδ with the infinitesimal value δ are the retarded (R) z = E + iδ and advanced (A) parts z = E −iδ.
By using the recursive Green’s function method [S11, S12], we get the surface Green’s functions:
˜GL
0(˜k, z) = [z −˜usc(˜k) −˜t†
sc ˜GL(˜k, z)˜tsc]−1,
(S72)
˜GR
1 (˜k, z) = [z −˜uNM(˜k) −˜ub −˜tNM ˜GR(˜k, z)˜t†
NM]−1,
(S73)
with the semi-infinite Green’s functions ˜GL(˜k, z) and ˜GR(˜k, z), and ˜ub = Ubˆσ0 ⊗ˆτ3 for the strength of the potential
barrier Ub. We obtain the semi-infinite Green’s functions for both retarded (z = E + iδ) and advanced (z = E −iδ)
parts by using the local ˆu and hopping terms ˆt, see Refs. [S11–S13].
For the semi-infinite Green’s function, when only NN hopping is included, we choose the local ˆu and the hopping
terms ˆt as ˆu →˜usc and ˆt →˜tsc1 with the NN hopping ˜tsc1. In the case of including the second NN hopping term ˜tsc2
like the gxy(x2−y2)-wave AM along the x-direction, we replace the local ˆu and the hopping term ˆt as [S11]
ˆu →
˜usc ˜tsc1
˜t†
sc1
˜usc
,
(S74)
ˆt →
˜tsc2
0
˜t†
sc1 ˜tsc2
.
(S75)
To connect ˆV10, if the second NN hopping term is included in the formulation, e.g., gxy(x2−y2)-wave AM along the
x-direction, we pick up the subspace of the semi-infinite Green’s function ˜GL
0 = ˜Gbb, which is defined as:
˜G =
˜Gaa
˜Gab
˜Gba
˜Gbb
!
,
(S76)
in the basis [ ˆψ†
a, ˆψ†
b] with ˆψ†
l=a,b = [c†
l↑, c†
l↓, cl↑, cl↓]. We have calculated the conductance in superconducting AM
junctions of the main text by using this formulation.
S2.3: Power law behavior in the momentum dependent conductance
Here, we demonstrate the zero-bias conductance ¯σ(˜k, eV
= 0) and the zero-bias normal state conductance
¯σN(˜k, eV = 0) ≡¯σN at ˜k along the z and x-directions. For the dx2−y2-wave AM, we plot ¯σ(˜k, eV = 0) and ¯σN
along the z-direction as a function of the potential barrier Ub in Figs. S3 (a,b,c) and ¯σ(˜k, eV = 0) along the z-
direction as a function of ¯σN(˜k, eV = 0) in Figs. S3 (d,e,f), at td2 = ∆. Then we choose the momentum as (a,d)
(kx, ky) = (0.2π, 0.2π) (crossed flat band), (b,e) (0.4π, 0) (BFS), and (c,f) (0.3π, 0) (gapped region).
In Fig. S3
(a), on the crossed flat bands, ¯σ is almost 2, and ¯σN is suppressed for Ub. In Fig. S3 (b), on the BFS, we obtain
¯σ(˜k, eV = 0) ∼¯σN. On the gapped region shown in Fig. S3 (c), both ¯σ(˜k, eV = 0) and ¯σN strongly suppressed.
When we see the power law of the zero-bias conductance for each momentum, we obtain ¯σ(˜k, eV = 0) ∼2¯σ0
N on the
crossed flat band [Fig. S3 (d)], ¯σ(˜k, eV = 0) ∼¯σ1
N on the BFS [Fig. S3 (e)], ¯σ(˜k, eV = 0) ∼¯σ2
N [Fig. S3 (f)]. We
note that the power law in the gapped region deviates from ¯σ(˜k, eV = 0) ∼¯σ2
N owing to the smearing factor δ in the
lattice Green’s function method. Along the x-direction, for the dxy-wave AM, we also show ¯σ(˜k, eV = 0) and ¯σN as a
function of Ub in Figs. S4 (a,b,c) and ¯σ(˜k, eV = 0) as a function of ¯σN in Figs. S4 (d,e,f), at td1 = ∆. As well as along
the z-direction, on the surface arc state at (ky, kz) = (0, 0.3π), ¯σ(˜k, eV = 0) becomes almost 2, and ¯σN is suppressed
by Ub [Fig. S4 (a)], on the BFS at (ky, kz) = (0.1π, 0), we obtain ¯σ(˜k, eV = 0) ∼¯σN [Fig. S4 (b)], and on the gapped
region at (ky, kz) = (0.1π, 0.35π), both ¯σ(˜k, eV = 0) and ¯σN strongly suppressed shown in [Fig. S4 (c)]. Then we also
obtain the power law as ¯σ(˜k, eV = 0) ∼2¯σ0
N on the surface arc state [Fig. S4 (d)], ¯σ(˜k, eV = 0) ∼¯σ1
N on the BFS
[Fig. S4 (e)], ¯σ(˜k, eV = 0) ∼¯σ2
N [Fig. S4 (f)]. It is noted that the deviation in the gapped region originates from δ in
the Green’s function. Thus, three distinct types of power laws 2¯σ0
N (zero-energy surface Andreev bound states: perfect
resonance of Andreev reflection), ¯σ1
N (BFS), and ¯σ2
N (gapped region: Andreev reflection without resonance) coexist
in the zero-bias momentum dependent conductance in superconducting AM junctions. We have also commented on
this discussion in the main text.
9
0
1
2
0
0.5
1
0
1
2
0
5
10
15
20
0
1
2
0
5
10
15
20
0
1
2
0
5
10
15
20
(kx, ky) = (0.2⇡, 0.2⇡)
(kx, ky) = (0.4⇡, 0)
Crossed flat bands
Bogoliubov-Fermi surface
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Conductance
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(d)
(e)
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E0ZtaEM/fJRH8naCuXCBdwpCB0IVHRxJQNX4/lTCkMV
iA/ZIJ9FsBWATky2SI+WV0z8HWwWpVqsZLDJ4DfTfQo
ObBoeBPswr+5K8NDA1aDYDFzCDNGjA5MRZqRnaKZnF
mi7KADjSoOBmkGJQYNYHyYMzgweDAEMIQC7W1lWM2wh
WErkxqTD1MwUyhEKRMjVI8wAwpgigMADyCcYA=</lat
exit>
¯σ(˜k, eV = 0)
¯σ(˜k, eV = 0)
vXuPy8Pv2Hf93Z8uS6o0pIEITI=">ACmnichVHLSg
MxFD2O7/poVQRBF2JRXEjJiKi4Et0ogviqClZKZkzb0H
kxkxZ08Af8AReuFyoH+AHuPEHXPgJ4lLBjQtvpwOiot
6Q5OTknpscruFZMlCMPTZojU3NLa1t7YmOzq7uZKqndz
twK74psqZruf6uwQNhSUdklVSW2PV8wW3DEjtGebF2v1
MVfiBdZ0sdemLf5kVHFqTJFVH5VH/O4H4ukEWb53M2Vy
XfDleP86k0y7Aohn8CPQZpxLHmpm6RwFcmKjAhoADRd
gCR0BjDzoYPOL2ERLnE5LRvcAxEqStUJagDE5smdYin
fZi1qFzrWYQqU16xaLpk3IYo+yBXbEXds9u2BN7/7VWG
NWo/eWQdqOuFV4+eTKw+favyqZdofSp+kNhUPbfnhQKm
I28SPLmRUzNpVmvXz06fdmc2xgNx9gFeyZ/5+yR3ZFDp
/pqXq6LjTMkqEH693b8BNuTGX06M70+lZ6fiFvVhkGMY
Jz6MYN5LGENWXo3xDmucaMNaQvasrZST9UaYk0fvoS29
QGyY5jt</latexit>¯σN
¯σN
¯σN
¯σN
FIG. S3. For the dx2−y2-wave AM, (a-c) zero-bias conductance ¯σ(˜k, eV = 0) and zero-bias conductance in the normal state
¯σN along the z-direction as a function of the potential barrier Ub at (kx, ky). (d-f) ¯σ(˜k, eV = 0) along the z-direction as
a function of ¯σN at (kx, ky). We choose the momentum as (a,d) (kx, ky) = (0.2π, 0.2π) (crossed flat band), (b,e) (0.4π, 0)
[Bogoliubov-Fermi surface (BFS)], and (c,f) (0.3π, 0) (gapped region). We select the parameters as µ = −4.5t, ∆= 0.01t,
td2 = ∆, and δ = 0.0001∆.
0
1
2
0
0.5
1
0
1
2
0
5
10
15
20
0
1
2
0
0.5
1
0
1
2
0
5
10
15
20
0
1
2
0
0.5
1
0
1
2
0
5
10
15
20
Surface arc states
Bogoliubov-Fermi surface
ZBeBtuTPKuz3P8A/2DNcnpnVzTo=">ACkXichVG7Sg
NBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb
7YnQR08Qf8AQUrBQvxA/wAG3/Awk8Qywg2Ft5sFkSD8Q
4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPj
u35TtXTuaI7puPta6rPTcPmijCEyfdj6uWZvKcVlv3
Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8K
MYTLMXCmGoFcgQSiGLiT8gjyM40FGFBQ4bgrAJFT6N
A8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTad
GzX9UK3TKyZNj5RTSLJndsfq7Inds1f2+WetIKzR+MsJ
7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTN
DZmGS71Zv3Z6Uc+u7iSDGXbD3sjfNXthj+TQr3rt9t
85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjL
YwBYUetfDJa5xI41LK1JGWmumSh2RZhw/Qtr8Aq0xlYQ
=</latexit>Ub/t
Conductance
p
nVzTo=">ACkXichVG7SgNBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb7YnQR08Qf8AQUrBQvxA/w
AG3/Awk8Qywg2Ft5sFkSD8Q4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju35TtXTuaI7puPta6rPT
cPmijCEyfdj6uWZvKcVlv3Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8KMYTLMXCmGoFcgQSiGL
iT8gjyM40FGFBQ4bgrAJFT6NA8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadGzX9UK3TKyZNj5RTSLJ
ndsfq7Inds1f2+WetIKzR+MsJ7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTNDZmGS71Zv3Z6Uc+u7i
SDGXbD3sjfNXthj+TQr3rt9t85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLYwBYUetfDJa5xI41L
K1JGWmumSh2RZhw/Qtr8Aq0xlYQ=</latexit>Ub/t
p
nVzTo=">ACkXichVG7SgNBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb7YnQR08Qf8AQUrBQvxA/w
AG3/Awk8Qywg2Ft5sFkSD8Q4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju35TtXTuaI7puPta6rPT
cPmijCEyfdj6uWZvKcVlv3Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8KMYTLMXCmGoFcgQSiGL
iT8gjyM40FGFBQ4bgrAJFT6NA8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadGzX9UK3TKyZNj5RTSLJ
ndsfq7Inds1f2+WetIKzR+MsJ7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTNDZmGS71Zv3Z6Uc+u7i
SDGXbD3sjfNXthj+TQr3rt9t85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLYwBYUetfDJa5xI41L
K1JGWmumSh2RZhw/Qtr8Aq0xlYQ=</latexit>Ub/t
(a)
(b)
(c)
(d)
(e)
(f)
Gapped region
(ky, kz) = (0, 0.3⇡)
(ky, kz) = (0.1⇡, 0)
(ky, kz) = (0.1⇡, 0.35⇡)
¯σ(˜k, eV = 0)
¯σ(˜k, eV = 0)
vXuPy8Pv2Hf93Z8uS6o0pIEITI=">ACmnichVHLSg
MxFD2O7/poVQRBF2JRXEjJiKi4Et0ogviqClZKZkzb0H
kxkxZ08Af8AReuFyoH+AHuPEHXPgJ4lLBjQtvpwOiot
6Q5OTknpscruFZMlCMPTZojU3NLa1t7YmOzq7uZKqndz
twK74psqZruf6uwQNhSUdklVSW2PV8wW3DEjtGebF2v
1MVfiBdZ0sdemLf5kVHFqTJFVH5VH/O4H4ukEWb53M2V
yXfDleP86k0y7Aohn8CPQZpxLHmpm6RwFcmKjAhoADR
dgCR0BjDzoYPOL2ERLnE5LRvcAxEqStUJagDE5smdYin
fZi1qFzrWYQqU16xaLpk3IYo+yBXbEXds9u2BN7/7VWG
NWo/eWQdqOuFV4+eTKw+favyqZdofSp+kNhUPbfnhQK
mI28SPLmRUzNpVmvXz06fdmc2xgNx9gFeyZ/5+yR3ZFD
p/pqXq6LjTMkqEH693b8BNuTGX06M70+lZ6fiFvVhkGM
YJz6MYN5LGENWXo3xDmucaMNaQvasrZST9UaYk0fvoS2
9QGyY5jt</latexit>¯σN
¯σN
¯σN
¯σN
FIG. S4. For the dxy-wave AM, (a-c) zero-bias conductance ¯σ(˜k, eV = 0) and zero-bias conductance in the normal state ¯σN
along the x-direction as a function of the potential barrier Ub at (ky, kz). (d-f) ¯σ(˜k, eV = 0) along the x-direction as a function
of ¯σN at (ky, kz). We choose the momentum as (a,d) (ky, kz) = (0, 0.3π) (surface arc state), (b,e) (0.1π, 0) (BFS), and (c,f)
(0.1π, 0.35π) (gapped region). We select the parameters as µ = −4.5t, ∆= 0.01t, td1 = ∆, and δ = 0.0001∆.
10
S3: MODEL AND CONDUCTANCE OF HYBRID JUNCTIONS BASED ON 3D SUPERCONDUCTORS
WITH 3D ALTERMAGNETS
In this section, we demonstrate the conductance and bulk density of states (DOS) by using the model of 3D SCs
with 3D AMs. Finally, we study the stability of zero-bias conductance in the case of 3D SCs with 3D AMs.
S3.1: Model of 3D superconductors with 3D altermagnets
In the main text, we focused on the case of 2D AMs. We now proceed to consider 3D AMs suggested in Ref. [S14]:
dz(x+y), gzx(x2−3y2), and gyz(3x2−y2)-wave, coexisting with superconducting orders. We follow the same approach
discussed in S2.1 and S2.2 of the SM, but make the following changes. Instead of Eq. (2) in the main text and Eqs.
(S4)-(S6), the Hamiltonian of 3D AM is rewritten as [S14]
ˆHα′
AM(k) = M α′
k ˆσ3ˆτ3,
(S77)
M d3
k = td3 sin kz(sin kx + sin ky),
(S78)
M g2
k = tg2 sin kz sin kx(sin2 kx −3 sin2 ky)
(S79)
M g3
k = tg3 sin ky sin kz(3 sin2 kx −sin2 ky)
(S80)
where tα′ with α′ = d3, g2, g3 are the strength of the dz(x+y), gzx(x2−3y2), and gyz(3x2−y2)-wave AMs, respectively.
For the Fourier transformation, we obtain the local and hopping terms:
ˆuα′
AM(kx, ky) = 0,
(S81)
ˆtd3
AM1(kx, ky) = i
2td3(sin kx + sin ky),
(S82)
ˆtg2
AM1(kx, ky) = tg2 sin kx(sin2 kx −3 sin2 ky),
(S83)
ˆtg3
AM1(kx, ky) = tg3 sin ky(3 sin2 kx −sin2 ky),
(S84)
along the z-direction, and
ˆud3
AM(ky, kz) = td3 sin kz sin ky,
(S85)
ˆug3
AM(ky, kz) = tg3 sin ky sin kz
3
2 −sin2 ky
,
(S86)
ˆtd3
AM1(ky, kz) = i
2td3 sin kz,
(S87)
ˆtg2
AM1(ky, kz) = i
2tg2
3
4 sin kz −3 sin2 ky sin kz
,
(S88)
ˆtg3
AM1(ky, kz) = 0,
(S89)
ˆtd3
AM2(ky, kz) = ˆtg2
AM2(ky, kz) = 0,
(S90)
ˆtg3
AM2(ky, kz) = −3
4tg3 sin ky sin kz,
(S91)
ˆtd3
AM3(ky, kz) = ˆtg3
AM3(ky, kz) = 0,
(S92)
ˆtg2
AM3(ky, kz) = −i
8tg2 sin kz,
(S93)
along the x-direction, with
˜tα′
sc3(˜k) =
ˆtα′
AM3(˜k)
0
0
−ˆtα′∗
AM3(−˜k)
!
.
(S94)
For including the second NN hopping term ˜tsc2 like the gyz(3x2−y2)-wave AM, we adopt Eqs. (S74) and (S75) to obtain
11
the semi-infinite Green’s function. When the third NN hopping term ˜tsc3 is included like for the gzx(x2−3y2)-wave AM,
then we replace:
ˆu →
˜usc ˜tsc1 ˜tsc2
˜t†
sc1
˜usc ˜tsc1
˜t†
sc2 ˜t†
sc1
˜usc
,
(S95)
ˆt →
˜tsc3
0
0
˜tsc2 ˜tsc3
0
˜tsc1 ˜tsc2 ˜tsc3
.
(S96)
We also choose the subspace of the semi-infinite Green’s function ˜GL
0 = ˜Gcc defined as:
˜G =
˜Gaa
˜Gab
˜Gac
˜Gba
˜Gbb
˜Gbc
˜Gca
˜Gcb
˜Gcc
,
(S97)
in the basis [ ˆψ†
a, ˆψ†
b, ˆψ†
c] with ˆψ†
l=a,b,c = [c†
l↑, c†
l↓, cl↑, cl↓], to connect ˆV10. This method is applicable when calculating
the conductance along the x-direction for the gzx(x2−3y2)-wave AM in S3.3 of the SM. We calculate the conductance
in 3D superconducting AM junctions by this formulation. The results are shown in the next subsections.
S3.2: DOS and conductance along the z-direction
We discussed the conductance in 2D AMs in the main text and the BFS in the End Matter (EM). In this section,
we present the additional results of the conductance caused by the 3D superconducting AMs: dz(x+y), gzx(x2−3y2),
and gyz(3x2−y2)-wave. This additional study aims to support the role of 3D superconducting AMs for hosting exotic
zero-energy flat states. To obtain the conductance in the case of 3D AMs, we use Eqs. (S85)-(S94), Eqs. (S74) and
(S75), and (S95)-(S97).
We plot the momentum dependent conductance at zero-bias voltage eV = 0 and the normalized conductance along
the z-directions for each 3D AM in Fig. S5 at Ub = 5t. Then we show the projected Fermi surface with up (magenta)
and down spins (cyan) for each transport direction in panels (d,e,f) of Fig. S5. Along the z-direction, for the dz(x+y)
AM, because the crossed flat bands do not appear [Fig. S5 (a)], the resulting line shape of the conductance does
not change [Fig. S5 (d)]. While for the gzx(x2−3y2) and gyz(3x2−y2)-wave AM, both BFS and crossed flat bands are
obtained [Figs. S5 (b,c)]. Then the number of corners of the crossed flat bands corresponds to that of the nodal lines
of AM. These behaviors are the same as those in the case of 2D AMs, as we worked in the main text. Nevertheless,
the zero-bias conductance peak (ZBCP) is not strongly suppressed with the increase of altermagnetism shown in
Figs. S5 (e,f). Indeed, the crossed flat band is also demonstrated for the gzx(x2−3y2) and gyz(3x2−y2)-wave AMs, not
dz(x+y)-wave AM. Next, we discuss the BFS, and then, we calculate the bulk DOS, as described in the case of 2D
AM in the EM. In Fig. S6, we also show how the BFS emerges in the case of 3D AM: dz(x+y), dzx(x2−3y2), and
dyz(3x2−y2)-wave, by using the bulk DOS. As we also mentioned in the EM, the bright (dark) color of the bulk DOS
corresponds to the BFS (the crossed flat band and the surface arc state), and the bright point at (kx, ky) = (0, 0)
is the nodal point where the energy gap closes. For the dz(x+y)-wave AM, the structure of DOS does not change
[Figs. S6 (a)] compared with that without altermagnetism. It means that the projected BFS does not appear along
the z-direction, and then the corresponding conductance is independent of td3 [Fig. S5 (b)]. While the BFS appears
for the gzx(x2−3y2) and gyz(3x2−y2)-wave AM by tα′ with α′ = g2, g3 [Figs. S6(b) and Figs. S6 (c)]. Then the number
of corners corresponds to the nodes of 3D AM. Hence, 3D AMs lead to the BFS for gzx(x2−3y2) and gyz(3x2−y2)-wave
AMs, except for the dz(x+y)-wave AM. We have shown the conductance in 2D superconducting AMs in the main text.
S3.3: DOS and conductance along the x-direction
Along the x-direction, we also show the momentum dependent conductance at zero-bias voltage eV = 0 and the
normalized conductance along the x-directions for each 3D AM in Fig. S7 at Ub = 5t. For the dz(x+y)-wave AM shown
12
FIG. S5.
(a,b,c) Conductance along the z-direction as a function of (kx, ky) at eV = 0 with (a) dz(x+y), (b) gzx(x2−3y2), and
(c) gyz(3x2−y2)-wave AM. We select the altermagnetic order as (a) td3 = ∆, (b) tg2 = ∆, and (c) tg3 = ∆. (d,e,f) Conductance
along the z-direction with (d) dz(x+y), (e) dzx(x2−3y2), and (f) gyz(3x2−y2)-wave AM. σN indicates the conductance in the normal
state at eV = 0. Gray, light-red, green, and blue lines indicate the strength of the altermagnetic order as tα′ = 0, 0.5∆, ∆, and
1.5∆with α′ = d3, g2, g3. In (d,e,f), we show the image of the normal state Fermi surface when we see along the z-direction.
We set parameters as µ = −4.5t, ∆= 0.01t, Ub = 5t, and δ = 0.01∆.
FIG. S6.
Bulk density of states (DOS) at E = 0 in 3D chiral d-wave SCs projected onto the z-direction for (a) dz(x+y),
(b) gzx(x2−3y2), and (c) gyz(3x2−y2)-wave AM. AM is set as tα′ = ∆with α′ = d3, g2, g3. Bright (dark) region indicates the
Bogoliubov Fermi surface (crossed flat bands). We set parameters as µ = −4.5t, ∆= 0.01t, and δ = 0.01∆.
in Figs. S7 (a), the intensity of surface arc states becomes weak due to the BFS [Figs. S8 (a)], and the conductance
at zero-bias voltage is first enhanced and suppressed by td3 in Fig. S7 (d). For the gzx(x2−3y2)-wave AM, because the
point nodes mainly become BFS, see also Figs. S8 (b), the region of surface arc states reduces by BFS [Fig. S7 (b)].
Indeed, as shown in Fig. S7 (e), the zero-bias conductance is suppressed by td2. While for the gyz(3x2−y2)-wave AM,
spin-split surface arc states appear [Fig. S7 (c)], and the resulting zero-bias conductance is strongly suppressed by td3
[Fig. S7 (f)]. Indeed, 3D AMs also lead to the modification of the surface states and the resulting conductance, as
we worked in the case of 2D AMs in the main text. The energy gap structure becomes gapless at td3 = ∆[Figs. S8
(a)] for the dz(x+y)-wave AM. While for the gzx(x2−3y2)-wave AM, at tg2 = ∆, not only line node at kz = 0 but also
point nodes at kz = kF with the Fermi vector kF mainly become the BFS [Fig. S8 (b)]. As a result, because the
13
FIG. S7.
(a,b,c) Conductance along the x-direction as a function of (ky, kz) at eV = 0 with (a) dz(x+y), (b) gzx(x2−3y2), and
(c) gyz(3x2−y2)-wave AM. We select the altermagnetic order as (a) td3 = ∆, (b) tg2 = ∆, and (c) tg3 = ∆. (d,e,f) Conductance
along the x-direction with (d) dz(x+y), (e) dzx(x2−3y2), and (f) gyz(3x2−y2)-wave AM. σN indicates the conductance in the normal
state at eV = 0. Gray, light-red, green, and blue lines indicate the strength of the altermagnetic order as tα′ = 0, 0.5∆, ∆, and
1.5∆with α′ = d3, g2, g3. In (d,e,f), we show the image of the normal state Fermi surface when we see along the x-direction.
We set parameters as µ = −4.5t, ∆= 0.01t, Ub = 5t, and δ = 0.01∆.
FIG. S8.
Bulk density of states (DOS) at E = 0 in 3D chiral d-wave SCs projected onto the x-direction for (a) dz(x+y),
(b) gzx(x2−3y2), and (c) gyz(3x2−y2)-wave AM. AM is set as tα′ = ∆with α′ = d3, g2, g3. Bright (dark) region indicates the
Bogoliubov Fermi surface (surface arc states). We set parameters as µ = −4.5t, ∆= 0.01t, and δ = 0.01∆.
region of the surface arc states is small for the BFS, the zero-bias conductance is suppressed by tg2 [Fig. S7 (e)]. For
gg3-wave AM, the line node at kz = 0 partly becomes the BFS at tg3 = ∆[Fig. S8 (c)]. Since surface arc states
are spin-split by avoiding the BFS [Fig. S7 (e)], the resulting zero-bias conductance is also suppressed, and the line
shape of the conductance is also split [Fig. S7 (f)]. Indeed, the BFS appears in 3D AMs, as well as 2D AM cases, and
surface arc states modified by the BFS contribute to the line shape of the conductance. We have also demonstrated
the conductance in the case of 2D AMs in the main text.
14
0
1
2
0
5
0
1
2
0
5
0
1
2
0
5
0
1
2
0
5
0
1
2
0
5
s
ha1_base64="xCB1wEd
I2KCm42LTYKuk2v8TqN
0=">ACkXichVHLSs
NAFD3GV62PVt0IbsRSc
SF1IlIfq6IbwY2vWMFq
SeK0Dc2LZFrQ4A/4Awq
uFyIH+AHuPEHXPQTxK
WCGxfepgFRsd4wmTNn7
rlzZq7moYvGt0SJ1d
3T29sb54/8DgUCI5PL
LrOzVP54rumI63p6k+N
w2bK8IQJt9zPa5amsnz
WnW1uZ+vc83HtHLv
8wFLtlEydFUQdagUC5
YqKp4VaKezophMsQwLY
+I3kCOQhQbTvIeBRz
BgY4aLHDYEIRNqPDp24
cMBpe4AwTEeYSMcJ/jF
HS1iLU4ZKbJX+ZVrt
R6xN62ZNP1TrdIpJwyP
lBNLsid2yV/bI7tgz+/
izVhDWaHo5plrablbT
JyNb/q7JoFqh8qdp
6FihMfRqkHc3ZJq30F
v6+sn56/byVjqYtfsh
fxfsQZ7oBvY9Tf9ZpNv
Xbxo5EXejFqkPyzHb/
B7lxGzmaym/Op3EzUqh
jGMYlp6scCcljDBhSq
7+ECV7iWRqUlKSetFK
ljkgzim8hrX8CvMyVhA
=</latexit>Ub/t
σ(eV = 0)/σN
(a)
(b)
(c)
T
YKuk2v8TqN0=">ACkXichVHLSsNAFD3GV62PVt0IbsRScSF1IlIfq6IbwY2vWMFqSeK0Dc2LZF
rQ4A/4AwquFyIH+AHuPEHXPQTxKWCGxfepgFRsd4wmTNn7rlzZq7moYvGt0SJ1d3T29sb54/
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dFUQdagUC5YqKp4VaKezophMsQwLY+I3kCOQhQbTvIeBRzBgY4aLHDYEIRNqPDp24cMBpe4AwTE
eYSMcJ/jFHS1iLU4ZKbJX+ZVrtR6xN62ZNP1TrdIpJwyPlBNLsid2yV/bI7tgz+/izVhDWaHo
5plrablbTJyNb/q7JoFqh8qdp6FihMfRqkHc3ZJq30Fv6+sn56/byVjqYtfshfxfsQZ7oB
vY9Tf9ZpNvXbxo5EXejFqkPyzHb/B7lxGzmaym/Op3EzUqhjGMYlp6scCcljDBhSq7+ECV7iWRq
UlKSetFKljkgzim8hrX8CvMyVhA=</latexit>Ub/t
(d)
(e)
(f)
z-direction
-wave AM
-wave AM
-wave AM
-wave AM
-wave AM
gyz(3x2−y2)
gzx(x2−3y2)
4="UPe0X7rR5MUoYnxa0ocr9oITZ7Q=">ACj
XichVFLSgNBEH2Ovxg/iboR3ASDoihIxo/iAR
c6FKNUTGMDPpxMH5MTMJxuAFXAsuREHBhXgA
D+DGC7jIEcSlghsX1kwGRSxmu6uel2vqh8lma
piO4zVWoTWtvaOzkBXsLunty8U7h/Yto2SJfO0
bKiGtSuJNlcVnacdxVH5rmlxUZNUviMdrjvO
2Vu2YqhbzkVk2c1sagrBUWHYL28rnq8fjRZGX
iJBeOstiCa7OReIx59teJwrd1I/yAfeRhQEYJG
jh0OSrEGHTyiAOBpOwLKqEWeQp3jvHCYLELVE
WpwyR0EM6ixRlfFSn2K1pe2yZuqi0LWJGMqe
2R17Y0/snr2wz4a1ql4N9y8VuqU6l5u50OlQ6u
Nflka3g4MfVhOGRNnNTkoYN7TopA20NclXK9
fvn4/C21uDlaHWM37JX0XbMaeySFevldvt3gm
xcI0oAaz+Xb2Z6OxROxMZMNDnljyqAYxgnOY
xhyTWsI409dVxhktcCSFhVlgSlupQovPGcQvE
1a/AEn0k5w=</latexit>dz(x+y)
gzx(x2−3y2)
gyz(3x2−y2)
0
1
2
0
5
x-direction
σ(eV = 0)/σN
-wave AM
dz(x+y)
td3/∆
td3/∆
tg2/∆
tg2/∆
tg3/∆
tg3/∆
4
0
0
2
σ(eV = 0)/σN
σ(eV = 0)/σN
σ(eV = 0)/σN
σ(eV = 0)/σN
FIG. S9. Normalized zero-bias conductance along z (a,b,c) and x (d,e,f) transport directions as functions of Ub and tα′. The
altermagnetic order: (a,d) dz(x+y), (b,e) gzx(x2−3y2), (c,f) gyz(3x2−y2)-wave. σN denotes the conductance in the normal state at
eV = 0, and we calculate it for each tα′ and Ub. Gray circles indicate the parameters shown in Fig. S5 and Fig. S7. Parameters:
µ = −4.5t, ∆= 0.01t, and δ = 0.01∆.
S3.4: Stability of zero-bias conductance in 3D superconducting AM junctions
In the main text, we discussed the normalized zero-bias conductance for the potential barrier Ub and the strength
of 2D AM tα space. We also demonstrate the normalized zero-bias conductance along z [Figs. S9 (a,b,c)] and x
[Figs. S9 (d,e,f)] directions for Ub and the strength of 3D AM tα′. For the dz(x+y)-wave AM, because the ZBCP is not
suppressed with the increase of tα′ along the z-direction, the white area does not decrease [Figs. S9 (a)]. Then, crossed
flat bands do not appear for the dz(x+y)-wave AM. While for the gzx(x2−3y2) and gyz(3x2−y2)-wave AM, even though
the crossed flat bands are realized by AM, the normalized zero-bias conductance is not suppressed [Figs. S9 (b,c)]. We
note that crossed flat bands emerge even at small Ub for the gzx(x2−3y2) and gyz(3x2−y2)-wave AM [Figs. S9 (a,b,c)].
Along the x-direction, for the dz(x+y) and gyz(3x2−y2)-wave AM, the normalized zero-bias conductance is enhanced
once for 0.25∆≤tα′ ≤∆with α′ = d3, g3, and it is suppressed for tα′ > ∆[Figs. S9 (d,f)], see also Figs. S7 (d,f).
While for the gzx(x2−3y2)-wave AM, the normalized zero-bias conductance gradually decreases with the increase of td2
[Figs. S7 (e)]. This behavior corresponds to the modification of surface arc states by the BFS. Therefore, zero-bias
conductance can also be modified by 3D AMs, as we have found in the case of 2D AMs in the main text.
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Crossed surface flat bands in three-dimensional superconducting altermagnets Yuri Fukaya,1 Bo Lu,2 Keiji Yada,3 Yukio Tanaka,3 and Jorge Cayao4 1Faculty of Environmental Life, Natural Science and Technology, Okayama University, 700-8530 Okayama, Japan 2 300354 Tianjin, China 3 464-8603 Nagoya, Japan 4 516, S-751 20 Uppsala, Sweden (Dated: October 17, 2025) Superconducting altermagnets have proven to be a promising ground for emergent phenomena but their study has involved two dimensional systems. In this work, we investigate three-dimensional dand g-wave altermagnets with chiral d-wave superconductivity and show the formation of crossed surface flat bands due to the underlying symmetries. We find that these crossed flat bands appear at zero energy in the surface along z due to the superconducting nodal lines in the xy-plane, while the number of corners is determined by the crystal symmetry of altermagnets. We also show that the superconducting nodal lines give rise to Bogoliubov-Fermi surfaces, which then affect the appearance of zero-energy arcs in the surface along x. Moreover, we demonstrate that the crossed surface flat bands, surface arcs, and Bogoliubov-Fermi surfaces give rise to distinct signals in charge conductance, hence offering a solid way for their detection and paving the way for realizing higher dimensional topological phases using altermagnets. The discovery of altermagnetism has recently triggered great interest due to its potential for realizing emergent phenomena and spintronic applications [1-4]. Altermagnets (AMs) are characterized by a zero net magnetization as a result of their momentum dependent spin splitting of energy bands and anisotropic spin-polarized Fermi surfaces [2, 5-14]. While these altermagnetic properties have proven crucial in the normal state [1], it has been recently shown that combining AMs with superconductivity offers unprecedented opportunities [4, 15]. For instance, altermagnetism has been shown to induce key bulk and surface phenomena [4, 15], including anomalous Josephson transport [16-22], Majorana physics [23-29], exotic superconducting pairs [30-39], nonreciprocal transport [40-43], nontrivial light-matter coupling [36, 37], as well as distinct types of stable subgap states [44, 45]. The discovery of stable subgap states in AMs is of particular relevance because they signal the emergence of unusual superconducting states protected by symmetry and topology. In this regard, altermagnetism has been shown to induce Bogoliubov-Fermi surfaces (BFSs), flat bands, and Andreev bound states (ABSs), all exhibiting strong ties to the underlying altermagnetic and superconducting symmetries [44, 45]. While these ideas indeed show the potential of AMs for designing topological superconducting phases, the few existing studies have only addressed two-dimensional (2D) systems, thus leaving unexplored a large family of 3D AMs. One of the most interesting systems is Sr2RuO4 superconductors, which have very recently been predicted to host altermagnetic order [46, 47] as a phase that coexists with a very likely 3D chiral d-wave superconductivity [48-51] probed by nuclear magnetic resonance [52, 53]. It is therefore natural to wonder about the design and detection of higher dimensional topological superconducting phases by exploiting the symmetry and topology of 3D superconducting AMs. In this work, we consider 3D superconducting AMs [Fig. 1] and demonstrate the emergence of crossed flat bands as a generic topological surface phenomenon due to the interplay between intrinsic superconducting and altermagnetic symmetries. In particular, we focus on d- and g-wave AMs with chiral d-wave superconductivity, which are very likely to appear in Sr2RuO4, see [46, 47] and [48-51]. We discover that the xy-plane nodal lines of chiral d-wave superconductivity ensure that the crossed flat bands appear at zero energy on the [001] surface, while their corners are determined by the altermagnetic nodes. The crossed surface flat bands represent the higher dimensional analog of zero-energy ABSs [54, 55] in 3D superconducting AMs, thus broadening their initial conception in Weyl superconductors [56] and the idea that flat bands inherit the shape of the projected Fermi surface [57]. Moreover, we find that the superconducting nodal lines induce BFSs, which then modify the formation of arcs on the [100] surface. We further show that the crossed flat bands, arc states, and BFSs produce very distinctive conductance signatures which can be used for their detection. Our findings establish 3D superconducting AMs as a fertile ground for designing symmetry protected topological phenomena. 3D superconducting AMs.- We consider a 3D superconducting AM modelled by the following Bogoliubov-de Gennes (BdG) Hamiltonian, ˆHα BdG(k) = ε(k)ˆσ0ˆτ3 + ˆHα AM(k) + ˆHSC(k) , (1) where ε(k) = -μ -2t cos kx -2t cos ky -2t cos kz, while ˆHα AM(k) and ˆHSC(k) describe the altermagnetic and superconducting orders, with α denoting the type of altermagnetism (see below). Here, k = (kx, ky, kz), μ is the chemical potential, t is the hopping integral, while ˆσj and ˆτj are the j-th Pauli matrices in spin and particle-hole 16 Oct 2025 2 Superconducting altermagnet x y z (a) Normal metal kx ky kz (c) (d) -wave altermagnet Crossed flat band dxy " # Superconducting altermagnet (b) " # Crossed flat band altermagnet point node Bogoliubov Fermi surface Surface arc -wave gxy(x2-y2) (f) (g) (e) FIG. 1. (a,b) Sketches of the studied junctions along z- (a) and x-directions (b), formed by a 3D superconducting altermagnet (light blue) and a normal metal (light red). (c-f) Bulk and surface properties of dxy- and gxy(x2-y2)-wave altermagnets with chiral d-wave superconductivity. (c,d) Bulk Fermi volumes for dxy- (c) and gxy(x2-y2)-wave (d) altermagnets, where the cyan and magenta colors indicate up and down spins. The line nodes of the Fermi volumes give rise to Bogoliubov-Fermi surfaces (gray) under chiral d-wave superconductivity. (e) A projection of the Fermi volumes on the [100] surface along x leads to surface arc states (orange), whose ends are marked by the point nodes. (f,g) When projecting the Fermi volumes on the [001] surface along z, crossed flat bands emerge (green) with their corners defined by the nodes of the 2D spin-polarized altermagnetic Fermi surfaces, indicated by cyan and magenta ellipses for down and up spins. The crossed flat bands appear at zero energy, which is ensured by the nodal lines of chiral d-wave of superconductivity. subspaces, respectively. Altermagnetism is modelled by ˆHα AM(k) = M α k ˆσ3ˆτ3 , (2) where M α k characterizes α-wave altermagnetic order with α = d, g. More precisely, we focus on dxy- and dx2-y2wave AMs described by M d1 k = 2td1 sin kx sin ky, M d2 k = td2(cos kx -cos ky). We also consider g-wave AMs with M g k = 4tg sin kx sin ky(cos kx-cos ky). Thus, td1, td2, and tg represent the strength of dxy-, dx2-y2-, and gxy(x2-y2)- wave AMs. While ˆHAM(k) is described by an effective 2D model, our findings remain in 3D AMs [58], see S3 in the Supplementary Material (SM) [59]. Moreover, 3D chiral superconductivity is modelled by ˆHSC(k) = -∆sin kz[sin kxˆτ2 + sin kyˆτ1]ˆσ2, with pair amplitude ∆. The considered models are motivated by recent studies on Sr2RuO4, which predict altermagnetic order [46, 47] and also a very likely 3D chiral d-wave superconductivity [48-51]. Even though we focus on a particular 3D nodal superconductor, our main analysis can be straightforwardly generalized to other types of superconducting AMs with unconventional superconductivity. Emergence of crossed surface flat bands.-We are interested in unveiling the impact of altermagnetism on superconductivity in a 3D superconducting AM modelled by Eq. (1). For this purpose, it is useful to first inspect the energy bands, which are obtained as Es,α k,± = sM α k ± p ε2(k) + |ψ(k)|2 , (3) where s = ± and ψ(k) = ∆sin kz[sin kx +i sin ky]. These energy bands help understand the appearance of bulk phases. In the absence of AMs, the 3D chiral superconductor has line nodes in the xy-plane, while point nodes in the z-plane [Fig. 1(c,d)]. These line and point nodes give rise, respectively, to zero-energy surface flat bands on the [001] surface and to surface arc states on the [100] surface [54, 55, 57, 60]; the shape of the flat bands is determined by that of the projected Fermi surface, with a disk shape for a spherical Fermi surface [60]. In the presence of altermagnetism, we obtain Es,α k,-= 0 for [M α k ]2 = ε2(k) + |ψ(k)|2. This defines the formation of a BFS, and shows that the line nodes of the chiral d-wave superconductor become BFSs due to altermagnetism, except for the nodal lines of AMs. Interestingly, altermagnetism strongly deforms the surface states of the chiral superconductor, giving rise to crossed flat bands and modified surface arcs in the [001] and [100] surfaces [Fig. 1(e,f,g). While the surface arcs are inherited from the chiral superconductor, the crossed flat bands are entirely due to the interplay between altermagnetism and chiral d-wave superconductivity: by projecting the altermagnetic Fermi surfaces, we identify that the altermagnetic nodes (spin-degenerate lines in 3D) define the number of corners of the crossed flat band. Moreover, due to the nodes of the 3D chiral d-wave superconductivity, the crossed flat bands appear at zero energy. This analysis also applies to spin-triplet pz-wave superconductivity [61]. Thus, the emergence of crossed flat bands is a generic phenomenon resulting from the interplay between altermagnetism and nodal superconductivity and hence tied to the inherent crystal symmetries. Topological origin of the crossed surface flat bands.- The symmetries required for the emergence of crossed flat bands also imply that they have a topological origin [57, 61-64], which we show here. In 3D superconductors with nodal lines but without altermagnetism, a nonzero 1D winding number can be obtained with the chiral operator by combining the pseudo-time-reversal (pTRS) and particle-hole (PH) symmetries [60], where pTRS is the product of conventional TRS and local gauge transformation. This local gauge transformation is introduced to cancel out the inner phase of the pair potential. In the presence of altermagnetism, the chiral symmetry is broken because of the broken TRS by AMs [1, 4]. Still, the Hamiltonian keeps the pseudo-magnetic mirror symmetry (pMMS), which is given by the product of pTRS 3 and mirror symmetry. For dxy- and dx2-y2-wave AMs, the pMMSs are given, respectively, by ˆΘ ′ k ˆHd1 BdG(kx, ky, kz)ˆΘ ′† k = [ ˆHd1 BdG(-νkx, νky, kz)]∗ (4) ˆΘ ′ k ˆHd2 BdG(kx, ky, kz)ˆΘ ′† k = [ ˆHd2 BdG(-νky, -νkx, kz)]∗(5) where ˆΘ ′ k = ˆU † φk ˆΘ ˆUφk denotes the pMMS, ˆΘ = iˆσ2ˆτ0 is the conventional TRS, ˆUφk = e-iφkˆσ0ˆτ3/2 is the local gauge transformation, and φk = tan-1(sin ky/ sin kx) is the phase of the pair potential. Here, ν = +1 and ν = -1 in Eq. (4) indicate pMMSs for dxy-wave AMs in the xz- and yz-planes, respectively. Also, ν = +1 and ν = -1 in Eq. (5) indicate pMMSs for dx2-y2-wave AMs in the diagonal planes on x = y and x = -y, respectively. For gxy(x2-y2)-wave AMs, the system has both pMMSs given by Eq. (4) and Eq. (5). In addition to the pMMSs, ˆHα BdG(k) has particle-hole symmetry given by ˆC ˆHα BdG(kx, ky, kz) ˆC† = -[ ˆHα BdG(-kx, -ky, -kz)]∗ (6) with ˆC = ˆσ0ˆτ1. Then, ˆHα BdG(k) anticommutes with the chiral operator ˆΓk = -i ˆC ˆΘ ′ k on the magnetic mirror plane. Concrete forms of ˆΓk on each mirror plane are given in S1.3 of [59]. Importantly, these magnetic mirror and chiral symmetries are still available on the smooth surfaces, which is perpendicular to the magnetic mirror planes, allowing for the symmetry-protection of surface states as a unique effect due to altermagnetism and nodal superconductivity. For example, a vertical pMMS is given in Eq. (4) and the corresponding chiral symmetry is accessible on the kx- and ky-axes in the surface Brillouin zone. Hence, a winding number can be obtained on the symmetric lines, whose nonzero values signal the appearance of topological phases with topologically protected surface states [59]. Thus, a nonzero winding number along the kx, ky-axis and/or the diagonal direction implies that the crossed flat bands along the [001] direction discussed in the previous section are topologically protected [Fig. 1(f,g)]. Also, because of the pMMS on the xz-plane, topologically protected zero-energy surface states appear on the [100] surface, which we identified as surface arcs [Fig. 1(e)]. As a result, both surface arcs and crossed flat bands are topologically protected and can be realized in 3D superconducting AMs [65]. Conductance signatures of crossed flat bands.-Having shown the emergence of crossed flat bands and surface arcs, as well as their topological origin, here we explore how they can be detected by means of conductance. For this purpose, we couple the 3D superconducting AM to a normal metal, such that we can assess transport along z and x directions [Fig. 1(a,b)]. We obtain the conductance following the common tunneling Hamiltonian approach based on the Lee-Fisher formula [44, 66-68]: σ(eV ) = 1 2π Z π -π d ̃k ̄σ( ̃k, eV ) , (7) FIG. 2. (a,c) Zero-bias conductance along the z-direction [001] as a function of momenta kx,y for a superconducting AM with dxy-wave (a) and gxy(x2-y2)-wave (c) altermagnetism. (b,d) Normalized total conductance along the z-direction as a function of eV for distinct values of the altermagnetic strength tα. σN indicates the conductance in the normal state at eV = 0. The insets in (b,d) indicate the normal state Fermi surfaces projected onto the [001] surface for up (magenta) and down spins (cyan). Parameters: tα = ∆(a,c), μ = -4.5t, ∆= 0.01t, Ub = 5t, and δ = 0.01∆. with ̃k = (ku, kv) for u = x, y and v = y, z, while ̄σ( ̃k, eV ) is the momentum dependent conductance obtained as ̄σ( ̃k, eV ) = Tr′ X i,j=0,1 [ ˆGii ˆV † ji ˆGjj ˆVji -ˆGij ˆVji ˆGij ˆVji] , (8) where we have dropped the momenta for simplicity and Tr′ means that the trace is taken in electron space. Here, ˆV10 = -ˆtJ ⊗ˆτ0, with ˆtJ characterising the coupling between the superconducting AM and metallic lead [Fig. 1(a,b)]. Moreover, ˆGnm are Green's functions numerically obtained by using the recursive Green's function method [44, 67-71], and the details of this formulation is presented in S2 of the SM [59]. We first assess the conductance signatures along z direction [001], see Fig. 2. We begin by exploring the momentum dependence conductance ̄σ( ̃k, eV = 0) at zero bias eV = 0, which is presented in Fig. 2(a,c) for superconducting AMs with dxy- and gxy(x2-y2)-wave altermagnetism. The corresponding total conductances σ(eV ) are then shown in Fig. 2(b,d) as a function of eV for distinct altermagnetic strengths. The most relevant signature of the momentum dependent conductance at zero bias is that the strongest intensities come from the contribution of the crossed surface flat bands [Fig. 2(a,c)], 4 which emerge at the [001] surface as discussed in the previous section. Depending on the type of underlying altermagnetism, the zero-bias conductance ̄σ( ̃k, eV = 0) unveils crossed flat bands with different number of corners [Fig. 2(a,c)], as expected since the corners of the emergent crossed flat bands are determined by the nodes of the AM. For dxy-wave AM, the zero-bias momentum dependent conductance reveals four corners [Fig. 2(a)], while eight corners for gxy(x2-y2)-wave AM [Fig. 2(c)]. For dx2-y2-wave AMs, ̄σ( ̃k, eV = 0) also exhibits four corners but with a crossed flat band rotated by π/4, see Sec. E1 and Fig. 4(a) in the End Matter. Moreover, ̄σ( ̃k, eV = 0) in Fig. 2(a,c) acquires lower but nonzero intensities around the crossed flat bands, which arise due to contributions from the BFS. This is confirmed by projecting the zero-energy density of states on the [001] surface, see Sec. E1 and Fig. 5 in the End Matter. There also appears a darkish area between the BFS and crossed flat band in Fig. 2(a,c) originating from a gapped dispersion where quasiparticles attain vanishing velocity along z, see S1.2 in SM [59]. Notably, we find that, for the crossed flat bands, BFSs and gapped regions, respectively, ̄σ( ̃k, eV = 0) is given by 2 ̄σ0 N, ̄σ1 N, and ̄σ2 N, unveiling three distinct types of power laws that coexist in 3D superconducting AMs [72]; see S2 in [59]. When inspecting the total conductance σ(eV ) as a function of eV in Fig. 2(b,d), it develops a zero-bias peak whose height decreases as the altermagnetic strength increases; see also Fig. 4(b). The presence of a large zero-bias conductance peak (ZBCP) without any altermagnetism is due to the zero-energy ABSs of the chiral d-wave superconductor [55, 64, 73]. Altermagnetism, however, reduces this ZBCP but, interestingly, its reduced value remains even at strong values of altermagnetic strengths [Fig. 2(b,d)]. While the same reduction occurs for dxy-, dx2-y2-, and gxy(x2-y2)-wave AMs, the latter exhibits a stronger and well-formed ZBCP; see also Sec. E5 and Fig. 8 of the End Matter. Nevertheless, in all cases, the ZBCP constitutes a direct evidence of the crossed surface flat bands formed at the [001] surface of 3D superconducting AMs. Conductance along x direction and signatures of surface arcs.-Conductance along x reveals distinct signatures of the 3D superconducting AM, giving rise to direction dependent transport in these systems. In Fig. 3(a,c), we present the momentum dependent conductance ̄σ( ̃k, eV = 0) at zero bias for superconducting AMs with dxy- and gxy(x2-y2)-wave altermagnetism. Fig. 3(b,d) shows the total conductances σ(eV ) as a function of eV for distinct strengths of altermagnetism. For completeness, in Sec. E3 and Fig. 6 of the End Matter we also show these conductances for the case of dx2-y2wave altermagnetism. From Fig. 3(a,c), the momentum dependent conductance ̄σ( ̃k, eV = 0) develops high intensities around ky = 0 due to the contribution of zeroenergy surface arcs states formed at the [100] surface. As discussed before, these zero-energy surface arcs originate FIG. 3. (a,c) Zero-bias conductance along the x-direction [100] as a function of ky,z for a superconducting AM with dxy-wave (a) and gxy(x2-y2)-wave (c) altermagnetism. (b,d) Normalized total conductance along the x-direction [100] as a function of eV for distinct values of the altermagnetic strength tα. σN indicates the conductance in the normal state at eV = 0. The insets in (b,d) indicate the normal state Fermi surfaces projected onto the [100] surface for up (magenta) and down spins (cyan). Parameters: tα = ∆(a,c), μ = -4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆. from the MMS on the xz-plane. Interestingly, the surface arc conductance signal is interrupted around kz = 0 due to a reduced contribution coming from the BFSs; see also Fig. 1(d). The role of the BFS can be indeed confirmed by projecting the density of states on the [100] surface, see Fig. 7. Also, ̄σ( ̃k, eV = 0) in Fig. 6(a) reveals the formation of spin-split surface arc states for dx2-y2wave altermagnetism, which disperse with ky,z and exhibit a vanishing effect due to BFSs. Here, we also find that ̄σ( ̃k, eV = 0) is given by 2 ̄σ0 N, 2 ̄σ1 N, and 2 ̄σ2 N in the presence of zero-energy surface arcs, BFS, and gapped region; see S2 in [59]. For the total conductance σ(eV ) as a function of eV , it exhibits a broad peaked profile around zero bias at zero altermagnetism [Fig. 3(b,d)]. This broad conductance profile sharpens at finite altermagnetism for dxy and gxy(x2-y2)-wave AMs, producing a clear ZBCP [Fig. 3(b,d)]. Interestingly, for dx2-y2-wave AMs, the total conductance splits into two resonances that broaden as the altermagnetic strength increases [Fig. 6(b)]; this spin-split total conductance originates from the spin-split surface arcs observed in ̄σ( ̃k, eV = 0) in Fig. 6(a). We note that this behavior is similar to the reported for transport in Sr2RuO4 Ref. [74], supporting the idea that conductance along x is determined by the surface arc states and modified by the BFS. 5 In conclusion, we have demonstrated that crossed surface flat bands emerge in three dimensional superconducting altermagnets as a generic topological phenomenon protected by intrinsic crystal symmetries. Also, we have shown that the crossed flat bands can be detected by means of conductance, which depends on the transport direction and can also assess the formation of topological surface arcs and Bogoliubov-Fermi surfaces. We have found that the zero-bias momentum resolved conductance exhibits three distinct types of power laws, offering a solid way for detecting the emergent topological phases e. g., by Doppler shift measurements as those predicted in Refs. [75-78]. Our findings also hold experimental relevance since there already exist NMR experiments in Sr2RuO4 [52, 53] suggesting three-dimensional chiral d-wave superconductivity [48-51] and predictions of altermagnetism in the same system [46, 47]. Our work, therefore, establishes three-dimensional superconducting altermagnets as an outstanding ground for realizing higher dimensional topological phases, which can also pave the way for future tunneling spectroscopy experiments in three dimensional superconductors. Y. F. acknowledges financial support from the Sumitomo Foundation and the calculation support from Okayama University. B. L. acknowledges financial support from the National Natural Science Foundation of China (project 12474049). K. Y., Y. 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In the case of the momentum dependent conductance at zero bias ̄σ( ̃k, eV = 0) shown in Fig. 4(a), it develops high intensity values along the diagonals kx = ±ky in contrast to the case of dxywave altermagnetism having large values at kx,y = 0, see Fig. 2(a). These large zero-bias conductance values reveal the emergence of crossed flat bands in the [001] surface of superconducting AMs with dx2-y2-wave altermagnetism. As noted in the main text, ̄σ( ̃k, eV = 0) also reflects the BFS, seen as an area that surrounds the crossed flat band in Fig. 4(a). Between the BFS and the area of the crossed flat band appears a darkish region, which is gapped and quasiparticles acquire vanishing velocity and hence vanishing conductance. Interestingly, ̄σ( ̃k, eV = 0) acquires a unique dependence on its zero-bias normal state counterpart: ̄σ( ̃k, eV = 0) is roughly given by 2 ̄σ0 N, ̄σ1 N, and ̄σ2 N for the regime with crossed flat bands, BFSs, and gapped region, respectively; see S2 in [59]. Also, the total conductance σ(eV ) varies from a very peaked profile at zero bias in the absence of dx2-y2-wave altermagnetism to a reduced but visible peak at finite as the altermagnetic strength increases and surpasses the pair potential [Fig. 4(b)]. Thus, the reduced peaked profile manifests the formation of crossed flat bands on the [001] surface. E2. Projected density of states on the [001] surface.- We have seen that the gapless structure of the energy dispersion in Eq. (3) originates from BFSs. It is thus expected that the density of states (DOS) enhances due to BFSs. To show this, we calculate the bulk DOS as D(E) = -1 π Z π -π dkTr′[Im ̃GR(k, E)], (9) where Tr′ indicates the trace taken in electron space, ̃GR(k, E) is the retarded Green's function ̃GR(k, E) = [E + iδ -ˆHBdG(k)]-1, with ˆHBdG(k) given by Eq. (1), E is the energy and δ is an infinitesimal positive number. Using Eq. (9), we obtain the projected DOS on the [001] surface by integrating along the transport direction. To unveil the BFS, in Fig. 5 we plot the projected DOS on the [001] surface at zero energy for dxy-, dx2-y2-, and gxy(x2-y2)-wave AMs. The most important feature is that the BFS produces the highest intensity in the projected zero-energy DOS, see Fig. 5. Interestingly, the BFSs surround regions with tiny values, which possess corners depending on the type of AM. These inner regions correspond to the formation of crossed flat bands, discussed in Figs. 2(a,c) and 4(a). The bright point at kx,y = 0 is due to the nodal point where the energy gap closes. The BFS signals in the projected DOS remain in more elaborated 3D AMs, see Sec. S3 in the SM [59]. E3. Conductance along x-direction with dx2-y2-wave FIG. 4. (a) Zero-bias conductance along the z-direction [001] for a superconducting AM with dx2-y2-wave altermagnetism as a function of kx,y. (b) Normalized total conductance along the z-direction as a function of eV for distinct values of the altermagnetic strength td2; σN indicates the conductance in the normal state at eV = 0. The inset in (b) shows the normal state Fermi surfaces projected onto the [001] surface for up (magenta) and down spins (cyan). Parameters: td2 = ∆(a), μ = -4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆. FIG. 5. (a-c) Projected zero-energy DOS on the [001] surface as a function of kx,y for a superconducting AM with dxy- (a), dx2-y2- (b), and gxy(x2-y2)-wave altermagnetism. Parameters: tα = ∆, μ = -4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆. AMs.-Conductance along x in a superconducting AM with dx2-y2-wave altermagnetism is shown in Fig. 6(a) and we observe the formation of arcs on the [100] surface. Here, the momentum dependent zero-bias conductance ̄σ( ̃k, eV = 0) shows that the arcs are spin-split and disperse with momentum, unlike those due to dxywave altermagnetism in Fig. 3(a). The region with tiny but nonzero intensities corresponds to BFSs, while even weaker values of ̄σ( ̃k, eV = 0) occur in a gapped region, see S1 in SM [59]. Here, we have also verified that ̄σ( ̃k, eV = 0) is given by 2 ̄σ0 N, ̄σ1 N, and ̄σ2 N for the surface arcs, BFS, and gapped region; see S2 in [59]. Moreover, a finite altermagnetism splits the zero-bias total conductance peak of σ(eV ) in Fig. 6(b), producing two broad resonances around zero bias as a signature of surface arcs. E4. Projected density of states on the [100] surface.- For the projected DOS on the [100] surface using Eq. (9), we find that BFSs appear around kz = 0 for all the cases of altermagnetism [Fig. 7]. Depending on the type of AM, the projected DOS shows the BFSs coexisting with 9 FIG. 6. (a) Zero-bias conductance along the x-direction [100] for a superconducting AM with dx2-y2-wave altermagnetism as a function of kx,y. (b) Normalized total conductance along the x-direction [100] as a function of eV for distinct values of the altermagnetic strength td2. σN is the conductance in the normal state at eV = 0. The inset in (b) shows the normal state Fermi surfaces projected onto the x-direction for up (magenta) and down spins (cyan). Parameters: td2 = ∆ (a), μ = -4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆. regions having tiny but nonzero intensities that correspond to the zero-energy surface arcs on the [100] surface [Fig. 3(a,c) and Fig. 7(a)]. We also confirm the BFS signatures in proper 3D AMs, see Sec. S2 in the SM [59]. E5. Stability of the ZBCP.-Having shown the formation of crossed flat bands, surface arcs and BFSs, we now explore the ZBCP σ(eV = 0) under variations of the altermagnetic strength tα and barrier potential Ub. This is shown in Fig. 8 for transport along z and x, in both cases for dxy-, dx2-y2-, and gxy(x2-y2)-wave altermagnetism. For transport along z, the largest zero-bias conductance σ(eV = 0) is concentrated below tα ∆, as we indeed show in Fig. 2(b,d). We also note FIG. 7. (a-c) Projected zero-energy DOS on the [100] surface as a function of ky,z for a superconducting AM with dxy- (a), dx2-y2- (b), and gxy(x2-y2)-wave altermagnetism. Parameters: tα = ∆, μ = -4.5t, ∆= 0.01t, Ub = 5t, δ = 0.01∆. FIG. 8. Normalized zero-bias conductance along z (a,c,e) and x (b,d,f) directions as a function of Ub and tα for superconducting AMs with dxy-, dx2-y2-, and gxy(x2-y2)-wave altermagnetism. Here, σN denotes the conductance in the normal state at eV = 0, and we calculate it for each tα and Ub. Insets: crossed flat bands and normal state Fermi surfaces for up (magenta) and down (cyan) spins. The gray circles mark the values of Ub and tα chosen for Fig. 2 and Fig. 3. Parameters: μ = -4.5t, ∆= 0.01t, δ = 0.01∆. that the crossed flat bands exist even at Ub = 0, since the corners of the crossed flat bands are already formed without any barrier. However, a finite Ub sharpens the tunneling conductance, and much larger conductance values occur along momenta of the altermagnetic nodal lines. Hence, the visibility of the crossed flat band is enhanced by large Ub, which corresponds to low transparencies. Along the x-direction shown in Figs. 8(b,d,f), the value of σ(eV = 0) takes values ranging from 1 to 2, and it does not change significantly for variations of Ub and tα, even though the surface arc signatures partly disappear under BFSs. Also, σ(eV = 0) becomes smaller as td2 increases because the surface arc states are split by dx2-y2-wave altermagnetism [Fig. 6]. The stability of the ZBCP discussed here also persists for 3D AMs, see S3 in [59]. Supplementary material for "Crossed surface flat bands in three-dimensional altermagnetic superconductors" Yuri Fukaya,1 Bo Lu,2 Keiji Yada,3 Yukio Tanaka,3 and Jorge Cayao4 1Faculty of Environmental Life, Natural Science and Technology, Okayama University, 700-8530 Okayama, Japan 2 300354 Tianjin, China 3 464-8603 Nagoya, Japan 4 516, S-751 20 Uppsala, Sweden In this supplementary material, we provide additional details of the bulk system studied in the main text, discuss how we calculate conductance, and demonstrate that our main findings remain for a system involving three-dimensional (3D) altermagnetism with 3D chiral d-superconductivity. S1: BULK PROPERTIES OF THREE-DIMENSIONAL SUPERCONDUCTING ALTERMAGNETS First of all, we present the details of the model Hamiltonian of 3D superconductors (SCs) with two-dimensional (2D) altermagnets (AMs). Then we will show their corresponding bulk properties. S1.1: Hamiltonian model with a 3D superconductor and 2D altermagnet In this subsection, we discuss further details of the model Hamiltonian. We start by considering the normal part of the Hamiltonian, which is given by ˆHα(k) = ˆH0(k) + M α k ˆσ3, (S1) where ˆH0(k) is the kinetic term and M α k characterizes altermagnetism. More precisely, the kinetic part is given by ˆH0(k) = ε(k)ˆσ0, (S2) ε(k) = -μ -2t cos kx -2t cos ky -2t cos kz, (S3) where k = (kx, ky, kz) is the momentum, ˆσ0,1,2,3 is the Pauli matrices in spin space, μ = -4.5t is the chemical potential, t is the hopping integral along the (x, y) and z-directions. We consider the d and g-wave altermagentism modelled by M d1 k = 2td1 sin kx sin ky, (S4) M d2 k = td2(cos kx -cos ky), (S5) M g k = 4tg sin kx sin ky(cos kx -cos ky), (S6) where td1, td2, and tg are the strength of dxy, dx2-y2 and gxy(x2-y2)-wave AMs, respectively. When adding the superconductivity, the system is described by the Bogoliubov-de Gennes (BdG) Hamiltonian: ˆHα BdG(k) = ε(k)ˆσ0ˆτ3 + ˆHα AM(k) + ˆHSC(k), (S7) ˆHα AM(k) = M α k ˆσ3ˆτ3, (S8) ˆHSC(k) = -∆sin kz[sin kxˆτ2 + sin kyˆτ1]ˆσ2 (S9) with ˆτ0,1,2,3 the Pauli matrices in Nambu space and ∆= 0.01t the pair amplitude of the chiral d-wave state. The eigenvalues of ˆHα BdG(k) are obtained as: Es,α k,±(k) = sM α k ± p ε2(k) + |ψ(k)|2, (S10) with s = ± and ψ(k) = ∆sin kz[sin kx + i sin ky]. Eq. (S10) is presented in the main text as Eq. (3) and helps us identify the formation of Bogoliubov-Fermi surfaces (BFSs). 2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 Fermi surface for up-spin Fermi surface for down-spin kx/⇡ kx/⇡ kx/⇡ kx/⇡ (a) (e) (b) (f) (c) (g) (d) (h) td2 = 0 h a1_base64="8LOq2YG5cR nqXy2mVygizAuRkZM="> ACi3ichVG9SsNQGD3Gv 1qtrboILsWiOEj5WkVFA QRHFu1WlApSbytwTQJyW2 hFl/A0cVBFwUH8QF8ABd fwKGPI4KLg5+TQOiYv1C 7j3O98yeFojml4kqjRo XR2df0hvrC/QORwWhsaH jbsyuLnK6bdpuXlM9YR qWyElDmiLvuEIta6bY0Y5 Wm/c7VeF6hm1tyZoj9stq yTKhq5KpvKyUD9InyxTI ZagJPkV/w1SAUgqIwdu 8ceDmBDRwVlCFiQjE2o8P jZRQoEh7l91JlzGRn+vcA JwqytcJfgDpXZI15LfNoN WIvPzZmer9b5Kya/Livjm KAnuqVXeqQ7eqaP2fV/ RnNf6nxrW0wilET0c3/ 9VlXmXOPxStVFo3N3ek0Q RC74Xg705PtN0qbfmV4/P XzcXNybqk3RNL+zvihr0 wA6t6pt+kxUbFwhzQKmfc fwG2+lkai45l51NrEwHUY UwhnFMcR7zWME6Msj5OZz hApdKRJlRFpWlVqvSEWh G8K2UtU/JnJKA td2 = 0 td2 = 0.5∆ td2 = 0.5∆ td2 = ∆ td2 = ∆ td2 = 1.5∆ td2 = 1.5∆ Es,d2 k,± /∆ Es,d2 k,± /∆ h a1_base64="bZ9NwHSNfc 5dkBETC3G8sYV+hgc="> ACmHicSyrIySwuMTC4y cjEzMLKxs7BycXNw8vHLy AoFacX1qUnBqanJ+TXx SRlFicmpOZlxpaklmSkxp RUJSamJuUkxqelO0Mkg8v Sy0qzszPCympLEiNzU1Mz 8tMy0xOLAEKxQsIu8ZVa +ukGNXGV8ck5VZn1+po18 YLKBvoWVqYGZkaKRjoGRi YGxmbgRhG5iZGxgqGQBEQ UGaAgoB8ge0MQwpDPkM yQylDLkMqQx5DCVAdg5DI kMxEYzGDIYMBQAxWIZqo FiRUBWJlg+laGWgQuotx SoKhWoIhEomg0k04G8aKh oHpAPMrMYrDsZaEsOEBcB dSowqBpcNVhp8NnghMFqg 5cGf3CaVQ02A+SWSiCdB NGbWhDP3yUR/J2grlwgXc KQgdCFR0cSUDV+P5UwpDF YgP2SCfRbAVgE5MtkiPl VdM/B1sFqVarGSwyeA30 30KDmwaHgT7MK/uSvDQwN Wg2AxcwgmCxoICbEWakZ2 imZxZouygA40qDgZpBi UGDWB8mDM4MHgwBDCEAu2 tZJjPsIJhJZMUkwOTO5Mn RCkTI1SPMAMKYAoCABmOl 1k= E+,d2 k,+ E-,d2 k,- = "i+Hedv0yjsbGITpbpT0mZIE4ncs=">ACmHicSyr IySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qU nBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkx qelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAE KxQsIu8ZVa+ukGNXGV8ck5VZn1+ro1sYLKBvoWVqY mpoYKRjoGRhYmpgaAxmWlpaGFoYKhkAREFBmgIKAfI HtDEMKQz5DMkMpQy5DKkMeQwlQHYOQyJDMRBGMxg yGDAUAMViGaqBYkVAViZYPpWhloELqLcUqCoVqCIRK JoNJNOBvGioaB6QDzKzGKw7GWhLDhAXAXUqMKgaXD VYafDZ4ITBaoOXBn9wmlUNgPklkognQTRm1oQz98l EfydoK5cIF3CkIHQhUdHElA1fj+VMKQxWID9kgn0W wFYBOTLZIj5ZVXTPwdbBalWqxksMngN9N9Cg5sGh4E +zCv7krw0MDVoNgMXMIJgsaCAmxFmpGdopmcWaKLs oAONKg4GaQYlBg1gfJgzODB4MAQwhALtrWSYz7CYS WTFJMDkzuTJ0QpEyNUjzADCmAKAgBMn5dy E+,d2 k,- E-,d2 k,+ FIG. S1. Es,d2 k,± for the dx2-y2-wave altermagnet (AM) as a function of kx at ky = 0 scanning on the Fermi surface with (a-d) up and (e-g) down spins. We choose the altermagnetic order as (a,d) td2 = 0, (b,e) td2 = 0.5∆, (c,f) td2 = ∆, and (d,g) td2 = 1.5∆. In the normal state, at ky = 0, kz is determined by the normal state energy dispersion ε(k) ± M α k = 0 for each kx. For each (kx, kz), we plot Es,d2 k,± as a function of kx at ky = 0. In (a,e), we obtain E+,d2 k,+ = E-,d2 k,+ and E+,d2 k,-= E-,d2 k,- for each kx, and then the shape of zero-energy flat bands becomes a disk. In the presence of the altermagnetism, if E+,d2 k,+ ∼E-,d2 k,+ and E+,d2 k,- ∼E-,d2 k,-, zero-energy flat bands remain, and the resulting zero-energy flat bands becomes crossed. We select the parameter as: μ = -4.5t and ∆= 0.01t. S1.2: Structure of quasiparticle energy dispersions in the bulk In the following, we show how to determine the width of crossed flat bands. We start with the quasiparticle energy dispersion Es,d2 k,± as a function of kx at ky = 0 by scanning on the Fermi surface with up [Fig. S1 (a-d)] and down spins [Fig. S1 (e-g)], for the dx2-y2-wave AM. Then kz is determined by the normal state energy dispersion ε(k) ± M d2 k = 0 for each kx at ky = 0. Because the spin splitting of the Fermi surface is small, the results for up and down spins are almost the same. Without altermagnetism (M α k = 0), we obtain E+,d2 k,+ = E-,d2 k,+ and E+,d2 k,- = E-,d2 k,-, that is, E+,d2 k,+ -E-,d2 k,+ = E+,d2 k,--E-,d2 k,-= 2M α k = 0 for each momentum, and then we also obtain a disk of zero-energy flat bands. In the presence of altermagnetism (M α k ̸= 0), we generally get we obtain E+,d2 k,+ ̸= E-,d2 k,+ and E+,d2 k,-̸= E-,d2 k,-. Nevertheless, the relations of E+,d2 k,+ ∼E-,d2 k,+ and E+,d2 k,-∼E-,d2 k,-, that is, E+,d2 k,+ -E-,d2 k,+ = E+,d2 k,--E-,d2 k,-= 2M α k ∼0 can be satisfied, and then zero-energy surface Andreev bound states remain. Along the z-direction, although the dispersions of E+,d2 k,+ = E-,d2 k,+ and E+,d2 k,- = E-,d2 k,- for each (kx, ky) are inexplicit, we can obtain their approximate expression at small value of M α k . In the case of the dx2-y2-wave AM, E+,d2 k,+ and E-,d2 k,-are partly split, and E+,d2 k,+ ∼ E-,d2 k,+ and E+,d2 k,- ∼E-,d2 k,- are obtained [Figs. S1 (b,c,d,f,g,h)]. Then the momentum of this overlapping region is ks ∼0.08π at td2 = 0.5∆[Figs. S1 (b,e)], ks ∼0.06π at td2 = ∆[Figs. S1 (c,f)], and ks ∼0.04π at td2 = 1.5∆ [Figs. S1 (d,g)]. If E+,d2 k,+ ∼E-,d2 k,+ and E+,d2 k,- ∼E-,d2 k,-, the zero-energy flat bands remain, and as a result, this momentum ks corresponds to the size of the crossed flat band as discussed in Fig. 4(a) of the main text. Indeed, the energy dispersion of the quasiparticle Es,d2 k,± remains the degeneracy within the momentum ks even with the altermagnetism, and it contributes to the size of the crossed flat band. Along the x-direction, for the dxy-wave AM, when ky = 0 (kx = 0), we obtain M α k = 0 for each kz. It means that surface arc states, which appear along the kz-axis, are not affected by the dxy-wave AM; however, they are only modified by the BFS. While, for the dx2-y2-wave AM, M α k = 0 satisfies at |kx| = |ky| for each kz, and arc surface states appears at (ky, kz) with |ky| = |kx|. For the gxy(x2-y2)-wave AM, if ky = 0 (kx = 0) or |kx| = |ky|, we obtain M α k = 0. As a result, arc surface states are partly determined as shown in Fig. 3(c) of the main text. Thus, crossed flat bands can have the width, and the surface arc states can remain shown in Figs. 2(a,c), 3(a,c), 4(a), and 6(a) of the main text. 3 FIG. S2. (a,c) E+,d2 k,+ = -E-,d2 k,-and (b,d) E+,d2 k,-= -E-,d2 k,+ for (kx, ky) space for the dx2-y2-wave AM at (a,b) kz = 0.15π and (c,d) 0.25π. (e) Es,d2 k,± for the dx2-y2-wave AM as a function of kx at ky = 0 by changing kz. (f) Es,d2 k,± for the dx2-y2-wave AM as a function of kz at ky = 0 by changing kx. We select the parameters as μ = -4.5t, ∆= 0.01t, and td2 = ∆. Next, we discuss the velocity of the quasiparticle along the z-direction. In Fig. S2, we plot Es,d2 k,± for the dx2-y2-wave AM at td2 = ∆with ∆= 0.01t. E+,d2 k,+ = -E-,d2 k,- and E+,d2 k,-= -E-,d2 k,+ for (kx, ky) space at kz = 0.15π, 0.25π are shown in Figs. S2 (a,b,c,d). The amplitude of E+,d2 k,± becomes the smallest at the Fermi level. To see more details, we plot Es,d2 k,± as a function of kx at ky = 0 for each kz. The energy gap is opening for 0.2π ≤kz ≤0.42π. The BFS appears for kz ≥0.2π, and a point node for kz = 0.42π. We also show Es,d2 k,± as a function of kz at ky = 0 by changing kx in Fig. S2 (f). The energy gap is also opening for 0 ≤kz ≤0.35π. The BFS also exhibits for kx ≥0.35π, and a point node for kx = 0. These behaviors indicate that the energy gap closes around kx0 = ±0.35π and the BFS is realized for kx ≥|kx0|. Es,d2 k,± at the minimal energy of the upper band can have zero velocity at kx = kx0. When the minimal energy is zero, the velocity along the z-direction of the quasiparticle also becomes zero, leading to zero current. Thus, the total conductance vanishes at kx = kx0. Hence, this idea explains the vanishing conductance in the gapped regions of Figs. 2(a,c) and 4(a), as well as Figs. 3(a,c) and 6(a) of the main text. S1.3: Details on the topological stability of zero-energy surface states Here, we discuss the details of the winding number, which protects the crossed flat bands and arc states. To define the winding number, a chiral operator that anticommutes with the Hamiltonian is required [S1-S5]. The most common definition of the chiral operator is the one constructed from time-reversal (TRS) and particle-hole symmetries (PHS) [S1-S3]. However, in the present system, both AM and pair potential break the TRS, and therefore, this chiral symmetry is broken. Nevertheless, to consider chiral symmetry in the present system, it is necessary to take into account the remaining symmetries of the system. For example, the breaking of time-reversal symmetry due to the pair potential can be recovered by the following local gauge transformation [S6]: ˆH′ SC(k) = ˆUφk ˆHSC(k) ˆU † φk = -∆ q sin2 kx + sin2 ky sin kzˆσ2ˆτ2, (S11) 4 with ˆUφk = e-iφkˆσ0ˆτ3/2 = cos(φk/2)ˆσ0ˆτ0 -i sin(φk/2)ˆσ0ˆτ3 and φk = tan-1(sin ky/ sin kx), i. e., cos φk = sin kx/ q sin2 kx + sin2 ky , sin φk = sin ky/ q sin2 kx + sin2 ky . (S12) This local gauge transformation does not affect the kinetic and altermagnetic parts: ˆUφkε(k)ˆσ0ˆτ3 ˆU † φk = ε(k)ˆσ0ˆτ3 and ˆUφk ˆHα AM(k) ˆU † φk = ˆHα AM(k). By this local gauge transformation, the normal part and pair potential term of the Hamiltonian have the following pseudo-time-reversal (pTRS) and particle-hole symmetries: ˆΘ ˆH′ NM+SC(k)ˆΘ† = [ ˆH′ NM+SC(-k)]∗, (S13) ˆC ˆH′ NM+SC(k) ˆC† = -[ ˆH′ NM+SC(-k)]∗, (S14) with ˆH′ NM+SC(k) = ε(k)ˆσ0ˆτ3 + ˆH′ SC(k), the conventional TRS ˆΘ = iˆσ2ˆτ0, and PHS ˆC = ˆσ0ˆτ1. We note that the conventional TRS ˆΘ and PHS ˆC are also defined in the main text. Here, we introduce a chiral operator ˆΓ ≡-i ˆC ˆΘ = ˆσ2ˆτ1. Using the Eqs. (S13) and (S14), we obtain the following chiral symmetry: ˆΓ ˆH′ NM+SC(k)ˆΓ† = -ˆH′ NM+SC(k), (S15) where ˆΓ anticommutes H′ NM+SC(k). Since H′ NM+SC(k) is the Hamiltonian obtained from the local gauge transformation, we go back to the original gauge: ˆU † φk ˆΓ ˆUφk ˆHNM+SC(k) ˆU † φk ˆΓ† ˆUφk = -ˆHNM+SC(k). (S16) Then, we obtain the chiral operator for the original gauge: ˆΓk = ˆU † φk ˆΓ ˆUφk. (S17) On the other hand, this anticommutation relation in Eq. (S16) is no longer satisfied when we consider ˆHα AM(k) due to the broken TRS: ˆΘ ˆHα AM(k)ˆΘ† = -[ ˆHα AM(-k)]∗, (S18) ˆC ˆHα AM(k) ˆC† = -[ ˆHα AM(-k)]∗. (S19) Nevertheless, ˆHα AM(k) has the magnetic mirror symmetry (MMS), which is the product of pTRS and the mirror symmetry. The corresponding mirror plane depends on the symmetry of AMs. In the main text, since we discuss the chiral operator ˆΓk by focusing on the total Hamiltonian ˆHα BdG(k), we introduce the pseudo-magnetic mirror symmetry (pMMS): ˆΘ ′ k = ˆU † φk ˆΘ ˆUφk. While here, we demonstrate the chiral operator ˆΓk for each term in ˆHα BdG(k). Thus, we note that we use MMS in the SM, not pMMS. In the case of the dxy-wave AM, ˆHα AM(k) has the following MMS: ˆHα AM(kx, ky, kz) = -ˆHα AM(-kx, ky, kz) = -ˆHα AM(kx, -ky, kz), (S20) where the first and the second line denote the mirror symmetry on the yz- and zx-plane, respectively. On the other hand, for the dx2-y2-wave AM, ˆHα AM(k) has diagonal magnetic mirror planes, ˆHα AM(kx, ky, kz) = -ˆHα AM(ky, kx, kz) = -ˆHα AM(-ky, -kx, kz). (S21) For the gxy(x2-y2)-wave AM, both Eqs. (S20) and (S21) is available. As a result, pTRS in Eq. (S18) becomes ˆΘ ˆHα AM(kx, ky, kz)ˆΘ† = ˆHα AM(kx, -ky, -kz) = ˆHα AM(-kx, ky, -kz), (S22) 5 for dxy and gxy(x2-y2)-wave AMs, and ˆΘ ˆHα AM(kx, ky, kz)ˆΘ† = ˆHα AM(-ky, -kx, -kz) = ˆHα AM(ky, kx, -kz), (S23) for dx2-y2- and gxy(x2-y2)-wave AMs. Finally, we obtain the chiral symmetry by the combination of local gauge symmetry, MMS, and PHS: ˆΓk ˆHα AM(kx, ky, kz)ˆΓ† k = ˆHα AM(-kx, ky, kz) = ˆHα AM(kx, -ky, kz), (S24) for dxy and gxy(x2-y2)-wave AMs, and ˆΓk ˆHα AM(kx, ky, kz)ˆΓ† k = ˆHα AM(ky, kx, kz) = ˆHα AM(-ky, -kx, kz), (S25) for dx2-y2- and gxy(x2-y2)-wave AMs. Eqs. (S24) and (S25) say that Hα AM(k) anticommutes with ˆΓk on the magnetic mirror plane. The chiral operators defined on each magnetic mirror plane are as follows: ˆΓkx=0 = ˆσ2ˆτ2, (S26) ˆΓky=0 = ˆσ2ˆτ1 (S27) ˆΓkx=ky = ˆσ2(τ1 + τ2)/ √ 2, (S28) ˆΓkx=-ky = ˆσ2(τ1 -τ2)/ √ 2. (S29) Using these chiral operators, we can calculate the winding number on each magnetic mirror plane. When the winding number is nonzero, the system becomes topologically nontrivial, and the crossed flat bands and/or surface arcs appear. Therefore, the analysis carried out in this subsection demonstrates the topological origin of the crossed flat bands and surface arcs discussed in the main text. S2: CALCULATION OF CONDUCTANCE IN A HYBRID SYSTEM FORMED BY A SUPERCONDUCTING ALTERMAGNET AND A NORMAL METAL In the main text, we have discussed the conductance in superconducting AM/normal metal junctions. Here, in this section, we show the details of how the conductance is calculated in superconducting AM junctions. We also discuss the details of the power law in the momentum dependent conductance. S2.1: Model of the hybrid junction We hereby provide the model and formulation in superconducting hybrid junctions. Then we assume the semiinfinite superconducting AM and normal metal regimes. In superconducting AM/normal metal junctions, we consider the Hamiltonian: ˆH = ˆHα SC+AM + ˆHJ + ˆHNM. (S30) In ˆHα SC+AM, when we consider the junction along the w = z, x-direction, the local ̃uα sc( ̃k), nearest-neighbor (NN) ̃tα sc1( ̃k), and second NN hopping terms ̃tα sc2( ̃k) with ̃k = (ku, kv) for u = x, y and v = y, z are given by ̃uα sc( ̃k) = ˆu0( ̃k) + ˆuα AM( ̃k) ˆu∆( ̃k) ˆu† ∆( ̃k) -ˆu∗ 0(- ̃k) -ˆuα∗ AM(- ̃k) , (S31) 6 ̃tα sc1( ̃k) = ˆt0( ̃k) + ˆtα AM1( ̃k) ˆt∆( ̃k) ̄t∆( ̃k) -ˆt∗ 0(- ̃k) -ˆtα∗ AM1(- ̃k) . (S32) ̃tα sc2( ̃k) = ˆtα AM2( ̃k) 0 0 -ˆtα∗ AM2(- ̃k) . (S33) Along the z-direction, we obtain ˆu0(kx, ky) = [-μ -2t cos kx -2t cos ky]ˆσ0, (S34) ˆuα AM(kx, ky) = ˆHα AM(kx, ky), (S35) ˆt0 = -tˆσ0, (S36) ˆtα AM1(kx, ky) = ˆtα AM2(kx, ky) = 0, (S37) ˆu∆= 0, (S38) ˆt∆= i∆ 2 [sin kx -i sin ky] iˆσ2, (S39) ̄t∆= -i∆ 2 [sin kx + i sin ky] iˆσ2, (S40) and along the x-direction, ˆu0(ky, kz) = [-μ -2t cos ky -2t cos kz]ˆσ0, (S41) ˆud1 AM(ky, kz) = 0, (S42) ˆud2 AM(ky, kz) = -td2 cos ky, (S43) ˆug AM(ky, kz) = 0, (S44) ˆt0 = -tˆσ0, (S45) ˆtd1 AM1(ky) = -itd1 2 sin ky, (S46) ˆtd2 AM1(ky) = td2 2 , (S47) ˆtg AM1(ky) = 2itg sin ky cos ky, (S48) ˆtd1 AM2(ky) = ˆtd2 AM2(ky) = 0, (S49) ˆtg AM2(ky) = -itg sin ky, (S50) ˆu∆= i∆sin ky sin kz, (S51) ˆt∆= i∆ 2 sin ky iˆσ2, (S52) ̄t∆= -i∆ 2 sin ky iˆσ2. (S53) In the normal metal ˆHNM, the local ̃uNM( ̃k) and nonlocal terms ̃tNM( ̃k) are given by ̃uNM( ̃k) = ˆu0( ̃k) 0 0 -ˆu∗ 0(- ̃k) , (S54) ̃tNM( ̃k) = ˆt0( ̃k) 0 0 -ˆt∗ 0(- ̃k) . (S55) 7 The tunneling Hamiltonian is defined as ˆHJ = ˆtJ ⊗ˆτ3, (S56) with the Pauli matrices in Nambu space ˆτi=0,1,2,3. Here, ˆtJ denotes the tunneling Hamiltonian in the electron space: ˆtJ = -tˆσ0, (S57) along the z and x-directions. These equations shown here are used in the next subsection to calculate the conductance. S2.2: Details of calculation of conductance In this subsection, we show the details of how we calculate the conductance in superconducting AM junctions. The conductance is obtained based on the Lee-Fisher formula [S7-S10]: σ(eV ) = 1 2π Z π -π d ̃k ̄σ( ̃k, eV ), (S58) with ̄σ( ̃k, eV ) = Tr′[ ̃ga( ̃k, eV ) + ̃gb( ̃k, eV ) - ̃gc( ̃k, eV ) - ̃gd( ̃k, eV )]. (S59) Eq. (S59) is given by Eq. (8) of the main text. Here, we present the explicit expression for Green's function ̃ga,b,c,d( ̃k, eV ). For E = eV with the energy E and the bias voltage V , ̃ga,b,c,d( ̃k, eV ) are calculated by ̃ga( ̃k, E) = ̃G11( ̃k, E) ˆV10 ̃G00( ̃k, E) ˆV † 10, (S60) ̃gb( ̃k, E) = ̃G00( ̃k, E) ˆV † 10 ̃G11( ̃k, E) ˆV10, (S61) ̃gc( ̃k, E) = ̃G01( ̃k, E) ˆV10 ̃G01( ̃k, E) ˆV10, (S62) ̃gd( ̃k, E) = ̃G10( ̃k, E) ˆV † 10 ̃G10( ̃k, E) ˆV † 10. (S63) Here, ˆV10 is defined as ˆV10 = -ˆtJ ⊗ˆτ0 for obtaining the conductance owing to the current operator, and this equation is already provided in the main text. ̃Gij( ̃k, E) with i, j = 0, 1 are numerically calculated by the retarded (R) and advanced (A) Green's function ̃GR ij( ̃k, E) and ̃GA ij( ̃k, E): ̃G00( ̃k, E) = -i 2[ ̃GA 00( ̃k, E) - ̃GR 00( ̃k, E)], (S64) ̃G11( ̃k, E) = -i 2[ ̃GA 11( ̃k, E) - ̃GR 11( ̃k, E)], (S65) ̃G01( ̃k, E) = -i 2[ ̃GA 01( ̃k, E) - ̃GR 01( ̃k, E)], (S66) ̃G10( ̃k, E) = -i 2[ ̃GA 10( ̃k, E) - ̃GR 10( ̃k, E)]. (S67) Local ̃Gi,i( ̃k, E) and nonlocal Green's functions ̃Gi,j( ̃k, E) are calculated by ̃G00( ̃k, z) = [{ ̃GL 0( ̃k, z)}-1 - ̃tJ ̃GR 1 ( ̃k, z) ̃t† J]-1, (S68) ̃G11( ̃k, z) = [{ ̃GR 1 ( ̃k, z)}-1 - ̃t† J ̃GL 0( ̃k, z) ̃tJ]-1, (S69) ̃G01( ̃k, z) = ̃GL 0( ̃k, z) ̃tJ ̃G11( ̃k, z), (S70) ̃G10( ̃k, z) = ̃GR 1 ( ̃k, z) ̃t† J ̃G00( ̃k, z), (S71) 8 where z = E ± iδ with the infinitesimal value δ are the retarded (R) z = E + iδ and advanced (A) parts z = E -iδ. By using the recursive Green's function method [S11, S12], we get the surface Green's functions: ̃GL 0( ̃k, z) = [z - ̃usc( ̃k) - ̃t† sc ̃GL( ̃k, z) ̃tsc]-1, (S72) ̃GR 1 ( ̃k, z) = [z - ̃uNM( ̃k) - ̃ub - ̃tNM ̃GR( ̃k, z) ̃t† NM]-1, (S73) with the semi-infinite Green's functions ̃GL( ̃k, z) and ̃GR( ̃k, z), and ̃ub = Ubˆσ0 ⊗ˆτ3 for the strength of the potential barrier Ub. We obtain the semi-infinite Green's functions for both retarded (z = E + iδ) and advanced (z = E -iδ) parts by using the local ˆu and hopping terms ˆt, see Refs. [S11-S13]. For the semi-infinite Green's function, when only NN hopping is included, we choose the local ˆu and the hopping terms ˆt as ˆu → ̃usc and ˆt → ̃tsc1 with the NN hopping ̃tsc1. In the case of including the second NN hopping term ̃tsc2 like the gxy(x2-y2)-wave AM along the x-direction, we replace the local ˆu and the hopping term ˆt as [S11] ˆu → ̃usc ̃tsc1 ̃t† sc1 ̃usc , (S74) ˆt → ̃tsc2 0 ̃t† sc1 ̃tsc2 . (S75) To connect ˆV10, if the second NN hopping term is included in the formulation, e.g., gxy(x2-y2)-wave AM along the x-direction, we pick up the subspace of the semi-infinite Green's function ̃GL 0 = ̃Gbb, which is defined as: ̃G = ̃Gaa ̃Gab ̃Gba ̃Gbb ! , (S76) in the basis [ ˆψ† a, ˆψ† b] with ˆψ† l=a,b = [c† l↑, c† l↓, cl↑, cl↓]. We have calculated the conductance in superconducting AM junctions of the main text by using this formulation. S2.3: Power law behavior in the momentum dependent conductance Here, we demonstrate the zero-bias conductance ̄σ( ̃k, eV = 0) and the zero-bias normal state conductance ̄σN( ̃k, eV = 0) ≡ ̄σN at ̃k along the z and x-directions. For the dx2-y2-wave AM, we plot ̄σ( ̃k, eV = 0) and ̄σN along the z-direction as a function of the potential barrier Ub in Figs. S3 (a,b,c) and ̄σ( ̃k, eV = 0) along the zdirection as a function of ̄σN( ̃k, eV = 0) in Figs. S3 (d,e,f), at td2 = ∆. Then we choose the momentum as (a,d) (kx, ky) = (0.2π, 0.2π) (crossed flat band), (b,e) (0.4π, 0) (BFS), and (c,f) (0.3π, 0) (gapped region). In Fig. S3 (a), on the crossed flat bands, ̄σ is almost 2, and ̄σN is suppressed for Ub. In Fig. S3 (b), on the BFS, we obtain ̄σ( ̃k, eV = 0) ∼ ̄σN. On the gapped region shown in Fig. S3 (c), both ̄σ( ̃k, eV = 0) and ̄σN strongly suppressed. When we see the power law of the zero-bias conductance for each momentum, we obtain ̄σ( ̃k, eV = 0) ∼2 ̄σ0 N on the crossed flat band [Fig. S3 (d)], ̄σ( ̃k, eV = 0) ∼ ̄σ1 N on the BFS [Fig. S3 (e)], ̄σ( ̃k, eV = 0) ∼ ̄σ2 N [Fig. S3 (f)]. We note that the power law in the gapped region deviates from ̄σ( ̃k, eV = 0) ∼ ̄σ2 N owing to the smearing factor δ in the lattice Green's function method. Along the x-direction, for the dxy-wave AM, we also show ̄σ( ̃k, eV = 0) and ̄σN as a function of Ub in Figs. S4 (a,b,c) and ̄σ( ̃k, eV = 0) as a function of ̄σN in Figs. S4 (d,e,f), at td1 = ∆. As well as along the z-direction, on the surface arc state at (ky, kz) = (0, 0.3π), ̄σ( ̃k, eV = 0) becomes almost 2, and ̄σN is suppressed by Ub [Fig. S4 (a)], on the BFS at (ky, kz) = (0.1π, 0), we obtain ̄σ( ̃k, eV = 0) ∼ ̄σN [Fig. S4 (b)], and on the gapped region at (ky, kz) = (0.1π, 0.35π), both ̄σ( ̃k, eV = 0) and ̄σN strongly suppressed shown in [Fig. S4 (c)]. Then we also obtain the power law as ̄σ( ̃k, eV = 0) ∼2 ̄σ0 N on the surface arc state [Fig. S4 (d)], ̄σ( ̃k, eV = 0) ∼ ̄σ1 N on the BFS [Fig. S4 (e)], ̄σ( ̃k, eV = 0) ∼ ̄σ2 N [Fig. S4 (f)]. It is noted that the deviation in the gapped region originates from δ in the Green's function. Thus, three distinct types of power laws 2 ̄σ0 N (zero-energy surface Andreev bound states: perfect resonance of Andreev reflection), ̄σ1 N (BFS), and ̄σ2 N (gapped region: Andreev reflection without resonance) coexist in the zero-bias momentum dependent conductance in superconducting AM junctions. We have also commented on this discussion in the main text. 9 0 1 2 0 0.5 1 0 1 2 0 5 10 15 20 0 1 2 0 5 10 15 20 0 1 2 0 5 10 15 20 (kx, ky) = (0.2⇡, 0.2⇡) (kx, ky) = (0.4⇡, 0) Crossed flat bands Bogoliubov-Fermi surface ZBeBtuTPKuz3P8A/2DNcnpnVzTo=">ACkXichVG7Sg NBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb 7YnQR08Qf8AQUrBQvxA/wAG3/Awk8Qywg2Ft5sFkSD8Q 4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju 35TtXTuaI7puPta6rPTcPmijCEyfdj6uWZvKcVlv3 Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8K MYTLMXCmGoFcgQSiGLiT8gjyM40FGFBQ4bgrAJFT6NA 8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadG zX9UK3TKyZNj5RTSLJndsfq7Inds1f2+WetIKzR+MsJ7 VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTN DZmGS71Zv3Z6Uc+u7iSDGXbD3sjfNXthj+TQr3rt9t8 5woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLY wBYUetfDJa5xI41LK1JGWmumSh2RZhw/Qtr8Aq0xlYQ= Ub/t Conductance p nVzTo=">ACkXichVG7SgNBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb7YnQR08Qf8AQUrBQvxA/w AG3/Awk8Qywg2Ft5sFkSD8Q4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju35TtXTuaI7puPta6rPTc PmijCEyfdj6uWZvKcVlv3Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8KMYTLMXCmGoFcgQSiGLi T8gjyM40FGFBQ4bgrAJFT6NA8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadGzX9UK3TKyZNj5RTSLJn dsfq7Inds1f2+WetIKzR+MsJ7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTNDZmGS71Zv3Z6Uc+u7iS DGXbD3sjfNXthj+TQr3rt9t85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLYwBYUetfDJa5xI41LK 1JGWmumSh2RZhw/Qtr8Aq0xlYQ= Ub/t p nVzTo=">ACkXichVG7SgNBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb7YnQR08Qf8AQUrBQvxA/w AG3/Awk8Qywg2Ft5sFkSD8Q4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju35TtXTuaI7puPta6rPTc PmijCEyfdj6uWZvKcVlv3Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8KMYTLMXCmGoFcgQSiGLi T8gjyM40FGFBQ4bgrAJFT6NA8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadGzX9UK3TKyZNj5RTSLJn dsfq7Inds1f2+WetIKzR+MsJ7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTNDZmGS71Zv3Z6Uc+u7iS DGXbD3sjfNXthj+TQr3rt9t85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLYwBYUetfDJa5xI41LK 1JGWmumSh2RZhw/Qtr8Aq0xlYQ= Ub/t (a) (b) (c) 0 1 2 0 0.5 1 0 1 2 0 0.5 1 (d) (e) (f) Gapped region (kx, ky) = (0.3⇡, 0) bMD0M+xMExR70Iat/sHWSE1JFUM=">ACpHicSyrIyS wuMTC4ycjEzMLKxs7BycXNw8vHLyAoFacX1qUnBqan J+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mk g8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQvIxyQlF sUZ6bnJmrElGTmpKRWxyTlVmfX1uqkhtkaMYLKBvo GYCBAibDEMpQZoCgHyB7QwxDCkM+QzJDKUMuQypDHkM JUB2DkMiQzEQRjMYMhgwFADFYhmqgWJFQFYmWD6VoZa BC6i3FKgqFagiESiaDSTgbxoqGgekA8ysxisOxloSw4 QFwF1KjCoGlw1WGnw2eCEwWqDlwZ/cJpVDTYD5JZKIJ0 E0ZtaEM/fJRH8naCuXCBdwpCB0IVHRxJQNX4/lTCkMV iA/ZIJ9FsBWATky2SI+WV0z8HWwWpVqsZLDJ4DfTfQo ObBoeBPswr+5K8NDA1aDYDFzCDNGjA5MRZqRnaKZnF mi7KADjSoOBmkGJQYNYHyYMzgweDAEMIQC7W1lWM2wh WErkxqTD1MwUyhEKRMjVI8wAwpgigMADyCcYA= ̄σ( ̃k, eV = 0) ̄σ( ̃k, eV = 0) vXuPy8Pv2Hf93Z8uS6o0pIEITI=">ACmnichVHLSg MxFD2O7/poVQRBF2JRXEjJiKi4Et0ogviqClZKZkzb0H kxkxZ08Af8AReuFyoH+AHuPEHXPgJ4lLBjQtvpwOiot 6Q5OTknpscruFZMlCMPTZojU3NLa1t7YmOzq7uZKqndz twK74psqZruf6uwQNhSUdklVSW2PV8wW3DEjtGebF2v1 MVfiBdZ0sdemLf5kVHFqTJFVH5VH/O4H4ukEWb53M2Vy XfDleP86k0y7Aohn8CPQZpxLHmpm6RwFcmKjAhoADRd gCR0BjDzoYPOL2ERLnE5LRvcAxEqStUJagDE5smdYin fZi1qFzrWYQqU16xaLpk3IYo+yBXbEXds9u2BN7/7VWG NWo/eWQdqOuFV4+eTKw+favyqZdofSp+kNhUPbfnhQKm I28SPLmRUzNpVmvXz06fdmc2xgNx9gFeyZ/5+yR3ZFDp /pqXq6LjTMkqEH693b8BNuTGX06M70+lZ6fiFvVhkGMY Jz6MYN5LGENWXo3xDmucaMNaQvasrZST9UaYk0fvoS29 QGyY5jt ̄σN ̄σN ̄σN ̄σN FIG. S3. For the dx2-y2-wave AM, (a-c) zero-bias conductance ̄σ( ̃k, eV = 0) and zero-bias conductance in the normal state ̄σN along the z-direction as a function of the potential barrier Ub at (kx, ky). (d-f) ̄σ( ̃k, eV = 0) along the z-direction as a function of ̄σN at (kx, ky). We choose the momentum as (a,d) (kx, ky) = (0.2π, 0.2π) (crossed flat band), (b,e) (0.4π, 0) [Bogoliubov-Fermi surface (BFS)], and (c,f) (0.3π, 0) (gapped region). We select the parameters as μ = -4.5t, ∆= 0.01t, td2 = ∆, and δ = 0.0001∆. 0 1 2 0 0.5 1 0 1 2 0 5 10 15 20 0 1 2 0 0.5 1 0 1 2 0 5 10 15 20 0 1 2 0 0.5 1 0 1 2 0 5 10 15 20 Surface arc states Bogoliubov-Fermi surface ZBeBtuTPKuz3P8A/2DNcnpnVzTo=">ACkXichVG7Sg NBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb 7YnQR08Qf8AQUrBQvxA/wAG3/Awk8Qywg2Ft5sFkSD8Q 4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPj u35TtXTuaI7puPta6rPTcPmijCEyfdj6uWZvKcVlv3 Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8K MYTLMXCmGoFcgQSiGLiT8gjyM40FGFBQ4bgrAJFT6N A8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTad GzX9UK3TKyZNj5RTSLJndsfq7Inds1f2+WetIKzR+MsJ 7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTN DZmGS71Zv3Z6Uc+u7iSDGXbD3sjfNXthj+TQr3rt9t 85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjL YwBYUetfDJa5xI41LK1JGWmumSh2RZhw/Qtr8Aq0xlYQ = Ub/t Conductance p nVzTo=">ACkXichVG7SgNBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb7YnQR08Qf8AQUrBQvxA/w AG3/Awk8Qywg2Ft5sFkSD8Q4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju35TtXTuaI7puPta6rPT cPmijCEyfdj6uWZvKcVlv3Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8KMYTLMXCmGoFcgQSiGL iT8gjyM40FGFBQ4bgrAJFT6NA8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadGzX9UK3TKyZNj5RTSLJ ndsfq7Inds1f2+WetIKzR+MsJ7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTNDZmGS71Zv3Z6Uc+u7i SDGXbD3sjfNXthj+TQr3rt9t85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLYwBYUetfDJa5xI41L K1JGWmumSh2RZhw/Qtr8Aq0xlYQ= Ub/t p nVzTo=">ACkXichVG7SgNBFD2u7/iK2g2YohYSJwViY8qaCPYqHGNYGLYXSdxyb7YnQR08Qf8AQUrBQvxA/w AG3/Awk8Qywg2Ft5sFkSD8Q4zc+bMPXfmcDXNHzB2EuH1NnV3dPb1x8bGBwaHomPju35TtXTuaI7puPta6rPT cPmijCEyfdj6uWZvKcVlv3Odq3PMNx94VJy4vWGrZNkqGrgqiDpVi3lLFsWcF2tm8KMYTLMXCmGoFcgQSiGL iT8gjyM40FGFBQ4bgrAJFT6NA8hgcIkrICDOI2SE9xniJG2SlmcMlRiK7SW6XQsTadGzX9UK3TKyZNj5RTSLJ ndsfq7Inds1f2+WetIKzR+MsJ7VpTy93iyPlE9uNflUW7wPG3qo1Co+z2ngRKWA69GOTNDZmGS71Zv3Z6Uc+u7i SDGXbD3sjfNXthj+TQr3rt9t85woxapD8ux2tYG8hJadT6e3FRGYualUfJjGNWerHEjLYwBYUetfDJa5xI41L K1JGWmumSh2RZhw/Qtr8Aq0xlYQ= Ub/t (a) (b) (c) (d) (e) (f) Gapped region (ky, kz) = (0, 0.3⇡) (ky, kz) = (0.1⇡, 0) (ky, kz) = (0.1⇡, 0.35⇡) ̄σ( ̃k, eV = 0) ̄σ( ̃k, eV = 0) vXuPy8Pv2Hf93Z8uS6o0pIEITI=">ACmnichVHLSg MxFD2O7/poVQRBF2JRXEjJiKi4Et0ogviqClZKZkzb0H kxkxZ08Af8AReuFyoH+AHuPEHXPgJ4lLBjQtvpwOiot 6Q5OTknpscruFZMlCMPTZojU3NLa1t7YmOzq7uZKqndz twK74psqZruf6uwQNhSUdklVSW2PV8wW3DEjtGebF2v 1MVfiBdZ0sdemLf5kVHFqTJFVH5VH/O4H4ukEWb53M2V yXfDleP86k0y7Aohn8CPQZpxLHmpm6RwFcmKjAhoADR dgCR0BjDzoYPOL2ERLnE5LRvcAxEqStUJagDE5smdYin fZi1qFzrWYQqU16xaLpk3IYo+yBXbEXds9u2BN7/7VWG NWo/eWQdqOuFV4+eTKw+favyqZdofSp+kNhUPbfnhQK mI28SPLmRUzNpVmvXz06fdmc2xgNx9gFeyZ/5+yR3ZFD p/pqXq6LjTMkqEH693b8BNuTGX06M70+lZ6fiFvVhkGM YJz6MYN5LGENWXo3xDmucaMNaQvasrZST9UaYk0fvoS2 9QGyY5jt ̄σN ̄σN ̄σN ̄σN FIG. S4. For the dxy-wave AM, (a-c) zero-bias conductance ̄σ( ̃k, eV = 0) and zero-bias conductance in the normal state ̄σN along the x-direction as a function of the potential barrier Ub at (ky, kz). (d-f) ̄σ( ̃k, eV = 0) along the x-direction as a function of ̄σN at (ky, kz). We choose the momentum as (a,d) (ky, kz) = (0, 0.3π) (surface arc state), (b,e) (0.1π, 0) (BFS), and (c,f) (0.1π, 0.35π) (gapped region). We select the parameters as μ = -4.5t, ∆= 0.01t, td1 = ∆, and δ = 0.0001∆. 10 S3: MODEL AND CONDUCTANCE OF HYBRID JUNCTIONS BASED ON 3D SUPERCONDUCTORS WITH 3D ALTERMAGNETS In this section, we demonstrate the conductance and bulk density of states (DOS) by using the model of 3D SCs with 3D AMs. Finally, we study the stability of zero-bias conductance in the case of 3D SCs with 3D AMs. S3.1: Model of 3D superconductors with 3D altermagnets In the main text, we focused on the case of 2D AMs. We now proceed to consider 3D AMs suggested in Ref. [S14]: dz(x+y), gzx(x2-3y2), and gyz(3x2-y2)-wave, coexisting with superconducting orders. We follow the same approach discussed in S2.1 and S2.2 of the SM, but make the following changes. Instead of Eq. (2) in the main text and Eqs. (S4)-(S6), the Hamiltonian of 3D AM is rewritten as [S14] ˆHα′ AM(k) = M α′ k ˆσ3ˆτ3, (S77) M d3 k = td3 sin kz(sin kx + sin ky), (S78) M g2 k = tg2 sin kz sin kx(sin2 kx -3 sin2 ky) (S79) M g3 k = tg3 sin ky sin kz(3 sin2 kx -sin2 ky) (S80) where tα′ with α′ = d3, g2, g3 are the strength of the dz(x+y), gzx(x2-3y2), and gyz(3x2-y2)-wave AMs, respectively. For the Fourier transformation, we obtain the local and hopping terms: ˆuα′ AM(kx, ky) = 0, (S81) ˆtd3 AM1(kx, ky) = i 2td3(sin kx + sin ky), (S82) ˆtg2 AM1(kx, ky) = tg2 sin kx(sin2 kx -3 sin2 ky), (S83) ˆtg3 AM1(kx, ky) = tg3 sin ky(3 sin2 kx -sin2 ky), (S84) along the z-direction, and ˆud3 AM(ky, kz) = td3 sin kz sin ky, (S85) ˆug3 AM(ky, kz) = tg3 sin ky sin kz 3 2 -sin2 ky , (S86) ˆtd3 AM1(ky, kz) = i 2td3 sin kz, (S87) ˆtg2 AM1(ky, kz) = i 2tg2 3 4 sin kz -3 sin2 ky sin kz , (S88) ˆtg3 AM1(ky, kz) = 0, (S89) ˆtd3 AM2(ky, kz) = ˆtg2 AM2(ky, kz) = 0, (S90) ˆtg3 AM2(ky, kz) = -3 4tg3 sin ky sin kz, (S91) ˆtd3 AM3(ky, kz) = ˆtg3 AM3(ky, kz) = 0, (S92) ˆtg2 AM3(ky, kz) = -i 8tg2 sin kz, (S93) along the x-direction, with ̃tα′ sc3( ̃k) = ˆtα′ AM3( ̃k) 0 0 -ˆtα′∗ AM3(- ̃k) ! . (S94) For including the second NN hopping term ̃tsc2 like the gyz(3x2-y2)-wave AM, we adopt Eqs. (S74) and (S75) to obtain 11 the semi-infinite Green's function. When the third NN hopping term ̃tsc3 is included like for the gzx(x2-3y2)-wave AM, then we replace: ˆu → ̃usc ̃tsc1 ̃tsc2 ̃t† sc1 ̃usc ̃tsc1 ̃t† sc2 ̃t† sc1 ̃usc , (S95) ˆt → ̃tsc3 0 0 ̃tsc2 ̃tsc3 0 ̃tsc1 ̃tsc2 ̃tsc3 . (S96) We also choose the subspace of the semi-infinite Green's function ̃GL 0 = ̃Gcc defined as: ̃G = ̃Gaa ̃Gab ̃Gac ̃Gba ̃Gbb ̃Gbc ̃Gca ̃Gcb ̃Gcc , (S97) in the basis [ ˆψ† a, ˆψ† b, ˆψ† c] with ˆψ† l=a,b,c = [c† l↑, c† l↓, cl↑, cl↓], to connect ˆV10. This method is applicable when calculating the conductance along the x-direction for the gzx(x2-3y2)-wave AM in S3.3 of the SM. We calculate the conductance in 3D superconducting AM junctions by this formulation. The results are shown in the next subsections. S3.2: DOS and conductance along the z-direction We discussed the conductance in 2D AMs in the main text and the BFS in the End Matter (EM). In this section, we present the additional results of the conductance caused by the 3D superconducting AMs: dz(x+y), gzx(x2-3y2), and gyz(3x2-y2)-wave. This additional study aims to support the role of 3D superconducting AMs for hosting exotic zero-energy flat states. To obtain the conductance in the case of 3D AMs, we use Eqs. (S85)-(S94), Eqs. (S74) and (S75), and (S95)-(S97). We plot the momentum dependent conductance at zero-bias voltage eV = 0 and the normalized conductance along the z-directions for each 3D AM in Fig. S5 at Ub = 5t. Then we show the projected Fermi surface with up (magenta) and down spins (cyan) for each transport direction in panels (d,e,f) of Fig. S5. Along the z-direction, for the dz(x+y) AM, because the crossed flat bands do not appear [Fig. S5 (a)], the resulting line shape of the conductance does not change [Fig. S5 (d)]. While for the gzx(x2-3y2) and gyz(3x2-y2)-wave AM, both BFS and crossed flat bands are obtained [Figs. S5 (b,c)]. Then the number of corners of the crossed flat bands corresponds to that of the nodal lines of AM. These behaviors are the same as those in the case of 2D AMs, as we worked in the main text. Nevertheless, the zero-bias conductance peak (ZBCP) is not strongly suppressed with the increase of altermagnetism shown in Figs. S5 (e,f). Indeed, the crossed flat band is also demonstrated for the gzx(x2-3y2) and gyz(3x2-y2)-wave AMs, not dz(x+y)-wave AM. Next, we discuss the BFS, and then, we calculate the bulk DOS, as described in the case of 2D AM in the EM. In Fig. S6, we also show how the BFS emerges in the case of 3D AM: dz(x+y), dzx(x2-3y2), and dyz(3x2-y2)-wave, by using the bulk DOS. As we also mentioned in the EM, the bright (dark) color of the bulk DOS corresponds to the BFS (the crossed flat band and the surface arc state), and the bright point at (kx, ky) = (0, 0) is the nodal point where the energy gap closes. For the dz(x+y)-wave AM, the structure of DOS does not change [Figs. S6 (a)] compared with that without altermagnetism. It means that the projected BFS does not appear along the z-direction, and then the corresponding conductance is independent of td3 [Fig. S5 (b)]. While the BFS appears for the gzx(x2-3y2) and gyz(3x2-y2)-wave AM by tα′ with α′ = g2, g3 [Figs. S6(b) and Figs. S6 (c)]. Then the number of corners corresponds to the nodes of 3D AM. Hence, 3D AMs lead to the BFS for gzx(x2-3y2) and gyz(3x2-y2)-wave AMs, except for the dz(x+y)-wave AM. We have shown the conductance in 2D superconducting AMs in the main text. S3.3: DOS and conductance along the x-direction Along the x-direction, we also show the momentum dependent conductance at zero-bias voltage eV = 0 and the normalized conductance along the x-directions for each 3D AM in Fig. S7 at Ub = 5t. For the dz(x+y)-wave AM shown 12 FIG. S5. (a,b,c) Conductance along the z-direction as a function of (kx, ky) at eV = 0 with (a) dz(x+y), (b) gzx(x2-3y2), and (c) gyz(3x2-y2)-wave AM. We select the altermagnetic order as (a) td3 = ∆, (b) tg2 = ∆, and (c) tg3 = ∆. (d,e,f) Conductance along the z-direction with (d) dz(x+y), (e) dzx(x2-3y2), and (f) gyz(3x2-y2)-wave AM. σN indicates the conductance in the normal state at eV = 0. Gray, light-red, green, and blue lines indicate the strength of the altermagnetic order as tα′ = 0, 0.5∆, ∆, and 1.5∆with α′ = d3, g2, g3. In (d,e,f), we show the image of the normal state Fermi surface when we see along the z-direction. We set parameters as μ = -4.5t, ∆= 0.01t, Ub = 5t, and δ = 0.01∆. FIG. S6. Bulk density of states (DOS) at E = 0 in 3D chiral d-wave SCs projected onto the z-direction for (a) dz(x+y), (b) gzx(x2-3y2), and (c) gyz(3x2-y2)-wave AM. AM is set as tα′ = ∆with α′ = d3, g2, g3. Bright (dark) region indicates the Bogoliubov Fermi surface (crossed flat bands). We set parameters as μ = -4.5t, ∆= 0.01t, and δ = 0.01∆. in Figs. S7 (a), the intensity of surface arc states becomes weak due to the BFS [Figs. S8 (a)], and the conductance at zero-bias voltage is first enhanced and suppressed by td3 in Fig. S7 (d). For the gzx(x2-3y2)-wave AM, because the point nodes mainly become BFS, see also Figs. S8 (b), the region of surface arc states reduces by BFS [Fig. S7 (b)]. Indeed, as shown in Fig. S7 (e), the zero-bias conductance is suppressed by td2. While for the gyz(3x2-y2)-wave AM, spin-split surface arc states appear [Fig. S7 (c)], and the resulting zero-bias conductance is strongly suppressed by td3 [Fig. S7 (f)]. Indeed, 3D AMs also lead to the modification of the surface states and the resulting conductance, as we worked in the case of 2D AMs in the main text. The energy gap structure becomes gapless at td3 = ∆[Figs. S8 (a)] for the dz(x+y)-wave AM. While for the gzx(x2-3y2)-wave AM, at tg2 = ∆, not only line node at kz = 0 but also point nodes at kz = kF with the Fermi vector kF mainly become the BFS [Fig. S8 (b)]. As a result, because the 13 FIG. S7. (a,b,c) Conductance along the x-direction as a function of (ky, kz) at eV = 0 with (a) dz(x+y), (b) gzx(x2-3y2), and (c) gyz(3x2-y2)-wave AM. We select the altermagnetic order as (a) td3 = ∆, (b) tg2 = ∆, and (c) tg3 = ∆. (d,e,f) Conductance along the x-direction with (d) dz(x+y), (e) dzx(x2-3y2), and (f) gyz(3x2-y2)-wave AM. σN indicates the conductance in the normal state at eV = 0. Gray, light-red, green, and blue lines indicate the strength of the altermagnetic order as tα′ = 0, 0.5∆, ∆, and 1.5∆with α′ = d3, g2, g3. In (d,e,f), we show the image of the normal state Fermi surface when we see along the x-direction. We set parameters as μ = -4.5t, ∆= 0.01t, Ub = 5t, and δ = 0.01∆. FIG. S8. Bulk density of states (DOS) at E = 0 in 3D chiral d-wave SCs projected onto the x-direction for (a) dz(x+y), (b) gzx(x2-3y2), and (c) gyz(3x2-y2)-wave AM. AM is set as tα′ = ∆with α′ = d3, g2, g3. Bright (dark) region indicates the Bogoliubov Fermi surface (surface arc states). We set parameters as μ = -4.5t, ∆= 0.01t, and δ = 0.01∆. region of the surface arc states is small for the BFS, the zero-bias conductance is suppressed by tg2 [Fig. S7 (e)]. For gg3-wave AM, the line node at kz = 0 partly becomes the BFS at tg3 = ∆[Fig. S8 (c)]. Since surface arc states are spin-split by avoiding the BFS [Fig. S7 (e)], the resulting zero-bias conductance is also suppressed, and the line shape of the conductance is also split [Fig. S7 (f)]. Indeed, the BFS appears in 3D AMs, as well as 2D AM cases, and surface arc states modified by the BFS contribute to the line shape of the conductance. We have also demonstrated the conductance in the case of 2D AMs in the main text. 14 0 1 2 0 5 0 1 2 0 5 0 1 2 0 5 0 1 2 0 5 0 1 2 0 5 s ha1_base64="xCB1wEd I2KCm42LTYKuk2v8TqN 0=">ACkXichVHLSs NAFD3GV62PVt0IbsRSc SF1IlIfq6IbwY2vWMFq SeK0Dc2LZFrQ4A/4Awq uFyIH+AHuPEHXPQTxK WCGxfepgFRsd4wmTNn7 rlzZq7moYvGt0SJ1d 3T29sb54/8DgUCI5PL LrOzVP54rumI63p6k+N w2bK8IQJt9zPa5amsnz WnW1uZ+vc83HtHLv 8wFLtlEydFUQdagUC5 YqKp4VaKezophMsQwLY +I3kCOQhQbTvIeBRz BgY4aLHDYEIRNqPDp24 cMBpe4AwTEeYSMcJ/jF HS1iLU4ZKbJX+ZVrt R6xN62ZNP1TrdIpJwyP lBNLsid2yV/bI7tgz+/ izVhDWaHo5plrablbT JyNb/q7JoFqh8qdp 6FihMfRqkHc3ZJq30F v6+sn56/byVjqYtfsh fxfsQZ7oBvY9Tf9ZpNv Xbxo5EXejFqkPyzHb/ B7lxGzmaym/Op3EzUqh jGMYlp6scCcljDBhSq 7+ECV7iWRqUlKSetFK ljkgzim8hrX8CvMyVhA = Ub/t σ(eV = 0)/σN (a) (b) (c) T YKuk2v8TqN0=">ACkXichVHLSsNAFD3GV62PVt0IbsRScSF1IlIfq6IbwY2vWMFqSeK0Dc2LZF rQ4A/4AwquFyIH+AHuPEHXPQTxKWCGxfepgFRsd4wmTNn7rlzZq7moYvGt0SJ1d3T29sb54/ 8DgUCI5PLrOzVP54rumI63p6k+Nw2bK8IQJt9zPa5amsnzWnW1uZ+vc83HtHLv8wFLtlEy dFUQdagUC5YqKp4VaKezophMsQwLY+I3kCOQhQbTvIeBRzBgY4aLHDYEIRNqPDp24cMBpe4AwTE eYSMcJ/jFHS1iLU4ZKbJX+ZVrtR6xN62ZNP1TrdIpJwyPlBNLsid2yV/bI7tgz+/izVhDWaHo 5plrablbTJyNb/q7JoFqh8qdp6FihMfRqkHc3ZJq30Fv6+sn56/byVjqYtfshfxfsQZ7oB vY9Tf9ZpNvXbxo5EXejFqkPyzHb/B7lxGzmaym/Op3EzUqhjGMYlp6scCcljDBhSq7+ECV7iWRq UlKSetFKljkgzim8hrX8CvMyVhA= Ub/t (d) (e) (f) z-direction -wave AM -wave AM -wave AM -wave AM -wave AM gyz(3x2-y2) gzx(x2-3y2) 4="UPe0X7rR5MUoYnxa0ocr9oITZ7Q=">ACj XichVFLSgNBEH2Ovxg/iboR3ASDoihIxo/iAR c6FKNUTGMDPpxMH5MTMJxuAFXAsuREHBhXgA D+DGC7jIEcSlghsX1kwGRSxmu6uel2vqh8lma piO4zVWoTWtvaOzkBXsLunty8U7h/Yto2SJfO0 bKiGtSuJNlcVnacdxVH5rmlxUZNUviMdrjvO 2Vu2YqhbzkVk2c1sagrBUWHYL28rnq8fjRZGX iJBeOstiCa7OReIx59teJwrd1I/yAfeRhQEYJG jh0OSrEGHTyiAOBpOwLKqEWeQp3jvHCYLELVE WpwyR0EM6ixRlfFSn2K1pe2yZuqi0LWJGMqe 2R17Y0/snr2wz4a1ql4N9y8VuqU6l5u50OlQ6u Nflka3g4MfVhOGRNnNTkoYN7TopA20NclXK9 fvn4/C21uDlaHWM37JX0XbMaeySFevldvt3gm xcI0oAaz+Xb2Z6OxROxMZMNDnljyqAYxgnOY xhyTWsI409dVxhktcCSFhVlgSlupQovPGcQvE 1a/AEn0k5w= dz(x+y) gzx(x2-3y2) gyz(3x2-y2) 0 1 2 0 5 x-direction σ(eV = 0)/σN -wave AM dz(x+y) td3/∆ td3/∆ tg2/∆ tg2/∆ tg3/∆ tg3/∆ 4 0 0 2 σ(eV = 0)/σN σ(eV = 0)/σN σ(eV = 0)/σN σ(eV = 0)/σN FIG. S9. Normalized zero-bias conductance along z (a,b,c) and x (d,e,f) transport directions as functions of Ub and tα′. The altermagnetic order: (a,d) dz(x+y), (b,e) gzx(x2-3y2), (c,f) gyz(3x2-y2)-wave. σN denotes the conductance in the normal state at eV = 0, and we calculate it for each tα′ and Ub. Gray circles indicate the parameters shown in Fig. S5 and Fig. S7. Parameters: μ = -4.5t, ∆= 0.01t, and δ = 0.01∆. S3.4: Stability of zero-bias conductance in 3D superconducting AM junctions In the main text, we discussed the normalized zero-bias conductance for the potential barrier Ub and the strength of 2D AM tα space. We also demonstrate the normalized zero-bias conductance along z [Figs. S9 (a,b,c)] and x [Figs. S9 (d,e,f)] directions for Ub and the strength of 3D AM tα′. For the dz(x+y)-wave AM, because the ZBCP is not suppressed with the increase of tα′ along the z-direction, the white area does not decrease [Figs. S9 (a)]. Then, crossed flat bands do not appear for the dz(x+y)-wave AM. While for the gzx(x2-3y2) and gyz(3x2-y2)-wave AM, even though the crossed flat bands are realized by AM, the normalized zero-bias conductance is not suppressed [Figs. S9 (b,c)]. We note that crossed flat bands emerge even at small Ub for the gzx(x2-3y2) and gyz(3x2-y2)-wave AM [Figs. S9 (a,b,c)]. Along the x-direction, for the dz(x+y) and gyz(3x2-y2)-wave AM, the normalized zero-bias conductance is enhanced once for 0.25∆≤tα′ ≤∆with α′ = d3, g3, and it is suppressed for tα′ > ∆[Figs. S9 (d,f)], see also Figs. S7 (d,f). While for the gzx(x2-3y2)-wave AM, the normalized zero-bias conductance gradually decreases with the increase of td2 [Figs. S7 (e)]. This behavior corresponds to the modification of surface arc states by the BFS. Therefore, zero-bias conductance can also be modified by 3D AMs, as we have found in the case of 2D AMs in the main text. [S1] K. Yada, M. Sato, Y. Tanaka, and T. Yokoyama, Surface density of states and topological edge states in noncentrosymmetric superconductors, Phys. Rev. B 83, 064505 (2011). [S2] M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Topology of Andreev bound states with flat dispersion, Phys. Rev. B 83, 224511 (2011). [S3] P. M. R. Brydon, A. P. Schnyder, and C. Timm, Topologically protected flat zero-energy surface bands in noncentrosymmetric superconductors, Phys. Rev. B 84, 020501 (2011). [S4] Y. Tanaka, M. Sato, and N. Nagaosa, Symmetry and topology in superconductors-odd-frequency pairing and edge states-, J. Phys. Soc. Jpn. 81, 011013 (2012). 15 [S5] Y. Tanaka, S. Tamura, and J. Cayao, Theory of Majorana zero modes in unconventional superconductors, Prog. Theor. Exp. Phys. 2024, 08C105 (2024). [S6] S. Kobayashi, Y. Tanaka, and M. Sato, Fragile surface zero-energy flat bands in three-dimensional chiral superconductors, Phys. Rev. B 92, 214514 (2015). [S7] P. A. Lee and D. S. Fisher, Anderson localization in two dimensions, Phys. Rev. Lett. 47, 882 (1981). [S8] R. Ohashi, S. Kobayashi, S. Kanazawa, Y. Tanaka, and Y. Kawaguchi, Surface density of states and tunneling spectroscopy of a spin- 3 2 superconductor with Bogoliubov-Fermi surfaces, Phys. Rev. B 110, 104515 (2024). [S9] Y. Fukaya, K. Yada, and Y. Tanaka, Tunneling conductance in superconducting junctions with p-wave unconventional magnets breaking time-reversal symmetry, arXiv , 2506.13372 (2025). [S10] B. Lu, P. Mercebach, P. Burset, K. Yada, J. Cayao, Y. Tanaka, and Y. Fukaya, Engineering subgap states in superconductors by altermagnetism, arXiv: 2508.03364 (2025). [S11] A. Umerski, Closed-form solutions to surface Green's functions, Phys. Rev. B 55, 5266 (1997). [S12] D. Takagi, S. Tamura, and Y. Tanaka, Odd-frequency pairing and proximity effect in Kitaev chain systems including a topological critical point, Phys. Rev. B 101, 024509 (2020). [S13] P. San-Jose, J. Cayao, E. Prada, and R. Aguado, Multiple Andreev reflection and critical current in topological superconducting nanowire junctions, New J. Phys. 15, 075019 (2013). [S14] M. Ezawa, Third-order and fifth-order nonlinear spin-current generation in g-wave and i-wave altermagnets and perfectly nonreciprocal spin current in f-wave magnets, Phys. Rev. B 111, 125420 (2025).
|
2510.14723
|
Bayes-ically fair: A Bayesian Ranking of the Olympic
Medal Table
C. MacDermott, C.J. Scarrott, J. Ferguson
School of Mathematical and Statistical Sciences, University of Galway
October 17, 2025
Abstract
Evaluating a country’s sporting success provides insight into its decision-making
and infrastructure for developing athletic talent.
The Olympic Games serve as a
global benchmark, yet conventional medal rankings can be unduly influenced by pop-
ulation size.
We propose a Bayesian ranking scheme to rank the performance of
National Olympic Committees by their ‘long-run’ medals to population ratio. The
algorithm aims to mitigate the influence of large populations and reduce the stochas-
tic fluctuations for smaller nations by applying shrinkage. These long-run rankings
provide a more stable and interpretable ordering of national sporting performance
across games compared to existing methods.
Keywords: hierarchical models, sports statistics, medal probability
1
Introduction
The Olympic Games represent the pinnacle of international sporting competition, offering
a global stage to evaluate national athletic success. Every four years, elite athletes from na-
tions and territories represented by National Olympic Committees (NOCs) compete across
a wide variety of sports for ‘Olympic glory’. As such, medal tables, comparing the rela-
tive performance of differing NOCs, are examined by media outlets and sports authorities
world-wide. Particular scrutiny goes on how many medals their home country has won,
where they rank compared to other nations, and how their performance compares with
previous Games.
During the 2024 Olympic Games in Paris, one of the key narratives centered on whether
the United States or China would top the standard lexicographic ordering of the medal
table, which ranks nations primarily by gold medal count. Both finished with 40 gold
medals, but the United States was placed ahead due to its higher overall medal tally.
However, while total medal counts are a common point of media focus, they inherently
favor more populous NOCs, as these have a larger pool of athletes to draw from. For
example, Ireland will never top a medal table based purely on total counts, as the sheer
1
arXiv:2510.14723v1 [stat.AP] 16 Oct 2025
disparity in population makes it virtually impossible to compete with nations hundreds of
times larger. To address this imbalance, an alternative approach is to scale medal counts by
population, producing the so-called ‘per-capita’ table. However, these tables are also crude
in that they fail to recognize inherent stochasticity over repeated Olympic cycles, which
on a relative scale will be large for smaller countries. The result is that the per-capita
table is typically led by small population countries. For instance, Grenada, Dominica and
Saint Lucia, with respective populations of 112,579; 67,408 and 184,100 occupied the top
3 positions on the 2024 per-capita table.
In response to these limitations, approaches have been proposed in the scientific liter-
ature to create alternative rankings. These approaches can be divided into methods that
challenge the standard ranking produced by the number of gold medals (or total medals) by
(1) reweighing the contributions of gold, silver and bronze medals in various ways (Ogwang
and Cho, 2021), (2) producing composite tables combining medal tables ordered by differ-
ing criteria (Cao et al., 2016), and (3) ranking performance relative to collective resources
available to a particular country, as specified by variables such as population and GDP
using approaches like Data Envelope Analysis (Lozano et al., 2002; Bernard and Busse,
2004; Churilov and Flitman, 2006; Wu et al., 2009; Li et al., 2015; Flegl and Andrade,
2018), and linear regression (Halsey, 2009).
Finally, the method recently proposed by Duncan and Parece (2024) has received sub-
stantial attention in various popular media outlets around the world, like the The New
York Times (Coy, 2024) and The Guardian (Ryan and Nicholas, 2024) who referred to it
as the Goldilocks method due to not favoring large or small countries. The Duncan-Parece
method ranks according to a novel probabilistic index that adjusts for population effects,
which we will compare and contrast to our own approach below.
In this article, we propose Bayesian ranking as an alternative method to order medal
tables.
Unlike the traditional approaches, Bayesian ranking focuses on the most likely
ordering of the key parameters of interest that drive the observed data (Laird and Louis,
1989). In the Olympic medal setting, we can informally define these key parameters as
the long-run per-capita medal ratios that would be observed in differing countries if the
Olympics were repeated several times under differing conditions. In other words, the medal
table we produce via Bayesian ranking does not aim to summarize what happened in a
particular Olympic Games – but instead to infer which countries are most efficient in
producing Olympic medalists, relative to their population size over the long term.
In the case that the data from a single Olympics is used, the ranking produced could
be considered a modification of the ranking by the observed per-capita medal ratios. For
large countries such as the USA and China, the Bayesian estimates of long-term per-capita
ratios will closely resemble observed values, given their large populations and relatively
stable performance. In contrast, for smaller countries, the Bayesian estimates may diverge
significantly from observed ratios, as their medal counts are more susceptible to stochastic
variation within a single Olympic cycle.
A key consideration in this approach is the choice of prior distribution for these un-
derlying long-term ratios, as the reliability of the rankings relies on how reasonable the
prior is. Hierarchical Bayesian modeling, as used in the methods discussed later, mitigates
some subjectivity by leveraging the observed distribution of medal counts across countries
2
to help estimate the prior, thereby enhancing the robustness of the resulting rankings.
In Section 2, we will describe our Bayesian ranking model in detail, with careful dis-
cussion regarding the plausible validity of our modeling assumptions. In Section 3, we
examine the medal tables produced by our approach over the last six Olympic cycles, and
contrast with the results produced by the recent algorithm proposed by Duncan and Parece
(2024) mentioned above. In Section 4, we provide additional evaluation and justification of
our modeling assumptions, and discuss both caveats as well as possible extensions of our
approach.
2
Methods
The National Olympic Committees (NOCs) represent each country, as well as the individual
athletes from the Refugee Olympic Team. The C NOCs that competed at a games are
indexed by c = 1, . . . , C and the total population size per NOC is given by nc. These vary
in each Olympic Games, but for notational convenience we’ll ignore this in the following
presentation which will focus on the NOCs in Paris 2024 (C = 204). As with the usual
per-capita analysis, only NOCs with a measurable population are included.
The total
number of medals won by country c is the sum of the number of medals won by their
individual athletes Mc = 1 × M1,c + 2 × M2,c + · · · , where Mi,c is the random variable for
the number of their athletes that won exactly i medals. Here each medal-winning country
in team competitions such as relays or field sports is considered a nominal athlete that wins
a single medal, which is in keeping with the official lexiographic ranking compiled by the
International Olympic Committee. The total number of medals achieved by each country
c will be modeled using a combination of interrelated Poisson processes representing the
number of athletes Mi,c winning exactly i medals.
If we momentarily ignore the number of total medals, Mc, and consider only the num-
ber of unique medal winning athletes, Uc = P
i Mi,c, we can assume Uc to be roughly
Poisson distributed.
The Poisson distribution is used as an approximation to the con-
ceptual idea that the number of unique medal winners from country c is approximately a
Uc ∼Binomial(nc, pc) distribution, where a unique medal winner is considered a rare event
from a large number nc of independent Bernoulli trials representing each person within a
population with probability of success pc. The binomial/Poisson assumptions are considered
in Section 4.
However, care has to be taken with modeling total medals won, Mc, as the world’s
highest achieving athletes that win more than one medal would invalidate the Poisson
limiting assumption that it should be impossible to have two or more events at the same
instance. Hence, we will instead model the number of athletes winning i = 1, 2, . . . as
conditionally independent Poisson processes, that are interlinked via a prescribed relation-
ship between the country-specific impacts on the probability of winning that number of
medals.
Note that this assumption implies that the number of unique medal winners,
Uc = P
i Mi,c will follow a Poisson distribution, whereas the total number of medals won
Mc = 1 × M1,c + 2 × M2,c + · · · , will not.
In Paris 2024 the highest number of medals won by any individual athlete was four
3
medals, so for notational convenience we will denote M4,c as the number of athletes winning
four or more medals, and so we only consider the finite sum to M4,c. It is straightforward
to adapt the modeling framework to allow for a different number of medals won by an
individual athlete. The number of athletes winning i = 1, 2, 3 and 4 or more medals in
each competing country are assumed to be Poisson distributed:
M1,c
∼
Poisson(λ1,c),
M2,c
∼
Poisson(λ2,c),
M3,c
∼
Poisson(λ3,c), and
M4,c
∼
Poisson(λ4,c).
The expected number (and variance) of athletes winning each number of medals is then
λi,c = nc pi,c, where pi,c = P(Xc = i) is the probability of an individual Xc from competing
country c winning exactly i medals. Again, for convenience in notation p4,c = P(Xc ≥4).
The country-specific Poisson distributions are equivalent to a classical frequentist inference
of a country-specific random effect on the probability of a winning that number of medals.
These Poisson distributions are treated as independent of each other, conditional on
those country-specific probabilities p1,c, p2,c, p3,c and p4,c of an individual winning a given
number of medals. The relationship between these probabilities is defined through param-
eters for the following conditional probabilities:
pc
=
P(Xc ≥1) i.e., the probability of athlete from c being a unique medal winner,
q2
=
P (X ≥2 | X ≥1) ,
q3
=
P (X ≥3 | X ≥2) , and
q4
=
P (X ≥4 | X ≥3) .
The probability of being a unique medal winner pc retains the country-specific subscript
notation. In contrast, the consecutive conditional probabilities qi of winning at least one
additional medal having already won i medals are not country-specific, once the condition-
ing on being a unique medal winner from that NOC is taken into account. There is a slight
abuse of notation here in that these conditional probabilities depend on the country-specific
random variable Xc ≥1 for being unique medal winner, but we use the random variable
X to represent the number of medals won by an individual from a generic NOC winning
that number of medals.
The country-specific effect of population biology, demography, and the ability to develop
sporting talent is assumed to be entirely driven through the parameter for the chance of
producing a unique medal winner pc = P(Xc ≥1). The above framework then assumes that
conditional on someone being a medal winner, the likelihood of winning further medals has
no further country-specific effects. This assumption seems reasonable as when conditioning
on the rare event of winning one or more medals, we are implicitly conditioning on an
athlete that has the talent, training environment and funding to perform at a world class
level in their chosen sport; the conditional probability of winning more than one medal
given this context might be largely independent of the athlete’s home country.
These
4
multi-medal winning athletes are also sufficiently rare that they will not be very influential
on the rankings inference, except for the small countries that have been fortunate enough
to host such talent. Indeed we see no substantive change in the results presented when
only the unique medal winning athletes are modelled.
From a computational view this construction has the benefit of not needing country
specific random effects for these multi-medal winning probabilities which would be hard
to inform from the data as the majority of countries have no such athletes. The country-
specific probabilities of being a unique medal winner are more reliably estimated as there
are based on a larger number of such athletes.
The unconditional probability of an individual from country c of winning a certain
number of medals will vary by country according to:
p1,c = P(Xc = 1)
=
P(Xc ≥1) −P(Xc ≥2) = pc (1 −q2),
p2,c = P(Xc = 2)
=
P(Xc ≥2) −P(Xc ≥3) = pc q2 (1 −q3),
p3,c = P(Xc = 3)
=
P(Xc ≥3) −P(Xc ≥4) = pc q2 q3 (1 −q4), and
p4,c = P(Xc ≥4)
=
pc q2 q3 q4.
These parameters are given uninformative and independent prior distributions of
q2
∼
Uniform(0,1),
q3
∼
Uniform(0,1), and
q4
∼
Uniform(0,1),
so that they are predominantly learnt from the data. The country-specific probability of
an individual becoming a unique medal winner is assumed to follow a prior distribution:
pc ∼Beta(α, β),
where α and β are given independent uniform hyperprior distributions:
α
∼
Uniform(0, 1), and
β
∼
Uniform(0, 108).
Note that sensible values for α and β should correspond to an expected medal prob-
ability from the associated beta-prior: E(pc) =
α
α+β, that is relatively close to the total
number of medals divided by the global population (1039/8, 000, 000, 000 = 1.3 × 10−7 for
Paris 2024). However, this information has not been hardcoded into the model: instead,
the non-informative hyperpriors give the model the flexibility to learn the most plausible
values for α and β based on the observed medal counts across differing countries.
The priors induce shrinkage to reduce the uncertainty for small countries which have
limited sample information by pulling their estimates towards the global average, while
preserving the data-driven estimates for larger countries. The pooling of information across
all countries for the multi-medal winning athletes also ensures they do not have undue
influences on the estimates for small countries. The shrinkage provides a better estimate
of the ‘long-run’ probability of winning medals, which would provide a more stable ranking
5
of each nation’s performance. Alternative prior specifications are considered in the online
Appendix, with little sensitivity to this choice found.
In summary, we have just four parameters to obtain a posterior distribution for each
country; namely the country-specific random effect for the probability of a unique medal
winner pc = P(Xc ≥1), and the three conditional probabilities of such an athlete winning
further medals qi = P (X ≥i | X ≥i −1) for i = 2, 3 and 4.
The population size data are obtained from the United Nations Nations (2025) which
are updated for each Olympic year. The Olympics data are obtained from the International
Olympic Committee (2025). Gibbs sampling using the rJAGS package (Plummer, 2024) is
used to draw posterior samples of (pc, q2, q3, q4) for each country, which are then transformed
as above to country-specific medal probabilities, and subsequently to expected per-capita
medal ratios: E(Mc/nc) = p1,c+2 p2,c+· · · . The rank for the expected per-capita number of
medals won in a particular country, among all medal-winning countries, is then calculated
for each posterior sample. Finally, the estimated posterior mean of these ranks is used to
order the medal table as is commonplace in the Bayesian ranking literature. Alternative
statistics, like the posterior median rank, that could be used for the ranking are considered
in Section 4.
Finally, it is worth noting that while we base our ranking on the posterior distribution
of the expected medal rate per capita E(Mc/nc) in the above, this is equivalent to ranking
on the probability of an individual from that country winning at least one medal pc =
P(Xc ≥1), which is easily seen by expansion of expectation:
E(Mc/nc)
=
E(M1,c/nc) + 2 E(M2,c/nc) + 3 E(M3,c/nc) + 4 E(M4,c/nc)
=
pc (1 −q2 + 2 q2 (1 −q3) + 3 q2 q3 (1 −q4) + 4 q2 q3 q4)
where the terms in the bracket are not country specific.
3
Results
Table 1 presents the Bayesian ranking of the top ten and selected countries from Paris
2024. The posterior mean rank over those countries that won at least one medal is used to
order the table. As per usual approaches, the NOCs that did not win any medals are not
ranked. Population size, total medals won, observed per-capita medal winning rates and
rankings by these observed rates, as well as the Duncan and Parece (2024) U-index are also
provided for comparison. The posterior median medal winning rate per million and 95%
credible intervals for uncertainty estimation are provided.
The posterior median is utilized due to the strong skewness in the credible intervals
for smaller countries to be discussed in Section 3.3. It is worth noting that the posterior
median rate does not follow exactly the same ordering as the posterior ranks due their
skewness. A Shiny app is available from MacDermott et al. (2025), which allows users
to explore the different rankings (including the posterior median rank, as an alternative)
across all NOCs, including demographic comparisons for the last six Olympic Games going
back to Athens 2004.
6
Ranks
Country
Medals per million
Credible Interval
Posterior
Mean
Per-
capita
U-Index
Name
Code Population
(thousands)
Medal
Total
Observed Posterior
Median
95%
Lower
95%
Upper
1
4
6
New Zealand
NZL
5,214
20
3.84
3.31
2.05
4.89
2
7
1
Australia
AUS
26,713
53
1.98
1.80
1.32
2.31
3
8
8
Hungary
HUN
9,676
19
1.96
1.83
1.06
2.89
4
10
4
Netherlands
NLD
18,229
34
1.87
1.68
1.15
2.40
5
1
27
Grenada
GRD
117
2
17.09
2.42
0.34
8.16
6
6
18
Jamaica
JAM
2,839
6
2.11
1.71
0.67
3.54
7
5
25
Bahrain
BHR
1,607
4
2.49
1.74
0.53
4.25
8
11
15
Georgia
GEO
3,808
7
1.84
1.58
0.65
3.00
9
12
17
Croatia
HRV
3,875
7
1.81
1.60
0.65
3.10
10
13
13
Denmark
DNK
5,977
9
1.51
1.35
0.64
2.50
15
18
20
Ireland
IRL
5,255
7
1.33
1.01
0.42
2.04
16
24
2
France
FRA
66,549
64
0.96
0.95
0.72
1.22
17
25
3
Great Britain
GBR
69,138
65
0.94
0.95
0.72
1.21
21
2
41
Dominica
DMA
66
1
15.15
1.10
0.06
5.64
24
3
29
Saint Lucia
LCA
180
2
11.11
0.96
0.05
4.98
46
42
43
Hong Kong
HKG
7,415
4
0.54
0.36
0.09
0.96
51
48
5
United States
USA
345,427
126
0.36
0.34
0.29
0.41
76
76
114
China
CHN
1,419,321
91
0.06
0.06
0.05
0.08
Table 1: Bayesian ranking of the top 10 and selected countries in Paris 2024 by posterior
estimates of medal-winning probabilities, normalized per million population, including 95%
credible intervals. The Duncan-Parece U-index based ranks are provided.
3.1
Bayesian Shrinkage and Ranking Stability
As mentioned, a key feature of the Bayesian ranking approach is its shrinkage effect, which
adjusts for small-sample uncertainty. It prevents countries with exceptionally high per-
capita medal counts from being overemphasized from a single-event which may not persist
over the long-term. The shrinkage can be seen for Grenada in Table 1 having an observed
medals per million of 17 (rank 1 per-capita), which decreased to 2.42 (Bayesian rank 5).
Similarly, Dominica and Saint Lucia were ranked second and third on a per-capita basis,
but have been shrunk to 21 and 24 respectively in the Bayesian ranking. Using Bayesian
ranking, New Zealand, Australia and Hungary are ranked as the top 3, but under the per-
capita rate it would be Grenada, Dominica and then Saint Lucia, a combination which is
unlikely to be seen again in future Olympics. The Bayesian shrinkage results in a more
stable and interpretable ranking compared to naive per-capita methods, which tend to
disproportionately favor smaller nations.
The performances of larger countries are well informed by the sample data, so see
relatively little change in the posterior inference. The USA and China achieved the same
number of gold medals, the USA won both more silver and bronze medals so are ranked first
in the official lexicographic medal table. But in the per-capita rankings they are placed as
48 and 76 respectively, due to their very large population sizes. The shrinkage is minimal
for such large countries with the the USA moving down 3 positions to 51 in the Bayesian
rank and China only 1 rank up to 75. The slightly larger change for the USA is discussed
further in Section 3.2.
In Figure 1 we can see the shrinkage effect across all countries, by comparing the
7
Figure 1: Comparison of posterior median and observed medal winning rates by population
size for each country with at least one medallist. Directed arrows indicate the shrinkage
effect from observed per-capita rates to posterior estimates. The 95% credible intervals
for the medal rate per million are shown by the gray lines, with boundary range ticks.
Those countries with any multi-medal athletes are shown as a different color and flags
mark countries listed in Table 1.
observed and posterior median per-capita medal probabilities. The smaller nations on the
left of the graph are pulled closer to the global mean, reflecting the underlying uncertainty
in their success rates. In contrast, larger countries on the right see little effect of the prior,
as their rankings are more data-driven.
It is worth noting that the level of shrinkage is a function of three factors: (1) the
population size, (2) the number of medals won by each NOC (note that these first two
factors pertain to the observed medal winning rate) and (3) whether there are any multi-
medal winning athletes. This feature can be seen by the shrinkage arrows having different
lengths for NOCs with similar population sizes which are close to each other on the x-axis.
Generally, if two NOCs have same population size and medal counts, the one with more
multi-medal winners will see more shrinkage to the global average which will be discussed
in Section 3.2. The NOCs with an unusually high medal count, for a given population
size, will see more shrinkage as is expected when providing more stable medal winning rate
estimates.
8
It should be noted that only medal winning countries are shown in Figure 1. However, a
consequence of the shrinkage is that NOCs that won no medals exhibit shrinkage upwards,
which is again to be expected to provide more stable medal winning rate estimates. This
shrinkage effect for non-medallist NOCs is discussed in more depth in the online Appendix.
It is of course important to include the non-medal winning NOCs in the inference, as they
inform the global average rates and their exclusion would lead to a bias.
3.2
Impact of Multi-Medal Winners on Bayesian Rankings
Another feature of the Bayesian ranking model is its emphasis on the number of unique
medal winners (and the number of medals they won) rather than the total medal count.
This feature of the model is a notable improvement over Duncan and Parece (2024) which
ignores the athletes that compete in several events and break the binomial assumption that
medals won in differing events are independent. The focus on the unique medal winners
also helps prevent a small number of extraordinary multi-medalists from disproportionately
inflating a country’s ranking. One must not diminish the achievement of such athletes, but
a goal here is to provide a more stable estimate of each NOC’s medal winning rate which
may be more reflective of the NOC’s ability to develop sporting talent. The outstanding
ability of such athletes is likely more driven by their individual genetic factors, and the
particular events they compete in (e.g. more multi-medal winners are seen in swimming
events), as opposed to being a strong indicator of an NOC’s talent development capability.
This effect is illustrated in Figure 1, where countries with similar populations and
nearly identical total medal counts so are near each other on the graph (in both x and
y direction) exhibit different levels of shrinkage.
The countries colored in purple have
multi-medal winners exhibit more shrinkage than those with unique medal winners only
which are colored in gold.
For instance, in 2024 Ireland and Norway with essentially
the same populations won 7 and 8 medals respectively.
Ireland’s medals were won by
six unique medalists, with one athlete winning multiple medals, while Norway’s medals
were distributed across eight different athletes. As another example, Grenada and Saint
Lucia are both small nations with similar populations that won two medals in Paris 2024.
They ranked 1 and 3 on the observed per capita table. However, because Saint Lucia’s
two medals were won by the same athlete, Julien Alfred, their estimated medal rate is
shrunk more than Grenada’s, whose two medals were won by separate athletes. Finally,
Hong Kong won four medals in 2024, but two of those were won by the swimmer Siobh´an
Haughey, which explains the larger shrinkage applied to Hong Kong’s observed medal ratio
compared to similar size countries. This feature underscores the model’s tendency to favor
broader medal distribution across athletes rather than concentrated success among a few
high achieving individuals. This feature can also effect the degree of shrinkage applied to
bigger countries, albeit to a lesser extent as the absolute amount of shrinkage is smaller for
larger countries. For instance, USA moves by 3 positions in the ranking due to their 10
multi-medal winners winning 23 out of a total 126 medals, whereas China only moves by 1
position as they have only 3 multi-medal winners winning 7 out of there 91 medals.
9
3.3
Ranking Uncertainty and Credible Intervals
On their own, the posterior mean ranks should not be used to compare the performance of
different countries, just like p-values should not be used to judge effect sizes, as they do not
account for uncertainty. To address this, credible intervals for every medal winning NOC
from Paris 2024 in Figure 2, in addition to the selection in Table 1, which provide a more
nuanced assessment of each NOC’s ranking stability.
Figure 2: Average Bayesian rank with 80% credible intervals for Paris 2024, ordered by
rank. Smaller nations exhibit wider credible intervals due to greater uncertainty in their
medal counts, while larger countries have more stable rankings.
As expected, smaller countries exhibit wider credible intervals, reflecting greater ranking
uncertainty due to their limited sample sizes, while larger nations have more stable rankings.
Further, for well performing countries the rank distributions are right skewed due to the
natural bound at first rank and vice-verse for weakest performing countries bounded by
the bottom rank. The asymmetry is most clear for the smallest countries with the most
uncertainty.
For example, the USA at posterior mean rank 48 has a very short (and symmetric)
credible intervals reflecting the large sample of competing athletes. New Zealand’s credible
interval does not overlap those of Australia and Hungary, reinforcing its standout position
as the top-performing country according to the Bayesian ranking. In contrast, Grenada’s
10
credible interval overlaps substantially with those of many other small nations, suggest-
ing that its superior per-capita medal count may not represent a statistically meaningful
performance difference when accounting for sample uncertainty.
In general, countries with fewer medal winners (and smaller populations) exhibit wider
credible intervals, while those with consistent performances across multiple events (typi-
cally larger nations) have more stable rankings. This highlights a key advantage of the
Bayesian approach with the credible intervals - it provides a probabilistic measure of rank-
ing uncertainty, ensuring that countries with overlapping intervals are not misinterpreted
as significantly different in performance despite differences in their rank positions.
3.4
Comparison with Duncan-Parece Method
The Bayesian posterior mean ranks based on the estimated medal-winning probabilities
presented in Table 1 differ significantly from the Duncan and Parece (2024) U-index rank-
ings. Under the Bayesian ranking, New Zealand, Australia, and Hungary rank among the
top-performing nations, while Grenada, despite its high observed per-capita medal count,
is ranked slightly lower due to the shrinkage effects. The USA and China have seen little
change in the Bayesian ranking compared to a per-capita rating.
The Duncan-Parece method implicitly penalizes smaller countries in the rankings, as
their approach is essentially p-value based. It is well known that p-values are sample size
dependent (Wasserstein and Lazar, 2016). A small sample limits the evidence available,
even for a large effect size, so smaller NOCs cannot achieve the small p-values required to be
ranked highly, unless they attain an extraordinarily large number of medals relative to their
population size. As such the larger countries are given an advantage in the Duncan-Parece
method.
The advantage given to large NOCs in the Duncan-Parece method is seen in Table 1
and Figure 3 by it assigning higher ranks to large countries like the USA at rank 5, Great
Britain at rank 3 and France at rank 2, which benefit from their overall representation of
athletes competing across events. In general, the Duncan-Parece U-index ranked countries
which are ranked highly have large population sizes. The only ranking scheme shown that
places higher priority on larger NOCs is the total medals based ranking in gray. The per-
capita based ranking generally gives priority to smaller NOCs. The Bayesian ranking can
be reasonably described as achieving a balance between smaller and larger nations, with a
mix of such nations achieving high ranks.
3.5
Long-term Stability of Bayesian Ranking
One of the key motivators for the Bayesian ranking approach is to provide a stable long-term
estimate of each NOC’s performance, based on a single Olympic Games, using shrinkage
effects whilst also accounting for population size effects.
Section 3.1 demonstrated the
shrinkage effects and how they approach accounts for population size. Here we will evaluate
its ability to provide an interpretable long-term measure of performance by considering its
stability and behavior when applied to many past Olympic Games.
The Bayesian ranking method is applied entirely separately for each of the last 6
11
Figure 3: Comparison of rankings across different methods for Paris 2024 as a function of
population size. Bayesian rank (purple), U-index rank (orange), total medals rank (dashed
dark gray), and per-capita medals rank (dashed light gray) are shown. Notably, the U-
index closely follows the total medal count ranking, highlighting its tendency to favor larger
nations, while the Bayesian ranking adjusts for population size.
Olympic Games, so there is no carryover of information across Olympic Games in the
inference.
Notice the strong consistency in the Bayesian ranking for each NOC shown
across these 6 Games, which provides comfort in the interpretation of the ranks as a stable
long-term estimate of each NOC’s performance, due to the shrinkage effects. In particular,
New Zealand topped the ranks in the last 4 Olympic Games after a notable improvement
since Athens 2004 which is known to be influenced by increases in funding and strat-
egy/policy priority changes resulting from the creation of Sparc (now Sports NZ) and its
subsidiary High Performance Sports NZ which were formed in 2002 and 2012 respectively.
The Netherlands have made consistent gains since 2008. Hungary did not perform so well
in Beijing 2008, but otherwise ranked very highly. Similarly, Georgia has performed con-
sistently well. Jamaica dropped from a consistent rank of 2-3 in size Beijing 2008 to rank
6 in Paris 2024. Interestingly, both Grenada and Bahrain performed sufficiently well in
Paris 2024 to appear in the Top 10 for the first time, despite the shrinkage applied to their
estimates.
12
Figure 4: Top 10 ranked countries across the last 6 Olympic Games. Every country that
is within the Top 10 in any Game is joined to their rank across all 6 games.
4
Discussion
In this final section, we discuss some of the assumptions behind our model, possible ways
it might be extended and more detailed comparison to the Duncan and Parece (2024)
method. We first examine the equal treatment of medal types and the validity of the Poisson
assumption. We then discuss the assumption of constant conditional probabilities for multi-
medal winners across countries. Finally, we compare our Bayesian ranking algorithm to
existing probabilistic approaches, particularly the Duncan-Parece method, and highlight
key advantages of our approach in capturing long-term national Olympic performance.
Throughout the analysis we have treated all medals equally, but we could have consid-
ered either separate Poisson processes for each type of medal or having a weighting scheme
for each type of medal. The former medal type specific Poisson process would have many
zeroes and therefore limit the information to the likelihood. In this case, the inference
would be strongly influenced by the priors which is best avoided. Previous literature has
justified treating gold, silver and bronze medals equally. For instance, Bian et al. (2005)
has argued that the performances winning gold, silver and bronze medals are often very
similar, with a degree of luck being the crucial determining factor, which partially justifies
13
equal weighting. Ogwang and Cho (2021) also found that the first principal component
estimated from the joint distribution of gold, silver and bronze medal totals across coun-
tries had approximately equal weight on the number of gold, silver and bronze medals and
explains over 94% of the total variance, indicating very strong correlation in the gold, silver
and bronze medal counts across different countries.
The Poisson assumption, as a limiting approximation to the binomial distribution for
the number of unique medal winners per country, can be empirically tested using observed
medal counts over multiple Olympic Games. Assuming that a country’s true medal-winning
rate remains stable across successive Olympic cycles, we have considered diagnostic plots of
estimated mean and variance, and hypothesis tests to detect possible excess-Poisson varia-
tion (see the online Appendix). These tests provide support for the Poisson approximation.
While there are obvious physical limitations to the binomial argument that total medals
can be regarded as a sum of equal probability Bernoulli trials, these are likely negligible
in practice. For example, factors such age, socioeconomic conditions, and race may influ-
ence the likelihood of individual success in different events. However, even if individual
probabilities vary the total medal counts can still be reasonably modeled using a Poisson
distribution (with rate scaled by population size) as long as individual successes or failures
can be treated as independent events. The assumption of independence is somewhat more
difficult to justify. On one hand, given the rarity of athletes with the talent to win an
Olympic medal, it seems reasonable to treat these individuals as independent rare events.
On the other hand, the notion that ‘a rising tide lifts all boats’ should be acknowledged -
training environments, national sports culture, and psychological factors can create perfor-
mance spillover effects, particularly in elite competitive sports.
Recall the model assumes that the conditional probabilities of an individual winning
two or more medals, given that they have won at least one medal are constant across
all countries. We have partially justified this assumption under the premise that when
conditioning on an Olympic medal winner, one is implicitly conditioning on having the
training environment, funding and genetic talent to compete as a world class athletes (i.e.
we are mostly conditioning away those factors that might cause differing levels of success
per-capita across countries). However, if a country focuses their resources on particular
sports (such as swimming) where multi-medal winners are more likely, the true conditional
probabilities for that country of winning multiple medals given a single medal may be larger
than the global average. Unfortunately, there is not enough data from a single Olympic
Games to estimate these conditional probabilities in a coherent way at a per-country level,
particularly for smaller countries and, as such, complete pooling represents an attractive
modeling assumption from a practical standpoint. In practice, the effects of incorporating
such corrections in our ranking (which would in any case be driven primarily by single
medal winners who constitute the vast majority of medals) would be unlikely to change
the conclusions. To elaborate further, since the conditional probabilities are not country
specific, the shrinkage applied to a country’s raw per-capita medal rate by our model will
tend to be slightly larger if its medals were won by disproportionally few athletes compared
to a scenario where the same number of medals were won by separate athletes. This effect
may have resulted in the USA doing slightly worse in the Bayesian ranking for Paris 2024
(position 51) compared to the observed per-capita ranking (position 48).
However, as
14
shown in Figure 2, the practical differences in performance between countries ranked as
51 and 48 by Bayesian ranking is statistically negligible due to the strongly overlapping
credible intervals.
To produce a ranking, we have summarized the posterior distribution of the rank for a
country’s medal rate by its mean. As demonstrated in Figure 2, the posterior distributions
of highly-placed smaller countries tend to be quite right skewed compared to larger coun-
tries, with the result that they may be ranked less advantageously using the posterior mean
rank compared to using the posterior median rank. For instance, Grenada would be ranked
as the 2nd best country in Paris 2024 according to posterior median rank, as opposed to
5th with posterior mean rank, although the other positions in the top 10 are identical to
those in Table 1, having accounted for that shift. Effectively, the posterior mean rank ap-
plies a relative penalty to the ranking of smaller countries with asymmetric high-variability
posterior distributions, compared to countries that have symmetric posteriors. We have
chosen the posterior mean rank as it is in keeping with previous applications of Bayesian
ranking, such as Laird and Louis (1989), but posterior median rank could equally be used
to create the ranking table. Alternative rankings (such as rankings based on the posterior
median) can be explored using the Shiny app at MacDermott et al. (2025)
The goal of our model is to infer which countries are most efficient in producing Olympic
medalists relative to their population size. This population-scaled efficiency is a subjective
definition of success, and may not be something that a country strives for given that some
Olympic events have more prestige and glamour than others.
For instance, while the
United States places at 51 on our Bayesian ranking table (despite winning the table on
standard lexicographic medal counts), it does particularly well in many of the ‘marquee’
track and field events that enjoy particular media attention. Many factors might influence
this relative Olympic success such as GDP per capita, the proportion of GDP allocated to
sports funding, climate, demographic and social factors. If the aim was to instead infer
which countries were most successful given fixed values of such variables, such factors could
be accounted for in a Bayesian Poisson regression, which again used log-population size as
an offset in modeling the probability of a unique medal, and incorporated a normal random
effect for each country, with the inferred value of this random effect used to rank differing
NOCs. The implicit prior for medal winning probabilities across countries would be log-
normal (as opposed to beta) in this modified model, but with shifts in the mean parameter
based on the covariate information for a NOC.
This brings up another point of discussion. How much influence does the hierarchically
estimated beta prior influence the overall results and ranking? The choice of beta rather
than log-normal or some other choice is a subjective choice, and could be considered as a
weakness of our model by some. It turns out that the exact choice of hierarchical prior
has relatively little influence in the final results. We have also investigated hierarchical
specifications of log-normal, logit normal and mixture-beta priors for the probability of a
unique medal (see the online Appendix for details). In all models, New Zealand is ranked as
the best performing NOC, and the top 5 and top 10 ranked countries are consistent across
all models, with slightly differing orderings based on the implied shrinkage. In particular,
the results from the mixture beta prior are almost identical to the results above. Given
that the mixture beta has theoretical advantages in that it can adapt to the most general
15
distribution of probabilities across countries, it is comforting that the results from the beta
and mixture beta priors are so similar.
Finally we return to the approach of Duncan and Parece (2024), which ranks by the
likelihood of achieving a country’s observed medal count (or higher) under a binomial
model. That is, the model considers possible awarded medals to a country as independent
Bernoulli trials with probability success proportional to the country’s population. One nice
feature of their approach is their binomial probability implicitly respects the total quota
of possible Olympic medals that a country can win. For instance, in Tokyo 2021 the total
medals that a country could possibly win was 579 due to restrictions on the total number
of athletes that an NOC could enter. While this isn’t respected by our Poisson distribution
(which has infinite support), we note that in practice the probability distributions for total
medals corresponding to Poisson and binomial approaches are very similar again based on
the large n and small p approximation (See the online Appendix for more details).
The Duncan-Parece method is effectively a transformed p-value that tests the null hy-
pothesis that the probability of a country winning a particular Olympic medal equals the
proportion of the aggregate population of countries that have previously won an Olympic
medal that resides in that country. The index tends to favorably rank higher population
countries. For example, in the Tokyo 2020 Olympics, the USA ranked 6th using Duncan-
Parece’s U-index, with a score of 15.2. However, if we imagine that both the population of
the USA and its medal count doubled, the U-index for the USA increases to 34, in which
case they would would be ranked as the top performing country. This is despite the USA’s
relative performance (in terms of medals per-capita) remaining unchanged. The effect just
described is due to a well known feature of p-values (Wasserstein and Lazar, 2016). Small
observed effects (such as small observed differences between country-specific and global
per-capita medal ratio) can correspond to extremely small p-values when the effect is es-
timated with small standard error (roughly the standard error for estimated per-capita
ratios are proportional to the square root of population). These issues have been previ-
ously discussed in the context of ranking effect sizes in genetics and genomics in Ferguson
and Chang (2020), and sheds some light on the differences in rankings produced by our
Bayesian ranking algorithm and their probabilistic index. To advocate for our approach,
as demonstrated in Section 3, New Zealand has led our Bayesian ranking algorithm con-
sistently over the previous four Olympic cycles, and they have also ‘topped’ the empirical
per-capita medal table over this collective period when restricting to countries that have
a population of at least 1,000,000; indicating that our algorithm is better at inferring the
countries with the best long-run performance even though we infer this performance from
a single Olympic Games.
5
Conclusion
In conclusion, our Bayesian ranking algorithm is a new approach way to rank each country’s
performance in international competitions such as the Olympic Games. It strikes a balance
between consideration of total medals, which would give an advantage to larger countries,
and empirical per-capita medal ratios which are overly influenced by stochastic factors
16
pertaining to a particular Olympics and notable advantage the large number of smaller
nations.
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18
Appendix
In this appendix we consider secondary details about the method and results. A Shiny
app is available from MacDermott et al. (2025) for interactively exploring the results of the
Bayesian ranking, including how the ranking might vary by country-specific demographic
factors and comparison to other ranking methods.
A
Bayesian Shrinkage and Ranking Stability
In the main paper we presented the shrinkage results for the medal winning countries in
Paris 2024. The shrinkage brings each country closer to the global average, as a means
to provide a more stable long-run estimate of their performance. The large countries see
little shrinkage as they are well informed by the data, while the smaller nations see more
shrinkage. The countries with one or more multi-medal winning athletes also saw more
shrinkage than those with only single medal winning athletes of a similar size, as the
multi-medal winning rate is assumed constant across all nations. Here we will explore the
(reverse) impact of the shrinkage of the countries who won no medals in Paris 2024
In the main paper we presented the shrinkage results for the medal winning countries
in Paris 2024. The shrinkage brings each country closer to the global average, in order
to provide a more stable long-run estimate of their performance. The large countries see
little shrinkage as they are well informed by the data, whereas the smaller nations see
more shrinkage. The countries with one or more multi-medal winning athletes also saw
more shrinkage than those with single medal winning athletes of a similar size, as the
multi-medal winning rate is assumed constant across all nations. Here we will explore the
(reverse) impact of the shrinkage of the countries who won no medals in Paris 2024.
The uplift of the medal winning rate for NOCs with no medalists can be seen in Figure 5
by the vertical arrows. The shrinkage for these NOCs is only affected by their population
size, as they have zero medal count and no multi-medal winners. Broadly speaking, there is
a monotonically decreasing inflation effect of the shrinkage with population size. The small-
est countries receive the biggest uplift towards the global average, as the larger populations’
estimates are more data-driven.
The uplift of the medal winning rate for NOCs with no medalists can be seen in Figure 5
by the vertical arrows. The shrinkage for these NOCs is only affected by their population
size, as they have zero medal count and no multi-medal winners. Broadly speaking, there
is a monotonically decreasing inflation effect of the shrinkage with population size. The
smallest countries receive the biggest uplift towards the global average, as the larger popu-
lations’ estimates are more data-driven. The non-medal winning countries are not assigned
a rank in the Bayesian ranking, as per usual protocol.
E(Mc/nc) = pc (1 −q2 + 2 q2 (1 −q3) + 3 q2 q3 (1 −q4) + 4 q2 q3 q4) ≈pc
The non-medal winning countries are not assigned a rank in the Bayesian ranking,
as per usual protocol. However, if they were ranked they would be amongst the bottom
19
Figure 5: Effect of shrinkage on the medal winning rate against population size for each
country that won a medal (as per Figure 1 of the main paper, and the inflation effect for
those who won no medals.
countries, which can be seen by the low height of their arrows compared to most medal
winning NOCs.
B
Approximate Equivalence to Duncan-Parece Method
The model underlying the proposed Bayesian ranking can be considered as an approxima-
tion to the Duncan and Parece (2024) method in the circumstance that the probabilities
pc, which we define in the Methods section of the main manuscript, are equal across com-
peting NOCs. The Duncan-Parece method models the number of medals won by a NOC
using a binomial distribution where each trial is one of the medals that an NOC could
win, up to the maximum possible number of medals, with a country specific probabil-
ity of winning. Under the quota system the maximum number of medals that could be
achieved was M = 559 trials for Paris 2024, out of a total of T = 1039 medals. Under
the assumption of equal medal winning capability for any individual around the world, the
expected number of medals won by a country is proportional to the country’s population
(see equation (4) of Duncan and Parece (2024)). But due to the close equivalence between
20
the Poisson/binomial, you can view our per-capita specification as simply changing the
denominator term.
The model underlying the proposed Bayesian ranking can be considered as an approxi-
mation to the Duncan and Parece (2024) method in the circumstance that the probabilities
pc, which we define in the Methods section of the main manuscript are equal across com-
peting NOCs. The Duncan-Parece method models the number of medals won by a NOC
using a binomial distribution where each trial is one of the medals that an NOC could
win, up to the maximum possible number of medals, with a country specific probabil-
ity of winning. Under the quota system the maximum number of medals that could be
achieved was M = 559 trials for Paris 2024, out of a total of T = 1039 medals. Under
the assumption of equal medal winning capability for any individual around the world, the
expected number of medals won by a country is proportional to the country’s population
(see equation (4) of Duncan and Parece (2024)). But due to the close equivalence between
the Poisson/binomial, you can view our per-capita specification as simply changing the
denominator term of the binomial probability.
The population size for country c is nc and the total population of the world is N.
The Duncan-Parece model is that the number of medals won by any country is Mc ∼
Binomial(M, πc) where πc is probability of that country winning a given medal. Under the
equal capability assumption (per person) across all NOCs, the expected number of medals
won should be proportional to that countries population so M × πc = nc
N T. Rearranging
we obtain:
πc = nc
N
T
M
so under the equi-capability reference model the number of potential medals follows a
Binomial
M, nc
N
T
M
. Such a binomial is approximately a Poisson
nc
N T
which in turn is
approximately Binomial
nc, T
N
. The latter probability pc = T/N is per-capita probability
of an individual from a given NOC winning a medal underlying our model, assuming it
does not change over countries (that is pc is constant).
Although the two underlying models approximate each other under the null equi-
capability reference model, our inference approach does not have the same restriction on
small countries due to the sample size impacts on the (transformed) p-values which un-
derlie the U-index of Duncan and Parece (2024). Another substantial improvement in our
method is the explicit handling of multi-medal winning athletes, which would break the
usual independence assumption of the binomial. A drawback of our approach compared to
Duncan and Parece (2024) is that we do not explicitly handle each country’s quota. How-
ever, even for the most successful countries, medal winning is a rare event so no country
get sufficiently close to their quota for this to be of concern.
C
Sensitivity to Prior Choice
The shrinkage which stabilizes the long-run estimates of expected medal winning perfor-
mance is mostly imposed by the choice of prior distribution (with only mild impacts from
the assumed common multi-medal winning probabilities across all NOCs). In the main
21
Country
Posterior Mean Ranks
Medals per million
Beta CI
Population
(thousands)
Beta
Log-
Normal
Mixed-
Beta
Observed
Beta
Median
Log-Normal
Median
Mixed-Beta
Median
95%
Lower
95%
Upper
NZL
5,214
1
1
1
3.84
3.31
3.61
3.27
2.05
4.89
AUS
26,713
2
3
2
1.98
1.80
1.82
1.79
1.32
2.31
HUN
9,676
3
4
3
1.96
1.83
1.84
1.79
1.06
2.89
NLD
18,229
4
5
4
1.87
1.68
1.73
1.71
1.15
2.40
GRD
117
5
2
6
17.09
2.42
7.14
2.13
0.34
8.16
JAM
2,839
6
7
5
2.11
1.71
1.82
1.68
0.67
3.54
BHR
1,607
7
6
7
2.49
1.74
2.00
1.66
0.53
4.25
GEO
3,808
8
8
8
1.84
1.58
1.68
1.56
0.65
3.00
HRV
3,875
9
9
9
1.81
1.60
1.65
1.54
0.65
3.10
DNK
5,977
10
10
10
1.51
1.35
1.42
1.40
0.64
2.50
IRL
5,255
15
15
15
1.33
1.01
1.05
1.02
0.42
2.04
FRA
66,549
16
16
16
0.96
0.95
0.95
0.95
0.72
1.22
GBR
69,138
17
19
17
0.94
0.95
0.95
0.95
0.72
1.21
USA
345,427
51
47
51
0.36
0.34
0.34
0.34
0.29
0.41
CHN
1,419,321
76
78
77
0.06
0.06
0.06
0.06
0.05
0.08
Table 2: Bayesian ranking of the top 10 and selected countries in Paris 2024 based on poste-
rior estimates of medal-winning probabilities, normalized per million population. Rankings
are shown under different prior specifications: the beta prior, a log-normal prior, and a flex-
ible mixed-beta prior. The ranks are based on mean ranking of posterior mean probabilites.
Corresponding 95% credible intervals for the beta prior are also provided supporting the
appropriateness of the beta prior in the analysis.
paper, we present the results when a beta prior distribution is used to inform the country-
specific probability of an individual becoming a unique medal winner, pc = P(Xc ≥1),:
pc ∼Beta(α, β)
with hyperparameters:
α
∼
Uniform(0, 1), and
β
∼
Uniform(0, 108).
This prior is a natural choice and appears to produce sensible Bayesian rankings, but we
have also explored the sensitivity to alternative priors and their hyperparameters.
Firstly, we considered a truncated log-normal prior distribution which allows for a heav-
ier upper tail for the prior distribution of the unique medal winner probability
pc ∼Log-Normal(µ, σ)I(0, 1)
with hyperparameters:
µ
∼
Normal(−15, 0.323), and
σ
∼
Gamma(0.001, 0.001)
where the gamma is parameterized by the shape and rate. The hyperprior for the mean
is chosen to be close to the overall medal winning rate over the world population. The
22
Figure 6: Effect of shrinkage on the medal-winning rate against population size for each
country that won a medal (as per Figure 1 of the main paper), and the inflation effect for
those who won no medals. for a log-normal prior with colours indicating how the shrinkage
compares under each model
resulting shrinkage is shown in Figure 6.
The shrinkage is somewhat flatter compared
to the beta prior, so that the smaller countries like Grenada are not shrunk as much
as for the beta prior, and the larger countries experience notably more shrinkage. The
trajectory of shrinkage vs population for the non-medal winning countries is also notably
flatter compared to with the beta prior. The resultant rankings are shown in Table 2.
The beta prior is used as the reference, where a black font means no change, green is an
improvement in the ranked position and orange is a decline. The resulting ranks do not
change much compared to the beta prior. The key difference in the Top 10 is that Grenada
moves from rank 5 with the beta prior to 2nd place. But notice that the shrinkage for the
larger countries leads to the UK being slightly pulled down the ranks, and the USA and
Japan brought slightly up the ranks.
The shrinkage for larger countries is indicative of the prior being informative over the
large amount of sample information, which is not desirable. In addition, it appears that
the heavy tail of the log-normal is allowing substantial variation in the probability of being
a unique medal winner, and subsequently higher posterior mass on extreme performance,
for countries with small populations. However, when one considers the uncertainty via the
23
95% credible intervals for the medal winning rate (calculated from the beta specification)
these changes are not significant, as the estimated rates from the log-normal prior are well
within these intervals (although it is worth noting that the medal winning rate for Grenada
is close to the upper bound).
A logit-normal prior:
pc ∼Logit-Normal(µ, σ)
with hyperparameters:
µ
∼
Normal(−15, 0.323), and
σ
∼
Gamma(0.001, 0.001)
gave near identical results to the log-normal, so are not discussed for brevity.
Figure 7: Effect of shrinkage on the medal-winning rate against population size for each
country that won a medal (as per Figure 1 of the main paper), and the inflation effect for
those who won no medals. for a mixed-beta prior with colours indicating how the shrinkage
compares under each model
A very flexible mixture of beta components was also explored to assess the robustness of
the prior to heavy-tailed behavior. Three components were used, offering enough flexibility
to approximate characteristics of priors such as the log-normal, while including the standard
24
beta prior as a special case. The three components
pj
c ∼Beta(αj, βj)
for i = 1, 2, 3 have the same hyperparameters as for usual single beta prior as above. A
Dirichlet(1,1,1) distribution is used for the mixture weights for the three beta components.
The resulting shrinkage is shown in Figure 6 is very similar to using simple one component
beta prior which provides comfort in the results presented the main paper, as it appears
the choice of beta prior (and hyperparameters) is not strongly influential on the results
and the changes seen with the log-normal prior above are likely due to this prior specifica-
tion enforcing a unrealistically heavy tail that fails to adequately shrink highly stochastic
extreme rates for small NOCs.
One subtle distinction, however, emerges at the tails: at large population sizes for non-
medal winners, the mixed-beta prior results in slightly less shrinkage than the regular beta.
This can occur because the mixture model may allocate higher-population observations
more weight from flatter components, which exert less shrinkage. However, these deviations
are minimal and affect only non-medal-winning countries, which are excluded from the final
rankings.
D
Poisson assumptions
Using the medal winning results across multiple Olympic Games we can evaluate the equal
mean/variance assumption of the Poisson model. Figure 8 plots the sample mean against
the variance of the number of medals for each NOC across all seven of the Olympic Games
from Athens 2004 to Paris 2024, shown on a log-scale. A non-medal winning country is
included as a zero for that Games, but a country which did not compete in the Games are
set to missing. You can clearly see a strong mean-variance relationship as the NOCs are
close to the line of equality. Some of the notable deviations from the line of equality are
due to host NOCs (Greece, China, UK, Brazil, Japan and France). But overall the equal
mean/variance assumption of the Poisson appears extremely reasonable.
There is a mild crudeness to this evaluation due to the multi-medal winning athletes.
In our model, the total number of medals won by country c is Mc = 1 × M1,c + 2 × M2,c +
· · · P
i i Mi.c. Each of the Mi,c is assumed to be Poisson distributed. But the total number
of medals won Mc is not exactly Poisson due to the scaling by the number of medals i, but
will be close to Poisson as the multi-medal winning athletes are sufficiently rare especially
as the total Mc will be dominated by the 1 × M1,c term which is exactly Poisson.
Another aspect of approximation in this diagnostic plot is that it assumes the number of
medals on offer (including quotas of those available), population size, and the underlying
performance of each NOC (as measured by their long-term medal rate: E(Mc/nc)) are
constant over time. As we are considering only Games following Athens 2004 these factors
will not have changed much, and so is of little concern.
As an alternative test of the Poisson assumption, we defined p-values to test for excess
Poisson variation in the medal counts over two successive Olympic Games, assuming perfor-
mance, the total number of medals on offer and population all remain stable. Under these
25
assumptions (in addition to independence) conditional on R, that is the total medals won
over two sucessive games, the number of medals won by a country in a particular Games
has a binomial distribution with R trials and probability of success 0.5. The results are not
shown here, but the country-specific p-values demonstrated no evidence for super-Poisson
variability (with the notable exception of host countries).
Figure 8: Average medals over the seven previous Olympic games are plotted against the
empirical variance of the number of medals (y-axis), by country. Gross departures from
the line of equality signify possible violations of the Poisson assumption. The shaded area
represents a 90% predictive band for the empirical variance of a random sample of size
seven from the Poisson distribution.
26
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Bayes-ically fair: A Bayesian Ranking of the Olympic Medal Table C. MacDermott, C.J. Scarrott, J. Ferguson 17, 2025 Abstract Evaluating a country's sporting success provides insight into its decision-making and infrastructure for developing athletic talent. The Olympic Games serve as a global benchmark, yet conventional medal rankings can be unduly influenced by population size. We propose a Bayesian ranking scheme to rank the performance of National Olympic Committees by their 'long-run' medals to population ratio. The algorithm aims to mitigate the influence of large populations and reduce the stochastic fluctuations for smaller nations by applying shrinkage. These long-run rankings provide a more stable and interpretable ordering of national sporting performance across games compared to existing methods. Keywords: hierarchical models, sports statistics, medal probability 1 Introduction The Olympic Games represent the pinnacle of international sporting competition, offering a global stage to evaluate national athletic success. Every four years, elite athletes from nations and territories represented by National Olympic Committees (NOCs) compete across a wide variety of sports for 'Olympic glory'. As such, medal tables, comparing the relative performance of differing NOCs, are examined by media outlets and sports authorities world-wide. Particular scrutiny goes on how many medals their home country has won, where they rank compared to other nations, and how their performance compares with previous Games. During the 2024 Olympic Games in Paris, one of the key narratives centered on whether the United States or China would top the standard lexicographic ordering of the medal table, which ranks nations primarily by gold medal count. Both finished with 40 gold medals, but the United States was placed ahead due to its higher overall medal tally. However, while total medal counts are a common point of media focus, they inherently favor more populous NOCs, as these have a larger pool of athletes to draw from. For example, Ireland will never top a medal table based purely on total counts, as the sheer 1 16 Oct 2025 disparity in population makes it virtually impossible to compete with nations hundreds of times larger. To address this imbalance, an alternative approach is to scale medal counts by population, producing the so-called 'per-capita' table. However, these tables are also crude in that they fail to recognize inherent stochasticity over repeated Olympic cycles, which on a relative scale will be large for smaller countries. The result is that the per-capita table is typically led by small population countries. For instance, Grenada, Dominica and Saint Lucia, with respective populations of 112,579; 67,408 and 184,100 occupied the top 3 positions on the 2024 per-capita table. In response to these limitations, approaches have been proposed in the scientific literature to create alternative rankings. These approaches can be divided into methods that challenge the standard ranking produced by the number of gold medals (or total medals) by (1) reweighing the contributions of gold, silver and bronze medals in various ways (Ogwang and Cho, 2021), (2) producing composite tables combining medal tables ordered by differing criteria (Cao et al., 2016), and (3) ranking performance relative to collective resources available to a particular country, as specified by variables such as population and GDP using approaches like Data Envelope Analysis (Lozano et al., 2002; Bernard and Busse, 2004; Churilov and Flitman, 2006; Wu et al., 2009; Li et al., 2015; Flegl and Andrade, 2018), and linear regression (Halsey, 2009). Finally, the method recently proposed by Duncan and Parece (2024) has received substantial attention in various popular media outlets around the world, like the The New York Times (Coy, 2024) and The Guardian (Ryan and Nicholas, 2024) who referred to it as the Goldilocks method due to not favoring large or small countries. The Duncan-Parece method ranks according to a novel probabilistic index that adjusts for population effects, which we will compare and contrast to our own approach below. In this article, we propose Bayesian ranking as an alternative method to order medal tables. Unlike the traditional approaches, Bayesian ranking focuses on the most likely ordering of the key parameters of interest that drive the observed data (Laird and Louis, 1989). In the Olympic medal setting, we can informally define these key parameters as the long-run per-capita medal ratios that would be observed in differing countries if the Olympics were repeated several times under differing conditions. In other words, the medal table we produce via Bayesian ranking does not aim to summarize what happened in a particular Olympic Games - but instead to infer which countries are most efficient in producing Olympic medalists, relative to their population size over the long term. In the case that the data from a single Olympics is used, the ranking produced could be considered a modification of the ranking by the observed per-capita medal ratios. For large countries such as the USA and China, the Bayesian estimates of long-term per-capita ratios will closely resemble observed values, given their large populations and relatively stable performance. In contrast, for smaller countries, the Bayesian estimates may diverge significantly from observed ratios, as their medal counts are more susceptible to stochastic variation within a single Olympic cycle. A key consideration in this approach is the choice of prior distribution for these underlying long-term ratios, as the reliability of the rankings relies on how reasonable the prior is. Hierarchical Bayesian modeling, as used in the methods discussed later, mitigates some subjectivity by leveraging the observed distribution of medal counts across countries 2 to help estimate the prior, thereby enhancing the robustness of the resulting rankings. In Section 2, we will describe our Bayesian ranking model in detail, with careful discussion regarding the plausible validity of our modeling assumptions. In Section 3, we examine the medal tables produced by our approach over the last six Olympic cycles, and contrast with the results produced by the recent algorithm proposed by Duncan and Parece (2024) mentioned above. In Section 4, we provide additional evaluation and justification of our modeling assumptions, and discuss both caveats as well as possible extensions of our approach. 2 Methods The National Olympic Committees (NOCs) represent each country, as well as the individual athletes from the Refugee Olympic Team. The C NOCs that competed at a games are indexed by c = 1, . . . , C and the total population size per NOC is given by nc. These vary in each Olympic Games, but for notational convenience we'll ignore this in the following presentation which will focus on the NOCs in Paris 2024 (C = 204). As with the usual per-capita analysis, only NOCs with a measurable population are included. The total number of medals won by country c is the sum of the number of medals won by their individual athletes Mc = 1 × M1,c + 2 × M2,c + · · · , where Mi,c is the random variable for the number of their athletes that won exactly i medals. Here each medal-winning country in team competitions such as relays or field sports is considered a nominal athlete that wins a single medal, which is in keeping with the official lexiographic ranking compiled by the International Olympic Committee. The total number of medals achieved by each country c will be modeled using a combination of interrelated Poisson processes representing the number of athletes Mi,c winning exactly i medals. If we momentarily ignore the number of total medals, Mc, and consider only the number of unique medal winning athletes, Uc = P i Mi,c, we can assume Uc to be roughly Poisson distributed. The Poisson distribution is used as an approximation to the conceptual idea that the number of unique medal winners from country c is approximately a Uc ∼Binomial(nc, pc) distribution, where a unique medal winner is considered a rare event from a large number nc of independent Bernoulli trials representing each person within a population with probability of success pc. The binomial/Poisson assumptions are considered in Section 4. However, care has to be taken with modeling total medals won, Mc, as the world's highest achieving athletes that win more than one medal would invalidate the Poisson limiting assumption that it should be impossible to have two or more events at the same instance. Hence, we will instead model the number of athletes winning i = 1, 2, . . . as conditionally independent Poisson processes, that are interlinked via a prescribed relationship between the country-specific impacts on the probability of winning that number of medals. Note that this assumption implies that the number of unique medal winners, Uc = P i Mi,c will follow a Poisson distribution, whereas the total number of medals won Mc = 1 × M1,c + 2 × M2,c + · · · , will not. In Paris 2024 the highest number of medals won by any individual athlete was four 3 medals, so for notational convenience we will denote M4,c as the number of athletes winning four or more medals, and so we only consider the finite sum to M4,c. It is straightforward to adapt the modeling framework to allow for a different number of medals won by an individual athlete. The number of athletes winning i = 1, 2, 3 and 4 or more medals in each competing country are assumed to be Poisson distributed: M1,c ∼ Poisson(λ1,c), M2,c ∼ Poisson(λ2,c), M3,c ∼ Poisson(λ3,c), and M4,c ∼ Poisson(λ4,c). The expected number (and variance) of athletes winning each number of medals is then λi,c = nc pi,c, where pi,c = P(Xc = i) is the probability of an individual Xc from competing country c winning exactly i medals. Again, for convenience in notation p4,c = P(Xc ≥4). The country-specific Poisson distributions are equivalent to a classical frequentist inference of a country-specific random effect on the probability of a winning that number of medals. These Poisson distributions are treated as independent of each other, conditional on those country-specific probabilities p1,c, p2,c, p3,c and p4,c of an individual winning a given number of medals. The relationship between these probabilities is defined through parameters for the following conditional probabilities: pc = P(Xc ≥1) i.e., the probability of athlete from c being a unique medal winner, q2 = P (X ≥2 | X ≥1) , q3 = P (X ≥3 | X ≥2) , and q4 = P (X ≥4 | X ≥3) . The probability of being a unique medal winner pc retains the country-specific subscript notation. In contrast, the consecutive conditional probabilities qi of winning at least one additional medal having already won i medals are not country-specific, once the conditioning on being a unique medal winner from that NOC is taken into account. There is a slight abuse of notation here in that these conditional probabilities depend on the country-specific random variable Xc ≥1 for being unique medal winner, but we use the random variable X to represent the number of medals won by an individual from a generic NOC winning that number of medals. The country-specific effect of population biology, demography, and the ability to develop sporting talent is assumed to be entirely driven through the parameter for the chance of producing a unique medal winner pc = P(Xc ≥1). The above framework then assumes that conditional on someone being a medal winner, the likelihood of winning further medals has no further country-specific effects. This assumption seems reasonable as when conditioning on the rare event of winning one or more medals, we are implicitly conditioning on an athlete that has the talent, training environment and funding to perform at a world class level in their chosen sport; the conditional probability of winning more than one medal given this context might be largely independent of the athlete's home country. These 4 multi-medal winning athletes are also sufficiently rare that they will not be very influential on the rankings inference, except for the small countries that have been fortunate enough to host such talent. Indeed we see no substantive change in the results presented when only the unique medal winning athletes are modelled. From a computational view this construction has the benefit of not needing country specific random effects for these multi-medal winning probabilities which would be hard to inform from the data as the majority of countries have no such athletes. The countryspecific probabilities of being a unique medal winner are more reliably estimated as there are based on a larger number of such athletes. The unconditional probability of an individual from country c of winning a certain number of medals will vary by country according to: p1,c = P(Xc = 1) = P(Xc ≥1) -P(Xc ≥2) = pc (1 -q2), p2,c = P(Xc = 2) = P(Xc ≥2) -P(Xc ≥3) = pc q2 (1 -q3), p3,c = P(Xc = 3) = P(Xc ≥3) -P(Xc ≥4) = pc q2 q3 (1 -q4), and p4,c = P(Xc ≥4) = pc q2 q3 q4. These parameters are given uninformative and independent prior distributions of q2 ∼ Uniform(0,1), q3 ∼ Uniform(0,1), and q4 ∼ Uniform(0,1), so that they are predominantly learnt from the data. The country-specific probability of an individual becoming a unique medal winner is assumed to follow a prior distribution: pc ∼Beta(α, β), where α and β are given independent uniform hyperprior distributions: α ∼ Uniform(0, 1), and β ∼ Uniform(0, 108). Note that sensible values for α and β should correspond to an expected medal probability from the associated beta-prior: E(pc) = α α+β, that is relatively close to the total number of medals divided by the global population (1039/8, 000, 000, 000 = 1.3 × 10-7 for Paris 2024). However, this information has not been hardcoded into the model: instead, the non-informative hyperpriors give the model the flexibility to learn the most plausible values for α and β based on the observed medal counts across differing countries. The priors induce shrinkage to reduce the uncertainty for small countries which have limited sample information by pulling their estimates towards the global average, while preserving the data-driven estimates for larger countries. The pooling of information across all countries for the multi-medal winning athletes also ensures they do not have undue influences on the estimates for small countries. The shrinkage provides a better estimate of the 'long-run' probability of winning medals, which would provide a more stable ranking 5 of each nation's performance. Alternative prior specifications are considered in the online Appendix, with little sensitivity to this choice found. In summary, we have just four parameters to obtain a posterior distribution for each country; namely the country-specific random effect for the probability of a unique medal winner pc = P(Xc ≥1), and the three conditional probabilities of such an athlete winning further medals qi = P (X ≥i | X ≥i -1) for i = 2, 3 and 4. The population size data are obtained from the United Nations Nations (2025) which are updated for each Olympic year. The Olympics data are obtained from the International Olympic Committee (2025). Gibbs sampling using the rJAGS package (Plummer, 2024) is used to draw posterior samples of (pc, q2, q3, q4) for each country, which are then transformed as above to country-specific medal probabilities, and subsequently to expected per-capita medal ratios: E(Mc/nc) = p1,c+2 p2,c+· · · . The rank for the expected per-capita number of medals won in a particular country, among all medal-winning countries, is then calculated for each posterior sample. Finally, the estimated posterior mean of these ranks is used to order the medal table as is commonplace in the Bayesian ranking literature. Alternative statistics, like the posterior median rank, that could be used for the ranking are considered in Section 4. Finally, it is worth noting that while we base our ranking on the posterior distribution of the expected medal rate per capita E(Mc/nc) in the above, this is equivalent to ranking on the probability of an individual from that country winning at least one medal pc = P(Xc ≥1), which is easily seen by expansion of expectation: E(Mc/nc) = E(M1,c/nc) + 2 E(M2,c/nc) + 3 E(M3,c/nc) + 4 E(M4,c/nc) = pc (1 -q2 + 2 q2 (1 -q3) + 3 q2 q3 (1 -q4) + 4 q2 q3 q4) where the terms in the bracket are not country specific. 3 Results Table 1 presents the Bayesian ranking of the top ten and selected countries from Paris 2024. The posterior mean rank over those countries that won at least one medal is used to order the table. As per usual approaches, the NOCs that did not win any medals are not ranked. Population size, total medals won, observed per-capita medal winning rates and rankings by these observed rates, as well as the Duncan and Parece (2024) U-index are also provided for comparison. The posterior median medal winning rate per million and 95% credible intervals for uncertainty estimation are provided. The posterior median is utilized due to the strong skewness in the credible intervals for smaller countries to be discussed in Section 3.3. It is worth noting that the posterior median rate does not follow exactly the same ordering as the posterior ranks due their skewness. A Shiny app is available from MacDermott et al. (2025), which allows users to explore the different rankings (including the posterior median rank, as an alternative) across all NOCs, including demographic comparisons for the last six Olympic Games going back to Athens 2004. 6 Ranks Country Medals per million Credible Interval Posterior Mean Percapita U-Index Name Code Population (thousands) Medal Total Observed Posterior Median 95% Lower 95% Upper 1 4 6 New Zealand NZL 5,214 20 3.84 3.31 2.05 4.89 2 7 1 Australia AUS 26,713 53 1.98 1.80 1.32 2.31 3 8 8 Hungary HUN 9,676 19 1.96 1.83 1.06 2.89 4 10 4 Netherlands NLD 18,229 34 1.87 1.68 1.15 2.40 5 1 27 Grenada GRD 117 2 17.09 2.42 0.34 8.16 6 6 18 Jamaica JAM 2,839 6 2.11 1.71 0.67 3.54 7 5 25 Bahrain BHR 1,607 4 2.49 1.74 0.53 4.25 8 11 15 Georgia GEO 3,808 7 1.84 1.58 0.65 3.00 9 12 17 Croatia HRV 3,875 7 1.81 1.60 0.65 3.10 10 13 13 Denmark DNK 5,977 9 1.51 1.35 0.64 2.50 15 18 20 Ireland IRL 5,255 7 1.33 1.01 0.42 2.04 16 24 2 France FRA 66,549 64 0.96 0.95 0.72 1.22 17 25 3 Great Britain GBR 69,138 65 0.94 0.95 0.72 1.21 21 2 41 Dominica DMA 66 1 15.15 1.10 0.06 5.64 24 3 29 Saint Lucia LCA 180 2 11.11 0.96 0.05 4.98 46 42 43 Hong Kong HKG 7,415 4 0.54 0.36 0.09 0.96 51 48 5 United States USA 345,427 126 0.36 0.34 0.29 0.41 76 76 114 China CHN 1,419,321 91 0.06 0.06 0.05 0.08 Table 1: Bayesian ranking of the top 10 and selected countries in Paris 2024 by posterior estimates of medal-winning probabilities, normalized per million population, including 95% credible intervals. The Duncan-Parece U-index based ranks are provided. 3.1 Bayesian Shrinkage and Ranking Stability As mentioned, a key feature of the Bayesian ranking approach is its shrinkage effect, which adjusts for small-sample uncertainty. It prevents countries with exceptionally high percapita medal counts from being overemphasized from a single-event which may not persist over the long-term. The shrinkage can be seen for Grenada in Table 1 having an observed medals per million of 17 (rank 1 per-capita), which decreased to 2.42 (Bayesian rank 5). Similarly, Dominica and Saint Lucia were ranked second and third on a per-capita basis, but have been shrunk to 21 and 24 respectively in the Bayesian ranking. Using Bayesian ranking, New Zealand, Australia and Hungary are ranked as the top 3, but under the percapita rate it would be Grenada, Dominica and then Saint Lucia, a combination which is unlikely to be seen again in future Olympics. The Bayesian shrinkage results in a more stable and interpretable ranking compared to naive per-capita methods, which tend to disproportionately favor smaller nations. The performances of larger countries are well informed by the sample data, so see relatively little change in the posterior inference. The USA and China achieved the same number of gold medals, the USA won both more silver and bronze medals so are ranked first in the official lexicographic medal table. But in the per-capita rankings they are placed as 48 and 76 respectively, due to their very large population sizes. The shrinkage is minimal for such large countries with the the USA moving down 3 positions to 51 in the Bayesian rank and China only 1 rank up to 75. The slightly larger change for the USA is discussed further in Section 3.2. In Figure 1 we can see the shrinkage effect across all countries, by comparing the 7 Figure 1: Comparison of posterior median and observed medal winning rates by population size for each country with at least one medallist. Directed arrows indicate the shrinkage effect from observed per-capita rates to posterior estimates. The 95% credible intervals for the medal rate per million are shown by the gray lines, with boundary range ticks. Those countries with any multi-medal athletes are shown as a different color and flags mark countries listed in Table 1. observed and posterior median per-capita medal probabilities. The smaller nations on the left of the graph are pulled closer to the global mean, reflecting the underlying uncertainty in their success rates. In contrast, larger countries on the right see little effect of the prior, as their rankings are more data-driven. It is worth noting that the level of shrinkage is a function of three factors: (1) the population size, (2) the number of medals won by each NOC (note that these first two factors pertain to the observed medal winning rate) and (3) whether there are any multimedal winning athletes. This feature can be seen by the shrinkage arrows having different lengths for NOCs with similar population sizes which are close to each other on the x-axis. Generally, if two NOCs have same population size and medal counts, the one with more multi-medal winners will see more shrinkage to the global average which will be discussed in Section 3.2. The NOCs with an unusually high medal count, for a given population size, will see more shrinkage as is expected when providing more stable medal winning rate estimates. 8 It should be noted that only medal winning countries are shown in Figure 1. However, a consequence of the shrinkage is that NOCs that won no medals exhibit shrinkage upwards, which is again to be expected to provide more stable medal winning rate estimates. This shrinkage effect for non-medallist NOCs is discussed in more depth in the online Appendix. It is of course important to include the non-medal winning NOCs in the inference, as they inform the global average rates and their exclusion would lead to a bias. 3.2 Impact of Multi-Medal Winners on Bayesian Rankings Another feature of the Bayesian ranking model is its emphasis on the number of unique medal winners (and the number of medals they won) rather than the total medal count. This feature of the model is a notable improvement over Duncan and Parece (2024) which ignores the athletes that compete in several events and break the binomial assumption that medals won in differing events are independent. The focus on the unique medal winners also helps prevent a small number of extraordinary multi-medalists from disproportionately inflating a country's ranking. One must not diminish the achievement of such athletes, but a goal here is to provide a more stable estimate of each NOC's medal winning rate which may be more reflective of the NOC's ability to develop sporting talent. The outstanding ability of such athletes is likely more driven by their individual genetic factors, and the particular events they compete in (e.g. more multi-medal winners are seen in swimming events), as opposed to being a strong indicator of an NOC's talent development capability. This effect is illustrated in Figure 1, where countries with similar populations and nearly identical total medal counts so are near each other on the graph (in both x and y direction) exhibit different levels of shrinkage. The countries colored in purple have multi-medal winners exhibit more shrinkage than those with unique medal winners only which are colored in gold. For instance, in 2024 Ireland and Norway with essentially the same populations won 7 and 8 medals respectively. Ireland's medals were won by six unique medalists, with one athlete winning multiple medals, while Norway's medals were distributed across eight different athletes. As another example, Grenada and Saint Lucia are both small nations with similar populations that won two medals in Paris 2024. They ranked 1 and 3 on the observed per capita table. However, because Saint Lucia's two medals were won by the same athlete, Julien Alfred, their estimated medal rate is shrunk more than Grenada's, whose two medals were won by separate athletes. Finally, Hong Kong won four medals in 2024, but two of those were won by the swimmer Siobh ́an Haughey, which explains the larger shrinkage applied to Hong Kong's observed medal ratio compared to similar size countries. This feature underscores the model's tendency to favor broader medal distribution across athletes rather than concentrated success among a few high achieving individuals. This feature can also effect the degree of shrinkage applied to bigger countries, albeit to a lesser extent as the absolute amount of shrinkage is smaller for larger countries. For instance, USA moves by 3 positions in the ranking due to their 10 multi-medal winners winning 23 out of a total 126 medals, whereas China only moves by 1 position as they have only 3 multi-medal winners winning 7 out of there 91 medals. 9 3.3 Ranking Uncertainty and Credible Intervals On their own, the posterior mean ranks should not be used to compare the performance of different countries, just like p-values should not be used to judge effect sizes, as they do not account for uncertainty. To address this, credible intervals for every medal winning NOC from Paris 2024 in Figure 2, in addition to the selection in Table 1, which provide a more nuanced assessment of each NOC's ranking stability. Figure 2: Average Bayesian rank with 80% credible intervals for Paris 2024, ordered by rank. Smaller nations exhibit wider credible intervals due to greater uncertainty in their medal counts, while larger countries have more stable rankings. As expected, smaller countries exhibit wider credible intervals, reflecting greater ranking uncertainty due to their limited sample sizes, while larger nations have more stable rankings. Further, for well performing countries the rank distributions are right skewed due to the natural bound at first rank and vice-verse for weakest performing countries bounded by the bottom rank. The asymmetry is most clear for the smallest countries with the most uncertainty. For example, the USA at posterior mean rank 48 has a very short (and symmetric) credible intervals reflecting the large sample of competing athletes. New Zealand's credible interval does not overlap those of Australia and Hungary, reinforcing its standout position as the top-performing country according to the Bayesian ranking. In contrast, Grenada's 10 credible interval overlaps substantially with those of many other small nations, suggesting that its superior per-capita medal count may not represent a statistically meaningful performance difference when accounting for sample uncertainty. In general, countries with fewer medal winners (and smaller populations) exhibit wider credible intervals, while those with consistent performances across multiple events (typically larger nations) have more stable rankings. This highlights a key advantage of the Bayesian approach with the credible intervals - it provides a probabilistic measure of ranking uncertainty, ensuring that countries with overlapping intervals are not misinterpreted as significantly different in performance despite differences in their rank positions. 3.4 Comparison with Duncan-Parece Method The Bayesian posterior mean ranks based on the estimated medal-winning probabilities presented in Table 1 differ significantly from the Duncan and Parece (2024) U-index rankings. Under the Bayesian ranking, New Zealand, Australia, and Hungary rank among the top-performing nations, while Grenada, despite its high observed per-capita medal count, is ranked slightly lower due to the shrinkage effects. The USA and China have seen little change in the Bayesian ranking compared to a per-capita rating. The Duncan-Parece method implicitly penalizes smaller countries in the rankings, as their approach is essentially p-value based. It is well known that p-values are sample size dependent (Wasserstein and Lazar, 2016). A small sample limits the evidence available, even for a large effect size, so smaller NOCs cannot achieve the small p-values required to be ranked highly, unless they attain an extraordinarily large number of medals relative to their population size. As such the larger countries are given an advantage in the Duncan-Parece method. The advantage given to large NOCs in the Duncan-Parece method is seen in Table 1 and Figure 3 by it assigning higher ranks to large countries like the USA at rank 5, Great Britain at rank 3 and France at rank 2, which benefit from their overall representation of athletes competing across events. In general, the Duncan-Parece U-index ranked countries which are ranked highly have large population sizes. The only ranking scheme shown that places higher priority on larger NOCs is the total medals based ranking in gray. The percapita based ranking generally gives priority to smaller NOCs. The Bayesian ranking can be reasonably described as achieving a balance between smaller and larger nations, with a mix of such nations achieving high ranks. 3.5 Long-term Stability of Bayesian Ranking One of the key motivators for the Bayesian ranking approach is to provide a stable long-term estimate of each NOC's performance, based on a single Olympic Games, using shrinkage effects whilst also accounting for population size effects. Section 3.1 demonstrated the shrinkage effects and how they approach accounts for population size. Here we will evaluate its ability to provide an interpretable long-term measure of performance by considering its stability and behavior when applied to many past Olympic Games. The Bayesian ranking method is applied entirely separately for each of the last 6 11 Figure 3: Comparison of rankings across different methods for Paris 2024 as a function of population size. Bayesian rank (purple), U-index rank (orange), total medals rank (dashed dark gray), and per-capita medals rank (dashed light gray) are shown. Notably, the Uindex closely follows the total medal count ranking, highlighting its tendency to favor larger nations, while the Bayesian ranking adjusts for population size. Olympic Games, so there is no carryover of information across Olympic Games in the inference. Notice the strong consistency in the Bayesian ranking for each NOC shown across these 6 Games, which provides comfort in the interpretation of the ranks as a stable long-term estimate of each NOC's performance, due to the shrinkage effects. In particular, New Zealand topped the ranks in the last 4 Olympic Games after a notable improvement since Athens 2004 which is known to be influenced by increases in funding and strategy/policy priority changes resulting from the creation of Sparc (now Sports NZ) and its subsidiary High Performance Sports NZ which were formed in 2002 and 2012 respectively. The Netherlands have made consistent gains since 2008. Hungary did not perform so well in Beijing 2008, but otherwise ranked very highly. Similarly, Georgia has performed consistently well. Jamaica dropped from a consistent rank of 2-3 in size Beijing 2008 to rank 6 in Paris 2024. Interestingly, both Grenada and Bahrain performed sufficiently well in Paris 2024 to appear in the Top 10 for the first time, despite the shrinkage applied to their estimates. 12 Figure 4: Top 10 ranked countries across the last 6 Olympic Games. Every country that is within the Top 10 in any Game is joined to their rank across all 6 games. 4 Discussion In this final section, we discuss some of the assumptions behind our model, possible ways it might be extended and more detailed comparison to the Duncan and Parece (2024) method. We first examine the equal treatment of medal types and the validity of the Poisson assumption. We then discuss the assumption of constant conditional probabilities for multimedal winners across countries. Finally, we compare our Bayesian ranking algorithm to existing probabilistic approaches, particularly the Duncan-Parece method, and highlight key advantages of our approach in capturing long-term national Olympic performance. Throughout the analysis we have treated all medals equally, but we could have considered either separate Poisson processes for each type of medal or having a weighting scheme for each type of medal. The former medal type specific Poisson process would have many zeroes and therefore limit the information to the likelihood. In this case, the inference would be strongly influenced by the priors which is best avoided. Previous literature has justified treating gold, silver and bronze medals equally. For instance, Bian et al. (2005) has argued that the performances winning gold, silver and bronze medals are often very similar, with a degree of luck being the crucial determining factor, which partially justifies 13 equal weighting. Ogwang and Cho (2021) also found that the first principal component estimated from the joint distribution of gold, silver and bronze medal totals across countries had approximately equal weight on the number of gold, silver and bronze medals and explains over 94% of the total variance, indicating very strong correlation in the gold, silver and bronze medal counts across different countries. The Poisson assumption, as a limiting approximation to the binomial distribution for the number of unique medal winners per country, can be empirically tested using observed medal counts over multiple Olympic Games. Assuming that a country's true medal-winning rate remains stable across successive Olympic cycles, we have considered diagnostic plots of estimated mean and variance, and hypothesis tests to detect possible excess-Poisson variation (see the online Appendix). These tests provide support for the Poisson approximation. While there are obvious physical limitations to the binomial argument that total medals can be regarded as a sum of equal probability Bernoulli trials, these are likely negligible in practice. For example, factors such age, socioeconomic conditions, and race may influence the likelihood of individual success in different events. However, even if individual probabilities vary the total medal counts can still be reasonably modeled using a Poisson distribution (with rate scaled by population size) as long as individual successes or failures can be treated as independent events. The assumption of independence is somewhat more difficult to justify. On one hand, given the rarity of athletes with the talent to win an Olympic medal, it seems reasonable to treat these individuals as independent rare events. On the other hand, the notion that 'a rising tide lifts all boats' should be acknowledged - training environments, national sports culture, and psychological factors can create performance spillover effects, particularly in elite competitive sports. Recall the model assumes that the conditional probabilities of an individual winning two or more medals, given that they have won at least one medal are constant across all countries. We have partially justified this assumption under the premise that when conditioning on an Olympic medal winner, one is implicitly conditioning on having the training environment, funding and genetic talent to compete as a world class athletes (i.e. we are mostly conditioning away those factors that might cause differing levels of success per-capita across countries). However, if a country focuses their resources on particular sports (such as swimming) where multi-medal winners are more likely, the true conditional probabilities for that country of winning multiple medals given a single medal may be larger than the global average. Unfortunately, there is not enough data from a single Olympic Games to estimate these conditional probabilities in a coherent way at a per-country level, particularly for smaller countries and, as such, complete pooling represents an attractive modeling assumption from a practical standpoint. In practice, the effects of incorporating such corrections in our ranking (which would in any case be driven primarily by single medal winners who constitute the vast majority of medals) would be unlikely to change the conclusions. To elaborate further, since the conditional probabilities are not country specific, the shrinkage applied to a country's raw per-capita medal rate by our model will tend to be slightly larger if its medals were won by disproportionally few athletes compared to a scenario where the same number of medals were won by separate athletes. This effect may have resulted in the USA doing slightly worse in the Bayesian ranking for Paris 2024 (position 51) compared to the observed per-capita ranking (position 48). However, as 14 shown in Figure 2, the practical differences in performance between countries ranked as 51 and 48 by Bayesian ranking is statistically negligible due to the strongly overlapping credible intervals. To produce a ranking, we have summarized the posterior distribution of the rank for a country's medal rate by its mean. As demonstrated in Figure 2, the posterior distributions of highly-placed smaller countries tend to be quite right skewed compared to larger countries, with the result that they may be ranked less advantageously using the posterior mean rank compared to using the posterior median rank. For instance, Grenada would be ranked as the 2nd best country in Paris 2024 according to posterior median rank, as opposed to 5th with posterior mean rank, although the other positions in the top 10 are identical to those in Table 1, having accounted for that shift. Effectively, the posterior mean rank applies a relative penalty to the ranking of smaller countries with asymmetric high-variability posterior distributions, compared to countries that have symmetric posteriors. We have chosen the posterior mean rank as it is in keeping with previous applications of Bayesian ranking, such as Laird and Louis (1989), but posterior median rank could equally be used to create the ranking table. Alternative rankings (such as rankings based on the posterior median) can be explored using the Shiny app at MacDermott et al. (2025) The goal of our model is to infer which countries are most efficient in producing Olympic medalists relative to their population size. This population-scaled efficiency is a subjective definition of success, and may not be something that a country strives for given that some Olympic events have more prestige and glamour than others. For instance, while the United States places at 51 on our Bayesian ranking table (despite winning the table on standard lexicographic medal counts), it does particularly well in many of the 'marquee' track and field events that enjoy particular media attention. Many factors might influence this relative Olympic success such as GDP per capita, the proportion of GDP allocated to sports funding, climate, demographic and social factors. If the aim was to instead infer which countries were most successful given fixed values of such variables, such factors could be accounted for in a Bayesian Poisson regression, which again used log-population size as an offset in modeling the probability of a unique medal, and incorporated a normal random effect for each country, with the inferred value of this random effect used to rank differing NOCs. The implicit prior for medal winning probabilities across countries would be lognormal (as opposed to beta) in this modified model, but with shifts in the mean parameter based on the covariate information for a NOC. This brings up another point of discussion. How much influence does the hierarchically estimated beta prior influence the overall results and ranking? The choice of beta rather than log-normal or some other choice is a subjective choice, and could be considered as a weakness of our model by some. It turns out that the exact choice of hierarchical prior has relatively little influence in the final results. We have also investigated hierarchical specifications of log-normal, logit normal and mixture-beta priors for the probability of a unique medal (see the online Appendix for details). In all models, New Zealand is ranked as the best performing NOC, and the top 5 and top 10 ranked countries are consistent across all models, with slightly differing orderings based on the implied shrinkage. In particular, the results from the mixture beta prior are almost identical to the results above. Given that the mixture beta has theoretical advantages in that it can adapt to the most general 15 distribution of probabilities across countries, it is comforting that the results from the beta and mixture beta priors are so similar. Finally we return to the approach of Duncan and Parece (2024), which ranks by the likelihood of achieving a country's observed medal count (or higher) under a binomial model. That is, the model considers possible awarded medals to a country as independent Bernoulli trials with probability success proportional to the country's population. One nice feature of their approach is their binomial probability implicitly respects the total quota of possible Olympic medals that a country can win. For instance, in Tokyo 2021 the total medals that a country could possibly win was 579 due to restrictions on the total number of athletes that an NOC could enter. While this isn't respected by our Poisson distribution (which has infinite support), we note that in practice the probability distributions for total medals corresponding to Poisson and binomial approaches are very similar again based on the large n and small p approximation (See the online Appendix for more details). The Duncan-Parece method is effectively a transformed p-value that tests the null hypothesis that the probability of a country winning a particular Olympic medal equals the proportion of the aggregate population of countries that have previously won an Olympic medal that resides in that country. The index tends to favorably rank higher population countries. For example, in the Tokyo 2020 Olympics, the USA ranked 6th using DuncanParece's U-index, with a score of 15.2. However, if we imagine that both the population of the USA and its medal count doubled, the U-index for the USA increases to 34, in which case they would would be ranked as the top performing country. This is despite the USA's relative performance (in terms of medals per-capita) remaining unchanged. The effect just described is due to a well known feature of p-values (Wasserstein and Lazar, 2016). Small observed effects (such as small observed differences between country-specific and global per-capita medal ratio) can correspond to extremely small p-values when the effect is estimated with small standard error (roughly the standard error for estimated per-capita ratios are proportional to the square root of population). These issues have been previously discussed in the context of ranking effect sizes in genetics and genomics in Ferguson and Chang (2020), and sheds some light on the differences in rankings produced by our Bayesian ranking algorithm and their probabilistic index. To advocate for our approach, as demonstrated in Section 3, New Zealand has led our Bayesian ranking algorithm consistently over the previous four Olympic cycles, and they have also 'topped' the empirical per-capita medal table over this collective period when restricting to countries that have a population of at least 1,000,000; indicating that our algorithm is better at inferring the countries with the best long-run performance even though we infer this performance from a single Olympic Games. 5 Conclusion In conclusion, our Bayesian ranking algorithm is a new approach way to rank each country's performance in international competitions such as the Olympic Games. It strikes a balance between consideration of total medals, which would give an advantage to larger countries, and empirical per-capita medal ratios which are overly influenced by stochastic factors 16 pertaining to a particular Olympics and notable advantage the large number of smaller nations. References Bernard, A. B. and M. R. Busse (2004). Who wins the Olympic Games: Economic resources and medal totals. Review of Economics and Statistics 86(1), 413-417. Bian, X. et al. (2005). Predicting Olympic medal counts: The effects of economic development on Olympic performance. The Park Place Economist 13(1), 37-44. Cao, X., Y. Fu, J. Du, J. Sun, and M. Wang (2016). Measuring Olympics performance based on a distance-based approach. International Transactions in Operational Research 23(5), 979-990. Churilov, L. and A. Flitman (2006). Towards fair ranking of Olympics achievements: The case of Sydney 2000. Computers & Operations Research 33(7), 2057-2082. Coy, P. (2024). Which country will win the Paris Olympics? Don't just count medals. The New York Times 07/22/24. Duncan, R. C. and A. Parece (2024). Population-adjusted national rankings in the Olympics. Journal of Sports Analytics 10(1), 87-104. Ferguson, J. and J. Chang (2020). An empirical Bayesian ranking method, with applications to high throughput biology. Bioinformatics 36(1), 177-185. Flegl, M. and L. Andrade (2018). Measuring countries' performance at the Summer Olympic Games in Rio 2016. Opsearch 55(3), 823-846. Halsey, L. G. (2009). The true success of nations at recent Olympic Games: Comparing actual versus expected medal success. Sport in Society 12(10), 1353-1368. International Olympic Committee (2025). Official Olympic Games Data. Laird, N. M. and T. A. Louis (1989). Empirical Bayes ranking methods. Journal of Educational Statistics 14(1), 29-46. Li, Y., X. Lei, Q. Dai, and L. Liang (2015). Performance evaluation of participating nations at the 2012 London Summer Olympics by a two-stage data envelopment analysis. European Journal of Operational Research 243(3), 964-973. Lozano, S., G. Villa, F. Guerrero, and P. Cort ́es (2002). Measuring the performance of nations at the Summer Olympics using data envelopment analysis. Journal of the Operational Research Society 53(5), 501-511. 17 MacDermott, C., C. J. Scarrott, and J. Ferguson (2025). Interactive dashboard exploring medal rankings. https://mq698p-carl-scarrott.shinyapps.io/Bayesian_ Medal_Ranking_app/. Nations, U. (2025). Un department of economics, population division - data downloads. Ogwang, T. and D. I. Cho (2021). Olympic rankings based on objective weighting schemes. Journal of Applied Statistics 48(3), 573-582. Plummer, M. (2024). rjags: Bayesian Graphical Models using MCMC. R package version 4-16. Ryan, J. and J. Nicholas (2024). Ranking the medal table by gold, total, or most medals per capita - Who actually won at the Olympics? The Guardian 09/12/24. Wasserstein, R. L. and N. A. Lazar (2016). The ASA statement on p-values: Context, process, and purpose. The American Statistician 70(2), 129-133. Wu, J., L. Liang, and Y. Chen (2009). DEA game cross-efficiency approach to Olympic rankings. Omega 37(4), 909-918. 18 Appendix In this appendix we consider secondary details about the method and results. A Shiny app is available from MacDermott et al. (2025) for interactively exploring the results of the Bayesian ranking, including how the ranking might vary by country-specific demographic factors and comparison to other ranking methods. A Bayesian Shrinkage and Ranking Stability In the main paper we presented the shrinkage results for the medal winning countries in Paris 2024. The shrinkage brings each country closer to the global average, as a means to provide a more stable long-run estimate of their performance. The large countries see little shrinkage as they are well informed by the data, while the smaller nations see more shrinkage. The countries with one or more multi-medal winning athletes also saw more shrinkage than those with only single medal winning athletes of a similar size, as the multi-medal winning rate is assumed constant across all nations. Here we will explore the (reverse) impact of the shrinkage of the countries who won no medals in Paris 2024 In the main paper we presented the shrinkage results for the medal winning countries in Paris 2024. The shrinkage brings each country closer to the global average, in order to provide a more stable long-run estimate of their performance. The large countries see little shrinkage as they are well informed by the data, whereas the smaller nations see more shrinkage. The countries with one or more multi-medal winning athletes also saw more shrinkage than those with single medal winning athletes of a similar size, as the multi-medal winning rate is assumed constant across all nations. Here we will explore the (reverse) impact of the shrinkage of the countries who won no medals in Paris 2024. The uplift of the medal winning rate for NOCs with no medalists can be seen in Figure 5 by the vertical arrows. The shrinkage for these NOCs is only affected by their population size, as they have zero medal count and no multi-medal winners. Broadly speaking, there is a monotonically decreasing inflation effect of the shrinkage with population size. The smallest countries receive the biggest uplift towards the global average, as the larger populations' estimates are more data-driven. The uplift of the medal winning rate for NOCs with no medalists can be seen in Figure 5 by the vertical arrows. The shrinkage for these NOCs is only affected by their population size, as they have zero medal count and no multi-medal winners. Broadly speaking, there is a monotonically decreasing inflation effect of the shrinkage with population size. The smallest countries receive the biggest uplift towards the global average, as the larger populations' estimates are more data-driven. The non-medal winning countries are not assigned a rank in the Bayesian ranking, as per usual protocol. E(Mc/nc) = pc (1 -q2 + 2 q2 (1 -q3) + 3 q2 q3 (1 -q4) + 4 q2 q3 q4) ≈pc The non-medal winning countries are not assigned a rank in the Bayesian ranking, as per usual protocol. However, if they were ranked they would be amongst the bottom 19 Figure 5: Effect of shrinkage on the medal winning rate against population size for each country that won a medal (as per Figure 1 of the main paper, and the inflation effect for those who won no medals. countries, which can be seen by the low height of their arrows compared to most medal winning NOCs. B Approximate Equivalence to Duncan-Parece Method The model underlying the proposed Bayesian ranking can be considered as an approximation to the Duncan and Parece (2024) method in the circumstance that the probabilities pc, which we define in the Methods section of the main manuscript, are equal across competing NOCs. The Duncan-Parece method models the number of medals won by a NOC using a binomial distribution where each trial is one of the medals that an NOC could win, up to the maximum possible number of medals, with a country specific probability of winning. Under the quota system the maximum number of medals that could be achieved was M = 559 trials for Paris 2024, out of a total of T = 1039 medals. Under the assumption of equal medal winning capability for any individual around the world, the expected number of medals won by a country is proportional to the country's population (see equation (4) of Duncan and Parece (2024)). But due to the close equivalence between 20 the Poisson/binomial, you can view our per-capita specification as simply changing the denominator term. The model underlying the proposed Bayesian ranking can be considered as an approximation to the Duncan and Parece (2024) method in the circumstance that the probabilities pc, which we define in the Methods section of the main manuscript are equal across competing NOCs. The Duncan-Parece method models the number of medals won by a NOC using a binomial distribution where each trial is one of the medals that an NOC could win, up to the maximum possible number of medals, with a country specific probability of winning. Under the quota system the maximum number of medals that could be achieved was M = 559 trials for Paris 2024, out of a total of T = 1039 medals. Under the assumption of equal medal winning capability for any individual around the world, the expected number of medals won by a country is proportional to the country's population (see equation (4) of Duncan and Parece (2024)). But due to the close equivalence between the Poisson/binomial, you can view our per-capita specification as simply changing the denominator term of the binomial probability. The population size for country c is nc and the total population of the world is N. The Duncan-Parece model is that the number of medals won by any country is Mc ∼ Binomial(M, πc) where πc is probability of that country winning a given medal. Under the equal capability assumption (per person) across all NOCs, the expected number of medals won should be proportional to that countries population so M × πc = nc N T. Rearranging we obtain: πc = nc N T M so under the equi-capability reference model the number of potential medals follows a Binomial M, nc N T M . Such a binomial is approximately a Poisson nc N T which in turn is approximately Binomial nc, T N . The latter probability pc = T/N is per-capita probability of an individual from a given NOC winning a medal underlying our model, assuming it does not change over countries (that is pc is constant). Although the two underlying models approximate each other under the null equicapability reference model, our inference approach does not have the same restriction on small countries due to the sample size impacts on the (transformed) p-values which underlie the U-index of Duncan and Parece (2024). Another substantial improvement in our method is the explicit handling of multi-medal winning athletes, which would break the usual independence assumption of the binomial. A drawback of our approach compared to Duncan and Parece (2024) is that we do not explicitly handle each country's quota. However, even for the most successful countries, medal winning is a rare event so no country get sufficiently close to their quota for this to be of concern. C Sensitivity to Prior Choice The shrinkage which stabilizes the long-run estimates of expected medal winning performance is mostly imposed by the choice of prior distribution (with only mild impacts from the assumed common multi-medal winning probabilities across all NOCs). In the main 21 Country Posterior Mean Ranks Medals per million Beta CI Population (thousands) Beta LogNormal MixedBeta Observed Beta Median Log-Normal Median Mixed-Beta Median 95% Lower 95% Upper NZL 5,214 1 1 1 3.84 3.31 3.61 3.27 2.05 4.89 AUS 26,713 2 3 2 1.98 1.80 1.82 1.79 1.32 2.31 HUN 9,676 3 4 3 1.96 1.83 1.84 1.79 1.06 2.89 NLD 18,229 4 5 4 1.87 1.68 1.73 1.71 1.15 2.40 GRD 117 5 2 6 17.09 2.42 7.14 2.13 0.34 8.16 JAM 2,839 6 7 5 2.11 1.71 1.82 1.68 0.67 3.54 BHR 1,607 7 6 7 2.49 1.74 2.00 1.66 0.53 4.25 GEO 3,808 8 8 8 1.84 1.58 1.68 1.56 0.65 3.00 HRV 3,875 9 9 9 1.81 1.60 1.65 1.54 0.65 3.10 DNK 5,977 10 10 10 1.51 1.35 1.42 1.40 0.64 2.50 IRL 5,255 15 15 15 1.33 1.01 1.05 1.02 0.42 2.04 FRA 66,549 16 16 16 0.96 0.95 0.95 0.95 0.72 1.22 GBR 69,138 17 19 17 0.94 0.95 0.95 0.95 0.72 1.21 USA 345,427 51 47 51 0.36 0.34 0.34 0.34 0.29 0.41 CHN 1,419,321 76 78 77 0.06 0.06 0.06 0.06 0.05 0.08 Table 2: Bayesian ranking of the top 10 and selected countries in Paris 2024 based on posterior estimates of medal-winning probabilities, normalized per million population. Rankings are shown under different prior specifications: the beta prior, a log-normal prior, and a flexible mixed-beta prior. The ranks are based on mean ranking of posterior mean probabilites. Corresponding 95% credible intervals for the beta prior are also provided supporting the appropriateness of the beta prior in the analysis. paper, we present the results when a beta prior distribution is used to inform the countryspecific probability of an individual becoming a unique medal winner, pc = P(Xc ≥1),: pc ∼Beta(α, β) with hyperparameters: α ∼ Uniform(0, 1), and β ∼ Uniform(0, 108). This prior is a natural choice and appears to produce sensible Bayesian rankings, but we have also explored the sensitivity to alternative priors and their hyperparameters. Firstly, we considered a truncated log-normal prior distribution which allows for a heavier upper tail for the prior distribution of the unique medal winner probability pc ∼Log-Normal(μ, σ)I(0, 1) with hyperparameters: μ ∼ Normal(-15, 0.323), and σ ∼ Gamma(0.001, 0.001) where the gamma is parameterized by the shape and rate. The hyperprior for the mean is chosen to be close to the overall medal winning rate over the world population. The 22 Figure 6: Effect of shrinkage on the medal-winning rate against population size for each country that won a medal (as per Figure 1 of the main paper), and the inflation effect for those who won no medals. for a log-normal prior with colours indicating how the shrinkage compares under each model resulting shrinkage is shown in Figure 6. The shrinkage is somewhat flatter compared to the beta prior, so that the smaller countries like Grenada are not shrunk as much as for the beta prior, and the larger countries experience notably more shrinkage. The trajectory of shrinkage vs population for the non-medal winning countries is also notably flatter compared to with the beta prior. The resultant rankings are shown in Table 2. The beta prior is used as the reference, where a black font means no change, green is an improvement in the ranked position and orange is a decline. The resulting ranks do not change much compared to the beta prior. The key difference in the Top 10 is that Grenada moves from rank 5 with the beta prior to 2nd place. But notice that the shrinkage for the larger countries leads to the UK being slightly pulled down the ranks, and the USA and Japan brought slightly up the ranks. The shrinkage for larger countries is indicative of the prior being informative over the large amount of sample information, which is not desirable. In addition, it appears that the heavy tail of the log-normal is allowing substantial variation in the probability of being a unique medal winner, and subsequently higher posterior mass on extreme performance, for countries with small populations. However, when one considers the uncertainty via the 23 95% credible intervals for the medal winning rate (calculated from the beta specification) these changes are not significant, as the estimated rates from the log-normal prior are well within these intervals (although it is worth noting that the medal winning rate for Grenada is close to the upper bound). A logit-normal prior: pc ∼Logit-Normal(μ, σ) with hyperparameters: μ ∼ Normal(-15, 0.323), and σ ∼ Gamma(0.001, 0.001) gave near identical results to the log-normal, so are not discussed for brevity. Figure 7: Effect of shrinkage on the medal-winning rate against population size for each country that won a medal (as per Figure 1 of the main paper), and the inflation effect for those who won no medals. for a mixed-beta prior with colours indicating how the shrinkage compares under each model A very flexible mixture of beta components was also explored to assess the robustness of the prior to heavy-tailed behavior. Three components were used, offering enough flexibility to approximate characteristics of priors such as the log-normal, while including the standard 24 beta prior as a special case. The three components pj c ∼Beta(αj, βj) for i = 1, 2, 3 have the same hyperparameters as for usual single beta prior as above. A Dirichlet(1,1,1) distribution is used for the mixture weights for the three beta components. The resulting shrinkage is shown in Figure 6 is very similar to using simple one component beta prior which provides comfort in the results presented the main paper, as it appears the choice of beta prior (and hyperparameters) is not strongly influential on the results and the changes seen with the log-normal prior above are likely due to this prior specification enforcing a unrealistically heavy tail that fails to adequately shrink highly stochastic extreme rates for small NOCs. One subtle distinction, however, emerges at the tails: at large population sizes for nonmedal winners, the mixed-beta prior results in slightly less shrinkage than the regular beta. This can occur because the mixture model may allocate higher-population observations more weight from flatter components, which exert less shrinkage. However, these deviations are minimal and affect only non-medal-winning countries, which are excluded from the final rankings. D Poisson assumptions Using the medal winning results across multiple Olympic Games we can evaluate the equal mean/variance assumption of the Poisson model. Figure 8 plots the sample mean against the variance of the number of medals for each NOC across all seven of the Olympic Games from Athens 2004 to Paris 2024, shown on a log-scale. A non-medal winning country is included as a zero for that Games, but a country which did not compete in the Games are set to missing. You can clearly see a strong mean-variance relationship as the NOCs are close to the line of equality. Some of the notable deviations from the line of equality are due to host NOCs (Greece, China, UK, Brazil, Japan and France). But overall the equal mean/variance assumption of the Poisson appears extremely reasonable. There is a mild crudeness to this evaluation due to the multi-medal winning athletes. In our model, the total number of medals won by country c is Mc = 1 × M1,c + 2 × M2,c + · · · P i i Mi.c. Each of the Mi,c is assumed to be Poisson distributed. But the total number of medals won Mc is not exactly Poisson due to the scaling by the number of medals i, but will be close to Poisson as the multi-medal winning athletes are sufficiently rare especially as the total Mc will be dominated by the 1 × M1,c term which is exactly Poisson. Another aspect of approximation in this diagnostic plot is that it assumes the number of medals on offer (including quotas of those available), population size, and the underlying performance of each NOC (as measured by their long-term medal rate: E(Mc/nc)) are constant over time. As we are considering only Games following Athens 2004 these factors will not have changed much, and so is of little concern. As an alternative test of the Poisson assumption, we defined p-values to test for excess Poisson variation in the medal counts over two successive Olympic Games, assuming performance, the total number of medals on offer and population all remain stable. Under these 25 assumptions (in addition to independence) conditional on R, that is the total medals won over two sucessive games, the number of medals won by a country in a particular Games has a binomial distribution with R trials and probability of success 0.5. The results are not shown here, but the country-specific p-values demonstrated no evidence for super-Poisson variability (with the notable exception of host countries). Figure 8: Average medals over the seven previous Olympic games are plotted against the empirical variance of the number of medals (y-axis), by country. Gross departures from the line of equality signify possible violations of the Poisson assumption. The shaded area represents a 90% predictive band for the empirical variance of a random sample of size seven from the Poisson distribution. 26
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2510.14719
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Tawa: Automatic Warp Specialization for
Modern GPUs with Asynchronous References
Hongzheng Chen∗†, Bin Fan, Alexander Collins, Bastian Hagedorn, Evghenii Gaburov,
Masahiro Masuda, Matthew Brookhart, Chris Sullivan, Jason Knight, Zhiru Zhang†, Vinod Grover
NVIDIA
†Cornell University
hzchen@cs.cornell.edu, vgrover@nvidia.com
Abstract—Modern GPUs feature specialized hardware units
that enable high-performance, asynchronous dataflow execution.
However, the conventional SIMT programming model is funda-
mentally misaligned with this task-parallel hardware, creating
a significant programmability gap. While hardware-level warp
specialization is the key to unlocking peak performance, it
forces developers to manually orchestrate complex, low-level
communication and software pipelines—a process that is labor-
intensive, error-prone, and unsustainable. To address this chal-
lenge, we present Tawa, an automated compiler that systemat-
ically generates high-performance, warp-specialized code from
a high-level, tile-based program. Central to our approach is a
novel IR abstraction, asynchronous references (aref), which
expresses warp-level communication without exposing low-level
hardware details. Using this abstraction, Tawa automatically
partitions programs into producer-consumer roles and manages
the intricate dataflow pipeline, relieving developers of invasive
kernel rewriting. Evaluation on NVIDIA H100 GPUs across rep-
resentative LLM kernels shows that Tawa delivers high hardware
utilization, achieving up to 1.1× speedup over highly optimized
cuBLAS GEMM kernels. For attention workloads, Tawa attains
1.2× speedup over Triton and matches the performance of the
hand-optimized CUTLASS C++ FlashAttention-3 kernel with far
less programming effort.
I. INTRODUCTION
Modern GPU architectures have evolved from homogeneous
processors into complex heterogeneous systems, integrating
multiple specialized hardware units to maximize performance.
Alongside traditional CUDA cores, modern GPUs integrate
dedicated engines such as Tensor Cores for dense matrix
computations and the Tensor Memory Accelerator (TMA)
for asynchronous data movement. This architectural design
enables a fine-grained, asynchronous dataflow execution model
where data movement can be explicitly orchestrated and
overlapped with computation. Exploiting this capability is crit-
ical for latency-sensitive workloads such as Large Language
Models (LLMs), whose efficiency is often limited more by
data orchestration than by raw FLOPs [1]–[5].
However, the prevailing programming paradigm for GPUs,
Single-Instruction Multiple-Thread (SIMT), which underpins
CUDA, is fundamentally misaligned with this heterogeneous,
task-parallel reality. The SIMT model assumes that all threads
within a group (a warp) execute the same instructions on
different data, excelling at problems with uniform control
flow. This model is ill-suited for managing diverse hardware
∗Work partially done while interning at NVIDIA.
units that must be driven by different, concurrent tasks. For
example, one task might involve generating addresses and
initiating a bulk copy with the TMA, while another performs
a matrix multiplication on the Tensor Cores. This creates a
paradigm mismatch: the hardware is capable of asynchronous,
task-parallel execution, but the software model is built for
synchronous, data-parallel execution.
To resolve this conflict, modern architectures like NVIDIA’s
Hopper and Blackwell GPUs introduce hardware-level warp
specialization [6], [7]. This mechanism reflects the hardware’s
task-parallel nature by allowing different groups of threads
(warps) to take on distinct roles. For instance, producer warps
can be dedicated to managing data transfers via the TMA,
feeding data to consumer warps that execute computations
on the Tensor Cores. While this new model unlocks the
hardware’s potential, programming for it remains difficult
because the surrounding software ecosystem and abstractions
are still rooted in the SIMT paradigm. This programmabil-
ity gap makes developing high-performance kernels a labor-
intensive endeavor, often requiring over a thousand lines of
hand-optimized code even for a basic GEMM kernel. This
difficulty manifests in three primary challenges:
Challenge 1: Coordinating concurrent warp roles. Be-
cause the SIMT model lacks first-class abstractions for assign-
ing heterogeneous roles, developers must manually restructure
kernels, encapsulating role-specific logic within complex con-
ditional branches. They must then orchestrate the execution of
these specialized warps to ensure that producer and consumer
tasks remain correctly synchronized and efficiently overlapped.
This manual process is inherently fragile. Even modern, high-
level domain-specific languages (DSLs) like Triton [8] retain
SIMT-centric abstractions, which preclude the explicit assign-
ment of warp roles and thus cannot naturally express warp-
specialized programs.
Challenge 2: Low-level communication management.
CUDA does not currently expose C++-level interfaces for
TMA and transaction barrier (mbarriers) operations. Conse-
quently, programmers must rely on inline PTX to configure
TMA descriptors, initialize and manipulate mbarriers, and
orchestrate synchronization. This requires reasoning explic-
itly about arrival counts, barrier phases, and shared-memory
aliasing, where subtle errors can easily lead to deadlock.
The difficulty is exacerbated when multiple mbarriers and
input parameters are involved. Such low-level orchestration
arXiv:2510.14719v1 [cs.LG] 16 Oct 2025
undermines code readability and substantially complicates
debugging and maintenance. High-level libraries like CUT-
LASS [9] and ThunderKittens [10] offer pre-written kernels
but still require users to carefully compose them to satisfy the
underlying communication requirements.
Challenge 3: Resource allocation and pipeline orchestra-
tion. Effectively overlapping the work of specialized producer
and consumer warps necessitates the manual implementation
of deep software pipelines. Developers must explicitly manage
multi-buffering schemes in shared memory, partition kernels
into prologue, main body, and epilogue stages, and carefully
balance the allocation of hardware resources like registers
and shared memory to respect occupancy limits. These design
choices are tightly coupled and create a vast, complex opti-
mization space. For example, increasing the pipeline depth af-
fects shared memory capacity, which in turn constrains thread
block size and impacts TMA descriptor layouts. Navigating
this space requires repeated, labor-intensive kernel refactoring,
making performance tuning both difficult and unsustainable.
To address these challenges, we propose Tawa1, an au-
tomated compiler that systematically analyzes computation
graphs generated from a tile-based DSL, partitions programs
into warp-specialized components, and manages both commu-
nication and pipelining. Our contributions are as follows:
• We propose asynchronous references (aref), an ab-
straction for warp-level communication. aref expresses
asynchronous data transfer intent without exposing low-
level hardware primitives, providing a clean and portable
representation at the IR level.
• We present Tawa, an automatic compilation flow for warp
specialization on NVIDIA GPUs. Tawa takes in a Triton
program, performs task-aware partitioning across warp
groups, applies multi-granularity pipelining, and gener-
ates high-performance PTX code for GPU execution.
• We evaluate Tawa on an H100 GPU using representative
LLM kernels. Our approach delivers high hardware uti-
lization with concise kernel code, matching or surpassing
hand-optimized baselines. For GEMM, Tawa achieves
up to 1.1× speedup over cuBLAS, and for attention
kernels, it attains 1.2× speedup over Triton and matches
the performance of the hand-optimized CUTLASS C++
FlashAttention-3 [11] kernel.
II. BACKGROUND AND RELATED WORK
In this section, we first discuss the modern GPU architecture
and briefly summarize the existing GPU programming models.
A. Modern NVIDIA GPU Architecture
As shown in Fig. 1, the Streaming Multiprocessor (SM)
forms the fundamental compute unit of modern GPUs, en-
capsulating both computation and memory resources. Each
SM integrates a large register file, a warp scheduler that
can issue instructions to one warp (32 threads) per clock
1A shorthand for Task-Aware Warp Specialization with Aref. The imple-
mentation has been upstreamed to OpenAI Triton: https://github.com/triton-
lang/triton/tree/aref auto ws
Fig. 1: A simplified NVIDIA H100 SXM5 GPU architecture.
cycle, and multiple CUDA cores that handle general-purpose
arithmetic. Additionally, SMs include specialized Tensor Cores
for accelerating matrix computations and a memory block
that serves as software-managed shared memory (SMEM). A
thread block, or Cooperative Thread Array (CTA), is scheduled
on an SM and shares its registers and SMEM. Recent GPU
generations, such as Hopper, have increased their reliance on
Tensor Cores as the dominant source of compute throughput,
where 90% of the total compute throughput is contributed by
Tensor Cores [6]. As such, achieving high performance on
modern GPUs is to fully saturate these Tensor Cores with
sufficient and timely data.
To this end, Hopper introduces two key hardware in-
novations aimed at improving Tensor Core utilization: the
Tensor Memory Accelerator (TMA) and Warp Group Ma-
trix Multiply-Accumulate (WGMMA). The TMA facili-
tates hardware-managed, multidimensional data transfers from
global memory (GMEM) to shared memory, offloading the
address generation and transfer coordination tasks from soft-
ware to a dedicated unit. To further enable this overlap,
Hopper extends the programming model with asynchronous
transaction barriers, which allow threads to synchronize not
just based on arrival but also on the volume of data exchanged.
Each transaction carries a byte count, and the barrier only
releases threads once both all producers have arrived and the
expected total transaction count is reached. This mechanism
is especially powerful in coordinating asynchronous memory
copies and collective data exchanges across thread blocks.
In parallel, WGMMA introduces warp-group-level matrix
multiply operations, allowing multiple warps (4 warps as one
warp group) to cooperatively execute large-tile GEMM kernels
in an asynchronous and efficient manner. Together, these
mechanisms provide a foundation for warp specialization,
where different warps within an SM assume distinct roles
in data communication and computation, thereby improv-
ing throughput for LLM workloads [10]–[12]. These trends
illustrate a broader architectural shift toward asynchronous
dataflow execution, a direction continued in subsequent ar-
chitectures such as Blackwell [7].
2
(a) Tawa compilation
1 @triton.jit
2 def matmul(a_desc, b_desc, c_ptr,
3
M, N, K,
4
stride_cm: tl.const, stride_cn: tl.const,
5
Mt: tl.const, Nt: tl.const, Kt: tl.const):
6
pid = tl.program_id(axis=0)
7
num_pid_m = tl.cdiv(M, Mt)
8
pid_m = pid % num_pid_m
9
pid_n = pid // num_pid_m
10
o_am = pid_m * Mt
11
o_bn = pid_n * Nt
12
o_k = 0
13
acc = tl.zeros((Mt, Nt), dtype=tl.float32)
14
for k in range(0, tl.cdiv(K, Kt)):
15
# Data loading
16
a = tl.tma_load(a_desc, [o_am, o_k], [Mt, Kt])
17
b = tl.tma_load(b_desc, [o_bn, o_k], [Nt, Kt])
18
# Computation
19
acc = tl.dot(a, b.T, acc=acc)
20
o_k += Kt
21
offs_cm = pid_m * Mt + tl.arange(0, Mt)
22
offs_cn = pid_n * Nt + tl.arange(0, Nt)
23
c_ptrs = c_ptr + stride_cm * offs_cm[:, None] \
24
+ stride_cn * offs_cn[None, :]
25
tl.store(c_ptrs, acc)
(b) A simplified Triton GEMM program
1 module attributes {"num-warps" = 8 : i32} {
2
tt.func public @matmul(
3
%arg0: !tt.ptr<i8>, %arg1: !tt.ptr<i8>,
4
%arg2: !tt.ptr<f16>, %arg3: i32,
5
%arg4: i32, %arg5: i32) {
6
%ABaref = tawa.create_aref : ()
7
-> tensor<2x!tawa.aref<tuple
8
<tensor<128x64xf16>, tensor<128x64xf16>>>
9
// ... more initialization
10
tawa.warp_group {
11
scf.for %k = %c0 to %K step %c1
12
iter_args(%arg8 = %c0) {
13
%ta = tt.tma_load %arg0[%offset_am, %arg8]
14
%tb = tt.tma_load %arg1[%offset_bn, %arg8]
15
%extracted = %ABaref[%k mod %cst_D]
16
tawa.put(%extracted, %ta, %tb)
17
%kI = arith.addi %arg8, %c64
18
scf.yield %kI : i32
19
}} {partition = 0 : i32}
20
tawa.warp_group {
21
%tc = scf.for %k = %c0 to %K step %c1
22
iter_args(%acc = %cst_0) {
23
%extracted = %ABaref[%k mod %cst_D]
24
%ta, %tb = tawa.get(%extracted)
25
%part_ret = tt.dot_wait %ta, %tb, %acc
26
%res = tt.dot_wait %part_ret {pendings = 1}
27
tawa.consumed(%extracted)
28
scf.yield %res : tensor<128x128xf32>
29
}
30
// epilogue: calculate C address and store
31
tt.store %c_ptr, %tc
32
} {partition = 1 : i32}
33 }}
(c) Highly simplified Tawa MLIR code (D = 2)
Fig. 2: Example Tawa compilation flow from the Triton frontend to the internal MLIR representation. tl is the shorthand for
the Triton language. tt is the Triton MLIR dialect. Code snippets are simplified for demonstration purposes.
B. Existing GPU Programming Models
CUDA is the dominant interface for GPU program-
ming [13], but as architectures evolve, programming complex-
ity has increased substantially. NVIDIA’s CUTLASS C++ [9]
library provides templated kernels for GEMM and related
computations, but leveraging it to write neural network op-
erators requires intricate configuration and hand-tuning. To
reduce engineering effort for domain workloads, specialized
libraries have emerged. ThunderKittens [10] introduces a
compact C++ interface based on 16×16 tile abstractions,
offering performance-competitive GEMM and attention ker-
nels. FlashInfer [12] focuses on efficient attention and KV-
cache management with JIT specialization to support large-
scale LLM inference. These systems simplify access to high-
performance kernels but remain library-centric: they provide
optimized operator templates rather than a general-purpose
strategy for different applications, which may cause extensive
maintenance issues for hand-tuning performance for different
sizes of matrices.
To further raise the abstraction level, domain-specific com-
piler infrastructures have been proposed. TVM [14] pioneered
an end-to-end compilation framework with automated schedul-
ing [15], [16], while Relax [17] extends it with a graph-level IR
for dynamic workloads. Fireiron [18] also leverages a schedule
language for specifying the tiling and data movement, and
Graphene [19] further enhances this idea by providing an
IR for mapping tensors with various layouts to Tensor Core
instructions. While powerful, these systems were developed
prior to Hopper and do not directly exploit TMA and hardware
warp specialization, leaving significant performance untapped.
Similarly, lots of efforts on GPU warp specialization
[20]–
[23] were proposed before the Hopper architecture, which do
not effectively use the latest TMA features.
Motivated by the need for finer control, a new class of
tile-based kernel languages has emerged. Triton [8], built on
MLIR [24], allows programmers to write efficient kernels
in Python while abstracting tiling and memory management.
Despite its success in democratizing kernel development,
Triton still lacks inherent abstractions for warp specialization,
leaving a performance gap on the Hopper architecture. Recent
work such as TileLang [25] and Mosaic [26] extends this
direction by introducing explicit APIs for dataflow scheduling
and pipeline composition. Cypress [27] proposes a task-based
programming model that abstracts complex data movement
and synchronization through logical computation descriptions
and mapping specifications. However, programmers are still
required to manually manage the multi-level hierarchy of
threads, warps, and SMs, along with the corresponding data
decomposition. Gluon [28] proposes an IR for Triton that adds
layout abstractions but still requires programmers to explicitly
incorporate communication operations. These approaches pro-
vide expert users with expressive control, but they continue
to impose significant responsibility for synchronization and
resource management, which limits programmer productivity.
To this end, we argue that users should not have to cope with
low-level hardware details. Instead, warp-specialized programs
3
should be generated automatically. Tawa bridges these gaps by
introducing aref as an IR-level abstraction for warp commu-
nication and automatically generating warp-specialized, TMA-
enabled pipelines, thereby achieving high performance without
imposing low-level programming burden.
III. TAWA COMPILER
In this section, we first give an overview of the Tawa
compiler and discuss details of the asynchronous reference
abstraction and the compilation process.
A. Overview
Fig. 2a illustrates the overall design of Tawa. Since Tri-
ton [8] has become the de facto backend for PyTorch [29],
[30] and is widely adopted for writing efficient GPU kernels,
we select Triton as the frontend interface of our compiler.
Programmers write kernels in Triton-Python with tiled compu-
tation and TMA communication in a CTA (Fig. 2b), which are
first translated into the standard Triton-MLIR representation.
Building upon this infrastructure, Tawa introduces a series
of compiler passes that automate the generation of warp-
specialized programs directly from high-level Python code.
To address the challenges outlined in § I, Tawa applies three
key transformations: (1) Task-aware partitioning (§ III-C).
We introduce a novel algorithm that partitions the program
into producer and consumer warp groups, ensuring correct role
assignment and communication boundaries, directly address-
ing Challenge 1. (2) Asynchronous reference abstraction
(§ III-B). We design aref, an intermediate representation that
captures inter-warp dataflow explicitly. By expressing com-
munication intent at the IR level, aref allows the compiler
to automatically insert and coordinate the necessary synchro-
nization and data-movement operations, resolving Challenge
2. We also propose a new Tawa MLIR dialect to encode those
aref operations. (3) Multi-granularity software pipelining
(§ III-D). On top of the aref program, we apply a software
pipelining pass that orchestrates overlap between communica-
tion and computation across multiple granularities, resolving
Challenge 3.
The resulting program is shown in Fig. 2c. Once these
passes are applied, the aref representation is lowered to
the standard Triton-MLIR dialect, after which the existing
Triton compilation pipeline proceeds unchanged: the code is
progressively lowered to LLVM IR and finally to PTX, ready
for execution on NVIDIA GPUs.
B. Asynchronous Reference
Fig. 3: aref abstraction and associated operations.
PUT
σ(a).E = 1
⟨σ, put(a, v)⟩→⟨σ[ a 7→⟨buf = v, F = 1, E = 0⟩], ϵ⟩
GET
σ(a).F = 1
⟨σ, get(a)⟩→⟨σ[ a 7→⟨buf = σ(a).buf, F = 0, E = 0⟩], σ(a).buf⟩
CONSUMED
⟨σ, consumed(a)⟩→⟨σ[ a 7→⟨buf = σ(a).buf, F = 0, E = 1⟩], ϵ⟩
Fig. 4: Operational semantics of the aref operations. σ is the
store map from an aref identifier a to the actual structure
σ(a) = ⟨buf, F, E⟩, where F is the full mbarrier flag and
E is the empty mbarrier flag. Initially E = 1, F = 0.
To better model communication between warps, we intro-
duce asynchronous reference [31], or aref, an IR abstraction
that models a one-slot channel between producer and con-
sumer on the GPU as shown in Fig. 3. It packages a data buffer
together with two synchronization primitives implemented by
hardware mbarriers, conventionally named empty and full.
At any moment exactly one of these barriers encodes the state
of the slot: when the empty mbarrier holds a credit, the slot
may be written by the producer; when the full mbarrier
holds a credit, the slot contains a published value ready to be
read by the consumer.
The aref interface exposes three operations: put, get,
and consumed, whose formal semantics are defined in Fig. 4.
These semantics provide a rigorous foundation for lowering
the operations to GPU-specific instructions, as discussed in
§ III-E. put is the producer’s publication step: it requires
the slot to be empty (the empty mbarrier E holds a credit),
writes the payload to the buffer, and flips the state to full
by arriving on the full mbarrier F with release ordering.
This transition makes the data visible to the consumer and
prevents subsequent writes from reusing the slot prematurely.
get is the consumer’s acquisition step: it requires the slot
to be full (the full mbarrier F is observed), performs a
read of the buffer into a program variable, and transitions
the aref into a borrowed state in which neither E nor F
holds a credit, reflecting that the value is in use but the slot
is not yet reusable. consumed closes the handshake: once
the consumer no longer needs the value, it arrives on the
empty mbarrier E, restoring the empty credit and enabling
the next put; this induces the happens-before chain from
producer’s writes to consumer’s reads and on to the producer’s
subsequent reuse. The aref therefore serves as a first-class
IR value representing a concrete communication path that
enforces correct ordering of data movement and computation
across warp groups.
Compared to constructs such as the OpenCL pipe [32] or
similar mechanisms in other accelerator programming mod-
els [33]–[37], the aref abstraction is specifically designed
for GPU inter-warp communication. Its semantics are rooted
4
(a) Computation graph of Fig. 2b.
(b) Partitioned computation graph.
(c) Achieved warp specialization. Suppose D = 2.
Fig. 5: Task-aware partition for the GEMM kernel in Fig. 2b. Notice warp specialization is just one way of achieving the
overlapping in (c), and the actual latency is not shown to scale.
in hardware-supported mbarrier synchronization, providing
a clear ordering model. In addition to its synchronization
semantics, aref is also type-generic, supporting buffers of
type <T>, which allows it to carry structured payloads such as
tensors or tuples while reusing the same mbarriers. To facilitate
deep pipelining, multiple aref instances can be grouped into
a cyclic buffer of depth D, effectively forming a ring structure
that supports the storage and transfer of intermediate data
across pipeline stages.
C. Task-Aware Partitioning
We begin by traversing the MLIR computation graph as in
Fig. 5a, where each node is an MLIR operation and each edge
captures the use-def dependencies. We take two steps of the
partitioning process: partition annotation and loop distribution.
1) Partition Annotation
We perform a backward traversal along the use-def chains
starting at the kernel’s side-effecting sinks (e.g., Store).
During the traversal, we attach a semantic tag to every node
by inspecting its effects. Nodes that contribute to address
computation, such as the pointer arithmetic for data transfers,
are marked as iteration statements (orange edges). Nodes that
transform or consume a tile for actual computation, such
as WGMMA, are marked as tile statements (blue edges).
Importantly, iteration statements are not necessarily contiguous
in the IR, as highlighted in Fig. 2b.
A typical division on Hopper is to create a load WG as
the producer and a compute WG as the consumer, as shown
in Fig. 4. Tawa annotates the iteration statements and their
dominated TMALoads in the producer partition (WG0), and
the remaining tile statements in the consumer partition (WG1).
Notice the epilogue is also attached to WG1 to make sure it
is actually executed by some WGs. For Blackwell and later
generations of GPUs that support more WGs, Tawa can also
partition the IR based on available warp roles on GPUs by
annotating the semantic roles during graph traversal.
2) Loop Distribution
After the operations are annotated, we need to construct
the actual IR that divides the workload for each WG. For
each cross-partition edge (e.g., TMALoad→LocalAlloc in
Fig. 5a), we create an aref tensor with size D, denoting a D-
slot cyclic buffer. A put operation is created on the producer
side that places each TMALoad result into an aref slot. The
slot index is computed as the iteration k modulo the buffer size
D, allowing the buffer slots to be reused (L15, L23 in Fig. 2c).
Similarly, a get operation is created on the consumer side
that gets the data from the same aref slot. After the data are
used for computation in WGMMA, an additional consumed
operation is inserted at the end to indicate the slot has been
freed. As mentioned in § III-B, aref supports grouping
multiple values in the buffer to share the mbarrier. Therefore, a
straightforward optimization when creating aref is to check
whether the loaded tensors are used in the same WGMMA.
If so, we can put a tuple of tensors in the aref buffer
and share the channel for communication between WGs. The
resulting computation graph is shown in Fig. 5b. Since aref
has already carried buffers, the LocalAlloc operations can
also be eliminated but to directly read from the aref buffer.
To make both partitions independently progress through the
original loop nest, we also need to perform loop distribution
around the cut. Two WG regions are created in the IR as
shown in Fig. 2c (L10-32). The scf.for loop is cloned so
that WG0 and WG1 each carry an isomorphic loop over k.
Iteration statements remain in WG0, and tile statements remain
in WG1. The inter-warp communication is done by aref
highlighted in the code (L16, L24). Finally, the computed
result is yielded from the for loop and stored back to global
memory in the epilogue. The resulting execution is the warp-
specialized timeline in Fig. 5c, which shows how TMA loads
are overlapped with Tensor Core computation.
D. Multi-Granularity Pipelining
After we have the warp-specialized program, we conduct
further optimization on the compute warp group, where we in-
5
troduce two pipelining mechanisms to make sure the hardware
components are working in parallel. Specifically, we consider
a fine-grained pipeline that overlaps MMA address calculation
and computation, and a coarse-grained pipeline that overlaps
CUDA and Tensor Core computation.
1) Fine-Grained Pipeline
Fig. 6: Fine-grained pipeline overlapping MMA address cal-
culation and computation. Actual latency not shown to scale.
Modern GPUs expose both CUDA Cores, which handle
scalar and address computations, and Tensor Cores, which
accelerate MMA operations; however, a na¨ıve sequential ex-
ecution forces these units to operate in sequence, such that
address generation stalls tensor execution and vice versa,
leading to poor utilization. To alleviate this inefficiency, we
introduce an automatic pipelining algorithm that overlaps
address calculation on CUDA Cores with MMA execution
on Tensor Cores through a bounded pipeline of depth P. At
each iteration, the compiler issues the next MMA instruction
asynchronously (WGMMA.issue(k)), and only stalls when
the number of outstanding operations reaches the maximum
pipeline depth (WGMMA.wait() {pendings=P}). As the
k-th wait operation does not need to be done immedi-
ately after each MMA issue, enabling more MMA instruc-
tions to be in flight before waiting. Once the pipeline is
filled, the result reference from iteration k −P is released
(aref.consumed(k-P)), which maintains correctness by
enforcing dataflow dependencies and freeing resources for
reuse. After the final iteration, the compiler drains the pipeline
in an epilogue by consuming the remaining pending opera-
tions. This design effectively decouples address computation
and matrix multiplication, sustaining a continuous flow of
work across both functional units, and effectively hides latency
and reduces idle time compared to the sequential baseline.
2) Coarse-Grained Pipeline
The fine-grained pipeline is most effective when the loop
primarily consists of matrix multiplication operations. How-
ever, when additional computations are performed on CUDA
Cores, these operations can also be overlapped with MMA.
Therefore, we further propose a coarse-grained three-stage
producer-transform-(optional) consumer assembly line in Al-
gorithm 1. Each loop iteration j has a Tensor Core stage
Tj that produces an intermediate tile (e.g., a matmul frag-
ment), a CUDA Core stage Cj that transforms that inter-
mediate (e.g., normalization, activation, reduction), and an
optional downstream Tensor Core stage Uj that consumes
Algorithm 1: Coarse-Grained CUDA and Tensor Core
Pipelining
Input: Tile count N; Boolean USE_U
1 Prologue;
2 MAYBEAREFGET(T0);
3 ISSUEANDCOMMIT(T0);
4 DOTWAIT(T0) ;
// materialize T0 output
5 MAYBEAREFCONSUMED(T0);
6 COMPUTE(C0);
7 Steady state;
8 for j ←1 to N −1 do
9
MAYBEAREFGET(Tj); ISSUEANDCOMMIT(Tj);
10
if USE_U then
11
MAYBEAREFGET(Uj−1);
12
ISSUEANDCOMMIT(Uj−1);
13
DOTWAIT(Tj−1) ; // ensure Tj−1 results are
visible
14
MAYBEAREFCONSUMED(Tj−1); COMPUTE(Cj−1);
15
if USE_U then
16
DOTWAIT(Uj−1);
17
MAYBEAREFCONSUMED(Uj−1)
18 Epilogue;
19 DOTWAIT(TN−1);
20 MAYBEAREFCONSUMED(TN−1);
21 COMPUTE(CN−1);
22 if USE_U then
23
MAYBEAREFGET(UN−1);
24
ISSUEANDCOMMIT(UN−1);
25
DOTWAIT(UN−1);
26
MAYBEAREFCONSUMED(UN−1)
27 FINALEPILOGUEANDSTORE();
the transformed result (e.g., a second matmul/epilogue). The
prologue boots the pipeline by running T0 to completion
and immediately computing C0. Steady-state then overlaps Tj
with Cj−1 (and, if present, Uj−1): in each iteration we first
enqueue and commit the Tensor Core work for the current
tile, optionally enqueue and commit the downstream U for
the previous tile, then perform a precise completion wait on
the dot-product group(s) that produced Tj−1’s results and run
the CUDA Core transform Cj−1. The epilogue drains the
last CUDA Core transform and, if U is used, issues and
retires UN−1. Correctness depends on two synchronization
mechanisms: DOTWAIT(Tj −1) (and DOTWAIT(Uj −1))
ensures compute-group completion at Tensor Core boundaries,
while MAYBEAREFGET(·) are wrappers that perform the get
operation only when a stage actually consumes a cross-WG
aref object. When no external data needs to be read, the
wrappers compile to no-ops, keeping the template minimal
and avoiding unnecessary stalls.
From an MLIR computation graph, the schedule is con-
structed by several principled analyses followed by a me-
chanical synthesis step. Stage identification partitions the per-
iteration subgraph into Tj, Cj, and (optionally) Uj using
dialect- and type-level cues: Tensor Core micro-tiles and their
glue form T (and possibly U if a second tensor-core phase ex-
6
ists), whereas arithmetic, normalization, activation, reduction,
or layout transformation that reads T’s outputs forms C. This
yields a tiny intra-iteration DAG with edges Tj →Cj and, if
present, Cj →Uj. An example of attention [38] is that the
QKT forms T, the softmax operation forms C, and the second
GEMM PV constructs U. Finally, an aref-use inspection
marks any stage that reads data produced by a different warp-
group; only for those marked stages will the wrappers MAY-
BEAREFGET(·) and MAYBEAREFCONSUMED(·) be emitted.
E. aref Lowering
The lowering of aref transforms high-level asynchronous
references into explicit synchronization and memory-transfer
instructions that execute directly on the GPU. At the IR level,
create_aref declares a buffer and allocates the necessary
mbarriers, serving as the root of a communication subgraph.
The compiler then rewrites put, get, and consumed oper-
ations into concrete instructions: put expands into an asyn-
chronous TMA load guarded by empty/full barriers, get
becomes a blocking wait on the corresponding full barrier,
and consumed signals buffer availability by arriving at the
empty barrier. This pattern-matching ensures that once the
entire subgraph is rewritten, the abstract aref nodes can be
safely erased, leaving only executable low-level synchroniza-
tion and data-movement instructions.
A critical aspect of this transformation is correctness under
pipelining, where multiple tiles are loaded, computed, and
consumed in flight. To avoid deadlock from circular waits,
the lowering introduces a parity mechanism: each operation
alternates between two sets of barriers indexed by iteration
parity. When parity toggles, a consumer may skip waiting if
data has already been produced, and producers can reuse buffer
slots without overwriting values still in use. This mechanism
not only guarantees liveness but also enables efficient multi-
buffering, allowing communication and computation to overlap
across multiple stages. In this way, aref lowering provides
a systematic bridge between an abstract, declarative view of
asynchronous dataflow and the precise TMA and mbarrier
instructions required for high-performance GPU execution.
IV. FURTHER OPTIMIZATIONS
To further increase the Tensor Core utilization in warp-
specialized scenarios, Tawa further integrates additional op-
timizations that are incorporated into the compilation flow,
ensuring that programs automatically benefit without requiring
manual restructuring.
A. Cooperative Compute Warp Groups
A key limitation of warp-specialized execution is the re-
stricted register budget available to each warp group, which
constrains the maximum tile size and can limit overall compute
intensity. The cooperative warp group optimization alleviates
this bottleneck by enabling multiple warp groups to collabo-
ratively compute the same tile. As illustrated in Fig. 7a, two
consumer warp groups (WG1 and WG2) jointly consume data
Fig. 7: (a) Cooperative warp groups; (b) Persistent kernel.
produced by a TMA warp group (WG0). By pooling their reg-
isters, the cooperating warps can form larger tiles that increase
arithmetic intensity, improve data reuse, and reduce memory
traffic. Conceptually, this optimization reintroduces a form of
standard warp parallelism that multiple warps performing the
same computation on different fragments of the same tile, into
the warp-specialized setting. Tawa handles this transparently
by informing the backend code generator of the cooperative
mapping so that thread indices are consistently assigned.
Importantly, no changes are needed to the aref abstraction,
since communication semantics remain unchanged.
B. Persistent Kernels
Another major source of inefficiency in GPU pipelines
stems from repeated kernel launches, each introduces signifi-
cant CTA launch overhead. Persistent kernels reduce this over-
head by launching only as many CTAs as there are SMs and
keeping them resident throughout execution. As depicted in
Fig. 7b, Tawa transforms the execution model into a software-
managed pipeline, where resident CTAs are responsible for
iterating over tiles. This enables overlapping of TMA-driven
loads, Tensor Core computation, and CUDA Core epilogues
across iterations, eliminating relaunch latency and reducing
idle time. Tawa automatically inserts synchronization before
each epilogue to preserve correctness. This optimization is
especially effective in deep pipelines with large K dimensions,
where cumulative launch costs would otherwise dominate.
V. EVALUATION
We first describe our experimental setting and then show
the results on different benchmarks in this section.
A. Experimental Setup
Tawa is built on top of Triton [8] (commit: 0c7edf)
with about 4K lines of C++ and introduces a compact MLIR
dialect that encapsulates aref operations. We compare Tawa
against a set of state-of-the-art GPU programming frameworks
and libraries, including the closed-source commercial library
cuBLAS [39] (v12.7), the NVIDIA open-source CUDA linear
algebra library CUTLASS [9] (v4.0.0), and the baseline Tri-
ton under the same commit. We also include two academic
frameworks: ThunderKittens [10] (6c27e2), a CUDA-based
AI kernel library, and TileLang [25] (bcfc83), a TVM [14]-
based DSL. To ensure a fair comparison, all baselines are
hand-tuned with a fixed set of tile sizes (combinations of
7
256
512
1024
2048
4096
8192
16384
0
200
400
600
800
1000
1200
FP16
256
512
1024
2048
4096
8192
16384
0
500
1000
1500
2000
FP8
GEMM K size
Throughput (TFLOPs/s)
Theoretical Peak
cuBLAS
Tawa
Triton
TileLang
ThunderKittens
Fig. 8: GEMM performance results for different frameworks. M = N = 8192, and varied K.
{64, 128, 256}). In the default Triton implementation, the
official software pipelining is enabled, with pipeline depth
predetermined to balance overlap between communication and
computation. For kernels generated by Tawa, the size of the
aref and the depth of the MMA pipeline are selected manu-
ally to maximize performance, providing each approach with
equivalent opportunity to exploit the underlying hardware.
All experiments are conducted on an NVIDIA H100 SXM5
GPU equipped with 80GB of HBM3e memory, using CUDA
12.7 as the software stack. Experiments are conducted using
25 warp-up and 1000 measurement runs, and we compute
the average execution time from these. We benchmark several
representative workloads. The first is GEMM and its variants,
evaluated with both FP16 and FP8 precision. The second is
an attention kernel, with various sequence lengths to reflect
realistic LLM workloads. These benchmarks represent diverse
computational patterns and provide a comprehensive assess-
ment of the effectiveness of automatic warp specialization rel-
ative to hand-optimized libraries and DSL-based approaches.
B. Matrix Multiplication
We evaluate GEMM kernels with matrix dimensions of
M = N = 8192 while sweeping the inner dimension K
from 256 to 16384, and report the throughput for FP16
and FP8. As shown in Fig. 8, across both precisions, Tawa
achieves the same level of performance as the highly optimized
kernel library cuBLAS while outperforming the other general-
purpose frameworks on most shapes. Tawa reaches up to 79%
hardware utilization in both FP16 and FP8.
In FP16, Tawa delivers average speedups of 1.01×, 1.13×,
1.23×, and 1.09× over cuBLAS, Triton, TileLang, and Thun-
derKittens, respectively. The improvements over Triton high-
light the advantage of overlapping TMA-driven data move-
ment with WGMMA execution. In contrast, Triton employs
an Ampere-style software pipelining scheme for asynchronous
copies, which is less effective on Hopper than the hardware-
backed warp-specialization strategy in Tawa. TileLang and
ThunderKittens are extensively tuned for large K, leading to
stronger performance than Tawa when K ≥8192.
In FP8, the benefits of automatic warp specialization are
more pronounced, particularly at small K. On average, Tawa
achieves 1.06×, 1.02×, 2.40×, and 1.24× speedup over
1K
2K
4K
8K
16K
GEMM size M = N = K
0
200
400
600
800
Batched GEMM
2
4
6
8
# of groups G
Grouped GEMM
Throughput (TFLOPs/s)
Tawa
Triton
Fig. 9: FP16 batched GEMM and grouped GEMM results.
cuBLAS, Triton, TileLang, and ThunderKittens, respectively.
TileLang encounters layout-management challenges for FP8
WGMMA, yielding an inferior implementation that lags other
frameworks; at K = 256, Tawa is 3.99× faster than TileLang.
ThunderKittens appears less carefully tuned for FP8, resulting
in up to 1.61× lower throughput at small K. Overall, these
results indicate that Tawa’s aref abstraction and automatic
warp specialization provide a robust orchestration mechanism
that overlaps TMA loads and WGMMA across precisions.
C. GEMM Variants
To assess the generalizability of Tawa across diverse kernel
patterns, we evaluate two GEMM variants implemented in
Triton. The first is batched GEMM, which executes multiple
small GEMMs of identical shape within a single kernel. The
second is grouped GEMM, which processes multiple GEMMs
of varying shapes in one kernel. Both patterns are pervasive
in Mixture of Experts models [40]–[42]. For batched GEMM,
we fix the batch size to 8 and sweep M, N, and K across
a range of values. For grouped GEMM, we vary the number
and shapes of GEMMs. As shown in Fig. 9, Tawa consistently
outperforms Triton baselines in both cases, achieving speedups
of up to 7%. The improvements arise from Tawa’s aref-
based partitioning and automatic warp specialization, which
allow data movement for one GEMM to be overlapped with
computation from another. In batched GEMM, this overlap
reduces idle time between consecutive small multiplications,
8
1024
2048
4096
8192
16384
0
200
400
600
800
FP16, causal=false
1024
2048
4096
8192
16384
0
200
400
600
800
FP16, causal=true
1024
2048
4096
8192
16384
0
250
500
750
1000
FP8, causal=false
1024
2048
4096
8192
16384
0
250
500
750
1000
FP8, causal=true
Context length
Throughput (TFLOPs/s)
FA3 (CUTLASS)
Tawa
Triton
TileLang
ThunderKittens
Fig. 10: FP16 and FP8 MHA performance results for different frameworks.
1
2
3
MMA Depth (P)
1
2
3
Aref Size (D)
159
0
0
567
437
0
657
652
433
Non-Persistent GEMM
1
2
3
MMA Depth (P)
1
2
3
Aref Size (D)
167
0
0
610
519
0
704
720
513
Persistent GEMM
0
200
400
600
Performance (TFLOPs/sec)
Fig. 11: Impact of different aref and MMA pipeline sizes.
while in grouped GEMM, it schedules heterogeneous shapes
more efficiently within a single kernel.
D. Multi-Head Attention
We evaluate (non-)causal multi-head attention (MHA)
across sequence lengths L ∈[1024, 16384] with batch size
4 and head dimension 128. As shown in Fig. 10, Tawa is
compared against strong reference implementations, including
handwritten FlashAttention-3 (FA3) [11] built on CUTLASS
C++, as well as Triton, TileLang, and ThunderKittens.
For FP16, Tawa attains up to 96% of FA3 throughput and
delivers a consistent 1.21× speedup over Triton. This gap
largely stems from the Triton baseline being effectively a
FlashAttention-2 style implementation [43], which does not
fully exploit Hopper warp specialized producer and consumer
pipelines, in particular TMA driven global to shared trans-
fers overlapped with WGMMA on Tensor Cores. At short
sequences, the advantage of warp specialization is muted
because prologue, epilogue, and barrier costs are not yet
amortized. As L grows to at least 4K, memory traffic and
synchronization become dominant; in this regime Tawa out-
performs TileLang and ThunderKittens by up to 1.13× and
1.23×, respectively, indicating more effective overlap between
data movement and compute. Although causal MHA reduces
effective Tensor Core utilization due to mask induced control
and data hazards, Tawa sustains strong performance at long
sequences and reaches its peak at L = 16K.
For FP8, the benefits of automatic warp specialization are
even more pronounced at large L. Tawa reaches up to 89% of
FA3 and achieves 1.11× speedup over Triton. TileLang and
ThunderKittens failed to execute our FP8 attention configu-
rations, suggesting that their kernels are tuned primarily for
FP16 and lack the layout management and pipeline scheduling
needed to handle FP8 WGMMA paths robustly.
Overall, the results support two conclusions. First, aref
provides a principled abstraction that converts generic kernels
without asynchronous semantics into dataflow pipelines: by
treating tiles as references with explicit get and put phases,
Tawa overlaps TMA transfers, shared-memory materialization,
and WGMMA while maintaining warp-group occupancy and
steady MMA throughput. Second, these gains hold across
precision from FP16 to FP8 and semantics from noncausal
to causal, showing that Tawa automatic warp specialization
policies generalize well.
E. Hyperparameter Selection
To study the impact of hyperparameters in our compilation
flow, we conduct experiments on the aref size D, the number
of disjoint shared-memory staging buffers governed by asyn-
chronous barriers, and the MMA depth P, how many compute
tiles (WGMMA issue groups) are overlapped in flight, with
(non-)persistent GEMM kernels (K = 16384). As shown in
Fig. 11, across the feasible region (D ≥P), performance
consistently increases when increasing D. A larger D lets
TMA producers prefetch more, smoothing latency variation
and better hiding global-to-shared transfer under compute.
9
Triton w/o WS
+Auto WS
+Cooperative WGs
+Large Tile Size
+Persistent Kernel
+Better Aref Size
0
200
400
600
800
Throughput (TFLOPs/s)
104
393 395
572
632
718
GEMM
Triton w/o WS
+Auto WS
+Cooperative WGs
+Pipeline
+Better Aref Size
0
200
400
600
800
209
232
593
645
654
MHA
Fig. 12: Impact of different optimizations on FP16 kernels.
Further increasing D requires more shared memory to be
allocated, which is not achievable for large tile size.
Varying P at fixed D shows the classic over-pipelining
trade-off. Increasing P from 1 to 2 generally helps (persistent
peaks at D = 3, P = 2), because MMA reduces stalls on
accumulator availability and amortizes instruction-level laten-
cies. Pushing to P = 3 lowers throughput: deeper compute
pipelines increase live fragments and accumulator registers and
can depress WG occupancy, so the marginal latency hiding is
outweighed by pressure on registers and shared memory.
Comparing the two panels, the persistent version is consis-
tently faster (roughly +5-10% on most feasible points). Persis-
tence keeps CTAs resident to pull multiple tiles from a global
work queue, which eliminates grid-scheduling and tail effects,
improves cache locality for weights and epilogue metadata,
and maintains a steady-state TMA and MMA rhythm with
fewer disruptions. Taken together, the figure suggests using a
larger D and moderate P (1-2) to balance overlap and resource
pressure, and uses persistent kernels to stabilize the pipeline
and harvest cache/reuse benefits.
F. Ablation Study
To further study the effect of different optimization tech-
niques in Tawa, we conduct an ablation study on both GEMM
and MHA kernels. We use the largest GEMM (K = 16384)
and MHA kernels (L = 16384) in our previous experiments,
and the results are shown in Fig. 12. The baseline Triton
without adding warp specialization (WS) delivers a throughput
of only 104 TFLOPs/s. By enabling WS with a single warp
group while maintaining the same tiling configuration (+Auto
WS), performance improves 3.78× to 393 TFLOPs/s. This
substantial jump reflects the efficiency gains from decoupling
producer and consumer roles at the warp level, which improves
parallelism and mitigates pipeline stalls.
The benefits of warp specialization become even more
evident as the configuration is extended to two compute warp
groups. With the same tile size of 128×128×64, throughput
does not change a lot. However, the increased number of warps
reduces the register pressure and can enable a larger tile size
for the warps to cooperatively work on. When enlarging the tile
size to 128×256×64 (+Large Tile Size), it boosts the perfor-
mance by 1.46×. The enlarged tile shape improves data reuse
and reduces redundant memory traffic, showing that tiling
interacts synergistically with warp specialization to exploit
the hardware more effectively. Similar trends can be found in
MHA, where the automatic warp specialization (+Auto WS)
and cooperative warp groups (+Cooperative WGs) together
boost the performance by 2.84×.
Beyond warp specialization and tiling, the introduction
of persistent kernels further amplifies performance. In this
mode, thread blocks are retained on SMs across multiple
iterations, thereby avoiding costly kernel launch overheads and
improving data locality. With persistence enabled (+Persistent
Kernel), the system further improves the performance by
10%. Finally, fine-tuning the scheduling depth parameter in
conjunction with persistence leads to a peak throughput of 718
TFLOPs/s (+Better Aref Size), representing nearly a seven-
fold improvement over the baseline. For MHA, adding a
coarse-grained pipeline and fine-tuning the aref size leads to
the final performance of 654 TFLOPs/s. This progression high-
lights how Tawa carefully incorporates warp specialization,
cooperative warp groups, and pipelining, which can unlock
better utilization of modern GPU hardware.
VI. FUTURE WORK
One natural extension of Tawa is to broaden the set
of communication patterns supported by the compiler. Cur-
rently, Tawa primarily targets producer-consumer pipelines
built around double-buffering, but more advanced schemes
such as ping-pong kernels and multicast communication could
further improve efficiency. Ping-pong kernels allow alternating
roles between warp groups across iterations, thereby balancing
workload and alleviating bottlenecks in cases where computa-
tion and data movement demands shift dynamically. Similarly,
multicasting enables a single producer warp to distribute
data to multiple consumer warps simultaneously, reducing
redundant memory transfers and better utilizing shared buffers.
Looking further ahead, we aim to generalize Tawa be-
yond Hopper to support next-generation architectures such as
Blackwell. In particular, Blackwell introduces tensor memory
(tmem) as a new hardware-managed memory hierarchy that
complements TMA by providing more flexible data orches-
tration between shared and global memory. Leveraging tmem
effectively will require extending aref to model multi-level
data movement and revisiting scheduling decisions across
heterogeneous memory resources. In addition, future work-
loads demand even greater scalability, motivating support for
multiple producers and consumers for multiple warp groups,
as well as more sophisticated graph partitioning algorithms
that account for load balancing, register pressure, and memory
footprint simultaneously. By incorporating these extensions,
Tawa could evolve into a general-purpose compiler framework
for warp-specialized, asynchronous GPU programming across
successive hardware generations.
10
VII. CONCLUSION
In this paper, we presented Tawa, an automated compiler
that generates efficient warp-specialized kernels for modern
GPUs through the introduction of the asynchronous reference.
Our evaluation demonstrates that Tawa attains performance
comparable to highly optimized handwritten libraries across
a range of benchmarks, while significantly reducing program-
ming effort. We believe this work offers a step toward prin-
cipled compiler support for asynchronous GPU programming
and can inspire future research on automatically generating
high-performance kernels for next-generation architectures.
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Tawa: Automatic Warp Specialization for Modern GPUs with Asynchronous References Hongzheng Chen∗†, Bin Fan, Alexander Collins, Bastian Hagedorn, Evghenii Gaburov, Masahiro Masuda, Matthew Brookhart, Chris Sullivan, Jason Knight, Zhiru Zhang†, Vinod Grover NVIDIA †Cornell University , Abstract-Modern GPUs feature specialized hardware units that enable high-performance, asynchronous dataflow execution. However, the conventional SIMT programming model is fundamentally misaligned with this task-parallel hardware, creating a significant programmability gap. While hardware-level warp specialization is the key to unlocking peak performance, it forces developers to manually orchestrate complex, low-level communication and software pipelines-a process that is laborintensive, error-prone, and unsustainable. To address this challenge, we present Tawa, an automated compiler that systematically generates high-performance, warp-specialized code from a high-level, tile-based program. Central to our approach is a novel IR abstraction, asynchronous references (aref), which expresses warp-level communication without exposing low-level hardware details. Using this abstraction, Tawa automatically partitions programs into producer-consumer roles and manages the intricate dataflow pipeline, relieving developers of invasive kernel rewriting. Evaluation on NVIDIA H100 GPUs across representative LLM kernels shows that Tawa delivers high hardware utilization, achieving up to 1.1× speedup over highly optimized cuBLAS GEMM kernels. For attention workloads, Tawa attains 1.2× speedup over Triton and matches the performance of the hand-optimized CUTLASS C++ FlashAttention-3 kernel with far less programming effort. I. INTRODUCTION Modern GPU architectures have evolved from homogeneous processors into complex heterogeneous systems, integrating multiple specialized hardware units to maximize performance. Alongside traditional CUDA cores, modern GPUs integrate dedicated engines such as Tensor Cores for dense matrix computations and the Tensor Memory Accelerator (TMA) for asynchronous data movement. This architectural design enables a fine-grained, asynchronous dataflow execution model where data movement can be explicitly orchestrated and overlapped with computation. Exploiting this capability is critical for latency-sensitive workloads such as Large Language Models (LLMs), whose efficiency is often limited more by data orchestration than by raw FLOPs [1]-[5]. However, the prevailing programming paradigm for GPUs, Single-Instruction Multiple-Thread (SIMT), which underpins CUDA, is fundamentally misaligned with this heterogeneous, task-parallel reality. The SIMT model assumes that all threads within a group (a warp) execute the same instructions on different data, excelling at problems with uniform control flow. This model is ill-suited for managing diverse hardware ∗Work partially done while interning at NVIDIA. units that must be driven by different, concurrent tasks. For example, one task might involve generating addresses and initiating a bulk copy with the TMA, while another performs a matrix multiplication on the Tensor Cores. This creates a paradigm mismatch: the hardware is capable of asynchronous, task-parallel execution, but the software model is built for synchronous, data-parallel execution. To resolve this conflict, modern architectures like NVIDIA's Hopper and Blackwell GPUs introduce hardware-level warp specialization [6], [7]. This mechanism reflects the hardware's task-parallel nature by allowing different groups of threads (warps) to take on distinct roles. For instance, producer warps can be dedicated to managing data transfers via the TMA, feeding data to consumer warps that execute computations on the Tensor Cores. While this new model unlocks the hardware's potential, programming for it remains difficult because the surrounding software ecosystem and abstractions are still rooted in the SIMT paradigm. This programmability gap makes developing high-performance kernels a laborintensive endeavor, often requiring over a thousand lines of hand-optimized code even for a basic GEMM kernel. This difficulty manifests in three primary challenges: Challenge 1: Coordinating concurrent warp roles. Because the SIMT model lacks first-class abstractions for assigning heterogeneous roles, developers must manually restructure kernels, encapsulating role-specific logic within complex conditional branches. They must then orchestrate the execution of these specialized warps to ensure that producer and consumer tasks remain correctly synchronized and efficiently overlapped. This manual process is inherently fragile. Even modern, highlevel domain-specific languages (DSLs) like Triton [8] retain SIMT-centric abstractions, which preclude the explicit assignment of warp roles and thus cannot naturally express warpspecialized programs. Challenge 2: Low-level communication management. CUDA does not currently expose C++-level interfaces for TMA and transaction barrier (mbarriers) operations. Consequently, programmers must rely on inline PTX to configure TMA descriptors, initialize and manipulate mbarriers, and orchestrate synchronization. This requires reasoning explicitly about arrival counts, barrier phases, and shared-memory aliasing, where subtle errors can easily lead to deadlock. The difficulty is exacerbated when multiple mbarriers and input parameters are involved. Such low-level orchestration 16 Oct 2025 undermines code readability and substantially complicates debugging and maintenance. High-level libraries like CUTLASS [9] and ThunderKittens [10] offer pre-written kernels but still require users to carefully compose them to satisfy the underlying communication requirements. Challenge 3: Resource allocation and pipeline orchestration. Effectively overlapping the work of specialized producer and consumer warps necessitates the manual implementation of deep software pipelines. Developers must explicitly manage multi-buffering schemes in shared memory, partition kernels into prologue, main body, and epilogue stages, and carefully balance the allocation of hardware resources like registers and shared memory to respect occupancy limits. These design choices are tightly coupled and create a vast, complex optimization space. For example, increasing the pipeline depth affects shared memory capacity, which in turn constrains thread block size and impacts TMA descriptor layouts. Navigating this space requires repeated, labor-intensive kernel refactoring, making performance tuning both difficult and unsustainable. To address these challenges, we propose Tawa1, an automated compiler that systematically analyzes computation graphs generated from a tile-based DSL, partitions programs into warp-specialized components, and manages both communication and pipelining. Our contributions are as follows: • We propose asynchronous references (aref), an abstraction for warp-level communication. aref expresses asynchronous data transfer intent without exposing lowlevel hardware primitives, providing a clean and portable representation at the IR level. • We present Tawa, an automatic compilation flow for warp specialization on NVIDIA GPUs. Tawa takes in a Triton program, performs task-aware partitioning across warp groups, applies multi-granularity pipelining, and generates high-performance PTX code for GPU execution. • We evaluate Tawa on an H100 GPU using representative LLM kernels. Our approach delivers high hardware utilization with concise kernel code, matching or surpassing hand-optimized baselines. For GEMM, Tawa achieves up to 1.1× speedup over cuBLAS, and for attention kernels, it attains 1.2× speedup over Triton and matches the performance of the hand-optimized CUTLASS C++ FlashAttention-3 [11] kernel. II. BACKGROUND AND RELATED WORK In this section, we first discuss the modern GPU architecture and briefly summarize the existing GPU programming models. A. Modern NVIDIA GPU Architecture As shown in Fig. 1, the Streaming Multiprocessor (SM) forms the fundamental compute unit of modern GPUs, encapsulating both computation and memory resources. Each SM integrates a large register file, a warp scheduler that can issue instructions to one warp (32 threads) per clock 1A shorthand for Task-Aware Warp Specialization with Aref. The implementation has been upstreamed to OpenAI Triton: https://github.com/tritonlang/triton/tree/aref auto ws Fig. 1: A simplified NVIDIA H100 SXM5 GPU architecture. cycle, and multiple CUDA cores that handle general-purpose arithmetic. Additionally, SMs include specialized Tensor Cores for accelerating matrix computations and a memory block that serves as software-managed shared memory (SMEM). A thread block, or Cooperative Thread Array (CTA), is scheduled on an SM and shares its registers and SMEM. Recent GPU generations, such as Hopper, have increased their reliance on Tensor Cores as the dominant source of compute throughput, where 90% of the total compute throughput is contributed by Tensor Cores [6]. As such, achieving high performance on modern GPUs is to fully saturate these Tensor Cores with sufficient and timely data. To this end, Hopper introduces two key hardware innovations aimed at improving Tensor Core utilization: the Tensor Memory Accelerator (TMA) and Warp Group Matrix Multiply-Accumulate (WGMMA). The TMA facilitates hardware-managed, multidimensional data transfers from global memory (GMEM) to shared memory, offloading the address generation and transfer coordination tasks from software to a dedicated unit. To further enable this overlap, Hopper extends the programming model with asynchronous transaction barriers, which allow threads to synchronize not just based on arrival but also on the volume of data exchanged. Each transaction carries a byte count, and the barrier only releases threads once both all producers have arrived and the expected total transaction count is reached. This mechanism is especially powerful in coordinating asynchronous memory copies and collective data exchanges across thread blocks. In parallel, WGMMA introduces warp-group-level matrix multiply operations, allowing multiple warps (4 warps as one warp group) to cooperatively execute large-tile GEMM kernels in an asynchronous and efficient manner. Together, these mechanisms provide a foundation for warp specialization, where different warps within an SM assume distinct roles in data communication and computation, thereby improving throughput for LLM workloads [10]-[12]. These trends illustrate a broader architectural shift toward asynchronous dataflow execution, a direction continued in subsequent architectures such as Blackwell [7]. 2 (a) Tawa compilation 1 @triton.jit 2 def matmul(a_desc, b_desc, c_ptr, 3 M, N, K, 4 stride_cm: tl.const, stride_cn: tl.const, 5 Mt: tl.const, Nt: tl.const, Kt: tl.const): 6 pid = tl.program_id(axis=0) 7 num_pid_m = tl.cdiv(M, Mt) 8 pid_m = pid % num_pid_m 9 pid_n = pid // num_pid_m 10 o_am = pid_m * Mt 11 o_bn = pid_n * Nt 12 o_k = 0 13 acc = tl.zeros((Mt, Nt), dtype=tl.float32) 14 for k in range(0, tl.cdiv(K, Kt)): 15 # Data loading 16 a = tl.tma_load(a_desc, [o_am, o_k], [Mt, Kt]) 17 b = tl.tma_load(b_desc, [o_bn, o_k], [Nt, Kt]) 18 # Computation 19 acc = tl.dot(a, b.T, acc=acc) 20 o_k += Kt 21 offs_cm = pid_m * Mt + tl.arange(0, Mt) 22 offs_cn = pid_n * Nt + tl.arange(0, Nt) 23 c_ptrs = c_ptr + stride_cm * offs_cm[:, None] \ 24 + stride_cn * offs_cn[None, :] 25 tl.store(c_ptrs, acc) (b) A simplified Triton GEMM program 1 module attributes {"num-warps" = 8 : i32} { 2 tt.func public @matmul( 3 %arg0: !tt.ptr , %arg1: !tt.ptr , 4 %arg2: !tt.ptr , %arg3: i32, 5 %arg4: i32, %arg5: i32) { 6 %ABaref = tawa.create_aref : () 7 -> tensor , tensor >> 9 // ... more initialization 10 tawa.warp_group { 11 scf.for %k = %c0 to %K step %c1 12 iter_args(%arg8 = %c0) { 13 %ta = tt.tma_load %arg0[%offset_am, %arg8] 14 %tb = tt.tma_load %arg1[%offset_bn, %arg8] 15 %extracted = %ABaref[%k mod %cst_D] 16 tawa.put(%extracted, %ta, %tb) 17 %kI = arith.addi %arg8, %c64 18 scf.yield %kI : i32 19 }} {partition = 0 : i32} 20 tawa.warp_group { 21 %tc = scf.for %k = %c0 to %K step %c1 22 iter_args(%acc = %cst_0) { 23 %extracted = %ABaref[%k mod %cst_D] 24 %ta, %tb = tawa.get(%extracted) 25 %part_ret = tt.dot_wait %ta, %tb, %acc 26 %res = tt.dot_wait %part_ret {pendings = 1} 27 tawa.consumed(%extracted) 28 scf.yield %res : tensor 29 } 30 // epilogue: calculate C address and store 31 tt.store %c_ptr, %tc 32 } {partition = 1 : i32} 33 }} (c) Highly simplified Tawa MLIR code (D = 2) Fig. 2: Example Tawa compilation flow from the Triton frontend to the internal MLIR representation. tl is the shorthand for the Triton language. tt is the Triton MLIR dialect. Code snippets are simplified for demonstration purposes. B. Existing GPU Programming Models CUDA is the dominant interface for GPU programming [13], but as architectures evolve, programming complexity has increased substantially. NVIDIA's CUTLASS C++ [9] library provides templated kernels for GEMM and related computations, but leveraging it to write neural network operators requires intricate configuration and hand-tuning. To reduce engineering effort for domain workloads, specialized libraries have emerged. ThunderKittens [10] introduces a compact C++ interface based on 16×16 tile abstractions, offering performance-competitive GEMM and attention kernels. FlashInfer [12] focuses on efficient attention and KVcache management with JIT specialization to support largescale LLM inference. These systems simplify access to highperformance kernels but remain library-centric: they provide optimized operator templates rather than a general-purpose strategy for different applications, which may cause extensive maintenance issues for hand-tuning performance for different sizes of matrices. To further raise the abstraction level, domain-specific compiler infrastructures have been proposed. TVM [14] pioneered an end-to-end compilation framework with automated scheduling [15], [16], while Relax [17] extends it with a graph-level IR for dynamic workloads. Fireiron [18] also leverages a schedule language for specifying the tiling and data movement, and Graphene [19] further enhances this idea by providing an IR for mapping tensors with various layouts to Tensor Core instructions. While powerful, these systems were developed prior to Hopper and do not directly exploit TMA and hardware warp specialization, leaving significant performance untapped. Similarly, lots of efforts on GPU warp specialization [20]- [23] were proposed before the Hopper architecture, which do not effectively use the latest TMA features. Motivated by the need for finer control, a new class of tile-based kernel languages has emerged. Triton [8], built on MLIR [24], allows programmers to write efficient kernels in Python while abstracting tiling and memory management. Despite its success in democratizing kernel development, Triton still lacks inherent abstractions for warp specialization, leaving a performance gap on the Hopper architecture. Recent work such as TileLang [25] and Mosaic [26] extends this direction by introducing explicit APIs for dataflow scheduling and pipeline composition. Cypress [27] proposes a task-based programming model that abstracts complex data movement and synchronization through logical computation descriptions and mapping specifications. However, programmers are still required to manually manage the multi-level hierarchy of threads, warps, and SMs, along with the corresponding data decomposition. Gluon [28] proposes an IR for Triton that adds layout abstractions but still requires programmers to explicitly incorporate communication operations. These approaches provide expert users with expressive control, but they continue to impose significant responsibility for synchronization and resource management, which limits programmer productivity. To this end, we argue that users should not have to cope with low-level hardware details. Instead, warp-specialized programs 3 should be generated automatically. Tawa bridges these gaps by introducing aref as an IR-level abstraction for warp communication and automatically generating warp-specialized, TMAenabled pipelines, thereby achieving high performance without imposing low-level programming burden. III. TAWA COMPILER In this section, we first give an overview of the Tawa compiler and discuss details of the asynchronous reference abstraction and the compilation process. A. Overview Fig. 2a illustrates the overall design of Tawa. Since Triton [8] has become the de facto backend for PyTorch [29], [30] and is widely adopted for writing efficient GPU kernels, we select Triton as the frontend interface of our compiler. Programmers write kernels in Triton-Python with tiled computation and TMA communication in a CTA (Fig. 2b), which are first translated into the standard Triton-MLIR representation. Building upon this infrastructure, Tawa introduces a series of compiler passes that automate the generation of warpspecialized programs directly from high-level Python code. To address the challenges outlined in § I, Tawa applies three key transformations: (1) Task-aware partitioning (§ III-C). We introduce a novel algorithm that partitions the program into producer and consumer warp groups, ensuring correct role assignment and communication boundaries, directly addressing Challenge 1. (2) Asynchronous reference abstraction (§ III-B). We design aref, an intermediate representation that captures inter-warp dataflow explicitly. By expressing communication intent at the IR level, aref allows the compiler to automatically insert and coordinate the necessary synchronization and data-movement operations, resolving Challenge 2. We also propose a new Tawa MLIR dialect to encode those aref operations. (3) Multi-granularity software pipelining (§ III-D). On top of the aref program, we apply a software pipelining pass that orchestrates overlap between communication and computation across multiple granularities, resolving Challenge 3. The resulting program is shown in Fig. 2c. Once these passes are applied, the aref representation is lowered to the standard Triton-MLIR dialect, after which the existing Triton compilation pipeline proceeds unchanged: the code is progressively lowered to LLVM IR and finally to PTX, ready for execution on NVIDIA GPUs. B. Asynchronous Reference Fig. 3: aref abstraction and associated operations. PUT σ(a).E = 1 ⟨σ, put(a, v)⟩→⟨σ[ a 7→⟨buf = v, F = 1, E = 0⟩], ε⟩ GET σ(a).F = 1 ⟨σ, get(a)⟩→⟨σ[ a 7→⟨buf = σ(a).buf, F = 0, E = 0⟩], σ(a).buf⟩ CONSUMED ⟨σ, consumed(a)⟩→⟨σ[ a 7→⟨buf = σ(a).buf, F = 0, E = 1⟩], ε⟩ Fig. 4: Operational semantics of the aref operations. σ is the store map from an aref identifier a to the actual structure σ(a) = ⟨buf, F, E⟩, where F is the full mbarrier flag and E is the empty mbarrier flag. Initially E = 1, F = 0. To better model communication between warps, we introduce asynchronous reference [31], or aref, an IR abstraction that models a one-slot channel between producer and consumer on the GPU as shown in Fig. 3. It packages a data buffer together with two synchronization primitives implemented by hardware mbarriers, conventionally named empty and full. At any moment exactly one of these barriers encodes the state of the slot: when the empty mbarrier holds a credit, the slot may be written by the producer; when the full mbarrier holds a credit, the slot contains a published value ready to be read by the consumer. The aref interface exposes three operations: put, get, and consumed, whose formal semantics are defined in Fig. 4. These semantics provide a rigorous foundation for lowering the operations to GPU-specific instructions, as discussed in § III-E. put is the producer's publication step: it requires the slot to be empty (the empty mbarrier E holds a credit), writes the payload to the buffer, and flips the state to full by arriving on the full mbarrier F with release ordering. This transition makes the data visible to the consumer and prevents subsequent writes from reusing the slot prematurely. get is the consumer's acquisition step: it requires the slot to be full (the full mbarrier F is observed), performs a read of the buffer into a program variable, and transitions the aref into a borrowed state in which neither E nor F holds a credit, reflecting that the value is in use but the slot is not yet reusable. consumed closes the handshake: once the consumer no longer needs the value, it arrives on the empty mbarrier E, restoring the empty credit and enabling the next put; this induces the happens-before chain from producer's writes to consumer's reads and on to the producer's subsequent reuse. The aref therefore serves as a first-class IR value representing a concrete communication path that enforces correct ordering of data movement and computation across warp groups. Compared to constructs such as the OpenCL pipe [32] or similar mechanisms in other accelerator programming models [33]-[37], the aref abstraction is specifically designed for GPU inter-warp communication. Its semantics are rooted 4 (a) Computation graph of Fig. 2b. (b) Partitioned computation graph. (c) Achieved warp specialization. Suppose D = 2. Fig. 5: Task-aware partition for the GEMM kernel in Fig. 2b. Notice warp specialization is just one way of achieving the overlapping in (c), and the actual latency is not shown to scale. in hardware-supported mbarrier synchronization, providing a clear ordering model. In addition to its synchronization semantics, aref is also type-generic, supporting buffers of type , which allows it to carry structured payloads such as tensors or tuples while reusing the same mbarriers. To facilitate deep pipelining, multiple aref instances can be grouped into a cyclic buffer of depth D, effectively forming a ring structure that supports the storage and transfer of intermediate data across pipeline stages. C. Task-Aware Partitioning We begin by traversing the MLIR computation graph as in Fig. 5a, where each node is an MLIR operation and each edge captures the use-def dependencies. We take two steps of the partitioning process: partition annotation and loop distribution. 1) Partition Annotation We perform a backward traversal along the use-def chains starting at the kernel's side-effecting sinks (e.g., Store). During the traversal, we attach a semantic tag to every node by inspecting its effects. Nodes that contribute to address computation, such as the pointer arithmetic for data transfers, are marked as iteration statements (orange edges). Nodes that transform or consume a tile for actual computation, such as WGMMA, are marked as tile statements (blue edges). Importantly, iteration statements are not necessarily contiguous in the IR, as highlighted in Fig. 2b. A typical division on Hopper is to create a load WG as the producer and a compute WG as the consumer, as shown in Fig. 4. Tawa annotates the iteration statements and their dominated TMALoads in the producer partition (WG0), and the remaining tile statements in the consumer partition (WG1). Notice the epilogue is also attached to WG1 to make sure it is actually executed by some WGs. For Blackwell and later generations of GPUs that support more WGs, Tawa can also partition the IR based on available warp roles on GPUs by annotating the semantic roles during graph traversal. 2) Loop Distribution After the operations are annotated, we need to construct the actual IR that divides the workload for each WG. For each cross-partition edge (e.g., TMALoad→LocalAlloc in Fig. 5a), we create an aref tensor with size D, denoting a Dslot cyclic buffer. A put operation is created on the producer side that places each TMALoad result into an aref slot. The slot index is computed as the iteration k modulo the buffer size D, allowing the buffer slots to be reused (L15, L23 in Fig. 2c). Similarly, a get operation is created on the consumer side that gets the data from the same aref slot. After the data are used for computation in WGMMA, an additional consumed operation is inserted at the end to indicate the slot has been freed. As mentioned in § III-B, aref supports grouping multiple values in the buffer to share the mbarrier. Therefore, a straightforward optimization when creating aref is to check whether the loaded tensors are used in the same WGMMA. If so, we can put a tuple of tensors in the aref buffer and share the channel for communication between WGs. The resulting computation graph is shown in Fig. 5b. Since aref has already carried buffers, the LocalAlloc operations can also be eliminated but to directly read from the aref buffer. To make both partitions independently progress through the original loop nest, we also need to perform loop distribution around the cut. Two WG regions are created in the IR as shown in Fig. 2c (L10-32). The scf.for loop is cloned so that WG0 and WG1 each carry an isomorphic loop over k. Iteration statements remain in WG0, and tile statements remain in WG1. The inter-warp communication is done by aref highlighted in the code (L16, L24). Finally, the computed result is yielded from the for loop and stored back to global memory in the epilogue. The resulting execution is the warpspecialized timeline in Fig. 5c, which shows how TMA loads are overlapped with Tensor Core computation. D. Multi-Granularity Pipelining After we have the warp-specialized program, we conduct further optimization on the compute warp group, where we in5 troduce two pipelining mechanisms to make sure the hardware components are working in parallel. Specifically, we consider a fine-grained pipeline that overlaps MMA address calculation and computation, and a coarse-grained pipeline that overlaps CUDA and Tensor Core computation. 1) Fine-Grained Pipeline Fig. 6: Fine-grained pipeline overlapping MMA address calculation and computation. Actual latency not shown to scale. Modern GPUs expose both CUDA Cores, which handle scalar and address computations, and Tensor Cores, which accelerate MMA operations; however, a na ̈ıve sequential execution forces these units to operate in sequence, such that address generation stalls tensor execution and vice versa, leading to poor utilization. To alleviate this inefficiency, we introduce an automatic pipelining algorithm that overlaps address calculation on CUDA Cores with MMA execution on Tensor Cores through a bounded pipeline of depth P. At each iteration, the compiler issues the next MMA instruction asynchronously (WGMMA.issue(k)), and only stalls when the number of outstanding operations reaches the maximum pipeline depth (WGMMA.wait() {pendings=P}). As the k-th wait operation does not need to be done immediately after each MMA issue, enabling more MMA instructions to be in flight before waiting. Once the pipeline is filled, the result reference from iteration k -P is released (aref.consumed(k-P)), which maintains correctness by enforcing dataflow dependencies and freeing resources for reuse. After the final iteration, the compiler drains the pipeline in an epilogue by consuming the remaining pending operations. This design effectively decouples address computation and matrix multiplication, sustaining a continuous flow of work across both functional units, and effectively hides latency and reduces idle time compared to the sequential baseline. 2) Coarse-Grained Pipeline The fine-grained pipeline is most effective when the loop primarily consists of matrix multiplication operations. However, when additional computations are performed on CUDA Cores, these operations can also be overlapped with MMA. Therefore, we further propose a coarse-grained three-stage producer-transform-(optional) consumer assembly line in Algorithm 1. Each loop iteration j has a Tensor Core stage Tj that produces an intermediate tile (e.g., a matmul fragment), a CUDA Core stage Cj that transforms that intermediate (e.g., normalization, activation, reduction), and an optional downstream Tensor Core stage Uj that consumes Algorithm 1: Coarse-Grained CUDA and Tensor Core Pipelining Input: Tile count N; Boolean USE_U 1 Prologue; 2 MAYBEAREFGET(T0); 3 ISSUEANDCOMMIT(T0); 4 DOTWAIT(T0) ; // materialize T0 output 5 MAYBEAREFCONSUMED(T0); 6 COMPUTE(C0); 7 Steady state; 8 for j ←1 to N -1 do 9 MAYBEAREFGET(Tj); ISSUEANDCOMMIT(Tj); 10 if USE_U then 11 MAYBEAREFGET(Uj-1); 12 ISSUEANDCOMMIT(Uj-1); 13 DOTWAIT(Tj-1) ; // ensure Tj-1 results are visible 14 MAYBEAREFCONSUMED(Tj-1); COMPUTE(Cj-1); 15 if USE_U then 16 DOTWAIT(Uj-1); 17 MAYBEAREFCONSUMED(Uj-1) 18 Epilogue; 19 DOTWAIT(TN-1); 20 MAYBEAREFCONSUMED(TN-1); 21 COMPUTE(CN-1); 22 if USE_U then 23 MAYBEAREFGET(UN-1); 24 ISSUEANDCOMMIT(UN-1); 25 DOTWAIT(UN-1); 26 MAYBEAREFCONSUMED(UN-1) 27 FINALEPILOGUEANDSTORE(); the transformed result (e.g., a second matmul/epilogue). The prologue boots the pipeline by running T0 to completion and immediately computing C0. Steady-state then overlaps Tj with Cj-1 (and, if present, Uj-1): in each iteration we first enqueue and commit the Tensor Core work for the current tile, optionally enqueue and commit the downstream U for the previous tile, then perform a precise completion wait on the dot-product group(s) that produced Tj-1's results and run the CUDA Core transform Cj-1. The epilogue drains the last CUDA Core transform and, if U is used, issues and retires UN-1. Correctness depends on two synchronization mechanisms: DOTWAIT(Tj -1) (and DOTWAIT(Uj -1)) ensures compute-group completion at Tensor Core boundaries, while MAYBEAREFGET(·) are wrappers that perform the get operation only when a stage actually consumes a cross-WG aref object. When no external data needs to be read, the wrappers compile to no-ops, keeping the template minimal and avoiding unnecessary stalls. From an MLIR computation graph, the schedule is constructed by several principled analyses followed by a mechanical synthesis step. Stage identification partitions the periteration subgraph into Tj, Cj, and (optionally) Uj using dialect- and type-level cues: Tensor Core micro-tiles and their glue form T (and possibly U if a second tensor-core phase ex6 ists), whereas arithmetic, normalization, activation, reduction, or layout transformation that reads T's outputs forms C. This yields a tiny intra-iteration DAG with edges Tj →Cj and, if present, Cj →Uj. An example of attention [38] is that the QKT forms T, the softmax operation forms C, and the second GEMM PV constructs U. Finally, an aref-use inspection marks any stage that reads data produced by a different warpgroup; only for those marked stages will the wrappers MAYBEAREFGET(·) and MAYBEAREFCONSUMED(·) be emitted. E. aref Lowering The lowering of aref transforms high-level asynchronous references into explicit synchronization and memory-transfer instructions that execute directly on the GPU. At the IR level, create_aref declares a buffer and allocates the necessary mbarriers, serving as the root of a communication subgraph. The compiler then rewrites put, get, and consumed operations into concrete instructions: put expands into an asynchronous TMA load guarded by empty/full barriers, get becomes a blocking wait on the corresponding full barrier, and consumed signals buffer availability by arriving at the empty barrier. This pattern-matching ensures that once the entire subgraph is rewritten, the abstract aref nodes can be safely erased, leaving only executable low-level synchronization and data-movement instructions. A critical aspect of this transformation is correctness under pipelining, where multiple tiles are loaded, computed, and consumed in flight. To avoid deadlock from circular waits, the lowering introduces a parity mechanism: each operation alternates between two sets of barriers indexed by iteration parity. When parity toggles, a consumer may skip waiting if data has already been produced, and producers can reuse buffer slots without overwriting values still in use. This mechanism not only guarantees liveness but also enables efficient multibuffering, allowing communication and computation to overlap across multiple stages. In this way, aref lowering provides a systematic bridge between an abstract, declarative view of asynchronous dataflow and the precise TMA and mbarrier instructions required for high-performance GPU execution. IV. FURTHER OPTIMIZATIONS To further increase the Tensor Core utilization in warpspecialized scenarios, Tawa further integrates additional optimizations that are incorporated into the compilation flow, ensuring that programs automatically benefit without requiring manual restructuring. A. Cooperative Compute Warp Groups A key limitation of warp-specialized execution is the restricted register budget available to each warp group, which constrains the maximum tile size and can limit overall compute intensity. The cooperative warp group optimization alleviates this bottleneck by enabling multiple warp groups to collaboratively compute the same tile. As illustrated in Fig. 7a, two consumer warp groups (WG1 and WG2) jointly consume data Fig. 7: (a) Cooperative warp groups; (b) Persistent kernel. produced by a TMA warp group (WG0). By pooling their registers, the cooperating warps can form larger tiles that increase arithmetic intensity, improve data reuse, and reduce memory traffic. Conceptually, this optimization reintroduces a form of standard warp parallelism that multiple warps performing the same computation on different fragments of the same tile, into the warp-specialized setting. Tawa handles this transparently by informing the backend code generator of the cooperative mapping so that thread indices are consistently assigned. Importantly, no changes are needed to the aref abstraction, since communication semantics remain unchanged. B. Persistent Kernels Another major source of inefficiency in GPU pipelines stems from repeated kernel launches, each introduces significant CTA launch overhead. Persistent kernels reduce this overhead by launching only as many CTAs as there are SMs and keeping them resident throughout execution. As depicted in Fig. 7b, Tawa transforms the execution model into a softwaremanaged pipeline, where resident CTAs are responsible for iterating over tiles. This enables overlapping of TMA-driven loads, Tensor Core computation, and CUDA Core epilogues across iterations, eliminating relaunch latency and reducing idle time. Tawa automatically inserts synchronization before each epilogue to preserve correctness. This optimization is especially effective in deep pipelines with large K dimensions, where cumulative launch costs would otherwise dominate. V. EVALUATION We first describe our experimental setting and then show the results on different benchmarks in this section. A. Experimental Setup Tawa is built on top of Triton [8] (commit: 0c7edf) with about 4K lines of C++ and introduces a compact MLIR dialect that encapsulates aref operations. We compare Tawa against a set of state-of-the-art GPU programming frameworks and libraries, including the closed-source commercial library cuBLAS [39] (v12.7), the NVIDIA open-source CUDA linear algebra library CUTLASS [9] (v4.0.0), and the baseline Triton under the same commit. We also include two academic frameworks: ThunderKittens [10] (6c27e2), a CUDA-based AI kernel library, and TileLang [25] (bcfc83), a TVM [14]- based DSL. To ensure a fair comparison, all baselines are hand-tuned with a fixed set of tile sizes (combinations of 7 256 512 1024 2048 4096 8192 16384 0 200 400 600 800 1000 1200 FP16 256 512 1024 2048 4096 8192 16384 0 500 1000 1500 2000 FP8 GEMM K size Throughput (TFLOPs/s) Theoretical Peak cuBLAS Tawa Triton TileLang ThunderKittens Fig. 8: GEMM performance results for different frameworks. M = N = 8192, and varied K. {64, 128, 256}). In the default Triton implementation, the official software pipelining is enabled, with pipeline depth predetermined to balance overlap between communication and computation. For kernels generated by Tawa, the size of the aref and the depth of the MMA pipeline are selected manually to maximize performance, providing each approach with equivalent opportunity to exploit the underlying hardware. All experiments are conducted on an NVIDIA H100 SXM5 GPU equipped with 80GB of HBM3e memory, using CUDA 12.7 as the software stack. Experiments are conducted using 25 warp-up and 1000 measurement runs, and we compute the average execution time from these. We benchmark several representative workloads. The first is GEMM and its variants, evaluated with both FP16 and FP8 precision. The second is an attention kernel, with various sequence lengths to reflect realistic LLM workloads. These benchmarks represent diverse computational patterns and provide a comprehensive assessment of the effectiveness of automatic warp specialization relative to hand-optimized libraries and DSL-based approaches. B. Matrix Multiplication We evaluate GEMM kernels with matrix dimensions of M = N = 8192 while sweeping the inner dimension K from 256 to 16384, and report the throughput for FP16 and FP8. As shown in Fig. 8, across both precisions, Tawa achieves the same level of performance as the highly optimized kernel library cuBLAS while outperforming the other generalpurpose frameworks on most shapes. Tawa reaches up to 79% hardware utilization in both FP16 and FP8. In FP16, Tawa delivers average speedups of 1.01×, 1.13×, 1.23×, and 1.09× over cuBLAS, Triton, TileLang, and ThunderKittens, respectively. The improvements over Triton highlight the advantage of overlapping TMA-driven data movement with WGMMA execution. In contrast, Triton employs an Ampere-style software pipelining scheme for asynchronous copies, which is less effective on Hopper than the hardwarebacked warp-specialization strategy in Tawa. TileLang and ThunderKittens are extensively tuned for large K, leading to stronger performance than Tawa when K ≥8192. In FP8, the benefits of automatic warp specialization are more pronounced, particularly at small K. On average, Tawa achieves 1.06×, 1.02×, 2.40×, and 1.24× speedup over 1K 2K 4K 8K 16K GEMM size M = N = K 0 200 400 600 800 Batched GEMM 2 4 6 8 # of groups G Grouped GEMM Throughput (TFLOPs/s) Tawa Triton Fig. 9: FP16 batched GEMM and grouped GEMM results. cuBLAS, Triton, TileLang, and ThunderKittens, respectively. TileLang encounters layout-management challenges for FP8 WGMMA, yielding an inferior implementation that lags other frameworks; at K = 256, Tawa is 3.99× faster than TileLang. ThunderKittens appears less carefully tuned for FP8, resulting in up to 1.61× lower throughput at small K. Overall, these results indicate that Tawa's aref abstraction and automatic warp specialization provide a robust orchestration mechanism that overlaps TMA loads and WGMMA across precisions. C. GEMM Variants To assess the generalizability of Tawa across diverse kernel patterns, we evaluate two GEMM variants implemented in Triton. The first is batched GEMM, which executes multiple small GEMMs of identical shape within a single kernel. The second is grouped GEMM, which processes multiple GEMMs of varying shapes in one kernel. Both patterns are pervasive in Mixture of Experts models [40]-[42]. For batched GEMM, we fix the batch size to 8 and sweep M, N, and K across a range of values. For grouped GEMM, we vary the number and shapes of GEMMs. As shown in Fig. 9, Tawa consistently outperforms Triton baselines in both cases, achieving speedups of up to 7%. The improvements arise from Tawa's arefbased partitioning and automatic warp specialization, which allow data movement for one GEMM to be overlapped with computation from another. In batched GEMM, this overlap reduces idle time between consecutive small multiplications, 8 1024 2048 4096 8192 16384 0 200 400 600 800 FP16, causal=false 1024 2048 4096 8192 16384 0 200 400 600 800 FP16, causal=true 1024 2048 4096 8192 16384 0 250 500 750 1000 FP8, causal=false 1024 2048 4096 8192 16384 0 250 500 750 1000 FP8, causal=true Context length Throughput (TFLOPs/s) FA3 (CUTLASS) Tawa Triton TileLang ThunderKittens Fig. 10: FP16 and FP8 MHA performance results for different frameworks. 1 2 3 MMA Depth (P) 1 2 3 Aref Size (D) 159 0 0 567 437 0 657 652 433 Non-Persistent GEMM 1 2 3 MMA Depth (P) 1 2 3 Aref Size (D) 167 0 0 610 519 0 704 720 513 Persistent GEMM 0 200 400 600 Performance (TFLOPs/sec) Fig. 11: Impact of different aref and MMA pipeline sizes. while in grouped GEMM, it schedules heterogeneous shapes more efficiently within a single kernel. D. Multi-Head Attention We evaluate (non-)causal multi-head attention (MHA) across sequence lengths L ∈[1024, 16384] with batch size 4 and head dimension 128. As shown in Fig. 10, Tawa is compared against strong reference implementations, including handwritten FlashAttention-3 (FA3) [11] built on CUTLASS C++, as well as Triton, TileLang, and ThunderKittens. For FP16, Tawa attains up to 96% of FA3 throughput and delivers a consistent 1.21× speedup over Triton. This gap largely stems from the Triton baseline being effectively a FlashAttention-2 style implementation [43], which does not fully exploit Hopper warp specialized producer and consumer pipelines, in particular TMA driven global to shared transfers overlapped with WGMMA on Tensor Cores. At short sequences, the advantage of warp specialization is muted because prologue, epilogue, and barrier costs are not yet amortized. As L grows to at least 4K, memory traffic and synchronization become dominant; in this regime Tawa outperforms TileLang and ThunderKittens by up to 1.13× and 1.23×, respectively, indicating more effective overlap between data movement and compute. Although causal MHA reduces effective Tensor Core utilization due to mask induced control and data hazards, Tawa sustains strong performance at long sequences and reaches its peak at L = 16K. For FP8, the benefits of automatic warp specialization are even more pronounced at large L. Tawa reaches up to 89% of FA3 and achieves 1.11× speedup over Triton. TileLang and ThunderKittens failed to execute our FP8 attention configurations, suggesting that their kernels are tuned primarily for FP16 and lack the layout management and pipeline scheduling needed to handle FP8 WGMMA paths robustly. Overall, the results support two conclusions. First, aref provides a principled abstraction that converts generic kernels without asynchronous semantics into dataflow pipelines: by treating tiles as references with explicit get and put phases, Tawa overlaps TMA transfers, shared-memory materialization, and WGMMA while maintaining warp-group occupancy and steady MMA throughput. Second, these gains hold across precision from FP16 to FP8 and semantics from noncausal to causal, showing that Tawa automatic warp specialization policies generalize well. E. Hyperparameter Selection To study the impact of hyperparameters in our compilation flow, we conduct experiments on the aref size D, the number of disjoint shared-memory staging buffers governed by asynchronous barriers, and the MMA depth P, how many compute tiles (WGMMA issue groups) are overlapped in flight, with (non-)persistent GEMM kernels (K = 16384). As shown in Fig. 11, across the feasible region (D ≥P), performance consistently increases when increasing D. A larger D lets TMA producers prefetch more, smoothing latency variation and better hiding global-to-shared transfer under compute. 9 Triton w/o WS +Auto WS +Cooperative WGs +Large Tile Size +Persistent Kernel +Better Aref Size 0 200 400 600 800 Throughput (TFLOPs/s) 104 393 395 572 632 718 GEMM Triton w/o WS +Auto WS +Cooperative WGs +Pipeline +Better Aref Size 0 200 400 600 800 209 232 593 645 654 MHA Fig. 12: Impact of different optimizations on FP16 kernels. Further increasing D requires more shared memory to be allocated, which is not achievable for large tile size. Varying P at fixed D shows the classic over-pipelining trade-off. Increasing P from 1 to 2 generally helps (persistent peaks at D = 3, P = 2), because MMA reduces stalls on accumulator availability and amortizes instruction-level latencies. Pushing to P = 3 lowers throughput: deeper compute pipelines increase live fragments and accumulator registers and can depress WG occupancy, so the marginal latency hiding is outweighed by pressure on registers and shared memory. Comparing the two panels, the persistent version is consistently faster (roughly +5-10% on most feasible points). Persistence keeps CTAs resident to pull multiple tiles from a global work queue, which eliminates grid-scheduling and tail effects, improves cache locality for weights and epilogue metadata, and maintains a steady-state TMA and MMA rhythm with fewer disruptions. Taken together, the figure suggests using a larger D and moderate P (1-2) to balance overlap and resource pressure, and uses persistent kernels to stabilize the pipeline and harvest cache/reuse benefits. F. Ablation Study To further study the effect of different optimization techniques in Tawa, we conduct an ablation study on both GEMM and MHA kernels. We use the largest GEMM (K = 16384) and MHA kernels (L = 16384) in our previous experiments, and the results are shown in Fig. 12. The baseline Triton without adding warp specialization (WS) delivers a throughput of only 104 TFLOPs/s. By enabling WS with a single warp group while maintaining the same tiling configuration (+Auto WS), performance improves 3.78× to 393 TFLOPs/s. This substantial jump reflects the efficiency gains from decoupling producer and consumer roles at the warp level, which improves parallelism and mitigates pipeline stalls. The benefits of warp specialization become even more evident as the configuration is extended to two compute warp groups. With the same tile size of 128×128×64, throughput does not change a lot. However, the increased number of warps reduces the register pressure and can enable a larger tile size for the warps to cooperatively work on. When enlarging the tile size to 128×256×64 (+Large Tile Size), it boosts the performance by 1.46×. The enlarged tile shape improves data reuse and reduces redundant memory traffic, showing that tiling interacts synergistically with warp specialization to exploit the hardware more effectively. Similar trends can be found in MHA, where the automatic warp specialization (+Auto WS) and cooperative warp groups (+Cooperative WGs) together boost the performance by 2.84×. Beyond warp specialization and tiling, the introduction of persistent kernels further amplifies performance. In this mode, thread blocks are retained on SMs across multiple iterations, thereby avoiding costly kernel launch overheads and improving data locality. With persistence enabled (+Persistent Kernel), the system further improves the performance by 10%. Finally, fine-tuning the scheduling depth parameter in conjunction with persistence leads to a peak throughput of 718 TFLOPs/s (+Better Aref Size), representing nearly a sevenfold improvement over the baseline. For MHA, adding a coarse-grained pipeline and fine-tuning the aref size leads to the final performance of 654 TFLOPs/s. This progression highlights how Tawa carefully incorporates warp specialization, cooperative warp groups, and pipelining, which can unlock better utilization of modern GPU hardware. VI. FUTURE WORK One natural extension of Tawa is to broaden the set of communication patterns supported by the compiler. Currently, Tawa primarily targets producer-consumer pipelines built around double-buffering, but more advanced schemes such as ping-pong kernels and multicast communication could further improve efficiency. Ping-pong kernels allow alternating roles between warp groups across iterations, thereby balancing workload and alleviating bottlenecks in cases where computation and data movement demands shift dynamically. Similarly, multicasting enables a single producer warp to distribute data to multiple consumer warps simultaneously, reducing redundant memory transfers and better utilizing shared buffers. Looking further ahead, we aim to generalize Tawa beyond Hopper to support next-generation architectures such as Blackwell. In particular, Blackwell introduces tensor memory (tmem) as a new hardware-managed memory hierarchy that complements TMA by providing more flexible data orchestration between shared and global memory. Leveraging tmem effectively will require extending aref to model multi-level data movement and revisiting scheduling decisions across heterogeneous memory resources. In addition, future workloads demand even greater scalability, motivating support for multiple producers and consumers for multiple warp groups, as well as more sophisticated graph partitioning algorithms that account for load balancing, register pressure, and memory footprint simultaneously. 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A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND
AGRICULTURAL SUSTAINABILITY
ELIA MORETTI1,2, MICHEL LOREAU3,4 AND MICHAEL BENZAQUEN1,2,5
1Chair of Econophysics and Complex Systems, École polytechnique, 91128 Palaiseau Cedex, France.
2LadHyX, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France.
3Station d’écologie théorique et expérimentale, CNRS, 09200 Moulis, France.
4Institute of Ecology, College of Urban and Environmental Sciences, Peking University, Beijing 100871, China.
5Capital Fund Management, 23 rue de l’Université, 75007 Paris, France.
Abstract. Feeding a larger and wealthier global population without transgressing ecological limits is in-
creasingly challenging, as rising food demand—especially for animal products—intensifies pressure on ecosys-
tems, accelerates deforestation, and erodes biodiversity and soil health. We develop a stylized, spatially
explicit global model that links exogenous food-demand trajectories to crop and livestock production, land
conversion, and feedbacks from ecosystem integrity that, in turn, shape future yields and land needs. Cali-
brated to post-1960 trends in population, income, yields, input use, and land use, the model reproduces the
joint rise of crop and meat demand and the associated expansion and intensification of agriculture. We use
it to compare business-as-usual, supply-side, demand-side, and mixed-policy scenarios. Three results stand
out. First, productivity-oriented supply-side measures (e.g. reduced chemical inputs, organic conversion,
lower livestock density) often trigger compensatory land expansion that undermines ecological gains—so that
supply-side action alone cannot halt deforestation or widespread degradation. Second, demand-side change,
particularly reduced meat consumption, consistently relieves both intensification and expansion pressures;
in our simulations, only substantial demand reductions (on the order of 40% of projected excess demand by
2100) deliver simultaneous increases in forest area and declines in degraded land. Third, integrated policy
portfolios that jointly constrain land conversion, temper input intensification, and curb demand outperform
any single lever. Together, these findings clarify the system-level trade-offs that frustrate piecemeal inter-
ventions and identify the policy combinations most likely to keep global food provision within ecological
limits.
1. Introduction
Feeding a growing and wealthier population in the 21st century presents one of the biggest challenges
to planetary sustainability. Rapid income growth and global population expansion are driving up the total
demand for crops and for animal products, thereby escalating the pressures on the natural ecosystem. This
increasing food demand often comes at the expense of forests, semi-natural ecosystems, and long-term soil
health, fueling deforestation, biodiversity loss, and land degradation at unprecedented scales (26; 81; 59).
As a result, reconciling food security with the safe operating space for humanity—maintaining ecological
integrity while meeting human needs—has emerged as a global policy and scientific imperative (68; 85; 45).
A central question in this debate is whether ongoing increases in agricultural productivity, alongside
the adoption of more sustainable farming practices—such as organic and conservation agriculture, reduc-
tions in chemical inputs, or lower livestock densities—can suffice to meet rising demand while preserving
ecosystems (65; 74). Or, conversely, is a fundamental transformation of demand, especially through dietary
shifts and coordinated, system-wide policy measures, necessary to avoid crossing more critical environmen-
tal thresholds? In light of clear trade-offs between food production, land use, and environmental services,
understanding the boundaries and synergies of supply-side versus demand-side strategies remains an open
challenge (31; 2).
The literature tackling these questions spans empirical, scenario-based, and theoretical approaches. Global
integrated assessment models (IAMs) such as GLOBIOM (34), IMAGE (80), and MAgPIE (46) have illumi-
nated the possible consequences of alternative technology, land use, and diet scenarios for GHG emissions,
biodiversity, and food security. Yet, their complexity and extensive parameterization can make it difficult to
parse out causal mechanisms and feedback loops, or to transparently compare policy levers on a conceptual
1
arXiv:2510.14720v1 [econ.TH] 16 Oct 2025
2A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
level. On the theoretical side, dynamic and system-analytic models have provided key insights into food
system tipping points, feedback loops between agricultural intensification and land expansion, and the im-
plications of self-reinforcing land degradation (6; 17; 7; 70). Land use and food system theories drawing on
non-linear dynamics, spatially explicit frameworks, and ecosystem service feedback loops have highlighted the
risks of unintended consequences and compensatory effects across spatial and management scales (48; 24; 4).
Notably, archetype-based and stylized models have clarified how yield gains can sometimes induce addi-
tional land clearing (“rebound” or Jevons effects) rather than sparing nature, depending on feedback loops
among value chains, governance, and behavioral factors (1; 41; 36). Complementing this are explorations
of “land sparing” versus “land sharing” paradigms, which have underlined the limitations of both simplistic
intensification and extensive conservation solutions in isolation (61; 25).
Building on this theoretical tradition, our study makes several distinctive contributions to the debate.
We present a transparent, parsimonious model that connects exogenous food demand trajectories to crop
and livestock production, land conversion, and feedback from ecosystem integrity that influences yields and
future land uses. Such an approach allows us to systematically explore the interplay of demand growth,
management intensity, and policy intervention in shaping landscape outcomes and ecological stability.
Key strengths of our work include a calibration to observed, contemporary global trends—most notably
the rapid growth in meat demand occurring alongside sustained improvements in average crop yields. By
adopting a global perspective, our analysis controls for indirect land-use change effects (88). Furthermore, our
framework enables both a realistic assessment of future baseline trajectories and an exploration of plausible
supply- and demand-side alternatives, moving beyond purely theoretical scenarios or country case studies.
Furthermore, our spatially explicit landscape component captures the heterogeneity and emergent proper-
ties of coupled socio-ecological dynamics, allowing us to assess how context-specific degradation, and restora-
tion processes aggregate into world-scale indicators. This integrated framework is particularly well suited to
investigating “policy mixes” and their emergent synergies or trade-offs—issues that remain underexplored
yet are highly salient for real-world sustainability transitions (69; 16).
Within this framework, we directly address whether supply-side productivity gains can meet future food
demand while protecting global ecosystems, or if coordinated policy interventions that also target consump-
tion behaviors are essential to achieving long-term sustainability. By illuminating the dynamic, cross-scale
feedbacks underpinning both successes and failures in historical and prospective management strategies,
our findings contribute to a growing, more theoretically grounded understanding of the pathways—and
pitfalls—toward a sustainable food future.
2. Model
2.1. Model Overview
We have developed a spatially explicit, discrete-time model that dynamically links the growth in food
demand to agricultural production, land conversion among primary land-cover types — namely cropland,
pasture, and natural land — and crucially incorporates feedback loops from natural ecosystem integrity
(Fig. 1). The model landscape is represented as a square grid where each cell’s land-cover class changes over
time, allowing us to track both spatial configuration and aggregate stocks of land-use types. This enables
robust quantification of processes such as deforestation, natural regeneration, and degradation pressure as
they unfold over time.
The drivers of food demand are exogenous time series that combine projected population growth, changes
in per-capita income, and a parameter reflecting dietary preference for meat. This framework is flexible
enough to accommodate demand-side policy interventions, such as shifts to lower-meat diets, by perturbing
these demand trajectories.
Agricultural production in our model is governed by both land area and the intensity of management.
Cropland yields respond to chemical intensification and mechanization while livestock output depends on
available pasture area and on an adjustable livestock-density variable.
If, at any time step, aggregate
production fails to meet demand, the system responds first by intensifying production (for example, by
increasing mechanization and chemical inputs, or livestock density). Persistent shortfalls inevitably trigger
the conversion of natural land to cropland or pasture, resulting in deforestation and loss of critical habitat.
Central to the model is the ecosystem integrity index, a composite indicator reflecting both soil health
and biodiversity. This index decreases with intensified mechanization and chemical use but recovers through
periods of natural restoration. The integrity trajectories follow exponential decay and recovery processes,
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY3
DEMAND
MEAT
PRODUCTION
CROP
PRODUCTION
LANDSCAPE
Crop Demand
Meat Deforestation
Natural
Ecosystem
Integrity
Crop for Meat
Production
Unsatisfied
Demand
Unsatisfied
Demand
Meat Demand
Crop Deforestation
Population growth
Increasing
Cereal
Demand
Increasing
Meat
Demand
Economic growth
Natural
Ecosystem
Integrity
Mechanization
Chemical
Inputs
Technology
Organic
Production
Natural
Ecosystem
Integrity
Livestock
Density
Crop Inputs
Technology
Organic
Production
Calibrated
Exogenous
Policy-Shocked
Agricultural
Land
Natural
Land
Forest
Land
Natural
Ecosystem
Integrity
Degraded
Land
Cropland
Pasture
Land
0
1
2
Figure 1. Schematic overview of the our dynamic simulation model organized into four main sec-
tions: the left panel introduces the demand drivers; the top and bottom panels detail the production
and ecological feedback processes; and the right panel summarizes the resulting landscape structure
and transitions. Specifically, the left panel shows how external drivers—specifically, population and
economic growth (marked with stars for exogenous variables)—jointly determine increases in both
cereal and meat demand. Central panels illustrate how these demand trajectories feed into crop
and meat production systems, which are influenced by intensification variables such as technology
advancement, chemical inputs, mechanization, livestock density, and the adoption of organic pro-
duction (with targets indicating calibrated variables and lightning bolts indicating policy-shocked
variables). Meat and crop production outcomes depend on and feedback to the integrity of natural
ecosystems (a synthetic measure of local biodiversity and soil health, see Section 2.4), with low
integrity reducing yields and increasing future land needs. The model tracks unsatisfied demand
for both crops and meat, which triggers further land conversion—expanding either cropland or pas-
ture at the expense of natural land and semi-natural vegetation. The right panel details landscape
structure and transitions among key land categories: cropland, pastureland, natural land, forest,
and semi-natural vegetation. Feedback loops connect ecosystem integrity not only to landscape
structure, but also to production system performance, thus enabling the model to capture cross-
scale dynamics and the effects of multiple policy levers.
each with a characteristic time scale. Importantly, ecosystem integrity feeds back into agricultural yields,
while overexploitation undermines future productivity, thereby creating self-limiting dynamics. In addition,
the remaining intact vegetation modulates the pace of ecosystem degradation and restoration at the land-
scape level through the provision of vital ecosystem services, such as water filtration, climate stability, and
biodiversity storage.
The model incorporates a sustainable alternative to conventional farming through the inclusion of organic
management. The choice to focus on organic systems is motivated by their clear institutional definition
and the availability of consistent global datasets on organically managed land (67; 50).
In the model,
the transition to organic farming is treated as an exogenous process—that is, the share of organic land is
calibrated to empirical data and provided as an input rather than emerging endogenously from the system’s
dynamics. Organic management is characterized by the absence of synthetic chemical inputs and stricter
limits on livestock density, thereby reducing pressure on ecosystems, albeit often at the cost of short-term
yield reductions. The model explores the implications of policy-driven increases in organic adoption and
4A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
broader reductions in chemical input use, assessing their combined effects on production and ecosystem
integrity.
Our framework is capable of contrasting a business-as-usual development trajectory with a suite of alterna-
tive policy scenarios, where changes are implemented as constraints or shifts to key model parameters. These
include limiting livestock density, restricting deforestation, reducing chemical inputs, and implementing
demand-side interventions. This architecture allows us to systematically assess the synergies and trade-offs
inherent in: (i) supply-side sustainability measures, (ii) combinations of supply- and demand-side interven-
tions, and (iii) the overall scale of consumption habit transformation needed to achieve a transition onto
ecologically sustainable pathways for global land use and food production.
2.2. Demand: Economic and Population Growth
Population (Nt) and per-capita income (It) are specified as exogenous drivers that change throught time
t. Income determines both the total caloric requirement and the composition of human diets. Total caloric
demand is modeled using a logarithmic (Engel-type) relationship (12) :
(2.1)
Dt = (a + b log It)Nt,
where Dt is total caloric demand, and a and b are positive parameters. The logarithmic form is widely used
in empirical studies of food demand because it captures the diminishing marginal increase of total caloric
intake with rising income (3).
Demand for animal products is modeled as a power-law function of income:
(2.2)
Dm,t = c I d
t Nt,
with c, d > 0, where d represents the income elasticity of meat demand. Power-law (or constant-elasticity)
functions are commonly adopted in agricultural economics to represent the empirically observed sublinear
increase in meat and high-value food consumption with income(13; 29).
The demand for crops directly consumed by humans is the residual once the caloric contribution of meat
is subtracted:
(2.3)
Dfood
c,t
= Dt −Dm,t.
Then, total crop demand in the system comprises both food and feed uses:
(2.4)
Dc,t = Dfood
c,t
+ Dfeed
c,t ,
where Dfeed
c,t
is the requirement of the livestock sector for feed inputs (see Section 2.3). To mimic real-world
adaptive demand responses, the model incorporates a feedback mechanism in which consumption is partially
suppressed if prior output falls short of demand—analogous to price feedback in markets:
(2.5)
Dm,t ←Dm,t
ωm
1 −αm
Dm,t−1 −Qm,t−1
Dm,t−1
,
(2.6)
Dfood
c,t
←Dfood
c,t
ωc
1 −αc
Dc,t−1 −Qc,t−1
Dc,t−1
,
where Qm,t−1 and Qc,t−1 denote last period’s meat and crop output, respectively, and ωm, ωfood
c
are nor-
malization constants.
The notation ←indicates that the exogenously determined demand at time t is
endogenously adjusted in response to supply shortfalls in t −1. Lastly, normalizing by ωm and ωc ensures
appropriate scaling of global demand within our stylized representation of the Earth system, while preserving
the overall trend – see Appendix A.1 for further details.
2.3. Production: Crop and Livestock Systems
Our model incorporates both conventional and organic production systems, which together determine the
total outputs of crops and livestock at each time step:
Qc,t = qc,t + qo
c,t
Qm,t = qm,t + qo
m,t,
(2.7)
where q·,t and qo
·,t denote conventional and organic outputs respectively. The share of land under organic
management increases exogenously, while both systems are managed by adaptive rules simulating a represen-
tative farmer who adjusts input intensities in response to changing economic and environmental conditions.
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY5
For conventional crop production, total yield (qc,t) is modeled as a multiplicative function of cultivated
area (Ac,t), technological level (Tt), chemical inputs (Φc,t ≥1), mechanization (Mc,t ≥1), and the condition
of the natural ecosystem (0 < ¯εc,t ≤1):
(2.8)
qc,t = Ac,t(1 + Tt)(Φc,t)k(Mc,t)f(¯εc,t)1−k−f
The exponents k < 1 and f < 1 determine the sensitivities of yield to chemical and mechanical intensification,
while ¯εc,t captures the impact of average ecosystem integrity on conventional cropland.
This multiplicative function follows the long tradition of agricultural production modeling (e.g., Cobb–Douglas
and Mitscherlich-type functions, see (57)), which assume proportional rather than additive contributions of
inputs. Including chemical inputs and mechanization reflects historical evidence that these were the dominant
drivers of yield growth during the Green Revolution (22; 63). In contrast, Tt is modeled as an efficiency shifter
that increases the productivity of inputs without proportionally raising environmental degradation—for in-
stance, through innovations in precision agriculture or improved crop varieties (see Eq. 2.20). This follows
the common interpretation of technology as a residual factor capturing yield gains not directly attributable
to input intensification (62; 76).
Total organic crop yield, which forgo chemical inputs, is given by:
(2.9)
qo
c,t = Ac,t(1 + Tt)(Mc,t)f(¯ε o
c,t)1−k−f
where ¯ε o
c,t is averaged only over organic cropland. Since Φc,t ≥1, organic yields are typically lower than
those from conventional management.
Conventional meat production follows a similar structure, with intensification achieved through increased
livestock density (Λm,t):
(2.10)
qm,t = Am,t(1 + Tt)(Λm,t)h(¯εm,t)1−h
where Am,t is pasture area, Λm,t is livestock density, and h modulates sensitivity to ecosystem integrity.
Under organic livestock management, livestock densities are restricted, and yields are determined as:
(2.11)
qo
m,t = Am,t(1 + Tt)(min{Λmax, Λm,t})h(¯ε o
m,t)1−h.
Planned meat output generates a demand for crop-derived feed, which is passed to the crop module as:
(2.12)
Dfeed
c,t
= r
qm,t + qo
m,t
where r is the crop-to-meat feed conversion factor. If total crop production is insufficient to meet both food
and feed demands, livestock production is scaled back accordingly:
Qm,t ←Qm,t min
1,
Qc,t
Dfood
c,t
+ Dfeed
c,t
!
(2.13)
This minimization ensures that meat output does not exceed the level achievable with available crop re-
sources after accounting for both direct human consumption and animal feed requirements. The symbol ←
indicates that this constraint applies within the same time period, i.e. livestock are assumed to rely on crop
production from the current period. While this represents a simplification—real systems include storage and
trade—introducing such mechanisms would substantially increase model complexity.
Input intensities (mechanization, chemical use, livestock density) are updated dynamically for both organic
and conventional systems as if managed by a representative farmer who adjusts input intensities to reduce
gaps between demand and output in each period. This assumption is consistent with standard adjustment-
cost frameworks in agricultural economics and ecological-economic modeling (17; 35).
Formally, adjustment follows a proportional correction rule:
(2.14)
Mt = Mt−1 + βMt−1
Dc,t −Qc,t−1
Dc,t
(2.15)
Φt = Φt−1 + δΦt−1
Dc,t −Qc,t−1
Dc,t
(2.16)
Λt = Λt−1 + γΛt−1
Dm,t −Qm,t−1
Dm,t
6A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
where β, δ, and γ set the speed of adaptive intensification. Technology advances exogenously via logistic
growth, aligning with standard practice in stylized food–environment models (40):
(2.17)
Tt = Tt−1 + νTt−1
1 −Tt−1
Tmax
where ν and Tmax respectively determine the rate and the upper bound of technological advance.
2.4. Landscape and Natural Ecosystem Integrity
The simulated landscape is represented as a 100 × 100 grid1, where each cell is designated as cropland,
pasture, or natural land, providing a stylized analogue of real-world land use (Fig. 2). The initial allocation
reflects global 1960 conditions; over time, land transitions between states as economic and environmental
pressures change.
Natural land may be converted to agriculture if production consistently falls short of demand, whereas
underutilized or low-productivity agricultural plots can be abandoned and gradually restored through natural
processes. In addition to these demand-driven adjustments, the model incorporates a baseline rate of land
expansion and contraction to capture natural turnover in the agricultural sector. This reflects the reality
that some farmers exit or enter farming independently of production shortfalls or market conditions, due to
personal, institutional, or contextual factors (38; 8). Specifically, the rates of expansion (φ+
·,t) and contraction
(φ−
·,t) of agricultural land are modeled as:
(2.18)
φ+
·,t = φ
1 + ζ+
·
D·,t −Q·,t−1
D·,t
(2.19)
φ−
·,t = φ
1 + ζ−
·
Q·,t−1 −D·,t
D·,t
Here, φ+· and φ−· represent baseline expansion and contraction rates, while ζ+· and ζ−· scale the response
to unmet demand or excess production. The notation · is a placeholder for the relevant land type (crop or
pasture).
Sites prioritized for conversion to agriculture are those with highest ecosystem integrity (thus promising
higher yields), while marginal, less productive plots tend to be abandoned first, consistent with empirical
patterns of agricultural abandonment in more degraded areas (66; 10).
Each site is assigned a natural ecosystem integrity index (ε), which serves as a synthetic indicator quantify-
ing both above-ground and below-ground ecosystem conditions. Specifically, ε reflects the multi-dimensional
state of ecosystem health, including biodiversity as well as soil health attributes. This approach draws on
established ecological frameworks (20; 84; 5), where ecosystem integrity is understood as the capacity of
an ecosystem to maintain its characteristic structure, composition, and functional processes in the face of
human disturbance. In our model, natural sites have ε values between 0 and 2, reflecting a gradient from
degraded to pristine conditions, whereas agricultural plots are constrained between 0 and 1. Conversion
from high-integrity natural to agricultural land causes a sharp drop in ε, representing rapid loss of habitat
complexity (14). In contrast, restoration or abandonment of agriculture allows for gradual recovery, with
sites retaining their existing ε and accumulating integrity over time. For analytical purposes, we refer to
cells with ε > 1.9 as forest and those with ε < 0.1 as degraded—these labels facilitate interpretation but do
not alter the underlying dynamics.
Ecosystem integrity in each cell (εi,t) is updated at every time step using exponential decay, a standard
approach in stylized ecological models (54; 53). This formulation captures both degradation from agricul-
tural use and natural restoration when land is left fallow. Exponential decay is widely used to represent
gradual ecosystem degradation due to sustained pressures, while logistic-style recovery functions mimic the
diminishing returns of restoration as ecosystems approach their maximum integrity (44). Let τi denote the
land-use type of cell i—m for pasture, c for cropland, and n for natural land—and let Et represent the
system-wide provision of ecosystem services at time t (see 2.21). The update rule is:
1This system size was chosen to be sufficiently large to smooth out effects of random initialization, while remaining computa-
tionally tractable. Results remain consistent when varying grid size, since no variable explicitly depends on system dimensions.
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY7
(2.20)
εi,t =
εi,t−1
1 −θi,m
Λt
Et
if τi = m
εi,t−1
1 −θi,c
Mt + Φt
Et
if τi = c
εi,t−1
1 −θi,nEt
1 −εi,t−1
εmax
if τi = n
Here, Λt, Mt, and Φt are the intensities of livestock density, mechanization, and chemical inputs, respectively.
The parameters θi,m, θi,c, and θi,n determine the rates of degradation (for agricultural land) or restoration
(for natural land), modulated by overall system intensification and the supply of ecosystem services. εmax
represents the maximum possible ecosystem integrity.
Ecosystem service supply is captured by a non-linear function of the area and integrity of natural patches
in the landscape, drawing on empirical evidence for sublinear (concave) scaling (44; 52):
(2.21)
Et =
P
i∈N εi,t
P
i∈N εi,0
p
where N stands for all natural cells and p < 1 ensures diminishing returns as the natural area increases.
2.5. Parameter Specification and Numerical Experiments
The model parameters were determined by combining data-driven estimation, calibration against observed
agricultural trends, and fixed assignments based on commonly-used values found in the literature (Fig. 3).
Most ecological parameters, including the average degradation and restoration rates (θi,·), and the ecosystem
service exponent (n), were primarily set using empirical data or well-established values from previous studies.
The same approach was used for the power exponents in the production functions. For these parameters, a
more detailed discussion of choice, context specificity, and inherent complexity is provided in the Appendix.
Of the remaining parameters, those with influence limited to exogenous variables were fitted directly to
corresponding empirical data (e.g. demand parameters and the rates of sustainable and organic production
adoption). Those that depended on the dynamic realization of the system were calibrated to reproduce
observed land use and input trends, particularly using the 1960–2022 FAO data (28).
We explored our model through extensive numerical simulations. Unless stated otherwise, results are
presented as averages over 500 simulation runs, and standard errors are not shown in the figures as their
magnitude was found to be negligible (smaller than line width). It should be noted that the model does
not include stochasticity in the production functions; thus, the observed variation among simulations arises
solely from the randomly generated landscape configurations, the stochasticity of restoration and degradation
rates, and the random nature of the cells targeted for land-use change.
We evaluated a business-as-usual trajectory alongside a suite of alternative policy scenarios. The baseline
scenario was produced by running the model up to the year 2100 with the parameter values listed in Table
1 of the Appendix, while the policy scenarios targeted modifications in the form of constraints or adjusted
model parameters. Unless otherwise noted (as in Fig. 6), all parameter variations in these scenarios represent
a 10% increase or decrease relative to the corresponding calibrated baseline value. For scenarios involving
changes in demand, we implemented a 10% reduction with respect to the additional demand attributed to
projected population and income growth between 2022 and 2100. To maintain consistency and comparability
between scenarios, every simulation was initialized with all natural ecosystem integrity values set to their
maximum value for both natural and agricultural land, and assumed that all land not allocated to agriculture
is classified as forest, and thereby excluding the presence of semi-natural vegetation. Although this initial
condition may not fully reflect levels of natural degradation in 1960, it is reasonable given that the global
expansion of intensive agricultural practices has occurred mainly in recent decades. Moreover, because our
primary focus is on outcomes relative to the baseline scenario, this assumption does not substantially affect
our overall conclusions. Further information on the initialization procedure can be found in the Appendix.
8A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
Figure 2. Spatial patterns of the initial landscape and subsequent land-use changes. Panel (a)
shows the initial arrangement of natural land (green), cropland (yellow), and pastureland (brown),
capturing the mosaic structure at the start of the simulation. Panels (b) to (d) illustrate land-
use transitions by 2100: green marks new natural land arising from abandonment and restoration,
yellow corresponds to recent cropland expansion, brown highlights pastureland conversion, and gray
indicates cells that switched from natural land, cropland or pasture land land-use types, respectively,
to another over the simulation period. As agriculture expands, cropland and pasture increasingly
encroach upon forest edges, concentrating in more fertile zones and leaving behind scattered, less
productive patches that begin natural restoration.
Panels (e) to (g) present histograms of the
final distribution of ecosystem integrity for natural land, cropland, and pastureland, respectively.
These distributions reveal pronounced degradation in croplands due to continued intensification via
mechanization and chemical inputs. Simultaneously, abandoned sites emerge in the natural land
distribution, showing early signs of ecological recovery, although these seminatural patches remain
well below the ecosystem integrity threshold necessary to qualify as forest (εi,t > 1.9, see Eq. 2.20).
3. Results
The business-as-usual scenario provides a reference trajectory under current trends in population, con-
sumption habits, and land management practices.
This scenario serves as a baseline against which the
effectiveness and trade-offs of alternative policy interventions can be assessed.
3.1. Business-as-usual Scenario
The business-as-usual scenario highlights a persistent transformation of the landscape, primarily driven
by increasing demand for food and livestock products (see Fig. 4). As agricultural activity intensifies and
expands, natural and semi-natural ecosystems are progressively converted or degraded (see Fig. 4a and
Fig. 4d). Indeed, the expansion of agricultural land directly reduces both the area and overall integrity of
natural habitats, resulting in a notable decline in forest land and average ecosystem health within each land
use category.
This transformation is particularly evident in the case of cropland (see Fig. 4b and Fig. 4e). Here, both
land consumption and ecological degradation reach their highest levels, largely as a consequence of the high
degree of intensification and mechanization required to meet rising food demands. A substantial proportion
(≈40% in recent years) of these crops is allocated to livestock feed, and thus the ecological footprint of crop
expansion is strongly tied to meat production and associated changes in dietary habits.
Pastureland dynamics further illustrate the complex feedback mechanisms in the system. While advances
in technology and animal husbandry—such as vaccines and growth hormones—have enabled higher livestock
densities and contributed to a recent decline in the area under pasture, our model projects that this trend
will ultimately reverse. As technological improvements plateau and can no longer keep pace with rapidly
rising demand for meat, the pressure for additional pasture expansion resurfaces.
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY9
Figure 3. Model calibration against major historical trends in agricultural, land use, and socioeco-
nomic indicators. Each panel displays the model’s simulated results (solid orange lines, normalized
to 1 in 1960 to emphasize rates of change) and historical observations from 1960 to 2022 (black
diamonds; data from FAO (28; 27)). The model parameters were calibrated to reproduce empirical
trends, and all curves represent averages over 500 Monte Carlo simulations. In particular, the model
closely captures key historical patterns, in particular sustained increases in agricultural production
and yield, driven by a significant expansion in total demand for both crop and animal products. It
also reproduces the peak and subsequent decline in the rate of pasture land expansion, which results
from land use efficiency gains as intensification has offset some demand growth. Stable or rising
trends in production inputs, such as fertilizer use and mechanization, are also well represented.
Gray-shaded panels indicate exogenous input variables (e.g., population, per capita income): here,
dashed orange lines represent fitted values matched directly to observed data, which are then used
as fixed, absolute model inputs.
Taken together, these trends accelerate the fragmentation of remaining high-integrity habitats and the
spread of degraded patches across the landscape. As demonstrated by spatial analyses (Fig. 2), contigu-
ous forest edges are eroded, creating a more fragmented matrix of land uses, while marginal agricultural
areas are often abandoned in favor of clearing more productive forested sites. This mosaic of change not
only undermines biodiversity but also weakens the ecosystem services that support long-term agricultural
productivity.
Collectively, these findings underscore that, without targeted interventions, current trajectories reinforce a
self-perpetuating cycle of ecological decline and agricultural expansion. Under this business-as-usual scenario,
we observe reduced landscape resilience, increasing dependence on intensification solutions. The baseline
scenario therefore highlights the urgent need to explore alternative policy actions.
3.2. Effects of Single Policy Measures
We next assess the effects of implementing individual policy interventions, each focused on a specific
lever within the food and land system. The complete set of considered scenarios is presented in Fig. 5,
with corresponding parameter details provided in the Appendix. This analysis not only evaluates the direct
impact of each measure but also reveals how compensatory system responses can undercut their effectiveness,
producing trade-offs or unintended outcomes.
10A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
Figure 4. Land use and ecosystem integrity trajectories under the business-as-usual scenario.
Top panels (a–c) show the temporal trends of the various land-use types: panel (a) tracks the
ongoing decline in forest area, panel (b) captures the steady expansion of cropland, and panel
(c) traces changes in pastureland.
The gray-shaded background indicates the historical period,
with projected dynamics shown to the right of the dashed vertical line. Pastureland area initially
decreases due to advances in technology and more efficient animal husbandry, which allow for higher
livestock densities. However, as technological improvements plateau and can no longer match the
rising demand for meat, our model anticipates a renewed expansion of pasture area in the future.
Bottom panels (d–f) depict temporal changes in the mean ecosystem integrity index for natural
land (d), cropland (e), and pastureland (f). These plots reveal the rapid and sustained degradation
of ecosystems associated with agricultural expansion and the ongoing loss of ecological integrity
across all land types. Of particular interest is the fact that declines in natural ecosystem integrity
lag behind the decrease in forest area; this is because deforestation initially diminishes the supply of
ecosystem services vital for maintaining ecosystem stability (see Eq. 2.21). As these services decline,
the degradation of agricultural land accelerates (see Eq. 2.20), leading to land abandonment and a
reduction in the average natural ecosystems integrity.
Fig. 5a maps single-policy scenarios in a space defined by forest land area variation and degraded land
variation, both calculated relative to the baseline. These two metrics were selected to capture the dual
dimensions of land system change: forest land variation reflects shifts in the extent of intact natural habitats
(landscape-scale expansion or protection), while degraded land variation quantifies changes in ecosystem
integrity within managed areas due to intensification or restoration efforts. By presenting results in this way,
we can compare policies that limit expansion (horizontal direction) with those that primarily aim to improve
land already in production (vertical direction), clearly illustrating system-wide trade-offs.
The analysis reveals considerable variability in policy outcomes, much of which stems from compensatory
dynamics—where gains in one area trigger pressures elsewhere. For example, supply-side measures such as
reducing chemical inputs or expanding organic crop production are effective at lowering land degradation,
but typically cause declines in forest area. This is because reduced yields on existing farmland (from chemical
restrictions or organic conversion) require compensatory expansion, driving new deforestation and landscape
fragmentation to maintain overall production levels. Similarly, implementing deforestation bans for crop or
meat production often leads to increased land degradation, because production pressures shift onto existing
agricultural land, resulting in more intensive management. Over time, such intensification can diminish
yields and degrade ecosystem integrity, which may ultimately drive further land expansion and limit the
effectiveness of deforestation policies.
Conversely, demand-focused interventions—particularly reductions in crop or meat consumption—deliver
more comprehensive positive outcomes, as they simultaneously relieve pressure for both intensification and
expansion. Policies that reduce meat consumption result in the largest gains for forest area and the most
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY11
Figure 5. Impact of individual and combined policy interventions on forest and degraded area
variation. Variations in forest area (horizontal axis) and degraded area (vertical axis) are shown as
percentage changes with respect to the baseline scenario. Points in the upper-right green quadrant
indicate policy strategies that succeed in both expanding forest area and reducing degraded land,
while those in the lower-left quadrant are detrimental on all grounds. Panel (a) shows the effects of
individual (single) policy interventions, each marked by a labeled point. Only the reduction in meat
consumption achieves positive outcomes on both axes, illustrating that most single interventions
are insufficient due to persistent compensatory dynamics within the system.
Panel (b) reports
outcomes for policy mixes that specifically include increased organic crop production and decreased
livestock density—two interventions often viewed as hallmarks of sustainability. Here, each point
represents a different mix with other measures.
The results demonstrate that these strategies,
even when paired with moderate reductions (10%) in crop and meat demand, consistently fail to
halt forest loss, often resulting in further declines in forest land area. Panel (c) highlights selected
policy mixes containing key interventions, such as chemical use reduction or restrictions on crop
and livestock-driven" deforestation, with distinct symbols denoting which measures are included.
This panel shows that policy mixes combining interventions that target both intensification and
expansion (i.e. orthogonal strategies) are situated in the green quadrant, signifying that integrated
approaches are most effective at overcoming compensatory effects.
pronounced reductions in degraded land.
This is because a substantial share of cropland is devoted to
growing feed for livestock, and thus plant-based diets reduce both direct and indirect land pressures
4. Synergies and Trade-Offs in Policy Portfolios
We then turn to the analysis of policy combinations addressing multiple drivers at once, exploring how
different mixes of interventions impact ecosystem integrity. Figures 5b, 5c, 7 and 6 summarize these results,
allowing direct comparison of policy portfolios.
12A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
Figure 6. Effect of demand reductions on forest and degraded land areas under enhanced organic
crop production and reduced livestock density policies. Each panel shows the percentage change
in forest land (green, left axis) and degraded land (orange, right axis) compared to the baseline,
across scenarios with progressive demand reductions. In particular, the x-axis represents increasing
reductions in projected crop and meat demand by 2100, where 0 means no change from business-as-
usual and 1 means no demand increase after 2022. The results demonstrate that only when demand
reductions surpass roughly 40% do both forest expansion and significant decreases in degraded land
occur.
An initial finding, visible in Figure 5b, is that policies often regarded as highly sustainable—namely,
widespread adoption of organic crop production (eliminating chemical inputs) and reduced livestock inten-
sification (for improved animal welfare)—consistently result in a declining forest land area, regardless of
whether they are combined with moderate reductions (10%) in crop and meat demand.
The underlying mechanisms behind this result differ between crop and meat systems. For organic crop
production, the primary constraint is related to the strong diminishing returns (k = 0.2 in Eq. 2.8 and
Eq. 2.9) from chemical inputs. Moderate chemical use can maintain relatively high yields, but if chemical
inputs are eliminated entirely, yield losses grow disproportionately and become very difficult to offset. Most
farmers are unable to fully compensate for these losses through additional intensification or mechanization,
as the required extra effort would be prohibitive—leading the system to expand cultivation into natural areas
to maintain production targets.
The dynamics differ for organic meat production. Here, shifting to organic practices does not generate the
same drastic loss of productivity per unit area because industrial animal production is less constrained by
input intensification—modern techniques like vertical livestock integration (as seen recently in China (33; 83))
support continued output growth. Consequently, organic meat policy mixes exert only modest negative effects
on forest land. However, if livestock production is broadly de-intensified without such adaptive capacity,
the pressure to maintain meat supply quickly translates into greater expansion of pasture or feed crops,
significantly amplifying deforestation.
While some past studies have stopped at these findings (25)—concluding that sustainable production is
unfeasible—our model enables quantification of the scale of demand reduction necessary to reverse these
negative trends. As shown in Figure 6, only scenarios with at least a 40% reduction in projected crop and
meat demand by 2100 achieve net gains in forest cover and notable reductions in degraded land when either
an increase of organic crop production or a reduction in livestock density is promoted. This highlights the
scale of lifestyle and societal change required for promoting animal welfare and eliminating chemical use.
Focusing next on policy mixes that exclude organic crop expansion and livestock intensity reduction,
the analysis reveals that combining interventions yield more favorable outcomes than isolated policies. As
demonstrated in Fig. 5c, mixes incorporating at least two different policies outperform single interventions.
When policies target both degradation (e.g.
reduced chemical use) and deforestation (e.g.
conversion
controls) together, their effects reinforce one another and mitigate compensatory expansion or intensification.
Furthermore, these policies can initiate a positive feedback loop: by halting deforestation, ecosystem service
provision is enhanced, and when combined with reduced intensification, this leads to improved natural
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY13
ecosystem integrity. As ecosystem health recovers, agricultural yields are boosted, which in turn reduces the
need for future land expansion.
Examining the highest-performing mixes (Figure 7), we see that the top portfolios for reducing degra-
dation mainly involve a combination of reduction in chemical inputs alongside decrease in crop and meat
consumption—pointing again to the necessity of shifting dietary habits. Conversely, the best-performing
portfolios for forest preservation include deforestation restrictions and meat demand reduction, while mea-
sures aimed solely at reducing chemicals or crop demand are less central—since they contribute minimally
to curbing deforestation, especially that driven by livestock.
Ultimately, only a single portfolio achieves top-tier outcomes for both forest recovery and land degradation,
combining deforestation limiting measures, intensification limits, and reductions in both meat and crop con-
sumption. While additional trade-offs may exist beyond the scope of this study, these results highlight that
well-designed policy portfolios aligning both supply- and landscape-level actions—with a strong emphasis on
demand reduction—are essential for achieving “win-win” gains and real progress toward sustainability.
5. Discussion
5.1. Beyond the Supply-Side: Systemic Barriers to Sustainability
Our results highlight important limits to achieving a sustainable food system through supply-side reforms
alone, echoing findings across the literature on land use, agricultural intensification, and food system sustain-
ability (26; 69; 73). Measures such as reducing chemical inputs and adopting more animal-friendly or organic
production systems provide meaningful improvements at the farm scale—ranging from reductions in local
pollution to gains in animal welfare and soil health (65; 78). These changes are valuable in themselves, and
often come with additional social benefits, including positive impact on human health and responsiveness to
consumer demand for more sustainable food options.
However, our model suggests that when per-hectare productivity is deliberately limited—whether by ban-
ning chemical inputs or reducing livestock density—the system tends to compensate by converting additional
natural or semi-natural land to agriculture. This compensatory expansion can reduce or even offset the en-
vironmental benefits achieved on existing farmland, and in some cases may exacerbate pressures such as
deforestation, landscape fragmentation, and ecosystem degradation, consistent with previous observations
about indirect land-use change (72; 41). Even when multiple supply-side measures are combined, our analysis
indicates that their cumulative impacts alone are unlikely to halt forest loss or secure broad-based ecosystem
integrity.
Let us stress that these findings do not imply that supply-side reforms are undesirable; rather, they
highlight the need to complement them with demand-side transformations and broader systemic changes if
the goal is to achieve consistent sustainability.
5.2. Structural Need for Demand Reductions
A second, and more profound, implication of our findings is that only strong demand-side interven-
tions are capable of delivering transformative change.
Substantial reductions in total food—especially
meat—consumption relieve system-wide pressure by cutting both the intensification and expansion drivers
simultaneously. This is consistent with a growing body of research indicating that dietary shifts toward
reduced animal-source foods (i.e., more plant-based diets) are critical to staying within planetary boundaries
and achieving global climate, biodiversity, and public health targets (85; 64; 77).
Our quantitative analysis provides a concrete threshold: unless at least 40% of projected excess demand
for crops and meat is curbed by 2100, even the most ambitious supply-side or reforestation interventions are
unlikely to reverse deforestation or halt land degradation (see also (2)). Crucially, demand modification is not
only more effective at limiting land use changes but also makes it easier for other sustainability policies (like
reduced chemical use or organic expansion) to succeed—a synergy recognized in other modeling exercises
and global sustainability assessments (16).
5.3. Integrated Approaches and Systemic Solutions
Despite the importance of demand-side measures, our results do not dismiss the value of integrated,
multi-faceted policy mixes.
Rather, they demonstrate that combinations of carefully designed interven-
tions—addressing both production and consumption—yield the most robust improvements for both forest
and degraded land areas variation. Such an integrative approach directly mitigates compensatory effects,
14A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
Figure 7. Comparative effectiveness of top-performing policy mixes for reducing deforestation
and land degradation. Panels (a) and (b) display, for each of the ten best policy combinations,
the relative variation in forest land and degraded land areas as a percentage change from the
baseline scenario. Policy mixes are ranked separately by their success in reducing deforestation (a)
and degradation (b). Dashed bars indicate policy mixes that appear among the top performers
for both objectives together. Panel (c) details the intervention composition of each policy mix.
Each row represents a ranked policy mix (from most to least effective; top to bottom), and each
circle indicates the inclusion of specific interventions for the corresponding scenario (green for
deforestation-focused, orange for degradation-focused). Despite some overlap, the policy measures
most effective at reducing deforestation are not always the same as those best at lowering land
degradation. Only a policy portfolio achieves high performance in both dimensions, highlighting
the importance—and challenge—of integrated strategies capable of delivering simultaneous gains
for forest conservation and reduced agricultural ecosystem degradation.
wherein policies addressing one part of the system inadvertently shift environmental pressures elsewhere
unless implemented holistically (41; 48; 75). For instance, coupling landscape-level deforestation controls
with intensity reduction measures (e.g., limits on chemical inputs) can prevent rebound effects and increase
overall ecosystem integrity (55; 9).
It is worth noting, however, that many popular narratives around sustainable intensification or organic
conversion tend to overemphasize the benefits of any single intervention, without sufficient attention to the
cross-scale feedbacks and compensatory land dynamics illuminated by both our results and other studies (31;
60). A successful transition to a food system that maintains ecosystem integrity and food security together
will likely require not just technical or agronomic innovation, but deep societal, economic, and behavioral
changes (69).
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY15
5.4. Limitations and Unaccounted Aspects
While we believe our study advances the understanding of systemic trade-offs and leverage points in the
pursuit of sustainable food and land systems, it is important to acknowledge several limitations that may
influence the interpretation and applicability of our findings—both in terms of modeling assumptions and
the practicalities of real-world policy implementation.
Our framework adopts a stylized, spatially explicit global representation. This abstraction allows us to
distill essential system dynamics and identify general insights on policy interactions, while avoiding the
computational and data demands of fully resolving geographic, ecological, and socio-economic heterogeneity.
But, local variations—shaped by climate variability, land tenure, market integration, and governance (41;
48; 82)—are not explicitly represented. However, these features could be incorporated in future extensions
through nested multi-scale models or spatially explicit downscaling approaches, enabling more context-
sensitive analyses (43).
Similarly, our land conversion and abandonment rules respond primarily to aggregate supply-demand
imbalances. This simplification abstracts from mediating effects such as access to infrastructure, transaction
costs, or governance quality. While this allows transparent exploration of system-wide drivers, future model
iterations could incorporate heterogeneous decision-making rules or probabilistic entry/exit dynamics to
capture local socio-institutional variation (31).
Crop and livestock production modules rely on generic functional forms calibrated to global data. While
sufficient for identifying broad trends, these formulations omit local adaptation, site-specific constraints (e.g.,
soil degradation, pests, water stress), and stochastic variability in yields. Likewise, our management sys-
tems currently exclude agroecological or diversified strategies, focusing instead on conventional and organic
monocultural systems. Future work could integrate heterogeneity in production technologies, mixed-cropping
systems, or resilience-enhancing practices to more realistically capture the spectrum of sustainable manage-
ment options (39).
The socio-political dimension of policy implementation is currently represented exogenously, assuming
instantaneous and uniform enactment. While this simplification enables clear analysis of policy potential,
it neglects real-world constraints such as political resistance, institutional capacity, stakeholder engagement,
and uneven enforcement (49; 56). Incorporating adaptive policy implementation, delays, and compliance
heterogeneity would allow more realistic assessment of feasible outcomes and the robustness of interventions.
Our model does not differentiate actors in the food system, and therefore omits social and equity di-
mensions. While this enables focus on aggregate system-level trade-offs, it excludes potential distributional
consequences such as food insecurity, livelihood impacts, or shifts in nutrition.
Future extensions could
integrate agent-based or household-level modules to explore these outcomes under alternative policy scenar-
ios (86; 15).
Climate-related risks and long-term ecological processes are simplified. Feedbacks from extreme events,
biodiversity loss, tipping points, and delayed ecosystem recovery are not dynamically represented.
This
choice was made to retain tractability for global simulations while highlighting general mechanisms. Future
work could include stochastic climate shocks, ecological inertia, and recovery lags to capture more realistic
ecological dynamics (71; 47; 18).
Behavioral and demographic dynamics are also exogenous. Food demand does not adapt through social
learning, policy feedback, or price-mediated behavior, and population changes and migration are not modeled.
These processes are known to shape long-term land-use trajectories (37; 4).
Incorporating endogenous
behavioral adaptation and demographic dynamics would allow more robust exploration of feasible pathways
and potential rebound effects (42; 79).
Finally, as with all global modeling efforts, results are subject to parameter uncertainty, data limitations,
and simplifications of real-world complexity. Our stylized approach enables identification of core system-
level interactions and policy synergies, but should be interpreted as generating hypotheses rather than
prescriptive solutions. Future work should prioritize multi-scale, participatory, and transdisciplinary models
that incorporate climate risk, social equity, behavioral feedbacks, and realistic policy processes, providing
the basis for actionable, locally calibrated insights (42; 58).
Conclusion
The results of this study highlight the urgent need for a paradigm shift in how we address food system
sustainability. Technological and agronomic improvements, while necessary, are insufficient in isolation to
16A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
meet the challenges posed by rising global food demand and intensifying environmental pressures. Our sys-
temic approach reveals that only integrated strategies—combining ambitious dietary changes with reformed
agricultural practices and strengthened landscape conservation—can reconcile the competing demands of
productivity, ecosystem integrity, and social equity.
Our research supports the growing consensus in the literature that demand-side interventions must be
placed at the center of sustainability and climate agendas (85; 32). Avoiding counterproductive trade-offs
requires a holistic, system-based approach that aligns policies across supply, demand, and governance. At
the same time, future progress depends on robust efforts to tailor and implement these solutions to diverse
regional, cultural, and socio-economic contexts, as well as on building public support for transformative
change.
Going forward, further refinement of integrated models like ours is needed. In particular, future research
should focus on incorporating more explicit representations of climate impact, equity dimensions, market and
policy dynamics, and the complex processes underlying behavior and dietary shifts. Advancing multi-scale,
participatory, and interdisciplinary modeling will be crucial to develop actionable roadmaps for policymakers
and stakeholders.
Acknowledgments
This research was conducted within the Econophysics & Complex Systems Research Chair, under the aegis
of the Fondation du Risque, the Fondation de l’Ecole polytechnique, the Ecole polytechnique and Capital
Fund Management. Michel Loreau was supported by the TULIP Laboratory of Excellence (ANR-10-LABX-
41).
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A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY21
Appendix
A.1. Initialization
The system initialization proceeds as follows:
• The model employs a 100×100 grid to represent the landscape, initially allocating 50% of the area to
forest land. This proportion reflects estimates for the year 1960 based on FAO data (28). To spatially
distribute this fraction, a fractal field with a characteristic correlation length ξ is generated. Previous
research suggests that fractal patterns realistically capture empirical land use distributions (51; 87).
The value for ξ, chosen to produce a visually realistic arrangement, is reported in Table 1. After
allocating forested cells, the remaining landscape is randomly assigned to pasture (35%) and cropland
(15%).
• Next, each cell receives specific degradation and restoration rates. These are assigned using the
concept of lifespan, which aligns well with the properties of exponential distributions. This approach
has been motivated by soil science literature, where the exponential lifetime of soil layers is tied to
degradation and erosion processes (21). Following results this stream of research, each parameter
θi,· is sampled as:
θi,· =
1
N(µ·, σ·),
where · stands for natural (n), pasture (m), or cropland (c), and N denotes a lognormal distribu-
tion. The adoption of a lognormal distribution ensures the statistical characteristics of the rates are
consistent with those reported by Evans et al. (21).
• Initial real demand for both crops and meat is computed following Eq. 2.1 and Eq. 2.2. To translate
this demand into the model’s internal units, we normalize by assigning constants ω· such that the
initial yield for both livestock and crops is set to q·,0 = 1. Assuming that at the starting point total
production matches demand, we have Qc,0 = 1500 = Dc,0 and Qm,0 = 3500 = Dm,0. Since this step
only rescales the reference point, the overall system dynamics are unaffected.
• Intensification factors are then initialized to have q·,0 = 1 in the initial year, which requires livestock
density, mechanization, chemical input intensity, and average ecosystem integrity to all be set to
one at initialization. In contrast, we assign the initial technology level a small value (T0 = 0.001),
reflecting the limited state of agricultural innovation prior to the Green Revolution. While this may
result in yields slightly above one, the difference is negligible for trend analysis.
• With all state variables initialized, the model is ready to proceed with the simulation.
A.2. Calibration procedure and parameter values
Model parameters were categorized into four distinct groups (see Table 1): (i) measured parameters (M),
which are directly fitted from empirical datasets; (ii) estimated parameters (E), sourced from values reported
in the literature; (iii) calibrated parameters (C), which are adjusted during model calibration to achieve the
best agreement between simulated and observed data; and (iv) fixed parameters (F), which are manually
specified due to the lack of real-world analogs or because of model simplifications.
Measured parameters (M) are defined as those whose values are determined using empirical data cor-
responding directly to the model functions they influence.
This group mainly encompasses parameters
governing demand and the initial state of the landscape, where reliable datasets are readily available. Pa-
rameter values were estimated by fitting model outputs to observed data from FAO(28; 27), using a least
squares minimization approach to ensure alignment with real-world patterns. It should be emphasized that
the aim of this calibration was not to achieve exact reproduction of the full distribution of empirical curves,
but rather to capture the essential trends relevant to the aims of this study. Within this context, all fit-
ted functions yield a robust agreement with the empirical data, adequately representing the characteristic
patterns for each parameter considered (see Fig. 3).
Estimated parameters (E) were assigned values drawn from empirical studies and widely reported liter-
ature benchmarks. This category chiefly includes ecological parameters such as rates of degradation and
restoration, for which precise values are not directly measurable within the model context. As described in
Section A.1, these rates were informed by global analyses and meta-analyses covering soil degradation and
forest recovery dynamics(21; 30; 19). For parameters associated with the yield function, values were chosen
based on extensive agricultural economics and ecological economics literature(23; 11; 57). While reported
22A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY
Section
Notation
Description
Value
Group
Landscape
Grid size
100x100
F
Initialization
Natural land share
50%
M
Cropland share
15%
M
Pasture land share
35%
M
ξ
Landscape fragmentation
10
F
µc
Crop degradation distribution shift
1
E
µm
Pasture degradation distribution shift
1
E
σc
Crop degradation distribution scale
10000
E
σm
Pasture degradation distribution scale
5000
E
µn
Nature restoration distribution shift
1
E
σn
Pasture degradation distribution scale
1000
E
Demand
a
Calories demand calculation
-138.2
M
[kcal · year−1 · per capita]
b
Calories demand calculation
744.4
M
[kcal · $−1 · per capita]
c
Baseline demand for animal products
210
M
[Kg · year−1 · per capita]
r
Conversion demand for animal products to calories
3000
E
[kcal · Kg−1]
d
Income elasticity of meat demand
0.65
M
αm
Feedback strength for meat demand
0.5
C
αc
Feedback strength for crop demand
0.1
C
Production
k
Yield elasticity to chemical inputs
0.2
E
f
Yield elasticity to mechanization
0.5
E
h
Sensitivity of meat yield to intensification
0.95
E
Λmax
Maximum livestock density (organic)
3
E
β
Speed of mechanization adjustment
0.95
C
δ
Speed of chemical input adjustment
1.1
C
γ
Speed of livestock density adjustment
0.95
C
ν
Technological progress rate
0.1
E
Tmax
Technological ceiling
0.2
E
Landscape
φ
Baseline land expansion rate (crop/pasture)
3e-4
C
dynamics
ζ+
c
Expansion rate (cropland)
130
C
ζ+
m
Expansion rate (pasture land)
500
C
ζ−
c
Contraction rate (cropland)
120
C
ζ−
m
Contraction rate (pasture land)
500
C
εmax
Maximum natural ecosystem integrity
2
F
p
Diminishing return for ecosystem service
0.25
E
Table 1. Parameters of the model. Group codes can be: measured (M), estimated (E), calibrated
(C), or fixed (F).
values do vary between studies and there is no universal consensus—often due to differences in scale, crop
types, or methodological approaches—our selections are consistent with established ranges. Importantly,
the chosen parameter values are supported by the model’s ability to reproduce historical patterns of yield
growth observed in real-world data, lending further confidence to their plausibility within the framework of
this study.
Parameters classified as calibrated (C) are primarily associated with the model’s adjustment and feedback
mechanisms. These parameters cannot be directly inferred from empirical data as they are fundamentally
linked to internal model behavior—that is, they govern logical or process-based linkages for which data
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY23
are sparse or context-specific. Accordingly, they were tuned manually to best reproduce the broad pat-
terns present in observed system dynamics, using an iterative, trial-and-error approach. The goal here, as
elsewhere, was not to exactly fit every empirical curve but to recapitulate the essential dynamic behavior
pertinent to our research questions. This calibration approach ensures the model captures the relevant sys-
tem variability while maintaining enough flexibility to allow exploration of alternative scenarios within the
stylized earth system structure.
|
A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY ELIA MORETTI1,2, MICHEL LOREAU3,4 AND MICHAEL BENZAQUEN1,2,5 1Chair of Econophysics and Complex Systems, École polytechnique, 91128 Palaiseau Cedex, France. 2LadHyX, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France. 3Station d'écologie théorique et expérimentale, CNRS, 09200 Moulis, France. 4 100871, China. 5Capital Fund Management, 23 rue de l'Université, 75007 Paris, France. Abstract. Feeding a larger and wealthier global population without transgressing ecological limits is increasingly challenging, as rising food demand-especially for animal products-intensifies pressure on ecosystems, accelerates deforestation, and erodes biodiversity and soil health. We develop a stylized, spatially explicit global model that links exogenous food-demand trajectories to crop and livestock production, land conversion, and feedbacks from ecosystem integrity that, in turn, shape future yields and land needs. Calibrated to post-1960 trends in population, income, yields, input use, and land use, the model reproduces the joint rise of crop and meat demand and the associated expansion and intensification of agriculture. We use it to compare business-as-usual, supply-side, demand-side, and mixed-policy scenarios. Three results stand out. First, productivity-oriented supply-side measures (e.g. reduced chemical inputs, organic conversion, lower livestock density) often trigger compensatory land expansion that undermines ecological gains-so that supply-side action alone cannot halt deforestation or widespread degradation. Second, demand-side change, particularly reduced meat consumption, consistently relieves both intensification and expansion pressures; in our simulations, only substantial demand reductions (on the order of 40% of projected excess demand by 2100) deliver simultaneous increases in forest area and declines in degraded land. Third, integrated policy portfolios that jointly constrain land conversion, temper input intensification, and curb demand outperform any single lever. Together, these findings clarify the system-level trade-offs that frustrate piecemeal interventions and identify the policy combinations most likely to keep global food provision within ecological limits. 1. Introduction Feeding a growing and wealthier population in the 21st century presents one of the biggest challenges to planetary sustainability. Rapid income growth and global population expansion are driving up the total demand for crops and for animal products, thereby escalating the pressures on the natural ecosystem. This increasing food demand often comes at the expense of forests, semi-natural ecosystems, and long-term soil health, fueling deforestation, biodiversity loss, and land degradation at unprecedented scales (26; 81; 59). As a result, reconciling food security with the safe operating space for humanity-maintaining ecological integrity while meeting human needs-has emerged as a global policy and scientific imperative (68; 85; 45). A central question in this debate is whether ongoing increases in agricultural productivity, alongside the adoption of more sustainable farming practices-such as organic and conservation agriculture, reductions in chemical inputs, or lower livestock densities-can suffice to meet rising demand while preserving ecosystems (65; 74). Or, conversely, is a fundamental transformation of demand, especially through dietary shifts and coordinated, system-wide policy measures, necessary to avoid crossing more critical environmental thresholds? In light of clear trade-offs between food production, land use, and environmental services, understanding the boundaries and synergies of supply-side versus demand-side strategies remains an open challenge (31; 2). The literature tackling these questions spans empirical, scenario-based, and theoretical approaches. Global integrated assessment models (IAMs) such as GLOBIOM (34), IMAGE (80), and MAgPIE (46) have illuminated the possible consequences of alternative technology, land use, and diet scenarios for GHG emissions, biodiversity, and food security. Yet, their complexity and extensive parameterization can make it difficult to parse out causal mechanisms and feedback loops, or to transparently compare policy levers on a conceptual 1 16 Oct 2025 2A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY level. On the theoretical side, dynamic and system-analytic models have provided key insights into food system tipping points, feedback loops between agricultural intensification and land expansion, and the implications of self-reinforcing land degradation (6; 17; 7; 70). Land use and food system theories drawing on non-linear dynamics, spatially explicit frameworks, and ecosystem service feedback loops have highlighted the risks of unintended consequences and compensatory effects across spatial and management scales (48; 24; 4). Notably, archetype-based and stylized models have clarified how yield gains can sometimes induce additional land clearing ("rebound" or Jevons effects) rather than sparing nature, depending on feedback loops among value chains, governance, and behavioral factors (1; 41; 36). Complementing this are explorations of "land sparing" versus "land sharing" paradigms, which have underlined the limitations of both simplistic intensification and extensive conservation solutions in isolation (61; 25). Building on this theoretical tradition, our study makes several distinctive contributions to the debate. We present a transparent, parsimonious model that connects exogenous food demand trajectories to crop and livestock production, land conversion, and feedback from ecosystem integrity that influences yields and future land uses. Such an approach allows us to systematically explore the interplay of demand growth, management intensity, and policy intervention in shaping landscape outcomes and ecological stability. Key strengths of our work include a calibration to observed, contemporary global trends-most notably the rapid growth in meat demand occurring alongside sustained improvements in average crop yields. By adopting a global perspective, our analysis controls for indirect land-use change effects (88). Furthermore, our framework enables both a realistic assessment of future baseline trajectories and an exploration of plausible supply- and demand-side alternatives, moving beyond purely theoretical scenarios or country case studies. Furthermore, our spatially explicit landscape component captures the heterogeneity and emergent properties of coupled socio-ecological dynamics, allowing us to assess how context-specific degradation, and restoration processes aggregate into world-scale indicators. This integrated framework is particularly well suited to investigating "policy mixes" and their emergent synergies or trade-offs-issues that remain underexplored yet are highly salient for real-world sustainability transitions (69; 16). Within this framework, we directly address whether supply-side productivity gains can meet future food demand while protecting global ecosystems, or if coordinated policy interventions that also target consumption behaviors are essential to achieving long-term sustainability. By illuminating the dynamic, cross-scale feedbacks underpinning both successes and failures in historical and prospective management strategies, our findings contribute to a growing, more theoretically grounded understanding of the pathways-and pitfalls-toward a sustainable food future. 2. Model 2.1. Model Overview We have developed a spatially explicit, discrete-time model that dynamically links the growth in food demand to agricultural production, land conversion among primary land-cover types - namely cropland, pasture, and natural land - and crucially incorporates feedback loops from natural ecosystem integrity (Fig. 1). The model landscape is represented as a square grid where each cell's land-cover class changes over time, allowing us to track both spatial configuration and aggregate stocks of land-use types. This enables robust quantification of processes such as deforestation, natural regeneration, and degradation pressure as they unfold over time. The drivers of food demand are exogenous time series that combine projected population growth, changes in per-capita income, and a parameter reflecting dietary preference for meat. This framework is flexible enough to accommodate demand-side policy interventions, such as shifts to lower-meat diets, by perturbing these demand trajectories. Agricultural production in our model is governed by both land area and the intensity of management. Cropland yields respond to chemical intensification and mechanization while livestock output depends on available pasture area and on an adjustable livestock-density variable. If, at any time step, aggregate production fails to meet demand, the system responds first by intensifying production (for example, by increasing mechanization and chemical inputs, or livestock density). Persistent shortfalls inevitably trigger the conversion of natural land to cropland or pasture, resulting in deforestation and loss of critical habitat. Central to the model is the ecosystem integrity index, a composite indicator reflecting both soil health and biodiversity. This index decreases with intensified mechanization and chemical use but recovers through periods of natural restoration. The integrity trajectories follow exponential decay and recovery processes, A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY3 DEMAND MEAT PRODUCTION CROP PRODUCTION LANDSCAPE Crop Demand Meat Deforestation Natural Ecosystem Integrity Crop for Meat Production Unsatisfied Demand Unsatisfied Demand Meat Demand Crop Deforestation Population growth Increasing Cereal Demand Increasing Meat Demand Economic growth Natural Ecosystem Integrity Mechanization Chemical Inputs Technology Organic Production Natural Ecosystem Integrity Livestock Density Crop Inputs Technology Organic Production Calibrated Exogenous Policy-Shocked Agricultural Land Natural Land Forest Land Natural Ecosystem Integrity Degraded Land Cropland Pasture Land 0 1 2 Figure 1. Schematic overview of the our dynamic simulation model organized into four main sections: the left panel introduces the demand drivers; the top and bottom panels detail the production and ecological feedback processes; and the right panel summarizes the resulting landscape structure and transitions. Specifically, the left panel shows how external drivers-specifically, population and economic growth (marked with stars for exogenous variables)-jointly determine increases in both cereal and meat demand. Central panels illustrate how these demand trajectories feed into crop and meat production systems, which are influenced by intensification variables such as technology advancement, chemical inputs, mechanization, livestock density, and the adoption of organic production (with targets indicating calibrated variables and lightning bolts indicating policy-shocked variables). Meat and crop production outcomes depend on and feedback to the integrity of natural ecosystems (a synthetic measure of local biodiversity and soil health, see Section 2.4), with low integrity reducing yields and increasing future land needs. The model tracks unsatisfied demand for both crops and meat, which triggers further land conversion-expanding either cropland or pasture at the expense of natural land and semi-natural vegetation. The right panel details landscape structure and transitions among key land categories: cropland, pastureland, natural land, forest, and semi-natural vegetation. Feedback loops connect ecosystem integrity not only to landscape structure, but also to production system performance, thus enabling the model to capture crossscale dynamics and the effects of multiple policy levers. each with a characteristic time scale. Importantly, ecosystem integrity feeds back into agricultural yields, while overexploitation undermines future productivity, thereby creating self-limiting dynamics. In addition, the remaining intact vegetation modulates the pace of ecosystem degradation and restoration at the landscape level through the provision of vital ecosystem services, such as water filtration, climate stability, and biodiversity storage. The model incorporates a sustainable alternative to conventional farming through the inclusion of organic management. The choice to focus on organic systems is motivated by their clear institutional definition and the availability of consistent global datasets on organically managed land (67; 50). In the model, the transition to organic farming is treated as an exogenous process-that is, the share of organic land is calibrated to empirical data and provided as an input rather than emerging endogenously from the system's dynamics. Organic management is characterized by the absence of synthetic chemical inputs and stricter limits on livestock density, thereby reducing pressure on ecosystems, albeit often at the cost of short-term yield reductions. The model explores the implications of policy-driven increases in organic adoption and 4A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY broader reductions in chemical input use, assessing their combined effects on production and ecosystem integrity. Our framework is capable of contrasting a business-as-usual development trajectory with a suite of alternative policy scenarios, where changes are implemented as constraints or shifts to key model parameters. These include limiting livestock density, restricting deforestation, reducing chemical inputs, and implementing demand-side interventions. This architecture allows us to systematically assess the synergies and trade-offs inherent in: (i) supply-side sustainability measures, (ii) combinations of supply- and demand-side interventions, and (iii) the overall scale of consumption habit transformation needed to achieve a transition onto ecologically sustainable pathways for global land use and food production. 2.2. Demand: Economic and Population Growth Population (Nt) and per-capita income (It) are specified as exogenous drivers that change throught time t. Income determines both the total caloric requirement and the composition of human diets. Total caloric demand is modeled using a logarithmic (Engel-type) relationship (12) : (2.1) Dt = (a + b log It)Nt, where Dt is total caloric demand, and a and b are positive parameters. The logarithmic form is widely used in empirical studies of food demand because it captures the diminishing marginal increase of total caloric intake with rising income (3). Demand for animal products is modeled as a power-law function of income: (2.2) Dm,t = c I d t Nt, with c, d > 0, where d represents the income elasticity of meat demand. Power-law (or constant-elasticity) functions are commonly adopted in agricultural economics to represent the empirically observed sublinear increase in meat and high-value food consumption with income(13; 29). The demand for crops directly consumed by humans is the residual once the caloric contribution of meat is subtracted: (2.3) Dfood c,t = Dt -Dm,t. Then, total crop demand in the system comprises both food and feed uses: (2.4) Dc,t = Dfood c,t + Dfeed c,t , where Dfeed c,t is the requirement of the livestock sector for feed inputs (see Section 2.3). To mimic real-world adaptive demand responses, the model incorporates a feedback mechanism in which consumption is partially suppressed if prior output falls short of demand-analogous to price feedback in markets: (2.5) Dm,t ←Dm,t ωm 1 -αm Dm,t-1 -Qm,t-1 Dm,t-1 , (2.6) Dfood c,t ←Dfood c,t ωc 1 -αc Dc,t-1 -Qc,t-1 Dc,t-1 , where Qm,t-1 and Qc,t-1 denote last period's meat and crop output, respectively, and ωm, ωfood c are normalization constants. The notation ←indicates that the exogenously determined demand at time t is endogenously adjusted in response to supply shortfalls in t -1. Lastly, normalizing by ωm and ωc ensures appropriate scaling of global demand within our stylized representation of the Earth system, while preserving the overall trend - see Appendix A.1 for further details. 2.3. Production: Crop and Livestock Systems Our model incorporates both conventional and organic production systems, which together determine the total outputs of crops and livestock at each time step: Qc,t = qc,t + qo c,t Qm,t = qm,t + qo m,t, (2.7) where q·,t and qo ·,t denote conventional and organic outputs respectively. The share of land under organic management increases exogenously, while both systems are managed by adaptive rules simulating a representative farmer who adjusts input intensities in response to changing economic and environmental conditions. A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY5 For conventional crop production, total yield (qc,t) is modeled as a multiplicative function of cultivated area (Ac,t), technological level (Tt), chemical inputs (Φc,t ≥1), mechanization (Mc,t ≥1), and the condition of the natural ecosystem (0 1.9 as forest and those with ε 1.9, see Eq. 2.20). 3. Results The business-as-usual scenario provides a reference trajectory under current trends in population, consumption habits, and land management practices. This scenario serves as a baseline against which the effectiveness and trade-offs of alternative policy interventions can be assessed. 3.1. Business-as-usual Scenario The business-as-usual scenario highlights a persistent transformation of the landscape, primarily driven by increasing demand for food and livestock products (see Fig. 4). As agricultural activity intensifies and expands, natural and semi-natural ecosystems are progressively converted or degraded (see Fig. 4a and Fig. 4d). Indeed, the expansion of agricultural land directly reduces both the area and overall integrity of natural habitats, resulting in a notable decline in forest land and average ecosystem health within each land use category. This transformation is particularly evident in the case of cropland (see Fig. 4b and Fig. 4e). Here, both land consumption and ecological degradation reach their highest levels, largely as a consequence of the high degree of intensification and mechanization required to meet rising food demands. A substantial proportion (≈40% in recent years) of these crops is allocated to livestock feed, and thus the ecological footprint of crop expansion is strongly tied to meat production and associated changes in dietary habits. Pastureland dynamics further illustrate the complex feedback mechanisms in the system. While advances in technology and animal husbandry-such as vaccines and growth hormones-have enabled higher livestock densities and contributed to a recent decline in the area under pasture, our model projects that this trend will ultimately reverse. As technological improvements plateau and can no longer keep pace with rapidly rising demand for meat, the pressure for additional pasture expansion resurfaces. A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY9 Figure 3. Model calibration against major historical trends in agricultural, land use, and socioeconomic indicators. Each panel displays the model's simulated results (solid orange lines, normalized to 1 in 1960 to emphasize rates of change) and historical observations from 1960 to 2022 (black diamonds; data from FAO (28; 27)). The model parameters were calibrated to reproduce empirical trends, and all curves represent averages over 500 Monte Carlo simulations. In particular, the model closely captures key historical patterns, in particular sustained increases in agricultural production and yield, driven by a significant expansion in total demand for both crop and animal products. It also reproduces the peak and subsequent decline in the rate of pasture land expansion, which results from land use efficiency gains as intensification has offset some demand growth. Stable or rising trends in production inputs, such as fertilizer use and mechanization, are also well represented. Gray-shaded panels indicate exogenous input variables (e.g., population, per capita income): here, dashed orange lines represent fitted values matched directly to observed data, which are then used as fixed, absolute model inputs. Taken together, these trends accelerate the fragmentation of remaining high-integrity habitats and the spread of degraded patches across the landscape. As demonstrated by spatial analyses (Fig. 2), contiguous forest edges are eroded, creating a more fragmented matrix of land uses, while marginal agricultural areas are often abandoned in favor of clearing more productive forested sites. This mosaic of change not only undermines biodiversity but also weakens the ecosystem services that support long-term agricultural productivity. Collectively, these findings underscore that, without targeted interventions, current trajectories reinforce a self-perpetuating cycle of ecological decline and agricultural expansion. Under this business-as-usual scenario, we observe reduced landscape resilience, increasing dependence on intensification solutions. The baseline scenario therefore highlights the urgent need to explore alternative policy actions. 3.2. Effects of Single Policy Measures We next assess the effects of implementing individual policy interventions, each focused on a specific lever within the food and land system. The complete set of considered scenarios is presented in Fig. 5, with corresponding parameter details provided in the Appendix. This analysis not only evaluates the direct impact of each measure but also reveals how compensatory system responses can undercut their effectiveness, producing trade-offs or unintended outcomes. 10A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY Figure 4. Land use and ecosystem integrity trajectories under the business-as-usual scenario. Top panels (a-c) show the temporal trends of the various land-use types: panel (a) tracks the ongoing decline in forest area, panel (b) captures the steady expansion of cropland, and panel (c) traces changes in pastureland. The gray-shaded background indicates the historical period, with projected dynamics shown to the right of the dashed vertical line. Pastureland area initially decreases due to advances in technology and more efficient animal husbandry, which allow for higher livestock densities. However, as technological improvements plateau and can no longer match the rising demand for meat, our model anticipates a renewed expansion of pasture area in the future. Bottom panels (d-f) depict temporal changes in the mean ecosystem integrity index for natural land (d), cropland (e), and pastureland (f). These plots reveal the rapid and sustained degradation of ecosystems associated with agricultural expansion and the ongoing loss of ecological integrity across all land types. Of particular interest is the fact that declines in natural ecosystem integrity lag behind the decrease in forest area; this is because deforestation initially diminishes the supply of ecosystem services vital for maintaining ecosystem stability (see Eq. 2.21). As these services decline, the degradation of agricultural land accelerates (see Eq. 2.20), leading to land abandonment and a reduction in the average natural ecosystems integrity. Fig. 5a maps single-policy scenarios in a space defined by forest land area variation and degraded land variation, both calculated relative to the baseline. These two metrics were selected to capture the dual dimensions of land system change: forest land variation reflects shifts in the extent of intact natural habitats (landscape-scale expansion or protection), while degraded land variation quantifies changes in ecosystem integrity within managed areas due to intensification or restoration efforts. By presenting results in this way, we can compare policies that limit expansion (horizontal direction) with those that primarily aim to improve land already in production (vertical direction), clearly illustrating system-wide trade-offs. The analysis reveals considerable variability in policy outcomes, much of which stems from compensatory dynamics-where gains in one area trigger pressures elsewhere. For example, supply-side measures such as reducing chemical inputs or expanding organic crop production are effective at lowering land degradation, but typically cause declines in forest area. This is because reduced yields on existing farmland (from chemical restrictions or organic conversion) require compensatory expansion, driving new deforestation and landscape fragmentation to maintain overall production levels. Similarly, implementing deforestation bans for crop or meat production often leads to increased land degradation, because production pressures shift onto existing agricultural land, resulting in more intensive management. Over time, such intensification can diminish yields and degrade ecosystem integrity, which may ultimately drive further land expansion and limit the effectiveness of deforestation policies. Conversely, demand-focused interventions-particularly reductions in crop or meat consumption-deliver more comprehensive positive outcomes, as they simultaneously relieve pressure for both intensification and expansion. Policies that reduce meat consumption result in the largest gains for forest area and the most A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY11 Figure 5. Impact of individual and combined policy interventions on forest and degraded area variation. Variations in forest area (horizontal axis) and degraded area (vertical axis) are shown as percentage changes with respect to the baseline scenario. Points in the upper-right green quadrant indicate policy strategies that succeed in both expanding forest area and reducing degraded land, while those in the lower-left quadrant are detrimental on all grounds. Panel (a) shows the effects of individual (single) policy interventions, each marked by a labeled point. Only the reduction in meat consumption achieves positive outcomes on both axes, illustrating that most single interventions are insufficient due to persistent compensatory dynamics within the system. Panel (b) reports outcomes for policy mixes that specifically include increased organic crop production and decreased livestock density-two interventions often viewed as hallmarks of sustainability. Here, each point represents a different mix with other measures. The results demonstrate that these strategies, even when paired with moderate reductions (10%) in crop and meat demand, consistently fail to halt forest loss, often resulting in further declines in forest land area. Panel (c) highlights selected policy mixes containing key interventions, such as chemical use reduction or restrictions on crop and livestock-driven" deforestation, with distinct symbols denoting which measures are included. This panel shows that policy mixes combining interventions that target both intensification and expansion (i.e. orthogonal strategies) are situated in the green quadrant, signifying that integrated approaches are most effective at overcoming compensatory effects. pronounced reductions in degraded land. This is because a substantial share of cropland is devoted to growing feed for livestock, and thus plant-based diets reduce both direct and indirect land pressures 4. Synergies and Trade-Offs in Policy Portfolios We then turn to the analysis of policy combinations addressing multiple drivers at once, exploring how different mixes of interventions impact ecosystem integrity. Figures 5b, 5c, 7 and 6 summarize these results, allowing direct comparison of policy portfolios. 12A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY Figure 6. Effect of demand reductions on forest and degraded land areas under enhanced organic crop production and reduced livestock density policies. Each panel shows the percentage change in forest land (green, left axis) and degraded land (orange, right axis) compared to the baseline, across scenarios with progressive demand reductions. In particular, the x-axis represents increasing reductions in projected crop and meat demand by 2100, where 0 means no change from business-asusual and 1 means no demand increase after 2022. The results demonstrate that only when demand reductions surpass roughly 40% do both forest expansion and significant decreases in degraded land occur. An initial finding, visible in Figure 5b, is that policies often regarded as highly sustainable-namely, widespread adoption of organic crop production (eliminating chemical inputs) and reduced livestock intensification (for improved animal welfare)-consistently result in a declining forest land area, regardless of whether they are combined with moderate reductions (10%) in crop and meat demand. The underlying mechanisms behind this result differ between crop and meat systems. For organic crop production, the primary constraint is related to the strong diminishing returns (k = 0.2 in Eq. 2.8 and Eq. 2.9) from chemical inputs. Moderate chemical use can maintain relatively high yields, but if chemical inputs are eliminated entirely, yield losses grow disproportionately and become very difficult to offset. Most farmers are unable to fully compensate for these losses through additional intensification or mechanization, as the required extra effort would be prohibitive-leading the system to expand cultivation into natural areas to maintain production targets. The dynamics differ for organic meat production. Here, shifting to organic practices does not generate the same drastic loss of productivity per unit area because industrial animal production is less constrained by input intensification-modern techniques like vertical livestock integration (as seen recently in China (33; 83)) support continued output growth. Consequently, organic meat policy mixes exert only modest negative effects on forest land. However, if livestock production is broadly de-intensified without such adaptive capacity, the pressure to maintain meat supply quickly translates into greater expansion of pasture or feed crops, significantly amplifying deforestation. While some past studies have stopped at these findings (25)-concluding that sustainable production is unfeasible-our model enables quantification of the scale of demand reduction necessary to reverse these negative trends. As shown in Figure 6, only scenarios with at least a 40% reduction in projected crop and meat demand by 2100 achieve net gains in forest cover and notable reductions in degraded land when either an increase of organic crop production or a reduction in livestock density is promoted. This highlights the scale of lifestyle and societal change required for promoting animal welfare and eliminating chemical use. Focusing next on policy mixes that exclude organic crop expansion and livestock intensity reduction, the analysis reveals that combining interventions yield more favorable outcomes than isolated policies. As demonstrated in Fig. 5c, mixes incorporating at least two different policies outperform single interventions. When policies target both degradation (e.g. reduced chemical use) and deforestation (e.g. conversion controls) together, their effects reinforce one another and mitigate compensatory expansion or intensification. Furthermore, these policies can initiate a positive feedback loop: by halting deforestation, ecosystem service provision is enhanced, and when combined with reduced intensification, this leads to improved natural A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY13 ecosystem integrity. As ecosystem health recovers, agricultural yields are boosted, which in turn reduces the need for future land expansion. Examining the highest-performing mixes (Figure 7), we see that the top portfolios for reducing degradation mainly involve a combination of reduction in chemical inputs alongside decrease in crop and meat consumption-pointing again to the necessity of shifting dietary habits. Conversely, the best-performing portfolios for forest preservation include deforestation restrictions and meat demand reduction, while measures aimed solely at reducing chemicals or crop demand are less central-since they contribute minimally to curbing deforestation, especially that driven by livestock. Ultimately, only a single portfolio achieves top-tier outcomes for both forest recovery and land degradation, combining deforestation limiting measures, intensification limits, and reductions in both meat and crop consumption. While additional trade-offs may exist beyond the scope of this study, these results highlight that well-designed policy portfolios aligning both supply- and landscape-level actions-with a strong emphasis on demand reduction-are essential for achieving "win-win" gains and real progress toward sustainability. 5. Discussion 5.1. Beyond the Supply-Side: Systemic Barriers to Sustainability Our results highlight important limits to achieving a sustainable food system through supply-side reforms alone, echoing findings across the literature on land use, agricultural intensification, and food system sustainability (26; 69; 73). Measures such as reducing chemical inputs and adopting more animal-friendly or organic production systems provide meaningful improvements at the farm scale-ranging from reductions in local pollution to gains in animal welfare and soil health (65; 78). These changes are valuable in themselves, and often come with additional social benefits, including positive impact on human health and responsiveness to consumer demand for more sustainable food options. However, our model suggests that when per-hectare productivity is deliberately limited-whether by banning chemical inputs or reducing livestock density-the system tends to compensate by converting additional natural or semi-natural land to agriculture. This compensatory expansion can reduce or even offset the environmental benefits achieved on existing farmland, and in some cases may exacerbate pressures such as deforestation, landscape fragmentation, and ecosystem degradation, consistent with previous observations about indirect land-use change (72; 41). Even when multiple supply-side measures are combined, our analysis indicates that their cumulative impacts alone are unlikely to halt forest loss or secure broad-based ecosystem integrity. Let us stress that these findings do not imply that supply-side reforms are undesirable; rather, they highlight the need to complement them with demand-side transformations and broader systemic changes if the goal is to achieve consistent sustainability. 5.2. Structural Need for Demand Reductions A second, and more profound, implication of our findings is that only strong demand-side interventions are capable of delivering transformative change. Substantial reductions in total food-especially meat-consumption relieve system-wide pressure by cutting both the intensification and expansion drivers simultaneously. This is consistent with a growing body of research indicating that dietary shifts toward reduced animal-source foods (i.e., more plant-based diets) are critical to staying within planetary boundaries and achieving global climate, biodiversity, and public health targets (85; 64; 77). Our quantitative analysis provides a concrete threshold: unless at least 40% of projected excess demand for crops and meat is curbed by 2100, even the most ambitious supply-side or reforestation interventions are unlikely to reverse deforestation or halt land degradation (see also (2)). Crucially, demand modification is not only more effective at limiting land use changes but also makes it easier for other sustainability policies (like reduced chemical use or organic expansion) to succeed-a synergy recognized in other modeling exercises and global sustainability assessments (16). 5.3. Integrated Approaches and Systemic Solutions Despite the importance of demand-side measures, our results do not dismiss the value of integrated, multi-faceted policy mixes. Rather, they demonstrate that combinations of carefully designed interventions-addressing both production and consumption-yield the most robust improvements for both forest and degraded land areas variation. Such an integrative approach directly mitigates compensatory effects, 14A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY Figure 7. Comparative effectiveness of top-performing policy mixes for reducing deforestation and land degradation. Panels (a) and (b) display, for each of the ten best policy combinations, the relative variation in forest land and degraded land areas as a percentage change from the baseline scenario. Policy mixes are ranked separately by their success in reducing deforestation (a) and degradation (b). Dashed bars indicate policy mixes that appear among the top performers for both objectives together. Panel (c) details the intervention composition of each policy mix. Each row represents a ranked policy mix (from most to least effective; top to bottom), and each circle indicates the inclusion of specific interventions for the corresponding scenario (green for deforestation-focused, orange for degradation-focused). Despite some overlap, the policy measures most effective at reducing deforestation are not always the same as those best at lowering land degradation. Only a policy portfolio achieves high performance in both dimensions, highlighting the importance-and challenge-of integrated strategies capable of delivering simultaneous gains for forest conservation and reduced agricultural ecosystem degradation. wherein policies addressing one part of the system inadvertently shift environmental pressures elsewhere unless implemented holistically (41; 48; 75). For instance, coupling landscape-level deforestation controls with intensity reduction measures (e.g., limits on chemical inputs) can prevent rebound effects and increase overall ecosystem integrity (55; 9). It is worth noting, however, that many popular narratives around sustainable intensification or organic conversion tend to overemphasize the benefits of any single intervention, without sufficient attention to the cross-scale feedbacks and compensatory land dynamics illuminated by both our results and other studies (31; 60). A successful transition to a food system that maintains ecosystem integrity and food security together will likely require not just technical or agronomic innovation, but deep societal, economic, and behavioral changes (69). A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY15 5.4. Limitations and Unaccounted Aspects While we believe our study advances the understanding of systemic trade-offs and leverage points in the pursuit of sustainable food and land systems, it is important to acknowledge several limitations that may influence the interpretation and applicability of our findings-both in terms of modeling assumptions and the practicalities of real-world policy implementation. Our framework adopts a stylized, spatially explicit global representation. This abstraction allows us to distill essential system dynamics and identify general insights on policy interactions, while avoiding the computational and data demands of fully resolving geographic, ecological, and socio-economic heterogeneity. But, local variations-shaped by climate variability, land tenure, market integration, and governance (41; 48; 82)-are not explicitly represented. However, these features could be incorporated in future extensions through nested multi-scale models or spatially explicit downscaling approaches, enabling more contextsensitive analyses (43). Similarly, our land conversion and abandonment rules respond primarily to aggregate supply-demand imbalances. This simplification abstracts from mediating effects such as access to infrastructure, transaction costs, or governance quality. While this allows transparent exploration of system-wide drivers, future model iterations could incorporate heterogeneous decision-making rules or probabilistic entry/exit dynamics to capture local socio-institutional variation (31). Crop and livestock production modules rely on generic functional forms calibrated to global data. While sufficient for identifying broad trends, these formulations omit local adaptation, site-specific constraints (e.g., soil degradation, pests, water stress), and stochastic variability in yields. Likewise, our management systems currently exclude agroecological or diversified strategies, focusing instead on conventional and organic monocultural systems. Future work could integrate heterogeneity in production technologies, mixed-cropping systems, or resilience-enhancing practices to more realistically capture the spectrum of sustainable management options (39). The socio-political dimension of policy implementation is currently represented exogenously, assuming instantaneous and uniform enactment. While this simplification enables clear analysis of policy potential, it neglects real-world constraints such as political resistance, institutional capacity, stakeholder engagement, and uneven enforcement (49; 56). Incorporating adaptive policy implementation, delays, and compliance heterogeneity would allow more realistic assessment of feasible outcomes and the robustness of interventions. Our model does not differentiate actors in the food system, and therefore omits social and equity dimensions. While this enables focus on aggregate system-level trade-offs, it excludes potential distributional consequences such as food insecurity, livelihood impacts, or shifts in nutrition. Future extensions could integrate agent-based or household-level modules to explore these outcomes under alternative policy scenarios (86; 15). Climate-related risks and long-term ecological processes are simplified. Feedbacks from extreme events, biodiversity loss, tipping points, and delayed ecosystem recovery are not dynamically represented. This choice was made to retain tractability for global simulations while highlighting general mechanisms. Future work could include stochastic climate shocks, ecological inertia, and recovery lags to capture more realistic ecological dynamics (71; 47; 18). Behavioral and demographic dynamics are also exogenous. Food demand does not adapt through social learning, policy feedback, or price-mediated behavior, and population changes and migration are not modeled. These processes are known to shape long-term land-use trajectories (37; 4). Incorporating endogenous behavioral adaptation and demographic dynamics would allow more robust exploration of feasible pathways and potential rebound effects (42; 79). Finally, as with all global modeling efforts, results are subject to parameter uncertainty, data limitations, and simplifications of real-world complexity. Our stylized approach enables identification of core systemlevel interactions and policy synergies, but should be interpreted as generating hypotheses rather than prescriptive solutions. Future work should prioritize multi-scale, participatory, and transdisciplinary models that incorporate climate risk, social equity, behavioral feedbacks, and realistic policy processes, providing the basis for actionable, locally calibrated insights (42; 58). Conclusion The results of this study highlight the urgent need for a paradigm shift in how we address food system sustainability. Technological and agronomic improvements, while necessary, are insufficient in isolation to 16A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY meet the challenges posed by rising global food demand and intensifying environmental pressures. Our systemic approach reveals that only integrated strategies-combining ambitious dietary changes with reformed agricultural practices and strengthened landscape conservation-can reconcile the competing demands of productivity, ecosystem integrity, and social equity. Our research supports the growing consensus in the literature that demand-side interventions must be placed at the center of sustainability and climate agendas (85; 32). Avoiding counterproductive trade-offs requires a holistic, system-based approach that aligns policies across supply, demand, and governance. At the same time, future progress depends on robust efforts to tailor and implement these solutions to diverse regional, cultural, and socio-economic contexts, as well as on building public support for transformative change. Going forward, further refinement of integrated models like ours is needed. In particular, future research should focus on incorporating more explicit representations of climate impact, equity dimensions, market and policy dynamics, and the complex processes underlying behavior and dietary shifts. Advancing multi-scale, participatory, and interdisciplinary modeling will be crucial to develop actionable roadmaps for policymakers and stakeholders. Acknowledgments This research was conducted within the Econophysics & Complex Systems Research Chair, under the aegis of the Fondation du Risque, the Fondation de l'Ecole polytechnique, the Ecole polytechnique and Capital Fund Management. Michel Loreau was supported by the TULIP Laboratory of Excellence (ANR-10-LABX41). 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Fractal geometry in forestry applications. Forest Ecology and Management, 46(3-4):179188, 1991. [88] Erasmus KHJ zu Ermgassen, Cécile Renier, Andrea Garcia, Tomás Carvalho, and Patrick Meyfroidt. Sustainable commodity sourcing requires measuring and governing land use change at multiple scales. Conservation Letters, 17(3):e13016, 2024. A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY21 Appendix A.1. Initialization The system initialization proceeds as follows: • The model employs a 100×100 grid to represent the landscape, initially allocating 50% of the area to forest land. This proportion reflects estimates for the year 1960 based on FAO data (28). To spatially distribute this fraction, a fractal field with a characteristic correlation length ξ is generated. Previous research suggests that fractal patterns realistically capture empirical land use distributions (51; 87). The value for ξ, chosen to produce a visually realistic arrangement, is reported in Table 1. After allocating forested cells, the remaining landscape is randomly assigned to pasture (35%) and cropland (15%). • Next, each cell receives specific degradation and restoration rates. These are assigned using the concept of lifespan, which aligns well with the properties of exponential distributions. This approach has been motivated by soil science literature, where the exponential lifetime of soil layers is tied to degradation and erosion processes (21). Following results this stream of research, each parameter θi,· is sampled as: θi,· = 1 N(μ·, σ·), where · stands for natural (n), pasture (m), or cropland (c), and N denotes a lognormal distribution. The adoption of a lognormal distribution ensures the statistical characteristics of the rates are consistent with those reported by Evans et al. (21). • Initial real demand for both crops and meat is computed following Eq. 2.1 and Eq. 2.2. To translate this demand into the model's internal units, we normalize by assigning constants ω· such that the initial yield for both livestock and crops is set to q·,0 = 1. Assuming that at the starting point total production matches demand, we have Qc,0 = 1500 = Dc,0 and Qm,0 = 3500 = Dm,0. Since this step only rescales the reference point, the overall system dynamics are unaffected. • Intensification factors are then initialized to have q·,0 = 1 in the initial year, which requires livestock density, mechanization, chemical input intensity, and average ecosystem integrity to all be set to one at initialization. In contrast, we assign the initial technology level a small value (T0 = 0.001), reflecting the limited state of agricultural innovation prior to the Green Revolution. While this may result in yields slightly above one, the difference is negligible for trend analysis. • With all state variables initialized, the model is ready to proceed with the simulation. A.2. Calibration procedure and parameter values Model parameters were categorized into four distinct groups (see Table 1): (i) measured parameters (M), which are directly fitted from empirical datasets; (ii) estimated parameters (E), sourced from values reported in the literature; (iii) calibrated parameters (C), which are adjusted during model calibration to achieve the best agreement between simulated and observed data; and (iv) fixed parameters (F), which are manually specified due to the lack of real-world analogs or because of model simplifications. Measured parameters (M) are defined as those whose values are determined using empirical data corresponding directly to the model functions they influence. This group mainly encompasses parameters governing demand and the initial state of the landscape, where reliable datasets are readily available. Parameter values were estimated by fitting model outputs to observed data from FAO(28; 27), using a least squares minimization approach to ensure alignment with real-world patterns. It should be emphasized that the aim of this calibration was not to achieve exact reproduction of the full distribution of empirical curves, but rather to capture the essential trends relevant to the aims of this study. Within this context, all fitted functions yield a robust agreement with the empirical data, adequately representing the characteristic patterns for each parameter considered (see Fig. 3). Estimated parameters (E) were assigned values drawn from empirical studies and widely reported literature benchmarks. This category chiefly includes ecological parameters such as rates of degradation and restoration, for which precise values are not directly measurable within the model context. As described in Section A.1, these rates were informed by global analyses and meta-analyses covering soil degradation and forest recovery dynamics(21; 30; 19). For parameters associated with the yield function, values were chosen based on extensive agricultural economics and ecological economics literature(23; 11; 57). While reported 22A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY Section Notation Description Value Group Landscape Grid size 100x100 F Initialization Natural land share 50% M Cropland share 15% M Pasture land share 35% M ξ Landscape fragmentation 10 F μc Crop degradation distribution shift 1 E μm Pasture degradation distribution shift 1 E σc Crop degradation distribution scale 10000 E σm Pasture degradation distribution scale 5000 E μn Nature restoration distribution shift 1 E σn Pasture degradation distribution scale 1000 E Demand a Calories demand calculation -138.2 M [kcal · year-1 · per capita] b Calories demand calculation 744.4 M [kcal · $-1 · per capita] c Baseline demand for animal products 210 M [Kg · year-1 · per capita] r Conversion demand for animal products to calories 3000 E [kcal · Kg-1] d Income elasticity of meat demand 0.65 M αm Feedback strength for meat demand 0.5 C αc Feedback strength for crop demand 0.1 C Production k Yield elasticity to chemical inputs 0.2 E f Yield elasticity to mechanization 0.5 E h Sensitivity of meat yield to intensification 0.95 E Λmax Maximum livestock density (organic) 3 E β Speed of mechanization adjustment 0.95 C δ Speed of chemical input adjustment 1.1 C γ Speed of livestock density adjustment 0.95 C ν Technological progress rate 0.1 E Tmax Technological ceiling 0.2 E Landscape φ Baseline land expansion rate (crop/pasture) 3e-4 C dynamics ζ+ c Expansion rate (cropland) 130 C ζ+ m Expansion rate (pasture land) 500 C ζc Contraction rate (cropland) 120 C ζm Contraction rate (pasture land) 500 C εmax Maximum natural ecosystem integrity 2 F p Diminishing return for ecosystem service 0.25 E Table 1. Parameters of the model. Group codes can be: measured (M), estimated (E), calibrated (C), or fixed (F). values do vary between studies and there is no universal consensus-often due to differences in scale, crop types, or methodological approaches-our selections are consistent with established ranges. Importantly, the chosen parameter values are supported by the model's ability to reproduce historical patterns of yield growth observed in real-world data, lending further confidence to their plausibility within the framework of this study. Parameters classified as calibrated (C) are primarily associated with the model's adjustment and feedback mechanisms. These parameters cannot be directly inferred from empirical data as they are fundamentally linked to internal model behavior-that is, they govern logical or process-based linkages for which data A GLOBAL SYSTEMS PERSPECTIVE ON FOOD DEMAND, DEFORESTATION AND AGRICULTURAL SUSTAINABILITY23 are sparse or context-specific. Accordingly, they were tuned manually to best reproduce the broad patterns present in observed system dynamics, using an iterative, trial-and-error approach. The goal here, as elsewhere, was not to exactly fit every empirical curve but to recapitulate the essential dynamic behavior pertinent to our research questions. This calibration approach ensures the model captures the relevant system variability while maintaining enough flexibility to allow exploration of alternative scenarios within the stylized earth system structure.
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2510.14726
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Cross-Layer Feature Self-Attention Module for Multi-Scale Object Detection
Dingzhou Xiea, Rushi Lanc, Cheng Pangc,∗, Enhao Ningb, Jiahao Zengb, Wei Zhengd,∗
aGuangzhou Huashang College, Hua shang Road, Guangzhou, 511300, Guangdong, China
bGuangxi Normal University, Yu cai Road, Guilin, 541004, Gangxi, China
cGuilin University of Electronic Technology, Jin ji Road, Guilin, 541004, Gangxi, China
dJinling Institute of Technology, Hong jing Road, Nanjing, 211169, Jiangsu, China
Abstract
Recent object detection methods have made remarkable progress by leveraging attention mechanisms to improve feature
discriminability. However, most existing approaches are confined to refining single-layer or fusing dual-layer features,
overlooking the rich inter-layer dependencies across multi-scale representations.
This limits their ability to capture
comprehensive contextual information essential for detecting objects with large scale variations.
In this paper, we
propose a novel Cross-Layer Feature Self-Attention Module (CFSAM), which holistically models both local and global
dependencies within multi-scale feature maps. CFSAM consists of three key components: a convolutional local feature
extractor, a Transformer-based global modeling unit that efficiently captures cross-layer interactions, and a feature fusion
mechanism to restore and enhance the original representations. When integrated into the SSD300 framework, CFSAM
significantly boosts detection performance, achieving 78.6% mAP on PASCAL VOC (vs. 75.5% baseline) and 52.1% mAP
on COCO (vs. 43.1% baseline), outperforming existing attention modules. Moreover, the module accelerates convergence
during training without introducing substantial computational overhead. Our work highlights the importance of explicit
cross-layer attention modeling in advancing multi-scale object detection.
Keywords:
attention mechanism, object detection, feature fusion, feature enhancement
1. Introduction
The escalating demand for intelligent automation and
industrial inspection has positioned robust multi-scale ob-
ject detection as a cornerstone technology.
While deep
learning has driven significant progress, detecting objects
with extreme scale variation remains a formidable chal-
lenge, primarily due to the semantic and resolution gaps
between features at different levels of the network hierar-
chy [1, 2]. Effectively integrating these multi-scale features
∗Corresponding author:
Email addresses: dingzhouxie@163.com (Dingzhou Xie),
rslan2016@163.com (Rushi Lan), pangcheng3@guet.edu.cn (Cheng
Pang), ningenhao@xs.ustb.edu.cn (Enhao Ning),
zeng_jiah@163.com (Jiahao Zeng), zhengwei@jit.edu.cn (Wei
Zheng)
to achieve consistent performance across small, medium,
and large objects is thus a critical and ongoing research
focus.
Attention mechanisms have become instrumental in ad-
vancing this field by enabling models to dynamically focus
on more informative spatial regions or feature channels
[3, 4, 5]. A pivotal aspect of this is modeling long-range de-
pendencies to capture global contextual information. Two
dominant paradigms for this purpose are structured con-
volutional operators [6, 7, 8, 9, 10] and self-attention mech-
anisms [11, 12]. While enhanced convolutions can enlarge
the receptive field, their capacity for global modeling is
often constrained by local connectivity and may require
careful hyperparameter tuning to avoid artifacts. In con-
trast, self-attention mechanisms, catalyzed by the Vision
arXiv:2510.14726v1 [cs.CV] 16 Oct 2025
Transformer (ViT) [13], inherently model pairwise inter-
actions across all positions in a feature map, providing
superior flexibility in building global context. Subsequent
architectures like the Swin Transformer [14] have further
enhanced their efficiency and applicability to vision tasks.
Notwithstanding
their
achievements,
these
meth-
ods have some limits in multi-scale object detection.
Convolution-based methods often struggle to capture de-
pendencies outside their fixed receptive field. Also, many
modern self-attention modules are made for single-scale
feature maps [15, 16, 17, 18]. They process patches inde-
pendently and lack a clear way to model connections across
different layers of a feature pyramid. This restricts the in-
tegration of contextual information across semantic levels
and hinders the model from jointly using fine details and
high-level semantics, which is very important for accurate
multi-scale detection.
To bridge this gap, we propose a Cross-Layer Feature
Self-Attention Module (CFSAM). CFSAM is designed to
holistically model interactions across multi-scale features
within a unified and efficient self-attention framework. It
comprises three components: 1) a local feature extrac-
tor using convolutional layers to preserve spatial details,
2) a global feature modeling unit that employs a novel
partition-based Transformer to efficiently capture long-
range dependencies across all input scales simultaneously,
and 3) a feature fusion and restoration unit that coher-
ently integrates the refined features back to their original
dimensions. The principal contributions of this work are:
• To effectively exploit long-range dependencies be-
tween multi-layer feature maps, we designed the
SSD300-CFSAM model.
This design enables the
model to better understand both the relationships be-
tween objects in an image and the contextual infor-
mation of the scene.
• The CFSAM module achieves notable performance
improvements without a significant increase in model
complexity or the number of parameters.
Further-
more, its plug-and-play design allows for seamless and
flexible integration into various existing neural net-
work architectures.
• Quantitative and qualitative experimental results on
PASCAL VOC and COCO datasets show that our
method not only achieves significant performance im-
provements over the baseline but also maintains su-
perior computational efficiency compared to other en-
hancement modules.
The remainder of this paper is structured as follows:
Section 2 reviews related work. Section 3 details the CF-
SAM methodology. Section 4 presents experimental set-
tings, results, and ablation studies. Finally, Section 5 con-
cludes the paper.
2.
Related work
multi-scale object detection requires different attention
to targets at different scales in the same image, while the
attention mechanism focuses on weighting important in-
formation on the feature map, which allows neural net-
works to pay more attention to or prioritize specific input
features, thus improving the performance of visual tasks
such as object detection. There are complementarities be-
tween the two, so we conduct an in-depth study on how
attention mechanisms can facilitate multi-scale object de-
tection. Among them, channel attention and spatial atten-
tion are two commonly used attention mechanisms when
processing image data.
Channel attention models include SENet [19], GSoP-
Net [20], and SRM [21]. In 2017, Hu et al. introduced
the concept of channel attention and proposed the SENet
model. The core idea of this model is to dynamically adjust
the importance of each feature channel using Squeeze and
Excitation operations. In 2019, Gao et al. proposed the
GSoP-Net model, which utilizes the Global Second-Order
2
Pooling (GSoP) Block to model higher-order statistical in-
formation. The GSoP Block also consists of Squeeze and
Excitation modules.
In the Squeeze module, the GSoP
Block first reduces the number of channels with a 1 × 1
convolution, then computes the covariance matrix to cap-
ture the correlations between different channels. The co-
variance matrix is then row-normalized to establish con-
nections between channels.
By employing second-order
pooling, the GSoP Block enhances the ability to gather
global information. Also in 2019, Lee et al. introduced
the Style-Based Recalibration Module (SRM). The SRM
consists of two main parts: style pooling and style inte-
gration. The style pooling performs global average pooling
and calculates the standard deviation on the input features
to extract style features from each channel. The style in-
tegration replaces the original fully connected layer with
a lightweight channel-wise fully connected layer to reduce
computational requirements. It estimates style weights for
each channel based on the results of the style pooling and
then recalibrates the feature maps using these style weights
to amplify or suppress their information.
Classic spatial attention methods have evolved signifi-
cantly, with recent research focusing on efficiently integrat-
ing and enhancing spatial feature representations. For in-
stance, ACmix[22] elegantly combines the local feature ex-
traction power of convolutional operations with the global
receptive field of self-attention, offering a unified computa-
tional structure that adaptively fuses the outputs of both
paradigms. Meanwhile, for dense prediction tasks such as
object detection, Deformable Attention (DAttn) [23] has
emerged as a prominent solution. It avoids the excessive
computational cost of global self-attention by allowing a
sparse set of key sampling points to dynamically deform
based on the input features, thereby enabling the model to
focus on more relevant spatial locations efficiently. These
methods exemplify the modern trend of building spatially-
aware models that are both powerful and computationally
feasible.
Convolutional neural networks excel at handling image
data due to their properties such as translation invariance,
making them widely used in image processing. However,
CNNs lack scale and distortion invariance. In 2015, Jader-
berg et al. proposed Spatial Transformer Networks, which
learn an affine transformation for each input image to im-
prove the model’s ability to handle targets in different po-
sitions, orientations, and scales. Integrating spatial trans-
formers into neural network models allows the models to
automatically focus on the regions of interest, thereby en-
hancing the accuracy and robustness of the models in ob-
ject recognition tasks.
Subsequent research efforts have
achieved even greater success in this area [7, 24, 25, 26].
The self-attention mechanism is a method for comput-
ing attention within a single input sequence. Unlike tradi-
tional attention mechanisms [27, 28, 29, 30], self-attention
considers not only the relevance between each position in
the input sequence and a target but also the relevance
between each position and other positions.
This allows
self-attention to capture interactions between all parts of
the input sequence, better handling long-range dependen-
cies.
In 2017, Vaswani et al.
[31] proposed the Trans-
former model, which adopts self-attention to model rela-
tionships between different positional elements in the input
sequence. The Transformer achieved impressive results in
natural language processing tasks, enabling long sequence
modeling and parallel computation. Besides natural lan-
guage processing, the Transformer model has also been
applied in computer vision, audio processing, medical di-
agnosis, and other fields [32, 33, 34, 35]. In 2020, Doso-
vitskiy et al. introduced the Vision Transformer model,
which consists of an encoder composed of multiple Trans-
former modules and a linear layer for classification. Unlike
traditional CNNs, ViT takes a two-dimensional image as
input but transforms it into a one-dimensional sequence
to perform self-attention computations on the sequence,
learning relationships between different positions. To pre-
vent information loss, ViT adds trainable position embed-
3
Figure 1: Illustration of the whole network. By inserting the CFSAM into the SSD network model, the module takes six predicted feature
maps as input. After modeling the dependency relationships, the module outputs the same size and number of predicted feature maps, which
are used for object prediction.
dings at each position of the sequence to provide positional
information. The emergence of ViT offers a new perspec-
tive in the field of deep learning, applying self-attention to
other domains.
Using the core ideas of these methods, we propose to
construct a multi-scale object detection model based on
cross-layer feature self-attention module to strengthen the
ability to model the relationship between feature maps at
different levels and enhance the accuracy of multi-scale
object detection.
3.
Methodology
Multi-scale object detection networks are deep neural
network models used for object detection.
They utilize
multi-scale prediction feature maps, which have the advan-
tage of effectively detecting objects of different sizes. Com-
mon multi-scale object detection networks include Faster
R-CNN [36], YOLO [37, 38, 39], and SSD [40]. The SSD
model predicts objects using anchor boxes of different sizes
and aspect ratios on multi-scale feature maps. This re-
duces the sensitivity of the SSD network to changes in
object sizes and improves detection accuracy.
We have
chosen the SSD object detection network as the baseline
model with an input image size of 300 × 300. Figure 1 il-
lustrates how the CFSAM is applied in the SSD network.
For multi-scale object detection networks, this module pro-
vides a plug-and-play solution.
The CFSAM we propose consists of three parts: local
feature extraction, global feature extraction, and feature
fusion restoration.
As shown in Figure 2, F1, F2, and
F3 represent the multi-scale predicted feature maps. The
local feature extraction part utilizes convolutional opera-
tions. Specifically, it first performs local feature extraction
using a convolutional layer with a kernel size of 3 × 3, and
then adjusts the channel count using a convolutional layer
with a kernel size of 1 × 1, resulting in feature maps L1,
L2, and L3. The global feature extraction part employs
Flatten, Partition, and Combine operations for global fea-
ture modeling. The Flatten operation is used to flatten
the H × W regions of L1, L2, and L3 feature maps into
vectors and concatenate them along the vector dimension
to obtain the cross-layer fused feature vector. The Parti-
tion operation divides the feature vector into two parts by
sampling with a certain interval. During the self-attention
calculation, each feature point of a feature vector only in-
teracts with other feature points within itself, reducing
4
Figure 2: Illustrates the structural diagram of the CFSAM, which includes three parts: local feature extraction, global feature extraction,
and feature fusion restoration.
the computational complexity. The Combine operation is
applied to restore the feature vector back to its original
shape.
In the feature fusion restoration part, a short-
cut branch is first used to concatenate the feature vec-
tor obtained after global feature extraction with the cross-
layer feature vector before global feature extraction along
the channel dimension, doubling the number of channels.
Then, a convolutional layer with a kernel size of 1 × 1 is
used to adjust the channel count back to the original size.
Finally, the feature vector is split and restored to three
feature maps R1, R2, and R3, which have the same scale
as F1, F2, and F3, respectively. The Transformer unit can
capture long-distance dependency relationships in the in-
put sequence and weight the information at each position.
Therefore, the resulting three feature maps contain richer
contextual information.
3.1. Local feature extraction
Local features are crucial for object classification. For
example, when detecting an image of a cat, convolutional
layers may recognize local features such as the ears, eyes,
and nose of the cat. These local features have a significant
impact on determining the target category in the image.
To capture the local information of the input feature map,
a local feature extraction structure is designed using con-
volutional layers with kernel sizes of 1 × 1 and 3 × 3. This
design effectively promotes the detection efficiency of the
target detection model while adding only a small amount
of computation.
For the given input feature maps F1, F2, and F3, their
channel numbers and sizes are different. They are sepa-
rately input into a convolutional layer with a kernel size of
3 × 3 and a convolutional layer with a kernel size of 1 × 1.
The 3 × 3 convolutional layer is used to extract local spa-
tial features from the image, including edge and texture
information.
These local features are crucial for object
classification. For example, when detecting an image of
a cat, the convolutional layer may identify local features
such as ears, eyes, and nose, which greatly influence the
determination of the object category in the image. The
1 × 1 convolutional layer reduces the channel number of
the input feature maps, thereby reducing computational
costs and improving detection efficiency.
Since the pre-
dicted feature maps have a significant difference in channel
numbers, with some reaching 1024 dimensions while oth-
ers have only 256 dimensions, inputting them directly into
the Transformer unit without processing would result in
higher computational and memory costs for feature maps
with larger channel numbers. This would slow down the
5
network’s operation speed. Therefore, using a 1 × 1 con-
volutional layer to uniformly adjust the channel number
to 256 dimensions significantly reduces the computational
load. Experimental results have shown that the detection
accuracy of the network is not compromised.
After ex-
tracting the local information, output feature maps L1,
L2, and L3 are obtained, with the same channel number
for all three feature maps.
3.2.
Global feature extraction
The SSD object detection model extracts features of dif-
ferent scales from various layers of the network for predic-
tion, and there is a close relationship between these feature
maps at different levels. In object detection tasks, there
exists interdependence among different objects. Therefore,
in the object detection network, it is necessary to consider
the dependency between feature maps at different levels in
order to better locate and detect target objects. Before ex-
tracting global features, feature fusion is performed. Fea-
ture fusion methods include direct addition, channel con-
catenation, and layer-wise fusion, among others.
These
methods require upsampling to align the scale of the fea-
ture maps, which increases the computational load of the
model. We achieve cross-layer fusion of all predicted fea-
ture maps by flattening and concatenating them, without
using upsampling operations, thereby avoiding an increase
in computational load for the model.
As shown in the upper left half of Figure 2, the input
feature maps are L1 ∈RH1×W 1×C, L2 ∈RH2×W 2×C,
and L3 ∈RH3×W 3×C.
Firstly, the feature maps are
flattened to L1′ ∈RC×H1W 1, L2′ ∈RC×H2W 2, and
L3′ ∈RC×H3W 3.
Then, they are concatenated into a
feature matrix L ∈RC×(H1W 1+H2W 2+H3W 3), which pre-
serves a significant amount of low-level feature map infor-
mation, benefiting the network in detecting small objects.
The formula is as follows:
L =
n
X
i=1
Flatten(Conv1×1(Conv3×3(Fi)))
(1)
By simply flattening and concatenating, we can obtain
a cross-layer feature matrix that can effectively save com-
putational resources.
As shown in the upper right half of Figure 2, three oper-
ations are involved in extracting global features: Partition,
Transformer, and Combine operations. The model needs
to input the feature matrix into the Transformer unit to
compute attention. If self-attention is computed for each
feature point with every other feature point, it would re-
quire a significant amount of computational resources. To
address this issue, the feature matrix can be divided into
a specified number of parts before being input into the
Transformer unit, which significantly reduces the compu-
tational cost.
First, the Partition operation is performed by setting
the value of the partition count. In this case, Part is set to
2, meaning that the feature matrix is split into two feature
matrices. By using interval sampling, the feature matrix
L ∈RC×Length can be divided into new feature matrices
LP ∈RP art×(Length/P art)×C. This splitting method pre-
serves the spatial order of the pixels. If the length of the
feature matrix L is not divisible by Part, interpolation is
applied to the specified dimension using bilinear interpo-
lation. Compared to the patch-based nonlinear splitting
method used in MobileViT Block, the Partition operation
enables linear splitting and offers greater flexibility in the
splitting approach.The formula is as follows:
LP = Partition(L)
(2)
Next, the Transformer unit is used to model long-distance
dependencies. When performing self-attention computa-
tion, each feature point in the LP feature matrix only
calculates self-attention with the feature points in the
same row. This achieves the goal of reducing computa-
tional cost. Assuming the height, width, and channel di-
mensions of the feature map are H, W, and C respec-
tively, without using the Partition operation, the compu-
tational cost is denoted as O(WHC) .
After applying
6
the Partition operation with a value of 2 and using in-
terval sampling to select every other feature point within
each block for self-attention computation, the computa-
tional cost becomes O(WHC/2). Theoretically, the com-
putational cost is reduced by half. This approach is pos-
sible because image data itself contains a large amount
of data redundancy. For lower-level feature maps, adja-
cent pixels generally have similar information, and calcu-
lating self-attention for each pixel and its neighboring pix-
els would be computationally wasteful. On larger feature
maps, the computational cost of calculating self-attention
for neighboring pixels far exceeds the benefits in accu-
racy. After the Transformer unit computation, we obtain
LP ′ ∈RP art×(Length/P art)×C, which is represented by the
following formula:
LP ′(p) = Transformer(LP(p)), 1 ⩽p ⩽Part
(3)
Finally, through the Combine operation, the feature ma-
trix LP ′ is restored to its original arrangement. After this
process, we obtain L′ ∈RC×Length, where L′ has the same
dimensions as L.The formula is as follows:
L′ = Combine(LP ′(P))
(4)
3.3.
Feature fusion restoration
In order to obtain richer contextual information and
achieve the plug-and-play capability of modules, the re-
inforced feature vectors are fused with the original fea-
ture vectors. By performing convolutional operations, the
feature vectors are restored to their original scale with-
out modifying the structure of the target detection net-
work. The processed feature maps have stronger expres-
sive power as they not only capture the local and global
relationships between feature maps at different levels but
also model the dependencies between different levels and
different-sized objects. This improvement in capturing ca-
pabilities enhances the performance of the model.
L ∈RC×Length represents the input feature matrix to
the Transformer unit, while L′ ∈RC×Length represents the
output feature matrix from the Transformer unit. Both L
and L′ have the same scale. By concatenating these two
feature maps along the channel dimension, a new feature
matrix is generated, providing more feature information.
To restore the feature matrix back to the original multi-
scale feature maps, a 1x1 convolution layer is used to re-
duce the channel dimension of the feature matrix, resulting
in the feature matrix R ∈RC×Length. If bilinear interpo-
lation was used during the splitting of the feature matrix,
it is also necessary to use bilinear interpolation here to
restore the scale. Then, the feature matrix is sliced and
reshaped to obtain three feature maps R1 ∈RH1×W 1×C1,
R2 ∈RH2×W 2×C2, and R3 ∈RH3×W 3×C3 with the same
scale as the initial feature maps. The formula is as follows:
Ri = Recover(Conv1×1(Concat(L, L′)))
(5)
The processed feature maps possess stronger expressive
power, capturing the local and global connections between
different hierarchical feature maps and modeling the de-
pendencies between different levels and different-sized ob-
jects. This helps improve the model’s performance.
4.
Experiments
4.1.
Experimental settings
For our experiments, we trained our model on the PAS-
CAL VOC dataset [41], which contains 20 different ob-
ject classes. The training set consisted of 16,551 images.
Our model was implemented using the PyTorch framework
and trained and tested on an NVIDIA GeForce RTX 3090
GPU. During the training phase, we used the SGD opti-
mizer to update the parameters of the network model. The
batch size was set to 48, and we employed a technique for
dynamically adjusting the learning rate. The initial learn-
ing rate was set to 1e-3 and was changed to 1e-4 at 130
epochs and 1e-5 at 220 epochs. The momentum parameter
was set to 0.9, and the weight decay was set to 5e-4. The
total number of training epochs was set to 260.
7
Figure 3: Qualitative comparison of the prediction results between our method and other methods on the PASCAL VOC dataset.
Also, to better validate our method, we train on the
COCO dataset[42], which consists of 80 different object
classes with more complex object classification scenarios.
The dataset contains 82,783 training images and 40,504
test images.
Training and testing were performed on
NVIDIA GeForce RTX 4060 GPU. The batch size was
set to 16, the number of training epochs was set to 120,
and the rest of the parameters were set as in the PASCAL
VOC dataset.
We selected the SSD multi-scale object detection net-
work as the baseline model. The input image size was set
to 300 × 300. We qualitatively and quantitatively com-
pared our proposed modules with SE Block, CFAM [43],
and MobileViT Block algorithms. By comparing the dif-
ferences between various attention mechanisms, we can
demonstrate the effectiveness of our proposed method. SE
Block controls the contribution of each channel to the fi-
nal output by computing channel-wise weights, thereby
helping the model focus on important channel informa-
tion and improve accuracy in classification or regression
tasks. CFAM explicitly models the interdependencies be-
tween different-level feature maps, allowing the network
to focus on higher-level feature maps for detecting large
objects and lower-level feature maps for detecting small
objects.
MobileViT Block is a lightweight visual model
based on Transformer, designed specifically for resource-
constrained scenarios such as mobile devices.
4.2.
Qualitative comparisons
First, we present the prediction result images of different
object detection algorithms, comparing them using test
images from various scenarios, including dense object im-
ages, complex scene object images, and small object im-
ages. The results demonstrate that our proposed model
is capable of detecting targets with greater accuracy com-
pared to other algorithms (shown in Figure 4).
To visualize the decision process of the neural network,
we used the EigenCAM [44] method, which effectively vi-
sualizes the contribution of features at different levels of
the object detection network to the final detection results.
As shown in Figure 5, the results generated by SSD300-SE,
SSD300-CFAM, and SSD300-MViT are depicted in Fig.
7(c), (d), and (e) respectively, highlighting the importance
of different regions of pixels for the network’s detection re-
sults.
We compared the results using test images from
various scenarios, including images with complex scenes,
8
Figure 4: Qualitative comparison of class activation maps between our method and other methods on the PASCAL VOC dataset.
images containing salient objects, and images with dense
objects. The first image showcases the detection results in
a complex scene, while the second and third images repre-
sent the detection of salient objects. The fourth image is
an example of detecting dense objects. From the results, it
is evident that our proposed model exhibits the most accu-
rate class activation maps compared to other algorithms.
By observing the SSD class activation diagram as in
Fig. 7(b), we can find that although the SSD network ex-
tracts feature maps at different scales for detection, and
the large-scale feature maps can be used to detect small ob-
jects while the small-scale feature maps are used to detect
large objects; it does not effectively establish the local-
global connection of the feature maps at different scales.
By introducing the CFSAM module, we can find that the
SSD-CFSAM class activation map as in Fig. 7(f) creates
a connection between local and global features, and it can
be inferred that the CFSAM module effectively enhances
the ability of SSD missing.
4.3.
Quantitative comparisons
We conducted numerical evaluations of different network
models on the VOC2007 test set. The evaluation results
are shown in Table 1, which presents the quantitative com-
parison of various methods, including SSD300, HyperNet
[45], Ion [46], SSD300-SE, SSD300-CFAM, and SSD300-
MViT. Compared to other networks, our proposed model
achieves a higher mAP. By introducing the CFSAM mod-
ule into the SSD300 model, the mAP value is improved
by 3.1%, leading to a significant enhancement in detection
performance. As a comparative experiment, the SSD300-
MViT model, despite having a significantly higher num-
ber of parameters due to the addition of multiple Mobile-
ViT Blocks, obtains a lower mAP value. This is because
the MobileViT Block focuses on refining individual feature
maps without considering the relationships among multi-
ple prediction feature layers, which leads to inferior perfor-
mance in multi-scale object detection networks compared
to the CFSAM module. Table 1 also displays the AP val-
9
Table 1: Quantitative comparison of detection accuracy between our method and other methods on the PASCAL VOC dataset. Bold values
indicate the highest AP for each category.
Model
SSD300
HyperNet
Ion
SSD300-SE
SSD300-CFAM
SSD300-MViT
Ours
mAP
75.5
76.3
76.5
77.2
77.6
78.1
78.6
aero
81.2
77.4
79.2
83.4
79.6
84.1
82.7
bic.
82.4
83.3
79.2
83.5
84.6
85.5
85.4
bird
72.4
75.0
77.4
75.3
76.3
77.0
76.1
boat
66.0
69.1
69.8
69.9
71.8
70.0
73.3
bot.
47.2
62.4
55.7
50.0
50.6
51.6
52.7
bus
84.2
83.1
85.2
84.6
85.1
86.5
85.4
car
85.1
87.4
84.2
86.0
86.1
86.6
86.9
cat
85.3
87.4
89.8
86.6
89.1
88.1
88.8
cha.
57.4
57.1
57.5
60.7
62.4
62.8
62.2
cow
80.9
79.8
78.5
82.1
84.6
85.3
87.2
din.
76.0
71.4
73.8
76.0
75.9
74.5
76.6
dog
82.1
85.1
87.8
85.5
86.2
85.8
86.1
hor.
85.9
85.1
85.9
85.7
87.6
86.7
88.0
mot.
82.9
80.0
81.3
83.4
83.3
84.0
85.1
per.
77.3
79.1
75.3
78.2
77.5
79.6
79.3
pot.
49.9
51.2
49.7
53.4
50.4
51.2
52.1
she.
76.7
79.1
76.9
77.1
77.9
78.0
78.9
sofa
76.7
75.7
74.6
79.4
79.3
79.6
79.1
tra.
86.5
80.9
85.2
86.4
87.8
87.8
88.1
tv.
73.8
76.5
82.1
77.1
76.3
77.1
78.1
Table 2:
Quantitative comparison of the detection accuracy of
our method with SSD-Det and BoundConvNet on PASCAL VOC
dataset.
Model
SSD300
SSD-Det
BoundConvNet
Ours
mAP
75.5
77.1
78.5
78.6
ues of these network models for the 20 object categories.
From the data in the table, it can be observed that our pro-
posed model outperforms existing methods, particularly in
the category of small objects.
Meanwhile,
we compare the performance of SSD-
Det[47]and BoundConvNet [48], the target detection mod-
els proposed in recent years, on the VOC2007 dataset, and
Table 3: Quantitative comparison of detection accuracy between our
method and other methods on the COCO dataset.
Model
SSD300
RCNN-FPN
EfficientDet
Ours
AP@0.5
41.2
56.9
53.0
52.1
as shown in Table 2, our proposed SSD-CFSAM outper-
forms them.
In order to better evaluate our model, we perform nu-
merical evaluation on the COCO2014 test set.The number
of image targets in the COCO dataset is higher and the
target size is smaller, so the task on the COCO dataset
is more challenging. The evaluation results are shown in
10
Table 4: Quantitative Comparison of Detection Speed, Computa-
tional Complexity, and Parameter Count between Our Method and
Other Methods on the PASCAL VOC Dataset.
Model
FPS↑
GFLOPs↓
Params/MB↓
SSD300
38
31.37
26.29
SSD300-SE
36
31.38
26.51
SSD300-CFAM
44
30.93
25.69
SSD300-MViT
44
53.74
76.75
Ours
57
45.22
45.16
the table 3, by quantitatively comparing with SSD300,
RCNN-FPN[49], and EfficientDet [50].Compared with the
baseline, our proposed CFSAM module significantly im-
proves the performance of SSD300 on the COCO dataset,
and the AP@0.5 value is improved by 10.9% compared
with the baseline model.Although the SSD model is one
of the pioneers in the field of single-stage target detec-
tion, with the iteration of target detection technology, it
is being surpassed by models such as Faster RCNN-FPN
and efficientDet surpassed, as a comparison experiment we
can see that our proposed CFSAM module applied to the
SSD model closes the distance with Faster RCNN-FPN
and EfficientDet-D0 and achieves a good performance.
Table 4 shows the comparison of FPS, GFLOPs, and
parameter count for five network models.
SSD300-SE
and SSD300-CFAM models use channel attention meth-
ods, while SSD300-MViT model and our proposed model
use self-attention methods, therefore, the SSD300-MViT
model and our proposed model based on self-attention
mechanism have higher floating-point operations per sec-
ond (GFLOPs) and more parameters in the Params indi-
cator. Because our model uses the Partition operation to
reduce the computation of self-attention, our model has
a significant reduction in GFLOPs and Params compared
to the SSD300-MViT model, and performs better. Our
model has the best performance in the FPS indicator with
the fastest detection speed. Although the FPS of object
detection models is usually related to the model parame-
Figure 5: Comparison of training loss between our method and other
methods on the PASCAL VOC dataset.
ter count and computational complexity, it is not absolute,
and many factors affect the FPS of the network model.
FPS is more dependent on the speed of forward propaga-
tion of the object detection model.
We also compared the training losses of the models. To
make a fair comparison, we set the number of training
epochs to 260 and used dynamic learning rate adjustment.
The initial learning rate was set to 1e-3 and changed to 1e-
4 and 1e-5 at 130 and 220 epochs, respectively. As shown
in Figure 6, the baseline SSD300 model converged the slow-
est, while our proposed model converged the fastest. The
learning rate adjustment at 130 epochs resulted in a signif-
icant decrease in loss value, and our model had the lowest
loss value at convergence. These results demonstrate that
the CFSAM module can speed up the training process and
reduce the convergence loss of the network model.
4.4.
Ablation study
A comprehensive ablation study was conducted to
meticulously evaluate the impact of each component
within the proposed CFSAM module.
The experiments
were performed on the PASCAL VOC dataset using the
SSD300 as the baseline model. To clearly illustrate the
contribution of each component, we report the Aver-
age Precision (AP) for three representative object cate-
gories—bird (small objects), bus (large objects), and cow
(medium-sized objects)—in Table 5. These categories were
11
selected to reflect the model’s performance across objects
of different scales.
The experiments commenced with the baseline SSD300
model.
We then progressively integrated the three core
components of CFSAM: Local Feature Extraction (LFE),
Global Feature Extraction (GFE), and Feature Fusion
Restoration (FFR). The results are encapsulated in Ta-
ble 5. The baseline model achieved APs of 72.4%, 84.2%,
and 80.9% for bird, bus, and cow, respectively. The intro-
duction of the LFE component, responsible for capturing
spatial details and unifying channel dimensions, brought
noticeable improvements across all categories, with the AP
for cow rising to 82.1%. This underscores the importance
of preserving fine-grained local information.
Subsequently, adding the GFE component, which mod-
els long-range, cross-layer dependencies via our novel
Partition-based Transformer, led to a further performance
boost. The AP for the challenging bird category increased
to 75.3%, demonstrating that capturing global contextual
relationships is crucial for detecting objects with extreme
scale variations, particularly small targets.
The final integration of the FFR component, which co-
herently fuses the original and enhanced features, resulted
in the highest AP scores for all three categories, reaching
76.1%, 85.4%, and 87.2% for bird, bus, and cow, respec-
tively. This confirms that the synergistic operation of all
three components enables the model to construct a more
powerful and discriminative feature representation, effec-
tively combining local precision with global context.
Table 5:
Ablation study of the proposed CFSAM components on
the PASCAL VOC 2007 test set.
Model
bird
bus
cow
SSD300 (Baseline)
72.4
84.2
80.9
SSD300+LEF
73.2
84.7
79.4
SSD300+LEF+GFE
75.4
84.5
85.7
SSD300+CFSAM(Ours)
76.1
85.4
87.2
To further investigate the impact of CFSAM’s place-
ment within the feature pyramid, we conduct an ablation
study by inserting the module into different subsets of
the SSD300 prediction layers.
Specifically, we compare
three configurations: applying CFSAM only to the first
three layers (shallow features), only to the last three lay-
ers (deep features), and to all six prediction layers. We
evaluate the performance on three representative object
categories—bird (small objects), bus (large objects), and
cow (medium-sized objects)—to analyze the scale-specific
effects. Results on the PASCAL VOC 2007 test set are
summarized in Table 6.
Table 6:
Ablation study on the insertion positions of CFSAM in
SSD300 on PASCAL VOC 2007 test set.
Model
bird
bus
cow
SSD300 (Baseline)
72.4
84.2
80.9
+CFSAM(Shallow)
74.8
84.6
83.2
+CFSAM(Deep)
73.5
85.0
84.7
+CFSAM(All)
76.1
85.4
87.2
We observe that inserting CFSAM at any level brings
improvements over the baseline across all object scales.
However, the full configuration (CFSAM-All) achieves the
highest AP scores for all three categories, demonstrat-
ing the importance of cross-layer attention across the en-
tire feature hierarchy. Notably, applying CFSAM only to
the first three layers shows the strongest improvement for
small objects, while applying it only to the last three lay-
ers benefits large and medium-sized objects detection the
most. This aligns with the intuition that shallow features
contain more detailed information beneficial for small ob-
jects, while deep features capture higher-level semantics
crucial for large and medium-sized objects. Nevertheless,
only the complete integration across all six layers maxi-
mizes performance uniformly across all object scales, en-
abling comprehensive modeling of dependencies from fine-
grained details to high-level semantics.
12
5.
Discussion
In this paper we propose a cross-layer feature retention
module (CFSAM) and also construct a modified version
of the SSD300 multiscale target detection model based
on this module. The experimental results verify that the
proposed cross-layer feature retention module (CFSAM)
with plug-and-play capability is more performant relative
to other enhancement modules, and the CFSAM-based
multiscale target detection model is greatly improved in
the baseline.
However, due to the use of a transformer
computational unit in CFSAM, the approach cannot be
applied in object detection models for mobile devices. In
future work, we will explore optimizing CFSAM in terms
of reducing feature dimensions and compressing model pa-
rameters to address this limitation.
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Cross-Layer Feature Self-Attention Module for Multi-Scale Object Detection Dingzhou Xiea, Rushi Lanc, Cheng Pangc,∗, Enhao Ningb, Jiahao Zengb, Wei Zhengd,∗ aGuangzhou Huashang College, Hua shang Road, Guangzhou, 511300, Guangdong, China bGuangxi Normal University, Yu cai Road, Guilin, 541004, Gangxi, China cGuilin 541004, Gangxi, China dJinling 211169, Jiangsu, China Abstract Recent object detection methods have made remarkable progress by leveraging attention mechanisms to improve feature discriminability. However, most existing approaches are confined to refining single-layer or fusing dual-layer features, overlooking the rich inter-layer dependencies across multi-scale representations. This limits their ability to capture comprehensive contextual information essential for detecting objects with large scale variations. In this paper, we propose a novel Cross-Layer Feature Self-Attention Module (CFSAM), which holistically models both local and global dependencies within multi-scale feature maps. CFSAM consists of three key components: a convolutional local feature extractor, a Transformer-based global modeling unit that efficiently captures cross-layer interactions, and a feature fusion mechanism to restore and enhance the original representations. When integrated into the SSD300 framework, CFSAM significantly boosts detection performance, achieving 78.6% mAP on PASCAL VOC (vs. 75.5% baseline) and 52.1% mAP on COCO (vs. 43.1% baseline), outperforming existing attention modules. Moreover, the module accelerates convergence during training without introducing substantial computational overhead. Our work highlights the importance of explicit cross-layer attention modeling in advancing multi-scale object detection. Keywords: attention mechanism, object detection, feature fusion, feature enhancement 1. Introduction The escalating demand for intelligent automation and industrial inspection has positioned robust multi-scale object detection as a cornerstone technology. While deep learning has driven significant progress, detecting objects with extreme scale variation remains a formidable challenge, primarily due to the semantic and resolution gaps between features at different levels of the network hierarchy [1, 2]. Effectively integrating these multi-scale features ∗Corresponding author: Email addresses: (Dingzhou Xie), (Rushi Lan), (Cheng Pang), (Enhao Ning), (Jiahao Zeng), (Wei Zheng) to achieve consistent performance across small, medium, and large objects is thus a critical and ongoing research focus. Attention mechanisms have become instrumental in advancing this field by enabling models to dynamically focus on more informative spatial regions or feature channels [3, 4, 5]. A pivotal aspect of this is modeling long-range dependencies to capture global contextual information. Two dominant paradigms for this purpose are structured convolutional operators [6, 7, 8, 9, 10] and self-attention mechanisms [11, 12]. While enhanced convolutions can enlarge the receptive field, their capacity for global modeling is often constrained by local connectivity and may require careful hyperparameter tuning to avoid artifacts. In contrast, self-attention mechanisms, catalyzed by the Vision 16 Oct 2025 Transformer (ViT) [13], inherently model pairwise interactions across all positions in a feature map, providing superior flexibility in building global context. Subsequent architectures like the Swin Transformer [14] have further enhanced their efficiency and applicability to vision tasks. Notwithstanding their achievements, these methods have some limits in multi-scale object detection. Convolution-based methods often struggle to capture dependencies outside their fixed receptive field. Also, many modern self-attention modules are made for single-scale feature maps [15, 16, 17, 18]. They process patches independently and lack a clear way to model connections across different layers of a feature pyramid. This restricts the integration of contextual information across semantic levels and hinders the model from jointly using fine details and high-level semantics, which is very important for accurate multi-scale detection. To bridge this gap, we propose a Cross-Layer Feature Self-Attention Module (CFSAM). CFSAM is designed to holistically model interactions across multi-scale features within a unified and efficient self-attention framework. It comprises three components: 1) a local feature extractor using convolutional layers to preserve spatial details, 2) a global feature modeling unit that employs a novel partition-based Transformer to efficiently capture longrange dependencies across all input scales simultaneously, and 3) a feature fusion and restoration unit that coherently integrates the refined features back to their original dimensions. The principal contributions of this work are: • To effectively exploit long-range dependencies between multi-layer feature maps, we designed the SSD300-CFSAM model. This design enables the model to better understand both the relationships between objects in an image and the contextual information of the scene. • The CFSAM module achieves notable performance improvements without a significant increase in model complexity or the number of parameters. Furthermore, its plug-and-play design allows for seamless and flexible integration into various existing neural network architectures. • Quantitative and qualitative experimental results on PASCAL VOC and COCO datasets show that our method not only achieves significant performance improvements over the baseline but also maintains superior computational efficiency compared to other enhancement modules. The remainder of this paper is structured as follows: Section 2 reviews related work. Section 3 details the CFSAM methodology. Section 4 presents experimental settings, results, and ablation studies. Finally, Section 5 concludes the paper. 2. Related work multi-scale object detection requires different attention to targets at different scales in the same image, while the attention mechanism focuses on weighting important information on the feature map, which allows neural networks to pay more attention to or prioritize specific input features, thus improving the performance of visual tasks such as object detection. There are complementarities between the two, so we conduct an in-depth study on how attention mechanisms can facilitate multi-scale object detection. Among them, channel attention and spatial attention are two commonly used attention mechanisms when processing image data. Channel attention models include SENet [19], GSoPNet [20], and SRM [21]. In 2017, Hu et al. introduced the concept of channel attention and proposed the SENet model. The core idea of this model is to dynamically adjust the importance of each feature channel using Squeeze and Excitation operations. In 2019, Gao et al. proposed the GSoP-Net model, which utilizes the Global Second-Order 2 Pooling (GSoP) Block to model higher-order statistical information. The GSoP Block also consists of Squeeze and Excitation modules. In the Squeeze module, the GSoP Block first reduces the number of channels with a 1 × 1 convolution, then computes the covariance matrix to capture the correlations between different channels. The covariance matrix is then row-normalized to establish connections between channels. By employing second-order pooling, the GSoP Block enhances the ability to gather global information. Also in 2019, Lee et al. introduced the Style-Based Recalibration Module (SRM). The SRM consists of two main parts: style pooling and style integration. The style pooling performs global average pooling and calculates the standard deviation on the input features to extract style features from each channel. The style integration replaces the original fully connected layer with a lightweight channel-wise fully connected layer to reduce computational requirements. It estimates style weights for each channel based on the results of the style pooling and then recalibrates the feature maps using these style weights to amplify or suppress their information. Classic spatial attention methods have evolved significantly, with recent research focusing on efficiently integrating and enhancing spatial feature representations. For instance, ACmix[22] elegantly combines the local feature extraction power of convolutional operations with the global receptive field of self-attention, offering a unified computational structure that adaptively fuses the outputs of both paradigms. Meanwhile, for dense prediction tasks such as object detection, Deformable Attention (DAttn) [23] has emerged as a prominent solution. It avoids the excessive computational cost of global self-attention by allowing a sparse set of key sampling points to dynamically deform based on the input features, thereby enabling the model to focus on more relevant spatial locations efficiently. These methods exemplify the modern trend of building spatiallyaware models that are both powerful and computationally feasible. Convolutional neural networks excel at handling image data due to their properties such as translation invariance, making them widely used in image processing. However, CNNs lack scale and distortion invariance. In 2015, Jaderberg et al. proposed Spatial Transformer Networks, which learn an affine transformation for each input image to improve the model's ability to handle targets in different positions, orientations, and scales. Integrating spatial transformers into neural network models allows the models to automatically focus on the regions of interest, thereby enhancing the accuracy and robustness of the models in object recognition tasks. Subsequent research efforts have achieved even greater success in this area [7, 24, 25, 26]. The self-attention mechanism is a method for computing attention within a single input sequence. Unlike traditional attention mechanisms [27, 28, 29, 30], self-attention considers not only the relevance between each position in the input sequence and a target but also the relevance between each position and other positions. This allows self-attention to capture interactions between all parts of the input sequence, better handling long-range dependencies. In 2017, Vaswani et al. [31] proposed the Transformer model, which adopts self-attention to model relationships between different positional elements in the input sequence. The Transformer achieved impressive results in natural language processing tasks, enabling long sequence modeling and parallel computation. Besides natural language processing, the Transformer model has also been applied in computer vision, audio processing, medical diagnosis, and other fields [32, 33, 34, 35]. In 2020, Dosovitskiy et al. introduced the Vision Transformer model, which consists of an encoder composed of multiple Transformer modules and a linear layer for classification. Unlike traditional CNNs, ViT takes a two-dimensional image as input but transforms it into a one-dimensional sequence to perform self-attention computations on the sequence, learning relationships between different positions. To prevent information loss, ViT adds trainable position embed3 Figure 1: Illustration of the whole network. By inserting the CFSAM into the SSD network model, the module takes six predicted feature maps as input. After modeling the dependency relationships, the module outputs the same size and number of predicted feature maps, which are used for object prediction. dings at each position of the sequence to provide positional information. The emergence of ViT offers a new perspective in the field of deep learning, applying self-attention to other domains. Using the core ideas of these methods, we propose to construct a multi-scale object detection model based on cross-layer feature self-attention module to strengthen the ability to model the relationship between feature maps at different levels and enhance the accuracy of multi-scale object detection. 3. Methodology Multi-scale object detection networks are deep neural network models used for object detection. They utilize multi-scale prediction feature maps, which have the advantage of effectively detecting objects of different sizes. Common multi-scale object detection networks include Faster R-CNN [36], YOLO [37, 38, 39], and SSD [40]. The SSD model predicts objects using anchor boxes of different sizes and aspect ratios on multi-scale feature maps. This reduces the sensitivity of the SSD network to changes in object sizes and improves detection accuracy. We have chosen the SSD object detection network as the baseline model with an input image size of 300 × 300. Figure 1 illustrates how the CFSAM is applied in the SSD network. For multi-scale object detection networks, this module provides a plug-and-play solution. The CFSAM we propose consists of three parts: local feature extraction, global feature extraction, and feature fusion restoration. As shown in Figure 2, F1, F2, and F3 represent the multi-scale predicted feature maps. The local feature extraction part utilizes convolutional operations. Specifically, it first performs local feature extraction using a convolutional layer with a kernel size of 3 × 3, and then adjusts the channel count using a convolutional layer with a kernel size of 1 × 1, resulting in feature maps L1, L2, and L3. The global feature extraction part employs Flatten, Partition, and Combine operations for global feature modeling. The Flatten operation is used to flatten the H × W regions of L1, L2, and L3 feature maps into vectors and concatenate them along the vector dimension to obtain the cross-layer fused feature vector. The Partition operation divides the feature vector into two parts by sampling with a certain interval. During the self-attention calculation, each feature point of a feature vector only interacts with other feature points within itself, reducing 4 Figure 2: Illustrates the structural diagram of the CFSAM, which includes three parts: local feature extraction, global feature extraction, and feature fusion restoration. the computational complexity. The Combine operation is applied to restore the feature vector back to its original shape. In the feature fusion restoration part, a shortcut branch is first used to concatenate the feature vector obtained after global feature extraction with the crosslayer feature vector before global feature extraction along the channel dimension, doubling the number of channels. Then, a convolutional layer with a kernel size of 1 × 1 is used to adjust the channel count back to the original size. Finally, the feature vector is split and restored to three feature maps R1, R2, and R3, which have the same scale as F1, F2, and F3, respectively. The Transformer unit can capture long-distance dependency relationships in the input sequence and weight the information at each position. Therefore, the resulting three feature maps contain richer contextual information. 3.1. Local feature extraction Local features are crucial for object classification. For example, when detecting an image of a cat, convolutional layers may recognize local features such as the ears, eyes, and nose of the cat. These local features have a significant impact on determining the target category in the image. To capture the local information of the input feature map, a local feature extraction structure is designed using convolutional layers with kernel sizes of 1 × 1 and 3 × 3. This design effectively promotes the detection efficiency of the target detection model while adding only a small amount of computation. For the given input feature maps F1, F2, and F3, their channel numbers and sizes are different. They are separately input into a convolutional layer with a kernel size of 3 × 3 and a convolutional layer with a kernel size of 1 × 1. The 3 × 3 convolutional layer is used to extract local spatial features from the image, including edge and texture information. These local features are crucial for object classification. For example, when detecting an image of a cat, the convolutional layer may identify local features such as ears, eyes, and nose, which greatly influence the determination of the object category in the image. The 1 × 1 convolutional layer reduces the channel number of the input feature maps, thereby reducing computational costs and improving detection efficiency. Since the predicted feature maps have a significant difference in channel numbers, with some reaching 1024 dimensions while others have only 256 dimensions, inputting them directly into the Transformer unit without processing would result in higher computational and memory costs for feature maps with larger channel numbers. This would slow down the 5 network's operation speed. Therefore, using a 1 × 1 convolutional layer to uniformly adjust the channel number to 256 dimensions significantly reduces the computational load. Experimental results have shown that the detection accuracy of the network is not compromised. After extracting the local information, output feature maps L1, L2, and L3 are obtained, with the same channel number for all three feature maps. 3.2. Global feature extraction The SSD object detection model extracts features of different scales from various layers of the network for prediction, and there is a close relationship between these feature maps at different levels. In object detection tasks, there exists interdependence among different objects. Therefore, in the object detection network, it is necessary to consider the dependency between feature maps at different levels in order to better locate and detect target objects. Before extracting global features, feature fusion is performed. Feature fusion methods include direct addition, channel concatenation, and layer-wise fusion, among others. These methods require upsampling to align the scale of the feature maps, which increases the computational load of the model. We achieve cross-layer fusion of all predicted feature maps by flattening and concatenating them, without using upsampling operations, thereby avoiding an increase in computational load for the model. As shown in the upper left half of Figure 2, the input feature maps are L1 ∈RH1×W 1×C, L2 ∈RH2×W 2×C, and L3 ∈RH3×W 3×C. Firstly, the feature maps are flattened to L1′ ∈RC×H1W 1, L2′ ∈RC×H2W 2, and L3′ ∈RC×H3W 3. Then, they are concatenated into a feature matrix L ∈RC×(H1W 1+H2W 2+H3W 3), which preserves a significant amount of low-level feature map information, benefiting the network in detecting small objects. The formula is as follows: L = n X i=1 Flatten(Conv1×1(Conv3×3(Fi))) (1) By simply flattening and concatenating, we can obtain a cross-layer feature matrix that can effectively save computational resources. As shown in the upper right half of Figure 2, three operations are involved in extracting global features: Partition, Transformer, and Combine operations. The model needs to input the feature matrix into the Transformer unit to compute attention. If self-attention is computed for each feature point with every other feature point, it would require a significant amount of computational resources. To address this issue, the feature matrix can be divided into a specified number of parts before being input into the Transformer unit, which significantly reduces the computational cost. First, the Partition operation is performed by setting the value of the partition count. In this case, Part is set to 2, meaning that the feature matrix is split into two feature matrices. By using interval sampling, the feature matrix L ∈RC×Length can be divided into new feature matrices LP ∈RP art×(Length/P art)×C. This splitting method preserves the spatial order of the pixels. If the length of the feature matrix L is not divisible by Part, interpolation is applied to the specified dimension using bilinear interpolation. Compared to the patch-based nonlinear splitting method used in MobileViT Block, the Partition operation enables linear splitting and offers greater flexibility in the splitting approach.The formula is as follows: LP = Partition(L) (2) Next, the Transformer unit is used to model long-distance dependencies. When performing self-attention computation, each feature point in the LP feature matrix only calculates self-attention with the feature points in the same row. This achieves the goal of reducing computational cost. Assuming the height, width, and channel dimensions of the feature map are H, W, and C respectively, without using the Partition operation, the computational cost is denoted as O(WHC) . After applying 6 the Partition operation with a value of 2 and using interval sampling to select every other feature point within each block for self-attention computation, the computational cost becomes O(WHC/2). Theoretically, the computational cost is reduced by half. This approach is possible because image data itself contains a large amount of data redundancy. For lower-level feature maps, adjacent pixels generally have similar information, and calculating self-attention for each pixel and its neighboring pixels would be computationally wasteful. On larger feature maps, the computational cost of calculating self-attention for neighboring pixels far exceeds the benefits in accuracy. After the Transformer unit computation, we obtain LP ′ ∈RP art×(Length/P art)×C, which is represented by the following formula: LP ′(p) = Transformer(LP(p)), 1 ⩽p ⩽Part (3) Finally, through the Combine operation, the feature matrix LP ′ is restored to its original arrangement. After this process, we obtain L′ ∈RC×Length, where L′ has the same dimensions as L.The formula is as follows: L′ = Combine(LP ′(P)) (4) 3.3. Feature fusion restoration In order to obtain richer contextual information and achieve the plug-and-play capability of modules, the reinforced feature vectors are fused with the original feature vectors. By performing convolutional operations, the feature vectors are restored to their original scale without modifying the structure of the target detection network. The processed feature maps have stronger expressive power as they not only capture the local and global relationships between feature maps at different levels but also model the dependencies between different levels and different-sized objects. This improvement in capturing capabilities enhances the performance of the model. L ∈RC×Length represents the input feature matrix to the Transformer unit, while L′ ∈RC×Length represents the output feature matrix from the Transformer unit. Both L and L′ have the same scale. By concatenating these two feature maps along the channel dimension, a new feature matrix is generated, providing more feature information. To restore the feature matrix back to the original multiscale feature maps, a 1x1 convolution layer is used to reduce the channel dimension of the feature matrix, resulting in the feature matrix R ∈RC×Length. If bilinear interpolation was used during the splitting of the feature matrix, it is also necessary to use bilinear interpolation here to restore the scale. Then, the feature matrix is sliced and reshaped to obtain three feature maps R1 ∈RH1×W 1×C1, R2 ∈RH2×W 2×C2, and R3 ∈RH3×W 3×C3 with the same scale as the initial feature maps. The formula is as follows: Ri = Recover(Conv1×1(Concat(L, L′))) (5) The processed feature maps possess stronger expressive power, capturing the local and global connections between different hierarchical feature maps and modeling the dependencies between different levels and different-sized objects. This helps improve the model's performance. 4. Experiments 4.1. Experimental settings For our experiments, we trained our model on the PASCAL VOC dataset [41], which contains 20 different object classes. The training set consisted of 16,551 images. Our model was implemented using the PyTorch framework and trained and tested on an NVIDIA GeForce RTX 3090 GPU. During the training phase, we used the SGD optimizer to update the parameters of the network model. The batch size was set to 48, and we employed a technique for dynamically adjusting the learning rate. The initial learning rate was set to 1e-3 and was changed to 1e-4 at 130 epochs and 1e-5 at 220 epochs. The momentum parameter was set to 0.9, and the weight decay was set to 5e-4. The total number of training epochs was set to 260. 7 Figure 3: Qualitative comparison of the prediction results between our method and other methods on the PASCAL VOC dataset. Also, to better validate our method, we train on the COCO dataset[42], which consists of 80 different object classes with more complex object classification scenarios. The dataset contains 82,783 training images and 40,504 test images. Training and testing were performed on NVIDIA GeForce RTX 4060 GPU. The batch size was set to 16, the number of training epochs was set to 120, and the rest of the parameters were set as in the PASCAL VOC dataset. We selected the SSD multi-scale object detection network as the baseline model. The input image size was set to 300 × 300. We qualitatively and quantitatively compared our proposed modules with SE Block, CFAM [43], and MobileViT Block algorithms. By comparing the differences between various attention mechanisms, we can demonstrate the effectiveness of our proposed method. SE Block controls the contribution of each channel to the final output by computing channel-wise weights, thereby helping the model focus on important channel information and improve accuracy in classification or regression tasks. CFAM explicitly models the interdependencies between different-level feature maps, allowing the network to focus on higher-level feature maps for detecting large objects and lower-level feature maps for detecting small objects. MobileViT Block is a lightweight visual model based on Transformer, designed specifically for resourceconstrained scenarios such as mobile devices. 4.2. Qualitative comparisons First, we present the prediction result images of different object detection algorithms, comparing them using test images from various scenarios, including dense object images, complex scene object images, and small object images. The results demonstrate that our proposed model is capable of detecting targets with greater accuracy compared to other algorithms (shown in Figure 4). To visualize the decision process of the neural network, we used the EigenCAM [44] method, which effectively visualizes the contribution of features at different levels of the object detection network to the final detection results. As shown in Figure 5, the results generated by SSD300-SE, SSD300-CFAM, and SSD300-MViT are depicted in Fig. 7(c), (d), and (e) respectively, highlighting the importance of different regions of pixels for the network's detection results. We compared the results using test images from various scenarios, including images with complex scenes, 8 Figure 4: Qualitative comparison of class activation maps between our method and other methods on the PASCAL VOC dataset. images containing salient objects, and images with dense objects. The first image showcases the detection results in a complex scene, while the second and third images represent the detection of salient objects. The fourth image is an example of detecting dense objects. From the results, it is evident that our proposed model exhibits the most accurate class activation maps compared to other algorithms. By observing the SSD class activation diagram as in Fig. 7(b), we can find that although the SSD network extracts feature maps at different scales for detection, and the large-scale feature maps can be used to detect small objects while the small-scale feature maps are used to detect large objects; it does not effectively establish the localglobal connection of the feature maps at different scales. By introducing the CFSAM module, we can find that the SSD-CFSAM class activation map as in Fig. 7(f) creates a connection between local and global features, and it can be inferred that the CFSAM module effectively enhances the ability of SSD missing. 4.3. Quantitative comparisons We conducted numerical evaluations of different network models on the VOC2007 test set. The evaluation results are shown in Table 1, which presents the quantitative comparison of various methods, including SSD300, HyperNet [45], Ion [46], SSD300-SE, SSD300-CFAM, and SSD300MViT. Compared to other networks, our proposed model achieves a higher mAP. By introducing the CFSAM module into the SSD300 model, the mAP value is improved by 3.1%, leading to a significant enhancement in detection performance. As a comparative experiment, the SSD300MViT model, despite having a significantly higher number of parameters due to the addition of multiple MobileViT Blocks, obtains a lower mAP value. This is because the MobileViT Block focuses on refining individual feature maps without considering the relationships among multiple prediction feature layers, which leads to inferior performance in multi-scale object detection networks compared to the CFSAM module. Table 1 also displays the AP val9 Table 1: Quantitative comparison of detection accuracy between our method and other methods on the PASCAL VOC dataset. Bold values indicate the highest AP for each category. Model SSD300 HyperNet Ion SSD300-SE SSD300-CFAM SSD300-MViT Ours mAP 75.5 76.3 76.5 77.2 77.6 78.1 78.6 aero 81.2 77.4 79.2 83.4 79.6 84.1 82.7 bic. 82.4 83.3 79.2 83.5 84.6 85.5 85.4 bird 72.4 75.0 77.4 75.3 76.3 77.0 76.1 boat 66.0 69.1 69.8 69.9 71.8 70.0 73.3 bot. 47.2 62.4 55.7 50.0 50.6 51.6 52.7 bus 84.2 83.1 85.2 84.6 85.1 86.5 85.4 car 85.1 87.4 84.2 86.0 86.1 86.6 86.9 cat 85.3 87.4 89.8 86.6 89.1 88.1 88.8 cha. 57.4 57.1 57.5 60.7 62.4 62.8 62.2 cow 80.9 79.8 78.5 82.1 84.6 85.3 87.2 din. 76.0 71.4 73.8 76.0 75.9 74.5 76.6 dog 82.1 85.1 87.8 85.5 86.2 85.8 86.1 hor. 85.9 85.1 85.9 85.7 87.6 86.7 88.0 mot. 82.9 80.0 81.3 83.4 83.3 84.0 85.1 per. 77.3 79.1 75.3 78.2 77.5 79.6 79.3 pot. 49.9 51.2 49.7 53.4 50.4 51.2 52.1 she. 76.7 79.1 76.9 77.1 77.9 78.0 78.9 sofa 76.7 75.7 74.6 79.4 79.3 79.6 79.1 tra. 86.5 80.9 85.2 86.4 87.8 87.8 88.1 tv. 73.8 76.5 82.1 77.1 76.3 77.1 78.1 Table 2: Quantitative comparison of the detection accuracy of our method with SSD-Det and BoundConvNet on PASCAL VOC dataset. Model SSD300 SSD-Det BoundConvNet Ours mAP 75.5 77.1 78.5 78.6 ues of these network models for the 20 object categories. From the data in the table, it can be observed that our proposed model outperforms existing methods, particularly in the category of small objects. Meanwhile, we compare the performance of SSDDet[47]and BoundConvNet [48], the target detection models proposed in recent years, on the VOC2007 dataset, and Table 3: Quantitative comparison of detection accuracy between our method and other methods on the COCO dataset. Model SSD300 RCNN-FPN EfficientDet Ours 41.2 56.9 53.0 52.1 as shown in Table 2, our proposed SSD-CFSAM outperforms them. In order to better evaluate our model, we perform numerical evaluation on the COCO2014 test set.The number of image targets in the COCO dataset is higher and the target size is smaller, so the task on the COCO dataset is more challenging. The evaluation results are shown in 10 Table 4: Quantitative Comparison of Detection Speed, Computational Complexity, and Parameter Count between Our Method and Other Methods on the PASCAL VOC Dataset. Model FPS↑ GFLOPs↓ Params/MB↓ SSD300 38 31.37 26.29 SSD300-SE 36 31.38 26.51 SSD300-CFAM 44 30.93 25.69 SSD300-MViT 44 53.74 76.75 Ours 57 45.22 45.16 the table 3, by quantitatively comparing with SSD300, RCNN-FPN[49], and EfficientDet [50].Compared with the baseline, our proposed CFSAM module significantly improves the performance of SSD300 on the COCO dataset, and the value is improved by 10.9% compared with the baseline model.Although the SSD model is one of the pioneers in the field of single-stage target detection, with the iteration of target detection technology, it is being surpassed by models such as Faster RCNN-FPN and efficientDet surpassed, as a comparison experiment we can see that our proposed CFSAM module applied to the SSD model closes the distance with Faster RCNN-FPN and EfficientDet-D0 and achieves a good performance. Table 4 shows the comparison of FPS, GFLOPs, and parameter count for five network models. SSD300-SE and SSD300-CFAM models use channel attention methods, while SSD300-MViT model and our proposed model use self-attention methods, therefore, the SSD300-MViT model and our proposed model based on self-attention mechanism have higher floating-point operations per second (GFLOPs) and more parameters in the Params indicator. Because our model uses the Partition operation to reduce the computation of self-attention, our model has a significant reduction in GFLOPs and Params compared to the SSD300-MViT model, and performs better. Our model has the best performance in the FPS indicator with the fastest detection speed. Although the FPS of object detection models is usually related to the model parameFigure 5: Comparison of training loss between our method and other methods on the PASCAL VOC dataset. ter count and computational complexity, it is not absolute, and many factors affect the FPS of the network model. FPS is more dependent on the speed of forward propagation of the object detection model. We also compared the training losses of the models. To make a fair comparison, we set the number of training epochs to 260 and used dynamic learning rate adjustment. The initial learning rate was set to 1e-3 and changed to 1e4 and 1e-5 at 130 and 220 epochs, respectively. As shown in Figure 6, the baseline SSD300 model converged the slowest, while our proposed model converged the fastest. The learning rate adjustment at 130 epochs resulted in a significant decrease in loss value, and our model had the lowest loss value at convergence. These results demonstrate that the CFSAM module can speed up the training process and reduce the convergence loss of the network model. 4.4. Ablation study A comprehensive ablation study was conducted to meticulously evaluate the impact of each component within the proposed CFSAM module. The experiments were performed on the PASCAL VOC dataset using the SSD300 as the baseline model. To clearly illustrate the contribution of each component, we report the Average Precision (AP) for three representative object categories-bird (small objects), bus (large objects), and cow (medium-sized objects)-in Table 5. These categories were 11 selected to reflect the model's performance across objects of different scales. The experiments commenced with the baseline SSD300 model. We then progressively integrated the three core components of CFSAM: Local Feature Extraction (LFE), Global Feature Extraction (GFE), and Feature Fusion Restoration (FFR). The results are encapsulated in Table 5. The baseline model achieved APs of 72.4%, 84.2%, and 80.9% for bird, bus, and cow, respectively. The introduction of the LFE component, responsible for capturing spatial details and unifying channel dimensions, brought noticeable improvements across all categories, with the AP for cow rising to 82.1%. This underscores the importance of preserving fine-grained local information. Subsequently, adding the GFE component, which models long-range, cross-layer dependencies via our novel Partition-based Transformer, led to a further performance boost. The AP for the challenging bird category increased to 75.3%, demonstrating that capturing global contextual relationships is crucial for detecting objects with extreme scale variations, particularly small targets. The final integration of the FFR component, which coherently fuses the original and enhanced features, resulted in the highest AP scores for all three categories, reaching 76.1%, 85.4%, and 87.2% for bird, bus, and cow, respectively. This confirms that the synergistic operation of all three components enables the model to construct a more powerful and discriminative feature representation, effectively combining local precision with global context. Table 5: Ablation study of the proposed CFSAM components on the PASCAL VOC 2007 test set. Model bird bus cow SSD300 (Baseline) 72.4 84.2 80.9 SSD300+LEF 73.2 84.7 79.4 SSD300+LEF+GFE 75.4 84.5 85.7 SSD300+CFSAM(Ours) 76.1 85.4 87.2 To further investigate the impact of CFSAM's placement within the feature pyramid, we conduct an ablation study by inserting the module into different subsets of the SSD300 prediction layers. Specifically, we compare three configurations: applying CFSAM only to the first three layers (shallow features), only to the last three layers (deep features), and to all six prediction layers. We evaluate the performance on three representative object categories-bird (small objects), bus (large objects), and cow (medium-sized objects)-to analyze the scale-specific effects. Results on the PASCAL VOC 2007 test set are summarized in Table 6. Table 6: Ablation study on the insertion positions of CFSAM in SSD300 on PASCAL VOC 2007 test set. Model bird bus cow SSD300 (Baseline) 72.4 84.2 80.9 +CFSAM(Shallow) 74.8 84.6 83.2 +CFSAM(Deep) 73.5 85.0 84.7 +CFSAM(All) 76.1 85.4 87.2 We observe that inserting CFSAM at any level brings improvements over the baseline across all object scales. However, the full configuration (CFSAM-All) achieves the highest AP scores for all three categories, demonstrating the importance of cross-layer attention across the entire feature hierarchy. Notably, applying CFSAM only to the first three layers shows the strongest improvement for small objects, while applying it only to the last three layers benefits large and medium-sized objects detection the most. This aligns with the intuition that shallow features contain more detailed information beneficial for small objects, while deep features capture higher-level semantics crucial for large and medium-sized objects. Nevertheless, only the complete integration across all six layers maximizes performance uniformly across all object scales, enabling comprehensive modeling of dependencies from finegrained details to high-level semantics. 12 5. Discussion In this paper we propose a cross-layer feature retention module (CFSAM) and also construct a modified version of the SSD300 multiscale target detection model based on this module. The experimental results verify that the proposed cross-layer feature retention module (CFSAM) with plug-and-play capability is more performant relative to other enhancement modules, and the CFSAM-based multiscale target detection model is greatly improved in the baseline. However, due to the use of a transformer computational unit in CFSAM, the approach cannot be applied in object detection models for mobile devices. In future work, we will explore optimizing CFSAM in terms of reducing feature dimensions and compressing model parameters to address this limitation. 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2510.14716
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Approaching the Continuous from the Discrete:
an Infinite Tensor Product Construction
Antonio Lorenzin1 and Fabio Zanasi2
1 a.lorenzin.95@gmail.com
2 University College London, Computer Science Department, UK
f.zanasi@ucl.ac.uk
Abstract Increasingly in recent years probabilistic computation has
been investigated through the lenses of categorical algebra, especially via
string diagrammatic calculi. Whereas categories of discrete and Gaussian
probabilistic processes have been thoroughly studied, with various axio-
matisation results, more expressive classes of continuous probability are
less understood, because of the intrinsic difficulty of describing infinite
behaviour by algebraic means.
In this work, we establish a universal construction that adjoins infinite
tensor products, allowing continuous probability to be investigated from
discrete settings. Our main result applies this construction to FinStoch,
the category of finite sets and stochastic matrices, obtaining a category of
locally constant Markov kernels, where the objects are finite sets plus the
Cantor space 2N. Any probability measure on the reals can be reasoned
about in this category. Furthermore, we show how to lift axiomatisation
results through the infinite tensor product construction. This way we ob-
tain an axiomatic presentation of continuous probability over countable
powers of 2 = {0, 1}.
Keywords: Categorical Probability · String Diagrams · Infinite Tensor
Products · Cantor Space
1
Introduction
Category-theoretic approaches to probabilistic computation have attracted grow-
ing interest in recent years, with applications ranging from evidential decision
theory ([8]) to random graphs ([1]) and active inference ([28]). Their ability to
highlight the underlying algebraic structures provides a rigorous semantics that
enhances formal clarity, compositional methodologies, and admits an intuitive
description in terms of string diagrams ([22,24]). In particular, fundamental no-
tions such as determinism, Bayesian inversion, and Bayesian updates can be
studied algebraically in the diagrammatic language ([11,19,6]).
Following this emerging line of research, recent developments have focused
on providing complete axiomatisations of categorical models of probability, via
string diagrammatic theories. Completeness is key to guarantee that any se-
mantic equality, and thus algebraic reasoning about the aforementioned notions,
arXiv:2510.14716v1 [math.CT] 16 Oct 2025
2
A. Lorenzin and F. Zanasi
can in principle be derived in the diagrammatic calculus. In this context, ax-
iomatising a (symmetric monoidal) category C, which expresses our semantic
domain, means to identify a set of generating string diagrams Σ and equations
E such that the category Free(Σ, E) freely obtained by the theory (Σ, E) is iso-
morphic (or just equivalent) to C. In other words, C-morphisms may be regarded
as Σ-diagrams quotiented by E. A first such result is the axiomatisation of the
category of finite sets and stochastic matrices (with monoidal product given by
disjoint sum) via the theory of convex algebras, presented in [10] following [26].
Another such result is contained in [25], which studies a category of Gaussian
probability using string diagrams expressing affine, relational, and quadratic be-
haviour. Most relevant to the present scope is the complete axiomatisation of
a category of discrete probabilistic processes given in [21]. Specifically, the cat-
egory under investigation, denoted by BinStoch, is the category of stochastic
matrices between powers of 2 = {0, 1}, and the associated syntax is generated
by probability distributions and Boolean operations. Two related efforts general-
ise this description beyond BinStoch: [23] extends the framework by introducing
a graded structure to encompass imprecise probabilities, while [4] notes that the
use of an additional monoidal product enables an axiomatisation of FinStoch,
the category of stochastic matrices between arbitrary finite sets.
To date, a fundamental gap in this research area is the absence of a string
diagrammatic axiomatisation for categories of continuous probability beyond the
Gaussian case. A chief example is BorelStoch, whose objects are standard Borel
spaces and whose morphisms are Markov kernels (generalisations of stochastic
matrices to the continuous setting). It is generally unclear how to encode infinite
behaviour directly via a choice of generators and equations within the existing
framework of string diagrammatic algebra.
In this work we address such challenge through the use of infinite tensor
products, which are special limits of tensor (monoidal) products.3 This approach
is inspired by [16], where it is shown that R is the infinite tensor product of finite
sets in BorelStoch. In particular, the celebrated Kolmogorov extension theorem
ensures that all probability measures on R can be seen as limits of probability
measures on finite sets. Our main contributions are the following:
1. Introduce a universal construction that adjoins infinite tensor products (The-
orem 1);
2. Employ this construction to describe freely generated categories with an
infinite tensor product, which admit a diagrammatic representation through
plate notation (Section 4);
3. Provide a characterisation of the category FinStoch⊗∞obtained by adjoining
infinite tensor products to FinStoch (Theorem 2) in terms of Stone spaces
and locally constant Markov kernels.
4. Provide an axiomatic presentation of CantorStochlc, the restriction of FinStoch⊗∞
to powers of 2 together with the Cantor space 2N, its (countably) infinite
power (Corollary 2).
3 Throughout, we use “tensor” and “monoidal” synonymously, in order to both follow
the terminology of [16] and adhere to the terminology of categorical probability.
Approaching the Continuous from the Discrete
3
Regarding the third contribution, one might hope that the whole of BorelStoch
could be recovered as FinStoch⊗∞, i.e. by adjoining infinite tensor products to its
discrete counterpart FinStoch. However, this is overly ambitious: not because of
probability measures, which can always be described as above, but because of the
freeness of measurable functions, whose behaviour cannot be fully captured by
finite operations. What we obtain instead is a subcategory of BorelStoch, which
is nonetheless quite expressive: probability measures on R are in bijective cor-
respondence with probability measures on 2N, thus showing that CantorStochlc
is rich enough to encompass all probability measures on R. Additionally, this
category includes nontrivial kernels that capture biased behaviour with respect
to their inputs (see Example 7 for a detailed discussion).
Outline First we recall semicartesian categories, which provide the necessary
structure to define infinite tensor products (Section 2). In Section 3 we give
a universal construction of the category obtained by adjoining infinite tensor
products to a semicartesian category. String diagrams and monoidal theories are
discussed in Section 4, while Section 5 restricts the focus to Markov categories.
Section 6 presents the main result, giving an explicit description of the univer-
sal construction for FinStoch and BinStoch. The developed syntax is applied in
Section 7 to a short use case on Markov chains. Concluding remarks and fu-
ture directions are briefly discussed in Section 8. Proofs of auxiliary results are
deferred to the appendices.
2
Semicartesian Categories and Infinite Tensor Products
For the sake of generality, we will first consider semicartesian categories (some-
times known as affine symmetric monoidal categories), as their theory is rich
enough to allow the study of infinite tensor products ([16]). From Section 5
onward, we will restrict focus to Markov categories, broadly adopted for model-
ling probabilistic and statistical behaviour — see, e.g., [11,7,20,18]. In fact, the
examples introduced below are all Markov categories.
Definition 1. A semicartesian category is a symmetric monoidal category
(C, ⊗, I) in which the monoidal unit I is terminal. The unique morphism of type
X →I is denoted by delX (in string diagrams,
) and called delete morphism.
Example 1. The main example for our purposes is the semicartesian category
FinStoch [11, Example 2.5], whose objects are finite sets and whose morphisms
f : X →Y are functions Y ×X →R≥0 satisfying Σy∈Y f(y, x) = 1, also known as
stochastic functions. To better highlight the connection with probability theory,
one generally employs the “conditional notation” and write f(y | x) in place of
f(y, x). In particular, 1 = Σy∈Y f(y, x) = Σy∈Y f(y | x) means that f(−| x) is
a probability distribution on Y for each x ∈X. Composition is defined by
g f(z | x) := P
y∈Y g(z | y)f(y | x) for any composable morphisms f : X →Y
and g: Y →Z, . while the tensor product is given by f ⊗h(y, w | x, z) :=
f(y | x)h(w | z) for any f : X →Y and h: Z →W. In particular, the delete maps
are given by del( | x) = 1.
4
A. Lorenzin and F. Zanasi
Example 2. Another important example is BinStoch, the full subcategory of
FinStoch given by finite powers of 2 = {0, 1}. This has become especially relevant
after the work [21] provided a complete axiomatisation for such a category.
Example 3. In [16], where infinite tensor products were originally defined, the
main example is BorelStoch, which generalizes FinStoch outside the finite context.
BorelStoch is the category whose objects are standard Borel spaces (measurably
isomorphic to a finite set, Z, or R by Kuratowski’s theorem) and whose morph-
isms are Markov kernels f : X →Y , given by honest functions f : ΣY × X →
[0, 1], where ΣY is the σ-algebra of Y , such that f(U | −): X →[0, 1] is measur-
able and f(−| x): ΣY →[0, 1] is a probability measure.
The measurability condition allows us to define a composition by integration:
g f(U | x) :=
R
Y g(U | dy)f(dy | x). The tensor product is determined by the rule
f ⊗h(U × V | x, z) := f(U | x)h(V | z). As before, the delete maps are given by
delX(−| x) = 1 for all x ∈X.
Example 4. A more general version of BorelStoch is Stoch, where objects are all
measurable spaces. See [11] for details.
Infinite tensor products were introduced in [16] to formulate zero–one laws
in categorical language. They have since become central in category-theoretic
approaches to continuous probability; for instance, [13] employs them in discuss-
ing laws of large numbers. We now recall the basic setup and introduce some
new terminology that will be convenient for later use. Throughout, let Pfin(J)
denote the category whose objects are finite subsets of a countable set J, with a
morphism F →F ′ whenever F ⊆F ′.
Definition 2. Let C be a semicartesian category. A functor X : Pfin(J)op →C
is an abstract infinite tensor product4 if there exists a family of objects
(Xj)j∈J such that X sends F to XF := N
j∈F Xj and the inclusion F ⊆F ′ to
the morphism
πF ′,F : XF ′ →XF
which deletes all j ∈F ′ \ F, and keeps the rest unchanged. Up to permutation,
πF ′,F = idXF ⊗delXF ′\F .
Intuitively, an abstract infinite tensor product is a choice of an infinite number
of objects in C.
Notation 1 Whenever X is an abstract infinite tensor product, its associated
countable set J may be denoted by JX for clarity.
4 To be precise, we should say abstract countable infinite tensor product. To avoid
overloading notation, we will simply omit the adjective, and throughout infinite will
always mean countable infinite. Nonetheless, the full generality is expected to work,
and this restriction is important only to our main application.
Approaching the Continuous from the Discrete
5
Definition 3. Let C be a semicartesian category. A (concrete) infinite tensor
product X = N
j∈J Xj for a family of objects (Xj)j∈J in C is a limit of the
associated abstract infinite tensor product, provided this limit is preserved by
−⊗Y for all Y ∈C.
The morphisms X →XF given by the universal property will be called finite
marginalizations and denoted by πF .
Note that such products are not strictly unique; rather, they are unique up to
isomorphism. This subtle distinction mirrors the one between strict and strong
monoidal functors, and it is the reason we will work with the latter throughout.
Definition 4. A semicartesian category has infinite tensor products if every
abstract infinite tensor is realised, i.e. it is associated to a concrete infinite tensor
product.
In [16], it was shown that BorelStoch has infinite tensor products, and moreover
R is isomorphic to any nontrivial infinite tensor product (this follows by the
mentioned Kuratowski’s theorem, since R is measurably isomorphic to any un-
countable standard Borel space).
Throughout, symmetric monoidal functors are always meant to be strong,
meaning that their coherence morphisms are always isomorphisms.
Definition 5. A symmetric monoidal functor ϕ: C →D between two semi-
cartesian categories with infinite tensor products is ITP-preserving if ϕ(N
j∈J Xj)
is an infinite tensor product of the family (ϕ(Xj))j∈J.5
Before proceeding, we also consider the following technical requirement, neces-
sary to establish the forthcoming results. In [17], semicartesian categories are
called theories and the property below is called normality. Following a now-
established tradition in categorical modelling of probability [6,11,22], we will
often depict morphisms of these categories using string diagrams.
Definition 6. A semicartesian category C has cancellative deletions if f ⊗
delX = g ⊗delX implies f = g for all parallel morphisms f, g and all objects X.
In string diagrams,
f
=
g
=⇒
f
=
g
for all f and g.
In particular, surjectiveness of the marginalizations πF ′,F : XF ′ →XF and
πF : X →XF above is ensured.
The fact that FinStoch, BinStoch, BorelStoch and Stoch satisfy this property
follows by direct check: whenever f ⊗delX(U | a, x) = g ⊗delX(U | a, x), then
f(U | a) = f(U | a) delX( | x) = g(U | a) delX( | x) = g(U | a) concludes the proof
since delX( | x) = 1 for all x. Note that to ensure this property, the emptyset is
not an object of these categories.
5 In particular, the finite marginalizations πF are preserved by ϕ. As πF is a notation
for any infinite tensor product, we will write ϕ(πF ) = πF for simplicity.
6
A. Lorenzin and F. Zanasi
3
A Universal Construction for Infinite Tensor Products
We now focus on proving the following statement, which intuitively states that
adjoining infinite tensor products satisfies a natural universal property. This
result is propaedeutic for our aims, but it is also of independent interest for
other works using infinite tensor products [5,14,16].
Theorem 1. Let C be a semicartesian category with cancellative deletions. Then
there exist a semicartesian category C⊗∞with infinite tensor products and a
strict symmetric monoidal functor C →C⊗∞, such that for every semicartesian
category D with infinite tensor products and every symmetric monoidal func-
tor ϕ: C →D, there exists an ITP-preserving symmetric monoidal functor
˜ϕ: C⊗∞→D such that the following diagram commutes
C
D
C⊗∞
ϕ
˜ϕ
Moreover, such a ˜ϕ is unique up to natural isomorphism.
First, we define what C⊗∞is. Its objects will be abstract infinite tensor
products. However, morphisms are a little bit more subtle, as they should allow
these tensor products to be limits in C⊗∞. This leads to the following.
Definition 7. Let X : Pfin(JX) →C and Y : Pfin(JY ) →C be two abstract in-
finite tensor products. A compatible family f : X →Y is given by an index
set Λf ⊆Pfin(JX) × Pfin(JY ) and a collection (fF,G : X(F) →Y (G))(F,G)∈Λf
subject to the following requests:
– (Naturality) For every F ′ ⊇F and G′ ⊇G, if (F, G), (F ′, G′) ∈Λf, then
the diagram
X(F ′)
Y (G′)
X(F)
Y (G)
fF ′,G′
πF ′,F
πG′,G
fF,G
commutes;
– (Covering condition) For every G ∈Pfin(JY ), there exists F ∈Pfin(JX) such
that (F, G) ∈Λf;
– (Hereditariness) If (F, G) ∈Λf, then (F ′, G) ∈Λf for every F ′ ⊇F.
We moreover say that two compatible families f, f′ : X →Y are equivalent if
fF,G = f ′
F,G for all (F, G) in the intersection of the index sets Λf ∩Λf ′.
Hereditariness above (together with cancellativity of deletions) ensures transitiv-
ity of the equivalence defined for compatible families; in particular, the covering
condition also holds for Λf ∩Λf ′.
We are finally ready to define C⊗∞. To obtain a strict monoidal functor
C →C⊗∞, we single out an element ∗, common to all countable sets J considered.
Approaching the Continuous from the Discrete
7
Definition 8. Let C be a semicartesian category with cancellative deletion, and
let ∗be a set. Then C⊗∞is the semicartesian category defined as follows:
– Each object is an abstract infinite tensor product X : Pfin(J)op →C, with
∗∈J;
– Morphisms f : X →Y are equivalence classes of compatible families;
– Composition is defined pointwise: gf is the compatible family given by
gfF,H := gG,HfF,G
whenever there is a finite subset G;
– The monoidal product of X and Y is given by
X ⊗Y : Pfin(JX ∪JY ) −→
C
F
7−→X(F ∩JX) ⊗Y (F ∩JY ),
and consequently, given f : X →Z and g: Y →W, f ⊗g is the morphism
such that
(f ⊗g)F,G := fF ∩JX,G∩JZ ⊗gF ∩JY ,G∩JW .
(The monoidal unit is given by Pfin({∗}) ∋{∗} 7→I ∈C).
The composition is thoroughly studied in Section B, where it is shown that
C⊗∞is a semicartesian category. In particular, the request of having cancellative
deletions ensures that the composition is well-defined.
By construction, there exists a fully faithful symmetric monoidal functor
C →C⊗∞, given by considering each object X of C as a functor Pfin({∗}) →C
defined by {∗} 7→X. We note that C⊗∞is strict if C is, since the monoidal
product is described pointwise.
Proposition 1. C⊗∞has infinite tensor products.
Proof. We start by claiming that X : Pfin(JX) →C is the infinite tensor product
of the family X(G), with G ∈Pfin(JX), in C⊗∞. Indeed, consider a morphism
f : A →X. By definition, this yields a family of morphisms fG : A →XG indexed
by G. Conversely, if one has a family of morphisms fG : A →XG such that
πG′,GfG′ = fG for all G′ ⊇G, then one can easily construct a compatible family
fF,G : A(F) →X(G), uniquely defined up to equivalence. The fact that the
infinite tensor product is preserved under tensoring follows by the generality of
the argument.
Now, given any countable family (Xk)k∈K in C⊗∞, consider the set H :=
F
k∈K JXk, which is countable because it is a countable union of countable sets.
We claim that the abstract infinite tensor product Y : Pfin(H) →C given by
F 7→N
k∈K Xk(F ∩JXk) is indeed the infinite tensor product of (Xk)k∈K.
Starting with f : A →Y , by definition of morphisms in C⊗∞, one obtains
fF : A →N
k∈F Xk for all F ∈Pfin(K). Conversely, given a family (fF : A →
N
k∈F Xk)F ∈Pfin(K) such that πF ′,F fF ′ = fF , the fact that every Xk is an infinite
tensor product ensures that each fF is determined by (f G
F : A →N
k∈F XG∩JXk ),
with G a finite subset of F
k∈F JXk. In particular, f determines a family of
morphisms indexed by all finite subsets of H = F
k∈K JXk, giving rise to a com-
patible family A →Y . As wanted, Y is the infinite tensor product of (Xk)k∈K.
8
A. Lorenzin and F. Zanasi
Finally, we prove of the main statement of the section.
Proof (Proof of Theorem 1). Given ϕ: C →D a symmetric monoidal functor,
then an abstract infinite tensor product X : Pfin(J) →C immediately gives rise
to an abstract infinite tensor product on D by composition and coherence iso-
morphisms. Since D has infinite tensor products, this is realized as a concrete
infinite tensor product, denoted by ˜ϕ(X) and equipped with finite marginaliza-
tions πF : ˜ϕ(X) →ϕ(X)F .
Now, given a morphism f : X →Y in C⊗∞, by definition this is a compat-
ible family (fF,G : X(F) →Y (G)). By the properties, ϕ(fF,G) gives rise to a
morphism ˜ϕ(f)G : ˜ϕ(X) →ϕ(Y )G given by the composition
˜ϕ(X)
ϕ(X)F
ϕ(Y )G
πF
ϕ(fF,G)
(1)
and naturality ensures that ˜ϕ(f)G is a family that factors through the limit
of ϕ(Y )G, which by definition is ˜ϕ(Y ); in other words, we obtain a morphism
˜ϕ(X) →˜ϕ(Y ), which we denote by ˜ϕ(f).
Proving that ˜ϕ so defined is a symmetric monoidal functor is now a simple
routine check, and it is ITP-preserving by construction. Moreover, its uniqueness
up to natural isomorphism follows from the universal property of limits and the
description of morphisms provided by (1) (note it is not only sufficient to describe
a choice of morphisms, but necessary).
Proposition 2. Let C and D be two semicartesian categories with cancellative
deletions, and assume D has infinite tensor products. If a symmetric monoidal
functor ϕ: C →D is faithful, then so is ˜ϕ: C⊗∞→D.
Proof. Let f, g: X →Y be two parallel morphisms in C⊗∞such that ˜ϕ(f) =
˜ϕ(g). Consider their marginalizations πG ˜ϕ(f) = πF ˜ϕ(g): ˜ϕ(X) →˜ϕ(Y )G. For
any G, there exists F ⊆JX such that πGf = fF,GπF because f is a compatible
family, and analogously for g. Moreover, F can be chosen for both f and g by
covering condition and hereditariness, so also πGg = gF,GπF . Using this equality
via ˜ϕ, we obtain
ϕ(fF,G)πF = ˜ϕ(fF,GπF ) = ˜ϕ(πGf) = πG ˜ϕ(f) =
πG ˜ϕ(g) = ˜ϕ(πGg) = ˜ϕ(gF,GπF ) = ϕ(gF,G)πF
Since D has cancellative deletions, πF is an epimorphism (note that πF = idXF ⊗
delXJX \F up to permutations), and therefore ϕ(fF,G) = ϕ(gF,G). By faithfulness
of ϕ, we conclude that fF,G = gF,G. Arbitrariness of G, together with naturality
of compatible families, implies that f and g are equivalent compatible families,
i.e. f = g.
4
String Diagrams and Monoidal Theories for Infinite
Tensor Products
In this section we study how to derive presentations by generators and equations
for C⊗∞from those of C. In doing so, we also introduce a string diagrammatic
Approaching the Continuous from the Discrete
9
gG,H
fF,G
=
fF,G
gG,H
fF,G
gH,K
=
gH,K
fF,G
Figure 1. Sequential and parallel composition of the plate notation.
representation for morphisms of C⊗∞which highlights their formulation as com-
patible families of C-morphisms (Definitions 7 and 8). Our approach is reminis-
cent of the ‘plate’ notation commonly used in the study of Bayesian networks, see
e.g. [3]. In category-theoretic approaches to probability, this has recently been
employed in [5]. Our use of plate notation is similar in spirit, but it is tailored
to ensure a formal representation of compatible families.
We recall that a morphism f : X →Y in C⊗∞is given by a compatible family
(fF,G : X(F) →Y (G))(F,G)∈Λf , with fF,G in C, which we write as
fF,G
(F, G) ∈Λf
X
Y
in plate notation. This morphism may also be denoted by
f
X
Y
for brevity,
where the double wire of the input and output of f indicate that X and Y
are infinite tensor products. For simplicity, we will generally omit (F, G) ∈Λf,
since the index set is determined by the compatible family. This notation is
compatible in the obvious way with sequential and parallel composition in C⊗∞
(see Fig. 1). Moreover, it will be convenient for graphical reasoning to allow
plates to disappear when infinite tensors are not involved, i.e. (fF,G : X(F) →
Y (G))(F,G)∈Λf is a bona fide morphism of C. More precisely,
fF,G
(F, G) ∈Λf
X
Y
=
f
X
Y
(2)
whenever X and Y belong to the image of C →C⊗∞.
We can use plate notation to define a variation of the construction of a
symmetric monoidal category from generators and equations, which encompasses
infinite tensor products. This will be useful in the following sections to derive
axiomatisation results.
First, recall that a symmetric monoidal theory (SMT) is a pair (Σ, E), where
Σ is a signature of generators o: m →n with an arity m ∈N and coarity n ∈N,
and E is a set of equations between Σ-terms. A Σ-term c of type m →n will
be represented graphically as a box with m dangling wires on the left and n
10
A. Lorenzin and F. Zanasi
on the right, also written
c
m
n. Formally, Σ-terms are freely obtained by
sequential and parallel compositions of the generators in Σ together with the
identity
: 1 →1, the symmetry
: 2 →2, and the ‘empty’ diagram
: 0 →0. Sequential composition of Σ-terms
c
m
n and
d
v
n
is depicted
as
c
m
n
d
v , of type m →v. Parallel composition of Σ-terms
c1
m1
n1
and
c2
m2
n2 is depicted as
c1
m1
n1
c2
m2
n2, of type m1 + m2 →n1 + n2. Given an
SMT (Σ, E), the symmetric monoidal category Free(Σ, E) freely generated by
(Σ, E) has objects the natural numbers and morphisms m →n the Σ-terms
of type m →n quotiented by E and by the laws of symmetric strict monoidal
categories (Appendix A), with sequential and parallel composition defined as on
the corresponding Σ-terms. More details can be found e.g. in [2,29].
Example 5. The SMC freely obtained by generators
and
p
for all p ∈[0, 1],
and set of equations as in [21, Figure 4] is named CausCirc in [21], as we may
regard its string diagrams as causal circuits. Also in [21] it is proven that
CausCirc ∼= BinStoch, meaning that such an SMT axiomatises BinStoch.
We note that Free(Σ, E) is semicartesian with cancellative deletions if and
only if both of the following are satisfied:
1. There is only one Σ-term (up to =E) of type n →0, for any n ∈N;
2. Given two Σ-terms f, g, f ⊗deln =E g ⊗deln if and only if f =E g.
Under these assumptions, we may define how to freely obtain a category with
infinite tensor products from (Σ, E) as follows.
Definition 9. Given an SMT (Σ, E) satisfying the two assumptions above, let
Free∞(Σ, E) be defined as the category (with infinite tensor products) whose
objects are natural numbers together with an additional object ∞, and whose
morphisms are given by compatible families of Σ-terms, subject to the equations
of Fig. 1 and (2).
The proof that Free∞(Σ, E) is indeed a category follows by the same argument
used to prove that C⊗∞is a category (see Section B).
Regarding the tensor product, one sets ∞⊗N = ∞for any N natural number
or ∞, and the tensor product is obtained by reindexing the compatible families
according to chosen fixed bijections N ⊔{0, 1, . . . , n −1} ∼= N and N ⊔N ∼= N.
These choices are necessary to ensure that associators, unitors and permutations
are well-defined: for example, if we call ϕ the bijection N ⊔N
∼
=
−→N, then the
associator is a reindexing with respect to the bijection
N
N ⊔N
N ⊔N ⊔N
N ⊔N
N.
ϕ−1
id⊔ϕ−1
ϕ⊔id
ϕ
Approaching the Continuous from the Discrete
11
Moreover, we can verify that Free∞(Σ, E) indeed has infinite tensor products,
by an argument analogous to the one showing Proposition 1.
We now state that the infinite tensor product construction of Definition 8,
when applied to a freely generated SMC Free(Σ, E), is presented by the same
(Σ, E) via the Free∞(−) construction above.
Proposition 3. Let (Σ, E) be a symmetric monoidal theory such that Free(Σ, E)
is a semicartesian category with cancellative deletions. Then Free∞(Σ, E) ≃
(Free(Σ, E))⊗∞via an ITP-preserving symmetric monoidal functor.
Proof. By applying Theorem 1 and Proposition 2, we have a faithful functor
Free(Σ, E)⊗∞→Free∞(Σ, E). Fullness is immediate by definition of Free∞(Σ, E).
We emphasise that Free∞(Σ, E) and (Free(Σ, E))⊗∞are not isomorphic. In
fact, these two categories reflect different trade-offs. The former, Free∞(Σ, E),
admits a single infinite tensor product ∞, which may appear more natural at
first glance, but then ∞⊗∞= ∞requires to fix a chosen bijection N ⊔N ∼= N,
as mentioned above. The latter, (Free(Σ, E))⊗∞, instead allows multiple infinite
tensor products, with the advantage that its tensor product does not rely on any
choice of bijections. Regardless of this distinction, in both categories one can
reason equationally using the SMT (Σ, E) together with the equations of Fig. 1
and (2).
Combining Theorem 1 and Proposition 3, one can prove the expected uni-
versal property for the free construction.
Corollary 1. Let (Σ, E) be an SMT satisfying the two assumptions preced-
ing Definition 9, and consider a semicartesian category D with infinite tensor
products. Then a symmetric monoidal functor Free(Σ, E) →D extends to an
ITP-preserving symmetric monoidal functor Free∞(Σ, E) →D.
In other words, Free∞(Σ, E) is initial among the semicartesian categories with
infinite tensor products in which (Σ, E) can be interpreted.
Proof. By Theorem 1, Free(Σ, E) →D extends to (Free(Σ, E))⊗∞→D. By
composing this functor with the equivalence Free∞(Σ, E) ≃(Free(Σ, E))⊗∞
provided in Proposition 3, the statement follows.
5
Infinite Tensor Products and Markov Categories
As mentioned, our motivating examples are not just semicartesian but also
Markov categories. Given the relevance of these structures in the categorical
probability literature, it is of interest to show that whenever C is a Markov
category, then so is C⊗∞.
Definition 10. A Markov category is a semicartesian category C where every
object X comes equipped with a cocommutative comonoid copyX : X →X ⊗X
(whose counit is given by delX) compatible with the monoidal product. In string
12
A. Lorenzin and F. Zanasi
diagrams, copyX is usually denoted by
and the cocommutative comonoid
equations read as follows.
=
=
=
=
Compatibility with the monoidal product is instead written as follows.
X ⊗Y
X ⊗Y
X ⊗Y
=
X
X
X
Y
Y
Y
(The delete maps are obviously compatible since they are the determined by ter-
minality of I).
As mentioned, FinStoch, BinStoch, BorelStoch and Stoch are Markov categories.
In FinStoch, copyX(y, z | x) := 1 if x = y = z and 0 otherwise. More details can
be found in [6,11].
Proposition 4. Let C be a Markov category with cancellative deletions. Then
C⊗∞is a Markov category as well.
Proof. For every abstract infinite tensor product X, the collection of copy maps
(copyX(F ) : X(F) →X(F) ⊗X(F))F ∈Pfin(JX) gives rise to a compatible family,
thanks to compatibility with the monoidal product. We then define copyX to be
(the equivalence class of) such compatible family. Graphically,
X
X
X
:=
X
X
X
X(F)
X(F)
X(F)
By Fig. 1, the cocommutative comonoid equations hold because they hold inside
the plate. Similarly, compatibility with the monoidal product holds, therefore
concluding the proof.
In particular, by construction of the copy of X, a direct computation shows that
X is a Kolmogorov product in the sense of [16].
6
A Stone Space Characterisation of FinStoch⊗∞
In this section we explicitly describe what FinStoch⊗∞and BinStoch⊗∞are,
therefore highlighting what information about the continuous setting can be
recovered from FinStoch and BinStoch. By Proposition 2, these infinite tensor
products extensions can be viewed as subcategories of BorelStoch. With this in
Approaching the Continuous from the Discrete
13
mind, it suffices to understand what is the relevant notion of Markov kernels
associated to FinStoch⊗∞and BinStoch⊗∞.
To this end, we seek to find a category where everything is seemingly “finitely-
determined”, in a way that resembles the notion of compatible families. For this
reason, we focus on Stone spaces. We recall that these spaces are compact Haus-
dorff spaces with a base consisting of clopen sets. In particular, we will denote
by Clopen(X) the clopen sets of a Stone space X. These spaces are particularly
well-known in the theory due to Stone duality, which states that Clopen(−) yields
a contravariant equivalence between the category of Stone spaces and continuous
maps and the category of Boolean algebras and algebra homomorphisms.
In this setting, the most important Stone space for our purposes is the Can-
tor space 2N, which, as the notation suggests, is given by the product of count-
ably many copies of 2 = {0, 1}. By Brouwer’s theorem, the Cantor space is
the only non-empty Stone space that is second-countable and without isolated
points. We now equip Stone spaces with a special type of Markov kernels.
Definition 11. Let X and Y be two Stone spaces. A locally constant Markov
kernel X to Y , denoted by f : X
Y , is given by a function
f : Clopen(Y ) × X →
[0, 1]
(U, x)
7→f(U | x)
such that
– For all U ∈Clopen(Y ), the function f(U | −): X →[0, 1] is locally constant,
i.e. for every x ∈X there exists an open A ∋x such that f(U | x) = f(U | y)
for all y ∈A;
– For all x ∈X, the function f(−| x): Clopen(Y ) →[0, 1] is a finitely-additive
probability measure, i.e. f(Y | x) = 1 and for every finite collection of disjoint
clopen sets {Ui}n
i=1,
f
n
[
i=1
Ui
x
!
=
n
X
i=1
f(Ui | x).
Stone spaces with locally constant Markov kernels form a Markov category de-
noted by StoneStochlc (see Section C for details).
Remark 1. By Carathéodory’s extension theorem, a finitely-additive probability
measure on clopen sets gives rise to a σ-additive probability measure on the
generated σ-algebra, often called the Baire σ-algebra. In particular, this allows
us to integrate against the Markov kernels described above, as they are particular
instances of the standard Markov kernels recalled in Example 3, where f(−| x)
is required to be a probability measure on a σ-algebra.
Example 6. Not all Markov kernels are locally constant. For the sake of an ex-
ample, we define a Markov kernel from the Cantor space 2N to 2 given by
f(U | (xn)) :=
(P∞
n=0
2xn
3n+1
if U = {1}
1 −P∞
n=0
2xn
3n+1
if U = {0}
14
A. Lorenzin and F. Zanasi
This is clearly not locally constant, since any choice of (xn) changes the value of
f(U | (xn)), although (xn) 7→P∞
n=0
2xn
3n+1 is the standard continuous embedding
2N ,→[0, 1].
Nonetheless, all continuous functions yield locally constant Markov kernels by
means of delta measures (see Section C for details). We now present two examples
of locally constant Markov kernels that demonstrate the expressive power of this
concept.
Example 7. Note that Clopen(2N) is the collections of sets of the form UF ×2N\F ,
where F ⊆N is finite and UF ⊆2F is arbitrary.
1. A simple example of a locally constant Markov kernel is p: I
2N, defined
by
p(UF × 2N\F | ) := #(UF )
2#(F )
where #(A) denotes the size of A. This construction is similar to the Le-
besgue measure on the real line, in that it is translation-invariant: p(UF ×
2N\F | ) = p(UG × 2N\G | ) whenever UF and UG are in bijection. The fact
that it is locally constant is immediate, since I has only one element.
2. A more involved example of a locally constant Markov kernel f : 2N
2N
can be defined using the Markov kernel p: I
2N introduced in Item 1. We
set
f(U | x) :=
(
1
2p(U | ) + 1
2
if x ∈U
1
2p(U | )
if x /∈U
Intuitively, the probability is biased toward the conditioning point x. The
fact that f(−| x) is a finitely-additive probability measure follows from the
same property of p(−| ). Moreover, f(U | −) is constant on U and on its
complement, hence f is locally constant.
Proposition 5. StoneStochlc, the category of Stone spaces and locally constant
Markov kernels, has all infinite tensor products. More explicitly, the infinite
tensor product of (Xj)j∈J is given by Q
j∈J Xj.
Proof. First of all, we note that X := Q
j∈J Xj is a Stone space, with com-
pactness ensured by Tychonoff’s theorem. Now, let us consider A another Stone
space, and consider f : A
X a locally constant Markov kernel. We claim that
f is determined by the marginals fF , given by the composition
A
X
XF = Q
j∈F Xj,
f
πF
with F any finite subset of J. By compactness, for every clopen U in X there
exist F ∈Pfin(J) and a clopen UF ⊆XF such that U = UF × Q
j∈J\F Xj. So
f(U | a) = fF (UF | a). In particular, f is determined by its marginals, as claimed.
Conversely, if we consider a family (fF : A
XF )F ∈Pfin(J) that is compatible
in the sense that πF,F ′ fF = fF ′, for every clopen set U ⊆Q
j∈J Xj, we can
Approaching the Continuous from the Discrete
15
define f(U | a) := fF (UF | a), and compatibility ensures that this definition is
independent of F. The limit property is therefore ensured. The preservation
under tensoring follows by arbitrariness of the family of Stone spaces.
Finally, we obtain our second main theorem, yielding a characterisation of
infinite tensor products over FinStoch.
Theorem 2. The ITP-preserving symmetric monoidal functor
ϕ: FinStoch⊗∞→StoneStochlc,
obtained by the inclusion FinStoch ,→StoneStochlc, is fully faithful. Moreover,
its essential image is given by infinite tensor products of finite sets in the sense
of topological spaces.
In particular, by Brouwer’s theorem, any object in the essential image is iso-
morphic either to a finite set or to the Cantor space. Moreover, ϕ preserves the
copy map because in both settings the finite marginalizations are deterministic.
In other words, this functor is strong gs-monoidal in the sense of [12].
Proof. First, recall that the functor is obtained by Theorem 1, and faithfulness is
immediate by Proposition 2. We therefore only need to show fullness. Now, given
a morphism between infinite tensor product of finite sets X = Q
j∈JX Xj and
Y = Q
j∈JY Yj, we wish to show that f : X
Y is uniquely determined by a
compatible family fF,G : XF
YG. By Proposition 5, f is uniquely determined
by (fG)G∈Pfin(JY ), with fG := πGf. Let us now focus on fG. Since it is locally
constant, fG(U | −) is constant on each member of a finite family (AU
i )i=1,...,nU .
Since YG is finite, its subsets U are finite, so we obtain a finite family of clopens
(AU
i )i,...,nU,U⊆YG.
As any clopen in X is of the form AF × Q
j∈J\F Xj for some F, there is a
finite set F common to all members of the finite family (AU
i )i,U, such that each
AU
i can be written in such a cylindrical form. In particular, fG(U | a) = fG(U | b)
for any b such that πF (a) = πF (b). In other words, fG factors through πF by
definining, for any ˜a ∈XF , fF,G(U | ˜a) := fG(U | a), where a ∈X is any element
such that πF (a) = ˜a. It is now a direct check that the given definition of fF,G
results in a compatible family, therefore ensuring fullness.
Corollary 2. BinStoch⊗∞is equivalent to CantorStochlc, the full subcategory of
StoneStochlc in which objects are 2N = {0, 1}N, where N is any natural number
or N.
In particular, by combining Proposition 3, Corollary 2, and Example 5, we
obtain a characterisation by generators and equations of CantorStochlc.
Corollary 3. CantorStochlc is isomorphic to Free∞(Σ, E), where (Σ, E) is the
symmetric monoidal theory of CausCirc ( cf. Example 5).
16
A. Lorenzin and F. Zanasi
Another important consequence of our results is that CantorStochlc includes
all probability measures on R. This follows from the fact that R is the infinite
tensor power of 2 in BorelStoch [16]. Indeed, since both R in BorelStoch and
2N in CantorStochlc are associated to compatible families of morphisms I →2n
in FinStoch, we obtain BorelStoch(I, R) ∼= CantorStochlc(I, 2N). This observation
suggests that CantorStochlc is a rather expressive setting for studying continuous
probability. We provide a simple use case below.
7
A Use Case of Plate Notation: Markov Chains
As an example of diagrammatic reasoning using plate notation, we briefly discuss
Markov chains, which have been recently studied via Markov categories in [15].
We restrict here to the time-homogenous case, which defines a Markov chain
with n steps in BinStoch as an endomorphism f : X →X inductively, as follows:
c1
X
X
:=
f
X
X
cn
Xn
X
:=
cn−1
f
X
Xn−1
X
For example, c3 is given by the following string diagram:
f
X
X
f
X
f
X
The idea is that each copy of X represents a random variable of the chain, and
the use of the same f for all steps translates to the request that P(Xn | Xn−1) =
P(Xn+1 | Xn), i.e. that the obtained Markov chain is time-homogeneous. Addi-
tional details can be found in [15].
It can be verified that the cns defines a compatible family, hence a morphism
c: X →XN, the infinite tensor power of X. As one may expect, c is invariant un-
der the addition of a precedent step in the inductive construction. Note this prop-
erty cannot be formulated in the discrete setting of BinStoch (and of CausCirc);
however, we can easily prove it by diagrammatic reasoning in CausCirc⊗∞, as
justified by Corollary 3:
cn
=
cn−1
f
=
cn−1
f
=
cn−1
f
=
cn
f
where the last step is a simple reindexing of the compatible family defining c.
8
Conclusion and Future Work
This paper focuses on introducing categorical methods for studying continuous
probabilistic processes as limits of discrete ones. We do so by introducing a uni-
versal construction that associates to any semicartesian category C with cancel-
lative deletions a semicartesian category C⊗∞with all (countably) infinite tensor
Approaching the Continuous from the Discrete
17
products. We discuss how axiomatic presentations for C can be lifted to C⊗∞,
and introduce plate notation to manipulate as string diagrams C⊗∞-morphisms.
As main case study, we focus on FinStoch, whose infinite tensor construc-
tion FinStoch⊗∞we characterise in terms of Stone spaces and locally constant
Markov kernels. Furthermore, the existence of an axiomatic presentation of the
subcategory BinStoch allows us to derive one for BinStoch⊗∞. We can effectively
use diagrammatic reasoning in BinStoch⊗∞to study any probability measure on
R. We give a very simple use case regarding Markov chains; clearly, this only
scratches the surface of what can be studied in BinStoch⊗∞, which remains to
be explored in follow-up work.
Moreover, our approach suggests ways towards axiomatising Markov kernels
beyond local constancy. One such direction may be based on describing general
Markov kernels in terms of locally constant ones, possibly restricting to standard
Borel spaces. Another direction may be to investigate a universal construction
that guarantees disintegration of measures, in the sense of having conditionals
in Markov categories ([11]).
Additionally, studying locally constant Markov kernels could yield a descrip-
tion of probabilistic morphisms between Boolean algebras via Stone duality,
which may constitute an insightful contribution in its own right.
Another interesting question is studying examples different from FinStoch
and BinStoch. Natural candidates are Gaussian probability (Gauss, as studied
in [25]), Gaussian mixtures ([27]), or even non probabilistic examples of Markov
categories, such as the category of finite sets and multivalued functions ([11,
Example 2.6]).
Finally, we are interested in integrations of our approach (and of plate nota-
tion) with categorical approaches to probability that go beyond Markov cat-
egories, such as partial Markov categories [9], tape diagrams [4], and graded
diagrams [23].
Acknowledgments. This work was partially supported by the ARIA Safeguarded AI
TA1.1 programme. During the preparation of this article, we became aware that Sean
Moss and Sam Staton had independently considered locally constant Markov kernels
and infinite tensor products, although their work is not publicly available at present.
Disclosure of Interests. The authors have no competing interests to declare that
are relevant to the content of this article.
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A
Laws of Symmetric Strict Monoidal Categories
c1
c2
c3
=
c1
c2
c3
c
=
c
=
c
c1
c2
c3
=
c1
c2
c3
c
=
c
=
c
c1
d1
d2
c2
=
c1
d1
d2
c2
c
=
c
=
B
Compatible Families Form a Semicartesian Category
This appendix shows that the universal construction C⊗∞offered in Section 2
indeed yields a semicartesian category. We start with a precise definition of the
composition, briefly sketched in Definition 8.
Definition 12. Let C be a semicartesian category with cancellative deletion. Let
X, Y and Z be abstract infinite tensor products and let f : X →Y and g: Y →Z
be compatible families.
We define the composition of compatible families gf : X →Y by setting
Λgf ⊆Pfin(JX) × Pfin(JZ) to be the set of pairs (F, H) for which there exists
G ∈Pfin(JY ) such that (F, G) ∈Λf and (G, H) ∈Λg, and
gfF,H := gG,HfF,G
for any meaningful choice of G.
The proof that this is indeed a good notion of composition is easily shown. Given
two choices G and G′, we can consider the following commutative diagram given
by naturality
X( ˜F)
Y (G ∪G′)
Z(H)
X(F)
Y (G)
Z(H)
f ˜
F ,G∪G′
gG∪G′,H
fF,G
gG,H
Approaching the Continuous from the Discrete
21
where ˜F is obtained by the covering condition. Cancellativity of deletions then
implies surjectivity of X( ˜F) →X(F), which in turn shows that the lower com-
position is completely determined by the upper composition, therefore conclud-
ing that gG,HfF,G = gG′,HfF,G′.
Lemma 1. Let C be a semicartesian category with cancellative deletion. The
equivalence of compatible families respects pointwise composition.
Proof. This follows from a reasoning similar to the one above: let f, f′ : X →Y
and g, g′ : Y →Z be compatible families such that f is equivalent to f ′ and g is
equivalent to g′. Then, for each H ∈Pfin(JZ), there exists G and G′ such that
gG,H and g′
G′,H are defined. By equivalence of g and g′ and hereditariness, we
have gG∪G′,H = g′
G∪G′,H. Now, a similar reasoning for f and f′ starting from
G ∪G′ gives rise to a set ˜F such that f ˜
F ,G∪G′ = f ′
˜
F ,G∪G′. Combining the two
together, we proved that for every H there exists ˜F such that gf ˜
F ,H = g′f ′
˜
F ,H.
In particular, for each (F, H) ∈Λgf ∩Λg′f ′, we have ˜F ⊇F such that gf ˜
F ,H =
g′f ′
˜
F ,H, and by cancellativity of deletion we conclude that gfF,H = g′f ′
F,H as
well.
Proposition 6. Let C be a semicartesian category with cancellative deletions.
Then C⊗∞is a semicartesian category.
Proof. By Lemma 1, C⊗∞is a category. The remaining properties follow by dir-
ect check: its symmetric monoidal structure is inherited from that of C, with
associators, unitors and permutations described pointwise, while semicartesian-
ity follows from terminality of I in C, by definition of compatible families.
C
Locally Constant Markov Kernels Form a Markov
Category
This appendix complements Section 6 and establishes that StoneStochlc, the cat-
egory of Stone spaces and locally constant Markov kernels, is a Markov category.
We first ensure that the composition is well-behaved.
Lemma 2. Let f : X
Y and g: Y
Z be locally constant Markov kernels.
Then their composition
gf(V | x) :=
Z
Y
g(V | y)f(dy | x)
is locally constant.
The proof is instructive as it sheds light on the “finiteness” of the notion.
22
A. Lorenzin and F. Zanasi
Proof. Since g(V | −) is locally constant on a Stone space, we can construct a
finite clopen cover {Ui}n
i=1 of Y where all clopens are disjoint and g(V | −)|Ui is
constant. Then,
Z
Y
g(V | y)f(dy | x) =
n
X
i=1
g(V | Ui)f(Ui | x)
we now define Ai the clopen set such that x ∈Ai and f(Ui | −) is constant on
Ai. Then, on Tn
i=1 Ai, the whole summation is constant, therefore concluding
the proof.
To show that StoneStochlc is a Markov category, we need to ensure the ex-
istence of copy maps. In Stoch, these are determined by the diagonal maps
X →X × X. With this in mind, we show the following.
Lemma 3. Any continuous function X →Y gives rise to a locally constant
Markov kernel X
Y .
Proof. That a continuous function yields a (continuous) Markov kernel is well-
known, and follows by setting ˜f(U | x) := δf(x)(U) for any continuous function
f : X →Y . Then for any clopen U, f −1(U) and its complement are sufficient
to ensure that ˜f(U | −) is locally constant. The fact that ˜f(−| x) is a finitely-
additive probability measure is immediate.
Proposition 7. The category StoneStochlc, whose objects are Stone spaces and
whose morphisms are locally constant Markov kernels, is a Markov subcategory
of Stoch.
In particular, since Stoch has cancellative deletions, so does StoneStochlc.
Proof. By Lemma 2, StoneStochlc is indeed a category. The tensor product of
two locally constant Markov kernels is clearly a locally constant Markov kernel.
Moreover, since all continuous maps give rise to locally constant Markov kernels
(Lemma 3), copy and delete maps also belong to StoneStochlc, thus concluding
the proof.
|
Approaching the Continuous from the Discrete: an Infinite Tensor Product Construction Antonio Lorenzin1 and Fabio Zanasi2 1 2 University College London, Computer Science Department, UK Abstract Increasingly in recent years probabilistic computation has been investigated through the lenses of categorical algebra, especially via string diagrammatic calculi. Whereas categories of discrete and Gaussian probabilistic processes have been thoroughly studied, with various axiomatisation results, more expressive classes of continuous probability are less understood, because of the intrinsic difficulty of describing infinite behaviour by algebraic means. In this work, we establish a universal construction that adjoins infinite tensor products, allowing continuous probability to be investigated from discrete settings. Our main result applies this construction to FinStoch, the category of finite sets and stochastic matrices, obtaining a category of locally constant Markov kernels, where the objects are finite sets plus the Cantor space 2N. Any probability measure on the reals can be reasoned about in this category. Furthermore, we show how to lift axiomatisation results through the infinite tensor product construction. This way we obtain an axiomatic presentation of continuous probability over countable powers of 2 = {0, 1}. Keywords: Categorical Probability · String Diagrams · Infinite Tensor Products · Cantor Space 1 Introduction Category-theoretic approaches to probabilistic computation have attracted growing interest in recent years, with applications ranging from evidential decision theory ([8]) to random graphs ([1]) and active inference ([28]). Their ability to highlight the underlying algebraic structures provides a rigorous semantics that enhances formal clarity, compositional methodologies, and admits an intuitive description in terms of string diagrams ([22,24]). In particular, fundamental notions such as determinism, Bayesian inversion, and Bayesian updates can be studied algebraically in the diagrammatic language ([11,19,6]). Following this emerging line of research, recent developments have focused on providing complete axiomatisations of categorical models of probability, via string diagrammatic theories. Completeness is key to guarantee that any semantic equality, and thus algebraic reasoning about the aforementioned notions, 16 Oct 2025 2 A. Lorenzin and F. Zanasi can in principle be derived in the diagrammatic calculus. In this context, axiomatising a (symmetric monoidal) category C, which expresses our semantic domain, means to identify a set of generating string diagrams Σ and equations E such that the category Free(Σ, E) freely obtained by the theory (Σ, E) is isomorphic (or just equivalent) to C. In other words, C-morphisms may be regarded as Σ-diagrams quotiented by E. A first such result is the axiomatisation of the category of finite sets and stochastic matrices (with monoidal product given by disjoint sum) via the theory of convex algebras, presented in [10] following [26]. Another such result is contained in [25], which studies a category of Gaussian probability using string diagrams expressing affine, relational, and quadratic behaviour. Most relevant to the present scope is the complete axiomatisation of a category of discrete probabilistic processes given in [21]. Specifically, the category under investigation, denoted by BinStoch, is the category of stochastic matrices between powers of 2 = {0, 1}, and the associated syntax is generated by probability distributions and Boolean operations. Two related efforts generalise this description beyond BinStoch: [23] extends the framework by introducing a graded structure to encompass imprecise probabilities, while [4] notes that the use of an additional monoidal product enables an axiomatisation of FinStoch, the category of stochastic matrices between arbitrary finite sets. To date, a fundamental gap in this research area is the absence of a string diagrammatic axiomatisation for categories of continuous probability beyond the Gaussian case. A chief example is BorelStoch, whose objects are standard Borel spaces and whose morphisms are Markov kernels (generalisations of stochastic matrices to the continuous setting). It is generally unclear how to encode infinite behaviour directly via a choice of generators and equations within the existing framework of string diagrammatic algebra. In this work we address such challenge through the use of infinite tensor products, which are special limits of tensor (monoidal) products.3 This approach is inspired by [16], where it is shown that R is the infinite tensor product of finite sets in BorelStoch. In particular, the celebrated Kolmogorov extension theorem ensures that all probability measures on R can be seen as limits of probability measures on finite sets. Our main contributions are the following: 1. Introduce a universal construction that adjoins infinite tensor products (Theorem 1); 2. Employ this construction to describe freely generated categories with an infinite tensor product, which admit a diagrammatic representation through plate notation (Section 4); 3. Provide a characterisation of the category FinStoch⊗∞obtained by adjoining infinite tensor products to FinStoch (Theorem 2) in terms of Stone spaces and locally constant Markov kernels. 4. Provide an axiomatic presentation of CantorStochlc, the restriction of FinStoch⊗∞ to powers of 2 together with the Cantor space 2N, its (countably) infinite power (Corollary 2). 3 Throughout, we use "tensor" and "monoidal" synonymously, in order to both follow the terminology of [16] and adhere to the terminology of categorical probability. Approaching the Continuous from the Discrete 3 Regarding the third contribution, one might hope that the whole of BorelStoch could be recovered as FinStoch⊗∞, i.e. by adjoining infinite tensor products to its discrete counterpart FinStoch. However, this is overly ambitious: not because of probability measures, which can always be described as above, but because of the freeness of measurable functions, whose behaviour cannot be fully captured by finite operations. What we obtain instead is a subcategory of BorelStoch, which is nonetheless quite expressive: probability measures on R are in bijective correspondence with probability measures on 2N, thus showing that CantorStochlc is rich enough to encompass all probability measures on R. Additionally, this category includes nontrivial kernels that capture biased behaviour with respect to their inputs (see Example 7 for a detailed discussion). Outline First we recall semicartesian categories, which provide the necessary structure to define infinite tensor products (Section 2). In Section 3 we give a universal construction of the category obtained by adjoining infinite tensor products to a semicartesian category. String diagrams and monoidal theories are discussed in Section 4, while Section 5 restricts the focus to Markov categories. Section 6 presents the main result, giving an explicit description of the universal construction for FinStoch and BinStoch. The developed syntax is applied in Section 7 to a short use case on Markov chains. Concluding remarks and future directions are briefly discussed in Section 8. Proofs of auxiliary results are deferred to the appendices. 2 Semicartesian Categories and Infinite Tensor Products For the sake of generality, we will first consider semicartesian categories (sometimes known as affine symmetric monoidal categories), as their theory is rich enough to allow the study of infinite tensor products ([16]). From Section 5 onward, we will restrict focus to Markov categories, broadly adopted for modelling probabilistic and statistical behaviour - see, e.g., [11,7,20,18]. In fact, the examples introduced below are all Markov categories. Definition 1. A semicartesian category is a symmetric monoidal category (C, ⊗, I) in which the monoidal unit I is terminal. The unique morphism of type X →I is denoted by delX (in string diagrams, ) and called delete morphism. Example 1. The main example for our purposes is the semicartesian category FinStoch [11, Example 2.5], whose objects are finite sets and whose morphisms f : X →Y are functions Y ×X →R≥0 satisfying Σy∈Y f(y, x) = 1, also known as stochastic functions. To better highlight the connection with probability theory, one generally employs the "conditional notation" and write f(y | x) in place of f(y, x). In particular, 1 = Σy∈Y f(y, x) = Σy∈Y f(y | x) means that f(-| x) is a probability distribution on Y for each x ∈X. Composition is defined by g f(z | x) := P y∈Y g(z | y)f(y | x) for any composable morphisms f : X →Y and g: Y →Z, . while the tensor product is given by f ⊗h(y, w | x, z) := f(y | x)h(w | z) for any f : X →Y and h: Z →W. In particular, the delete maps are given by del( | x) = 1. 4 A. Lorenzin and F. Zanasi Example 2. Another important example is BinStoch, the full subcategory of FinStoch given by finite powers of 2 = {0, 1}. This has become especially relevant after the work [21] provided a complete axiomatisation for such a category. Example 3. In [16], where infinite tensor products were originally defined, the main example is BorelStoch, which generalizes FinStoch outside the finite context. BorelStoch is the category whose objects are standard Borel spaces (measurably isomorphic to a finite set, Z, or R by Kuratowski's theorem) and whose morphisms are Markov kernels f : X →Y , given by honest functions f : ΣY × X → [0, 1], where ΣY is the σ-algebra of Y , such that f(U | -): X →[0, 1] is measurable and f(-| x): ΣY →[0, 1] is a probability measure. The measurability condition allows us to define a composition by integration: g f(U | x) := R Y g(U | dy)f(dy | x). The tensor product is determined by the rule f ⊗h(U × V | x, z) := f(U | x)h(V | z). As before, the delete maps are given by delX(-| x) = 1 for all x ∈X. Example 4. A more general version of BorelStoch is Stoch, where objects are all measurable spaces. See [11] for details. Infinite tensor products were introduced in [16] to formulate zero-one laws in categorical language. They have since become central in category-theoretic approaches to continuous probability; for instance, [13] employs them in discussing laws of large numbers. We now recall the basic setup and introduce some new terminology that will be convenient for later use. Throughout, let Pfin(J) denote the category whose objects are finite subsets of a countable set J, with a morphism F →F ′ whenever F ⊆F ′. Definition 2. Let C be a semicartesian category. A functor X : Pfin(J)op →C is an abstract infinite tensor product4 if there exists a family of objects (Xj)j∈J such that X sends F to XF := N j∈F Xj and the inclusion F ⊆F ′ to the morphism πF ′,F : XF ′ →XF which deletes all j ∈F ′ \ F, and keeps the rest unchanged. Up to permutation, πF ′,F = idXF ⊗delXF ′ . Intuitively, an abstract infinite tensor product is a choice of an infinite number of objects in C. Notation 1 Whenever X is an abstract infinite tensor product, its associated countable set J may be denoted by JX for clarity. 4 To be precise, we should say abstract countable infinite tensor product. To avoid overloading notation, we will simply omit the adjective, and throughout infinite will always mean countable infinite. Nonetheless, the full generality is expected to work, and this restriction is important only to our main application. Approaching the Continuous from the Discrete 5 Definition 3. Let C be a semicartesian category. A (concrete) infinite tensor product X = N j∈J Xj for a family of objects (Xj)j∈J in C is a limit of the associated abstract infinite tensor product, provided this limit is preserved by -⊗Y for all Y ∈C. The morphisms X →XF given by the universal property will be called finite marginalizations and denoted by πF . Note that such products are not strictly unique; rather, they are unique up to isomorphism. This subtle distinction mirrors the one between strict and strong monoidal functors, and it is the reason we will work with the latter throughout. Definition 4. A semicartesian category has infinite tensor products if every abstract infinite tensor is realised, i.e. it is associated to a concrete infinite tensor product. In [16], it was shown that BorelStoch has infinite tensor products, and moreover R is isomorphic to any nontrivial infinite tensor product (this follows by the mentioned Kuratowski's theorem, since R is measurably isomorphic to any uncountable standard Borel space). Throughout, symmetric monoidal functors are always meant to be strong, meaning that their coherence morphisms are always isomorphisms. Definition 5. A symmetric monoidal functor φ: C →D between two semicartesian categories with infinite tensor products is ITP-preserving if φ(N j∈J Xj) is an infinite tensor product of the family (φ(Xj))j∈J.5 Before proceeding, we also consider the following technical requirement, necessary to establish the forthcoming results. In [17], semicartesian categories are called theories and the property below is called normality. Following a nowestablished tradition in categorical modelling of probability [6,11,22], we will often depict morphisms of these categories using string diagrams. Definition 6. A semicartesian category C has cancellative deletions if f ⊗ delX = g ⊗delX implies f = g for all parallel morphisms f, g and all objects X. In string diagrams, f = g =⇒ f = g for all f and g. In particular, surjectiveness of the marginalizations πF ′,F : XF ′ →XF and πF : X →XF above is ensured. The fact that FinStoch, BinStoch, BorelStoch and Stoch satisfy this property follows by direct check: whenever f ⊗delX(U | a, x) = g ⊗delX(U | a, x), then f(U | a) = f(U | a) delX( | x) = g(U | a) delX( | x) = g(U | a) concludes the proof since delX( | x) = 1 for all x. Note that to ensure this property, the emptyset is not an object of these categories. 5 In particular, the finite marginalizations πF are preserved by φ. As πF is a notation for any infinite tensor product, we will write φ(πF ) = πF for simplicity. 6 A. Lorenzin and F. Zanasi 3 A Universal Construction for Infinite Tensor Products We now focus on proving the following statement, which intuitively states that adjoining infinite tensor products satisfies a natural universal property. This result is propaedeutic for our aims, but it is also of independent interest for other works using infinite tensor products [5,14,16]. Theorem 1. Let C be a semicartesian category with cancellative deletions. Then there exist a semicartesian category C⊗∞with infinite tensor products and a strict symmetric monoidal functor C →C⊗∞, such that for every semicartesian category D with infinite tensor products and every symmetric monoidal functor φ: C →D, there exists an ITP-preserving symmetric monoidal functor ̃φ: C⊗∞→D such that the following diagram commutes C D C⊗∞ φ ̃φ Moreover, such a ̃φ is unique up to natural isomorphism. First, we define what C⊗∞is. Its objects will be abstract infinite tensor products. However, morphisms are a little bit more subtle, as they should allow these tensor products to be limits in C⊗∞. This leads to the following. Definition 7. Let X : Pfin(JX) →C and Y : Pfin(JY ) →C be two abstract infinite tensor products. A compatible family f : X →Y is given by an index set Λf ⊆Pfin(JX) × Pfin(JY ) and a collection (fF,G : X(F) →Y (G))(F,G)∈Λf subject to the following requests: - (Naturality) For every F ′ ⊇F and G′ ⊇G, if (F, G), (F ′, G′) ∈Λf, then the diagram X(F ′) Y (G′) X(F) Y (G) fF ′,G′ πF ′,F πG′,G fF,G commutes; - (Covering condition) For every G ∈Pfin(JY ), there exists F ∈Pfin(JX) such that (F, G) ∈Λf; - (Hereditariness) If (F, G) ∈Λf, then (F ′, G) ∈Λf for every F ′ ⊇F. We moreover say that two compatible families f, f′ : X →Y are equivalent if fF,G = f ′ F,G for all (F, G) in the intersection of the index sets Λf ∩Λf ′. Hereditariness above (together with cancellativity of deletions) ensures transitivity of the equivalence defined for compatible families; in particular, the covering condition also holds for Λf ∩Λf ′. We are finally ready to define C⊗∞. To obtain a strict monoidal functor C →C⊗∞, we single out an element ∗, common to all countable sets J considered. Approaching the Continuous from the Discrete 7 Definition 8. Let C be a semicartesian category with cancellative deletion, and let ∗be a set. Then C⊗∞is the semicartesian category defined as follows: - Each object is an abstract infinite tensor product X : Pfin(J)op →C, with ∗∈J; - Morphisms f : X →Y are equivalence classes of compatible families; - Composition is defined pointwise: gf is the compatible family given by gfF,H := gG,HfF,G whenever there is a finite subset G; - The monoidal product of X and Y is given by X ⊗Y : Pfin(JX ∪JY ) -→ C F 7-→X(F ∩JX) ⊗Y (F ∩JY ), and consequently, given f : X →Z and g: Y →W, f ⊗g is the morphism such that (f ⊗g)F,G := fF ∩JX,G∩JZ ⊗gF ∩JY ,G∩JW . (The monoidal unit is given by Pfin({∗}) ∋{∗} 7→I ∈C). The composition is thoroughly studied in Section B, where it is shown that C⊗∞is a semicartesian category. In particular, the request of having cancellative deletions ensures that the composition is well-defined. By construction, there exists a fully faithful symmetric monoidal functor C →C⊗∞, given by considering each object X of C as a functor Pfin({∗}) →C defined by {∗} 7→X. We note that C⊗∞is strict if C is, since the monoidal product is described pointwise. Proposition 1. C⊗∞has infinite tensor products. Proof. We start by claiming that X : Pfin(JX) →C is the infinite tensor product of the family X(G), with G ∈Pfin(JX), in C⊗∞. Indeed, consider a morphism f : A →X. By definition, this yields a family of morphisms fG : A →XG indexed by G. Conversely, if one has a family of morphisms fG : A →XG such that πG′,GfG′ = fG for all G′ ⊇G, then one can easily construct a compatible family fF,G : A(F) →X(G), uniquely defined up to equivalence. The fact that the infinite tensor product is preserved under tensoring follows by the generality of the argument. Now, given any countable family (Xk)k∈K in C⊗∞, consider the set H := F k∈K JXk, which is countable because it is a countable union of countable sets. We claim that the abstract infinite tensor product Y : Pfin(H) →C given by F 7→N k∈K Xk(F ∩JXk) is indeed the infinite tensor product of (Xk)k∈K. Starting with f : A →Y , by definition of morphisms in C⊗∞, one obtains fF : A →N k∈F Xk for all F ∈Pfin(K). Conversely, given a family (fF : A → N k∈F Xk)F ∈Pfin(K) such that πF ′,F fF ′ = fF , the fact that every Xk is an infinite tensor product ensures that each fF is determined by (f G F : A →N k∈F XG∩JXk ), with G a finite subset of F k∈F JXk. In particular, f determines a family of morphisms indexed by all finite subsets of H = F k∈K JXk, giving rise to a compatible family A →Y . As wanted, Y is the infinite tensor product of (Xk)k∈K. 8 A. Lorenzin and F. Zanasi Finally, we prove of the main statement of the section. Proof (Proof of Theorem 1). Given φ: C →D a symmetric monoidal functor, then an abstract infinite tensor product X : Pfin(J) →C immediately gives rise to an abstract infinite tensor product on D by composition and coherence isomorphisms. Since D has infinite tensor products, this is realized as a concrete infinite tensor product, denoted by ̃φ(X) and equipped with finite marginalizations πF : ̃φ(X) →φ(X)F . Now, given a morphism f : X →Y in C⊗∞, by definition this is a compatible family (fF,G : X(F) →Y (G)). By the properties, φ(fF,G) gives rise to a morphism ̃φ(f)G : ̃φ(X) →φ(Y )G given by the composition ̃φ(X) φ(X)F φ(Y )G πF φ(fF,G) (1) and naturality ensures that ̃φ(f)G is a family that factors through the limit of φ(Y )G, which by definition is ̃φ(Y ); in other words, we obtain a morphism ̃φ(X) → ̃φ(Y ), which we denote by ̃φ(f). Proving that ̃φ so defined is a symmetric monoidal functor is now a simple routine check, and it is ITP-preserving by construction. Moreover, its uniqueness up to natural isomorphism follows from the universal property of limits and the description of morphisms provided by (1) (note it is not only sufficient to describe a choice of morphisms, but necessary). Proposition 2. Let C and D be two semicartesian categories with cancellative deletions, and assume D has infinite tensor products. If a symmetric monoidal functor φ: C →D is faithful, then so is ̃φ: C⊗∞→D. Proof. Let f, g: X →Y be two parallel morphisms in C⊗∞such that ̃φ(f) = ̃φ(g). Consider their marginalizations πG ̃φ(f) = πF ̃φ(g): ̃φ(X) → ̃φ(Y )G. For any G, there exists F ⊆JX such that πGf = fF,GπF because f is a compatible family, and analogously for g. Moreover, F can be chosen for both f and g by covering condition and hereditariness, so also πGg = gF,GπF . Using this equality via ̃φ, we obtain φ(fF,G)πF = ̃φ(fF,GπF ) = ̃φ(πGf) = πG ̃φ(f) = πG ̃φ(g) = ̃φ(πGg) = ̃φ(gF,GπF ) = φ(gF,G)πF Since D has cancellative deletions, πF is an epimorphism (note that πF = idXF ⊗ delXJX up to permutations), and therefore φ(fF,G) = φ(gF,G). By faithfulness of φ, we conclude that fF,G = gF,G. Arbitrariness of G, together with naturality of compatible families, implies that f and g are equivalent compatible families, i.e. f = g. 4 String Diagrams and Monoidal Theories for Infinite Tensor Products In this section we study how to derive presentations by generators and equations for C⊗∞from those of C. In doing so, we also introduce a string diagrammatic Approaching the Continuous from the Discrete 9 gG,H fF,G = fF,G gG,H fF,G gH,K = gH,K fF,G Figure 1. Sequential and parallel composition of the plate notation. representation for morphisms of C⊗∞which highlights their formulation as compatible families of C-morphisms (Definitions 7 and 8). Our approach is reminiscent of the 'plate' notation commonly used in the study of Bayesian networks, see e.g. [3]. In category-theoretic approaches to probability, this has recently been employed in [5]. Our use of plate notation is similar in spirit, but it is tailored to ensure a formal representation of compatible families. We recall that a morphism f : X →Y in C⊗∞is given by a compatible family (fF,G : X(F) →Y (G))(F,G)∈Λf , with fF,G in C, which we write as fF,G (F, G) ∈Λf X Y in plate notation. This morphism may also be denoted by f X Y for brevity, where the double wire of the input and output of f indicate that X and Y are infinite tensor products. For simplicity, we will generally omit (F, G) ∈Λf, since the index set is determined by the compatible family. This notation is compatible in the obvious way with sequential and parallel composition in C⊗∞ (see Fig. 1). Moreover, it will be convenient for graphical reasoning to allow plates to disappear when infinite tensors are not involved, i.e. (fF,G : X(F) → Y (G))(F,G)∈Λf is a bona fide morphism of C. More precisely, fF,G (F, G) ∈Λf X Y = f X Y (2) whenever X and Y belong to the image of C →C⊗∞. We can use plate notation to define a variation of the construction of a symmetric monoidal category from generators and equations, which encompasses infinite tensor products. This will be useful in the following sections to derive axiomatisation results. First, recall that a symmetric monoidal theory (SMT) is a pair (Σ, E), where Σ is a signature of generators o: m →n with an arity m ∈N and coarity n ∈N, and E is a set of equations between Σ-terms. A Σ-term c of type m →n will be represented graphically as a box with m dangling wires on the left and n 10 A. Lorenzin and F. Zanasi on the right, also written c m n. Formally, Σ-terms are freely obtained by sequential and parallel compositions of the generators in Σ together with the identity : 1 →1, the symmetry : 2 →2, and the 'empty' diagram : 0 →0. Sequential composition of Σ-terms c m n and d v n is depicted as c m n d v , of type m →v. Parallel composition of Σ-terms c1 m1 n1 and c2 m2 n2 is depicted as c1 m1 n1 c2 m2 n2, of type m1 + m2 →n1 + n2. Given an SMT (Σ, E), the symmetric monoidal category Free(Σ, E) freely generated by (Σ, E) has objects the natural numbers and morphisms m →n the Σ-terms of type m →n quotiented by E and by the laws of symmetric strict monoidal categories (Appendix A), with sequential and parallel composition defined as on the corresponding Σ-terms. More details can be found e.g. in [2,29]. Example 5. The SMC freely obtained by generators and p for all p ∈[0, 1], and set of equations as in [21, Figure 4] is named CausCirc in [21], as we may regard its string diagrams as causal circuits. Also in [21] it is proven that CausCirc ∼= BinStoch, meaning that such an SMT axiomatises BinStoch. We note that Free(Σ, E) is semicartesian with cancellative deletions if and only if both of the following are satisfied: 1. There is only one Σ-term (up to =E) of type n →0, for any n ∈N; 2. Given two Σ-terms f, g, f ⊗deln =E g ⊗deln if and only if f =E g. Under these assumptions, we may define how to freely obtain a category with infinite tensor products from (Σ, E) as follows. Definition 9. Given an SMT (Σ, E) satisfying the two assumptions above, let Free∞(Σ, E) be defined as the category (with infinite tensor products) whose objects are natural numbers together with an additional object ∞, and whose morphisms are given by compatible families of Σ-terms, subject to the equations of Fig. 1 and (2). The proof that Free∞(Σ, E) is indeed a category follows by the same argument used to prove that C⊗∞is a category (see Section B). Regarding the tensor product, one sets ∞⊗N = ∞for any N natural number or ∞, and the tensor product is obtained by reindexing the compatible families according to chosen fixed bijections N ⊔{0, 1, . . . , n -1} ∼= N and N ⊔N ∼= N. These choices are necessary to ensure that associators, unitors and permutations are well-defined: for example, if we call φ the bijection N ⊔N ∼ = -→N, then the associator is a reindexing with respect to the bijection N N ⊔N N ⊔N ⊔N N ⊔N N. φ-1 id⊔φ-1 φ⊔id φ Approaching the Continuous from the Discrete 11 Moreover, we can verify that Free∞(Σ, E) indeed has infinite tensor products, by an argument analogous to the one showing Proposition 1. We now state that the infinite tensor product construction of Definition 8, when applied to a freely generated SMC Free(Σ, E), is presented by the same (Σ, E) via the Free∞(-) construction above. Proposition 3. Let (Σ, E) be a symmetric monoidal theory such that Free(Σ, E) is a semicartesian category with cancellative deletions. Then Free∞(Σ, E) ≃ (Free(Σ, E))⊗∞via an ITP-preserving symmetric monoidal functor. Proof. By applying Theorem 1 and Proposition 2, we have a faithful functor Free(Σ, E)⊗∞→Free∞(Σ, E). Fullness is immediate by definition of Free∞(Σ, E). We emphasise that Free∞(Σ, E) and (Free(Σ, E))⊗∞are not isomorphic. In fact, these two categories reflect different trade-offs. The former, Free∞(Σ, E), admits a single infinite tensor product ∞, which may appear more natural at first glance, but then ∞⊗∞= ∞requires to fix a chosen bijection N ⊔N ∼= N, as mentioned above. The latter, (Free(Σ, E))⊗∞, instead allows multiple infinite tensor products, with the advantage that its tensor product does not rely on any choice of bijections. Regardless of this distinction, in both categories one can reason equationally using the SMT (Σ, E) together with the equations of Fig. 1 and (2). Combining Theorem 1 and Proposition 3, one can prove the expected universal property for the free construction. Corollary 1. Let (Σ, E) be an SMT satisfying the two assumptions preceding Definition 9, and consider a semicartesian category D with infinite tensor products. Then a symmetric monoidal functor Free(Σ, E) →D extends to an ITP-preserving symmetric monoidal functor Free∞(Σ, E) →D. In other words, Free∞(Σ, E) is initial among the semicartesian categories with infinite tensor products in which (Σ, E) can be interpreted. Proof. By Theorem 1, Free(Σ, E) →D extends to (Free(Σ, E))⊗∞→D. By composing this functor with the equivalence Free∞(Σ, E) ≃(Free(Σ, E))⊗∞ provided in Proposition 3, the statement follows. 5 Infinite Tensor Products and Markov Categories As mentioned, our motivating examples are not just semicartesian but also Markov categories. Given the relevance of these structures in the categorical probability literature, it is of interest to show that whenever C is a Markov category, then so is C⊗∞. Definition 10. A Markov category is a semicartesian category C where every object X comes equipped with a cocommutative comonoid copyX : X →X ⊗X (whose counit is given by delX) compatible with the monoidal product. In string 12 A. Lorenzin and F. Zanasi diagrams, copyX is usually denoted by and the cocommutative comonoid equations read as follows. = = = = Compatibility with the monoidal product is instead written as follows. X ⊗Y X ⊗Y X ⊗Y = X X X Y Y Y (The delete maps are obviously compatible since they are the determined by terminality of I). As mentioned, FinStoch, BinStoch, BorelStoch and Stoch are Markov categories. In FinStoch, copyX(y, z | x) := 1 if x = y = z and 0 otherwise. More details can be found in [6,11]. Proposition 4. Let C be a Markov category with cancellative deletions. Then C⊗∞is a Markov category as well. Proof. For every abstract infinite tensor product X, the collection of copy maps (copyX(F ) : X(F) →X(F) ⊗X(F))F ∈Pfin(JX) gives rise to a compatible family, thanks to compatibility with the monoidal product. We then define copyX to be (the equivalence class of) such compatible family. Graphically, X X X := X X X X(F) X(F) X(F) By Fig. 1, the cocommutative comonoid equations hold because they hold inside the plate. Similarly, compatibility with the monoidal product holds, therefore concluding the proof. In particular, by construction of the copy of X, a direct computation shows that X is a Kolmogorov product in the sense of [16]. 6 A Stone Space Characterisation of FinStoch⊗∞ In this section we explicitly describe what FinStoch⊗∞and BinStoch⊗∞are, therefore highlighting what information about the continuous setting can be recovered from FinStoch and BinStoch. By Proposition 2, these infinite tensor products extensions can be viewed as subcategories of BorelStoch. With this in Approaching the Continuous from the Discrete 13 mind, it suffices to understand what is the relevant notion of Markov kernels associated to FinStoch⊗∞and BinStoch⊗∞. To this end, we seek to find a category where everything is seemingly "finitelydetermined", in a way that resembles the notion of compatible families. For this reason, we focus on Stone spaces. We recall that these spaces are compact Hausdorff spaces with a base consisting of clopen sets. In particular, we will denote by Clopen(X) the clopen sets of a Stone space X. These spaces are particularly well-known in the theory due to Stone duality, which states that Clopen(-) yields a contravariant equivalence between the category of Stone spaces and continuous maps and the category of Boolean algebras and algebra homomorphisms. In this setting, the most important Stone space for our purposes is the Cantor space 2N, which, as the notation suggests, is given by the product of countably many copies of 2 = {0, 1}. By Brouwer's theorem, the Cantor space is the only non-empty Stone space that is second-countable and without isolated points. We now equip Stone spaces with a special type of Markov kernels. Definition 11. Let X and Y be two Stone spaces. A locally constant Markov kernel X to Y , denoted by f : X Y , is given by a function f : Clopen(Y ) × X → [0, 1] (U, x) 7→f(U | x) such that - For all U ∈Clopen(Y ), the function f(U | -): X →[0, 1] is locally constant, i.e. for every x ∈X there exists an open A ∋x such that f(U | x) = f(U | y) for all y ∈A; - For all x ∈X, the function f(-| x): Clopen(Y ) →[0, 1] is a finitely-additive probability measure, i.e. f(Y | x) = 1 and for every finite collection of disjoint clopen sets {Ui}n i=1, f n [ i=1 Ui x ! = n X i=1 f(Ui | x). Stone spaces with locally constant Markov kernels form a Markov category denoted by StoneStochlc (see Section C for details). Remark 1. By Carathéodory's extension theorem, a finitely-additive probability measure on clopen sets gives rise to a σ-additive probability measure on the generated σ-algebra, often called the Baire σ-algebra. In particular, this allows us to integrate against the Markov kernels described above, as they are particular instances of the standard Markov kernels recalled in Example 3, where f(-| x) is required to be a probability measure on a σ-algebra. Example 6. Not all Markov kernels are locally constant. For the sake of an example, we define a Markov kernel from the Cantor space 2N to 2 given by f(U | (xn)) := (P∞ n=0 2xn 3n+1 if U = {1} 1 -P∞ n=0 2xn 3n+1 if U = {0} 14 A. Lorenzin and F. Zanasi This is clearly not locally constant, since any choice of (xn) changes the value of f(U | (xn)), although (xn) 7→P∞ n=0 2xn 3n+1 is the standard continuous embedding 2N ,→[0, 1]. Nonetheless, all continuous functions yield locally constant Markov kernels by means of delta measures (see Section C for details). We now present two examples of locally constant Markov kernels that demonstrate the expressive power of this concept. Example 7. Note that Clopen(2N) is the collections of sets of the form UF ×2N , where F ⊆N is finite and UF ⊆2F is arbitrary. 1. A simple example of a locally constant Markov kernel is p: I 2N, defined by p(UF × 2N | ) := #(UF ) 2#(F ) where #(A) denotes the size of A. This construction is similar to the Lebesgue measure on the real line, in that it is translation-invariant: p(UF × 2N | ) = p(UG × 2N | ) whenever UF and UG are in bijection. The fact that it is locally constant is immediate, since I has only one element. 2. A more involved example of a locally constant Markov kernel f : 2N 2N can be defined using the Markov kernel p: I 2N introduced in Item 1. We set f(U | x) := ( 1 2p(U | ) + 1 2 if x ∈U 1 2p(U | ) if x /∈U Intuitively, the probability is biased toward the conditioning point x. The fact that f(-| x) is a finitely-additive probability measure follows from the same property of p(-| ). Moreover, f(U | -) is constant on U and on its complement, hence f is locally constant. Proposition 5. StoneStochlc, the category of Stone spaces and locally constant Markov kernels, has all infinite tensor products. More explicitly, the infinite tensor product of (Xj)j∈J is given by Q j∈J Xj. Proof. First of all, we note that X := Q j∈J Xj is a Stone space, with compactness ensured by Tychonoff's theorem. Now, let us consider A another Stone space, and consider f : A X a locally constant Markov kernel. We claim that f is determined by the marginals fF , given by the composition A X XF = Q j∈F Xj, f πF with F any finite subset of J. By compactness, for every clopen U in X there exist F ∈Pfin(J) and a clopen UF ⊆XF such that U = UF × Q j∈J Xj. So f(U | a) = fF (UF | a). In particular, f is determined by its marginals, as claimed. Conversely, if we consider a family (fF : A XF )F ∈Pfin(J) that is compatible in the sense that πF,F ′ fF = fF ′, for every clopen set U ⊆Q j∈J Xj, we can Approaching the Continuous from the Discrete 15 define f(U | a) := fF (UF | a), and compatibility ensures that this definition is independent of F. The limit property is therefore ensured. The preservation under tensoring follows by arbitrariness of the family of Stone spaces. Finally, we obtain our second main theorem, yielding a characterisation of infinite tensor products over FinStoch. Theorem 2. The ITP-preserving symmetric monoidal functor φ: FinStoch⊗∞→StoneStochlc, obtained by the inclusion FinStoch ,→StoneStochlc, is fully faithful. Moreover, its essential image is given by infinite tensor products of finite sets in the sense of topological spaces. In particular, by Brouwer's theorem, any object in the essential image is isomorphic either to a finite set or to the Cantor space. Moreover, φ preserves the copy map because in both settings the finite marginalizations are deterministic. In other words, this functor is strong gs-monoidal in the sense of [12]. Proof. First, recall that the functor is obtained by Theorem 1, and faithfulness is immediate by Proposition 2. We therefore only need to show fullness. Now, given a morphism between infinite tensor product of finite sets X = Q j∈JX Xj and Y = Q j∈JY Yj, we wish to show that f : X Y is uniquely determined by a compatible family fF,G : XF YG. By Proposition 5, f is uniquely determined by (fG)G∈Pfin(JY ), with fG := πGf. Let us now focus on fG. Since it is locally constant, fG(U | -) is constant on each member of a finite family (AU i )i=1,...,nU . Since YG is finite, its subsets U are finite, so we obtain a finite family of clopens (AU i )i,...,nU,U⊆YG. As any clopen in X is of the form AF × Q j∈J Xj for some F, there is a finite set F common to all members of the finite family (AU i )i,U, such that each AU i can be written in such a cylindrical form. In particular, fG(U | a) = fG(U | b) for any b such that πF (a) = πF (b). In other words, fG factors through πF by definining, for any ̃a ∈XF , fF,G(U | ̃a) := fG(U | a), where a ∈X is any element such that πF (a) = ̃a. It is now a direct check that the given definition of fF,G results in a compatible family, therefore ensuring fullness. Corollary 2. BinStoch⊗∞is equivalent to CantorStochlc, the full subcategory of StoneStochlc in which objects are 2N = {0, 1}N, where N is any natural number or N. In particular, by combining Proposition 3, Corollary 2, and Example 5, we obtain a characterisation by generators and equations of CantorStochlc. Corollary 3. CantorStochlc is isomorphic to Free∞(Σ, E), where (Σ, E) is the symmetric monoidal theory of CausCirc ( cf. Example 5). 16 A. Lorenzin and F. Zanasi Another important consequence of our results is that CantorStochlc includes all probability measures on R. This follows from the fact that R is the infinite tensor power of 2 in BorelStoch [16]. Indeed, since both R in BorelStoch and 2N in CantorStochlc are associated to compatible families of morphisms I →2n in FinStoch, we obtain BorelStoch(I, R) ∼= CantorStochlc(I, 2N). This observation suggests that CantorStochlc is a rather expressive setting for studying continuous probability. We provide a simple use case below. 7 A Use Case of Plate Notation: Markov Chains As an example of diagrammatic reasoning using plate notation, we briefly discuss Markov chains, which have been recently studied via Markov categories in [15]. We restrict here to the time-homogenous case, which defines a Markov chain with n steps in BinStoch as an endomorphism f : X →X inductively, as follows: c1 X X := f X X cn Xn X := cn-1 f X Xn-1 X For example, c3 is given by the following string diagram: f X X f X f X The idea is that each copy of X represents a random variable of the chain, and the use of the same f for all steps translates to the request that P(Xn | Xn-1) = P(Xn+1 | Xn), i.e. that the obtained Markov chain is time-homogeneous. Additional details can be found in [15]. It can be verified that the cns defines a compatible family, hence a morphism c: X →XN, the infinite tensor power of X. As one may expect, c is invariant under the addition of a precedent step in the inductive construction. Note this property cannot be formulated in the discrete setting of BinStoch (and of CausCirc); however, we can easily prove it by diagrammatic reasoning in CausCirc⊗∞, as justified by Corollary 3: cn = cn-1 f = cn-1 f = cn-1 f = cn f where the last step is a simple reindexing of the compatible family defining c. 8 Conclusion and Future Work This paper focuses on introducing categorical methods for studying continuous probabilistic processes as limits of discrete ones. We do so by introducing a universal construction that associates to any semicartesian category C with cancellative deletions a semicartesian category C⊗∞with all (countably) infinite tensor Approaching the Continuous from the Discrete 17 products. We discuss how axiomatic presentations for C can be lifted to C⊗∞, and introduce plate notation to manipulate as string diagrams C⊗∞-morphisms. As main case study, we focus on FinStoch, whose infinite tensor construction FinStoch⊗∞we characterise in terms of Stone spaces and locally constant Markov kernels. Furthermore, the existence of an axiomatic presentation of the subcategory BinStoch allows us to derive one for BinStoch⊗∞. We can effectively use diagrammatic reasoning in BinStoch⊗∞to study any probability measure on R. We give a very simple use case regarding Markov chains; clearly, this only scratches the surface of what can be studied in BinStoch⊗∞, which remains to be explored in follow-up work. Moreover, our approach suggests ways towards axiomatising Markov kernels beyond local constancy. One such direction may be based on describing general Markov kernels in terms of locally constant ones, possibly restricting to standard Borel spaces. Another direction may be to investigate a universal construction that guarantees disintegration of measures, in the sense of having conditionals in Markov categories ([11]). Additionally, studying locally constant Markov kernels could yield a description of probabilistic morphisms between Boolean algebras via Stone duality, which may constitute an insightful contribution in its own right. Another interesting question is studying examples different from FinStoch and BinStoch. Natural candidates are Gaussian probability (Gauss, as studied in [25]), Gaussian mixtures ([27]), or even non probabilistic examples of Markov categories, such as the category of finite sets and multivalued functions ([11, Example 2.6]). Finally, we are interested in integrations of our approach (and of plate notation) with categorical approaches to probability that go beyond Markov categories, such as partial Markov categories [9], tape diagrams [4], and graded diagrams [23]. Acknowledgments. This work was partially supported by the ARIA Safeguarded AI TA1.1 programme. During the preparation of this article, we became aware that Sean Moss and Sam Staton had independently considered locally constant Markov kernels and infinite tensor products, although their work is not publicly available at present. Disclosure of Interests. The authors have no competing interests to declare that are relevant to the content of this article. References 1. Ackerman, N., Freer, C.E., Kaddar, Y., Karkwowski, J., Moss, S., Roy, D., Staton, S., Yang, H.: Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets. In: Proceedings of the ACM on Programming Languages, vol. 8, pp. 1819-1849. ACM (2024). https://doi.org/10.1145/ 3632903 2. Baez, J.C., Coya, B., Rebro, F.: Props in network theory. Theory Appl. Categ. 33, 727-783 (2018), www.tac.mta.ca/tac/volumes/33/25/33-25abs.html 18 A. Lorenzin and F. Zanasi 3. Barber, D.: Bayesian Reasoning and Machine Learning. Cambridge University Press (2012) 4. Bonchi, F., Cioffo, C.J., Di Giorgio, A., Di Lavore, E.: Tape Diagrams for Monoidal Monads. In: Cîrstea, C., Knapp, A. (eds.) 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), vol. 342, pp. 11:1-11:24. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2025). https://doi.org/10.4230/LIPIcs.CALCO.2025.11, https:// drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.11 5. Chen, L., Fritz, T., Gonda, T., Klingler, A., Lorenzin, A.: The Aldous-Hoover theorem in categorical probability, 6. Cho, K., Jacobs, B.: Disintegration and Bayesian inversion via string diagrams. Math. Structures Comput. Sci. 29, 938-971 (2019). https://doi.org/10.1017/ S0960129518000488 7. Cornish, R.: Stochastic neural network symmetrisation in markov categories, 8. Di Lavore, E., Román, M.: Evidential decision theory via partial Markov categories. In: Proceedings of LICS. pp. 1-14 (2023). https://doi.org/10.1109/LICS56636. 2023.10175776 9. Di Lavore, E., Román, M., Sobociński, P.: Partial Markov categories, 10. Fritz, T.: A presentation of the category of stochastic matrices (2009), https: //arxiv.org/abs/0902.2554 11. Fritz, T.: A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Advances in Mathematics 370, 107239 (2020). https://doi.org/https://doi.org/10.1016/j.aim.2020.107239 12. Fritz, T., Gadducci, F., Trotta, D., Corradini, A.: From gs-monoidal to oplax cartesian categories: Constructions and functorial completeness. Applied Categorical Structures 31(5), 42 (2023). https://doi.org/10.1007/ s10485-023-09750-z, https://doi.org/10.1007/s10485-023-09750-z 13. Fritz, T., Gonda, T., Lorenzin, A., Perrone, P., Mohammed, A.S.: Empirical measures and strong laws of large numbers in categorical probability (2025), 14. Fritz, T., Gonda, T., Perrone, P.: De Finetti's theorem in categorical probability. J. Stoch. Anal. 2(4) (2021). https://doi.org/10.31390.josa.2.4.06, 15. Fritz, T., Klingler, A., McNeely, D., Shah Mohammed, A., Wang, Y.: Hidden markov models and the bayes filter in categorical probability. IEEE Transactions on Information Theory 71(9), 7052-7075 (2025). https://doi.org/10.1109/TIT. 2025.3584695 16. Fritz, T., Rischel, E.F.: Infinite products and zero-one laws in categorical probability. Compositionality 2, 3 (2020). https://doi.org/10.32408/ compositionality-2-3, 17. Houghton-Larsen, N.G.: A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing. Ph.D. thesis, - hagen (2021), 18. Jacobs, B., Kissinger, A., Zanasi, F.: Causal inference by string diagram surgery. In: Bojańczyk, M., Simpson, A. (eds.) Foundations of Software Science and Computation Structures. pp. 313-329. Springer International Publishing, Cham (2019), https://doi.org/10.1007/978-3-030-17127-8_18 Approaching the Continuous from the Discrete 19 19. Jacobs, B., Kissinger, A., Zanasi, F.: Causal inference via string diagram surgery: A diagrammatic approach to interventions and counterfactuals. Mathematical Structures in Computer Science 31(5), 553-574 (2021). https://doi.org/10. 1017/S096012952100027X 20. Jacobs, B., Zanasi, F.: The logical essentials of Bayesian reasoning. In: Foundations of probabilistic programming, pp. 295-332. Cambridge: Cambridge University Press (2021). https://doi.org/10.1017/9781108770750.010 21. Piedeleu, R., Torres-Ruiz, M., Silva, A., Zanasi, F.: A complete axiomatisation of equivalence for discrete probabilistic programming. In: Programming Languages and Systems: 34th European Symposium on Programming, ESOP 2025, Held as Part of the International Joint Conferences on Theory and Practice of Software, ETAPS 2025, Hamilton, ON, Canada, May 3-8, 2025, Proceedings, Part II. pp. 202-229. Springer-Verlag, Berlin, Heidelberg (2025). https://doi.org/10.1007/ 978-3-031-91121-7_9, https://doi.org/10.1007/978-3-031-91121-7_9 22. Piedeleu, R., Zanasi, F.: An Introduction to String Diagrams for Computer Scientists. Elements in Applied Category Theory, Cambridge University Press (2025) 23. Sarkis, R., Zanasi, F.: String diagrams for graded monoidal theories, with an application to imprecise probability. In: Cîrstea, C., Knapp, A. (eds.) 11th Conference on Algebra and Coalgebra in Computer Science, CALCO 2025, June 1618, 2025, . LIPIcs, vol. 342, pp. 5:1-5:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2025). https://doi.org/10.4230/ LIPICS.CALCO.2025.5, https://doi.org/10.4230/LIPIcs.CALCO.2025.5 24. Selinger, P.: A survey of graphical languages for monoidal categories. In: New structures for physics, pp. 289-355. Berlin: Springer (2011). https://doi.org/10. 1007/978-3-642-12821-9_4, 25. Stein, D., Zanasi, F., Piedeleu, R., Samuelson, R.: Graphical quadratic algebra, 26. Stone, M.H.: Postulates for the barycentric calculus. Annali di Matematica Pura ed Applicata 29, 25-30 (1949), https://api.semanticscholar.org/CorpusID: 122252152 27. Torres-Ruiz, M., Piedeleu, R., Silva, A., Zanasi, F.: A complete diagrammatic calculus for conditional gaussian mixtures (2025), https://arxiv.org/abs/2510. 04649 28. Tull, S., Kleiner, J., St Clere Smithe, T.: Active inference in string diagrams: A categorical account of predictive processing and free energy, 29. Zanasi, F.: Interacting Hopf algebras: the theory of linear systems. CoRR abs/1805.03032 (2018) 20 A. Lorenzin and F. Zanasi A Laws of Symmetric Strict Monoidal Categories c1 c2 c3 = c1 c2 c3 c = c = c c1 c2 c3 = c1 c2 c3 c = c = c c1 d1 d2 c2 = c1 d1 d2 c2 c = c = B Compatible Families Form a Semicartesian Category This appendix shows that the universal construction C⊗∞offered in Section 2 indeed yields a semicartesian category. We start with a precise definition of the composition, briefly sketched in Definition 8. Definition 12. Let C be a semicartesian category with cancellative deletion. Let X, Y and Z be abstract infinite tensor products and let f : X →Y and g: Y →Z be compatible families. We define the composition of compatible families gf : X →Y by setting Λgf ⊆Pfin(JX) × Pfin(JZ) to be the set of pairs (F, H) for which there exists G ∈Pfin(JY ) such that (F, G) ∈Λf and (G, H) ∈Λg, and gfF,H := gG,HfF,G for any meaningful choice of G. The proof that this is indeed a good notion of composition is easily shown. Given two choices G and G′, we can consider the following commutative diagram given by naturality X( ̃F) Y (G ∪G′) Z(H) X(F) Y (G) Z(H) f ̃ F ,G∪G′ gG∪G′,H fF,G gG,H Approaching the Continuous from the Discrete 21 where ̃F is obtained by the covering condition. Cancellativity of deletions then implies surjectivity of X( ̃F) →X(F), which in turn shows that the lower composition is completely determined by the upper composition, therefore concluding that gG,HfF,G = gG′,HfF,G′. Lemma 1. Let C be a semicartesian category with cancellative deletion. The equivalence of compatible families respects pointwise composition. Proof. This follows from a reasoning similar to the one above: let f, f′ : X →Y and g, g′ : Y →Z be compatible families such that f is equivalent to f ′ and g is equivalent to g′. Then, for each H ∈Pfin(JZ), there exists G and G′ such that gG,H and g′ G′,H are defined. By equivalence of g and g′ and hereditariness, we have gG∪G′,H = g′ G∪G′,H. Now, a similar reasoning for f and f′ starting from G ∪G′ gives rise to a set ̃F such that f ̃ F ,G∪G′ = f ′ ̃ F ,G∪G′. Combining the two together, we proved that for every H there exists ̃F such that gf ̃ F ,H = g′f ′ ̃ F ,H. In particular, for each (F, H) ∈Λgf ∩Λg′f ′, we have ̃F ⊇F such that gf ̃ F ,H = g′f ′ ̃ F ,H, and by cancellativity of deletion we conclude that gfF,H = g′f ′ F,H as well. Proposition 6. Let C be a semicartesian category with cancellative deletions. Then C⊗∞is a semicartesian category. Proof. By Lemma 1, C⊗∞is a category. The remaining properties follow by direct check: its symmetric monoidal structure is inherited from that of C, with associators, unitors and permutations described pointwise, while semicartesianity follows from terminality of I in C, by definition of compatible families. C Locally Constant Markov Kernels Form a Markov Category This appendix complements Section 6 and establishes that StoneStochlc, the category of Stone spaces and locally constant Markov kernels, is a Markov category. We first ensure that the composition is well-behaved. Lemma 2. Let f : X Y and g: Y Z be locally constant Markov kernels. Then their composition gf(V | x) := Z Y g(V | y)f(dy | x) is locally constant. The proof is instructive as it sheds light on the "finiteness" of the notion. 22 A. Lorenzin and F. Zanasi Proof. Since g(V | -) is locally constant on a Stone space, we can construct a finite clopen cover {Ui}n i=1 of Y where all clopens are disjoint and g(V | -)|Ui is constant. Then, Z Y g(V | y)f(dy | x) = n X i=1 g(V | Ui)f(Ui | x) we now define Ai the clopen set such that x ∈Ai and f(Ui | -) is constant on Ai. Then, on Tn i=1 Ai, the whole summation is constant, therefore concluding the proof. To show that StoneStochlc is a Markov category, we need to ensure the existence of copy maps. In Stoch, these are determined by the diagonal maps X →X × X. With this in mind, we show the following. Lemma 3. Any continuous function X →Y gives rise to a locally constant Markov kernel X Y . Proof. That a continuous function yields a (continuous) Markov kernel is wellknown, and follows by setting ̃f(U | x) := δf(x)(U) for any continuous function f : X →Y . Then for any clopen U, f -1(U) and its complement are sufficient to ensure that ̃f(U | -) is locally constant. The fact that ̃f(-| x) is a finitelyadditive probability measure is immediate. Proposition 7. The category StoneStochlc, whose objects are Stone spaces and whose morphisms are locally constant Markov kernels, is a Markov subcategory of Stoch. In particular, since Stoch has cancellative deletions, so does StoneStochlc. Proof. By Lemma 2, StoneStochlc is indeed a category. The tensor product of two locally constant Markov kernels is clearly a locally constant Markov kernel. Moreover, since all continuous maps give rise to locally constant Markov kernels (Lemma 3), copy and delete maps also belong to StoneStochlc, thus concluding the proof.
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2510.14717
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Seesaw: Accelerating Training by Balancing Learning
Rate and Batch Size Scheduling
Alexandru Meterez*,1,2, Depen Morwani*,1,2, Jingfeng Wu3, Costin-Andrei Oncescu1,
Cengiz Pehlevan1,2, and Sham Kakade1,2
1Harvard University
2Kempner Institute at Harvard University
3University of California, Berkeley
Abstract
Increasing the batch size during training — a “batch ramp” — is a promising strategy to accelerate
large language model pretraining. While for SGD, doubling the batch size can be equivalent to halving the
learning rate, the optimal strategy for adaptive optimizers like Adam is less clear. As a result, any batch-ramp
scheduling, if used at all, is typically tuned heuristically.
This work develops a principled framework for batch-size scheduling and introduces Seesaw: whenever
a standard scheduler would halve the learning rate, Seesaw instead multiplies it by 1/
√
2 and doubles the
batch size, preserving loss dynamics while reducing serial steps. Theoretically, we provide, to our knowledge,
the first finite-sample proof of equivalence between learning-rate decay and batch-size ramp-up for SGD on
noisy linear regression, and we extend this equivalence to normalized SGD, a tractable proxy for Adam, under
a variance-dominated regime observed in practice. Empirically, on 150M/300M/600M-parameter models
trained at Chinchilla scale using a constant (critical) batch size, Seesaw matches cosine decay at equal FLOPs
while reducing wall-clock time by ≈36%, approaching the theoretical limit implied by our analysis.
1
Introduction
In recent years, large language models (LLMs) have demonstrated remarkable progress across
diverse tasks, including outperforming humans in competitive benchmarks and international
competitions (Huang and Yang, 2025; Petrov et al., 2025; El-Kishky et al., 2025). A central driver of
this progress has been the steady increase in pre-training compute (Kaplan et al., 2020; Hoffmann
et al., 2022). However, hardware improvements have not kept pace with the rapid escalation of
training requirements, resulting in wall-clock times extending to several months for state-of-the-art
models (Erdil and Schneider-Joseph, 2024).
A widely studied strategy to reduce wall clock time is increasing the batch size (You et al., 2017;
Goyal et al., 2017). Empirical studies show that larger batches can proportionally reduce the number
of optimization steps required for convergence (Zhang et al., 2024; McCandlish et al., 2018; Shallue
et al., 2019), while maintaining comparable per-step runtime through parallelization. However,
beyond a maximum batch size termed as critical batch size (CBS), further scaling reduces sample
efficiency and limits gains in training speed.
While most prior work focuses on training with a fixed batch size, recent large-scale LLM train-
ing runs employ batch size schedules that gradually increase batch size over the course of train-
⋆: Equal contribution.
Correspondence to: ameterez@g.harvard.edu, dmorwani@g.harvard.edu
1
arXiv:2510.14717v1 [cs.LG] 16 Oct 2025
ing (Dubey et al., 2024; Touvron et al., 2023; Adler et al., 2024; OLMo et al., 2024; Team, 2025). This
practice has been observed to further reduce training times without compromising model perfor-
mance. However, to the best of our knowledge, the “batch ramp” schedules are not theoretically
grounded and instead tuned heuristically. The lack of theoretical justification leaves open whether
these heuristics are close to optimal, motivating the central question of our study: what is the optimal
batch size schedule for minimizing serial runtime while not sacrificing performance?
1.1
Theoretical Contributions
We theoretically prove, to the best our knowledge, the first non-asymptotic equivalence result
between learning rate decay and batch size ramp up in SGD in linear regression with additive
noise. Further, we extend our equivalence result to normalized SGD (considered a proxy for Adam),
leading to the batch size scheduling algorithm Seesaw (Algorithm 1). We introduce an informal
version of our main theorem here, as well as the corollary leading up to Seesaw, and we formalize
the statements in Section 5.
Theorem (Informal version of Theorem 1). Consider a base SGD process that runs for k phases, using a
fixed learning rate throughout, while the batch size doubles after each phase. Now consider an alternative
process where the batch size is fixed but the learning rate halves after each phase, and where each phase
processes the same number of data points as in the base process. Then, the excess risk of the base process is
within a constant factor of that of the alternative process.
Corollary (Informal version of Corollary 1). Consider a base normalized SGD process (Equation 4) that
runs for k phases, where the batch size doubles after each phase while the learning rate decays by a factor of
√
2. Consider an alternative normalized SGD process where the batch size is fixed but the learning rate halves
after each phase, and where each phase processes the same number of data points as in the base process. Then,
the excess risk of the base process is within a constant factor of that of the alternative process.
1.2
Empirical Contributions
Based on the theoretical analysis, we introduce Seesaw (Algorithm 1), a learning rate and batch
size scheduler that reduces the serial runtime of LLM pre-training runs by approximately 36% via
increasing the batch size during training at specific points. We provide empirical results in Figure 1
and show that at (or below) the critical batch size, our method achieves a significant serial runtime
acceleration across several model and data scales, while maintaining the same performance as
training with cosine decay. We also empirically show that Seesaw also works even when using
AdamW with tuned weight decay in Figure 4 of Appendix C, making Seesaw a practical solution for
reducing the wall-clock time of LLM pretraining.
2
Related Work
Role of batch size in scaling.
Understanding batch size ramp up schemes during training has
been a topic of interest in recent years due to its crucial role in decreasing wall clock runtime. Various
methods of increasing the batch size have been used in common LLMs such as LLaMA (Dubey et al.,
2024; Touvron et al., 2023), Nemotron (Adler et al., 2024), OLMo (OLMo et al., 2024; Groeneveld et al.,
2024), Apertus (Team, 2025). The reason behind ramping up the batch size is to take advantage of the
parallel computation of samples and thus reducing the total number of sequential steps. However,
since increasing the batch size reduces the total number of gradient steps taken by the model during
training, there is a maximal batch size which can be achieved without becoming data inefficient,
called the critical batch size (CBS) (Erdil and Schneider-Joseph, 2024; Zhang et al., 2024; Shallue et al.,
2
105
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300M
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600M
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10000
15000
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Steps
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4.0
Validation Loss
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Batch size:
128
256
512
1024
Scheduler:
Cosine
Seesaw
Figure 1: Seesaw comparison with cosine decay in 150M (left), 300M (middle) and 600M (right)
models trained at Chinchilla scale. Seesaw matches the loss dynamics of cosine annealing in FLOPs
(top row), but achieves a significant speed up in terms of serial runtime (bottom row). Runs are
swept over learning rates and plotted at the best learning rate for cosine annealing in terms of
validation loss, at each batch size. The validation losses at the end of training are provided in Table 1.
Note the axes: the top plots are on a logarithmic scale while the bottom are on a linear scale. For
more experimental details, see Section 4.
2019; Jain et al., 2018). Recent work also looks at the effect of batch size on SGD optimization in
LLMs (Sre´ckovi´c et al., 2025; Marek et al., 2025), following previously established theoretical results
in noisy quadratic models (Zhang et al., 2019).
SGD for linear regression.
Recently, Zhang et al. (2024) have analyzed the CBS using weight
averaging in linear regression and established scaling laws as a function of data and model size. The
bias-variance analysis used by Zhang et al. (2024) has a longstanding history in the literature (Jain
et al., 2017) and has been used to study batch ramp-up schemes in SGD (Jain et al., 2018). These
rates have been recently made tight by (Zou et al., 2021; Wu et al., 2022a,b) for general spectra of the
data covariance. Recently, (Meterez et al., 2025) have used a simplified mathematical framework
for rederiving the same bounds by rotating the dynamics in the eigenbasis of the data. A similar
diagonalizing idea has also been previously used in literature by Bordelon and Pehlevan (2021); Wu
et al. (2023a,b).
Stochastic Differential Equations (SDEs).
Another point of view for studying the interaction
between batch size and learning rate in optimization is through SDEs (Li et al., 2021; Xie et al., 2020;
Compagnoni et al., 2024; Jastrz˛ebski et al., 2017). Malladi et al. (2022) study how to scale the learning
rate as a function of the batch size in adaptive algorithms, extending previous work that introduced
the square root scaling rule (Granziol et al., 2022; You et al., 2019).
Empirical work.
Scaling laws for the CBS and the optimal batch size have also been recently
observed by (Bergsma et al., 2025). In line with our conclusions regarding SGD, the linear scaling
3
rule for SGD has been observed by (Smith et al., 2017), showing that in SGD, linearly increasing the
batch size is equivalent to decreasing the learning rate. McCandlish et al. (2018) propose a metric
based on the Hessian and the noise that correlates with the CBS over training. While their proposed
metric is based on having access to the Hessian, which is prohibitive for current large-scale runs,
they find that the noise scale increases during a training run, which aligns with our theoretical
predictions. Lastly, perhaps the most similar to our work is Merrill et al. (2025), who propose a
batch size warmup scheme based on starting from a checkpoint with various multiples k of the
current batch size, and pick the largest k⋆where the loss is ϵ-close to the original loss. Based on this
methodology, they propose the scaling rule Bt+1 = 2Bt and ηt+1 =
√
2η. In contrast, we propose a
simple drop-in replacement for existing cosine schedulers, motivated rigorously by (normalized)
SGD on quadratics. Moreover, we argue that the scheduler proposed by (Merrill et al., 2025) will
lead to instabilities and divergence after a fixed number of steps, based on our theoretical analysis in
Lemma 4.
3
Seesaw: Algorithmic Details
We begin by providing an intuitive derivation of Seesaw, and the practical implementation of our
algorithm. To build intuition, consider 2 different SGD processes. In one process we take 2 steps
at learning rate η/2 and batch size B, and in the other we take 1 step at learning rate η and batch
size 2B. Consider a general smooth loss function L(x) and let g0 = ∇L(x0). Then, through a simple
Taylor expansion up to first order in η, we have the loss of the (η, 2B) process and the loss of the 2
half step process (η/2, B) respectively:
L(x1) = L(x0) −ηg⊤
0 (g0 + ξ′) + O(η2)
Cov(ξ′) = σ2
2BId
L(x2) = L(x0) −η
2g⊤
0 (2g0 + ξ0 + ξ1) + O(η2)
Cov(ξi) = σ2
B Id.
Note that the 2 processes are equivalent up to first order both in the deterministic part and in the
noise terms up to O(η2), an argument which has been previously shown by Malladi et al. (2022). We
formalize this SGD intuition in Theorem 1 and extend it to normalized SGD as an analytical proxy
to Adam in Corollary 1.
3.1
Extension to Normalized SGD
From the previous subsection, intuitively, for SGD, cutting the learning rate by a factor of α should
be equivalent to increasing the batch size by a factor of α. To design a practical training algorithm
based on the SGD analysis and arrive at Seesaw, we begin with the Adam update rule and simplify
until we obtain normalized SGD (NSGD), which is a commonly used tractable analytical proxy
for Adam (Jelassi et al., 2022; Zhao et al., 2024; Xie et al., 2024). Suppose we are optimizing over
parameters θ and denote the gradients at each time step gt. Then, for learning rate η and ignoring
the bias correction, the parameter update for Adam is given by:
mt = β1mt−1 + (1 −β1)gt
(1)
vt = β2vt−1 + (1 −β2)g2
t
(2)
θt = θt −η
mt
√vt + ϵ
(3)
where mt is the momentum term, vt is the second moment term, β1, β2 are their respective exponen-
tial decay rates, and ϵ ensures stability. For NSGD, we approximate the per-coordinate preconditioner
of Adam will a single scalar preconditioner, set β1 = β2 = 0 and replace the denominator with the
4
true expected value of the squared gradient norms over the population:
θt = θt −η
gt
pE∥gt∥2
(4)
Algorithm 1: Seesaw
Inputs: η0 (initial learning rate), B0 (initial
batch size), α > 1 (step decay factor), S (an
array of steps at which input scheduler cuts η
by α), T (total training steps)
η ←η0, B ←B0
for t ←1 to T do
if t ∈S then
η ←η/√α;
B ←B · α;
end
end
Equation 4 describes the NSGD update
rule, which is a crucial component of de-
signing Seesaw. While the full analysis is
deferred to Appendix B, the expected gra-
dient norms can be decomposed as:
E∥gt∥2 = mean + variance
(5)
where the variance scales down with the
batch size. To design Seesaw, we assume
that the variance dominates the expected
gradient squared norms (Assumption 2),
and we motivate why this assumption is
reasonable in Appendix B. This step re-
duces (up to constant factors) the NSGD
update rule to SGD with a rescaled learn-
ing rate (Equation 7), allowing us to extend
risk equivalence to NSGD (Corollary 1) in Section 5. For NSGD, informally, Corollary 1 shows that
any learning rate cut by a factor of α and batch size increase by a factor of β are equivalent as long
as α√β is held constant. We further empirically comapre Seesaw with other possible schedulers in
Figure 5.
3.2
Achievable Speedups
While our theory is established for step decay schedulers, in practice we approximate cosine decay
with a step decay by considering a decay of α, and passing the times (as measured in tokens) where
the cosine would cut the learning rate by α as input to Seesaw. Then, at these points, we instead
cut the learning rate by √α and increase the batch size by β, where the schedulers are equivalent in
terms of loss as long as we keep the product α√β fixed. However, we cannot arbitrarily increase the
batch size at time t and expect the risk to match the underlying process. Lemma 4 quantifies this
and the main takeaway is stated below:
Remark 1. The most aggressive ramp up scheme we can use is given by α = √β. (for a formal argument see
Lemma 4)
In Section 4.1 we empirically verify this constraint and show that α = √β is the most aggressive
scheme we can choose without divergence. The corresponding algorithm is provided in Algorithm 1.
At the most aggressive limit, we can compute the theoretical speedup we would hope to achieve
where the standard scheduler is the cosine decay.
Lemma 1 (Maximum Theoretical Speedup under Cosine Decay). Consider a baseline training process of
T total steps using a constant batch size and a cosine learning rate schedule η(t) = η0 cos( πt
2T ). An equivalent
process run with a batch ramping schedule like Seesaw, in the continuous limit 1, will have a total of 2T
π steps.
This yields a maximum theoretical serial runtime reduction of (1 −2
π) ≈36.3%.
1In the continuous-time limit, we consider an aggressive (non-divergent) batch size ramp that maintains the relationship
α = √β. Consequently, the total number of sequential steps is given by the integral of the normalized learning rate
schedule: R T
0
η(t)
η0 dt = R T
0 cos( πt
2T )dt = 2T
π .
5
Lemma 1 provides an upper bound on the acceleration from Seesaw. In Figure 1, we can indeed see
that the total number of steps are reduced approximately by 36% as predicted.
4
Empirical Findings
In this section, we present the experimental details and methodology for evaluating Seesaw. We
denote by D the dataset size, N the number of parameters.
B=128
B=256
B=512
B=1024
150M (cosine)
3.0282
3.0353
3.0696
3.1214
150M (Seesaw)
3.0208
3.0346
3.0687
3.1318
300M (cosine)
2.8531
2.8591
2.8696
2.9369
300M (Seesaw)
2.8452
2.8561
2.8700
2.9490
600M (cosine)
-
2.6904
2.6988
2.7128
600M (Seesaw)
-
2.6883
2.6944
2.7132
Table 1: Final validation losses picked at the best learning rate (for the cosine annealing scheduler)
for each batch size, for α = 1.1. Note that the dynamics match robustly across the 2 schedulers when
trained at CBS.
Model and dataset.
We pretrain models of size 150M, 300M and 600M (non-embedding) pa-
rameters at Chinchilla scaling i.e. D = 20N (Hoffmann et al., 2022). We use the OLMo (Groen-
eveld et al., 2024) codebase to train all of our models. For each experiment, we do learning rate
warmup for 10% of the total amount of tokens, followed by learning rate decay following co-
sine scheduling or Seesaw. We report the architectural details of each model as a tuple (depth,
# heads, width), and thus we have for 150M (12, 16, 1024), 300M (24, 16, 1024) and for 600M
(24, 22, 1408). Unless mentioned otherwise, each model is trained using AdamW, with weight
decay λ = 0.0 (with the exception of the weight decay experiments in Appendix C, where we
sweep over λ ∈{0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0}), β1 = 0.9, β2 = 0.95, ϵ = 10−8.
For each run we sweep over learning rates η ∈{0.001, 0.003, 0.01, 0.03} and initial batch sizes
B ∈{128, 256, 512, 1024}, at sequence length L = 1024. Similar to the OLMo training codebase, we
enable z-loss during training, but provide ablations over it in Appendix E showing that it does not
affect the model performance at our scales. All our models are pretrained on the C4 dataset (Raffel
et al., 2020), tokenized with the T5 tokenizer.
Experimental design.
We compare Seesaw with cosine annealing by training models at the
critical batch size (CBS) B⋆, approximated based on (Zhang et al., 2024), namely B⋆≈256k (150M),
B⋆≈512k (300M) and B⋆≈1024k (600M) tokens. The main results comparing Seesaw and cosine
annealing at equal FLOPs are provided in Figure 1. The precise final losses obtained by the 2
schedulers are provided in Table 1.
4.1
Can We Do Better?
Recall that based on Corollary 1 and Lemma 4, we have a family of equivalent schedules in NSGD,
given by a fixed product α√β, under the constraint that α ≥√β. Ideally, we would like to make β as
large as possible, since this would lead to larger batch sizes, and thus assuming enough devices are
available, the lowest serial runtime. Crucially, the constraint prevents us from using a too agressive
6
2.00
2.25
2.50
2.75
3.00
3.25
Tokens
1e9
3.025
3.050
3.075
3.100
3.125
Validation Loss
B=256
2.00
2.25
2.50
2.75
3.00
3.25
Tokens
1e9
3.05
3.10
3.15
Validation Loss
B=512
α = 2, β = 1
α = 23/4, β =
p
2
α =
p
2, β = 2
α = 21/4, β = 23/2
α = 1, β = 4
Figure 2: 150M models trained at batch size 256 (left) and 512 (right) with α and β values following
the line of equivalence α√β = 2 described in Table 2. Note that the target to match is the blue trace,
and our theory (Lemma 4) predicts that the red and purple traces should not match the baseline
(blue trace) due to instabilities.
batch size scheduler. In this section, we empirically verify our theoretical prediction by testing
schedulers positioned at various points on the (α, β) axis.
α
2
23/4
21/2
21/4
1
β
1
21/2
2
23/2
22
Table 2: α, β values used to test the extreme values of the equivalence.
Namely, we train 150M models at fixed batch size and Chinchilla scale, and we approximate cosine
decay with a step decay scheduler that halves the learning rate at the token counts where the cosine
schedule’s learning rate would halve. This gives us the baseline α = 2 and β = 1, with the product
α√β = 2. Based on the theoretical constraint and the equivalence line, the most aggressive scheduler
we could use is α =
√
2 and β = 2. To validate our hypothesis, we compare with α = 1 and β = 4,
and points in between at geometric intervals. Table 2 gives an overview of the experimental design,
and Figure 2 shows that indeed the most aggressive schedules tend to underperform.
4.2
When Does Assumption 2 Fail?
Up to this point, a crucial assumption for the development of our theory and the design of Seesaw
has been Assumption 2. Recall that Assumption 2 states that the expected gradient norms – namely,
the denominator of the NSGD update step, is dominated by the additive noise. Intuitively, since
the noise variance decreases with the batch size as O(1/B), one can see that past a certain batch the
additive noise will become small, and thus Assumption 2 will fail. In Figure 3, we can see that at
sufficiently large batch sizes, indeed Seesaw starts to perform worse as compared to the underlying
cosine schedule. The first hypothesis could be that it is still possible to match the underlying
schedule, but with a learning rate equivalence as given by mean dominating in the denominator. As
mean does not scale with batch size, therefore, using the equivalence schedule of SGD could be a
promising candidate. We explore this option in Figure 3, and it turns out that this schedule performs
even worse than the Seesaw schedule. We hypothesise that beyond a certain batch size, it is not
7
109
3.2
3.4
3.6
3.8
4.0
Validation Loss
B=1024
109
3.2
3.4
3.6
3.8
4.0
Validation Loss
B=2048
109
Tokens
3.5
4.0
4.5
5.0
5.5
6.0
Validation Loss
B=4096
109
Tokens
3.5
4.0
4.5
5.0
5.5
6.0
Validation Loss
B=8192
Scheduler:
Cosine
Fixed LR
Seesaw
Figure 3: 150M models trained past CBS (roughly 256), at batch sizes 1024, 2048, 4096 and 8192, for 3
schedulers: cosine decay (blue), constant learning rate with increasing batch size based on Seesaw
(orange) and Seesaw (green). Note that none of the proposed schedules is able to match the cosine
curve, with the discrepancy increasing as the batch size grows more.
possible to match the performance of learning rate decay by any equivalent batch size ramp up for
Adam or normalized SGD, which we motivate using the following toy example.
For simplicity, we look at NGD (normalized gradient descent) in 1D, for the quadratic loss L(x) =
1
2hx2, where x, h ∈R and h ≥0. Training with NGD, we have the loss gradients with respect to the
parameters and the update rule:
∇xL = hx
xt+1 := xt −ηh sign(xt)
where sign(xt) =
xt
|xt|. Note that if xt > 0, then xt+1 = xt −ηh and if xt < 0, then xt+1 = xt + ηh,
implying that the model does not reach the minimizer and instead converges to a stable cycle of
O(ηh) around the minimizer. In order to escape this stable cycle and reach the minimizer, it is thus
necessary to decay the learning rate. Therefore, if we slightly relax the setup and think of large batch
training as being close to NGD regime, we can see that further increasing the batch size does not
change the dynamics. Therefore, past a certain batch size, learning rate decay is crucial for achieving
a lower loss with NGD.
8
5
Theoretical Analysis
In this section we introduce the main theoretical contributions of our work. Namely, under mild
assumptions, we establish a formal equivalence between learning rate decay and batch size ramp up
in SGD and normalized SGD.
Setup and notation.
We use the notation f ≲g to mean that there exists some constant a > 0 such
that f(x) ≤ag(x) for any x. We also use the notation f ≂g if f(x) ≲g(x) ≲f(x) for all x. We denote
the samples (x, y) where x ∈Rd and y ∈R, with the distribution and risk:
x ∼N(0, H)
y|x ∼N(⟨w⋆, x⟩, σ2)
R(w) = 1
2E(⟨w, x⟩−y)2
where the expectation is over the (x, y), w⋆is the minimizer, and σ2 is the variance of the additive
noise. We also use R(wt, η) to denote the risk at time t for a process trained with η, but we drop
the η parameter when it is clear from context. We consider step decay schedules for the learning
rate, where, the learning rate in the kth phase is denoted by ηk and Pk denotes the total number
of data samples used in the kth phase. Similarly, for batch ramp schedules, Bk denotes the batch
size in the kth phase. For discussion, we will use the bias-variance decomposition terminology of
risk (Jain et al., 2018, 2017; Zou et al., 2021; Wu et al., 2022a,b; Meterez et al., 2025). Informally,
bias corresponds to the risk of the averaged iterates, while variance corresponds to the noise in the
iterates, and R(wt) = biast + variancet. We will denote the stochastic gradient at time t by gt and
let E∥gt∥2 represent its expected squared norm under the population distribution.
5.1
Main Results
In this section, we first introduce the main assumptions and discuss their implications, followed by
the main theoretical results. Our first assumption states that the risk of the SGD process is bounded.
Assumption 1 (Bounded risk.). Consider a SGD process with a given scheduling scheme for the batch size
and learning rate. Let t0 denote the first time when the scheduler changes either the learning rate or the batch
size. Then, we assume that there exists a constant c > 1 such that R(wt) ≤cσ2 for all t > t0.
In general, we expect every “well tuned” scheduler to start cutting when R(wt0) ≲σ2, as we want
to minimize the bias component of the risk before cutting down the learning rate to reduce noise
in the iterates. Moreover, for a well-behaved schedule, as we expect the risk to decrease over time,
this condition should hold throughout the process. Our second assumption states that the expected
gradient squared norms of the NSGD update rule are dominated by the additive noise term.
Assumption 2 (Variance dominated.). Assume that E∥gt∥2 ≂σ2
Bt .
Under Assumption 2, the NSGD process effectively reduces to SGD with a rescaled learning rate
(Equation 7), up to constant factors. Based on the previously established assumptions, we can now
state the equivalence result. We use the notation R(ηt, Bt) to denote the risk at time t of an SGD
process trained with the learning rate scheduler ηt and batch size scheduler Bt.
Theorem 1 (SGD Equivalence). Fix
0.01
Tr(H) ≥η > 0, B > 0, and parameters α1, α2 > 1, β1, β2 > 1 with
α1β1 = α2β2. Define the two phase-indexed schedules
(ηk, Bk) :=
η α−k
1 , B βk
1
,
(η′
k, B′
k) :=
η α−k
2 , B βk
2
,
k = 0, 1, 2, . . .
and run two SGD procedures in phases k = 0, 1, . . . so that, in phase k, each procedure processes the same
number of samples (possibly depending on k) under its respective schedule. Let R(ηk, Bk) and R(η′
k, B′
k)
9
denote the (population) risk of the two procedures at the end of phase k. If Assumption 1 holds (for both
procedures) with constant c, then
R(1.01 · η′
k, B′
k) ≲c R(ηk, Bk) ≲c R(η′
k, B′
k),
where R(λ·η′
k, B′
k) denotes the risk of the second procedure when its entire learning-rate schedule is multiplied
by a uniform factor λ > 0, and A ≲c B means A ≤C(c) B for a numerical constant C(c) depending only on
c (and absolute constants).
We defer the full proof to Appendix A.1. Now, we extend this result to Normalized SGD. Under
Assumption 2, NSGD reduces to SGD with a rescaled learning rate ˜η ≂η
√
B
σ√
Tr(H) (Equation 7).
Consequently, we can extend Theorem 1 to the normalized SGD case. We formalize this in the
following corollary:
Corollary 1 (Normalized SGD Equivalence). Fix
0.01
Tr(H) ≥η > 0, B > 0, and parameters α1, α2 > 1,
β1, β2 > 1 with α1
√β1 = α2
√β2. Define the two phase-indexed schedules
(ηk, Bk) :=
η α−k
1 , B βk
1
,
(η′
k, B′
k) :=
η α−k
2 , B βk
2
,
k = 0, 1, 2, . . .
and run two normalized SGD procedures in phases k = 0, 1, . . . so that, in phase k, each procedure processes
the same number of samples (possibly depending on k) under its respective schedule. Let R(ηk, Bk) and
R(η′
k, B′
k) denote the (population) risk of the two procedures at the end of phase k. If Assumption 1 and 2
holds (for both procedures) with constant c, then
R(1.01 · η′
k, B′
k) ≲c R(ηk, Bk) ≲c R(η′
k, B′
k),
where R(λ·η′
k, B′
k) denotes the risk of the second procedure when its entire learning-rate schedule is multiplied
by a uniform factor λ > 0, and A ≲c B means A ≤C(c) B for a numerical constant C(c) depending only on
c (and absolute constants).
6
Discussion and Conclusions
In this work, we provide a rigorous analysis of batch ramp through the lens of noisy linear regression,
offering theoretical insight into a practice that has thus far been guided largely by empirical heuristics.
Building upon this analysis, we introduce Seesaw (Algorithm 1), a principled batch size scheduling
algorithm that can serve as a drop-in replacement for commonly used learning-rate schedules in
adaptive optimizers such as Adam.
Our empirical evaluation demonstrates that Seesaw matches the performance of standard cosine
annealing schedules while reducing the serial training runtime by approximately 36%. These results
indicate that dynamically balancing the learning rate and batch size provides an effective means to
accelerate training without compromising performance.
Acknowledgements
AM would like to thank Jacob Zavatone-Veth and Alex Damian for helpful discussions. The authors
would also like to thank Max Shad and Bala Desinghu for their help with the cluster. AM, DM
acknowledge the support of a Kempner Institute Graduate Research Fellowship. CP is supported
by an NSF CAREER Award (IIS-2239780), DARPA grants DIAL-FP-038 and AIQ-HR00112520041,
the Simons Collaboration on the Physics of Learning and Neural Computation, and the William F.
Milton Fund from Harvard University. AM, SK and DM acknowledge that this work has been made
possible in part by a gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner
10
Institute for the Study of Natural and Artificial Intelligence. SK and DM acknowledge support from
the Office of Naval Research under award N0001422-1-2377 and the National Science Foundation
Grant under award #IIS 2229881. DM is also supported by a Simons Investigator Fellowship, NSF
grant DMS-2134157, DARPA grant W911NF2010021,and DOE grant DE-SC0022199.
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14
A
Proofs for Section 5
A.1
Preliminaries
We take as a convention for eigenvalues ordering λmax = λ1 ≥λ2 ≥· · · > 0. For two matrices A
and B we use the notation A ⪯B to denote that B −A is positive semi-definite (PSD). We denote
⟨u, v⟩for the inner product between u and v. Moreover, with a slight abuse of notation, we use the
notation ≤as elementwise comparison, namely u ≤v if ui ≤vi for all i and A ≤B if Aij ≤Bij for
all i, j. To simplify the analysis, we will follow the approach of Meterez et al. (2025) and work in the
eigenbasis of the data covariance H. For the sake of completeness, we restate the main derivation for
the bias and variance iterates in the case of constant learning rate and constant batch size, starting
from the SGD update rule:
wt+1 −w⋆=
I −η
B
B
X
i=1
xix⊤
i
(wt −w⋆) −η
B
B
X
i=1
xiϵi
=⇒Σt+1 = Σt −ηΣtH −ηHΣt + η2
1 + 1
B
HΣtH + η2
B Tr(HΣt)H + η2
B σ2I
=⇒Mt+1 = Mt −ηMtΛ −ηΛMt + η2
1 + 1
B
ΛMtΛ + η2
B Tr(ΛMt)Λ + η2
B σ2I
(6)
where in the last equation Mt = QΣtQ⊤is the iterate covariance matrix rotated in the eigenbasis of
H. Since we can write the excess risk as:
R(wt) −R(w⋆) = 1
2Tr(ΛMt) = 1
2⟨λ, mt⟩
where mt = diag(Mt), it suffices to push a diag operator through equation equation 6. Finally, we
get:
mt+1 =
"
I −2ηΛ + η2
1 + 1
B
Λ2 + η2
B λλ⊤
#
|
{z
}
A
mt + η2σ2
B λ = Atm0 + η2σ2
B
t−1
X
i=0
Aiλ
where f
mt := Atm0 and mt := η2σ2
B
Pt−1
i=0 Aiλ are the bias and variance iterates respectively.
Before we begin proving the main statements, we introduce several helpful lemmas that we will use.
Lemma 2. For η ≤0.01/Tr(H) and α ≥1, we have the elementwise inequality:
αk
η 1 ≥
I −
I −η
αk Λ
2!−1
λ ≥αk
2η 1
Proof. We have:
I −
I −η
αk Λ
2!−1
=
I −
I + η2
α2k Λ2 −2 η
α2 Λ
!!−1
=
η
αk Λ
2 −η
αk Λ
−1
≥
2η
αk Λ
−1
Note that trivially we also have the other direction by noticing that
1
2−η
αk λ ≤1. Multiplying by λ
gives us the conclusion.
15
Lemma 3. For η ≤0.01/Tr(H) and α1, α2, β1, β2 ≥1 such that α1β1 = α2β2 and α1 ≤α2, we have:
I −1.01 η
αk
2
Λ
2βk
1
⪯
I −η
αk
1
Λ
2βk
2
⪯
I −η
αk
2
Λ
2βk
1
.
Proof. RHS bound. Since both sides are diagonal matrices, it suffices to prove the scalar inequality
for every x = ηλi:
1 −x
αk
1
2βk
2
≤
1 −x
αk
2
2βk
1
.
Taking logarithms and defining
f(x) = 2βk
2 log(1 −x/αk
1)
2βk
1 log(1 −x/αk
2) = αk
1 log(1 −x/αk
1)
αk
2 log(1 −x/αk
2) = g(α1)
g(α2),
where g(y) = y log(1 −x/y). For 0 < x < 1 and y > 1, g(y) is monotonically increasing, so for
α1 ≤α2, we have g(α1) ≤g(α2) and hence g(α1)/g(α2) ≥1 (since g(α2) < 0). Thus f(x) ≥1, which
proves the RHS inequality.
LHS bound. Similarly, we use the scalar inequality and the bounds
−x −x2
2 ≥ln(1 −x) ≥−x −x2.
Since ln(·) is monotone, we apply it to both sides:
βk
1 ln
1 −1.01
αk
2
x
≤βk
1
−1.01
αk
2
x −1.012
2α2k
2
x2
!
,
βk
2 ln
1 −1
αk
1
x
≥βk
2
−1
αk
1
x −
1
α2k
1
x2
.
It suffices to prove that:
βk
1
−1.01
αk
2
x −1.012
2α2k
2
x2
!
≥βk
2
−1
αk
1
x −
1
α2k
1
x2
.
Using β1
α2 = β2
α1 and β1
α2
2 =
β2
α1α2 , we obtain:
1
αk
1
(1.01) +
1
2αk
1αk
2
(1.01)2x −1
αk
1
−
1
α2k
1
x ≥0,
⇐⇒x ≤
0.01
1
αk
1 −1.012
2αk
2
.
Using α1 ≤α2, we get
x ≤αk
1 · 0.01
1 −1.012
2
,
which holds automatically under η ≤0.01/Tr(H) and x = ηλi. This concludes the proof.
A.2
Proofs of Main Statements
Proof of Theorem 1. Consider 2 processes: process 1 will have a learning rate step decay factor of α1
and a batch size ramp up factor of β1 and process 2 will have α2 and β2 respectively. Define the
transition matrices:
16
Ak =
"
I −η
αk
1
Λ
2
+
η2
Bα2k
1 βk
1
(Λ2 + λλ⊤)
#
Ck =
"
I −η
αk
2
Λ
2
+
η2
Bα2k
2 βk
2
(Λ2 + λλ⊤)
#
Denote process 1 as mk(η) and process 2 as rk(η) where they depend on the base learning rate η -
note that we skip the indexing on η when it is clear from context. In order to keep both the per stage
data count, mk does βk
2Pk steps per stage, and rk does βk
1Pk steps per stage. We begin by establishing
the upper bound first. Note that we assume that α1β1 = α2β2, and without loss of generality due to
symmetry, that β1 ≥β2 (and consequently α1 ≤α2).
Upper bound.
Before we begin, we introduce the idea behind the proof. We define Mk = βk
1Pk
and Nk = βk
2Pk. The derivation proceeds by unrolling the recurrence first over a single step, then
over βk
2 steps, and finally over Pk stages.
mN1:k ≤AkmN1:k−1 +
η2σ2
Bα2k
1 βk
1
λ
≤
I −η
αk
1
Λ
2
mN1:k−1 + (1 + 2c) η2σ2
Bα2k
1 βk
1
λ,
which follows from Assumption 1.
mN1:k ≤
I −η
αk
1
Λ
2βk
2
mN1:k−βk
2 + (1 + 2c) η2σ2
Bα2k
1 βk
1
βk
2 −1
X
i=0
I −η
αk
1
Λ
2i
λ
≤
I −η
αk
1
Λ
2βk
2
mN1:k−βk
2 + (1 + 2c) η2σ2
Bα2k
1 βk
1
"
I −
I −η
αk
1
Λ
2βk
2
# "
I −
I −η
αk
1
Λ
2#−1
λ.
Applying Lemma 2, we have:
mN1:k ≤
I −η
αk
1
Λ
2βk
2
mN1:k−βk
2 + (1 + 2c)
ησ2
Bαk
1βk
1
"
I −
I −η
αk
1
Λ
2βk
2
#
1
≤
I −η
αk
1
Λ
2βk
2
mN1:k−βk
2 + 2(1 + 2c)η2σ2
B
β2
α2
1β1
k
λ.
By Lemma 3, we can replace the term with one involving (α2, β1):
mN1:k ≤
I −η
αk
2
Λ
2βk
1
mN1:k−βk
2 + 2(1 + 2c)η2σ2
B
β2
α2
1β1
k
λ.
Following, we can unroll over Pk:
mN1:k ≤
I −η
αk
2
Λ
2Mk
mN1:k−1 + 2(1 + 2c)η2σ2
B
β2
α2
1β1
k Pk−1
X
i=0
I −η
αk
2
Λ
2βk
1 i
λ.
17
Finally, recursively unrolling across k yields:
mN1:k ≤
" k
Y
s=1
I −η
αs
2
Λ
2Ms#
m0
+ 2(1 + 2c)η2σ2
B
k
X
r=1
1
α1α2
r "
k
Y
s=r+1
I −η
αs
2
Λ
2Ms# Pr−1
X
i=0
I −η
αk
2
Λ
2βr
1i
λ.
For the lower bound, we follow a similar strategy, by bounding the term λλ⊤≥0:
rM1:k ≥
I −η
αk
2
Λ
2
rM1:k−1 +
η2σ2
Bα2k
2 βk
2
λ
≥
I −η
αk
2
Λ
2·βk
1
rM1:k−βk
1 +
η2σ2
Bα2k
2 βk
2
βk
1 −1
X
i=0
I −η
αk
2
Λ
2i
λ
=
I −η
αk
2
Λ
2·βk
1
rM1:k−βk
1 +
η2σ2
Bα2k
2 βk
2
"
I −
I −η
αk
2
Λ
2·βk
1
# "
I −
I −η
αk
2
Λ
2#−1
λ
≥
I −η
αk
2
Λ
2·βk
1
rM1:k−βk
1 + 1
2
ησ2
Bαk
2βk
2
"
I −
I −η
αk
2
Λ
2·βk
1
#
1
Lemma 2
≥
I −η
αk
2
Λ
2·βk
1
rM1:k−βk
1 + 1
4
η2σ2
B
β1
α2
2β2
k
λ
≥
I −η
αk
2
Λ
2·Mk
rM1:k−1 + 1
4
η2σ2
B
β1
α2
2β2
k Pk−1
X
i=0
I −η
αk
2
Λ
2βk
1 i
λ
≥
" k
Y
s=1
I −η
αs
2
Λ
2·Ms#
r0
+ 1
4
η2σ2
B
k
X
r=1
1
α1α2
r "
k
Y
s=r+1
I −η
αs
2
Λ
2·Ms# Pr−1
X
i=0
I −η
αk
2
Λ
2βr
1i
λ
Note that the bias terms are equal erM1:k = f
mN1:k, and the variance terms are mN1:k ≥4(1 + 2c)rM1:k.
Dotting the terms into λ gives us the upper bound from Theorem 1.
Lower bound.
We now turn our attention towards proving the lower bound in Theorem 1. Note
that the bias terms have an exponentially decaying dominating term. In order to obtain an inequality
in the reverse direction for these terms, we compare m(η) with r(1.01η). We begin with lower
bounding m:
18
mN1:k(η) ≥
I −η
αk
1
Λ
2
mN1:k−1 +
η2σ2
Bα2k
1 βk
1
λ
≥
I −η
αk
1
Λ
2βk
2
mN1:k−βk
2 + 1
4
η2σ2
B
1
α1α2
k
λ
≥
I −η
αk
1
Λ
2Nk
mN1:k−1 + 1
4
η2σ2
B
1
α1α2
k Pk−1
X
i=0
I −η
αk
1
Λ
2βk
2 i
λ
≥
" k
Y
s=1
I −η
αs
1
Λ
2Ns#
m0
+ 1
4
η2σ2
B
k
X
r=1
1
α1α2
r "
k
Y
s=r+1
I −η
αs
1
Λ
2Ns# Pr−1
X
i=0
I −η
αk
1
Λ
2βr
2i
λ
Now we need to establish an upper bound for r(1.01η). We follow a similar analysis as we did for
the upper bound subsection:
rM1:k(1.01η)
≤
I −1.01η
αk
2
Λ
2
rM1:k−1 + 1.012 · (1 + 2c) η2σ2
Bα2k
1 βk
1
λ
≤
I −1.01η
αk
2
Λ
2βk
1
rM1:k−βk
1 + 2 · 1.012 · (1 + 2c)η2σ2
B
1
α1α2
k
λ
≤
I −η
αk
1
Λ
2βk
2
rM1:k−βk
1 + 2 · 1.012 · (1 + 2c)η2σ2
B
1
α1α2
k
λ
Lemma 3
≤
I −η
αk
1
Λ
2Nk
rM1:k−1 + 2 · 1.012 · (1 + 2c)η2σ2
B
1
α1α2
k Pk−1
X
i=0
I −η
αk
1
Λ
2βk
2 i
λ
≤
" k
Y
s=1
I −η
αs
1
Λ
2Ns#
r0
+ 2 · 1.012 · (1 + 2c)η2σ2
B
k
X
r=1
1
α1α2
r "
k
Y
s=r+1
I −η
αs
1
Λ
2Ns# Pr−1
X
i=0
I −η
αk
1
Λ
2βr
2i
λ
Comparing the bias and variance terms gives us the conclusion.
19
B
Normalized SGD Analysis
Under the setup introduced in Section 5, we have the update rule for normalized SGD is:
wt+1 = wt −η
1
p
E∥gt∥2 gt
where gt = 1
B
PB
i=1 g(i)
t
for i indexing the sample and batch size B.
For MSE and y = (w⋆)⊤x + ϵ, the loss is:
L(wt) = 1
2B
B
X
i=1
(w⊤
t x(i) −y(i))2
= 1
2B
B
X
i=1
((wt −w⋆)⊤x(i) −ϵ)2
If we look at the risk at time t we have:
R(wt) = 1
2B
B
X
i=1
E[(wt −w⋆)⊤x(i)x(i),⊤(wt −w⋆) + ϵ2]
= 1
2B
B
X
i=1
E[(wt −w⋆)⊤x(i)x(i),⊤(wt −w⋆)] + σ2
2
= 1
2E[(wt −w⋆)⊤xx⊤(wt −w⋆)] + σ2
2
= 1
2Tr(HΣt) + σ2
2
So the risk is equal to:
R(wt) = 1
2Tr(HΣt) + σ2
2
=⇒R(wt) −R(w⋆) = 1
2Tr(HΣt)
Analyzing the gradients.
Taking the gradient for 1 sample:
g(i)
t
:= ∇wtL = (w⊤
t x(i) −y(i))x(i) = x(i)(x(i))⊤(wt −w⋆) −ϵx(i)
So we have:
gt = 1
B
B
X
i=1
x(i)(x(i))⊤(wt −w⋆) −1
B
B
X
i=1
ϵx(i)
Moving forwards, we need to calculate the term in the denominator. Skipping the time index in
order to simplify the notation, we have:
20
E∥g∥2 = 1
B2 E
B
X
i,j=1
g(i),⊤g(j)
= 1
B2
B
X
i=j
E[g(i),⊤g(i)] + 1
B2
B
X
i̸=j
E[g(i),⊤g(j)]
= 1
B E∥g(i)∥2 +
1 −1
B
E[g(i)]⊤E[g(j)]
First term.
If we look at each of these 2 terms we have:
E∥g(i)∥2 = E[(wt −w⋆)⊤xx⊤xx⊤(wt −w⋆)] + σ2E[x⊤x]
= E[Tr(xx⊤xx⊤(wt −w⋆)(wt −w⋆)⊤)] + σ2Tr(E[xx⊤])
= Tr(E[xx⊤xx⊤]Σt) + σ2Tr(H)
= Tr((2H2 + HTr(H))Σt) + σ2Tr(H)
= 2Tr(H2Σt) + Tr(H)Tr(HΣt) + σ2Tr(H)
Second term.
For the other term, let δt = wt −w⋆and we have:
E[g(i)]⊤E[g(j)] = E[x(i)(x(i))⊤δt]⊤E[x(j)(x(j))⊤δt] + σ2δijTr(H)
= E[δt]⊤H2E[δt] + σ2δijTr(H)
= Tr(H2E[δt]E[δt]⊤)
i ̸= j
So the denominator is equal to:
E∥gt∥2 = 1
B
h
2Tr(H2Σt) + Tr(H)Tr(HΣt) + σ2Tr(H)
i
+
1 −1
B
Tr(H2E[δt]E[δt]⊤)
= σ2
B Tr(H) + 1
B
h
2Tr(H2Σt) + Tr(H)Tr(HΣt)
i
+
1 −1
B
Tr(H2E[δt]E[δt]⊤)
Since E[δt] decays to 0 exponentially fast, and Σt ⪯O(σ2I) (Lemma 8) (Jain et al., 2018), then for
large enough t, we have that the gradient norms are dominated by the additive variance, which
is captured in Assumption 2. For the remainder of this paper we will assume t is large enough
for this assumption to hold, and with a slight abuse of notation we will write = (as opposed to ≈):
E∥gt∥2 = σ2
B Tr(H).
Under Assumption 2, we have the following update rule:
wt+1 = wt −η
√
B
σ
pTr(H)∇wtL
(7)
Note that this is simply SGD with a learning rate ˜η = η
√
B
σ√
Tr(H).
B.1
How Aggressive Can the Scheduler Be?
In this section we provide a short lemma explaining what is the most aggressive scheduler we could
possibly used, based on hard contraints on α, β.
21
Lemma 4 (Divergence conditions.). Suppose we are in the same setting as Corollary 1. For a fixed initial
learning rate η, the training dynamics diverge asymptotically if α < √β as the training time goes to infinity,
for α and β constants independent of time.
Proof. To see this, we focus on the scaling of ˜ηk ≂η
√
β
α
k
. Note that if √β > α, then at every cut
we are effectively increasing the learning rate. Thus, there must exist k > 0 such that ˜ηk > ηmax,
where ηmax is the maximum convergent learning rate for SGD (Jain et al., 2018; Wu et al., 2022b),
leading to divergence.
22
C
Weight Decay
In this section we provide experiments on 150M models trained with AdamW, sweeping weight de-
cay λ ∈{0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0} and learning rate η ∈{0.001, 0.003, 0.01, 0.03},
and the rest of the parameters are as explained in Section 4. For every figure we pick the best (η, λ)
pair on cosine annealing, and we use the values for Seesaw. Across all batch sizes (128, 256, 512),
the optimal (η, λ) pair from the sweep turned out to be (η, λ) = (0.003, 0.0001). Figure 4 shows the
results:
2 × 109
3 × 109
Tokens
3.00
3.05
3.10
3.15
3.20
3.25
Validation Loss
Batch Size: 128
Scheduler
Seesaw
Cosine Decay
2 × 109
3 × 109
Tokens
3.00
3.05
3.10
3.15
3.20
3.25
Validation Loss
Batch Size: 256
2 × 109
3 × 109
Tokens
3.00
3.05
3.10
3.15
3.20
3.25
3.30
Validation Loss
Batch Size: 512
Figure 4: 150M experiments with weight decay across different batch sizes (128, 256, 512) for cosine
annealing and Seesaw, for learning rate and weight decay values (η, λ) = (0.003, 0.0001). Note that
the losses overlap during training. We provide the final validation losses in Table 3.
Table 3 shows the final validation losses:
B=128
B=256
B=512
150M (cosine)
3.0125
3.0220
3.0559
150M (Seesaw)
3.0027
3.0210
3.0588
Table 3: Final validation losses picked at the best learning rate (for the cosine annealing scheduler)
for each batch size, for α = 1.1 and weight decay 0.003. Note that the dynamics match robustly.
23
D
Comparison to other schedulers
We compare our scheme with other schedulers in this section.
109
2 × 109
3 × 109
Tokens
3.00
3.05
3.10
3.15
3.20
Validation Loss
Batch Size: 128
109
2 × 109
3 × 109
Tokens
3.00
3.05
3.10
3.15
3.20
Validation Loss
Batch Size: 256
η, B · 2
η, B · 4
η/2, B
η/
p
2, B · 2
Figure 5: 150M models trained with 4 different schedules, at CBS (right) and just below (left). Blue
trace keeps learning rate fixed and doubles batch size, orange trace keeps learning rate fixed and
quadruples batch size, green trace halves learning rate at fixed batch size, and red trace is Seesaw.
Note that the naive scheduling (blue) severely underperforms the baseline (green) and Seesaw (red).
24
E
Auxiliary Losses
In this section we ablate over the effect of z-loss on the training dynamics (OLMo et al., 2024). We
observe no difference in the training stability of our models at 150M scale in Figure 6:
105
106
107
108
109
Tokens
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
Validation Loss
Learning rate: 0.003
105
106
107
108
109
Tokens
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
Validation Loss
Learning rate: 0.01
105
106
107
108
109
Tokens
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
Validation Loss
Learning rate: 0.03
Batch size:
128
256
512
Z-Loss:
False
True
Figure 6: 150M models trained with cosine decay in Chinchilla scale, across 3 learning rates and 3
batch sizes. Note that the final validation losses are equal whether Z-Loss is enabled or not.
However, while the final validation loss does not change as an effect of z-loss at our scale, we have
observed certain instabilities in the z-loss towards the end of training when using Seesaw in Figure 7.
We speculate that the way we are scaling the learning rate and batch size might not be the proper
way to do it for z-loss, and we leave this study for future work.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Tokens
1e10
0.0005
0.0010
0.0015
0.0020
0.0025
Z-Loss
Figure 7: 600M models trained with Seesaw decay in Chinchilla scale, with Z-Loss.
25
|
Seesaw: Accelerating Training by Balancing Learning Rate and Batch Size Scheduling Alexandru Meterez*,1,2, Depen Morwani*,1,2, Jingfeng Wu3, Costin-Andrei Oncescu1, Cengiz Pehlevan1,2, and Sham Kakade1,2 1Harvard University 2Kempner Institute at Harvard University 3 - a "batch ramp" - is a promising strategy to accelerate large language model pretraining. While for SGD, doubling the batch size can be equivalent to halving the learning rate, the optimal strategy for adaptive optimizers like Adam is less clear. As a result, any batch-ramp scheduling, if used at all, is typically tuned heuristically. This work develops a principled framework for batch-size scheduling and introduces Seesaw: whenever a standard scheduler would halve the learning rate, Seesaw instead multiplies it by 1/ √ 2 and doubles the batch size, preserving loss dynamics while reducing serial steps. Theoretically, we provide, to our knowledge, the first finite-sample proof of equivalence between learning-rate decay and batch-size ramp-up for SGD on noisy linear regression, and we extend this equivalence to normalized SGD, a tractable proxy for Adam, under a variance-dominated regime observed in practice. Empirically, on 150M/300M/600M-parameter models trained at Chinchilla scale using a constant (critical) batch size, Seesaw matches cosine decay at equal FLOPs while reducing wall-clock time by ≈36%, approaching the theoretical limit implied by our analysis. 1 Introduction In recent years, large language models (LLMs) have demonstrated remarkable progress across diverse tasks, including outperforming humans in competitive benchmarks and international competitions (Huang and Yang, 2025; Petrov et al., 2025; El-Kishky et al., 2025). A central driver of this progress has been the steady increase in pre-training compute (Kaplan et al., 2020; Hoffmann et al., 2022). However, hardware improvements have not kept pace with the rapid escalation of training requirements, resulting in wall-clock times extending to several months for state-of-the-art models (Erdil and Schneider-Joseph, 2024). A widely studied strategy to reduce wall clock time is increasing the batch size (You et al., 2017; Goyal et al., 2017). Empirical studies show that larger batches can proportionally reduce the number of optimization steps required for convergence (Zhang et al., 2024; McCandlish et al., 2018; Shallue et al., 2019), while maintaining comparable per-step runtime through parallelization. However, beyond a maximum batch size termed as critical batch size (CBS), further scaling reduces sample efficiency and limits gains in training speed. While most prior work focuses on training with a fixed batch size, recent large-scale LLM training runs employ batch size schedules that gradually increase batch size over the course of train- ⋆: Equal contribution. Correspondence to: , 1 16 Oct 2025 ing (Dubey et al., 2024; Touvron et al., 2023; Adler et al., 2024; OLMo et al., 2024; Team, 2025). This practice has been observed to further reduce training times without compromising model performance. However, to the best of our knowledge, the "batch ramp" schedules are not theoretically grounded and instead tuned heuristically. The lack of theoretical justification leaves open whether these heuristics are close to optimal, motivating the central question of our study: what is the optimal batch size schedule for minimizing serial runtime while not sacrificing performance? 1.1 Theoretical Contributions We theoretically prove, to the best our knowledge, the first non-asymptotic equivalence result between learning rate decay and batch size ramp up in SGD in linear regression with additive noise. Further, we extend our equivalence result to normalized SGD (considered a proxy for Adam), leading to the batch size scheduling algorithm Seesaw (Algorithm 1). We introduce an informal version of our main theorem here, as well as the corollary leading up to Seesaw, and we formalize the statements in Section 5. Theorem (Informal version of Theorem 1). Consider a base SGD process that runs for k phases, using a fixed learning rate throughout, while the batch size doubles after each phase. Now consider an alternative process where the batch size is fixed but the learning rate halves after each phase, and where each phase processes the same number of data points as in the base process. Then, the excess risk of the base process is within a constant factor of that of the alternative process. Corollary (Informal version of Corollary 1). Consider a base normalized SGD process (Equation 4) that runs for k phases, where the batch size doubles after each phase while the learning rate decays by a factor of √ 2. Consider an alternative normalized SGD process where the batch size is fixed but the learning rate halves after each phase, and where each phase processes the same number of data points as in the base process. Then, the excess risk of the base process is within a constant factor of that of the alternative process. 1.2 Empirical Contributions Based on the theoretical analysis, we introduce Seesaw (Algorithm 1), a learning rate and batch size scheduler that reduces the serial runtime of LLM pre-training runs by approximately 36% via increasing the batch size during training at specific points. We provide empirical results in Figure 1 and show that at (or below) the critical batch size, our method achieves a significant serial runtime acceleration across several model and data scales, while maintaining the same performance as training with cosine decay. We also empirically show that Seesaw also works even when using AdamW with tuned weight decay in Figure 4 of Appendix C, making Seesaw a practical solution for reducing the wall-clock time of LLM pretraining. 2 Related Work Role of batch size in scaling. Understanding batch size ramp up schemes during training has been a topic of interest in recent years due to its crucial role in decreasing wall clock runtime. Various methods of increasing the batch size have been used in common LLMs such as LLaMA (Dubey et al., 2024; Touvron et al., 2023), Nemotron (Adler et al., 2024), OLMo (OLMo et al., 2024; Groeneveld et al., 2024), Apertus (Team, 2025). The reason behind ramping up the batch size is to take advantage of the parallel computation of samples and thus reducing the total number of sequential steps. However, since increasing the batch size reduces the total number of gradient steps taken by the model during training, there is a maximal batch size which can be achieved without becoming data inefficient, called the critical batch size (CBS) (Erdil and Schneider-Joseph, 2024; Zhang et al., 2024; Shallue et al., 2 105 106 107 108 109 Tokens 4 6 8 10 Validation Loss 150M 105 106 107 108 109 1010 Tokens 4 6 8 10 Validation Loss 300M 106 107 108 109 1010 Tokens 4 6 8 10 Validation Loss 600M 0 5000 10000 15000 20000 25000 Steps 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Validation Loss 0 10000 20000 30000 40000 50000 Steps 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Validation Loss 0 10000 20000 30000 40000 50000 Steps 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Validation Loss Batch size: 128 256 512 1024 Scheduler: Cosine Seesaw Figure 1: Seesaw comparison with cosine decay in 150M (left), 300M (middle) and 600M (right) models trained at Chinchilla scale. Seesaw matches the loss dynamics of cosine annealing in FLOPs (top row), but achieves a significant speed up in terms of serial runtime (bottom row). Runs are swept over learning rates and plotted at the best learning rate for cosine annealing in terms of validation loss, at each batch size. The validation losses at the end of training are provided in Table 1. Note the axes: the top plots are on a logarithmic scale while the bottom are on a linear scale. For more experimental details, see Section 4. 2019; Jain et al., 2018). Recent work also looks at the effect of batch size on SGD optimization in LLMs (Sre ́ckovi ́c et al., 2025; Marek et al., 2025), following previously established theoretical results in noisy quadratic models (Zhang et al., 2019). SGD for linear regression. Recently, Zhang et al. (2024) have analyzed the CBS using weight averaging in linear regression and established scaling laws as a function of data and model size. The bias-variance analysis used by Zhang et al. (2024) has a longstanding history in the literature (Jain et al., 2017) and has been used to study batch ramp-up schemes in SGD (Jain et al., 2018). These rates have been recently made tight by (Zou et al., 2021; Wu et al., 2022a,b) for general spectra of the data covariance. Recently, (Meterez et al., 2025) have used a simplified mathematical framework for rederiving the same bounds by rotating the dynamics in the eigenbasis of the data. A similar diagonalizing idea has also been previously used in literature by Bordelon and Pehlevan (2021); Wu et al. (2023a,b). Stochastic Differential Equations (SDEs). Another point of view for studying the interaction between batch size and learning rate in optimization is through SDEs (Li et al., 2021; Xie et al., 2020; Compagnoni et al., 2024; Jastrz ̨ebski et al., 2017). Malladi et al. (2022) study how to scale the learning rate as a function of the batch size in adaptive algorithms, extending previous work that introduced the square root scaling rule (Granziol et al., 2022; You et al., 2019). Empirical work. Scaling laws for the CBS and the optimal batch size have also been recently observed by (Bergsma et al., 2025). In line with our conclusions regarding SGD, the linear scaling 3 rule for SGD has been observed by (Smith et al., 2017), showing that in SGD, linearly increasing the batch size is equivalent to decreasing the learning rate. McCandlish et al. (2018) propose a metric based on the Hessian and the noise that correlates with the CBS over training. While their proposed metric is based on having access to the Hessian, which is prohibitive for current large-scale runs, they find that the noise scale increases during a training run, which aligns with our theoretical predictions. Lastly, perhaps the most similar to our work is Merrill et al. (2025), who propose a batch size warmup scheme based on starting from a checkpoint with various multiples k of the current batch size, and pick the largest k⋆where the loss is ε-close to the original loss. Based on this methodology, they propose the scaling rule Bt+1 = 2Bt and ηt+1 = √ 2η. In contrast, we propose a simple drop-in replacement for existing cosine schedulers, motivated rigorously by (normalized) SGD on quadratics. Moreover, we argue that the scheduler proposed by (Merrill et al., 2025) will lead to instabilities and divergence after a fixed number of steps, based on our theoretical analysis in Lemma 4. 3 Seesaw: Algorithmic Details We begin by providing an intuitive derivation of Seesaw, and the practical implementation of our algorithm. To build intuition, consider 2 different SGD processes. In one process we take 2 steps at learning rate η/2 and batch size B, and in the other we take 1 step at learning rate η and batch size 2B. Consider a general smooth loss function L(x) and let g0 = ∇L(x0). Then, through a simple Taylor expansion up to first order in η, we have the loss of the (η, 2B) process and the loss of the 2 half step process (η/2, B) respectively: L(x1) = L(x0) -ηg⊤ 0 (g0 + ξ′) + O(η2) Cov(ξ′) = σ2 2BId L(x2) = L(x0) -η 2g⊤ 0 (2g0 + ξ0 + ξ1) + O(η2) Cov(ξi) = σ2 B Id. Note that the 2 processes are equivalent up to first order both in the deterministic part and in the noise terms up to O(η2), an argument which has been previously shown by Malladi et al. (2022). We formalize this SGD intuition in Theorem 1 and extend it to normalized SGD as an analytical proxy to Adam in Corollary 1. 3.1 Extension to Normalized SGD From the previous subsection, intuitively, for SGD, cutting the learning rate by a factor of α should be equivalent to increasing the batch size by a factor of α. To design a practical training algorithm based on the SGD analysis and arrive at Seesaw, we begin with the Adam update rule and simplify until we obtain normalized SGD (NSGD), which is a commonly used tractable analytical proxy for Adam (Jelassi et al., 2022; Zhao et al., 2024; Xie et al., 2024). Suppose we are optimizing over parameters θ and denote the gradients at each time step gt. Then, for learning rate η and ignoring the bias correction, the parameter update for Adam is given by: mt = β1mt-1 + (1 -β1)gt (1) vt = β2vt-1 + (1 -β2)g2 t (2) θt = θt -η mt √vt + ε (3) where mt is the momentum term, vt is the second moment term, β1, β2 are their respective exponential decay rates, and ε ensures stability. For NSGD, we approximate the per-coordinate preconditioner of Adam will a single scalar preconditioner, set β1 = β2 = 0 and replace the denominator with the 4 true expected value of the squared gradient norms over the population: θt = θt -η gt pE∥gt∥2 (4) Algorithm 1: Seesaw Inputs: η0 (initial learning rate), B0 (initial batch size), α > 1 (step decay factor), S (an array of steps at which input scheduler cuts η by α), T (total training steps) η ←η0, B ←B0 for t ←1 to T do if t ∈S then η ←η/√α; B ←B · α; end end Equation 4 describes the NSGD update rule, which is a crucial component of designing Seesaw. While the full analysis is deferred to Appendix B, the expected gradient norms can be decomposed as: E∥gt∥2 = mean + variance (5) where the variance scales down with the batch size. To design Seesaw, we assume that the variance dominates the expected gradient squared norms (Assumption 2), and we motivate why this assumption is reasonable in Appendix B. This step reduces (up to constant factors) the NSGD update rule to SGD with a rescaled learning rate (Equation 7), allowing us to extend risk equivalence to NSGD (Corollary 1) in Section 5. For NSGD, informally, Corollary 1 shows that any learning rate cut by a factor of α and batch size increase by a factor of β are equivalent as long as α√β is held constant. We further empirically comapre Seesaw with other possible schedulers in Figure 5. 3.2 Achievable Speedups While our theory is established for step decay schedulers, in practice we approximate cosine decay with a step decay by considering a decay of α, and passing the times (as measured in tokens) where the cosine would cut the learning rate by α as input to Seesaw. Then, at these points, we instead cut the learning rate by √α and increase the batch size by β, where the schedulers are equivalent in terms of loss as long as we keep the product α√β fixed. However, we cannot arbitrarily increase the batch size at time t and expect the risk to match the underlying process. Lemma 4 quantifies this and the main takeaway is stated below: Remark 1. The most aggressive ramp up scheme we can use is given by α = √β. (for a formal argument see Lemma 4) In Section 4.1 we empirically verify this constraint and show that α = √β is the most aggressive scheme we can choose without divergence. The corresponding algorithm is provided in Algorithm 1. At the most aggressive limit, we can compute the theoretical speedup we would hope to achieve where the standard scheduler is the cosine decay. Lemma 1 (Maximum Theoretical Speedup under Cosine Decay). Consider a baseline training process of T total steps using a constant batch size and a cosine learning rate schedule η(t) = η0 cos( πt 2T ). An equivalent process run with a batch ramping schedule like Seesaw, in the continuous limit 1, will have a total of 2T π steps. This yields a maximum theoretical serial runtime reduction of (1 -2 π) ≈36.3%. 1In the continuous-time limit, we consider an aggressive (non-divergent) batch size ramp that maintains the relationship α = √β. Consequently, the total number of sequential steps is given by the integral of the normalized learning rate schedule: R T 0 η(t) η0 dt = R T 0 cos( πt 2T )dt = 2T π . 5 Lemma 1 provides an upper bound on the acceleration from Seesaw. In Figure 1, we can indeed see that the total number of steps are reduced approximately by 36% as predicted. 4 Empirical Findings In this section, we present the experimental details and methodology for evaluating Seesaw. We denote by D the dataset size, N the number of parameters. B=128 B=256 B=512 B=1024 150M (cosine) 3.0282 3.0353 3.0696 3.1214 150M (Seesaw) 3.0208 3.0346 3.0687 3.1318 300M (cosine) 2.8531 2.8591 2.8696 2.9369 300M (Seesaw) 2.8452 2.8561 2.8700 2.9490 600M (cosine) - 2.6904 2.6988 2.7128 600M (Seesaw) - 2.6883 2.6944 2.7132 Table 1: Final validation losses picked at the best learning rate (for the cosine annealing scheduler) for each batch size, for α = 1.1. Note that the dynamics match robustly across the 2 schedulers when trained at CBS. Model and dataset. We pretrain models of size 150M, 300M and 600M (non-embedding) parameters at Chinchilla scaling i.e. D = 20N (Hoffmann et al., 2022). We use the OLMo (Groeneveld et al., 2024) codebase to train all of our models. For each experiment, we do learning rate warmup for 10% of the total amount of tokens, followed by learning rate decay following cosine scheduling or Seesaw. We report the architectural details of each model as a tuple (depth, # heads, width), and thus we have for 150M (12, 16, 1024), 300M (24, 16, 1024) and for 600M (24, 22, 1408). Unless mentioned otherwise, each model is trained using AdamW, with weight decay λ = 0.0 (with the exception of the weight decay experiments in Appendix C, where we sweep over λ ∈{0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0}), β1 = 0.9, β2 = 0.95, ε = 10-8. For each run we sweep over learning rates η ∈{0.001, 0.003, 0.01, 0.03} and initial batch sizes B ∈{128, 256, 512, 1024}, at sequence length L = 1024. Similar to the OLMo training codebase, we enable z-loss during training, but provide ablations over it in Appendix E showing that it does not affect the model performance at our scales. All our models are pretrained on the C4 dataset (Raffel et al., 2020), tokenized with the T5 tokenizer. Experimental design. We compare Seesaw with cosine annealing by training models at the critical batch size (CBS) B⋆, approximated based on (Zhang et al., 2024), namely B⋆≈256k (150M), B⋆≈512k (300M) and B⋆≈1024k (600M) tokens. The main results comparing Seesaw and cosine annealing at equal FLOPs are provided in Figure 1. The precise final losses obtained by the 2 schedulers are provided in Table 1. 4.1 Can We Do Better? Recall that based on Corollary 1 and Lemma 4, we have a family of equivalent schedules in NSGD, given by a fixed product α√β, under the constraint that α ≥√β. Ideally, we would like to make β as large as possible, since this would lead to larger batch sizes, and thus assuming enough devices are available, the lowest serial runtime. Crucially, the constraint prevents us from using a too agressive 6 2.00 2.25 2.50 2.75 3.00 3.25 Tokens 1e9 3.025 3.050 3.075 3.100 3.125 Validation Loss B=256 2.00 2.25 2.50 2.75 3.00 3.25 Tokens 1e9 3.05 3.10 3.15 Validation Loss B=512 α = 2, β = 1 α = 23/4, β = p 2 α = p 2, β = 2 α = 21/4, β = 23/2 α = 1, β = 4 Figure 2: 150M models trained at batch size 256 (left) and 512 (right) with α and β values following the line of equivalence α√β = 2 described in Table 2. Note that the target to match is the blue trace, and our theory (Lemma 4) predicts that the red and purple traces should not match the baseline (blue trace) due to instabilities. batch size scheduler. In this section, we empirically verify our theoretical prediction by testing schedulers positioned at various points on the (α, β) axis. α 2 23/4 21/2 21/4 1 β 1 21/2 2 23/2 22 Table 2: α, β values used to test the extreme values of the equivalence. Namely, we train 150M models at fixed batch size and Chinchilla scale, and we approximate cosine decay with a step decay scheduler that halves the learning rate at the token counts where the cosine schedule's learning rate would halve. This gives us the baseline α = 2 and β = 1, with the product α√β = 2. Based on the theoretical constraint and the equivalence line, the most aggressive scheduler we could use is α = √ 2 and β = 2. To validate our hypothesis, we compare with α = 1 and β = 4, and points in between at geometric intervals. Table 2 gives an overview of the experimental design, and Figure 2 shows that indeed the most aggressive schedules tend to underperform. 4.2 When Does Assumption 2 Fail? Up to this point, a crucial assumption for the development of our theory and the design of Seesaw has been Assumption 2. Recall that Assumption 2 states that the expected gradient norms - namely, the denominator of the NSGD update step, is dominated by the additive noise. Intuitively, since the noise variance decreases with the batch size as O(1/B), one can see that past a certain batch the additive noise will become small, and thus Assumption 2 will fail. In Figure 3, we can see that at sufficiently large batch sizes, indeed Seesaw starts to perform worse as compared to the underlying cosine schedule. The first hypothesis could be that it is still possible to match the underlying schedule, but with a learning rate equivalence as given by mean dominating in the denominator. As mean does not scale with batch size, therefore, using the equivalence schedule of SGD could be a promising candidate. We explore this option in Figure 3, and it turns out that this schedule performs even worse than the Seesaw schedule. We hypothesise that beyond a certain batch size, it is not 7 109 3.2 3.4 3.6 3.8 4.0 Validation Loss B=1024 109 3.2 3.4 3.6 3.8 4.0 Validation Loss B=2048 109 Tokens 3.5 4.0 4.5 5.0 5.5 6.0 Validation Loss B=4096 109 Tokens 3.5 4.0 4.5 5.0 5.5 6.0 Validation Loss B=8192 Scheduler: Cosine Fixed LR Seesaw Figure 3: 150M models trained past CBS (roughly 256), at batch sizes 1024, 2048, 4096 and 8192, for 3 schedulers: cosine decay (blue), constant learning rate with increasing batch size based on Seesaw (orange) and Seesaw (green). Note that none of the proposed schedules is able to match the cosine curve, with the discrepancy increasing as the batch size grows more. possible to match the performance of learning rate decay by any equivalent batch size ramp up for Adam or normalized SGD, which we motivate using the following toy example. For simplicity, we look at NGD (normalized gradient descent) in 1D, for the quadratic loss L(x) = 1 2hx2, where x, h ∈R and h ≥0. Training with NGD, we have the loss gradients with respect to the parameters and the update rule: ∇xL = hx xt+1 := xt -ηh sign(xt) where sign(xt) = xt |xt|. Note that if xt > 0, then xt+1 = xt -ηh and if xt 0 such that f(x) ≤ag(x) for any x. We also use the notation f ≂g if f(x) ≲g(x) ≲f(x) for all x. We denote the samples (x, y) where x ∈Rd and y ∈R, with the distribution and risk: x ∼N(0, H) y|x ∼N(⟨w⋆, x⟩, σ2) R(w) = 1 2E(⟨w, x⟩-y)2 where the expectation is over the (x, y), w⋆is the minimizer, and σ2 is the variance of the additive noise. We also use R(wt, η) to denote the risk at time t for a process trained with η, but we drop the η parameter when it is clear from context. We consider step decay schedules for the learning rate, where, the learning rate in the kth phase is denoted by ηk and Pk denotes the total number of data samples used in the kth phase. Similarly, for batch ramp schedules, Bk denotes the batch size in the kth phase. For discussion, we will use the bias-variance decomposition terminology of risk (Jain et al., 2018, 2017; Zou et al., 2021; Wu et al., 2022a,b; Meterez et al., 2025). Informally, bias corresponds to the risk of the averaged iterates, while variance corresponds to the noise in the iterates, and R(wt) = biast + variancet. We will denote the stochastic gradient at time t by gt and let E∥gt∥2 represent its expected squared norm under the population distribution. 5.1 Main Results In this section, we first introduce the main assumptions and discuss their implications, followed by the main theoretical results. Our first assumption states that the risk of the SGD process is bounded. Assumption 1 (Bounded risk.). Consider a SGD process with a given scheduling scheme for the batch size and learning rate. Let t0 denote the first time when the scheduler changes either the learning rate or the batch size. Then, we assume that there exists a constant c > 1 such that R(wt) ≤cσ2 for all t > t0. In general, we expect every "well tuned" scheduler to start cutting when R(wt0) ≲σ2, as we want to minimize the bias component of the risk before cutting down the learning rate to reduce noise in the iterates. Moreover, for a well-behaved schedule, as we expect the risk to decrease over time, this condition should hold throughout the process. Our second assumption states that the expected gradient squared norms of the NSGD update rule are dominated by the additive noise term. Assumption 2 (Variance dominated.). Assume that E∥gt∥2 ≂σ2 Bt . Under Assumption 2, the NSGD process effectively reduces to SGD with a rescaled learning rate (Equation 7), up to constant factors. Based on the previously established assumptions, we can now state the equivalence result. We use the notation R(ηt, Bt) to denote the risk at time t of an SGD process trained with the learning rate scheduler ηt and batch size scheduler Bt. Theorem 1 (SGD Equivalence). Fix 0.01 Tr(H) ≥η > 0, B > 0, and parameters α1, α2 > 1, β1, β2 > 1 with α1β1 = α2β2. Define the two phase-indexed schedules (ηk, Bk) := η α-k 1 , B βk 1 , (η′ k, B′ k) := η α-k 2 , B βk 2 , k = 0, 1, 2, . . . and run two SGD procedures in phases k = 0, 1, . . . so that, in phase k, each procedure processes the same number of samples (possibly depending on k) under its respective schedule. Let R(ηk, Bk) and R(η′ k, B′ k) 9 denote the (population) risk of the two procedures at the end of phase k. If Assumption 1 holds (for both procedures) with constant c, then R(1.01 · η′ k, B′ k) ≲c R(ηk, Bk) ≲c R(η′ k, B′ k), where R(λ·η′ k, B′ k) denotes the risk of the second procedure when its entire learning-rate schedule is multiplied by a uniform factor λ > 0, and A ≲c B means A ≤C(c) B for a numerical constant C(c) depending only on c (and absolute constants). We defer the full proof to Appendix A.1. Now, we extend this result to Normalized SGD. Under Assumption 2, NSGD reduces to SGD with a rescaled learning rate ̃η ≂η √ B σ√ Tr(H) (Equation 7). Consequently, we can extend Theorem 1 to the normalized SGD case. We formalize this in the following corollary: Corollary 1 (Normalized SGD Equivalence). Fix 0.01 Tr(H) ≥η > 0, B > 0, and parameters α1, α2 > 1, β1, β2 > 1 with α1 √β1 = α2 √β2. Define the two phase-indexed schedules (ηk, Bk) := η α-k 1 , B βk 1 , (η′ k, B′ k) := η α-k 2 , B βk 2 , k = 0, 1, 2, . . . and run two normalized SGD procedures in phases k = 0, 1, . . . so that, in phase k, each procedure processes the same number of samples (possibly depending on k) under its respective schedule. Let R(ηk, Bk) and R(η′ k, B′ k) denote the (population) risk of the two procedures at the end of phase k. If Assumption 1 and 2 holds (for both procedures) with constant c, then R(1.01 · η′ k, B′ k) ≲c R(ηk, Bk) ≲c R(η′ k, B′ k), where R(λ·η′ k, B′ k) denotes the risk of the second procedure when its entire learning-rate schedule is multiplied by a uniform factor λ > 0, and A ≲c B means A ≤C(c) B for a numerical constant C(c) depending only on c (and absolute constants). 6 Discussion and Conclusions In this work, we provide a rigorous analysis of batch ramp through the lens of noisy linear regression, offering theoretical insight into a practice that has thus far been guided largely by empirical heuristics. Building upon this analysis, we introduce Seesaw (Algorithm 1), a principled batch size scheduling algorithm that can serve as a drop-in replacement for commonly used learning-rate schedules in adaptive optimizers such as Adam. Our empirical evaluation demonstrates that Seesaw matches the performance of standard cosine annealing schedules while reducing the serial training runtime by approximately 36%. These results indicate that dynamically balancing the learning rate and batch size provides an effective means to accelerate training without compromising performance. Acknowledgements AM would like to thank Jacob Zavatone-Veth and Alex Damian for helpful discussions. The authors would also like to thank Max Shad and Bala Desinghu for their help with the cluster. AM, DM acknowledge the support of a Kempner Institute Graduate Research Fellowship. CP is supported by an NSF CAREER Award (IIS-2239780), DARPA grants DIAL-FP-038 and AIQ-HR00112520041, the Simons Collaboration on the Physics of Learning and Neural Computation, and the William F. Milton Fund from Harvard University. AM, SK and DM acknowledge that this work has been made possible in part by a gift from the Chan Zuckerberg Initiative Foundation to establish the Kempner 10 Institute for the Study of Natural and Artificial Intelligence. SK and DM acknowledge support from the Office of Naval Research under award N0001422-1-2377 and the National Science Foundation Grant under award #IIS 2229881. DM is also supported by a Simons Investigator Fellowship, NSF grant DMS-2134157, DARPA grant W911NF2010021,and DOE grant DE-SC0022199. References Yichen Huang and Lin F Yang. Gemini 2.5 pro capable of winning gold at imo 2025. arXiv preprint , 2025. Ivo Petrov, Jasper Dekoninck, Lyuben Baltadzhiev, Maria Drencheva, Kristian Minchev, Mislav Balunovi ́c, Nikola Jovanovi ́c, and Martin Vechev. Proof or bluff? evaluating llms on 2025 usa math olympiad. arXiv preprint , 2025. 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Learning rates as a function of batch size: A random matrix theory approach to neural network training. Journal of Machine Learning Research, 23(173):1-65, 2022. Yang You, Jing Li, Sashank Reddi, Jonathan Hseu, Sanjiv Kumar, Srinadh Bhojanapalli, Xiaodan Song, James Demmel, Kurt Keutzer, and Cho-Jui Hsieh. Large batch optimization for deep learning: Training bert in 76 minutes. arXiv preprint , 2019. Shane Bergsma, Nolan Dey, Gurpreet Gosal, Gavia Gray, Daria Soboleva, and Joel Hestness. Power lines: Scaling laws for weight decay and batch size in llm pre-training. arXiv preprint , 2025. Samuel L Smith, Pieter-Jan Kindermans, Chris Ying, and Quoc V Le. Don't decay the learning rate, increase the batch size. arXiv preprint , 2017. William Merrill, Shane Arora, Dirk Groeneveld, and Hannaneh Hajishirzi. Critical batch size revisited: A simple empirical approach to large-batch language model training. arXiv preprint , 2025. Samy Jelassi, David Dobre, Arthur Mensch, Yuanzhi Li, and Gauthier Gidel. Dissecting adaptive methods in gans. arXiv preprint , 2022. Rosie Zhao, Depen Morwani, David Brandfonbrener, Nikhil Vyas, and Sham Kakade. Deconstructing what makes a good optimizer for language models. arXiv preprint , 2024. Shuo Xie, Mohamad Amin Mohamadi, and Zhiyuan Li. Adam exploits l∞-geometry of loss landscape via coordinate-wise adaptivity. arXiv preprint , 2024. 13 Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of machine learning research, 21(140):1-67, 2020. 14 A Proofs for Section 5 A.1 Preliminaries We take as a convention for eigenvalues ordering λmax = λ1 ≥λ2 ≥· · · > 0. For two matrices A and B we use the notation A ⪯B to denote that B -A is positive semi-definite (PSD). We denote ⟨u, v⟩for the inner product between u and v. Moreover, with a slight abuse of notation, we use the notation ≤as elementwise comparison, namely u ≤v if ui ≤vi for all i and A ≤B if Aij ≤Bij for all i, j. To simplify the analysis, we will follow the approach of Meterez et al. (2025) and work in the eigenbasis of the data covariance H. For the sake of completeness, we restate the main derivation for the bias and variance iterates in the case of constant learning rate and constant batch size, starting from the SGD update rule: wt+1 -w⋆= I -η B B X i=1 xix⊤ i (wt -w⋆) -η B B X i=1 xiεi =⇒Σt+1 = Σt -ηΣtH -ηHΣt + η2 1 + 1 B HΣtH + η2 B Tr(HΣt)H + η2 B σ2I =⇒Mt+1 = Mt -ηMtΛ -ηΛMt + η2 1 + 1 B ΛMtΛ + η2 B Tr(ΛMt)Λ + η2 B σ2I (6) where in the last equation Mt = QΣtQ⊤is the iterate covariance matrix rotated in the eigenbasis of H. Since we can write the excess risk as: R(wt) -R(w⋆) = 1 2Tr(ΛMt) = 1 2⟨λ, mt⟩ where mt = diag(Mt), it suffices to push a diag operator through equation equation 6. Finally, we get: mt+1 = " I -2ηΛ + η2 1 + 1 B Λ2 + η2 B λλ⊤ # | {z } A mt + η2σ2 B λ = Atm0 + η2σ2 B t-1 X i=0 Aiλ where f mt := Atm0 and mt := η2σ2 B Pt-1 i=0 Aiλ are the bias and variance iterates respectively. Before we begin proving the main statements, we introduce several helpful lemmas that we will use. Lemma 2. For η ≤0.01/Tr(H) and α ≥1, we have the elementwise inequality: αk η 1 ≥ I - I -η αk Λ 2!-1 λ ≥αk 2η 1 Proof. We have: I - I -η αk Λ 2!-1 = I - I + η2 α2k Λ2 -2 η α2 Λ !!-1 = η αk Λ 2 -η αk Λ -1 ≥ 2η αk Λ -1 Note that trivially we also have the other direction by noticing that 1 2-η αk λ ≤1. Multiplying by λ gives us the conclusion. 15 Lemma 3. For η ≤0.01/Tr(H) and α1, α2, β1, β2 ≥1 such that α1β1 = α2β2 and α1 ≤α2, we have: I -1.01 η αk 2 Λ 2βk 1 ⪯ I -η αk 1 Λ 2βk 2 ⪯ I -η αk 2 Λ 2βk 1 . Proof. RHS bound. Since both sides are diagonal matrices, it suffices to prove the scalar inequality for every x = ηλi: 1 -x αk 1 2βk 2 ≤ 1 -x αk 2 2βk 1 . Taking logarithms and defining f(x) = 2βk 2 log(1 -x/αk 1) 2βk 1 log(1 -x/αk 2) = αk 1 log(1 -x/αk 1) αk 2 log(1 -x/αk 2) = g(α1) g(α2), where g(y) = y log(1 -x/y). For 0 1, g(y) is monotonically increasing, so for α1 ≤α2, we have g(α1) ≤g(α2) and hence g(α1)/g(α2) ≥1 (since g(α2) α, then at every cut we are effectively increasing the learning rate. Thus, there must exist k > 0 such that ̃ηk > ηmax, where ηmax is the maximum convergent learning rate for SGD (Jain et al., 2018; Wu et al., 2022b), leading to divergence. 22 C Weight Decay In this section we provide experiments on 150M models trained with AdamW, sweeping weight decay λ ∈{0.000001, 0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0} and learning rate η ∈{0.001, 0.003, 0.01, 0.03}, and the rest of the parameters are as explained in Section 4. For every figure we pick the best (η, λ) pair on cosine annealing, and we use the values for Seesaw. Across all batch sizes (128, 256, 512), the optimal (η, λ) pair from the sweep turned out to be (η, λ) = (0.003, 0.0001). Figure 4 shows the results: 2 × 109 3 × 109 Tokens 3.00 3.05 3.10 3.15 3.20 3.25 Validation Loss Batch Size: 128 Scheduler Seesaw Cosine Decay 2 × 109 3 × 109 Tokens 3.00 3.05 3.10 3.15 3.20 3.25 Validation Loss Batch Size: 256 2 × 109 3 × 109 Tokens 3.00 3.05 3.10 3.15 3.20 3.25 3.30 Validation Loss Batch Size: 512 Figure 4: 150M experiments with weight decay across different batch sizes (128, 256, 512) for cosine annealing and Seesaw, for learning rate and weight decay values (η, λ) = (0.003, 0.0001). Note that the losses overlap during training. We provide the final validation losses in Table 3. Table 3 shows the final validation losses: B=128 B=256 B=512 150M (cosine) 3.0125 3.0220 3.0559 150M (Seesaw) 3.0027 3.0210 3.0588 Table 3: Final validation losses picked at the best learning rate (for the cosine annealing scheduler) for each batch size, for α = 1.1 and weight decay 0.003. Note that the dynamics match robustly. 23 D Comparison to other schedulers We compare our scheme with other schedulers in this section. 109 2 × 109 3 × 109 Tokens 3.00 3.05 3.10 3.15 3.20 Validation Loss Batch Size: 128 109 2 × 109 3 × 109 Tokens 3.00 3.05 3.10 3.15 3.20 Validation Loss Batch Size: 256 η, B · 2 η, B · 4 η/2, B η/ p 2, B · 2 Figure 5: 150M models trained with 4 different schedules, at CBS (right) and just below (left). Blue trace keeps learning rate fixed and doubles batch size, orange trace keeps learning rate fixed and quadruples batch size, green trace halves learning rate at fixed batch size, and red trace is Seesaw. Note that the naive scheduling (blue) severely underperforms the baseline (green) and Seesaw (red). 24 E Auxiliary Losses In this section we ablate over the effect of z-loss on the training dynamics (OLMo et al., 2024). We observe no difference in the training stability of our models at 150M scale in Figure 6: 105 106 107 108 109 Tokens 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 Validation Loss Learning rate: 0.003 105 106 107 108 109 Tokens 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 Validation Loss Learning rate: 0.01 105 106 107 108 109 Tokens 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 Validation Loss Learning rate: 0.03 Batch size: 128 256 512 Z-Loss: False True Figure 6: 150M models trained with cosine decay in Chinchilla scale, across 3 learning rates and 3 batch sizes. Note that the final validation losses are equal whether Z-Loss is enabled or not. However, while the final validation loss does not change as an effect of z-loss at our scale, we have observed certain instabilities in the z-loss towards the end of training when using Seesaw in Figure 7. We speculate that the way we are scaling the learning rate and batch size might not be the proper way to do it for z-loss, and we leave this study for future work. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Tokens 1e10 0.0005 0.0010 0.0015 0.0020 0.0025 Z-Loss Figure 7: 600M models trained with Seesaw decay in Chinchilla scale, with Z-Loss. 25
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Speculative Model Risk in Healthcare AI: Using Storytelling to Surface
Unintended Harms
Xingmeng Zhao, Dan Schumacher, Veronica Rammouz, and Anthony Rios
The University of Texas at San Antonio
{xingmeng.zhao, anthony.rios}@utsa.edu
Abstract
Artificial intelligence (AI) is rapidly transform-
ing healthcare, enabling fast development of
tools like stress monitors, wellness trackers,
and mental health chatbots. However, rapid and
low-barrier development can introduce risks of
bias, privacy violations, and unequal access,
especially when systems ignore real-world con-
texts and diverse user needs. Many recent meth-
ods use AI to detect risks automatically, but this
can reduce human engagement in understand-
ing how harms arise and who they affect. We
present a human-centered framework that gen-
erates user stories and supports multi-agent dis-
cussions to help people think creatively about
potential benefits and harms before deploy-
ment. In a user study, participants who read
stories recognized a broader range of harms,
distributing their responses more evenly across
all 13 harm types. In contrast, those who did
not read stories focused primarily on privacy
and well-being (58.3%). Our findings show
that storytelling helped participants speculate
about a broader range of harms and benefits
and think more creatively about AI’s impact
on users. Dataset and code are available at
https: https://anonymous.4open.science/
r/storytelling-healthcare/README.md.
1
Introduction
AI is integrated into many aspects of daily life,
including finance, healthcare, law, and recommen-
dation systems (Ashurst et al., 2020). In healthcare,
tools such as stress monitors (Kargarandehkordi
et al., 2025), wellness trackers (Fabbrizio et al.,
2023), and mental health chatbots (MacNeill et al.,
2024) can directly influence well-being. Recent
prompting methods like vibe coding (Chow and
Ng, 2025) let people describe how an AI sys-
tem should behave or sound, making it easier for
non-experts to prototype AI tools. For instance,
CareYaya allows caregivers to build stress-tracking
applications using short natural language descrip-
tions (Kenny, 2023). While this innovation speeds
Story 1
Story 2
Story 3
Story 4
Story 5
What if these harms became real? How might they
unfold, and what could we do now to prevent them?
Figure 1: Illustration of using speculative stories to help
people imagine potential harms and benefits of health-
care AI and foster more creative and ethical thinking.
up AI development, it also introduces risks related
to fairness, bias, and accountability (Weidinger
et al., 2023). These risks are serious in health-
care, where even small model errors can cause
harm, including delayed treatment, privacy loss,
or health inequities (Roller et al., 2020; Chinta
et al., 2025). Systems lacking safeguards for fair-
ness and accountability may ultimately harm the
very users they are meant to help (Shelby et al.,
2022, 2023). Governments are responding through
measures such as the EU AI Act (European Par-
liament, 2023) and a U.S. Executive Order (Biden,
2023), which call for greater transparency, safety,
and accountability (Khan et al., 2025). Yet regu-
lation remains region-specific and slow to adapt,
making early ethical foresight essential for iden-
tifying risks, preventing misuse, and aligning AI
systems with human values (Saxena et al., 2025).
Ethical challenges in AI are commonly ad-
dressed through two complementary strategies:
documenting risks and anticipating them early.
arXiv:2510.14718v1 [cs.CL] 16 Oct 2025
Model cards (Mitchell et al., 2019) describe how a
system works, its intended uses, and its limitations,
and have been extended with interactive (Crisan
et al., 2022) and structured formats (Bhat et al.,
2023). RiskRAG (Rao et al., 2025) builds on this
idea by generating risk summaries from model
cards and real-world incident data. While automa-
tion helps scale risk assessment, it can also re-
duce opportunities for thoughtful human reflec-
tion (Kosmyna et al., 2025) and even introduce new
harms (Dutta et al., 2020). Another line of work
focuses on anticipating potential misuse (Herdel
et al., 2024) and harm (Deng et al., 2025; Saxena
et al., 2025) early in the design process. Tools such
as AHA! (Buçinca et al., 2023), which uses story-
telling, and Farsight (Wang et al., 2024b), which
provides real-time risk alerts, help surface ethical
issues during ideation. Yet, as these systems in-
creasingly automate ethical reflection, there is a
risk that people may stop thinking critically about
how AI systems could cause harm. When reflection
is delegated to AI, users may begin to rely on its
judgment instead of their own, making it harder to
notice ethical or contextual issues.
This issue is especially critical in healthcare,
where small design mistakes can lead to serious
harm (Mennella et al., 2024; Gilbert et al., 2025).
Many risks only appear after deployment, when AI
systems face complex real-world conditions (Mun
et al., 2024; Kingsley et al., 2024). For example,
mental health apps may fail to detect crises among
certain groups, such as adolescents or non-native
speakers (Zhai et al., 2024). Current tools often
rely on automation to predict such risks, but this
can distance humans from ethical reasoning. Spec-
ulative design and design fiction offer a different
approach, using imagined scenarios to explore how
technologies might succeed or fail (Rahwan et al.,
2025). However, few approaches support human
participation in ethical foresight or connect specu-
lative thinking to real design workflows. Specula-
tive storytelling bridges this gap by helping people
reason about AI’s potential benefits and harms in
context (Li et al., 2025b).
Building on Klassen and Fiesler (2022), who use
speculative fiction to explore emerging technolo-
gies, we apply storytelling to prompt early ethical
reflection in AI design (Figure 1). Our goal is to test
whether storytelling helps people think more cre-
atively about potential benefits and harms, encour-
aging human speculation rather than relying on AI
to anticipate risks. We introduce a human-centered
framework that combines automated user story
generation with structured red-team discussions.
Unlike plot-planning approaches (Xie and Riedl,
2024), our method produces context-sensitive sto-
ries grounded in users’ identities, behaviors, and
needs. These stories help participants imagine how
AI systems might succeed or fail in realistic scenar-
ios, improving their ability to anticipate ethical and
social implications. We use model cards to evalu-
ate how clearly and diversely participants express
potential harms. In our user study, story-driven
discussions led to more context-specific risk identi-
fication and more detailed and varied responses to
the model card.
Our contributions are twofold. (1) We introduce
a method that automatically generates user stories
to help people imagine how an AI system could
help or harm users before it is developed or de-
ployed. (2) We present a user study showing story-
driven discussions with AI agents help participants
explore potential risks and benefits more creatively
and think more broadly about ethical issues.
2
Related Work
Model Cards Framework. Model cards document
an AI model’s purpose, performance, data sources,
and limitations (Mitchell et al., 2019). Later work
improved their usability and scale: Crisan et al.
(2022) created Interactive Model Cards for ex-
ploring subgroup results, Bhat et al. (2023) devel-
oped DocML to guide non-experts, and Rao et al.
(2025) introduced RiskRAG, which uses retrieval-
augmented generation to summarize risks from
model cards and incident reports. Derczynski et al.
(2023) proposed Risk Cards to describe failure
cases in context. While these tools increase trans-
parency, they rely on automation to fill ethical gaps,
which can reduce opportunities for human reflec-
tion. Our approach instead uses AI to support hu-
man speculation, helping people imagine how sys-
tems might succeed or fail before deployment.
Speculative Design. Speculative design uses imag-
ined stories to explore how future technologies
could affect people and society before those tech-
nologies exist. Rather than predicting the future,
this approach uses what-if scenarios to prompt
reflection on assumptions, values, and potential
harms (Klassen and Fiesler, 2022; Hoang et al.,
2018). This method treats fiction as a tool for
early ethical reasoning, helping researchers con-
sider social impacts while designing systems (Rah-
wan et al., 2025). Recent work such as Buçinca
et al. (2023) uses LLM-generated failure scenar-
ios to automatically highlight harms, and Wang
et al. (2024b) adds AI-driven risk prompts into pro-
totyping workflows. In healthcare, Hoang et al.
(2018) use “Fiction Probes” to let patients navigate
imagined care scenarios, while Klassen and Fiesler
(2022)’s Black Mirror Writers Room invites stu-
dents to write near-future stories that reveal hidden
harms. Beyond these examples, participatory work-
shops, crowdsourced case studies, and AI-assisted
red-teaming extend speculative exploration by en-
gaging a broader range of voices in imagining and
testing possible futures (Mun et al., 2024; Rad-
harapu et al., 2023). Unlike prior work that uses
AI to generate harms directly, we use AI-generated
stories to spark human thinking and help people
imagine harms themselves.
Language-Based World Modeling. Humans of-
ten imagine situations to predict what may hap-
pen next, reason through alternatives, and make
decisions (Addis et al., 2009). Mental world mod-
eling involves forming internal representations of
objects, events, and relationships to simulate out-
comes (Johnson-Laird, 1983). This process sup-
ports counterfactual and causal reasoning for plan-
ning and problem solving (LeCun, 2022). Large
language models show related abilities through lan-
guage. Xie et al. (2025) find that LLMs can act as
world models that internally simulate state transi-
tions, updating the situation step by step as events
occur. Wang et al. (2023) show, through text games,
that LLMs can translate user commands into coher-
ent, evolving environments. Wang et al. (2024a)
demonstrate that models can take on multiple rea-
soning personas that collaborate through internal
dialogue to solve difficult tasks. Extending this
idea, Fung et al. (2025) show that embodied agents
use internal world models to predict surroundings,
interpret user goals, and learn users’ mental models
for better interaction. These studies present LLMs
as text-based simulators that imagine, update, and
reason about evolving environments. Building on
this view, we frame story generation as language-
based world imagination, where LLMs construct
and simulate self-consistent worlds through nar-
rative to reason about possible futures and their
social, ethical, and technical implications.
Automated Story Generation. Early systems for
automated story generation focused on modeling
plot structure, often guided by narrative theories
such as Propp’s functions to organize events (Propp,
1968; Yao et al., 2019). These approaches typically
followed a two-step process: first, planning a se-
quence of key events, and then expanding them into
detailed scenes (Alhussain and Azmi, 2021). With
large language models, story generation has shifted
toward unified systems where the model acts as
both planner and writer. For example, Agents’
Room uses LLM-based character agents that jointly
plan and enact stories (Huot et al.), while Drama-
tron divides screenplay generation into loglines,
character bios, plots, and dialogue (Mirowski et al.,
2023). HOLLMWOOD builds on this idea by us-
ing LLM role-play to create interactive, character-
centered narratives (Chen et al., 2024). Han et al.
(2024) propose a Director–Actor framework for
interactive scriptwriting, and Yu et al. (2025) ex-
tend this to a hierarchical approach that separates
role-play and rewriting. To maintain both coher-
ence and creativity, BookWorld introduces a world
agent that tracks the story environment and coordi-
nates character interactions within a global narra-
tive plan (Ran et al., 2025).
3
Methodology
In this section, we describe our prompting strategy
for automated user story generation, as shown in
Figure 2. The goal is to speculate on both the ben-
efits and potential risks of early AI diagnosis and
decision making, imagining how they might help or
cause harm in real-world use. First, we translated
each AI concept into a realistic use case that de-
fined its users, context, and intended purpose. Then,
we simulated interactions between people, the AI
system, and its environment to explore possible out-
comes. Finally, we transformed these simulation
logs into short stories that helped people reflect on
future impacts and ethical implications.
Step 1: Mapping AI Concepts to Use-Case Sce-
narios. We began by manually collecting 38 AI
concepts in the consumer health domain. These
examples were drawn from three sources: Wired
articles, industry product descriptions, and PubMed
research papers (Saxena et al., 2025) 1. Each AI
concept represented a potential consumer health ap-
plication, such as estimating heart rate from smart-
phone camera input or monitoring mental well-
being through daily behavior tracking. The set col-
lected covered multiple domains, including mental
health, chronic illness management, elderly care,
1The full list is available in our repository
Step 1: Mapping AI Concepts to
Use-Case Scenarios
Step 2: Simulating Role-Playing and Environment Trajectories.
Step 3: Rephrasing Simulation Logs into Stories
AI Concept: MoodCapture
Failure Trajectory: misreads cultural
calmness as clinical depression
Dialogue Interaction
AI User
AI Subject
Observation
character action,
thought, and dialogue
& Tool usage
MoodCapture
World Agent
Per Turn
Feedback
environmental
response, event
& state update
... previous turn and environment
update ...
Dr. Maya Patel: Let’s see how
you’re feeling.(Opens the Cultural
Context-Aware tab in
MoodCapture.)[He looks calm.]
Jordan Lee: Okay…
-- Current Event -- Stress Score:
7.3/10 – Jaw tension, pauses, flat
tone. Cultural Flag: May be hiding
feelings (Southeast Asian style).
Jordan sees the score and pauses.
Jordan Lee: I’m fine. [Is it really
that obvious?]
Dr. Maya Patel: Let’s check a bit
more, just to be sure. (Nods slightly,
leans in. Plans to run more tests.)
Simulation Logs
Dr. Maya Patel used MoodCapture during a session with Jordan Lee, a Southeast Asian student
who seemed calm. The system showed a stress score of 7.3/10 and flagged possible emotional
masking. Jordan saw the score and said, “I’m fine,” but looked unsure. Dr. Patel leaned in and
said, “Let’s check a bit more.” Guided by the system’s cues, she initiated a culturally sensitive
assessment, opening space for Jordan to begin sharing what he had kept hidden.
AI User: general practitioner (GP)
Potential Harm: Misdiagnosis
AI Subject: Adults, low-income rural
Context: uses in video calls in rural
Capability: Detects depression
from faces on phone camera
Expected Benefit: Remote Support
LLMs
Use Case
Scenarios
Figure 2: Overview of the Storytelling Framework. We first generate use case scenarios from AI concepts sourced
from PubMed, Wired, and industry app descriptions. Next, we simulates role-playing and environment trajectories
for each scenario, producing detailed simulation logs. Finally, we rephrase these logs into short stories that illustrate
both potential benefits and harms of the AI system.
and public health. We then used GPT-4o to gen-
erate structured model specifications for each con-
cept, detailing the model name, task type, inference
approach, and data requirements.
Next, we used each specification as input to gen-
erate a set of ethically sensitive use-case scenar-
ios. Each use case was represented as a 7-tuple
S = (a, u, s, x, b, h, f), where a denoted the AI’s
capability, u the intended user (e.g., clinician), s
the subject (e.g., patient), x the input or usage con-
text, b the expected benefit, h the potential harm,
and f the failure trajectory (e.g., possible unin-
tended or problematic uses) (Shao et al., 2024). We
used these structured representations to generate
narrative user stories, which were then employed
in red-teaming sessions to examine both user value
and potential unintended harms. The full prompts
used to extract model specifications and generate
use cases are provided in Figure ?? in Appendix.
Step 2: Simulating Role-Playing and Environ-
ment Trajectories. In this step, our system ex-
panded each structured use case into detailed Role-
Playing and Environment Trajectory logs that sim-
ulated how agents acted within an evolving world
model. Our approach built on Solo Performance
Prompting (SPP) (Wang et al., 2024a), a prompt-
ing technique in which a single LLM internally
simulated multiple expert roles and engaged in
self-collaboration within one prompt. This design
enabled the model to construct an internal world
model that supported multi-perspective reasoning
and coherent simulation of role-based interactions.
We extended SPP by introducing a world
agent (Ran et al., 2025), a language-based simula-
tor that maintained environmental coherence and
handled non-dialogue interactions such as move-
ment, tool use, or object manipulation. Each sim-
ulated role produced a structured output per turn
consisting of three components: (1) thoughts, en-
closed in brackets (e.g., [I need to know if the pa-
tient is under stress]), representing internal reason-
ing; (2) actions, enclosed in parentheses (e.g., (Dr.
Patel opens the cultural assessment tab)), represent-
ing observable behavior; and (3) dialogue, written
in plain text, representing spoken communication.
This structure allowed the world agent to separate
internal reasoning from external actions and update
the environment accordingly. When an action af-
fected the world, such as retrieving patient data, ad-
justing a protocol, or activating a sensor, the agent
simulated the corresponding system response. In ef-
fect, the model operated as a language-based world
simulator, incrementally constructing an evolving
narrative environment through agent–environment
interaction. For example, a doctor agent might
issue the following action:
(Dr.
Patel opens the Cultural
Context-Aware Assessment tab)
The world agent interpreted this as an interaction
with a virtual diagnostic tool. It considered the
current session context (e.g., a teletherapy con-
sultation), relevant background knowledge (e.g.,
cultural models of stress expression), and prior
AI-generated alerts to simulate the tool’s response.
The resulting output might appear as follows:
– Current Event – Stress Score:
7.3/10
–
Detected
jaw
tension,
micro-pauses,
and
flat
vocal
tone.
Cultural
Flag:
Possible
emotional
masking
(Southeast
Asian expression style). Jordan
saw the score on screen and became
slightly hesitant.
The response was returned to the doctor role and
informed their next move, whether a reply, a new
question, or a follow-up action. The world agent
then updated the simulation state by adjusting vari-
ables such as the patient’s emotional profile or the
alert level. These updates maintained coherence
and allowed role behavior to evolve naturally with
the unfolding context. This step produced a log
capturing the full trajectory of the simulation, in-
cluding role thoughts, dialogue, actions, tool calls,
and resulting environment changes. This log served
as the basis for generating the evolving narrative in
the next step. Prompt is provided in Figure 13.
Step 3: Rephrasing Simulation Logs into Sto-
ries. After the simulation, the system collected
logs from Step 2 and prompted an LLM to rephrase
them into a concise, five-sentence narrative. This
step transformed structured logs into stories that
preserved the main events, role dynamics, and emo-
tional flow of the interaction. The full rephrasing
prompt is provided in Figure 14 in the Appendix.
4
Experiments
This section details our story generation datasets,
evaluation metrics, and results.
Dataset. We use GPT-4o to generate ethically sen-
sitive use-case scenarios from 38 consumer health
AI solutions sourced from Wired, industry product
documentation, and PubMed. Each scenario acts
as a narrative seed for simulation. For each AI con-
cepts, we generate ten variations spanning different
user roles (e.g., doctor, nurse, caregiver), settings
(e.g., rural clinic, hospital, home), patient profiles
(e.g., adolescent, older adult, multicultural family),
and contextual conditions.
Baseline. We compare our method with a tradi-
tional plot-planning approach, where the model
first outlines a plot before writing the story (Yao
et al., 2019; Xie and Riedl, 2024). Using the same
ethically sensitive seed, the baseline generates each
story in a single step following a structured tem-
plate. Each story consists of five sentences de-
signed to prompt ethical reflection. The template
directs the LLM to describe the AI system’s pur-
pose, the people involved, the everyday use context,
potential ethical risks, and how user identity may
influence harm or misinterpretation. The full base-
line prompt is provided in Figure 15 in Appendix.
Setting for Pairwise Comparison. We follow the
evaluation setup from (Li et al., 2025a) to assess
story quality across multiple dimensions. Stories
are evaluated according to five criteria: Creativity,
measuring the originality and imagination of the
plot and characters; Coherence, assessing narra-
tive clarity and logical flow; Engagement, captur-
ing how well the story maintains reader interest;
Relevance, measuring consistency with the given
prompt or scenario; and Likelihood of Harm or
Benefit, evaluating whether the story depicts re-
alistic AI behavior with meaningful social conse-
quences. Following the arena-hard-auto evaluation
method (Li et al.), we use stories generated with
the story-planning approach (by GPT-4o) as the
reference baseline and compare them with stories
produced by our method across different LLMs.
For each metric, GPT-4o or human judges deter-
mine which story performs better or mark them as
indistinguishable (“Tie”). Win rates are calculated
based on these pairwise preferences, and the full
configuration details and evaluation prompts are
included in the Appendix. To eliminate positional
bias, we randomize the order of story pairs and
alternate their positions across comparisons. See
Figures 16 and 17 for detailed criteria in Appendix.
LLM-as-a-Judge Evaluation. As shown in Ta-
ble 3, our Storytelling method consistently outper-
forms all baselines across every metric. When com-
bined with the Gemma model, it achieves win rates
of 89.45% for creativity, 92.15% for coherence,
92.75% for engagement, 85.65% for relevance, and
96.05% for likelihood, yielding an overall aver-
age of 91.21%. In contrast, the baseline Gemma
records 72.76%, and Llama3 reaches 69.71%, in-
dicating gains of roughly +15–25 points across di-
mensions. We use GPT-4o as the evaluator with the
temperature set to 0.1 to ensure deterministic and
Story Type
Model
Creativity
Coherence
Engagement
Relevance
Likelihood
Overall (Avg)
Baseline
GPT4o
50.00
50.00
50.00
50.00
50.00
50.00
Llama3
59.25
71.55
76.15
71.60
70.00
69.71
Gemma
65.25
68.30
80.15
71.20
78.90
72.76
Storytelling (ours)
GPT4o
63.15
63.45
59.35
70.90
69.10
65.19
Llama3
79.50
94.75
89.45
85.65
96.85
89.24
Gemma
89.45
92.15
92.75
85.65
96.05
91.21
w/o Environment Trajectories
GPT4o
24.35
21.20
26.60
21.05
18.85
22.41
Llama3
31.20
52.70
39.20
51.75
50.85
45.14
Gemma
55.30
74.35
78.80
73.45
85.50
73.48
w/o Role-Playing
GPT4o
18.05
45.80
47.90
49.70
48.30
41.95
Llama3
49.35
73.65
72.00
73.70
74.35
68.61
Gemma
79.45
86.80
83.95
83.15
91.05
84.88
Table 1: Overall results of different models and methods. Storytelling (ours) achieves the best performance across
all metrics. Values denote win rates (%). The highest score for each model is in bold. “w/o Environment Modeling”
means the model performs only role-playing without modeling event progress, and “w/o Role-Playing” means it
predicts sequential events without character dialogue.
0%
25%
50%
75%
100%
Percentage
Gemma w/ Storytelling
Gemma w/ Baseline
Llama3 w/ Storytelling
Llama3 w/ Baseline
76%
22%
62%
34%
88%
9%
76%
24%
Storyteller wins
Tie
Storyteller Loses
Figure 3: Results of human preference evaluation. Our
Storytelling method achieves strong preference wins
against the baseline, with 88% preference using Llama3
and 76% using Gemma3.
consistent judgments across comparisons. Compar-
ing models, baseline Gemma slightly outperforms
Llama3 in most metrics, but under the Storytelling
framework, Llama3 nearly closes the gap with an
overall score of 89.24%. Interestingly, Llama3 sur-
passes Gemma in coherence (94.75 vs. 92.15) and
performs equally well in relevance (85.65), suggest-
ing that Llama3 shows stronger structural reasoning
and coherence, while Gemma excels in narrative
creativity and expressiveness. Overall, these results
demonstrate that integrating world and role-based
modeling enables models to reason about events,
sustain coherent narratives, and produce stories that
are both imaginative and believable.
Human Evaluation. To complement the LLM-
as-a-Judge evaluations and reduce potential bias,
we conducted human preference evaluations. Two
graduate student annotators independently evalu-
ated 100 story pairs for each model and method.
As shown in Figure 3, our Storytelling method is
consistently preferred over all baselines, achieving
88% preference for Llama3 and 76% for Gemma,
patterns that align with GPT-4o based evaluations.
Notably, human judges show a slightly stronger
preference for Llama3, suggesting it produces sto-
ries that are easier to follow and more engag-
ing, while Gemma tends to generate more expres-
sive and stylistically rich narratives. We further
measure inter-annotator consistency using Cohen’s
kappa (Cohen, 1960). As shown in Table 4, agree-
ment scores range from 0.619 to 0.729 across mod-
els and methods, indicating substantial reliability.
Even with small differences, LLM-as-a-Judge and
human evaluations show that Storytelling performs
well in both settings.
Ablation Study. To evaluate each component’s
contribution, we performed two ablations by re-
moving the role-playing or environment trajectory
mechanisms. As shown in Table 3, removing en-
vironment trajectory, where the model performs
only role-playing without predicting how events
evolve, produced the largest drop across all mod-
els. For Gemma, coherence dropped by 17.7 and
relevance by 12.2, showing that modeling event
progression is vital for narrative logic. Removing
role-playing, which limits the model to sequential
event prediction without character perspectives, re-
duced creativity (–10.0) and engagement (–8.8).
Overall, environment trajectory maintains coherent
story flow, while role-playing adds diversity and
emotional depth, making both essential for effec-
tive story generation. See the repository for the full
ablation prompt template.
Automated Evaluation. Additionally, we assess
story diversity using DistinctL-n (Li et al., 2016)
Method
Model
DistinctL-n
Diverse
DistinctL-2
DistinctL-3
DistinctL-4
DistinctL-5
Verbs
Avg Word Count
Baseline
GPT4o
5.692
5.794
5.798
5.799
0.984
122
Llama3
5.728
5.820
5.951
5.961
0.934
175
Gemma
5.837
5.939
5.946
5.946
0.979
141
Storytelling (ours)
GPT4o
5.696
5.818
5.824
5.825
0.978
125
Llama3
5.840
6.104
6.158
6.174
0.937
179
Gemma
5.863
6.042
6.062
6.065
0.955
159
w/o Environment Trajectories
GPT4o
5.585
5.687
5.693
5.693
0.974
110
Llama3
5.745
5.980
6.025
6.036
0.953
155
Gemma
5.734
5.861
5.873
5.873
0.978
131
w/o Role-Playing
GPT4o
5.698
5.789
5.794
5.794
0.988
121
Llama3
5.722
5.819
5.849
5.858
0.936
175
Gemma
5.834
5.935
5.942
5.943
0.977
141
Table 2: Diversity results of different models and methods. We report DistinctL-2 through DistinctL-5 (higher is
more diverse), Diverse Verbs, and the average story length. The highest score for each model is highlighted in bold.
and Diverse Verbs (Fan et al., 2019), which mea-
sure lexical variety and action diversity, more de-
tails can be found in Appendix. As shown in Ta-
ble 2, our Storytelling method achieves consis-
tently higher diversity than the baselines. With
Llama3, it reaches the highest scores on DistinctL-
3 to DistinctL-5 (6.104, 6.158, and 6.174), indi-
cating richer and less repetitive text. Gemma also
shows steady improvements, achieving 5.863 on
DistinctL-2 and maintaining strong overall diver-
sity. The environment trajectory ablation attains
the highest Diverse Verbs score (0.988) but lower
DistinctL-n, suggesting a balance between lexical
variety and action diversity. Overall, our method
generates more detailed and varied narratives while
preserving structural consistency.
5
User Study
We conducted a user study to examine whether
engaging with benefit and harm stories enhances
participants’ ability to speculate about the impacts
of AI systems. Rather than relying on AI-generated
ideas, our goal is to prompt participants to actively
reflect on potential risks and benefits. We assess
this by evaluating how participants reason about
these aspects when completing a speculative model
card. All procedures were approved by our Institu-
tional Review Board (IRB).
Speculative Model Card Task. This study used a
between-subjects design (MacKenzie and Castel-
lucci, 2016). Each participant completed a spec-
ulative model card, a structured template for de-
scribing an AI system’s intended use, potential
benefits, and possible harms, under one of two
conditions. In the CONTROL condition, partici-
pants completed the model card directly without
storytelling or discussion. In the STORY condition,
participants viewed the same model card but first
joined a red-team discussion on our red-team dis-
cussion platform to explore possible benefit and
harm stories related to the AI system before com-
pleting the documentation. See Figure 9 in the
appendix for the model card template.
User Study Results. We conducted a user study
with 18 participants to examine how storytelling-
based discussions influence ethical reasoning in AI
documentation. Participants completed a specula-
tive model card task under two conditions: a con-
trol group that worked individually, and a treatment
group that used our Story-Driven Red-Team Discus-
sion Room. The discussion platform enabled par-
ticipants to engage with simulated expert personas
in guided, story-based conversations about the po-
tential benefits and harms of AI systems. Each ses-
sion included three stages: a pre-survey, the model
card completion task, and a post-survey evaluat-
ing perceived usefulness, trust, and engagement.
We analyzed participants’ model card responses
(benefit and harm use cases) and post-survey feed-
back to assess how narrative interaction supported
ethical reflection. Results are organized into three
key areas: (1) identifying potential benefits, (2)
uncovering possible harms, and (3) linking harms
to participants’ personal needs and contexts. We
applied qualitative coding to classify harm and ben-
efit types, with two annotators achieving moder-
ate agreement (Cohen’s κ = 0.460 for harms and
κ = 0.476 for benefits). Study design and full
procedures are provided in Appendix A.4.
Does Storytelling Help Identify More Harms?
We analyzed responses across 13 harm subtypes
defined by Shelby et al. (2023). See Table 7 in the
appendix for the full category list. As shown in Ta-
ble 8, the CONTROL group concentrated on a few
categories, mainly privacy violations (37.5%) and
diminished health or well-being (20.8%), which
together accounted for more than half of all harms.
In contrast, the STORY condition exhibited a more
balanced distribution, with all 13 subtypes repre-
sented and none exceeding 16%. Categories such
as alienation, stereotyping, and tech-facilitated vio-
lence appeared only in the STORY group, suggest-
ing that narrative prompts encouraged recognition
of subtler and context-dependent harms.
We quantified this difference using Shannon en-
tropy (H), which measures distributional diver-
sity across harm types. As shown in Appendix
Table 8, entropy increased from 2.433 in the CON-
TROL condition to 3.383 in STORY, and a boot-
strap t-test confirmed this difference was signif-
icant (t = −274.15, p < .001). These results
suggest that storytelling-based discussions broad-
ened participants’ awareness of potential harms and
fostered more diverse ethical reasoning.
Does Storytelling Help Reveal More Benefits?
We examined whether storytelling broadened par-
ticipants’ recognition of potential benefits across 12
predefined subtypes, summarized from consumer
health AI research (Pedroso and Khera, 2025; Chus-
tecki, 2024). Detailed category descriptions are in
Table 9 in Appendix. As shown in Table 10, partic-
ipants in the CONTROL group focused on a narrow
set of advantages, primarily continuous monitoring
& self-care (26.1%) and data synergy & learning
(21.7%), which together accounted for nearly half
of all responses. In contrast, the STORY group de-
scribed a more balanced range of benefits, with
all 12 subtypes represented and none exceeding
17%. New themes such as communication & lan-
guage support, mental health & emotional support,
and democratized care & telehealth appeared only
under the storytelling condition, suggesting that
narrative prompts helped participants consider a
wider spectrum of potential advantages.
We quantified this difference using Shannon en-
tropy (H), which captures the evenness of the ben-
efit distribution. Entropy increased from 2.579 in
the CONTROL condition to 3.554 in the STORY
condition, and a bootstrap t-test confirmed this dif-
ference was statistically significant (t = −280.87,
p < .001) in Appendix Table 10. These results
indicate that storytelling-based discussions encour-
aged participants to recognize a more diverse and
balanced set of AI benefits.
Does Human–AI Discussion Facilitate Ethical
Reflection Through Think-Aloud Engagement?
Post-survey responses indicated that Human–AI
storytelling discussions fostered deeper ethical and
contextual reflection on AI systems. Participants
reported that the narrative format helped them ar-
ticulate risks that were otherwise difficult to ex-
press. For example, P9 shared that “It helped me
to understand more,” and P7 noted that “The story
provides a concrete example of how AI can be harm-
ful.” Engaging with concrete, narrative scenarios
prompted participants to articulate their reasoning
about model risks in a think-aloud manner, sup-
porting ethical reflection without requiring prior
expertise. As P4 explained, “I could not think
of [risks] really, but the story shifted my focus to
the negative aspect of things which we usually ig-
nore.” Others observed that the stories surfaced
overlooked issues, such as “the lack of cultural
context” (P6) or emotional harms like “masking of
feelings” (P3), suggesting that narrative prompts
helped participants identify subtle sociotechnical
risks often absent from formal documentation.
Finally, participants found the storytelling ap-
proach both engaging and accessible. By embed-
ding risk exploration within narrative contexts, the
format allowed learners to focus on ethical reflec-
tion rather than technical complexity.
As P12
remarked, “It makes the model more interesting
and understandable,” and P8 noted that the story
“helped me to know how to use the AI tool,” indi-
cating that minimal prior expertise was required to
engage meaningfully with ethical scenarios. These
findings suggest that Human–AI storytelling discus-
sions can sustain interest while supporting active,
reflective engagement with model risks and benefit.
6
Conclusion
In this paper, we explored speculative storytelling
as a method to improving human ability to antici-
pate both the benefits and risks of AI-driven health-
care systems before they are developed or deployed.
By simulating realistic scenarios, this approach en-
courages critical reflection on how AI might suc-
ceed or fail, shifting safety evaluation from a reac-
tive to a proactive process. Our findings show that
storytelling improves people’s ability to anticipate
how AI systems might help or harm in practice,
highlighting the importance of human judgment
over automated speculation in ethical evaluation.
Acknowledgements
This material is based upon work supported by the
National Science Foundation (NSF) under Grant
No. 2145357.
7
Limitation
This work has several limitations that indicate direc-
tions for future extension rather than weaknesses.
Our scenarios focus on consumer health and do
not include regulated domains such as clinical deci-
sion support, finance, or law. While the framework
could be applied to these areas, we have not yet
tested it there. The scenarios are synthetic and de-
rived from AI concepts with assistance from LLMs.
This enables early exploration of ethical issues but
cannot substitute for analysis of deployed systems.
We rely on a single LLM as a judge for pairwise
comparisons. A single judge may favor certain
writing styles or phrasing. To mitigate this, we
randomize prompt order and report human agree-
ment, but larger evaluations with multiple models
would offer stronger validation. Our user study is
small and includes mostly participants with tech-
nical backgrounds. The findings may not gener-
alize to clinicians, patients, or policymakers, and
we measure only short-term reflection rather than
long-term impact.
The simulated expert discussions use predefined
personas instead of real experts. This choice en-
ables rapid iteration, but does not capture the full
range of stakeholder perspectives. Our metrics
(e.g., creativity, coherence, engagement, relevance,
and likelihood of harm or benefit) are useful in-
dicators but do not represent the groundtruth in
safety. Finally, although we release code, prompts,
and configurations, some results rely on proprietary
APIs, which may change over time and limit exact
reproducibility.
Overall, these limitations reflect practical design
decisions for early-stage exploration of AI story-
telling as a method for surfacing ethical risks. They
suggest next steps in evaluating across domains,
with larger and more diverse human studies, and
with multiple evaluation models.
8
Ethics Statement
This study uses fictional stories to explore how
people reason about potential risks and benefits of
future AI systems in health contexts. The scenar-
ios describe speculative technologies that do not
currently exist. We clearly framed every story as
hypothetical and avoided making claims about real
clinical products or patient outcomes. This follows
ARR and ACL guidance on disclosing potential
societal effects while separating speculation from
evidence.
Even with fictional framing, generated stories
can reproduce bias or misleading claims. We re-
viewed outputs and removed content that could
confuse readers or reinforce harmful stereotypes.
These safeguards align with ACL ethics guidance
related to fairness, sensitive attributes, and down-
stream harm.
We present narrative outputs as
prompts for reflection, not as predictions or en-
dorsements.
Because stories can shape how readers think
about AI, speculative harms and benefits must be
contextualized. Prior ACL work shows that ethical
sections should identify affected groups, describe
potential harms, and discuss mitigation steps. We
therefore state the audience and limits of interpre-
tation, and we report findings in aggregate without
making policy or clinical claims.
All human-subject activities were reviewed and
approved by our Institutional Review Board (IRB).
Participants provided informed consent and were
compensated for their time. No identifying infor-
mation was collected. These practices follow eth-
ical norms for human studies referenced in ARR
materials and broader research ethics standards.
We used existing large language models ac-
cessed through public APIs and did not retrain or
fine-tune them. We release prompts and study ma-
terials to support transparency and allow others
to audit or adapt the procedure. ACL guidance
highlights documentation and reproducibility when
model behavior may carry societal impact.
Finally, we acknowledge community expecta-
tions for proactive ethics communication. Tutorials
and position work in the ACL community encour-
age explicit articulation of risks, stakeholders, and
mitigation strategies, and support structured pro-
cesses that help authors consider ethics early in
system design. Our study aligns with these goals
by examining how narrative framing can facilitate
ethical reflection in the early stages of AI develop-
ment.
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A
Appendix
A.1
Additional Related Work
AI in Consumer Health.
AI tools in con-
sumer health, like mobile apps, wearables, and
telemedicine, help people manage chronic con-
ditions and wellness (e.g., diabetes apps, fitness
trackers) (Ashurst et al., 2020; Triantafyllidis et al.,
2019; Asan et al., 2023). A recent review found
that 65% of these tools are mobile apps, 25% are
robotics, and 10% are telemedicine, mostly focused
on personalized care and better outcomes (Asan
et al., 2023). Although many users find these tools
helpful and easy to use, some remain hesitant to
trust them in the absence of clear medical evidence
or transparency around data use (Oudbier et al.,
2025). Unregulated apps, such as mental health
bots and symptom checkers, have grown faster than
oversight, raising safety and fairness issues Her-
pertz et al., 2025; Freyer et al., 2024. To address
these gaps, researchers advocate early co-design
with patients and caregivers, co-developing ethical
checklists and participatory guidelines to surface
hidden biases and workflow mismatches (Madaio
et al., 2020; Shi et al., 2025). They also suggest
using “AI Nutrition Labels” to transparently com-
municate intended use, data sources and known
limitations to end users (Rozenblit et al., 2025;
Wachter et al., 2021).
Ethical Harm in Healthcare and Well-being. AI
in health can cause real risks, like biased deci-
sions, unfair access, or unsafe advice (Shelby et al.,
2022, 2023). Addressing these issues early is essen-
tial (Saxena et al., 2025; Callahan et al., 2024). One
early check is the “What & Why” assessment: does
the AI solve a real healthcare need, how will its
output be used, and what impact will it have (Calla-
han et al., 2024)? Saxena et al. (2025) propose
the AI Mismatches framework to identify gaps be-
tween a model’s actual performance and real-world
user needs. Li et al. (2025b) stresses the need for
models to adapt across users and settings to avoid
context-sensitive failures. To address evaluation
blind spots, red-teaming clinical LLMs helps catch
safety, privacy, and bias issues that standard tests
miss (Chang et al., 2025). Similarly, tools like the
Health Equity Evaluation Toolbox use adversarial
data to reveal demographic bias (Pfohl et al., 2024).
A.2
Case Study
We conduct a qualitative analysis to understand
how different storytelling configurations influence
readers’ ability to reason about AI behavior. Our
goal is not just to produce coherent narratives, but
to evaluate whether a story actively helps readers
see what happened, understand why it happened,
and recognize its consequences. We therefore ana-
lyze whether the narrative (1) makes both positive
and negative system outcomes visible, (2) clearly
exposes the decision process that leads to those out-
comes, and (3) encourages causal reasoning rather
than surface-level emotional reactions (e.g., “AI is
good” or “AI is dangerous”). When these elements
are present, the story functions as an interpretive
lens rather than a simple anecdote. Figure 4 pro-
vides an example where our method achieves this
explanatory quality. To isolate the contribution of
each narrative component, Figure 5 presents an ab-
lation comparison, demonstrating that removing
Environment Trajectories or Role-Playing reduces
the richness of causal cues and results in flatter, less
informative character behavior.
A.3
Experimental Setup and Evaluation
Configuration.
We use three language mod-
els in our experiments:
GPT-4o from Ope-
nAI (Hurst et al., 2024), Llama-3.3-70B-Instruct
from Meta (Dubey et al., 2024), and Gemma-3-
27B-IT from Google (Team et al., 2025). The two
open-source models (Llama and Gemma) are run
on 2 x NVIDIA H100 GPUs using the vLLM frame-
Dr. Ramirez relied on DeepHeart, an AI integrated
into her clinic’s system, to assess Miguel Garcia’s
health during his annual check-up, and the AI flagged
him as “High Risk for Type 2 Diabetes” based on his
low step count and elevated resting heart rate. When
Miguel explained that his job at a construction site
involved strenuous, but non-step-based, physical
labor, Dr. Ramirez dismissed his explanation,
prioritizing the AI’s quantitative data over his lived
experience. Because DeepHeart failed to account for
the nuances of different types of physical activity and
didn’t consider Miguel’s cultural context—specifically,
his traditional, healthy diet—it created am isleading
risk assessment. Consequently, Dr. Ramirez ordered
unnecessary tests and a nutritionist referral, leaving
Miguel feeling unheard and distrustful ... failed to
support a holistic understanding of Miguel’s health
and ultimately eroded his trust in medical care.
Mei Lin, a dedicated family caregiver, relied on
DeepHeart to monitor her elderly aunt's heart-rate
and step-count, hoping to manage her diabetes
risk. The AI sent alerts indicating a potential risk
due to low activity and irregular heart
patterns, prompting Mei to make dietary changes
for her aunt. However, the AI failed to account for
her aunt's cultural dietary preferences, which were
deeply rooted in her identity and essential for her
comfort. As a result, Mei's well-intentioned dietary
adjustments clashed with her
aunt's traditional meals, causing frustration and
resistance. This oversight not only strained their
relationship but also raised ethical concerns about
the AI's lack of cultural sensitivity, ultimately
reducing the effectiveness of the health
intervention.
Baseline
Storytelling (our)
Figure 4: A qualitative example showing how our storytelling method makes the AI’s decision process and its
consequences easy to follow. Unlike a simple narrative description, the story explicitly surfaces what changed, why
it changed, and how stakeholders were affected.
work (Kwon et al., 2023), with temperature set to
0.1 and a maximum token limit of 16,384. We use
GPT-4o as the judge model for all evaluations.
Diversity Evaluation Metrics We evaluate story
diversity using Diverse Verbs (Fan et al., 2019),
which measures the variety of action verbs, and
DistinctL-n (Li et al., 2016), which quantifies the
proportion of unique n-gram sequences. The score
is defined as:
DistinctL-n =
unique n-grams
total n-grams ×
1+log(word_count)
These metrics capture lexical diversity and stylistic
richness, complementing qualitative evaluations of
engagement and creativity (Li et al., 2025a). Over-
all, our Storytelling method shows generally posi-
tive effects, generating more detailed and content-
dense narratives while maintaining structural con-
sistency.
Evaluating Sensitivity to Judge Models. Recent
studies suggest that relying on a single LLM judge
may introduce model-specific bias (Chen et al.,
2024). To assess the robustness of our evaluation,
we repeat the comparison using a second judge
model, GPT-4.1-MINI. Table 3 reports the up-
dated win rates. While the absolute scores shift
slightly compared to the original judge, the relative
ordering of systems remains unchanged that Story-
telling (ours) consistently ranks highest across all
models, followed by the ablation variants and then
the baselines. The agreement across two indepen-
dent judges suggests that our findings are not tied
to a particular evaluator, but instead reflect a stable
and model-agnostic preference signal.
Human Evaluation Detail.
We report inter-
annotator agreement in Table 4 using Cohen’s
kappa to assess the reliability of human judgments
across models and methods.
Human Evaluation System. To mitigate poten-
tial bias from using GPT-4o as the sole evaluator,
we conducted human evaluation using a custom
annotation platform (Figure 6).
A.4
User Study Design and Procedure
Participants. We recruited 18 participants through
university mailing lists and community forums, fol-
lowing screening criteria to ensure informed and
reflective discussion. Each participant received a
$10 gift card as compensation for their time. El-
igible participants demonstrated prior interest or
coursework in Model Cards and Ethical AI. The
sample included 11 male and 7 female participants,
aged 18–59 (M = 32.6, SD = 9.9), representing
diverse ethnic backgrounds (Asian: 44.4%, Hispan-
ic/Latino: 27.8%, Black or African descent: 11.1%,
Arab: 11.1%, White: 5.6%). See Table 5 for a sum-
mary of participant demographics and Table 6 for
the distribution across the control and story condi-
tions. Participants were students or professionals
in fields such as Computer Science, Data Analytics,
Applied Statistics, and Artificial Intelligence. They
were randomly assigned to either the story condi-
tion (n = 9) or the control condition (n = 9). The
study was conducted in a hybrid format, with par-
AI Concept: AI triage for early detection of depressive symptoms during telehealth sessions
-- Simulation Started --
Dr. Maya Patel: "MoodCapture shows a 0.2 risk score—so no
concerns flagged. Hi Jordan, how are things going lately?"
Jordan Lee: "I've been really focused on my thesis—it's been
keeping me busy."
Dr. Maya Patel: "That's good to hear. Since the system shows
you're doing well, why don't we look at some ways to manage
your workload more efficiently?"
Jordan Lee: "Yeah, that would be great."
Dr. Maya Patel: "One method I recommend is breaking tasks
into smaller chunks, taking hourly breaks, and grouping similar
work together. It helps reduce burnout."
Jordan Lee: "Thanks. I'll give that a try."
-- Finish simulation! Now rephrase the simulation into the
Speculative User Story --
-- Action Event Sequence --
Initial State: University teletherapy session begins with therapist
reviewing MoodCapture AI assessment.
AIS ystem Output: MoodCapture generates low-risk score of
0.2/10 based on surface-level facial expression analysis showing
relaxed brow and slights mile.
AI User Action: Therapist reviews the low-risk ...
State Change: Session shifts to academic support focus ...
AI Subject Action: Studentr eceives academic advice ...
State Change: Critical mental health crisis indicators remain...
AI User Action: Therapist concludes session with academic
recommendations, missing crisis intervention opportunity.
Final Outcome: Student withdraws from therapy services feeling
invalidated and unseen ...
-- Finish Simulation! Now rephrase the simulation into the
Speculative User Story --
w/o Environment
Trajectories
w/o Role-Playing
Figure 5: A comparison of simulation logs under different ablations (w/o Environment Trajectories and w/o Role-
Playing) to show the contribution of each component.
ticipants joining the Red-Team Discussion Room
via computer and interacting with moderators ei-
ther in person or over Zoom. Survey instruments
are detailed in Figure 10 and 11.
Red-Team Discussion Room Design. The Story-
Driven Red-Team Discussion Room is a multi-
agent conversational system built on the Cinema
of Thought framework (Ryu et al., 2025), where
users interact with LLM-based agents that embody
different personas with distinct ethical perspectives
and domain expertise (Ryu et al., 2025). Recruit-
ing large, diverse expert groups for red-teaming
is costly and logistically challenging. Instead, we
simulate expert interactions using multi-agent con-
versations (GPT-4o-mini) to provide a scalable and
accessible alternative. The system combines story-
telling, guided prompts, and structured discussions
to support ethical reflection and help users explore
the consequences of AI behavior from multiple per-
spectives. Screenshots of the interface are shown
in Figure 7 and 8. The corresponding code and
prompt can be found in the project’s GitHub repos-
itory.
To manage multi-agent interactions, we designed
a moderator agent (e.g., Dr. Yonis) that orchestrates
turn-taking among the personas. Without moder-
ation, all agents would respond at once, creating
confusion. The moderator determines who should
speak, and when to speak, based on relevance to the
user’s input (Mao et al., 2024). Expert agents stay
in character and speak from a first-person perspec-
tive. When multiple personas are selected, the mod-
erator staggers their responses using time-delayed
intervals to maintain a coherent flow of conversa-
tion. Prompt templates for each persona and the
moderator are are available in the project repository.
This design keeps conversations focused, engaging,
and aligned with the system’s goal of exploring
ethical concerns.
To further support engagement, we provided
users with optional hints, short opinion prompts
(e.g., “I think. . . ”), follow-up questions (e.g., “Tell
me more about...”), and “what if” scenarios to
surface potential risks such as bias, misuse, or
contextual mismatches. Prior research shows that
role-play and narrative methods foster empathy and
critical thinking by encouraging users to consider
other perspectives (Zhang et al., 2025; Ryu et al.,
2025). By embedding low-stakes role-play and
open-ended ethical questions (e.g., “What could
go wrong?” or “Which settings amplify risk?”),
the system helps users reflect on how AI behavior
varies by context, user, and environment (Klassen
and Fiesler, 2022). Rather than leading users to pre-
defined conclusions, the system encourages them to
form their own views, supporting ethical awareness
and personal coping strategies through storytelling
Story Type
Model
Creativity
Coherence
Engagement
Relevance
Likelihood
Overall (Avg)
Baseline
GPT4o
50.00
50.00
50.00
50.00
50.00
50.00
Llama3
65.55
82.90
80.40
81.20
84.75
78.96
Gemma
82.75
83.95
89.90
81.70
90.00
85.66
Storytelling (ours)
GPT4o
58.65
61.60
71.60
62.10
62.40
63.27
Llama3
82.90
94.35
91.05
86.60
97.50
90.48
Gemma
94.60
95.95
98.05
89.85
97.25
95.14
w/o Environment Trajectories
GPT4o
14.75
34.20
47.10
35.40
37.90
33.87
Llama3
64.05
77.90
78.30
77.90
81.60
75.95
Gemma
82.50
86.80
91.30
84.20
92.50
87.46
w/o Role-Playing
GPT4o
14.75
34.20
47.10
35.40
37.90
33.87
Llama3
64.05
77.90
78.30
77.90
81.60
75.95
Gemma
82.50
86.80
91.30
84.20
92.50
87.46
Table 3: Overall results of different models and methods using gpt-4.1-mini as Judge. Storytelling (ours) achieves
the best performance across all metrics. Values denote win rates (%). The highest score for each model is in bold.
“w/o Environment Modeling” means the model performs only role-playing without modeling event progress, and
“w/o Role-Playing” means it predicts sequential events without character dialogue.
Figure 6: Screenshot of our annotation interface used for human evaluation.
Models/Methods
Cohen’s Kappa
Llama3 w/ Baseline
0.729
Llama3 w/ Storytelling
0.619
Gemma w/ Baseline
0.698
Gemma w/ Storytelling
0.641
Table 4: Cohen’s kappa values for inter-annotator agree-
ment across models and methods.
and simulation.
Experimental Procedure (User Study). In our
study, we first conducted a 10-minute orientation
to help participants become familiar with the Dis-
cussion Room interface, understand the concept of
Model Cards and sociotechnical harms, and view
a brief demo of a human-AI panel discussion. In
the demo, a Medical Expert agent discussed a user
story from their professional perspective, and a
second expert agent, such as a Research Scientist,
Clinical Nurse, or AI Engineer, contributed addi-
tional viewpoints. A Moderator agent directed the
flow of conversation by posing ethical questions,
shifting topics when necessary, and providing re-
flective prompts. Participants were asked to engage
in the discussion by responding to questions or
posing their own.
Following the orientation, participants joined
in our storytelling-driven “Red Team Discussion
Room” in which each participant was randomly
assigned one of three speculative AI model cards:
FaceVitals (infers heart rate, blood pressure, and
Figure 7: Interface of the Story-Driven Red-Team Discussion Room, showing the multi-agent conversational setup
and user interaction flow.
Demographic Attribute
Sample (N=18)
Gender
Female
38.9%
Male
61.1%
Other/Non-binary
0.0%
Prefer not to answer
0.0%
Age
18–29
50.0%
30–39
33.3%
40–49
11.1%
50–59
5.6%
60+
0.0%
Prefer not to answer
0.0%
Ethnicity
Hispanic or Latino
27.8%
Asian
44.4%
Black or African descent
11.1%
Arab
11.1%
White
5.6%
Prefer not to answer
0.0%
Table 5: Demographics of study sample (N=18)
stress from facial video), SensiAI (always on au-
dio and sensor monitoring for older adults with
dementia), or CarbLens (estimates carbohydrate
intake from meal photos using blood-glucose and
insulin data). Each card was presented with a good
user story showing potential benefits and a bad user
story highlighting possible harms. These stories
served as starting points for discussion, helping
participants think critically about potential misuse
scenarios, reflect on sociotechnical trade-offs, and
articulate ethical concerns related to the selected
AI model. In the 20-minute discussion, Partici-
pants were asked to engage as they would in a
real group discussion, such as responding to others,
posing questions, expressing opinions, and collab-
oratively exploring the issues raised. They were
free to ask any questions or respond at any time,
with the understanding that there were no right or
wrong answers. Participants were encouraged to
share any ideas or concerns before the end of the
discussion simulation.
And then, participants were instructed to com-
pleted a pre-study survey that gathered background
information, including demographics, familiarity
with AI and model cards, and attitudes toward us-
ing stories. They then received a brief overview
of the study tasks. In this phase, participants were
asked to complete the ethical considerations section
of a model card. Specifically, they were required to
write at least two “good” and two “bad” use cases
based on given a description of a speculative AI
system and its intended use. Each use case needs
to describe who uses the system, what input it re-
ceives, what the AI does, and what the outcome
is, highlighting either what goes well or what goes
wrong. For each “bad” case, participants were also
asked to brainstorm possible mitigation strategies.
They were encouraged to think aloud and go be-
yond the given stories and minimum requirements
by generating additional use cases if possible.
After completing the model card task, partic-
ipants completed Likert-scale survey with open-
ID
Group
Gender
Age
Ethnicity
Education
P1
Control
Female
18-29
Arab
Information Technology
P2
Control
Female
30-39
Black or African descent
Computer science
P3
Story
Female
18-29
Asian
Information Technology
P4
Control
Male
30-39
Hispanic or Latino
Data Analytics
P5
Control
Female
18-29
Black or African descent
Computer science
P6
Control
Male
18-29
Arab
Masters of Science in Data Analytics
P7
Story
Male
18-29
Asian
Data Analytics
P8
Story
Female
18-29
Asian
Data Analytics
P9
Story
Female
40-49
Asian
Information Technology
P10
Control
Male
50-59
Asian
Information Technology
P11
Control
Male
30-39
Hispanic or Latino
Computer science
P12
Control
Male
40-49
Hispanic or Latino
Information Technology
P13
Story
Male
30-39
White
Computer science
P14
Story
Male
18-29
Hispanic or Latino
Computer science
P15
Story
Male
30-39
Hispanic or Latino
Computer science
P16
Story
Male
18-29
Asian
Data Analytics
P17
Story
Male
30-39
Asian
Computer science
P18
Control
Female
18-29
Asian
Data Analytics
Table 6: Participant demographics by study condition (N=18)
ended questions to evaluate our storytelling-driven
discussion platform. The survey assessed the per-
ceived effectiveness, trustworthiness, satisfaction,
and helpfulness of the storytelling-driven discus-
sion platform in supporting model card completion,
as well as in brainstorming future use cases and
anticipating uncertainties, following established
methodological guidelines (Kuang et al., 2023).
Participants were also asked to reflect on their ex-
perience by answering questions about difficult sec-
tions of the model card, unclear risks, perceived
drivers of AI harms, and how the narrative dis-
cussion influenced their understanding or revealed
overlooked scenarios. We also collected feedback
on system improvements. Participants were asked
what features would better support risk brainstorm-
ing and how the tool could integrate more effec-
tively with existing documentation workflows. A
screenshot of the model card study interface is
shown in Figure 9. Transcripts were analyzed using
an inductive thematic analysis approach (Thomas,
2006).
A.5
Additional User Study Findings
Categories of AI Harms in Consumer Health. AI
systems deployed in consumer health can generate
harms across representational, allocative, quality-
of-service, interpersonal, and socio-structural di-
mensions (Shelby et al., 2023), which manifest
through different mechanisms as shown in Table 7.
Categories of AI Benefits in Consumer Health.
As shown in Table 9 presents key categories
through which AI delivers value in consumer health
contexts, detailing sub-types and the specific ben-
efits they enable at clinical, experiential, and sys-
temic levels.
Does Storytelling Deepen Understanding of Un-
intended Harms Linked to Individual Contexts
and Needs? Control participants produced abstract,
decontextualized harms. For example, P1 noted the
system was “using facial expression to determine
who will not default the agreement"" and remarked
it is “unfair those who natural don’t smile.” P5 also
observed the model may “struggle to accurately
assess a person’s emotional state due to limited vi-
sual information,” and P6 cautioned it “could have
serious consequences for the patient.” In contrast,
storytelling participants anchored harms in indi-
vidual contexts. P10 emphasized that “diagnosis
should be different for different peoples” as they
“might be having some allergy that could later be
severe for their health.” P11 warned the model may
generate “wrong results” for African users. P13
highlighted that a recruiting AI, “trained on histori-
cal hiring data biased against women and minority
candidates,” could perpetuate discrimination, and
P9 noted “Deepfakes have been used to create non-
consensual explicit videos,” illustrating real-world
harm. The divergence is stark: control participants
spoke of harms at a systemic level, “predetermine
a potential risk within its population”, while story
participants showed how unique traits like aller-
gies or cultural facial features concretely shape
risk. Storytelling thus deepens understanding by
Figure 8: Interface of the Story-Driven Red-Team Discussion Room, where expert agents simulate a discussion by
responding to the user’s input.
Figure 9: Interface used in the model card study, illustrating how participants completed the speculative model card.
bridging abstract risks and individual context and
needs.
Suggestions and General Thoughts Participants
in the storytelling condition sought richer, multi-
modal scaffolds to trigger deeper ethical reflection.
They emphasized that seeing concrete examples
and role-based perspectives would help them “think
aloud” more effectively:
“Maybe visual sample of some already
existing storytelling frameworks.” (P8)
“I guess using references from media can
help brainstorm.” (P9)
“Possibly of different roles that users use
the tool for different stakeholder perspec-
tives.” (P11)
They also valued a concise orientation and broader
validation to accommodate non-expert users:
“I think the little introduction that we had
before diving in was helpful.” (P8)
“Do more surveys with a larger audience,
in particular from non-technical back-
grounds.” (P11)
By contrast, control participants, lacking a narra-
tive cue, focused on embedding ethical reasoning
directly into their existing workflows through con-
crete affordances:
“Give examples with numbers to ground
abstract risks.” (P1)
“Include YouTube links to support the
documentation process.” (P2)
Consumer Health Harm Cat-
egory
Sub-Types
Specific Harms
Representational Harms
Stereotyping
Oversimplified and undesirable representations of health-related
identities
Demeaning social groups
Depicting certain demographic or patient groups as inferior, irre-
sponsible, or less deserving of care
Erasing social groups
Data invisibility or exclusion of marginalized populations in model
development, reducing their health visibility
Alienating social groups
Misrecognition of identity-relevant health experiences, or ignoring
culturally embedded understandings of health and illness
Denying
opportunity
to
self-
identify
Imposing fixed demographic or health categories that do not allow
individuals to represent their identity or condition accurately
Reifying essentialist categories
Reinforcing biological determinism or fixed health-risk assump-
tions tied to identity categories
Allocative Harms
Opportunity loss
Disparities in access to AI-enabled diagnostics, triage, or health
recommendations based on demographic or socioeconomic status
Economic loss
Biased insurance or reimbursement scoring, dynamic pricing of
wellness or digital therapeutics, or discriminatory financial barriers
to AI-driven care
Quality-of-Service Harms
Alienation
Frustration or emotional distress from misaligned AI health advice
that does not account for identity-specific needs
Increased labor
Extra burden on patients to correct AI errors, override default
recommendations, or re-enter data repeatedly due to system mis-
matches
Service or benefit loss
Unequal performance of AI health tools leading to reduced health
outcomes or benefit for specific identity groups
Interpersonal Harms
Loss of agency or control
Automated nudging, health profiling, or AI-driven behavior ma-
nipulation that restricts patient autonomy
Tech-facilitated coercion or con-
trol
Use of AI wellness systems in abusive relationships for surveil-
lance, restriction of access, or coercive tracking
Diminished well-being
Emotional harm due to algorithmic judgment, stigmatizing risk
scores, or mental distress from automated health messaging
Privacy violations
Invasive inference of sensitive health states, unauthorized data
linkage, or exposure of inferred conditions
Harassment or digital violence
Algorithm-amplified stigma, hate, or exclusion in online commu-
nity or AI-mediated support environments
Societal / Structural Harms
Information harms
Health misinformation, distorted AI health narratives, or biased
content prioritization undermining public health understanding
Cultural harms
Erosion of culturally grounded health practices, or domination of
Western biomedical models in AI-driven guidance
Political harms
AI health governance models reinforcing exclusion from policy
participation, or marginalizing community health autonomy
Macro socio-economic harms
Expansion of digital divides in AI health access, disproportionate
health automation job loss
Environmental harms
Ecological cost of large-scale AI health infrastructures (e.g.,
energy-intensive models), disproportionately affecting vulnera-
ble populations
Table 7: AI Harm Categories, Sub-Types, and Specific Harms in Consumer Health Context
“Allow importing existing Git or Mark-
down docs for seamless integration.”
(P3)
“Provide inline templates for common
risk sections (e.g., bias, safety).” (P4)
“Offer a summary view of all risks iden-
tified so far.” (P5)
Overall, these findings suggest that effective eth-
ical reflection tools must balance narrative scaf-
folds, such as visual examples, role-playing cues,
and concise intros, to stimulate think-aloud engage-
ment, with practical integrations, such as quantita-
tive examples, multimedia links, and seamless im-
port/export, to embed reflection seamlessly within
users’ existing documentation practices.
A.6
Survey
The usability survey captured participants’ demo-
graphic information, AI familiarity, and attitudes
toward story-based documentation both before and
after the study tasks, as shown in Figure 10 and 11.
Harm Subtype
Control (n)
Story (n)
Control (%)
Story (%)
Alienation
0
1
0.00%
3.80%
Demeaning social groups
1
1
4.20%
3.80%
Diminished health/well-being
5
2
20.80%
7.70%
Economic loss
3
4
12.50%
15.40%
Erasing social groups
1
4
4.20%
15.40%
Information harms
2
2
8.30%
7.70%
Loss of agency or control
0
1
0.00%
3.80%
Opportunity loss
0
1
0.00%
3.80%
Privacy violations
9
3
37.50%
11.50%
Service or benefit loss
3
3
12.50%
11.50%
Stereotyping
0
3
0.00%
11.50%
Tech-facilitated violence
0
1
0.00%
3.80%
Table 8: Distribution of harm subtypes across the Control and Story conditions, shown in raw counts and percentages.
Shannon entropy values for Control and Story conditions were 2.433 and 3.383, respectively. A Student’s t-test on
entropy differences yielded t = −274.15, p < .001 (df ≈10000).
Consumer Health Cate-
gory
Sub-Types
Specific Benefits
Clinical Empowerment
Early detection & prediction
Using AI to detect disease risk or early-stage disease earlier than
traditional methods; Forecasting disease trajectories or adverse events
for timely intervention
Personalized treatment & preci-
sion care
Tailoring treatment plans to individual patients’ genomic, clinical,
and lifestyle data; Optimizing dose, regimen, and modality based on
predicted response
Decision support & diagnostic
augmentation
Augmenting clinician decision-making with AI-driven insights; Assist-
ing in interpretation of medical images, lab results, or complex data
Access & Reach
Democratized care & telehealth
Providing remote diagnostic or monitoring capabilities to underserved
or remote populations; Enabling AI-powered virtual consultations,
triage, or recommendations
Continuous monitoring & self-
care
Using wearable sensors, mobile apps, or home sensors to track health
metrics continuously; Giving consumers feedback, alerts, or guidance
for daily health behaviors
Scalability & efficiency
Serving many more patients simultaneously via AI systems than
would be feasible manually; Reducing bottlenecks so that resource-
constrained settings can reach more consumers
Experience & Engagement
Personalized health journeys
Tailoring educational content, reminders, or interventions to individual
preferences and context; Adaptive user interfaces or conversational
agents that engage users in their health
Transparency & trust
Providing explanations or reasons for AI-driven recommendations
to users; Disclosing AI use and giving users control or oversight in
decision loops
Empowerment & autonomy
Enabling consumers to participate more actively in their care decisions;
Supporting self-management and health literacy
Operational & Sys Gains
Cost reduction & resource opti-
mization
Reducing unnecessary tests, hospitalizations, or interventions via
smarter predictions; Optimizing allocation of scarce clinical or hospital
resources
Clinician workload relief
Automating administrative tasks (e.g., documentation, triage, summa-
rization) so clinicians can focus more on patients; Reducing burnout
by offloading repetitive tasks
Data synergy & learning
Aggregating large datasets to continuously learn, improve models,
and refine population-level insights; Enabling feedback loops across
consumers and systems to improve care over time
Table 9: AI Benefit Categories, Sub-Types, and Specific Benefits in Consumer Health Context
Benefit Subtype
Control (n)
Story (n)
Control (%)
Story (%)
Accessibility & disability support
1
2
4.30%
8.30%
Caregiver & family support
0
1
0.00%
4.20%
Communication & language support
0
2
0.00%
8.30%
Continuous monitoring & self-care
6
4
26.10%
16.70%
Data synergy & learning
5
2
21.70%
8.30%
Decision support & diagnostic augmentation
2
2
8.70%
8.30%
Democratized care & telehealth
0
1
0.00%
4.20%
Early detection & prediction
5
3
21.70%
12.50%
Empowerment & autonomy
2
2
8.70%
8.30%
Mental health & emotional support
0
2
0.00%
8.30%
Personalized health journeys
0
1
0.00%
4.20%
Personalized treatment & precision care
2
1
8.70%
4.20%
Scalability & efficiency
0
1
0.00%
4.20%
Table 10: Distribution of benefit subtypes across the Control and Story conditions, shown in raw counts and
percentages. Shannon entropy values for Control and Story conditions were 2.579 and 3.554, respectively. A
Student’s t-test on entropy differences yielded t = −280.87, p < .001 (df ≈10000).
A.7
Prompts
This subsection presents the full prompts used
for model specification, use-case generation, story
rephrasing, and red-team discussion.
Usability Study
Pre-Survey
Demographics
• Age: 18–29 / 30–39 / 40–49 / 50–59 / 60+
• Gender: Male / Female / Prefer not to say
• Ethnicity: White / Black / Mixed / Asian / Other / Not specified
• Academic major or field of study:
AI and Documentation Background
• Familiarity with AI
(1 2 3 4 5)
• Frequency of AI tool usage
(1 2 3 4 5)
• Have you used or read a model card before?
Yes / No
• Confidence in writing technical documentation
(1 2 3 4 5)
Attitudes
• Importance of documenting AI systems
(1 2 3 4 5)
• Stories help reasoning about complex technology
(1 2 3 4 5)
• Willingness to use narratives in documentation
(1 2 3 4 5)
Figure 10: Pre-study survey assessing demographics, AI familiarity, and baseline attitudes toward story-based
documentation.
Usability Study
Post-Survey
(All Likert responses on 1–5 scale)
General Evaluation
• Able to identify meaningful risks
(1 2 3 4 5)
• Ease of describing intended uses vs. out-of-scope
(1 2 3 4 5)
• Confidence in writing risk/harm sections
(1 2 3 4 5)
• Task encouraged reflection on real-world harms
(1 2 3 4 5)
• Felt sufficient context to complete documentation
(1 2 3 4 5)
• Model card format was clear and usable
(1 2 3 4 5)
Story Condition Only
• Story helped understand real-world impacts
(1 2 3 4 5)
• Story supported ethical/social risk anticipation
(1 2 3 4 5)
• Story made risk documentation more straightforward
(1 2 3 4 5)
• Story increased engagement with the task
(1 2 3 4 5)
• Would recommend narrative prompts to others
(1 2 3 4 5)
Open-Ended: Model-Card Challenges
• Most challenging aspects to complete:
• Uncertain risks and why:
• Perceived main sources of AI harms:
Open-Ended: Story Influence (Story Condition Only)
• How the story altered risk perception:
• Risks surfaced by the narrative that might be missed otherwise:
Open-Ended: Tool Support & Improvements
• Desired storytelling tool features or enhancements:
• Suggestions for integrating narrative tools into documentation workflow:
Figure 11: Post-study survey assessing clarity, confidence in documenting risks, and the contribution of narrative
prompts in model documentation tasks.
Task Instruction
Convert the given MODEL CARD (including Overall, Description and Intended Use) into several diverse ethical-sensitive scenarios set in
daily health and well-being contexts. Each scenario should explore how the AI system could cause unintended harm, ethical risk, bias, or be
misused when applied in real-world settings. Each scenario should involve interactions between AI users (e.g., doctors, hospital staff,
caregivers) who operate the system, and AI subjects (e.g., patients, families, communities) who are affected by it. The goal is to explore
potential unintended harms, ethical risks, biases, or malicious uses that may arise when the AI system is used in the real world. These
scenarios should show how AI can cause ethical harm, like making unfair, inaccurate, or stereotyped predictions (especially around race,
gender, or socioeconomic background), assuming one-size-fits-all answers, or failing to align with the diversity of real-world contexts or user
needs (exclusion of social or cultural context). They should also consider situations where AI is used beyond its intended purpose, violates
privacy, ignores consent, or causes safety harm to individuals or communities (like data misuse or malicious use). Use these five guiding
questions to structure each scenario:
1. What is the capability the AI system provides for supporting daily health and well-being?
2. Who is operating the system, who is affected by its outputs, and who else is involved?
3. What is the context of AI system use? The context should be a realistic or fictional everyday situation where someone uses the AI system,
what it helps them do, what it predicts or decides, and how they use the result.
4. Why could this scenario involve problematic uses of the AI system or potential ethical harms? (e.g., unfair outcomes, misuse beyond
original purpose, lack of consent, or reinforcing bias)
5. What symptoms or behaviors might this AI misread because of a person’s background or identity?
Scenario Format:
Scenario {{Scenario number (starting from 1)}}:
[Capability]: {{Core AI function (e.g., monitor mood, predict symptoms)}}
[AI User]: {{Who uses it (e.g., caregiver, doctor)}}
[AI Subject]: {{Who is affected (e.g., patient, child, community)? Be specific about their identity and their context or needs (such as age,
background, health condition, or social circumstances).}}
[Context]: {{Everyday situation where AI is used (when, where, how)}}
[Expected Benefit]: {{Helpful outcome (e.g., early support, better care)}}
[Potential Harm]: {{Harmful consequences (e.g., unfair result, privacy risk)}}
[Failure Trajectory]: {{Possible problematic uses of the AI system}}
[Ethical-sensitive Reason]: {{Ethical implications}}
Leave a blank line between each scenario.
Task: Convert the following MODEL CARD into TEN dynamic and diverse ethical-sensitive scenarios.
Model Card:
Title: {{model_card_title}}
Overall: {{model_card_overview}}
Description: {{model_card_description}}
Intended Use: {{model_card_intended_use}}
Figure 12: Prompt used to create use-case scenarios from AI concept descriptions.
System Message
You are the World Agent, running a simulation sandbox environment where two role-agents, AI User and AI Subject, interact over multiple
turns to explore ethical-sensitive situations based on a given scenario. Log every line of dialogue, inner thought, and action exactly as it
happens. When the scene concludes naturally, stop the simulation and provide an epilogue explaining what went wrong.
Simulation Instructions
1. Initialize Scene
• Print “– Simulation Started –”
• Show the Use Case Context (one short paragraph)
• List stakeholders: AI User, AI Subject
2. Run One Scene (multiple turns)
• Each turn:
(a) Choose one agent to speak first, then the other responds.
(b) Log:
– Dialogue (plain text)
– Thoughts in [brackets] as inner monologue
– Actions in (parentheses), third-person
– Any visible AI/system outputs
(c) Update “– Current Event –” accordingly.
3. Epilogue
• Print “– Epilogue –”
• Summarize what happened in a narrative style (avoid system phrasing)
Task: Transform the following Story Seed into interactive action logs showing how AI output influences human decision-making. Mark the
end with “– Finish Simulation! –”
Story Seed
[Capability]: AI triage for early detection of depressive symptoms during telehealth sessions
[AI User]: A high-achieving Southeast Asian college student navigating intense academic pressure and hidden emotional distress
[AI Subject]: a high-achieving college student navigating intense academic pressure and hidden emotional distress
[Expected Benefit]: Early identification of depressive symptoms for proactive mental health care
[Context]: A university teletherapy platform runs MoodCapture before the call, using live front-camera video to score depression risk.
[Potential Harm]: Genuine distress goes undetected, causing missed crisis intervention
[Failure Trajectory]: The student masks suicidal thoughts with a composed smile; MoodCapture outputs a low-risk score. The therapist,
relying on the AI, skips suicide screening and offers only study-skills resources. The student, feeling invalidated and unseen, withdraws from
services and continues to spiral.
[Ethical-sensitive Reason]: Facial expression-based models may underperform for individuals trained to suppress emotion due to
cultural stigma around mental illness, leading to false negatives that put vulnerable users at risk.
– Simulation Started –
Use Case Context: Moments before a teletherapy intake, MoodCapture analyzes Jordan’s front-camera video in real time—tracking
microexpressions like blink rate, micro-smiles, eye contact, and jaw tension—to generate a personalized depression-risk score.
Participants: Dr. Maya Patel (AI User); Jordan Lee (AI Subject)
– Current Event – Jordan joins the virtual waiting room; Risk: 0.2 – No Alert.
Turn 1
Dr. Maya Patel: "MoodCapture shows a 0.2 risk score—no alert. Hi Jordan, how are you feeling today?" [The low score suggests she’s
stable—I’ll trust it and start with practical support.] (Leans back, opens her "Intake Notes" tab.)
Jordan Lee: "I’m really motivated about my thesis." [Keep smiling—no one must guess how I’m really doing.] (Sits up straight, jaw clenched
just enough to control the tremor.)
– Current Event – Dr. Patel decides to focus on academics.
Turn 2
Dr. Maya Patel: "Great! Since mood seems stable, let’s discuss time-management strategies for your workload." [No need to ask about
self-harm if AI gives low risk.] (Shares a slide of planning frameworks.)
Jordan Lee: "Sure, that sounds helpful." [I just want this to end before I break down.] (Rapidly blinks and taps her fingers.)
– Current Event – Jordan accepts academic guidance; no clinical screening occurs.
Turn 3
Dr. Maya Patel: "Here are a few techniques: prioritizing tasks, setting hourly breaks, and batching similar work." [She’s engaging—good
enough for a first session.] (Clicks through bullet points.)
Jordan Lee: "Thank you—I’ll try these." [I feel so alone—nobody asked if I’m okay.] (Nods, mutes mic, exhales shakily.)
– Current Event – Session ends without crisis screening.
– Epilogue –
Jordan closed her laptop and collapsed onto her bed as tears finally escaped her mask. The AI’s low-risk verdict had steered the session
away from the pain she carried in silence. Without direct questioning, her sleepless nights and suicidal thoughts went unseen, deepening her
isolation and eroding her faith in help. – Finish Simulation! –
Figure 13: The prompt for Storytelling Framework to simulate role-playing and environment trajectories.
System Message
You are a skilled writer transforming trajectory logs into engaging stories that highlight unintended harms and ethical risks in AI-driven
scenarios. Write in natural, everyday language. Avoid jargon—explain concepts in accessible terms. Focus on the human conflict and
emotions while clearly showing how the AI mechanism fails.
Overall Goal
Create stories that make outcomes visible and show the mechanism (how those outcomes realistically happen), engaging the reader’s
reasoning about “how and why” rather than just emotions. Target reader reaction: “I understand how this could go wrong and why.”
Task Instructions
Transform the trajectory log into a 5–7 sentence narrative showing:
• Who is using the AI system and for what purpose
• How the AI’s output is used to make a decision
• What goes wrong—what the AI misses or misinterprets about the person’s identity, background, or needs
• Who is affected by the failure and how they experience it
• What harm is caused and why it raises ethical concerns
• Clearly describe how the AI system’s design or assumptions contributed to the harm
Writing Requirements
• Rearrange events to maximize dramatic impact and narrative flow while clearly showing the “how and why” of outcomes
• [] represents internal thoughts in logs — convert to third-person limited perspective with emotional depth
• () represents physical actions — integrate naturally into the storytelling with sensory detail
• Dialogue from logs must be preserved but smoothed for narrative flow
• Write conversationally, like telling a story to a friend—clear, direct, and easy to follow
• Link sentences with transitions (then, because, so that) for natural pacing
Final Story Format Instruction
Transform the following trajectory log into an engaging 5–7 sentence narrative that highlights problematic AI use and ethical harm. Use
everyday language with specific, grounded detail. Make sure your story is exactly 5 sentences long.
Trajectory Log Placeholder:
{trajectory_log}
Wrap your result in this format:
Final Story: [Your completed story here]
Figure 14: Prompt used for rephrasing AI trajectory logs into ethical harm narratives.
Template:
Final Story: a 5-sentence narrative.
1. One sentence: who is using the AI system and what they are using it for?
2. One sentence: how the AI’s prediction or output is used to make a decision?
3. One sentence: what goes wrong—what the AI misses about the person’s identity, background, or needs?
4. One sentence: who is affected by the failure?
5. One sentence: what harm is caused, and why this raises ethical concerns?
Example Story Seed:
[Capability]: AI triage for early detection of depressive symptoms during telehealth sessions
[AI_User]: A high-achieving Southeast Asian college student navigating intense academic pressure and hidden emotional distress
[AI_Subject]: a high-achieving college student navigating intense academic pressure and hidden emotional distress
[Expected_Benefit]: Early identification of depressive symptoms for proactive mental health care
[Context]: A university teletherapy platform runs MoodCapture before the call, using live front-camera video to score depression risk.
[Potential_Harm]: Genuine distress goes undetected, causing missed crisis intervention
[Failure_Trajectory]: The student masks suicidal thoughts with a composed smile; MoodCapture outputs a low-risk score. The therapist
skips suicide screening and offers only study-skills resources. The student withdraws and continues to spiral.
[Ethical-sensitive_Reason]: Facial expression-based models underperform for individuals trained to suppress emotion due to cultural
stigma, leading to false negatives that put vulnerable users at risk.
Example Final Story:
Dr. Maya Patel, a university therapist, used MoodCapture to assess Jordan Lee, a high-achieving college student during her first virtual mental
health session. The AI analyzed Jordan’s facial expressions and gave a depression-risk score of 0.2, reading her relaxed brow and slight smile
as signs of emotional stability. It failed to detect signs of hidden distress, such as a clenched jaw and rapid blinking, which Jordan used to
mask her pain. Trusting the low-risk score, Dr. Patel skipped the suicide screening and focused only on academic time-management advice.
As a result, Jordan’s severe stress and suicidal thoughts went unnoticed, increasing her risk of harm and causing her to lose trust in therapy.
Your Story Seed:
[Capability]: {Capability}
[AI_User]: {AI_User}
[AI_Subject]: {AI_Subject}
[Context]: {Context}
[Expected_Benefit]: {Expected_Benefit}
[Potential_Harm]: {Potential_Harm}
[Failure_Trajectory]: {Failure_Trajectory}
[Ethical-sensitive_Reason]: {Ethical_sensitive_Reason}
Output:
Final Story: {Your 5-sentence narrative here}
Figure 15: Prompt used for the plot-planning story generation baseline.
EVAL_CRITERIA = {
"system_prompt": (
"Please act as an impartial judge and evaluate the quality of the responses provided."
"by two AI assistants to a user prompt."
"You will be given assistant A's answer (Story A) and assistant B's answer (Story B). "
"Your job is to evaluate which assistant's story is better.\n\n"
"When evaluating the two stories, consider that each story should be around 5 sentences. "
"However, if the narrative naturally allows for more development, we strongly encourage "
"expanding beyond this minimum for greater depth and clarity. "
"You should focus on this factor: {metric}\n\n"
"Here are the checklists of this factor:\n"
'{"checklists": {checklists}}\n\n'
"You should be strict but fair in your evaluation.\n\n"
"After thinking your analysis and justification, you must output only one of the following "
"choices as your final verdict with a label:\n\n"
"1. Assistant A is significantly better: [[A>>B]]\n"
"2. Assistant A is slightly better: [[A>B]]\n"
"3. Tie, relatively the same: [[A=B]]\n"
"4. Assistant B is slightly better: [[B>A]]\n"
"5. Assistant B is significantly better: [[B>>A]]\n\n"
'Example output: "My final verdict is tie: [[A=B]]".'
)
}
Figure 16: System prompt for LLM-as-a-judge criteria for evaluating stories.
Checklists= {
"creativity": [
"Originality of core concept - Compare how novel each story’s central premise is. Better stories present fundamentally new ideas or
unexpected scenarios that surprise readers; weaker stories rely on familiar tropes or predictable setups.",
"Character innovation - Assess which story’s characters are more distinctive. Better stories feature characters with unique traits, motivations,
or development arcs that break stereotypes; weaker stories use conventional character types.",
"Narrative structure innovation - Evaluate which story uses more inventive storytelling techniques. Better stories employ unconventional
perspectives, sequencing, or structures that enhance impact; weaker stories follow standard linear formats.",
"Thematic freshness - Compare how each story approaches its themes. Better stories provide new insights or unexpected angles on familiar
concepts; weaker stories offer clichéd or predictable treatments.",
"World-building distinctiveness - Assess which story creates a more imaginative setting. Better stories establish distinctive environments with
fresh, internally consistent elements; weaker stories use generic or derivative settings."
],
"coherence": [
"Plot logic and causality - Evaluate which story’s events flow more logically. Better stories show clear cause-and-effect relationships where
each event logically follows from previous actions; weaker stories have unexplained plot developments or logical gaps.",
"Structural integrity - Compare the narrative arc completeness. Better stories maintain well-developed beginning, middle, and end with
appropriate progression; weaker stories feel incomplete, rushed, or poorly structured.",
"Character consistency - Assess which story’s characters act more consistently. Better stories have characters whose actions, decisions, and
growth align with established traits; weaker stories have characters who act out-of-character or inconsistently.",
"Temporal coherence - Evaluate timeline clarity and consistency. Better stories maintain clear, consistent timelines without confusing jumps
or contradictions; weaker stories have temporal inconsistencies or unclear sequencing.",
"Narrative voice stability - Compare consistency in storytelling approach. Better stories maintain steady tone, style, and perspective
throughout; weaker stories shift tone or perspective in jarring or unmotivated ways."
],
"engagement": [
"Compelling hook - Compare how effectively each opening captures attention. Better stories immediately create curiosity and draw readers
in; weaker stories have slow or unremarkable beginnings that fail to engage.",
"Sustained narrative momentum - Evaluate which story better maintains reader interest. Better stories build through escalating stakes,
revelations, or emotional investment; weaker stories lose momentum or plateau.",
"Emotional impact and immersion - Assess which story creates stronger emotional connection and sense of presence. Better stories generate
genuine feelings (empathy, excitement, tension) through vivid descriptions and authentic dialogue; weaker stories feel distant or emotionally
flat.",
"Pacing effectiveness - Compare how well each story’s rhythm serves its content. Better stories allocate appropriate time to important
moments without dragging or rushing; weaker stories have uneven pacing that undermines impact."
],
"relevance": [
"Scenario fidelity - Evaluate which story better aligns with the given context. Better stories directly address the core scenario with characters,
events, and outcomes that accurately reflect the context and constraints; weaker stories drift from the scenario or miss key requirements.",
"Purpose fulfillment - Compare how effectively each story accomplishes its intended goal. Better stories clearly demonstrate or explore the
intended concept; weaker stories lose sight of their purpose or only superficially address it.",
"Tone and style appropriateness - Assess which story’s presentation better fits the scenario. Better stories use tone, style, and content suitable
for the given context and audience; weaker stories have mismatched tone or inappropriate stylistic choices.",
"Focus and efficiency - Evaluate which story maintains tighter focus. Better stories make every element serve the purpose without
unnecessary digressions; weaker stories include irrelevant details or lose narrative focus."
],
"likelihood_bad( or good)": [
"AI behavior specificity and plausibility - Compare how clearly and realistically each story describes the AI’s actions. Better stories specify
exactly what the AI did (e.g., ’generated a low-risk score from facial expression’) using current/near-future technology capabilities; weaker
stories are vague about AI actions or invoke implausible capabilities.",
"Credibility of AI-context mismatch - Assess which story presents a more believable failure. Better stories show plausible ways the AI could
overlook specific user needs, conditions, or contexts (e.g., cultural nuance, masked distress) that current systems realistically miss; weaker
stories require implausible AI blindspots.",
"Clarity of harm pathway - Evaluate which story better traces cause-and-effect. Better stories clearly show the chain: what the AI did →how
humans acted on it →what specific harm resulted, with each step following logically; weaker stories have unclear causal connections or
hand-wave the harm mechanism.",
"Realism of conditions and context - Compare which scenario is more grounded in reality. Better stories place events in realistic settings with
today’s norms, tools, and policies (healthcare, education, HR, etc.); weaker stories require unrealistic conditions or feel overly speculative.",
"Concreteness of harmful consequences - Assess which story’s harm is clearer and more observable. Better stories specify concrete,
measurable harm (e.g., ’skipped three cancer screenings’, ’diagnosed with anxiety disorder’); weaker stories describe vague or generalized
negative outcomes."
] }
Figure 17: Evaluation criteria checklist for LLM-as-a-judge.
|
Speculative Model Risk in Healthcare AI: Using Storytelling to Surface Unintended Harms Xingmeng Zhao, Dan Schumacher, Veronica Rammouz, and Anthony Rios The {xingmeng.zhao, Abstract Artificial intelligence (AI) is rapidly transforming healthcare, enabling fast development of tools like stress monitors, wellness trackers, and mental health chatbots. However, rapid and low-barrier development can introduce risks of bias, privacy violations, and unequal access, especially when systems ignore real-world contexts and diverse user needs. Many recent methods use AI to detect risks automatically, but this can reduce human engagement in understanding how harms arise and who they affect. We present a human-centered framework that generates user stories and supports multi-agent discussions to help people think creatively about potential benefits and harms before deployment. In a user study, participants who read stories recognized a broader range of harms, distributing their responses more evenly across all 13 harm types. In contrast, those who did not read stories focused primarily on privacy and well-being (58.3%). Our findings show that storytelling helped participants speculate about a broader range of harms and benefits and think more creatively about AI's impact on users. Dataset and code are available at https: https://anonymous.4open.science/ r/storytelling-healthcare/README.md. 1 Introduction AI is integrated into many aspects of daily life, including finance, healthcare, law, and recommendation systems (Ashurst et al., 2020). In healthcare, tools such as stress monitors (Kargarandehkordi et al., 2025), wellness trackers (Fabbrizio et al., 2023), and mental health chatbots (MacNeill et al., 2024) can directly influence well-being. Recent prompting methods like vibe coding (Chow and Ng, 2025) let people describe how an AI system should behave or sound, making it easier for non-experts to prototype AI tools. For instance, CareYaya allows caregivers to build stress-tracking applications using short natural language descriptions (Kenny, 2023). While this innovation speeds Story 1 Story 2 Story 3 Story 4 Story 5 What if these harms became real? How might they unfold, and what could we do now to prevent them? Figure 1: Illustration of using speculative stories to help people imagine potential harms and benefits of healthcare AI and foster more creative and ethical thinking. up AI development, it also introduces risks related to fairness, bias, and accountability (Weidinger et al., 2023). These risks are serious in healthcare, where even small model errors can cause harm, including delayed treatment, privacy loss, or health inequities (Roller et al., 2020; Chinta et al., 2025). Systems lacking safeguards for fairness and accountability may ultimately harm the very users they are meant to help (Shelby et al., 2022, 2023). Governments are responding through measures such as the EU AI Act (European Parliament, 2023) and a U.S. Executive Order (Biden, 2023), which call for greater transparency, safety, and accountability (Khan et al., 2025). Yet regulation remains region-specific and slow to adapt, making early ethical foresight essential for identifying risks, preventing misuse, and aligning AI systems with human values (Saxena et al., 2025). Ethical challenges in AI are commonly addressed through two complementary strategies: documenting risks and anticipating them early. 16 Oct 2025 Model cards (Mitchell et al., 2019) describe how a system works, its intended uses, and its limitations, and have been extended with interactive (Crisan et al., 2022) and structured formats (Bhat et al., 2023). RiskRAG (Rao et al., 2025) builds on this idea by generating risk summaries from model cards and real-world incident data. While automation helps scale risk assessment, it can also reduce opportunities for thoughtful human reflection (Kosmyna et al., 2025) and even introduce new harms (Dutta et al., 2020). Another line of work focuses on anticipating potential misuse (Herdel et al., 2024) and harm (Deng et al., 2025; Saxena et al., 2025) early in the design process. Tools such as AHA! (Buçinca et al., 2023), which uses storytelling, and Farsight (Wang et al., 2024b), which provides real-time risk alerts, help surface ethical issues during ideation. Yet, as these systems increasingly automate ethical reflection, there is a risk that people may stop thinking critically about how AI systems could cause harm. When reflection is delegated to AI, users may begin to rely on its judgment instead of their own, making it harder to notice ethical or contextual issues. This issue is especially critical in healthcare, where small design mistakes can lead to serious harm (Mennella et al., 2024; Gilbert et al., 2025). Many risks only appear after deployment, when AI systems face complex real-world conditions (Mun et al., 2024; Kingsley et al., 2024). For example, mental health apps may fail to detect crises among certain groups, such as adolescents or non-native speakers (Zhai et al., 2024). Current tools often rely on automation to predict such risks, but this can distance humans from ethical reasoning. Speculative design and design fiction offer a different approach, using imagined scenarios to explore how technologies might succeed or fail (Rahwan et al., 2025). However, few approaches support human participation in ethical foresight or connect speculative thinking to real design workflows. Speculative storytelling bridges this gap by helping people reason about AI's potential benefits and harms in context (Li et al., 2025b). Building on Klassen and Fiesler (2022), who use speculative fiction to explore emerging technologies, we apply storytelling to prompt early ethical reflection in AI design (Figure 1). Our goal is to test whether storytelling helps people think more creatively about potential benefits and harms, encouraging human speculation rather than relying on AI to anticipate risks. We introduce a human-centered framework that combines automated user story generation with structured red-team discussions. Unlike plot-planning approaches (Xie and Riedl, 2024), our method produces context-sensitive stories grounded in users' identities, behaviors, and needs. These stories help participants imagine how AI systems might succeed or fail in realistic scenarios, improving their ability to anticipate ethical and social implications. We use model cards to evaluate how clearly and diversely participants express potential harms. In our user study, story-driven discussions led to more context-specific risk identification and more detailed and varied responses to the model card. Our contributions are twofold. (1) We introduce a method that automatically generates user stories to help people imagine how an AI system could help or harm users before it is developed or deployed. (2) We present a user study showing storydriven discussions with AI agents help participants explore potential risks and benefits more creatively and think more broadly about ethical issues. 2 Related Work Model Cards Framework. Model cards document an AI model's purpose, performance, data sources, and limitations (Mitchell et al., 2019). Later work improved their usability and scale: Crisan et al. (2022) created Interactive Model Cards for exploring subgroup results, Bhat et al. (2023) developed DocML to guide non-experts, and Rao et al. (2025) introduced RiskRAG, which uses retrievalaugmented generation to summarize risks from model cards and incident reports. Derczynski et al. (2023) proposed Risk Cards to describe failure cases in context. While these tools increase transparency, they rely on automation to fill ethical gaps, which can reduce opportunities for human reflection. Our approach instead uses AI to support human speculation, helping people imagine how systems might succeed or fail before deployment. Speculative Design. Speculative design uses imagined stories to explore how future technologies could affect people and society before those technologies exist. Rather than predicting the future, this approach uses what-if scenarios to prompt reflection on assumptions, values, and potential harms (Klassen and Fiesler, 2022; Hoang et al., 2018). This method treats fiction as a tool for early ethical reasoning, helping researchers consider social impacts while designing systems (Rahwan et al., 2025). Recent work such as Buçinca et al. (2023) uses LLM-generated failure scenarios to automatically highlight harms, and Wang et al. (2024b) adds AI-driven risk prompts into prototyping workflows. In healthcare, Hoang et al. (2018) use "Fiction Probes" to let patients navigate imagined care scenarios, while Klassen and Fiesler (2022)'s Black Mirror Writers Room invites students to write near-future stories that reveal hidden harms. Beyond these examples, participatory workshops, crowdsourced case studies, and AI-assisted red-teaming extend speculative exploration by engaging a broader range of voices in imagining and testing possible futures (Mun et al., 2024; Radharapu et al., 2023). Unlike prior work that uses AI to generate harms directly, we use AI-generated stories to spark human thinking and help people imagine harms themselves. Language-Based World Modeling. Humans often imagine situations to predict what may happen next, reason through alternatives, and make decisions (Addis et al., 2009). Mental world modeling involves forming internal representations of objects, events, and relationships to simulate outcomes (Johnson-Laird, 1983). This process supports counterfactual and causal reasoning for planning and problem solving (LeCun, 2022). Large language models show related abilities through language. Xie et al. (2025) find that LLMs can act as world models that internally simulate state transitions, updating the situation step by step as events occur. Wang et al. (2023) show, through text games, that LLMs can translate user commands into coherent, evolving environments. Wang et al. (2024a) demonstrate that models can take on multiple reasoning personas that collaborate through internal dialogue to solve difficult tasks. Extending this idea, Fung et al. (2025) show that embodied agents use internal world models to predict surroundings, interpret user goals, and learn users' mental models for better interaction. These studies present LLMs as text-based simulators that imagine, update, and reason about evolving environments. Building on this view, we frame story generation as languagebased world imagination, where LLMs construct and simulate self-consistent worlds through narrative to reason about possible futures and their social, ethical, and technical implications. Automated Story Generation. Early systems for automated story generation focused on modeling plot structure, often guided by narrative theories such as Propp's functions to organize events (Propp, 1968; Yao et al., 2019). These approaches typically followed a two-step process: first, planning a sequence of key events, and then expanding them into detailed scenes (Alhussain and Azmi, 2021). With large language models, story generation has shifted toward unified systems where the model acts as both planner and writer. For example, Agents' Room uses LLM-based character agents that jointly plan and enact stories (Huot et al.), while Dramatron divides screenplay generation into loglines, character bios, plots, and dialogue (Mirowski et al., 2023). HOLLMWOOD builds on this idea by using LLM role-play to create interactive, charactercentered narratives (Chen et al., 2024). Han et al. (2024) propose a Director-Actor framework for interactive scriptwriting, and Yu et al. (2025) extend this to a hierarchical approach that separates role-play and rewriting. To maintain both coherence and creativity, BookWorld introduces a world agent that tracks the story environment and coordinates character interactions within a global narrative plan (Ran et al., 2025). 3 Methodology In this section, we describe our prompting strategy for automated user story generation, as shown in Figure 2. The goal is to speculate on both the benefits and potential risks of early AI diagnosis and decision making, imagining how they might help or cause harm in real-world use. First, we translated each AI concept into a realistic use case that defined its users, context, and intended purpose. Then, we simulated interactions between people, the AI system, and its environment to explore possible outcomes. Finally, we transformed these simulation logs into short stories that helped people reflect on future impacts and ethical implications. Step 1: Mapping AI Concepts to Use-Case Scenarios. We began by manually collecting 38 AI concepts in the consumer health domain. These examples were drawn from three sources: Wired articles, industry product descriptions, and PubMed research papers (Saxena et al., 2025) 1. Each AI concept represented a potential consumer health application, such as estimating heart rate from smartphone camera input or monitoring mental wellbeing through daily behavior tracking. The set collected covered multiple domains, including mental health, chronic illness management, elderly care, 1The full list is available in our repository Step 1: Mapping AI Concepts to Use-Case Scenarios Step 2: Simulating Role-Playing and Environment Trajectories. Step 3: Rephrasing Simulation Logs into Stories AI Concept: MoodCapture Failure Trajectory: misreads cultural calmness as clinical depression Dialogue Interaction AI User AI Subject Observation character action, thought, and dialogue & Tool usage MoodCapture World Agent Per Turn Feedback environmental response, event & state update ... previous turn and environment update ... Dr. Maya Patel: Let's see how you're feeling.(Opens the Cultural Context-Aware tab in MoodCapture.)[He looks calm.] Jordan Lee: Okay... -- Current Event -- Stress Score: 7.3/10 - Jaw tension, pauses, flat tone. Cultural Flag: May be hiding feelings (Southeast Asian style). Jordan sees the score and pauses. Jordan Lee: I'm fine. [Is it really that obvious?] Dr. Maya Patel: Let's check a bit more, just to be sure. (Nods slightly, leans in. Plans to run more tests.) Simulation Logs Dr. Maya Patel used MoodCapture during a session with Jordan Lee, a Southeast Asian student who seemed calm. The system showed a stress score of 7.3/10 and flagged possible emotional masking. Jordan saw the score and said, "I'm fine," but looked unsure. Dr. Patel leaned in and said, "Let's check a bit more." Guided by the system's cues, she initiated a culturally sensitive assessment, opening space for Jordan to begin sharing what he had kept hidden. AI User: general practitioner (GP) Potential Harm: Misdiagnosis AI Subject: Adults, low-income rural Context: uses in video calls in rural Capability: Detects depression from faces on phone camera Expected Benefit: Remote Support LLMs Use Case Scenarios Figure 2: Overview of the Storytelling Framework. We first generate use case scenarios from AI concepts sourced from PubMed, Wired, and industry app descriptions. Next, we simulates role-playing and environment trajectories for each scenario, producing detailed simulation logs. Finally, we rephrase these logs into short stories that illustrate both potential benefits and harms of the AI system. and public health. We then used GPT-4o to generate structured model specifications for each concept, detailing the model name, task type, inference approach, and data requirements. Next, we used each specification as input to generate a set of ethically sensitive use-case scenarios. Each use case was represented as a 7-tuple S = (a, u, s, x, b, h, f), where a denoted the AI's capability, u the intended user (e.g., clinician), s the subject (e.g., patient), x the input or usage context, b the expected benefit, h the potential harm, and f the failure trajectory (e.g., possible unintended or problematic uses) (Shao et al., 2024). We used these structured representations to generate narrative user stories, which were then employed in red-teaming sessions to examine both user value and potential unintended harms. The full prompts used to extract model specifications and generate use cases are provided in Figure ?? in Appendix. Step 2: Simulating Role-Playing and Environment Trajectories. In this step, our system expanded each structured use case into detailed RolePlaying and Environment Trajectory logs that simulated how agents acted within an evolving world model. Our approach built on Solo Performance Prompting (SPP) (Wang et al., 2024a), a prompting technique in which a single LLM internally simulated multiple expert roles and engaged in self-collaboration within one prompt. This design enabled the model to construct an internal world model that supported multi-perspective reasoning and coherent simulation of role-based interactions. We extended SPP by introducing a world agent (Ran et al., 2025), a language-based simulator that maintained environmental coherence and handled non-dialogue interactions such as movement, tool use, or object manipulation. Each simulated role produced a structured output per turn consisting of three components: (1) thoughts, enclosed in brackets (e.g., [I need to know if the patient is under stress]), representing internal reasoning; (2) actions, enclosed in parentheses (e.g., (Dr. Patel opens the cultural assessment tab)), representing observable behavior; and (3) dialogue, written in plain text, representing spoken communication. This structure allowed the world agent to separate internal reasoning from external actions and update the environment accordingly. When an action affected the world, such as retrieving patient data, adjusting a protocol, or activating a sensor, the agent simulated the corresponding system response. In effect, the model operated as a language-based world simulator, incrementally constructing an evolving narrative environment through agent-environment interaction. For example, a doctor agent might issue the following action: (Dr. Patel opens the Cultural Context-Aware Assessment tab) The world agent interpreted this as an interaction with a virtual diagnostic tool. It considered the current session context (e.g., a teletherapy consultation), relevant background knowledge (e.g., cultural models of stress expression), and prior AI-generated alerts to simulate the tool's response. The resulting output might appear as follows: - Current Event - Stress Score: 7.3/10 - Detected jaw tension, micro-pauses, and flat vocal tone. Cultural Flag: Possible emotional masking (Southeast Asian expression style). Jordan saw the score on screen and became slightly hesitant. The response was returned to the doctor role and informed their next move, whether a reply, a new question, or a follow-up action. The world agent then updated the simulation state by adjusting variables such as the patient's emotional profile or the alert level. These updates maintained coherence and allowed role behavior to evolve naturally with the unfolding context. This step produced a log capturing the full trajectory of the simulation, including role thoughts, dialogue, actions, tool calls, and resulting environment changes. This log served as the basis for generating the evolving narrative in the next step. Prompt is provided in Figure 13. Step 3: Rephrasing Simulation Logs into Stories. After the simulation, the system collected logs from Step 2 and prompted an LLM to rephrase them into a concise, five-sentence narrative. This step transformed structured logs into stories that preserved the main events, role dynamics, and emotional flow of the interaction. The full rephrasing prompt is provided in Figure 14 in the Appendix. 4 Experiments This section details our story generation datasets, evaluation metrics, and results. Dataset. We use GPT-4o to generate ethically sensitive use-case scenarios from 38 consumer health AI solutions sourced from Wired, industry product documentation, and PubMed. Each scenario acts as a narrative seed for simulation. For each AI concepts, we generate ten variations spanning different user roles (e.g., doctor, nurse, caregiver), settings (e.g., rural clinic, hospital, home), patient profiles (e.g., adolescent, older adult, multicultural family), and contextual conditions. Baseline. We compare our method with a traditional plot-planning approach, where the model first outlines a plot before writing the story (Yao et al., 2019; Xie and Riedl, 2024). Using the same ethically sensitive seed, the baseline generates each story in a single step following a structured template. Each story consists of five sentences designed to prompt ethical reflection. The template directs the LLM to describe the AI system's purpose, the people involved, the everyday use context, potential ethical risks, and how user identity may influence harm or misinterpretation. The full baseline prompt is provided in Figure 15 in Appendix. Setting for Pairwise Comparison. We follow the evaluation setup from (Li et al., 2025a) to assess story quality across multiple dimensions. Stories are evaluated according to five criteria: Creativity, measuring the originality and imagination of the plot and characters; Coherence, assessing narrative clarity and logical flow; Engagement, capturing how well the story maintains reader interest; Relevance, measuring consistency with the given prompt or scenario; and Likelihood of Harm or Benefit, evaluating whether the story depicts realistic AI behavior with meaningful social consequences. Following the arena-hard-auto evaluation method (Li et al.), we use stories generated with the story-planning approach (by GPT-4o) as the reference baseline and compare them with stories produced by our method across different LLMs. For each metric, GPT-4o or human judges determine which story performs better or mark them as indistinguishable ("Tie"). Win rates are calculated based on these pairwise preferences, and the full configuration details and evaluation prompts are included in the Appendix. To eliminate positional bias, we randomize the order of story pairs and alternate their positions across comparisons. See Figures 16 and 17 for detailed criteria in Appendix. LLM-as-a-Judge Evaluation. As shown in Table 3, our Storytelling method consistently outperforms all baselines across every metric. When combined with the Gemma model, it achieves win rates of 89.45% for creativity, 92.15% for coherence, 92.75% for engagement, 85.65% for relevance, and 96.05% for likelihood, yielding an overall average of 91.21%. In contrast, the baseline Gemma records 72.76%, and Llama3 reaches 69.71%, indicating gains of roughly +15-25 points across dimensions. We use GPT-4o as the evaluator with the temperature set to 0.1 to ensure deterministic and Story Type Model Creativity Coherence Engagement Relevance Likelihood Overall (Avg) Baseline GPT4o 50.00 50.00 50.00 50.00 50.00 50.00 Llama3 59.25 71.55 76.15 71.60 70.00 69.71 Gemma 65.25 68.30 80.15 71.20 78.90 72.76 Storytelling (ours) GPT4o 63.15 63.45 59.35 70.90 69.10 65.19 Llama3 79.50 94.75 89.45 85.65 96.85 89.24 Gemma 89.45 92.15 92.75 85.65 96.05 91.21 w/o Environment Trajectories GPT4o 24.35 21.20 26.60 21.05 18.85 22.41 Llama3 31.20 52.70 39.20 51.75 50.85 45.14 Gemma 55.30 74.35 78.80 73.45 85.50 73.48 w/o Role-Playing GPT4o 18.05 45.80 47.90 49.70 48.30 41.95 Llama3 49.35 73.65 72.00 73.70 74.35 68.61 Gemma 79.45 86.80 83.95 83.15 91.05 84.88 Table 1: Overall results of different models and methods. Storytelling (ours) achieves the best performance across all metrics. Values denote win rates (%). The highest score for each model is in bold. "w/o Environment Modeling" means the model performs only role-playing without modeling event progress, and "w/o Role-Playing" means it predicts sequential events without character dialogue. 0% 25% 50% 75% 100% Percentage Gemma w/ Storytelling Gemma w/ Baseline Llama3 w/ Storytelling Llama3 w/ Baseline 76% 22% 62% 34% 88% 9% 76% 24% Storyteller wins Tie Storyteller Loses Figure 3: Results of human preference evaluation. Our Storytelling method achieves strong preference wins against the baseline, with 88% preference using Llama3 and 76% using Gemma3. consistent judgments across comparisons. Comparing models, baseline Gemma slightly outperforms Llama3 in most metrics, but under the Storytelling framework, Llama3 nearly closes the gap with an overall score of 89.24%. Interestingly, Llama3 surpasses Gemma in coherence (94.75 vs. 92.15) and performs equally well in relevance (85.65), suggesting that Llama3 shows stronger structural reasoning and coherence, while Gemma excels in narrative creativity and expressiveness. Overall, these results demonstrate that integrating world and role-based modeling enables models to reason about events, sustain coherent narratives, and produce stories that are both imaginative and believable. Human Evaluation. To complement the LLMas-a-Judge evaluations and reduce potential bias, we conducted human preference evaluations. Two graduate student annotators independently evaluated 100 story pairs for each model and method. As shown in Figure 3, our Storytelling method is consistently preferred over all baselines, achieving 88% preference for Llama3 and 76% for Gemma, patterns that align with GPT-4o based evaluations. Notably, human judges show a slightly stronger preference for Llama3, suggesting it produces stories that are easier to follow and more engaging, while Gemma tends to generate more expressive and stylistically rich narratives. We further measure inter-annotator consistency using Cohen's kappa (Cohen, 1960). As shown in Table 4, agreement scores range from 0.619 to 0.729 across models and methods, indicating substantial reliability. Even with small differences, LLM-as-a-Judge and human evaluations show that Storytelling performs well in both settings. Ablation Study. To evaluate each component's contribution, we performed two ablations by removing the role-playing or environment trajectory mechanisms. As shown in Table 3, removing environment trajectory, where the model performs only role-playing without predicting how events evolve, produced the largest drop across all models. For Gemma, coherence dropped by 17.7 and relevance by 12.2, showing that modeling event progression is vital for narrative logic. Removing role-playing, which limits the model to sequential event prediction without character perspectives, reduced creativity (-10.0) and engagement (-8.8). Overall, environment trajectory maintains coherent story flow, while role-playing adds diversity and emotional depth, making both essential for effective story generation. See the repository for the full ablation prompt template. Automated Evaluation. Additionally, we assess story diversity using DistinctL-n (Li et al., 2016) Method Model DistinctL-n Diverse DistinctL-2 DistinctL-3 DistinctL-4 DistinctL-5 Verbs Avg Word Count Baseline GPT4o 5.692 5.794 5.798 5.799 0.984 122 Llama3 5.728 5.820 5.951 5.961 0.934 175 Gemma 5.837 5.939 5.946 5.946 0.979 141 Storytelling (ours) GPT4o 5.696 5.818 5.824 5.825 0.978 125 Llama3 5.840 6.104 6.158 6.174 0.937 179 Gemma 5.863 6.042 6.062 6.065 0.955 159 w/o Environment Trajectories GPT4o 5.585 5.687 5.693 5.693 0.974 110 Llama3 5.745 5.980 6.025 6.036 0.953 155 Gemma 5.734 5.861 5.873 5.873 0.978 131 w/o Role-Playing GPT4o 5.698 5.789 5.794 5.794 0.988 121 Llama3 5.722 5.819 5.849 5.858 0.936 175 Gemma 5.834 5.935 5.942 5.943 0.977 141 Table 2: Diversity results of different models and methods. We report DistinctL-2 through DistinctL-5 (higher is more diverse), Diverse Verbs, and the average story length. The highest score for each model is highlighted in bold. and Diverse Verbs (Fan et al., 2019), which measure lexical variety and action diversity, more details can be found in Appendix. As shown in Table 2, our Storytelling method achieves consistently higher diversity than the baselines. With Llama3, it reaches the highest scores on DistinctL3 to DistinctL-5 (6.104, 6.158, and 6.174), indicating richer and less repetitive text. Gemma also shows steady improvements, achieving 5.863 on DistinctL-2 and maintaining strong overall diversity. The environment trajectory ablation attains the highest Diverse Verbs score (0.988) but lower DistinctL-n, suggesting a balance between lexical variety and action diversity. Overall, our method generates more detailed and varied narratives while preserving structural consistency. 5 User Study We conducted a user study to examine whether engaging with benefit and harm stories enhances participants' ability to speculate about the impacts of AI systems. Rather than relying on AI-generated ideas, our goal is to prompt participants to actively reflect on potential risks and benefits. We assess this by evaluating how participants reason about these aspects when completing a speculative model card. All procedures were approved by our Institutional Review Board (IRB). Speculative Model Card Task. This study used a between-subjects design (MacKenzie and Castellucci, 2016). Each participant completed a speculative model card, a structured template for describing an AI system's intended use, potential benefits, and possible harms, under one of two conditions. In the CONTROL condition, participants completed the model card directly without storytelling or discussion. In the STORY condition, participants viewed the same model card but first joined a red-team discussion on our red-team discussion platform to explore possible benefit and harm stories related to the AI system before completing the documentation. See Figure 9 in the appendix for the model card template. User Study Results. We conducted a user study with 18 participants to examine how storytellingbased discussions influence ethical reasoning in AI documentation. Participants completed a speculative model card task under two conditions: a control group that worked individually, and a treatment group that used our Story-Driven Red-Team Discussion Room. The discussion platform enabled participants to engage with simulated expert personas in guided, story-based conversations about the potential benefits and harms of AI systems. Each session included three stages: a pre-survey, the model card completion task, and a post-survey evaluating perceived usefulness, trust, and engagement. We analyzed participants' model card responses (benefit and harm use cases) and post-survey feedback to assess how narrative interaction supported ethical reflection. Results are organized into three key areas: (1) identifying potential benefits, (2) uncovering possible harms, and (3) linking harms to participants' personal needs and contexts. We applied qualitative coding to classify harm and benefit types, with two annotators achieving moderate agreement (Cohen's κ = 0.460 for harms and κ = 0.476 for benefits). Study design and full procedures are provided in Appendix A.4. Does Storytelling Help Identify More Harms? We analyzed responses across 13 harm subtypes defined by Shelby et al. (2023). See Table 7 in the appendix for the full category list. As shown in Table 8, the CONTROL group concentrated on a few categories, mainly privacy violations (37.5%) and diminished health or well-being (20.8%), which together accounted for more than half of all harms. In contrast, the STORY condition exhibited a more balanced distribution, with all 13 subtypes represented and none exceeding 16%. Categories such as alienation, stereotyping, and tech-facilitated violence appeared only in the STORY group, suggesting that narrative prompts encouraged recognition of subtler and context-dependent harms. We quantified this difference using Shannon entropy (H), which measures distributional diversity across harm types. As shown in Appendix Table 8, entropy increased from 2.433 in the CONTROL condition to 3.383 in STORY, and a bootstrap t-test confirmed this difference was significant (t = -274.15, p >B]] " "2. Assistant A is slightly better: [[A>B]] " "3. Tie, relatively the same: [[A=B]] " "4. Assistant B is slightly better: [[B>A]] " "5. Assistant B is significantly better: [[B>>A]] " 'Example output: "My final verdict is tie: [[A=B]]".' ) } Figure 16: System prompt for LLM-as-a-judge criteria for evaluating stories. Checklists= { "creativity": [ "Originality of core concept - Compare how novel each story's central premise is. Better stories present fundamentally new ideas or unexpected scenarios that surprise readers; weaker stories rely on familiar tropes or predictable setups.", "Character innovation - Assess which story's characters are more distinctive. Better stories feature characters with unique traits, motivations, or development arcs that break stereotypes; weaker stories use conventional character types.", "Narrative structure innovation - Evaluate which story uses more inventive storytelling techniques. Better stories employ unconventional perspectives, sequencing, or structures that enhance impact; weaker stories follow standard linear formats.", "Thematic freshness - Compare how each story approaches its themes. Better stories provide new insights or unexpected angles on familiar concepts; weaker stories offer clichéd or predictable treatments.", "World-building distinctiveness - Assess which story creates a more imaginative setting. Better stories establish distinctive environments with fresh, internally consistent elements; weaker stories use generic or derivative settings." ], "coherence": [ "Plot logic and causality - Evaluate which story's events flow more logically. Better stories show clear cause-and-effect relationships where each event logically follows from previous actions; weaker stories have unexplained plot developments or logical gaps.", "Structural integrity - Compare the narrative arc completeness. Better stories maintain well-developed beginning, middle, and end with appropriate progression; weaker stories feel incomplete, rushed, or poorly structured.", "Character consistency - Assess which story's characters act more consistently. Better stories have characters whose actions, decisions, and growth align with established traits; weaker stories have characters who act out-of-character or inconsistently.", "Temporal coherence - Evaluate timeline clarity and consistency. Better stories maintain clear, consistent timelines without confusing jumps or contradictions; weaker stories have temporal inconsistencies or unclear sequencing.", "Narrative voice stability - Compare consistency in storytelling approach. Better stories maintain steady tone, style, and perspective throughout; weaker stories shift tone or perspective in jarring or unmotivated ways." ], "engagement": [ "Compelling hook - Compare how effectively each opening captures attention. Better stories immediately create curiosity and draw readers in; weaker stories have slow or unremarkable beginnings that fail to engage.", "Sustained narrative momentum - Evaluate which story better maintains reader interest. Better stories build through escalating stakes, revelations, or emotional investment; weaker stories lose momentum or plateau.", "Emotional impact and immersion - Assess which story creates stronger emotional connection and sense of presence. Better stories generate genuine feelings (empathy, excitement, tension) through vivid descriptions and authentic dialogue; weaker stories feel distant or emotionally flat.", "Pacing effectiveness - Compare how well each story's rhythm serves its content. Better stories allocate appropriate time to important moments without dragging or rushing; weaker stories have uneven pacing that undermines impact." ], "relevance": [ "Scenario fidelity - Evaluate which story better aligns with the given context. Better stories directly address the core scenario with characters, events, and outcomes that accurately reflect the context and constraints; weaker stories drift from the scenario or miss key requirements.", "Purpose fulfillment - Compare how effectively each story accomplishes its intended goal. Better stories clearly demonstrate or explore the intended concept; weaker stories lose sight of their purpose or only superficially address it.", "Tone and style appropriateness - Assess which story's presentation better fits the scenario. Better stories use tone, style, and content suitable for the given context and audience; weaker stories have mismatched tone or inappropriate stylistic choices.", "Focus and efficiency - Evaluate which story maintains tighter focus. Better stories make every element serve the purpose without unnecessary digressions; weaker stories include irrelevant details or lose narrative focus." ], "likelihood_bad( or good)": [ "AI behavior specificity and plausibility - Compare how clearly and realistically each story describes the AI's actions. Better stories specify exactly what the AI did (e.g., 'generated a low-risk score from facial expression') using current/near-future technology capabilities; weaker stories are vague about AI actions or invoke implausible capabilities.", "Credibility of AI-context mismatch - Assess which story presents a more believable failure. Better stories show plausible ways the AI could overlook specific user needs, conditions, or contexts (e.g., cultural nuance, masked distress) that current systems realistically miss; weaker stories require implausible AI blindspots.", "Clarity of harm pathway - Evaluate which story better traces cause-and-effect. Better stories clearly show the chain: what the AI did →how humans acted on it →what specific harm resulted, with each step following logically; weaker stories have unclear causal connections or hand-wave the harm mechanism.", "Realism of conditions and context - Compare which scenario is more grounded in reality. Better stories place events in realistic settings with today's norms, tools, and policies (healthcare, education, HR, etc.); weaker stories require unrealistic conditions or feel overly speculative.", "Concreteness of harmful consequences - Assess which story's harm is clearer and more observable. Better stories specify concrete, measurable harm (e.g., 'skipped three cancer screenings', 'diagnosed with anxiety disorder'); weaker stories describe vague or generalized negative outcomes." ] } Figure 17: Evaluation criteria checklist for LLM-as-a-judge.
|
2510.14713
|
Camera Movement Classification in Historical
Footage: A Comparative Study of Deep Video Models
Tingyu Lin1
Armin Dadras1,2
Florian Kleber1
Robert Sablatnig1
1Computer Vision Lab, TU Wien,
1040 Vienna, Austria
{tylin, adadras, kleber, sab}@cvl.tuwien.ac.at
2Institute of Creative\Media/Technologies, St. Pölten University of Applied Sciences,
3100 St. Pölten, Austria
Abstract
Camera movement conveys spatial and narrative information essential for under-
standing video content. While recent camera movement classification (CMC)
methods perform well on modern datasets, their generalization to historical footage
remains unexplored. This paper presents the first systematic evaluation of deep
video CMC models on archival film material. We summarize representative meth-
ods and datasets, highlighting differences in model design and label definitions.
Five standard video classification models are assessed on the HISTORIAN dataset,
which includes expert-annotated World War II footage. The best-performing model,
Video Swin Transformer, achieves 80.25% accuracy, showing strong convergence
despite limited training data. Our findings highlight the challenges and potential of
adapting existing models to low-quality video and motivate future work combining
diverse input modalities and temporal architectures.
1
Introduction
Camera movement is central to cinematic expression, shaping narrative structure, visual rhythm, and
audience engagement [1, 2]. Camera movement classification (CMC) assigns semantic labels to short
video segments based on the type of camera-induced motion, typically including categories such as
pan, tilt, track, dolly, truck, and zoom. Figure 1 shows a typical track movement in historical footage,
where the camera follows a moving object to maintain framing. The background displacement reveals
global motion, reflecting the semantic structure that CMC models aim to capture.
Figure 1: Example of a track camera movement from the HISTORIAN [7] dataset. Frames are
sampled every 20 frames to illustrate the motion.
Preprint. Under review.
arXiv:2510.14713v1 [cs.CV] 16 Oct 2025
Recent advances in CMC have explored both handcrafted descriptors, such as those based on motion
vectors or optical flow [6, 11], and data-driven approaches using deep neural networks [4, 8, 12]. Most
of these methods are trained and evaluated on modern video datasets (see Table 1). Historical footage
poses distinct challenges for computational models, often exhibiting noise, blur, exposure shifts, and
irregular motion. These conditions violate common assumptions in modern video processing, such
as clean appearance, consistent frame quality, and smoothly captured motion. Consequently, the
generalization of existing CMC techniques to historical material remains unexplored.
Table 1: Comparison of publicly available datasets for CMC.
Dataset
Video Source
Scale (Shots / Videos)
Types
HISTORIAN [7]
WWII archival films
838 movements / 98 films
8
MovieShots [12]
Modern movie trailers
46857 shots / 7858 videos
4
MOVE-SET [4]
Multi-domain video content
100K+ frame pairs / 448 videos
9
Petrogianni et al.’s dataset [10]
Feature films across decades
1803 shots / 48 films
10
This study contributes in two directions. First, we provide a structured summary of representative
CMC methods and publicly available datasets, highlighting architectural differences, input features,
and label definitions. As many existing methods lack open-source implementations, this survey
addresses reproducibility gaps and supports future benchmarking. Second, we evaluate the feasibil-
ity of applying general-purpose video classification models, initially developed for human action
recognition, to the CMC task in historical footage.
Beyond a methodological investigation, this work is part of a broader effort to develop automated
tools for analyzing historical film material within visual heritage pipelines. CMC is a fundamental
step in this context, supporting applications such as automatic video summarization, content retrieval,
and narrative reconstruction. In particular, our experiments are conducted on the HISTORIAN dataset,
which is designed for sustainable film preservation and semantic annotation of World War II archival
documentaries. To the best of our knowledge, this is the first attempt to apply deep learning-based
CMC models to degraded historical footage. Our findings offer insights into model robustness under
domain shift and provide a reproducible benchmark that aligns with the goals of applied computer
vision in cultural heritage contexts.
2
Related Work
Key representative CMC models are summarized in Table 2, with a focus on differences in architecture,
input features, and labeling schemes.
Table 2: Comparison of representative CMC methods.
Method
Model Type
Input Features
Types
Wang & Cheong [15]
Rule-based + MRF
Optical flow, motion entropy, attention maps
7
CAMHID [6]
Rule-based + SVM
Macroblock motion vectors
4
2D Histogram [11]
Rule-based + matching
2D histograms of flow direction and magnitude
10
SGNet [12]
Multi-branch CNN
RGB, saliency, segmentation
4
MUL-MOVE-Net [4]
CNN + BiLSTM
Optical flow histograms
9
Petrogianni et al. [10]
CNN + LSTM / SVM
Low-level visual statistics
10
LWSRNet [8]
Lightweight 3D CNN
RGB, flow, saliency, segmentation
8
Early approaches to CMC relied primarily on handcrafted motion descriptors derived from motion
vector fields or optical flow analysis. Wang and Cheong [15] proposed a semantically-informed
taxonomy, differentiating seven directing styles using foreground-background segmentation and
temporal smoothness constraints. Hasan et al. [6] introduced CAMHID, which computes histograms
of macroblock-based motion vectors and classifies them into four categories using support vector
machines. Prasertsakul et al. [11] extended this direction by constructing two-dimensional motion
direction and magnitude histograms and applying template-matching rules to classify ten movement
types. While computationally efficient, these rule-based methods often face difficulty generalizing to
unconstrained or noisy conditions, particularly in scenes dominated by foreground motion or nonrigid
elements, as noted in [6, 11].
2
With the rise of deep learning, CMC has seen significant improvements. SGNet [12] pioneered this
transition, modeling CMC as a four-category classification task (static, motion, push, pull). SGNet
employed multi-branch convolutional neural networks (CNNs), integrating visual features from RGB
frames, saliency maps, and semantic segmentation. Chen et al.[4] introduced MUL-MOVE-Net,
employing bidirectional long short-term memory (BiLSTM) modules over optical flow histograms,
expanding classification to nine camera movements, including directional and rotational motions.
Petrogianni et al.[10] explored interpretable low-level visual features (e.g., shot length, motion
strength) with both SVM and LSTM classifiers across ten motion categories. Recently, Li et al. [8]
presented LWSRNet, a lightweight 3D CNN architecture fusing multiple input modalities for joint
camera motion and scale prediction, achieving state-of-the-art results on their dataset.
Several datasets with camera movement annotations have been made publicly available, including
MovieShots [12], MOVE-SET [4], the dataset by Petrogianni et al. [10], and HISTORIAN [7]. These
datasets differ significantly in their source material, scale, and movement types, as summarized in
Table 1. Among them, HISTORIAN is the only dataset focused on historical video content. It contains
annotated segments extracted from 183 World War II archival film shots, with frame-level annotations
across eight camera movement types. Another important challenge is the lack of standardized
movement definitions across datasets. As shown in Table 3, each dataset adopts its own set of
camera motion categories, differing in granularity and terminology. This inconsistency complicates
cross-dataset evaluation and poses challenges for fine-tuning models pretrained on modern footage
for use in historical contexts.
Table 3: Comparison of camera movement types defined in each dataset.
Dataset
Camera Movement Types
HISTORIAN [7]
pan, tilt, track, truck, dolly, zoom, pedestal, pan_tilt
MovieShots [12]
static, motion, push, pull
MOVE-SET [4]
static, up, down, left, right, zoom in, zoom out, rotate left, rotate right
Petrogianni et al.’s
dataset [10]
static, vertical movement, tilt, panoramic, panoramic lateral, travelling
in, travelling out, zoom in, aerial, handheld
3
Method
Although CMC differs from human action recognition regarding semantic focus and motion locality,
the two tasks share important structural properties. Both involve learning to model temporal dynamics
and to distinguish between fine-grained motion patterns from raw video input. This suggests that
general-purpose video classification models developed initially for action recognition can serve as
effective baselines for CMC. In particular, their ability to capture spatiotemporal dependencies from
appearance and motion cues aligns well with the needs of CMC, where frame-to-frame movement
consistency plays a central role. At the same time, camera motion introduces distinct modeling
challenges. Unlike human actions, which are often spatially localized and semantically interpretable,
camera movements influence the entire frame in a globally coherent yet visually less distinctive
manner. The associated motion cues are often subtle and exhibit lower visual variance across classes.
This issue is further amplified in historical footage, where degradation, overscan, and unstable
cinematography are prevalent. As a result, CMC requires models to rely more on low-level temporal
motion patterns than on high-level object semantics.
To examine the adaptability of established video classification models to the CMC task, we select five
widely used architectures that represent different design paradigms. These include 3D convolutional
networks such as C3D [13] and I3D [3], which directly encode short-term spatiotemporal motion
from RGB inputs; factorized 3D CNNs like R(2+1)D [14], which decouple spatial and temporal
learning; 2D CNNs with segmental consensus such as TSN [16], which aggregate information across
sparsely sampled frames; and hierarchical spatiotemporal transformers exemplified by the Video
Swin Transformer [9], which model long-range dependencies through local attention blocks.
3
4
Experiments
Our experiments are based on the HISTORIAN dataset [7], which contains 767 manually annotated
movement segments extracted from 183 historical film shots. The original annotations include
eight categories, but we exclude underrepresented classes such as zoom (4 instances) and pedestal
(1 instance), retaining six categories with sufficient sample sizes: pan, tilt, track, truck, dolly,
and pan_tilt. Each annotated movement segment is converted into a fixed-length clip, with input
resolution, temporal stride, and preprocessing tailored to each model’s default configuration. To
maximize training data given the small dataset, we adopt a 9:1 train-validation split, grouping all
segments from the same shot in the same partition to avoid leakage. We acknowledge the small
validation size and plan to explore cross-validation in future work. All models are trained on RGB
inputs only, without additional flow or multimodal streams. Pretrained weights are used where
applicable to facilitate convergence: C3D is initialized from Sports1M, while the other models use
ImageNet pretraining. We report standard classification metrics: top-1 accuracy, macro-averaged F1
score, and top-2 accuracy to account for near-miss predictions. Table 4 presents the results.
Table 4: Performance of each model on the HISTORIAN validation set (6-class).
Model
Top-1 Accuracy (%)
Top-2 Accuracy (%)
Weighted F1 (%)
C3D
64.20
81.48
59.16
R(2+1)D
48.15
64.20
37.28
TSN
50.62
75.31
40.19
I3D
74.07
77.78
69.50
Video Swin
80.25
87.65
76.24
Across all models, we observe a consistent performance gap between architectures with stronger
temporal modeling capacity and those relying on static or sparsely sampled features. I3D and Video
Swin, which incorporate 3D convolutions and spatiotemporal attention mechanisms, outperform
simpler models such as TSN and R(2+1)D. These results support our hypothesis that modeling
temporal continuity is essential for recognizing subtle and globally coherent camera movement
patterns, particularly in degraded historical footage. Note that due to the limited size of the annotated
dataset, all results should be interpreted with caution. Class imbalance and scarce examples may
introduce training dynamics and model generalization variance. One consistent observation is that
the Video Swin Transformer achieves the highest accuracy and F1 score, demonstrating strong
convergence and generalization even with relatively few training samples.
For comparison, we reference the traditional baseline reported in the HISTORIAN paper [7], which
combines dense optical flow estimation [5] with rule-based filtering and angular binning following
the CAMHID method [6]. Their evaluation was conducted on a subset containing only pan and
tilt categories, along with numerous static segments not included in the released dataset. In this
restricted setting, the reported accuracy reached 82%. While our evaluation includes six movement
categories and uses a different partition of the data, our best model achieves a comparable accuracy
of 80.25%, suggesting that standard video classification architectures offer a competitive alternative
to handcrafted methods under the challenging conditions of historical footage.
5
Conclusions and Outlook
This work presents a structured investigation of CMC in historical footage. We review representative
CMC models and datasets and empirically evaluate five deep video classification architectures
designed initially for human action recognition. Our experiments on the HISTORIAN dataset
demonstrate that these models can achieve reasonable performance despite the challenges of degraded
archival content, with the best model reaching 80.25% accuracy.
Several directions remain open for future research. First, input modalities can be extended beyond
RGB to include optical flow, depth, or learned motion representations, which may improve robustness
to visual degradation. Second, due to the lack of open-source implementations for most CMC methods,
reimplementing and benchmarking these systems would enable fairer and more comprehensive
comparisons. Finally, transfer learning strategies using modern CMC datasets for pretraining before
fine-tuning on historical footage could help improve generalization under domain shift.
4
Acknowledgments and Disclosure of Funding
This work was supported by the Austrian Science Fund (FWF) – doc.funds.connect, under project
grant no. DFH 37-N: "Visual Heritage: Visual Analytics and Computer Vision Meet Cultural
Heritage.".
References
[1] David Bordwell. On the History of Film Style. Harvard University Press, 1997.
[2] David Bordwell, Kristin Thompson, and Jeff Smith. Film art: An introduction, volume 7. McGraw-Hill,
New York, 2010.
[3] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset.
In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017.
[4] Zeyu Chen, Yana Zhang, Lianyi Zhang, and Cheng Yang. Ro-textcnn based mul-move-net for camera
motion classification. In 2021 IEEE/ACIS 20th International Fall Conference on Computer and Information
Science (ICIS Fall), pages 182–186. IEEE, 2021.
[5] Gunnar Farnebäck. Two-frame motion estimation based on polynomial expansion. In Proceedings of the
Scandinavian Conference on Image Analysis, pages 363–370. Springer, 2003.
[6] Muhammad Abul Hasan, Min Xu, Xiangjian He, and Changsheng Xu. Camhid: Camera motion histogram
descriptor and its application to cinematographic shot classification. IEEE Transactions on Circuits and
Systems for Video Technology, 24(10):1682–1695, 2014.
[7] Daniel Helm, Fabian Jogl, and Martin Kampel. Historian: A large-scale historical film dataset with
cinematographic annotation. In 2022 IEEE International Conference on Image Processing (ICIP), pages
2087–2091, 2022.
[8] Yuzhi Li, Tianfeng Lu, and Feng Tian. A lightweight weak semantic framework for cinematographic shot
classification. Scientific Reports, 13(1):16089, 2023.
[9] Ze Liu, Jia Ning, Yue Cao, Yixuan Wei, Zheng Zhang, Stephen Lin, and Han Hu. Video swin transformer.
In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages
3202–3211, June 2022.
[10] Antonia Petrogianni, Panagiotis Koromilas, and Theodoros Giannakopoulos. Film shot type classification
based on camera movement styles. In Iberian Conference on Pattern Recognition and Image Analysis,
pages 602–615. Springer, 2022.
[11] Pawin Prasertsakul, Toshiaki Kondo, and Hiroyuki Iida. Video shot classification using 2d motion
histogram. In 2017 14th International Conference on Electrical Engineering/Electronics, Computer,
Telecommunications and Information Technology (ECTI-CON), pages 202–205. IEEE, 2017.
[12] Anyi Rao, Jiaze Wang, Linning Xu, Xuekun Jiang, Qingqiu Huang, Bolei Zhou, and Dahua Lin. A unified
framework for shot type classification based on subject centric lens. In The European Conference on
Computer Vision (ECCV), pages 17–34. Springer, 2020.
[13] Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal
features with 3d convolutional networks. In Proceedings of the IEEE International Conference on Computer
Vision (ICCV), December 2015.
[14] Du Tran, Heng Wang, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at
spatiotemporal convolutions for action recognition. In Proceedings of the IEEE Conference on Computer
Vision and Pattern Recognition (CVPR), June 2018.
[15] Hee Lin Wang and Loong-Fah Cheong. Taxonomy of directing semantics for film shot classification. IEEE
Transactions on Circuits and Systems for Video Technology, 19(10):1529–1542, 2009.
[16] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua Lin, Xiaoou Tang, and Luc Van Gool. Temporal
segment networks: Towards good practices for deep action recognition. In The European Conference on
Computer Vision (ECCV), pages 20–36. Springer, 2016.
5
|
Camera Movement Classification in Historical Footage: A Comparative Study of Deep Video Models Tingyu Lin1 Armin Dadras1,2 Florian Kleber1 Robert Sablatnig1 1Computer Vision Lab, TU Wien, 1040 Vienna, Austria {tylin, adadras, kleber, 2 /Technologies, St. Pölten 3100 St. Pölten, Austria Abstract Camera movement conveys spatial and narrative information essential for understanding video content. While recent camera movement classification (CMC) methods perform well on modern datasets, their generalization to historical footage remains unexplored. This paper presents the first systematic evaluation of deep video CMC models on archival film material. We summarize representative methods and datasets, highlighting differences in model design and label definitions. Five standard video classification models are assessed on the HISTORIAN dataset, which includes expert-annotated World War II footage. The best-performing model, Video Swin Transformer, achieves 80.25% accuracy, showing strong convergence despite limited training data. Our findings highlight the challenges and potential of adapting existing models to low-quality video and motivate future work combining diverse input modalities and temporal architectures. 1 Introduction Camera movement is central to cinematic expression, shaping narrative structure, visual rhythm, and audience engagement [1, 2]. Camera movement classification (CMC) assigns semantic labels to short video segments based on the type of camera-induced motion, typically including categories such as pan, tilt, track, dolly, truck, and zoom. Figure 1 shows a typical track movement in historical footage, where the camera follows a moving object to maintain framing. The background displacement reveals global motion, reflecting the semantic structure that CMC models aim to capture. Figure 1: Example of a track camera movement from the HISTORIAN [7] dataset. Frames are sampled every 20 frames to illustrate the motion. Preprint. Under review. 16 Oct 2025 Recent advances in CMC have explored both handcrafted descriptors, such as those based on motion vectors or optical flow [6, 11], and data-driven approaches using deep neural networks [4, 8, 12]. Most of these methods are trained and evaluated on modern video datasets (see Table 1). Historical footage poses distinct challenges for computational models, often exhibiting noise, blur, exposure shifts, and irregular motion. These conditions violate common assumptions in modern video processing, such as clean appearance, consistent frame quality, and smoothly captured motion. Consequently, the generalization of existing CMC techniques to historical material remains unexplored. Table 1: Comparison of publicly available datasets for CMC. Dataset Video Source Scale (Shots / Videos) Types HISTORIAN [7] WWII archival films 838 movements / 98 films 8 MovieShots [12] Modern movie trailers 46857 shots / 7858 videos 4 MOVE-SET [4] Multi-domain video content 100K+ frame pairs / 448 videos 9 Petrogianni et al.'s dataset [10] Feature films across decades 1803 shots / 48 films 10 This study contributes in two directions. First, we provide a structured summary of representative CMC methods and publicly available datasets, highlighting architectural differences, input features, and label definitions. As many existing methods lack open-source implementations, this survey addresses reproducibility gaps and supports future benchmarking. Second, we evaluate the feasibility of applying general-purpose video classification models, initially developed for human action recognition, to the CMC task in historical footage. Beyond a methodological investigation, this work is part of a broader effort to develop automated tools for analyzing historical film material within visual heritage pipelines. CMC is a fundamental step in this context, supporting applications such as automatic video summarization, content retrieval, and narrative reconstruction. In particular, our experiments are conducted on the HISTORIAN dataset, which is designed for sustainable film preservation and semantic annotation of World War II archival documentaries. To the best of our knowledge, this is the first attempt to apply deep learning-based CMC models to degraded historical footage. Our findings offer insights into model robustness under domain shift and provide a reproducible benchmark that aligns with the goals of applied computer vision in cultural heritage contexts. 2 Related Work Key representative CMC models are summarized in Table 2, with a focus on differences in architecture, input features, and labeling schemes. Table 2: Comparison of representative CMC methods. Method Model Type Input Features Types Wang & Cheong [15] Rule-based + MRF Optical flow, motion entropy, attention maps 7 CAMHID [6] Rule-based + SVM Macroblock motion vectors 4 2D Histogram [11] Rule-based + matching 2D histograms of flow direction and magnitude 10 SGNet [12] Multi-branch CNN RGB, saliency, segmentation 4 MUL-MOVE-Net [4] CNN + BiLSTM Optical flow histograms 9 Petrogianni et al. [10] CNN + LSTM / SVM Low-level visual statistics 10 LWSRNet [8] Lightweight 3D CNN RGB, flow, saliency, segmentation 8 Early approaches to CMC relied primarily on handcrafted motion descriptors derived from motion vector fields or optical flow analysis. Wang and Cheong [15] proposed a semantically-informed taxonomy, differentiating seven directing styles using foreground-background segmentation and temporal smoothness constraints. Hasan et al. [6] introduced CAMHID, which computes histograms of macroblock-based motion vectors and classifies them into four categories using support vector machines. Prasertsakul et al. [11] extended this direction by constructing two-dimensional motion direction and magnitude histograms and applying template-matching rules to classify ten movement types. While computationally efficient, these rule-based methods often face difficulty generalizing to unconstrained or noisy conditions, particularly in scenes dominated by foreground motion or nonrigid elements, as noted in [6, 11]. 2 With the rise of deep learning, CMC has seen significant improvements. SGNet [12] pioneered this transition, modeling CMC as a four-category classification task (static, motion, push, pull). SGNet employed multi-branch convolutional neural networks (CNNs), integrating visual features from RGB frames, saliency maps, and semantic segmentation. Chen et al.[4] introduced MUL-MOVE-Net, employing bidirectional long short-term memory (BiLSTM) modules over optical flow histograms, expanding classification to nine camera movements, including directional and rotational motions. Petrogianni et al.[10] explored interpretable low-level visual features (e.g., shot length, motion strength) with both SVM and LSTM classifiers across ten motion categories. Recently, Li et al. [8] presented LWSRNet, a lightweight 3D CNN architecture fusing multiple input modalities for joint camera motion and scale prediction, achieving state-of-the-art results on their dataset. Several datasets with camera movement annotations have been made publicly available, including MovieShots [12], MOVE-SET [4], the dataset by Petrogianni et al. [10], and HISTORIAN [7]. These datasets differ significantly in their source material, scale, and movement types, as summarized in Table 1. Among them, HISTORIAN is the only dataset focused on historical video content. It contains annotated segments extracted from 183 World War II archival film shots, with frame-level annotations across eight camera movement types. Another important challenge is the lack of standardized movement definitions across datasets. As shown in Table 3, each dataset adopts its own set of camera motion categories, differing in granularity and terminology. This inconsistency complicates cross-dataset evaluation and poses challenges for fine-tuning models pretrained on modern footage for use in historical contexts. Table 3: Comparison of camera movement types defined in each dataset. Dataset Camera Movement Types HISTORIAN [7] pan, tilt, track, truck, dolly, zoom, pedestal, pan_tilt MovieShots [12] static, motion, push, pull MOVE-SET [4] static, up, down, left, right, zoom in, zoom out, rotate left, rotate right Petrogianni et al.'s dataset [10] static, vertical movement, tilt, panoramic, panoramic lateral, travelling in, travelling out, zoom in, aerial, handheld 3 Method Although CMC differs from human action recognition regarding semantic focus and motion locality, the two tasks share important structural properties. Both involve learning to model temporal dynamics and to distinguish between fine-grained motion patterns from raw video input. This suggests that general-purpose video classification models developed initially for action recognition can serve as effective baselines for CMC. In particular, their ability to capture spatiotemporal dependencies from appearance and motion cues aligns well with the needs of CMC, where frame-to-frame movement consistency plays a central role. At the same time, camera motion introduces distinct modeling challenges. Unlike human actions, which are often spatially localized and semantically interpretable, camera movements influence the entire frame in a globally coherent yet visually less distinctive manner. The associated motion cues are often subtle and exhibit lower visual variance across classes. This issue is further amplified in historical footage, where degradation, overscan, and unstable cinematography are prevalent. As a result, CMC requires models to rely more on low-level temporal motion patterns than on high-level object semantics. To examine the adaptability of established video classification models to the CMC task, we select five widely used architectures that represent different design paradigms. These include 3D convolutional networks such as C3D [13] and I3D [3], which directly encode short-term spatiotemporal motion from RGB inputs; factorized 3D CNNs like R(2+1)D [14], which decouple spatial and temporal learning; 2D CNNs with segmental consensus such as TSN [16], which aggregate information across sparsely sampled frames; and hierarchical spatiotemporal transformers exemplified by the Video Swin Transformer [9], which model long-range dependencies through local attention blocks. 3 4 Experiments Our experiments are based on the HISTORIAN dataset [7], which contains 767 manually annotated movement segments extracted from 183 historical film shots. The original annotations include eight categories, but we exclude underrepresented classes such as zoom (4 instances) and pedestal (1 instance), retaining six categories with sufficient sample sizes: pan, tilt, track, truck, dolly, and pan_tilt. Each annotated movement segment is converted into a fixed-length clip, with input resolution, temporal stride, and preprocessing tailored to each model's default configuration. To maximize training data given the small dataset, we adopt a 9:1 train-validation split, grouping all segments from the same shot in the same partition to avoid leakage. We acknowledge the small validation size and plan to explore cross-validation in future work. All models are trained on RGB inputs only, without additional flow or multimodal streams. Pretrained weights are used where applicable to facilitate convergence: C3D is initialized from Sports1M, while the other models use ImageNet pretraining. We report standard classification metrics: top-1 accuracy, macro-averaged F1 score, and top-2 accuracy to account for near-miss predictions. Table 4 presents the results. Table 4: Performance of each model on the HISTORIAN validation set (6-class). Model Top-1 Accuracy (%) Top-2 Accuracy (%) Weighted F1 (%) C3D 64.20 81.48 59.16 R(2+1)D 48.15 64.20 37.28 TSN 50.62 75.31 40.19 I3D 74.07 77.78 69.50 Video Swin 80.25 87.65 76.24 Across all models, we observe a consistent performance gap between architectures with stronger temporal modeling capacity and those relying on static or sparsely sampled features. I3D and Video Swin, which incorporate 3D convolutions and spatiotemporal attention mechanisms, outperform simpler models such as TSN and R(2+1)D. These results support our hypothesis that modeling temporal continuity is essential for recognizing subtle and globally coherent camera movement patterns, particularly in degraded historical footage. Note that due to the limited size of the annotated dataset, all results should be interpreted with caution. Class imbalance and scarce examples may introduce training dynamics and model generalization variance. One consistent observation is that the Video Swin Transformer achieves the highest accuracy and F1 score, demonstrating strong convergence and generalization even with relatively few training samples. For comparison, we reference the traditional baseline reported in the HISTORIAN paper [7], which combines dense optical flow estimation [5] with rule-based filtering and angular binning following the CAMHID method [6]. Their evaluation was conducted on a subset containing only pan and tilt categories, along with numerous static segments not included in the released dataset. In this restricted setting, the reported accuracy reached 82%. While our evaluation includes six movement categories and uses a different partition of the data, our best model achieves a comparable accuracy of 80.25%, suggesting that standard video classification architectures offer a competitive alternative to handcrafted methods under the challenging conditions of historical footage. 5 Conclusions and Outlook This work presents a structured investigation of CMC in historical footage. We review representative CMC models and datasets and empirically evaluate five deep video classification architectures designed initially for human action recognition. Our experiments on the HISTORIAN dataset demonstrate that these models can achieve reasonable performance despite the challenges of degraded archival content, with the best model reaching 80.25% accuracy. Several directions remain open for future research. First, input modalities can be extended beyond RGB to include optical flow, depth, or learned motion representations, which may improve robustness to visual degradation. Second, due to the lack of open-source implementations for most CMC methods, reimplementing and benchmarking these systems would enable fairer and more comprehensive comparisons. Finally, transfer learning strategies using modern CMC datasets for pretraining before fine-tuning on historical footage could help improve generalization under domain shift. 4 Acknowledgments and Disclosure of Funding This work was supported by the Austrian Science Fund (FWF) - doc.funds.connect, under project grant no. DFH 37-N: "Visual Heritage: Visual Analytics and Computer Vision Meet Cultural Heritage.". References [1] David Bordwell. On the History of Film Style. Harvard University Press, 1997. [2] David Bordwell, Kristin Thompson, and Jeff Smith. Film art: An introduction, volume 7. McGraw-Hill, New York, 2010. [3] Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. [4] Zeyu Chen, Yana Zhang, Lianyi Zhang, and Cheng Yang. Ro-textcnn based mul-move-net for camera motion classification. In 2021 IEEE/ACIS 20th International Fall Conference on Computer and Information Science (ICIS Fall), pages 182-186. IEEE, 2021. [5] Gunnar Farnebäck. Two-frame motion estimation based on polynomial expansion. In Proceedings of the Scandinavian Conference on Image Analysis, pages 363-370. Springer, 2003. [6] Muhammad Abul Hasan, Min Xu, Xiangjian He, and Changsheng Xu. Camhid: Camera motion histogram descriptor and its application to cinematographic shot classification. IEEE Transactions on Circuits and Systems for Video Technology, 24(10):1682-1695, 2014. [7] Daniel Helm, Fabian Jogl, and Martin Kampel. Historian: A large-scale historical film dataset with cinematographic annotation. In 2022 IEEE International Conference on Image Processing (ICIP), pages 2087-2091, 2022. [8] Yuzhi Li, Tianfeng Lu, and Feng Tian. A lightweight weak semantic framework for cinematographic shot classification. Scientific Reports, 13(1):16089, 2023. [9] Ze Liu, Jia Ning, Yue Cao, Yixuan Wei, Zheng Zhang, Stephen Lin, and Han Hu. Video swin transformer. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 3202-3211, June 2022. [10] Antonia Petrogianni, Panagiotis Koromilas, and Theodoros Giannakopoulos. Film shot type classification based on camera movement styles. In Iberian Conference on Pattern Recognition and Image Analysis, pages 602-615. Springer, 2022. [11] Pawin Prasertsakul, Toshiaki Kondo, and Hiroyuki Iida. Video shot classification using 2d motion histogram. In 2017 14th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTI-CON), pages 202-205. IEEE, 2017. [12] Anyi Rao, Jiaze Wang, Linning Xu, Xuekun Jiang, Qingqiu Huang, Bolei Zhou, and Dahua Lin. A unified framework for shot type classification based on subject centric lens. In The European Conference on Computer Vision (ECCV), pages 17-34. Springer, 2020. [13] Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal features with 3d convolutional networks. In Proceedings of the IEEE International Conference on Computer Vision (ICCV), December 2015. [14] Du Tran, Heng Wang, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018. [15] Hee Lin Wang and Loong-Fah Cheong. Taxonomy of directing semantics for film shot classification. IEEE Transactions on Circuits and Systems for Video Technology, 19(10):1529-1542, 2009. [16] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua Lin, Xiaoou Tang, and Luc Van Gool. Temporal segment networks: Towards good practices for deep action recognition. In The European Conference on Computer Vision (ECCV), pages 20-36. Springer, 2016. 5
|
2510.14712
|
J. Fluid Mech.
1
Lattice Boltzmann model for non-ideal
compressible fluid dynamics
S. A. Hosseini1†, M. Feinberg1‡, I. V. Karlin1¶
1Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland.
(Received xx; revised xx; accepted xx)
We present a lattice Boltzmann formulation for the simulation of compressible, non-ideal
fluid flows. The method employs first-neighbor lattices and introduces a consistent set
of correction terms through quasi-equilibrium attractors, ensuring positive-definite and
Galilean-invariant Navier–Stokes dissipation rates. This construction circumvents the
need for extended stencils or ad hoc regularization, while maintaining numerical stability
and thermodynamic consistency across a broad range of flow regimes. The resulting model
accurately reproduces both Euler- and Navier–Stokes-level hydrodynamics. As a stringent
validation, we demonstrate, for the first time within a lattice Boltzmann framework,
quantitatively accurate simulations of drop–shock interactions at Mach numbers up to
1.47. The proposed approach thus extends the applicability of lattice Boltzmann methods
to high-speed, non-ideal compressible flows with a minimal kinetic stencil.
Key words:
1. Introduction
Non-ideal compressible fluid dynamics is a novel and rapidly developing branch of fluid
mechanics, mainly due to the emergence of methods and technologies operating in the
near-, trans- and super-critical regimes. It is, in part, concerned with gas-dynamics of
single phase fluids in non-ideal thermodynamic states, i.e. states where compressibility
factor differs from unity (Guardone et al. 2024). These states are illustrated in a P-T
diagram for CO2 in Figure 1.
Non-ideal compressible fluid dynamics comprises the
dynamics of supercritical fluids, dense vapors, and liquids. The latter two are separated
by the Widom line, an extension beyond the critical point that delineates vapor-like
and liquid-like behavior, identified by maxima in the constant-pressure specific heat
(Simeoni et al. 2010). In addition it also includes fluids of higher molecular complexity
with negative fundamental gas dynamics derivative (Colonna et al. 2009), also known
as Bethe-Zel’dovich-Thompson (BZT) fluids (Zel’Dovich 1946; Thompson & Lambrakis
1973). The rapidly growing interest in the energy industry for fluids operating in those
thermodynamic states along with their radically different behavior as compared to
fluids in ideal states are two important factors demonstrating the need for systematic
studies of such flows. However, as noted in the literature, while considerable effort
has been put in developing experimental set-ups for non-ideal fluid dynamics in recent
years (Lettieri et al. 2015; Zocca et al. 2019; Gallarini et al. 2021; Head et al. 2022),
† Email address for correspondence: shosseini@ethz.ch
‡ Email address for correspondence: mfeinberg@ethz.ch
¶ Email address for correspondence: ikarlin@ethz.ch
arXiv:2510.14712v1 [physics.flu-dyn] 16 Oct 2025
2
S. A. Hosseini, M. Feinberg, I. V. Karlin
Figure
1:
Pressure-temperature
diagram
for
CO2.
The
color-bar
indicates
the
compressibility factor Z = P/ρRT. Figure reproduced from Guardone et al. (2024).
experimental data remains scarce and is complicated to acquire (Guardone et al. 2024).
This means that development of consistent and efficient numerical tools for the simulation
of such flows is of the utmost importance in the current state of development of both
our understanding of non-ideal compressible fluid dynamics physics and technologies
such as organic Rankine cycles or supercritical CO2 turbines (Guardone et al. 2024;
Gunawan et al. 2023; Chen et al. 2023). More specifically, direct numerical simulations
are necessary both to understand the complex physics of non-ideal compressible fluid
dynamics and to generate engineering databases. Examples include skin friction and
dissipation coefficients (Cramer et al. 1996; Pini & De Servi 2018), heat transfer in
wall-bounded turbulent supercritical flows (Peeters et al. 2016; Kawai 2019), and the
critical mass flow rate and pressure in nozzles and turbines involving flashing flows. In
addition, while direct numerical simulations can be at times cost-prohibitive for full-scale
simulations, they are a necessary tool for development of the subgrid-scale closure models
required for the more cost effective large eddy and Reynolds-averaged Navier-Stokes
simulations (Garnier et al. 2009). Recent studies on dense vapor effects on fluid structures
in homogeneous isotropic turbulence have shown that while large-scale flow structures are
mostly affected by molecular complexity, smaller-scale flow structures are impacted by
local variations in sound speed (Sciacovelli et al. 2017; Duan et al. 2021). In addition, the
dense vapor has been shown to display modified shocklet structures, with considerably
reduced jumps in pressure, density, and entropy in compression shocklets, and emergence
of expansion shocklets (Giauque et al. 2017, 2020). Similar importance has been reported
for wall bounded turbulence where substantially different – from ideal gas – dynamics
at smaller scales have been reported (Vadrot et al. 2021) and strong normal gradients in
density and viscosity –and therefore local speed of sound and Mach number– are primary
elements affecting turbulence (Pecnik & Patel 2017; Peeters et al. 2016). Reliable direct
numerical simulation studies of non-ideal compressible fluid dynamics can fill existing
gaps in understanding of physics of such flows. A reliable numerical scheme pre-supposes a
physical model valid and consistently covering all regime of interest here, i.e. super-, trans-
and near-critical flows involving pronounced non-ideal and compressibility effects. The
development and validation of efficient numerical schemes for Navier-Stokes-Korteweg
(NSK) dynamics, regardless of method, can have a significant impact on the broader
literature and knowledge on non-ideal compressible flows.
The lattice Boltzmann method (LBM) is an excellent candidate for the flow regimes
of interest here. Already well established as an efficient method for resolved simula-
tion of flows in the incompressible limit, the distinct numerical features of LBM offer
Lattice Boltzmann model for non-ideal compressible fluid dynamics
3
advantages for compressible, non-ideal fluid dynamics. Specifically, the decoupling of
non-linearity and non-locality noted by S. Succi, in LBM ”nonlinearity is local, non-
locality is linear” (Succi 2001; Succi & Succi 2018). LBM solvers are conservative and low-
dissipative schemes with excellent spectral properties –both dissipation and dispersion–
as compared to conventional solvers of the same order (Hosseini 2020; Hosseini et al.
2019, 2022b; Martinez et al. 1994; Wissocq et al. 2019; Peng et al. 2010), especially for
normal propagation modes, i.e. acoustic modes (Bres et al. 2009; Viggen 2011, 2014).
Along with incompressible ideal fluid dynamics, two-phase flow simulation has witnessed
major success and growth in popularity in LBM, starting with formulations such as
the color-gradient (Gunstensen et al. 1991), pseudo-potential (Shan & Chen 1993) and
free energy (Swift et al. 1996) in the early 90’s. The latter two are of special interest
in the context of non-ideal fluid dynamics as in effect they solve a form of the NSK.
The large majority of the literature that focuses or uses this class of models has been
tailored towards becoming essentially highly efficient interface tracking models for multi-
phase flows (Luo et al. 2021; Mazloomi et al. 2015; Hosseini & Karlin 2023; Chen et al.
2014; Li et al. 2016). Non-ideal compressible thermodynamics have long been neglected
in LBM applications. The free energy model for instance, which is the only one out
of the previously-listed approaches which results from minimization of a free energy
functional under constraint on total mass, can readily be shown to be related to mean-
field kinetic models such as the Boltzmann-Enskog-Vlasov (Hosseini et al. 2022a; Hosseini
& Karlin 2023) equations and recover, at the hydrodynamic limit and under specific
scaling, the NSK equations (Hosseini et al. 2022a) and mean-field van der Waals fluid
thermodynamics. This means that it can not only model two-phase flow dynamics and
phase separation, but also properly recover all mean-field, near-, trans- and super-critical
behaviors associated to non-ideal fluids, as demonstrated for instance through studies of
properties such as the Tollman length (Hosseini et al. 2022a; Hosseini & Karlin 2023;
Reyhanian et al. 2020, 2021; Lulli et al. 2022a,b; H´azi & M´arkus 2008).
Use and extension of this class of thermodynamically consistent model to compressible
non-ideal fluid dynamics is a largely under-explored area that can considerably impact
research on non-ideal compressible fluid dynamics. First attempts at proposing LBM
for compressible non-ideal flows were documented in (He & Doolen 2002). Since then,
thermal multi-phase models based on the pseudo-potential approach have witnessed
steady growth. However, the vast majority of the models and studies in the literature
have been tailored to boiling applications, see for instance (Li et al. 2015; Fei et al.
2020; Saito et al. 2021; Fang et al. 2017; Huang et al. 2021). To the author’s knowledge,
only documented attempts at modeling compressible non-ideal flows beyond evaporation
are (Vienne et al. 2024) where a hybrid lattice Boltzmann finite differences scheme –
finite differences for the energy equation– was proposed and (Reyhanian et al. 2020,
2021) where the authors proposed a numerical model based on the Particles-on-Demand
realization of the lattice Boltzmann method Dorschner et al. (2018). We propose to
address this gap in the literature with a lattice Boltzmann model for non-ideal fluids in
the compressible regime. To retain the main advantages of the lattice Boltzmann method
the proposed model relies on classical first neighbor lattices, taking advantage of a second
distribution function for the energy balance equations (Rykov 1975; Saadat et al. 2021;
Prasianakis & Karlin 2007, 2008). In addition, the model removes the Galilean-variant
errors in the diagonal components of the third-order equilibrium moments tensor and
provides for independent control over the bulk viscosity; The latter is a critical point
as the bulk viscosity dictated by the BGK structure, for equilibria carrying a pressure
other than the ideal gas pressure, can take on negative values in the hydordynamic
limit, see (Hosseini et al. 2022a). Together with a consistent second-order discretization
4
S. A. Hosseini, M. Feinberg, I. V. Karlin
and treatment of source terms, the model will be shown to correctly recover the target
hydrodynamic limit.
The article starts with a brief introduction of the target hydrodynamic limit followed
by presentation of the proposed lattice Boltzmann model. The proposed model is then
validated with a list of configurations of increasing complexity.
2. Balance equations for compressible non-ideal fluid
We begin with a brief overview of the compressible non-ideal fluid system. To that
end, we introduce the macroscopic fields of fluid density ρ(x, t), momentum ρu(x, t) and
bulk energy ρE(x, t). The latter is the sum of the flow kinetic energy and the internal
energy ρe,
ρE = ρe + 1
2ρu2.
(2.1)
The specific internal energy e(v, T) per unit mass is a function of absolute temperature
T and specific volume v = 1/ρ, and is defined by a familiar thermodynamic relation for
its differential,
de = cvdT +
T
∂P
∂T
−P
dv,
(2.2)
where P(v, T) is the thermodynamic pressure and cv is specific heat at constant volume,
cv = ∂e
∂T .
(2.3)
Below, it will be convenient to consider the thermodynamic equation of state as a function
of the density ρ rather than of the specific volume v,
P(ρ, T) = P(v, T)|v=1/ρ ,
(2.4)
so that the differential of the internal energy (2.2) is written,
de = cvdT −
T
∂P
∂T
−P
dρ
ρ2 .
(2.5)
The D-dimensional mass, momentum and bulk energy balance equations are,
∂tρ + ∇· ρu = 0,
(2.6)
∂t(ρu) + ∇· ρu ⊗u + ∇P + ∇· TNS + ∇· TK = 0,
(2.7)
∂t(ρE) + ∇· (ρEu + Pu) + u · (∇· TK) + ∇· (u · TNS) + ∇· q = 0.
(2.8)
The viscous stress tensor is defined as,
TNS = −µ
∇u + ∇u† −2
D(∇· u)I
−η(∇· u)I,
(2.9)
where µ and η are the dynamic and the bulk viscosity coefficients, respectively. Further-
more, the Korteweg surface tension tensor is defined as,
TK = κ∇ρ ⊗∇ρ −κ
ρ∇2ρ + 1
2|∇ρ|2
I,
(2.10)
where κ is the capillarity coefficient. Finally, the Fourier heat flux is defined as,
q = −k∇T,
(2.11)
Lattice Boltzmann model for non-ideal compressible fluid dynamics
5
where k is the thermal conductivity coefficient. All transport coefficients are considered
constants below.
While the bulk energy balance equation (2.8) shall be our primary target equation
in the following, we mention several other related forms that are implied by the system
(2.6), (2.7) and (2.8). Denoting K = (1/2)ρu2 the flow kinetic energy, and computing its
time derivative using the continuity equation (2.6) and the momentum balance (2.7), we
find,
∂tK + ∇· (Ku) = −u · ∇P −(u∇) : TK −(u∇) : TNS.
(2.12)
Using the latter in the bulk energy balance (2.8), one obtains the balance equation for
the internal energy,
ρ(∂te + u · ∇e) = −P(∇· u) −TNS : ∇u −∇· q.
(2.13)
Furthermore, using the differential of the internal energy (2.5) and the continuity equation
(2.6), one obtains the temperature equation,
ρcv(∂tT + u · ∇T) = −T
∂P
∂T
(∇· u) −TNS : ∇u −∇· q,
(2.14)
and similarly the pressure equation,
∂tP + u · ∇P = −ρc2
s(∇· u) −
1
ρcv
∂P
∂T
(TNS : ∇u + ∇· q),
(2.15)
where cs is the speed of sound,
c2
s = ∂P
∂ρ +
T
ρ2cv
∂P
∂T
2
.
(2.16)
Yet another form is for the total energy E which takes into account the energy of the
liquid-vapour interface,
E = ρE + 1
2κ|∇ρ|2.
(2.17)
Introducing the interface energy, Eκ = (1/2)κ|∇ρ|2, we first compute its time derivative
using the continuity equation (2.6) to get,
∂tEκ + ∇· (Eκu) + TK : ∇u + ∇· (κρ(∇· u)∇ρ) = 0.
(2.18)
Summing up the time derivatives of the bulk energy (2.8) and of the interface energy
(2.18), we obtain the balance equation for the total energy (2.17) in the standard form,
∂tE + ∇· (Eu + Pu + TNS · u + TK · u + κρ(∇· u)∇ρ + q) = 0.
(2.19)
In summary, the various energy balance forms listed above are implied by the set of
balance equations for mass (2.6), momentum (2.7) and bulk energy (2.8), and which
shall be the target equations for the lattice Boltzmann realization. While the total energy
balance form (2.19) seems to be preferred for conventional finite volume methods, the
bulk energy version of the energy balance is more convenient in the lattice Boltzmann
setting due to locality of the corresponding bulk energy field.
Furthermore, we mention a special case where the specific heat at constant volume
is a function of the absolute temperature only, cv = cv(T). This implies that the
thermodynamic equation of state is a linear function of temperature,
P(v, T) = Tb(v) + a(v),
(2.20)
where a and b are functions of specific volume (or density) only. Indeed, since (2.2) is
6
S. A. Hosseini, M. Feinberg, I. V. Karlin
complete differential, equality of mixed derivatives implies,
∂cv
∂v = ∂
∂T
T
∂P
∂T
−P
= T ∂2P
∂T∂T = 0.
(2.21)
Commonly used examples are the van der Waals and the Carnahan–Starling equations of
state. From a more microscopic viewpoint, corresponding balance equations are derived,
in particular, from the Enskog–Vlasov kinetic equation. In order to link the above
mentioned energy equations to those appearing in the kinetic theory, we introduce the
pressure due to excluded volume effect of hard spheres (Enskog), Phs = TB(ρ) = Tb(1/ρ),
and the pressure due to mean-field interaction (Vlasov), Pmf = A(ρ) = a(1/ρ), so that
the above equation of state (2.20) is written,
P = Phs + Pmf.
(2.22)
The specific internal energy is partitioned accordingly,
de = dehs + demf,
(2.23)
dehs = cv(T)dT,
(2.24)
demf = Pmf
dρ
ρ2 .
(2.25)
Here ehs is the specific thermal energy and emf is the specific molecular potential energy
(in the mean-field approximation). For the Enskog model of hard spheres, the specific
heat is that of the ideal monatomic gas, cv = (3/2)R. Balance equation for the thermal
energy ρehs is obtained upon excluding the molecular potential energy from the balance
of the internal energy (2.13),
∂t(ρehs) + ∇· (ρehsu) + Phs(∇· u) + TNS : ∇u + ∇· q = 0.
(2.26)
Introducing the total kinetic energy,
ρEhs = ρehs + 1
2ρu2,
(2.27)
we obtain its balance upon adding the flow kinetic energy equation (2.12) to the thermal
energy balance (2.26),
∂t(ρEhs) + ∇· (ρEhsu + Phsu) + u · (∇Pmf + ∇· TK) + ∇· (u · TNS) + ∇· q = 0. (2.28)
The latter balance equation can be derived directly from the Enskog–Vlasov kinetic
equation under appropriate scaling in the hydrodynamic limit. Conversely, starting
with the total kinetic energy balance (2.28) and adding the balance for the potential
energy (2.25) (since emf depends only on the density, the latter follows by the continuity
equation),
∂t(ρemf) + ∇· (ρemfu + Pmfu) −u · ∇Pmf = 0,
(2.29)
we arrive at the bulk energy balance (2.8) (and, consequently, after adding the balance of
the interface energy (2.18), at the total energy balance (2.19)), for the special case of the
equation of state (2.20). These considerations show that a more general phenomenological
bulk energy balance (2.8) is consistent with the special case derived directly from kinetic
theory and thus gives us a further reason for using (2.8) as the target energy balance
equation in the lattice Boltzmann context. Consequently, and without loss of generality,
we use the van der Waals equation of state in the numerical examples below,
P(ρ, T) = ρRT
1 −bρ −aρ2.
(2.30)
Lattice Boltzmann model for non-ideal compressible fluid dynamics
7
The excluded volume parameter b and the long-range molecular attraction parameter
a are defined in terms of the critical state thermodynamic data, critical point density
ρc, critical temperature Tc and critical pressure Pc, as follows: a = 27R2T 2
c /64Pc, b =
RTc/8Pc. The differentials of the specific internal energy (2.5) and of the specific entropy
s for the van der Waals fluid are, respectively,
de = cvdT −adρ,
(2.31)
ds = cvdT
T
−
Rdρ
ρ(1 −bρ).
(2.32)
In the next section we shall introduce a lattice Boltzmann model that recovers the above
system of balance equations (2.6), (2.7) and (2.8) in the hydrodynamic limit. Before so
doing, we comment on a motivation to proceed with the lattice Boltzmann modeling in
the context of compressible non-ideal fluid. By looking at the pressure equation (2.15),
one realizes that, for the non-ideal fluid, the adiabatic speed of sound squared (2.16)
becomes negative for a van der Waals-type equation of state in the thermodynamically
unstable spinodal region of the density-temperature diagram. Consequently, the change of
type of the evolution equation from hyperbolic to parabolic occurs, and special treatment
invoking Maxwell’s equal area rule has to be invoked in the conventional CFD methods.
On the contrary, the lattice Boltzmann method, while being a kinetic theory, is able
to circumvent this issue as it inherits propagation along fixed characteristics - discrete
velocities. Thus, the to be introduced lattice Boltzmann model should not be viewed as
yet another numerical scheme but rather as a reduced kinetic theory.
3. Lattice Boltzmann model for non-ideal compressible flows
We consider the standard D3Q27 discrete velocity set vi = cci in D = 3 dimensions,
with Q = 27 velocities,
ci = (cix, ciy, ciz), ciα ∈{−1, 0, 1}.
(3.1)
The D3Q27 lattice (3.1) is characterized with the lattice speed of sound,
ς =
1
√
3c.
(3.2)
Below, we consider lattice units by setting c = 1. Motivated by the generic approach
initialized in Hosseini et al. (2022a); Karlin & Hosseini (2025), we consider the lattice
Boltzmann equations for the two sets of populations fi(x, t) and gi(x, t), i = 1, . . . , Q,
fi(x + ciδt, t + δt) = fi + 2β (f eq
i
−fi) + (1 −β) (f ∗
i −f eq
i ) ,
(3.3)
gi(x + ciδt, t + δt) = gi + 2β (geq
i
−gi) + (1 −β) (g∗
i −geq
i ) .
(3.4)
The first term in the right-hand side of (3.3) and (3.4) represents the lattice Bhatnagar–
Gross–Krook (LBGK) relaxation to the local equilibrium feq
i
and geq
i , respectively, with
a relaxation parameter β tied to the viscosity,
β =
Pδt
2µ + Pδt.
(3.5)
The last term in (3.3) and (3.4) is a generalized forcing, characterized by shifted-
equilibrium populations f∗
i and g∗
i , respectively.
To define the discrete equilibrium and shifted equilibrium, we first introduce functions
8
S. A. Hosseini, M. Feinberg, I. V. Karlin
in two variables, ξα and ζαα,
Ψiα(ξα, ζαα) = 1 −c2
iα + 1
2
(3c2
iα −2)ζαα + ciαξα
.
(3.6)
The equilibrium populations feq
i
are defined by setting the parameters as follows:
ξeq
α = uα,
(3.7)
ζeq
αα = P
ρ + u2
α.
(3.8)
With the definitions (3.7) and (3.8) in the functions (3.6), the local equilibrium popula-
tions are represented with a product-form,
f eq
i
= ρ
Y
α
Ψiα
uα, P
ρ + u2
α
.
(3.9)
For the shifted-equilibrium populations f∗
i , the parameters ξα and ζαα in the functions
(3.6) are set as follows:
ξ∗
α = uα + δtFα
ρ ,
(3.10)
ζ∗
αα = P
ρ +
uα + δtFα
ρ
2
+ δtΦαα
ρ .
(3.11)
When compared to their equilibrium counterparts, the added terms in (3.10) and (3.11)
are, respectively,
Fα = −κρ∂α∇· ∇ρ,
(3.12)
Φαα = −∂α
ρuα
u2
α + 3P
ρ −3ς2
+ Φ′,
(3.13)
Φ′ = P
D + 2
D
−ρc2
s
P −η
µ
(∇· u),
(3.14)
where cs is the speed of sound (2.16). We finalize the construction of the shifted-
equilibrium populations by using again the products of functions (3.6) but with the
parameters set according to (3.10) and (3.11),
f ∗
i = ρ
Y
α
Ψiα
uα + δtFα
ρ , P
ρ +
uα + δtFα
ρ
2
+ δtΦαα
ρ
!
,
(3.15)
which allows us to define the forcing term in the lattice Boltzmann equation (3.4). Before
moving on to the second distribution function a few comments are in order:
• Eq. (3.13) is made up of two contributions. The first term is introduced to correct
errors associated with diagonal components of the viscous stress tensor stemming from
the c3
i = ci bias of the D3Q27 lattice. This correction fixes the bulk viscosity to η′ =
D+2
D
−ρc2
s
P
µ.
• The above-mentioned bulk viscosity is only positive-definite under D+2
D
⩾ρc2
s
P which
is not guaranteed to hold. The second term in Eq. (3.13) is introduced to set the
hydrodynamic dissipation rate of normal modes to an independent value of η guaranteeing
positive-definiteness.
Moving onto the g-populations, we consider a generating function (bulk energy (2.1)
Lattice Boltzmann model for non-ideal compressible fluid dynamics
9
per unit mass),
E(ρ, u, T) = e(ρ, T) + u2
2 ,
(3.16)
and introduce operators Oα acting on the generating function E as,
OαE =
P
ρ
∂E
∂uα
+ uαE.
(3.17)
The energy moments can be computed by repeated application of the operators (3.17)
on the generating function (3.16). For clarity, let us compute the first two moments
explicitly,
OαE =
P
ρ + E
uα,
(3.18)
O2
αE = u2
α
2P
ρ + E
+ P
ρ
P
ρ + E
.
(3.19)
The discrete equilibrium geq
i
can now be defined by setting ξα = Oα and ζαα = O2
α in
the functions (3.6) and using the product-form as an operator-function acting on the
generating function (3.16),
geq
i (ρ, u, T) = ρ
Y
α
Ψiα
Oα, O2
α
E(ρ, u, T).
(3.20)
Furthermore, let us introduce the following notation for the shifted flow velocity u∗and
shifted temperature T ∗,
u∗= u + δtF
ρ ,
(3.21)
T ∗= T −
δt2
2ρ2cv
F · F .
(3.22)
Shifted-equilibrium populations g∗
i are defined using the equilibrium product-form (3.20)
but evaluated at shifted values (3.21) and (3.22), and adding a correction,
g∗
i (ρ, u, T) = geq
i (ρ, u∗, T ∗) +
1
2ci · q′,
if c2
i = 1,
0,
otherwise.
(3.23)
where the non-equilibrium energy flux correction is the vector q′ with the components,
q′
α = δt
P∂α
e + P
ρ
−
k
µ
T∂αT + uαΦ′
.
(3.24)
The moments of orders zero and one of g∗
i are as follows:
Q
X
i=1
g∗
i = ρE (ρ, u∗, T ∗) ,
(3.25)
Q
X
i=1
cig∗
i = u∗(P(ρ, T ∗) + ρE(ρ, u∗, T ∗)) + q′.
(3.26)
A few comments about g∗
i are in order:
• The equilibrium of Eq. (3.16) would have naturally resulted in a Navier-Stokes level
heat flux that is a function of specific enthalpy, i.e. ∝∂α(e+P/ρ). In the case of an ideal
10
S. A. Hosseini, M. Feinberg, I. V. Karlin
gas equation of ∂α(e + P/ρ) = (cv + R)∂αT since both internal energy and pressure are
only functions of temperature. For a non-ideal equation of state however, since the latter
are functions of temperature and density this would lead to an additional flux driven
by density gradients. The first two terms in Eq. (3.24) restore the correct form of the
Fourier heat flux to the model while the last term accounts for viscous heating brought
in by the bulk viscosity in the f-population quasi-equilibrium, see Eq. (3.13).
• Looking at g∗
i as an attractor towards a shifted velocity u∗and temperature T ∗,
we note that the shifted temperature is systematically lower than T, see Eq. (3.22).
Developing the moment of order zero as,
Q
X
i=1
g∗
i = 1
2ρu2 + δtu · F + δt2
2ρ F · F + ρe(ρ, T ∗),
(3.27)
with,
e(ρ, T ∗) = e(ρ, T) + ∂e(ρ, T)
∂T
T
δT + O
δT 2
,
(3.28)
where we introduced δT = T ∗−T for readability. To correctly recover fluxes up to order
δt2 one must have,
∂e(ρ, T)
∂T
T
δT = −δt2
2ρ2 F · F ,
(3.29)
which explains Eq. (3.22) and why T ∗is systematically lower than T. Note that while
higher order terms appear for non-linear-in-temperature equations of state, such as
Peng–Robinson (Peng & Robinson 1976), for linear equations such as van der Waals
there are no higher order terms, see Eq. (2.20) and (2.21).
Finally, the locally conserved fields elsewhere in the equilibrium and shifted-equilibrium
populations, the density ρ, the momentum ρu and the bulk energy ρE, are consistently
defined via the zeroth- and the first-order moments of the populations,
ρ =
Q
X
i=1
fi,
(3.30)
ρu =
Q
X
i=1
cifi + δt
2 F ,
(3.31)
ρE =
Q
X
i=1
gi + δt
2 u · F .
(3.32)
This concludes the proposed lattice Boltzmann model for compressible non-ideal flow
simulations. Detailed multi-scale analysis of the hydrodynamic limit is presented in
Appendix A and demonstrates that the present lattice Boltzmann model recovers the
hydrodynamic equations (2.6), (2.7) and (2.8). Numerical applications and validation of
the model are presented in the next section.
4. Numerical applications
4.1. Consistency: Dispersion and dissipation of hydrodynamic eigen-modes
As a first step, we shall probe the dispersion and dissipation properties of hydrody-
namic modes comprising the shear, the normal and the entropic modes, in the limit of
resolved flows simulation. This benchmark will involve: (a) speed of sound, (b) dissipation
Lattice Boltzmann model for non-ideal compressible fluid dynamics
11
Table 1: Critical properties of nitrogen N2.
Substance R/cv
Pc[Pa]
ρc[kg/m3] Tc[K]
Nitrogen
0.4
3.4 × 106
241.96
126.2
0.7
0.8
0.9
1
Tr
200
300
400
500
cs[m/s]
Figure 2: Speed of sound for nitrogen N2 on the saturated liquid and vapor branches.
Line: analytical solution from Eq. (4.4), Markers: simulations.
of shear waves, (c) shear stress viscous heating and entropic mode dissipation, and (d)
normal mode dissipation rate. For all cases, for the sake of readability and without loss
of generality, we consider Nitrogen with properties listed in Table 1.
The setup to investigate the speed of sound consists of a one-dimensional domain of
size Lx = 0.1 [m] discretized with the spacing δx = 10[µm]. The initial conditions are,
P(x, t) =
(
P0, ∀x ⩽Lx/2
P0 + δP, ∀x > Lx/2
,
(4.1)
ux(x, t) = ux0,
(4.2)
T(x, t) = T0.
(4.3)
Once the initial conditions are set, the system is left to evolve, resulting in two opposite-
moving pressure fronts propagating at a speed that becomes constant after a short initial
transition time. For sufficiently weak perturbations, this corresponds to the speed of
sound in the system. A series of such cases for different initial conditions on both liquid
and vapor branches of the saturation curve for Tr ∈[0.7, 1] were run and compared to
the analytical speed of sound (2.16) for the van der Waals equation of state (2.30),
cs =
s
RT(1 + R/cv)
(1 −bρ)2
−2aρ.
(4.4)
The results are shown in Figure 2 and point to excellent agreement.
The next configuration is set to measure the effective shear viscosity. To that end, we
set up a one-dimensional domain of size Lx = 0.1 [m] with periodic boundary conditions
in both x−and y−directions and discretized with Nx = 100 grid points. Initial conditions
12
S. A. Hosseini, M. Feinberg, I. V. Karlin
0
0.5
1
Ma
0
2
4
6
[m2/s]
10-4
Figure 3: Kinematic viscosity as measured from shear wave decay simulations at different
Mach numbers. Plain black line: analytical viscosity, square markers: viscosity measured
from simulations.
are set as,
uy(x) = Maycs(ρ0, T0) + δMaycs(ρ0, T0) sin
2πx
Lx
,
(4.5)
ux(x) = 0,
(4.6)
ρ(x) = ρ0,
(4.7)
T(x) = T0.
(4.8)
The maximum of the deviation uy−Maycs(ρ0, T0) in the domain is monitored throughout
the simulation and its evolution in time is fitted with an exponential decay function,
umax
y
(t) ∝exp
−4π2
L2x
µ
ρ t
.
(4.9)
The viscosity measured from simulations is compared to those predicted from the multi-
scale analysis. The results are shown in Fig. 3 and point to excellent agreement and
Galilean invariance of the measured viscosity.
As the next case, to validate both viscous heating and entropic modes dissipation rate,
we consider the two-dimensional thermal Couette flow. The case consists of a pseudo one-
dimensional domain of size Lx with walls at x = 0 and x = Lx and periodic boundary
conditions along the y-direction. The flow is subject to the following boundary conditions,
{ux, uy, T}(x = 0) = {0, 0, Tw},
(4.10)
{ux, uy, T}(x = Lx) = {0, Uw, Tw}.
(4.11)
The analytical steady-state solution to this configuration can readily be derived as,
uy(x) = Uw
x
Lx
,
(4.12)
T(x) = Tw + µU 2
w
2k
x
Lx
1 −x
Lx
.
(4.13)
To validate the model, we consider a domain of size Lx = 1 [mm] discretized with 100
grid-points. Simulations were run for Pr ∈{0.6, 1.2, 4.9} and Ma ∈{0.8, 1.2, 1.6}. Results
are displayed in Fig. 4 and show excellent agreement with analytical solutions.
Lattice Boltzmann model for non-ideal compressible fluid dynamics
13
0
0.5
1
x/Lx
1.6
1.65
1.7
1.75
1.8
1.85
/ c
1.04
1.06
1.08
1.1
1.12
1.14
1.16
T/Tc
0
0.5
1
x/Lx
1.6
1.65
1.7
1.75
1.8
1.85
/ c
1.04
1.06
1.08
1.1
1.12
1.14
1.16
T/Tc
Figure 4: Left panel: Temperature and density distribution along channel for thermal
Couette flow considering different Prandtl numbers. Triangle, square and circular markers
are analytical results for Pr ∈{0.6, 1.2, 4.9} respectively. Plain and dashed lines are
temperature and density profiles from simulations. Here Ma=0.8 for all cases. Right
panel: Temperature and density distribution for different Ma numbers. Triangle, square
and circular markers are analytical results for Ma ∈{0.8, 1.2, 1.6} respectively. Plain and
dashed lines are temperature and density profiles from simulations. Here Pr=1.2 for all
cases.
Finally, we move on to probing dissipation rate of normal modes, i.e. acoustics. To
that end we set up a 1-D domain of size Lx with periodic boundary conditions in both
x−and y−directions. Defining a uniform background state, (ρ0, T0, P0), we add a small
perturbation to it at t = 0,
ux(x) = Maxcs(ρ0, T0),
(4.14)
uy(x) = 0,
(4.15)
P(x) = P0 + δP.
(4.16)
The density and temperature fields can be readily computed using isentropic relations
for van der Waals fluid (Kouremenos et al. 1988; Nederstigt & Pecnik 2023),
P0
ρ
γ0
P ρ
0
=
P
ργP ρ ,
(4.17)
T0
ρ
γ0
T ρ−1
0
=
T
ργT ρ−1 ,
(4.18)
where,
γP ρ = ρ
P
cp
cv
∂P
∂ρ
T
,
(4.19)
γT ρ =
1
ρcv
∂P
∂T
ρ
+ 1.
(4.20)
Here cp is,
cp = cv + T
∂P
∂T
v
∂v
∂T
P
,
(4.21)
14
S. A. Hosseini, M. Feinberg, I. V. Karlin
0
0.5
1
Ma
0
0.2
0.4
0.6
0.8
1
[m2/s]
10-3
Figure 5: Normal dissipation rate α as measured from normal wave decay simulations
at different Mach numbers. Plain black line: analytical dissipation rate, square markers:
dissipation rate measured from simulations.
which for the van der Waals equation of state leads to,
cp = cv +
R2T
RT −2aρ(1 −bρ)2 .
(4.22)
One can readily show that for P →ρRT, i.e. the limit of an ideal gas, the following
holds (γP ρ, γT ρ) →cp
cv . We leave the system to evolve over time and monitor the acoustic
energy (Landau & Lifshitz 1987),
Eacoustic = 1
2
Z "
ρ0(u −u0)2 + ρ′2c2
s(ρ0, P0)
ρ0
#
dx,
(4.23)
where ρ′ = ρ −ρ0. It can readily be shown that the decay for a propagating plane wave
is proportional to,
Eacoustic ∝exp
−4π2
L2x
αt
.
(4.24)
with (Landau & Lifshitz 1987),
α = 2(D −1)
D
µ
ρ + η
ρ + k
ρcp
cp
cv
−1
.
(4.25)
Simulations were conducted for different initial values of velocity and effective dissipation
rates measured. Results are shown in Fig. 5. The data points to very good agreement
with analytical predictions.
4.2. Multi-phase regime
4.2.1. Liquid-vapour coexistence
Here we look into the multiphase regime; as a standard validation of thermodynamic
and mechanical consistency we first probe the co-existence densities. For that purpose
we consider the van der Waals equation of state fitted to the N2 critical point. See table
1 for properties. Simulations are conducted in a 1-D domain of size Lx = 0.4 [mm] with
periodic boundary conditions. The domain is filled with saturated vapor with a column
of saturated liquid at the center. Simulations are left to evolve until convergence of the
Lattice Boltzmann model for non-ideal compressible fluid dynamics
15
0.4
10
100
600
[kg/m3]
40
60
80
100
120
140
T[K]
Figure 6: Liquid-vapor co-existence densities for N2. Black lines are from the Maxwell
construction while markers are from simulations.
Table 2: Grid properties used to model N2 interface at different resolutions.
Case
δx
δt
1
0.1 [µm]
2.5 × 10−11[s]
2
0.5 [µm] 1.25 × 10−10[s]
3
1 [µm]
2.5 × 10−10[s]
4
5 [µm]
1.25 × 10−9[s]
density field. Simulations were run for Tr ∈[0.3 0.99]. Results are shown in Fig. 6 and
show excellent agreement with theory. Theoretical results are obtained using the Maxwell
equal area construction.
4.2.2. Interface consistency and convergence
To illustrate the consistency and convergence of the proposed solver to the target
hydrodynamic system we compare the density profile of Nitrogen liquid vapor interface.
Nitrogen system critical properties are listed in table 1. The capillarity coefficient is
set to κ = 10−10 [m7/kg.s2]. The simulation consists of a periodic 1-D domain of size
Lx = 0.5[mm] with a liquid column at the center surrounded with vapor. We consider a
system at Tr = 0.9. The simulations are initialized with the corresponding co-existence
densities in each phase, i.e. (ρl, ρv) = (399.8, 102.71) [kg/m3]. Simulations are conducted
at various resolutions detailed in Table 2. The results obtained from simulations are
compared to data from a high resolution iterative finite differences solver for,
∂xP = κρ∂3
xρ.
(4.26)
The results are shown in Fig. 7. Results point to excellent agreement with reference
solution and convergent behavior as resolution increases.
16
S. A. Hosseini, M. Feinberg, I. V. Karlin
0
5
10
x[ m]
0.5
1
1.5
r
0
5
10
x[ m]
0.5
1
1.5
Figure 7: Liquid-vapor interface for nitrogen N2 at Tr = 0.9. Black lines are converged
results from implicit finite differences solver and red markers from LBM simulations.
Bottom left panel: δx = 0.1 [µm], Bottom right panel: δx = 0.5 [µm], top left panel:
δx = 1 [µm] and top right panel: δx = 5 [µm].
4.3. Compressible configurations
4.3.1. 1-D non-ideal shock tubes
Until the early 1980s, shock tube experiments were limited to gases exhibiting classical
wave behavior. Borisov et al. (1983) first reported a shock tube experiment aimed at
investigating the non-classical wave phenomena in a dense gas, i.e. near the thermody-
namic critical point. The nonlinear dynamics of gases are fundamentally governed by the
fundamental derivative of gas dynamics (Thompson 1971):
Γ = 1 + ρ
cs
∂cs
∂ρ
s
.
(4.27)
For simple waves, Γ represents the rate of change of the convected sound speed with
respect to density. When Γ > 0, the flow exhibits positive nonlinearity, i.e. disturbances
steepen forward to form compression shocks. Conversely, when Γ < 0, negative nonlin-
earity occurs and disturbances steepen backward, leading to expansion shocks. In regions
of negative nonlinearity, gases display distinct non-classical phenomena.
Following (Argrow 1996), we illustrate non-classical wave fields using three non-ideal
shock tube cases, each with flow regions wholly or partly in the regime of negative
nonlinearity. In all three cases, simulations consist of a 1-D domain of size Lx = 1[m],
initially divided into right and left halves. The initial conditions set for each half are listed
in table 3. In all presented cases the grid-size is set to δx = 0.001 [m]. Other simulation-
specific parameters are listed in table 4. In all cases the shock front is initially located
on the half-length of the domain. Results for case I are shown and compared to reference
data in Fig. 8. This configuration is a typical example of non-classical gas dynamics
as both left and right half-domains lead to Γ < 0. In Fig. 8 one can clearly observe a
rarefaction shock moving from right to left, i.e. low to high pressure. Additionally, one
also observes a compression waves propagating into the low pressure region.
Lattice Boltzmann model for non-ideal compressible fluid dynamics
17
Table 3: List of initial conditions for shock tube cases.
Case
R/cv
(Pr, ρr)left
(Pr, ρr)right
I
0.0125
(1.09,0.885)
(0.879,0.562)
II
0.329
(1.6077,1.01) (0.8957,0.594)
III
0.0125
(3.00,1.818)
(0.575,0.275)
Table 4: List of initial conditions for shock tube cases.
Case
δx
δt
maximum CFL maximum Ma
I
0.001[m]
8.3[µs]
0.4998
0.45
II
0.001[m]
4[µs]
0.4226
0.196
III
0.001[m] 2.86[µs]
0.4538
1.846
0
0.5
1
0.5
0.6
0.7
0.8
0.9
0
0.5
1
0.85
0.9
0.95
1
1.05
1.1
Figure 8: Reduced density and pressure fields for shock tube I at time t = 0.45Lx
p
Pc/ρc.
Plain lines are reference data from (Guardone & Vigevano 2002) and markers from
simulations.
The second configuration does not fall in the non-classical regime. As shown in Fig. 9,
in agreement with reference data, both pressure and density fields show a compression
front moving towards the low pressure side and a rarefaction wave moving in the opposite
direction. Finally, the last tube is different from previous cases in the sense that it
dynamically crosses the Γ = 0 line during the simulations. Both sides of the initial
conditions are in the classical region. The crossing of Γ = 0 results in the formation
of mixed rarefaction wave composed of a rarefaction shock connected to a rarefaction
fan. Additionally, as shown in Fig. 10 a compression shock is traveling toward the low
pressure area. All three configurations show excellent agreement with reference data and
show that the model properly captures non-classical compressible gas dynamic behavior.
18
S. A. Hosseini, M. Feinberg, I. V. Karlin
0
0.5
1
0.6
0.8
1
0
0.5
1
0.8
1
1.2
1.4
1.6
Figure 9: Reduced density and pressure fields for shock tube II at time t = 0.2Lx
p
Pc/ρc.
Plain lines are reference data from (Guardone & Vigevano 2002) and markers from
simulations.
0
0.5
1
0
0.5
1
1.5
2
0
0.5
1
0.5
1
1.5
2
2.5
3
Figure 10: Reduced density and pressure fields for shock tube III at time t
=
0.15Lx
p
Pc/ρc. Plain lines are reference data from (Guardone & Vigevano 2002) and
markers from simulations.
4.4. Shock-liquid column interaction
As our final configuration, to further showcase the consistency and suitability of the
model for compressible regimes we consider the case of a circular liquid column interacting
with a planar shock-wave. The case consists of a 2-D domain of size Lx×Ly, here resolved
with 800×800 grid-points, divided into subdomains via a shock positioned at xs. On the
right hand side of the shock the pre-shock state (ρ1, T1, ux,1) is set to that of a saturated
vapor at Tr = 0.9, following (Reyhanian et al. 2020). The post-shock state, to the left of
the shock, (ρ2, T2, ux,2) is derived using the Rankine-Hugoniot conditions. Furthermore,
a saturated liquid column at Tr = 0.9 of radius R, here resolved with 65 grid-points,
is placed at (xc, yc) in the pre-shock domain. Times are non-dimensionalized using the
characteristic time t0 = 2R0ςv
usςl
q
ρl
ρv where us is the shock speed, ρv,l are the vapor/liquid
densities and ςv,l are the saturated vapor/liquid speed of sound. The shock speed is
Lattice Boltzmann model for non-ideal compressible fluid dynamics
19
defined via the shock Mach number Mas as us = Masςv.
To further stabilize simulations especially near sharp fronts, we use a non-linear numerical
viscosity scheme as devised in (Cook & Cabot 2004; Fiorina & Lele 2005). This consists
in adding a numerical contribution to dissipation coefficients as,
{µ, η, k}eff = {µ, η, k} + {µ, η, k}num,
(4.28)
where the numerical contributions are defined as,
{µ, η}num = C{µ,η,k}δxr+1|∇r−1S|,
(4.29)
where,
S = 1
2
q
(∇u + ∇u†) : (∇u + ∇u†),
(4.30)
and the overbar in Eq. (4.29) indicates a Gaussian filter. Furthermore (Fiorina & Lele
2005),
knum = ρςCkδxr+1|∇r−1|∇s||,
(4.31)
with the gradient of entropy of the fluid defined in accord with (2.32),
∇s = cv
T ∇T −
R
ρ(1 −bρ)∇ρ.
(4.32)
Here to minimize the numerical dissipation we have set r = 5. The evolution of the
field, represented via a Schlieren image for Mas = 1.47 is shown in Fig. 11. The wave
structures that arise during the initial stages of shock-droplet interaction are commonly
used to validate numerical schemes. In the present study, representative wave patterns
are extracted and illustrated in Figure 11. Only the early-phase interaction between a
planar shock wave and a cylindrical water column is considered here. As the incident
shock wave travels from left to right across the liquid column, both a transmitted wave
and a reflected shock wave are generated. The reflected shock wave propagates upstream
into the surrounding vapor, while the transmitted wave moves downstream within the
liquid column. Note that the transmitted shock wave moves faster than the incident shock
wave, as speed of sound in the liquid is larger than in the vapor phase. Upon reaching the
downstream wall of the column, the transmitted wave re-emerges into the downstream
vapor. Simultaneously, expansion waves reflect repeatedly within the liquid column. At
the upper lateral edge of the water column, the incident shock wave, Mach stem, and
slip line intersect to form a triple point. These wave structures along with the two-phase
interface represent characteristic features of early-stage shock-droplet interaction. They
appear as discontinuities of varying intensity, posing significant challenges for numerical
modeling. The liquid column then starts flattening in the flow direction and expanding in
the radial direction. As further quantitative validation Figure 12 represents the evolution
of the width of the column, W on the center-line along the x-axis as compared to
experiments and numerical simulations as reported in (Reyhanian et al. 2020) for three
different Mach numbers. The results are in good agreement with both the experiments
and simulations showing that the deformation of the droplet was accurately captured by
the proposed scheme.
5. Conclusion
Development and validation of lattice Boltzmann models for non-ideal fluids in the
compressible regime is an area that has been left under-developed. Given the growing
interest in technologies involving flows in compressible regime where non-ideal effects are
20
S. A. Hosseini, M. Feinberg, I. V. Karlin
Figure 11: Schlieren images of shock/liquid column interaction case at (top to bottom):
t/t0
= 0, t/t0
= 0.3 and t/t0
= 0.7. Schlieren images are generated as ϕ =
exp
−a
||∇ρ||
max(||∇ρ||)
with a = 100. This numerical Schlieren representation is taken from
(Quirk & Karni 1996; Meng & Colonius 2015). ISW: Incident Shock Wave, LC: Liquid
Column, TSW: Transmitted Shock Wave, RSW: Reflected Shock Wave, MS: Mach Stem,
R-TW: Re-transmitted Wave and REW: Reflected Expansion Wave.
quite relevant, development of numerical tools able to capture non-ideal effects in the
compressible regime is of critical importance. In the present contribution, we presented a
lattice Boltzmann model for such flows. The model uses two double distribution functions,
allowing it to be consistently realized on a classical first-neighbor lattice and providing for
the possibility to freely tune the specific heat capacity at constant volume. In addition, to
maintain positive-definiteness the bulk viscosity is made independent from the relaxation
coefficient and isentropic sound speed (Hosseini et al. 2022a; Hosseini & Karlin 2025).
Surface tension is introduced into the system via a consistent realization of the force
contribution, in both distribution functions. All of these ingredients led to a numerical
model that properly captures the target physics, as evidenced from various studied test-
cases, and allow for high speed simulation within the efficient framework of the lattice
Boltzmann method. In the present work a simple single relaxation time formulation was
Lattice Boltzmann model for non-ideal compressible fluid dynamics
21
100
t/t0
0.2
0.4
0.6
0.8
1
1.2
W/D0
Figure 12: Evolution of of drop size along x-axis W over time for three different
Mach numbers. Simulations: (Black plain line) Ma=1.47, (Red dashed line) Ma=1.3
and (Dotted blue line) Ma=1.18. Experiments (Igra & Takayama 2001): (Black filled
circular markers) Ma=1.47, (Red filled triangle markers) Ma=1.3 and (Blue filled square
markers) Ma=1.18. Numerical results from (Reyhanian et al. 2020): (Black unfilled
circular markers) Ma=1.47, (Red unfilled triangle markers) Ma=1.3 and (Blue unfilled
square markers) Ma=1.18
used to carry out simulations. Extension of the solver to more robust collision models
better suited to carry out turbulent configurations will be the topic of future publications.
Acknowledgment
This work was supported by European Research Council (ERC) Advanced Grant No.
834763-PonD and by the Swiss National Science Foundation (SNSF) Grants 200021-
228065 and 200021-236715. Computational resources at the Swiss National Super Com-
puting Center (CSCS) were provided under Grants No. s1286, sm101 and s1327. Authors
would like to thank Patrick Jenny for his support and fruitful discussions.
Declaration of interests
The authors report that they do not have a conflict of interest.
Data Availability Statement
The data that support the findings of this study are available from the corresponding
author upon request.
22
S. A. Hosseini, M. Feinberg, I. V. Karlin
Appendix A. Multi-scale analysis of lattice Boltzmann model for
compressible non-ideal flows
The first step in the multi-scale analysis is a Taylor expansion of the lattice Boltzmann
equations,
{fi, gi}(x + ciδt, t + δt) = {fi, gi}(x, t) + 2β ({f eq
i , geq
i }(x, t) −{fi, gi}(x, t))
+ (1 −β) ({f ∗
i , g∗
i }(x, t) −{f eq
i , geq
i }(x, t)) ,
(A 1)
around (x, t), leading to the following space and time-evolution equations,
δtDt{fi, gi} + δt2
2 Dt
2{fi, gi} + O
δt3
= 2β ({f eq
i , geq
i } −{fi, gi})
+ (1 −β) ({f ∗
i , g∗
i } −{f eq
i , geq
i }) .
(A 2)
Introducing the flow characteristic size and time, L and T the equations are made non-
dimensional as,
δx
L D′
t{fi, gi} + δx2
2L2 D′
t
2{fi, gi} = 2β ({f eq
i , geq
i } −{fi, gi})
+ (1 −β) ({f ∗
i , g∗
i } −{f eq
i , geq
i }) ,
(A 3)
where,
D′
t = L/T
δx/δt (∂′
t + c′
i · ∇′) .
(A 4)
Assuming acoustic scaling and hydrodynamic scaling, ε ∼δx/L ∼δt/T and dropping
the primes,
εDt{fi, gi} + ε2
2 Dt
2{fi, gi} = 2β ({f eq
i , geq
i } −{fi, gi}) + (1 −β) ({f ∗
i , g∗
i } −{f eq
i , geq
i }) ,
(A 5)
We introduce the following multi-scale expansions:
fi = f (0)
i
+ εf (1)
i
+ ε2f (2)
i
+ O(ε3),
(A 6a)
gi = g(0)
i
+ εg(1)
i
+ ε2g(2)
i
+ O(ε3),
(A 6b)
f ∗
i = f ∗
i
(0) + εf ∗
i
(1) + ε2f ∗
i
(2) + O(ε3),
(A 6c)
g∗
i = g∗
i
(0) + εg∗
i
(1) + ε2g∗
i
(2) + O(ε3),
(A 6d)
∂t = ε∂(1)
t
+ ε2∂(2)
t
+ O(ε3).
(A 6e)
Noting that for the definition of f∗
i in (3.15) divergence from the equilibrium will arise
only at the ε1 level (i.e. f ∗
i
(0) = f eq
i ), we separate by orders of the smallness parameter,
ε0 : {f (0)
i
, g(0)
i
} = {f eq
i , geq
i },
(A 7a)
ε : D(1)
t {f (0)
i
, g(0)
i
} = −2β{f (1)
i
, g(1)
i
} + (1 −β) {f ∗(1)
i , g∗(1)
i },
(A 7b)
ε2 : ∂(2)
t
{f (0)
i
, g(0)
i
} + D(1)
t (1 −β)
{f (1)
i
, g(1)
i
} + 1
2{f ∗(1)
i , g∗(1)
i }
=
−2β{f (2)
i
, g(2)
i
} + (1 −β) {f ∗(2)
i , g∗(2)
i }.
(A 7c)
Lattice Boltzmann model for non-ideal compressible fluid dynamics
23
The following solvability conditions apply,
Q
X
i=1
f (k)
i
= 0, ∀k > 0,
(A 8)
Q
X
i=1
cif (1)
i
+ 1
2
Q
X
i=1
cif ∗(1)
i
= 0,
(A 9)
Q
X
i=1
cif (k)
i
= 0, ∀k > 1,
(A 10)
Q
X
i=1
g(1)
i
+ 1
2
Q
X
i=1
g∗(1)
i
= 0,
(A 11)
Q
X
i=1
g(k)
i
= 0, ∀k > 0.
(A 12)
Taking the zeroth-order moment of Eq. (A 7b) for fi,
∂(1)
t
ρ + ∇· ρu = 0,
(A 13)
where we have used,
Q
X
i=1
f ∗
i
(1) = 0,
(A 14)
and solvability condition (A 8). For the first-order moment of (A 7b) for fi,
∂(1)
t
ρu + ∇· ρu ⊗u + ∇P = −2β
=0
z
}|
{
Q
X
i=1
cif (1)
i
+ 1
2cif ∗
i
(1)
!
+
Q
X
i=1
cif ∗
i
(1)
|
{z
}
F
,
(A 15)
where we used (A 9) and further set,
F = −∇· TK,
(A 16)
where,
TK = κ∇ρ ⊗∇ρ −κ
ρ∇2ρ + 1
2|∇ρ|2
I.
(A 17)
The force can also be shown to simplify to,
F = κρ∇∇2ρ.
(A 18)
Finally taking the zeroth-order moment for gi,
∂(1)
t
ρE + ∇· u (ρE + P) = −2β
=0
z
}|
{
Q
X
i=1
g(1)
i
+ 1
2g∗
i
(1)
!
+
Q
X
i=1
g∗
i
(1)
|
{z
}
u·F
.
(A 19)
24
S. A. Hosseini, M. Feinberg, I. V. Karlin
Summing up balance equations at order ε,
∂(1)
t
ρ + ∇· ρu = 0,
(A 20)
∂(1)
t
ρu + ∇· (ρu ⊗u + PI) −F = 0,
(A 21)
∂(1)
t
ρE + ∇· ρu (E + P/ρ) −u · F = 0.
(A 22)
The last equation Eq. (A 22) can be transformed into a balance equation for internal
energy, using ,
∂(1)
t
K + ∇· uK + u · ∇P −u · F = 0,
(A 23)
as,
∂(1)
t
ρe + ∇· ρue + P∇· u = 0.
(A 24)
Furthermore, using
de = cvdT −
T
∂P
∂T
ρ
−P
!
dρ
ρ2 ,
(A 25)
and Eq. (A 20) a balance equation for temperature can be derived as,
∂(1)
t
T + u · ∇T + T
ρcv
∂P
∂T
ρ
∇· u = 0.
(A 26)
Finally, using
dP =
∂P
∂ρ
T
dρ +
∂P
∂T
ρ
dT,
(A 27)
we can also write a balance equation for pressure as,
∂(1)
t
P + u · ∇P + ρc2
s∇· u = 0,
(A 28)
where,
c2
s =
∂P
∂ρ
T
+
T
cvρ2
∂P
∂T
2
ρ
.
(A 29)
At order ε2, the zeroth order moment of fi leads to,
∂(2)
t
ρ = 0,
(A 30)
and the first order moments,
∂(2)
t
ρu + ∇· (1 −β)
" Q
X
i=1
ci ⊗cifi
(1)
!
+ 1
2
Q
X
i=1
ci ⊗cif ∗(1)
i
!#
= 0.
(A 31)
Here we can use Eq. (A 7b) to obtain,
∂(2)
t
ρu + ∇·
1
2 −1
2β
∂(1)
t
Π2(f (0)) + ∇Π3(f (0)) −
Q
X
i=1
ci ⊗cif ∗(1)
i
!
= 0.
(A 32)
Here, Π(0)
2 (f) and Π(0)
3 (f) are the second and third order moments tensor of the
equilibrium distribution function,
Π2(f (0)) = ρu ⊗u + PI,
(A 33)
Π3(f (0)) = ρu ⊗(u ⊗u + PI) ◦(1 −J) + 3ρς2u ⊗I ◦J.
(A 34)
Lattice Boltzmann model for non-ideal compressible fluid dynamics
25
Using Eqs. (A 20) and (A 21) we can write,
∂(1)
t
(ρu ⊗u + PI) = u ⊗F + F ⊗u −∇· ρu ⊗u ⊗u −∇Pu −(∇Pu)†
P
∇u + ∇u†
+ ∂(1)
t
PI.
(A 35)
For the last term, i.e. ∂(1)
t
P, we write a balance equation for P,
∂(1)
t
P =
P −ρc2
s
∇· u −∇· Pu,
(A 36)
while,
∇· Π3(f (0)) =
∇· ρu ⊗u ⊗u + ∇Pu + ∇Pu†
+ Ψ + I∇· Pu,
(A 37)
where,
Ψαα = −∂αuα
u2
α + 3(P −ρς2)
.
(A 38)
Adding up all terms,
∂(1)
t
Π2(f (0)) + ∇· Π3(f (0)) = u ⊗F + F ⊗u + P
∇u + ∇u†
+
P −ρc2
s
∇· uI −Ψ.
(A 39)
Setting,
Q
X
i=1
ci ⊗cif ∗(1)
i
=
u ⊗F + F ⊗u + Ψ + P
D + 2
D
−ρc2
s
P −η
µ
∇· uI
(A 40)
and plugging it back into Eq. (A 32),
∂(2)
t
ρu −∇·
µ
∇u + ∇u† −2
D∇· uI
+ η∇· uI
= 0,
(A 41)
Where we used,
µ =
1
2β −1
2
Pδt.
(A 42)
For the second population, at order ε2,
∂(2)
t
ρE + ∇·
1
2 −1
2β
∂(1)
t
Π1(g(0)
i
) + ∇· Π2(g(0)
i
) −
Q
X
i=1
cig∗
i
(1)
!
= 0,
(A 43)
where,
Π1(g(0)
i
) = u(ρE + P),
(A 44)
Π2(g(0)
i
) = u ⊗u (ρE + 2P) + (E + P/ρ) PI.
(A 45)
Here we can use,
∂(1)
t
Pu + ∇· Pu ⊗u + P
ρ ∇P −P
ρ F + u
ρc2
s −P
∇· u = 0,
(A 46)
and
∂(1)
t
ρEu + ∇· (ρE + P)u ⊗u + E∇P −EF −Pu · ∇u −u(u · F ) = 0,
(A 47)
26
S. A. Hosseini, M. Feinberg, I. V. Karlin
Adding up both contributions,
∂(1)
t
Π1(g(0)
i
)+∇·Π2(g(0)
i
) = −2∇· Pu ⊗u + 2∇· Pu ⊗u
|
{z
}
=0
+∇· ρEu ⊗u −∇· ρEu ⊗u
|
{z
}
=0
+∇P(P/ρ + E) −(E + P/ρ)∇P
|
{z
}
P ∇(E+P/ρ)
+Pu · ∇u −u
ρc2
s −P
∇· u
+ (P/ρ + E)F + u(u · F ).
(A 48)
Further expanding,
∂(1)
t
Π1(g(0)
i
) + ∇· Π2(g(0)
i
) = P∇h
=P u·(∇u+∇u†)
z
}|
{
+P∇(u2/2) + Pu · ∇u +u
P −ρc2
s
∇· u
+ (P/ρ + E)F + u(u · F ),
(A 49)
where h = H −K. Plugging this back into the balance equation,
∂(2)
t
ρE −∇· u ·
µ
∇u + ∇u −2
D∇· uI
+ η∇· uI
|
{z
}
−u·TNS
+ ∇·
1
2 −1
2β
P∇h + u · P
D + 2
D
−ρc2
s
P −η
µ
∇· uI
+(P/ρ + E)F + u(u · F ) −
Q
X
i=1
cig∗
i
(1)
!
= 0.
(A 50)
Using the definition of the shifted equilibrium,
Π0(g∗(1)
i
) = u · F ,
(A 51)
Π1(g∗(1)
i
) =
u(u · F ) + F
P
ρ + E
+P
∇h −κ
µ∇T + u ·
D + 2
D
−ρc2
s
P −η
µ
∇· uI
.
(A 52)
we recover,
∂(2)
t
ρE + ∇· u · TNS −∇· k∇T = 0.
(A 53)
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J. Fluid Mech. 1 Lattice Boltzmann model for non-ideal compressible fluid dynamics S. A. Hosseini1†, M. Feinberg1‡, I. V. Karlin1¶ 1 8092 Zurich, Switzerland. (Received xx; revised xx; accepted xx) We present a lattice Boltzmann formulation for the simulation of compressible, non-ideal fluid flows. The method employs first-neighbor lattices and introduces a consistent set of correction terms through quasi-equilibrium attractors, ensuring positive-definite and Galilean-invariant Navier-Stokes dissipation rates. This construction circumvents the need for extended stencils or ad hoc regularization, while maintaining numerical stability and thermodynamic consistency across a broad range of flow regimes. The resulting model accurately reproduces both Euler- and Navier-Stokes-level hydrodynamics. As a stringent validation, we demonstrate, for the first time within a lattice Boltzmann framework, quantitatively accurate simulations of drop-shock interactions at Mach numbers up to 1.47. The proposed approach thus extends the applicability of lattice Boltzmann methods to high-speed, non-ideal compressible flows with a minimal kinetic stencil. Key words: 1. Introduction Non-ideal compressible fluid dynamics is a novel and rapidly developing branch of fluid mechanics, mainly due to the emergence of methods and technologies operating in the near-, trans- and super-critical regimes. It is, in part, concerned with gas-dynamics of single phase fluids in non-ideal thermodynamic states, i.e. states where compressibility factor differs from unity (Guardone et al. 2024). These states are illustrated in a P-T diagram for CO2 in Figure 1. Non-ideal compressible fluid dynamics comprises the dynamics of supercritical fluids, dense vapors, and liquids. The latter two are separated by the Widom line, an extension beyond the critical point that delineates vapor-like and liquid-like behavior, identified by maxima in the constant-pressure specific heat (Simeoni et al. 2010). In addition it also includes fluids of higher molecular complexity with negative fundamental gas dynamics derivative (Colonna et al. 2009), also known as Bethe-Zel'dovich-Thompson (BZT) fluids (Zel'Dovich 1946; Thompson & Lambrakis 1973). The rapidly growing interest in the energy industry for fluids operating in those thermodynamic states along with their radically different behavior as compared to fluids in ideal states are two important factors demonstrating the need for systematic studies of such flows. However, as noted in the literature, while considerable effort has been put in developing experimental set-ups for non-ideal fluid dynamics in recent years (Lettieri et al. 2015; Zocca et al. 2019; Gallarini et al. 2021; Head et al. 2022), † Email address for correspondence: ‡ Email address for correspondence: ¶ Email address for correspondence: 16 Oct 2025 2 S. A. Hosseini, M. Feinberg, I. V. Karlin Figure 1: Pressure-temperature diagram for CO2. The color-bar indicates the compressibility factor Z = P/ρRT. Figure reproduced from Guardone et al. (2024). experimental data remains scarce and is complicated to acquire (Guardone et al. 2024). This means that development of consistent and efficient numerical tools for the simulation of such flows is of the utmost importance in the current state of development of both our understanding of non-ideal compressible fluid dynamics physics and technologies such as organic Rankine cycles or supercritical CO2 turbines (Guardone et al. 2024; Gunawan et al. 2023; Chen et al. 2023). More specifically, direct numerical simulations are necessary both to understand the complex physics of non-ideal compressible fluid dynamics and to generate engineering databases. Examples include skin friction and dissipation coefficients (Cramer et al. 1996; Pini & De Servi 2018), heat transfer in wall-bounded turbulent supercritical flows (Peeters et al. 2016; Kawai 2019), and the critical mass flow rate and pressure in nozzles and turbines involving flashing flows. In addition, while direct numerical simulations can be at times cost-prohibitive for full-scale simulations, they are a necessary tool for development of the subgrid-scale closure models required for the more cost effective large eddy and Reynolds-averaged Navier-Stokes simulations (Garnier et al. 2009). Recent studies on dense vapor effects on fluid structures in homogeneous isotropic turbulence have shown that while large-scale flow structures are mostly affected by molecular complexity, smaller-scale flow structures are impacted by local variations in sound speed (Sciacovelli et al. 2017; Duan et al. 2021). In addition, the dense vapor has been shown to display modified shocklet structures, with considerably reduced jumps in pressure, density, and entropy in compression shocklets, and emergence of expansion shocklets (Giauque et al. 2017, 2020). Similar importance has been reported for wall bounded turbulence where substantially different - from ideal gas - dynamics at smaller scales have been reported (Vadrot et al. 2021) and strong normal gradients in density and viscosity -and therefore local speed of sound and Mach number- are primary elements affecting turbulence (Pecnik & Patel 2017; Peeters et al. 2016). Reliable direct numerical simulation studies of non-ideal compressible fluid dynamics can fill existing gaps in understanding of physics of such flows. A reliable numerical scheme pre-supposes a physical model valid and consistently covering all regime of interest here, i.e. super-, transand near-critical flows involving pronounced non-ideal and compressibility effects. The development and validation of efficient numerical schemes for Navier-Stokes-Korteweg (NSK) dynamics, regardless of method, can have a significant impact on the broader literature and knowledge on non-ideal compressible flows. The lattice Boltzmann method (LBM) is an excellent candidate for the flow regimes of interest here. Already well established as an efficient method for resolved simulation of flows in the incompressible limit, the distinct numerical features of LBM offer Lattice Boltzmann model for non-ideal compressible fluid dynamics 3 advantages for compressible, non-ideal fluid dynamics. Specifically, the decoupling of non-linearity and non-locality noted by S. Succi, in LBM "nonlinearity is local, nonlocality is linear" (Succi 2001; Succi & Succi 2018). LBM solvers are conservative and lowdissipative schemes with excellent spectral properties -both dissipation and dispersionas compared to conventional solvers of the same order (Hosseini 2020; Hosseini et al. 2019, 2022b; Martinez et al. 1994; Wissocq et al. 2019; Peng et al. 2010), especially for normal propagation modes, i.e. acoustic modes (Bres et al. 2009; Viggen 2011, 2014). Along with incompressible ideal fluid dynamics, two-phase flow simulation has witnessed major success and growth in popularity in LBM, starting with formulations such as the color-gradient (Gunstensen et al. 1991), pseudo-potential (Shan & Chen 1993) and free energy (Swift et al. 1996) in the early 90's. The latter two are of special interest in the context of non-ideal fluid dynamics as in effect they solve a form of the NSK. The large majority of the literature that focuses or uses this class of models has been tailored towards becoming essentially highly efficient interface tracking models for multiphase flows (Luo et al. 2021; Mazloomi et al. 2015; Hosseini & Karlin 2023; Chen et al. 2014; Li et al. 2016). Non-ideal compressible thermodynamics have long been neglected in LBM applications. The free energy model for instance, which is the only one out of the previously-listed approaches which results from minimization of a free energy functional under constraint on total mass, can readily be shown to be related to meanfield kinetic models such as the Boltzmann-Enskog-Vlasov (Hosseini et al. 2022a; Hosseini & Karlin 2023) equations and recover, at the hydrodynamic limit and under specific scaling, the NSK equations (Hosseini et al. 2022a) and mean-field van der Waals fluid thermodynamics. This means that it can not only model two-phase flow dynamics and phase separation, but also properly recover all mean-field, near-, trans- and super-critical behaviors associated to non-ideal fluids, as demonstrated for instance through studies of properties such as the Tollman length (Hosseini et al. 2022a; Hosseini & Karlin 2023; Reyhanian et al. 2020, 2021; Lulli et al. 2022a,b; H ́azi & M ́arkus 2008). Use and extension of this class of thermodynamically consistent model to compressible non-ideal fluid dynamics is a largely under-explored area that can considerably impact research on non-ideal compressible fluid dynamics. First attempts at proposing LBM for compressible non-ideal flows were documented in (He & Doolen 2002). Since then, thermal multi-phase models based on the pseudo-potential approach have witnessed steady growth. However, the vast majority of the models and studies in the literature have been tailored to boiling applications, see for instance (Li et al. 2015; Fei et al. 2020; Saito et al. 2021; Fang et al. 2017; Huang et al. 2021). To the author's knowledge, only documented attempts at modeling compressible non-ideal flows beyond evaporation are (Vienne et al. 2024) where a hybrid lattice Boltzmann finite differences scheme - finite differences for the energy equation- was proposed and (Reyhanian et al. 2020, 2021) where the authors proposed a numerical model based on the Particles-on-Demand realization of the lattice Boltzmann method Dorschner et al. (2018). We propose to address this gap in the literature with a lattice Boltzmann model for non-ideal fluids in the compressible regime. To retain the main advantages of the lattice Boltzmann method the proposed model relies on classical first neighbor lattices, taking advantage of a second distribution function for the energy balance equations (Rykov 1975; Saadat et al. 2021; Prasianakis & Karlin 2007, 2008). In addition, the model removes the Galilean-variant errors in the diagonal components of the third-order equilibrium moments tensor and provides for independent control over the bulk viscosity; The latter is a critical point as the bulk viscosity dictated by the BGK structure, for equilibria carrying a pressure other than the ideal gas pressure, can take on negative values in the hydordynamic limit, see (Hosseini et al. 2022a). Together with a consistent second-order discretization 4 S. A. Hosseini, M. Feinberg, I. V. Karlin and treatment of source terms, the model will be shown to correctly recover the target hydrodynamic limit. The article starts with a brief introduction of the target hydrodynamic limit followed by presentation of the proposed lattice Boltzmann model. The proposed model is then validated with a list of configurations of increasing complexity. 2. Balance equations for compressible non-ideal fluid We begin with a brief overview of the compressible non-ideal fluid system. To that end, we introduce the macroscopic fields of fluid density ρ(x, t), momentum ρu(x, t) and bulk energy ρE(x, t). The latter is the sum of the flow kinetic energy and the internal energy ρe, ρE = ρe + 1 2ρu2. (2.1) The specific internal energy e(v, T) per unit mass is a function of absolute temperature T and specific volume v = 1/ρ, and is defined by a familiar thermodynamic relation for its differential, de = cvdT + T ∂P ∂T -P dv, (2.2) where P(v, T) is the thermodynamic pressure and cv is specific heat at constant volume, cv = ∂e ∂T . (2.3) Below, it will be convenient to consider the thermodynamic equation of state as a function of the density ρ rather than of the specific volume v, P(ρ, T) = P(v, T)|v=1/ρ , (2.4) so that the differential of the internal energy (2.2) is written, de = cvdT - T ∂P ∂T -P dρ ρ2 . (2.5) The D-dimensional mass, momentum and bulk energy balance equations are, ∂tρ + ∇· ρu = 0, (2.6) ∂t(ρu) + ∇· ρu ⊗u + ∇P + ∇· TNS + ∇· TK = 0, (2.7) ∂t(ρE) + ∇· (ρEu + Pu) + u · (∇· TK) + ∇· (u · TNS) + ∇· q = 0. (2.8) The viscous stress tensor is defined as, TNS = -μ ∇u + ∇u† -2 D(∇· u)I -η(∇· u)I, (2.9) where μ and η are the dynamic and the bulk viscosity coefficients, respectively. Furthermore, the Korteweg surface tension tensor is defined as, TK = κ∇ρ ⊗∇ρ -κ ρ∇2ρ + 1 2|∇ρ|2 I, (2.10) where κ is the capillarity coefficient. Finally, the Fourier heat flux is defined as, q = -k∇T, (2.11) Lattice Boltzmann model for non-ideal compressible fluid dynamics 5 where k is the thermal conductivity coefficient. All transport coefficients are considered constants below. While the bulk energy balance equation (2.8) shall be our primary target equation in the following, we mention several other related forms that are implied by the system (2.6), (2.7) and (2.8). Denoting K = (1/2)ρu2 the flow kinetic energy, and computing its time derivative using the continuity equation (2.6) and the momentum balance (2.7), we find, ∂tK + ∇· (Ku) = -u · ∇P -(u∇) : TK -(u∇) : TNS. (2.12) Using the latter in the bulk energy balance (2.8), one obtains the balance equation for the internal energy, ρ(∂te + u · ∇e) = -P(∇· u) -TNS : ∇u -∇· q. (2.13) Furthermore, using the differential of the internal energy (2.5) and the continuity equation (2.6), one obtains the temperature equation, ρcv(∂tT + u · ∇T) = -T ∂P ∂T (∇· u) -TNS : ∇u -∇· q, (2.14) and similarly the pressure equation, ∂tP + u · ∇P = -ρc2 s(∇· u) - 1 ρcv ∂P ∂T (TNS : ∇u + ∇· q), (2.15) where cs is the speed of sound, c2 s = ∂P ∂ρ + T ρ2cv ∂P ∂T 2 . (2.16) Yet another form is for the total energy E which takes into account the energy of the liquid-vapour interface, E = ρE + 1 2κ|∇ρ|2. (2.17) Introducing the interface energy, Eκ = (1/2)κ|∇ρ|2, we first compute its time derivative using the continuity equation (2.6) to get, ∂tEκ + ∇· (Eκu) + TK : ∇u + ∇· (κρ(∇· u)∇ρ) = 0. (2.18) Summing up the time derivatives of the bulk energy (2.8) and of the interface energy (2.18), we obtain the balance equation for the total energy (2.17) in the standard form, ∂tE + ∇· (Eu + Pu + TNS · u + TK · u + κρ(∇· u)∇ρ + q) = 0. (2.19) In summary, the various energy balance forms listed above are implied by the set of balance equations for mass (2.6), momentum (2.7) and bulk energy (2.8), and which shall be the target equations for the lattice Boltzmann realization. While the total energy balance form (2.19) seems to be preferred for conventional finite volume methods, the bulk energy version of the energy balance is more convenient in the lattice Boltzmann setting due to locality of the corresponding bulk energy field. Furthermore, we mention a special case where the specific heat at constant volume is a function of the absolute temperature only, cv = cv(T). This implies that the thermodynamic equation of state is a linear function of temperature, P(v, T) = Tb(v) + a(v), (2.20) where a and b are functions of specific volume (or density) only. Indeed, since (2.2) is 6 S. A. Hosseini, M. Feinberg, I. V. Karlin complete differential, equality of mixed derivatives implies, ∂cv ∂v = ∂ ∂T T ∂P ∂T -P = T ∂2P ∂T∂T = 0. (2.21) Commonly used examples are the van der Waals and the Carnahan-Starling equations of state. From a more microscopic viewpoint, corresponding balance equations are derived, in particular, from the Enskog-Vlasov kinetic equation. In order to link the above mentioned energy equations to those appearing in the kinetic theory, we introduce the pressure due to excluded volume effect of hard spheres (Enskog), Phs = TB(ρ) = Tb(1/ρ), and the pressure due to mean-field interaction (Vlasov), Pmf = A(ρ) = a(1/ρ), so that the above equation of state (2.20) is written, P = Phs + Pmf. (2.22) The specific internal energy is partitioned accordingly, de = dehs + demf, (2.23) dehs = cv(T)dT, (2.24) demf = Pmf dρ ρ2 . (2.25) Here ehs is the specific thermal energy and emf is the specific molecular potential energy (in the mean-field approximation). For the Enskog model of hard spheres, the specific heat is that of the ideal monatomic gas, cv = (3/2)R. Balance equation for the thermal energy ρehs is obtained upon excluding the molecular potential energy from the balance of the internal energy (2.13), ∂t(ρehs) + ∇· (ρehsu) + Phs(∇· u) + TNS : ∇u + ∇· q = 0. (2.26) Introducing the total kinetic energy, ρEhs = ρehs + 1 2ρu2, (2.27) we obtain its balance upon adding the flow kinetic energy equation (2.12) to the thermal energy balance (2.26), ∂t(ρEhs) + ∇· (ρEhsu + Phsu) + u · (∇Pmf + ∇· TK) + ∇· (u · TNS) + ∇· q = 0. (2.28) The latter balance equation can be derived directly from the Enskog-Vlasov kinetic equation under appropriate scaling in the hydrodynamic limit. Conversely, starting with the total kinetic energy balance (2.28) and adding the balance for the potential energy (2.25) (since emf depends only on the density, the latter follows by the continuity equation), ∂t(ρemf) + ∇· (ρemfu + Pmfu) -u · ∇Pmf = 0, (2.29) we arrive at the bulk energy balance (2.8) (and, consequently, after adding the balance of the interface energy (2.18), at the total energy balance (2.19)), for the special case of the equation of state (2.20). These considerations show that a more general phenomenological bulk energy balance (2.8) is consistent with the special case derived directly from kinetic theory and thus gives us a further reason for using (2.8) as the target energy balance equation in the lattice Boltzmann context. Consequently, and without loss of generality, we use the van der Waals equation of state in the numerical examples below, P(ρ, T) = ρRT 1 -bρ -aρ2. (2.30) Lattice Boltzmann model for non-ideal compressible fluid dynamics 7 The excluded volume parameter b and the long-range molecular attraction parameter a are defined in terms of the critical state thermodynamic data, critical point density ρc, critical temperature Tc and critical pressure Pc, as follows: a = 27R2T 2 c /64Pc, b = RTc/8Pc. The differentials of the specific internal energy (2.5) and of the specific entropy s for the van der Waals fluid are, respectively, de = cvdT -adρ, (2.31) ds = cvdT T - Rdρ ρ(1 -bρ). (2.32) In the next section we shall introduce a lattice Boltzmann model that recovers the above system of balance equations (2.6), (2.7) and (2.8) in the hydrodynamic limit. Before so doing, we comment on a motivation to proceed with the lattice Boltzmann modeling in the context of compressible non-ideal fluid. By looking at the pressure equation (2.15), one realizes that, for the non-ideal fluid, the adiabatic speed of sound squared (2.16) becomes negative for a van der Waals-type equation of state in the thermodynamically unstable spinodal region of the density-temperature diagram. Consequently, the change of type of the evolution equation from hyperbolic to parabolic occurs, and special treatment invoking Maxwell's equal area rule has to be invoked in the conventional CFD methods. On the contrary, the lattice Boltzmann method, while being a kinetic theory, is able to circumvent this issue as it inherits propagation along fixed characteristics - discrete velocities. Thus, the to be introduced lattice Boltzmann model should not be viewed as yet another numerical scheme but rather as a reduced kinetic theory. 3. Lattice Boltzmann model for non-ideal compressible flows We consider the standard D3Q27 discrete velocity set vi = cci in D = 3 dimensions, with Q = 27 velocities, ci = (cix, ciy, ciz), ciα ∈{-1, 0, 1}. (3.1) The D3Q27 lattice (3.1) is characterized with the lattice speed of sound, ς = 1 √ 3c. (3.2) Below, we consider lattice units by setting c = 1. Motivated by the generic approach initialized in Hosseini et al. (2022a); Karlin & Hosseini (2025), we consider the lattice Boltzmann equations for the two sets of populations fi(x, t) and gi(x, t), i = 1, . . . , Q, fi(x + ciδt, t + δt) = fi + 2β (f eq i -fi) + (1 -β) (f ∗ i -f eq i ) , (3.3) gi(x + ciδt, t + δt) = gi + 2β (geq i -gi) + (1 -β) (g∗ i -geq i ) . (3.4) The first term in the right-hand side of (3.3) and (3.4) represents the lattice BhatnagarGross-Krook (LBGK) relaxation to the local equilibrium feq i and geq i , respectively, with a relaxation parameter β tied to the viscosity, β = Pδt 2μ + Pδt. (3.5) The last term in (3.3) and (3.4) is a generalized forcing, characterized by shiftedequilibrium populations f∗ i and g∗ i , respectively. To define the discrete equilibrium and shifted equilibrium, we first introduce functions 8 S. A. Hosseini, M. Feinberg, I. V. Karlin in two variables, ξα and ζαα, Ψiα(ξα, ζαα) = 1 -c2 iα + 1 2 (3c2 iα -2)ζαα + ciαξα . (3.6) The equilibrium populations feq i are defined by setting the parameters as follows: ξeq α = uα, (3.7) ζeq αα = P ρ + u2 α. (3.8) With the definitions (3.7) and (3.8) in the functions (3.6), the local equilibrium populations are represented with a product-form, f eq i = ρ Y α Ψiα uα, P ρ + u2 α . (3.9) For the shifted-equilibrium populations f∗ i , the parameters ξα and ζαα in the functions (3.6) are set as follows: ξ∗ α = uα + δtFα ρ , (3.10) ζ∗ αα = P ρ + uα + δtFα ρ 2 + δtΦαα ρ . (3.11) When compared to their equilibrium counterparts, the added terms in (3.10) and (3.11) are, respectively, Fα = -κρ∂α∇· ∇ρ, (3.12) Φαα = -∂α ρuα u2 α + 3P ρ -3ς2 + Φ′, (3.13) Φ′ = P D + 2 D -ρc2 s P -η μ (∇· u), (3.14) where cs is the speed of sound (2.16). We finalize the construction of the shiftedequilibrium populations by using again the products of functions (3.6) but with the parameters set according to (3.10) and (3.11), f ∗ i = ρ Y α Ψiα uα + δtFα ρ , P ρ + uα + δtFα ρ 2 + δtΦαα ρ ! , (3.15) which allows us to define the forcing term in the lattice Boltzmann equation (3.4). Before moving on to the second distribution function a few comments are in order: • Eq. (3.13) is made up of two contributions. The first term is introduced to correct errors associated with diagonal components of the viscous stress tensor stemming from the c3 i = ci bias of the D3Q27 lattice. This correction fixes the bulk viscosity to η′ = D+2 D -ρc2 s P μ. • The above-mentioned bulk viscosity is only positive-definite under D+2 D ⩾ρc2 s P which is not guaranteed to hold. The second term in Eq. (3.13) is introduced to set the hydrodynamic dissipation rate of normal modes to an independent value of η guaranteeing positive-definiteness. Moving onto the g-populations, we consider a generating function (bulk energy (2.1) Lattice Boltzmann model for non-ideal compressible fluid dynamics 9 per unit mass), E(ρ, u, T) = e(ρ, T) + u2 2 , (3.16) and introduce operators Oα acting on the generating function E as, OαE = P ρ ∂E ∂uα + uαE. (3.17) The energy moments can be computed by repeated application of the operators (3.17) on the generating function (3.16). For clarity, let us compute the first two moments explicitly, OαE = P ρ + E uα, (3.18) O2 αE = u2 α 2P ρ + E + P ρ P ρ + E . (3.19) The discrete equilibrium geq i can now be defined by setting ξα = Oα and ζαα = O2 α in the functions (3.6) and using the product-form as an operator-function acting on the generating function (3.16), geq i (ρ, u, T) = ρ Y α Ψiα Oα, O2 α E(ρ, u, T). (3.20) Furthermore, let us introduce the following notation for the shifted flow velocity u∗and shifted temperature T ∗, u∗= u + δtF ρ , (3.21) T ∗= T - δt2 2ρ2cv F · F . (3.22) Shifted-equilibrium populations g∗ i are defined using the equilibrium product-form (3.20) but evaluated at shifted values (3.21) and (3.22), and adding a correction, g∗ i (ρ, u, T) = geq i (ρ, u∗, T ∗) + 1 2ci · q′, if c2 i = 1, 0, otherwise. (3.23) where the non-equilibrium energy flux correction is the vector q′ with the components, q′ α = δt P∂α e + P ρ - k μ T∂αT + uαΦ′ . (3.24) The moments of orders zero and one of g∗ i are as follows: Q X i=1 g∗ i = ρE (ρ, u∗, T ∗) , (3.25) Q X i=1 cig∗ i = u∗(P(ρ, T ∗) + ρE(ρ, u∗, T ∗)) + q′. (3.26) A few comments about g∗ i are in order: • The equilibrium of Eq. (3.16) would have naturally resulted in a Navier-Stokes level heat flux that is a function of specific enthalpy, i.e. ∝∂α(e+P/ρ). In the case of an ideal 10 S. A. Hosseini, M. Feinberg, I. V. Karlin gas equation of ∂α(e + P/ρ) = (cv + R)∂αT since both internal energy and pressure are only functions of temperature. For a non-ideal equation of state however, since the latter are functions of temperature and density this would lead to an additional flux driven by density gradients. The first two terms in Eq. (3.24) restore the correct form of the Fourier heat flux to the model while the last term accounts for viscous heating brought in by the bulk viscosity in the f-population quasi-equilibrium, see Eq. (3.13). • Looking at g∗ i as an attractor towards a shifted velocity u∗and temperature T ∗, we note that the shifted temperature is systematically lower than T, see Eq. (3.22). Developing the moment of order zero as, Q X i=1 g∗ i = 1 2ρu2 + δtu · F + δt2 2ρ F · F + ρe(ρ, T ∗), (3.27) with, e(ρ, T ∗) = e(ρ, T) + ∂e(ρ, T) ∂T T δT + O δT 2 , (3.28) where we introduced δT = T ∗-T for readability. To correctly recover fluxes up to order δt2 one must have, ∂e(ρ, T) ∂T T δT = -δt2 2ρ2 F · F , (3.29) which explains Eq. (3.22) and why T ∗is systematically lower than T. Note that while higher order terms appear for non-linear-in-temperature equations of state, such as Peng-Robinson (Peng & Robinson 1976), for linear equations such as van der Waals there are no higher order terms, see Eq. (2.20) and (2.21). Finally, the locally conserved fields elsewhere in the equilibrium and shifted-equilibrium populations, the density ρ, the momentum ρu and the bulk energy ρE, are consistently defined via the zeroth- and the first-order moments of the populations, ρ = Q X i=1 fi, (3.30) ρu = Q X i=1 cifi + δt 2 F , (3.31) ρE = Q X i=1 gi + δt 2 u · F . (3.32) This concludes the proposed lattice Boltzmann model for compressible non-ideal flow simulations. Detailed multi-scale analysis of the hydrodynamic limit is presented in Appendix A and demonstrates that the present lattice Boltzmann model recovers the hydrodynamic equations (2.6), (2.7) and (2.8). Numerical applications and validation of the model are presented in the next section. 4. Numerical applications 4.1. Consistency: Dispersion and dissipation of hydrodynamic eigen-modes As a first step, we shall probe the dispersion and dissipation properties of hydrodynamic modes comprising the shear, the normal and the entropic modes, in the limit of resolved flows simulation. This benchmark will involve: (a) speed of sound, (b) dissipation Lattice Boltzmann model for non-ideal compressible fluid dynamics 11 Table 1: Critical properties of nitrogen N2. Substance R/cv Pc[Pa] ρc[kg/m3] Tc[K] Nitrogen 0.4 3.4 × 106 241.96 126.2 0.7 0.8 0.9 1 Tr 200 300 400 500 cs[m/s] Figure 2: Speed of sound for nitrogen N2 on the saturated liquid and vapor branches. Line: analytical solution from Eq. (4.4), Markers: simulations. of shear waves, (c) shear stress viscous heating and entropic mode dissipation, and (d) normal mode dissipation rate. For all cases, for the sake of readability and without loss of generality, we consider Nitrogen with properties listed in Table 1. The setup to investigate the speed of sound consists of a one-dimensional domain of size Lx = 0.1 [m] discretized with the spacing δx = 10[μm]. The initial conditions are, P(x, t) = ( P0, ∀x ⩽Lx/2 P0 + δP, ∀x > Lx/2 , (4.1) ux(x, t) = ux0, (4.2) T(x, t) = T0. (4.3) Once the initial conditions are set, the system is left to evolve, resulting in two oppositemoving pressure fronts propagating at a speed that becomes constant after a short initial transition time. For sufficiently weak perturbations, this corresponds to the speed of sound in the system. A series of such cases for different initial conditions on both liquid and vapor branches of the saturation curve for Tr ∈[0.7, 1] were run and compared to the analytical speed of sound (2.16) for the van der Waals equation of state (2.30), cs = s RT(1 + R/cv) (1 -bρ)2 -2aρ. (4.4) The results are shown in Figure 2 and point to excellent agreement. The next configuration is set to measure the effective shear viscosity. To that end, we set up a one-dimensional domain of size Lx = 0.1 [m] with periodic boundary conditions in both x-and y-directions and discretized with Nx = 100 grid points. Initial conditions 12 S. A. Hosseini, M. Feinberg, I. V. Karlin 0 0.5 1 Ma 0 2 4 6 [m2/s] 10-4 Figure 3: Kinematic viscosity as measured from shear wave decay simulations at different Mach numbers. Plain black line: analytical viscosity, square markers: viscosity measured from simulations. are set as, uy(x) = Maycs(ρ0, T0) + δMaycs(ρ0, T0) sin 2πx Lx , (4.5) ux(x) = 0, (4.6) ρ(x) = ρ0, (4.7) T(x) = T0. (4.8) The maximum of the deviation uy-Maycs(ρ0, T0) in the domain is monitored throughout the simulation and its evolution in time is fitted with an exponential decay function, umax y (t) ∝exp -4π2 L2x μ ρ t . (4.9) The viscosity measured from simulations is compared to those predicted from the multiscale analysis. The results are shown in Fig. 3 and point to excellent agreement and Galilean invariance of the measured viscosity. As the next case, to validate both viscous heating and entropic modes dissipation rate, we consider the two-dimensional thermal Couette flow. The case consists of a pseudo onedimensional domain of size Lx with walls at x = 0 and x = Lx and periodic boundary conditions along the y-direction. The flow is subject to the following boundary conditions, {ux, uy, T}(x = 0) = {0, 0, Tw}, (4.10) {ux, uy, T}(x = Lx) = {0, Uw, Tw}. (4.11) The analytical steady-state solution to this configuration can readily be derived as, uy(x) = Uw x Lx , (4.12) T(x) = Tw + μU 2 w 2k x Lx 1 -x Lx . (4.13) To validate the model, we consider a domain of size Lx = 1 [mm] discretized with 100 grid-points. Simulations were run for Pr ∈{0.6, 1.2, 4.9} and Ma ∈{0.8, 1.2, 1.6}. Results are displayed in Fig. 4 and show excellent agreement with analytical solutions. Lattice Boltzmann model for non-ideal compressible fluid dynamics 13 0 0.5 1 x/Lx 1.6 1.65 1.7 1.75 1.8 1.85 / c 1.04 1.06 1.08 1.1 1.12 1.14 1.16 T/Tc 0 0.5 1 x/Lx 1.6 1.65 1.7 1.75 1.8 1.85 / c 1.04 1.06 1.08 1.1 1.12 1.14 1.16 T/Tc Figure 4: Left panel: Temperature and density distribution along channel for thermal Couette flow considering different Prandtl numbers. Triangle, square and circular markers are analytical results for Pr ∈{0.6, 1.2, 4.9} respectively. Plain and dashed lines are temperature and density profiles from simulations. Here Ma=0.8 for all cases. Right panel: Temperature and density distribution for different Ma numbers. Triangle, square and circular markers are analytical results for Ma ∈{0.8, 1.2, 1.6} respectively. Plain and dashed lines are temperature and density profiles from simulations. Here Pr=1.2 for all cases. Finally, we move on to probing dissipation rate of normal modes, i.e. acoustics. To that end we set up a 1-D domain of size Lx with periodic boundary conditions in both x-and y-directions. Defining a uniform background state, (ρ0, T0, P0), we add a small perturbation to it at t = 0, ux(x) = Maxcs(ρ0, T0), (4.14) uy(x) = 0, (4.15) P(x) = P0 + δP. (4.16) The density and temperature fields can be readily computed using isentropic relations for van der Waals fluid (Kouremenos et al. 1988; Nederstigt & Pecnik 2023), P0 ρ γ0 P ρ 0 = P ργP ρ , (4.17) T0 ρ γ0 T ρ-1 0 = T ργT ρ-1 , (4.18) where, γP ρ = ρ P cp cv ∂P ∂ρ T , (4.19) γT ρ = 1 ρcv ∂P ∂T ρ + 1. (4.20) Here cp is, cp = cv + T ∂P ∂T v ∂v ∂T P , (4.21) 14 S. A. Hosseini, M. Feinberg, I. V. Karlin 0 0.5 1 Ma 0 0.2 0.4 0.6 0.8 1 [m2/s] 10-3 Figure 5: Normal dissipation rate α as measured from normal wave decay simulations at different Mach numbers. Plain black line: analytical dissipation rate, square markers: dissipation rate measured from simulations. which for the van der Waals equation of state leads to, cp = cv + R2T RT -2aρ(1 -bρ)2 . (4.22) One can readily show that for P →ρRT, i.e. the limit of an ideal gas, the following holds (γP ρ, γT ρ) →cp cv . We leave the system to evolve over time and monitor the acoustic energy (Landau & Lifshitz 1987), Eacoustic = 1 2 Z " ρ0(u -u0)2 + ρ′2c2 s(ρ0, P0) ρ0 # dx, (4.23) where ρ′ = ρ -ρ0. It can readily be shown that the decay for a propagating plane wave is proportional to, Eacoustic ∝exp -4π2 L2x αt . (4.24) with (Landau & Lifshitz 1987), α = 2(D -1) D μ ρ + η ρ + k ρcp cp cv -1 . (4.25) Simulations were conducted for different initial values of velocity and effective dissipation rates measured. Results are shown in Fig. 5. The data points to very good agreement with analytical predictions. 4.2. Multi-phase regime 4.2.1. Liquid-vapour coexistence Here we look into the multiphase regime; as a standard validation of thermodynamic and mechanical consistency we first probe the co-existence densities. For that purpose we consider the van der Waals equation of state fitted to the N2 critical point. See table 1 for properties. Simulations are conducted in a 1-D domain of size Lx = 0.4 [mm] with periodic boundary conditions. The domain is filled with saturated vapor with a column of saturated liquid at the center. Simulations are left to evolve until convergence of the Lattice Boltzmann model for non-ideal compressible fluid dynamics 15 0.4 10 100 600 [kg/m3] 40 60 80 100 120 140 T[K] Figure 6: Liquid-vapor co-existence densities for N2. Black lines are from the Maxwell construction while markers are from simulations. Table 2: Grid properties used to model N2 interface at different resolutions. Case δx δt 1 0.1 [μm] 2.5 × 10-11[s] 2 0.5 [μm] 1.25 × 10-10[s] 3 1 [μm] 2.5 × 10-10[s] 4 5 [μm] 1.25 × 10-9[s] density field. Simulations were run for Tr ∈[0.3 0.99]. Results are shown in Fig. 6 and show excellent agreement with theory. Theoretical results are obtained using the Maxwell equal area construction. 4.2.2. Interface consistency and convergence To illustrate the consistency and convergence of the proposed solver to the target hydrodynamic system we compare the density profile of Nitrogen liquid vapor interface. Nitrogen system critical properties are listed in table 1. The capillarity coefficient is set to κ = 10-10 [m7/kg.s2]. The simulation consists of a periodic 1-D domain of size Lx = 0.5[mm] with a liquid column at the center surrounded with vapor. We consider a system at Tr = 0.9. The simulations are initialized with the corresponding co-existence densities in each phase, i.e. (ρl, ρv) = (399.8, 102.71) [kg/m3]. Simulations are conducted at various resolutions detailed in Table 2. The results obtained from simulations are compared to data from a high resolution iterative finite differences solver for, ∂xP = κρ∂3 xρ. (4.26) The results are shown in Fig. 7. Results point to excellent agreement with reference solution and convergent behavior as resolution increases. 16 S. A. Hosseini, M. Feinberg, I. V. Karlin 0 5 10 x[ m] 0.5 1 1.5 r 0 5 10 x[ m] 0.5 1 1.5 Figure 7: Liquid-vapor interface for nitrogen N2 at Tr = 0.9. Black lines are converged results from implicit finite differences solver and red markers from LBM simulations. Bottom left panel: δx = 0.1 [μm], Bottom right panel: δx = 0.5 [μm], top left panel: δx = 1 [μm] and top right panel: δx = 5 [μm]. 4.3. Compressible configurations 4.3.1. 1-D non-ideal shock tubes Until the early 1980s, shock tube experiments were limited to gases exhibiting classical wave behavior. Borisov et al. (1983) first reported a shock tube experiment aimed at investigating the non-classical wave phenomena in a dense gas, i.e. near the thermodynamic critical point. The nonlinear dynamics of gases are fundamentally governed by the fundamental derivative of gas dynamics (Thompson 1971): Γ = 1 + ρ cs ∂cs ∂ρ s . (4.27) For simple waves, Γ represents the rate of change of the convected sound speed with respect to density. When Γ > 0, the flow exhibits positive nonlinearity, i.e. disturbances steepen forward to form compression shocks. Conversely, when Γ 0, (A 8) Q X i=1 cif (1) i + 1 2 Q X i=1 cif ∗(1) i = 0, (A 9) Q X i=1 cif (k) i = 0, ∀k > 1, (A 10) Q X i=1 g(1) i + 1 2 Q X i=1 g∗(1) i = 0, (A 11) Q X i=1 g(k) i = 0, ∀k > 0. (A 12) Taking the zeroth-order moment of Eq. (A 7b) for fi, ∂(1) t ρ + ∇· ρu = 0, (A 13) where we have used, Q X i=1 f ∗ i (1) = 0, (A 14) and solvability condition (A 8). For the first-order moment of (A 7b) for fi, ∂(1) t ρu + ∇· ρu ⊗u + ∇P = -2β =0 z }| { Q X i=1 cif (1) i + 1 2cif ∗ i (1) ! + Q X i=1 cif ∗ i (1) | {z } F , (A 15) where we used (A 9) and further set, F = -∇· TK, (A 16) where, TK = κ∇ρ ⊗∇ρ -κ ρ∇2ρ + 1 2|∇ρ|2 I. (A 17) The force can also be shown to simplify to, F = κρ∇∇2ρ. (A 18) Finally taking the zeroth-order moment for gi, ∂(1) t ρE + ∇· u (ρE + P) = -2β =0 z }| { Q X i=1 g(1) i + 1 2g∗ i (1) ! + Q X i=1 g∗ i (1) | {z } u·F . (A 19) 24 S. A. Hosseini, M. Feinberg, I. V. Karlin Summing up balance equations at order ε, ∂(1) t ρ + ∇· ρu = 0, (A 20) ∂(1) t ρu + ∇· (ρu ⊗u + PI) -F = 0, (A 21) ∂(1) t ρE + ∇· ρu (E + P/ρ) -u · F = 0. (A 22) The last equation Eq. (A 22) can be transformed into a balance equation for internal energy, using , ∂(1) t K + ∇· uK + u · ∇P -u · F = 0, (A 23) as, ∂(1) t ρe + ∇· ρue + P∇· u = 0. (A 24) Furthermore, using de = cvdT - T ∂P ∂T ρ -P ! dρ ρ2 , (A 25) and Eq. (A 20) a balance equation for temperature can be derived as, ∂(1) t T + u · ∇T + T ρcv ∂P ∂T ρ ∇· u = 0. (A 26) Finally, using dP = ∂P ∂ρ T dρ + ∂P ∂T ρ dT, (A 27) we can also write a balance equation for pressure as, ∂(1) t P + u · ∇P + ρc2 s∇· u = 0, (A 28) where, c2 s = ∂P ∂ρ T + T cvρ2 ∂P ∂T 2 ρ . (A 29) At order ε2, the zeroth order moment of fi leads to, ∂(2) t ρ = 0, (A 30) and the first order moments, ∂(2) t ρu + ∇· (1 -β) " Q X i=1 ci ⊗cifi (1) ! + 1 2 Q X i=1 ci ⊗cif ∗(1) i !# = 0. (A 31) Here we can use Eq. (A 7b) to obtain, ∂(2) t ρu + ∇· 1 2 -1 2β ∂(1) t Π2(f (0)) + ∇Π3(f (0)) - Q X i=1 ci ⊗cif ∗(1) i ! = 0. (A 32) Here, Π(0) 2 (f) and Π(0) 3 (f) are the second and third order moments tensor of the equilibrium distribution function, Π2(f (0)) = ρu ⊗u + PI, (A 33) Π3(f (0)) = ρu ⊗(u ⊗u + PI) ◦(1 -J) + 3ρς2u ⊗I ◦J. (A 34) Lattice Boltzmann model for non-ideal compressible fluid dynamics 25 Using Eqs. (A 20) and (A 21) we can write, ∂(1) t (ρu ⊗u + PI) = u ⊗F + F ⊗u -∇· ρu ⊗u ⊗u -∇Pu -(∇Pu)† P ∇u + ∇u† + ∂(1) t PI. (A 35) For the last term, i.e. ∂(1) t P, we write a balance equation for P, ∂(1) t P = P -ρc2 s ∇· u -∇· Pu, (A 36) while, ∇· Π3(f (0)) = ∇· ρu ⊗u ⊗u + ∇Pu + ∇Pu† + Ψ + I∇· Pu, (A 37) where, Ψαα = -∂αuα u2 α + 3(P -ρς2) . (A 38) Adding up all terms, ∂(1) t Π2(f (0)) + ∇· Π3(f (0)) = u ⊗F + F ⊗u + P ∇u + ∇u† + P -ρc2 s ∇· uI -Ψ. (A 39) Setting, Q X i=1 ci ⊗cif ∗(1) i = u ⊗F + F ⊗u + Ψ + P D + 2 D -ρc2 s P -η μ ∇· uI (A 40) and plugging it back into Eq. (A 32), ∂(2) t ρu -∇· μ ∇u + ∇u† -2 D∇· uI + η∇· uI = 0, (A 41) Where we used, μ = 1 2β -1 2 Pδt. (A 42) For the second population, at order ε2, ∂(2) t ρE + ∇· 1 2 -1 2β ∂(1) t Π1(g(0) i ) + ∇· Π2(g(0) i ) - Q X i=1 cig∗ i (1) ! = 0, (A 43) where, Π1(g(0) i ) = u(ρE + P), (A 44) Π2(g(0) i ) = u ⊗u (ρE + 2P) + (E + P/ρ) PI. (A 45) Here we can use, ∂(1) t Pu + ∇· Pu ⊗u + P ρ ∇P -P ρ F + u ρc2 s -P ∇· u = 0, (A 46) and ∂(1) t ρEu + ∇· (ρE + P)u ⊗u + E∇P -EF -Pu · ∇u -u(u · F ) = 0, (A 47) 26 S. A. Hosseini, M. Feinberg, I. V. Karlin Adding up both contributions, ∂(1) t Π1(g(0) i )+∇·Π2(g(0) i ) = -2∇· Pu ⊗u + 2∇· Pu ⊗u | {z } =0 +∇· ρEu ⊗u -∇· ρEu ⊗u | {z } =0 +∇P(P/ρ + E) -(E + P/ρ)∇P | {z } P ∇(E+P/ρ) +Pu · ∇u -u ρc2 s -P ∇· u + (P/ρ + E)F + u(u · F ). (A 48) Further expanding, ∂(1) t Π1(g(0) i ) + ∇· Π2(g(0) i ) = P∇h =P u·(∇u+∇u†) z }| { +P∇(u2/2) + Pu · ∇u +u P -ρc2 s ∇· u + (P/ρ + E)F + u(u · F ), (A 49) where h = H -K. Plugging this back into the balance equation, ∂(2) t ρE -∇· u · μ ∇u + ∇u -2 D∇· uI + η∇· uI | {z } -u·TNS + ∇· 1 2 -1 2β P∇h + u · P D + 2 D -ρc2 s P -η μ ∇· uI +(P/ρ + E)F + u(u · F ) - Q X i=1 cig∗ i (1) ! = 0. (A 50) Using the definition of the shifted equilibrium, Π0(g∗(1) i ) = u · F , (A 51) Π1(g∗(1) i ) = u(u · F ) + F P ρ + E +P ∇h -κ μ∇T + u · D + 2 D -ρc2 s P -η μ ∇· uI . 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2510.14714
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Loss functions arising from the index of agreement
Hristos Tyralis1*, Georgia Papacharalampous2
1Support Command, Hellenic Air Force, Elefsina Air Base, 19 200, Elefsina, Greece
(montchrister@gmail.com,
hristos@itia.ntua.gr,
https://orcid.org/0000-0002-8932-
4997)
2Department of Land, Environment, Agriculture and Forestry, University of Padova, Viale
dell'Università 16, 35020, Legnaro, Italy (papacharalampous.georgia@gmail.com,
georgia.papacharalampous@unipd.it, https://orcid.org/0000-0001-5446-954X)
*Corresponding author
Abstract: We examine the theoretical properties of the index of agreement loss function
𝐿𝑊, the negatively oriented counterpart of Willmott’s index of agreement, a common
metric in environmental sciences and engineering. We prove that 𝐿𝑊 is bounded within
[0,1], translation and scale invariant, and estimates the parameter 𝔼𝐹[𝑦] ± 𝕍𝐹
1/2[𝑦] when
fitting a distribution. We propose 𝐿NR2 as a theoretical improvement, which replaces the
denominator of 𝐿𝑊 with the sum of Euclidean distances, better aligning with the
underlying geometric intuition. This new loss function retains the appealing properties of
𝐿𝑊 but also admits closed-form solutions for linear model parameter estimation. We show
that as the correlation between predictors and the dependent variable approaches 1,
parameter estimates from squared error, 𝐿NR2 and 𝐿𝑊 converge. This behavior is
mirrored in hydrologic model calibration (a core task in water resources engineering),
where performance becomes nearly identical across these loss functions. Finally, we
suggest potential improvements for existing 𝐿𝑝-norm variants of the index of agreement.
Keywords: extremum estimation; index of agreement; loss function; model calibration;
model evaluation; point forecasting
1.
Introduction
Loss functions are fundamental tools in machine learning, statistical science and
engineering, used for model estimation (also referred to as fitting, calibrating or training)
and model ranking (evaluation) [1]. During fitting, model parameters are estimated by
minimizing the average discrepancy between predictions and observations, quantified by
the loss function. For ranking, a model’s performance is evaluated by comparing the
2
average loss of predictions against observations in a test set.
Domain-specific loss functions developed in environmental sciences and engineering
([2], [3], [4]) hold potential for broader application. One established metric is Willmotts’s
[5] index of agreement, defined as
𝑑(𝒛, 𝒚) ∶= 1 −∑
(𝑧𝑖−𝑦𝑖)2
𝑛
𝑖=1
/ ∑
(|𝑧𝑖−𝜇(𝒚)| + |𝜇(𝒚) −𝑦𝑖|)2
𝑛
𝑖=1
(1.1)
where 𝒛= (𝑧1, … , 𝑧𝑛)T represents model predictions, 𝒚= (𝑦1, … , 𝑦𝑛)T represents
realizations and 𝜇(𝒚) is the sample mean of realizations (defined in eq. (A.19)). Since 𝑑 is
positively oriented (higher values indicate better performance), we study its negatively
oriented counterpart without loss of generality, which we name the index of agreement
loss function 𝐿𝑊 (subscript 𝑊 for Willmott):
𝐿𝑊(𝒛, 𝒚) ∶= ∑
(𝑧𝑖−𝑦𝑖)2
𝑛
𝑖=1
/ ∑
(|𝑧𝑖−𝜇(𝒚)| + |𝜇(𝒚) −𝑦𝑖|)2
𝑛
𝑖=1
(1.2)
Although the index of agreement has been extensively researched, most studies are
empirical [6] and a theoretical analysis is lacking. Here we aim to provide a theoretical
foundation for the index of agreement and to propose variants with improved theoretical
properties, building upon its fundamental construction principles. Our contributions are
as follows:
(i)
We examine the theoretical properties of 𝐿𝑊(𝒛, 𝒚) as an extremum estimator [7],
demonstrating that it is bounded in [0,1], and is translation and scale invariant. When
fitting a sample of realizations 𝒚, it estimates the parameter 𝜃̂ = 𝜇(𝒚) ± 𝜎(𝒚), where 𝜎(𝒚)
is the sample standard deviation defined in eq. (A.21). For linear models, no closed-form
solution has been identified, which remains an open problem.
(ii)
We propose an alternative definition for 𝐿𝑊, which we call 𝐿NR2 (subscript NR for
Norm Ratio, subscript 2 for 𝐿2-norm), defined in Section 3.2. This function maintains
consistency with the distance-based arguments underlying the index of agreement, but
uses 𝐿2-norm-based constructions. Thus, 𝐿NR2 represents a theoretical improvement over
the original index. It preserves the [0,1] bound and translation and scale invariance. When
fitting a sample 𝒚, it estimates 𝜃̂ = 𝜇(𝒚) ± 𝜎(𝒚), analogous to the original index. For linear
models 𝒛= 𝑎𝒙+ 𝑏𝟏, we provide closed-form estimates for parameters 𝑎 and 𝑏.
(iii) Constructions based on 𝐿𝑝-norms (in Section 3.3) generalize the new index of
agreement, enabling a theoretically consistent variant of Legates and McCabe Jr [8] and
Krause et al. [9] index of agreement variants.
3
(iv) Although our theoretical developments focus on estimation, we empirically extend
these insights to model evaluation. Such an investigation is necessary because a theory
linking estimation and evaluation for non-additive extremum estimators (unlike additive
M-estimators; see [10]) is currently lacking in statistical science. Empirical results based
on predictions from simulated data and real-world case studies (the latter using
hydrologic models) estimated with 𝐿𝑊 and 𝐿NR2, will facilitate a reassessment of prior
studies that employed the index of agreement.
The remainder of the manuscript is structured as follows. Section 2 reviews essential
background on prediction evaluation, estimation theory and the index of agreement.
Section 3 presents new material and the main theoretical results, focusing on the original
index of agreement and our subsequent developments. Empirical demonstrations using
both simulated and real-world data modeled by hydrologic models is included in Section
4. The implications of these theoretical contributions are discussed in Section 5, followed
by concluding remarks in Section 6. Given the extensive vector representations and
calculus required for the theoretical analysis and proofs, Appendix A introduces
consistent notation and definitions for vector calculus. Common identification [1] and loss
functions used throughout the manuscript, are detailed in Appendix B, while the
probability distributions used are listed in Appendix C. All proofs are supplied in
Appendix D. Appendix E details the statistical software to reproduce the analysis.
Supplementary material to reproduce all empirical analyses and figures is available.
2.
Theoretical background
Section 2.1 outlines the theoretical framework for prediction evaluation, which applies to
additive formulations of a loss function for both assessing a single model and comparing
multiple models. As the index of agreement is non-additive, its application to parameter
estimation places it within the broader category of extremum estimators. The theoretical
foundations for this general class of estimators, including the specific case of M-estimators
(which align directly to the framework in Section 2.1), are presented in Section 2.2.
Section 2.3 then describes the index of agreement and develops the formulations required
for its subsequent analysis in Section 3.
4
2.1 Comparison and adequacy of predictions
2.1.1 Comparing predictions from multiple models
Let 𝒛= (𝑧1, … , 𝑧𝑛)T represent a vector of model predictions and 𝒚= (𝑦1, … , 𝑦𝑛)T a vector
of corresponding observed values. Model comparisons are performed using a loss
function 𝐿(𝑧, 𝑦) which quantifies the discrepancy between a single prediction 𝑧 and its
observation 𝑦 ([21], [1]). Standard loss functions include the absolute error (AE) loss 𝐿AE
(eq. (B.1)) and the squared error (SE) loss 𝐿SE (eq. (B.2)). A model is preferred over
another if it achieves a lower average loss 𝐿̅ across all observations. This average is
defined as [1]:
𝐿̅(𝒛, 𝒚) = (1/𝑛) ∑
𝐿(𝑧𝑖, 𝑦𝑖)
𝑛
𝑖=1
(2.1)
Commonly used averages of these loss functions include the mean absolute error (MAE;
eq. (B.3)), which corresponds to the AE loss and the mean squared error (MSE; eq. (B.4)),
which corresponds to the SE loss.
2.1.2 Strictly consistent loss functions
A key property a loss function can have is known as (strict) consistency. To explain this
concept, we must first define the notion of a statistical functional. Let 𝑦 be a random
variable (hereinafter, random variables are underlined). A materialization of 𝑦 is written
as 𝑦. We write 𝑦~𝐹 to indicate that 𝑦 follows a cumulative distribution function (CDF) 𝐹,
defined as:
𝐹(𝑦) ∶= 𝑃(𝑦≤𝑦)
(2.2)
A one-dimensional statistical functional (or simply a functional) 𝑇 is a mapping [1]
𝑇: ℱ→𝒫(𝐷), 𝐹↦𝑇(𝐹) ⊆𝐷
(2.3)
where ℱ is a class of probability distributions. The functional maps each distribution 𝐹∈
ℱ to a subset 𝑇(𝐹) of the domain 𝐷⊆ℝ (i.e., an element of the power set 𝒫(𝐷)). This
subset 𝑇(𝐹) can be single-valued or set-valued. Common examples of statistical
functionals include the median 𝑄1/2(𝐹), the 𝜏-quantile 𝑄𝜏(𝐹) and the mean 𝔼𝐹𝑦[𝑦].
A loss function 𝐿 is ℱ-consistent for the functional 𝑇 if the following inequality holds
for all distributions in the class ([1], [11])
𝔼𝐹[𝐿(𝑡, 𝑦)] ≤𝔼𝐹[𝐿(𝑧, 𝑦)] ∀ 𝐹∈ℱ, 𝑡∈𝑇(𝐹), 𝑧∈𝐷
(2.4)
5
The loss function is strictly ℱ-consistent if it is ℱ-consistent and equality in eq. (2.4)
implies that 𝑧∈𝑇(𝐹).
The practical importance of strict consistency can be understood from the following
example. Suppose a modeler is tasked with predicting the mean of the random variable 𝑦.
The SE loss 𝐿SE is strictly consistent for the mean [1]. This means that if the modeler
follows the directive, i.e. predicts the true mean, her/his predictions will, in expectation,
minimize the loss when evaluated using 𝐿SE. Eq. (2.4) is a population-level property,
therefore, its natural empirical counterpart for comparing predictions is the sample
average loss defined in eq. (2.1). 𝐿SE belongs to the broader family of Bregman loss
functions, all of which are strictly consistent for the mean ([1], [12], [13], [14], [15], [16]).
2.1.3 Skill scores
When evaluating predictions from multiple competing models, skill scores provide a
standardized framework for comparison. The general form of a skill score, as given by [1]
is:
𝐿̅skill(𝒛, 𝒚, 𝒛ref) ∶= (𝐿̅(𝒛, 𝒚) −𝐿̅(𝒛ref, 𝒚))/(𝐿̅(𝒛optimal, 𝒚) −𝐿̅(𝒛ref, 𝒚))
(2.5)
where 𝒛 represents the predictions from the model of interest, 𝒛ref represents the
predictions from the reference model and 𝒛optimal represents the theoretically optimal
predictions that achieve the minimum possible loss. Skill scores of this form are positively
oriented.
A common simplification arises when the loss function 𝐿 has a minimum of 0, as this
implies 𝐿̅(𝒛optimal, 𝒚) = 0. The skill score then simplifies to:
𝐿̅skill(𝒛, 𝒚, 𝒛ref) ∶= 1 −𝐿̅(𝒛, 𝒚)/𝐿̅(𝒛ref, 𝒚)
(2.6)
This ratio-based form is intuitively interpretable as the relative improvement over the
reference method.
A prominent example is the Nash-Sutcliffe efficiency (NSE) [17], a skill score based on
the squared error loss relative to a climatological mean [18]. The NSE is constructed by
taking 𝐿SE as the evaluation loss and using the mean climatology as the reference
prediction. Substituting these into eq. (2.6) yields the NSE, which is defined in eq. (B.5).
2.1.4 Adequacy of predictions from a single model
While loss functions are instrumental for the comparative ranking of models, a distinct
6
framework is required to assess the absolute adequacy of a single set of predictions. This
assessment is formalized through the concept of identification functions (1], [19]).
An identification function 𝑉: 𝐷× 𝐼→ℝ, 𝐼⊆ℝ serves as a tool for evaluating whether a
prediction 𝑧 equals a specified functional 𝑇(𝐹). The function 𝑉 is termed a strict ℱ-
identification function for the functional 𝑇(𝐹) if it satisfies the following condition
𝔼𝐹[𝑉(𝑧, 𝑦)] = 0 ⇔𝑧∈𝑇(𝐹) ∀ 𝐹∈ℱ, 𝑡∈𝑇(𝐹), 𝑧∈𝐷
(2.7)
If a strict identification function exists for the functional 𝑇, then 𝑇 is called identifiable.
The practical significance of identifiability lies in its ability to enable an absolute check of
predictive performance. For instance, if a model claims to predict the median, the average
of its identification function values over a test set can be used to statistically verify this
claim, a concept aligned with calibration in the statistical literature ([18], [20]).
It is crucial to distinguish the roles of loss and identification functions. The former are
used for comparative model ranking, while the latter are used for the absolute assessment
of predictions for a specific functional [21].
A notable connection between these concepts is established by Osband’s principle [22].
This principle states that a strictly ℱ-consistent loss function 𝐿 and a strict ℱ-
identification function 𝑉 are linked via the derivative ([1], [21], [22])
𝜕𝐿(z,𝑦)
𝜕𝑧
= ℎ(𝑧)𝑉(𝑧, 𝑦)
(2.8)
where ℎ is a nonnegative function and 𝑉 is an oriented function. The function 𝑉 is an
oriented strict ℱ-identification function if it is a strict ℱ-identification function for the
functional 𝑇(𝐹) and, moreover 𝔼𝐹[𝑉(𝑧, 𝑦)] > 0 if and only if 𝑧> 𝑇(𝐹) ∀ 𝐹∈ℱ, z ∈𝐷⊆ℝ.
([21], [23]).
Common identification function examples include 𝑉median (eq. (B.6)) for the median
and 𝑉mean (eq. (B.7)) for the mean. In practice, the empirical average identification
function 𝑉̅ is computed on a test set
𝑉̅(𝒛, 𝒚) = (1/𝑛) ∑
𝑉(𝑧𝑖, 𝑦𝑖)
𝑛
𝑖=1
(2.9)
Commonly used empirical averages include 𝑉̅median (eq. (B.8)) and 𝑉̅mean (eq. (B.9)), the
latter also known as mean error (ME). Adequate predictions for the functional 𝑇 should
yield 𝑉̅ ≈0 (subject to scaling of the predicted variable). For example, a median
prediction is adequate if roughly half the predictions exceed their corresponding
observations.
7
Finally, it is possible for multiple models to be adequate for the same functional (e.g.,
both identifying the mean) yet exhibit different average losses. In such cases, the model
with the lower average loss (e.g. lower MSE for mean predictions) is still preferred, as it
is more accurate even though both are calibrated.
2.1.5 Prespecifying the loss function in model estimation and prediction evaluation
Theoretical work establishes that the loss function used for ranking must be disclosed a
priori, as model rankings depend critically on this choice ([1], [24]). This dependency can
be illustrated with an example. Consider Model #1, trained to predict conditional medians
using the absolute error (AE) loss, and Model #2, trained to predict conditional means
using the squared error (SE) loss. If the dependent variable follows an asymmetric
distribution, their relative performance reverses depending on the loss function, since
Model #1 will outperform Model #2 under AE loss, while the opposite holds under SE loss
[10]. This follows directly from the equivalence between the consistency of an M-
estimator (used in training) and evaluation with a strictly consistent loss function.
2.2 Consistent extremum and M-estimators
The general class of extremum estimators was formalized by [7] [25] and [26] and.
Consider a loss function 𝐿(𝜽, 𝒚) that quantifies the discrepancy for a parameter vector 𝜽
given a realization 𝒚= (𝑦1, … , 𝑦𝑛)T. Assuming the true parameter 𝜽0 lies in the parameter
space 𝜣, an extremum estimator 𝜽̂(𝒚) for 𝜽0 is defined as:
𝜽̂(𝒚) ∶= argmin
𝜽∈𝜣
𝐿(𝜽, 𝒚)
(2.10)
The properties of 𝜽̂(𝒚) depend on the sample size 𝑛 through the realizations 𝒚. While
𝐿 can take general forms, the additive case has been extensively studied. When 𝐿 is
additive, the resulting estimators are called M-estimators ([27], [28]). For a one-
dimensional parameter 𝜃0, this takes the form:
𝜃̂(𝒚) ∶= argmin
𝜃∈𝛩
∑
𝐿(𝜃, 𝑦𝑖)
𝑛
𝑖=1
(2.11)
A canonical example is the SE loss, which yields the M-estimator for the mean:
𝜇̂(𝒚) ∶= argmin
𝜃∈𝛩
∑
(𝜃−𝑦𝑖)2
𝑛
𝑖=1
, 𝛩⊆ℝ
(2.12)
Consistency is a key property for an estimator. An estimator 𝜽̂ is consistent if it
converges in probability to the true parameter value 𝜽0, i.e. 𝜽̂(𝒚)
𝑃→𝜽0. For instance, 𝜇̂(𝒚)
8
is consistent for 𝔼𝐹𝑦[𝑦], where 𝐹𝑦 is the cumulative distribution function (CDF) of 𝑦, and
𝔼𝐹𝑦[𝑦] is the expectation of 𝑦.
A fundamental connection exists between the two distinct procedures of estimation
and prediction evaluation for the case of M-estimators. Under general conditions, an M-
estimator is consistent for the parameter 𝜃0 if and only if the loss function 𝐿 in eq. (2.11)
is strictly consistent for the functional 𝑇 corresponding to that parameter [10].
2.3 The original index of agreement loss function
The index of agreement loss function admits the following vectorized representation:
𝐿𝑊(𝒛, 𝒚) = ‖𝒛−𝒚‖2
2/‖(|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|)‖2
2
(2.13)
where 𝒛= (𝑧1, … , 𝑧𝑛)T represents the vector of model predictions, 𝒚= (𝑦1, … , 𝑦𝑛)T
represents the vector of observed values, ‖𝒙‖2 is the Euclidean norm (eq. (A.9)), |𝒙|
indicates the element-wise absolute value (eq. (A.4)) and 𝟏 denotes the all-ones vector
defined in eq. (A.3). 𝐿𝑊 is defined when 𝒚 is non-constant (i.e., has at least one element
different from the others) or 𝒛≠𝒚 (where ≠ indicates pairwise inequality for at least one
vector element; see eq. (A.6)).
This vectorized form simplifies theoretical analysis. For instance, Proof D.1 in
Appendix D establishes that 𝐿𝑊 is bounded:
0 ≤𝐿𝑊(𝒛, 𝒚) ≤1
(2.14)
As noted in the Introduction, 𝐿𝑊 is negatively oriented. Furthermore, 𝐿𝑊(𝒛, 𝒚) = 0, if and
only if 𝒛= 𝒚 and 𝒚 is non-constant.
3.
The index of agreement
This section examines the original index of agreement and introduces theoretically
consistent variants developed from its foundational construction principles. Formal
proofs for the presented results are given in Appendix D.
3.1 The index of agreement loss function
This section details key properties of the original index of agreement loss function 𝐿𝑊.
3.1.1 Translation invariance
The loss function 𝐿𝑊 is translation invariant, meaning that shifting predictions and
realizations by a constant 𝑐∈ℝ its value does not change (see Proof D.2):
9
𝐿𝑊(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = 𝐿𝑊(𝒛, 𝒚), 𝑐∈ℝ
(3.1)
3.1.2 Scale invariance
The loss function 𝐿𝑊 exhibits scale invariance, meaning its value remains unchanged
when both predictions and observations are scaled by a non-zero-constant 𝑐∈ℝ\{0}:
𝐿𝑊(𝑐𝒛, 𝑐𝒚) = 𝐿𝑊(𝒛, 𝒚), 𝑐∈ℝ\{0}
(3.2)
This property follows directly from the definition in eq. (2.13). Therefore, its behavior
remains unchanged across different measurement units.
3.1.3 Extremum estimation
To analyze 𝐿𝑊 as an extremum estimator for a scalar parameter 𝜃, we minimize it for a
constant vector 𝜃𝟏, given non-constant observations 𝒚:
𝜃̂(𝒚) ∶= argmin
𝜃∈𝛩
𝐿𝑊(𝜃𝟏, 𝒚)
(3.3)
As shown in Proof D.3, this minimization yields two distinct estimators:
𝜃̂(𝒚) = 𝜇(𝒚) ± 𝜎(𝒚)
(3.4)
Figure 1 illustrates 𝐿𝑊(𝜃𝟏, 𝒚) as a function of 𝜃 for a realization 𝒚 of 𝒚, indicating the
points where it attains its minimum. The minimum value of 𝐿𝑊(𝜃𝟏, 𝒚) is (𝜎(𝒚))/(𝜎(𝒚) +
mad(𝒚)), where mad(𝒚) is the mean absolute deviation defined in eq. (A.20).
Figure 1. Illustration of 𝐿𝑊(𝜃𝟏, 𝒚) function.
The non-uniqueness precludes consistency, as the estimator does not converge to a
single parameter value. To resolve this, we define two modified estimators, i.e. the upper-
10
bound estimator
𝜃̂+(𝒚) = max{argmin
𝜃∈𝛩
𝐿𝑊(𝜃𝟏, 𝒚)} = 𝜇(𝒚) + 𝜎(𝒚)
(3.5)
and the lower-bound estimator
𝜃̂−(𝒚) = min{argmin
𝜃∈𝛩
𝐿𝑊(𝜃𝟏, 𝒚)} = 𝜇(𝒚) −𝜎(𝒚)
(3.6)
Proof D.4 establishes that, under independent identically distributed (IID) variables
assumption, 𝜃̂+(𝒚) and 𝜃̂−(𝒚) are consistent for 𝔼𝐹[𝑦] + 𝕍𝐹
1/2[𝑦] and 𝔼𝐹[𝑦] −𝕍𝐹
1/2[𝑦],
respectively, where 𝕍𝐹[𝑦] is the variance of 𝑦.
3.1.4 Estimation of a linear model
An open problem is the estimation of the true parameters 𝜽0 = (𝑎0, 𝑏0) ∈ℝ2 for the linear
model 𝒛= 𝑎𝒙+ 𝑏𝟏, using the 𝐿𝑊 loss. The estimator
𝜽̂(𝒙, 𝒚) ∶= argmin
𝜽∈ℝ2
𝐿𝑊(𝑎𝒙+ 𝑏𝟏, 𝒚)
(3.7)
lacks a known closed-form solution.
3.2 The new agreement loss function
The original index of agreement (eq. (2.13)) is fundamentally a ratio of two squared
Euclidean norms. While the numerator ‖𝒛−𝒚‖2
2 directly measures the magnitude of
prediction errors, the denominator involves the norm of the composite vector
|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|. This construction, which applies element-wise absolute
values, complicates geometric interpretation.
To address this, we propose an alternative loss function 𝐿NR2 based on the Euclidean
distances between the vectors 𝒛, 𝒚 and the mean vector 𝟏𝜇(𝒚). These vectors form a
triangle in ℝ𝑛, satisfying the triangle inequality in eq. (A.31)
0 ≤‖𝒛−𝒚‖2 ≤‖𝒛−𝟏𝜇(𝒚)‖2 + ‖𝟏𝜇(𝒚) −𝒚‖2
(3.8)
Leveraging this inequality, we define:
𝐿NR2(𝒛, 𝒚) ∶= ‖𝒛−𝒚‖2
2/(‖𝒛−𝟏𝜇(𝒚)‖2 + ‖𝟏𝜇(𝒚) −𝒚‖2)2
(3.9)
where the subscript NR stands for norm ratio. 𝐿NR2 is defined when 𝒚 is non-constant or
𝒛≠𝒚. An equivalent expression is:
𝐿NR2(𝒛, 𝒚) = (∑
(𝑧𝑖−𝑦𝑖)2
𝑛
𝑖=1
)/(√∑
(𝑧𝑖−𝜇(𝒚))2
𝑛
𝑖=1
+ √∑
(𝜇(𝒚) −𝑦𝑖)2
𝑛
𝑖=1
)2 (3.10)
𝐿NR2 has the following geometric and theoretical advantages:
11
(i)
(Intuitive geometric foundation): Unlike the original index in eq. (1.1), where the
denominator sums element-wise absolute deviations before applying the Euclidean norm,
𝐿NR2 operates directly on the vectors 𝒛−𝟏𝜇(𝒚) and 𝟏𝜇(𝒚) −𝒚. This aligns with the
Euclidean triangle inequality in eq. (3.8), treating prediction errors as vector
displacements in ℝ𝑛. The denominator represents the total path length from 𝒛 to 𝒚 via
𝟏𝜇(𝒚), while the numerator is the direct displacement.
(ii)
(Theoretical consistency): The element-wise absolute values in the original
denominator disrupt vector-space properties. By eliminating these, 𝐿NR2 preserves
translation and scale invariance (inherited from 𝐿𝑊, see Sections 3.2.3 and 3.2.2) and
maintains compatibility with norm axioms avoiding ad hoc element-wise operations. This
enables cleaner analysis (e.g. closed-form solutions for linear models; see Section 3.2.5).
(iii) (Interpretability): 𝐿NR2 quantifies the inefficiency of predictions relative to a
“baseline path” through 𝟏𝜇(𝒚).
3.2.1 Bounds
From eqs. (3.8) and (3.9), it follows immediately that:
0 ≤𝐿NR2(𝒛, 𝒚) ≤1
(3.11)
Furthermore, 𝐿NR2(𝒛, 𝒚) = 0, if and only if 𝒛= 𝒚 and 𝒚 is non-constant. Applying eq.
(A.32) shows that 𝐿NR2 attains its upper bound of 1 in the following cases:
(i)
When 𝒚= 𝟏𝜇(𝒚) and 𝒛≠𝒚.
(ii)
When 𝒛= 𝟏𝜇(𝒚) and 𝒚 is not constant.
(iii) When 𝒛−𝟏𝜇(𝒚) = 𝑎(𝟏𝜇(𝒚) −𝒚) for some 𝑎> 0 and 𝒚 is not constant.
3.2.2 Translation invariance
Analogous to 𝐿𝑊, the loss function 𝐿NR2 is translation invariant (see Proof D.5):
𝐿NR2(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = 𝐿NR2(𝒛, 𝒚), 𝑐∈ℝ
(3.12)
3.2.3 Scale invariance
The scale invariance of 𝐿NR2
𝐿NR2(𝑐𝒛, 𝑐𝒚) = 𝐿NR2(𝒛, 𝒚), 𝑐∈ℝ\{0}
(3.13)
follows directly from its definition in eq. (3.9).
12
3.2.4 Extremum estimation
To analyze 𝐿NR2 as an extremum estimator for a scalar parameter 𝜃, we minimize it for a
constant vector 𝜃𝟏, given non-constant observations 𝒚:
𝜃̂(𝒚) ∶= argmin
𝜃∈𝛩
𝐿NR2(𝜃𝟏, 𝒚)
(3.14)
As shown in Proof D.6, this minimization yields two distinct estimators:
𝜃̂(𝒚) = 𝜇(𝒚) ± 𝜎(𝒚)
(3.15)
Figure 2 illustrates 𝐿NR2(𝜃𝟏, 𝒚) as a function of 𝜃 for a realization 𝒚 of 𝒚, indicating the
points where it attains its minimum. The minimum value of 𝐿NR2(𝜃𝟏, 𝒚) is 1/2.
Figure 2. Illustration of 𝐿NR2(𝜃𝟏, 𝒚) function.
The non-uniqueness precludes consistency, as the estimator does not converge to a single
parameter value. To resolve this, we define the upper-bound estimator
𝜃̂+(𝒚) = max{argmin
𝜃∈𝛩
𝐿NR2(𝜃𝟏, 𝒚)} = 𝜇(𝒚) + 𝜎(𝒚)
(3.16)
and the lower-bound estimator
𝜃̂−(𝒚) = min{argmin
𝜃∈𝛩
𝐿NR2(𝜃𝟏, 𝒚)} = 𝜇(𝒚) −𝜎(𝒚)
(3.17)
Under the IID assumption, Proof D.4 establishes that 𝜃̂+(𝒚) and 𝜃̂−(𝒚) are consistent for
𝔼𝐹[𝑦] + 𝕍𝐹
1/2[𝑦] and 𝔼𝐹[𝑦] −𝕍𝐹
1/2[𝑦], respectively.
3.2.5 Estimation of a linear model
For the linear model 𝒛= 𝑎𝒙+ 𝑏𝟏, we estimate the true parameters 𝜽0 = (𝑎0, 𝑏0) ∈ℝ2,
13
using the 𝐿NR2 loss, via the extremum estimator 𝜽̂ = (𝑎̂, 𝑏̂)
𝜽̂(𝒙, 𝒚) ∶= argmin
𝜽∈ℝ2
𝐿NR2(𝑎𝒙+ 𝑏𝟏, 𝒚)
(3.18)
As shown in Proof D.7 (eqs. (D.123) and (D.124)), the solution is:
𝜽̂(𝒙, 𝒚) = (sign(𝜌(𝒙, 𝒚))(||𝒚𝑐||2/||𝒙𝑐||2), 𝜇(𝒚) −(sign(𝜌(𝒙, 𝒚))𝜇(𝒙)||𝒚𝑐||2)/
||𝒙𝑐||2)
(3.19)
where 𝒙𝑐 and 𝒚𝑐 are centered vectors defined in eq. (A.23), 𝜌(𝒙, 𝒚) is the Pearson
correlation defined in eq. (A.28) and sign(𝑥) is the scalar sign function, defined in eq.
(A.16). At this point, 𝐿NR2 attains its minimum value (see eq. (D.125)):
𝐿NR2(𝑎̂𝒙+ 𝑏̂𝟏, 𝒚) = (1 −|𝜌(𝒙, 𝒚)|)/2
(3.20)
When a correctly specified linear model is trained using the SE loss, its predictions
correspond to the conditional mean 𝔼𝐹𝑦|𝒙[𝑦|𝒙]. This occurs because the SE-derived
estimator is an M-estimator, and the SE loss is strictly consistent for the mean (see Section
2.1.2), i.e., it is uniquely minimized in expectation by the mean prediction [10]. While such
results are well-established for M-estimators, their extension to the broader class of
extremum estimators remains an open problem. For completeness, we also present the
standard least-squares regression parameter estimates:
𝜽̂(𝒙, 𝒚) = (𝜌(𝒙, 𝒚))(||𝒚𝑐||2/||𝒙𝑐||2), 𝜇(𝒚) −(𝜌(𝒙, 𝒚)𝜇(𝒙)||𝒚𝑐||2)/||𝒙𝑐||2) (3.21)
3.3 Generalization of the new index of agreement
Legates and McCabe Jr [8] proposed the following loss function as an improvement of the
original index of agreement
𝐿LMC(𝒛, 𝒚) ∶= ∑
|𝑧𝑖−𝑦𝑖|
𝑛
𝑖=1
/ ∑
(|𝑧𝑖−𝑓(𝒚)| + |𝑓(𝒚) −𝑦𝑖|)
𝑛
𝑖=1
(3.22)
where 𝑓(𝒚) is a benchmark method (e.g. climatology or persistence). 𝐿LMC is the equal-
weight special case of a weighted variant of the index of agreement introduced by [29]. It
admits the vectorized formulation:
𝐿LMC(𝒛, 𝒚) = ‖𝒛−𝒚‖1/‖(|𝒛−𝟏𝑓(𝒚)| + |𝟏𝑓(𝒚) −𝒚|)‖1
(3.23)
where ‖𝒙‖1 is the Manhattan norm, defined in eq. (A.8).
Krause et al. [9] later introduced a generalization of the form
𝐿KBB(𝒛, 𝒚) ∶= ∑
|𝑧𝑖−𝑦𝑖|𝑝
𝑛
𝑖=1
/ ∑
(|𝑧𝑖−𝜇(𝒚)| + |𝜇(𝒚) −𝑦𝑖|)𝑝
𝑛
𝑖=1
, 𝑝= 1,2, … (3.24)
This admits the vectorized formulation
14
𝐿KBB(𝒛, 𝒚) = ‖𝒛−𝒚‖𝑝
𝑝/‖(|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|)‖𝑝
𝑝, 𝑝= 1,2, …
(3.25)
where ‖𝒙‖𝑝 is the 𝐿𝑝-norm, defined in eq. (A.10), and the Chebyshev norm as the special
case for 𝑝= ∞, defined in eq. (A.11). 𝐿KBB is another equal-weight special case of the
weighted variant introduced by [29], for 𝑝= 1, 2, …. However, this definition shares the
theoretical limitations of 𝐿𝑊 (see Section 3.2). Furthermore, it employs the mean
climatology 𝟏𝜇(𝒚) as the benchmark, which aligns naturally with Euclidean norms but not
necessarily with other 𝐿𝑝-norms. This is less flexible than the definition of 𝐿LMC, which
allows for a tailored climatology 𝑓(𝒚). For instance, the Manhattan norm ‖𝒙‖1
corresponds more naturally to the median climatology 𝟏median(𝒚) [1].
Analogous to 𝐿NR2, we propose the loss function 𝐿NR𝑝 (defined for non-constant 𝒚 or
𝒛≠𝒚)
𝐿NR𝑝(𝒛, 𝒚) ∶= ‖𝒛−𝒚‖𝑝
𝑝/(‖𝒛−𝟏𝐿𝑝−mean(𝒚)‖𝑝+ ‖𝟏𝐿𝑝−mean(𝒚) −𝒚‖𝑝)𝑝
(3.26)
where 𝐿𝑝−mean(𝒚) is the 𝐿𝑝 mean of 𝒚 [30]. Examples of 𝐿𝑝−mean are the sample median,
which equals the 𝐿1 mean, and the sample mean, which equals the 𝐿2 mean. The proposed
𝐿NR𝑝 retains the advantages of 𝐿NR2 (see Section 3.2), including its intuitive geometric
foundation, theoretical consistency and interpretability, while adapting the benchmark
method to the chosen norm. For 𝑝= 1 and setting 𝑓(𝒚) = 𝐿𝑝−mean(𝒚), the 𝐿LMC loss is a
special case of the 𝐿NR𝑝 loss. Although a complete theoretical analysis of 𝐿NR𝑝 is left for
subsequent work, the following bound follows directly from its construction:
0 ≤𝐿NR𝑝(𝒛, 𝒚) ≤1
(3.27)
4.
Applications
This section presents applications designed to explore the implications of our theoretical
insights. We analyze both simulated data and real-world case studies using hydrologic
models. All computations were performed in R (version 4.5.1, [31]) within the RStudio
integrated development environment (version 2025.09.0+387). Supplementary
materials provide reproducible code and datasets for these analyses, and the specific R
packages used are listed in Appendix E.
4.1 Predicting a Gaussian random variable
In the first example, we simulate data from a Gaussian random variable 𝑦~𝑁(𝜇, 𝜎) (whose
probability density function (PDF) is defined in eq. (C.1)). A sample of 𝑛= 1 000
15
realizations was simulated with true parameters 𝜇= 0 and 𝜎= 1. Using the first 500
samples as the training set, we computed three climatology-based predictive models:
(i)
Model #1: 𝑧= 𝜇(𝒚1:500).
(ii)
Model #2: 𝑧= 𝜇(𝒚1:500) −𝜎(𝒚1:500).
(iii) Model #3: 𝑧= 𝜇(𝒚1:500) + 𝜎(𝒚1:500).
These three models were then used to forecast the remaining 500 samples (the test set),
as illustrated in Figure 3.
Figure 3. 500 observations from the test set of a random variable 𝑦~𝑁(0,1). The values
𝜇(𝒚), 𝜇(𝒚) −𝜎(𝒚), 𝜇(𝒚) + 𝜎(𝒚) were computed from the 500 obsevations in the training
set and are used as constant predictions in the test set.
We evaluate the performance of these three models on the test set in Table 1. To align
with the negative orientation of the other metrics (MSE, 𝐿NR2 and 𝐿𝑊), we use 1 −
NSE(𝒛, 𝒚), instead of the conventional NSE. This transformed metric is bounded below by
0.
16
Table 2. Performance of models on the test set. Predictions 𝒛 are the three climatologies
computed from the training set (models #1, #2 and #3), while observations 𝒚= 𝒚501:1000
are from the test set.
Metric
Model #1
Model #2
Model #3
MSE(𝒛, 𝒚)
1.020768
1.858657
2.146621
1 −NSE(𝒛, 𝒚) 1.005198
1.830307
2.113878
𝐿NR2(𝒛, 𝒚)
0.8745487
0.5010791 0.5003633
𝐿𝑊(𝒛, 𝒚)
0.8979654
0.5590595 0.5583535
Due to random sampling, the results on the test set deviate slightly from the theoretical
values expected from the training set (omitted for brevity). However, the sample is
sufficiently large to confirm the theoretical findings from Section 3. As noted in Section
2.1.3, using MSE or 1 −NSE does not alter the model rankings. Model #1 outperforms the
others with respect to MSE, as expected since the MSE is based on the 𝐿SE loss, strictly
consistent for the mean. Models #2 and #3 perform similarly under MSE, a result subject
to random variation but expected given the symmetry of both the 𝐿SE loss and the
Gaussian distribution.
In contrast, the rankings are completely reversed when comparing models using the
𝐿NR2 and 𝐿𝑊 losses. These results still align with the theory developed in Section 3. Models
#2 and #3 demonstrate similar performance and significantly outperform model #1
under both metrics. The value of 𝐿NR2 approaches its theoretical minimum of 0.50 (subject
to randomness), confirming the finding illustrated in Figure 2. Similarly, the values of 𝐿𝑊
are near their theoretical minimum ((𝜎(𝒚))/(𝜎(𝒚) + mad(𝒚))), as shown in Figure 1.
These results may appear paradoxical at first glance. Based on Figure 3 and prior
experience with the 𝐿𝑊 loss, one might expect model #1 to outperform the others on all
metrics. However, the superior performance of models #2 and #3 under the 𝐿NR2 and 𝐿𝑊
losses is explained in Sections 3.1.3 and 3.2.4. These sections establish that the
climatologies 𝜇(𝒚) ± 𝜎(𝒚) are the minimizers for these specific loss functions, when
fitting to a sample from a probability distribution.
4.2 Predicting data from a linear model with error
In the second example, we simulate data from a Gamma-distributed predictor random
variable 𝑥~Gamma(𝜃, 𝑎) (whose PDF is defined in eq. (C.2)) and a log-normal error
random variable 𝜀~Lognormal(𝜇, 𝜎) (whose PDF is defined in eq. (C.3)). We generate 𝑛=
4 000 realizations using the true parameters 𝜃= 1.8, 𝑎= 0.4, 𝜇= 0 and 𝜎= 2. We then
use these simulations to generate 4 000 realizations of 𝑦, from the linear model 𝑦= 𝑎0 +
17
𝑎1𝑥+ 𝜀, with a fixed intercept 𝑎0 = 2.1. We examine three cases for the slope coefficient
𝑎1: 𝑎1 = 0.6 (Figure 4a), 𝑎1 = 6.0 (Figure 4b) and 𝑎1 = 20.0 (Figure 4c). We intentionally
introduced highly asymmetric variables in this simulation to ensure the results are
sufficiently general, despite their empirical nature.
18
Figure 4. Predictions of linear models fitted using different loss functions. Black dots
represent 2 000 test observations of 𝑦= 2.1 + 𝑎1𝑥+ 𝜀, where 𝑥~Gamma(1.8, 0.4) and
𝜀~Lognormal(0, 2). Results are shown for three slope values: (a) 𝑎1 = 0.6, (b) 𝑎1 = 6.0,
(c) 𝑎1 = 20.0. Overlaid are predictions from linear models 𝑧= 𝑎̂0 + 𝑎̂1𝑥 fitted on a 2 000-
observation training set (not shown) using the SE loss (red lines), the 𝐿NR2 loss (blue
lines) and the 𝐿𝑊 loss (green lines).
We first detail the procedure for the dataset corresponding to 𝑎1 = 0.6. We split the
data into a training set of 2 000 observations and a test set of 2 000 observations. We then
19
fit linear models of the form z = 𝑎0 + 𝑎1𝑥 to the training set using three different loss
functions.
(i)
Model #1: Fit using the SE loss (least squares regression).
(ii)
Model #2: Fit using the 𝐿NR2 loss.
(iii) Model #3: Fit using the 𝐿𝑊 loss.
The estimated parameters for model #1 are 𝑎̂01, 𝑎̂11 (given by eq. (3.21)), for model #2,
they are 𝑎̂02, 𝑎̂12 (given by eq. (3.19)) and for model #3, they are 𝑎̂03, 𝑎̂13 (obtained by
numerical minimization, as explicit estimators have not been derived; see Section 3.1.4).
We then generate predictions for the test set using the equation 𝑧= 𝑎̂0 + 𝑎̂1𝑥. We repeat
this entire procedure for the other two true values of 𝑎1, while keeping the parameters of
the probability distributions and the intercept fixed. Figure 4 presents the predictions
from all models in the test set and Table 3 summarizes their numerical performance.
Table 3. Estimated parameters and test set performance for the three linear models from
Section 4.2. For each value of the slope coefficient 𝑎1, the table shows the parameters
estimated on the training set and the subsequent performance metrics computed on the
test set observations 𝒚= 𝒚2 001:4 000. Predictions 𝒛 are generated by each of the three
fitted models.
model #1
model #2
model #3
model #1
model #2
model #3
model #1
model #2
model #3
Parameter/
Metric
𝑎1 = 0.6
𝑎1 = 0.6
𝑎1 = 0.6
𝑎1 = 6.0
𝑎1 = 6.0
𝑎1 = 6.0
𝑎1 = 2.1
𝑎1 = 2.1
𝑎1 = 2.1
𝑎̂0
8.0655037 -5.10243
-16.74804 8.065504
-2.011659 -6.295647 8.065504
2.456986
1.750401
𝑎̂1
0.8935388 19.82837
12.47191
6.293539
20.783999 20.352048 20.293539 28.358309 28.347614
MSE(𝒛, 𝒚) 580.3332
996.0642
1002.823
580.3332
813.2273
818.57
580.3332
641.3484
641.8254
𝐿NR2(𝒛, 𝒚) 0.9096833 0.4514271 0.4692202 0.5399942 0.332705
0.335796
0.1736741 0.1425356 0.1426502
𝐿𝑊(𝒛, 𝒚)
0.9657116 0.7132495 0.7077607 0.6732447 0.468934
0.466139
0.1963293 0.1620158 0.1619878
𝑉̅mean(𝒛, 𝒚) -0.367136 0.4266737 -16.5665
-0.367136 0.371751
-4.226231 -0.367136 0.3898335 -0.3245264
The predictions in Figure 4 and the parameter estimates in Table 3 show that the
estimated models differ for lower values of 𝑎1, but converge toward identical solutions as
𝑎1 increases. For models #1 and #2, this behavior is explained by eqs. (3.19) and (3.21).
As 𝑎1 increases, the correlation 𝜌(𝒙, 𝒚) tends toward 1, causing sign(𝜌(𝒙, 𝒚) to become
equal to 𝜌(𝒙, 𝒚).
The model comparisons under each loss function are consistent with the theoretical
framework established by [1], [24], as discussed in Section 2.1.5. Specifically, each model
outperforms the others with respect to the loss function used for its training. Model #1
achieves the lowest MSE, model #2 the lowest empirical 𝐿NR2 loss and model #3 the
lowest empirical 𝐿𝑊 loss. As 𝑎1 increases, the models become increasingly similar, and
their performance metrics converge. The values of the 𝐿NR2 and 𝐿𝑊 losses are very similar,
20
which we attribute to the limited range [0,1] of these functions, as well as a potential
equivalence for predictor-dependent variables with specific properties.
The numerical experiment also confirms that model #1 accurately identifies the
conditional mean, as theoretically expected from Sections 2.1.4 and 2.1.5) (the value of
𝑉̅mean is nearly 0, consistently across all values of 𝑎1). Notably, model #2 also exhibits this
property, although this does not imply it can theoretically identify the conditional mean.
In contrast, model #3 does not appear to identify the mean, at least for low values of 𝑎1.
As 𝑎1 increases, its behavior becomes similar to that of #1 and #2.
4.3 Application to hydrologic modeling
To assess the practical relevance of our theoretical analysis in a hydrologic context, we
applied the methods using the Génie Rural (GR) GR4J daily lumped hydrologic model [32].
The model was implemented via the airGR R package ([33], [34]). We modified the
software to include the SE, 𝐿NR2 and 𝐿𝑊 loss functions as options for model extremum
estimation as outlined in see Section 2.2.
We applied the model to daily precipitation, potential evapotranspiration and
streamflow data from 10 catchments in France. These catchments were selected from the
airGRdatasets R package [35] based on the availability of complete records without
missing values. Since these results are broadly applicable to various hydrologic models
and catchments as well as machine learning regression models, we refrain from providing
exhaustive details on the models and data herein. The software implementation is
available as Supplementary Material.
The implementation was straightforward. We used the year 1999 as a warm-up period,
the period from 2000-2008 for model calibration, and 2009-2018 for model testing
(validation). For each catchment, we estimated three model instances, each using a
different loss function:
(i)
Model #1: Estimated using the SE loss (least squares regression).
(ii)
Model #2: Estimated using the 𝐿NR2 loss.
(iii) Model #3: Estimated using the 𝐿𝑊 loss.
Figure 5 presents examples of streamflow predictions from all three models for the
year 2018 at catchment A273011002 (other years are omitted for visual clarity). The
predictions from all models are highly similar. This visual similarity is confirmed by
21
Figure 6, which plots observations against predictions for each model, showing a close
alignment to the 1:1 line for all three.
Figure 5. Predictions from the three models detailed in Section 4.3 for the year 2018 at
catchment A273011002.
22
Figure 6. Observations plotted against predictions for the three models detailed in Section
4.3 for the year 2018 at catchment A273011002.
To quantify the differences between the models, we report their performance metrics
for the calibration (estimation) period in Table 4 and for the validation (testing) period in
Table 5. Presenting results for both periods is important due to the potential for statistical
dissimilarity between the calibration and validation datasets, a factor not present in the
controlled simulation setting of Section 4.2.
Table 4. Average losses (MSE, 𝐿NR2, 𝐿𝑊) for all three models for each catchment in the
calibration set. The best-performing model for each metric and catchment is highlighted
in bold.
Model
Model #1
Model #2
Model #3
Model #1
Model #2
Model #3
Model #1
Model #2
Model #3
Basin/loss
MSE
MSE
MSE
𝐿NR2
𝐿NR2
𝐿NR2
𝐿𝑊
𝐿𝑊
𝐿𝑊
A273011002 0.8004038 0.8213370 0.8206815 0.0412526 0.0402055 0.0402058 0.0428323 0.0417655 0.0417652
A605102001 0.6815371 0.7118592 0.7116614 0.0574751 0.0549629 0.0549630 0.0604838 0.0578356 0.0578355
B222001001 0.1452777 0.1473876 0.1472875 0.0240881 0.0237482 0.0237484 0.0246229 0.0242898 0.0242896
F439000101 0.0156214 0.0158345 0.0158534 0.0220948 0.0217949 0.0217955 0.0225355 0.0222006 0.0222001
H010002001 0.1543760 0.1565821 0.1565649 0.0280440 0.0276462 0.0276462 0.0287580 0.0283501 0.0283500
H120101001 0.0987467 0.0999243 0.0998982 0.0225813 0.0223170 0.0223171 0.0230437 0.0227771 0.0227770
H622101001 0.0719342 0.0725890 0.0725868 0.0151390 0.0150013 0.0150013 0.0153548 0.0152148 0.0152148
J171171001 0.0999581 0.1008159 0.1008084 0.0159707 0.0158368 0.0158368 0.0162061 0.0160709 0.0160709
J421191001 0.1908086 0.1919245 0.1919046 0.0109564 0.0108926 0.0108926 0.0110576 0.0109942 0.0109942
K134181001 0.1093134 0.1104352 0.1103770 0.0167198 0.0165522 0.0165523 0.0169642 0.0168023 0.0168021
The results in the validation period (Table 5) were less uniform. For instance, Model
#2 outperformed Model #1 with respect to MSE in 7 out of 10 catchments. This is
expected, given the close similarity of the parameter estimates across all models and the
potential for statistical dissimilarity between the calibration and validation periods.
23
Table 5. Average losses (MSE, 𝐿NR2, 𝐿𝑊) for all three models for each catchment in the
validation (test) set. The best-performing model for each metric and catchment is
highlighted in bold.
Model
Model #1
Model #2
Model #3
Model #1
Model #2
Model #3
Model #1
Model #2
Model #3
Basin/loss
MSE
MSE
MSE
𝐿NR2
𝐿NR2
𝐿NR2
𝐿𝑊
𝐿𝑊
𝐿𝑊
A273011002 0.7685125 0.7506903 0.7507642 0.0454044 0.0422782 0.0423143 0.0471956 0.0439308 0.0439685
A605102001 0.4491356 0.4854598 0.4855521 0.0443461 0.0437923 0.0438100 0.0461854 0.0456384 0.0456576
B222001001 0.1360638 0.1306809 0.1307524 0.0243684 0.0227379 0.0227663 0.0248823 0.0232137 0.0232427
F439000101 0.0315423 0.0320811 0.0320874 0.0343641 0.0339732 0.0339406 0.0355261 0.0350892 0.0350533
H010002001 0.1590732 0.1455477 0.1455789 0.0211302 0.0188218 0.0188279 0.0214761 0.0191156 0.0191217
H120101001 0.2413662 0.2227167 0.2229445 0.0348958 0.0316126 0.0316505 0.0356209 0.0322373 0.0322764
H622101001 0.1128062 0.1053143 0.1053167 0.0299121 0.0274271 0.0274283 0.0305397 0.0279809 0.0279821
J171171001 0.1318598 0.1262418 0.1262628 0.0182125 0.0171466 0.0171507 0.0184918 0.0174056 0.0174098
J421191001 0.2334413 0.2259492 0.2259939 0.0110676 0.0105920 0.0105951 0.0111783 0.0106949 0.0106981
K134181001 0.0996014 0.1009891 0.1009341 0.0136642 0.0135830 0.0135825 0.0138255 0.0137501 0.0137495
During the calibration period (Table 4), each model performed best with respect to the
specific loss function used for its training, as expected. However, the empirical loss values
differed only minimally between models. This suggests that for this catchment dataset,
the relationship between inputs and outputs is sufficiently strong that the different
estimators converge to similar parameter sets, analogous to the high-correlation scenario
illustrated in Figure 4c. Notably, in some cases the parameter estimates were so similar
that models #2 and #3 achieved nearly identical performance under both the 𝐿NR2 and
𝐿𝑊 losses.
Table 6 shows the empirical average identification function 𝑉̅mean for the mean,
computed on the validation set. As expected from theory (Section 2.1.4), model #1,
estimated with the SE loss, correctly identifies the mean (values near zero), indicating
adequate absolute performance for mean prediction. Although not proven theoretically,
Models #2 and #3 also produced values of 𝑉̅mean near zero. This is a consequence of their
parameter estimates being very similar to those of model #1, as noted in the discussion
of Table 4.
Table 6. Empirical average identification function 𝑉̅mean for all three models for each
catchment in the validation set. The value closest to zero (indicating best identification of
the mean) for each catchment is highlighted in bold.
Model
Basin
Model #1
Model #2
Model #3
A273011002 -0.1749492 -0.1325392 -0.1332546
A605102001 -0.0438793 -0.0091700 -0.0101047
B222001001 -0.0272470 -0.0061254 -0.0065730
F439000101 -0.0570816 -0.0526462 -0.0523539
H010002001 -0.0591676 -0.0388627 -0.0389624
H120101001 -0.1592403 -0.1394463 -0.1397120
H622101001 -0.1033602 -0.0914236 -0.0913975
J171171001 -0.0452491 -0.0329591 -0.0330311
J421191001 -0.1414669 -0.1233617 -0.1234942
K134181001 0.0775674 0.0939171
0.0936123
24
5.
Discussion
At first glance, it may appear counterintuitive that 𝐿NR2, when used for parameter
estimation of a distribution, yields two distinct estimators, that differ from the mean. This
contrasts with the empirical finding that, when used to fit a linear model, its predictions
appear to equal the marginal mean of the dependent variable 𝒚. This apparent
contradiction may be explained by the fact that 𝐿NR2 is defined only when 𝒚 is non-
constant or 𝒛≠𝒚. Consequently, the expectation 𝔼𝐹[𝐿NR2(𝑧, 𝒚)] is defined only for
distributions where 𝒚 is non-constant. This restriction may introduce irregularities if one
attempts its minimization as in eq. (2.4). This restriction may be circumvented when
fitting a linear model, which allows for non-constant predictions 𝒛 (for which 𝐿NR2 is
defined irrespectively of the distribution of 𝒚). Similarly, 𝐿𝑊 does not identify a specific
functional when fitting a probability distribution and its predictions do not necessarily
have a mean equal to the marginal mean of 𝒚.
In environmental sciences, it is sometimes assumed that predictions yielding a loss
function value of 0 will perfectly predict the environmental process of interest, implying
that a lower average loss invariably corresponds to a better prediction of the process
itself. This concept must be understood precisely. Under the framework of expectation
minimization of the loss function (eq. (2.4)), models predict specific functionals of the
process rather than the process itself. For example, a linear model fitted with a quantile
loss will predict conditional quantiles, not the process, even though the quantile loss
equals 0 for a perfect prediction 𝒛= 𝒚 [36]. Applying this to our study, consider the
analysis in Section 4.1. The mean climatology appears to better predict the process itself,
yet it achieves higher (worse) 𝐿NR2 and 𝐿𝑊 losses compared to predictions of the form
𝜇(𝒚) ± 𝜎(𝒚). All losses equal 0 only for a perfect prediction 𝒛= 𝒚. This result is less
paradoxical when considering that models predict functionals of the predictive
distribution rather than the process itself. That principle already applied in hydrologic
science and engineering to predict quantiles ([37], [38], [39]) and expectiles (an extension
of the mean functional [40]) using hydrologic models [41].
In environmental sciences and engineering, it is common practice to assess predictions
using multiple metrics, and debates continue over which metrics are superior [8]. This
perspective has driven the introduction of numerous new metrics. This practice should
25
be contextualized within statistical theory, which advocates for the a priori specification
of the loss function used for model assessment ([1], [24]), as model rankings are critically
dependent on this choice. Our work underscores the importance of this approach, while
establishing the theoretical properties of the 𝐿NR2 and 𝐿𝑊 loss functions. In this context,
we do not argue for the universal superiority of one loss function. Each loss function has
distinct theoretical merits and, more importantly, each is intrinsically linked to the
specific functional we aim to predict. The SE loss is simpler to use and highly
interpretable, as it elicits the mean. The 𝐿NR2 loss yields specific estimates when used as
an extremum estimator on a sample from a probability distribution, though it lacks
consistency due to non-uniqueness. However, when applied to a linear model, it provides
unique parameter estimates, and our empirical evaluation suggests that the expectation
of its predictions equals the mean of the marginal distribution of the dependent variable.
A significant advantage of 𝐿NR2 is its translation and scale invariance, coupled with its
bounded range of [0,1]. In environmental science and engineering, such boundedness is
often perceived to enhance the interpretability of prediction assessments. 𝐿𝑊 also
possesses translation and scale invariance and is bounded in [0,1]. However, it likely lacks
the property of predicting the marginal mean of the dependent variable, when used to fit
a linear model and does not yield explicit parameter estimates.
It would be interesting to examine the consistency of 𝐿NR2 and 𝐿𝑊 for multidimensional
functionals (our analysis was restricted to one-dimensional functionals), a topic discussed
by [1]. For instance, the general form of a consistent loss function for the component-wise
expectation is a Bregman loss function ([12], [16], [42],)
𝐿(𝒛, 𝒚) ∶= 𝜑(𝒚) −𝜑(𝒛) −〈∇𝜑(𝒛), 𝒚−𝒛〉
(5.1)
where 𝜑: ℝ𝑑→ℝ is convex with gradient ∇𝜑: ℝ𝑑→ℝ𝑑. The SE loss is a special case of eq.
(5.1). In this regard, a challenge for both 𝐿NR2 and 𝐿𝑊 is that they must be examined within
the class of distributions for which 𝒚 is not constant.
Preliminary theoretical and empirical work by the authors on 𝐿NR𝑝 has not identified
specific properties related to existing functionals. Determining what functional, if any,
models fitted with 𝐿NR𝑝 identify remains an open problem. Such functionals, if they exist,
are termed hydrograph functionals by [43] due to their significance in hydrologic science
and engineering.
26
6.
Conclusions
In this study, we examined the theoretical properties of the index of agreement loss
function, 𝐿𝑊 (the negatively oriented index of agreement). Our analysis established that
𝐿𝑊is translation and scale invariant, bounded within [0,1], and, when employed as an
extremum estimator for a sample from a probability distribution, yields the parameter
estimates 𝔼𝐹[𝑦] ± 𝕍𝐹
1/2[𝑦].
We proposed 𝐿NR2 as a theoretically consistent improvement over the original index of
agreement. Unlike 𝐿𝑊, whose denominator relies on component-wise vector operations,
𝐿NR2 is constructed using Euclidean distances in both its numerator and denominator.
This new formulation retains the appealing properties of translation and scale invariance
and a [0,1] bound and it produces the same 𝔼𝐹[𝑦] ± 𝕍𝐹
1/2[𝑦] estimates when fitting a
distribution. A significant strength of 𝐿NR2 is that it provides closed-form solutions when
used as an extremum estimator to fit a linear model.
Empirical analysis demonstrated that predictions of linear models fitted with 𝐿NR2
have mean equal to the mean of the dependent variable, a property shared with the
squared error loss function (though the latter identifies the conditional mean). This
represents a notable benefit over 𝐿𝑊, for which we found no evidence of a similar
identification property.
When applied to calibrate hydrologic models, both loss functions resulted in parameter
sets that produced remarkably similar performance across all empirical average loss
metrics, including MSE. This behavior is analogous to the high-correlation scenario in
linear regression, where the models converge as the correlation 𝜌(𝒙, 𝒚) approaches 1.
Finally, while variants of the 𝐿𝑊 loss based on norms other than the Euclidean norm
have been proposed, we introduced a corresponding generalization, 𝐿NR𝑝, based on the
principles underlying 𝐿NR2. The theoretical properties of this generalized function remain
an open area of research. Preliminary results did not indicate that models trained with
the 𝐿NR1 loss, for instance, can identify the median or mean functional.
Appendix A
Vector notation
This appendix defines the vector operations, norms, and statistical functions used
throughout the manuscript. All vectors are column vectors in ℝ𝑛.
27
Basic vector definitions
Vector notation:
𝒙= (𝑥1, … , 𝑥𝑛)T
(A.1)
where the superscript
T denotes the transpose.
Zero vector (all elements zero):
𝟎= (0, … ,0)T
(A.2)
All-ones vector (all elements unity):
𝟏= (1, … ,1)T
(A.3)
Element-wise operations
Absolute value vector:
|𝒙| ∶= (|𝑥1|, … , |𝑥𝑛|)T
(A.4)
Element-wise comparison (holds for all 𝑖∈{1, … , 𝑛}):
𝒙> 𝒚⇔𝑥𝑖> 𝑦𝑖 (similarly for ≥, =, ≤, <)
(A.5)
Element-wise inequality:
𝒙≠𝒚⇔∃𝑖∈{1, … , 𝑛}: 𝑥𝑖≠𝑦𝑖
(A.6)
Inner products and norms
Euclidean inner product:
〈𝒙, 𝒚〉∶= 𝒙T𝒚= ∑
𝑥𝑖𝑦𝑖
𝑛
𝑖=1
= 〈𝒚, 𝒙〉
(A.7)
𝐿𝑝-norms (𝑝≥1):
‖𝒙‖1 = ∑
|𝑥𝑖|
𝑛
𝑖=1
(Manhattan norm)
(A.8)
‖𝒙‖2 = (∑
𝑥𝑖
2
𝑛
𝑖=1
)1/2 (Euclidean norm)
(A.9)
‖𝒙‖𝑝∶= (∑
|𝑥𝑖|𝑝
𝑛
𝑖=1
)1/𝑝 (𝐿𝑝-norm)
(A.10)
‖𝒙‖∞= max{|𝑥1|, … , |𝑥𝑛|} (Chebyshev norm)
(A.11)
Squared Euclidean norm:
‖𝒙‖2
2 = 〈𝒙, 𝒙〉= 𝒙T𝒙= ∑
𝑥𝑖
2
𝑛
𝑖=1
(A.12)
Scaling invariance:
‖𝑎𝒙‖2
2 = 𝑎2‖𝒙‖2
2
(A.13)
Norm equivalences:
‖|𝒙|‖2 = ‖𝒙‖2
(A.14)
28
〈|𝒙|, |𝒙|〉= 〈𝒙, 𝒙〉= ‖𝒙‖2
2
(A.15)
Sign and indicator functions
Scalar sign:
sign(𝑥) ∶= { 1, 𝑥> 0
−1, 𝑥< 0
(A.16)
Vector extension:
sign(𝒙) ∶= (sign(𝑥1), … , sign(𝑥𝑛))T
(A.17)
Indicator function:
𝟙𝐴{𝑥} ∶= {
1, 𝑥∈𝐴
0, otherwise
(A.18)
Statistical functions for vectors
Sample mean:
𝜇(𝒙) ∶= (1/𝑛)𝟏T𝒙= (1/𝑛) ∑
𝑥𝑖
𝑛
𝑖=1
= (1/𝑛)〈𝒙, 𝟏〉
(A.19)
Mean absolute deviation:
mad(𝒙) ∶= 𝜇(|𝟏𝜇(𝒙) −𝒙|)
(A.20)
Sample standard deviation:
𝜎(𝒙) ∶= √(‖𝒙‖2
2 −𝑛𝜇2(𝒙))/√𝑛
(A.21)
Sample variance:
𝜎2(𝒙) = (‖𝒙‖2
2 −𝑛𝜇2(𝒙))/𝑛
(A.22)
Centered vectors and key identities:
Centered vector (mean-zero):
𝒙𝑐∶= 𝒙−𝟏𝜇(𝒙)
(A.23)
Pythagorean decomposition:
‖𝒙‖2
2 = ‖𝟏𝜇(𝒙)‖2
2 + ‖𝒙c‖2
2
(A.24)
Sum of squared deviations:
‖𝒙c‖2
2 = ∑
(𝑥𝑖−𝜇(𝒙))2
𝑛
𝑖=1
(A.25)
Norm expansion:
‖𝒙+ 𝒚‖2
2 = ‖𝒙‖2
2 + ‖𝒚‖2
2 + 2〈𝒙, 𝒚〉
(A.26)
Orthogonality to unity:
〈𝒙c, 𝟏〉= 0
(A.27)
29
Pearson correlation:
𝜌(𝒙, 𝒚) ∶= 〈𝒙c, 𝒚c〉/(‖𝒙c‖2‖𝒚c‖2)
(A.28)
Inequalities:
Cauchy-Schwarz inequality ([44], p.33):
|〈𝒙, 𝒚〉| ≤‖𝒙‖2‖𝒚‖2
(A.29)
with equality ([44], p.34):
|〈𝒙, 𝒚〉| = ‖𝒙‖2‖𝒚‖2, iff 𝒙= 𝟎, or 𝒚= 𝟎, or 𝒙= 𝑎𝒚
(A.30)
Triangle inequality ([44], p.35):
‖𝒙+ 𝒚‖2 ≤‖𝒙‖2 + ‖𝒚‖2
(A.31)
with equality
‖𝒙+ 𝒚‖2 = ‖𝒙‖2 + ‖𝒚‖2, iff 𝒙= 𝟎, or 𝒚= 𝟎, or 𝒙= 𝑎𝒚, 𝑎> 0
(A.32)
Norm bounds:
‖𝒙‖2 ≤‖𝒙‖1 ≤√𝑛‖𝒙‖2
(A.33)
Variance identity
‖𝟏𝜇(𝒙) −𝒙‖2
2 = 〈𝟏𝜇(𝒙) −𝒙, 𝟏𝜇(𝒙) −𝒙〉⇒
(A.34)
‖𝟏𝜇(𝒙) −𝒙‖2
2 = ‖𝟏‖2
2 𝜇2(𝒙) + ‖𝒙‖2
2 −2𝜇(𝒙)〈𝟏, 𝒙〉⇒
(A.35)
‖𝟏𝜇(𝒙) −𝒙‖2
2 = 𝑛𝜇2(𝒙) + ‖𝒙‖2
2 −2𝑛𝜇2(𝒙) ⇒
(A.36)
‖𝟏𝜇(𝒙) −𝒙‖2
2 = ‖𝒙‖2
2 −𝑛𝜇2(𝒙) ⇒
(A.37)
‖𝟏𝜇(𝒙) −𝒙‖2
2 = 𝑛𝜎2(𝒙)
(A.38)
Appendix B
Loss and identification functions
This appendix defines the established loss and identification functions referenced
throughout this paper. Let 𝒛= (𝑧1, … , 𝑧𝑛)T be a vector of predictions and 𝒚= (𝑦1, … , 𝑦𝑛)T
be a vector of corresponding observed values.
The absolute error loss function is defined as:
𝐿AE(𝑧, 𝑦) ∶= |𝑧−𝑦|
(B.1)
The squared error loss function is defined as:
𝐿SE(𝑧, 𝑦) ∶= (𝑧−𝑦)2
(B.2)
The mean absolute error (MAE) is defined as:
MAE(𝒛, 𝒚) ∶= (1/𝑛) ∑
𝐿AE(𝑧𝑖, 𝑦𝑖)
𝑛
𝑖=1
(B.3)
30
The mean squared error (MSE) is defined as:
MSE(𝒛, 𝒚) ∶= (1/𝑛) ∑
𝐿SE(𝑧𝑖, 𝑦𝑖)
𝑛
𝑖=1
(B.4)
The Nash-Sutcliffe efficiency (NSE) [17] is defined as:
NSE(𝒛, 𝒚) ∶= 1 −MSE(𝒛, 𝒚)/MSE(𝟏𝜇(𝒚), 𝒚)
(B.5)
A strict identification function for the median [1] is defined as:
𝑉median(𝑧, 𝑦) ∶= 𝟙[0,∞){𝑧−𝑦} −1/2
(B.6)
where 𝟙𝐴{𝑥} is the indicator function defined in eq. (A.18).
A strict identification function for the mean [1] is defined as:
𝑉mean(𝑧, 𝑦) ∶= 𝑧−𝑦
(B.7)
The empirical average identification function 𝑉̅median is defined as:
𝑉̅median(𝒛, 𝒚) = (1/𝑛) ∑
𝑉median(𝑧𝑖, 𝑦𝑖)
𝑛
𝑖=1
(B.8)
The empirical average identification function 𝑉̅mean is defined as:
𝑉̅mean(𝒛, 𝒚) = (1/𝑛) ∑
𝑉mean(𝑧𝑖, 𝑦𝑖)
𝑛
𝑖=1
(B.9)
Appendix C
Probability distributions
This appendix defines the probability density functions (PDFs) for the probability
distributions referenced in the main text.
The PDF of a Gaussian (normal) random variable 𝑦~𝑁(𝜇, 𝜎) is defined as:
𝑓𝑁(𝑦; 𝜇, 𝜎) ∶=
1
𝜎√2𝜋exp (−
(𝑦 − 𝜇)2
2𝜎2 ) , 𝑦∈ℝ, 𝜇∈ℝ, 𝜎> 0
(C.1)
The PDF of a Gamma distributed random variable 𝑦~Gamma(𝜃, 𝑎) is defined as:
𝑓Gamma(𝑦; 𝜃, 𝑎) ∶=
1
𝜃𝑎Γ(𝑎) 𝑦𝑎−1exp(−𝑦/𝜃), 𝑦≥0, 𝜃> 0, 𝑎> 0
(C.2)
The PDF of a log-normally distributed random variable 𝑦~Lognormal(𝜇, 𝜎) is defined as:
𝑓Lognormal(𝑦; 𝜇, 𝜎) ∶=
1
𝑦𝜎√2𝜋exp (−
(log (𝑦) − 𝜇)2
2𝜎2
) , 𝑦> 0, 𝜇∈ℝ, 𝜎> 0
(C.3)
Appendix D
Proofs
Proof D.1 (Bound of 𝐿𝑊, adapted from [45]): The bound 0 ≤𝐿𝑊(𝒛, 𝒚) ≤1 is established
as follows:
(i)
Lower bound
The lower bound follows immediately, since both numerator and denominator are
31
squared Euclidean norms, which are non-negative.
(ii)
Upper bound
Starting from the element-wise triangle inequality for vectors (eq. (A.5)):
|𝒛−𝒚| = |𝒛−𝟏𝜇(𝒚) + 𝟏𝜇(𝒚) −𝒚| ≤|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|,
(D.1)
where the inequality holds element-wise. Applying the Euclidian norm (which preserves
order for non-negative vectors), squaring and by leveraging eq. (A.14) yields
‖𝒛−𝒚‖2
2 ≤‖(|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|)‖2
2
(D.2)
Thus 𝐿𝑊(𝒛, 𝒚) ≤1.∎
Proof D.2 (Translation invariance of 𝐿𝑊): Substituting translation vectors into eq. (2.13):
𝐿𝑊(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = ‖(𝒛+ 𝑐𝟏) −(𝒚+ 𝑐𝟏)‖2
2/‖(|(𝒛+ 𝑐𝟏) −𝟏𝜇(𝒚+ 𝑐𝟏)| + |𝟏𝜇(𝒚+
𝑐𝟏) −(𝒚+ 𝑐𝟏)|)‖2
2 ⇒
(D.3)
𝐿𝑊(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = ‖𝒛−𝒚‖2
2/‖(|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|)‖2
2 ⇒
(D.4)
𝐿𝑊(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = 𝐿𝑊(𝒛, 𝒚), 𝑐∈ℝ
(D.5)
Thus 𝐿𝑊 is translation invariant.∎
Proof D.3 (𝐿𝑊 for extremum estimation): We show that argmin
𝜃∈𝛩
𝐿𝑊(𝜃𝟏, 𝒚) = 𝜇(𝒚) ± 𝜎(𝒚).
For 𝒛= 𝜃𝟏 and 𝒚 not constant, the nominator of 𝐿𝑊 is:
‖𝒛−𝒚‖2
2 = ‖𝜃𝟏−𝒚‖2
2
(D.6)
By expanding the norm (eq. (A.26)) it follows that:
‖𝒛−𝒚‖2
2 = ‖𝜃𝟏‖2
2 + ‖𝒚‖2
2 −2〈𝜃𝟏, 𝒚〉⇒
(D.7)
‖𝒛−𝒚‖2
2 = 𝑛𝜃2 + ‖𝒚‖2
2 −2𝑛𝜃𝜇(𝒚) ⇒
(D.8)
‖𝒛−𝒚‖2
2 = 𝑛(𝜃−𝜇(𝒚))2 + ‖𝒚‖2
2 −𝑛𝜇2(𝒚)
(D.9)
By substituting with the sample variance (eq. (A.22)) it follows that:
‖𝒛−𝒚‖2
2 = 𝑛(𝜃−𝜇(𝒚))2 + 𝑛𝜎2(𝒚)
(D.10)
The denominator expands to:
‖(|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|)‖2
2 = ‖|𝒛−𝟏𝜇(𝒚)|‖2
2 + ‖|𝟏𝜇(𝒚) −𝒚|‖2
2 + 2〈|𝒛−
𝟏𝜇(𝒚)|, |𝟏𝜇(𝒚) −𝒚|〉⇒
(D.11)
‖(|𝒛−𝟏𝜇(𝒚)| + |𝟏𝜇(𝒚) −𝒚|)‖2
2 = ‖|𝜃𝟏−𝟏𝜇(𝒚)|‖2
2 + ‖|𝟏𝜇(𝒚) −𝒚|‖2
2 + 2〈|𝜃𝟏−
𝟏𝜇(𝒚)|, |𝟏𝜇(𝒚) −𝒚|〉
(D.12)
The first term in eq. (D.12) is
32
‖|𝜃𝟏−𝟏𝜇(𝒚)|‖2
2 = (𝜃−𝜇(𝒚))2‖𝟏‖2
2 ⇒
(D.13)
‖|𝜃𝟏−𝟏𝜇(𝒚)|‖2
2 = 𝑛(𝜃−𝜇(𝒚))2
(D.14)
From eq. (A.38), the second term in eq. (D.12) is
‖|𝟏𝜇(𝒚) −𝒚|‖2
2 = 𝑛𝜎2(𝒚)
(D.15)
The last term in eq. (D.12) is
〈|𝜃𝟏−𝟏𝜇(𝒚)|, |𝟏𝜇(𝒚) −𝒚|〉= |𝜃−𝜇(𝒚)|〈|𝟏|, |𝟏𝜇(𝒚) −𝒚|〉⇒
(D.16)
〈|𝜃𝟏−𝟏𝜇(𝒚)|, |𝟏𝜇(𝒚) −𝒚|〉= |𝜃−𝜇(𝒚)|〈𝟏, |𝟏𝜇(𝒚) −𝒚|〉⇒
(D.17)
〈|𝜃𝟏−𝟏𝜇(𝒚)|, |𝟏𝜇(𝒚) −𝒚|〉= |𝜃−𝜇(𝒚)|‖𝟏𝜇(𝒚) −𝒚‖1 ⇒
(D.18)
〈|𝜃𝟏−𝟏𝜇(𝒚)|, |𝟏𝜇(𝒚) −𝒚|〉= 𝑛|𝜃−𝜇(𝒚)|mad(𝒚)
(D.19)
where mad(𝒚) is the sample mean absolute deviation of 𝒚 defined in (A.20). Then
𝐿𝑊(𝜃𝟏, 𝒚) equals:
𝐿𝑊(𝜃𝟏, 𝒚) = (𝑛(𝜃−𝜇(𝒚))2 + 𝑛𝜎2(𝒚))/(𝑛(𝜃−𝜇(𝒚))2 + 𝑛𝜎2(𝒚) + 2𝑛|𝜃−
𝜇(𝒚)|mad(𝒚)) ⇒
(D.20)
𝐿𝑊(𝜃𝟏, 𝒚) = ((𝜃−𝜇(𝒚))2 + 𝜎2(𝒚))/((𝜃−𝜇(𝒚))2 + 𝜎2(𝒚) + 2|𝜃−
𝜇(𝒚)|mad(𝒚))
(D.21)
Let
𝑢1 = 𝜃−𝜇(𝒚)
(D.22)
𝑢2 = 𝜎(𝒚)
(D.23)
𝑢3 = mad(𝒚)
(D.24)
After substituting eqs. (D.22), (D.23) and (D.24) in (D.21) we have:
𝐿𝑊(𝜃𝟏, 𝒚) = (𝑢1
2 + 𝑢2
2)/(𝑢1
2 + 𝑢2
2 + 2|𝑢1|𝑢3)
(D.25)
The partial derivative
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
is defined for 𝜃≠𝜇(𝒚). For 𝜃> 𝜇(𝒚):
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= (2𝑢1(𝑢1
2 + 𝑢2
2 + 2𝑢1𝑢3) −2(𝑢1 + 𝑢3)(𝑢1
2 + 𝑢2
2))/(𝑢1
2 + 𝑢2
2 +
2𝑢1𝑢3)2 ⇒
(D.26)
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= 2𝑢3(𝑢1
2 −𝑢2
2)/(𝑢1
2 + 𝑢2
2 + 2𝑢1𝑢3)2
(D.27)
For 𝜃< 𝜇(𝒚):
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= (2𝑢1(𝑢1
2 + 𝑢2
2 −2𝑢1𝑢3) −2(𝑢1 −𝑢3)(𝑢1
2 + 𝑢2
2))/(𝑢1
2 + 𝑢2
2 −
2𝑢1𝑢3)2 ⇒
(D.28)
33
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= −2𝑢3(𝑢1
2 −𝑢2
2)/(𝑢1
2 + 𝑢2
2 −2𝑢1𝑢3)2
(D.29)
Combining the two partial derivatives, we have
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= (2𝑢3(𝑢1
2 −𝑢2
2)sign(𝑢1))/(𝑢1
2 + 𝑢2
2 + 2|𝑢1|𝑢3)2
(D.30)
which is a function of 𝜃 only through 𝑢1. The sign function is defined in eq. (A.16). The
partial derivative
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
is not defined at 𝜃= 𝜇(𝒚) because (see Figure 1)
lim
𝜃→𝜇(𝒚)+
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= −2𝑢3/𝑢2
2
(D.31)
and
lim
𝜃→𝜇(𝒚)−
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
= 2𝑢3/𝑢2
2
(D.32)
The second order partial derivative
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
is defined for 𝜃≠𝜇(𝒚).
We set
𝑢4 = 𝑢1
2 + 𝑢2
2 + 2𝑢1𝑢3
(D.33)
𝑢5 = 𝑢1
2 + 𝑢2
2 −2𝑢1𝑢3
(D.34)
For 𝜃> 𝜇(𝒚):
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢1𝑢3𝑢4
2 −8𝑢3(𝑢1
2 −𝑢2
2)𝑢4(𝑢1 + 𝑢3))/𝑢4
4 ⇒
(D.35)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(𝑢1𝑢4 −2(𝑢1
2 −𝑢2
2)(𝑢1 + 𝑢3)))/𝑢4
3 ⇒
(D.36)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(𝑢1
3 + 𝑢1𝑢2
2 + 2𝑢1
2𝑢3 −2(𝑢1
3 + 𝑢1
2𝑢3 −𝑢1𝑢2
2 −𝑢3𝑢2
2)))/
𝑢4
3 ⇒
(D.37)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(−𝑢1
3 + 3𝑢1𝑢2
2 + 2𝑢3𝑢2
2))/𝑢4
3 ⇒
(D.38)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(2𝑢3𝑢2
2 + 𝑢1(3𝑢2
2 −𝑢1
2)))/𝑢4
3
(D.39)
For 𝜃< 𝜇(𝒚):
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (−4𝑢1𝑢3𝑢5
2 + 8𝑢3(𝑢1
2 −𝑢2
2)𝑢5(𝑢1 −𝑢3))/𝑢5
4 ⇒
(D.40)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(−𝑢1𝑢5 + 2(𝑢1
2 −𝑢2
2)(𝑢1 −𝑢3)))/𝑢5
3 ⇒
(D.41)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(−𝑢1
3 −𝑢1𝑢2
2 + 2𝑢1
2𝑢3 + 2(𝑢1
3 −𝑢1
2𝑢3 −𝑢1𝑢2
2 + 𝑢3𝑢2
2)))/
𝑢5
3 ⇒
(D.42)
34
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(𝑢1
3 −3𝑢1𝑢2
2 + 2𝑢3𝑢2
2))/𝑢5
3 ⇒
(D.43)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(2𝑢3𝑢2
2 −𝑢1(3𝑢2
2 −𝑢1
2)))/𝑢5
3
(D.44)
Combining the two partial derivatives, we have
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝑢3(2𝑢3𝑢2
2 + 𝑢1(3𝑢2
2 −𝑢1
2)sign(𝑢1)))/(𝑢1
2 + 𝑢2
2 + 2|𝑢1|𝑢3)3 (D.45)
which is a function of 𝜃 only through 𝑢1. The partial derivative
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
is not defined at
𝜃= 𝜇(𝒚) because
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
is not defined at this point.
The function 𝐿𝑊(𝜃𝟏, 𝒚) has two global minima at
𝜃̂ = 𝜇(𝒚) ± 𝜎(𝒚)
(D.46)
because
(i)
The function 𝐿𝑊(𝜃𝟏, 𝒚) is a continuous function of 𝜃 and
𝜕𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃
𝜃=𝜃̂ = 0
(D.47)
(ii)
It has two local minima at 𝜃̂, because
𝐿𝑊(𝜃̂𝟏, 𝒚) = (𝜎(𝒚))/(𝜎(𝒚) + mad(𝒚)) < 1
(D.48)
𝜕2𝐿𝑊(𝜃𝟏,𝒚)
𝜕𝜃2
𝜃=𝜃̂ = mad(𝒚)/(𝜎(𝒚)(𝜎(𝒚) + mad(𝒚))2) > 0
(D.49)
(iii) Its value at 𝜃= 𝜇(𝒚) equals
𝐿𝑊(𝜇(𝒚)𝟏, 𝒚) = 1
(D.50)
(iv) The limits equal
lim
𝜃→∞𝐿𝑊(𝜃𝟏, 𝒚) = lim
𝜃→−∞𝐿𝑊(𝜃𝟏, 𝒚) = 1
(D.51)
(v)
Therefore, the two local minima of 𝐿𝑊(𝜃𝟏, 𝒚) are global minima for 𝜃> 𝜇(𝒚) and
𝜃< 𝜇(𝒚) respectively. Since the value of the function 𝐿𝑊(𝜃𝟏, 𝒚) at the two local minima
is equal, then those local minima should be also global minima in the set of real numbers.∎
Proof D.4 (Consistency of estimators based on 𝐿𝑊): We prove that 𝜃̂+(𝒚) = 𝜇(𝒚) + 𝜎(𝒚)
is a consistent estimator of 𝔼𝐹[𝑦] + 𝕍𝐹
1/2[𝑦]. First, we note that 𝜇(𝒚) converges in
probability to 𝔼𝐹[𝑦] for IID 𝑦1, … , 𝑦𝑛 with CDF 𝐹 ([46], page 55). Secondly, we note that
𝜎2(𝒚) converges in probability to 𝕍𝐹[𝑦] for IID 𝑦1, … , 𝑦𝑛 ([46], page 55). By the continuous
mapping theorem, 𝜎(𝒚) converges in probability to 𝕍𝐹
1/2[𝑦]. By Slutsky’s theorem we have
35
that 𝜃̂+(𝒚)
𝑃𝜃
→ 𝔼𝐹[𝑦] + 𝕍𝐹
1/2[𝑦], therefore 𝜃̂+(𝒚) is a consistent estimator. The proof for
𝜃̂−(𝒚) is analogous.∎
Proof D.5 (Translation invariance of 𝐿NR2): Substituting translation vectors into (2.13):
𝐿NR2(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = ‖(𝒛+ 𝑐𝟏) −(𝒚+ 𝑐𝟏)‖2
2/(‖𝒛+ 𝑐𝟏−𝟏𝜇(𝒚+ 𝑐𝟏)‖2 + ‖𝟏𝜇(𝒚+
𝑐𝟏) −𝒚−𝑐𝟏‖2)2 ⇒
(D.52)
𝐿NR2(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = ‖𝒛−𝒚‖2
2/(‖𝒛−𝟏𝜇(𝒚)‖2 + ‖𝟏𝜇(𝒚) −𝒚‖2)2 ⇒
(D.53)
𝐿NR2(𝒛+ 𝑐𝟏, 𝒚+ 𝑐𝟏) = 𝐿NR2(𝒛, 𝒚), 𝑐∈ℝ
(D.54)
Thus 𝐿NR2 is translation invariant.∎
Proof D.6 (𝐿NR2 for extremum estimation): We show that argmin
𝜃∈𝛩
𝐿NR2(𝜃𝟏, 𝒚) = 𝜇(𝒚) ±
𝜎(𝒚). For 𝒛= 𝜃𝟏 and 𝒚 not constant, the nominator of 𝐿NR2 is (see eq. (D.10))
‖𝒛−𝒚‖2
2 = 𝑛(𝜃−𝜇(𝒚))2 + 𝑛𝜎2(𝒚)
(D.55)
The first part of the denominator is:
‖𝒛−𝟏𝜇(𝒚)‖2
2 = ‖𝜃𝟏−𝟏𝜇(𝒚)‖2
2 ⇒
(D.56)
‖𝒛−𝟏𝜇(𝒚)‖2
2 = 𝑛(𝜃−𝜇(𝒚))2
(D.57)
Therefore 𝐿NR2(𝜃𝟏, 𝒚) equals
𝐿NR2(𝜃𝟏, 𝒚) = (𝑛(𝜃−𝜇(𝒚))2 + 𝑛𝜎2(𝒚))/(√𝑛|𝜃−𝜇(𝒚)| + √𝑛𝜎(𝒚))2 ⇒ (D.58)
𝐿NR2(𝜃𝟏, 𝒚) = ((𝜃−𝜇(𝒚))2 + 𝜎2(𝒚))/(|𝜃−𝜇(𝒚)| + 𝜎(𝒚))2
(D.59)
The first partial derivative of 𝐿𝑎2(𝜃𝟏, 𝒚) is
𝜕𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃
= {2𝜎(𝒚)(𝜃−𝜇(𝒚) −𝜎(𝒚))/(𝜎(𝒚) + 𝜃−𝜇(𝒚))3, 𝜃> 𝜇(𝒚)
2𝜎(𝒚)(𝜃−𝜇(𝒚) + 𝜎(𝒚))/(𝜎(𝒚) + 𝜇(𝒚) −𝜃)3, 𝜃< 𝜇(𝒚)
(D.60)
or equivalently
𝜕𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃
= 2𝜎(𝒚)(𝜃−𝜇(𝒚) −sign(𝜃−𝜇(𝒚))𝜎(𝒚))/(𝜎(𝒚) + |𝜃−𝜇(𝒚)|)3 (D.61)
The second partial derivative of 𝐿𝑎2(𝜃𝟏, 𝒚) is
𝜕2𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃2
= {(−4𝜎(𝒚)(𝜃−𝜇(𝒚) −2𝜎(𝒚)))/(𝜎(𝒚) + 𝜃−𝜇(𝒚))4, 𝜃> 𝜇(𝒚)
(−4𝜎(𝒚)(𝜇(𝒚) −𝜃−2𝜎(𝒚)))/(𝜎(𝒚) + 𝜇(𝒚) −𝜃)4, 𝜃< 𝜇(𝒚) (D.62)
or equivalently
𝜕2𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃2
= (4𝜎(𝒚)(2𝜎(𝒚) −|𝜃−𝜇(𝒚)|))/(𝜎(𝒚) + |𝜃−𝜇(𝒚)|)4, 𝜃∈ℝ\
{𝜇(𝒚)}
(D.63)
36
The derivative
𝜕𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃
is not defined at 𝜃= 𝜇(𝒚), because (see Figure 2)
lim
𝜃→𝜇(𝒚)−
𝜕𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃
= 2/𝜎(𝒚)
(D.64)
and
lim
𝜃→𝜇(𝒚)+
𝜕𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃
= −2/𝜎(𝒚)
(D.65)
The function 𝐿NR2(𝜃𝟏, 𝒚) has two global minima at
𝜃̂ = 𝜇(𝒚) ± 𝜎(𝒚)
(D.66)
because
(i)
The function 𝐿NR2(𝜃𝟏, 𝒚) is a continuous function of 𝜃 and
𝜕𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃
𝜃=𝜃̂ = 0
(D.67)
(ii)
It has two local minima at 𝜃̂, because
𝐿NR2(𝜃̂𝟏, 𝒚) = 1/2 < 1
(D.68)
𝜕2𝐿NR2(𝜃𝟏,𝒚)
𝜕𝜃2
𝜃=𝜃̂ = 1/(4𝜎2(𝒚)) > 0
(D.69)
(iii) Its value at 𝜃= 𝜇(𝒚) equals
𝐿NR2(𝜇(𝒚)𝟏, 𝒚) = 1
(D.70)
(iv) The limits equal
lim
𝜃→∞𝐿NR2(𝜃𝟏, 𝒚) = lim
𝜃→−∞𝐿NR2(𝜃𝟏, 𝒚) = 1
(D.71)
(v)
Therefore, the two local minima of 𝐿NR2(𝜃𝟏, 𝒚) are global minima for 𝜃> 𝜇(𝒚) and
𝜃< 𝜇(𝒚) respectively. Since the values of the function 𝐿NR2(𝜃𝟏, 𝒚) at the two local minima
are equal, then those local minima should be also global minima in the set of real
numbers.∎
Proof D.7 (𝐿NR2 for extremum estimation of a linear model): We show that
argmin
𝜽∈ℝ2
𝐿NR2(𝑎𝒙+ 𝑏𝟏, 𝒚) = (sign(𝜌(𝒙, 𝒚))(‖𝒚𝑐‖2/‖𝒙𝑐‖2), 𝜇(𝒚) −
(sign(𝜌(𝒙, 𝒚))𝜇(𝒙)‖𝒚𝑐‖2)/‖𝒙𝑐‖2), where 𝒙𝑐 and 𝒚𝑐 are centered vectors defined in eq.
(A.23) and 𝜌(𝒙, 𝒚) is Pearson correlation defined in eq. (A.28).
We assume that 𝒚 is not constant and 𝒛= 𝑎𝒙+ 𝑏𝟏. Then due to the translation
invariance property (eq. (3.12)):
𝐿NR2(𝒛, 𝒚) = 𝐿NR2(𝒛−𝜇(𝒚)𝟏, 𝒚−𝜇(𝒚)𝟏) ⇒
(D.72)
37
𝐿NR2(𝒛, 𝒚) = 𝐿NR2(𝑎𝒙+ 𝑏𝟏−𝜇(𝒚)𝟏, 𝒚𝑐)
(D.73)
We set:
𝑐= 𝑎𝜇(𝒙) + 𝑏−𝜇(𝒚)
(D.74)
Therefore, the first argument of the left part function in eq. (D.73), after substituting b
with its transformation in eq. (D.74) is
𝑎𝒙+ 𝑏𝟏−𝜇(𝒚)𝟏= 𝑎𝒙+ (𝑐−𝑎𝜇(𝒙) + 𝜇(𝒚))𝟏−𝜇(𝒚)𝟏⇒
(D.75)
𝑎𝒙+ 𝑏𝟏−𝜇(𝒚)𝟏= 𝑎𝒙+ 𝑐𝟏−𝑎𝜇(𝒙)𝟏⇒
(D.76)
𝑎𝒙+ 𝑏𝟏−𝜇(𝒚)𝟏= 𝑎𝒙𝑐+ 𝑐𝟏
(D.77)
Therefore, the arguments of 𝐿NR2 are transformed to include centered vectors that ease
the analytical procedure to find the minima:
𝐿NR2(𝒛, 𝒚) = 𝐿NR2(𝑎𝒙𝑐+ 𝑐𝟏, 𝒚𝑐) ⇒
(D.78)
𝐿NR2(𝒛, 𝒚) = ‖𝑎𝒙𝑐+ 𝑐𝟏−𝒚𝑐‖2
2/(‖𝑎𝒙𝑐+ 𝑐𝟏‖2 + ‖𝒚𝑐‖2)2
(D.79)
The 𝐿NR2 nominator equals
‖𝑎𝒙𝑐+ 𝑐𝟏−𝒚𝑐‖2
2 = 〈(𝑎𝒙𝑐−𝒚𝑐) + 𝑐𝟏, (𝑎𝒙𝑐−𝒚𝑐) + 𝑐𝟏〉⇒
(D.80)
‖𝑎𝒙𝑐+ 𝑐𝟏−𝒚𝑐‖2
2 = ‖𝑎𝒙𝑐−𝒚𝑐‖2
2 + 𝑛𝑐2 + 2〈𝑎𝒙𝑐−𝒚𝑐, 𝑐𝟏〉
(D.81)
By leveraging orthogonality to unity by the eq. (A.27), applied to the last term in the right
part of eq. (D.81) we have
‖𝑎𝒙𝑐+ 𝑐𝟏−𝒚𝑐‖2
2 = ‖𝑎𝒙𝑐−𝒚𝑐‖2
2 + 𝑛𝑐2 ⇒
(D.82)
‖𝑎𝒙𝑐+ 𝑐𝟏−𝒚𝑐‖2
2 = 𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 −2𝑎〈𝒙𝑐, 𝒚𝑐〉+ 𝑛𝑐2
(D.83)
Furthermore, the first part of the denominator of 𝐿NR2 in eq. (D.79) equals
‖𝑎𝒙𝑐+ 𝑐𝟏‖2
2 = 〈𝑎𝒙𝑐+ 𝑐𝟏, 𝑎𝒙𝑐+ 𝑐𝟏〉⇒
(D.84)
‖𝑎𝒙𝑐+ 𝑐𝟏‖2
2 = 𝑎2‖𝒙𝑐‖2
2 + 𝑛𝑐2
(D.85)
We set 𝑤1 equal to the left part of eq. (D.83) and 𝑤2 equal to the left part of eq. (D.85)
𝑤1 = 𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 −2𝑎〈𝒙𝑐, 𝒚𝑐〉+ 𝑛𝑐2
(D.86)
𝑤2 = 𝑎2‖𝒙𝑐‖2
2 + 𝑛𝑐2
(D.87)
We also set 𝑤3 equal to
𝑤3 = 𝑤2 + ‖𝒚𝑐‖2
2 + 2‖𝒚𝑐‖2√𝑤2
(D.88)
Therefore,
𝑤1 = 𝑤2 + ‖𝒚𝑐‖2
2 −2𝑎〈𝒙𝑐, 𝒚𝑐〉
(D.89)
38
and,
𝐿NR2(𝒛, 𝒚) = 𝑤1/𝑤3
(D.90)
or equivalently
𝐿NR2(𝒛, 𝒚) = (𝑤2 + ‖𝒚𝑐‖2
2 −2𝑎〈𝒙𝑐, 𝒚𝑐〉)/(𝑤2 + ‖𝒚𝑐‖2
2 + 2‖𝒚𝑐‖2√𝑤2)
(D.91)
Partial derivatives:
The first order partial derivative of 𝐿𝑎2 with respect to 𝑐 is
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑐
= (2𝑛𝑐𝑤3 −𝑤1(2𝑛𝑐+ 2𝑛𝑐‖𝒚𝑐‖2/√𝑤2))/𝑤3
2 ⇒
(D.92)
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑐
= 2𝑛𝑐(√𝑤2𝑤3 −√𝑤2𝑤1 −𝑤1‖𝒚𝑐‖2)/(√𝑤2𝑤3
2)
(D.93)
or equivalently
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑐
= (2𝑛𝑐𝑤4)/(√𝑤2𝑤3
2)
(D.94)
where
𝑤4 = 2𝑎〈𝒙𝑐, 𝒚𝑐〉(√𝑎2‖𝒙𝑐‖2
2 + 𝑛𝑐2 + ‖𝒚𝑐‖2) + ‖𝒚𝑐‖2(𝑎2‖𝒙𝑐‖2
2 + 𝑛𝑐2 −
‖𝒚𝑐‖2
2)
(D.95)
or equivalently
𝑤4 = 2𝑎〈𝒙𝑐, 𝒚𝑐〉(√𝑤2 + ‖𝒚𝑐‖2) + ‖𝒚𝑐‖2(𝑤2 −‖𝒚𝑐‖2
2) ⇒
(D.96)
𝑤4 = (√𝑤2 + ‖𝒚𝑐‖2)(2𝑎〈𝒙𝑐, 𝒚𝑐〉+ ‖𝒚𝑐‖2(√𝑤2 −‖𝒚𝑐‖2))
(D.97)
The first order partial derivative of 𝐿NR2 with respect to 𝑎 is
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑎
= (2(𝑎‖𝒙𝑐‖2
2 −〈𝒙𝑐, 𝒚𝑐〉)𝑤3) −(2𝑎‖𝒙𝑐‖2
2𝑤1(1 + ‖𝒚𝑐‖2/√𝑤2))/
𝑤3
2 ⇒
(D.98)
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑎
= 𝑤5/(√𝑤2𝑤3
2)
(D.99)
where
𝑤5 = 2√𝑤2(𝑎‖𝒙𝑐‖2
2 −〈𝒙𝑐, 𝒚𝑐〉)(√𝑤2 + ‖𝒚𝑐‖2)2 −2𝑎‖𝒙𝑐‖2
2(𝑤2 + ‖𝒚𝑐‖2
2 −
2𝑎〈𝒙𝑐, 𝒚𝑐〉)(√𝑤2 + ‖𝒚𝑐‖2) ⇒
(D.100)
𝑤5 = (√𝑤2 + ‖𝒚𝑐‖2)(2√𝑤2(𝑎‖𝒙𝑐‖2
2 −〈𝒙𝑐, 𝒚𝑐〉)(√𝑤2 + ‖𝒚𝑐‖2) −2𝑎‖𝒙𝑐‖2
2(𝑤2 + ‖𝒚𝑐‖2
2 −
2𝑎〈𝒙𝑐, 𝒚𝑐〉))
(D.101)
Minimization:
To minimize 𝐿NR2(𝑎𝒙+ 𝑏𝟏, 𝒚) we observe from eq. (D.94) that the partial derivative
39
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑐
equals 0 if and only if 𝑐= 0 or 𝑤4 = 0.
Case 1 (𝑐= 0):
𝑐= 0 ⇔𝑎𝜇(𝒙) + 𝑏−𝜇(𝒚) = 0 ⇔𝑏= 𝜇(𝒚) −𝑎𝜇(𝒙)
(D.102)
We have to find solutions of the equation
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑎
= 0 (where the partial derivative is
defined in eq. (D.99)), i.e. equivalently solutions of the equation 𝑤5 = 0. First, we compute
𝑤2 from eq. (D.87):
𝑤2 = 𝑎2‖𝒙𝑐‖2
2
(D.103)
Then we compute 𝑤5 from eq. (D.101):
𝑤5 = (|𝑎|‖𝒙𝑐‖2 + ‖𝒚𝑐‖2)(2|𝑎|‖𝒙𝑐‖2(𝑎‖𝒙𝑐‖2
2 −〈𝒙𝑐, 𝒚𝑐〉)(|𝑎|‖𝒙𝑐‖2 + ‖𝒚𝑐‖2) −
2𝑎‖𝒙𝑐‖2
2(𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 −2𝑎〈𝒙𝑐, 𝒚𝑐〉)) ⇒
(D.104)
𝑤5 = 2‖𝒙𝑐‖2(|𝑎|‖𝒙𝑐‖2 + ‖𝒚𝑐‖2)((𝑎‖𝒙𝑐‖2
2 −〈𝒙𝑐, 𝒚𝑐〉)(𝑎2‖𝒙𝑐‖2 + |𝑎|‖𝒚𝑐‖2) −
‖𝒙𝑐‖2(𝑎3‖𝒙𝑐‖2
2 + 𝑎‖𝒚𝑐‖2
2 −2𝑎2〈𝒙𝑐, 𝒚𝑐〉)) ⇒
(D.105)
𝑤5 = 2‖𝒙𝑐‖2(|𝑎|‖𝒙𝑐‖2 + ‖𝒚𝑐‖2)((𝑎3‖𝒙𝑐‖2
3 + 𝑎|𝑎|‖𝒙𝑐‖2
2‖𝒚𝑐‖2 −𝑎2‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉−
|𝑎|‖𝒚𝑐‖2〈𝒙𝑐, 𝒚𝑐〉) −𝑎3‖𝒙𝑐‖2
3 −𝑎‖𝒙𝑐‖2‖𝒚𝑐‖2
2 + 2𝑎2‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉) ⇒
(D.106)
𝑤5 = 2‖𝒙𝑐‖2(|𝑎|‖𝒙𝑐‖2 + ‖𝒚𝑐‖2)(𝑎|𝑎|‖𝒙𝑐‖2
2‖𝒚𝑐‖2 −|𝑎|‖𝒚𝑐‖2〈𝒙𝑐, 𝒚𝑐〉−𝑎‖𝒙𝑐‖2‖𝒚𝑐‖2
2 +
𝑎2‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉)
(D.107)
𝑤5 is a function of |𝑎|, so to find where it equals 0 we examine two cases for 𝑎. The first
case is 𝑎> 0, when 𝑤5 simplifies to:
𝑤5 = 2𝑎‖𝒙𝑐‖2(𝑎‖𝒙𝑐‖2 + ‖𝒚𝑐‖2)(𝑎‖𝒙𝑐‖2
2‖𝒚𝑐‖2 −‖𝒚𝑐‖2〈𝒙𝑐, 𝒚𝑐〉−‖𝒙𝑐‖2‖𝒚𝑐‖2
2 +
𝑎‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉)
(D.108)
Therefore, 𝑤5 = 0 when:
𝑎= (‖𝒚𝑐‖2〈𝒙𝑐, 𝒚𝑐〉+ ‖𝒙𝑐‖2‖𝒚𝑐‖2
2)/(‖𝒙𝑐‖2
2‖𝒚𝑐‖2 + ‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉) ⇒
(D.109)
𝑎= ‖𝒚𝑐‖2/‖𝒙𝑐‖2
(D.110)
The first case is 𝑎< 0, when 𝑤5 simplifies to:
𝑤5 = 2𝑎‖𝒙𝑐‖2(𝑎‖𝒙𝑐‖2 + ‖𝒚𝑐‖2)(−𝑎‖𝒙𝑐‖2
2‖𝒚𝑐‖2 + ‖𝒚𝑐‖2〈𝒙𝑐, 𝒚𝑐〉−‖𝒙𝑐‖2‖𝒚𝑐‖2
2 +
𝑎‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉)
(D.111)
Therefore, 𝑤5 = 0 when:
𝑎= (−‖𝒚𝑐‖2〈𝒙𝑐, 𝒚𝑐〉+ ‖𝒙𝑐‖2‖𝒚𝑐‖2
2)/(−‖𝒙𝑐‖2
2‖𝒚𝑐‖2 + ‖𝒙𝑐‖2〈𝒙𝑐, 𝒚𝑐〉) ⇒ (D.112)
𝑎= −‖𝒚𝑐‖2/‖𝒙𝑐‖2
(D.113)
40
Summarizing the two cases (𝑎> 0 and 𝑎< 0), we have that 𝑤5 = 0 for values of 𝑎= 𝑎̂,
where 𝑎̂:
𝑎̂ = ±‖𝒚𝑐‖2/‖𝒙𝑐‖2
(D.114)
From eq. (D.102), after substituting 𝑎 with 𝑎̂, we obtain the respective values of 𝑏:
𝑏̂ = 𝜇(𝒚) ∓𝜇(𝒙)‖𝒚𝑐‖2/‖𝒙𝑐‖2
(D.115)
The value of 𝐿NR2(𝒛, 𝒚) at (𝑎̂, 𝑏̂) can be computed from eq. (D.90) after first computing 𝑤2
(based on eqs. (D.87) and (D.114)) and 𝑤3 (based on eqs. (D.88) and (D.114)):
𝑤2 = ‖𝒚𝑐‖2
2
(D.116)
𝑤3 = 4‖𝒚𝑐‖2
2
(D.117)
Therefore 𝐿NR2(𝒛, 𝒚) at (𝑎̂, 𝑏̂) equals
𝐿NR2(𝒛, 𝒚) = (‖𝒚𝑐‖2
2 −𝑎̂〈𝒙𝑐, 𝒚𝑐〉)/(2‖𝒚𝑐‖2
2) ⇒
(D.118)
𝐿NR2(𝒛, 𝒚) = (‖𝒚𝑐‖2
2 ∓(‖𝒚𝑐‖2/‖𝒙𝑐‖2)〈𝒙𝑐, 𝒚𝑐〉)/(2‖𝒚𝑐‖2
2) ⇒
(D.119)
𝐿NR2(𝒛, 𝒚) = (1 ∓𝜌(𝒙, 𝒚))/2
(D.120)
Therefore 𝐿NR2 has two local optima. Which one of them is the minimum depends on the
sign of 𝜌(𝒙, 𝒚). If 𝜌(𝒙, 𝒚) > 0 then 𝐿NR2(𝒛, 𝒚) attains its minimum at (1 −𝜌(𝒙, 𝒚)/2 when
𝑎̂ = ‖𝒚𝑐‖2/‖𝒙𝑐‖2
(D.121)
If 𝜌(𝒙, 𝒚) < 0 then 𝐿NR2(𝒛, 𝒚) attains its minimum at (1 + 𝜌(𝒙, 𝒚)/2 when
𝑎̂ = −‖𝒚𝑐‖2/‖𝒙𝑐‖2
(D.122)
Those two cases can be combined to obtain a unique estimate for 𝑎:
𝑎̂ = sign(𝜌(𝒙, 𝒚))(‖𝒚𝑐‖2/‖𝒙𝑐‖2)
(D.123)
From eq. (D.102), after substituting 𝑎 with 𝑎̂, we obtain the respective values of 𝑏:
𝑏̂ = 𝜇(𝒚) −(sign(𝜌(𝒙, 𝒚))𝜇(𝒙)‖𝒚𝑐‖2)/‖𝒙𝑐‖2
(D.124)
At (𝑎̂, 𝑏̂), 𝐿NR2(𝒛, 𝒚) attains its minimum value
𝐿NR2(𝒛, 𝒚) = (1 −|𝜌(𝒙, 𝒚)|)/2
(D.125)
To complete the proof, one possible way is to compute the second derivatives. However
this is complicated. Yet, the global optimality of 𝐿NR2(𝒛, 𝒚) at 𝑐= 0 can also be proven by
observing that for fixed 𝑎:
(i)
𝐿NR2(𝑎𝒙𝑐+ 𝑐𝟏, 𝒚𝑐) is even in 𝑐, as it depends on 𝑐 only through 𝑤2 (see eqs. (D.90),
(D.89) and (D.88)):
41
𝐿NR2(𝑎𝒙𝑐+ 𝑐𝟏, 𝒚𝑐) = 𝐿NR2(𝑎𝒙𝑐−𝑐𝟏, 𝒚𝑐)
(D.126)
because from eq. (D.87), 𝑤2 = 𝑎2‖𝒙𝑐‖2
2 + 𝑛𝑐2.
(ii)
It is straightforward to prove that:
lim
𝑐→∞𝐿NR2(𝑎𝒙𝑐+ 𝑐𝟏, 𝒚𝑐) = lim
𝑐→−∞𝐿NR2(𝑎𝒙𝑐+ 𝑐𝟏, 𝒚𝑐) = 1 ≥(1 −|𝜌(𝒙, 𝒚)|)/
2
(D.127)
(iii) 𝐿NR2(𝑎𝒙𝑐+ 𝑐𝟏, 𝒚𝑐) is continuous and differentiable for every 𝑐.
(iv) 𝑐= 0 is a unique critical point since
𝜕𝐿NR2(𝒛,𝒚)
𝜕𝑐
≠0, otherwise (see case 2 that proves
that 𝑤4 ≠0).
Due to properties (iii) and (iv), there is no other local optimum, therefore 𝑐= 0 should be
a global optimum. Then property (ii) implies that 𝑐= 0 is a global minimum.
To prove the global optimality for 𝑎̂, we note that from eqs. (D.87) – (D.90), at 𝑐= 0:
𝐿NR2(𝒛, 𝒚) = ‖𝑎𝒙𝑐−𝒚𝑐‖2
2/(‖𝑎𝒙𝑐‖2 + ‖𝒚𝑐‖2)2 ⇒
(D.128)
𝐿NR2(𝒛, 𝒚) = (𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 −2𝑎〈𝒙𝑐, 𝒚𝑐〉)/(𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 +
2|𝑎|‖𝒙𝑐‖2‖𝒚𝑐‖2) ⇒
(D.129)
𝐿NR2(𝒛, 𝒚) = (𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 −2𝑎𝜌(𝒙, 𝒚)‖𝒙𝑐‖2‖𝒚𝑐‖2)/(𝑎2‖𝒙𝑐‖2
2 + ‖𝒚𝑐‖2
2 +
2|𝑎|‖𝒙𝑐‖2‖𝒚𝑐‖2) ⇒
(D.130)
𝐿NR2(𝒛, 𝒚) = ((𝑎/𝑎̂)2 + 1 −2(𝑎/𝑎̂)|𝜌(𝒙, 𝒚)|)/((𝑎/𝑎̂)2 + 1 + 2sign(𝜌(𝒙, 𝒚))|𝑎|/
𝑎̂)
(D.131)
We set
𝑡= 𝑎/𝑎̂
(D.132)
Then we have four cases:
(i)
If 𝑎> 0 and 𝜌(𝒙, 𝒚) > 0, then 𝑡> 0 and:
𝐿NR2(𝒛, 𝒚) = 1 −(2𝑡(1 + 𝜌(𝒙, 𝒚)))/(𝑡+ 1)2
(D.133)
This is minimized for 𝑡= 1, i.e. when 𝑎= 𝑎̂ and 𝐿NR2(𝒛, 𝒚) = (1 −|𝜌(𝒙, 𝒚)|)/2, which is a
global minimum.
(ii)
If 𝑎< 0 and 𝜌(𝒙, 𝒚) > 0, then 𝑡< 0 and:
𝐿NR2(𝒛, 𝒚) = 1 + (2𝑡(1 −𝜌(𝒙, 𝒚)))/(𝑡−1)2
(D.134)
This is minimized for 𝑡= −1, i.e. when 𝑎= 𝑎̂ and 𝐿NR2(𝒛, 𝒚) = (1 + |𝜌(𝒙, 𝒚)|)/2, which is
not a global minimum.
42
(iii) If 𝑎> 0 and 𝜌(𝒙, 𝒚) < 0, then 𝑡< 0 and:
𝐿NR2(𝒛, 𝒚) = 1 + (2𝑡(1 + 𝜌(𝒙, 𝒚)))/(𝑡−1)2
(D.135)
This is minimized for 𝑡= −1, i.e. when 𝑎= 𝑎̂ and 𝐿NR2(𝒛, 𝒚) = (1 + |𝜌(𝒙, 𝒚)|)/2, which is
not a global minimum.
(iv) If 𝑎< 0 and 𝜌(𝒙, 𝒚) < 0, then 𝑡> 0 and:
𝐿NR2(𝒛, 𝒚) = 1 −(2𝑡(1 −𝜌(𝒙, 𝒚)))/(𝑡+ 1)2
(D.136)
This is minimized for 𝑡= 1, i.e. when 𝑎= 𝑎̂ and 𝐿NR2(𝒛, 𝒚) = (1 −|𝜌(𝒙, 𝒚)|)/2, which is a
global minimum.
This completes the proof for the case of 𝑎.
Case 2 (𝑤4 = 0):
We prove that 𝐿NR2(𝒛, 𝒚) > (1 −|𝜌(𝒙, 𝒚)|)/2, when 𝑤4 = 0, where 𝑤4 is defined in eq.
(D.97). We try to solve the equation:
𝑤4 = 0 ⇒
(D.137)
(√𝑤2 + ‖𝒚𝑐‖2)(2𝑎〈𝒙𝑐, 𝒚𝑐〉+ ‖𝒚𝑐‖2(√𝑤2 −‖𝒚𝑐‖2)) = 0 ⇒
(D.138)
−2𝑎〈𝒙𝑐, 𝒚𝑐〉= ‖𝒚𝑐‖2(√𝑤2 −‖𝒚𝑐‖2)
(D.139)
This can also be written as:
√𝑤2 = ‖𝒚𝑐‖2 −2𝑎𝜌(𝒙, 𝒚)‖𝒙c‖2
(D.140)
Therefore, from eq. (D.89):
𝑤1 = 𝑤2 + ‖𝒚𝑐‖2
2 + ‖𝒚𝑐‖2(√𝑤2 −‖𝒚𝑐‖2) ⇒
(D.141)
𝑤1 = 𝑤2 + ‖𝒚𝑐‖2√𝑤2 ⇒
(D.142)
𝑤1 = √𝑤2(√𝑤2 + ‖𝒚𝑐‖2)
(D.143)
Furthermore, from eq. (D.88):
𝑤3 = (√𝑤2 + ‖𝒚𝑐‖2)2
(D.144)
Returning to eq. (D.90) and substituting the values of 𝑤1, √𝑤2 and 𝑤3:
𝐿NR2(𝒛, 𝒚) = √𝑤2/(√𝑤2 + ‖𝒚𝑐‖2) ⇒
(D.145)
𝐿NR2(𝒛, 𝒚) = (‖𝒚𝑐‖2 −2𝑎𝜌(𝒙, 𝒚)‖𝒙c‖2)/(2(‖𝒚𝑐‖2 −𝑎𝜌(𝒙, 𝒚)‖𝒙c‖2))
(D.146)
We set,
𝑤6 = (𝑎‖𝒙c‖2)/‖𝒚c‖2
(D.147)
and substitute in eq. (D.146):
43
𝐿NR2(𝒛, 𝒚) = (1 −2𝜌(𝒙, 𝒚)𝑤6)/(2(1 −𝜌(𝒙, 𝒚)𝑤6))
(D.148)
Furthermore, the following inequalities hold (based on eqs. (D.87), (D.140) and (D.147)):
√𝑤2 ≥|𝑎|‖𝒙c‖2 ≥0 ⇒
(D.149)
‖𝒚𝑐‖2 −2𝑎𝜌(𝒙, 𝒚)‖𝒙c‖2 ≥|𝑎|‖𝒙c‖2 ≥0 ⇒
(D.150)
1 −2𝜌(𝒙, 𝒚)𝑤6 ≥|𝑤6| ≥0
(D.151)
Now we have to prove that 𝐿NR2(𝒛, 𝒚) > (1 −|𝜌(𝒙, 𝒚)|)/2 or equivalently (based on eq.
(D.148)) that:
(1 −2𝜌(𝒙, 𝒚)𝑤6)/(2(1 −𝜌(𝒙, 𝒚)𝑤6)) ≥(1 −|𝜌(𝒙, 𝒚)|)/2
(D.152)
or equivalently
(1 −2𝜌(𝒙, 𝒚)𝑤6)/(1 −𝜌(𝒙, 𝒚)𝑤6) ≥1 −|𝜌(𝒙, 𝒚)| ⇒
(D.153)
(−𝜌(𝒙, 𝒚)𝑤6 + |𝜌(𝒙, 𝒚)| −𝜌(𝒙, 𝒚)|𝜌(𝒙, 𝒚)|𝑤6)/(1 −𝜌(𝒙, 𝒚)𝑤6) ≥0
(D.154)
We examine two cases for proving the inequality in eq. (D.154), depending on the sign of
𝜌(𝒙, 𝒚):
Case 2.a:
If 𝜌(𝒙, 𝒚) > 0 we have to prove:
(𝜌(𝒙, 𝒚)(−𝑤6 + 1 −𝜌(𝒙, 𝒚)𝑤6))/(1 −𝜌(𝒙, 𝒚)𝑤6) ≥0 ⇒
(D.155)
(𝜌(𝒙, 𝒚)(1 −𝑤6(1 + 𝜌(𝒙, 𝒚))))/(1 −𝜌(𝒙, 𝒚)𝑤6) ≥0
(D.156)
(i)
Sign of the denominator:
If 1 −𝜌(𝒙, 𝒚)𝑤6 ≤0, then 𝑤6 ≥1/𝜌(𝒙, 𝒚). However, the constraint 1 −2𝜌(𝒙, 𝒚)𝑤6 ≥|𝑤6|
would imply 1 ≥(2𝜌(𝒙, 𝒚) + 1)𝑤6, which in turn would imply that 1 ≥(2𝜌(𝒙, 𝒚) +
1)/𝜌(𝒙, 𝒚) or equivalently −1 ≥1/𝜌(𝒙, 𝒚) which in turn is impossible, therefore 1 −
𝜌(𝒙, 𝒚)𝑤6 > 0.
(ii)
Sign of the nominator:
To prove that the nominator is positive, it suffices to prove that 𝑤6 ≤1/(1 + 𝜌(𝒙, 𝒚)).
If 𝑤6 ≥0, then 1 −2𝜌(𝒙, 𝒚)𝑤6 ≥|𝑤6| implies that 𝑤6 ≤1/(1 + 2𝜌(𝒙, 𝒚)), which implies
that 𝑤6 ≤1/(1 + 𝜌(𝒙, 𝒚)).
If 𝑤6 < 0, then 𝑤6 < 0 ≤1/(1 + 𝜌(𝒙, 𝒚)).
Case 2.b:
If 𝜌(𝒙, 𝒚) < 0 we have to prove:
(−𝜌(𝒙, 𝒚)(𝑤6 + 1 −𝜌(𝒙, 𝒚)𝑤6))/(1 −𝜌(𝒙, 𝒚)𝑤6) ≥0 ⇒
(D.157)
44
(|𝜌(𝒙, 𝒚)|(1 + (1 + |𝜌(𝒙, 𝒚)|)𝑤6))/(1 + |𝜌(𝒙, 𝒚)|𝑤6) ≥0
(D.158)
(i)
Sign of the denominator:
If 𝑤6 ≥0, then 1 + |𝜌(𝒙, 𝒚)|𝑤6 ≥0.
If 𝑤6 < 0, then 1 −2𝜌(𝒙, 𝒚)𝑤6 ≥|𝑤6| implies that 𝑤6 ≥−1/(2|𝜌(𝒙, 𝒚)| + 1), which in
turn implies that 𝑤6 ≥−1/(2|𝜌(𝒙, 𝒚)| + 1) ≥−1/|𝜌(𝒙, 𝒚)|, therefore 1 + |𝜌(𝒙, 𝒚)|𝑤6 ≥
0.
(ii)
Sign of the nominator:
To prove that the nominator is positive, it suffices to prove that 𝑤6 ≥−1/(1 + |𝜌(𝒙, 𝒚)|).
If 𝑤6 > 0, then 𝑤6 > 0 > −1/(1 + |𝜌(𝒙, 𝒚)|).
If 𝑤6 < 0, then we have proved that 𝑤6 ≥−1/(2|𝜌(𝒙, 𝒚)| + 1), that implies 𝑤6 ≥
−1/(2|𝜌(𝒙, 𝒚)| + 1) ≥−1/(1 + |𝜌(𝒙, 𝒚)|) that completes the proof.∎
Appendix E
Statistical software
All analyses were conducted using the R programming language [31].
Data processing and visualization were performed using the data.table [47] and
tidyverse ([48], [49]) packages.
Hydrologic modeling was carried out with the airGR package ([33], [34]), which we
modified to implement the SE, 𝐿NR2 and 𝐿𝑊 loss functions as additional calibration
options. The required hydrologic data were retrieved from the airGRdatasets package
[35].
Model performance metrics were computed using the scoringfunctions package
([50], [51]).
Finally, all reports were generated using the devtools [52], knitr ([53], [54], [55])
and rmarkdown ([56], [57], [58]) packages.
Supplementary material: We provide the following files to reproduce results of Sections
3 and 4:
Section
3.1.3:
01_figure_Lw_extremum_estimator.Rmd,
01_figure_Lw_extremum_estimator.html.
Section
3.2.4:
02_figure_LNR2_extremum_estimator.Rmd,
02_figure_LNR2_extremum_estimator.html.
Section
4.1:
03_figure_forecasting_Gaussian_distribution.Rmd,
45
03_figure_forecasting_Gaussian_distribution.html.
Section
4.2:
04a_linear_model.Rmd,
04a_linear_model.html,
04b_linear_model.Rmd, 04b_linear_model.html, 04c_linear_model.Rmd,
04c_linear_model.html.
Section
4.3:
05a_hydro_data.Rmd,
05a_hydro_data.html,
05a_hydro_data.RData,
05b_hydro_modelling.Rmd,
05b_hydro_modelling.html,
05b_hydro_modelling.RData,
05c_hydro_modelling.Rmd,
05c_hydro_modelling.html,
05d_hydro_modelling.Rmd, 05d_hydro_modelling.html.
Conflicts of interest: The authors declare no conflict of interest.
Statement: During the preparation of this work, the authors used DeepSeek-V.3.2 to
assist with language polishing and to enhance readability. After using this service, the
authors reviewed and edited the content as needed and take full responsibility for the
content of the published article.
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Loss functions arising from the index of agreement Hristos Tyralis1*, Georgia Papacharalampous2 1Support Command, Hellenic Air Force, Elefsina Air Base, 19 200, Elefsina, Greece ( , , https://orcid.org/0000-0002-89324997) 2 'Università 16, 35020, Legnaro, Italy ( , , https://orcid.org/0000-0001-5446-954X) *Corresponding author Abstract: We examine the theoretical properties of the index of agreement loss function LW, the negatively oriented counterpart of Willmott's index of agreement, a common metric in environmental sciences and engineering. We prove that LW is bounded within [0,1], translation and scale invariant, and estimates the parameter EF[y] ± VF 1/2[y] when fitting a distribution. We propose LNR2 as a theoretical improvement, which replaces the denominator of LW with the sum of Euclidean distances, better aligning with the underlying geometric intuition. This new loss function retains the appealing properties of LW but also admits closed-form solutions for linear model parameter estimation. We show that as the correlation between predictors and the dependent variable approaches 1, parameter estimates from squared error, LNR2 and LW converge. This behavior is mirrored in hydrologic model calibration (a core task in water resources engineering), where performance becomes nearly identical across these loss functions. Finally, we suggest potential improvements for existing Lp-norm variants of the index of agreement. Keywords: extremum estimation; index of agreement; loss function; model calibration; model evaluation; point forecasting 1. Introduction Loss functions are fundamental tools in machine learning, statistical science and engineering, used for model estimation (also referred to as fitting, calibrating or training) and model ranking (evaluation) [1]. During fitting, model parameters are estimated by minimizing the average discrepancy between predictions and observations, quantified by the loss function. For ranking, a model's performance is evaluated by comparing the 2 average loss of predictions against observations in a test set. Domain-specific loss functions developed in environmental sciences and engineering ([2], [3], [4]) hold potential for broader application. One established metric is Willmotts's [5] index of agreement, defined as d(z, y) ∶= 1 -∑ (zi-yi)2 n i=1 / ∑ (|zi-μ(y)| + |μ(y) -yi|)2 n i=1 (1.1) where z= (z1, ... , zn)T represents model predictions, y= (y1, ... , yn)T represents realizations and μ(y) is the sample mean of realizations (defined in eq. (A.19)). Since d is positively oriented (higher values indicate better performance), we study its negatively oriented counterpart without loss of generality, which we name the index of agreement loss function LW (subscript W for Willmott): LW(z, y) ∶= ∑ (zi-yi)2 n i=1 / ∑ (|zi-μ(y)| + |μ(y) -yi|)2 n i=1 (1.2) Although the index of agreement has been extensively researched, most studies are empirical [6] and a theoretical analysis is lacking. Here we aim to provide a theoretical foundation for the index of agreement and to propose variants with improved theoretical properties, building upon its fundamental construction principles. Our contributions are as follows: (i) We examine the theoretical properties of LW(z, y) as an extremum estimator [7], demonstrating that it is bounded in [0,1], and is translation and scale invariant. When fitting a sample of realizations y, it estimates the parameter θ̂ = μ(y) ± σ(y), where σ(y) is the sample standard deviation defined in eq. (A.21). For linear models, no closed-form solution has been identified, which remains an open problem. (ii) We propose an alternative definition for LW, which we call LNR2 (subscript NR for Norm Ratio, subscript 2 for L2-norm), defined in Section 3.2. This function maintains consistency with the distance-based arguments underlying the index of agreement, but uses L2-norm-based constructions. Thus, LNR2 represents a theoretical improvement over the original index. It preserves the [0,1] bound and translation and scale invariance. When fitting a sample y, it estimates θ̂ = μ(y) ± σ(y), analogous to the original index. For linear models z= ax+ b1, we provide closed-form estimates for parameters a and b. (iii) Constructions based on Lp-norms (in Section 3.3) generalize the new index of agreement, enabling a theoretically consistent variant of Legates and McCabe Jr [8] and Krause et al. [9] index of agreement variants. 3 (iv) Although our theoretical developments focus on estimation, we empirically extend these insights to model evaluation. Such an investigation is necessary because a theory linking estimation and evaluation for non-additive extremum estimators (unlike additive M-estimators; see [10]) is currently lacking in statistical science. Empirical results based on predictions from simulated data and real-world case studies (the latter using hydrologic models) estimated with LW and LNR2, will facilitate a reassessment of prior studies that employed the index of agreement. The remainder of the manuscript is structured as follows. Section 2 reviews essential background on prediction evaluation, estimation theory and the index of agreement. Section 3 presents new material and the main theoretical results, focusing on the original index of agreement and our subsequent developments. Empirical demonstrations using both simulated and real-world data modeled by hydrologic models is included in Section 4. The implications of these theoretical contributions are discussed in Section 5, followed by concluding remarks in Section 6. Given the extensive vector representations and calculus required for the theoretical analysis and proofs, Appendix A introduces consistent notation and definitions for vector calculus. Common identification [1] and loss functions used throughout the manuscript, are detailed in Appendix B, while the probability distributions used are listed in Appendix C. All proofs are supplied in Appendix D. Appendix E details the statistical software to reproduce the analysis. Supplementary material to reproduce all empirical analyses and figures is available. 2. Theoretical background Section 2.1 outlines the theoretical framework for prediction evaluation, which applies to additive formulations of a loss function for both assessing a single model and comparing multiple models. As the index of agreement is non-additive, its application to parameter estimation places it within the broader category of extremum estimators. The theoretical foundations for this general class of estimators, including the specific case of M-estimators (which align directly to the framework in Section 2.1), are presented in Section 2.2. Section 2.3 then describes the index of agreement and develops the formulations required for its subsequent analysis in Section 3. 4 2.1 Comparison and adequacy of predictions 2.1.1 Comparing predictions from multiple models Let z= (z1, ... , zn)T represent a vector of model predictions and y= (y1, ... , yn)T a vector of corresponding observed values. Model comparisons are performed using a loss function L(z, y) which quantifies the discrepancy between a single prediction z and its observation y ([21], [1]). Standard loss functions include the absolute error (AE) loss LAE (eq. (B.1)) and the squared error (SE) loss LSE (eq. (B.2)). A model is preferred over another if it achieves a lower average loss L̅ across all observations. This average is defined as [1]: L̅(z, y) = (1/n) ∑ L(zi, yi) n i=1 (2.1) Commonly used averages of these loss functions include the mean absolute error (MAE; eq. (B.3)), which corresponds to the AE loss and the mean squared error (MSE; eq. (B.4)), which corresponds to the SE loss. 2.1.2 Strictly consistent loss functions A key property a loss function can have is known as (strict) consistency. To explain this concept, we must first define the notion of a statistical functional. Let y be a random variable (hereinafter, random variables are underlined). A materialization of y is written as y. We write y~F to indicate that y follows a cumulative distribution function (CDF) F, defined as: F(y) ∶= P(y≤y) (2.2) A one-dimensional statistical functional (or simply a functional) T is a mapping [1] T: F→P(D), F↦T(F) ⊆D (2.3) where F is a class of probability distributions. The functional maps each distribution F∈ F to a subset T(F) of the domain D⊆R (i.e., an element of the power set P(D)). This subset T(F) can be single-valued or set-valued. Common examples of statistical functionals include the median Q1/2(F), the τ-quantile Qτ(F) and the mean EFy[y]. A loss function L is F-consistent for the functional T if the following inequality holds for all distributions in the class ([1], [11]) EF[L(t, y)] ≤EF[L(z, y)] ∀ F∈F, t∈T(F), z∈D (2.4) 5 The loss function is strictly F-consistent if it is F-consistent and equality in eq. (2.4) implies that z∈T(F). The practical importance of strict consistency can be understood from the following example. Suppose a modeler is tasked with predicting the mean of the random variable y. The SE loss LSE is strictly consistent for the mean [1]. This means that if the modeler follows the directive, i.e. predicts the true mean, her/his predictions will, in expectation, minimize the loss when evaluated using LSE. Eq. (2.4) is a population-level property, therefore, its natural empirical counterpart for comparing predictions is the sample average loss defined in eq. (2.1). LSE belongs to the broader family of Bregman loss functions, all of which are strictly consistent for the mean ([1], [12], [13], [14], [15], [16]). 2.1.3 Skill scores When evaluating predictions from multiple competing models, skill scores provide a standardized framework for comparison. The general form of a skill score, as given by [1] is: L̅skill(z, y, zref) ∶= (L̅(z, y) -L̅(zref, y))/(L̅(zoptimal, y) -L̅(zref, y)) (2.5) where z represents the predictions from the model of interest, zref represents the predictions from the reference model and zoptimal represents the theoretically optimal predictions that achieve the minimum possible loss. Skill scores of this form are positively oriented. A common simplification arises when the loss function L has a minimum of 0, as this implies L̅(zoptimal, y) = 0. The skill score then simplifies to: L̅skill(z, y, zref) ∶= 1 -L̅(z, y)/L̅(zref, y) (2.6) This ratio-based form is intuitively interpretable as the relative improvement over the reference method. A prominent example is the Nash-Sutcliffe efficiency (NSE) [17], a skill score based on the squared error loss relative to a climatological mean [18]. The NSE is constructed by taking LSE as the evaluation loss and using the mean climatology as the reference prediction. Substituting these into eq. (2.6) yields the NSE, which is defined in eq. (B.5). 2.1.4 Adequacy of predictions from a single model While loss functions are instrumental for the comparative ranking of models, a distinct 6 framework is required to assess the absolute adequacy of a single set of predictions. This assessment is formalized through the concept of identification functions (1], [19]). An identification function V: D× I→R, I⊆R serves as a tool for evaluating whether a prediction z equals a specified functional T(F). The function V is termed a strict Fidentification function for the functional T(F) if it satisfies the following condition EF[V(z, y)] = 0 ⇔z∈T(F) ∀ F∈F, t∈T(F), z∈D (2.7) If a strict identification function exists for the functional T, then T is called identifiable. The practical significance of identifiability lies in its ability to enable an absolute check of predictive performance. For instance, if a model claims to predict the median, the average of its identification function values over a test set can be used to statistically verify this claim, a concept aligned with calibration in the statistical literature ([18], [20]). It is crucial to distinguish the roles of loss and identification functions. The former are used for comparative model ranking, while the latter are used for the absolute assessment of predictions for a specific functional [21]. A notable connection between these concepts is established by Osband's principle [22]. This principle states that a strictly F-consistent loss function L and a strict Fidentification function V are linked via the derivative ([1], [21], [22]) ∂L(z,y) ∂z = h(z)V(z, y) (2.8) where h is a nonnegative function and V is an oriented function. The function V is an oriented strict F-identification function if it is a strict F-identification function for the functional T(F) and, moreover EF[V(z, y)] > 0 if and only if z> T(F) ∀ F∈F, z ∈D⊆R. ([21], [23]). Common identification function examples include Vmedian (eq. (B.6)) for the median and Vmean (eq. (B.7)) for the mean. In practice, the empirical average identification function V̅ is computed on a test set V̅(z, y) = (1/n) ∑ V(zi, yi) n i=1 (2.9) Commonly used empirical averages include V̅median (eq. (B.8)) and V̅mean (eq. (B.9)), the latter also known as mean error (ME). Adequate predictions for the functional T should yield V̅ ≈0 (subject to scaling of the predicted variable). For example, a median prediction is adequate if roughly half the predictions exceed their corresponding observations. 7 Finally, it is possible for multiple models to be adequate for the same functional (e.g., both identifying the mean) yet exhibit different average losses. In such cases, the model with the lower average loss (e.g. lower MSE for mean predictions) is still preferred, as it is more accurate even though both are calibrated. 2.1.5 Prespecifying the loss function in model estimation and prediction evaluation Theoretical work establishes that the loss function used for ranking must be disclosed a priori, as model rankings depend critically on this choice ([1], [24]). This dependency can be illustrated with an example. Consider Model #1, trained to predict conditional medians using the absolute error (AE) loss, and Model #2, trained to predict conditional means using the squared error (SE) loss. If the dependent variable follows an asymmetric distribution, their relative performance reverses depending on the loss function, since Model #1 will outperform Model #2 under AE loss, while the opposite holds under SE loss [10]. This follows directly from the equivalence between the consistency of an Mestimator (used in training) and evaluation with a strictly consistent loss function. 2.2 Consistent extremum and M-estimators The general class of extremum estimators was formalized by [7] [25] and [26] and. Consider a loss function L(θ, y) that quantifies the discrepancy for a parameter vector θ given a realization y= (y1, ... , yn)T. Assuming the true parameter θ0 lies in the parameter space Θ, an extremum estimator θ̂(y) for θ0 is defined as: θ̂(y) ∶= argmin θ∈Θ L(θ, y) (2.10) The properties of θ̂(y) depend on the sample size n through the realizations y. While L can take general forms, the additive case has been extensively studied. When L is additive, the resulting estimators are called M-estimators ([27], [28]). For a onedimensional parameter θ0, this takes the form: θ̂(y) ∶= argmin θ∈Θ ∑ L(θ, yi) n i=1 (2.11) A canonical example is the SE loss, which yields the M-estimator for the mean: μ̂(y) ∶= argmin θ∈Θ ∑ (θ-yi)2 n i=1 , Θ⊆R (2.12) Consistency is a key property for an estimator. An estimator θ̂ is consistent if it converges in probability to the true parameter value θ0, i.e. θ̂(y) P→θ0. For instance, μ̂(y) 8 is consistent for EFy[y], where Fy is the cumulative distribution function (CDF) of y, and EFy[y] is the expectation of y. A fundamental connection exists between the two distinct procedures of estimation and prediction evaluation for the case of M-estimators. Under general conditions, an Mestimator is consistent for the parameter θ0 if and only if the loss function L in eq. (2.11) is strictly consistent for the functional T corresponding to that parameter [10]. 2.3 The original index of agreement loss function The index of agreement loss function admits the following vectorized representation: LW(z, y) = ‖z-y‖2 2/‖(|z-1μ(y)| + |1μ(y) -y|)‖2 2 (2.13) where z= (z1, ... , zn)T represents the vector of model predictions, y= (y1, ... , yn)T represents the vector of observed values, ‖x‖2 is the Euclidean norm (eq. (A.9)), |x| indicates the element-wise absolute value (eq. (A.4)) and 1 denotes the all-ones vector defined in eq. (A.3). LW is defined when y is non-constant (i.e., has at least one element different from the others) or z≠y (where ≠ indicates pairwise inequality for at least one vector element; see eq. (A.6)). This vectorized form simplifies theoretical analysis. For instance, Proof D.1 in Appendix D establishes that LW is bounded: 0 ≤LW(z, y) ≤1 (2.14) As noted in the Introduction, LW is negatively oriented. Furthermore, LW(z, y) = 0, if and only if z= y and y is non-constant. 3. The index of agreement This section examines the original index of agreement and introduces theoretically consistent variants developed from its foundational construction principles. Formal proofs for the presented results are given in Appendix D. 3.1 The index of agreement loss function This section details key properties of the original index of agreement loss function LW. 3.1.1 Translation invariance The loss function LW is translation invariant, meaning that shifting predictions and realizations by a constant c∈R its value does not change (see Proof D.2): 9 LW(z+ c1, y+ c1) = LW(z, y), c∈R (3.1) 3.1.2 Scale invariance The loss function LW exhibits scale invariance, meaning its value remains unchanged when both predictions and observations are scaled by a non-zero-constant c∈R\{0}: LW(cz, cy) = LW(z, y), c∈R\{0} (3.2) This property follows directly from the definition in eq. (2.13). Therefore, its behavior remains unchanged across different measurement units. 3.1.3 Extremum estimation To analyze LW as an extremum estimator for a scalar parameter θ, we minimize it for a constant vector θ1, given non-constant observations y: θ̂(y) ∶= argmin θ∈Θ LW(θ1, y) (3.3) As shown in Proof D.3, this minimization yields two distinct estimators: θ̂(y) = μ(y) ± σ(y) (3.4) Figure 1 illustrates LW(θ1, y) as a function of θ for a realization y of y, indicating the points where it attains its minimum. The minimum value of LW(θ1, y) is (σ(y))/(σ(y) + mad(y)), where mad(y) is the mean absolute deviation defined in eq. (A.20). Figure 1. Illustration of LW(θ1, y) function. The non-uniqueness precludes consistency, as the estimator does not converge to a single parameter value. To resolve this, we define two modified estimators, i.e. the upper10 bound estimator θ̂+(y) = max{argmin θ∈Θ LW(θ1, y)} = μ(y) + σ(y) (3.5) and the lower-bound estimator θ̂-(y) = min{argmin θ∈Θ LW(θ1, y)} = μ(y) -σ(y) (3.6) Proof D.4 establishes that, under independent identically distributed (IID) variables assumption, θ̂+(y) and θ̂-(y) are consistent for EF[y] + VF 1/2[y] and EF[y] -VF 1/2[y], respectively, where VF[y] is the variance of y. 3.1.4 Estimation of a linear model An open problem is the estimation of the true parameters θ0 = (a0, b0) ∈R2 for the linear model z= ax+ b1, using the LW loss. The estimator θ̂(x, y) ∶= argmin θ∈R2 LW(ax+ b1, y) (3.7) lacks a known closed-form solution. 3.2 The new agreement loss function The original index of agreement (eq. (2.13)) is fundamentally a ratio of two squared Euclidean norms. While the numerator ‖z-y‖2 2 directly measures the magnitude of prediction errors, the denominator involves the norm of the composite vector |z-1μ(y)| + |1μ(y) -y|. This construction, which applies element-wise absolute values, complicates geometric interpretation. To address this, we propose an alternative loss function LNR2 based on the Euclidean distances between the vectors z, y and the mean vector 1μ(y). These vectors form a triangle in Rn, satisfying the triangle inequality in eq. (A.31) 0 ≤‖z-y‖2 ≤‖z-1μ(y)‖2 + ‖1μ(y) -y‖2 (3.8) Leveraging this inequality, we define: LNR2(z, y) ∶= ‖z-y‖2 2/(‖z-1μ(y)‖2 + ‖1μ(y) -y‖2)2 (3.9) where the subscript NR stands for norm ratio. LNR2 is defined when y is non-constant or z≠y. An equivalent expression is: LNR2(z, y) = (∑ (zi-yi)2 n i=1 )/(√∑ (zi-μ(y))2 n i=1 + √∑ (μ(y) -yi)2 n i=1 )2 (3.10) LNR2 has the following geometric and theoretical advantages: 11 (i) (Intuitive geometric foundation): Unlike the original index in eq. (1.1), where the denominator sums element-wise absolute deviations before applying the Euclidean norm, LNR2 operates directly on the vectors z-1μ(y) and 1μ(y) -y. This aligns with the Euclidean triangle inequality in eq. (3.8), treating prediction errors as vector displacements in Rn. The denominator represents the total path length from z to y via 1μ(y), while the numerator is the direct displacement. (ii) (Theoretical consistency): The element-wise absolute values in the original denominator disrupt vector-space properties. By eliminating these, LNR2 preserves translation and scale invariance (inherited from LW, see Sections 3.2.3 and 3.2.2) and maintains compatibility with norm axioms avoiding ad hoc element-wise operations. This enables cleaner analysis (e.g. closed-form solutions for linear models; see Section 3.2.5). (iii) (Interpretability): LNR2 quantifies the inefficiency of predictions relative to a "baseline path" through 1μ(y). 3.2.1 Bounds From eqs. (3.8) and (3.9), it follows immediately that: 0 ≤LNR2(z, y) ≤1 (3.11) Furthermore, LNR2(z, y) = 0, if and only if z= y and y is non-constant. Applying eq. (A.32) shows that LNR2 attains its upper bound of 1 in the following cases: (i) When y= 1μ(y) and z≠y. (ii) When z= 1μ(y) and y is not constant. (iii) When z-1μ(y) = a(1μ(y) -y) for some a> 0 and y is not constant. 3.2.2 Translation invariance Analogous to LW, the loss function LNR2 is translation invariant (see Proof D.5): LNR2(z+ c1, y+ c1) = LNR2(z, y), c∈R (3.12) 3.2.3 Scale invariance The scale invariance of LNR2 LNR2(cz, cy) = LNR2(z, y), c∈R\{0} (3.13) follows directly from its definition in eq. (3.9). 12 3.2.4 Extremum estimation To analyze LNR2 as an extremum estimator for a scalar parameter θ, we minimize it for a constant vector θ1, given non-constant observations y: θ̂(y) ∶= argmin θ∈Θ LNR2(θ1, y) (3.14) As shown in Proof D.6, this minimization yields two distinct estimators: θ̂(y) = μ(y) ± σ(y) (3.15) Figure 2 illustrates LNR2(θ1, y) as a function of θ for a realization y of y, indicating the points where it attains its minimum. The minimum value of LNR2(θ1, y) is 1/2. Figure 2. Illustration of LNR2(θ1, y) function. The non-uniqueness precludes consistency, as the estimator does not converge to a single parameter value. To resolve this, we define the upper-bound estimator θ̂+(y) = max{argmin θ∈Θ LNR2(θ1, y)} = μ(y) + σ(y) (3.16) and the lower-bound estimator θ̂-(y) = min{argmin θ∈Θ LNR2(θ1, y)} = μ(y) -σ(y) (3.17) Under the IID assumption, Proof D.4 establishes that θ̂+(y) and θ̂-(y) are consistent for EF[y] + VF 1/2[y] and EF[y] -VF 1/2[y], respectively. 3.2.5 Estimation of a linear model For the linear model z= ax+ b1, we estimate the true parameters θ0 = (a0, b0) ∈R2, 13 using the LNR2 loss, via the extremum estimator θ̂ = (â, b̂) θ̂(x, y) ∶= argmin θ∈R2 LNR2(ax+ b1, y) (3.18) As shown in Proof D.7 (eqs. (D.123) and (D.124)), the solution is: θ̂(x, y) = (sign(ρ(x, y))(||yc||2/||xc||2), μ(y) -(sign(ρ(x, y))μ(x)||yc||2)/ ||xc||2) (3.19) where xc and yc are centered vectors defined in eq. (A.23), ρ(x, y) is the Pearson correlation defined in eq. (A.28) and sign(x) is the scalar sign function, defined in eq. (A.16). At this point, LNR2 attains its minimum value (see eq. (D.125)): LNR2(âx+ b̂1, y) = (1 -|ρ(x, y)|)/2 (3.20) When a correctly specified linear model is trained using the SE loss, its predictions correspond to the conditional mean EFy|x[y|x]. This occurs because the SE-derived estimator is an M-estimator, and the SE loss is strictly consistent for the mean (see Section 2.1.2), i.e., it is uniquely minimized in expectation by the mean prediction [10]. While such results are well-established for M-estimators, their extension to the broader class of extremum estimators remains an open problem. For completeness, we also present the standard least-squares regression parameter estimates: θ̂(x, y) = (ρ(x, y))(||yc||2/||xc||2), μ(y) -(ρ(x, y)μ(x)||yc||2)/||xc||2) (3.21) 3.3 Generalization of the new index of agreement Legates and McCabe Jr [8] proposed the following loss function as an improvement of the original index of agreement LLMC(z, y) ∶= ∑ |zi-yi| n i=1 / ∑ (|zi-f(y)| + |f(y) -yi|) n i=1 (3.22) where f(y) is a benchmark method (e.g. climatology or persistence). LLMC is the equalweight special case of a weighted variant of the index of agreement introduced by [29]. It admits the vectorized formulation: LLMC(z, y) = ‖z-y‖1/‖(|z-1f(y)| + |1f(y) -y|)‖1 (3.23) where ‖x‖1 is the Manhattan norm, defined in eq. (A.8). Krause et al. [9] later introduced a generalization of the form LKBB(z, y) ∶= ∑ |zi-yi|p n i=1 / ∑ (|zi-μ(y)| + |μ(y) -yi|)p n i=1 , p= 1,2, ... (3.24) This admits the vectorized formulation 14 LKBB(z, y) = ‖z-y‖p p/‖(|z-1μ(y)| + |1μ(y) -y|)‖p p, p= 1,2, ... (3.25) where ‖x‖p is the Lp-norm, defined in eq. (A.10), and the Chebyshev norm as the special case for p= ∞, defined in eq. (A.11). LKBB is another equal-weight special case of the weighted variant introduced by [29], for p= 1, 2, .... However, this definition shares the theoretical limitations of LW (see Section 3.2). Furthermore, it employs the mean climatology 1μ(y) as the benchmark, which aligns naturally with Euclidean norms but not necessarily with other Lp-norms. This is less flexible than the definition of LLMC, which allows for a tailored climatology f(y). For instance, the Manhattan norm ‖x‖1 corresponds more naturally to the median climatology 1median(y) [1]. Analogous to LNR2, we propose the loss function LNRp (defined for non-constant y or z≠y) LNRp(z, y) ∶= ‖z-y‖p p/(‖z-1Lp-mean(y)‖p+ ‖1Lp-mean(y) -y‖p)p (3.26) where Lp-mean(y) is the Lp mean of y [30]. Examples of Lp-mean are the sample median, which equals the L1 mean, and the sample mean, which equals the L2 mean. The proposed LNRp retains the advantages of LNR2 (see Section 3.2), including its intuitive geometric foundation, theoretical consistency and interpretability, while adapting the benchmark method to the chosen norm. For p= 1 and setting f(y) = Lp-mean(y), the LLMC loss is a special case of the LNRp loss. Although a complete theoretical analysis of LNRp is left for subsequent work, the following bound follows directly from its construction: 0 ≤LNRp(z, y) ≤1 (3.27) 4. Applications This section presents applications designed to explore the implications of our theoretical insights. We analyze both simulated data and real-world case studies using hydrologic models. All computations were performed in R (version 4.5.1, [31]) within the RStudio integrated development environment (version 2025.09.0+387). Supplementary materials provide reproducible code and datasets for these analyses, and the specific R packages used are listed in Appendix E. 4.1 Predicting a Gaussian random variable In the first example, we simulate data from a Gaussian random variable y~N(μ, σ) (whose probability density function (PDF) is defined in eq. (C.1)). A sample of n= 1 000 15 realizations was simulated with true parameters μ= 0 and σ= 1. Using the first 500 samples as the training set, we computed three climatology-based predictive models: (i) Model #1: z= μ(y1:500). (ii) Model #2: z= μ(y1:500) -σ(y1:500). (iii) Model #3: z= μ(y1:500) + σ(y1:500). These three models were then used to forecast the remaining 500 samples (the test set), as illustrated in Figure 3. Figure 3. 500 observations from the test set of a random variable y~N(0,1). The values μ(y), μ(y) -σ(y), μ(y) + σ(y) were computed from the 500 obsevations in the training set and are used as constant predictions in the test set. We evaluate the performance of these three models on the test set in Table 1. To align with the negative orientation of the other metrics (MSE, LNR2 and LW), we use 1 - NSE(z, y), instead of the conventional NSE. This transformed metric is bounded below by 0. 16 Table 2. Performance of models on the test set. Predictions z are the three climatologies computed from the training set (models #1, #2 and #3), while observations y= y501:1000 are from the test set. Metric Model #1 Model #2 Model #3 MSE(z, y) 1.020768 1.858657 2.146621 1 -NSE(z, y) 1.005198 1.830307 2.113878 LNR2(z, y) 0.8745487 0.5010791 0.5003633 LW(z, y) 0.8979654 0.5590595 0.5583535 Due to random sampling, the results on the test set deviate slightly from the theoretical values expected from the training set (omitted for brevity). However, the sample is sufficiently large to confirm the theoretical findings from Section 3. As noted in Section 2.1.3, using MSE or 1 -NSE does not alter the model rankings. Model #1 outperforms the others with respect to MSE, as expected since the MSE is based on the LSE loss, strictly consistent for the mean. Models #2 and #3 perform similarly under MSE, a result subject to random variation but expected given the symmetry of both the LSE loss and the Gaussian distribution. In contrast, the rankings are completely reversed when comparing models using the LNR2 and LW losses. These results still align with the theory developed in Section 3. Models #2 and #3 demonstrate similar performance and significantly outperform model #1 under both metrics. The value of LNR2 approaches its theoretical minimum of 0.50 (subject to randomness), confirming the finding illustrated in Figure 2. Similarly, the values of LW are near their theoretical minimum ((σ(y))/(σ(y) + mad(y))), as shown in Figure 1. These results may appear paradoxical at first glance. Based on Figure 3 and prior experience with the LW loss, one might expect model #1 to outperform the others on all metrics. However, the superior performance of models #2 and #3 under the LNR2 and LW losses is explained in Sections 3.1.3 and 3.2.4. These sections establish that the climatologies μ(y) ± σ(y) are the minimizers for these specific loss functions, when fitting to a sample from a probability distribution. 4.2 Predicting data from a linear model with error In the second example, we simulate data from a Gamma-distributed predictor random variable x~Gamma(θ, a) (whose PDF is defined in eq. (C.2)) and a log-normal error random variable ε~Lognormal(μ, σ) (whose PDF is defined in eq. (C.3)). We generate n= 4 000 realizations using the true parameters θ= 1.8, a= 0.4, μ= 0 and σ= 2. We then use these simulations to generate 4 000 realizations of y, from the linear model y= a0 + 17 a1x+ ε, with a fixed intercept a0 = 2.1. We examine three cases for the slope coefficient a1: a1 = 0.6 (Figure 4a), a1 = 6.0 (Figure 4b) and a1 = 20.0 (Figure 4c). We intentionally introduced highly asymmetric variables in this simulation to ensure the results are sufficiently general, despite their empirical nature. 18 Figure 4. Predictions of linear models fitted using different loss functions. Black dots represent 2 000 test observations of y= 2.1 + a1x+ ε, where x~Gamma(1.8, 0.4) and ε~Lognormal(0, 2). Results are shown for three slope values: (a) a1 = 0.6, (b) a1 = 6.0, (c) a1 = 20.0. Overlaid are predictions from linear models z= â0 + â1x fitted on a 2 000observation training set (not shown) using the SE loss (red lines), the LNR2 loss (blue lines) and the LW loss (green lines). We first detail the procedure for the dataset corresponding to a1 = 0.6. We split the data into a training set of 2 000 observations and a test set of 2 000 observations. We then 19 fit linear models of the form z = a0 + a1x to the training set using three different loss functions. (i) Model #1: Fit using the SE loss (least squares regression). (ii) Model #2: Fit using the LNR2 loss. (iii) Model #3: Fit using the LW loss. The estimated parameters for model #1 are â01, â11 (given by eq. (3.21)), for model #2, they are â02, â12 (given by eq. (3.19)) and for model #3, they are â03, â13 (obtained by numerical minimization, as explicit estimators have not been derived; see Section 3.1.4). We then generate predictions for the test set using the equation z= â0 + â1x. We repeat this entire procedure for the other two true values of a1, while keeping the parameters of the probability distributions and the intercept fixed. Figure 4 presents the predictions from all models in the test set and Table 3 summarizes their numerical performance. Table 3. Estimated parameters and test set performance for the three linear models from Section 4.2. For each value of the slope coefficient a1, the table shows the parameters estimated on the training set and the subsequent performance metrics computed on the test set observations y= y2 001:4 000. Predictions z are generated by each of the three fitted models. model #1 model #2 model #3 model #1 model #2 model #3 model #1 model #2 model #3 Parameter/ Metric a1 = 0.6 a1 = 0.6 a1 = 0.6 a1 = 6.0 a1 = 6.0 a1 = 6.0 a1 = 2.1 a1 = 2.1 a1 = 2.1 â0 8.0655037 -5.10243 -16.74804 8.065504 -2.011659 -6.295647 8.065504 2.456986 1.750401 â1 0.8935388 19.82837 12.47191 6.293539 20.783999 20.352048 20.293539 28.358309 28.347614 MSE(z, y) 580.3332 996.0642 1002.823 580.3332 813.2273 818.57 580.3332 641.3484 641.8254 LNR2(z, y) 0.9096833 0.4514271 0.4692202 0.5399942 0.332705 0.335796 0.1736741 0.1425356 0.1426502 LW(z, y) 0.9657116 0.7132495 0.7077607 0.6732447 0.468934 0.466139 0.1963293 0.1620158 0.1619878 V̅mean(z, y) -0.367136 0.4266737 -16.5665 -0.367136 0.371751 -4.226231 -0.367136 0.3898335 -0.3245264 The predictions in Figure 4 and the parameter estimates in Table 3 show that the estimated models differ for lower values of a1, but converge toward identical solutions as a1 increases. For models #1 and #2, this behavior is explained by eqs. (3.19) and (3.21). As a1 increases, the correlation ρ(x, y) tends toward 1, causing sign(ρ(x, y) to become equal to ρ(x, y). The model comparisons under each loss function are consistent with the theoretical framework established by [1], [24], as discussed in Section 2.1.5. Specifically, each model outperforms the others with respect to the loss function used for its training. Model #1 achieves the lowest MSE, model #2 the lowest empirical LNR2 loss and model #3 the lowest empirical LW loss. As a1 increases, the models become increasingly similar, and their performance metrics converge. The values of the LNR2 and LW losses are very similar, 20 which we attribute to the limited range [0,1] of these functions, as well as a potential equivalence for predictor-dependent variables with specific properties. The numerical experiment also confirms that model #1 accurately identifies the conditional mean, as theoretically expected from Sections 2.1.4 and 2.1.5) (the value of V̅mean is nearly 0, consistently across all values of a1). Notably, model #2 also exhibits this property, although this does not imply it can theoretically identify the conditional mean. In contrast, model #3 does not appear to identify the mean, at least for low values of a1. As a1 increases, its behavior becomes similar to that of #1 and #2. 4.3 Application to hydrologic modeling To assess the practical relevance of our theoretical analysis in a hydrologic context, we applied the methods using the Génie Rural (GR) GR4J daily lumped hydrologic model [32]. The model was implemented via the airGR R package ([33], [34]). We modified the software to include the SE, LNR2 and LW loss functions as options for model extremum estimation as outlined in see Section 2.2. We applied the model to daily precipitation, potential evapotranspiration and streamflow data from 10 catchments in France. These catchments were selected from the airGRdatasets R package [35] based on the availability of complete records without missing values. Since these results are broadly applicable to various hydrologic models and catchments as well as machine learning regression models, we refrain from providing exhaustive details on the models and data herein. The software implementation is available as Supplementary Material. The implementation was straightforward. We used the year 1999 as a warm-up period, the period from 2000-2008 for model calibration, and 2009-2018 for model testing (validation). For each catchment, we estimated three model instances, each using a different loss function: (i) Model #1: Estimated using the SE loss (least squares regression). (ii) Model #2: Estimated using the LNR2 loss. (iii) Model #3: Estimated using the LW loss. Figure 5 presents examples of streamflow predictions from all three models for the year 2018 at catchment A273011002 (other years are omitted for visual clarity). The predictions from all models are highly similar. This visual similarity is confirmed by 21 Figure 6, which plots observations against predictions for each model, showing a close alignment to the 1:1 line for all three. Figure 5. Predictions from the three models detailed in Section 4.3 for the year 2018 at catchment A273011002. 22 Figure 6. Observations plotted against predictions for the three models detailed in Section 4.3 for the year 2018 at catchment A273011002. To quantify the differences between the models, we report their performance metrics for the calibration (estimation) period in Table 4 and for the validation (testing) period in Table 5. Presenting results for both periods is important due to the potential for statistical dissimilarity between the calibration and validation datasets, a factor not present in the controlled simulation setting of Section 4.2. Table 4. Average losses (MSE, LNR2, LW) for all three models for each catchment in the calibration set. The best-performing model for each metric and catchment is highlighted in bold. Model Model #1 Model #2 Model #3 Model #1 Model #2 Model #3 Model #1 Model #2 Model #3 Basin/loss MSE MSE MSE LNR2 LNR2 LNR2 LW LW LW A273011002 0.8004038 0.8213370 0.8206815 0.0412526 0.0402055 0.0402058 0.0428323 0.0417655 0.0417652 A605102001 0.6815371 0.7118592 0.7116614 0.0574751 0.0549629 0.0549630 0.0604838 0.0578356 0.0578355 B222001001 0.1452777 0.1473876 0.1472875 0.0240881 0.0237482 0.0237484 0.0246229 0.0242898 0.0242896 F439000101 0.0156214 0.0158345 0.0158534 0.0220948 0.0217949 0.0217955 0.0225355 0.0222006 0.0222001 H010002001 0.1543760 0.1565821 0.1565649 0.0280440 0.0276462 0.0276462 0.0287580 0.0283501 0.0283500 H120101001 0.0987467 0.0999243 0.0998982 0.0225813 0.0223170 0.0223171 0.0230437 0.0227771 0.0227770 H622101001 0.0719342 0.0725890 0.0725868 0.0151390 0.0150013 0.0150013 0.0153548 0.0152148 0.0152148 J171171001 0.0999581 0.1008159 0.1008084 0.0159707 0.0158368 0.0158368 0.0162061 0.0160709 0.0160709 J421191001 0.1908086 0.1919245 0.1919046 0.0109564 0.0108926 0.0108926 0.0110576 0.0109942 0.0109942 K134181001 0.1093134 0.1104352 0.1103770 0.0167198 0.0165522 0.0165523 0.0169642 0.0168023 0.0168021 The results in the validation period (Table 5) were less uniform. For instance, Model #2 outperformed Model #1 with respect to MSE in 7 out of 10 catchments. This is expected, given the close similarity of the parameter estimates across all models and the potential for statistical dissimilarity between the calibration and validation periods. 23 Table 5. Average losses (MSE, LNR2, LW) for all three models for each catchment in the validation (test) set. The best-performing model for each metric and catchment is highlighted in bold. Model Model #1 Model #2 Model #3 Model #1 Model #2 Model #3 Model #1 Model #2 Model #3 Basin/loss MSE MSE MSE LNR2 LNR2 LNR2 LW LW LW A273011002 0.7685125 0.7506903 0.7507642 0.0454044 0.0422782 0.0423143 0.0471956 0.0439308 0.0439685 A605102001 0.4491356 0.4854598 0.4855521 0.0443461 0.0437923 0.0438100 0.0461854 0.0456384 0.0456576 B222001001 0.1360638 0.1306809 0.1307524 0.0243684 0.0227379 0.0227663 0.0248823 0.0232137 0.0232427 F439000101 0.0315423 0.0320811 0.0320874 0.0343641 0.0339732 0.0339406 0.0355261 0.0350892 0.0350533 H010002001 0.1590732 0.1455477 0.1455789 0.0211302 0.0188218 0.0188279 0.0214761 0.0191156 0.0191217 H120101001 0.2413662 0.2227167 0.2229445 0.0348958 0.0316126 0.0316505 0.0356209 0.0322373 0.0322764 H622101001 0.1128062 0.1053143 0.1053167 0.0299121 0.0274271 0.0274283 0.0305397 0.0279809 0.0279821 J171171001 0.1318598 0.1262418 0.1262628 0.0182125 0.0171466 0.0171507 0.0184918 0.0174056 0.0174098 J421191001 0.2334413 0.2259492 0.2259939 0.0110676 0.0105920 0.0105951 0.0111783 0.0106949 0.0106981 K134181001 0.0996014 0.1009891 0.1009341 0.0136642 0.0135830 0.0135825 0.0138255 0.0137501 0.0137495 During the calibration period (Table 4), each model performed best with respect to the specific loss function used for its training, as expected. However, the empirical loss values differed only minimally between models. This suggests that for this catchment dataset, the relationship between inputs and outputs is sufficiently strong that the different estimators converge to similar parameter sets, analogous to the high-correlation scenario illustrated in Figure 4c. Notably, in some cases the parameter estimates were so similar that models #2 and #3 achieved nearly identical performance under both the LNR2 and LW losses. Table 6 shows the empirical average identification function V̅mean for the mean, computed on the validation set. As expected from theory (Section 2.1.4), model #1, estimated with the SE loss, correctly identifies the mean (values near zero), indicating adequate absolute performance for mean prediction. Although not proven theoretically, Models #2 and #3 also produced values of V̅mean near zero. This is a consequence of their parameter estimates being very similar to those of model #1, as noted in the discussion of Table 4. Table 6. Empirical average identification function V̅mean for all three models for each catchment in the validation set. The value closest to zero (indicating best identification of the mean) for each catchment is highlighted in bold. Model Basin Model #1 Model #2 Model #3 A273011002 -0.1749492 -0.1325392 -0.1332546 A605102001 -0.0438793 -0.0091700 -0.0101047 B222001001 -0.0272470 -0.0061254 -0.0065730 F439000101 -0.0570816 -0.0526462 -0.0523539 H010002001 -0.0591676 -0.0388627 -0.0389624 H120101001 -0.1592403 -0.1394463 -0.1397120 H622101001 -0.1033602 -0.0914236 -0.0913975 J171171001 -0.0452491 -0.0329591 -0.0330311 J421191001 -0.1414669 -0.1233617 -0.1234942 K134181001 0.0775674 0.0939171 0.0936123 24 5. Discussion At first glance, it may appear counterintuitive that LNR2, when used for parameter estimation of a distribution, yields two distinct estimators, that differ from the mean. This contrasts with the empirical finding that, when used to fit a linear model, its predictions appear to equal the marginal mean of the dependent variable y. This apparent contradiction may be explained by the fact that LNR2 is defined only when y is nonconstant or z≠y. Consequently, the expectation EF[LNR2(z, y)] is defined only for distributions where y is non-constant. This restriction may introduce irregularities if one attempts its minimization as in eq. (2.4). This restriction may be circumvented when fitting a linear model, which allows for non-constant predictions z (for which LNR2 is defined irrespectively of the distribution of y). Similarly, LW does not identify a specific functional when fitting a probability distribution and its predictions do not necessarily have a mean equal to the marginal mean of y. In environmental sciences, it is sometimes assumed that predictions yielding a loss function value of 0 will perfectly predict the environmental process of interest, implying that a lower average loss invariably corresponds to a better prediction of the process itself. This concept must be understood precisely. Under the framework of expectation minimization of the loss function (eq. (2.4)), models predict specific functionals of the process rather than the process itself. For example, a linear model fitted with a quantile loss will predict conditional quantiles, not the process, even though the quantile loss equals 0 for a perfect prediction z= y [36]. Applying this to our study, consider the analysis in Section 4.1. The mean climatology appears to better predict the process itself, yet it achieves higher (worse) LNR2 and LW losses compared to predictions of the form μ(y) ± σ(y). All losses equal 0 only for a perfect prediction z= y. This result is less paradoxical when considering that models predict functionals of the predictive distribution rather than the process itself. That principle already applied in hydrologic science and engineering to predict quantiles ([37], [38], [39]) and expectiles (an extension of the mean functional [40]) using hydrologic models [41]. In environmental sciences and engineering, it is common practice to assess predictions using multiple metrics, and debates continue over which metrics are superior [8]. This perspective has driven the introduction of numerous new metrics. This practice should 25 be contextualized within statistical theory, which advocates for the a priori specification of the loss function used for model assessment ([1], [24]), as model rankings are critically dependent on this choice. Our work underscores the importance of this approach, while establishing the theoretical properties of the LNR2 and LW loss functions. In this context, we do not argue for the universal superiority of one loss function. Each loss function has distinct theoretical merits and, more importantly, each is intrinsically linked to the specific functional we aim to predict. The SE loss is simpler to use and highly interpretable, as it elicits the mean. The LNR2 loss yields specific estimates when used as an extremum estimator on a sample from a probability distribution, though it lacks consistency due to non-uniqueness. However, when applied to a linear model, it provides unique parameter estimates, and our empirical evaluation suggests that the expectation of its predictions equals the mean of the marginal distribution of the dependent variable. A significant advantage of LNR2 is its translation and scale invariance, coupled with its bounded range of [0,1]. In environmental science and engineering, such boundedness is often perceived to enhance the interpretability of prediction assessments. LW also possesses translation and scale invariance and is bounded in [0,1]. However, it likely lacks the property of predicting the marginal mean of the dependent variable, when used to fit a linear model and does not yield explicit parameter estimates. It would be interesting to examine the consistency of LNR2 and LW for multidimensional functionals (our analysis was restricted to one-dimensional functionals), a topic discussed by [1]. For instance, the general form of a consistent loss function for the component-wise expectation is a Bregman loss function ([12], [16], [42],) L(z, y) ∶= φ(y) -φ(z) -〈∇φ(z), y-z〉 (5.1) where φ: Rd→R is convex with gradient ∇φ: Rd→Rd. The SE loss is a special case of eq. (5.1). In this regard, a challenge for both LNR2 and LW is that they must be examined within the class of distributions for which y is not constant. Preliminary theoretical and empirical work by the authors on LNRp has not identified specific properties related to existing functionals. Determining what functional, if any, models fitted with LNRp identify remains an open problem. Such functionals, if they exist, are termed hydrograph functionals by [43] due to their significance in hydrologic science and engineering. 26 6. Conclusions In this study, we examined the theoretical properties of the index of agreement loss function, LW (the negatively oriented index of agreement). Our analysis established that LWis translation and scale invariant, bounded within [0,1], and, when employed as an extremum estimator for a sample from a probability distribution, yields the parameter estimates EF[y] ± VF 1/2[y]. We proposed LNR2 as a theoretically consistent improvement over the original index of agreement. Unlike LW, whose denominator relies on component-wise vector operations, LNR2 is constructed using Euclidean distances in both its numerator and denominator. This new formulation retains the appealing properties of translation and scale invariance and a [0,1] bound and it produces the same EF[y] ± VF 1/2[y] estimates when fitting a distribution. A significant strength of LNR2 is that it provides closed-form solutions when used as an extremum estimator to fit a linear model. Empirical analysis demonstrated that predictions of linear models fitted with LNR2 have mean equal to the mean of the dependent variable, a property shared with the squared error loss function (though the latter identifies the conditional mean). This represents a notable benefit over LW, for which we found no evidence of a similar identification property. When applied to calibrate hydrologic models, both loss functions resulted in parameter sets that produced remarkably similar performance across all empirical average loss metrics, including MSE. This behavior is analogous to the high-correlation scenario in linear regression, where the models converge as the correlation ρ(x, y) approaches 1. Finally, while variants of the LW loss based on norms other than the Euclidean norm have been proposed, we introduced a corresponding generalization, LNRp, based on the principles underlying LNR2. The theoretical properties of this generalized function remain an open area of research. Preliminary results did not indicate that models trained with the LNR1 loss, for instance, can identify the median or mean functional. Appendix A Vector notation This appendix defines the vector operations, norms, and statistical functions used throughout the manuscript. All vectors are column vectors in Rn. 27 Basic vector definitions Vector notation: x= (x1, ... , xn)T (A.1) where the superscript T denotes the transpose. Zero vector (all elements zero): 0= (0, ... ,0)T (A.2) All-ones vector (all elements unity): 1= (1, ... ,1)T (A.3) Element-wise operations Absolute value vector: |x| ∶= (|x1|, ... , |xn|)T (A.4) Element-wise comparison (holds for all i∈{1, ... , n}): x> y⇔xi> yi (similarly for ≥, =, ≤, 0 -1, x 0 (A.32) Norm bounds: ‖x‖2 ≤‖x‖1 ≤√n‖x‖2 (A.33) Variance identity ‖1μ(x) -x‖2 2 = 〈1μ(x) -x, 1μ(x) -x〉⇒ (A.34) ‖1μ(x) -x‖2 2 = ‖1‖2 2 μ2(x) + ‖x‖2 2 -2μ(x)〈1, x〉⇒ (A.35) ‖1μ(x) -x‖2 2 = nμ2(x) + ‖x‖2 2 -2nμ2(x) ⇒ (A.36) ‖1μ(x) -x‖2 2 = ‖x‖2 2 -nμ2(x) ⇒ (A.37) ‖1μ(x) -x‖2 2 = nσ2(x) (A.38) Appendix B Loss and identification functions This appendix defines the established loss and identification functions referenced throughout this paper. Let z= (z1, ... , zn)T be a vector of predictions and y= (y1, ... , yn)T be a vector of corresponding observed values. The absolute error loss function is defined as: LAE(z, y) ∶= |z-y| (B.1) The squared error loss function is defined as: LSE(z, y) ∶= (z-y)2 (B.2) The mean absolute error (MAE) is defined as: MAE(z, y) ∶= (1/n) ∑ LAE(zi, yi) n i=1 (B.3) 30 The mean squared error (MSE) is defined as: MSE(z, y) ∶= (1/n) ∑ LSE(zi, yi) n i=1 (B.4) The Nash-Sutcliffe efficiency (NSE) [17] is defined as: NSE(z, y) ∶= 1 -MSE(z, y)/MSE(1μ(y), y) (B.5) A strict identification function for the median [1] is defined as: Vmedian(z, y) ∶= 1[0,∞){z-y} -1/2 (B.6) where 1A{x} is the indicator function defined in eq. (A.18). A strict identification function for the mean [1] is defined as: Vmean(z, y) ∶= z-y (B.7) The empirical average identification function V̅median is defined as: V̅median(z, y) = (1/n) ∑ Vmedian(zi, yi) n i=1 (B.8) The empirical average identification function V̅mean is defined as: V̅mean(z, y) = (1/n) ∑ Vmean(zi, yi) n i=1 (B.9) Appendix C Probability distributions This appendix defines the probability density functions (PDFs) for the probability distributions referenced in the main text. The PDF of a Gaussian (normal) random variable y~N(μ, σ) is defined as: fN(y; μ, σ) ∶= 1 σ√2πexp (- (y - μ)2 2σ2 ) , y∈R, μ∈R, σ> 0 (C.1) The PDF of a Gamma distributed random variable y~Gamma(θ, a) is defined as: fGamma(y; θ, a) ∶= 1 θaΓ(a) ya-1exp(-y/θ), y≥0, θ> 0, a> 0 (C.2) The PDF of a log-normally distributed random variable y~Lognormal(μ, σ) is defined as: fLognormal(y; μ, σ) ∶= 1 yσ√2πexp (- (log (y) - μ)2 2σ2 ) , y> 0, μ∈R, σ> 0 (C.3) Appendix D Proofs Proof D.1 (Bound of LW, adapted from [45]): The bound 0 ≤LW(z, y) ≤1 is established as follows: (i) Lower bound The lower bound follows immediately, since both numerator and denominator are 31 squared Euclidean norms, which are non-negative. (ii) Upper bound Starting from the element-wise triangle inequality for vectors (eq. (A.5)): |z-y| = |z-1μ(y) + 1μ(y) -y| ≤|z-1μ(y)| + |1μ(y) -y|, (D.1) where the inequality holds element-wise. Applying the Euclidian norm (which preserves order for non-negative vectors), squaring and by leveraging eq. (A.14) yields ‖z-y‖2 2 ≤‖(|z-1μ(y)| + |1μ(y) -y|)‖2 2 (D.2) Thus LW(z, y) ≤1.∎ Proof D.2 (Translation invariance of LW): Substituting translation vectors into eq. (2.13): LW(z+ c1, y+ c1) = ‖(z+ c1) -(y+ c1)‖2 2/‖(|(z+ c1) -1μ(y+ c1)| + |1μ(y+ c1) -(y+ c1)|)‖2 2 ⇒ (D.3) LW(z+ c1, y+ c1) = ‖z-y‖2 2/‖(|z-1μ(y)| + |1μ(y) -y|)‖2 2 ⇒ (D.4) LW(z+ c1, y+ c1) = LW(z, y), c∈R (D.5) Thus LW is translation invariant.∎ Proof D.3 (LW for extremum estimation): We show that argmin θ∈Θ LW(θ1, y) = μ(y) ± σ(y). For z= θ1 and y not constant, the nominator of LW is: ‖z-y‖2 2 = ‖θ1-y‖2 2 (D.6) By expanding the norm (eq. (A.26)) it follows that: ‖z-y‖2 2 = ‖θ1‖2 2 + ‖y‖2 2 -2〈θ1, y〉⇒ (D.7) ‖z-y‖2 2 = nθ2 + ‖y‖2 2 -2nθμ(y) ⇒ (D.8) ‖z-y‖2 2 = n(θ-μ(y))2 + ‖y‖2 2 -nμ2(y) (D.9) By substituting with the sample variance (eq. (A.22)) it follows that: ‖z-y‖2 2 = n(θ-μ(y))2 + nσ2(y) (D.10) The denominator expands to: ‖(|z-1μ(y)| + |1μ(y) -y|)‖2 2 = ‖|z-1μ(y)|‖2 2 + ‖|1μ(y) -y|‖2 2 + 2〈|z1μ(y)|, |1μ(y) -y|〉⇒ (D.11) ‖(|z-1μ(y)| + |1μ(y) -y|)‖2 2 = ‖|θ1-1μ(y)|‖2 2 + ‖|1μ(y) -y|‖2 2 + 2〈|θ11μ(y)|, |1μ(y) -y|〉 (D.12) The first term in eq. (D.12) is 32 ‖|θ1-1μ(y)|‖2 2 = (θ-μ(y))2‖1‖2 2 ⇒ (D.13) ‖|θ1-1μ(y)|‖2 2 = n(θ-μ(y))2 (D.14) From eq. (A.38), the second term in eq. (D.12) is ‖|1μ(y) -y|‖2 2 = nσ2(y) (D.15) The last term in eq. (D.12) is 〈|θ1-1μ(y)|, |1μ(y) -y|〉= |θ-μ(y)|〈|1|, |1μ(y) -y|〉⇒ (D.16) 〈|θ1-1μ(y)|, |1μ(y) -y|〉= |θ-μ(y)|〈1, |1μ(y) -y|〉⇒ (D.17) 〈|θ1-1μ(y)|, |1μ(y) -y|〉= |θ-μ(y)|‖1μ(y) -y‖1 ⇒ (D.18) 〈|θ1-1μ(y)|, |1μ(y) -y|〉= n|θ-μ(y)|mad(y) (D.19) where mad(y) is the sample mean absolute deviation of y defined in (A.20). Then LW(θ1, y) equals: LW(θ1, y) = (n(θ-μ(y))2 + nσ2(y))/(n(θ-μ(y))2 + nσ2(y) + 2n|θμ(y)|mad(y)) ⇒ (D.20) LW(θ1, y) = ((θ-μ(y))2 + σ2(y))/((θ-μ(y))2 + σ2(y) + 2|θμ(y)|mad(y)) (D.21) Let u1 = θ-μ(y) (D.22) u2 = σ(y) (D.23) u3 = mad(y) (D.24) After substituting eqs. (D.22), (D.23) and (D.24) in (D.21) we have: LW(θ1, y) = (u1 2 + u2 2)/(u1 2 + u2 2 + 2|u1|u3) (D.25) The partial derivative ∂LW(θ1,y) ∂θ is defined for θ≠μ(y). For θ> μ(y): ∂LW(θ1,y) ∂θ = (2u1(u1 2 + u2 2 + 2u1u3) -2(u1 + u3)(u1 2 + u2 2))/(u1 2 + u2 2 + 2u1u3)2 ⇒ (D.26) ∂LW(θ1,y) ∂θ = 2u3(u1 2 -u2 2)/(u1 2 + u2 2 + 2u1u3)2 (D.27) For θ μ(y): ∂2LW(θ1,y) ∂θ2 = (4u1u3u4 2 -8u3(u1 2 -u2 2)u4(u1 + u3))/u4 4 ⇒ (D.35) ∂2LW(θ1,y) ∂θ2 = (4u3(u1u4 -2(u1 2 -u2 2)(u1 + u3)))/u4 3 ⇒ (D.36) ∂2LW(θ1,y) ∂θ2 = (4u3(u1 3 + u1u2 2 + 2u1 2u3 -2(u1 3 + u1 2u3 -u1u2 2 -u3u2 2)))/ u4 3 ⇒ (D.37) ∂2LW(θ1,y) ∂θ2 = (4u3(-u1 3 + 3u1u2 2 + 2u3u2 2))/u4 3 ⇒ (D.38) ∂2LW(θ1,y) ∂θ2 = (4u3(2u3u2 2 + u1(3u2 2 -u1 2)))/u4 3 (D.39) For θ 0 (D.49) (iii) Its value at θ= μ(y) equals LW(μ(y)1, y) = 1 (D.50) (iv) The limits equal lim θ→∞LW(θ1, y) = lim θ→-∞LW(θ1, y) = 1 (D.51) (v) Therefore, the two local minima of LW(θ1, y) are global minima for θ> μ(y) and θ μ(y) 2σ(y)(θ-μ(y) + σ(y))/(σ(y) + μ(y) -θ)3, θ μ(y) (-4σ(y)(μ(y) -θ-2σ(y)))/(σ(y) + μ(y) -θ)4, θ 0 (D.69) (iii) Its value at θ= μ(y) equals LNR2(μ(y)1, y) = 1 (D.70) (iv) The limits equal lim θ→∞LNR2(θ1, y) = lim θ→-∞LNR2(θ1, y) = 1 (D.71) (v) Therefore, the two local minima of LNR2(θ1, y) are global minima for θ> μ(y) and θ 0, when w5 simplifies to: w5 = 2a‖xc‖2(a‖xc‖2 + ‖yc‖2)(a‖xc‖2 2‖yc‖2 -‖yc‖2〈xc, yc〉-‖xc‖2‖yc‖2 2 + a‖xc‖2〈xc, yc〉) (D.108) Therefore, w5 = 0 when: a= (‖yc‖2〈xc, yc〉+ ‖xc‖2‖yc‖2 2)/(‖xc‖2 2‖yc‖2 + ‖xc‖2〈xc, yc〉) ⇒ (D.109) a= ‖yc‖2/‖xc‖2 (D.110) The first case is a 0 and a 0 then LNR2(z, y) attains its minimum at (1 -ρ(x, y)/2 when â = ‖yc‖2/‖xc‖2 (D.121) If ρ(x, y) 0 and ρ(x, y) > 0, then t> 0 and: LNR2(z, y) = 1 -(2t(1 + ρ(x, y)))/(t+ 1)2 (D.133) This is minimized for t= 1, i.e. when a= â and LNR2(z, y) = (1 -|ρ(x, y)|)/2, which is a global minimum. (ii) If a 0, then t 0 and ρ(x, y) 0 and: LNR2(z, y) = 1 -(2t(1 -ρ(x, y)))/(t+ 1)2 (D.136) This is minimized for t= 1, i.e. when a= â and LNR2(z, y) = (1 -|ρ(x, y)|)/2, which is a global minimum. This completes the proof for the case of a. Case 2 (w4 = 0): We prove that LNR2(z, y) > (1 -|ρ(x, y)|)/2, when w4 = 0, where w4 is defined in eq. (D.97). We try to solve the equation: w4 = 0 ⇒ (D.137) (√w2 + ‖yc‖2)(2a〈xc, yc〉+ ‖yc‖2(√w2 -‖yc‖2)) = 0 ⇒ (D.138) -2a〈xc, yc〉= ‖yc‖2(√w2 -‖yc‖2) (D.139) This can also be written as: √w2 = ‖yc‖2 -2aρ(x, y)‖xc‖2 (D.140) Therefore, from eq. (D.89): w1 = w2 + ‖yc‖2 2 + ‖yc‖2(√w2 -‖yc‖2) ⇒ (D.141) w1 = w2 + ‖yc‖2√w2 ⇒ (D.142) w1 = √w2(√w2 + ‖yc‖2) (D.143) Furthermore, from eq. (D.88): w3 = (√w2 + ‖yc‖2)2 (D.144) Returning to eq. (D.90) and substituting the values of w1, √w2 and w3: LNR2(z, y) = √w2/(√w2 + ‖yc‖2) ⇒ (D.145) LNR2(z, y) = (‖yc‖2 -2aρ(x, y)‖xc‖2)/(2(‖yc‖2 -aρ(x, y)‖xc‖2)) (D.146) We set, w6 = (a‖xc‖2)/‖yc‖2 (D.147) and substitute in eq. (D.146): 43 LNR2(z, y) = (1 -2ρ(x, y)w6)/(2(1 -ρ(x, y)w6)) (D.148) Furthermore, the following inequalities hold (based on eqs. (D.87), (D.140) and (D.147)): √w2 ≥|a|‖xc‖2 ≥0 ⇒ (D.149) ‖yc‖2 -2aρ(x, y)‖xc‖2 ≥|a|‖xc‖2 ≥0 ⇒ (D.150) 1 -2ρ(x, y)w6 ≥|w6| ≥0 (D.151) Now we have to prove that LNR2(z, y) > (1 -|ρ(x, y)|)/2 or equivalently (based on eq. (D.148)) that: (1 -2ρ(x, y)w6)/(2(1 -ρ(x, y)w6)) ≥(1 -|ρ(x, y)|)/2 (D.152) or equivalently (1 -2ρ(x, y)w6)/(1 -ρ(x, y)w6) ≥1 -|ρ(x, y)| ⇒ (D.153) (-ρ(x, y)w6 + |ρ(x, y)| -ρ(x, y)|ρ(x, y)|w6)/(1 -ρ(x, y)w6) ≥0 (D.154) We examine two cases for proving the inequality in eq. (D.154), depending on the sign of ρ(x, y): Case 2.a: If ρ(x, y) > 0 we have to prove: (ρ(x, y)(-w6 + 1 -ρ(x, y)w6))/(1 -ρ(x, y)w6) ≥0 ⇒ (D.155) (ρ(x, y)(1 -w6(1 + ρ(x, y))))/(1 -ρ(x, y)w6) ≥0 (D.156) (i) Sign of the denominator: If 1 -ρ(x, y)w6 ≤0, then w6 ≥1/ρ(x, y). 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2510.14711
|
Fast and Scalable Score-Based Kernel Calibration Tests
Pierre Glaser1
David Widmann2
Fredrik Lindsten3
Arthur Gretton1
1Gatsby Computational Neuroscience Unit, University College London, London, UK
2Department of Information Technology, Uppsala University, Sweden
3Division of Statistics and Machine Learning, Linköping University, Sweden
Abstract
We introduce the Kernel Calibration Conditional
Stein Discrepancy test (KCCSD test), a non-
parametric, kernel-based test for assessing the cali-
bration of probabilistic models with well-defined
scores. In contrast to previous methods, our test
avoids the need for possibly expensive expecta-
tion approximations while providing control over
its type-I error. We achieve these improvements
by using a new family of kernels for score-based
probabilities that can be estimated without proba-
bility density samples, and by using a conditional
goodness-of-fit criterion for the KCCSD test’s U-
statistic. We demonstrate the properties of our test
on various synthetic settings.
1
INTRODUCTION
Calibration is a statistical property of predictive probabilis-
tic models that ensures that a model’s prediction matches
the conditional distribution of the predicted variable given
the prediction. A calibrated model expresses the uncertainty
about its predictions reliably by being neither over- nor un-
derconfident, and hence can be useful even if its accuracy
is suboptimal. This property is essential in safety-critical
applications such as autonomous driving. Unfortunately,
empirical studies revealed that popular machine learning
models such as deep neural networks tend to trade off cali-
bration for accuracy [Guo et al., 2017]. This has lead to an
increased interest in the study of calibrated models in recent
years.
Calibration has been studied in the meteorological and statis-
tical literature for many decades [e.g., Murphy and Winkler,
1977, DeGroot and Fienberg, 1983]. For a long time, re-
search on calibration has been focused on different notions
of calibration for probabilistic classifiers [e.g., Murphy and
Winkler, 1977, DeGroot and Fienberg, 1983, Platt, 2000,
Zadrozny and Elkan, 2001, Bröcker, 2009, Naeini et al.,
2015, Guo et al., 2017, Kull et al., 2017, Kumar et al., 2018,
Kull et al., 2019, Vaicenavicius et al., 2019, Widmann et al.,
2019] and on calibration of quantiles and confidence in-
tervals for real-valued regression problems [e.g., Ho and
Lee, 2005, Rueda et al., 2006, Taillardat et al., 2016, Song
et al., 2019, Fasiolo et al., 2020]. Regarding the calibra-
tion of classification models, different hypothesis tests have
been proposed [e.g., Cox, 1958, Bröcker and Smith, 2007,
Vaicenavicius et al., 2019, Widmann et al., 2019, Gweon,
2022, Lee et al., 2022]. Given a predictive model and a
validation dataset, these tests output whether a model is
likely to be uncalibrated. The recent work of Widmann et al.
[2021] generalized the calibration-framework introduced
for classification in [Widmann et al., 2019] to (possibly
multi-dimensional) continuous-valued predictive models. In
particular, Widmann et al. [2021] introduced a kernel-based
hypothesis test for such general classes of models.
An important potential consumer of calibration tests is
Bayesian inference, and in particular simulation-based in-
ference (SBI), for which miscalibration is particularly un-
desirable. SBI [Cranmer et al., 2020] lies at the intersection
of machine learning and domain sciences, and refers to the
set of methods that train probabilistic models to estimate
the posterior over scientific parameters of interest given
some observed data. The models are trained using pairs
composed of parameters drawn from a prior distribution,
and their associated “synthetic” observed data, obtained by
running a probabilistic program called the simulator, taking
a parameter value as input, and that faithfully mimics the
physical generative process of interest. The increasing num-
ber of use cases combined with advances in probabilistic
modeling has elevated SBI to a critical role in solving com-
plex scientific problems such as particle physics [Gilman
et al., 2018] and neuroscience [Glöckler et al., 2022, Glaser
et al., 2022]. However, as discussed in [Hermans et al.,
2021], overconfidence in SBI models can conceal credible
alternative scientific hypotheses, and result in incorrect dis-
coveries [Hermans et al., 2021], highlighting the need for
Accepted for the 39th Conference on Uncertainty in Artificial Intelligence (UAI 2023).
arXiv:2510.14711v1 [stat.ML] 16 Oct 2025
principled and performant calibration tests suitable for such
models.
While the theoretical framework of Widmann et al. [2021]
describes the calibration of any probabilistic model, ap-
plying its associated calibration test to Bayesian inference
remains challenging: indeed, the test statistics require com-
puting expectations against the probabilistic models of inter-
est, for reasons bearing both to the calibration setting, and
to the limitations of currently available kernel-based tools
for probabilistic models. Although such expectations can
be computed exactly for classification models, expectations
against generic probabilistic models are usually intractable
and must be approximated. In cases where the models are un-
normalized, these approximations are both computationally
expensive—sometimes, prohibitively—and biased, thereby
compromising theoretical guarantees of the calibration tests
of Widmann et al. [2021], including type-I error control.
Contributions In this paper, we introduce the kernel
calibration-conditional Stein discrepancy (or KCCSD) test,
a new nonparametric, score-based kernel calibration test
which addresses the limitations of existing methods. The
KCCSD test builds on the insight that the definition of cali-
bration given by Vaicenavicius et al. [2019] is a conditional
goodness of fit property, as we remark in Section 3. This
fact allows us to leverage the kernel conditional goodness of
fit test proposed by Jitkrittum et al. [2020] as the backbone
of the KCCSD test. Unlike the test-statistics of Widmann
et al. [2021], the KCCSD test statistic does not contain
explicit expectations against the probabilistic models; how-
ever, as in [Widmann et al., 2021], it requires evaluating a
kernel between probabilities densities, which in most cases
of interest introduces an (intractable) expectation against
the densities. To eliminate this limitation, we construct two
new kernels between probability distributions that do not
involve expectations against its input distributions, while
remaining suitable for statistical testing. These kernels rely
on a generalized version of the Fisher divergence and are of
independent interest. We investigate a connection between
these kernels and diffusion, akin to Stein methods, and dis-
cuss the relationships with other kernels on distributions. By
using such kernels in the KCCSD test statistic, we obtain
a fast and scalable calibration test that remains consistent
and calibrated for unnormalized models, answering the need
for such tests discussed above. We confirm in Section 6 the
properties and benefits of the KCCSD test against alterna-
tives on synthetic experiments.
2
BACKGROUND
Notation
We consider probabilistic systems character-
ized by a joint distribution P(X, Y ) of random variables
(X, Y ) taking values in X × Y, and study probabilistic
models P|· : x ∈X 7−→P|x(·) ∈P(Y) approximating
the unknown conditional distribution of Y given X = x,
P|x(·) ≃P(Y ∈· | X = x). The target variable Y is typ-
ically a parameter of a probabilistic system of interest—
like synaptic weights in biological neural networks—while
the input variable X is observed data—like neuron voltage
traces measured using electrophysiology.
2.1
CALIBRATION OF PREDICTIVE MODELS
Calibration: General Definition A probabilistic model P|·
is called calibrated or reliable [Bröcker, 2008, Vaicenavicius
et al., 2019, Widmann et al., 2021] if it satisfies
P|X = P
Y ∈· | P|X
P(X)-a.s..
(1)
Note that this definition applies to general predictive proba-
bilistic models, also beyond classification, and only assumes
that the conditional distributions on the right-hand side exist.
Hypothesis Testing: Kernel Calibration Error There are
multiple ways to test whether a given predictive probabilis-
tic model is calibrated. In this section, we introduce the
kernel-based tests of Widmann et al. [2019] and their later
generalization [Widmann et al., 2021], since our KCCSD
test is built on these approaches. These tests turn the equality
between conditional distributions present in Equation (1)
into a more classical equality between two joint distributions.
The transformation is achieved by noting that
P|X = P
Y ∈· | P|X
P(X)-a.s.
⇐⇒(P|X, Y )
d= (P|X, Z)
where Z is an “auxiliary” variable such that Z | P|X ∼
P|X(·). This identity was used by Widmann et al. [2021]
to construct an MMD-type calibration test based on the
(squared) kernel calibration error (SKCE) criterion
sup
h∈B(0H,1)
E(x,y,z)∼P(X,Y,Z)
h(P|x, y) −h(P|x, z)
. (2)
Here, B(0H, 1) is the unit ball of a reproducing kernel
Hilbert space (RKHS) H of functions with positive defi-
nite kernel kH : (P|X × Y)2 →R. As noted by Widmann
et al. [2021], the SKCE generalizes the (squared) kernel clas-
sification calibration error (SKCCE) defined for the special
case of discrete output spaces Y = {1, . . . , d} [Widmann
et al., 2019], to continuous ones. Given n pairs of sam-
ples {(P|xi, yi)}
n
i=1
i.i.d.
∼P(P|X, Y ), Widmann et al. [2021]
consider the following SKCE estimator
\
SKCE =
2
n(n −1)
X
1≤i<j≤n
G((P|xi, yi), (P|xj, yj))
(3)
where
G((p, y), (p′, y′)) :=k((p, y), (p′, y′))
−Ez∼p k((p, z), (p′, y′))
−Ez′∼p′ k((p, y), (p′, z′))
+ Ez∼p Ez′∼p′ k((p, z), (p′, z′)).
(4)
For a target false rejection rate α ∈(0, 1), the test of
Widmann et al. [2021] follows standard methodology in
recent nonparametric testing [Gretton et al., 2012, 2007,
Chwialkowski et al., 2016] by rejecting the null hypothesis
that the model is calibrated if \
SKCE > γ1−α, where γ1−α
denotes the (1−α)-quantile of \
SKCE under the null. While
various methods are available to estimate this quantile, all
tests experiments in this paper use a bootstrap approach
[Arcones and Giné, 1992]. As discussed by Widmann et al.
[2021], Equation (3) contains two important sources of pos-
sible intractability:
First Problem The last three terms in the sum are expecta-
tions under predictions of the probabilistic model of interest.
However, closed-form expressions for these expectations
are only available in restricted cases, such as for classi-
fication and for Gaussian models coupled with Gaussian
kernels. When these expectations are not available, they
must be approximated numerically. If the distributions P|X
are given in the form of unnormalized models, this approxi-
mation requires running expensive approximation methods
that often take the form of an MCMC algorithm and must
be performed for every sample of P|X used to estimate the
test statistic.
Second Problem The second source is the evaluation
of the kernel function k. We restrict our attention to
the conventional form of tensor-product type kernels
k((p, y), (p′, y′)) = kP (p, p′)kY (y, y′) chosen in this set-
ting. While typically many tractable choices for the kernel
kY exist (taking as input discrete or Euclidean values), the
choices for kP , taking as input two probability distributions
p and p′, are more limited and require expensive approxima-
tions methods when working with unnormalized models.
A popular approach to design kernels on distributions [Sz-
abó et al., 2015, 2016] is to first embed the probability
distributions in a Hilbert space H using a map ϕ, and then
compose it with a kernel kH on H:
kP (p, p′) = kH(ϕ(p), ϕ(p′)).
Any valid kernel on H, like the linear kernel kH(z, z′) =
⟨z, z′⟩H, the Gaussian kernel kH(z, z′) = e−∥z−z′∥
2
H,
or the inverse multiquadric kernel kH(z, z′)
=
(1 +
∥z −z′∥2
H)−1 can be used. In practice, the map ϕ can be
set to be the mean embedding map to an RKHS H, e.g.,
ϕ(µ) =
R
kH(z, ·) µ(dz). Kernels kH that are functions of
∥ϕ(µ) −ϕ(ν)∥2
H := MMD2(µ, ν), are often referred to as
MMD-type kernels [Meunier et al., 2022]. Other distances,
like the Wasserstein distance in 1 dimension or the sliced
Wasserstein distance [Bonneel et al., 2015] in multiple di-
mensions, also take this form for some choice of ϕ and
H, and can thus be used to construct kernels on distribu-
tions [Meunier et al., 2022]. In general, however, computing
kP (p, p′) becomes intractable apart from special cases such
as when p and p′ are Gaussian distributions. While there
exist finite-samples estimators for such kernels, a fast cal-
ibration estimation method based on Equation (2) would
require an estimator that does not require samples from p
and p′.
2.2
KERNEL CONDITIONAL GOODNESS-OF-FIT
TEST
We briefly introduce the background on goodness-of-fit
methods relevant to our new test. Conditional goodness-
of-fit (or CGOF) testing adapts the familiar goodness of
fit tests to the conditional case. In particular, CGOF tests
whether
H0 : Q|Z = P(Y ∈· | Z)
P(Z)-a.s.
(5)
given a candidate Q|z for the conditional distribution
P(Y ∈· | Z = z) and samples {(zi, yi)}
n
i=1
i.i.d
∼P(Z, Y ).
This problem was studied by Jitkrittum et al. [2020] for the
case Z × Y ⊂Rdz × Rdy and models Q|z with a differen-
tiable, strictly positive density fQ|z. They proposed a kernel
CGOF test for Equation (5) based on the (squared) kernel
conditional Stein discrepancy (KCSD)
DQ|·(P) :=
E(z,y)∼P(Z,Y )
KzξQ|z(y, ·)
2
FK
(6)
Here, FK is an Fdy
l
(e.g
dy times
z
}|
{
Fl × · · · × Fl)-vector-valued
RKHS with kernel K : Z × Z →L(Fdy
l , Fdy
l ), Kz is its
associated linear operator on Fdy
l
with Kzg := K(z, ·)g ∈
L(Z, Fdy
l ) for g ∈Fdy
l , Fl is an RKHS on Y with kernel
l: Y × Y →R and ξQ|z is the “kernelized score”:
ξQ|z(y, ·) = l(y, ·)∇y log fQ|z(y) + ∇yl(y, ·) ∈Fdy
l .
We refer to Jitkrittum et al. [2020, Section 2 and 3] for an
intuition behind the KCSD formula, and its relationship to
the more familiar Kernel Stein Discrepancy Chwialkowski
et al. [2016], Gorham and Mackey [2017]. Under certain
assumptions, the null hypothesis in Equation (5) is true
if and only if DQ|·(P) = 0. In particular, the latter will
hold Jitkrittum et al. [2020, Theorem 1] if Y and Z are
compact and the kernels K and l are universal, meaning
that FK (resp. Fl) is dense with respect to C(Z, Fdy
l ) (resp.
C(Y, R)), the space of continuous functions from Z to Fdy
l
(resp. Y to R) 1. An instance of a universal Fdy-reproducing
kernel is given by
K(z, z′) = k(z, z′)IF
dy
l
(7)
1These statements hold for noncompact Y, Z by replacing
continuous functions by continuous functions vanishing at infinity
[Jitkrittum et al., 2020, Theorem 1].
where IF
dy
l
∈L(Fdy
l , Fdy
l ) is the identity operator and k is
a real-valued universal kernel [Carmeli et al., 2010]. Jitkrit-
tum et al. [2020] showed that the CGOF statistic DQ|·(P)
admits an unbiased consistent estimator and used it to con-
struct hypothesis tests of Equation (5) with operator-valued
kernels of the form in Equation (7).
3
KERNEL
CALIBRATION-CONDITIONAL STEIN
DISCREPANCY
Calibration testing in the sense of Equation (1) is an in-
stance of conditional goodness-of-fit testing of Equation (5)
with input Z = P|X, target Y , and models Q|z = z = P|x.
Assuming that Y ⊂Rdy and that distributions P|x have
a differentiable, strictly positive density fP|x. In that case,
the (squared) kernel conditional Stein discrepancy in Equa-
tion (6) becomes
CP|·(P) :=
E(x,y)∼P(X,Y )
KP|xξP|x(y, ·)
2
FK ,
(8)
where now K is a kernel on P|X . To emphasize the calibra-
tion setting, we call CP|· the kernel calibration-conditional
Stein discrepancy (KCCSD). Similar to the KCSD, given
samples {P|xi, yi}
n
i=1
i.i.d.
∼P(P|X, Y ) and assuming a ker-
nel K of the form in Equation (7), statistic CP|·(P) has an
unbiased consistent estimator
d
CP|· =
2
n(n −1)
X
1≤i<j≤n
H((P|xi, yi), (P|xj, yj))
where
H((p, y), (p′, y′)) := k(p, p′)h((p, y), (p′, y′))
(9)
with
h((p, y), (p′, y′)) := l(y, y′)sp(y)⊤sp′(y′)
+
dy
X
i=1
∂2
∂yi∂y′
i
l(y, y′) + sp(y)⊤∇y′l(y, y′)
+ sp′(y′)⊤∇yl(y, y′),
(10)
where sp(y) := ∇y log fp(y) (resp. sp′(y)) is the score of
p (resp. p′). In Section A in the supplement we discuss how
the formula of bCP|· generalizes to operator-valued kernels
that are not of the form in Equation (7).
The above framing of the calibration problem conveniently
avoids the first source of possible intractability present in the
SKCE. For instance, for Gaussian models the test statistic
can be evaluated exactly for arbitrary kernels l on Y whereas
a closed-form expression of the SKCE is known only in the
special case where l is a Gaussian kernel.
Proposition 3.1 shows that the KCCSD can be viewed as
a special case of the SKCE. More generally, as shown in
Section B, the KCSD is a special form of the MMD.
Proposition 3.1 (Special case of Lemma B.1). Under weak
assumptions (see Lemma B.1), the KCCSD with respect to
kernels l: Y ×Y →R and k: P|X ×P|X →R is equivalent
to the SKCE with kernel H : (P|X × Y) × (P|X × Y) →R
defined in Equation (9).
The full testing procedure is outlined in Algorithm 1. The
computations can be performed with kernels K of the form
in Equation (7) or more general operator-valued kernels, but
crucially the method requires that K is tractable. Thus for
general models of probability distributions, such as energy-
based models and other unnormalized density models, it re-
mains to address the second source of intractability, namely
to construct a kernel K that can be evaluated efficiently.
Algorithm 1: CGOF Calibration Test (Tractable Kernel)
Data: Pairs {(P|xi, yi)}
n
i=1
i.i.d.
∼P(P|X, Y )
Result: Whether to reject H0 : “model is calibrated”
Parameters: Number of data samples n, kernel
l: Y2 →R, kernel k: (P|X )2 →R, level α
/* Estimate KCCSD using Equation (10) or (A.1)
*/
1 bC ←
2
n(n−1)
P
1≤i<j≤n
H((P|xi, yi), (P|xj, yj))
/* Use e.g. bootstrap [Arcones and Giné, 1992]
*/
2 bCα ←approximate (1 −α)-quantile of bC
3 if bC < bCα then
4
return Fail to reject H0
5 else
6
return Reject H0
7 end
4
TRACTABLE KERNELS FOR
GENERAL UNNORMALIZED
DENSITIES
In this section, we introduce two kernels between (density-
based) probability distributions that admit unbiased esti-
mates that neither require samples from the said distribu-
tions nor require access their normalizing constant. Cru-
cially, the properties of these new kernels allow to extend
the scope of calibration tests to a more general setting, in-
cluding Bayesian inference.
General Recipe
As in prior work on kernels for distribu-
tions [Meunier et al., 2022, Szabó et al., 2016], our proposed
kernels take the form of exponentiated Hilbertian metrics
k(p, q) = e−∥ϕ(p)−ϕ(q)∥2
H/(2σ2)
for two probability densities p and q, defined on some set
X ⊂Rd, where H is some separable Hilbert space, ϕ: p 7→
ϕ(p) ∈H is a feature map, and σ > 0 is a bandwidth
parameter. Our contributions in this section consist in pairs
of carefully designed ϕ and H that will allow approximating
k easily.
4.1
THE GENERALIZED FISHER DIVERGENCE
(KERNEL)
Our starting point is the Fisher Divergence [Lyu, 2012,
Sriperumbudur et al., 2017, Hyvärinen, 2005], also known
as the Relative Fisher Information [Otto and Villani, 2000],
between two probability densities p and q, which is given
by
FD(p, q) :=
Z
X
∥sp(x) −sq(x)∥2 p(x) dx.
The Fisher Divergence is a convenient tool to compare un-
normalized densities of the form
p(x) :=
tractable
z}|{
f(x)
Zf
|{z}
intractable
where
Zf :=
Z
X
f(x) dx
as the score of p can be evaluated without knowing Zf:
sp(x) = ∇x(log f(x)/Zf) = ∇x log f(x).
This property confers to the (squared) Fisher Divergence a
tractable unbiased estimator given n i.i.d. samples {Xi}n
i=1
from p, which takes the form:
\
FD(p, q) = 1
n
n
X
i=1
∥sp(Xi) −sq(Xi)∥2.
While the assumption ensuring access to samples from p is
realistic in the unsupervised learning literature [Hyvärinen,
2005], or when dealing with special instances of unnormal-
ized densities such as truncated densities f(x) = p(x)1x∈C,
it does not hold in the context of unnormalized models. We
overcome this issue by constructing a generalized version
of the Fisher Divergence:
Definition 4.1 (Generalized Fisher Divergence). Let p, q be
two probability densities on X, and ν a probability measure
on X. The Generalized Fisher Divergence between p and q
is defined as
GFDν(p, q) :=
Z
X
∥sp(x) −sq(x)∥2 ν(dx)
if Eν ∥sp∥2 , Eν ∥sq∥2 < +∞, and +∞otherwise.
The Generalized Fisher Divergence differs from the Fisher
Divergence in that the integration is performed with re-
spect to some given base measure ν instead of p. If the
support of ν covers the support of p and q, then we have
that GFDν(p, q) = 0 iff. p = q. Moreover, if ν can be sam-
pled from in a tractable manner, then GFDν(p, q) admits
a tractable estimator given samples {Zi}n
i=1 from ν of the
form
\
GFDν(p, q) = 1
n
n
X
i=1
∥sp(Zi) −sq(Zi)∥2.
In practice, the tractability assumption as well as the support
assumption for any p, q are verified by setting ν to be a
standard Gaussian distribution.
The
Exponentiated-GFD
Kernel
Importantly,
the
(square root of the) Generalized Fisher Divergence is a
Hilbertian metric on the space of probability densities.
Indeed, for p, q such that Eν ∥sp∥2 , Eν ∥sq∥2 < +∞, we
have that
GFDν(p, q) = ∥ϕ(p) −ϕ(q)∥2
L2(ν)
where ϕ: p 7→sp(·) ∈L2(ν) can be checked to be injective.
The latter fact allows to construct a kernel Kν on the space
of probability densities based on the Generalized Fisher
Divergence as follows:
Definition 4.2 (Exponentiated GFD Kernel). Let p, q be
two probability densities on X, and ν a probability measure
on X. The exponentiated GFD kernel between p and q is
defined as
Kν(p, q) := e−GFDν(p,q)/(2σ2)
Since the (square root of the) GFD is a Hilbertian met-
ric, Kν is positive definite [Meunier et al., 2022], and can
be estimated given samples of ν by replacing GFDν with
its empirical counterpart. We summarize the computation
method for Kν in Algorithm 2.
Algorithm 2: Exponentiated GFD Kernel
Data: Probability densities p, q on X
Result: Approx.
\
Kν(p, q) of Kν(p, q) in Theorem 4.2
Parameters: Base measure ν, num. of base samples m
1 for i ←1 to m do
2
Draw Zi ∼ν
3 end
4 return exp
−
1
2mσ2
Pm
i=1 ∥sp(Zi) −sq(Zi)∥2
Use in hypothesis testing In addition to being tractable to
estimate, we show that when X is compact (for instance, a
bounded subset of Rd), the exponentiated GFD kernels Kν
are universal. As a consequence, our KCCSD test, which
is an instance of a KCSD test, will be able to distinguish
the null-hypothesis from any alternative satisfying mild
smoothness assumptions, as guaranteed by Jitkrittum et al.
[2020, Theorem 1].
Proposition 4.1. Assume that X is compact, ν has full
support on X, and let PX be the set of twice-differentiable
probability densities on X equipped with the norm ∥p∥2 =
∥p∥2
L2(ν)+Pd
i=1 ∥∂ip∥2
L2(ν)+Pd
i,j=1 ∥∂i∂jp∥2
L2(ν). Then
Kν is universal for any bounded subset of PX .
Proof. The proof is given in Section D.2.
4.2
THE KERNELIZED GENERALIZED FISHER
DIVERGENCE (KERNEL)
While the recipe given above suffices to obtain a valid kernel
on the space of probability densities, the approximation
error arising from the discretization of the base measure ν
may scale unfavorably with the dimension of the underlying
space X. To address this issue, it is possible to apply a
kernel-smoothing step to the GFD feature map ϕ(p) by
composing it with an integral operator TK,ν associated with
a X-vector-valued kernel K and its RKHS HK
TK,ν : f ∈L(X, Rd) 7−→
Z
X
Kxf(x) ν(dx) ∈HK
and comparing the difference in feature map using the
squared RKHS norm ∥·∥2
HK. This choice of feature map
yields another metric, which we call the “kernelized” GFD:
KGFD(p, q) := ∥TK,νsp −Tk,νsq∥2
HK .
which, like the GFD, admits a tractable, unbiased estimator:
1
m2
m
X
i,j=1
K(Zi, Zj)(sp −sq)(Zi), (sp −sq)(Zj)
X .
Since the KGFD is also a Hilbertian metric, we build upon
it to construct our second proposal kernel:
Definition 4.3 (Exponentiated KGFD Kernel). Consider
the setting of Theorem 4.2, and let k be a bounded positive
definite kernel. The exponentiated KGFD kernel is given
by:
KK,ν := e−KGFD(p,q)/(2σ2)
For characteristic kernels K, the integral operator TK,ν is a
Hilbertian isometry between L2(ν, Rd) and HK, making the
exponentiated KGFD kernel positive definite. Additionally,
KK,ν enjoys a similar universality property as its GFD
analogue, as discussed in the next proposition.
Proposition 4.2. Assume that X is compact, ν has full-
support on X, and let PX be the set of twice-differentiable
probability densities equipped with the norm ∥p∥2
=
∥p∥2
L2(ν)+Pd
i=1 ∥∂ip∥2
L2(ν)+Pd
i,j=1 ∥∂i∂jp∥2
L2(ν). Then
KK,ν is universal for any bounded subset of PX .
A diffusion interpretation of the KGFD
In this section,
we establish a relationship between the KGFD and diffusion
processes [Rogers and Williams, 2000], further anchoring
the KGFD to the array of previously known divergences
while opening the door for possible refinements and gener-
alizations. Diffusion processes are well-known instances of
stochastic processes (Xt)t≥0 that evolve from some initial
distribution µ0 towards a target distribution p according to
the differential update rule
dXt = sp(Xt) dt +
√
2dWt,
X0 ∼µ0,
where Wt is a standard Brownian motion. For any time
t ≥0, the probability density of Xt is the solution µµ0,p(·, t)
of the so-called Fokker-Planck equation
∂µ(x, t)
∂t
= div(−µ(x, t)sp(x)) + ∆xµ(x, t)
(11)
with initial condition µ(·, 0) = µ0. Proposition 4.3 estab-
lishes a link between these solutions and the KGFD:
Proposition 4.3 (Diffusion interpretation of the KGFD). Let
µν,p (resp. µν,q) be the solution of Equation (11) with initial
condition ν and target p (resp. q). Let k be a real-valued,
twice-differentiable kernel. Then, we have that
lim
t→0
1
t MMD(µν,p(·, t), µν,q(·, t)) =
p
KGFD(p, q)
where the MMD is w.r.t. the kernel k, and the KGFD is
with respect to the matrix-valued kernel ∇x∇yk(x, y).
Proof. See Section D of the Appendix.
Proposition 4.3 frames the exponentiated KGFD kernel as
the t →0 limit of the kernel obtained by setting
ϕt : p 7−→∇x log µν,p(·, t)
which is the score of the solution of the Fokker-Planck
equation Equation (11) with target p and initial measure ν,
and setting H = H. Interestingly, the other limit case t →
∞recovers the exponentiated MMD kernel. Indeed, under
mild conditions, the Fokker-Planck solution converges to the
target and thus we have that limt→∞ϕt(p) = p: the feature
map converges to the identity. Thus, the diffusion framework
introduced above allows to recover both the KGFD and the
MMD as special cases. However, while the limit t →0 and
t →∞both yield Hilbertian metrics, it is an open question
whether for a given time 0 < t < ∞, ϕt is also Hilbertian.
A positive answer to this question would allow to construct
positive definite kernels that can possibly overcome the
pitfalls of score-based tools [Wenliang and Kanagawa, 2020,
Zhang et al., 2022], while being computable in finite time.
5
FAST AND SCALABLE CALIBRATION
TESTS
The framing of the calibration testing problem of Section 3
alongside with the GFD-based kernels of Section 4 allows
us to design a fast and scalable alternative to the pioneering
tests of Widmann et al. [2019]. The full testing procedure is
outlined in Algorithm 3.
Algorithm 3: CGOF Calibration Test (GFD Kernel)
Data: Pairs {(P|xi, yi)}
n
i=1
i.i.d.
∼P(P|X, Y )
Result: Whether to reject H0 : “model is calibrated”
Parameters: Base measure ν, num. of base samples m,
number of data samples n, kernel l: Y2 →R,
significance level α
1 for i ←1 to m do
2
Draw zi ∼ν
3 end
4 for 1 ≤i < j ≤n do
/* Use Algorithm 2 with base samples {zk}m
k=1
*/
5
κi,j ←
\
Kν(P|xi, P|xj)
6 end
7 Run Algorithm 1 with kernel k(P|xi, P|xj) := κi,j
Calibration tests as a reliability tests in Bayesian infer-
ence As one main motivation for studying calibration of
generic probabilistic models is Bayesian inference, it is im-
portant to note that reliability metrics traditionally used in
Bayesian inference such as conservativeness [Hermans et al.,
2021] differ from the notion of calibration in Equation (1).
We first briefly recall the notion of posterior coverage:
Definition 5.1 (Conservativeness of a Bayesian model [Her-
mans et al., 2021]). Let P|x(·) be a conditional distribution
model for P(Y ∈· | X = x), and assume that P|x has a den-
sity fP|x for P(X)-almost every x. For level 1 −α ∈[0, 1],
let ΘP|x(1 −α) be the highest density region of P|x.2 Then
P|· is said to be conservative if
E(x,y)∼P(X,Y ) 1ΘP|x(1−α)(y) ≥1 −α.
In the following proposition, we show that a probabilistic
model that is calibrated according to Equation (1) is also
conservative in the sense of Hermans et al. [2021], ground-
ing the use of our tests in Bayesian inference.
Proposition 5.1 (Calibrated models are conservative). If a
model P|· is calibrated in the sense of Equation (1), then it
is conservative.
The proof is given in Section C of the appendix.
2The highest density region of a probabilistic model P|x with
density fP|x is defined [see, e.g., Hyndman, 1996] by ΘP|x(1 −
α) := {y: fP|x(y) ≥cP|x(1 −α)} where cP|x(1 −α) :=
sup{c:
R
1[c,∞)(fP|x(y)) P|x(dy) ≥1 −α}.
6
EXPERIMENTS
We validate the properties of our proposed calibration tests
with synthetic data and compare them with existing tests
based on the SKCE.3 More concretely, we run KCCSD tests
using either a exponentiated GFD kernel or kernelized ex-
ponentiated GFD kernel with a matrix-valued kernel of the
form in Equation (7) with real-valued Gaussian kernel k;
and compare them with SKCE tests using two already in-
vestigated kernels on distributions: the exponentiated MMD
kernel with a Gaussian kernel on the ground space, and, for
isotropic Gaussian distributions, the exponentiated Wasser-
stein kernel with closed-form expression
kW
N(µ, σ2Id), N(µ′, σ′2Id)
= exp
−(∥µ −µ′∥2
2 + d(σ2 −σ′2))/(2ℓ2)
.
We set the base measure ν of the GFD and kernelized GFD
kernels to be a standard Gaussian. On Y, we study the
Gaussian and the inverse multi-quadric (IMQ) kernel.
We repeated all experiments with 100 resampled datasets
and used a wild bootstrap with 500 samples for approximat-
ing the quantiles of the test statistic with a prescribed sig-
nificance level of α = 0.05. The bandwidths of the kernels
are selected with the median heuristic. A "second-order"
median heuristic is used for the ground-space kernels of the
KGFD and the exponentiated MMD kernel: For each pair of
distributions, we compute the median distance between sam-
ples from an equally weighted mixture of these distributions
(numerically for tractable cases such as Gaussian distribu-
tions and using samples otherwise), and then the bandwidth
of the kernel is set to the median of these evaluations.
We repeatedly generate datasets {(P|xi, yi)}i in a two-step
procedure: First we sample distributions P|xi and then we
draw a corresponding target yi for each P|xi. We compare
different setups of targets Y and Gaussian distributions P|X
with varying degree δ ≥0 of miscalibration (models are
calibrated for δ = 0 and miscalibrated otherwise):4
Mean Gaussian Model (MGM)
Here X = Y = R5,
P(X) = N(0, I5), P(Y | X = x) = N(x, I5), and P|x =
N(x + δc, I5) for c ∈{15, e1} ⊂R5 (miscalibration of all
dimensions or only the first one).
Linear Gaussian Model (LGM)
Here X = R5, Y = R,
P(X) = N(0, I5), and P|x = N(δ + P5
i=1 ixi, 1).
Heteroscedastic Gaussian Model (HGM)
Here X = R3,
Y = R, P(X) = N(0, I3), P(Y | X = x) = N(m(x), 1),
and P|x = N(m(x), σ2(x)) with m(x) = P3
i=1 xi and
3The code to reproduce the experiments is available at https:
//github.com/pierreglaser/kccsd.
4MGM is adapted from a model used by Widmann et al. [2021],
and LGM, HGM, and QGM were used by Jitkrittum et al. [2020].
27
29
num samples
0.0
0.5
1.0
rejection rate
MGM (H0: = 0)
27
29
num samples
LGM (H0: = 0)
27
29
num samples
HGM (H1: = 1)
27
29
num samples
QGM (H1: = 1)
Fisher (KCCSD)
Kernelized Fisher (KCCSD)
MMD (SKCE)
Wasserstein (SKCE)
Figure 1: Rejection rates of the KCCSD and SKCE tests with a Gaussian kernel on the target space Y (significance level
α = 0.05). All kernels and test statistics are evaluated exactly using closed-form expressions.
24
26
28
210
warm-up steps
0.0
0.5
1.0
rejection rate
LGM (H0: = 0)
24
26
28
210
warm-up steps
HGM (H0: = 0)
24
26
28
210
warm-up steps
QGM (H0: = 0)
Fisher (SKCE)
MMD (SKCE)
Figure 2: False rejection rates of the SKCE tests for the calibrated LGM, HMC, and QGM (n = 200 data points, significance
level α = 0.05). The expectations in the test statistic are estimated with 2 samples obtained with the Metropolis-adjusted
Langevin algorithm (MALA) without step size tuning.
σ2(x) = 1 + 10δ exp (−∥x −c∥2
2/(2 · 0.82)) for c =
2/3 13.
Quadratic Gaussian Model (QGM)
Here X = Y = R,
P(X) = U(−2, 2), P(Y | X = x) = N(0.1x2 + x + 1, 1),
and P|x = N(0.1(1 −δ)x2 + x + 1, 1).
Figure 1 demonstrates that the proposed KCCSD tests are
calibrated: The false rejection rates (type I errors) of the
calibrated MGM and LGM do not exceed the set signifi-
cance level, apart from sampling noise. Figures F.1 and F.7
in the suppplementary material confirm empirically that this
is the case also when we approximate the Fisher and MMD
kernels using samples.
Moreover, we see in Figure 1 that for the miscalibrated
HGM the SKCE tests exhibit larger rejection rates, and
hence test power, than the KCCSD tests in the small sample
regime, regardless of the kernel choice. This specific setting
with Gaussian distributions and a Gaussian kernel on the
target space Y is favourable to the SKCE test as both its test
statistic, as well as the exponentiated MMD or Wasserstein
kernel evaluations are available in closed-form. In such
analytical scenarios we expect the score-based KCCSD tests
to perform worse [Wenliang and Kanagawa, 2020, Zhang
et al., 2022]. However, the KCCSD tests present themselves
as a practically useful alternative even in this example: For
the miscalibrated HGM their rejection rates are close to
100% with ≥256 data points, and for the miscalibrated
QGM they show very similar performance as the SKCE
tests. Overall, as expected, we see in Figure 1 that for all
studied tests rejection rates for the miscalibrated models
increases with increasing number of samples.
One main advantage of the KCCSD over the SKCE is that
it has first-class support for unnormalized models for which
only the score function is available: In contrast to the SKCE
its test statistic only involves scores but no expectations. In
principle, for unnormalized models these expectations in the
test statistic of the SKCE can be approximated with, e.g.,
MCMC sampling. However, Figure 2 shows that there is a
major caveat: If the MCMC method is not tuned sufficiently
well (e.g., if the chain is too short or the proposal step size
is not tuned properly), it might return biased samples which
causes the SKCE tests to be miscalibrated. On the other
hand, increasing the number of MCMC samples increases
the computational advantage of the KCCSD even more.
Another difference between the KCCSD and SKCE is high-
lighted in Figures F.1 and F.2: The number of combinations
of kernels for which the test statistic can be evaluated ex-
actly is smaller for the SKCE (in these Gaussian examples,
it requires Gaussian kernels on the target space).
One limitation of the (kernelized) exponentiated GFD Ker-
nel is that it necessitates setting an additional hyperparame-
ter: the base measure ν, which weights the score differences
between its two input distributions p and q at all points of
the ground space X. While our experiments have set ν to be
a Gaussian measure in order to obtain closed-form expres-
sions for Gaussian p, q, other choices may be more adequate
depending on the problem at hand. For instance, when p and
q are posterior models for a given prior π, we hypothesize
that setting ν to π constitutes a better default choice.
7
CONCLUSION
In this paper, we introduced the Kernel Calibration Condi-
tional Stein Discrepancy test, a fast and reliable alternative
to prior calibration tests for general, density-based proba-
bilistic models, thereby addressing an important need in the
Bayesian inference community. In doing so, we introduced
kernels for density-based inputs, which we believe are of
independent interest and could be used in other domains
such as distribution regression [Szabó et al., 2016] or meta-
learning [Denevi et al., 2020]. Moreover, while the set of
experiments conducted in this paper focused on “offline”
calibration testing, its low computational cost opens the
door to promising new use cases. One particularly interest-
ing avenue would consist in using the KCCSD test criterion
as a regularizer directly within the training procedure of a
probabilistic model, allowing not only to detect miscalibra-
tion but also to prevent it in the first place. We look forward
to seeing extensions and applications of the tools introduced
in this paper.
Acknowledgements
This research was financially supported by the Centre for
Interdisciplinary Mathematics (CIM) at Uppsala University,
Sweden, by the projects NewLEADS - New Directions in
Learning Dynamical Systems (contract number: 621-2016-
06079) and Handling Uncertainty in Machine Learning Sys-
tems (contract number: 2020-04122), funded by the Swedish
Research Council, by the Kjell och Märta Beijer Foundation,
by the Wallenberg AI, Autonomous Systems and Software
Program (WASP) funded by the Knut and Alice Wallenberg
Foundation, and by the Excellence Center at Linköping-
Lund in Information Technology (ELLIIT). Pierre Glaser
and Arthur Gretton acknowledge support from the Gatsby
Charitable Foundation.
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Supplementary Material
A
CONDITIONAL GOODNESS-OF-FIT: GENERAL OPERATOR-VALUED KERNEL
Assume that
• kernel l ∈C2(Y × Y, R),
• densities P|x ∈C1(Y, R) for P(X)-almost all x, and that
• E(x,y)∼P(X,Y )
KP|xξP|x(y, ·)
FK < ∞.
Due to the Bochner integrability of (x, y) 7→KP|xξP|x(y, ·) expectation and inner product commute [see Andreas Christ-
mann, 2008, Definition A.5.20], and hence we have
CP|·(P) =
E(x,y)∼P(X,Y )
KP|xξP|x(y, ·)
2
FK
=
E(x,y)∼P(X,Y )
KP|xξP|x(y, ·)
, E(x′,y′)∼P(X,Y )
h
KP|x′ξP|x′(y′, ·)
i
FK
= E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y )
KP|xξP|x(y, ·), KP|x′ξP|x′(y′, ·)
FK
= E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y )
K∗
P|x′KP|xξP|x(y, ·), ξP|x′(y′, ·)
F
dy
l
,
where K∗
P|x′ is the adjoint of KP|x′. The reproducing property implies K∗
P|x′KP|x = K(P|x, P|x′), and therefore we get
CP|·(P) = E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y )
K(P|x, P|x′)ξP|x(y, ·), ξP|x′(y′, ·)
F
dy
l
= E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y ) H((P|x, y), (P|x′, y′))
where
H((p, y), (p′, y′)) :=
K(p, p′)ξp(y, ·), ξp′(y′, ·)
F
dy
l
=
K(p, p′)ξp(y, ·), l(y′, ·)∇y′ log fp′(y′) + ∇y′l(y′, ·)
F
dy
l
.
For i ∈{1, . . . , dy}, let proji : Fdy
l
→Fl be the projection map to the ith subspace of the product space Fdy
l , and similarly
let ιi : Fl →Fdy
l
be the embedding of Fl in the ith subspace of Fdy
l
via x 7→(0, . . . , 0, x, 0, . . . , 0). Then we can write
H((p, y), (p′, y′)) =
dy
X
i=1
proji K(p, p′)ξp(y, ·), l(y′, ·) ∂
∂y′
i
log fp′(y′) + ∂
∂y′
i
l(y′, ·)
Fl
=
dy
X
i=1
(proji K(p, p′)ξp(y, ·))(y′) ∂
∂y′
i
log fp′(y′) + ∂
∂y′
i
(proji K(p, p′)ξp(y, ·))(y′)
.
Since K(p, p′) ∈L(Fdy
l ) is a linear operator, we have
K(p, p′)ξp(y, ·) = K(p, p′)(l(y, ·)∇y log fp(y)) + K(p, p′)∇yl(y, ·).
For 1 ≤i, j ≤dy, define Ki,j(p, p′): Fl →Fl as the continuous linear operator
Ki,j(p, p′) := proji K(p, p′)ιj.
Thus we have
proji K(p, p′)ξp(y, ·) =
dy
X
j=1
∂
∂yj
log fp(y)
Ki,j(p, p′)l(y, ·) +
dy
X
j=1
∂
∂yj
Ki,j(p, p′)l(y, ·),
and therefore
(proji K(p, p′)ξp(y, ·))(y′) =
dy
X
j=1
∂
∂yj
log p(y)
(Ki,j(p, p′)l(y, ·))(y′) +
dy
X
j=1
∂
∂yj
(Ki,j(p, p′)l(y, ·))(y′).
Due to the differentiability of kernel l we can interchange inner product and differentiation [Andreas Christmann, 2008,
Lemma 4.34], and thus we obtain
H((p, y), (p′, y′)) =
dy
X
i,j=1
∂
∂yj
log fp(y)
∂
∂y′
i
log fp′(y′)
(Ki,j(p, p′)l(y, ·))(y′)
+
dy
X
i,j=1
∂
∂y′
i
log fp′(y′)
∂
∂yj
(Ki,j(p, p′)l(y, ·))(y′)
+
dy
X
i,j=1
∂
∂yj
log fp(y)
∂
∂y′
i
(Ki,j(p, p′)l(y, ·))(y′)
+
dy
X
i,j=1
∂
∂y′
i
∂
∂yj
(Ki,j(p, p′)l(y, ·))(y′),
Define A: (P|X × Y)2 →Rdy×dy by
[A((p, y), (p′, y′))]i,j := (Ki,j(p, p′)l(y, ·))(y′)
(1 ≤i, j ≤dy).
Thus we obtain
H((p, y), (p′, y′)) = (sp′(y′) + ∇y′)⊤A((p, y), (p′, y′))(sp(y) + ∇y),
(A.1)
where for x, x′ ∈Rd, M(x, x′) ∈Rd×d we use the notation
∇⊤
x M(x, x′) =
∇⊤
x [M(x, x′)]:,1
· · ·
∇⊤
x [M(x, x′)]:,d
=
divx[M(x, x′)]:,1
· · ·
divx[M(x, x′)]:,d
,
and similarly
M(x, x′)∇x′ =
∇⊤
x′M(x, x′)⊤⊤=
divx′[M(x, x′)]1,:
· · ·
divx′[M(x, x′)]d,:
⊤
and
∇⊤
x M(x, x′)∇x′ = ∇⊤
x (M(x, x′)∇⊤
x′) =
d
X
i,j=1
∂2
∂xi∂x′
j
[M(x, x′)]i,j.
Thus, given samples {(P|xi, yi)}n
i=1
i.i.d.
∼P(P|X, Y ), an unbiased estimator of statistic CP|·(P) is
d
CP|· =
2
n(n −1)
X
1≤i<j≤n
H((P|xi, yi), (P|xj, yj)),
where H is given by Equation (A.1).
If kernel K is of the form in Equation (7), we recover the simpler formula in Equation (9). In this case A((p, y), (p′, y′)) =
k(p, p′)l(y, y′)Idy ∈Rdy×dy, i.e., A is a scaled identity matrix.
B
KCCSD AS A SPECIAL CASE OF SKCE
We prove the following general lemma that establishes the KCSD as a special case of the MMD. Then Proposition 3.1
follows immediately by considering random variables Z = P|X and Y , and models Q|z = z = P|x.
Lemma B.1 (KCSD as a special case of the MMD). Let Q|z be models of the conditional distributions P(Y ∈· | Z = z).
Moreover, we assume that
• Q|z has a density fQ|z ∈C1(Y, R) for P(Z)-almost all z,
• kernel l ∈C2(Y × Y, R),
• E(z,y)∼P(Z,Y )
KzξQ|z(y, ·)
FK < ∞, and
•
H
∂Y l(y, y′)fQ|z(y)n(y) dS(y′) = 0 and
H
∂Y ∇yl(y, y′)fQ|z(y′)n(y′) dS(y′) = 0 for P(Z)-almost all z,
where n(y) is the unit vector normal to the boundary ∂Y of Y at y ∈Y.5
Then
DQ|·(P) = MMD2
kQ|· (P(Z, Y ), PQ|·(Z, Y ))
where we define distribution PQ|· by
PQ|·(Z ∈A, Y ∈B) :=
Z
A
Q|z(Y ∈B) P(Z ∈dz)
and kernel kQ|· : (Z × Y) × (Z × Y) →R as
kQ|·((z, y), (z′, y′)) := (sQ|z′(y′) + ∇y′)TA((z, y), (z′, y′))(sQ|z(y) + ∇y),
using the same notation as in Section A and similarly defining A((z, y), (z′, y′)) ∈Rdy×dy by
[A((z, y), (z′, y′))]i,j := (Ki,j(z, z′)l(y, ·))(y′)
(1 ≤i, j ≤dy).
If K is of the form k(·, ·)IF
dy
l , function A simplifies to
A((z, y), (z′, y′)) = k(z, z′)l(y, y′)Idy
and kernel kQ|· is given by
kQ|·((z, y), (z′, y′))
= k(z, z′)
l(y, y′)sQ|z(y)TsQ|z′(y′) + sQ|z(y)T∇y′l(y, y′) + sQ|z′(y′)T∇yl(y, y′) +
dy
X
i=1
∂2
∂yi∂y′
i
l(y, y′)
.
Proof. From a similar calculation as in Section A [cf. Jitkrittum et al., 2020, Section A.2] we obtain that
kQ|·((z, y), (z′, y′)) =
KzξQ|z(y, ·), Kz′ξQ|z′(y′, ·)
FK
.
Thus kQ|· is an inner product of the features of (z, y) and (z′, y′) given by the feature map (z, y) 7→KzξQ|z(y, ·) ∈FK,
and therefore kQ|· is a positive-definite kernel. Moreover, from our assumption we obtain
E(z,y)∼P(Z,Y ) |kQ|·((z, y), (z, y))|1/2 = E(z,y)∼P(Z,Y )
KzξQ|z(y, ·)
FK < ∞.
Thus the mean embedding µP(Z,Y ) ∈FK of P(Z, Y ) exists [Gretton et al., 2012, Lemma 3].
5These assumptions are not restrictive in practice since they are satisfied if the conditions of [Jitkrittum et al., 2020, Theorem 1] hold
which are required to ensure that DQ|·(P) = 0 if and only if Q|Z(·) = P(Y ∈·|Z) P(Z)-almost surely.
Due to the Bochner integrability of (z, y) 7→KzξQ|z(y, ·) expectation and inner product commute [see Andreas Christmann,
2008, Definition A.5.20], and hence we have
E(z,y)∼PQ|·(Z,Y ) E(z′,y′)∼PQ|·(Z,Y ) kQ|·((z, y), (z′, y′)) =
E(z,y)∼PQ|·(Z,Y ) KzξQ|z(y, ·)
2
FK
=
Ez∼P(Z) Ey∼Q|z KzξQ|z(y, ·)
2
FK
=
Ez∼P(Z) Kz Ey∼Q|z ξQ|z(y, ·)
2
FK .
Due to the last assumption [Chwialkowski et al., 2016, Lemma 5.1] we know that
Ey∼Q|z ξQ|z(y, ·) = 0,
which implies
E(z,y)∼PQ|·(Z,Y ) E(z′,y′)∼PQ|·(Z,Y ) kQ|·((z, y), (z′, y′)) = 0.
Thus the mean embedding µPQ|·(Z,Y ) ∈FK of PQ|·(Z,Y ) exists and satisfies ∥µPQ|·(Z,Y )∥2
FK = 0, and hence µPQ|·(Z,Y ) =
0. We obtain [Gretton et al., 2012, Lemma 4] that
MMD2
kQ|· (P(Z, Y ), PQ|·(Z, Y )) = ∥µP(Z,Y ) −µPQ|·(Z,Y )∥2
FK
= ∥µP(Z,Y )∥2
FK
= E(z,y)∼P(Z,Y ) E(z′,y′)∼P(Z,Y ) kQ|·((z, y), (z′, y′))
= E(z,y)∼P(Z,Y ) E(z′,y′)∼P(Z,Y )
KzξQ|z(y, ·), Kz′ξQ|z′(y′, ·)
FK
= DQ|·(P),
where the last equality follows from [Jitkrittum et al., 2020, Section A.2].
C
CALIBRATION IMPLIES EXPECTED COVERAGE
We show that the sense of calibration employed by our tests implies posterior coverage in the sense of Hermans et al. [2021].
Again let us note P|x(·) for a model of the conditional distribution P(Y ∈· | X = x). Moreover, we assume that P|x has a
density fP|x for P(X)-almost every x.
For level 1 −α ∈[0, 1], let ΘP|x(1 −α) be the highest density region of a probabilistic model P|x with density fP|. It is
defined [see, e.g., Hyndman, 1996] by
ΘP|x(1 −α) :=
y: fP|x(y) ≥cP|x(1 −α)
where
cP|x(1 −α) := sup
(
c:
Z
n
˜y : fP|x(˜y)≥c
o P|x(dy) ≥1 −α
)
.
Hence, by definition [see, e.g., Hermans et al., 2021]
Ey∼P|x 1
y ∈ΘP|x(1 −α)
=
Z
ΘP|x(1−α)
P|x(dy) ≥1 −α.
Assume that model P|· is calibrated. By definition, it satisfies
P(Y ∈· | P|X) = P|X
P(X)-almost surely.
Hence, for all α ∈[0, 1], we obtain
E(x,y)∼P(X,Y ) 1
y ∈ΘP|x(1 −α)
= E(P|x,y)∼P(P|X,Y ) 1
y ∈ΘP|x(1 −α)
= EP|x∼P(P|X) Ey∼P|x 1
y ∈ΘP|x(1 −α)
≥EP|x∼P(P|X)
1 −α
= 1 −α.
Thus model P|· has expected coverage for all α ∈[0, 1].
D
DIFFUSION-LIMIT AND UNIVERSALITY
D.1
FISHER DIVERGENCE AS A DIFFUSION LIMIT
We recall that for a map f and a measure µ, the push-forward measure of µ by f, noted f#µ, is the measure on the image
space of f which verifies, for any measurable function g
Z
g(x) f#µ(dx) =
Z
g(f(x)) µ(dx).
To prove the differential inequality linking the MMD and the KGFD, we rely on the following reformulation of the
Fokker-Planck equation:
∂µ(x, t)
∂t
= divx(−µ(x, t)sp(x)) + ∆xµ(x, t)
= divx(−µ(x, t)sp(x)) + divx ∇xµ(x, t)
= divx(−µ(x, t)sp(x)) + divx(µ(x, t)∇x log µ(x, t))
= divx(−µ(x, t)(sp(x) −∇x log µ(x, t)).
We remark that since the density µ(x, t) is twice differentiable in x and differentiable in t [Johnson, 2004], this equation
holds in the strong sense, and not only in the sense of distributions. Because of that, one has
∂tµ(x, t) = lim
∆→0
µ(x, t + ∆) −µ(x, t)
∆
.
Let us consider an RKHS H with kernel k, and let h ∈H. Let us define mt(x) := m(x, t) := µν,p(x, t) −µν,q(x, t) and
we note MMD(mt) the function given by
MMD(mt) =
ZZ
k(x, y)mt(x)mt(y) dx dy
1/2
= MMD(µν,p(·, t), µν,q(·, t)).
To show that limt→0 d
dt MMD(mt) = KGFD(p, q), we first analyze the differential properties of the easier to handle MMD2
and complete the proof using a chain rule argument. The first variation (also called Gateaux Derivative) of m 7→MMD2(m)
is a linear functional on the space of functions
f −g
f, g: X × [0, ∞) →R
with
∀t ≥0:
Z
X
f(x, t) dx =
Z
X
g(x, t) dx = 1
,
given by
δ MMD2
δm
: f 7→
Z
2k(x, y)mt(x)f(y) dx dy.
Using the chain rule for Gateaux derivatives, we have that
d MMD2(m)
dt
= d MMD2
dm
(m)dm
dt
=
Z
2k(x, y)mt(x)dm
dt (y) dx dy.
From the Fokker-Planck Equation, we have that
dm
dt = ∂tµν,p −∂tµν,q
= divx(µν,p∇x log
p
µν,p
) −divx(µν,q∇x log
q
µν,q
)
= divx(ν∇x log p
ν ) −divx(ν∇x log q
ν ) + o(1)
= divx(ν∇x log p
q ) + o(1)
Plugging the last equation in the chain rule, we have:
d MMD2(m)
dt
=
Z
2mt(x)divyν(y)∇y log p
q (y)k(x, y)dxdy + o(1)
=
Z
2mt(x)
∇yk(x, y), ν(y)∇y log p
q (y)
dx dy + o(1).
Similarly, since m0 = µν,p(·, 0) −µν,q(·, 0) = ν −ν = 0, we have mt(x) = t∂tm(x, 0) + ox(t). The calculation follows
as:
d MMD2(m)
dt
=
Z
2t × ∂tm(x, t)
∇yk(x, y), ν(y)∇y log p
q (y)
dx dy + o(t)
=
Z
2t × divx ν(x)∇x log p
q (x)
∇yk(x, y), ν(y)∇y log p
q (y)
dx dy + o(t)
=
Z
2t ×
ν(x)∇x log p
q (x), ∇x
∇yk(x, y), ν(y)∇y log p
q (y)
dx dy + o(t)
=
Z
2t ×
ν(x)∇x log p
q (x), ∇x∇yk(x, y), ν(y)∇y log p
q (y)
dx dy + o(t).
To get rid of the degenerate scaling as t →0, we now focus on (the derivative of)
q
MMD2(mt) as t →0. Notice that since
MMD(m0) = 0, the derivative of
q
MMD2(mt) does not exist a priori for t = 0: we consider instead d
dt
q
MMD2(mt)
t=t,
and extend it by continuity by setting t →0. We have:
d
q
MMD2(mt)
dt
=
1
2
q
MMD2(mt)
d MMD2(mt)
dt
.
As
MMD2(mt) =
Z
k(x, y)mt(x)mt(y) dx dy
we obtain through similar calculations that
MMD2(mt) =
Z Z
t2
ν(x)∇x log p
q (x), ∇x∇yk(x, y), ν(y)∇y log p
q (y)
dx dy + o(t)
from which the results follows. Note that the matrix-valued kernel (K(x, y))ij = (∇x∇yk(x, y))ij is positive definite, a
result akin to one of Zhou [2008] but for the matrix-valued case. Indeed, for all x, y ∈X, z, t ∈Rd,
zK(x, y)t =
* d
X
i=1
zi∂ik(x, ·),
d
X
i=1
ti∂ik(y, ·)
+
H
where ∂ik(x, ·) ∈H [Zhou, 2008]. In the following, we write ϕ(x, y) = Pd
i=1 yi∂ik(xi, ·). Now, for all sets of {xi}n
i=1 ∈X,
{yj}n
i=1 ∈Rd, we have
n
X
i,j=1
K(xi, xj)yj, yi
Rd =
n
X
i,j=1
ϕ(xi, yi), ϕ(xj, yj)
H
=
* n
X
i=1
ϕ(xi, yi),
n
X
i=1
ϕ(xi, yi)
+
H
≥0
from which it follows that K is indeed positive definite [Micchelli and Pontil, 2005, Theorem 2.1].
D.2
UNIVERSALITY OF THE EXPONENTIATED-GFD AND EXPONENTIATED-KGFD KERNEL
To prove the universality of Kν and Kν,K under the assumptions discussed in the related propositions, we rely on the
following theorem [Christmann and Steinwart, 2010, Theorem 2.2].
Theorem D.1. On a compact metric space (Z, dZ ) and for a continuous and injective map ϕ : Z 7→H, where H is a
separable Hilbert space, the kernel K(z, z′) = e−γ∥ϕ(z)−ϕ(z′)∥2
H is universal.
We first focus on the universality of Kν. We set as our goal to apply that theorem to our setting, in which Z := PX is a
(sub)set of probability densities, which needs to be associated with a suitably chosen metric in order to make PX to be
compact, and ϕ continuous. As bounded subsets of differentiable densities, whose elements can be framed as elements of the
Sobolev space of first order W2,1(ν) [Taylor, 1996]), are not compact a priori, we restrict ourselves to twice-differentiable
densities with bounded Sobolev norm of second order, i.e., to W2,2(ν) with norm ∥p∥2
W2,2 := ∥p∥2
L2(ν)+Pd
i=1 ∥∂ip∥2
L2(ν)+
Pd
i,j=1 ∥∂i∂jp∥2
L2(ν). From the Rellich-Kondrachov theorem [Taylor, 1996], we know that when ν has compact support,
the canonical canonical injection I : W2,2(ν) →W2,1(ν) is a compact operator. As a consequence, for any bounded subset
A of PX we thus have that I(A) is compact for ∥f∥2
W2,1 := ∥f∥2
L2(ν) + Pd
i=1 ∥∂if∥2
L2(ν), which implies that any bounded
subset A of PX is compact for d(z, z′) = ∥z −z′∥W2,1. To apply the above theorem, it remains to prove the continuity and
injectivity of ϕ : p 7→∇log p under this metric (in that case the separable Hilbert space H is set to L2(ν)). And indeed,
for such a choice of d, ϕ and H, ϕ is continuous. To prove this fact, remark that differentiable densities with full support
on X are bounded away from 0, making the use of a ϕ: p 7→∇log p = ∇p/p continuous. Moreover, ϕ is injective as
dW 2,1(p, q) := ∥p −q∥W 2,1 ̸= 0 implies ∥∇log p −∇log q∥L2(ν) ̸= 0. Thus, all conditions of [Christmann and Steinwart,
2010, Theorem 2.2] are satisfied, and the result follows as a consequence.
We now move on to prove the universality of Kν,K. The proof follows the same reasoning as the proof of the universality
of Kν, the only difference being the fact that the feature map ˜ϕ of KK,ν is given by Tν ◦ϕ, where ϕ: p 7→∇log p and
TK,ν : L(X, Rd) →HK is given by
TK,ν : f 7→
Z
X
Kxf(x) ν(dx).
However, if ν is a probability measure and K is bounded, then TK,ν is a bounded operator, and thus continuous, making ˜ϕ
continuous. Moreover, if K is characteristic, TK,ν is injective. Thus ˜ϕ is injective and continuous, from which the result
follows by Christmann and Steinwart [2010].
E
BACKGROUND ON STEIN AND FISHER DIVERGENCES
The Fisher Divergence
Consider two continuously differentiable densities p and q on Rd. Then the Fisher diver-
gence [Sriperumbudur et al., 2017, Johnson, 2004] between p and q is defined as:
FD(p||q) =
Z
Rd ∥∇log p(x) −∇log q(x)∥2
2 p(x) dx.
We refer to Sriperumbudur et al. [2017] for an overview of the properties of the Fisher divergence, including its relative
strength w.r.t. other divergences, and other formulations. The Fisher divergence was used for learning statistical models of
some training data in Hyvärinen [2005], Sriperumbudur et al. [2017], and more recently in Song and Ermon [2019].
Stein Discrepancies
Of proximity to the Fisher divergence is the family of Stein discrepancies [Anastasiou et al., 2022].
Stein discrepancies build upon the concept of Stein operators, which are operators AP such that
EQ [APf] = 0 ⇐⇒Q = P
for any f within a set G(AP) ⊂dom(AP) called the Stein class of AP. Following this definition, the AP-stein discrepancy
is defined as
SDAP(P, Q) =
sup
f∈G(A)
∥EQ Af∥
which satisfies by construction the axioms of a dissimilarity (or divergence) measure between P and Q.
Link Between the Fisher divergence and Diffusion Stein Discrepancies
Perhaps the most famous Stein discrepancy is
the one that sets AP to be the infinitesimal generator of the isotropic diffusion process toward P [Gorham et al., 2019]:
(
dXt
= ∇log p(Xt) dt +
√
2 dWt
(Ad,Pf)(·)
= ⟨∇log p(·), ∇f⟩+ ⟨∇, ∇f⟩
Recalling that EP [Ad,Pf] = 0 for all f ∈G(Ad,P), we obtain the following formulation for the diffusion Stein discrepancy
SDAd,P(P, Q) := sup
f
∥EQ Ad,Pf∥= sup
f
EQ(∇log p −∇log q)⊤∇f
= sup
g=∇f
EQ(∇log p −∇log q)⊤g
,
highlighting the connection between the Fisher divergence and the diffusion Stein discrepancy.
Link Between the Fisher divergence and the Kernelized Stein Discrepancy
Given a RKHS H such that BH⊗d(0H⊗d, 1)
is a Stein class for Ad,P, the kernelized Stein discrepancy [Gorham and Mackey, 2017] is given by
KSD(P, Q) :=
sup
h=∇f∈H⊗d:∥h∥H⊗d≤1
∥EQ ⟨∇log p(x) −∇log q(x), h(x)⟩∥
=
sup
h=∇f∈H⊗d:∥h∥H⊗d≤1
⟨h, EQ(∇log p(x) −∇log q(x))k(x, ·)⟩1/2
H⊗d
= ∥EQ [(∇log p(x) −∇log q(x))k(x, ·)]∥H⊗d
=
I⋆
k,Q(∇log p −∇log q)
H⊗d
where I⋆
k,Q is the adjoint of the canonical injection from H⊗d to (L2(Q))⊗d, also known as the kernel integral operator.
This derivation shows that the KSD can be seen as a kernelized version of the Fisher divergence.
Link between MMD and KSD
It is possible [Gorham and Mackey, 2017] to reframe the KSD as an MMD with a specific
kernel. Indeed, given some base kernel k(x, y), define the following “Stein” kernel
˜k(x, y) = ⟨∇log p(x)k(x, ·) + ∇k(x, ·), ∇log p(y)k(y, ·) + ∇log k(y, ·)⟩H⊗d
which is positive definite as an inner product of a feature map of x. Then H˜k = Ad,P(H) and ∥f∥H˜k = ∥Af∥H⊗d
k .
Moreover, we have that EP ˜h = 0 for all ˜h ∈H˜k. By the definition of the KSD, we have that
KSD(P, Q) =
sup
h∈H⊗d : ∥h∥H⊗d≤1
∥EQ Ad,Ph∥
=
sup
h∈H⊗d : ∥h∥H⊗d≤1
∥EQ Ad,Ph −EP Ad,Ph∥H
=
sup
h∈H˜k : ∥h∥H˜k ≤1
∥EQ h −EP h∥H˜k
= MMD˜k(P, Q).
Differential Inequalities between the KL and the Fisher Divergence
It is well known [Carrillo et al., 2003] that the KL
divergence can be related to the Fisher divergence by considering the evolution of KL(Pt||Q) when Pt evolves according to
the Fokker-Planck equation
∂tpt(x) = div(pt(x)(∇log qt(x) −∇log pt(x))),
P0 = P.
(E.1)
(Two relevant side notes: for any t ≥0, Pt is the law at time t of the Markov process (Xt)t≥0 such that X0 ∼P and
undergoing an isotropic diffusion towards Q. Moreover, Equation (E.1) is also the Wasserstein gradient flow equation
of KL(·||Q) starting from P). Recalling that Equation (E.1) is satisfied in the sense of distributions, and relying on
Gateaux-Derivative formulas for Free Energy-type functionals [see Ambrosio et al., 2005, for more precise statements], we
have:
dKL(Pt||Q)
dt
= ∂KL
dP
Pt
dPt
dt
=
Z
⟨∇(log pt(x) −log qt(x)), (∇log qt −∇log pt)⟩dPt(x)
= −FD(Pt, Q).
Figure E.1: Relationships between the Fisher divergence, the KL divergence, the MMD, and the KSD [Liu, 2016].
F
EXPERIMENTAL RESULTS
This section contains visualizations of all experiments discussed in 6, including figures contained in the main text. In all
experiments we set the significance level to α = 0.05. Every experiment is repeated for 100 randomly sampled datasets and
with 500 bootstrap iterations for estimating the quantile of the test statistic.
We use Gaussian distributions and compare the KCCSD and the SKCE with different combinations of kernels. For the
KCCSD, for Gaussian distributions all considered test statistics can be evaluated exactly. Alternatively, for the exponentiated
(kernelized) Fisher kernel and the exponentiated MMD kernel one can resort to approximations using samples from the base
measure. For the SKCE, however, the test statistic can be evaluated exactly on in special cases such as Gaussian kernels on
the target space. All approximate evaluations are performed with 10 samples.
F.1
MEAN GAUSSIAN MODEL
26
27
28
29
210
num samples
0.00
0.25
0.50
0.75
1.00
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Fisher + IMQ
Exp. Kernelized Fisher + Gaussian
Exp. Kernelized Fisher + IMQ
Exp. MMD + Gaussian
Exp. MMD + IMQ
Exp. Wasserstein + Gaussian
Exp. Wasserstein + IMQ
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.1: False rejection rate of the KCCSD for MGM (δ = 0).
26
27
28
29
210
num samples
0.0
0.5
1.0
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Kernelized Fisher + Gaussian
Exp. MMD + Gaussian
Exp. Wasserstein + Gaussian
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.2: False rejection rate of the SKCE for MGM (δ = 0).
26
27
28
29
210
num samples
0.0
0.2
0.4
0.6
0.8
1.0
rejction rate
exact
Exp. Fisher + Gaussian
Exp. Fisher + IMQ
Exp. Kernelized Fisher + Gaussian
Exp. Kernelized Fisher + IMQ
Exp. MMD + Gaussian
Exp. MMD + IMQ
Exp. Wasserstein + Gaussian
Exp. Wasserstein + IMQ
26
27
28
29
210
num samples
rejction rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.3: Rejection rate of the KCCSD for MGM (δ = 0.1, c = 1d).
26
27
28
29
210
num samples
0.0
0.2
0.4
0.6
0.8
1.0
rejction rate
exact
Exp. Fisher + Gaussian
Exp. Kernelized Fisher + Gaussian
Exp. MMD + Gaussian
Exp. Wasserstein + Gaussian
26
27
28
29
210
num samples
rejction rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.4: Rejection rate of the SKCE for MGM (δ = 0.1, c = 1d).
26
27
28
29
210
num samples
0.0
0.2
0.4
0.6
0.8
1.0
rejction rate
exact
Exp. Fisher + Gaussian
Exp. Fisher + IMQ
Exp. Kernelized Fisher + Gaussian
Exp. Kernelized Fisher + IMQ
Exp. MMD + Gaussian
Exp. MMD + IMQ
Exp. Wasserstein + Gaussian
Exp. Wasserstein + IMQ
26
27
28
29
210
num samples
rejction rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.5: Rejection rate of the KCCSD for MGM (δ = 0.1, c = e1).
26
27
28
29
210
num samples
0.00
0.25
0.50
0.75
1.00
rejction rate
exact
Exp. Fisher + Gaussian
Exp. Kernelized Fisher + Gaussian
Exp. MMD + Gaussian
Exp. Wasserstein + Gaussian
26
27
28
29
210
num samples
rejction rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.6: Rejection rate of the SKCE for MGM (δ = 0.1, c = e1).
F.2
LINEAR GAUSSIAN MODEL
26
27
28
29
210
num samples
0.00
0.25
0.50
0.75
1.00
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Fisher + IMQ
Exp. Kernelized Fisher + Gaussian
Exp. Kernelized Fisher + IMQ
Exp. MMD + Gaussian
Exp. MMD + IMQ
Exp. Wasserstein + Gaussian
Exp. Wasserstein + IMQ
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.7: False rejection rate of the KCCSD for LGM (δ = 0).
26
27
28
29
210
num samples
0.0
0.5
1.0
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Kernelized Fisher + Gaussian
Exp. MMD + Gaussian
Exp. Wasserstein + Gaussian
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.8: False rejection rate of the SKCE for LGM (δ = 0).
F.3
HETEROSCEDASTIC GAUSSIAN MODEL
26
27
28
29
210
num samples
0.0
0.2
0.4
0.6
0.8
1.0
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Fisher + IMQ
Exp. Kernelized Fisher + Gaussian
Exp. Kernelized Fisher + IMQ
Exp. MMD + Gaussian
Exp. MMD + IMQ
Exp. Wasserstein + Gaussian
Exp. Wasserstein + IMQ
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.9: Rejection rate of the KCCSD for HGM (δ = 1).
26
27
28
29
210
num samples
0.0
0.5
1.0
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Kernelized Fisher + Gaussian
Exp. MMD + Gaussian
Exp. Wasserstein + Gaussian
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.10: Rejection rate of the SKCE for HGM (δ = 1).
F.4
QUADRATIC GAUSSIAN MODEL
26
27
28
29
210
num samples
0.0
0.2
0.4
0.6
0.8
1.0
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Fisher + IMQ
Exp. Kernelized Fisher + Gaussian
Exp. Kernelized Fisher + IMQ
Exp. MMD + Gaussian
Exp. MMD + IMQ
Exp. Wasserstein + Gaussian
Exp. Wasserstein + IMQ
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.11: Rejection rate of the KCCSD for QGM (δ = 1).
26
27
28
29
210
num samples
0.0
0.2
0.4
0.6
0.8
1.0
rejection rate
exact
Exp. Fisher + Gaussian
Exp. Kernelized Fisher + Gaussian
Exp. MMD + Gaussian
Exp. Wasserstein + Gaussian
26
27
28
29
210
num samples
rejection rate
approximate
Exp. Fisher (discrete) + Gaussian
Exp. Fisher (discrete) + IMQ
Exp. Kernelized Fisher (discrete) + Gaussian
Exp. Kernelized Fisher (discrete) + IMQ
Exp. MMD (discrete) + Gaussian
Exp. MMD (discrete) + IMQ
Figure F.12: Rejection rate of the SKCE for QGM (δ = 1).
|
Fast and Scalable Score-Based Kernel Calibration Tests Pierre Glaser1 David Widmann2 Fredrik Lindsten3 Arthur Gretton1 1Gatsby Computational Neuroscience Unit, University College London, London, UK 2 3Division of Statistics and Machine Learning, Linköping University, Sweden Abstract We introduce the Kernel Calibration Conditional Stein Discrepancy test (KCCSD test), a nonparametric, kernel-based test for assessing the calibration of probabilistic models with well-defined scores. In contrast to previous methods, our test avoids the need for possibly expensive expectation approximations while providing control over its type-I error. We achieve these improvements by using a new family of kernels for score-based probabilities that can be estimated without probability density samples, and by using a conditional goodness-of-fit criterion for the KCCSD test's Ustatistic. We demonstrate the properties of our test on various synthetic settings. 1 INTRODUCTION Calibration is a statistical property of predictive probabilistic models that ensures that a model's prediction matches the conditional distribution of the predicted variable given the prediction. A calibrated model expresses the uncertainty about its predictions reliably by being neither over- nor underconfident, and hence can be useful even if its accuracy is suboptimal. This property is essential in safety-critical applications such as autonomous driving. Unfortunately, empirical studies revealed that popular machine learning models such as deep neural networks tend to trade off calibration for accuracy [Guo et al., 2017]. This has lead to an increased interest in the study of calibrated models in recent years. Calibration has been studied in the meteorological and statistical literature for many decades [e.g., Murphy and Winkler, 1977, DeGroot and Fienberg, 1983]. For a long time, research on calibration has been focused on different notions of calibration for probabilistic classifiers [e.g., Murphy and Winkler, 1977, DeGroot and Fienberg, 1983, Platt, 2000, Zadrozny and Elkan, 2001, Bröcker, 2009, Naeini et al., 2015, Guo et al., 2017, Kull et al., 2017, Kumar et al., 2018, Kull et al., 2019, Vaicenavicius et al., 2019, Widmann et al., 2019] and on calibration of quantiles and confidence intervals for real-valued regression problems [e.g., Ho and Lee, 2005, Rueda et al., 2006, Taillardat et al., 2016, Song et al., 2019, Fasiolo et al., 2020]. Regarding the calibration of classification models, different hypothesis tests have been proposed [e.g., Cox, 1958, Bröcker and Smith, 2007, Vaicenavicius et al., 2019, Widmann et al., 2019, Gweon, 2022, Lee et al., 2022]. Given a predictive model and a validation dataset, these tests output whether a model is likely to be uncalibrated. The recent work of Widmann et al. [2021] generalized the calibration-framework introduced for classification in [Widmann et al., 2019] to (possibly multi-dimensional) continuous-valued predictive models. In particular, Widmann et al. [2021] introduced a kernel-based hypothesis test for such general classes of models. An important potential consumer of calibration tests is Bayesian inference, and in particular simulation-based inference (SBI), for which miscalibration is particularly undesirable. SBI [Cranmer et al., 2020] lies at the intersection of machine learning and domain sciences, and refers to the set of methods that train probabilistic models to estimate the posterior over scientific parameters of interest given some observed data. The models are trained using pairs composed of parameters drawn from a prior distribution, and their associated "synthetic" observed data, obtained by running a probabilistic program called the simulator, taking a parameter value as input, and that faithfully mimics the physical generative process of interest. The increasing number of use cases combined with advances in probabilistic modeling has elevated SBI to a critical role in solving complex scientific problems such as particle physics [Gilman et al., 2018] and neuroscience [Glöckler et al., 2022, Glaser et al., 2022]. However, as discussed in [Hermans et al., 2021], overconfidence in SBI models can conceal credible alternative scientific hypotheses, and result in incorrect discoveries [Hermans et al., 2021], highlighting the need for Accepted for the 39th Conference on Uncertainty in Artificial Intelligence (UAI 2023). 16 Oct 2025 principled and performant calibration tests suitable for such models. While the theoretical framework of Widmann et al. [2021] describes the calibration of any probabilistic model, applying its associated calibration test to Bayesian inference remains challenging: indeed, the test statistics require computing expectations against the probabilistic models of interest, for reasons bearing both to the calibration setting, and to the limitations of currently available kernel-based tools for probabilistic models. Although such expectations can be computed exactly for classification models, expectations against generic probabilistic models are usually intractable and must be approximated. In cases where the models are unnormalized, these approximations are both computationally expensive-sometimes, prohibitively-and biased, thereby compromising theoretical guarantees of the calibration tests of Widmann et al. [2021], including type-I error control. Contributions In this paper, we introduce the kernel calibration-conditional Stein discrepancy (or KCCSD) test, a new nonparametric, score-based kernel calibration test which addresses the limitations of existing methods. The KCCSD test builds on the insight that the definition of calibration given by Vaicenavicius et al. [2019] is a conditional goodness of fit property, as we remark in Section 3. This fact allows us to leverage the kernel conditional goodness of fit test proposed by Jitkrittum et al. [2020] as the backbone of the KCCSD test. Unlike the test-statistics of Widmann et al. [2021], the KCCSD test statistic does not contain explicit expectations against the probabilistic models; however, as in [Widmann et al., 2021], it requires evaluating a kernel between probabilities densities, which in most cases of interest introduces an (intractable) expectation against the densities. To eliminate this limitation, we construct two new kernels between probability distributions that do not involve expectations against its input distributions, while remaining suitable for statistical testing. These kernels rely on a generalized version of the Fisher divergence and are of independent interest. We investigate a connection between these kernels and diffusion, akin to Stein methods, and discuss the relationships with other kernels on distributions. By using such kernels in the KCCSD test statistic, we obtain a fast and scalable calibration test that remains consistent and calibrated for unnormalized models, answering the need for such tests discussed above. We confirm in Section 6 the properties and benefits of the KCCSD test against alternatives on synthetic experiments. 2 BACKGROUND Notation We consider probabilistic systems characterized by a joint distribution P(X, Y ) of random variables (X, Y ) taking values in X × Y, and study probabilistic models P|· : x ∈X 7-→P|x(·) ∈P(Y) approximating the unknown conditional distribution of Y given X = x, P|x(·) ≃P(Y ∈· | X = x). The target variable Y is typically a parameter of a probabilistic system of interestlike synaptic weights in biological neural networks-while the input variable X is observed data-like neuron voltage traces measured using electrophysiology. 2.1 CALIBRATION OF PREDICTIVE MODELS Calibration: General Definition A probabilistic model P|· is called calibrated or reliable [Bröcker, 2008, Vaicenavicius et al., 2019, Widmann et al., 2021] if it satisfies P|X = P Y ∈· | P|X P(X)-a.s.. (1) Note that this definition applies to general predictive probabilistic models, also beyond classification, and only assumes that the conditional distributions on the right-hand side exist. Hypothesis Testing: Kernel Calibration Error There are multiple ways to test whether a given predictive probabilistic model is calibrated. In this section, we introduce the kernel-based tests of Widmann et al. [2019] and their later generalization [Widmann et al., 2021], since our KCCSD test is built on these approaches. These tests turn the equality between conditional distributions present in Equation (1) into a more classical equality between two joint distributions. The transformation is achieved by noting that P|X = P Y ∈· | P|X P(X)-a.s. ⇐⇒(P|X, Y ) d= (P|X, Z) where Z is an "auxiliary" variable such that Z | P|X ∼ P|X(·). This identity was used by Widmann et al. [2021] to construct an MMD-type calibration test based on the (squared) kernel calibration error (SKCE) criterion sup h∈B(0H,1) E(x,y,z)∼P(X,Y,Z) h(P|x, y) -h(P|x, z) . (2) Here, B(0H, 1) is the unit ball of a reproducing kernel Hilbert space (RKHS) H of functions with positive definite kernel kH : (P|X × Y)2 →R. As noted by Widmann et al. [2021], the SKCE generalizes the (squared) kernel classification calibration error (SKCCE) defined for the special case of discrete output spaces Y = {1, . . . , d} [Widmann et al., 2019], to continuous ones. Given n pairs of samples {(P|xi, yi)} n i=1 i.i.d. ∼P(P|X, Y ), Widmann et al. [2021] consider the following SKCE estimator \ SKCE = 2 n(n -1) X 1≤i γ1-α, where γ1-α denotes the (1-α)-quantile of \ SKCE under the null. While various methods are available to estimate this quantile, all tests experiments in this paper use a bootstrap approach [Arcones and Giné, 1992]. As discussed by Widmann et al. [2021], Equation (3) contains two important sources of possible intractability: First Problem The last three terms in the sum are expectations under predictions of the probabilistic model of interest. However, closed-form expressions for these expectations are only available in restricted cases, such as for classification and for Gaussian models coupled with Gaussian kernels. When these expectations are not available, they must be approximated numerically. If the distributions P|X are given in the form of unnormalized models, this approximation requires running expensive approximation methods that often take the form of an MCMC algorithm and must be performed for every sample of P|X used to estimate the test statistic. Second Problem The second source is the evaluation of the kernel function k. We restrict our attention to the conventional form of tensor-product type kernels k((p, y), (p′, y′)) = kP (p, p′)kY (y, y′) chosen in this setting. While typically many tractable choices for the kernel kY exist (taking as input discrete or Euclidean values), the choices for kP , taking as input two probability distributions p and p′, are more limited and require expensive approximations methods when working with unnormalized models. A popular approach to design kernels on distributions [Szabó et al., 2015, 2016] is to first embed the probability distributions in a Hilbert space H using a map φ, and then compose it with a kernel kH on H: kP (p, p′) = kH(φ(p), φ(p′)). Any valid kernel on H, like the linear kernel kH(z, z′) = ⟨z, z′⟩H, the Gaussian kernel kH(z, z′) = e-∥z-z′∥ 2 H, or the inverse multiquadric kernel kH(z, z′) = (1 + ∥z -z′∥2 H)-1 can be used. In practice, the map φ can be set to be the mean embedding map to an RKHS H, e.g., φ(μ) = R kH(z, ·) μ(dz). Kernels kH that are functions of ∥φ(μ) -φ(ν)∥2 H := MMD2(μ, ν), are often referred to as MMD-type kernels [Meunier et al., 2022]. Other distances, like the Wasserstein distance in 1 dimension or the sliced Wasserstein distance [Bonneel et al., 2015] in multiple dimensions, also take this form for some choice of φ and H, and can thus be used to construct kernels on distributions [Meunier et al., 2022]. In general, however, computing kP (p, p′) becomes intractable apart from special cases such as when p and p′ are Gaussian distributions. While there exist finite-samples estimators for such kernels, a fast calibration estimation method based on Equation (2) would require an estimator that does not require samples from p and p′. 2.2 KERNEL CONDITIONAL GOODNESS-OF-FIT TEST We briefly introduce the background on goodness-of-fit methods relevant to our new test. Conditional goodnessof-fit (or CGOF) testing adapts the familiar goodness of fit tests to the conditional case. In particular, CGOF tests whether H0 : Q|Z = P(Y ∈· | Z) P(Z)-a.s. (5) given a candidate Q|z for the conditional distribution P(Y ∈· | Z = z) and samples {(zi, yi)} n i=1 i.i.d ∼P(Z, Y ). This problem was studied by Jitkrittum et al. [2020] for the case Z × Y ⊂Rdz × Rdy and models Q|z with a differentiable, strictly positive density fQ|z. They proposed a kernel CGOF test for Equation (5) based on the (squared) kernel conditional Stein discrepancy (KCSD) DQ|·(P) := E(z,y)∼P(Z,Y ) KzξQ|z(y, ·) 2 FK (6) Here, FK is an Fdy l (e.g dy times z }| { Fl × · · · × Fl)-vector-valued RKHS with kernel K : Z × Z →L(Fdy l , Fdy l ), Kz is its associated linear operator on Fdy l with Kzg := K(z, ·)g ∈ L(Z, Fdy l ) for g ∈Fdy l , Fl is an RKHS on Y with kernel l: Y × Y →R and ξQ|z is the "kernelized score": ξQ|z(y, ·) = l(y, ·)∇y log fQ|z(y) + ∇yl(y, ·) ∈Fdy l . We refer to Jitkrittum et al. [2020, Section 2 and 3] for an intuition behind the KCSD formula, and its relationship to the more familiar Kernel Stein Discrepancy Chwialkowski et al. [2016], Gorham and Mackey [2017]. Under certain assumptions, the null hypothesis in Equation (5) is true if and only if DQ|·(P) = 0. In particular, the latter will hold Jitkrittum et al. [2020, Theorem 1] if Y and Z are compact and the kernels K and l are universal, meaning that FK (resp. Fl) is dense with respect to C(Z, Fdy l ) (resp. C(Y, R)), the space of continuous functions from Z to Fdy l (resp. Y to R) 1. An instance of a universal Fdy-reproducing kernel is given by K(z, z′) = k(z, z′)IF dy l (7) 1These statements hold for noncompact Y, Z by replacing continuous functions by continuous functions vanishing at infinity [Jitkrittum et al., 2020, Theorem 1]. where IF dy l ∈L(Fdy l , Fdy l ) is the identity operator and k is a real-valued universal kernel [Carmeli et al., 2010]. Jitkrittum et al. [2020] showed that the CGOF statistic DQ|·(P) admits an unbiased consistent estimator and used it to construct hypothesis tests of Equation (5) with operator-valued kernels of the form in Equation (7). 3 KERNEL CALIBRATION-CONDITIONAL STEIN DISCREPANCY Calibration testing in the sense of Equation (1) is an instance of conditional goodness-of-fit testing of Equation (5) with input Z = P|X, target Y , and models Q|z = z = P|x. Assuming that Y ⊂Rdy and that distributions P|x have a differentiable, strictly positive density fP|x. In that case, the (squared) kernel conditional Stein discrepancy in Equation (6) becomes CP|·(P) := E(x,y)∼P(X,Y ) KP|xξP|x(y, ·) 2 FK , (8) where now K is a kernel on P|X . To emphasize the calibration setting, we call CP|· the kernel calibration-conditional Stein discrepancy (KCCSD). Similar to the KCSD, given samples {P|xi, yi} n i=1 i.i.d. ∼P(P|X, Y ) and assuming a kernel K of the form in Equation (7), statistic CP|·(P) has an unbiased consistent estimator d CP|· = 2 n(n -1) X 1≤i 0 is a bandwidth parameter. Our contributions in this section consist in pairs of carefully designed φ and H that will allow approximating k easily. 4.1 THE GENERALIZED FISHER DIVERGENCE (KERNEL) Our starting point is the Fisher Divergence [Lyu, 2012, Sriperumbudur et al., 2017, Hyvärinen, 2005], also known as the Relative Fisher Information [Otto and Villani, 2000], between two probability densities p and q, which is given by FD(p, q) := Z X ∥sp(x) -sq(x)∥2 p(x) dx. The Fisher Divergence is a convenient tool to compare unnormalized densities of the form p(x) := tractable z}|{ f(x) Zf |{z} intractable where Zf := Z X f(x) dx as the score of p can be evaluated without knowing Zf: sp(x) = ∇x(log f(x)/Zf) = ∇x log f(x). This property confers to the (squared) Fisher Divergence a tractable unbiased estimator given n i.i.d. samples {Xi}n i=1 from p, which takes the form: \ FD(p, q) = 1 n n X i=1 ∥sp(Xi) -sq(Xi)∥2. While the assumption ensuring access to samples from p is realistic in the unsupervised learning literature [Hyvärinen, 2005], or when dealing with special instances of unnormalized densities such as truncated densities f(x) = p(x)1x∈C, it does not hold in the context of unnormalized models. We overcome this issue by constructing a generalized version of the Fisher Divergence: Definition 4.1 (Generalized Fisher Divergence). Let p, q be two probability densities on X, and ν a probability measure on X. The Generalized Fisher Divergence between p and q is defined as GFDν(p, q) := Z X ∥sp(x) -sq(x)∥2 ν(dx) if Eν ∥sp∥2 , Eν ∥sq∥2 < +∞, and +∞otherwise. The Generalized Fisher Divergence differs from the Fisher Divergence in that the integration is performed with respect to some given base measure ν instead of p. If the support of ν covers the support of p and q, then we have that GFDν(p, q) = 0 iff. p = q. Moreover, if ν can be sampled from in a tractable manner, then GFDν(p, q) admits a tractable estimator given samples {Zi}n i=1 from ν of the form \ GFDν(p, q) = 1 n n X i=1 ∥sp(Zi) -sq(Zi)∥2. In practice, the tractability assumption as well as the support assumption for any p, q are verified by setting ν to be a standard Gaussian distribution. The Exponentiated-GFD Kernel Importantly, the (square root of the) Generalized Fisher Divergence is a Hilbertian metric on the space of probability densities. Indeed, for p, q such that Eν ∥sp∥2 , Eν ∥sq∥2 < +∞, we have that GFDν(p, q) = ∥φ(p) -φ(q)∥2 L2(ν) where φ: p 7→sp(·) ∈L2(ν) can be checked to be injective. The latter fact allows to construct a kernel Kν on the space of probability densities based on the Generalized Fisher Divergence as follows: Definition 4.2 (Exponentiated GFD Kernel). Let p, q be two probability densities on X, and ν a probability measure on X. The exponentiated GFD kernel between p and q is defined as Kν(p, q) := e-GFDν(p,q)/(2σ2) Since the (square root of the) GFD is a Hilbertian metric, Kν is positive definite [Meunier et al., 2022], and can be estimated given samples of ν by replacing GFDν with its empirical counterpart. We summarize the computation method for Kν in Algorithm 2. Algorithm 2: Exponentiated GFD Kernel Data: Probability densities p, q on X Result: Approx. \ Kν(p, q) of Kν(p, q) in Theorem 4.2 Parameters: Base measure ν, num. of base samples m 1 for i ←1 to m do 2 Draw Zi ∼ν 3 end 4 return exp - 1 2mσ2 Pm i=1 ∥sp(Zi) -sq(Zi)∥2 Use in hypothesis testing In addition to being tractable to estimate, we show that when X is compact (for instance, a bounded subset of Rd), the exponentiated GFD kernels Kν are universal. As a consequence, our KCCSD test, which is an instance of a KCSD test, will be able to distinguish the null-hypothesis from any alternative satisfying mild smoothness assumptions, as guaranteed by Jitkrittum et al. [2020, Theorem 1]. Proposition 4.1. Assume that X is compact, ν has full support on X, and let PX be the set of twice-differentiable probability densities on X equipped with the norm ∥p∥2 = ∥p∥2 L2(ν)+Pd i=1 ∥∂ip∥2 L2(ν)+Pd i,j=1 ∥∂i∂jp∥2 L2(ν). Then Kν is universal for any bounded subset of PX . Proof. The proof is given in Section D.2. 4.2 THE KERNELIZED GENERALIZED FISHER DIVERGENCE (KERNEL) While the recipe given above suffices to obtain a valid kernel on the space of probability densities, the approximation error arising from the discretization of the base measure ν may scale unfavorably with the dimension of the underlying space X. To address this issue, it is possible to apply a kernel-smoothing step to the GFD feature map φ(p) by composing it with an integral operator TK,ν associated with a X-vector-valued kernel K and its RKHS HK TK,ν : f ∈L(X, Rd) 7-→ Z X Kxf(x) ν(dx) ∈HK and comparing the difference in feature map using the squared RKHS norm ∥·∥2 HK. This choice of feature map yields another metric, which we call the "kernelized" GFD: KGFD(p, q) := ∥TK,νsp -Tk,νsq∥2 HK . which, like the GFD, admits a tractable, unbiased estimator: 1 m2 m X i,j=1 K(Zi, Zj)(sp -sq)(Zi), (sp -sq)(Zj) X . Since the KGFD is also a Hilbertian metric, we build upon it to construct our second proposal kernel: Definition 4.3 (Exponentiated KGFD Kernel). Consider the setting of Theorem 4.2, and let k be a bounded positive definite kernel. The exponentiated KGFD kernel is given by: KK,ν := e-KGFD(p,q)/(2σ2) For characteristic kernels K, the integral operator TK,ν is a Hilbertian isometry between L2(ν, Rd) and HK, making the exponentiated KGFD kernel positive definite. Additionally, KK,ν enjoys a similar universality property as its GFD analogue, as discussed in the next proposition. Proposition 4.2. Assume that X is compact, ν has fullsupport on X, and let PX be the set of twice-differentiable probability densities equipped with the norm ∥p∥2 = ∥p∥2 L2(ν)+Pd i=1 ∥∂ip∥2 L2(ν)+Pd i,j=1 ∥∂i∂jp∥2 L2(ν). Then KK,ν is universal for any bounded subset of PX . A diffusion interpretation of the KGFD In this section, we establish a relationship between the KGFD and diffusion processes [Rogers and Williams, 2000], further anchoring the KGFD to the array of previously known divergences while opening the door for possible refinements and generalizations. Diffusion processes are well-known instances of stochastic processes (Xt)t≥0 that evolve from some initial distribution μ0 towards a target distribution p according to the differential update rule dXt = sp(Xt) dt + √ 2dWt, X0 ∼μ0, where Wt is a standard Brownian motion. For any time t ≥0, the probability density of Xt is the solution μμ0,p(·, t) of the so-called Fokker-Planck equation ∂μ(x, t) ∂t = div(-μ(x, t)sp(x)) + ∆xμ(x, t) (11) with initial condition μ(·, 0) = μ0. Proposition 4.3 establishes a link between these solutions and the KGFD: Proposition 4.3 (Diffusion interpretation of the KGFD). Let μν,p (resp. μν,q) be the solution of Equation (11) with initial condition ν and target p (resp. q). Let k be a real-valued, twice-differentiable kernel. Then, we have that lim t→0 1 t MMD(μν,p(·, t), μν,q(·, t)) = p KGFD(p, q) where the MMD is w.r.t. the kernel k, and the KGFD is with respect to the matrix-valued kernel ∇x∇yk(x, y). Proof. See Section D of the Appendix. Proposition 4.3 frames the exponentiated KGFD kernel as the t →0 limit of the kernel obtained by setting φt : p 7-→∇x log μν,p(·, t) which is the score of the solution of the Fokker-Planck equation Equation (11) with target p and initial measure ν, and setting H = H. Interestingly, the other limit case t → ∞recovers the exponentiated MMD kernel. Indeed, under mild conditions, the Fokker-Planck solution converges to the target and thus we have that limt→∞φt(p) = p: the feature map converges to the identity. Thus, the diffusion framework introduced above allows to recover both the KGFD and the MMD as special cases. However, while the limit t →0 and t →∞both yield Hilbertian metrics, it is an open question whether for a given time 0 < t < ∞, φt is also Hilbertian. A positive answer to this question would allow to construct positive definite kernels that can possibly overcome the pitfalls of score-based tools [Wenliang and Kanagawa, 2020, Zhang et al., 2022], while being computable in finite time. 5 FAST AND SCALABLE CALIBRATION TESTS The framing of the calibration testing problem of Section 3 alongside with the GFD-based kernels of Section 4 allows us to design a fast and scalable alternative to the pioneering tests of Widmann et al. [2019]. The full testing procedure is outlined in Algorithm 3. Algorithm 3: CGOF Calibration Test (GFD Kernel) Data: Pairs {(P|xi, yi)} n i=1 i.i.d. ∼P(P|X, Y ) Result: Whether to reject H0 : "model is calibrated" Parameters: Base measure ν, num. of base samples m, number of data samples n, kernel l: Y2 →R, significance level α 1 for i ←1 to m do 2 Draw zi ∼ν 3 end 4 for 1 ≤i < j ≤n do /* Use Algorithm 2 with base samples {zk}m k=1 */ 5 κi,j ← \ Kν(P|xi, P|xj) 6 end 7 Run Algorithm 1 with kernel k(P|xi, P|xj) := κi,j Calibration tests as a reliability tests in Bayesian inference As one main motivation for studying calibration of generic probabilistic models is Bayesian inference, it is important to note that reliability metrics traditionally used in Bayesian inference such as conservativeness [Hermans et al., 2021] differ from the notion of calibration in Equation (1). We first briefly recall the notion of posterior coverage: Definition 5.1 (Conservativeness of a Bayesian model [Hermans et al., 2021]). Let P|x(·) be a conditional distribution model for P(Y ∈· | X = x), and assume that P|x has a density fP|x for P(X)-almost every x. For level 1 -α ∈[0, 1], let ΘP|x(1 -α) be the highest density region of P|x.2 Then P|· is said to be conservative if E(x,y)∼P(X,Y ) 1ΘP|x(1-α)(y) ≥1 -α. In the following proposition, we show that a probabilistic model that is calibrated according to Equation (1) is also conservative in the sense of Hermans et al. [2021], grounding the use of our tests in Bayesian inference. Proposition 5.1 (Calibrated models are conservative). If a model P|· is calibrated in the sense of Equation (1), then it is conservative. The proof is given in Section C of the appendix. 2The highest density region of a probabilistic model P|x with density fP|x is defined [see, e.g., Hyndman, 1996] by ΘP|x(1 - α) := {y: fP|x(y) ≥cP|x(1 -α)} where cP|x(1 -α) := sup{c: R 1[c,∞)(fP|x(y)) P|x(dy) ≥1 -α}. 6 EXPERIMENTS We validate the properties of our proposed calibration tests with synthetic data and compare them with existing tests based on the SKCE.3 More concretely, we run KCCSD tests using either a exponentiated GFD kernel or kernelized exponentiated GFD kernel with a matrix-valued kernel of the form in Equation (7) with real-valued Gaussian kernel k; and compare them with SKCE tests using two already investigated kernels on distributions: the exponentiated MMD kernel with a Gaussian kernel on the ground space, and, for isotropic Gaussian distributions, the exponentiated Wasserstein kernel with closed-form expression kW N(μ, σ2Id), N(μ′, σ′2Id) = exp -(∥μ -μ′∥2 2 + d(σ2 -σ′2))/(2l2) . We set the base measure ν of the GFD and kernelized GFD kernels to be a standard Gaussian. On Y, we study the Gaussian and the inverse multi-quadric (IMQ) kernel. We repeated all experiments with 100 resampled datasets and used a wild bootstrap with 500 samples for approximating the quantiles of the test statistic with a prescribed significance level of α = 0.05. The bandwidths of the kernels are selected with the median heuristic. A "second-order" median heuristic is used for the ground-space kernels of the KGFD and the exponentiated MMD kernel: For each pair of distributions, we compute the median distance between samples from an equally weighted mixture of these distributions (numerically for tractable cases such as Gaussian distributions and using samples otherwise), and then the bandwidth of the kernel is set to the median of these evaluations. We repeatedly generate datasets {(P|xi, yi)}i in a two-step procedure: First we sample distributions P|xi and then we draw a corresponding target yi for each P|xi. We compare different setups of targets Y and Gaussian distributions P|X with varying degree δ ≥0 of miscalibration (models are calibrated for δ = 0 and miscalibrated otherwise):4 Mean Gaussian Model (MGM) Here X = Y = R5, P(X) = N(0, I5), P(Y | X = x) = N(x, I5), and P|x = N(x + δc, I5) for c ∈{15, e1} ⊂R5 (miscalibration of all dimensions or only the first one). Linear Gaussian Model (LGM) Here X = R5, Y = R, P(X) = N(0, I5), and P|x = N(δ + P5 i=1 ixi, 1). Heteroscedastic Gaussian Model (HGM) Here X = R3, Y = R, P(X) = N(0, I3), P(Y | X = x) = N(m(x), 1), and P|x = N(m(x), σ2(x)) with m(x) = P3 i=1 xi and 3The code to reproduce the experiments is available at https: //github.com/pierreglaser/kccsd. 4MGM is adapted from a model used by Widmann et al. [2021], and LGM, HGM, and QGM were used by Jitkrittum et al. [2020]. 27 29 num samples 0.0 0.5 1.0 rejection rate MGM (H0: = 0) 27 29 num samples LGM (H0: = 0) 27 29 num samples HGM (H1: = 1) 27 29 num samples QGM (H1: = 1) Fisher (KCCSD) Kernelized Fisher (KCCSD) MMD (SKCE) Wasserstein (SKCE) Figure 1: Rejection rates of the KCCSD and SKCE tests with a Gaussian kernel on the target space Y (significance level α = 0.05). All kernels and test statistics are evaluated exactly using closed-form expressions. 24 26 28 210 warm-up steps 0.0 0.5 1.0 rejection rate LGM (H0: = 0) 24 26 28 210 warm-up steps HGM (H0: = 0) 24 26 28 210 warm-up steps QGM (H0: = 0) Fisher (SKCE) MMD (SKCE) Figure 2: False rejection rates of the SKCE tests for the calibrated LGM, HMC, and QGM (n = 200 data points, significance level α = 0.05). The expectations in the test statistic are estimated with 2 samples obtained with the Metropolis-adjusted Langevin algorithm (MALA) without step size tuning. σ2(x) = 1 + 10δ exp (-∥x -c∥2 2/(2 · 0.82)) for c = 2/3 13. Quadratic Gaussian Model (QGM) Here X = Y = R, P(X) = U(-2, 2), P(Y | X = x) = N(0.1x2 + x + 1, 1), and P|x = N(0.1(1 -δ)x2 + x + 1, 1). Figure 1 demonstrates that the proposed KCCSD tests are calibrated: The false rejection rates (type I errors) of the calibrated MGM and LGM do not exceed the set significance level, apart from sampling noise. Figures F.1 and F.7 in the suppplementary material confirm empirically that this is the case also when we approximate the Fisher and MMD kernels using samples. Moreover, we see in Figure 1 that for the miscalibrated HGM the SKCE tests exhibit larger rejection rates, and hence test power, than the KCCSD tests in the small sample regime, regardless of the kernel choice. This specific setting with Gaussian distributions and a Gaussian kernel on the target space Y is favourable to the SKCE test as both its test statistic, as well as the exponentiated MMD or Wasserstein kernel evaluations are available in closed-form. In such analytical scenarios we expect the score-based KCCSD tests to perform worse [Wenliang and Kanagawa, 2020, Zhang et al., 2022]. However, the KCCSD tests present themselves as a practically useful alternative even in this example: For the miscalibrated HGM their rejection rates are close to 100% with ≥256 data points, and for the miscalibrated QGM they show very similar performance as the SKCE tests. Overall, as expected, we see in Figure 1 that for all studied tests rejection rates for the miscalibrated models increases with increasing number of samples. One main advantage of the KCCSD over the SKCE is that it has first-class support for unnormalized models for which only the score function is available: In contrast to the SKCE its test statistic only involves scores but no expectations. In principle, for unnormalized models these expectations in the test statistic of the SKCE can be approximated with, e.g., MCMC sampling. However, Figure 2 shows that there is a major caveat: If the MCMC method is not tuned sufficiently well (e.g., if the chain is too short or the proposal step size is not tuned properly), it might return biased samples which causes the SKCE tests to be miscalibrated. On the other hand, increasing the number of MCMC samples increases the computational advantage of the KCCSD even more. Another difference between the KCCSD and SKCE is highlighted in Figures F.1 and F.2: The number of combinations of kernels for which the test statistic can be evaluated exactly is smaller for the SKCE (in these Gaussian examples, it requires Gaussian kernels on the target space). One limitation of the (kernelized) exponentiated GFD Kernel is that it necessitates setting an additional hyperparameter: the base measure ν, which weights the score differences between its two input distributions p and q at all points of the ground space X. While our experiments have set ν to be a Gaussian measure in order to obtain closed-form expressions for Gaussian p, q, other choices may be more adequate depending on the problem at hand. For instance, when p and q are posterior models for a given prior π, we hypothesize that setting ν to π constitutes a better default choice. 7 CONCLUSION In this paper, we introduced the Kernel Calibration Conditional Stein Discrepancy test, a fast and reliable alternative to prior calibration tests for general, density-based probabilistic models, thereby addressing an important need in the Bayesian inference community. In doing so, we introduced kernels for density-based inputs, which we believe are of independent interest and could be used in other domains such as distribution regression [Szabó et al., 2016] or metalearning [Denevi et al., 2020]. Moreover, while the set of experiments conducted in this paper focused on "offline" calibration testing, its low computational cost opens the door to promising new use cases. One particularly interesting avenue would consist in using the KCCSD test criterion as a regularizer directly within the training procedure of a probabilistic model, allowing not only to detect miscalibration but also to prevent it in the first place. We look forward to seeing extensions and applications of the tools introduced in this paper. Acknowledgements This research was financially supported by the Centre for Interdisciplinary Mathematics (CIM) at Uppsala University, Sweden, by the projects NewLEADS - New Directions in Learning Dynamical Systems (contract number: 621-201606079) and Handling Uncertainty in Machine Learning Systems (contract number: 2020-04122), funded by the Swedish Research Council, by the Kjell och Märta Beijer Foundation, by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation, and by the Excellence Center at LinköpingLund in Information Technology (ELLIIT). Pierre Glaser and Arthur Gretton acknowledge support from the Gatsby Charitable Foundation. References L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2005. A. Anastasiou, A. Barp, F.-X. Briol, B. Ebner, R. E. Gaunt, F. Ghaderinezhad, J. Gorham, A. Gretton, C. Ley, Q. Liu, et al. 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Journal of computational and Applied Mathematics, 2008. Supplementary Material A CONDITIONAL GOODNESS-OF-FIT: GENERAL OPERATOR-VALUED KERNEL Assume that • kernel l ∈C2(Y × Y, R), • densities P|x ∈C1(Y, R) for P(X)-almost all x, and that • E(x,y)∼P(X,Y ) KP|xξP|x(y, ·) FK < ∞. Due to the Bochner integrability of (x, y) 7→KP|xξP|x(y, ·) expectation and inner product commute [see Andreas Christmann, 2008, Definition A.5.20], and hence we have CP|·(P) = E(x,y)∼P(X,Y ) KP|xξP|x(y, ·) 2 FK = E(x,y)∼P(X,Y ) KP|xξP|x(y, ·) , E(x′,y′)∼P(X,Y ) h KP|x′ξP|x′(y′, ·) i FK = E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y ) KP|xξP|x(y, ·), KP|x′ξP|x′(y′, ·) FK = E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y ) K∗ P|x′KP|xξP|x(y, ·), ξP|x′(y′, ·) F dy l , where K∗ P|x′ is the adjoint of KP|x′. The reproducing property implies K∗ P|x′KP|x = K(P|x, P|x′), and therefore we get CP|·(P) = E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y ) K(P|x, P|x′)ξP|x(y, ·), ξP|x′(y′, ·) F dy l = E(x,y)∼P(X,Y ) E(x′,y′)∼P(X,Y ) H((P|x, y), (P|x′, y′)) where H((p, y), (p′, y′)) := K(p, p′)ξp(y, ·), ξp′(y′, ·) F dy l = K(p, p′)ξp(y, ·), l(y′, ·)∇y′ log fp′(y′) + ∇y′l(y′, ·) F dy l . For i ∈{1, . . . , dy}, let proji : Fdy l →Fl be the projection map to the ith subspace of the product space Fdy l , and similarly let ιi : Fl →Fdy l be the embedding of Fl in the ith subspace of Fdy l via x 7→(0, . . . , 0, x, 0, . . . , 0). Then we can write H((p, y), (p′, y′)) = dy X i=1 proji K(p, p′)ξp(y, ·), l(y′, ·) ∂ ∂y′ i log fp′(y′) + ∂ ∂y′ i l(y′, ·) Fl = dy X i=1 (proji K(p, p′)ξp(y, ·))(y′) ∂ ∂y′ i log fp′(y′) + ∂ ∂y′ i (proji K(p, p′)ξp(y, ·))(y′) . Since K(p, p′) ∈L(Fdy l ) is a linear operator, we have K(p, p′)ξp(y, ·) = K(p, p′)(l(y, ·)∇y log fp(y)) + K(p, p′)∇yl(y, ·). For 1 ≤i, j ≤dy, define Ki,j(p, p′): Fl →Fl as the continuous linear operator Ki,j(p, p′) := proji K(p, p′)ιj. Thus we have proji K(p, p′)ξp(y, ·) = dy X j=1 ∂ ∂yj log fp(y) Ki,j(p, p′)l(y, ·) + dy X j=1 ∂ ∂yj Ki,j(p, p′)l(y, ·), and therefore (proji K(p, p′)ξp(y, ·))(y′) = dy X j=1 ∂ ∂yj log p(y) (Ki,j(p, p′)l(y, ·))(y′) + dy X j=1 ∂ ∂yj (Ki,j(p, p′)l(y, ·))(y′). Due to the differentiability of kernel l we can interchange inner product and differentiation [Andreas Christmann, 2008, Lemma 4.34], and thus we obtain H((p, y), (p′, y′)) = dy X i,j=1 ∂ ∂yj log fp(y) ∂ ∂y′ i log fp′(y′) (Ki,j(p, p′)l(y, ·))(y′) + dy X i,j=1 ∂ ∂y′ i log fp′(y′) ∂ ∂yj (Ki,j(p, p′)l(y, ·))(y′) + dy X i,j=1 ∂ ∂yj log fp(y) ∂ ∂y′ i (Ki,j(p, p′)l(y, ·))(y′) + dy X i,j=1 ∂ ∂y′ i ∂ ∂yj (Ki,j(p, p′)l(y, ·))(y′), Define A: (P|X × Y)2 →Rdy×dy by [A((p, y), (p′, y′))]i,j := (Ki,j(p, p′)l(y, ·))(y′) (1 ≤i, j ≤dy). Thus we obtain H((p, y), (p′, y′)) = (sp′(y′) + ∇y′)⊤A((p, y), (p′, y′))(sp(y) + ∇y), (A.1) where for x, x′ ∈Rd, M(x, x′) ∈Rd×d we use the notation ∇⊤ x M(x, x′) = ∇⊤ x [M(x, x′)]:,1 · · · ∇⊤ x [M(x, x′)]:,d = divx[M(x, x′)]:,1 · · · divx[M(x, x′)]:,d , and similarly M(x, x′)∇x′ = ∇⊤ x′M(x, x′)⊤ ⊤= divx′[M(x, x′)]1,: · · · divx′[M(x, x′)]d,: ⊤ and ∇⊤ x M(x, x′)∇x′ = ∇⊤ x (M(x, x′)∇⊤ x′) = d X i,j=1 ∂2 ∂xi∂x′ j [M(x, x′)]i,j. Thus, given samples {(P|xi, yi)}n i=1 i.i.d. ∼P(P|X, Y ), an unbiased estimator of statistic CP|·(P) is d CP|· = 2 n(n -1) X 1≤i<j≤n H((P|xi, yi), (P|xj, yj)), where H is given by Equation (A.1). If kernel K is of the form in Equation (7), we recover the simpler formula in Equation (9). In this case A((p, y), (p′, y′)) = k(p, p′)l(y, y′)Idy ∈Rdy×dy, i.e., A is a scaled identity matrix. B KCCSD AS A SPECIAL CASE OF SKCE We prove the following general lemma that establishes the KCSD as a special case of the MMD. Then Proposition 3.1 follows immediately by considering random variables Z = P|X and Y , and models Q|z = z = P|x. Lemma B.1 (KCSD as a special case of the MMD). Let Q|z be models of the conditional distributions P(Y ∈· | Z = z). Moreover, we assume that • Q|z has a density fQ|z ∈C1(Y, R) for P(Z)-almost all z, • kernel l ∈C2(Y × Y, R), • E(z,y)∼P(Z,Y ) KzξQ|z(y, ·) FK < ∞, and • H ∂Y l(y, y′)fQ|z(y)n(y) dS(y′) = 0 and H ∂Y ∇yl(y, y′)fQ|z(y′)n(y′) dS(y′) = 0 for P(Z)-almost all z, where n(y) is the unit vector normal to the boundary ∂Y of Y at y ∈Y.5 Then DQ|·(P) = MMD2 kQ|· (P(Z, Y ), PQ|·(Z, Y )) where we define distribution PQ|· by PQ|·(Z ∈A, Y ∈B) := Z A Q|z(Y ∈B) P(Z ∈dz) and kernel kQ|· : (Z × Y) × (Z × Y) →R as kQ|·((z, y), (z′, y′)) := (sQ|z′(y′) + ∇y′)TA((z, y), (z′, y′))(sQ|z(y) + ∇y), using the same notation as in Section A and similarly defining A((z, y), (z′, y′)) ∈Rdy×dy by [A((z, y), (z′, y′))]i,j := (Ki,j(z, z′)l(y, ·))(y′) (1 ≤i, j ≤dy). If K is of the form k(·, ·)IF dy l , function A simplifies to A((z, y), (z′, y′)) = k(z, z′)l(y, y′)Idy and kernel kQ|· is given by kQ|·((z, y), (z′, y′)) = k(z, z′) l(y, y′)sQ|z(y)TsQ|z′(y′) + sQ|z(y)T∇y′l(y, y′) + sQ|z′(y′)T∇yl(y, y′) + dy X i=1 ∂2 ∂yi∂y′ i l(y, y′) . Proof. From a similar calculation as in Section A [cf. Jitkrittum et al., 2020, Section A.2] we obtain that kQ|·((z, y), (z′, y′)) = KzξQ|z(y, ·), Kz′ξQ|z′(y′, ·) FK . Thus kQ|· is an inner product of the features of (z, y) and (z′, y′) given by the feature map (z, y) 7→KzξQ|z(y, ·) ∈FK, and therefore kQ|· is a positive-definite kernel. Moreover, from our assumption we obtain E(z,y)∼P(Z,Y ) |kQ|·((z, y), (z, y))|1/2 = E(z,y)∼P(Z,Y ) KzξQ|z(y, ·) FK < ∞. Thus the mean embedding μP(Z,Y ) ∈FK of P(Z, Y ) exists [Gretton et al., 2012, Lemma 3]. 5These assumptions are not restrictive in practice since they are satisfied if the conditions of [Jitkrittum et al., 2020, Theorem 1] hold which are required to ensure that DQ|·(P) = 0 if and only if Q|Z(·) = P(Y ∈·|Z) P(Z)-almost surely. Due to the Bochner integrability of (z, y) 7→KzξQ|z(y, ·) expectation and inner product commute [see Andreas Christmann, 2008, Definition A.5.20], and hence we have E(z,y)∼PQ|·(Z,Y ) E(z′,y′)∼PQ|·(Z,Y ) kQ|·((z, y), (z′, y′)) = E(z,y)∼PQ|·(Z,Y ) KzξQ|z(y, ·) 2 FK = Ez∼P(Z) Ey∼Q|z KzξQ|z(y, ·) 2 FK = Ez∼P(Z) Kz Ey∼Q|z ξQ|z(y, ·) 2 FK . Due to the last assumption [Chwialkowski et al., 2016, Lemma 5.1] we know that Ey∼Q|z ξQ|z(y, ·) = 0, which implies E(z,y)∼PQ|·(Z,Y ) E(z′,y′)∼PQ|·(Z,Y ) kQ|·((z, y), (z′, y′)) = 0. Thus the mean embedding μPQ|·(Z,Y ) ∈FK of PQ|·(Z,Y ) exists and satisfies ∥μPQ|·(Z,Y )∥2 FK = 0, and hence μPQ|·(Z,Y ) = 0. We obtain [Gretton et al., 2012, Lemma 4] that MMD2 kQ|· (P(Z, Y ), PQ|·(Z, Y )) = ∥μP(Z,Y ) -μPQ|·(Z,Y )∥2 FK = ∥μP(Z,Y )∥2 FK = E(z,y)∼P(Z,Y ) E(z′,y′)∼P(Z,Y ) kQ|·((z, y), (z′, y′)) = E(z,y)∼P(Z,Y ) E(z′,y′)∼P(Z,Y ) KzξQ|z(y, ·), Kz′ξQ|z′(y′, ·) FK = DQ|·(P), where the last equality follows from [Jitkrittum et al., 2020, Section A.2]. C CALIBRATION IMPLIES EXPECTED COVERAGE We show that the sense of calibration employed by our tests implies posterior coverage in the sense of Hermans et al. [2021]. Again let us note P|x(·) for a model of the conditional distribution P(Y ∈· | X = x). Moreover, we assume that P|x has a density fP|x for P(X)-almost every x. For level 1 -α ∈[0, 1], let ΘP|x(1 -α) be the highest density region of a probabilistic model P|x with density fP|. It is defined [see, e.g., Hyndman, 1996] by ΘP|x(1 -α) := y: fP|x(y) ≥cP|x(1 -α) where cP|x(1 -α) := sup ( c: Z n ̃y : fP|x( ̃y)≥c o P|x(dy) ≥1 -α ) . Hence, by definition [see, e.g., Hermans et al., 2021] Ey∼P|x 1 y ∈ΘP|x(1 -α) = Z ΘP|x(1-α) P|x(dy) ≥1 -α. Assume that model P|· is calibrated. By definition, it satisfies P(Y ∈· | P|X) = P|X P(X)-almost surely. Hence, for all α ∈[0, 1], we obtain E(x,y)∼P(X,Y ) 1 y ∈ΘP|x(1 -α) = E(P|x,y)∼P(P|X,Y ) 1 y ∈ΘP|x(1 -α) = EP|x∼P(P|X) Ey∼P|x 1 y ∈ΘP|x(1 -α) ≥EP|x∼P(P|X) 1 -α = 1 -α. Thus model P|· has expected coverage for all α ∈[0, 1]. D DIFFUSION-LIMIT AND UNIVERSALITY D.1 FISHER DIVERGENCE AS A DIFFUSION LIMIT We recall that for a map f and a measure μ, the push-forward measure of μ by f, noted f#μ, is the measure on the image space of f which verifies, for any measurable function g Z g(x) f#μ(dx) = Z g(f(x)) μ(dx). To prove the differential inequality linking the MMD and the KGFD, we rely on the following reformulation of the Fokker-Planck equation: ∂μ(x, t) ∂t = divx(-μ(x, t)sp(x)) + ∆xμ(x, t) = divx(-μ(x, t)sp(x)) + divx ∇xμ(x, t) = divx(-μ(x, t)sp(x)) + divx(μ(x, t)∇x log μ(x, t)) = divx(-μ(x, t)(sp(x) -∇x log μ(x, t)). We remark that since the density μ(x, t) is twice differentiable in x and differentiable in t [Johnson, 2004], this equation holds in the strong sense, and not only in the sense of distributions. Because of that, one has ∂tμ(x, t) = lim ∆→0 μ(x, t + ∆) -μ(x, t) ∆ . Let us consider an RKHS H with kernel k, and let h ∈H. Let us define mt(x) := m(x, t) := μν,p(x, t) -μν,q(x, t) and we note MMD(mt) the function given by MMD(mt) = ZZ k(x, y)mt(x)mt(y) dx dy 1/2 = MMD(μν,p(·, t), μν,q(·, t)). To show that limt→0 d dt MMD(mt) = KGFD(p, q), we first analyze the differential properties of the easier to handle MMD2 and complete the proof using a chain rule argument. The first variation (also called Gateaux Derivative) of m 7→MMD2(m) is a linear functional on the space of functions f -g f, g: X × [0, ∞) →R with ∀t ≥0: Z X f(x, t) dx = Z X g(x, t) dx = 1 , given by δ MMD2 δm : f 7→ Z 2k(x, y)mt(x)f(y) dx dy. Using the chain rule for Gateaux derivatives, we have that d MMD2(m) dt = d MMD2 dm (m)dm dt = Z 2k(x, y)mt(x)dm dt (y) dx dy. From the Fokker-Planck Equation, we have that dm dt = ∂tμν,p -∂tμν,q = divx(μν,p∇x log p μν,p ) -divx(μν,q∇x log q μν,q ) = divx(ν∇x log p ν ) -divx(ν∇x log q ν ) + o(1) = divx(ν∇x log p q ) + o(1) Plugging the last equation in the chain rule, we have: d MMD2(m) dt = Z 2mt(x)divyν(y)∇y log p q (y)k(x, y)dxdy + o(1) = Z 2mt(x) ∇yk(x, y), ν(y)∇y log p q (y) dx dy + o(1). Similarly, since m0 = μν,p(·, 0) -μν,q(·, 0) = ν -ν = 0, we have mt(x) = t∂tm(x, 0) + ox(t). The calculation follows as: d MMD2(m) dt = Z 2t × ∂tm(x, t) ∇yk(x, y), ν(y)∇y log p q (y) dx dy + o(t) = Z 2t × divx ν(x)∇x log p q (x) ∇yk(x, y), ν(y)∇y log p q (y) dx dy + o(t) = Z 2t × ν(x)∇x log p q (x), ∇x ∇yk(x, y), ν(y)∇y log p q (y) dx dy + o(t) = Z 2t × ν(x)∇x log p q (x), ∇x∇yk(x, y), ν(y)∇y log p q (y) dx dy + o(t). To get rid of the degenerate scaling as t →0, we now focus on (the derivative of) q MMD2(mt) as t →0. Notice that since MMD(m0) = 0, the derivative of q MMD2(mt) does not exist a priori for t = 0: we consider instead d dt q MMD2(mt) t=t, and extend it by continuity by setting t →0. We have: d q MMD2(mt) dt = 1 2 q MMD2(mt) d MMD2(mt) dt . As MMD2(mt) = Z k(x, y)mt(x)mt(y) dx dy we obtain through similar calculations that MMD2(mt) = Z Z t2 ν(x)∇x log p q (x), ∇x∇yk(x, y), ν(y)∇y log p q (y) dx dy + o(t) from which the results follows. Note that the matrix-valued kernel (K(x, y))ij = (∇x∇yk(x, y))ij is positive definite, a result akin to one of Zhou [2008] but for the matrix-valued case. Indeed, for all x, y ∈X, z, t ∈Rd, zK(x, y)t = * d X i=1 zi∂ik(x, ·), d X i=1 ti∂ik(y, ·) + H where ∂ik(x, ·) ∈H [Zhou, 2008]. In the following, we write φ(x, y) = Pd i=1 yi∂ik(xi, ·). Now, for all sets of {xi}n i=1 ∈X, {yj}n i=1 ∈Rd, we have n X i,j=1 K(xi, xj)yj, yi Rd = n X i,j=1 φ(xi, yi), φ(xj, yj) H = * n X i=1 φ(xi, yi), n X i=1 φ(xi, yi) + H ≥0 from which it follows that K is indeed positive definite [Micchelli and Pontil, 2005, Theorem 2.1]. D.2 UNIVERSALITY OF THE EXPONENTIATED-GFD AND EXPONENTIATED-KGFD KERNEL To prove the universality of Kν and Kν,K under the assumptions discussed in the related propositions, we rely on the following theorem [Christmann and Steinwart, 2010, Theorem 2.2]. Theorem D.1. On a compact metric space (Z, dZ ) and for a continuous and injective map φ : Z 7→H, where H is a separable Hilbert space, the kernel K(z, z′) = e-γ∥φ(z)-φ(z′)∥2 H is universal. We first focus on the universality of Kν. We set as our goal to apply that theorem to our setting, in which Z := PX is a (sub)set of probability densities, which needs to be associated with a suitably chosen metric in order to make PX to be compact, and φ continuous. As bounded subsets of differentiable densities, whose elements can be framed as elements of the Sobolev space of first order W2,1(ν) [Taylor, 1996]), are not compact a priori, we restrict ourselves to twice-differentiable densities with bounded Sobolev norm of second order, i.e., to W2,2(ν) with norm ∥p∥2 W2,2 := ∥p∥2 L2(ν)+Pd i=1 ∥∂ip∥2 L2(ν)+ Pd i,j=1 ∥∂i∂jp∥2 L2(ν). From the Rellich-Kondrachov theorem [Taylor, 1996], we know that when ν has compact support, the canonical canonical injection I : W2,2(ν) →W2,1(ν) is a compact operator. As a consequence, for any bounded subset A of PX we thus have that I(A) is compact for ∥f∥2 W2,1 := ∥f∥2 L2(ν) + Pd i=1 ∥∂if∥2 L2(ν), which implies that any bounded subset A of PX is compact for d(z, z′) = ∥z -z′∥W2,1. To apply the above theorem, it remains to prove the continuity and injectivity of φ : p 7→∇log p under this metric (in that case the separable Hilbert space H is set to L2(ν)). And indeed, for such a choice of d, φ and H, φ is continuous. To prove this fact, remark that differentiable densities with full support on X are bounded away from 0, making the use of a φ: p 7→∇log p = ∇p/p continuous. Moreover, φ is injective as dW 2,1(p, q) := ∥p -q∥W 2,1 ̸= 0 implies ∥∇log p -∇log q∥L2(ν) ̸= 0. Thus, all conditions of [Christmann and Steinwart, 2010, Theorem 2.2] are satisfied, and the result follows as a consequence. We now move on to prove the universality of Kν,K. The proof follows the same reasoning as the proof of the universality of Kν, the only difference being the fact that the feature map ̃φ of KK,ν is given by Tν ◦φ, where φ: p 7→∇log p and TK,ν : L(X, Rd) →HK is given by TK,ν : f 7→ Z X Kxf(x) ν(dx). However, if ν is a probability measure and K is bounded, then TK,ν is a bounded operator, and thus continuous, making ̃φ continuous. Moreover, if K is characteristic, TK,ν is injective. Thus ̃φ is injective and continuous, from which the result follows by Christmann and Steinwart [2010]. E BACKGROUND ON STEIN AND FISHER DIVERGENCES The Fisher Divergence Consider two continuously differentiable densities p and q on Rd. Then the Fisher divergence [Sriperumbudur et al., 2017, Johnson, 2004] between p and q is defined as: FD(p||q) = Z Rd ∥∇log p(x) -∇log q(x)∥2 2 p(x) dx. We refer to Sriperumbudur et al. [2017] for an overview of the properties of the Fisher divergence, including its relative strength w.r.t. other divergences, and other formulations. The Fisher divergence was used for learning statistical models of some training data in Hyvärinen [2005], Sriperumbudur et al. [2017], and more recently in Song and Ermon [2019]. Stein Discrepancies Of proximity to the Fisher divergence is the family of Stein discrepancies [Anastasiou et al., 2022]. Stein discrepancies build upon the concept of Stein operators, which are operators AP such that EQ [APf] = 0 ⇐⇒Q = P for any f within a set G(AP) ⊂dom(AP) called the Stein class of AP. Following this definition, the AP-stein discrepancy is defined as SDAP(P, Q) = sup f∈G(A) ∥EQ Af∥ which satisfies by construction the axioms of a dissimilarity (or divergence) measure between P and Q. Link Between the Fisher divergence and Diffusion Stein Discrepancies Perhaps the most famous Stein discrepancy is the one that sets AP to be the infinitesimal generator of the isotropic diffusion process toward P [Gorham et al., 2019]: ( dXt = ∇log p(Xt) dt + √ 2 dWt (Ad,Pf)(·) = ⟨∇log p(·), ∇f⟩+ ⟨∇, ∇f⟩ Recalling that EP [Ad,Pf] = 0 for all f ∈G(Ad,P), we obtain the following formulation for the diffusion Stein discrepancy SDAd,P(P, Q) := sup f ∥EQ Ad,Pf∥= sup f EQ(∇log p -∇log q)⊤∇f = sup g=∇f EQ(∇log p -∇log q)⊤g , highlighting the connection between the Fisher divergence and the diffusion Stein discrepancy. Link Between the Fisher divergence and the Kernelized Stein Discrepancy Given a RKHS H such that BH⊗d(0H⊗d, 1) is a Stein class for Ad,P, the kernelized Stein discrepancy [Gorham and Mackey, 2017] is given by KSD(P, Q) := sup h=∇f∈H⊗d:∥h∥H⊗d≤1 ∥EQ ⟨∇log p(x) -∇log q(x), h(x)⟩∥ = sup h=∇f∈H⊗d:∥h∥H⊗d≤1 ⟨h, EQ(∇log p(x) -∇log q(x))k(x, ·)⟩1/2 H⊗d = ∥EQ [(∇log p(x) -∇log q(x))k(x, ·)]∥H⊗d = I⋆ k,Q(∇log p -∇log q) H⊗d where I⋆ k,Q is the adjoint of the canonical injection from H⊗d to (L2(Q))⊗d, also known as the kernel integral operator. This derivation shows that the KSD can be seen as a kernelized version of the Fisher divergence. Link between MMD and KSD It is possible [Gorham and Mackey, 2017] to reframe the KSD as an MMD with a specific kernel. Indeed, given some base kernel k(x, y), define the following "Stein" kernel ̃k(x, y) = ⟨∇log p(x)k(x, ·) + ∇k(x, ·), ∇log p(y)k(y, ·) + ∇log k(y, ·)⟩H⊗d which is positive definite as an inner product of a feature map of x. Then H ̃k = Ad,P(H) and ∥f∥H ̃k = ∥Af∥H⊗d k . Moreover, we have that EP ̃h = 0 for all ̃h ∈H ̃k. By the definition of the KSD, we have that KSD(P, Q) = sup h∈H⊗d : ∥h∥H⊗d≤1 ∥EQ Ad,Ph∥ = sup h∈H⊗d : ∥h∥H⊗d≤1 ∥EQ Ad,Ph -EP Ad,Ph∥H = sup h∈H ̃k : ∥h∥H ̃k ≤1 ∥EQ h -EP h∥H ̃k = MMD ̃k(P, Q). Differential Inequalities between the KL and the Fisher Divergence It is well known [Carrillo et al., 2003] that the KL divergence can be related to the Fisher divergence by considering the evolution of KL(Pt||Q) when Pt evolves according to the Fokker-Planck equation ∂tpt(x) = div(pt(x)(∇log qt(x) -∇log pt(x))), P0 = P. (E.1) (Two relevant side notes: for any t ≥0, Pt is the law at time t of the Markov process (Xt)t≥0 such that X0 ∼P and undergoing an isotropic diffusion towards Q. Moreover, Equation (E.1) is also the Wasserstein gradient flow equation of KL(·||Q) starting from P). Recalling that Equation (E.1) is satisfied in the sense of distributions, and relying on Gateaux-Derivative formulas for Free Energy-type functionals [see Ambrosio et al., 2005, for more precise statements], we have: dKL(Pt||Q) dt = ∂KL dP Pt dPt dt = Z ⟨∇(log pt(x) -log qt(x)), (∇log qt -∇log pt)⟩dPt(x) = -FD(Pt, Q). Figure E.1: Relationships between the Fisher divergence, the KL divergence, the MMD, and the KSD [Liu, 2016]. F EXPERIMENTAL RESULTS This section contains visualizations of all experiments discussed in 6, including figures contained in the main text. In all experiments we set the significance level to α = 0.05. Every experiment is repeated for 100 randomly sampled datasets and with 500 bootstrap iterations for estimating the quantile of the test statistic. We use Gaussian distributions and compare the KCCSD and the SKCE with different combinations of kernels. For the KCCSD, for Gaussian distributions all considered test statistics can be evaluated exactly. Alternatively, for the exponentiated (kernelized) Fisher kernel and the exponentiated MMD kernel one can resort to approximations using samples from the base measure. For the SKCE, however, the test statistic can be evaluated exactly on in special cases such as Gaussian kernels on the target space. All approximate evaluations are performed with 10 samples. F.1 MEAN GAUSSIAN MODEL 26 27 28 29 210 num samples 0.00 0.25 0.50 0.75 1.00 rejection rate exact Exp. Fisher + Gaussian Exp. Fisher + IMQ Exp. Kernelized Fisher + Gaussian Exp. Kernelized Fisher + IMQ Exp. MMD + Gaussian Exp. MMD + IMQ Exp. Wasserstein + Gaussian Exp. Wasserstein + IMQ 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.1: False rejection rate of the KCCSD for MGM (δ = 0). 26 27 28 29 210 num samples 0.0 0.5 1.0 rejection rate exact Exp. Fisher + Gaussian Exp. Kernelized Fisher + Gaussian Exp. MMD + Gaussian Exp. Wasserstein + Gaussian 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.2: False rejection rate of the SKCE for MGM (δ = 0). 26 27 28 29 210 num samples 0.0 0.2 0.4 0.6 0.8 1.0 rejction rate exact Exp. Fisher + Gaussian Exp. Fisher + IMQ Exp. Kernelized Fisher + Gaussian Exp. Kernelized Fisher + IMQ Exp. MMD + Gaussian Exp. MMD + IMQ Exp. Wasserstein + Gaussian Exp. Wasserstein + IMQ 26 27 28 29 210 num samples rejction rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.3: Rejection rate of the KCCSD for MGM (δ = 0.1, c = 1d). 26 27 28 29 210 num samples 0.0 0.2 0.4 0.6 0.8 1.0 rejction rate exact Exp. Fisher + Gaussian Exp. Kernelized Fisher + Gaussian Exp. MMD + Gaussian Exp. Wasserstein + Gaussian 26 27 28 29 210 num samples rejction rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.4: Rejection rate of the SKCE for MGM (δ = 0.1, c = 1d). 26 27 28 29 210 num samples 0.0 0.2 0.4 0.6 0.8 1.0 rejction rate exact Exp. Fisher + Gaussian Exp. Fisher + IMQ Exp. Kernelized Fisher + Gaussian Exp. Kernelized Fisher + IMQ Exp. MMD + Gaussian Exp. MMD + IMQ Exp. Wasserstein + Gaussian Exp. Wasserstein + IMQ 26 27 28 29 210 num samples rejction rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.5: Rejection rate of the KCCSD for MGM (δ = 0.1, c = e1). 26 27 28 29 210 num samples 0.00 0.25 0.50 0.75 1.00 rejction rate exact Exp. Fisher + Gaussian Exp. Kernelized Fisher + Gaussian Exp. MMD + Gaussian Exp. Wasserstein + Gaussian 26 27 28 29 210 num samples rejction rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.6: Rejection rate of the SKCE for MGM (δ = 0.1, c = e1). F.2 LINEAR GAUSSIAN MODEL 26 27 28 29 210 num samples 0.00 0.25 0.50 0.75 1.00 rejection rate exact Exp. Fisher + Gaussian Exp. Fisher + IMQ Exp. Kernelized Fisher + Gaussian Exp. Kernelized Fisher + IMQ Exp. MMD + Gaussian Exp. MMD + IMQ Exp. Wasserstein + Gaussian Exp. Wasserstein + IMQ 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.7: False rejection rate of the KCCSD for LGM (δ = 0). 26 27 28 29 210 num samples 0.0 0.5 1.0 rejection rate exact Exp. Fisher + Gaussian Exp. Kernelized Fisher + Gaussian Exp. MMD + Gaussian Exp. Wasserstein + Gaussian 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.8: False rejection rate of the SKCE for LGM (δ = 0). F.3 HETEROSCEDASTIC GAUSSIAN MODEL 26 27 28 29 210 num samples 0.0 0.2 0.4 0.6 0.8 1.0 rejection rate exact Exp. Fisher + Gaussian Exp. Fisher + IMQ Exp. Kernelized Fisher + Gaussian Exp. Kernelized Fisher + IMQ Exp. MMD + Gaussian Exp. MMD + IMQ Exp. Wasserstein + Gaussian Exp. Wasserstein + IMQ 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.9: Rejection rate of the KCCSD for HGM (δ = 1). 26 27 28 29 210 num samples 0.0 0.5 1.0 rejection rate exact Exp. Fisher + Gaussian Exp. Kernelized Fisher + Gaussian Exp. MMD + Gaussian Exp. Wasserstein + Gaussian 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.10: Rejection rate of the SKCE for HGM (δ = 1). F.4 QUADRATIC GAUSSIAN MODEL 26 27 28 29 210 num samples 0.0 0.2 0.4 0.6 0.8 1.0 rejection rate exact Exp. Fisher + Gaussian Exp. Fisher + IMQ Exp. Kernelized Fisher + Gaussian Exp. Kernelized Fisher + IMQ Exp. MMD + Gaussian Exp. MMD + IMQ Exp. Wasserstein + Gaussian Exp. Wasserstein + IMQ 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.11: Rejection rate of the KCCSD for QGM (δ = 1). 26 27 28 29 210 num samples 0.0 0.2 0.4 0.6 0.8 1.0 rejection rate exact Exp. Fisher + Gaussian Exp. Kernelized Fisher + Gaussian Exp. MMD + Gaussian Exp. Wasserstein + Gaussian 26 27 28 29 210 num samples rejection rate approximate Exp. Fisher (discrete) + Gaussian Exp. Fisher (discrete) + IMQ Exp. Kernelized Fisher (discrete) + Gaussian Exp. Kernelized Fisher (discrete) + IMQ Exp. MMD (discrete) + Gaussian Exp. MMD (discrete) + IMQ Figure F.12: Rejection rate of the SKCE for QGM (δ = 1).
|
2510.14708
|
SLIE: A Secure and Lightweight Cryptosystem
for Data Sharing in IoT Healthcare Services⋆
Ha Xuan Son1
Nguyen Quoc Anh2
, Phat T. Tran-Truong3,4⋆⋆
, Le Thanh
Tuan5, and Pham Thanh Nghiem6
1 RMIT University, Ho Chi Minh City, Vietnam
ha.son@rmit.edu.vn
2 Hitachi Digital Services, Ho Chi Minh City, Vietnam
anh.quocnguyen@hitachids.com
3 Faculty of Computer Science and Engineering, Ho Chi Minh City University of
Technology (HCMUT), Ho Chi Minh City, Vietnam
4 Vietnam National University Ho Chi Minh City (VNU-HCM), Ho Chi Minh City,
Vietnam
⋆phatttt@hcmut.edu.vn
5 NLS Tech Vietnam, Ho Chi Minh City, Vietnam
tuan@nlstech.net
6 Polarista A.I, Ho Chi Minh City, Vietnam
nghiem.pham@polarista.vn
Abstract. The Internet of Medical Things (IoMT) has revolutionized
healthcare by transforming medical operations into standardized, inter-
operable services. However, this service-oriented model introduces signif-
icant security vulnerabilities in device management and communication,
which are especially critical given the sensitivity of medical data. To ad-
dress these risks, this paper proposes SLIE (Secure and Lightweight Iden-
tity Encryption), a novel cryptosystem based on Wildcard Key Deriva-
tion Identity-Based Encryption (WKD-IBE). SLIE ensures scalable trust
and secure omnidirectional communication through end-to-end encryp-
tion, hierarchical access control, and a lightweight key management sys-
tem designed for resource-constrained devices. It incorporates constant-
time operations, memory obfuscation, and expiry-based key revocation
to counter side-channel, man-in-the-middle, and unauthorized access at-
tacks, thereby ensuring compliance with standards like HIPAA and GDPR.
Evaluations show that SLIE significantly outperforms RSA, with encryp-
tion and decryption times of 0.936ms and 0.217ms for 1KB of data, an
84.54% improvement in encryption speed, a 99.70% improvement in de-
cryption speed, and an energy efficiency of 0.014 J/KB.
Keywords: WKD-IBE Encryption · IoT Services · Healthcare Services
· IoT Security · Data Sharing
⋆Paper has been accepted for publication in the Proceedings of the 23th International
Conference on Service-Oriented Computing (ICSOC 2025, https: // icsoc2025.
hit. edu. cn/ )
⋆⋆Corresponding author.
arXiv:2510.14708v1 [cs.CR] 16 Oct 2025
2
H. X. Son et al.
1
Introduction
The growing diversity of IoMT devices has enabled novel telehealth and telemedicine
business models through the servitization of medical operations within service-
oriented architectures. These architectures facilitate remote healthcare delivery
by dynamically composing services to maximize value for stakeholders, includ-
ing providers and patients [1]. By transforming IoT devices into IoT service
providers, the IoMT ecosystem enables crowdsourced functionality, where de-
vices not only collect data but also share resources and coordinate actuation
across distinguished networks.
Despite these advancements, healthcare’s unique security challenges present
critical adoption barriers. Medical datasets contain 50% more sensitive informa-
tion than other sectors, making them high-priority targets for cyberattacks. This
is reflected in a 45% increase in ransomware incidents and an average breach
cost of $10.93 million [2, 3]. While centralized systems like Electronic Health
Records (EHRs) and cloud platforms enhance accessibility, their API-dependent
architectures introduce vulnerabilities to unauthorized access [4]. Existing cryp-
tographic solutions, though promising, often fail to address the fundamental
heterogeneity of IoMT ecosystems, which encompass devices ranging from wear-
ables to implantables with disparate energy, storage, and computational capa-
bilities [5]. This diversity necessitates adaptive service models that holistically
address both functional (e.g., data acquisition) and non-functional (e.g., real-
time performance, security) requirements.
Current security solutions remain inadequate as they either: (1) address com-
munication security in isolation, or (2) fail to accommodate the heterogeneous
nature of IoMT devices. While symmetric encryption (e.g., AES) offers efficiency,
it suffers from key distribution vulnerabilities [6]. Emerging solutions based
on Attribute-Based Encryption (ABE), Hardware Security Modules, Trusted
Execution Environments, and Blockchain provide promising directions for de-
centralized key management, but require careful adaptation to IoMT’s unique
constraints of heterogeneous devices and strict regulatory requirements [7]. To
overcome these limitations, we propose SLIE (Secure and Lightweight Identity
Encryption) - a novel cryptosystem optimized for omnidirectional IoMT service
communication. Our proposed cryptosystem makes three fundamental advances
in IoMT security:
– Hierarchical Key Management: A novel WKD-IBE framework that en-
ables efficient, fine-grained access control across distributed IoMT devices
while maintaining minimal computational overhead.
– Resource-Aware Access Control: Non-commutative authorization mech-
anisms specifically optimized for constrained devices, ensuring secure com-
munication without compromising performance.
– Automated Compliance: An expiry-based key revocation that enforces
HIPAA/GDPR requirements while minimizing management overhead.
Our comprehensive experiments demonstrate SLIE’s robust performance across
diverse IoMT device capabilities (see Figure 1), delivering three key security
SLIE: Secure IoT Healthcare Services Communication
3
Fig. 1: SLIE compatibility spectrum across healthcare IoT devices
advantages: (1) end-to-end encryption with proven resistance against MITM
attacks through DBDH-secure cryptographic primitives, (2) comprehensive pro-
tection against differential cryptanalysis and chosen-plaintext attacks, and (3)
built-in compliance with HIPAA/GDPR regulatory requirements.
2
Literature Review
The sensitive nature of healthcare data makes it a prime target for cyberat-
tacks, with breaches risking patient privacy, safety, and institutional reputation.
Securing this data requires efficient, many-to-many, end-to-end encryption across
low-power IoT devices. An ideal cryptographic framework must therefore pro-
vide robust identity authentication, key management, and access control while
remaining adaptable to fast-paced emergency scenarios.
ABE presents a promising solution by integrating access policies into encryp-
tion, allowing decryption only when user attributes satisfy predefined criteria [8].
This approach simplifies key management compared to traditional PKI and en-
ables granular access control. For instance, Satheesh and Sree [9] utilized bilinear
mapping in ABE for secure decentralized cloud storage, Zhang et al. [10] applied
Ciphertext-Policy ABE to streamline healthcare data encryption, and Obiri et
al. [11] adopted Key-Policy ABE (KP-ABE) to enable flexible policy adjust-
ments. However, ABE-based systems often rely on computationally intensive
bilinear pairing operations, rendering them less suitable for dynamic, resource-
constrained IoT environments.
As an alternative, IBE simplifies key distribution by using a user’s identity
(e.g., an email) as their public key, eliminating the need for pre-shared certifi-
cates [12]. Li et al. [13] further enhanced IBE with wildcard key derivation for
improved access control and integrated Blockchain for decentralized authoriza-
tion. While such approaches reduce overhead, they often retain a dependence on
a CA, introducing a potential security bottleneck. Conversely, while lightweight
cryptographic algorithms are designed for resource-constrained devices, they of-
ten lack end-to-end security structures, making them vulnerable to attacks.
Reflecting on these trade-offs, it is clear that the expansion of healthcare IoT
demands a cryptographic framework that simultaneously addresses three criti-
4
H. X. Son et al.
cal dimensions for cross-device data sharing: (1) strong security guarantees, (2)
efficient resource utilization, and (3) compliance with domain-specific policies.
3
Methodology
3.1
Hierarchical Access Control with Time-Bound Encryp-
tion
Our proposed framework integrates lightweight IBE with an efficient revocation
mechanism, specifically designed for resource-constrained IoT devices handling
sensitive cross-sectional healthcare data. Unlike existing IBE schemes that focus
on general access control, SLIE addresses the unique requirements of healthcare
hierarchies through role-specific temporal constraints and automatic delegation
patterns that align with clinical workflows. Particularly, in Fig. 2:
– Doctor Alice can access /HC/Data/EHR/full for entire departmental records
– Nurse Charlie is restricted to specific patient records (patient_1/record)
– Third-party providers only receive access to diagnostic outcomes (diagnosis)
All data undergoes end-to-end encryption before cloud upload, eliminating plain-
text exposure risks. The system implements temporally-graded access, with key
validity periods scaled by role sensitivity to balance security and operational
flexibility in time-sensitive cases (e.g., stroke, myocardial infarction).
Fig. 2: Hierarchical structure of health-
care communication
Fig. 3: IBE-based key generation (RSA
for double protection is optional)
Keys have a validity period τ that is automatically determined based on
healthcare role requirements. This approach addresses a critical limitation in
existing IBE schemes, where revocation typically requires maintaining central-
ized blacklists or complex re-encryption procedures. Our time-bounded approach
eliminates these overheads through automatic expiration mechanisms. A key is
considered valid or expired based on the current time t (1):
Expire(SKAlice, t) =
(
Valid
if t ≤τ
Invalid
if t > τ
(1)
The expiration or validity period (τ) for data access should be set through
a risk assessment considering data sensitivity, role changes, regulations (e.g.,
HIPAA, GDPR), and institutional policies, as current regulations lack specific
SLIE: Secure IoT Healthcare Services Communication
5
timeframes for healthcare roles. Particularly, in the U.S., with an average hospital
stay of 5.5 days [14], nurses providing direct care should have a τ of 6–7 days,
doctors expecting follow-ups a τ of up to 21 days, and third-party contractors
(e.g., IT, medical services), with less 24/7 monitoring responsibility, a τ of up
to 2 days. Family members may receive 1-day portal access. Nevertheless, this
highly depends on the aforementioned risk assessment factors. These durations
reflect the varying levels of access privilege and security risk associated with
different roles. Specifically, the key lifecycle consists of the following steps:
1. Initial Key Assignment: Alice receives a private key SKAlice with an
expiry time τ0.
2. Periodic Key Refresh: When τ0 approaches, a new key SK′
Alice is issued
with an extended expiry τ1 > τ0 (2):
Refresh(SKAlice, τ0) = (SK′
Alice, τ1)
(2)
3. Revocation by Non-Renewal: If no new key is issued after expiry, Alice’s
access is revoked by default.
Access keys for staff are automatically renewed based on active employment
and patient responsibilities, as verified by hospital systems. This automated
revocation eliminates centralized infrastructure, ensures immediate access ter-
mination, and scales with organizational complexity.
3.2
Identity-Based Key Generation
SLIE leverages WKD-IBE [15] for key management (Fig. 3), but extends the
standard approach through healthcare-specific pattern designs over (Z∗
p ∪{⊥})ℓ.
Our contribution lies in the development of patterns that encode clinical hierar-
chies and role relationships, allowing fine-grained, non-commutative delegation
and efficient key derivation across multiple concurrent healthcare departments
encoded as disjoint slot partitions. An initial setup entity generates the param-
eters and a master secret key (MSK) (3). For Alice, with identity IDAlice, a
private key is derived (4). Messages are encrypted (5) and decrypted using Al-
ice’s private key (6). In addition, keys are preserved on serve database.
(Params, MSK) = Setup(1κ)
(3)
SKAlice = KeyDer(Params, MSK, Pattern(IDAlice))
(4)
C = Encrypt(Params, Pattern(IDAlice), M)
(5)
M = Decrypt(Params, SKAlice, C)
(6)
Unlike traditional IBE, which uses generic identity strings, SLIE’s WKD-
IBE framework integrates healthcare workflow patterns directly into the crypto-
graphic structure. This encodes role hierarchy and temporal constraints, enabling
automatic policy enforcement without external mechanisms.
6
H. X. Son et al.
4
Experiments and Results
The SLIE evaluation was conducted on the LifeSnaps Fitbit Dataset [16], a
longitudinal repository containing over 71 million multi-modal health records
from 71 participants collected over four months. The dataset’s capture of 35+
clinical biomarkers at varying temporal resolutions (seconds to daily) provides
a realistic benchmark for validating the security of healthcare IoT systems.
4.1
Performance on Diverse Resource-constrained Devices
Table 1 illustrates the compatibility and performance of SLIE across a diverse
range of resource-constrained healthcare IoT devices. It demonstrates SLIE’s
adaptability to real-world healthcare applications by evaluating key operation,
such as key generation, encryption, decryption, and delegation, which are av-
eraged over 10 runs to ensure statistical reliability. Resource-rich devices like
the MacBook Pro (2016) demonstrate exceptional efficiency, with encryption
and decryption times of 0.056 ms and 0.0755 ms, respectively, and a notably low
delegation time of 14 ms. In contrast, ultra-low-power devices such as the Ap-
ple Watch Series 6 achieve encryption and decryption in 0.1 ms and 0.03 ms,
respectively, showcasing SLIE’s optimization for constrained environments. The
Raspberry Pi 4 strikes a balance, with encryption and decryption times of 0.3 ms
and 0.4 ms, suitable for edge computing in IoMT networks.
Table 1: Performance evaluation of SLIE on various devices
Devices
Generate key Encrypt Decrypt Delegate
Apple Watch Series 6
122ms
0.1ms
0.03ms
82ms
Fitbit Versa 2
182ms
42ms
53ms
83ms
Raspberry Pi 4
122ms
0.3ms
0.4ms
82ms
Macbook Pro 2016 MLH42
99ms
0.056ms 0.0755ms
14ms
Table 2 details the power consumption and resource utilization of SLIE’s
core operations on representative IoT devices. Encryption and decryption oper-
ations exhibit remarkable energy efficiency, consuming 0.014 J and 0.009 J per
1 KB data respectively. These metrics highlight SLIE’s suitability for battery-
constrained devices, enabling prolonged operation in wearable and implantable
medical devices. The delegation operation, critical for secure access control in
IoMT systems, consumes 0.052 J/KB. While delegation is more resource-intensive,
its moderate power usage (1.45 W) and CPU utilization (35.6 %) ensure feasi-
bility on low-power platforms. Memory usage remains modest, ranging from
12.5 MB (decryption) to 22.3 MB (delegation), further supporting SLIE’s lightweight
design. Collectively, these attributes demonstrate SLIE as an energy-efficient
and scalable solution for secure data sharing for IoMT services, balancing per-
formance, power, and security requirements.
SLIE: Secure IoT Healthcare Services Communication
7
Table 2: Power consumption analysis of SLIE operations
Operation Exec. time (ms) Power Usage (W) Mem. Usage (MB) CPU Util. (%) Energy Eff. (J/KB)
Encrypt
12
1.20
16.80
27.50
0.014
Decrypt
9
0.95
12.50
19.20
0.009
Delegate
18
1.45
22.30
35.60
0.052
4.2
Scalability and Efficiency in IoT Data Encryption
Table 3 quantifies SLIE’s performance across data sizes from 1 kB to 10 MB,
averaged over 10 runs. For smaller data sizes (1 KB), SLIE achieves encryption
and decryption times of 0.936 ms and 0.217 ms, respectively, with minimal CPU
usage (0.5 % and 0.7 %) and RAM consumption (1520 kB and 1300 kB). As data
sizes increase, SLIE metrics increase linearly, demonstrating its suitability for
resource-constrained IoMT devices, such as wearables and edge nodes.
Table 3: Performance metrics of SLIE under different scenarios
Data Size Key Creation
Time (ms)
Encryption
Time (ms)
Encryption
CPU Usage (%)
Encryption
RAM Usage (KB)
Decryption
Time (ms)
Decryption
CPU Usage (%)
Decryption
RAM Usage (KB)
1KB
81
0.936
0.5
1520
0.217
0.7
1300
100KB
81
3.500
1.8
5200
1.184
1.9
4500
500KB
85
4.769
2.8
9650
1.846
3.0
8800
1MB
83
12.677
3.2
12000
3.318
4.2
11500
5MB
95
15.530
4.5
35600
8.993
5.5
30400
10MB
83
25.036
8.3
55120
15.119
9.2
50800
As benchmarked in Table 4, SLIE demonstrates a compelling balance of
performance and security. It drastically outperforms RSA, achieving an 84.54%
faster encryption and a 99.70% faster decryption. While its raw speed is lower
than the symmetric cipher ChaCha20, SLIE provides essential, built-in secu-
rity features—such as hierarchical access control—that ChaCha20 lacks. These
suggest that SLIE a more optimal solution for IoMT services.
Table 4: Average performance metrics comparison
Algorithm Key Creation (ms) Enc (ms) Dec (ms) CPU Usage (%) RAM Usage (KB)
SLIE
82.3
9.73
4.49
3.66
17,535
RSA
121.85
62.93
1476.57
5.86
31,388
ChaCha20
None
3.57
2.81
2.33
14,775
5
Conclusion
In conclusion, the proposed SLIE cryptosystem establishes a robust foundation
for secure IoMT services by combining the efficiency of a lightweight WKD-IBE
framework with scalable trust management. It achieves a compelling perfor-
mance profile, surpassing RSA in speed by over 84% for encryption and 99% for
decryption while operating with high energy efficiency (0.014 J/KB). Crucially,
SLIE overcomes the key management shortcomings of symmetric systems like
ChaCha20 by natively supporting hierarchical access control and automatic key
revocation, ensuring both security and regulatory compliance. Looking ahead,
our research will evolve to incorporate post-quantum security, optimize for next-
generation implantables, and demonstrate real-world interoperability with stan-
dards such as FHIR in clinical settings.
8
H. X. Son et al.
6
Code Availability
The source code developed for this study and the scripts used to generate the
results can be found at https://github.com/SonHaXuan/SLIE.
Acknowledgement
The authors acknowledge Ho Chi Minh City University of Technology (HC-
MUT), VNU-HCM, for supporting this study.
References
1. Athman Bouguettaya et al. An internet of things service roadmap. Communica-
tions of the ACM, 64(9):86–95, 2021.
2. The HIPAA Journal. Healthcare ransomware attacks involve 20 percents of stored
sensitive data, 2024.
3. World Economic Forum. Healthcare pays the highest price of any sector for cy-
berattacks — that’s why cyber resilience is key, 2024.
4. Lancer G Gates. Cyber attacks on interoperable electronic health records: A clear
and present danger. Missouri medicine, 121 1:6–9, 2024.
5. Athman Bouguettaya et al. A service computing manifesto: the next 10 years.
Communications of the ACM, 60(4):64–72, 2017.
6. M Shankar Lingam et al.
Data encryption as security measure in iot-enabled
healthcare. In Smart Trends in Computing and Communications: Proceedings of
SmartCom 2020, pages 69–81. Springer, 2020.
7. Mohamed Nabeel. The many faces of end-to-end encryption and their security
analysis. 2017 IEEE International Conference on Edge Computing (EDGE), pages
252–259, 2017.
8. J. Li et al. Attribute based encryption with privacy protection and accountability
for cloudiot. IEEE Transactions on Cloud Computing, 10:762–773, 2022.
9. Kavuri KSVA Satheesh and T Krishna Sree. Ab-dam: attribute-based data access
model in blockchain for healthcare applications. Multimedia Tools and Applica-
tions, 81(17):23567–23588, 2022.
10. Leyou Zhang et al. Hidden ciphertext policy attribute-based encryption with fast
decryption for personal health record system. IEEE Access, 7:33202–33213, 2019.
11. Isaac Amankona Obiri et al. A fully secure kp-abe scheme on prime-order bilin-
ear groups through selective techniques. Security and Communication Networks,
2020(1):8869057, 2020.
12. Yinxia Sun et al.
Efficient identity-based encryption with revocation for data
privacy in internet of things. IEEE IoT-J, 9(4):2734–2743, 2021.
13. Minghui Li et al. Nacda: naming-based access control and decentralized authoriza-
tion for secure many-to-many data sharing. Electronics, 12(7):1651, 2023.
14. Kelley Tipton et al. Interventions to decrease hospital length of stay. 2021.
15. Sam Kumar et al.
{JEDI}:{Many-to-Many}{End-to-End} encryption and key
delegation for {IoT}.
In 28th USENIX security symposium (USENIX Security
19), pages 1519–1536, 2019.
16. Sofia Yfantidou et al. Lifesnaps, a 4-month multi-modal dataset capturing unob-
trusive snapshots of our lives in the wild. Scientific Data, 9, 2022.
|
SLIE: A Secure and Lightweight Cryptosystem for Data Sharing in IoT Healthcare Services⋆ Ha Xuan Son1 Nguyen Quoc Anh2 , Phat T. Tran-Truong3,4⋆⋆ , Le Thanh Tuan5, and Pham Thanh Nghiem6 1 RMIT University, Ho Chi Minh City, Vietnam 2 Hitachi Digital Services, Ho Chi Minh City, Vietnam 3 Faculty of Computer Science and Engineering, Ho Chi Minh City (HCMUT), Ho Chi Minh City, Vietnam 4 Vietnam National University Ho Chi Minh City (VNU-HCM), Ho Chi Minh City, Vietnam ⋆ 5 NLS Tech Vietnam, Ho Chi Minh City, Vietnam 6 Polarista A.I, Ho Chi Minh City, Vietnam Abstract. The Internet of Medical Things (IoMT) has revolutionized healthcare by transforming medical operations into standardized, interoperable services. However, this service-oriented model introduces significant security vulnerabilities in device management and communication, which are especially critical given the sensitivity of medical data. To address these risks, this paper proposes SLIE (Secure and Lightweight Identity Encryption), a novel cryptosystem based on Wildcard Key Derivation Identity-Based Encryption (WKD-IBE). SLIE ensures scalable trust and secure omnidirectional communication through end-to-end encryption, hierarchical access control, and a lightweight key management system designed for resource-constrained devices. It incorporates constanttime operations, memory obfuscation, and expiry-based key revocation to counter side-channel, man-in-the-middle, and unauthorized access attacks, thereby ensuring compliance with standards like HIPAA and GDPR. Evaluations show that SLIE significantly outperforms RSA, with encryption and decryption times of 0.936ms and 0.217ms for 1KB of data, an 84.54% improvement in encryption speed, a 99.70% improvement in decryption speed, and an energy efficiency of 0.014 J/KB. Keywords: WKD-IBE Encryption · IoT Services · Healthcare Services · IoT Security · Data Sharing ⋆Paper has been accepted for publication in the Proceedings of the 23th International Conference on Service-Oriented Computing (ICSOC 2025, https: // icsoc2025. hit. edu. cn/ ) ⋆⋆Corresponding author. 16 Oct 2025 2 H. X. Son et al. 1 Introduction The growing diversity of IoMT devices has enabled novel telehealth and telemedicine business models through the servitization of medical operations within serviceoriented architectures. These architectures facilitate remote healthcare delivery by dynamically composing services to maximize value for stakeholders, including providers and patients [1]. By transforming IoT devices into IoT service providers, the IoMT ecosystem enables crowdsourced functionality, where devices not only collect data but also share resources and coordinate actuation across distinguished networks. Despite these advancements, healthcare's unique security challenges present critical adoption barriers. Medical datasets contain 50% more sensitive information than other sectors, making them high-priority targets for cyberattacks. This is reflected in a 45% increase in ransomware incidents and an average breach cost of $10.93 million [2, 3]. While centralized systems like Electronic Health Records (EHRs) and cloud platforms enhance accessibility, their API-dependent architectures introduce vulnerabilities to unauthorized access [4]. Existing cryptographic solutions, though promising, often fail to address the fundamental heterogeneity of IoMT ecosystems, which encompass devices ranging from wearables to implantables with disparate energy, storage, and computational capabilities [5]. This diversity necessitates adaptive service models that holistically address both functional (e.g., data acquisition) and non-functional (e.g., realtime performance, security) requirements. Current security solutions remain inadequate as they either: (1) address communication security in isolation, or (2) fail to accommodate the heterogeneous nature of IoMT devices. While symmetric encryption (e.g., AES) offers efficiency, it suffers from key distribution vulnerabilities [6]. Emerging solutions based on Attribute-Based Encryption (ABE), Hardware Security Modules, Trusted Execution Environments, and Blockchain provide promising directions for decentralized key management, but require careful adaptation to IoMT's unique constraints of heterogeneous devices and strict regulatory requirements [7]. To overcome these limitations, we propose SLIE (Secure and Lightweight Identity Encryption) - a novel cryptosystem optimized for omnidirectional IoMT service communication. Our proposed cryptosystem makes three fundamental advances in IoMT security: - Hierarchical Key Management: A novel WKD-IBE framework that enables efficient, fine-grained access control across distributed IoMT devices while maintaining minimal computational overhead. - Resource-Aware Access Control: Non-commutative authorization mechanisms specifically optimized for constrained devices, ensuring secure communication without compromising performance. - Automated Compliance: An expiry-based key revocation that enforces HIPAA/GDPR requirements while minimizing management overhead. Our comprehensive experiments demonstrate SLIE's robust performance across diverse IoMT device capabilities (see Figure 1), delivering three key security SLIE: Secure IoT Healthcare Services Communication 3 Fig. 1: SLIE compatibility spectrum across healthcare IoT devices advantages: (1) end-to-end encryption with proven resistance against MITM attacks through DBDH-secure cryptographic primitives, (2) comprehensive protection against differential cryptanalysis and chosen-plaintext attacks, and (3) built-in compliance with HIPAA/GDPR regulatory requirements. 2 Literature Review The sensitive nature of healthcare data makes it a prime target for cyberattacks, with breaches risking patient privacy, safety, and institutional reputation. Securing this data requires efficient, many-to-many, end-to-end encryption across low-power IoT devices. An ideal cryptographic framework must therefore provide robust identity authentication, key management, and access control while remaining adaptable to fast-paced emergency scenarios. ABE presents a promising solution by integrating access policies into encryption, allowing decryption only when user attributes satisfy predefined criteria [8]. This approach simplifies key management compared to traditional PKI and enables granular access control. For instance, Satheesh and Sree [9] utilized bilinear mapping in ABE for secure decentralized cloud storage, Zhang et al. [10] applied Ciphertext-Policy ABE to streamline healthcare data encryption, and Obiri et al. [11] adopted Key-Policy ABE (KP-ABE) to enable flexible policy adjustments. However, ABE-based systems often rely on computationally intensive bilinear pairing operations, rendering them less suitable for dynamic, resourceconstrained IoT environments. As an alternative, IBE simplifies key distribution by using a user's identity (e.g., an email) as their public key, eliminating the need for pre-shared certificates [12]. Li et al. [13] further enhanced IBE with wildcard key derivation for improved access control and integrated Blockchain for decentralized authorization. While such approaches reduce overhead, they often retain a dependence on a CA, introducing a potential security bottleneck. Conversely, while lightweight cryptographic algorithms are designed for resource-constrained devices, they often lack end-to-end security structures, making them vulnerable to attacks. Reflecting on these trade-offs, it is clear that the expansion of healthcare IoT demands a cryptographic framework that simultaneously addresses three criti4 H. X. Son et al. cal dimensions for cross-device data sharing: (1) strong security guarantees, (2) efficient resource utilization, and (3) compliance with domain-specific policies. 3 Methodology 3.1 Hierarchical Access Control with Time-Bound Encryption Our proposed framework integrates lightweight IBE with an efficient revocation mechanism, specifically designed for resource-constrained IoT devices handling sensitive cross-sectional healthcare data. Unlike existing IBE schemes that focus on general access control, SLIE addresses the unique requirements of healthcare hierarchies through role-specific temporal constraints and automatic delegation patterns that align with clinical workflows. Particularly, in Fig. 2: - Doctor Alice can access /HC/Data/EHR/full for entire departmental records - Nurse Charlie is restricted to specific patient records (patient_1/record) - Third-party providers only receive access to diagnostic outcomes (diagnosis) All data undergoes end-to-end encryption before cloud upload, eliminating plaintext exposure risks. The system implements temporally-graded access, with key validity periods scaled by role sensitivity to balance security and operational flexibility in time-sensitive cases (e.g., stroke, myocardial infarction). Fig. 2: Hierarchical structure of healthcare communication Fig. 3: IBE-based key generation (RSA for double protection is optional) Keys have a validity period τ that is automatically determined based on healthcare role requirements. This approach addresses a critical limitation in existing IBE schemes, where revocation typically requires maintaining centralized blacklists or complex re-encryption procedures. Our time-bounded approach eliminates these overheads through automatic expiration mechanisms. A key is considered valid or expired based on the current time t (1): Expire(SKAlice, t) = ( Valid if t ≤τ Invalid if t > τ (1) The expiration or validity period (τ) for data access should be set through a risk assessment considering data sensitivity, role changes, regulations (e.g., HIPAA, GDPR), and institutional policies, as current regulations lack specific SLIE: Secure IoT Healthcare Services Communication 5 timeframes for healthcare roles. Particularly, in the U.S., with an average hospital stay of 5.5 days [14], nurses providing direct care should have a τ of 6-7 days, doctors expecting follow-ups a τ of up to 21 days, and third-party contractors (e.g., IT, medical services), with less 24/7 monitoring responsibility, a τ of up to 2 days. Family members may receive 1-day portal access. Nevertheless, this highly depends on the aforementioned risk assessment factors. These durations reflect the varying levels of access privilege and security risk associated with different roles. Specifically, the key lifecycle consists of the following steps: 1. Initial Key Assignment: Alice receives a private key SKAlice with an expiry time τ0. 2. Periodic Key Refresh: When τ0 approaches, a new key SK′ Alice is issued with an extended expiry τ1 > τ0 (2): Refresh(SKAlice, τ0) = (SK′ Alice, τ1) (2) 3. Revocation by Non-Renewal: If no new key is issued after expiry, Alice's access is revoked by default. Access keys for staff are automatically renewed based on active employment and patient responsibilities, as verified by hospital systems. This automated revocation eliminates centralized infrastructure, ensures immediate access termination, and scales with organizational complexity. 3.2 Identity-Based Key Generation SLIE leverages WKD-IBE [15] for key management (Fig. 3), but extends the standard approach through healthcare-specific pattern designs over (Z∗ p ∪{⊥})l. Our contribution lies in the development of patterns that encode clinical hierarchies and role relationships, allowing fine-grained, non-commutative delegation and efficient key derivation across multiple concurrent healthcare departments encoded as disjoint slot partitions. An initial setup entity generates the parameters and a master secret key (MSK) (3). For Alice, with identity IDAlice, a private key is derived (4). Messages are encrypted (5) and decrypted using Alice's private key (6). In addition, keys are preserved on serve database. (Params, MSK) = Setup(1κ) (3) SKAlice = KeyDer(Params, MSK, Pattern(IDAlice)) (4) C = Encrypt(Params, Pattern(IDAlice), M) (5) M = Decrypt(Params, SKAlice, C) (6) Unlike traditional IBE, which uses generic identity strings, SLIE's WKDIBE framework integrates healthcare workflow patterns directly into the cryptographic structure. This encodes role hierarchy and temporal constraints, enabling automatic policy enforcement without external mechanisms. 6 H. X. Son et al. 4 Experiments and Results The SLIE evaluation was conducted on the LifeSnaps Fitbit Dataset [16], a longitudinal repository containing over 71 million multi-modal health records from 71 participants collected over four months. The dataset's capture of 35+ clinical biomarkers at varying temporal resolutions (seconds to daily) provides a realistic benchmark for validating the security of healthcare IoT systems. 4.1 Performance on Diverse Resource-constrained Devices Table 1 illustrates the compatibility and performance of SLIE across a diverse range of resource-constrained healthcare IoT devices. It demonstrates SLIE's adaptability to real-world healthcare applications by evaluating key operation, such as key generation, encryption, decryption, and delegation, which are averaged over 10 runs to ensure statistical reliability. Resource-rich devices like the MacBook Pro (2016) demonstrate exceptional efficiency, with encryption and decryption times of 0.056 ms and 0.0755 ms, respectively, and a notably low delegation time of 14 ms. In contrast, ultra-low-power devices such as the Apple Watch Series 6 achieve encryption and decryption in 0.1 ms and 0.03 ms, respectively, showcasing SLIE's optimization for constrained environments. The Raspberry Pi 4 strikes a balance, with encryption and decryption times of 0.3 ms and 0.4 ms, suitable for edge computing in IoMT networks. Table 1: Performance evaluation of SLIE on various devices Devices Generate key Encrypt Decrypt Delegate Apple Watch Series 6 122ms 0.1ms 0.03ms 82ms Fitbit Versa 2 182ms 42ms 53ms 83ms Raspberry Pi 4 122ms 0.3ms 0.4ms 82ms Macbook Pro 2016 MLH42 99ms 0.056ms 0.0755ms 14ms Table 2 details the power consumption and resource utilization of SLIE's core operations on representative IoT devices. Encryption and decryption operations exhibit remarkable energy efficiency, consuming 0.014 J and 0.009 J per 1 KB data respectively. These metrics highlight SLIE's suitability for batteryconstrained devices, enabling prolonged operation in wearable and implantable medical devices. The delegation operation, critical for secure access control in IoMT systems, consumes 0.052 J/KB. While delegation is more resource-intensive, its moderate power usage (1.45 W) and CPU utilization (35.6 %) ensure feasibility on low-power platforms. Memory usage remains modest, ranging from 12.5 MB (decryption) to 22.3 MB (delegation), further supporting SLIE's lightweight design. Collectively, these attributes demonstrate SLIE as an energy-efficient and scalable solution for secure data sharing for IoMT services, balancing performance, power, and security requirements. SLIE: Secure IoT Healthcare Services Communication 7 Table 2: Power consumption analysis of SLIE operations Operation Exec. time (ms) Power Usage (W) Mem. Usage (MB) CPU Util. (%) Energy Eff. (J/KB) Encrypt 12 1.20 16.80 27.50 0.014 Decrypt 9 0.95 12.50 19.20 0.009 Delegate 18 1.45 22.30 35.60 0.052 4.2 Scalability and Efficiency in IoT Data Encryption Table 3 quantifies SLIE's performance across data sizes from 1 kB to 10 MB, averaged over 10 runs. For smaller data sizes (1 KB), SLIE achieves encryption and decryption times of 0.936 ms and 0.217 ms, respectively, with minimal CPU usage (0.5 % and 0.7 %) and RAM consumption (1520 kB and 1300 kB). As data sizes increase, SLIE metrics increase linearly, demonstrating its suitability for resource-constrained IoMT devices, such as wearables and edge nodes. Table 3: Performance metrics of SLIE under different scenarios Data Size Key Creation Time (ms) Encryption Time (ms) Encryption CPU Usage (%) Encryption RAM Usage (KB) Decryption Time (ms) Decryption CPU Usage (%) Decryption RAM Usage (KB) 1KB 81 0.936 0.5 1520 0.217 0.7 1300 100KB 81 3.500 1.8 5200 1.184 1.9 4500 500KB 85 4.769 2.8 9650 1.846 3.0 8800 1MB 83 12.677 3.2 12000 3.318 4.2 11500 5MB 95 15.530 4.5 35600 8.993 5.5 30400 10MB 83 25.036 8.3 55120 15.119 9.2 50800 As benchmarked in Table 4, SLIE demonstrates a compelling balance of performance and security. It drastically outperforms RSA, achieving an 84.54% faster encryption and a 99.70% faster decryption. While its raw speed is lower than the symmetric cipher ChaCha20, SLIE provides essential, built-in security features-such as hierarchical access control-that ChaCha20 lacks. These suggest that SLIE a more optimal solution for IoMT services. Table 4: Average performance metrics comparison Algorithm Key Creation (ms) Enc (ms) Dec (ms) CPU Usage (%) RAM Usage (KB) SLIE 82.3 9.73 4.49 3.66 17,535 RSA 121.85 62.93 1476.57 5.86 31,388 ChaCha20 None 3.57 2.81 2.33 14,775 5 Conclusion In conclusion, the proposed SLIE cryptosystem establishes a robust foundation for secure IoMT services by combining the efficiency of a lightweight WKD-IBE framework with scalable trust management. It achieves a compelling performance profile, surpassing RSA in speed by over 84% for encryption and 99% for decryption while operating with high energy efficiency (0.014 J/KB). Crucially, SLIE overcomes the key management shortcomings of symmetric systems like ChaCha20 by natively supporting hierarchical access control and automatic key revocation, ensuring both security and regulatory compliance. Looking ahead, our research will evolve to incorporate post-quantum security, optimize for nextgeneration implantables, and demonstrate real-world interoperability with standards such as FHIR in clinical settings. 8 H. X. Son et al. 6 Code Availability The source code developed for this study and the scripts used to generate the results can be found at https://github.com/SonHaXuan/SLIE. Acknowledgement The authors acknowledge Ho Chi Minh City (HCMUT), VNU-HCM, for supporting this study. References 1. Athman Bouguettaya et al. An internet of things service roadmap. Communications of the ACM, 64(9):86-95, 2021. 2. The HIPAA Journal. Healthcare ransomware attacks involve 20 percents of stored sensitive data, 2024. 3. World Economic Forum. Healthcare pays the highest price of any sector for cyberattacks - that's why cyber resilience is key, 2024. 4. Lancer G Gates. Cyber attacks on interoperable electronic health records: A clear and present danger. Missouri medicine, 121 1:6-9, 2024. 5. Athman Bouguettaya et al. A service computing manifesto: the next 10 years. Communications of the ACM, 60(4):64-72, 2017. 6. M Shankar Lingam et al. Data encryption as security measure in iot-enabled healthcare. In Smart Trends in Computing and Communications: Proceedings of SmartCom 2020, pages 69-81. Springer, 2020. 7. Mohamed Nabeel. The many faces of end-to-end encryption and their security analysis. 2017 IEEE International Conference on Edge Computing (EDGE), pages 252-259, 2017. 8. J. Li et al. Attribute based encryption with privacy protection and accountability for cloudiot. IEEE Transactions on Cloud Computing, 10:762-773, 2022. 9. Kavuri KSVA Satheesh and T Krishna Sree. Ab-dam: attribute-based data access model in blockchain for healthcare applications. Multimedia Tools and Applications, 81(17):23567-23588, 2022. 10. Leyou Zhang et al. Hidden ciphertext policy attribute-based encryption with fast decryption for personal health record system. IEEE Access, 7:33202-33213, 2019. 11. Isaac Amankona Obiri et al. A fully secure kp-abe scheme on prime-order bilinear groups through selective techniques. Security and Communication Networks, 2020(1):8869057, 2020. 12. Yinxia Sun et al. Efficient identity-based encryption with revocation for data privacy in internet of things. IEEE IoT-J, 9(4):2734-2743, 2021. 13. Minghui Li et al. Nacda: naming-based access control and decentralized authorization for secure many-to-many data sharing. Electronics, 12(7):1651, 2023. 14. Kelley Tipton et al. Interventions to decrease hospital length of stay. 2021. 15. Sam Kumar et al. {JEDI}:{Many-to-Many}{End-to-End} encryption and key delegation for {IoT}. In 28th USENIX security symposium (USENIX Security 19), pages 1519-1536, 2019. 16. Sofia Yfantidou et al. Lifesnaps, a 4-month multi-modal dataset capturing unobtrusive snapshots of our lives in the wild. Scientific Data, 9, 2022.
|
2510.14709
|
Proceedings of TerraBytes: Towards global datasets and models for Earth Observation Workshop at ICML 1–12
Where are the Whales: A Human-in-the-loop Detection
Method for Identifying Whales in High-resolution Satellite
Imagery
Caleb Robinson
caleb.robinson@microsoft.com
Microsoft AI for Good Research Lab, Redmond, Washington, USA
Kimberly T. Goetz
Marine Mammal Laboratory, Alaska Fisheries Science Center, National Marine Fisheries Service,
NOAA, Seattle, Washington, USA
Christin B. Khan
Northeast Fisheries Science Center, National Marine Fisheries Service, NOAA, Woods Hole, Mas-
sachusetts, USA
Meredith Sackett
Azura Consulting, under contract to NOAA Fisheries, Northeast Fisheries Science Center, Woods
Hole, MA USA
Kathleen Leonard
Protected Resources Division, Alaska Regional Office, National Marine Fisheries Service, NOAA,
Anchorage, AK
Rahul Dodhia and Juan M. Lavista Ferres
Microsoft AI for Good Research Lab, Redmond, Washington, USA
Abstract
Effective monitoring of whale popula-
tions is critical for conservation, but tra-
ditional survey methods are expensive
and difficult to scale. While prior work
has shown that whales can be identified
in very high-resolution (VHR) satellite
imagery, large-scale automated detec-
tion remains challenging due to a lack
of annotated imagery, variability in im-
age quality and environmental condi-
tions, and the cost of building robust
machine learning pipelines over massive
remote sensing archives. We present a
semi-automated approach for surfacing
possible whale detections in VHR im-
agery using a statistical anomaly de-
tection method that flags spatial out-
liers, i.e. “interesting points”. We pair
this detector with a web-based labeling
interface designed to enable experts to
quickly annotate the interesting points.
We evaluate our system on three bench-
mark scenes with known whale anno-
tations and achieve recalls of 90.3% to
96.4%, while reducing the area requir-
ing expert inspection by up to 99.8%
— from over 1,000 sq km to less than
2 sq km in some cases.
Our method
does not rely on labeled training data
and offers a scalable first step toward fu-
ture machine-assisted marine mammal
monitoring from space. We have open
sourced the entire pipeline at https:
//github.com/microsoft/whales.
Keywords:
whales, anomaly detec-
tion, labeling interface, VHR imagery
1. Introduction
Understanding the spatial distribution of
whales is essential for guiding conservation
action, informing marine spatial planning,
© C. Robinson, K.T. Goetz, C.B. Khan, M. Sackett, K. Leonard, R. Dodhia & J.M.L. Ferres.
arXiv:2510.14709v1 [cs.CV] 16 Oct 2025
Robinson Goetz Khan Sackett Leonard Dodhia Ferres
and upholding mandates such as the Ma-
rine Mammal Protection Act and the En-
dangered Species Act. The critically endan-
gered North Atlantic right whale (Eubalaena
glacialis) has been experiencing a decline for
over 10 years — with an estimated 372 indi-
viduals remaining (95% probability interval
360 to 383) as of 2024 [9].
In the Pacific,
the Cook Inlet beluga is also critically en-
dangered, with an estimated population size
of 381 individuals (95% probability interval
317 to 473) as of 2022 [7].
Despite this,
our ability to monitor these animals across
broad spatial and temporal scales remains
constrained by the cost and logistical com-
plexity of traditional survey methods such as
aerial, vessel-based, and acoustic monitoring.
Recent studies have shown that whales can
be visually identified in very high-resolution
(VHR) satellite imagery (e.g., 0.3 m/px or
better) [5; 4; 10].
These findings have led
to exploratory work applying automated de-
tection approaches, including deep learning
models trained on satellite and aerial im-
agery datasets [1; 8; 2].
Such models typ-
ically require large labeled datasets and do
not generalize well to new environments, sen-
sors, or ocean conditions. Recent efforts have
consolidated all previous annotated satellite
imagery into a single data archive [3], how-
ever whale identification in novel satellite im-
agery is still largely driven by exhaustive ex-
pert annotation in desktop GIS software [11].
Manually annotating 100 km2 of 31 cm/px
imagery takes 3 hours and 20 minutes ac-
cording to previous estimates [4], limiting the
feasibility of scaling these efforts as satellite
imagery archives grow.
In this work, we propose a simple alterna-
tive: using statistical anomaly detection to
automatically identify outliers in VHR satel-
lite scenes, and presenting these to experts
through a browser-based labeling interface.
Our method detects anomalous regions based
on localized standardization of spectral in-
tensities and aggregates these into a set of
candidate detections – “interesting points”.
These points can then be reviewed by hu-
man annotators, enabling more scalable an-
notation workflows and bootstrapping larger
datasets for training deep learning models.
We evaluate our approach on three previ-
ously annotated Maxar VHR satellite scenes:
from Cape Cod Bay in 2021 [10], Pen´ınsula
Vald´es in 2012 [3], and Pen´ınsula Vald´es in
2014 [3], which collectively contain 174 man-
ually annotated whales.
Over imagery of
calm water, our method achieves high recall
while substantially reducing the search area
that require manual inspection by an order of
magnitude. While not a complete solution to
whale detection, our system provides a useful
first step in bridging the gap between manual
annotation and future automated pipelines.
2. Problem Formulation and
Motivation
We aim to develop a semi-automated frame-
work for detecting whales in VHR optical
satellite imagery of open water.
Let X
denote a large satellite image scene cov-
ering hundreds of square kilometers with
sub-meter spatial resolution (e.g., 0.3 m/px).
The objective is to identify all observable ma-
rine mammals within X while minimizing the
amount of expert labeling time required.
We assume that no prior dataset of an-
notated marine mammal instances is avail-
able for the specific region, ocean, or imag-
ing conditions covered by X. We do not as-
sume access to a pre-trained vision model for
detecting marine mammals in satellite im-
agery. Importantly, we assume that the im-
agery may be captured under non-ideal en-
vironmental conditions.
The ocean surface
may exhibit high texture variation due to
wind-induced whitecaps, and the scene may
contain confounding artifacts such as ves-
sels, buoys, aquaculture, floating debris, or
2
Where are the Whales
Figure 1: Web-based labeling application for quickly annotating “interesting points”. For
each candidate region of imagery flagged as statistically anomalous, labelers can inspect the
corresponding imagery (the scale of each pixel is overlaid on the image), assign confidence
levels (Possible, Probable, Definite), and classify object type (e.g., whale, ship, debris, zoo-
plankton). Metadata such as location, date, and chips ID are displayed alongside adjustable
image rendering controls (zoom, brightness, contrast). Once a class button is pressed the
interface automatically loads a new image/interesting point to minimize the time per an-
notation.
even dense zooplankton swarms. Addition-
ally, partial occlusion from clouds or haze
may further complicate interpretation. Any
practical method for solving this problem
must therefore be robust to these sources
of noise and variability, operating effectively
under “real-world” constraints.
We frame the task of finding marine mam-
mals in the ocean as one of anomaly detec-
tion. Our hypothesis is that marine mam-
mals, while rare, will appear as local statisti-
cal outliers in an otherwise spatially homoge-
neous imagery1. Using this assumption, we
define a function fanomaly : X →P that maps
1. And further, imagery that doesn’t fit this criteria
– e.g. images captured with a high frequency of
white-capped waves – will be identifiable by the
relatively high of outliers and can be ignored.
the input image X to a set of spatial locations
P ⊂X corresponding to candidate points of
interest. These points are selected based on
deviations from local statistical norms.
The anomaly detection function fanomaly
may be instantiated via classical statistical
descriptors (e.g., local variance or z-scores
within fixed-radius windows) or learned fea-
ture representations derived from deep mod-
els. The resulting candidate set P serves as a
focus of attention for downstream expert re-
view or targeted labeling, enabling a path for
human-in-the-loop verification and training
of object detectors in data-sparse settings.
3
Robinson Goetz Khan Sackett Leonard Dodhia Ferres
Figure 2:
Comparison of numerical error
in variance estimation under 32-bit preci-
sion for the naive and mean-shifted formu-
lations of computing standard deviation in
the rolling window approach. The x-axis
shows the ratio of variance to mean inten-
sity for synthetic image chips.
The y-axis
shows the absolute error in standard devi-
ation relative to a double-precision ground
truth.
The naive formulation suffers from
catastrophic cancellation when variance is
small compared to the mean (which hap-
pens when using the rolling window approach
over homogeneous ocean regions), while the
shifted method achieves consistently lower
error across all regimes.
3. Methods
3.1. Detection of Statistically
Interesting Pixels
Let X ∈RC×H×W denote a pansharpened
and orthorectified VHR multispectral satel-
lite image2, where C is the number of spec-
tral channels and H × W are the spatial di-
mensions. Our objective is to identify pixels
in X that exhibit statistically anomalous be-
havior relative to their local spatial context.
2. Assuming trivial access to such imagery hides a
huge amount of complexity, see [6] for an in-depth
discussion of the topic. For the purposes of the
discussion of anomaly detection methods we ab-
stract this out, however ‘real-world’ deployments
of such methods require significant engineering.
Our proposed method is simple – we com-
pute a per-pixel deviation score for each
location in X, identifying extreme outliers
pixels across the scene.
The following sec-
tions describe two standardization methods
for computing these deviation scores. High-
deviation pixels are grouped into spatially
contiguous regions and filtered based on geo-
metric (i.e. size or outlier region) and statis-
tical criteria. The resulting centroids of these
regions, i.e. the “interesting points”, form a
candidate set P ⊂X suitable for expert re-
view or active labeling.
To focus the analysis and mitigate false
positives,
optional
vector
overlays
(e.g.,
coastlines or land masks) can be applied
to constrain the spatial domain.
This
is especially important in open-water ap-
plications,
where land features can con-
tain high-contrast structures that falsely
trigger the anomaly detection mechanism.
By masking out land and limiting analy-
sis to known ocean extents, we increase
the specificity of detection.
Specifically,
we use the Global Self-consistent, Hierar-
chical, High-resolution Geography Database
(GSHHG) [12] for land/water masking.
3.2. Standardization Methods
We
compute
a
deviation
tensor
D
∈
RC×H×W where each element Dcij captures
the standardized anomaly of pixel (i, j) in
channel c.
We implement two methods
for deriving D: chunked standardization,
which uses coarse statistics, but is efficient
to compute and rolling window standard-
ization which uses local statistics, but is
more computationally demanding.
3.2.1. Chunked Standardization
The first method partitions X into non-
overlapping square chunks of size s × s (typ-
ically s = 1024 pixels). For each chunk, we
compute the mean, µc, and standard devia-
4
Where are the Whales
tion, σc, for each channel. The deviation at
each pixel is then computed as:
Dcij = Xcij −µc
σc
This method is simple to parallelize and effi-
cient (with time complexity O(CHW)), but
may overlook fine-grained deviations when
anomalies are diluted across large homoge-
neous windows.
3.2.2. Rolling Window
Standardization
The second method employs a sliding win-
dow approach that computes localized statis-
tics centered at each pixel. For each location
(i, j), the local mean, µcij, and variance, σ2
cij,
are estimated using a square window of size
k × k (e.g. k = 31 pixels):
Dcij = Xcij −µcij
q
σ2
cij + ε
We implement this approach via channel-
wise convolution operations using a uniform
averaging kernel, see Listing 1 for a PyTorch
implementation. This allows us to run the
computation in parallel utilizing GPUs for
processing.
Numerical stability considerations for
the
rolling
window
standardization.
We estimate local variance using the identity
σ2
cij = E[X2] −(E[X])2, as both terms can
be implemented as convolutions.
However,
in regions where the variance is small (e.g.,
calm water), this formulation suffers from
catastrophic cancellation due to the limited
precision of 32-bit floating point arithmetic.
To address this, we subtract a global per-
channel mean, ¯Xc, from X before convolu-
tion:
X′
cij = Xcij −¯Xc
This is a standard method for stabilizing the
computation of σ2 by reducing the absolute
magnitude of the squared and squared-mean
terms.
Figure 2 shows the effect that this
shifted method has on the error of the com-
putation as a function of the similarity of the
variance and mean. Finally, we add a small
constant, ε = 10−8, to the denominator to
ensure positive values and avoid undefined
behavior under the square root.
3.3. Anomaly Aggregation and
Feature Extraction
We aggregate the channel-wise deviations, D,
across a large input scene over the channel
dimension in a scalar anomaly map, A ∈
RH×W , as follows:
Aij =
C
X
c=1
|Dcij|
We then threshold A either at a fixed value
or using a high quantile (e.g., the 99.99th
percentile) to produce a binary mask of
anomalous pixels. We then extract spatially
contiguous connected components (using 8-
connected neighborhood logic) and compute
their average anomaly intensity.
We filter out regions whose area falls be-
low a minimum threshold (e.g., less than 1.5
square meters).
We then represent surviv-
ing features as points using their geometric
centroids and record their area and average
aggregate deviation value.
4. Experiments and Results
Our proposed methods have several free pa-
rameters that influence the quality and num-
ber of “interesting points” that a labeler
must examine: the size of the window over
which statistics are computed — s in the
chunked standardization approach and k
in the rolling window standardization
approach — the anomaly threshold used for
binarizing A (determining which pixels are
5
Robinson Goetz Khan Sackett Leonard Dodhia Ferres
Scene
Annotated Whales
True Positives
False Positives
Recall
Cape Cod Bay 2021
31
28
280
90.3%
Pen´ınsula Vald´es 2012
84
81
276
96.4%
Pen´ınsula Vald´es 2014
59
54
3622
91.5%
Table 1: Detection performance across three benchmark scenes.
anomalous), and the area threshold (deter-
mining how large an anomalous region must
be to be considered “interesting”). A desir-
able configuration is one in which all of the
whales in a scene are marked as “interesting”
(i.e. high recall) and few non-whale points
are marked as “interesting” (i.e. high preci-
sion).
We evaluate the effectiveness of our meth-
ods by applying them to three satellite scenes
previously studied in the literature, each con-
taining manually annotated whale locations:
Cape Cod Bay 2021 [10] Two
WorldView-3
scenes
(Catalog
IDs
1040010067D36B00
and
10400100674B2100)
captured
on
April 24, 2021 covering ∼200 km2 of
Cape Cod Bay3 at a 0.3 m/px spatial
resolution.
It contains 31 annotated
North Atlantic right whales, 25 of which
are considered definite, and 6 of which
are ambiguous.
Pen´ınsula Vald´es 2012 [3] A
WorldView-2
scene
(Catalog
ID
103001001C8C0300)
captured
on
September 19, 2012, covering 120 km2
of the Vald´es Peninsula in Argentina
with a spatial resolution of 0.56 m/px.
It includes 84 Southern right whales
(Eubalaena australis) labels categorized
by confidence: 15 definite, 32 probable,
and 37 possible sightings.
3. We approximate the study area used in [10] by
buffering the extent of the annotated points by
500m.
Pen´ınsula Vald´es 2014 [3] A
WorldView-3
scene
(Catalog
ID
10400100032A3700) captured on Octo-
ber 16, 2014 covering 560 km2, also from
the Vald´es Peninsula, with a spatial
resolution of 0.37 m/px. It includes 59
Southern right whale labels: 23 definite,
12 probable, and 24 possible.
Window size
We run the chunked and
rolling
window
standardization
methods
with different window sizes for the Cape Cod
Bay 2021 scene and show the distribution of
aggregated anomaly scores in the top panel
of Figure 3.
We find similar distributions
of anomaly values among window sizes of
256, 512, 1024, and 2048 for the chunked
standardization method and kernel sizes of
41 and 51 for the rolling window method.
Kernel sizes of less than 41 see a drop-off
in tail values as there is less context with
which to determine whether a given value is
an anomaly or not. Practically, on an V100
GPU, there is little difference in execution
time of the rolling window standardization
computation with larger window sizes.
Threshold values
We run the rolling win-
dow standardization method with k = 51
on each of the three evaluation scenes and
plot the distribution of anomaly values in
the bottom panel of Figure 3.
We find
that the distributions from the Cape Cod
Bay 2021 and Vald´es 2012 scenes — both of
which have very still water — have shorter
tails, while the Vald´es 2014 scene — with a
large number of white-capped waves — has
a long tailed distribution with 99.99th per-
6
Where are the Whales
Figure 3: (Top) The distribution of anomaly
values (summed absolute deviations) on the
Cape Cod Bay scene [10] using the two stan-
dardization methods with varying kernel (k)
or window (s) sizes.
We find that both
methods result in similar distributions with
a window or kernel size greater than 50.
(Bottom) Comparison of anomaly value dis-
tributions (using rolling window standardiza-
tion with a k = 51 kernel) across the three
evaluation scenes.
The Vald´es 2014 scene
has a high density of high-contrast features
(mostly white-capped waves) which shows up
as a longer tail in the anomaly distribution.
centile values of 11.37, 11.90 and 55.25, re-
spectively (computed over the RGB channels
only).
We always use a conservative value
of 1.5 square for area thresholding based
on conversations with experts and observing
that positive identifications may only high-
light parts of a whale. In general, we find
that it is possible to run the interesting point
methods with a range of anomaly thresholds
and choose the result that returns a reason-
able number of points per square kilometer
after area filtering (using < 2 points/sq km
as a rough cutoff).
Results
We evaluate our methods using
the 99.99th percentile anomaly threshold for
Cape Cod Bay 2021 and Vald´es 2012 and
the 99.9th percentile anomaly threshold for
Vald´es 2014 (trading off a larger number of
false positives for higher true positives). We
define an “interesting point” as a true posi-
tive if it falls within a 100 meter radius of an
annotated whale location and a false positive
otherwise.
Results are summarized in Ta-
ble 1. We find that most waves are identified
as anomalous – ‘false positives’.
In scenes
with still water, we find recall values greater
than 90% with relatively few false positives.
5. Labeling interface
To facilitate rapid annotation of “interesting
point” detections in large amounts of satel-
lite imagery, we developed a browser-based
labeling tool designed for expert reviewers –
shown in Figure 1.
Each session presents the user with a se-
quence of 100m×100m image chips centered
on geographic centroids of the interesting re-
gions outputted from the anomaly detection
pipeline.
These chips are accompanied by
contextual metadata, including the imagery
acquisition date and geographic coordinates,
which are also visualized on an interactive
map.
Users can select one of sixteen pre-
defined semantic classes (e.g., whale, ship,
debris, oil), and for whale detections also
assign species, confidence (possible, probable,
or definite), and add free-form comments.
After a class is selected, the interface will
immediately load the next chip, facilitating
rapid review.
7
Robinson Goetz Khan Sackett Leonard Dodhia Ferres
Figure 4: Example whale detections from the May 22, 2020 scene (left) and two detections
from the larger Cape Cod Bay 2021 scene that were not considered in the Hodul et al. 2022
study (middle, right). The May 2020 scene (1,056 sq km) produced 220 interesting points,
each of which were labeled by 3 experts in a total of 31 minutes. The Cape Cod Bay scene
(1,083 sq km) produced 555 interesting points, each was also labeled by 3 experts in a total
of 139 minutes.
The tool provides controls for adjusting
image zoom, brightness, and contrast in real
time.
These settings allow users to opti-
mize visual clarity when interpreting difficult
scenes, such as those with haze, glint, or low
contrast.
The labeling system itself is also deliber-
ately simple and is implemented as a sin-
gle multi-threaded Python HTTP server ap-
plication that operates in a stateless mode
with a single page web frontend. No login
credentials are required, and labeler identity
is tracked solely through self-supplied ids to
avoid duplicate sampling. Each chips is as-
signed to multiple labelers and is automat-
ically removed from circulation once it has
been annotated by three distinct labelers.
This redundancy provides a simple mecha-
nism for quality control through label con-
sensus, while allowing for validation in down-
stream workflows. All labels and associated
metadata are saved in a CSV file by the
server application.
The anomaly detection pipeline, labeling
interface and associated backend are open-
sourced with demo data and setup instruc-
tions at https://github.com/microsoft/
whales.
6. Case Study and Discussion
To assess the practical deployment of our
system
in
new,
unlabeled
imagery,
we
applied the anomaly detector and label-
ing
interface
to
a
previously
unstudied
scene captured on May 22,
2020,
over
Cape Cod Bay (WorldView-3 catalog ID
10400100585EFA00). The scene covers 1,056
sq km at a 0.3 m/px resolution and generally
has calm water throughout the scene.
We used the rolling window standardiza-
tion approach with deviation threshold of
the 99.99th percentile and an area thresh-
old of 1.5 sq meters which found 220 “in-
teresting points”, corresponding to approxi-
mately 0.21% of the total pixels4. Each of
these were subsequently annotated by three
expert labelers using our the user interface
which took 31 minutes of total human effort
4. Calculated by using a 50m buffer around each
interesting point which corresponds to the area
shown to the labeler in the user interface.
8
Where are the Whales
class LocalContextStandardization(Module):
def __init__(self, in_channels: int = 3, kernel_size: int = 9,
shift_val=None):
,→
super().__init__()
self.shift_val = shift_val
weights = torch.nn.Parameter(
torch.zeros(
in_channels, in_channels, kernel_size, kernel_size,
dtype=torch.float32
,→
),
requires_grad=False,
)
for i in range(in_channels):
weights[i, i] = (
torch.ones(kernel_size, kernel_size, dtype=torch.float32)
/ kernel_size**2.0
)
self.conv = Conv2d(
in_channels,
in_channels,
kernel_size=kernel_size,
padding="same",
padding_mode="replicate",
bias=False,
)
self.conv.weight = weights
def forward(self, x: Tensor) -> Tensor:
if self.shift_val is not None:
x = x - self.shift_val
else:
x = x - x.mean(dim=(0, 2, 3), keepdim=True)
mu = self.conv(x)
squares = self.conv(x**2.0)
variance = squares - mu**2.0
return (x - mu) / (torch.sqrt(variance) + 1e-8)
Listing 1: Implementation of the rolling window standardization approach, which ap-
plies a channel-wise local standardization to imagery using convolution-based estimates of
mean and variance.
(as measured by the total time between the
interface serving an image chip to be labeled
and the subsequent response with class an-
notation over the three labelers). Using the
approximation from [4] — that manually an-
notating 100 sq km of imagery takes approxi-
mately 3 hours and 20 minutes — annotating
9
Robinson Goetz Khan Sackett Leonard Dodhia Ferres
Figure 5: A map of 555 “interesting points”
in red found over the two WorldView-3
scenes from used in the [10] study. The whale
points from the Hodul et al. study are shown
in green and the study area (buffered from
the extent of the labeled points) is shown
in yellow. Each of the “interesting points”
was annotated by three expert labelers which
took a total of 139 minutes. Manual anno-
tation of the entire scene would take approx-
imately ∼35 hours. The interesting points
covered 28 out of the 31 whales found in the
Hodul et al. study (which were all correctly
identified by the labelers). Further, 5 addi-
tional whale points were found outside of the
previously examined study area.
this scene would have taken a single labeler
∼35 hours.
Most
detections
were
attributed
to
whitecap (n=420) and unsure (n=234),
reflecting the presence of breaking waves
near the shore. Only two points were anno-
tated as a whale, of which, all three labelers
agreed on one point as a whale (see Figure
4). We observe that the whale annotations
took
substantially
longer
to
process
—
ranging from 15 to 106 seconds each — due
to increased scrutiny and zooming behavior
by the annotators.
In contrast, non-whale
categories required only 2.8 seconds per
chips on average (with a standard deviation
of 8.5 seconds).
Further, we applied the same procedure to
the two scenes used in the Hodul et al. 2022
study [10] (see the Cape Cod Bay 2021 de-
scription). This study focused on a ∼200 sq
km area taken from two larger WorldView-3
strips, however the entire area from the two
strips covers 1,083 sq km. Figure 5 shows this
larger extent, with the approximate study
area shown as a yellow box. We find a total
of 555 interesting points in the larger scene,
which took expert labelers 139 minutes in to-
tal to annotate. The average time per non-
whale annotation was 5.1 seconds while the
average time per whale annotation was 11.23
seconds. The 555 interesting points covered
28 out of the 31 interesting points found
in the Hodul et al.
study, and the anno-
tators correctly annotated each as a whale.
Additionally, the annotators flagged 5 other
points outside of the original study area as
whales, two of which are shown in Figure 4.
In both cases, the cost of pre-processing the
scenes and applying our methods is negligi-
ble compared to the cost of having marine
biologists annotate them.
Finally, we observe – and emphasize as
a limitation – that applying our methods
to scenes that contain white-capped waves
from high winds results in a large num-
10
Where are the Whales
ber of false positives.
The simple meth-
ods we propose here are not able to dis-
tinguish between different classes of anoma-
lous groupings of pixels (i.e. between white-
caps, whales, buoys, or otherwise), and fur-
ther work and labeled datasets are needed to
find whales in these challenging conditions.
Nevertheless, satellite imagery archives con-
tain unstudied imagery of calm seas that can
be mined for whale detections and used to
bootstrap larger modeling efforts.
Impact Statement
This work contributes tools and methodol-
ogy for identifying whales in satellite imagery
to support marine mammal monitoring and
conservation at scale. By reducing the need
for exhaustive manual annotation, our semi-
automated pipeline enables domain experts
to focus their attention on anomalous “in-
teresting points”, accelerating detection of
whales in vast ocean areas.
We emphasize that these methods are not
fully automated and require expert interpre-
tation to ensure reliable use. The approach
is sensitive to imaging conditions, scene com-
plexity, and spectral variability, and may
produce false positives in challenging envi-
ronments.
As such, it is intended as a
decision-support system to augment — not
replace — expert-driven analysis.
All system components are open-sourced
to promote transparency,
reproducibility,
and community adoption. This work marks a
key step toward scalable, expert-in-the-loop
remote sensing for conservation. By enabling
rapid review of vast ocean imagery for spec-
trally distinct features, it allows experts to
label whales and other surface objects, creat-
ing high-quality training data. This data can
power future model development, accelerat-
ing geospatial insights into whale presence
and enhancing both conservation efforts and
maritime domain awareness.
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Proceedings of TerraBytes: Towards global datasets and models for Earth Observation Workshop at ICML 1-12 Where are the Whales: A Human-in-the-loop Detection Method for Identifying Whales in High-resolution Satellite Imagery Caleb Robinson Microsoft AI for Good Research Lab, Redmond, Washington, USA Kimberly T. Goetz Marine Mammal Laboratory, Alaska Fisheries Science Center, National Marine Fisheries Service, NOAA, Seattle, Washington, USA Christin B. Khan Northeast Fisheries Science Center, National Marine Fisheries Service, NOAA, Woods Hole, Massachusetts, USA Meredith Sackett Azura Consulting, under contract to NOAA Fisheries, Northeast Fisheries Science Center, Woods Hole, MA USA Kathleen Leonard Protected Resources Division, Alaska Regional Office, National Marine Fisheries Service, NOAA, Anchorage, AK Rahul Dodhia and Juan M. Lavista Ferres Microsoft AI for Good Research Lab, Redmond, Washington, USA Abstract Effective monitoring of whale populations is critical for conservation, but traditional survey methods are expensive and difficult to scale. While prior work has shown that whales can be identified in very high-resolution (VHR) satellite imagery, large-scale automated detection remains challenging due to a lack of annotated imagery, variability in image quality and environmental conditions, and the cost of building robust machine learning pipelines over massive remote sensing archives. We present a semi-automated approach for surfacing possible whale detections in VHR imagery using a statistical anomaly detection method that flags spatial outliers, i.e. "interesting points". We pair this detector with a web-based labeling interface designed to enable experts to quickly annotate the interesting points. We evaluate our system on three benchmark scenes with known whale annotations and achieve recalls of 90.3% to 96.4%, while reducing the area requiring expert inspection by up to 99.8% - from over 1,000 sq km to less than 2 sq km in some cases. Our method does not rely on labeled training data and offers a scalable first step toward future machine-assisted marine mammal monitoring from space. We have open sourced the entire pipeline at https: //github.com/microsoft/whales. Keywords: whales, anomaly detection, labeling interface, VHR imagery 1. Introduction Understanding the spatial distribution of whales is essential for guiding conservation action, informing marine spatial planning, © C. Robinson, K.T. Goetz, C.B. Khan, M. Sackett, K. Leonard, R. Dodhia & J.M.L. Ferres. 16 Oct 2025 Robinson Goetz Khan Sackett Leonard Dodhia Ferres and upholding mandates such as the Marine Mammal Protection Act and the Endangered Species Act. The critically endangered North Atlantic right whale (Eubalaena glacialis) has been experiencing a decline for over 10 years - with an estimated 372 individuals remaining (95% probability interval 360 to 383) as of 2024 [9]. In the Pacific, the Cook Inlet beluga is also critically endangered, with an estimated population size of 381 individuals (95% probability interval 317 to 473) as of 2022 [7]. Despite this, our ability to monitor these animals across broad spatial and temporal scales remains constrained by the cost and logistical complexity of traditional survey methods such as aerial, vessel-based, and acoustic monitoring. Recent studies have shown that whales can be visually identified in very high-resolution (VHR) satellite imagery (e.g., 0.3 m/px or better) [5; 4; 10]. These findings have led to exploratory work applying automated detection approaches, including deep learning models trained on satellite and aerial imagery datasets [1; 8; 2]. Such models typically require large labeled datasets and do not generalize well to new environments, sensors, or ocean conditions. Recent efforts have consolidated all previous annotated satellite imagery into a single data archive [3], however whale identification in novel satellite imagery is still largely driven by exhaustive expert annotation in desktop GIS software [11]. Manually annotating 100 km2 of 31 cm/px imagery takes 3 hours and 20 minutes according to previous estimates [4], limiting the feasibility of scaling these efforts as satellite imagery archives grow. In this work, we propose a simple alternative: using statistical anomaly detection to automatically identify outliers in VHR satellite scenes, and presenting these to experts through a browser-based labeling interface. Our method detects anomalous regions based on localized standardization of spectral intensities and aggregates these into a set of candidate detections - "interesting points". These points can then be reviewed by human annotators, enabling more scalable annotation workflows and bootstrapping larger datasets for training deep learning models. We evaluate our approach on three previously annotated Maxar VHR satellite scenes: from Cape Cod Bay in 2021 [10], Pen ́ınsula Vald ́es in 2012 [3], and Pen ́ınsula Vald ́es in 2014 [3], which collectively contain 174 manually annotated whales. Over imagery of calm water, our method achieves high recall while substantially reducing the search area that require manual inspection by an order of magnitude. While not a complete solution to whale detection, our system provides a useful first step in bridging the gap between manual annotation and future automated pipelines. 2. Problem Formulation and Motivation We aim to develop a semi-automated framework for detecting whales in VHR optical satellite imagery of open water. Let X denote a large satellite image scene covering hundreds of square kilometers with sub-meter spatial resolution (e.g., 0.3 m/px). The objective is to identify all observable marine mammals within X while minimizing the amount of expert labeling time required. We assume that no prior dataset of annotated marine mammal instances is available for the specific region, ocean, or imaging conditions covered by X. We do not assume access to a pre-trained vision model for detecting marine mammals in satellite imagery. Importantly, we assume that the imagery may be captured under non-ideal environmental conditions. The ocean surface may exhibit high texture variation due to wind-induced whitecaps, and the scene may contain confounding artifacts such as vessels, buoys, aquaculture, floating debris, or 2 Where are the Whales Figure 1: Web-based labeling application for quickly annotating "interesting points". For each candidate region of imagery flagged as statistically anomalous, labelers can inspect the corresponding imagery (the scale of each pixel is overlaid on the image), assign confidence levels (Possible, Probable, Definite), and classify object type (e.g., whale, ship, debris, zooplankton). Metadata such as location, date, and chips ID are displayed alongside adjustable image rendering controls (zoom, brightness, contrast). Once a class button is pressed the interface automatically loads a new image/interesting point to minimize the time per annotation. even dense zooplankton swarms. Additionally, partial occlusion from clouds or haze may further complicate interpretation. Any practical method for solving this problem must therefore be robust to these sources of noise and variability, operating effectively under "real-world" constraints. We frame the task of finding marine mammals in the ocean as one of anomaly detection. Our hypothesis is that marine mammals, while rare, will appear as local statistical outliers in an otherwise spatially homogeneous imagery1. Using this assumption, we define a function fanomaly : X →P that maps 1. And further, imagery that doesn't fit this criteria - e.g. images captured with a high frequency of white-capped waves - will be identifiable by the relatively high of outliers and can be ignored. the input image X to a set of spatial locations P ⊂X corresponding to candidate points of interest. These points are selected based on deviations from local statistical norms. The anomaly detection function fanomaly may be instantiated via classical statistical descriptors (e.g., local variance or z-scores within fixed-radius windows) or learned feature representations derived from deep models. The resulting candidate set P serves as a focus of attention for downstream expert review or targeted labeling, enabling a path for human-in-the-loop verification and training of object detectors in data-sparse settings. 3 Robinson Goetz Khan Sackett Leonard Dodhia Ferres Figure 2: Comparison of numerical error in variance estimation under 32-bit precision for the naive and mean-shifted formulations of computing standard deviation in the rolling window approach. The x-axis shows the ratio of variance to mean intensity for synthetic image chips. The y-axis shows the absolute error in standard deviation relative to a double-precision ground truth. The naive formulation suffers from catastrophic cancellation when variance is small compared to the mean (which happens when using the rolling window approach over homogeneous ocean regions), while the shifted method achieves consistently lower error across all regimes. 3. Methods 3.1. Detection of Statistically Interesting Pixels Let X ∈RC×H×W denote a pansharpened and orthorectified VHR multispectral satellite image2, where C is the number of spectral channels and H × W are the spatial dimensions. Our objective is to identify pixels in X that exhibit statistically anomalous behavior relative to their local spatial context. 2. Assuming trivial access to such imagery hides a huge amount of complexity, see [6] for an in-depth discussion of the topic. For the purposes of the discussion of anomaly detection methods we abstract this out, however 'real-world' deployments of such methods require significant engineering. Our proposed method is simple - we compute a per-pixel deviation score for each location in X, identifying extreme outliers pixels across the scene. The following sections describe two standardization methods for computing these deviation scores. Highdeviation pixels are grouped into spatially contiguous regions and filtered based on geometric (i.e. size or outlier region) and statistical criteria. The resulting centroids of these regions, i.e. the "interesting points", form a candidate set P ⊂X suitable for expert review or active labeling. To focus the analysis and mitigate false positives, optional vector overlays (e.g., coastlines or land masks) can be applied to constrain the spatial domain. This is especially important in open-water applications, where land features can contain high-contrast structures that falsely trigger the anomaly detection mechanism. By masking out land and limiting analysis to known ocean extents, we increase the specificity of detection. Specifically, we use the Global Self-consistent, Hierarchical, High-resolution Geography Database (GSHHG) [12] for land/water masking. 3.2. Standardization Methods We compute a deviation tensor D ∈ RC×H×W where each element Dcij captures the standardized anomaly of pixel (i, j) in channel c. We implement two methods for deriving D: chunked standardization, which uses coarse statistics, but is efficient to compute and rolling window standardization which uses local statistics, but is more computationally demanding. 3.2.1. Chunked Standardization The first method partitions X into nonoverlapping square chunks of size s × s (typically s = 1024 pixels). For each chunk, we compute the mean, μc, and standard devia4 Where are the Whales tion, σc, for each channel. The deviation at each pixel is then computed as: Dcij = Xcij -μc σc This method is simple to parallelize and efficient (with time complexity O(CHW)), but may overlook fine-grained deviations when anomalies are diluted across large homogeneous windows. 3.2.2. Rolling Window Standardization The second method employs a sliding window approach that computes localized statistics centered at each pixel. For each location (i, j), the local mean, μcij, and variance, σ2 cij, are estimated using a square window of size k × k (e.g. k = 31 pixels): Dcij = Xcij -μcij q σ2 cij + ε We implement this approach via channelwise convolution operations using a uniform averaging kernel, see Listing 1 for a PyTorch implementation. This allows us to run the computation in parallel utilizing GPUs for processing. Numerical stability considerations for the rolling window standardization. We estimate local variance using the identity σ2 cij = E[X2] -(E[X])2, as both terms can be implemented as convolutions. However, in regions where the variance is small (e.g., calm water), this formulation suffers from catastrophic cancellation due to the limited precision of 32-bit floating point arithmetic. To address this, we subtract a global perchannel mean, ̄Xc, from X before convolution: X′ cij = Xcij - ̄Xc This is a standard method for stabilizing the computation of σ2 by reducing the absolute magnitude of the squared and squared-mean terms. Figure 2 shows the effect that this shifted method has on the error of the computation as a function of the similarity of the variance and mean. Finally, we add a small constant, ε = 10-8, to the denominator to ensure positive values and avoid undefined behavior under the square root. 3.3. Anomaly Aggregation and Feature Extraction We aggregate the channel-wise deviations, D, across a large input scene over the channel dimension in a scalar anomaly map, A ∈ RH×W , as follows: Aij = C X c=1 |Dcij| We then threshold A either at a fixed value or using a high quantile (e.g., the 99.99th percentile) to produce a binary mask of anomalous pixels. We then extract spatially contiguous connected components (using 8connected neighborhood logic) and compute their average anomaly intensity. We filter out regions whose area falls below a minimum threshold (e.g., less than 1.5 square meters). We then represent surviving features as points using their geometric centroids and record their area and average aggregate deviation value. 4. Experiments and Results Our proposed methods have several free parameters that influence the quality and number of "interesting points" that a labeler must examine: the size of the window over which statistics are computed - s in the chunked standardization approach and k in the rolling window standardization approach - the anomaly threshold used for binarizing A (determining which pixels are 5 Robinson Goetz Khan Sackett Leonard Dodhia Ferres Scene Annotated Whales True Positives False Positives Recall Cape Cod Bay 2021 31 28 280 90.3% Pen ́ınsula Vald ́es 2012 84 81 276 96.4% Pen ́ınsula Vald ́es 2014 59 54 3622 91.5% Table 1: Detection performance across three benchmark scenes. anomalous), and the area threshold (determining how large an anomalous region must be to be considered "interesting"). A desirable configuration is one in which all of the whales in a scene are marked as "interesting" (i.e. high recall) and few non-whale points are marked as "interesting" (i.e. high precision). We evaluate the effectiveness of our methods by applying them to three satellite scenes previously studied in the literature, each containing manually annotated whale locations: Cape Cod Bay 2021 [10] Two WorldView-3 scenes (Catalog IDs 1040010067D36B00 and 10400100674B2100) captured on April 24, 2021 covering ∼200 km2 of Cape Cod Bay3 at a 0.3 m/px spatial resolution. It contains 31 annotated North Atlantic right whales, 25 of which are considered definite, and 6 of which are ambiguous. Pen ́ınsula Vald ́es 2012 [3] A WorldView-2 scene (Catalog ID 103001001C8C0300) captured on September 19, 2012, covering 120 km2 of the Vald ́es Peninsula in Argentina with a spatial resolution of 0.56 m/px. It includes 84 Southern right whales (Eubalaena australis) labels categorized by confidence: 15 definite, 32 probable, and 37 possible sightings. 3. We approximate the study area used in [10] by buffering the extent of the annotated points by 500m. Pen ́ınsula Vald ́es 2014 [3] A WorldView-3 scene (Catalog ID 10400100032A3700) captured on October 16, 2014 covering 560 km2, also from the Vald ́es Peninsula, with a spatial resolution of 0.37 m/px. It includes 59 Southern right whale labels: 23 definite, 12 probable, and 24 possible. Window size We run the chunked and rolling window standardization methods with different window sizes for the Cape Cod Bay 2021 scene and show the distribution of aggregated anomaly scores in the top panel of Figure 3. We find similar distributions of anomaly values among window sizes of 256, 512, 1024, and 2048 for the chunked standardization method and kernel sizes of 41 and 51 for the rolling window method. Kernel sizes of less than 41 see a drop-off in tail values as there is less context with which to determine whether a given value is an anomaly or not. Practically, on an V100 GPU, there is little difference in execution time of the rolling window standardization computation with larger window sizes. Threshold values We run the rolling window standardization method with k = 51 on each of the three evaluation scenes and plot the distribution of anomaly values in the bottom panel of Figure 3. We find that the distributions from the Cape Cod Bay 2021 and Vald ́es 2012 scenes - both of which have very still water - have shorter tails, while the Vald ́es 2014 scene - with a large number of white-capped waves - has a long tailed distribution with 99.99th per6 Where are the Whales Figure 3: (Top) The distribution of anomaly values (summed absolute deviations) on the Cape Cod Bay scene [10] using the two standardization methods with varying kernel (k) or window (s) sizes. We find that both methods result in similar distributions with a window or kernel size greater than 50. (Bottom) Comparison of anomaly value distributions (using rolling window standardization with a k = 51 kernel) across the three evaluation scenes. The Vald ́es 2014 scene has a high density of high-contrast features (mostly white-capped waves) which shows up as a longer tail in the anomaly distribution. centile values of 11.37, 11.90 and 55.25, respectively (computed over the RGB channels only). We always use a conservative value of 1.5 square for area thresholding based on conversations with experts and observing that positive identifications may only highlight parts of a whale. In general, we find that it is possible to run the interesting point methods with a range of anomaly thresholds and choose the result that returns a reasonable number of points per square kilometer after area filtering (using Tensor: if self.shift_val is not None: x = x - self.shift_val else: x = x - x.mean(dim=(0, 2, 3), keepdim=True) mu = self.conv(x) squares = self.conv(x**2.0) variance = squares - mu**2.0 return (x - mu) / (torch.sqrt(variance) + 1e-8) Listing 1: Implementation of the rolling window standardization approach, which applies a channel-wise local standardization to imagery using convolution-based estimates of mean and variance. (as measured by the total time between the interface serving an image chip to be labeled and the subsequent response with class annotation over the three labelers). Using the approximation from [4] - that manually annotating 100 sq km of imagery takes approximately 3 hours and 20 minutes - annotating 9 Robinson Goetz Khan Sackett Leonard Dodhia Ferres Figure 5: A map of 555 "interesting points" in red found over the two WorldView-3 scenes from used in the [10] study. The whale points from the Hodul et al. study are shown in green and the study area (buffered from the extent of the labeled points) is shown in yellow. Each of the "interesting points" was annotated by three expert labelers which took a total of 139 minutes. Manual annotation of the entire scene would take approximately ∼35 hours. The interesting points covered 28 out of the 31 whales found in the Hodul et al. study (which were all correctly identified by the labelers). Further, 5 additional whale points were found outside of the previously examined study area. this scene would have taken a single labeler ∼35 hours. Most detections were attributed to whitecap (n=420) and unsure (n=234), reflecting the presence of breaking waves near the shore. Only two points were annotated as a whale, of which, all three labelers agreed on one point as a whale (see Figure 4). We observe that the whale annotations took substantially longer to process - ranging from 15 to 106 seconds each - due to increased scrutiny and zooming behavior by the annotators. In contrast, non-whale categories required only 2.8 seconds per chips on average (with a standard deviation of 8.5 seconds). Further, we applied the same procedure to the two scenes used in the Hodul et al. 2022 study [10] (see the Cape Cod Bay 2021 description). This study focused on a ∼200 sq km area taken from two larger WorldView-3 strips, however the entire area from the two strips covers 1,083 sq km. Figure 5 shows this larger extent, with the approximate study area shown as a yellow box. We find a total of 555 interesting points in the larger scene, which took expert labelers 139 minutes in total to annotate. The average time per nonwhale annotation was 5.1 seconds while the average time per whale annotation was 11.23 seconds. The 555 interesting points covered 28 out of the 31 interesting points found in the Hodul et al. study, and the annotators correctly annotated each as a whale. Additionally, the annotators flagged 5 other points outside of the original study area as whales, two of which are shown in Figure 4. In both cases, the cost of pre-processing the scenes and applying our methods is negligible compared to the cost of having marine biologists annotate them. Finally, we observe - and emphasize as a limitation - that applying our methods to scenes that contain white-capped waves from high winds results in a large num10 Where are the Whales ber of false positives. The simple methods we propose here are not able to distinguish between different classes of anomalous groupings of pixels (i.e. between whitecaps, whales, buoys, or otherwise), and further work and labeled datasets are needed to find whales in these challenging conditions. Nevertheless, satellite imagery archives contain unstudied imagery of calm seas that can be mined for whale detections and used to bootstrap larger modeling efforts. Impact Statement This work contributes tools and methodology for identifying whales in satellite imagery to support marine mammal monitoring and conservation at scale. By reducing the need for exhaustive manual annotation, our semiautomated pipeline enables domain experts to focus their attention on anomalous "interesting points", accelerating detection of whales in vast ocean areas. We emphasize that these methods are not fully automated and require expert interpretation to ensure reliable use. The approach is sensitive to imaging conditions, scene complexity, and spectral variability, and may produce false positives in challenging environments. As such, it is intended as a decision-support system to augment - not replace - expert-driven analysis. All system components are open-sourced to promote transparency, reproducibility, and community adoption. This work marks a key step toward scalable, expert-in-the-loop remote sensing for conservation. By enabling rapid review of vast ocean imagery for spectrally distinct features, it allows experts to label whales and other surface objects, creating high-quality training data. This data can power future model development, accelerating geospatial insights into whale presence and enhancing both conservation efforts and maritime domain awareness. References [1] Alex Borowicz, Hieu Le, Grant Humphries, Georg Nehls, Caroline H ̈oschle, Vladislav Kosarev, and Heather J Lynch. Aerial-trained deep learning networks for surveying cetaceans from satellite imagery. PloS one, 14(10):e0212532, 2019. [2] Justine Boulent, Bertrand Charry, Malcolm McHugh Kennedy, Emily Tissier, Raina Fan, Marianne Marcoux, Cortney A Watt, and Antoine Gagn ́eTurcotte. Scaling whale monitoring using deep learning: A human-in-the-loop solution for analyzing aerial datasets. Frontiers in Marine Science, 10:370, 2023. [3] Hannah C Cubaynes and Peter T Fretwell. Whales from space dataset, an annotated satellite image dataset of whales for training machine learning models. Scientific data, 9(1):245, 2022. [4] Hannah C Cubaynes, Peter T Fretwell, Connor Bamford, Laura Gerrish, and Jennifer A Jackson. Whales from space: Four mysticete species described using new vhr satellite imagery. Marine Mammal Science, 35(2):466-491, 2019. [5] Peter T Fretwell, Iain J Staniland, and Jaume Forcada. Whales from space: counting southern right whales by satellite. PloS one, 9(2):e88655, 2014. [6] Kimberly T. Goetz, Lauren Connor, Christin Khan, John Wall, Meredith Sackett, Caleb Robinson, Juan M. Lavista Ferres, Claire Porter, and Michelle A. LaRue. From pixels to marine mammals: Open-source preprocessing workflow for satellite imagery to enhance animal detection and identification. Manuscript in preparation, 2025. 11 Robinson Goetz Khan Sackett Leonard Dodhia Ferres [7] KT Goetz, KEW Shelden, CL Sims, JM Waite, and PR Wade. Abundance of belugas (delphinapterus leucas) in cook inlet, alaska, june 2021 and june 2022. 2023. [8] Emilio Guirado, Siham Tabik, Marga L Rivas, Domingo Alcaraz-Segura, and Francisco Herrera. Whale counting in satellite and aerial images with deep learning. Scientific reports, 9(1):1-12, 2019. [9] Sean A. Hayes, Elizabeth Josephson, Katherine Maze-Foley, Patricia E. Rosel, Jessica McCordic, Amy Brossard, Samuel Chavez-Rosales, Timothy V.N. Cole, Lance P. Garrison, Joshua Hatch, Allison Henry, Daniel Linden, Jenny Litz, Marjorie C. Lyssikatos, Keith D. Mullin, Kimberly Murray, Christopher Orphanides, Richard M. Pace, Debra L. Palka, Jessica Powell, Kristin Precoda, Melissa Soldevilla, and Frederick W. Wenzel. U.s. atlantic and gulf of mexico marine mammal stock assessments 2023. Noaa technical memorandum, National Oceanic and Atmospheric Administration, National Marine Fisheries Service, Northeast Fisheries Science Center, Woods Hole, MA, December 2024. US . [10] Matus Hodul, Anders Knudby, Brigid McKenna, Amy James, Charles Mayo, Moira Brown, Delphine Durette-Morin, and Stephen Bird. Individual north atlantic right whales identified from space. Marine Mammal Science, 2022. [11] Christin B Khan, Kimberly T Goetz, Hannah C Cubaynes, Caleb Robinson, Erin Murnane, Tyler Aldrich, Meredith Sackett, Penny J Clarke, Michelle A LaRue, Timothy White, et al. A biologist's guide to the galaxy: Leveraging artificial intelligence and very highresolution satellite imagery to monitor marine mammals from space. Journal of Marine Science and Engineering, 11 (3):595, 2023. [12] P ̊al Wessel and Walter HF Smith. A global, self-consistent, hierarchical, high-resolution shoreline database. Journal of Geophysical Research: Solid Earth, 101(B4):8741-8743, 1996. 12
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2510.14707
|
Toward a unified view of agnostic parametrizations for deformed black holes
Manuel Del Piano
,1, 2, 3, ˚ Ciro De Simone
,4, 2, : Mattia Damia Paciarini
,3, ; Vittorio De
Falco
,5, § Miko laj Myszkowski
,3, ¶ Francesco Sannino
,3, 1, 2, 4, ˚˚ and Vania Vellucci
3, ::
1Scuola Superiore Meridionale, Largo S. Marcellino, 10, 80138 Napoli, Italy.
2INFN sezione di Napoli, via Cintia, 80126 Napoli, Italy.
3Quantum Theory Center (ℏQTC) & D-IAS, Southern Denmark University, Campusvej 55, 5230 Odense M, Denmark.
4Department of Physics E. Pancini, Universit`a di Napoli Federico II, via Cintia, 80126 Napoli, Italy.
5Ministero dell’Istruzione e del Merito (M.I.M., ex M.I.U.R.)
A variety of robust and effective descriptions have been devised to extract model-independent in-
formation about the fundamental properties of black holes from observational data when searching
for deviations from general relativity. In this work, we construct explicit transformation maps es-
tablishing the equivalence among three relevant parametrizations for different spacetime patches:
Johannsen–Psaltis, Rezzolla–Zhidenko, and Effective Metric Description. We then select represen-
tative black hole geometries to determine the minimal number of parameters required within each
scheme to reproduce the associated quasi-normal mode spectra with a prescribed degree of accuracy.
Our analysis shows that, for the given observables, a finite set of coefficients suffices to attain the
desired precision in the three frameworks. Finally, we emphasize how the individual strengths of
these effective descriptions can be exploited to probe complementary aspects of black hole physics.
I.
INTRODUCTION
Astrophysical black holes (BHs) constitute natural lab-
oratories to test gravity in extreme regimes. They are
among the most fascinating compact objects, as they are
endowed with an event horizon hiding a region which is
causally disconnected from the exterior. Although they
may appear to be rather complex physical systems from
an observational point of view, they are characterized
only by their mass, charge, and spin, as predicted by
general relativity (GR) [1, 2]. This makes BHs remark-
ably simple objects despite their tangled origins. How-
ever, the fundamental laws governing them have not yet
been robustly confirmed, requiring further investigations
to encompass potential deviations from the classical GR
metrics.
A strategy to explore such deformations was first
introduced by Johannsen and Psaltis, incorporating a
quadrupole moment independent of both mass and spin
[3, 4]. This approach, based on previous developments
[5], allows one to test the no-hair conjecture via the ex-
amination of different physical phenomena, such as the
location of the innermost stable circular orbit (ISCO)
[6, 7] or gravitational lensing effects [8] through the de-
tection of potential modifications of the quadrupole mo-
ment. This model has been revised by Cardoso, Pani, and
Rico [9] to address some of its weaknesses. Indeed, the
original metric accounts only for corrections preserving
the horizon area–mass relation. This limitation can be
˚ manuel.delpiano-ssm@unina.it
: ciro.desimone@unina.it
; damiapaciarinim@qtc.sdu.dk
§ deltafi.mat@live.it
¶ mikolaj@qtc.sdu.dk
˚˚ sannino@qtc.sdu.dk
:: vellucci@qtc.sdu.dk
removed by introducing additional parameters, dominat-
ing in both weak- and strong-field regimes. The resulting
framework yields the most general static and spherically
symmetric BH geometry, characterized by twice the num-
ber of parameters of the original approach. This implies
that, in the strong-field limit, all parameters contribute
with comparable significance.
Building upon this idea, Rezzolla and Zhidenko pro-
posed a parametrization for static and spherically sym-
metric BHs within metric theories of gravity [10, 11].
Their approach utilizes a continued-fraction expansion
in terms of a compactified radial coordinate, offering bet-
ter convergence properties compared to traditional Tay-
lor series techniques. This method enables efficient ap-
proximations of various metrics with a reduced set of
coefficients, with the hope of facilitating the comparison
of observational data against predictions from different
gravitational models (see e.g. [12–15]).
More recently, in [16–19] an independent alternative
model-independent framework was developed to describe
modifications of classical GR for BH metrics, termed the
Effective Metric Description (EMD). Here metric defor-
mations are parametrized in terms of physical quanti-
ties, such as the radial proper distance.
Focusing on
static and spherically symmetric spacetimes, explicit ex-
pansions of the metric near the event horizon were con-
structed in terms of physical coefficients, allowing for
the general, coordinate-independent parametrization of
thermodynamic quantities, such as the Hawking temper-
ature. In addition, the asymptotic behavior was analyzed
and provided further constraints on the metric.
Alto-
gether, these analyses lead to model-independent consis-
tency conditions on metric deformations. The approach
not only offers insights into the interplay between quan-
tum effects and BH thermodynamics, or more generally
on different sources of corrections to GR [20, 21] but also
in testing the mathematical consistency of generic BH
models [16–19] across different space-time dimensions.
arXiv:2510.14707v1 [gr-qc] 16 Oct 2025
2
Collectively, these parametrizations serve to accommo-
date potential deviations from classical BH solutions that
can help interpret GR precision tests. Ultimately, these
frameworks should be used to bridge the gap between al-
ternative theories of gravity and empirical observations.
It is therefore desirable and timely to build a unified view
of different agnostic parametrizations for deformed BH
metrics. This is the overarching goal of this work. To this
end, we will construct the transformation maps connect-
ing the three aforementioned frameworks, highlighting
the advantages and limitations of each approach. Estab-
lishing a coherent and concrete correspondence among
the proposed parametrizations is crucial for a compara-
tive analysis and a deeper understanding of their intrinsic
features.
In recent years, BH parametrizations have also at-
tracted remarkable attention in the field of BH pertur-
bation theory [22–25].
Once a BH is excited, it will
reach an equilibrium configuration through the emission
of gravitational waves. This relaxation process can be
divided into three phases: an early response, depending
on the initial conditions of the disturbance [26]; a ring-
down phase, where the fluctuation can be modeled as a
superposition of complex frequencies called quasi-normal
modes (QNMs) [27–29]; finally a late-time phase, where
the perturbation decays as a power-law tail [30–32]. It
turns out that modifications of the near-horizon metric
can have non-negligible effects on the response of a BH
to perturbations, especially in the QNM spectrum, which
we will analyze in detail within the three aforementioned
frameworks.
However, the attention is focused on the
so-called eikonal QNMs, i.e., the limit of large angular
momentum or multipole index ℓ" 1 [33, 34].
There
are several reasons behind the choice to investigate this
observable in this limit. One is that, under suitable as-
sumptions, the eikonal QNMs are directly related to the
intrinsic properties of the underlying BH geometry [35],
such as the orbital frequency Ω(being also related to the
BH shadow radius bsh [36–39]) and the Lyapunov expo-
nent λ of the photon sphere. The orbital frequency Ω
and the Lyapunov exponent λ (associated with the in-
stability of photon sphere orbits) represent the real and
imaginary parts of the QNM frequency, respectively.
This work offers the opportunity to better understand
the mathematical and physical link between the different
parametrizations [9, 10, 40], while simultaneously allow-
ing a comparison between the QNM spectrum and the
BH shadow results [41].
The manuscript is structured as follows.
In Section
II, we introduce the various parametrizations. Specifi-
cally, we discuss the Johannsen-Psaltis (JP) [3] in subsec-
tion II A, the Rezzolla-Zhidenko (RZ) [10] in II B, and the
EMD [17] in II C. In Section III we provide the transfor-
mation maps relating the aforementioned parametriza-
tions, where we also discuss their mathematical aspects
and connections.
Finally, in Sec.
IV, we consider the
QNMs in the eikonal limit to compare the different
parametrizations. Throughout this work, we adopt geo-
metric units, namely G “ c “ 1. We offer our conclusions
in Sec. V and in the appendices we provide helpful math-
ematical details for the EMD framework relevant for this
work.
II.
AGNOSTIC PARAMETRIZATIONS
Consider a generic static and spherically symmetric
spacetime, whose line element in Schwarzschild coordi-
nates pt, r, θ, φq reads
ds2 “ gttprqdt2 ` grrprqdr2 ` r2 `
dθ2 ` sin2 θdφ2˘
, (1)
where gtt and grr are functions of the radial coordi-
nate r only. This is our starting point to introduce the
Johannsen-Psaltis (JP) [3], the Rezzolla-Zhidenko (RZ)
[10], and the EMD [17] parametrizations.
We further
require the parametrizations to abide the experimental
bounds existing on the deviations from GR in the weak-
field regime, also known as parametrized post-Newtonian
(PPN) constraints [42, 43].
A.
Johannsen-Psaltis parametrization
The JP parametrization originates from the need to
test the no-hair conjecture with observations of BHs
in the electromagnetic spectrum.
This approach is
expressed through a parametric spacetime containing
a quadrupole moment independent of both mass and
spin [3].
Therefore, any deviation from GR manifests
in anomalous contributions to the quadrupole moment
[3, 5]. However, this framework has been generalized by
the authors in [9] to overcome a series of issues.
The
original JP parametrization reads (cf. Eq. (1)):
gJP
tt “ ´r1 ` gprqs
˜
1 ´ 2 ˜
M
r
¸
,
gJP
rr “ 1 ` gprq
1 ´ 2 ˜
M
r
, (2)
where the function gprq is defined by the series:
gprq :“
8
ÿ
n“1
ϵn
˜ ˜
M
r
¸n
“
8
ÿ
n“1
ϵn
2n
´rH
r
¯n
,
(3)
with ϵn real constants to be determined from observa-
tions, and ˜
M is related to the horizon position rH “ 2 ˜
M
and to the gravitational mass-energy of the spacetime M
through:
M “ ˜
M
´
1 ´ ϵ1
2
¯
.
(4)
The revised JP parametrization becomes [9]
gJP
tt “ ´
“
1 ` gtprq
‰
˜
1 ´ 2 ˜
M
r
¸
,
gJP
rr “ 1 ` grprq
1 ´ 2 ˜
M
r
,
(5)
3
where the functions gt and gr are expanded as
giprq “
8
ÿ
n“1
ϵi
n
˜ ˜
M
r
¸n
with
i “ t, r ,
(6)
and the set of parameters tϵt
i, ϵr
i uiě1 has doubled with
respect to the previous version of the JP parametrization.
The comparison with the PPN parameters entails
M “ ˜
M
ˆ
1 ´ ϵt
1
2
˙
,
(7a)
ϵr
1 “ γ p2 ´ ϵt
1q ´ 2 ,
(7b)
2ϵt
2 “ pβ ´ γqpϵt
1 ´ 2q2 ` 4ϵt
1 .
(7c)
Hence, the parameters ϵt
1, ϵr
2, and tϵt
i, ϵr
i uiě3 are not con-
strained, even in the case of GR, where β “ γ “ 1.
B.
Rezzolla-Zhidenko parametrization
The RZ parametrization aims to describe BH space-
times in generic metric theories of gravity using a model-
independent approach [10]. The line element of the RZ
parametrization reads (cf. Eq. (1))
gRZ
tt
“ ´N 2prq
and
gRZ
rr “ B2prq
N 2prq .
(8)
The BH event horizon is located at r “ rH ą 0 and is
defined by NprHq “ 0.
The radial coordinate is then
compactified by introducing the dimensionless variable
x :“ 1 ´ rH{r, such that x “ 0 marks the location of the
event horizon, while x “ 1 corresponds to spatial infinity.
We rewrite N as N 2 “ xApxq, where Apxq ą 0 for
0 ď x ď 1. We further express the functions A and B in
terms of the parameters ϵ, a0, and b0, as follows:
Apxq “ 1 ´ ϵp1 ´ xq ` pa0 ´ ϵqp1 ´ xq2 ` ˜Apxqp1 ´ xq3 ,
(9a)
Bpxq “ 1 ` b0p1 ´ xq ` ˜Bpxqp1 ´ xq2 ,
(9b)
where ϵ encodes deviations of rH from the Schwarzschild
radius ϵ :“ 2M´rH
rH
and the functions ˜A and ˜B describe
the metric near the horizon (for x » 0) and are finite
there, as well as at spatial infinity (for x » 1).
To achieve rapid convergence, these two functions are
modeled via the Pad´e approximants in the form of con-
tinued fractions as
˜Apxq “
a1
1 `
a2x
1 `
a3x
1 ` . . .
,
(10a)
˜Bpxq “
b1
1 `
b2x
1 `
b3x
1 ` . . .
,
(10b)
where a1, a2, a3 . . . and b1, b2, b3 . . . are dimensionless con-
stants determined from observations.
Note that by comparing the large-distance expansion
in Eq. (8) with the PPN one, we can put experimental
bound on the first coefficients:
a0 “ pβ ´ γqp1 ` ϵq2
2
À 10´5 ,
(11a)
b0 “ pγ ´ 1qp1 ` ϵq
2
À 10´5 .
(11b)
C.
Effective Metric Description parametrization
The EMD parametrizes the deformations of the classi-
cal Schwarzschild metric in terms of spacetime invariants,
which can be measured by observers independently from
the set of coordinates, and preserve the same symmetries
as GR. A natural choice for such a physical quantity is
the radial proper distance to the BH horizon [16, 17, 40].
The metric components can be written as
gEMD
tt
“ ´hprq
and
gEMD
rr
“
1
fprq ,
(12)
where
hprq “ 1 ´ ΨpXq
r
and
fprq “ 1 ´ ΦpXq
r
,
(13)
are positive definite for r ą rH (rH being the BH event
horizon) and the deformation functions ΨpXq and ΦpXq
are parametrized by a physical quantity X that is mono-
tonic in r and invariant under coordinate reparametriza-
tions. The specific choice of X does not affect the physical
conclusions, since different parametrization schemes can
always be locally mapped into each other [17].
We divide the discussion into regions close to (see
Sec. II C 1) and far from (see Sec. II C 2) the BH hori-
zon.
1.
EMD near the BH horizon
We consider Ψpρq and Φpρq, where ρ “ ρprq, with
r ě rH, is the radial proper distance measured from the
BH horizon, which is fixed by the following differential
equation
dρ
dr “
ˆ
1 ´ Φpρq
r
˙´1{2
with
ρprHq “ 0 .
(14)
The solution can be expressed as a series expansion close
to the BH horizon through
Φpρq “ rH ` 2M
8
ÿ
n“1
ξn
´ ρ
2M
¯n
,
(15a)
Ψpρq “ rH ` 2M
8
ÿ
n“1
θn
´ ρ
2M
¯n
,
(15b)
4
where ξn and θn are real constants.1 The series (15) are
assumed to have a non-vanishing radius of convergence,
and all derivatives of h and f with respect to r evaluated
in rH are well defined. This implies ξ2n´1 “ θ2n´1 “ 0
for all n P N, and ξ2 ď
M
8rH [17]. We will adopt these
assumptions for the rest of this work.
The proper distance ρprq and its inverse rpρq can be
obtained via Eq. (15) as follows:
rpρq “ rH ` 2M
8
ÿ
n“1
an
´ ρ
2M
¯n
,
(16a)
ρprq “ 2M
8
ÿ
n“1
bn
ˆr ´ rH
2M
˙n{2
,
(16b)
where the coefficients an and bn can be found iteratively
by solving the differential problem (14). Explicit expres-
sions for those coefficients are reported in [16].
The derivatives of the metric functions at the horizon
can be written in terms of tξ2n, θ2nuně1 as2
2M f p1q
H
“ 1 ` ?1 ´ 16yHξ2
2yH
,
(17a)
2M hp1q
H “ 1
yH
„
1 ´ θ2
2ξ2
´
1 ´
a
1 ´ 16yHξ2
¯ȷ
,
(17b)
where yH :“ rH{p2Mq. The deformation to the higher-
order derivatives take the form
p2Mqn ´
f pnq
H
´ pf pnq
H qclass
¯
9 ξ2n`n.l.pyH, ξ2, . . . , ξ2n´2q ,
(18)
where pf pnq
H qclass is the classical expression for the deriva-
tive of the metric at the horizon (for Schwarzschild yH “
1 and pf pnq
H qclass “ p´1qn`1p2Mq´n) and n.l. indicates a
non-linear dependence on yH and tξ2pu1ďpăn. The same
holds for hpnq
H
with respect to θ2n and pyH, tθ2pu1ďpănq.
2.
EMD at large distance from the BH
We now consider the spacetime regions far from the
BH horizon, where the spacetime is weakly curved, but
still affected by the presence of the BH. Hence, we can
asymptotically expand the deformation functions [16, 21].
The full radial proper distance dprq from r “ 0 is then
given by the differential problem
dd
dr “
ˇˇˇˇ1 ´ Φpdq
r
ˇˇˇˇ
´1{2
with
dp0q “ 0 .
(19)
1 Let us remark that the constants introduced in this way are
coinciding with the parameters xn and tn defined in [40].
2 Following the notation in [16], we denote the derivatives of the
function ϕ with respect to the radial coordinate r evaluated at
the horizon rH as ϕpnq
H
:“ dnϕ{drnˇˇ
r“rH .
In Appendix A, we show the steps to obtain the expres-
sions of the metric functions as reported below
fprq “ 1 ´ 2M
r
´ 2M 2ω1
r2
` pkω1 ´ ω2q2M 3
r3
´
„ˆ
k2 ` k ` 3
2
˙
ω1 ` ω2
1 ´ 2kω2 ` ω3
ȷ 2M 4
r4
` O
ˆM 5
r5
˙
,
(20a)
hprq “ 1 ´ 2M
r
´ 2M 2γ1
r2
` pkγ1 ´ γ2q2M 3
r3 `
´
„ˆ
k2 ` k ` 3
2
˙
γ1 ` γ1ω1 ´ 2kγ2 ` γ3
ȷ 2M 4
r4
` O
ˆM 5
r5
˙
.
(20b)
It is important to emphasize that the asymptotic se-
ries in Eqs. (20) must be interpreted with care. When
truncating the parametrization to a finite number of
deformation parameters, for instance pω1, . . . , ωnq and
pγ1, . . . , γnq, only the terms up to order 1{rn`1 should
be retained. Coefficients at higher orders in 1{r are not
reliable, because they implicitly depend on additional pa-
rameters that have been set to zero by the truncation. As
a consequence, the apparent 1{rm term with m ą n ` 1
does not represent the true asymptotic behavior of the
full parametrization, but rather a spurious artifact of the
cutoff.
This issue is absent in the Johannsen–Psaltis
parametrization, where truncating the expansion auto-
matically suppresses all higher-order terms, ensuring in-
ternal consistency. In contrast, in the EMD approach,
truncation must be imposed manually by discarding all
terms beyond the order supported by the retained num-
ber of parameters.
The combinations kω1 ´ ω2 and kγ1 ´ γ2, as well as
those appearing at higher orders, are independent of the
choice of k. Indeed, observable quantities do not depend
on the choice of this constant, see [17, 20] for details.
We also note that when the expansion is truncated at
O
`
M 4{r4˘
, the PN-expansions of f and h are equal when
tγ1, γ2u and tω1, ω2u are exchanged, whereas at the next
leading order this is spoiled by the presence of mixed
terms involving γ1ω1.
By comparing (20) with the PPN expansion, we ob-
tain:
|γ1| “ |β ´ γ| À 10´5
with
γ “ 1 .
(21)
At leading order in the PPN framework, we are able to
map only γ1 to β and γ, while ω1 remains unconstrained.
Conversely, the EMD expansion (20) is only able to cap-
ture a subset of PPN corrections, constrained by γ “ 1.
5
III.
TRANSFORMATION MAPS RELATING
THE DIFFERENT PARAMETRIZATIONS
In this section, we establish the relationships among
the three distinct frameworks. Notably, while the event
horizons in these parametrizations are defined through
different formalisms, they all depend on free parameters.
However, in all cases, they must uniquely identify the
BH event horizon rH. This observation is significant, as
it facilitates the subsequent computational analyses.
1.
JP to EMD (near horizon) parametrizations
First, we compare the temporal metric coefficients gtt
in the EMD (12) and JP (5) parametrizations:
gEMD
tt
“ ´
8
ÿ
n“1
hpnq
H
n! pr ´ rHqn
“ ´1 ` gtprq
r
pr ´ rHq “ gJP
tt .
(22)
Dividing both sides by pr ´ rHq we obtain
8
ÿ
n“1
hpnq
H
n! pr ´ rHqn´1 “ 1 ` gtprq
r
“
“ 1
r
«
1 `
8
ÿ
j“1
ϵt
j
2j
´rH
r
¯j
ff
”
8
ÿ
j“0
ϵt
j
2j
rj
H
rj`1 ,
(23)
where we included additional coefficient ϵt
0 “ 1 in the
sum. Evaluating Eq. (23) at r “ rH, we obtain for n “ 1
hp1q
H “ 1 ` gtprHq
rH
“ 1
rH
8
ÿ
j“0
ϵt
j
2j .
(24)
The coefficients hpnq
H can be computed by taking n deriva-
tives of Eq. (23) and then evaluating it at r “ rH. The
general expression for the coefficients hpnq
H
is thus:
hpnq
H “ n dn
drn
˜ 8
ÿ
j“1
ϵt
j
2j
rj
H
rj`1
¸ ˇˇˇˇ
r“rH
“
“ p´1qn n
rn`1
H
8
ÿ
j“1
ϵt
j
2j
pj ` nq!
j!
.
(25)
A similar approach can be applied to the radial metric
component, yielding:
gEMD
rr
“
1
fprq “ 1 ` grprq
1 ´ rH{r “ gJP
rr ,
(26)
where
fprq “
8
ÿ
n“1
f pnq
H
n! pr ´ rHqn .
(27)
This can be also written as
f p1q
H
“
1
rHr1 ` grprHqs ,
(28a)
f pnq
H
“ n dn
drn
„
1
rp1 ` grprqq
ȷ ˇˇˇˇ
r“rH
,
(28b)
where the general term f pnq
H
cannot be easily found and
then written in a closed form.
Comparing Eqs. (28a) with (17a), Eqs. (24) with (17b),
and recalling that rH “ 2M{p1 ´ ϵt
1{2q, we finally obtain
ξ2 “ grprHqp2 ´ ϵt
1q
8r1 ` grprHqs2 ,
θ2 “ gtprHqpϵt
1 ´ 2q
8r1 ` grprHqs ,
(29)
with grprHq ‰ ´1. The case grprHq “ 1 saturates the
bound on ξ2 ď p2 ´ ϵt
1q{32.
The expressions for the
higher-order coefficients tθ2n, ξ2nuně2 are involved and
we refrain from reporting them here. We note, however,
that the higher-order derivatives of the metric functions
depend linearly on these parameters, making their com-
putation straightforward.
From these calculations we note that the EMD coef-
ficients depend on an infinite number of JP terms, and
vice versa. The JP coefficients can be written in terms
of the EMD ones by inverting the linear system (25).
2.
JP to EMD (large distance) parametrizations
The expansion of JP metric at large distance reads:
gJP
tt “ ´1 `
˜
Mp2 ´ ϵt
1q
r
`
˜
M 2p2ϵt
1 ´ ϵt
2q
r2
` O
´
˜
M 3{r3¯
,
(30a)
gJP
rr “ 1 `
˜
Mp2 ` ϵr
1q
r
`
˜
M 2pϵr
2 ` 2ϵr
1 ` 4q
r2
` O
´
˜
M 3{r3¯
.
(30b)
Comparing (30) with the EMD expansion (20), and
taking into account the relation (7a), we finally obtain:
ϵr
1 “ ´ϵt
1 ,
(31a)
ω1 “ 2rϵr
2 ` ϵt
1p2 ´ ϵt
1qs
pϵt
1 ´ 2q2
,
γ1 “ 2p2ϵt
1 ´ ϵt
2q
pϵt
1 ´ 2q2 ,
(31b)
ω2 “ ´
2
pϵt
1 ´ 2q3
␣
2ϵr
3 ` rkpϵt
1 ´ 2q ´ 2ϵt
1spϵt
1 ´ 2qϵt
1
`ϵr
2r4pϵt
1 ´ 1q ` kp2 ´ ϵt
1qs
(
,
(31c)
γ2 “ 2kpϵt
1 ´ 2qp2ϵt
1 ´ ϵt
2q ´ 8ϵt
2 ` 4ϵt
3
pϵt
1 ´ 2q3
.
(31d)
where we write only the first few terms.
We note that this comparison can be easily carried out,
because there is a one-to-one correspondence among the
JP and EMD coefficients. This is due to the asymptotic
expansion of the JP metric, which establishes a direct
link among the two parametrizations.
6
3.
JP to RZ parametrizations
We note that the map between Eqs. (8) and (2) has
already been presented in Ref. [10]. Here, we adapt it
to the revised JP parametrization (5). Comparing that
with Eqs. (9a) and (9b), we have
´ϵrH
r ` pa0 ´ ϵq
´rH
r
¯2
`
´rH
r
¯3 ˜A
´
1 ´ rH
r
¯
“ gtprq ,
(32a)
b0
rH
r `
´rH
r
¯2 ˜B
´
1 ´ rH
r
¯
“ gBprq .
(32b)
where we introduced the auxiliary function
gBprq :“
a
r1 ` gtprqsr1 ` grprqs ´ 1 ,
(33)
The first few coefficients can be extracted as
ϵ “ ´ϵt
1
2 , a0 “ 1
2
ˆϵt
2
2 ´ ϵt
1
˙
, b0 “ ϵt
1 ` ϵr
1
4
.
(34)
By matching the two parametrizations near the horizon,
we obtain algebraic relations between the RZ coefficients
tan, bnuně1 and the JP coefficients tϵt
n, ϵr
nuně1. The first
few expressions are displayed below:
a1 “ gtprHq ´ a0 ` 2ϵ “
8
ÿ
n“3
ϵt
n
2n ,
(35a)
b1 “ gBprHq ´ b0 ,
(35b)
a2 “ ´pr gtq1 ` gt ` ϵ
a1
ˇˇˇ
r“rH
´ 1
“
8
ÿ
n“4
ϵt
npn ´ 3q
2n
M
8
ÿ
n“3
ϵt
n
2n ,
(35c)
b2 “ ´pr gBq1
b1
ˇˇˇˇ r“rH ´ 1 ,
(35d)
a3 “ pr2gtq2 ´ 2a1r1 ` a2pa2 ` 2qs
2a1a2
ˇˇˇ
r“rH
,
(35e)
b3 “ pr2gBq2
2b1b2
ˇˇˇˇ r“rH ´ pb2 ` 1q ,
(35f)
where the prime indicates the derivative with respect to
r. Naturally, in Eq. (35) we can easily extend to higher
orders if needed.
Finally, we remark that the terms a1, a2, a3 . . . do not
depend on ϵt
1 and ϵt
2, whereas the terms b1, b2, b3 . . . follow
a more complicated expression, which we chose not to
display here. In the simplest case when ϵt
3 ‰ 0 and ϵt
n “ 0
for n ą 3, we have a2 “ 0 and the approximant for
the function Nprq reproduces it exactly. This illustrates
how the RZ coefficients can be related to the JP ones.
The inverse map can be achieved by inverting the linear
system (35).
4.
RZ to EMD (near horizon) parametrizations
The final map to consider is between the RZ and EMD
parametrizations. Throughout the calculations we make
use of the results from Sec. III 3. By employing Eqs.
(22) and (28b), we obtain the following two relations:
gtprq “ r
« 8
ÿ
n“1
hpnq
H
n! pr ´ rHqn´1
ff
´ 1 ,
(36a)
grprq “
1
ř8
n“1
f pnq
H
n! pr ´ rHqn´1
´ 1 .
(36b)
We can now exploit Eq.
(35) to express the RZ coef-
ficients in terms of the derivatives of gtprq and grprq,
which, in turn, are related to the EMD coefficients.
We can obtain the direct link between the first-order
coefficients pθ2, ξ2q and pa0, a1, b0, b1q of the EMD and
RZ parametrizations, respectively, by using hp1q
H and f p1q
H
(cf. Eqs. (24) and (28a)) and
hprq “ Nprq2
and
fprq “ Nprq2
Bprq2 .
(37)
In order to determine the relation between the coeffi-
cients, let us first define y “ r{p2Mq and then expand
the RZ metric functions up to the first order in y ´ yH:
Npyq2 “ p3 ` a0 ` a1qyH ´ 2
y2
H
py ´ yHq ` Opy ´ yHq2 ,
(38a)
Npyq2
Bpyq2 “ p3 ` a0 ` a1qyH ´ 2
p1 ` b0 ` b1q2 y2
H
py ´ yHq ` Opy ´ yHq2 .
(38b)
Comparing these expansions with the first derivatives of
Eq. (17) and solving for ξ2 and θ2, we obtain
ξ2 “ p1 ` ϵqp1 ` A ´ 2ϵqpBpB ` 2q ´ A ` 2ϵq
4p1 ` Bq4
,
(39a)
θ2 “ p1 ` ϵqp2ϵ ´ Aq
8
ˆ
1 ` |1 ` 2A ´ BpB ` 2q ´ 4ϵ|
p1 ` Bq2
˙
,
(39b)
where A :“ a0 ` a1 and B :“ b0 ` b1. It is worth noticing
that the parameters ξ2 and θ2 depend on the RZ param-
eters only through the combinations A and B. Moreover,
the coefficients an and bn with n ą 1 do not contribute
to the first-order parameters ξ2 and θ2.
Additional constraints on the parameters can be ob-
tained by observing that, in the limit b0 “ b1 “ 0, we
have B “ 0. At the horizon, Bpyq ” 1 ` O
`
py ´ yHq2˘
,
so the first derivatives at the horizon in Eqs. (38) must co-
incide with (17a) and (17b). This implies the additional
condition ξ2 “ θ2. Furthermore, when setting B “ 0, the
absolute value in Eq. (39b) must be taken into account,
which requires imposing
1 ` 2A ´ 4ϵ ą 0 ñ 2ϵ ´ a0 ´ a1 ă 1
2 .
(40)
The relations in Eq. (39) can be inverted to obtain the
coefficients A and B as functions of θ2 and ξ2. As previ-
ously noted, if θ2 “ ξ2, then the first derivatives of fpyq
7
and hpyq at yH must coincide. Moreover, in the limit
ϵ Ñ 0 (see under Eq. (9)) the EMD and RZ parametriza-
tions should recover the Schwarzschild BH. Therefore, we
have:
A “
1
2
?
2 ξ2
”
8θ2ξ2 ` p1 ` ϵqpξ2 ´ θ2q
`
?
1 ` ϵ pξ2 ´ θ2q
a
1 ` ϵ ´ 16ξ2
ı1{2
´ 1 ,
(41a)
B “ 2ϵ ´ θ2
2ξ2
˜
1 ´
c
1 ´ 16ξ2
1 ` ϵ
¸
.
(41b)
This approach can be straightforwardly generalized to
higher-order coefficients by following the procedure al-
ready described for the first-order terms. The expressions
beyond first order become rather cumbersome, and hence
we choose not to display them here. We conclude there
exists a well-defined and direct correspondence between
the coefficients of the EMD and RZ parametrizations.
5.
RZ to EMD (large distance) parametrization
The comparison between RZ and EMD parametriza-
tions at large distance can be obtained by considering
the following expansion for the RZ metric components:
gRZ
tt
“ ´1 ` 2M
r
´
4a0M 2
pϵ ` 1q2r2
´
8M 3 ”
˜Ap1q ´ a0 ` ϵ
ı
pϵ ` 1q3r3
` O
ˆM 4
r4
˙
,
(42a)
gRZ
rr “ 1 ` 2Mp2b0 ` ϵ ` 1q
pϵ ` 1qr
` O
ˆM 2
r2
˙
.
(42b)
Comparing these expansions with Eq. (20), we obtain
b0 “ 0 and the following conditions on the first few pa-
rameters:
ω1 “ 4 ˜Bp1q ´ 2a0
pϵ ` 1q2
,
γ1 “ ´
2a0
pϵ ` 1q2 ,
(43a)
γ2 “ ´
2
”
a0pkp1 ` ϵq ´ 2q ` 2p ˜Ap1q ` ϵq
ı
pϵ ` 1q3
,
(43b)
ω2 “ ´
2
pϵ ` 1q3
!
a0rkp1 ` ϵq ´ 2s ` 2
”
ϵ ` ˜Ap1q
` p2 ´ kqp1 ` ϵq ˜Bp1q ` 2 ˜B1p1q
ı)
,
(43c)
where we note that the full continued fractions ˜Ap1q and
˜Bp1q (cf. Eq. (10)) and their derivatives appear, with the
prime denoting differentiation with respect to x defined
in Sec. II B. As mentioned previously, the coefficients of
RZ and EMD parametrizations at large distance are in
one-to-one correspondence.
IV.
EIKONAL LIMIT
As an application of the mapping among the three
parametrizations, we now consider the eikonal limit of
QNMs within the framework of BH perturbation theory.
QNMs are the characteristic complex oscillation modes
of perturbed compact objects [44]; their real and imag-
inary parts correspond to the oscillation frequency and
the inverse damping time, respectively. For a static and
spherically symmetric BH metric (1), the QNMs ω satisfy
the following Schr¨odinger-like equation:
d2Ψ
dr2˚
` pω2 ´ V prqqΨ “ 0 ,
(44)
with outgoing boundary conditions e˘iωr˚ at the BH
event horizon and at spatial infinity, expressed in terms
of the tortoise coordinate r˚, defined through dr˚{dr “
a
´grr{gtt. Here, V prq denotes the perturbation poten-
tial, which depends on the spin of the perturbing field
Ψ. Within the QNM spectrum, a notable role is played
by the eikonal limit, corresponding to the regime of very
large angular momentum, or multipole index, ℓ" 1. In
this limit, the QNM frequency reads
ωnℓ“ Ωℓ´ i
ˆ
n ` 1
2
˙
|λ| ,
(45)
where n P N is the overtone number, Ωis the orbital
frequency of light rays at the photon-sphere radius rps,
and λ is the Lyapunov exponent [34].
The eikonal limit establishes a correspondence between
the QNMs and the properties of the photon sphere, un-
der the following two criteria [45]: (1) the perturbation
potential is positive definite (to avoid instabilities), single
peaked, and decays to zero at the boundaries; and (2) the
perturbation is a test scalar field or other field minimally
coupled to gravity. This implies that the eikonal QNMs
associated with gravitational perturbations may not, in
general, be directly related to the properties of the un-
stable photon orbit. For this reason, we will restrict our
analysis to perturbations of test scalar and electromag-
netic fields.
Given a static and spherically symmetric spacetime in
Eq. (1), the photon dynamics can be described using the
effective potential U 2prq “ ´gttprq{r2. The radius, or-
bital frequency [46], and Lyapunov exponent of the pho-
ton sphere are then given by [34]3
rps g1
ttprpsq “ 2 gttprpsq ,
(46a)
Ω2 “ U 2prpsq “ ´gttprq
r2
ˇˇˇˇ
r“rps
,
(46b)
λ2 “ ´
r2
ps
2 grrprpsq
d2U 2prq
dr2
ˇˇˇˇ
r“rps
,
(46c)
3 Note the change of the metric signature with respect to
Eqs. (35), (37), and (40) reported in Ref. [34].
8
respectively.
We note that Ωis closely related to the
radius of the BH shadow through bsh :“ Uprpsq´1 ” Ω´1
[39].4
A.
Eikonal limit in different parametrizations
In this subsection, we investigate the eikonal limit
within the three BH parametrizations introduced in
Sec. II. For a generic parametrization, the correspond-
ing set of infinite coefficients is treated as independent.
However, once a specific model is considered, these coef-
ficients become functions of a finite number of free pa-
rameters.
In the next subsection, we examine small deviations
from the Schwarzschild solution using models specified
by the mass M and a single additional l, which quan-
tifies the departure from the GR geometry. This intro-
duces a displacement of the BH horizon radius rH from
its Schwarzschild value 2M. To describe this effect, we
adopt the perturbative parameter ϵ (see below Eq. (9)),
which allows for a Taylor expansion of the quantities that
enter the eikonal QNM limit. Other choices, such as ϵt
1
in the JP parametrization, are equally possible.
We then expand the physically relevant quantities in
Eq. (46) in the eikonal limit up to linear order in ϵ for
each parametrization. Higher-order terms in ϵ are subse-
quently used to estimate the truncation error.
1.
Rezzolla-Zhidenko parametrization
We start with the RZ parametrization, where the ϵ
parameter emerges naturally. We assume the coefficients
an and bn admit a power-series expansion in ϵ of the form
anpϵq “
ÿ
p“0
An,p ϵp
and
bnpϵq “
ÿ
p“0
Bn,p ϵp ,
(47)
for n ě 0 and A0,0 “ A1,0 “ B0,0 “ B1,0 “ 0. The last
condition ensures that the leading contributions to the
metric functions from a0pϵq, a1pϵq, b0pϵq, and b1pϵq vanish
in the Schwarzschild limit ϵ Ñ 0, thereby guaranteeing
that the deformations associated with Eqs. (9a) and (9b)
are continuously switched off.
The values of the coefficients An,p and Bn,p depend
on the specific model under consideration. In general,
these coefficients appear in the series expansion of the
quantities in Eqs. (46) through nonlinear combinations at
each order in ϵ. These expressions simplify dramatically
4 In Ref. [34], the authors use a different potential Vrprq instead
of U2prq.
when A2,0 “ B2,0 “ 0:
rRZ
ps
M “ 3 ´ 4
9 p5 ` A0,1 ` A1,1q ϵ ` O
`
ϵ2˘
,
(48a)
ΩRZM “
1
3
?
3 `
2
81
?
3 p6 ` 3A0,1 ` 2A1,1q ϵ ` O
`
ϵ2˘
,
(48b)
λRZM “
1
3
?
3 `
2
81
?
3 p´4 ` 7A0,1 ` 4A1,1
`9B0,1 ´ 6B1,1q ϵ ` O
`
ϵ2˘
.
(48c)
In the following sections, we restrict our attention to BH
metrics satisfying the aforementioned condition.
2.
Johannsen-Psaltis parametrization
Following
the
same
reasoning
as
for
the
RZ
parametrization, we note that the deformation functions
gtprq and grprq must vanish in the Schwarzschild limit.
This implies that all JP parameters tϵt
n, ϵr
nuně1 scale ho-
mogeneously with ϵ. Indeed, as shown in Eq. (34), we
know that ϵt
1 “ ´2 ϵ exactly.
Therefore, for ϵ Ñ 0
and the scaling behavior of Eq. (47), we generally have
ϵr
1, ϵt
2 „ O
`
ϵk˘
, with k ě 1. Hence, we assume the fol-
lowing general power series expansion
ϵt
npϵq “
ÿ
p“1
pϵt
nqp ϵp
for
n ě 2 ,
(49a)
ϵr
npϵq “
ÿ
p“1
pϵr
nqp ϵp
for
n ě 1 ,
(49b)
which could be equivalently read as a power series in
ϵt
1. The expressions for the coefficients pϵt
nqp and pϵr
nqp
are determined by the model under investigation. Insert-
ing the expansions (49) into Eq. (46) written in the JP
parametrization (truncated for brevity at ϵt
4 and ϵr
4), we
obtain
rJP
ps
M “ 3 ´
ˆ8
3 ` pϵt
2q1
9
` pϵt
3q1
18
` 2pϵt
4q1
81
˙
ϵ ` O
`
ϵ2˘
,
(50a)
ΩJPM “
1
3
?
3 `
ˆ2
9 ` pϵt
2q1
54
` pϵt
3q1
162 ` 2pϵt
4q1
486
˙ ϵ
?
3 ` O
`
ϵ2˘
,
(50b)
λJPM “
1
3
?
3 `
ˆ 5
27 ` 2pϵt
2q1
81
` pϵt
3q1
162 ` pϵt
4q1
1458
´ pϵr
1q1
18
´ pϵr
2q1
54
´ pϵr
3q1
162 ´ pϵr
4q1
486
˙ ϵ
?
3 ` O
`
ϵ2˘
.
(50c)
As expected from Eqs. (35) and (48), all JP parameters
enter linearly at the first order in ϵ.
9
3.
EMD (near horizon) parametrization
As shown in [40], the EMD parametrization near the
horizon can be extended up to the photon sphere by em-
ploying the Pad´e approximation [47] (see Appendix B for
details). In this case, the position of the event horizon is
written as
rH “ 2Mp1 ` cq ,
(51)
and hence, the expressions for the quantities reported in
Eq. (46) can be compactly written as
rEMDpnhq
ps
M
“ σ0 ` σ1 c ` O
`
c2˘
,
(52a)
ΩEMDpnhqM “ η0 ` η1 c ` O
`
c2˘
,
(52b)
λEMDpnhqM “ Υ0 ` Υ1 c ` O
`
c2˘
.
(52c)
where
c
quantifies
the
departure
of
rH
from
the
Schwarzschild radius and is related to the RZ parame-
ter as
c “ ´
ϵ
ϵ ` 1 .
(53)
The coefficients are linear combinations of the param-
eters tθ2n, ξ2nuně1 and their explicit form depends on
the order of the employed Pad´e approximant, see Table I
in [40] and Table III in Appendix B. A Pad´e approxima-
tion of order pN, Mq involves a total of N ` M parame-
ters. We note that the effective parameters tθ2n, ξ2nuně1
must still be expanded in power series of ϵ, as we will see
in the subsection dedicated to specific BH models.
4.
EMD (at large distance) parametrization
The EMD parametrization at large distances must re-
cover the Schwarzschild metric when the deformation pa-
rameters tγi, ωiuiě1 vanish. However, the position of the
event horizon does not explicitly appear as an input, and
therefore there is no direct connection to the ϵ parame-
ter. Nevertheless, by inspecting the relations Eqs. (43)
and ansatz (47) (or equivalently by considering Eq. (31)
and ansatz (49)), we find that tγi, ωiuiě1 indeed admit a
power series expansion in ϵ.
Therefore, we assume that tγi, ωiuiě1 can be expanded
similarly to the previous cases, namely
γnpϵq “
ÿ
p“1
γn,p ϵp
and
ωnpϵq “
ÿ
p“1
ωn,p ϵp ,
(54)
where tγn,p, ωn,puně1, pě1 assume different values de-
pending on the model under consideration.
Using the
metric functions in Eqs. (20), we can express Eqs. (46)
as
rEMDpldq
ps
M
“ 3 ` 1
9
”
p15 ´ 3k ` 2k2qγ1,1
` p5 ´ 4kqγ2,1 ` 2γ3,1
ı
ϵ ` O
`
ϵ2˘
, (55a)
ΩEMDpldqM “
1
3
?
3 `
1
162
?
3
”
p´2k2 ` 4k ´ 21qγ1,1
` p4k ´ 6qγ2,2 ´ 2γ3,1
ı
ϵ ` O
`
ϵ2˘
, (55b)
λEMDpldqM “
1
3
?
3 `
1
162
?
3
”
p6k2 ´ 4k ` 21qγ1,1
` 2p5k ´ 6qγ2,2 ` 6γ3,1 ´ p2k2 ´ 4k ` 21qω1,1
` p4k ´ 6qω2,1 ´ 2ω3,1
ı
ϵ ` O
`
ϵ2˘
,
(55c)
Above we kept the dependence on the gauge parameter
k to keep the expressions as general as possible.
B.
Application to BH models
As an application of the formalism developed in the
previous subsection, let us consider the following BH
spacetimes: Hayward [48], Bardeen [49], and Simpson-
Visser II [50].
These are regular BH candidates,
which can be effectively described as deformations of
the Schwarzschild geometry and satisfy the condition
gtt grr “ ´1. Their gtt metric components read
´gH
ttprq “ 1 ´
2Mr2
r3 ` 2M l2
H
,
(56a)
´gB
ttprq “ 1 ´
2Mr2
pr2 ` l2
Bq3{2 ,
(56b)
´gSV2
tt
prq “ 1 ´ 2M
r
exp
"
´lSV2
r
*
.
(56c)
The regularization length-scale parameters plH, lB, lSV2q
have different physical interpretations in each model, are
allowed to vary over a finite range, and can be expressed
in terms of ϵ as follows:
lH “ 2M
c
ϵ
p1 ` ϵq3 ,
(57a)
lB “ 2M
a
p1 ` ϵq2{3 ´ 1
1 ` ϵ
,
(57b)
lSV2 “ 2M logp1 ` ϵq
1 ` ϵ
.
(57c)
We analytically expand the quantities in Eqs. (46)
around ϵ “ 0 up to quadratic order for each BH model
and compare the results with those obtained in the Rez-
zolla–Zhidenko, Johannsen–Psaltis, near-horizon EMD,
and large-distance EMD parametrizations, fixing the
number of free parameters to four in Eqs. (48), (50),
(52), and (55). The outcome of this comparison is sum-
marized in Table I. We observe that, at linear order, all
10
BH models
Parametrizations
rps
M ´ 3
ΩM ´
1
3
?
3
λM ´
1
3
?
3
Hayward
From the model
´16
9 ϵ ` 80
27ϵ2
8
81
?
3ϵ ´
40
243
?
3ϵ2
´
16
81
?
3ϵ `
16
243
?
3ϵ2
Rezzolla-Zhidenko
´16
9 ϵ ` 28892
9801 ϵ2
8
81
?
3ϵ ´
4012
24057
?
3ϵ2
´
16
81
?
3ϵ `
173960
2910897
?
3ϵ2
Johannsen-Psaltis
´16
9 ϵ ` 32
9 ϵ2
8
81
?
3ϵ ´
440
2187
?
3ϵ2
´
16
81
?
3ϵ `
32
243
?
3ϵ2
EMD near horizon
´16
9 ϵ ` 799
243ϵ2
8
81
?
3ϵ ´
110
729
?
3ϵ2
´
16
81
?
3ϵ ´
107
2187
?
3ϵ2
EMD large distance
´16
9 ϵ ` 176
81 ϵ2
8
81
?
3ϵ ´
296
2187
?
3ϵ2
´
16
81
?
3ϵ ´
368
2187
?
3ϵ2
Bardeen
From the model
´20
9 ϵ ` 650
243ϵ2
4
27
?
3ϵ ´
98
729
?
3ϵ2
´
8
81
?
3ϵ ´
68
729
?
3ϵ2
Rezzolla-Zhidenko
´20
9 ϵ ` 650
243ϵ2
4
27
?
3ϵ ´
98
729
?
3ϵ2
´
8
81
?
3ϵ ´
68
729
?
3ϵ2
Johannsen-Psaltis
´20
9 ϵ ` 650
243ϵ2
4
27
?
3ϵ ´
98
729
?
3ϵ2
´
8
81
?
3ϵ ´
68
729
?
3ϵ2
EMD near horizon
´20
9 ϵ ` 652
243ϵ2
4
27
?
3ϵ ´
391
2916
?
3ϵ2
´
8
81
?
3ϵ ´
421
4374
?
3ϵ2
EMD large distance
´20
9 ϵ ` 370
243ϵ2
4
27
?
3ϵ ´
98
729
?
3ϵ2
´
8
81
?
3ϵ ´
68
729
?
3ϵ2
Simpson-Visser II
All parametrizations
´8
3ϵ ` 74
27ϵ2
2
9
?
3ϵ ´
7
81
?
3ϵ2
2
27
?
3ϵ ´
11
81
?
3ϵ2
TABLE I: Expressions of the photon sphere radius rps, orbital frequency Ω, and Lyapunov exponent λ. The values
are rescaled by the BH mass M, shifted with respect to the corresponding Schwarzschild result, and computed up to
quadratic order in ϵ in the case of Hayward, Bardeen, and Simpson-Visser II models using four parameters in each
parametrization. For the last BH spacetime, we report only the expressions for the exact metric, as they are identical
across all parametrizations.
parametrizations reproduce the same result for all phys-
ical quantities in all BH models. Aside from the Simp-
son–Visser II case, where all parametrizations trivially
yield identical expansions, no single parametrization con-
sistently performs better than the others across all ob-
servables and BH models.
A complementary analysis is presented in Table II,
where, instead of fixing the number of parameters a pri-
ori, we set the desired tolerance and let the required num-
ber of parameters vary to match the exact expressions. In
the case of the EMD near-horizon parametrization, this
procedure coincides with a Pad´e approximation of order
pN, 3q, where the number of free parameters is controlled
by the integer N. Also in this setting, no parametrization
emerges as universally favored across all BH models.
Finally, in Fig. 1 we illustrate the convergence of the
linear and quadratic coefficients for the Bardeen BH
as a representative example.
Similar behavior is ob-
served for all other BH models. The plots highlight that
all parametrizations perform on a comparable footing.
Therefore, the choice of parametrization should be guided
by the desired accuracy and the number of parameters
one is willing to introduce. This has to be considered on
a case-by-case basis for each physical observable and BH
model.
V.
DISCUSSION AND CONCLUSIONS
This
work
begins
by
examining
three
model-
independent parametrizations of BH spacetimes:
Jo-
hannsen–Psaltis,
which
describes
deformations
as
quadrupolar deviations from GR; Rezzolla–Zhidenko,
which introduces departures from the Schwarzschild solu-
tion and employs Pad´e-like continued-fraction expansions
valid far from the event horizon; and EMD, which pro-
vides local BH metric descriptions in terms of physical
quantities such as the physical distance. We then estab-
lish, for the first time, explicit mappings among these
three frameworks, thereby offering a unified perspective
on agnostic parametrizations of deformed BHs.
As a relevant application, we investigate the QNMs of
11
BH models
Hayward
Bardeen
Simpson-Visser II
order
ϵ
ϵ2
ϵ
ϵ2
ϵ
ϵ2
Rezzolla-Zhidenko
2
7
2
4
1
1
Johannsen-Psaltis
3
6
2
4
1
2
EMD near horizon
4
ą 7
4
ą 7
4
4
EMD large distance
3
6
2
4
1
2
TABLE II: Number of parameters required to reproduce the exact results for the three observables at orders ϵ and ϵ2.
2
3
4
5
6
2.400
2.375
2.350
2.325
2.300
2.275
2.250
2.225
O( )
rps
2
3
4
5
6
0.081
0.082
0.083
0.084
0.085
2
3
4
5
6
0.10
0.09
0.08
0.07
0.06
2
3
4
5
6
number of parameters
1.5
2.0
2.5
3.0
3.5
O(
2)
2
3
4
5
6
number of parameters
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05
2
3
4
5
6
number of parameters
0.16
0.14
0.12
0.10
0.08
0.06
Bardeen
Rezzolla-Zhidenko
Johannsen-Psalits
EMD near horizon
EMD far region
FIG. 1: Comparison of the convergence to the linear coefficient (top row) and quadratic coefficient (bottom row)
for the Bardeen BH model, varying the number of the related parameters. The horizontal dashed black line is the
value computed from the model, reported in Table I. For the near-horizon EMD with four parameters, two Pad´e
approximants were used: (2,2) and (1,3), the latter being more accurate.
BHs in the eikonal limit.
We analyse the QNM spec-
trum across the different parametrizations, considering
small deviations from the Schwarzschild case by treating
the deformation parameter (ϵ) as a perturbative quan-
tity. This allows us to expand the effective coefficients
of each parametrization in power series of ϵ. To demon-
strate the effectiveness of our framework, we apply it to
three regular BH models: Hayward, Bardeen, and Simp-
son–Visser II. We summarize our results in several tables
and figures carrying different information: in Table I we
report the expansions up to quadratic order in ϵ using
only four parameters for each parametrization and for
each BH model; in Table II we fix the accuracy and let
the number of coefficients vary freely; in Fig. 1 we show
the convergence of the different expansions in terms of
the number of retained parameters and also check the
values of the attained tolerances. Across all analyses, we
find that the three parametrizations are fundamentally
equivalent: none systematically outperforms the others.
Superiority is context-dependent and emerges only once
the target accuracy, the observable under consideration,
and the number of parameters are specified within a given
BH model.
Our astrophysical case study supports a democratic
perspective: parametrizations should be viewed as com-
plementary tools rather than competing prescriptions.
This philosophy naturally paves the way for developing
new parametrizations tailored to specific observables or
regimes. Moreover, recent works [19, 51] have empha-
sized the usefulness of proper-time EMD parametriza-
tions in capturing effective quantum corrections near
the horizon. In this light, the mappings established in
our work offer promising opportunities to translate such
quantum-gravity–inspired deformations across different
parametrization schemes.
As a natural extension, the
same methodology could be applied to other problems
in BH physics and generalized to stationary and axially
symmetric spacetimes.
12
ACKNOWLEDGEMENTS
V. De Falco is grateful to Gruppo Nazionale di Fisica
Matematica of Istituto Nazionale di Alta Matematica
for support. C. De Simone acknowledges the support of
INFN sez. di Napoli, iniziativa specifica QGSKY. M. Del
Piano acknowledges support from INFN sez. di Napoli,
iniziativa specifica ST&FI and expresses gratitude to the
Southern Denmark University, the Danish Institute for
Advanced Study, and the Quantum Theory Center in
Odense (Denmark) for hosting during the initial stages of
the work. The work of F. Sannino and M. Damia Pacia-
rini is partially supported by the Carlsberg Foundation,
grant CF22-0922. This work contributes to COST Ac-
tion CA23130 – Bridging high and low energies in search
of quantum gravity (BridgeQG).
Appendix A: Treatment of logarithmic functions for EMD at large distance
In order to determine the metric functions, we could start by assuming that Φpdq and Ψpdq can be expanded in
power series of 1{d. However, such an approach will lead to logarithmic terms in the asymptotic expansions of hprq
and fprq. Therefore, to avoid that consequence, we can reabsorb the logarithms in Φpdq and Ψpdq by defining them
as
Φpdq “ 2M
«
1 `
8
ÿ
n“1
n´1
ÿ
m“0
Ωn,mM n
dn
logm
ˆ d
2M
˙ff
,
Ψpdq “ 2M
«
1 `
8
ÿ
n“1
n´1
ÿ
m“0
Γn,mM n
dn
logm
ˆ d
2M
˙ff
,
(A1)
where Ωn,m and Γn,m are real constants and ωn ” Ωn,0 and γn ” Γn,0.
We consider the post-Newtonian (PN) expansion M{r ! 1, (the weak-field limit M{r ! 1 is related to low velocity
limit v{c ! 1 via the virial theorem [52], namely M{r „ v2{c2). In this regime, we have 1{pridjq „ 1{pri`jq and
the function dprq admits the logarithmic corrections: pM{rqj logkpr{Mq, with tk, ju P N0 and 0 ă k ď j ´ 1. We
PN-expand Eq. (19) via Eq. (A1), which gives [17, 20]
dprq
M
“ r
M ` log
´ r
2M
¯
` k ´
ˆ3
2 ` ω1
˙ M
r ´
„5
4 ` ω1p5 ´ 2kq
4
` ω2 ` Ω2,1
2
ȷ M 2
r2 ` M 2pω1 ´ Ω2,1q
2r2
log
´ r
2M
¯
´
„35
24 ` p74 ´ 21k ` 9k2qω1
27
` 45ω2
1 ` 6p7 ´ 6kqω2 ` 2p8 ` 3kqΩ2,1
54
` 9ω3 ` 3Ω3,1 ` 2Ω3,2
27
ȷ M 3
r3 `
«
p7 ´ 6kqω1
` 6ω2 ´ 2p4 ´ 3kqΩ2,1 ´ 3Ω3,1 ´ 2Ω3,2
ff
M 3
9r3 log
´ r
2M
¯
´ rω1 ` 2Ω2,1 ´ Ω3,2sM 3
3r3 log2 ´ r
2M
¯
` O
ˆM 4
r4
˙
,
(A2)
where k is a real integration constant, kept as a free parameter. It is important to note that k plays the role of the
gauge for d, as its value determines the position at which d “ 0. Inserting Eq. (A2) into the metric functions modified
with the series in Eq. (A1), we obtain the conditions to balance and cancel out (up to a given order) the logarithmic
functions in Eq. (20), which requires
Ω2,1 “ ω1 , Ω3,1 “ ´ω1 ` 2ω2 , Ω3,2 “ ω1 , Ω4,3 “ ω1 , Ω4,2 “ ´5
2ω1 ` 6
2ω2 , Ω4,1 “ ω1 ´ 2ω2 ` 3ω3 ,
Ω5,4 “ ω1 , Ω5,3 “ ´13
3 ω1 ` 12
3 ω2 , Ω5,2 “ 9
2ω1 ´ 14
2 ω2 ` 12
2 ω3 , Ω5,1 “ ´ω1 ` 2ω2 ´ 3ω3 ` 4ω4 ,
for the first few terms. Although it is difficult to generalize the above relations, they can be computed to arbitrary
order and applied to the set tΓn,mu0ďmďn´1
ně1
. Using the above conditions, we are able to obtain the metric functions
as in Eq. (20) and eliminate the logarithmic functions appearing in Eq. (A2).
Appendix B: Extending the EMD near horizon in the eikonal limit
In the EMD framework near the horizon, the potential U2prq can be expressed as a power series around rH:
U 2prq “
8
ÿ
n“1
vnpr ´ rHqn,
(B1)
where vn depend on tθ2ku and tξ2ku (see [40] for the explicit form of these coefficients). It has been shown in [40] that,
by employing Pad´e approximations at a sufficiently high order pN, Mq, the convergence radius of the series increases
13
N
M
η0
η1
2
1
1
4
`?
3 ´ 1
˘
´1
4
`?
3 ´ 1
˘
`
1
135
`
61 ´ 68
?
3
˘ θ2
c ` 2
9
`
2
?
3 ´ 7
˘ θ4
c ` 8
3
`?
3 ´ 2
˘ θ6
c
2
2
1
4
a
9
?
3 ´ 15
´1
4
a
9
?
3 ´ 15 ´
1
630
c
1274659
?
3 ´ 6386663
3
θ2
c ´ 11
15
c
1
6
`
627
?
3 ´ 1075
˘θ4
c
´
2444
?
2
b`
14
?
3 ´ 9
˘ `
85675
?
3 ` 148401
˘
θ6
c ´
16
c
51
?
3 ` 265
3
θ8
c
1
3
1
3
?
3
´
1
3
?
3 ´
19676
25515
?
3
θ2
c ´
656
405
?
3
θ4
c ´
64
27
?
3
θ6
c ´ 128
81
?
3
θ8
c
2
3
1
3
?
3
´
1
3
?
3 ´
98837
127575
?
3
θ2
c ´ 136096
76545
?
3
θ4
c ´
4384
1215
?
3
θ6
c ´ 1216
243
?
3
θ8
c ´ 256
81
?
3
θ10
c
3
3
1
3
?
3
´
1
3
?
3 ´
652294
841995
?
3
θ2
c ´
2066276
1148175
?
3
θ4
c ´
62528
15309
?
3
θ6
c ´ 29152
3645
?
3
θ8
c
´ 2560
243
?
3
θ10
c
´ 512
81
?
3
θ12
c
N
M
Υ0
Υ1
2
1
`?
3 ´ 1
˘ `?
3 ` 1
˘2
16
4?
3
´
`?
3 ´ 1
˘ `?
3 ` 1
˘2
16
4?
3
`
`
13
?
3 ´ 116
˘ `?
3 ` 1
˘
180
4?
3
θ2
c `
`?
3 ´ 2
˘ `?
3 ` 1
˘
2
4?
3
θ4
c
´2
`?
3 ´ 2
˘ `?
3 ` 1
˘
4?
3
θ6
c
2
2
1
24
c
1
2
`
963 ´ 529
?
3
˘
´ 1
24
c
1
2
`
963 ´ 529
?
3
˘
´
a
187519473 ´ 102837005
?
3
3780
θ2
c ` 1
60
a
162387 ´ 92399
?
3θ4
c
+
78
`
9
?
3 ` 23
˘
a
61617
?
3 ` 106731
θ6
c ` 4796
c
2
241053
?
3 ` 417609
θ8
c
1
3
1
3
?
3
´
1
3
?
3 ´
35396
25515
?
3
θ2
c `
112
1215
?
3
θ4
c ` 448
81
?
3
θ6
c ` 512
81
?
3
θ8
c
2
3
1
3
?
3
´
1
3
?
3 ´
171496
127575
?
3
θ2
c ` 10160
5103
?
3
θ4
c ` 2752
135
?
3
θ6
c ` 1280
27
?
3
θ8
c ` 1024
27
?
3
θ10
c
3
1
1
3
?
3
´
1
3
?
3 ´
5662841
4209975
?
3
θ2
c `
2857228
1148175
?
3
θ4
c ` 2398688
76545
?
3
θ6
c ` 423776
3645
?
3
θ8
c
` 50432
243
?
3
θ10
c
` 11776
81
?
3
θ12
c
TABLE III: Orbital frequency pη0, η1q and Lyapunov exponent pΥ0, Υ1q coefficients of the photon sphere obtained
from Eqs. (46b) and (46c) in terms of the Pad´e approximants of the potential U2
N,M.
significantly up to the photon sphere location. This approach allows for the computation of the BH shadow radius bsh,
orbital frequency Ω, and Lyapunov exponent λ. Assuming that the position of the BH horizon is defined by Eq. (51),
for small deviations c ! 1 from the Schwarzschild horizon, we can write Ωand λ in mass units M as:
ΩM “ η0 ` η1 c ` Opc2q,
(B2a)
λ M “ Υ0 ` Υ1 c ` Opc2q.
(B2b)
The coefficients pη0, η1q and pΥ0, Υ1q are reported in Table III and depend on the order of the Pad´e approximation
pN, Mq. The Schwarzschild limit is already recovered at order p1, 3q. From Eq. (B2), the eikonal QNMs can be
directly reconstructed at any order pN, Mq of the Pad´e approximation using Eq. (45).
14
The relation between bsh and Ωcan be further exploited to investigate the large-N limit in the Pad´e approximation
of order pN, 3q. First, let us recall the expression of the shadow radius in terms of c from [40]:
bsh
M “ τ0 ` τ1 c ` Opc2q,
(B3)
as well as the explicit dependence of τ1 on c and on the order N of the Pad´e approximation:
τ1pNq “ 3
?
3 `
N`3
ÿ
k“1
β2kpNqθ2k
c
with
β2kpNq “
B
B θ2k
1
UN,3
ˇˇˇˇ
prpsqN,3
,
(B4)
where prpsqN,3 and UN,3 correspond to the photon sphere radius and potential computed at the order pN, 3q of the
Pad´e approximation, respectively. Similarly, the coefficient η1 of the photon sphere frequency can be expressed as
η1pNq “ ´ 1
3
?
3 `
N`3
ÿ
k“1
γ2kpNqθ2k
c
with
γ2kpNq “ B UN,3
B θ2k
ˇˇˇˇ
prpsqN,3
.
(B5)
Recalling that, at the lowest order, UN,3 “ 1{p3
?
3q, the coefficients γ2kpNq and β2kpNq are related by
β2kpNq “
B
B θ2k
1
UN,3
ˇˇˇˇ
prpsqN,3
“ ´
1
U 2
N,3
γ2kpNq “ ´p3
?
3q2γ2kpNq,
(B6)
and the asymptotic behavior of γ2kpNq can be derived from the N Ñ 8 limit of β2kpNq obtained in [40]
lim
NÑ8
β2kpNq
3
?
3
Ñ s2k,
s „ 1.5245.
(B7)
This result is compatible with Fig. 2, where the first few coefficients γ2k have been computed up to N “ 9. Moreover,
the asymptotic limit of η1pNq becomes
lim
NÑ8 c η1pNq “ ´
c
3
?
3 ´
8
ÿ
k“1
s2k
3
?
3θ2k “ ´
c
3
?
3 ´
1
3
?
3
1
2M rΨp2Msq ´ rHs,
(B8)
and substituting it into the full expression of Ω“ η0 ` η1c, we obtain
lim
NÑ8 Ω“ lim
NÑ8pη0 ` η1cq “
2
3
?
3 ´
1
3
?
3
1
2M Ψp2Msq.
(B9)
In the case of the Lyapunov exponent, the asymptotic N limit is more intricate, since λ depends on both the photon
sphere radius and the second-order derivatives of the potential. However, by analogy with Ω, we adopt a similar
ansatz for Υ1pNq, which can be written as
Υ1pNq “
1
3
?
3 `
N`3
ÿ
k“1
δ2kpNqθ2k
c ,
(B10)
where δ2k is a suitable set of numerical coefficients. In general, δ2k depend on the order of the Pad´e approximation
and are related to the derivative of the Lyapunov exponent with respect to the parameters θ2k, evaluated at the radius
of the photon sphere. The coefficients δ2k computed up to k “ 7 and N “ 9 are shown in Fig. 2. Although the
individual values of δ2k seem to converge to a constant value, it is not clear whether resummation similar to the BH
shadow radius and photon sphere frequency is possible. We note that the coefficient δ8 reaches the largest value.
Finally, we remark on the second order ϵ expansion of the EMD near-horizon parametrization. Since the ϵ depen-
dence is encapsulated in the c coefficient in Eq. (51), it is important to study how this relation changes at second
order. We have thus checked that for all BH models of interest, Eq. (51) does not receive corrections at O
`
c2˘
.
Therefore, rH acquires O
`
ϵ2˘
contributions only from the expansion of c “ ´ϵ{p1 ` ϵq. Moreover, the second-order
expansion involves significantly more terms than the first-order one, namely linear and quadratic terms in tθnu, as well
as possible mixing among the linear coefficients. We find that, for a given order pN, Mq of the Pad´e approximation,
the number of coefficients at second order scales quadratically with K “ N ` M as pK2 ` 5K ` 6q{2. Similarly, at
order ϵN, the number of coefficients will grow as „ KN.
15
1
2
3
4
5
6
7
8
9
N
1.20
1.25
1.30
1.35
1.40
1.45
1.50
(
3 3
2)1/2
(
3 3
4)1/4
(
3 3
6)1/6
(
3 3
8)1/8
(
3 3
10)1/10
(
3 3
12)1/12
(
3 3
14)1/14
1
2
3
4
5
6
7
8
9
N
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
(
3 3
2)1/2
(3 3
4)1/4
(3 3
6)1/6
(3 3
8)1/8
(3 3
10)1/10
(3 3
12)1/12
(3 3
14)1/14
FIG. 2: Coefficients γ2k (left) and δ2k (right) for different orders of the Pad´e approximation pN, 3q.
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Toward a unified view of agnostic parametrizations for deformed black holes Manuel Del Piano ,1, 2, 3, ̊ Ciro De Simone ,4, 2, : Mattia Damia Paciarini ,3, ; Vittorio De Falco ,5, § Miko laj Myszkowski ,3, ¶ Francesco Sannino ,3, 1, 2, 4, ̊ ̊ and Vania Vellucci 3, :: 1Scuola Superiore Meridionale, Largo S. Marcellino, 10, 80138 Napoli, Italy. 2INFN sezione di Napoli, via Cintia, 80126 Napoli, Italy. 3Quantum Theory Center (ħQTC) & D-IAS, Southern Denmark University, Campusvej 55, 5230 Odense M, Denmark. 4 . Pancini, Universit`a di Napoli Federico II, via Cintia, 80126 Napoli, Italy. 5Ministero dell'Istruzione e del Merito (M.I.M., ex M.I.U.R.) A variety of robust and effective descriptions have been devised to extract model-independent information about the fundamental properties of black holes from observational data when searching for deviations from general relativity. In this work, we construct explicit transformation maps establishing the equivalence among three relevant parametrizations for different spacetime patches: Johannsen-Psaltis, Rezzolla-Zhidenko, and Effective Metric Description. We then select representative black hole geometries to determine the minimal number of parameters required within each scheme to reproduce the associated quasi-normal mode spectra with a prescribed degree of accuracy. Our analysis shows that, for the given observables, a finite set of coefficients suffices to attain the desired precision in the three frameworks. Finally, we emphasize how the individual strengths of these effective descriptions can be exploited to probe complementary aspects of black hole physics. I. INTRODUCTION Astrophysical black holes (BHs) constitute natural laboratories to test gravity in extreme regimes. They are among the most fascinating compact objects, as they are endowed with an event horizon hiding a region which is causally disconnected from the exterior. Although they may appear to be rather complex physical systems from an observational point of view, they are characterized only by their mass, charge, and spin, as predicted by general relativity (GR) [1, 2]. This makes BHs remarkably simple objects despite their tangled origins. However, the fundamental laws governing them have not yet been robustly confirmed, requiring further investigations to encompass potential deviations from the classical GR metrics. A strategy to explore such deformations was first introduced by Johannsen and Psaltis, incorporating a quadrupole moment independent of both mass and spin [3, 4]. This approach, based on previous developments [5], allows one to test the no-hair conjecture via the examination of different physical phenomena, such as the location of the innermost stable circular orbit (ISCO) [6, 7] or gravitational lensing effects [8] through the detection of potential modifications of the quadrupole moment. This model has been revised by Cardoso, Pani, and Rico [9] to address some of its weaknesses. Indeed, the original metric accounts only for corrections preserving the horizon area-mass relation. This limitation can be ̊ : ; § ¶ ̊ ̊ :: removed by introducing additional parameters, dominating in both weak- and strong-field regimes. The resulting framework yields the most general static and spherically symmetric BH geometry, characterized by twice the number of parameters of the original approach. This implies that, in the strong-field limit, all parameters contribute with comparable significance. Building upon this idea, Rezzolla and Zhidenko proposed a parametrization for static and spherically symmetric BHs within metric theories of gravity [10, 11]. Their approach utilizes a continued-fraction expansion in terms of a compactified radial coordinate, offering better convergence properties compared to traditional Taylor series techniques. This method enables efficient approximations of various metrics with a reduced set of coefficients, with the hope of facilitating the comparison of observational data against predictions from different gravitational models (see e.g. [12-15]). More recently, in [16-19] an independent alternative model-independent framework was developed to describe modifications of classical GR for BH metrics, termed the Effective Metric Description (EMD). Here metric deformations are parametrized in terms of physical quantities, such as the radial proper distance. Focusing on static and spherically symmetric spacetimes, explicit expansions of the metric near the event horizon were constructed in terms of physical coefficients, allowing for the general, coordinate-independent parametrization of thermodynamic quantities, such as the Hawking temperature. In addition, the asymptotic behavior was analyzed and provided further constraints on the metric. Altogether, these analyses lead to model-independent consistency conditions on metric deformations. The approach not only offers insights into the interplay between quantum effects and BH thermodynamics, or more generally on different sources of corrections to GR [20, 21] but also in testing the mathematical consistency of generic BH models [16-19] across different space-time dimensions. 16 Oct 2025 2 Collectively, these parametrizations serve to accommodate potential deviations from classical BH solutions that can help interpret GR precision tests. Ultimately, these frameworks should be used to bridge the gap between alternative theories of gravity and empirical observations. It is therefore desirable and timely to build a unified view of different agnostic parametrizations for deformed BH metrics. This is the overarching goal of this work. To this end, we will construct the transformation maps connecting the three aforementioned frameworks, highlighting the advantages and limitations of each approach. Establishing a coherent and concrete correspondence among the proposed parametrizations is crucial for a comparative analysis and a deeper understanding of their intrinsic features. In recent years, BH parametrizations have also attracted remarkable attention in the field of BH perturbation theory [22-25]. Once a BH is excited, it will reach an equilibrium configuration through the emission of gravitational waves. This relaxation process can be divided into three phases: an early response, depending on the initial conditions of the disturbance [26]; a ringdown phase, where the fluctuation can be modeled as a superposition of complex frequencies called quasi-normal modes (QNMs) [27-29]; finally a late-time phase, where the perturbation decays as a power-law tail [30-32]. It turns out that modifications of the near-horizon metric can have non-negligible effects on the response of a BH to perturbations, especially in the QNM spectrum, which we will analyze in detail within the three aforementioned frameworks. However, the attention is focused on the so-called eikonal QNMs, i.e., the limit of large angular momentum or multipole index l" 1 [33, 34]. There are several reasons behind the choice to investigate this observable in this limit. One is that, under suitable assumptions, the eikonal QNMs are directly related to the intrinsic properties of the underlying BH geometry [35], such as the orbital frequency Ω(being also related to the BH shadow radius bsh [36-39]) and the Lyapunov exponent λ of the photon sphere. The orbital frequency Ω and the Lyapunov exponent λ (associated with the instability of photon sphere orbits) represent the real and imaginary parts of the QNM frequency, respectively. This work offers the opportunity to better understand the mathematical and physical link between the different parametrizations [9, 10, 40], while simultaneously allowing a comparison between the QNM spectrum and the BH shadow results [41]. The manuscript is structured as follows. In Section II, we introduce the various parametrizations. Specifically, we discuss the Johannsen-Psaltis (JP) [3] in subsection II A, the Rezzolla-Zhidenko (RZ) [10] in II B, and the EMD [17] in II C. In Section III we provide the transformation maps relating the aforementioned parametrizations, where we also discuss their mathematical aspects and connections. Finally, in Sec. IV, we consider the QNMs in the eikonal limit to compare the different parametrizations. Throughout this work, we adopt geometric units, namely G " c " 1. We offer our conclusions in Sec. V and in the appendices we provide helpful mathematical details for the EMD framework relevant for this work. II. AGNOSTIC PARAMETRIZATIONS Consider a generic static and spherically symmetric spacetime, whose line element in Schwarzschild coordinates pt, r, θ, φq reads ds2 " gttprqdt2 ` grrprqdr2 ` r2 ` dθ2 ` sin2 θdφ2 ̆ , (1) where gtt and grr are functions of the radial coordinate r only. This is our starting point to introduce the Johannsen-Psaltis (JP) [3], the Rezzolla-Zhidenko (RZ) [10], and the EMD [17] parametrizations. We further require the parametrizations to abide the experimental bounds existing on the deviations from GR in the weakfield regime, also known as parametrized post-Newtonian (PPN) constraints [42, 43]. A. Johannsen-Psaltis parametrization The JP parametrization originates from the need to test the no-hair conjecture with observations of BHs in the electromagnetic spectrum. This approach is expressed through a parametric spacetime containing a quadrupole moment independent of both mass and spin [3]. Therefore, any deviation from GR manifests in anomalous contributions to the quadrupole moment [3, 5]. However, this framework has been generalized by the authors in [9] to overcome a series of issues. The original JP parametrization reads (cf. Eq. (1)): gJP tt " ́r1 ` gprqs ̃ 1 ́ 2 ̃ M r ̧ , gJP rr " 1 ` gprq 1 ́ 2 ̃ M r , (2) where the function gprq is defined by the series: gprq :" 8 ÿ n"1 εn ̃ ̃ M r ̧n " 8 ÿ n"1 εn 2n ́rH r ̄n , (3) with εn real constants to be determined from observations, and ̃ M is related to the horizon position rH " 2 ̃ M and to the gravitational mass-energy of the spacetime M through: M " ̃ M ́ 1 ́ ε1 2 ̄ . (4) The revised JP parametrization becomes [9] gJP tt " ́ " 1 ` gtprq ‰ ̃ 1 ́ 2 ̃ M r ̧ , gJP rr " 1 ` grprq 1 ́ 2 ̃ M r , (5) 3 where the functions gt and gr are expanded as giprq " 8 ÿ n"1 εi n ̃ ̃ M r ̧n with i " t, r , (6) and the set of parameters tεt i, εr i uiě1 has doubled with respect to the previous version of the JP parametrization. The comparison with the PPN parameters entails M " ̃ M ˆ 1 ́ εt 1 2 ̇ , (7a) εr 1 " γ p2 ́ εt 1q ́ 2 , (7b) 2εt 2 " pβ ́ γqpεt 1 ́ 2q2 ` 4εt 1 . (7c) Hence, the parameters εt 1, εr 2, and tεt i, εr i uiě3 are not constrained, even in the case of GR, where β " γ " 1. B. Rezzolla-Zhidenko parametrization The RZ parametrization aims to describe BH spacetimes in generic metric theories of gravity using a modelindependent approach [10]. The line element of the RZ parametrization reads (cf. Eq. (1)) gRZ tt " ́N 2prq and gRZ rr " B2prq N 2prq . (8) The BH event horizon is located at r " rH ą 0 and is defined by NprHq " 0. The radial coordinate is then compactified by introducing the dimensionless variable x :" 1 ́ rH{r, such that x " 0 marks the location of the event horizon, while x " 1 corresponds to spatial infinity. We rewrite N as N 2 " xApxq, where Apxq ą 0 for 0 ď x ď 1. We further express the functions A and B in terms of the parameters ε, a0, and b0, as follows: Apxq " 1 ́ εp1 ́ xq ` pa0 ́ εqp1 ́ xq2 ` ̃Apxqp1 ́ xq3 , (9a) Bpxq " 1 ` b0p1 ́ xq ` ̃Bpxqp1 ́ xq2 , (9b) where ε encodes deviations of rH from the Schwarzschild radius ε :" 2M ́rH rH and the functions ̃A and ̃B describe the metric near the horizon (for x » 0) and are finite there, as well as at spatial infinity (for x » 1). To achieve rapid convergence, these two functions are modeled via the Pad ́e approximants in the form of continued fractions as ̃Apxq " a1 1 ` a2x 1 ` a3x 1 ` . . . , (10a) ̃Bpxq " b1 1 ` b2x 1 ` b3x 1 ` . . . , (10b) where a1, a2, a3 . . . and b1, b2, b3 . . . are dimensionless constants determined from observations. Note that by comparing the large-distance expansion in Eq. (8) with the PPN one, we can put experimental bound on the first coefficients: a0 " pβ ́ γqp1 ` εq2 2 À 10 ́5 , (11a) b0 " pγ ́ 1qp1 ` εq 2 À 10 ́5 . (11b) C. Effective Metric Description parametrization The EMD parametrizes the deformations of the classical Schwarzschild metric in terms of spacetime invariants, which can be measured by observers independently from the set of coordinates, and preserve the same symmetries as GR. A natural choice for such a physical quantity is the radial proper distance to the BH horizon [16, 17, 40]. The metric components can be written as gEMD tt " ́hprq and gEMD rr " 1 fprq , (12) where hprq " 1 ́ ΨpXq r and fprq " 1 ́ ΦpXq r , (13) are positive definite for r ą rH (rH being the BH event horizon) and the deformation functions ΨpXq and ΦpXq are parametrized by a physical quantity X that is monotonic in r and invariant under coordinate reparametrizations. The specific choice of X does not affect the physical conclusions, since different parametrization schemes can always be locally mapped into each other [17]. We divide the discussion into regions close to (see Sec. II C 1) and far from (see Sec. II C 2) the BH horizon. 1. EMD near the BH horizon We consider Ψpρq and Φpρq, where ρ " ρprq, with r ě rH, is the radial proper distance measured from the BH horizon, which is fixed by the following differential equation dρ dr " ˆ 1 ́ Φpρq r ̇ ́1{2 with ρprHq " 0 . (14) The solution can be expressed as a series expansion close to the BH horizon through Φpρq " rH ` 2M 8 ÿ n"1 ξn ́ ρ 2M ̄n , (15a) Ψpρq " rH ` 2M 8 ÿ n"1 θn ́ ρ 2M ̄n , (15b) 4 where ξn and θn are real constants.1 The series (15) are assumed to have a non-vanishing radius of convergence, and all derivatives of h and f with respect to r evaluated in rH are well defined. This implies ξ2n ́1 " θ2n ́1 " 0 for all n P N, and ξ2 ď M 8rH [17]. We will adopt these assumptions for the rest of this work. The proper distance ρprq and its inverse rpρq can be obtained via Eq. (15) as follows: rpρq " rH ` 2M 8 ÿ n"1 an ́ ρ 2M ̄n , (16a) ρprq " 2M 8 ÿ n"1 bn ˆr ́ rH 2M ̇n{2 , (16b) where the coefficients an and bn can be found iteratively by solving the differential problem (14). Explicit expressions for those coefficients are reported in [16]. The derivatives of the metric functions at the horizon can be written in terms of tξ2n, θ2nuně1 as2 2M f p1q H " 1 ` ?1 ́ 16yHξ2 2yH , (17a) 2M hp1q H " 1 yH „ 1 ́ θ2 2ξ2 ́ 1 ́ a 1 ́ 16yHξ2 ̄ȷ , (17b) where yH :" rH{p2Mq. The deformation to the higherorder derivatives take the form p2Mqn ́ f pnq H ́ pf pnq H qclass ̄ 9 ξ2n`n.l.pyH, ξ2, . . . , ξ2n ́2q , (18) where pf pnq H qclass is the classical expression for the derivative of the metric at the horizon (for Schwarzschild yH " 1 and pf pnq H qclass " p ́1qn`1p2Mq ́n) and n.l. indicates a non-linear dependence on yH and tξ2pu1ďpăn. The same holds for hpnq H with respect to θ2n and pyH, tθ2pu1ďpănq. 2. EMD at large distance from the BH We now consider the spacetime regions far from the BH horizon, where the spacetime is weakly curved, but still affected by the presence of the BH. Hence, we can asymptotically expand the deformation functions [16, 21]. The full radial proper distance dprq from r " 0 is then given by the differential problem dd dr " ˇˇˇˇ1 ́ Φpdq r ˇˇˇˇ ́1{2 with dp0q " 0 . (19) 1 Let us remark that the constants introduced in this way are coinciding with the parameters xn and tn defined in [40]. 2 Following the notation in [16], we denote the derivatives of the function φ with respect to the radial coordinate r evaluated at the horizon rH as φpnq H :" dnφ{drnˇˇ r"rH . In Appendix A, we show the steps to obtain the expressions of the metric functions as reported below fprq " 1 ́ 2M r ́ 2M 2ω1 r2 ` pkω1 ́ ω2q2M 3 r3 ́ „ˆ k2 ` k ` 3 2 ̇ ω1 ` ω2 1 ́ 2kω2 ` ω3 ȷ 2M 4 r4 ` O ˆM 5 r5 ̇ , (20a) hprq " 1 ́ 2M r ́ 2M 2γ1 r2 ` pkγ1 ́ γ2q2M 3 r3 ` ́ „ˆ k2 ` k ` 3 2 ̇ γ1 ` γ1ω1 ́ 2kγ2 ` γ3 ȷ 2M 4 r4 ` O ˆM 5 r5 ̇ . (20b) It is important to emphasize that the asymptotic series in Eqs. (20) must be interpreted with care. When truncating the parametrization to a finite number of deformation parameters, for instance pω1, . . . , ωnq and pγ1, . . . , γnq, only the terms up to order 1{rn`1 should be retained. Coefficients at higher orders in 1{r are not reliable, because they implicitly depend on additional parameters that have been set to zero by the truncation. As a consequence, the apparent 1{rm term with m ą n ` 1 does not represent the true asymptotic behavior of the full parametrization, but rather a spurious artifact of the cutoff. This issue is absent in the Johannsen-Psaltis parametrization, where truncating the expansion automatically suppresses all higher-order terms, ensuring internal consistency. In contrast, in the EMD approach, truncation must be imposed manually by discarding all terms beyond the order supported by the retained number of parameters. The combinations kω1 ́ ω2 and kγ1 ́ γ2, as well as those appearing at higher orders, are independent of the choice of k. Indeed, observable quantities do not depend on the choice of this constant, see [17, 20] for details. We also note that when the expansion is truncated at O ` M 4{r4 ̆ , the PN-expansions of f and h are equal when tγ1, γ2u and tω1, ω2u are exchanged, whereas at the next leading order this is spoiled by the presence of mixed terms involving γ1ω1. By comparing (20) with the PPN expansion, we obtain: |γ1| " |β ́ γ| À 10 ́5 with γ " 1 . (21) At leading order in the PPN framework, we are able to map only γ1 to β and γ, while ω1 remains unconstrained. Conversely, the EMD expansion (20) is only able to capture a subset of PPN corrections, constrained by γ " 1. 5 III. TRANSFORMATION MAPS RELATING THE DIFFERENT PARAMETRIZATIONS In this section, we establish the relationships among the three distinct frameworks. Notably, while the event horizons in these parametrizations are defined through different formalisms, they all depend on free parameters. However, in all cases, they must uniquely identify the BH event horizon rH. This observation is significant, as it facilitates the subsequent computational analyses. 1. JP to EMD (near horizon) parametrizations First, we compare the temporal metric coefficients gtt in the EMD (12) and JP (5) parametrizations: gEMD tt " ́ 8 ÿ n"1 hpnq H n! pr ́ rHqn " ́1 ` gtprq r pr ́ rHq " gJP tt . (22) Dividing both sides by pr ́ rHq we obtain 8 ÿ n"1 hpnq H n! pr ́ rHqn ́1 " 1 ` gtprq r " " 1 r « 1 ` 8 ÿ j"1 εt j 2j ́rH r ̄j ff " 8 ÿ j"0 εt j 2j rj H rj`1 , (23) where we included additional coefficient εt 0 " 1 in the sum. Evaluating Eq. (23) at r " rH, we obtain for n " 1 hp1q H " 1 ` gtprHq rH " 1 rH 8 ÿ j"0 εt j 2j . (24) The coefficients hpnq H can be computed by taking n derivatives of Eq. (23) and then evaluating it at r " rH. The general expression for the coefficients hpnq H is thus: hpnq H " n dn drn ̃ 8 ÿ j"1 εt j 2j rj H rj`1 ̧ ˇˇˇˇ r"rH " " p ́1qn n rn`1 H 8 ÿ j"1 εt j 2j pj ` nq! j! . (25) A similar approach can be applied to the radial metric component, yielding: gEMD rr " 1 fprq " 1 ` grprq 1 ́ rH{r " gJP rr , (26) where fprq " 8 ÿ n"1 f pnq H n! pr ́ rHqn . (27) This can be also written as f p1q H " 1 rHr1 ` grprHqs , (28a) f pnq H " n dn drn „ 1 rp1 ` grprqq ȷ ˇˇˇˇ r"rH , (28b) where the general term f pnq H cannot be easily found and then written in a closed form. Comparing Eqs. (28a) with (17a), Eqs. (24) with (17b), and recalling that rH " 2M{p1 ́ εt 1{2q, we finally obtain ξ2 " grprHqp2 ́ εt 1q 8r1 ` grprHqs2 , θ2 " gtprHqpεt 1 ́ 2q 8r1 ` grprHqs , (29) with grprHq ‰ ́1. The case grprHq " 1 saturates the bound on ξ2 ď p2 ́ εt 1q{32. The expressions for the higher-order coefficients tθ2n, ξ2nuně2 are involved and we refrain from reporting them here. We note, however, that the higher-order derivatives of the metric functions depend linearly on these parameters, making their computation straightforward. From these calculations we note that the EMD coefficients depend on an infinite number of JP terms, and vice versa. The JP coefficients can be written in terms of the EMD ones by inverting the linear system (25). 2. JP to EMD (large distance) parametrizations The expansion of JP metric at large distance reads: gJP tt " ́1 ` ̃ Mp2 ́ εt 1q r ` ̃ M 2p2εt 1 ́ εt 2q r2 ` O ́ ̃ M 3{r3 ̄ , (30a) gJP rr " 1 ` ̃ Mp2 ` εr 1q r ` ̃ M 2pεr 2 ` 2εr 1 ` 4q r2 ` O ́ ̃ M 3{r3 ̄ . (30b) Comparing (30) with the EMD expansion (20), and taking into account the relation (7a), we finally obtain: εr 1 " ́εt 1 , (31a) ω1 " 2rεr 2 ` εt 1p2 ́ εt 1qs pεt 1 ́ 2q2 , γ1 " 2p2εt 1 ́ εt 2q pεt 1 ́ 2q2 , (31b) ω2 " ́ 2 pεt 1 ́ 2q3 ␣ 2εr 3 ` rkpεt 1 ́ 2q ́ 2εt 1spεt 1 ́ 2qεt 1 `εr 2r4pεt 1 ́ 1q ` kp2 ́ εt 1qs ( , (31c) γ2 " 2kpεt 1 ́ 2qp2εt 1 ́ εt 2q ́ 8εt 2 ` 4εt 3 pεt 1 ́ 2q3 . (31d) where we write only the first few terms. We note that this comparison can be easily carried out, because there is a one-to-one correspondence among the JP and EMD coefficients. This is due to the asymptotic expansion of the JP metric, which establishes a direct link among the two parametrizations. 6 3. JP to RZ parametrizations We note that the map between Eqs. (8) and (2) has already been presented in Ref. [10]. Here, we adapt it to the revised JP parametrization (5). Comparing that with Eqs. (9a) and (9b), we have ́εrH r ` pa0 ́ εq ́rH r ̄2 ` ́rH r ̄3 ̃A ́ 1 ́ rH r ̄ " gtprq , (32a) b0 rH r ` ́rH r ̄2 ̃B ́ 1 ́ rH r ̄ " gBprq . (32b) where we introduced the auxiliary function gBprq :" a r1 ` gtprqsr1 ` grprqs ́ 1 , (33) The first few coefficients can be extracted as ε " ́εt 1 2 , a0 " 1 2 ˆεt 2 2 ́ εt 1 ̇ , b0 " εt 1 ` εr 1 4 . (34) By matching the two parametrizations near the horizon, we obtain algebraic relations between the RZ coefficients tan, bnuně1 and the JP coefficients tεt n, εr nuně1. The first few expressions are displayed below: a1 " gtprHq ́ a0 ` 2ε " 8 ÿ n"3 εt n 2n , (35a) b1 " gBprHq ́ b0 , (35b) a2 " ́pr gtq1 ` gt ` ε a1 ˇˇˇ r"rH ́ 1 " 8 ÿ n"4 εt npn ́ 3q 2n M 8 ÿ n"3 εt n 2n , (35c) b2 " ́pr gBq1 b1 ˇˇˇˇ r"rH ́ 1 , (35d) a3 " pr2gtq2 ́ 2a1r1 ` a2pa2 ` 2qs 2a1a2 ˇˇˇ r"rH , (35e) b3 " pr2gBq2 2b1b2 ˇˇˇˇ r"rH ́ pb2 ` 1q , (35f) where the prime indicates the derivative with respect to r. Naturally, in Eq. (35) we can easily extend to higher orders if needed. Finally, we remark that the terms a1, a2, a3 . . . do not depend on εt 1 and εt 2, whereas the terms b1, b2, b3 . . . follow a more complicated expression, which we chose not to display here. In the simplest case when εt 3 ‰ 0 and εt n " 0 for n ą 3, we have a2 " 0 and the approximant for the function Nprq reproduces it exactly. This illustrates how the RZ coefficients can be related to the JP ones. The inverse map can be achieved by inverting the linear system (35). 4. RZ to EMD (near horizon) parametrizations The final map to consider is between the RZ and EMD parametrizations. Throughout the calculations we make use of the results from Sec. III 3. By employing Eqs. (22) and (28b), we obtain the following two relations: gtprq " r « 8 ÿ n"1 hpnq H n! pr ́ rHqn ́1 ff ́ 1 , (36a) grprq " 1 ř8 n"1 f pnq H n! pr ́ rHqn ́1 ́ 1 . (36b) We can now exploit Eq. (35) to express the RZ coefficients in terms of the derivatives of gtprq and grprq, which, in turn, are related to the EMD coefficients. We can obtain the direct link between the first-order coefficients pθ2, ξ2q and pa0, a1, b0, b1q of the EMD and RZ parametrizations, respectively, by using hp1q H and f p1q H (cf. Eqs. (24) and (28a)) and hprq " Nprq2 and fprq " Nprq2 Bprq2 . (37) In order to determine the relation between the coefficients, let us first define y " r{p2Mq and then expand the RZ metric functions up to the first order in y ́ yH: Npyq2 " p3 ` a0 ` a1qyH ́ 2 y2 H py ́ yHq ` Opy ́ yHq2 , (38a) Npyq2 Bpyq2 " p3 ` a0 ` a1qyH ́ 2 p1 ` b0 ` b1q2 y2 H py ́ yHq ` Opy ́ yHq2 . (38b) Comparing these expansions with the first derivatives of Eq. (17) and solving for ξ2 and θ2, we obtain ξ2 " p1 ` εqp1 ` A ́ 2εqpBpB ` 2q ́ A ` 2εq 4p1 ` Bq4 , (39a) θ2 " p1 ` εqp2ε ́ Aq 8 ˆ 1 ` |1 ` 2A ́ BpB ` 2q ́ 4ε| p1 ` Bq2 ̇ , (39b) where A :" a0 ` a1 and B :" b0 ` b1. It is worth noticing that the parameters ξ2 and θ2 depend on the RZ parameters only through the combinations A and B. Moreover, the coefficients an and bn with n ą 1 do not contribute to the first-order parameters ξ2 and θ2. Additional constraints on the parameters can be obtained by observing that, in the limit b0 " b1 " 0, we have B " 0. At the horizon, Bpyq " 1 ` O ` py ́ yHq2 ̆ , so the first derivatives at the horizon in Eqs. (38) must coincide with (17a) and (17b). This implies the additional condition ξ2 " θ2. Furthermore, when setting B " 0, the absolute value in Eq. (39b) must be taken into account, which requires imposing 1 ` 2A ́ 4ε ą 0 ñ 2ε ́ a0 ́ a1 ă 1 2 . (40) The relations in Eq. (39) can be inverted to obtain the coefficients A and B as functions of θ2 and ξ2. As previously noted, if θ2 " ξ2, then the first derivatives of fpyq 7 and hpyq at yH must coincide. Moreover, in the limit ε Ñ 0 (see under Eq. (9)) the EMD and RZ parametrizations should recover the Schwarzschild BH. Therefore, we have: A " 1 2 ? 2 ξ2 " 8θ2ξ2 ` p1 ` εqpξ2 ́ θ2q ` ? 1 ` ε pξ2 ́ θ2q a 1 ` ε ́ 16ξ2 ı1{2 ́ 1 , (41a) B " 2ε ́ θ2 2ξ2 ̃ 1 ́ c 1 ́ 16ξ2 1 ` ε ̧ . (41b) This approach can be straightforwardly generalized to higher-order coefficients by following the procedure already described for the first-order terms. The expressions beyond first order become rather cumbersome, and hence we choose not to display them here. We conclude there exists a well-defined and direct correspondence between the coefficients of the EMD and RZ parametrizations. 5. RZ to EMD (large distance) parametrization The comparison between RZ and EMD parametrizations at large distance can be obtained by considering the following expansion for the RZ metric components: gRZ tt " ́1 ` 2M r ́ 4a0M 2 pε ` 1q2r2 ́ 8M 3 " ̃Ap1q ́ a0 ` ε ı pε ` 1q3r3 ` O ˆM 4 r4 ̇ , (42a) gRZ rr " 1 ` 2Mp2b0 ` ε ` 1q pε ` 1qr ` O ˆM 2 r2 ̇ . (42b) Comparing these expansions with Eq. (20), we obtain b0 " 0 and the following conditions on the first few parameters: ω1 " 4 ̃Bp1q ́ 2a0 pε ` 1q2 , γ1 " ́ 2a0 pε ` 1q2 , (43a) γ2 " ́ 2 " a0pkp1 ` εq ́ 2q ` 2p ̃Ap1q ` εq ı pε ` 1q3 , (43b) ω2 " ́ 2 pε ` 1q3 ! a0rkp1 ` εq ́ 2s ` 2 " ε ` ̃Ap1q ` p2 ́ kqp1 ` εq ̃Bp1q ` 2 ̃B1p1q ı) , (43c) where we note that the full continued fractions ̃Ap1q and ̃Bp1q (cf. Eq. (10)) and their derivatives appear, with the prime denoting differentiation with respect to x defined in Sec. II B. As mentioned previously, the coefficients of RZ and EMD parametrizations at large distance are in one-to-one correspondence. IV. EIKONAL LIMIT As an application of the mapping among the three parametrizations, we now consider the eikonal limit of QNMs within the framework of BH perturbation theory. QNMs are the characteristic complex oscillation modes of perturbed compact objects [44]; their real and imaginary parts correspond to the oscillation frequency and the inverse damping time, respectively. For a static and spherically symmetric BH metric (1), the QNMs ω satisfy the following Schr ̈odinger-like equation: d2Ψ dr2 ̊ ` pω2 ́ V prqqΨ " 0 , (44) with outgoing boundary conditions e ̆iωr ̊ at the BH event horizon and at spatial infinity, expressed in terms of the tortoise coordinate r ̊, defined through dr ̊{dr " a ́grr{gtt. Here, V prq denotes the perturbation potential, which depends on the spin of the perturbing field Ψ. Within the QNM spectrum, a notable role is played by the eikonal limit, corresponding to the regime of very large angular momentum, or multipole index, l" 1. In this limit, the QNM frequency reads ωnl" Ωl ́ i ˆ n ` 1 2 ̇ |λ| , (45) where n P N is the overtone number, Ωis the orbital frequency of light rays at the photon-sphere radius rps, and λ is the Lyapunov exponent [34]. The eikonal limit establishes a correspondence between the QNMs and the properties of the photon sphere, under the following two criteria [45]: (1) the perturbation potential is positive definite (to avoid instabilities), single peaked, and decays to zero at the boundaries; and (2) the perturbation is a test scalar field or other field minimally coupled to gravity. This implies that the eikonal QNMs associated with gravitational perturbations may not, in general, be directly related to the properties of the unstable photon orbit. For this reason, we will restrict our analysis to perturbations of test scalar and electromagnetic fields. Given a static and spherically symmetric spacetime in Eq. (1), the photon dynamics can be described using the effective potential U 2prq " ́gttprq{r2. The radius, orbital frequency [46], and Lyapunov exponent of the photon sphere are then given by [34]3 rps g1 ttprpsq " 2 gttprpsq , (46a) Ω2 " U 2prpsq " ́gttprq r2 ˇˇˇˇ r"rps , (46b) λ2 " ́ r2 ps 2 grrprpsq d2U 2prq dr2 ˇˇˇˇ r"rps , (46c) 3 Note the change of the metric signature with respect to Eqs. (35), (37), and (40) reported in Ref. [34]. 8 respectively. We note that Ωis closely related to the radius of the BH shadow through bsh :" Uprpsq ́1 " Ω ́1 [39].4 A. Eikonal limit in different parametrizations In this subsection, we investigate the eikonal limit within the three BH parametrizations introduced in Sec. II. For a generic parametrization, the corresponding set of infinite coefficients is treated as independent. However, once a specific model is considered, these coefficients become functions of a finite number of free parameters. In the next subsection, we examine small deviations from the Schwarzschild solution using models specified by the mass M and a single additional l, which quantifies the departure from the GR geometry. This introduces a displacement of the BH horizon radius rH from its Schwarzschild value 2M. To describe this effect, we adopt the perturbative parameter ε (see below Eq. (9)), which allows for a Taylor expansion of the quantities that enter the eikonal QNM limit. Other choices, such as εt 1 in the JP parametrization, are equally possible. We then expand the physically relevant quantities in Eq. (46) in the eikonal limit up to linear order in ε for each parametrization. Higher-order terms in ε are subsequently used to estimate the truncation error. 1. Rezzolla-Zhidenko parametrization We start with the RZ parametrization, where the ε parameter emerges naturally. We assume the coefficients an and bn admit a power-series expansion in ε of the form anpεq " ÿ p"0 An,p εp and bnpεq " ÿ p"0 Bn,p εp , (47) for n ě 0 and A0,0 " A1,0 " B0,0 " B1,0 " 0. The last condition ensures that the leading contributions to the metric functions from a0pεq, a1pεq, b0pεq, and b1pεq vanish in the Schwarzschild limit ε Ñ 0, thereby guaranteeing that the deformations associated with Eqs. (9a) and (9b) are continuously switched off. The values of the coefficients An,p and Bn,p depend on the specific model under consideration. In general, these coefficients appear in the series expansion of the quantities in Eqs. (46) through nonlinear combinations at each order in ε. These expressions simplify dramatically 4 In Ref. [34], the authors use a different potential Vrprq instead of U2prq. when A2,0 " B2,0 " 0: rRZ ps M " 3 ́ 4 9 p5 ` A0,1 ` A1,1q ε ` O ` ε2 ̆ , (48a) ΩRZM " 1 3 ? 3 ` 2 81 ? 3 p6 ` 3A0,1 ` 2A1,1q ε ` O ` ε2 ̆ , (48b) λRZM " 1 3 ? 3 ` 2 81 ? 3 p ́4 ` 7A0,1 ` 4A1,1 `9B0,1 ́ 6B1,1q ε ` O ` ε2 ̆ . (48c) In the following sections, we restrict our attention to BH metrics satisfying the aforementioned condition. 2. Johannsen-Psaltis parametrization Following the same reasoning as for the RZ parametrization, we note that the deformation functions gtprq and grprq must vanish in the Schwarzschild limit. This implies that all JP parameters tεt n, εr nuně1 scale homogeneously with ε. Indeed, as shown in Eq. (34), we know that εt 1 " ́2 ε exactly. Therefore, for ε Ñ 0 and the scaling behavior of Eq. (47), we generally have εr 1, εt 2 „ O ` εk ̆ , with k ě 1. Hence, we assume the following general power series expansion εt npεq " ÿ p"1 pεt nqp εp for n ě 2 , (49a) εr npεq " ÿ p"1 pεr nqp εp for n ě 1 , (49b) which could be equivalently read as a power series in εt 1. The expressions for the coefficients pεt nqp and pεr nqp are determined by the model under investigation. Inserting the expansions (49) into Eq. (46) written in the JP parametrization (truncated for brevity at εt 4 and εr 4), we obtain rJP ps M " 3 ́ ˆ8 3 ` pεt 2q1 9 ` pεt 3q1 18 ` 2pεt 4q1 81 ̇ ε ` O ` ε2 ̆ , (50a) ΩJPM " 1 3 ? 3 ` ˆ2 9 ` pεt 2q1 54 ` pεt 3q1 162 ` 2pεt 4q1 486 ̇ ε ? 3 ` O ` ε2 ̆ , (50b) λJPM " 1 3 ? 3 ` ˆ 5 27 ` 2pεt 2q1 81 ` pεt 3q1 162 ` pεt 4q1 1458 ́ pεr 1q1 18 ́ pεr 2q1 54 ́ pεr 3q1 162 ́ pεr 4q1 486 ̇ ε ? 3 ` O ` ε2 ̆ . (50c) As expected from Eqs. (35) and (48), all JP parameters enter linearly at the first order in ε. 9 3. EMD (near horizon) parametrization As shown in [40], the EMD parametrization near the horizon can be extended up to the photon sphere by employing the Pad ́e approximation [47] (see Appendix B for details). In this case, the position of the event horizon is written as rH " 2Mp1 ` cq , (51) and hence, the expressions for the quantities reported in Eq. (46) can be compactly written as rEMDpnhq ps M " σ0 ` σ1 c ` O ` c2 ̆ , (52a) ΩEMDpnhqM " η0 ` η1 c ` O ` c2 ̆ , (52b) λEMDpnhqM " Υ0 ` Υ1 c ` O ` c2 ̆ . (52c) where c quantifies the departure of rH from the Schwarzschild radius and is related to the RZ parameter as c " ́ ε ε ` 1 . (53) The coefficients are linear combinations of the parameters tθ2n, ξ2nuně1 and their explicit form depends on the order of the employed Pad ́e approximant, see Table I in [40] and Table III in Appendix B. A Pad ́e approximation of order pN, Mq involves a total of N ` M parameters. We note that the effective parameters tθ2n, ξ2nuně1 must still be expanded in power series of ε, as we will see in the subsection dedicated to specific BH models. 4. EMD (at large distance) parametrization The EMD parametrization at large distances must recover the Schwarzschild metric when the deformation parameters tγi, ωiuiě1 vanish. However, the position of the event horizon does not explicitly appear as an input, and therefore there is no direct connection to the ε parameter. Nevertheless, by inspecting the relations Eqs. (43) and ansatz (47) (or equivalently by considering Eq. (31) and ansatz (49)), we find that tγi, ωiuiě1 indeed admit a power series expansion in ε. Therefore, we assume that tγi, ωiuiě1 can be expanded similarly to the previous cases, namely γnpεq " ÿ p"1 γn,p εp and ωnpεq " ÿ p"1 ωn,p εp , (54) where tγn,p, ωn,puně1, pě1 assume different values depending on the model under consideration. Using the metric functions in Eqs. (20), we can express Eqs. (46) as rEMDpldq ps M " 3 ` 1 9 " p15 ́ 3k ` 2k2qγ1,1 ` p5 ́ 4kqγ2,1 ` 2γ3,1 ı ε ` O ` ε2 ̆ , (55a) ΩEMDpldqM " 1 3 ? 3 ` 1 162 ? 3 " p ́2k2 ` 4k ́ 21qγ1,1 ` p4k ́ 6qγ2,2 ́ 2γ3,1 ı ε ` O ` ε2 ̆ , (55b) λEMDpldqM " 1 3 ? 3 ` 1 162 ? 3 " p6k2 ́ 4k ` 21qγ1,1 ` 2p5k ́ 6qγ2,2 ` 6γ3,1 ́ p2k2 ́ 4k ` 21qω1,1 ` p4k ́ 6qω2,1 ́ 2ω3,1 ı ε ` O ` ε2 ̆ , (55c) Above we kept the dependence on the gauge parameter k to keep the expressions as general as possible. B. Application to BH models As an application of the formalism developed in the previous subsection, let us consider the following BH spacetimes: Hayward [48], Bardeen [49], and SimpsonVisser II [50]. These are regular BH candidates, which can be effectively described as deformations of the Schwarzschild geometry and satisfy the condition gtt grr " ́1. Their gtt metric components read ́gH ttprq " 1 ́ 2Mr2 r3 ` 2M l2 H , (56a) ́gB ttprq " 1 ́ 2Mr2 pr2 ` l2 Bq3{2 , (56b) ́gSV2 tt prq " 1 ́ 2M r exp " ́lSV2 r * . (56c) The regularization length-scale parameters plH, lB, lSV2q have different physical interpretations in each model, are allowed to vary over a finite range, and can be expressed in terms of ε as follows: lH " 2M c ε p1 ` εq3 , (57a) lB " 2M a p1 ` εq2{3 ́ 1 1 ` ε , (57b) lSV2 " 2M logp1 ` εq 1 ` ε . (57c) We analytically expand the quantities in Eqs. (46) around ε " 0 up to quadratic order for each BH model and compare the results with those obtained in the Rezzolla-Zhidenko, Johannsen-Psaltis, near-horizon EMD, and large-distance EMD parametrizations, fixing the number of free parameters to four in Eqs. (48), (50), (52), and (55). The outcome of this comparison is summarized in Table I. We observe that, at linear order, all 10 BH models Parametrizations rps M ́ 3 ΩM ́ 1 3 ? 3 λM ́ 1 3 ? 3 Hayward From the model ́16 9 ε ` 80 27ε2 8 81 ? 3ε ́ 40 243 ? 3ε2 ́ 16 81 ? 3ε ` 16 243 ? 3ε2 Rezzolla-Zhidenko ́16 9 ε ` 28892 9801 ε2 8 81 ? 3ε ́ 4012 24057 ? 3ε2 ́ 16 81 ? 3ε ` 173960 2910897 ? 3ε2 Johannsen-Psaltis ́16 9 ε ` 32 9 ε2 8 81 ? 3ε ́ 440 2187 ? 3ε2 ́ 16 81 ? 3ε ` 32 243 ? 3ε2 EMD near horizon ́16 9 ε ` 799 243ε2 8 81 ? 3ε ́ 110 729 ? 3ε2 ́ 16 81 ? 3ε ́ 107 2187 ? 3ε2 EMD large distance ́16 9 ε ` 176 81 ε2 8 81 ? 3ε ́ 296 2187 ? 3ε2 ́ 16 81 ? 3ε ́ 368 2187 ? 3ε2 Bardeen From the model ́20 9 ε ` 650 243ε2 4 27 ? 3ε ́ 98 729 ? 3ε2 ́ 8 81 ? 3ε ́ 68 729 ? 3ε2 Rezzolla-Zhidenko ́20 9 ε ` 650 243ε2 4 27 ? 3ε ́ 98 729 ? 3ε2 ́ 8 81 ? 3ε ́ 68 729 ? 3ε2 Johannsen-Psaltis ́20 9 ε ` 650 243ε2 4 27 ? 3ε ́ 98 729 ? 3ε2 ́ 8 81 ? 3ε ́ 68 729 ? 3ε2 EMD near horizon ́20 9 ε ` 652 243ε2 4 27 ? 3ε ́ 391 2916 ? 3ε2 ́ 8 81 ? 3ε ́ 421 4374 ? 3ε2 EMD large distance ́20 9 ε ` 370 243ε2 4 27 ? 3ε ́ 98 729 ? 3ε2 ́ 8 81 ? 3ε ́ 68 729 ? 3ε2 Simpson-Visser II All parametrizations ́8 3ε ` 74 27ε2 2 9 ? 3ε ́ 7 81 ? 3ε2 2 27 ? 3ε ́ 11 81 ? 3ε2 TABLE I: Expressions of the photon sphere radius rps, orbital frequency Ω, and Lyapunov exponent λ. The values are rescaled by the BH mass M, shifted with respect to the corresponding Schwarzschild result, and computed up to quadratic order in ε in the case of Hayward, Bardeen, and Simpson-Visser II models using four parameters in each parametrization. For the last BH spacetime, we report only the expressions for the exact metric, as they are identical across all parametrizations. parametrizations reproduce the same result for all physical quantities in all BH models. Aside from the Simpson-Visser II case, where all parametrizations trivially yield identical expansions, no single parametrization consistently performs better than the others across all observables and BH models. A complementary analysis is presented in Table II, where, instead of fixing the number of parameters a priori, we set the desired tolerance and let the required number of parameters vary to match the exact expressions. In the case of the EMD near-horizon parametrization, this procedure coincides with a Pad ́e approximation of order pN, 3q, where the number of free parameters is controlled by the integer N. Also in this setting, no parametrization emerges as universally favored across all BH models. Finally, in Fig. 1 we illustrate the convergence of the linear and quadratic coefficients for the Bardeen BH as a representative example. Similar behavior is observed for all other BH models. The plots highlight that all parametrizations perform on a comparable footing. Therefore, the choice of parametrization should be guided by the desired accuracy and the number of parameters one is willing to introduce. This has to be considered on a case-by-case basis for each physical observable and BH model. V. DISCUSSION AND CONCLUSIONS This work begins by examining three modelindependent parametrizations of BH spacetimes: Johannsen-Psaltis, which describes deformations as quadrupolar deviations from GR; Rezzolla-Zhidenko, which introduces departures from the Schwarzschild solution and employs Pad ́e-like continued-fraction expansions valid far from the event horizon; and EMD, which provides local BH metric descriptions in terms of physical quantities such as the physical distance. We then establish, for the first time, explicit mappings among these three frameworks, thereby offering a unified perspective on agnostic parametrizations of deformed BHs. As a relevant application, we investigate the QNMs of 11 BH models Hayward Bardeen Simpson-Visser II order ε ε2 ε ε2 ε ε2 Rezzolla-Zhidenko 2 7 2 4 1 1 Johannsen-Psaltis 3 6 2 4 1 2 EMD near horizon 4 ą 7 4 ą 7 4 4 EMD large distance 3 6 2 4 1 2 TABLE II: Number of parameters required to reproduce the exact results for the three observables at orders ε and ε2. 2 3 4 5 6 2.400 2.375 2.350 2.325 2.300 2.275 2.250 2.225 O( ) rps 2 3 4 5 6 0.081 0.082 0.083 0.084 0.085 2 3 4 5 6 0.10 0.09 0.08 0.07 0.06 2 3 4 5 6 number of parameters 1.5 2.0 2.5 3.0 3.5 O( 2) 2 3 4 5 6 number of parameters 0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 2 3 4 5 6 number of parameters 0.16 0.14 0.12 0.10 0.08 0.06 Bardeen Rezzolla-Zhidenko Johannsen-Psalits EMD near horizon EMD far region FIG. 1: Comparison of the convergence to the linear coefficient (top row) and quadratic coefficient (bottom row) for the Bardeen BH model, varying the number of the related parameters. The horizontal dashed black line is the value computed from the model, reported in Table I. For the near-horizon EMD with four parameters, two Pad ́e approximants were used: (2,2) and (1,3), the latter being more accurate. BHs in the eikonal limit. We analyse the QNM spectrum across the different parametrizations, considering small deviations from the Schwarzschild case by treating the deformation parameter (ε) as a perturbative quantity. This allows us to expand the effective coefficients of each parametrization in power series of ε. To demonstrate the effectiveness of our framework, we apply it to three regular BH models: Hayward, Bardeen, and Simpson-Visser II. We summarize our results in several tables and figures carrying different information: in Table I we report the expansions up to quadratic order in ε using only four parameters for each parametrization and for each BH model; in Table II we fix the accuracy and let the number of coefficients vary freely; in Fig. 1 we show the convergence of the different expansions in terms of the number of retained parameters and also check the values of the attained tolerances. Across all analyses, we find that the three parametrizations are fundamentally equivalent: none systematically outperforms the others. Superiority is context-dependent and emerges only once the target accuracy, the observable under consideration, and the number of parameters are specified within a given BH model. Our astrophysical case study supports a democratic perspective: parametrizations should be viewed as complementary tools rather than competing prescriptions. This philosophy naturally paves the way for developing new parametrizations tailored to specific observables or regimes. Moreover, recent works [19, 51] have emphasized the usefulness of proper-time EMD parametrizations in capturing effective quantum corrections near the horizon. In this light, the mappings established in our work offer promising opportunities to translate such quantum-gravity-inspired deformations across different parametrization schemes. As a natural extension, the same methodology could be applied to other problems in BH physics and generalized to stationary and axially symmetric spacetimes. 12 ACKNOWLEDGEMENTS V. De Falco is grateful to Gruppo Nazionale di Fisica Matematica of Istituto Nazionale di Alta Matematica for support. C. De Simone acknowledges the support of INFN sez. di Napoli, iniziativa specifica QGSKY. M. Del Piano acknowledges support from INFN sez. di Napoli, iniziativa specifica ST&FI and expresses gratitude to the Southern Denmark University, the Danish Institute for Advanced Study, and the Quantum Theory Center in Odense (Denmark) for hosting during the initial stages of the work. The work of F. Sannino and M. Damia Paciarini is partially supported by the Carlsberg Foundation, grant CF22-0922. This work contributes to COST Action CA23130 - Bridging high and low energies in search of quantum gravity (BridgeQG). Appendix A: Treatment of logarithmic functions for EMD at large distance In order to determine the metric functions, we could start by assuming that Φpdq and Ψpdq can be expanded in power series of 1{d. However, such an approach will lead to logarithmic terms in the asymptotic expansions of hprq and fprq. Therefore, to avoid that consequence, we can reabsorb the logarithms in Φpdq and Ψpdq by defining them as Φpdq " 2M « 1 ` 8 ÿ n"1 n ́1 ÿ m"0 Ωn,mM n dn logm ˆ d 2M ̇ff , Ψpdq " 2M « 1 ` 8 ÿ n"1 n ́1 ÿ m"0 Γn,mM n dn logm ˆ d 2M ̇ff , (A1) where Ωn,m and Γn,m are real constants and ωn " Ωn,0 and γn " Γn,0. We consider the post-Newtonian (PN) expansion M{r ! 1, (the weak-field limit M{r ! 1 is related to low velocity limit v{c ! 1 via the virial theorem [52], namely M{r „ v2{c2). In this regime, we have 1{pridjq „ 1{pri`jq and the function dprq admits the logarithmic corrections: pM{rqj logkpr{Mq, with tk, ju P N0 and 0 ă k ď j ́ 1. We PN-expand Eq. (19) via Eq. (A1), which gives [17, 20] dprq M " r M ` log ́ r 2M ̄ ` k ́ ˆ3 2 ` ω1 ̇ M r ́ „5 4 ` ω1p5 ́ 2kq 4 ` ω2 ` Ω2,1 2 ȷ M 2 r2 ` M 2pω1 ́ Ω2,1q 2r2 log ́ r 2M ̄ ́ „35 24 ` p74 ́ 21k ` 9k2qω1 27 ` 45ω2 1 ` 6p7 ́ 6kqω2 ` 2p8 ` 3kqΩ2,1 54 ` 9ω3 ` 3Ω3,1 ` 2Ω3,2 27 ȷ M 3 r3 ` « p7 ́ 6kqω1 ` 6ω2 ́ 2p4 ́ 3kqΩ2,1 ́ 3Ω3,1 ́ 2Ω3,2 ff M 3 9r3 log ́ r 2M ̄ ́ rω1 ` 2Ω2,1 ́ Ω3,2sM 3 3r3 log2 ́ r 2M ̄ ` O ˆM 4 r4 ̇ , (A2) where k is a real integration constant, kept as a free parameter. It is important to note that k plays the role of the gauge for d, as its value determines the position at which d " 0. Inserting Eq. (A2) into the metric functions modified with the series in Eq. (A1), we obtain the conditions to balance and cancel out (up to a given order) the logarithmic functions in Eq. (20), which requires Ω2,1 " ω1 , Ω3,1 " ́ω1 ` 2ω2 , Ω3,2 " ω1 , Ω4,3 " ω1 , Ω4,2 " ́5 2ω1 ` 6 2ω2 , Ω4,1 " ω1 ́ 2ω2 ` 3ω3 , Ω5,4 " ω1 , Ω5,3 " ́13 3 ω1 ` 12 3 ω2 , Ω5,2 " 9 2ω1 ́ 14 2 ω2 ` 12 2 ω3 , Ω5,1 " ́ω1 ` 2ω2 ́ 3ω3 ` 4ω4 , for the first few terms. Although it is difficult to generalize the above relations, they can be computed to arbitrary order and applied to the set tΓn,mu0ďmďn ́1 ně1 . Using the above conditions, we are able to obtain the metric functions as in Eq. (20) and eliminate the logarithmic functions appearing in Eq. (A2). Appendix B: Extending the EMD near horizon in the eikonal limit In the EMD framework near the horizon, the potential U2prq can be expressed as a power series around rH: U 2prq " 8 ÿ n"1 vnpr ́ rHqn, (B1) where vn depend on tθ2ku and tξ2ku (see [40] for the explicit form of these coefficients). It has been shown in [40] that, by employing Pad ́e approximations at a sufficiently high order pN, Mq, the convergence radius of the series increases 13 N M η0 η1 2 1 1 4 `? 3 ́ 1 ̆ ́1 4 `? 3 ́ 1 ̆ ` 1 135 ` 61 ́ 68 ? 3 ̆ θ2 c ` 2 9 ` 2 ? 3 ́ 7 ̆ θ4 c ` 8 3 `? 3 ́ 2 ̆ θ6 c 2 2 1 4 a 9 ? 3 ́ 15 ́1 4 a 9 ? 3 ́ 15 ́ 1 630 c 1274659 ? 3 ́ 6386663 3 θ2 c ́ 11 15 c 1 6 ` 627 ? 3 ́ 1075 ̆θ4 c ́ 2444 ? 2 b` 14 ? 3 ́ 9 ̆ ` 85675 ? 3 ` 148401 ̆ θ6 c ́ 16 c 51 ? 3 ` 265 3 θ8 c 1 3 1 3 ? 3 ́ 1 3 ? 3 ́ 19676 25515 ? 3 θ2 c ́ 656 405 ? 3 θ4 c ́ 64 27 ? 3 θ6 c ́ 128 81 ? 3 θ8 c 2 3 1 3 ? 3 ́ 1 3 ? 3 ́ 98837 127575 ? 3 θ2 c ́ 136096 76545 ? 3 θ4 c ́ 4384 1215 ? 3 θ6 c ́ 1216 243 ? 3 θ8 c ́ 256 81 ? 3 θ10 c 3 3 1 3 ? 3 ́ 1 3 ? 3 ́ 652294 841995 ? 3 θ2 c ́ 2066276 1148175 ? 3 θ4 c ́ 62528 15309 ? 3 θ6 c ́ 29152 3645 ? 3 θ8 c ́ 2560 243 ? 3 θ10 c ́ 512 81 ? 3 θ12 c N M Υ0 Υ1 2 1 `? 3 ́ 1 ̆ `? 3 ` 1 ̆2 16 4? 3 ́ `? 3 ́ 1 ̆ `? 3 ` 1 ̆2 16 4? 3 ` ` 13 ? 3 ́ 116 ̆ `? 3 ` 1 ̆ 180 4? 3 θ2 c ` `? 3 ́ 2 ̆ `? 3 ` 1 ̆ 2 4? 3 θ4 c ́2 `? 3 ́ 2 ̆ `? 3 ` 1 ̆ 4? 3 θ6 c 2 2 1 24 c 1 2 ` 963 ́ 529 ? 3 ̆ ́ 1 24 c 1 2 ` 963 ́ 529 ? 3 ̆ ́ a 187519473 ́ 102837005 ? 3 3780 θ2 c ` 1 60 a 162387 ́ 92399 ? 3θ4 c + 78 ` 9 ? 3 ` 23 ̆ a 61617 ? 3 ` 106731 θ6 c ` 4796 c 2 241053 ? 3 ` 417609 θ8 c 1 3 1 3 ? 3 ́ 1 3 ? 3 ́ 35396 25515 ? 3 θ2 c ` 112 1215 ? 3 θ4 c ` 448 81 ? 3 θ6 c ` 512 81 ? 3 θ8 c 2 3 1 3 ? 3 ́ 1 3 ? 3 ́ 171496 127575 ? 3 θ2 c ` 10160 5103 ? 3 θ4 c ` 2752 135 ? 3 θ6 c ` 1280 27 ? 3 θ8 c ` 1024 27 ? 3 θ10 c 3 1 1 3 ? 3 ́ 1 3 ? 3 ́ 5662841 4209975 ? 3 θ2 c ` 2857228 1148175 ? 3 θ4 c ` 2398688 76545 ? 3 θ6 c ` 423776 3645 ? 3 θ8 c ` 50432 243 ? 3 θ10 c ` 11776 81 ? 3 θ12 c TABLE III: Orbital frequency pη0, η1q and Lyapunov exponent pΥ0, Υ1q coefficients of the photon sphere obtained from Eqs. (46b) and (46c) in terms of the Pad ́e approximants of the potential U2 N,M. significantly up to the photon sphere location. This approach allows for the computation of the BH shadow radius bsh, orbital frequency Ω, and Lyapunov exponent λ. Assuming that the position of the BH horizon is defined by Eq. (51), for small deviations c ! 1 from the Schwarzschild horizon, we can write Ωand λ in mass units M as: ΩM " η0 ` η1 c ` Opc2q, (B2a) λ M " Υ0 ` Υ1 c ` Opc2q. (B2b) The coefficients pη0, η1q and pΥ0, Υ1q are reported in Table III and depend on the order of the Pad ́e approximation pN, Mq. The Schwarzschild limit is already recovered at order p1, 3q. From Eq. (B2), the eikonal QNMs can be directly reconstructed at any order pN, Mq of the Pad ́e approximation using Eq. (45). 14 The relation between bsh and Ωcan be further exploited to investigate the large-N limit in the Pad ́e approximation of order pN, 3q. First, let us recall the expression of the shadow radius in terms of c from [40]: bsh M " τ0 ` τ1 c ` Opc2q, (B3) as well as the explicit dependence of τ1 on c and on the order N of the Pad ́e approximation: τ1pNq " 3 ? 3 ` N`3 ÿ k"1 β2kpNqθ2k c with β2kpNq " B B θ2k 1 UN,3 ˇˇˇˇ prpsqN,3 , (B4) where prpsqN,3 and UN,3 correspond to the photon sphere radius and potential computed at the order pN, 3q of the Pad ́e approximation, respectively. Similarly, the coefficient η1 of the photon sphere frequency can be expressed as η1pNq " ́ 1 3 ? 3 ` N`3 ÿ k"1 γ2kpNqθ2k c with γ2kpNq " B UN,3 B θ2k ˇˇˇˇ prpsqN,3 . (B5) Recalling that, at the lowest order, UN,3 " 1{p3 ? 3q, the coefficients γ2kpNq and β2kpNq are related by β2kpNq " B B θ2k 1 UN,3 ˇˇˇˇ prpsqN,3 " ́ 1 U 2 N,3 γ2kpNq " ́p3 ? 3q2γ2kpNq, (B6) and the asymptotic behavior of γ2kpNq can be derived from the N Ñ 8 limit of β2kpNq obtained in [40] lim NÑ8 β2kpNq 3 ? 3 Ñ s2k, s „ 1.5245. (B7) This result is compatible with Fig. 2, where the first few coefficients γ2k have been computed up to N " 9. Moreover, the asymptotic limit of η1pNq becomes lim NÑ8 c η1pNq " ́ c 3 ? 3 ́ 8 ÿ k"1 s2k 3 ? 3θ2k " ́ c 3 ? 3 ́ 1 3 ? 3 1 2M rΨp2Msq ́ rHs, (B8) and substituting it into the full expression of Ω" η0 ` η1c, we obtain lim NÑ8 Ω" lim NÑ8pη0 ` η1cq " 2 3 ? 3 ́ 1 3 ? 3 1 2M Ψp2Msq. (B9) In the case of the Lyapunov exponent, the asymptotic N limit is more intricate, since λ depends on both the photon sphere radius and the second-order derivatives of the potential. However, by analogy with Ω, we adopt a similar ansatz for Υ1pNq, which can be written as Υ1pNq " 1 3 ? 3 ` N`3 ÿ k"1 δ2kpNqθ2k c , (B10) where δ2k is a suitable set of numerical coefficients. In general, δ2k depend on the order of the Pad ́e approximation and are related to the derivative of the Lyapunov exponent with respect to the parameters θ2k, evaluated at the radius of the photon sphere. The coefficients δ2k computed up to k " 7 and N " 9 are shown in Fig. 2. Although the individual values of δ2k seem to converge to a constant value, it is not clear whether resummation similar to the BH shadow radius and photon sphere frequency is possible. We note that the coefficient δ8 reaches the largest value. Finally, we remark on the second order ε expansion of the EMD near-horizon parametrization. Since the ε dependence is encapsulated in the c coefficient in Eq. (51), it is important to study how this relation changes at second order. We have thus checked that for all BH models of interest, Eq. (51) does not receive corrections at O ` c2 ̆ . Therefore, rH acquires O ` ε2 ̆ contributions only from the expansion of c " ́ε{p1 ` εq. Moreover, the second-order expansion involves significantly more terms than the first-order one, namely linear and quadratic terms in tθnu, as well as possible mixing among the linear coefficients. 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Parametrized quasinormal mode framework for modified Teukolsky equations. Phys. Rev. D, 110(10):104007, 2024. [26] Ameya Chavda, Macarena Lagos, and Lam Hui. The impact of initial conditions on quasi-normal modes. JCAP, 07:084, 2025. [27] R. A. Konoplya and A. Zhidenko. Quasinormal modes of black holes: From astrophysics to string theory. Rev. Mod. Phys., 83:793-836, 2011. [28] Edgardo Franzin, Stefano Liberati, and Vania Vellucci. From regular black holes to horizonless objects: quasinormal modes, instabilities and spectroscopy. JCAP, 01: 020, 2024. [29] Vitor Cardoso, Masashi Kimura, Andrea Maselli, Emanuele Berti, Caio F. B. Macedo, and Ryan McManus. Parametrized black hole quasinormal ringdown: Decoupled equations for nonrotating black holes. Phys. Rev. D, 99(10):104077, 2019. [30] Satoshi Okuzumi, Kunihito Ioka, and Masa-aki Sakagami. Possible Discovery of Nonlinear Tail and Quasinormal Modes in Black Hole Ringdown. Phys. Rev. D, 77:124018, 2008. [31] Ermis Mitsou. 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2510.14710
|
MCbiF: Measuring Topological Autocorrelation in Multiscale
Clusterings via 2-Parameter Persistent Homology
Juni Schindler∗and
Mauricio Barahona†
Department of Mathematics, Imperial College London, UK
Abstract
Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of
coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily
hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools
from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter
filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The
MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram
for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF
yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise
the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH
captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures
higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use
of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF
feature maps outperform information-based baseline features on both regression and classification tasks
on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to
real-world data to measure non-hierarchies in wild mice social grouping patterns across time.
Keywords: topological data analysis, multiparameter persistent homology, multiscale clustering, non-
hierarchical clustering, Sankey diagrams
1
Introduction
In many applications, datasets possess an intrinsic multiscale structure, whereby meaningful descriptions
exist at different scales, i.e., at different resolutions or levels of coarseness. Think, for instance, of the
multi-resolution structure in commuter mobility patterns [Alessandretti et al., 2020, Schindler et al., 2023],
communities in social networks [Beguerisse-Díaz et al., 2017] and thematic groups of documents [Blei et al.,
2003, Grootendorst, 2022]; the subgroupings in single-cell data [Hoekzema et al., 2022] or phylogenetic
trees [Chan et al., 2013]; and the functional substructures in proteins [Delvenne et al., 2010, Delmotte et al.,
2011]. In such cases, the natural description of the dataset goes beyond a single clustering and consists of
a multi-resolution sequence of partitions across scales parametrised by a scale parameter t. Traditionally,
multiscale descriptions have emerged from hierarchical clustering, where t corresponds to the depth of the
dendrogram [Carlsson and Mémoli, 2010, Murtagh and Contreras, 2012]. However, in many important
real-world applications, the data structure is multiscale, yet non-hierarchical. Examples include temporal
clustering, where t corresponds to physical time [Rosvall and Bergstrom, 2010, Liechti and Bonhoeffer, 2020,
Bovet et al., 2022]; topic modelling and document classification, where t captures the coarseness of the topic
groupings [Altuncu et al., 2019, Fukuyama et al., 2023, Liu et al., 2025]; and generic multiscale clusterings
for data that result from exploiting a diffusion on the data geometry, where t is the increasing time horizon
of the diffusion [Coifman et al., 2005, Azran and Ghahramani, 2006, Lambiotte et al., 2014].
A natural problem is then how to analyse and compare non-hierarchical multi-resolution sequences of
partitions that are organised by the scale t. Here we address this question from the perspective of topological
data analysis [Carlsson and Zomorodian, 2009, Carlsson et al., 2009, Botnan and Lesnick, 2023] by introducing
the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that
encodes the patterns of cluster intersections across all scales.
∗Corresponding author: juni.schindler19@imperial.ac.uk,
ORCID ID: 0000-0002-8728-9286
†Corresponding author: m.barahona@imperial.ac.uk,
ORCID ID: 0000-0002-1089-5675
1
arXiv:2510.14710v1 [math.AT] 16 Oct 2025
Problem definition.
A partition π of a finite set X “ tx1, x2, ..., xNu is a collection of mutually exclusive
subsets Ci Ď X (here called clusters) that cover X, i.e., π “ tC1, . . . , Ccu such that X “ Ťc
i“1 Ci, and
Ci
Ş Cj “ H, @i ‰ j. The cardinality |π| “ c is the number of clusters in π and, for notational convenience,
we use πi to denote the i-th cluster Ci of π.
Let ΠX denote the space of partitions of X. We write π ď π1 if every cluster in π is contained in a cluster
of π1. This refinement relation constitutes a partial order and leads to the partition lattice pΠX, ďq with
lower bound ˆ0 :“ ttx1u, . . . , txNuu and upper bound ˆ1 :“ tXu [Birkhoff, 1967].
Here, we consider a sequence of partitions defined as a piecewise-constant function θ : rt1, 8q Ñ ΠX, t ÞÑ
θptq P ΠX such that a partition of X is assigned to each t, and the scale index t P rt1, 8q has M change
points t1 ă t2 ă ... ă tM. In particular, θptq “ θptmq for t P rtm, tm`1q, m “ 1, . . . , M ´ 1, and θptq “ θptMq
for t P rtM, 8q. The sequence θ is called hierarchical if either θpsq ď θptq, @s ď t, or θpsq ě θptq, @s ď t.
The sequence θ is called coarse-graining if |θpsq| ě |θptq|, @s ď t. 1 Conversely, θ is called fine-graining if
|θpsq| ď |θptq|, @s ď t.
Our goal is to characterise and analyse arbitrary sequences of partitions θ, including non-hierarchical
ones, in an integrated manner, taking account of memory effects across the scale t.
Remark 1. Here, we are not concerned with the task of computing the multiscale clustering (i.e., the
sequence of partitions θ) from dataset X, for which several methods exist. Rather, we take θ as a given, and
we aim to analyse its structure.
Remark 2. This problem is distinct from consensus clustering, which aims to produce a summary partition
by combining a set of partitions obtained, e.g., from different optimisations or clustering algorithms [Strehl
and Ghosh, 2002, Vega-Pons and Ruiz-Shulcloper, 2011].
1
Input: Sequence of partitions
Output: Multiscale Clustering Bifiltration
0
2
3
4
0
1
2
3
4
(a)
(b)
Figure 1: (a) Illustration of how the MCbiF encodes the structure of a non-hierarchical sequence of partitions θ as a
bifiltration of abstract simplicial complexes Ks,t. See Example 42 for a detailed description. (b) The Hilbert functions
HFkps, tq of the MCbiF are invariants that capture the topological autocorrelation of θ: violations of the refinement
order at dimension k “ 0, and higher-order cluster inconsistencies at dimension k “ 1. The Hilbert functions can be
used as feature maps for downstream machine learning tasks.
Contributions.
To address this problem, we define the MCbiF, a bifiltration of abstract simplicial
complexes, which represents the clusters and their intersection patterns in the sequence θ for varying starting
scale s and lag t ´ s (Fig. 1). Using the machinery of multiparameter persistent homology (MPH) [Carlsson
and Zomorodian, 2009, Carlsson et al., 2009, Botnan and Lesnick, 2023], we prove that the MCbiF leads to
1Coarse-graining is equivalent to non-decreasing mean cluster size (see Remark 37 in Appendix A.1).
2
a block decomposable persistence module with stable Hilbert functions HFkps, tq, and we show that these
invariants serve as measures of the topological autocorrelation of the sequence of partitions θ across the scale t.
In particular, HFkps, tq quantifies the non-hierarchy in θ in two complementary ways: at dimension k “ 0, it
detects the lack of a maximal partition in the subposet θprs, tsq with respect to refinement, and at dimension
k “ 1, it quantifies the higher-order inconsistencies of cluster assignments across scales. In contrast, baseline
methods such as ultrametrics [Carlsson and Mémoli, 2010] or information-based measures [Meilă, 2003] are
restricted to pairwise comparisons between, respectively, elements or clusters; hence these methods cannot
detect higher-order cluster inconsistencies. Furthermore, we provide an equivalent nerve-based construction
of the MCbiF that can be interpreted as a higher-order extension of the Sankey diagram of the sequence of
partitions. In the hierarchical case, the 1-dimensional MPH of the MCbiF is trivial, and the Sankey diagram
reduces to a dendrogram such that the 0-dimensional MCbiF Hilbert function can be obtained from the
number of branches in the dendrogram.
The Hilbert functions of the MCbiF provide interpretable feature maps that can be used in downstream
machine learning tasks. In our experiments, the MCbiF feature maps outperform information-based baseline
features [Meilă, 2007] on both regression and classification tasks on non-hierarchical sequences of partitions.
We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social
grouping patterns across time [Bovet et al., 2022].
2
Related Work
Dendrograms and Ultrametrics.
A hierarchical, coarse-graining sequence θ with θpt1 “ 0q “ ˆ0 and
θptMq “ ˆ1 is called an agglomerative dendrogram, and can be represented by an acyclic rooted merge
tree [Jain et al., 1999, Carlsson and Mémoli, 2010]. One can define an ultrametric Dθ from the first-merge
times, which corresponds to the depth in the dendrogram. Carlsson and Mémoli [2010] showed that there
is a one-to-one correspondence between agglomerative dendrograms and ultrametrics, which can be used
to efficiently compare two such dendrograms via the Gromov-Hausdorff distance between the ultrametric
spaces [Mémoli et al., 2023]. When θ is non-hierarchical, however, the first-merge times no longer define the
sequence uniquely because clusters that have merged can split off again. In this case, θ cannot be represented
by a tree and Dθ does not fulfil the triangle inequality in general. Hence, ultrametrics cannot be used to
analyse and compare non-hierarchical sequences of partitions (see Section 5).
Information-based Comparison of Clusterings.
Information-based measures can be used to compare
a pair of partitions. Assuming a uniform distribution on X, one can derive probability distributions for
partitions interpreted as random variables and thus measure the information gain and loss between two
partitions using the conditional entropy (CE) or the variation of information (VI), which is a metric on
ΠX [Meilă, 2003, 2007]. See Appendix E for detailed formulas. Extending information-based measures to
more than two partitions is non-trivial. In consensus clustering, the average VI is used as a consensus index
(CI) [Vinh and Epps, 2009, Vinh et al., 2010] for multiple partitions. However, the CI is independent of the
ordering in the sequence and so cannot capture memory effects in sequences of partitions. Another limitation
of these measures is that they rely only on the joint probability between pairs of random variables, hence
higher-order cluster inconsistencies are not captured (see Section 5).
3
Background
3.1
Sankey Diagrams
Non-hierarchical sequences of partitions θ are visualised by M-layered flow graphs Spθq “ pV “ V1 Z ... Z
VM, E “ E1 Z ... Z EM´1q called Sankey diagrams [Sankey, 1898, Zarate et al., 2018], where each level
m “ 1, . . . , M corresponds to a partition and vertices Vm represent its clusters while the directed edges Em
between levels indicate the overlap between clusters:
Vm :“ tpm, iq | 1 ď i ď |θptmq|u
and
Em “ trpm, iq, pm ` 1, jqs | θptmqi X θptm`1qj ‰ Hu ,
(1)
where ru, vs P Em denotes a directed edge from u P Vm to v P Vm`1. If θ is hierarchical, the Sankey diagram
Spθq is a directed tree—a merge-tree if θ is coarse-graining, or a split-tree if θ is fine-graining. The graph
Spθq is sometimes also called an alluvial diagram [Rosvall and Bergstrom, 2010].
Sankey diagrams are studied in computer graphics as they allow for the visualisation of complex relational
data. In this context, a Sankey diagram is represented as a layout on the plane, whereby the nodes in each
3
layer Vm are vertically ordered according to a ranking τm : Vm Ñ t1, . . . , |Vm|u, and the layout of the Sankey
diagram is then defined by the collection of such rankings, τ :“ pτ1, . . . , τMq. For visualisation purposes, the
layered layout should ideally minimise the number of crossings between consecutive layers, where a crossing
between two edges ru, vs, ru1, v1s P Em occurs if τmpuq ą τmpu1q and τm`1pvq ă τm`1pv1q or vice versa, and
the crossing number [Warfield, 1977] is given by:
κθpτq :“
M´1
ÿ
m“1
ÿ
ru,vs,ru1,v1sPEm
1τmpuqąτmpu1q^τm`1pvqăτm`1pv1q,
(2)
where 1 denotes the indicator function. The crossing number κθpτq of the layout of the Sankey diagram
Spθq can be minimised by permuting the rankings in the layers, τm, and we denote the minimum crossing
number for the layout as:
κθ :“ min
τ
κθpτq.
(3)
This problem is known to be NP-complete [Garey and Johnson, 2006] and finding efficient optimisation
algorithms is an active research area [Zarate et al., 2018, Li et al., 2025].
3.2
Multiparameter Persistent Homology
Multiparameter persistent homology (MPH) is an extension of standard persistent homology to n ą 1
parameters, first introduced by Carlsson and Zomorodian [2009]. We present here basic definitions, see
Carlsson and Zomorodian [2009], Carlsson et al. [2009], Botnan and Lesnick [2023] for details.
Simplicial Complex.
Let K be a simplicial complex defined for the set X, such that K Ď 2X and
K is closed under the operation of building subsets. The elements of σ P K are called simplices and a
k-dimensional simplex (or k-simplex) can be represented as σ “ rx1, ..., xk`1s where x1, . . . , xk`1 P X and
we have fixed an arbitrary order on X. Note that k “ 0 corresponds to vertices, k “ 1 to edges, and k “ 2
to triangles. We define the k-skeleton Kk of K as the union of its n-simplices for n ď k. We also define
dimpKq as the largest dimension of any simplex in K.
Multiparameter Filtration.
Let us define the parameter space pP, ďq as the product of n ě 1 partially
ordered sets P “ P1 ˆ ¨ ¨ ¨ ˆ Pn, i.e., a ď b for a, b P P if and only if ai ď bi in Pi for i “ 1, . . . , n. A
collection of subcomplexes pKaqaPRn with K “ Ť
aPRn Ka and inclusion maps tia,b : Ka Ñ Kbuaďb that
yield a commutative diagram is called a multiparameter filtration (or bifiltration for n “ 2). We denote by
birthpσq Ď P the set of parameters, called multigrades (or bigrades for n “ 2), at which simplex σ P K
first appears in the filtration. For example, the sublevel filtration Ka “ tσ P K | fpσq ď au for a filtration
function f : K Ñ P maps each simplex σ to a unique multigrade fpσq, i.e., |birthpσq| “ 1. A filtration is
called one-critical if it is isomorphic to a sublevel filtration, and multi-critical otherwise.
Multiparameter Persistent Homology.
Let Hk for k P t0, . . . , dimpKqu denote the k-dimensional
homology functor with coefficients in a field [Hatcher, 2002], see Appendix C for details. Then Hk applied
to the multiparameter filtration leads to a multiparameter persistence module, i.e., a collection of vector
spaces pHkpKaqqaPRn, which are the homology groups whose elements are the generators of k-dimensional
non-bounding cycles, and linear maps tıa,b :“ Hkpia,bq : HkpKaq Ñ HkpKbquaďb that yield a commutative
diagram called multiparameter persistent homology (MPH). For dimension k “ 0, Hk captures the number of
disconnected components and for k “ 1, the number of holes. Note that, for n “ 1, we recover standard
persistent homology (PH) [Edelsbrunner et al., 2002].
Hilbert Function.
While barcodes are complete invariants of 1-parameter PH (n “ 1), the more
complicated algebraic structure of MPH (n ě 2) does not allow for such simple invariants in general; hence,
various non-complete invariants of the MPH are used in practice. We focus on the k-dimensional Hilbert
function [Botnan and Lesnick, 2023] defined as
HFk : P Ñ N0, a ÞÑ rankrHkpia,aqs “ dimrHkpKaqs,
(4)
which maps each filtration index a to the k-dimensional Betti number of the corresponding complex Ka. We
call the k-dimensional MPH trivial if HFk “ 0. The Hilbert distance is then defined as the L2 norm on the
space of Hilbert functions and can be used to compare multiparameter persistence modules.
4
4
The Multiscale Clustering Bifiltration (MCbiF)
The central object of our paper is a novel bifiltration of abstract simplicial complexes that encodes cluster
intersection patterns in the sequence of partitions θ across the scale t.
Definition 3 (Multiscale Clustering Bifiltration). Let θ : rt1, 8q Ñ ΠX be a sequence of partitions. We
define M, the Multiscale Clustering Bifiltration (MCbiF) as a bifiltration of abstract simplicial complexes:
M :“ pKs,tqt1ďsďt
where
Ks,t :“
ď
t1ďsďrďt
ď
CPθprq
∆C,
t1 ď s ď t.
(5)
In this construction, each cluster C corresponds to a p|C| ´ 1q-dimensional solid simplex ∆C :“ 2C,
which, by definition, contains all its lower dimensional simplices [Schindler and Barahona, 2025]. This
echoes natural concepts of data clustering as information compression or lumping [Rosvall and Bergstrom,
2008, 2011, Lambiotte et al., 2014], and of clusters as equivalence classes [Brualdi, 2010]. The MCbiF
then aggregates all clusters (simplices) from partition θpsq to θptq through the union operators, such that
a k-simplex σ “ rx1, . . . , xk`1s P Ks,t consists of elements that are assigned to the same cluster (at least
once) in the interval rs, ts, i.e., x1, . . . , xk`1 P C for some cluster C P θprq and r P rs, ts. The bifiltration
thus depends not only on the lag |t ´ s| but also on the starting scale s, and captures the topological
autocorrelation in the sequence of partitions, see Fig. 1.
We first show that the MCbiF is indeed a well-defined bifiltration.
Proposition 4. M is a multi-critical bifiltration uniquely defined by its values on the finite grid P “
tps, tq P rt1, . . . , tMs ˆ rt1, . . . , tMs | s ď tu with partial order ps, tq ď ps1, t1q if s ě s1, t ď t1.
The proof is straightforward and can be found in Appendix A.2.
The MCbiF leads to a triangular commutative diagram where the arrows indicate inclusion maps between
abstract simplicial complexes (see Fig. 1). The sequence of partitions θptq is encoded by the complexes Kt,t
on the diagonal of the diagram. Moving along horizontal arrows corresponds to fixing a starting scale s and
going forward in the sequence θ, thus capturing the coarse-graining of partitions. Moving along vertical
arrows corresponds to fixing an end scale t and aggregating θ going backwards, capturing the fine-graining
of partitions.
Remark 5. By fixing s :“ t1 (i.e., the top row in the commutative MCbiF diagram), we recover the
1-parameter Multiscale Clustering Filtration (MCF) defined by Schindler and Barahona [2025]. MCF was
designed to quantify non-hierarchies in coarse-graining sequences of partitions and thus cannot capture
fine-graining. For example, a large cluster C P θps1q prevents MCF from detecting cluster assignment conflicts
between elements x, y P C for t ě s1, see Section 4.1. In contrast, MCbiF encodes the full topological
autocorrelation contained in θ by varying both the starting scale s and the lag t ´ s.
Applying MPH to the bifiltration M at dimensions k ď dim K, for K “ KtM,tM , leads to a triangular
diagram of simplicial complexes HkpKs,tq called persistence module (see Section 3.2). This persistence
module has strong algebraic properties, as stated in the following proposition.
Proposition 6. For any k ď dim K, the MCbiF persistence module HkpKs,tq is pointwise finite-dimensional,
finitely presented and block-decomposable.
See Botnan and Lesnick [2023] for definitions and Appendix A.2 for details and a full proof.
The properties in Proposition 6 are important because they guarantee algebraic stability of the
MCbiF [Bjerkevik, 2021]. In particular, the finite presentation property implies stability of the MCbiF
Hilbert functions HFkps, tq (Eq. 4) with respect to small changes in the module [Oudot and Scoccola, 2024,
Corollary 8.2.]. This justifies the use of HFkps, tq as simple interpretable invariants for the topological
autocorrelation captured by MCbiF, as exploited in Section 4.1.
4.1
Measuring Topological Autocorrelation with MCbiF
We now show how topological autocorrelation as measured by HFkps, tq can be used to detect cluster-
assignment conflicts. We focus on dimensions k “ 0, 1, for which MPH is implemented in RIVET [Lesnick and
Wright, 2015]. Note that HF0ps, tq counts the number of connected components and HF1ps, tq the number
of 1-dimensional holes in Ks,t, see Section 3.2 for details. We show below that the computation of these
invariants reveals different aspects of the non-hierarchy in the sequence of partitions.
5
4.1.1
Low-order Non-Hierarchy in Sequences of Partitions
Hierarchy in a sequence of partitions can be understood as a refinement of partitions captured by the
partition lattice.
Definition 7 (Hierarchy). The sequence of partitions θ is hierarchical in rs, ts if we have a strict sequence of
refinements: either θpr1q ď θpr2q, @r1, r2 P rs, ts with r1 ď r2 (agglomerative) or θpr1q ě θpr2q, @r1, r2 P rs, ts
with r1 ď r2 (divisive). We say that θ is strictly hierarchical if it is hierarchical in rt1, 8q.
One important aspect of hierarchy is the nestedness of the clusters in the sequence.
Definition 8 (Nestedness). We say that θ is nested in rs, ts when @r1, r2 P rs, ts, we have that @C P
θpr1q, C1 P θpr2q, one of the sets CzC1, C1zC or C X C1 is empty. See [Korte and Vygen, 2012, Definition
2.12]. We say that θ is strictly nested when θ is nested in rt1, 8q.
Remark 9. It follows directly from the definitions that a hierarchical sequence θ is always nested. However,
nestedness does not necessarily imply hierarchy, as illustrated by the example in Fig 3b.
We can quantify the low-order non-hierarchy in a sequence θ through the computation of the invariant
HF0ps, tq and the associated notion of 0-conflicts defined next.
Remark 10. Each partition θptq can be interpreted as an equivalence relation „t given by the property of
belonging to the same cluster, i.e., x „t y if D C P θptq such that x, y P C [Brualdi, 2010].
Definition 11 (0-conflict and triangle 0-conflict). a) We say that θ has a 0-conflict in rs, ts if the subposet
θprs, tsq has no maximum, i.e., Er P rs, ts such that θpr1q ď θprq, @r1 P rs, ts.
b) We say that θ has a triangle 0-conflict in rs, ts if D x, y, z P X such that Dr1, r2 P rs, ts: x „r1 y „r2 z and
Er P rs, ts: x ȷr y ȷr z.
Next, we show that all triangle 0-conflicts are also 0-conflicts. Moreover, all 0-conflicts break hierarchy
and triangle 0-conflicts additionally break nestedness.
Proposition 12.
(i) Every triangle 0-conflict is a 0-conflict, but the opposite is not true.
(ii) If θ has a 0-conflict in rs, ts, then θ is non-hierarchical in rs, ts.
(iii) If θ is either coarse- or fine-graining but non-hierarchical in rs, ts, then θ has a 0-conflict in rs, ts.
(iv) If θ has a triangle 0-conflict in rs, ts, then θ is non-nested in rs, ts.
See Appendix A.2.1 for the simple proof. Fig. 3b illustrates a 0-conflict that is not a triangle 0-conflict,
and Fig. 3c shows a triangle 0-conflict.
Remark 13. Non-nestedness and non-hierarchy do not imply the presence of a 0-conflict. To see this,
consider the simple counter-example given by θp0q “ ttx, yu, tzuu, θp1q “ ˆ1, θp2q “ ttxu, ty, zuu, which is
non-nested but the partition θp1q is the maximum of the subposet θpr0, 1, 2sq. This illustrates the need
for the additional assumption of coarse- or fine-graining of θ in Proposition 12 (iii) for the condition of no
0-conflict to imply hierarchy.
The following proposition develops a sharp upper bound for HF0 that can be used to capture 0-conflicts.
Proposition 14.
(i) HF0ps, tq ď minrPrs,ts |θprq|, @rs, ts Ď rt1, 8q.
(ii) HF0ps, tq ă minrPrs,ts |θprq| iff θ has a 0-conflict in rs, ts.
(iii) HF0ps, tq “ |θprq| for r P rs, ts iff θprq is the maximum of the subposet θprs, tsq.
See Appendix A.2.1 for a full proof. Proposition 14 shows that HF0 measures low-order non-hierarchy in
θ by capturing 0-conflicts. To quantify this, we introduce a global normalised measure for the sequence θ,
defined as follows.
Definition 15 (Average 0-conflict). Let T :“ tM ` tM´t1
M´1 . The average 0-conflict is defined as:
0 ď ¯c0pθq :“ 1 ´
2
|T ´ t1|2
ż T
t1
ż T
s
HF0ps, tq
minrPrs,ts HF0pr, rqds dt ď 1.
(6)
6
Higher values of ¯c0pθq indicate a high level of 0-conflicts and increased low-order non-hierarchy, as shown
by the next corollary.
Corollary 16.
(i) If θ is hierarchical in rs, ts, then HF0ps, tq “ minp|θpsq|, |θptq|q. As a special case, this
implies HF0pt, tq “ |θptq|, @t ě t1.
(ii) ¯c0pθq ą 0 iff θ has a 0-conflict.
(iii) Let θ be either coarse- or fine-graining. Then, ¯c0pθq “ 0 iff θ is strictly hierarchical.
A proof can be found in Appendix A.2.1.
We can further detect triangle 0-conflicts by analysing the graph-theoretic properties of the MCbiF
1-skeleton Ks,t
1 . Recall that the clustering coefficient C of a graph is defined as the ratio of the number of
triangles to the number of paths of length 2 in the graph [Luce and Perry, 1949, Newman, 2018].
Proposition 17. CpKs,t
1 q ă 1 iff there is a triple x, y, z P X that leads to a triangle 0-conflict for rs, ts, and
which is not a cycle, i.e., additionally to property b) in Definition 11 we also have Er3 P rs, ts: x „r3 z.
See Appendix A.2.1 for a proof. Let us consider the graph generated as the disjoint union of all clusters
from partitions in [s,t] as cliques. This graph is equivalent to the MCbiF 1-skeleton Ks,t
1 . Proposition 17
shows that the clustering coefficient of this graph can be used to detect triangle 0-conflicts that are not
cycles. To be able to detect triangle 0-conflicts that correspond to non-bounding cycles, we turn to the
1-dimensional homology in the next section.
4.1.2
Higher-order Inconsistencies between Clusters in Sequences of Partitions
Measuring 0-conflicts in θ is only one way of capturing non-hierarchy. An additional phenomenon that can
arise in non-hierarchical sequences is higher-order inconsistencies of cluster assignments across scales. These
are captured by the 1-dimensional homology groups [Schindler and Barahona, 2025] and the associated
notion of 1-conflict, which we define next.
Recall the definition of 1-cycles Z1pKs,tq and non-bounding cycles H1pKs,tq (Eq.14) summarised in
Appendix C.
Definition 18 (1-conflict). We say that θ has a 1-conflict in rs, ts if D x1, . . . , xn P X such that the 1-cycle
z “ rx1, x2s ` ¨ ¨ ¨ ` rxn´1, xns ` rxn, x1s P Z1pKs,tq is non-bounding; in other words, rzs P H1pKs,tq with
rzs ‰ 0.
The number of distinct 1-conflicts for the interval rs, ts (up to equivalence of the homology classes) is
given by HF1ps, tq. We first show that 1-conflicts also lead to triangle 0-conflicts and thus break hierarchy
and nestedness of θ.
Proposition 19.
(i) HF1ps, tq ě 1 iff θ has a 1-conflict in rs, ts.
(ii) If θ has a 1-conflict in rs, ts, then it also has a triangle 0-conflict.
(iii) If θ is hierarchical in rs, ts, then HF1ps, tq “ 0.
See Appendix A.2.1 for a proof. Proposition 19 shows that a 1-conflict is a special kind of triangle
0-conflict arising from higher-order cluster inconsistencies across scales. This is illustrated in Fig. 12 and
more details are provided in Examples 42 and 43 in Appendix B, which present sequences of partitions where
different 1-conflicts emerge across scales. Moreover, Proposition 19 (iii) shows that the MCbiF has a trivial
1-dimensional MPH if θ is strictly hierarchical.
Remark 20. Proposition 19 states that every 1-conflict is a triangle 0-conflict. However, not every (triangle)
0-conflict is a 1-conflict, see Example 42. Note also that several triangle 0-conflicts in the sequence θ can
lead to a 1-conflict, when the triangle 0-conflicts are linked together in such a way as to form a non-bounding
cycle, see Example 42. We can test for these systematically using HF1.
Remark 21. The presence of a 1-conflict for the interval rs, ts signals the fact that assigning all the elements
involved in the conflict to a shared cluster would increase the consistency of the sequence. Hence, when a
1-conflict gets resolved, e.g., the corresponding homology generator dies in the MPH at ps, t1q, t ă t1, then
we say that θpt1q is a conflict-resolving partition, see Schindler and Barahona [2025]. As two illustrations,
consider Example 42, where a 1-conflict emerges through the interplay of partitions at scales t “ 1, 2, 3 and
gets resolved by θp4q “ ˆ1, and Example 43, where three different 1-conflicts first emerge between scales t “ 1
and t “ 6 and then get resolved one by one by partitions θp7q, θp8q and θp9q.
7
To quantify the presence of 1-conflicts in θ we introduce an unnormalised global measure for the sequence
θ, as follows.
Definition 22 (Average 1-conflict). Let T be defined as in Definition 15. The average 1-conflict is defined
as:
0 ď ¯c1pθq :“
2
|T ´ t1|2
ż T
t1
ż T
s
HF1ps, tqds dt.
(7)
Corollary 23. c1pθq ą 0 iff θ has a 1-conflict. In particular, if θ is strictly nested, then c1pθq “ 0.
Remark 24. While 0-conflicts (¯c0pθq ą 0) can be defined in relation to the refinement order that gives rise
to the partition lattice, the partition lattice cannot be used to detect higher-order cluster inconsistencies
(1-conflicts), which can be captured and quantified instead by HF1 and the average measure ¯c1pθq.
In Figure 2, we provide a summary of our theoretical results and their relationships.
Proposition 12,
Remark 13
0-conflict in
Proposition 12,
Remark 13
Proposition 12
Triangle 0-conflict in
Definition 18
Proposition 19
1-conflict in
Non-hierarchical in
Remark 9
Non-nested in
Proposition 19
Higher-order
inconsistencies in
Definition 18
Corollary 23
1-dim. Hilbert function
Proposition 14
0-dim. Hilbert function
Corollary 16
Average 0-conflict
Clustering Coefficient
Average 1-conflict
Proposition 31
Strong triangle inequality
of
violated
Proposition 17
Figure 2: Summary of key theoretical results and their relationships indicated by arrows. Double-headed arrows
represent equivalences (iff), whereas single-headed arrows represent implications (if).
4.2
MCbiF as a Higher-Order Sankey Diagram
Recall the definition of the Sankey diagram of the sequence of partitions introduced in Section 3.1, and its
associated representation in terms of an M-layered graph with vertices Vm at each layer representing the
clusters of θptmq, see Eq. (1). Let us define the disjoint union Apℓ, mq :“ VℓZ ... Z Vm, 1 ď ℓď m, which
assigns an index to each cluster in θptq for t P rtℓ, tms. Furthermore, recall that θptqi denotes the i-th cluster
Ci of θptq. The nerve-based MCbiF can then be defined as follows.
Definition 25 (Nerve-based MCbiF). Let s P rtℓ, tℓ`1q, ℓ“ 1, ..., M ´1, and t P rtm, tm`1q, m “ ℓ, ..., M ´1
or t ě tm for m “ M. We define the nerve-based MCbiF as
˜
M :“ p ˜Ks,tqt1ďsďt,
where
˜Ks,t :“ tσ Ď Apℓ, mq :
č
pn,iqPσ
θptnqi ‰ Hu.
(8)
The nerve-based MCbiF
˜
M is a 1-critical bifiltration with simplices representing clusters and their
intersections, in contrast to the original MCbiF M (Eq (5)) in which the simplices represent elements in X
and their equivalence relations. Despite these different perspectives, Proposition 40 in the appendix shows
that
˜
M and M lead to the same MPH and can be considered as equivalent. The proof of this equivalence
follows from an extension of results by Schindler and Barahona [2025].
However, the dimensionality of M and
˜
M can differ, as shown in the following proposition.
Proposition 26.
(i) dim Ks,t “ maxsďrďt maxcPθprq |C| ´ 1, @t1 ď s ď t.
(ii) dim ˜Ktm,tm`n “ n, @1 ď m ď M, 0 ď n ď M ´ m.
8
See proof in Appendix A.2.2. The nerve-based MCbiF is therefore computationally advantageous when
M ă maxt1ďt maxCPθptq |C| ´ 1, making it preferable in applications where the clusters are large.
The nerve-based MCbiF
˜
M can be interpreted as a higher-order extension of the Sankey diagram Spθq
(Eq. 1) in the sense that
˜
M not only records pairwise intersections between clusters in consecutive partitions
of θ, like Spθq), but also takes into account higher-order intersections between clusters in sub-sequences of θ.
More formally, we can state the following proposition that relates Spθq to the nerve-based MCbiF.
Proposition 27. The Sankey diagram graph Spθq is a strict 1-dimensional subcomplex of ˜K :“ ˜Kt1,tM . In
particular, Vm “ Ktm,tm and Em “ Ktm,tm`1, @m “ 1, . . . , M ´ 1. Hence, we can retrieve Spθq from the
zigzag filtration
¨ ¨ ¨ Ðâ ˜Ktm,tm ãÑ ˜Ktm,tm`1 Ðâ ˜Ktm`1,tm`1 ãÑ . . . ,
(9)
which is a subfiltration of the nerve-based MCbiF.
A proof can be found in Appendix A.2.2. For details on zigzag persistence, see Carlsson and de Silva
[2010] and Appendix D.
Furthermore, the 0- and 1-conflicts that can arise in a single layer Em of the Sankey diagram can be fully
characterised as follows.
Proposition 28.
(i) There is a 0-conflict in rtm, tm`1s iff Du P Vm and v P Vm`1 with degpuq ě 2 and
degpvq ě 2, where deg denotes the node degree in the bipartite graph pVm Z Vm`1, Emq associated
with the Sankey diagram.
(ii) There is a triangle 0-conflict in rtm, tm`1s iff there is a path of length at least 3 in Em.
(iii) There is 1-conflict in rtm, tm`1s iff there is an (even) cycle in Em.
See Appendix A.2.2 for the proof and Fig. 3 for an illustration.
(b) 0-conflict
(c) Triangle 0-conflict
MCbiF
Sankey diagram
(d) 1-conflict
(a) No conflict
Figure 3: Relationship between different types of conflicts and the crossings in a single-layer Sankey diagram.
Importantly, a cycle in Em leads to a crossing in Em that cannot be undone, see Fig. 3d for an illustration.
Hence, Proposition 28 (iii) implies that the sum of the elements of the superdiagonal of HF1 provides a lower
bound for the minimal crossing number of the Sankey diagram, κθ (defined in Eq. 2). We state this as the
following corollary.
Corollary 29. řM´1
m“1 HF1ptm, tm`1q ď κθ.
Remark 30. Note that 1-conflicts that arise across multiple partitions in the sequence (i.e., across multiple
layers) do not necessarily lead to crossings. See Fig. 1, where the chosen ordering of the elements does not
lead to a crossing in the layout of the Sankey diagram despite the presence of a 1-conflict. However, we
hypothesise that the full HF0 and HF1 feature maps capture more complicated crossings that arise in the
Sankey layout across many layers. This insight is exploited in our computational tasks below.
9
5
Mathematical Links of MCbiF to Other Methods
We now present some mathematical connections of MCbiF to the related methods introduced in Section 2.
Ultrametrics.
Given a sequence of partitions θ with θpt1 “ 0q “ ˆ0 and θptMq “ ˆ1, let us define the matrix
of first-merge times conditioned on the starting scale s:
Dθ,spxi, xjq :“ mintt ě s | D C P θptq : xi, xj P Cu.
(10)
Clearly, when s “ 0, this recovers the standard matrix of first-merge times Dθ :“ Dθ,0 discussed in Section 2.
If θ is hierarchical, i.e., an agglomerative dendrogram, then Dθ is an ultrametric, i.e., it fulfils the strong
triangle inequality: Dθpx, zq ď max pDθpx, yq, Dθpx, zqq @x, y, z P X.
Corollary 16 states that the number of branches in the agglomerative dendrogram at level t, which is
given by |θptq|, is equal to HF0ps, tq for any s ď t. Hence, HF0ps, tq contains the same information as the
ultrametric in the hierarchical case, see also Schindler and Barahona [2025] and Proposition 32 below.
If, on the other hand, θ is non-hierarchical, triangle 0-conflicts can lead to violations of the (strong)
triangle inequality:
Proposition 31. The triplet x, y, z P X leads to a triangle 0-conflict in rs, ts iff x, y, z violate the strong
triangle inequality for Dθ,s, i.e., Dθ,spx, zq ą maxpDθ,spx, yq, Dθ,spy, zqq.
See Appendix A.3 for a proof and Fig. 8a for an illustration. Proposition 31 shows that ¯c0pθq measures
how much the ultrametric property of Dθ is violated.
Recall that Dθ,s is a dissimilarity measure that can be used to define a filtration [Chazal et al., 2014].
Next, we show that the 0-dimensional MPH of MCbiF corresponds to the 0-dimensional MPH of a Rips-based
bifiltration constructed from Dθ,s.
Proposition 32. Let us define the Merge-Rips bifiltration L based on Dθ,s as
L “ pLs,tqt1ďsďt
where
Ls,t “ tσ Ă X | @x, y P σ :
Dθ,spx, yq ď tu.
(11)
Then the 0-dimensional MPH of the Merge-Rips bifiltration, L, and of the MCbiF, M, are equivalent, but
the 1-dimensional MPH of L and M are generally not equivalent. Furthermore, if θ is strictly hierarchical,
then L has a trivial 1-dimensional MPH.
A proof is presented in Appendix A.3 and follows from an argument in Schindler and Barahona [2025].
In the hierarchical case, the 1-dimensional MPH of the Merge-Rips bifiltration is trivial because Dθ fulfils
the strong triangle inequality, and is thus equivalent to the MCbiF, whose 1-dimensional MPH is also trivial
in the hierarchical case, see Proposition 19.
Conditional Entropy.
The conditional entropy (CE) is only defined for pairs of partitions. CE is defined
as the expected Shannon information of the conditional probability of a partition θptq “ tC1, . . . , Cnu given
θpsq “ tC1
1, . . . , C1
mu:
Pt|sri|js “ |Ci X C1
j|
|C1
j|
.
For the special case of M “ 2 (i.e., only two partitions in the sequence θ), it can be shown that HF0pt1, t2q
follows directly from the spectral properties of the matrix Pt2|t1P T
t2|t1 interpreted as an undirected graph, as
shown by the following proposition.
Proposition 33. HF0pt1, t2q “ dimpker Lq, where L “ diagpPt2|t11q ´ Pt2|t1P T
t2|t1 is a weighted Laplacian.
The proof can be found in Appendix A.3.
Remark 34. Note that Pt|s only encodes the pairwise relationship between clusters, and does not capture
higher-order cluster inconsistencies. In particular, CE cannot detect 1-conflicts arising across more than two
scales, as seen in Example 44 in Appendix B.
6
Experiments
6.1
Regression Task: Minimal Crossing Number of Sankey Layout
Task
In our first experiment, we consider a task of relevance in computer graphics and data visualisation:
the minimisation of the crossing number of Sankey diagram layouts [Zarate et al., 2018, Li et al., 2025]. This
minimisation is NP-complete, and here we use our MCbiF topological feature maps to predict the minimal
crossing number κθ of the Sankey diagram Spθq of a given sequence of partitions θ (see Section 3.1).
10
Data
We test our measures on synthetic datasets generated by sampling randomly from the space of
coarse-graining sequences of partitions.
Definition 35 (Space of coarse-graining sequences of partitions). The space of coarse-graining sequences of
partitions, denoted ΠM
N , is defined as the set of coarse-graining sequences θ : r0, 8q Ñ ΠX with |X| “ N
and M change points tm “ 0, . . . , M ´ 1, such that |θpsq| ě |θptq|, @s ď t, which start with the partition of
singletons θpt1 “ 0q “ ˆ0 and end with the full set θptM “ M ´ 1q “ ˆ1.
Setting M “ 20, we generate two datasets of 20,000 random samples θ P ΠM
N for N “ 5 and N “ 10.
For each of the generated θ, we compute three feature maps: the information-based pairwise conditional
entropy matrix CE [Meilă, 2003] (see Eq. 15) and our MCbiF Hilbert functions (HF0 and HF1). In addition,
as a baseline feature map, we also consider the (non-unique) raw label encoding of θ given by the N ˆ M
matrix whose m-th column contains the labels of the clusters in θptmq assigned to the elements in X. As
our prediction target, we take y “ κθ (Eq. 3), the minimal crossing number of the layout of the Sankey
diagram, which we computed with the OmicsSankey algorithm [Li et al., 2025]. See Section 3.1 for details.
We expect that predicting y will be harder for N “ 10 because the increased complexity of ΠN
M allows for
more complicated crossings in the Sankey diagram.
Table 1: Regression task. Test R2 score of LR, CNN and MLP models trained on different features for N “ 5 and
N “ 10. See Appendix F.1 for train R2 scores.
N
Method
Raw label encoding
HF0
HF1
HF0 & HF1
CE
5
LR
0.001
0.147
0.486
0.539
0.392
CNN
-0.006
0.155
0.504
0.544
0.492
MLP
-0.002
0.150
0.491
0.541
0.409
10
LR
-0.012
0.214
0.448
0.516
0.457
CNN
0.000
0.211
0.448
0.507
0.454
MLP
0.000
0.212
0.450
0.514
0.458
Results
As a preliminary assessment, we first compute the Pearson correlation, r, between the crossing
number y and the three measures under investigation: the information-theoretical measure CI (i.e., the
average CE, see Eq. (16)) and the MCbiF topological average measures ¯c0 and ¯c1. The correlation between
CI and y is low (r “ 0.20 for N “ 5 and r “ 0.06 for N “ 10), higher for ¯c0, and highest for ¯c1 (r “ 0.47
for N “ 5, 10) (see Fig. 11 in Appendix F.1). This is consistent with our theoretical results in Section 4.2,
which show the relation between the crossing number and HF1 (see, e.g., Corollary 29).
We then proceed to the regression task of predicting κθ. We split each dataset into training (64%),
validation (16%) and test (20%). For each feature map (or their combinations), we train three different
models: linear regression (LR), multilayer perceptron (MLP), and convolutional neural network (CNN).We
use the mean-squared error (MSE) as our loss function and employ the validation set for hyperparameter
tuning. See Appendix F.1 for details of all the models. We then evaluate the model performance on the
unseen test data using the coefficient of determination (R2).
We find that the raw label encoding of θ does not improve upon the mean prediction (R2 « 0) and that
MCbiF feature maps outperform the information-based feature map (Table 1). In particular, the combined
HF0 and HF1 features lead to a significantly better model performance than CE (p ă 0.0001, t-test on the
residuals). Furthermore, HF0 and HF1 yield R2 “ 0.544 for N “ 5 and R2 “ 0.516 for N “ 10 whereas CE
only achieves R2 “ 0.492 and R2 “ 0.458, respectively. The strong performance of the simple LR model
demonstrates the interpretability of the MCbiF features, important for explainable AI (XAI) [Adadi and
Berrada, 2018].
6.2
Classification Task: Non-Order-Preserving Sequences of Partitions
Task
In our second experiment, we classify whether a sequence of partitions is order-preserving or not, i.e.,
whether a sequence θ is compatible with a total ordering on the set X. This task is of relevance in several
areas, from the study of preference relations in utility theory in social sciences [Roberts, 2009] to the analysis
of weak orderings and partition refinement algorithms in computer science [Habib et al., 1999].
Definition 36 (Order-preserving sequence of partitions). When a partition θptmq is equipped with a total
order ăm on the clusters it is called an ordered partition.2 Such a partition induces a total preorder Àm on
2The ranking τm : Vm Ñ t1, . . . , |Vm|u of the vertices Vm in the Sankey diagram Spθq is one example of a total order ăm on
the clusters, see Section 3.1.
11
X [Stanley, 2011], i.e., if rxst ăm ryst then x Àm y. We call the sequence of partitions θ order-preserving if
there exist total orders pă1, . . . , ăMq such that the total preorders pÀ1, . . . , ÀMq are compatible across the
sequence, i.e., @ℓ, m we have x Àℓy iff x Àm y, @x, y P X.
0.26
0.26
0.27
0.27
0.27
0.27
0.28
CI
ns
0.80
0.80
0.81
0.81
0.81
c0
****
0.00
2.00
4.00
6.00
c1
****
y = 0
y = 1
Figure 4:
Difference between order-
preserving (y
“
0) and non-order-
preserving (y “ 1) sequences (**** indi-
cates p ă 0.0001, Mann-Whitney U test).
According to this definition, a sequence θ is non-order-preserving
if there is no total order on X that is consistent with all the total
preorders induced by the partitions θptq.
Data
We carry out this classification task on synthetic data for
which we have a ground truth. From the space of coarse-graining
sequences of partitions ΠM
N , introduced in Definition 35, we generate
a balanced dataset of 3,700 partitions θ P Π30
500, half of which are
order-preserving (y “ 0) and the other half are non-order-preserving
(y “ 1). The loss of order-preservation is induced by introducing
random swaps in the node labels across layers. See Appendix F.2
for details. For each of the generated θ we compute CE, HF0 and
HF1 using the computationally advantageous nerve-based MCbiF.
We choose N “ 500 and M “ 30 to demonstrate the scalability of
our method.
Table 2: Classification task. Test accu-
racy of logistic regression trained on dif-
ferent features.
Raw label
encoding
HF0
HF1
CE
0.53
0.56
0.97
0.50
Results
Firstly, whereas we find no significant difference between
the information-theoretical CI of order-preserving (y “ 0) and non-
order-preserving (y “ 1) sequences, we observe a statistically signif-
icant increase of ¯c0 and ¯c1 for order-preserving sequences (Fig. 4).
For the classification task, we split our data into training (80%)
and test (20%).
For each feature map, we then train a logistic
regression on the training split, and evaluate the accuracy on the
test split, see Appendix F.2. We find that HF1 predicts the label
y “ t0, 1u encoding the (lack of) order preservation with high accuracy (0.95). In contrast, CE and the raw
label encoding of θ cannot improve on a random classifier (Table 2). Our results thus demonstrate the high
sensitivity of MCbiF to order-preservation in θ because non-order-preserving sequences induce 1-conflicts
that we capture with HF1.
6.3
Application to Real-World Temporal Data
In our final experiment, we apply MCbiF to temporal sequences of partitions computed from real-world
contact data of free-ranging house mice that capture the changes in the social network structure of the
rodents over time [Bovet et al., 2022].
Data
Each partition θτptq describes mice social groupings for N “ 281 individual mice at weeks t P r1, . . . , 9s,
i.e., the nine weeks in the study period (28 February-1 May 2017). Hence, each sequence captures the
fine-graining of social groups over the transition from winter to spring. Each partition sequence is computed
at temporal resolution τ ą 0, where the parameter τ modulates how fine the temporal community structure
is (Fig. 13). See Bovet et al. [2022] for details.
We use MCbiF to compare the temporal sequences θτi for nine parameters τi, i “ 1, . . . , 9, as provided
in Bovet et al. [2022]. See Appendix F.3 for details. For each of the nine partition sequences θτi, we compute
HF0 and HF1 using the computationally advantageous nerve-based MCbiF, which induces a 50-fold reduction
in computation time due to a much lower number of simplices (260 simplices for the nerve-based MCbiF
instead of 116,700 for the original MCbiF).
Results
Bovet et al. [2022] identified that the temporal resolutions τ2 “ 1 s, τ4 “ 60 s and τ8 “ 24 h lead
to robust sequences of partitions. Using the Hilbert distance, i.e., the L2-norm on the 0- and 1-dimensional
MCbiF Hilbert functions, we find these temporal resolutions to be representative for three distinct temporal
regimes characterised by different degrees of non-hierarchy, as measured by ¯c0 and ¯c1 (Fig. 5). In particular,
high ¯c0 indicates that mice tend to split off groups over time, and high ¯c1 indicates that mice meet in
overlapping subgroups but never jointly in one nest box. Note that θτ2 has a strong non-hierarchical structure
because the large-scale mice social clusters get disrupted in the transition to spring. In contrast, θτ8 is more
hierarchical as it captures the underlying stable social groups revealed by the higher temporal resolution.
However, θτ4 has the strongest hierarchy as indicated by a lower ¯c0 and an absence of 1-conflicts (¯c1 “ 0) and
12
(a)
(b)
τ9
τ8
τ7
τ5
τ4
τ1
τ2
τ3
τ6
Figure 5: (a) Analysis of non-hierarchical sequences of partitions θτi compiled from the temporal social interactions
of a mice population over a period of 9 weeks. Each θτi is formed by a sequence of social groupings θτiptq over week t.
Different sequences of partitions were computed as a function of the parameter τi. We display Sankey diagrams and
MCbiF feature maps for θτi at three parameters τi (i “ 2, 4, 8) identified as robust in the original work by Bovet
et al. [2022]. These three sequences θτi exhibit different types of non-hierarchy, as shown by our topological feature
maps and our measures of average 0-conflict (¯c0) and average 1-conflict (¯c1). (b) The θτi (i “ 2, 4, 8) found in Bovet
et al. [2022] as robust behaviours correspond to distinct topological characteristics of the sequences of partitions, as
captured by the block structure in the distance between MCbiF Hilbert functions.
thus corresponds to a sweet spot in hierarchical organisation between the low and high temporal resolutions.
Finally, the Hilbert distances also capture an increased time reversibility in the sequence θτ8 due to the
increased stability of social groupings over time, see Fig. 14 in Appendix F.3.
7
Conclusion
We have introduced the MCbiF, a novel bifiltration that encodes the cluster intersection patterns in a
multiscale, non-hierarchical sequence of partitions, θ. Its stable Hilbert functions HFk quantify the topological
autocorrelation of θ and measure non-hierarchy in two complementary ways: the Hilbert function at dimension
k “ 0 captures the absence of a maximum with respect to the refinement order (0-conflicts), whereas the
Hilbert function at dimension k “ 1 captures the emergence of higher-order cluster inconsistencies (1-conflicts).
This is summarised by the measures of average 0-conflict ¯c0pθq and average 1-conflict ¯c1pθq, which are global,
history-dependent and sensitive to the ordering of the partitions in θ. The MCbiF extends the 1-parameter
MCF defined by Schindler and Barahona [2025] to a 2-parameter filtration, leading to richer algebraic
invariants that describe the full topological information in θ. We remark that the MCbiF is independent of
the chosen clustering algorithm and can be applied to any (non-hierarchical) sequence of partitions θ.
We demonstrate with numerical experiments that the MCbiF Hilbert functions provide topological feature
maps that can be used for downstream machine learning tasks, and are shown to outperform information-
based features on regression and classification tasks on non-hierarchical sequences of partitions. Moreover,
the grounding of MCbiF features in algebraic topology enhances interpretability, a crucial attribute for XAI
and applications to real-world data.
Limitations and future work
Our analysis of the MCbiF MPH is restricted to dimensions 0 and 1 due
to current limitations of the RIVET software [Lesnick and Wright, 2015] used in our numerical experiments.
Analysing topological autocorrelation for higher dimensions would allow us to capture more complex
higher-order cluster inconsistencies and could be the object of future research.
Furthermore, we focused here on Hilbert functions as our topological invariants because of their compu-
tational efficiency and analytical simplicity, which facilitates our theoretical analysis. In future work, we
plan to use richer feature maps by exploiting the block decomposition of the MCbiF persistence module,
which leads to barcodes [Bjerkevik, 2021], or by using multiparameter persistence landscapes [Vipond, 2020].
Another future direction is to use MCbiF to evaluate the consistency of assignments in consensus
clustering [Strehl and Ghosh, 2002, Vega-Pons and Ruiz-Shulcloper, 2011]. Indeed, it can be shown that the
values of the Hilbert function HFkps, tq that are further away from the diagonal (s “ t) are more robust to
permutations of the ordering of partitions in θ (see Proposition 41 in Appendix A.4), and, in particular,
HFkpt1, tMq only depends on the set of distinct partitions in the sequence θprt1, 8qq and is independent to
13
any permutation in their order. Hence, in future work, HFkpt1, tMq could be used as an overall measure of
consistency in θ in the context of consensus clustering.
Finally, we plan to analyse minimal cycle representatives of the MPH [Li et al., 2021] to localise 1-conflicts
in the sequence of partitions, which is of interest to compute conflict-resolving partitions in consensus
clustering, or to identify inconsistent assignments in temporal clustering [Liechti and Bonhoeffer, 2020].
Reproducibility Statement
Detailed proofs of all theoretical results can be found in Appendix A. Extensive documentation of our
experiments is presented in Appendix F. The dataset studied in Section 6.3 is publicly available at: https:
//dataverse.harvard.edu/file.xhtml?fileId=5657692. Python code for the MCbiF method is publicly
available at: https://github.com/barahona-research-group/MCF
Acknowledgments
JS acknowledges support from the EPSRC (PhD studentship through the Department of Mathematics
at Imperial College London). MB acknowledges support from EPSRC grant EP/N014529/1 supporting
the EPSRC Centre for Mathematics of Precision Healthcare. We thank Kevin Michalewicz, Asem Alaa
and Christian Schindler for valuable discussions on the computational aspects of this project. We thank
Alex Bovet for discussions about the temporal dataset studied in this paper. We also thank Arne Wolf for
discussions about the stability of multiparameter persistence modules. This work benefited from discussions
at the London-Oxford TDA seminar in November 2024.
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17
Appendices
A
Proofs of Theoretical Results
A.1
Proofs for Section 1
We first state a simple fact about coarse-graining sequences of partitions.
Remark 37. Let θptqi denote the i-th cluster Ci of θptq. It is a simple fact that θ is coarse-graining iff the
mean cluster size is non-decreasing, i.e.,
1
|θpsq|
ř|θpsq|
i
|θpsqi| ď
1
|θptq|
ř|θptq|
j
|θptqj| for s ď t. The proof follows
directly from the fact that ř|θpsq|
i
|θpsqi| “ ř|θptq|
i
|θptqi| “ N.
A.2
Proofs for Section 4
We provide a proof for the multi-criticality of the MCbiF filtration stated in Proposition 4.
Proof of Proposition 4. The MCbiF is indeed a bifiltration because Ks,t Ď Ks1,t1 if s ě s1 and t ď t1. See
Fig. 6 for the triangular diagram of the MCbiF filtration, where arrows indicate inclusion maps. The
MCbiF is uniquely defined by its values on the finite grid rt1, . . . , tMs ˆ rt1, . . . , tMs because θ has change
points t1 ă ¨ ¨ ¨ ă tM.
It is a multi-critical bifiltration because for x P X we have rxs P Ks,t for all
s, t P rt1, 8qop ˆ rt1, 8q. In particular, x P Kt1,t1 and x P Kt1`δ,t1`δ for δ ą 0 but pt1, t1q and pt1 ` δ, t1 ` δq
are incomparable in the poset rt1, 8qop ˆ rt1, 8q.
. . .
. . .
. . .
Kt,t
Kt,t1
Kt,t2
. . .
Kt1,t1
Kt1,t2
. . .
Kt2,t2
. . .
Figure 6: Triangular commutative diagram of the MCbiF for t1 ď t ď t1 ď t2. The arrows indicate inclusion maps
between simplicial complexes.
Next, we provide formal definitions for algebraic properties of persistence modules, see Botnan and
Lesnick [2023] for details.
Definition 38. For partially ordered sets P1, P2, we call an interval I Ď P1 ˆ P2 a block if it can be written
as one of the following types:
1. Birth quadrant: I “ S1 ˆ S2 for downsets S1 Ď P1 and S2 Ď P2.
2. Death quadrant: I “ S1 ˆ S2 for upsets S1 Ď P1 and S2 Ď P2.
3. Vertical band: I “ S1 ˆ P2 for an interval S1 Ď P1.
4. Horizontal band: I “ P1 ˆ S2 for an interval S2 Ď P2.
Definition 39. Let Vect denote the category of k-vector spaces for a fixed field k. For a partially ordered
set P, a P-indexed persistence module is a functor F : P ÞÑ Vect. We say that:
a) F is called pointwise finite-dimensional if dimpFaq ă 8 for all a P P.
b) F is called finitely presented if there exists a morphism of free modules ϕ1 : F1 Ñ F1 such that
cokerpϕ1q – F and F0 and F1 are finitely generated.
c) F is called block-decomposable if it decomposes into blocks F À
BPBpF q kB where BpFq is a multiset of
blocks that depends on F.
We can now provide the proof for Proposition 6, which shows that the MCbiF persistence module (see
Fig. 7) is pointwise finite-dimensional, finitely presented and block-decomposable. The proof relies on the
equivalent nerve-based construction of the MCbiF (see Proposition 40), and the exactness of the persistence
module from which block-decomposability follows [Cochoy and Oudot, 2020].
18
Proof of Proposition 6. The MCbiF module is pointwise finite-dimensional because the homology groups
of finite simplicial complexes are finite. As the MCbiF is defined uniquely by its values on a finite grid
(Proposition 4), its persistence module consists of finitely many vector spaces and finitely many linear maps
between them, hence it is finitely presented.
To prove block-decomposability, we use the nerve-based MCbiF p ˜Ks,tqt1ďsďt, which leads to the same
persistence module, see Proposition 40. As the module is uniquely defined by its values on a finite grid,
we can use Theorem 9.6 by Cochoy and Oudot [2020] that implies block-decomposability if the persistence
module is exact. Hence, it suffices to show that for all t1 ď t ď t1 ď t2 ď t3 the diagram
Hkp ˜Kt,t2q
Hkp ˜Kt,t3q
Hkp ˜Kt1,t2q
Hkp ˜Kt1,t3q
induces an exact sequence:
Hkp ˜Kt1,t2q Ñ Hkp ˜Kt,t2q ‘ Hkp ˜Kt1,t3q Ñ Hkp ˜Kt,t3q
(12)
By construction of the MCbiF, ˜Kt,t3 “ ˜Kt,t2 Y ˜Kt1,t3. Furthermore, ˜Kt,t2 “ ˜Kt,t1 Y ˜Kt1,t2 and ˜Kt1,t3 “
˜Kt1,t2Y ˜Kt2,t3. Without loss of generality, t “ tk, t1 “ tℓ, t2 “ tm, t3 “ tn for change points tk ă tℓă tm ă tn
of θ such that Apk, ℓq X Apm, nq “ H. Hence, ˜Kt,t1 X ˜Kt2,t3 “ H and ˜Kt1,t2 “ ˜Kt,t2 X ˜Kt1,t3. This means
that Eq. (12) is a Mayer-Vietoris sequence for all k ě 0, implying exactness [Hatcher, 2002, p. 149] and
proving the block decomposability [Cochoy and Oudot, 2020, Theorem 9.6].
. . .
. . .
. . .
HkpKt,tq
HkpKt,t1q
HkpKt,t2q
. . .
HkpKt1,t1q
HkpKt1,t2q
. . .
HkpKt2,t2q
. . .
Figure 7: Multiparameter persistence module of the MCbiF for t1 ď t ď t1 ď t2. The arrows indicate linear maps
between vector spaces.
A.2.1
Proofs for Section 4.1
We continue with the proof of Proposition 12 that relates 0-conflicts to hierarchy and triangle 0-conflicts to
nestedness.
Proof of Proposition 12. (ii) If θ has a 0-conflict then Dr1, r2 P rs, ts such that θpr1q ę θpr2q and θpr1q ğ θpr2q,
otherwise θprs, tsq would have a maximum. Hence, θ is not hierarchical in rs, ts.
(iii) Let us first assume that θ is coarse-graining, i.e., |θptq| ď |θprq| for all r P rs, ts. We show that no
0-conflict in rs, ts implies that θ is hierarchical in rs, ts. Let r1, r2 P rs, ts with r1 ď r2, then the subposet
θprr1, r2sq has a maximum because of the absence of a 0-conflict, and the maximum is given by θpr2q due to
coarse-graining. Hence, θpr1q ď θpr2q. As r1, r2 were chosen arbitrarily, this implies that θ is hierarchical in
rs, ts. The argument is analogous for the case that θ is fine-graining.
(iv) Let x, y, z P X be in a triangle 0-conflict. In particular, x „r1 y „r2 z with x ‰ y, x ‰ z and y ‰ z.
Hence, there are C P θpr1q and C1 P θpr2q such that x, y P C and y, z P C1, as well as z R C and x R C1.
This implies txu P CzC1, tzu P C1zC and tyu P C X C1, showing that C and C1 are non-nested. Hence, θ is
non-nested in rs, ts.
(i) Moreover, Er P rs, ts such that x „r„r y. In particular, Er P rs, ts such that EC2 P θprq with C Ď C2
and C Ď C2. Hence, Er P rs, ts such that θpr1q ď θprq and θpr2q ď θprq, implying that the subposet θprs, tsq
has no maximum. This shows that every triangle 0-conflict is also a 0-conflict, proving statement (i). Note
that the opposite is not true as illustrated by the example in Fig. 3b.
We now provide a proof for Proposition 14 on properties of the 0-dimensional Hilbert function of the
MCbiF.
19
Proof of Proposition 14. (i) HF0ps, tq is equal to the number of connected components of Ks,t. Let r1 P rs, ts
such that c “ |θpr1q| “ minrPrs,ts |θprq|. We can represent θprq “ tC1, . . . , Ccu and by construction ∆C P Ks,t
for all C P θprq. Hence, if two elements x, y P X are in the same cluster C P θprq then rx, ys P Ks,t and the
0-simplices rxs, rys P Ks,t are in the same connected component. As θprq has c mutually disjoint clusters,
this means that there cannot be more than c disconnected components in Ks,t and HF0ps, tq ď c “ |θpr1q|.
As r1 P rs, ts was chosen arbitrarily, this implies HF0ps, tq ď minrPrs,ts |θprq|.
We prove statement (ii) by the contrapositive and show that the following two conditions are equivalent:
C1: HF0ps, tq “ minrPrs,ts |θprq|.
C2: Dr P rs, ts such that θpr1q ď θprq, @r1 P rs, ts.
Note that C2 is equivalent to there is no 0-conflict in rs, ts. “ðù” consider first that C2 is true and θprq is
an upper bound for the partitions θpr1q, r1 P rs, ts. This implies that @r1 P rs, ts we have that @C1 P θpr1q
there DC P θprq such that C1 P C. By construction of the MCbiF (Eq. 5) this implies @σ1 P Ks,t there
Dσ P Kr,r such that σ1 Ď σ. This means Ks,t Ď Kr,r and thus Ks,t “ Kr,r. As Kr,r has |θprq| disconnected
components this implies HF0ps, tq “ |θprq|, showing C1.
“ùñ” To prove the other direction, assume that C1 is true. Then there exists r P rs, ts such that
c :“ HF0ps, tq “ |θprq| with |θprq| “ minqPrs,ts |θpqq|. In particular, the disconnected components of Ks,t
are given by the clusters of θprq denoted by C1, . . . , Cc. Let r1 P rs, ts and C1 P θprq. Then Di P r1, . . . , cs
such that C1 Ď Ci P θprq because otherwise the solid simplex ∆C1 would connect two solid simplices in
t∆C1, . . . , ∆Ccu, contradicting that they are disconnected in Ks,t. Hence, the clusters of θpr1q are all subsets
of cluster of θprq, implying θpr1q ď θprq. As r1 P rs, ts was chosen arbitrary this shows C2.
We finally prove statement (iii). “ùñ” Note that HF0ps, tq “ |θprq| implies |θprq| “ minr1Prs,ts |θpr1q|
according to (i). Then (ii) shows that C2 is true for r, i.e., θprq is the maximum of the subposet θprs, tsq.
“ðù” The other direction follows directly from the proof of (ii).
We next prove Corollary 16 about some properties of the average 0-conflict, which follows immediately
from Proposition 14.
Proof of Corollary 16. We begin with the proof of statement (i). If θ is hierarchical in rs, ts then θ is either
coarse- or fine-graining. Assume first that θ is coarse-graining, then θps1q ď θptq for all s1 P rs, ts and together
with hierarchy, this implies that θptq is an upper bound of the subposet θprs, tsq. Hence, Proposition 14 (iii)
shows that HF0ps, tq “ |θptq|. Moreover, HF0ps, tq “ minp|θpsq|, |θptq|q because coarse-graining implies
|θpsq| ě |θptq|. A similar argument also shows HF0ps, tq “ |θpsq| “ minp|θpsq|, |θptq|q if θ is fine-graining.
We continue with proving (ii).
¯c0pθq ą 0 is equivalent to Ds, t P rt1, tMs such that HF0ps, tq ă
minrPrs,ts |θprq|, according to Definition 15. This is again equivalent to Ds, t P rt1, tMs such that θ has
a 0-conflict in rs, ts, according to Proposition 14 (ii).
We finally prove statement (iii). “ùñ” ¯c0pθq means that θ has no 0-conflict in rt1, 8q. As θ is also
coarse- or finge-graining, Proposition 12 (iii) then shows that θ is strictly hierarchical. “ðù” If θ is strictly
hierarchical, then it has no 0-conflicts according to Proposition 12 (ii) and statement (ii) implies that
¯c0pθq “ 0.
Next, we provide the proof of Proposition 17 about the relation between triangle 0-conflicts and the
clustering coefficient.
Proof of Proposition 17. Assume that CpKs,t
1 q ă 1. Then there exist x, y, z P X that form a path of length
2 but no triangle, see Newman [2018] for details on the clustering coefficient. Without loss of generality,
rx, ys, ry, zs P Ks,t
1
but rx, zs R Ks,t
1 . This implies Dr1, r2 P rs, ts: x „r1 y „r2 z and Er P rs, ts: x „r z.
Hence, x, y, z lead to a triangle 0-conflict.
We can now prove Proposition 19 on 1-conflicts.
Proof of Proposition 19. Statement (i) follows directly from the definition of 1-conflicts that HF1ps, tq “
dimrHkpKs,tqs ě 1 iff θ has a 1-conflict.
We next prove statement (ii): If HF1ps, tq ě 1 there exists a 1-cycle z “ rx1, x2s`¨ ¨ ¨`rxn´1, xns`rxn, x1s
that is non-bounding, i.e., h :“ rzs ‰ 0 in H1pKs,tq, see Appendix C for details. Case 1: Assume Er P rs, ts:
x1 „r x2 „r x3, then it follows immediately that x1, x2, x3 lead to a triangle 0-conflict. Case 2: Assume
Dr P rs, ts: x1 „r x2 „r x3. As rzs ‰ 0 there exists a 1-cycle ˜z “ r˜x1, ˜x2s ` ¨ ¨ ¨ ` `r˜xm´1, ˜xms ` r˜xm, ˜x1s P
20
Z1pKs,tq such that ˜z is homologous to z, i.e., ˜z “ z ` B2w for w P C2pKs,tq, and such that Er P rs, ts:
˜x1 „r ˜x2 „r ˜x3. In particular, ˜x1, ˜x2, ˜x3 lead to a triangle 0-conflict.
We finally prove statement (iii): If θ is hierarchical, then it has no 0-conflicts according to Corollary 16.
Hence, θ also has no triangle 0-conflict in rs, ts and so (i) implies that HF1ps, tq “ 0.
A.2.2
Proofs for Section 4.2
Next, we provide the proof about the equivalence between MCbiF and nerve-based MCbiF.
Proposition 40. The bifiltrations M and
˜
M lead to the same persistence module.
Proof. The proof follows from Proposition 30 in Schindler and Barahona [2025], which extends directly to
the 2-parameter case.
Next, we prove Proposition 26 about the dimension of the nerve-based MCbiF.
Proof of Proposition 26. Statement (i) follows directly from the definition in Eq. (5). We show statement (ii)
by induction. Base case: From the definition of the nerve-based MCbiF, it follows directly that dim N tm,tm “ 0
because the indices in Apm, mq correspond to mutually exclusive clusters. Induction step: Let us assume that
dim N tm,tm`n “ n, then there exist C0, . . . , Cn P θprtm, tm`nsq such that C0 X ¨ ¨ ¨ X Cn ‰ H. As the clusters
in partition θptm`n`1q cover the set X there exist a cluster C P θptm`n`1q such that C X C0 X ¨ ¨ ¨ X Cn ‰ H.
Hence, dim N tm,tm`n ě n ` 1. If dim N tm,tm`n ą n ` 1 there would exist a second cluster C1 P θptm`n`1q
with C1 X C X C0 X ¨ ¨ ¨ X Cn ‰ H but C1 X C ‰ H contradicts that clusters of θptm`n`1q are mutually
exclusive. Hence, dim N tm,tm`n “ n ` 1, proving statement (ii) by induction.
We provide a proof for the connection between Sankey diagrams and the nerve-based MCbiF.
Proof of Proposition 27. The Sankey diagram graph Spθq “ pV “ V1 Z ... Z VM, E “ E1 Z ... Z EM´1q is
a strict 1-dimensional subcomplex of ˜K “ ˜Kt1,tM because ˜Ktm,tm “ Vm Ď ˜K and ˜Ktm,tm`1 “ Em Ď ˜K.
This also shows that the zigzag filtration (9) contains exactly the same vertices (0-simplices) and edges
(1-simplices) as Spθq.
We next prove Proposition 28 that characterises conflicts that can arise in a single layer of the Sankey
diagram.
Proof of Proposition 28. (i) Suppose that θ has a 0-conflict in rtm, tm`1s.
Then θptmq ę θptm`1q and
θptmq ğ θptm`1q. This means that there exists C P θpr1q such that DC1, C2 P θpr2q with C X C1 ‰ H,
C X C2 ‰ H and C1 X C2 “ H, otherwise θpr1q ď θpr2q. Hence, θpr1q ę θpr2q is equivalent to Du P Vm
(the node corresponding to cluster C) with degree degpuq ě 2 in Em. An analogous argument shows that
θpr1q ğ θpr2q is equivalent to Dv P Vm`1 with degpvq ě 2. This proves the statement.
(ii) Let x, y, z P X form a triangle 0-conflict for the interval rtm, tm`1s, i.e., x „tm y „tm`1 z but
x ȷtm`1 y ȷtm z.
In particular, the elements x, y, z are mutually distinct.
This means there exist
C1, C2 P θptmq and C1
1, C1
2 P θptm`1q such that x, y P C1, z P C2, y, z P C1
1 and x P C1
2. This is equivalent to
C1 X C2 “ H, C1
1 X C1
2 “ H and C1 X C1
1 ‰ H, C1 X C1
2 ‰ H, C2 X C1
2 ‰ H. Let u, u1 P Vm correspond to C1
and C2, respectively, and v, v1 P Vm`1 correspond to C1
1 and C1
2, respectively. Then the above is equivalent
to ru1, vs, rv, us, ru, v1s P Em, which is again equivalent to the existence of a path in Em that has length at
least 3.
(iii) The statement follows from the fact that every cycle in Em is even because the graph pVmZVm`1, Emq
is bipartite and the fact that every cycle in Em “ Ktm,tm`1 is non-bounding because dim Ktm,tm`1 “ 1.
A.3
Proofs for Section 5
We continue by proving that 0-conflicts can induce violations of the strong triangle inequality as stated in
Proposition 31.
Proof of Proposition 31. Let x, y, z P X lead to a triangle 0-conflict for the interval rs, ts, i.e., Dr1, r2 P rs, ts:
x „r1 y „r2 z and Er P rs, ts: x ȷr y ȷr z. This means Dθ,spx, yq ď r1 and Dθ,spy, zq ď r2. Let us
define r3 :“ Dθ,spxzq ď tM. We know that r3 ‰ r1 and r3 ‰ r2 as otherwise x „r3 y „r3 z for r3 P rs, ts,
contradicting the lack of transitivity. Hence, without loss of generality, r1 ă r2 ă r3. Then the above is
equivalent to Dθ,spxzq “ r3 ą r1 ě minpDθ,spx, yq, Dθ,spy, zqq, which is a violation of the strong triangle
inequality.
21
Next, we provide a proof of Proposition 32 that establishes the connection between the MPH of the
MCbiF and that of the Merge-Rips bifiltration constructed from the matrix of first-merge times, Dθ,s.
Proof of Proposition 32. First note that L “ pLs,tqt1ďsďt is indeed a well-defined bifiltration because Ls,t Ď
Ls1,t1 if s ě s1 and t ď t1.
In particular, L is also defined uniquely on the finite grid P “ tps, tq P
rt1, . . . , tMs ˆ rt1, . . . , tMs | s ď tu with partial order ps, tq ď ps1, t1q if s ě s1, t ď t1.
The proof of the proposition then follows from a simple extension of Proposition 32 in Schindler and
Barahona [2025] to the 2-parameter case. To see that the 0-dimensional MPH of L and M are equivalent,
note that both bifiltrations have the same 1-skeleton. Moreover, the 1-dimensional MPH is generally not
equivalent because L is a Rips-based bifiltration and thus 2-determined, whereas M is not 2-determined.
If θ is strictly hierarchical, then Dθ,s fulfils the strong-triangle inequality and thus the Rips-based
bifiltration leads to a trivial 1-dimensional homology, see [Schindler and Barahona, 2025, Corollary 33].
Hence, the 1-dimensional MPH of L is trivial.
Finally, we provide a brief proof for Proposition 33 linking the 0-dimensional Hilbert function of a pair of
partitions and the graph Laplacian built from the conditional entropy matrix between both partitions.
Proof of Proposition 33. Using Proposition 27, we prove the statement with the equivalent nerve-based
MCbiF. Note that the graph G :“ Pt2|t1P T
t2|t1 has the same vertices and edges as the simplicial complex
˜Kt1,t2, which is 1-dimensional and thus also a graph according to Proposition 26. This shows that HFpt1, t2q
is given by the number of connected components in G. Furthermore, observe that P T
t2|t11 “ 1, and that the
resulting matrix L is the Laplacian of the undirected graph G. Hence, dimpker Lq is equal to the number of
connected graph components [Chung, 1997], proving the statement.
A.4
Proofs for Section 7
It follows from the construction of MCbiF that the Hilbert functions are invariant to certain swaps of
partitions in θ.
Proposition 41. HFkps, tq is invariant to swaps of partitions in sequence θ between s and t, for t1 ď s ď t.
Proof. Let us denote the change points of θ by t1 ă t2 ă ¨ ¨ ¨ ă tM. Without loss of generality, s “ tm and
t “ tm`n for m ` n ď M. Let us now consider a permutation τ : r1, . . . , Ms Ñ r1, . . . , Ms such that τpiq “ i
for 1 ď i ă m and m ` n ă i ď M and define the permuted sequence of partitions θτ as θτptmq “ θptτpmqq.
Despite the permutation we still get the same MCbiF for θ and θτ for parameters s ď t because
ď
sďrďt
ď
CPθprq
∆C “
ď
sďrďt
ď
CPθτ prq
∆C.
This implies that HFkps, tq is the same for θ and θτ.
B
Additional Examples
Our first example corresponds to the sequence of partitions analysed in Fig.1.
Example 42 (3-element example). Let X “ tx1, x2, x3u and we define θp0q “ ˆ0, θp1q “ ttx1, x2u, tx3uu,
θp2q “ ttx1u, tx2, x3uu, θp3q “ ttx1, x3u, tx2uu and θp4q “ ˆ1 so that θ is coarse-graining with M “ 5 change
points. This example corresponds to Fig. 1a, and the 0- and 1-dimensional Hilbert functions are provided in
Fig. 1b.
Note that HF0p1, 2q ă |θp2q| and HF0p2, 3q ă |θp3q| indicates the presence of two triangle 0-conflicts,
which are not 1-conflicts because HF1p1, 2q “ HF1p2, 3q “ 0. See Fig. 8a for an illustration. As shown in
Proposition 31, the triangle 0-conflicts violate the strong triangle inequality of the matrix of first merge
times Dθ (Eq 10), e.g., Dθpx1, x3q “ 3 ą maxpDθpx1, x2q, Dθpx2, x3qq “ 2.
In addition, HF1p1, 3q “ 1 indicates the presence of a 1-conflict that arises from the higher-order
inconsistencies of cluster assignments across partitions θp1q, θp2q and θp3q. See Fig. 8b for an illustration.
In particular, the equivalence relations x1 „1 x2, x2 „2 x3 and x3 „3 x1 induce a 1-cycle z “ rx1, x2s `
rx2, x3s ` rx3, x1s P Z1pK1,3q and due to the lack of transitivity on the interval r1, 3s, the 1-cycle z is also
non-bounding, yielding a 1-conflict. The 1-conflict then gets resolved at t “ 4 because θp4q “ ˆ1 restores
transitivity on the interval r1, 4s. See Fig. 8c for an illustration.
22
(a) connected 0-conflict
(b) 1-conflict
(c) no conflict
Figure 8: (a) Illustration of a triangle 0-conflict that violates the strong triangle inequality of the matrix of first
merge times Dθ,s (Eq 10), (b) a 1-conflict and (c) three elements that are in no conflict due to global transitivity. If
we choose r1 “ 1, r2 “ 2, r3 “ 3 and r “ 4, the conflicts depicted here correspond to the conflicts in Example 42.
Example 43 (4-element example). We now consider the more complex case of a 4-element set X “
tx1, x2, x3, x4u. Let us start with θp0q “ ˆ0 and append in sequence the 6 distinct partitions that contain
two singletons and one cluster of size 2, i.e., θp1q “ ttx1, x2u, tx3u, tx4uu, θp2q “ ttx1u, tx2, x3u, tx4uu, θp3q “
ttx1u, tx2u, tx3, x4uu, θp4q “ ttx1, x3u, tx2u, tx4uu, θp5q “ ttx1, x4u, tx2u, tx3uu and θp6q “ ttx1u, tx2, x4u, tx3uu.
Finally, we append consecutively three partitions, each of which contains a cluster of size 3, i.e., θp7q “
ttx1, x2, x3u, tx4uu, θp8q “ ttx1u, tx2, x3, x4uu and θp9q “ ttx1, x3, x4u, tx2uu. θ is a coarse-graining, non-
hierarchical sequence with M “ 10 change points. See Fig. 9 for a Sankey diagram of θ.
Figure 9: Hilbert functions and Sankey diagram for the sequence of partitions θ defined in Example 43.
To analyse the topological autocorrelation of θ, we compute the MCbiF Hilbert functions HFk for
dimensions k “ 0, 1 (see Fig. 9). We observe that HF0ps, s ` 3q “ 1 ă minrPrs,s`3s |θprq| for all 0 ď s ď 8,
which implies that the hierarchy of θ is broken after no-less than three steps in the sequence when starting
at scale s. Moreover, we can detect that θ is non-nested and has higher-order cluster inconsistencies because
1-conflicts emerge at scales t “ 4, 5, 6, as indicated by non-zero values in HF1. The 1-conflicts get resolved
one-by-one through the partitions that contain clusters of size 3, and at t “ 9, when the third such partition
appears in θ, all 1-conflicts are resolved.
Finally, we show an example that demonstrates how conditional entropy does not detect 1-conflicts in
general.
Example 44 (CE cannot detect 1-conflicts). Let X “ tx1, x2, x3, x4u for which we consider two differ-
ent sequences of partitions θptq and ηptq such that θp1q “ ηp1q “ ttx1, x2u, tx3u, tx4uu, θp2q “ ηp2q “
ttx1u, tx2, x3u, tx4uu but θp3q “ ttx1, x3u, tx2u, tx4uu ‰ θp3q “ ttx1u, tx2u, tx3, x4uu. See Fig. 10 for a
Sankey diagram representation of the two diagrams. Note that θ and η only differ at scale t “ 3. However,
this difference is crucial because a 1-conflict emerges in θ at scale t “ 0, whereas η has only triangle 0-conflicts
and no 1-conflict. Note that θ corresponds to the toy example in Fig. 1 with one additional isolated element.
In accordance with our theoretical results developed in Section 4.1, we can use the 1-dimensional Hilbert
function HF1 to detect the 1-conflict in θ and distinguish the two sequences. In particular, HF1pθp1q, θp3qq “ 1
but HF1pηpiq, ηpjqq “ 0 for all i, j P r1, 2, 3s, i ď j. In contrast, the conditional entropy H (see Eq. 15)
23
(a) Sequence
(1-conflict)
(b) Sequence
(no 1-conflict)
Figure 10: Sankey diagrams for sequences θ and η defined in Example 44. Note that a 1-conflict emerges in θ at scale
t “ 3, but η has no 1-conflict.
cannot distinguish between the two sequences as they yield the same pairwise conditional entropies. In
particular, Hpθpiq|θpjqq “ Hpηpiq|ηpjqq “ 1
2 log 2 for i ‰ j. This demonstrates that the conditional entropy
cannot detect higher-order cluster inconsistencies in sequences of partitions.
C
Details on the Homology Functor
We provide additional background on simplicial homology and its functoriality, following Hatcher [2002].
Simplicial Homology.
Let K be a simplicial complex defined on the finite set X. For a fixed field
k (the RIVET software uses the finite filed k “ Z2 [Wright and Zheng, 2020]) and for all dimensions
k P t0, 1, ..., dimpKqu we define the k-vector space CkpKq whose elements z are given by a formal sum
z “
ÿ
σPK
dimpσq“k
aσσ
(13)
with coefficients aσ P k, called a k-chain. Note that the k-dimensional simplices σ “ rx0, x1, ..., xks P K form
a basis of CkpKq. For a fixed total order on X, the boundary operator is the linear map Bk : Ck ÝÑ Ck´1
defined through an alternating sum operation on the basis vectors σ “ rx0, x1, ..., xks given by
Bkpσq “
kÿ
i“0
p´1qirx0, x1, ..., ˆxi, ..., xks,
where ˆxi means that vertex xi is deleted from the simplex σ. The boundary operator fulfils the property
im Bk`1 Ă ker Bk. Hence, it connects the vector spaces Ck, k P t0, 1, ..., dimpKqu, through linear maps
. . .
Bk`1
ÝÝÝÑ Ck
Bk
ÝÑ Ck´1
Bk´1
ÝÝÝÑ . . . B2
ÝÑ C1
B1
ÝÑ C0
B0
ÝÑ 0,
leading to a sequence of vector spaces called chain complex. The elements in Zk :“ ker Bk are called k-cycles
and the elements in Bk :“ im Bk`1 are called k-boundaries. Finally, the k-th homology group Hk is defined
as the quotient of vector spaces
Hk :“ Zk{Bk,
(14)
whose elements are equivalence classes rzs of k-cycles z P Zk. Each equivalence class rzs ‰ 0 corresponds to a
generator of non-bounding cycles, i.e., k-cycles that are not the k-boundaries of k ` 1-dimensional simplices.
This captures connected components at dimension k “ 0, holes at k “ 1 and voids at k “ 2.
Functoriality of Hk.
For fixed k, Hk can be considered as a functor Hk : Top Ñ Vect, where Top
denotes the category of topological spaces whose morphisms are continuous maps and Vect the category of
vector spaces whose morphisms are linear maps. In particular, each topological space K is sent to a vector
space HkpKq and a continuous map g : K Ñ K1 is sent to a linear map Hkpgq : HkpKq Ñ HkpK1q such that
compositions of morphisms are preserved, i.e., Hkpg ˝ fq “ Hkpgq ˝ Hkpfq for two continuous maps f and g.
D
Details on Zigzag Persistence
We provide background on zigzag persistence, which was first introduced by Carlsson and de Silva [2010].
For additional details, see Dey and Wang [2022].
24
Zigzag Filtration.
Let t1 ă ¨ ¨ ¨ ă tM be a sequence of real-valued parameter values. For simplicity
we assume tm “ m for m “ 1, . . . , M. Let Km be a simplicial complex defined on the set X for every
m “ 1, . . . , M. If either Km Ď Km`1 or Km`1 Ď Km, for all m “ 1, . . . , M, we call the following diagram
a zigzag filtration:
K1 Ø K2 Ø ¨ ¨ ¨ Ø KM´1 Ø KM,
where Km Ø Km`1 is either a forward inclusion Km ãÑ Km`1 or a backward inclusion Km Ðâ Km`1.
While forward inclusion corresponds to simplex addition, backward inclusion can be interpreted as simplex
deletion.
Zigzag Persistence.
Applying the homology functor Hk to the zigzag filtration leads to a so called zigzag
persistence module given by:
HkpK1q Ø HkpK2q Ø ¨ ¨ ¨ Ø HkpKM´1q Ø HkpKMq,
where HkpKmq Ø HkpKm`1q is either a forward or backward linear map. Using quiver theory, it can be
shown that a zigzag persistence module has a unique interval decomposition that provides a barcode as a
simple invariant.
E
Details on Information-based Baseline Methods
Information-based measures can be used to compare arbitrary pairs of partitions in the sequence θ [Meilă,
2007]. Assuming a uniform distribution on X, the conditional probability distribution of θptq “ tC1, . . . , Cnu
given θpsq “ tC1
1, . . . , C1
mu is:
Pt|sri|js “ |Ci X C1
j|
|C1
j|
,
and the joint probability Ps,tri, js is defined similarly. The conditional entropy (CE) Hpt|sq is then given by
the expected Shannon information:
Hpt|sq “ ´
|θptq|
ÿ
i“1
|θpsq|
ÿ
j“1
Ps,tri, js logpPt|sri|jsq
(15)
It measures how much information about θptq we gain by knowing θpsq. If θpsq ď θptq there is no
information gain and Hpt|sq “ 0. We denote the conditional entropy matrix CEs,t “ Hpt|sq. Furthermore,
we can compute the variation of information (VI) VIps, tq “ Hps|tq ` Hpt|sq, which is a metric. Both CE
and VI are bounded by log N.
Extending information-based measures for the analysis and comparison of more than two partitions is
non-trivial. However, the pairwise comparisons can be summarised with the consensus index (CI) [Vinh
et al., 2010] which can be computed as the average VI:
CIpθq :“
řM
i“1,iăj VIpti, tjq
MpM ´ 1q{2
(16)
F
Details on Experiments
F.1
Regression Task
Figure 11 shows the correlation between the minimal crossing number y “ κpθq (Eq. 3) and information-
and MCbiF-based summary statistics. In addition to the results already described in the main text, we also
observe that the correlation between CI and ¯c0 (r “ ´0.32 for N “ 5, r “ ´0.48 for N “ 10) is stronger
than with ¯c1 (r “ ´0.12 for N “ 5, r “ ´0.34 for N “ 10). This can be explained by the fact that CI and
¯c0 can both be computed from pairwise interactions of clusters in contrast to ¯c1, see Section 5. Furthermore,
we observe a strong correlation between ¯c0 and ¯c1 (r “ 0.52 for N “ 5 and r “ 0.43 for N “ 10) because of
the dependencies between 0- and 1-conflicts, see Section 4.1
Note that we can consider our information- and MCbiF-based feature maps as M ˆ M greyscale images,
where HF0 and HF1 are symmetric and CE is asymmetric. The raw label encoding of θ is also similarly
interpreted as an N ˆ M greyscale image. For our regression task, we train a simple CNN [LeCun and
Bengio, 1998] with one convolution and max-pool layer and one fully connected layer and also a simple
25
N = 5
N = 10
Figure 11: Pearson correlation (r) between crossing number y, information-based consensus index CI and MCbiF-based
conflict measures ¯c0 and ¯c1 for N “ 5 and N “ 10.
MLP [Bishop, 2006] with one or two hidden layers and dropout [Srivastava et al., 2014]. For each feature
map (or their combinations) separately, we perform hyperparameter optimisation for the number of filters
(ranging from 2 to 6) and kernel size (chosen as 4, 8, 16, 32 or 64) in the CNN and the number of nodes
(chosen as 4, 8, 16, 32, 64, 128 or 256), number of layers (1 or 2) and dropout rate (chosen as 0.00, 0.25 or
0.50) in the MLP. We use the Adam optimiser [Kingma and Ba, 2017] with learning rate chosen as 0.01,
0.005, 0.001, 0.0005 or 0.0001 for training.
We perform a full grid search of the hyperparameter space for the three different models and the different
feature maps (or their combinations). We used the train split of our data for training and the validation
split for evaluation and hyperparameter selection. Below, we detail the hyperparameters for the best MCbiF-
and CE-based models, which were chosen according to the performance on the validation split.
• Optimal model for HF0 & HF1 at N “ 5: CNN with 4 filters, kernel size 3, and learning rate 0.001.
• Optimal model for CE at N “ 5: CNN with 8 filters, kernel size 2, and learning rate 0.005.
• Optimal model for HF0 & HF1 at N “ 10: LR.
• Optimal model for CE at N “ 10: MLP with a single layer of 256 nodes, no dropout and a learning
rate of 0.001.
We present the train R2 scores for the optimised LR, CNN and MLP models trained on the different
features in Tables 3. The test R2 scores are presented in Table 1 in the main text.
Table 3: Train R2 scores of LR, CNN and MLP models trained on different features for N “ 5 and N “ 10.
N
Method
Raw label encoding
HF0
HF1
HF0 & HF1
CE
5
LR
0.005
0.163
0.493
0.550
0.409
CNN
0.000
0.170
0.509
0.562
0.515
MLP
0.006
0.160
0.499
0.547
0.439
10
LR
0.013
0.230
0.456
0.522
0.464
CNN
0.009
0.220
0.456
0.519
0.476
MLP
0.003
0.218
0.453
0.515
0.468
F.2
Classification Task
Details on Synthetic Data.
We generate order-preserving (y “ 0) sequences θ P ΠM
N through the
following scheme: Let us assume that we have a total order X “ tx1, . . . , xNu given by the element labels,
i.e., xi ă xj if i ă j. We construct each θptmq, m “ 0, . . . , M ´ 1, by cutting X into clusters of the form
C “ txi, xi`1, . . . , xi`nu. It is easy to verify that θ is indeed order-preserving. We adapt this scheme to
generate sequences θ P ΠM
N that are non-order-preserving (y “ 0): Again, we start by constructing each
sequence θptmq through cutting the ordered set X as before. Additionally, with probability p “ 0.1, we
swap the cluster assignments in θptmq for two arbitrary elements x, y P X. If N and M are large enough,
the so-generated sequence θ is almost surely non-order-preserving. We chose N “ 500 and M “ 30 to
demonstrate the scalability of the MCbiF method.
26
The number of clusters of all our generated sequences of partitions θ P ΠM
N for both classes is decreasing
linearly, see Fig. 12 (a). Moreover, the average number of swaps for sequences with y “ 1 is 2.98 for our
choice of p “ 0.1, see Fig. 12 (b).
Figure 12: Classification task: Histogram of the number of swaps in non-order-preserving sequences θ (class y “ 1).
See text for the scheme to introduce random swaps in the node labels as a means to break order-preservation.
F.3
Application to Real-World Temporal Data
Data Preprocessing.
The temporal sequences of partitions computed by Bovet et al. [2022] are available
at: https://dataverse.harvard.edu/file.xhtml?fileId=5657692. We restricted the partitions to the
N “ 281 mice that were present throughout the full study period to ensure well-defined sequences of
partitions, and considered the first nine temporal resolution values τi, i “ 1, . . . , 9, since θτ10 is an outlier.
Note that the sequences tend to be fine-graining, see Fig. 13.
0
1
2
3
4
5
6
7
8
Scale t [in weeks]
5
10
15
20
25
30
|
(t)|
Temporal resolution
1 = 0.1 s
2 = 1 s
3 = 8 s
4 = 60 s
5 = 1 h
6 = 5 h
7 = 8 h
8 = 24 h
9 = 7 d
Figure 13: Number of clusters over weeks t for different temporal resolutions τ, where larger values of τ produce a
higher number of clusters because of the increased temporal resolution.
τ9
τ8
τ7
τ5
τ4
τ1
τ2
τ3
τ6
Figure 14: Hilbert distance between forward and backward sequences θf
τ and θb
τ for different temporal resolutions τ.
27
Time Reversibility.
In the main text, we restricted our analysis to the so-called forward Flow Stability
sequences of partitions. However, by reversing time direction, Bovet et al. [2022] computed a second set of
backward sequences. For each temporal resolution τi, we thus get a forward and backward sequence denoted
by θf
τi and θb
τi, respectively. Here we use the MCbiF to compare the forward and backward sequences of
partitions for different τi and we compute the Hilbert distance ∥HFkpθf
τiq ´ HFkpθb
τiq ∥2 for k “ 0, 1, see
Fig. 14.
We observe that the Hilbert distance between forward and backward sequences is high for τ2 because the
large-scale group structure changes significantly over the study period, so that the temporal flows at low
resolution τ2 are not reversible. In contrast, the Hilbert distance between forward and backward sequences is
low for τ8 because the underlying social groups are more stable over the study period, leading to increased
time reversibility at the high temporal resolution.
28
|
MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology Juni Schindler∗and Mauricio Barahona† . Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF feature maps outperform information-based baseline features on both regression and classification tasks on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time. Keywords: topological data analysis, multiparameter persistent homology, multiscale clustering, nonhierarchical clustering, Sankey diagrams 1 Introduction In many applications, datasets possess an intrinsic multiscale structure, whereby meaningful descriptions exist at different scales, i.e., at different resolutions or levels of coarseness. Think, for instance, of the multi-resolution structure in commuter mobility patterns [Alessandretti et al., 2020, Schindler et al., 2023], communities in social networks [Beguerisse-Díaz et al., 2017] and thematic groups of documents [Blei et al., 2003, Grootendorst, 2022]; the subgroupings in single-cell data [Hoekzema et al., 2022] or phylogenetic trees [Chan et al., 2013]; and the functional substructures in proteins [Delvenne et al., 2010, Delmotte et al., 2011]. In such cases, the natural description of the dataset goes beyond a single clustering and consists of a multi-resolution sequence of partitions across scales parametrised by a scale parameter t. Traditionally, multiscale descriptions have emerged from hierarchical clustering, where t corresponds to the depth of the dendrogram [Carlsson and Mémoli, 2010, Murtagh and Contreras, 2012]. However, in many important real-world applications, the data structure is multiscale, yet non-hierarchical. Examples include temporal clustering, where t corresponds to physical time [Rosvall and Bergstrom, 2010, Liechti and Bonhoeffer, 2020, Bovet et al., 2022]; topic modelling and document classification, where t captures the coarseness of the topic groupings [Altuncu et al., 2019, Fukuyama et al., 2023, Liu et al., 2025]; and generic multiscale clusterings for data that result from exploiting a diffusion on the data geometry, where t is the increasing time horizon of the diffusion [Coifman et al., 2005, Azran and Ghahramani, 2006, Lambiotte et al., 2014]. A natural problem is then how to analyse and compare non-hierarchical multi-resolution sequences of partitions that are organised by the scale t. Here we address this question from the perspective of topological data analysis [Carlsson and Zomorodian, 2009, Carlsson et al., 2009, Botnan and Lesnick, 2023] by introducing the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes the patterns of cluster intersections across all scales. ∗Corresponding author: , ORCID ID: 0000-0002-8728-9286 †Corresponding author: , ORCID ID: 0000-0002-1089-5675 1 16 Oct 2025 Problem definition. A partition π of a finite set X " tx1, x2, ..., xNu is a collection of mutually exclusive subsets Ci Ď X (here called clusters) that cover X, i.e., π " tC1, . . . , Ccu such that X " Ťc i"1 Ci, and Ci Ş Cj " H, @i ‰ j. The cardinality |π| " c is the number of clusters in π and, for notational convenience, we use πi to denote the i-th cluster Ci of π. Let ΠX denote the space of partitions of X. We write π ď π1 if every cluster in π is contained in a cluster of π1. This refinement relation constitutes a partial order and leads to the partition lattice pΠX, ďq with lower bound ˆ0 :" ttx1u, . . . , txNuu and upper bound ˆ1 :" tXu [Birkhoff, 1967]. Here, we consider a sequence of partitions defined as a piecewise-constant function θ : rt1, 8q Ñ ΠX, t ÞÑ θptq P ΠX such that a partition of X is assigned to each t, and the scale index t P rt1, 8q has M change points t1 ă t2 ă ... ă tM. In particular, θptq " θptmq for t P rtm, tm`1q, m " 1, . . . , M ́ 1, and θptq " θptMq for t P rtM, 8q. The sequence θ is called hierarchical if either θpsq ď θptq, @s ď t, or θpsq ě θptq, @s ď t. The sequence θ is called coarse-graining if |θpsq| ě |θptq|, @s ď t. 1 Conversely, θ is called fine-graining if |θpsq| ď |θptq|, @s ď t. Our goal is to characterise and analyse arbitrary sequences of partitions θ, including non-hierarchical ones, in an integrated manner, taking account of memory effects across the scale t. Remark 1. Here, we are not concerned with the task of computing the multiscale clustering (i.e., the sequence of partitions θ) from dataset X, for which several methods exist. Rather, we take θ as a given, and we aim to analyse its structure. Remark 2. This problem is distinct from consensus clustering, which aims to produce a summary partition by combining a set of partitions obtained, e.g., from different optimisations or clustering algorithms [Strehl and Ghosh, 2002, Vega-Pons and Ruiz-Shulcloper, 2011]. 1 Input: Sequence of partitions Output: Multiscale Clustering Bifiltration 0 2 3 4 0 1 2 3 4 (a) (b) Figure 1: (a) Illustration of how the MCbiF encodes the structure of a non-hierarchical sequence of partitions θ as a bifiltration of abstract simplicial complexes Ks,t. See Example 42 for a detailed description. (b) The Hilbert functions HFkps, tq of the MCbiF are invariants that capture the topological autocorrelation of θ: violations of the refinement order at dimension k " 0, and higher-order cluster inconsistencies at dimension k " 1. The Hilbert functions can be used as feature maps for downstream machine learning tasks. Contributions. To address this problem, we define the MCbiF, a bifiltration of abstract simplicial complexes, which represents the clusters and their intersection patterns in the sequence θ for varying starting scale s and lag t ́ s (Fig. 1). Using the machinery of multiparameter persistent homology (MPH) [Carlsson and Zomorodian, 2009, Carlsson et al., 2009, Botnan and Lesnick, 2023], we prove that the MCbiF leads to 1Coarse-graining is equivalent to non-decreasing mean cluster size (see Remark 37 in Appendix A.1). 2 a block decomposable persistence module with stable Hilbert functions HFkps, tq, and we show that these invariants serve as measures of the topological autocorrelation of the sequence of partitions θ across the scale t. In particular, HFkps, tq quantifies the non-hierarchy in θ in two complementary ways: at dimension k " 0, it detects the lack of a maximal partition in the subposet θprs, tsq with respect to refinement, and at dimension k " 1, it quantifies the higher-order inconsistencies of cluster assignments across scales. In contrast, baseline methods such as ultrametrics [Carlsson and Mémoli, 2010] or information-based measures [Meilă, 2003] are restricted to pairwise comparisons between, respectively, elements or clusters; hence these methods cannot detect higher-order cluster inconsistencies. Furthermore, we provide an equivalent nerve-based construction of the MCbiF that can be interpreted as a higher-order extension of the Sankey diagram of the sequence of partitions. In the hierarchical case, the 1-dimensional MPH of the MCbiF is trivial, and the Sankey diagram reduces to a dendrogram such that the 0-dimensional MCbiF Hilbert function can be obtained from the number of branches in the dendrogram. The Hilbert functions of the MCbiF provide interpretable feature maps that can be used in downstream machine learning tasks. In our experiments, the MCbiF feature maps outperform information-based baseline features [Meilă, 2007] on both regression and classification tasks on non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time [Bovet et al., 2022]. 2 Related Work Dendrograms and Ultrametrics. A hierarchical, coarse-graining sequence θ with θpt1 " 0q " ˆ0 and θptMq " ˆ1 is called an agglomerative dendrogram, and can be represented by an acyclic rooted merge tree [Jain et al., 1999, Carlsson and Mémoli, 2010]. One can define an ultrametric Dθ from the first-merge times, which corresponds to the depth in the dendrogram. Carlsson and Mémoli [2010] showed that there is a one-to-one correspondence between agglomerative dendrograms and ultrametrics, which can be used to efficiently compare two such dendrograms via the Gromov-Hausdorff distance between the ultrametric spaces [Mémoli et al., 2023]. When θ is non-hierarchical, however, the first-merge times no longer define the sequence uniquely because clusters that have merged can split off again. In this case, θ cannot be represented by a tree and Dθ does not fulfil the triangle inequality in general. Hence, ultrametrics cannot be used to analyse and compare non-hierarchical sequences of partitions (see Section 5). Information-based Comparison of Clusterings. Information-based measures can be used to compare a pair of partitions. Assuming a uniform distribution on X, one can derive probability distributions for partitions interpreted as random variables and thus measure the information gain and loss between two partitions using the conditional entropy (CE) or the variation of information (VI), which is a metric on ΠX [Meilă, 2003, 2007]. See Appendix E for detailed formulas. Extending information-based measures to more than two partitions is non-trivial. In consensus clustering, the average VI is used as a consensus index (CI) [Vinh and Epps, 2009, Vinh et al., 2010] for multiple partitions. However, the CI is independent of the ordering in the sequence and so cannot capture memory effects in sequences of partitions. Another limitation of these measures is that they rely only on the joint probability between pairs of random variables, hence higher-order cluster inconsistencies are not captured (see Section 5). 3 Background 3.1 Sankey Diagrams Non-hierarchical sequences of partitions θ are visualised by M-layered flow graphs Spθq " pV " V1 Z ... Z VM, E " E1 Z ... Z EM ́1q called Sankey diagrams [Sankey, 1898, Zarate et al., 2018], where each level m " 1, . . . , M corresponds to a partition and vertices Vm represent its clusters while the directed edges Em between levels indicate the overlap between clusters: Vm :" tpm, iq | 1 ď i ď |θptmq|u and Em " trpm, iq, pm ` 1, jqs | θptmqi X θptm`1qj ‰ Hu , (1) where ru, vs P Em denotes a directed edge from u P Vm to v P Vm`1. If θ is hierarchical, the Sankey diagram Spθq is a directed tree-a merge-tree if θ is coarse-graining, or a split-tree if θ is fine-graining. The graph Spθq is sometimes also called an alluvial diagram [Rosvall and Bergstrom, 2010]. Sankey diagrams are studied in computer graphics as they allow for the visualisation of complex relational data. In this context, a Sankey diagram is represented as a layout on the plane, whereby the nodes in each 3 layer Vm are vertically ordered according to a ranking τm : Vm Ñ t1, . . . , |Vm|u, and the layout of the Sankey diagram is then defined by the collection of such rankings, τ :" pτ1, . . . , τMq. For visualisation purposes, the layered layout should ideally minimise the number of crossings between consecutive layers, where a crossing between two edges ru, vs, ru1, v1s P Em occurs if τmpuq ą τmpu1q and τm`1pvq ă τm`1pv1q or vice versa, and the crossing number [Warfield, 1977] is given by: κθpτq :" M ́1 ÿ m"1 ÿ ru,vs,ru1,v1sPEm 1τmpuqąτmpu1q^τm`1pvqăτm`1pv1q, (2) where 1 denotes the indicator function. The crossing number κθpτq of the layout of the Sankey diagram Spθq can be minimised by permuting the rankings in the layers, τm, and we denote the minimum crossing number for the layout as: κθ :" min τ κθpτq. (3) This problem is known to be NP-complete [Garey and Johnson, 2006] and finding efficient optimisation algorithms is an active research area [Zarate et al., 2018, Li et al., 2025]. 3.2 Multiparameter Persistent Homology Multiparameter persistent homology (MPH) is an extension of standard persistent homology to n ą 1 parameters, first introduced by Carlsson and Zomorodian [2009]. We present here basic definitions, see Carlsson and Zomorodian [2009], Carlsson et al. [2009], Botnan and Lesnick [2023] for details. Simplicial Complex. Let K be a simplicial complex defined for the set X, such that K Ď 2X and K is closed under the operation of building subsets. The elements of σ P K are called simplices and a k-dimensional simplex (or k-simplex) can be represented as σ " rx1, ..., xk`1s where x1, . . . , xk`1 P X and we have fixed an arbitrary order on X. Note that k " 0 corresponds to vertices, k " 1 to edges, and k " 2 to triangles. We define the k-skeleton Kk of K as the union of its n-simplices for n ď k. We also define dimpKq as the largest dimension of any simplex in K. Multiparameter Filtration. Let us define the parameter space pP, ďq as the product of n ě 1 partially ordered sets P " P1 ˆ ̈ ̈ ̈ ˆ Pn, i.e., a ď b for a, b P P if and only if ai ď bi in Pi for i " 1, . . . , n. A collection of subcomplexes pKaqaPRn with K " Ť aPRn Ka and inclusion maps tia,b : Ka Ñ Kbuaďb that yield a commutative diagram is called a multiparameter filtration (or bifiltration for n " 2). We denote by birthpσq Ď P the set of parameters, called multigrades (or bigrades for n " 2), at which simplex σ P K first appears in the filtration. For example, the sublevel filtration Ka " tσ P K | fpσq ď au for a filtration function f : K Ñ P maps each simplex σ to a unique multigrade fpσq, i.e., |birthpσq| " 1. A filtration is called one-critical if it is isomorphic to a sublevel filtration, and multi-critical otherwise. Multiparameter Persistent Homology. Let Hk for k P t0, . . . , dimpKqu denote the k-dimensional homology functor with coefficients in a field [Hatcher, 2002], see Appendix C for details. Then Hk applied to the multiparameter filtration leads to a multiparameter persistence module, i.e., a collection of vector spaces pHkpKaqqaPRn, which are the homology groups whose elements are the generators of k-dimensional non-bounding cycles, and linear maps tıa,b :" Hkpia,bq : HkpKaq Ñ HkpKbquaďb that yield a commutative diagram called multiparameter persistent homology (MPH). For dimension k " 0, Hk captures the number of disconnected components and for k " 1, the number of holes. Note that, for n " 1, we recover standard persistent homology (PH) [Edelsbrunner et al., 2002]. Hilbert Function. While barcodes are complete invariants of 1-parameter PH (n " 1), the more complicated algebraic structure of MPH (n ě 2) does not allow for such simple invariants in general; hence, various non-complete invariants of the MPH are used in practice. We focus on the k-dimensional Hilbert function [Botnan and Lesnick, 2023] defined as HFk : P Ñ N0, a ÞÑ rankrHkpia,aqs " dimrHkpKaqs, (4) which maps each filtration index a to the k-dimensional Betti number of the corresponding complex Ka. We call the k-dimensional MPH trivial if HFk " 0. The Hilbert distance is then defined as the L2 norm on the space of Hilbert functions and can be used to compare multiparameter persistence modules. 4 4 The Multiscale Clustering Bifiltration (MCbiF) The central object of our paper is a novel bifiltration of abstract simplicial complexes that encodes cluster intersection patterns in the sequence of partitions θ across the scale t. Definition 3 (Multiscale Clustering Bifiltration). Let θ : rt1, 8q Ñ ΠX be a sequence of partitions. We define M, the Multiscale Clustering Bifiltration (MCbiF) as a bifiltration of abstract simplicial complexes: M :" pKs,tqt1ďsďt where Ks,t :" ď t1ďsďrďt ď CPθprq ∆C, t1 ď s ď t. (5) In this construction, each cluster C corresponds to a p|C| ́ 1q-dimensional solid simplex ∆C :" 2C, which, by definition, contains all its lower dimensional simplices [Schindler and Barahona, 2025]. This echoes natural concepts of data clustering as information compression or lumping [Rosvall and Bergstrom, 2008, 2011, Lambiotte et al., 2014], and of clusters as equivalence classes [Brualdi, 2010]. The MCbiF then aggregates all clusters (simplices) from partition θpsq to θptq through the union operators, such that a k-simplex σ " rx1, . . . , xk`1s P Ks,t consists of elements that are assigned to the same cluster (at least once) in the interval rs, ts, i.e., x1, . . . , xk`1 P C for some cluster C P θprq and r P rs, ts. The bifiltration thus depends not only on the lag |t ́ s| but also on the starting scale s, and captures the topological autocorrelation in the sequence of partitions, see Fig. 1. We first show that the MCbiF is indeed a well-defined bifiltration. Proposition 4. M is a multi-critical bifiltration uniquely defined by its values on the finite grid P " tps, tq P rt1, . . . , tMs ˆ rt1, . . . , tMs | s ď tu with partial order ps, tq ď ps1, t1q if s ě s1, t ď t1. The proof is straightforward and can be found in Appendix A.2. The MCbiF leads to a triangular commutative diagram where the arrows indicate inclusion maps between abstract simplicial complexes (see Fig. 1). The sequence of partitions θptq is encoded by the complexes Kt,t on the diagonal of the diagram. Moving along horizontal arrows corresponds to fixing a starting scale s and going forward in the sequence θ, thus capturing the coarse-graining of partitions. Moving along vertical arrows corresponds to fixing an end scale t and aggregating θ going backwards, capturing the fine-graining of partitions. Remark 5. By fixing s :" t1 (i.e., the top row in the commutative MCbiF diagram), we recover the 1-parameter Multiscale Clustering Filtration (MCF) defined by Schindler and Barahona [2025]. MCF was designed to quantify non-hierarchies in coarse-graining sequences of partitions and thus cannot capture fine-graining. For example, a large cluster C P θps1q prevents MCF from detecting cluster assignment conflicts between elements x, y P C for t ě s1, see Section 4.1. In contrast, MCbiF encodes the full topological autocorrelation contained in θ by varying both the starting scale s and the lag t ́ s. Applying MPH to the bifiltration M at dimensions k ď dim K, for K " KtM,tM , leads to a triangular diagram of simplicial complexes HkpKs,tq called persistence module (see Section 3.2). This persistence module has strong algebraic properties, as stated in the following proposition. Proposition 6. For any k ď dim K, the MCbiF persistence module HkpKs,tq is pointwise finite-dimensional, finitely presented and block-decomposable. See Botnan and Lesnick [2023] for definitions and Appendix A.2 for details and a full proof. The properties in Proposition 6 are important because they guarantee algebraic stability of the MCbiF [Bjerkevik, 2021]. In particular, the finite presentation property implies stability of the MCbiF Hilbert functions HFkps, tq (Eq. 4) with respect to small changes in the module [Oudot and Scoccola, 2024, Corollary 8.2.]. This justifies the use of HFkps, tq as simple interpretable invariants for the topological autocorrelation captured by MCbiF, as exploited in Section 4.1. 4.1 Measuring Topological Autocorrelation with MCbiF We now show how topological autocorrelation as measured by HFkps, tq can be used to detect clusterassignment conflicts. We focus on dimensions k " 0, 1, for which MPH is implemented in RIVET [Lesnick and Wright, 2015]. Note that HF0ps, tq counts the number of connected components and HF1ps, tq the number of 1-dimensional holes in Ks,t, see Section 3.2 for details. We show below that the computation of these invariants reveals different aspects of the non-hierarchy in the sequence of partitions. 5 4.1.1 Low-order Non-Hierarchy in Sequences of Partitions Hierarchy in a sequence of partitions can be understood as a refinement of partitions captured by the partition lattice. Definition 7 (Hierarchy). The sequence of partitions θ is hierarchical in rs, ts if we have a strict sequence of refinements: either θpr1q ď θpr2q, @r1, r2 P rs, ts with r1 ď r2 (agglomerative) or θpr1q ě θpr2q, @r1, r2 P rs, ts with r1 ď r2 (divisive). We say that θ is strictly hierarchical if it is hierarchical in rt1, 8q. One important aspect of hierarchy is the nestedness of the clusters in the sequence. Definition 8 (Nestedness). We say that θ is nested in rs, ts when @r1, r2 P rs, ts, we have that @C P θpr1q, C1 P θpr2q, one of the sets CzC1, C1zC or C X C1 is empty. See [Korte and Vygen, 2012, Definition 2.12]. We say that θ is strictly nested when θ is nested in rt1, 8q. Remark 9. It follows directly from the definitions that a hierarchical sequence θ is always nested. However, nestedness does not necessarily imply hierarchy, as illustrated by the example in Fig 3b. We can quantify the low-order non-hierarchy in a sequence θ through the computation of the invariant HF0ps, tq and the associated notion of 0-conflicts defined next. Remark 10. Each partition θptq can be interpreted as an equivalence relation „t given by the property of belonging to the same cluster, i.e., x „t y if D C P θptq such that x, y P C [Brualdi, 2010]. Definition 11 (0-conflict and triangle 0-conflict). a) We say that θ has a 0-conflict in rs, ts if the subposet θprs, tsq has no maximum, i.e., Er P rs, ts such that θpr1q ď θprq, @r1 P rs, ts. b) We say that θ has a triangle 0-conflict in rs, ts if D x, y, z P X such that Dr1, r2 P rs, ts: x „r1 y „r2 z and Er P rs, ts: x ȷr y ȷr z. Next, we show that all triangle 0-conflicts are also 0-conflicts. Moreover, all 0-conflicts break hierarchy and triangle 0-conflicts additionally break nestedness. Proposition 12. (i) Every triangle 0-conflict is a 0-conflict, but the opposite is not true. (ii) If θ has a 0-conflict in rs, ts, then θ is non-hierarchical in rs, ts. (iii) If θ is either coarse- or fine-graining but non-hierarchical in rs, ts, then θ has a 0-conflict in rs, ts. (iv) If θ has a triangle 0-conflict in rs, ts, then θ is non-nested in rs, ts. See Appendix A.2.1 for the simple proof. Fig. 3b illustrates a 0-conflict that is not a triangle 0-conflict, and Fig. 3c shows a triangle 0-conflict. Remark 13. Non-nestedness and non-hierarchy do not imply the presence of a 0-conflict. To see this, consider the simple counter-example given by θp0q " ttx, yu, tzuu, θp1q " ˆ1, θp2q " ttxu, ty, zuu, which is non-nested but the partition θp1q is the maximum of the subposet θpr0, 1, 2sq. This illustrates the need for the additional assumption of coarse- or fine-graining of θ in Proposition 12 (iii) for the condition of no 0-conflict to imply hierarchy. The following proposition develops a sharp upper bound for HF0 that can be used to capture 0-conflicts. Proposition 14. (i) HF0ps, tq ď minrPrs,ts |θprq|, @rs, ts Ď rt1, 8q. (ii) HF0ps, tq ă minrPrs,ts |θprq| iff θ has a 0-conflict in rs, ts. (iii) HF0ps, tq " |θprq| for r P rs, ts iff θprq is the maximum of the subposet θprs, tsq. See Appendix A.2.1 for a full proof. Proposition 14 shows that HF0 measures low-order non-hierarchy in θ by capturing 0-conflicts. To quantify this, we introduce a global normalised measure for the sequence θ, defined as follows. Definition 15 (Average 0-conflict). Let T :" tM ` tM ́t1 M ́1 . The average 0-conflict is defined as: 0 ď ̄c0pθq :" 1 ́ 2 |T ́ t1|2 ż T t1 ż T s HF0ps, tq minrPrs,ts HF0pr, rqds dt ď 1. (6) 6 Higher values of ̄c0pθq indicate a high level of 0-conflicts and increased low-order non-hierarchy, as shown by the next corollary. Corollary 16. (i) If θ is hierarchical in rs, ts, then HF0ps, tq " minp|θpsq|, |θptq|q. As a special case, this implies HF0pt, tq " |θptq|, @t ě t1. (ii) ̄c0pθq ą 0 iff θ has a 0-conflict. (iii) Let θ be either coarse- or fine-graining. Then, ̄c0pθq " 0 iff θ is strictly hierarchical. A proof can be found in Appendix A.2.1. We can further detect triangle 0-conflicts by analysing the graph-theoretic properties of the MCbiF 1-skeleton Ks,t 1 . Recall that the clustering coefficient C of a graph is defined as the ratio of the number of triangles to the number of paths of length 2 in the graph [Luce and Perry, 1949, Newman, 2018]. Proposition 17. CpKs,t 1 q ă 1 iff there is a triple x, y, z P X that leads to a triangle 0-conflict for rs, ts, and which is not a cycle, i.e., additionally to property b) in Definition 11 we also have Er3 P rs, ts: x „r3 z. See Appendix A.2.1 for a proof. Let us consider the graph generated as the disjoint union of all clusters from partitions in [s,t] as cliques. This graph is equivalent to the MCbiF 1-skeleton Ks,t 1 . Proposition 17 shows that the clustering coefficient of this graph can be used to detect triangle 0-conflicts that are not cycles. To be able to detect triangle 0-conflicts that correspond to non-bounding cycles, we turn to the 1-dimensional homology in the next section. 4.1.2 Higher-order Inconsistencies between Clusters in Sequences of Partitions Measuring 0-conflicts in θ is only one way of capturing non-hierarchy. An additional phenomenon that can arise in non-hierarchical sequences is higher-order inconsistencies of cluster assignments across scales. These are captured by the 1-dimensional homology groups [Schindler and Barahona, 2025] and the associated notion of 1-conflict, which we define next. Recall the definition of 1-cycles Z1pKs,tq and non-bounding cycles H1pKs,tq (Eq.14) summarised in Appendix C. Definition 18 (1-conflict). We say that θ has a 1-conflict in rs, ts if D x1, . . . , xn P X such that the 1-cycle z " rx1, x2s ` ̈ ̈ ̈ ` rxn ́1, xns ` rxn, x1s P Z1pKs,tq is non-bounding; in other words, rzs P H1pKs,tq with rzs ‰ 0. The number of distinct 1-conflicts for the interval rs, ts (up to equivalence of the homology classes) is given by HF1ps, tq. We first show that 1-conflicts also lead to triangle 0-conflicts and thus break hierarchy and nestedness of θ. Proposition 19. (i) HF1ps, tq ě 1 iff θ has a 1-conflict in rs, ts. (ii) If θ has a 1-conflict in rs, ts, then it also has a triangle 0-conflict. (iii) If θ is hierarchical in rs, ts, then HF1ps, tq " 0. See Appendix A.2.1 for a proof. Proposition 19 shows that a 1-conflict is a special kind of triangle 0-conflict arising from higher-order cluster inconsistencies across scales. This is illustrated in Fig. 12 and more details are provided in Examples 42 and 43 in Appendix B, which present sequences of partitions where different 1-conflicts emerge across scales. Moreover, Proposition 19 (iii) shows that the MCbiF has a trivial 1-dimensional MPH if θ is strictly hierarchical. Remark 20. Proposition 19 states that every 1-conflict is a triangle 0-conflict. However, not every (triangle) 0-conflict is a 1-conflict, see Example 42. Note also that several triangle 0-conflicts in the sequence θ can lead to a 1-conflict, when the triangle 0-conflicts are linked together in such a way as to form a non-bounding cycle, see Example 42. We can test for these systematically using HF1. Remark 21. The presence of a 1-conflict for the interval rs, ts signals the fact that assigning all the elements involved in the conflict to a shared cluster would increase the consistency of the sequence. Hence, when a 1-conflict gets resolved, e.g., the corresponding homology generator dies in the MPH at ps, t1q, t ă t1, then we say that θpt1q is a conflict-resolving partition, see Schindler and Barahona [2025]. As two illustrations, consider Example 42, where a 1-conflict emerges through the interplay of partitions at scales t " 1, 2, 3 and gets resolved by θp4q " ˆ1, and Example 43, where three different 1-conflicts first emerge between scales t " 1 and t " 6 and then get resolved one by one by partitions θp7q, θp8q and θp9q. 7 To quantify the presence of 1-conflicts in θ we introduce an unnormalised global measure for the sequence θ, as follows. Definition 22 (Average 1-conflict). Let T be defined as in Definition 15. The average 1-conflict is defined as: 0 ď ̄c1pθq :" 2 |T ́ t1|2 ż T t1 ż T s HF1ps, tqds dt. (7) Corollary 23. c1pθq ą 0 iff θ has a 1-conflict. In particular, if θ is strictly nested, then c1pθq " 0. Remark 24. While 0-conflicts ( ̄c0pθq ą 0) can be defined in relation to the refinement order that gives rise to the partition lattice, the partition lattice cannot be used to detect higher-order cluster inconsistencies (1-conflicts), which can be captured and quantified instead by HF1 and the average measure ̄c1pθq. In Figure 2, we provide a summary of our theoretical results and their relationships. Proposition 12, Remark 13 0-conflict in Proposition 12, Remark 13 Proposition 12 Triangle 0-conflict in Definition 18 Proposition 19 1-conflict in Non-hierarchical in Remark 9 Non-nested in Proposition 19 Higher-order inconsistencies in Definition 18 Corollary 23 1-dim. Hilbert function Proposition 14 0-dim. Hilbert function Corollary 16 Average 0-conflict Clustering Coefficient Average 1-conflict Proposition 31 Strong triangle inequality of violated Proposition 17 Figure 2: Summary of key theoretical results and their relationships indicated by arrows. Double-headed arrows represent equivalences (iff), whereas single-headed arrows represent implications (if). 4.2 MCbiF as a Higher-Order Sankey Diagram Recall the definition of the Sankey diagram of the sequence of partitions introduced in Section 3.1, and its associated representation in terms of an M-layered graph with vertices Vm at each layer representing the clusters of θptmq, see Eq. (1). Let us define the disjoint union Apl, mq :" VlZ ... Z Vm, 1 ď lď m, which assigns an index to each cluster in θptq for t P rtl, tms. Furthermore, recall that θptqi denotes the i-th cluster Ci of θptq. The nerve-based MCbiF can then be defined as follows. Definition 25 (Nerve-based MCbiF). Let s P rtl, tl`1q, l" 1, ..., M ́1, and t P rtm, tm`1q, m " l, ..., M ́1 or t ě tm for m " M. We define the nerve-based MCbiF as ̃ M :" p ̃Ks,tqt1ďsďt, where ̃Ks,t :" tσ Ď Apl, mq : č pn,iqPσ θptnqi ‰ Hu. (8) The nerve-based MCbiF ̃ M is a 1-critical bifiltration with simplices representing clusters and their intersections, in contrast to the original MCbiF M (Eq (5)) in which the simplices represent elements in X and their equivalence relations. Despite these different perspectives, Proposition 40 in the appendix shows that ̃ M and M lead to the same MPH and can be considered as equivalent. The proof of this equivalence follows from an extension of results by Schindler and Barahona [2025]. However, the dimensionality of M and ̃ M can differ, as shown in the following proposition. Proposition 26. (i) dim Ks,t " maxsďrďt maxcPθprq |C| ́ 1, @t1 ď s ď t. (ii) dim ̃Ktm,tm`n " n, @1 ď m ď M, 0 ď n ď M ́ m. 8 See proof in Appendix A.2.2. The nerve-based MCbiF is therefore computationally advantageous when M ă maxt1ďt maxCPθptq |C| ́ 1, making it preferable in applications where the clusters are large. The nerve-based MCbiF ̃ M can be interpreted as a higher-order extension of the Sankey diagram Spθq (Eq. 1) in the sense that ̃ M not only records pairwise intersections between clusters in consecutive partitions of θ, like Spθq), but also takes into account higher-order intersections between clusters in sub-sequences of θ. More formally, we can state the following proposition that relates Spθq to the nerve-based MCbiF. Proposition 27. The Sankey diagram graph Spθq is a strict 1-dimensional subcomplex of ̃K :" ̃Kt1,tM . In particular, Vm " Ktm,tm and Em " Ktm,tm`1, @m " 1, . . . , M ́ 1. Hence, we can retrieve Spθq from the zigzag filtration ̈ ̈ ̈ Ðâ ̃Ktm,tm ãÑ ̃Ktm,tm`1 Ðâ ̃Ktm`1,tm`1 ãÑ . . . , (9) which is a subfiltration of the nerve-based MCbiF. A proof can be found in Appendix A.2.2. For details on zigzag persistence, see Carlsson and de Silva [2010] and Appendix D. Furthermore, the 0- and 1-conflicts that can arise in a single layer Em of the Sankey diagram can be fully characterised as follows. Proposition 28. (i) There is a 0-conflict in rtm, tm`1s iff Du P Vm and v P Vm`1 with degpuq ě 2 and degpvq ě 2, where deg denotes the node degree in the bipartite graph pVm Z Vm`1, Emq associated with the Sankey diagram. (ii) There is a triangle 0-conflict in rtm, tm`1s iff there is a path of length at least 3 in Em. (iii) There is 1-conflict in rtm, tm`1s iff there is an (even) cycle in Em. See Appendix A.2.2 for the proof and Fig. 3 for an illustration. (b) 0-conflict (c) Triangle 0-conflict MCbiF Sankey diagram (d) 1-conflict (a) No conflict Figure 3: Relationship between different types of conflicts and the crossings in a single-layer Sankey diagram. Importantly, a cycle in Em leads to a crossing in Em that cannot be undone, see Fig. 3d for an illustration. Hence, Proposition 28 (iii) implies that the sum of the elements of the superdiagonal of HF1 provides a lower bound for the minimal crossing number of the Sankey diagram, κθ (defined in Eq. 2). We state this as the following corollary. Corollary 29. řM ́1 m"1 HF1ptm, tm`1q ď κθ. Remark 30. Note that 1-conflicts that arise across multiple partitions in the sequence (i.e., across multiple layers) do not necessarily lead to crossings. See Fig. 1, where the chosen ordering of the elements does not lead to a crossing in the layout of the Sankey diagram despite the presence of a 1-conflict. However, we hypothesise that the full HF0 and HF1 feature maps capture more complicated crossings that arise in the Sankey layout across many layers. This insight is exploited in our computational tasks below. 9 5 Mathematical Links of MCbiF to Other Methods We now present some mathematical connections of MCbiF to the related methods introduced in Section 2. Ultrametrics. Given a sequence of partitions θ with θpt1 " 0q " ˆ0 and θptMq " ˆ1, let us define the matrix of first-merge times conditioned on the starting scale s: Dθ,spxi, xjq :" mintt ě s | D C P θptq : xi, xj P Cu. (10) Clearly, when s " 0, this recovers the standard matrix of first-merge times Dθ :" Dθ,0 discussed in Section 2. If θ is hierarchical, i.e., an agglomerative dendrogram, then Dθ is an ultrametric, i.e., it fulfils the strong triangle inequality: Dθpx, zq ď max pDθpx, yq, Dθpx, zqq @x, y, z P X. Corollary 16 states that the number of branches in the agglomerative dendrogram at level t, which is given by |θptq|, is equal to HF0ps, tq for any s ď t. Hence, HF0ps, tq contains the same information as the ultrametric in the hierarchical case, see also Schindler and Barahona [2025] and Proposition 32 below. If, on the other hand, θ is non-hierarchical, triangle 0-conflicts can lead to violations of the (strong) triangle inequality: Proposition 31. The triplet x, y, z P X leads to a triangle 0-conflict in rs, ts iff x, y, z violate the strong triangle inequality for Dθ,s, i.e., Dθ,spx, zq ą maxpDθ,spx, yq, Dθ,spy, zqq. See Appendix A.3 for a proof and Fig. 8a for an illustration. Proposition 31 shows that ̄c0pθq measures how much the ultrametric property of Dθ is violated. Recall that Dθ,s is a dissimilarity measure that can be used to define a filtration [Chazal et al., 2014]. Next, we show that the 0-dimensional MPH of MCbiF corresponds to the 0-dimensional MPH of a Rips-based bifiltration constructed from Dθ,s. Proposition 32. Let us define the Merge-Rips bifiltration L based on Dθ,s as L " pLs,tqt1ďsďt where Ls,t " tσ Ă X | @x, y P σ : Dθ,spx, yq ď tu. (11) Then the 0-dimensional MPH of the Merge-Rips bifiltration, L, and of the MCbiF, M, are equivalent, but the 1-dimensional MPH of L and M are generally not equivalent. Furthermore, if θ is strictly hierarchical, then L has a trivial 1-dimensional MPH. A proof is presented in Appendix A.3 and follows from an argument in Schindler and Barahona [2025]. In the hierarchical case, the 1-dimensional MPH of the Merge-Rips bifiltration is trivial because Dθ fulfils the strong triangle inequality, and is thus equivalent to the MCbiF, whose 1-dimensional MPH is also trivial in the hierarchical case, see Proposition 19. Conditional Entropy. The conditional entropy (CE) is only defined for pairs of partitions. CE is defined as the expected Shannon information of the conditional probability of a partition θptq " tC1, . . . , Cnu given θpsq " tC1 1, . . . , C1 mu: Pt|sri|js " |Ci X C1 j| |C1 j| . For the special case of M " 2 (i.e., only two partitions in the sequence θ), it can be shown that HF0pt1, t2q follows directly from the spectral properties of the matrix Pt2|t1P T t2|t1 interpreted as an undirected graph, as shown by the following proposition. Proposition 33. HF0pt1, t2q " dimpker Lq, where L " diagpPt2|t11q ́ Pt2|t1P T t2|t1 is a weighted Laplacian. The proof can be found in Appendix A.3. Remark 34. Note that Pt|s only encodes the pairwise relationship between clusters, and does not capture higher-order cluster inconsistencies. In particular, CE cannot detect 1-conflicts arising across more than two scales, as seen in Example 44 in Appendix B. 6 Experiments 6.1 Regression Task: Minimal Crossing Number of Sankey Layout Task In our first experiment, we consider a task of relevance in computer graphics and data visualisation: the minimisation of the crossing number of Sankey diagram layouts [Zarate et al., 2018, Li et al., 2025]. This minimisation is NP-complete, and here we use our MCbiF topological feature maps to predict the minimal crossing number κθ of the Sankey diagram Spθq of a given sequence of partitions θ (see Section 3.1). 10 Data We test our measures on synthetic datasets generated by sampling randomly from the space of coarse-graining sequences of partitions. Definition 35 (Space of coarse-graining sequences of partitions). The space of coarse-graining sequences of partitions, denoted ΠM N , is defined as the set of coarse-graining sequences θ : r0, 8q Ñ ΠX with |X| " N and M change points tm " 0, . . . , M ́ 1, such that |θpsq| ě |θptq|, @s ď t, which start with the partition of singletons θpt1 " 0q " ˆ0 and end with the full set θptM " M ́ 1q " ˆ1. Setting M " 20, we generate two datasets of 20,000 random samples θ P ΠM N for N " 5 and N " 10. For each of the generated θ, we compute three feature maps: the information-based pairwise conditional entropy matrix CE [Meilă, 2003] (see Eq. 15) and our MCbiF Hilbert functions (HF0 and HF1). In addition, as a baseline feature map, we also consider the (non-unique) raw label encoding of θ given by the N ˆ M matrix whose m-th column contains the labels of the clusters in θptmq assigned to the elements in X. As our prediction target, we take y " κθ (Eq. 3), the minimal crossing number of the layout of the Sankey diagram, which we computed with the OmicsSankey algorithm [Li et al., 2025]. See Section 3.1 for details. We expect that predicting y will be harder for N " 10 because the increased complexity of ΠN M allows for more complicated crossings in the Sankey diagram. Table 1: Regression task. Test R2 score of LR, CNN and MLP models trained on different features for N " 5 and N " 10. See Appendix F.1 for train R2 scores. N Method Raw label encoding HF0 HF1 HF0 & HF1 CE 5 LR 0.001 0.147 0.486 0.539 0.392 CNN -0.006 0.155 0.504 0.544 0.492 MLP -0.002 0.150 0.491 0.541 0.409 10 LR -0.012 0.214 0.448 0.516 0.457 CNN 0.000 0.211 0.448 0.507 0.454 MLP 0.000 0.212 0.450 0.514 0.458 Results As a preliminary assessment, we first compute the Pearson correlation, r, between the crossing number y and the three measures under investigation: the information-theoretical measure CI (i.e., the average CE, see Eq. (16)) and the MCbiF topological average measures ̄c0 and ̄c1. The correlation between CI and y is low (r " 0.20 for N " 5 and r " 0.06 for N " 10), higher for ̄c0, and highest for ̄c1 (r " 0.47 for N " 5, 10) (see Fig. 11 in Appendix F.1). This is consistent with our theoretical results in Section 4.2, which show the relation between the crossing number and HF1 (see, e.g., Corollary 29). We then proceed to the regression task of predicting κθ. We split each dataset into training (64%), validation (16%) and test (20%). For each feature map (or their combinations), we train three different models: linear regression (LR), multilayer perceptron (MLP), and convolutional neural network (CNN).We use the mean-squared error (MSE) as our loss function and employ the validation set for hyperparameter tuning. See Appendix F.1 for details of all the models. We then evaluate the model performance on the unseen test data using the coefficient of determination (R2). We find that the raw label encoding of θ does not improve upon the mean prediction (R2 « 0) and that MCbiF feature maps outperform the information-based feature map (Table 1). In particular, the combined HF0 and HF1 features lead to a significantly better model performance than CE (p ă 0.0001, t-test on the residuals). Furthermore, HF0 and HF1 yield R2 " 0.544 for N " 5 and R2 " 0.516 for N " 10 whereas CE only achieves R2 " 0.492 and R2 " 0.458, respectively. The strong performance of the simple LR model demonstrates the interpretability of the MCbiF features, important for explainable AI (XAI) [Adadi and Berrada, 2018]. 6.2 Classification Task: Non-Order-Preserving Sequences of Partitions Task In our second experiment, we classify whether a sequence of partitions is order-preserving or not, i.e., whether a sequence θ is compatible with a total ordering on the set X. This task is of relevance in several areas, from the study of preference relations in utility theory in social sciences [Roberts, 2009] to the analysis of weak orderings and partition refinement algorithms in computer science [Habib et al., 1999]. Definition 36 (Order-preserving sequence of partitions). When a partition θptmq is equipped with a total order ăm on the clusters it is called an ordered partition.2 Such a partition induces a total preorder Àm on 2The ranking τm : Vm Ñ t1, . . . , |Vm|u of the vertices Vm in the Sankey diagram Spθq is one example of a total order ăm on the clusters, see Section 3.1. 11 X [Stanley, 2011], i.e., if rxst ăm ryst then x Àm y. We call the sequence of partitions θ order-preserving if there exist total orders pă1, . . . , ăMq such that the total preorders pÀ1, . . . , ÀMq are compatible across the sequence, i.e., @l, m we have x Àly iff x Àm y, @x, y P X. 0.26 0.26 0.27 0.27 0.27 0.27 0.28 CI ns 0.80 0.80 0.81 0.81 0.81 c0 **** 0.00 2.00 4.00 6.00 c1 **** y = 0 y = 1 Figure 4: Difference between orderpreserving (y " 0) and non-orderpreserving (y " 1) sequences (**** indicates p ă 0.0001, Mann-Whitney U test). According to this definition, a sequence θ is non-order-preserving if there is no total order on X that is consistent with all the total preorders induced by the partitions θptq. Data We carry out this classification task on synthetic data for which we have a ground truth. From the space of coarse-graining sequences of partitions ΠM N , introduced in Definition 35, we generate a balanced dataset of 3,700 partitions θ P Π30 500, half of which are order-preserving (y " 0) and the other half are non-order-preserving (y " 1). The loss of order-preservation is induced by introducing random swaps in the node labels across layers. See Appendix F.2 for details. For each of the generated θ we compute CE, HF0 and HF1 using the computationally advantageous nerve-based MCbiF. We choose N " 500 and M " 30 to demonstrate the scalability of our method. Table 2: Classification task. Test accuracy of logistic regression trained on different features. Raw label encoding HF0 HF1 CE 0.53 0.56 0.97 0.50 Results Firstly, whereas we find no significant difference between the information-theoretical CI of order-preserving (y " 0) and nonorder-preserving (y " 1) sequences, we observe a statistically significant increase of ̄c0 and ̄c1 for order-preserving sequences (Fig. 4). For the classification task, we split our data into training (80%) and test (20%). For each feature map, we then train a logistic regression on the training split, and evaluate the accuracy on the test split, see Appendix F.2. We find that HF1 predicts the label y " t0, 1u encoding the (lack of) order preservation with high accuracy (0.95). In contrast, CE and the raw label encoding of θ cannot improve on a random classifier (Table 2). Our results thus demonstrate the high sensitivity of MCbiF to order-preservation in θ because non-order-preserving sequences induce 1-conflicts that we capture with HF1. 6.3 Application to Real-World Temporal Data In our final experiment, we apply MCbiF to temporal sequences of partitions computed from real-world contact data of free-ranging house mice that capture the changes in the social network structure of the rodents over time [Bovet et al., 2022]. Data Each partition θτptq describes mice social groupings for N " 281 individual mice at weeks t P r1, . . . , 9s, i.e., the nine weeks in the study period (28 February-1 May 2017). Hence, each sequence captures the fine-graining of social groups over the transition from winter to spring. Each partition sequence is computed at temporal resolution τ ą 0, where the parameter τ modulates how fine the temporal community structure is (Fig. 13). See Bovet et al. [2022] for details. We use MCbiF to compare the temporal sequences θτi for nine parameters τi, i " 1, . . . , 9, as provided in Bovet et al. [2022]. See Appendix F.3 for details. For each of the nine partition sequences θτi, we compute HF0 and HF1 using the computationally advantageous nerve-based MCbiF, which induces a 50-fold reduction in computation time due to a much lower number of simplices (260 simplices for the nerve-based MCbiF instead of 116,700 for the original MCbiF). Results Bovet et al. [2022] identified that the temporal resolutions τ2 " 1 s, τ4 " 60 s and τ8 " 24 h lead to robust sequences of partitions. Using the Hilbert distance, i.e., the L2-norm on the 0- and 1-dimensional MCbiF Hilbert functions, we find these temporal resolutions to be representative for three distinct temporal regimes characterised by different degrees of non-hierarchy, as measured by ̄c0 and ̄c1 (Fig. 5). In particular, high ̄c0 indicates that mice tend to split off groups over time, and high ̄c1 indicates that mice meet in overlapping subgroups but never jointly in one nest box. Note that θτ2 has a strong non-hierarchical structure because the large-scale mice social clusters get disrupted in the transition to spring. In contrast, θτ8 is more hierarchical as it captures the underlying stable social groups revealed by the higher temporal resolution. However, θτ4 has the strongest hierarchy as indicated by a lower ̄c0 and an absence of 1-conflicts ( ̄c1 " 0) and 12 (a) (b) τ9 τ8 τ7 τ5 τ4 τ1 τ2 τ3 τ6 Figure 5: (a) Analysis of non-hierarchical sequences of partitions θτi compiled from the temporal social interactions of a mice population over a period of 9 weeks. Each θτi is formed by a sequence of social groupings θτiptq over week t. Different sequences of partitions were computed as a function of the parameter τi. We display Sankey diagrams and MCbiF feature maps for θτi at three parameters τi (i " 2, 4, 8) identified as robust in the original work by Bovet et al. [2022]. These three sequences θτi exhibit different types of non-hierarchy, as shown by our topological feature maps and our measures of average 0-conflict ( ̄c0) and average 1-conflict ( ̄c1). (b) The θτi (i " 2, 4, 8) found in Bovet et al. [2022] as robust behaviours correspond to distinct topological characteristics of the sequences of partitions, as captured by the block structure in the distance between MCbiF Hilbert functions. thus corresponds to a sweet spot in hierarchical organisation between the low and high temporal resolutions. Finally, the Hilbert distances also capture an increased time reversibility in the sequence θτ8 due to the increased stability of social groupings over time, see Fig. 14 in Appendix F.3. 7 Conclusion We have introduced the MCbiF, a novel bifiltration that encodes the cluster intersection patterns in a multiscale, non-hierarchical sequence of partitions, θ. Its stable Hilbert functions HFk quantify the topological autocorrelation of θ and measure non-hierarchy in two complementary ways: the Hilbert function at dimension k " 0 captures the absence of a maximum with respect to the refinement order (0-conflicts), whereas the Hilbert function at dimension k " 1 captures the emergence of higher-order cluster inconsistencies (1-conflicts). This is summarised by the measures of average 0-conflict ̄c0pθq and average 1-conflict ̄c1pθq, which are global, history-dependent and sensitive to the ordering of the partitions in θ. The MCbiF extends the 1-parameter MCF defined by Schindler and Barahona [2025] to a 2-parameter filtration, leading to richer algebraic invariants that describe the full topological information in θ. We remark that the MCbiF is independent of the chosen clustering algorithm and can be applied to any (non-hierarchical) sequence of partitions θ. We demonstrate with numerical experiments that the MCbiF Hilbert functions provide topological feature maps that can be used for downstream machine learning tasks, and are shown to outperform informationbased features on regression and classification tasks on non-hierarchical sequences of partitions. Moreover, the grounding of MCbiF features in algebraic topology enhances interpretability, a crucial attribute for XAI and applications to real-world data. Limitations and future work Our analysis of the MCbiF MPH is restricted to dimensions 0 and 1 due to current limitations of the RIVET software [Lesnick and Wright, 2015] used in our numerical experiments. Analysing topological autocorrelation for higher dimensions would allow us to capture more complex higher-order cluster inconsistencies and could be the object of future research. Furthermore, we focused here on Hilbert functions as our topological invariants because of their computational efficiency and analytical simplicity, which facilitates our theoretical analysis. In future work, we plan to use richer feature maps by exploiting the block decomposition of the MCbiF persistence module, which leads to barcodes [Bjerkevik, 2021], or by using multiparameter persistence landscapes [Vipond, 2020]. Another future direction is to use MCbiF to evaluate the consistency of assignments in consensus clustering [Strehl and Ghosh, 2002, Vega-Pons and Ruiz-Shulcloper, 2011]. Indeed, it can be shown that the values of the Hilbert function HFkps, tq that are further away from the diagonal (s " t) are more robust to permutations of the ordering of partitions in θ (see Proposition 41 in Appendix A.4), and, in particular, HFkpt1, tMq only depends on the set of distinct partitions in the sequence θprt1, 8qq and is independent to 13 any permutation in their order. Hence, in future work, HFkpt1, tMq could be used as an overall measure of consistency in θ in the context of consensus clustering. Finally, we plan to analyse minimal cycle representatives of the MPH [Li et al., 2021] to localise 1-conflicts in the sequence of partitions, which is of interest to compute conflict-resolving partitions in consensus clustering, or to identify inconsistent assignments in temporal clustering [Liechti and Bonhoeffer, 2020]. Reproducibility Statement Detailed proofs of all theoretical results can be found in Appendix A. Extensive documentation of our experiments is presented in Appendix F. The dataset studied in Section 6.3 is publicly available at: https: //dataverse.harvard.edu/file.xhtml?fileId=5657692. Python code for the MCbiF method is publicly available at: https://github.com/barahona-research-group/MCF Acknowledgments JS acknowledges support from the EPSRC (PhD studentship through the ). MB acknowledges support from EPSRC grant EP/N014529/1 supporting the EPSRC Centre for Mathematics of Precision Healthcare. We thank Kevin Michalewicz, Asem Alaa and Christian Schindler for valuable discussions on the computational aspects of this project. We thank Alex Bovet for discussions about the temporal dataset studied in this paper. We also thank Arne Wolf for discussions about the stability of multiparameter persistence modules. This work benefited from discussions at the London-Oxford TDA seminar in November 2024. References Amina Adadi and Mohammed Berrada. 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It is a simple fact that θ is coarse-graining iff the mean cluster size is non-decreasing, i.e., 1 |θpsq| ř|θpsq| i |θpsqi| ď 1 |θptq| ř|θptq| j |θptqj| for s ď t. The proof follows directly from the fact that ř|θpsq| i |θpsqi| " ř|θptq| i |θptqi| " N. A.2 Proofs for Section 4 We provide a proof for the multi-criticality of the MCbiF filtration stated in Proposition 4. Proof of Proposition 4. The MCbiF is indeed a bifiltration because Ks,t Ď Ks1,t1 if s ě s1 and t ď t1. See Fig. 6 for the triangular diagram of the MCbiF filtration, where arrows indicate inclusion maps. The MCbiF is uniquely defined by its values on the finite grid rt1, . . . , tMs ˆ rt1, . . . , tMs because θ has change points t1 ă ̈ ̈ ̈ ă tM. It is a multi-critical bifiltration because for x P X we have rxs P Ks,t for all s, t P rt1, 8qop ˆ rt1, 8q. In particular, x P Kt1,t1 and x P Kt1`δ,t1`δ for δ ą 0 but pt1, t1q and pt1 ` δ, t1 ` δq are incomparable in the poset rt1, 8qop ˆ rt1, 8q. . . . . . . . . . Kt,t Kt,t1 Kt,t2 . . . Kt1,t1 Kt1,t2 . . . Kt2,t2 . . . Figure 6: Triangular commutative diagram of the MCbiF for t1 ď t ď t1 ď t2. The arrows indicate inclusion maps between simplicial complexes. Next, we provide formal definitions for algebraic properties of persistence modules, see Botnan and Lesnick [2023] for details. Definition 38. For partially ordered sets P1, P2, we call an interval I Ď P1 ˆ P2 a block if it can be written as one of the following types: 1. Birth quadrant: I " S1 ˆ S2 for downsets S1 Ď P1 and S2 Ď P2. 2. Death quadrant: I " S1 ˆ S2 for upsets S1 Ď P1 and S2 Ď P2. 3. Vertical band: I " S1 ˆ P2 for an interval S1 Ď P1. 4. Horizontal band: I " P1 ˆ S2 for an interval S2 Ď P2. Definition 39. Let Vect denote the category of k-vector spaces for a fixed field k. For a partially ordered set P, a P-indexed persistence module is a functor F : P ÞÑ Vect. We say that: a) F is called pointwise finite-dimensional if dimpFaq ă 8 for all a P P. b) F is called finitely presented if there exists a morphism of free modules φ1 : F1 Ñ F1 such that cokerpφ1q - F and F0 and F1 are finitely generated. c) F is called block-decomposable if it decomposes into blocks F À BPBpF q kB where BpFq is a multiset of blocks that depends on F. We can now provide the proof for Proposition 6, which shows that the MCbiF persistence module (see Fig. 7) is pointwise finite-dimensional, finitely presented and block-decomposable. The proof relies on the equivalent nerve-based construction of the MCbiF (see Proposition 40), and the exactness of the persistence module from which block-decomposability follows [Cochoy and Oudot, 2020]. 18 Proof of Proposition 6. The MCbiF module is pointwise finite-dimensional because the homology groups of finite simplicial complexes are finite. As the MCbiF is defined uniquely by its values on a finite grid (Proposition 4), its persistence module consists of finitely many vector spaces and finitely many linear maps between them, hence it is finitely presented. To prove block-decomposability, we use the nerve-based MCbiF p ̃Ks,tqt1ďsďt, which leads to the same persistence module, see Proposition 40. As the module is uniquely defined by its values on a finite grid, we can use Theorem 9.6 by Cochoy and Oudot [2020] that implies block-decomposability if the persistence module is exact. Hence, it suffices to show that for all t1 ď t ď t1 ď t2 ď t3 the diagram Hkp ̃Kt,t2q Hkp ̃Kt,t3q Hkp ̃Kt1,t2q Hkp ̃Kt1,t3q induces an exact sequence: Hkp ̃Kt1,t2q Ñ Hkp ̃Kt,t2q ' Hkp ̃Kt1,t3q Ñ Hkp ̃Kt,t3q (12) By construction of the MCbiF, ̃Kt,t3 " ̃Kt,t2 Y ̃Kt1,t3. Furthermore, ̃Kt,t2 " ̃Kt,t1 Y ̃Kt1,t2 and ̃Kt1,t3 " ̃Kt1,t2Y ̃Kt2,t3. Without loss of generality, t " tk, t1 " tl, t2 " tm, t3 " tn for change points tk ă tlă tm ă tn of θ such that Apk, lq X Apm, nq " H. Hence, ̃Kt,t1 X ̃Kt2,t3 " H and ̃Kt1,t2 " ̃Kt,t2 X ̃Kt1,t3. This means that Eq. (12) is a Mayer-Vietoris sequence for all k ě 0, implying exactness [Hatcher, 2002, p. 149] and proving the block decomposability [Cochoy and Oudot, 2020, Theorem 9.6]. . . . . . . . . . HkpKt,tq HkpKt,t1q HkpKt,t2q . . . HkpKt1,t1q HkpKt1,t2q . . . HkpKt2,t2q . . . Figure 7: Multiparameter persistence module of the MCbiF for t1 ď t ď t1 ď t2. The arrows indicate linear maps between vector spaces. A.2.1 Proofs for Section 4.1 We continue with the proof of Proposition 12 that relates 0-conflicts to hierarchy and triangle 0-conflicts to nestedness. Proof of Proposition 12. (ii) If θ has a 0-conflict then Dr1, r2 P rs, ts such that θpr1q ę θpr2q and θpr1q ğ θpr2q, otherwise θprs, tsq would have a maximum. Hence, θ is not hierarchical in rs, ts. (iii) Let us first assume that θ is coarse-graining, i.e., |θptq| ď |θprq| for all r P rs, ts. We show that no 0-conflict in rs, ts implies that θ is hierarchical in rs, ts. Let r1, r2 P rs, ts with r1 ď r2, then the subposet θprr1, r2sq has a maximum because of the absence of a 0-conflict, and the maximum is given by θpr2q due to coarse-graining. Hence, θpr1q ď θpr2q. As r1, r2 were chosen arbitrarily, this implies that θ is hierarchical in rs, ts. The argument is analogous for the case that θ is fine-graining. (iv) Let x, y, z P X be in a triangle 0-conflict. In particular, x „r1 y „r2 z with x ‰ y, x ‰ z and y ‰ z. Hence, there are C P θpr1q and C1 P θpr2q such that x, y P C and y, z P C1, as well as z R C and x R C1. This implies txu P CzC1, tzu P C1zC and tyu P C X C1, showing that C and C1 are non-nested. Hence, θ is non-nested in rs, ts. (i) Moreover, Er P rs, ts such that x „r„r y. In particular, Er P rs, ts such that EC2 P θprq with C Ď C2 and C Ď C2. Hence, Er P rs, ts such that θpr1q ď θprq and θpr2q ď θprq, implying that the subposet θprs, tsq has no maximum. This shows that every triangle 0-conflict is also a 0-conflict, proving statement (i). Note that the opposite is not true as illustrated by the example in Fig. 3b. We now provide a proof for Proposition 14 on properties of the 0-dimensional Hilbert function of the MCbiF. 19 Proof of Proposition 14. (i) HF0ps, tq is equal to the number of connected components of Ks,t. Let r1 P rs, ts such that c " |θpr1q| " minrPrs,ts |θprq|. We can represent θprq " tC1, . . . , Ccu and by construction ∆C P Ks,t for all C P θprq. Hence, if two elements x, y P X are in the same cluster C P θprq then rx, ys P Ks,t and the 0-simplices rxs, rys P Ks,t are in the same connected component. As θprq has c mutually disjoint clusters, this means that there cannot be more than c disconnected components in Ks,t and HF0ps, tq ď c " |θpr1q|. As r1 P rs, ts was chosen arbitrarily, this implies HF0ps, tq ď minrPrs,ts |θprq|. We prove statement (ii) by the contrapositive and show that the following two conditions are equivalent: C1: HF0ps, tq " minrPrs,ts |θprq|. C2: Dr P rs, ts such that θpr1q ď θprq, @r1 P rs, ts. Note that C2 is equivalent to there is no 0-conflict in rs, ts. "ðù" consider first that C2 is true and θprq is an upper bound for the partitions θpr1q, r1 P rs, ts. This implies that @r1 P rs, ts we have that @C1 P θpr1q there DC P θprq such that C1 P C. By construction of the MCbiF (Eq. 5) this implies @σ1 P Ks,t there Dσ P Kr,r such that σ1 Ď σ. This means Ks,t Ď Kr,r and thus Ks,t " Kr,r. As Kr,r has |θprq| disconnected components this implies HF0ps, tq " |θprq|, showing C1. "ùñ" To prove the other direction, assume that C1 is true. Then there exists r P rs, ts such that c :" HF0ps, tq " |θprq| with |θprq| " minqPrs,ts |θpqq|. In particular, the disconnected components of Ks,t are given by the clusters of θprq denoted by C1, . . . , Cc. Let r1 P rs, ts and C1 P θprq. Then Di P r1, . . . , cs such that C1 Ď Ci P θprq because otherwise the solid simplex ∆C1 would connect two solid simplices in t∆C1, . . . , ∆Ccu, contradicting that they are disconnected in Ks,t. Hence, the clusters of θpr1q are all subsets of cluster of θprq, implying θpr1q ď θprq. As r1 P rs, ts was chosen arbitrary this shows C2. We finally prove statement (iii). "ùñ" Note that HF0ps, tq " |θprq| implies |θprq| " minr1Prs,ts |θpr1q| according to (i). Then (ii) shows that C2 is true for r, i.e., θprq is the maximum of the subposet θprs, tsq. "ðù" The other direction follows directly from the proof of (ii). We next prove Corollary 16 about some properties of the average 0-conflict, which follows immediately from Proposition 14. Proof of Corollary 16. We begin with the proof of statement (i). If θ is hierarchical in rs, ts then θ is either coarse- or fine-graining. Assume first that θ is coarse-graining, then θps1q ď θptq for all s1 P rs, ts and together with hierarchy, this implies that θptq is an upper bound of the subposet θprs, tsq. Hence, Proposition 14 (iii) shows that HF0ps, tq " |θptq|. Moreover, HF0ps, tq " minp|θpsq|, |θptq|q because coarse-graining implies |θpsq| ě |θptq|. A similar argument also shows HF0ps, tq " |θpsq| " minp|θpsq|, |θptq|q if θ is fine-graining. We continue with proving (ii). ̄c0pθq ą 0 is equivalent to Ds, t P rt1, tMs such that HF0ps, tq ă minrPrs,ts |θprq|, according to Definition 15. This is again equivalent to Ds, t P rt1, tMs such that θ has a 0-conflict in rs, ts, according to Proposition 14 (ii). We finally prove statement (iii). "ùñ" ̄c0pθq means that θ has no 0-conflict in rt1, 8q. As θ is also coarse- or finge-graining, Proposition 12 (iii) then shows that θ is strictly hierarchical. "ðù" If θ is strictly hierarchical, then it has no 0-conflicts according to Proposition 12 (ii) and statement (ii) implies that ̄c0pθq " 0. Next, we provide the proof of Proposition 17 about the relation between triangle 0-conflicts and the clustering coefficient. Proof of Proposition 17. Assume that CpKs,t 1 q ă 1. Then there exist x, y, z P X that form a path of length 2 but no triangle, see Newman [2018] for details on the clustering coefficient. Without loss of generality, rx, ys, ry, zs P Ks,t 1 but rx, zs R Ks,t 1 . This implies Dr1, r2 P rs, ts: x „r1 y „r2 z and Er P rs, ts: x „r z. Hence, x, y, z lead to a triangle 0-conflict. We can now prove Proposition 19 on 1-conflicts. Proof of Proposition 19. Statement (i) follows directly from the definition of 1-conflicts that HF1ps, tq " dimrHkpKs,tqs ě 1 iff θ has a 1-conflict. We next prove statement (ii): If HF1ps, tq ě 1 there exists a 1-cycle z " rx1, x2s` ̈ ̈ ̈`rxn ́1, xns`rxn, x1s that is non-bounding, i.e., h :" rzs ‰ 0 in H1pKs,tq, see Appendix C for details. Case 1: Assume Er P rs, ts: x1 „r x2 „r x3, then it follows immediately that x1, x2, x3 lead to a triangle 0-conflict. Case 2: Assume Dr P rs, ts: x1 „r x2 „r x3. As rzs ‰ 0 there exists a 1-cycle ̃z " r ̃x1, ̃x2s ` ̈ ̈ ̈ ` `r ̃xm ́1, ̃xms ` r ̃xm, ̃x1s P 20 Z1pKs,tq such that ̃z is homologous to z, i.e., ̃z " z ` B2w for w P C2pKs,tq, and such that Er P rs, ts: ̃x1 „r ̃x2 „r ̃x3. In particular, ̃x1, ̃x2, ̃x3 lead to a triangle 0-conflict. We finally prove statement (iii): If θ is hierarchical, then it has no 0-conflicts according to Corollary 16. Hence, θ also has no triangle 0-conflict in rs, ts and so (i) implies that HF1ps, tq " 0. A.2.2 Proofs for Section 4.2 Next, we provide the proof about the equivalence between MCbiF and nerve-based MCbiF. Proposition 40. The bifiltrations M and ̃ M lead to the same persistence module. Proof. The proof follows from Proposition 30 in Schindler and Barahona [2025], which extends directly to the 2-parameter case. Next, we prove Proposition 26 about the dimension of the nerve-based MCbiF. Proof of Proposition 26. Statement (i) follows directly from the definition in Eq. (5). We show statement (ii) by induction. Base case: From the definition of the nerve-based MCbiF, it follows directly that dim N tm,tm " 0 because the indices in Apm, mq correspond to mutually exclusive clusters. Induction step: Let us assume that dim N tm,tm`n " n, then there exist C0, . . . , Cn P θprtm, tm`nsq such that C0 X ̈ ̈ ̈ X Cn ‰ H. As the clusters in partition θptm`n`1q cover the set X there exist a cluster C P θptm`n`1q such that C X C0 X ̈ ̈ ̈ X Cn ‰ H. Hence, dim N tm,tm`n ě n ` 1. If dim N tm,tm`n ą n ` 1 there would exist a second cluster C1 P θptm`n`1q with C1 X C X C0 X ̈ ̈ ̈ X Cn ‰ H but C1 X C ‰ H contradicts that clusters of θptm`n`1q are mutually exclusive. Hence, dim N tm,tm`n " n ` 1, proving statement (ii) by induction. We provide a proof for the connection between Sankey diagrams and the nerve-based MCbiF. Proof of Proposition 27. The Sankey diagram graph Spθq " pV " V1 Z ... Z VM, E " E1 Z ... Z EM ́1q is a strict 1-dimensional subcomplex of ̃K " ̃Kt1,tM because ̃Ktm,tm " Vm Ď ̃K and ̃Ktm,tm`1 " Em Ď ̃K. This also shows that the zigzag filtration (9) contains exactly the same vertices (0-simplices) and edges (1-simplices) as Spθq. We next prove Proposition 28 that characterises conflicts that can arise in a single layer of the Sankey diagram. Proof of Proposition 28. (i) Suppose that θ has a 0-conflict in rtm, tm`1s. Then θptmq ę θptm`1q and θptmq ğ θptm`1q. This means that there exists C P θpr1q such that DC1, C2 P θpr2q with C X C1 ‰ H, C X C2 ‰ H and C1 X C2 " H, otherwise θpr1q ď θpr2q. Hence, θpr1q ę θpr2q is equivalent to Du P Vm (the node corresponding to cluster C) with degree degpuq ě 2 in Em. An analogous argument shows that θpr1q ğ θpr2q is equivalent to Dv P Vm`1 with degpvq ě 2. This proves the statement. (ii) Let x, y, z P X form a triangle 0-conflict for the interval rtm, tm`1s, i.e., x „tm y „tm`1 z but x ȷtm`1 y ȷtm z. In particular, the elements x, y, z are mutually distinct. This means there exist C1, C2 P θptmq and C1 1, C1 2 P θptm`1q such that x, y P C1, z P C2, y, z P C1 1 and x P C1 2. This is equivalent to C1 X C2 " H, C1 1 X C1 2 " H and C1 X C1 1 ‰ H, C1 X C1 2 ‰ H, C2 X C1 2 ‰ H. Let u, u1 P Vm correspond to C1 and C2, respectively, and v, v1 P Vm`1 correspond to C1 1 and C1 2, respectively. Then the above is equivalent to ru1, vs, rv, us, ru, v1s P Em, which is again equivalent to the existence of a path in Em that has length at least 3. (iii) The statement follows from the fact that every cycle in Em is even because the graph pVmZVm`1, Emq is bipartite and the fact that every cycle in Em " Ktm,tm`1 is non-bounding because dim Ktm,tm`1 " 1. A.3 Proofs for Section 5 We continue by proving that 0-conflicts can induce violations of the strong triangle inequality as stated in Proposition 31. Proof of Proposition 31. Let x, y, z P X lead to a triangle 0-conflict for the interval rs, ts, i.e., Dr1, r2 P rs, ts: x „r1 y „r2 z and Er P rs, ts: x ȷr y ȷr z. This means Dθ,spx, yq ď r1 and Dθ,spy, zq ď r2. Let us define r3 :" Dθ,spxzq ď tM. We know that r3 ‰ r1 and r3 ‰ r2 as otherwise x „r3 y „r3 z for r3 P rs, ts, contradicting the lack of transitivity. Hence, without loss of generality, r1 ă r2 ă r3. Then the above is equivalent to Dθ,spxzq " r3 ą r1 ě minpDθ,spx, yq, Dθ,spy, zqq, which is a violation of the strong triangle inequality. 21 Next, we provide a proof of Proposition 32 that establishes the connection between the MPH of the MCbiF and that of the Merge-Rips bifiltration constructed from the matrix of first-merge times, Dθ,s. Proof of Proposition 32. First note that L " pLs,tqt1ďsďt is indeed a well-defined bifiltration because Ls,t Ď Ls1,t1 if s ě s1 and t ď t1. In particular, L is also defined uniquely on the finite grid P " tps, tq P rt1, . . . , tMs ˆ rt1, . . . , tMs | s ď tu with partial order ps, tq ď ps1, t1q if s ě s1, t ď t1. The proof of the proposition then follows from a simple extension of Proposition 32 in Schindler and Barahona [2025] to the 2-parameter case. To see that the 0-dimensional MPH of L and M are equivalent, note that both bifiltrations have the same 1-skeleton. Moreover, the 1-dimensional MPH is generally not equivalent because L is a Rips-based bifiltration and thus 2-determined, whereas M is not 2-determined. If θ is strictly hierarchical, then Dθ,s fulfils the strong-triangle inequality and thus the Rips-based bifiltration leads to a trivial 1-dimensional homology, see [Schindler and Barahona, 2025, Corollary 33]. Hence, the 1-dimensional MPH of L is trivial. Finally, we provide a brief proof for Proposition 33 linking the 0-dimensional Hilbert function of a pair of partitions and the graph Laplacian built from the conditional entropy matrix between both partitions. Proof of Proposition 33. Using Proposition 27, we prove the statement with the equivalent nerve-based MCbiF. Note that the graph G :" Pt2|t1P T t2|t1 has the same vertices and edges as the simplicial complex ̃Kt1,t2, which is 1-dimensional and thus also a graph according to Proposition 26. This shows that HFpt1, t2q is given by the number of connected components in G. Furthermore, observe that P T t2|t11 " 1, and that the resulting matrix L is the Laplacian of the undirected graph G. Hence, dimpker Lq is equal to the number of connected graph components [Chung, 1997], proving the statement. A.4 Proofs for Section 7 It follows from the construction of MCbiF that the Hilbert functions are invariant to certain swaps of partitions in θ. Proposition 41. HFkps, tq is invariant to swaps of partitions in sequence θ between s and t, for t1 ď s ď t. Proof. Let us denote the change points of θ by t1 ă t2 ă ̈ ̈ ̈ ă tM. Without loss of generality, s " tm and t " tm`n for m ` n ď M. Let us now consider a permutation τ : r1, . . . , Ms Ñ r1, . . . , Ms such that τpiq " i for 1 ď i ă m and m ` n ă i ď M and define the permuted sequence of partitions θτ as θτptmq " θptτpmqq. Despite the permutation we still get the same MCbiF for θ and θτ for parameters s ď t because ď sďrďt ď CPθprq ∆C " ď sďrďt ď CPθτ prq ∆C. This implies that HFkps, tq is the same for θ and θτ. B Additional Examples Our first example corresponds to the sequence of partitions analysed in Fig.1. Example 42 (3-element example). Let X " tx1, x2, x3u and we define θp0q " ˆ0, θp1q " ttx1, x2u, tx3uu, θp2q " ttx1u, tx2, x3uu, θp3q " ttx1, x3u, tx2uu and θp4q " ˆ1 so that θ is coarse-graining with M " 5 change points. This example corresponds to Fig. 1a, and the 0- and 1-dimensional Hilbert functions are provided in Fig. 1b. Note that HF0p1, 2q ă |θp2q| and HF0p2, 3q ă |θp3q| indicates the presence of two triangle 0-conflicts, which are not 1-conflicts because HF1p1, 2q " HF1p2, 3q " 0. See Fig. 8a for an illustration. As shown in Proposition 31, the triangle 0-conflicts violate the strong triangle inequality of the matrix of first merge times Dθ (Eq 10), e.g., Dθpx1, x3q " 3 ą maxpDθpx1, x2q, Dθpx2, x3qq " 2. In addition, HF1p1, 3q " 1 indicates the presence of a 1-conflict that arises from the higher-order inconsistencies of cluster assignments across partitions θp1q, θp2q and θp3q. See Fig. 8b for an illustration. In particular, the equivalence relations x1 „1 x2, x2 „2 x3 and x3 „3 x1 induce a 1-cycle z " rx1, x2s ` rx2, x3s ` rx3, x1s P Z1pK1,3q and due to the lack of transitivity on the interval r1, 3s, the 1-cycle z is also non-bounding, yielding a 1-conflict. The 1-conflict then gets resolved at t " 4 because θp4q " ˆ1 restores transitivity on the interval r1, 4s. See Fig. 8c for an illustration. 22 (a) connected 0-conflict (b) 1-conflict (c) no conflict Figure 8: (a) Illustration of a triangle 0-conflict that violates the strong triangle inequality of the matrix of first merge times Dθ,s (Eq 10), (b) a 1-conflict and (c) three elements that are in no conflict due to global transitivity. If we choose r1 " 1, r2 " 2, r3 " 3 and r " 4, the conflicts depicted here correspond to the conflicts in Example 42. Example 43 (4-element example). We now consider the more complex case of a 4-element set X " tx1, x2, x3, x4u. Let us start with θp0q " ˆ0 and append in sequence the 6 distinct partitions that contain two singletons and one cluster of size 2, i.e., θp1q " ttx1, x2u, tx3u, tx4uu, θp2q " ttx1u, tx2, x3u, tx4uu, θp3q " ttx1u, tx2u, tx3, x4uu, θp4q " ttx1, x3u, tx2u, tx4uu, θp5q " ttx1, x4u, tx2u, tx3uu and θp6q " ttx1u, tx2, x4u, tx3uu. Finally, we append consecutively three partitions, each of which contains a cluster of size 3, i.e., θp7q " ttx1, x2, x3u, tx4uu, θp8q " ttx1u, tx2, x3, x4uu and θp9q " ttx1, x3, x4u, tx2uu. θ is a coarse-graining, nonhierarchical sequence with M " 10 change points. See Fig. 9 for a Sankey diagram of θ. Figure 9: Hilbert functions and Sankey diagram for the sequence of partitions θ defined in Example 43. To analyse the topological autocorrelation of θ, we compute the MCbiF Hilbert functions HFk for dimensions k " 0, 1 (see Fig. 9). We observe that HF0ps, s ` 3q " 1 ă minrPrs,s`3s |θprq| for all 0 ď s ď 8, which implies that the hierarchy of θ is broken after no-less than three steps in the sequence when starting at scale s. Moreover, we can detect that θ is non-nested and has higher-order cluster inconsistencies because 1-conflicts emerge at scales t " 4, 5, 6, as indicated by non-zero values in HF1. The 1-conflicts get resolved one-by-one through the partitions that contain clusters of size 3, and at t " 9, when the third such partition appears in θ, all 1-conflicts are resolved. Finally, we show an example that demonstrates how conditional entropy does not detect 1-conflicts in general. Example 44 (CE cannot detect 1-conflicts). Let X " tx1, x2, x3, x4u for which we consider two different sequences of partitions θptq and ηptq such that θp1q " ηp1q " ttx1, x2u, tx3u, tx4uu, θp2q " ηp2q " ttx1u, tx2, x3u, tx4uu but θp3q " ttx1, x3u, tx2u, tx4uu ‰ θp3q " ttx1u, tx2u, tx3, x4uu. See Fig. 10 for a Sankey diagram representation of the two diagrams. Note that θ and η only differ at scale t " 3. However, this difference is crucial because a 1-conflict emerges in θ at scale t " 0, whereas η has only triangle 0-conflicts and no 1-conflict. Note that θ corresponds to the toy example in Fig. 1 with one additional isolated element. In accordance with our theoretical results developed in Section 4.1, we can use the 1-dimensional Hilbert function HF1 to detect the 1-conflict in θ and distinguish the two sequences. In particular, HF1pθp1q, θp3qq " 1 but HF1pηpiq, ηpjqq " 0 for all i, j P r1, 2, 3s, i ď j. In contrast, the conditional entropy H (see Eq. 15) 23 (a) Sequence (1-conflict) (b) Sequence (no 1-conflict) Figure 10: Sankey diagrams for sequences θ and η defined in Example 44. Note that a 1-conflict emerges in θ at scale t " 3, but η has no 1-conflict. cannot distinguish between the two sequences as they yield the same pairwise conditional entropies. In particular, Hpθpiq|θpjqq " Hpηpiq|ηpjqq " 1 2 log 2 for i ‰ j. This demonstrates that the conditional entropy cannot detect higher-order cluster inconsistencies in sequences of partitions. C Details on the Homology Functor We provide additional background on simplicial homology and its functoriality, following Hatcher [2002]. Simplicial Homology. Let K be a simplicial complex defined on the finite set X. For a fixed field k (the RIVET software uses the finite filed k " Z2 [Wright and Zheng, 2020]) and for all dimensions k P t0, 1, ..., dimpKqu we define the k-vector space CkpKq whose elements z are given by a formal sum z " ÿ σPK dimpσq"k aσσ (13) with coefficients aσ P k, called a k-chain. Note that the k-dimensional simplices σ " rx0, x1, ..., xks P K form a basis of CkpKq. For a fixed total order on X, the boundary operator is the linear map Bk : Ck ÝÑ Ck ́1 defined through an alternating sum operation on the basis vectors σ " rx0, x1, ..., xks given by Bkpσq " kÿ i"0 p ́1qirx0, x1, ..., ˆxi, ..., xks, where ˆxi means that vertex xi is deleted from the simplex σ. The boundary operator fulfils the property im Bk`1 Ă ker Bk. Hence, it connects the vector spaces Ck, k P t0, 1, ..., dimpKqu, through linear maps . . . Bk`1 ÝÝÝÑ Ck Bk ÝÑ Ck ́1 Bk ́1 ÝÝÝÑ . . . B2 ÝÑ C1 B1 ÝÑ C0 B0 ÝÑ 0, leading to a sequence of vector spaces called chain complex. The elements in Zk :" ker Bk are called k-cycles and the elements in Bk :" im Bk`1 are called k-boundaries. Finally, the k-th homology group Hk is defined as the quotient of vector spaces Hk :" Zk{Bk, (14) whose elements are equivalence classes rzs of k-cycles z P Zk. Each equivalence class rzs ‰ 0 corresponds to a generator of non-bounding cycles, i.e., k-cycles that are not the k-boundaries of k ` 1-dimensional simplices. This captures connected components at dimension k " 0, holes at k " 1 and voids at k " 2. Functoriality of Hk. For fixed k, Hk can be considered as a functor Hk : Top Ñ Vect, where Top denotes the category of topological spaces whose morphisms are continuous maps and Vect the category of vector spaces whose morphisms are linear maps. In particular, each topological space K is sent to a vector space HkpKq and a continuous map g : K Ñ K1 is sent to a linear map Hkpgq : HkpKq Ñ HkpK1q such that compositions of morphisms are preserved, i.e., Hkpg ̋ fq " Hkpgq ̋ Hkpfq for two continuous maps f and g. D Details on Zigzag Persistence We provide background on zigzag persistence, which was first introduced by Carlsson and de Silva [2010]. For additional details, see Dey and Wang [2022]. 24 Zigzag Filtration. Let t1 ă ̈ ̈ ̈ ă tM be a sequence of real-valued parameter values. For simplicity we assume tm " m for m " 1, . . . , M. Let Km be a simplicial complex defined on the set X for every m " 1, . . . , M. If either Km Ď Km`1 or Km`1 Ď Km, for all m " 1, . . . , M, we call the following diagram a zigzag filtration: K1 Ø K2 Ø ̈ ̈ ̈ Ø KM ́1 Ø KM, where Km Ø Km`1 is either a forward inclusion Km ãÑ Km`1 or a backward inclusion Km Ðâ Km`1. While forward inclusion corresponds to simplex addition, backward inclusion can be interpreted as simplex deletion. Zigzag Persistence. Applying the homology functor Hk to the zigzag filtration leads to a so called zigzag persistence module given by: HkpK1q Ø HkpK2q Ø ̈ ̈ ̈ Ø HkpKM ́1q Ø HkpKMq, where HkpKmq Ø HkpKm`1q is either a forward or backward linear map. Using quiver theory, it can be shown that a zigzag persistence module has a unique interval decomposition that provides a barcode as a simple invariant. E Details on Information-based Baseline Methods Information-based measures can be used to compare arbitrary pairs of partitions in the sequence θ [Meilă, 2007]. Assuming a uniform distribution on X, the conditional probability distribution of θptq " tC1, . . . , Cnu given θpsq " tC1 1, . . . , C1 mu is: Pt|sri|js " |Ci X C1 j| |C1 j| , and the joint probability Ps,tri, js is defined similarly. The conditional entropy (CE) Hpt|sq is then given by the expected Shannon information: Hpt|sq " ́ |θptq| ÿ i"1 |θpsq| ÿ j"1 Ps,tri, js logpPt|sri|jsq (15) It measures how much information about θptq we gain by knowing θpsq. If θpsq ď θptq there is no information gain and Hpt|sq " 0. We denote the conditional entropy matrix CEs,t " Hpt|sq. Furthermore, we can compute the variation of information (VI) VIps, tq " Hps|tq ` Hpt|sq, which is a metric. Both CE and VI are bounded by log N. Extending information-based measures for the analysis and comparison of more than two partitions is non-trivial. However, the pairwise comparisons can be summarised with the consensus index (CI) [Vinh et al., 2010] which can be computed as the average VI: CIpθq :" řM i"1,iăj VIpti, tjq MpM ́ 1q{2 (16) F Details on Experiments F.1 Regression Task Figure 11 shows the correlation between the minimal crossing number y " κpθq (Eq. 3) and informationand MCbiF-based summary statistics. In addition to the results already described in the main text, we also observe that the correlation between CI and ̄c0 (r " ́0.32 for N " 5, r " ́0.48 for N " 10) is stronger than with ̄c1 (r " ́0.12 for N " 5, r " ́0.34 for N " 10). This can be explained by the fact that CI and ̄c0 can both be computed from pairwise interactions of clusters in contrast to ̄c1, see Section 5. Furthermore, we observe a strong correlation between ̄c0 and ̄c1 (r " 0.52 for N " 5 and r " 0.43 for N " 10) because of the dependencies between 0- and 1-conflicts, see Section 4.1 Note that we can consider our information- and MCbiF-based feature maps as M ˆ M greyscale images, where HF0 and HF1 are symmetric and CE is asymmetric. The raw label encoding of θ is also similarly interpreted as an N ˆ M greyscale image. For our regression task, we train a simple CNN [LeCun and Bengio, 1998] with one convolution and max-pool layer and one fully connected layer and also a simple 25 N = 5 N = 10 Figure 11: Pearson correlation (r) between crossing number y, information-based consensus index CI and MCbiF-based conflict measures ̄c0 and ̄c1 for N " 5 and N " 10. MLP [Bishop, 2006] with one or two hidden layers and dropout [Srivastava et al., 2014]. For each feature map (or their combinations) separately, we perform hyperparameter optimisation for the number of filters (ranging from 2 to 6) and kernel size (chosen as 4, 8, 16, 32 or 64) in the CNN and the number of nodes (chosen as 4, 8, 16, 32, 64, 128 or 256), number of layers (1 or 2) and dropout rate (chosen as 0.00, 0.25 or 0.50) in the MLP. We use the Adam optimiser [Kingma and Ba, 2017] with learning rate chosen as 0.01, 0.005, 0.001, 0.0005 or 0.0001 for training. We perform a full grid search of the hyperparameter space for the three different models and the different feature maps (or their combinations). We used the train split of our data for training and the validation split for evaluation and hyperparameter selection. Below, we detail the hyperparameters for the best MCbiFand CE-based models, which were chosen according to the performance on the validation split. • Optimal model for HF0 & HF1 at N " 5: CNN with 4 filters, kernel size 3, and learning rate 0.001. • Optimal model for CE at N " 5: CNN with 8 filters, kernel size 2, and learning rate 0.005. • Optimal model for HF0 & HF1 at N " 10: LR. • Optimal model for CE at N " 10: MLP with a single layer of 256 nodes, no dropout and a learning rate of 0.001. We present the train R2 scores for the optimised LR, CNN and MLP models trained on the different features in Tables 3. The test R2 scores are presented in Table 1 in the main text. Table 3: Train R2 scores of LR, CNN and MLP models trained on different features for N " 5 and N " 10. N Method Raw label encoding HF0 HF1 HF0 & HF1 CE 5 LR 0.005 0.163 0.493 0.550 0.409 CNN 0.000 0.170 0.509 0.562 0.515 MLP 0.006 0.160 0.499 0.547 0.439 10 LR 0.013 0.230 0.456 0.522 0.464 CNN 0.009 0.220 0.456 0.519 0.476 MLP 0.003 0.218 0.453 0.515 0.468 F.2 Classification Task Details on Synthetic Data. We generate order-preserving (y " 0) sequences θ P ΠM N through the following scheme: Let us assume that we have a total order X " tx1, . . . , xNu given by the element labels, i.e., xi ă xj if i ă j. We construct each θptmq, m " 0, . . . , M ́ 1, by cutting X into clusters of the form C " txi, xi`1, . . . , xi`nu. It is easy to verify that θ is indeed order-preserving. We adapt this scheme to generate sequences θ P ΠM N that are non-order-preserving (y " 0): Again, we start by constructing each sequence θptmq through cutting the ordered set X as before. Additionally, with probability p " 0.1, we swap the cluster assignments in θptmq for two arbitrary elements x, y P X. If N and M are large enough, the so-generated sequence θ is almost surely non-order-preserving. We chose N " 500 and M " 30 to demonstrate the scalability of the MCbiF method. 26 The number of clusters of all our generated sequences of partitions θ P ΠM N for both classes is decreasing linearly, see Fig. 12 (a). Moreover, the average number of swaps for sequences with y " 1 is 2.98 for our choice of p " 0.1, see Fig. 12 (b). Figure 12: Classification task: Histogram of the number of swaps in non-order-preserving sequences θ (class y " 1). See text for the scheme to introduce random swaps in the node labels as a means to break order-preservation. F.3 Application to Real-World Temporal Data Data Preprocessing. The temporal sequences of partitions computed by Bovet et al. [2022] are available at: https://dataverse.harvard.edu/file.xhtml?fileId=5657692. We restricted the partitions to the N " 281 mice that were present throughout the full study period to ensure well-defined sequences of partitions, and considered the first nine temporal resolution values τi, i " 1, . . . , 9, since θτ10 is an outlier. Note that the sequences tend to be fine-graining, see Fig. 13. 0 1 2 3 4 5 6 7 8 Scale t [in weeks] 5 10 15 20 25 30 | (t)| Temporal resolution 1 = 0.1 s 2 = 1 s 3 = 8 s 4 = 60 s 5 = 1 h 6 = 5 h 7 = 8 h 8 = 24 h 9 = 7 d Figure 13: Number of clusters over weeks t for different temporal resolutions τ, where larger values of τ produce a higher number of clusters because of the increased temporal resolution. τ9 τ8 τ7 τ5 τ4 τ1 τ2 τ3 τ6 Figure 14: Hilbert distance between forward and backward sequences θf τ and θb τ for different temporal resolutions τ. 27 Time Reversibility. In the main text, we restricted our analysis to the so-called forward Flow Stability sequences of partitions. However, by reversing time direction, Bovet et al. [2022] computed a second set of backward sequences. For each temporal resolution τi, we thus get a forward and backward sequence denoted by θf τi and θb τi, respectively. Here we use the MCbiF to compare the forward and backward sequences of partitions for different τi and we compute the Hilbert distance ∥HFkpθf τiq ́ HFkpθb τiq ∥2 for k " 0, 1, see Fig. 14. We observe that the Hilbert distance between forward and backward sequences is high for τ2 because the large-scale group structure changes significantly over the study period, so that the temporal flows at low resolution τ2 are not reversible. In contrast, the Hilbert distance between forward and backward sequences is low for τ8 because the underlying social groups are more stable over the study period, leading to increased time reversibility at the high temporal resolution. 28
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2510.14715
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Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX631
Numerical Studies on the Radio Afterglows in TDE (I): Forward Shock
Guobin Mou (牟国斌)1, 2
1Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, China; gbmou@njnu.edu.cn
2Nanjing key laboratory of particle physics and astrophysics, China
ABSTRACT
Recent long-term radio monitoring of tidal disruption events (TDEs) suggests that radio afterglows are
common.
Most studies argue that these afterglows may arise from forward shocks (FS) produced by the
interaction between the TDE outflow and the hot, diffuse circumnuclear medium (CNM). Current theoretical
models do not model the evolution of relativistic electrons in space, which introduces uncertainties. Here
we conducted hydrodynamic simulations to study the hydrodynamic evolution of relativistic electrons, and
calculated the synchrotron spectra via radiative transfer. We focus on the FS scenario with non-relativistic
outflows, and various parameters of the outflow and CNM are explored. A moderate outflow with kinetic
energy of several 1050 erg in a Galactic center - like CNM can produce mJy-level radio afterglows at a distance
of 100 Mpc. The self-absorption frequency exhibits a slow decline at early times and a rapid decrease at
late times. We derived the temporal evolution of the high-frequency radio flux, revealing its characteristic
rise and decline pattern. We also find that: (1) the radio spectra for narrow outflows are clearly anisotropic
along different sight lines; (2) the FS parameters inferred from radio spectra using conventional analytical
formulas deviate significantly from those in simulations, in which the inferred shock radii are half of those
from simulations, and the inferred energies are an order of magnitude lower.
Keywords:
radio continuum: transients - radiation mechanisms: non-thermal - galaxies: active -
(galaxies:) quasars: supermassive black holes
1. INTRODUCTION
Some TDE candidates exhibit radio emission with
luminosities of 1036−42 erg s−1 and time lags span-
ning from days to years (Alexander et al. 2020). Re-
cent studies indicate that about half of all optically-
selected TDEs exhibit show radio emission that rises
on timescales of hundreds of days (Cendes et al. 2024),
suggesting that TDEs with radio emission delayed by
years are prevalent.
These radio afterglows of TDEs arises from syn-
chrotron radiation of cosmic ray electrons (CRes),
though there are several possible sources for these elec-
trons. Some studies argue that they could come from
shocks driven by relativistic jets (Bloom et al. 2011;
Burrows et al. 2011; Zauderer et al. 2011; Giannios &
Metzger 2011; Lei et al. 2016). However, observations
over the past decade show that jetted TDEs are rare
(Andreoni et al. 2022). In most current observational
studies, CRes are interpreted as originating from the
forward shock (FS) when wide or narrow nonrelativistic
outflow interacting with hot and diffuse CNM (Alexan-
der et al. 2016). Alternatively, some studies suggest that
CRes may originate from shocks produced by interac-
tion between the unbound debris and CNM (Krolik et
al. 2016; Yalinewich et al. 2019) or dense torus (Lei et al.
2024), or collisions between TDE outflows and ambient
clouds (Mou & Wang 2021; Mou et al. 2022; Zhuang et
al. 2025). The former scenario essentially belongs to the
FS scenario, focusing on the shock sweeping through the
material ahead of the shock front. In contrast, the lat-
ter corresponds to the bow shock (BS) scenario, which
focuses on the shock interacting sweeping through the
outflow. We leave the hydrodynamic simulation results
for the BS scenario to a forthcoming paper.
Although the origin of the radio emission is still de-
bated, the presence of TDE outflows inevitably leads
to the formation of a FS. The TDE outflow has been
demonstrated to be both powerful and high-velocity.
The outflow could be generated in both self-interaction
process due to GR reprocessing (Sadowski et al. 2016;
Lu & Bonnerot 2020), circularization process of infalling
debris (Steinberg et al. 2022), or the finally accretion
process (Dai et al. 2018; Curd & Narayan 2019; Bu et
al. 2023; Hu et al. 2024).
The ratio of mass outflow
rate to the mass accretion rate could be considerable
arXiv:2510.14715v1 [astro-ph.HE] 16 Oct 2025
2
Mou
(up to 1/10, e.g., Steinberg et al. 2022; Thomsen et al.
2022), and accordingly, the mass of the outflow could
reach 10−2−10−1M⊙for disrupting a sun-like star. The
existence of high speed outflow has been demonstrated
by X-ray and UV observations of TDEs (Kara et al.
2018; Hung et al. 2019; Xiang et al. 2024), with some
blueshifted absorption lines indicating velocities of up
to 0.2c. In this context, we first focus on investigating
the FS scenario to explore the general characteristics of
its radio emission.
Theoretical investigations usually employ analytic
methods (Barniol Duran et al. 2013; Matsumoto & Pi-
ran 2021), or conduct hydrodynamic simulations (Hu et
al. 2025) that do not incorporate relativistic electrons
in simulations. Due to the simplifications, the spatial
distribution and evolution of relativistic electrons re-
mains largely unknown, which introduces uncertainties
in connecting shock physics with the radio spectra. The
complexity of the fluid behavior therefore makes numer-
ical simulations incorporating relativistic electrons in-
dispensable.
In this study, we employ a shock-injected CRe ap-
proach, utilizing hydrodynamical simulations to inves-
tigate the spatial evolution of CRe after the shock ac-
celeration. Radiative transfer calculations are then used
to calculate the radio spectra along different directions.
This study reveals the relationship between the physical
parameters of the FS and the resulting radio emission,
and highlights the errors that cannot be overlooked in
previous studies.
In Section 2, we introduce the physics related to the
models. We introduce the settings of the simulations in
Section 3, and present the data processing methods in
Section 4. Results are presented in Section 5 and we
give a brief summary in Section 6.
2. PHYSICS OF THE MODELS
2.1. CNM
Up to now, observational constraints on the hot
CNM are available for two sources: Sgr A* and M87.
The density of CNM around Sgr A* follows ρ(r) ≃
30 mH cm−3r−1
−1 (r−1
≡r/0.1pc, Xu et al. 2006;
Gillessen et al. 2019). For M87, although the BH mass is
three orders of magnitude higher than Sgr A*, its CNM
density at a given distance in parsec is only one order
of magnitude higher than that of Sgr A* (Russell et al.
2015). Here we simply assume a CNM density following
a power-law form similar to that of Sgr A*:
ρ(r) = Csgr × 30 mH cm−3r−n
−1
(1)
where Csgr is set to be 3.0 in the fiducial case, indicating
a density 3 times that at 0.1 parsec from Sgr A*, and n
is the density index which is set to be 1.0 in the fiducial
case. We also investigated different power-law indices
as well (Table 1). Furthermore, the density profile may
presents different slopes inside and outside the position
where the stellar wind could marginally escape the grav-
itation potential of the SMBH (Generozov et al. 2017).
Thus, we examined broken power-law cases, where the
dividing radius between the “inner” and “outer” regions
is denoted as rturn (runs Knbr, Lnbr).
2.2. TDE Outflow
TDE is a transient phenomenon, and we assume that
the outflow persists for 1 year. We constrain the out-
flow to a biconical structure with a half-opening angle
of θ0, and inject it into the simulation domain along
the two polar axes from the inner boundary of r. We
parameterize the injected outflow with its velocity vout
and mass outflow rate
˙Mout (Table 1), which remaining
constant during the outflow ejection epoch. In the fidu-
cial model, the velocity is set to be 0.2 c, and the total
outflow mass is 0.01M⊙, with a corresponding energy
(kinetically dominated) of 3.5 × 1050 erg. We also ex-
amined a more powerful case and two weaker cases for
comparison.
2.3. Shocks and Relativistic Electrons
The electron acceleration efficiency ϵe is assumed to
be the fraction of the energy flux that can be dissipated
at the shock (i.e., the change in the kinetic energy flux
across the shock) channelled into the accelerated rela-
tivistic electrons in the downstream.
In the frame of
shock front, the expression of ϵe is
ϵe =
e2vd
1
2ρiv3s(1 −C−2)
(2)
where e2 is the energy density of CRe in the downstream,
vd is the downstream velocity, ρi is the pre-shock density,
and C ≡4M2/(M2+3) is the compression ratio. When
the Mach number M ≫1, we have vd = vs/4 and C =
4. In this case, ϵe can be simplified to 0.6e2/ed, where
ed is thermal pressure in the downstream.
The acceleration efficiency ϵe is highly uncertain, and
conventionally it should be lower than that of CRp. Sim-
ulations suggest that the acceleration efficiency of CRp
of ∼10% for high Mach number shocks (e.g., Capri-
oli & Spitkovsky 2014), and ϵe should not exceed 10%.
Here we adopt ϵe = 0.03 as the fiducial value.
Note
that the definition of ϵe here is slightly different from
the one adopted in some literature where it is defined
as ϵ′
e = Ee/Es, i.e., the total energy of relativistic elec-
trons to the shock energy. The ratio between the two is
ϵ′
e/ϵe = (1 −C−2) = 0.94 for M ≫1, and thus one does
not need to distinguish them.
Radio Afterglows I: Forward Shock
3
2nd
3rd
4th
1st
pre-shock
post-shock
pre-shock
post-shock
Field Quantity ( arbitrary unit )
Distance ( x10-3 )
e2 x10
e1
Figure 1. Schematic diagram of the CRe injection method.
The energy density distributions (e1 and e2) refer to those
in the nine meshes intersected by the blue line segment in
the upper right panel. The simulation code uses 4 grid levels
to capture the shock, which stabilizes at the 4th mesh (see
the orange line). The CRe component is injected at the 4th
mesh.
Due to numerical diffusion, the values of e2 in the
1st – 3rd mesh subsequently become nonzero, but this has a
negligible effect on the results.
In general, the spectral energy distribution of CRes
follow a power-law form:
dncr
dγe = A0γ−p
e , where p is the
power-law index and we set p = 2.5 in most cases. This
distribution is related to e2 through:
e2 =
Z 2000
2
γmec2 dncr
dγe
dγe ,
(3)
where the integration upper limit (1 GeV) depends on
the cooling and shock acceleration processes, but has
little effect on the radio emission below 50 GHz discussed
here. Once e2 is specified in simulations, the coefficient
A0 and the spectral energy distribution of CRe can be
determined accordingly.
2.4. Magnetic Field
The relativistic electrons and protons can induce vari-
ous instabilities, and drive magnetic perturbations, and
amplify the magnetic field (Bell & Lucek 2001; Schure
et al. 2012). The magnetic pressure can be enhanced
to a fraction of the ram pressure. Given the complexity
of this process, we do not include the magnetic field in
simulations, and simply assume that the ratio of mag-
netic field energy density to that of CRe in each mesh
remains a constant, i.e.,
B2(t)/8π
e2(t)
= ϵB
ϵe
.
(4)
We set ϵB = 0.10 in this study, which is higher than ϵe.
3. NUMERICAL SETTINGS
We conduct the two-fluid simulation with ZEUS-3D
code (Clarke 2010). For simplicity, we did not incorpo-
rate the magnetic field and physical diffusion of CRe.
The timescale of radiative cooling of the CNM and out-
flow is much longer than the simulation timescale, and
thus radiative cooling is negligible. The hydrodynamic
equations are
∂ρ
∂t + ∇· (ρv) = 0,
(5)
ρdv
dt = −∇(p1 + p2) −ρ∇Φ,
(6)
∂e1
∂t + ∇· (e1v) = −p1∇· v,
(7)
∂e2
∂t + ∇· (e2v) = −p2∇· v,
(8)
where p1 ≡(γ1−1)e1 is the thermal pressure (γ1 = 5/3),
p2 ≡(γ2 −1)e2 is the pressure of CRe (γ2 = 4/3),
and Φ is the gravitation potential (Φ = −GMbh/r,
Mbh = 5 × 106 M⊙in this work). We did not simu-
late the acceleration process of electrons, but instead,
directly injected the CRe component as the second fluid
which is embodied as the energy density e2. As men-
tioned in Section 2.3, e2 is assigned the value of 5
3ϵeed,
where ed is thermal pressure in the downstream. The
shock front in ZEUS-3D is typically captured with 4
meshes, and thus we inject the CRe at the 4th-mesh
(the final mesh capturing shocks) when the parameters
of the post-shock gas are stabilized (figure 1).
After
that, we subtract the CRe energy density from that of
the thermal gas to help maintain energy conservation.
We adopt 2.5 dimensional spherical coordinates, in
which the system is symmetric in ϕ−direction.
The
biconical outflow is ejected along the two polar axes.
The inner and the outer boundary of r is set to be
0.0061 pc and 0.60 pc, respectively.
Exceptions are
run Bm01, Cv03, and Mm01, where the outer boundary
is extended to 1.07 pc to better encompass the radio
decay phase. The computation domain is divided into
2560 (or 2880 for exceptions) pieces in r−direction with
dri+1/dri = 1.0018, and 256 pieces in θ−direction with
dθj+1/dθj = 1.004. The high resolution adopted here is
sufficient to ensure the results are convergent.
4. CALCULATING THE RADIO SPECTRA
The simulations provide the distributions of the CRe
energy density and the magnetic field strength in the
study region. These parameters can be used to calcu-
late the emitting synchrotron spectrum. As shown in
Figure 2, we map the radiation zone enclosed by the FS
front into a 3D Cylindrical coordinate (for polar direc-
tion spectra) and 3D Cartesian coordinate system (for
4
Mou
Table 1. Parameters for modeling the radio afterglows. θ0 is the half-opening angle of the outflow. Ek is the kinetic energy of
the outflow. Csgr is the ratio of the density at r = 0.1 pc to that at the Galactic Center. F0.88GHz,max represents the maximum
value of F0.88GHz over time which is observed along the equatorial direction with a luminosity distance of dL = 100 Mpc.
F1.4GHz,max and F6GHz,max are the same, but for 1.4 GHz and 6 GHz, respectively. For Knbr and Lnbr, we adopt two different
density indices inside and outside a transition radius of 0.1 pc: n = 1.5 (2.5) for r < 0.1 pc, and n = 0.5 (1.0) for r ≥0.1 pc.
The value of F6GHz,max in run Lnbr represents the second flare.
Run
vout
Mout
θ0
Ek
Csgr
n
F0.88GHz,max
F1.4GHz,max
F6GHz,max
units
c
M⊙
deg.
erg
mJy
mJy
mJy
A
0.2
0.01
30
3.5 × 1050
3.0
1.0
3.9
4.0
1.7
Bm10
0.2
0.10
30
3.5 × 1051
3.0
1.0
20
19
7.2
Cv03
0.3
0.01
30
8.0 × 1050
3.0
1.0
10
11
6.1
Dv01
0.1
0.01
30
8.9 × 1049
3.0
1.0
0.60
0.46
0.16
Ev01c1
0.1
0.01
30
8.9 × 1049
1.0
1.0
0.17
0.12
0.04
Fo10
0.2
0.01
10
3.5 × 1050
3.0
1.0
1.4
1.2
0.40
Go60
0.2
0.01
60
3.5 × 1050
3.0
1.0
5.6
6.2
2.8
Ho90
0.2
0.01
90
3.5 × 1050
3.0
1.0
7.3
8.2
3.4
In15
0.2
0.01
30
3.5 × 1050
3.0
1.5
3.1
3.2
1.3
Jn20
0.2
0.01
30
3.5 × 1050
3.0
2.0
2.6
2.8
2.8
Knbr
0.2
0.01
30
3.5 × 1050
3.0
(1.5, 0.5)
4.7
5.0
2.4
*Lnbr
0.2
0.10
90
3.5 × 1051
3.0
(2.5, 1.0)
45
49
25 (2nd)
*Mm01
0.2
0.10
30
3.5 × 1051
10.0
1.0
36
43
34
*Nn25
0.2
0.01
30
3.5 × 1050
3.0
2.5
2.5
3.0
5.2
Runs marked with an asterisk (*) indicate that their results are presented in the Appendix.
Polar Direction
Equatorial
Direction
Spherical
Cylindrical or Cartesian
Quadrant
Reconst-
ruction
Re-gridded
Figure 2. Schematic diagram of data processing and spectral calculation. First, the 2.5-dimensional spherical coordinates
are re-gridded into a 3D coordinate system, and the CRe energy density distribution in the new coordinates is obtained via
interpolation. Next, quadrant reconstruction is carried out to recover the full 3D physical scenario. Finally, synchrotron radiation
transfer is performed along the polar and equatorial directions to compute the emitting spectra in both directions.
Radio Afterglows I: Forward Shock
5
equatorial-direction spectra) via interpolation. The Z-
axis of the Cartesian coordinate is aligned with the polar
axis and the origin is coincident with that in the spher-
ical coordinates.
Subsequently, we employ quadrant reconstruction to
build the three-dimensional shock structure in the Cylin-
drical and Cartesian coordinate system. For 3D Cylin-
drical coordinates, we employ 180 meshes along Z direc-
tion, and 64 meshes along R-direction. For 3D Cartesian
coordinates, we employed 128, 32, 72 meshes along the
X, Y and Z directions, respectively. We have verified the
convergence of the spectrum results by resolution tests.
We solve the radiation transfer equation in the polar
direction (along Z-axis, similar to “face-on” view) and
the equatorial direction (along X-axis, “edge-on” view),
respectively. Within one mesh, each physical parameter
can be treated as uniform, and the emitting intensity
can be derived from the radiation transfer equation:
∆Iν
∆x = −ανIν + jν
(9)
where αν is the absorption coefficient, ∆x is the mesh
length along the sight line, and jν is the volume emis-
sivity. The solution of the intensity at the beginning of
the mesh (i+1) or the end of the mesh i is:
Iν(i + 1) = Iν(i)e−∆τν(i) + jν(i)
αν(i)
h
1 −e−∆τν(i)i
(10)
where ∆τν(i) = αν(i)∆xi is the optical depth across the
i −th zone. We continue the calculation iteratively to
obtain the intensity from the final edge cell xN along
the line of sight.
The emitting spectra are derived by integrating the
intensity from all above final edge cells of the radiation
zone:
Fν =
Z
k
Iν,N(k)dΩcos θ
(11)
where Iν,N(k) is intensity from the k-th edge cell in the
radiation zone, and θ is the angle between the normal di-
rection of the radiative zone and the orientation towards
the observers. In the polar and equatorial directions, the
emitting radio spectra are calculated by:
Fν(pol) =
X
R
Iν,N2πR∆R/d2
L ,
(12)
Fν(eqt) =
X
y,z
Iν,N∆y∆z/d2
L .
(13)
where dL is the luminous distance. In this paper, we
normalize the value of Fν at dL = 100 Mpc, correspond-
ing to a cosmological redshift of z = 0.023.
In the polar direction, the separation distance between
the two forward shocks moving toward and away from
us results in a significant light-travel time difference
(LTTD, see Appendix Figure 1A). Therefore, we incor-
porated this correction in calculating the polar-direction
spectra. For instance, in run A, we find that the radi-
ation from the +Z FS of tsource = 2 yr arrives at the
observer at approximately the same time as that from
the −Z FS of tsource = 1.4 yr. Hence, for the polar ra-
dio spectrum of t = 2 yr, we used simulation data of the
+Z shock at t = 2 yr and the −Z shock at t = 1.4 yr.
Of course, the impact of the LTTD is not substantial;
for the fiducial model, its effect on the monochromatic
flux is within 20%. The LTTD is less significant in the
equatorial direction; hence, we did not incorporate such
a correction in this direction.
5. RESULTS AND DISCUSSIONS
5.1. Basic feature of the absorption frequency and
monochromatic luminosity
In the fiducial case (run A), we find that the FS sce-
nario can produce a self-absorption frequency νp around
GHz, with a peak monochromatic flux of several mJy.
This corresponds to a peak luminosity (defined as νpFνp)
of 1037 erg s−1. Such a luminosity is consistent with that
observed in some radio TDEs (Cendes et al. 2024).
The self-absorption frequency of run A decreases over
time (Figure 4), with νp(t) roughly following a t−0.8
trend overall. However, the decline is more gradual at
early times (t−0.5) and becomes steeper at later times
(t−1.25). This two-stage evolutionary feature is also ob-
served in other models. The turnover point corresponds
to the moment when the swept CNM mass becomes com-
parable to the ejecta mass. However, the discrepancies
in temporal indices among different models reflect the
complexity of the underlying hydrodynamic behavior.
The radio flux is affected by the FS parameters, which
are determined by both the outflow and the CNM. When
the ejecta mass is increased by a factor of 10 (run Bm01),
the monochromatic fluxes at three frequencies (0.88, 1.4
and 6 GHz) rise to 4–5 times those in the fiducial case.
Correspondingly, the maximum value of the peak lumi-
nosity increases to 3 × 1038 erg s−1. The highest flux
among the single density index models is from model
Mm01, in which the CNM density is 10 times that of
the Galactic Center. The maximal flux reaches several
tens of mJy (Table 1).
Compared to the fiducial model, increasing the out-
flow velocity (Cv03), enlarging the outflow opening an-
gle (Go60, Ho90), or raising the CNM density all lead to
significantly higher radio fluxes. However, our simula-
tions indicate that cases where the monochromatic flux
exceeds 10 mJy are uncommon. Generally, when the flux
exceeds 10 mJy (dL/100Mpc)−2, or the peak luminos-
6
Mou
0.3
0.2
0.1
0.0
0.1
0.2
0.3
X/parsec
0.0
0.1
0.2
0.3
0.4
0.5
Y/parsec
0
1
2
3
log10 (ne/cm
3)
6
5
4
log10 (e2/erg cm
3)
0.3
0.2
0.1
0.0
0.1
0.2
0.3
X/parsec
0.0
0.1
0.2
0.3
0.4
0.5
Y/parsec
0
1
2
3
log10 (ne/cm
3)
6
5
4
log10 (e2/erg cm
3)
0.3
0.2
0.1
0.0
0.1
0.2
0.3
X/parsec
0.0
0.1
0.2
0.3
0.4
0.5
Y/parsec
0
1
2
3
log10 (ne/cm
3)
6
5
4
log10 (e2/erg cm
3)
Figure 3. Snapshots of run A (fiducial run) at t = 2 yr (left), 5 yr (middle) and 10 yr (right). In each panel, the left half
window shows the density distribution, and the right half window shows the energy density of CRe.
108
109
1010
Frequency/Hz
10
1
100
101
F /mJy (dL = 100 Mpc)
2.5
1.0
2 yr (Eqt)
5 yr (Eqt)
10 yr (Eqt)
15 yr (Eqt)
2 yr (Pol)
5 yr (Pol)
10 yr (Pol)
15 yr (Pol)
100
101
Time (yr)
10
1
100
p/GHz
run A p (Eqt)
run A p (Pol)
Fo10 p (Eqt)
Fo10 p (Pol)
Ho90 p
100
101
Time/yr
10
1
100
101
F /mJy (dL = 100 Mpc)
=0.88 GHz (Eqt)
=1.4 GHz (Eqt)
=6.0 GHz (Eqt)
=0.88 GHz (Pol)
=1.4 GHz (Pol)
=6.0 GHz (Pol)
Figure 4. The left panel shows the synthetic radio spectra along the equatorial direction (Eqt) and polar direction (Pol). Note
that we have considered the light-travel time difference in the polar direction. The middle panel presents the time evolution of
the self-absorption frequency νp. The right panel presents the light curves of Fν for three frequencies.
ity νpLνp exceeds 1038 erg s−1, it may indicate that the
outflow is quite powerful (exceeding ∼1051 erg), or that
the CNM density is relatively high (significantly above
that in the Galactic center). Moreover, none of these
simulations produce high luminosities of 1039 erg s−1
or large fluxes of 100 mJy (dL/100Mpc)−2, which have
been detected in certain candidates such as ASASSN-
15oi (Horesh et al. 2021) and AT2018hyz (Cendes et al.
2022). Within the FS scenario, achieving such a high lu-
minosity would require outflow’s energy or CNM density
significantly higher than those assumed in our simula-
tions. Alternatively, other mechanisms beyond the FS
scenario may be responsible. For moderate conditions,
however, the maximal peak luminosity (νpLνp) drops
below 1037 erg s−1 (e.g., run Ev01c1 with a velocity of
0.1c and a Milky Way-like density).
5.2. Properties of the radio spectra
The spectra are predominantly contributed by the
CRes at “cap” of the FS (the region within the out-
flow’s opening angle), while the lateral FS also hosts a
substantial population of CRes (Figure 3, A2).
Such
a distribution of CRes leads to anisotropy of the ra-
dio spectra (Figure 4). Along the equatorial direction,
CRes in the lateral region of the FS contribute to low-
frequency emission, leading to a broadening of the spec-
trum at ν < νp.
Meanwhile, the longer path of syn-
chrotron emission through “cap” region increases the
optical depth, resulting in a self-absorption frequency
νp that is higher than in the polar direction.
The anisotropy becomes increasingly significant as the
outflow opening angle decreases (Figure 5). In particu-
lar, when θ0 is small θ0 ≲10◦, the spectra in the equato-
rial direction exhibit a “flat–top” feature – that is, below
the self-absorption frequency, the flux decreases slowly
as the frequency decreases, largely deviate from the ν5/2
feature. When θ0 is large, the “flat–top” feature disap-
pears, and the emitting spectra along the two directions
become nearly identical. Therefore, multi-frequency ob-
servations could help to constrain the viewing angle and
outflow opening angle.
Simulations provide detailed data that can be used
to assess the deviations of the widely adopted energy
equipartition method (Barniol Duran et al. 2013). We
extract the values of νp and Fνp from the synthetic spec-
Radio Afterglows I: Forward Shock
7
108
109
1010
Frequency/Hz
10
2
10
1
100
F /mJy (dL = 100 Mpc)
2.5
1.0
1 yr (Eqt)
2 yr (Eqt)
5 yr (Eqt)
11 yr (Eqt)
1 yr (Pol)
2 yr (Pol)
5 yr (Pol)
11 yr (Pol)
108
109
1010
Frequency/Hz
10
1
100
101
F /mJy (dL = 100 Mpc)
2 yr (Eqt)
4 yr (Eqt)
11 yr (Eqt)
2 yr (Pol)
4 yr (Pol)
11 yr (Pol)
Figure 5. The emitting radio spectra for different outflow opening angles θ0. The left panel shows the spectra at different
epochs for run Fo10 (θ0 = 10◦), and the right panel is for run Go60 (θ0 = 60◦). Obviously when the outflow’s opening angle is
smaller, the anisotropy of the spectra is more significant.
108
109
1010
Frequency/Hz
10
2
10
1
100
101
F /mJy (dL = 100 Mpc)
Bm01 (5yr)
Cv03 (5yr)
Dv01 (5yr)
Ev01 (5yr)
Ho90 (4yr)
In15 (4yr)
Jn20 (4yr)
Knbr (4yr)
Figure 6. The synthetic spectra generated by our model for
different models.
tra (e.g., Figure 6), and then apply the equipartition
method (see Appendix B for more details) to calculate
parameters including the forward shock radius Req, ve-
locity v′
s, total number of thermal electrons N ′
e, the den-
sity of the post-shocked CNM n′
e, and various energy
components including relativistic electron’s energy E′
e,
the magnetic field energy E′
B, and the shock energy E′
s.
Our results are listed in Table A1. It should be noted
that N ′
e refers to the number of thermal electrons of
the Maxwellian distribution, rather than the number of
relativistic electrons accounting for the radio emission.
Many observational studies mistakenly treat the num-
ber or number density of relativistic electrons as that
of thermal electrons (as pointed out in Matsumoto &
Piran 2021). Our N ′
e is derived following the sequence:
E′
e →E′
s →proton/electron mass →N ′
e (see Appendix
B).
For comparison,
the corresponding hydrodynamic
data are presented in Table 2. The shock energy here is
the sum of the kinetic and internal energies of the post-
shock CNM. We calculate the shock energy both over the
entire domain marked as Es, and within the outflow’s
opening angle marked as Es(cone). It should be noted
that at late stages, as shown in the t = 10 yr panel of
Figure 2, the CRe has clearly extended beyond the out-
flow cone. Moreover, the CNM swept up by the forward
shock has undergone significant expansion, resulting in
a thickness-to-radius ratio that substantially exceeds its
early-time value and is also significantly greater than the
commonly adopted value of 0.1. Therefore, applying the
equipartition method at late times introduces additional
uncertainty, and here we mainly focus on the data for
t ≤5 yr in most cases.
First, the analytical estimates of the FS radius are sig-
nificantly lower than those in simulations. Specifically,
based on the polar direction spectra, the estimated radii
are about half of those in simulation; for the equatorial
direction spectra, the estimated radii are approximately
1/3 ∼1/2 of those in simulations (the column Req/Rs
in Table 2).
Second, the estimated shock energies E′
s are on av-
erage, 9 times lower than those from simulations (see
the column E′
s/Es in Table 2). The estimated total en-
ergy of CRe are on average 9-10 times lower than those
from simulations. The energies derived from the analyt-
ical method are systematically underestimated, primar-
ily due to the fact that derivation in the equipartition
method involves multiple approximations, not the spec-
ified values of the ϵe or ϵB parameter.
Third, the analytically estimated densities of the post-
shock CNM (n′
e) are higher by a factor of a few to dozens
compared to those in simulations (the column n′
e/ne in
Table 2). Note that, since the derived shock radius cor-
8
Mou
responding to n′
e is about half of the simulated value,
the difference n′
e and ne is expected to be further re-
duced when compared at the same radius. However, a
discrepancy of several times (averaged a factor of 4-5)
still remains.
Finally, we also investigate the influence of different
power law indices of CRe. We perform calculations for
both p = 2.2 and p = 2.8, and find that the above
conclusions still hold (Table 2).
In summary, we recommend that when estimating
shock parameters using the equipartition method with
specified ϵe and ϵB, the derived Req should be multiplied
by 2, the Ee value by 9 ∼10, the estimated shock energy
(Ee/ϵe) by 9, and the CNM density by 4 ∼5. Since the
magnetic field energy should be multiplied by a factor of
9 ∼10 while the radius doubles, magnetic field strength
remains essentially unchanged.
5.3. Properties of light curves
First, we considered the early stage in which the mass
of the post-shock CNM is much lower than the ejecta
mass, and the forward shock velocity vs can be treated
as a constant. The mass of the post-shock CNM by the
FS is ˙MCNM = ρCNM(r)vs Ωr2 where Ωis the solid angle
of the outflow. The total shock energy is
Es(t) =
Z t
0
1
2
˙MCNMv2
s dt =
1
2(3 −n)ρ0Ωv5−n
s
t3−n .
(14)
The total energy of CRe is Ee(t) ≃ϵeEs(t) ∝t3−n.
As the forward shock expands, the density of the pre-
shock CNM decreases and thus the thermal pressure
of the post-shock CNM decreases with time:
P2 ≃
3
4ρCNM(r)v2
s ∝t−n.
The post-shock CNM undergoes
adiabatic expansion with its volume increasing as V ∝
p−3/5
2
∝t3n/5. This adiabatic cooling causes the energy
of each CRe to decrease as V −1/3. Consequently, the to-
tal energy of CRes evolves as E2(t) ∝t3−6n/5, which is
slower than t3−n due to adiabatic cooling. On the other
hand, the magnetic field strength can be estimated by
B = (8πϵBρCNMv2
s)1/2 = (8πϵBρ0v2−n
s
t−n)1/2 ∝t−n/2 .
(15)
The typical frequency of synchrotron radiation from an
electron with the Lorentz factor γm is
νm = γ2
meB
2πmec ∝B ∝t−n/2 .
(16)
The spectral power at frequency νm from those electrons
with the Lorentz factor γm is
Lνm ≃Ne
νm
4
3σT cγ2
m
B2
8π ∝t3−n .
(17)
Thus, for a fixed frequency νi > νp, the monochromatic
luminosity is
Lνi = Lνm
νi
νm
1−p
2
∝t Γ1
(18)
where the time index Γ1 ≡3 −0.25np −1.45n.
We compared this analytical result with the simu-
lation results (Figure 4 and 7) and found that they
are generally in agreement.
Based on these results,
for most parameter values (p > 2, n ≥1), we have
Γ1 ≲1, suggesting that the high frequency radio flux
generated by the FS exhibits a more gradual evolution
over time than t1.0. Interestingly, for density index of
n > 3/(0.25p + 1.45), Lνi is expected to decrease mono-
tonically with time – the radio flux would only become
fainter and fainter after the very early epoch (Figure 7,
run Jn20).
Second, we consider the late stage in which the post-
shock CNM’s mass significantly exceeds the outflow
mass.
In this stage, the kinetic energy of post-shock
CNM approaches saturation:
1
2Mcnm(r)
h
3vs(r)
4
i2
≃
constant,
where Mcnm
is the swept mass by the
FS. The velocity variation and the newly swept-up
CNM mass satisfy the relationship:
dvs(r)/vs(r) =
−1
2dMcnm(r)/Mcnm(r). From this equation, we derived
that vs ∝r
n−3
2
s
, rs ∝t
2
5−n , and vs ∝t
n−3
5−n . The total
shock energy in this stage can be roughly regarded as
a constant, and the same is true for the total energy of
CRe. On the other hand, since the magnetic pressure
is proportional to the shock ram pressure, the magnetic
field strength scales as B ∝t−
3
5−n .
Thus in the late
stage, Lνi scales as
Lνi ∝A0(t)B
p+1
2 (t) ∝t Γ2
(19)
where the index Γ2 ≡−3p+3
2(5−n).
We compared this analytical result with the simula-
tion results and found that they are generally in agree-
ment.
Specifically, values of Γ2 from simulations are
slightly lower (in terms of absolute magnitude, slightly
larger), meaning that real decline of flux is slightly faster
than the above expression.
Such a temporal evolving law of the high frequency
(ν > νp) radio flux provides a way for constraining the
CNM density index n.
In certain cases with particular density distributions,
a second flare in high-frequency flux may emerge. For
example, in run Lnbr (n = 2.5 for r < 0.1 pc and n = 1.0
for r ≥0.1 pc) which is similar to the case explored in
Matsumoto & Piran (2024), as the shock enters the outer
region with a more gradually declining density (t ≃3
yr), the radio emission begins to increase, exhibiting a
Radio Afterglows I: Forward Shock
9
rise timescale of one years. However, the second flare
not obvious, which is only enhanced by 10% (Figure
A4). Another example is run Knbr (n = 1.5 for r <
0.1 pc and n = 0.5 for r ≥0.1 pc), exhibiting a rise
timescale of two years and a flux enhancement of 73%.
This suggests that if the second flare is more prominent,
the density index n in the outer region should be lower
n ≲0.5 under the FS scenario. Similar phenomena – a
second brightening at high frequencies (ν > νp) – have
been reported in some sources, such as AT2019azh (15.5
GHz, Goodwin et al. 2022) and AT2020vwl (5.5 GHz,
Goodwin et al. 2024). However, in these observations,
the rise and decay timescales of the second brightening
are short, with the flux halving timescale of 0.6 year,
which differs from our simulations. Therefore, the light
curves obtained from current simulations indicate that
the FS light curves generally exhibit slow variability.
Rapid and large–amplitude variations (at least twice)
are likely caused by other mechanisms.
In Figure A3 and A4 of the Appendix, we present the
temporal evolution of the flux at two additional single
frequencies: 0.88 GHz and 1.4 GHz. These two frequen-
cies are also commonly used in observations. For these
low-frequency flux, we do not observe any second flare
in any of the models.
Finally, we also present the specific values of the shock
energy and outflow energy in Table 2. Based on simu-
lations, we find that when the high-frequency radio flux
begins to decline, the shock energy starts to saturate,
and its value at that turnover moment is approximately
half of the outflow energy. This empirical relation can
be used to estimate the kinetic energy of the outflow.
If long-term monitoring can capture the rising – decline
phase of the flux at ν > νp, one can infer that the out-
flow’s kinetic energy is about 2ϵ−1
e
times the value of Ee,
which corresponds to 20Ee if ϵe = 0.1.
6. CONCLUSIONS
In this study, we conducted hydrodynamic simulations
to explored the FS scenario of the radio afterglows fre-
quently observed in TDEs.
We inject the CRe com-
ponent into the grids near the shock front and simu-
late its hydrodynamic evolution after the shock acceler-
ation. Based on the hydrodynamic data, we calculated
the emitting radio spectra along two different directions
using radiative transfer.
We demonstrate that within a CNM environment sim-
ilar to that of the Galactic Center, a moderate outflow
(with kinetic energies around 1050 erg) can produce ra-
dio afterglows at the mJy level for dL = 100 Mpc, with
the self-absorption frequency near the GHz range. The
corresponding luminosity is on the order of 1037 erg s−1.
100
101
Time/yr
10
2
10
1
100
101
F /mJy (dL = 100 Mpc)
t1.0
t
1.0
Bm01
Cv03
Dv01
Ev01c1
Fo10
Go60
In15
Jn20
Knbr
Figure 7. Temporal evolution of the monochromatic flux
(6 GHz) in the equatorial direction. In most cases, the flux
exhibit a rising-then-falling trend. The maximum flux occurs
when the mass of the post-shock CNM becomes comparable
to the outflow mass, specifically in the range of 0.4 to 1.7
times the ejecta mass based on our different simulation runs.
Although the flux depends on the outflow strength and
CNM density, we find that fluxes above the 10 mJy level
requires either a very strong outflow (1051 erg) or a den-
sity significantly higher than that of the Galactic Center.
The self-absorption frequency νp shows a two-stage
power-law evolution, in which νp decreases slowly in the
early stage and rapidly in the late stage.
We find that the synthetic radio spectra exhibit
anisotropic feature that has not been previously re-
ported. In particular, when the outflow opening angle
is small, the radio spectrum in the equatorial direction
shows a “flat-top” feature.
Previous observational studies commonly employ ana-
lytical formulas of the equipartition method to estimate
the shock parameters.
After taking into account the
specific distribution of CRe and radiative transfer, we
reveal a significant discrepancy between the analytically
estimated values and the actual values: (1) the energies
of CRe in simulations (Ee in Table 2) can be, on aver-
age, as much as 9-10 times higher than the estimated
ones (E′
e in Table A1); (2) the shock energies (Es) are
on average 9 times higher than the estimated ones (E′
s);
(3) the shock radii are twice of those estimated values;
(4) the post-shock CNM densities (ne) are lower by a
factor of several compared to the estimated values (n′
e).
Under most parameter settings, the flux exhibits an
initial rising phase followed by a subsequent decline
10
Mou
Table 2. Physical parameters from simulations, and the ratios of the values obtained using the equipartition method (Table
A1) to those from the simulation data. Vs the instantaneous shock velocity, and Ne is the total amount of electrons in the
post-shock CNM within the opening angle of the outflow. In the “Direc” column, the labels p2.2 and p2.8 indicate that spectral
indices of p = 2.2 and 2.8 for the CRe are explored.
Run
∆T
Direc
νp
Fνp
Rs
Vs
Ne
ne
Ee
Es(cones)
Es
Req/Rs
n′
e/ne
E′
s/Es
units
yr
–
GHz
mJy
pc
c
cm−3
erg
erg
erg
A
2
Pol
1.6
2.0
0.109
0.163
1.6 × 1054
306
1.2 × 1048
3.9 × 1049
4.9 × 1049
0.45
6
0.11
A
2
Eqt
2.0
2.0
0.109
0.163
1.6 × 1054
306
1.2 × 1048
3.9 × 1049
4.9 × 1049
0.36
14.
0.084
A
2
Eqt p2.2
2.3
3.1
0.109
0.163
1.6 × 1054
306
1.2 × 1048
3.9 × 1049
4.9 × 1049
0.39
9
0.067
A
2
Eqt p2.8
1.7
1.1
0.109
0.163
1.6 × 1054
306
1.2 × 1048
3.9 × 1049
4.9 × 1049
0.32
22
0.086
A
5
Pol
0.98
4.4
0.25
0.144
3.3 × 1054
88
5.1 × 1048
1.4 × 1050
1.9 × 1050
0.46
7
0.12
A
5
Eqt
1.11
4.1
0.25
0.144
3.3 × 1054
88
5.1 × 1048
1.4 × 1050
1.9 × 1050
0.40
12
0.095
A
5
Eqt p2.2
1.3
6.2
0.25
0.144
3.3 × 1054
88
5.1 × 1048
1.4 × 1050
1.9 × 1050
0.41
9
0.068
A
5
Eqt p2.8
1.0
2.3
0.25
0.144
3.3 × 1054
88
5.1 × 1048
1.4 × 1050
1.9 × 1050
0.34
24
0.089
A
10
Pol
0.48
3.4
0.422
0.087
2.5 × 1055
30
8.3 × 1048
1.4 × 1050
2.2 × 1050
0.50
6
0.15
A
10
Eqt
0.46
2.9
0.422
0.087
2.5 × 1055
30
8.3 × 1048
1.4 × 1050
2.2 × 1050
0.48
7
0.13
A
10
Eqt p2.2
0.52
4.5
0.422
0.087
2.5 × 1055
30
8.3 × 1048
1.4 × 1050
2.2 × 1050
0.52
4
0.10
A
10
Eqt p2.8
0.4
1.7
0.422
0.087
2.5 × 1055
30
8.3 × 1048
1.4 × 1050
2.2 × 1050
0.43
10
0.14
Bm01
5
Pol
1.25
10.7
0.301
0.186
1.1 × 1055
110
1.1 × 1049
3.7 × 1050
4.5 × 1050
0.47
5
0.11
Bm01
5
Eqt
1.6
10.5
0.301
0.186
1.1 × 1055
110
1.1 × 1049
3.7 × 1050
4.5 × 1050
0.36
14
0.082
Bm01
10
Pol
0.87
20
0.579
0.173
4.5 × 1055
38
3.5 × 1049
1.1 × 1051
1.4 × 1051
0.47
7
0.11
Bm01
10
Eqt
1.0
19
0.579
0.173
4.5 × 1055
38
3.5 × 1049
1.1 × 1051
1.4 × 1051
0.40
13
0.086
Cv03
5
Pol
1.2
12.5
0.338
0.178
1.6 × 1055
51
1.6 × 1049
3.5 × 1050
4.9 × 1050
0.46
8
0.12
Cv03
5
Eqt
1.2
10.5
0.338
0.178
1.6 × 1055
51
1.6 × 1049
3.5 × 1050
4.9 × 1050
0.43
10
0.10
Dv01
5
Pol
0.65
0.54
0.142
0.086
2.6 × 1054
213
5.6 × 1047
1.8 × 1049
2.2 × 1049
0.46
7
0.12
Dv01
5
Eqt
0.82
0.49
0.142
0.086
2.6 × 1054
213
5.6 × 1047
1.8 × 1049
2.2 × 1049
0.35
19
0.10
Ev01c1
5
Pol
0.35
0.16
0.152
0.093
9.6 × 1053
76
2.4 × 1047
8.4 × 1048
9.6 × 1048
0.45
6
0.13
Ev01c1
5
Eqt
0.45
0.15
0.152
0.093
9.6 × 1053
76
2.4 × 1047
8.4 × 1048
9.6 × 1048
0.34
17
0.088
Fo10
2
Pol
1.4
0.26
0.118
0.188
1.2 × 1053
238
2.2 × 1047
3.7 × 1048
6.5 × 1048
0.52
4
0.12
Fo10
2
Eqt
1.7
0.28
0.118
0.188
1.2 × 1053
238
2.2 × 1047
3.7 × 1048
6.5 × 1048
0.44
7
0.11
Fo10
5
Pol
0.9
0.8
0.283
0.172
7.0 × 1053
101
1.2 × 1048
1.9 × 1049
3.6 × 1049
0.57
2
0.13
Fo10
5
Eqt
1.1
0.83
0.283
0.172
7.0 × 1053
101
1.2 × 1048
1.9 × 1049
3.6 × 1049
0.47
5
0.11
Fo10
11
Pol
0.56
1.5
0.572
0.127
3.8 × 1054
20
4.2 × 1048
5.2 × 1049
1.1 × 1050
0.61
4
0.15
Fo10
11
Eqt
0.58
1.4
0.572
0.127
3.8 × 1054
20
4.2 × 1048
5.2 × 1049
1.1 × 1050
0.57
5
0.13
Go60
2
Pol
1.55
4.4
0.097
0.137
5.5 × 1054
350
2.5 × 1048
9.9 × 1049
1.1 × 1050
0.46
5
0.12
Go60
2
Eqt
1.8
4.4
0.097
0.137
5.5 × 1054
350
2.5 × 1048
9.9 × 1049
1.1 × 1050
0.39
10
0.10
Go60
4
Pol
0.98
6.7
0.177
0.124
1.9 × 1055
136
6.5 × 1048
2.2 × 1050
2.5 × 1050
0.49
5
0.13
Go60
4
Eqt
1.0
5.9
0.177
0.124
1.9 × 1055
136
6.5 × 1048
2.2 × 1050
2.5 × 1050
0.45
7
0.11
Go60
11
Pol
0.34
3.2
0.357
0.063
7.5 × 1055
38
1.0 × 1049
2.0 × 1050
2.5 × 1050
0.49
5
0.16
Go60
11
Eqt
0.34
2.8
0.357
0.063
7.5 × 1055
38
1.0 × 1049
2.0 × 1050
2.5 × 1050
0.46
6
0.14
Ho90
2
–
1.58
6
0.088
0.122
1.0 × 1055
402
3.4 × 1048
1.6 × 1050
1.6 × 1050
0.50
4
0.13
Ho90
5
–
0.68
5.8
0.188
0.088
4.9 × 1055
110
7.9 × 1048
2.9 × 1050
2.9 × 1050
0.53
3
0.15
Ho90
10
–
0.32
3.1
0.29
0.055
1.2 × 1056
53
1.2 × 1049
2.9 × 1050
2.9 × 1050
0.54
3
0.15
In15
4
Pol
1.0
3.4
0.204
0.157
4.9 × 1054
105
3.5 × 1048
1.1 × 1050
1.4 × 1050
0.50
5
0.11
In15
4
Eqt
1.2
3.1
0.204
0.157
4.9 × 1054
105
3.5 × 1048
1.1 × 1050
1.4 × 1050
0.40
13
0.086
Jn20
4
Pol
1.0
3.2
0.202
0.163
4.5 × 1054
101
3.5 × 1048
9.7 × 1049
1.4 × 1050
0.49
6
0.10
Jn20
4
Eqt
1.2
2.7
0.202
0.163
4.5 × 1054
101
3.5 × 1048
9.7 × 1049
1.4 × 1050
0.38
15
0.071
Knbr
4
Pol
1.3
4.7
0.198
0.144
6.5 × 1054
163
4.2 × 1048
1.3 × 1050
1.6 × 1050
0.46
7
0.11
Knbr
4
Eqt
1.6
4.6
0.198
0.144
6.5 × 1054
163
4.2 × 1048
1.3 × 1050
1.6 × 1050
0.37
16
0.088
phase.
An exception occurs when the density index
n > 1.5: after a brief rise lasting several months, only
the declining phase remains. The radio flux rises rel-
atively slowly (flatter than t1.0), and its decay is also
gradual, with the flux halving timescale of several years.
These variation features can be used to test the applica-
bility of the FS scenario, which are quite different from
the bow shock scenario, in which the radio flux can ex-
hibit a sharp rise followed by a rapid decline (Mou et al.
2022; Zhuang et al. 2025). We also find that when the
high-frequency radio flux begins to decline, the shock
energy at that turnover moment is approximately one
half of the outflow’s kinetic energy.
Finally, we would like to emphasize that there are sev-
eral uncertainties in the interaction between the outflow
and the CNM, such as the angular distribution of the
outflow, the possibility of abrupt changes in the radial
distribution of the hot/diffuse CNM, and the evolution
of the magnetic field.
Given the current limited con-
straints on the physics of the outflow, CNM and shock,
Radio Afterglows I: Forward Shock
11
our results are based on simplified assumptions. Further
progress in understanding the radio afterglows will ne-
cessitate a coordinated development between theoretical
modeling and observations.
ACKNOWLEDGEMENTS
G.M. was supported by the National Key R&D Pro-
gram of China (Grant No. 2023YFA1607904), and the
NSFC (No. 12473013).
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12
Mou
108
109
1010
Frequency/Hz
10
1
100
101
F /mJy (dL = 100 Mpc)
2 yr (Eqt)
5 yr (Eqt)
10 yr (Eqt)
2 yr (Pol)
5 yr (Pol)
10 yr (Pol)
2 yr (Pol without LTTD)
5 yr (Pol without LTTD)
10 yr (Pol without LTTD)
Figure A1. Comparison of the spectra in the polar direction with and without considering light-travel time difference (LTTD)
under the fiducial model (run A). The results suggest that the inclusion of LTTD could slightly affect the spectra, as evidenced
by the comparison between the dashed and dotted lines with the same colors.
APPENDIX
A. THE INFLUENCE OF THE LIGHT-TRAVEL TIME DIFFERENCE
We show in Figure A1 the impact of considering versus neglecting LTTD on the spectra along the polar direction
for run A. The results indicate that LTTD can affect the emergent spectrum in the polar direction, although the effect
is not particularly significant. In the main text, all spectral calculations along the polar direction have taken LTTD
into account.
B. THE USAGE PROCEDURE OF THE ENERGY EQUIPARTITION/MINIMAL METHOD
Barniol Duran et al. (2013) developed the energy equipartition/minimal method for synchrotron emission, which
provides estimate for the size of the radiation zone, and the energies of magnetic field and CRe. However, some
observational articles failed to apply this method correctly, resulting in that first, the number of CRe components were
mistakenly regarded as the whole gas including thermal components (as pointed out by Matsumoto & Piran 2021),
and second, the derived magnetic field energy and CRe energy were inconsistent with the specified values of ϵB and ϵe.
The reason for the second point is that, many observational studies implicitly employ the three–component minimal
energy method when a factor ξ11/(13+2p) is applied to the Eeq calculation formula (ξ ≡1+ϵ−1
e ) (equation 28 in Barniol
Duran et al. 2013, see Section 4.2.2 therein for details). In this case, Eeq actually is Ee + EB + Ethp (Ethp is the
total energy of thermal protons), in which these three components satisfy Ee : EB : Ethp = 11
17ξ−1 :
6
17 : 11
17(1 −ξ−1).
In other words, once ξ or ϵe is specified, the relationship among EB, Ee, and Ethp is already determined, which is
therefore inconsistent with further specifying the value of ϵB. As a result, the values of Ee and EB reported in many
studies often do not match their adopted values of ϵe and ϵB.
The procedure of the equipartition method should follow the way to find equipartition radius Req when (Ee + EB)
reaches the minimal value, and then derive the values under the specified value of assumed ϵB and ϵe. Following this
procedure, each calculation step has well-defined physical meaning. All the following parameters in Step 1 – 3 are the
same as Barniol Duran et al. (2013), and all the parameters in the formula are unchanged.
Step 1: Calculating the equipartition radius Req.
Radio Afterglows I: Forward Shock
13
When EB = (6/11)Ee, Ee + EB reaches the minimal value. In this case, the radius of the radiation zone is also
determined.
Req ≈(1 × 1017cm) [21.8(525)p−1]
1
13+2p γ
2−p
13+2p
m
×
h
F
6+p
13+2p
p,mJy d
2(p+6)
13+2p
L,28
ν−1
p,10 (1 + z)−19+3p
13+2p
i
× f
−
5+p
13+2p
A
f
−
1
13+2p
V
Γ
p+8
13+2p 4
1
13+2p .
(B1)
This is just equation 27 in (Barniol Duran et al. 2013) except for an extra correction term 4
1
13+2p for the isotropic
number of radiating electrons. In this formula, γm = χe(Γ−1), where χe = p−2
p−1ϵe
mp
me , and if γm is found to be γm < 2,
one should use γm = 2. Note the value of Γ is the bulk Lorentz factor of the radiation zone, and one can set it to be
Γ = 1 in Step 1–3 before iteration.
Step 2: Calculating R, Ee, EB, Ne and B.
After setting the values of ϵB and ϵe, the condition for the minimal energy is usually broken, and the radius of the
radiation zone slightly deviates from Req by a factor of ϵ1/17:
R = Reqϵ1/17 ,
(B2)
where ϵ ≡11ϵB/(6ϵe). The energy in relativistic electrons within the observed region with γe ≥γm is
Ee =Nemec2γeΓ = 4(γe/γm)2−p 27c3F 4
ν,pd8
Lη5Γ2
16
√
3π3e2m2eν7p(1 + z)11f 3
AR6
≈4(γe/γm)2−p × 4.4 × 1050erg ×
h
F 4
p,mJy d8
L,28 ν−7
p,10 η5 (1 + z)−11i
Γ2
f 3
A R6
17
,
(B3)
while the energy in the magnetic field is
EB = (BΓ)2
8π
V = 8π6m6
ec2ν10
p (1 + z)14f 4
AfV R11
81e2F 4ν,pd8
Lη
20
3 Γ8
≈(2.1 × 1046erg)
h
F −4
p,mJy d−8
L,28 ν10
p,10 η−20
3 (1 + z)14i
× f 4
A fV R11
17
Γ8
,
(B4)
where the volume of the radiation zone V = fV πR3/Γ4. These are equation 17 and 18 in (Barniol Duran et al. 2013),
except for the an extra correction term 4(γe/γm)2−p in equation B3 for the isotropic number of radiating electrons
and those electrons with γe ≥γm.
Subsequently, one can derive the values of B and Ne from the values of Ee, EB and R.
Ne = 4(γe/γm)2−p 9cF 3
ν,pd6
Lη
10
3
8
√
3π2e2m2eν5p(1 + z)8f 2
AR4
≈4 × 1054 (γe/γm)2−p h
F 3
p,mJy d6
L,28 ν−5
p,10 η
10
3 (1 + z)−8i
×
1
f 2
A R4
17
,
(B5)
B = 8π3m3
ecν5
p(1 + z)7f 2
AR4
9eF 2ν,pd4
Lη
10
3 Γ3
≈(1.3 × 10−2 G)
h
F −2
p,mJy d−4
L,28 ν5
p,10 η−10
3 (1 + z)7i
× f 2
A R4
17
Γ3
.
(B6)
These are equation 15 and 16 in (Barniol Duran et al. 2013), except for the an extra correction term 4(γe/γm)2−p in
equation B5. Note that Ne is the number of relativistic electrons with γe ≥γm, which does not include the thermal
components. Thus, one should not use Ne directly to calculate the density of hot CNM. The Lorentz factor of the
relativistic electrons radiating at νp is (equation 14 in Barniol Duran et al. 2013):
γa =
3Fν,pd2
Lη
5
3 Γ
2πν2p(1 + z)3mefAR2 ≈525
h
Fp,mJy d2
L,28 ν−2
p,10 η
5
3 (1 + z)−3i
Γ
fA R2
17
.
(B7)
14
Mou
0.2
0.1
0.0
0.1
0.2
X/parsec
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Y/parsec
0
1
2
3
log10 (ne/cm
3)
6
5
4
log10 (e2/erg cm
3)
0.2
0.1
0.0
0.1
0.2
X/parsec
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Y/parsec
0
1
2
3
log10 (ne/cm
3)
6
5
4
log10 (e2/erg cm
3)
Figure A2.
Snapshots of run Fo10 and Go60 at t = 5 yr.
Step 3: Finding out Rs(t) and checking the self-consistency.
According to the radio data achieved in different periods, one can derive the relationship between shock radius and
time: Rs(t), from which the shock velocity vs and Γ−value can be obtained. Subsequently, do iterations and bring
the Γ−value back to equations B1–B7 in Step 1–2 to obtain the parameter values.
Step 4: Calculating the shock energy and the CNM density.
The shock energy is calculated as Es = Ee/ϵe. The enclosed mass of the hot CNM swept up by the FS can be
derived as Mcnm(Rs) = Es/[(Γ −1)c2] in the relativistic regime, or Mcnm(Rs) = 2Es/v2
s in the nonrelativistic regime.
The averaged post-shock CNM density can be estimated by ρ(Rs) ≃Mcnm(Rs)/V , while the density of the the
pre-shock CNM ahead of the shock front should be further reduced by a factor of 4 (density jump). It should be noted
that the shock energy Es is not equivalent to the outflow energy Ek; rather, Es represents a robust lower limit for Ek.
When the swept-up CNM mass is significantly less than the outflow mass, the outflow energy can be approximated by
Ek ≃Es × Mout
Mcnm
(B8)
Due to the lack of constraints on Mout, it is challenging to estimate Ek, and it may be approximated as Es when
the forward shock velocity decreases significantly for the FS scenario.
A program implementing the above procedures is available on GitHub 1. Please note: The above program and
web-based code are intended only to illustrate the workflow of the energy equipartition method, and do not account
for the errors introduced by this approach. One should note that the equipartition method provides rough estimates of
the shock radius, Ee and EB, since the derivation in Barniol Duran et al. (2013) involves multiple approximations, and
could lead to considerable derivation. For better matching the results of radiation transfer calculations, one should
multiply the Req by a factor of 2, Ee and EB by a factor of 9 (on average).
C. SHOCK PARAMETERS DERIVED FROM THE EQUIPARTITION METHOD
Based on the synthetic radio spectra obtained from the hydrodynamic simulations, we extract νp and Fνp, and then
follow the steps of the equipartition method to derive the “estimated” shock radius Req, density n′
e, and various energy
components (E′
e, E′
B, E′
s) in Table A1. These parameters are used for comparison with the actual values in simulations
(Table 2).
1 https://github.com/G-Mou/RadioTDE
Radio Afterglows I: Forward Shock
15
Table A1. Shock parameters derived by using the equipartition method based on the synthetic radio spectra. We assume that
the emitting region is a shell of thickness 0.1Req, and opening angle of the shock is equal to the outflow’s opening angle in each
model.
Run
∆T
Direc
νp
Fνp
Req
V
′
s ≡Req/∆T
N′
e
n′
e
E′
e
E′
B
E′
shk
(units)
yr
–
GHz
mJy
pc
c
cm−3
erg
erg
erg
A
2
Pol
1.6
2
0.049
0.080
9.1 × 1053
1762
1.5 × 1047
5.8 × 1047
5.2 × 1048
A
2
Eqt
2.0
2.0
0.039
0.064
1.1 × 1054
4300
1.2 × 1047
4.6 × 1047
4.1 × 1048
A
2
Eqt (p2.2)
2.3
3.1
0.042
0.069
7.7 × 1053
2870
9.8 × 1046
3.4 × 1047
3.3 × 1048
A
2
Eqt (p2.8)
1.7
1.1
0.035
0.057
1.5 × 1054
6644
1.3 × 1047
5.0 × 1047
4.2 × 1048
A
5
Pol
0.98
4.4
0.116
0.076
4.2 × 1054
615
6.5 × 1047
2.4 × 1048
2.2 × 1049
A
5
Eqt
1.1
4.1
0.099
0.065
4.7 × 1054
1094
5.3 × 1047
2.0 × 1048
1.8 × 1049
A
5
Eqt (p2.2)
1.3
6.2
0.103
0.067
3.3 × 1054
829
3.9 × 1047
1.4 × 1048
1.3 × 1049
A
5
Eqt (p2.8)
1.0
2.3
0.084
0.055
6.5 × 1054
2107
5.2 × 1047
2.1 × 1048
1.7 × 1049
A
10
Pol
0.48
3.4
0.21
0.069
7.7 × 1054
189
9.7 × 1047
3.6 × 1048
3.2 × 1049
A
10
Eqt
0.46
2.9
0.203
0.066
7.3 × 1054
197
8.4 × 1047
3.1 × 1048
2.8 × 1049
A
10
Eqt (p2.2)
0.52
4.5
0.221
0.072
4.9 × 1054
124
6.7 × 1047
2.3 × 1048
2.2 × 1049
A
10
Eqt (p2.8)
0.4
1.7
0.182
0.060
9.8 × 1054
313
9.1 × 1047
3.6 × 1048
3.0 × 1049
Bm01
5
Pol
1.25
10.7
0.140
0.092
6.5 × 1054
560
1.5 × 1048
5.5 × 1048
4.9 × 1049
Bm01
5
Eqt
1.6
10.5
0.108
0.071
8.4 × 1054
1547
1.1 × 1048
4.2 × 1048
3.7 × 1049
Bm01
10
Pol
0.87
20.0
0.270
0.088
2.2 × 1055
258
4.5 × 1048
1.7 × 1049
1.5 × 1060
Bm01
10
Eqt
1.0
19.0
0.230
0.075
2.4 × 1055
475
3.7 × 1048
1.4 × 1049
1.2 × 1050
Cv03
5
Pol
1.2
12.5
0.157
0.103
6.5 × 1054
398
1.8 × 1048
6.8 × 1048
6.1 × 1049
Cv03
5
Eqt
1.2
10.5
0.145
0.095
6.2 × 1054
487
1.5 × 1048
5.5 × 1048
5.0 × 1049
Dv01
5
Pol
0.65
0.54
0.065
0.042
1.7 × 1054
1409
8.0 × 1046
3.0 × 1047
2.7 × 1048
Dv01
5
Eqt
0.82
0.49
0.049
0.032
2.1 × 1054
3947
5.7 × 1046
2.1 × 1047
1.9 × 1048
Ev01c1
5
Pol
0.35
0.16
0.069
0.045
6.2 × 1053
446
3.5 × 1046
1.3 × 1047
1.2 × 1048
Ev01c1
5
Eqt
0.45
0.15
0.052
0.034
7.7 × 1053
1293
2.5 × 1046
9.3 × 1046
8.4 × 1047
Fo10
2
Pol
1.4
0.26
0.061
0.10
8.7 × 1052
860
2.3 × 1046
8.6 × 1046
7.7 × 1047
Fo10
2
Eqt
1.7
0.28
0.052
0.085
1.1 × 1053
1723
2.1 × 1046
7.7 × 1046
6.9 × 1047
Fo10
5
Pol
0.9
0.8
0.161
0.105
4.6 × 1053
246
1.4 × 1047
5.1 × 1047
4.6 × 1048
Fo10
5
Eqt
1.1
0.83
0.134
0.088
5.7 × 1053
529
1.2 × 1047
4.4 × 1047
3.9 × 1048
Fo10
11
Pol
0.56
1.5
0.347
0.103
1.6 × 1054
86
4.7 × 1047
1.7 × 1048
1.6 × 1049
Fo10
11
Eqt
0.58
1.4
0.324
0.096
1.7 × 1054
108
4.2 × 1047
1.5 × 1048
1.4 × 1049
Go60
2
Pol
1.55
4.4
0.045
0.074
2.6 × 1054
1781
3.8 × 1047
1.4 × 1048
1.3 × 1049
Go60
2
Eqt
1.8
4.4
0.038
0.062
3.2 × 1054
3419
3.3 × 1047
1.2 × 1048
1.1 × 1049
Go60
4
Pol
0.98
6.7
0.086
0.070
7.6 × 1054
724
9.9 × 1047
3.3 × 1048
3.3 × 1049
Go60
4
Eqt
1.0
5.9
0.079
0.065
7.4 × 1054
900
8.3 × 1047
2.8 × 1048
2.8 × 1049
Go60
11
Pol
0.34
3.2
0.175
0.052
1.6 × 1055
186
1.2 × 1048
3.9 × 1048
3.9 × 1049
Go60
11
Eqt
0.34
2.8
0.164
0.049
1.6 × 1055
216
1.0 × 1048
3.4 × 1048
3.4 × 1049
Ho90
2
–
1.58
6.0
0.044
0.072
4.3 × 1044
1624
6.0 × 1047
2.2 × 1048
2.0 × 1049
Ho90
5
–
0.68
5.8
0.100
0.065
1.2 × 1055
370
1.3 × 1048
4.9 × 1048
4.4 × 1049
Ho90
10
–
0.32
3.1
0.158
0.052
1.9 × 1055
148
1.3 × 1048
5.0 × 1048
4.5 × 1049
In15
4
Pol
1.0
3.4
0.102
0.083
2.5 × 1054
568
4.7 × 1047
1.7 × 1048
1.6 × 1049
In15
4
Eqt
1.2
3.1
0.081
0.066
3.0 × 1054
1320
3.5 × 1047
1.3 × 1048
1.2 × 1049
Jn20
4
Pol
1.0
3.2
0.099
0.081
2.4 × 1054
605
4.3 × 1047
1.6 × 1048
1.4 × 1049
Jn20
4
Eqt
1.2
2.7
0.076
0.062
2.8 × 1054
1542
3.0 × 1047
1.1 × 1048
9.9 × 1048
Knbr
4
Pol
1.3
4.7
0.091
0.074
2.5 × 1054
1124
5.3 × 1047
2.0 × 1048
1.8 × 1049
Knbr
4
Eqt
1.6
4.6
0.073
0.060
3.0 × 1054
2601
4.2 × 1047
1.6 × 1048
1.4 × 1049
16
Mou
100
101
Time/yr
10
2
10
1
100
101
F /mJy (dL = 100 Mpc)
Bm01
Cv03
Dv01
Ev01c1
Fo10
Go60
In15
Jn20
Knbr
100
101
Time/yr
10
2
10
1
100
101
F /mJy (dL = 100 Mpc)
Bm01
Cv03
Dv01
Ev01c1
Fo10
Go60
In15
Jn20
Knbr
Figure A3. Temporal evolution of the monochromatic flux Fν in the equatorial direction. The left panel is for ν = 0.88 GHz,
and the right panel is for 1.4 GHz.
100
101
Time/yr
10
1
100
101
102
F /mJy (dL = 100 Mpc)
t1.0
t
1.0
t
2.0
Lnbr
Mm01
Nn25
100
101
Time/yr
10
1
100
101
102
F /mJy (dL = 100 Mpc)
Lnbr: 0.88 GHz
Lnbr: 1.4 GHz
Mm01: 0.88 GHz
Mm01: 1.4 GHz
Nn25: 0.88 GHz
Nn25: 1.4 GHz
Figure A4. Temporal evolution of the monochromatic flux Fν in the equatorial direction for model Lnbr, Mn01 and Nn25.
The left panel is for ν = 6 GHz, and the right panel is for 0.88 and 1.4 GHz.
|
Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX631 Numerical Studies on the Radio Afterglows in TDE (I): Forward Shock Guobin Mou (牟国斌)1, 2 1 210023, China; 2Nanjing key laboratory of particle physics and astrophysics, China ABSTRACT Recent long-term radio monitoring of tidal disruption events (TDEs) suggests that radio afterglows are common. Most studies argue that these afterglows may arise from forward shocks (FS) produced by the interaction between the TDE outflow and the hot, diffuse circumnuclear medium (CNM). Current theoretical models do not model the evolution of relativistic electrons in space, which introduces uncertainties. Here we conducted hydrodynamic simulations to study the hydrodynamic evolution of relativistic electrons, and calculated the synchrotron spectra via radiative transfer. We focus on the FS scenario with non-relativistic outflows, and various parameters of the outflow and CNM are explored. A moderate outflow with kinetic energy of several 1050 erg in a Galactic center - like CNM can produce mJy-level radio afterglows at a distance of 100 Mpc. The self-absorption frequency exhibits a slow decline at early times and a rapid decrease at late times. We derived the temporal evolution of the high-frequency radio flux, revealing its characteristic rise and decline pattern. We also find that: (1) the radio spectra for narrow outflows are clearly anisotropic along different sight lines; (2) the FS parameters inferred from radio spectra using conventional analytical formulas deviate significantly from those in simulations, in which the inferred shock radii are half of those from simulations, and the inferred energies are an order of magnitude lower. Keywords: radio continuum: transients - radiation mechanisms: non-thermal - galaxies: active - (galaxies:) quasars: supermassive black holes 1. INTRODUCTION Some TDE candidates exhibit radio emission with luminosities of 1036-42 erg s-1 and time lags spanning from days to years (Alexander et al. 2020). Recent studies indicate that about half of all opticallyselected TDEs exhibit show radio emission that rises on timescales of hundreds of days (Cendes et al. 2024), suggesting that TDEs with radio emission delayed by years are prevalent. These radio afterglows of TDEs arises from synchrotron radiation of cosmic ray electrons (CRes), though there are several possible sources for these electrons. Some studies argue that they could come from shocks driven by relativistic jets (Bloom et al. 2011; Burrows et al. 2011; Zauderer et al. 2011; Giannios & Metzger 2011; Lei et al. 2016). However, observations over the past decade show that jetted TDEs are rare (Andreoni et al. 2022). In most current observational studies, CRes are interpreted as originating from the forward shock (FS) when wide or narrow nonrelativistic outflow interacting with hot and diffuse CNM (Alexander et al. 2016). Alternatively, some studies suggest that CRes may originate from shocks produced by interaction between the unbound debris and CNM (Krolik et al. 2016; Yalinewich et al. 2019) or dense torus (Lei et al. 2024), or collisions between TDE outflows and ambient clouds (Mou & Wang 2021; Mou et al. 2022; Zhuang et al. 2025). The former scenario essentially belongs to the FS scenario, focusing on the shock sweeping through the material ahead of the shock front. In contrast, the latter corresponds to the bow shock (BS) scenario, which focuses on the shock interacting sweeping through the outflow. We leave the hydrodynamic simulation results for the BS scenario to a forthcoming paper. Although the origin of the radio emission is still debated, the presence of TDE outflows inevitably leads to the formation of a FS. The TDE outflow has been demonstrated to be both powerful and high-velocity. The outflow could be generated in both self-interaction process due to GR reprocessing (Sadowski et al. 2016; Lu & Bonnerot 2020), circularization process of infalling debris (Steinberg et al. 2022), or the finally accretion process (Dai et al. 2018; Curd & Narayan 2019; Bu et al. 2023; Hu et al. 2024). The ratio of mass outflow rate to the mass accretion rate could be considerable 16 Oct 2025 2 Mou (up to 1/10, e.g., Steinberg et al. 2022; Thomsen et al. 2022), and accordingly, the mass of the outflow could reach 10-2-10-1M⊙for disrupting a sun-like star. The existence of high speed outflow has been demonstrated by X-ray and UV observations of TDEs (Kara et al. 2018; Hung et al. 2019; Xiang et al. 2024), with some blueshifted absorption lines indicating velocities of up to 0.2c. In this context, we first focus on investigating the FS scenario to explore the general characteristics of its radio emission. Theoretical investigations usually employ analytic methods (Barniol Duran et al. 2013; Matsumoto & Piran 2021), or conduct hydrodynamic simulations (Hu et al. 2025) that do not incorporate relativistic electrons in simulations. Due to the simplifications, the spatial distribution and evolution of relativistic electrons remains largely unknown, which introduces uncertainties in connecting shock physics with the radio spectra. The complexity of the fluid behavior therefore makes numerical simulations incorporating relativistic electrons indispensable. In this study, we employ a shock-injected CRe approach, utilizing hydrodynamical simulations to investigate the spatial evolution of CRe after the shock acceleration. Radiative transfer calculations are then used to calculate the radio spectra along different directions. This study reveals the relationship between the physical parameters of the FS and the resulting radio emission, and highlights the errors that cannot be overlooked in previous studies. In Section 2, we introduce the physics related to the models. We introduce the settings of the simulations in Section 3, and present the data processing methods in Section 4. Results are presented in Section 5 and we give a brief summary in Section 6. 2. PHYSICS OF THE MODELS 2.1. CNM Up to now, observational constraints on the hot CNM are available for two sources: Sgr A* and M87. The density of CNM around Sgr A* follows ρ(r) ≃ 30 mH cm-3r-1 -1 (r-1 ≡r/0.1pc, Xu et al. 2006; Gillessen et al. 2019). For M87, although the BH mass is three orders of magnitude higher than Sgr A*, its CNM density at a given distance in parsec is only one order of magnitude higher than that of Sgr A* (Russell et al. 2015). Here we simply assume a CNM density following a power-law form similar to that of Sgr A*: ρ(r) = Csgr × 30 mH cm-3r-n -1 (1) where Csgr is set to be 3.0 in the fiducial case, indicating a density 3 times that at 0.1 parsec from Sgr A*, and n is the density index which is set to be 1.0 in the fiducial case. We also investigated different power-law indices as well (Table 1). Furthermore, the density profile may presents different slopes inside and outside the position where the stellar wind could marginally escape the gravitation potential of the SMBH (Generozov et al. 2017). Thus, we examined broken power-law cases, where the dividing radius between the "inner" and "outer" regions is denoted as rturn (runs Knbr, Lnbr). 2.2. TDE Outflow TDE is a transient phenomenon, and we assume that the outflow persists for 1 year. We constrain the outflow to a biconical structure with a half-opening angle of θ0, and inject it into the simulation domain along the two polar axes from the inner boundary of r. We parameterize the injected outflow with its velocity vout and mass outflow rate ̇Mout (Table 1), which remaining constant during the outflow ejection epoch. In the fiducial model, the velocity is set to be 0.2 c, and the total outflow mass is 0.01M⊙, with a corresponding energy (kinetically dominated) of 3.5 × 1050 erg. We also examined a more powerful case and two weaker cases for comparison. 2.3. Shocks and Relativistic Electrons The electron acceleration efficiency εe is assumed to be the fraction of the energy flux that can be dissipated at the shock (i.e., the change in the kinetic energy flux across the shock) channelled into the accelerated relativistic electrons in the downstream. In the frame of shock front, the expression of εe is εe = e2vd 1 2ρiv3s(1 -C-2) (2) where e2 is the energy density of CRe in the downstream, vd is the downstream velocity, ρi is the pre-shock density, and C ≡4M2/(M2+3) is the compression ratio. When the Mach number M ≫1, we have vd = vs/4 and C = 4. In this case, εe can be simplified to 0.6e2/ed, where ed is thermal pressure in the downstream. The acceleration efficiency εe is highly uncertain, and conventionally it should be lower than that of CRp. Simulations suggest that the acceleration efficiency of CRp of ∼10% for high Mach number shocks (e.g., Caprioli & Spitkovsky 2014), and εe should not exceed 10%. Here we adopt εe = 0.03 as the fiducial value. Note that the definition of εe here is slightly different from the one adopted in some literature where it is defined as ε′ e = Ee/Es, i.e., the total energy of relativistic electrons to the shock energy. The ratio between the two is ε′ e/εe = (1 -C-2) = 0.94 for M ≫1, and thus one does not need to distinguish them. Radio Afterglows I: Forward Shock 3 2nd 3rd 4th 1st pre-shock post-shock pre-shock post-shock Field Quantity ( arbitrary unit ) Distance ( x10-3 ) e2 x10 e1 Figure 1. Schematic diagram of the CRe injection method. The energy density distributions (e1 and e2) refer to those in the nine meshes intersected by the blue line segment in the upper right panel. The simulation code uses 4 grid levels to capture the shock, which stabilizes at the 4th mesh (see the orange line). The CRe component is injected at the 4th mesh. Due to numerical diffusion, the values of e2 in the 1st - 3rd mesh subsequently become nonzero, but this has a negligible effect on the results. In general, the spectral energy distribution of CRes follow a power-law form: dncr dγe = A0γ-p e , where p is the power-law index and we set p = 2.5 in most cases. This distribution is related to e2 through: e2 = Z 2000 2 γmec2 dncr dγe dγe , (3) where the integration upper limit (1 GeV) depends on the cooling and shock acceleration processes, but has little effect on the radio emission below 50 GHz discussed here. Once e2 is specified in simulations, the coefficient A0 and the spectral energy distribution of CRe can be determined accordingly. 2.4. Magnetic Field The relativistic electrons and protons can induce various instabilities, and drive magnetic perturbations, and amplify the magnetic field (Bell & Lucek 2001; Schure et al. 2012). The magnetic pressure can be enhanced to a fraction of the ram pressure. Given the complexity of this process, we do not include the magnetic field in simulations, and simply assume that the ratio of magnetic field energy density to that of CRe in each mesh remains a constant, i.e., B2(t)/8π e2(t) = εB εe . (4) We set εB = 0.10 in this study, which is higher than εe. 3. NUMERICAL SETTINGS We conduct the two-fluid simulation with ZEUS-3D code (Clarke 2010). For simplicity, we did not incorporate the magnetic field and physical diffusion of CRe. The timescale of radiative cooling of the CNM and outflow is much longer than the simulation timescale, and thus radiative cooling is negligible. The hydrodynamic equations are ∂ρ ∂t + ∇· (ρv) = 0, (5) ρdv dt = -∇(p1 + p2) -ρ∇Φ, (6) ∂e1 ∂t + ∇· (e1v) = -p1∇· v, (7) ∂e2 ∂t + ∇· (e2v) = -p2∇· v, (8) where p1 ≡(γ1-1)e1 is the thermal pressure (γ1 = 5/3), p2 ≡(γ2 -1)e2 is the pressure of CRe (γ2 = 4/3), and Φ is the gravitation potential (Φ = -GMbh/r, Mbh = 5 × 106 M⊙in this work). We did not simulate the acceleration process of electrons, but instead, directly injected the CRe component as the second fluid which is embodied as the energy density e2. As mentioned in Section 2.3, e2 is assigned the value of 5 3εeed, where ed is thermal pressure in the downstream. The shock front in ZEUS-3D is typically captured with 4 meshes, and thus we inject the CRe at the 4th-mesh (the final mesh capturing shocks) when the parameters of the post-shock gas are stabilized (figure 1). After that, we subtract the CRe energy density from that of the thermal gas to help maintain energy conservation. We adopt 2.5 dimensional spherical coordinates, in which the system is symmetric in φ-direction. The biconical outflow is ejected along the two polar axes. The inner and the outer boundary of r is set to be 0.0061 pc and 0.60 pc, respectively. Exceptions are run Bm01, Cv03, and Mm01, where the outer boundary is extended to 1.07 pc to better encompass the radio decay phase. The computation domain is divided into 2560 (or 2880 for exceptions) pieces in r-direction with dri+1/dri = 1.0018, and 256 pieces in θ-direction with dθj+1/dθj = 1.004. The high resolution adopted here is sufficient to ensure the results are convergent. 4. CALCULATING THE RADIO SPECTRA The simulations provide the distributions of the CRe energy density and the magnetic field strength in the study region. These parameters can be used to calculate the emitting synchrotron spectrum. As shown in Figure 2, we map the radiation zone enclosed by the FS front into a 3D Cylindrical coordinate (for polar direction spectra) and 3D Cartesian coordinate system (for 4 Mou Table 1. Parameters for modeling the radio afterglows. θ0 is the half-opening angle of the outflow. Ek is the kinetic energy of the outflow. Csgr is the ratio of the density at r = 0.1 pc to that at the Galactic Center. F0.88GHz,max represents the maximum value of F0.88GHz over time which is observed along the equatorial direction with a luminosity distance of dL = 100 Mpc. F1.4GHz,max and F6GHz,max are the same, but for 1.4 GHz and 6 GHz, respectively. For Knbr and Lnbr, we adopt two different density indices inside and outside a transition radius of 0.1 pc: n = 1.5 (2.5) for r νp, the monochromatic luminosity is Lνi = Lνm νi νm 1-p 2 ∝t Γ1 (18) where the time index Γ1 ≡3 -0.25np -1.45n. We compared this analytical result with the simulation results (Figure 4 and 7) and found that they are generally in agreement. Based on these results, for most parameter values (p > 2, n ≥1), we have Γ1 ≲1, suggesting that the high frequency radio flux generated by the FS exhibits a more gradual evolution over time than t1.0. Interestingly, for density index of n > 3/(0.25p + 1.45), Lνi is expected to decrease monotonically with time - the radio flux would only become fainter and fainter after the very early epoch (Figure 7, run Jn20). Second, we consider the late stage in which the postshock CNM's mass significantly exceeds the outflow mass. In this stage, the kinetic energy of post-shock CNM approaches saturation: 1 2Mcnm(r) h 3vs(r) 4 i2 ≃ constant, where Mcnm is the swept mass by the FS. The velocity variation and the newly swept-up CNM mass satisfy the relationship: dvs(r)/vs(r) = -1 2dMcnm(r)/Mcnm(r). From this equation, we derived that vs ∝r n-3 2 s , rs ∝t 2 5-n , and vs ∝t n-3 5-n . The total shock energy in this stage can be roughly regarded as a constant, and the same is true for the total energy of CRe. On the other hand, since the magnetic pressure is proportional to the shock ram pressure, the magnetic field strength scales as B ∝t3 5-n . Thus in the late stage, Lνi scales as Lνi ∝A0(t)B p+1 2 (t) ∝t Γ2 (19) where the index Γ2 ≡-3p+3 2(5-n). We compared this analytical result with the simulation results and found that they are generally in agreement. Specifically, values of Γ2 from simulations are slightly lower (in terms of absolute magnitude, slightly larger), meaning that real decline of flux is slightly faster than the above expression. Such a temporal evolving law of the high frequency (ν > νp) radio flux provides a way for constraining the CNM density index n. In certain cases with particular density distributions, a second flare in high-frequency flux may emerge. For example, in run Lnbr (n = 2.5 for r νp) - have been reported in some sources, such as AT2019azh (15.5 GHz, Goodwin et al. 2022) and AT2020vwl (5.5 GHz, Goodwin et al. 2024). However, in these observations, the rise and decay timescales of the second brightening are short, with the flux halving timescale of 0.6 year, which differs from our simulations. Therefore, the light curves obtained from current simulations indicate that the FS light curves generally exhibit slow variability. Rapid and large-amplitude variations (at least twice) are likely caused by other mechanisms. In Figure A3 and A4 of the Appendix, we present the temporal evolution of the flux at two additional single frequencies: 0.88 GHz and 1.4 GHz. These two frequencies are also commonly used in observations. For these low-frequency flux, we do not observe any second flare in any of the models. Finally, we also present the specific values of the shock energy and outflow energy in Table 2. Based on simulations, we find that when the high-frequency radio flux begins to decline, the shock energy starts to saturate, and its value at that turnover moment is approximately half of the outflow energy. This empirical relation can be used to estimate the kinetic energy of the outflow. If long-term monitoring can capture the rising - decline phase of the flux at ν > νp, one can infer that the outflow's kinetic energy is about 2ε-1 e times the value of Ee, which corresponds to 20Ee if εe = 0.1. 6. CONCLUSIONS In this study, we conducted hydrodynamic simulations to explored the FS scenario of the radio afterglows frequently observed in TDEs. We inject the CRe component into the grids near the shock front and simulate its hydrodynamic evolution after the shock acceleration. Based on the hydrodynamic data, we calculated the emitting radio spectra along two different directions using radiative transfer. We demonstrate that within a CNM environment similar to that of the Galactic Center, a moderate outflow (with kinetic energies around 1050 erg) can produce radio afterglows at the mJy level for dL = 100 Mpc, with the self-absorption frequency near the GHz range. The corresponding luminosity is on the order of 1037 erg s-1. 100 101 Time/yr 10 2 10 1 100 101 F /mJy (dL = 100 Mpc) t1.0 t 1.0 Bm01 Cv03 Dv01 Ev01c1 Fo10 Go60 In15 Jn20 Knbr Figure 7. Temporal evolution of the monochromatic flux (6 GHz) in the equatorial direction. In most cases, the flux exhibit a rising-then-falling trend. The maximum flux occurs when the mass of the post-shock CNM becomes comparable to the outflow mass, specifically in the range of 0.4 to 1.7 times the ejecta mass based on our different simulation runs. Although the flux depends on the outflow strength and CNM density, we find that fluxes above the 10 mJy level requires either a very strong outflow (1051 erg) or a density significantly higher than that of the Galactic Center. The self-absorption frequency νp shows a two-stage power-law evolution, in which νp decreases slowly in the early stage and rapidly in the late stage. We find that the synthetic radio spectra exhibit anisotropic feature that has not been previously reported. In particular, when the outflow opening angle is small, the radio spectrum in the equatorial direction shows a "flat-top" feature. Previous observational studies commonly employ analytical formulas of the equipartition method to estimate the shock parameters. After taking into account the specific distribution of CRe and radiative transfer, we reveal a significant discrepancy between the analytically estimated values and the actual values: (1) the energies of CRe in simulations (Ee in Table 2) can be, on average, as much as 9-10 times higher than the estimated ones (E′ e in Table A1); (2) the shock energies (Es) are on average 9 times higher than the estimated ones (E′ s); (3) the shock radii are twice of those estimated values; (4) the post-shock CNM densities (ne) are lower by a factor of several compared to the estimated values (n′ e). Under most parameter settings, the flux exhibits an initial rising phase followed by a subsequent decline 10 Mou Table 2. Physical parameters from simulations, and the ratios of the values obtained using the equipartition method (Table A1) to those from the simulation data. Vs the instantaneous shock velocity, and Ne is the total amount of electrons in the post-shock CNM within the opening angle of the outflow. In the "Direc" column, the labels p2.2 and p2.8 indicate that spectral indices of p = 2.2 and 2.8 for the CRe are explored. Run ∆T Direc νp Fνp Rs Vs Ne ne Ee Es(cones) Es Req/Rs n′ e/ne E′ s/Es units yr - GHz mJy pc c cm-3 erg erg erg A 2 Pol 1.6 2.0 0.109 0.163 1.6 × 1054 306 1.2 × 1048 3.9 × 1049 4.9 × 1049 0.45 6 0.11 A 2 Eqt 2.0 2.0 0.109 0.163 1.6 × 1054 306 1.2 × 1048 3.9 × 1049 4.9 × 1049 0.36 14. 0.084 A 2 Eqt p2.2 2.3 3.1 0.109 0.163 1.6 × 1054 306 1.2 × 1048 3.9 × 1049 4.9 × 1049 0.39 9 0.067 A 2 Eqt p2.8 1.7 1.1 0.109 0.163 1.6 × 1054 306 1.2 × 1048 3.9 × 1049 4.9 × 1049 0.32 22 0.086 A 5 Pol 0.98 4.4 0.25 0.144 3.3 × 1054 88 5.1 × 1048 1.4 × 1050 1.9 × 1050 0.46 7 0.12 A 5 Eqt 1.11 4.1 0.25 0.144 3.3 × 1054 88 5.1 × 1048 1.4 × 1050 1.9 × 1050 0.40 12 0.095 A 5 Eqt p2.2 1.3 6.2 0.25 0.144 3.3 × 1054 88 5.1 × 1048 1.4 × 1050 1.9 × 1050 0.41 9 0.068 A 5 Eqt p2.8 1.0 2.3 0.25 0.144 3.3 × 1054 88 5.1 × 1048 1.4 × 1050 1.9 × 1050 0.34 24 0.089 A 10 Pol 0.48 3.4 0.422 0.087 2.5 × 1055 30 8.3 × 1048 1.4 × 1050 2.2 × 1050 0.50 6 0.15 A 10 Eqt 0.46 2.9 0.422 0.087 2.5 × 1055 30 8.3 × 1048 1.4 × 1050 2.2 × 1050 0.48 7 0.13 A 10 Eqt p2.2 0.52 4.5 0.422 0.087 2.5 × 1055 30 8.3 × 1048 1.4 × 1050 2.2 × 1050 0.52 4 0.10 A 10 Eqt p2.8 0.4 1.7 0.422 0.087 2.5 × 1055 30 8.3 × 1048 1.4 × 1050 2.2 × 1050 0.43 10 0.14 Bm01 5 Pol 1.25 10.7 0.301 0.186 1.1 × 1055 110 1.1 × 1049 3.7 × 1050 4.5 × 1050 0.47 5 0.11 Bm01 5 Eqt 1.6 10.5 0.301 0.186 1.1 × 1055 110 1.1 × 1049 3.7 × 1050 4.5 × 1050 0.36 14 0.082 Bm01 10 Pol 0.87 20 0.579 0.173 4.5 × 1055 38 3.5 × 1049 1.1 × 1051 1.4 × 1051 0.47 7 0.11 Bm01 10 Eqt 1.0 19 0.579 0.173 4.5 × 1055 38 3.5 × 1049 1.1 × 1051 1.4 × 1051 0.40 13 0.086 Cv03 5 Pol 1.2 12.5 0.338 0.178 1.6 × 1055 51 1.6 × 1049 3.5 × 1050 4.9 × 1050 0.46 8 0.12 Cv03 5 Eqt 1.2 10.5 0.338 0.178 1.6 × 1055 51 1.6 × 1049 3.5 × 1050 4.9 × 1050 0.43 10 0.10 Dv01 5 Pol 0.65 0.54 0.142 0.086 2.6 × 1054 213 5.6 × 1047 1.8 × 1049 2.2 × 1049 0.46 7 0.12 Dv01 5 Eqt 0.82 0.49 0.142 0.086 2.6 × 1054 213 5.6 × 1047 1.8 × 1049 2.2 × 1049 0.35 19 0.10 Ev01c1 5 Pol 0.35 0.16 0.152 0.093 9.6 × 1053 76 2.4 × 1047 8.4 × 1048 9.6 × 1048 0.45 6 0.13 Ev01c1 5 Eqt 0.45 0.15 0.152 0.093 9.6 × 1053 76 2.4 × 1047 8.4 × 1048 9.6 × 1048 0.34 17 0.088 Fo10 2 Pol 1.4 0.26 0.118 0.188 1.2 × 1053 238 2.2 × 1047 3.7 × 1048 6.5 × 1048 0.52 4 0.12 Fo10 2 Eqt 1.7 0.28 0.118 0.188 1.2 × 1053 238 2.2 × 1047 3.7 × 1048 6.5 × 1048 0.44 7 0.11 Fo10 5 Pol 0.9 0.8 0.283 0.172 7.0 × 1053 101 1.2 × 1048 1.9 × 1049 3.6 × 1049 0.57 2 0.13 Fo10 5 Eqt 1.1 0.83 0.283 0.172 7.0 × 1053 101 1.2 × 1048 1.9 × 1049 3.6 × 1049 0.47 5 0.11 Fo10 11 Pol 0.56 1.5 0.572 0.127 3.8 × 1054 20 4.2 × 1048 5.2 × 1049 1.1 × 1050 0.61 4 0.15 Fo10 11 Eqt 0.58 1.4 0.572 0.127 3.8 × 1054 20 4.2 × 1048 5.2 × 1049 1.1 × 1050 0.57 5 0.13 Go60 2 Pol 1.55 4.4 0.097 0.137 5.5 × 1054 350 2.5 × 1048 9.9 × 1049 1.1 × 1050 0.46 5 0.12 Go60 2 Eqt 1.8 4.4 0.097 0.137 5.5 × 1054 350 2.5 × 1048 9.9 × 1049 1.1 × 1050 0.39 10 0.10 Go60 4 Pol 0.98 6.7 0.177 0.124 1.9 × 1055 136 6.5 × 1048 2.2 × 1050 2.5 × 1050 0.49 5 0.13 Go60 4 Eqt 1.0 5.9 0.177 0.124 1.9 × 1055 136 6.5 × 1048 2.2 × 1050 2.5 × 1050 0.45 7 0.11 Go60 11 Pol 0.34 3.2 0.357 0.063 7.5 × 1055 38 1.0 × 1049 2.0 × 1050 2.5 × 1050 0.49 5 0.16 Go60 11 Eqt 0.34 2.8 0.357 0.063 7.5 × 1055 38 1.0 × 1049 2.0 × 1050 2.5 × 1050 0.46 6 0.14 Ho90 2 - 1.58 6 0.088 0.122 1.0 × 1055 402 3.4 × 1048 1.6 × 1050 1.6 × 1050 0.50 4 0.13 Ho90 5 - 0.68 5.8 0.188 0.088 4.9 × 1055 110 7.9 × 1048 2.9 × 1050 2.9 × 1050 0.53 3 0.15 Ho90 10 - 0.32 3.1 0.29 0.055 1.2 × 1056 53 1.2 × 1049 2.9 × 1050 2.9 × 1050 0.54 3 0.15 In15 4 Pol 1.0 3.4 0.204 0.157 4.9 × 1054 105 3.5 × 1048 1.1 × 1050 1.4 × 1050 0.50 5 0.11 In15 4 Eqt 1.2 3.1 0.204 0.157 4.9 × 1054 105 3.5 × 1048 1.1 × 1050 1.4 × 1050 0.40 13 0.086 Jn20 4 Pol 1.0 3.2 0.202 0.163 4.5 × 1054 101 3.5 × 1048 9.7 × 1049 1.4 × 1050 0.49 6 0.10 Jn20 4 Eqt 1.2 2.7 0.202 0.163 4.5 × 1054 101 3.5 × 1048 9.7 × 1049 1.4 × 1050 0.38 15 0.071 Knbr 4 Pol 1.3 4.7 0.198 0.144 6.5 × 1054 163 4.2 × 1048 1.3 × 1050 1.6 × 1050 0.46 7 0.11 Knbr 4 Eqt 1.6 4.6 0.198 0.144 6.5 × 1054 163 4.2 × 1048 1.3 × 1050 1.6 × 1050 0.37 16 0.088 phase. An exception occurs when the density index n > 1.5: after a brief rise lasting several months, only the declining phase remains. The radio flux rises relatively slowly (flatter than t1.0), and its decay is also gradual, with the flux halving timescale of several years. These variation features can be used to test the applicability of the FS scenario, which are quite different from the bow shock scenario, in which the radio flux can exhibit a sharp rise followed by a rapid decline (Mou et al. 2022; Zhuang et al. 2025). We also find that when the high-frequency radio flux begins to decline, the shock energy at that turnover moment is approximately one half of the outflow's kinetic energy. Finally, we would like to emphasize that there are several uncertainties in the interaction between the outflow and the CNM, such as the angular distribution of the outflow, the possibility of abrupt changes in the radial distribution of the hot/diffuse CNM, and the evolution of the magnetic field. Given the current limited constraints on the physics of the outflow, CNM and shock, Radio Afterglows I: Forward Shock 11 our results are based on simplified assumptions. Further progress in understanding the radio afterglows will necessitate a coordinated development between theoretical modeling and observations. ACKNOWLEDGEMENTS G.M. was supported by the National Key R&D Program of China (Grant No. 2023YFA1607904), and the NSFC (No. 12473013). REFERENCES Alexander, K. D., Berger, E., Guillochon, J., Zauderer, B. A., & Williams, P. K. G. 2016, ApJL, 819, L25 Alexander, K. D., van Velzen, S., Horesh, A., & Zauderer, B. A. 2020, Space Sci. Rev., 216, 81 Andreoni, I., Coughlin, M. W., Perley, D. A., et al. 2022, Nature, 612, 430 Barniol Duran, R., Nakar, E., & Piran, T. 2013, ApJ, 772, 78 Bell, A. R. &Lucek, S. G. 2001, MNRAS, 321, 433 Bloom, J. S., Giannios, D., Metzger, B. 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THE INFLUENCE OF THE LIGHT-TRAVEL TIME DIFFERENCE We show in Figure A1 the impact of considering versus neglecting LTTD on the spectra along the polar direction for run A. The results indicate that LTTD can affect the emergent spectrum in the polar direction, although the effect is not particularly significant. In the main text, all spectral calculations along the polar direction have taken LTTD into account. B. THE USAGE PROCEDURE OF THE ENERGY EQUIPARTITION/MINIMAL METHOD Barniol Duran et al. (2013) developed the energy equipartition/minimal method for synchrotron emission, which provides estimate for the size of the radiation zone, and the energies of magnetic field and CRe. However, some observational articles failed to apply this method correctly, resulting in that first, the number of CRe components were mistakenly regarded as the whole gas including thermal components (as pointed out by Matsumoto & Piran 2021), and second, the derived magnetic field energy and CRe energy were inconsistent with the specified values of εB and εe. The reason for the second point is that, many observational studies implicitly employ the three-component minimal energy method when a factor ξ11/(13+2p) is applied to the Eeq calculation formula (ξ ≡1+ε-1 e ) (equation 28 in Barniol Duran et al. 2013, see Section 4.2.2 therein for details). In this case, Eeq actually is Ee + EB + Ethp (Ethp is the total energy of thermal protons), in which these three components satisfy Ee : EB : Ethp = 11 17ξ-1 : 6 17 : 11 17(1 -ξ-1). In other words, once ξ or εe is specified, the relationship among EB, Ee, and Ethp is already determined, which is therefore inconsistent with further specifying the value of εB. As a result, the values of Ee and EB reported in many studies often do not match their adopted values of εe and εB. The procedure of the equipartition method should follow the way to find equipartition radius Req when (Ee + EB) reaches the minimal value, and then derive the values under the specified value of assumed εB and εe. Following this procedure, each calculation step has well-defined physical meaning. All the following parameters in Step 1 - 3 are the same as Barniol Duran et al. (2013), and all the parameters in the formula are unchanged. Step 1: Calculating the equipartition radius Req. Radio Afterglows I: Forward Shock 13 When EB = (6/11)Ee, Ee + EB reaches the minimal value. In this case, the radius of the radiation zone is also determined. Req ≈(1 × 1017cm) [21.8(525)p-1] 1 13+2p γ 2-p 13+2p m × h F 6+p 13+2p p,mJy d 2(p+6) 13+2p L,28 ν-1 p,10 (1 + z)-19+3p 13+2p i × f - 5+p 13+2p A f - 1 13+2p V Γ p+8 13+2p 4 1 13+2p . (B1) This is just equation 27 in (Barniol Duran et al. 2013) except for an extra correction term 4 1 13+2p for the isotropic number of radiating electrons. In this formula, γm = χe(Γ-1), where χe = p-2 p-1εe mp me , and if γm is found to be γm < 2, one should use γm = 2. Note the value of Γ is the bulk Lorentz factor of the radiation zone, and one can set it to be Γ = 1 in Step 1-3 before iteration. Step 2: Calculating R, Ee, EB, Ne and B. After setting the values of εB and εe, the condition for the minimal energy is usually broken, and the radius of the radiation zone slightly deviates from Req by a factor of ε1/17: R = Reqε1/17 , (B2) where ε ≡11εB/(6εe). The energy in relativistic electrons within the observed region with γe ≥γm is Ee =Nemec2γeΓ = 4(γe/γm)2-p 27c3F 4 ν,pd8 Lη5Γ2 16 √ 3π3e2m2eν7p(1 + z)11f 3 AR6 ≈4(γe/γm)2-p × 4.4 × 1050erg × h F 4 p,mJy d8 L,28 ν-7 p,10 η5 (1 + z)-11i Γ2 f 3 A R6 17 , (B3) while the energy in the magnetic field is EB = (BΓ)2 8π V = 8π6m6 ec2ν10 p (1 + z)14f 4 AfV R11 81e2F 4ν,pd8 Lη 20 3 Γ8 ≈(2.1 × 1046erg) h F -4 p,mJy d-8 L,28 ν10 p,10 η-20 3 (1 + z)14i × f 4 A fV R11 17 Γ8 , (B4) where the volume of the radiation zone V = fV πR3/Γ4. These are equation 17 and 18 in (Barniol Duran et al. 2013), except for the an extra correction term 4(γe/γm)2-p in equation B3 for the isotropic number of radiating electrons and those electrons with γe ≥γm. Subsequently, one can derive the values of B and Ne from the values of Ee, EB and R. Ne = 4(γe/γm)2-p 9cF 3 ν,pd6 Lη 10 3 8 √ 3π2e2m2eν5p(1 + z)8f 2 AR4 ≈4 × 1054 (γe/γm)2-p h F 3 p,mJy d6 L,28 ν-5 p,10 η 10 3 (1 + z)-8i × 1 f 2 A R4 17 , (B5) B = 8π3m3 ecν5 p(1 + z)7f 2 AR4 9eF 2ν,pd4 Lη 10 3 Γ3 ≈(1.3 × 10-2 G) h F -2 p,mJy d-4 L,28 ν5 p,10 η-10 3 (1 + z)7i × f 2 A R4 17 Γ3 . (B6) These are equation 15 and 16 in (Barniol Duran et al. 2013), except for the an extra correction term 4(γe/γm)2-p in equation B5. Note that Ne is the number of relativistic electrons with γe ≥γm, which does not include the thermal components. Thus, one should not use Ne directly to calculate the density of hot CNM. The Lorentz factor of the relativistic electrons radiating at νp is (equation 14 in Barniol Duran et al. 2013): γa = 3Fν,pd2 Lη 5 3 Γ 2πν2p(1 + z)3mefAR2 ≈525 h Fp,mJy d2 L,28 ν-2 p,10 η 5 3 (1 + z)-3i Γ fA R2 17 . (B7) 14 Mou 0.2 0.1 0.0 0.1 0.2 X/parsec 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Y/parsec 0 1 2 3 log10 (ne/cm 3) 6 5 4 log10 (e2/erg cm 3) 0.2 0.1 0.0 0.1 0.2 X/parsec 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Y/parsec 0 1 2 3 log10 (ne/cm 3) 6 5 4 log10 (e2/erg cm 3) Figure A2. Snapshots of run Fo10 and Go60 at t = 5 yr. Step 3: Finding out Rs(t) and checking the self-consistency. According to the radio data achieved in different periods, one can derive the relationship between shock radius and time: Rs(t), from which the shock velocity vs and Γ-value can be obtained. Subsequently, do iterations and bring the Γ-value back to equations B1-B7 in Step 1-2 to obtain the parameter values. Step 4: Calculating the shock energy and the CNM density. The shock energy is calculated as Es = Ee/εe. The enclosed mass of the hot CNM swept up by the FS can be derived as Mcnm(Rs) = Es/[(Γ -1)c2] in the relativistic regime, or Mcnm(Rs) = 2Es/v2 s in the nonrelativistic regime. The averaged post-shock CNM density can be estimated by ρ(Rs) ≃Mcnm(Rs)/V , while the density of the the pre-shock CNM ahead of the shock front should be further reduced by a factor of 4 (density jump). It should be noted that the shock energy Es is not equivalent to the outflow energy Ek; rather, Es represents a robust lower limit for Ek. When the swept-up CNM mass is significantly less than the outflow mass, the outflow energy can be approximated by Ek ≃Es × Mout Mcnm (B8) Due to the lack of constraints on Mout, it is challenging to estimate Ek, and it may be approximated as Es when the forward shock velocity decreases significantly for the FS scenario. A program implementing the above procedures is available on GitHub 1. Please note: The above program and web-based code are intended only to illustrate the workflow of the energy equipartition method, and do not account for the errors introduced by this approach. One should note that the equipartition method provides rough estimates of the shock radius, Ee and EB, since the derivation in Barniol Duran et al. (2013) involves multiple approximations, and could lead to considerable derivation. For better matching the results of radiation transfer calculations, one should multiply the Req by a factor of 2, Ee and EB by a factor of 9 (on average). C. SHOCK PARAMETERS DERIVED FROM THE EQUIPARTITION METHOD Based on the synthetic radio spectra obtained from the hydrodynamic simulations, we extract νp and Fνp, and then follow the steps of the equipartition method to derive the "estimated" shock radius Req, density n′ e, and various energy components (E′ e, E′ B, E′ s) in Table A1. These parameters are used for comparison with the actual values in simulations (Table 2). 1 https://github.com/G-Mou/RadioTDE Radio Afterglows I: Forward Shock 15 Table A1. Shock parameters derived by using the equipartition method based on the synthetic radio spectra. We assume that the emitting region is a shell of thickness 0.1Req, and opening angle of the shock is equal to the outflow's opening angle in each model. Run ∆T Direc νp Fνp Req V ′ s ≡Req/∆T N′ e n′ e E′ e E′ B E′ shk (units) yr - GHz mJy pc c cm-3 erg erg erg A 2 Pol 1.6 2 0.049 0.080 9.1 × 1053 1762 1.5 × 1047 5.8 × 1047 5.2 × 1048 A 2 Eqt 2.0 2.0 0.039 0.064 1.1 × 1054 4300 1.2 × 1047 4.6 × 1047 4.1 × 1048 A 2 Eqt (p2.2) 2.3 3.1 0.042 0.069 7.7 × 1053 2870 9.8 × 1046 3.4 × 1047 3.3 × 1048 A 2 Eqt (p2.8) 1.7 1.1 0.035 0.057 1.5 × 1054 6644 1.3 × 1047 5.0 × 1047 4.2 × 1048 A 5 Pol 0.98 4.4 0.116 0.076 4.2 × 1054 615 6.5 × 1047 2.4 × 1048 2.2 × 1049 A 5 Eqt 1.1 4.1 0.099 0.065 4.7 × 1054 1094 5.3 × 1047 2.0 × 1048 1.8 × 1049 A 5 Eqt (p2.2) 1.3 6.2 0.103 0.067 3.3 × 1054 829 3.9 × 1047 1.4 × 1048 1.3 × 1049 A 5 Eqt (p2.8) 1.0 2.3 0.084 0.055 6.5 × 1054 2107 5.2 × 1047 2.1 × 1048 1.7 × 1049 A 10 Pol 0.48 3.4 0.21 0.069 7.7 × 1054 189 9.7 × 1047 3.6 × 1048 3.2 × 1049 A 10 Eqt 0.46 2.9 0.203 0.066 7.3 × 1054 197 8.4 × 1047 3.1 × 1048 2.8 × 1049 A 10 Eqt (p2.2) 0.52 4.5 0.221 0.072 4.9 × 1054 124 6.7 × 1047 2.3 × 1048 2.2 × 1049 A 10 Eqt (p2.8) 0.4 1.7 0.182 0.060 9.8 × 1054 313 9.1 × 1047 3.6 × 1048 3.0 × 1049 Bm01 5 Pol 1.25 10.7 0.140 0.092 6.5 × 1054 560 1.5 × 1048 5.5 × 1048 4.9 × 1049 Bm01 5 Eqt 1.6 10.5 0.108 0.071 8.4 × 1054 1547 1.1 × 1048 4.2 × 1048 3.7 × 1049 Bm01 10 Pol 0.87 20.0 0.270 0.088 2.2 × 1055 258 4.5 × 1048 1.7 × 1049 1.5 × 1060 Bm01 10 Eqt 1.0 19.0 0.230 0.075 2.4 × 1055 475 3.7 × 1048 1.4 × 1049 1.2 × 1050 Cv03 5 Pol 1.2 12.5 0.157 0.103 6.5 × 1054 398 1.8 × 1048 6.8 × 1048 6.1 × 1049 Cv03 5 Eqt 1.2 10.5 0.145 0.095 6.2 × 1054 487 1.5 × 1048 5.5 × 1048 5.0 × 1049 Dv01 5 Pol 0.65 0.54 0.065 0.042 1.7 × 1054 1409 8.0 × 1046 3.0 × 1047 2.7 × 1048 Dv01 5 Eqt 0.82 0.49 0.049 0.032 2.1 × 1054 3947 5.7 × 1046 2.1 × 1047 1.9 × 1048 Ev01c1 5 Pol 0.35 0.16 0.069 0.045 6.2 × 1053 446 3.5 × 1046 1.3 × 1047 1.2 × 1048 Ev01c1 5 Eqt 0.45 0.15 0.052 0.034 7.7 × 1053 1293 2.5 × 1046 9.3 × 1046 8.4 × 1047 Fo10 2 Pol 1.4 0.26 0.061 0.10 8.7 × 1052 860 2.3 × 1046 8.6 × 1046 7.7 × 1047 Fo10 2 Eqt 1.7 0.28 0.052 0.085 1.1 × 1053 1723 2.1 × 1046 7.7 × 1046 6.9 × 1047 Fo10 5 Pol 0.9 0.8 0.161 0.105 4.6 × 1053 246 1.4 × 1047 5.1 × 1047 4.6 × 1048 Fo10 5 Eqt 1.1 0.83 0.134 0.088 5.7 × 1053 529 1.2 × 1047 4.4 × 1047 3.9 × 1048 Fo10 11 Pol 0.56 1.5 0.347 0.103 1.6 × 1054 86 4.7 × 1047 1.7 × 1048 1.6 × 1049 Fo10 11 Eqt 0.58 1.4 0.324 0.096 1.7 × 1054 108 4.2 × 1047 1.5 × 1048 1.4 × 1049 Go60 2 Pol 1.55 4.4 0.045 0.074 2.6 × 1054 1781 3.8 × 1047 1.4 × 1048 1.3 × 1049 Go60 2 Eqt 1.8 4.4 0.038 0.062 3.2 × 1054 3419 3.3 × 1047 1.2 × 1048 1.1 × 1049 Go60 4 Pol 0.98 6.7 0.086 0.070 7.6 × 1054 724 9.9 × 1047 3.3 × 1048 3.3 × 1049 Go60 4 Eqt 1.0 5.9 0.079 0.065 7.4 × 1054 900 8.3 × 1047 2.8 × 1048 2.8 × 1049 Go60 11 Pol 0.34 3.2 0.175 0.052 1.6 × 1055 186 1.2 × 1048 3.9 × 1048 3.9 × 1049 Go60 11 Eqt 0.34 2.8 0.164 0.049 1.6 × 1055 216 1.0 × 1048 3.4 × 1048 3.4 × 1049 Ho90 2 - 1.58 6.0 0.044 0.072 4.3 × 1044 1624 6.0 × 1047 2.2 × 1048 2.0 × 1049 Ho90 5 - 0.68 5.8 0.100 0.065 1.2 × 1055 370 1.3 × 1048 4.9 × 1048 4.4 × 1049 Ho90 10 - 0.32 3.1 0.158 0.052 1.9 × 1055 148 1.3 × 1048 5.0 × 1048 4.5 × 1049 In15 4 Pol 1.0 3.4 0.102 0.083 2.5 × 1054 568 4.7 × 1047 1.7 × 1048 1.6 × 1049 In15 4 Eqt 1.2 3.1 0.081 0.066 3.0 × 1054 1320 3.5 × 1047 1.3 × 1048 1.2 × 1049 Jn20 4 Pol 1.0 3.2 0.099 0.081 2.4 × 1054 605 4.3 × 1047 1.6 × 1048 1.4 × 1049 Jn20 4 Eqt 1.2 2.7 0.076 0.062 2.8 × 1054 1542 3.0 × 1047 1.1 × 1048 9.9 × 1048 Knbr 4 Pol 1.3 4.7 0.091 0.074 2.5 × 1054 1124 5.3 × 1047 2.0 × 1048 1.8 × 1049 Knbr 4 Eqt 1.6 4.6 0.073 0.060 3.0 × 1054 2601 4.2 × 1047 1.6 × 1048 1.4 × 1049 16 Mou 100 101 Time/yr 10 2 10 1 100 101 F /mJy (dL = 100 Mpc) Bm01 Cv03 Dv01 Ev01c1 Fo10 Go60 In15 Jn20 Knbr 100 101 Time/yr 10 2 10 1 100 101 F /mJy (dL = 100 Mpc) Bm01 Cv03 Dv01 Ev01c1 Fo10 Go60 In15 Jn20 Knbr Figure A3. Temporal evolution of the monochromatic flux Fν in the equatorial direction. The left panel is for ν = 0.88 GHz, and the right panel is for 1.4 GHz. 100 101 Time/yr 10 1 100 101 102 F /mJy (dL = 100 Mpc) t1.0 t 1.0 t 2.0 Lnbr Mm01 Nn25 100 101 Time/yr 10 1 100 101 102 F /mJy (dL = 100 Mpc) Lnbr: 0.88 GHz Lnbr: 1.4 GHz Mm01: 0.88 GHz Mm01: 1.4 GHz Nn25: 0.88 GHz Nn25: 1.4 GHz Figure A4. Temporal evolution of the monochromatic flux Fν in the equatorial direction for model Lnbr, Mn01 and Nn25. The left panel is for ν = 6 GHz, and the right panel is for 0.88 and 1.4 GHz.
|
2510.14703
|
ToolPRM: Fine-Grained Inference Scaling of Structured Outputs
for Function Calling
Jianghao Lin1∗, Yuanyuan Shi1∗, Xin Peng2, Renjie Ding2, Hairui Wang2, Yuxuan Peng2, Bizhe Bai3,
Weixi Song3, Fengshuo Bai1, Huacan Chai1, Weinan Zhang1, Fei Huang2, Ying Wen1
1Shanghai Jiao Tong University, 2Longshine AI Research Institute, 3Shanghai Innovation Institute
Abstract
Large language models (LLMs) are increasingly demonstrating
strong capabilities as autonomous agents, with function calling
serving as a core mechanism for interaction with the environment.
Meanwhile, inference scaling has become a cutting-edge technique
to enhance LLM performance by allocating more computational re-
sources during the inference process. However, current research on
inference scaling primarily focuses on unstructured output genera-
tion tasks, leaving its application in structured outputs, like function
calling, largely underexplored. To bridge this gap, we propose an in-
ference scaling framework that combines fine-grained beam search
with a process reward model, ToolPRM, which scores the internal
steps of each single function call. To train ToolPRM, we construct
the first fine-grained intra-call process supervision dataset, auto-
matically annotated with function-masking techniques to provide
step-level rewards for structured tool-use reasoning. Extensive ex-
periments demonstrate that ToolPRM beats the coarse-grained and
outcome reward models in terms of predictive accuracy, indicating
its stronger capability in supervising the function calling inference
process. Inference scaling technique equipped with ToolPRM also
significantly improves the backbone model performance across
various function calling tasks and benchmarks. More importantly,
we reveal a key principle for applying inference scaling techniques
to structured outputs: “explore more but retain less” due to
the unrecoverability characteristics of structured function calling
generation.
Keywords
LLM Agents, Function Calling, Inference Scaling
1
Introduction
Large language models (LLMs) have demonstrated remarkable ca-
pabilities in a diverse range of tasks [6, 9, 16, 19, 44, 45, 52], rapidly
evolving from general-purpose text generators to complex rea-
soning engines that can follow instructions, perform step-by-step
reasoning, and integrate domain-specific knowledge into practical
workflows. Building upon these advances, the concept of LLM-
based Agents1 has emerged [38, 49], extending LLMs from passive
text generators to active decision-making entities capable of per-
ceiving their environment, reasoning over context, and performing
actions through external tools. At the core of such agents lies the
function calling mechanism, which enables them to interact with
external tools, retrieve information, and perform actions, allow-
ing them to bridge linguistic reasoning with real-world operations.
∗Both authors contributed equally to this research.
1In this paper, the term “LLM-based Agents” is interchangeable with “LLM Agents”.
This mechanism marks a fundamental shift in how LLMs interact
with their surroundings, i.e., transforming static text prediction
into dynamic action execution, where the correctness and structure
of the outputs directly impact downstream system reliability.
In this broader view, inference scaling plays a crucial role in
enhancing an agent’s reasoning and decision-making process. It
provides a flexible way to trade off computational cost against
reasoning quality, enabling LLMs to deliberate more deeply or ex-
plore alternative reasoning paths without retraining the base model.
Rather than generating a single, greedily decoded output, inference
scaling strategies allocate additional computational resources at
inference time, to explore a wider range of possibilities and multi-
ple reasoning trajectories, and select the best trajectory for better
performance [15, 55].
Existing research on inference scaling predominantly focuses
on unstructured output generation tasks, such as mathemat-
ical problems [29, 42] and other intricate reasoning tasks [24],
where each sentence or discrete unit serves as a discrete step
of thought. Many recent approaches combine Tree of thoughts
(ToT) [50] with tree-based search algorithms, generalizing the lin-
ear Chain-of-Thought [41] (CoT) by maintaining multiple potential
reasoning branches from a given state. To efficiently navigate the
expanded reasoning space, beam search and Monte Carlo Tree
Search (MCTS) have become common inference scaling techniques,
and several studies have adapted them for LLM inference scal-
ing [8, 21, 42, 51, 57]. These approaches typically rely on either
self-evaluation [46] or process reward models (PRMs) [12, 24] to
score intermediate reasoning paths and guide the search toward
optimal outcomes.
Despite its importance for constructing reliable and adaptive
LLM agents, the application of inference scaling techniques to
structured function calling scenarios remains significantly un-
derexplored. Current studies that incorporate process reward mech-
anisms and inference scaling of function calling capabilities for
agents typically operate at a coarse-grained level [25, 40], treating
an entire round of function calling response as a singular, mono-
lithic step. For example, Wang et al. [40] adopt a Best-of-N approach,
where multiple function call candidates are scored and ranked by
an outcome reward model (ORM). The final output is then selected
based on the overall quality without applying distinct fine-grained
rewards to the sub-components within each function call. However,
this overlooks the inherent multi-stage nature of the function call-
ing process, which involves solving a sequence of sub-tasks such
as selecting the appropriate function name, identifying relevant pa-
rameters, and determining parameter values. Treating the function
calling process as a single atomic step overlooks opportunities for
fine-grained inference scaling and intra-call reward optimization
arXiv:2510.14703v1 [cs.AI] 16 Oct 2025
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Trovato et al.
that could substantially improve the agent’s reasoning efficiency
and reliability.2
To this end, we propose a novel fine-grained process reward
mechanism tailored for function calling capabilities of LLM agents
(dubbed ToolPRM). In contrast to prior approaches that treat each
function call as an indivisible unit, our method decomposes the
agent’s tool-call process into semantically interpretable intermedi-
ate reasoning steps. We design a dedicated process reward model,
ToolPRM, trained under fine-grained intra-call supervision with
meticulously curated step-level annotations constructed from both
the xlam-function-calling-60k [23] and xlam-irrelevance-7.5k [20]
datasets. Additionally, we further introduce fine-grained beam
search guided by ToolPRM, which substantially outperforms exist-
ing function calling baselines in our experiments.
Beyond establishing the empirical advantages of our approach,
we further analyze the conceptual implications of inference scaling
when applied to agents on structured versus unstructured reasoning
tasks. In traditional inference scaling scenarios such as mathemati-
cal reasoning, the unstructured nature of outputs allows for flexible
expansion — simply maintaining more candidate reasoning paths
can enhance the quality of outcome, since erroneous intermediate
steps may be revised later through reflection and process super-
vision. In contrast, for structured function calling outputs of LLM
agents, maintaining a large number of concurrent trajectories of-
ten degrades performance, as early missteps cannot be corrected
downstream, leading to wasted subsequent computational budget.
Therefore, we introduce a task-aligned inference scaling principle
of function calling for LLM agents: allocate computation to explor-
ing a wider range of decisions (i.e., increasing beam width), while
aggressively pruning redundant or low-quality candidates (i.e., re-
ducing the number of active beams). We summarize this principle as
“explore more, but retain less”, emphasizing an efficient trade-off
between exploration and reliability within agentic function calling
processes.
The major contributions of this paper are as follows:
• Fine-Grained Intra-Call Process Supervision Dataset:
We construct and release a novel annotated dataset specif-
ically designed for fine-grained step-level supervision of
structured function calling in LLM agents. This dataset, along
with its annotation scripts, is provided via the anonymous
link and will be made publicly available, facilitating future
research and benchmarking in fine-grained reward modeling
for agent function calling process.
• ToolPRM: We introduce ToolPRM, a fine-grained process
reward modeling framework specifically tailored for infer-
ence scaling in function calling LLM agents. ToolPRM could
effectively assist different backbone models in conducting
test-time scaling for superior performance. We also make the
code and model checkpoints publicly accessible to support
the community’s further research.
• Inference Scaling Principle for Structured Outputs: We
identify and formalize a critical principle for inference scal-
ing in structured output tasks: Explore more but retain less.
This principle underscores the importance of broadening
exploration over decision space while aggressively pruning
2In this paper, the term “function” is interchangeable with “tool” and “API”.
unpromising or unrecoverable steps, leading to more reliable
function-calls by agents.
Collectively, these contributions aim to advance the understanding
and practical realization of scalable, trustworthy, and fine-grained
reasoning mechanisms in the structured function calling capabilities
of LLM agents.
2
Related Works
Our work is closely related to (1) function callin for LLM agents,
(2) inference scaling srategies, and (3) process reward models.
2.1
Function Calling for LLM Agents
Function calling has become a fundamental mechanism that en-
ables LLM-based agents to move beyond passive text generation
and interact with external tools, databases, and environments. This
capability effectively bridges reasoning and action, empowering
LLMs to function as autonomous agents capable of executing com-
plex tasks in structured, verifiable ways [23, 27, 30, 33, 35].
Recent research has explored various dimensions of this function-
calling paradigm. For instance, from the training side, IBM’s Granite-
20B-FunctionCalling [1] employs multi-task learning to jointly train
across seven core sub-tasks, demonstrating the benefit of holistic
learning for function calling. Hammer [20] and Octopus [7] fo-
cus on improving syntactic and semantic robustness, respectively
using function masking technique and conditional masking tech-
nique to ensure correctness under diverse contexts. Meanwhile,
ToolACE [22] introduces a self-evolution synthesis framework that
automatically generates high-quality function-calling data, high-
lighting the importance of scalable data construction for LLM-agent
training on function calling task. TinyAgent [10] integrates a novel
tool retrieval technique to reduce input context length and enhance
inference efficiency in LLM edge deployment scenarios. Together,
these studies form a growing foundation for structured, controllable,
and verifiable decision-making within LLM agents.
To evaluate such capabilities, several benchmarks have been
proposed, including BFCL3, API-Bank [17], NESTful [4], and Com-
plexFuncBench [56], etc. Among them, for example, NESTful em-
phasizes nested and compositional API calls, whereas Complex-
FuncBench targets multi-step and constrained scenarios, both cru-
cial for assessing multi-hop reasoning and hierarchical planning of
LLM agents.
However, existing work primarily focuses on overall validity of
the outputs rather than intra-process reasoning quality. In this pa-
per, we take a step further by disassembling complex function calls
into fine-grained decision steps and designing a process-level re-
ward model that guides the trajectory selection in inference-scaling
process. Ultimately, we aim to achieve more reliable, interpretable,
and controllable agentic behavior in function calling.
2.2
Inference Scaling Strategies
Inference scaling has recently emerged as a crucial mechanism
for enhancing the reasoning and decision-making capabilities of
LLM-based agents [3], and could be more computationally efficient
than scaling model parameters [43]. Instead of relying on a single
3https://gorilla.cs.berkeley.edu/leaderboard.html
ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
forward pass, scaling strategies amplify inference-time computa-
tion to improve both accuracy and robustness in complex problem
solving [42].
Early studies introduced straightforward sampling-based meth-
ods. The self-consistency strategy [39] samples multiple indepen-
dent reasoning paths via temperature sampling [2] and aggregates
their outcomes through majority voting, improving stability in
open-ended reasoning. Similarly, the Best-of-N (BoN) approach [5]
evaluates a batch of candidate outputs using a outcome reward
model to select the optimal one. More recently, search-based scaling
has become the dominant paradigm, especially for complex multi-
step reasoning tasks. Tree-structured inference methods, such as
beam search, Monte Carlo Tree Search (MCTS), and their agent-
oriented variants [8, 21, 51, 57], allow the model to explore multiple
reasoning trajectories, prune low-value branches, and dynamically
refine decisions based on intermediate feedback.
While studies have validated inference scaling in unstructured
tasks, its role in structured output generation, such as function
calling, remains relatively underexplored. In this paper, we extend
inference scaling into the structured function-calling domain, treat-
ing each function call as a compositional reasoning process. We
analyze how intra-call inference scaling improves decision relia-
bility at the step level and discuss the computational trade-offs
between search depth, width, and overall performance.
2.3
Process Reward Models
In complex, multi-step reasoning tasks, the final output of an agent
may be incorrect due to an error at any intermediate step. Outcome
Reward Models (ORMs), which evaluate only the final result, pro-
vide a sparse and delayed reward signal that makes it difficult to
perform credit assignment and identify the specific point of fail-
ure. To address this, Process Reward Models (PRMs) have been
developed to provide more granular feedback by evaluating the
correctness or quality of each intermediate step within a reasoning
trajectory [37, 54? , 55].
PRMs are initially prominent as a component in Reinforcement
Learning from Human Feedback (RLHF) pipelines [26]. Instead of
relying on a single reward for a final answer, researchers trained
PRMs on fine-grained human feedback for each step of a model’s
reasoning process. For instance, in the domain of mathematical rea-
soning, Lightman et al. [18] demonstrate that supervising a model
with step-by-step rewards is a more data-efficient and effective
method for teaching complex reasoning than rewarding only the
final, correct answer. This dense reward signal allows the model to
learn more robust and generalizable problem-solving procedures.
More recently, PRM has been adapted as a heuristic to guide search
algorithms over the model’s reasoning paths. For example, Ma et
al. [24] employ a PRM to score the intermediate thoughts generated
within a Tree-of-Thoughts (ToT) framework, enabling the search
algorithm to prune unpromising branches and focus computational
resources on more plausible reasoning trajectories. To improve the
reliability of the feedback itself, innovations like the Reasoning-
Driven PRM (R-PRM) have been proposed, which require the reward
model to generate its own reasoning to justify its evaluation of each
step, leading to more robust guidance [32].
In this paper, we explore the PRM paradigm in a fine-grained
manner to the structured function calling capabilities of LLM agents.
We introduce a method to automatically curate the first large-scale,
fine-grained, intra-call process supervision dataset for agent tool
use. Using this dataset, we train ToolPRM, a specialized model that
excels at verifying the intermediate steps involved in constructing a
valid tool call. By integrating ToolPRM with a beam search strategy,
we provide agents with a powerful mechanism for deliberation,
enabling them to achieve state-of-the-art performance in complex
tool-use scenarios.
3
Methodology
This section introduces the fine-grained process reward model for
function calling (i.e., ToolPRM). We first present fine-grained de-
composition for each turn of function callings, and present the data
collection and annotation for reward modeling training. Then, we
design a state transition machenism for function calling, and apply
beam search with fine-grained process supervision with ToolPRM.
3.1
Fine-Grained Decomposition of Function
Calls
The essence of ToolPRM lies in its fine-grained decomposition
of the function calling process. Traditional process supervision
approaches for tool invocations primarily rely on coarse-grained
response-level rewards, which evaluate the function call as a mono-
lithic unit. However, such coarse granularity limits the interpretabil-
ity, debuggability, and optimization potential.
To this end, ToolPRM decomposes each function call into fine-
grained but semantically meaningful steps. Each LLM-generated
response comprises a sequence of function calls tailored to a given
user query. The construction of each function call is further broken
down into (a) selecting the appropriate function name and (b) iter-
atively identifying parameter names and assigning corresponding
values. Rather than evaluating a function call as a whole, ToolPRM
provides fine-grained supervision that assesses the correctness of
each constituent decision. As shown in Figure 1, this decomposition
serves as the foundation for data collection and annotation.
3.2
Data Collection and Annotation
To enable robust reward modeling under our fine-grained pro-
cess supervision paradigm, we construct a high-quality annotated
dataset that reflects the intricacies of individual decision points.
3.2.1
Data Collection. We begin by collecting natural language
queries paired with their corresponding structured function calls,
using xlam-function-calling-60k [23] and xlam-irrelevance-7.5k [20]
datasets as the foundations. As shown in Figure 1, we apply function
masking that selectively replaces function names and parameter
identifiers with random strings. The masking mechanism intro-
duces function ambiguity, encouraging the model to rely on contex-
tual understanding of descriptions instead of simple memorization
of the tool names, thus enhancing model robustness and general-
ization [20]. We adopt Hammer2.1-3b and Hammer2.1-7b as policy
models to perform the rollout for data collection. Given a user query
and a set of masked function candidates, each rollout generates a
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Trovato et al.
Function Calls with Step Labels
Candidate Functions
[
{
"name": "get_stock_price",
"description": "Fetches stock price",
"parameters": {
"ticker": {
"type": "string",
"description": "The stock symbol",
"default": "AAPL"
},
... (More Parameters)
}
},
... (More Functions)
]
Masked Candidate Functions
[
{
"name": "xkz9qvnb8",
"description": "Fetches stock price",
"parameters": {
"y8z7x6": {
"type": "string",
"description": "The stock symbol",
"default": "AAPL"
},
... (More Parameters)
}
},
... (More Functions)
]
Function Calling
Policy Model
Function
Masking
Query
Can you tell me the stock
price of Google today?
[
{
"name": "xkz9qvnb8", <FUNC_NAME>
"parameters": {
"y8z7x6": "GOOGLE", <ARG_VALUE>
"start_date": "250506", <ARG_VALUE>
"end_date": "250509", <ARG_VALUE>
"include_volume": true, <ARG_VALUE>
... (More Parameters)
} <PARAM_FINISH>
} <FUNC_FINISH> ,
... (More Function Calls if Needed)
] <TOTAL_FINISH>
Figure 1: The illustration of data collection for ToolPRM.
sequence of function calls, which is subsequently annotated with
fine-grained step labels.
3.2.2
Result Annotation. As shown in Figure 1, the structured func-
tion call generated by the policy model is usually in JSON format.
We can provide fine-grained step labels with the following types:
• <FUNC_NAME>: Whether the selected function name at this
round is correct or not.
• <ARG_VALUE>: Whether the one pair of parameter name and
value is correctly filled in, which can repeat multiple times.
• <PARAM_FINISH>: Whether all the parameters and values are
correctly assigned.
• <FUNC_FINISH>: Whether the single function call (one ele-
ment of the list) is correct or not.
• <TOTAL_FINISH>: Whether the overall response (a list of
function calls) is correct or not.
Each symbol is followed by a binary label annotated by exact match
with any of the possible ground truths, to indicate the correctness
of the corresponding fine-grained step. Some of the step labels
seem redundant (e.g., <ARG_VALUE> and <PARAM_FINISH>), but our
experiments show that such a hierarchical step label redundancy
can help the reward model generalize and perform better.
3.3
State Transition Mechanism and ToolPRM
Training
Based on the fine-grained step labels above, we can further formal-
ize the function calling generation process as a dynamic decision
process with a series of state transitions. As illustrated in Figure 2,
we define the following five fine-grained states:
• State #0 (Initial State): Input context and masked function
candidates are presented.
• State #1 (To Select Function Name): The model is required
to select the function name from the candidates that are
aligned with the query intent.
• State #2 (To Select Parameter Name): The model should
choose one necessary parameter to be filled in for the se-
lected function.
• State #3 (To Fill in Parameter Value): The model has to
assign the proper value to the previously selected function
parameter.
• State #4 (Terminated State): One LLM response for func-
tion calls completes.
Each transition between states can be explicitly supervised by the
fine-grained step labels discussed in Section "Data Collection and
Annotation for Reward Modeling".
To train ToolPRM under fine-grained process supervision, we
represent the function calling generation as a trajectory of decision
steps, each labeled with a binary process reward.
Let T = {(𝑠𝑡,𝑎𝑡,𝑟𝑡)}𝑇
𝑡=1 denote a trajectory, where 𝑠𝑡is the state
at step 𝑡, 𝑎𝑡is the action (e.g., selecting a function or filling a pa-
rameter via language token generation), and 𝑟𝑡∈{0, 1} indicates
its correctness. Each state encodes the current decision context,
including the input query, masked function candidates, as well as
the partially generated function calls.
In this paper, since the backbone of ToolPRM is also a large
language model, we adopt the tokens “+” and “-” for positive and
negative reward labels, respectively. Given 𝑁annotated trajecto-
ries {T𝑖}𝑁
𝑖=1, ToolPRM is trained to predict 𝑟𝑡for each (𝑠𝑡,𝑎𝑡) via
generative process reward modeling:
LToolPRM = −E𝜏∈D,(𝑠𝑡,𝑎𝑡,𝑟𝑡)∈𝜏log𝑝𝜃(𝑟𝑡|𝑠𝑡,𝑎𝑡),
(1)
where𝑟(𝑖)
𝑡
∈{+, −} is the binary label token, and𝜃is the parameters
of backbone LLM.
3.4
Beam Search with Fine-Grained Process
Supervision
As shown in Figure 2, we apply beam search guided by ToolPRM to
generate high-quality structured outputs. At each step, ToolPRM
assigns a fine-grained process reward to candidate actions based on
their local state-action pair, allowing us to prune incorrect partial
trajectories early. Specifically, we compute the score𝑠for each beam
candidate with ToolPRM i.e., 𝑠= 𝑒𝑠+/(𝑒𝑠+ + 𝑒𝑠−), where 𝑠+ and 𝑠−
are the predictive logits on label tokens {+, −}. We maintain the
top-𝑁highest-scoring candidates, where 𝑁indicates the number
of beams. At each step, every preserved candidate can generate 𝑀
ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
TODO
Choose Func. Name
TODO
Choose Param. Name
TODO
Fill in Param. Value
Terminated State
Initial State
Initial State
State #0
State #1
State #2
State #3
State #4
Candidate Output #1
Evaluate with
ToolPRM
...
Figure 2: The state transition of function calling (left) and beam search with ToolPRM (right).
subsequent steps for ToolPRM evaluation, where 𝑀denotes the
beam width.
We propose a core principle for inference scaling in structured
output generation: explore more but retain less. Concretely, we
increase beam width 𝑀to explore a wider range of candidate trajec-
tories, but retain only a small number 𝑁of highly promising ones.
This design reflects the unrecoverability of structured outputs, such
as function calls in JSON format, where a single incorrect decision
(e.g., a wrong function name or argument value) can invalidate the
entire trajectory.
This stands in contrast to inference scaling in unstructured tasks
such as math reasoning [34] or free-form text generation [31, 53],
where early errors can often be corrected or compensated for in
later steps. In such settings, preserving a diverse set of candidates
throughout decoding (i.e., larger 𝑁) is beneficial. However, in struc-
tured generation, most decision points admit only a single or very
few valid actions. Retaining incorrect candidates leads to inefficient
use of the generation budget, as future steps cannot recover from
early structural mistakes.
Therefore, expanding the search space (larger beam width 𝑀)
while aggressively pruning based on ToolPRM’s step-wise super-
vision (smaller number of beams 𝑁) ensures that computational
resources are concentrated on valid, high-quality structured out-
puts. This principle underpins the effectiveness of ToolPRM in
scaling inference for structured generation tasks.
4
Experiments
4.1
Experiment Setups
4.1.1
Datasets. To train and validate our proposed ToolPRM for
function calling, we process and annotate the xlam-function-calling-
60k [23] and xlam-irrelevance-7.5k [20] datasets using a well-designed
pipeline, as detailed in Section 3. We report the dataset statistics in
Table 1. The resulting annotated dataset exhibits an average of 6.25
step labels per sample, culminating in a total of 192,061 samples.
The integration of function masking techniques further expands the
Table 1: The statistics of our constructed ToolPRM dataset.
Sample Granularity
(Data Split)
Positive
Negative
Total
Step (Train)
4,380,323
731,665
5,111,988
Trajectory (Train)
466,786
127,648
594,434
Step (Test)
488,611
81,366
569,977
Trajectory (Test)
52,030
14,019
66,049
dataset size to 4 5 times, bringing it to a scale that surpasses typical
datasets for Math PRM development, such as OpenAI’s publicly
available prm800k dataset [18].
Note that our constructed dataset above is used to train and vali-
date the predictive accuracy of reward models. To further validate
the effectiveness of our inference scaling strategy using ToolPRM,
we adopt two function calling benchmarks: BFCL (Berkeley Func-
tion Calling Leaderboard) [47] and ToolAlpaca [36]. Following pre-
vious works [1, 20], we use abstract-syntax-tree-based (AST-based)
accuracy as the metric for BFCL, and employ F1 scores of both API
selection and parameter value assignment for ToolAlpaca.
4.1.2
Baselines. We compare our proposed ToolPRM to different
sets of baselines from two aspects.
To evaluate the predictive accuracy improvement brought by
fine-grained intra-call process supervision, we train two baseline
models: outcome reward model (ORM) and coarse-grained process
reward model (C-PRM). Compared with our ToolPRM, i.e., fine-
grained process reward model using all the step labels, they are
trained on the same dataset with different choices of step labels:
• ORM is trained to discriminate the final result of the en-
tire function calling sequence with <TOTAL_FINISH>. All the
other intermediate step labels are disregarded.
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Trovato et al.
580
1160
1740
2320
2900
Training Step
0.00
0.05
0.10
0.15
0.20
Loss
580
1160
1740
2320
2900
Training Step
0.90
0.95
1.00
Step Acc
580
1160
1740
2320
2900
Training Step
0.90
0.95
1.00
Trajectory Acc
ORM
C-PRM
ToolPRM
Figure 3: The learning curves of different reward models (i.e., ORM, C-PRM, ToolPRM) on five-epoch training, in terms of loss,
step-level accuracy, and trajectory-level accuracy.
• C-PRM offers a more granular assessment of function call
correctness than ORM by considering all the step labels
except <ARG_VALUE>.
To evaluate the function calling capabilities, we adopt the fol-
lowing three types of baselines:
• General Purpose Models. This group comprises large lan-
guage models that have not undergone specific finetuning
for function calling tasks, nor do they employ the inference
scaling strategies. Their performance serves as a baseline rep-
resenting general capabilities. The models include GPT-4o,
GPT-4o-mini [13], Llama-3.1-8B-Instruct [11], Mistral-Nemo-
Instruct [14], and the Qwen2.5-Instruct series (72B, 32B, 7B,
3B, 1.5B) [48].
• Function Calling Models. This category includes models
that have been specifically designed or finetuned for function
calling tasks. These models include GRANITE (GRANITE-
20B-FUNCTIONCALLING) [1], xLAM-fc-r series (7B, 1B) [23],
OpenFunctions-v2 [28], and Hammer2.1 series (7B, 3B, 1.5B,
0.5B) [20].
• Inference Scaling Strategies. We evaluate the performance
enhancements afforded by different inference scaling meth-
ods. We choose the Hammer2.1 series (7B, 3B, and 1.5B vari-
ants) as base policy models. Three distinct inference scaling
strategies are applied and compared against our ToolPRM:
token-level beam search, majority voting, and best of N.
4.1.3
Implementation Details. All the experiments are conducted
on NVIDIA 8xH100 GPU Clusters. We use Hammer2.1-3b as the
reward model backbone and adopt SFT to train ToolPRM for 5
epochs with Adam optimizer. The batch size is 1024. The learning
rate is 1e-3 with a warmup ratio of 0.008 followed by linear learning
rate decay. The weight decay is 1e-5. As for the fine-grained beam
search with ToolPRM, we set the temperature as 0.8, and select the
number of beams 𝑁and the beam width 𝑀from {1, 2, 4, 8, 16}.
4.2
Predictive Accuracy of Reward Model
High predictive accuracy of the reward model is vital for increas-
ing the probability of selecting and retaining correct function call
Table 2: Predicting accuracy of RMs of different granularities
Reward Model
Loss
Step Acc
Trajectory Acc
ORM
0.0536
98.39%
98.39%
C-PRM
0.0371
98.87%
99.06%
ToolPRM
0.0286
99.11%
99.38%
sequences during inference scaling, ultimately leading to superior
overall inference outcomes. Hence, this section investigates the
performance of reward models for function calls with varying gran-
ularities.
We use three metrics for this evaluation: model loss, step-level
accuracy (Step Acc), and trajectory-level accuracy (Trajectory Acc).
Step Acc quantifies the correctness at each discrete process super-
vision step. It is worth noting that since each reward model (ORM,
C-PRM, and ToolPRM) is supervised at its distinct level of granular-
ity, the inherent difficulty of accurately predicting each step differs.
Hence, we introduce trajectory-level accuracy to facilitate an apple-
to-apple comparison of models trained with differing supervisory
step labels. The trajectory accuracy evaluates the correctness of
the entire function call sequence. A trajectory is deemed accurate
if the final assessment of the entire sequence is correct. This means
that for C-PRM and ToolPRM, even if their evaluations of some
intermediate steps are incorrect, the trajectory can still be classified
accurately if their overall judgment of the function call’s success or
failure is correct.
We report the predictive accuracy results in Table 2 and illus-
trate the learning curves over the five-epoch training in Figure 4.
We can observe that reward models with finer granularity consis-
tently achieve higher predictive accuracy across all three metrics.
Specifically, our proposed ToolPRM outperforms both C-PRM and
ORM in this regard. This suggests that fine-grained process reward
modeling, such as our proposed ToolPRM, offers superior perfor-
mance not only in terms of the precision of step-level supervision
but also in their ultimate effectiveness in judging overall sequence
correctness compared to their coarser-grained counterparts.
ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Table 3: Performance comparison of the general purpose models, function calling models, and different inference scaling
techniques applied on Hammer2.1 series of models, evaluated on BFCL and ToolAlpaca. Multi., Paral., Mul.P. represents the
Multiple split, Parallel split, and Multiple Parallel split of BFCL, separately. And Avg. represents an unweighted average of all
the sub-categories of that benchmark. The best results of each model type (i.e., general purpose, function calling, and inference
scaling) is given in bold and the second best is underlined.
Type
Model
Size
BFCL
ToolAlpaca
Simple
Multi.
Paral.
Mul.P.
Avg.
F1-API
F1-Args
Avg.
General
Purpose
GPT-4o
-
77.17
95.00
93.50
85.00
87.67
88.64
66.67
77.66
GPT-4o-mini
-
80.08
90.50
89.50
87.00
86.77
64.34
54.69
59.52
Llama-3.1-8B-Instruct
8B
72.83
93.50
87.00
83.50
84.21
75.64
55.12
65.38
Mistral-Nemo-Instruct
12B
77.00
93.50
89.50
84.50
86.13
87.31
66.18
76.75
Qwen2.5-72B-Instruct
72B
80.25
97.50
93.50
92.00
90.81
83.21
65.56
74.39
Qwen2.5-32B-Instruct
32B
72.83
94.00
93.50
88.50
87.21
85.82
61.11
73.47
Qwen2.5-7B-Instruct
7B
75.33
94.50
91.50
84.50
86.46
83.70
60.00
71.85
Qwen2.5-3B-Instruct
3B
74.17
90.50
79.50
79.00
80.79
70.80
51.63
61.22
Qwen2.5-1.5B-Instruct
1.5B
72.42
87.00
81.50
75.50
79.11
62.07
43.23
52.65
Function
Calling
GRANITE
20B
72.83
91.50
84.00
81.50
82.46
77.27
58.00
67.64
xLAM-7b-fc-r
7B
73.08
93.50
87.00
84.00
84.40
67.26
58.96
63.11
xLAM-1b-fc-r
1.3B
69.67
89.50
79.00
66.50
76.17
64.86
50.58
57.72
OpenFunctions-v2
7B
83.27
93.00
85.50
66.00
81.94
72.93
51.26
62.10
Hammer2.1-7B
7B
78.08
95.00
93.50
88.00
88.65
80.93
64.60
72.77
Hammer2.1-3B
3B
81.42
95.00
89.50
81.50
86.86
80.31
62.83
71.57
Hammer2.1-1.5B
1.5B
74.67
92.00
84.50
80.00
82.79
77.42
61.17
69.30
Hammer2.1-0.5B
0.5B
68.00
83.00
71.50
54.00
69.13
77.10
60.67
68.89
Inference
Scaling
Hammer2.1-7B (Base)
7B
78.08
95.00
93.50
88.00
88.65
80.93
64.60
72.77
+ Token-level Beam Search
74.58
93.50
91.50
82.50
85.52
79.69
62.37
71.03
+ Marjority
79.58
95.00
93.50
85.00
88.27
79.03
65.26
72.15
+ Best of N (ORM)
76.58
94.50
91.50
87.00
87.40
78.86
67.73
73.30
+ ToolPRM (Ours)
79.08
95.50
94.50
89.00
89.52
81.42
65.30
73.36
Hammer2.1-3B (Base)
3B
81.42
95.00
89.50
81.50
86.86
80.31
62.83
71.57
+ Token-level Beam Search
74.50
91.00
87.50
77.00
82.50
78.29
59.32
68.81
+ Marjority
79.50
95.00
89.00
81.00
86.13
76.00
58.27
67.14
+ Best of N (ORM)
77.08
93.50
89.50
84.00
86.02
77.29
60.16
68.73
+ ToolPRM (Ours)
80.50
95.50
91.50
88.00
88.88
80.78
63.13
71.96
Hammer2.1-1.5B (Base)
1.5B
74.67
92.00
84.50
80.00
82.79
77.42
61.17
69.30
+ Token-level Beam Search
72.33
91.50
81.00
73.50
79.58
73.03
56.68
64.86
+ Marjority
77.50
91.50
84.50
80.00
83.38
72.88
63.61
68.25
+ Best of N (ORM)
75.25
90.00
86.50
84.00
83.94
77.42
61.66
69.54
+ ToolPRM (Ours)
78.42
92.00
87.50
84.50
85.61
82.68
63.18
72.93
4.3
Inference Scaling Performance
In this section, we conduct a comprehensive examination and com-
parative analysis of our proposed ToolPRM, as well as various
function calling baseline models and inference scaling strategies
on two datasets.
As evidenced in Table 3, we can observe that our proposed Tool-
PRM can generally achieve the best performance compared with
other inference scaling strategies given base policy models of differ-
ent sizes. Besides, the performance of other inference scaling strate-
gies is fairly unstable and can sometimes perform worse than the
base model. The potential reason is that the non-greedy sampling
might cause minor errors during the function calling generation,
which could directly ruin the entire trajectory. This suggests the
necessity of intra-call process supervision through the fine-grained
beam search for structured function calling generation.
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Trovato et al.
20
21
22
23
24
Generation Budget
0.81
0.82
0.83
0.84
0.85
0.86
Averaged Accuracy
Hammer2.1-1.5B on BFCL
20
21
22
23
24
Generation Budget
0.83
0.84
0.85
0.86
0.87
0.88
0.89
Averaged Accuracy
Hammer2.1-3B on BFCL
20
21
22
23
24
Generation Budget
0.66
0.68
0.70
0.72
Averaged F1 Score
Hammer2.1-1.5B on Toolalpaca
20
21
22
23
24
Generation Budget
0.66
0.67
0.68
0.69
0.70
0.71
Averaged F1 Score
Hammer2.1-3B on Toolalpaca
ToolPRM; N=4
ToolPRM; M=4
Best of N
Majority
Figure 4: The performance on BFCL (Averaged Accuracy) and ToolAlpaca (Averaged F1 Score) w.r.t. different generation budgets
and different inference scaling strategies. We conduct experiments on Hammer2.1-1.5B and Hammer2.1-3B policy models.
Notably, the performance uplift brought by our ToolPRM is more
pronounced for smaller policy models. This characteristic renders
ToolPRM particularly advantageous for on-device inference scenar-
ios, which is of considerable importance for applications such as
function calling that are frequently deployed in edge environments
and are critical for real-world utility.
Illustratively, the Hammer2.1-1.5B model, when augmented with
our ToolPRM method, achieves performance comparable to the
baseline 3B model. Similarly, applying ToolPRM to the Hammer2.1-
3B model elevates its performance to a level on par with the base-
line 7B model. Furthermore, the integration of ToolPRM with the
Hammer2.1-7B model enables it to outperform several significantly
larger models that previously held a performance advantage, in-
cluding state-of-the-art models such as Qwen2.5-32B-Instruct.
These empirical results validate the effectiveness of our pro-
posed inference scaling framework for structured function calling
generation. Applying fine-grained beam search with ToolPRM can
substantially improve the capability of base models at inference
time through computational scaling.
4.4
In-Depth Analysis
In this section, we analyze how allocating additional computational
budget affects the function calling performance, providing an em-
pirical validation of our “explore more, but retain less” principle.
Using the Hammer2.1-1.5B and Hammer2.1-3B as policy models, we
conducted experiments on both the BFCL and ToolAlpaca bench-
marks, with the results plotted in Figure 4.
For baseline methods like Best-of-N and majority voting, the
generation budget is the number of sampled candidate trajectories.
For our method, the budget is controlled by two key hyperparame-
ters: the beam width 𝑀(how many options to explore at each step)
and the number of beams 𝑁(how many active beams to retain).
To test our principle, we created two variants: one where we fix
𝑁= 4 and scale the exploration width 𝑀, and another where we
fix 𝑀= 4 and scale the retention number 𝑁.
The results provide a clear validation of our hypothesis. As
shown in Figure 4, consistently increasing the exploration width (𝑀)
leads to generally monotonic performance improvements across
both models and benchmarks. This demonstrates that allowing the
agent to consider a broader set of possibilities at each decision
point is highly beneficial. Conversely, increasing the number of
retained active beams (𝑁) yields minimal gains and, in several cases,
leads to a significant degradation in performance. This is because
retaining faulty intermediate paths in structured action generation
is uniquely detrimental; an early error cannot be corrected, and
keeping the invalid trajectory alive misguides subsequent steps and
wastes computation.
Our ToolPRM approach, when configured to “explore more”, con-
sistently and significantly outperforms both Best-of-N and majority
voting, which show only modest or erratic gains with a larger bud-
get. This analysis confirms that for agents performing structured
actions like function calling, the most effective use of computational
resources is not to maintain more parallel plans, but to more thor-
oughly explore the options at each critical decision point, guided
by a precise, fine-grained process reward model.
5
Conclusion
In this paper, we propose a fine-grained inference scaling frame-
work to enhance the performance of LLM agents on structured
function calling tasks. We construct an intra-call step-annotated
dataset to train ToolPRM, a process reward model that supervises
each intermediate step. Integrated with beam search, ToolPRM
achieves the highest supervision accuracy and enables base LLMs
to attain state-of-the-art results. We also identify a key principle for
structured inference scaling: “explore more but retain less” based
on the unrecoverability of structured output generation. However,
the optimal trade-off in this principle is not yet dynamically ad-
justable. Future work can explore adaptive strategies to calibrate
exploration and retention based on input complexity or ToolPRM-
derived confidence.
ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
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ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling Jianghao Lin1∗, Yuanyuan Shi1∗, Xin Peng2, Renjie Ding2, Hairui Wang2, Yuxuan Peng2, Bizhe Bai3, Weixi Song3, Fengshuo Bai1, Huacan Chai1, Weinan Zhang1, Fei Huang2, Ying Wen1 1Shanghai Jiao Tong University, 2Longshine AI Research Institute, 3Shanghai Innovation Institute Abstract Large language models (LLMs) are increasingly demonstrating strong capabilities as autonomous agents, with function calling serving as a core mechanism for interaction with the environment. Meanwhile, inference scaling has become a cutting-edge technique to enhance LLM performance by allocating more computational resources during the inference process. However, current research on inference scaling primarily focuses on unstructured output generation tasks, leaving its application in structured outputs, like function calling, largely underexplored. To bridge this gap, we propose an inference scaling framework that combines fine-grained beam search with a process reward model, ToolPRM, which scores the internal steps of each single function call. To train ToolPRM, we construct the first fine-grained intra-call process supervision dataset, automatically annotated with function-masking techniques to provide step-level rewards for structured tool-use reasoning. Extensive experiments demonstrate that ToolPRM beats the coarse-grained and outcome reward models in terms of predictive accuracy, indicating its stronger capability in supervising the function calling inference process. Inference scaling technique equipped with ToolPRM also significantly improves the backbone model performance across various function calling tasks and benchmarks. More importantly, we reveal a key principle for applying inference scaling techniques to structured outputs: "explore more but retain less" due to the unrecoverability characteristics of structured function calling generation. Keywords LLM Agents, Function Calling, Inference Scaling 1 Introduction Large language models (LLMs) have demonstrated remarkable capabilities in a diverse range of tasks [6, 9, 16, 19, 44, 45, 52], rapidly evolving from general-purpose text generators to complex reasoning engines that can follow instructions, perform step-by-step reasoning, and integrate domain-specific knowledge into practical workflows. Building upon these advances, the concept of LLMbased Agents1 has emerged [38, 49], extending LLMs from passive text generators to active decision-making entities capable of perceiving their environment, reasoning over context, and performing actions through external tools. At the core of such agents lies the function calling mechanism, which enables them to interact with external tools, retrieve information, and perform actions, allowing them to bridge linguistic reasoning with real-world operations. ∗Both authors contributed equally to this research. 1In this paper, the term "LLM-based Agents" is interchangeable with "LLM Agents". This mechanism marks a fundamental shift in how LLMs interact with their surroundings, i.e., transforming static text prediction into dynamic action execution, where the correctness and structure of the outputs directly impact downstream system reliability. In this broader view, inference scaling plays a crucial role in enhancing an agent's reasoning and decision-making process. It provides a flexible way to trade off computational cost against reasoning quality, enabling LLMs to deliberate more deeply or explore alternative reasoning paths without retraining the base model. Rather than generating a single, greedily decoded output, inference scaling strategies allocate additional computational resources at inference time, to explore a wider range of possibilities and multiple reasoning trajectories, and select the best trajectory for better performance [15, 55]. Existing research on inference scaling predominantly focuses on unstructured output generation tasks, such as mathematical problems [29, 42] and other intricate reasoning tasks [24], where each sentence or discrete unit serves as a discrete step of thought. Many recent approaches combine Tree of thoughts (ToT) [50] with tree-based search algorithms, generalizing the linear Chain-of-Thought [41] (CoT) by maintaining multiple potential reasoning branches from a given state. To efficiently navigate the expanded reasoning space, beam search and Monte Carlo Tree Search (MCTS) have become common inference scaling techniques, and several studies have adapted them for LLM inference scaling [8, 21, 42, 51, 57]. These approaches typically rely on either self-evaluation [46] or process reward models (PRMs) [12, 24] to score intermediate reasoning paths and guide the search toward optimal outcomes. Despite its importance for constructing reliable and adaptive LLM agents, the application of inference scaling techniques to structured function calling scenarios remains significantly underexplored. Current studies that incorporate process reward mechanisms and inference scaling of function calling capabilities for agents typically operate at a coarse-grained level [25, 40], treating an entire round of function calling response as a singular, monolithic step. For example, Wang et al. [40] adopt a Best-of-N approach, where multiple function call candidates are scored and ranked by an outcome reward model (ORM). The final output is then selected based on the overall quality without applying distinct fine-grained rewards to the sub-components within each function call. However, this overlooks the inherent multi-stage nature of the function calling process, which involves solving a sequence of sub-tasks such as selecting the appropriate function name, identifying relevant parameters, and determining parameter values. Treating the function calling process as a single atomic step overlooks opportunities for fine-grained inference scaling and intra-call reward optimization 16 Oct 2025 Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Trovato et al. that could substantially improve the agent's reasoning efficiency and reliability.2 To this end, we propose a novel fine-grained process reward mechanism tailored for function calling capabilities of LLM agents (dubbed ToolPRM). In contrast to prior approaches that treat each function call as an indivisible unit, our method decomposes the agent's tool-call process into semantically interpretable intermediate reasoning steps. We design a dedicated process reward model, ToolPRM, trained under fine-grained intra-call supervision with meticulously curated step-level annotations constructed from both the xlam-function-calling-60k [23] and xlam-irrelevance-7.5k [20] datasets. Additionally, we further introduce fine-grained beam search guided by ToolPRM, which substantially outperforms existing function calling baselines in our experiments. Beyond establishing the empirical advantages of our approach, we further analyze the conceptual implications of inference scaling when applied to agents on structured versus unstructured reasoning tasks. In traditional inference scaling scenarios such as mathematical reasoning, the unstructured nature of outputs allows for flexible expansion - simply maintaining more candidate reasoning paths can enhance the quality of outcome, since erroneous intermediate steps may be revised later through reflection and process supervision. In contrast, for structured function calling outputs of LLM agents, maintaining a large number of concurrent trajectories often degrades performance, as early missteps cannot be corrected downstream, leading to wasted subsequent computational budget. Therefore, we introduce a task-aligned inference scaling principle of function calling for LLM agents: allocate computation to exploring a wider range of decisions (i.e., increasing beam width), while aggressively pruning redundant or low-quality candidates (i.e., reducing the number of active beams). We summarize this principle as "explore more, but retain less", emphasizing an efficient trade-off between exploration and reliability within agentic function calling processes. The major contributions of this paper are as follows: • Fine-Grained Intra-Call Process Supervision Dataset: We construct and release a novel annotated dataset specifically designed for fine-grained step-level supervision of structured function calling in LLM agents. This dataset, along with its annotation scripts, is provided via the anonymous link and will be made publicly available, facilitating future research and benchmarking in fine-grained reward modeling for agent function calling process. • ToolPRM: We introduce ToolPRM, a fine-grained process reward modeling framework specifically tailored for inference scaling in function calling LLM agents. ToolPRM could effectively assist different backbone models in conducting test-time scaling for superior performance. We also make the code and model checkpoints publicly accessible to support the community's further research. • Inference Scaling Principle for Structured Outputs: We identify and formalize a critical principle for inference scaling in structured output tasks: Explore more but retain less. This principle underscores the importance of broadening exploration over decision space while aggressively pruning 2In this paper, the term "function" is interchangeable with "tool" and "API". unpromising or unrecoverable steps, leading to more reliable function-calls by agents. Collectively, these contributions aim to advance the understanding and practical realization of scalable, trustworthy, and fine-grained reasoning mechanisms in the structured function calling capabilities of LLM agents. 2 Related Works Our work is closely related to (1) function callin for LLM agents, (2) inference scaling srategies, and (3) process reward models. 2.1 Function Calling for LLM Agents Function calling has become a fundamental mechanism that enables LLM-based agents to move beyond passive text generation and interact with external tools, databases, and environments. This capability effectively bridges reasoning and action, empowering LLMs to function as autonomous agents capable of executing complex tasks in structured, verifiable ways [23, 27, 30, 33, 35]. Recent research has explored various dimensions of this functioncalling paradigm. For instance, from the training side, IBM's Granite20B-FunctionCalling [1] employs multi-task learning to jointly train across seven core sub-tasks, demonstrating the benefit of holistic learning for function calling. Hammer [20] and Octopus [7] focus on improving syntactic and semantic robustness, respectively using function masking technique and conditional masking technique to ensure correctness under diverse contexts. Meanwhile, ToolACE [22] introduces a self-evolution synthesis framework that automatically generates high-quality function-calling data, highlighting the importance of scalable data construction for LLM-agent training on function calling task. TinyAgent [10] integrates a novel tool retrieval technique to reduce input context length and enhance inference efficiency in LLM edge deployment scenarios. Together, these studies form a growing foundation for structured, controllable, and verifiable decision-making within LLM agents. To evaluate such capabilities, several benchmarks have been proposed, including BFCL3, API-Bank [17], NESTful [4], and ComplexFuncBench [56], etc. Among them, for example, NESTful emphasizes nested and compositional API calls, whereas ComplexFuncBench targets multi-step and constrained scenarios, both crucial for assessing multi-hop reasoning and hierarchical planning of LLM agents. However, existing work primarily focuses on overall validity of the outputs rather than intra-process reasoning quality. In this paper, we take a step further by disassembling complex function calls into fine-grained decision steps and designing a process-level reward model that guides the trajectory selection in inference-scaling process. Ultimately, we aim to achieve more reliable, interpretable, and controllable agentic behavior in function calling. 2.2 Inference Scaling Strategies Inference scaling has recently emerged as a crucial mechanism for enhancing the reasoning and decision-making capabilities of LLM-based agents [3], and could be more computationally efficient than scaling model parameters [43]. Instead of relying on a single 3https://gorilla.cs.berkeley.edu/leaderboard.html ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling Conference acronym 'XX, June 03-05, 2018, Woodstock, NY forward pass, scaling strategies amplify inference-time computation to improve both accuracy and robustness in complex problem solving [42]. Early studies introduced straightforward sampling-based methods. The self-consistency strategy [39] samples multiple independent reasoning paths via temperature sampling [2] and aggregates their outcomes through majority voting, improving stability in open-ended reasoning. Similarly, the Best-of-N (BoN) approach [5] evaluates a batch of candidate outputs using a outcome reward model to select the optimal one. More recently, search-based scaling has become the dominant paradigm, especially for complex multistep reasoning tasks. Tree-structured inference methods, such as beam search, Monte Carlo Tree Search (MCTS), and their agentoriented variants [8, 21, 51, 57], allow the model to explore multiple reasoning trajectories, prune low-value branches, and dynamically refine decisions based on intermediate feedback. While studies have validated inference scaling in unstructured tasks, its role in structured output generation, such as function calling, remains relatively underexplored. In this paper, we extend inference scaling into the structured function-calling domain, treating each function call as a compositional reasoning process. We analyze how intra-call inference scaling improves decision reliability at the step level and discuss the computational trade-offs between search depth, width, and overall performance. 2.3 Process Reward Models In complex, multi-step reasoning tasks, the final output of an agent may be incorrect due to an error at any intermediate step. Outcome Reward Models (ORMs), which evaluate only the final result, provide a sparse and delayed reward signal that makes it difficult to perform credit assignment and identify the specific point of failure. To address this, Process Reward Models (PRMs) have been developed to provide more granular feedback by evaluating the correctness or quality of each intermediate step within a reasoning trajectory [37, 54? , 55]. PRMs are initially prominent as a component in Reinforcement Learning from Human Feedback (RLHF) pipelines [26]. Instead of relying on a single reward for a final answer, researchers trained PRMs on fine-grained human feedback for each step of a model's reasoning process. For instance, in the domain of mathematical reasoning, Lightman et al. [18] demonstrate that supervising a model with step-by-step rewards is a more data-efficient and effective method for teaching complex reasoning than rewarding only the final, correct answer. This dense reward signal allows the model to learn more robust and generalizable problem-solving procedures. More recently, PRM has been adapted as a heuristic to guide search algorithms over the model's reasoning paths. For example, Ma et al. [24] employ a PRM to score the intermediate thoughts generated within a Tree-of-Thoughts (ToT) framework, enabling the search algorithm to prune unpromising branches and focus computational resources on more plausible reasoning trajectories. To improve the reliability of the feedback itself, innovations like the ReasoningDriven PRM (R-PRM) have been proposed, which require the reward model to generate its own reasoning to justify its evaluation of each step, leading to more robust guidance [32]. In this paper, we explore the PRM paradigm in a fine-grained manner to the structured function calling capabilities of LLM agents. We introduce a method to automatically curate the first large-scale, fine-grained, intra-call process supervision dataset for agent tool use. Using this dataset, we train ToolPRM, a specialized model that excels at verifying the intermediate steps involved in constructing a valid tool call. By integrating ToolPRM with a beam search strategy, we provide agents with a powerful mechanism for deliberation, enabling them to achieve state-of-the-art performance in complex tool-use scenarios. 3 Methodology This section introduces the fine-grained process reward model for function calling (i.e., ToolPRM). We first present fine-grained decomposition for each turn of function callings, and present the data collection and annotation for reward modeling training. Then, we design a state transition machenism for function calling, and apply beam search with fine-grained process supervision with ToolPRM. 3.1 Fine-Grained Decomposition of Function Calls The essence of ToolPRM lies in its fine-grained decomposition of the function calling process. Traditional process supervision approaches for tool invocations primarily rely on coarse-grained response-level rewards, which evaluate the function call as a monolithic unit. However, such coarse granularity limits the interpretability, debuggability, and optimization potential. To this end, ToolPRM decomposes each function call into finegrained but semantically meaningful steps. Each LLM-generated response comprises a sequence of function calls tailored to a given user query. The construction of each function call is further broken down into (a) selecting the appropriate function name and (b) iteratively identifying parameter names and assigning corresponding values. Rather than evaluating a function call as a whole, ToolPRM provides fine-grained supervision that assesses the correctness of each constituent decision. As shown in Figure 1, this decomposition serves as the foundation for data collection and annotation. 3.2 Data Collection and Annotation To enable robust reward modeling under our fine-grained process supervision paradigm, we construct a high-quality annotated dataset that reflects the intricacies of individual decision points. 3.2.1 Data Collection. We begin by collecting natural language queries paired with their corresponding structured function calls, using xlam-function-calling-60k [23] and xlam-irrelevance-7.5k [20] datasets as the foundations. As shown in Figure 1, we apply function masking that selectively replaces function names and parameter identifiers with random strings. The masking mechanism introduces function ambiguity, encouraging the model to rely on contextual understanding of descriptions instead of simple memorization of the tool names, thus enhancing model robustness and generalization [20]. We adopt Hammer2.1-3b and Hammer2.1-7b as policy models to perform the rollout for data collection. Given a user query and a set of masked function candidates, each rollout generates a Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Trovato et al. Function Calls with Step Labels Candidate Functions [ { "name": "get_stock_price", "description": "Fetches stock price", "parameters": { "ticker": { "type": "string", "description": "The stock symbol", "default": "AAPL" }, ... (More Parameters) } }, ... (More Functions) ] Masked Candidate Functions [ { "name": "xkz9qvnb8", "description": "Fetches stock price", "parameters": { "y8z7x6": { "type": "string", "description": "The stock symbol", "default": "AAPL" }, ... (More Parameters) } }, ... (More Functions) ] Function Calling Policy Model Function Masking Query Can you tell me the stock price of Google today? [ { "name": "xkz9qvnb8", "parameters": { "y8z7x6": "GOOGLE", "start_date": "250506", "end_date": "250509", "include_volume": true, ... (More Parameters) } } , ... (More Function Calls if Needed) ] Figure 1: The illustration of data collection for ToolPRM. sequence of function calls, which is subsequently annotated with fine-grained step labels. 3.2.2 Result Annotation. As shown in Figure 1, the structured function call generated by the policy model is usually in JSON format. We can provide fine-grained step labels with the following types: • : Whether the selected function name at this round is correct or not. • : Whether the one pair of parameter name and value is correctly filled in, which can repeat multiple times. • : Whether all the parameters and values are correctly assigned. • : Whether the single function call (one element of the list) is correct or not. • : Whether the overall response (a list of function calls) is correct or not. Each symbol is followed by a binary label annotated by exact match with any of the possible ground truths, to indicate the correctness of the corresponding fine-grained step. Some of the step labels seem redundant (e.g., and ), but our experiments show that such a hierarchical step label redundancy can help the reward model generalize and perform better. 3.3 State Transition Mechanism and ToolPRM Training Based on the fine-grained step labels above, we can further formalize the function calling generation process as a dynamic decision process with a series of state transitions. As illustrated in Figure 2, we define the following five fine-grained states: • State #0 (Initial State): Input context and masked function candidates are presented. • State #1 (To Select Function Name): The model is required to select the function name from the candidates that are aligned with the query intent. • State #2 (To Select Parameter Name): The model should choose one necessary parameter to be filled in for the selected function. • State #3 (To Fill in Parameter Value): The model has to assign the proper value to the previously selected function parameter. • State #4 (Terminated State): One LLM response for function calls completes. Each transition between states can be explicitly supervised by the fine-grained step labels discussed in Section "Data Collection and Annotation for Reward Modeling". To train ToolPRM under fine-grained process supervision, we represent the function calling generation as a trajectory of decision steps, each labeled with a binary process reward. Let T = {(st,at,rt)}T t=1 denote a trajectory, where stis the state at step t, atis the action (e.g., selecting a function or filling a parameter via language token generation), and rt∈{0, 1} indicates its correctness. Each state encodes the current decision context, including the input query, masked function candidates, as well as the partially generated function calls. In this paper, since the backbone of ToolPRM is also a large language model, we adopt the tokens "+" and "-" for positive and negative reward labels, respectively. Given Nannotated trajectories {Ti}N i=1, ToolPRM is trained to predict rtfor each (st,at) via generative process reward modeling: LToolPRM = -Eτ∈D,(st,at,rt)∈τlogpθ(rt|st,at), (1) wherer(i) t ∈{+, -} is the binary label token, andθis the parameters of backbone LLM. 3.4 Beam Search with Fine-Grained Process Supervision As shown in Figure 2, we apply beam search guided by ToolPRM to generate high-quality structured outputs. At each step, ToolPRM assigns a fine-grained process reward to candidate actions based on their local state-action pair, allowing us to prune incorrect partial trajectories early. Specifically, we compute the scoresfor each beam candidate with ToolPRM i.e., s= es+/(es+ + es-), where s+ and sare the predictive logits on label tokens {+, -}. We maintain the top-Nhighest-scoring candidates, where Nindicates the number of beams. At each step, every preserved candidate can generate M ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling Conference acronym 'XX, June 03-05, 2018, Woodstock, NY TODO Choose Func. Name TODO Choose Param. Name TODO Fill in Param. Value Terminated State Initial State Initial State State #0 State #1 State #2 State #3 State #4 Candidate Output #1 Evaluate with ToolPRM ... Figure 2: The state transition of function calling (left) and beam search with ToolPRM (right). subsequent steps for ToolPRM evaluation, where Mdenotes the beam width. We propose a core principle for inference scaling in structured output generation: explore more but retain less. Concretely, we increase beam width Mto explore a wider range of candidate trajectories, but retain only a small number Nof highly promising ones. This design reflects the unrecoverability of structured outputs, such as function calls in JSON format, where a single incorrect decision (e.g., a wrong function name or argument value) can invalidate the entire trajectory. This stands in contrast to inference scaling in unstructured tasks such as math reasoning [34] or free-form text generation [31, 53], where early errors can often be corrected or compensated for in later steps. In such settings, preserving a diverse set of candidates throughout decoding (i.e., larger N) is beneficial. However, in structured generation, most decision points admit only a single or very few valid actions. Retaining incorrect candidates leads to inefficient use of the generation budget, as future steps cannot recover from early structural mistakes. Therefore, expanding the search space (larger beam width M) while aggressively pruning based on ToolPRM's step-wise supervision (smaller number of beams N) ensures that computational resources are concentrated on valid, high-quality structured outputs. This principle underpins the effectiveness of ToolPRM in scaling inference for structured generation tasks. 4 Experiments 4.1 Experiment Setups 4.1.1 Datasets. To train and validate our proposed ToolPRM for function calling, we process and annotate the xlam-function-calling60k [23] and xlam-irrelevance-7.5k [20] datasets using a well-designed pipeline, as detailed in Section 3. We report the dataset statistics in Table 1. The resulting annotated dataset exhibits an average of 6.25 step labels per sample, culminating in a total of 192,061 samples. The integration of function masking techniques further expands the Table 1: The statistics of our constructed ToolPRM dataset. Sample Granularity (Data Split) Positive Negative Total Step (Train) 4,380,323 731,665 5,111,988 Trajectory (Train) 466,786 127,648 594,434 Step (Test) 488,611 81,366 569,977 Trajectory (Test) 52,030 14,019 66,049 dataset size to 4 5 times, bringing it to a scale that surpasses typical datasets for Math PRM development, such as OpenAI's publicly available prm800k dataset [18]. Note that our constructed dataset above is used to train and validate the predictive accuracy of reward models. To further validate the effectiveness of our inference scaling strategy using ToolPRM, we adopt two function calling benchmarks: BFCL (Berkeley Function Calling Leaderboard) [47] and ToolAlpaca [36]. Following previous works [1, 20], we use abstract-syntax-tree-based (AST-based) accuracy as the metric for BFCL, and employ F1 scores of both API selection and parameter value assignment for ToolAlpaca. 4.1.2 Baselines. We compare our proposed ToolPRM to different sets of baselines from two aspects. To evaluate the predictive accuracy improvement brought by fine-grained intra-call process supervision, we train two baseline models: outcome reward model (ORM) and coarse-grained process reward model (C-PRM). Compared with our ToolPRM, i.e., finegrained process reward model using all the step labels, they are trained on the same dataset with different choices of step labels: • ORM is trained to discriminate the final result of the entire function calling sequence with . All the other intermediate step labels are disregarded. Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Trovato et al. 580 1160 1740 2320 2900 Training Step 0.00 0.05 0.10 0.15 0.20 Loss 580 1160 1740 2320 2900 Training Step 0.90 0.95 1.00 Step Acc 580 1160 1740 2320 2900 Training Step 0.90 0.95 1.00 Trajectory Acc ORM C-PRM ToolPRM Figure 3: The learning curves of different reward models (i.e., ORM, C-PRM, ToolPRM) on five-epoch training, in terms of loss, step-level accuracy, and trajectory-level accuracy. • C-PRM offers a more granular assessment of function call correctness than ORM by considering all the step labels except . To evaluate the function calling capabilities, we adopt the following three types of baselines: • General Purpose Models. This group comprises large language models that have not undergone specific finetuning for function calling tasks, nor do they employ the inference scaling strategies. Their performance serves as a baseline representing general capabilities. The models include GPT-4o, GPT-4o-mini [13], Llama-3.1-8B-Instruct [11], Mistral-NemoInstruct [14], and the Qwen2.5-Instruct series (72B, 32B, 7B, 3B, 1.5B) [48]. • Function Calling Models. This category includes models that have been specifically designed or finetuned for function calling tasks. These models include GRANITE (GRANITE20B-FUNCTIONCALLING) [1], xLAM-fc-r series (7B, 1B) [23], OpenFunctions-v2 [28], and Hammer2.1 series (7B, 3B, 1.5B, 0.5B) [20]. • Inference Scaling Strategies. We evaluate the performance enhancements afforded by different inference scaling methods. We choose the Hammer2.1 series (7B, 3B, and 1.5B variants) as base policy models. Three distinct inference scaling strategies are applied and compared against our ToolPRM: token-level beam search, majority voting, and best of N. 4.1.3 Implementation Details. All the experiments are conducted on NVIDIA 8xH100 GPU Clusters. We use Hammer2.1-3b as the reward model backbone and adopt SFT to train ToolPRM for 5 epochs with Adam optimizer. The batch size is 1024. The learning rate is 1e-3 with a warmup ratio of 0.008 followed by linear learning rate decay. The weight decay is 1e-5. As for the fine-grained beam search with ToolPRM, we set the temperature as 0.8, and select the number of beams Nand the beam width Mfrom {1, 2, 4, 8, 16}. 4.2 Predictive Accuracy of Reward Model High predictive accuracy of the reward model is vital for increasing the probability of selecting and retaining correct function call Table 2: Predicting accuracy of RMs of different granularities Reward Model Loss Step Acc Trajectory Acc ORM 0.0536 98.39% 98.39% C-PRM 0.0371 98.87% 99.06% ToolPRM 0.0286 99.11% 99.38% sequences during inference scaling, ultimately leading to superior overall inference outcomes. Hence, this section investigates the performance of reward models for function calls with varying granularities. We use three metrics for this evaluation: model loss, step-level accuracy (Step Acc), and trajectory-level accuracy (Trajectory Acc). Step Acc quantifies the correctness at each discrete process supervision step. It is worth noting that since each reward model (ORM, C-PRM, and ToolPRM) is supervised at its distinct level of granularity, the inherent difficulty of accurately predicting each step differs. Hence, we introduce trajectory-level accuracy to facilitate an appleto-apple comparison of models trained with differing supervisory step labels. The trajectory accuracy evaluates the correctness of the entire function call sequence. A trajectory is deemed accurate if the final assessment of the entire sequence is correct. This means that for C-PRM and ToolPRM, even if their evaluations of some intermediate steps are incorrect, the trajectory can still be classified accurately if their overall judgment of the function call's success or failure is correct. We report the predictive accuracy results in Table 2 and illustrate the learning curves over the five-epoch training in Figure 4. We can observe that reward models with finer granularity consistently achieve higher predictive accuracy across all three metrics. Specifically, our proposed ToolPRM outperforms both C-PRM and ORM in this regard. This suggests that fine-grained process reward modeling, such as our proposed ToolPRM, offers superior performance not only in terms of the precision of step-level supervision but also in their ultimate effectiveness in judging overall sequence correctness compared to their coarser-grained counterparts. ToolPRM: Fine-Grained Inference Scaling of Structured Outputs for Function Calling Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Table 3: Performance comparison of the general purpose models, function calling models, and different inference scaling techniques applied on Hammer2.1 series of models, evaluated on BFCL and ToolAlpaca. Multi., Paral., Mul.P. represents the Multiple split, Parallel split, and Multiple Parallel split of BFCL, separately. And Avg. represents an unweighted average of all the sub-categories of that benchmark. The best results of each model type (i.e., general purpose, function calling, and inference scaling) is given in bold and the second best is underlined. Type Model Size BFCL ToolAlpaca Simple Multi. Paral. Mul.P. Avg. F1-API F1-Args Avg. General Purpose GPT-4o - 77.17 95.00 93.50 85.00 87.67 88.64 66.67 77.66 GPT-4o-mini - 80.08 90.50 89.50 87.00 86.77 64.34 54.69 59.52 Llama-3.1-8B-Instruct 8B 72.83 93.50 87.00 83.50 84.21 75.64 55.12 65.38 Mistral-Nemo-Instruct 12B 77.00 93.50 89.50 84.50 86.13 87.31 66.18 76.75 Qwen2.5-72B-Instruct 72B 80.25 97.50 93.50 92.00 90.81 83.21 65.56 74.39 Qwen2.5-32B-Instruct 32B 72.83 94.00 93.50 88.50 87.21 85.82 61.11 73.47 Qwen2.5-7B-Instruct 7B 75.33 94.50 91.50 84.50 86.46 83.70 60.00 71.85 Qwen2.5-3B-Instruct 3B 74.17 90.50 79.50 79.00 80.79 70.80 51.63 61.22 Qwen2.5-1.5B-Instruct 1.5B 72.42 87.00 81.50 75.50 79.11 62.07 43.23 52.65 Function Calling GRANITE 20B 72.83 91.50 84.00 81.50 82.46 77.27 58.00 67.64 xLAM-7b-fc-r 7B 73.08 93.50 87.00 84.00 84.40 67.26 58.96 63.11 xLAM-1b-fc-r 1.3B 69.67 89.50 79.00 66.50 76.17 64.86 50.58 57.72 OpenFunctions-v2 7B 83.27 93.00 85.50 66.00 81.94 72.93 51.26 62.10 Hammer2.1-7B 7B 78.08 95.00 93.50 88.00 88.65 80.93 64.60 72.77 Hammer2.1-3B 3B 81.42 95.00 89.50 81.50 86.86 80.31 62.83 71.57 Hammer2.1-1.5B 1.5B 74.67 92.00 84.50 80.00 82.79 77.42 61.17 69.30 Hammer2.1-0.5B 0.5B 68.00 83.00 71.50 54.00 69.13 77.10 60.67 68.89 Inference Scaling Hammer2.1-7B (Base) 7B 78.08 95.00 93.50 88.00 88.65 80.93 64.60 72.77 + Token-level Beam Search 74.58 93.50 91.50 82.50 85.52 79.69 62.37 71.03 + Marjority 79.58 95.00 93.50 85.00 88.27 79.03 65.26 72.15 + Best of N (ORM) 76.58 94.50 91.50 87.00 87.40 78.86 67.73 73.30 + ToolPRM (Ours) 79.08 95.50 94.50 89.00 89.52 81.42 65.30 73.36 Hammer2.1-3B (Base) 3B 81.42 95.00 89.50 81.50 86.86 80.31 62.83 71.57 + Token-level Beam Search 74.50 91.00 87.50 77.00 82.50 78.29 59.32 68.81 + Marjority 79.50 95.00 89.00 81.00 86.13 76.00 58.27 67.14 + Best of N (ORM) 77.08 93.50 89.50 84.00 86.02 77.29 60.16 68.73 + ToolPRM (Ours) 80.50 95.50 91.50 88.00 88.88 80.78 63.13 71.96 Hammer2.1-1.5B (Base) 1.5B 74.67 92.00 84.50 80.00 82.79 77.42 61.17 69.30 + Token-level Beam Search 72.33 91.50 81.00 73.50 79.58 73.03 56.68 64.86 + Marjority 77.50 91.50 84.50 80.00 83.38 72.88 63.61 68.25 + Best of N (ORM) 75.25 90.00 86.50 84.00 83.94 77.42 61.66 69.54 + ToolPRM (Ours) 78.42 92.00 87.50 84.50 85.61 82.68 63.18 72.93 4.3 Inference Scaling Performance In this section, we conduct a comprehensive examination and comparative analysis of our proposed ToolPRM, as well as various function calling baseline models and inference scaling strategies on two datasets. As evidenced in Table 3, we can observe that our proposed ToolPRM can generally achieve the best performance compared with other inference scaling strategies given base policy models of different sizes. Besides, the performance of other inference scaling strategies is fairly unstable and can sometimes perform worse than the base model. The potential reason is that the non-greedy sampling might cause minor errors during the function calling generation, which could directly ruin the entire trajectory. This suggests the necessity of intra-call process supervision through the fine-grained beam search for structured function calling generation. Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Trovato et al. 20 21 22 23 24 Generation Budget 0.81 0.82 0.83 0.84 0.85 0.86 Averaged Accuracy Hammer2.1-1.5B on BFCL 20 21 22 23 24 Generation Budget 0.83 0.84 0.85 0.86 0.87 0.88 0.89 Averaged Accuracy Hammer2.1-3B on BFCL 20 21 22 23 24 Generation Budget 0.66 0.68 0.70 0.72 Averaged F1 Score Hammer2.1-1.5B on Toolalpaca 20 21 22 23 24 Generation Budget 0.66 0.67 0.68 0.69 0.70 0.71 Averaged F1 Score Hammer2.1-3B on Toolalpaca ToolPRM; N=4 ToolPRM; M=4 Best of N Majority Figure 4: The performance on BFCL (Averaged Accuracy) and ToolAlpaca (Averaged F1 Score) w.r.t. different generation budgets and different inference scaling strategies. We conduct experiments on Hammer2.1-1.5B and Hammer2.1-3B policy models. Notably, the performance uplift brought by our ToolPRM is more pronounced for smaller policy models. This characteristic renders ToolPRM particularly advantageous for on-device inference scenarios, which is of considerable importance for applications such as function calling that are frequently deployed in edge environments and are critical for real-world utility. Illustratively, the Hammer2.1-1.5B model, when augmented with our ToolPRM method, achieves performance comparable to the baseline 3B model. Similarly, applying ToolPRM to the Hammer2.13B model elevates its performance to a level on par with the baseline 7B model. Furthermore, the integration of ToolPRM with the Hammer2.1-7B model enables it to outperform several significantly larger models that previously held a performance advantage, including state-of-the-art models such as Qwen2.5-32B-Instruct. These empirical results validate the effectiveness of our proposed inference scaling framework for structured function calling generation. Applying fine-grained beam search with ToolPRM can substantially improve the capability of base models at inference time through computational scaling. 4.4 In-Depth Analysis In this section, we analyze how allocating additional computational budget affects the function calling performance, providing an empirical validation of our "explore more, but retain less" principle. Using the Hammer2.1-1.5B and Hammer2.1-3B as policy models, we conducted experiments on both the BFCL and ToolAlpaca benchmarks, with the results plotted in Figure 4. For baseline methods like Best-of-N and majority voting, the generation budget is the number of sampled candidate trajectories. For our method, the budget is controlled by two key hyperparameters: the beam width M(how many options to explore at each step) and the number of beams N(how many active beams to retain). To test our principle, we created two variants: one where we fix N= 4 and scale the exploration width M, and another where we fix M= 4 and scale the retention number N. The results provide a clear validation of our hypothesis. As shown in Figure 4, consistently increasing the exploration width (M) leads to generally monotonic performance improvements across both models and benchmarks. This demonstrates that allowing the agent to consider a broader set of possibilities at each decision point is highly beneficial. Conversely, increasing the number of retained active beams (N) yields minimal gains and, in several cases, leads to a significant degradation in performance. This is because retaining faulty intermediate paths in structured action generation is uniquely detrimental; an early error cannot be corrected, and keeping the invalid trajectory alive misguides subsequent steps and wastes computation. Our ToolPRM approach, when configured to "explore more", consistently and significantly outperforms both Best-of-N and majority voting, which show only modest or erratic gains with a larger budget. This analysis confirms that for agents performing structured actions like function calling, the most effective use of computational resources is not to maintain more parallel plans, but to more thoroughly explore the options at each critical decision point, guided by a precise, fine-grained process reward model. 5 Conclusion In this paper, we propose a fine-grained inference scaling framework to enhance the performance of LLM agents on structured function calling tasks. We construct an intra-call step-annotated dataset to train ToolPRM, a process reward model that supervises each intermediate step. Integrated with beam search, ToolPRM achieves the highest supervision accuracy and enables base LLMs to attain state-of-the-art results. 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[42] Yangzhen Wu, Zhiqing Sun, Shanda Li, Sean Welleck, and Yiming Yang. 2024. Inference scaling laws: An empirical analysis of compute-optimal inference for problem-solving with language models. arXiv preprint (2024). Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Trovato et al. [43] Yangzhen Wu, Zhiqing Sun, Shanda Li, Sean Welleck, and Yiming Yang. 2024. Inference scaling laws: An empirical analysis of compute-optimal inference for problem-solving with language models. arXiv preprint (2024). [44] Yunjia Xi, Jianghao Lin, Yongzhao Xiao, Zheli Zhou, Rong Shan, Te Gao, Jiachen Zhu, Weiwen Liu, Yong Yu, and Weinan Zhang. 2025. A survey of llm-based deep search agents: Paradigm, optimization, evaluation, and challenges. arXiv preprint (2025). [45] Yunjia Xi, Weiwen Liu, Jianghao Lin, Bo Chen, Ruiming Tang, Weinan Zhang, and Yong Yu. 2024. Memocrs: Memory-enhanced sequential conversational recommender systems with large language models. 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[50] Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimhan. 2023. Tree of thoughts: Deliberate problem solving with large language models. Advances in neural information processing systems 36 (2023), 11809-11822. [51] Shun Zhang, Zhenfang Chen, Yikang Shen, Mingyu Ding, Joshua B Tenenbaum, and Chuang Gan. 2023. Planning with large language models for code generation. arXiv preprint (2023). [52] Weinan Zhang, Junwei Liao, Ning Li, Kounianhua Du, and Jianghao Lin. 2024. Agentic information retrieval. arXiv preprint (2024). [53] Zhuosheng Zhang, Yao Yao, Aston Zhang, Xiangru Tang, Xinbei Ma, Zhiwei He, Yiming Wang, Mark Gerstein, Rui Wang, Gongshen Liu, et al. 2025. Igniting language intelligence: The hitchhiker's guide from chain-of-thought reasoning to language agents. Comput. Surveys 57, 8 (2025), 1-39. [54] Congmin Zheng, Jiachen Zhu, Jianghao Lin, Xinyi Dai, Yong Yu, Weinan Zhang, and Mengyue Yang. 2025. Cold: Counterfactually-guided length debiasing for process reward models. arXiv preprint (2025). [55] Congming Zheng, Jiachen Zhu, Zhuoying Ou, Yuxiang Chen, Kangning Zhang, Rong Shan, Zeyu Zheng, Mengyue Yang, Jianghao Lin, Yong Yu, et al. 2025. A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models. arXiv preprint (2025). [56] Lucen Zhong, Zhengxiao Du, Xiaohan Zhang, Haiyi Hu, and Jie Tang. 2025. ComplexFuncBench: Exploring Multi-Step and Constrained Function Calling under Long-Context Scenario. arXiv preprint (2025). [57] Andy Zhou, Kai Yan, Michal Shlapentokh-Rothman, Haohan Wang, and Yu-Xiong Wang. 2023. Language agent tree search unifies reasoning acting and planning in language models. arXiv preprint (2023). Received 20 February 2007; revised 12 March 2009; accepted 5 June 2009
|
2510.14704
|
Intelligent Systems Group
University of Siegen
Dataset Pruning in RecSys and ML:
Best Practice or Mal-Practice?
Bachelor’s Thesis
Computer Science
Leonie Winter
1097962
30.09.2025
Examiners
Prof. Dr. Joeran Beel
Intelligent Systems Group
M. Sc. Tobias Vente
Intelligent Systems Group
This Bachelor’s Thesis is handed in according to the requirements of the University of
Siegen for the study program Bachelor Computer Science of the year 2012 (PO 2012).
Process period
12.05.2025 to 30.09.2025
Examiners
Prof. Dr. Joeran Beel
M. Sc. Tobias Vente
Eigenständigkeitserklärung
Ich versichere, dass ich die schriftliche Ausarbeitung selbständig angefertigt und keine
anderen als die angegebenen Hilfsmittel benutzt habe. Alle Stellen, die dem Wortlaut
oder dem Sinn nach (inkl. Übersetzungen) anderen Werken entnommen sind, habe ich
in jedem einzelnen Fall unter genauer Angabe der Quelle (einschließlich des World Wide
Web sowie generativer KI und anderer elektronischer Datensammlungen) deutlich als
Entlehnung kenntlich gemacht. Dies gilt auch für angefügte Zeichnungen, bildliche
Darstellungen, Skizzen und dergleichen. Ich nehme zur Kenntnis, dass die
nachgewiesene Unterlassung der Herkunftsangabe als versuchte Täuschung gewertet
wird.
Ich bestätige, dass die elektronische Version inhaltlich mit der gedruckten Version
übereinstimmt.
____________________
__________________________________________
Datum
Unterschrift
Declaration of Authorship
I certify that I have prepared this written work independently and have not used any aids
other than those specified. I have clearly identified all passages that are taken from
other works in terms of wording or meaning (including translations) as borrowed material
in each individual case, stating the exact source (including the World Wide Web as well
as generative AI and other electronic data collections). This also applies to attached
drawings, pictorial representations, sketches and the like. I acknowledge that the proven
omission of the indication of origin will be regarded as attempted deception.
I confirm that the content of the electronic version is the same as the printed version.
I
I.
Contents
I.
Contents ............................................................................................................................... I
II.
Abstract ............................................................................................................................ III
III.
Acknowledgements ...................................................................................................... IV
1.
Introduction ........................................................................................................................ 1
1.1
Background ................................................................................................................ 1
1.2
Research Problem ....................................................................................................... 1
1.3
Research Question ...................................................................................................... 3
1.4
Research Objective ..................................................................................................... 3
2.
Background ........................................................................................................................ 3
3.
Related Work ...................................................................................................................... 4
4.
Methodology ...................................................................................................................... 7
4.1
Datasets ...................................................................................................................... 7
4.1.1
Dataset Selection ................................................................................................ 7
4.1.2
Preprocessing of Datasets ................................................................................... 8
4.2
Algorithm Selection ................................................................................................. 10
4.3
Training and Testing ................................................................................................ 11
5.
Results .............................................................................................................................. 13
5.1
Dataset Analysis ....................................................................................................... 13
5.1.1
Unprocessed Dataset Characteristics ............................................................... 13
5.1.2
Core-Pruned Dataset Characteristics ................................................................ 15
5.1.3
Structural Characteristics ................................................................................. 16
5.1.4
Distributional Characteristics ........................................................................... 17
5.2
Algorithm Performance Analysis ............................................................................. 22
5.2.1
Overall Performance Trends ............................................................................ 22
5.2.2
Training and Testing on Pruned Datasets ........................................................ 24
5.2.3
Comparison of Testing on Pruned and Unpruned Testsets .............................. 27
6.
Discussion and Interpretation ........................................................................................... 32
6.1.1
Dataset Characteristics ..................................................................................... 32
6.1.2
Algorithm Performance .................................................................................... 36
7.
Conclusion ........................................................................................................................ 45
8.
Future Work and Limitations ........................................................................................... 48
9.
Summary .......................................................................................................................... 49
II
10.
Bibliography ................................................................................................................. 51
11.
Appendix ...................................................................................................................... 57
III
II.
Abstract
Offline evaluations in recommender system research depend heavily on datasets, many of
which are pruned, such as the widely used MovieLens collections. This thesis examines the
impact of data pruning – specifically, removing users with fewer than a specified number of
interactions – on both dataset characteristics and algorithm performance. Five benchmark
datasets were analysed in their unpruned form and at five successive pruning levels (5, 10, 20,
50, 100). For each coreset, we examined structural and distributional characteristics and
trained and tested eleven representative algorithms. To further assess if pruned datasets lead
to artificially inflated performance results, we also evaluated models trained on the pruned
train sets but tested on unpruned data.
Results show that commonly applied core pruning can be highly selective, leaving as little as
2% of the original users in some datasets. Traditional algorithms achieved higher nDCG@10
scores when both training and testing on pruned data; however, this advantage largely
disappeared when evaluated on unpruned test sets. Across all algorithms, performance
declined with increasing pruning levels when tested on unpruned data, highlighting the impact
of dataset reduction on the performance of recommender algorithms.
IV
III. Acknowledgements
I would like to express my gratitude to Prof. Dr Ing. Beel, as well as to Lukas Wegmeth and
Tobias Vente, for their supervision and valuable guidance and insight throughout the creation
of this thesis.
Throughout the course of writing this thesis, ChatGPT was used as an auxiliary writing tool,
particularly for phrasing suggestions and proofreading. All suggestions from ChatGPT were
carefully reviewed, edited and adapted to ensure correctness and consistency with my own
voice. This practice aligns with the guideline that generative models can be used to support
writing, but their use should be disclosed transparently. [7]
The code for all experiments conducted during this thesis is openly accessible at GitHub [51].
1
1. Introduction
1.1 Background
The performance of recommender systems is typically assessed through offline evaluations,
online evaluations, or user studies. Offline evaluations – training and testing algorithms on
datasets – are the most common first step to narrow down promising algorithms. [14, 41, 54]
Ideally, the datasets for offline evaluations should match the intended application domain [15,
25]; nonetheless, many studies rely on a small set of benchmark datasets, some of which have
been pruned, meaning specific interactions have been removed [6, 8, 26, 27, 39, 47].
Data pruning has been shown to alter structural characteristics of datasets, which are key
factors in explaining performance variance across datasets (e.g. density) [1, 15, 18, 20]. A
commonly applied pruning technique is the systematic removal of low-activity users from
datasets [20, 54, 55]. These so-called cold-start users are a recognised challenge in
recommender-system research, and removing them from both the training and test sets
effectively avoids evaluating recommender performance on this group [20, 24, 30]. This can
lead to artificially inflated performance scores [8] and limit generalizability [6, 15, 20], as
pruning causes the test data to diverge from real-world scenarios, resulting in datasets
composed primarily of high-activity users with rich interaction histories [20].
As more complex models emerge and the demand for sustainable recommender systems
grows, interest in data reduction techniques such as data pruning has increased [5, 9, 43, 45,
49, 58], even though data pruning is recognised as a barrier to comparability and
reproducibility in recommender-system research [17, 22, 27, 47, 54]. These contrasting
developments warrant closer examination of how data pruning affects both dataset
characteristics and algorithm performance.
1.2 Research Problem
Data pruning has been established as a popular step in dataset preprocessing, as demonstrated
in reviews. Sun et al. found that of more than 42 examined papers that explicitly stated
preprocessing datasets, 60% applied Data Pruning in this process [47]. This aligns with Beel
and Brunel[8], who found that 65% of the papers they examined that employed offline
evaluations used either a pruned dataset or pruned a dataset during the preprocessing steps.
2
While data pruning has been proven to positively impact runtime, computational load, and
dataset density, seemingly achieving improved performance values [4], a good result in
performance evaluation on a pruned dataset is not generally transferable to other datasets.
In recommender-system research, so-called cold-start users or items present a well-known
challenge, as many algorithms struggle to generate reliable recommendations for users
without a rich interaction history. Newly introduced items lacking similarity to existing ones
pose the same challenge. [2, 29, 30, 42]
These challenging user groups are effectively eliminated when user-based data pruning is
applied. As a result, algorithm evaluations on pruned datasets are biased toward high-activity
users, since the challenging low-activity group is progressively excluded from both training
and testing. [24] In practice, applying the MovieLens-standard 20-core pruning can exclude
up to 42% of users while still retaining about 95% of interactions [8].
Algorithm performance partly depends on the dataset structure. For instance, density is a key
factor in explaining performance variance. [14, 16] Therefore, it is possible to influence
algorithm performance through the deliberate selection of datasets or coresets. However,
higher levels of pruning cause a greater divergence from the raw data that a recommender is
likely to encounter in real-world scenarios. [20] Consequently, pruned datasets can hinder
generalisation and comparability when evaluating algorithms.
Although data pruning is a commonly applied procedure, 44% of researchers still do not
report the preprocessing applied to the datasets used in their studies, as a recent review
showed [55]. This lack of transparency is problematic because it greatly hinders the
reproducibility of algorithm performance and the ability to make fair comparisons.
In summary, while dataset pruning is a widely acknowledged tool, it poses three key
problems: reproducibility suffers when pruning is undocumented, comparability is
compromised when coresets are deliberately selected, and generalisability declines as pruned
datasets diverge from real-world data.
Despite growing awareness of these issues, the combined effects of progressive data pruning
on dataset characteristics and algorithm performance remain insufficiently examined. The
lack of systematic analysis of how progressive data pruning alters dataset characteristics and
affects the measured performance and ranking of recommender algorithms creates uncertainty
3
in evaluating and comparing recommendation models. Addressing this gap is essential for
reproducibility, fairness, and the reliable development of recommender systems.
1.3 Research Question
Building on the previous section, this thesis aims to mitigate the gap identified in research on
data pruning and its impact on algorithm performance and the structural characteristics of
datasets. Unlike previous studies, we will investigate the changes in distributional
characteristics of the pruned datasets, with a focus on user, item, and interaction retention. By
progressively increasing the pruning core-level of datasets, comparing the performance of 11
algorithms on these pruned variations of the datasets and examining the changes in dataset
characteristics, we aim to address the following research question:
How does progressive dataset pruning by increasing core thresholds affect the performance
of recommender systems, the structural and distributional characteristics of the dataset?
1.4 Research Objective
The objective of this thesis is two-fold. First, we aim to assess how progressive pruning
affects dataset characteristics and to quantify the proportion of the original data that remains
at each pruning level. Second, we strive to determine whether progressive pruning alters the
ranking of algorithms and whether it artificially inflates their performance scores.
2. Background
Recommender systems have become indispensable for delivering personalised
recommendations across diverse application areas, including e-commerce, streaming services,
social networks, and education [29, 31, 35, 42]. The development of recommender systems
requires both models and datasets, yet it is the datasets on which different models are trained,
evaluated and compared in offline studies. Consequently, datasets play a foundational role in
recommender-system research, and selecting the right dataset is a critical step in the
development process [47]. Some datasets have become benchmarks, widely used for training
4
and evaluation in recommender system research. For example, the MovieLens datasets [53],
which a recent review found to be used in approximately 50% of studies [2].
The most commonly used variants of the MovieLens datasets are pruned. The term data
pruning refers to setting a threshold t for a group (e.g., users or items) and eliminating all
interactions from any user or item whose total interactions fall below that threshold [20]. A
well-known example is the MovieLens datasets (except ML Full Latest), which apply t = 20
to users, retaining only interactions from users with at least 20 interactions in the original
dataset. Such a subset is called 20-core pruned, where core denotes the minimum interaction
count required for inclusion. In this thesis, 0-core refers to the full, unpruned dataset, which
has only been randomly downsampled.
Although data pruning is sometimes also referred to as data filtering, it differs fundamentally
from downsampling, in which a subset is drawn (e.g., randomly) without intentionally altering
the dataset’s inherent characteristics [43]. In contrast, subjecting datasets to pruning methods
alters the structural characteristics of the datasets, such as the average number of ratings per
user or item, or the density of the dataset, which describes the fraction of ratings in the dataset
relative to all possible user-item interactions [1, 18]. These characteristics are part of our
analysis and are therefore briefly outlined in the Methodology chapter.
3. Related Work
The question of how to create sound evaluation for recommender systems [6, 54] and
machine learning algorithms is an active field of research that spans multiple factors,
including the impact of reduction strategies (e.g., dataset pruning) on dataset composition and
algorithm performance [13, 34, 58]. Various strategies aim to decrease the computational load
of evaluations or improve model performance and efficiency by altering the datasets or
models, including model pruning (e.g., by removing weights or nodes) [11, 53],
downsampling, and the intentional removal of targeted interactions. Removing interactions
from a dataset to alter its composition or quality can take several forms, such as data pruning,
noise reduction (e.g., via sampling techniques) [28], or the removal of users with a
detrimental impact on performance [36].
5
To the best of our knowledge, Beel and Brunel [8] were the first to review a set of conference
papers on dataset usage to determine the prevalence of pruned datasets in recommender-
system evaluations. Their review found that almost 50% of offline evaluations were
conducted on at least partially pruned datasets, with the 20-core pruned MovieLens datasets
being the most frequently used. In assessing the impact of 20-core pruning on algorithm
performance, they compared six collaborative algorithms on an unpruned version of
MovieLens with a version containing only users who exceeded the 20-interaction threshold.
Their results show increased performance on the pruned dataset; however, it is worth noting
that the test sets were also pruned. While this paper provides the foundation for our work, its
performance comparison was limited to a single dataset and reported only error-based metrics
(RMSE, MAE). In this thesis, we extend the analysis to ranking-based metrics and evaluate a
broader variety of algorithms across a wider range of datasets and pruning levels.
A recent review on standard practices in recommender evaluation has found that dataset
selection and preprocessing techniques, including pruning, vary widely across the field [55].
Zhao et al. found that 44% of studies did not report any preprocessing details and 34%
implicitly applied n-core filtering with thresholds of 5 or 10. Thresholds varied widely, with
some papers using 25- or 30-core pruning, but the authors’ own experiments focused only on
5- and 10-core filters. They observed that these two thresholds produced significantly
different rankings on some datasets (e.g., the sparse Amazon datasets). Although they
reported the proportion of users excluded, exploring a broader range of core thresholds and
conducting a more thorough analysis of changes in the datasets’ structural and distributional
characteristics would be valuable.
Sun et al. [47] had earlier taken a similar approach, finding that 60% of studies applied 5- or
10-core filtering, with some using thresholds as high as 20 or 30. The authors also examined
the 5- and 10-core filters, which Zhao et al. later confirmed to be aligned with.
Bentzer and Thulin [10] investigated a user-based approach of data pruning by limiting the
number of users included in their training sets. The base dataset was MovieLens-25M, already
20-core pruned for users, and it was further filtered to retain only items that had been
interacted with by at least 10% of users. The authors then evaluated SVD and IBCF on
progressively smaller test sets and found IBCF to achieve higher accuracy on small datasets,
whereas SVD was preferable for larger datasets, despite a slight loss in accuracy, because of
6
its lower execution time. Their results highlight that, in certain scenarios, accepting a slight
decrease in performance can be a worthwhile trade-off for faster execution. However, because
their experiment only included two algorithms, the findings have limited generalisability.
Driven by the goal of developing more sustainable and efficient recommender systems, data
pruning has also become a focus in studies on data-reduction techniques. Spillo et al. [45]
investigated the trade-offs between data reduction and algorithm performance, as well as the
impact on the carbon footprint of recommender systems. In their study, the authors reduced
the data only after splitting the datasets into training and test sets, thereby manipulating only
the size of the training data. They confirmed that data reduction decreases both emissions and
performance, while some algorithms showed scenario-dependent potential for achieving
acceptable trade-offs.
Similarly, Arabzadeh et al. [4, 5] examined whether a trade-off between energy efficiency and
accuracy is possible by applying two user-centred downsampling approaches to datasets.
Unlike Spillo et al., they also incorporated core pruning of both users and items at levels 10
and 30, which left only a fraction of the datasets viable for evaluation. Their results showed
that more data generally had a positive impact on accuracy, but aligned with Spillo et al., that
scenario-dependent trade-offs can be found for individual algorithms. While this study makes
a valuable contribution to more sustainable recommender systems, the analysis of dataset
characteristics did not extend beyond a basic examination of structural properties.
Regarding dataset characteristics, Deldjoo et al. [18] proposed a rich framework of structural
and distributional properties, which we partially adopt in our analysis. Their focus was on
assessing the impact of individual dataset characteristics and identifying those with the most
significant influence. Shape, Interactions per User or Item, and Density have emerged as
impactful characteristics in their results, showing that deliberate dataset selection, focusing on
key characteristics, can directly impact performance. In this thesis, we adopt some of the
proposed explanatory variables of their study and examine their evolution across increasingly
pruned coresets.
Fan et al. [24] explored the MovieLens datasets by selectively removing interactions per user
(e.g., the first 15) to assess which interactions are most informative for preference elicitation.
They confirmed that MovieLens is a statistical outlier in its distributional characteristics.
Their results also showed that, because users were initially encouraged to rate popular items
7
until a threshold was reached during the collection of interactions for the MovieLens datasets,
these early 15 interactions contributed little to shaping user preferences.
Further research on the influence of the datasets on algorithm performance was also
conducted by Chin et al. [15]; however, their dataset selection did not include pruned
variations. Doerfel et al. [20] examined how dataset pruning affects the performance ranking
of recommender systems in the context of social bookmarking. They found that density
increased with higher core levels, but the low-core settings of 2 and 3 were most consistent
with the raw data.
4. Methodology
4.1 Datasets
This subsection outlines the dataset selection, all preprocessing steps and the train-test
splitting strategy applied. Additionally, we outline the key characteristics in our dataset
analysis.
4.1.1 Dataset Selection
The selection of datasets is based on several reviews, identifying the most commonly used
datasets in recommender-system research. The popularity of the MovieLens datasets is
undisputed, with a range of 40% [8], 50% [2], and up to 84% [6] of studied papers employing
at least one version of MovieLens in their experiments [12, 17, 25, 26, 39, 47, 15]. There are
multiple versions of the MovieLens dataset available, with only the MovieLens Full Latest
being unpruned and usable for our experiment. [27]
In recommender research, the Amazon [38] datasets are also widely adopted, with 35% [8] to
42% [6] of examined papers including at least one version in their experimentation [25, 39,
47].
Another benchmark dataset in recommender-systems research is the Yelp dataset[59], which
is employed in multiple versions from different timeframes [47, 17, 26, 15].
The aforementioned datasets are all classed as explicit datasets, expressing users’ preferences
for items by numerical ratings ranging from 1 (lowest) to 5 (highest). To employ an implicit
dataset in our experiment as well, we chose the Gowalla [16] dataset, which consists of point-
8
of-interest data where every interaction with an item is recognised as a positive preference
value, and all non-interacted items are considered negative [25].
For readability, shortened dataset names are used throughout this thesis. The Amazon Toys &
Games and Amazon CD & Vinyl datasets are referred to as Amazon Toys and Amazon CD,
respectively. Similarly, all references to MovieLens denote the MovieLens Full Latest dataset
unless otherwise specified.
As the analysis of structural and distributional characteristics of the datasets is part of this
study, please refer to Chapter 5.1 for detailed statistics of each pruned version and the non-
pre-processed datasets.
4.1.2 Preprocessing of Datasets
All datasets selected for the experiment have been submitted to the same procedure of
preprocessing, consisting of transformation into an implicit dataset for originally explicit
datasets, removal of duplicate interactions, downsampling to a random subset of fixed size,
pruning to six different core-subsets, data splitting and the creation of atomic files. The
specific actions taken in each of these steps are listed in detail below, in the same sequence as
they were executed.
Transforming explicit datasets to implicit datasets All datasets, except for the inherently
implicit Gowalla dataset, have been transformed into implicit datasets by retaining all
interactions with rating values equal to or greater than 4.0 [3, 44]. These ratings were assigned
the binary value of 1.
Downsampling For all implicit datasets with more than 3,000,000 interactions, a random
subset has been extracted using Pandas’ sample function on the interaction dataframe. The
resulting subset had a length of 3,000,000 for each dataset. Hereinafter, this downsampled
dataset will be referred to as the unpruned or 0-core dataset.
Coreset Creation Each dataset was retained in an unpruned version, referred to as 0-core,
which retained all interactions by all users and with all items included in the downsampled
set. Subsequently, the dataset was pruned in five additional coresets, where each core
represented the minimum count of interactions a user had to have in the dataset to be retained
9
in the respective coreset. For example, the 5-coreset retained all interactions by users with at
least five interactions in the dataset. Each of the selected datasets was pruned into 5-core, 10-
core, 20-core, 50-core, and 100-core subsets.
Please note that the MovieLens datasets are commonly described as 20-core pruned [8], but in
practice, the threshold of 20 is only applied to users [27]. Items remain in the dataset even if
they have been interacted with only once. Strictly speaking, the term core filtering would be
more accurate, since core pruning can imply a recursive process that continues removing
interactions until both users and items meet the minimum interaction threshold. For clarity, in
this thesis, we use the term '20-core pruned' to denote datasets filtered by a 20-interaction user
threshold, which may still include items with as few as one interaction.
Data Splitting. For each core subset and the unpruned variant of each dataset, a user-based
split into train and test sets is performed, with 80% of interactions per user assigned to the
train set and the remaining 20% of interactions per user assigned to the test set. The data
splitting was applied via LensKit’s user-based partitioning and the SampleFrac partition
method, which randomly selects a fraction of test rows per user1.
Creating Atomic Files The train and test sets have been transformed into the atomic file
format required for use by the RecBole library2.
Analysis of Dataset Characteristics
One focus of this study is to analyse the dataset characteristics and the impact Data Pruning
has on the structural and distributional characteristics of the datasets. Therefore, all datasets
intended for the training and testing of the Recommender algorithms were analysed under the
following factors to allow further comparison following the framework employed by Deldjoo
et al. [18]:
Structural Characteristics
Distributional Characteristics
Total Number of Interactions (# Interactions) = |Int|
Total Number of Users (# Users) = |U|
Total Number of Items (# Items) = |I|
Space Size = |𝑈𝑈| × |𝐼𝐼|
Average Number of Interactions per User:
(Int /U) =
|𝐼𝐼𝐼𝐼𝐼𝐼|
|𝑈𝑈|
Average Number of Interactions per Item:
(Int / I) =
|𝐼𝐼𝐼𝐼𝐼𝐼|
|𝐼𝐼|
1 https://lkpy.lenskit.org/0.14.4/crossfold#lenskit.crossfold.partition_users
2 https://recbole.io/docs/user_guide/data/atomic_files.html
10
Shape =
|𝑈𝑈|
|𝐼𝐼|
Density =
|𝐼𝐼𝐼𝐼𝐼𝐼|
|𝑈𝑈|×|𝐼𝐼|
Sparsity = 1 −𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷
GiniItem = 1 −2 ∑
|𝐼𝐼|+1−𝑖𝑖
|𝐼𝐼|+1 ×
|𝐼𝐼𝐼𝐼𝐼𝐼𝑖𝑖|
|𝐼𝐼𝐼𝐼𝐼𝐼|
|𝐼𝐼|
𝑖𝑖=1
GiniUser = 1 −2 ∑
|𝑈𝑈|+1−𝑢𝑢
|𝑈𝑈|+1 ×
|𝐼𝐼𝐼𝐼𝐼𝐼𝑈𝑈|
|𝐼𝐼𝐼𝐼𝐼𝐼|
|𝑈𝑈|
𝑢𝑢=1
These equations assume items and users are sorted according to
the number of interactions associated with them (IntI and IntU)
Table 1 Structural and Distributional Characteristics: Overview of unprocessed datasets showing total counts (#),
interactions (Int), users (U), and items (I), along with key characteristics (Avg. Int/User, Avg. Int/Item, shape, sparsity, Gini
coefficients, and feedback type). Abbreviations are as indicated.
Basic dataset characteristics are described by the total number of interactions (all user-item
events), the total number of users (all unique individuals) and the total number of items (all
unique entities available for interaction).
Space Size measures the capacity of the Interaction Matrix, allowing for the comparison of
datasets in terms of the maximum number of preferences that can be collected from users.
Shape indicates the user-to-item ratio, which can help foresee if item-based or user-based
algorithms might be advantageous.
Density measures the number of observed interactions in relation to the maximum number of
possible interactions.
Sparsity describes the proportion of missing interactions in a dataset by subtracting the
Density from 1.
Average Number of Interactions per User / Item describes the mean number of interactions a
user or an item has in this dataset.
GiniItem measures the rating frequency distribution for items, ranging from 0, which signifies
that all items receive the same number of interactions, to 1, which indicates absolute
inequality in interaction distribution, e.g., one item receiving all interactions.
GiniUser describes the rating frequency distribution for users, where 0 signifies that all users
contribute the same number of interactions in the dataset, and 1 represents that only one user
is the sole contributor of interactions.
4.2 Algorithm Selection
We selected a diverse set of algorithms from the LensKit [21] and RecBole [56] libraries.
Random and PopScore served as baselines. Biased MF and Implicit MF represented
traditional matrix-factorisation methods [48, 57], while User KNN [23] and Item KNN [19,
11
23] covered neighbourhood-based approaches. From RecBole, we included the BPR
algorithm [40], SimpleX [33], and several deep-learning recommenders chosen with a focus
on minimising energy consumption: DiffRec [50], DMF [52], and MultiVAE [32].
Hereinafter, we clustered the models into three groups.
Group 1 consists of traditional recommender algorithms: BiasedMF, ImplicitMF, UserKNN,
ItemKNN
Group 2 is composed of modern recommender models: DiffRec, DMF, MultiVAE, SimpleX
Group 3 holds a baseline from each library: PopScore, Random
Library
Models
LensKit
PopScore
Biased MF
Implicit MF
User KNN
Item KNN
RecBole
Random
BPR
DiffRec
DMF
MultiVAE
SimpleX
Table 2: Overview of Models and Libraries included in the Experiment
4.3 Training and Testing
In the first phase, the training and test sets were created from the respective core-pruned
dataset. That means that both the training data and the test data were core-pruned.
Unpruned
Dataset
Splitting
[80%, 20%]
Train Set
Unpruned
Test Set
Unpruned
Core-Pruned
Dataset
Splitting
[80%, 20%]
Core Train
Set
Core Test Set
Figure 1 Dataset Splitting for Phase 1 and 2
12
Figure 2 Training and Testing of Recommender Algorithms employed in 2 Phases
In the second phase, the recommenders were trained on data from the core pruned datasets,
while the test data was taken from the unpruned test set. To achieve this, a test set was created
from the unpruned data, which was disjoint from the core pruned training data and
downsampled to the same size as the core pruned test data. By excluding possibly overlapping
interactions and downsampling, a fair comparison of evaluation results on unseen data under
similar conditions was enabled.
This comparison procedure was executed for each core subset of each dataset.
All recommender algorithms were trained with their default parameter as set by their
respective library. The neighbourhood-based algorithms were trained with the maximum
number of neighbours set to 20, and the Matrix Factorisation algorithms, BiasedMF and
ImplicitMF, were trained with 50 features.
As the primary objective of this thesis was not to achieve the highest possible performance
scores, but to examine changes in rank across coresets, no hyperparameter tuning was applied.
Ranking-Based Metrics
The evaluation of Recommender algorithms’ performance on ranking tasks can be achieved
through a wide variety of metrics. Research surrounding ranking tasks typically employs at
least one accuracy metric, e.g. Recall or Precision, or nDCG@k in evaluation. nDCG
(normalised Discounted Cumulative Gain) measures how well relevant items are ranked near
the top of the recommendation list, with scores ranging from 0 for a poor list of
recommendations to 1 for a perfect list of recommendations with every item at the ideal rank.
Recall measures the proportion of all relevant items that appear in the list of
recommendations, indicating how completely the algorithm retrieves the items of interest.
Phase 1:
Train on Core Train Set
Test on Core Train Set
Create Unpruned Test Set
- equal size of Core Test
Set
- disjunct to Core Train Set
Phase 2:
Train on Core Train Set
Test on Unpruned Test Set
13
Precision reflects the proportion of recommended items that are actually relevant, assessing
the accuracy of the top-ranked results. When either Precision, Recall, or nDCG is evaluated at
“@k”, this means the top k recommended items are considered when computing the metric.
Due to the scope of this thesis, the results will only be reported on the NDCG@10 metric. The
results for Precision@10 and Recall@10 are included in the Appendix.
5. Results
5.1 Dataset Analysis
In this section, we will provide the results of analysing the dataset characteristics for all
datasets. As a baseline, we will first present the characteristics of the unprocessed datasets to
allow a comparison with the pre-processed, but unpruned, 0-core datasets. Subsequently, we
will show the dataset characteristics for each core-pruned version of the datasets.
5.1.1 Unprocessed Dataset Characteristics
Dataset
Interactions
Users
Items
Avg. Int
/ User
Avg. Int
/ Item
Shape
Sparsity
in %
Gini
User
Gini
Item
Feedback
Amazon CD
4458901
434060
1944316
10.27
2.29
0.22
99.9995
0.75
0.50 Explicit
Amazon Toys
7998969
624792
4204994
12.80
1.90
0.15
99.9997
0.80
0.40 Explicit
Gowalla
3981334
107092
1280969
37.18
3.11
0.08
99.9971
0.66
0.54 Implicit
MovieLens
FL
33832162
330975
83239
102.22
406.45
3.98
99.8772
0.70
0.95 Explicit
Yelp
5261667
1326101
174567
3.97
30.14
7.60
99.9977
0.63
0.72 Explicit
Table 3 Baseline Dataset Characteristics: Overview of the unprocessed datasets, reporting average interactions per user
and per item (Avg. Int / User; Avg. Int / Item), matrix shape (users × items), sparsity, Gini coefficients for users and items,
and feedback
Analysing the unprocessed datasets confirms MovieLens as the densest. The total number of
users exceeds the total number of items, and the average number of ratings, both per item and
per user, are the highest of all tested datasets. While the GiniUser value ranges in the mid ranks
across datasets, the MovieLens dataset shows the highest GiniItem value of 0.95, exhibiting the
highest inequality in rating frequency across items. This confirms the short head and long tail
conformation, with a few popular items receiving large numbers of interactions and a large
group of items receiving very few interactions.
14
The Amazon datasets are the sparsest in the experiment, exhibiting high levels of inequality in
rating frequency across users. Still, they have the lowest GiniItem values among the tested
datasets. In terms of the average number of interactions, both Amazon datasets rank in the
mid-range for interactions per user but show the lowest average number of interactions per
item in the experiment.
The Gowalla dataset displays the smallest Shape, with the total number of users being
exceeded by the total number of items. While the average number of interactions per user is
the second highest in the experiment, the average number of interactions per item is lower
compared to the other datasets.
Besides MovieLens, the Yelp dataset is the only one displaying high values for both Shape
and GiniItem, as well as the second-highest average number of interactions per item. In terms
of rating frequency per user, the Yelp dataset has a lower rating frequency than the compared
datasets, with a GiniUser of 0.63.
Comparing the characteristics of the unprocessed dataset to the downsampled, yet unpruned,
versions of the 0-Core datasets, the relative ranking based on these statistics remained largely
consistent.
15
5.1.2 Core-Pruned Dataset Characteristics
0-Core Pruning
Dataset
Interactions
Users
Items
Avg. Int. /
User
Avg. Int.
/ Item
Space Size
Shape
Sparsity
Gini
User
Gini
Item
Amazon CD
3000000
376115 1471494
7.98
2.04
553450965810
0.26
99.9995%
0.73
0.45
Amazon Toys
3000000
406436 1956864
7.38
1.53
795339976704
0.21
99.9996%
0.74
0.30
Gowalla
3000000
104439 1105866
28.72
2.71
115495539174
0.09
99.9974%
0.66
0.51
MovieLens FL
3000000
268342
34889
11.18
85.99
9362184038
7.69
99.9680%
0.64
0.91
Yelp
3000000
949214
163998
3.16
18.29
155669197572
5.79
99.9981%
0.57
0.73
5-Core Pruning
Dataset
Interactions Users
Items
Avg. Int. /
User
Avg. Int.
/ Item
Space Size
Shape Sparsity
Gini
User
Gini
Item
Amazon CD
2515024
107132 1308076
23.48
1.92
140136798032
0.08
99.9982%
0.60
0.42
Amazon Toys
2499239
101882 1698279
24.53
1.47
173024061078
0.06
99.9986%
0.59
0.28
Gowalla
2942493
77873 1089959
37.79
2.70
84878377207
0.07
99.9965%
0.57
0.51
MovieLens FL
2718416
132958
34416
20.45
78.99
4575882528
3.86
99.9406%
0.48
0.91
Yelp
1730318
129820
143934
13.33
12.02
18685511880
0.90
99.9907%
0.45
0.72
10-Core Pruning
Dataset
Interactions Users
Items
Avg. Int. /
User
Avg. Int.
/ Item
Space Size
Shape Sparsity
Gini
User
Gini
Item
Amazon CD
2180293
55519 1177234
39.27
1.85
65358854446
0.05
99.9967%
0.54
0.41
Amazon Toys
2197395
55611 1532748
39.51
1.43
85237649028
0.04
99.9974%
0.53
0.26
Gowalla
2830783
61579 1061525
45.97
2.67
65367647975
0.06
99.9957%
0.53
0.50
MovieLens FL
2377431
81663
34034
29.11
69.85
2779318542
2.40
99.9145%
0.40
0.90
Yelp
1214677
49057
129533
24.76
9.38
6354500381
0.38
99.9809%
0.41
0.71
20-Core Pruning
Dataset
Interactions Users
Items
Avg. Int. /
User
Avg. Int.
/ Item
Space Size
Shape Sparsity
Gini
User
Gini
Item
Amazon CD
1804688
27582 1019721
65.43
1.77
28125944622
0.03
99.9936%
0.47
0.38
Amazon Toys
1830233
28463 1325512
64.30
1.38
37728048056
0.02
99.9951%
0.47
0.24
Gowalla
2478446
36732
976549
67.47
2.54
35870597868
0.04
99.9931%
0.46
0.49
MovieLens FL
1826386
41403
33289
44.11
54.86
1378264467
1.24
99.8675%
0.32
0.89
Yelp
802206
17652
111692
45.45
7.18
1971587184
0.16
99.9593%
0.36
0.68
50-Core Pruning
Dataset
Interactions Users
Items
Avg. Int. /
User
Avg. Int.
/ Item
Space Size
Shape Sparsity
Gini
User
Gini
Item
Amazon CD
1268938
9860
775156
128.70
1.64
7643038160
0.01
99.9834%
0.39
0.34
Amazon Toys
1276790
10073
987503
126.75
1.29
9947117719
0.01
99.9872%
0.39
0.20
Gowalla
1776980
14295
793831
124.31
2.24
11347814145
0.02
99.9843%
0.40
0.45
MovieLens FL
894887
10512
31124
85.13
28.75
327175488
0.34
99.7265%
0.24
0.84
Yelp
412749
4317
85747
95.61
4.81
370169799
0.05
99.8885%
0.28
0.62
100-Core Pruning
Dataset
Interactions Users
Items
Avg. Int. /
User
Avg. Int.
/ Item
Space Size
Shape Sparsity
Gini
User
Gini
Item
Amazon CD
870960
4053
573131
214.89
1.52
2322899943
0.01
99.9625%
0.33
0.30
Amazon Toys
855740
3911
702727
218.80
1.22
2748365297
0.01
99.9689%
0.34
0.16
Gowalla
1156429
5210
601281
221.96
1.92
3132674010
0.01
99.9631%
0.36
0.39
MovieLens FL
348196
2329
27076
149.50
12.86
63060004
0.09
99.4478%
0.18
0.76
Yelp
200169
1182
61321
169.35
3.26
72481422
0.02
99.7238%
0.23
0.54
Table 4 Characteristics of the Coresets
16
5.1.3 Structural Characteristics
The following section provides a brief overview of the observed changes in structural
characteristics after pruning the datasets.
5.1.3.1
Density / Sparsity
Across all pruning levels, the ranking of datasets by their density remained largely unaltered
from the unpruned versions. 5-Core pruning merely switched the order of the rank-neighbours
Yelp and Gowalla, similarly to 50-Core pruning for Gowalla and Amazon CD. The density of
all datasets continued to increase with each Core-level.
5.1.3.2
Space Size
The Space Size continuously decreased for all datasets with each stage of pruning, with only
adjacently ranking datasets switching order for 10-core pruning and 50-core pruning.
5.1.3.3
Shape
Across all pruning levels, the Shape values of all datasets decreased continuously, with the
largest difference between the unpruned and 5-core pruned datasets reported.
5.1.3.4
Average Number of Interactions per Item
The average number of interactions per item varied for the unprocessed datasets, with values
ranging from 1.53 for Amazon Toys up to 85.99 for MovieLens. The span between the
maximum and minimum values stayed considerably large across all pruning levels, with the
rank of datasets by this characteristic entirely unchanged.
5.1.3.5
Average Number of Interactions per User
The ranking by Average Number of Interactions per User fluctuated notably more than per
Item. Pruning at the lower level of 5-core, the sparse Amazon Toys dataset, ranked fourth for
0-core, switched ranks with the formerly ranked second MovieLens dataset when ranked in
descending order by Average Number of Interactions per User.
20-Core pruning resulted in every dataset switching rank with an adjacent dataset, except for
the Gowalla dataset, which consistently displayed the highest average number of interactions
per user across all core levels. Pruning at thresholds above 50 relegated the MovieLens
dataset, which ranked second highest in the unpruned version, to the last rank of all datasets.
17
5.1.4 Distributional Characteristics
Figure 3: Boxplots showing the distribution of Gini Coefficients for users and items across pruning levels, where 1 signifies
total inequality in interaction distribution and 0 stands for a uniform distribution of interactions
5.1.4.1
GiniUser
All datasets had a GiniUser value of more than 0.5 for the downsampled but unpruned version,
with the Amazon datasets displaying the maximum values in the experiment of 0.7. For all
core-pruned versions, the highest values continued to be those of the Amazon datasets, and
MovieLens had the lowest values for all thresholds above 5. The Amazon and Gowalla
datasets showed similar decreasing trends and values across all pruning levels, as did Yelp
and MovieLens, but their GiniUser values were lower throughout all levels.
5.1.4.2
GiniItem
The values for GiniItem consistently stayed in the same order for all pruning levels, with
MovieLens displaying the maximum value of 0.9 for the unpruned version and 0.8 for the
100-core pruned version. While each was about 0.2 lower than MovieLens, both the Yelp and
Gowalla datasets showed a similar trend, reaching minimum values of 0.5 and 0.4,
respectively, in the 100-core version for GiniItem. Overall, the datasets displayed a drop of a
maximum of 0.2 from 0-core to 100-core pruned in terms of GiniItem, thus a much lower
fluctuation than GiniUser.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
Item Gini of 5 Datasets across 6 Core -Pruned
Versions
0-Core Pruning
5-Core Pruning
10-Core Pruning
20-Core Pruning
50-Core Pruning
100-Core Pruning
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
User Gini of 5 Datasets across 6 Core -Pruned
Versions
0-Core Pruning
5-Core Pruning
10-Core Pruning
20-Core Pruning
50-Core Pruning
100-Core Pruning
18
Relative Retention Rates
In this section, we provide an in-depth analysis of the impact of data pruning with increasing
core thresholds on dataset characteristics, with a focus on the extent to which the users, items,
and interactions of the unpruned dataset are retained in the pruned datasets.
5.1.4.3
User Retention Rate
Across all five examined datasets, pruning has affected the relative retention of users as early
as the first pruning level of 5-core. Only one dataset, namely Gowalla, retained more than half
of its users compared to the baseline, with a 75% relative retention rate. The lowest user
retention rate for 5-core pruning was observed for the Yelp dataset, retaining only 14% of its
user base after the first level of pruning was applied. Both Amazon datasets, CD & Vinyl and
Toys & Games, reported a relative user retention under 30%, with only a 3% difference
between them. The MovieLens dataset retained 50% of the users compared to the unpruned
version.
Applying 10-core pruning to the datasets only left the Gowalla dataset with more than half of
the users, while MovieLens dropped to less than a third of the original user-base, with 30% of
users retained. The remaining datasets reported retention rates below 15%, with Yelp
dropping as low as 5% relative user retention.
28%
15%
7%
3%
1%
25%
14%
7%
2%
0.96%
75%
59%
35%
14%
5%
50%
30%
15%
4%
0.87%
14%
5%
1.86%
0.45%
0.12%
5
10
20
50
100
CORE PRUNING APPLIED TO DATASETS
RELATIVE USER RETENTION
AMAZON CD & VINYL
AMAZON TOYS & GAMES
GOWALLA
MOVIELENS FL
YELP
Figure 4 User Retention Across Core Levels: Grouped bar chart showing, for each pruning level, the proportion of users
retained in each dataset relative to its 0-core (unpruned) set.
19
The commonly executed 20-core pruning reduced the user numbers for the MovieLens dataset
by half compared to the 10-core version, with a user retention rate of 15%. Both Amazon
datasets dropped to values below 10% and the Yelp dataset retained merely 1.86% of its
original user base. Gowalla continued to maintain the largest relative share of users
throughout the remaining core levels of 20, 50, and 100, with 35%, 14%, and 5%
respectively.
With the exception of the Gowalla dataset, pruning at the core level of 50 dropped all
remaining datasets to below 5% and pruning at the highest examined level of 100 dropped to
below 1% of users retained compared to the unpruned dataset. The lowest user retention rate
was examined for the 100-core pruned Yelp dataset, reporting 0.12% of the original user base
in the coreset.
5.1.4.4
Item Retention Rate
Figure 5 Item Retention Across Core Levels: Grouped bar chart showing, for each pruning level, the proportion of items
retained in each dataset relative to its 0-core (unpruned) set.
Pruning datasets affects the relative retention of items compared to the unpruned datasets.
Similar to the user retention rate, the item retention rate dropped continuously as the pruning
levels rose, but for items, the retention rates stayed higher than for users.
89%
80%
69%
53%
39%
87%
78%
68%
50%
36%
99%
96%
88%
72%
54%
99%
98%
95%
89%
78%
88%
79%
68%
52%
37%
5
10
20
50
100
CORE PRUNING APPLIED TO DATASETS
RELATIVE ITEM RETENTION
AMAZON CD & VINYL
AMAZON TOYS & GAMES
GOWALLA
MOVIELENS FL
YELP
20
Examining the 5-core pruned datasets, all datasets retained more than 85% of items compared
to the unpruned dataset, with MovieLens and Gowalla only dropping by 1% to 99% of item
retention. Both Amazon datasets and the Yelp dataset retained around 80% of items.
10-core pruning the datasets resulted in item retention rates dropping by between 1% for the
MovieLens dataset and 9% for both Amazon datasets, and the Yelp dataset.
The popular 20-core pruned versions of the datasets continued to show similar retention rates
for the two Amazon datasets and the Yelp dataset, dropping to just below 70% item retention.
The MovieLens dataset retained the most items with 95% item retention, and the Gowalla
dataset retained 88% of items.
Pruning with a core level of 50 left the Amazon datasets and the Yelp dataset with about 50%
of the original items. The Gowalla dataset reported a retention rate of 72% and the MovieLens
dataset dropped just below 90% of item retention.
The trend continued for Core 100: the Yelp and Amazon datasets dropped to rates between
36% and 39%, while Gowalla retained about half of the items compared to the unpruned
dataset. The MovieLens dataset showed an item retention rate of 78% for the highest level of
pruning examined in the study.
21
5.1.4.5
Interaction Retention Rate
Figure 6 Interaction Retention Across Core Levels: Grouped bar chart showing, for each pruning level, the proportion of
interactions retained in each dataset relative to its 0-core (unpruned) set.
Analysis of the retained interactions after pruning the datasets revealed differences in the
extent to which the datasets were affected by the pruning levels. For the first level of pruning
to core 5, the majority of datasets remained with more than 80% of interactions, with Gowalla
showing the highest interaction retention rate of 98%. The Yelp dataset was reduced by 42%
through 5-core pruning, retaining 58% of the interactions.
This trend continued for 10-core pruning: both Amazon datasets and the MovieLens dataset
retained around 70-80% of their interactions, while Gowalla retained the biggest relative
portion of 94% of interactions. Pruning the Yelp dataset with a threshold of 10 dropped the
interaction retention below the halfway mark, with 40% of interactions retained in the 10-
coreset.
Pruning with 20-core levels, all datasets dropped by between 10-20% of interaction retention
rates, with Yelp reaching as low as 27% and Gowalla retaining 83% of interactions.
The 50-core pruning had the most significant impact on the MovieLens dataset, resulting in a
30% drop in interaction retention, which led to lower retention rates compared to the Amazon
datasets, at 42% and 43%.
84%
73%
60%
42%
29%
83%
73%
61%
43%
29%
98%
94%
83%
59%
39%
91%
79%
61%
30%
12%
58%
40%
27%
14%
7%
5
10
20
50
100
CORE PRUNING APPLIED TO DATASETS
RELATIVE INTERACTION RETENTION
AMAZON CD & VINYL
AMAZON TOYS & GAMES
GOWALLA
MOVIELENS FL
YELP
22
Pruning with a threshold set to 100 only retained more than a third of the original interactions
in the Gowalla dataset and displayed retention rates below 15% for both the MovieLens and
Yelp datasets.
5.2 Algorithm Performance Analysis
This chapter is organised into three parts to provide a comprehensive overview of the results
obtained by training and testing 11 recommender algorithms on progressively pruned versions
of five datasets. The first section presents the overall performance trends observed for the
three algorithm groups of traditional, modern, and baseline algorithms. The second section
reports the results for each group when both training and testing were conducted on the
pruned datasets. The final section presents the outcomes of training performed on pruned
datasets while testing on unpruned datasets.
5.2.1 Overall Performance Trends
This section summarises the overall performance patterns observed for three algorithm groups
across all pruning levels, when trained and tested on pruned data. Group 1 consists of
traditional algorithms: User KNN, Item KNN, Biased MF, Implicit MF and BPR. Group 2
comprises the modern algorithms we have selected: DiffRec, DMF, MultiVAE, and SimpleX.
Group 3 comprises the two baseline algorithms chosen for the experiments: Random and
PopScore.
Figure 7 Average nDCG@10 Across Pruning Levels: Average nDCG@10 of traditional, modern, and baseline algorithms
on five datasets, trained and tested on equally pruned data.
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500
0
10
20
30
40
50
60
70
80
90
100
AVERAGE NDCG@10
CORE PRUNING APPLIED
AVERAGE NDCG@10 ACROSS PRUNING LEVELS
ON 5 DATASETS
(PRUNED TRAIN & TEST DATA)
Group 1 (Traditional)
Group 2 (Modern)
Group 3 (Baseline)
23
For the traditional algorithms (Group 1), the average nDCG@10 measured across five
datasets showed consistent values for 5-core and 10-core pruning, with an overall increase in
nDCG@10 for the 50-core and 100-core pruned versions of the datasets. The second group,
modern algorithms, on average, did not improve performance with progressively pruning the
datasets. The baseline algorithms in Group 3 overall showed almost no change in nDCG@10.
While the averaged nDCG@10 for the three groups displayed a maximum difference of 0.01,
the performance of the algorithms in these groups showed notable fluctuations.
Figure 8 Average Relative Change (%) in nDCG@10 vs. 0-core Baseline: Heatmap showing the percent change in
nDCG@10 at each pruning level relative to the unpruned (0-core) performance; positive values (blue) indicate improvement,
negative values(red) indicate decline.
Comparing the averaged nDCG@10 values per algorithm showed both positive and negative
relative changes compared to training and testing on the 0-core version of the datasets.
Implicit MF (+ 96%) and Random (+ > 100%) displayed high relative differences to their
results on the 0-coresets, while DiffRec had an average relative change of -80% for 100-core
pruned datasets. This variance in Average Relative Change highlights pronounced differences
between the reactions to pruning levels, prompting a more in-depth analysis of algorithm
performance. In the following subsection, we will analyse the performance of the algorithms
within the groups in more detail.
Algorithms
BiasedMF
100%
BPR
DiffRec
DMF
ImplicitMF
Item KNN
MultiVAE
PopScore
Random
SimpleX
User KNN
-100%
Cores
0
5
10
20
50 100
Average Relative Change in nDCG@10 across Pruning Levels
24
5.2.2 Training and Testing on Pruned Datasets
In this section, we will provide an overview of the algorithms’ performance, both trained and
tested on the datasets in the core-pruned versions (Phase 1). The algorithms are grouped into
three categories: Group 1, which includes traditional recommender algorithms such as User
KNN; Group 2, comprising modern algorithms, e.g., DiffRec and DMF; and Group 3,
consisting of the two included baselines: Random and PopScore.
5.2.2.1
Group 1: Traditional Recommender Algorithms Trained and Tested on Pruned
Data
Figure 9 Relative Change in nDCG@10 Across Pruning Levels for Traditional Algorithms: Percentage change in average
nDCG@10 of BiasedMF, ImplicitMF, UserKNN, ItemKNN, and BPR compared to their performance on the unpruned (0-
core) datasets.
Measuring the relative change in average nDCG@10 between each core version and the
baseline of 0-core, deviations from the baseline were reported for all included algorithms
(Figure 9).
Biased MF dropped by 6% for 5-core and 13% for 10-core pruning, when compared to the
average nDCG@10 achieved on the 0-core datasets. For 50-core and 100-core pruning,
Biased MF increased in average nDCG@10 by 23% and 77% respectively.
Implicit MF displayed negligible changes in performance for 5- and 10-core pruning (< 3%),
but improved performance for the remaining pruning levels of 20, 50 and 100. The highest
-20%
0%
20%
40%
60%
80%
100%
0
10
20
30
40
50
60
70
80
90
100
RELATIVE CHANGE IN AVERAGE NDCG@10 COMPARED
TO 0-CORE PRUNED DATASET
CORE PRUNING APPLIED
RELATIVE CHANGE IN NDCG@10 ACROSS PRUNING
LEVELS
GROUP 1: TRADITIONAL ALGORITHMS
Biased MF
Implicit MF
User KNN
Item KNN
BPR
25
measured values in relative change of this group were calculated for Implicit MF at 50-core
(52%) and 100-core (96%).
Across the five datasets, User KNN experienced an average decrease in nDCG@10 for 5-core
pruning (-3%) and 10-core pruning (-5%), while showing notable improvements in
performance for the 50-core (36%) and 100-core (70%) pruned datasets.
Item KNN displayed the highest increase for the first two pruning levels of 5 (7%) and 10
(13%), ranked second best for 50-core pruning (44%), but showed less improvement for 100-
core pruning than the majority of algorithms in the group (40%).
The final algorithm of Group 1, BPR, showed negligible improvement for the first two levels
of pruning, with under 5% improvement for both 5-core and 10-core pruning. The relative
change increased for 20-core pruning (17%), 50-core pruning (36%), and 100-core pruning
(21%).
5.2.2.2
Group 2: Modern Recommender Algorithms Trained and Tested on Pruned
Data
Figure 10 Relative Change in nDCG@10 Across Pruning Levels for Modern Algorithms: Percentage change in average
nDCG@10 of DiffRec, DMF, MultiVAE and SimpleX compared to their performance on the unpruned (0-core) datasets.
-100%
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
0
10
20
30
40
50
60
70
80
90
100
RELATIVE CHANGE IN AVERAGE NDCG@10 COMPARED
TO 0-CORE PRUNED DATASET
CORE PRUNING APPLIED
RELATIVE CHANGE IN NDCG@10 ACROSS PRUNING
LEVELS
GROUP 2: MODERN ALGORITHMS
DiffRec
DMF
MultiVAE
SimpleX
26
The second group of algorithms, comprised of the modern recommenders DiffRec, DMF,
MultiVAE, and SimpleX, deviated from the traditional recommender group in multiple
pruning stages.
Comparing the average nDCG@10 across five datasets, DiffRec improved by 25% relative to
the averaged performance on the 0-core versions of the datasets for 5-core pruning. The
second pruning level of 10-core yielded a 16% increase, while 20-core pruning displayed
negligible changes in performance (1%). For both 50-core and 100-core pruning, DiffRec’s
performance decreased by 52% and 80%, respectively.
In contrast to DiffRec, DMF displayed a decline in performance for pruning cores 5, 10 and
20, a negligible increase for 50-core pruning (< 5%) and overall improved most on the 100-
core pruned dataset by 48%.
MultiVAE was the only observed algorithm in this group to exhibit a continuous decline in
average nDCG@10 across all datasets for each pruning level. Even 5-core and 10-core
pruning decreased performance compared to the unpruned dataset by over 20%, and pruning
at levels equal to or higher than 50 reduced the performance by 50% on average.
As the final algorithm of this group, SimpleX displayed minor relative changes compared to
0-core datasets for 5-core and 10-core pruning (< -5%), but consistently increased in average
nDCG@10 for 20-core (16%), 50-core (56%) and 100-core (44%) pruned datasets. The
relative change for 20-core and 50-core was the highest relative improvement observed in this
group.
5.2.2.3
Group 3: Baseline Recommender Algorithms Trained and Tested on Pruned
Data
Group 3, containing only the two baseline algorithms PopScore and Random, displayed the
overall lowest nDCG@10.
Random only produced a measurable value for nDCG@10 on the MovieLens dataset for the
0-core and 5-core versions, as well as the Yelp dataset for the 10-core versions. Each
subsequent pruning stage added another dataset to the set of those generating a measurable
value for nDCG@10, resulting in relative improvements of 350% for 50-core and 850% for
27
100-core. Executing the Random algorithm on the Amazon Toys & Games dataset yielded no
observable value for nDCG@10 regardless of pruning intensity.
In contrast to Random, training and testing the PopScore algorithm achieved measurable
nDCG@10 values above zero across all core-pruned versions of all datasets. While
improvements were observed for the 5-core version of Gowalla, the overall nDCG@10
declined by 9% for the 5-core pruned versions of the datasets. Pruning at 10 and 20 decreased
the average performance by 18% and 17%, respectively, compared to the 0-core pruned
datasets. Only 100-core pruning achieved an improved average nDCG@10 (29%).
5.2.3 Comparison of Testing on Pruned and Unpruned Testsets
Building on the previous analyses, this section focuses on the setup for comparison. For
clarity, we will distinguish between two experimental phases: Phase 1, also represented by the
previous section of the Results chapter, refers to training and testing the algorithms on pruned
datasets. Phase 2 denotes the setting in which the training was executed on the pruned training
sets, whereas the testing was performed on the unpruned test sets. The 0-coresets were
excluded from Phase 2, as the training data in these coresets was not pruned.
To facilitate a direct comparison of the influence data pruning has on performance scores
under these two evaluation setups, a ranking of the algorithms by descending performance
(measured in nDCG@10) was created and visualised in a heatmap (Figure 11). Rank 1 (dark
blue) signifies the best performance, as measured by the average nDCG@10 of all algorithms
on this particular coreset of the respective dataset. In contrast, Rank 11 (dark red) marks the
lowest measured performance score.
28
Figure 11 Algorithm Ranking Across Two Evaluation Phases: Two-part heatmap showing algorithm ranks (based on
nDCG@10) across five datasets and six pruning levels. The upper panel depicts Phase 1 (training and testing on equally
pruned data), while the lower panel shows Phase 2 (training on pruned data and testing on unpruned data).
Datasets
Cores
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
Algorithms
BiasedMF
Rank 1
BPR
DiffRec
DMF
ImplicitMF
Item KNN
MultiVAE
PopScore
Random
SimpleX
User KNN
Rank 11
Cores
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
Datasets
Cores
5
10
20
50 100
5
10
20
50 100
5
10
20
50 100
5
10
20
50 100
5
10
20
50 100
Algorithms
BiasedMF
Rank 1
BPR
DiffRec
DMF
ImplicitMF
Item KNN
MultiVAE
PopScore
Random
SimpleX
User KNN
Rank 11
Cores
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
0
5
10
20
50 100
Ranking of Algorithms by NDCG@10 (Trained on Pruned and Tested on Unpruned Data)
Ranking of Algorithms by NDCG@10 (Trained and Tested on Pruned Data)
Amazon CD
Amazon Toys
Gowalla
MovieLens
Yelp
Amazon CD
Amazon Toys
Gowalla
MovieLens
Yelp
29
In Phase 1, User KNN ranked highest for the entirety of three datasets, on several coresets of
the remaining two datasets (MovieLens, Yelp) and displayed the highest consistency in
ranking of both traditional and modern algorithms. On the Yelp dataset, User KNN was
outperformed by Implicit MF on three coresets: 5-core, 10-core and 20-core. The MovieLens
dataset exhibited more fluctuations for the higher ranks, with User KNN ranking third or
fourth on all coresets above 5-core, and Item KNN rising to first rank for 5-core, 10-core, and
20-core pruning. DiffRec ranked second for the 5-core and 10-core subset, and DMF took the
lead for the 100-core pruned subset of MovieLens. PopScore ranked highest on the 50-core
subset of MovieLens.
Phase 1 was characterised by consistent top rankings of User KNN, Item KNN, and Implicit
MF, with Item KNN and Implicit MF exhibiting rank fluctuations primarily across entire
datasets and to a lesser extent on singular coresets.
Algorithms at mid-level ranks displayed a higher frequency of rank changes, often within a
dataset. One example of this behaviour would be MultiVAE across the MovieLens dataset,
where its rank fluctuated between fourth and ninth place. DiffRec achieved varying ranks
both across entire datasets and coresets, with ranking position changing from second to as low
as tenth across the Yelp dataset and almost a different rank for every coreset on MovieLens
(only 10-core and 20-core ranked second).
The only algorithm without any fluctuations in ranking in Phase 1 was Random, which
continuously ranked last (11th out of 11 algorithms). Almost no variation in rank was also
observed for Biased MF, which ranked 10th on all but three coresets in Phase 1.
In summary, rank changes were observed frequently in Phase 1, both per dataset (e.g.,
SimpleX on MovieLens) and per coreset, as evidenced by PopScore for both MovieLens and
Yelp.
In comparison to Phase 1, the top contender User KNN displayed both more frequent and
more extreme rank changes in Phase 2: While the lowest rank in Phase 1 was a singular fourth
rank on 50-core pruned MovieLens, User KNN held a rank equal to or below sixth for 48% of
coresets in Phase 2 (12 of 25). As the second neighbourhood-based algorithm, Item KNN also
30
consistently showed a lower rank than in Phase 1, ranking overall eighth for both entire
datasets and achieving its best ranking of fifth.
Implicit MF, which even ranked first for three coresets in Phase 1, mostly ranked fifth to
seventh, and on the 20-core subset of MovieLens, even ranked ninth. Biased MF was ranked
10th for 68% (17 of 25) of coresets in Phase 2, with a slightly higher frequency of rank
changes on Amazon Toys & Games and the Yelp dataset.
BPR was observed to rank fourth for 68% (17 of 25) in Phase 1, compared to ranking second
for 60% of coresets in Phase 2. On the 100-core pruned subset of Yelp, BPR even ranked best
of all algorithms.
MultiVAE exhibited difficulties in recommending on the Amazon Toys dataset in Phase 2,
consistently ranking 10th place. While the performance was in the lower ranges for the
Amazon datasets, on the remaining three datasets, MultiVAE consistently outperformed the
majority of algorithms: On Yelp, MultiVAE held the second rank for 50-core and 100-core,
on MovieLens for all but 50-core, and on Gowalla, the rank varied between fourth and fifth.
DiffRec placed higher in the ranks in Phase 2 compared to the first test phase, still overall
performing best on Yelp and MovieLens, as it did in Phase 1. Rank changes were observed
less frequently compared to Phase 1. DiffRec performance notably declined with each
progressive pruning stage on the Amazon Toys & Games dataset, ranking fifth on 5-core,
sixth on 10-core and 20-core, and ninth on 50-core and 100-core. However, the absolute
nDCG@10 values were observed to be higher in Phase 2 than in Phase 1. On the 5-core
pruned dataset, DiffRec achieved a nDCG@10 of 0.0153 in Phase 1, compared to 0.1155 in
Phase 2.
DMF was the only algorithm showing improvements in rank by performance on the more
severely pruned version of MovieLens. Contrary to the behaviour examined for the remaining
algorithms, DMF achieved a higher rank for each core level on MovieLens, with a first rank
for both the 50-core and 100-core models. On Amazon Toys & Games, no rank variation was
observed for DMF, which consistently ranked third best for all coresets.
SimpleX achieved an undisputed first rank on three datasets (both Amazon and Gowalla),
ranked between fifth and sixth on all coresets of MovieLens and showed minor fluctuations
within the top 3 ranks on the Yelp coresets. Its average nDCG@10 across all coresets was
31
1276% higher than in Phase 1, and the absolute nDCG@10 values were the highest observed
in the entire experiment.
Average nDCG@10 across 5 Pruning Levels (Trainset Pruned, Testset Unpruned)
Cores
5
10
20
50
100
Biased MF
0.0001
0.0001
0.0002
0.0001
0.0001
Implicit MF
0.0013
0.0013
0.0020
0.0014
0.0014
User KNN
0.0018
0.0018
0.0025
0.0019
0.0017
Item KNN
0.0010
0.0011
0.0018
0.0010
0.0009
BPR
0.0628
0.0509
0.0364
0.0213
0.0136
DiffRec
0.1155
0.0705
0.0194
0.0043
0.0022
DMF
0.0078
0.0073
0.0056
0.0059
0.0048
MultiVAE
0.0102
0.0087
0.0087
0.0045
0.0041
SimpleX
0.2216
0.1702
0.1123
0.0540
0.0253
PopScore
0.0006
0.0005
0.0011
0.0005
0.0003
Random
0.0001
0.0000
0.0000
0.0001
0.0000
Table 5 Average nDCG@10 of Algorithms over 5 Datasets Across 5-Core Pruning Levels in Phase 2, with the highest
overall values in the experiment underlined
In summary, the traditional algorithms in Group 1 overall displayed a lower performance in
Phase 2 compared to Phase 1, while the modern algorithms in Group 2 partially exhibited
substantial increases in measured nDCG@10 values. Notably, SimpleX and DiffRec showed
an improved performance under the second test configuration when tested on unpruned test
sets. In comparison with the 5-core baseline in Phase 2, each additional pruning level resulted
in a notable performance decrease observed for all Group 2 (Modern) algorithms. The
Interpretation and Discussion chapter will elaborate on this particular finding.
32
6. Discussion and Interpretation
6.1.1 Dataset Characteristics
In this section, we examine how progressive pruning influenced key characteristics of the
datasets. We begin by analysing the impact of data pruning on structural properties, e.g.
density and the average number of interactions per user and per item, as well as distributional
characteristics, such as the Gini coefficients of users and items. Secondly, we focus on
retention rates, discussing the extent of data loss observed across increasing levels of pruning.
6.1.1.1
Structural Dataset Characteristics
Our analysis of the datasets' characteristics revealed that pruning affected all of the examined
datasets, though to varying extents. The removal of users and their interactions from a dataset
resulted in an unsurprising decrease in the total number of interactions, users, and items for
each core-level pruning applied to the datasets. Subsequently, structural characteristics
dependent on at least one of these three total numbers were equally impacted by pruning:
Space Size, which describes the maximum possible interactions calculated as users × items,
and Shape, denoting the relation of the total number of users to the total number of items,
continuously decreased with each pruning stage.
Figure 12 Average nDCG@10 of Traditional and Modern Algorithms Across Datasets: Average nDCG@10 for five
datasets at different core-pruning levels when training and testing are both performed on equally pruned data.
Density, a key factor in explaining performance variance [1], was observed to increase with
each core level across all examined datasets in our study. While the sparser datasets, Amazon
CD and Amazon Toys, showed improved performance for both the traditional and baseline
0.0000
0.0050
0.0100
0.0150
0.0200
0.0250
0.0300
0.0350
0.0400
0
10
20
30
40
50
60
70
80
90
100
AVERAGE NDCG@10
CORE PRUNING APPLIED
AVERAGE NDCG@10 OF TRADITIONAL & MODERN
ALGORITHMS ACROSS DATASETS
(PRUNED TRAIN & TEST DATA)
Amazon CD
Amazon Toys
Gowalla
MovieLens
Yelp
33
algorithms on coresets with higher density, this was not the case for MovieLens. Being the
densest dataset in the experiment, MovieLens exhibited distinct behaviour: despite increasing
density with progressive pruning, both groups of traditional and modern algorithms showed
no performance gains. On the contrary, for most core levels, declining performance was
observed on the MovieLens datasets as seen in Figure 12. These findings indicate that density
alone cannot account for the performance variance observed under progressive pruning.
Researchers have recently identified "Interactions per User" as another key dataset
characteristic when examining performance variance [18]. In our experiment, the Yelp dataset
showed the most significant rise in Average Interactions per User, increasing by 322% (from
3.16 to 13.33) for 5-core pruning up to 5258% (from 3.16 to 169.35) for 100-core pruning. In
Phase 1, the average nDCG@10 indeed increased for the majority of algorithms alongside the
Average Interactions per User on the Yelp dataset, with a noticeable drop in performance for
100-core pruning. A similar trend was observed on the Gowalla dataset, which showed the
overall smallest increase in Interactions per User – from 32% (from 28.72 to 37.79) at 5-core
to 673% (from 28.72 to 221.96) at 100-core pruning. Algorithm performance did not decline
even for the 100-core subset of Gowalla, suggesting that additional factors influence both the
dataset characteristics and algorithm performance.
The impact of pruning on Average Interactions per Item was less pronounced than on
interactions per user, but differed considerably among datasets. At 5-core pruning, Gowalla
remained unaffected, while Yelp displayed a 34% (from 18.29 to 12.02) reduction. Across all
pruning levels, MovieLens and Yelp experienced the most potent effects, with decreases of up
to 85% (from 85.99 to 12.86) and 82% (from 18.29 to 3.26), respectively, for 100-core
pruning. This division between moderately and heavily affected datasets aligns with changes
in Space Size: MovieLens and Yelp consistently exhibited the lowest values, whereas
Gowalla and both Amazon datasets retained much higher portions of Space Size across
pruning levels.
Analysis of GiniItem divided the datasets into the same two groups. Amazon CD, Amazon
Toys, and Gowalla consistently exhibited lower GiniItem values, indicating a pattern closer to
equal interaction distribution across items. In contrast, MovieLens and Yelp displayed the
highest and second-highest GiniItem values at every pruning level, denoting a highly uneven
distribution of interactions. MovieLens, in particular, is known for a short-head, long-tail
34
distribution, where a few popular items attract many interactions, while most items receive
little interaction [8, 18, 27]. This trait is reflected in its GiniItem scores. Overall, GiniItem scores
varied little across pruning stages and preserved the dataset ranking observed for the 0-
coresets through to the 100-coresets.
GiniUser exhibited trends similar to GiniItem, with scores decreasing at each pruning level.
However, the difference between unpruned datasets and the 100-coresets was substantially
larger than the corresponding GiniItem differences. The most significant decline was observed
for MovieLens, where the 100-coreset showed a drop of -0.46 compared to the 0-coreset. All
other datasets followed the same pattern, though less markedly, with GiniUser reductions
ranging from -0.24 to -0.40. This outcome aligns with expectations: progressively excluding
low-activity users leaves a population of generally high-activity users, thereby driving the
interaction distribution closer to uniformity.
In summary, pruning affected all structural characteristics of the datasets, though not in
uniform severity. Density increased for all datasets; however, its link to performance was
inconsistent, as demonstrated by the declining performance scores on the already dense
MovieLens dataset. Interactions per User overall increased for each dataset per core level, yet
yielded varying outcomes as displayed in the comparison of Yelp and Gowalla: Gowalla, with
the lowest relative increase in interactions per user, showed a steady rise in algorithm
performance, while Yelp (highest relative increase in interactions per user) declined after 50-
core pruning. Additional properties, such as the Gini measures for items and users,
highlighted the impact on user-centred characteristics, while showing that item characteristics
are affected to a lesser extent.
6.1.1.2
Distributional Characteristics: Retention of Interactions, Items and Users
Pruning affected the datasets’ structural properties to differing degrees, while similar trends of
varying intensity were frequently observed. A comparable pattern emerged examining the
retention rates of interactions, items, and users across pruning levels. In this context, retention
is defined as the proportion (in per cent) of the original dataset that is preserved in a coreset.
User retention, therefore, describes the percentage of users retained in a coreset compared to
the 0-core baseline, and this definition applies equally to item and interaction retention.
Among the three retention measures, user retention showed the most pronounced variation,
differing substantially across both entire datasets and core levels. In our analysis, we observed
35
that the impact of data pruning on datasets varies already at low core levels: 5-core pruning
revealed a gap of 61% between the datasets with the highest and lowest user retention rates.
Sun et al. [47] reported 5 and 10 as common core-levels observed for 60% of experiments
applying data pruning. Comparing the Gowalla and the Yelp datasets side by side revealed
how varying the impact of each core level can be on two different datasets (Figure 13).
Pruning Yelp at 5-core removed 86% of users, while retaining 58% of interactions.
Meanwhile, Gowalla marked a 25% reduction in users, losing merely 2% of total interactions
in the dataset. 5-core pruning removed 86% of Yelp’s users while preserving 58% of its
interactions, whereas Gowalla experienced a 25% loss in users, retaining 98% of its
interactions. This difference in effect stems from structural differences between the two
datasets: Yelp contains a significantly higher number of users than items, with the lowest
GiniUser among the unpruned datasets. In contrast, Gowalla’s GiniUser is insignificantly higher,
but it contains more items than users. Therefore, pruning low-activity users removes a smaller
share of the total interactions for Gowalla, causing a steadier decline in user retention across
pruning levels.
Figure 13 User Exclusion Rates Across Pruning Levels for Gowalla and Yelp: Waterfall charts showing the percentage of
users removed at each core-pruning level (5, 10, 20, 50, 100) for the Gowalla (left) and Yelp (right) datasets, illustrating
their contrasting user-retention patterns per level.
The use of 20-core pruned datasets (e.g., MovieLens 100k, MovieLens 1M, …) is a common
design choice in recommender research, but Beel and Brunel [8] already established in 2018
that this results in the exclusion of 42% of users compared to an unpruned version of the
dataset. In our experiments, this rate was considerably higher: the examined datasets
25.44%
15.60%
23.79%
21.48%
8.70%
5
10
20
50
100
Core Pruning Applied
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Percent of Users Excluded
EXCLUSION RATE OF USERS ACROSS 5 PRUNING
LEVELS ON GOWALLA
86.32%
8.51%
1.40%
0.33%
3.31%
5
10
20
50
100
Core Pruning Applied
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Percent of Users Excluded
EXCLUSION RATE OF USERS
ACROSS 5 PRUNING SETTINGS ON YELP
36
experienced user loss ranging from 65% for Gowalla, over 93% (both Amazon datasets), up
to 98.14% (Yelp) when 20-core pruning was applied. Even the popular MovieLens dataset
retained merely 15% of users and 61% of interactions, highlighting the uneven distribution of
interactions per user.
At the highest level of pruning examined in our experiments, 100-core, a singular dataset
(Gowalla) retained more than 1% of the users compared to the 0-coreset, while
simultaneously retaining almost 40% of interactions and 54% of items. Whereas 100-core
pruning may not be a common design choice, it highlights how far a dataset can be pruned
while still retaining almost half of the interactions and items.
Data Pruning is acknowledged as a process that can reintroduce bias [22, 54], even if the
original dataset was bias-free. Comparing algorithm performance on a dataset heavily altered,
for example, by 50-core or 100-core pruning, would inevitably lead to results skewed in
favour of high-activity users. Recommendations for cold-start users are known to be
challenging [30], but a comparison of algorithm performance on these users is typically
avoided by using pruned datasets. The impact of including cold-start users in evaluation
scenarios will be discussed in more detail in the second part of this chapter.
In summary, we analysed the impact of data pruning even at low levels on the structural and
distributional characteristics. We found that while item-centred characteristics were affected,
it was to a lesser extent than user-centred characteristics. Overall, datasets reacted to the same
levels of pruning in varying intensities; for example, Yelp experienced an 86% loss of users
compared to Gowalla’s 25% loss when 5-core pruning was applied. We observed that both
structural and distributional characteristics were substantially changed by data pruning.
6.1.2 Algorithm Performance
The preceding analysis of dataset characteristics provides essential context for interpreting
variance in algorithmic performance. Understanding how progressive pruning altered
structural and distributional properties clarifies the conditions under which the algorithms
were evaluated. In the following section, we build on these findings to examine how altered
dataset characteristics influenced the performance of the tested algorithms.
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In our experiment, we employed two complementary evaluation strategies to examine
algorithm performance on progressively pruned datasets. In Phase 1, algorithms were trained
and tested on equally pruned train and test sets, reflecting the common practice of using
pruned datasets throughout recommender system research. In Phase 2, the algorithms were
trained on the same training sets, while evaluation was conducted on test sets derived from
unpruned data. In the discussion that follows, we first examine the results from Phase 1 and
then compare them with those from Phase 2, highlighting what these contrasts reveal about
the implications of training and testing on pruned datasets.
6.1.2.1
Phase 1: Training and Testing on Pruned Datasets
According to Beel and Brunel [8], the key question when examining the effects of data
pruning is whether pruning leads to changes in the performance ranking of algorithms. During
Phase 1, we observed that the performance ranks of most algorithms varied across the
progressive pruning levels. The Random algorithm represented the sole exception, ranking
last across every coreset examined in Phase 1, with multiple 0-values observed for average
nDCG@10.
PopScore, as the second baseline algorithm, displayed improvements for higher levels of
pruning, especially on denser datasets, such as MovieLens. In one case, on the 50-core pruned
MovieLens dataset, PopScore even outperformed all remaining algorithms. The dataset’s
inherent structure can likely explain this ranking. As shown in the previous section,
MovieLens contains a small number of very popular items with which many users have
interacted. Because PopScore recommends items based on popularity, its improved
performance on extensively pruned data is not unexpected. Examining the characteristics of
50-core pruned MovieLens reveals conditions particularly favourable towards a popularity-
based algorithm: both the item retention rate and the GiniItem were observed to be much higher
for MovieLens than for the compared datasets after 50-core pruning.
In general, we observed rank changes to be more frequent and more extreme in two datasets:
the MovieLens and Yelp datasets. The first indication that these datasets differed was that
User KNN, which ranked undisputedly first on the remaining three datasets across all cores,
only experienced rank changes on MovieLens and Yelp. Analysing the absolute nDCG@10
values for these datasets, however, paints a different picture: Even the Yelp dataset’s
progressive pruning led to an increase in absolute nDCG@10 values with every core level,
38
whereas the performance on the MovieLens dataset declined with each successive core level.
On Yelp pruned to cores 5 to 20, User KNN consistently ranked second, supported by a
steady rise in nDCG@10 scores per core, and was surpassed only by Implicit MF, which
achieved its peak performance on these specific coresets in Phase 1.
Structural characteristics of the MovieLens coresets account for the rank changes observed
between the two neighbourhood-based approaches, Item KNN and User KNN. For the
unpruned dataset (0-core) and the heavily pruned 100-coreset, User KNN outperformed Item
KNN, whereas the medium levels of pruning favoured Item KNN. This observation can be
attributed to the structural characteristics observed when pruning the MovieLens dataset: the
long-tail item distribution in the unpruned version makes recommending based on item
similarity difficult, as many items receive very few interactions, leading to better performance
for User KNN over Item KNN. Because the 100-coreset retains only users with extensive
interaction histories, it provides ideal conditions for a user-similarity algorithm and therefore,
explains User KNN’s advantage on this coreset.
The Matrix Factorisation-based algorithms, Biased MF and Implicit MF, both increased in
performance scores as pruning levels rose, exhibiting the most significant performance
improvement compared to the 0-core datasets on average on the 100-coreset. This behaviour
is not surprising, as it has been acknowledged that algorithms employing Matrix Factorisation
display performance improvements as Sparsity decreases [46]. As we have shown in the
analysis of the coreset characteristics, Density increased with each level of pruning applied,
thus Sparsity (𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆= 1 −𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷) decreased and performance for Biased MF and
Implicit MF improved.
BPR profited to an extent of rising density by progressive pruning, exhibiting enhanced
performance scores with each core-level for Gowalla and both Amazon datasets, though to a
lesser degree for Amazon Toys. The performance on the denser datasets MovieLens and Yelp
dropped sharply at 100-core pruning. For example, BPR’s score declined by 44% (from 0.025
to 0.014) compared to the 50-core level when trained and tested on the 100-coreset. This
decline aligns with the sharp reduction in interactions both datasets experienced when pruned
to 100 cores, as neither retained more than 15% of interactions at this pruning level.
Interaction loss poses a disadvantage for algorithms like BPR, which depend on positive-
negative item pairs to learn preferences [37, 40].
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Overall, we observed positive trends for the group of traditional algorithms under
progressively increased pruning conditions. The group containing the modern algorithms
displayed less homogeneous behaviour across pruning levels, with individual algorithms
reacting differently to data reduction. For example, DiffRec responded to low pruning levels
with an average initial performance improvement of 25% (5-core) and 15% (10-core)
compared to the unpruned baseline. Further pruning continuously diminished this effect to the
point of 100-core pruning, exhibiting an average performance decline of 80%.
MultiVAE exhibited a similar trend to DiffRec, with a decrease in performance scores across
higher pruning levels in response to increased pruning intensity. Compared to the rest of the
modern algorithms, MultiVAE reacted most sensitively to even low-level pruning, showing
performance declines of 22% and 27% at the 5-core and 10-core levels, respectively. We
observed that MultiVAE ranked higher on the unpruned versions of Yelp and MovieLens,
with a sharp decline on Yelp as early as when 5-core pruning was applied. This correlates
with the data loss Yelp experienced when 5-core pruning was applied: relative user retention
measured 14%, while interaction retention dropped to 58%. MultiVAE achieved higher
performance scores on the dense datasets; however, it exhibited a drastic decline in
performance with data reduction.
DMF appeared less sensitive to data reduction, with only minor changes in performance
compared to the unpruned baseline; however, the MovieLens dataset elicited steeper declines
across pruning levels. Although the highest score on the MovieLens coresets was obtained
when training and testing on the unpruned version, a comparison of the 50-core and 100-core
results showed a performance improvement. This trend was observed for all examined
datasets and resembled the behaviour of the traditional Matrix Factorisation algorithms.
Overall, DMF improved on average by 48% on the 100-coresets.
SimpleX responded differently to 100-core pruning for the MovieLens and Yelp datasets than
DMF. For MovieLens, a steady decline was observed across core levels. SimpleX achieved
the highest performance score on Yelp when 50-core pruning was applied. When 100-core
pruned, the Amazon datasets and Gowalla yielded the highest nDCG@10 values for SimpleX.
This division between the two dataset groups, contrasting in density and shape, resulting in
differing scores for algorithms, was a recurring observation throughout the first phase of the
experiment.
40
To summarise, we observed the algorithm groups to react differently to data pruning,
resulting in various rank changes both for algorithms performing better on entire datasets as
well as rank changes across pruning levels for a singular dataset. The baseline algorithms
improved towards the higher levels of pruning, while the modern algorithms tended to react
more sensitively to data reduction. This aligns with the commonly accepted assumption that
deep learning algorithms perform better when the amount of data increases [45]. In contrast,
the traditional algorithms showed an overall performance improvement with each level of
pruning applied. Additionally, we observed that the traditional algorithms, on average,
outperformed the group of modern algorithms across all core levels.
6.1.2.2
Phase 2: Training on Pruned Datasets, Testing on Unpruned Datasets
We demonstrated, through the examination of Phase 1, that training and evaluating algorithms
on pruned versions of datasets can lead to performance rank changes between algorithms,
both across different pruning levels and across entire datasets. In Phase 2, we trained the
algorithms on the same training sets as in Phase 1, but the evaluation was performed on test
sets derived from an unpruned dataset. To enable a fair comparison, these test sets had the
same size as the pruned test sets from Phase 1, but included low-activity users. Furthermore, it
was ensured that the test sets only included unseen data by choosing disjoint subsets of the
training sets.
The baseline algorithms, PopScore and Random, exhibited notably different behaviour
compared to Phase 1, when evaluation included recommendations for low-activity users. For
instance, PopScore’s overall positive trend, especially pronounced at core levels 20 and
above, flattened almost entirely, showing only a slight increase for the 20-core pruned
MovieLens dataset. Random fared even worse: on three datasets (both Amazon and Gowalla),
it failed to produce a single non-zero average nDCG@10 value.
The majority of traditional algorithms (80%) experienced a similar trend as the baseline
algorithms in Phase 2. While there were minimal differences in average nDCG@10 for the
datasets, the overall trend was the same: the algorithms only improved their performance on
the 20-core pruned MovieLens training set. Some of the algorithms in this group showed
slight nDCG@10 increases on individual coresets when compared to the lowest core-level (5-
core), but the absolute differences never exceeded 0.0007 and are thereby deemed
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insignificant. Compared to the performance scores achieved in Phase 1 on the pruned test sets,
the traditional algorithms (Biased MF, Implicit MF, Item KNN, User KNN) suffered a decline
of an average 95% across the pruning levels.
The only exception to this decline was BPR, which overall performed better under the
evaluation conditions in Phase 2 and displayed an average increase in nDCG@10 of 143%
across all pruned training sets when evaluated on unpruned test sets. Compared to Phase 1,
BPR’s improvement in the performance-based ranking of algorithms was only second to
SimpleX: on the entirety of all coresets for both Amazon datasets and the Gowalla datasets,
BPR ranked in second place in Phase 2. The lowest rank we observed was for the MovieLens
datasets, when pruned to 10 cores and higher, where BPR ranked 4th. For the highly
challenging 100-core pruned Yelp dataset, BPR performed best of all algorithms, displaying a
stronger tolerance towards heavy data reductions than the remaining algorithms. While BPR
did not respond to data reduction as strongly, comparing the nDCG@10 values across the
pruning levels in Phase 2 still showed a decrease as the pruning intensity increased. Applying
10-core pruning (0.051) lowered the average nDCG@10 by 19% compared to 5-core pruning
(0.063), and 100-core pruning caused an even steeper decline of 78%, down to 0.014. The
steady decline shows that BPR’s performance was progressively hindered as the pruning
levels increased.
Figure 14 Figure X – Average nDCG@10 Across Pruning Levels with Pruned Training and Unpruned Test Data:
Average nDCG@10 for traditional, modern, and baseline algorithms across five datasets when models are trained on pruned
datasets but evaluated on unpruned test sets.
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0
10
20
30
40
50
60
70
80
90
100
AVERAGE NDCG@10
CORE PRUNING APPLIED
AVERAGE NDCG@10 ACROSS PRUNING LEVELS
ON 5 DATASETS
(PRUNED TRAIN & UNPRUNED TEST DATA)
Group 1 (Traditional)
Group 2 (Modern)
Group 3 (Baseline)
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The modern algorithms achieved much higher absolute performance scores under Phase 2
evaluation compared to the traditional algorithms. Across all examined coresets, the
traditional algorithms attained a mean nDCG@10 of 0.010, while the modern algorithms
achieved a mean of 0.038. All modern algorithms exhibited a declining performance trend as
pruning levels increased (Figure 14), though their absolute nDCG@10 in some cases were
higher than the corresponding Phase 1 values. For example, DiffRec achieved values for 5- to
20-core pruning, which were an average 370% higher than the scores for the corresponding
training sets in Phase 1. However, increasing the pruning level from 5-core to 10-core resulted
in a 39% drop in mean nDCG@10 (0.116 to 0.071), and 100-core pruning even dropped by
98% to 0.002.
It is worth mentioning that training SimpleX on pruned and testing it on unpruned data
yielded the highest observed nDCG@10 scores in the entire experiment. On both Amazon
and the Gowalla datasets, SimpleX was the only algorithm we observed to reach mean
nDCG@10 values above 0.25, with a maximum score attained on the Amazon Toys dataset
pruned at core-level 5 (0.49). Nonetheless, a declining trend in these scores was observed as
pruning intensified, which demonstrates the detrimental impact that progressive data pruning
had on the modern algorithms examined in this thesis.
Overall, the findings contrasted sharply with those of Phase 1, as pruning the datasets did not
yield any improvement in mean performance for either algorithm group. This aligns with the
claim made by Beel and Brunel [8] in their initial study: Data Pruning artificially inflates the
performance score when evaluated on test sets that have also undergone pruning. Algorithms
previously experiencing upward trends on the pruned test sets did not display the same
improvements on the unpruned test sets. These findings suggest that the improved
performance on progressively pruned datasets stems from the test set composition, as the test
sets were the only element that differed between the two evaluation settings.
The test sets in Phase 1 were derived by splitting the pruned core datasets into a train and test
set, to resemble the usage of a ready-to-use pruned dataset (e.g. most of the MovieLens
datasets). If algorithm evaluation is conducted on such a dataset, the test set will underly
similarly altered structural and distributional characteristics as the dataset it was split off
from. As we have shown in the analysis of the dataset characteristics, pruning users according
43
to their number of interactions in a dataset can lead to coresets that differ substantially in
composition and structural characteristics.
Taken together, these observations allow us to return to the core research question and
examine it in light of the results.
RQ: How does progressive dataset pruning by increasing core thresholds affect the
performance of recommender systems, the structural and distributional characteristics of
the dataset?
In order to respond to the first part of the question, it is necessary to consider the second part
first. We have examined a variety of structural and distributional characteristics across pruned
coresets and found all analysed characteristics to be altered by progressively pruning, though
not to the same degree. Pruning users according to their total number of interactions in the
datasets had a decreasing effect on the total number of items and interactions in the pruned
coresets, and consequently, on all characteristics that depend on at least one of these total
numbers. The continuous decrease in these characteristics with rising pruning levels was
unsurprising: what stood out was the variability in dataset responses.
Structural and distributional characteristics were affected unevenly by pruning levels, but the
clearest difference appeared in the retention of users, items and interactions at each stage
(Figure 13). For instance, as early as just 5-core pruning, the gap between the highest user
retention (Gowalla) and the lowest (Yelp) exceeded 60 percentage points. This consequently
resulted in vastly different compositions for the core-pruned datasets, as some datasets
retained much higher rates of their interactions than others. Even though all datasets had the
same number of interactions in the 0-coresets, merely 5-core pruning led to almost halving the
dataset size for individual datasets (Yelp), while nearly retaining the full number of
interactions (Gowalla, MovieLens).
Our results additionally highlighted that the user retention rates dropped drastically for
commonly chosen core-levels of 5, 10 and 20: only a single dataset retained more than 30%
of users in its 20-coreset, while the majority of datasets only contained 7-15% of users
compared to their unpruned versions. These varying degrees of user retention were also
visible in the distributional characteristics when comparing the GiniUser. Especially dense
datasets with an extremely short head, long tail distribution of interactions, such as the
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MovieLens dataset, displayed a trend toward uniformity across pruning levels in GiniUser.
Whereas GiniItem remained considerably high, leaning towards an unequal distribution of
interactions across items.
In summary, progressive pruning altered both structural and distributional characteristics
across all datasets; however, the extent of these changes was strongly influenced by the
inherent composition of the unpruned (0-core) datasets.
Dataset characteristics, such as density, shape, and the average number of interactions per
user/item, are known to influence algorithm performance [18]. Therefore, the fluctuations
observed in our experiments when comparing algorithm performance across datasets and their
core-pruned versions were expected after analysing the effect pruning had on the dataset
characteristics. Nonetheless, some algorithms showed distinctly different behaviour on
individual coresets, a pattern we hypothesise arises from their underlying operating
mechanisms. For example, Matrix Factorisation-based algorithms tend to improve
performance as data sparsity decreases, a trend mirrored in the results of our experimentation
in Phase 1 and the dataset analysis.
The key question of whether progressive data pruning leads to changes in the performance
ranking of algorithms was answered affirmatively for all examined datasets. Rank changes
occurred more frequently and with greater magnitude on the two datasets with the highest
density, largest values for shape and average number of interactions per item: MovieLens and
Yelp. This observation remained true for both Phase 1 and Phase 2.
In Phase 1, overall trends emerged across the three algorithm groups: traditional algorithms
responded positively to higher pruning levels, whereas modern algorithms tended to show
declining performance as pruning intensified. In Phase 2 – where test sets still included low-
activity users while training was performed on pruned data – the performance rankings
changed markedly, and traditional algorithms experienced a substantial drop in performance.
This suggests that the improvements observed for traditional algorithms in Phase 1 were
largely due to the progressively easier test sets, as higher pruning levels increasingly retained
only high-activity users in the coresets.
The superior Phase 2 performance of modern algorithms supports the assumption that they are
better equipped than traditional algorithms to handle typical recommender system challenges,
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such as data sparsity and the user cold-start problem [2, 31]. A scenario with cold-start users
was simulated in Phase 2 by including low-activity users in the test sets.
To revisit the core research question: progressive pruning altered all examined dataset
characteristics, yet the underlying composition of the original datasets remained reflected in
these altered characteristics. Both the inherent dataset composition and the pruning-induced
changes influenced algorithm performance, confirming that data pruning affects
performance ranking. However, rank changes cannot be explained solely by the
progressively pruned training sets, as the composition of the test sets and the intrinsic
mechanisms of the algorithms also play significant roles.
7. Conclusion
This thesis set out to analyse the impact of progressive data pruning on dataset characteristics
and algorithm performance. To this end, we first analysed five datasets in both unpruned and
progressively pruned states (5, 10, 20, 50, and 100-core pruning) to evaluate how their
characteristics evolved under increasing pruning intensity.
The results show that inherent dataset characteristics observed in the unpruned versions shape
the extent to which they are altered by progressive pruning. For example, datasets with a high
average number of interactions per user retained a much larger proportion of their original
user base as pruning intensified. In contrast, datasets with lower average interactions per user,
e.g. Yelp, quickly experienced heavy user loss and preserved fewer total interactions than
datasets with higher interaction averages per user. Consequently, the resulting coresets varied
significantly in terms of characteristics, total size, and user retention. For example, when 5-
core pruning was applied, Gowalla retained as many as 75% of its users and 98% of its
interactions, whereas Yelp lost 86% of its users and only retained 58% of its interactions. This
contrast widened at higher pruning levels: under 100-core pruning on Yelp retained merely
0.12% of its original users, while Gowalla still maintained about 5%.
When datasets were pruned at the commonly used 20-core threshold, 80% of them retained
15% or fewer of their original users. This pruning strategy resembles the procedure applied to
the popular MovieLens datasets, which are widely regarded as benchmark data for evaluating
recommender systems. Using datasets pruned in this severity means that roughly 85% of users
are ignored when evaluating an algorithm, thereby excluding the very group – low-activity or
46
cold-start users – that is generally considered challenging for generating suitable
recommendations.
Pruned datasets differ substantially from data encountered in real-world settings, as they
primarily contain users with comparatively rich interaction histories. Consequently, good
results achieved on such datasets cannot necessarily be generalised to other scenarios.
Furthermore, pruning can introduce bias into datasets, in our case by skewing performance
towards high-activity users. This raises concerns about fairness in recommender systems and
has become the subject of recent research.
The influence of pruning on performance scores and, ultimately, the performance ranking of
algorithms was assessed by training and testing eleven algorithms on increasingly pruned
datasets. The selected algorithms – covering baseline, traditional, and modern methods –
allowed us to analyse whether pruning affected these groups to different extents.
We found that specific datasets as a whole, as well as particular coresets, provided favourable
conditions for individual algorithms. For example, denser datasets benefited matrix
factorisation-based methods, while coresets with few low-activity items favoured Item KNN.
Overall, the results suggest that traditional algorithms tended to improve as pruning levels
increased, whereas modern algorithms showed no corresponding performance increase on
progressively pruned coresets. In summary, changes in the performance rank of algorithms
were observed across all datasets, with a higher frequency and greater magnitude on the
denser datasets, MovieLens and Yelp.
To determine whether the “easier” test sets – those in which low-activity users are
successively removed, leaving only high-activity users – contributed to the positive trend
observed for some algorithms across pruning levels, we conducted a second evaluation. This
evaluation utilised the same training sets as before, but the test sets were derived from the
unpruned datasets, thereby reintroducing low-activity users. While rank changes occurred
across coresets and entire datasets under these evaluation conditions as well, the overall trends
and performance values declined sharply, particularly for the traditional algorithms. Whereas
traditional algorithms had outperformed modern ones across most coresets in the first
evaluation phase, testing on unpruned datasets yielded higher overall performance scores for
the modern algorithms. A comparison of score evolution across increasing pruning levels
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showed an overall decline in performance as pruning intensified for all examined algorithms
of both groups.
Combining our findings from the dataset-characteristic analysis, the experiments with training
and testing on pruned data, and the experiments with training on pruned data while testing on
unpruned data, we conclude that data pruning can artificially inflate performance scores. This
effect occurs when evaluation is carried out on test sets pruned to the same level as the
training data. Furthermore, specific datasets and pruning levels favoured individual
algorithms, indicating that evaluating only on single datasets or individual coresets is not
advisable for a balanced comparison of algorithm performance. Results based on such narrow
evaluations cannot necessarily be generalised to other scenarios that involve different
datasets. One such scenario is the data a recommender system encounters in real-world use.
Choosing an unsuitable algorithm in this context could lead to performance that differs
significantly from what offline evaluations suggested.
Besides failing to deliver generally applicable results, data pruning also hampers effective
comparisons between algorithms in research. Selecting a coreset that is especially favourable
to a particular algorithm can skew performance results in its favour, while making other
comparable algorithms appear less suitable.
Data pruning can help reduce computational load and training time, thereby supporting
greener recommender-system research, provided acceptable trade-offs between data reduction
and performance are possible. However, we concur with evaluation frameworks that
recommend using pruned datasets only as a supplement to unpruned datasets. Alternatively,
training on pruned data while evaluating on unpruned data could offer a compromise that
lowers training time and computational cost while still testing on a heterogeneous group of
users. Additionally, our results highlight the importance of clearly documenting all
preprocessing steps applied to datasets in recommender-systems research. Although many
studies apply data pruning during pre-processing or use pre-pruned datasets, such as
MovieLens, crucial documentation of dataset choice and preprocessing steps, including data
pruning, is frequently absent. Without precise knowledge of a dataset’s preprocessing history,
replicating experiments and comparing algorithms under equal conditions becomes nearly
impossible.
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In conclusion, this thesis demonstrates that progressive data pruning can strongly influence
both dataset composition and algorithm evaluation, leading to inflated performance scores and
limited generalisability if not handled with care. By combining detailed dataset analyses with
extensive algorithmic assessment, we provide evidence that pruning should be applied only as
a complement to unpruned data and documented transparently. Future recommender-system
research can build on these insights by developing frameworks and preprocessing standards
that balance computational efficiency with fairness and real-world applicability.
8. Future Work and Limitations
Due to computational limitations encountered during the experiment, several concessions had
to be made to facilitate the evaluation of 11 algorithms on five datasets.
Hyperparameter-Optimisation
All models were trained with their default parameter values, respectively set by either the
LensKit or the RecBole library, as the focus of this study was not to achieve the highest
possible performance for all algorithms but to compare how the increasingly pruned versions
of the datasets affected the performance of the models relatively. Future work should consider
implementing hyperparameter-tuning.
K-Fold Cross Validation
Due to the extensive computation times, it was not possible to include K-Fold Cross-
Validation in this study. Future research could consider implementing cross-validation with k
= 5 or k = 10.
Algorithm Selection
To further reduce the computational cost, the deep learning models investigated in this study
were chosen for their lower computational complexity and shorter training times. It would be
of interest to see how more complex deep learning models are impacted by data pruning.
Metrics
Our research focused on ranking-based performance metrics, researchers could examine the
impact data pruning has on, e.g. beyond-accuracy metrics for deep learning models.
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Datasets
Due to the scope of this work, only five datasets could be included in the experiments; further
research should focus on a more diverse variety of data sources.
User-Based and Item-Based Pruning
In our experiment, we pruned interactions based on the total number of interactions the
corresponding user had in the dataset. It is interesting to examine how pruning based on item
interactions, e.g. pruning all interactions with an item if it is interacted with less than a
specific number of times, impacts both dataset characteristics and algorithm performance.
Addressing these limitations in future research will not only strengthen the validity and
generalisability of our findings but also provide a richer foundation for developing fair,
efficient, and reproducible recommender-system evaluations. By building on the proposed
directions, subsequent studies can refine methods, broaden datasets, and deepen
understanding of how data pruning shapes both dataset characteristics and algorithm
performance.
9. Summary
This thesis investigates how data pruning – removing users whose total number of
interactions falls below a defined threshold – affects both datasets' characteristics and
algorithm performance in recommender systems. Five widely used datasets were examined in
their unpruned form and at 5-, 10-, 20-, 50-, and 100-core pruning levels. For each stage, we
analysed structural characteristics, such as density, shape, and the average number of
interactions per user/item, as well as distributional characteristics measured by Gini
coefficients for users and items. Retention rates of users, items, and interactions were tracked
to reveal how dataset composition evolves as low-activity users are successively removed.
Dataset characteristics were found to be influenced by pruning to varying degrees. Even at the
lowest pruning level of 5-core, the Yelp dataset retained only 14% of its users and 58% of its
interactions. In contrast, Gowalla preserved 75% of its users and 98% of its interactions,
illustrating how differently datasets can react to identical pruning thresholds. At the
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commonly used 20-core level, all but one dataset were reduced to 15% or fewer of their
original users, and at the highest pruning level (100-core), Yelp retained a mere 0.12% of its
user base. These findings highlight the heterogeneous impact of progressive pruning on
structural and distributional characteristics across datasets.
Building on this analysis, algorithm performance was evaluated across the same progressive
pruning levels in two complementary settings. Phase 1 involved training and testing on
increasingly pruned datasets, while Phase 2 used the same pruned training sets but tested on
unpruned datasets to reintroduce low-activity users to the evaluation. While traditional
algorithms overall showed improving performance with rising pruning levels in Phase 1, this
trend diminished significantly in Phase 2, where modern algorithms continuously
outperformed the majority of traditional algorithms. The modern algorithms exhibited a
general trend of declining performance as pruning levels increased during both phases,
despite average nDCG@10 values being higher in Phase 2. Rank changes occurred between
algorithms in both phases, across entire datasets and for specific coresets. On the denser
datasets, Yelp and MovieLens, both frequency and magnitude of rank changes increased
across core levels.
The two-phase evaluation revealed how pruning can artificially inflate performance scores
and alter algorithm rankings when testing is conducted on equally pruned data, and how these
effects change when testing occurs instead on unpruned datasets.
Together, these analyses of algorithm performance and dataset characteristics provide a
detailed picture of how progressive data pruning influences both the statistical structure of
recommender-systems datasets and the measured performance of baseline, traditional, and
modern algorithms. The results highlight that pruning levels have a strong impact on dataset
composition and algorithmic evaluation, and the careful selection of datasets and coresets can
provide beneficial conditions for individual algorithms. The findings underline the importance
of selecting diverse datasets, using core-pruned datasets only as supplementary analysis to
unpruned datasets, and carefully documenting every preprocessing step to which datasets are
subjected.
51
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57
11. Appendix
Phase 1 (Training & Testing on Pruned Datasets) nDCG@10
Phase 2 (Testing on Unpruned Datasets) nDCG@10
Biased MF
Biased MF
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0000
0.0000
0.0002
0.0070
0.0007
5
0.0000
0.0000
0.0002
0.0065
0.0007
5
0.0000
0.0000
0.0000
0.0004
0.0000
10
0.0000
0.0000
0.0003
0.0059
0.0006
10
0.0000
0.0000
0.0000
0.0003
0.0000
20
0.0000
0.0000
0.0004
0.0063
0.0008
20
0.0000
0.0000
0.0000
0.0010
0.0000
50
0.0006
0.0000
0.0008
0.0070
0.0013
50
0.0000
0.0000
0.0000
0.0003
0.0000
100
0.0008
0.0000
0.0013
0.0097
0.0022
100
0.0000
0.0000
0.0000
0.0003
0.0000
Min
0.0000
0.0000
0.0002
0.0059
0.0006 Min
0.0000
0.0000
0.0000
0.0003
0.0000
Max
0.0008
0.0000
0.0013
0.0097
0.0022 Max
0.0000
0.0000
0.0000
0.0010
0.0000
Average
0.0003
0.0000
0.0005
0.0071
0.0011 Average
0.0000
0.0000
0.0000
0.0005
0.0000
Implicit MF
ImplicitMF
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0204
0.0021
0.0282
0.0424
0.0286
5
0.0202
0.0018
0.0298
0.0380
0.0332
5
0.0010
0.0001
0.0017
0.0021
0.0017
10
0.0215
0.0024
0.0328
0.0310
0.0366
10
0.0012
0.0001
0.0019
0.0017
0.0019
20
0.0266
0.0029
0.0398
0.0271
0.0452
20
0.0013
0.0002
0.0023
0.0042
0.0021
50
0.0410
0.0044
0.0555
0.0234
0.0606
50
0.0015
0.0002
0.0028
0.0011
0.0016
100
0.0565
0.0068
0.0789
0.0210
0.0751
100
0.0021
0.0002
0.0029
0.0004
0.0012
Min
0.0202
0.0018
0.0282
0.0210
0.0286 Min
0.0010
0.0001
0.0017
0.0004
0.0012
Max
0.0565
0.0068
0.0789
0.0424
0.0751 Max
0.0021
0.0002
0.0029
0.0042
0.0021
Average
0.0310
0.0034
0.0442
0.0305
0.0465 Average
0.0014
0.0002
0.0023
0.0019
0.0017
PopScore
PopScore
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0057
0.0007
0.0056
0.0396
0.0075
5
0.0047
0.0007
0.0060
0.0353
0.0069
5
0.0002
0.0000
0.0004
0.0019
0.0004
10
0.0050
0.0010
0.0067
0.0298
0.0062
10
0.0002
0.0001
0.0004
0.0017
0.0003
20
0.0063
0.0009
0.0078
0.0272
0.0071
20
0.0003
0.0000
0.0005
0.0042
0.0004
50
0.0092
0.0014
0.0120
0.0251
0.0095
50
0.0004
0.0001
0.0006
0.0011
0.0004
100
0.0149
0.0019
0.0154
0.0278
0.0161
100
0.0004
0.0000
0.0004
0.0004
0.0004
Min
0.0047
0.0007
0.0056
0.0251
0.0062 Min
0.0002
0.0000
0.0004
0.0004
0.0003
Max
0.0149
0.0019
0.0154
0.0396
0.0161 Max
0.0004
0.0001
0.0006
0.0042
0.0004
Average
0.0076
0.0011
0.0089
0.0308
0.0089 Average
0.0003
0.0000
0.0005
0.0018
0.0004
User KNN
User KNN
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0337
0.0082
0.0512
0.0504
0.0312
5
0.0341
0.0081
0.0523
0.0428
0.0317
5
0.0018
0.0004
0.0031
0.0022
0.0017
10
0.0364
0.0083
0.0546
0.0340
0.0332
10
0.0019
0.0004
0.0030
0.0017
0.0018
20
0.0450
0.0092
0.0621
0.0279
0.0419
20
0.0023
0.0005
0.0036
0.0043
0.0021
50
0.0621
0.0095
0.0795
0.0239
0.0628
50
0.0026
0.0005
0.0040
0.0009
0.0015
100
0.0820
0.0103
0.1013
0.0251
0.0780
100
0.0026
0.0003
0.0036
0.0006
0.0014
Min
0.0337
0.0081
0.0512
0.0239
0.0312 Min
0.0018
0.0003
0.0030
0.0006
0.0014
Max
0.0820
0.0103
0.1013
0.0504
0.0780 Max
0.0026
0.0005
0.0040
0.0043
0.0021
Average
0.0489
0.0089
0.0668
0.0340
0.0464 Average
0.0022
0.0004
0.0034
0.0019
0.0017
Item KNN
Item KNN
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0107
0.0030
0.0216
0.0448
0.0066
5
0.0130
0.0037
0.0223
0.0440
0.0102
5
0.0008
0.0002
0.0014
0.0023
0.0006
10
0.0176
0.0041
0.0246
0.0371
0.0146
10
0.0010
0.0003
0.0014
0.0020
0.0008
20
0.0245
0.0053
0.0296
0.0314
0.0227
20
0.0012
0.0003
0.0017
0.0049
0.0011
50
0.0331
0.0060
0.0379
0.0246
0.0229
50
0.0014
0.0002
0.0019
0.0008
0.0005
100
0.0413
0.0068
0.0424
0.0145
0.0161
100
0.0015
0.0002
0.0019
0.0003
0.0004
Min
0.0107
0.0030
0.0216
0.0145
0.0066 Min
0.0008
0.0002
0.0014
0.0003
0.0004
Max
0.0413
0.0068
0.0424
0.0448
0.0229 Max
0.0015
0.0003
0.0019
0.0049
0.0011
Average
0.0234
0.0048
0.0297
0.0327
0.0155 Average
0.0012
0.0003
0.0017
0.0021
0.0007
BPR
BPR
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0050
0.0003
0.0146
0.0367
0.0132
5
0.0050
0.0003
0.0150
0.0350
0.0150
5
0.0448
0.2004
0.0230
0.0319
0.0141
10
0.0064
0.0005
0.0180
0.0304
0.0172
10
0.0324
0.1655
0.0211
0.0249
0.0106
20
0.0091
0.0005
0.0216
0.0273
0.0229
20
0.0211
0.1234
0.0183
0.0131
0.0062
50
0.0151
0.0012
0.0296
0.0243
0.0248
50
0.0147
0.0702
0.0103
0.0075
0.0036
100
0.0264
0.0025
0.0401
0.0138
0.0019
100
0.0117
0.0417
0.0068
0.0027
0.0049
Min
0.0050
0.0003
0.0146
0.0138
0.0019 Min
0.0117
0.0417
0.0068
0.0027
0.0036
Max
0.0264
0.0025
0.0401
0.0367
0.0248 Max
0.0448
0.2004
0.0230
0.0319
0.0141
Average
0.0112
0.0009
0.0232
0.0279
0.0158 Average
0.0249
0.1202
0.0159
0.0160
0.0079
58
Phase 1 (Training & Testing on Pruned Datasets) nDCG@10
Phase 2 (Testing on Unpruned Datasets) nDCG@10
DiffRec
DiffRec
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0078
0.0000
0.0097
0.0329
0.0108
5
0.0050
0.0004
0.0105
0.0395
0.0212
5
0.0054
0.0004
0.0111
0.4842
0.0765
10
0.0050
0.0004
0.0094
0.0370
0.0191
10
0.0049
0.0001
0.0104
0.3142
0.0229
20
0.0046
0.0006
0.0069
0.0304
0.0196
20
0.0038
0.0002
0.0079
0.0766
0.0086
50
0.0014
0.0007
0.0015
0.0213
0.0042
50
0.0015
0.0000
0.0053
0.0125
0.0023
100
0.0023
0.0005
0.0029
0.0060
0.0005
100
0.0008
0.0000
0.0025
0.0065
0.0012
Min
0.0014
0.0000
0.0015
0.0060
0.0005 Min
0.0008
0.0000
0.0025
0.0065
0.0012
Max
0.0078
0.0007
0.0105
0.0395
0.0212 Max
0.0054
0.0004
0.0111
0.4842
0.0765
Average
0.0044
0.0004
0.0068
0.0279
0.0126 Average
0.0033
0.0001
0.0074
0.1788
0.0223
DMF
DMF
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0016
0.0005
0.0045
0.0328
0.0063
5
0.0024
0.0006
0.0037
0.0285
0.0056
5
0.0018
0.0007
0.0046
0.0268
0.0052
10
0.0029
0.0009
0.0028
0.0274
0.0051
10
0.0028
0.0007
0.0030
0.0272
0.0030
20
0.0025
0.0008
0.0035
0.0229
0.0057
20
0.0013
0.0007
0.0063
0.0170
0.0029
50
0.0061
0.0011
0.0112
0.0204
0.0083
50
0.0022
0.0005
0.0047
0.0191
0.0031
100
0.0117
0.0013
0.0125
0.0280
0.0142
100
0.0022
0.0004
0.0037
0.0146
0.0031
Min
0.0016
0.0005
0.0028
0.0204
0.0051 Min
0.0013
0.0004
0.0030
0.0146
0.0029
Max
0.0117
0.0013
0.0125
0.0328
0.0142 Max
0.0028
0.0007
0.0063
0.0272
0.0052
Average
0.0045
0.0009
0.0064
0.0267
0.0075 Average
0.0021
0.0006
0.0045
0.0209
0.0035
MultiVAE
MultiVAE
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0055
0.0003
0.0054
0.0391
0.0143
5
0.0037
0.0005
0.0060
0.0339
0.0063
5
0.0031
0.0000
0.0058
0.0351
0.0068
10
0.0035
0.0006
0.0066
0.0303
0.0064
10
0.0021
0.0000
0.0057
0.0313
0.0044
20
0.0036
0.0002
0.0063
0.0274
0.0032
20
0.0013
0.0000
0.0063
0.0304
0.0054
50
0.0055
0.0005
0.0045
0.0105
0.0114
50
0.0007
0.0000
0.0049
0.0123
0.0044
100
0.0058
0.0006
0.0127
0.0115
0.0020
100
0.0004
0.0000
0.0034
0.0132
0.0034
Min
0.0035
0.0002
0.0045
0.0105
0.0020 Min
0.0004
0.0000
0.0034
0.0123
0.0034
Max
0.0058
0.0006
0.0127
0.0391
0.0143 Max
0.0031
0.0000
0.0063
0.0351
0.0068
Average
0.0046
0.0005
0.0069
0.0255
0.0073 Average
0.0015
0.0000
0.0052
0.0245
0.0049
Random
Random
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0000
0.0000
0.0000
0.0002
0.0000
5
0.0000
0.0000
0.0000
0.0002
0.0000
5
0.0000
0.0000
0.0000
0.0002
0.0001
10
0.0000
0.0000
0.0000
0.0002
0.0001
10
0.0000
0.0000
0.0000
0.0002
0.0000
20
0.0000
0.0000
0.0000
0.0003
0.0001
20
0.0000
0.0000
0.0000
0.0002
0.0000
50
0.0001
0.0000
0.0000
0.0004
0.0004
50
0.0000
0.0000
0.0000
0.0002
0.0001
100
0.0002
0.0000
0.0001
0.0011
0.0005
100
0.0000
0.0000
0.0000
0.0001
0.0000
Min
0.0000
0.0000
0.0000
0.0002
0.0000 Min
0.0000
0.0000
0.0000
0.0001
0.0000
Max
0.0002
0.0000
0.0001
0.0011
0.0005 Max
0.0000
0.0000
0.0000
0.0002
0.0001
Average
0.0001
0.0000
0.0000
0.0004
0.0002 Average
0.0000
0.0000
0.0000
0.0002
0.0000
SimpleX
SimpleX
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
Core
Amazon CD Amazon ToyGowalla
MovieLens Yelp
0
0.0028
0.0004
0.0071
0.0233
0.0064
5
0.0028
0.0004
0.0082
0.0195
0.0078
5
0.3057
0.4901
0.2683
0.0228
0.0211
10
0.0030
0.0006
0.0079
0.0182
0.0098
10
0.2396
0.3387
0.2454
0.0140
0.0135
20
0.0054
0.0005
0.0098
0.0176
0.0132
20
0.1543
0.2083
0.1854
0.0044
0.0093
50
0.0103
0.0011
0.0117
0.0193
0.0200
50
0.0690
0.0927
0.0996
0.0024
0.0065
100
0.0164
0.0019
0.0179
0.0154
0.0059
100
0.0323
0.0455
0.0449
0.0005
0.0032
Min
0.0028
0.0004
0.0071
0.0154
0.0059 Min
0.0323
0.0455
0.0449
0.0005
0.0032
Max
0.0164
0.0019
0.0179
0.0233
0.0200 Max
0.3057
0.4901
0.2683
0.0228
0.0211
Average
0.0068
0.0008
0.0104
0.0189
0.0105 Average
0.1602
0.2351
0.1687
0.0088
0.0107
59
PRECISION@10 (PHASE 1)
Amazon CD
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0000
0.0060
0.0016
0.0106
0.0046
0.0019
0.0028
0.0008
0.0015
0.0000
0.0010
5
0.0000
0.0083
0.0020
0.0143
0.0066
0.0024
0.0023
0.0010
0.0015
0.0000
0.0013
10
0.0000
0.0130
0.0032
0.0222
0.0117
0.0041
0.0031
0.0024
0.0022
0.0000
0.0018
20
0.0000
0.0199
0.0049
0.0345
0.0197
0.0072
0.0038
0.0024
0.0034
0.0000
0.0043
50
0.0009
0.0359
0.0085
0.0546
0.0308
0.0132
0.0014
0.0064
0.0055
0.0000
0.0092
100
0.0012
0.0518
0.0131
0.0741
0.0391
0.0246
0.0021
0.0115
0.0053
0.0002
0.0151
Amazon Toys
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0000
0.0008
0.0002
0.0027
0.0013
0.0001
0.0000
0.0002
0.0001
0.0000
0.0002
5
0.0000
0.0009
0.0003
0.0036
0.0020
0.0002
0.0002
0.0003
0.0002
0.0000
0.0003
10
0.0000
0.0016
0.0005
0.0052
0.0029
0.0003
0.0002
0.0005
0.0003
0.0000
0.0004
20
0.0000
0.0023
0.0007
0.0071
0.0044
0.0004
0.0003
0.0006
0.0001
0.0000
0.0004
50
0.0000
0.0040
0.0012
0.0088
0.0057
0.0012
0.0007
0.0010
0.0005
0.0000
0.0010
100
0.0000
0.0065
0.0017
0.0094
0.0064
0.0025
0.0005
0.0011
0.0007
0.0001
0.0017
Gowalla
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0002
0.0159
0.0035
0.0277
0.0125
0.0084
0.0060
0.0030
0.0033
0.0000
0.0045
5
0.0002
0.0174
0.0039
0.0300
0.0136
0.0090
0.0069
0.0025
0.0037
0.0000
0.0054
10
0.0002
0.0210
0.0047
0.0356
0.0169
0.0114
0.0062
0.0023
0.0046
0.0000
0.0058
20
0.0004
0.0299
0.0066
0.0478
0.0237
0.0158
0.0054
0.0032
0.0054
0.0000
0.0084
50
0.0007
0.0478
0.0106
0.0693
0.0349
0.0251
0.0015
0.0101
0.0042
0.0000
0.0112
100
0.0013
0.0695
0.0141
0.0894
0.0400
0.0352
0.0029
0.0127
0.0120
0.0001
0.0174
MovieLens
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0020
0.0141
0.0130
0.0160
0.0149
0.0127
0.0120
0.0114
0.0137
0.0001
0.0089
5
0.0023
0.0156
0.0146
0.0174
0.0177
0.0147
0.0165
0.0123
0.0145
0.0001
0.0090
10
0.0028
0.0184
0.0175
0.0200
0.0215
0.0172
0.0210
0.0156
0.0173
0.0002
0.0110
20
0.0037
0.0212
0.0208
0.0219
0.0242
0.0206
0.0232
0.0174
0.0206
0.0003
0.0140
50
0.0056
0.0229
0.0245
0.0236
0.0244
0.0240
0.0210
0.0197
0.0098
0.0004
0.0191
100
0.0075
0.0219
0.0270
0.0249
0.0144
0.0134
0.0057
0.0268
0.0111
0.0011
0.0150
Yelp
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0001
0.0077
0.0018
0.0079
0.0018
0.0040
0.0033
0.0017
0.0044
0.0000
0.0022
5
0.0002
0.0118
0.0022
0.0111
0.0038
0.0057
0.0083
0.0019
0.0022
0.0000
0.0032
10
0.0003
0.0208
0.0034
0.0188
0.0080
0.0098
0.0113
0.0026
0.0034
0.0000
0.0059
20
0.0005
0.0348
0.0054
0.0318
0.0164
0.0178
0.0152
0.0042
0.0028
0.0001
0.0106
50
0.0010
0.0569
0.0093
0.0561
0.0208
0.0238
0.0039
0.0081
0.0114
0.0003
0.0192
100
0.0013
0.0708
0.0152
0.0739
0.0155
0.0020
0.0003
0.0129
0.0019
0.0004
0.0059
60
PRECISION@10 (PHASE 2)
Amazon CD
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0005
0.0001
0.0009
0.0004
0.0170
0.0026
0.0010
0.0014
0.0000
0.0977
10
0.0000
0.0007
0.0002
0.0013
0.0006
0.0165
0.0025
0.0017
0.0013
0.0000
0.0993
20
0.0000
0.0010
0.0002
0.0018
0.0010
0.0142
0.0019
0.0009
0.0010
0.0000
0.0830
50
0.0000
0.0013
0.0004
0.0023
0.0012
0.0123
0.0008
0.0013
0.0007
0.0000
0.0476
100
0.0000
0.0019
0.0003
0.0024
0.0016
0.0102
0.0004
0.0010
0.0004
0.0000
0.0243
Amazon Toys
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0001
0.0000
0.0005
0.0002
0.1943
0.0005
0.0009
0.0000
0.0000
0.4327
10
0.0000
0.0001
0.0001
0.0005
0.0004
0.1387
0.0002
0.0009
0.0000
0.0000
0.2680
20
0.0000
0.0002
0.0000
0.0005
0.0003
0.0837
0.0001
0.0009
0.0000
0.0000
0.1384
50
0.0000
0.0002
0.0001
0.0004
0.0002
0.0343
0.0001
0.0005
0.0000
0.0000
0.0464
100
0.0000
0.0002
0.0000
0.0003
0.0003
0.0156
0.0000
0.0005
0.0000
0.0001
0.0172
Gowalla
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0011
0.0003
0.0019
0.0009
0.0136
0.0071
0.0034
0.0038
0.0000
0.1144
10
0.0000
0.0012
0.0003
0.0021
0.0010
0.0132
0.0069
0.0022
0.0039
0.0000
0.1189
20
0.0000
0.0017
0.0004
0.0027
0.0014
0.0132
0.0053
0.0047
0.0042
0.0000
0.1104
50
0.0000
0.0023
0.0005
0.0035
0.0017
0.0083
0.0033
0.0032
0.0029
0.0000
0.0730
100
0.0000
0.0025
0.0004
0.0034
0.0017
0.0055
0.0013
0.0018
0.0017
0.0000
0.0348
MovieLens
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0002
0.0010
0.0009
0.0011
0.0011
0.0142
0.1479
0.0127
0.0151
0.0001
0.0109
10
0.0002
0.0011
0.0010
0.0011
0.0012
0.0130
0.1156
0.0132
0.0141
0.0001
0.0084
20
0.0005
0.0026
0.0026
0.0026
0.0030
0.0055
0.0289
0.0059
0.0096
0.0001
0.0027
50
0.0002
0.0009
0.0009
0.0008
0.0007
0.0034
0.0054
0.0059
0.0047
0.0001
0.0021
100
0.0003
0.0004
0.0003
0.0006
0.0003
0.0011
0.0020
0.0038
0.0034
0.0001
0.0004
Yelp
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0008
0.0002
0.0007
0.0003
0.0056
0.0218
0.0018
0.0023
0.0000
0.0070
10
0.0000
0.0010
0.0002
0.0010
0.0005
0.0050
0.0094
0.0013
0.0017
0.0000
0.0057
20
0.0000
0.0013
0.0002
0.0013
0.0007
0.0036
0.0046
0.0011
0.0017
0.0000
0.0049
50
0.0000
0.0012
0.0003
0.0011
0.0004
0.0024
0.0013
0.0008
0.0012
0.0000
0.0039
100
0.0000
0.0009
0.0003
0.0011
0.0003
0.0032
0.0005
0.0008
0.0008
0.0000
0.0018
61
RECALL@10 (PHASE 1)
Amazon CD
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0000
0.0281
0.0096
0.0438
0.0126
0.0064
0.0111
0.0030
0.0095
0.0000
0.0038
5
0.0001
0.0262
0.0069
0.0428
0.0147
0.0053
0.0070
0.0027
0.0061
0.0000
0.0034
10
0.0001
0.0240
0.0059
0.0407
0.0190
0.0050
0.0049
0.0035
0.0039
0.0000
0.0025
20
0.0001
0.0249
0.0062
0.0435
0.0244
0.0053
0.0037
0.0017
0.0031
0.0000
0.0032
50
0.0009
0.0359
0.0085
0.0546
0.0308
0.0055
0.0004
0.0030
0.0024
0.0000
0.0039
100
0.0012
0.0518
0.0131
0.0741
0.0391
0.0060
0.0003
0.0032
0.0012
0.0000
0.0039
Amazon Toys
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0000
0.0029
0.0010
0.0112
0.0040
0.0003
0.0000
0.0009
0.0005
0.0000
0.0005
5
0.0000
0.0022
0.0009
0.0106
0.0047
0.0003
0.0005
0.0008
0.0006
0.0000
0.0005
10
0.0000
0.0027
0.0011
0.0099
0.0049
0.0004
0.0005
0.0010
0.0007
0.0000
0.0005
20
0.0000
0.0029
0.0009
0.0092
0.0056
0.0002
0.0004
0.0007
0.0001
0.0000
0.0003
50
0.0000
0.0040
0.0012
0.0088
0.0057
0.0003
0.0003
0.0005
0.0002
0.0000
0.0003
100
0.0000
0.0065
0.0017
0.0094
0.0064
0.0005
0.0000
0.0003
0.0001
0.0000
0.0003
Gowalla
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0003
0.0326
0.0072
0.0604
0.0254
0.0141
0.0089
0.0053
0.0062
0.0000
0.0077
5
0.0003
0.0333
0.0073
0.0601
0.0259
0.0138
0.0093
0.0038
0.0063
0.0000
0.0085
10
0.0003
0.0345
0.0077
0.0594
0.0271
0.0150
0.0077
0.0029
0.0063
0.0000
0.0070
20
0.0004
0.0368
0.0081
0.0591
0.0293
0.0141
0.0050
0.0028
0.0050
0.0000
0.0069
50
0.0007
0.0478
0.0106
0.0693
0.0349
0.0120
0.0005
0.0051
0.0017
0.0000
0.0046
100
0.0013
0.0695
0.0141
0.0894
0.0400
0.0093
0.0004
0.0035
0.0032
0.0000
0.0042
MovieLens
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0089
0.0698
0.0639
0.0839
0.0736
0.0626
0.0565
0.0575
0.0668
0.0003
0.0405
5
0.0078
0.0606
0.0560
0.0702
0.0709
0.0561
0.0639
0.0469
0.0550
0.0003
0.0312
10
0.0061
0.0451
0.0426
0.0498
0.0542
0.0408
0.0503
0.0368
0.0411
0.0003
0.0239
20
0.0054
0.0322
0.0318
0.0332
0.0379
0.0294
0.0333
0.0242
0.0293
0.0003
0.0190
50
0.0056
0.0229
0.0245
0.0236
0.0244
0.0161
0.0140
0.0131
0.0063
0.0002
0.0126
100
0.0075
0.0219
0.0270
0.0249
0.0144
0.0049
0.0019
0.0099
0.0040
0.0004
0.0054
Yelp
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
0
0.0008
0.0449
0.0123
0.0481
0.0104
0.0229
0.0179
0.0112
0.0256
0.0001
0.0116
5
0.0007
0.0497
0.0110
0.0475
0.0157
0.0233
0.0321
0.0091
0.0109
0.0001
0.0127
10
0.0007
0.0493
0.0085
0.0432
0.0196
0.0205
0.0228
0.0061
0.0080
0.0001
0.0128
20
0.0007
0.0489
0.0080
0.0440
0.0250
0.0205
0.0172
0.0051
0.0031
0.0001
0.0125
50
0.0010
0.0569
0.0093
0.0561
0.0208
0.0121
0.0021
0.0043
0.0059
0.0002
0.0101
100
0.0013
0.0708
0.0152
0.0739
0.0155
0.0005
0.0001
0.0037
0.0007
0.0001
0.0015
62
RECALL@10 (PHASE 2)
Amazon CD
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0013
0.0003
0.0022
0.0009
0.0547
0.0073
0.0028
0.0043
0.0000
0.2751
10
0.0000
0.0012
0.0003
0.0021
0.0011
0.0326
0.0061
0.0038
0.0022
0.0000
0.1898
20
0.0000
0.0013
0.0003
0.0023
0.0013
0.0165
0.0048
0.0020
0.0011
0.0000
0.1028
50
0.0000
0.0014
0.0005
0.0024
0.0013
0.0080
0.0015
0.0033
0.0004
0.0000
0.0344
100
0.0000
0.0019
0.0003
0.0024
0.0016
0.0048
0.0011
0.0038
0.0002
0.0000
0.0120
Amazon Toys
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0001
0.0000
0.0005
0.0002
0.1943
0.0005
0.0009
0.0000
0.0000
0.4327
10
0.0000
0.0001
0.0001
0.0005
0.0004
0.1387
0.0002
0.0009
0.0000
0.0000
0.2680
20
0.0000
0.0002
0.0000
0.0005
0.0003
0.0837
0.0001
0.0009
0.0000
0.0000
0.1384
50
0.0000
0.0002
0.0001
0.0004
0.0002
0.0343
0.0001
0.0005
0.0000
0.0000
0.0464
100
0.0000
0.0002
0.0000
0.0003
0.0003
0.0156
0.0000
0.0005
0.0000
0.0001
0.0172
Gowalla
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0018
0.0004
0.0034
0.0015
0.0224
0.0097
0.0050
0.0062
0.0000
0.2267
10
0.0000
0.0019
0.0004
0.0032
0.0016
0.0188
0.0086
0.0032
0.0060
0.0000
0.1930
20
0.0000
0.0021
0.0005
0.0034
0.0017
0.0134
0.0075
0.0066
0.0067
0.0000
0.1234
50
0.0000
0.0024
0.0005
0.0036
0.0018
0.0061
0.0050
0.0054
0.0053
0.0000
0.0494
100
0.0000
0.0025
0.0004
0.0034
0.0017
0.0032
0.0020
0.0055
0.0043
0.0000
0.0153
MovieLens
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0004
0.0031
0.0028
0.0034
0.0034
0.0491
0.4559
0.0454
0.0554
0.0003
0.0352
10
0.0004
0.0024
0.0023
0.0024
0.0028
0.0368
0.2940
0.0436
0.0487
0.0003
0.0193
20
0.0010
0.0059
0.0060
0.0061
0.0070
0.0225
0.0854
0.0280
0.0544
0.0003
0.0058
50
0.0003
0.0011
0.0011
0.0010
0.0009
0.0122
0.0178
0.0347
0.0244
0.0003
0.0022
100
0.0003
0.0006
0.0004
0.0007
0.0003
0.0050
0.0119
0.0303
0.0260
0.0003
0.0005
Yelp
Core
BiasedMF
ImplicitMF PopScore
User-KNN
Item-KNN
BPR
DiffRec
DMF
MultiVAE Random
SimpleX
5
0.0000
0.0025
0.0006
0.0024
0.0008
0.0213
0.0941
0.0082
0.0112
0.0001
0.0303
10
0.0000
0.0026
0.0005
0.0022
0.0011
0.0149
0.0299
0.0058
0.0083
0.0001
0.0163
20
0.0000
0.0026
0.0005
0.0025
0.0014
0.0081
0.0102
0.0055
0.0087
0.0001
0.0094
50
0.0000
0.0016
0.0004
0.0017
0.0006
0.0046
0.0031
0.0049
0.0083
0.0001
0.0055
100
0.0000
0.0015
0.0004
0.0014
0.0006
0.0046
0.0021
0.0060
0.0064
0.0001
0.0026
|
Intelligent Systems Group : Best Practice or Mal-Practice? Bachelor's Thesis Computer Science Leonie Winter 1097962 30.09.2025 Examiners Prof. Dr. Joeran Beel Intelligent Systems Group M. Sc. Tobias Vente Intelligent Systems Group This Bachelor's Thesis is handed in according to the requirements of the 2012 (PO 2012). Process period 12.05.2025 to 30.09.2025 Examiners Prof. Dr. Joeran Beel M. Sc. Tobias Vente Eigenständigkeitserklärung Ich versichere, dass ich die schriftliche Ausarbeitung selbständig angefertigt und keine anderen als die angegebenen Hilfsmittel benutzt habe. Alle Stellen, die dem Wortlaut oder dem Sinn nach (inkl. Übersetzungen) anderen Werken entnommen sind, habe ich in jedem einzelnen Fall unter genauer Angabe der Quelle (einschließlich des World Wide Web sowie generativer KI und anderer elektronischer Datensammlungen) deutlich als Entlehnung kenntlich gemacht. Dies gilt auch für angefügte Zeichnungen, bildliche Darstellungen, Skizzen und dergleichen. Ich nehme zur Kenntnis, dass die nachgewiesene Unterlassung der Herkunftsangabe als versuchte Täuschung gewertet wird. Ich bestätige, dass die elektronische Version inhaltlich mit der gedruckten Version übereinstimmt. ____________________ __________________________________________ Datum Unterschrift Declaration of Authorship I certify that I have prepared this written work independently and have not used any aids other than those specified. I have clearly identified all passages that are taken from other works in terms of wording or meaning (including translations) as borrowed material in each individual case, stating the exact source (including the World Wide Web as well as generative AI and other electronic data collections). This also applies to attached drawings, pictorial representations, sketches and the like. I acknowledge that the proven omission of the indication of origin will be regarded as attempted deception. I confirm that the content of the electronic version is the same as the printed version. I I. Contents I. Contents ............................................................................................................................... I II. Abstract ............................................................................................................................ III III. Acknowledgements ...................................................................................................... IV 1. Introduction ........................................................................................................................ 1 1.1 Background ................................................................................................................ 1 1.2 Research Problem ....................................................................................................... 1 1.3 Research Question ...................................................................................................... 3 1.4 Research Objective ..................................................................................................... 3 2. Background ........................................................................................................................ 3 3. Related Work ...................................................................................................................... 4 4. Methodology ...................................................................................................................... 7 4.1 Datasets ...................................................................................................................... 7 4.1.1 Dataset Selection ................................................................................................ 7 4.1.2 Preprocessing of Datasets ................................................................................... 8 4.2 Algorithm Selection ................................................................................................. 10 4.3 Training and Testing ................................................................................................ 11 5. Results .............................................................................................................................. 13 5.1 Dataset Analysis ....................................................................................................... 13 5.1.1 Unprocessed Dataset Characteristics ............................................................... 13 5.1.2 Core-Pruned Dataset Characteristics ................................................................ 15 5.1.3 Structural Characteristics ................................................................................. 16 5.1.4 Distributional Characteristics ........................................................................... 17 5.2 Algorithm Performance Analysis ............................................................................. 22 5.2.1 Overall Performance Trends ............................................................................ 22 5.2.2 Training and Testing on Pruned Datasets ........................................................ 24 5.2.3 Comparison of Testing on Pruned and Unpruned Testsets .............................. 27 6. Discussion and Interpretation ........................................................................................... 32 6.1.1 Dataset Characteristics ..................................................................................... 32 6.1.2 Algorithm Performance .................................................................................... 36 7. Conclusion ........................................................................................................................ 45 8. Future Work and Limitations ........................................................................................... 48 9. Summary .......................................................................................................................... 49 II 10. Bibliography ................................................................................................................. 51 11. Appendix ...................................................................................................................... 57 III II. Abstract Offline evaluations in recommender system research depend heavily on datasets, many of which are pruned, such as the widely used MovieLens collections. This thesis examines the impact of data pruning - specifically, removing users with fewer than a specified number of interactions - on both dataset characteristics and algorithm performance. Five benchmark datasets were analysed in their unpruned form and at five successive pruning levels (5, 10, 20, 50, 100). For each coreset, we examined structural and distributional characteristics and trained and tested eleven representative algorithms. To further assess if pruned datasets lead to artificially inflated performance results, we also evaluated models trained on the pruned train sets but tested on unpruned data. Results show that commonly applied core pruning can be highly selective, leaving as little as 2% of the original users in some datasets. Traditional algorithms achieved higher nDCG@10 scores when both training and testing on pruned data; however, this advantage largely disappeared when evaluated on unpruned test sets. Across all algorithms, performance declined with increasing pruning levels when tested on unpruned data, highlighting the impact of dataset reduction on the performance of recommender algorithms. IV III. Acknowledgements I would like to express my gratitude to Prof. Dr Ing. Beel, as well as to Lukas Wegmeth and Tobias Vente, for their supervision and valuable guidance and insight throughout the creation of this thesis. Throughout the course of writing this thesis, ChatGPT was used as an auxiliary writing tool, particularly for phrasing suggestions and proofreading. All suggestions from ChatGPT were carefully reviewed, edited and adapted to ensure correctness and consistency with my own voice. This practice aligns with the guideline that generative models can be used to support writing, but their use should be disclosed transparently. [7] The code for all experiments conducted during this thesis is openly accessible at GitHub [51]. 1 1. Introduction 1.1 Background The performance of recommender systems is typically assessed through offline evaluations, online evaluations, or user studies. Offline evaluations - training and testing algorithms on datasets - are the most common first step to narrow down promising algorithms. [14, 41, 54] Ideally, the datasets for offline evaluations should match the intended application domain [15, 25]; nonetheless, many studies rely on a small set of benchmark datasets, some of which have been pruned, meaning specific interactions have been removed [6, 8, 26, 27, 39, 47]. Data pruning has been shown to alter structural characteristics of datasets, which are key factors in explaining performance variance across datasets (e.g. density) [1, 15, 18, 20]. A commonly applied pruning technique is the systematic removal of low-activity users from datasets [20, 54, 55]. These so-called cold-start users are a recognised challenge in recommender-system research, and removing them from both the training and test sets effectively avoids evaluating recommender performance on this group [20, 24, 30]. This can lead to artificially inflated performance scores [8] and limit generalizability [6, 15, 20], as pruning causes the test data to diverge from real-world scenarios, resulting in datasets composed primarily of high-activity users with rich interaction histories [20]. As more complex models emerge and the demand for sustainable recommender systems grows, interest in data reduction techniques such as data pruning has increased [5, 9, 43, 45, 49, 58], even though data pruning is recognised as a barrier to comparability and reproducibility in recommender-system research [17, 22, 27, 47, 54]. These contrasting developments warrant closer examination of how data pruning affects both dataset characteristics and algorithm performance. 1.2 Research Problem Data pruning has been established as a popular step in dataset preprocessing, as demonstrated in reviews. Sun et al. found that of more than 42 examined papers that explicitly stated preprocessing datasets, 60% applied Data Pruning in this process [47]. This aligns with Beel and Brunel[8], who found that 65% of the papers they examined that employed offline evaluations used either a pruned dataset or pruned a dataset during the preprocessing steps. 2 While data pruning has been proven to positively impact runtime, computational load, and dataset density, seemingly achieving improved performance values [4], a good result in performance evaluation on a pruned dataset is not generally transferable to other datasets. In recommender-system research, so-called cold-start users or items present a well-known challenge, as many algorithms struggle to generate reliable recommendations for users without a rich interaction history. Newly introduced items lacking similarity to existing ones pose the same challenge. [2, 29, 30, 42] These challenging user groups are effectively eliminated when user-based data pruning is applied. As a result, algorithm evaluations on pruned datasets are biased toward high-activity users, since the challenging low-activity group is progressively excluded from both training and testing. [24] In practice, applying the MovieLens-standard 20-core pruning can exclude up to 42% of users while still retaining about 95% of interactions [8]. Algorithm performance partly depends on the dataset structure. For instance, density is a key factor in explaining performance variance. [14, 16] Therefore, it is possible to influence algorithm performance through the deliberate selection of datasets or coresets. However, higher levels of pruning cause a greater divergence from the raw data that a recommender is likely to encounter in real-world scenarios. [20] Consequently, pruned datasets can hinder generalisation and comparability when evaluating algorithms. Although data pruning is a commonly applied procedure, 44% of researchers still do not report the preprocessing applied to the datasets used in their studies, as a recent review showed [55]. This lack of transparency is problematic because it greatly hinders the reproducibility of algorithm performance and the ability to make fair comparisons. In summary, while dataset pruning is a widely acknowledged tool, it poses three key problems: reproducibility suffers when pruning is undocumented, comparability is compromised when coresets are deliberately selected, and generalisability declines as pruned datasets diverge from real-world data. Despite growing awareness of these issues, the combined effects of progressive data pruning on dataset characteristics and algorithm performance remain insufficiently examined. The lack of systematic analysis of how progressive data pruning alters dataset characteristics and affects the measured performance and ranking of recommender algorithms creates uncertainty 3 in evaluating and comparing recommendation models. Addressing this gap is essential for reproducibility, fairness, and the reliable development of recommender systems. 1.3 Research Question Building on the previous section, this thesis aims to mitigate the gap identified in research on data pruning and its impact on algorithm performance and the structural characteristics of datasets. Unlike previous studies, we will investigate the changes in distributional characteristics of the pruned datasets, with a focus on user, item, and interaction retention. By progressively increasing the pruning core-level of datasets, comparing the performance of 11 algorithms on these pruned variations of the datasets and examining the changes in dataset characteristics, we aim to address the following research question: How does progressive dataset pruning by increasing core thresholds affect the performance of recommender systems, the structural and distributional characteristics of the dataset? 1.4 Research Objective The objective of this thesis is two-fold. First, we aim to assess how progressive pruning affects dataset characteristics and to quantify the proportion of the original data that remains at each pruning level. Second, we strive to determine whether progressive pruning alters the ranking of algorithms and whether it artificially inflates their performance scores. 2. Background Recommender systems have become indispensable for delivering personalised recommendations across diverse application areas, including e-commerce, streaming services, social networks, and education [29, 31, 35, 42]. The development of recommender systems requires both models and datasets, yet it is the datasets on which different models are trained, evaluated and compared in offline studies. Consequently, datasets play a foundational role in recommender-system research, and selecting the right dataset is a critical step in the development process [47]. Some datasets have become benchmarks, widely used for training 4 and evaluation in recommender system research. For example, the MovieLens datasets [53], which a recent review found to be used in approximately 50% of studies [2]. The most commonly used variants of the MovieLens datasets are pruned. The term data pruning refers to setting a threshold t for a group (e.g., users or items) and eliminating all interactions from any user or item whose total interactions fall below that threshold [20]. A well-known example is the MovieLens datasets (except ML Full Latest), which apply t = 20 to users, retaining only interactions from users with at least 20 interactions in the original dataset. Such a subset is called 20-core pruned, where core denotes the minimum interaction count required for inclusion. In this thesis, 0-core refers to the full, unpruned dataset, which has only been randomly downsampled. Although data pruning is sometimes also referred to as data filtering, it differs fundamentally from downsampling, in which a subset is drawn (e.g., randomly) without intentionally altering the dataset's inherent characteristics [43]. In contrast, subjecting datasets to pruning methods alters the structural characteristics of the datasets, such as the average number of ratings per user or item, or the density of the dataset, which describes the fraction of ratings in the dataset relative to all possible user-item interactions [1, 18]. These characteristics are part of our analysis and are therefore briefly outlined in the Methodology chapter. 3. Related Work The question of how to create sound evaluation for recommender systems [6, 54] and machine learning algorithms is an active field of research that spans multiple factors, including the impact of reduction strategies (e.g., dataset pruning) on dataset composition and algorithm performance [13, 34, 58]. Various strategies aim to decrease the computational load of evaluations or improve model performance and efficiency by altering the datasets or models, including model pruning (e.g., by removing weights or nodes) [11, 53], downsampling, and the intentional removal of targeted interactions. Removing interactions from a dataset to alter its composition or quality can take several forms, such as data pruning, noise reduction (e.g., via sampling techniques) [28], or the removal of users with a detrimental impact on performance [36]. 5 To the best of our knowledge, Beel and Brunel [8] were the first to review a set of conference papers on dataset usage to determine the prevalence of pruned datasets in recommendersystem evaluations. Their review found that almost 50% of offline evaluations were conducted on at least partially pruned datasets, with the 20-core pruned MovieLens datasets being the most frequently used. In assessing the impact of 20-core pruning on algorithm performance, they compared six collaborative algorithms on an unpruned version of MovieLens with a version containing only users who exceeded the 20-interaction threshold. Their results show increased performance on the pruned dataset; however, it is worth noting that the test sets were also pruned. While this paper provides the foundation for our work, its performance comparison was limited to a single dataset and reported only error-based metrics (RMSE, MAE). In this thesis, we extend the analysis to ranking-based metrics and evaluate a broader variety of algorithms across a wider range of datasets and pruning levels. A recent review on standard practices in recommender evaluation has found that dataset selection and preprocessing techniques, including pruning, vary widely across the field [55]. Zhao et al. found that 44% of studies did not report any preprocessing details and 34% implicitly applied n-core filtering with thresholds of 5 or 10. Thresholds varied widely, with some papers using 25- or 30-core pruning, but the authors' own experiments focused only on 5- and 10-core filters. They observed that these two thresholds produced significantly different rankings on some datasets (e.g., the sparse Amazon datasets). Although they reported the proportion of users excluded, exploring a broader range of core thresholds and conducting a more thorough analysis of changes in the datasets' structural and distributional characteristics would be valuable. Sun et al. [47] had earlier taken a similar approach, finding that 60% of studies applied 5- or 10-core filtering, with some using thresholds as high as 20 or 30. The authors also examined the 5- and 10-core filters, which Zhao et al. later confirmed to be aligned with. Bentzer and Thulin [10] investigated a user-based approach of data pruning by limiting the number of users included in their training sets. The base dataset was MovieLens-25M, already 20-core pruned for users, and it was further filtered to retain only items that had been interacted with by at least 10% of users. The authors then evaluated SVD and IBCF on progressively smaller test sets and found IBCF to achieve higher accuracy on small datasets, whereas SVD was preferable for larger datasets, despite a slight loss in accuracy, because of 6 its lower execution time. Their results highlight that, in certain scenarios, accepting a slight decrease in performance can be a worthwhile trade-off for faster execution. However, because their experiment only included two algorithms, the findings have limited generalisability. Driven by the goal of developing more sustainable and efficient recommender systems, data pruning has also become a focus in studies on data-reduction techniques. Spillo et al. [45] investigated the trade-offs between data reduction and algorithm performance, as well as the impact on the carbon footprint of recommender systems. In their study, the authors reduced the data only after splitting the datasets into training and test sets, thereby manipulating only the size of the training data. They confirmed that data reduction decreases both emissions and performance, while some algorithms showed scenario-dependent potential for achieving acceptable trade-offs. Similarly, Arabzadeh et al. [4, 5] examined whether a trade-off between energy efficiency and accuracy is possible by applying two user-centred downsampling approaches to datasets. Unlike Spillo et al., they also incorporated core pruning of both users and items at levels 10 and 30, which left only a fraction of the datasets viable for evaluation. Their results showed that more data generally had a positive impact on accuracy, but aligned with Spillo et al., that scenario-dependent trade-offs can be found for individual algorithms. While this study makes a valuable contribution to more sustainable recommender systems, the analysis of dataset characteristics did not extend beyond a basic examination of structural properties. Regarding dataset characteristics, Deldjoo et al. [18] proposed a rich framework of structural and distributional properties, which we partially adopt in our analysis. Their focus was on assessing the impact of individual dataset characteristics and identifying those with the most significant influence. Shape, Interactions per User or Item, and Density have emerged as impactful characteristics in their results, showing that deliberate dataset selection, focusing on key characteristics, can directly impact performance. In this thesis, we adopt some of the proposed explanatory variables of their study and examine their evolution across increasingly pruned coresets. Fan et al. [24] explored the MovieLens datasets by selectively removing interactions per user (e.g., the first 15) to assess which interactions are most informative for preference elicitation. They confirmed that MovieLens is a statistical outlier in its distributional characteristics. Their results also showed that, because users were initially encouraged to rate popular items 7 until a threshold was reached during the collection of interactions for the MovieLens datasets, these early 15 interactions contributed little to shaping user preferences. Further research on the influence of the datasets on algorithm performance was also conducted by Chin et al. [15]; however, their dataset selection did not include pruned variations. Doerfel et al. [20] examined how dataset pruning affects the performance ranking of recommender systems in the context of social bookmarking. They found that density increased with higher core levels, but the low-core settings of 2 and 3 were most consistent with the raw data. 4. Methodology 4.1 Datasets This subsection outlines the dataset selection, all preprocessing steps and the train-test splitting strategy applied. Additionally, we outline the key characteristics in our dataset analysis. 4.1.1 Dataset Selection The selection of datasets is based on several reviews, identifying the most commonly used datasets in recommender-system research. The popularity of the MovieLens datasets is undisputed, with a range of 40% [8], 50% [2], and up to 84% [6] of studied papers employing at least one version of MovieLens in their experiments [12, 17, 25, 26, 39, 47, 15]. There are multiple versions of the MovieLens dataset available, with only the MovieLens Full Latest being unpruned and usable for our experiment. [27] In recommender research, the Amazon [38] datasets are also widely adopted, with 35% [8] to 42% [6] of examined papers including at least one version in their experimentation [25, 39, 47]. Another benchmark dataset in recommender-systems research is the Yelp dataset[59], which is employed in multiple versions from different timeframes [47, 17, 26, 15]. The aforementioned datasets are all classed as explicit datasets, expressing users' preferences for items by numerical ratings ranging from 1 (lowest) to 5 (highest). To employ an implicit dataset in our experiment as well, we chose the Gowalla [16] dataset, which consists of point8 of-interest data where every interaction with an item is recognised as a positive preference value, and all non-interacted items are considered negative [25]. For readability, shortened dataset names are used throughout this thesis. The Amazon Toys & Games and Amazon CD & Vinyl datasets are referred to as Amazon Toys and Amazon CD, respectively. Similarly, all references to MovieLens denote the MovieLens Full Latest dataset unless otherwise specified. As the analysis of structural and distributional characteristics of the datasets is part of this study, please refer to Chapter 5.1 for detailed statistics of each pruned version and the nonpre-processed datasets. 4.1.2 Preprocessing of Datasets All datasets selected for the experiment have been submitted to the same procedure of preprocessing, consisting of transformation into an implicit dataset for originally explicit datasets, removal of duplicate interactions, downsampling to a random subset of fixed size, pruning to six different core-subsets, data splitting and the creation of atomic files. The specific actions taken in each of these steps are listed in detail below, in the same sequence as they were executed. Transforming explicit datasets to implicit datasets All datasets, except for the inherently implicit Gowalla dataset, have been transformed into implicit datasets by retaining all interactions with rating values equal to or greater than 4.0 [3, 44]. These ratings were assigned the binary value of 1. Downsampling For all implicit datasets with more than 3,000,000 interactions, a random subset has been extracted using Pandas' sample function on the interaction dataframe. The resulting subset had a length of 3,000,000 for each dataset. Hereinafter, this downsampled dataset will be referred to as the unpruned or 0-core dataset. Coreset Creation Each dataset was retained in an unpruned version, referred to as 0-core, which retained all interactions by all users and with all items included in the downsampled set. Subsequently, the dataset was pruned in five additional coresets, where each core represented the minimum count of interactions a user had to have in the dataset to be retained 9 in the respective coreset. For example, the 5-coreset retained all interactions by users with at least five interactions in the dataset. Each of the selected datasets was pruned into 5-core, 10core, 20-core, 50-core, and 100-core subsets. Please note that the MovieLens datasets are commonly described as 20-core pruned [8], but in practice, the threshold of 20 is only applied to users [27]. Items remain in the dataset even if they have been interacted with only once. Strictly speaking, the term core filtering would be more accurate, since core pruning can imply a recursive process that continues removing interactions until both users and items meet the minimum interaction threshold. For clarity, in this thesis, we use the term '20-core pruned' to denote datasets filtered by a 20-interaction user threshold, which may still include items with as few as one interaction. Data Splitting. For each core subset and the unpruned variant of each dataset, a user-based split into train and test sets is performed, with 80% of interactions per user assigned to the train set and the remaining 20% of interactions per user assigned to the test set. The data splitting was applied via LensKit's user-based partitioning and the SampleFrac partition method, which randomly selects a fraction of test rows per user1. Creating Atomic Files The train and test sets have been transformed into the atomic file format required for use by the RecBole library2. Analysis of Dataset Characteristics One focus of this study is to analyse the dataset characteristics and the impact Data Pruning has on the structural and distributional characteristics of the datasets. Therefore, all datasets intended for the training and testing of the Recommender algorithms were analysed under the following factors to allow further comparison following the framework employed by Deldjoo et al. [18]: Structural Characteristics Distributional Characteristics Total Number of Interactions (# Interactions) = |Int| Total Number of Users (# Users) = |U| Total Number of Items (# Items) = |I| Space Size = |UU| × |II| Average Number of Interactions per User: (Int /U) = |IIIIII| |UU| Average Number of Interactions per Item: (Int / I) = |IIIIII| |II| 1 https://lkpy.lenskit.org/0.14.4/crossfold#lenskit.crossfold.partition_users 2 https://recbole.io/docs/user_guide/data/atomic_files.html 10 Shape = |UU| |II| Density = |IIIIII| |UU|×|II| Sparsity = 1 -DDDDDDDDDDDDDD GiniItem = 1 -2 ∑ |II|+1-ii |II|+1 × |IIIIIIii| |IIIIII| |II| ii=1 GiniUser = 1 -2 ∑ |UU|+1-uu |UU|+1 × |IIIIIIUU| |IIIIII| |UU| uu=1 These equations assume items and users are sorted according to the number of interactions associated with them (IntI and IntU) Table 1 Structural and Distributional Characteristics: Overview of unprocessed datasets showing total counts (#), interactions (Int), users (U), and items (I), along with key characteristics (Avg. Int/User, Avg. Int/Item, shape, sparsity, Gini coefficients, and feedback type). Abbreviations are as indicated. Basic dataset characteristics are described by the total number of interactions (all user-item events), the total number of users (all unique individuals) and the total number of items (all unique entities available for interaction). Space Size measures the capacity of the Interaction Matrix, allowing for the comparison of datasets in terms of the maximum number of preferences that can be collected from users. Shape indicates the user-to-item ratio, which can help foresee if item-based or user-based algorithms might be advantageous. Density measures the number of observed interactions in relation to the maximum number of possible interactions. Sparsity describes the proportion of missing interactions in a dataset by subtracting the Density from 1. Average Number of Interactions per User / Item describes the mean number of interactions a user or an item has in this dataset. GiniItem measures the rating frequency distribution for items, ranging from 0, which signifies that all items receive the same number of interactions, to 1, which indicates absolute inequality in interaction distribution, e.g., one item receiving all interactions. GiniUser describes the rating frequency distribution for users, where 0 signifies that all users contribute the same number of interactions in the dataset, and 1 represents that only one user is the sole contributor of interactions. 4.2 Algorithm Selection We selected a diverse set of algorithms from the LensKit [21] and RecBole [56] libraries. Random and PopScore served as baselines. Biased MF and Implicit MF represented traditional matrix-factorisation methods [48, 57], while User KNN [23] and Item KNN [19, 11 23] covered neighbourhood-based approaches. From RecBole, we included the BPR algorithm [40], SimpleX [33], and several deep-learning recommenders chosen with a focus on minimising energy consumption: DiffRec [50], DMF [52], and MultiVAE [32]. Hereinafter, we clustered the models into three groups. Group 1 consists of traditional recommender algorithms: BiasedMF, ImplicitMF, UserKNN, ItemKNN Group 2 is composed of modern recommender models: DiffRec, DMF, MultiVAE, SimpleX Group 3 holds a baseline from each library: PopScore, Random Library Models LensKit PopScore Biased MF Implicit MF User KNN Item KNN RecBole Random BPR DiffRec DMF MultiVAE SimpleX Table 2: Overview of Models and Libraries included in the Experiment 4.3 Training and Testing In the first phase, the training and test sets were created from the respective core-pruned dataset. That means that both the training data and the test data were core-pruned. Unpruned Dataset Splitting [80%, 20%] Train Set Unpruned Test Set Unpruned Core-Pruned Dataset Splitting [80%, 20%] Core Train Set Core Test Set Figure 1 Dataset Splitting for Phase 1 and 2 12 Figure 2 Training and Testing of Recommender Algorithms employed in 2 Phases In the second phase, the recommenders were trained on data from the core pruned datasets, while the test data was taken from the unpruned test set. To achieve this, a test set was created from the unpruned data, which was disjoint from the core pruned training data and downsampled to the same size as the core pruned test data. By excluding possibly overlapping interactions and downsampling, a fair comparison of evaluation results on unseen data under similar conditions was enabled. This comparison procedure was executed for each core subset of each dataset. All recommender algorithms were trained with their default parameter as set by their respective library. The neighbourhood-based algorithms were trained with the maximum number of neighbours set to 20, and the Matrix Factorisation algorithms, BiasedMF and ImplicitMF, were trained with 50 features. As the primary objective of this thesis was not to achieve the highest possible performance scores, but to examine changes in rank across coresets, no hyperparameter tuning was applied. Ranking-Based Metrics The evaluation of Recommender algorithms' performance on ranking tasks can be achieved through a wide variety of metrics. Research surrounding ranking tasks typically employs at least one accuracy metric, e.g. Recall or Precision, or nDCG@k in evaluation. nDCG (normalised Discounted Cumulative Gain) measures how well relevant items are ranked near the top of the recommendation list, with scores ranging from 0 for a poor list of recommendations to 1 for a perfect list of recommendations with every item at the ideal rank. Recall measures the proportion of all relevant items that appear in the list of recommendations, indicating how completely the algorithm retrieves the items of interest. Phase 1: Train on Core Train Set Test on Core Train Set Create Unpruned Test Set - equal size of Core Test Set - disjunct to Core Train Set Phase 2: Train on Core Train Set Test on Unpruned Test Set 13 Precision reflects the proportion of recommended items that are actually relevant, assessing the accuracy of the top-ranked results. When either Precision, Recall, or nDCG is evaluated at "@k", this means the top k recommended items are considered when computing the metric. Due to the scope of this thesis, the results will only be reported on the NDCG@10 metric. The results for Precision@10 and Recall@10 are included in the Appendix. 5. Results 5.1 Dataset Analysis In this section, we will provide the results of analysing the dataset characteristics for all datasets. As a baseline, we will first present the characteristics of the unprocessed datasets to allow a comparison with the pre-processed, but unpruned, 0-core datasets. Subsequently, we will show the dataset characteristics for each core-pruned version of the datasets. 5.1.1 Unprocessed Dataset Characteristics Dataset Interactions Users Items Avg. Int / User Avg. Int / Item Shape Sparsity in % Gini User Gini Item Feedback Amazon CD 4458901 434060 1944316 10.27 2.29 0.22 99.9995 0.75 0.50 Explicit Amazon Toys 7998969 624792 4204994 12.80 1.90 0.15 99.9997 0.80 0.40 Explicit Gowalla 3981334 107092 1280969 37.18 3.11 0.08 99.9971 0.66 0.54 Implicit MovieLens FL 33832162 330975 83239 102.22 406.45 3.98 99.8772 0.70 0.95 Explicit Yelp 5261667 1326101 174567 3.97 30.14 7.60 99.9977 0.63 0.72 Explicit Table 3 Baseline Dataset Characteristics: Overview of the unprocessed datasets, reporting average interactions per user and per item (Avg. Int / User; Avg. Int / Item), matrix shape (users × items), sparsity, Gini coefficients for users and items, and feedback Analysing the unprocessed datasets confirms MovieLens as the densest. The total number of users exceeds the total number of items, and the average number of ratings, both per item and per user, are the highest of all tested datasets. While the GiniUser value ranges in the mid ranks across datasets, the MovieLens dataset shows the highest GiniItem value of 0.95, exhibiting the highest inequality in rating frequency across items. This confirms the short head and long tail conformation, with a few popular items receiving large numbers of interactions and a large group of items receiving very few interactions. 14 The Amazon datasets are the sparsest in the experiment, exhibiting high levels of inequality in rating frequency across users. Still, they have the lowest GiniItem values among the tested datasets. In terms of the average number of interactions, both Amazon datasets rank in the mid-range for interactions per user but show the lowest average number of interactions per item in the experiment. The Gowalla dataset displays the smallest Shape, with the total number of users being exceeded by the total number of items. While the average number of interactions per user is the second highest in the experiment, the average number of interactions per item is lower compared to the other datasets. Besides MovieLens, the Yelp dataset is the only one displaying high values for both Shape and GiniItem, as well as the second-highest average number of interactions per item. In terms of rating frequency per user, the Yelp dataset has a lower rating frequency than the compared datasets, with a GiniUser of 0.63. Comparing the characteristics of the unprocessed dataset to the downsampled, yet unpruned, versions of the 0-Core datasets, the relative ranking based on these statistics remained largely consistent. 15 5.1.2 Core-Pruned Dataset Characteristics 0-Core Pruning Dataset Interactions Users Items Avg. Int. / User Avg. Int. / Item Space Size Shape Sparsity Gini User Gini Item Amazon CD 3000000 376115 1471494 7.98 2.04 553450965810 0.26 99.9995% 0.73 0.45 Amazon Toys 3000000 406436 1956864 7.38 1.53 795339976704 0.21 99.9996% 0.74 0.30 Gowalla 3000000 104439 1105866 28.72 2.71 115495539174 0.09 99.9974% 0.66 0.51 MovieLens FL 3000000 268342 34889 11.18 85.99 9362184038 7.69 99.9680% 0.64 0.91 Yelp 3000000 949214 163998 3.16 18.29 155669197572 5.79 99.9981% 0.57 0.73 5-Core Pruning Dataset Interactions Users Items Avg. Int. / User Avg. Int. / Item Space Size Shape Sparsity Gini User Gini Item Amazon CD 2515024 107132 1308076 23.48 1.92 140136798032 0.08 99.9982% 0.60 0.42 Amazon Toys 2499239 101882 1698279 24.53 1.47 173024061078 0.06 99.9986% 0.59 0.28 Gowalla 2942493 77873 1089959 37.79 2.70 84878377207 0.07 99.9965% 0.57 0.51 MovieLens FL 2718416 132958 34416 20.45 78.99 4575882528 3.86 99.9406% 0.48 0.91 Yelp 1730318 129820 143934 13.33 12.02 18685511880 0.90 99.9907% 0.45 0.72 10-Core Pruning Dataset Interactions Users Items Avg. Int. / User Avg. Int. / Item Space Size Shape Sparsity Gini User Gini Item Amazon CD 2180293 55519 1177234 39.27 1.85 65358854446 0.05 99.9967% 0.54 0.41 Amazon Toys 2197395 55611 1532748 39.51 1.43 85237649028 0.04 99.9974% 0.53 0.26 Gowalla 2830783 61579 1061525 45.97 2.67 65367647975 0.06 99.9957% 0.53 0.50 MovieLens FL 2377431 81663 34034 29.11 69.85 2779318542 2.40 99.9145% 0.40 0.90 Yelp 1214677 49057 129533 24.76 9.38 6354500381 0.38 99.9809% 0.41 0.71 20-Core Pruning Dataset Interactions Users Items Avg. Int. / User Avg. Int. / Item Space Size Shape Sparsity Gini User Gini Item Amazon CD 1804688 27582 1019721 65.43 1.77 28125944622 0.03 99.9936% 0.47 0.38 Amazon Toys 1830233 28463 1325512 64.30 1.38 37728048056 0.02 99.9951% 0.47 0.24 Gowalla 2478446 36732 976549 67.47 2.54 35870597868 0.04 99.9931% 0.46 0.49 MovieLens FL 1826386 41403 33289 44.11 54.86 1378264467 1.24 99.8675% 0.32 0.89 Yelp 802206 17652 111692 45.45 7.18 1971587184 0.16 99.9593% 0.36 0.68 50-Core Pruning Dataset Interactions Users Items Avg. Int. / User Avg. Int. / Item Space Size Shape Sparsity Gini User Gini Item Amazon CD 1268938 9860 775156 128.70 1.64 7643038160 0.01 99.9834% 0.39 0.34 Amazon Toys 1276790 10073 987503 126.75 1.29 9947117719 0.01 99.9872% 0.39 0.20 Gowalla 1776980 14295 793831 124.31 2.24 11347814145 0.02 99.9843% 0.40 0.45 MovieLens FL 894887 10512 31124 85.13 28.75 327175488 0.34 99.7265% 0.24 0.84 Yelp 412749 4317 85747 95.61 4.81 370169799 0.05 99.8885% 0.28 0.62 100-Core Pruning Dataset Interactions Users Items Avg. Int. / User Avg. Int. / Item Space Size Shape Sparsity Gini User Gini Item Amazon CD 870960 4053 573131 214.89 1.52 2322899943 0.01 99.9625% 0.33 0.30 Amazon Toys 855740 3911 702727 218.80 1.22 2748365297 0.01 99.9689% 0.34 0.16 Gowalla 1156429 5210 601281 221.96 1.92 3132674010 0.01 99.9631% 0.36 0.39 MovieLens FL 348196 2329 27076 149.50 12.86 63060004 0.09 99.4478% 0.18 0.76 Yelp 200169 1182 61321 169.35 3.26 72481422 0.02 99.7238% 0.23 0.54 Table 4 Characteristics of the Coresets 16 5.1.3 Structural Characteristics The following section provides a brief overview of the observed changes in structural characteristics after pruning the datasets. 5.1.3.1 Density / Sparsity Across all pruning levels, the ranking of datasets by their density remained largely unaltered from the unpruned versions. 5-Core pruning merely switched the order of the rank-neighbours Yelp and Gowalla, similarly to 50-Core pruning for Gowalla and Amazon CD. The density of all datasets continued to increase with each Core-level. 5.1.3.2 Space Size The Space Size continuously decreased for all datasets with each stage of pruning, with only adjacently ranking datasets switching order for 10-core pruning and 50-core pruning. 5.1.3.3 Shape Across all pruning levels, the Shape values of all datasets decreased continuously, with the largest difference between the unpruned and 5-core pruned datasets reported. 5.1.3.4 Average Number of Interactions per Item The average number of interactions per item varied for the unprocessed datasets, with values ranging from 1.53 for Amazon Toys up to 85.99 for MovieLens. The span between the maximum and minimum values stayed considerably large across all pruning levels, with the rank of datasets by this characteristic entirely unchanged. 5.1.3.5 Average Number of Interactions per User The ranking by Average Number of Interactions per User fluctuated notably more than per Item. Pruning at the lower level of 5-core, the sparse Amazon Toys dataset, ranked fourth for 0-core, switched ranks with the formerly ranked second MovieLens dataset when ranked in descending order by Average Number of Interactions per User. 20-Core pruning resulted in every dataset switching rank with an adjacent dataset, except for the Gowalla dataset, which consistently displayed the highest average number of interactions per user across all core levels. Pruning at thresholds above 50 relegated the MovieLens dataset, which ranked second highest in the unpruned version, to the last rank of all datasets. 17 5.1.4 Distributional Characteristics Figure 3: Boxplots showing the distribution of Gini Coefficients for users and items across pruning levels, where 1 signifies total inequality in interaction distribution and 0 stands for a uniform distribution of interactions 5.1.4.1 GiniUser All datasets had a GiniUser value of more than 0.5 for the downsampled but unpruned version, with the Amazon datasets displaying the maximum values in the experiment of 0.7. For all core-pruned versions, the highest values continued to be those of the Amazon datasets, and MovieLens had the lowest values for all thresholds above 5. The Amazon and Gowalla datasets showed similar decreasing trends and values across all pruning levels, as did Yelp and MovieLens, but their GiniUser values were lower throughout all levels. 5.1.4.2 GiniItem The values for GiniItem consistently stayed in the same order for all pruning levels, with MovieLens displaying the maximum value of 0.9 for the unpruned version and 0.8 for the 100-core pruned version. While each was about 0.2 lower than MovieLens, both the Yelp and Gowalla datasets showed a similar trend, reaching minimum values of 0.5 and 0.4, respectively, in the 100-core version for GiniItem. Overall, the datasets displayed a drop of a maximum of 0.2 from 0-core to 100-core pruned in terms of GiniItem, thus a much lower fluctuation than GiniUser. 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Item Gini of 5 Datasets across 6 Core -Pruned Versions 0-Core Pruning 5-Core Pruning 10-Core Pruning 20-Core Pruning 50-Core Pruning 100-Core Pruning 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 User Gini of 5 Datasets across 6 Core -Pruned Versions 0-Core Pruning 5-Core Pruning 10-Core Pruning 20-Core Pruning 50-Core Pruning 100-Core Pruning 18 Relative Retention Rates In this section, we provide an in-depth analysis of the impact of data pruning with increasing core thresholds on dataset characteristics, with a focus on the extent to which the users, items, and interactions of the unpruned dataset are retained in the pruned datasets. 5.1.4.3 User Retention Rate Across all five examined datasets, pruning has affected the relative retention of users as early as the first pruning level of 5-core. Only one dataset, namely Gowalla, retained more than half of its users compared to the baseline, with a 75% relative retention rate. The lowest user retention rate for 5-core pruning was observed for the Yelp dataset, retaining only 14% of its user base after the first level of pruning was applied. Both Amazon datasets, CD & Vinyl and Toys & Games, reported a relative user retention under 30%, with only a 3% difference between them. The MovieLens dataset retained 50% of the users compared to the unpruned version. Applying 10-core pruning to the datasets only left the Gowalla dataset with more than half of the users, while MovieLens dropped to less than a third of the original user-base, with 30% of users retained. The remaining datasets reported retention rates below 15%, with Yelp dropping as low as 5% relative user retention. 28% 15% 7% 3% 1% 25% 14% 7% 2% 0.96% 75% 59% 35% 14% 5% 50% 30% 15% 4% 0.87% 14% 5% 1.86% 0.45% 0.12% 5 10 20 50 100 CORE PRUNING APPLIED TO DATASETS RELATIVE USER RETENTION AMAZON CD & VINYL AMAZON TOYS & GAMES GOWALLA MOVIELENS FL YELP Figure 4 User Retention Across Core Levels: Grouped bar chart showing, for each pruning level, the proportion of users retained in each dataset relative to its 0-core (unpruned) set. 19 The commonly executed 20-core pruning reduced the user numbers for the MovieLens dataset by half compared to the 10-core version, with a user retention rate of 15%. Both Amazon datasets dropped to values below 10% and the Yelp dataset retained merely 1.86% of its original user base. Gowalla continued to maintain the largest relative share of users throughout the remaining core levels of 20, 50, and 100, with 35%, 14%, and 5% respectively. With the exception of the Gowalla dataset, pruning at the core level of 50 dropped all remaining datasets to below 5% and pruning at the highest examined level of 100 dropped to below 1% of users retained compared to the unpruned dataset. The lowest user retention rate was examined for the 100-core pruned Yelp dataset, reporting 0.12% of the original user base in the coreset. 5.1.4.4 Item Retention Rate Figure 5 Item Retention Across Core Levels: Grouped bar chart showing, for each pruning level, the proportion of items retained in each dataset relative to its 0-core (unpruned) set. Pruning datasets affects the relative retention of items compared to the unpruned datasets. Similar to the user retention rate, the item retention rate dropped continuously as the pruning levels rose, but for items, the retention rates stayed higher than for users. 89% 80% 69% 53% 39% 87% 78% 68% 50% 36% 99% 96% 88% 72% 54% 99% 98% 95% 89% 78% 88% 79% 68% 52% 37% 5 10 20 50 100 CORE PRUNING APPLIED TO DATASETS RELATIVE ITEM RETENTION AMAZON CD & VINYL AMAZON TOYS & GAMES GOWALLA MOVIELENS FL YELP 20 Examining the 5-core pruned datasets, all datasets retained more than 85% of items compared to the unpruned dataset, with MovieLens and Gowalla only dropping by 1% to 99% of item retention. Both Amazon datasets and the Yelp dataset retained around 80% of items. 10-core pruning the datasets resulted in item retention rates dropping by between 1% for the MovieLens dataset and 9% for both Amazon datasets, and the Yelp dataset. The popular 20-core pruned versions of the datasets continued to show similar retention rates for the two Amazon datasets and the Yelp dataset, dropping to just below 70% item retention. The MovieLens dataset retained the most items with 95% item retention, and the Gowalla dataset retained 88% of items. Pruning with a core level of 50 left the Amazon datasets and the Yelp dataset with about 50% of the original items. The Gowalla dataset reported a retention rate of 72% and the MovieLens dataset dropped just below 90% of item retention. The trend continued for Core 100: the Yelp and Amazon datasets dropped to rates between 36% and 39%, while Gowalla retained about half of the items compared to the unpruned dataset. The MovieLens dataset showed an item retention rate of 78% for the highest level of pruning examined in the study. 21 5.1.4.5 Interaction Retention Rate Figure 6 Interaction Retention Across Core Levels: Grouped bar chart showing, for each pruning level, the proportion of interactions retained in each dataset relative to its 0-core (unpruned) set. Analysis of the retained interactions after pruning the datasets revealed differences in the extent to which the datasets were affected by the pruning levels. For the first level of pruning to core 5, the majority of datasets remained with more than 80% of interactions, with Gowalla showing the highest interaction retention rate of 98%. The Yelp dataset was reduced by 42% through 5-core pruning, retaining 58% of the interactions. This trend continued for 10-core pruning: both Amazon datasets and the MovieLens dataset retained around 70-80% of their interactions, while Gowalla retained the biggest relative portion of 94% of interactions. Pruning the Yelp dataset with a threshold of 10 dropped the interaction retention below the halfway mark, with 40% of interactions retained in the 10coreset. Pruning with 20-core levels, all datasets dropped by between 10-20% of interaction retention rates, with Yelp reaching as low as 27% and Gowalla retaining 83% of interactions. The 50-core pruning had the most significant impact on the MovieLens dataset, resulting in a 30% drop in interaction retention, which led to lower retention rates compared to the Amazon datasets, at 42% and 43%. 84% 73% 60% 42% 29% 83% 73% 61% 43% 29% 98% 94% 83% 59% 39% 91% 79% 61% 30% 12% 58% 40% 27% 14% 7% 5 10 20 50 100 CORE PRUNING APPLIED TO DATASETS RELATIVE INTERACTION RETENTION AMAZON CD & VINYL AMAZON TOYS & GAMES GOWALLA MOVIELENS FL YELP 22 Pruning with a threshold set to 100 only retained more than a third of the original interactions in the Gowalla dataset and displayed retention rates below 15% for both the MovieLens and Yelp datasets. 5.2 Algorithm Performance Analysis This chapter is organised into three parts to provide a comprehensive overview of the results obtained by training and testing 11 recommender algorithms on progressively pruned versions of five datasets. The first section presents the overall performance trends observed for the three algorithm groups of traditional, modern, and baseline algorithms. The second section reports the results for each group when both training and testing were conducted on the pruned datasets. The final section presents the outcomes of training performed on pruned datasets while testing on unpruned datasets. 5.2.1 Overall Performance Trends This section summarises the overall performance patterns observed for three algorithm groups across all pruning levels, when trained and tested on pruned data. Group 1 consists of traditional algorithms: User KNN, Item KNN, Biased MF, Implicit MF and BPR. Group 2 comprises the modern algorithms we have selected: DiffRec, DMF, MultiVAE, and SimpleX. Group 3 comprises the two baseline algorithms chosen for the experiments: Random and PopScore. Figure 7 Average nDCG@10 Across Pruning Levels: Average nDCG@10 of traditional, modern, and baseline algorithms on five datasets, trained and tested on equally pruned data. 0.0000 0.0100 0.0200 0.0300 0.0400 0.0500 0 10 20 30 40 50 60 70 80 90 100 AVERAGE NDCG@10 CORE PRUNING APPLIED AVERAGE NDCG@10 ACROSS PRUNING LEVELS ON 5 DATASETS (PRUNED TRAIN & TEST DATA) Group 1 (Traditional) Group 2 (Modern) Group 3 (Baseline) 23 For the traditional algorithms (Group 1), the average nDCG@10 measured across five datasets showed consistent values for 5-core and 10-core pruning, with an overall increase in nDCG@10 for the 50-core and 100-core pruned versions of the datasets. The second group, modern algorithms, on average, did not improve performance with progressively pruning the datasets. The baseline algorithms in Group 3 overall showed almost no change in nDCG@10. While the averaged nDCG@10 for the three groups displayed a maximum difference of 0.01, the performance of the algorithms in these groups showed notable fluctuations. Figure 8 Average Relative Change (%) in nDCG@10 vs. 0-core Baseline: Heatmap showing the percent change in nDCG@10 at each pruning level relative to the unpruned (0-core) performance; positive values (blue) indicate improvement, negative values(red) indicate decline. Comparing the averaged nDCG@10 values per algorithm showed both positive and negative relative changes compared to training and testing on the 0-core version of the datasets. Implicit MF (+ 96%) and Random (+ > 100%) displayed high relative differences to their results on the 0-coresets, while DiffRec had an average relative change of -80% for 100-core pruned datasets. This variance in Average Relative Change highlights pronounced differences between the reactions to pruning levels, prompting a more in-depth analysis of algorithm performance. In the following subsection, we will analyse the performance of the algorithms within the groups in more detail. Algorithms BiasedMF 100% BPR DiffRec DMF ImplicitMF Item KNN MultiVAE PopScore Random SimpleX User KNN -100% Cores 0 5 10 20 50 100 Average Relative Change in nDCG@10 across Pruning Levels 24 5.2.2 Training and Testing on Pruned Datasets In this section, we will provide an overview of the algorithms' performance, both trained and tested on the datasets in the core-pruned versions (Phase 1). The algorithms are grouped into three categories: Group 1, which includes traditional recommender algorithms such as User KNN; Group 2, comprising modern algorithms, e.g., DiffRec and DMF; and Group 3, consisting of the two included baselines: Random and PopScore. 5.2.2.1 Group 1: Traditional Recommender Algorithms Trained and Tested on Pruned Data Figure 9 Relative Change in nDCG@10 Across Pruning Levels for Traditional Algorithms: Percentage change in average nDCG@10 of BiasedMF, ImplicitMF, UserKNN, ItemKNN, and BPR compared to their performance on the unpruned (0core) datasets. Measuring the relative change in average nDCG@10 between each core version and the baseline of 0-core, deviations from the baseline were reported for all included algorithms (Figure 9). Biased MF dropped by 6% for 5-core and 13% for 10-core pruning, when compared to the average nDCG@10 achieved on the 0-core datasets. For 50-core and 100-core pruning, Biased MF increased in average nDCG@10 by 23% and 77% respectively. Implicit MF displayed negligible changes in performance for 5- and 10-core pruning (< 3%), but improved performance for the remaining pruning levels of 20, 50 and 100. The highest -20% 0% 20% 40% 60% 80% 100% 0 10 20 30 40 50 60 70 80 90 100 RELATIVE CHANGE IN AVERAGE NDCG@10 COMPARED TO 0-CORE PRUNED DATASET CORE PRUNING APPLIED RELATIVE CHANGE IN NDCG@10 ACROSS PRUNING LEVELS GROUP 1: TRADITIONAL ALGORITHMS Biased MF Implicit MF User KNN Item KNN BPR 25 measured values in relative change of this group were calculated for Implicit MF at 50-core (52%) and 100-core (96%). Across the five datasets, User KNN experienced an average decrease in nDCG@10 for 5-core pruning (-3%) and 10-core pruning (-5%), while showing notable improvements in performance for the 50-core (36%) and 100-core (70%) pruned datasets. Item KNN displayed the highest increase for the first two pruning levels of 5 (7%) and 10 (13%), ranked second best for 50-core pruning (44%), but showed less improvement for 100core pruning than the majority of algorithms in the group (40%). The final algorithm of Group 1, BPR, showed negligible improvement for the first two levels of pruning, with under 5% improvement for both 5-core and 10-core pruning. The relative change increased for 20-core pruning (17%), 50-core pruning (36%), and 100-core pruning (21%). 5.2.2.2 Group 2: Modern Recommender Algorithms Trained and Tested on Pruned Data Figure 10 Relative Change in nDCG@10 Across Pruning Levels for Modern Algorithms: Percentage change in average nDCG@10 of DiffRec, DMF, MultiVAE and SimpleX compared to their performance on the unpruned (0-core) datasets. -100% -80% -60% -40% -20% 0% 20% 40% 60% 80% 100% 0 10 20 30 40 50 60 70 80 90 100 RELATIVE CHANGE IN AVERAGE NDCG@10 COMPARED TO 0-CORE PRUNED DATASET CORE PRUNING APPLIED RELATIVE CHANGE IN NDCG@10 ACROSS PRUNING LEVELS GROUP 2: MODERN ALGORITHMS DiffRec DMF MultiVAE SimpleX 26 The second group of algorithms, comprised of the modern recommenders DiffRec, DMF, MultiVAE, and SimpleX, deviated from the traditional recommender group in multiple pruning stages. Comparing the average nDCG@10 across five datasets, DiffRec improved by 25% relative to the averaged performance on the 0-core versions of the datasets for 5-core pruning. The second pruning level of 10-core yielded a 16% increase, while 20-core pruning displayed negligible changes in performance (1%). For both 50-core and 100-core pruning, DiffRec's performance decreased by 52% and 80%, respectively. In contrast to DiffRec, DMF displayed a decline in performance for pruning cores 5, 10 and 20, a negligible increase for 50-core pruning (< 5%) and overall improved most on the 100core pruned dataset by 48%. MultiVAE was the only observed algorithm in this group to exhibit a continuous decline in average nDCG@10 across all datasets for each pruning level. Even 5-core and 10-core pruning decreased performance compared to the unpruned dataset by over 20%, and pruning at levels equal to or higher than 50 reduced the performance by 50% on average. As the final algorithm of this group, SimpleX displayed minor relative changes compared to 0-core datasets for 5-core and 10-core pruning (< -5%), but consistently increased in average nDCG@10 for 20-core (16%), 50-core (56%) and 100-core (44%) pruned datasets. The relative change for 20-core and 50-core was the highest relative improvement observed in this group. 5.2.2.3 Group 3: Baseline Recommender Algorithms Trained and Tested on Pruned Data Group 3, containing only the two baseline algorithms PopScore and Random, displayed the overall lowest nDCG@10. Random only produced a measurable value for nDCG@10 on the MovieLens dataset for the 0-core and 5-core versions, as well as the Yelp dataset for the 10-core versions. Each subsequent pruning stage added another dataset to the set of those generating a measurable value for nDCG@10, resulting in relative improvements of 350% for 50-core and 850% for 27 100-core. Executing the Random algorithm on the Amazon Toys & Games dataset yielded no observable value for nDCG@10 regardless of pruning intensity. In contrast to Random, training and testing the PopScore algorithm achieved measurable nDCG@10 values above zero across all core-pruned versions of all datasets. While improvements were observed for the 5-core version of Gowalla, the overall nDCG@10 declined by 9% for the 5-core pruned versions of the datasets. Pruning at 10 and 20 decreased the average performance by 18% and 17%, respectively, compared to the 0-core pruned datasets. Only 100-core pruning achieved an improved average nDCG@10 (29%). 5.2.3 Comparison of Testing on Pruned and Unpruned Testsets Building on the previous analyses, this section focuses on the setup for comparison. For clarity, we will distinguish between two experimental phases: Phase 1, also represented by the previous section of the Results chapter, refers to training and testing the algorithms on pruned datasets. Phase 2 denotes the setting in which the training was executed on the pruned training sets, whereas the testing was performed on the unpruned test sets. The 0-coresets were excluded from Phase 2, as the training data in these coresets was not pruned. To facilitate a direct comparison of the influence data pruning has on performance scores under these two evaluation setups, a ranking of the algorithms by descending performance (measured in nDCG@10) was created and visualised in a heatmap (Figure 11). Rank 1 (dark blue) signifies the best performance, as measured by the average nDCG@10 of all algorithms on this particular coreset of the respective dataset. In contrast, Rank 11 (dark red) marks the lowest measured performance score. 28 Figure 11 Algorithm Ranking Across Two Evaluation Phases: Two-part heatmap showing algorithm ranks (based on nDCG@10) across five datasets and six pruning levels. The upper panel depicts Phase 1 (training and testing on equally pruned data), while the lower panel shows Phase 2 (training on pruned data and testing on unpruned data). Datasets Cores 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 Algorithms BiasedMF Rank 1 BPR DiffRec DMF ImplicitMF Item KNN MultiVAE PopScore Random SimpleX User KNN Rank 11 Cores 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 Datasets Cores 5 10 20 50 100 5 10 20 50 100 5 10 20 50 100 5 10 20 50 100 5 10 20 50 100 Algorithms BiasedMF Rank 1 BPR DiffRec DMF ImplicitMF Item KNN MultiVAE PopScore Random SimpleX User KNN Rank 11 Cores 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 0 5 10 20 50 100 Ranking of Algorithms by NDCG@10 (Trained on Pruned and Tested on Unpruned Data) Ranking of Algorithms by NDCG@10 (Trained and Tested on Pruned Data) Amazon CD Amazon Toys Gowalla MovieLens Yelp Amazon CD Amazon Toys Gowalla MovieLens Yelp 29 In Phase 1, User KNN ranked highest for the entirety of three datasets, on several coresets of the remaining two datasets (MovieLens, Yelp) and displayed the highest consistency in ranking of both traditional and modern algorithms. On the Yelp dataset, User KNN was outperformed by Implicit MF on three coresets: 5-core, 10-core and 20-core. The MovieLens dataset exhibited more fluctuations for the higher ranks, with User KNN ranking third or fourth on all coresets above 5-core, and Item KNN rising to first rank for 5-core, 10-core, and 20-core pruning. DiffRec ranked second for the 5-core and 10-core subset, and DMF took the lead for the 100-core pruned subset of MovieLens. PopScore ranked highest on the 50-core subset of MovieLens. Phase 1 was characterised by consistent top rankings of User KNN, Item KNN, and Implicit MF, with Item KNN and Implicit MF exhibiting rank fluctuations primarily across entire datasets and to a lesser extent on singular coresets. Algorithms at mid-level ranks displayed a higher frequency of rank changes, often within a dataset. One example of this behaviour would be MultiVAE across the MovieLens dataset, where its rank fluctuated between fourth and ninth place. DiffRec achieved varying ranks both across entire datasets and coresets, with ranking position changing from second to as low as tenth across the Yelp dataset and almost a different rank for every coreset on MovieLens (only 10-core and 20-core ranked second). The only algorithm without any fluctuations in ranking in Phase 1 was Random, which continuously ranked last (11th out of 11 algorithms). Almost no variation in rank was also observed for Biased MF, which ranked 10th on all but three coresets in Phase 1. In summary, rank changes were observed frequently in Phase 1, both per dataset (e.g., SimpleX on MovieLens) and per coreset, as evidenced by PopScore for both MovieLens and Yelp. In comparison to Phase 1, the top contender User KNN displayed both more frequent and more extreme rank changes in Phase 2: While the lowest rank in Phase 1 was a singular fourth rank on 50-core pruned MovieLens, User KNN held a rank equal to or below sixth for 48% of coresets in Phase 2 (12 of 25). As the second neighbourhood-based algorithm, Item KNN also 30 consistently showed a lower rank than in Phase 1, ranking overall eighth for both entire datasets and achieving its best ranking of fifth. Implicit MF, which even ranked first for three coresets in Phase 1, mostly ranked fifth to seventh, and on the 20-core subset of MovieLens, even ranked ninth. Biased MF was ranked 10th for 68% (17 of 25) of coresets in Phase 2, with a slightly higher frequency of rank changes on Amazon Toys & Games and the Yelp dataset. BPR was observed to rank fourth for 68% (17 of 25) in Phase 1, compared to ranking second for 60% of coresets in Phase 2. On the 100-core pruned subset of Yelp, BPR even ranked best of all algorithms. MultiVAE exhibited difficulties in recommending on the Amazon Toys dataset in Phase 2, consistently ranking 10th place. While the performance was in the lower ranges for the Amazon datasets, on the remaining three datasets, MultiVAE consistently outperformed the majority of algorithms: On Yelp, MultiVAE held the second rank for 50-core and 100-core, on MovieLens for all but 50-core, and on Gowalla, the rank varied between fourth and fifth. DiffRec placed higher in the ranks in Phase 2 compared to the first test phase, still overall performing best on Yelp and MovieLens, as it did in Phase 1. Rank changes were observed less frequently compared to Phase 1. DiffRec performance notably declined with each progressive pruning stage on the Amazon Toys & Games dataset, ranking fifth on 5-core, sixth on 10-core and 20-core, and ninth on 50-core and 100-core. However, the absolute nDCG@10 values were observed to be higher in Phase 2 than in Phase 1. On the 5-core pruned dataset, DiffRec achieved a nDCG@10 of 0.0153 in Phase 1, compared to 0.1155 in Phase 2. DMF was the only algorithm showing improvements in rank by performance on the more severely pruned version of MovieLens. Contrary to the behaviour examined for the remaining algorithms, DMF achieved a higher rank for each core level on MovieLens, with a first rank for both the 50-core and 100-core models. On Amazon Toys & Games, no rank variation was observed for DMF, which consistently ranked third best for all coresets. SimpleX achieved an undisputed first rank on three datasets (both Amazon and Gowalla), ranked between fifth and sixth on all coresets of MovieLens and showed minor fluctuations within the top 3 ranks on the Yelp coresets. Its average nDCG@10 across all coresets was 31 1276% higher than in Phase 1, and the absolute nDCG@10 values were the highest observed in the entire experiment. Average nDCG@10 across 5 Pruning Levels (Trainset Pruned, Testset Unpruned) Cores 5 10 20 50 100 Biased MF 0.0001 0.0001 0.0002 0.0001 0.0001 Implicit MF 0.0013 0.0013 0.0020 0.0014 0.0014 User KNN 0.0018 0.0018 0.0025 0.0019 0.0017 Item KNN 0.0010 0.0011 0.0018 0.0010 0.0009 BPR 0.0628 0.0509 0.0364 0.0213 0.0136 DiffRec 0.1155 0.0705 0.0194 0.0043 0.0022 DMF 0.0078 0.0073 0.0056 0.0059 0.0048 MultiVAE 0.0102 0.0087 0.0087 0.0045 0.0041 SimpleX 0.2216 0.1702 0.1123 0.0540 0.0253 PopScore 0.0006 0.0005 0.0011 0.0005 0.0003 Random 0.0001 0.0000 0.0000 0.0001 0.0000 Table 5 Average nDCG@10 of Algorithms over 5 Datasets Across 5-Core Pruning Levels in Phase 2, with the highest overall values in the experiment underlined In summary, the traditional algorithms in Group 1 overall displayed a lower performance in Phase 2 compared to Phase 1, while the modern algorithms in Group 2 partially exhibited substantial increases in measured nDCG@10 values. Notably, SimpleX and DiffRec showed an improved performance under the second test configuration when tested on unpruned test sets. In comparison with the 5-core baseline in Phase 2, each additional pruning level resulted in a notable performance decrease observed for all Group 2 (Modern) algorithms. The Interpretation and Discussion chapter will elaborate on this particular finding. 32 6. Discussion and Interpretation 6.1.1 Dataset Characteristics In this section, we examine how progressive pruning influenced key characteristics of the datasets. We begin by analysing the impact of data pruning on structural properties, e.g. density and the average number of interactions per user and per item, as well as distributional characteristics, such as the Gini coefficients of users and items. Secondly, we focus on retention rates, discussing the extent of data loss observed across increasing levels of pruning. 6.1.1.1 Structural Dataset Characteristics Our analysis of the datasets' characteristics revealed that pruning affected all of the examined datasets, though to varying extents. The removal of users and their interactions from a dataset resulted in an unsurprising decrease in the total number of interactions, users, and items for each core-level pruning applied to the datasets. Subsequently, structural characteristics dependent on at least one of these three total numbers were equally impacted by pruning: Space Size, which describes the maximum possible interactions calculated as users × items, and Shape, denoting the relation of the total number of users to the total number of items, continuously decreased with each pruning stage. Figure 12 Average nDCG@10 of Traditional and Modern Algorithms Across Datasets: Average nDCG@10 for five datasets at different core-pruning levels when training and testing are both performed on equally pruned data. Density, a key factor in explaining performance variance [1], was observed to increase with each core level across all examined datasets in our study. While the sparser datasets, Amazon CD and Amazon Toys, showed improved performance for both the traditional and baseline 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 0.0350 0.0400 0 10 20 30 40 50 60 70 80 90 100 AVERAGE NDCG@10 CORE PRUNING APPLIED AVERAGE NDCG@10 OF TRADITIONAL & MODERN ALGORITHMS ACROSS DATASETS (PRUNED TRAIN & TEST DATA) Amazon CD Amazon Toys Gowalla MovieLens Yelp 33 algorithms on coresets with higher density, this was not the case for MovieLens. Being the densest dataset in the experiment, MovieLens exhibited distinct behaviour: despite increasing density with progressive pruning, both groups of traditional and modern algorithms showed no performance gains. On the contrary, for most core levels, declining performance was observed on the MovieLens datasets as seen in Figure 12. These findings indicate that density alone cannot account for the performance variance observed under progressive pruning. Researchers have recently identified "Interactions per User" as another key dataset characteristic when examining performance variance [18]. In our experiment, the Yelp dataset showed the most significant rise in Average Interactions per User, increasing by 322% (from 3.16 to 13.33) for 5-core pruning up to 5258% (from 3.16 to 169.35) for 100-core pruning. In Phase 1, the average nDCG@10 indeed increased for the majority of algorithms alongside the Average Interactions per User on the Yelp dataset, with a noticeable drop in performance for 100-core pruning. A similar trend was observed on the Gowalla dataset, which showed the overall smallest increase in Interactions per User - from 32% (from 28.72 to 37.79) at 5-core to 673% (from 28.72 to 221.96) at 100-core pruning. Algorithm performance did not decline even for the 100-core subset of Gowalla, suggesting that additional factors influence both the dataset characteristics and algorithm performance. The impact of pruning on Average Interactions per Item was less pronounced than on interactions per user, but differed considerably among datasets. At 5-core pruning, Gowalla remained unaffected, while Yelp displayed a 34% (from 18.29 to 12.02) reduction. Across all pruning levels, MovieLens and Yelp experienced the most potent effects, with decreases of up to 85% (from 85.99 to 12.86) and 82% (from 18.29 to 3.26), respectively, for 100-core pruning. This division between moderately and heavily affected datasets aligns with changes in Space Size: MovieLens and Yelp consistently exhibited the lowest values, whereas Gowalla and both Amazon datasets retained much higher portions of Space Size across pruning levels. Analysis of GiniItem divided the datasets into the same two groups. Amazon CD, Amazon Toys, and Gowalla consistently exhibited lower GiniItem values, indicating a pattern closer to equal interaction distribution across items. In contrast, MovieLens and Yelp displayed the highest and second-highest GiniItem values at every pruning level, denoting a highly uneven distribution of interactions. MovieLens, in particular, is known for a short-head, long-tail 34 distribution, where a few popular items attract many interactions, while most items receive little interaction [8, 18, 27]. This trait is reflected in its GiniItem scores. Overall, GiniItem scores varied little across pruning stages and preserved the dataset ranking observed for the 0coresets through to the 100-coresets. GiniUser exhibited trends similar to GiniItem, with scores decreasing at each pruning level. However, the difference between unpruned datasets and the 100-coresets was substantially larger than the corresponding GiniItem differences. The most significant decline was observed for MovieLens, where the 100-coreset showed a drop of -0.46 compared to the 0-coreset. All other datasets followed the same pattern, though less markedly, with GiniUser reductions ranging from -0.24 to -0.40. This outcome aligns with expectations: progressively excluding low-activity users leaves a population of generally high-activity users, thereby driving the interaction distribution closer to uniformity. In summary, pruning affected all structural characteristics of the datasets, though not in uniform severity. Density increased for all datasets; however, its link to performance was inconsistent, as demonstrated by the declining performance scores on the already dense MovieLens dataset. Interactions per User overall increased for each dataset per core level, yet yielded varying outcomes as displayed in the comparison of Yelp and Gowalla: Gowalla, with the lowest relative increase in interactions per user, showed a steady rise in algorithm performance, while Yelp (highest relative increase in interactions per user) declined after 50core pruning. Additional properties, such as the Gini measures for items and users, highlighted the impact on user-centred characteristics, while showing that item characteristics are affected to a lesser extent. 6.1.1.2 Distributional Characteristics: Retention of Interactions, Items and Users Pruning affected the datasets' structural properties to differing degrees, while similar trends of varying intensity were frequently observed. A comparable pattern emerged examining the retention rates of interactions, items, and users across pruning levels. In this context, retention is defined as the proportion (in per cent) of the original dataset that is preserved in a coreset. User retention, therefore, describes the percentage of users retained in a coreset compared to the 0-core baseline, and this definition applies equally to item and interaction retention. Among the three retention measures, user retention showed the most pronounced variation, differing substantially across both entire datasets and core levels. In our analysis, we observed 35 that the impact of data pruning on datasets varies already at low core levels: 5-core pruning revealed a gap of 61% between the datasets with the highest and lowest user retention rates. Sun et al. [47] reported 5 and 10 as common core-levels observed for 60% of experiments applying data pruning. Comparing the Gowalla and the Yelp datasets side by side revealed how varying the impact of each core level can be on two different datasets (Figure 13). Pruning Yelp at 5-core removed 86% of users, while retaining 58% of interactions. Meanwhile, Gowalla marked a 25% reduction in users, losing merely 2% of total interactions in the dataset. 5-core pruning removed 86% of Yelp's users while preserving 58% of its interactions, whereas Gowalla experienced a 25% loss in users, retaining 98% of its interactions. This difference in effect stems from structural differences between the two datasets: Yelp contains a significantly higher number of users than items, with the lowest GiniUser among the unpruned datasets. In contrast, Gowalla's GiniUser is insignificantly higher, but it contains more items than users. Therefore, pruning low-activity users removes a smaller share of the total interactions for Gowalla, causing a steadier decline in user retention across pruning levels. Figure 13 User Exclusion Rates Across Pruning Levels for Gowalla and Yelp: Waterfall charts showing the percentage of users removed at each core-pruning level (5, 10, 20, 50, 100) for the Gowalla (left) and Yelp (right) datasets, illustrating their contrasting user-retention patterns per level. The use of 20-core pruned datasets (e.g., MovieLens 100k, MovieLens 1M, ...) is a common design choice in recommender research, but Beel and Brunel [8] already established in 2018 that this results in the exclusion of 42% of users compared to an unpruned version of the dataset. In our experiments, this rate was considerably higher: the examined datasets 25.44% 15.60% 23.79% 21.48% 8.70% 5 10 20 50 100 Core Pruning Applied 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent of Users Excluded EXCLUSION RATE OF USERS ACROSS 5 PRUNING LEVELS ON GOWALLA 86.32% 8.51% 1.40% 0.33% 3.31% 5 10 20 50 100 Core Pruning Applied 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Percent of Users Excluded EXCLUSION RATE OF USERS ACROSS 5 PRUNING SETTINGS ON YELP 36 experienced user loss ranging from 65% for Gowalla, over 93% (both Amazon datasets), up to 98.14% (Yelp) when 20-core pruning was applied. Even the popular MovieLens dataset retained merely 15% of users and 61% of interactions, highlighting the uneven distribution of interactions per user. At the highest level of pruning examined in our experiments, 100-core, a singular dataset (Gowalla) retained more than 1% of the users compared to the 0-coreset, while simultaneously retaining almost 40% of interactions and 54% of items. Whereas 100-core pruning may not be a common design choice, it highlights how far a dataset can be pruned while still retaining almost half of the interactions and items. Data Pruning is acknowledged as a process that can reintroduce bias [22, 54], even if the original dataset was bias-free. Comparing algorithm performance on a dataset heavily altered, for example, by 50-core or 100-core pruning, would inevitably lead to results skewed in favour of high-activity users. Recommendations for cold-start users are known to be challenging [30], but a comparison of algorithm performance on these users is typically avoided by using pruned datasets. The impact of including cold-start users in evaluation scenarios will be discussed in more detail in the second part of this chapter. In summary, we analysed the impact of data pruning even at low levels on the structural and distributional characteristics. We found that while item-centred characteristics were affected, it was to a lesser extent than user-centred characteristics. Overall, datasets reacted to the same levels of pruning in varying intensities; for example, Yelp experienced an 86% loss of users compared to Gowalla's 25% loss when 5-core pruning was applied. We observed that both structural and distributional characteristics were substantially changed by data pruning. 6.1.2 Algorithm Performance The preceding analysis of dataset characteristics provides essential context for interpreting variance in algorithmic performance. Understanding how progressive pruning altered structural and distributional properties clarifies the conditions under which the algorithms were evaluated. In the following section, we build on these findings to examine how altered dataset characteristics influenced the performance of the tested algorithms. 37 In our experiment, we employed two complementary evaluation strategies to examine algorithm performance on progressively pruned datasets. In Phase 1, algorithms were trained and tested on equally pruned train and test sets, reflecting the common practice of using pruned datasets throughout recommender system research. In Phase 2, the algorithms were trained on the same training sets, while evaluation was conducted on test sets derived from unpruned data. In the discussion that follows, we first examine the results from Phase 1 and then compare them with those from Phase 2, highlighting what these contrasts reveal about the implications of training and testing on pruned datasets. 6.1.2.1 Phase 1: Training and Testing on Pruned Datasets According to Beel and Brunel [8], the key question when examining the effects of data pruning is whether pruning leads to changes in the performance ranking of algorithms. During Phase 1, we observed that the performance ranks of most algorithms varied across the progressive pruning levels. The Random algorithm represented the sole exception, ranking last across every coreset examined in Phase 1, with multiple 0-values observed for average nDCG@10. PopScore, as the second baseline algorithm, displayed improvements for higher levels of pruning, especially on denser datasets, such as MovieLens. In one case, on the 50-core pruned MovieLens dataset, PopScore even outperformed all remaining algorithms. The dataset's inherent structure can likely explain this ranking. As shown in the previous section, MovieLens contains a small number of very popular items with which many users have interacted. Because PopScore recommends items based on popularity, its improved performance on extensively pruned data is not unexpected. Examining the characteristics of 50-core pruned MovieLens reveals conditions particularly favourable towards a popularitybased algorithm: both the item retention rate and the GiniItem were observed to be much higher for MovieLens than for the compared datasets after 50-core pruning. In general, we observed rank changes to be more frequent and more extreme in two datasets: the MovieLens and Yelp datasets. The first indication that these datasets differed was that User KNN, which ranked undisputedly first on the remaining three datasets across all cores, only experienced rank changes on MovieLens and Yelp. Analysing the absolute nDCG@10 values for these datasets, however, paints a different picture: Even the Yelp dataset's progressive pruning led to an increase in absolute nDCG@10 values with every core level, 38 whereas the performance on the MovieLens dataset declined with each successive core level. On Yelp pruned to cores 5 to 20, User KNN consistently ranked second, supported by a steady rise in nDCG@10 scores per core, and was surpassed only by Implicit MF, which achieved its peak performance on these specific coresets in Phase 1. Structural characteristics of the MovieLens coresets account for the rank changes observed between the two neighbourhood-based approaches, Item KNN and User KNN. For the unpruned dataset (0-core) and the heavily pruned 100-coreset, User KNN outperformed Item KNN, whereas the medium levels of pruning favoured Item KNN. This observation can be attributed to the structural characteristics observed when pruning the MovieLens dataset: the long-tail item distribution in the unpruned version makes recommending based on item similarity difficult, as many items receive very few interactions, leading to better performance for User KNN over Item KNN. Because the 100-coreset retains only users with extensive interaction histories, it provides ideal conditions for a user-similarity algorithm and therefore, explains User KNN's advantage on this coreset. The Matrix Factorisation-based algorithms, Biased MF and Implicit MF, both increased in performance scores as pruning levels rose, exhibiting the most significant performance improvement compared to the 0-core datasets on average on the 100-coreset. This behaviour is not surprising, as it has been acknowledged that algorithms employing Matrix Factorisation display performance improvements as Sparsity decreases [46]. As we have shown in the analysis of the coreset characteristics, Density increased with each level of pruning applied, thus Sparsity (SSSSSSSSSSSSSSSS= 1 -DDDDDDDDDDDDDD) decreased and performance for Biased MF and Implicit MF improved. BPR profited to an extent of rising density by progressive pruning, exhibiting enhanced performance scores with each core-level for Gowalla and both Amazon datasets, though to a lesser degree for Amazon Toys. The performance on the denser datasets MovieLens and Yelp dropped sharply at 100-core pruning. For example, BPR's score declined by 44% (from 0.025 to 0.014) compared to the 50-core level when trained and tested on the 100-coreset. This decline aligns with the sharp reduction in interactions both datasets experienced when pruned to 100 cores, as neither retained more than 15% of interactions at this pruning level. Interaction loss poses a disadvantage for algorithms like BPR, which depend on positivenegative item pairs to learn preferences [37, 40]. 39 Overall, we observed positive trends for the group of traditional algorithms under progressively increased pruning conditions. The group containing the modern algorithms displayed less homogeneous behaviour across pruning levels, with individual algorithms reacting differently to data reduction. For example, DiffRec responded to low pruning levels with an average initial performance improvement of 25% (5-core) and 15% (10-core) compared to the unpruned baseline. Further pruning continuously diminished this effect to the point of 100-core pruning, exhibiting an average performance decline of 80%. MultiVAE exhibited a similar trend to DiffRec, with a decrease in performance scores across higher pruning levels in response to increased pruning intensity. Compared to the rest of the modern algorithms, MultiVAE reacted most sensitively to even low-level pruning, showing performance declines of 22% and 27% at the 5-core and 10-core levels, respectively. We observed that MultiVAE ranked higher on the unpruned versions of Yelp and MovieLens, with a sharp decline on Yelp as early as when 5-core pruning was applied. This correlates with the data loss Yelp experienced when 5-core pruning was applied: relative user retention measured 14%, while interaction retention dropped to 58%. MultiVAE achieved higher performance scores on the dense datasets; however, it exhibited a drastic decline in performance with data reduction. DMF appeared less sensitive to data reduction, with only minor changes in performance compared to the unpruned baseline; however, the MovieLens dataset elicited steeper declines across pruning levels. Although the highest score on the MovieLens coresets was obtained when training and testing on the unpruned version, a comparison of the 50-core and 100-core results showed a performance improvement. This trend was observed for all examined datasets and resembled the behaviour of the traditional Matrix Factorisation algorithms. Overall, DMF improved on average by 48% on the 100-coresets. SimpleX responded differently to 100-core pruning for the MovieLens and Yelp datasets than DMF. For MovieLens, a steady decline was observed across core levels. SimpleX achieved the highest performance score on Yelp when 50-core pruning was applied. When 100-core pruned, the Amazon datasets and Gowalla yielded the highest nDCG@10 values for SimpleX. This division between the two dataset groups, contrasting in density and shape, resulting in differing scores for algorithms, was a recurring observation throughout the first phase of the experiment. 40 To summarise, we observed the algorithm groups to react differently to data pruning, resulting in various rank changes both for algorithms performing better on entire datasets as well as rank changes across pruning levels for a singular dataset. The baseline algorithms improved towards the higher levels of pruning, while the modern algorithms tended to react more sensitively to data reduction. This aligns with the commonly accepted assumption that deep learning algorithms perform better when the amount of data increases [45]. In contrast, the traditional algorithms showed an overall performance improvement with each level of pruning applied. Additionally, we observed that the traditional algorithms, on average, outperformed the group of modern algorithms across all core levels. 6.1.2.2 Phase 2: Training on Pruned Datasets, Testing on Unpruned Datasets We demonstrated, through the examination of Phase 1, that training and evaluating algorithms on pruned versions of datasets can lead to performance rank changes between algorithms, both across different pruning levels and across entire datasets. In Phase 2, we trained the algorithms on the same training sets as in Phase 1, but the evaluation was performed on test sets derived from an unpruned dataset. To enable a fair comparison, these test sets had the same size as the pruned test sets from Phase 1, but included low-activity users. Furthermore, it was ensured that the test sets only included unseen data by choosing disjoint subsets of the training sets. The baseline algorithms, PopScore and Random, exhibited notably different behaviour compared to Phase 1, when evaluation included recommendations for low-activity users. For instance, PopScore's overall positive trend, especially pronounced at core levels 20 and above, flattened almost entirely, showing only a slight increase for the 20-core pruned MovieLens dataset. Random fared even worse: on three datasets (both Amazon and Gowalla), it failed to produce a single non-zero average nDCG@10 value. The majority of traditional algorithms (80%) experienced a similar trend as the baseline algorithms in Phase 2. While there were minimal differences in average nDCG@10 for the datasets, the overall trend was the same: the algorithms only improved their performance on the 20-core pruned MovieLens training set. Some of the algorithms in this group showed slight nDCG@10 increases on individual coresets when compared to the lowest core-level (5core), but the absolute differences never exceeded 0.0007 and are thereby deemed 41 insignificant. Compared to the performance scores achieved in Phase 1 on the pruned test sets, the traditional algorithms (Biased MF, Implicit MF, Item KNN, User KNN) suffered a decline of an average 95% across the pruning levels. The only exception to this decline was BPR, which overall performed better under the evaluation conditions in Phase 2 and displayed an average increase in nDCG@10 of 143% across all pruned training sets when evaluated on unpruned test sets. Compared to Phase 1, BPR's improvement in the performance-based ranking of algorithms was only second to SimpleX: on the entirety of all coresets for both Amazon datasets and the Gowalla datasets, BPR ranked in second place in Phase 2. The lowest rank we observed was for the MovieLens datasets, when pruned to 10 cores and higher, where BPR ranked 4th. For the highly challenging 100-core pruned Yelp dataset, BPR performed best of all algorithms, displaying a stronger tolerance towards heavy data reductions than the remaining algorithms. While BPR did not respond to data reduction as strongly, comparing the nDCG@10 values across the pruning levels in Phase 2 still showed a decrease as the pruning intensity increased. Applying 10-core pruning (0.051) lowered the average nDCG@10 by 19% compared to 5-core pruning (0.063), and 100-core pruning caused an even steeper decline of 78%, down to 0.014. The steady decline shows that BPR's performance was progressively hindered as the pruning levels increased. Figure 14 Figure X - Average nDCG@10 Across Pruning Levels with Pruned Training and Unpruned Test Data: Average nDCG@10 for traditional, modern, and baseline algorithms across five datasets when models are trained on pruned datasets but evaluated on unpruned test sets. 0.0000 0.0200 0.0400 0.0600 0.0800 0.1000 0 10 20 30 40 50 60 70 80 90 100 AVERAGE NDCG@10 CORE PRUNING APPLIED AVERAGE NDCG@10 ACROSS PRUNING LEVELS ON 5 DATASETS (PRUNED TRAIN & UNPRUNED TEST DATA) Group 1 (Traditional) Group 2 (Modern) Group 3 (Baseline) 42 The modern algorithms achieved much higher absolute performance scores under Phase 2 evaluation compared to the traditional algorithms. Across all examined coresets, the traditional algorithms attained a mean nDCG@10 of 0.010, while the modern algorithms achieved a mean of 0.038. All modern algorithms exhibited a declining performance trend as pruning levels increased (Figure 14), though their absolute nDCG@10 in some cases were higher than the corresponding Phase 1 values. For example, DiffRec achieved values for 5- to 20-core pruning, which were an average 370% higher than the scores for the corresponding training sets in Phase 1. However, increasing the pruning level from 5-core to 10-core resulted in a 39% drop in mean nDCG@10 (0.116 to 0.071), and 100-core pruning even dropped by 98% to 0.002. It is worth mentioning that training SimpleX on pruned and testing it on unpruned data yielded the highest observed nDCG@10 scores in the entire experiment. On both Amazon and the Gowalla datasets, SimpleX was the only algorithm we observed to reach mean nDCG@10 values above 0.25, with a maximum score attained on the Amazon Toys dataset pruned at core-level 5 (0.49). Nonetheless, a declining trend in these scores was observed as pruning intensified, which demonstrates the detrimental impact that progressive data pruning had on the modern algorithms examined in this thesis. Overall, the findings contrasted sharply with those of Phase 1, as pruning the datasets did not yield any improvement in mean performance for either algorithm group. This aligns with the claim made by Beel and Brunel [8] in their initial study: Data Pruning artificially inflates the performance score when evaluated on test sets that have also undergone pruning. Algorithms previously experiencing upward trends on the pruned test sets did not display the same improvements on the unpruned test sets. These findings suggest that the improved performance on progressively pruned datasets stems from the test set composition, as the test sets were the only element that differed between the two evaluation settings. The test sets in Phase 1 were derived by splitting the pruned core datasets into a train and test set, to resemble the usage of a ready-to-use pruned dataset (e.g. most of the MovieLens datasets). If algorithm evaluation is conducted on such a dataset, the test set will underly similarly altered structural and distributional characteristics as the dataset it was split off from. As we have shown in the analysis of the dataset characteristics, pruning users according 43 to their number of interactions in a dataset can lead to coresets that differ substantially in composition and structural characteristics. Taken together, these observations allow us to return to the core research question and examine it in light of the results. RQ: How does progressive dataset pruning by increasing core thresholds affect the performance of recommender systems, the structural and distributional characteristics of the dataset? In order to respond to the first part of the question, it is necessary to consider the second part first. We have examined a variety of structural and distributional characteristics across pruned coresets and found all analysed characteristics to be altered by progressively pruning, though not to the same degree. Pruning users according to their total number of interactions in the datasets had a decreasing effect on the total number of items and interactions in the pruned coresets, and consequently, on all characteristics that depend on at least one of these total numbers. The continuous decrease in these characteristics with rising pruning levels was unsurprising: what stood out was the variability in dataset responses. Structural and distributional characteristics were affected unevenly by pruning levels, but the clearest difference appeared in the retention of users, items and interactions at each stage (Figure 13). For instance, as early as just 5-core pruning, the gap between the highest user retention (Gowalla) and the lowest (Yelp) exceeded 60 percentage points. This consequently resulted in vastly different compositions for the core-pruned datasets, as some datasets retained much higher rates of their interactions than others. Even though all datasets had the same number of interactions in the 0-coresets, merely 5-core pruning led to almost halving the dataset size for individual datasets (Yelp), while nearly retaining the full number of interactions (Gowalla, MovieLens). Our results additionally highlighted that the user retention rates dropped drastically for commonly chosen core-levels of 5, 10 and 20: only a single dataset retained more than 30% of users in its 20-coreset, while the majority of datasets only contained 7-15% of users compared to their unpruned versions. These varying degrees of user retention were also visible in the distributional characteristics when comparing the GiniUser. Especially dense datasets with an extremely short head, long tail distribution of interactions, such as the 44 MovieLens dataset, displayed a trend toward uniformity across pruning levels in GiniUser. Whereas GiniItem remained considerably high, leaning towards an unequal distribution of interactions across items. In summary, progressive pruning altered both structural and distributional characteristics across all datasets; however, the extent of these changes was strongly influenced by the inherent composition of the unpruned (0-core) datasets. Dataset characteristics, such as density, shape, and the average number of interactions per user/item, are known to influence algorithm performance [18]. Therefore, the fluctuations observed in our experiments when comparing algorithm performance across datasets and their core-pruned versions were expected after analysing the effect pruning had on the dataset characteristics. Nonetheless, some algorithms showed distinctly different behaviour on individual coresets, a pattern we hypothesise arises from their underlying operating mechanisms. For example, Matrix Factorisation-based algorithms tend to improve performance as data sparsity decreases, a trend mirrored in the results of our experimentation in Phase 1 and the dataset analysis. The key question of whether progressive data pruning leads to changes in the performance ranking of algorithms was answered affirmatively for all examined datasets. Rank changes occurred more frequently and with greater magnitude on the two datasets with the highest density, largest values for shape and average number of interactions per item: MovieLens and Yelp. This observation remained true for both Phase 1 and Phase 2. In Phase 1, overall trends emerged across the three algorithm groups: traditional algorithms responded positively to higher pruning levels, whereas modern algorithms tended to show declining performance as pruning intensified. In Phase 2 - where test sets still included lowactivity users while training was performed on pruned data - the performance rankings changed markedly, and traditional algorithms experienced a substantial drop in performance. This suggests that the improvements observed for traditional algorithms in Phase 1 were largely due to the progressively easier test sets, as higher pruning levels increasingly retained only high-activity users in the coresets. The superior Phase 2 performance of modern algorithms supports the assumption that they are better equipped than traditional algorithms to handle typical recommender system challenges, 45 such as data sparsity and the user cold-start problem [2, 31]. A scenario with cold-start users was simulated in Phase 2 by including low-activity users in the test sets. To revisit the core research question: progressive pruning altered all examined dataset characteristics, yet the underlying composition of the original datasets remained reflected in these altered characteristics. Both the inherent dataset composition and the pruning-induced changes influenced algorithm performance, confirming that data pruning affects performance ranking. However, rank changes cannot be explained solely by the progressively pruned training sets, as the composition of the test sets and the intrinsic mechanisms of the algorithms also play significant roles. 7. Conclusion This thesis set out to analyse the impact of progressive data pruning on dataset characteristics and algorithm performance. To this end, we first analysed five datasets in both unpruned and progressively pruned states (5, 10, 20, 50, and 100-core pruning) to evaluate how their characteristics evolved under increasing pruning intensity. The results show that inherent dataset characteristics observed in the unpruned versions shape the extent to which they are altered by progressive pruning. For example, datasets with a high average number of interactions per user retained a much larger proportion of their original user base as pruning intensified. In contrast, datasets with lower average interactions per user, e.g. Yelp, quickly experienced heavy user loss and preserved fewer total interactions than datasets with higher interaction averages per user. Consequently, the resulting coresets varied significantly in terms of characteristics, total size, and user retention. For example, when 5core pruning was applied, Gowalla retained as many as 75% of its users and 98% of its interactions, whereas Yelp lost 86% of its users and only retained 58% of its interactions. This contrast widened at higher pruning levels: under 100-core pruning on Yelp retained merely 0.12% of its original users, while Gowalla still maintained about 5%. When datasets were pruned at the commonly used 20-core threshold, 80% of them retained 15% or fewer of their original users. This pruning strategy resembles the procedure applied to the popular MovieLens datasets, which are widely regarded as benchmark data for evaluating recommender systems. Using datasets pruned in this severity means that roughly 85% of users are ignored when evaluating an algorithm, thereby excluding the very group - low-activity or 46 cold-start users - that is generally considered challenging for generating suitable recommendations. Pruned datasets differ substantially from data encountered in real-world settings, as they primarily contain users with comparatively rich interaction histories. Consequently, good results achieved on such datasets cannot necessarily be generalised to other scenarios. Furthermore, pruning can introduce bias into datasets, in our case by skewing performance towards high-activity users. This raises concerns about fairness in recommender systems and has become the subject of recent research. The influence of pruning on performance scores and, ultimately, the performance ranking of algorithms was assessed by training and testing eleven algorithms on increasingly pruned datasets. The selected algorithms - covering baseline, traditional, and modern methods - allowed us to analyse whether pruning affected these groups to different extents. We found that specific datasets as a whole, as well as particular coresets, provided favourable conditions for individual algorithms. For example, denser datasets benefited matrix factorisation-based methods, while coresets with few low-activity items favoured Item KNN. Overall, the results suggest that traditional algorithms tended to improve as pruning levels increased, whereas modern algorithms showed no corresponding performance increase on progressively pruned coresets. In summary, changes in the performance rank of algorithms were observed across all datasets, with a higher frequency and greater magnitude on the denser datasets, MovieLens and Yelp. To determine whether the "easier" test sets - those in which low-activity users are successively removed, leaving only high-activity users - contributed to the positive trend observed for some algorithms across pruning levels, we conducted a second evaluation. This evaluation utilised the same training sets as before, but the test sets were derived from the unpruned datasets, thereby reintroducing low-activity users. While rank changes occurred across coresets and entire datasets under these evaluation conditions as well, the overall trends and performance values declined sharply, particularly for the traditional algorithms. Whereas traditional algorithms had outperformed modern ones across most coresets in the first evaluation phase, testing on unpruned datasets yielded higher overall performance scores for the modern algorithms. A comparison of score evolution across increasing pruning levels 47 showed an overall decline in performance as pruning intensified for all examined algorithms of both groups. Combining our findings from the dataset-characteristic analysis, the experiments with training and testing on pruned data, and the experiments with training on pruned data while testing on unpruned data, we conclude that data pruning can artificially inflate performance scores. This effect occurs when evaluation is carried out on test sets pruned to the same level as the training data. Furthermore, specific datasets and pruning levels favoured individual algorithms, indicating that evaluating only on single datasets or individual coresets is not advisable for a balanced comparison of algorithm performance. Results based on such narrow evaluations cannot necessarily be generalised to other scenarios that involve different datasets. One such scenario is the data a recommender system encounters in real-world use. Choosing an unsuitable algorithm in this context could lead to performance that differs significantly from what offline evaluations suggested. Besides failing to deliver generally applicable results, data pruning also hampers effective comparisons between algorithms in research. Selecting a coreset that is especially favourable to a particular algorithm can skew performance results in its favour, while making other comparable algorithms appear less suitable. Data pruning can help reduce computational load and training time, thereby supporting greener recommender-system research, provided acceptable trade-offs between data reduction and performance are possible. However, we concur with evaluation frameworks that recommend using pruned datasets only as a supplement to unpruned datasets. Alternatively, training on pruned data while evaluating on unpruned data could offer a compromise that lowers training time and computational cost while still testing on a heterogeneous group of users. Additionally, our results highlight the importance of clearly documenting all preprocessing steps applied to datasets in recommender-systems research. Although many studies apply data pruning during pre-processing or use pre-pruned datasets, such as MovieLens, crucial documentation of dataset choice and preprocessing steps, including data pruning, is frequently absent. Without precise knowledge of a dataset's preprocessing history, replicating experiments and comparing algorithms under equal conditions becomes nearly impossible. 48 In conclusion, this thesis demonstrates that progressive data pruning can strongly influence both dataset composition and algorithm evaluation, leading to inflated performance scores and limited generalisability if not handled with care. By combining detailed dataset analyses with extensive algorithmic assessment, we provide evidence that pruning should be applied only as a complement to unpruned data and documented transparently. Future recommender-system research can build on these insights by developing frameworks and preprocessing standards that balance computational efficiency with fairness and real-world applicability. 8. Future Work and Limitations Due to computational limitations encountered during the experiment, several concessions had to be made to facilitate the evaluation of 11 algorithms on five datasets. Hyperparameter-Optimisation All models were trained with their default parameter values, respectively set by either the LensKit or the RecBole library, as the focus of this study was not to achieve the highest possible performance for all algorithms but to compare how the increasingly pruned versions of the datasets affected the performance of the models relatively. Future work should consider implementing hyperparameter-tuning. K-Fold Cross Validation Due to the extensive computation times, it was not possible to include K-Fold CrossValidation in this study. Future research could consider implementing cross-validation with k = 5 or k = 10. Algorithm Selection To further reduce the computational cost, the deep learning models investigated in this study were chosen for their lower computational complexity and shorter training times. It would be of interest to see how more complex deep learning models are impacted by data pruning. Metrics Our research focused on ranking-based performance metrics, researchers could examine the impact data pruning has on, e.g. beyond-accuracy metrics for deep learning models. 49 Datasets Due to the scope of this work, only five datasets could be included in the experiments; further research should focus on a more diverse variety of data sources. User-Based and Item-Based Pruning In our experiment, we pruned interactions based on the total number of interactions the corresponding user had in the dataset. It is interesting to examine how pruning based on item interactions, e.g. pruning all interactions with an item if it is interacted with less than a specific number of times, impacts both dataset characteristics and algorithm performance. Addressing these limitations in future research will not only strengthen the validity and generalisability of our findings but also provide a richer foundation for developing fair, efficient, and reproducible recommender-system evaluations. By building on the proposed directions, subsequent studies can refine methods, broaden datasets, and deepen understanding of how data pruning shapes both dataset characteristics and algorithm performance. 9. Summary This thesis investigates how data pruning - removing users whose total number of interactions falls below a defined threshold - affects both datasets' characteristics and algorithm performance in recommender systems. Five widely used datasets were examined in their unpruned form and at 5-, 10-, 20-, 50-, and 100-core pruning levels. For each stage, we analysed structural characteristics, such as density, shape, and the average number of interactions per user/item, as well as distributional characteristics measured by Gini coefficients for users and items. Retention rates of users, items, and interactions were tracked to reveal how dataset composition evolves as low-activity users are successively removed. Dataset characteristics were found to be influenced by pruning to varying degrees. Even at the lowest pruning level of 5-core, the Yelp dataset retained only 14% of its users and 58% of its interactions. In contrast, Gowalla preserved 75% of its users and 98% of its interactions, illustrating how differently datasets can react to identical pruning thresholds. At the 50 commonly used 20-core level, all but one dataset were reduced to 15% or fewer of their original users, and at the highest pruning level (100-core), Yelp retained a mere 0.12% of its user base. These findings highlight the heterogeneous impact of progressive pruning on structural and distributional characteristics across datasets. Building on this analysis, algorithm performance was evaluated across the same progressive pruning levels in two complementary settings. Phase 1 involved training and testing on increasingly pruned datasets, while Phase 2 used the same pruned training sets but tested on unpruned datasets to reintroduce low-activity users to the evaluation. While traditional algorithms overall showed improving performance with rising pruning levels in Phase 1, this trend diminished significantly in Phase 2, where modern algorithms continuously outperformed the majority of traditional algorithms. The modern algorithms exhibited a general trend of declining performance as pruning levels increased during both phases, despite average nDCG@10 values being higher in Phase 2. Rank changes occurred between algorithms in both phases, across entire datasets and for specific coresets. On the denser datasets, Yelp and MovieLens, both frequency and magnitude of rank changes increased across core levels. The two-phase evaluation revealed how pruning can artificially inflate performance scores and alter algorithm rankings when testing is conducted on equally pruned data, and how these effects change when testing occurs instead on unpruned datasets. Together, these analyses of algorithm performance and dataset characteristics provide a detailed picture of how progressive data pruning influences both the statistical structure of recommender-systems datasets and the measured performance of baseline, traditional, and modern algorithms. The results highlight that pruning levels have a strong impact on dataset composition and algorithmic evaluation, and the careful selection of datasets and coresets can provide beneficial conditions for individual algorithms. 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Appendix Phase 1 (Training & Testing on Pruned Datasets) nDCG@10 Phase 2 (Testing on Unpruned Datasets) nDCG@10 Biased MF Biased MF Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0000 0.0000 0.0002 0.0070 0.0007 5 0.0000 0.0000 0.0002 0.0065 0.0007 5 0.0000 0.0000 0.0000 0.0004 0.0000 10 0.0000 0.0000 0.0003 0.0059 0.0006 10 0.0000 0.0000 0.0000 0.0003 0.0000 20 0.0000 0.0000 0.0004 0.0063 0.0008 20 0.0000 0.0000 0.0000 0.0010 0.0000 50 0.0006 0.0000 0.0008 0.0070 0.0013 50 0.0000 0.0000 0.0000 0.0003 0.0000 100 0.0008 0.0000 0.0013 0.0097 0.0022 100 0.0000 0.0000 0.0000 0.0003 0.0000 Min 0.0000 0.0000 0.0002 0.0059 0.0006 Min 0.0000 0.0000 0.0000 0.0003 0.0000 Max 0.0008 0.0000 0.0013 0.0097 0.0022 Max 0.0000 0.0000 0.0000 0.0010 0.0000 Average 0.0003 0.0000 0.0005 0.0071 0.0011 Average 0.0000 0.0000 0.0000 0.0005 0.0000 Implicit MF ImplicitMF Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0204 0.0021 0.0282 0.0424 0.0286 5 0.0202 0.0018 0.0298 0.0380 0.0332 5 0.0010 0.0001 0.0017 0.0021 0.0017 10 0.0215 0.0024 0.0328 0.0310 0.0366 10 0.0012 0.0001 0.0019 0.0017 0.0019 20 0.0266 0.0029 0.0398 0.0271 0.0452 20 0.0013 0.0002 0.0023 0.0042 0.0021 50 0.0410 0.0044 0.0555 0.0234 0.0606 50 0.0015 0.0002 0.0028 0.0011 0.0016 100 0.0565 0.0068 0.0789 0.0210 0.0751 100 0.0021 0.0002 0.0029 0.0004 0.0012 Min 0.0202 0.0018 0.0282 0.0210 0.0286 Min 0.0010 0.0001 0.0017 0.0004 0.0012 Max 0.0565 0.0068 0.0789 0.0424 0.0751 Max 0.0021 0.0002 0.0029 0.0042 0.0021 Average 0.0310 0.0034 0.0442 0.0305 0.0465 Average 0.0014 0.0002 0.0023 0.0019 0.0017 PopScore PopScore Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0057 0.0007 0.0056 0.0396 0.0075 5 0.0047 0.0007 0.0060 0.0353 0.0069 5 0.0002 0.0000 0.0004 0.0019 0.0004 10 0.0050 0.0010 0.0067 0.0298 0.0062 10 0.0002 0.0001 0.0004 0.0017 0.0003 20 0.0063 0.0009 0.0078 0.0272 0.0071 20 0.0003 0.0000 0.0005 0.0042 0.0004 50 0.0092 0.0014 0.0120 0.0251 0.0095 50 0.0004 0.0001 0.0006 0.0011 0.0004 100 0.0149 0.0019 0.0154 0.0278 0.0161 100 0.0004 0.0000 0.0004 0.0004 0.0004 Min 0.0047 0.0007 0.0056 0.0251 0.0062 Min 0.0002 0.0000 0.0004 0.0004 0.0003 Max 0.0149 0.0019 0.0154 0.0396 0.0161 Max 0.0004 0.0001 0.0006 0.0042 0.0004 Average 0.0076 0.0011 0.0089 0.0308 0.0089 Average 0.0003 0.0000 0.0005 0.0018 0.0004 User KNN User KNN Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0337 0.0082 0.0512 0.0504 0.0312 5 0.0341 0.0081 0.0523 0.0428 0.0317 5 0.0018 0.0004 0.0031 0.0022 0.0017 10 0.0364 0.0083 0.0546 0.0340 0.0332 10 0.0019 0.0004 0.0030 0.0017 0.0018 20 0.0450 0.0092 0.0621 0.0279 0.0419 20 0.0023 0.0005 0.0036 0.0043 0.0021 50 0.0621 0.0095 0.0795 0.0239 0.0628 50 0.0026 0.0005 0.0040 0.0009 0.0015 100 0.0820 0.0103 0.1013 0.0251 0.0780 100 0.0026 0.0003 0.0036 0.0006 0.0014 Min 0.0337 0.0081 0.0512 0.0239 0.0312 Min 0.0018 0.0003 0.0030 0.0006 0.0014 Max 0.0820 0.0103 0.1013 0.0504 0.0780 Max 0.0026 0.0005 0.0040 0.0043 0.0021 Average 0.0489 0.0089 0.0668 0.0340 0.0464 Average 0.0022 0.0004 0.0034 0.0019 0.0017 Item KNN Item KNN Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0107 0.0030 0.0216 0.0448 0.0066 5 0.0130 0.0037 0.0223 0.0440 0.0102 5 0.0008 0.0002 0.0014 0.0023 0.0006 10 0.0176 0.0041 0.0246 0.0371 0.0146 10 0.0010 0.0003 0.0014 0.0020 0.0008 20 0.0245 0.0053 0.0296 0.0314 0.0227 20 0.0012 0.0003 0.0017 0.0049 0.0011 50 0.0331 0.0060 0.0379 0.0246 0.0229 50 0.0014 0.0002 0.0019 0.0008 0.0005 100 0.0413 0.0068 0.0424 0.0145 0.0161 100 0.0015 0.0002 0.0019 0.0003 0.0004 Min 0.0107 0.0030 0.0216 0.0145 0.0066 Min 0.0008 0.0002 0.0014 0.0003 0.0004 Max 0.0413 0.0068 0.0424 0.0448 0.0229 Max 0.0015 0.0003 0.0019 0.0049 0.0011 Average 0.0234 0.0048 0.0297 0.0327 0.0155 Average 0.0012 0.0003 0.0017 0.0021 0.0007 BPR BPR Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0050 0.0003 0.0146 0.0367 0.0132 5 0.0050 0.0003 0.0150 0.0350 0.0150 5 0.0448 0.2004 0.0230 0.0319 0.0141 10 0.0064 0.0005 0.0180 0.0304 0.0172 10 0.0324 0.1655 0.0211 0.0249 0.0106 20 0.0091 0.0005 0.0216 0.0273 0.0229 20 0.0211 0.1234 0.0183 0.0131 0.0062 50 0.0151 0.0012 0.0296 0.0243 0.0248 50 0.0147 0.0702 0.0103 0.0075 0.0036 100 0.0264 0.0025 0.0401 0.0138 0.0019 100 0.0117 0.0417 0.0068 0.0027 0.0049 Min 0.0050 0.0003 0.0146 0.0138 0.0019 Min 0.0117 0.0417 0.0068 0.0027 0.0036 Max 0.0264 0.0025 0.0401 0.0367 0.0248 Max 0.0448 0.2004 0.0230 0.0319 0.0141 Average 0.0112 0.0009 0.0232 0.0279 0.0158 Average 0.0249 0.1202 0.0159 0.0160 0.0079 58 Phase 1 (Training & Testing on Pruned Datasets) nDCG@10 Phase 2 (Testing on Unpruned Datasets) nDCG@10 DiffRec DiffRec Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0078 0.0000 0.0097 0.0329 0.0108 5 0.0050 0.0004 0.0105 0.0395 0.0212 5 0.0054 0.0004 0.0111 0.4842 0.0765 10 0.0050 0.0004 0.0094 0.0370 0.0191 10 0.0049 0.0001 0.0104 0.3142 0.0229 20 0.0046 0.0006 0.0069 0.0304 0.0196 20 0.0038 0.0002 0.0079 0.0766 0.0086 50 0.0014 0.0007 0.0015 0.0213 0.0042 50 0.0015 0.0000 0.0053 0.0125 0.0023 100 0.0023 0.0005 0.0029 0.0060 0.0005 100 0.0008 0.0000 0.0025 0.0065 0.0012 Min 0.0014 0.0000 0.0015 0.0060 0.0005 Min 0.0008 0.0000 0.0025 0.0065 0.0012 Max 0.0078 0.0007 0.0105 0.0395 0.0212 Max 0.0054 0.0004 0.0111 0.4842 0.0765 Average 0.0044 0.0004 0.0068 0.0279 0.0126 Average 0.0033 0.0001 0.0074 0.1788 0.0223 DMF DMF Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0016 0.0005 0.0045 0.0328 0.0063 5 0.0024 0.0006 0.0037 0.0285 0.0056 5 0.0018 0.0007 0.0046 0.0268 0.0052 10 0.0029 0.0009 0.0028 0.0274 0.0051 10 0.0028 0.0007 0.0030 0.0272 0.0030 20 0.0025 0.0008 0.0035 0.0229 0.0057 20 0.0013 0.0007 0.0063 0.0170 0.0029 50 0.0061 0.0011 0.0112 0.0204 0.0083 50 0.0022 0.0005 0.0047 0.0191 0.0031 100 0.0117 0.0013 0.0125 0.0280 0.0142 100 0.0022 0.0004 0.0037 0.0146 0.0031 Min 0.0016 0.0005 0.0028 0.0204 0.0051 Min 0.0013 0.0004 0.0030 0.0146 0.0029 Max 0.0117 0.0013 0.0125 0.0328 0.0142 Max 0.0028 0.0007 0.0063 0.0272 0.0052 Average 0.0045 0.0009 0.0064 0.0267 0.0075 Average 0.0021 0.0006 0.0045 0.0209 0.0035 MultiVAE MultiVAE Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0055 0.0003 0.0054 0.0391 0.0143 5 0.0037 0.0005 0.0060 0.0339 0.0063 5 0.0031 0.0000 0.0058 0.0351 0.0068 10 0.0035 0.0006 0.0066 0.0303 0.0064 10 0.0021 0.0000 0.0057 0.0313 0.0044 20 0.0036 0.0002 0.0063 0.0274 0.0032 20 0.0013 0.0000 0.0063 0.0304 0.0054 50 0.0055 0.0005 0.0045 0.0105 0.0114 50 0.0007 0.0000 0.0049 0.0123 0.0044 100 0.0058 0.0006 0.0127 0.0115 0.0020 100 0.0004 0.0000 0.0034 0.0132 0.0034 Min 0.0035 0.0002 0.0045 0.0105 0.0020 Min 0.0004 0.0000 0.0034 0.0123 0.0034 Max 0.0058 0.0006 0.0127 0.0391 0.0143 Max 0.0031 0.0000 0.0063 0.0351 0.0068 Average 0.0046 0.0005 0.0069 0.0255 0.0073 Average 0.0015 0.0000 0.0052 0.0245 0.0049 Random Random Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0000 0.0000 0.0000 0.0002 0.0000 5 0.0000 0.0000 0.0000 0.0002 0.0000 5 0.0000 0.0000 0.0000 0.0002 0.0001 10 0.0000 0.0000 0.0000 0.0002 0.0001 10 0.0000 0.0000 0.0000 0.0002 0.0000 20 0.0000 0.0000 0.0000 0.0003 0.0001 20 0.0000 0.0000 0.0000 0.0002 0.0000 50 0.0001 0.0000 0.0000 0.0004 0.0004 50 0.0000 0.0000 0.0000 0.0002 0.0001 100 0.0002 0.0000 0.0001 0.0011 0.0005 100 0.0000 0.0000 0.0000 0.0001 0.0000 Min 0.0000 0.0000 0.0000 0.0002 0.0000 Min 0.0000 0.0000 0.0000 0.0001 0.0000 Max 0.0002 0.0000 0.0001 0.0011 0.0005 Max 0.0000 0.0000 0.0000 0.0002 0.0001 Average 0.0001 0.0000 0.0000 0.0004 0.0002 Average 0.0000 0.0000 0.0000 0.0002 0.0000 SimpleX SimpleX Core Amazon CD Amazon ToyGowalla MovieLens Yelp Core Amazon CD Amazon ToyGowalla MovieLens Yelp 0 0.0028 0.0004 0.0071 0.0233 0.0064 5 0.0028 0.0004 0.0082 0.0195 0.0078 5 0.3057 0.4901 0.2683 0.0228 0.0211 10 0.0030 0.0006 0.0079 0.0182 0.0098 10 0.2396 0.3387 0.2454 0.0140 0.0135 20 0.0054 0.0005 0.0098 0.0176 0.0132 20 0.1543 0.2083 0.1854 0.0044 0.0093 50 0.0103 0.0011 0.0117 0.0193 0.0200 50 0.0690 0.0927 0.0996 0.0024 0.0065 100 0.0164 0.0019 0.0179 0.0154 0.0059 100 0.0323 0.0455 0.0449 0.0005 0.0032 Min 0.0028 0.0004 0.0071 0.0154 0.0059 Min 0.0323 0.0455 0.0449 0.0005 0.0032 Max 0.0164 0.0019 0.0179 0.0233 0.0200 Max 0.3057 0.4901 0.2683 0.0228 0.0211 Average 0.0068 0.0008 0.0104 0.0189 0.0105 Average 0.1602 0.2351 0.1687 0.0088 0.0107 59 PRECISION@10 (PHASE 1) Amazon CD Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0000 0.0060 0.0016 0.0106 0.0046 0.0019 0.0028 0.0008 0.0015 0.0000 0.0010 5 0.0000 0.0083 0.0020 0.0143 0.0066 0.0024 0.0023 0.0010 0.0015 0.0000 0.0013 10 0.0000 0.0130 0.0032 0.0222 0.0117 0.0041 0.0031 0.0024 0.0022 0.0000 0.0018 20 0.0000 0.0199 0.0049 0.0345 0.0197 0.0072 0.0038 0.0024 0.0034 0.0000 0.0043 50 0.0009 0.0359 0.0085 0.0546 0.0308 0.0132 0.0014 0.0064 0.0055 0.0000 0.0092 100 0.0012 0.0518 0.0131 0.0741 0.0391 0.0246 0.0021 0.0115 0.0053 0.0002 0.0151 Amazon Toys Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0000 0.0008 0.0002 0.0027 0.0013 0.0001 0.0000 0.0002 0.0001 0.0000 0.0002 5 0.0000 0.0009 0.0003 0.0036 0.0020 0.0002 0.0002 0.0003 0.0002 0.0000 0.0003 10 0.0000 0.0016 0.0005 0.0052 0.0029 0.0003 0.0002 0.0005 0.0003 0.0000 0.0004 20 0.0000 0.0023 0.0007 0.0071 0.0044 0.0004 0.0003 0.0006 0.0001 0.0000 0.0004 50 0.0000 0.0040 0.0012 0.0088 0.0057 0.0012 0.0007 0.0010 0.0005 0.0000 0.0010 100 0.0000 0.0065 0.0017 0.0094 0.0064 0.0025 0.0005 0.0011 0.0007 0.0001 0.0017 Gowalla Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0002 0.0159 0.0035 0.0277 0.0125 0.0084 0.0060 0.0030 0.0033 0.0000 0.0045 5 0.0002 0.0174 0.0039 0.0300 0.0136 0.0090 0.0069 0.0025 0.0037 0.0000 0.0054 10 0.0002 0.0210 0.0047 0.0356 0.0169 0.0114 0.0062 0.0023 0.0046 0.0000 0.0058 20 0.0004 0.0299 0.0066 0.0478 0.0237 0.0158 0.0054 0.0032 0.0054 0.0000 0.0084 50 0.0007 0.0478 0.0106 0.0693 0.0349 0.0251 0.0015 0.0101 0.0042 0.0000 0.0112 100 0.0013 0.0695 0.0141 0.0894 0.0400 0.0352 0.0029 0.0127 0.0120 0.0001 0.0174 MovieLens Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0020 0.0141 0.0130 0.0160 0.0149 0.0127 0.0120 0.0114 0.0137 0.0001 0.0089 5 0.0023 0.0156 0.0146 0.0174 0.0177 0.0147 0.0165 0.0123 0.0145 0.0001 0.0090 10 0.0028 0.0184 0.0175 0.0200 0.0215 0.0172 0.0210 0.0156 0.0173 0.0002 0.0110 20 0.0037 0.0212 0.0208 0.0219 0.0242 0.0206 0.0232 0.0174 0.0206 0.0003 0.0140 50 0.0056 0.0229 0.0245 0.0236 0.0244 0.0240 0.0210 0.0197 0.0098 0.0004 0.0191 100 0.0075 0.0219 0.0270 0.0249 0.0144 0.0134 0.0057 0.0268 0.0111 0.0011 0.0150 Yelp Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0001 0.0077 0.0018 0.0079 0.0018 0.0040 0.0033 0.0017 0.0044 0.0000 0.0022 5 0.0002 0.0118 0.0022 0.0111 0.0038 0.0057 0.0083 0.0019 0.0022 0.0000 0.0032 10 0.0003 0.0208 0.0034 0.0188 0.0080 0.0098 0.0113 0.0026 0.0034 0.0000 0.0059 20 0.0005 0.0348 0.0054 0.0318 0.0164 0.0178 0.0152 0.0042 0.0028 0.0001 0.0106 50 0.0010 0.0569 0.0093 0.0561 0.0208 0.0238 0.0039 0.0081 0.0114 0.0003 0.0192 100 0.0013 0.0708 0.0152 0.0739 0.0155 0.0020 0.0003 0.0129 0.0019 0.0004 0.0059 60 PRECISION@10 (PHASE 2) Amazon CD Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0005 0.0001 0.0009 0.0004 0.0170 0.0026 0.0010 0.0014 0.0000 0.0977 10 0.0000 0.0007 0.0002 0.0013 0.0006 0.0165 0.0025 0.0017 0.0013 0.0000 0.0993 20 0.0000 0.0010 0.0002 0.0018 0.0010 0.0142 0.0019 0.0009 0.0010 0.0000 0.0830 50 0.0000 0.0013 0.0004 0.0023 0.0012 0.0123 0.0008 0.0013 0.0007 0.0000 0.0476 100 0.0000 0.0019 0.0003 0.0024 0.0016 0.0102 0.0004 0.0010 0.0004 0.0000 0.0243 Amazon Toys Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0001 0.0000 0.0005 0.0002 0.1943 0.0005 0.0009 0.0000 0.0000 0.4327 10 0.0000 0.0001 0.0001 0.0005 0.0004 0.1387 0.0002 0.0009 0.0000 0.0000 0.2680 20 0.0000 0.0002 0.0000 0.0005 0.0003 0.0837 0.0001 0.0009 0.0000 0.0000 0.1384 50 0.0000 0.0002 0.0001 0.0004 0.0002 0.0343 0.0001 0.0005 0.0000 0.0000 0.0464 100 0.0000 0.0002 0.0000 0.0003 0.0003 0.0156 0.0000 0.0005 0.0000 0.0001 0.0172 Gowalla Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0011 0.0003 0.0019 0.0009 0.0136 0.0071 0.0034 0.0038 0.0000 0.1144 10 0.0000 0.0012 0.0003 0.0021 0.0010 0.0132 0.0069 0.0022 0.0039 0.0000 0.1189 20 0.0000 0.0017 0.0004 0.0027 0.0014 0.0132 0.0053 0.0047 0.0042 0.0000 0.1104 50 0.0000 0.0023 0.0005 0.0035 0.0017 0.0083 0.0033 0.0032 0.0029 0.0000 0.0730 100 0.0000 0.0025 0.0004 0.0034 0.0017 0.0055 0.0013 0.0018 0.0017 0.0000 0.0348 MovieLens Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0002 0.0010 0.0009 0.0011 0.0011 0.0142 0.1479 0.0127 0.0151 0.0001 0.0109 10 0.0002 0.0011 0.0010 0.0011 0.0012 0.0130 0.1156 0.0132 0.0141 0.0001 0.0084 20 0.0005 0.0026 0.0026 0.0026 0.0030 0.0055 0.0289 0.0059 0.0096 0.0001 0.0027 50 0.0002 0.0009 0.0009 0.0008 0.0007 0.0034 0.0054 0.0059 0.0047 0.0001 0.0021 100 0.0003 0.0004 0.0003 0.0006 0.0003 0.0011 0.0020 0.0038 0.0034 0.0001 0.0004 Yelp Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0008 0.0002 0.0007 0.0003 0.0056 0.0218 0.0018 0.0023 0.0000 0.0070 10 0.0000 0.0010 0.0002 0.0010 0.0005 0.0050 0.0094 0.0013 0.0017 0.0000 0.0057 20 0.0000 0.0013 0.0002 0.0013 0.0007 0.0036 0.0046 0.0011 0.0017 0.0000 0.0049 50 0.0000 0.0012 0.0003 0.0011 0.0004 0.0024 0.0013 0.0008 0.0012 0.0000 0.0039 100 0.0000 0.0009 0.0003 0.0011 0.0003 0.0032 0.0005 0.0008 0.0008 0.0000 0.0018 61 RECALL@10 (PHASE 1) Amazon CD Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0000 0.0281 0.0096 0.0438 0.0126 0.0064 0.0111 0.0030 0.0095 0.0000 0.0038 5 0.0001 0.0262 0.0069 0.0428 0.0147 0.0053 0.0070 0.0027 0.0061 0.0000 0.0034 10 0.0001 0.0240 0.0059 0.0407 0.0190 0.0050 0.0049 0.0035 0.0039 0.0000 0.0025 20 0.0001 0.0249 0.0062 0.0435 0.0244 0.0053 0.0037 0.0017 0.0031 0.0000 0.0032 50 0.0009 0.0359 0.0085 0.0546 0.0308 0.0055 0.0004 0.0030 0.0024 0.0000 0.0039 100 0.0012 0.0518 0.0131 0.0741 0.0391 0.0060 0.0003 0.0032 0.0012 0.0000 0.0039 Amazon Toys Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0000 0.0029 0.0010 0.0112 0.0040 0.0003 0.0000 0.0009 0.0005 0.0000 0.0005 5 0.0000 0.0022 0.0009 0.0106 0.0047 0.0003 0.0005 0.0008 0.0006 0.0000 0.0005 10 0.0000 0.0027 0.0011 0.0099 0.0049 0.0004 0.0005 0.0010 0.0007 0.0000 0.0005 20 0.0000 0.0029 0.0009 0.0092 0.0056 0.0002 0.0004 0.0007 0.0001 0.0000 0.0003 50 0.0000 0.0040 0.0012 0.0088 0.0057 0.0003 0.0003 0.0005 0.0002 0.0000 0.0003 100 0.0000 0.0065 0.0017 0.0094 0.0064 0.0005 0.0000 0.0003 0.0001 0.0000 0.0003 Gowalla Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0003 0.0326 0.0072 0.0604 0.0254 0.0141 0.0089 0.0053 0.0062 0.0000 0.0077 5 0.0003 0.0333 0.0073 0.0601 0.0259 0.0138 0.0093 0.0038 0.0063 0.0000 0.0085 10 0.0003 0.0345 0.0077 0.0594 0.0271 0.0150 0.0077 0.0029 0.0063 0.0000 0.0070 20 0.0004 0.0368 0.0081 0.0591 0.0293 0.0141 0.0050 0.0028 0.0050 0.0000 0.0069 50 0.0007 0.0478 0.0106 0.0693 0.0349 0.0120 0.0005 0.0051 0.0017 0.0000 0.0046 100 0.0013 0.0695 0.0141 0.0894 0.0400 0.0093 0.0004 0.0035 0.0032 0.0000 0.0042 MovieLens Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0089 0.0698 0.0639 0.0839 0.0736 0.0626 0.0565 0.0575 0.0668 0.0003 0.0405 5 0.0078 0.0606 0.0560 0.0702 0.0709 0.0561 0.0639 0.0469 0.0550 0.0003 0.0312 10 0.0061 0.0451 0.0426 0.0498 0.0542 0.0408 0.0503 0.0368 0.0411 0.0003 0.0239 20 0.0054 0.0322 0.0318 0.0332 0.0379 0.0294 0.0333 0.0242 0.0293 0.0003 0.0190 50 0.0056 0.0229 0.0245 0.0236 0.0244 0.0161 0.0140 0.0131 0.0063 0.0002 0.0126 100 0.0075 0.0219 0.0270 0.0249 0.0144 0.0049 0.0019 0.0099 0.0040 0.0004 0.0054 Yelp Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 0 0.0008 0.0449 0.0123 0.0481 0.0104 0.0229 0.0179 0.0112 0.0256 0.0001 0.0116 5 0.0007 0.0497 0.0110 0.0475 0.0157 0.0233 0.0321 0.0091 0.0109 0.0001 0.0127 10 0.0007 0.0493 0.0085 0.0432 0.0196 0.0205 0.0228 0.0061 0.0080 0.0001 0.0128 20 0.0007 0.0489 0.0080 0.0440 0.0250 0.0205 0.0172 0.0051 0.0031 0.0001 0.0125 50 0.0010 0.0569 0.0093 0.0561 0.0208 0.0121 0.0021 0.0043 0.0059 0.0002 0.0101 100 0.0013 0.0708 0.0152 0.0739 0.0155 0.0005 0.0001 0.0037 0.0007 0.0001 0.0015 62 RECALL@10 (PHASE 2) Amazon CD Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0013 0.0003 0.0022 0.0009 0.0547 0.0073 0.0028 0.0043 0.0000 0.2751 10 0.0000 0.0012 0.0003 0.0021 0.0011 0.0326 0.0061 0.0038 0.0022 0.0000 0.1898 20 0.0000 0.0013 0.0003 0.0023 0.0013 0.0165 0.0048 0.0020 0.0011 0.0000 0.1028 50 0.0000 0.0014 0.0005 0.0024 0.0013 0.0080 0.0015 0.0033 0.0004 0.0000 0.0344 100 0.0000 0.0019 0.0003 0.0024 0.0016 0.0048 0.0011 0.0038 0.0002 0.0000 0.0120 Amazon Toys Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0001 0.0000 0.0005 0.0002 0.1943 0.0005 0.0009 0.0000 0.0000 0.4327 10 0.0000 0.0001 0.0001 0.0005 0.0004 0.1387 0.0002 0.0009 0.0000 0.0000 0.2680 20 0.0000 0.0002 0.0000 0.0005 0.0003 0.0837 0.0001 0.0009 0.0000 0.0000 0.1384 50 0.0000 0.0002 0.0001 0.0004 0.0002 0.0343 0.0001 0.0005 0.0000 0.0000 0.0464 100 0.0000 0.0002 0.0000 0.0003 0.0003 0.0156 0.0000 0.0005 0.0000 0.0001 0.0172 Gowalla Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0018 0.0004 0.0034 0.0015 0.0224 0.0097 0.0050 0.0062 0.0000 0.2267 10 0.0000 0.0019 0.0004 0.0032 0.0016 0.0188 0.0086 0.0032 0.0060 0.0000 0.1930 20 0.0000 0.0021 0.0005 0.0034 0.0017 0.0134 0.0075 0.0066 0.0067 0.0000 0.1234 50 0.0000 0.0024 0.0005 0.0036 0.0018 0.0061 0.0050 0.0054 0.0053 0.0000 0.0494 100 0.0000 0.0025 0.0004 0.0034 0.0017 0.0032 0.0020 0.0055 0.0043 0.0000 0.0153 MovieLens Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0004 0.0031 0.0028 0.0034 0.0034 0.0491 0.4559 0.0454 0.0554 0.0003 0.0352 10 0.0004 0.0024 0.0023 0.0024 0.0028 0.0368 0.2940 0.0436 0.0487 0.0003 0.0193 20 0.0010 0.0059 0.0060 0.0061 0.0070 0.0225 0.0854 0.0280 0.0544 0.0003 0.0058 50 0.0003 0.0011 0.0011 0.0010 0.0009 0.0122 0.0178 0.0347 0.0244 0.0003 0.0022 100 0.0003 0.0006 0.0004 0.0007 0.0003 0.0050 0.0119 0.0303 0.0260 0.0003 0.0005 Yelp Core BiasedMF ImplicitMF PopScore User-KNN Item-KNN BPR DiffRec DMF MultiVAE Random SimpleX 5 0.0000 0.0025 0.0006 0.0024 0.0008 0.0213 0.0941 0.0082 0.0112 0.0001 0.0303 10 0.0000 0.0026 0.0005 0.0022 0.0011 0.0149 0.0299 0.0058 0.0083 0.0001 0.0163 20 0.0000 0.0026 0.0005 0.0025 0.0014 0.0081 0.0102 0.0055 0.0087 0.0001 0.0094 50 0.0000 0.0016 0.0004 0.0017 0.0006 0.0046 0.0031 0.0049 0.0083 0.0001 0.0055 100 0.0000 0.0015 0.0004 0.0014 0.0006 0.0046 0.0021 0.0060 0.0064 0.0001 0.0026
|
2510.14705
|
Leveraging Learned Image Prior for 3D Gaussian Compression
Seungjoo Shin
POSTECH
seungjoo.shin@postech.ac.kr
Jaesik Park
Seoul National University
jaesik.park@snu.ac.kr
Sunghyun Cho
POSTECH
s.cho@postech.ac.kr
Abstract
Compression techniques for 3D Gaussian Splatting (3DGS)
have recently achieved considerable success in minimiz-
ing storage overhead for 3D Gaussians while preserving
high rendering quality. Despite the impressive storage re-
duction, the lack of learned priors restricts further ad-
vances in the rate-distortion trade-off for 3DGS compres-
sion tasks. To address this, we introduce a novel 3DGS
compression framework that leverages the powerful rep-
resentational capacity of learned image priors to recover
compression-induced quality degradation. Built upon ini-
tially compressed Gaussians, our restoration network effec-
tively models the compression artifacts in the image space
between degraded and original Gaussians. To enhance the
rate-distortion performance, we provide coarse rendering
residuals into the restoration network as side information.
By leveraging the supervision of restored images, the com-
pressed Gaussians are refined, resulting in a highly compact
representation with enhanced rendering performance. Our
framework is designed to be compatible with existing Gaus-
sian compression methods, making it broadly applicable
across different baselines. Extensive experiments validate
the effectiveness of our framework, demonstrating superior
rate-distortion performance and outperforming the render-
ing quality of state-of-the-art 3DGS compression methods
while requiring substantially less storage.
1. Introduction
Recent advances in 3D representations have boosted the
performance benchmarks of novel-view synthesis, which
aims to model volumetric scenes to synthesize photo-
realistic unseen views. One of the representative repre-
sentations is Neural Radiance Fields (NeRFs) [26], which
exploits coordinate-based neural networks to enable high-
fidelity volumetric rendering. More recently, 3D Gaussian
Splatting (3DGS) [17] has emerged as a prominent rep-
resentation offering real-time rendering performance with
high-quality outputs. Specifically, it introduces 3D Gaus-
sians as point-based primitives parameterized by learnable
parameters that determine their shapes and appearances.
The remarkable capabilities of 3DGS have inspired numer-
ous subsequent approaches, focusing on improving render-
ing quality [30, 46], dynamic scene modeling [19, 39, 42,
43], and scene generation [3, 7, 40].
Despite its impressive performance, 3DGS suffers from
excessive storage requirements, posing significant chal-
lenges for real-world applications. This overhead mainly
arises from its fine-grained parameterization, where each
Gaussian is defined by 59 learnable attributes, including po-
sition, covariance, color, opacity, and spherical harmonics
coefficients. To achieve high-fidelity scene representation,
3DGS is initialized from Structure-from-Motion (SfM) [32]
points and progressively increases the number of Gaus-
sians, often scaling up to millions of Gaussians. While it
enables high-quality rendering, it inevitably leads to sub-
stantial storage overhead due to the large number of Gaus-
sians and their associated parameters. As a result, the stor-
age and memory demands are further amplified, limiting the
practicality of 3DGS for scalable deployment, particularly
in resource-constrained environments.
To alleviate the storage burden of Gaussians, recent
3DGS approaches propose several compression techniques
that aim to minimize the number of Gaussian primi-
tives [11, 25], the redundancy in their attributes [8, 27, 29],
or both objectives [6, 10, 13, 18, 33, 38]. In particular,
while attribute-level compression effectively reduces stor-
age overhead, it inevitably introduces information loss,
leading to discrepancies between the original Gaussians and
their compressed counterparts. Such distortion in attribute
values often results in visible compression artifacts in the
rendered images, degrading the overall rendering fidelity.
To mitigate this degradation, we aim to restore the com-
pressed Gaussian attributes to synthesize clean renderings
comparable to those obtained from uncompressed Gaus-
sians. Toward this goal, we leverage the strong represen-
tational power of learned priors, trained on large-scale
data and supported by an effective network architecture.
However, constructing learned priors directly in the high-
dimensional attribute space is challenging due to the non-
trivial distributions of different attribute types and their de-
arXiv:2510.14705v1 [cs.CV] 16 Oct 2025
Reference image
JPEG
Gaussian
Figure 1. Visualization of compression artifacts in JPEG and
Gaussian representation. We visualize the pixel-wise error maps
(MSE) induced by each artifact to highlight the degradation.
pendence on scene complexity. To address this, we pro-
pose to learn image-level priors from rendered outputs in-
stead of modeling attribute distributions directly. Specifi-
cally, our method restores degraded images acquired from
compressed attributes by exploiting learned image priors
and subsequently employs those restored images as refer-
ences to refine the degraded Gaussian attributes.
In this paper, we propose a novel 3DGS framework that
leverages the impressive representational power of learned
image priors to restore degraded Gaussians resulting from
attribute compression. Given a set of optimized Gaussians,
our method first applies a feed-forward Gaussian compres-
sion method, FCGS [8], to build an initial compressed rep-
resentation with minimal storage and rendering quality. We
then restore the degraded images rendered from these com-
pressed Gaussians by employing an image restoration net-
work trained to remove visual artifacts introduced by 3DGS
compression. Additionally, we provide the coarse resid-
ual between the degraded and original renderings to the
restoration network as side information. This residual guid-
ance enables the network to capture scene-specific degra-
dation patterns, thereby enhancing the restoration of com-
pressed Gaussians and improving rate-distortion perfor-
mance. The restored images, containing enhanced visual in-
formation, subsequently serve as reference targets to refine
the compressed Gaussians for a few additional optimization
steps. As a result, experimental evaluations demonstrate
that our framework achieves superior rendering quality un-
der comparable storage requirements, compared to baseline
methods. Moreover, our approach achieves state-of-the-art
(SOTA) rate-distortion performance, outperforming exist-
ing 3DGS compression approaches.
We summarize our contributions as follows:
• We present a novel 3D Gaussian compression framework
that exploits learned image priors to remove Gaussian
compression artifacts in rendered images.
• Our framework refines compressed Gaussian attributes
using restored images to obtain high-quality Gaussians.
• Our restoration network successfully models the com-
pression artifacts between the compressed and original
Gaussians and further enhances rate-distortion perfor-
mance by leveraging coarse residuals as side information.
• Experimental results show that our framework achieves
superior rendering quality under comparable storage re-
quirements and demonstrates SOTA rate-distortion per-
formance over existing 3DGS compression approaches.
2. Related Work
2.1. 3D Gaussian Splatting and Compression
3DGS [17] represents a volumetric scene using a set of
anisotropic 3D Gaussians, each of which is a point-based
primitive defined by learnable attributes that describe its
shape and appearance. By employing tile-based rasteriza-
tion, this point-based representation enables real-time, high-
fidelity novel-view synthesis. Despite its remarkable render-
ing performance, representing volumetric scenes requires
substantial storage space, since a large number of Gaus-
sians are necessary to achieve high-quality rendering, and
each Gaussian is associated with several attributes to de-
fine intricate shape and appearance. To address this lim-
itation, recent 3DGS approaches have proposed primitive
pruning methods [11, 18, 25] and attribute encoding meth-
ods [6, 8, 10, 13, 18, 27, 29, 33, 38].
Primitive-level compression.
Primitive pruning meth-
ods [11, 17, 25] aim to minimize the number of Gaus-
sian primitives by reducing those with less contribution, us-
ing importance score-based pruning. Following the prun-
ing scheme of 3DGS [17], several methods [24, 27, 28]
adopt opacity-based pruning criteria often combined with
additional strategies for more efficient pruning. Other ap-
proaches exploit carefully designed pruning criteria that ac-
count for rendering contribution [11, 13], Gaussian vol-
ume [10], learnable binary masks [6, 18, 33, 34, 38], and
multiple components including gradient, per-pixel saliency,
and attribute values [25]. In particular, Mini-Splatting [11]
achieves high rendering quality with a minimal number of
primitives by leveraging global contribution scores in com-
bination with a deblurring strategy.
Attribute-level
compression.
Attribute
compression
methods [6, 8, 10, 13, 18, 27, 29, 33, 38] focus on en-
coding Gaussian attributes in a compact format, which
can be efficiently structured in various representations.
Several approaches adopt quantization-based techniques
that leverage global attribute statistics, including vector
quantization [10, 18, 28, 29], latent quantization [13],
and image codec [27]. Other methods utilize hash grids
to capture the local similarity of attributes [18, 33]. Built
upon Scaffold-GS [24], an anchor-based Gaussian repre-
sentation, HAC [6] and ContextGS [38] suggest context
modeling to minimize the entropy of anchor features.
Despite impressive encoding capability, lacking general-
izable priors results in limited compression performance
for these per-scene encoding schemes. Recently, FCGS [8]
introduced a feed-forward compression network, enabling
optimization-free compression within a few seconds. Al-
though the compression network constructs generalizable
priors from a large-scale 3DGS dataset, the limited effec-
tiveness of these priors makes it challenging to preserve
rendering quality when compressing Gaussian attributes
into highly compact storage, due to the unstructured design
and high variability of Gaussian representations.
Despite achieving significant storage reductions, most
approaches remain limited by their reliance on per-scene in-
formation, without exploiting any generalizable priors. To
overcome this limitation, we propose a novel 3DGS com-
pression framework that leverages learned image priors to
restore compression-induced degradation, thereby improv-
ing rate-distortion performance across diverse scenes.
2.2. Image Restoration
Image restoration aims to restore a high-quality image from
its degraded counterpart. Recently, deep learning meth-
ods have significantly improved the performance of image
restoration tasks, which include sub-problems such as de-
noising [48, 49], deblurring [15], super-resolution [35, 50],
and JPEG compression artifact reduction [16, 45]. Various
neural network architectures, especially U-Net [31] vari-
ants, are designed to capture specific degradation artifacts
and learn generalizable image priors, achieving remark-
able performance in image restoration tasks. With the in-
troduction of diverse convolutional neural network (CNN)-
based [5, 9, 45, 48] and transformer-based [4, 20, 47] archi-
tectures, SOTA performance has continued to be advanced.
To exploit the strong synthesis capabilities of modern
generative models (e.g., GANs or diffusion models), sev-
eral approaches [21, 35–37, 50] incorporate generative pri-
ors learned in those models, enabling high-quality im-
age restoration despite severe degradation. Recently, sev-
eral approaches have adopted generative priors from im-
age restoration models for 3D reconstruction, particularly
in tasks such as super-resolution [12] and few-shot recon-
struction [23, 41]. While these methods demonstrate effec-
tiveness in severely degraded reconstruction settings, they
might suffer from reduced 3D consistency. In such cases,
the generative priors may generate new patterns for each
viewpoint, making them less suitable for scenarios where
the scene is not severely degraded and geometric consis-
tency must be preserved.
In this manner, we exploit an image restoration model
to benefit from the highly expressive learned priors for the
3DGS compression. By employing the restoration network,
we successfully recover the degraded images rendered from
compressed Gaussians and leverage these restored images
to enhance the overall quality of compressed Gaussians.
3. Preliminaries: 3D Gaussian Splatting
Our framework is built upon 3DGS [17], which models vol-
umetric scenes using a set of point-based anisotropic Gaus-
sian primitives G = (p, s, r, o, k). Each Gaussian Gi is de-
fined by a set of attributes consisting of a position pi ∈R3,
a scale vector si ∈R3, a quaternion-based rotation vector
ri ∈R4, an opacity oi ∈[0, 1], and spherical harmonics
(SH) coefficients ki ∈R3×(L+1)2, where L is the maxi-
mum SH degree. The covariance matrix Σi ∈R3×3 is de-
fined by a rotation matrix derived from ri, and a scaling
matrix derived from si.
For rasterization, 3D Gaussians are projected into 2D
space for a given view. The RGB pixel value ˆC(·) at view-
space coordinate x ∈R2 is acquired by blending N depth-
sorted Gaussians with a view-dependent Gaussian color
ci ∈R3 derived from SH coefficients ki as:
ˆC(x) =
N
X
i=1
ciαi
i−1
Y
j=1
(1 −αj),
(1)
αi = oi · exp (−1
2(x −p′
i)⊤Σ
′−1
i
(x −p′
i)),
(2)
where p′
i and Σ
′−1
i
denote projected Gaussian position and
covariance, respectively. Initialized from SfM points, the
Gaussians G are optimized to synthesize photo-realistic ren-
dered images for the given set of training viewpoints.
4. Method
Fig. 2 illustrates the overall pipeline of our approach. Given
a set of optimized Gaussians G, we obtain a set of com-
pressed Gaussians ˜G by applying a compression method
(Sec. 4.1). To mitigate the quality degradation, we construct
effective side information by computing the coarse resid-
ual between the rendered images of the original and com-
pressed Gaussians (Sec. 4.2). Given the degraded render-
ings and corresponding coarse residuals, we adopt an im-
age restoration network that leverages learned image priors
to estimate original rendering images comparable to those
produced by the original Gaussians (Sec. 4.3). Finally, these
restored rendering images are utilized to refine the com-
pressed Gaussian attributes, resulting in improved rendering
fidelity while maintaining compact storage usage (Sec. 4.4).
4.1. Initial Gaussian Compression
The initial compression step builds compressed Gaussians
˜G that achieve significant storage savings while introducing
rendering distortion as a trade-off. Given a set of optimized
Gaussians G, we formulate the compression process as:
˜G = H(G),
˜G = (˜p,˜s, ˜r, ˜o, ˜k),
| ˜G| ≤|G|,
(3)
where H(·) denotes a lossy Gaussian compression opera-
tion that produces compressed attributes by reducing the
3D Gaussians
Initial Gaussian
Compression
Compressed Gaussians
Rasterize
Refined Gaussians
Gaussian
Refinement
Side information
Image Restoration
!
"!
#$
"$
#!
Figure 2. Illustration of overall pipeline. Given a set of Gaussians as input, we perform initial compression, followed by an image
restoration network that restores the rendered images. Finally, we refine the compressed Gaussians using the restored images to enhance
the rendering quality.
number of Gaussians or removing attribute redundancy,
at the cost of information loss. Specifically, we adopt
FCGS [8], a feed-forward 3DGS compression method that
achieves remarkable compression performance with mini-
mal computational overhead. Notably, other Gaussian com-
pression methods satisfying Eq. (3) are also compatible with
our pipeline as the initial compression module. The lossy
Gaussian compression process inevitably produces infor-
mation loss, which manifests as noticeable compression ar-
tifacts in the rendered outputs, as demonstrated in Fig. 1.
Mitigating this degradation while preserving storage effi-
ciency remains the core challenge of our framework.
We aim to recover high-quality Gaussians from their
compressed counterparts. However, directly modeling the
compression-induced distortion is challenging due to the
lack of effective priors that can be learned through a spe-
cialized network architecture capable of predicting restored
Gaussians. To address this, we reformulate the problem as
image-space restoration, where compression artifacts are
modeled in the rendered image domain. Specifically, we
leverage the rendered images obtained from the original
Gaussians G, denoted as I = {I1, ..., IK}, and the de-
graded images from the compressed Gaussians ˜G, denoted
as ˜I = {˜I1, ..., ˜IK}, to indirectly estimate the attribute
residuals using an image restoration network.
4.2. Side Information
To achieve more effective restoration for compressed Gaus-
sian renderings, we incorporate side information to support
the network to overcome the performance limitations. The
degraded images exhibit both compression artifacts with
aliasing patterns introduced by the 3DGS rendering process,
which differ from typical compression degradations such as
JPEG, as shown in Fig. 1. Specifically, unlike block-based
or frequency-domain artifacts in JPEG, these artifacts often
appear as view-dependent, high-frequency distortions that
are spatially inconsistent across images. These artifacts re-
sult in less effective restoration performance when applying
a conventional image restoration framework, indicating the
need for additional side information.
Coarse rendering residuals.
To address this challenge,
we aim to improve restoration performance by incorporat-
ing side information. Specifically, we exploit the residual
between the original and degraded images as auxiliary guid-
ance for restoration. However, fully leveraging these high-
resolution and high-precision residuals incurs substantial
storage overhead. To mitigate this, we compress the resid-
ual information R = I −˜I into coarse residuals ˜R using a
compact image codec, JPEG-XL [1], defined as:
˜R = Q(R; λrate) = Q(I −˜I; λrate),
(4)
where Q(·) denotes residual compression operation, and
λrate is a hyperparameter that controls the rate-distortion
trade-off. By adjusting λrate, our framework can flexibly
support multiple quality levels, allowing users to balance
storage efficiency and restoration quality according to their
specific requirements. During the decoding process, we
condition the restoration network on the compressed coarse
residual ˜R as decoder side information, providing addi-
tional guidance to recover high-fidelity renderings from
the degraded inputs. Furthermore, we uniformly sample a
minimal set of essential training views for side informa-
tion, leveraging the observation that dataset-provided train-
ing views often contain substantial overlap. This allows our
framework to operate more efficiently by reducing unnec-
essary computational and memory overhead.
4.3. Image Restoration
Fig. 3 illustrates the proposed restoration framework. Built
upon the image restoration network, we opt to model the
pixel-level residuals R = I −˜I between original renderings
I and their corresponding degraded renderings ˜I with the
support of side information ˜R as:
ˆI = ˜I + ˆR = ˜I + F(˜I, ˜R),
(5)
where F(·) represents the image restoration network, and
ˆI and ˆR denote the restored image and residuals, respec-
tively. For implementation, we adopt NAFNet [5], an im-
age restoration architecture that has demonstrated effective-
Degraded image
Original image
Residual
JPEG-XL
−
C
Restored image
Restoration network
C
−subtraction
concatenation
Figure 3. Illustration of the proposed image restoration framework. To train the image restoration network for Gaussian compression,
we construct a paired dataset consisting of degraded and original images rendered from Gaussians optimized on real-world 3D scenes.
Given a degraded image and the corresponding original image, we first compute their residual. The residual is then quantized and concate-
nated with the degraded image. This concatenated input is fed into the image restoration network, which is designed to remove Gaussian
compression artifacts and recover a high-quality image.
ness in handling various degradation types while maintain-
ing computational efficiency. To incorporate additional side
information, we modify the input layer of the network to
take a concatenation of the two images, a degraded image
˜I, and an auxiliary guidance ˜R.
To train the image restoration network, we construct a
3DGS compression artifact dataset consisting of paired im-
ages rendered from both original and compressed 3DGS
representations. Specifically, we first optimize 3DGS rep-
resentations on diverse real-world 3D scenes from the
DL3DV-10K dataset [22] and subsequently compress the
optimized Gaussians using FCGS [8] to obtain compressed
Gaussians. We then render images from both the original
and compressed Gaussians at multiple training views to
generate paired image samples for compression artifact re-
moval. These paired images are finally leveraged to train the
image restoration network with the following objectives:
Lrestore = L1(R, ˆR) + λLPIPSLLPIPS(I, ˆI),
(6)
where L1 and LLPIPS denote L1 loss and LPIPS [51] loss,
and λLPIPS controls the contribution of the perceptual term.
4.4. Gaussian Refinement
While the restoration network improves the visual fidelity of
degraded renderings, the final objective of our framework is
to refine the compressed Gaussian attributes to restore their
high rendering ability. We achieve this by re-optimizing the
compressed Gaussians ˜G into refined Gaussians ˆG under the
supervision of the restored images ˆI, effectively propagat-
ing the information obtained from the learned image priors.
Since the refinement aims to adjust the compressed Gaus-
sians rather than learning from scratch, it is designed to per-
form with only a few optimization iterations, minimizing
computational overhead. The following objective guides the
refinement process:
Lrefine = (1 −λ)L1(ˆI, ˆI′) + λLSSIM(ˆI, ˆI′),
(7)
where ˆI′ denotes the rendering output from the refined
Gaussians ˆG, and L1 and LSSIM denote the L1 loss and
SSIM loss, respectively. λ is a balancing weight. During op-
timization, we update all Gaussian attributes, whose values
are compressed in the initial compression step. This choice
can be adjusted based on the specific compression targets of
the initial compression method. Also, we preserve the num-
ber of Gaussians during the refinement stage.
5. Experiments
5.1. Experimental Setting
Datasets.
To build a 3DGS compression artifact dataset,
we leverage 495 real-world scenes from the DL3DV-10K
dataset [22]. As a result, we construct 18,555 image pairs
containing visual artifacts introduced by Gaussian compres-
sion, which are used to train the image restoration network.
For 3DGS evaluation, we benchmark on two representa-
tive novel-view synthesis datasets, Mip-NeRF 360 [2], and
Deep Blending [14]. Specifically, we evaluate nine scenes
from Mip-NeRF 360, and two from Deep Blending. Follow-
ing the evaluation protocol of 3DGS [17], we report PSNR,
SSIM, and LPIPS to assess rendering quality. In addition,
we also evaluate the storage size of each method to com-
pare the compression efficiency.
Implementation details.
For the image restoration net-
work, we adopt NAFNet [5], a CNN-based image restora-
tion architecture. We train it for 200K iterations on a single
NVIDIA RTX 6000 Ada GPU (48GB). During training, we
augment the auxiliary guidance using JPEG compression to
simulate diverse quality levels for robust optimization. Fol-
lowing the optimization scheme of Mini-Splatting [11], we
optimize the 3DGS representation for 30K iterations with a
minimal number of Gaussians. After optimization, we com-
press the Gaussian attributes using FCGS [8] and render
LightGaussian
Mini-C
C3DGS
PNG
FCGS
Ours
Ours
LightGaussian
Mini-C
C3DGS
PNG
FCGS
Ours
Ours
LightGaussian
Mini-C
C3DGS
PNG
FCGS
Ours
Ours
Size (MB)
Size (MB)
Size (MB)
PSNR
SSIM
LPIPS
Mip-NeRF 360
Mip-NeRF 360
Mip-NeRF 360
Figure 4. Rate-distortion curves comparing our method with existing post-compression approaches. We optimize Gaussians following
Mini-Splatting [11] and compress them using each post-compression method. Results are averaged over all scenes of Mip-NeRF 360 [2].
Table 1. Comparison with post-compression approaches. We
evaluate the rendering quality and storage size (MB) on Mip-NeRF
360 [2] and Deep Blending [14].
Method
Mip-NeRF 360 [2]
Deep Blending [14]
PSNR ↑
SSIM ↑
LPIPS ↓
Size ↓
PSNR ↑
SSIM ↑
LPIPS ↓
Size ↓
Mini-Splatting [11]
27.39
0.827
0.196
202.9
30.06
0.910
0.239
138.3
C3DGS [29]
27.03
0.813
0.220
13.87
29.80
0.903
0.257
12.10
LightGaussian [10]
27.00
0.816
0.213
39.13
29.56
0.900
0.253
26.48
Mini-C [11]
27.43
0.826
0.201
48.42
29.93
0.905
0.244
30.63
PNG [44]
27.23
0.822
0.205
13.65
29.87
0.906
0.245
8.39
FCGS-low [8]
27.09
0.817
0.211
11.78
29.53
0.897
0.252
7.21
FCGS-high [8]
27.32
0.824
0.203
14.08
29.74
0.900
0.247
8.48
Ours-low
27.24
0.821
0.205
11.53
29.96
0.906
0.248
7.21
Ours-high
27.33
0.824
0.202
13.63
30.01
0.907
0.247
8.14
them to acquire rendering image pairs. We further quan-
tize residual images using a standard image codec, JPEG-
XL [1], and employ them as side information for degraded
image restoration. To balance storage efficiency and restora-
tion performance, we uniformly sample 40% of the training
views for side information. With supervision from the re-
stored reference images, Gaussian refinement is performed
by optimizing the compressed Gaussians for 5K iterations,
which takes less than three minutes per scene on a single
NVIDIA RTX 6000 Ada GPU (48GB). Furthermore, we
provide multiple bitrate variants by adjusting the rate bal-
ancing weight λrate, which controls the storage cost of the
side information. Specifically, ‘Ours-low’ uses a smaller
λrate to achieve lower bitrate, while ‘Ours-high’ adopts a
larger λrate to allocate more storage for improved quality.
5.2. Comparison
To validate the effectiveness of the proposed compression
framework, we compare our approach with existing 3DGS
compression methods, categorized into post-compression
approaches and compact representation approaches.
Post-compression approaches aim to reduce storage
costs by compressing pre-optimized Gaussians, including
C3DGS [29], LightGaussian [10], Mini-Splatting-C [11],
PNG [44], and FCGS [8]. In particular, C3DGS and Light-
Table 2. Comparison with compact representation approaches.
We evaluate rendering quality and storage size (MB) on Mip-
NeRF 360 [2] and Deep Blending [14].
Method
Mip-NeRF 360 [2]
Deep Blending [14]
PSNR ↑
SSIM ↑
LPIPS ↓
Size ↓
PSNR ↑
SSIM ↑
LPIPS ↓
Size ↓
3DGS [17]
27.44
0.813
0.218
822.6
29.48
0.900
0.246
692.5
CompGS [28]
27.04
0.804
0.243
22.93
29.89
0.907
0.253
15.15
Compact-3DGS [18]
26.95
0.797
0.244
26.31
29.71
0.901
0.257
21.75
EAGLES [13]
27.10
0.807
0.234
59.49
29.72
0.906
0.249
54.45
SOG [27]
27.02
0.800
0.226
43.77
29.21
0.891
0.271
19.32
HAC [6]
27.49
0.807
0.236
16.95
29.99
0.902
0.268
4.51
LocoGS [33]
27.33
0.814
0.219
13.89
30.06
0.904
0.249
7.64
Ours-low
27.24
0.821
0.205
11.53
29.96
0.906
0.248
7.21
Ours-high
27.33
0.824
0.202
13.63
30.01
0.907
0.247
8.14
Gaussian require additional optimization steps for fine-
tuning, whereas Mini-Splatting-C, PNG, and FCGS are
optimization-free methods. For a fair comparison, we equal-
ize the overall optimization budget by performing an addi-
tional 5K iterations, matching the refinement setting of our
method, before compression for Mini-Splatting-C, PNG,
and FCGS. This results in a total of 35K iterations for
their initial optimization. In contrast, compact represen-
tation approaches directly learn efficient representations
from scratch without relying on pre-optimized Gaussians,
including SOTA compact Gaussian representations such
as CompGS [28], Compact-3DGS [18], EAGLES [13],
HAC [6], and LocoGS [33].
Post-compression approaches.
Tab. 1 and Fig. 4 show
the quantitative comparison with existing post-compression
approaches. Built upon the same initial Gaussians, our
compression method demonstrates a superior rate-distortion
trade-off, consistently outperforming the rendering quality
of baseline methods while achieving significant storage re-
duction up to 19.2× compared to the uncompressed rep-
resentation, Mini-Splatting. Specifically, in the low bitrate
setting, our method outperforms the rendering quality of
FCGS-low with comparable or even smaller storage con-
sumption. In the high bitrate setting, our method shows su-
perior rendering performance, except for Mini-Splatting-C,
Figure 5. Ablation study on view-sampling. Results are averaged over all scenes of Mip-NeRF 360 [2].
PSNR on Mip-NeRF 360
SSIM on Mip-NeRF 360
Size (MB)
Size (MB)
PSNR
SSIM
Figure 6. Ablation study on learned image prior. Results are
averaged over all scenes of Mip-NeRF 360 [2].
while maintaining the minimal storage cost. Our method
achieves comparable rendering quality to Mini-Splatting-
C, which requires up to 3.8× storage usage. Meanwhile,
vector-quantization-based approaches, C3DGS and Light-
Gaussian, suffer from lower rendering performance despite
their relatively large storage sizes. These results highlight
the remarkable performance of our method as a leading
post-compression solution, achieving high-quality render-
ing and substantial storage savings.
Compact representation approaches.
Tab. 2 presents
a quantitative comparison against recent compact Gaus-
sian representations, which optimize Gaussians under com-
pressed designs. Overall, our method achieves superior
rendering quality against SOTA compact representations
while maintaining the lowest storage overhead, with the ex-
ception of HAC on the Deep Blending. Notably, our ap-
proach demonstrates considerable perceptual improvements
in terms of SSIM and LPIPS, contributing to more faithful
and realistic rendering outputs. In contrast, quantization-
based methods such as CompGS, Compact-3DGS, EA-
GLES, and SOG suffer from noticeable degradation in ren-
dering quality despite requiring larger storage sizes. Com-
pared to SOTA compact baselines, HAC, and LocoGS, our
method achieves better rate-distortion trade-offs, deliver-
ing high-fidelity rendering while preserving minimal stor-
age costs. Specifically, on the Mip-NeRF 360, our method
achieves the best performance across all rendering metrics
while requiring the smallest storage footprint.
PNG
FCGS
FCGS
PNG
PNG + Ours
FCGS + Ours
FCGS + Ours
PNG + Ours
Size (MB)
Size (MB)
PSNR
SSIM
PSNR on Mip-NeRF 360
SSIM on Mip-NeRF 360
Figure 7. Ablation study on initial compression method. Results
are averaged over all scenes of Mip-NeRF 360 [2].
5.3. Ablation Study
Learned image prior.
We investigate the effectiveness of
learned image priors for rendering image restoration. To this
end, we evaluate the rendering quality of refined Gaussians
with and without the use of the trained image restoration
model. Fig. 6 shows the rendering quality of refined Gaus-
sians according to the utilization of learned image priors to
the image restoration process. Note that ‘w/o image prior’
refers to the setting where image restoration is performed
by simply adding quantized residuals to the degraded im-
age. In contrast, ‘Ours’ leverages learned image priors em-
bedded in the trained image restoration network to refine
the degraded image, as described in Sec. 4.3. As a result,
our method achieves noticeable improvements in rendering
quality for all evaluation metrics with the support of learned
image priors. Overall, these results clearly demonstrate that
leveraging learned image priors is crucial for recovering the
visual fidelity of compressed Gaussian renderings.
View-sampling.
We also validate the effectiveness of
view sampling in balancing storage efficiency and restora-
tion quality. Fig. 5 presents the Gaussian refinement perfor-
mance with respect to the sampling ratio of training views
used as side information. Sampling a smaller set of essen-
tial views reduces storage consumption by minimizing re-
dundancy while maintaining comparable rendering quality
with only minor degradation. Based on this observation, we
adopt a 40% sampling ratio to achieve a favorable balance
between storage cost and rendering quality.
Table 3. Ablation study on side information. ‘Image prior’ and
‘Side info.’ denote the learned image prior and side information,
respectively. Results are averaged over all scenes of Mip-NeRF
360 [2].
Image prior
Side info.
PSNR ↑
SSIM ↑
LPIPS ↓
Size ↓
✗
✗
26.98
0.816
0.212
10.89
✗
✓
27.16
0.819
0.208
11.53
✓
✗
27.00
0.817
0.211
10.89
✓
✓
27.24
0.821
0.205
11.53
Side information.
Moreover, we investigate the effective-
ness of side information in enhancing restoration perfor-
mance. As shown in Fig. 1, Gaussian compression arti-
facts are non-trivial to remove using the restoration net-
work alone, even with a sufficiently large network scale
and extensive training dataset. Tab. 3 reports the perfor-
mance of our framework under the difference configura-
tion of the learned image prior and side information. We
observe that even a minimal increase in storage size can
lead to an evident improvement in rendering quality when
accompanied by side information. Also, incorporating side
information enables more effective utilization of the learned
image prior. Without side information, the learned prior
yields only marginal benefits in rendering quality. In con-
trast, when side information is leveraged, the learned prior
provides significantly greater benefits, resulting in enhanced
rendering quality.
Initial compression method.
Furthermore, we demon-
strate the compatibility of our method with various Gaus-
sian compression approaches. To this end, we incorporate
PNG [44], a representative post-compression method, into
our compression scheme, as shown in Fig. 7. Despite the
suboptimal rendering quality of the initial compressed re-
sults, our method consistently delivers significant improve-
ments, particularly achieving noticeable gains in PSNR, re-
gardless of the underlying compression method. These re-
sults indicate that our framework can generalize well and
support a wide range of future compression methods.
6. Conclusion
In this paper, we introduced a novel 3D Gaussian com-
pression framework that leverages an image restoration net-
work to recover the visual quality of compressed Gaus-
sians. Our approach tackles compression-induced artifacts
by reformulating Gaussian restoration as an image-space
problem, enabling the use of powerful restoration networks
trained on large-scale data. Experimental results demon-
strate that our method effectively captures scene-specific
degradation patterns and removes compression artifacts,
achieving enhanced rendering fidelity and state-of-the-art
rate-distortion performance compared to existing 3DGS
compression methods, including both post-compression ap-
proaches and compact representation approaches.
Limitations.
Despite the impressive restoration perfor-
mance and broad applicability of our method, it introduces
additional computational overhead due to the restoration
process. While the additional process of our framework
is more efficient compared to existing post-compression
methods, it still requires non-negligible computational costs
compared to optimization-free approaches. Moreover, our
current design relies on separately quantizing the side in-
formation using an external image codec before feeding it
into the restoration network. A promising future direction
is to improve both computational efficiency and restoration
quality by jointly optimizing residual quantization and im-
age restoration within a unified, end-to-end trainable frame-
work.
Acknowledgments
This work was supported by the National Research Foun-
dation of Korea (NRF) grant funded by Korea government
(Ministry of Education) (No.2022R1A6A1A03052954, Ba-
sic Science Research Program), and Institute of In-
formation & communications Technology Planning &
Evaluation (IITP) grant funded by Korea government
(MSIT) (No.RS-2019-II191906, AI Graduate School Pro-
gram (POSTECH); No.RS-2021-II211343, AI Graduate
School Program (Seoul National University)).
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Leveraging Learned Image Prior for 3D Gaussian Compression Seungjoo Shin POSTECH Jaesik Park Seoul National University Sunghyun Cho POSTECH Abstract Compression techniques for 3D Gaussian Splatting (3DGS) have recently achieved considerable success in minimizing storage overhead for 3D Gaussians while preserving high rendering quality. Despite the impressive storage reduction, the lack of learned priors restricts further advances in the rate-distortion trade-off for 3DGS compression tasks. To address this, we introduce a novel 3DGS compression framework that leverages the powerful representational capacity of learned image priors to recover compression-induced quality degradation. Built upon initially compressed Gaussians, our restoration network effectively models the compression artifacts in the image space between degraded and original Gaussians. To enhance the rate-distortion performance, we provide coarse rendering residuals into the restoration network as side information. By leveraging the supervision of restored images, the compressed Gaussians are refined, resulting in a highly compact representation with enhanced rendering performance. Our framework is designed to be compatible with existing Gaussian compression methods, making it broadly applicable across different baselines. Extensive experiments validate the effectiveness of our framework, demonstrating superior rate-distortion performance and outperforming the rendering quality of state-of-the-art 3DGS compression methods while requiring substantially less storage. 1. Introduction Recent advances in 3D representations have boosted the performance benchmarks of novel-view synthesis, which aims to model volumetric scenes to synthesize photorealistic unseen views. One of the representative representations is Neural Radiance Fields (NeRFs) [26], which exploits coordinate-based neural networks to enable highfidelity volumetric rendering. More recently, 3D Gaussian Splatting (3DGS) [17] has emerged as a prominent representation offering real-time rendering performance with high-quality outputs. Specifically, it introduces 3D Gaussians as point-based primitives parameterized by learnable parameters that determine their shapes and appearances. The remarkable capabilities of 3DGS have inspired numerous subsequent approaches, focusing on improving rendering quality [30, 46], dynamic scene modeling [19, 39, 42, 43], and scene generation [3, 7, 40]. Despite its impressive performance, 3DGS suffers from excessive storage requirements, posing significant challenges for real-world applications. This overhead mainly arises from its fine-grained parameterization, where each Gaussian is defined by 59 learnable attributes, including position, covariance, color, opacity, and spherical harmonics coefficients. To achieve high-fidelity scene representation, 3DGS is initialized from Structure-from-Motion (SfM) [32] points and progressively increases the number of Gaussians, often scaling up to millions of Gaussians. While it enables high-quality rendering, it inevitably leads to substantial storage overhead due to the large number of Gaussians and their associated parameters. As a result, the storage and memory demands are further amplified, limiting the practicality of 3DGS for scalable deployment, particularly in resource-constrained environments. To alleviate the storage burden of Gaussians, recent 3DGS approaches propose several compression techniques that aim to minimize the number of Gaussian primitives [11, 25], the redundancy in their attributes [8, 27, 29], or both objectives [6, 10, 13, 18, 33, 38]. In particular, while attribute-level compression effectively reduces storage overhead, it inevitably introduces information loss, leading to discrepancies between the original Gaussians and their compressed counterparts. Such distortion in attribute values often results in visible compression artifacts in the rendered images, degrading the overall rendering fidelity. To mitigate this degradation, we aim to restore the compressed Gaussian attributes to synthesize clean renderings comparable to those obtained from uncompressed Gaussians. Toward this goal, we leverage the strong representational power of learned priors, trained on large-scale data and supported by an effective network architecture. However, constructing learned priors directly in the highdimensional attribute space is challenging due to the nontrivial distributions of different attribute types and their de16 Oct 2025 Reference image JPEG Gaussian Figure 1. Visualization of compression artifacts in JPEG and Gaussian representation. We visualize the pixel-wise error maps (MSE) induced by each artifact to highlight the degradation. pendence on scene complexity. To address this, we propose to learn image-level priors from rendered outputs instead of modeling attribute distributions directly. Specifically, our method restores degraded images acquired from compressed attributes by exploiting learned image priors and subsequently employs those restored images as references to refine the degraded Gaussian attributes. In this paper, we propose a novel 3DGS framework that leverages the impressive representational power of learned image priors to restore degraded Gaussians resulting from attribute compression. Given a set of optimized Gaussians, our method first applies a feed-forward Gaussian compression method, FCGS [8], to build an initial compressed representation with minimal storage and rendering quality. We then restore the degraded images rendered from these compressed Gaussians by employing an image restoration network trained to remove visual artifacts introduced by 3DGS compression. Additionally, we provide the coarse residual between the degraded and original renderings to the restoration network as side information. This residual guidance enables the network to capture scene-specific degradation patterns, thereby enhancing the restoration of compressed Gaussians and improving rate-distortion performance. The restored images, containing enhanced visual information, subsequently serve as reference targets to refine the compressed Gaussians for a few additional optimization steps. As a result, experimental evaluations demonstrate that our framework achieves superior rendering quality under comparable storage requirements, compared to baseline methods. Moreover, our approach achieves state-of-the-art (SOTA) rate-distortion performance, outperforming existing 3DGS compression approaches. We summarize our contributions as follows: • We present a novel 3D Gaussian compression framework that exploits learned image priors to remove Gaussian compression artifacts in rendered images. • Our framework refines compressed Gaussian attributes using restored images to obtain high-quality Gaussians. • Our restoration network successfully models the compression artifacts between the compressed and original Gaussians and further enhances rate-distortion performance by leveraging coarse residuals as side information. • Experimental results show that our framework achieves superior rendering quality under comparable storage requirements and demonstrates SOTA rate-distortion performance over existing 3DGS compression approaches. 2. Related Work 2.1. 3D Gaussian Splatting and Compression 3DGS [17] represents a volumetric scene using a set of anisotropic 3D Gaussians, each of which is a point-based primitive defined by learnable attributes that describe its shape and appearance. By employing tile-based rasterization, this point-based representation enables real-time, highfidelity novel-view synthesis. Despite its remarkable rendering performance, representing volumetric scenes requires substantial storage space, since a large number of Gaussians are necessary to achieve high-quality rendering, and each Gaussian is associated with several attributes to define intricate shape and appearance. To address this limitation, recent 3DGS approaches have proposed primitive pruning methods [11, 18, 25] and attribute encoding methods [6, 8, 10, 13, 18, 27, 29, 33, 38]. Primitive-level compression. Primitive pruning methods [11, 17, 25] aim to minimize the number of Gaussian primitives by reducing those with less contribution, using importance score-based pruning. Following the pruning scheme of 3DGS [17], several methods [24, 27, 28] adopt opacity-based pruning criteria often combined with additional strategies for more efficient pruning. Other approaches exploit carefully designed pruning criteria that account for rendering contribution [11, 13], Gaussian volume [10], learnable binary masks [6, 18, 33, 34, 38], and multiple components including gradient, per-pixel saliency, and attribute values [25]. In particular, Mini-Splatting [11] achieves high rendering quality with a minimal number of primitives by leveraging global contribution scores in combination with a deblurring strategy. Attribute-level compression. Attribute compression methods [6, 8, 10, 13, 18, 27, 29, 33, 38] focus on encoding Gaussian attributes in a compact format, which can be efficiently structured in various representations. Several approaches adopt quantization-based techniques that leverage global attribute statistics, including vector quantization [10, 18, 28, 29], latent quantization [13], and image codec [27]. Other methods utilize hash grids to capture the local similarity of attributes [18, 33]. Built upon Scaffold-GS [24], an anchor-based Gaussian representation, HAC [6] and ContextGS [38] suggest context modeling to minimize the entropy of anchor features. Despite impressive encoding capability, lacking generalizable priors results in limited compression performance for these per-scene encoding schemes. Recently, FCGS [8] introduced a feed-forward compression network, enabling optimization-free compression within a few seconds. Although the compression network constructs generalizable priors from a large-scale 3DGS dataset, the limited effectiveness of these priors makes it challenging to preserve rendering quality when compressing Gaussian attributes into highly compact storage, due to the unstructured design and high variability of Gaussian representations. Despite achieving significant storage reductions, most approaches remain limited by their reliance on per-scene information, without exploiting any generalizable priors. To overcome this limitation, we propose a novel 3DGS compression framework that leverages learned image priors to restore compression-induced degradation, thereby improving rate-distortion performance across diverse scenes. 2.2. Image Restoration Image restoration aims to restore a high-quality image from its degraded counterpart. Recently, deep learning methods have significantly improved the performance of image restoration tasks, which include sub-problems such as denoising [48, 49], deblurring [15], super-resolution [35, 50], and JPEG compression artifact reduction [16, 45]. Various neural network architectures, especially U-Net [31] variants, are designed to capture specific degradation artifacts and learn generalizable image priors, achieving remarkable performance in image restoration tasks. With the introduction of diverse convolutional neural network (CNN)- based [5, 9, 45, 48] and transformer-based [4, 20, 47] architectures, SOTA performance has continued to be advanced. To exploit the strong synthesis capabilities of modern generative models (e.g., GANs or diffusion models), several approaches [21, 35-37, 50] incorporate generative priors learned in those models, enabling high-quality image restoration despite severe degradation. Recently, several approaches have adopted generative priors from image restoration models for 3D reconstruction, particularly in tasks such as super-resolution [12] and few-shot reconstruction [23, 41]. While these methods demonstrate effectiveness in severely degraded reconstruction settings, they might suffer from reduced 3D consistency. In such cases, the generative priors may generate new patterns for each viewpoint, making them less suitable for scenarios where the scene is not severely degraded and geometric consistency must be preserved. In this manner, we exploit an image restoration model to benefit from the highly expressive learned priors for the 3DGS compression. By employing the restoration network, we successfully recover the degraded images rendered from compressed Gaussians and leverage these restored images to enhance the overall quality of compressed Gaussians. 3. Preliminaries: 3D Gaussian Splatting Our framework is built upon 3DGS [17], which models volumetric scenes using a set of point-based anisotropic Gaussian primitives G = (p, s, r, o, k). Each Gaussian Gi is defined by a set of attributes consisting of a position pi ∈R3, a scale vector si ∈R3, a quaternion-based rotation vector ri ∈R4, an opacity oi ∈[0, 1], and spherical harmonics (SH) coefficients ki ∈R3×(L+1)2, where L is the maximum SH degree. The covariance matrix Σi ∈R3×3 is defined by a rotation matrix derived from ri, and a scaling matrix derived from si. For rasterization, 3D Gaussians are projected into 2D space for a given view. The RGB pixel value ˆC(·) at viewspace coordinate x ∈R2 is acquired by blending N depthsorted Gaussians with a view-dependent Gaussian color ci ∈R3 derived from SH coefficients ki as: ˆC(x) = N X i=1 ciαi i-1 Y j=1 (1 -αj), (1) αi = oi · exp (-1 2(x -p′ i)⊤Σ ′-1 i (x -p′ i)), (2) where p′ i and Σ ′-1 i denote projected Gaussian position and covariance, respectively. Initialized from SfM points, the Gaussians G are optimized to synthesize photo-realistic rendered images for the given set of training viewpoints. 4. Method Fig. 2 illustrates the overall pipeline of our approach. Given a set of optimized Gaussians G, we obtain a set of compressed Gaussians ̃G by applying a compression method (Sec. 4.1). To mitigate the quality degradation, we construct effective side information by computing the coarse residual between the rendered images of the original and compressed Gaussians (Sec. 4.2). Given the degraded renderings and corresponding coarse residuals, we adopt an image restoration network that leverages learned image priors to estimate original rendering images comparable to those produced by the original Gaussians (Sec. 4.3). Finally, these restored rendering images are utilized to refine the compressed Gaussian attributes, resulting in improved rendering fidelity while maintaining compact storage usage (Sec. 4.4). 4.1. Initial Gaussian Compression The initial compression step builds compressed Gaussians ̃G that achieve significant storage savings while introducing rendering distortion as a trade-off. Given a set of optimized Gaussians G, we formulate the compression process as: ̃G = H(G), ̃G = ( ̃p, ̃s, ̃r, ̃o, ̃k), | ̃G| ≤|G|, (3) where H(·) denotes a lossy Gaussian compression operation that produces compressed attributes by reducing the 3D Gaussians Initial Gaussian Compression Compressed Gaussians Rasterize Refined Gaussians Gaussian Refinement Side information Image Restoration ! "! # #! Figure 2. Illustration of overall pipeline. Given a set of Gaussians as input, we perform initial compression, followed by an image restoration network that restores the rendered images. Finally, we refine the compressed Gaussians using the restored images to enhance the rendering quality. number of Gaussians or removing attribute redundancy, at the cost of information loss. Specifically, we adopt FCGS [8], a feed-forward 3DGS compression method that achieves remarkable compression performance with minimal computational overhead. Notably, other Gaussian compression methods satisfying Eq. (3) are also compatible with our pipeline as the initial compression module. The lossy Gaussian compression process inevitably produces information loss, which manifests as noticeable compression artifacts in the rendered outputs, as demonstrated in Fig. 1. Mitigating this degradation while preserving storage efficiency remains the core challenge of our framework. We aim to recover high-quality Gaussians from their compressed counterparts. However, directly modeling the compression-induced distortion is challenging due to the lack of effective priors that can be learned through a specialized network architecture capable of predicting restored Gaussians. To address this, we reformulate the problem as image-space restoration, where compression artifacts are modeled in the rendered image domain. Specifically, we leverage the rendered images obtained from the original Gaussians G, denoted as I = {I1, ..., IK}, and the degraded images from the compressed Gaussians ̃G, denoted as ̃I = { ̃I1, ..., ̃IK}, to indirectly estimate the attribute residuals using an image restoration network. 4.2. Side Information To achieve more effective restoration for compressed Gaussian renderings, we incorporate side information to support the network to overcome the performance limitations. The degraded images exhibit both compression artifacts with aliasing patterns introduced by the 3DGS rendering process, which differ from typical compression degradations such as JPEG, as shown in Fig. 1. Specifically, unlike block-based or frequency-domain artifacts in JPEG, these artifacts often appear as view-dependent, high-frequency distortions that are spatially inconsistent across images. These artifacts result in less effective restoration performance when applying a conventional image restoration framework, indicating the need for additional side information. Coarse rendering residuals. To address this challenge, we aim to improve restoration performance by incorporating side information. Specifically, we exploit the residual between the original and degraded images as auxiliary guidance for restoration. However, fully leveraging these highresolution and high-precision residuals incurs substantial storage overhead. To mitigate this, we compress the residual information R = I - ̃I into coarse residuals ̃R using a compact image codec, JPEG-XL [1], defined as: ̃R = Q(R; λrate) = Q(I - ̃I; λrate), (4) where Q(·) denotes residual compression operation, and λrate is a hyperparameter that controls the rate-distortion trade-off. By adjusting λrate, our framework can flexibly support multiple quality levels, allowing users to balance storage efficiency and restoration quality according to their specific requirements. During the decoding process, we condition the restoration network on the compressed coarse residual ̃R as decoder side information, providing additional guidance to recover high-fidelity renderings from the degraded inputs. Furthermore, we uniformly sample a minimal set of essential training views for side information, leveraging the observation that dataset-provided training views often contain substantial overlap. This allows our framework to operate more efficiently by reducing unnecessary computational and memory overhead. 4.3. Image Restoration Fig. 3 illustrates the proposed restoration framework. Built upon the image restoration network, we opt to model the pixel-level residuals R = I - ̃I between original renderings I and their corresponding degraded renderings ̃I with the support of side information ̃R as: ˆI = ̃I + ˆR = ̃I + F( ̃I, ̃R), (5) where F(·) represents the image restoration network, and ˆI and ˆR denote the restored image and residuals, respectively. For implementation, we adopt NAFNet [5], an image restoration architecture that has demonstrated effectiveDegraded image Original image Residual JPEG-XL - C Restored image Restoration network C -subtraction concatenation Figure 3. Illustration of the proposed image restoration framework. To train the image restoration network for Gaussian compression, we construct a paired dataset consisting of degraded and original images rendered from Gaussians optimized on real-world 3D scenes. Given a degraded image and the corresponding original image, we first compute their residual. The residual is then quantized and concatenated with the degraded image. This concatenated input is fed into the image restoration network, which is designed to remove Gaussian compression artifacts and recover a high-quality image. ness in handling various degradation types while maintaining computational efficiency. To incorporate additional side information, we modify the input layer of the network to take a concatenation of the two images, a degraded image ̃I, and an auxiliary guidance ̃R. To train the image restoration network, we construct a 3DGS compression artifact dataset consisting of paired images rendered from both original and compressed 3DGS representations. Specifically, we first optimize 3DGS representations on diverse real-world 3D scenes from the DL3DV-10K dataset [22] and subsequently compress the optimized Gaussians using FCGS [8] to obtain compressed Gaussians. We then render images from both the original and compressed Gaussians at multiple training views to generate paired image samples for compression artifact removal. These paired images are finally leveraged to train the image restoration network with the following objectives: Lrestore = L1(R, ˆR) + λLPIPSLLPIPS(I, ˆI), (6) where L1 and LLPIPS denote L1 loss and LPIPS [51] loss, and λLPIPS controls the contribution of the perceptual term. 4.4. Gaussian Refinement While the restoration network improves the visual fidelity of degraded renderings, the final objective of our framework is to refine the compressed Gaussian attributes to restore their high rendering ability. We achieve this by re-optimizing the compressed Gaussians ̃G into refined Gaussians ˆG under the supervision of the restored images ˆI, effectively propagating the information obtained from the learned image priors. Since the refinement aims to adjust the compressed Gaussians rather than learning from scratch, it is designed to perform with only a few optimization iterations, minimizing computational overhead. The following objective guides the refinement process: Lrefine = (1 -λ)L1(ˆI, ˆI′) + λLSSIM(ˆI, ˆI′), (7) where ˆI′ denotes the rendering output from the refined Gaussians ˆG, and L1 and LSSIM denote the L1 loss and SSIM loss, respectively. λ is a balancing weight. During optimization, we update all Gaussian attributes, whose values are compressed in the initial compression step. This choice can be adjusted based on the specific compression targets of the initial compression method. Also, we preserve the number of Gaussians during the refinement stage. 5. Experiments 5.1. Experimental Setting Datasets. To build a 3DGS compression artifact dataset, we leverage 495 real-world scenes from the DL3DV-10K dataset [22]. As a result, we construct 18,555 image pairs containing visual artifacts introduced by Gaussian compression, which are used to train the image restoration network. For 3DGS evaluation, we benchmark on two representative novel-view synthesis datasets, Mip-NeRF 360 [2], and Deep Blending [14]. Specifically, we evaluate nine scenes from Mip-NeRF 360, and two from Deep Blending. Following the evaluation protocol of 3DGS [17], we report PSNR, SSIM, and LPIPS to assess rendering quality. In addition, we also evaluate the storage size of each method to compare the compression efficiency. Implementation details. For the image restoration network, we adopt NAFNet [5], a CNN-based image restoration architecture. We train it for 200K iterations on a single NVIDIA RTX 6000 Ada GPU (48GB). During training, we augment the auxiliary guidance using JPEG compression to simulate diverse quality levels for robust optimization. Following the optimization scheme of Mini-Splatting [11], we optimize the 3DGS representation for 30K iterations with a minimal number of Gaussians. After optimization, we compress the Gaussian attributes using FCGS [8] and render LightGaussian Mini-C C3DGS PNG FCGS Ours Ours LightGaussian Mini-C C3DGS PNG FCGS Ours Ours LightGaussian Mini-C C3DGS PNG FCGS Ours Ours Size (MB) Size (MB) Size (MB) PSNR SSIM LPIPS Mip-NeRF 360 Mip-NeRF 360 Mip-NeRF 360 Figure 4. Rate-distortion curves comparing our method with existing post-compression approaches. We optimize Gaussians following Mini-Splatting [11] and compress them using each post-compression method. Results are averaged over all scenes of Mip-NeRF 360 [2]. Table 1. Comparison with post-compression approaches. We evaluate the rendering quality and storage size (MB) on Mip-NeRF 360 [2] and Deep Blending [14]. Method Mip-NeRF 360 [2] Deep Blending [14] PSNR ↑ SSIM ↑ LPIPS ↓ Size ↓ PSNR ↑ SSIM ↑ LPIPS ↓ Size ↓ Mini-Splatting [11] 27.39 0.827 0.196 202.9 30.06 0.910 0.239 138.3 C3DGS [29] 27.03 0.813 0.220 13.87 29.80 0.903 0.257 12.10 LightGaussian [10] 27.00 0.816 0.213 39.13 29.56 0.900 0.253 26.48 Mini-C [11] 27.43 0.826 0.201 48.42 29.93 0.905 0.244 30.63 PNG [44] 27.23 0.822 0.205 13.65 29.87 0.906 0.245 8.39 FCGS-low [8] 27.09 0.817 0.211 11.78 29.53 0.897 0.252 7.21 FCGS-high [8] 27.32 0.824 0.203 14.08 29.74 0.900 0.247 8.48 Ours-low 27.24 0.821 0.205 11.53 29.96 0.906 0.248 7.21 Ours-high 27.33 0.824 0.202 13.63 30.01 0.907 0.247 8.14 them to acquire rendering image pairs. We further quantize residual images using a standard image codec, JPEGXL [1], and employ them as side information for degraded image restoration. To balance storage efficiency and restoration performance, we uniformly sample 40% of the training views for side information. With supervision from the restored reference images, Gaussian refinement is performed by optimizing the compressed Gaussians for 5K iterations, which takes less than three minutes per scene on a single NVIDIA RTX 6000 Ada GPU (48GB). Furthermore, we provide multiple bitrate variants by adjusting the rate balancing weight λrate, which controls the storage cost of the side information. Specifically, 'Ours-low' uses a smaller λrate to achieve lower bitrate, while 'Ours-high' adopts a larger λrate to allocate more storage for improved quality. 5.2. Comparison To validate the effectiveness of the proposed compression framework, we compare our approach with existing 3DGS compression methods, categorized into post-compression approaches and compact representation approaches. Post-compression approaches aim to reduce storage costs by compressing pre-optimized Gaussians, including C3DGS [29], LightGaussian [10], Mini-Splatting-C [11], PNG [44], and FCGS [8]. In particular, C3DGS and LightTable 2. Comparison with compact representation approaches. We evaluate rendering quality and storage size (MB) on MipNeRF 360 [2] and Deep Blending [14]. Method Mip-NeRF 360 [2] Deep Blending [14] PSNR ↑ SSIM ↑ LPIPS ↓ Size ↓ PSNR ↑ SSIM ↑ LPIPS ↓ Size ↓ 3DGS [17] 27.44 0.813 0.218 822.6 29.48 0.900 0.246 692.5 CompGS [28] 27.04 0.804 0.243 22.93 29.89 0.907 0.253 15.15 Compact-3DGS [18] 26.95 0.797 0.244 26.31 29.71 0.901 0.257 21.75 EAGLES [13] 27.10 0.807 0.234 59.49 29.72 0.906 0.249 54.45 SOG [27] 27.02 0.800 0.226 43.77 29.21 0.891 0.271 19.32 HAC [6] 27.49 0.807 0.236 16.95 29.99 0.902 0.268 4.51 LocoGS [33] 27.33 0.814 0.219 13.89 30.06 0.904 0.249 7.64 Ours-low 27.24 0.821 0.205 11.53 29.96 0.906 0.248 7.21 Ours-high 27.33 0.824 0.202 13.63 30.01 0.907 0.247 8.14 Gaussian require additional optimization steps for finetuning, whereas Mini-Splatting-C, PNG, and FCGS are optimization-free methods. For a fair comparison, we equalize the overall optimization budget by performing an additional 5K iterations, matching the refinement setting of our method, before compression for Mini-Splatting-C, PNG, and FCGS. This results in a total of 35K iterations for their initial optimization. In contrast, compact representation approaches directly learn efficient representations from scratch without relying on pre-optimized Gaussians, including SOTA compact Gaussian representations such as CompGS [28], Compact-3DGS [18], EAGLES [13], HAC [6], and LocoGS [33]. Post-compression approaches. Tab. 1 and Fig. 4 show the quantitative comparison with existing post-compression approaches. Built upon the same initial Gaussians, our compression method demonstrates a superior rate-distortion trade-off, consistently outperforming the rendering quality of baseline methods while achieving significant storage reduction up to 19.2× compared to the uncompressed representation, Mini-Splatting. Specifically, in the low bitrate setting, our method outperforms the rendering quality of FCGS-low with comparable or even smaller storage consumption. In the high bitrate setting, our method shows superior rendering performance, except for Mini-Splatting-C, Figure 5. Ablation study on view-sampling. Results are averaged over all scenes of Mip-NeRF 360 [2]. PSNR on Mip-NeRF 360 SSIM on Mip-NeRF 360 Size (MB) Size (MB) PSNR SSIM Figure 6. Ablation study on learned image prior. Results are averaged over all scenes of Mip-NeRF 360 [2]. while maintaining the minimal storage cost. Our method achieves comparable rendering quality to Mini-SplattingC, which requires up to 3.8× storage usage. Meanwhile, vector-quantization-based approaches, C3DGS and LightGaussian, suffer from lower rendering performance despite their relatively large storage sizes. These results highlight the remarkable performance of our method as a leading post-compression solution, achieving high-quality rendering and substantial storage savings. Compact representation approaches. Tab. 2 presents a quantitative comparison against recent compact Gaussian representations, which optimize Gaussians under compressed designs. Overall, our method achieves superior rendering quality against SOTA compact representations while maintaining the lowest storage overhead, with the exception of HAC on the Deep Blending. Notably, our approach demonstrates considerable perceptual improvements in terms of SSIM and LPIPS, contributing to more faithful and realistic rendering outputs. In contrast, quantizationbased methods such as CompGS, Compact-3DGS, EAGLES, and SOG suffer from noticeable degradation in rendering quality despite requiring larger storage sizes. Compared to SOTA compact baselines, HAC, and LocoGS, our method achieves better rate-distortion trade-offs, delivering high-fidelity rendering while preserving minimal storage costs. Specifically, on the Mip-NeRF 360, our method achieves the best performance across all rendering metrics while requiring the smallest storage footprint. PNG FCGS FCGS PNG PNG + Ours FCGS + Ours FCGS + Ours PNG + Ours Size (MB) Size (MB) PSNR SSIM PSNR on Mip-NeRF 360 SSIM on Mip-NeRF 360 Figure 7. Ablation study on initial compression method. Results are averaged over all scenes of Mip-NeRF 360 [2]. 5.3. Ablation Study Learned image prior. We investigate the effectiveness of learned image priors for rendering image restoration. To this end, we evaluate the rendering quality of refined Gaussians with and without the use of the trained image restoration model. Fig. 6 shows the rendering quality of refined Gaussians according to the utilization of learned image priors to the image restoration process. Note that 'w/o image prior' refers to the setting where image restoration is performed by simply adding quantized residuals to the degraded image. In contrast, 'Ours' leverages learned image priors embedded in the trained image restoration network to refine the degraded image, as described in Sec. 4.3. As a result, our method achieves noticeable improvements in rendering quality for all evaluation metrics with the support of learned image priors. Overall, these results clearly demonstrate that leveraging learned image priors is crucial for recovering the visual fidelity of compressed Gaussian renderings. View-sampling. We also validate the effectiveness of view sampling in balancing storage efficiency and restoration quality. Fig. 5 presents the Gaussian refinement performance with respect to the sampling ratio of training views used as side information. Sampling a smaller set of essential views reduces storage consumption by minimizing redundancy while maintaining comparable rendering quality with only minor degradation. Based on this observation, we adopt a 40% sampling ratio to achieve a favorable balance between storage cost and rendering quality. Table 3. Ablation study on side information. 'Image prior' and 'Side info.' denote the learned image prior and side information, respectively. Results are averaged over all scenes of Mip-NeRF 360 [2]. Image prior Side info. PSNR ↑ SSIM ↑ LPIPS ↓ Size ↓ ✗ ✗ 26.98 0.816 0.212 10.89 ✗ ✓ 27.16 0.819 0.208 11.53 ✓ ✗ 27.00 0.817 0.211 10.89 ✓ ✓ 27.24 0.821 0.205 11.53 Side information. Moreover, we investigate the effectiveness of side information in enhancing restoration performance. As shown in Fig. 1, Gaussian compression artifacts are non-trivial to remove using the restoration network alone, even with a sufficiently large network scale and extensive training dataset. Tab. 3 reports the performance of our framework under the difference configuration of the learned image prior and side information. We observe that even a minimal increase in storage size can lead to an evident improvement in rendering quality when accompanied by side information. Also, incorporating side information enables more effective utilization of the learned image prior. Without side information, the learned prior yields only marginal benefits in rendering quality. In contrast, when side information is leveraged, the learned prior provides significantly greater benefits, resulting in enhanced rendering quality. Initial compression method. Furthermore, we demonstrate the compatibility of our method with various Gaussian compression approaches. To this end, we incorporate PNG [44], a representative post-compression method, into our compression scheme, as shown in Fig. 7. Despite the suboptimal rendering quality of the initial compressed results, our method consistently delivers significant improvements, particularly achieving noticeable gains in PSNR, regardless of the underlying compression method. These results indicate that our framework can generalize well and support a wide range of future compression methods. 6. Conclusion In this paper, we introduced a novel 3D Gaussian compression framework that leverages an image restoration network to recover the visual quality of compressed Gaussians. Our approach tackles compression-induced artifacts by reformulating Gaussian restoration as an image-space problem, enabling the use of powerful restoration networks trained on large-scale data. Experimental results demonstrate that our method effectively captures scene-specific degradation patterns and removes compression artifacts, achieving enhanced rendering fidelity and state-of-the-art rate-distortion performance compared to existing 3DGS compression methods, including both post-compression approaches and compact representation approaches. Limitations. Despite the impressive restoration performance and broad applicability of our method, it introduces additional computational overhead due to the restoration process. While the additional process of our framework is more efficient compared to existing post-compression methods, it still requires non-negligible computational costs compared to optimization-free approaches. Moreover, our current design relies on separately quantizing the side information using an external image codec before feeding it into the restoration network. A promising future direction is to improve both computational efficiency and restoration quality by jointly optimizing residual quantization and image restoration within a unified, end-to-end trainable framework. Acknowledgments This work was supported by the National Research Foundation of Korea (NRF) grant funded by Korea government (Ministry of Education) (No.2022R1A6A1A03052954, Basic Science Research Program), and - formation & communications Technology Planning & Evaluation (IITP) grant funded by Korea government (MSIT) (No.RS-2019-II191906, AI Graduate School Program (POSTECH); No.RS-2021-II211343, AI Graduate School Program (Seoul National University)). References [1] Jyrki Alakuijala, Ruud Van Asseldonk, Sami Boukortt, Martin Bruse, Iulia-Maria Coms,a, Moritz Firsching, Thomas Fischbacher, Evgenii Kliuchnikov, Sebastian Gomez, Robert Obryk, et al. Jpeg xl next-generation image compression architecture and coding tools. 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2510.14701
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Identification of formation of amorphous Si phase in SiOxNy films produced by plasma
enhanced chemical vapor deposition
M. V. Voitovycha, A. Sarikova, b, c, *, V. O. Yukhymchuka, V. V. Voitovychd, M. O. Semenenkoa, c, d
a V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, 41
Nauky Avenue, 03028 Kyiv, Ukraine
b Educational Scientific Institute of High Technologies, Taras Shevchenko National University of
Kyiv, 4-g Hlushkova Avenue, 03022 Kyiv, Ukraine
c National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 37
Beresteiskyi Avenue, 03056 Kyiv, Ukraine
d Institute of Physics, National Academy of Sciences of Ukraine, 46 Nauky Avenue, 03028 Kyiv,
Ukraine
*Corresponding author’s email: sarikov@isp.kiev.ua
Abstract
Peculiarities of formation of inclusions of amorphous Si (a-Si) phase in Si-rich Si oxynitride
films grown by plasma-enhanced chemical vapor deposition (PECVD) are studied by combined
Raman scattering and infrared (IR) absorption spectroscopy. The Raman scattering results identify
presence of a-Si phase in the studied films at the relative Si content exceeding a threshold value of
about 0.4. The a-Si amount correlates with the concentration of hydrogen in the films, the presence
of which is detected by characteristic IR absorption bands corresponding to Si–H bending (~660
cm–1) and stretching (a composite band in the range of ~1900-2400 cm–1) vibrations. The method of
deconvolution of IR absorbance spectra in the range of ~600 to 1300 cm–1 developed earlier is used
to reliably separate the IR band at ~660 cm–1. This band is identified to origin from the amorphous
Si phase within the studied Si oxynitride films. This makes it possible to propose IR spectroscopy
with analysis of the low-wavenumber part of the spectra as an efficient method of identifying phase
composition of Si-rich Si oxynitride films. The obtained results contribute to understanding of the
regularities of formation of phase compositions of PECVD grown Si oxynitride films and are useful
for controlling the films properties for practical applications.
Keywords: Si oxynitride films, nanocomposite, hydrogen, FTIR spectroscopy, Raman scattering,
plasma-enhanced chemical vapor deposition
1. Introduction
Si oxynitride films (SiOxNy, 0 ≤ x ≤ 2, 0 ≤ y ≤ 4/3) have a great significance for fabricating
modern microelectronic and optoelectronic devices [1, 2]. As the structure of such films defines
their properties, their study attracted great attention in recent decades [3, 4]. Naturally, the main
focus was to study the behavior of atoms of the main film forming elements, namely silicon, oxygen
and nitrogen. It may be considered well-established today that structural arrangement of Si, O and
N atoms in the SiOxNy films depends on the film composition. In the films with less than ~50% of
Si, the chemical composition is dominated by Si suboxide bonding (set of silicon-oxygen tetrahedra
obeying the random bonding statistics) with N present as a significant impurity [5, 6]. Considering
the N atoms entering into the network to substitute O atoms maintaining the same structure as that
of a stoichiometric SiO2 material, the proper chemical formula for such oxynitride materials should
be SiO2–yNy [7].
Increase of the relative Si concentration principally changes the SiOxNy film morphology
and structure. In [8], Si-rich films were considered as composites, the microstructure of which may
be represented as a mixture of silicon-oxygen tetrahedra (Si–OaSi4-a, a = 0…4) and silicon-nitrogen
pyramids (Si3N) in the framework of the mixture model. Along with formation of such phase
composition, existence of variable-size amorphous Si (a-Si) clusters embedded in a dielectric matrix
in as-deposited Si-rich SiOxNy films was inferred based on photoluminescence (PL) and Raman
spectroscopy studies [9, 10]. In particular, these clusters were responsible for room-temperature PL
in the range of 1.5-2.0 eV. Manifestation of two luminescence peaks [4] was attributed to two
different average cluster sizes. The PL origin from the amorphous Si inclusions in this case was
supported by the presence of the features of elemental Si in the X-ray photoelectron spectra of the
samples with x < 0.96 and by a lack of crystalline features in the glancing angle X-ray diffraction
patterns.
Infrared (IR) spectroscopy with mathematical deconvolution of the IR absorption band is
successfully used for study of the microstructure of Si oxide films with excess Si content [11, 12].
In our previous publication [13], the characteristics of the elementary bands corresponding to Si–N
and Si–H bond vibrations in the spectral range of ~600 to 1300 cm–1 have been determined that
enable reliable deconvolution of the IR absorbance spectra with separation of the Si–O, Si–N, and
Si–H components. It is known that hydrogen in Si oxynitride films is contained in high (of the order
of 1022 cm–3 [7, 14]) concentrations and is responsible for quite noticeable absorption of IR
radiation in well-known spectral regions. On the other hand, the Si–H bond vibration frequency is
very sensitive to the local structural arrangement, which is evidenced by the results of the studies of
such materials as Si oxides and amorphous Si [15-17]. This opens a way to investigate IR
absorption spectroscopy as a method for identifying not only Si oxynitride, oxide and nitride phases
but also amorphous Si phase in Si oxynitride films. In this work, we use combined Raman
scattering and Fourier transform IR (FTIR) spectroscopy with analysis of various Si–H vibration
bands to investigate the peculiarities of the formation of a-Si phase in the Si-rich Si oxynitride films
obtained by plasma-enhanced chemical vapor deposition (PECVD).
2. Experimental
In our experiments, 300 ± 5 nm thick SiOxNy films with different compositions
(stoichiometry indices x and y) obtained by PECVD were used. The values of the N2O/SiH4 flow
ratio during the film deposition and the respective film stoichiometries are presented in Table 1.
Double-side polished boron-doped CZ Si wafers with the thickness of 300 µm and 1 mm thick
sapphire plates were used as substrates for infrared (IR) and Raman spectroscopy investigations,
respectively. More details about the film deposition procedure and determination of the film
composition can be found in our previous publication [13].
Table 1. Deposition conditions and composition of SiOxNy films
Sample
N2O/SiH4
flow
ratio
Composition
x
y
Relative Si
сontent
#1
9
1.95
0.01
0.34
#2
3
1.3
0.28
0.39
#3
1.5
1.09
0.32
0.41
#4
0.6
0.85
0.27
0.48
#5
0.4
0.59
0.16
0.57
#6
0.3
0.42
0.14
0.64
#7
0.2
0.37
0.11
0.68
#8
0.1
0.23
0.07
0.77
#9
0.06
0.18
0.05
0.80
Raman spectra of the studied films were excited by the emission of a CNI Model PSU-H-
FDA solid-state laser with λ = 457 nm at room temperature and recorded using an MDR-23
spectrometer equipped with an Andor iDus 401A CCD detector (UK). In order to prevent the film
structure from being affected during the measurements, the excitation radiation power density was
fairly small, not exceeding 103 W/cm2. For correct comparison of the bands in different Raman
spectra, the latter were normalized to the intensity of the band corresponding to amorphous Si (~480
cm–1).
FTIR transmittance spectra in the range of 400−4000 cm–1 were measured at room
temperature using a PerkinElmer BX-II spectrometer. The measurement resolution was 2 cm−1, the
number of scans was 100, and the measurement accuracy was ~0.5%. A Si substrate was used as a
reference sample for all the measurements.
IR absorbance spectra were recalculated from the transmittance spectra using the Beer–
Bouguer–Lambert law. In this work, the bands related to Si–H stretching (a wide composite band at
around 2200 cm–1) and bending (peaked near 660 cm–1) vibrations were analyzed. To separate the
contributions from different Si–H containing complexes, the IR absorption bands corresponding to
the Si–H stretching vibrations were mathematically deconvoluted into elementary Gaussian profiles
with the parameters (the maximum position and full width at half-maximum) determined earlier
[14, 15]. The details of this procedure were described in [14].
3. Results and discussion
3.1. Raman scattering investigation of SiOxNy films
Phase structure of the SiOxNy films with different compositions (see Table 1) was analyzed
by Raman spectroscopy. Figure 1(a) shows a number of representative Raman spectra of these
films. As can be seen from this figure, the spectra contain two asymmetric bands in the spectral
range of 300-800 cm–1 with the maxima at ~476 cm–1 and ~660 cm–1. The first band is associated
with Si–Si and Si–O bond vibrations, while the second band corresponds to vibrations of Si–H
bonds in an amorphous phase [18-20]. The shape and the frequency position of these bands
evidence amorphous structure of our films and presence of silicon-hydrogen bonds, which is typical
of the films obtained by PECVD using SiH4 gas [9, 14, 18, 21].
As can be further seen from Figure 1(a), the wide asymmetric band in the frequency range ~
300-550 cm–1 is the most intense one in the measured Raman spectra. The maximum position of
this band is always fixed at ~476 cm–1, while its half-width Γ decreases with the increase in the
relative Si content in the films. We analyzed the shape of this band by decomposing it into Gaussian
components (see inset in Figure 1(a)) according to the procedure described in [18, 19, 22, 23]. As a
result, two Gaussian components with the parameters ω1 = 476 cm–1 (maximum position) and Γ1 =
66 cm–1 (half-width), and ω2 = 420 cm–1 and Γ2 = 124 cm–1 were identified.
As is well known, in Raman spectroscopy, spectral position and shape of a band (width,
symmetry, and frequency at maximum) are characteristic parameters that provide important
information about the structural state of a studied material. For Si oxide based structures containing
amorphous and crystalline clusters, Raman peaks at ~480 cm–1 and with a half-width of ~60-80 cm–
1 corresponding to amorphous silicon TO mode, together with very narrow peaks with the maximum
at ~520 cm–1 and a half-width of ~5 cm–1 corresponding to crystalline Si were observed [20, 22].
Amorphous Si dioxide (a-SiO2) exhibited a broad peak at ~495 cm–1 corresponding to the
vibrational mode of four-member SiO2 rings [24]. A shift of the spectrum peak from that typical of
SiO2 to the one typical of a-Si, its broadening and appearance of a one-side asymmetry were
observed for SiOx films with excess Si content (x < 2) at decreasing x, indicating heterogeneity of
the film structure and change in its phase composition [20, 22]. Raman spectrum of amorphous Si is
characterized by broad, unstructured bands in the range of 100-550 cm–1 associated with transverse
acoustic (TA, ~145 cm–1), transverse optical (TO, ~475 cm–1), longitudinal optical (LO, ~365 cm–1)
and longitudinal acoustic (LA, ~305 cm–1) phonon vibrations of Si–Si bonds [19, 22, 25].
Therefore, the elementary band at ω1 ≈ 476 cm–1 observed in our structures (see inset in Figure 1(a))
may be attributed to the contribution of the TO mode of Si–Si bond vibrations in amorphous silicon.
The broader band at ω2 ≈ 420 cm–1 may be caused by Raman scattering of light due to Si–O bond
vibrations in the amorphous Si oxide matrix [20, 26].
Figure 1. (a) – Raman spectra of SiOxNy, films obtained at different N2O/SiH4 gas ratios. Inset –
Raman spectrum of the #6 sample deconvoluted into Gaussian components. (b) – Integrated
intensity of the ω(TO) band of the Raman spectra versus Si content in the SiOxNy films.
Figure 1(b) shows the dependence of the relative contribution of the integrated intensity of
the TO band to the analyzed Raman spectrum in the range of ~300 to 550 cm–1, on the relative Si
content in the studied SiOxNy films. As can be seen from this figure, there is a direct correlation
between this intensity and the amount of Si in the films. It should be concluded therefore that an
increase in the relative Si content in non-stoichiometric Si oxynitride films gives rise to an increase
in the amount of amorphous Si in them.
3.2. IR spectroscopy study of SiOxNy film structure
IR transmittance spectra in the frequency range of 400-4000 cm–1 of the SiOxNy films with
different compositions are presented in Figure 2. It can be seen from this figure that the spectra
contain transmission bands peaked at ~452, 660, 870, and 1060 cm–1 in the low-frequency region
and at ~2100 and 3600 cm–1 in the high-frequency region. The bands in the range between 900 and
1300 cm–1 as well as at ~452 cm–1 correspond to Si–O–Si valence and rocking vibrations,
respectively [1]. The band peaked at ~870 cm–1 is associated with Si–N bonds and the band at ~660
cm–1 corresponds to bending vibrations of Si–H bonds [2]. It should be noted at this that the bands
in the range of ~600 to 1300 cm–1 corresponding to Si–O, Si–N and Si–H bond vibrations can
overlap, especially for the films with relatively high Si contents, which greatly complicates analysis
of the IR spectra and, hence, the SiOxNy film structure. The high-frequency bands with the maxima
at ~2100 and 3600 cm–1 in the measured IR transmittance spectra are caused by valence vibrations
of Si–H bonds and Si–OH groups, respectively [1].
Figure 2. FTIR transmittance spectra of SiOxNy films with different compositions.
In this work, we focus on a study of IR absorption bands associated with Si–H bond
vibrations. Figure 3 shows the high-frequency Si–H related band for a number of the studied
SiOxNy films. It can be seen from this figure that the shape of this band, its maximum position,
intensity and area strongly depend on the SiOxNy film composition. Namely, increase of the relative
Si content in the films leads to the shift of the band peak to lower frequencies, and this shift can be
quite large. It should be noted as well that no high-frequency hydrogen-related band is detected in
the IR absorbance spectra for the sample with the highest oxygen content and, hence, the lowest Si
content (sample #1, see Table 1).
Figure 3. IR absorbance spectra of the SiOxNy films in the range of Si–H stretching vibrations. The
arrow shows the direction of the shift of the maximum position of the integrated band with the
increase in the relative Si content in the films.
By integrating the area under the IR spectra in the range of ~1900-2400 cm–1, the amount of
hydrogen bound in the studied films can be calculated [14, 15]. The calculated hydrogen
concentration as a function of the relative Si content in the films is presented in Figure 4. It can be
seen from this figure that the hydrogen concentration is of the order of ~1021-1022 cm–3, which
coincides by the order of magnitude with the known literature data (see e.g. [7]). This concentration
increases with the relative amount of Si in the films, which looks quite natural, since growth of the
latter is associated with an increase in the SiH4 concentration in the gas mixture during the film
deposition. For the samples with the highest Si content, a tendency to saturation is noticeable.
Composition of the IR spectral band in the range of ~1900-2400 cm–1 allows one to draw
fundamental conclusions about the structural arrangement of hydrogen atoms in the Si oxynitride
network. It is well-known that the considered wide band is a superposition of elementary bands
corresponding to stretching modes of H–Si(Si3–nOn) structural units, where n = 0…3 [14, 15].
Hence, we deconvoluted this band into elementary Gaussian components for different Si oxynitride
film stoichiometries taking into account both the nature of the elementary components [15] as well
as our previous experience gained during the study of the kinetics of hydrogen effusion from Si-rich
Si oxynitride films induced by heat treatments [14]. The deconvolution results for the films depicted
in Figure 3 are presented in Figure 5. As can be seen from this figure, three elementary components,
namely H1, H2 and H3 with the characteristics presented in Table 2, were separated. As can be
further seen from Figure 5, the spectrum of the sample #2 having low Si content (see Table 1) is
described by the only component Н1. Increase in the relative Si content leads to redistribution of the
main contribution from H1 to H2 (for the sample #3) and H3 (for the samples #7 and #9) with a
tendency of gradual disappearance of the previous bands. Figure 6 provides a more detailed
information about the dependence of the relative contributions of the bands H1, H2 and H3 to the
FTIR spectra of the SiOxNy films on the relative Si content in the films for all the samples presented
in Table 1.
Figure 4. Hydrogen concentration versus relative Si content in the SiOxNy films.
It is known [4, 15] that the quantity and type of the neighboring atoms to a hydrogen atom
correlate with the Si–H bond length, and, hence, influence on the Si–H stretching vibrations
frequency. For each of the separated elementary bands H1 to H3, the Si–H bond length was
determined using the following expression [4]:
νSi–H (dSi–H)3 = 7074 cm2
(1)
where νSi–H is the band peak position and dSi–H is the Si–H bond length, respectively. The obtained
bond lengths together with the respective H–Si(Si3–nOn) complexes identified according to [15] are
presented in Table 2. It should be noted that the Si–H bond length for the bands H1 (0.146 nm) and
H3 (0.151 nm) determined by the expression (1) practically do not differ from the respective values
for Si dioxide (0.144 nm) and amorphous Si (0.152 nm) as indicated in [4]. We may conclude
therefore that the results of the analysis of the high-frequency parts of the IR spectra agree well with
existence of amorphous Si phase within the Si oxynitride structure at the relative Si content above
about 0.4 (samples #3 to #9, see Figure 6).
Figure 5. Deconvolution of the IR absorption band corresponding to Si–H stretching vibrations on
elementary Gaussian components corresponding to contributions from H–Si(O3) (H1), H–Si(Si2O)
(H2) and H–Si(Si3) (H3) complexes.
We have mentioned above that another Si–H related band in the measured IR absorbance
spectra corresponds to the peak position of about 660 cm-1. This band is related to Si–H bending
vibrations [9, 14-16]. Analysis of the integrated intensity of this band would provide additional
Table 2. Characteristics of the Gaussian components of the Si–H related IR absorption band
Band
Maximum position, cm–1
Width, cm–1
Oscillation type
Complex
dSi–H, nm
Н1
2252±2
75±3
stretching
H–Si(O3)
0.146
Н2
2150±4
100±2
stretching
H–Si(Si2O)
0.149
Н3
2050±4
85±4
stretching
H–Si(Si3)
0.151
Figure 6. Integrated intensity of IR absorbance bands corresponding to Si–H vibrations versus Si
content in the SiOxNy films. The maximum position of the IR bands: 1 – ~660 cm–1, 2 – ~2050 cm–
1, 3 – ~2150 cm–1, and 4 – ~2250 cm–1.
information about structural arrangement of hydrogen in the Si oxynitride films as well as highlight
the Si oxynitride phase composition. However, analysis of this band is generally complicated as the
band may overlap with the bands corresponding to Si–O and Si–N bond vibrations also present in
the range of ~600 to 1300 cm–1. In our previous publication [13], we determined the parameters of
the elementary Gaussian components of the IR absorbance spectra of Si-rich Si oxynitride films in
the mentioned frequency range thus enabling separation of the Si–O, Si–N and Si–H related bands.
Figure 7(a) shows the IR absorbance spectra of a number of SiOxNy samples in the range of 500 to
800 cm–1 with separated absorption band near ~660 cm–1 corresponding to Si–H bond vibrations.
This band is described by a single Gaussian with the mean half-width of ~82 ± 2 cm–1. It can be
seen from Figure 7(a) that the intensity and maximum position of this band depend on the N2O/SiH4
flow ratio during Si oxynitride film growth and, hence, the relative Si content in the films. The
dependence of the integrated intensity of this band on the relative Si content in the studied films is
also presented in Figure 6. As can be seen from Figures 6 and 7(a), the mentioned band is absent in
the IR spectra of the films with the smallest Si contents (samples #1 and #2) and increases its
integrated intensity with subsequent growth of the relative Si amount (samples #3 to #9). The
position of the band maximum shifts at this from ~665 cm–1 for the sample #3 (relative Si content of
0.41) to ~635 cm–1 for the sample #9 (relative Si content of 0.8), as shown by the data provided in
Figure 7(b). These findings agree well with the results of [4, 18], where the low-frequency IR band
corresponding to Si–H bending vibrations in PECVD grown Si oxynitride films had no definite
peak position. We conclude therefore that there is a clear correlation between the position of the
peak and the integrated intensity of the IR absorption band at ~660 cm–1 corresponding to Si–H
bond bending vibrations on the one hand, and the Si content in the SiOxNy samples on the other
hand.
We have established above that the IR absorption band at ~2050 cm–1 corresponds to the
Si–H bond stretching vibrations in amorphous Si phase. It can be seen from Figure 6 that the
integrated intensity of this band has a direct correlation with that of the band at ~660 cm–1.
Moreover, as can be seen from Figure 8, the integrated intensity of the latter band is directly
proportional to the integrated intensity of the TO mode of Si–Si bond vibrations in the measured
Raman spectra. Therefore, the IR absorption band peaked at ~640-660 cm–1 may be confidently
Figure 7. (a) – FTIR absorbance spectra of the SiOxNy films corresponding to Si–H bending
vibrations and (b) – dependence of the maximum position of the band at ~660 cm–1 on the relative
Si content in the SiOxNy films.
Figure 8. Integrated intensity of the IR absorption band at ~660 cm–1 versus integrated intensity of
the ω(TO) band of the Raman spectra of the SiOxNy films.
attributed to the presence of amorphous Si phase in the non-stoichiometric Si oxynitride films. This
conclusion is supported by the published results by other researchers. It is reported in particular that
the IR absorption band with the peak position around 640 cm–1 is typical for a-Si layers [27] and is
considered to origin from Si–H bond vibrations in H–Si(Si3) complexes [16, 17]. In Si oxynitride
films, this band was reliably detected only in the case of high relative Si contents. It was not
detected in the spectra of the samples deposited by PECVD technique, which had a rather high
oxygen concentration and a composition much closer to SiO2 than to amorphous Si [15]. On the
other hand, a band peaked at ~640 cm–1 corresponding to Si–H bending vibrations was present in
the IR spectra of as-deposited Si-rich Si oxide films [13]. Moreover, an absorption band with the
maximum near 670 cm–1 clearly manifested itself in the IR spectra of the SiOxNy samples with high
Si fraction [4]. In both latter cases, presence of a-Si inclusions in the studied films was established
independently.
3.3. Comparative analysis of IR and Raman scattering results
Comparative analysis of the evolution of the Si–H related IR bands with the SiOxNy
composition along with the respective Raman scattering results allows one to get a deeper insight
into the peculiarities of the formation of phase composition of Si oxynitride films grown by plasma-
enhanced chemical vapor deposition.
As demonstrated by the data from [4, 9, 15, 21] as well as our Raman and IR spectroscopy
investigations, formation of amorphous Si clusters in an oxynitride matrix is possible in Si
oxynitride films deposited by PECVD at different N2O/SiH4 gas flow ratios. In particular, presence
of a-Si phase is detected by Raman measurements (observation of the TO mode of Si–Si bond
vibrations at ω1 ~ 476 cm–1, see Figure 1(a)). At this, the amount of amorphous Si grows with the
increase in the relative Si content in the films.
Using the approach to deconvolution of IR absorbance spectra of Si oxynitride films
presented in [13], a low-frequency IR band corresponding to Si–H bending vibrations (~640-660
cm–1) is reliably separated. As was already mentioned above, the integrated intensity of this band
correlates both with the intensity of the IR absorption band at ~2050 cm–1 identified to origin from
the Si–H stretching vibrations in amorphous Si (see Figure 6), as well as with the integrated
intensity of the TO mode in the Raman scattering spectra (see Figure 8), which made it possible to
reliably attribute this band to the presence of a-Si phase. As can be further seen from Figure 6, no a-
Si related signal is detected in the IR spectra of the samples with small relative Si concentrations
(below about 0.4). This points to a certain threshold value of excess Si content, at which amorphous
Si phase begins to form. This is further demonstrated by Figure 9, where the dependence of the
integrated intensity of the IR absorption band at ~660 cm–1 on the excess Si content in the
investigated Si oxynitride films calculated as 1 – x/2 – 3y/4 is shown. As can be seen from this
figure, the threshold excess Si value is about 0.2, and the amount of the amorphous Si in the Si
oxynitride films, which is proportional to the integrated intensity of the IR band at ~660 cm–1,
linearly grows with the excess Si concentration above this value.
Figure 9. Integrated intensity of the IR absorption band at ~660 cm–1 versus excess relative Si
content in the SiOxNy films.
The properties of the IR absorption band at ~660 cm–1 allow one to consider IR
spectroscopy as an efficient method for tracking Si agglomeration into amorphous Si phase in an
oxynitride matrix and formation of a particular phase composition of PECVD grown films.
According to this method, IR transmittance spectra of SiOxNy films in the low-frequency region
should be only measured. Using the deconvolution procedure and the characteristics of the
elementary IR bands corresponding to Si–O, Si–N and Si–H (~660 cm–1) bond vibrations [13],
phase composition of the studied films, including the presence or absence of amorphous Si
inclusions in the films, may be inferred. We believe that this method is one of the simplest and most
non-destructive ones for monitoring Si agglomeration in SiOxNy films produced by PECVD
technology and determining their phase composition and structure.
The results obtained in this study extend our understanding of the peculiarities of formation
of phase compositions of PECVD grown Si-rich Si oxynitride films. Recently, a thermodynamic
theory describing phase composition of SiOxNy films with different stoichiometries grown at
different temperatures was proposed [28], which, however, considered only a random bonding
model distribution of Si, O and N atoms. Identification of the possibility of presence of amorphous
Si clusters in such films should be used for updating the theoretical model, in particular, to improve
its potential for tailoring the phase composition properties of Si oxynitride films for practical
applications.
4. Соnclusions
In this work, peculiarities of formation of amorphous Si phase in Si oxynitride films grown
by plasma-enhanced chemical vapor deposition are studied by combined Raman scattering and IR
spectroscopy investigations. The results of the Raman spectroscopy demonstrate that at a certain
ratio of precursor gases in the growth chamber during SiOxNy film deposition, excess Si content
gives rise to formation of amorphous Si phase within the films. This is evidenced by the appearance
of the peak at ~476 cm–1 corresponding to the transverse optical phonon mode characteristic of a-Si.
Large half-width of this band (Γ = 66 cm–1) indicates the absence of long-range order in the film
structure, which is typical for amorphous materials. The results of our Raman spectroscopy
investigations are in good agreement with the XPS, XRD, and PL results reported in [4, 6, 9, 21] for
similar Si oxynitride films prepared by PECVD technology.
Using the method of deconvolution of IR absorbance spectra in the wavelength range of
~600 to 1300 cm–1 into elementary components, proposed in [13], the elementary band at ~660 cm–1
corresponding to Si–H bending vibrations was separated. The comparative analysis of the IR and
Raman spectroscopy results enabled identification of this band as related to the presence of
amorphous Si inclusions in the studied films. It was demonstrated that formation of amorphous Si
clusters in Si oxynitride films takes place at the relative excess Si concentrations exceeding a
threshold value of about 0.2. After the threshold, the amount of the a-Si phase linearly grows with
the excess Si concentration. This result makes it possible to consider IR spectroscopy with analysis
of the low-wavelength (~600 to 1300 cm–1) part of spectra as an efficient method of identifying
phase composition of Si-rich Si oxynitride films.
The results of this study enhance our understanding of the peculiarities of the formation of
phase compositions of PECVD grown Si oxynitride films and may be used for developing methods
of creating the films with required phase compositions for practical applications.
Acknowledgement
This work has been supported by the budget project no. III-4-21 “Electrical, optical and
photoelectrical characteristics of the systems with nanostructured surfaces and nanocrystals;
physical mechanisms of emission conversion in modern optoelectronic structures for developing
novel optoelectronic, sensoric, energy saving illumination, and information recording and storage
devices” of the National Academy of Sciences of Ukraine.
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2025 IEEE 15th International Conference “Nanomaterials: Applications & Properties” (IEEE
NAP-2025) Bratislava, Slovakia, Sep. 7-12, 2025 (in press).
|
Identification of formation of amorphous Si phase in SiOxNy films produced by plasma enhanced chemical vapor deposition M. V. Voitovycha, A. Sarikova, b, c, *, V. O. Yukhymchuka, V. V. Voitovychd, M. O. Semenenkoa, c, d a V. Lashkaryov 41 Nauky Avenue, 03028 Kyiv, Ukraine b Educational Scientific 4-g Hlushkova Avenue, 03022 Kyiv, Ukraine c National Technical "Igor Sikorsky Kyiv Polytechnic Institute", 37 Beresteiskyi Avenue, 03056 Kyiv, Ukraine d 46 Nauky Avenue, 03028 Kyiv, Ukraine *Corresponding author's email: Abstract Peculiarities of formation of inclusions of amorphous Si (a-Si) phase in Si-rich Si oxynitride films grown by plasma-enhanced chemical vapor deposition (PECVD) are studied by combined Raman scattering and infrared (IR) absorption spectroscopy. The Raman scattering results identify presence of a-Si phase in the studied films at the relative Si content exceeding a threshold value of about 0.4. The a-Si amount correlates with the concentration of hydrogen in the films, the presence of which is detected by characteristic IR absorption bands corresponding to Si-H bending (~660 cm-1) and stretching (a composite band in the range of ~1900-2400 cm-1) vibrations. The method of deconvolution of IR absorbance spectra in the range of ~600 to 1300 cm-1 developed earlier is used to reliably separate the IR band at ~660 cm-1. This band is identified to origin from the amorphous Si phase within the studied Si oxynitride films. This makes it possible to propose IR spectroscopy with analysis of the low-wavenumber part of the spectra as an efficient method of identifying phase composition of Si-rich Si oxynitride films. The obtained results contribute to understanding of the regularities of formation of phase compositions of PECVD grown Si oxynitride films and are useful for controlling the films properties for practical applications. Keywords: Si oxynitride films, nanocomposite, hydrogen, FTIR spectroscopy, Raman scattering, plasma-enhanced chemical vapor deposition 1. Introduction Si oxynitride films (SiOxNy, 0 ≤ x ≤ 2, 0 ≤ y ≤ 4/3) have a great significance for fabricating modern microelectronic and optoelectronic devices [1, 2]. As the structure of such films defines their properties, their study attracted great attention in recent decades [3, 4]. Naturally, the main focus was to study the behavior of atoms of the main film forming elements, namely silicon, oxygen and nitrogen. It may be considered well-established today that structural arrangement of Si, O and N atoms in the SiOxNy films depends on the film composition. In the films with less than ~50% of Si, the chemical composition is dominated by Si suboxide bonding (set of silicon-oxygen tetrahedra obeying the random bonding statistics) with N present as a significant impurity [5, 6]. Considering the N atoms entering into the network to substitute O atoms maintaining the same structure as that of a stoichiometric SiO2 material, the proper chemical formula for such oxynitride materials should be SiO2-yNy [7]. Increase of the relative Si concentration principally changes the SiOxNy film morphology and structure. In [8], Si-rich films were considered as composites, the microstructure of which may be represented as a mixture of silicon-oxygen tetrahedra (Si-OaSi4-a, a = 0...4) and silicon-nitrogen pyramids (Si3N) in the framework of the mixture model. Along with formation of such phase composition, existence of variable-size amorphous Si (a-Si) clusters embedded in a dielectric matrix in as-deposited Si-rich SiOxNy films was inferred based on photoluminescence (PL) and Raman spectroscopy studies [9, 10]. In particular, these clusters were responsible for room-temperature PL in the range of 1.5-2.0 eV. Manifestation of two luminescence peaks [4] was attributed to two different average cluster sizes. The PL origin from the amorphous Si inclusions in this case was supported by the presence of the features of elemental Si in the X-ray photoelectron spectra of the samples with x < 0.96 and by a lack of crystalline features in the glancing angle X-ray diffraction patterns. Infrared (IR) spectroscopy with mathematical deconvolution of the IR absorption band is successfully used for study of the microstructure of Si oxide films with excess Si content [11, 12]. In our previous publication [13], the characteristics of the elementary bands corresponding to Si-N and Si-H bond vibrations in the spectral range of ~600 to 1300 cm-1 have been determined that enable reliable deconvolution of the IR absorbance spectra with separation of the Si-O, Si-N, and Si-H components. It is known that hydrogen in Si oxynitride films is contained in high (of the order of 1022 cm-3 [7, 14]) concentrations and is responsible for quite noticeable absorption of IR radiation in well-known spectral regions. On the other hand, the Si-H bond vibration frequency is very sensitive to the local structural arrangement, which is evidenced by the results of the studies of such materials as Si oxides and amorphous Si [15-17]. This opens a way to investigate IR absorption spectroscopy as a method for identifying not only Si oxynitride, oxide and nitride phases but also amorphous Si phase in Si oxynitride films. In this work, we use combined Raman scattering and Fourier transform IR (FTIR) spectroscopy with analysis of various Si-H vibration bands to investigate the peculiarities of the formation of a-Si phase in the Si-rich Si oxynitride films obtained by plasma-enhanced chemical vapor deposition (PECVD). 2. Experimental In our experiments, 300 ± 5 nm thick SiOxNy films with different compositions (stoichiometry indices x and y) obtained by PECVD were used. The values of the N2O/SiH4 flow ratio during the film deposition and the respective film stoichiometries are presented in Table 1. Double-side polished boron-doped CZ Si wafers with the thickness of 300 μm and 1 mm thick sapphire plates were used as substrates for infrared (IR) and Raman spectroscopy investigations, respectively. More details about the film deposition procedure and determination of the film composition can be found in our previous publication [13]. Table 1. Deposition conditions and composition of SiOxNy films Sample N2O/SiH4 flow ratio Composition x y Relative Si сontent #1 9 1.95 0.01 0.34 #2 3 1.3 0.28 0.39 #3 1.5 1.09 0.32 0.41 #4 0.6 0.85 0.27 0.48 #5 0.4 0.59 0.16 0.57 #6 0.3 0.42 0.14 0.64 #7 0.2 0.37 0.11 0.68 #8 0.1 0.23 0.07 0.77 #9 0.06 0.18 0.05 0.80 Raman spectra of the studied films were excited by the emission of a CNI Model PSU-HFDA solid-state laser with λ = 457 nm at room temperature and recorded using an MDR-23 spectrometer equipped with an Andor iDus 401A CCD detector (UK). In order to prevent the film structure from being affected during the measurements, the excitation radiation power density was fairly small, not exceeding 103 W/cm2. For correct comparison of the bands in different Raman spectra, the latter were normalized to the intensity of the band corresponding to amorphous Si (~480 cm-1). FTIR transmittance spectra in the range of 400-4000 cm-1 were measured at room temperature using a PerkinElmer BX-II spectrometer. The measurement resolution was 2 cm-1, the number of scans was 100, and the measurement accuracy was ~0.5%. A Si substrate was used as a reference sample for all the measurements. IR absorbance spectra were recalculated from the transmittance spectra using the BeerBouguer-Lambert law. In this work, the bands related to Si-H stretching (a wide composite band at around 2200 cm-1) and bending (peaked near 660 cm-1) vibrations were analyzed. To separate the contributions from different Si-H containing complexes, the IR absorption bands corresponding to the Si-H stretching vibrations were mathematically deconvoluted into elementary Gaussian profiles with the parameters (the maximum position and full width at half-maximum) determined earlier [14, 15]. The details of this procedure were described in [14]. 3. Results and discussion 3.1. Raman scattering investigation of SiOxNy films Phase structure of the SiOxNy films with different compositions (see Table 1) was analyzed by Raman spectroscopy. Figure 1(a) shows a number of representative Raman spectra of these films. As can be seen from this figure, the spectra contain two asymmetric bands in the spectral range of 300-800 cm-1 with the maxima at ~476 cm-1 and ~660 cm-1. The first band is associated with Si-Si and Si-O bond vibrations, while the second band corresponds to vibrations of Si-H bonds in an amorphous phase [18-20]. The shape and the frequency position of these bands evidence amorphous structure of our films and presence of silicon-hydrogen bonds, which is typical of the films obtained by PECVD using SiH4 gas [9, 14, 18, 21]. As can be further seen from Figure 1(a), the wide asymmetric band in the frequency range ~ 300-550 cm-1 is the most intense one in the measured Raman spectra. The maximum position of this band is always fixed at ~476 cm-1, while its half-width Γ decreases with the increase in the relative Si content in the films. We analyzed the shape of this band by decomposing it into Gaussian components (see inset in Figure 1(a)) according to the procedure described in [18, 19, 22, 23]. As a result, two Gaussian components with the parameters ω1 = 476 cm-1 (maximum position) and Γ1 = 66 cm-1 (half-width), and ω2 = 420 cm-1 and Γ2 = 124 cm-1 were identified. As is well known, in Raman spectroscopy, spectral position and shape of a band (width, symmetry, and frequency at maximum) are characteristic parameters that provide important information about the structural state of a studied material. For Si oxide based structures containing amorphous and crystalline clusters, Raman peaks at ~480 cm-1 and with a half-width of ~60-80 cm1 corresponding to amorphous silicon TO mode, together with very narrow peaks with the maximum at ~520 cm-1 and a half-width of ~5 cm-1 corresponding to crystalline Si were observed [20, 22]. Amorphous Si dioxide (a-SiO2) exhibited a broad peak at ~495 cm-1 corresponding to the vibrational mode of four-member SiO2 rings [24]. A shift of the spectrum peak from that typical of SiO2 to the one typical of a-Si, its broadening and appearance of a one-side asymmetry were observed for SiOx films with excess Si content (x < 2) at decreasing x, indicating heterogeneity of the film structure and change in its phase composition [20, 22]. Raman spectrum of amorphous Si is characterized by broad, unstructured bands in the range of 100-550 cm-1 associated with transverse acoustic (TA, ~145 cm-1), transverse optical (TO, ~475 cm-1), longitudinal optical (LO, ~365 cm-1) and longitudinal acoustic (LA, ~305 cm-1) phonon vibrations of Si-Si bonds [19, 22, 25]. Therefore, the elementary band at ω1 ≈ 476 cm-1 observed in our structures (see inset in Figure 1(a)) may be attributed to the contribution of the TO mode of Si-Si bond vibrations in amorphous silicon. The broader band at ω2 ≈ 420 cm-1 may be caused by Raman scattering of light due to Si-O bond vibrations in the amorphous Si oxide matrix [20, 26]. Figure 1. (a) - Raman spectra of SiOxNy, films obtained at different N2O/SiH4 gas ratios. Inset - Raman spectrum of the #6 sample deconvoluted into Gaussian components. (b) - Integrated intensity of the ω(TO) band of the Raman spectra versus Si content in the SiOxNy films. Figure 1(b) shows the dependence of the relative contribution of the integrated intensity of the TO band to the analyzed Raman spectrum in the range of ~300 to 550 cm-1, on the relative Si content in the studied SiOxNy films. As can be seen from this figure, there is a direct correlation between this intensity and the amount of Si in the films. It should be concluded therefore that an increase in the relative Si content in non-stoichiometric Si oxynitride films gives rise to an increase in the amount of amorphous Si in them. 3.2. IR spectroscopy study of SiOxNy film structure IR transmittance spectra in the frequency range of 400-4000 cm-1 of the SiOxNy films with different compositions are presented in Figure 2. It can be seen from this figure that the spectra contain transmission bands peaked at ~452, 660, 870, and 1060 cm-1 in the low-frequency region and at ~2100 and 3600 cm-1 in the high-frequency region. The bands in the range between 900 and 1300 cm-1 as well as at ~452 cm-1 correspond to Si-O-Si valence and rocking vibrations, respectively [1]. The band peaked at ~870 cm-1 is associated with Si-N bonds and the band at ~660 cm-1 corresponds to bending vibrations of Si-H bonds [2]. It should be noted at this that the bands in the range of ~600 to 1300 cm-1 corresponding to Si-O, Si-N and Si-H bond vibrations can overlap, especially for the films with relatively high Si contents, which greatly complicates analysis of the IR spectra and, hence, the SiOxNy film structure. The high-frequency bands with the maxima at ~2100 and 3600 cm-1 in the measured IR transmittance spectra are caused by valence vibrations of Si-H bonds and Si-OH groups, respectively [1]. Figure 2. FTIR transmittance spectra of SiOxNy films with different compositions. In this work, we focus on a study of IR absorption bands associated with Si-H bond vibrations. Figure 3 shows the high-frequency Si-H related band for a number of the studied SiOxNy films. It can be seen from this figure that the shape of this band, its maximum position, intensity and area strongly depend on the SiOxNy film composition. Namely, increase of the relative Si content in the films leads to the shift of the band peak to lower frequencies, and this shift can be quite large. It should be noted as well that no high-frequency hydrogen-related band is detected in the IR absorbance spectra for the sample with the highest oxygen content and, hence, the lowest Si content (sample #1, see Table 1). Figure 3. IR absorbance spectra of the SiOxNy films in the range of Si-H stretching vibrations. The arrow shows the direction of the shift of the maximum position of the integrated band with the increase in the relative Si content in the films. By integrating the area under the IR spectra in the range of ~1900-2400 cm-1, the amount of hydrogen bound in the studied films can be calculated [14, 15]. The calculated hydrogen concentration as a function of the relative Si content in the films is presented in Figure 4. It can be seen from this figure that the hydrogen concentration is of the order of ~1021-1022 cm-3, which coincides by the order of magnitude with the known literature data (see e.g. [7]). This concentration increases with the relative amount of Si in the films, which looks quite natural, since growth of the latter is associated with an increase in the SiH4 concentration in the gas mixture during the film deposition. For the samples with the highest Si content, a tendency to saturation is noticeable. Composition of the IR spectral band in the range of ~1900-2400 cm-1 allows one to draw fundamental conclusions about the structural arrangement of hydrogen atoms in the Si oxynitride network. It is well-known that the considered wide band is a superposition of elementary bands corresponding to stretching modes of H-Si(Si3-nOn) structural units, where n = 0...3 [14, 15]. Hence, we deconvoluted this band into elementary Gaussian components for different Si oxynitride film stoichiometries taking into account both the nature of the elementary components [15] as well as our previous experience gained during the study of the kinetics of hydrogen effusion from Si-rich Si oxynitride films induced by heat treatments [14]. The deconvolution results for the films depicted in Figure 3 are presented in Figure 5. As can be seen from this figure, three elementary components, namely H1, H2 and H3 with the characteristics presented in Table 2, were separated. As can be further seen from Figure 5, the spectrum of the sample #2 having low Si content (see Table 1) is described by the only component Н1. Increase in the relative Si content leads to redistribution of the main contribution from H1 to H2 (for the sample #3) and H3 (for the samples #7 and #9) with a tendency of gradual disappearance of the previous bands. Figure 6 provides a more detailed information about the dependence of the relative contributions of the bands H1, H2 and H3 to the FTIR spectra of the SiOxNy films on the relative Si content in the films for all the samples presented in Table 1. Figure 4. Hydrogen concentration versus relative Si content in the SiOxNy films. It is known [4, 15] that the quantity and type of the neighboring atoms to a hydrogen atom correlate with the Si-H bond length, and, hence, influence on the Si-H stretching vibrations frequency. For each of the separated elementary bands H1 to H3, the Si-H bond length was determined using the following expression [4]: νSi-H (dSi-H)3 = 7074 cm2 (1) where νSi-H is the band peak position and dSi-H is the Si-H bond length, respectively. The obtained bond lengths together with the respective H-Si(Si3-nOn) complexes identified according to [15] are presented in Table 2. It should be noted that the Si-H bond length for the bands H1 (0.146 nm) and H3 (0.151 nm) determined by the expression (1) practically do not differ from the respective values for Si dioxide (0.144 nm) and amorphous Si (0.152 nm) as indicated in [4]. We may conclude therefore that the results of the analysis of the high-frequency parts of the IR spectra agree well with existence of amorphous Si phase within the Si oxynitride structure at the relative Si content above about 0.4 (samples #3 to #9, see Figure 6). Figure 5. Deconvolution of the IR absorption band corresponding to Si-H stretching vibrations on elementary Gaussian components corresponding to contributions from H-Si(O3) (H1), H-Si(Si2O) (H2) and H-Si(Si3) (H3) complexes. We have mentioned above that another Si-H related band in the measured IR absorbance spectra corresponds to the peak position of about 660 cm-1. This band is related to Si-H bending vibrations [9, 14-16]. Analysis of the integrated intensity of this band would provide additional Table 2. Characteristics of the Gaussian components of the Si-H related IR absorption band Band Maximum position, cm-1 Width, cm-1 Oscillation type Complex dSi-H, nm Н1 2252±2 75±3 stretching H-Si(O3) 0.146 Н2 2150±4 100±2 stretching H-Si(Si2O) 0.149 Н3 2050±4 85±4 stretching H-Si(Si3) 0.151 Figure 6. Integrated intensity of IR absorbance bands corresponding to Si-H vibrations versus Si content in the SiOxNy films. The maximum position of the IR bands: 1 - ~660 cm-1, 2 - ~2050 cm1, 3 - ~2150 cm-1, and 4 - ~2250 cm-1. information about structural arrangement of hydrogen in the Si oxynitride films as well as highlight the Si oxynitride phase composition. However, analysis of this band is generally complicated as the band may overlap with the bands corresponding to Si-O and Si-N bond vibrations also present in the range of ~600 to 1300 cm-1. In our previous publication [13], we determined the parameters of the elementary Gaussian components of the IR absorbance spectra of Si-rich Si oxynitride films in the mentioned frequency range thus enabling separation of the Si-O, Si-N and Si-H related bands. Figure 7(a) shows the IR absorbance spectra of a number of SiOxNy samples in the range of 500 to 800 cm-1 with separated absorption band near ~660 cm-1 corresponding to Si-H bond vibrations. This band is described by a single Gaussian with the mean half-width of ~82 ± 2 cm-1. It can be seen from Figure 7(a) that the intensity and maximum position of this band depend on the N2O/SiH4 flow ratio during Si oxynitride film growth and, hence, the relative Si content in the films. The dependence of the integrated intensity of this band on the relative Si content in the studied films is also presented in Figure 6. As can be seen from Figures 6 and 7(a), the mentioned band is absent in the IR spectra of the films with the smallest Si contents (samples #1 and #2) and increases its integrated intensity with subsequent growth of the relative Si amount (samples #3 to #9). The position of the band maximum shifts at this from ~665 cm-1 for the sample #3 (relative Si content of 0.41) to ~635 cm-1 for the sample #9 (relative Si content of 0.8), as shown by the data provided in Figure 7(b). These findings agree well with the results of [4, 18], where the low-frequency IR band corresponding to Si-H bending vibrations in PECVD grown Si oxynitride films had no definite peak position. We conclude therefore that there is a clear correlation between the position of the peak and the integrated intensity of the IR absorption band at ~660 cm-1 corresponding to Si-H bond bending vibrations on the one hand, and the Si content in the SiOxNy samples on the other hand. We have established above that the IR absorption band at ~2050 cm-1 corresponds to the Si-H bond stretching vibrations in amorphous Si phase. It can be seen from Figure 6 that the integrated intensity of this band has a direct correlation with that of the band at ~660 cm-1. Moreover, as can be seen from Figure 8, the integrated intensity of the latter band is directly proportional to the integrated intensity of the TO mode of Si-Si bond vibrations in the measured Raman spectra. Therefore, the IR absorption band peaked at ~640-660 cm-1 may be confidently Figure 7. (a) - FTIR absorbance spectra of the SiOxNy films corresponding to Si-H bending vibrations and (b) - dependence of the maximum position of the band at ~660 cm-1 on the relative Si content in the SiOxNy films. Figure 8. Integrated intensity of the IR absorption band at ~660 cm-1 versus integrated intensity of the ω(TO) band of the Raman spectra of the SiOxNy films. attributed to the presence of amorphous Si phase in the non-stoichiometric Si oxynitride films. This conclusion is supported by the published results by other researchers. It is reported in particular that the IR absorption band with the peak position around 640 cm-1 is typical for a-Si layers [27] and is considered to origin from Si-H bond vibrations in H-Si(Si3) complexes [16, 17]. In Si oxynitride films, this band was reliably detected only in the case of high relative Si contents. It was not detected in the spectra of the samples deposited by PECVD technique, which had a rather high oxygen concentration and a composition much closer to SiO2 than to amorphous Si [15]. On the other hand, a band peaked at ~640 cm-1 corresponding to Si-H bending vibrations was present in the IR spectra of as-deposited Si-rich Si oxide films [13]. Moreover, an absorption band with the maximum near 670 cm-1 clearly manifested itself in the IR spectra of the SiOxNy samples with high Si fraction [4]. In both latter cases, presence of a-Si inclusions in the studied films was established independently. 3.3. Comparative analysis of IR and Raman scattering results Comparative analysis of the evolution of the Si-H related IR bands with the SiOxNy composition along with the respective Raman scattering results allows one to get a deeper insight into the peculiarities of the formation of phase composition of Si oxynitride films grown by plasmaenhanced chemical vapor deposition. As demonstrated by the data from [4, 9, 15, 21] as well as our Raman and IR spectroscopy investigations, formation of amorphous Si clusters in an oxynitride matrix is possible in Si oxynitride films deposited by PECVD at different N2O/SiH4 gas flow ratios. In particular, presence of a-Si phase is detected by Raman measurements (observation of the TO mode of Si-Si bond vibrations at ω1 ~ 476 cm-1, see Figure 1(a)). At this, the amount of amorphous Si grows with the increase in the relative Si content in the films. Using the approach to deconvolution of IR absorbance spectra of Si oxynitride films presented in [13], a low-frequency IR band corresponding to Si-H bending vibrations (~640-660 cm-1) is reliably separated. As was already mentioned above, the integrated intensity of this band correlates both with the intensity of the IR absorption band at ~2050 cm-1 identified to origin from the Si-H stretching vibrations in amorphous Si (see Figure 6), as well as with the integrated intensity of the TO mode in the Raman scattering spectra (see Figure 8), which made it possible to reliably attribute this band to the presence of a-Si phase. As can be further seen from Figure 6, no aSi related signal is detected in the IR spectra of the samples with small relative Si concentrations (below about 0.4). This points to a certain threshold value of excess Si content, at which amorphous Si phase begins to form. This is further demonstrated by Figure 9, where the dependence of the integrated intensity of the IR absorption band at ~660 cm-1 on the excess Si content in the investigated Si oxynitride films calculated as 1 - x/2 - 3y/4 is shown. As can be seen from this figure, the threshold excess Si value is about 0.2, and the amount of the amorphous Si in the Si oxynitride films, which is proportional to the integrated intensity of the IR band at ~660 cm-1, linearly grows with the excess Si concentration above this value. Figure 9. Integrated intensity of the IR absorption band at ~660 cm-1 versus excess relative Si content in the SiOxNy films. The properties of the IR absorption band at ~660 cm-1 allow one to consider IR spectroscopy as an efficient method for tracking Si agglomeration into amorphous Si phase in an oxynitride matrix and formation of a particular phase composition of PECVD grown films. According to this method, IR transmittance spectra of SiOxNy films in the low-frequency region should be only measured. Using the deconvolution procedure and the characteristics of the elementary IR bands corresponding to Si-O, Si-N and Si-H (~660 cm-1) bond vibrations [13], phase composition of the studied films, including the presence or absence of amorphous Si inclusions in the films, may be inferred. We believe that this method is one of the simplest and most non-destructive ones for monitoring Si agglomeration in SiOxNy films produced by PECVD technology and determining their phase composition and structure. The results obtained in this study extend our understanding of the peculiarities of formation of phase compositions of PECVD grown Si-rich Si oxynitride films. Recently, a thermodynamic theory describing phase composition of SiOxNy films with different stoichiometries grown at different temperatures was proposed [28], which, however, considered only a random bonding model distribution of Si, O and N atoms. Identification of the possibility of presence of amorphous Si clusters in such films should be used for updating the theoretical model, in particular, to improve its potential for tailoring the phase composition properties of Si oxynitride films for practical applications. 4. Соnclusions In this work, peculiarities of formation of amorphous Si phase in Si oxynitride films grown by plasma-enhanced chemical vapor deposition are studied by combined Raman scattering and IR spectroscopy investigations. The results of the Raman spectroscopy demonstrate that at a certain ratio of precursor gases in the growth chamber during SiOxNy film deposition, excess Si content gives rise to formation of amorphous Si phase within the films. This is evidenced by the appearance of the peak at ~476 cm-1 corresponding to the transverse optical phonon mode characteristic of a-Si. Large half-width of this band (Γ = 66 cm-1) indicates the absence of long-range order in the film structure, which is typical for amorphous materials. The results of our Raman spectroscopy investigations are in good agreement with the XPS, XRD, and PL results reported in [4, 6, 9, 21] for similar Si oxynitride films prepared by PECVD technology. Using the method of deconvolution of IR absorbance spectra in the wavelength range of ~600 to 1300 cm-1 into elementary components, proposed in [13], the elementary band at ~660 cm-1 corresponding to Si-H bending vibrations was separated. The comparative analysis of the IR and Raman spectroscopy results enabled identification of this band as related to the presence of amorphous Si inclusions in the studied films. It was demonstrated that formation of amorphous Si clusters in Si oxynitride films takes place at the relative excess Si concentrations exceeding a threshold value of about 0.2. After the threshold, the amount of the a-Si phase linearly grows with the excess Si concentration. This result makes it possible to consider IR spectroscopy with analysis of the low-wavelength (~600 to 1300 cm-1) part of spectra as an efficient method of identifying phase composition of Si-rich Si oxynitride films. The results of this study enhance our understanding of the peculiarities of the formation of phase compositions of PECVD grown Si oxynitride films and may be used for developing methods of creating the films with required phase compositions for practical applications. Acknowledgement This work has been supported by the budget project no. 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Zacharias, Characteristics of hydrogen effusion from the Si-H bonds in Si rich silicon oxynitride films for nanocrystalline silicon based photovoltaic applications, Adv. Mater. Res. 854 (2014) 69-74. https://doi.org/10.4028/www.scientific.net/AMR.854.69. 15. A. Sassella, A. Borghesi, F. Corni, A. Monelli, G. Ottaviani, R. Tonini, B. Pivac, M. Bacchetta, L. Zanotti, Infrared study of Si-rich silicon oxide films deposited by plasma-enhanced chemical vapor deposition, J. Vac. Sci. Technol. A 15 (1997) 377-389. https://doi.org/10.1116/1.580495. 16. G. Lucovsky, J. Yang, S. S. Chao, J. E. Tyler, W. Czubatyj, Oxygen-bonding environments in glow-discharge-deposited amorphous silicon-hydrogen alloy films Phys. Rev. B 28 (1983) 3225-3223. https://doi.org/10.1103/PhysRevB.28.3225. 17. B. Fischer, A. Lambertz, M. Nuys, W. Beyer, W. Duan, K. Bittkau, K. Ding, U. Rau, Insights into the Si-H bonding configuration at the amorphous/crystalline silicon interface of silicon heterojunction solar cells by Raman and FTIR spectroscopy, Adv. Mater. 35 (2023) 2306351 (9 pages). https://doi.org/10.1002/adma.202306351. 18. V. A. Volodina, D. I. Koshelev, Quantitative analysis of hydrogen in amorphous silicon using Raman scattering spectroscopy, J. Raman Spectrosc. 44 (2013) 1760-1764. https://doi.org/10.1002/jrs.4408. 19. C. Vichery, V. Le Nader, C. Frantz, Y. Zhang, J. Michlera, L. Philippe. Stabilization mechanism of electrodeposited silicon thin films, Phys. Chem. Chem. Phys. 16 (2014) 22222-22228. https://doi.org/10.1039/C4CP02797C. 20. V. Ya. Bratus', V. A. Yukhimchuk, L. I. Berezhinsky, M. Ya. Valakh, I. P. Vorona, I. Z. Indutnyi, T. T. Petrenko, P. E. Shepeliavyœ, I. B. Yanchuk, Structural transformations and silicon nanocrystalline formation in SiOx films, Semicond. 35 (2001) 821-826. https://doi.org/10.1134/1.1385719. 21. A. Hartel, D. Hiller, S. Gutsch, P. Löper, S. Estrad ́e, F. Peir ́o, B. Garrido, M. Zacharias, Formation of size-controlled silicon nanocrystals in plasma enhanced chemical vapor deposition grown SiOxNy/SiO2 superlattices, Thin Solid Films 520 (2011) 121-125. https://doi.org/10.1016/j.tsf.2011.06.084. 22. V. V. Voitovych, R. M. Rudenko, A. G. Kolosiuk, M. M. Krasko, V. O. Juhimchuk, M. V. Voitovych, S. S. Ponomarov, A. M. Kraitchinskii, V. Yu. Povarchuk, V. A. Makara, Effect of thin on the processes of silicon-nanocrystal formation in amorphous SiOx thin-film matrices, Semicond. 48 (2014) 73-76. https://doi.org/10.1134/S1063782614010242. 23. I. G. Neizvestniy, V. A. Volodin, A. A. Gismatulin, G. N. Kamaev, A. H. Antonenko, A. G. Cherkov, V. G. Litovchenko, I. P. Lisovsky, I. Yu. Maidanchuk, Formation of Si nanocrystals in SiOx, SiOx:C:H films and Si/SiO2 multilayer nano-heterostructures by pulse laser treatments, Proc. SPIE 9440 (2014) 94400F. https://doi.org/10.1117/12.2179349. 24. A. A. Sirenko, J. R. Fox, L. A. Akimov, X. X. Xi, S. Ruvimov, Z. Liliental-Weber, In situ Raman scattering studies of the amorphous and crystalline Si nanoparticles, Solid State Commun. 113 (2000) 553-558. https://doi.org/10.1016/S0038-1098(99)00539-6. 25. Y. Zhu, C. Yao, J. Wang, H. Zhu, T. Shen, X. Gao, J. Sun, K. Wei, D. Wang, Y. Sheng, Z. Wang, Micro-Raman spectroscopy characterization of silicon with different structures irradiated with energetic Bi-ions, Nucl. Instr. Meth. Phys. Res. B: Beam Interactions with Materials and Atoms 365 (2015) 123-127. https://doi.org/10.1016/j.nimb.2015.08.041. 26. A. Rahmani, M. Benoit, C. Benoit, Signature of small rings in the Raman spectra of normal and compressed amorphous silica: A combined classical and ab initio study, Phys. Rev. B 68 (2003) 184202 (12 pages). https://doi.org/10.1103/PhysRevB.68.184202. 27. A. A. Langford, M. L. Fleet, B. P. Nelson, W. A. Lanford, M. Maley, Infrared absorption strength and hydrogen content of hydrogenated amorphous silicon, Phys. Rev. B 45 (1992) 13367-13377. https://doi.org/10.1103/PhysRevB.45.13367. 28. M. Babiichuk, A. Sarikov, M. Voitovych, V. Voitovych, Phase composition versus stoichiometry and fabrication temperature of Si oxynitride films by thermodynamic modeling, 2025 IEEE 15th International Conference "Nanomaterials: Applications & Properties" (IEEE NAP-2025) Bratislava, Slovakia, Sep. 7-12, 2025 (in press).
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2510.14706
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1
Layered Bimetal Nanoporous Platforms for SERS Sensing
Yanqiu Zou1±, Anastasiia Sapunova2,3±, Tommaso Giovannini4, Chen Wang5, Huaizhou Jin6, Vincenzo
Caligiuri7, Andrea Schirato8,10, Luca Bursi9, Alessandro Alabastri10, Shukun Weng2,3, Ali Douaki2,11,
German Lanzavecchia2,11, Ivan Marri11, Roman Krahne2, Nicolò Maccaferri12,13, Zhenrong Zheng1*,
Shangzhong Jin5* and Denis Garoli*2,5,11
1. State Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering,
Zhejiang University, Hangzhou 310027, China.
2. Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy.
3. Università degli Studi di Milano-Bicocca, Piazza dell'Ateneo Nuovo 1, 20126, Milano, Italy
4. Department of Physics, University of Rome Tor Vergata, and INFN, Via della Ricerca Scientifica 1, I-
00133, Rome, Italy
5. College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, China
6. Key Laboratory of Quantum Precision Measurement, College of Physics, Zhejiang University of
Technology, Hangzhou, China.
7. Dipartimento di Fisica, Università della Calabria, via P. Bucci 33b, 87036 Rende, CS, Italy
8. Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
9. Department of Physical, Computer, and Mathematical Sciences (FIM), Università degli Studi di
Modena e Reggio Emilia, and Nanoscience Institute, CNR-NANO, via Campi 213/a, 41125, Modena,
Italy
10. Department of Physics and Astronomy, Rice University, Houston, Texas 77005, United States
11. Università degli Studi di Modena e Reggio Emilia, Via Amendola 2, 42122, Reggio Emilia (Italy)
12. Department of Physics, Umeå University, Linnaeus väg 24, 901 87 Umeå, Sweden
13. Umeå Centre for Microbial Research, Umeå University, 901 87 Umeå, Sweden
2
±Equally contribution
* Corresponding author: Denis Garoli, denis.garoli@unimore.it; Zhenrong Zheng, zzr@zju.edu.cn;
Shangzhong Jin, Jinsz@cjlu.edu.cn
Keywords: plasmonics, nanoporous materials, Raman, gold, silver, copper
Abstract
Nanoporous metals are extensively investigated as platforms for applications in plasmonics. They present
high surface areas and strong local electric fields that can be tuned at different energies, playing with the
choice of the metals and the morphology of the porous layers. Until recently, research in the field of
plasmonics has primarily focused on porous metals composed of a single element, with limited attention
given to the impact of alloy composition. The investigation of bi-metallic systems has only just begun to
emerge in the literature. In particular, combining two or more different plasmonic metals, it could be
possible to explore the interactions between two metals excited at specific energies. This involves
plasmonic coupling, electron transfer, band hybridization at the interface, electromagnetic field
interactions, and possibly thermal and electronic energy transfer depending on separation, size, and
materials involved. The analysis of bi-metal systems can also be interesting in biomolecule detection,
such as in the case of Surface Enhanced Raman Scattering (SERS). Here we report, for the first time, a
detailed study (comprising morphological analyses, numerical modelling, and optical spectroscopies) on
bi-metal nanoporous platforms prepared with a dry-synthesis method enabling the easy and controllable
fabrication of bilayers combining different metals such as Au, Ag, and Cu.
Materials and Methods:
Samples fabrication
The samples preparation is based on the original methods proposed by Kwon et al.1. In brief, poly (methyl
methacrylate) (PMMA) was spin-coated on a Si substrate at 4000 rpm for 2 min. Each metal (>99.99 %
3
purity) was subsequently deposited by electron-beam evaporation onto the PMMA thin film at room
temperature with an oblique incidence angle of 80◦, a deposition rate of 0.1 nm/s, and a target thickness
of about 12 nm. The deposited Au film was plasma treated in O2 at a power of 200W until the entire
PMMA layer was removed. In the case of Ag and Cu films, in order to avoid fast oxidation, the plasma
treatment was performed in N2 at 200W until the PMMA layer was completely removed (as previously
demonstrated, this ensures that the oxidation is kept to a very low level2). The bi-layer structure was
obtained by repeating the preparation using the first porous layer as substrate for the deposition of the
second porous layer on top.
Spectroscopic measurements
Raman spectroscopy measurements were performed using a HORIBA LabRAM HR Evolution Raman
spectrometer (HORIBA Jobin Yvon, Kyoto, Japan) with a 50X long-focal-length objective (NA = 0.75)
, a 100μm-hole aperture, and a 600 gr/mm grating. SERS spectra of Rhodamine 6G (R6G) were acquired
under different conditions to characterize system performance: 100 nM R6G with laser power attenuated
to 10% at 532 nm excitation; 1 μM R6G with 5% laser power at 633 nm; and 0.1 mM R6G with full laser
power (no attenuation) at 785 nm. For each substrate, a Raman map was collected over a 20 × 20 μm²
area with a step size of 5 μm, resulting in 25 individual spectra. The final spectrum for each substrate
was generated by averaging these 25 spectra after pre-processing. Spectra of the protein ADAMTS3 were
obtained by averaging five randomly collected spectra, using a 5% ND filter and 60s acquisition time for
both 532nm and 633nm excitations. For 785nm excitation, a 100% ND filter (no attenuation) and 120s
acquisition time were used.
Data Analysis
The raw spectra were baseline-corrected and smoothed using HORIBA LabSpec6 software.
Subsequently, all data averaging, normalization, spectral deconvolution, peak fitting, and graphical
plotting were performed using OriginPro 2025b.
Numerical modelling
4
To explore the plasmonic properties of porous NPMs, we performed a numerical investigation of the
electromagnetic response using a finite element method (FEM) commercial software, employing
COMSOL Multiphysics. In particular, following a procedure reported in detail in our recently reported
works1,2 nanometric pores and irregularities of NPMs have been numerically reproduced from scanning
electron microscopy (SEM) images of the experimental samples. However, continuum modeling, such
as FEM, might fail in describing the response properties of bimetallic nanostructured materials3–5.
Therefore, to validate the FEM procedure, the optical response of NPMs has also been calculated using
a fully atomistic approach, named frequency-dependent fluctuating charges and dipoles (ωFQFμ)5–7. This
method provides a reliable description of noble metal nanoparticles, comparable to ab initio5 but at
significantly lower computational cost, and can thus handle systems composed of thousands of atoms8–
11. Remarkably, it can also be applied to bimetallic systems and, in the specific case of Au/Ag
nanoparticles, yields results in almost perfect agreement with experiment12. For cross-validation of the
COMSOL results, we have adopted an idealized picture in which the same porous structure is generated
and then considered either as Ag or Au, which are characterized by the same lattice constant (4.08 Å).
Such structures were created by introducing randomly generated nanometric pores, thereby generating
realistic surface roughness and porosity. In particular, we modeled a 7 × 7 × 4 (nm3) structure, composed
of 6584 atoms, a size unfeasible for ab initio atomistic models (see SI for a graphical representation).
The same nanoporous geometry was then used to simulate Ag/Ag, Au/Au, and mixed Au/Ag bilayers,
allowing for a direct comparison of their optical response and a detailed analysis of the different
compositional effects. All ωFQFμ calculations are performed using the Ag/Au parameters reported in
Ref. 12, with the open-source plasmonX software6.
Dielectric Permittivity and Ellipsometry.
Spectroscopic ellipsometry was employed to determine the dielectric permittivity of nanoporous Ag/Ag,
Ag/Au, and Ag/Cu bilayers. The measurements provided both the real and imaginary components of the
dielectric function over the visible range. The experimental data were fitted using a model composed of
Lorentzian and Gaussian oscillators to capture the main features of each alloy’s optical response. The
resulting dielectric permittivities accurately reproduce the experimental ellipsometric parameters Ψ and
5
Δ. Complete fitting details and oscillator parameters are reported in the Supporting Information (Table
S1).
Introduction
Over the last decades, research on nanoporous metals (NPMs) has grown steadily, driven by their
broad range of applications in sensing13–15, catalysys16,17, photonics18,19 and biomedicine20. The main
characteristic of NPMs is their high surface-to-volume ratio that, combined with unique chemical
reactivity and mass transport properties, makes this family of nanostructured materials highly interesting
for advanced research. NPMs can be prepared following different strategies such as template-assisted
physical vapor deposition, electrochemical processes, and chemical dealloying18,21. The latter is probably
the most used method to prepare NPMs through selective dissolution of the more chemically active
component from precursor alloys. Dealloying methods have been applied to prepare NPMs as gold13,18,22,
copper23,24, silver25,26, and aluminum27,28. A specific aspect of dealloying is the presence of residual less
noble metal at the end of the process. This effect is difficult to avoid and leads to resulting nanoporous
structures with varying alloy compositions29,30. Interestingly, this residual “contamination” in the final
porous film, long considered as a limitation in the preparation of pure porous structures, has recently
been shown to enhance plasmonic properties of nanoporous gold containing residual amount of the less
noble metal due to modulation of its permittivity by the alloying components.31–33 The combination of
two or more metals in alloys is a strategy to provide a way to tune the electronic state through charge
transfer between the metals, making it possible to alter the chemical interaction between molecules and
the surface of the alloy. In particular, the adsorption behavior of molecules having either high
electronegativity (e.g., oxygen and halogens), strong polarity (e.g., carbonyl and nitro groups), partial
charge (e.g., carboxylate and hydride-like hydrogen), or high electron density (e.g., alkenes and arenes)
is affected by the alloy composition. For instance, palladium-silver alloy promoted the adsorption of
negatively charged molecules (carboxylate intermediate) due to the high electronegativity of silver. The
alteration of the electronic state by alloying on the molecular detection capability has been recently
studied by La et al.33 In particular, a SERS enhancement of up to 6 orders of magnitude for nanoporous
gold containing silver was observed and attributed to the narrowing of the electronic structure and its
6
alignment close to the Fermi level. These results suggest that the electronic state modulation of the metal
structure can affect molecule-surface interactions.
Here, we decided to explore bi-metallic nanoporous systems from a different point of view.
Instead of using chemical dealloying, we used a recently reported dry-synthesis method to prepare highly
pure layers of plasmonic metals such as Au, Ag, and Cu1–3. The peculiarity of this approach lies in the
possibility to produce stacked nanoporous layers, either composed of the same metal or of different
metals deposited sequentially on top of one another, as demonstrated previously in refs. 2,3 At the best of
our knowledge, this is the first example of perfectly stacked nanoporous metal layers, and it represents a
valuable platform to investigate how the use of two different metals can impact the performance in terms
of SERS enhancements at different wavelengths across the visible spectral range. Experimental results
and numerical simulations will be presented to shed light on the behavior of these novel plasmonic
nanoporous platforms.
Results and discussion
Figure 1 illustrates the planar morphologies of the samples used for these experiments. While the
first row (Fig. 1(a-c)) reports the single NPM films, respectively Au, Ag, and Cu, the other micrographs
show the obtained morphologies for the bi-metallic layered films. As can be seen, the single metal films
present a nanoporous structure with fully connected ligaments of diameters of a few tens of nm. In the
case of bi-metallic layered films, the morphologies vary significantly among the samples. In all the bi-
metallic samples (Fig. 1(d-i)), it is possible to observe the two layers of metals stacked one on the other,
but not mixed, with a gap between the 2 metallic nanostructures that can only be due to a potential small
oxide layer surrounding the Ag and Cu nanoporous structures.
7
Figure 1: (a-e) SEM micrographs of the investigated samples; porous layers of (a) Au; (b) Ag;
(c) Cu; (d) Au/Ag; (e) Ag/Au; (f) Au/Cu; (g) Cu/Au; (h) Ag/Cu; (i) Cu/Ag.
Following the same approach used in our recent works3,4, we performed numerical studies of the
electromagnetic response of the porous bi-layer system. This enables us to obtain an overview of the
electromagnetic (EM) field distributions at the wavelengths of excitation used in the experiments. The
nanometric structural features of the layers were reproduced directly from the SEM images of the
experimental samples in the finite element method (FEM)-based model, developed using COMSOL
Multiphysics. Specifically, the top-view SEM micrographs of the Au, Ag, and Cu films used in the
experiments were imported into the FEM solver and extruded along the thickness direction of the layer
to build the 3D numerical geometry. As previously shown3, this method enabled us to numerically
8
account for the actual discontinuities and profile of the materials down to the nanoscale. Figures 2, 3, 4,
and 5 report an estimation of the EM field enhancement distribution along the surfaces of the two (the
top and bottom) porous layers in the different cases, taking into consideration three wavelengths of
excitation (550, 630, and 780 nm).
Figure 2: (a,b) EM field enhancement distributions for (a) homometallic nanoporous bilayer structures
composed of Cu (top), Au (middle), and Ag (bottom), calculated with FEM at excitation wavelengths of
550 nm, 630 nm, and 780 nm (from left to right); (b) homometallic nanoporous bilayers composed of
Au (top) and Ag (bottom), obtained using the ωFQFμ approach at the same excitation wavelengths. All
9
maps are normalized to the same color scale to highlight spatial regions of strongest field enhancement
localization. (c,d) Maximum field enhancement values extracted from FEM (c) and ωFQFμ (d)
simulations for each structure.
Figure 2 shows the EM field distributions at wavelengths of 550, 630, and 780 nm for bi-layer
nanoporous structures made of a single metal: Cu (a, top), Au (a, middle), and Ag (a, bottom). For the
specific case of Au and Ag, the EM distributions are also computed using the ωFQFμ method, for
comparison (Fig. 2b). In all cases, the EM enhancement profiles were evaluated on the surface of the bi-
layer structure. The color scale indicates a normalized intensity of the EM field, calculated as the ratio
of the electric field in the presence of nanoporous metals to the incident field. All field maps are
normalized to the same values, pointing out the regions where the EM field is highly localized.
Nevertheless, at the FEM level (Fig. 2a), each homometallic sample is characterized by its own
geometry and plasmonic properties; therefore, the diagram of the maximum intensities of the EM field
is shown for each case (Fig. 2c). The strongest enhancement across all studied wavelengths is observed
for Ag at 630 nm, whereas Cu shows the weakest response, with almost negligible activity at 550 nm.
For Cu and Au, the localized enhanced fields are mainly observed in the pore gaps, whereas in
nanoporous Ag, there is a large number of “hot spots" strongly affecting the distribution of the field.
Similar results are also obtained at the ωFQFμ level (see Fig. 2b) for Ag and Au NPMs. In this case, the
electric field shows a different enhancement distribution depending on the metal and the laser frequency,
reflecting a different induced density on the surface. In particular, as for FEM results, the response of Ag
NPM is significantly larger than that of Au NPM, and also more hot-spots are observed. Nevertheless,
the trends of the maximum electric field enhancements do not correlate exactly with those computed by
using FEM (see Fig. 2c-d): in fact, in both Ag and Au NPMs, the maximum enhanced field is reported
at 780 nm. This can, however, be due to the diverse plasmonic response provided by the two models, and
by the small model structure considered in ωFQFμ calculations.
Understanding the optical response of homometallic nanoporous structures at different excitation
wavelengths is crucial for a deeper comprehension of the processes occurring across the layers in bi-
metallic configurations and for disentangling the individual contribution of each metal. Accordingly, new
plasmonic effects emerge in Au/Ag, Ag/Au, Au/Cu, Cu/Au, Ag/Cu, and Cu/Ag NPM systems as a result
10
of interlayer coupling and interactions between the constituent metals. Figure 3 depicts the EM field
enhancement distributions for Au/Ag and Ag/Au NPMs (a-b) and field enhancement maxima (c-d) at
different wavelengths computed by using FEM (a-c) and at the ωFQFμ level (b-d). At both levels, in the
Au/Ag bilayer, the silver layer (Figure 3a), which is placed on top of Au displays significantly reduced
EM field intensity at all studied wavelengths relative to the homometallic Ag layer (Figure 2a-b).
Nevertheless, the Ag layer exhibits stronger plasmonic activity, despite the less intense localization of
hot spots caused by the electrodynamical interaction between two porous layers consisting of two
different plasmonic materials. Interestingly, at the FEM level, illumination of the Au/Ag NPM at 780
nm produces a field enhancement comparable to that at 630 nm (Fig. 3c, top), which can likely be
associated with the interaction between plasmonic modes of silver and gold. This is also confirmed by
the field maps computed at the ωFQFμ level and by the maximum field enhancements, which are similar
at all considered frequencies (Fig. 3d, top). The same interaction induces a stronger enhancement for Au
layer (Figure 3c-d) at 550 nm than in the homometallic layer (Figure 2c-d) and rises with increasing
wavelength, a trend that is perfectly reproduced qualitatively by both methods.
11
Figure 3: (a-b) The enhanced EM field distribution of Au and Ag layers for Au/Ag (top) and Ag/Au
(bottom) NPMs calculated at the FEM (a) and ωFQFμ (b) levels of theory. (c-d) FEM c) and ωFQFμ
(d) field enhancement maxima for each structure on top and bottom sides.
In contrast, when the Au layer is on top of Ag (Figure 3a-b, bottom), the plasmonic response
remains weaker at 550 and 780 nm, but increases significantly at 630 nm because of interlayer coupling.
This remarkable result, reproduced by both numerical methods (see Fig. 3c-d, top), indicates that the
stacking sequence strongly affects the resonance conditions. As a result, at 630 nm, silver has a weaker
plasmonic response than in the homometallic Ag case. Between two systems, Au/Ag and Ag/Au
configurations at 550 nm, the enhancement is better in the first configuration when the Ag layer is on
top, consistent with the superior visible range plasmonic activity of Ag and its more efficient near-field
localization. The wavelength-dependent trends of the enhanced field maxima predicted by the two
numerical methods are qualitatively identical, validating the FEM-based approach for the subsequent
analyses.
12
Although Au and Cu do not show the strong plasmonic response of Ag (Figure 2), their coupling
still produces noteworthy effects. Figure 4 compares the EM field enhancement distributions and maxima
for Au/Cu and Cu/Au bilayer configurations. In the case of Cu as upper layer (Figure 4 (a-c)), low
plasmonic enhancement of Cu is compensated by coupling with underlying Au, especially at longer
wavelengths (780 nm). When Au is on top of Cu (Figure 4 (g-i)), instead, the enhancement is mainly
determined by the Au layer itself, while Cu acts as a substrate preventing additional EM field
enhancement, possibly because of significant losses in the Cu layer. Interestingly, the field EM
enhancement intensities of Cu are mostly the same in combination with Ag (Figure 5).
Figure 4: The enhanced EM field distribution of Au and Cu layers for(a-f) Au/Cu and (g-l) Cu/Au
NPMs calculated as a ratio of the electric field in the presence of nanoporous metals to the incident
field for different wavelengths.
13
In bimetallic structures Ag/Cu (Fig. 5 (a-f)) and Cu/Ag (Fig. 5(g-l)), the EM field enhancement
is dominated by silver, while copper mainly shows negligible activity, as can be seen from a comparison
between the EM field enhancement maxima of these configurations and those of the homometallic
structure composed of Ag (Fig. 2a, bottom). For Cu located on top of Ag, the lower Ag layer still supports
a strong EM field localization, though slightly reduced due to copper-induced losses. In contrast, when
Ag is deposited on top of Cu, the EM field enhancement is intensively concentrated in the upper Ag
layer, but the overall response, as mentioned earlier, is reduced compared to Ag alone due to the lossy
copper layer. These results confirm that the stacking order determines whether the Cu layer suppresses
the Ag layer’s resonances or favors the interlayer EM interactions.
These conclusions are also supported by ellipsometric investigations that allowed us to retrieve the
dielectric permittivities of the bi-metallic combinations (see Supporting Information). In particular,
homometallic bi-layer Ag/Ag exhibits a pronounced metallic response with a negative real permittivity
throughout the visible range, confirming its validity as a plasmonic material and efficient field
confinement. Ag/Au shows broader dispersion and larger optical losses, reflecting its intrinsically
damped plasmonic character. Conversely, Ag/Cu displays a largely suppressed plasmonic response, with
the real part of ε only weakly negative and significant absorption across the visible spectrum.
14
Figure 5: The enhanced EM field distribution of Ag and Cu layers for(a-f) Ag/Cu and (g-l) Cu/Ag
NPMs calculated as a ratio of the electric field in the presence of nanoporous metals to the incident
field for different wavelengths.
The numerical results provide a framework for understanding and analyzing data during SERS
experiments on the investigated bi-metallic NPMs. As predicted by our calculations, the efficiency of
EM field enhancement and distribution in bi-layer systems is determined not only by the type of metal,
but also by the order of the layers and the coupling efficiency between them. In homometallic structures,
the enhancement is defined by plasmonic properties of each metal; moreover, the Fermi level remains
the same. At the excitation energies explored here, in Au–Ag systems, silver dominates in terms of
15
enhancement intensity, whereas the enhancement of gold is explained by the coupling of plasmonic
modes. In the case of Ag–Cu, the interlayer coupling is weak, and the enhancement is determined mainly
by silver. For Au–Cu, gold dominates in intensity. Additionally, the observed enhancement is still higher
than in homometallic structures but remains significantly lower compared to configurations containing
Ag.
Starting from the numerical prediction of the EM field distributions in our samples, the
experimental performance in terms of SERS enhancement of the NPM films was examined under
conditions as close as possible to the simulated ones. In particular, three excitation wavelengths very
close to those analyzed in the simulations, i.e., 532 nm, 633 nm, and 785 nm, were employed
experimentally, and Rhodamine 6G (R6G) was used as the molecular probe. The average SERS spectra
of R6G, measured across the series of NPMs configurations with distinct bi-layer stacking sequences
(e.g., Au/Cu, Ag/Au), are compared in Fig. 6 (a-c) to evaluate their corresponding SERS enhancement
performances. Fig. 6 (d-f) presents the intensities of five characteristic R6G Raman bands at 620, 1280,
1360, 1509, and 1649 cm⁻¹ for the top three NPMs configurations, which were selected based on their
highest overall SERS intensities under identical measurement conditions, as marked by asterisks in
Figures 6 (a-c). To amid potential peak shifts, each intensity value was determined by taking the
maximum within a ±10 cm⁻¹ window around the nominal wavenumber. The SERS spectra and
corresponding peak intensities of R6G in Fig. 6 vary markedly across the various substrates and
excitation wavelengths. Specifically, under 532 nm excitation, the average spectra of 100 nM R6G show
that samples containing Ag as the bottom layer, combined with either Ag or Cu layers on top, yield the
highest signal enhancement. The Au NPMs rank next in terms of enhancement efficacy, whereas other
configurations of homometallic NPMs demonstrate remarkably inferior performances. Notably, the
homometallic Cu structure yields the weakest R6G signal. As illustrated by the histogram in Fig. 6(d),
the Ag/Cu NPM configuration delivers the most intense peaks for nearly all extracted characteristic bands
of R6G, except for that at 1360 cm-1. Under 633 nm excitation, with 1 μM R6G, it is evident that the Ag
NPM significantly outperforms the Au nanoporous structure. In contrast, the Cu/Ag and other
configurations show weaker enhancements, with the homometallic Cu structure yielding the lowest
overall enhancement. These observations are further supported by the extracted intensities of the five
16
characteristic Raman bands of R6G in Fig. 6(e), which clearly show that the homometallic Ag NPM
produces signals approximately 2.5 times stronger than those obtained from the homometallic Au
substrate. These results are consistent with the simulation data presented in Fig. 2.
Figure 6. SERS spectra of R6G on various substrates under excitation wavelengths of (a)
532nm, (b) 633nm and (c) 785nm; (d-f): Comparison of the five characteristic R6G band intensities for
the top three NPMs configurations identified in Figures 6 (a-c), respectively.
Under 785 nm excitation with 100 μM R6G, a different trend emerges: the Ag/Au structure (with Ag
beneath Au) exhibits the strongest R6G signal enhancement, despite its negligible performance at shorter
wavelengths. This result highlights the significant influence of the excitation wavelength on the SERS
efficiency across different materials. Conversely, under 785 nm laser excitation, the enhancement from
the homometallic Ag structure is even weaker than that of the Cu/Au structure, which appears
inconsistent with the simulation results in Fig. 2. We attribute this discrepancy to the potential oxidation
of the exposed Ag nanostructured top layer, as there was a significant time gap (more than one month)
17
between substrate fabrication and measurement. In contrast, the Ag/Au structure, protected by the
chemically inert Au top layer, likely maintained its plasmonic properties. To verify this hypothesis, a
supplementary experiment was conducted using NPM substrates within two weeks from fabrication (see
Supplementary Information), showing that the homometallic Ag structure demonstrates significantly
stronger enhancement than Ag/Au, which is consistent with the simulations in Fig. 2.
To further assess SERS performance, based on the above discussion, we evaluated the signal
uniformity, limit of detection (LOD), and concentration dependence of Ag/Cu, homometallic Ag (bi-
layer Ag/Ag), and Ag/Au NPMs under 532 nm, 633 nm, and 785 nm excitation, respectively. Results are
summarized in Figure 7. Fig. 7 (a-c) reports the averaged SERS spectra of R6G at varying concentrations
measured under consistent experimental conditions. The lowest detectable concentrations of R6G were
determined to be as low as 10 nM, 0.1 μM, and 10 μM for 532, 633, and 785 nm excitation wavelengths,
respectively. Concentration calibration curves are also shown in Fig. 7(d-f) left panels, derived from the
five characteristic SERS bands at 620, 1280, 1360, 1509, and 1649 cm⁻¹ in the SERS spectra shown in
Fig. 7 (a-c). We found that the intensities of five characteristic SERS bands of R6G exhibit an essentially
linear relationship with the logarithm of the corresponding R6G concentrations, which is consistent with
the results in the literature34. Fig. 7 (d-f) Right panels displayed the intensity distributions of these five
SERS bands across multiple randomly selected spectra on different nanoporous substrates. For each R6G
characteristic peak, the relative standard deviation (RSD) of the intensities across the spectra was
calculated. Subsequently, the average RSD (mean RSD) for the five peaks was determined, yielding
values of 9.98%, 7.76%, and 15.53% for Ag/Cu, Ag, and Ag/Au substrates, respectively. These mean
18
RSD values demonstrate excellent reproducibility and uniformity of the SERS response across the
different nanoporous substrates.
Figure 7: (a-c) Concentration-dependent SERS spectra of R6G on Ag/Cu, Ag/Ag and Ag/Au NPMs
under 532, 633 and 785nm, respectively. (d-f) Left panels: Calibration curves for the five
characteristic bands (620, 1280, 1360, 1509 and 1649 cm-1) derived from the corresponding SERS
spectra shown in (a-c); Right panels: Intensity distribution of these bands from randomly selected
spectra for 1μM, 10 μM and 1 mM R6G on the respective substrates.
The long-term stability of the fabricated porous substrates was evaluated by measuring the SERS
spectra of R6G on films stored for several months (Fig. S2, Supporting Information). Furthermore, the
mechanical durability was assessed by testing the SERS enhancement after ultrasonic washing. The
substrates maintained significant SERS activity even after 40 minutes of ultrasonication, demonstrating
robust performance under harsh conditions (Fig. S3, Supporting Information). This combination of long-
19
term and mechanical stability makes these substrates highly suitable for practical applications, such as
photocatalytic or biochemical testing.
Figure 8: (a-c) Normalized SERS spectra of ADAMTS3 at various concentrations on Ag/Cu, Ag/Ag,
and Ag/Au NPMs under 532, 633, and 785nm excitation, respectively. (d-f) Fit-integrated areas under
the corresponding characteristics bands marked with dot lines in (a-c).
The translation of SERS substrates from conceptual fabrication to practical application is a critical
step in biosensing, particularly for the early screening and diagnosis of cancers and related biomarkers35–
37. The ability to detect low-concentration biomarkers with high specificity and sensitivity provides a
crucial, time-sensitive advantage for early diagnosis and subsequent therapeutic intervention. In a recent
work, we reported a DNA origami-based SERS platform for the sensitive detection of ADAMTS3
protein, a promising biomarker associated with hepatocellular carcinoma (HCC)38. Here, we build upon
our previous findings by deploying the novel bi-layer metal nanoporous films as a superior SERS
20
substrate to re-investigate the detection of the same ADAMTS3 protein. As discussed above, these
substrates offer large, uniform active area and long-term stability, therefore they can be interesting for
applied SERS biosensing.
Figure 8 presents the analytical performance of the bi-layer NPM SERS substrates for the
detection of ADAMTS3. Fig. 8 (a-c) displays the normalized mean SERS spectra of ADAMTS3 protein
at various concentrations, acquired using 532, 633, and 785 nm laser excitation wavelengths,
respectively. Each set of spectra corresponds to the optimally designed bi-layer configuration (e.g.
Ag/Cu, homometallic Ag/Ag, and Ag/Au) for the respective laser wavelength, as detailed in the previous
sections. The fitted areas under the characteristic SERS bands of ADAMTS3 protein (marked by dotted
lines in a-c) at 1098, 1258, and 1328 cm-1 were plotted in Fig. 8(d-f) as a function of logarithmic
concentration with error bars. The peak fitting was performed using a Gaussian function, consistent with
our previous methodology39.
As shown in the figure, the characteristic SERS peaks of the ADAMTS3 protein remain clearly
distinguishable at a concentration of 10⁻⁹ M using the three optimal enhancement substrates with the
magnification of Ag/Cu, Ag/Ag, and Ag/Au under 532, 633, and 785 nm excitation, respectively.
Notably, with 633 nm excitation, these characteristic peaks are still detectable even at a lower
concentration of 10⁻¹⁰ M, representing an enhancement factor comparable to that achieved in our recent
work using an optimized Au-dimer/DNA origami as a SERS platform38. Furthermore, the fitted areas of
the characteristic peaks at 1098, 1258, and 1328 cm⁻¹ were plotted against the logarithmic concentration.
Under 633 nm excitation, the data points exhibit smaller error bars compared to those at 532 and 785 nm,
and follow an approximately linear relationship. This indicates that the peak areas increase nearly linearly
with increasing protein concentration, which is consistent with our previous findings38.
In contrast, this linear relationship is not always observed under 532 nm and 785 nm excitation.
The fitted areas at 1098 and 1258cm-1 exhibit an approximate linear trend ranging from 10-7 to 10-9 M,
but with a notable deviation occurring at 10-6 M at 785nm excitation, and all these fitted areas are
associated with relatively large error bars. Under 532nm laser excitation, the relationship between the
fitted area and logarithmic concentration deviates entirely from linearity for all three characteristic peaks,
accompanied by significantly larger error bars compared to other wavelengths. This pronounced
nonlinearity and data scatter can be attributed to the wavelength-specific electronic interaction between
21
the protein and the corresponding bi-metallic substrate. As highlighted in the introduction section, the
adsorption behavior of molecules with polar or partially charged groups—such as those present in the
ADAMTS3 protein—is highly sensitive to the electronic state of the alloy surface. We hypothesize that
under 532 nm excitation, the specific bi-layer configuration (e.g., Ag/Cu) exhibits an electronic structure
that leads to non-uniform or saturable adsorption of protein molecules across the concentration series.
This results in inconsistent SERS responses and larger variability in measured intensities, thereby
disrupting the linear correlation and increasing observational uncertainties.
Conclusions
In summary, we conducted a comprehensive investigation of novel plasmonic platforms composed of
stacked layers of noble plasmonic metals (NPMs). By preparing multiple configurations (Au/Ag, Ag/Au,
Au/Cu, Cu/Au, Ag/Cu, and Cu/Ag), we were able to systematically explore how the choice of metals
and their stacking order influence metal–metal interactions. Extensive numerical simulations, carried out
using both finite element methods (FEM, COMSOL) and fully atomistic models, provided valuable
insights into the optical properties of these nanostructures under excitation at different energies within
the visible range. These models proved effective in describing the behavior of porous systems,
highlighting their potential not only for enhanced spectroscopy but also for a variety of additional
applications. To experimentally validate the platforms, we performed SERS measurements on both a
standard probe molecule (R6G) and a biologically relevant target (the ADAMTS3 biomarker). These
experiments demonstrated the strong performance of the bilayer systems as SERS substrates and offered
initial evidence of how bimetallic interactions may facilitate interlayer interactions at the metal–metal
interface. Altogether, this work provides a promising first step toward the study of complex multi-
materials plasmonic platforms for SERS, combining the advantages of reproducibility, simplicity, and
low fabrication cost.
22
Author contribution
YZ performed the Raman experiments and analyzed the data, AS and TG performed the numerical
simulations, AS, AA, LB, NM and IM supported the numerical simulations and results discussion, CW
and HJ supported the experimental measurements, AD, GL, SW and RK supported the samples
fabrication, ZJ, ZZ, and DG supervised the work. YZ and AS equally contributed to the present work.
All authors contributed to the writing of the manuscript.
Supporting Information
Supporting Information is available free of charge at: ……..
Acknowledgment
The authors thank the National Natural Science Foundation of China (No.22202167), the National Key
Research and Development Project of China (No. 2023YFF0613603), and the HORIZON-MSCADN-
2022: DYNAMO, grant Agreement 101072818. NM and BJ acknowledge the ‘Excellence by Choice’
Programme at Umeà University funded by Kempestielserna (grant no. JCK-2130.3). The authors thank
the Clean Room Facility of IIT.
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1. Additional SERS Experiments
26
- R6G test under 785nm excitation with new set of substrates
Fig. S1. SERS spectra of R6G at different concentrations obtained from NPMs with varying
configuration (Ag/Cu, Ag/Au and Ag/Ag) under 785nm excitation.
The SERS performance of different bi-layer nanoporous substrates (Ag/Ag, Ag/Au,
Ag/Cu) was re-evaluated using R6G under 785 nm excitation, as shown in Fig. S1. A
direct comparison reveals that the Ag/Ag substrate possesses the highest enhancement
capability, which is different from the previous results in Figure 6(c). It successfully
detected R6G at a concentration of 0.1 μM, whereas under identical conditions, the Ag/Au
and Ag/Cu substrates only yielded measurable signals at a higher concentration of 1 μM.
27
This result clearly indicates that the Ag/Ag bi-layer structure offers superior SERS
sensitivity over the other two configurations at 785nm excitation.
- Long-term stability of the substrates
After initial characterization, we re-evaluated the SERS performance of 100nM R6G
under 532 nm excitation on various bi-layer nanoporous substrates following a two-month
storage period under ambient conditions. The four substrates exhibiting the highest
enhancement were selected for further analysis, as shown in Fig. S2. Both the bi-layer Ag
and bi-layer Au substrates maintained the strongest SERS enhancement after storage. In
contrast, the Au/Cu and Cu/Au configurations also demonstrated considerable
enhancement following the two-month period. Notably, the Ag/Cu substrate, which
initially ranked third in enhancement intensity in Figure 6(a) and Figure 6(d), showed a
significant reduction in SERS performance after storage. We attribute these observations
to the protective role of gold against oxidation, which helps preserve the nano-structural
integrity of the underlying porous film. In comparison, silver and copper are more
susceptible to oxidation over time, leading to diminished SERS activity in Ag/Cu bilayers
after prolonged exposure to air.
28
Fig. S2. Normalized SERS spectra of 100nM R6G acquired from the selected bilayer nanoporous
substrates, before (short dot line) and after (solid line) two months storage period. Spectra were
collected under 532nm excitation with all measurements using a 10% laser filter except for the post-
storage Au/Cu and Cu/Au samples, for which a 100% laser power was employed. All spectra are
normalized to [0,1] range for comparative purposes.
29
Fig. S3 Evolution of SERS performance after storage. Mean SERS spectra of 0.1mM R6G on bi-layer
nanoporous substrates with different configurations under 633nm excitation (5% laser filter), acquired
after a two-month storage period.
30
Fig. S4 Evolution of SERS performance after storage under 785nm laser excitation. a) Mean SERS
spectra of 0.1mM R6G obtained initially (dotted line) and after storage (solid line); b) Corresponding
normalized post-storage spectra.
Having established the variation in initial SERS enhancement across substrates in Figure
6(b), we proceeded to evaluate its subsequent evolution of the same substrates following
a two-month storage period under 633nm excitation. As shown in Fig. S3, Ag/Au bilayer
emerges as the most effective substrate, exhibiting the highest SERS intensity, followed
by Au/Ag configuration. In contrast, the initially superior configurations (Ag/Ag, Au/Au,
and Cu/Ag, marked by asterisks) displayed a considerable decline after storage. This
marked contrast underscores the significant role of structural stability and oxidation
resistance in maintaining SERS activity over time.
The results obtained under 785 nm excitation differed from those under 532 nm and 633
nm, as shown in Fig. S4. For comparison, we evaluated the top four enhanced SERS
configurations—Ag/Au, Cu/Au, Au/Ag, and Au/Cu—before and after a two-month
storage period. As illustrated in Fig. S4(a), the SERS intensity of R6G decreased in all
configurations between the initial and post-storage measurements. However,
configurations with Ag/Au and Cu/Au consistently exhibited stronger enhancement than
the others, both initially and after storage. In Fig. S4(b), after normalization of the post-
storage spectra, it can be observed that Ag/Au and Cu/Au configurations maintained
superior enhancement performance before and after storage. In contrast, the double-layer
silver porous structure, which initially ranked third in enhancement (as shown in the main
text Figure 6(c)), exhibited weaker signals than both Au/Ag and Au/Cu after two months
of storage.
31
- Influence of washing to the substrates
Fig. S5 Evaluation of the washing effect on substrate stability. Mean SERS spectra of the background
signal and 10⁻⁶ M ADAMTS3 were acquired under 633 nm excitation after 40 minutes of sequential
ultrasonication in ethanol and ultrapure water. The results for both Ag/Au and Ag/Ag porous substrates
are shown, with error bars representing the standard deviation across multiple measurements.
2. Additional data calculations
32
Figure
S6.
Ag/Ag
(a),
Ag/Au
(b),
Au/Ag
(c),
Au/Au
(d)
NPM
structures exploited in ωFQFμ calculations.
33
Figure S7. The enhanced EM field distribution of Au and Ag bottom layers for Au/Ag (top) and
Ag/Au (bottom) NPMs calculated at the FEM (a) and ωFQFμ (b) levels of theory.
3. Ellipsometric characterization
Figure S8: Real (black) and imaginary (red) parts of the dielectric permittivity for Ag/Ag (a), Ag/Au
(d), and Ag/Cu (g) nanoporous bilayers, obtained by fitting ellipsometric data using a combination of
Lorentzian and Gaussian oscillators. Corresponding experimental (dots) and fitted (solid lines)
ellipsometric angles Ψ and Δ are shown in panels (b-c), (e-f), and (h-i), respectively. The oscillator
parameters are reported in Table S1.
The dielectric functions of the three nanoporous bilayers, Ag/Ag, Ag/Au, and Ag/Cu,
shown in Figure S8.a,d,g, respectively, were modeled using a set of Lorentzian and
34
Gaussian oscillators to capture the key features of their optical response. For Ag/Ag
(Figure S8a), the response is dominated by a single Lorentzian centered at 1.297 eV,
accounting for the low-energy plasmonic resonance typical of nanostructured silver. A
Gaussian oscillator with negative amplitude was included at 2.84 eV to compensate for
the overestimated ε’’ background induced by the Lorentzian tail, ensuring better
agreement with the experimental data in the near-UV region. In the case of Ag/Au (Figure
S8d), a broader Lorentzian (En0 = 1.20 eV, Br = 1.44 eV) and two Gaussians were
required to account for the increased damping and the presence of interband transitions
characteristic of gold. The higher complexity of this system reflects the hybridized
electronic structure of the alloy. Finally, the Ag/Cu (Figure S8g) sample shows a markedly
suppressed optical response, with three weak oscillators spread across the visible range.
The low oscillator strengths and absence of sharp features are consistent with stronger
damping and reduced plasmonic activity due to the presence of copper. In all cases, the
fitted dielectric functions reproduce the experimental ellipsometric parameters Ψ and Δ,
shown in Figures S8b,c, S8e,f, and S8h,i with high fidelity across the measured spectral
range.
35
Table S1: Parameters of the Lorentzian and Gaussian oscillators used to model the dielectric functions
of the Ag/Ag, Ag/Au, and Ag/Cu bilayers. En0: central energy (eV), Br: broadening (eV), A0: amplitude
(dimensionless).
Sample
Oscillator Type
A0
En0 (eV)
Br (eV)
Ag/Ag
Lorentzian
10.01
1.297
0.200
Gaussian
–1.40
2.840
1.023
Ag/Au
Lorentzian
13.17
1.200
1.440
Gaussian
1.07
2.880
0.644
Gaussian
2.98
4.580
1.745
Ag/Cu
Lorentzian
0.115
3.210
0.960
Gaussian
0.021
1.970
0.440
|
1 Layered Bimetal Nanoporous Platforms for SERS Sensing Yanqiu Zou1±, Anastasiia Sapunova2,3±, Tommaso Giovannini4, Chen Wang5, Huaizhou Jin6, Vincenzo Caligiuri7, Andrea Schirato8,10, Luca Bursi9, Alessandro Alabastri10, Shukun Weng2,3, Ali Douaki2,11, German Lanzavecchia2,11, Ivan Marri11, Roman Krahne2, Nicolò Maccaferri12,13, Zhenrong Zheng1*, Shangzhong Jin5* and Denis Garoli*2,5,11 1. State Key Laboratory of Modern Optical Instrumentation, 310027, China. 2. Istituto Italiano di Tecnologia, Via Morego 30, 16163 Genova, Italy. 3. Università degli Studi di Milano-Bicocca, Piazza dell'Ateneo Nuovo 1, 20126, Milano, Italy 4. 1, I00133, Rome, Italy 5. 310018, China 6. Key Laboratory of Quantum Precision Measurement, . 7. Dipartimento di Fisica, Università della Calabria, via P. Bucci 33b, 87036 Rende, CS, Italy 8. Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy 9. (FIM), Università degli Studi di Modena e Reggio Emilia, and Nanoscience Institute, CNR-NANO, via Campi 213/a, 41125, Modena, Italy 10. 77005, United States 11. Università degli Studi di Modena e Reggio Emilia, Via Amendola 2, 42122, Reggio Emilia (Italy) 12. å University, Linnaeus väg 24, 901 87 Umeå, Sweden 13. Umeå Centre for Microbial Research, Umeå University, 901 87 Umeå, Sweden 2 ±Equally contribution * Corresponding author: Denis Garoli, ; Zhenrong Zheng, ; Shangzhong Jin, Keywords: plasmonics, nanoporous materials, Raman, gold, silver, copper Abstract Nanoporous metals are extensively investigated as platforms for applications in plasmonics. They present high surface areas and strong local electric fields that can be tuned at different energies, playing with the choice of the metals and the morphology of the porous layers. Until recently, research in the field of plasmonics has primarily focused on porous metals composed of a single element, with limited attention given to the impact of alloy composition. The investigation of bi-metallic systems has only just begun to emerge in the literature. In particular, combining two or more different plasmonic metals, it could be possible to explore the interactions between two metals excited at specific energies. This involves plasmonic coupling, electron transfer, band hybridization at the interface, electromagnetic field interactions, and possibly thermal and electronic energy transfer depending on separation, size, and materials involved. The analysis of bi-metal systems can also be interesting in biomolecule detection, such as in the case of Surface Enhanced Raman Scattering (SERS). Here we report, for the first time, a detailed study (comprising morphological analyses, numerical modelling, and optical spectroscopies) on bi-metal nanoporous platforms prepared with a dry-synthesis method enabling the easy and controllable fabrication of bilayers combining different metals such as Au, Ag, and Cu. Materials and Methods: Samples fabrication The samples preparation is based on the original methods proposed by Kwon et al.1. In brief, poly (methyl methacrylate) (PMMA) was spin-coated on a Si substrate at 4000 rpm for 2 min. Each metal (>99.99 % 3 purity) was subsequently deposited by electron-beam evaporation onto the PMMA thin film at room temperature with an oblique incidence angle of 80◦, a deposition rate of 0.1 nm/s, and a target thickness of about 12 nm. The deposited Au film was plasma treated in O2 at a power of 200W until the entire PMMA layer was removed. In the case of Ag and Cu films, in order to avoid fast oxidation, the plasma treatment was performed in N2 at 200W until the PMMA layer was completely removed (as previously demonstrated, this ensures that the oxidation is kept to a very low level2). The bi-layer structure was obtained by repeating the preparation using the first porous layer as substrate for the deposition of the second porous layer on top. Spectroscopic measurements Raman spectroscopy measurements were performed using a HORIBA LabRAM HR Evolution Raman spectrometer (HORIBA Jobin Yvon, Kyoto, Japan) with a 50X long-focal-length objective (NA = 0.75) , a 100μm-hole aperture, and a 600 gr/mm grating. SERS spectra of Rhodamine 6G (R6G) were acquired under different conditions to characterize system performance: 100 nM R6G with laser power attenuated to 10% at 532 nm excitation; 1 μM R6G with 5% laser power at 633 nm; and 0.1 mM R6G with full laser power (no attenuation) at 785 nm. For each substrate, a Raman map was collected over a 20 × 20 μm2 area with a step size of 5 μm, resulting in 25 individual spectra. The final spectrum for each substrate was generated by averaging these 25 spectra after pre-processing. Spectra of the protein ADAMTS3 were obtained by averaging five randomly collected spectra, using a 5% ND filter and 60s acquisition time for both 532nm and 633nm excitations. For 785nm excitation, a 100% ND filter (no attenuation) and 120s acquisition time were used. Data Analysis The raw spectra were baseline-corrected and smoothed using HORIBA LabSpec6 software. Subsequently, all data averaging, normalization, spectral deconvolution, peak fitting, and graphical plotting were performed using OriginPro 2025b. Numerical modelling 4 To explore the plasmonic properties of porous NPMs, we performed a numerical investigation of the electromagnetic response using a finite element method (FEM) commercial software, employing COMSOL Multiphysics. In particular, following a procedure reported in detail in our recently reported works1,2 nanometric pores and irregularities of NPMs have been numerically reproduced from scanning electron microscopy (SEM) images of the experimental samples. However, continuum modeling, such as FEM, might fail in describing the response properties of bimetallic nanostructured materials3-5. Therefore, to validate the FEM procedure, the optical response of NPMs has also been calculated using a fully atomistic approach, named frequency-dependent fluctuating charges and dipoles (ωFQFμ)5-7. This method provides a reliable description of noble metal nanoparticles, comparable to ab initio5 but at significantly lower computational cost, and can thus handle systems composed of thousands of atoms811. Remarkably, it can also be applied to bimetallic systems and, in the specific case of Au/Ag nanoparticles, yields results in almost perfect agreement with experiment12. For cross-validation of the COMSOL results, we have adopted an idealized picture in which the same porous structure is generated and then considered either as Ag or Au, which are characterized by the same lattice constant (4.08 Å). Such structures were created by introducing randomly generated nanometric pores, thereby generating realistic surface roughness and porosity. In particular, we modeled a 7 × 7 × 4 (nm3) structure, composed of 6584 atoms, a size unfeasible for ab initio atomistic models (see SI for a graphical representation). The same nanoporous geometry was then used to simulate Ag/Ag, Au/Au, and mixed Au/Ag bilayers, allowing for a direct comparison of their optical response and a detailed analysis of the different compositional effects. All ωFQFμ calculations are performed using the Ag/Au parameters reported in Ref. 12, with the open-source plasmonX software6. Dielectric Permittivity and Ellipsometry. Spectroscopic ellipsometry was employed to determine the dielectric permittivity of nanoporous Ag/Ag, Ag/Au, and Ag/Cu bilayers. The measurements provided both the real and imaginary components of the dielectric function over the visible range. The experimental data were fitted using a model composed of Lorentzian and Gaussian oscillators to capture the main features of each alloy's optical response. The resulting dielectric permittivities accurately reproduce the experimental ellipsometric parameters Ψ and 5 Δ. Complete fitting details and oscillator parameters are reported in the Supporting Information (Table S1). Introduction Over the last decades, research on nanoporous metals (NPMs) has grown steadily, driven by their broad range of applications in sensing13-15, catalysys16,17, photonics18,19 and biomedicine20. The main characteristic of NPMs is their high surface-to-volume ratio that, combined with unique chemical reactivity and mass transport properties, makes this family of nanostructured materials highly interesting for advanced research. NPMs can be prepared following different strategies such as template-assisted physical vapor deposition, electrochemical processes, and chemical dealloying18,21. The latter is probably the most used method to prepare NPMs through selective dissolution of the more chemically active component from precursor alloys. Dealloying methods have been applied to prepare NPMs as gold13,18,22, copper23,24, silver25,26, and aluminum27,28. A specific aspect of dealloying is the presence of residual less noble metal at the end of the process. This effect is difficult to avoid and leads to resulting nanoporous structures with varying alloy compositions29,30. Interestingly, this residual "contamination" in the final porous film, long considered as a limitation in the preparation of pure porous structures, has recently been shown to enhance plasmonic properties of nanoporous gold containing residual amount of the less noble metal due to modulation of its permittivity by the alloying components.31-33 The combination of two or more metals in alloys is a strategy to provide a way to tune the electronic state through charge transfer between the metals, making it possible to alter the chemical interaction between molecules and the surface of the alloy. In particular, the adsorption behavior of molecules having either high electronegativity (e.g., oxygen and halogens), strong polarity (e.g., carbonyl and nitro groups), partial charge (e.g., carboxylate and hydride-like hydrogen), or high electron density (e.g., alkenes and arenes) is affected by the alloy composition. For instance, palladium-silver alloy promoted the adsorption of negatively charged molecules (carboxylate intermediate) due to the high electronegativity of silver. The alteration of the electronic state by alloying on the molecular detection capability has been recently studied by La et al.33 In particular, a SERS enhancement of up to 6 orders of magnitude for nanoporous gold containing silver was observed and attributed to the narrowing of the electronic structure and its 6 alignment close to the Fermi level. These results suggest that the electronic state modulation of the metal structure can affect molecule-surface interactions. Here, we decided to explore bi-metallic nanoporous systems from a different point of view. Instead of using chemical dealloying, we used a recently reported dry-synthesis method to prepare highly pure layers of plasmonic metals such as Au, Ag, and Cu1-3. The peculiarity of this approach lies in the possibility to produce stacked nanoporous layers, either composed of the same metal or of different metals deposited sequentially on top of one another, as demonstrated previously in refs. 2,3 At the best of our knowledge, this is the first example of perfectly stacked nanoporous metal layers, and it represents a valuable platform to investigate how the use of two different metals can impact the performance in terms of SERS enhancements at different wavelengths across the visible spectral range. Experimental results and numerical simulations will be presented to shed light on the behavior of these novel plasmonic nanoporous platforms. Results and discussion Figure 1 illustrates the planar morphologies of the samples used for these experiments. While the first row (Fig. 1(a-c)) reports the single NPM films, respectively Au, Ag, and Cu, the other micrographs show the obtained morphologies for the bi-metallic layered films. As can be seen, the single metal films present a nanoporous structure with fully connected ligaments of diameters of a few tens of nm. In the case of bi-metallic layered films, the morphologies vary significantly among the samples. In all the bimetallic samples (Fig. 1(d-i)), it is possible to observe the two layers of metals stacked one on the other, but not mixed, with a gap between the 2 metallic nanostructures that can only be due to a potential small oxide layer surrounding the Ag and Cu nanoporous structures. 7 Figure 1: (a-e) SEM micrographs of the investigated samples; porous layers of (a) Au; (b) Ag; (c) Cu; (d) Au/Ag; (e) Ag/Au; (f) Au/Cu; (g) Cu/Au; (h) Ag/Cu; (i) Cu/Ag. Following the same approach used in our recent works3,4, we performed numerical studies of the electromagnetic response of the porous bi-layer system. This enables us to obtain an overview of the electromagnetic (EM) field distributions at the wavelengths of excitation used in the experiments. The nanometric structural features of the layers were reproduced directly from the SEM images of the experimental samples in the finite element method (FEM)-based model, developed using COMSOL Multiphysics. Specifically, the top-view SEM micrographs of the Au, Ag, and Cu films used in the experiments were imported into the FEM solver and extruded along the thickness direction of the layer to build the 3D numerical geometry. As previously shown3, this method enabled us to numerically 8 account for the actual discontinuities and profile of the materials down to the nanoscale. Figures 2, 3, 4, and 5 report an estimation of the EM field enhancement distribution along the surfaces of the two (the top and bottom) porous layers in the different cases, taking into consideration three wavelengths of excitation (550, 630, and 780 nm). Figure 2: (a,b) EM field enhancement distributions for (a) homometallic nanoporous bilayer structures composed of Cu (top), Au (middle), and Ag (bottom), calculated with FEM at excitation wavelengths of 550 nm, 630 nm, and 780 nm (from left to right); (b) homometallic nanoporous bilayers composed of Au (top) and Ag (bottom), obtained using the ωFQFμ approach at the same excitation wavelengths. All 9 maps are normalized to the same color scale to highlight spatial regions of strongest field enhancement localization. (c,d) Maximum field enhancement values extracted from FEM (c) and ωFQFμ (d) simulations for each structure. Figure 2 shows the EM field distributions at wavelengths of 550, 630, and 780 nm for bi-layer nanoporous structures made of a single metal: Cu (a, top), Au (a, middle), and Ag (a, bottom). For the specific case of Au and Ag, the EM distributions are also computed using the ωFQFμ method, for comparison (Fig. 2b). In all cases, the EM enhancement profiles were evaluated on the surface of the bilayer structure. The color scale indicates a normalized intensity of the EM field, calculated as the ratio of the electric field in the presence of nanoporous metals to the incident field. All field maps are normalized to the same values, pointing out the regions where the EM field is highly localized. Nevertheless, at the FEM level (Fig. 2a), each homometallic sample is characterized by its own geometry and plasmonic properties; therefore, the diagram of the maximum intensities of the EM field is shown for each case (Fig. 2c). The strongest enhancement across all studied wavelengths is observed for Ag at 630 nm, whereas Cu shows the weakest response, with almost negligible activity at 550 nm. For Cu and Au, the localized enhanced fields are mainly observed in the pore gaps, whereas in nanoporous Ag, there is a large number of "hot spots" strongly affecting the distribution of the field. Similar results are also obtained at the ωFQFμ level (see Fig. 2b) for Ag and Au NPMs. In this case, the electric field shows a different enhancement distribution depending on the metal and the laser frequency, reflecting a different induced density on the surface. In particular, as for FEM results, the response of Ag NPM is significantly larger than that of Au NPM, and also more hot-spots are observed. Nevertheless, the trends of the maximum electric field enhancements do not correlate exactly with those computed by using FEM (see Fig. 2c-d): in fact, in both Ag and Au NPMs, the maximum enhanced field is reported at 780 nm. This can, however, be due to the diverse plasmonic response provided by the two models, and by the small model structure considered in ωFQFμ calculations. Understanding the optical response of homometallic nanoporous structures at different excitation wavelengths is crucial for a deeper comprehension of the processes occurring across the layers in bimetallic configurations and for disentangling the individual contribution of each metal. Accordingly, new plasmonic effects emerge in Au/Ag, Ag/Au, Au/Cu, Cu/Au, Ag/Cu, and Cu/Ag NPM systems as a result 10 of interlayer coupling and interactions between the constituent metals. Figure 3 depicts the EM field enhancement distributions for Au/Ag and Ag/Au NPMs (a-b) and field enhancement maxima (c-d) at different wavelengths computed by using FEM (a-c) and at the ωFQFμ level (b-d). At both levels, in the Au/Ag bilayer, the silver layer (Figure 3a), which is placed on top of Au displays significantly reduced EM field intensity at all studied wavelengths relative to the homometallic Ag layer (Figure 2a-b). Nevertheless, the Ag layer exhibits stronger plasmonic activity, despite the less intense localization of hot spots caused by the electrodynamical interaction between two porous layers consisting of two different plasmonic materials. Interestingly, at the FEM level, illumination of the Au/Ag NPM at 780 nm produces a field enhancement comparable to that at 630 nm (Fig. 3c, top), which can likely be associated with the interaction between plasmonic modes of silver and gold. This is also confirmed by the field maps computed at the ωFQFμ level and by the maximum field enhancements, which are similar at all considered frequencies (Fig. 3d, top). The same interaction induces a stronger enhancement for Au layer (Figure 3c-d) at 550 nm than in the homometallic layer (Figure 2c-d) and rises with increasing wavelength, a trend that is perfectly reproduced qualitatively by both methods. 11 Figure 3: (a-b) The enhanced EM field distribution of Au and Ag layers for Au/Ag (top) and Ag/Au (bottom) NPMs calculated at the FEM (a) and ωFQFμ (b) levels of theory. (c-d) FEM c) and ωFQFμ (d) field enhancement maxima for each structure on top and bottom sides. In contrast, when the Au layer is on top of Ag (Figure 3a-b, bottom), the plasmonic response remains weaker at 550 and 780 nm, but increases significantly at 630 nm because of interlayer coupling. This remarkable result, reproduced by both numerical methods (see Fig. 3c-d, top), indicates that the stacking sequence strongly affects the resonance conditions. As a result, at 630 nm, silver has a weaker plasmonic response than in the homometallic Ag case. Between two systems, Au/Ag and Ag/Au configurations at 550 nm, the enhancement is better in the first configuration when the Ag layer is on top, consistent with the superior visible range plasmonic activity of Ag and its more efficient near-field localization. The wavelength-dependent trends of the enhanced field maxima predicted by the two numerical methods are qualitatively identical, validating the FEM-based approach for the subsequent analyses. 12 Although Au and Cu do not show the strong plasmonic response of Ag (Figure 2), their coupling still produces noteworthy effects. Figure 4 compares the EM field enhancement distributions and maxima for Au/Cu and Cu/Au bilayer configurations. In the case of Cu as upper layer (Figure 4 (a-c)), low plasmonic enhancement of Cu is compensated by coupling with underlying Au, especially at longer wavelengths (780 nm). When Au is on top of Cu (Figure 4 (g-i)), instead, the enhancement is mainly determined by the Au layer itself, while Cu acts as a substrate preventing additional EM field enhancement, possibly because of significant losses in the Cu layer. Interestingly, the field EM enhancement intensities of Cu are mostly the same in combination with Ag (Figure 5). Figure 4: The enhanced EM field distribution of Au and Cu layers for(a-f) Au/Cu and (g-l) Cu/Au NPMs calculated as a ratio of the electric field in the presence of nanoporous metals to the incident field for different wavelengths. 13 In bimetallic structures Ag/Cu (Fig. 5 (a-f)) and Cu/Ag (Fig. 5(g-l)), the EM field enhancement is dominated by silver, while copper mainly shows negligible activity, as can be seen from a comparison between the EM field enhancement maxima of these configurations and those of the homometallic structure composed of Ag (Fig. 2a, bottom). For Cu located on top of Ag, the lower Ag layer still supports a strong EM field localization, though slightly reduced due to copper-induced losses. In contrast, when Ag is deposited on top of Cu, the EM field enhancement is intensively concentrated in the upper Ag layer, but the overall response, as mentioned earlier, is reduced compared to Ag alone due to the lossy copper layer. These results confirm that the stacking order determines whether the Cu layer suppresses the Ag layer's resonances or favors the interlayer EM interactions. These conclusions are also supported by ellipsometric investigations that allowed us to retrieve the dielectric permittivities of the bi-metallic combinations (see Supporting Information). In particular, homometallic bi-layer Ag/Ag exhibits a pronounced metallic response with a negative real permittivity throughout the visible range, confirming its validity as a plasmonic material and efficient field confinement. Ag/Au shows broader dispersion and larger optical losses, reflecting its intrinsically damped plasmonic character. Conversely, Ag/Cu displays a largely suppressed plasmonic response, with the real part of ε only weakly negative and significant absorption across the visible spectrum. 14 Figure 5: The enhanced EM field distribution of Ag and Cu layers for(a-f) Ag/Cu and (g-l) Cu/Ag NPMs calculated as a ratio of the electric field in the presence of nanoporous metals to the incident field for different wavelengths. The numerical results provide a framework for understanding and analyzing data during SERS experiments on the investigated bi-metallic NPMs. As predicted by our calculations, the efficiency of EM field enhancement and distribution in bi-layer systems is determined not only by the type of metal, but also by the order of the layers and the coupling efficiency between them. In homometallic structures, the enhancement is defined by plasmonic properties of each metal; moreover, the Fermi level remains the same. At the excitation energies explored here, in Au-Ag systems, silver dominates in terms of 15 enhancement intensity, whereas the enhancement of gold is explained by the coupling of plasmonic modes. In the case of Ag-Cu, the interlayer coupling is weak, and the enhancement is determined mainly by silver. For Au-Cu, gold dominates in intensity. Additionally, the observed enhancement is still higher than in homometallic structures but remains significantly lower compared to configurations containing Ag. Starting from the numerical prediction of the EM field distributions in our samples, the experimental performance in terms of SERS enhancement of the NPM films was examined under conditions as close as possible to the simulated ones. In particular, three excitation wavelengths very close to those analyzed in the simulations, i.e., 532 nm, 633 nm, and 785 nm, were employed experimentally, and Rhodamine 6G (R6G) was used as the molecular probe. The average SERS spectra of R6G, measured across the series of NPMs configurations with distinct bi-layer stacking sequences (e.g., Au/Cu, Ag/Au), are compared in Fig. 6 (a-c) to evaluate their corresponding SERS enhancement performances. Fig. 6 (d-f) presents the intensities of five characteristic R6G Raman bands at 620, 1280, 1360, 1509, and 1649 cm-1 for the top three NPMs configurations, which were selected based on their highest overall SERS intensities under identical measurement conditions, as marked by asterisks in Figures 6 (a-c). To amid potential peak shifts, each intensity value was determined by taking the maximum within a ±10 cm-1 window around the nominal wavenumber. The SERS spectra and corresponding peak intensities of R6G in Fig. 6 vary markedly across the various substrates and excitation wavelengths. Specifically, under 532 nm excitation, the average spectra of 100 nM R6G show that samples containing Ag as the bottom layer, combined with either Ag or Cu layers on top, yield the highest signal enhancement. The Au NPMs rank next in terms of enhancement efficacy, whereas other configurations of homometallic NPMs demonstrate remarkably inferior performances. Notably, the homometallic Cu structure yields the weakest R6G signal. As illustrated by the histogram in Fig. 6(d), the Ag/Cu NPM configuration delivers the most intense peaks for nearly all extracted characteristic bands of R6G, except for that at 1360 cm-1. Under 633 nm excitation, with 1 μM R6G, it is evident that the Ag NPM significantly outperforms the Au nanoporous structure. In contrast, the Cu/Ag and other configurations show weaker enhancements, with the homometallic Cu structure yielding the lowest overall enhancement. These observations are further supported by the extracted intensities of the five 16 characteristic Raman bands of R6G in Fig. 6(e), which clearly show that the homometallic Ag NPM produces signals approximately 2.5 times stronger than those obtained from the homometallic Au substrate. These results are consistent with the simulation data presented in Fig. 2. Figure 6. SERS spectra of R6G on various substrates under excitation wavelengths of (a) 532nm, (b) 633nm and (c) 785nm; (d-f): Comparison of the five characteristic R6G band intensities for the top three NPMs configurations identified in Figures 6 (a-c), respectively. Under 785 nm excitation with 100 μM R6G, a different trend emerges: the Ag/Au structure (with Ag beneath Au) exhibits the strongest R6G signal enhancement, despite its negligible performance at shorter wavelengths. This result highlights the significant influence of the excitation wavelength on the SERS efficiency across different materials. Conversely, under 785 nm laser excitation, the enhancement from the homometallic Ag structure is even weaker than that of the Cu/Au structure, which appears inconsistent with the simulation results in Fig. 2. We attribute this discrepancy to the potential oxidation of the exposed Ag nanostructured top layer, as there was a significant time gap (more than one month) 17 between substrate fabrication and measurement. In contrast, the Ag/Au structure, protected by the chemically inert Au top layer, likely maintained its plasmonic properties. To verify this hypothesis, a supplementary experiment was conducted using NPM substrates within two weeks from fabrication (see Supplementary Information), showing that the homometallic Ag structure demonstrates significantly stronger enhancement than Ag/Au, which is consistent with the simulations in Fig. 2. To further assess SERS performance, based on the above discussion, we evaluated the signal uniformity, limit of detection (LOD), and concentration dependence of Ag/Cu, homometallic Ag (bilayer Ag/Ag), and Ag/Au NPMs under 532 nm, 633 nm, and 785 nm excitation, respectively. Results are summarized in Figure 7. Fig. 7 (a-c) reports the averaged SERS spectra of R6G at varying concentrations measured under consistent experimental conditions. The lowest detectable concentrations of R6G were determined to be as low as 10 nM, 0.1 μM, and 10 μM for 532, 633, and 785 nm excitation wavelengths, respectively. Concentration calibration curves are also shown in Fig. 7(d-f) left panels, derived from the five characteristic SERS bands at 620, 1280, 1360, 1509, and 1649 cm-1 in the SERS spectra shown in Fig. 7 (a-c). We found that the intensities of five characteristic SERS bands of R6G exhibit an essentially linear relationship with the logarithm of the corresponding R6G concentrations, which is consistent with the results in the literature34. Fig. 7 (d-f) Right panels displayed the intensity distributions of these five SERS bands across multiple randomly selected spectra on different nanoporous substrates. For each R6G characteristic peak, the relative standard deviation (RSD) of the intensities across the spectra was calculated. Subsequently, the average RSD (mean RSD) for the five peaks was determined, yielding values of 9.98%, 7.76%, and 15.53% for Ag/Cu, Ag, and Ag/Au substrates, respectively. These mean 18 RSD values demonstrate excellent reproducibility and uniformity of the SERS response across the different nanoporous substrates. Figure 7: (a-c) Concentration-dependent SERS spectra of R6G on Ag/Cu, Ag/Ag and Ag/Au NPMs under 532, 633 and 785nm, respectively. (d-f) Left panels: Calibration curves for the five characteristic bands (620, 1280, 1360, 1509 and 1649 cm-1) derived from the corresponding SERS spectra shown in (a-c); Right panels: Intensity distribution of these bands from randomly selected spectra for 1μM, 10 μM and 1 mM R6G on the respective substrates. The long-term stability of the fabricated porous substrates was evaluated by measuring the SERS spectra of R6G on films stored for several months (Fig. S2, Supporting Information). Furthermore, the mechanical durability was assessed by testing the SERS enhancement after ultrasonic washing. The substrates maintained significant SERS activity even after 40 minutes of ultrasonication, demonstrating robust performance under harsh conditions (Fig. S3, Supporting Information). This combination of long19 term and mechanical stability makes these substrates highly suitable for practical applications, such as photocatalytic or biochemical testing. Figure 8: (a-c) Normalized SERS spectra of ADAMTS3 at various concentrations on Ag/Cu, Ag/Ag, and Ag/Au NPMs under 532, 633, and 785nm excitation, respectively. (d-f) Fit-integrated areas under the corresponding characteristics bands marked with dot lines in (a-c). The translation of SERS substrates from conceptual fabrication to practical application is a critical step in biosensing, particularly for the early screening and diagnosis of cancers and related biomarkers3537. The ability to detect low-concentration biomarkers with high specificity and sensitivity provides a crucial, time-sensitive advantage for early diagnosis and subsequent therapeutic intervention. In a recent work, we reported a DNA origami-based SERS platform for the sensitive detection of ADAMTS3 protein, a promising biomarker associated with hepatocellular carcinoma (HCC)38. Here, we build upon our previous findings by deploying the novel bi-layer metal nanoporous films as a superior SERS 20 substrate to re-investigate the detection of the same ADAMTS3 protein. As discussed above, these substrates offer large, uniform active area and long-term stability, therefore they can be interesting for applied SERS biosensing. Figure 8 presents the analytical performance of the bi-layer NPM SERS substrates for the detection of ADAMTS3. Fig. 8 (a-c) displays the normalized mean SERS spectra of ADAMTS3 protein at various concentrations, acquired using 532, 633, and 785 nm laser excitation wavelengths, respectively. Each set of spectra corresponds to the optimally designed bi-layer configuration (e.g. Ag/Cu, homometallic Ag/Ag, and Ag/Au) for the respective laser wavelength, as detailed in the previous sections. The fitted areas under the characteristic SERS bands of ADAMTS3 protein (marked by dotted lines in a-c) at 1098, 1258, and 1328 cm-1 were plotted in Fig. 8(d-f) as a function of logarithmic concentration with error bars. The peak fitting was performed using a Gaussian function, consistent with our previous methodology39. As shown in the figure, the characteristic SERS peaks of the ADAMTS3 protein remain clearly distinguishable at a concentration of 10-9 M using the three optimal enhancement substrates with the magnification of Ag/Cu, Ag/Ag, and Ag/Au under 532, 633, and 785 nm excitation, respectively. Notably, with 633 nm excitation, these characteristic peaks are still detectable even at a lower concentration of 10-10 M, representing an enhancement factor comparable to that achieved in our recent work using an optimized Au-dimer/DNA origami as a SERS platform38. Furthermore, the fitted areas of the characteristic peaks at 1098, 1258, and 1328 cm-1 were plotted against the logarithmic concentration. Under 633 nm excitation, the data points exhibit smaller error bars compared to those at 532 and 785 nm, and follow an approximately linear relationship. This indicates that the peak areas increase nearly linearly with increasing protein concentration, which is consistent with our previous findings38. In contrast, this linear relationship is not always observed under 532 nm and 785 nm excitation. The fitted areas at 1098 and 1258cm-1 exhibit an approximate linear trend ranging from 10-7 to 10-9 M, but with a notable deviation occurring at 10-6 M at 785nm excitation, and all these fitted areas are associated with relatively large error bars. Under 532nm laser excitation, the relationship between the fitted area and logarithmic concentration deviates entirely from linearity for all three characteristic peaks, accompanied by significantly larger error bars compared to other wavelengths. This pronounced nonlinearity and data scatter can be attributed to the wavelength-specific electronic interaction between 21 the protein and the corresponding bi-metallic substrate. As highlighted in the introduction section, the adsorption behavior of molecules with polar or partially charged groups-such as those present in the ADAMTS3 protein-is highly sensitive to the electronic state of the alloy surface. We hypothesize that under 532 nm excitation, the specific bi-layer configuration (e.g., Ag/Cu) exhibits an electronic structure that leads to non-uniform or saturable adsorption of protein molecules across the concentration series. This results in inconsistent SERS responses and larger variability in measured intensities, thereby disrupting the linear correlation and increasing observational uncertainties. Conclusions In summary, we conducted a comprehensive investigation of novel plasmonic platforms composed of stacked layers of noble plasmonic metals (NPMs). By preparing multiple configurations (Au/Ag, Ag/Au, Au/Cu, Cu/Au, Ag/Cu, and Cu/Ag), we were able to systematically explore how the choice of metals and their stacking order influence metal-metal interactions. Extensive numerical simulations, carried out using both finite element methods (FEM, COMSOL) and fully atomistic models, provided valuable insights into the optical properties of these nanostructures under excitation at different energies within the visible range. These models proved effective in describing the behavior of porous systems, highlighting their potential not only for enhanced spectroscopy but also for a variety of additional applications. To experimentally validate the platforms, we performed SERS measurements on both a standard probe molecule (R6G) and a biologically relevant target (the ADAMTS3 biomarker). These experiments demonstrated the strong performance of the bilayer systems as SERS substrates and offered initial evidence of how bimetallic interactions may facilitate interlayer interactions at the metal-metal interface. Altogether, this work provides a promising first step toward the study of complex multimaterials plasmonic platforms for SERS, combining the advantages of reproducibility, simplicity, and low fabrication cost. 22 Author contribution YZ performed the Raman experiments and analyzed the data, AS and TG performed the numerical simulations, AS, AA, LB, NM and IM supported the numerical simulations and results discussion, CW and HJ supported the experimental measurements, AD, GL, SW and RK supported the samples fabrication, ZJ, ZZ, and DG supervised the work. YZ and AS equally contributed to the present work. All authors contributed to the writing of the manuscript. Supporting Information Supporting Information is available free of charge at: ........ Acknowledgment The authors thank the National Natural Science Foundation of China (No.22202167), the National Key Research and Development Project of China (No. 2023YFF0613603), and the HORIZON-MSCADN2022: DYNAMO, grant Agreement 101072818. NM and BJ acknowledge the 'Excellence by Choice' Programme at Umeà University funded by Kempestielserna (grant no. JCK-2130.3). The authors thank the Clean Room Facility of IIT. References (1) Kwon, H.; Barad, H.-N.; Silva Olaya, A. R.; Alarcón-Correa, M.; Hahn, K.; Richter, G.; Wittstock, G.; Fischer, P. Dry Synthesis of Pure and Ultrathin Nanoporous Metallic Films. ACS Appl. Mater. Interfaces 2023, 15 (4), 5620-5627. https://doi.org/10.1021/acsami.2c19584. (2) Tapani, T.; Caligiuri, V.; Zou, Y.; Griesi, A.; Ivanov, Y. P.; Cuscunà, M.; Balestra, G.; Lin, H.; Sapunova, A.; Franceschini, P.; Tognazzi, A.; De Angelis, C.; Divitini, G.; Carzino, R.; Kwon, H.; Fischer, P.; Krahne, R.; Maccaferri, N.; Garoli, D. Disordered Plasmonic System with Dense Copper Nano-Island Morphology. Nanophotonics 2025, 14 (12), 2151-2160. https://doi.org/10.1515/nanoph-2024-0743. (3) Caligiuri, V.; Kwon, H.; Griesi, A.; Ivanov, Y. 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Nanoscale Adv. 2025, 7 (17), 5212-5220. https://doi.org/10.1039/D5NA00486A. 1. Additional SERS Experiments 26 - R6G test under 785nm excitation with new set of substrates Fig. S1. SERS spectra of R6G at different concentrations obtained from NPMs with varying configuration (Ag/Cu, Ag/Au and Ag/Ag) under 785nm excitation. The SERS performance of different bi-layer nanoporous substrates (Ag/Ag, Ag/Au, Ag/Cu) was re-evaluated using R6G under 785 nm excitation, as shown in Fig. S1. A direct comparison reveals that the Ag/Ag substrate possesses the highest enhancement capability, which is different from the previous results in Figure 6(c). It successfully detected R6G at a concentration of 0.1 μM, whereas under identical conditions, the Ag/Au and Ag/Cu substrates only yielded measurable signals at a higher concentration of 1 μM. 27 This result clearly indicates that the Ag/Ag bi-layer structure offers superior SERS sensitivity over the other two configurations at 785nm excitation. - Long-term stability of the substrates After initial characterization, we re-evaluated the SERS performance of 100nM R6G under 532 nm excitation on various bi-layer nanoporous substrates following a two-month storage period under ambient conditions. The four substrates exhibiting the highest enhancement were selected for further analysis, as shown in Fig. S2. Both the bi-layer Ag and bi-layer Au substrates maintained the strongest SERS enhancement after storage. In contrast, the Au/Cu and Cu/Au configurations also demonstrated considerable enhancement following the two-month period. Notably, the Ag/Cu substrate, which initially ranked third in enhancement intensity in Figure 6(a) and Figure 6(d), showed a significant reduction in SERS performance after storage. We attribute these observations to the protective role of gold against oxidation, which helps preserve the nano-structural integrity of the underlying porous film. In comparison, silver and copper are more susceptible to oxidation over time, leading to diminished SERS activity in Ag/Cu bilayers after prolonged exposure to air. 28 Fig. S2. Normalized SERS spectra of 100nM R6G acquired from the selected bilayer nanoporous substrates, before (short dot line) and after (solid line) two months storage period. Spectra were collected under 532nm excitation with all measurements using a 10% laser filter except for the poststorage Au/Cu and Cu/Au samples, for which a 100% laser power was employed. All spectra are normalized to [0,1] range for comparative purposes. 29 Fig. S3 Evolution of SERS performance after storage. Mean SERS spectra of 0.1mM R6G on bi-layer nanoporous substrates with different configurations under 633nm excitation (5% laser filter), acquired after a two-month storage period. 30 Fig. S4 Evolution of SERS performance after storage under 785nm laser excitation. a) Mean SERS spectra of 0.1mM R6G obtained initially (dotted line) and after storage (solid line); b) Corresponding normalized post-storage spectra. Having established the variation in initial SERS enhancement across substrates in Figure 6(b), we proceeded to evaluate its subsequent evolution of the same substrates following a two-month storage period under 633nm excitation. As shown in Fig. S3, Ag/Au bilayer emerges as the most effective substrate, exhibiting the highest SERS intensity, followed by Au/Ag configuration. In contrast, the initially superior configurations (Ag/Ag, Au/Au, and Cu/Ag, marked by asterisks) displayed a considerable decline after storage. This marked contrast underscores the significant role of structural stability and oxidation resistance in maintaining SERS activity over time. The results obtained under 785 nm excitation differed from those under 532 nm and 633 nm, as shown in Fig. S4. For comparison, we evaluated the top four enhanced SERS configurations-Ag/Au, Cu/Au, Au/Ag, and Au/Cu-before and after a two-month storage period. As illustrated in Fig. S4(a), the SERS intensity of R6G decreased in all configurations between the initial and post-storage measurements. However, configurations with Ag/Au and Cu/Au consistently exhibited stronger enhancement than the others, both initially and after storage. In Fig. S4(b), after normalization of the poststorage spectra, it can be observed that Ag/Au and Cu/Au configurations maintained superior enhancement performance before and after storage. In contrast, the double-layer silver porous structure, which initially ranked third in enhancement (as shown in the main text Figure 6(c)), exhibited weaker signals than both Au/Ag and Au/Cu after two months of storage. 31 - Influence of washing to the substrates Fig. S5 Evaluation of the washing effect on substrate stability. Mean SERS spectra of the background signal and 10-6 M ADAMTS3 were acquired under 633 nm excitation after 40 minutes of sequential ultrasonication in ethanol and ultrapure water. The results for both Ag/Au and Ag/Ag porous substrates are shown, with error bars representing the standard deviation across multiple measurements. 2. Additional data calculations 32 Figure S6. Ag/Ag (a), Ag/Au (b), Au/Ag (c), Au/Au (d) NPM structures exploited in ωFQFμ calculations. 33 Figure S7. The enhanced EM field distribution of Au and Ag bottom layers for Au/Ag (top) and Ag/Au (bottom) NPMs calculated at the FEM (a) and ωFQFμ (b) levels of theory. 3. Ellipsometric characterization Figure S8: Real (black) and imaginary (red) parts of the dielectric permittivity for Ag/Ag (a), Ag/Au (d), and Ag/Cu (g) nanoporous bilayers, obtained by fitting ellipsometric data using a combination of Lorentzian and Gaussian oscillators. Corresponding experimental (dots) and fitted (solid lines) ellipsometric angles Ψ and Δ are shown in panels (b-c), (e-f), and (h-i), respectively. The oscillator parameters are reported in Table S1. The dielectric functions of the three nanoporous bilayers, Ag/Ag, Ag/Au, and Ag/Cu, shown in Figure S8.a,d,g, respectively, were modeled using a set of Lorentzian and 34 Gaussian oscillators to capture the key features of their optical response. For Ag/Ag (Figure S8a), the response is dominated by a single Lorentzian centered at 1.297 eV, accounting for the low-energy plasmonic resonance typical of nanostructured silver. A Gaussian oscillator with negative amplitude was included at 2.84 eV to compensate for the overestimated ε'' background induced by the Lorentzian tail, ensuring better agreement with the experimental data in the near-UV region. In the case of Ag/Au (Figure S8d), a broader Lorentzian (En0 = 1.20 eV, Br = 1.44 eV) and two Gaussians were required to account for the increased damping and the presence of interband transitions characteristic of gold. The higher complexity of this system reflects the hybridized electronic structure of the alloy. Finally, the Ag/Cu (Figure S8g) sample shows a markedly suppressed optical response, with three weak oscillators spread across the visible range. The low oscillator strengths and absence of sharp features are consistent with stronger damping and reduced plasmonic activity due to the presence of copper. In all cases, the fitted dielectric functions reproduce the experimental ellipsometric parameters Ψ and Δ, shown in Figures S8b,c, S8e,f, and S8h,i with high fidelity across the measured spectral range. 35 Table S1: Parameters of the Lorentzian and Gaussian oscillators used to model the dielectric functions of the Ag/Ag, Ag/Au, and Ag/Cu bilayers. En0: central energy (eV), Br: broadening (eV), A0: amplitude (dimensionless). Sample Oscillator Type A0 En0 (eV) Br (eV) Ag/Ag Lorentzian 10.01 1.297 0.200 Gaussian -1.40 2.840 1.023 Ag/Au Lorentzian 13.17 1.200 1.440 Gaussian 1.07 2.880 0.644 Gaussian 2.98 4.580 1.745 Ag/Cu Lorentzian 0.115 3.210 0.960 Gaussian 0.021 1.970 0.440
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LLM Agents for Automated Web Vulnerability Reproduction:
Are We There Yet?
BIN LIU, Harbin Institute of Technology, Shenzhen, China
YANJIE ZHAO, Huazhong University of Science and Technology, China
GUOAI XU, Harbin Institute of Technology, Shenzhen, China
HAOYU WANG, Huazhong University of Science and Technology, China
Large language model (LLM) agents have demonstrated remarkable capabilities in software engineering and
cybersecurity tasks, including code generation, vulnerability discovery and automated testing. One critical
but underexplored application is automated web vulnerability reproduction, which transforms vulnerability
reports into working exploits. Although recent advances suggest promising potential, significant challenges
remain in applying LLM agents to real-world web vulnerability reproduction scenarios.
In this paper, we present the first comprehensive evaluation of state-of-the-art LLM agents for automated
web vulnerability reproduction. We first systematically assess 20 agents from software engineering, cyberse-
curity, and general domains across 16 dimensions, including technical capabilities, environment adaptability,
and user experience factors, on 3 representative web vulnerabilities. Based on the results, we select three top-
performing agents (OpenHands, SWE-agent, and CAI) for in-depth evaluation on our constructed benchmark
dataset of 80 real-world CVEs spanning 7 vulnerability types and 6 web technologies. Our results reveal that
while LLM agents achieve reasonable success on simple library-based vulnerabilities, they consistently fail
on complex service-based vulnerabilities requiring multi-component environments. Furthermore, complex
environment configurations and authentication barriers create a critical gap where agents can execute exploit
code but fail to trigger actual vulnerabilities. We observe that agents demonstrate high sensitivity to input
guidance, with performance degrading by over 33.3% under incomplete authentication information. Our
findings highlight the significant gap between current LLM agent capabilities and the demands of reliable
automated vulnerability reproduction, emphasizing the need for advances in environmental adaptation and
autonomous problem-solving capabilities.
CCS Concepts: • Software and its engineering;
Additional Key Words and Phrases: Web Vulnerability, LLM Agents, Vulnerability Management
ACM Reference Format:
Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang. 2025. LLM Agents for Automated Web Vulnerability
Reproduction: Are We There Yet?. 1, 1 (October 2025), 22 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn
1
Introduction
Recently, Large language model (LLM) agents have achieved remarkable success across software
engineering and cybersecurity domains. In software engineering, these agents have demonstrated
strong capabilities in code generation [15, 19, 31], bug fixing [36, 83, 90], issue resolution [37, 86]
Authors’ Contact Information: Bin Liu, Harbin Institute of Technology, Shenzhen, China, liubin1999@stu.hit.edu.cn; Yanjie
Zhao, Huazhong University of Science and Technology, China, yanjie_zhao@hust.edu.cn; Guoai Xu, Harbin Institute of
Technology, Shenzhen, China, xga@hit.edu.cn; Haoyu Wang, Huazhong University of Science and Technology, China,
haoyuwang@hust.edu.cn.
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, Vol. 1, No. 1, Article . Publication date: October 2025.
arXiv:2510.14700v1 [cs.SE] 16 Oct 2025
2
Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
and code review [24, 71]. In cybersecurity, they have shown impressive performance in vulnerability
detection [30, 41, 68], fuzzing [18, 47, 74], exploit generation [17, 80], malware analysis [11, 49]
and penetration testing [28, 40]. These successes demonstrate their ability to understand complex
technical contexts, reason about software behavior, and automate sophisticated workflows.
Web applications have become the backbone of modern digital infrastructure, representing
the dominant form of deployed applications in enterprise environments [69]. This widespread
adoption has made them prime targets for cyberattacks, with web vulnerabilities consistently
dominating security incident reports [72]. The OWASP Top 10 identifies critical issues such as
SQL injection, Cross-Site Scripting (XSS), and Cross-Site Request Forgery (CSRF) as the majority
of reported vulnerabilities [55], while the National Vulnerability Database confirms that web
application vulnerabilities represent the fastest-growing category of security threats [52].
However, effectively managing these web vulnerabilities relies heavily on successful reproduction,
which involves transforming vulnerability reports into working Proof-of-Concept (PoC) exploits
that validate threats and enable effective patching. This process presents unique challenges that are
different from other security tasks where LLM agents have succeeded. Manual reproduction requires
expertise in environment configuration, dependency resolution, authentication mechanisms, and
exploit crafting, often taking days or weeks per vulnerability [20, 61]. Automated reproduction
faces additional complexities: software version incompatibilities, incomplete PoC code, complex
deployment, and network setups. These factors lead to low success rates in existing automated
reproduction methods, highlighting the substantial technical barriers.
While LLM agents have remarkable capabilities in related security and software engineering
tasks, their effectiveness for comprehensive web vulnerability reproduction remains underexplored.
Existing cybersecurity benchmarks for LLM agents suffer from significant limitations that fail
to capture real-world complexity. Many evaluations rely on simplified Capture-the-Flag (CTF)
scenarios [10, 64, 73, 85, 88] that focus on individual challenges rather than end-to-end reproduction
processes. Additionally, current benchmarks often evaluate agents on small, isolated code snippets
or fragments, limiting their scope to specific programming languages such as C/C++ [7, 38, 50, 79],
Java [42] or JavaScript [14], rather than the diverse components of production web applications.
However, end-to-end vulnerability reproduction requires agents to handle complex multi-step
workflows involving environment setup, dependency management, and real-world deployment
uncertainties. These capabilities extend far beyond current simplified evaluation scenarios, creating
a significant gap in agent ability for practical vulnerability reproduction applications.
To address this gap, we present the first comprehensive empirical study examining LLM agents’
ability for automated web vulnerability reproduction. We systematically evaluate 20 representative
agents across 16 core dimensions on 3 representative CVEs, then conduct in-depth assessment
of the top 3 agents using our constructed benchmark dataset of 80 real-world CVEs. We assess
current agent capabilities and identify fundamental limitations that prevent reliable automated
vulnerability reproduction. In summary, our main contributions include:
• We construct a comprehensive benchmark dataset comprising 80 real-world CVEs covering 7
vulnerability types and 6 web technologies with complete reproduction environments.
• We conduct the first systematic evaluation of 20 state-of-the-art LLM agents for automated
web vulnerability reproduction across 16 core dimensions on 3 representative CVEs, establish-
ing a rigorous evaluation framework. From this evaluation, we select the three top-performing
agents (OpenHands, SWE-agent, and CAI) for in-depth analysis.
• We provide critical empirical insights revealing the significant gap between current LLM
agent capabilities and practical requirements, identifying key failure modes and documenting
over 33.3% performance degradation under incomplete authentication information.
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LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
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2
Background
2.1
LLM Agents
LLMs exhibit impressive performance across multiple domains, including natural language compre-
hension, code synthesis, and reasoning processes. Based on these foundational abilities, researchers
have developed LLM-based agent systems [39, 65, 75] that leverage LLMs as core reasoning engines
for autonomous decision-making, while utilizing various tools and operations to interact with
external environments in order to accomplish sophisticated tasks.
2.1.1
Core Mechanisms. LLM agents combine LLM reasoning with action execution capabilities
through several key mechanisms. ReAct [87] introduces iterative reasoning-action cycles, where
models generate reasoning traces before executing actions, with results integrated into context
for subsequent cycles. Toolformer [63] demonstrates autonomous tool usage, enhancing factual
accuracy and computational abilities.
For complex tasks, simple reactive loops prove insufficient, requiring explicit planning capabili-
ties. Early systems like BabyAGI [51] and Auto-GPT [26] introduced dynamic task lists where agents
execute, reorderand generate tasks continuously. Advanced frameworks such as AgentVerse [16]
and ChatDev [62] adopt multi-agent architectures where controller agents decompose goals into
subtasks distributed to task-specific agents, enabling coordination across different domains. Re-
flexion [66] addresses failure handling through reflection mechanisms that analyze past mistakes
and store self-reflections in memory for improving future planning decisions. This self-correction
capability enables iterative problem-solving and strategy refinement over time.
2.1.2
Representative Agents and Frameworks. LLM agents have emerged across diverse application
domains. Software engineering agents assist with code generation, debugging, and repository-level
analysis. Cybersecurity agents focus on threat detection, vulnerability analysis, and automated
penetration testing. General-purpose agents aim for broad applicability, emphasizing flexible tool
use and adaptive reasoning across various tasks. Additionally, agent orchestration frameworks like
LangChain [13] and AutoGen [81] provide infrastructure for building and coordinating multi-agent
systems, enabling developers to construct domain-specific agents with standardized tooling and
communication protocols. A summary of representative agents and frameworks is shown in Table 1.
Table 1. Representative LLM agents and orchestration frameworks across different domains.
Domain
Agent
Year
Domain
Agent
Year
Software
Engineering
AGENTLESS [82]
2024
Cybersecurity
PentestGPT [17]
2023
OpenHands [78]
2024
CAI [48]
2024
SWE-agent [84]
2024
PentestAgent [59]
2024
AutoCodeRover [89]
2024
Nebula [53]
2024
Aider [23]
2023
HackingBuddyGPT [27]
2023
RepoAgent [45]
2024
PentAGI [57]
2024
ChatDev [62]
2023
EnIGMA [2]
2024
MetaGPT [29]
2024
AI-OPS [4]
2024
General-Purpose
Agents
AutoGPT [26]
2023
Orchestration
Frameworks
LangChain [13]
2022
BabyAGI [51]
2023
AutoGen [81]
2023
AgentVerse [16]
2023
CrewAI [32]
2023
AIlice [5]
2024
LangGraph [33]
2024
2.2
Web Vulnerability Reproduction
Web vulnerability reproduction involves constructing operational PoC exploits from natural lan-
guage vulnerability reports to validate security flaws and guide remediation efforts.
, Vol. 1, No. 1, Article . Publication date: October 2025.
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Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
The common approach relies on manual analysis by security researchers using tools like Burp
Suite [60] or ZAP [56] to intercept, analyze and craft HTTP/S requests. While effective for complex
logic-based flaws, this method suffers from inefficiency and high labor costs that limit scalability.
Semi-automated systems target common vulnerability classes like XSS or SQL Injection by using
NLP to extract key entities from reports and inserting them into predefined attack templates [9].
Advanced works like Wang et al. [77] extract contextual information for targeted web fuzzing.
Commercial tools like Nessus [70] and Acunetix [34] represent this approach. While faster for simple
cases, these methods lack robustness for vulnerabilities requiring complex, multi-step interactions.
Program analysis based automated exploit generation (AEG) [8] achieved significant success
in binary applications through symbolic execution. Frameworks like Mayhem [12] and angr [67]
exemplify this approach. However, their application to web vulnerabilities is limited due to the
immense state space of web applications and inability to reason about high-level logic flaws.
LLM-based methods represent the newest paradigm, framing reproduction as an agentic problem
where LLMs act as reasoning engines. LLMs have been explored for exploiting vulnerabilities in
web applications based on known vulnerability descriptions with one-day setting [21, 22] and
investigating agent teams with hierarchical planning for zero-day scenarios [91]. Recent work [80]
showed AI systems can generate working exploits for published CVEs in 10-15 minutes, while
systems like PentestGPT [17] and PentestAgent [40] demonstrate autonomous penetration testing
capabilities. This methodology combines high-level contextual reasoning with machine execution
speed and scalability. All the above methods are shown in Table 2.
Table 2. Web vulnerability reproduction methods and tools, including manual, semi-automated, program
analysis-based, and LLM-based methods.
Tool
Category
Year
Tool
Category
Year
Burp Suite [60]
Manual
2003
AEG [8]
Program analysis
based
2011
OWASP ZAP [56]
2010
Mayhem [12]
2012
Nessus [70]
Semi-automated
2024
angr [67]
2016
Acunetix [34]
2024
AutoExploit [46]
2018
Website Vulnerability Scanner [58]
2024
PentestGPT [17]
LLM-based
2023
Wang et al. [77]
2023
PentestAgent [40]
2024
Aydin et al. [9]
2014
2.3
Motivation
Automated web vulnerability reproduction remains a critical bottleneck in security operations.
Manual analysis lacks scalability, semi-automated tools are brittle with poor comprehension, and
formal methods like AEG cannot handle web application complexity. This validation gap prevents
security teams from efficiently confirming and prioritizing threats.
LLM-based agents offer a promising solution through semantic comprehension, logical reasoning,
and tool invocation capabilities that address existing limitations. Recent advances have produced
general-purpose agents (as shown in Table 1) claiming broad problem-solving capabilities, and
specialized vulnerability analysis agents (as illustrated in Table 2) targeting security workflows
specifically. However, despite theoretical promise and agent proliferation, the practical effectiveness
of LLM agents in web vulnerability reproduction lacks empirical validation. This creates a critical
gap between claimed potential and demonstrated capability in this demanding security domain.
We conduct the first systematic empirical study evaluating LLM agents’ ability to repro-
duce web vulnerabilities from natural language descriptions. Our comprehensive framework
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LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
5
assesses success rates, reasoning processes, and failure modes, establishing empirical foundations
for understanding current capabilities and limitations in automated vulnerability validation.
3
Empirical Study Design
We design a comprehensive empirical study to systematically evaluate LLM agents for web vulner-
ability reproduction. Our study follows the workflow illustrated in Figure 1.
Our study evaluates agents according to the following four criteria:
C1: Effectiveness. We compare the effectiveness of different agents on web vulnerability
reproduction by measuring success rates across individual stages (environment setup, vulnerability
localization, and PoC generation with verification) and full end-to-end reproduction.
C2: Compatibility. We evaluate the adaptability of different agents across various vulnerability
types and web technologies.
C3: Efficiency. We compare the efficiency of different agents by measuring execution time, USD
cost, and token consumption, which are critical factors for the practical deployment of LLM agents.
C4: Robustness. We evaluate the robustness of different agents by analyzing their performance
under varying conditions, including model capabilities and input quality.
§ 3.1 Data Preparation
Vulnerability Database & Code Repository
§ 3.2 Agent Selection
§ 3.3 Task Specification
80 Web Vulnerability-PoC Datasets
7 Vulnerability Types
API Fetch
Vulnerability Report
Pre-patch Codebase
Manual Construction
/source
docker
-compose.yml
Dockerfile
PoC exploits
& Oracle
Metadata
Scripts
GPT-4.1
Claude-Sonnet-4
Gemini-2.5-pro
Foundation LLMs
Integration
§ 4 & § 5 Experimental
Setup & Results
Target Web Application Container
Library Application
Web Application
Other
Configurations
Database
Input
Env setup task
Localization task
Generation & Attack
PoC gen task
Reproduction task
Output Example
{
"stage": "e2e",
"success": <true|false>,
"files": [{"path": "./.bench/e2e/Dockerfile",…},
...
{"path": “./.bench/e2e/steps.md”,…}],
"script_files": {
"./.bench/e2e/run.sh": "commands"},
"run_commands": [
"bash ./.bench/e2e/run.sh"],
}
C1: Effectiveness
C2: Compatibility
C3: Efficiency
C4: Robustness
Evaluation Criteria
Agents
3 agents selected
from 20 candidates
for further analysis
Fig. 1. Systematic workflow for evaluating LLM agents in web vulnerability reproduction tasks.
3.1
Data Preparation
To effectively evaluate the performance of LLM agents in vulnerability reproduction, we have
curated a benchmark dataset comprising real-world web application vulnerabilities. Our dataset con-
struction follows a structured multi-round filtering strategy to ensure reliability and reproducibility
across diverse security scenarios in the real world.
Data Collection Process. We employ a systematic multi-round filtering strategy for data
collection. Initially, we search the OSV database’s Git ecosystem for service-based web vulnerability
keywords, including SSRF, CSRF, SQL Injection, Prototype Pollution, Path Traversal, Remote Code
Execution, and Cross-Site Scripting, limiting our scope to vulnerabilities disclosed after 2024. This
initial search yields 744 candidate entries. To ensure manageable experimental environments,
we further filter out projects with repository sizes exceeding 30MB, resulting in 224 remaining
, Vol. 1, No. 1, Article . Publication date: October 2025.
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Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
candidates. Finally, we apply three stringent criteria for final selection: (1) only vulnerabilities
affecting web-based applications are included, (2) vulnerabilities must be available in accessible Git
repositories where the specific vulnerable commit can be checked out with complete source code
access, and (3) vulnerability reports must be disclosed after 2024 to reflect current threat landscapes.
Through this systematic filtering process, we construct a high-quality dataset comprising 80
vulnerabilities.
Vulnerability Type Selection. Our curation process focuses on 7 critical web vulnerability
types strategically selected to comprehensively evaluate agent capabilities across different vulnera-
bility paradigms. We include six mainstream service-based vulnerability types: Cross-Site Request
Forgery (CSRF), Path Traversal, Remote Code Execution (RCE), SQL Injection (SQLI), Server-Side
Request Forgery (SSRF) and Cross-Site Scripting (XSS). These represent the most prevalent web
application security flaws that occur at the service level. Additionally, to provide a more holistic
assessment of agent performance, we incorporate Prototype Pollution as a representative library-
based vulnerability type that manifests through dependencies and third-party packages rather than
direct application code. This approach ensures our evaluation captures both traditional web service
vulnerabilities and modern JavaScript ecosystem threats.
Environment Construction. For each selected vulnerability, we execute a comprehensive data
preparation pipeline. After fetching the vulnerability reports via APIs, we identify the corresponding
pre-patch codebase from the linked Git repositories and check out the exact vulnerable commit.
We utilize Git’s diff capabilities to identify the precise code changes between vulnerable and
patched versions, allowing us to pinpoint the exact vulnerable code sections, including specific files,
functions and line numbers. We then construct fully containerized environments using Docker,
manually developing Dockerfile and docker-compose.yml configurations for each vulnerability to
ensure reproducible deployment across different systems. Each environment is thoroughly tested
to verify successful vulnerability reproduction before exploit development.
PoC Development. For PoC exploit construction, we develop PoC exploits based on existing
vulnerability reports. When existing PoCs are available in the reports, we adapt and refine them for
our containers; otherwise, we manually create new exploits from scratch. Our PoC development
follows vulnerability-specific methodologies: CSRF vulnerabilities are implemented using HTML-
based attack vectors, Prototype Pollution exploits are crafted in JavaScript to demonstrate object
manipulation, while other vulnerability categories utilize Python-based HTTP request construction
for systematic exploitation. Each PoC is accompanied by automated verification oracles that confirm
successful exploitation, enabling reliable automated assessment.
Our construction process generates dataset entries containing vulnerable source code directories,
Docker configuration files, environment setup scripts, executable PoC exploit code with verification
oracles, and structured metadata including CVE identifier, vulnerability description, affected file
paths, vulnerable function names, specific line numbers, corresponding GitHub repository URLs,
vulnerable commit hashes and temporal information. Our final dataset comprises 80 vulnerabilities
spanning 6 popular web technologies including Python, PHP, Java, JavaScript, TypeScript, and Go,
covering 7 vulnerability types.
3.2
Agent Selection
Recall that in Table 1, we listed 20 representative agents. These agents share common charac-
teristics: they are open-source projects available on GitHub with over 1,000 stars for software
engineering/general-purpose agents and over 100 stars for cybersecurity agents, released or up-
dated after 2023, and demonstrating active maintenance. All agents demonstrate basic operational
capabilities for automated task execution.
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LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
7
Initial Capability Assessment. We first conduct a preliminary capability assessment of these
20 representative agents using 16 core dimensions and 3 representative CVEs for testing. The 16
evaluation dimensions include source code access, tool integration, script execution, file operations,
web requests, security domain focus, custom structured output, custom task support, runtime
stability, error handling, session management, efficiency metrics, log persistence, latest LLM support,
user interface, and installation convenience. We select 3 representative CVEs for testing: CVE-
2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and CVE-2024-41112 (RCE), which represent
different vulnerability types and complexity levels.
Agent Selection Results. We conduct a comprehensive evaluation of these 20 agents spanning
multiple architectures and application domains, with detailed results presented in Table 3 (§ 5.1).
Through this evaluation round, we identify the 3 best-performing agents. Subsequently, we conduct
an in-depth empirical evaluation of these 3 selected agents using the complete 80-vulnerability
dataset described in § 3.1. Therefore, our further empirical evaluation focuses on these three
promising agents that demonstrate adequate capabilities and performance.
Outputs (all saved under ./.bench/e2e/):
- Intermediate artifacts
- steps.md (the reproduction steps and
key commands).
- result.json (consolidated end-to-end
result).
User Prompt
You are participating in web vulnerability reproduction in a safe container.
Rules:
- Never invent placeholder hostnames. Derive service_url from docker-compose
port mapping (localhost) or explicit run.sh output.
- You can edit/create files under the workspace and execute shell commands.
- Do NOT modify any source code files; only write under ${stage_dir}.
- Do NOT access files outside the current working directory.
Category: RCE
Vulnerability description: MailDev 2 through 2.1.0 allows Remote Code
Execution via a crafted Content-ID header for an e-mail attachment, leading to
lib/mailserver.js writing arbitrary code into the routes.js file.
Codebase Directory: ${codebase_dir}
Stage: End-to-End Reproduction.
Steps: Starting from the CVE folder and the codebase, autonomously:
1) Build/run the environment from scratch based on c
2) Localize the vulnerable code region (files/functions/lines)
3) Generate a PoC and an in-code verification oracle under ./.bench/e2e/ to
assess exploit success.
4) Execute the PoC, collect evidence/logs/state, and decide success/failure.
5) Produce a concise step-by-step reproduction document
(./.bench/e2e/steps.md).
Outputs (all saved under ./.bench/e2e/):
- Intermediate artifacts
- steps.md (the reproduction steps and key commands).
- result.json (consolidated end-to-end result).
Response.png
Agent
{
"stage" : "e2e",
"success": <true|false>,
"files": [
{"path": "./.bench/e2e/Dockerfile", "status":
"created|updated|copied"},
],
"run_commands": [
"bash ./.bench/e2e/run.sh"
],
"artifacts": {
"service_url": "http://localhost:<port>",
"exec_success": <true|false>,
"signals": {"status": 200, "log_regex": "..."},
"observed": {"status": 200, "log": "..."}
},
"steps": [
{"name": "env_setup", "success": <true|false>},
{"name": "localization", "success":
<true|false>},
{"name": "poc_and_verify", "success":
<true|false>, "notes": "..."}
],
…
}
JSON Response
Vulnerability Information
Reproduction Requirements
Output Schema
Fig. 2. Prompt template and example agent response.
3.3
Task Specification
We design four evaluation tasks of increasing complexity to systematically assess agent capabilities
across the vulnerability reproduction pipeline. The first three tasks are independent, single-stage
evaluations where each task receives expert-curated ground truth from prerequisite stages as
input, ensuring that only the current stage’s capabilities are measured. The final task represents a
comprehensive end-to-end evaluation requiring agents to autonomously navigate the complete
vulnerability reproduction workflow.
Environment Setup. This foundational task evaluates an agent’s ability to establish a functional
testing environment from source code alone. Given only the vulnerable application’s codebase,
agents must generate the necessary containerization files (Dockerfile, docker-compose.yml) and
deployment scripts (run.sh) to create a running instance container of the application.
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Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
Vulnerability Localization. With access to both the running environment and source code,
agents must perform code analysis to pinpoint the vulnerability’s location. Given the CVE descrip-
tion, agents analyze the codebase to produce a ranked list of suspicious files, functions, and specific
line numbers that correspond to the security flaw.
Proof-of-Concept (PoC) Exploits Generation. This task tests an agent’s exploit development
capabilities. Provided with the vulnerability’s exact location (file and line number), agents must
craft a working Proof-of-Concept exploits that successfully triggers the exploit and includes a
verification oracle to programmatically confirm successful exploitation.
End-to-End Reproduction. As the most challenging evaluation, this task provides agents only
with the source code and original CVE description. Agents must autonomously perform all previous
stages—from environment setup through PoC execution—and produce a complete, documented
workflow of their reproduction process.
To ensure fair evaluation across different agents, we establish a standardized input-output
interface that provides consistent task specifications and response formats. Each agent receives
identical prompts and must produce outputs in the same structured format, enabling objective
comparison of capabilities across all web vulnerability reproduction tasks.
As illustrated in Figure 2, the prompt contains structured vulnerability information including
CVE category, detailed description, codebase directory, and reproduction steps. The prompt also
specifies operational constraints such as workspace restrictions and file access limitations. Agents
must respond with structured JSON outputs that include stage indicators, success flags, file paths,
execution commands, and artifact metadata. This standardized format enables automated assessment
of task completion. It also allows verification that agents produce the required code artifacts
including Dockerfiles, execution scripts, PoC exploits, and verification oracles.
4
Experimental Setup
4.1
Studied Dataset and Agents
As introduced in § 3.1, we evaluate agent performance on a curated benchmark of 80 real-world
vulnerabilities across diverse categories and complexity levels. Following the setup described in § 3.2,
we assess the performance of 20 popular LLM agents across software engineering, cybersecurity, and
general-purpose, each integrated with three foundation LLMs: GPT-4.1 [54], Claude-Sonnet-4 [6],
and Gemini-2.5-Pro [25].
4.2
Metrics
To comprehensively evaluate agent performance across the four criteria (C1-C4), we measure
success rates at different granularities and additional efficiency metrics:
4.2.1
Success Rate Metrics. We measure the percentage of successful completions across different
stages of the vulnerability reproduction pipeline, with all results verified through manual inspection:
• Environment Setup Success Rate: The percentage of vulnerabilities for which an agent
successfully generates and deploys a functional vulnerable environment, verified through
manual testing of application accessibility and configuration correctness.
• Vulnerability Localization Accuracy (Top-K): The percentage of cases where the ground-
truth vulnerability location appears in the agent’s top-K predictions. We evaluate this metric
at three granularity levels:
– File-level Top-K: Ground-truth vulnerable file present in top-K file predictions.
– Function-level Top-K: Ground-truth vulnerable function present in top-K function predic-
tions.
– Line-level Top-K: Ground-truth vulnerable line present in top-K line predictions.
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LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
9
• PoC Generation Success Rate: We decompose this into two complementary sub-metrics:
– Execution Success Rate: The percentage of generated PoCs that execute without runtime
errors, verified through both automated testing and manual execution.
– Vulnerability Trigger Success Rate: The percentage of executable PoCs that successfully
trigger the vulnerability, confirmed through manual verification of exploitation effects.
• End-to-End Success Rate: The percentage of vulnerabilities for which an agent successfully
completes the entire reproduction workflow from environment setup to verified exploitation,
with manual confirmation of successful vulnerability triggering.
4.2.2
Efficiency Metrics. To assess operational costs and practical deployment considerations, we
also measure:
• Execution Time: Total wall-clock time required for task completion.
• Token Consumption: Total API tokens consumed during execution.
• Financial Cost: Estimated USD cost based on publicly available API pricing.
4.3
Implementation Details
All experiments are conducted on a standardized Linux machine with Ubuntu 20.04 LTS, equipped
with an Intel Xeon Platinum 8358P CPU (2.60GHz, 128 cores) and 2TB memory. We use Docker En-
gine 26.1.0 to containerize environments, ensuring isolation and reproducibility. Agent frameworks
are executed using Python 3.11 with dependencies installed from their official repositories.
To mitigate inherent randomness in LLM outputs, each experiment undergoes three independent
executions, with final metrics calculated as averages across all. A strict 60-minute timeout is enforced
per execution, where any run exceeding this limit is automatically terminated and classified as a
failure. All generated artifacts, execution logs, and API interactions are systematically captured and
stored for comprehensive post-hoc analysis. To maintain experimental feasibility, we impose budget
constraints of $2 per individual stage and $5 for complete end-to-end reproduction workflows.
To ensure fair comparison across all agents, browser-based interactions are strictly prohibited,
restricting all agents to command-line interfaces only.
5
Experimental Results
5.1
C1: Effectiveness
Motivation. Manual reproduction of security vulnerabilities represents a well-documented
bottleneck in software development, consuming substantial expert time and delaying remediation
efforts. Previous approaches relied heavily on manual processes where security researchers must
interpret vulnerability reports and develop proof-of-concept code. However, recent advances in LLM
agents present opportunities to automate these complex tasks. Therefore, this criterion evaluates
the effectiveness of current agents in performing core vulnerability reproduction tasks.
Approach. As described in § 4, we evaluate agent effectiveness across the four-stage vulnerability
reproduction pipeline using the success rate metrics defined in § 4.2. Each agent-model combination
is assessed on environment setup, vulnerability localization (measured by file-level, function-level
and line-level Top-3 accuracy), PoC generation with execution verification, vulnerability trigger
verification, and end-to-end reproduction success, which is defined in § 3.3.
Results. Table 3 shows the capability assessment results of 20 agents across 16 dimensions
using three representative CVEs: CVE-2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and
CVE-2024-41112 (RCE). The analysis reveals severe capability gaps across the agent landscape.
Software engineering agents show poor performance with only 2 out of 8 agents meeting basic
requirements. Most fail in critical areas like tool integaration and session management, with agents
, Vol. 1, No. 1, Article . Publication date: October 2025.
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Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
Table 3. Comparison of 20 agents for web vulnerability reproduction with CVE-2024-7065, CVE-2024-7458
and CVE-2024-41112.
Agent
SCA TI
SE
FO
WR
SDF
CSO CTS RS
EH SM EM
LP
LLS
UI
EI
Software Engineering Agents
AGENTLESS
✓
✗
✗
✗
✗
✗
✗
✗
G
#
✗
✗
✗
✗
G
#
C
✓
OpenHands
✓
✓
✓
✓
✓
✗
✓
✓
✓
✓
✓
✓
✓
✓
C/W
✓
SWE-agent
✓
G
#
✓
✓
✓
✗
G
#
✓
✓
✓
✓
✓
✓
✓
C
✓
AutoCodeRover
✓
✗
G
#
✓
✗
✗
✗
G
#
✓
✗
✗
✗
✓
G
#
C
✓
Aider
G
#
✓
G
#
✓
✗
✗
G
#
✓
✓
✓
✗
✓
✓
✓
C
✓
RepoAgent
✓
✗
G
#
G
#
✗
✗
✗
G
#
✓
✗
✗
✗
✗
G
#
C
✓
ChatDev
✗
✗
✗
G
#
✗
✗
G
#
G
#
G
#
✗
✗
✗
✗
✗
C/W
G
#
MetaGPT
G
#
✓
G
#
G
#
✗
✗
G
#
G
#
✓
✓
✗
✗
✓
G
#
C
✓
Cybersecurity Agents
PentestGPT
G
#
✓
✓
✓
✓
✓
G
#
✗
✓
✓
✓
✗
✓
✓
C
✓
CAI
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
C
✓
PentestAgent
✗
✓
✓
✓
✓
✓
✗
✗
✗
✗
✗
✗
✗
G
#
C
✓
Nebula
✓
✓
✓
G
#
✗
✓
✗
✗
✗
✗
✗
✗
✗
✗
C
G
#
HackingBuddyGPT
✗
✓
✓
✓
✓
✓
G
#
G
#
✓
✓
✓
✗
✓
✓
C
G
#
PentAGI
✗
G
#
✓
✓
✓
✓
✗
✗
✓
✓
✓
✗
✓
✓
C
✗
EnIGMA
✗
✓
✓
✓
✓
✓
✗
✗
✓
✓
✓
✓
✓
✗
C
G
#
AI-OPS
✗
✗
✗
✗
G
#
G
#
✗
✗
✓
✗
✓
✗
✗
✗
C
G
#
General-purpose Agents
AutoGPT
✓
G
#
G
#
✓
✓
✗
G
#
G
#
✗
✗
✓
✗
✓
✓
C/W
✗
BabyAGI
✓
G
#
G
#
G
#
✗
✗
✗
✗
G
#
G
#
✓
✗
✗
G
#
C
✓
AgentVerse
✓
✗
✗
✗
✗
✗
G
#
G
#
G
#
✗
✓
✗
✗
✗
C/W
G
#
AIlice
✓
✓
✓
G
#
✗
✗
✓
✓
G
#
✗
✓
✗
✓
✗
C/W
✗
Note: Symbols represent Good (✓), Average (G
#), and Poor (✗) qualitative assessment.
Abbreviations:
SCA: Source Code Access
RS:
Runtime Stability
TI:
Tool Integration
EH:
Error Handling
SE:
Script Execution
SM:
Session Management
FO:
File Operation
EM:
Efficiency Metrics
WR:
Web Request
LP:
Log Persistence
SDF:
Security Domain Focus
LLS: Latest LLM Support
CSO: Custom Structured Output
UI:
User Interface (C: CLI, C/W: CLI/WEB)
CTS:
Custom Task Support
EI:
Ease of Install
like AGENTLESS [82] and ChatDev [62] lacking over 10 essential capabilities. Cybersecurity agents
demonstrate domain knowledge but face limitations in technical implementation capabilities, most
lack essential technical features such as structured output and custom task support. Except for CAI
which achieves full capability, all other cybersecurity agents fail to support custom task types and
structured output formats, limiting their adaptability to diverse web vulnerability reproduction
scenarios. General-purpose agents offer broad applicability and flexible task handling but struggle
with specialized technical requirements. For example, AutoGPT [26] demonstrates reasonable
performance in 8-9 dimensions but lacks security domain focus and has installation challenges.
Only three agents demonstrate comprehensive capabilities: SWE-agent [84], OpenHands [78],
and CAI [48]. These agents form the foundation of our further empirical evaluation:
- SWE-agent employs a carefully designed Agent-Computer Interface (ACI) that enables au-
tonomous interaction with software repositories through specialized commands for code navigation,
editing, and testing. The framework demonstrates exceptional performance in repository-level vul-
nerability discovery through its integrated file viewer, search capabilities, and context management
system that maintains concise yet informative interaction histories.
, Vol. 1, No. 1, Article . Publication date: October 2025.
LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
11
- OpenHands provides a comprehensive platform for AI software development agents, featuring
an event-stream architecture that facilitates complex multi-turn interactions. The framework
excels in code generation, debugging, and repository manipulation through its sandboxed Docker
environment and extensive tool integration, including IPython execution, web browsing capabilities,
and multi-agent delegation patterns.
- CAI represents a specialized cybersecurity AI framework designed for autonomous penetration
testing and vulnerability assessment. Built around six fundamental pillars (Agents, Tools, Handoffs,
Patterns, Turns, and Human-In-The-Loop), CAI demonstrates superior performance in offensive se-
curity operations, achieving human-competitive results across diverse Capture-The-Flag challenges
and real-world bug bounty scenarios.
To ensure experimental consistency and fair comparison across all subsequent evaluations, we
configure each framework with identical LLM foundations: GPT-4.1[54], Claude-Sonnet-
4[6] and Gemini-2.5-Pro[25] for the following experiments.
Table 4 presents the comprehensive effectiveness evaluation results across these three selected
agent-model combinations. The results reveal significant challenges in automated vulnerability
reproduction, with substantial performance variations across different pipeline stages and agent-
model configurations.
Table 4. Effectiveness results: success rates (%) across vulnerability reproduction pipeline.
Agent
Model
Env Setup
Vulnerability Localization
PoC Generation
End-to-End
Setup
File Level
Func Level
Line Level
Execution
Trigger
Success@1
Success@3
OpenHands
Claude-Sonnet-4
28.8
46.3
36.3
23.8
70.0
21.3
10.0
22.5
Gemini-2.5-Pro
26.3
40.0
27.5
17.5
52.5
10.0
6.3
13.8
GPT-4.1
32.5
53.8
43.8
31.3
56.3
16.3
13.8
20.0
SWE-agent
Claude-Sonnet-4
22.5
50.0
40.0
27.5
58.8
11.3
8.8
15.0
Gemini-2.5-Pro
20.0
37.5
30.0
21.3
56.3
8.8
5.0
11.3
GPT-4.1
25.0
46.3
36.3
25.0
66.3
13.8
11.3
18.8
CAI
Claude-Sonnet-4
35.0
56.3
46.3
32.5
56.3
15.0
7.5
12.5
Gemini-2.5-Pro
38.8
50.0
38.8
26.3
33.8
17.5
3.8
10.0
GPT-4.1
36.3
58.8
48.8
35.0
25.0
20.0
10.0
16.3
Examining the overall end-to-end performance, OpenHands demonstrates the strongest capabili-
ties, particularly when paired with Claude-Sonnet-4, achieving the highest Success@3 rate of 22.5%.
OpenHands maintains consistent superiority across different models, with Success@3 rates of
22.5% (Claude-Sonnet-4), 20.0% (GPT-4.1), and 13.8% (Gemini-2.5-Pro). In comparison, SWE-agent
achieves maximum Success@3 rates of 18.8% (GPT-4.1), while CAI peaks at 16.3% (GPT-4.1). The
Success@1 results show similar patterns, with OpenHands + GPT-4.1 leading at 13.8%, followed
by OpenHands + Claude-Sonnet-4 at 10.0%. Notably, even the best-performing combinations fail
to achieve successful vulnerability reproduction in more than three-quarters of cases, indicating
substantial room for improvement in current agent capabilities.
Environment setup performance varies significantly across agents, with CAI demonstrating the
most consistent results, achieving rates above 35% across all models and peaking at 38.8% with
Gemini-2.5-Pro. OpenHands shows moderate performance ranging from 26.3% to 32.5%, while
SWE-agent exhibits the lowest and most variable setup success rates, spanning 20.0% to 25.0%.
Vulnerability localization reveals a clear performance hierarchy and degradation pattern. CAI
paired with GPT-4.1 leads in localization precision across all granularities: 58.8% file-level, 48.8%
function-level, and 35.0% line-level accuracy. However, the degradation from file-level to line-level
represents a consistent 35–40% relative decrease across all combinations. GPT-4.1 consistently
outperforms other models in localization tasks, while Gemini-2.5-Pro shows the weakest localization
capabilities across all agents.
, Vol. 1, No. 1, Article . Publication date: October 2025.
12
Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
The PoC generation stage reveals OpenHands’ key strength and a critical bottleneck. OpenHands
+ Claude-Sonnet-4 achieves the highest PoC execution rate at 70.0%, significantly outperforming
other combinations. However, a substantial execution-to-trigger gap exists across all agents: while
execution rates range from 25.0% to 70.0%, trigger rates remain uniformly low at 8.8% to 21.3%.
Remarkably, OpenHands + Claude-Sonnet-4 also leads in trigger success at 21.3%, demonstrating
superior PoC quality despite the overall low trigger rates.
Model-specific patterns emerge clearly. GPT-4.1 excels in localization precision but shows moder-
ate end-to-end performance. Claude-Sonnet-4 demonstrates exceptional synergy with OpenHands,
particularly in PoC generation and end-to-end orchestration. Gemini-2.5-Pro consistently under-
performs across most metrics, especially in PoC-related tasks.
The results highlight OpenHands’ superior workflow orchestration capabilities. While CAI
achieves the highest individual stage performance in environment setup and localization, Open-
Hands translates moderate individual performance into the strongest end-to-end success rates,
suggesting more effective inter-stage coordination and error recovery mechanisms.
We further analyze the underlying causes of these performance differences by examining the
tool utilization patterns across agents. Table 5 reveals that while all agents share basic execution
capabilities, OpenHands possesses unique workflow control tools that fundamentally enhance its
vulnerability reproduction effectiveness. The think tool enables structured reasoning processes
that strengthen agent memory and decision-making throughout the multi-stage pipeline, while the
finish tool provides systematic task summarization and completion mechanisms. These capabilities
explain OpenHands’ superior ability to maintain context across pipeline stages and achieve higher
end-to-end success rates despite comparable basic tool availability.
Unexpectedly, CAI’s extensive domain-specific security tool suite remains largely unutilized
in our task specifications, revealing a critical gap between tool availability and practical applica-
tion. While CAI provides numerous specialized cybersecurity tools (such as exploitation, privilege
scalation [48]) theoretically suitable for vulnerability reproduction, the agent fails to effectively
leverage these capabilities within the structured task framework, resulting in lower end-to-end
performance despite superior individual stage metrics in environment setup and localization. Addi-
tionally, OpenHands demonstrates superior compliance with output formatting requirements by
correctly generating artifacts and JSON outputs as specified in task prompts, indicating more robust
instruction following capabilities. This analysis reveals that tool ecosystem effectiveness of agent
depends not merely on capability breadth but on workflow integration, reasoning enhancement,
and systematic task completion mechanisms that enable consistent execution across complex
multi-stage security tasks.
Table 5. The comparison of tool utilization across agent frameworks during task execution.
Tool Type
Tool Capability
CAI
OpenHands
SWE-agent
Basic Execution
execute_bash
✓
✓
✓
str_replace_editor
✓
✓
✓
File Operations
✓
✓
✓
Workflow Control
think (Reasoning Enhancement)
×
✓
×
finish (Task Summary)
×
✓
×
Domain-Specific
Security/Vulnerability Tools
9
×
×
Used in Tasks
0
N/A
N/A
Task Execution
Clear Task Termination
✓
✓
✓
Structured Reasoning
×
✓
×
Correct Artifacts Output
×
✓
×
Proper JSON Format
✓
✓
×
, Vol. 1, No. 1, Article . Publication date: October 2025.
LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
13
Findings for C1: Current LLM agents achieve limited web vulnerability reproduction effective-
ness with end-to-end success rates below 25%. OpenHands demonstrates superior performance
through structured reasoning and systematic completion mechanisms, while tool availability
alone does not ensure effectiveness. The primary bottleneck remains the exploitation gap
where agents can execute PoCs but struggle to trigger actual vulnerabilities.
5.2
C2: Compatibility
Motivation. Real-world vulnerability reproduction environments exhibit significant diversity
across vulnerability types and web technologies. Different vulnerability categories require distinct
understanding patterns, exploitation techniques, and verification approaches. Similarly, vulnerabili-
ties manifest differently across web technologies due to language-specific features, frameworks,
and coding patterns. A practical vulnerability reproduction agent must demonstrate consistent
performance across this diversity rather than excelling only in specific domains. Understanding
compatibility helps assess whether current agents can serve as general tools or require specialized
adaptations.
Approach. As shown in § 3.1, our dataset covers diverse vulnerability types and web technolo-
gies. We conduct categorical evaluation across these dimensions to assess agent compatibility. For
vulnerability type compatibility, we analyze reproduction success rates across the seven vulnerabil-
ity categories in our dataset. For programming language compatibility, we evaluate performance
across the six primary languages. To ensure fairness and reproducibility, we use the top-performing
combination from § 5.1, specifically OpenHands with Claude-Sonnet-4.
Results. Table 6 presents stage-wise success rates across different vulnerability types. The results
reveal substantial performance variations, with end-to-end success rates (Success@3) ranging from
0% to 63%. Library-based Prototype Pollution achieves the highest reproduction success rate at
63%. Among service-based web vulnerabilities, CSRF performs best with 60%, followed by Path
Traversal at 36%. Other service-based vulnerabilities demonstrate considerably lower success rates:
XSS at 16%, SSRF at 15%, RCE at 8%, and SQL Injection with the lowest at 0%.
The low success rates for XSS and SSRF reflect agents’ difficulties with complex multi-component
environments requiring database systems, frontend-backend configurations, and middleware ser-
vices. Agents struggle to extract configurations from large repositories, perform data imports, and
handle intricate deployment procedures. SQL Injection faces similar environment setup challenges,
compounded by database-dependent architectures. Beyond environment setup, a critical bottleneck
emerges in vulnerability triggering due to API authentication requirements, as agents without
browser interaction capabilities cannot perform automated login procedures like human users.
In contrast, CSRF achieves perfect file localization due to clear vulnerability descriptions enabling
precise interface targeting. Prototype Pollution maintains consistent performance across stages,
benefiting from simpler library-based repositories requiring only dependency installation rather
than complex infrastructure.
While PoC execution rates remain high across vulnerability types, the dramatic drop from
execution to trigger phases reveals that agents can run exploit code but struggle with the precise
environmental conditions and interaction sequences needed for actual vulnerability manifestation.
Figure 3 demonstrates end-to-end reproduction success rates across web technologies. PHP
achieves the highest success rate at 38.9%, while TypeScript shows the lowest at 11.1%. PHP’s
superior performance stems from its prevalence in web security research, providing agents with
abundant training data and well-documented exploit patterns, combined with its simpler syntax and
direct web-oriented nature. JavaScript demonstrates strong performance through extensive client-
side vulnerability examples and straightforward DOM manipulation patterns. TypeScript’s poor
, Vol. 1, No. 1, Article . Publication date: October 2025.
14
Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
Table 6. Vulnerability reproduction success rates by vulnerability types.
Vulnerability Type
Environment
Code Localization
PoC Generation
End-to-End
Setup
File
Function
Line
Execution
Trigger
Success@1
Success@3
PROTOTYPE_POLLUTION
75%
88%
75%
63%
88%
50%
38%
63%
CSRF
80%
100%
80%
60%
80%
60%
40%
60%
PATH_TRAVERSAL
45%
82%
73%
45%
82%
27%
18%
36%
XSS
21%
53%
37%
21%
79%
11%
5%
16%
SSRF
15%
31%
23%
8%
77%
8%
0%
15%
RCE
8%
17%
8%
8%
67%
8%
0%
8%
SQLI
8%
0%
0%
0%
25%
0%
0%
0%
TOTAL
29%
46%
36%
24%
70%
18%
10%
23%
PHP
JavaScript
Java
Python
TypeScript
Others
0
5
10
15
20
25
30
35
40
End-to-End Success Rate (%)
38.9%
23.5%
18.2%
22.2%
11.1%
14.3%
Average: 22.5%
Fig. 3. Comparison of vulnerability reproduction success rates between different web technologies.
performance shows that its additional complexity and stricter syntax requirements hinder agents
despite its similarity to JavaScript. The results indicate that agent performance varies significantly
across both vulnerability types and web technologies, with success rates influenced by environment
complexity and the specific requirements of each vulnerability category.
Findings for C2: Agent performance exhibits substantial variation across vulnerability types
and web technologies. Service-based vulnerabilities face significant challenges from complex
environment setup, authentication requirements, and multi-component dependencies, while
library-based vulnerabilities achieve higher success through simpler deployment requirements.
Language compatibility depends on security research prevalence and syntactic complexity,
with web-oriented languages showing better agent adaptation.
5.3
C3: Efficiency
Motivation. While effectiveness measures reproduction success rates, efficiency evaluates the
computational and economic costs of achieving successful reproductions. Understanding resource
requirements is crucial for practical deployment, as organizations must balance reproduction
capabilities with operational costs. We analyze efficiency across successful reproduction attempts
to assess real-world viability of different agent-model combinations.
Approach. We adopt the efficiency evaluation methodology from § 4.2, measuring token con-
sumption, execution time, and monetary cost across the vulnerability reproduction pipeline. To
do so, we focus exclusively on the successful reproduction cases achieved by our three founda-
tional agents as identified in § 5.1. To ensure a fair comparison, all tests are conducted using the
Claude-Sonnet-4 model, and measurements are averaged across successful attempts for each agent.
, Vol. 1, No. 1, Article . Publication date: October 2025.
LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
15
Results. Table 7 presents efficiency metrics for all three agents calculated from successful
reproduction attempts. Environment setup demonstrates the highest resource requirements across
all agents, while vulnerability localization proves most economical. End-to-end reproduction costs
range from $1.68 to $2.19 with execution times between 25.7-34.0 minutes. OpenHands consistently
requires the highest resources ($2.19 total), followed by SWE-agent ($1.90), while CAI demonstrates
the most efficient utilization ($1.68).
Table 7. Efficiency analysis of successful vulnerability reproduction cases.
Agent
Env
Loc
PoC
E2E
T
Ti
C
T
Ti
C
T
Ti
C
T
Ti
C
OpenHands
78.3
18.2
1.17
15.2
4.3
0.23
52.7
11.5
0.79
146.2
34.0
2.19
SWE-agent
65.4
15.8
0.98
12.8
3.7
0.19
48.9
10.2
0.73
127.1
29.7
1.90
CAI
56.8
13.1
0.85
11.1
3.2
0.17
44.2
9.4
0.66
112.1
25.7
1.68
Env: Environment Setup, Loc: Vulnerability Localization, PoC: PoC Generation, E2E: End-to-End Reproduction.
T: Tokens (K), Ti: Time (min), C: Cost ($). Averages from successful cases only.
Costs based on Claude Sonnet pricing ($3/1M input, $15/1M output tokens).
The cost variations stem from architectural differences between agents. OpenHands incorpo-
rates extensive internal reasoning including explicit "think" and "finish" operations that generate
substantial token overhead. SWE-agent employs numerous system prompts to enforce tool usage
compliance, creating intermediate communication costs. CAI, despite having the most comprehen-
sive tool suite, exhibits minimal tool invocation in vulnerability reproduction tasks, resulting in
streamlined execution. These design choices create a trade-off between computational efficiency and
reproduction capability, with OpenHands prioritizing thorough reasoning, SWE-agent emphasizing
tool compliance, and CAI optimizing for direct execution.
Findings for C3: Web vulnerability reproduction requires substantial computational resources,
with end-to-end costs ranging from $1.68 to $2.19 per successful case. Environment setup
consumes the most resources across all agents, while agent architectural differences create
significant efficiency variations in the reproduction pipeline.
5.4
C4: Robustness Analysis
Motivation. Robustness measures how well agents perform under challenging conditions. In
vulnerability reproduction, agents face obstacles like limited processing capabilities, browser secu-
rity restrictions, or incomplete information. These challenges are inherent to real-world security
research environments where perfect conditions rarely exist. A robust agent should maintain rea-
sonable performance across different scenarios, adapting to unexpected failures and environmental
constraints. Understanding robustness helps us assess agent reliability for practical deployment
and identify which architectures handle reproduction challenges most effectively.
Approach. We design two experiments to test the robustness of our three selected agents
across key challenge areas: First, we compare agent performance using Claude-Sonnet-4 with and
without enhanced reasoning across different vulnerability reproduction stages in 10 vulnerabilities.
Second, we test agent performance under different authentication scenarios using 12 SQL Injection
vulnerabilities, comparing three cases: manually providing login tokens, prompting agents to login
themselves, and providing no authentication help.
Results. Figure 4 shows mixed results when comparing models with and without enhanced rea-
soning. Enhanced reasoning helps in early stages, with OpenHands improving from 60% to 70% in
, Vol. 1, No. 1, Article . Publication date: October 2025.
16
Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
Environment Setup and from 40% to 70% in File Localization, while improvements decrease in later
stages with PoC Execution showing smaller gaps and Trigger Success showing no improvement
across all agents. SWE-Agent shows the most consistent benefit from enhanced reasoning across
localization stages, while OpenHands demonstrates strong baseline performance even without
reasoning due to its built-in "think" operations, and CAI shows more variable results with minimal
reasoning benefits. The limited reasoning impact reflects both agent design differences and task
complexity variations, as reasoning enhancements prove most effective for analytical tasks like code
localization but provide little benefit for execution-oriented activities like vulnerability triggering.
Within our 10-vulnerability sample, agents demonstrate robustness across both reasoning config-
urations, suggesting that architectural design choices may be more influential than underlying
model reasoning capabilities for practical vulnerability reproduction tasks.
Environment
File Localization
Function Localization
Line Localization
PoC Execution
Trigger Success
0
10
20
30
40
50
60
70
80
Success Rate (%)
Openhands (w/ reasonning)
Openhands (w/o reasonning)
SWE-Agent (w/ reasonning)
SWE-Agent (w/o reasonning)
CAI (w/ reasonning)
CAI (w/o reasonning)
Fig. 4. Performance comparison between agents using Claude-Sonnet-4 with and without reasoning across
different vulnerability reproduction stages.
Table 8 shows authentication context impacts vulnerability reproduction success. Manual token
provision works best, with Openhands + Claude-Sonnet-4 achieving 67% success. Requiring agents
to handle login autonomously results in an average performance degradation of 33.3% compared to
manual token provision, with success rates dropping to 25-50% across agent-model combinations.
Removing authentication guidance causes near-complete failure, with only three configurations
maintaining 8% success rates. This reveals that current agents struggle with autonomous authentica-
tion discovery and rely heavily on explicit guidance. The consistent degradation across architectures
and models indicates that authentication handling is a widespread limitation.
Findings for C4: Current agents show robustness to reasoning changes but high sensitivity to
prompt variations. While reasoning versus non-reasoning modes produce similar performance,
incomplete authentication results in severe performance drops, highlighting agents’ limited
capacity for problem-solving without explicit guidance.
6
Discussion
Our empirical evaluation reveals that current open-source agents face substantial limitations
in vulnerability reproduction beyond specific ecosystems. While agents demonstrate reasonable
success in library-based JavaScript environments and relatively straightforward CSRF scenarios,
they struggle significantly with complex web vulnerabilities that require intricate environment
, Vol. 1, No. 1, Article . Publication date: October 2025.
LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?
17
Table 8. Comparison of trigger rate in authentication robustness analysis.
Agent
Model
Manual Token
Agent Login
No Auth Info
Openhands
Claude-Sonnet-4
67%
50%
8%
Gemini-2.5-Pro
58%
42%
8%
GPT-4.1
50%
33%
0%
SWE-Agent
Claude-Sonnet-4
58%
33%
8%
Gemini-2.5-Pro
50%
25%
0%
GPT-4.1
42%
17%
0%
CAI
Claude-Sonnet-4
50%
25%
0%
Gemini-2.5-Pro
42%
17%
0%
GPT-4.1
33%
8%
0%
setup and multi-step exploitation chains. The stark performance gaps—ranging from 0% for SQL
injection to 63% for prototype pollution—indicate that pure code-based approaches are insufficient
for comprehensive vulnerability reproduction. Traditional injection vulnerabilities, despite their
well-documented nature, consistently fail due to environment configuration challenges and the
complex interplay between application deployment, database setup, and exploitation context. This
suggests that current agent architectures fundamentally lack the systems-level understanding
necessary for realistic security testing scenarios.
The integration of Model Context Protocol (MCP)[1] with browser automation tools like Play-
wright or Chrome presents a breakthrough for vulnerability reproduction. By enabling agents
to control browser instances, capture network traffic, and monitor application state changes in
MCP-based approaches can bridge the gap between static code analysis and dynamic exploitation.
This browser-centric methodology allows agents to observe runtime behavior of web applications,
intercept HTTP requests and responses, and validate success through DOM manipulation and state
inspection. The ability to interact with web interfaces while monitoring network activity represents
a shift from traditional code-only approaches, unlocking automated reproduction capabilities for
complex web vulnerabilities that have resisted agent-based solutions.
7
Threats to Validity
Internal Validity. The primary internal threat stems from potential data leakage in LLM training
datasets. Many CVEs in our benchmark were disclosed before the training cutoff dates of evaluated
models, creating a risk that agents may leverage memorized exploit patterns rather than demon-
strating genuine reasoning capabilities. Although we deliberately avoided referencing specific
PoCs in our prompts, models may still access historical vulnerability data through their pre-trained
knowledge, potentially inflating performance metrics. Additionally, our evaluation employs binary
success metrics that only capture whether vulnerability reproduction succeeds or fails, providing a
single-dimensional assessment that may miss important aspects of the reproduction process.
External Validity. Our findings may have limited generalizability due to several factors. First,
our CVE selection, while diverse across vulnerability types and web technologies, represents a
curated subset that may not fully reflect the complexity distribution of real-world security scenarios.
Second, the containerized evaluation environment, despite efforts toward realism, may still differ
from production deployments in ways that affect agent performance. Finally, the rapid evolution of
LLM capabilities means our findings represent a temporal snapshot that may not predict future
agent performance across different models or architectures.
, Vol. 1, No. 1, Article . Publication date: October 2025.
18
Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang
8
Related Work
8.1
Software Engineering Benchmarks for LLM Agents
Software engineering represents a critical application domain for evaluating LLM agents, with
benchmarks spanning different complexity levels and task types. SWE-bench [37] evaluates agents’
abilities to solve real-world GitHub issues by generating patches for bug fixes, representing one
of the most challenging repository-level tasks that require understanding complex codebases and
coordinating changes across multiple files. Traditional coding benchmarks like HumanEval fo-
cus on isolated function generation from natural language descriptions, while MBPP evaluates
basic Python programming tasks. Additional benchmarks including BigCodeBench [92], Live-
CodeBench [35], and EvalPlus [43] extend evaluation across multiple languages and enhanced test
coverage. However, these SE benchmarks exhibit significant limitations: many focus on narrow
task scopes rather than diverse professional tasks, traditional benchmarks often involve only local
edits to individual functions rather than repository-wide reasoning, and they frequently ignore
autonomous environment setup and end-to-end task solving capabilities essential for practical soft-
ware engineering. Recent advances like AgentBench [44] provide more comprehensive evaluation
across diverse environments, highlighting the evolution toward more realistic and challenging
evaluation frameworks for software engineering agents.
8.2
Cybersecurity Benchmarks for LLM Agents
Various specialized benchmarks have been developed to evaluate LLM agents in cybersecurity.
InterCode-CTF [85] manually evaluated LLMs on Capture-the-Flag challenges requiring reverse
engineering and exploitation skills, while NYU CTF Bench [64] expanded this with an automated
framework for diverse offensive security tasks across cryptography, forensics and binary exploita-
tion. The CyberSecEval series [10, 73] focuses on evaluating cybersecurity risks including prompt
injection, code interpreter abuse and automated social engineering. ARVO [50] provides repro-
ducible memory vulnerabilities with triggering inputs and developer patches. SEC-bench [38]
evaluates PoC generation and vulnerability patching tasks using data from OSS-FUZZ project,
SecVulEval [3] benchmarks statement-level vulnerability detection in C/C++ code. CyberGym [79]
assesses vulnerability reproduction across large software projects. Most existing benchmarks exhibit
limitations in capturing real-world cybersecurity complexity, often focusing on isolated tasks or
synthetic environments. The most closely related work is CVE-bench [76], which remains limited
in both scope and scale, capturing only a narrow slice of agent performance in web vulnerability
reproduction scenarios.
9
Conclusion
Our evaluation of 20 LLM agents across 16 core dimensions on 3 representative CVEs identified three
top performers (OpenHands, SWE-agent, and CAI) for testing on 80 real-world CVEs spanning 6
web technologies and 7 vulnerability types. Results show agents handle simple library-based vulner-
abilities well but consistently fail on complex service-based scenarios requiring multi-component
environments and authentication. The gap between executing exploit code and triggering actual vul-
nerabilities reveals fundamental limitations in environmental adaptation. These findings highlight
the need for significant advances before automated vulnerability reproduction becomes practical.
Current agents lack the robustness required for real-world deployment, showing high sensitivity
to input guidance and poor performance under incomplete information.
Data Availability
The artifact of this paper can be accessed via https://figshare.com/s/7e55eaaaca0b0146ee62.
, Vol. 1, No. 1, Article . Publication date: October 2025.
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19
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LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? BIN LIU, Harbin (LLM) agents have demonstrated remarkable capabilities in software engineering and cybersecurity tasks, including code generation, vulnerability discovery and automated testing. One critical but underexplored application is automated web vulnerability reproduction, which transforms vulnerability reports into working exploits. Although recent advances suggest promising potential, significant challenges remain in applying LLM agents to real-world web vulnerability reproduction scenarios. In this paper, we present the first comprehensive evaluation of state-of-the-art LLM agents for automated web vulnerability reproduction. We first systematically assess 20 agents from software engineering, cybersecurity, and general domains across 16 dimensions, including technical capabilities, environment adaptability, and user experience factors, on 3 representative web vulnerabilities. Based on the results, we select three topperforming agents (OpenHands, SWE-agent, and CAI) for in-depth evaluation on our constructed benchmark dataset of 80 real-world CVEs spanning 7 vulnerability types and 6 web technologies. Our results reveal that while LLM agents achieve reasonable success on simple library-based vulnerabilities, they consistently fail on complex service-based vulnerabilities requiring multi-component environments. Furthermore, complex environment configurations and authentication barriers create a critical gap where agents can execute exploit code but fail to trigger actual vulnerabilities. We observe that agents demonstrate high sensitivity to input guidance, with performance degrading by over 33.3% under incomplete authentication information. Our findings highlight the significant gap between current LLM agent capabilities and the demands of reliable automated vulnerability reproduction, emphasizing the need for advances in environmental adaptation and autonomous problem-solving capabilities. CCS Concepts: • Software and its engineering; Additional Key Words and Phrases: Web Vulnerability, LLM Agents, Vulnerability Management ACM Reference Format: Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang. 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet?. 1, 1 (October 2025), 22 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn 1 Introduction Recently, Large language model (LLM) agents have achieved remarkable success across software engineering and cybersecurity domains. In software engineering, these agents have demonstrated strong capabilities in code generation [15, 19, 31], bug fixing [36, 83, 90], issue resolution [37, 86] Authors' Contact Information: Bin Liu, Harbin ; Yanjie Zhao, Huazhong ; Guoai Xu, Harbin ; Haoyu Wang, Huazhong . Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from . Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM XXXX-XXXX/2025/10-ART https://doi.org/10.1145/nnnnnnn.nnnnnnn , Vol. 1, No. 1, Article . Publication date: October 2025. 16 Oct 2025 2 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang and code review [24, 71]. In cybersecurity, they have shown impressive performance in vulnerability detection [30, 41, 68], fuzzing [18, 47, 74], exploit generation [17, 80], malware analysis [11, 49] and penetration testing [28, 40]. These successes demonstrate their ability to understand complex technical contexts, reason about software behavior, and automate sophisticated workflows. Web applications have become the backbone of modern digital infrastructure, representing the dominant form of deployed applications in enterprise environments [69]. This widespread adoption has made them prime targets for cyberattacks, with web vulnerabilities consistently dominating security incident reports [72]. The OWASP Top 10 identifies critical issues such as SQL injection, Cross-Site Scripting (XSS), and Cross-Site Request Forgery (CSRF) as the majority of reported vulnerabilities [55], while the National Vulnerability Database confirms that web application vulnerabilities represent the fastest-growing category of security threats [52]. However, effectively managing these web vulnerabilities relies heavily on successful reproduction, which involves transforming vulnerability reports into working Proof-of-Concept (PoC) exploits that validate threats and enable effective patching. This process presents unique challenges that are different from other security tasks where LLM agents have succeeded. Manual reproduction requires expertise in environment configuration, dependency resolution, authentication mechanisms, and exploit crafting, often taking days or weeks per vulnerability [20, 61]. Automated reproduction faces additional complexities: software version incompatibilities, incomplete PoC code, complex deployment, and network setups. These factors lead to low success rates in existing automated reproduction methods, highlighting the substantial technical barriers. While LLM agents have remarkable capabilities in related security and software engineering tasks, their effectiveness for comprehensive web vulnerability reproduction remains underexplored. Existing cybersecurity benchmarks for LLM agents suffer from significant limitations that fail to capture real-world complexity. Many evaluations rely on simplified Capture-the-Flag (CTF) scenarios [10, 64, 73, 85, 88] that focus on individual challenges rather than end-to-end reproduction processes. Additionally, current benchmarks often evaluate agents on small, isolated code snippets or fragments, limiting their scope to specific programming languages such as C/C++ [7, 38, 50, 79], Java [42] or JavaScript [14], rather than the diverse components of production web applications. However, end-to-end vulnerability reproduction requires agents to handle complex multi-step workflows involving environment setup, dependency management, and real-world deployment uncertainties. These capabilities extend far beyond current simplified evaluation scenarios, creating a significant gap in agent ability for practical vulnerability reproduction applications. To address this gap, we present the first comprehensive empirical study examining LLM agents' ability for automated web vulnerability reproduction. We systematically evaluate 20 representative agents across 16 core dimensions on 3 representative CVEs, then conduct in-depth assessment of the top 3 agents using our constructed benchmark dataset of 80 real-world CVEs. We assess current agent capabilities and identify fundamental limitations that prevent reliable automated vulnerability reproduction. In summary, our main contributions include: • We construct a comprehensive benchmark dataset comprising 80 real-world CVEs covering 7 vulnerability types and 6 web technologies with complete reproduction environments. • We conduct the first systematic evaluation of 20 state-of-the-art LLM agents for automated web vulnerability reproduction across 16 core dimensions on 3 representative CVEs, establishing a rigorous evaluation framework. From this evaluation, we select the three top-performing agents (OpenHands, SWE-agent, and CAI) for in-depth analysis. • We provide critical empirical insights revealing the significant gap between current LLM agent capabilities and practical requirements, identifying key failure modes and documenting over 33.3% performance degradation under incomplete authentication information. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 3 2 Background 2.1 LLM Agents LLMs exhibit impressive performance across multiple domains, including natural language comprehension, code synthesis, and reasoning processes. Based on these foundational abilities, researchers have developed LLM-based agent systems [39, 65, 75] that leverage LLMs as core reasoning engines for autonomous decision-making, while utilizing various tools and operations to interact with external environments in order to accomplish sophisticated tasks. 2.1.1 Core Mechanisms. LLM agents combine LLM reasoning with action execution capabilities through several key mechanisms. ReAct [87] introduces iterative reasoning-action cycles, where models generate reasoning traces before executing actions, with results integrated into context for subsequent cycles. Toolformer [63] demonstrates autonomous tool usage, enhancing factual accuracy and computational abilities. For complex tasks, simple reactive loops prove insufficient, requiring explicit planning capabilities. Early systems like BabyAGI [51] and Auto-GPT [26] introduced dynamic task lists where agents execute, reorderand generate tasks continuously. Advanced frameworks such as AgentVerse [16] and ChatDev [62] adopt multi-agent architectures where controller agents decompose goals into subtasks distributed to task-specific agents, enabling coordination across different domains. Reflexion [66] addresses failure handling through reflection mechanisms that analyze past mistakes and store self-reflections in memory for improving future planning decisions. This self-correction capability enables iterative problem-solving and strategy refinement over time. 2.1.2 Representative Agents and Frameworks. LLM agents have emerged across diverse application domains. Software engineering agents assist with code generation, debugging, and repository-level analysis. Cybersecurity agents focus on threat detection, vulnerability analysis, and automated penetration testing. General-purpose agents aim for broad applicability, emphasizing flexible tool use and adaptive reasoning across various tasks. Additionally, agent orchestration frameworks like LangChain [13] and AutoGen [81] provide infrastructure for building and coordinating multi-agent systems, enabling developers to construct domain-specific agents with standardized tooling and communication protocols. A summary of representative agents and frameworks is shown in Table 1. Table 1. Representative LLM agents and orchestration frameworks across different domains. Domain Agent Year Domain Agent Year Software Engineering AGENTLESS [82] 2024 Cybersecurity PentestGPT [17] 2023 OpenHands [78] 2024 CAI [48] 2024 SWE-agent [84] 2024 PentestAgent [59] 2024 AutoCodeRover [89] 2024 Nebula [53] 2024 Aider [23] 2023 HackingBuddyGPT [27] 2023 RepoAgent [45] 2024 PentAGI [57] 2024 ChatDev [62] 2023 EnIGMA [2] 2024 MetaGPT [29] 2024 AI-OPS [4] 2024 General-Purpose Agents AutoGPT [26] 2023 Orchestration Frameworks LangChain [13] 2022 BabyAGI [51] 2023 AutoGen [81] 2023 AgentVerse [16] 2023 CrewAI [32] 2023 AIlice [5] 2024 LangGraph [33] 2024 2.2 Web Vulnerability Reproduction Web vulnerability reproduction involves constructing operational PoC exploits from natural language vulnerability reports to validate security flaws and guide remediation efforts. , Vol. 1, No. 1, Article . Publication date: October 2025. 4 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang The common approach relies on manual analysis by security researchers using tools like Burp Suite [60] or ZAP [56] to intercept, analyze and craft HTTP/S requests. While effective for complex logic-based flaws, this method suffers from inefficiency and high labor costs that limit scalability. Semi-automated systems target common vulnerability classes like XSS or SQL Injection by using NLP to extract key entities from reports and inserting them into predefined attack templates [9]. Advanced works like Wang et al. [77] extract contextual information for targeted web fuzzing. Commercial tools like Nessus [70] and Acunetix [34] represent this approach. While faster for simple cases, these methods lack robustness for vulnerabilities requiring complex, multi-step interactions. Program analysis based automated exploit generation (AEG) [8] achieved significant success in binary applications through symbolic execution. Frameworks like Mayhem [12] and angr [67] exemplify this approach. However, their application to web vulnerabilities is limited due to the immense state space of web applications and inability to reason about high-level logic flaws. LLM-based methods represent the newest paradigm, framing reproduction as an agentic problem where LLMs act as reasoning engines. LLMs have been explored for exploiting vulnerabilities in web applications based on known vulnerability descriptions with one-day setting [21, 22] and investigating agent teams with hierarchical planning for zero-day scenarios [91]. Recent work [80] showed AI systems can generate working exploits for published CVEs in 10-15 minutes, while systems like PentestGPT [17] and PentestAgent [40] demonstrate autonomous penetration testing capabilities. This methodology combines high-level contextual reasoning with machine execution speed and scalability. All the above methods are shown in Table 2. Table 2. Web vulnerability reproduction methods and tools, including manual, semi-automated, program analysis-based, and LLM-based methods. Tool Category Year Tool Category Year Burp Suite [60] Manual 2003 AEG [8] Program analysis based 2011 OWASP ZAP [56] 2010 Mayhem [12] 2012 Nessus [70] Semi-automated 2024 angr [67] 2016 Acunetix [34] 2024 AutoExploit [46] 2018 Website Vulnerability Scanner [58] 2024 PentestGPT [17] LLM-based 2023 Wang et al. [77] 2023 PentestAgent [40] 2024 Aydin et al. [9] 2014 2.3 Motivation Automated web vulnerability reproduction remains a critical bottleneck in security operations. Manual analysis lacks scalability, semi-automated tools are brittle with poor comprehension, and formal methods like AEG cannot handle web application complexity. This validation gap prevents security teams from efficiently confirming and prioritizing threats. LLM-based agents offer a promising solution through semantic comprehension, logical reasoning, and tool invocation capabilities that address existing limitations. Recent advances have produced general-purpose agents (as shown in Table 1) claiming broad problem-solving capabilities, and specialized vulnerability analysis agents (as illustrated in Table 2) targeting security workflows specifically. However, despite theoretical promise and agent proliferation, the practical effectiveness of LLM agents in web vulnerability reproduction lacks empirical validation. This creates a critical gap between claimed potential and demonstrated capability in this demanding security domain. We conduct the first systematic empirical study evaluating LLM agents' ability to reproduce web vulnerabilities from natural language descriptions. Our comprehensive framework , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 5 assesses success rates, reasoning processes, and failure modes, establishing empirical foundations for understanding current capabilities and limitations in automated vulnerability validation. 3 Empirical Study Design We design a comprehensive empirical study to systematically evaluate LLM agents for web vulnerability reproduction. Our study follows the workflow illustrated in Figure 1. Our study evaluates agents according to the following four criteria: C1: Effectiveness. We compare the effectiveness of different agents on web vulnerability reproduction by measuring success rates across individual stages (environment setup, vulnerability localization, and PoC generation with verification) and full end-to-end reproduction. C2: Compatibility. We evaluate the adaptability of different agents across various vulnerability types and web technologies. C3: Efficiency. We compare the efficiency of different agents by measuring execution time, USD cost, and token consumption, which are critical factors for the practical deployment of LLM agents. C4: Robustness. We evaluate the robustness of different agents by analyzing their performance under varying conditions, including model capabilities and input quality. § 3.1 Data Preparation Vulnerability Database & Code Repository § 3.2 Agent Selection § 3.3 Task Specification 80 Web Vulnerability-PoC Datasets 7 Vulnerability Types API Fetch Vulnerability Report Pre-patch Codebase Manual Construction /source docker -compose.yml Dockerfile PoC exploits & Oracle Metadata Scripts GPT-4.1 Claude-Sonnet-4 Gemini-2.5-pro Foundation LLMs Integration § 4 & § 5 Experimental Setup & Results Target Web Application Container Library Application Web Application Other Configurations Database Input Env setup task Localization task Generation & Attack PoC gen task Reproduction task Output Example { "stage": "e2e", "success": , "files": [{"path": "./.bench/e2e/Dockerfile",...}, ... {"path": "./.bench/e2e/steps.md",...}], "script_files": { "./.bench/e2e/run.sh": "commands"}, "run_commands": [ "bash ./.bench/e2e/run.sh"], } C1: Effectiveness C2: Compatibility C3: Efficiency C4: Robustness Evaluation Criteria Agents 3 agents selected from 20 candidates for further analysis Fig. 1. Systematic workflow for evaluating LLM agents in web vulnerability reproduction tasks. 3.1 Data Preparation To effectively evaluate the performance of LLM agents in vulnerability reproduction, we have curated a benchmark dataset comprising real-world web application vulnerabilities. Our dataset construction follows a structured multi-round filtering strategy to ensure reliability and reproducibility across diverse security scenarios in the real world. Data Collection Process. We employ a systematic multi-round filtering strategy for data collection. Initially, we search the OSV database's Git ecosystem for service-based web vulnerability keywords, including SSRF, CSRF, SQL Injection, Prototype Pollution, Path Traversal, Remote Code Execution, and Cross-Site Scripting, limiting our scope to vulnerabilities disclosed after 2024. This initial search yields 744 candidate entries. To ensure manageable experimental environments, we further filter out projects with repository sizes exceeding 30MB, resulting in 224 remaining , Vol. 1, No. 1, Article . Publication date: October 2025. 6 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang candidates. Finally, we apply three stringent criteria for final selection: (1) only vulnerabilities affecting web-based applications are included, (2) vulnerabilities must be available in accessible Git repositories where the specific vulnerable commit can be checked out with complete source code access, and (3) vulnerability reports must be disclosed after 2024 to reflect current threat landscapes. Through this systematic filtering process, we construct a high-quality dataset comprising 80 vulnerabilities. Vulnerability Type Selection. Our curation process focuses on 7 critical web vulnerability types strategically selected to comprehensively evaluate agent capabilities across different vulnerability paradigms. We include six mainstream service-based vulnerability types: Cross-Site Request Forgery (CSRF), Path Traversal, Remote Code Execution (RCE), SQL Injection (SQLI), Server-Side Request Forgery (SSRF) and Cross-Site Scripting (XSS). These represent the most prevalent web application security flaws that occur at the service level. Additionally, to provide a more holistic assessment of agent performance, we incorporate Prototype Pollution as a representative librarybased vulnerability type that manifests through dependencies and third-party packages rather than direct application code. This approach ensures our evaluation captures both traditional web service vulnerabilities and modern JavaScript ecosystem threats. Environment Construction. For each selected vulnerability, we execute a comprehensive data preparation pipeline. After fetching the vulnerability reports via APIs, we identify the corresponding pre-patch codebase from the linked Git repositories and check out the exact vulnerable commit. We utilize Git's diff capabilities to identify the precise code changes between vulnerable and patched versions, allowing us to pinpoint the exact vulnerable code sections, including specific files, functions and line numbers. We then construct fully containerized environments using Docker, manually developing Dockerfile and docker-compose.yml configurations for each vulnerability to ensure reproducible deployment across different systems. Each environment is thoroughly tested to verify successful vulnerability reproduction before exploit development. PoC Development. For PoC exploit construction, we develop PoC exploits based on existing vulnerability reports. When existing PoCs are available in the reports, we adapt and refine them for our containers; otherwise, we manually create new exploits from scratch. Our PoC development follows vulnerability-specific methodologies: CSRF vulnerabilities are implemented using HTMLbased attack vectors, Prototype Pollution exploits are crafted in JavaScript to demonstrate object manipulation, while other vulnerability categories utilize Python-based HTTP request construction for systematic exploitation. Each PoC is accompanied by automated verification oracles that confirm successful exploitation, enabling reliable automated assessment. Our construction process generates dataset entries containing vulnerable source code directories, Docker configuration files, environment setup scripts, executable PoC exploit code with verification oracles, and structured metadata including CVE identifier, vulnerability description, affected file paths, vulnerable function names, specific line numbers, corresponding GitHub repository URLs, vulnerable commit hashes and temporal information. Our final dataset comprises 80 vulnerabilities spanning 6 popular web technologies including Python, PHP, Java, JavaScript, TypeScript, and Go, covering 7 vulnerability types. 3.2 Agent Selection Recall that in Table 1, we listed 20 representative agents. These agents share common characteristics: they are open-source projects available on GitHub with over 1,000 stars for software engineering/general-purpose agents and over 100 stars for cybersecurity agents, released or updated after 2023, and demonstrating active maintenance. All agents demonstrate basic operational capabilities for automated task execution. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 7 Initial Capability Assessment. We first conduct a preliminary capability assessment of these 20 representative agents using 16 core dimensions and 3 representative CVEs for testing. The 16 evaluation dimensions include source code access, tool integration, script execution, file operations, web requests, security domain focus, custom structured output, custom task support, runtime stability, error handling, session management, efficiency metrics, log persistence, latest LLM support, user interface, and installation convenience. We select 3 representative CVEs for testing: CVE2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and CVE-2024-41112 (RCE), which represent different vulnerability types and complexity levels. Agent Selection Results. We conduct a comprehensive evaluation of these 20 agents spanning multiple architectures and application domains, with detailed results presented in Table 3 (§ 5.1). Through this evaluation round, we identify the 3 best-performing agents. Subsequently, we conduct an in-depth empirical evaluation of these 3 selected agents using the complete 80-vulnerability dataset described in § 3.1. Therefore, our further empirical evaluation focuses on these three promising agents that demonstrate adequate capabilities and performance. Outputs (all saved under ./.bench/e2e/): - Intermediate artifacts - steps.md (the reproduction steps and key commands). - result.json (consolidated end-to-end result). User Prompt You are participating in web vulnerability reproduction in a safe container. Rules: - Never invent placeholder hostnames. Derive service_url from docker-compose port mapping (localhost) or explicit run.sh output. - You can edit/create files under the workspace and execute shell commands. - Do NOT modify any source code files; only write under {codebase_dir} Stage: End-to-End Reproduction. Steps: Starting from the CVE folder and the codebase, autonomously: 1) Build/run the environment from scratch based on c 2) Localize the vulnerable code region (files/functions/lines) 3) Generate a PoC and an in-code verification oracle under ./.bench/e2e/ to assess exploit success. 4) Execute the PoC, collect evidence/logs/state, and decide success/failure. 5) Produce a concise step-by-step reproduction document (./.bench/e2e/steps.md). Outputs (all saved under ./.bench/e2e/): - Intermediate artifacts - steps.md (the reproduction steps and key commands). - result.json (consolidated end-to-end result). Response.png Agent { "stage" : "e2e", "success": , "files": [ {"path": "./.bench/e2e/Dockerfile", "status": "created|updated|copied"}, ], "run_commands": [ "bash ./.bench/e2e/run.sh" ], "artifacts": { "service_url": "http://localhost: ", "exec_success": , "signals": {"status": 200, "log_regex": "..."}, "observed": {"status": 200, "log": "..."} }, "steps": [ {"name": "env_setup", "success": }, {"name": "localization", "success": }, {"name": "poc_and_verify", "success": , "notes": "..."} ], ... } JSON Response Vulnerability Information Reproduction Requirements Output Schema Fig. 2. Prompt template and example agent response. 3.3 Task Specification We design four evaluation tasks of increasing complexity to systematically assess agent capabilities across the vulnerability reproduction pipeline. The first three tasks are independent, single-stage evaluations where each task receives expert-curated ground truth from prerequisite stages as input, ensuring that only the current stage's capabilities are measured. The final task represents a comprehensive end-to-end evaluation requiring agents to autonomously navigate the complete vulnerability reproduction workflow. Environment Setup. This foundational task evaluates an agent's ability to establish a functional testing environment from source code alone. Given only the vulnerable application's codebase, agents must generate the necessary containerization files (Dockerfile, docker-compose.yml) and deployment scripts (run.sh) to create a running instance container of the application. , Vol. 1, No. 1, Article . Publication date: October 2025. 8 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Vulnerability Localization. With access to both the running environment and source code, agents must perform code analysis to pinpoint the vulnerability's location. Given the CVE description, agents analyze the codebase to produce a ranked list of suspicious files, functions, and specific line numbers that correspond to the security flaw. Proof-of-Concept (PoC) Exploits Generation. This task tests an agent's exploit development capabilities. Provided with the vulnerability's exact location (file and line number), agents must craft a working Proof-of-Concept exploits that successfully triggers the exploit and includes a verification oracle to programmatically confirm successful exploitation. End-to-End Reproduction. As the most challenging evaluation, this task provides agents only with the source code and original CVE description. Agents must autonomously perform all previous stages-from environment setup through PoC execution-and produce a complete, documented workflow of their reproduction process. To ensure fair evaluation across different agents, we establish a standardized input-output interface that provides consistent task specifications and response formats. Each agent receives identical prompts and must produce outputs in the same structured format, enabling objective comparison of capabilities across all web vulnerability reproduction tasks. As illustrated in Figure 2, the prompt contains structured vulnerability information including CVE category, detailed description, codebase directory, and reproduction steps. The prompt also specifies operational constraints such as workspace restrictions and file access limitations. Agents must respond with structured JSON outputs that include stage indicators, success flags, file paths, execution commands, and artifact metadata. This standardized format enables automated assessment of task completion. It also allows verification that agents produce the required code artifacts including Dockerfiles, execution scripts, PoC exploits, and verification oracles. 4 Experimental Setup 4.1 Studied Dataset and Agents As introduced in § 3.1, we evaluate agent performance on a curated benchmark of 80 real-world vulnerabilities across diverse categories and complexity levels. Following the setup described in § 3.2, we assess the performance of 20 popular LLM agents across software engineering, cybersecurity, and general-purpose, each integrated with three foundation LLMs: GPT-4.1 [54], Claude-Sonnet-4 [6], and Gemini-2.5-Pro [25]. 4.2 Metrics To comprehensively evaluate agent performance across the four criteria (C1-C4), we measure success rates at different granularities and additional efficiency metrics: 4.2.1 Success Rate Metrics. We measure the percentage of successful completions across different stages of the vulnerability reproduction pipeline, with all results verified through manual inspection: • Environment Setup Success Rate: The percentage of vulnerabilities for which an agent successfully generates and deploys a functional vulnerable environment, verified through manual testing of application accessibility and configuration correctness. • Vulnerability Localization Accuracy (Top-K): The percentage of cases where the groundtruth vulnerability location appears in the agent's top-K predictions. We evaluate this metric at three granularity levels: - File-level Top-K: Ground-truth vulnerable file present in top-K file predictions. - Function-level Top-K: Ground-truth vulnerable function present in top-K function predictions. - Line-level Top-K: Ground-truth vulnerable line present in top-K line predictions. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 9 • PoC Generation Success Rate: We decompose this into two complementary sub-metrics: - Execution Success Rate: The percentage of generated PoCs that execute without runtime errors, verified through both automated testing and manual execution. - Vulnerability Trigger Success Rate: The percentage of executable PoCs that successfully trigger the vulnerability, confirmed through manual verification of exploitation effects. • End-to-End Success Rate: The percentage of vulnerabilities for which an agent successfully completes the entire reproduction workflow from environment setup to verified exploitation, with manual confirmation of successful vulnerability triggering. 4.2.2 Efficiency Metrics. To assess operational costs and practical deployment considerations, we also measure: • Execution Time: Total wall-clock time required for task completion. • Token Consumption: Total API tokens consumed during execution. • Financial Cost: Estimated USD cost based on publicly available API pricing. 4.3 Implementation Details All experiments are conducted on a standardized Linux machine with Ubuntu 20.04 LTS, equipped with an Intel Xeon Platinum 8358P CPU (2.60GHz, 128 cores) and 2TB memory. We use Docker Engine 26.1.0 to containerize environments, ensuring isolation and reproducibility. Agent frameworks are executed using Python 3.11 with dependencies installed from their official repositories. To mitigate inherent randomness in LLM outputs, each experiment undergoes three independent executions, with final metrics calculated as averages across all. A strict 60-minute timeout is enforced per execution, where any run exceeding this limit is automatically terminated and classified as a failure. All generated artifacts, execution logs, and API interactions are systematically captured and stored for comprehensive post-hoc analysis. To maintain experimental feasibility, we impose budget constraints of 5 for complete end-to-end reproduction workflows. To ensure fair comparison across all agents, browser-based interactions are strictly prohibited, restricting all agents to command-line interfaces only. 5 Experimental Results 5.1 C1: Effectiveness Motivation. Manual reproduction of security vulnerabilities represents a well-documented bottleneck in software development, consuming substantial expert time and delaying remediation efforts. Previous approaches relied heavily on manual processes where security researchers must interpret vulnerability reports and develop proof-of-concept code. However, recent advances in LLM agents present opportunities to automate these complex tasks. Therefore, this criterion evaluates the effectiveness of current agents in performing core vulnerability reproduction tasks. Approach. As described in § 4, we evaluate agent effectiveness across the four-stage vulnerability reproduction pipeline using the success rate metrics defined in § 4.2. Each agent-model combination is assessed on environment setup, vulnerability localization (measured by file-level, function-level and line-level Top-3 accuracy), PoC generation with execution verification, vulnerability trigger verification, and end-to-end reproduction success, which is defined in § 3.3. Results. Table 3 shows the capability assessment results of 20 agents across 16 dimensions using three representative CVEs: CVE-2024-7065 (CSRF), CVE-2024-7458 (Path Traversal), and CVE-2024-41112 (RCE). The analysis reveals severe capability gaps across the agent landscape. Software engineering agents show poor performance with only 2 out of 8 agents meeting basic requirements. Most fail in critical areas like tool integaration and session management, with agents , Vol. 1, No. 1, Article . Publication date: October 2025. 10 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Table 3. Comparison of 20 agents for web vulnerability reproduction with CVE-2024-7065, CVE-2024-7458 and CVE-2024-41112. Agent SCA TI SE FO WR SDF CSO CTS RS EH SM EM LP LLS UI EI Software Engineering Agents AGENTLESS ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ G # ✗ ✗ ✗ ✗ G # C ✓ OpenHands ✓ ✓ ✓ ✓ ✓ ✗ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ C/W ✓ SWE-agent ✓ G # ✓ ✓ ✓ ✗ G # ✓ ✓ ✓ ✓ ✓ ✓ ✓ C ✓ AutoCodeRover ✓ ✗ G # ✓ ✗ ✗ ✗ G # ✓ ✗ ✗ ✗ ✓ G # C ✓ Aider G # ✓ G # ✓ ✗ ✗ G # ✓ ✓ ✓ ✗ ✓ ✓ ✓ C ✓ RepoAgent ✓ ✗ G # G # ✗ ✗ ✗ G # ✓ ✗ ✗ ✗ ✗ G # C ✓ ChatDev ✗ ✗ ✗ G # ✗ ✗ G # G # G # ✗ ✗ ✗ ✗ ✗ C/W G # MetaGPT G # ✓ G # G # ✗ ✗ G # G # ✓ ✓ ✗ ✗ ✓ G # C ✓ Cybersecurity Agents PentestGPT G # ✓ ✓ ✓ ✓ ✓ G # ✗ ✓ ✓ ✓ ✗ ✓ ✓ C ✓ CAI ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ C ✓ PentestAgent ✗ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ G # C ✓ Nebula ✓ ✓ ✓ G # ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✗ C G # HackingBuddyGPT ✗ ✓ ✓ ✓ ✓ ✓ G # G # ✓ ✓ ✓ ✗ ✓ ✓ C G # PentAGI ✗ G # ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✓ ✗ ✓ ✓ C ✗ EnIGMA ✗ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✓ ✓ ✓ ✗ C G # AI-OPS ✗ ✗ ✗ ✗ G # G # ✗ ✗ ✓ ✗ ✓ ✗ ✗ ✗ C G # General-purpose Agents AutoGPT ✓ G # G # ✓ ✓ ✗ G # G # ✗ ✗ ✓ ✗ ✓ ✓ C/W ✗ BabyAGI ✓ G # G # G # ✗ ✗ ✗ ✗ G # G # ✓ ✗ ✗ G # C ✓ AgentVerse ✓ ✗ ✗ ✗ ✗ ✗ G # G # G # ✗ ✓ ✗ ✗ ✗ C/W G # AIlice ✓ ✓ ✓ G # ✗ ✗ ✓ ✓ G # ✗ ✓ ✗ ✓ ✗ C/W ✗ Note: Symbols represent Good (✓), Average (G #), and Poor (✗) qualitative assessment. Abbreviations: SCA: Source Code Access RS: Runtime Stability TI: Tool Integration EH: Error Handling SE: Script Execution SM: Session Management FO: File Operation EM: Efficiency Metrics WR: Web Request LP: Log Persistence SDF: Security Domain Focus LLS: Latest LLM Support CSO: Custom Structured Output UI: User Interface (C: CLI, C/W: CLI/WEB) CTS: Custom Task Support EI: Ease of Install like AGENTLESS [82] and ChatDev [62] lacking over 10 essential capabilities. Cybersecurity agents demonstrate domain knowledge but face limitations in technical implementation capabilities, most lack essential technical features such as structured output and custom task support. Except for CAI which achieves full capability, all other cybersecurity agents fail to support custom task types and structured output formats, limiting their adaptability to diverse web vulnerability reproduction scenarios. General-purpose agents offer broad applicability and flexible task handling but struggle with specialized technical requirements. For example, AutoGPT [26] demonstrates reasonable performance in 8-9 dimensions but lacks security domain focus and has installation challenges. Only three agents demonstrate comprehensive capabilities: SWE-agent [84], OpenHands [78], and CAI [48]. These agents form the foundation of our further empirical evaluation: - SWE-agent employs a carefully designed Agent-Computer Interface (ACI) that enables autonomous interaction with software repositories through specialized commands for code navigation, editing, and testing. The framework demonstrates exceptional performance in repository-level vulnerability discovery through its integrated file viewer, search capabilities, and context management system that maintains concise yet informative interaction histories. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 11 - OpenHands provides a comprehensive platform for AI software development agents, featuring an event-stream architecture that facilitates complex multi-turn interactions. The framework excels in code generation, debugging, and repository manipulation through its sandboxed Docker environment and extensive tool integration, including IPython execution, web browsing capabilities, and multi-agent delegation patterns. - CAI represents a specialized cybersecurity AI framework designed for autonomous penetration testing and vulnerability assessment. Built around six fundamental pillars (Agents, Tools, Handoffs, Patterns, Turns, and Human-In-The-Loop), CAI demonstrates superior performance in offensive security operations, achieving human-competitive results across diverse Capture-The-Flag challenges and real-world bug bounty scenarios. To ensure experimental consistency and fair comparison across all subsequent evaluations, we configure each framework with identical LLM foundations: GPT-4.1[54], Claude-Sonnet4[6] and Gemini-2.5-Pro[25] for the following experiments. Table 4 presents the comprehensive effectiveness evaluation results across these three selected agent-model combinations. The results reveal significant challenges in automated vulnerability reproduction, with substantial performance variations across different pipeline stages and agentmodel configurations. Table 4. Effectiveness results: success rates (%) across vulnerability reproduction pipeline. Agent Model Env Setup Vulnerability Localization PoC Generation End-to-End Setup File Level Func Level Line Level Execution Trigger Success@1 Success@3 OpenHands Claude-Sonnet-4 28.8 46.3 36.3 23.8 70.0 21.3 10.0 22.5 Gemini-2.5-Pro 26.3 40.0 27.5 17.5 52.5 10.0 6.3 13.8 GPT-4.1 32.5 53.8 43.8 31.3 56.3 16.3 13.8 20.0 SWE-agent Claude-Sonnet-4 22.5 50.0 40.0 27.5 58.8 11.3 8.8 15.0 Gemini-2.5-Pro 20.0 37.5 30.0 21.3 56.3 8.8 5.0 11.3 GPT-4.1 25.0 46.3 36.3 25.0 66.3 13.8 11.3 18.8 CAI Claude-Sonnet-4 35.0 56.3 46.3 32.5 56.3 15.0 7.5 12.5 Gemini-2.5-Pro 38.8 50.0 38.8 26.3 33.8 17.5 3.8 10.0 GPT-4.1 36.3 58.8 48.8 35.0 25.0 20.0 10.0 16.3 Examining the overall end-to-end performance, OpenHands demonstrates the strongest capabilities, particularly when paired with Claude-Sonnet-4, achieving the highest Success@3 rate of 22.5%. OpenHands maintains consistent superiority across different models, with Success@3 rates of 22.5% (Claude-Sonnet-4), 20.0% (GPT-4.1), and 13.8% (Gemini-2.5-Pro). In comparison, SWE-agent achieves maximum Success@3 rates of 18.8% (GPT-4.1), while CAI peaks at 16.3% (GPT-4.1). The Success@1 results show similar patterns, with OpenHands + GPT-4.1 leading at 13.8%, followed by OpenHands + Claude-Sonnet-4 at 10.0%. Notably, even the best-performing combinations fail to achieve successful vulnerability reproduction in more than three-quarters of cases, indicating substantial room for improvement in current agent capabilities. Environment setup performance varies significantly across agents, with CAI demonstrating the most consistent results, achieving rates above 35% across all models and peaking at 38.8% with Gemini-2.5-Pro. OpenHands shows moderate performance ranging from 26.3% to 32.5%, while SWE-agent exhibits the lowest and most variable setup success rates, spanning 20.0% to 25.0%. Vulnerability localization reveals a clear performance hierarchy and degradation pattern. CAI paired with GPT-4.1 leads in localization precision across all granularities: 58.8% file-level, 48.8% function-level, and 35.0% line-level accuracy. However, the degradation from file-level to line-level represents a consistent 35-40% relative decrease across all combinations. GPT-4.1 consistently outperforms other models in localization tasks, while Gemini-2.5-Pro shows the weakest localization capabilities across all agents. , Vol. 1, No. 1, Article . Publication date: October 2025. 12 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang The PoC generation stage reveals OpenHands' key strength and a critical bottleneck. OpenHands + Claude-Sonnet-4 achieves the highest PoC execution rate at 70.0%, significantly outperforming other combinations. However, a substantial execution-to-trigger gap exists across all agents: while execution rates range from 25.0% to 70.0%, trigger rates remain uniformly low at 8.8% to 21.3%. Remarkably, OpenHands + Claude-Sonnet-4 also leads in trigger success at 21.3%, demonstrating superior PoC quality despite the overall low trigger rates. Model-specific patterns emerge clearly. GPT-4.1 excels in localization precision but shows moderate end-to-end performance. Claude-Sonnet-4 demonstrates exceptional synergy with OpenHands, particularly in PoC generation and end-to-end orchestration. Gemini-2.5-Pro consistently underperforms across most metrics, especially in PoC-related tasks. The results highlight OpenHands' superior workflow orchestration capabilities. While CAI achieves the highest individual stage performance in environment setup and localization, OpenHands translates moderate individual performance into the strongest end-to-end success rates, suggesting more effective inter-stage coordination and error recovery mechanisms. We further analyze the underlying causes of these performance differences by examining the tool utilization patterns across agents. Table 5 reveals that while all agents share basic execution capabilities, OpenHands possesses unique workflow control tools that fundamentally enhance its vulnerability reproduction effectiveness. The think tool enables structured reasoning processes that strengthen agent memory and decision-making throughout the multi-stage pipeline, while the finish tool provides systematic task summarization and completion mechanisms. These capabilities explain OpenHands' superior ability to maintain context across pipeline stages and achieve higher end-to-end success rates despite comparable basic tool availability. Unexpectedly, CAI's extensive domain-specific security tool suite remains largely unutilized in our task specifications, revealing a critical gap between tool availability and practical application. While CAI provides numerous specialized cybersecurity tools (such as exploitation, privilege scalation [48]) theoretically suitable for vulnerability reproduction, the agent fails to effectively leverage these capabilities within the structured task framework, resulting in lower end-to-end performance despite superior individual stage metrics in environment setup and localization. Additionally, OpenHands demonstrates superior compliance with output formatting requirements by correctly generating artifacts and JSON outputs as specified in task prompts, indicating more robust instruction following capabilities. This analysis reveals that tool ecosystem effectiveness of agent depends not merely on capability breadth but on workflow integration, reasoning enhancement, and systematic task completion mechanisms that enable consistent execution across complex multi-stage security tasks. Table 5. The comparison of tool utilization across agent frameworks during task execution. Tool Type Tool Capability CAI OpenHands SWE-agent Basic Execution execute_bash ✓ ✓ ✓ str_replace_editor ✓ ✓ ✓ File Operations ✓ ✓ ✓ Workflow Control think (Reasoning Enhancement) × ✓ × finish (Task Summary) × ✓ × Domain-Specific Security/Vulnerability Tools 9 × × Used in Tasks 0 N/A N/A Task Execution Clear Task Termination ✓ ✓ ✓ Structured Reasoning × ✓ × Correct Artifacts Output × ✓ × Proper JSON Format ✓ ✓ × , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 13 Findings for C1: Current LLM agents achieve limited web vulnerability reproduction effectiveness with end-to-end success rates below 25%. OpenHands demonstrates superior performance through structured reasoning and systematic completion mechanisms, while tool availability alone does not ensure effectiveness. The primary bottleneck remains the exploitation gap where agents can execute PoCs but struggle to trigger actual vulnerabilities. 5.2 C2: Compatibility Motivation. Real-world vulnerability reproduction environments exhibit significant diversity across vulnerability types and web technologies. Different vulnerability categories require distinct understanding patterns, exploitation techniques, and verification approaches. Similarly, vulnerabilities manifest differently across web technologies due to language-specific features, frameworks, and coding patterns. A practical vulnerability reproduction agent must demonstrate consistent performance across this diversity rather than excelling only in specific domains. Understanding compatibility helps assess whether current agents can serve as general tools or require specialized adaptations. Approach. As shown in § 3.1, our dataset covers diverse vulnerability types and web technologies. We conduct categorical evaluation across these dimensions to assess agent compatibility. For vulnerability type compatibility, we analyze reproduction success rates across the seven vulnerability categories in our dataset. For programming language compatibility, we evaluate performance across the six primary languages. To ensure fairness and reproducibility, we use the top-performing combination from § 5.1, specifically OpenHands with Claude-Sonnet-4. Results. Table 6 presents stage-wise success rates across different vulnerability types. The results reveal substantial performance variations, with end-to-end success rates (Success@3) ranging from 0% to 63%. Library-based Prototype Pollution achieves the highest reproduction success rate at 63%. Among service-based web vulnerabilities, CSRF performs best with 60%, followed by Path Traversal at 36%. Other service-based vulnerabilities demonstrate considerably lower success rates: XSS at 16%, SSRF at 15%, RCE at 8%, and SQL Injection with the lowest at 0%. The low success rates for XSS and SSRF reflect agents' difficulties with complex multi-component environments requiring database systems, frontend-backend configurations, and middleware services. Agents struggle to extract configurations from large repositories, perform data imports, and handle intricate deployment procedures. SQL Injection faces similar environment setup challenges, compounded by database-dependent architectures. Beyond environment setup, a critical bottleneck emerges in vulnerability triggering due to API authentication requirements, as agents without browser interaction capabilities cannot perform automated login procedures like human users. In contrast, CSRF achieves perfect file localization due to clear vulnerability descriptions enabling precise interface targeting. Prototype Pollution maintains consistent performance across stages, benefiting from simpler library-based repositories requiring only dependency installation rather than complex infrastructure. While PoC execution rates remain high across vulnerability types, the dramatic drop from execution to trigger phases reveals that agents can run exploit code but struggle with the precise environmental conditions and interaction sequences needed for actual vulnerability manifestation. Figure 3 demonstrates end-to-end reproduction success rates across web technologies. PHP achieves the highest success rate at 38.9%, while TypeScript shows the lowest at 11.1%. PHP's superior performance stems from its prevalence in web security research, providing agents with abundant training data and well-documented exploit patterns, combined with its simpler syntax and direct web-oriented nature. JavaScript demonstrates strong performance through extensive clientside vulnerability examples and straightforward DOM manipulation patterns. TypeScript's poor , Vol. 1, No. 1, Article . Publication date: October 2025. 14 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Table 6. Vulnerability reproduction success rates by vulnerability types. Vulnerability Type Environment Code Localization PoC Generation End-to-End Setup File Function Line Execution Trigger Success@1 Success@3 PROTOTYPE_POLLUTION 75% 88% 75% 63% 88% 50% 38% 63% CSRF 80% 100% 80% 60% 80% 60% 40% 60% PATH_TRAVERSAL 45% 82% 73% 45% 82% 27% 18% 36% XSS 21% 53% 37% 21% 79% 11% 5% 16% SSRF 15% 31% 23% 8% 77% 8% 0% 15% RCE 8% 17% 8% 8% 67% 8% 0% 8% SQLI 8% 0% 0% 0% 25% 0% 0% 0% TOTAL 29% 46% 36% 24% 70% 18% 10% 23% PHP JavaScript Java Python TypeScript Others 0 5 10 15 20 25 30 35 40 End-to-End Success Rate (%) 38.9% 23.5% 18.2% 22.2% 11.1% 14.3% Average: 22.5% Fig. 3. Comparison of vulnerability reproduction success rates between different web technologies. performance shows that its additional complexity and stricter syntax requirements hinder agents despite its similarity to JavaScript. The results indicate that agent performance varies significantly across both vulnerability types and web technologies, with success rates influenced by environment complexity and the specific requirements of each vulnerability category. Findings for C2: Agent performance exhibits substantial variation across vulnerability types and web technologies. Service-based vulnerabilities face significant challenges from complex environment setup, authentication requirements, and multi-component dependencies, while library-based vulnerabilities achieve higher success through simpler deployment requirements. Language compatibility depends on security research prevalence and syntactic complexity, with web-oriented languages showing better agent adaptation. 5.3 C3: Efficiency Motivation. While effectiveness measures reproduction success rates, efficiency evaluates the computational and economic costs of achieving successful reproductions. Understanding resource requirements is crucial for practical deployment, as organizations must balance reproduction capabilities with operational costs. We analyze efficiency across successful reproduction attempts to assess real-world viability of different agent-model combinations. Approach. We adopt the efficiency evaluation methodology from § 4.2, measuring token consumption, execution time, and monetary cost across the vulnerability reproduction pipeline. To do so, we focus exclusively on the successful reproduction cases achieved by our three foundational agents as identified in § 5.1. To ensure a fair comparison, all tests are conducted using the Claude-Sonnet-4 model, and measurements are averaged across successful attempts for each agent. , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 15 Results. Table 7 presents efficiency metrics for all three agents calculated from successful reproduction attempts. Environment setup demonstrates the highest resource requirements across all agents, while vulnerability localization proves most economical. End-to-end reproduction costs range from 2.19 with execution times between 25.7-34.0 minutes. OpenHands consistently requires the highest resources ( 1.90), while CAI demonstrates the most efficient utilization ( ). Averages from successful cases only. Costs based on Claude Sonnet pricing ( 15/1M output tokens). The cost variations stem from architectural differences between agents. OpenHands incorporates extensive internal reasoning including explicit "think" and "finish" operations that generate substantial token overhead. SWE-agent employs numerous system prompts to enforce tool usage compliance, creating intermediate communication costs. CAI, despite having the most comprehensive tool suite, exhibits minimal tool invocation in vulnerability reproduction tasks, resulting in streamlined execution. These design choices create a trade-off between computational efficiency and reproduction capability, with OpenHands prioritizing thorough reasoning, SWE-agent emphasizing tool compliance, and CAI optimizing for direct execution. Findings for C3: Web vulnerability reproduction requires substantial computational resources, with end-to-end costs ranging from 2.19 per successful case. Environment setup consumes the most resources across all agents, while agent architectural differences create significant efficiency variations in the reproduction pipeline. 5.4 C4: Robustness Analysis Motivation. Robustness measures how well agents perform under challenging conditions. In vulnerability reproduction, agents face obstacles like limited processing capabilities, browser security restrictions, or incomplete information. These challenges are inherent to real-world security research environments where perfect conditions rarely exist. A robust agent should maintain reasonable performance across different scenarios, adapting to unexpected failures and environmental constraints. Understanding robustness helps us assess agent reliability for practical deployment and identify which architectures handle reproduction challenges most effectively. Approach. We design two experiments to test the robustness of our three selected agents across key challenge areas: First, we compare agent performance using Claude-Sonnet-4 with and without enhanced reasoning across different vulnerability reproduction stages in 10 vulnerabilities. Second, we test agent performance under different authentication scenarios using 12 SQL Injection vulnerabilities, comparing three cases: manually providing login tokens, prompting agents to login themselves, and providing no authentication help. Results. Figure 4 shows mixed results when comparing models with and without enhanced reasoning. Enhanced reasoning helps in early stages, with OpenHands improving from 60% to 70% in , Vol. 1, No. 1, Article . Publication date: October 2025. 16 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang Environment Setup and from 40% to 70% in File Localization, while improvements decrease in later stages with PoC Execution showing smaller gaps and Trigger Success showing no improvement across all agents. SWE-Agent shows the most consistent benefit from enhanced reasoning across localization stages, while OpenHands demonstrates strong baseline performance even without reasoning due to its built-in "think" operations, and CAI shows more variable results with minimal reasoning benefits. The limited reasoning impact reflects both agent design differences and task complexity variations, as reasoning enhancements prove most effective for analytical tasks like code localization but provide little benefit for execution-oriented activities like vulnerability triggering. Within our 10-vulnerability sample, agents demonstrate robustness across both reasoning configurations, suggesting that architectural design choices may be more influential than underlying model reasoning capabilities for practical vulnerability reproduction tasks. Environment File Localization Function Localization Line Localization PoC Execution Trigger Success 0 10 20 30 40 50 60 70 80 Success Rate (%) Openhands (w/ reasonning) Openhands (w/o reasonning) SWE-Agent (w/ reasonning) SWE-Agent (w/o reasonning) CAI (w/ reasonning) CAI (w/o reasonning) Fig. 4. Performance comparison between agents using Claude-Sonnet-4 with and without reasoning across different vulnerability reproduction stages. Table 8 shows authentication context impacts vulnerability reproduction success. Manual token provision works best, with Openhands + Claude-Sonnet-4 achieving 67% success. Requiring agents to handle login autonomously results in an average performance degradation of 33.3% compared to manual token provision, with success rates dropping to 25-50% across agent-model combinations. Removing authentication guidance causes near-complete failure, with only three configurations maintaining 8% success rates. This reveals that current agents struggle with autonomous authentication discovery and rely heavily on explicit guidance. The consistent degradation across architectures and models indicates that authentication handling is a widespread limitation. Findings for C4: Current agents show robustness to reasoning changes but high sensitivity to prompt variations. While reasoning versus non-reasoning modes produce similar performance, incomplete authentication results in severe performance drops, highlighting agents' limited capacity for problem-solving without explicit guidance. 6 Discussion Our empirical evaluation reveals that current open-source agents face substantial limitations in vulnerability reproduction beyond specific ecosystems. While agents demonstrate reasonable success in library-based JavaScript environments and relatively straightforward CSRF scenarios, they struggle significantly with complex web vulnerabilities that require intricate environment , Vol. 1, No. 1, Article . Publication date: October 2025. LLM Agents for Automated Web Vulnerability Reproduction: Are We There Yet? 17 Table 8. Comparison of trigger rate in authentication robustness analysis. Agent Model Manual Token Agent Login No Auth Info Openhands Claude-Sonnet-4 67% 50% 8% Gemini-2.5-Pro 58% 42% 8% GPT-4.1 50% 33% 0% SWE-Agent Claude-Sonnet-4 58% 33% 8% Gemini-2.5-Pro 50% 25% 0% GPT-4.1 42% 17% 0% CAI Claude-Sonnet-4 50% 25% 0% Gemini-2.5-Pro 42% 17% 0% GPT-4.1 33% 8% 0% setup and multi-step exploitation chains. The stark performance gaps-ranging from 0% for SQL injection to 63% for prototype pollution-indicate that pure code-based approaches are insufficient for comprehensive vulnerability reproduction. Traditional injection vulnerabilities, despite their well-documented nature, consistently fail due to environment configuration challenges and the complex interplay between application deployment, database setup, and exploitation context. This suggests that current agent architectures fundamentally lack the systems-level understanding necessary for realistic security testing scenarios. The integration of Model Context Protocol (MCP)[1] with browser automation tools like Playwright or Chrome presents a breakthrough for vulnerability reproduction. By enabling agents to control browser instances, capture network traffic, and monitor application state changes in MCP-based approaches can bridge the gap between static code analysis and dynamic exploitation. This browser-centric methodology allows agents to observe runtime behavior of web applications, intercept HTTP requests and responses, and validate success through DOM manipulation and state inspection. The ability to interact with web interfaces while monitoring network activity represents a shift from traditional code-only approaches, unlocking automated reproduction capabilities for complex web vulnerabilities that have resisted agent-based solutions. 7 Threats to Validity Internal Validity. The primary internal threat stems from potential data leakage in LLM training datasets. Many CVEs in our benchmark were disclosed before the training cutoff dates of evaluated models, creating a risk that agents may leverage memorized exploit patterns rather than demonstrating genuine reasoning capabilities. Although we deliberately avoided referencing specific PoCs in our prompts, models may still access historical vulnerability data through their pre-trained knowledge, potentially inflating performance metrics. Additionally, our evaluation employs binary success metrics that only capture whether vulnerability reproduction succeeds or fails, providing a single-dimensional assessment that may miss important aspects of the reproduction process. External Validity. Our findings may have limited generalizability due to several factors. First, our CVE selection, while diverse across vulnerability types and web technologies, represents a curated subset that may not fully reflect the complexity distribution of real-world security scenarios. Second, the containerized evaluation environment, despite efforts toward realism, may still differ from production deployments in ways that affect agent performance. Finally, the rapid evolution of LLM capabilities means our findings represent a temporal snapshot that may not predict future agent performance across different models or architectures. , Vol. 1, No. 1, Article . Publication date: October 2025. 18 Bin Liu, Yanjie Zhao, Guoai Xu, and Haoyu Wang 8 Related Work 8.1 Software Engineering Benchmarks for LLM Agents Software engineering represents a critical application domain for evaluating LLM agents, with benchmarks spanning different complexity levels and task types. SWE-bench [37] evaluates agents' abilities to solve real-world GitHub issues by generating patches for bug fixes, representing one of the most challenging repository-level tasks that require understanding complex codebases and coordinating changes across multiple files. Traditional coding benchmarks like HumanEval focus on isolated function generation from natural language descriptions, while MBPP evaluates basic Python programming tasks. Additional benchmarks including BigCodeBench [92], LiveCodeBench [35], and EvalPlus [43] extend evaluation across multiple languages and enhanced test coverage. However, these SE benchmarks exhibit significant limitations: many focus on narrow task scopes rather than diverse professional tasks, traditional benchmarks often involve only local edits to individual functions rather than repository-wide reasoning, and they frequently ignore autonomous environment setup and end-to-end task solving capabilities essential for practical software engineering. Recent advances like AgentBench [44] provide more comprehensive evaluation across diverse environments, highlighting the evolution toward more realistic and challenging evaluation frameworks for software engineering agents. 8.2 Cybersecurity Benchmarks for LLM Agents Various specialized benchmarks have been developed to evaluate LLM agents in cybersecurity. InterCode-CTF [85] manually evaluated LLMs on Capture-the-Flag challenges requiring reverse engineering and exploitation skills, while NYU CTF Bench [64] expanded this with an automated framework for diverse offensive security tasks across cryptography, forensics and binary exploitation. The CyberSecEval series [10, 73] focuses on evaluating cybersecurity risks including prompt injection, code interpreter abuse and automated social engineering. ARVO [50] provides reproducible memory vulnerabilities with triggering inputs and developer patches. SEC-bench [38] evaluates PoC generation and vulnerability patching tasks using data from OSS-FUZZ project, SecVulEval [3] benchmarks statement-level vulnerability detection in C/C++ code. CyberGym [79] assesses vulnerability reproduction across large software projects. Most existing benchmarks exhibit limitations in capturing real-world cybersecurity complexity, often focusing on isolated tasks or synthetic environments. The most closely related work is CVE-bench [76], which remains limited in both scope and scale, capturing only a narrow slice of agent performance in web vulnerability reproduction scenarios. 9 Conclusion Our evaluation of 20 LLM agents across 16 core dimensions on 3 representative CVEs identified three top performers (OpenHands, SWE-agent, and CAI) for testing on 80 real-world CVEs spanning 6 web technologies and 7 vulnerability types. 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2510.14699
|
Efficient Verification of Metric Temporal
Properties with Past in Pointwise Semantics
S Akshay #
Department of CSE, Indian Institute of Technology Bombay, Mumbai, India
Prerak Contractor #
Department of CSE, Indian Institute of Technology Bombay, Mumbai, India
Paul Gastin #
Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMF, 91190, Gif-sur-Yvette, France
CNRS, ReLaX, IRL 2000, Siruseri, India
R Govind #
Uppsala University, Sweden
B Srivathsan #
Chennai Mathematical Institute, India
CNRS, ReLaX, IRL 2000, Siruseri, India
Abstract
Model checking for real-timed systems is a rich and diverse topic. Among the different logics
considered, Metric Interval Temporal Logic (MITL) is a powerful and commonly used logic, which
can succinctly encode many interesting timed properties especially when past and future modalities
are used together. In this work, we develop a new approach for MITL model checking in the pointwise
semantics, where our focus is on integrating past and maximizing determinism in the translated
automata.
Towards this goal, we define synchronous networks of timed automata with shared variables
and show that the past fragment of MITL can be translated in linear time to synchronous networks
of deterministic timed automata. Moreover determinism can be preserved even when the logic is
extended with future modalities at the top-level of the formula. We further extend this approach to
the full MITL with past, translating it into networks of generalized timed automata (GTA) with
future clocks (which extend timed automata and event clock automata). We present an SCC-based
liveness algorithm to analyse GTA. We implement our translation in a prototype tool which handles
both finite and infinite timed words and supports past modalities. Our experimental evaluation
demonstrates that our approach significantly outperforms the state-of-the-art in MITL satisfiability
checking in pointwise semantics on a benchmark suite of 72 formulas. Finally, we implement an
end-to-end model checking algorithm for pointwise semantics and demonstrate its effectiveness on
two well-known benchmarks.
2012 ACM Subject Classification Theory of computation →Timed and hybrid models; Theory of
computation →Quantitative automata; Theory of computation →Logic and verification
Keywords and phrases Real-time systems, Timed automata, Liveness, Event-clock automata, Clocks,
Timers, Verification, Zones, Simulations
1
Introduction
Linear Temporal Logic (LTL) is a standard specification formalism. A central ingredient
in the model-checking approach is a translation from LTL to automata.
The study of
algorithmic techniques to optimize LTL-to-automata translations spans almost four decades
– see [57] for a recent survey of this topic. Many of the theoretical techniques have also
been incorporated in practical model-checking tools [36, 22, 28, 56, 37]. For model-checking
systems with timing constraints, the de-facto automaton model would be timed automata
(TA) [4, 3]. State-of-the-art timed automata tools [41, 34, 56, 37] implement analysis methods
arXiv:2510.14699v1 [cs.FL] 16 Oct 2025
2
Efficient Verification of MTL with Past in Pointwise Semantics
for models represented as (networks of) timed automata (see Section 7 of [25] for the practical
applications of these tools). Properties checked on timed models are typically reachability,
liveness, fragments of timed CTL [41] or (untimed) LTL. On the other hand, there is a very
large body of foundational work on timed logics [53, 35, 54]. The two main timed logics are
Metric Temporal Logic [40] (MTL) and Timed Propositional Temporal Logic [8] (TPTL). In
order to use these logics for real-time model-checking, we need an efficient translation from
timed logics to timed automata. The foundations for MTL-to-TA (resp. TPTL-to-TA) have
been laid out in [5] (resp. [8]).
Broadly, there are two ways to interpret MTL formulae: over continuous (dense/super-
dense) timed signals [5, 45] or pointwise timed words [6, 58, 20]. In continuous (state-based)
semantics, the formula is interpreted over signals, which are functions that associate the
valuation of the propositions to all timestamps. Over the pointwise semantics, the formula
is interpreted over sequences of events (timed words), each consisting of a timestamp and a
set of propositions. Thus, in the continuous semantics, the system is under observation at
all times, while in the pointwise semantics, it is observed only when it executes a discrete
transition. However, as noted by Ferrére et al. [30], in the continuous semantics, arbitrary
MTL formulae can be simplified to the [0, ∞] fragment, while as pointed out in [2], this
simplification is not possible in the pointwise semantics. In this work, we only consider
pointwise semantics as our focus is on models represented in tools like UPPAAL, PAT,
TChecker whose behaviours are given as timed words, as in the classical timed automata
model of Alur and Dill [4].
In the pointwise semantics, it is known that MTL model checking is undecidable over
infinite words, and over finite words it is decidable but the complexity is Ackermanian [7, 50,
48, 49]. Thus, the literature often considers the more efficient MITL fragment, where Until
and Since operators must use non-punctual intervals (either [0, 0] or non-singleton intervals).
The classical approach for model checking for MITL involves translating MITL formula into
timed automata (TA) and then using state-of-the-art TA tools such as UPPAAL [41, 12]
or TChecker [34]. The best known tool that accomplishes this translation in the pointwise
semantics, called MightyL [20], goes via alternating 1-clock automata and compiles to
timed automata. Recently, an extension of timed automata called generalized timed automata
(GTA) was introduced in [1]. GTA are equipped with richer features than TA, and enable
a different translation from MITL to GTA, which can then be analyzed using a GTA
analysis tool from [1]. The theoretical foundations for such an MITL-to-GTA translation
which goes via transducers was presented in [2], where it was argued that the MITL-to-
GTA translation has clear theoretical benefits over the MITL to 1-clock Alternating Timed
Automata translation [18, 19] on which MightyL is based on. However, this latter approach
was not implemented. Moreover, neither of these approaches considered Past modalities, and
were restricted only to Future.
On the other hand, from the classical work of Maler et al [44], in the signal semantics, it
is known that having only Past modalities leads to efficient algorithms and in particular it
gives a translation to deterministic (timed) automata albeit over signals. Our first question
is whether the same holds under the pointwise semantics, and whether determinism can be
extended beyond just the past fragment. Next, if we cannot guarantee determinism, can we
reduce non-deterministic branching? Indeed, even if it does not change the expressiveness,
several works have highlighted the usefulness and succinctness of combining Past and Future
operators, both in the untimed setting [43, 46, 10] and in the timed setting [39, 30].
In this work, we address these questions both for past and future in the pointwise
semantics and to build the foundations for a new efficient model checking algorithm for
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
3
MITL with past that exploits determinism as much as possible. We show that for the past
fragment of MITL, we can obtain a linear-time translation to networks of deterministic timed
automata. Further, even when we have to handle future operators, where non-determinism
cannot be avoided, we propose techniques to reduce non-deterministic branching in the
resulting automata, using various ideas including sharing components, automata, clocks and
predictions. We start by defining a formal semantics for a model of synchronous network
of timed automata with shared variables, where each component of the network owns some
clocks which only it can reset but can be tested by all components. Moreover it allows for
instantaneous programs on transitions which have sequences of tests and updates, like in [2].
Our main contributions are the following:
1. We identify a fragment of metric temporal logic with past and future for which we provide
a linear time translation to networks of deterministic timed automata. In this fragment,
that we call detMITL+p, the outermost temporal modalities are future (next, until) and
can even have punctual intervals, while inner temporal modalities should be past: Since
with non-punctual interval or Yesterday. This fragment already subsumes many safety
properties, extending e.g., the G(pLTL) fragment defined in [9].
2. We present a new linear-time algorithm to translate detMITL+p formula to networks
of deterministic timed automata. This algorithm is in two steps. First we show how
the outermost temporal modalities can be handled with respect to initial satisfiability.
Next we show how the past temporal modalities are handled for general satisfiability.
We obtain a synchronous network of deterministic automata with shared variables that
captures the semantics. We start with Yesterday allowing an arbitrary interval constraint.
Then, we deal with Since operators with one-sided intervals of the form [0, a] or [a, ∞).
For these simpler cases, we construct networks where each automaton has at most 1
clock and 2 states, but critically uses the sharing feature across components. Finally,
the automaton for Since with a general two-sided interval [a,b] is the most complicated
part of the construction and results in a network whose one component uses O(a/(b−a))
clocks. Our construction works for both finite and infinite words.
3. Next, we extend this algorithm to handle future modalities. To achieve this we move
from networks of timed automata to networks of generalized timed automata (GTA) [1],
which have future clocks in addition to usual (history) clocks. We obtain a translation
from general MITL+p (with Past and allowing punctual intervals at the outermost level)
to networks of GTA with shared variable. While non-determinism cannot be avoided
here, we show that we can reduce non-deterministic branching in two ways: first by using
shared variables to share predictions, and second, by using non-Zenoness assumption and
the semantics of future clocks to obtain a 2-state automaton for Until (which results in
an exponential improvement in running time for the model checking algorithm). Again
our model checking algorithm works for infinite words and is easily adaptable for finite
timed words.
4. Since our model checking approach ultimately reduces to checking reachability and liveness
for GTAs, we propose an improved liveness-checking algorithm that extends Couvreur’s
SCC algorithm [26] (which is the state-of-the-art for timed automata) to the setting of
GTAs, marking the first adaptation of Couvreur’s procedure to automata with event
clocks or diagonal constraints.
5. Finally, we implement our algorithms in a prototype tool, called Tempora, built on top of
the open-source timed automata library Tchecker. Tempora supports satisfiability and
model checking for detMITL+p and general MITL+p, for both finite and infinite words, in
the pointwise semantics. We compare against MightyL which translates MITL formula
4
Efficient Verification of MTL with Past in Pointwise Semantics
to timed automata (and uses UPPAAL to check satisfiability with finite words and
OPAAL for infinite words). Our comparison on a suite of 72 benchmarks (22 new and 50
taken from the literature) shows that our pipeline (Tempora+Tchecker) significantly
outperforms MightyL pipeline; it is faster in all benchmarks for finite words and all but
2 for infinite words, and most often explores a smaller state space (in zones). We also
implement the theoretical translation to GTA presented in [2] and demonstrate an order
of magnitude improvement. Finally, we also implement an end-to-end model checking
algorithm for MITL+p over the pointwise semantics and evaluate our model-checking
pipeline on two models: Fischer’s mutual exclusion algorithm and the Dining Philosophers
problem.
Related works. Model checking for real-time systems has been extensively studied, with
Metric Interval Temporal Logic (MITL) emerging as a widely used specification language
due to its expressiveness for capturing timed properties. A central challenge lies in efficiently
translating MITL formulas into automata-based representations, enabling algorithmic veri-
fication. For a comprehensive overview of the topic, we refer the reader to the survey by
Bouyer [14].
Over continuous/signal semantics, we already discussed several works that focus on
obtaining networks of timed automata for various fragments of MITL with and without
past including [30, 44, 45]. Most of these employ construction of compositional testers or
transducers for timed modalities, but do not focus on sharing e.g., of networks and subformula.
A different purely logic based approach was considered in [47], where the timed automata
and MITL formula, still with signal semantics, are translated to an intermediate logical
language, which is then encoded into an SMT solver. This work also included synchronization
primitives and was implemented in a Java tool.
Coming back to pointwise semantics, in [5] that introduced MITL, the authors propose
the first such translation, which is known to be notoriously complicated, spanning several
pages, and acknowledged to be difficult in the follow-up works by different authors [45]. The
current state-of-the-art for MITL model checking is the procedure proposed by Brihaye et al.
in [20], where they translate MITL formulae to a network of timed automata, which can be
plugged in a standard timed automata tool implementing non-emptiness. Their procedure
and the associated tool, called MightyL [20] is based on a series of works that study a
translation of MITL to timed automata [18, 19], via alternating timed automata [49]. [17]
studies a zone-based algorithm directly for 1-clock alternating timed automata, and focusses
on a fragment of MTL.
Several tools support MITL model checking under continuous semantics, including [38,
23, 24, 52, 42, 47]. Under pointwise semantics, to the best of our knowledge, MightyL [20]
is the only tool that can handle the full future fragment of MITL (even if it only solves
satisfiability and does not handle past operators). The other works (to our knowledge) that
offer tool support are for restricted fragments such as MITL(0.∞) [21], or MITL over untimed
words [59].
Structure of the paper. In Section 2, we formalize the notion of a synchronous network
of timed automata with shared variables, whose transitions are instantaneous programs, and
define the model checking problem in terms of such networks. In Section 3 we describe the
detMITL+p logic followed by the translation to synchronous networks of deterministic timed
automata in Section 4. In Section 5, we explain how to integrate future modalities to get
a translation from MITL+p to generalized timed automata with future clocks. Section 6
describes our adaptation of Couvreur’s SCC algorithm to the setting of GTAs. Section 7
contains our experimental evaluation and comparisons and we end with a brief conclusion in
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
5
Section 8.
To improve the flow and readability of the paper, we have moved some proofs and
technical details to the Appendix. Moreover, some benchmark details and extended results
are presented in the Appendix.
2
Synchronous network of timed automata with shared variables
We define the model of network of timed systems that we use in this paper. The presentation
is different from previous works [11, 16, 33, 2], and is geared towards providing a clean and
concise explanation of our logic-to-automata translation. Before giving the formal syntax,
we highlight the main aspects in our definition.
Each component in the network is an automaton with a finite set of locations, real
valued variables (called clocks) and finitely valued variables (mainly boolean variables and
the state variable ranging over the finite set of locations). The semantics of the network is
synchronous: either time elapses in all components of the network (all clocks are increased
by the same amount), or all components execute simultaneously an instantaneous transition.
The components of the system communicate via shared variables (clocks, states, boolean
variables, . . . ). The paradigm is concurrent-read owner-write (CROW). Hence, each variable
is owned by a unique component of the system. A component may read all variables of the
system, but it may only modify its owned variables.
A transition of a component A is of the form (ℓ, prog, ℓ′) where ℓ, ℓ′ are locations of A
and prog is an instantaneous program, which is a sequence of tests and updates. For instance,
if x, y are clocks owned by A and out is a boolean variable owned by A, then a purely
local program could be A.x ∈(3, 5]; A.y := 2; A.out := 0 where A.x ∈(3, 5] is a test which
succeeds if the current value of clock x is within the interval, and A.y := 2 and A.out := 0
are updates. Tests may also use variables owned by other components. For instance, if
component B owns clock z, then a program of A could be of the form
A.x ∈(3, 5] ∧B.out• = 1 ∧B.z > 2 ∧B.st• = ℓ1; A.y := 2; A.out := 0
where B.st means the state of B. Since we use a synchronous semantics, all components of the
network simultaneously take a transition and execute the associated programs. By default,
a variable such as A.x or B.z used in a test refers to the value of the variable before the
synchronous transition is taken. It is sometimes convenient to refer to the value a variable
will have after the synchronous transition is taken. We do so by writing B.out• or B.st•.
We give now the formal syntax and semantics for a network of timed systems. A concrete
example follows in Figure 1 and Example 3.
▶Definition 1 (Timed automaton with shared variables). A timed automaton with shared
variables is a tuple A = (Q, X, V, ∆, init) where A is the name, Q is a finite set of locations
over which ranges the state variable A.st, X is a finite set of clock variables owned by A, V
is a finite set of boolean variables owned by A, ∆is the finite set of transitions, and init is
an initial condition for the variables owned by A.
A transition is a triple (ℓ, prog, ℓ′) where ℓ, ℓ′ ∈Q are locations and prog is an instantan-
eous program. A program is a sequence of tests and updates. An atomic update for a boolean
variable v ∈V owned by A has the form A.v := c (or simply v := c when A is clear from the
context) with c ∈{0, 1}. An atomic update for a clock variable x ∈X owned by A has the
form A.x := c (or simply x := c) with c ∈N = N ∪{+∞}. We assume that each variable is
updated at most once in a program. An atomic test for a clock variable x ∈X owned by A
has the form A.x ∈I (or simply x ∈I) where I is an integer bounded interval whose end
6
Efficient Verification of MTL with Past in Pointwise Semantics
ℓ0
A1
x = +∞
q•; x := 0
¬q•
ℓ0
ℓ1
A2
y = +∞
¬q•
q•; y := 0
¬p• ∧q•; y := 0
p•
¬p• ∧¬q•
ℓ0
ℓ1
ℓ2
A3
z = 0
z ≤100 ∧¬r•
z ≤100 ∧r• ∧A1.x < 2 ∧A2.st• = ℓ1 ∧A2.y• ≥5
z > 100
z ≤100 ∧r• ∧(A1.x ≥2 ∨A2.st• = ℓ0 ∨A2.y• < 5)
ℓ0
Mp
p := 0
p := 1
Figure 1 Synchronous Network of timed automata with shared variables. The model M is not
drawn and owns the boolean variables Prop = {p, q, r}. To lighten the figures above, we simply write
p•, q• and r• instead of M.p•, M.q• and M.r•. The automata A1, A2, A3 own respectively the
clock variables x, y and z. The names and initial conditions are depicted with an incoming arrow to
a location. For instance, the initial condition of A2 is st = ℓ0 ∧y = +∞. Notice that A3 refers to
the post values of the state A2.st• and clock A2.y• of A2.
points come from N (e.g., [0, 3) or [2, +∞) or (2, 5], etc). A may also test a clock variable
owned by another component with N.x ∈I or N.x• ∈I where N is a name and I is an
integer bounded interval. Similarly, an atomic test for a boolean variable b owned by A has
the form b = 0 or b = 1 (also written b and ¬b) and if the variable is not owned by A we
use N.b or N.b•. We write N.st = ℓ, N.st ̸= ℓ, N.st• = ℓand N.st• ̸= ℓfor testing whether
component N is in location ℓor not. Finally, a test in a program is a boolean combination
of atomic tests.
The initialisation init of A is a conjunction of atomic tests, one for each variable of A,
including the state variable. For instance, init could be st = ℓ0 ∧y = +∞∧out = 0.
▶Definition 2 (Synchronous network of timed automata). A network is a tuple A =
(A1, . . . , An) where each Ai = (Qi, Xi, Vi, ∆i, initi) is a timed automaton with shared vari-
ables. The network is closed if the references to external variables in programs are all of
the form Ai.st or Ai.st•, Ai.x or Ai.x• with x ∈Xi, Ai.v or Ai.v• with v ∈Vi, for some
1 ≤i ≤n. The network is partially ordered if there is a partial order ⪯on its components
such that whenever component A refers to a variable owned by component B, we have B ⪯A.
In this paper, we will only use partially ordered networks.
We describe now the semantics of a network. A configuration C of A is a valuation of
all variables of the network: for all 1 ≤i ≤n, we have C(Ai.st) ∈Qi, C(Ai.x) ∈R+ =
R+∪{+∞} for x ∈Xi, and C(Ai.v) ∈{0, 1} for v ∈Vi. A configuration is initial if it satisfies
all initial conditions of its components. For instance, if init2 is st = ℓ0 ∧y = +∞∧out = 0,
then it is satisfied at C if C(A2.st) = ℓ0, C(A2.y) = +∞and C(A2.out) = 0.
The network is synchronous, i.e., all components move simultaneously, and has two types
of transitions: time elapse and discrete. A time elapse by some non-negative real number
δ ≥0 is written C
δ−→C′ where C′ coincides with C on all variables other than clocks, and
the value of each clock is advanced by the same quantity δ: C′(Ai.x) = C(Ai.x) + δ for all
1 ≤i ≤n and x ∈Xi (with +∞+ δ = +∞).
A discrete transition occurs in the network when each component executes simultaneously
a transition. Let t = (t1, . . . , tn) with ti = (ℓi, progi, ℓ′
i) ∈∆i for all 1 ≤i ≤n. Let C, C′ be
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
7
two configurations. There is a discrete transition C
t−→C′ if the following holds:
C(Ai.st) = ℓi and C′(Ai.st) = ℓ′
i for all 1 ≤i ≤n,
for all i and x ∈Xi ∪Vi, we have C′(Ai.x) = c if progi contains an update Ai.x := c and
C′(Ai.x) = C(Ai.x) otherwise (recall that each variable is updated at most once),
all tests occurring in the programs evaluate to true:
for all i and clock x ∈Xi and integer bounded interval I, an atomic test Ai.x ∈I (resp.
Ai.x• ∈I) evaluates to true if C(Ai.x) ∈I (resp. C′(Ai.x) ∈I),
for all i and boolean variables v ∈Vi and c ∈{0, 1}, an atomic test Ai.v = c (resp.
Ai.v• = c) evaluates to true if C(Ai.v) = c (resp. C′(Ai.v) = c),
for all i, an atomic test Ai.st = ℓ(resp. Ai.st• = ℓ) evaluates to true if C(Ai.st) = ℓ(resp.
C′(Ai.st) = ℓ).
▶Example 3. We consider the specification given by the sentence
Φ = G≤100
r −→(YP<2 q ∧p S≥5 q)
.
We assume that the model M owns the boolean variables Prop = {p, q, r}, also called atomic
propositions. The sentence Φ says that during the first 100 units of time, whenever an event
satisfies r (i.e., M sets r to true, or r was true and is not updated by M) then some event
less than 2 time units in the strict past satisfies q (written as YP<2 q), and some other event
at least 5 time units in the past satisfies q and since then all events satisfy p (written as
p S≥5 q).
We give in Figure 1 a network of timed automata which implements the specification
Φ. Component A1 records with clock x the time elapsed since the latest q event occurred.
Clock x is initialized to +∞and remains so until the first q event is seen. Then it is reset to
0 whenever a q event occurs.
Component A2 is in location ℓ1 iff p since q holds (there is a q event in the past such
that since then all events satisfy p). When A2 is in location ℓ1, clock y records the time
elapsed since the earliest witness of p S q, i.e., the earliest q event such that since then all
events satisfy p.
Finally, A3 is in location ℓ2 if the sequence of events seen so far violates Φ: within the
first 100 time units, there is an r event such that the closest q event is more than 2 time
units in the strict past (A1.x ≥2) or p S q does not hold (A2.st• = ℓ0) or the earliest witness
of p S q is less than 5 time units in the past. Component A3 is in location ℓ1 if Φ is definitely
satisfied, more than 100 time units has elapsed without violating the property. Finally, A3
stays in location ℓ0 if the sequence of events seen so far satisfies Φ and at most 100 time
units has elapsed.
◀
2.1
The model checking problem
In this paper, we are interested in the model checking problem: given a model M owning a
set Prop of atomic propositions, and a specification Φ on Prop, do all behaviours of M satisfy
Φ. Behaviours of a model are formalized as timed words over sets of atomic propositions.
The alphabet Σ that we will consider is 2Prop, the set of subsets of Prop. A timed word over
Σ, denoted by w ∈(Σ × R)∞, is an infinite sequence of pairs of letter and timestamp of the
form (a0, τ0)(a1, τ1)(a2, τ2) . . . , where each ai is a subset of Prop and τi is the timestamp of
the i-th letter; timestamps are assumed to be non-decreasing, i.e., 0 ≤τ0 ≤τ1 ≤τ2 · · · .
8
Efficient Verification of MTL with Past in Pointwise Semantics
A run ρ of the model M is an alternating sequence of delay and discrete transitions,
starting from an initial configuration. Formally, a run is an infinite sequence
ρ := C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · ·
where C0 is an initial configuration, each δi ≥0 is a time delay, and each ti is a tuple
of transitions (one from each component of the model) as defined above, such that each
step respects the semantics of time elapse and discrete transitions. The timed word wρ =
(a0, τ0)(a1, τ1) · · · associated to the above run ρ is given as follows: for all i ≥0
τi = δ0 + δ1 + · · · + δi (the timestamp of occurrence of ti) and
ai is the set of atomic propositions p such that Ci+1(p) = 1 (the propositions that are
true after executing ti).
The timed word wρ is a behaviour of the model M.
The model checking problem asks whether all (infinite, non-zeno) behaviours generated
by all possible runs of the model M satisfy a given specification Φ, written as M |= Φ.
In this work, we are interested in specifications given by fragments of MTL, which we will
define in the later sections. To solve the model checking problem, we will construct from the
specification Φ a network of timed automata A = (A1, . . . , An). Then, we will consider the
closed network (M, A) and check whether some (bad) cycle may be iterated infinitely often.
For instance, to check whether a given model M satisfies the specification Φ from
Example 3, we consider the closed network N = (M, A1, A2, A3) where the automata
A1, A2, A3 are given in Figure 1. Then, M ̸|= Φ iff a configuration C of N where A3 is in
location ℓ2 is visited infinitely often.
We may also check whether a specification is satisfiable by considering a universal
model allowing all sequences of events. This is achieved by considering, for each atomic
proposition p ∈Prop, an automaton Mp as shown in Figure 1. Then, the model is the
network M = (Mp)p∈Prop.
3
The detMITL+p fragment
We present a fragment of MTL for which we can construct a network (A1, A2, . . . , An) such
that each Ai is deterministic. For the formula in Example 3, the automata A1, A2 and A3
shown in Figure 1 are deterministic. When (A1, . . . , An) are deterministic, the product with
the model (M, A1, . . . , An) does not further “multiply” the runs of M: for every run σ of
M, there is at most one run ρ of (M, A1, . . . , An) whose projection leads to σ. On the other
hand, if some Ai is non-deterministic, the run σ can branch out into multiple runs in the
product, leading to an explosion in the state-space (of the zone graph that is computed).
The set of MTL formulae over the atomic propositions Prop is defined as
φ := p | φ ∧φ | ¬φ | YI φ | XI φ | φ UI φ | φ SI φ
where p ∈Prop, and I is an interval with end-points from N = N ∪{∞}. Note that, our
definition of MTL is with both past and future modalities. The pointwise semantics of MTL
formulae is defined inductively as follows. A timed word w = (a0, τ0)(a1, τ1)(a2, τ2) · · · is
said to satisfy the MTL formula φ at position i ≥0, denoted as (w, i) |= φ if (omitting the
classical boolean connectives)
(w, i) |= p if p ∈ai
(w, i) |= YI φ if i > 0, (w, i −1) |= φ and τi −τi−1 ∈I.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
9
(w, i) |= XI φ if (w, i + 1) |= φ and τi+1 −τi ∈I.
(w, i) |= φ1 UI φ2 if there exists j ≥i s.t. (w, j) |= φ2, τj −τi ∈I and (w, k) |= φ1 for all
i ≤k < j.
(w, i) |= φ1 SI φ2 if there exists 0 ≤j ≤i s.t. (w, j)|=φ2, τi −τj ∈I and (w, k)|=φ1 for
all j < k ≤i.
A word w initially satisfies φ, written as w |= φ if (w, 0) |= φ. Finally, a model M satisfies
MTL formula φ, denoted as M |= φ if wρ |= φ for every run ρ of M. We rewrite the
derived operators G, F, P, H in terms of U or S, in the standard manner: FI φ = true UI φ,
GI φ = ¬ FI ¬φ, PI φ = true SI φ and HI φ = ¬ PI ¬φ.
▶Definition 4 (detMITL+p). The fragment detMITL+p consists of formulas where the
outermost temporal operators are future operators XI, UI and may use arbitrary intervals
I, while all inner temporal operators are restricted to past operators. Inner Since operators
must use non-punctual intervals (either [0, 0] or non-singleton intervals). Formally, formulas
are built as follows:
(Sentences)
Φ := p | Φ ∧Φ | ¬Φ | XI φ | φ UI φ
(Point-formulas)
φ := p | φ ∧φ | ¬φ | YI φ | φ SI′ φ
where I, I′ are integer-bounded intervals, and I′ is either [0, 0] or a non-singleton interval.
Sentences are evaluated at the first position (w, 0) of a word w, whereas point-formulas
can be checked anywhere in the word. The formula in Example 3 is a sentence in the
detMITL+p fragment: the derived GI operator induces an outer UI formula, and all the inner
operators are past. The formula YP<2 q can be rewritten as P(0,2) q ∨Y[0,0] P[0,0] q.
4
From detMITL+p to network of deterministic timed automata
For a detMITL+p sentence Φ we build a deterministic network NΦ and specify a set BadΦ
of bad configurations. Our objective is to achieve a construction that entails the following
theorem for detMITL+p sentences.
▶Theorem 5. Let Φ be a detMITL+p sentence and M a timed model. Then: M ̸|= Φ iff
there exists an infinite run of the closed network (M, NΦ) which eventually remains in bad
configurations.
Given a sentence Φ, we construct the following deterministic automata:
Aisat
p
Aisat
XI
Aisat
UI
for every atomic proposition p ∈Prop that is not under the scope of a temporal operator,
and every subformula XI and UI appearing in the sentence. The automaton Aisat
p
takes the
atomic proposition p as input. The automaton Aisat
XI
takes as input the boolean variable b1
corresponding to the argument of XI. Automaton Aisat
UI
takes two boolean variables b1, b2 as
inputs corresponding to the first and second arguments of UI. Notice that these operators
appear only at the outermost level of Φ and will be evaluated at the first position in a word.
The automata for these outermost operations will be called initial satisfiability automata,
and are superscripted with an isat. Apart from these, we build an automaton Ainit which
resets a clock xinit at the first action. All the isat automata make use of this single clock xinit
in their tests. The isat automata do not own any other clocks. Finally, for every temporal
point-formula φ appearing as a subformula of Φ, we build a deterministic automaton Aφ.
For these point-formula automata we also allow for sharing of clocks, which we explain later.
10
Efficient Verification of MTL with Past in Pointwise Semantics
1
2
Ainit
xinit := 0
1
2
3
4
Aisat
XI
true
b•
1 ∧x•
init ∈I
¬b•
1 ∨x•
init /∈I
Figure 3 Left: Clock xinit is reset on the first event and measures time elapsed for all the initial
satisfiability automata (Aisat
XI
and Aisat
UI ).
Right: Automaton Aisat
XI
for initial satisfiability for XI b1. The boolean variable b1 is not owned by
Aisat
XI
and stands for the argument of XI. In Aisat
XI , we simply write xinit instead of Ainit.xinit. The
automaton is deterministic and complete. If a run reaches state 3 (resp. 4) then XI b1 was initially
true (resp. false).
Bad configurations.
The collection of all the isat automata and the point-formula automata for Φ form the
network NΦ. States of each of the isat automata will be marked either true or false. A
configuration C of NΦ is bad if Φ evaluates to false when we replace every outermost p, and
operators XI, UI of Φ by the truth value of the state the corresponding isat automata are in.
For example if Φ = p ∧XI q, a configuration C1 where Aisat
p
= true and Aisat
XI q = true is not
bad, whereas C2 with Aisat
p
= false will be bad. The set of bad configurations of Φ will be
called BadΦ. This notion naturally extends to the product (M, NΦ): a configuration of the
product is bad if its projection to NΦ evaluates to false.
1
2
3
Aisat
p
true
p•
¬p•
Figure 2 Automaton Aisat
p
.
We describe the initial satisfiability automata in
Section 4.1 and show some local correctness properties.
The automata for past operators are presented in
Sections 4.2, 4.2.2 and 4.2.3. Finally, in Section 4.4,
we present the proof of Theorem 5 which shows the
global correctness of the construction, by making use
of the intermediate local correctness lemmas shown
in the previous sections.
4.1
Initial satisfiability of sentences
Figure 2 depicts the automaton Aisat
p
which takes as input the atomic proposition p. The
property that it satisfies is stated in the following lemma, whose proof is straightforward.
▶Lemma 6. For every timed word w = (a0, τ0)(a1, τ1) · · · over Prop with p ∈Prop, there is
a unique run C0
δ0,t0
−−−→C1
δ1,t1
−−−→C2 · · · of Aisat
p
over w. Moreover, w |= p iff for all i ≥1,
configuration Ci satisfies Aisat
p
.st = 2.
Figure 3 describes Ainit and Aisat
XI . The invariant satisfied by Aisat
XI
is explained in the
caption of Figure 3 and formalized below.
▶Lemma 7. Let Prop′ = Prop ⊎{b1} be an augmented set of atomic propositions. For
every word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0
δ0,t0
−−−→C1
δ1,t1
−−−→C2 · · ·
of (Ainit, Aisat
XI ) over w. Moreover, w |= XI b1 iff for all i ≥2, configuration Ci satisfies
Aisat
XI .st = 3.
Finally, Figure 4 gives the deterministic automaton for the initial satisfiability of UI.
Correctness of the automaton is formalized below.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
11
1
2
3
Aisat
UI
true
(x•
init ∈I ∧b•
1 ∧¬b•
2)
∨(x•
init < I ∧b•
1)
x•
init ∈I ∧b•
2
x•
init > I ∨
(x•
init < I ∧¬b•
1) ∨
(x•
init ∈I ∧¬b•
1 ∧¬b•
2)
Figure 4 Automaton Aisat
UI
for the initial satisfiability for b1 UI b2 (with I ̸= ∅). The boolean
variables b1, b2 are not owned by Aisat
UI
and stand for the left and right arguments of UI. The
automaton is deterministic and complete. We write xinit < I (resp. xinit > I) to state that xinit is
smaller (resp. larger) than all values in I. For instance, if I = [b, c) with b < c then xinit < I means
xinit < b and xinit > I means xinit ≥c. If a run reaches state 2 (resp. 3) then we know for sure that
b1 UI b2 was initially true (resp. false). State 1 means we still do not know whether b1 UI b2 was
initially true. If the run stays forever in state 1, b1 UI b2 was initially false.
ℓ0
ℓ1
AY
¬b•
1
b•
1
b•
1
¬b•
1
1
xlast = +∞
Alast
xlast := 0
Figure 5 Left: Automaton AY for the untimed Yesterday operator Y b1. The boolean variable b1
is not owned by AY and stands for the argument of Y.
Right: Sharer automaton Alast with clock xlast used by all Yesterday operators.
▶Lemma 8. Let Prop′ = Prop ⊎{b1, b2} be an augmented set of propositions. For every
word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0
δ0,t0
−−−→C1
δ1,t1
−−−→C2 · · · of
(Ainit, Aisat
UI ) over w. Moreover, w |= b1 UI b2 iff there exists an i ≥1 s.t. for all k ≥i,
configuration Ck satisfies Aisat
UI .st = 2,
Proof. Suppose w |= b1 UI b2. Then, (1) there exists an i ≥0 such that b2 ∈ai, (2) for all
0 ≤j < i, we have b1 ∈aj, and (3) τi −τ0 ∈I.
We assume that i ≥0 is minimal with the above property. Hence, in the run of w,
transitions tj for 0 ≤j < i are self-loops on state 1. Indeed, b•
1 is true at Cj+1 and either
τj −τ0 < I or τj −τ0 ∈I and b•
2 is false at Cj+1 due to the minimality of i.
transition ti goes to state 2 since b•
2 is true at Ci+1 and τi −τ0 ∈I.
Hence configuration Ci+1 satisfies Aisat
UI .st = 2.
Conversely, suppose the automaton moves to state 2 for the first time after reading the
prefix (a0, τ0)(a1, τ1) · · · (ai, τi). Since Aisat
UI
starts at state 1, the run loops around 1 on all
events a0 upto ai−1 and then a transition is made to state 2. In the loop b1 is true, and
in the transition that moves to 2, we have b2 to be true. The transition to 2 also checks
for xinit ∈I. Since xinit is reset at a0, we get τi −τ0 ∈I. Hence, the run shows the three
properties listed above and w |= b1 UI b2.
◀
4.2
Past operators
4.2.1
Yesterday operator.
The automaton for the Yesterday operator Y, depicted in Figure 5 (left), has two states ℓ0 and
ℓ1. The states remember whether the argument b1 of Y was true or not at the previous letter:
all transitions reading a b1 enter ℓ1, and the ones that read ¬b1 go to ℓ0. To incorporate the
timing constraint, we use a single state automaton Alast, depicted in Figure 5 (right), which
12
Efficient Verification of MTL with Past in Pointwise Semantics
resets a clock xlast in every step. Now, for an arbitrary interval constraint I, the current
position satisfies YI b1 if the previous letter had a b1 and the value of clock xlast (which was
reset at the previous position) when the current position is read falls in the interval I.
▶Lemma 9. Let Prop′ = Prop ⊎{b1}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′,
we have a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (Alast, AY). Moreover,
for all i ≥0, (w, i) |= Y b1 iff Ci(AY.st) = C′
i(AY.st) = ℓ1,
for all i ≥1, τi −τi−1 ∈I iff C′
i(Alast.xlast) ∈I.
Let outYI = (AY.st = ℓ1 ∧Alast.xlast ∈I). Then, for all i ≥0, we may check if (w, i) |= YI b1
with the test outYI during transition ti from C′
i to Ci+1.
4.2.2
Since operator with a one-sided interval.
The formula b1 SI b2 essentially asks that there exists a position j in the past, such that b2 is
true at that position, and at all positions between j and the current position, b1 is true, and
the time elapsed between the position j and the current position lies in the interval I. Note
that there could be multiple candidates for the position j. Let us first consider the untimed
version of the Since operator given in Figure 6 (left). The transition from ℓ0 to ℓ1 on b2 is
taken with the first b2 that could be a witness. When the automaton remains in ℓ1, each
time a b2 is seen, the automaton encounters a fresh witness.
When we move to the timed setting, some of these witnesses may satisfy the timing
constraint while others may not, and so this information must be tracked carefully. We will
first consider the simpler case where the intervals are one-sided, i.e., of the form [0, c), [0, c],
[b, +∞) or (b, +∞).
When the interval is one-sided, it is sufficient to track the two extreme b2 witnesses: the
earliest witness and the last possible witness at each point. The idea is simply that if the
earliest witness does not satisfy a lower-bound constraint, then no later witness could satisfy
it. Analogously, if the last witness does not satisfy an upper-bound, no earlier witness could
have satisfied it.
In the automaton Alast
S
of Figure 6 (middle), clock x maintains the time since the latest
b2 (which may serve as a witness for b1 S b2 when AS is in state ℓ1). For an upper-bound
interval SI where I = [0, c) or I = [0, c], we check the value of x, i.e., the time since the last
b2. The output is 1 if the target state of AS is ℓ1 and x ∈I.
▶Lemma 10. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′
there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (AS, Alast
S ). Moreover, for all
i ≥0:
ℓ0
ℓ1
AS
¬b•
2
b•
2
b•
1 ∨b•
2
¬b•
1 ∧¬b•
2
ℓ0
Alast
S
x = +∞
b•
2; x := 0
¬b•
2
ℓ0
Afirst
S
y = +∞
b•
2 ∧(¬b•
1 ∨AS.st = ℓ0);
y := 0
¬b•
2 ∨(b•
1 ∧AS.st = ℓ1)
Figure 6 Left: Automaton for the untimed Since operator b1 S b2.
Middle: Automaton recording with clock x the time since the last (most recent) occurrence of b2.
Right: Automaton recording with clock y time since the first (earliest) b2 which may serve as a
witness of b1 SI b2.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
13
1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,
2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Alast
S .x) = τi −τj where j ≤i, b2 ∈aj and b2 /∈ak
for all j < k ≤i.
3. Let I be an upper-bound interval and outSI = (AS.st• = ℓ1 ∧Alast
S .x• ∈I). Then, for all
i ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′
i
to Ci+1.
In the automaton Afirst
S
of Figure 6 (right), clock y maintains the time since the earliest
b2 which may serve as a witness for b1 S b2: clock y is reset when reading b2 while AS is in
state ℓ0 (the previous position did not satisfy b1 S b2) or when reading ¬b1 ∧b2 while AS is
in state ℓ1. For a lower-bound interval SI where I = [b, +∞) or I = (b, +∞) with b ∈N, we
check the value of y, i.e., the time since the earliest q. The output is 1 if the target state of
AS is ℓ1 and y ∈I.
▶Lemma 11. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′
there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (AS, Afirst
S
). Moreover, for all
i ≥0:
1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,
2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Afirst
S
.y) = τi −τj where j is the earliest witness at
i of b1 S b2: j ≤i, b2 ∈aj and b1 ∈ak for all j < k ≤i, and for all 0 ≤j′ < j either
b2 /∈aj′ or b1 /∈ak′ for some j′ < k′ ≤i.
3. Let I be a lower-bound interval and outSI = (AS.st• = ℓ1 ∧Afirst
S
.y• ∈I). Then, for all
i ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′
i
to Ci+1.
4.2.3
Since with a two-sided interval.
The general case with a nonempty non-singleton interval I is much harder. We may have
a position satisfying b1 SI b2 but neither the earliest nor the last witness of b2 satisfy both
the upper and lower constraints of I. In such a case, there could still be another witness in
between which does satisfy the constraint. We need to change our perspective and look at
each b2 event as a potential witness for b1 SI b2. However, we cannot simply start a new clock
at each b2 event, as we only have finitely many clocks at our disposal. We discuss below our
solution for this general case.
Let I be a non-singleton interval with lower bound b and upper bound c. We assume
0 /∈I and c ̸= +∞, otherwise it is a simple upper-bound or lower-bound constraint which
was already solved in Section 4.2.2. Since both b, c are integers and I is non-singleton, we
have c −b ≥1.
t
t + (c −b)
t′
t + b
t + c
t′ + b
t′ + c
Figure 7 Illustrating the idea for SI
The idea of our approach is illustrated in
Figure 7. Assume there is a b2 at t, represen-
ted by the red dot. Assuming I = [b, c], this
b2 is a witness for the events happening in
the interval [t + b, t + c] (marked by the red
line), assuming all the intermediate points
contain b1. Now, consider the last b2-event in the interval [t, t + (c −b)]. This is marked
by the green dot in the figure, at time t′. This b2 event acts as a witness for the events in
[t′ + b, t′ + c], marked by the green line in the figure. For any point to the right of t′ + c,
a valid b2 witness should necessarily be to the right of t + (c −b). This observation leads
to the following construction: when the first b2 is seen (the red dot), as long as there is a
14
Efficient Verification of MTL with Past in Pointwise Semantics
continuous sequence of b1’s following it, break the timeline into blocks of c −b and store the
earliest and latest b2 within these blocks. When the values become sufficiently large, the
earliest block becomes irrelevant, allowing to reuse clocks. Coupled with c −b ≥1, we are
able to implement this idea with a bounded number of clocks.
Let I be a non-singleton interval with lower and upper bounds b, c ∈N and 0 /∈I. Since
I is non-singleton, we have c −b ≥1. Let k = 2 +
j
b
c −b
k
.1 The automaton Agen
SI
for
b1 SI b2 has a single location ℓ0 and uses clocks x1, y1, . . . , xk, yk. The initial condition at
ℓ0 is x1 = y1 = · · · = xk = yk = +∞(all clocks are initially inactive). Automaton Agen
SI is
synchronized with AS (untimed since) from Figure 6 (left). Its transitions are defined in
Algorithm 1 with ◁= ≤if b ∈I and ◁= < otherwise.
Algorithm 1 Transitions of automaton Agen
SI .
1: if AS.st• = ℓ0 then
▷b1 S b2 does not hold
2:
x1, y1, . . . , xk, yk := +∞
3: else if ¬b•
1 ∨AS.st = ℓ0 then
▷b1 S b2 holds (AS.st• = ℓ1), new earliest witness
4:
x1, y1 := 0; x2, y2, . . . , xk, yk := +∞
5: else
▷b1 S b2 holds (AS.st• = ℓ1) and there are past witnesses (b•
1 ∧AS.st = ℓ1)
6:
if b ◁xn ̸= +∞∧(n = k ∨¬(b ◁xn+1 ̸= +∞)) then
▷Clock shift with 1 < n ≤k
7:
[x1, y1, . . . , xk, yk] := [xn, yn, . . . , xk, yk, +∞, . . . , +∞]
8:
end if
9:
if b•
2 then
▷new witness (not earliest)
10:
if xj < c −b then yj := 0
▷1 ≤j ≤k
11:
if c −b ≤xj ̸= +∞∧xj+1 = +∞then xj+1, yj+1 := 0
▷1 ≤j < k
12:
else
▷not a new witness (¬b•
2)
13:
Skip
14:
end if
15: end if
Notice that the automaton is deterministic and complete. Consequently, every word will
have exactly one run in the automaton. We give some intuitions before the formal statement
in Lemma 12.
The test of Line 1 is satisfied when the current position does not satisfy b1 S b2. In this
case, the new clocks are not needed and we set them to +∞, meaning inactive.
When the test of Line 3 succeeds, the current position satisfies b1 S b2 and there are no
other witness in the past. So we set x1, y1 to 0 and the other clocks are inactive.
If we have b ◁xn for some active clock with n > 1 then xn satisfies the lower bound con-
straint of I and it may serve as a witness instead of the earlier clocks x1, y1, . . . , xn−1, yn−1
(notice that xn < yn−1 ≤xn−1 < · · · < y1 ≤x1 and if one of these clocks satisfies the
upper bound constraint of I then so does xn). In this case, clocks x1, y1, . . . , xn−1, yn−1
are not needed anymore and we reduce the set of active clocks with the clock shift in
Lines (6–8).
When the test of Line 9 succeeds, the existing witnesses stay and the current position is a
new witness. Let x1, y1, . . . , xj, yj be the active clocks.
1 If the interval I is not of the form (b, c) (i.e., open on both sides) then we can construct an automaton
Agen
SI with k = 1 +
l
b
c −b
m
by changing xj < c −b on Line 12 and c −b ≤xj on Line 13 to xj ≤c −b
and c −b < xj respectively.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
15
If xj < c −b (Line 10) then we simply reset yj to 0 since the current position is now
the last witness in the block of length c −b which started when xj was reset. Otherwise
xj ≥c −b (Line 11) and the current position starts a new block by setting xj+1, yj+1 to
0 (new pair of active clocks). When this occurs, we have j < k thanks to the clock shift
of Lines (6–8).
The last possibility (Lines 12,13), the current position satisfies b1 S b2 but is not a new
witness since b2 doest not hold. In that case, we do not update the clocks since the set of
witnesses does not change.
▶Lemma 12. Let I be a non-singleton interval with lower and upper bounds b, c ∈N and
0 /∈I. Let ◁= ≤if b ∈I and ◁= < otherwise. Let Prop′ = Prop ⊎{b1, b2}. For every word
w = (a0, τ0)(a1, τ1) · · · over Prop′ there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · ·
of (AS, Agen
SI ).
Moreover, let outSI = (AS.st• = ℓ1 ∧(Agen
SI .x•
1 ∈I ∨Agen
SI .y•
1 ∈I)). Then, for all i ≥0,
we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′
i to Ci+1.
The proof of Lemma 12 will be given after the following technical lemma which establishes
all the necessary invariants of the automaton Agen
SI .
▶Lemma 13. Let I be a non-singleton interval with lower and upper bounds b, c ∈N and
0 /∈I. Let ◁= ≤if b ∈I and ◁= < otherwise. Let Prop′ = Prop ⊎{b1, b2}. For every word
w = (a0, τ0)(a1, τ1) · · · over Prop′ there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · ·
of (AS, Agen
SI ). Moreover, for all i ≥0 we have:
1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1.
2. When Ci+1(AS.st) = ℓ0 then configuration Ci+1 satisfies x1 = y1 = · · · = xk = yk = +∞.
3. When Ci+1(AS.st) = ℓ1 then for some 1 ≤j ≤k, configuration Ci+1 satisfies the following:
yj ≤xj < yj−1 ≤xj−1 < · · · < y1 ≤x1 < +∞, xj+1 = yj+1 = · · · = xk = yk = +∞,
xj′ −yj′ < c −b for all 1 ≤j′ ≤j, and xj′ −xj′+1 ≥c −b for all 1 ≤j′ < j,
there are indices i1 ≤i′
1 < i2 ≤i′
2 < · · · < ij ≤i′
j ≤i such that
b1 ∈ai′ for all i′ ∈(i1, i],
b2 ∈ai′ for all i′ ∈{i1, i′
1, . . . , ij, i′
j},
b2 /∈ai′ for all i′ ∈(i′
1, i2) ∪· · · ∪(i′
j−1, ij) ∪(i′
j, i],
x1 = τi −τi1, y1 = τi −τi′
1, . . . , xj = τi −τij, yj = τi −τi′
j,
b ◁x1 or (w, i1) ̸|= b1 ∧Y(b1 S b2). Moreover, if j > 1 then b ̸ ◁x2.
Proof. 1. was already stated in Lemma 10. 2. follows directly from Lines (1–2) of Algorithm 1.
3. The proof is by induction on i. The base case is i = 0. The initial state of AS is ℓ0,
i.e., C0(AS.st) = ℓ0. Now, C1(AS.st) = ℓ1 iff b2 ∈a0. In this case, Line 4 of Algorithm 1 is
taken and we easily check that item (3) of Lemma 12 holds with j = 1 and i1 = i′
1 = 0.
Consider now i > 0 and assume that item (3) hold for i −1. If Ci(AS.st) = ℓ0 then we
argue exactly as in the base case above. The interesting case is Ci(AS.st) = ℓ1. By induction,
there is 1 ≤j ≤k and indices i1 ≤i′
1 < i2 ≤i′
2 < · · · < ij ≤i′
j ≤i −1 satisfying all the
conditions of item (3) for i −1.
If b1 /∈ai then Line 4 of Algorithm 1 is taken. Notice that b2 ∈ai since the selfloop on ℓ1
of AS is taken. We easily check that item (3) of Lemma 12 holds with j = 1 and i1 = i′
1 = i.
It remains to deal with the case Ci+1(AS.st) = Ci(AS.st) = ℓ1 and b1 ∈ai.
Assume first that no clock shift occurs: we have b ̸ ◁C′
i(xj′) for all 1 < j′ ≤j.
16
Efficient Verification of MTL with Past in Pointwise Semantics
If b2 ∈ai then Lines (10–11) of Algorithm 1 are executed.
Recall that C′
i satisfies xj′ −xj′+1 ≥c −b for all 1 ≤j′ < j. Hence, if C′
i satisfies
xj′ < c −b it must be with j′ = j. In this case, Line 10 resets clock yj to 0. We see easily
that Ci+1 satisfies Item 3 with the same j and the sequence of indices i1 ≤i′
1 < i2 ≤i′
2 <
· · · < ij ≤i′′
j = i. Note that the last index in the sequence has been updated: i′′
j = i. Since
xj < c −b when yj is reset, Ci+1 satisfies xj −yj < c −b.
Assume now that C′
i satisfies xj ≥c −b. Assume towards a contradiction that j = k.
Then, C′
i satisfies x2 −xk ≥(k −2)(c −b). Since we had no clock shift, we also have x2 ≤b.
Together with c −b ≤xk we get (k −1)(c −b) ≤x2 ≤b. This is a contradiction with
k = 2 +
j
b
c −b
k
. Hence, j < k and Line 11 will reset clocks xj+1, yj+1 to 0. We easily check
that Item 3 is satisfied with j + 1 and the sequence of indices i1 ≤i′
1 < i2 ≤i′
2 < · · · <
ij ≤i′
j ≤ij+1 = i′
j+1 = i. In particular, Cj+1 satisfies xj −xj+1 = xj ≥c −b. And also, if
j + 1 = 2, we have b ̸ ◁0 = x2 at Ci+1 since 0 /∈I.
If b2 /∈ai then Line 13 of Algorithm 1 is taken and we easily check that item (3) holds for
i with the same j and indices i1 ≤i′
1 < i2 ≤i′
2 < · · · < ij ≤i′
j. In particular, since clocks
are not updated, for all 1 ≤j′ ≤j, we get Ci+1(xj′) = Ci(xj′) + τi −τi−1 = τi −τij′ and
similarly for yj′.
Finally, assume that a clock shift occurs: the test of Line 6 evaluates to true with some
1 < n ≤j. Let j′′ = j −n + 1. Then, 1 ≤j′′ < j and after the clock shift we have xj′′ ̸= +∞
and xj′′+1 = +∞. We can easily check that, after the clock shift, Item 3 holds with j′′ and
the sequence of indices in ≤i′
n < in+1 ≤i′
n+1 < · · · < ij ≤i′
j ≤i. In particular, the value of
x1 after the clock shift is the value of xn at C′
i, and b is smaller than (◁) this value due to
the condition of the clock shift. Similarly, if j′′ > 1 then b is smaller than (◁) the value of x2
after the clock shift which is the value of xn+1 at C′
i. Therefore, the last bullet of Item 3
holds after the clock shift, hence also at Ci+1.
Now, we can prove as above, in the case of no clock shift, that after executing Lines (9–14)
of Algorithm 1, Item 3 holds at Ci+1. This concludes the inductive proof of Item (3).
◀
Proof of Lemma 12. Fix i ≥0. By Lemma 13, position i satisfies the untimed since b1 S b2
iff Ci+1(AS.st) = ℓ1 (corresponding to AS.st• = ℓ1 in outSI). We assume that we are in
this case. The only question is whether there is a witnessing b2 within the interval I. We
claim that this can be determined by looking solely at the values of clocks x1 and y1 at
configuration Ci+1 (corresponding to the test Agen
SI .x•
1 ∈I ∨Agen
SI .y•
1 ∈I in outSI).
We can view this as follows. Suppose there is in the past a b2 event witnessing b1 SI b2:
b2 ∈ai′′ for some i′′ ≤i and b1 ∈ai′ for all i′′ < i′ ≤i and τi −τi′′ ∈I. Let 1 ≤j ≤k and
i1, i′
1, . . . , ij, i′
j be the indices given by Lemma 13 (3).
If i′′ < i1 then (w, i1) |= b1 ∧Y(b1 S b2). Therefore in this case we have b ◁x1 at Ci+1.
Now, b ◁τi −τi1 ≤τi −τi′′ ∈I. We deduce that i1 is also a witness of (w, i) |= b1 SI b2.
If i′
1 < i′′ then we must have j ≥2. Moreover, i2 ≤i′′ since there are no b2 in (i′
1, i2).
Using b ̸ ◁x2 at Ci+1, we deduce that b ̸ ◁τi −τi2 and then b ̸ ◁τi −τi′′, a contradiction with
τi −τi′′ ∈I.
Finally, if i1 < i′′ < i′
1 then using x1 −y1 < c −b at Ci+1 we get τi′
1 −τi1 < c −b. Using
τi1 ≤τi′′ ≤τi′
1 and τi −τi′′ ∈I, we deduce that either i1 or i′
1 (or both) is also a witness of
(w, i) |= b1 SI b2.
◀
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
17
4.3
A full example
We explain the construction of a network NΦ associated with a sentence Φ on an ex-
ample. We use the template automata defined in the previous sections. We instantiate the
formal arguments b1, b2 of these template automata with actual test formulas. For instance,
AY[b•
1/(¬p• ∧q•)] denotes a copy of the automaton AY from Figure 5 (left) in which we
substitute the test ¬p• ∧q• for the formal argument b•
1.
We consider the sentence Φ = (Φ1 ∧Φ2) →Φ3 where Φ1 = r and
Φ2 = FI6(p ∧φ4) = true UI6 (p ∧φ4)
Φ3 = FI7(φ3 ∧φ5) = true UI7 (φ3 ∧φ5)
φ4 = (p ∨q) SI4 φ3
φ5 = (φ1 ∨φ2) SI5 (q ∨φ3)
φ1 = YI1(¬p ∧q)
φ2 = YI2(¬p ∧q)
φ3 = YI3 r
with I4 = [0, c4) an upper bound constraint and I5 = [c5, +∞) a lower bound constraint.
The network NΦ does not contain automata for the boolean connectives. Instead, we
use the power of tests in automata, which allow arbitrary boolean combinations of atomic
tests.
A temporal subformula is either an atomic proposition or a formula where the
outermost connective is a temporal modality (Yesterday, Since, neXt or Until). The temporal
subformulas of the sentence Φ are the atomic propositions p, q, r, the point formulas φ1, . . . , φ5
and the sentences Φ2, Φ3.
The network NΦ does not need automata for the atomic propositions since they are
owned by the model M. For each temporal subformula, we add to the network the automata
required to determine the truth value of the subformula. These automata are described
in the previous sections. For instance, the temporal subformula φ4 requires a copy of the
automata AS and Alast
S
from Figure 6 (with suitable substitutions of the formal arguments).
Lemma 10 provides the test expression which allows to determine the truth value of φ4.
For the sentence Φ = (Φ1 ∧Φ2) →Φ3, we construct the network
NΦ = (B0, B1, B2, B3, B4, B5, B6, B7, A1, A2, A3)
where
B1 := Alast
B2 := AY[b•
1/(¬p• ∧q•)]
out1 = (B2.st = ℓ1 ∧B1.xlast ∈I1)
out2 = (B2.st = ℓ1 ∧B1.xlast ∈I2)
B3 := AY[b•
1/r•]
out3 = (B3.st = ℓ1 ∧B1.xlast ∈I3)
B4 := AS[b•
1/(p• ∨q•), b•
2/out3]
B5 := Alast
S [b•
2/out3]
B6 := AS[b•
1/(out1 ∨out2), b•
2/(q• ∨out3)]
B7 := Afirst
S
[b•
1/(out1 ∨out2), b•
2/(q• ∨out3)]
out4 = (B4.st• = ℓ1 ∧B5.x• ∈I4)
out5 = (B6.st• = ℓ1 ∧B7.y• ∈I5)
B0 := Ainit
A1 := Aisat
r
A2 := Aisat
UI6 [b•
1/true, b•
2/(p• ∧out4)]
A3 := Aisat
UI7 [b•
1/true, b•
2/(out3 ∧out5)]
In the automata above, we use ghost variables (outi)1≤i≤5 to make the definitions easier to
read. When we expand these definitions, we get for instance
B4 = AS[b•
1/(p• ∨q•), b•
2/(B3.st = ℓ1 ∧B1.xlast ∈I3)]
A2 := Aisat
UI6 [b•
1/true, b•
2/(p• ∧(B4.st• = ℓ1 ∧B5.x• ∈I4))]
Notice that the automaton B2 is shared: it is used in out1, out2. Similarly, B0 and B1 are
shared.
18
Efficient Verification of MTL with Past in Pointwise Semantics
It remains to define the set BadΦ of bad configurations for the sentence Φ. A configuration
C is bad if Φ evaluates to false at C. Hence, BadΦ is the set of configurations C which satisfy
A1.st = 2 ∧A2.st = 2 ∧¬(A3.st = 2) (i.e., Φ1 ∧Φ2 ∧¬Φ3).
4.4
Proof of Theorem 5
We give now the general construction for an arbitrary sentence Φ. Let φ1, . . . , φn be the
temporal point subformulas of Φ other than the atomic propositions. We assume that if φj
is a subformula of φi then j ≤i. Let Φ1, . . . , Φm be the temporal subsentences of Φ. We
define the network
NΦ = (B0, B1, B2, . . . , B2n−1, B2n, A1, . . . , Am)
together with the ghost variables out1, . . . , outn. For each 1 ≤i ≤n, we define B2i−1, B2i
and outi as follows:
If φi = YIi φ′
i then we let B2i = Alast. The automaton B2i−1 is a copy of the untimed
AY with a suitable substitution of its formal argument b1 that we define now. The
formula φ′
i is a boolean combination of temporal subformulas. Let φ′′
i be the same boolean
combination in which an outermost temporal subformula φj is replaced with outj and an
outermost atomic proposition p is replaced with p•. For instance, if φ′
i = (q ∧φ2) ∨φ4
then φ′′
i = (q• ∧out2) ∨out4. Then, B2i−1 = AY[b•
1/φ′′
i ]. Finally, outi = (B2i−1.st =
ℓ1 ∧B2i.xlast ∈Ii).
If φi = φ′
i SIi ψ′
i and I is an upper-bound interval. Then we let B2i−1 = AS[b•
1/φ′′
i , b•
2/ψ′′
i ],
B2i = Alast
S [b•
2/ψ′′
i ] and outi = (B2i−1.st• = ℓ1 ∧B2i.x• ∈Ii).
If φi = φ′
i SIi ψ′
i and I is a lower-bound interval. Then we let B2i−1 = AS[b•
1/φ′′
i , b•
2/ψ′′
i ],
B2i = Afirst
S
[b•
1/φ′′
i , b•
2/ψ′′
i ] and outi = (B2i−1.st• = ℓ1 ∧B2i.y• ∈Ii).
If φi = φ′
i SIi ψ′
i and I is a non-singleton interval which is neither an upper-bound nor a
lower-bound. Then we let B2i−1 = AS[b•
1/φ′′
i , b•
2/ψ′′
i ], B2i = Agen
SIi [b•
1/φ′′
i , b•
2/ψ′′
i , AS.st/B2i−1.st]
and outi = (B2i−1.st• = ℓ1 ∧(B2i.x•
1 ∈Ii ∨B2i.y•
1 ∈Ii)).
Next, we let B0 = Ainit and for 1 ≤i ≤m we define
If Φi = p is an atomic proposition, then we let Ai = Aisat
p
.
If Φi = XJi φ′
i then we let Ai = Aisat
XJi [b•
1/φ′′
i ].
If Φi = φ′
i UJi ψ′
i then we let Ai = Aisat
UJi [b•
1/φ′′
i , b•
2/ψ′′
i ].
Finally, as explained in Section 4, the set BadΦ of bad configurations consists of all configur-
ations C such that Φ evaluates to false when we replace each temporal subsentence Φi of Φ
by Ai.st = ℓi where ℓi is the state labelled true in Ai (ℓi = 3 if Φi is a next subformula, and
ℓi = 2 otherwise).
Now that we have constructed the deterministic network NΦ, we prove its correctness.
▶Lemma 14. Let Φ be a sentence in detMITL+p. The network NΦ constructed above is
deterministic and complete. Hence, for every word w = (a0, τ0)(a1, τ1) · · · over Prop there is
a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of NΦ. Moreover,
1. for all temporal point subformula φj of Φ (with 1 ≤j ≤n) and for all position i ≥0 in
w, we have (w, i) |= φj iff evaluating the test outj during transition ti from C′
i to Ci+1
yields true.
2. w ̸|= Φ iff there is k ≥0 such that Ci ∈BadΦ for all i ≥k.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
19
Proof. All automata used in the network NΦ are deterministic and complete. Hence, so is
the network and each timed word has indeed a unique run.
1. The proof is by induction on j. It follows easily from the correctness lemmas in the
previous sections: Lemma 9 for yesterday with an arbitrary interval, Lemma 10 for since
with an upper-bounded interval, Lemma 11 for since with a lower-bounded interval and
Lemma 12 for since with a non-singleton interval which is neither upper-bounded nor
lower-bounded.
2. Using (1) and the lemmas for initial satisfiability (Lemma 6, Lemma 7 and Lemma 8), we
deduce that for each temporal subsentences Φj of Φ (with 1 ≤j ≤m), we have w |= Φj iff
the automaton Aj eventually stabilises in a state labelled true, i.e., state 2 for an atomic
proposition or for an until modality, and state 3 for next. Notice that each automaton
Aj must eventually stabilise in some state since each cycle in these automata are actually
self-loops. Now, using the definition of BadΦ, we deduce that w ̸|= Φ iff the run for w
eventually stays in bad configurations.
◀
▶Remark 15. The number of automata in the network NΦ is O(|Φ|). The automata B2i
have a single state and own clocks, the automata B2i−1 have two states and do not own
clocks, and the automata Ai have 3 or 4 states and do not own clocks. The number of clocks
owned by B2i is 1 unless it comes from some SI where I is a two-sided interval with lower
and upper bounds b, c ∈N such that c−b ≥1. In that case, the number of clocks is 2k where
k = 2 +
j
b
c −b
k
≤2 + b. Hence, the total number of clocks in the network is O(Φ) if Φ does
not use since with two-sided intervals, or if k = O(1) for all since with two-sided intervals, or
if the constants are given in unary (actually, only the lower constants of two-sided intervals).
▶Remark 16. We reduce further the size of the network NΦ by removing duplicates. For
instance, if Φ has several occurrences of Y, possibly with different arguments and intervals,
the automaton Alast occurs as many times in the network constructed above. In practice,
we use a single occurrence of Alast adapting accordingly the atomic tests xlast ∈Ii in the
corresponding ghost variables outi. We also remove duplicate occurrences of AY automata
and of since automata (AS, Alast
S , Afirst
S
).
5
Integrating the Future
Our goal in this section is to include future modalities in the logic and obtain an efficient
model checking procedure for a general fragment of MTL with past and future. To do so,
we consider the logic detMITL+p from Definition 4 and allow Point formulas to also have
XI and UI′ operators. This results in a restriction of MTL in which, except in the topmost
level, the intervals in Since and Until are restricted to be [0, 0] or non-singleton. We call this
fragment MITL+p. This fragment subsumes the well-known MITL fragment and even goes
beyond extending it with past.
We recall that in the pointwise semantics, MTL model checking is undecidable over infinite
words, and over finite words it is decidable but the complexity is Ackermanian [7, 50, 48, 49].
Hence, the literature often considers the MITL fragment in the pointwise semantics, with
multiple known approaches for building efficient model checking algorithms. Of these we
highlight two recent ones, both of which only work with future modalities. The first from [20],
works by translating MITL into a network of timed automata (via 1-clock alternating
automata) and was implemented in a state-of-the-art tool MightyL. The second from [2],
converts MITL into generalized timed transducers, which use future clocks defined in [1].
However, this approach has not yet been implemented.
20
Efficient Verification of MTL with Past in Pointwise Semantics
In the rest of this section, we combine ideas from [2] and from [20], with our ideas for
initial satisfiability and past modalities presented in the previous sections, to develop a new
construction from formulas in MITL+p directly to network of generalized timed automata
with shared variables, without the explicit use of transducers. As we are dealing with future
modalities, non-determinism cannot be avoided, but we show two algorithmic improvements,
both designed to reduce non-deterministic branching. Our experimental results later showcase
improved performance (compared to either of the two approaches above) in number/type
of benchmarks solved and time taken, thus highlighting the significant impact of these
improvements.
5.1
Including Future clocks in the Model
First, we lift the definitions to include future clocks. Thus the set of clocks X is now partitioned
into history clocks XH, which are just clocks as we saw till now and XF , future clocks, that
take non-positive values. A future clock takes values in the interval R−= (−∞, 0] ∪{−∞},
where value −∞signifies that this clock is undefined or inactive. Thus, a generalized timed
automaton with shared variables (GTA) is a tuple A = (Q, X, V, ∆, init), where X = XH ·∪XF
is the set of clocks owned by A.
Now, an atomic update for a future clock variable y ∈XF owned by A is non-deterministic
(unlike to normal/history clocks) and has the form A.y :∈I (or simply y :∈I), for some
integer bounded interval I with end-points in Z−= {0, −1, −2, . . . , −∞}. Further, an atomic
test for y ∈XF owned by A has the form A.y ∈I (or simply y ∈I) where I ⊆R−is an
integer bounded interval. For simplicity, in examples we often write −y ∈I with endpoints
in N instead. As before A may also test a future clock variable owned by another component
with N.y ∈I or N.y• ∈I where N is a name. As before a network of GTA with shared
variables is just a tuple A = (A1, . . . , An) where each Ai is a GTA with shared variables.
The semantics of a network of GTA is the same as for network of TA as before with two
key differences: First, in the definition of a configuration C for any 1 ≤i ≤n, we have in
addition C(Ai.y) ∈R−for y ∈XF
i . Second, a time elapse transition C
δ−→C′ can be taken
only if (i) C′ coincides with C on all variables other than clocks, (ii) the value of each clock
(regardless of whether it is future or history) is advanced by the same quantity δ and (iii)
C(Ai.y) + δ ≤0 for all 1 ≤i ≤n and y ∈XF
i (with −∞+ δ = −∞). The third condition is
new and means that an elapse can occur only if it does not force a future clock to take a
positive value (same requirement as in [1, 2] for future clocks).
Finally, we recall that our TA (and GTA) are always strongly non-Zeno, i.e., every
accepting run is non-Zeno (a timed run is Zeno if P
i≥0 δi is bounded). As remarked in
Lemma 5 of [2], this assumption (not a severe restriction, since every GTA can be converted
easily to a strongly non-Zeno one) has interesting consequence for future clocks: if they are
not ultimately −∞, they should be released infinitely often. If not, there is a last point
where a future clock is released to a finite value, and the entire suffix of the run lies within
this finite time, contradicting non-Zenoness.
5.2
Model checking the future
For the model checking procedure, the idea at the high level is as before. Given a formula in
MITL+p, we translate it to a network of GTA with shared variables and then build a closed
network with a model and check existence of a bad cycle of configurations. As explained
in [2], the definition of a bad cycle now also has to take into account future clocks. The goal
of this section is to prove the following theorem.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
21
ℓ0
ℓ1
AX
¬b•
1
b•
1
b•
1
¬b•
1
1
Anext
xnext = 0; release(xnext)
Figure 8 Left: Automaton AX for the untimed Next operator X b1. The boolean variable b1 is
not owned by AX and stands for the argument of X.
Right: Automaton Anext with clock xnext used (shared) by all Next operators.
▶Theorem 17. Let Φ be a MITL+p sentence and M a timed model with history and future
clocks. Then, we can construct a network of GTA with shared variables NΦ of size linear
in size of Φ (assuming unary encoding of constants in the formula)2 such that M ̸|= Φ iff
there exists a reachable cycle of the closed network (M, NΦ), which eventually remains in
bad configurations and where every future clock x of the network is either released during the
cycle or is undefined −∞.
5.2.1
Next with sharing.
As for Yesterday, for any occurrence of the Next operator XI, we have an automaton depicted
in Figure 8 (left), which has two states ℓ0 and ℓ1 and captures the untimed semantics of
Next. The states guess whether the argument b1 of X will be true or not at the next letter
without worrying about the timing constraints. Then, we use a single state automaton Anext,
depicted in Figure 8 (right), which releases clock xnext whenever it reaches 0. Thus, it always
predicts the time to the next event. It will be shared across all occurrences of Next in the
entire network, to check all the timing constraints. Now, for an arbitrary interval constraint
I, the current position satisfies XI b1 if the next letter has a b1 and the value of clock xnext
(released here) falls in the interval I. Formally,
▶Lemma 18. Let Prop′ = Prop ⊎{b1}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′,
we have a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (AX, Anext). Moreover, for all
i ≥0,
1. (w, i) |= X b1 iff Ci+1(AX.st) = ℓ1,
2. τi+1 −τi ∈I iff Ci+1(Anext.xnext) ∈I.
Let outXI = (AX.st• = ℓ1 ∧−Anext.x•
next ∈I).
Then, for all i ≥0, we may check if
(w, i) |= XI b1 with the test outXI during transition ti from C′
i to Ci+1.
5.2.2
Until with sharing.
The formula b1 UI b2 asks that there exists a position j in the future (there could be many
of them), such that b2 is true at that position, and b1 is true at all positions between the
current position and j, and the time elapsed between the current position and position j
lies in the interval I. As for Since, we first consider the simpler case where intervals are
one-sided, i.e., of the form [0, c), [0, c], [b, +∞) or (b, +∞). Again it is sufficient to track the
two extreme b2 witnesses: the earliest witness and the last possible witness at each point. As
2 In fact, we can also do the same with binary encoding of constants with one caveat: the lower bounds
on intervals used in Since and Until operators should be bounded by a constant.
22
Efficient Verification of MTL with Past in Pointwise Semantics
ℓ0
Afirst
U
b•
2;
x = 0; release(x)
¬b•
2
ℓ0
ℓ1
AU
Afirst
U .x ̸= −∞
¬b•
2
¬b•
1 ∧¬b•
2
b•
2
b•
1 ∨b•
2
ℓ0
Alast
U
b•
2 ∧(¬b•
1 ∨AU.st• = ℓ0);
y = 0; release(y)
¬b•
2 ∨(b•
1 ∧AU.st = ℓ1)
Figure 9 Left: Afirst
U
guessing time to the earliest b2 with future clock x. release(x) stands for the
non-deterministic update x :∈R−, allowing x to be set to any non-positive value, including −∞.
Middle: AU for untimed until but which uses future clock x in the invariant of state ℓ1.
Right: Alast
U
guessing with future clock y time to the latest witness of b1 UI b2.
shown in Figure 9, we introduce three automata: (i) Afirst
U
which owns and uses future clock
x to track the time to the first b2, (ii) Alast
U
which owns and uses future clock y and tracks
the time to the latest witness of b1 U b2 and (iii) a GTA AU which captures the semantics of
the untimed Until operator, using clock x in the invariant of state ℓ1.
As we are dealing with future, all three automata are non-deterministic. However, the
only non-determinism in Afirst
U
and Alast
U
lies in the non-deterministic updates (release). The
zone automata3 built from them are actually deterministic since the release operation results
in a unique zone.
Unlike in the previous section, we cannot hope to get a completely
deterministic network since with future modalities we must have non-determinism (or pay an
exponential price to remove non-determinism). But we can try to limit the non-deterministic
branching for better performance in practice. Towards this, our networks of GTA with shared
variables for each future modality allow sharing of clocks and predictions (compared e.g.,
to using combined transducers as done in [2]). The advantage of using different automata
is that we can now share the prediction of the earliest witness without insisting to share
the latest witness at a point. This results in increased sharing and thus further decrease in
non-deterministic branching. Our experimental results in the next section validate these
theoretical insights.
▶Remark 19. The most interesting automaton is AU in Figure 9 (middle), which uses clock x
from Afirst
U , but does not own any clock. Notice that it is a two-state automaton and does not
require any Büchi acceptance condition. In fact, in the untimed setting (i.e., in the absence
of clocks), Until requires some Büchi acceptance condition and 3 states if the acceptance
condition is state-based. Lifting this to the timed setting gives a 3-state transducer with a
Büchi acceptance condition in [2]. However, we show in this paper that, surprisingly, 2 states
are sufficient and no acceptance condition is required, by exploiting clocks and non-zenoness.
The main idea is that in state ℓ1, we employ an invariant4 implying that b2 will eventually
occur. Under the non-zenoness assumption, stating that clock x has a finite (not −∞) value,
forces clock x to be eventually released. Hence, Afirst
U
must eventually take the loop labeled
b2. We highlight that going from 3 states to 2 states represents an exponential improvement
in non-deterministic branching and runtime as we go from 3O(|Φ|) to 2O(|Φ|) in the number of
zones that we will have to explore during reachability checking. Also, reducing the number
of Büchi acceptance conditions makes the liveness algorithm more efficient. Further, we also
3 Zone automata are finite abstractions of (generalized) timed automata, used for efficient algorithms
checking reachability [27, 1]
4 Note that, though we did not introduce state invariant in the model, it can easily be simulated with
guards on all incoming transitions –for succinctness we chose to present it this way.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
23
note that if we remove the invariant, and use of the clock, we obtain the automaton for
untimed Weak-Until.
The following lemma shows correctness of the above construction:
▶Lemma 20. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′
there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (Afirst
U , AU, Alast
U ). Moreover, for
all i ≥0:
1. (w, i) |= b1 U b2 iff Ci(AU.st) = C′
i(AU.st) = ℓ1,
2. If I is an upper-bound interval, we set outUI = (AU.st = ℓ1 ∧−Afirst
U .x ∈I). Then, for all
i ≥0, we may check if (w, i) |= b1 UI b2 with the test outUI during transition ti from C′
i
to Ci+1.
3. If I is a lower-bound interval, we set outUI = (AU.st = ℓ1 ∧−Alast
U .y ∈I). Then, for all
i ≥0, we may check if (w, i) |= b1 UI b2 with the test outUI during transition ti from C′
i
to Ci+1.
For the most difficult case of the General Until operator, we note that it follows as a dual
of the Since case from Section 4.2.3. To avoid repetitiveness, we omit a formal description of
this. Finally, combining all the above lemmas, we may prove Theorem 17 following the same
lines of the proof of Theorem 5 in Section 4.4.
▶Remark 21. Note that Theorem 5 and Theorem 17 can be extended to the case of finite
words, where we want to reach a bad configuration instead of eventually remaining in bad
configurations. The only modification required concerns the future modalities operator. We
can use the same automata defined above, but the definition of a bad configuration C for
finite words asks, in addition to the existing conditions, that (1) for each occurrence of AX
the final state is ℓ0 (C(AX.st) = ℓ0) and (2) for each occurrence of AU the final state is ℓ0
(C(AU.st) = ℓ0).
6
Improved simulation check and liveness for GTA
In this section, we describe our technical contributions that have gone into the development
of the LiveRGTA tool. We start with an overview of these contributions.
A liveness algorithm for timed automata or GTA computes a symbolic abstraction called
the zone graph. Zone graphs are known to be infinite and termination mechanisms for zone
computations have seen a long line of work (see [15] for a survey). A node in the zone graph is
of the form (q, Z) where q is a control state of the network of automata and Z is a constraint
system on clocks that represents the set of configurations abstracted by the current node. For
reachability, termination is guaranteed by using finite simulation relations, which are binary
relations on nodes (q, Z) ⪯(q, Z′). For liveness, to ensure soundness, simulation needs to be
replaced with a mutual simulation: (q, Z) ⪯(q, Z′) and (q, Z′) ⪯(q, Z). The procedure for
checking (q, Z) ⪯(q, Z′) is the most used operation in the zone graph computation and it is
extremely crucial to have efficient algorithms for the same. When the automaton contains no
diagonal constraints (guards of the form x −y ◁c), the check can be done in O(|X|2) where
X is the total number of clocks. When diagonal constraints are present, the check is known
to be NP-hard, and it has a running time of O(|X|2 · 2d) where d is the number of diagonals
present. Essentially, the check calls the diagonal-free module at most 2d times.
In the LiveRGTA tool, we have implemented an improved simulation check for GTAs with
future clocks, compared to the simulation check of [1]. This results in coarser simulations and
smaller parts of the zone graph being explored. There are cases where the gain is exponential,
24
Efficient Verification of MTL with Past in Pointwise Semantics
as for instance with GTAs of the form of Figure 10. We also use a new direct algorithm for
the mutual simulation check, that bypasses the need to call the basic simulation check twice.
This reduces the computation time by half.
ℓ0
ℓ1
ℓ2
· · ·
ℓn−1
ℓn
z = −1
x1 = −3
x1 = −1
x2 = −3
x2 = −1
xn = −3
xn = −1
Figure 10 A GTA that shows the exponential improvement produced smaller G-sets.
Finally, another important contribution is that we lift and adapt Couvreur’s algorithm
[26] for Büchi liveness to the setting of generalized timed automata.
We present below our procedure for the liveness problem of safe GTAs. Decidability of
liveness for safe GTAs was recently shown in [2, Theorem 11]. A GTA is live, i.e., it admits
an infinite accepting non-Zeno run, iff in its zone graph there is a reachable accepting cycle,
i.e., a reachable zone (q, Z) and a path (q, Z)
t1
−→(q1, Z1)
t2
−→· · · (qk, Zk)
tk+1
−−−→(q, Z′) such
that (q, Z) ⪯(q, Z′) and (q, Z′) ⪯(q, Z), each Büchi condition is witnessed along this cycle,
and each future clock x ∈XF is either released along the cycle or could take value −∞
(there is a valuation vx ∈Z with vx(x) = −∞).
There are several ways to implement this liveness check.
Our choice is to extend
Couvreur’s SCC algorithm [26, 55, 31] (the state-of-the-art approach for checking liveness of
timed automata) to the setting of GTAs. Our procedure can be thought of as an adaptation
of the idea for checking liveness of GTA proposed in [2] to the well-studied framework of
Couvreur’s SCC algorithm. On top of the usual liveness algorithm, we need to perform extra
checks to determine if future clocks are released in SCCs.
The idea for checking liveness in [2] ultimately boils down to identifying cycles in zone
graphs that satisfy certain special properties. Couvreur’s SCC algorithm provides an optimal
method for detecting cycles. Algorithm 2, given in Appendix B, is an SCC-based liveness
algorithm to check if a GTA A has an accepting cycle with respect to some generalized Büchi
acceptance condition. It leverages the various optimizations of Couvreur’s SCC algorithm [26]
that makes liveness checking more efficient for timed automata. Notice that, to the best
of our knowledge, Couvreur’s algorithm has not been previously extended to event-clock
automata or timed automata with diagonal constraints. Hence, Algorithm 2 is the first
extension of Couvreur’s liveness procedure for these classes.
7
Implementation and Experimental evaluation
In this section, we present an experimental evaluation of our prototype implementation tool
Tempora for satisfiability and model checking for MITL+p formulas. The tool contains two
modules:
MITL2GTA that implements our optimized translation from detMITL+p to networks
of deterministic timed automata (from Section 4) and the optimized translation from
MITL+p to networks of GTA (from Section 5), using the improvements (sharing and
two-state until) for future modalities.
LiveRGTA where we have implemented the new SCC-based liveness algorithm (discussed
in Section 6) and an improved reachability algorithm for GTA. This is an enhanced version
of GTA from [1], built on top of the open-source timed automata library Tchecker [34].
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
25
By combining these two modules, we obtain a new pipeline Tempora(MITL2GTA+LiveRGTA)
for satisfiability and model checking for MITL+p formulas (with both past and future modal-
ities) over timed words under pointwise semantics.
We organize our evaluation into three parts as follows:
First, we evaluate the performance of the detMITL+p to deterministic timed automata
translation on a set of benchmarks, which include both standalone past modalities and
combinations with future modalities. To the best of our knowledge, in the pointwise
semantics, there are no other tools that tackle this fragment.
Next, we compare our MITL+p to GTA translation against two competing approaches in
the pointwise semantics, namely:
1. MightyL [20]: a state-of-the-art tool that translates MITL to OCATA and then to
networks of timed automata. The resulting automata are analyzed using UPPAAL [41]
(for finite words) and OPAAL [37] (for infinite words).
2. Baseline GTA Translation an implementation of the (theoretical) translation from
MITL to GTA proposed in [2].
Finally, to demonstrate the full scope of our implementation, we present our full model-
checking pipeline on two classical examples, namely Fischer and Dining Philosophers with
MITL+p formulae capturing some standard properties.
Experimental setup
Our tool Tempora takes as input a MITL+p formula and applies the appropriate translation
described in Sections 4 and 5, producing a network of GTA (which is deterministic, when
the formula is in detMITL+p). Then, checking satisfiability of a formula over finite timed
words in the pointwise semantics reduces to checking emptiness of the corresponding network
of GTA. We also adapted our translation to work with finite timed words as explained in
Remark 21. We use an improved version of GTA reachability algorithm from [1], which is
built on the open source tool Tchecker [34]. Over infinite words, checking satisfiability
reduces to checking Büchi acceptance in GTA. For this, we implement the liveness algorithm
from [2], again on top of TChecker. Finally, we implemented a prototype model-checking
algorithm for MITL+p formulae w.r.t. a network of GTA with the pointwise semantics.
All experiments were run on an Ubuntu 18.04 machine with an Intel i7 3.40GHz processor
and 32 GB RAM. In this section, we describe the benchmarks and present an analysis of the
results. Details of our implementation, including the various engineering optimisations that
we employ, are discussed in Appendix C.
Benchmarks. We conduct our experiments on a suite of 72 benchmarks, which we have
grouped into three groups: (1) benchmarks designed to demonstrate past operators (alongside
future modalities), (2) benchmarks from MightyL [20] that are themselves based on examples
from [32, 51], and (3) benchmarks from Acacia synthesis tool’s benchmark suite [13].
Group 1. This group consists of 22 MITL formulae designed to highlight various strengths of
our translation. These include formulae featuring conjunctions of temporal operators, the
simultaneous use of past and future modalities (for which, to the best of our knowledge, no
existing tool supports), and other representative constructs. We rewrite the derived operators
G, F, R in terms of U, in the standard way. We use the following shorthand for common
patterns: ρ(n) = X(n) ∧G(p1 ∨p2), where X(n) = X[2,3]
(p1
=⇒X(n −1)) ∧(p2
=⇒
¬ X(n −1))
with base case of X(1) = p1, η(n) = G(Vn
i=1(pi =⇒(ai S[0,2] bi))) ∧G(Wn
i=1 pi),
Req-grant = G[2,∞)
(R =⇒F[4,5] G)
, and Spot-ex = G[0,10] F[1,2](a ⇐⇒X b).
Group 2. This group contains all 33 benchmarks from [20], accompanying the tool MightyL.
These are based on temporal patterns from [32, 51], and include families of parameterized
26
Efficient Verification of MTL with Past in Pointwise Semantics
formulae: F(k, I) = Vk
i=1 FI pi, G(k, I) = Vk
i=1 GI pi, U(k, I) = (· · · (p1 UI p2) · · · ) UI pk,
and R(k, I) = (· · · (p1 RI p2) · · · ) RI pk, µ(k, I) = Vk
i=1 F[3i−1,3i] ti ∧G ¬p and θ(k, I) =
¬((Vk
i=1 G F pi) =⇒G(q =⇒FI r)).
Group 3.
This group includes 17 benchmarks drawn from the Acacia synthesis tool’s
benchmark suite [13], which are also featured in the SyntComp competition suite [29], with
added timing intervals. These include formulae such as
Acacia1 =
F[0,2] G p
⇐⇒(G F q) ,
Acacia2 =
G
F[0,2]
p =⇒X[1,2] X[1,2] X[1,2] q
⇐⇒(G F r) ,
Acacia3 =
((p U[0,2] q) U[0,2] r) ∧((q U[0,2] r) U[0,2] p) ∧((r U[0,2] p) U[0,2] q)
⇐⇒G F s,
Acacia4 =
G(p =⇒F[3,4] q) ∧G(¬p =⇒F[3,4] ¬q)
⇐⇒G F r.
The complete list of all 72 benchmark formulae are given in Tables A and B of Appendix C.
Our results
We present the results of our experiments in Tables 1 to 3, which show the number of
nodes stored and the time taken by each tool to check satisfiability of the formulae over
finite and infinite words, respectively. We also present a log-scale plot of the time taken
by MightyL and Tempora in Figure 11, which shows the performance of our tool on the
various benchmarks. We discuss our experimental results in detail below. In all the tables, we
highlight in green the rows corresponding to formulae belonging to the detMITL+p fragment.
Satisfiability for the past fragment of detMITL+p. The first set of experiments, presented
in Table 1, evaluates the performance of our tool Tempora on detMITL+p formulae involving
past operators. We find that Tempora successfully handles all benchmarks in this category.
We note that MightyL could not be run on these, as it does not support past modalities
(listed as part of future work in [20]). The last but two formulae are specifically designed to
stress-test the handling of multiple nested past operators. Tempora processes each of them
efficiently, storing only a small number of nodes and completing each run in under 30 seconds.
The final formula in this set is more complex, combining several past operators with a
conjunction of future modalities. Here too, Tempora performs efficiently, requiring only 186
nodes and less than 2 ms to check satisfiability. These results demonstrate the effectiveness of
our deterministic translation, which avoids the exponential blowup typically associated with
non-deterministic approaches and enables scalable analysis even for complicated formulae. To
see this, consider, for instance, the formula F(pU[1,2]q∧pU[2,3]q∧pU[3,4]q∧pU[4,5]q), obtained
by replacing the past operator S with the future operator U in the 5th entry Tempora
solves this by generating 28,030 nodes in 784 ms. Another example is the nested formula
F(p U[1,2] (p U[1,2] (p U[1,2] q))) which results in 299,130 nodes and takes 9 seconds, compared
to the past version F(p S[1,2] (p S[1,2] (p S[1,2] q))) which generates only 1,303 nodes in 29.9
milliseconds.
Satisfiability for full MITL+p and comparisons. For the second set of experiments, we
evaluate the performance of Tempora on the satisfiability problem for MITL+p formulae
over both finite and infinite timed words. We compare our tool against MightyL and
the baseline implementation of the translation proposed in [2]. For finite timed words, we
compare against MightyL using UPPAAL’s reachability analysis; for infinite timed words,
we use MightyL with OPAAL’s liveness checking algorithm.
Figure 11(left) compares the performance of our tool and the MightyL pipeline on all
73 benchmarks for finite timed words, and Figure 11(right) presents the same comparison
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
27
Formula
Tempora
Stored nodes
Time in ms.
F[0,20](Y[2,3] p1 ∨Y[4,5] p2) ∨Y[6,7] p3)
287
1.5
G[0,20](Y[2,3] p1 ∨Y[4,5] p2) ∨Y[6,7] p3)
262
3.1
F(p S[1,2] (p S[1,2] (p S[1,2] q)))
1,303
29.9
F(p S[1,∞) (p S[1,∞) (p S[1,∞) q)))
66
0.5
F(p S[1,2] q ∧p S[2,3] q ∧p S[3,4] q ∧p S[4,5] q)
1,571
50.9
F(p S[1,∞) q ∧p S[2,∞) q ∧p S[3,∞) q ∧p S[4,∞) q)
68
0.8
F(p S[1,2] q ∧p S[2,3] q ∧p S[3,4] q ∧p S[4,5] q ∧p S[5,6] q)
27,737
1,762.1
F(p S[1,2] q ∧p S[2,3] q ∧p S[3,4] q ∧p S[4,5] q ∧p S[5,6] q ∧p S[6,7] q)
167,077
21,490.5
G(¬p1 ∨(a1 S[0,2] b1)) ∧(¬p2 ∨(a2 S[0,2] b2))∧
(¬p3 ∨(a3 S[0,2] b3)) ∧(¬p4 ∨(a4 S[0,2] b4)) ∧G(p1 ∨p2 ∨p3 ∨p4)
186
1.67
Table 1 Experimental results obtained by running our prototype implementation Tempora for
satisfiability of detMITL+p formulae over finite timed words. Timeout is set to 300 seconds.
for infinite words. The time reported is the time after the translation (to TA in case of
MightyL and GTA for Tempora), that is, the time for zone graph exploration to find a
reachable node (for finite timed words) or a live cycle (for infinite timed words). This is
because, the time for the translation itself is rather small in most cases (of the order of
3-4ms), except for the cases shown in Table 4 in Appendix C. For OPAAL, we report the
sum of time taken by OPAAL to generate C++ code, the g++ compiler to compile the
generated code and LTSmin to check liveness. We note that OPAAL is a multi-core tool
and uses 8 cores on our machine. On 10 of the benchmarks, OPAAL gave a segmentation
fault (which we mark as timeout) and UPPAAL reported an overflow error on the output
generated by MightyL for 2 benchmarks.
Table 2 presents selected results reporting the number of nodes stored and time taken by
each tool to check satisfiability over infinite words, while the full set of results appears in
Appendix C
As can be seen from Figure 11, in the finite timed words case, Tempora is always faster
than MightyL + UPPAAL and for benchmarks where both tools take more than 1s, we are
significantly faster, as the plot is in log-scale. Further, UPPAAL times out in several cases,
while we never do so on the benchmarks tested. Thus, we scale better on larger and more
complex formulae. For the infinite case, except for 2 formulae we are always better. Note that
the plot is only on the 63 benchmarks on which all three tools work, and therefore, we exclude
the formulae containing past operators. Our performance is significantly better in terms
of time-taken as can be seen from the figure. Further, out of the 72 benchmarks, OPAAL
timed out or seg-faulted on 18 benchmarks whereas we timeout on only 4 benchmarks (of
Figure 11 Log-scale plot MightyL vs Tempora comparison for time taken to check satisfiability
on finite (left) and infinite (right) timed words, after the translation.
Group 1 refers to our
benchmarks, while Group 2 are MightyL benchmarks [20] are Group 3 benchmarks derived from
Acacia [13]. Timeout set to 300s.
28
Efficient Verification of MTL with Past in Pointwise Semantics
Formula
Baseline
MightyL+ OPAAL
Tempora
Stored
Time in
Stored
Time in
Stored
Time in
nodes
ms.
nodes
ms.
nodes
ms.
F(5, [2, ∞))
-
TO
3069
373 + 820 + 85
200
2.38
F(5, [0, 2])
952
5.99
2693
214 + 702 + 82
68
0.49
U(5, [2, ∞))
567,192
5,404
9,383
382 + 960 + 114
4,075
32.70
U(5, [1, 2])
-
TO
-
TO
465,185
8,750
µ(2)
953,492
24,977
964
68,462 + 20,584+ 103
158
0.90
µ(3)
-
TO
-
TO
315
3.34
θ(3)
-
TO
42,804
836 + 1,057 + 225
330,044
2977
G(5, [0, 2])
4,793
34.6
5,913
257 + 760+ 101
224
1.45
G(5, [1, 2])
-
TO
-
TO
225
2.45
R(5, [2, ∞))
29,685
278.3
1,059
231 + 726 + 77
3,119
27.75
R(5, [1, 2])
-
TO
-
TO
1,093,187
23,474
p U[11,12] q
1,484
53
-
TO
220
1.37
Req-grant
-
TO
-
TO
444
4.00
Spot-ex
117,319
1,141
7,921
74,585 + 20,184 + 238
504
3.50
η(4)
-
TO
-
NA
249
1.49
ρ(4)
-
TO
-
TO
1933
15.93
Acacia1
-
TO
4,943
258 + 755 + 96
390
2.54
Acacia2
-
TO
742,427
426 + 949 + 1,683
231
1.94
Acacia3
-
TO
7,271,276
611 + 1,143 + 21,349
918
7.41
Acacia4
-
TO
-
TO
4,159
55.71
Table 2 Selected set of experimental results obtained by running our prototype implementation
Tempora, MightyL and, the baseline implementation of the translation proposed in [2] for
satisfiability over infinite traces. Timeout is set to 300 seconds. The complete table is available
in Table 8 in Appendix C.
which OPAAL also times out/segfaults on 3). Finally, Table 2 (and the extended tables
in Appendix C) show that Tempora is also significantly better than the Baseline (in fact
the MightyL pipeline also outperforms Baseline significantly on most benchmarks) on all
benchmarks.
In summary, both for finite words and infinite words over the pointwise semantics,
Tempora significantly outperforms MightyL and the baseline for checking satisfiability,
both in terms of time taken and size of stored nodes.
Model checking. Finally, going beyond satisfiability, we evaluate our model-checking
pipeline on a set of MITL+p formulas and two models: Fischer’s mutual exclusion algorithm
and the Dining Philosophers problem.
The model-checking pipeline is implemented in
Tempora and uses Tchecker for the zone graph exploration. To the best of our knowledge,
there is no existing tool among UPPAAL, MightyL, and OPAAL that can perform complete
model checking for pointwise semantics with a model and a MITL+p formula. As a result,
we are unable to provide a comparison for our model-checking pipeline. Nevertheless, as a
proof of concept, we evaluate our pipeline on two models (Fischer and Dining Philosophers)
against a set of formulas, with results in Table 3.
For the Fischer’s mutual exclusion algorithm, we consider a model with 3 to 7 processes and
check various properties. The first property checks whether the first process can eventually
enter the waiting state after making a request, while the second property checks whether
it can enter the critical section. We also check the classical safety property for mutual
exclusion protocols that ensures that no two processes can be in the critical section at the
same time. Our experimental results show that for Fischer with 6 processes, the property
G(req1 →F[0,5] wait1) can be checked to hold in 4.6 ms, while for Fischer with 7 processes, it
takes 5.3 ms. We also check the property that no two processes can be in the critical section
at the same time, which is reported to hold true for Fischer with 5 processes in 15.51 seconds.
For the Dining Philosophers problem, we consider models with 2 and 8 philosophers. We
check properties such as whether at least one philosopher is eating at some point in time, and
whether both philosophers are guaranteed to eat within a certain window of time. For the
model with 8 philosophers, we find that the property F[0,20](W8
i=1 eatingi) does not hold true
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
29
Sl.
Models
Formula
Tempora+ Tchecker
No.
Stored nodes
Time (ms)
1
Fischer (3)
G(¬req1 ∨F[0,20] wait1)
32,817
221.3
2
Fischer (4)
G(¬req1 ∨F[0,20] wait1)
449,108
5,723
3
Fischer (6)
G(¬req1 ∨F[0,5] wait1)
362
4.6
4
Fischer (7)
G(¬req1 ∨F[0,5] wait1)
391
5.3
5
Fischer (3)
G(¬req1 ∨F[0,20] cs)
33,918
209.69
6
Fischer (4)
G(¬req1 ∨F[0,20] cs)
352,774
3,749.6
7
Fischer (5)
G(¬((cs1 ∧(cs2 ∨cs3 ∨cs4 ∨cs5))∨
(cs2 ∧(cs1 ∨cs3 ∨cs4 ∨cs5)) ∨(cs3 ∧(cs1 ∨cs2 ∨cs4 ∨cs5))∨
(cs4 ∧(cs1 ∨cs2 ∨cs3 ∨cs5)) ∨(cs5 ∧(cs1 ∨cs2 ∨cs3 ∨cs4))))
527,829
15,514
8
Din-Phil(8, 10, 3, 1)
F[0,20](eating1 ∨eating2 ∨eating3 ∨eating4
∨eating5 ∨eating6 ∨eating7 ∨eating8)
371
5.8
9
Din-Phil(2, 10, 3, 1)
G[10,∞)(F[0,30](eating1) ∧F[0,30](eating2))
4,904
31.35
Table 3 Experimental results obtained by running our prototype implementation Tempora, and
the GTA liveness algorithm implementation in Tchecker for checking liveness. The numbers in the
brackets represent parameterization, for Fischer’s it is #-processes and for Dining Philosophers, it
represents the number of philosophers, timeout before releasing fork, time to eat and delay time to
release fork.
within 0.01 seconds, indicating that no philosopher is guaranteed to eat within 20 time units.
For the model with 2 philosophers, we check whether both philosophers are guaranteed to
eat within a certain time window, which does not hold true within 0.03 seconds. The results
indicate that our model-checking pipeline can efficiently handle the verification of MITL+p
properties on these models, demonstrating its potential for practical applications in verifying
timed systems.
Additional results, including ablation studies, are presented in Section C.4 of Appendix C.
Summary of Experimental Evaluation.
In summary, our experiments demonstrate the practicality and effectiveness of Tempora
in various settings. Tempora handles all standard MITL benchmarks used in prior works,
and crucially extends support to the richer MITL+p fragment - including past operators and
extending timed modalities to allow punctual constraints at the top level. This is beyond
the capabilities of state-of-the-art tools for pointwise MITL logics such as MightyL, which
do not support past operators or the punctual timed operators. Further, on benchmarks
where MightyL does work, Tempora outperforms it significantly both in the size of
zone graph computed and time taken to check satisfiability. Finally, Tempora can be
effectively used for model checking full MITL+p specifications, including both safety and
liveness properties on standard benchmarks such as Fischer’s mutual exclusion protocol and
the Dining Philosophers.
8
Conclusion
In this paper, we developed a new synchronization-based model checking technique that
allows translating formulas in MITL with past modalities (and punctuality at the outermost
level) to networks of small (generalized) timed automata with the general theme of making
it “more deterministic”. While we get deterministic automata for a large fragment, we show
improvements even in the general case, especially a surprising 2-state until automaton that
takes advantage of future clocks. All the technical developments have resulted in the tool
Tempora, which outperforms the current state-of-the-art tools. As future work, two natural
directions would be towards monitoring of MITL+p properties in the pointwise semantics,
and to try to extend our approach to the signal/continuous semantics.
30
Efficient Verification of MTL with Past in Pointwise Semantics
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early and simple principle. In Proceedings of the 2021 International Conference on Model
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Thomas Wilke. Specifying timed state sequences in powerful decidable logics and timed
automata. In FTRTFT, volume 863 of LNCS, pages 694–715. Springer, 1994.
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34
Efficient Verification of MTL with Past in Pointwise Semantics
A
Appendix for missing proofs
In this section, we provide the missing proofs omitted in Section 4.
▶Lemma 6. For every timed word w = (a0, τ0)(a1, τ1) · · · over Prop with p ∈Prop, there is
a unique run C0
δ0,t0
−−−→C1
δ1,t1
−−−→C2 · · · of Aisat
p
over w. Moreover, w |= p iff for all i ≥1,
configuration Ci satisfies Aisat
p
.st = 2.
Proof. In the forward direction, suppose w |= p. Then, we know that p ∈a0, i.e., the atomic
proposition p holds at the first position of the word. By construction, Aisat
p
(Figure 2), the
automaton starts in state 1 and checks whether p holds at the first position. If p ∈a0, then
Aisat
p
takes the transition to state 2, and from then on, continues to remain in state 2 on all
subsequent inputs. Hence, configuration Ci satisfies Aisat
p
.st = 2 for i ≥1.
In the other direction, suppose that for all i ≥1, the configuration Ci satisfies Aisat
p
.st = 2.
Since the only way for the automaton to reach state 2 from the initial state 1 is via a transition
that checks p ∈a0, it must be the case that p ∈a0. Therefore, w |= p.
◀
▶Lemma 7. Let Prop′ = Prop ⊎{b1} be an augmented set of atomic propositions. For
every word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0
δ0,t0
−−−→C1
δ1,t1
−−−→C2 · · ·
of (Ainit, Aisat
XI ) over w. Moreover, w |= XI b1 iff for all i ≥2, configuration Ci satisfies
Aisat
XI .st = 3.
Proof. Suppose that w |= XI b1. Then, this means that the timed word w satisfies (1)
b1 ∈a1, and (2) δ1 = τ1 −τ0 ∈I. Consider the run of the automaton (Ainit, Aisat
XI ) over w.
The automaton starts in state 1 at configuration C0. From state 1, the automaton moves to
state 2 after reading the first letter (a0, τ0), resetting clock xinit. Then, from state 2, since b1
holds at the next letter a1 and the clock value of xinit is δ1 = τ1 −τ0 ∈I, the automaton
moves to state 3. Since state 3 is a sink-state with a self-loop, Aisat
XI
remains in state 3 for
all i ≥2. Therefore, for all i ≥2, the configuration Ci satisfies Aisat
XI .st = 3.
Conversely, suppose that for all i ≥2, the configuration Ci satisfies Aisat
XI .st = 3. By
construction of Aisat
XI , state 3 can only be reached from state 2 if (a1, τ1) satisfies the following:
b1 ∈a1 and xinit ∈I. Further, since xinit was reset on reading (a0, τ0), this implies that
τ1 −τ0 ∈I. Therefore, w |= XI b1.
◀
▶Lemma 9. Let Prop′ = Prop ⊎{b1}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′,
we have a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (Alast, AY). Moreover,
for all i ≥0, (w, i) |= Y b1 iff Ci(AY.st) = C′
i(AY.st) = ℓ1,
for all i ≥1, τi −τi−1 ∈I iff C′
i(Alast.xlast) ∈I.
Let outYI = (AY.st = ℓ1 ∧Alast.xlast ∈I). Then, for all i ≥0, we may check if (w, i) |= YI b1
with the test outYI during transition ti from C′
i to Ci+1.
Proof. Since the network is deterministic and complete, there is a unique run of the network
(Alast, AY) on the given input word w.
Note that the network (Alast, AY) maintains two components:
Alast stores the time elapsed since the previous position using the clock xlast.
AY allows to check whether the formula Y b1 holds by storing the value of b1 at the
previous position.
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
35
By construction, the state of AY is updated as follows. If b1 was true at ai−1, then after
processing (ai−1, τi−1), AY enters state ℓ1 and remains there at both Ci and C′
i. Similarly, if
b1 was not true at ai−1, then after processing (ai−1, τi−1), AY enters state ℓ0 and remains
there at both Ci and C′
i.
Similarly, Alast resets xlast upon reading each letter of the input word. Hence, at C′
i, we
have Alast.xlast = δi = τi −τi−1. Therefore, for all i ≥1, τi −τi−1 ∈I iff C′
i(Alast.xlast) ∈I.
The formula YI b1 is true at i iff
b1 held at position i −1, which translates to AY being in state ℓ1 at configuration C′
i,
τi −τi−1 ∈I (i.e., Alast.xlast ∈I at C′
i).
Therefore, the test AY.st = ℓ1 ∧Alast.xlast ∈I precisely checks (w, i) |= YI b1 during the
transition ti from C′
i to Ci+1.
◀
▶Lemma 10. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′
there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (AS, Alast
S ). Moreover, for all
i ≥0:
1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,
2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Alast
S .x) = τi −τj where j ≤i, b2 ∈aj and b2 /∈ak
for all j < k ≤i.
3. Let I be an upper-bound interval and outSI = (AS.st• = ℓ1 ∧Alast
S .x• ∈I). Then, for all
i ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′
i
to Ci+1.
Proof.
1. From the semantics of the untimed S operator, (w, i) |= b1 S b2 if there exists 0 ≤j ≤i
such that b2 ∈aj, and b1 ∈ak for all j < k ≤i.
From Figure 6, it can be seen that AS moves to state ℓ1 upon seeing b2, and stays in ℓ1 as
long as b1 ∨b2 holds. AS returns to ℓ0 if it sees a position for which b1 ∨b2 does not hold.
Thus, if Ci+1(AS.st) = ℓ1, there exists some j ≤i where b2 ∈aj and if we take the largest
such j, at all positions in (j, i], b1 holds, which implies that (w, i) |= b1 S b2.
Conversely, suppose that (w, i) |= b1 S b2. This means, that there was some position j ≤i
such that (w, j) |= b2, and at all positions between j and i, b1 holds. Then, from the
construction, AS must be in state ℓ1 at Ci+1.
2. Alast
S .x is reset whenever b2 is observed, and is not updated until the next occurrence of
b2. Thus, Alast
S
stores the total time elapsed since the most recent occurrence of b2. We
have already seen from the previous item that if Ci+1(AS.st) = ℓ1, then there exists some
j ≤i where b2 ∈aj. Therefore, when Ci+1(AS.st) = ℓ1, the last observed b2 was at some
position j ≤i with no b2 between j and i, then Ci+1(Alast
S .x) = τi −τj.
3. Recall that b1 SI b2 holds at (w, i) if and only if the following two conditions are satisfied
the untimed condition for b1 S b2 is true, which from (i) we know is the case if the
automaton AS is in state ℓ1 at configuration Ci+1, and
the timed constraint is satisfied, i.e., τi −τj ∈I, where j is the most recent position
where b2 was true. This holds because I is an upper-bounded interval. From (2), we
know that when Ci+1(AS.st) = ℓ1, we have Ci+1(Alast
S .x) = τi −τj.
From the above, it is easy to see that the test outSI = (AS.st• = ℓ1 ∧Alast
S .x• ∈I) is
satisfied during transition ti from C′
i to Ci+1 exactly when (w, i) |= b1 SI b2.
◀
36
Efficient Verification of MTL with Past in Pointwise Semantics
▶Lemma 11. Let Prop′ = Prop⊎{b1, b2}. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′
there is a unique run C0
δ0
−→C′
0
t0
−→C1
δ1
−→C′
1
t1
−→C2 · · · of (AS, Afirst
S
). Moreover, for all
i ≥0:
1. (w, i) |= b1 S b2 iff Ci+1(AS.st) = ℓ1,
2. When Ci+1(AS.st) = ℓ1 we have Ci+1(Afirst
S
.y) = τi −τj where j is the earliest witness at
i of b1 S b2: j ≤i, b2 ∈aj and b1 ∈ak for all j < k ≤i, and for all 0 ≤j′ < j either
b2 /∈aj′ or b1 /∈ak′ for some j′ < k′ ≤i.
3. Let I be a lower-bound interval and outSI = (AS.st• = ℓ1 ∧Afirst
S
.y• ∈I). Then, for all
i ≥0, we may check if (w, i) |= b1 SI b2 with the test outSI during transition ti from C′
i
to Ci+1.
Proof.
1. This is already shown in the proof of Lemma 10 given above.
2. First, notice that if Ci+1(AS.st) = ℓ1, the automaton AS moved from ℓ0 to ℓ1 in the
past, i.e., with some ti′ with i′ ≤i. During this transition ti′, the automaton Afirst
S
takes
the loop which resets y. Hence, y was reset at least once in the past of i. Let j ≤i
be the position where y was last reset by Afirst
S
. We have b2 ∈aj and either b1 /∈aj or
Cj(AS.st) = C′
j(AS.st) = ℓ0.
Now we focus on the two parts of the statement that we need to show.
We already saw that b2 ∈aj. Next, we show that b1 ∈ak for all j < k ≤i. Assume
that b1 /∈ak for some j < k ≤i. If b2 ∈ak then Afirst
S
takes the loop which resets y
during transition tk, a contradiction with the maximality of j. Hence, b2 /∈ak and
transition tk takes AS to state ℓ0. So we must have some k < i′ ≤i such that ti′ takes
AS from ℓ0 to ℓ1. As explained above, Afirst
S
must take the loop which resets y during
transition ti′, again a contradiction with the maximality of j. Therefore, b1 ∈ak for
all j < k ≤i.
For the second part, suppose towards a contradiction that there was a position
0 ≤j′ < j such that b2 ∈aj′ and b1 ∈ak′ for all j′ < k′ ≤i. Since b2 ∈aj′ and
b1 ∈ak′ for all j′ < k′ ≤i, automaton AS must be in state ℓ1 from Cj′+1 to Ci+1.
Since j′ < j ≤i, automaton AS is in state ℓ1 at configurations Cj and C′
j. Further,
we know that y was reset by Afirst
S
during transition tj. Since C′
j(AS.st) = ℓ1, this is
only possible if b1 ̸∈aj, a contradiction. Therefore, for all 0 ≤j′ < j either b2 /∈aj′ or
b1 /∈ak′ for some j′ < k′ ≤i.
3. The proof proceeds as in the proof of Lemma 10, as we have shown the untimed and
timed aspects separately above. For the sake of completeness, we give the arguments
below: b1 SI b2 holds at (w, i) if and only if the following two conditions are satisfied
the untimed aspect follows from the fact that automaton AS is in state ℓ1 at configur-
ation Ci+1 (shown in the first part), and
the timed constraint is satisfied, i.e., τi −τj ∈I, where j is the earliest position where
b2 is seen such that this b2 can act as a witness for b1 S b2: This holds because I is
a lower-bounded interval. From (ii), we know that when Ci+1(AS.st) = ℓ1, we have
Ci+1(Afirst
S
.y) = τi −τj.
From the above, it is easy to see that the test outSI = (AS.st• = ℓ1 ∧Afirst
S
.y• ∈I) is
satisfied during transition ti from C′
i to Ci+1 exactly when (w, i) |= b1 SI b2.
◀
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
37
B
Appendix for liveness
For the purpose of the algorithm, we consider the finite graph G defined as the zone graph
of A quotiented by the mutual simulation relation. The finiteness and soundness of this
abstraction for liveness was shown in [2].
Algorithm 2 SCC decomposition based algorithm for GTA liveness with generalized Büchi
acceptance condition.
1: Global variables
2: count: Integer initialized to 0
3: Roots: Stack storing tuples (η, labels, releasedclocks) where η is a potential root
node, labels is the set of labels seen in SCC rooted at η, and releasedclocks is the
set of clocks released in SCC rooted at η
4: CurrentReleasedClocks: Stack storing the set of future clocks that have been
released in the current SCC
5: Active: Stack storing the nodes which are active in the SCC exploration of the current
root node
6: Todo: Stack storing pairs (η, succ) where η is a node and succ is a list of successor
node and transition.
7: function GTASCC(G, L)
▷Outputs if there is a live component in G which visits all
labels from L
8:
for Each initial node η in G do Dfs(η, L) end for
9: end function
10: function Dfs(η, L)
▷Perform DFS starting from η
11:
Push(η, ∅)
12:
while Todo is not empty do
13:
(ν, succ) ←Todo.top()
14:
if succ is empty then
15:
if (ν, _, _) = Roots.top() then
▷SCC rooted at u is explored
16:
Close(s)
17:
end if
18:
Todo.pop()
19:
else
20:
(t, χ) ←succ.FirstAndRemove()
▷Explore along transition ν
t−→χ
21:
if χ.dfsnum = 0 then
▷New node
22:
Push(χ, released(t))
▷released(t) is the set of future clocks released in t
23:
else if χ.current then
▷Cycle detected
24:
CurrentReleasedClocks.push(released(t))
25:
MergeSCC(χ, L)
26:
end if
27:
end if
28:
end while
29: end function
38
Efficient Verification of MTL with Past in Pointwise Semantics
30: function Push(η, Xrel) ▷η is a new node, Xrel is the set of clocks released on transition
to η
31:
count ←count + 1; η.dfsnum ←count; η.current ←true
32:
Active.push(η)
33:
Todo.push((η, next(η)))
▷next(η) returns the successors of η
34:
CurrentReleasedClocks.push(Xrel)
35:
Roots.push((η, labels(η), ∅))
36: end function
37: function Close(η)
▷Closes the SCC of η
38:
Roots.Pop()
39:
CurrentReleasedClocks.Pop()
40:
µ ←null
41:
repeat
42:
µ ←Active.top()
43:
µ.current = false
44:
Active.pop()
45:
until µ ̸= η
46: end function
47: function MergeSCC(v, L)
48:
A ←∅, R ←∅
49:
(µ, _, _) ←Roots.top()
50:
repeat
51:
c ←CurrentReleasedClocks.Top()
52:
CurrentReleasedClocks.Pop()
53:
R ←R ∪c
54:
(µ, l, c) ←Roots.Top()
55:
Roots.Pop()
56:
A ←A ∪l, R ←R ∪c
57:
until µ.dfsnum > η.dfsnum
58:
if L ⊆A and XF = R ∪InfClocks(µ) = XF then
▷InfClocks(µ) is the set of
future clocks which can take value −∞in zone of u
59:
output Accepting SCC detected
60:
end if
61:
Roots.Push((µ, A, R))
62: end function
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
39
C
Appendix for Implementation and Experimental evaluation
C.1
Implementation details.
Our tool, Tempora, takes as input a MITL+p formula ψ and generates a synchronous network
of generalized timed automata with shared variables based on the translation described in
Sections 4 and 5. The execution of this network is simulated using an array of shared variables.
Each generalized timed automaton writes its output to a dedicated shared variable, which
can then be accessed by its parent automaton. Synchronization is enforced by sequentially
executing each automaton in bottom-to-top order. A shared control variable tracks the
active automaton during execution, and a transition in an automaton is executed when its
ID matches the value of the control variable. Upon taking the transition, the control variable
is updated to the ID of the next automaton. A controller is used to reset the execution
after every automaton takes a transition and alternates between time elapse and automaton
execution. A master automaton evaluates the top-level boolean subformula, ensuring the
formula is satisfied at the start and, for finite timed words, that all automata have performed
a transition in the last round.
To avoid constructing automata for boolean operators, we use symbolic labels to represent
temporal operators and atomic propositions, generating boolean expressions with these labels.
For model checking, we construct the network of automata for ¬ψ, where ψ is the input
specification. The input model and the network of automata generated by our translation
are executed alternately, with transitions of the model updating atomic propositions in the
shared variable array, which the network of timed automata reads.
C.2
Comparison of translation times
In Table 4, we discuss the time for the translation for a few instances. We remark that the
time for the translation is rather small in most cases.
Sl.
Formula
Time taken by
No.
MightyL (ms)
Tempora (ms)
1
(((((p0 U[1,2] q1) U[1,2] q2) U[1,2] q3) U[1,2] q4) U[1,2] q5)
50
5
2
G[0,10] F[1,2]((a ∧(X b)) ∨(¬a∧!(X b)))
15
3
3
G[2,∞)((¬request ∨F[4,5] grant))
42
4
4
F[1,2] p1 ∧F[1,2] p2 ∧F[1,2] p3 ∧F[1,2] p4 ∧F[1,2] p5
49
3
Table 4 Comparison of time taken by Tempora, and MightyL for translation of MITL formulae
to generalized timed automata (timed automata respectively).
C.3
Engineering optimizations.
We have implemented several optimizations to improve performance, of which we discuss one
such key optimization here. First, in addition to the locations of each automaton, the global
state in our translation also stores the output of each automaton in the array of shared
variables. Each shared variable doubles the global state space of the network. However, at
any point in the sequential execution, the subsequent behaviour of the network depends only
on the outputs of the previous automaton which are the direct dependencies of the remaining
automata in the round. Therefore, once the last parent of an automaton reads a shared
variable, its value can be discarded and the variable reused to store the output of a different
automaton. We use static liveness analysis to determine when shared variables become dead
40
Efficient Verification of MTL with Past in Pointwise Semantics
and reuse them, reducing the number of auxiliary variables. This optimization minimizes
the state size and the overall state space of the zone graph.
C.4
Ablation Study
To better understand the contribution of individual improvements in Tempora, we perform
ablation studies by selectively disabling them and measuring their impact on performance.
For the ablation study, we consider (a) one of our theoretical contributions, namely sharing
of predictions, (b) specialized automata for the F operator and (c) finally, an engineering
optimization, namely memory reuse discussed above.
For each feature, we compare the performance of Tempora with the feature, Tempora
with the feature turned off, and the state-of-the-art tool MightyL. Our results demonstrate
that each of these improvements plays a critical role in enabling scalability, often yielding
exponential improvements.
Sharing Predictions. In the absence of prediction sharing, each until subformula in the
network of automata independently guesses the time of satisfaction. As a result, multiple non-
deterministic predictions must be resolved during reachability analysis, significantly increasing
the complexity of the state space. With prediction sharing enabled, a single dedicated
automaton generates the prediction, and the remaining components deterministically consume
this information.
This reduces the overall non-determinism in the system and leads to a much smaller
product automaton. Empirical results confirm this effect: benchmark families involving
multiple until operators show exponential gains in verification time when prediction sharing
is enabled.
We depict the results in Figure 12 (left) for ϕ(n) = Vn
i=1 p U[0,i] q.
Figure 12 Comparison of performance of Tempora with and without features. The left figure
shows the impact of sharing predictions. The middle figure shows the impact of specialized automata
for F operator. The right figure shows the impact of memory reuse. The x-axis represents the
parameter in the formula, while the y-axis shows the time taken in milliseconds.
Specialized Automaton for F While the eventually operator FI p can be encoded as
true UI p, this transformation can be an overkill for untimed or simple timing intervals like
[0, ∞). Tempora implements specialized automata for such cases that are smaller and
structurally simpler than general-purpose until-based encodings.
For instance, the automaton for F[0,a] p simply waits up to a time units for p to become
true, without introducing intermediate obligations or control states.
These specialized
constructions avoid unnecessary non-determinism and reduce clock usage. Our benchmarks
show that formulas involving eventualities with small intervals benefit substantially from
this, often reducing the number of locations and transitions in the resulting automaton
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
41
by an order of magnitude. The results are shown in Figure 12 (middle) for the formula
ϕ(n) = VN
i=1 F[0,i] p.
Reusing Memory The memory optimization addresses inefficiencies in how truth values of
atomic propositions are stored during automaton execution. Consider the formula ϕ(n) =
Vn
i=1 X pi, where each X pi depends only on the value of pi at the previous timestamp. In
a naive implementation, each pi is assigned a unique index in the state array, even though
their lifetimes do not overlap.
Tempora incorporates a live-range analysis to reuse memory indices once a variable
becomes dead — i.e., no longer accessed in the remaining evaluation of the round. In the
above example, all pi can safely share a single index. This drastically reduces the size of the
memory array and thus the number of reachable states. Our experiments demonstrate that
this reuse yields exponential improvements in model-checking time for formulas with long
chains of independent next operators, as depicted in Figure 12 (right).
C.5
Benchmarks and Extended Experimental Results
We next briefly describe the additional benchmarks we use to evaluate our performance.
Our benchmarks. We constructed several MITL+p formulae to show the benefits of our
improvements. To show the benefit of sharing clocks, we use a disjunction of multiple X
operators: G[0,2](Wn
i=1 X[2i,2i+1] pi), where each Next automaton uses a single shared clock.
A conjunction of multiple U operators with the same left and right children: G(Vn
i=1(pi →
a U[0,i] b)) ∧G(Wn
i=1 pi) shows the benefit of using sharer automata, where each Until
automaton uses a common sharer automaton, as well as using a smaller 2 state automaton
for Until. The power of using deterministic automata for top level operators can be seen
for a formula with conjunction of multiple U operators having general intervals at the top
level: Vn
i=1(p U[2i,2i+1] q). We show the benefit of reusing shared variables for formulae where
atomic propositions have short live ranges: Wn
i=1 G[2,∞)(pi). In all these formulae, we try
to ensure that they cannot be easily rewritten to a simpler formula, and are not trivially
satisfiable/unsatisfiable.
Acacia Benchmarks. Each LTL formula in the original Acacia benchmark is of the form
ψ ⇐⇒G F acc, where ψ is an LTL specification. We modify these examples by adding
intervals to temporal operators of ψ. For several of these specifications, we add intervals to
the operators specifying occurrences of atomic propositions, which adds stricter constraints
for the atomic propositions to occur within an interval. For some of the benchmarks, we also
add general intervals to nested U operators to add complexity to the formula.
The complete experimental results are presented in Table A (for finite words) and Table
B for infinite words.
42
Efficient Verification of MTL with Past in Pointwise Semantics
Sl. No
Formula
Baseline
MightyL+UPPAAL
TEMPORA
Nodes Stored
Time (ms)
Nodes Stored
Time (ms)
Nodes Stored
Time (ms)
1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3)
32341
179.083
11239
66.00
2744
25.4233
2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5)
1927857
24478.2
243106
1,348.00
24076
209.238
3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3)
70615
480.662
3440
17.00
329
1.57934
4
G[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 || p2 ||
p3)
35079
230.365
3440
20.00
345
2.02975
5
(X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) ||
atomic) && (!p2) || !(atomic))))))&& G(p1 || p2)
25210
165.833
885
5.00
437
2.63777
6
(X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))
&& (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X
[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !
(atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2)
|| !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2,
3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic)
&& (!p2) || !(atomic))))))))))))&& G(p1 || p2)
-
TO
3517869
43,896.00
3509
21.3181
7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q
5441
225,448.00
1990
12.3429
8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5
2130
14.2971
3712
25.00
160
0.791932
9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5
80500
619.015
3872
23.00
20
0.133869
10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5)
118296
984.777
16182
129.00
10777
84.286
11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) ))
48083
416.384
479764
215,172.00
3367
33.0925
12 G[2, Inf) ( (!request || F[4, 5] grant))
643
4.67705
60
307.00
71
0.511877
13 p U[11, 12] q
144
2.10697
57
1,408.00
38
0.305787
14
G((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 || p3 ||
p4)
-
N/A
-
N/A
186
1.67275
15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q)))
-
N/A
-
N/A
1303
29.9574
16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q)
-
N/A
-
N/A
1571
50.9912
17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q)
-
N/A
-
N/A
27737
1762.13
18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q)
-
N/A
-
N/A
167077
21490.5
19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)
-
N/A
-
N/A
262
3.18676
20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)
-
N/A
-
N/A
286
1.53985
21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf) q && p S[4, Inf) q)
-
N/A
-
N/A
68
0.798897
22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q)))
-
N/A
-
N/A
66
0.49014
MightyL benchmarks
23 F p1 && F p2 && F p3 && F p4 && F p5
9292
44.3188
3263
15.00
157
0.530741
24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5
1241
9.18455
3263
16.00
160
0.900405
25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5
74516
556.001
6592
44.00
177
0.806017
26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3
47766
524.722
455
1,362.00
59
0.34313
27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5
-
TO
6623
31,749.00
191
1.73696
28 (((((p0 U q1) U q2) U q3) U q4) U q5)
25972
101.764
8133
50.00
4063
21.7138
29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5)
104149
549.412
8133
40.00
6303
33.4324
30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5)
48557
319.548
18204
136.00
12496
82.9332
31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3)
16160
185.278
1140
3,278.00
3238
35.6263
32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4)
162606
3074.69
5006
19,078.00
27758
434.539
33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5)
1629834
62186.7
22228
107,447.00
235536
6455.47
34 G(!p) && F[0, 3] t1
110
0.790953
55
2.00
21
0.118925
35 G(!p) && F[0, 3] t1 && F[3, 6] t2
2396
12.5292
627
254.00
174
1.31732
36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3
48222
364.599
2450
6,690.00
419
2.98024
37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4
-
TO
48383
274,012.00
2079
11.2824
38 !(!(G F p1) || (G(!q || (F[100, 1000] r))))
6916
38.5607
463
3.00
271
1.66032
39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r))))
65149
418.318
3423
32.00
1005
5.93152
40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r))))
561208
5684.74
26047
280.00
3397
18.338
41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r))))
4651945
126995
202623
2,359.00
10953
59.6345
42 F[0, 30] (!p || G[0, 20] p)
31
0.32008
31
2.00
17
0.118925
43 G[0, 30](!p) || F[0, 20] p
37
0.303894
32
2.00
12
0.152296
44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5)
179753
1216.62
8128
44.00
6318
37.332
45 (((((p0 R q1) R q2) R q3) R q4) R q5)
47841
250.798
8128
33.00
4079
19.3469
46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3)
7571
69.5354
560
15.00
546
5.74616
47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5)
375155
6503.65
9792
393.00
12042
230.663
48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4)
54947
678.074
2368
74.00
2628
38.1337
49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5)
78161
587.855
9792
55.00
5892
39.5866
50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5
2735492
78412.1
3232
132.00
20
0.252719
51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5
70930
578.802
3232
18.00
20
0.136226
52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5
1815
14.362
3134
19.00
160
0.809136
53 G p1 && G p2 && G p3 && G p4 && G p5
12897
75.8965
3134
14.00
158
1.06356
54 !((F[0, 30] p) || !(F[0, 20] p))
60
0.568014
69
2.00
49
0.272393
55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p))
1272
10.5949
495
221.00
224
0.991115
Acacia benchmarks
56 ((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F(a)))
3047327
70749.6
54777
82,277.00
530
3.40564
57
(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F a))) || (!
(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) && !((G (F a))))
-
TO
3193831
41,472.00
3455
19.531
58
((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q && (X
F[2, Inf) r))))) && !(G(F(a))))
-
TO
214577
96,763.00
1059
7.08385
59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a)))
466847
5451.49
13712
21,448.00
225
1.61587
60
((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r U[0, 3]
s)))) && !(G(F a)))
-
TO
-
Error
564
5.20393
61
((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G(Fa))) ||
(!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && !(G(Fa)))
-
TO
-
Error
3276
19.6802
62
((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2,
3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F
[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && !(G(F a)))
-
TO
-
TO
366144
3712.65
63
(((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && (G(F
(a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u)))
&& !(G(F(a))))
-
TO
-
TO
362442
8380.15
64
(((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, 2]q)) ||
(G(F[1, 2]r))) && !(G(F(a))))
-
TO
-
TO
7362
86.5996
65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a)))
51613
392.448
6641
72.00
61
0.507483
66
(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1,
2] (!r) )) && ! (G( F (a))))
-
TO
-
TO
3153
40.0667
67
(( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((!(G(2, Inf)
p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a))))
885283
14219.3
99857
945.00
87
0.747122
68
(((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (!((G[0, 2]
r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a))))
-
TO
-
TO
5053
39.7407
69
(((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) &&
!(G(F(a))))
6802985
269414
51641
19,852.00
444
2.52895
70
(((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2](X[1, 2]
(X[1, 2] q))))))) && !(G(F a)))
-
TO
430184
3,928.00
1061
4.97043
71
(((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) && (G(p
|| (F[3, 4] (!q))))) && !(G(F a)))
-
TO
-
TO
1917
23.3503
72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a)))
13546
82.0316
3511
20.00
96
0.430384
Table A: Experimental results obtained by running TEMPORA, MightyL + UPPAAL and, the baseline implementation of the translation for satisfiability for finite traces. (Timeout is set to 300s)
S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan
43
Sl. No
Formula
Baseline
MightyL + OPAAL
TEMPORA
Nodes Stored
Time (ms)
Nodes Stored
(Reporting state
space given by
opaal) -> Non
deterministic, since
multi threaded.
Numbers change
across different
experiments
Time (ms). Note: opaal is multicore, and hence
uses 8 cores of local machine
Total time (ms)
Nodes Stored
Time (ms)
opaal
g++
ltsmin -> the
actual
exploration
1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3)
2292
14.1087
1161
172.79
661.47
76.03
910.28
212
1.08509
2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5)
29455
193.183
1861
260.97
710.76
80.51
1,052.25
552
3.17967
3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3)
1310856
16076.9
1162
227.10
700.43
76.16
1,003.69
458
2.42018
4
G[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1
|| p2 || p3)
1587059
21828.1
1450
263.38
718.80
77.70
1,059.88
1044
8.24895
5
(X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!
p1) || atomic) && (!p2) || !(atomic))))))&& G(p1 || p2)
44857
350.62
1893
203.98
663.58
82.98
950.54
168
0.987026
6
(X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !
(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !
((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!
p1) || atomic) && (!p2) || !(atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) ||
(X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) &&
(!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !
(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))))))))&& G(p1 ||
p2)
-
TO
-
-
-
Segfault
0.00
1933
15.9299
7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q
-
TO
-
-
-
TO
0.00
456
4.17251
8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5
2346
16.1747
969
301.04
736.75
75.30
1,113.09
224
1.30602
9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5
-
TO
1143
272.19
721.28
78.98
1,072.45
178
1.58184
10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5)
-
TO
7003
404.46
851.76
106.54
1,362.75
1087
9.45395
11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) ))
117319
1140.97
7921
74,584.66
20,184.02
238.35
95,007.03
504
3.50233
12 G[2, Inf) ( (!request || F[4, 5] grant))
-
TO
-
-
-
TO
0.00
444
4.00833
13 p U[11, 12] q
1360
21.8606
-
-
-
TO
0.00
220
1.3685
14
G((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2
|| p3 || p4)
-
N/A
-
-
-
N/A
798
2.19275
15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q)))
-
N/A
-
-
-
N/A
14440
1721.87
16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q)
-
N/A
-
-
-
N/A
1410
51.2159
17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q)
-
N/A
-
-
-
N/A
2310
132.646
18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q)
-
N/A
-
-
-
N/A
3927
418.381
19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)
-
N/A
-
-
-
N/A
91
0.509345
20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3)
-
N/A
-
-
-
N/A
267
1.44564
21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf9 q && p S[4, Inf) q)
-
N/A
-
-
-
N/A
114
0.645279
22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q)))
-
N/A
-
-
-
N/A
108
0.73709
MightyL benchmarks
23 F p1 && F p2 && F p3 && F p4 && F p5
822
5.11449
2165
190.21
698.33
79.48
968.01
64
0.407546
24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5
952
5.99807
2693
214.01
701.74
82.35
998.09
68
0.485472
25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5
-
TO
3069
372.57
820.01
85.33
1,277.91
200
2.38852
26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3
-
TO
-
-
-
Segfault
0.00
102
0.528738
27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5
-
TO
-
-
-
TO
0.00
158
1.05082
28 (((((p0 U q1) U q2) U q3) U q4) U q5)
1828
10.7493
1560
196.70
704.35
78.12
979.17
18689
91.4055
29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5)
47166
298.922
1477
219.19
699.27
76.57
995.03
2668
19.0006
30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5)
567192
5404.53
9383
382.47
960.14
113.60
1,456.21
4075
32.6984
31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3)
-
TO
-
-
-
Segfault
0.00
3254
30.8566
32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4)
-
TO
-
-
-
TO
0.00
36124
451.716
33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5)
-
TO
-
-
-
TO
0.00
465185
8750.47
34 G(!p) && F[0, 3] t1
310
1.95935
184
117.86
612.55
72.49
802.89
32
0.235228
35 G(!p) && F[0, 3] t1 && F[3, 6] t2
953492
24977.1
964
68,462.12
20,584.03
102.84
89,149.00
158
0.901839
36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3
-
TO
-
-
-
Segfault
0.00
315
3.3407
37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4
-
TO
-
-
-
TO
0.00
522
4.21687
38 !(!(G F p1) || (G(!q || (F[100, 1000] r))))
-
TO
1935
692.31
954.65
79.29
1,726.25
2823
18.9429
39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r))))
-
TO
4212
749.15
1,006.85
89.55
1,845.55
28570
196.333
40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r))))
-
TO
42804
836.05
1,056.50
225.40
2,117.95
330044
2976.99
41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r))))
-
TO
149029
906.87
1,133.70
473.50
2,514.08
3819965
88366.7
42 F[0, 30] (!p || G[0, 20] p)
9432
52.7811
43
131.32
615.22
69.64
816.18
704
5.34067
43 G[0, 30](!p) || F[0, 20] p
61
0.756158
37
126.61
617.54
75.80
819.95
27
0.295317
44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5)
145353
1141.14
2267
260.88
746.28
86.45
1,093.60
113881
905.989
45 (((((p0 R q1) R q2) R q3) R q4) R q5)
22354
140.505
1521
185.82
690.21
79.60
955.64
11957
61.4713
46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3)
1086882
15332.4
453
1,588.79
1,496.63
78.06
3,163.48
5985
55.9789
47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5)
-
TO
-
-
-
Segfault
0.00
1093187
23474.1
48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4)
-
TO
-
-
-
Segfault
0.00
79258
1018.67
49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5)
29685
278.315
1059
230.60
725.70
77.18
1,033.48
3119
27.748
50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5
-
TO
-
-
-
Segfault
0.00
225
2.45304
51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5
-
TO
1549
226.81
724.91
82.29
1,034.01
178
1.07516
52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5
4793
34.5633
5913
257.33
760.33
100.74
1,118.41
224
1.45227
53 G p1 && G p2 && G p3 && G p4 && G p5
195653
1451.52
8413
186.71
694.95
111.65
993.31
64
0.61791
54 !((F[0, 30] p) || !(F[0, 20] p))
7069
37.444
100
126.35
622.49
72.87
821.71
629
10.9368
55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p))
3072735
56119.3
624
62,506.68
20,677.93
112.71
83,297.33
1669
19.924
Acacia benchmarks
56
((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F
(a)))
-
TO
76875
150,058.47
42,824.04
2,635.25
195,517.76
4561
37.7644
57
(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F
a))) || (!(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q))
&& !((G (F a))))
-
TO
7271276
611.46
1,143.21
21,348.99
23,103.66
918
7.40638
58
((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q
&& (X F[2, Inf) r))))) && !(G(F(a))))
-
TO
523958
67,992.90
22,837.11
10,320.40
101,150.41
4614
32.6728
59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a)))
-
TO
22945
116,791.31
39,078.93
840.78
156,711.03
1617
12.6589
60
((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r
U[0, 3] s)))) && !(G(F a)))
-
TO
-
-
-
Segfault
0.00
820
6.57777
61
((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G
(Fa))) || (!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]
s)))))))))) && !(G(Fa)))
-
TO
6733
183,618.00
14,086.56
323.78
198,028.34
4658
32.0218
62
((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v
&& (X[2, 3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2,
Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) &&
!(G(F a)))
-
TO
-
-
-
Segfault
0.00
54366
545.194
63
(((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u)))
&& (G(F(a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) &&
(G(F[2, 4] u))) && !(G(F(a))))
-
TO
-
-
-
TO
0.00
-
TO
64
(((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1,
2]q)) || (G(F[1, 2]r))) && !(G(F(a))))
-
TO
-
-
-
TO
0.00
-
TO
65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a)))
-
TO
11391
515.86
1,265.08
159.79
1,940.73
246
1.536
66
(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p)
) U[1, 2] (!r) )) && ! (G( F (a))))
-
TO
-
-
-
Segfault
0.00
-
TO
67
(( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((!
(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a))))
-
TO
250875
516.21
950.14
750.60
2,216.94
3484
25.3826
68
(((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (!
((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a))))
-
TO
-
-
-
Segfault
0.00
2598
25.7481
69
(((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) &&
(G r)) && !(G(F(a))))
-
TO
35945
63,521.58
21,527.32
788.87
85,837.77
785
5.94832
70
(((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2]
(X[1, 2](X[1, 2] q))))))) && !(G(F a)))
-
TO
742427
426.25
948.75
1,682.79
3,057.79
231
1.94063
71
(((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q)))
&& (G(p || (F[3, 4] (!q))))) && !(G(F a)))
-
TO
-
-
-
TO
0.00
4159
55.7087
72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a)))
1802521
22455.7
4943
257.75
755.07
95.59
1,108.41
390
2.54234
Table B: Experimental results obtained by running TEMPORA , MightyL + OPPAAL and, the baseline implementation of the translation for satisfiability for infinite traces. (Timeout is set to 300s)
|
Efficient Verification of Metric Temporal Properties with Past in Pointwise Semantics S Akshay # # # Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMF, 91190, Gif-sur-Yvette, France CNRS, ReLaX, IRL 2000, Siruseri, India R Govind # Uppsala University, Sweden B Srivathsan # Chennai Mathematical Institute, India CNRS, ReLaX, IRL 2000, Siruseri, India Abstract Model checking for real-timed systems is a rich and diverse topic. Among the different logics considered, Metric Interval Temporal Logic (MITL) is a powerful and commonly used logic, which can succinctly encode many interesting timed properties especially when past and future modalities are used together. In this work, we develop a new approach for MITL model checking in the pointwise semantics, where our focus is on integrating past and maximizing determinism in the translated automata. Towards this goal, we define synchronous networks of timed automata with shared variables and show that the past fragment of MITL can be translated in linear time to synchronous networks of deterministic timed automata. Moreover determinism can be preserved even when the logic is extended with future modalities at the top-level of the formula. We further extend this approach to the full MITL with past, translating it into networks of generalized timed automata (GTA) with future clocks (which extend timed automata and event clock automata). We present an SCC-based liveness algorithm to analyse GTA. We implement our translation in a prototype tool which handles both finite and infinite timed words and supports past modalities. Our experimental evaluation demonstrates that our approach significantly outperforms the state-of-the-art in MITL satisfiability checking in pointwise semantics on a benchmark suite of 72 formulas. Finally, we implement an end-to-end model checking algorithm for pointwise semantics and demonstrate its effectiveness on two well-known benchmarks. 2012 ACM Subject Classification Theory of computation →Timed and hybrid models; Theory of computation →Quantitative automata; Theory of computation →Logic and verification Keywords and phrases Real-time systems, Timed automata, Liveness, Event-clock automata, Clocks, Timers, Verification, Zones, Simulations 1 Introduction Linear Temporal Logic (LTL) is a standard specification formalism. A central ingredient in the model-checking approach is a translation from LTL to automata. The study of algorithmic techniques to optimize LTL-to-automata translations spans almost four decades - see [57] for a recent survey of this topic. Many of the theoretical techniques have also been incorporated in practical model-checking tools [36, 22, 28, 56, 37]. For model-checking systems with timing constraints, the de-facto automaton model would be timed automata (TA) [4, 3]. State-of-the-art timed automata tools [41, 34, 56, 37] implement analysis methods 16 Oct 2025 2 Efficient Verification of MTL with Past in Pointwise Semantics for models represented as (networks of) timed automata (see Section 7 of [25] for the practical applications of these tools). Properties checked on timed models are typically reachability, liveness, fragments of timed CTL [41] or (untimed) LTL. On the other hand, there is a very large body of foundational work on timed logics [53, 35, 54]. The two main timed logics are Metric Temporal Logic [40] (MTL) and Timed Propositional Temporal Logic [8] (TPTL). In order to use these logics for real-time model-checking, we need an efficient translation from timed logics to timed automata. The foundations for MTL-to-TA (resp. TPTL-to-TA) have been laid out in [5] (resp. [8]). Broadly, there are two ways to interpret MTL formulae: over continuous (dense/superdense) timed signals [5, 45] or pointwise timed words [6, 58, 20]. In continuous (state-based) semantics, the formula is interpreted over signals, which are functions that associate the valuation of the propositions to all timestamps. Over the pointwise semantics, the formula is interpreted over sequences of events (timed words), each consisting of a timestamp and a set of propositions. Thus, in the continuous semantics, the system is under observation at all times, while in the pointwise semantics, it is observed only when it executes a discrete transition. However, as noted by Ferrére et al. [30], in the continuous semantics, arbitrary MTL formulae can be simplified to the [0, ∞] fragment, while as pointed out in [2], this simplification is not possible in the pointwise semantics. In this work, we only consider pointwise semantics as our focus is on models represented in tools like UPPAAL, PAT, TChecker whose behaviours are given as timed words, as in the classical timed automata model of Alur and Dill [4]. In the pointwise semantics, it is known that MTL model checking is undecidable over infinite words, and over finite words it is decidable but the complexity is Ackermanian [7, 50, 48, 49]. Thus, the literature often considers the more efficient MITL fragment, where Until and Since operators must use non-punctual intervals (either [0, 0] or non-singleton intervals). The classical approach for model checking for MITL involves translating MITL formula into timed automata (TA) and then using state-of-the-art TA tools such as UPPAAL [41, 12] or TChecker [34]. The best known tool that accomplishes this translation in the pointwise semantics, called MightyL [20], goes via alternating 1-clock automata and compiles to timed automata. Recently, an extension of timed automata called generalized timed automata (GTA) was introduced in [1]. GTA are equipped with richer features than TA, and enable a different translation from MITL to GTA, which can then be analyzed using a GTA analysis tool from [1]. The theoretical foundations for such an MITL-to-GTA translation which goes via transducers was presented in [2], where it was argued that the MITL-toGTA translation has clear theoretical benefits over the MITL to 1-clock Alternating Timed Automata translation [18, 19] on which MightyL is based on. However, this latter approach was not implemented. Moreover, neither of these approaches considered Past modalities, and were restricted only to Future. On the other hand, from the classical work of Maler et al [44], in the signal semantics, it is known that having only Past modalities leads to efficient algorithms and in particular it gives a translation to deterministic (timed) automata albeit over signals. Our first question is whether the same holds under the pointwise semantics, and whether determinism can be extended beyond just the past fragment. Next, if we cannot guarantee determinism, can we reduce non-deterministic branching? Indeed, even if it does not change the expressiveness, several works have highlighted the usefulness and succinctness of combining Past and Future operators, both in the untimed setting [43, 46, 10] and in the timed setting [39, 30]. In this work, we address these questions both for past and future in the pointwise semantics and to build the foundations for a new efficient model checking algorithm for S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 3 MITL with past that exploits determinism as much as possible. We show that for the past fragment of MITL, we can obtain a linear-time translation to networks of deterministic timed automata. Further, even when we have to handle future operators, where non-determinism cannot be avoided, we propose techniques to reduce non-deterministic branching in the resulting automata, using various ideas including sharing components, automata, clocks and predictions. We start by defining a formal semantics for a model of synchronous network of timed automata with shared variables, where each component of the network owns some clocks which only it can reset but can be tested by all components. Moreover it allows for instantaneous programs on transitions which have sequences of tests and updates, like in [2]. Our main contributions are the following: 1. We identify a fragment of metric temporal logic with past and future for which we provide a linear time translation to networks of deterministic timed automata. In this fragment, that we call detMITL+p, the outermost temporal modalities are future (next, until) and can even have punctual intervals, while inner temporal modalities should be past: Since with non-punctual interval or Yesterday. This fragment already subsumes many safety properties, extending e.g., the G(pLTL) fragment defined in [9]. 2. We present a new linear-time algorithm to translate detMITL+p formula to networks of deterministic timed automata. This algorithm is in two steps. First we show how the outermost temporal modalities can be handled with respect to initial satisfiability. Next we show how the past temporal modalities are handled for general satisfiability. We obtain a synchronous network of deterministic automata with shared variables that captures the semantics. We start with Yesterday allowing an arbitrary interval constraint. Then, we deal with Since operators with one-sided intervals of the form [0, a] or [a, ∞). For these simpler cases, we construct networks where each automaton has at most 1 clock and 2 states, but critically uses the sharing feature across components. Finally, the automaton for Since with a general two-sided interval [a,b] is the most complicated part of the construction and results in a network whose one component uses O(a/(b-a)) clocks. Our construction works for both finite and infinite words. 3. Next, we extend this algorithm to handle future modalities. To achieve this we move from networks of timed automata to networks of generalized timed automata (GTA) [1], which have future clocks in addition to usual (history) clocks. We obtain a translation from general MITL+p (with Past and allowing punctual intervals at the outermost level) to networks of GTA with shared variable. While non-determinism cannot be avoided here, we show that we can reduce non-deterministic branching in two ways: first by using shared variables to share predictions, and second, by using non-Zenoness assumption and the semantics of future clocks to obtain a 2-state automaton for Until (which results in an exponential improvement in running time for the model checking algorithm). Again our model checking algorithm works for infinite words and is easily adaptable for finite timed words. 4. Since our model checking approach ultimately reduces to checking reachability and liveness for GTAs, we propose an improved liveness-checking algorithm that extends Couvreur's SCC algorithm [26] (which is the state-of-the-art for timed automata) to the setting of GTAs, marking the first adaptation of Couvreur's procedure to automata with event clocks or diagonal constraints. 5. Finally, we implement our algorithms in a prototype tool, called Tempora, built on top of the open-source timed automata library Tchecker. Tempora supports satisfiability and model checking for detMITL+p and general MITL+p, for both finite and infinite words, in the pointwise semantics. We compare against MightyL which translates MITL formula 4 Efficient Verification of MTL with Past in Pointwise Semantics to timed automata (and uses UPPAAL to check satisfiability with finite words and OPAAL for infinite words). Our comparison on a suite of 72 benchmarks (22 new and 50 taken from the literature) shows that our pipeline (Tempora+Tchecker) significantly outperforms MightyL pipeline; it is faster in all benchmarks for finite words and all but 2 for infinite words, and most often explores a smaller state space (in zones). We also implement the theoretical translation to GTA presented in [2] and demonstrate an order of magnitude improvement. Finally, we also implement an end-to-end model checking algorithm for MITL+p over the pointwise semantics and evaluate our model-checking pipeline on two models: Fischer's mutual exclusion algorithm and the Dining Philosophers problem. Related works. Model checking for real-time systems has been extensively studied, with Metric Interval Temporal Logic (MITL) emerging as a widely used specification language due to its expressiveness for capturing timed properties. A central challenge lies in efficiently translating MITL formulas into automata-based representations, enabling algorithmic verification. For a comprehensive overview of the topic, we refer the reader to the survey by Bouyer [14]. Over continuous/signal semantics, we already discussed several works that focus on obtaining networks of timed automata for various fragments of MITL with and without past including [30, 44, 45]. Most of these employ construction of compositional testers or transducers for timed modalities, but do not focus on sharing e.g., of networks and subformula. A different purely logic based approach was considered in [47], where the timed automata and MITL formula, still with signal semantics, are translated to an intermediate logical language, which is then encoded into an SMT solver. This work also included synchronization primitives and was implemented in a Java tool. Coming back to pointwise semantics, in [5] that introduced MITL, the authors propose the first such translation, which is known to be notoriously complicated, spanning several pages, and acknowledged to be difficult in the follow-up works by different authors [45]. The current state-of-the-art for MITL model checking is the procedure proposed by Brihaye et al. in [20], where they translate MITL formulae to a network of timed automata, which can be plugged in a standard timed automata tool implementing non-emptiness. Their procedure and the associated tool, called MightyL [20] is based on a series of works that study a translation of MITL to timed automata [18, 19], via alternating timed automata [49]. [17] studies a zone-based algorithm directly for 1-clock alternating timed automata, and focusses on a fragment of MTL. Several tools support MITL model checking under continuous semantics, including [38, 23, 24, 52, 42, 47]. Under pointwise semantics, to the best of our knowledge, MightyL [20] is the only tool that can handle the full future fragment of MITL (even if it only solves satisfiability and does not handle past operators). The other works (to our knowledge) that offer tool support are for restricted fragments such as MITL(0.∞) [21], or MITL over untimed words [59]. Structure of the paper. In Section 2, we formalize the notion of a synchronous network of timed automata with shared variables, whose transitions are instantaneous programs, and define the model checking problem in terms of such networks. In Section 3 we describe the detMITL+p logic followed by the translation to synchronous networks of deterministic timed automata in Section 4. In Section 5, we explain how to integrate future modalities to get a translation from MITL+p to generalized timed automata with future clocks. Section 6 describes our adaptation of Couvreur's SCC algorithm to the setting of GTAs. Section 7 contains our experimental evaluation and comparisons and we end with a brief conclusion in S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 5 Section 8. To improve the flow and readability of the paper, we have moved some proofs and technical details to the Appendix. Moreover, some benchmark details and extended results are presented in the Appendix. 2 Synchronous network of timed automata with shared variables We define the model of network of timed systems that we use in this paper. The presentation is different from previous works [11, 16, 33, 2], and is geared towards providing a clean and concise explanation of our logic-to-automata translation. Before giving the formal syntax, we highlight the main aspects in our definition. Each component in the network is an automaton with a finite set of locations, real valued variables (called clocks) and finitely valued variables (mainly boolean variables and the state variable ranging over the finite set of locations). The semantics of the network is synchronous: either time elapses in all components of the network (all clocks are increased by the same amount), or all components execute simultaneously an instantaneous transition. The components of the system communicate via shared variables (clocks, states, boolean variables, . . . ). The paradigm is concurrent-read owner-write (CROW). Hence, each variable is owned by a unique component of the system. A component may read all variables of the system, but it may only modify its owned variables. A transition of a component A is of the form (l, prog, l′) where l, l′ are locations of A and prog is an instantaneous program, which is a sequence of tests and updates. For instance, if x, y are clocks owned by A and out is a boolean variable owned by A, then a purely local program could be A.x ∈(3, 5]; A.y := 2; A.out := 0 where A.x ∈(3, 5] is a test which succeeds if the current value of clock x is within the interval, and A.y := 2 and A.out := 0 are updates. Tests may also use variables owned by other components. For instance, if component B owns clock z, then a program of A could be of the form A.x ∈(3, 5] ∧B.out• = 1 ∧B.z > 2 ∧B.st• = l1; A.y := 2; A.out := 0 where B.st means the state of B. Since we use a synchronous semantics, all components of the network simultaneously take a transition and execute the associated programs. By default, a variable such as A.x or B.z used in a test refers to the value of the variable before the synchronous transition is taken. It is sometimes convenient to refer to the value a variable will have after the synchronous transition is taken. We do so by writing B.out• or B.st•. We give now the formal syntax and semantics for a network of timed systems. A concrete example follows in Figure 1 and Example 3. ▶Definition 1 (Timed automaton with shared variables). A timed automaton with shared variables is a tuple A = (Q, X, V, ∆, init) where A is the name, Q is a finite set of locations over which ranges the state variable A.st, X is a finite set of clock variables owned by A, V is a finite set of boolean variables owned by A, ∆is the finite set of transitions, and init is an initial condition for the variables owned by A. A transition is a triple (l, prog, l′) where l, l′ ∈Q are locations and prog is an instantaneous program. A program is a sequence of tests and updates. An atomic update for a boolean variable v ∈V owned by A has the form A.v := c (or simply v := c when A is clear from the context) with c ∈{0, 1}. An atomic update for a clock variable x ∈X owned by A has the form A.x := c (or simply x := c) with c ∈N = N ∪{+∞}. We assume that each variable is updated at most once in a program. An atomic test for a clock variable x ∈X owned by A has the form A.x ∈I (or simply x ∈I) where I is an integer bounded interval whose end 6 Efficient Verification of MTL with Past in Pointwise Semantics l0 A1 x = +∞ q•; x := 0 ¬q• l0 l1 A2 y = +∞ ¬q• q•; y := 0 ¬p• ∧q•; y := 0 p• ¬p• ∧¬q• l0 l1 l2 A3 z = 0 z ≤100 ∧¬r• z ≤100 ∧r• ∧A1.x 100 z ≤100 ∧r• ∧(A1.x ≥2 ∨A2.st• = l0 ∨A2.y• 0, (w, i -1) |= φ and τi -τi-1 ∈I. S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 9 (w, i) |= XI φ if (w, i + 1) |= φ and τi+1 -τi ∈I. (w, i) |= φ1 UI φ2 if there exists j ≥i s.t. (w, j) |= φ2, τj -τi ∈I and (w, k) |= φ1 for all i ≤k I ∨ (x• init I) to state that xinit is smaller (resp. larger) than all values in I. For instance, if I = [b, c) with b I means xinit ≥c. If a run reaches state 2 (resp. 3) then we know for sure that b1 UI b2 was initially true (resp. false). State 1 means we still do not know whether b1 UI b2 was initially true. If the run stays forever in state 1, b1 UI b2 was initially false. l0 l1 AY ¬b• 1 b• 1 b• 1 ¬b• 1 1 xlast = +∞ Alast xlast := 0 Figure 5 Left: Automaton AY for the untimed Yesterday operator Y b1. The boolean variable b1 is not owned by AY and stands for the argument of Y. Right: Sharer automaton Alast with clock xlast used by all Yesterday operators. ▶Lemma 8. Let Prop′ = Prop ⊎{b1, b2} be an augmented set of propositions. For every word w = (a0, τ0)(a1, τ1) · · · over Prop′, there is a unique run C0 δ0,t0 ---→C1 δ1,t1 ---→C2 · · · of (Ainit, Aisat UI ) over w. Moreover, w |= b1 UI b2 iff there exists an i ≥1 s.t. for all k ≥i, configuration Ck satisfies Aisat UI .st = 2, Proof. Suppose w |= b1 UI b2. Then, (1) there exists an i ≥0 such that b2 ∈ai, (2) for all 0 ≤j 1 then xn satisfies the lower bound constraint of I and it may serve as a witness instead of the earlier clocks x1, y1, . . . , xn-1, yn-1 (notice that xn 1 then b ̸ ◁x2. Proof. 1. was already stated in Lemma 10. 2. follows directly from Lines (1-2) of Algorithm 1. 3. The proof is by induction on i. The base case is i = 0. The initial state of AS is l0, i.e., C0(AS.st) = l0. Now, C1(AS.st) = l1 iff b2 ∈a0. In this case, Line 4 of Algorithm 1 is taken and we easily check that item (3) of Lemma 12 holds with j = 1 and i1 = i′ 1 = 0. Consider now i > 0 and assume that item (3) hold for i -1. If Ci(AS.st) = l0 then we argue exactly as in the base case above. The interesting case is Ci(AS.st) = l1. By induction, there is 1 ≤j ≤k and indices i1 ≤i′ 1 1 then b is smaller than (◁) the value of x2 after the clock shift which is the value of xn+1 at C′ i. Therefore, the last bullet of Item 3 holds after the clock shift, hence also at Ci+1. Now, we can prove as above, in the case of no clock shift, that after executing Lines (9-14) of Algorithm 1, Item 3 holds at Ci+1. This concludes the inductive proof of Item (3). ◀ Proof of Lemma 12. Fix i ≥0. By Lemma 13, position i satisfies the untimed since b1 S b2 iff Ci+1(AS.st) = l1 (corresponding to AS.st• = l1 in outSI). We assume that we are in this case. The only question is whether there is a witnessing b2 within the interval I. We claim that this can be determined by looking solely at the values of clocks x1 and y1 at configuration Ci+1 (corresponding to the test Agen SI .x• 1 ∈I ∨Agen SI .y• 1 ∈I in outSI). We can view this as follows. Suppose there is in the past a b2 event witnessing b1 SI b2: b2 ∈ai′′ for some i′′ ≤i and b1 ∈ai′ for all i′′ η.dfsnum 58: if L ⊆A and XF = R ∪InfClocks(μ) = XF then ▷InfClocks(μ) is the set of future clocks which can take value -∞in zone of u 59: output Accepting SCC detected 60: end if 61: Roots.Push((μ, A, R)) 62: end function S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 39 C Appendix for Implementation and Experimental evaluation C.1 Implementation details. Our tool, Tempora, takes as input a MITL+p formula ψ and generates a synchronous network of generalized timed automata with shared variables based on the translation described in Sections 4 and 5. The execution of this network is simulated using an array of shared variables. Each generalized timed automaton writes its output to a dedicated shared variable, which can then be accessed by its parent automaton. Synchronization is enforced by sequentially executing each automaton in bottom-to-top order. A shared control variable tracks the active automaton during execution, and a transition in an automaton is executed when its ID matches the value of the control variable. Upon taking the transition, the control variable is updated to the ID of the next automaton. A controller is used to reset the execution after every automaton takes a transition and alternates between time elapse and automaton execution. A master automaton evaluates the top-level boolean subformula, ensuring the formula is satisfied at the start and, for finite timed words, that all automata have performed a transition in the last round. To avoid constructing automata for boolean operators, we use symbolic labels to represent temporal operators and atomic propositions, generating boolean expressions with these labels. For model checking, we construct the network of automata for ¬ψ, where ψ is the input specification. The input model and the network of automata generated by our translation are executed alternately, with transitions of the model updating atomic propositions in the shared variable array, which the network of timed automata reads. C.2 Comparison of translation times In Table 4, we discuss the time for the translation for a few instances. We remark that the time for the translation is rather small in most cases. Sl. Formula Time taken by No. MightyL (ms) Tempora (ms) 1 (((((p0 U[1,2] q1) U[1,2] q2) U[1,2] q3) U[1,2] q4) U[1,2] q5) 50 5 2 G[0,10] F[1,2]((a ∧(X b)) ∨(¬a∧!(X b))) 15 3 3 G[2,∞)((¬request ∨F[4,5] grant)) 42 4 4 F[1,2] p1 ∧F[1,2] p2 ∧F[1,2] p3 ∧F[1,2] p4 ∧F[1,2] p5 49 3 Table 4 Comparison of time taken by Tempora, and MightyL for translation of MITL formulae to generalized timed automata (timed automata respectively). C.3 Engineering optimizations. We have implemented several optimizations to improve performance, of which we discuss one such key optimization here. First, in addition to the locations of each automaton, the global state in our translation also stores the output of each automaton in the array of shared variables. Each shared variable doubles the global state space of the network. However, at any point in the sequential execution, the subsequent behaviour of the network depends only on the outputs of the previous automaton which are the direct dependencies of the remaining automata in the round. Therefore, once the last parent of an automaton reads a shared variable, its value can be discarded and the variable reused to store the output of a different automaton. We use static liveness analysis to determine when shared variables become dead 40 Efficient Verification of MTL with Past in Pointwise Semantics and reuse them, reducing the number of auxiliary variables. This optimization minimizes the state size and the overall state space of the zone graph. C.4 Ablation Study To better understand the contribution of individual improvements in Tempora, we perform ablation studies by selectively disabling them and measuring their impact on performance. For the ablation study, we consider (a) one of our theoretical contributions, namely sharing of predictions, (b) specialized automata for the F operator and (c) finally, an engineering optimization, namely memory reuse discussed above. For each feature, we compare the performance of Tempora with the feature, Tempora with the feature turned off, and the state-of-the-art tool MightyL. Our results demonstrate that each of these improvements plays a critical role in enabling scalability, often yielding exponential improvements. Sharing Predictions. In the absence of prediction sharing, each until subformula in the network of automata independently guesses the time of satisfaction. As a result, multiple nondeterministic predictions must be resolved during reachability analysis, significantly increasing the complexity of the state space. With prediction sharing enabled, a single dedicated automaton generates the prediction, and the remaining components deterministically consume this information. This reduces the overall non-determinism in the system and leads to a much smaller product automaton. Empirical results confirm this effect: benchmark families involving multiple until operators show exponential gains in verification time when prediction sharing is enabled. We depict the results in Figure 12 (left) for φ(n) = Vn i=1 p U[0,i] q. Figure 12 Comparison of performance of Tempora with and without features. The left figure shows the impact of sharing predictions. The middle figure shows the impact of specialized automata for F operator. The right figure shows the impact of memory reuse. The x-axis represents the parameter in the formula, while the y-axis shows the time taken in milliseconds. Specialized Automaton for F While the eventually operator FI p can be encoded as true UI p, this transformation can be an overkill for untimed or simple timing intervals like [0, ∞). Tempora implements specialized automata for such cases that are smaller and structurally simpler than general-purpose until-based encodings. For instance, the automaton for F[0,a] p simply waits up to a time units for p to become true, without introducing intermediate obligations or control states. These specialized constructions avoid unnecessary non-determinism and reduce clock usage. Our benchmarks show that formulas involving eventualities with small intervals benefit substantially from this, often reducing the number of locations and transitions in the resulting automaton S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 41 by an order of magnitude. The results are shown in Figure 12 (middle) for the formula φ(n) = VN i=1 F[0,i] p. Reusing Memory The memory optimization addresses inefficiencies in how truth values of atomic propositions are stored during automaton execution. Consider the formula φ(n) = Vn i=1 X pi, where each X pi depends only on the value of pi at the previous timestamp. In a naive implementation, each pi is assigned a unique index in the state array, even though their lifetimes do not overlap. Tempora incorporates a live-range analysis to reuse memory indices once a variable becomes dead - i.e., no longer accessed in the remaining evaluation of the round. In the above example, all pi can safely share a single index. This drastically reduces the size of the memory array and thus the number of reachable states. Our experiments demonstrate that this reuse yields exponential improvements in model-checking time for formulas with long chains of independent next operators, as depicted in Figure 12 (right). C.5 Benchmarks and Extended Experimental Results We next briefly describe the additional benchmarks we use to evaluate our performance. Our benchmarks. We constructed several MITL+p formulae to show the benefits of our improvements. To show the benefit of sharing clocks, we use a disjunction of multiple X operators: G[0,2](Wn i=1 X[2i,2i+1] pi), where each Next automaton uses a single shared clock. A conjunction of multiple U operators with the same left and right children: G(Vn i=1(pi → a U[0,i] b)) ∧G(Wn i=1 pi) shows the benefit of using sharer automata, where each Until automaton uses a common sharer automaton, as well as using a smaller 2 state automaton for Until. The power of using deterministic automata for top level operators can be seen for a formula with conjunction of multiple U operators having general intervals at the top level: Vn i=1(p U[2i,2i+1] q). We show the benefit of reusing shared variables for formulae where atomic propositions have short live ranges: Wn i=1 G[2,∞)(pi). In all these formulae, we try to ensure that they cannot be easily rewritten to a simpler formula, and are not trivially satisfiable/unsatisfiable. Acacia Benchmarks. Each LTL formula in the original Acacia benchmark is of the form ψ ⇐⇒G F acc, where ψ is an LTL specification. We modify these examples by adding intervals to temporal operators of ψ. For several of these specifications, we add intervals to the operators specifying occurrences of atomic propositions, which adds stricter constraints for the atomic propositions to occur within an interval. For some of the benchmarks, we also add general intervals to nested U operators to add complexity to the formula. The complete experimental results are presented in Table A (for finite words) and Table B for infinite words. 42 Efficient Verification of MTL with Past in Pointwise Semantics Sl. No Formula Baseline MightyL+UPPAAL TEMPORA Nodes Stored Time (ms) Nodes Stored Time (ms) Nodes Stored Time (ms) 1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3) 32341 179.083 11239 66.00 2744 25.4233 2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5) 1927857 24478.2 243106 1,348.00 24076 209.238 3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3) 70615 480.662 3440 17.00 329 1.57934 4 G[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 || p2 || p3) 35079 230.365 3440 20.00 345 2.02975 5 (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))&& G(p1 || p2) 25210 165.833 885 5.00 437 2.63777 6 (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X [2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || ! (atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))))))))&& G(p1 || p2) - TO 3517869 43,896.00 3509 21.3181 7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q 5441 225,448.00 1990 12.3429 8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5 2130 14.2971 3712 25.00 160 0.791932 9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5 80500 619.015 3872 23.00 20 0.133869 10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5) 118296 984.777 16182 129.00 10777 84.286 11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) )) 48083 416.384 479764 215,172.00 3367 33.0925 12 G[2, Inf) ( (!request || F[4, 5] grant)) 643 4.67705 60 307.00 71 0.511877 13 p U[11, 12] q 144 2.10697 57 1,408.00 38 0.305787 14 G((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 || p3 || p4) - N/A - N/A 186 1.67275 15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q))) - N/A - N/A 1303 29.9574 16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q) - N/A - N/A 1571 50.9912 17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q) - N/A - N/A 27737 1762.13 18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q) - N/A - N/A 167077 21490.5 19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - N/A 262 3.18676 20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - N/A 286 1.53985 21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf) q && p S[4, Inf) q) - N/A - N/A 68 0.798897 22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q))) - N/A - N/A 66 0.49014 MightyL benchmarks 23 F p1 && F p2 && F p3 && F p4 && F p5 9292 44.3188 3263 15.00 157 0.530741 24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5 1241 9.18455 3263 16.00 160 0.900405 25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5 74516 556.001 6592 44.00 177 0.806017 26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 47766 524.722 455 1,362.00 59 0.34313 27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5 - TO 6623 31,749.00 191 1.73696 28 (((((p0 U q1) U q2) U q3) U q4) U q5) 25972 101.764 8133 50.00 4063 21.7138 29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5) 104149 549.412 8133 40.00 6303 33.4324 30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5) 48557 319.548 18204 136.00 12496 82.9332 31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) 16160 185.278 1140 3,278.00 3238 35.6263 32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) 162606 3074.69 5006 19,078.00 27758 434.539 33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5) 1629834 62186.7 22228 107,447.00 235536 6455.47 34 G(!p) && F[0, 3] t1 110 0.790953 55 2.00 21 0.118925 35 G(!p) && F[0, 3] t1 && F[3, 6] t2 2396 12.5292 627 254.00 174 1.31732 36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 48222 364.599 2450 6,690.00 419 2.98024 37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4 - TO 48383 274,012.00 2079 11.2824 38 !(!(G F p1) || (G(!q || (F[100, 1000] r)))) 6916 38.5607 463 3.00 271 1.66032 39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r)))) 65149 418.318 3423 32.00 1005 5.93152 40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r)))) 561208 5684.74 26047 280.00 3397 18.338 41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r)))) 4651945 126995 202623 2,359.00 10953 59.6345 42 F[0, 30] (!p || G[0, 20] p) 31 0.32008 31 2.00 17 0.118925 43 G[0, 30](!p) || F[0, 20] p 37 0.303894 32 2.00 12 0.152296 44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5) 179753 1216.62 8128 44.00 6318 37.332 45 (((((p0 R q1) R q2) R q3) R q4) R q5) 47841 250.798 8128 33.00 4079 19.3469 46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) 7571 69.5354 560 15.00 546 5.74616 47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5) 375155 6503.65 9792 393.00 12042 230.663 48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) 54947 678.074 2368 74.00 2628 38.1337 49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5) 78161 587.855 9792 55.00 5892 39.5866 50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5 2735492 78412.1 3232 132.00 20 0.252719 51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5 70930 578.802 3232 18.00 20 0.136226 52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5 1815 14.362 3134 19.00 160 0.809136 53 G p1 && G p2 && G p3 && G p4 && G p5 12897 75.8965 3134 14.00 158 1.06356 54 !((F[0, 30] p) || !(F[0, 20] p)) 60 0.568014 69 2.00 49 0.272393 55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p)) 1272 10.5949 495 221.00 224 0.991115 Acacia benchmarks 56 ((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F(a))) 3047327 70749.6 54777 82,277.00 530 3.40564 57 (((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F a))) || (! (((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) && !((G (F a)))) - TO 3193831 41,472.00 3455 19.531 58 ((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && !(G(F(a)))) - TO 214577 96,763.00 1059 7.08385 59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a))) 466847 5451.49 13712 21,448.00 225 1.61587 60 ((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r U[0, 3] s)))) && !(G(F a))) - TO - Error 564 5.20393 61 ((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G(Fa))) || (!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && !(G(Fa))) - TO - Error 3276 19.6802 62 ((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F [2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && !(G(F a))) - TO - TO 366144 3712.65 63 (((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && (G(F (a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && !(G(F(a)))) - TO - TO 362442 8380.15 64 (((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && !(G(F(a)))) - TO - TO 7362 86.5996 65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a))) 51613 392.448 6641 72.00 61 0.507483 66 (G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) )) && ! (G( F (a)))) - TO - TO 3153 40.0667 67 (( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a)))) 885283 14219.3 99857 945.00 87 0.747122 68 (((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (!((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a)))) - TO - TO 5053 39.7407 69 (((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && !(G(F(a)))) 6802985 269414 51641 19,852.00 444 2.52895 70 (((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2](X[1, 2] (X[1, 2] q))))))) && !(G(F a))) - TO 430184 3,928.00 1061 4.97043 71 (((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && !(G(F a))) - TO - TO 1917 23.3503 72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a))) 13546 82.0316 3511 20.00 96 0.430384 Table A: Experimental results obtained by running TEMPORA, MightyL + UPPAAL and, the baseline implementation of the translation for satisfiability for finite traces. (Timeout is set to 300s) S. Akshay, P. Contractor, P. Gastin, R. Govind and B. Srivathsan 43 Sl. No Formula Baseline MightyL + OPAAL TEMPORA Nodes Stored Time (ms) Nodes Stored (Reporting state space given by opaal) -> Non deterministic, since multi threaded. Numbers change across different experiments Time (ms). Note: opaal is multicore, and hence uses 8 cores of local machine Total time (ms) Nodes Stored Time (ms) opaal g++ ltsmin -> the actual exploration 1 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3) 2292 14.1087 1161 172.79 661.47 76.03 910.28 212 1.08509 2 G[0, 20](X[2, 3] p1 || X[4, 5] p2 || X[6, 7] p3 || X[8, 9] p4 || X[10, 11] p5) 29455 193.183 1861 260.97 710.76 80.51 1,052.25 552 3.17967 3 G((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G(p1 || p2 || p3) 1310856 16076.9 1162 227.10 700.43 76.16 1,003.69 458 2.42018 4 G[0, 10]((!p1 || a U[0, 1] b) && (!p2 || a U[0, 2] b) && (!p3 || a U[0, 3] b)) && G[0, 10](p1 || p2 || p3) 1587059 21828.1 1450 263.38 718.80 77.70 1,059.88 1044 8.24895 5 (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((! p1) || atomic) && (!p2) || !(atomic))))))&& G(p1 || p2) 44857 350.62 1893 203.98 663.58 82.98 950.54 168 0.987026 6 (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || ! (atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || ! ((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((! p1) || atomic) && (!p2) || !(atomic)))))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || !(atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))) && (!p2) || !((X[2, 3](((!p1) || (X[2, 3](((!p1) || atomic) && (!p2) || ! (atomic)))) && (!p2) || !((X[2, 3](((!p1) || atomic) && (!p2) || !(atomic))))))))))))&& G(p1 || p2) - TO - - - Segfault 0.00 1933 15.9299 7 p U[2, 3] q && p U[4, 5] q && p U[6, 7] q && p U[8, 9] q && p U[10, 11] q - TO - - - TO 0.00 456 4.17251 8 G[0, 2] p1 || G[0, 2] p2 || G[0, 2] p3 || G[0, 2] p4 || G[0, 2] p5 2346 16.1747 969 301.04 736.75 75.30 1,113.09 224 1.30602 9 G[2, Inf) p1 || G[2, Inf) p2 || G[2, Inf) p3 || G[2, Inf) p4 || G[2, Inf) p5 - TO 1143 272.19 721.28 78.98 1,072.45 178 1.58184 10 G[0, 2](F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5) - TO 7003 404.46 851.76 106.54 1,362.75 1087 9.45395 11 G[0, 10] F[1, 2] ( (a && (X b) ) || (!a && !(X b) )) 117319 1140.97 7921 74,584.66 20,184.02 238.35 95,007.03 504 3.50233 12 G[2, Inf) ( (!request || F[4, 5] grant)) - TO - - - TO 0.00 444 4.00833 13 p U[11, 12] q 1360 21.8606 - - - TO 0.00 220 1.3685 14 G((!p1 || (a S b)) && (!p2 || (a S b) && (!p3 || (a S b) && (!p4 || (a S b))))) && G(p1 || p2 || p3 || p4) - N/A - - - N/A 798 2.19275 15 F (p S[1, 2] (p S[1, 2] (p S[1, 2] q))) - N/A - - - N/A 14440 1721.87 16 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q) - N/A - - - N/A 1410 51.2159 17 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q) - N/A - - - N/A 2310 132.646 18 F (p S[1, 2] q && p S[2, 3] q && p S[3, 4] q && p S[4, 5] q && p S[5, 6] q && p S[6, 7] q) - N/A - - - N/A 3927 418.381 19 G[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - - - N/A 91 0.509345 20 F[0, 20](Y[2, 3] p1 || Y[4, 5] p2 || Y[6, 7] p3) - N/A - - - N/A 267 1.44564 21 F (p S[1, Inf) q && p S[2, Inf) q && p S[3, Inf9 q && p S[4, Inf) q) - N/A - - - N/A 114 0.645279 22 F (p S[1, Inf) (p S[1, Inf) (p S[1, Inf) q))) - N/A - - - N/A 108 0.73709 MightyL benchmarks 23 F p1 && F p2 && F p3 && F p4 && F p5 822 5.11449 2165 190.21 698.33 79.48 968.01 64 0.407546 24 F[0, 2] p1 && F[0, 2] p2 && F[0, 2] p3 && F[0, 2] p4 && F[0, 2] p5 952 5.99807 2693 214.01 701.74 82.35 998.09 68 0.485472 25 F[2, Inf) p1 && F[2, Inf) p2 && F[2, Inf) p3 && F[2, Inf) p4 && F[2, Inf) p5 - TO 3069 372.57 820.01 85.33 1,277.91 200 2.38852 26 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 - TO - - - Segfault 0.00 102 0.528738 27 F[1, 2] p1 && F[1, 2] p2 && F[1, 2] p3 && F[1, 2] p4 && F[1, 2] p5 - TO - - - TO 0.00 158 1.05082 28 (((((p0 U q1) U q2) U q3) U q4) U q5) 1828 10.7493 1560 196.70 704.35 78.12 979.17 18689 91.4055 29 (((((p0 U[0, 2] q1) U[0, 2] q2) U[0, 2] q3) U[0, 2] q4) U[0, 2] q5) 47166 298.922 1477 219.19 699.27 76.57 995.03 2668 19.0006 30 (((((p0 U[2, Inf) q1) U[2, Inf) q2) U[2, Inf) q3) U[2, Inf) q4) U[2, Inf) q5) 567192 5404.53 9383 382.47 960.14 113.60 1,456.21 4075 32.6984 31 (((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) - TO - - - Segfault 0.00 3254 30.8566 32 ((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) - TO - - - TO 0.00 36124 451.716 33 (((((p0 U[1, 2] q1) U[1, 2] q2) U[1, 2] q3) U[1, 2] q4) U[1, 2] q5) - TO - - - TO 0.00 465185 8750.47 34 G(!p) && F[0, 3] t1 310 1.95935 184 117.86 612.55 72.49 802.89 32 0.235228 35 G(!p) && F[0, 3] t1 && F[3, 6] t2 953492 24977.1 964 68,462.12 20,584.03 102.84 89,149.00 158 0.901839 36 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 - TO - - - Segfault 0.00 315 3.3407 37 G(!p) && F[0, 3] t1 && F[3, 6] t2 && F[6, 9] t3 && F[9, 12] t4 - TO - - - TO 0.00 522 4.21687 38 !(!(G F p1) || (G(!q || (F[100, 1000] r)))) - TO 1935 692.31 954.65 79.29 1,726.25 2823 18.9429 39 !(!((G F p1) && (G F p2)) || (G(!q || (F[100, 1000] r)))) - TO 4212 749.15 1,006.85 89.55 1,845.55 28570 196.333 40 !(!((G F p1) && (G F p2) && (G F p3)) || (G(!q || (F[100, 1000] r)))) - TO 42804 836.05 1,056.50 225.40 2,117.95 330044 2976.99 41 !(!((G F p1) && (G F p2) && (G F p3) && (G F p4)) || (G(!q || (F[100, 1000] r)))) - TO 149029 906.87 1,133.70 473.50 2,514.08 3819965 88366.7 42 F[0, 30] (!p || G[0, 20] p) 9432 52.7811 43 131.32 615.22 69.64 816.18 704 5.34067 43 G[0, 30](!p) || F[0, 20] p 61 0.756158 37 126.61 617.54 75.80 819.95 27 0.295317 44 (((((p0 R[0, 2] q1) R[0, 2] q2) R[0, 2] q3) R[0, 2] q4) R[0, 2] q5) 145353 1141.14 2267 260.88 746.28 86.45 1,093.60 113881 905.989 45 (((((p0 R q1) R q2) R q3) R q4) R q5) 22354 140.505 1521 185.82 690.21 79.60 955.64 11957 61.4713 46 (((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) 1086882 15332.4 453 1,588.79 1,496.63 78.06 3,163.48 5985 55.9789 47 (((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) R[1, 2] q5) - TO - - - Segfault 0.00 1093187 23474.1 48 ((((p0 R[1, 2] q1) R[1, 2] q2) R[1, 2] q3) R[1, 2] q4) - TO - - - Segfault 0.00 79258 1018.67 49 (((((p0 R[2, infty) q1) R[2, infty) q2) R[2, infty) q3) R[2, infty) q4) R[2, infty) q5) 29685 278.315 1059 230.60 725.70 77.18 1,033.48 3119 27.748 50 G[1, 2] p1 && G[1, 2] p2 && G[1, 2] p3 && G[1, 2] p4 && G[1, 2] p5 - TO - - - Segfault 0.00 225 2.45304 51 G[2, Inf) p1 && G[2, Inf) p2 && G[2, Inf) p3 && G[2, Inf) p4 && G[2, Inf) p5 - TO 1549 226.81 724.91 82.29 1,034.01 178 1.07516 52 G[0, 2] p1 && G[0, 2] p2 && G[0, 2] p3 && G[0, 2] p4 && G[0, 2] p5 4793 34.5633 5913 257.33 760.33 100.74 1,118.41 224 1.45227 53 G p1 && G p2 && G p3 && G p4 && G p5 195653 1451.52 8413 186.71 694.95 111.65 993.31 64 0.61791 54 !((F[0, 30] p) || !(F[0, 20] p)) 7069 37.444 100 126.35 622.49 72.87 821.71 629 10.9368 55 !(!((F[20, 40] True && G[0, 40] p)) || G[0, 20] (F[0, 20] p)) 3072735 56119.3 624 62,506.68 20,677.93 112.71 83,297.33 1669 19.924 Acacia benchmarks 56 ((F(q && (X[0, 4](p U[2, 3] r)))) && (G F(a))) || (!(F(q && (X[0, 4](p U[2, 3] r)))) && !(G F (a))) - TO 76875 150,058.47 42,824.04 2,635.25 195,517.76 4561 37.7644 57 (((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q) && (G (F a))) || (!(((p U[0, 2] q) U[0, 2] r) && ((q U[0, 2] r) U[0, 2] p) && ((r U[0, 2] p) U[0, 2] q)) && !((G (F a)))) - TO 7271276 611.46 1,143.21 21,348.99 23,103.66 918 7.40638 58 ((F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && (G(F(a)))) || (!(F[1, 10](p && (X[2, 4](q && (X F[2, Inf) r))))) && !(G(F(a)))) - TO 523958 67,992.90 22,837.11 10,320.40 101,150.41 4614 32.6728 59 ((G(!p || (q U[2, 3] r))) && (G(F a))) || (!(G(!p || (q U[2, 3] r))) && !(G(F a))) - TO 22945 116,791.31 39,078.93 840.78 156,711.03 1617 12.6589 60 ((p U[3, 4] ( q && (X [2, Inf)(r U[0, 3] s)))) && (G(F a))) || (!(p U[3, 4] ( q && (X[2, Inf) (r U[0, 3] s)))) && !(G(F a))) - TO - - - Segfault 0.00 820 6.57777 61 ((F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2]s)))))))))) && (G (Fa))) || (!(F[0, 2](p && (X[1, 2](F[0, 2](q && (X[1, 2](F[0, 2](r && (X[1, 2](F[0, 2] s)))))))))) && !(G(Fa))) - TO 6733 183,618.00 14,086.56 323.78 198,028.34 4658 32.0218 62 ((p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && (G(F a))) || (!(p U[0, 2] (q && (X[2, 4](r && (F[2, Inf)(s && (X[2, 4](F[2, Inf)(u && (X[2, 4](F[2, Inf)(v && (X[2, 3](F[2, Inf) w)))))))))))))) && !(G(F a))) - TO - - - Segfault 0.00 54366 545.194 63 (((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && (G(F(a)))) || (!((G(F[2, 4] p)) && (G(F[2, 4] q)) && (G(F[2, 4] r)) && (G(F[2, 4] s)) && (G(F[2, 4] u))) && !(G(F(a)))) - TO - - - TO 0.00 - TO 64 (((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && (G(F(a)))) || (!((G(F[1, 2]p)) || (G(F[1, 2]q)) || (G(F[1, 2]r))) && !(G(F(a)))) - TO - - - TO 0.00 - TO 65 ((!(G(2, Inf) p) || (F[0, 2] q)) && (G (F a))) || (!(!(G(2, Inf) p) || (F[0, 2] q)) && !(G (F a))) - TO 11391 515.86 1,265.08 159.79 1,940.73 246 1.536 66 (G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) ) && G( F (a))) || (!(G(( (p U[1, 2] q) U[1, 2] (!p) ) U[1, 2] (!r) )) && ! (G( F (a)))) - TO - - - Segfault 0.00 - TO 67 (( (!(G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) ) ) && (G(F(a)))) || (!((! (G(2, Inf) p) || (F[0, 2] q)) && (!(G(2, Inf) (!p)) || (F[0, 2] (!q)) )) && !(G(F(a)))) - TO 250875 516.21 950.14 750.60 2,216.94 3484 25.3826 68 (((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && (G(F(a)))) || (! ((G[0, 2] r) && (G[0, 2](!p || (F[1, 2] q))) && (G[0, 2](!q || (F[1, 2] r)))) && !(G(F(a)))) - TO - - - Segfault 0.00 2598 25.7481 69 (((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && (G(F(a)))) || (!((G(2, Inf)(!p || (F[1, 2] q))) && (G r)) && !(G(F(a)))) - TO 35945 63,521.58 21,527.32 788.87 85,837.77 785 5.94832 70 (((G(F[0, 2](!p || (X[1, 2](X[1, 2](X[1, 2] q))))))) && (G(F a))) || (!((G(F[0, 2](!p || (X[1, 2] (X[1, 2](X[1, 2] q))))))) && !(G(F a))) - TO 742427 426.25 948.75 1,682.79 3,057.79 231 1.94063 71 (((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && (G(F a))) && (!((G(!p || (F[3, 4] q))) && (G(p || (F[3, 4] (!q))))) && !(G(F a))) - TO - - - TO 0.00 4159 55.7087 72 ((F[0, 2](G p)) && (G(F a))) || (!(F[0, 2](G p)) && !(G(F a))) 1802521 22455.7 4943 257.75 755.07 95.59 1,108.41 390 2.54234 Table B: Experimental results obtained by running TEMPORA , MightyL + OPPAAL and, the baseline implementation of the translation for satisfiability for infinite traces. (Timeout is set to 300s)
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2510.14697
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Preprint
PURIFYING TASK VECTORS IN KNOWLEDGE-AWARE
SUBSPACE FOR MODEL MERGING
Bang An1
Yibo Yang1∗
Philip Torr2
Bernard Ghanem1
1King Abdullah University of Science and Technology (KAUST)
2University of Oxford
{bang.an, yibo.yang, bernard.ghanem}@kaust.edu.sa
philip.torr@eng.ox.ac.uk
ABSTRACT
Model merging aims to integrate task-specific abilities from individually fine-
tuned models into a single model without extra training. In recent model merg-
ing methods, task vector has become a fundamental building block, as it can en-
capsulate the residual information from finetuning. However, the merged model
often suffers from notable performance degradation due to the conflicts caused
by task-irrelevant redundancy in task vectors. Existing efforts in overcoming re-
dundancy by randomly dropping elements in the parameter space involves ran-
domness and lacks knowledge awareness. To address these challenges, in this
study, we propose Purifying TAsk VEctors (PAVE) in knowledge-aware sub-
space. Concretely, we sample some training examples from each task, and feed
them into their corresponding fine-tuned models to acquire the covariance ma-
trices before linear layers. We then perform a context-oriented singular value
decomposition, which accentuates the weight components most relevant to the
target knowledge. As a result, we can split fine-tuned model weights into task-
relevant and redundant components in the knowledge-aware subspace, and pu-
rify the task vector by pruning the redundant components. To induce fair prun-
ing efforts across models, we further introduce a spectral rank allocation strat-
egy by optimizing a normalized activated pruning error. The task vector purifi-
cation by our method as a plug-and-play scheme is applicable across various
task vector-based merging methods to improve their performance.
In experi-
ments, we demonstrate the effectiveness of PAVE across a diverse set of merg-
ing methods, tasks, and model architectures. Remarkably, when integrated on
the state-of-the-art merging method EMR-Merging with RoBERTa, our method
yields a significant performance improvement of 4.1% (from 80.18% w/o PAVE
to 84.28%) on the GLUE benchmark, approaching to the average performance
of 8 individual models, 85.55%. Our code and models are released at: https:
//github.com/ABgit111/purified_task_vector.
1
INTRODUCTION
Transformer-based models, including large language models (LLMs) (Devlin et al., 2019; Liu et al.,
2019; He et al., 2020; Radford et al., 2019; Touvron et al., 2023; Achiam et al., 2023) and vi-
sion models (Dosovitskiy et al., 2020; Radford et al., 2021) have found widespread applications
across various domains. Supervised fine-tuning (Radford et al., 2018; Hu et al., 2022) is a common
technique to enhance downstream task performance. Despite the effectiveness of multi-task learn-
ing (MTL) in acquiring versatile abilities across multiple domains (Caruana, 1997; Zhang & Yang,
2018; Vandenhende et al., 2021), its applicability will be limited when only individually fine-tuned
models are accessible due to data privacy concerns or computational constraints. To address this
challenge, model merging has been proposed, aiming to construct a single model by combining
parameters from domain-specific models, without relying on any training or fine-tuning efforts.
Early model merging methods focus on performing weighted (Matena & Raffel, 2022) or regularized
(Jin et al., 2022) averaging in the parameter space of fine-tuned models, yet the conflicts inherent in
∗Corresponding author
1
arXiv:2510.14697v1 [cs.AI] 16 Oct 2025
Preprint
Knowledge-aware subspace
∆𝑊= 𝑊!" −𝑊#
Task vector
Naive
DARE
PAVE (ours)
Formulation
Address redundancy
Knowledge awareness
∆𝑊$%&'=
1
1 −𝑝Drop 𝑊!" −𝑊#
∆𝑊(%)'= SVD* 𝑊!"𝐶𝐶+, −𝑊#
𝑊!"
𝑊#
Fine-tune
SVD$ 𝑊!"𝐶𝐶%&
Redundancy w.r.t. the
target knowledge
❌
✅(in parameter space)
✅(in subspace)
❌
❌
✅
Target knowledge as
characterized by 𝐶
➕
SVD $:( 𝑊!"𝐶𝐶%&
Naive task vector
Purified task vector
Figure 1: An illustration of our method and a comparison between PAVE and prior task vectors.
DARE addresses redundancy by randomly dropping elements in the parameter space without knowl-
edge awareness. PAVE refines task vectors by identifying the target knowledge-relevant components
within knowledge-aware subspace and removing the redundant components.
naive parameter averaging often lead to performance degradation of merged models. Subsequently,
Task Arithmetic (Ilharco et al., 2022) has achieved great success by introducing task vector, defined
as the difference between the fine-tuned model weights for a certain task and the base model. Since
task vector can capture the update direction of each task, it allows for efficient composition of
knowledge from multiple fine-tuned models while preserving their distinctive features, and thus has
become a fundamental building block in the follow-up studies (Ilharco et al., 2022; Cheng et al.,
2025; Du et al., 2024; Huang et al., 2024). However, the task vector constructed by subtracting the
base model weights from the fine-tuned weights contains substantial redundancy, which means that
not all parameter shifts during fine-tuning contribute to task-specific abilities (Yu et al., 2024). The
redundant or noisy components may interfere with the valuable components from other task vectors
during merging. As a result, performing model merging based on such task vectors still encounters
conflicts and yields suboptimal performance (Yadav et al., 2023).
To address the redundancy in task vectors, DARE (Yu et al., 2024) proposes to randomly mask
task vectors by Dropout (Srivastava et al., 2014) to induce sparsity and reduce conflict. Despite the
improved performance, we point out that: (1) random masks lack knowledge awareness and may
fail to accurately locate the parameter elements most relevant to a specific task; (2) task-relevant and
redundant elements are not necessarily separable in the parameter space, and actually task-specific
ability lies in a low-rank subspace of parameters (Hu et al., 2022), which means that masking in
the parameter space may not effectively disentangle abilities and resolve conflicts. Therefore, how
to purify task vectors to make them precisely align with task-specific abilities is still a fundamental
challenge in the model merging problem.
To this end, in this paper, we propose purifying task vectors in knowledge-aware subspace (PAVE).
Instead of addressing redundant and noisy elements in the original parameter space, we aim to
decompose fine-tuned LLM parameters into subspace such that the decomposed components are
ordered by their contribution to a specific task. Naively performing singular value decomposition
(SVD) on the fine-tuned weights is inadequate because it only captures the spectral structure of the
weight matrix itself and the resulting subspace is not knowledge-aware. In contrast, we adopt the
context-oriented decomposition that is initially proposed in a recent study for low-rank adaptation
(Yang et al., 2024b). Specifically, given multiple fine-tuned models on different tasks, we randomly
sample a small number of training samples for each task, and feed them into the corresponding
fine-tuned model. After obtaining the activation of each layer X, we calculate the covariance matrix
C = XXT and then perform the context-oriented decomposition by applying SVD to the product of
the fine-tuned weight matrix WFT and the covariance matrix C from its input, i.e., SVD(WFTC). By
doing so, the decomposed subspace is spanned by singular vectors that are ordered by their contri-
butions to the task associated with C. It has been observed that the first several components account
for the majority of the learned task-relevant knowledge. Therefore, we filter out the redundant and
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noisy directions by retaining only the largest r singular values and their singular vectors. We then
post-multiply by C−1 to reconstruct the fine-tuned model weights, which yields our purified task
vector formulated as ∆WPAVE = SVDr(WFTC)C−1 −WB. The illustration of our method and its
comparison with existing task vector constructions are shown in Figure 1.
To induce fair pruning efforts across models, we further introduce a spectral rank allocation strategy
that allocates the number of preserved ranks based on their spectral distribution. It aims to minimize
the summation of a normalized activated pruning error from each individual model. This objective
can be efficiently optimized using a greedy algorithm that iteratively excludes components with the
smallest normalized singular value. It is noteworthy that PAVE as a plug-and-play method can be
integrated on any task vector-based merging methods to improve their performance. We evaluate
the performance of PAVE in combination with several popular merging strategies, including Task
Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav et al., 2023), and EMR-Merging (Huang
et al., 2024). Experiments are conducted on the GLUE benchmark with two widely used language
models, RoBERTa (Liu et al., 2019) and DeBERTa (He et al., 2020), and also on generation tasks,
Math and Coding, with LLaMA-2-7B (Touvron et al., 2023). Additionally, we apply our approach
to ViT-B/16, ViT-B/32, and ViT-L/14 (Dosovitskiy et al., 2020) on 8 vision tasks.
The contributions of this study are summarized as follows:
• We propose PAVE, a plug-and-play method that can improve the performance of modern
model merging methods by purifying task vectors and resolving task-irrelevant redundan-
cies in knowledge-aware subspace.
• We design a spectral rank allocation strategy to further refine the redundancy pruning pro-
cess and improve the performance.
• We demonstrate the effectiveness of PAVE with a wide range of merging methods, tasks,
and model architectures. Notably, PAVE improves the performance of EMR-Merging on
GLUE with RoBERTa from 80.18% to 84.28%, which is close to the average result of 8
individual models 85.55%.
2
RELATED WORK
Model Merging. Model merging has emerged as an active area due to its great potential in inte-
grating abilities from multiple specialized models into a single model (Wortsman et al., 2022; Yadav
et al., 2023; Yang et al., 2023; Xiao et al., 2023). Different approaches have been developed in-
cluding optimization-driven methods (Matena & Raffel, 2022; Jin et al., 2022; Wang et al., 2024a),
test-time adaptation methods (Yang et al., 2023; 2024a), and task vector-based methods (Ilharco
et al., 2022; Yadav et al., 2023; Wei et al., 2025). In particular, task vector has become an influen-
tial tool, as it encodes the essential information obtained during fine-tuning and can be effectively
injected to a merged model using appropriate techniques. However, during the merging process,
evidences have shown that there is noise and redundancy in the plain task vectors, which can cause
conflicts and performance degradation. To address this issue, Ties-Merging (Yadav et al., 2023)
identifies redundancy in model parameters and proposes a trimming and sign-selection scheme to
resolve conflicts. EMR-Merging (Huang et al., 2024) employs a strategy of electing, masking, and
rescaling to integrate lightweight task-specific components, aiming to minimize the divergence be-
tween the merged model and each individual model. DARE (Yu et al., 2024), as a plug-and-play
method, resolves conflict issues by introducing random Dropout (Srivastava et al., 2014) to the task
vectors. Nevertheless, former methods mainly focus on reducing redundancy by removing elements
from the task vectors, which can be considered as sparsity-based approaches. In contrast, our method
purifies task vectors in a knowledge-aware subspace by performing context-oriented decomposition
and pruning the task-irrelevant weight components.
Low-Rank Decomposition for LLMs. Low-rank decomposition based on SVD has been widely
adopted in LLMs across multiple applications. For example, in low-rank adaptation (Hu et al.,
2022; Liu et al., 2024; Zhang et al., 2023; Yang et al., 2024b; 2025), SVD is often leveraged to
create lightweight low-rank adapters for improving fine-tuning performance (Meng et al., 2024)
or preserving task-specific abilities (Yang et al., 2024b). In model compression, various methods
have been proposed to prune model parameters based on the importance provided by singular values
(Yuan et al., 2023; Wang et al., 2024b; Hsu et al., 2022; Li et al., 2025). Similar techniques have also
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been developed for quantization (Li et al., 2024b;a). However, when it comes to model merging,
redundancy purification still relies mostly on sparsity-based approaches—removing or rescaling task
vectors in the parameter space. While decomposition approach is adopted in Twin Merging (Lu et al.,
2024), TSV-Merge (Gargiulo et al., 2025), and KnOTS (Stoica et al., 2024) as a pre-processing step,
the implementations in these works still lack knowledge awareness. As a comparison, our proposed
method performs task vector purification in knowledge-aware subspace to extract and highlight the
parts within a task vector that truly matter for its corresponding task, leading to smoother and more
effective merging.
3
METHODOLOGY
3.1
PRELIMINARIES
Given K different models {M{W l
1}L
l=1, M{W l
2}L
l=1, · · · , M{W l
K}L
l=1} that are fine-tuned on K
different tasks {T1, T2, · · · , TK} from the same base model M{W l
B}L
l=1, where L represents the
number of layers, model merging aims to produce a single model that simultaneously achieves good
performance on these tasks without extra training or fine-tuning. The merged model M{W l
M}L
l=1
can be given by:
M{W l
M}L
l=1 = ϕ(M{W l
1}L
l=1, · · · , M{W l
K}L
l=1, M{W l
B}L
l=1),
(1)
where ϕ refers to a merging method.
Most model merging methods focus on combining the parameters of fine-tuned models. Task vector
captures the task-specific changes introduced during fine-tuning and has become a common and es-
sential tool in modern model merging methods. It is typically defined as the element-wise difference
between a fine-tuned model and the base model,
∆W = WFT −WB,
(2)
where WFT denotes the fine-tuned model weights. Based on task vector, different merging methods
have been proposed to integrate these task-specific parameter variations into a merged model. For
example, in Task Arithmetic (Ilharco et al., 2022), the merged model weights can be computed as:
WM = WB + λ
K
X
i=1
∆Wi,
(3)
where λ is a hyperparameter.
However, recent studies have shown that naively calculating the difference between WFT and WB
cannot precisely represent the shifting direction of a certain task in the parameter space, as the
weight change caused by fine-tuning contains redundant components, e.g., task-irrelevant abilities
or noisy elements (Yadav et al., 2023; Yu et al., 2024). As a result, there still exist conflicts in the
subsequent merging procedure using these task vectors. DARE (Yu et al., 2024) proposes to sparsify
task vectors by randomly dropping elements and rescaling. Their task vector can be formulated as:
∆WDARE =
1
1 −pDrop(WFT −WB),
(4)
where Drop denotes the Dropout operation with rate p, and
1
1−p is a rescaling coefficient. Despite its
effectiveness in reducing conflict, several limitations remain as follows. First, the Dropout operation
involves randomness and lacks knowledge awareness, and thus it cannot locate the valuable compo-
nents in a targeted manner. Second, task-specific abilities actually lie in the subspace of pre-trained
parameters (Raghu et al., 2017; Li et al., 2018), suggesting that binary masking in the parameter
space may not truly separate redundancy or irrelevant components from the required ability. This
analysis motivates us to resort to subspace to purify task vectors.
3.2
PURIFYING TASK VECTORS IN KNOWLEDGE-AWARE SUBSPACE
Directly performing singular value decomposition (SVD) on fine-tuned model weights constructs
a subspace, but such subspace is not necessarily aligned with task-specific knowledge, as it only
4
Preprint
×𝐶!
"!
×𝐶#
"!
×𝐶$
"!
SVD
𝑊!𝐶!
𝑊#𝐶#
𝑊$𝐶$
𝑊!
𝑊#
𝑊$
𝑈!
Rank Allocation
SVD
SVD
𝑈#
𝑈$
Σ!
Σ#
Σ$
𝑉!
%
𝑉#
%
𝑉$
%
𝑈![: 𝑟!]
𝑈#[: 𝑟#]
𝑈$[: 𝑟$]
Σ![: 𝑟!]
Σ#[: 𝑟#]
Σ$[: 𝑟$]
𝑉!
%[: 𝑟!]
𝑉#
%[: 𝑟#]
𝑉$
%[: 𝑟$]
𝐶!
𝐶#
𝐶$
SVD&! 𝑊!𝐶! 𝐶!
"!
SVD&" 𝑊#𝐶# 𝐶#
"!
SVD&# 𝑊$𝐶$ 𝐶$
"!
⋮
⋮
⋮
⋮
⋮
Truncate
Merging
Method
𝑊'
Task 2
Task 𝐾
Task 1
sample
sample
sample
Figure 2: An overall illustration of our proposed method, PAVE, which serves as a plug-and-play
method for task-vector based merging methods. It begins by identifying task-aware components
through a decomposition approach, and then eliminates noise and less effective components via
rank-based pruning. A rank allocation strategy is designed to optimize the preserved rank ratios
across individual models. Overall, rather than employing the plain task vector Wk −WB, PAVE
performs model merging based on purified task vectors SVDrk(WkCk)C−1
k
−WB, 1 ≤k ≤K.
reflects the spectral distribution of the parameter matrix itself and pre-trained LLM weights handle
a broad range of tasks. To this end, we leverage the context-oriented singular value decomposition
(CO-SVD) (Yang et al., 2024b) to decompose fine-tuned weights into a knowledge-aware subspace.
Concretely, for each task Tk, 1 ≤k ≤K, we sample some training examples from this task and feed
them into the fine-tuned model Mk. After acquiring the input activation of each layer X (layer index
is omitted for simplicity), we calculate the covariance matrix as C = XXT . Projecting the weight
matrix of a linear layer onto the covariance matrix of its input activation can accentuate the task-
specific components related to the input. Therefore, we apply SVD on the product of each weight
matrix and its corresponding covariance matrix, i.e., SVD(WFTC). The decomposed components
are ordered by the singular values that reflect their contributions to the task as characterized by C.
Such subspace enjoys a low-rank structure, which means that the the knowledge-aware abilities can
be condensed into the first several leading singular values and singular vectors, as will be shown
in Section 4.1. Hence, we filter out the task-irrelevant components by keeping only the first r
components, and then reconstruct the weight matrix through right multiplication with C−1, i.e., the
inverse of covariance matrix C. Consequently, the purified weight matrix can be formulated as:
W †
FT = SVDr(WFTC)C−1 =
r
X
i=1
σiuivT
i C−1,
(5)
where σi denotes the i-th singular value, ui and vi are the left and right singular vectors, respectively.
We use the purified fine-tuned weight matrix W †
FT to construct our task vector as:
∆WPAVE = W †
FT −WB = SVDr(WFTC)C−1 −WB,
(6)
where WB is the base model weights before fine-tuning, and ∆WPAVE represents our purified task
vector, which can be used for any task vector-based merging methods, such as Ties-Merging (Yadav
et al., 2023), Task Arithmetic (Ilharco et al., 2022), and EMR-Merging (Huang et al., 2024).
The effectiveness of CO-SVD in identifying task-specific directions of weight matrices has been
verified by Yang et al. (2024b) for model adaptation. In addition, C = XXT is a symmetric pos-
itive semi-definite matrix, which enables efficient inverse operation with numerical stability. When
C is not invertible, we repeatedly add positive elements on its diagonal until it is invertible. In exper-
iments, we show that our method can be integrated on various model merging methods to improve
the performance of the merged models.
3.3
SPECTRAL RANK ALLOCATION
5
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Algorithm 1 Spectral rank allocation strategy
Require: Individual models {M{W l
1}L
l=1, M{W l
2}L
l=1, · · · , M{W l
K}L
l=1}, data sampling from target tasks
{X1, X2, · · · , XK}, a preserved rank ratio ρ, a stopping ratio γ.
1: Feed Xi into M{W l
i }L
l=1. Calculate covariance matrices for all layers and for all models, {Cl
i}K,L
i,l , and
the rank of each layer {Rl}L
l=1.
2: for l = 1 to L do
3:
Apply SVD to W l
i Cl
i and compute {σl
i,j}K,Rl
i,j
, sort them in descending order.
4:
Normalize {σl
i,j}K,Rl
i,j
by the largest singular value, {sl
i,j}K,Rl
i,j
=
σl
i,j
σl
i,max
K,Rl
i,j
5:
Initialize {rl
i}K,L
i,l
= {Rl}K,L
i,l
6:
Initialize index set I = {1, 2, · · · , K}
7:
while
Pk
i=1 rl
i
KRl
> ρ do
8:
Pick up the smallest sl
i,rl
i across i ∈I, suppose it is sl
t,rl
i
9:
rl
t = rl
t −1
10:
if rl
t ≤γRl then
11:
remove t from I
12:
end if
13:
end while
14: end for
15: Output: the rank ratios for each individual model, {ρi}K
i =
n PL
l=1 rl
i
PL
l=1 Rl
oK
i .
We have introduced the construction of our purified task vector in Section 3.2. However, when
merging multiple fine-tuned models, assigning the rank r to each model is highly non-trivial. To
optimize rank allocation under a specified preserved rank ratio ρ, we propose a layer-wise optimiza-
tion framework that determines the preserved rank of linear layer l by minimizing a summation of
normalized activated pruning error from different models. This strategy enables an adaptive preser-
vation of task-specific knowledge during the merging process.
We begin by introduce the activated pruning error with rank rl
i as:
∥SVDrl
i(W l
i Cl
i) −W l
i Cl
i∥2
F ,
(7)
where i refers to the i-th model. In practice, we further normalize it with σmax(W l
i Cl
i) to ensure
comparability across different models with various spectral magnitudes. This normalization bounds
the scaled spectrum within a range from the inverse of its condition number to 1, thus enabling a fair
assignment of rank across models. The normalized activated pruning error is defined as:
∥SVDrl
i(W l
i Cl
i) −W l
i Cl
i∥2
F
(σmax(W l
i Cl
i))2
=
rl
i
X
j=1
(σj(W l
i Cl
i))2
(σmax(W l
i Cl
i))2 =
rl
i
X
j=1
σl
i,j
σl
i,max
!2
.
(8)
Building on this formulation, we define the overall objective for the l-th layer as:
min
PK
i=1 rl
i
KRl
<ρ
K
X
i=1
∥SVDrl
i(W l
i Cl
i) −W l
i Cl
i∥2
F
(σmax(W l
i Cl
i))2
,
(9)
where Rl refers to full rank. Minimizing the objective Eq. (9) can be effectively approached using a
greedy algorithm, and the whole procedure has been summarized in Algorithm 1. In this procedure,
for each layer l, we iteratively remove the component associated with the smallest normalized sin-
gular value, defined as sl
t,rl
i =
σl
i,j
σl
i,max , thereby achieving optimality of Eq. (9) through the greedy
process. To prevent excessive rank pruning that causes performance degradation, we introduce a
threshold γ which serves as a stopping ratio. This rank strategy enables spectrum-based rank allo-
cation for model merging while respecting the global rank constraint.
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96
192
384
512
Pruned ranks
0.5
0.6
0.7
0.8
0.9
Performance
SST-2
SVD
CO-SVD random
ASVD with SST-2
CO-SVD with SST-2
CO-SVD with COLA
96
192
384
512
Pruned ranks
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Performance
MRPC
SVD
CO-SVD random
ASVD with MRPC
CO-SVD with MRPC
CO-SVD with COLA
96
192
384
512
Pruned ranks
0.5
0.6
0.7
0.8
0.9
Performance
QQP
SVD
CO-SVD random
ASVD with QQP
CO-SVD with QQP
CO-SVD with COLA
96
192
384
512
Pruned ranks
0.3
0.4
0.5
0.6
0.7
0.8
Performance
MNLI
SVD
CO-SVD random
ASVD with MNLI
CO-SVD with MNLI
CO-SVD with COLA
96
192
384
512
Pruned ranks
0.5
0.6
0.7
0.8
0.9
Performance
QNLI
SVD
CO-SVD random
ASVD with QNLI
CO-SVD with QNLI
CO-SVD with COLA
96
192
384
512
Pruned ranks
0.45
0.50
0.55
0.60
0.65
0.70
0.75
Performance
RTE
SVD
CO-SVD random
ASVD with RTE
CO-SVD with RTE
CO-SVD with COLA
Figure 3: Model rank pruning with different decomposition methods. The results show that the
context-oriented decomposition (CO-SVD) is more effective in aligning the subspace with the target
knowledge. The pruned ranks, 96, 192, 384 and 512, correspond to removing 1/8, 1/4, 1/2 and 2/3
of the full rank, respectively.
4
EXPERIMENTS
In this section, we provide the results on the GLUE benchmarks using RoBERTa-base (Liu et al.,
2019) and DeBERTa-base (He et al., 2020), as well as the generation benchmarks with LLaMA2-7B-
hf (Touvron et al., 2023) on mathematical solving and code generation tasks. We evaluate PAVE with
various merging methods, including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav
et al., 2023), and EMR-Merging (Huang et al., 2024). For further detailed experimental settings and
hyperparameters, please refer to Appendix A.
4.1
EFFECT OF DECOMPOSITION IN KNOWLEDGE-AWARE SUBSPACES
We demonstrate the effectiveness of decomposition in knowledge-aware subspaces through a se-
ries of numerical experiments. As concluded in Figure 3, we prune individual RoBERTa models
fine-tuned on six GLUE tasks using various decomposition methods. The decomposition methods
mentioned in Figure 3 are summarized as follows: SVD (Meng et al., 2024) applies singular value
decomposition directly to the weights. ASVD (Yuan et al., 2023) introduces a diagonal scaling ma-
trix, where each diagonal entry corresponds to the average absolute value of the associated feature.
CO-SVD (Yang et al., 2024b) leverages covariance matrices. In the CO-SVD random variant, the
covariance matrix is replaced by a sampled random square matrix from [-1,1], serving as a task-
agnostic baseline. CO-SVD with CoLA derives the covariance matrix using samples from CoLA,
highlighting the impact of selecting samples from an appropriate task.
From Figure 3, it is evident that, for a given rank, decomposition methods that incorporate task
information (CO-SVD, ASVD) outperform those that do not (SVD, CO-SVD random). In par-
ticular, CO-SVD demonstrates superior performance compared to ASVD, as CO-SVD preserves
the context information more effectively through task-specific activations. This pattern holds con-
sistently across all datasets evaluated, reinforcing the importance of using the correct task data to
guide decomposition. Notably, performance advantages of CO-SVD become even more prominent
at pruning larger ranks, where other methods exhibit sharp degradation, indicating CO-SVD’s ef-
fectiveness in condensing knowledge awareness. It is noteworthy that the input must originate from
the corresponding task to construct knowledge-aware subspace. When using CoLA samples to gen-
erate the covariance matrix, performance drops significantly across all tasks. These results suggest
knowledge-awareness is crucial for successful rank preservation, and emphasize the importance of
introducing knowledge-aware subspaces for purifying task vectors.
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Table 1: Results of merging fine-tuned RoBERTa models on eight datasets from GLUE benchmark.
The number of examples used to generate context matrix is 4096.
Methods
CoLA
SST-2
MRPC
STSB
QQP
MNLI
QNLI
RTE
Average
Individual
60.18
94.04
89.22
90.63
91.41
87.20
92.71
79.06
85.55
EMR-Merging
39.96
93.35
86.27
82.77
89.72
85.45
89.57
74.37
80.18
EMR-Merging W/ DARE
40.58
93.23
86.52
82.86
89.93
85.98
89.68
75.09
80.48+0.30
EMR-Merging W/ PAVE
60.42
93.69
87.75
89.73
89.13
85.27
92.09
76.17
84.28+4.10
Task Arithmetic
18.78
85.89
79.90
74.03
83.78
59.08
69.67
62.09
66.65
Task Arithmetic W/ DARE
18.26
85.67
80.39
73.79
83.77
59.20
69.58
61.01
66.45−0.20
Task Arithmetic W/ PAVE
18.96
85.67
80.64
74.14
83.77
58.97
69.91
61.73
66.72+0.07
Ties-Merging
20.48
84.40
81.13
58.19
85.70
64.65
74.81
42.96
64.04
Ties-Merging W/ DARE
17.56
83.83
80.39
63.60
85.79
64.82
0.7324
44.77
64.25+0.21
Ties-Merging W/ PAVE
25.45
82.45
79.41
64.03
85.43
69.93
77.19
51.62
66.93+2.89
Table 2: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark.
The number of examples used to generate context matrix is 4096.
Methods
CoLA
SST-2
MRPC
STSB
QQP
MNLI
QNLI
RTE
Average
Individual
59.10
95.06
89.21
91.31
91.50
88.52
93.30
69.31
84.66
EMR-Merging
47.05
94.72
71.32
87.57
88.36
87.38
92.29
64.26
79.11
EMR-Merging W/ DARE
47.76
94.61
72.06
87.30
88.77
87.67
92.28
66.43
79.61+0.50
EMR-Merging W/ PAVE
61.32
94.50
84.31
89.54
90.34
87.50
93.01
60.65
82.64+3.53
Task Arithmetic
3.03
75.00
68.38
52.72
63.44
43.46
51.16
50.90
51.01
Task Arithmetic W/ DARE
2.84
68.81
68.38
54.24
64.38
42.23
55.15
49.46
50.68−0.33
Task Arithmetic W/ PAVE
2.18
75.46
68.38
55.90
64.00
43.48
61.25
51.99
52.83+1.82
Ties-Merging
5.36
66.28
68.63
19.09
63.68
40.44
55.39
48.74
45.95
Ties-Merging W/ DARE
6.03
56.42
68.87
0.61
66.28
42.11
54.00
51.26
43.19−2.76
Ties-Merging W/ PAVE
1.39
68.35
68.63
36.76
63.12
44.01
59.36
50.90
49.06+3.11
4.2
MODEL MERGING FOR GLUE TASKS
To evaluate the effectivness of PAVE, we first present results on GLUE tasks using RoBERTa as
the backbone model. The numerical results are summarized in Table 1. We observe that PAVE
yields a notable average performance gain of +4.10% when applied to EMR-Merging, reducing
the gap to just 1.27% from the individual models’ average performance. For specific tasks such
as CoLA, PAVE significantly improves the score from 39.96% to 60.42%, even surpassing the in-
dividual model’s score of 60.18%. These improvements are not limited to EMR-Merging. PAVE
also shows improvements with Ties-Merging and Task Arithmetic strategies. For instance, when
integrating on Ties-Merging, PAVE achieves an average gain of +2.89%, outperforming both the
baseline and the DARE-enhanced variant.
Similar to the experimental setting for RoBERTa, we evaluate our method on DeBERTa and present
the results in Table 2. PAVE cooperates effectively with EMR-Merging, achieving a substantial
average performance gain of +3.53%, narrowing the average gap to 2.02% compared to the aver-
aged performance (84.66%) of individual models. Notably, on CoLA, PAVE boosts performance
to a new high of 61.32%, even surpassing the individual model’s score of 59.10%, also achieving
an improvement of 14.27% over the EMR-Merging baseline. Beyond EMR-Merging, PAVE also
improves results in Task Arithmetic and Ties-Merging settings. In Task Arithmetic, it raises the
average performance by +1.82%, while in Ties-Merging, it achieves a considerable gain of +3.11%
over the original baseline.
4.3
MODEL MERGING FOR GENERATIVE TASKS
For generative tasks, we conduct experiments merging the math model WizardMath-7B-V1.0 (Luo
et al., 2023) and the code model LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). As shown
8
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Table 3: Results of merging fine-tuned LLaMA-2-7B models on Math and Code tasks. The number
of examples used to generate the context matrix is 512.
Methods
Mathematical Solving
Code Generating
GSM8K
MATH
Human Eval
MBPP
Individual Math
54.9
10.7
-
-
Individual Code
-
-
39.6
30.1
Average Merging
32.2
5.0
29.2
29.0
Average Merging w/ DARE
31.9−0.3
5.0+0.0
29.8+0.6
28.6−0.4
Average Merging w/ PAVE
32.4+0.2
5.0+0.0
31.1+1.9
29.0+0.0
Task Arithmetic
34.3
4.1
33.5
27.6
Task Arithmetic w/ DARE
32.3−2.0
4.4+0.3
32.3−1.2
28.6+1.0
Task Arithmetic w/ PAVE
34.5+0.2
4.2+0.1
35.4+1.9
28.0+0.4
Table 4: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark.
The number of examples used to generate context matrix is 4096.
Methods
CoLA
SST-2
MRPC
STSB
QQP
MNLI
QNLI
RTE
Average
Individual
59.10
95.06
89.21
91.31
91.50
88.52
93.30
69.31
84.66
EMR-Merging
47.05
94.72
71.32
87.57
88.36
87.38
92.29
64.26
79.11
EMR-Merging W/ PAVE plain
59.88
94.61
82.60
88.83
90.40
87.35
92.44
58.48
81.82
EMR-Merging W/ PAVE
61.32
94.50
84.31
89.54
90.34
87.50
93.01
60.65
82.64
Task Arithmetic
3.03
75.00
68.38
52.72
63.44
43.46
51.16
50.90
51.01
Task Arithmetic W/ PAVE plain
2.56
74.73
68.38
54.83
64.03
42.84
60.19
51.16
52.34
Task Arithmetic W/ PAVE
2.18
75.46
68.38
55.90
64.00
43.48
61.25
51.99
52.83
Ties-Merging
5.36
66.28
68.63
19.09
63.68
40.44
55.39
48.74
45.95
Ties-Merging W/ PAVE plain
1.04
70.07
68.87
31.33
63.16
43.89
57.79
50.18
48.29
Ties-Merging W/ PAVE
1.39
68.35
68.63
36.76
63.12
44.01
59.36
50.90
49.06
in Table 3, PAVE demonstrates a marked improvement of +1.9% on Human Eval compared to Aver-
age Merging and Task Arithmetic. Notably, PAVE achieves the highest Human Eval score (35.4%)
among all merging strategies, indicating its effectiveness of our purified task vectors in code gener-
ation tasks. In the mathematical solving tasks, PAVE consistently maintains or modestly improves
performance across both GSM8K (+0.2%) and MATH (+0.1%) compared to the Task Arithmetic
baseline. It is worth noting that, in contrast to the randomness in DARE, PAVE is more stable and
practical, as none of the merged models show any degradation on the individual tasks. This sta-
bility, combined with consistent improvements across multiple tasks, highlights the robustness and
practical advantage of PAVE.
4.4
ABLATION STUDIES
We evaluate our rank allocation strategy Algorithm 1 in Table 4, using the same DeBERTa ex-
periment settings on models and merging methods in Section 4.2. The plain method uses a fixed
preserved rank ratio across all layers and models. Compared to this baseline, our adaptive strategy
consistently improves performance across different merging settings. In EMR-Merging, the rank
strategy improves the average performance from 81.82% to 82.64% (+0.82%). In other merging
methods like Task Arithmetic and Ties-Merging, the gains are +0.49% (from 52.34% to 52.83%)
and +0.77% (from 48.29% to 49.06%), respectively. These results suggest that uniform pruning
represents a naive and suboptimal approach, while our rank allocation strategy enables better per-
formance under the same pruning budget. Additionally, we include further experiments, presenting
numerical results when PAVE is integrated with alternative decomposition methods, as well as an
analysis of sample size and runtime performance. Please refer to Appendix B for the results.
9
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5
CONCLUSION
In this paper, we propose the purified task vector (PAVE) as a plug-in method for task vector-based
merging approaches. PAVE employs context-oriented decomposition to identify and prune redun-
dancies of the task vectors in knowledge-aware subspace, thereby mitigating conflicts arising from
redundancy. To further optimize the rank allocation across different fine-tuned models and improve
the merging performance, we introduce a spectral rank allocation strategy that optimizes the sum-
mation of normalized activated pruning errors. In experiment, we demonstrate the effectiveness
of PAVE with a wide range of merging methods, tasks and model architectures. Furthermore, we
explore the integration of alternative decomposition techniques within the PAVE framework and
confirm its compatibility with existing model merging approaches.
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A
IMPLEMENTATION DETAILS
A.1
MODEL MERGING ON GLUE BENCHMARK
In Section 4.2, we employ RoBERTa-base (Liu et al., 2019) and DeBERTa-base (He et al., 2020) as
the base model. The performance of each method is evaluated by eight tasks in the GLUE bench-
mark tasks, including CoLA (Warstadt et al., 2019), SST-2 (Socher et al., 2013), MRPC (Dolan
& Brockett, 2005), STS-B (Cer et al., 2017), QQP (Iyer et al., 2017), MNLI (Williams et al.,
2017), QNLI (Rajpurkar et al., 2016), RTE (Giampiccolo et al., 2007). For evaluation, CoLA is
assessed using the Matthews correlation coefficient, STS-B is measured by the average of Pearson
and Spearman correlation coefficients, and the remaining tasks are evaluated based on accuracy. The
RoBERTa checkpoints used for model merging are publicly available via the links in (Huang et al.,
2024). The DeBERTa checkpoints are obtained by training DeBERTa-base individually on each
GLUE task for 5 epochs using the AdamW (Loshchilov & Hutter, 2017) optimizer with a learning
rate of 2 × 10−5, a warm-up phase of 100 steps, and a weight decay of 10−2.
Following the setting in Huang et al. (2024), we report the numerical experiments of PAVE with
various merging methods including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav
et al., 2023) and EMR-Merging (Huang et al., 2024). We also evaluate these methods with plug-in
method DARE (Yu et al., 2024). Following the standard setting of each method, the hyperparameter
λ in Task Arithmetic and Ties-Merging is selected from {0.1, 0.3, 0.5, 0.7 0.9} based on the best
performance. Specifically, for the RoBERTa experiments, we set λ = 0.3 for Task Arithmetic and
λ = 0.9 for Ties-Merging; for BERT experiments, we set λ = 0.1 and λ = 0.5, respectively. The
dropout ratio for DARE is fixed to 0.1 to avoid instability. The preserved rank ratio ρ of PAVE is
chosen from { 7
8, 15
16, 31
32, 63
64}, and the stopping ratio is set to γ = ρ −1−ρ
2 .
To attain optimal performance on GLUE benchmarks, PAVE progressively progressively assigns full
rank to ∆WPAVE according to the performance of the plain merging method. This design reflects the
observation that certain tasks are more sensitive and cannot tolerate aggressive pruning. At each
step, we apply full rank to the the task whose performance is furthest from that of its individual
model. This process continues until all task vectors are full rank. As a result, we obtain K merged
models and select the one with the highest average performance. Since model merging is not com-
putationally expensive and requires no training, the computational cost of such procedure remains
acceptable.
A.2
MODEL MERGING ON GENERATION BENCHMARKS
Following the setup of DARE presented in Yu et al. (2024), we evaluate PAVE on mathematical
solving and code generation tasks, as described in Section 4.3. For mathematical solving, we con-
sider GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021), while for code generation,
we use HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021). We adopt the LLaMA-
2-7B (Touvron et al., 2023) model as the base model. For mathematical solving, we utilize the
fine-tuned model WizardMath-7B-V1.0 (Luo et al., 2023), while for code generation, we employ
LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). Additionally, code samples are drawn
from the CodeFeedback dataset (Zheng et al., 2024) to support covariance matrix generation. The
Task Arithmetic hyperparameter λ is selected from the set {0.5, 1}, and the dropout ratio for DARE
is fixed at 0.2. The pruning rank for PAVE is chosen from {4, 6, 8, 16}.
B
ADDITIONAL ANALYSIS AND RESULTS
B.1
TASK VECTOR WITH VARIOUS DECOMPOSITION METHODS
In this section, we present numerical experiments evaluating the performance of various decompo-
sition methods applied to our decomposition-based task vector. The setting of models and merging
methods follow the DeBERTa settings in Section 4.2. The evaluated decomposition methods in-
clude: The basic SVD method (Meng et al., 2024), directly approximates the fine-tuned weights
via a plain SVD decomposition. ASVD (Yuan et al., 2023), incorporates an input-dependent di-
agonal scaling matrix derived from the average absolute values of the previous layer’s outputs.
SVD-LLM (Wang et al., 2024b), applies Cholesky decomposition to the previous layer’s output.
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Table 5: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark.
The number of examples used to generate context matrix is 4096.
Methods
CoLA
SST-2
MRPC
STSB
QQP
MNLI
QNLI
RTE
Average
Individual
60.18
94.04
89.22
90.63
91.41
87.20
92.71
79.06
85.55
EMR-Merging
47.05
94.72
71.32
87.57
88.36
87.38
92.29
64.26
79.11
EMR-Merging W/ SVD
62.20
93.58
83.82
89.40
89.45
76.72
92.53
58.48
80.77
EMR-Merging W/ ASVD
62.33
95.07
84.56
89.73
89.98
87.44
92.86
61.01
82.87
EMR-Merging W/ SVD-LLM
61.82
94.72
84.07
89.55
90.10
87.44
92.90
61.37
82.74
EMR-Merging W/ PAVE
61.32
94.50
84.31
89.54
90.34
87.50
93.01
60.65
82.64
Task Arithmetic
3.03
75.00
68.38
52.72
63.44
43.46
51.16
50.90
51.01
Task Arithmetic W/ SVD
2.96
70.30
68.38
37.90
64.30
42.88
67.01
53.07
50.85
Task Arithmetic W/ ASVD
2.53
76.03
66.38
55.60
64.08
43.23
61.27
51.62
52.59
Task Arithmetic W/ SVD-LLM
2.74
74.89
68.38
55.48
64.02
43.54
61.45
51.62
52.76
Task Arithmetic W/ PAVE
2.18
75.46
68.38
55.90
64.00
43.48
61.25
51.99
52.83
Ties-Merging
5.36
66.28
68.63
19.09
63.68
40.44
55.39
48.74
45.95
Ties-Merging W/ SVD
-3.29
53.90
68.38
6.15
62.98
35.34
54.22
52.71
41.29
Ties-Merging W/ ASVD
-1.74
66.86
68.63
17.19
62.84
40.33
53.10
46.57
44.22
Ties-Merging W/ SVD-LLM
1.13
67.89
68.63
23.33
63.09
43.02
57.84
48.74
46.70
Ties-Merging W/ PAVE
1.39
68.35
68.63
36.76
63.12
44.01
59.36
50.90
49.06
64
128
256
512
1024
2048
4096
Samples
88.7
88.8
88.9
89.0
89.1
89.2
Performance
EMR-Merging W/ PAVE
Std
EMR-Merging
64
128
256
512
1024
2048
4096
Samples
2.0
4.0
6.0
8.0
10.0
12.0
14.0
Time (min)
PAVE Runtime
Figure 4: Performance and runtime analysis of PAVE across different sample sizes. Left: Mean per-
formance (solid line) and standard deviation (shaded region) of EMR-Merging with PAVE, showing
consistent improvements as more samples are used. Right: PAVE runtime grows with number of
samples.
CO-SVD (Yang et al., 2024b), as the default decomposition method embedded in PAVE, utilizes the
covariance matrix of the output of the previous layer as the activation matrix. Among the evaluated
methods, CO-SVD demonstrates the most consistent performance on Task Arithmetic and Ties-
Merging, while also exhibiting performance comparable to EMR-Merging. These results highlight
the strong potential of our decomposition-based purified task vector framework, which is designed
to integrate seamlessly with a variety of decomposition methods.
B.2
ANALYSIS OF SAMPLE SIZE AND TIME EFFICIENCY
We next analyze the impact of sample size on the stability, efficiency, and reproducibility of PAVE.
As illustrated in Figure 4, increasing the sample size consistently reduces the variance and im-
proves the overall performance of EMR-Merging with PAVE, highlighting its stability and robust-
ness. When the sample size is very small, due to the regularization, the method degenerates to plain
SVD, which explains the lower performance observed in this regime. Importantly, we observe that
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Table 6: Results of merging fine-tuned ViT models on 8 image classification benchmarks. The
number of samples used to generate context matrix is 4096.
Methods
SUN397
Cars
RESISC45
EuroSAT
SVHN
GTSRB
MNIST
DTD
Average
Individual ViT-B/32
75.3
77.7
96.1
99.7
97.5
98.7
99.7
79.4
90.5
EMR-Merging
75.2
72.8
93.5
99.5
96.9
98.1
99.6
74.4
88.7
EMR-Merging W/PAVE
76.3
74.3
93.8
99.3
96.7
97.6
99.6
76.1
89.2+0.5
Individual ViT-B/16
79.0
87.0
96.4
99.1
97.7
99.0
99.7
82.3
92.5
EMR-Merging
78.5
82.6
95.4
99.1
97.6
98.8
99.6
78.3
91.2
EMR-Merging W/PAVE
80.2
82.8
96.1
99.2
97.6
98.7
99.6
78.9
91.7+0.5
Individual ViT-L/14
82.3
92.4
97.4
100
98.1
99.2
99.7
84.1
94.2
EMR-Merging
83.1
90.7
96.7
99.7
97.9
99.1
99.6
82.7
93.7
EMR-Merging W/PAVE
83.6
91.7
97.2
99.8
99.7
98.9
99.7
82.2
93.9+0.2
even a moderate sample size (256–512) is sufficient to yield substantial performance gains, offering
a practical trade-off between accuracy and computational cost. This finding suggests that PAVE
can be effectively deployed in practice without incurring excessive runtime overhead, while still
preserving most of the performance benefits of using larger sample sizes. Throughout our numer-
ical experiments, we increased the number of samples up to 4096 ensure a stable and consistent
demonstration of our method’s performance. Employing larger sample sizes also helps ensure the
reproducibility of our results. All runtime measurements were conducted on a workstation equipped
with a single NVIDIA A100 GPU and 8 CPUs, with 512 GB of RAM.
B.3
MODEL MERGING FOR VIT MODELS
Settings. Following the setup of EMR-Merging presented in Huang et al. (2024), we employ ViT-
B/32, ViT-B/16 and ViT-L/14 (Radford et al., 2021) as the base model to evaluate the performance
of PAVE across 8 image classification tasks: SUN397 (Xiao et al., 2010), Cars Krause et al. (2013),
RESISC45 (Cheng et al., 2017), EuroSAT (Helber et al., 2019), SVHN (Netzer et al., 2011), GT-
SRB (Stallkamp et al., 2011), MNIST (LeCun, 1998), and DTD (Cimpoi et al., 2014). The preserved
rank ratio ρ of PAVE is chosen from { 7
8, 15
16, 31
32, 63
64}, and the stopping ratio is set to γ = ρ −1−ρ
2 .
The checkpoints are publicly accessible via the link provided in Ilharco et al. (2022).
Results. As shown in Table 6, while the plain EMR-Merging method already demonstrates strong
performance, PAVE still consistently enhances the performance across all ViT variants. For ViT-
B/32, PAVE improves the average accuracy by +0.5% (from 88.7% to 89.2). On the more capable
ViT-B/16 model, PAVE again contributes an improvement of +0.5%, increasing the average from
91.2% to 91.7%.
17
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Preprint PURIFYING TASK VECTORS IN KNOWLEDGE-AWARE SUBSPACE FOR MODEL MERGING Bang An1 Yibo Yang1∗ Philip Torr2 Bernard Ghanem1 1King Abdullah (KAUST) 2 {bang.an, yibo.yang, ABSTRACT Model merging aims to integrate task-specific abilities from individually finetuned models into a single model without extra training. In recent model merging methods, task vector has become a fundamental building block, as it can encapsulate the residual information from finetuning. However, the merged model often suffers from notable performance degradation due to the conflicts caused by task-irrelevant redundancy in task vectors. Existing efforts in overcoming redundancy by randomly dropping elements in the parameter space involves randomness and lacks knowledge awareness. To address these challenges, in this study, we propose Purifying TAsk VEctors (PAVE) in knowledge-aware subspace. Concretely, we sample some training examples from each task, and feed them into their corresponding fine-tuned models to acquire the covariance matrices before linear layers. We then perform a context-oriented singular value decomposition, which accentuates the weight components most relevant to the target knowledge. As a result, we can split fine-tuned model weights into taskrelevant and redundant components in the knowledge-aware subspace, and purify the task vector by pruning the redundant components. To induce fair pruning efforts across models, we further introduce a spectral rank allocation strategy by optimizing a normalized activated pruning error. The task vector purification by our method as a plug-and-play scheme is applicable across various task vector-based merging methods to improve their performance. In experiments, we demonstrate the effectiveness of PAVE across a diverse set of merging methods, tasks, and model architectures. Remarkably, when integrated on the state-of-the-art merging method EMR-Merging with RoBERTa, our method yields a significant performance improvement of 4.1% (from 80.18% w/o PAVE to 84.28%) on the GLUE benchmark, approaching to the average performance of 8 individual models, 85.55%. Our code and models are released at: https: //github.com/ABgit111/purified_task_vector. 1 INTRODUCTION Transformer-based models, including large language models (LLMs) (Devlin et al., 2019; Liu et al., 2019; He et al., 2020; Radford et al., 2019; Touvron et al., 2023; Achiam et al., 2023) and vision models (Dosovitskiy et al., 2020; Radford et al., 2021) have found widespread applications across various domains. Supervised fine-tuning (Radford et al., 2018; Hu et al., 2022) is a common technique to enhance downstream task performance. Despite the effectiveness of multi-task learning (MTL) in acquiring versatile abilities across multiple domains (Caruana, 1997; Zhang & Yang, 2018; Vandenhende et al., 2021), its applicability will be limited when only individually fine-tuned models are accessible due to data privacy concerns or computational constraints. To address this challenge, model merging has been proposed, aiming to construct a single model by combining parameters from domain-specific models, without relying on any training or fine-tuning efforts. Early model merging methods focus on performing weighted (Matena & Raffel, 2022) or regularized (Jin et al., 2022) averaging in the parameter space of fine-tuned models, yet the conflicts inherent in ∗Corresponding author 1 16 Oct 2025 Preprint Knowledge-aware subspace ∆W= W!" -W# Task vector Naive DARE PAVE (ours) Formulation Address redundancy Knowledge awareness ∆W W!"CC%& Redundancy w.r.t. the target knowledge ❌ ✅(in parameter space) ✅(in subspace) ❌ ❌ ✅ Target knowledge as characterized by C ➕ SVD "! SVD W!C! W#C# W W! W# W Σ! Σ# Σ % U![: r!] U#[: r#] U ] Σ![: r!] Σ#[: r#] Σ ] V! %[: r!] V# %[: r#] V ] C! C# C C "! ⋮ ⋮ ⋮ ⋮ ⋮ Truncate Merging Method W' Task 2 Task K Task 1 sample sample sample Figure 2: An overall illustration of our proposed method, PAVE, which serves as a plug-and-play method for task-vector based merging methods. It begins by identifying task-aware components through a decomposition approach, and then eliminates noise and less effective components via rank-based pruning. A rank allocation strategy is designed to optimize the preserved rank ratios across individual models. Overall, rather than employing the plain task vector Wk -WB, PAVE performs model merging based on purified task vectors SVDrk(WkCk)C-1 k -WB, 1 ≤k ≤K. reflects the spectral distribution of the parameter matrix itself and pre-trained LLM weights handle a broad range of tasks. To this end, we leverage the context-oriented singular value decomposition (CO-SVD) (Yang et al., 2024b) to decompose fine-tuned weights into a knowledge-aware subspace. Concretely, for each task Tk, 1 ≤k ≤K, we sample some training examples from this task and feed them into the fine-tuned model Mk. After acquiring the input activation of each layer X (layer index is omitted for simplicity), we calculate the covariance matrix as C = XXT . Projecting the weight matrix of a linear layer onto the covariance matrix of its input activation can accentuate the taskspecific components related to the input. Therefore, we apply SVD on the product of each weight matrix and its corresponding covariance matrix, i.e., SVD(WFTC). The decomposed components are ordered by the singular values that reflect their contributions to the task as characterized by C. Such subspace enjoys a low-rank structure, which means that the the knowledge-aware abilities can be condensed into the first several leading singular values and singular vectors, as will be shown in Section 4.1. Hence, we filter out the task-irrelevant components by keeping only the first r components, and then reconstruct the weight matrix through right multiplication with C-1, i.e., the inverse of covariance matrix C. Consequently, the purified weight matrix can be formulated as: W † FT = SVDr(WFTC)C-1 = r X i=1 σiuivT i C-1, (5) where σi denotes the i-th singular value, ui and vi are the left and right singular vectors, respectively. We use the purified fine-tuned weight matrix W † FT to construct our task vector as: ∆WPAVE = W † FT -WB = SVDr(WFTC)C-1 -WB, (6) where WB is the base model weights before fine-tuning, and ∆WPAVE represents our purified task vector, which can be used for any task vector-based merging methods, such as Ties-Merging (Yadav et al., 2023), Task Arithmetic (Ilharco et al., 2022), and EMR-Merging (Huang et al., 2024). The effectiveness of CO-SVD in identifying task-specific directions of weight matrices has been verified by Yang et al. (2024b) for model adaptation. In addition, C = XXT is a symmetric positive semi-definite matrix, which enables efficient inverse operation with numerical stability. When C is not invertible, we repeatedly add positive elements on its diagonal until it is invertible. In experiments, we show that our method can be integrated on various model merging methods to improve the performance of the merged models. 3.3 SPECTRAL RANK ALLOCATION 5 Preprint Algorithm 1 Spectral rank allocation strategy Require: Individual models {M{W l 1}L l=1, M{W l 2}L l=1, · · · , M{W l K}L l=1}, data sampling from target tasks {X1, X2, · · · , XK}, a preserved rank ratio ρ, a stopping ratio γ. 1: Feed Xi into M{W l i }L l=1. Calculate covariance matrices for all layers and for all models, {Cl i}K,L i,l , and the rank of each layer {Rl}L l=1. 2: for l = 1 to L do 3: Apply SVD to W l i Cl i and compute {σl i,j}K,Rl i,j , sort them in descending order. 4: Normalize {σl i,j}K,Rl i,j by the largest singular value, {sl i,j}K,Rl i,j = σl i,j σl i,max K,Rl i,j 5: Initialize {rl i}K,L i,l = {Rl}K,L i,l 6: Initialize index set I = {1, 2, · · · , K} 7: while Pk i=1 rl i KRl > ρ do 8: Pick up the smallest sl i,rl i across i ∈I, suppose it is sl t,rl i 9: rl t = rl t -1 10: if rl t ≤γRl then 11: remove t from I 12: end if 13: end while 14: end for 15: Output: the rank ratios for each individual model, {ρi}K i = n PL l=1 rl i PL l=1 Rl oK i . We have introduced the construction of our purified task vector in Section 3.2. However, when merging multiple fine-tuned models, assigning the rank r to each model is highly non-trivial. To optimize rank allocation under a specified preserved rank ratio ρ, we propose a layer-wise optimization framework that determines the preserved rank of linear layer l by minimizing a summation of normalized activated pruning error from different models. This strategy enables an adaptive preservation of task-specific knowledge during the merging process. We begin by introduce the activated pruning error with rank rl i as: ∥SVDrl i(W l i Cl i) -W l i Cl i∥2 F , (7) where i refers to the i-th model. In practice, we further normalize it with σmax(W l i Cl i) to ensure comparability across different models with various spectral magnitudes. This normalization bounds the scaled spectrum within a range from the inverse of its condition number to 1, thus enabling a fair assignment of rank across models. The normalized activated pruning error is defined as: ∥SVDrl i(W l i Cl i) -W l i Cl i∥2 F (σmax(W l i Cl i))2 = rl i X j=1 (σj(W l i Cl i))2 (σmax(W l i Cl i))2 = rl i X j=1 σl i,j σl i,max !2 . (8) Building on this formulation, we define the overall objective for the l-th layer as: min PK i=1 rl i KRl <ρ K X i=1 ∥SVDrl i(W l i Cl i) -W l i Cl i∥2 F (σmax(W l i Cl i))2 , (9) where Rl refers to full rank. Minimizing the objective Eq. (9) can be effectively approached using a greedy algorithm, and the whole procedure has been summarized in Algorithm 1. In this procedure, for each layer l, we iteratively remove the component associated with the smallest normalized singular value, defined as sl t,rl i = σl i,j σl i,max , thereby achieving optimality of Eq. (9) through the greedy process. To prevent excessive rank pruning that causes performance degradation, we introduce a threshold γ which serves as a stopping ratio. This rank strategy enables spectrum-based rank allocation for model merging while respecting the global rank constraint. 6 Preprint 96 192 384 512 Pruned ranks 0.5 0.6 0.7 0.8 0.9 Performance SST-2 SVD CO-SVD random ASVD with SST-2 CO-SVD with SST-2 CO-SVD with COLA 96 192 384 512 Pruned ranks 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Performance MRPC SVD CO-SVD random ASVD with MRPC CO-SVD with MRPC CO-SVD with COLA 96 192 384 512 Pruned ranks 0.5 0.6 0.7 0.8 0.9 Performance QQP SVD CO-SVD random ASVD with QQP CO-SVD with QQP CO-SVD with COLA 96 192 384 512 Pruned ranks 0.3 0.4 0.5 0.6 0.7 0.8 Performance MNLI SVD CO-SVD random ASVD with MNLI CO-SVD with MNLI CO-SVD with COLA 96 192 384 512 Pruned ranks 0.5 0.6 0.7 0.8 0.9 Performance QNLI SVD CO-SVD random ASVD with QNLI CO-SVD with QNLI CO-SVD with COLA 96 192 384 512 Pruned ranks 0.45 0.50 0.55 0.60 0.65 0.70 0.75 Performance RTE SVD CO-SVD random ASVD with RTE CO-SVD with RTE CO-SVD with COLA Figure 3: Model rank pruning with different decomposition methods. The results show that the context-oriented decomposition (CO-SVD) is more effective in aligning the subspace with the target knowledge. The pruned ranks, 96, 192, 384 and 512, correspond to removing 1/8, 1/4, 1/2 and 2/3 of the full rank, respectively. 4 EXPERIMENTS In this section, we provide the results on the GLUE benchmarks using RoBERTa-base (Liu et al., 2019) and DeBERTa-base (He et al., 2020), as well as the generation benchmarks with LLaMA2-7Bhf (Touvron et al., 2023) on mathematical solving and code generation tasks. We evaluate PAVE with various merging methods, including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav et al., 2023), and EMR-Merging (Huang et al., 2024). For further detailed experimental settings and hyperparameters, please refer to Appendix A. 4.1 EFFECT OF DECOMPOSITION IN KNOWLEDGE-AWARE SUBSPACES We demonstrate the effectiveness of decomposition in knowledge-aware subspaces through a series of numerical experiments. As concluded in Figure 3, we prune individual RoBERTa models fine-tuned on six GLUE tasks using various decomposition methods. The decomposition methods mentioned in Figure 3 are summarized as follows: SVD (Meng et al., 2024) applies singular value decomposition directly to the weights. ASVD (Yuan et al., 2023) introduces a diagonal scaling matrix, where each diagonal entry corresponds to the average absolute value of the associated feature. CO-SVD (Yang et al., 2024b) leverages covariance matrices. In the CO-SVD random variant, the covariance matrix is replaced by a sampled random square matrix from [-1,1], serving as a taskagnostic baseline. CO-SVD with CoLA derives the covariance matrix using samples from CoLA, highlighting the impact of selecting samples from an appropriate task. From Figure 3, it is evident that, for a given rank, decomposition methods that incorporate task information (CO-SVD, ASVD) outperform those that do not (SVD, CO-SVD random). In particular, CO-SVD demonstrates superior performance compared to ASVD, as CO-SVD preserves the context information more effectively through task-specific activations. This pattern holds consistently across all datasets evaluated, reinforcing the importance of using the correct task data to guide decomposition. Notably, performance advantages of CO-SVD become even more prominent at pruning larger ranks, where other methods exhibit sharp degradation, indicating CO-SVD's effectiveness in condensing knowledge awareness. It is noteworthy that the input must originate from the corresponding task to construct knowledge-aware subspace. When using CoLA samples to generate the covariance matrix, performance drops significantly across all tasks. These results suggest knowledge-awareness is crucial for successful rank preservation, and emphasize the importance of introducing knowledge-aware subspaces for purifying task vectors. 7 Preprint Table 1: Results of merging fine-tuned RoBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 60.18 94.04 89.22 90.63 91.41 87.20 92.71 79.06 85.55 EMR-Merging 39.96 93.35 86.27 82.77 89.72 85.45 89.57 74.37 80.18 EMR-Merging W/ DARE 40.58 93.23 86.52 82.86 89.93 85.98 89.68 75.09 80.48+0.30 EMR-Merging W/ PAVE 60.42 93.69 87.75 89.73 89.13 85.27 92.09 76.17 84.28+4.10 Task Arithmetic 18.78 85.89 79.90 74.03 83.78 59.08 69.67 62.09 66.65 Task Arithmetic W/ DARE 18.26 85.67 80.39 73.79 83.77 59.20 69.58 61.01 66.45-0.20 Task Arithmetic W/ PAVE 18.96 85.67 80.64 74.14 83.77 58.97 69.91 61.73 66.72+0.07 Ties-Merging 20.48 84.40 81.13 58.19 85.70 64.65 74.81 42.96 64.04 Ties-Merging W/ DARE 17.56 83.83 80.39 63.60 85.79 64.82 0.7324 44.77 64.25+0.21 Ties-Merging W/ PAVE 25.45 82.45 79.41 64.03 85.43 69.93 77.19 51.62 66.93+2.89 Table 2: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 59.10 95.06 89.21 91.31 91.50 88.52 93.30 69.31 84.66 EMR-Merging 47.05 94.72 71.32 87.57 88.36 87.38 92.29 64.26 79.11 EMR-Merging W/ DARE 47.76 94.61 72.06 87.30 88.77 87.67 92.28 66.43 79.61+0.50 EMR-Merging W/ PAVE 61.32 94.50 84.31 89.54 90.34 87.50 93.01 60.65 82.64+3.53 Task Arithmetic 3.03 75.00 68.38 52.72 63.44 43.46 51.16 50.90 51.01 Task Arithmetic W/ DARE 2.84 68.81 68.38 54.24 64.38 42.23 55.15 49.46 50.68-0.33 Task Arithmetic W/ PAVE 2.18 75.46 68.38 55.90 64.00 43.48 61.25 51.99 52.83+1.82 Ties-Merging 5.36 66.28 68.63 19.09 63.68 40.44 55.39 48.74 45.95 Ties-Merging W/ DARE 6.03 56.42 68.87 0.61 66.28 42.11 54.00 51.26 43.19-2.76 Ties-Merging W/ PAVE 1.39 68.35 68.63 36.76 63.12 44.01 59.36 50.90 49.06+3.11 4.2 MODEL MERGING FOR GLUE TASKS To evaluate the effectivness of PAVE, we first present results on GLUE tasks using RoBERTa as the backbone model. The numerical results are summarized in Table 1. We observe that PAVE yields a notable average performance gain of +4.10% when applied to EMR-Merging, reducing the gap to just 1.27% from the individual models' average performance. For specific tasks such as CoLA, PAVE significantly improves the score from 39.96% to 60.42%, even surpassing the individual model's score of 60.18%. These improvements are not limited to EMR-Merging. PAVE also shows improvements with Ties-Merging and Task Arithmetic strategies. For instance, when integrating on Ties-Merging, PAVE achieves an average gain of +2.89%, outperforming both the baseline and the DARE-enhanced variant. Similar to the experimental setting for RoBERTa, we evaluate our method on DeBERTa and present the results in Table 2. PAVE cooperates effectively with EMR-Merging, achieving a substantial average performance gain of +3.53%, narrowing the average gap to 2.02% compared to the averaged performance (84.66%) of individual models. Notably, on CoLA, PAVE boosts performance to a new high of 61.32%, even surpassing the individual model's score of 59.10%, also achieving an improvement of 14.27% over the EMR-Merging baseline. Beyond EMR-Merging, PAVE also improves results in Task Arithmetic and Ties-Merging settings. In Task Arithmetic, it raises the average performance by +1.82%, while in Ties-Merging, it achieves a considerable gain of +3.11% over the original baseline. 4.3 MODEL MERGING FOR GENERATIVE TASKS For generative tasks, we conduct experiments merging the math model WizardMath-7B-V1.0 (Luo et al., 2023) and the code model LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). As shown 8 Preprint Table 3: Results of merging fine-tuned LLaMA-2-7B models on Math and Code tasks. The number of examples used to generate the context matrix is 512. Methods Mathematical Solving Code Generating GSM8K MATH Human Eval MBPP Individual Math 54.9 10.7 - - Individual Code - - 39.6 30.1 Average Merging 32.2 5.0 29.2 29.0 Average Merging w/ DARE 31.9-0.3 5.0+0.0 29.8+0.6 28.6-0.4 Average Merging w/ PAVE 32.4+0.2 5.0+0.0 31.1+1.9 29.0+0.0 Task Arithmetic 34.3 4.1 33.5 27.6 Task Arithmetic w/ DARE 32.3-2.0 4.4+0.3 32.3-1.2 28.6+1.0 Task Arithmetic w/ PAVE 34.5+0.2 4.2+0.1 35.4+1.9 28.0+0.4 Table 4: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 59.10 95.06 89.21 91.31 91.50 88.52 93.30 69.31 84.66 EMR-Merging 47.05 94.72 71.32 87.57 88.36 87.38 92.29 64.26 79.11 EMR-Merging W/ PAVE plain 59.88 94.61 82.60 88.83 90.40 87.35 92.44 58.48 81.82 EMR-Merging W/ PAVE 61.32 94.50 84.31 89.54 90.34 87.50 93.01 60.65 82.64 Task Arithmetic 3.03 75.00 68.38 52.72 63.44 43.46 51.16 50.90 51.01 Task Arithmetic W/ PAVE plain 2.56 74.73 68.38 54.83 64.03 42.84 60.19 51.16 52.34 Task Arithmetic W/ PAVE 2.18 75.46 68.38 55.90 64.00 43.48 61.25 51.99 52.83 Ties-Merging 5.36 66.28 68.63 19.09 63.68 40.44 55.39 48.74 45.95 Ties-Merging W/ PAVE plain 1.04 70.07 68.87 31.33 63.16 43.89 57.79 50.18 48.29 Ties-Merging W/ PAVE 1.39 68.35 68.63 36.76 63.12 44.01 59.36 50.90 49.06 in Table 3, PAVE demonstrates a marked improvement of +1.9% on Human Eval compared to Average Merging and Task Arithmetic. Notably, PAVE achieves the highest Human Eval score (35.4%) among all merging strategies, indicating its effectiveness of our purified task vectors in code generation tasks. In the mathematical solving tasks, PAVE consistently maintains or modestly improves performance across both GSM8K (+0.2%) and MATH (+0.1%) compared to the Task Arithmetic baseline. It is worth noting that, in contrast to the randomness in DARE, PAVE is more stable and practical, as none of the merged models show any degradation on the individual tasks. This stability, combined with consistent improvements across multiple tasks, highlights the robustness and practical advantage of PAVE. 4.4 ABLATION STUDIES We evaluate our rank allocation strategy Algorithm 1 in Table 4, using the same DeBERTa experiment settings on models and merging methods in Section 4.2. The plain method uses a fixed preserved rank ratio across all layers and models. Compared to this baseline, our adaptive strategy consistently improves performance across different merging settings. In EMR-Merging, the rank strategy improves the average performance from 81.82% to 82.64% (+0.82%). In other merging methods like Task Arithmetic and Ties-Merging, the gains are +0.49% (from 52.34% to 52.83%) and +0.77% (from 48.29% to 49.06%), respectively. These results suggest that uniform pruning represents a naive and suboptimal approach, while our rank allocation strategy enables better performance under the same pruning budget. Additionally, we include further experiments, presenting numerical results when PAVE is integrated with alternative decomposition methods, as well as an analysis of sample size and runtime performance. Please refer to Appendix B for the results. 9 Preprint 5 CONCLUSION In this paper, we propose the purified task vector (PAVE) as a plug-in method for task vector-based merging approaches. 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Yibo Yang, Xiaojie Li, Zhongzhu Zhou, Shuaiwen Song, Jianlong Wu, Liqiang Nie, and Bernard Ghanem. Corda: Context-oriented decomposition adaptation of large language models for taskaware parameter-efficient fine-tuning. Advances in Neural Information Processing Systems, 37: 71768-71791, 2024b. Yibo Yang, Sihao Liu, Chuan Rao, Bang An, Tiancheng Shen, Philip HS Torr, Ming-Hsuan Yang, and Bernard Ghanem. Dynamic context-oriented decomposition for task-aware low-rank adaptation with less forgetting and faster convergence. arXiv preprint , 2025. Le Yu, Bowen Yu, Haiyang Yu, Fei Huang, and Yongbin Li. Language models are super mario: Absorbing abilities from homologous models as a free lunch. In Forty-first International Conference on Machine Learning, 2024. 13 Preprint Zhihang Yuan, Yuzhang Shang, Yue Song, Qiang Wu, Yan Yan, and Guangyu Sun. Asvd: Activation-aware singular value decomposition for compressing large language models. arXiv preprint , 2023. Qingru Zhang, Minshuo Chen, Alexander Bukharin, Nikos Karampatziakis, Pengcheng He, Yu Cheng, Weizhu Chen, and Tuo Zhao. Adalora: Adaptive budget allocation for parameterefficient fine-tuning. arXiv preprint , 2023. Yu Zhang and Qiang Yang. An overview of multi-task learning. National Science Review, 5(1): 30-43, 2018. Tianyu Zheng, Ge Zhang, Tianhao Shen, Xueling Liu, Bill Yuchen Lin, Jie Fu, Wenhu Chen, and Xiang Yue. Opencodeinterpreter: Integrating code generation with execution and refinement. arXiv preprint , 2024. 14 Preprint A IMPLEMENTATION DETAILS A.1 MODEL MERGING ON GLUE BENCHMARK In Section 4.2, we employ RoBERTa-base (Liu et al., 2019) and DeBERTa-base (He et al., 2020) as the base model. The performance of each method is evaluated by eight tasks in the GLUE benchmark tasks, including CoLA (Warstadt et al., 2019), SST-2 (Socher et al., 2013), MRPC (Dolan & Brockett, 2005), STS-B (Cer et al., 2017), QQP (Iyer et al., 2017), MNLI (Williams et al., 2017), QNLI (Rajpurkar et al., 2016), RTE (Giampiccolo et al., 2007). For evaluation, CoLA is assessed using the Matthews correlation coefficient, STS-B is measured by the average of Pearson and Spearman correlation coefficients, and the remaining tasks are evaluated based on accuracy. The RoBERTa checkpoints used for model merging are publicly available via the links in (Huang et al., 2024). The DeBERTa checkpoints are obtained by training DeBERTa-base individually on each GLUE task for 5 epochs using the AdamW (Loshchilov & Hutter, 2017) optimizer with a learning rate of 2 × 10-5, a warm-up phase of 100 steps, and a weight decay of 10-2. Following the setting in Huang et al. (2024), we report the numerical experiments of PAVE with various merging methods including Task Arithmetic (Ilharco et al., 2022), Ties-Merging (Yadav et al., 2023) and EMR-Merging (Huang et al., 2024). We also evaluate these methods with plug-in method DARE (Yu et al., 2024). Following the standard setting of each method, the hyperparameter λ in Task Arithmetic and Ties-Merging is selected from {0.1, 0.3, 0.5, 0.7 0.9} based on the best performance. Specifically, for the RoBERTa experiments, we set λ = 0.3 for Task Arithmetic and λ = 0.9 for Ties-Merging; for BERT experiments, we set λ = 0.1 and λ = 0.5, respectively. The dropout ratio for DARE is fixed to 0.1 to avoid instability. The preserved rank ratio ρ of PAVE is chosen from { 7 8, 15 16, 31 32, 63 64}, and the stopping ratio is set to γ = ρ -1-ρ 2 . To attain optimal performance on GLUE benchmarks, PAVE progressively progressively assigns full rank to ∆WPAVE according to the performance of the plain merging method. This design reflects the observation that certain tasks are more sensitive and cannot tolerate aggressive pruning. At each step, we apply full rank to the the task whose performance is furthest from that of its individual model. This process continues until all task vectors are full rank. As a result, we obtain K merged models and select the one with the highest average performance. Since model merging is not computationally expensive and requires no training, the computational cost of such procedure remains acceptable. A.2 MODEL MERGING ON GENERATION BENCHMARKS Following the setup of DARE presented in Yu et al. (2024), we evaluate PAVE on mathematical solving and code generation tasks, as described in Section 4.3. For mathematical solving, we consider GSM8K (Cobbe et al., 2021) and MATH (Hendrycks et al., 2021), while for code generation, we use HumanEval (Chen et al., 2021) and MBPP (Austin et al., 2021). We adopt the LLaMA2-7B (Touvron et al., 2023) model as the base model. For mathematical solving, we utilize the fine-tuned model WizardMath-7B-V1.0 (Luo et al., 2023), while for code generation, we employ LLaMA-2-7B-EvolCodeAlpaca (Agarwalla et al., 2024). Additionally, code samples are drawn from the CodeFeedback dataset (Zheng et al., 2024) to support covariance matrix generation. The Task Arithmetic hyperparameter λ is selected from the set {0.5, 1}, and the dropout ratio for DARE is fixed at 0.2. The pruning rank for PAVE is chosen from {4, 6, 8, 16}. B ADDITIONAL ANALYSIS AND RESULTS B.1 TASK VECTOR WITH VARIOUS DECOMPOSITION METHODS In this section, we present numerical experiments evaluating the performance of various decomposition methods applied to our decomposition-based task vector. The setting of models and merging methods follow the DeBERTa settings in Section 4.2. The evaluated decomposition methods include: The basic SVD method (Meng et al., 2024), directly approximates the fine-tuned weights via a plain SVD decomposition. ASVD (Yuan et al., 2023), incorporates an input-dependent diagonal scaling matrix derived from the average absolute values of the previous layer's outputs. SVD-LLM (Wang et al., 2024b), applies Cholesky decomposition to the previous layer's output. 15 Preprint Table 5: Results of merging fine-tuned DeBERTa models on eight datasets from GLUE benchmark. The number of examples used to generate context matrix is 4096. Methods CoLA SST-2 MRPC STSB QQP MNLI QNLI RTE Average Individual 60.18 94.04 89.22 90.63 91.41 87.20 92.71 79.06 85.55 EMR-Merging 47.05 94.72 71.32 87.57 88.36 87.38 92.29 64.26 79.11 EMR-Merging W/ SVD 62.20 93.58 83.82 89.40 89.45 76.72 92.53 58.48 80.77 EMR-Merging W/ ASVD 62.33 95.07 84.56 89.73 89.98 87.44 92.86 61.01 82.87 EMR-Merging W/ SVD-LLM 61.82 94.72 84.07 89.55 90.10 87.44 92.90 61.37 82.74 EMR-Merging W/ PAVE 61.32 94.50 84.31 89.54 90.34 87.50 93.01 60.65 82.64 Task Arithmetic 3.03 75.00 68.38 52.72 63.44 43.46 51.16 50.90 51.01 Task Arithmetic W/ SVD 2.96 70.30 68.38 37.90 64.30 42.88 67.01 53.07 50.85 Task Arithmetic W/ ASVD 2.53 76.03 66.38 55.60 64.08 43.23 61.27 51.62 52.59 Task Arithmetic W/ SVD-LLM 2.74 74.89 68.38 55.48 64.02 43.54 61.45 51.62 52.76 Task Arithmetic W/ PAVE 2.18 75.46 68.38 55.90 64.00 43.48 61.25 51.99 52.83 Ties-Merging 5.36 66.28 68.63 19.09 63.68 40.44 55.39 48.74 45.95 Ties-Merging W/ SVD -3.29 53.90 68.38 6.15 62.98 35.34 54.22 52.71 41.29 Ties-Merging W/ ASVD -1.74 66.86 68.63 17.19 62.84 40.33 53.10 46.57 44.22 Ties-Merging W/ SVD-LLM 1.13 67.89 68.63 23.33 63.09 43.02 57.84 48.74 46.70 Ties-Merging W/ PAVE 1.39 68.35 68.63 36.76 63.12 44.01 59.36 50.90 49.06 64 128 256 512 1024 2048 4096 Samples 88.7 88.8 88.9 89.0 89.1 89.2 Performance EMR-Merging W/ PAVE Std EMR-Merging 64 128 256 512 1024 2048 4096 Samples 2.0 4.0 6.0 8.0 10.0 12.0 14.0 Time (min) PAVE Runtime Figure 4: Performance and runtime analysis of PAVE across different sample sizes. Left: Mean performance (solid line) and standard deviation (shaded region) of EMR-Merging with PAVE, showing consistent improvements as more samples are used. Right: PAVE runtime grows with number of samples. CO-SVD (Yang et al., 2024b), as the default decomposition method embedded in PAVE, utilizes the covariance matrix of the output of the previous layer as the activation matrix. Among the evaluated methods, CO-SVD demonstrates the most consistent performance on Task Arithmetic and TiesMerging, while also exhibiting performance comparable to EMR-Merging. These results highlight the strong potential of our decomposition-based purified task vector framework, which is designed to integrate seamlessly with a variety of decomposition methods. B.2 ANALYSIS OF SAMPLE SIZE AND TIME EFFICIENCY We next analyze the impact of sample size on the stability, efficiency, and reproducibility of PAVE. As illustrated in Figure 4, increasing the sample size consistently reduces the variance and improves the overall performance of EMR-Merging with PAVE, highlighting its stability and robustness. When the sample size is very small, due to the regularization, the method degenerates to plain SVD, which explains the lower performance observed in this regime. Importantly, we observe that 16 Preprint Table 6: Results of merging fine-tuned ViT models on 8 image classification benchmarks. The number of samples used to generate context matrix is 4096. Methods SUN397 Cars RESISC45 EuroSAT SVHN GTSRB MNIST DTD Average Individual ViT-B/32 75.3 77.7 96.1 99.7 97.5 98.7 99.7 79.4 90.5 EMR-Merging 75.2 72.8 93.5 99.5 96.9 98.1 99.6 74.4 88.7 EMR-Merging W/PAVE 76.3 74.3 93.8 99.3 96.7 97.6 99.6 76.1 89.2+0.5 Individual ViT-B/16 79.0 87.0 96.4 99.1 97.7 99.0 99.7 82.3 92.5 EMR-Merging 78.5 82.6 95.4 99.1 97.6 98.8 99.6 78.3 91.2 EMR-Merging W/PAVE 80.2 82.8 96.1 99.2 97.6 98.7 99.6 78.9 91.7+0.5 Individual ViT-L/14 82.3 92.4 97.4 100 98.1 99.2 99.7 84.1 94.2 EMR-Merging 83.1 90.7 96.7 99.7 97.9 99.1 99.6 82.7 93.7 EMR-Merging W/PAVE 83.6 91.7 97.2 99.8 99.7 98.9 99.7 82.2 93.9+0.2 even a moderate sample size (256-512) is sufficient to yield substantial performance gains, offering a practical trade-off between accuracy and computational cost. This finding suggests that PAVE can be effectively deployed in practice without incurring excessive runtime overhead, while still preserving most of the performance benefits of using larger sample sizes. Throughout our numerical experiments, we increased the number of samples up to 4096 ensure a stable and consistent demonstration of our method's performance. Employing larger sample sizes also helps ensure the reproducibility of our results. All runtime measurements were conducted on a workstation equipped with a single NVIDIA A100 GPU and 8 CPUs, with 512 GB of RAM. B.3 MODEL MERGING FOR VIT MODELS Settings. Following the setup of EMR-Merging presented in Huang et al. (2024), we employ ViTB/32, ViT-B/16 and ViT-L/14 (Radford et al., 2021) as the base model to evaluate the performance of PAVE across 8 image classification tasks: SUN397 (Xiao et al., 2010), Cars Krause et al. (2013), RESISC45 (Cheng et al., 2017), EuroSAT (Helber et al., 2019), SVHN (Netzer et al., 2011), GTSRB (Stallkamp et al., 2011), MNIST (LeCun, 1998), and DTD (Cimpoi et al., 2014). The preserved rank ratio ρ of PAVE is chosen from { 7 8, 15 16, 31 32, 63 64}, and the stopping ratio is set to γ = ρ -1-ρ 2 . The checkpoints are publicly accessible via the link provided in Ilharco et al. (2022). Results. As shown in Table 6, while the plain EMR-Merging method already demonstrates strong performance, PAVE still consistently enhances the performance across all ViT variants. For ViTB/32, PAVE improves the average accuracy by +0.5% (from 88.7% to 89.2). On the more capable ViT-B/16 model, PAVE again contributes an improvement of +0.5%, increasing the average from 91.2% to 91.7%. 17
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2510.14698
|
FedPPA: Progressive Parameter Alignment for
Personalized Federated Learning
Maulidi Adi Prasetia∗, Muhamad Risqi U. Saputra†, and Guntur Dharma Putra∗‡
∗Universitas Gadjah Mada, Indonesia †Monash University, Indonesia
maulidiadiprasetia2000@mail.ugm.ac.id, risqi.saputra@monash.edu, gdputra@ugm.ac.id
Abstract—Federated Learning (FL) is designed as a decen-
tralized, privacy-preserving machine learning paradigm that
enables multiple clients to collaboratively train a model without
sharing their data. In real-world scenarios, however, clients
often have heterogeneous computational resources and hold non-
independent and identically distributed data (non-IID), which
poses significant challenges during training. Personalized Fed-
erated Learning (PFL) has emerged to address these issues by
customizing models for each client based on their unique data dis-
tribution. Despite its potential, existing PFL approaches typically
overlook the coexistence of model and data heterogeneity arising
from clients with diverse computational capabilities. To overcome
this limitation, we propose a novel method, called Progressive
Parameter Alignment (FedPPA), which progressively aligns the
weights of common layers across clients with the global model’s
weights. Our approach not only mitigates inconsistencies between
global and local models during client updates, but also pre-
serves client’s local knowledge, thereby enhancing personalization
robustness in non-IID settings. To further enhance the global
model performance while retaining strong personalization, we
also integrate entropy-based weighted averaging into the FedPPA
framework. Experiments on three image classification datasets,
including MNIST, FMNIST, and CIFAR-10, demonstrate that
FedPPA consistently outperforms existing FL algorithms, achiev-
ing superior performance in personalized adaptation.
Index Terms—Distributed training, personalized Federated
learning, inconsistent knowledge, parameter alignment, and
weighted entropy-based average.
I. INTRODUCTION
Federated Learning (FL) is a distributed machine learning
framework in which multiple clients collaboratively train a
shared model without directly sharing their data, thereby
enhancing overall model performance while preserving data
privacy [1]. FL works through iterative training rounds be-
tween participating clients and a central aggregation server. In
each round, clients train models locally using their private data
and transmit the resulting model to the server. Subsequently,
the server aggregates these models to generate a global model,
which is then redistributed to the clients for further local train-
ing in the next round [2], [3]. However, in real-world scenarios,
participating clients often have incomplete or sparse data, i.e.,
non-independent and identically distributed data (non-IID),
which would degrade the overall model performance.
Personalized Federated Learning (PFL) addresses these data
heterogeneity issues in the clients by grouping the models into
a shared base layer and a personalized head layer [4]. The base
‡Corresponding author.
layer, shared by the server and all clients, encodes the common
deep learning layer of the entire clients, while the head layer
is locally customized and is fine-tuned to capture the unique
characteristics of the client’s data. These personalization layers
enable each client’s model to more accurately align with
local data while still gaining advantages from the collective
knowledge shared across the networks. While PFL yields
strong performance, these schemes typically demand all clients
to have identical architecture. However, in practice, clients
often possess heterogeneous computational resources, making
it impractical to enforce a uniform model architecture across
all clients, leading to each client having distinct architectures.
Consequently, it becomes challenging to implement this per-
sonalization strategy that demands consistent model structures.
Recent studies [5], [6] proposed PFL methods to address the
issue of heterogeneous model architectures between clients.
FlexiFed identifies common components within clients’ lo-
cal models and facilitates joint training on these layers to
effectively fuse knowledge while maintaining architectural
flexibility [5]. In addition, FedADP introduces a comprehen-
sive model strategy to manage a variety of heterogeneous
network architectures and ensures that devices with limited
computational capacity can still significantly contribute to the
global model’s training [6]. However, these approaches often
overlook the potential parameter misalignment between the
server (global) model and the individual clients’ model during
local updates, potentially leading to inconsistencies between
existing client’s parameters and the updated parameters from
the global model. To address this knowledge inconsistency,
FedAS proposes a solution to align the global model’s param-
eter with those of each client during local training, mitigating
inconsistencies and improving overall model performance [7].
However, this solution heavily relies on the assumption that
every client uses the same model architecture, which is
quite unlikely given the significant variations in computational
power commonly found in real-world scenarios.
In this paper, we propose FedPPA: a PFL approach to
simultaneously address heterogeneity in client models and
non-IID nature of client data, while also resolving the knowl-
edge inconsistency between the server and the client models
during local updates. FedPPA progressively aligns the weights
of common layers across clients with the global model’s
weights, mitigating inconsistencies between global and local
models during client updates. While our main target is to
arXiv:2510.14698v1 [cs.LG] 16 Oct 2025
improve personalization effectiveness and address the knowl-
edge discrepancy problem, our approach also preserves client’s
local knowledge, thereby enhancing personalization robustness
in non-IID settings. In addition, we propose FedPPA+, in
which we incorporate an entropy-based weighted averaging to
FedPPA, retaining strong personalization with a better trade-
off to global model performance. The entropy values are
calculated from each client’s variability in label distribution,
where more labels exist in client’s dataset equal to a higher
contribution weight.
In summary, we propose the following contributions:
• We develop a PFL approach, termed FedPPA, which
progressively aligns client’s common layers with the
global model parameters during local updates, which is
suitable for FL with heterogeneous model architectures
and non-IID data distribution.
• We incorporate an entropy-based weighted averaging in
the global aggregation mechanism to better adapt to non-
IID data distribution.
• We perform a comprehensive experiment and analysis on
publicly available data sets for image classification (i.e.,
MNIST, F-MNIST, and CIFAR-10) using heterogeneous
model architectures (i.e., VGG-11, VGG-13, VGG-16,
and VGG-19).
The remainder of this paper is structured as follows. We
present the related work in Section II and discuss our proposed
solution in Section III. The implementation of our solution
with the corresponding evaluation is presented in Section IV,
while the conclusion of the paper is presented in Section V.
II. RELATED WORK
FL was introduced as a modern distributed AI training
paradigm designed to preserve user privacy by keeping data
localized on user devices [8]. Instead of sharing raw data
with a central server, FL enables collaborative model training
across multiple clients, ensuring high levels of data privacy and
security. While FL has successfully introduced collaborative
training across clients, achieving high model performance
using this approach typically relies on the assumption that
the datasets used for training are independent and identically
distributed (IID). In real-world applications, IID assumption
is rarely met, resulting in non-IID data distributions across
clients, which can significantly degrade the performance of
FL model. Previous studies have shown that non-IID data
distribution can reduce FL model performance for up to 29%
[9].
In order to address the performance degradation of standard
FL model due to the non-IID data distribution among clients,
PFL was introduced by Arivazhagan et al. [4] by incorporating
personalization layers. These personalization layers allow each
client’s model to better fit local data while still benefiting from
shared knowledge across the networks. Since then, various
PFL methods have been developed to address the diverse
challenges in distributed and personalized model training. For
instance, Hierarchical PFL (HPFL) [10]–[14] was developed
to enable hierarchical weight aggregation among groups of
clients through edge servers, which act as an intermediate
aggregation points. In this approach, clients with similar
resources or data characteristics are clustered together, and
they only allow to communicate with the server via the edge
servers, avoiding direct communication of each client with the
cloud server. This approach not only reduces communication
costs and latency but also lessens the computational load on
the central server, especially when training involves a large
number of clients. However, clustering similar clients might
lead to knowledge inconsistency across clusters due to limited
information sharing, resulting in knowledge isolation. Addi-
tionally, the HPFL approach is best suited for edge computing
architectures and may be challenging to be implemented in
more general or less structured environments.
Conventional PFL personalizes each client’s model by di-
viding it into two parts: a shared base layer and a personalized
head layer. While this approach has demonstrated good per-
formance, it relies on the assumption that all clients use the
same model architecture. In practice, however, clients often
have varying computational resources, making it difficult to
enforce a uniform model structure. To address this limitation,
recent researches, such as [5], [6], [15]–[18], have begun
exploring PFL methods that support heterogeneous model
architectures across clients. One notable example is FlexiFed,
proposed by [5]. FlexiFed allows each client to use different
model architectures during PFL training by introducing three
model aggregation strategies. One such strategy, called Max-
Common, clusters a subset of model architectures based on
their common layers and aggregates weights only from these
common layers. This approach facilitates effective federated
training across clients with heterogeneous models, ensuring
both flexibility and consistent performance. Nevertheless, this
approach might overlook the possibility of parameter mis-
alignment between the global model and each client’s local
model during local updates, which can lead to inconsistencies
in shared knowledge. Previous work, such as FedAS [7],
has attempted to resolve this problem by aligning the global
model’s weights with those of the client during local training.
However, this solution is limited to scenarios where clients
share a uniform model architecture. In contrast, our work
aims to tackle the challenges of PFL in the presence of both
heterogeneous client models and non-IID data, while also
resolving the knowledge inconsistency between the server and
the client models during local updates.
III. METHODOLOGY
Figure 1 shows an overview of our proposed framework that
highlights our two key contributions (depicted as yellow box),
namely 1) Progressive Parameter Alignment (PPA) and 2)
entropy-based weighted averaging. Compared to the standard
PFL methods, which straightforwardly override the client’s
base model with the global (server) model’s weights during
client update, our method progressively aligns common (and
clustered) layers in the clients with the global model, address-
ing the knowledge-inconsistency problem between global and
clients’ models more effectively. By re-aligning the aggregated
Non-IID Data
Among
Clients
Heterogeneous client’s
model architecture
Client A
Client B
Client C
Client D
Entropy-based
Weighted
Averaging
Common Layers 1
Common Layers 2
Cloud Server
Progressive
Parameter
Alignment
Global
Aggregation
Client Local
Update
(ē: 0.35)
(ē: 0.2)
(ē: 0.25)
(ē: 0.2)
𝓂i
t
ℒ
Alignment for
common layers 1
Alignment for
common layers 2
Alignment for
common layers 2
Fig. 1: Overview of the proposed framework, where common layers are extracted from each client’s heterogeneous models.
Each client is assigned to an entropy-based contribution weight during global aggregation. The updated model is then returned
to the clients, after which parameter alignment is progressively carried out locally based on each client’s distinct dataset.
knowledge from the central server with each client’s local
data before overwriting the client’s existing local knowledge,
our method ensures that clients benefit from the latest server-
aggregated information without sacrificing the personalized
knowledge manifested in the local models. Note that this
alignment occurs iteratively from the first common layer until
the end, which matches and synchronizes the most similar
layers as much as possible.
To better handle the non-IID data distribution, we employ an
entropy-based weighting to determine the contribution weight
of each client based on the variability of its local data. Incorpo-
rating these weights yields more robust performance in highly
skewed non-IID environments compared to the conventional
PFL. The entropy-based weights ensure that clients with richer,
more diverse datasets play a significant role in the knowledge
aggregation process. In summary, the steps performed in each
training round are described as follows:
• Calculation of client’s weight: In this step, we calcu-
late client’s weight contribution via the entropy-based
method. Intuitively, we put higher weights to the clients
with more diverse datasets.
• Local training: Each client trains their model locally
using their personalized dataset. Note that the client’s
local dataset is divided into training and evaluation/testing
dataset.
• Clustering common layers: All layers from each client
are compared to identify a list of common layers. Both
the architecture and the order of layers must match, e.g.,
clients with the same layer types but arranged differently
are not considered to share common layers. Only clients
containing the identified common layers would contribute
to the aggregation process in the server.
• Global aggregation: In the server, the client’s model
parameters is aggregated based on the client’s layer
similarity. Note that the aggregation would leverage an
entropy-based method, applied to the corresponding com-
mon layers.
• Progressive parameter alignment: Before updating each
client’s common layer parameters, parameter alignment
process is performed. Using the list of common layers
identified in the previous step, the previous client’s model
features are extracted for each common layer. These
output common features is used to align the client’s model
parameters by minimizing its differences with the global
model features.
• Model distribution: In the final step, the aligned param-
eters of the common layers are distributed back to each
client.
A. Progressive Parameter Alignment — FedPPA
Conventional PFL methods typically split the client models
into the base and the head (personalization) layers and over-
write their parameters directly during local updates. However,
this approach could lead to inconsistent knowledge between
local and global models, as the client already is trained
with their personalized dataset. We introduce a progressive
parameter alignment approach, namely FedPPA, which aligns
the aggregated global knowledge with client’s personalized
knowledge progressively for each common layer. In practice,
FedPPA works by minimizing the differences between the
global features and the features of the client model before
performing local updates [7]. The goal is to allow each client to
share common knowledge for the same task without sacrificing
personalized capabilities tailored to their unique dataset.
We summarize the notations used in this paper in Table I
and describe the complete workflow of FedPPA in Algorithm
1. FedPPA begins by extracting the common layers of each
client using the following function
TABLE I: Notations used in this paper and their definitions
Notation
Definition
Notation
Definition
M
Number of clients
Cl
Number of contribu-
tors for layer l
t
Number of rounds
Li
Total layers of client
i
mt
i
Client’s model i at
round t
H(l)
i
Output features of
layer l in client i
mt+1
i
Updated
client’s
model
Di
Dataset of client i
l
Layer of model m
ei
Entropy
value
of
client i
lt+1
i
, Cl = ExtractMaxCommonLayers(mt
i, mt
i+1),
(1)
where lt+1
i
is a set of common layers extracted on the
client i and Cl represents the number of clients contributing
to lt+1
i
. We adopt a layer-level extraction method from [5]
into our ExtractMaxCommonLayers function. Subsequently,
each model parameter from the extracted common layers is
averaged using the following formula
mt+1
i
=
1
Cl
X
li∈lt+1
i
li
Li
,
(2)
where mt+1
i
is the updated parameter from client i and Li
is the number of layers in the model. This computation is
performed iteratively for each layer in the client’s local model.
If li is selected as the common layers, it is incorporated into
lt+1
i
, which means that the parameter (i.e., the "knowledge")
from li is added to the aggregated parameters. Before aligning
the aggregated client’s model parameters in each common
layer with the global, we need to extract the current client’s
local features using the following formula
H(l)
i
= {h(l)(x; mt
i)|x ∈Di} ∈Rdl×Ni,
(3)
where H(l)
i
is the feature extracted from layer l in the local
model mt
i using the personalized data set of the client Di.
This equation is also used to extract the features from the
updated and aggregated client model mt+1
i
to ensure that
the dimensions of the data are equal. Finally, each client’s
common layers are aligned with mt
i using the following
equations
mt+1
i
←mt+1
i
−∇L(f(mt+1
i
, Di)H(l)
i )
(4)
and
L(f(mt+1
i
, Di), H(l)
i ) = ∥f(mt+1
i
, Di) −H(l)
i ∥2
2,
(5)
where L is the L2 loss function used to align the features
between mt+1
i
and mt
i. The goal of this alignment is to
make mt+1
i
closer to mt
i by minimizing the differences of
the features using Equation (5).
B. Entropy-Based Client Weighting
Conventional FL methods typically assume equal contribu-
tion from all clients during global aggregation. However, under
non-IID data distributions and heterogeneous model architec-
tures, client contributions should be weighted differently rather
than treated uniformly. To this end, we proposed to leverage
Entropy-based weighting to treat each client differently based
on their dataset variability. Entropy, which quantifies the
uncertainty or randomness of information, provides a natural
measure for weighting each client’s influence.
The Shannon’s entropy [19] method is employed to quantify
how random and uncertain a piece of information is. In the
context of a non-IID data distribution, a higher entropy value
indicates that a client’s overall data are highly uncertain and
diverse. To ensure fair comparison across clients, we normalize
each client’s entropy-based weight so that the total sums
to one. Based on this procedure, we propose the following
method for computing client contribution weights using the
Shannon entropy formula
ei =
H(D)i
PM
i=1 H(D)i
,
(6)
where H(D)i is the Shannon Entropy of each client i, which
can be calculated from [20]–[22]. As can be seen, each client’s
entropy value is divided with the sum of all client’s entropy
values to obtain the normalized entropy for each client, which
is denoted as ei.
Figure 2 shows the distribution of the contributions based
on weighted entropy from each client, applied to different non-
IID data with different Dirichlet values of α (see Section 4.2
Algorithm 1 FedPPA Algorithm
1: t = 1
▷Round counter
2: W = ∅
▷Store aligned models
3:
4: for i = 1,..,M do:
5:
lt+1
i
, Cl = ExtractMaxCommonLayers(mt
i, mt
i+1)
cf. (1)
6:
Calculate mt+1
i
cf. (2)
▷Aggregate model
7:
Update mt+1
i
, cf. (4)
▷Parameter alignment
8:
Add aligned model mt+1
i
to W
9: end for
10:
11: return W
Client 1
11%
Client 2
14%
Client 3
12%
Client 4
14%
Client 5
14%
Client 6
12%
Client 7
14%
Client 8
10%
(a) α = 0.5
Client 1
11%
Client 2
6%
Client 3
17%
Client 4
11%
Client 5
13%
Client 6
13%
Client 7
17%
Client 8
11%
(b) α = 0.1
Client 1
15%
Client 2
10%
Client 3
5%
Client 4
15%
Client 5
10%
Client 6
15%
Client 7
10%
Client 8
20%
(c) α = 0.01
Fig. 2: Distribution of weighted contributions from each client, calculated using Shanon Entropy, applied to different non-IID
data with different α values.
for more information on how the non-IID data are generated).
It can be seen that each client exhibits a distinct contribution
weight that reflects the variability of its class labels. When a
large value of α is used, the data distribution approaches IID,
resulting in relatively uniform client contribution. On the other
hand, when α decreases and the data become more non-IID,
the variation in contribution weights across clients becomes
more pronounced. This demonstrates that the entropy-based
method effectively captures the degree of uncertainty and
randomness in the client’s data.
C. FedPPA with Entropy-Guided Weighted Averaging
We propose FedPPA+, an extension of FedPPA that incorpo-
rates the client’s entropy value as the weighting factor during
the global knowledge aggregation. Each client would have
different aggregation weights based on their data distribution,
as mentioned in the previous section. In general, FedPPA+
follows the same steps as FedPPA, which can be found in
Algorithm 2, except that the global aggregation approach is
now leveraging the entropy-based weighted averaging. The
global aggregation process of FedPPA+ is formulated as
follows:
mt+1
i
=
ei ×
1
Cl
X
li∈lt+1
i
li
Li
(7)
where ei is the entropy value of client i. The only difference
with FedPPA is that the averaged knowledge is not directly
used as the new global model. Instead, it must first be weighted
by ei.
IV. EXPERIMENTAL EVALUATION
A. Dataset, Baselines, and Metrics
We performed our experiments using publicly available
dataset for image classification: 1) MNIST [24], a widely used
dataset of 70K grayscale handwritten digit images (0–9); 2)
Fashion-MNIST (F-MNIST) [25], a dataset of 70K grayscale
images representing 10 categories of clothing items, serving
as a more challenging version of MNIST; and 3) CIFAR-
10 [26], which contains 60K 32×32 color images across 10
object categories and is considered as the most challenging
Algorithm 2 FedPPA+ Algorithm
1: t = 1
▷Round counter
2: W = ∅
▷Store aligned models
3:
4: Calculate e for all clients using cf. (6)
5: for i = 1,..,M do:
6:
lt+1
i
, Cl = ExtractMaxCommonLayers(mt
i, mt
i+1)
cf. (1)
7:
Calculate mt+1
i
with ei cf. 7
▷Aggregate model
8:
Update mt+1
i
cf. 4
▷Parameter Alignment
9:
Add aligned model mt+1
i
to W
10: end for
11:
12: return W
dataset compared to MNIST and F-MNIST. As baselines, we
compared our approach with Max-Common [5] and FedAvg
[23], which represent the FL and PFL approaches, respectively.
Finally, the classification accuracy was used as the primary
evaluation metric to assess and compare the performance of
each method. For a comprehensive evaluation, we measured
both personalization performance (the average accuracy across
individual clients) and global performance (the accuracy of the
global model evaluated on the full dataset on the cloud server).
B. Experimental Setup
We conducted our experiments under conditions where
clients had both non-IID data distributions and heterogeneous
model architectures. Table II summarizes the experimental
setup which shows the number of clients, the number of com-
TABLE II: Experimental Scenario Setup
Scenario
Dirichlet α
Dataset
Clients
Rounds
1
0.5
MNIST
8
51
F-MNIST
CIFAR-10
2
0.1
MNIST
F-MNIST
CIFAR-10
3
0.01
MNIST
F-MNIST
CIFAR-10
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(a) MNIST α = 0.5
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(b) F-MNIST α = 0.5
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(c) CIFAR-10 α = 0.5
Fig. 3: Model accuracy and convergence of our methods in Scenario 1 (α = 0.5) with two baseline models [5], [23]. While
all methods can achieve good results, FedPPA and FedPPA+ surpass the baseline performances after 10 training iterations.
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(a) MNIST α = 0.1
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(b) F-MNIST α = 0.1
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(c) CIFAR-10 α = 0.1
Fig. 4: Model accuracy and convergence in Scenario 2 (α = 0.1). FedPPA and FedPPA+ exhibit stable performance relative
to the baselines. As smaller α means more non-IIDness, our methods demonstrated superior performance in personalization.
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(a) MNIST α = 0.01
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(b) F-MNIST α = 0.01
0
10
20
30
40
50
Rounds
0
20
40
60
80
100
Accuracy (%)
Max Common
FedAvg
FedPPA
FedPPA+
(c) CIFAR-10 α = 0.01
Fig. 5: Model accuracy and convergence in Scenario 3 (α = 0.01). As α further decreases, the disparity between our methods
and baseline models widens, highlighting the advantages of our proposed methods in handling personalization.
munication rounds, the dataset, and the α value. To simulate
a non-IID data distribution for each dataset, we leveraged the
Dirichlet distribution-based method [7], in which the param-
eter α is used to characterize the distribution, influencing the
shape and concentration of probabilities between categories.
Figure 6 depicts the label distribution across clients with
α ∈{0.5, 0.1, 0.01}, where it represents label distributions
from almost IID to highly non-IID. Note that each scenario is
run for 10 epochs in each client’s local training.
To simulate heterogeneous client model architectures, we
utilized the VGG model for image classification with different
depth, namely VGG-11, VGG-13, VGG-16, and VGG-19.
Since we used a total of 8 clients and only 4 different
model architectures were available, we assigned every 2 clients
with the same model architecture. As one of the baselines,
since FedAVG aggregates weights by averaging across client
models, this implies that weight aggregation can only occur
among clients sharing the same architecture. On the other
hand, similar to our approach, Max-Common was executed
as is, since the model has the ability to handle architecture
heterogeneity.
We implemented our proposed framework using PyTorch
library, where some of our implementations were inspired
by the work of [7] and [5]. The experiment was conducted
using an Nvidia RTX3060 GPU with 12GB of memory, a
CUDA Version 12.3, Python v3.10.12, and PyTorch v2.2.0.
The configuration includes 8 clients, each of whom locally
trains their model for 10 epochs using the same hyperparam-
0.0
0.2
0.4
0.6
0.8
1.0
Proportion
1
2
3
4
5
6
7
8
Client ID
(a) α = 0.5
0.0
0.2
0.4
0.6
0.8
1.0
Proportion
1
2
3
4
5
6
7
8
Client ID
(b) α = 0.1
0.0
0.2
0.4
0.6
0.8
1.0
Proportion
1
2
3
4
5
6
7
8
Client ID
(c) α = 0.01
Fig. 6: Distribution of labels among clients in the MNIST dataset using the Dirichlet Distribution approach.
eters. Stochastic Gradient Descent (SGD) was applied as an
optimizer function with a learning rate of 0.01, a momentum
of 0.9, and a learning rate decay of 5e−4. The entire PFL
training process was executed for 51 rounds in total.
C. Experimental Results
Table III shows the personalization performance of our ap-
proach, in terms of the average prediction accuracy, compared
to the baselines in Scenario 1 (α = 0.5), 2 (α = 0.1), and
3 (α = 0.01), respectively. As we can see, in all datasets,
our approaches, both FedPPA and FedPPA+, show superior
personalization performance, except for MNIST with α = 0.5
although the differences with other approaches are relatively
small (less than 0.5%). As expected, Max-Common, repre-
senting a PFL approach, achieves overall better performance
compared to FedAVG, which serves as a baseline for standard
FL methods.
To gain a clearer understanding of each model’s perfor-
mance over time during training, Figure 3, 4, and 5 illustrate
the personalization performance of our approach across all
scenarios, from the first until the last round. From these
figures, our approaches generally converge faster than the
baselines, achieving peak accuracy earlier in most scenarios.
Notably, the performance gap between our model and the
baselines widens as the value of α decreases (highly non-IID),
reaching nearly 25% differences in the most extreme case (See
CIFAR-10 with α = 0.01 in Table III). On the other hand, our
model show only a slight performance improvement over the
baselines at higher value of α, where the data distribution is
TABLE III: Accuracy of Comparison in Personalized Evalua-
tion
Dirichlet α
Dataset
FedAvg
[23]
Max-
Common
[5]
FedPPA
(Ours)
FedPPA+
(Ours)
0.5
MNIST
99.75
99.62
99.37
99.37
F-MNIST
92.37
92.62
92.62
93.37
CIFAR-10
72.62
74.25
77.87
76.37
0.1
MNIST
99.75
99.62
99.37
99.37
F-MNIST
96.37
96.24
97.00
97.25
CIFAR-10
78.25
80.75
91.62
90.87
0.01
MNIST
93.25
89.00
100.0
100.0
F-MNIST
94.12
95.62
100.0
100.0
CIFAR-10
63.37
73.37
98.87
98.62
1 Green color indicates that our methods outperform the baselines.
2 Red color indicates that our method performance is below the baselines.
closer to IID, as seen in Figure 3. This indicates that our model
shows a strong robustness in personalization performance
in an extreme non-IID scenario compared to the baselines.
Finally, all compared models demonstrated relatively lower
performance on the CIFAR-10 dataset, which is expected given
that CIFAR-10 is considered more challenging than MNIST
and F-MNIST.
The performance of our methods improved significantly as
the level of non-IIDness increased in the dataset. As shown
in Scenario 3 (α = 0.01) of Table III, our approaches reach
a prediction accuracy of up to 100% for both MNIST and
F-MNIST. In such extreme non-IID settings, some clients
possess only one or two labels, as illustrated in Fig. 6(c).
Under these circumstances, our highly personalized approach
can achieve perfect predictions, given a minimal number of
classes.
To highlight the performance differences between FedPPA
and FedPPA+, Table IV presents the global performance
evaluation between both methods across all tested datasets
with α = 0.5. As shown, FedPPA+ outperforms FedPPA on
two of the datasets, indicating its enhanced capability in han-
dling non-IID data distributions. Although FedPPA+ achieves
slightly lower accuracy on the MNIST dataset, the difference
is minimal (less than 2%). Nevertheless, further research is
required to better balance personalized adaptation and global
model performance, aiming to improve both personalization
and global accuracy.
Limitations. We acknowledge two primary limitations in
the present study. First, since FedPPA and FedPPA+ were
evaluated only on VGG model variations, these evaluations
may not capture unforeseen heterogeneity in other scenarios,
which may yield different results, for instance, when the
proposed methods are tested on transformer-based models.
Secondly, we conducted our evaluations on only eight FL
TABLE IV: Global Performance of FedPPA and FedPPA+
with α = 0.5
Dataset
FedPPA
FedPPA+
Accuracy
Round
Accuracy
Round
MNIST
79.25
50
77.62
36
F-MNIST
62.0
50
63.0
51
CIFAR-10
38.87
49
40.62
51
1 Green color indicates that FedPPA+ performs better than FedPPA.
2 Red colored indicates that FedPPA+ performance is below FedPPA.
nodes, which may require further exploration in scenarios
involving hundreds of participating nodes to confirm the
convergence of the model performance. Further research may
be directed towards validating these methods across more
diverse model architectures and at a larger scale.
V. CONCLUSION
In this paper, we introduced a novel PFL method, termed
FedPPA, designed to operate effectively in the presence of
both heterogeneous client models and non-IID data, while also
addressing the knowledge inconsistency between server and
client models during local updates. FedPPA works by progres-
sively aligning the weights of common layers between clients
and the global model, thereby improving personalization per-
formance under non-IID conditions. We further extended our
approach with FedPPA+, which enhances model robustness by
incorporating an entropy-based weighting mechanism during
global model aggregation. Experimental results in three image
classification datasets, namely MNIST, F-MNIST, and CIFAR-
10, demonstrated that our methods consistently outperform
baseline models in terms of personalization performance. As
FedPPA and FedPPA+ primarily aim to improve person-
alization effectiveness and address the issue of knowledge
discrepancies, future work may be directed toward balancing
personalization and global performance.
ACKNOWLEDGMENT
We express our gratitude to the Indonesia Endowment
Fund for Education (LPDP) under the Ministry of Finance of
the Republic of Indonesia for providing the scholarship and
supporting this research.
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FedPPA: Progressive Parameter Alignment for Personalized Federated Learning Maulidi Adi Prasetia∗, Muhamad Risqi U. Saputra†, and Guntur Dharma Putra∗‡ ∗Universitas Gadjah Mada, Indonesia †Monash University, Indonesia , , Abstract-Federated Learning (FL) is designed as a decentralized, privacy-preserving machine learning paradigm that enables multiple clients to collaboratively train a model without sharing their data. In real-world scenarios, however, clients often have heterogeneous computational resources and hold nonindependent and identically distributed data (non-IID), which poses significant challenges during training. Personalized Federated Learning (PFL) has emerged to address these issues by customizing models for each client based on their unique data distribution. Despite its potential, existing PFL approaches typically overlook the coexistence of model and data heterogeneity arising from clients with diverse computational capabilities. To overcome this limitation, we propose a novel method, called Progressive Parameter Alignment (FedPPA), which progressively aligns the weights of common layers across clients with the global model's weights. Our approach not only mitigates inconsistencies between global and local models during client updates, but also preserves client's local knowledge, thereby enhancing personalization robustness in non-IID settings. To further enhance the global model performance while retaining strong personalization, we also integrate entropy-based weighted averaging into the FedPPA framework. Experiments on three image classification datasets, including MNIST, FMNIST, and CIFAR-10, demonstrate that FedPPA consistently outperforms existing FL algorithms, achieving superior performance in personalized adaptation. Index Terms-Distributed training, personalized Federated learning, inconsistent knowledge, parameter alignment, and weighted entropy-based average. I. INTRODUCTION Federated Learning (FL) is a distributed machine learning framework in which multiple clients collaboratively train a shared model without directly sharing their data, thereby enhancing overall model performance while preserving data privacy [1]. FL works through iterative training rounds between participating clients and a central aggregation server. In each round, clients train models locally using their private data and transmit the resulting model to the server. Subsequently, the server aggregates these models to generate a global model, which is then redistributed to the clients for further local training in the next round [2], [3]. However, in real-world scenarios, participating clients often have incomplete or sparse data, i.e., non-independent and identically distributed data (non-IID), which would degrade the overall model performance. Personalized Federated Learning (PFL) addresses these data heterogeneity issues in the clients by grouping the models into a shared base layer and a personalized head layer [4]. The base ‡Corresponding author. layer, shared by the server and all clients, encodes the common deep learning layer of the entire clients, while the head layer is locally customized and is fine-tuned to capture the unique characteristics of the client's data. These personalization layers enable each client's model to more accurately align with local data while still gaining advantages from the collective knowledge shared across the networks. While PFL yields strong performance, these schemes typically demand all clients to have identical architecture. However, in practice, clients often possess heterogeneous computational resources, making it impractical to enforce a uniform model architecture across all clients, leading to each client having distinct architectures. Consequently, it becomes challenging to implement this personalization strategy that demands consistent model structures. Recent studies [5], [6] proposed PFL methods to address the issue of heterogeneous model architectures between clients. FlexiFed identifies common components within clients' local models and facilitates joint training on these layers to effectively fuse knowledge while maintaining architectural flexibility [5]. In addition, FedADP introduces a comprehensive model strategy to manage a variety of heterogeneous network architectures and ensures that devices with limited computational capacity can still significantly contribute to the global model's training [6]. However, these approaches often overlook the potential parameter misalignment between the server (global) model and the individual clients' model during local updates, potentially leading to inconsistencies between existing client's parameters and the updated parameters from the global model. To address this knowledge inconsistency, FedAS proposes a solution to align the global model's parameter with those of each client during local training, mitigating inconsistencies and improving overall model performance [7]. However, this solution heavily relies on the assumption that every client uses the same model architecture, which is quite unlikely given the significant variations in computational power commonly found in real-world scenarios. In this paper, we propose FedPPA: a PFL approach to simultaneously address heterogeneity in client models and non-IID nature of client data, while also resolving the knowledge inconsistency between the server and the client models during local updates. FedPPA progressively aligns the weights of common layers across clients with the global model's weights, mitigating inconsistencies between global and local models during client updates. While our main target is to 16 Oct 2025 improve personalization effectiveness and address the knowledge discrepancy problem, our approach also preserves client's local knowledge, thereby enhancing personalization robustness in non-IID settings. In addition, we propose FedPPA+, in which we incorporate an entropy-based weighted averaging to FedPPA, retaining strong personalization with a better tradeoff to global model performance. The entropy values are calculated from each client's variability in label distribution, where more labels exist in client's dataset equal to a higher contribution weight. In summary, we propose the following contributions: • We develop a PFL approach, termed FedPPA, which progressively aligns client's common layers with the global model parameters during local updates, which is suitable for FL with heterogeneous model architectures and non-IID data distribution. • We incorporate an entropy-based weighted averaging in the global aggregation mechanism to better adapt to nonIID data distribution. • We perform a comprehensive experiment and analysis on publicly available data sets for image classification (i.e., MNIST, F-MNIST, and CIFAR-10) using heterogeneous model architectures (i.e., VGG-11, VGG-13, VGG-16, and VGG-19). The remainder of this paper is structured as follows. We present the related work in Section II and discuss our proposed solution in Section III. The implementation of our solution with the corresponding evaluation is presented in Section IV, while the conclusion of the paper is presented in Section V. II. RELATED WORK FL was introduced as a modern distributed AI training paradigm designed to preserve user privacy by keeping data localized on user devices [8]. Instead of sharing raw data with a central server, FL enables collaborative model training across multiple clients, ensuring high levels of data privacy and security. While FL has successfully introduced collaborative training across clients, achieving high model performance using this approach typically relies on the assumption that the datasets used for training are independent and identically distributed (IID). In real-world applications, IID assumption is rarely met, resulting in non-IID data distributions across clients, which can significantly degrade the performance of FL model. Previous studies have shown that non-IID data distribution can reduce FL model performance for up to 29% [9]. In order to address the performance degradation of standard FL model due to the non-IID data distribution among clients, PFL was introduced by Arivazhagan et al. [4] by incorporating personalization layers. These personalization layers allow each client's model to better fit local data while still benefiting from shared knowledge across the networks. Since then, various PFL methods have been developed to address the diverse challenges in distributed and personalized model training. For instance, Hierarchical PFL (HPFL) [10]-[14] was developed to enable hierarchical weight aggregation among groups of clients through edge servers, which act as an intermediate aggregation points. In this approach, clients with similar resources or data characteristics are clustered together, and they only allow to communicate with the server via the edge servers, avoiding direct communication of each client with the cloud server. This approach not only reduces communication costs and latency but also lessens the computational load on the central server, especially when training involves a large number of clients. However, clustering similar clients might lead to knowledge inconsistency across clusters due to limited information sharing, resulting in knowledge isolation. Additionally, the HPFL approach is best suited for edge computing architectures and may be challenging to be implemented in more general or less structured environments. Conventional PFL personalizes each client's model by dividing it into two parts: a shared base layer and a personalized head layer. While this approach has demonstrated good performance, it relies on the assumption that all clients use the same model architecture. In practice, however, clients often have varying computational resources, making it difficult to enforce a uniform model structure. To address this limitation, recent researches, such as [5], [6], [15]-[18], have begun exploring PFL methods that support heterogeneous model architectures across clients. One notable example is FlexiFed, proposed by [5]. FlexiFed allows each client to use different model architectures during PFL training by introducing three model aggregation strategies. One such strategy, called MaxCommon, clusters a subset of model architectures based on their common layers and aggregates weights only from these common layers. This approach facilitates effective federated training across clients with heterogeneous models, ensuring both flexibility and consistent performance. Nevertheless, this approach might overlook the possibility of parameter misalignment between the global model and each client's local model during local updates, which can lead to inconsistencies in shared knowledge. Previous work, such as FedAS [7], has attempted to resolve this problem by aligning the global model's weights with those of the client during local training. However, this solution is limited to scenarios where clients share a uniform model architecture. In contrast, our work aims to tackle the challenges of PFL in the presence of both heterogeneous client models and non-IID data, while also resolving the knowledge inconsistency between the server and the client models during local updates. III. METHODOLOGY Figure 1 shows an overview of our proposed framework that highlights our two key contributions (depicted as yellow box), namely 1) Progressive Parameter Alignment (PPA) and 2) entropy-based weighted averaging. Compared to the standard PFL methods, which straightforwardly override the client's base model with the global (server) model's weights during client update, our method progressively aligns common (and clustered) layers in the clients with the global model, addressing the knowledge-inconsistency problem between global and clients' models more effectively. By re-aligning the aggregated Non-IID Data Among Clients Heterogeneous client's model architecture Client A Client B Client C Client D Entropy-based Weighted Averaging Common Layers 1 Common Layers 2 Cloud Server Progressive Parameter Alignment Global Aggregation Client Local Update (ē: 0.35) (ē: 0.2) (ē: 0.25) (ē: 0.2) mi t L Alignment for common layers 1 Alignment for common layers 2 Alignment for common layers 2 Fig. 1: Overview of the proposed framework, where common layers are extracted from each client's heterogeneous models. Each client is assigned to an entropy-based contribution weight during global aggregation. The updated model is then returned to the clients, after which parameter alignment is progressively carried out locally based on each client's distinct dataset. knowledge from the central server with each client's local data before overwriting the client's existing local knowledge, our method ensures that clients benefit from the latest serveraggregated information without sacrificing the personalized knowledge manifested in the local models. Note that this alignment occurs iteratively from the first common layer until the end, which matches and synchronizes the most similar layers as much as possible. To better handle the non-IID data distribution, we employ an entropy-based weighting to determine the contribution weight of each client based on the variability of its local data. Incorporating these weights yields more robust performance in highly skewed non-IID environments compared to the conventional PFL. The entropy-based weights ensure that clients with richer, more diverse datasets play a significant role in the knowledge aggregation process. In summary, the steps performed in each training round are described as follows: • Calculation of client's weight: In this step, we calculate client's weight contribution via the entropy-based method. Intuitively, we put higher weights to the clients with more diverse datasets. • Local training: Each client trains their model locally using their personalized dataset. Note that the client's local dataset is divided into training and evaluation/testing dataset. • Clustering common layers: All layers from each client are compared to identify a list of common layers. Both the architecture and the order of layers must match, e.g., clients with the same layer types but arranged differently are not considered to share common layers. Only clients containing the identified common layers would contribute to the aggregation process in the server. • Global aggregation: In the server, the client's model parameters is aggregated based on the client's layer similarity. Note that the aggregation would leverage an entropy-based method, applied to the corresponding common layers. • Progressive parameter alignment: Before updating each client's common layer parameters, parameter alignment process is performed. Using the list of common layers identified in the previous step, the previous client's model features are extracted for each common layer. These output common features is used to align the client's model parameters by minimizing its differences with the global model features. • Model distribution: In the final step, the aligned parameters of the common layers are distributed back to each client. A. Progressive Parameter Alignment - FedPPA Conventional PFL methods typically split the client models into the base and the head (personalization) layers and overwrite their parameters directly during local updates. However, this approach could lead to inconsistent knowledge between local and global models, as the client already is trained with their personalized dataset. We introduce a progressive parameter alignment approach, namely FedPPA, which aligns the aggregated global knowledge with client's personalized knowledge progressively for each common layer. In practice, FedPPA works by minimizing the differences between the global features and the features of the client model before performing local updates [7]. The goal is to allow each client to share common knowledge for the same task without sacrificing personalized capabilities tailored to their unique dataset. We summarize the notations used in this paper in Table I and describe the complete workflow of FedPPA in Algorithm 1. FedPPA begins by extracting the common layers of each client using the following function TABLE I: Notations used in this paper and their definitions Notation Definition Notation Definition M Number of clients Cl Number of contributors for layer l t Number of rounds Li Total layers of client i mt i Client's model i at round t H(l) i Output features of layer l in client i mt+1 i Updated client's model Di Dataset of client i l Layer of model m ei Entropy value of client i lt+1 i , Cl = ExtractMaxCommonLayers(mt i, mt i+1), (1) where lt+1 i is a set of common layers extracted on the client i and Cl represents the number of clients contributing to lt+1 i . We adopt a layer-level extraction method from [5] into our ExtractMaxCommonLayers function. Subsequently, each model parameter from the extracted common layers is averaged using the following formula mt+1 i = 1 Cl X li∈lt+1 i li Li , (2) where mt+1 i is the updated parameter from client i and Li is the number of layers in the model. This computation is performed iteratively for each layer in the client's local model. If li is selected as the common layers, it is incorporated into lt+1 i , which means that the parameter (i.e., the "knowledge") from li is added to the aggregated parameters. Before aligning the aggregated client's model parameters in each common layer with the global, we need to extract the current client's local features using the following formula H(l) i = {h(l)(x; mt i)|x ∈Di} ∈Rdl×Ni, (3) where H(l) i is the feature extracted from layer l in the local model mt i using the personalized data set of the client Di. This equation is also used to extract the features from the updated and aggregated client model mt+1 i to ensure that the dimensions of the data are equal. Finally, each client's common layers are aligned with mt i using the following equations mt+1 i ←mt+1 i -∇L(f(mt+1 i , Di)H(l) i ) (4) and L(f(mt+1 i , Di), H(l) i ) = ∥f(mt+1 i , Di) -H(l) i ∥2 2, (5) where L is the L2 loss function used to align the features between mt+1 i and mt i. The goal of this alignment is to make mt+1 i closer to mt i by minimizing the differences of the features using Equation (5). B. Entropy-Based Client Weighting Conventional FL methods typically assume equal contribution from all clients during global aggregation. However, under non-IID data distributions and heterogeneous model architectures, client contributions should be weighted differently rather than treated uniformly. To this end, we proposed to leverage Entropy-based weighting to treat each client differently based on their dataset variability. Entropy, which quantifies the uncertainty or randomness of information, provides a natural measure for weighting each client's influence. The Shannon's entropy [19] method is employed to quantify how random and uncertain a piece of information is. In the context of a non-IID data distribution, a higher entropy value indicates that a client's overall data are highly uncertain and diverse. To ensure fair comparison across clients, we normalize each client's entropy-based weight so that the total sums to one. Based on this procedure, we propose the following method for computing client contribution weights using the Shannon entropy formula ei = H(D)i PM i=1 H(D)i , (6) where H(D)i is the Shannon Entropy of each client i, which can be calculated from [20]-[22]. As can be seen, each client's entropy value is divided with the sum of all client's entropy values to obtain the normalized entropy for each client, which is denoted as ei. Figure 2 shows the distribution of the contributions based on weighted entropy from each client, applied to different nonIID data with different Dirichlet values of α (see Section 4.2 Algorithm 1 FedPPA Algorithm 1: t = 1 ▷Round counter 2: W = ∅ ▷Store aligned models 3: 4: for i = 1,..,M do: 5: lt+1 i , Cl = ExtractMaxCommonLayers(mt i, mt i+1) cf. (1) 6: Calculate mt+1 i cf. (2) ▷Aggregate model 7: Update mt+1 i , cf. (4) ▷Parameter alignment 8: Add aligned model mt+1 i to W 9: end for 10: 11: return W Client 1 11% Client 2 14% Client 3 12% Client 4 14% Client 5 14% Client 6 12% Client 7 14% Client 8 10% (a) α = 0.5 Client 1 11% Client 2 6% Client 3 17% Client 4 11% Client 5 13% Client 6 13% Client 7 17% Client 8 11% (b) α = 0.1 Client 1 15% Client 2 10% Client 3 5% Client 4 15% Client 5 10% Client 6 15% Client 7 10% Client 8 20% (c) α = 0.01 Fig. 2: Distribution of weighted contributions from each client, calculated using Shanon Entropy, applied to different non-IID data with different α values. for more information on how the non-IID data are generated). It can be seen that each client exhibits a distinct contribution weight that reflects the variability of its class labels. When a large value of α is used, the data distribution approaches IID, resulting in relatively uniform client contribution. On the other hand, when α decreases and the data become more non-IID, the variation in contribution weights across clients becomes more pronounced. This demonstrates that the entropy-based method effectively captures the degree of uncertainty and randomness in the client's data. C. FedPPA with Entropy-Guided Weighted Averaging We propose FedPPA+, an extension of FedPPA that incorporates the client's entropy value as the weighting factor during the global knowledge aggregation. Each client would have different aggregation weights based on their data distribution, as mentioned in the previous section. In general, FedPPA+ follows the same steps as FedPPA, which can be found in Algorithm 2, except that the global aggregation approach is now leveraging the entropy-based weighted averaging. The global aggregation process of FedPPA+ is formulated as follows: mt+1 i = ei × 1 Cl X li∈lt+1 i li Li (7) where ei is the entropy value of client i. The only difference with FedPPA is that the averaged knowledge is not directly used as the new global model. Instead, it must first be weighted by ei. IV. EXPERIMENTAL EVALUATION A. Dataset, Baselines, and Metrics We performed our experiments using publicly available dataset for image classification: 1) MNIST [24], a widely used dataset of 70K grayscale handwritten digit images (0-9); 2) Fashion-MNIST (F-MNIST) [25], a dataset of 70K grayscale images representing 10 categories of clothing items, serving as a more challenging version of MNIST; and 3) CIFAR10 [26], which contains 60K 32×32 color images across 10 object categories and is considered as the most challenging Algorithm 2 FedPPA+ Algorithm 1: t = 1 ▷Round counter 2: W = ∅ ▷Store aligned models 3: 4: Calculate e for all clients using cf. (6) 5: for i = 1,..,M do: 6: lt+1 i , Cl = ExtractMaxCommonLayers(mt i, mt i+1) cf. (1) 7: Calculate mt+1 i with ei cf. 7 ▷Aggregate model 8: Update mt+1 i cf. 4 ▷Parameter Alignment 9: Add aligned model mt+1 i to W 10: end for 11: 12: return W dataset compared to MNIST and F-MNIST. As baselines, we compared our approach with Max-Common [5] and FedAvg [23], which represent the FL and PFL approaches, respectively. Finally, the classification accuracy was used as the primary evaluation metric to assess and compare the performance of each method. For a comprehensive evaluation, we measured both personalization performance (the average accuracy across individual clients) and global performance (the accuracy of the global model evaluated on the full dataset on the cloud server). B. Experimental Setup We conducted our experiments under conditions where clients had both non-IID data distributions and heterogeneous model architectures. Table II summarizes the experimental setup which shows the number of clients, the number of comTABLE II: Experimental Scenario Setup Scenario Dirichlet α Dataset Clients Rounds 1 0.5 MNIST 8 51 F-MNIST CIFAR-10 2 0.1 MNIST F-MNIST CIFAR-10 3 0.01 MNIST F-MNIST CIFAR-10 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (a) MNIST α = 0.5 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (b) F-MNIST α = 0.5 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (c) CIFAR-10 α = 0.5 Fig. 3: Model accuracy and convergence of our methods in Scenario 1 (α = 0.5) with two baseline models [5], [23]. While all methods can achieve good results, FedPPA and FedPPA+ surpass the baseline performances after 10 training iterations. 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (a) MNIST α = 0.1 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (b) F-MNIST α = 0.1 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (c) CIFAR-10 α = 0.1 Fig. 4: Model accuracy and convergence in Scenario 2 (α = 0.1). FedPPA and FedPPA+ exhibit stable performance relative to the baselines. As smaller α means more non-IIDness, our methods demonstrated superior performance in personalization. 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (a) MNIST α = 0.01 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (b) F-MNIST α = 0.01 0 10 20 30 40 50 Rounds 0 20 40 60 80 100 Accuracy (%) Max Common FedAvg FedPPA FedPPA+ (c) CIFAR-10 α = 0.01 Fig. 5: Model accuracy and convergence in Scenario 3 (α = 0.01). As α further decreases, the disparity between our methods and baseline models widens, highlighting the advantages of our proposed methods in handling personalization. munication rounds, the dataset, and the α value. To simulate a non-IID data distribution for each dataset, we leveraged the Dirichlet distribution-based method [7], in which the parameter α is used to characterize the distribution, influencing the shape and concentration of probabilities between categories. Figure 6 depicts the label distribution across clients with α ∈{0.5, 0.1, 0.01}, where it represents label distributions from almost IID to highly non-IID. Note that each scenario is run for 10 epochs in each client's local training. To simulate heterogeneous client model architectures, we utilized the VGG model for image classification with different depth, namely VGG-11, VGG-13, VGG-16, and VGG-19. Since we used a total of 8 clients and only 4 different model architectures were available, we assigned every 2 clients with the same model architecture. As one of the baselines, since FedAVG aggregates weights by averaging across client models, this implies that weight aggregation can only occur among clients sharing the same architecture. On the other hand, similar to our approach, Max-Common was executed as is, since the model has the ability to handle architecture heterogeneity. We implemented our proposed framework using PyTorch library, where some of our implementations were inspired by the work of [7] and [5]. The experiment was conducted using an Nvidia RTX3060 GPU with 12GB of memory, a CUDA Version 12.3, Python v3.10.12, and PyTorch v2.2.0. The configuration includes 8 clients, each of whom locally trains their model for 10 epochs using the same hyperparam0.0 0.2 0.4 0.6 0.8 1.0 Proportion 1 2 3 4 5 6 7 8 Client ID (a) α = 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Proportion 1 2 3 4 5 6 7 8 Client ID (b) α = 0.1 0.0 0.2 0.4 0.6 0.8 1.0 Proportion 1 2 3 4 5 6 7 8 Client ID (c) α = 0.01 Fig. 6: Distribution of labels among clients in the MNIST dataset using the Dirichlet Distribution approach. eters. Stochastic Gradient Descent (SGD) was applied as an optimizer function with a learning rate of 0.01, a momentum of 0.9, and a learning rate decay of 5e-4. The entire PFL training process was executed for 51 rounds in total. C. Experimental Results Table III shows the personalization performance of our approach, in terms of the average prediction accuracy, compared to the baselines in Scenario 1 (α = 0.5), 2 (α = 0.1), and 3 (α = 0.01), respectively. As we can see, in all datasets, our approaches, both FedPPA and FedPPA+, show superior personalization performance, except for MNIST with α = 0.5 although the differences with other approaches are relatively small (less than 0.5%). As expected, Max-Common, representing a PFL approach, achieves overall better performance compared to FedAVG, which serves as a baseline for standard FL methods. To gain a clearer understanding of each model's performance over time during training, Figure 3, 4, and 5 illustrate the personalization performance of our approach across all scenarios, from the first until the last round. From these figures, our approaches generally converge faster than the baselines, achieving peak accuracy earlier in most scenarios. Notably, the performance gap between our model and the baselines widens as the value of α decreases (highly non-IID), reaching nearly 25% differences in the most extreme case (See CIFAR-10 with α = 0.01 in Table III). On the other hand, our model show only a slight performance improvement over the baselines at higher value of α, where the data distribution is TABLE III: Accuracy of Comparison in Personalized Evaluation Dirichlet α Dataset FedAvg [23] MaxCommon [5] FedPPA (Ours) FedPPA+ (Ours) 0.5 MNIST 99.75 99.62 99.37 99.37 F-MNIST 92.37 92.62 92.62 93.37 CIFAR-10 72.62 74.25 77.87 76.37 0.1 MNIST 99.75 99.62 99.37 99.37 F-MNIST 96.37 96.24 97.00 97.25 CIFAR-10 78.25 80.75 91.62 90.87 0.01 MNIST 93.25 89.00 100.0 100.0 F-MNIST 94.12 95.62 100.0 100.0 CIFAR-10 63.37 73.37 98.87 98.62 1 Green color indicates that our methods outperform the baselines. 2 Red color indicates that our method performance is below the baselines. closer to IID, as seen in Figure 3. This indicates that our model shows a strong robustness in personalization performance in an extreme non-IID scenario compared to the baselines. Finally, all compared models demonstrated relatively lower performance on the CIFAR-10 dataset, which is expected given that CIFAR-10 is considered more challenging than MNIST and F-MNIST. The performance of our methods improved significantly as the level of non-IIDness increased in the dataset. As shown in Scenario 3 (α = 0.01) of Table III, our approaches reach a prediction accuracy of up to 100% for both MNIST and F-MNIST. In such extreme non-IID settings, some clients possess only one or two labels, as illustrated in Fig. 6(c). Under these circumstances, our highly personalized approach can achieve perfect predictions, given a minimal number of classes. To highlight the performance differences between FedPPA and FedPPA+, Table IV presents the global performance evaluation between both methods across all tested datasets with α = 0.5. As shown, FedPPA+ outperforms FedPPA on two of the datasets, indicating its enhanced capability in handling non-IID data distributions. Although FedPPA+ achieves slightly lower accuracy on the MNIST dataset, the difference is minimal (less than 2%). Nevertheless, further research is required to better balance personalized adaptation and global model performance, aiming to improve both personalization and global accuracy. Limitations. We acknowledge two primary limitations in the present study. First, since FedPPA and FedPPA+ were evaluated only on VGG model variations, these evaluations may not capture unforeseen heterogeneity in other scenarios, which may yield different results, for instance, when the proposed methods are tested on transformer-based models. Secondly, we conducted our evaluations on only eight FL TABLE IV: Global Performance of FedPPA and FedPPA+ with α = 0.5 Dataset FedPPA FedPPA+ Accuracy Round Accuracy Round MNIST 79.25 50 77.62 36 F-MNIST 62.0 50 63.0 51 CIFAR-10 38.87 49 40.62 51 1 Green color indicates that FedPPA+ performs better than FedPPA. 2 Red colored indicates that FedPPA+ performance is below FedPPA. nodes, which may require further exploration in scenarios involving hundreds of participating nodes to confirm the convergence of the model performance. Further research may be directed towards validating these methods across more diverse model architectures and at a larger scale. V. CONCLUSION In this paper, we introduced a novel PFL method, termed FedPPA, designed to operate effectively in the presence of both heterogeneous client models and non-IID data, while also addressing the knowledge inconsistency between server and client models during local updates. FedPPA works by progressively aligning the weights of common layers between clients and the global model, thereby improving personalization performance under non-IID conditions. We further extended our approach with FedPPA+, which enhances model robustness by incorporating an entropy-based weighting mechanism during global model aggregation. Experimental results in three image classification datasets, namely MNIST, F-MNIST, and CIFAR10, demonstrated that our methods consistently outperform baseline models in terms of personalization performance. As FedPPA and FedPPA+ primarily aim to improve personalization effectiveness and address the issue of knowledge discrepancies, future work may be directed toward balancing personalization and global performance. ACKNOWLEDGMENT We express our gratitude to the Indonesia Endowment Fund for Education (LPDP) under the Ministry of Finance of the Republic of Indonesia for providing the scholarship and supporting this research. REFERENCES [1] M. N. P. 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Su, "Towards Efficient Heterogeneous Multi-Modal Federated Learning with Hierarchical Knowledge Disentanglement," in Proceedings of the 22nd ACM Conference on Embedded Networked Sensor Systems, (Hangzhou China), pp. 592-605, ACM, Nov. 2024. [15] Y. Liu, S. Guo, J. Zhang, Z. Hong, Y. Zhan, and Q. Zhou, "Collaborative Neural Architecture Search for Personalized Federated Learning," IEEE Transactions on Computers, vol. 74, pp. 250-262, Jan. 2025. [16] X. Zhou, Q. Yang, X. Zheng, W. Liang, K. I.-K. Wang, J. Ma, Y. Pan, and Q. Jin, "Personalized Federated Learning With Model-Contrastive Learning for Multi-Modal User Modeling in Human-Centric Metaverse," IEEE Journal on Selected Areas in Communications, vol. 42, pp. 817831, Apr. 2024. [17] J. Yan, J. Liu, H. Xu, Z. Wang, and C. Qiao, "Peaches: Personalized Federated Learning With Neural Architecture Search in Edge Computing," IEEE Transactions on Mobile Computing, vol. 23, pp. 10296-10312, Nov. 2024. [18] J. Xu, S. Wan, Y. Li, S. Luo, Z. 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2510.14702
|
Cognitive-Aligned Spatio-Temporal Large Language Models For
Next Point-of-Interest Prediction
Penglong Zhai∗
AMAP, Alibaba Group
Beijing, China
zhaipenglong.zpl@alibaba-inc.com
Jie Li∗
AMAP, Alibaba Group
Beijing, China
lj313796@alibaba-inc.com
Fanyi Di∗
AMAP, Alibaba Group
Beijing, China
difanyi.dfy@alibaba-inc.com
Yue Liu
AMAP, Alibaba Group
Beijing, China
ly355576@alibaba-inc.com
Yifang Yuan
AMAP, Alibaba Group
Beijing, China
yuanyifang.yyf@alibaba-inc.com
Jie Huang
AMAP, Alibaba Group
Beijing, China
jielu.hj@alibaba-inc.com
Peng Wu
AMAP, Alibaba Group
Beijing, China
zhouxiao.wp@alibaba-inc.com
Sicong Wang
AMAP, Alibaba Group
Beijing, China
wsc488938@alibaba-inc.com
Mingyang Yin
AMAP, Alibaba Group
Beijing, China
hengyang.ymy@alibaba-inc.com
Tingting Hu
AMAP, Alibaba Group
Beijing, China
jingting.htt@alibaba-inc.com
Yao Xu
AMAP, Alibaba Group
Beijing, China
xuenuo.xy@alibaba-inc.com
Xin Li†
AMAP, Alibaba Group
Beijing, China
beilai.bl@alibaba-inc.com
Abstract
The next point-of-interest (POI) recommendation task aims to pre-
dict the users’ immediate next destinations based on their prefer-
ences and historical check-ins, holding significant value in location-
based services. Recently, large language models (LLMs) have shown
great potential in recommender systems, which treat the next POI
prediction in a generative manner. However, these LLMs, pretrained
primarily on vast corpora of unstructured text, lack the native un-
derstanding of structured geographical entities and sequential mo-
bility patterns required for next POI prediction tasks. Moreover, in
industrial-scale POI prediction applications, incorporating world
knowledge and alignment of human cognition, such as seasons,
weather conditions, holidays, and users’ profiles (such as habits,
occupation, and preferences), can enhance the user experience
while improving recommendation performance. To address these
issues, we propose CoAST (Cognitive-Aligned Spatial-Temporal
LLMs), a framework employing natural language as an interface,
allowing for the incorporation of world knowledge, spatio-temporal
trajectory patterns, profiles, and situational information. Specifi-
cally, CoAST mainly comprises of 2 stages: (1) Recommendation
∗Contributed equally to this research.
†Corresponding Author.
Permission to make digital or hard copies of all or part of this work for personal or
classroom use is granted without fee provided that copies are not made or distributed
for profit or commercial advantage and that copies bear this notice and the full citation
on the first page. Copyrights for components of this work owned by others than the
author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or
republish, to post on servers or to redistribute to lists, requires prior specific permission
and/or a fee. Request permissions from permissions@acm.org.
Conference acronym ’XX, Woodstock, NY
© 2018 Copyright held by the owner/author(s). Publication rights licensed to ACM.
ACM ISBN 978-1-4503-XXXX-X/2018/06
https://doi.org/XXXXXXX.XXXXXXX
Knowledge Acquisition through continued pretraining on the en-
riched spatial-temporal trajectory data of the desensitized users;
(2) Cognitive Alignment to align cognitive judgments with human
preferences using enriched training data through Supervised Fine-
Tuning (SFT) and a subsequent Reinforcement Learning (RL) phase.
Extensive offline experiments on various real-world datasets and
online experiments deployed in "Guess Where You Go" of AMAP1
App homepage demonstrate the effectiveness of CoAST.
CCS Concepts
• Information systems →Location based services.
Keywords
Generative Recommendation, Next POI Prediction, Cognitive Rec-
ommendation, Large Language Models
ACM Reference Format:
Penglong Zhai, Jie Li, Fanyi Di, Yue Liu, Yifang Yuan, Jie Huang, Peng
Wu, Sicong Wang, Mingyang Yin, Tingting Hu, Yao Xu, and Xin Li. 2018.
Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-
of-Interest Prediction. In Proceedings of Make sure to enter the correct con-
ference title from your rights confirmation email (Conference acronym ’XX).
ACM, New York, NY, USA, 12 pages. https://doi.org/XXXXXXX.XXXXXXX
1
Introduction
Next Point-of-Interest (POI) prediction, as a variant of POI recom-
mendation task, aiming to predict users’ future locations based
on their historical sequential check-ins and other contextual in-
formation, is a fundamental task in Location-Based Services (LBS)
[9, 10, 19, 24, 25, 32, 36, 38, 40, 43]. For large-scale navigation plat-
forms such as AMAP, which processes billions of daily requests
1https://amap.com/
arXiv:2510.14702v1 [cs.AI] 16 Oct 2025
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Zhai et al.
from hundreds of millions of active users in China, making accu-
rate next POI prediction a critical capability for efficient destination
planning and personalized recommendation. Thus, next POI predic-
tion has been a popular research direction, capitalizing on develop-
ments in mobile and localization techniques, as they provide rich
location-based information. Much of the existing research in next
POI recommendation has predominantly focused on sequential rec-
ommendation and graph neural networks, tackling the challenge of
data sparsity, particularly for cold-start users and those with sparse
historical trajectories [8, 29, 33, 45, 48].
However, these approaches often fail to encapsulate the rich
contextual information embedded within Location-Based Social
Network (LBSN) data, such as the world knowledge and contextual
information, which are critical for improving the user experience
in industrial applications. Recently, Large Language Models (LLMs)
have demonstrated remarkable capabilities across various domains
[1, 27, 31, 41], offering new opportunities for this direction. How-
ever, their direct application to POI prediction task remains several
key challenges. First, POIs are typically indexed by discrete iden-
tifiers outside the LLM vocabulary, impeding the model’s ability
to capture semantic relationships [2, 47, 50]. Second, the intricate
interactions between the billions of users and POIs, coupled with
highly personalized and context-dependent behavioral patterns,
introduce complex collaborative signals that are difficult for LLMs
to encode and reason effectively [23, 25].
Another critical challenge, stemming from the real-world deploy-
ment, is that recommended POIs must be cognitively plausible to
users. Recommendations that violate human intuition or common-
sense reasoning, which we term human cognitive preference in this
work, even if statistically likely, can appear nonsensical or irrele-
vant, severely degrading the user experience and eroding trust in
the system [25]. In industrial applications, these logical lapses are
critical failures, as they directly risk frustrating users and reducing
retention.
Based on these observations, we formulate the task of human
cognitive preference alignment for individual POI prediction as: to
investigate issues of personality and individual differences (who),
environmental and social impacts (what), spatio-temporal features
and relationships (when and where), and corresponding actions
(how).
To address the aforementioned challenges and build a POI rec-
ommendation system that aligns with human cognitive preferences,
we introduce CoAST (Cognitive-Aligned Spatio-Temporal LLM), a
novel framework that endows a base LLM with the ability to gen-
erate cognitively plausible POI recommendations. CoAST begins
by adapting the LLM to the mobility domain through continued
pretraining on an enriched corpus of spatio-temporal tokenized POI
sequences, allowing it to master complex user behavior patterns.
Furthermore, we explicitly align the model with human common-
sense. This is achieved through a multi-stage fine-tuning process,
utilizing a curated dataset of cognitively-aware samples for both
Supervised Fine-Tuning (SFT) and a subsequent Reinforcement
Learning (RL) phase.
Our main contributions as as follows:
• To our knowledge, this is the first generic LLM-based cognitive-
aligned framework for the field of next POI prediction. The
proposed framework enhances the comprehension of human
cognitive preference in recommendation.
• We propose CoAST, a novel end-to-end LLM-based frame-
work for large-scale next POI prediction that effectively
bridges the semantic gap between world and recommen-
dation knowledge, which demonstrates significant improve-
ment on human cognitive alignment and applicable to prac-
tical applicability of LLMs in LBS applications.
• Through a series of inference optimization strategy, CoAST
is deployed in AMAP’s online production environment, a
leading real-world LBS platform. Compared traditional cas-
cade ranking system, long term online A/B test achieves a
4.23% and 4.59% increase in P-CTR and U-CTR, which is a
substantial improvement.
The remainder of this paper is organized as follows: Section
2 reviews related literature, Section 3 details problem definition
and the detailed categories of cognitive recommendation, Section
4 elaborates our methodology, Section 5 reports empirical results,
and Section 6 concludes the paper.
2
Related Work
2.1
Next POI Prediction
Next POI recommendation extends beyond conventional item rec-
ommendation by incorporating inherent geographical and tempo-
ral characteristics. Early methods relied on user check-in histo-
ries using Markov chains, or RNNs to capture sequential behav-
iors. Spatial-temporal modeling has been extensively studied with
works like ST-GRAT [26] employing graph attention networks for
spatial-temporal dependencies, STAN [25] introducing hierarchical
attention mechanisms, and Sun et al. [30] focusing on periodic pat-
terns in human mobility through time-aware embeddings. Recent
LLM-based approaches have focused on addressing the semantic
gap in POI recommendation. For example, GNPR-SID [32] proposed
semantic POI IDs using residual quantized variational autoencoders,
achieving improvements by mapping POIs into discrete semantic
spaces. LLM4POI [19] leveraged pretrained LLMs to preserve het-
erogeneous data in original format, effectively utilizing contextual
information.
2.2
LLMs for Recommendation
The integration of Large Language Models (LLMs) into recommen-
dation systems has emerged as a transformative research direction.
P5 [12] pioneered the "Pretrain, Personalized Prompt, and Predict"
paradigm, establishing the foundation for treating recommenda-
tion as a language processing task by converting all recommen-
dation data into natural language sequences. Building upon this,
[2] addressed the knowledge gap between LLMs’ capabilities and
recommendation-specific requirements by incorporating collabo-
rative filtering operations like Masked Item Modeling (MIM) and
Bayesian Personalized Ranking (BPR) through natural language sim-
ulation. More recently, GenRank [17] advanced generative ranking
systems for large-scale deployment. OneRec [5, 53] proposed a rev-
olutionary end-to-end framework that replaces traditional cascaded
architectures, achieving significant improvements in large-scale
online deployments.
Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
3
Preliminaries
3.1
Problem Formulation
Given a collection of POIs I = {𝑝1, 𝑝2, ..., 𝑝𝑁} and a group of users
U = {𝑢1,𝑢2, ...,𝑢𝑀}. Each POI can be characterized by a tuple
𝑝𝑖= (𝑙𝑜𝑛,𝑙𝑎𝑡,𝑐), where 𝑐represents side information of the POI
including the descriptive texts and pictures, 𝑙𝑜𝑛and 𝑙𝑎𝑡denotes
the geographical coordinates (longitude and latitude). Each user 𝑢
has the check-ins history ℎ𝑢= {(𝑝1,𝑡1), . . . , (𝑝𝑚,𝑡𝑚)} of length 𝑚,
where 𝑝𝑖,𝑖= 1, . . . ,𝑚reflects the 𝑖-th POI recently interacted by
the user and 𝑡𝑖refers to the check-in timestamps. The objective of
next POI prediction task is to predict the next POI 𝑝that the user 𝑢
will visit at time 𝑡given the recent check-in record ℎ𝑢, the user’s
request location 𝑙and the contextual situation 𝑠.
3.2
Cognitive-Aligned POI Prediction
Compared to traditional accuracy-oriented recommender systems
that rely on rapidly capturing users’ immediate interests, next POI
prediction is a complex issue in simulating individual behaviors and
contextual information. Empirically, we category cognitive-aligned
recommendations into the following four scenarios. Recommended
POIs that do not align with these categories may result in a dimin-
ished user experience or decreased user engagement. To summa-
rize, our approach formulates the next POI recommendation task
as a decision-making problem. The model synthesizes a diverse set
of contextual factors—including temporal (when), spatial (where),
user-specific (who), and other situational inputs—to determine the
optimal user action (what to do and how).
3.2.1
Temporal Consistency. User behavior often exhibits certain
regularities. For example, office workers typically commute to work-
places in the morning and return home at noon. Recommending
irrelevant POIs during these commuting periods may lead to poor
user experiences. We define these observed regularity patterns in
user mobility as temporal consistency. Formally, at the given time
𝑡, the temporal consistency score (denoted as TCS) of the predicted
POI ˆ𝑝can be formulated as,
TCS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) =
Í
𝑖=𝑚I(𝑡𝑖= 𝑡) · sim(𝑝𝑖, ˆ𝑝∗)
Í𝑚
𝑖=1 I(𝑡𝑖= 𝑡)
(1)
where sim(·) denotes the function measuring the similarity of the
inputs pair.
3.2.2
Spatial Clustering. Check-ins of human are empirically demon-
strated to occur within a certain range [20, 21]; that is, the distance
of a user’s next visit POI is typically distributed as a negative expo-
nential relative to the user’s usual activity space. Recommending
a POI that is far from the user’s position (or their active area) is
inappropriate. We define the spatial clustering phenomenon of user
mobility as spatial consistency. Formally, given the location 𝑝𝑢of
the user 𝑢, the spatial consistency score of the predicted POI ˆ𝑝is
formulated as,
SCS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) = 𝑒−𝑑(𝑝𝑢, ˆ𝑝)/𝑑𝑢
(2)
where 𝑒denotes the exponential function, 𝑑(·) is the distance func-
tion, and 𝑑𝑢represents the user’s historical average movement
distance.
3.2.3
Profile Alignment. Behaviors of human often correlate with
their profile tags. For instance, suggesting family-oriented venues or
restaurants to university students represents a cognitive mismatch
in recommendation logic. We define the user profile alignment
score to reflect the alignment between recommended item ˆ𝑝and
user characteristics:
PAS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) = LLM𝑢(𝑢, ˆ𝑝)
(3)
where a LLM model is utilized to evaluate the reasonableness of
the recommended POI ˆ𝑝given the profile of user 𝑢and the score
∈{0, 1}.
3.2.4
Situational Awareness. Situations such as weather, traffic,
and the conditions of the POIs could have a significant impact
on human’s traveling decisions. For example, recommending an
outdoor tennis court on a rainy day is considered inappropriate. We
define the recommendation system’s Awareness to these contextual
situations as situational awareness. Similar to definition of profile
alignment, we also utilize a LLM to score this metric,
SAS(𝑢,𝑙,𝑠,𝑡, ˆ𝑝) = LLM𝑠(𝑠, ˆ𝑝)
(4)
where the score ∈{0, 1}.
The definitions articulated above serve a dual purpose in our
methodology. They directly guide the creation of a training set of
cognitively plausible examples for the model’s alignment phase.
Simultaneously, these definitions are translated into quantitative
metrics, providing a principled framework for evaluating the cog-
nitive capabilities of our proposed model.
4
Methodologies
The overall framework is presented in Figure 1. The framework
consists of two main modules: (1) Spatio-temporal Knowledge Ac-
quisition module enables the LLMs to incorporate the linguistic and
collaborative semantics and (2) Cognitive Alignment to adjusting
the ST-LLM obtained from step (1) to generate results that better
align with human cognition. We finally detail our inference opti-
mization strategy in online deployment since this is a bottleneck of
large-scale application of generative models.
4.1
Spatio-temporal Knowledge Acquisition
Considering the huge amount of the POI corpus in large industrial
LBS applications, we firstly convert POI data into Semantic IDs
(SIDs) to reduce the total number of tokens. Building on these SIDs,
we propose to enable the LLM to incorporate the recommendation
knowledge by continued pretraining on an enriched corpus.
4.1.1
Semantic ID generation. A well crafted SID representation
can boost the LLM’s recommendation efficiency by capturing the
semantic and spatial relationships among POIs. There has already
been extensive research in this area [22, 32, 34, 49, 51, 54], so we
do not elaborate on it here. In brief, we adopt the RQ-VAE, which
adopts the multi-modal information of the POIs to learning the
codebooks. Specifically, we incorporate the spatial location infor-
mation and descriptive titles/pictures of the POIs to generate the
identifiers with residual quantization process. As a result, we get a
set of codebooks
𝐶𝑙= {𝒆𝑙
𝑘|𝑘= 1, ..., 𝐾},𝑙∈[1, 𝐿]
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Zhai et al.
Figure 1: Offline Training and Online Service Procedure of proposed method CoAST. Offline Training: (1) POIs are tokenized as
Semantic IDs (SIDs); (2) a base LLM is continually pretrained on the collected corpus, yielding ST-LLM; (3) multi-level user
profiles are distilled from the global tracking database with the annotations of LLM; (4) cognitive-aligned data are produced
through LLM annotation and human screening; (6) ST-LLM is further fine-tuned and aligned to finish this procedure. Online
Serving: CoAST is stored in a model engine and invoked through a data-service engine that converts real-time context—user
profile, recent check-ins, current time, location, and weather—into an instruction prompt. The model predicts the next POI in
SID form, which is mapped back to a readable location and returned to the mobile client.
where 𝐿represents the number of codebooks and 𝐾denotes the
size of each codebook. Therefore, a POI can be tokenized as an
ordered sequence of tokens, such as "<a_17> <b_21><c_119>", in
which the number of the above tokens represents the indices of the
responding codebook.
4.1.2
Continued Pretraining on Spatio-temporal Corpus. After ac-
quiring SIDs of the POIs, a simple method is to incorporate these
SIDs into the vocabulary of large language models (LLMs). This
allows the LLMs to perform recommendation tasks in a generative
manner, ultimately moving away from using item indices. Never-
theless, there are two remaining concerns: (1) these newly added
indices are out-of-vocabulary tokens for LLMs, incorporating these
linguistic and collaborative semantics becomes necessary. (2) Rec-
ommendation tasks present unique technical challenges that distin-
guish them from conventional NLP tasks, particularly in modeling
spatio-temporal user behaviors and handling data sparsity.
To achieve this incorporation, we implemented continued pre-
training [14] on the recommendation corpus, aiming to adjust LLMs
to improve understanding of linguistic and collaborative semantics.
Our approach to recommendation domain-adaptive pretraining is
straightforward—we continue pretraining a base model (like Qwen3
[41]) on a large corpus of unlabeled spatio-temporal recommenda-
tion domain-specific text. Specifically, the corpus are designed and
collected mainly focusing on 2 objectives: clear index-language-
space aligning (finding the POI with its description, location and
its category, or outputting the description and location given the
POI) and sequential POI prediction (predicting the next POI given
the historical POI check-ins).
While understanding the spatial relationships between POIs is
fundamental to LBS recommendation, directly instilling this knowl-
edge into LLMs is difficult. To address this, our approach tokenizes
each POI’s Geohash and integrates these tokens into our pretraining
data. This process enables the model to learn a unified embedding
space where the POI’s identity (SID), its semantic meaning, and its
location (via Geohash) are jointly represented and aligned. Table 1
illustrates the utilized corpus in this stage.
Table 1: Recommendation corpus collected for continuous
pretraining based on a base model. The words in orange
denote the newly added tokens in the base LLM.
SID-Location-Description Alignment Corpus
The Temple of Heaven Park is the world’s largest surviving
ancient architectural complex dedicated to the worship of
heaven. Originally constructed during the Ming Dynasty, it
served as the site where emperors of the Ming and Qing dy-
nasties held ceremonies to “sacrifice to heaven” and “pray for
a good harvest.” Located in <wm6j0> of Beijing, and the ID is
<a_82><b_59><c_191>.
Spatio-Temporal Behavior Sequence Recommendation
Corpus
Here is user uid_256’s historical POI check-ins. At 7 AM
June 10, the user ride to company <a_124><b_192><c_41>,
at 21 PM June 11, the user navigated to the hotel
<a_82><b_59><c_191>, at 15 PM June 20, the user searched
an office <a_12><b_28><c_140>,..., at 19 PM July 20, the user
walked to home <a_124><b_192><c_41>.
Masked prediction has recently been applied to adapt pretrained
models to specific downstream tasks by continuing to pretrain
models on in-domain unlabelled data [6, 15, 37]. Inspired by these
works, we continue our pretraining with masking some original
tokens of the base LLM. However, a crucial consideration in our
pretraining phase is the token composition of the training corpus,
Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
which presents a delicate trade-off between newly introduced POI
tokens (SIDs and Geohash tokens) and the original tokens within
the pretraining corpus.
Through the comprehensive continued pretraining on the corpus
of spatio-temporal recommendation domain, the model is adapted
to learn information centered around generic behavioral patterns.
However, its outputs may exhibit cognitive mis-alignment with
human reasoning principles. This necessitates a specialized cog-
nitive calibration process to bridge the divergent representations
between machine-learned patterns and anthropic decision-making
frameworks.
4.2
Cognitive Preference Alignment
4.2.1
User Profile Generation. Understanding user historical POI
check-ins is crucial for modeling the trajectory patterns of users.
However, it is challenging to effectively utilize extended user be-
havior sequences, which often extends to lengths of 10,000 in large-
scale recommendation systems [11, 16]. Yet, LLMs face difficulties
in processing these long sequences, and this hinders their ability
to grasp the patterns. To counteract this, we have developed multi-
level profile of the users by condensing extensive user behavior
into refined profiles that are more amenable to LLMs. The utilized
profiles can be mainly divided into three main categories: (1) Static
Profile, (2) Long-term Preference Profile, and (3) Periodic Demand
Profile.
Static profile, such as POIs of the users’ home and company,
are critical for next POI prediction. However, the labeled data is
pretty limited. To expand the coverage of static profiles, we first
employ LLM-based summarization of user behavioral sequences to
generate preliminary annotations. These labeled patterns are then
utilized to train models (like a MOE-Transformer network [7]) that
subsequently perform inference across the entire user population,
producing comprehensive static profiles that maintain semantic
consistency with human cognitive patterns.
Long-term preference profile refers to stable and long-term in-
tentions derived from long-term behaviors, which focuses more
on stability, accuracy, and long-term relevance, requiring a more
rigorous reasoning and calibration process [39, 46].
Periodic demand profile identifies recurring consumption pat-
terns that exhibit temporal periodicity correlated with interval cy-
cles (e.g., weekly/monthly), holiday events, and workday rhythms.
We employs a LLM to summarize the extended behavioral sequences
through scheduled interval sampling, implementing an automated
periodicity and long-term preference detection mechanism.
4.2.2
Cognitive-aligned Data Generation. Within the vast amount
of POI check-in data left by users, a significant portion is considered
noise (such as accidental clicks), or does not align with the human
cognition process. Therefore, we intervene in data collection by
combining the world knowledge of LLMs and human annotations.
Following similar process of [39], we collect the data from tradi-
tional CTR-oriented recommender systems that align with human
cognition preference defined in Section 3.
In particular, LLMs is utilized to access the users’ historical check-
ins, profile information and other contextual information (like time
and weather) to conduct chain-of-thought (CoT) reasoning [35],
thereby evaluating the potential candidate items aligned with hu-
man cognition for individual users. Subsequently, human annota-
tors review the reasoning process and outcomes generated by the
LLMs to refine and improve the accuracy of serendipity labels. This
annotated dataset—which contains user history, profiles, candidate
items, contextual conditions, and cognition annotations—is then
utilized to further fine-tune the model after continued pretraining
phase that can identify the examples aligned with human cogni-
tion for the subsequent training. This approach enables scalable
discovery of cognitive-aligned items while reducing the need for
extensive manual annotation.
4.2.3
Supervised Fine-tuning. After obtaining cognitive-aligned
data, with each record consisting of user𝑢, their multi-level profiles,
historical POI check-ins ℎ𝑢, the current situation of the user (time,
location, weather and so on) and the selected cognitive-aligned POI
𝑝𝑖, we begin the process of fine-tuning by aligning the inputs. Ini-
tially, all of the inputs are transformed into a text-based supervised
fine-tuning (SFT) dataset, where each record is a pair (𝑥,𝑦), with
𝑦denoting the output SID of the target POI 𝑝𝑖, and 𝑥as the input
prompt. For example,
Instruction:
Based on the user’s profile: {user-profile}, and his historical
POI check-ins: {check-ins}. Now the time is {time}, he is in
{user-location}, and the weather is {weather}. Please recom-
mend the next possible POI that the user may expect?
Response:
<a_112><b_32><c_5>
Subsequently, LLM is fine-tuned under supervision to enhance
its ability to produce the desired responses.
4.2.4
Cognitive Preference Alignment. Following the standard par-
adigm for human preference alignment, we continue to leverage
Reinforcement Learning (RL) to further refine the model’s outputs
and explicitly align its recommendations with human cognitive
preferences.
After obtaining datasets comprising human-annotated prefer-
ences over pairs of model-generated responses, we train the model
with the preference alignment algorithm. We choose Direct Prefer-
ence Optimization (DPO) [28] in this phase, since it has been widely
employed to align the model’s outputs more closely with human
preference.
L = −E(𝑥,𝑦𝑤,𝑦𝑙)∼𝐷
log𝜎
𝛽𝜋𝜃(𝑦𝑤| 𝑥)
𝜋ref(𝑦𝑤| 𝑥) −𝛽𝜋𝜃(𝑦𝑙| 𝑥)
𝜋ref(𝑦𝑙| 𝑥)
(5)
where 𝐷denote the preference dataset, 𝑥denotes the input prompt,
and 𝑦𝑤and 𝑦𝑙represent the preferred and dis-preferred responses
annotated by human and LLM. 𝜋𝜃is the trainable LLM for pref-
erences alignment, while 𝜋ref is a fixed reference policy. 𝜏is the
hyper-parameter of the regularization term balancing learning hu-
man preferences and staying close to the reference policy, thereby
preventing overfitting.
In summary, after the Supervised Fine-Tuning (SFT) phase, our
model adheres to the required output format (a list of 𝐿valid SIDs)
and already possesses a baseline level of cognitive awareness from
the training data. The subsequent DPO phase then significantly
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Zhai et al.
enhances this cognitive reasoning while preserving the model’s
recommendation performance.
4.3
Online Deployment
A primary challenge for the large-scale application of generative
models is their inference latency, particularly in recommendation
scenarios with high Queries Per Second (QPS). To enable the large-
scale online deployment of our generative model, inspired by recent
works [13, 52], we implemented a dual-pronged optimization strat-
egy targeting both system architecture and algorithmic efficiency,
achieving significant breakthroughs in stability and performance.
4.3.1
Prefill-Decoding Decoupling. We employ Prefill-Decoding
(PD) Decoupling [52] to separate the inference process into two
distinct stages: a computationally-intensive Prefill stage for prompt
processing and a memory-bound Decoding stage for iterative token
generation. By applying dedicated optimization strategies to each
stage, this approach mitigates resource contention and latency
jitter common in unified pipelines. The resulting improvements in
resource scheduling and concurrency led to a 10-fold increase in
throughput, drastically reducing the per-inference cost on the same
hardware.
4.3.2
Multi-Token Prediction. Furthermore, we leverage Multi-Token
Prediction (MTP) to overcome the sequential dependencies of tra-
ditional auto-regressive generation [13]. Specifically, we adopt
MTP Eagle, an optimized variant of the Eagle speculative decoding
method. Its key innovation lies in a single, reusable MTP module
that shares the same KV cache for predicting multiple draft tokens,
thus eliminating the redundant computational overhead of vanilla
MTP. Its chained decoding strategy, which only requires the last
predicted token and its hidden state to proceed, significantly sim-
plifies implementation by obviating the need to store historical
states. MTP Eagle is fully compatible with existing checkpoints
(e.g., DeepSeek-V3) and substantially reduces both memory foot-
print and computational complexity, enhancing the engineering
feasibility and resource efficiency of our deployment.
Collectively, these optimizations create a highly efficient and
stable inference framework. This is crucial for deploying large-scale
generative models in production, ensuring both a responsive user
experience and the economic viability of the system in a real-world
advertising setting.
5
Experiments
5.1
Experimental Settings
5.1.1
Scenario. Our method is performed on AMAP, a prominent
navigation and mapping platform hosting billions of users and POIs.
We focus on the "Guess Where You Go" card presented on the
homepage of the AMAP App, aiming to forecast the next subse-
quent POI using a user’s previous check-ins and profile information.
The objective of this scenario is to predict the next POI the users
are most likely to visit. By displaying the POIs on the homepage,
we aim to reduce the users’ search effort, thereby enhancing user
experience and long-term retention. Critically, our goal transcends
mere prediction accuracy; we strive to build a model that captures
the underlying dynamics of user mobility patterns and intents. This
allows the system to develop a deeper "cognitive" understanding,
ultimately facilitating more intelligent and context-aware decision-
making for the user.
The current deployed framework, which serves as our primary
baseline, employs a traditional cascaded ranking system for recom-
mendation. As illustrated in Figure 2, the cascaded ranking system
includes three stages: (1) Retrieval stage to recall a coarse-grained
POI corpus based on the user’s preference, historical check-ins, and
location [18, 44]; (2) Ranking stage to obtain top 10 POIs the users
most likely visit with a ranking model based on behavior sequence
transformer [3]; and (3) Cognitive Re-ranking stage with distance,
quality, and exposure filters to yield the most cognitively plausible
destination.
Figure 2: Cascade ranking system of "Guess Where You Go"
deployed on AMAP, which includes three stages from the left
to the right: Retrieval, Ranking, and Cognitive Re-ranking.
We evaluate our approach on three real-world datasets: Foursquare-
NYC [42], Foursquare-TKY [42], and Gowalla-CA [4] and a large-
scale proprietary dataset by sampling desensitized users from AMAP
(named Beijing). The details of the utilized dataset is illustrated in
Appendix A.
5.1.3
Implementation Details. The quantization module consists
of 3 codebook layers and for the generative recommendation mod-
ule, we employ the Qwen [27, 41] as the base model. We employ a
constant learning rate schedule with a learning rate of 1e-5, com-
bined with a warm-up phase of 20 steps. The model is trained on
200× NVIDIA H20 GPUs, with a batch size of 16 per GPU, gradient
accumulation steps set to 8, and a sequence length of 4096 tokens.
Each input consists of the most recent 50 check-in records of the
user.
5.1.4
Evaluation Metric. Following previous works, we employ the
accuracy metric for the top-1 recommendation accuracy (Acc@1)
and the four cognitive metrics (TCS, SCS, PAS and SAS) defined
in 3.2 as our metrics to evaluate the effect of the cognitive recom-
mendation. Detailed information of the metrics is illustrated in
Appendix C.
5.1.5
Baseline Methods. The details of the baseline methods in this
part is illustrated in Appendix B.
5.1.6
Our Models. In our experiments we consider three versions
of our model: (i) CoAST-0.5B: the model trained with Qwen2.5-0.5B
[27] as our base LLM. (ii) CoAST-1.5B: with Qwen2.5-1.5B as our
base LLM. (iii) CoAST-3B: A variation with Qwen2.5-3B as our base
LLM. (iv) CoAST-7B: A second variation on Qwen2.5-7B. Below,
unless stated otherwise, when we say “our model,” we refer to the
CoAST-0.5B.
Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Table 2: Comparison of different models on three datasets:
NYC, TKY, and CA. We present the model inputs including
the POI and User representation approach and whether visit
timestamps are utilized. RID, SID, UID and UP represent
Random one-hot ID, Semantic ID, User one-hot ID and User
Profiles, respectively. The results for the baseline methods
are borrowed from [19, 32].
Model
Inputs
Acc@1
POI
User
Time
NYC
TKY
CA
PRME
RID
UID
×
0.1159
0.1052
0.0521
PLSPL
RID
UID
×
0.1917
0.1889
0.1072
STAN
RID
UID
×
0.2231
0.1963
0.1104
GETNext
RID
UID
×
0.2435
0.1829
0.1357
STHGCN
RID
UID
×
0.2734
0.2950
0.1730
TPG
RID
UID
✓
0.2555
0.1420
0.1749
ROTAN
RID
UID
✓
0.3106
0.2458
0.2199
LLM4POI
RID
UID
✓
0.3372
0.3035
0.2065
GNPR-SID
SID
UID
✓
0.3618
0.3062
0.2403
CoAST-0.5B
SID
UP
✓
0.3705
0.3125
0.2555
CoAST-1.5B
SID
UP
✓
0.3727
0.3170
0.2585
CoAST-3B
SID
UP
✓
0.3872
0.3202
0.2611
CoAST-7B
SID
UP
✓
0.4027
0.3310
0.2721
Table 3: Comparison results of different versions of CoAST
on Beijing dataset with the Cascade Ranking baseline system.
’a-’ denotes average.
Baseline
CoAST-0.5B
CoAST-1.5B
CoAST-3B
CoAST-7B
Acc@1
0.2675
0.2820
0.2862
0.2923
0.3071
a-TCS
0.5561
0.6071
0.6021
0.6632
0.7543
a-SCS
0.8022
0.8212
0.8231
0.8452
0.8838
a-PAS
0.6232
0.6285
0.6240
0.6531
0.6970
a-SAS
0.2472
0.2840
0.2988
0.3240
0.3725
5.2
Offline Experiments
5.2.1
Overall Performance. Our method substantially outperforms
all baseline methods across all of the evaluated datasets. Table 2
illustrate the comparison results on NYC, TKY, and CA datasets.
Specifically, compared to the state-of-the-art GSNR-SID which uti-
lizes LLaMA-7B as its base model, we observe a substantial im-
provement on all of the versions of CoAST. Notably, CoAST-7B
demonstrated the most significant improvements, achieving gains
of 11.3%, 8.1%, and 13.2% over the SOTA, respectively.
Table 3 details the performance comparison between our pro-
posed model and the cascade ranking system on the Beijing dataset.
In addition to the standard Acc@1 metric, we evaluate the models
on four cognitive-aware recommendation metrics (where ’a-’ de-
notes average). Consistent with our findings on other datasets, our
model significantly outperforms the traditional cascade ranking
system on Acc@1. More critically, it demonstrates substantial gains
on the cognitive metrics, achieving remarkable uplifts of 35.6% in
temporal consistence and 50.68% in situational awareness perfor-
mance. These results highlight a key limitation of conventional
collaborative filtering-based systems: their struggle to capture per-
sonalized, periodic user needs and inability to effectively model
situational or contextual information. This validates the superiority
of our generative approach, which excels in understanding these
complex, dynamic aspects of user behavior.
5.2.2
Ablation Study. To evaluate the impact of individual modules,
we developed various versions of CoAST. Initially, certain essential
modules were removed: "w/o UP" eliminates user profiling from
the prompt. "w/o S-ST" eliminates situational and spatio-temporal
descriptions from the prompt. "w/o SLD" exclude the SID-Location-
Description alignment corpus in the continued pretraining stage.
"w/o CPT" exclude the Continued Pretraining stage in the align-
ment stage, only applying SFT and DPO. As illustrated in Table 4,
removing any module leads to noticeable performance degradation,
emphasizing the importance of each component. Significantly, the
lack of SFT results in the most considerable decrease in all met-
rics, demonstrating the essential role of preference alignment in
recommendation efficiency.
Table 4: Ablation study of CoAST (with the online deploy-
ment version CoAST-3B).
Acc@1
a-TCS
a-SCS
a-PAS
a-SAS
w/o UP
0.2720
0.6435
0.8211
0.5920
0.2923
w/o S-ST
0.2708
0.5905
0.7938
0.6496
0.2822
w/o SLD
0.2825
0.6078
0.8240
0.6501
0.3020
w/o CPT
0.2632
0.5875
0.7820
0.6020
0.2875
CoAST
0.2923
0.6632
0.8452
0.6531
0.3040
5.2.3
Case Study. To provide a qualitative and intuitive demon-
stration of our model’s advantages, we conduct a case study that
contrasts the recommendation lists generated by our model with
those from a baseline model for a representative user. Detailed
information of the case is illustrated in Appendix D.
5.2.4
Further Analysis. In this section, we analyze the impact on
the performance of sequence length (input tokens) and number of
utilized tokens in continued pretraining stage. Figure 3 presents the
results. Our investigation into the impact of input sequence length
shows that while longer context is beneficial, its utility eventually
saturates. The model’s performance gains are most pronounced
up to a context window of 2048 tokens (∼30 user historical check-
ins) and largely stabilize beyond 4096 tokens (∼50 check-ins). This
suggests that a moderately-sized history is sufficient to encapsulate
the user’s core preferences. Further extending the context window
provides limited new signal, as this long-term information is already
implicitly encoded in the user’s learned representation.
The volume of tokens involved in pretraining phase also proved
to be a key determinant of performance. We observed a nearly linear
improvement in model effectiveness as the training corpus size
increased. Constrained by our training budget, the final model was
trained to convergence on a dataset of 10B tokens. The consistent,
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Zhai et al.
non-saturating growth suggests that performance is largely data-
bound, with significant potential for improvement as more data
becomes available.
Figure 3: Impact on the performance of sequence length
(input tokens) and number of tokens involved in contin-
ued pretraining stage. a-CAS denotes the average Cognition
Alignment score computed by averaging the four cognitive
scores.
5.3
Online Experiments
5.3.1
Efficiency Analysis. To evaluate the online inference effi-
ciency, we compare different variants of our model with the efficiency-
SOTA cascade ranking system. Before deploying our model, we con-
duct a load test under the settings with a 50 QPS on 2× NVIDIA H20
GPUs. Figure 4 presents the inference p99 latency results of different
methods. As shown, the Cascade Ranking system, a widely adopted
paradigm in industrial recommender systems, demonstrates ex-
ceptional efficiency with its latency consistently around 20ms. As
a comparison, while CoAST-7B achieved the best offline perfor-
mance among all our variants, its high computational cost posed
a significant challenge for online serving. Even after applying the
acceleration strategies described in Section 4, its latency remained
above 120ms, rendering large-scale deployment infeasible.
To strike a practical balance between efficacy and efficiency, we
identified CoAST-0.5B as the optimal candidate. It delivers compet-
itive performance while maintaining a manageable inference cost
(∼30ms). Consequently, we deployed CoAST-0.5B to production
and conducted a online A/B test against the state-of-the-art (SOTA)
Cascade Ranking baseline.
5.3.2
Online A/B Test. To assess the online effect of CoAST, we
implemented a short-term and long-term A/B testing on "Guess
Figure 4: Comparison of Model Inference p99 Latency (ms)
under a 50 QPS Load Test on 2× NVIDIA H20 GPUs.
Where You Go" card of AMAP’s homepage. We examined CoAST-
0.5B, trained based on a Qwen2.5-0.5B model, compared to the
traditional cascade ranking baseline as shown in Table 5.
Table 5: Online improvement over traditional Cascade Rank-
ing baseline of short and long term A/B test. % denotes rela-
tive improvement. short and long represents short-term and
long-term A/B online test, respectively.
P-CTR U-CTR SC-Rate RE-Rate AC-Rate NF-Rate
short +5.06% +4.65%
+0.69%
+0.31%
+0.77%
-5.11%
long
+4.23% +4.59%
+1.01%
+1.65%
+5.99%
-5.85%
It is worth mentioning that this baseline already includes user
profiles, trending topics and seasonal features for improving cogni-
tion in the cognitive re-ranking stage. For short-term (one week)
online test, CoAST demonstrate significant improvements in met-
rics like P-CTR (PV CTR) and U-CTR (UV CTR). This indicates that
CoAST has increased the share of cognitive-aligned recommen-
dations that effectively attracted user attention in terms of clicks.
Additionally, it slightly enhances the user engagement metrics,
such as SC-Rate (Scroll Rate), RE-Rate (User Retention Rate), and
AC-Rate (User Active Rate), implying heightened user involvement.
Notably, the effects on general utility metrics such as CTR and NF-
Rate (Users’ Negative Feedback Rate) are more modest, while the
improvements on these engagement-related metrics often become
apparent only over the long term.
To investigate the long-term effects of the proposed method
in large-scale RSs, we maintain a small traffic for our baselines
and compare it with CoAST for over one month. Compared to the
baseline and the performance on the short-term test, our method
has achieved a significant enhancement in the user engagement
metrics, with a 5.85% average decrease in NF-Rate (Users’ Negative
Feedback Rate). This demonstrates that CoAST could help to miti-
gate counter-intuitive recommendations and effectively improve
user engagement while also boosting CTR, thereby enhancing the
overall user experience.
6
Conclusion
In this paper, we proposed a LLM-based next POI prediction ap-
proach, named CoAST. In order to adapt LLMs to sequential recom-
mendation tasks, we focused on a comprehensive pretraining task
to adapt a base LLM to align language and collaborative semantics
for recommendation. This task include two categories of corpus:
sequential POI prediction and explicit index-language alignment.
Based on the learned item indices, our approach employed these
alignment tasks to effectively adapt LLMs for sequential recom-
mendation. Furthermore, we give complete definition and metrics
on cognitive-aligned next POI prediction task. For constructing
a cognitive-aligned POI recommendation system in industrial ap-
plication, we design a multi-stage fine-tuning process, utilizing
a curated dataset of cognitively-aware samples for both Super-
vised Fine-Tuning (SFT) and a subsequent Reinforcement Learning
(RL) phase. This procedure explicitly align the model with human
common-sense. Extensive offline experiments on three datasets
demonstrated the effectiveness of our approach and Online A/B
Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
test achieves substantial improvements compared to traditional
cascade ranking recommendation system.
Acknowledgments
We would like to acknowledge the discussions of Jiawei Xue, Ning
Wang, and Fangfang Chen throughout this work. Furthermore, we
are grateful for Tucheng Lin, Jian Song, Shulong Han, Jinhui Chen,
Xiongfei Fan, Zudan Cao, Jing sun, Yifei Fan and Yukun Liu, for
their engineering support.
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Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Figure 5: Case comparison of recommendation results of CoAST and the traditional basline RSs. The top row represents the
user’s real-time check-ins with different action types. The middle and bottom row represent the recommendation results of
traditional basline RSs and CoAST, respectively.
A
Dataset
To validate our model’s performance in a real-world application,
we constructed a large-scale proprietary dataset by sampling de-
sensitized users from AMAP (named Beijing). On this dataset, we
conducted both extensive offline evaluations and an online deploy-
ment. Furthermore, to ensure a comprehensive and fair comparison
against existing baselines, we also evaluate our approach on three
real-world datasets: Foursquare-NYC [42], Foursquare-TKY [42],
and Gowalla-CA [4].
The pre-processing follows the settings in [32]. we pre-process
each dataset by removing POIs with fewer than 10 interactions
and users with fewer than 10 check-ins. The data is then sorted by
time, with 80% used for training, 10% for validation, and 10% for
testing. Users and POIs that do not appear in the training set are
removed from the test set to ensure consistency between training
and evaluation. Subsequently, the check-in records are grouped by
user and ordered chronologically. For each user, the last visited POI
is held out as the ground truth during evaluation, while the preced-
ing visits are used as the input sequence. Note that in the training
set, the user’s visit records are also treated as historical sequences
and concatenated with the test set to the specified length before
being fed into the model. Detailed dataset statistics are provided in
Table 6.
Table 6: Statistics of the preprocessed datasets. Avg.len de-
notes the average length of item sequences.
Dataset
#Users
#Items
#Inter.
Avg.len
Sparsity
NYC
1,083
5,135
104,074
136
98.1013%
TKY
2,293
7,873
361,430
195
97.9798%
CA
6,592
14,027
349,375
53
99.9996%
Beijing
10,000k
1,520k
38,0000k
38
99.9999%
B
Baseline
Following the settings of [19, 32], we compare our proposed method
with various next POI recommendation baselines, which can be
specified as the following four categories: (1) Traditional meth-
ods: PRME [9], a ranking-based metric embedding method; and
PLSPL [38], a personalized sequential recommendation model. (2)
Transformer-based methods: STAN [25], a spatio-temporal atten-
tion network; and GETNext [43], a Transformer-based next POI
recommendation model. (3) GCN-based method: STHGCN [40],
a spatio-temporal hierarchical graph convolutional network. (4)
Time-aware methods: TPG [24], a timestamp-guided model; and
ROTAN [10], a time-aware POI recommendation framework. (5)
LLM-based method: LLM4POI [19], a method based on LLMs; and
GNPR-SID [32], a state-of-the-art based on Semantics codes and
generative recommendation framework, which is pretty closely
related to our work.
C
Evaluation Metric
Following previous works, we employ the accuracy metric for the
top-1 recommendation, denoted as Acc@1, at the specified time.
Acc@1 = 1
𝑛
𝑛
∑︁
𝑖=1
I(𝑦𝑖= ˆ𝑦𝑖)
where 𝑛is the number of test samples, 𝑦𝑖and ˆ𝑦𝑖represent the
ground-truth label and the predict label of the 𝑖-th sample, respec-
tively. I is the indicator function, which equals 1 when the condition
is true and 0 otherwise. Otherwise, we take the four cognitive cat-
egories (TCS, SCS, PAS and SAS) defined in 3.2 as our metrics to
evaluate the effect of the cognitive recommendation.
D
Case Study
Figure 5 illustrates the case comparison of recommendation results
of CoAST and the traditional basline RSs. In the traditional cas-
cade ranking RSs, the model collapses to the user’s dominant daily
rhythm: go to company at morning and back to home at afternoon.
However, during other times of the day, conventional CTR-oriented
models tend to be dominated by users’ most recent interactions.
This short-term focus often leads to recommendations that fail
to capture the user’s broader, long-term interests, resulting in a
counter-intuitive user experience.
Conference acronym ’XX, June 03–05, 2018, Woodstock, NY
Zhai et al.
In contrast, our proposed generative model achieves a better
alignment of the user’s temporal context with their holistic prefer-
ence profile, thereby fostering a more well-rounded and sustainable
recommendation ecosystem.
|
Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Penglong Zhai∗ AMAP, Alibaba Group Beijing, China Jie Li∗ AMAP, Alibaba Group Beijing, China Fanyi Di∗ AMAP, Alibaba Group Beijing, China Yue Liu AMAP, Alibaba Group Beijing, China Yifang Yuan AMAP, Alibaba Group Beijing, China Jie Huang AMAP, Alibaba Group Beijing, China Peng Wu AMAP, Alibaba Group Beijing, China Sicong Wang AMAP, Alibaba Group Beijing, China Mingyang Yin AMAP, Alibaba Group Beijing, China Tingting Hu AMAP, Alibaba Group Beijing, China Yao Xu AMAP, Alibaba Group Beijing, China Xin Li† AMAP, Alibaba Group Beijing, China Abstract The next point-of-interest (POI) recommendation task aims to predict the users' immediate next destinations based on their preferences and historical check-ins, holding significant value in locationbased services. Recently, large language models (LLMs) have shown great potential in recommender systems, which treat the next POI prediction in a generative manner. However, these LLMs, pretrained primarily on vast corpora of unstructured text, lack the native understanding of structured geographical entities and sequential mobility patterns required for next POI prediction tasks. Moreover, in industrial-scale POI prediction applications, incorporating world knowledge and alignment of human cognition, such as seasons, weather conditions, holidays, and users' profiles (such as habits, occupation, and preferences), can enhance the user experience while improving recommendation performance. To address these issues, we propose CoAST (Cognitive-Aligned Spatial-Temporal LLMs), a framework employing natural language as an interface, allowing for the incorporation of world knowledge, spatio-temporal trajectory patterns, profiles, and situational information. Specifically, CoAST mainly comprises of 2 stages: (1) Recommendation ∗Contributed equally to this research. †Corresponding Author. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from . Conference acronym 'XX, Woodstock, NY Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 978-1-4503-XXXX-X/2018/06 https://doi.org/XXXXXXX.XXXXXXX Knowledge Acquisition through continued pretraining on the enriched spatial-temporal trajectory data of the desensitized users; (2) Cognitive Alignment to align cognitive judgments with human preferences using enriched training data through Supervised FineTuning (SFT) and a subsequent Reinforcement Learning (RL) phase. Extensive offline experiments on various real-world datasets and online experiments deployed in "Guess Where You Go" of AMAP1 App homepage demonstrate the effectiveness of CoAST. CCS Concepts • Information systems →Location based services. Keywords Generative Recommendation, Next POI Prediction, Cognitive Recommendation, Large Language Models ACM Reference Format: Penglong Zhai, Jie Li, Fanyi Di, Yue Liu, Yifang Yuan, Jie Huang, Peng Wu, Sicong Wang, Mingyang Yin, Tingting Hu, Yao Xu, and Xin Li. 2018. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Pointof-Interest Prediction. In Proceedings of Make sure to enter the correct conference title from your rights confirmation email (Conference acronym 'XX). ACM, New York, NY, USA, 12 pages. https://doi.org/XXXXXXX.XXXXXXX 1 Introduction Next Point-of-Interest (POI) prediction, as a variant of POI recommendation task, aiming to predict users' future locations based on their historical sequential check-ins and other contextual information, is a fundamental task in Location-Based Services (LBS) [9, 10, 19, 24, 25, 32, 36, 38, 40, 43]. For large-scale navigation platforms such as AMAP, which processes billions of daily requests 1https://amap.com/ 16 Oct 2025 Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. from hundreds of millions of active users in China, making accurate next POI prediction a critical capability for efficient destination planning and personalized recommendation. Thus, next POI prediction has been a popular research direction, capitalizing on developments in mobile and localization techniques, as they provide rich location-based information. Much of the existing research in next POI recommendation has predominantly focused on sequential recommendation and graph neural networks, tackling the challenge of data sparsity, particularly for cold-start users and those with sparse historical trajectories [8, 29, 33, 45, 48]. However, these approaches often fail to encapsulate the rich contextual information embedded within Location-Based Social Network (LBSN) data, such as the world knowledge and contextual information, which are critical for improving the user experience in industrial applications. Recently, Large Language Models (LLMs) have demonstrated remarkable capabilities across various domains [1, 27, 31, 41], offering new opportunities for this direction. However, their direct application to POI prediction task remains several key challenges. First, POIs are typically indexed by discrete identifiers outside the LLM vocabulary, impeding the model's ability to capture semantic relationships [2, 47, 50]. Second, the intricate interactions between the billions of users and POIs, coupled with highly personalized and context-dependent behavioral patterns, introduce complex collaborative signals that are difficult for LLMs to encode and reason effectively [23, 25]. Another critical challenge, stemming from the real-world deployment, is that recommended POIs must be cognitively plausible to users. Recommendations that violate human intuition or commonsense reasoning, which we term human cognitive preference in this work, even if statistically likely, can appear nonsensical or irrelevant, severely degrading the user experience and eroding trust in the system [25]. In industrial applications, these logical lapses are critical failures, as they directly risk frustrating users and reducing retention. Based on these observations, we formulate the task of human cognitive preference alignment for individual POI prediction as: to investigate issues of personality and individual differences (who), environmental and social impacts (what), spatio-temporal features and relationships (when and where), and corresponding actions (how). To address the aforementioned challenges and build a POI recommendation system that aligns with human cognitive preferences, we introduce CoAST (Cognitive-Aligned Spatio-Temporal LLM), a novel framework that endows a base LLM with the ability to generate cognitively plausible POI recommendations. CoAST begins by adapting the LLM to the mobility domain through continued pretraining on an enriched corpus of spatio-temporal tokenized POI sequences, allowing it to master complex user behavior patterns. Furthermore, we explicitly align the model with human commonsense. This is achieved through a multi-stage fine-tuning process, utilizing a curated dataset of cognitively-aware samples for both Supervised Fine-Tuning (SFT) and a subsequent Reinforcement Learning (RL) phase. Our main contributions as as follows: • To our knowledge, this is the first generic LLM-based cognitivealigned framework for the field of next POI prediction. The proposed framework enhances the comprehension of human cognitive preference in recommendation. • We propose CoAST, a novel end-to-end LLM-based framework for large-scale next POI prediction that effectively bridges the semantic gap between world and recommendation knowledge, which demonstrates significant improvement on human cognitive alignment and applicable to practical applicability of LLMs in LBS applications. • Through a series of inference optimization strategy, CoAST is deployed in AMAP's online production environment, a leading real-world LBS platform. Compared traditional cascade ranking system, long term online A/B test achieves a 4.23% and 4.59% increase in P-CTR and U-CTR, which is a substantial improvement. The remainder of this paper is organized as follows: Section 2 reviews related literature, Section 3 details problem definition and the detailed categories of cognitive recommendation, Section 4 elaborates our methodology, Section 5 reports empirical results, and Section 6 concludes the paper. 2 Related Work 2.1 Next POI Prediction Next POI recommendation extends beyond conventional item recommendation by incorporating inherent geographical and temporal characteristics. Early methods relied on user check-in histories using Markov chains, or RNNs to capture sequential behaviors. Spatial-temporal modeling has been extensively studied with works like ST-GRAT [26] employing graph attention networks for spatial-temporal dependencies, STAN [25] introducing hierarchical attention mechanisms, and Sun et al. [30] focusing on periodic patterns in human mobility through time-aware embeddings. Recent LLM-based approaches have focused on addressing the semantic gap in POI recommendation. For example, GNPR-SID [32] proposed semantic POI IDs using residual quantized variational autoencoders, achieving improvements by mapping POIs into discrete semantic spaces. LLM4POI [19] leveraged pretrained LLMs to preserve heterogeneous data in original format, effectively utilizing contextual information. 2.2 LLMs for Recommendation The integration of Large Language Models (LLMs) into recommendation systems has emerged as a transformative research direction. P5 [12] pioneered the "Pretrain, Personalized Prompt, and Predict" paradigm, establishing the foundation for treating recommendation as a language processing task by converting all recommendation data into natural language sequences. Building upon this, [2] addressed the knowledge gap between LLMs' capabilities and recommendation-specific requirements by incorporating collaborative filtering operations like Masked Item Modeling (MIM) and Bayesian Personalized Ranking (BPR) through natural language simulation. More recently, GenRank [17] advanced generative ranking systems for large-scale deployment. OneRec [5, 53] proposed a revolutionary end-to-end framework that replaces traditional cascaded architectures, achieving significant improvements in large-scale online deployments. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY 3 Preliminaries 3.1 Problem Formulation Given a collection of POIs I = {p1, p2, ..., pN} and a group of users U = {u1,u2, ...,uM}. Each POI can be characterized by a tuple pi= (lon,lat,c), where crepresents side information of the POI including the descriptive texts and pictures, lonand latdenotes the geographical coordinates (longitude and latitude). Each user u has the check-ins history hu= {(p1,t1), . . . , (pm,tm)} of length m, where pi,i= 1, . . . ,mreflects the i-th POI recently interacted by the user and tirefers to the check-in timestamps. The objective of next POI prediction task is to predict the next POI pthat the user u will visit at time tgiven the recent check-in record hu, the user's request location land the contextual situation s. 3.2 Cognitive-Aligned POI Prediction Compared to traditional accuracy-oriented recommender systems that rely on rapidly capturing users' immediate interests, next POI prediction is a complex issue in simulating individual behaviors and contextual information. Empirically, we category cognitive-aligned recommendations into the following four scenarios. Recommended POIs that do not align with these categories may result in a diminished user experience or decreased user engagement. To summarize, our approach formulates the next POI recommendation task as a decision-making problem. The model synthesizes a diverse set of contextual factors-including temporal (when), spatial (where), user-specific (who), and other situational inputs-to determine the optimal user action (what to do and how). 3.2.1 Temporal Consistency. User behavior often exhibits certain regularities. For example, office workers typically commute to workplaces in the morning and return home at noon. Recommending irrelevant POIs during these commuting periods may lead to poor user experiences. We define these observed regularity patterns in user mobility as temporal consistency. Formally, at the given time t, the temporal consistency score (denoted as TCS) of the predicted POI ˆpcan be formulated as, TCS(u,l,s,t, ˆp) = Í i=mI(ti= t) · sim(pi, ˆp∗) Ím i=1 I(ti= t) (1) where sim(·) denotes the function measuring the similarity of the inputs pair. 3.2.2 Spatial Clustering. Check-ins of human are empirically demonstrated to occur within a certain range [20, 21]; that is, the distance of a user's next visit POI is typically distributed as a negative exponential relative to the user's usual activity space. Recommending a POI that is far from the user's position (or their active area) is inappropriate. We define the spatial clustering phenomenon of user mobility as spatial consistency. Formally, given the location puof the user u, the spatial consistency score of the predicted POI ˆpis formulated as, SCS(u,l,s,t, ˆp) = e-d(pu, ˆp)/du (2) where edenotes the exponential function, d(·) is the distance function, and durepresents the user's historical average movement distance. 3.2.3 Profile Alignment. Behaviors of human often correlate with their profile tags. For instance, suggesting family-oriented venues or restaurants to university students represents a cognitive mismatch in recommendation logic. We define the user profile alignment score to reflect the alignment between recommended item ˆpand user characteristics: PAS(u,l,s,t, ˆp) = LLMu(u, ˆp) (3) where a LLM model is utilized to evaluate the reasonableness of the recommended POI ˆpgiven the profile of user uand the score ∈{0, 1}. 3.2.4 Situational Awareness. Situations such as weather, traffic, and the conditions of the POIs could have a significant impact on human's traveling decisions. For example, recommending an outdoor tennis court on a rainy day is considered inappropriate. We define the recommendation system's Awareness to these contextual situations as situational awareness. Similar to definition of profile alignment, we also utilize a LLM to score this metric, SAS(u,l,s,t, ˆp) = LLMs(s, ˆp) (4) where the score ∈{0, 1}. The definitions articulated above serve a dual purpose in our methodology. They directly guide the creation of a training set of cognitively plausible examples for the model's alignment phase. Simultaneously, these definitions are translated into quantitative metrics, providing a principled framework for evaluating the cognitive capabilities of our proposed model. 4 Methodologies The overall framework is presented in Figure 1. The framework consists of two main modules: (1) Spatio-temporal Knowledge Acquisition module enables the LLMs to incorporate the linguistic and collaborative semantics and (2) Cognitive Alignment to adjusting the ST-LLM obtained from step (1) to generate results that better align with human cognition. We finally detail our inference optimization strategy in online deployment since this is a bottleneck of large-scale application of generative models. 4.1 Spatio-temporal Knowledge Acquisition Considering the huge amount of the POI corpus in large industrial LBS applications, we firstly convert POI data into Semantic IDs (SIDs) to reduce the total number of tokens. Building on these SIDs, we propose to enable the LLM to incorporate the recommendation knowledge by continued pretraining on an enriched corpus. 4.1.1 Semantic ID generation. A well crafted SID representation can boost the LLM's recommendation efficiency by capturing the semantic and spatial relationships among POIs. There has already been extensive research in this area [22, 32, 34, 49, 51, 54], so we do not elaborate on it here. In brief, we adopt the RQ-VAE, which adopts the multi-modal information of the POIs to learning the codebooks. Specifically, we incorporate the spatial location information and descriptive titles/pictures of the POIs to generate the identifiers with residual quantization process. As a result, we get a set of codebooks Cl= {el k|k= 1, ..., K},l∈[1, L] Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. Figure 1: Offline Training and Online Service Procedure of proposed method CoAST. Offline Training: (1) POIs are tokenized as Semantic IDs (SIDs); (2) a base LLM is continually pretrained on the collected corpus, yielding ST-LLM; (3) multi-level user profiles are distilled from the global tracking database with the annotations of LLM; (4) cognitive-aligned data are produced through LLM annotation and human screening; (6) ST-LLM is further fine-tuned and aligned to finish this procedure. Online Serving: CoAST is stored in a model engine and invoked through a data-service engine that converts real-time context-user profile, recent check-ins, current time, location, and weather-into an instruction prompt. The model predicts the next POI in SID form, which is mapped back to a readable location and returned to the mobile client. where Lrepresents the number of codebooks and Kdenotes the size of each codebook. Therefore, a POI can be tokenized as an ordered sequence of tokens, such as " ", in which the number of the above tokens represents the indices of the responding codebook. 4.1.2 Continued Pretraining on Spatio-temporal Corpus. After acquiring SIDs of the POIs, a simple method is to incorporate these SIDs into the vocabulary of large language models (LLMs). This allows the LLMs to perform recommendation tasks in a generative manner, ultimately moving away from using item indices. Nevertheless, there are two remaining concerns: (1) these newly added indices are out-of-vocabulary tokens for LLMs, incorporating these linguistic and collaborative semantics becomes necessary. (2) Recommendation tasks present unique technical challenges that distinguish them from conventional NLP tasks, particularly in modeling spatio-temporal user behaviors and handling data sparsity. To achieve this incorporation, we implemented continued pretraining [14] on the recommendation corpus, aiming to adjust LLMs to improve understanding of linguistic and collaborative semantics. Our approach to recommendation domain-adaptive pretraining is straightforward-we continue pretraining a base model (like Qwen3 [41]) on a large corpus of unlabeled spatio-temporal recommendation domain-specific text. Specifically, the corpus are designed and collected mainly focusing on 2 objectives: clear index-languagespace aligning (finding the POI with its description, location and its category, or outputting the description and location given the POI) and sequential POI prediction (predicting the next POI given the historical POI check-ins). While understanding the spatial relationships between POIs is fundamental to LBS recommendation, directly instilling this knowledge into LLMs is difficult. To address this, our approach tokenizes each POI's Geohash and integrates these tokens into our pretraining data. This process enables the model to learn a unified embedding space where the POI's identity (SID), its semantic meaning, and its location (via Geohash) are jointly represented and aligned. Table 1 illustrates the utilized corpus in this stage. Table 1: Recommendation corpus collected for continuous pretraining based on a base model. The words in orange denote the newly added tokens in the base LLM. SID-Location-Description Alignment Corpus The Temple of Heaven Park is the world's largest surviving ancient architectural complex dedicated to the worship of heaven. Originally constructed during the Ming Dynasty, it served as the site where emperors of the Ming and Qing dynasties held ceremonies to "sacrifice to heaven" and "pray for a good harvest." Located in of Beijing, and the ID is . Spatio-Temporal Behavior Sequence Recommendation Corpus Here is user uid_256's historical POI check-ins. At 7 AM June 10, the user ride to company , at 21 PM June 11, the user navigated to the hotel , at 15 PM June 20, the user searched an office ,..., at 19 PM July 20, the user walked to home . Masked prediction has recently been applied to adapt pretrained models to specific downstream tasks by continuing to pretrain models on in-domain unlabelled data [6, 15, 37]. Inspired by these works, we continue our pretraining with masking some original tokens of the base LLM. However, a crucial consideration in our pretraining phase is the token composition of the training corpus, Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY which presents a delicate trade-off between newly introduced POI tokens (SIDs and Geohash tokens) and the original tokens within the pretraining corpus. Through the comprehensive continued pretraining on the corpus of spatio-temporal recommendation domain, the model is adapted to learn information centered around generic behavioral patterns. However, its outputs may exhibit cognitive mis-alignment with human reasoning principles. This necessitates a specialized cognitive calibration process to bridge the divergent representations between machine-learned patterns and anthropic decision-making frameworks. 4.2 Cognitive Preference Alignment 4.2.1 User Profile Generation. Understanding user historical POI check-ins is crucial for modeling the trajectory patterns of users. However, it is challenging to effectively utilize extended user behavior sequences, which often extends to lengths of 10,000 in largescale recommendation systems [11, 16]. Yet, LLMs face difficulties in processing these long sequences, and this hinders their ability to grasp the patterns. To counteract this, we have developed multilevel profile of the users by condensing extensive user behavior into refined profiles that are more amenable to LLMs. The utilized profiles can be mainly divided into three main categories: (1) Static Profile, (2) Long-term Preference Profile, and (3) Periodic Demand Profile. Static profile, such as POIs of the users' home and company, are critical for next POI prediction. However, the labeled data is pretty limited. To expand the coverage of static profiles, we first employ LLM-based summarization of user behavioral sequences to generate preliminary annotations. These labeled patterns are then utilized to train models (like a MOE-Transformer network [7]) that subsequently perform inference across the entire user population, producing comprehensive static profiles that maintain semantic consistency with human cognitive patterns. Long-term preference profile refers to stable and long-term intentions derived from long-term behaviors, which focuses more on stability, accuracy, and long-term relevance, requiring a more rigorous reasoning and calibration process [39, 46]. Periodic demand profile identifies recurring consumption patterns that exhibit temporal periodicity correlated with interval cycles (e.g., weekly/monthly), holiday events, and workday rhythms. We employs a LLM to summarize the extended behavioral sequences through scheduled interval sampling, implementing an automated periodicity and long-term preference detection mechanism. 4.2.2 Cognitive-aligned Data Generation. Within the vast amount of POI check-in data left by users, a significant portion is considered noise (such as accidental clicks), or does not align with the human cognition process. Therefore, we intervene in data collection by combining the world knowledge of LLMs and human annotations. Following similar process of [39], we collect the data from traditional CTR-oriented recommender systems that align with human cognition preference defined in Section 3. In particular, LLMs is utilized to access the users' historical checkins, profile information and other contextual information (like time and weather) to conduct chain-of-thought (CoT) reasoning [35], thereby evaluating the potential candidate items aligned with human cognition for individual users. Subsequently, human annotators review the reasoning process and outcomes generated by the LLMs to refine and improve the accuracy of serendipity labels. This annotated dataset-which contains user history, profiles, candidate items, contextual conditions, and cognition annotations-is then utilized to further fine-tune the model after continued pretraining phase that can identify the examples aligned with human cognition for the subsequent training. This approach enables scalable discovery of cognitive-aligned items while reducing the need for extensive manual annotation. 4.2.3 Supervised Fine-tuning. After obtaining cognitive-aligned data, with each record consisting of useru, their multi-level profiles, historical POI check-ins hu, the current situation of the user (time, location, weather and so on) and the selected cognitive-aligned POI pi, we begin the process of fine-tuning by aligning the inputs. Initially, all of the inputs are transformed into a text-based supervised fine-tuning (SFT) dataset, where each record is a pair (x,y), with ydenoting the output SID of the target POI pi, and xas the input prompt. For example, Instruction: Based on the user's profile: {user-profile}, and his historical POI check-ins: {check-ins}. Now the time is {time}, he is in {user-location}, and the weather is {weather}. Please recommend the next possible POI that the user may expect? Response: Subsequently, LLM is fine-tuned under supervision to enhance its ability to produce the desired responses. 4.2.4 Cognitive Preference Alignment. Following the standard paradigm for human preference alignment, we continue to leverage Reinforcement Learning (RL) to further refine the model's outputs and explicitly align its recommendations with human cognitive preferences. After obtaining datasets comprising human-annotated preferences over pairs of model-generated responses, we train the model with the preference alignment algorithm. We choose Direct Preference Optimization (DPO) [28] in this phase, since it has been widely employed to align the model's outputs more closely with human preference. L = -E(x,yw,yl)∼D logσ βπθ(yw| x) πref(yw| x) -βπθ(yl| x) πref(yl| x) (5) where Ddenote the preference dataset, xdenotes the input prompt, and ywand ylrepresent the preferred and dis-preferred responses annotated by human and LLM. πθis the trainable LLM for preferences alignment, while πref is a fixed reference policy. τis the hyper-parameter of the regularization term balancing learning human preferences and staying close to the reference policy, thereby preventing overfitting. In summary, after the Supervised Fine-Tuning (SFT) phase, our model adheres to the required output format (a list of Lvalid SIDs) and already possesses a baseline level of cognitive awareness from the training data. The subsequent DPO phase then significantly Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. enhances this cognitive reasoning while preserving the model's recommendation performance. 4.3 Online Deployment A primary challenge for the large-scale application of generative models is their inference latency, particularly in recommendation scenarios with high Queries Per Second (QPS). To enable the largescale online deployment of our generative model, inspired by recent works [13, 52], we implemented a dual-pronged optimization strategy targeting both system architecture and algorithmic efficiency, achieving significant breakthroughs in stability and performance. 4.3.1 Prefill-Decoding Decoupling. We employ Prefill-Decoding (PD) Decoupling [52] to separate the inference process into two distinct stages: a computationally-intensive Prefill stage for prompt processing and a memory-bound Decoding stage for iterative token generation. By applying dedicated optimization strategies to each stage, this approach mitigates resource contention and latency jitter common in unified pipelines. The resulting improvements in resource scheduling and concurrency led to a 10-fold increase in throughput, drastically reducing the per-inference cost on the same hardware. 4.3.2 Multi-Token Prediction. Furthermore, we leverage Multi-Token Prediction (MTP) to overcome the sequential dependencies of traditional auto-regressive generation [13]. Specifically, we adopt MTP Eagle, an optimized variant of the Eagle speculative decoding method. Its key innovation lies in a single, reusable MTP module that shares the same KV cache for predicting multiple draft tokens, thus eliminating the redundant computational overhead of vanilla MTP. Its chained decoding strategy, which only requires the last predicted token and its hidden state to proceed, significantly simplifies implementation by obviating the need to store historical states. MTP Eagle is fully compatible with existing checkpoints (e.g., DeepSeek-V3) and substantially reduces both memory footprint and computational complexity, enhancing the engineering feasibility and resource efficiency of our deployment. Collectively, these optimizations create a highly efficient and stable inference framework. This is crucial for deploying large-scale generative models in production, ensuring both a responsive user experience and the economic viability of the system in a real-world advertising setting. 5 Experiments 5.1 Experimental Settings 5.1.1 Scenario. Our method is performed on AMAP, a prominent navigation and mapping platform hosting billions of users and POIs. We focus on the "Guess Where You Go" card presented on the homepage of the AMAP App, aiming to forecast the next subsequent POI using a user's previous check-ins and profile information. The objective of this scenario is to predict the next POI the users are most likely to visit. By displaying the POIs on the homepage, we aim to reduce the users' search effort, thereby enhancing user experience and long-term retention. Critically, our goal transcends mere prediction accuracy; we strive to build a model that captures the underlying dynamics of user mobility patterns and intents. This allows the system to develop a deeper "cognitive" understanding, ultimately facilitating more intelligent and context-aware decisionmaking for the user. The current deployed framework, which serves as our primary baseline, employs a traditional cascaded ranking system for recommendation. As illustrated in Figure 2, the cascaded ranking system includes three stages: (1) Retrieval stage to recall a coarse-grained POI corpus based on the user's preference, historical check-ins, and location [18, 44]; (2) Ranking stage to obtain top 10 POIs the users most likely visit with a ranking model based on behavior sequence transformer [3]; and (3) Cognitive Re-ranking stage with distance, quality, and exposure filters to yield the most cognitively plausible destination. Figure 2: Cascade ranking system of "Guess Where You Go" deployed on AMAP, which includes three stages from the left to the right: Retrieval, Ranking, and Cognitive Re-ranking. We evaluate our approach on three real-world datasets: FoursquareNYC [42], Foursquare-TKY [42], and Gowalla-CA [4] and a largescale proprietary dataset by sampling desensitized users from AMAP (named Beijing). The details of the utilized dataset is illustrated in Appendix A. 5.1.3 Implementation Details. The quantization module consists of 3 codebook layers and for the generative recommendation module, we employ the Qwen [27, 41] as the base model. We employ a constant learning rate schedule with a learning rate of 1e-5, combined with a warm-up phase of 20 steps. The model is trained on 200× NVIDIA H20 GPUs, with a batch size of 16 per GPU, gradient accumulation steps set to 8, and a sequence length of 4096 tokens. Each input consists of the most recent 50 check-in records of the user. 5.1.4 Evaluation Metric. Following previous works, we employ the accuracy metric for the top-1 recommendation accuracy (Acc@1) and the four cognitive metrics (TCS, SCS, PAS and SAS) defined in 3.2 as our metrics to evaluate the effect of the cognitive recommendation. Detailed information of the metrics is illustrated in Appendix C. 5.1.5 Baseline Methods. The details of the baseline methods in this part is illustrated in Appendix B. 5.1.6 Our Models. In our experiments we consider three versions of our model: (i) CoAST-0.5B: the model trained with Qwen2.5-0.5B [27] as our base LLM. (ii) CoAST-1.5B: with Qwen2.5-1.5B as our base LLM. (iii) CoAST-3B: A variation with Qwen2.5-3B as our base LLM. (iv) CoAST-7B: A second variation on Qwen2.5-7B. Below, unless stated otherwise, when we say "our model," we refer to the CoAST-0.5B. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Table 2: Comparison of different models on three datasets: NYC, TKY, and CA. We present the model inputs including the POI and User representation approach and whether visit timestamps are utilized. RID, SID, UID and UP represent Random one-hot ID, Semantic ID, User one-hot ID and User Profiles, respectively. The results for the baseline methods are borrowed from [19, 32]. Model Inputs Acc@1 POI User Time NYC TKY CA PRME RID UID × 0.1159 0.1052 0.0521 PLSPL RID UID × 0.1917 0.1889 0.1072 STAN RID UID × 0.2231 0.1963 0.1104 GETNext RID UID × 0.2435 0.1829 0.1357 STHGCN RID UID × 0.2734 0.2950 0.1730 TPG RID UID ✓ 0.2555 0.1420 0.1749 ROTAN RID UID ✓ 0.3106 0.2458 0.2199 LLM4POI RID UID ✓ 0.3372 0.3035 0.2065 GNPR-SID SID UID ✓ 0.3618 0.3062 0.2403 CoAST-0.5B SID UP ✓ 0.3705 0.3125 0.2555 CoAST-1.5B SID UP ✓ 0.3727 0.3170 0.2585 CoAST-3B SID UP ✓ 0.3872 0.3202 0.2611 CoAST-7B SID UP ✓ 0.4027 0.3310 0.2721 Table 3: Comparison results of different versions of CoAST on Beijing dataset with the Cascade Ranking baseline system. 'a-' denotes average. Baseline CoAST-0.5B CoAST-1.5B CoAST-3B CoAST-7B Acc@1 0.2675 0.2820 0.2862 0.2923 0.3071 a-TCS 0.5561 0.6071 0.6021 0.6632 0.7543 a-SCS 0.8022 0.8212 0.8231 0.8452 0.8838 a-PAS 0.6232 0.6285 0.6240 0.6531 0.6970 a-SAS 0.2472 0.2840 0.2988 0.3240 0.3725 5.2 Offline Experiments 5.2.1 Overall Performance. Our method substantially outperforms all baseline methods across all of the evaluated datasets. Table 2 illustrate the comparison results on NYC, TKY, and CA datasets. Specifically, compared to the state-of-the-art GSNR-SID which utilizes LLaMA-7B as its base model, we observe a substantial improvement on all of the versions of CoAST. Notably, CoAST-7B demonstrated the most significant improvements, achieving gains of 11.3%, 8.1%, and 13.2% over the SOTA, respectively. Table 3 details the performance comparison between our proposed model and the cascade ranking system on the Beijing dataset. In addition to the standard Acc@1 metric, we evaluate the models on four cognitive-aware recommendation metrics (where 'a-' denotes average). Consistent with our findings on other datasets, our model significantly outperforms the traditional cascade ranking system on Acc@1. More critically, it demonstrates substantial gains on the cognitive metrics, achieving remarkable uplifts of 35.6% in temporal consistence and 50.68% in situational awareness performance. These results highlight a key limitation of conventional collaborative filtering-based systems: their struggle to capture personalized, periodic user needs and inability to effectively model situational or contextual information. This validates the superiority of our generative approach, which excels in understanding these complex, dynamic aspects of user behavior. 5.2.2 Ablation Study. To evaluate the impact of individual modules, we developed various versions of CoAST. Initially, certain essential modules were removed: "w/o UP" eliminates user profiling from the prompt. "w/o S-ST" eliminates situational and spatio-temporal descriptions from the prompt. "w/o SLD" exclude the SID-LocationDescription alignment corpus in the continued pretraining stage. "w/o CPT" exclude the Continued Pretraining stage in the alignment stage, only applying SFT and DPO. As illustrated in Table 4, removing any module leads to noticeable performance degradation, emphasizing the importance of each component. Significantly, the lack of SFT results in the most considerable decrease in all metrics, demonstrating the essential role of preference alignment in recommendation efficiency. Table 4: Ablation study of CoAST (with the online deployment version CoAST-3B). Acc@1 a-TCS a-SCS a-PAS a-SAS w/o UP 0.2720 0.6435 0.8211 0.5920 0.2923 w/o S-ST 0.2708 0.5905 0.7938 0.6496 0.2822 w/o SLD 0.2825 0.6078 0.8240 0.6501 0.3020 w/o CPT 0.2632 0.5875 0.7820 0.6020 0.2875 CoAST 0.2923 0.6632 0.8452 0.6531 0.3040 5.2.3 Case Study. To provide a qualitative and intuitive demonstration of our model's advantages, we conduct a case study that contrasts the recommendation lists generated by our model with those from a baseline model for a representative user. Detailed information of the case is illustrated in Appendix D. 5.2.4 Further Analysis. In this section, we analyze the impact on the performance of sequence length (input tokens) and number of utilized tokens in continued pretraining stage. Figure 3 presents the results. Our investigation into the impact of input sequence length shows that while longer context is beneficial, its utility eventually saturates. The model's performance gains are most pronounced up to a context window of 2048 tokens (∼30 user historical checkins) and largely stabilize beyond 4096 tokens (∼50 check-ins). This suggests that a moderately-sized history is sufficient to encapsulate the user's core preferences. Further extending the context window provides limited new signal, as this long-term information is already implicitly encoded in the user's learned representation. The volume of tokens involved in pretraining phase also proved to be a key determinant of performance. We observed a nearly linear improvement in model effectiveness as the training corpus size increased. Constrained by our training budget, the final model was trained to convergence on a dataset of 10B tokens. The consistent, Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. non-saturating growth suggests that performance is largely databound, with significant potential for improvement as more data becomes available. Figure 3: Impact on the performance of sequence length (input tokens) and number of tokens involved in continued pretraining stage. a-CAS denotes the average Cognition Alignment score computed by averaging the four cognitive scores. 5.3 Online Experiments 5.3.1 Efficiency Analysis. To evaluate the online inference efficiency, we compare different variants of our model with the efficiencySOTA cascade ranking system. Before deploying our model, we conduct a load test under the settings with a 50 QPS on 2× NVIDIA H20 GPUs. Figure 4 presents the inference p99 latency results of different methods. As shown, the Cascade Ranking system, a widely adopted paradigm in industrial recommender systems, demonstrates exceptional efficiency with its latency consistently around 20ms. As a comparison, while CoAST-7B achieved the best offline performance among all our variants, its high computational cost posed a significant challenge for online serving. Even after applying the acceleration strategies described in Section 4, its latency remained above 120ms, rendering large-scale deployment infeasible. To strike a practical balance between efficacy and efficiency, we identified CoAST-0.5B as the optimal candidate. It delivers competitive performance while maintaining a manageable inference cost (∼30ms). Consequently, we deployed CoAST-0.5B to production and conducted a online A/B test against the state-of-the-art (SOTA) Cascade Ranking baseline. 5.3.2 Online A/B Test. To assess the online effect of CoAST, we implemented a short-term and long-term A/B testing on "Guess Figure 4: Comparison of Model Inference p99 Latency (ms) under a 50 QPS Load Test on 2× NVIDIA H20 GPUs. Where You Go" card of AMAP's homepage. We examined CoAST0.5B, trained based on a Qwen2.5-0.5B model, compared to the traditional cascade ranking baseline as shown in Table 5. Table 5: Online improvement over traditional Cascade Ranking baseline of short and long term A/B test. % denotes relative improvement. short and long represents short-term and long-term A/B online test, respectively. P-CTR U-CTR SC-Rate RE-Rate AC-Rate NF-Rate short +5.06% +4.65% +0.69% +0.31% +0.77% -5.11% long +4.23% +4.59% +1.01% +1.65% +5.99% -5.85% It is worth mentioning that this baseline already includes user profiles, trending topics and seasonal features for improving cognition in the cognitive re-ranking stage. For short-term (one week) online test, CoAST demonstrate significant improvements in metrics like P-CTR (PV CTR) and U-CTR (UV CTR). This indicates that CoAST has increased the share of cognitive-aligned recommendations that effectively attracted user attention in terms of clicks. Additionally, it slightly enhances the user engagement metrics, such as SC-Rate (Scroll Rate), RE-Rate (User Retention Rate), and AC-Rate (User Active Rate), implying heightened user involvement. Notably, the effects on general utility metrics such as CTR and NFRate (Users' Negative Feedback Rate) are more modest, while the improvements on these engagement-related metrics often become apparent only over the long term. To investigate the long-term effects of the proposed method in large-scale RSs, we maintain a small traffic for our baselines and compare it with CoAST for over one month. Compared to the baseline and the performance on the short-term test, our method has achieved a significant enhancement in the user engagement metrics, with a 5.85% average decrease in NF-Rate (Users' Negative Feedback Rate). This demonstrates that CoAST could help to mitigate counter-intuitive recommendations and effectively improve user engagement while also boosting CTR, thereby enhancing the overall user experience. 6 Conclusion In this paper, we proposed a LLM-based next POI prediction approach, named CoAST. In order to adapt LLMs to sequential recommendation tasks, we focused on a comprehensive pretraining task to adapt a base LLM to align language and collaborative semantics for recommendation. This task include two categories of corpus: sequential POI prediction and explicit index-language alignment. Based on the learned item indices, our approach employed these alignment tasks to effectively adapt LLMs for sequential recommendation. Furthermore, we give complete definition and metrics on cognitive-aligned next POI prediction task. For constructing a cognitive-aligned POI recommendation system in industrial application, we design a multi-stage fine-tuning process, utilizing a curated dataset of cognitively-aware samples for both Supervised Fine-Tuning (SFT) and a subsequent Reinforcement Learning (RL) phase. This procedure explicitly align the model with human common-sense. Extensive offline experiments on three datasets demonstrated the effectiveness of our approach and Online A/B Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY test achieves substantial improvements compared to traditional cascade ranking recommendation system. Acknowledgments We would like to acknowledge the discussions of Jiawei Xue, Ning Wang, and Fangfang Chen throughout this work. 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CoST: Contrastive Quantization based Semantic Tokenization for Generative Recommendation. In Proceedings of the 18th ACM Conference on Recommender Systems (Bari, Italy) (RecSys '24). Association for Computing Machinery, New York, NY, USA, 969-974. Cognitive-Aligned Spatio-Temporal Large Language Models For Next Point-of-Interest Prediction Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Figure 5: Case comparison of recommendation results of CoAST and the traditional basline RSs. The top row represents the user's real-time check-ins with different action types. The middle and bottom row represent the recommendation results of traditional basline RSs and CoAST, respectively. A Dataset To validate our model's performance in a real-world application, we constructed a large-scale proprietary dataset by sampling desensitized users from AMAP (named Beijing). On this dataset, we conducted both extensive offline evaluations and an online deployment. Furthermore, to ensure a comprehensive and fair comparison against existing baselines, we also evaluate our approach on three real-world datasets: Foursquare-NYC [42], Foursquare-TKY [42], and Gowalla-CA [4]. The pre-processing follows the settings in [32]. we pre-process each dataset by removing POIs with fewer than 10 interactions and users with fewer than 10 check-ins. The data is then sorted by time, with 80% used for training, 10% for validation, and 10% for testing. Users and POIs that do not appear in the training set are removed from the test set to ensure consistency between training and evaluation. Subsequently, the check-in records are grouped by user and ordered chronologically. For each user, the last visited POI is held out as the ground truth during evaluation, while the preceding visits are used as the input sequence. Note that in the training set, the user's visit records are also treated as historical sequences and concatenated with the test set to the specified length before being fed into the model. Detailed dataset statistics are provided in Table 6. Table 6: Statistics of the preprocessed datasets. Avg.len denotes the average length of item sequences. Dataset #Users #Items #Inter. Avg.len Sparsity NYC 1,083 5,135 104,074 136 98.1013% TKY 2,293 7,873 361,430 195 97.9798% CA 6,592 14,027 349,375 53 99.9996% Beijing 10,000k 1,520k 38,0000k 38 99.9999% B Baseline Following the settings of [19, 32], we compare our proposed method with various next POI recommendation baselines, which can be specified as the following four categories: (1) Traditional methods: PRME [9], a ranking-based metric embedding method; and PLSPL [38], a personalized sequential recommendation model. (2) Transformer-based methods: STAN [25], a spatio-temporal attention network; and GETNext [43], a Transformer-based next POI recommendation model. (3) GCN-based method: STHGCN [40], a spatio-temporal hierarchical graph convolutional network. (4) Time-aware methods: TPG [24], a timestamp-guided model; and ROTAN [10], a time-aware POI recommendation framework. (5) LLM-based method: LLM4POI [19], a method based on LLMs; and GNPR-SID [32], a state-of-the-art based on Semantics codes and generative recommendation framework, which is pretty closely related to our work. C Evaluation Metric Following previous works, we employ the accuracy metric for the top-1 recommendation, denoted as Acc@1, at the specified time. Acc@1 = 1 n n ∑︁ i=1 I(yi= ˆyi) where nis the number of test samples, yiand ˆyirepresent the ground-truth label and the predict label of the i-th sample, respectively. I is the indicator function, which equals 1 when the condition is true and 0 otherwise. Otherwise, we take the four cognitive categories (TCS, SCS, PAS and SAS) defined in 3.2 as our metrics to evaluate the effect of the cognitive recommendation. D Case Study Figure 5 illustrates the case comparison of recommendation results of CoAST and the traditional basline RSs. In the traditional cascade ranking RSs, the model collapses to the user's dominant daily rhythm: go to company at morning and back to home at afternoon. However, during other times of the day, conventional CTR-oriented models tend to be dominated by users' most recent interactions. This short-term focus often leads to recommendations that fail to capture the user's broader, long-term interests, resulting in a counter-intuitive user experience. Conference acronym 'XX, June 03-05, 2018, Woodstock, NY Zhai et al. In contrast, our proposed generative model achieves a better alignment of the user's temporal context with their holistic preference profile, thereby fostering a more well-rounded and sustainable recommendation ecosystem.
|
2510.14694
|
Response to Discussions of “Causal and Counterfactual Views of
Missing Data Models”
Razieh Nabi,1 Rohit Bhattacharya,2 Ilya Shpitser,3 James M. Robins4
1Department of Biostatistics and Bioinformatics, Emory University, Atlanta, GA, U.S.A.
2Department of Computer Science, Williams College, Williamstown, MA, U.S.A.
3Department of Computer Science, Johns Hopkins University, Baltimore, MD, U.S.A.
4Departments of Biostatistics and Epidemiology, Harvard University, Boston, MA, U.S.A.
1
Introduction
We are grateful to the discussants – Levis and Kennedy [2025], Luo and Geng [2025], Wang
and van der Laan [2025], and Yang and Kim [2025] – for their thoughtful comments on our
paper [Nabi et al., 2025]. Below we summarize our main contributions before responding to each
discussion in turn.
Graphical models have emerged as an important tool for clarifying identifying assumptions made
in both causal inference [Pearl and Robins, 1995, Pearl, 2000] and missing data [Robins and
Gill, 1997, Bhattacharya et al., 2019, Nabi et al., 2020, Malinsky et al., 2021, Mohan and Pearl,
2021]. Our paper [Nabi et al., 2025] shows how recent techniques motivated by causal graphical
modeling may be fruitfully applied to obtain identification in missing data models. These recent
techniques allow us to obtain novel identification results that would not be possible to obtain
in standard causal inference problems, except by imposing implausible additional assumptions,
such as rank preservation.
Specifically, we formalize how identifiability of the target (i.e., complete) data law can be viewed
as identification of a joint distribution over counterfactuals L(1), the variables that would have
been observed if all missingness indicators R were set to one (that is, if no data were missing).
This reframing yields a counterfactual analogue of the g-formula. When assumptions encoded in
the Markov properties of a missing data DAG (m-DAG) allow us to express the counterfactual
g-formula in terms of the factuals, we obtain (nonparametric) identification. That is, noting
that the target law p(l(1)) satisfies p(l(1)) = p(l, R = 1)/p(R = 1 | l(1)), it follows that when the
missingness selection model p(R = 1 | L(1)) is identified from the observed law, so is p(l(1)).
In our paper we used the word “nonparametric” in two different ways. The joint distribution
of (L, L(1), R) is Markov to a given m-DAG if it factorizes as the product of the conditional
densities of each variable given its parents. In the graphical causal modeling literature, the
model is said to be “nonparametric” just when these conditional densities are left unrestricted
(with the exception that L is a deterministic function of its parents). The term “nonparametric
identification” refers to identification in such a model. In contrast, in the missing data literature
in statistics, a model is said to be nonparametric if it places no restrictions on either the observed
data law or the target data law p(l(1)). It is said to be nonparametric just identified (NPI) if
the target law is identified from the observed data law. The permutation model [Robins, 1997]
we discuss in our paper is known to be NPI. In contrast, all the other identified missing data
models in our paper place testable restrictions on the joint distribution of the observed data. In
1
arXiv:2510.14694v1 [stat.ME] 16 Oct 2025
fact, we conjecture that the permutation model is the only m-DAG model that is NPI. In the
following, we use nonparametric in the graphical causal modeling sense.
A key message of our paper is that features specific to missing data models (in particular the
partial observability of the counterfactuals through the proxies L and the structural restriction
that R and L do not cause L(1)) can deliver parameter identification in settings where analogous
parameters in hidden variable causal DAGs are not identified. We catalog several such non-
parametric identification techniques in m-DAGs, and we clarify similarities and differences with
standard causal identification, including when additional assumptions like rank preservation
might be needed for causal analogues.
Each discussion extends or challenges our framework in important ways. Levis and Kennedy
[2025] highlight identification strategies complementary to ours, based on the existence of
instrumental and shadow variables, discuss semiparametric estimation theory for functionals
arising from such strategies, as well as discuss connections between m-DAGs and Single World
Intervention Graphs (SWIGs). Luo and Geng [2025] analyze self-censoring MNAR mechanisms
with binary variables, deriving identifiability results that leverage auxiliary variables. Wang and
van der Laan [2025] emphasize, as we do, viewing missingness as interventions, and connect our
perspective to censoring in survival models. Yang and Kim [2025] and Wang and van der Laan
[2025] both examine challenges of applying m-DAGs in applications, particularly assumption
validation, scalability, and sensitivity analysis.
2
Response to the discussions
2.1
On the discussion by Levis and Kennedy
Levis and Kennedy [2025] draw attention to additional causal identification tools and highlight
implications for estimation. In particular, they emphasize the relevance of instrumental variables,
shadow variables, and SWIGs as complementary devices for reasoning about identification in
missing data problems. They also consider how identification results derived from m-DAGs can
be translated into practical estimation procedures with desirable asymptotic properties. Their
discussion situates our contribution within a broader pipeline of causal inference methods and
points toward promising directions for future methodological development.
We appreciate Levis and Kennedy’s thoughtful remarks. On their first point, we agree that
instrumental variables [Sun et al., 2018], shadow variables [Miao et al., 2024], and related
tools [Li et al., 2023] are useful complements. Our emphasis in the paper, however, was on
a nonparametric identification framework: aside from the Markov restrictions encoded via
independence assumptions in m-DAGs, we make no further distributional assumptions. In
contrast, the use of IVs or shadow variables begins in settings where nonparametric identification
fails, and proceeds by imposing additional restrictions (such as functional form constraints e.g.,
homogeneity assumptions when using of IVs), relevance assumptions which result in generic
identification, or restrictions on the support of variables to restore identification. These extra
ingredients move the analysis outside the purely nonparametric domain that was our focus.
On estimation, we appreciate the worked example they provide. While our goal in this paper was
primarily conceptual, highlighting the philosophical parallels and differences between missing data
and causal identification, their discussion underscores the importance of connecting identification
results to practical estimation and efficiency theory. We agree this is important future work.
We also agree with Levis and Kennedy that causal inference problems in the presence of
confounding often occur together with censoring, and that obtaining identification when both
complications are present is challenging. We also appreciate their worked example illustrating
these complications.
2
A
Y
A
a
Y (a)
A
1
Y (1) + A
0
Y (0)
R
L
R
r = 1
L(1)
A
X
Y
A
a
X
Y (a)
A
1
X
Y (1) +
A
0
X
Y (0)
R
X
L
R
r = 1
X
L(1)
(a)
(b)
(c)
(d)
Figure 1: (a) Ignorable treatment model with SWIGs; (b) Its MCAR analogue, with a single
relevant SWIG; (c) Conditionally ignorable treatment model with SWIGs; (d) Its MAR analogue,
again with only one relevant SWIG.
We would like to offer two notes of caution regarding generalizing the construction presented by
Levis and Kennedy, first on the appropriate generalization of the SWIG splitting construction
from causal diagrams to m-DAGs, and second on the general utility of using SWIGs for obtaining
identification in missing data settings.
On the translation of m-DAGs to SWIGs
For simple MAR models where the m-DAG’s structure resembles that of the standard condition-
ally ignorable model in causal inference, the translation of an m-DAG to an equivalent SWIG
is fairly natural. There are, however, important distinctions even in these simple settings. In
causal inference, SWIGs provide a template indexed by treatment interventions (two templates
if the treatment is binary) [Richardson and Robins, 2013]; in the missing data analogue, only the
template for R = 1 is meaningful, since the counterfactual L(0) is not defined. This difference is
highlighted in Figures 1(a, b) for a causal model satisfying ignorability and an analogous missing
data model that is MCAR. Similarly, Figures 1(c, d) highlight the same for a conditionally
ignorable causal model and a MAR missing data model.
Caution must be exercised when SWIGs are constructed from m-DAGs representing MNAR
mechanisms. For instance, in a missing data model with self-censoring, a natural idea for what
the SWIG should look like would be the graph shown in Figure 2(a). However, if we try to
collapse the random and fixed nodes in this SWIG to reconstruct the observed data graph, we
obtain the graph shown in Figure 2(b), which erroneously implies a cycle in the underlying data
generating process.
The problem arises from conflating the full data variable L(1) in m-DAGs, and the observed
data variable L relabeled to be counterfactual in the SWIG construction. An appropriate view
of m-DAGs that avoids these types of difficulties is to view full data variables such as L(1)
as unobserved confounders U, with additional structure imposed by missing data consistency.
In this view, the SWIG corresponding to the self-censoring model is better represented by
Figure 2(c), where the variable L(1) is viewed as an unobserved confounder U with special
structure, which influences both the observability indicator R, and the observed proxy variable
L. The m-DAG corresponding to this view of the self-censoring model is shown in Figure 2(d).
SWIGs do not make identification arguments for MNAR problems clearer
The primary motivation for SWIGs in the causal inference context is to provide a graphical
representation for independences that arise in identification arguments. The function of m-DAGs
is to provide precisely the same graphical representation in missing data problems, rendering
SWIGs unnecessary.
3
R
r = 1
L(1)
(a)
R
L
(b)
R
r = 1
U ≡L(1)
L(1)
(c)
R
U ≡L(1)
L
(d)
Figure 2: (a) A possible SWIG representation of a self-censoring missing data model; (b)
Stitching SWIGs into an observed-data model produces a cycle; (c) A SWIG that draws a
distinction between the full data variable L(1), viewed as an unobserved confounder U with extra
restrictions, and the observed variable L under an intervention where R is set to 1; (d) The full
data graph equating the full data variable L(1) with an unobserved confounder U with special
structure.
U1 ≡L(1)
1
U2 ≡L(1)
2
R1
R2
L1
L2
(a)
U1 ≡L(1)
1
U2 ≡L(1)
2
R1
R2
L(1)
1
L(1)
2
r1 = 1
r2 = 1
(b)
Figure 3: (a) The block-parallel MNAR model, labeling missing variables as unmeasured for the
purposes of constructing a SWIG; (b) The corresponding SWIG obtained from (a).
Take, for example, the independences that can be read from the m-DAG in Figure 3(a) and the
corresponding SWIG in Figure 3(b). From either graph, we are able to extract independences
of the form Rk ⊥⊥L(1)
k , R−k | L(1)
−k, for k ∈{1, 2} that define the binary block-parallel MNAR
model. That is, the m-DAG itself is just as expressive as the SWIG for reading off independences
important for identification in missing data models. In fact, we now argue that attempting
identification using standard causal inference arguments from the SWIG may be problematic
for missing data problems due to the single counterfactual nature of missing data, which as we
discussed in Section 6 of our paper is more akin to the rank preservation assumption in causal
inference.
To obtain the identification of the target law p(l(1)
1 , l(1)
2 ) of the block-parallel MNAR model, we
showed in Section 5.2 of our paper that a parallel use of the g-formula was required, which never
arose in standard causal inference settings. If just the marginal p(l(1)
2 ) is desired, this is obtained
by marginalization. Now suppose instead, we attempted to perform identification arguments
for p(l(1)
2 ) using the SWIG, in a manner similar to the derivation of the adjustment formula for
causal inference. This would proceed as follows. First notice from the SWIG in Figure 3(b) that
L(1)
2
⊥⊥R2 | U1, R1 – this is analogous to conditional ignorability except U1 is observed only
when R1 = 1. Thus we have,
p(l(1)
2 ) =
X
r1,u1
p(r1, u1, l(1)
2 )
=
X
r1,u1
p(r1, u1) p(l(1)
2
| r1, u1)
=
X
r1,u1
p(r1, u1) p(l(1)
2
| r1, u1, r2 = 1)
=
X
r1,u1
p(r1, u1) p(l2 | r1, u1, r2 = 1).
Despite the final expression above appearing to be a function that is devoid of counterfactuals,
the appearance of the variable U1 in the expression prevents identification, as U1 is observed
4
only when R1 = 1. Since R1 cannot be set to the value 1 in the formula above, we are unable to
establish identification via the arguments presented. In fact, no sequential strategy that we are
aware of based on the SWIG in Figure 3(b) would help establish identification. That is, the
parallel fixing arguments presented in Section 5.2 of our paper are required here.
In short, while applying the SWIG construction to m-DAGs offers useful insights in causal
problems with simple types of missingness, such as MCAR or MAR, we caution that such
constructions must be done with care, as Levis and Kennedy have done.
To illustrate, Figure 4(a) extends the example of Levis and Kennedy to include missingness in
both the outcome Y and treatment A. The goal here is to identify the average causal effect of A
on Y in the absence of censoring.
Just as in the previous example shown in Figures 3(a, b), an argument based solely on SWIGs
does not yield identification. Instead, identification strategies that leverage restrictions encoded
in both m-DAGs and SWIGs are necessary.
In particular, we first obtain the target law
p(X, A(1), Y (1)) by applying inverse probability weighting with the product of the propensity
scores for R1 and R2, resulting in the graph in Figure 4(b). We then apply the g-formula on
A(1) to adjust for confounding by X, taking advantage of the restrictions in the SWIG shown in
Figure 4(c). This yields p(Y (1,a)), the distribution of the outcome Y , had it not been censored
and had treatment been set to value a, from which the average causal effect of interest may be
obtained.
Finally, we note that there is currently no complete graphical identification theory for causal
parameters associated with arbitrary m-DAGs that encode both MNAR and (possible) con-
founding by unmeasured common causes U. We expect this theory to employ both SWIG and
m-DAG constructions, as in the worked example of Levis and Kennedy, and our examples above.
2.2
On the discussion by Luo and Geng
Luo and Geng [2025] extend our framework by focusing on self-censoring MNAR mechanisms,
where the missingness of a variable depends directly on its unobserved value. They study
such models with binary outcomes and establish identifiability results under explicit, testable
conditions. Their discussion highlights how auxiliary variables, either baseline or follow-up
measurements, can be leveraged to restore identification. In doing so, they illustrate that
self-censoring structures, which were not emphasized in our paper, can still yield identification
in important cases. They also point to future directions for extending these ideas beyond the
binary case to more general settings.
We thank Luo and Geng for their examples of self-censoring mechanisms. In our paper, we noted
self-censoring as a form of MNAR but did not explore it further, since m-DAG models with
self centering are never nonparametrically identified [Mohan et al., 2013]. Since our emphasis
was on nonparametric identification, we did not employ any restrictions not implied by the
m-DAG factorization, including (i) relevant restrictions needed by instrumental variable or
proxy methods, or (ii) constraints implied by state space restrictions. As Luo and Geng [2025]
illustrated, such assumptions can sometimes yield identifiability, and their results highlight
concrete testable conditions under which this occurs.
We fully agree that self-censoring is of practical importance, with income surveys, sensitive
health questions, and other social science settings providing common examples. Their discussion
usefully demonstrates how auxiliary information, through baseline or follow-up variables, can be
leveraged to address such scenarios.
More broadly, we believe that in MNAR submodels that identify the target law, restrictions
implied by the m-DAG always yield testable implications for the observed data law, with the
5
A(1)
X
Y (1)
RA
RY
A
Y
(a)
A(1) = A
X
Y (1) = Y
rA = 1
rY = 1
A
Y
(b)
A(1) = A
a
X
Y (1,a) = Y (a)
rA = 1
rY = 1
A
Y
(c)
Figure 4: (a) Extension of Levis and Kennedy’s example to have two missing variables. (b)
Graph obtained after fixing RA and RY in parallel. (c) SWIG obtained by splitting the treatment
variable A after already having fixed RA and RY .
exception of the permutation model [Robins, 1997]. Examples of such tests have been developed
in prior work on graphical models [Mohan and Pearl, 2014, Nabi and Bhattacharya, 2023, Guo
et al., 2023, Chen et al., 2023].
Overall, we view their discussion as complementary to our focus: while our framework aimed to
catalog nonparametric identification results, their work illustrates how additional structure can
both sharpen identifiability and connect to practical applications.
2.3
On the discussion by Wang and van der Laan
Wang and van der Laan [2025], like us, approach missingness through the lens of interventions.
They emphasize the close relationship between missing data models and multivariate censoring
models in survival analysis. They carefully prove that the permutation missingness model
is MNAR rather than MAR and flesh out our example of why the permutation model is
substantively plausible in some real world applications. They also discuss substantive settings
where the block sequential model is plausible.
They are less certain that the other identifiable MNAR models discussed in our paper are
substantively plausible. In our view, it is difficult to establish plausibility of any missing data
model in practice, whether it is formulated graphically or not. One advantage of the sort of
theory we present is a single formulation for a large class of identifiable models (including existing
models in the literature such as those described in [Zhou et al., 2010, Robins, 1997]). Since no
single model may be plausible in applications, an alternative strategy is to perform the analysis
of interest under a wide class of identifiable missingness models, as a form of sensitivity analysis.
We make one final observation regarding the plausibility of (or the lack there of) the identifiable
MNAR models considered in the paper. Identification in every case we discuss relies on an
assumption that one or more counterfactuals L(1)
j
suffice to control confounding of the “effect”
of Rk on Lk where j ̸= k. But why should we privilege L(1)
j
(or a set of such variables) as a
set sufficient for adjustment? It seems much more plausible that there exist other unmeasured
common causes of Rk and Lk (equivalently L(1)
k ) that would also need to be adjusted for to
eliminate confounding. Are we assuming the L(1)
j
suffice to control confounding because we
believe it to be so or rather because we want to achieve identification for identification’s sake? In
fact, similar plausibility concerns arise in non-graphical identifiable models, such as non-monotone
MAR. In our view, plausible models of missingness feature a description of data generation
that follows a temporal order (often representable as a DAG). Unfortunately, to be realistic,
even models of this type would feature unobserved confounding of the sort that would prevent
identifiability.
6
2.4
On the discussion by Yang and Kim
Yang and Kim [2025] emphasize the practical challenges of applying the m-DAG framework.
They point out that while m-DAGs provide a principled way to encode assumptions and extend
identification theory, their utility in practice depends critically on correct graph specification,
which may be difficult to achieve in applied settings. They raise concerns about scalability in
high-dimensional problems, where the number of variables and missingness patterns grows quickly,
and about the feasibility of validating the conditional independence assumptions encoded in an
m-DAG. They further stress the importance of developing systematic approaches to sensitivity
analysis, noting that small misspecifications of the graph can have consequences for identification
and downstream tasks.
We thank Yang and Kim for raising important questions about the practical utility of m-DAGs,
and graphical modeling more broadly. Our goal in this paper was primarily conceptual: to
provide a general framework that connects causal and missing data perspectives and to catalog
identification results that arise from this connection. Their emphasis on practical feasibility
is appreciated, and complementary to our focus. That said, continuing work in the graphical
modeling community over the last two decades has rendered much of the critiques of the graphical
modeling approach to causal inference and missing data problems out of date.
Identification of causal and missing data parameters must, by necessity, rely on equality re-
strictions in the full data distribution. Graphical models aim to encode these restrictions in a
way that allows the construction of a plausible data generating mechanism consistent with a
temporal order of events. Models that obtain identifiability without a corresponding graphical
representation, for instance the non-monotone missing at random (MAR) model, often lack such
a plausible mechanism. Indeed, existing efforts that argue for the plausibility of models such as
non-monotone MAR embed them into a graphical model or a mixture of such models [Robins
and Gill, 1997].
Over the last decade, an explosion of easy to use open-source packages that take advantage of
graphical models have been developed for all tasks in causal inference and missing data, including
establishing identification, constructing estimators and applying them to data, conducting
sensitivity analyses, establishing bounds on non-identified parameters, and model selection. A
non-exhaustive list of these packages includes: Ananke [Lee et al., 2023], autobounds [Duarte
et al., 2024], dosearch [Tikka et al., 2021], DAGitty [Textor et al., 2016], software developed
as part of the Tetrad project (equipped with Python and R interfaces) [Ramsey and Andrews,
2023], as well the pcalg package [Kalisch et al., 2012], and extensions to deal with missing values
[Andrews et al., 2024].
Further, sensitivity analyses results can be fruitfully applied to graphical models. For example,
nongraphical results in Robins et al. [1999] can easily be represented in graphical structures that
encode MAR, conditional or sequential ignorability, and other types of Markov restrictions. For
instance, the permutation model in Robins [1997] is shown to be a graphical model in our paper.
In addition, graphical models allow a particularly powerful form of nonparametric sensitivity
analysis, where consistent inferences are made in union models defined over a potentially large
class of graphs that the analyst is uncertain about; see for example, [VanderWeele and Shpitser,
2011, Yang et al., 2024, Shpitser and VanderWeele, 2010, Shpitser et al., 2010, Chang et al.,
2024, Wang et al., 2025]. For instance, the result in [VanderWeele and Shpitser, 2011] states
that identification of the causal effect by covariate adjustment may be formulated without
precise knowledge of the graph, but only via the set of common causes of the treatment and
outcome, provided an adjustment set exists. The method developed by Yang et al. [2024] provide
similar robustness guarantees with more complex types of identifying functionals, including the
front-door functional, and the ratio functional arising in instrumental variable analysis.
7
We share the authors’ view that algorithmic and computational advances will be needed to bring
graphical identification methods to bear on high-dimensional problems with complex missingness
patterns. We see this as an exciting frontier for future work, and one where continued integration
of causal inference tools with missing data methodology will be especially fruitful. We believe
smoothing and sparsity methods, as well as sum-product algorithm type methods developed in
the graphical modeling literature may all be relevant for these advances [Lee et al., 2021].
3
Conclusion
We thank the discussants for their thoughtful contributions. Together, their contributions point
toward a roadmap: combine graphical identification, auxiliary information, intervention-based
censoring perspectives, efficiency-oriented estimation, and structured sensitivity analysis into a
practical toolkit for MNAR problems.
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Response to Discussions of "Causal and Counterfactual Views of Missing Data Models" Razieh Nabi,1 Rohit Bhattacharya,2 Ilya Shpitser,3 James M. Robins4 1 .S.A. 2 .S.A. 3 .S.A. 4Departments of Biostatistics and Epidemiology, Harvard University, Boston, MA, U.S.A. 1 Introduction We are grateful to the discussants - Levis and Kennedy [2025], Luo and Geng [2025], Wang and van der Laan [2025], and Yang and Kim [2025] - for their thoughtful comments on our paper [Nabi et al., 2025]. Below we summarize our main contributions before responding to each discussion in turn. Graphical models have emerged as an important tool for clarifying identifying assumptions made in both causal inference [Pearl and Robins, 1995, Pearl, 2000] and missing data [Robins and Gill, 1997, Bhattacharya et al., 2019, Nabi et al., 2020, Malinsky et al., 2021, Mohan and Pearl, 2021]. Our paper [Nabi et al., 2025] shows how recent techniques motivated by causal graphical modeling may be fruitfully applied to obtain identification in missing data models. These recent techniques allow us to obtain novel identification results that would not be possible to obtain in standard causal inference problems, except by imposing implausible additional assumptions, such as rank preservation. Specifically, we formalize how identifiability of the target (i.e., complete) data law can be viewed as identification of a joint distribution over counterfactuals L(1), the variables that would have been observed if all missingness indicators R were set to one (that is, if no data were missing). This reframing yields a counterfactual analogue of the g-formula. When assumptions encoded in the Markov properties of a missing data DAG (m-DAG) allow us to express the counterfactual g-formula in terms of the factuals, we obtain (nonparametric) identification. That is, noting that the target law p(l(1)) satisfies p(l(1)) = p(l, R = 1)/p(R = 1 | l(1)), it follows that when the missingness selection model p(R = 1 | L(1)) is identified from the observed law, so is p(l(1)). In our paper we used the word "nonparametric" in two different ways. The joint distribution of (L, L(1), R) is Markov to a given m-DAG if it factorizes as the product of the conditional densities of each variable given its parents. In the graphical causal modeling literature, the model is said to be "nonparametric" just when these conditional densities are left unrestricted (with the exception that L is a deterministic function of its parents). The term "nonparametric identification" refers to identification in such a model. In contrast, in the missing data literature in statistics, a model is said to be nonparametric if it places no restrictions on either the observed data law or the target data law p(l(1)). It is said to be nonparametric just identified (NPI) if the target law is identified from the observed data law. The permutation model [Robins, 1997] we discuss in our paper is known to be NPI. In contrast, all the other identified missing data models in our paper place testable restrictions on the joint distribution of the observed data. In 1 16 Oct 2025 fact, we conjecture that the permutation model is the only m-DAG model that is NPI. In the following, we use nonparametric in the graphical causal modeling sense. A key message of our paper is that features specific to missing data models (in particular the partial observability of the counterfactuals through the proxies L and the structural restriction that R and L do not cause L(1)) can deliver parameter identification in settings where analogous parameters in hidden variable causal DAGs are not identified. We catalog several such nonparametric identification techniques in m-DAGs, and we clarify similarities and differences with standard causal identification, including when additional assumptions like rank preservation might be needed for causal analogues. Each discussion extends or challenges our framework in important ways. Levis and Kennedy [2025] highlight identification strategies complementary to ours, based on the existence of instrumental and shadow variables, discuss semiparametric estimation theory for functionals arising from such strategies, as well as discuss connections between m-DAGs and Single World Intervention Graphs (SWIGs). Luo and Geng [2025] analyze self-censoring MNAR mechanisms with binary variables, deriving identifiability results that leverage auxiliary variables. Wang and van der Laan [2025] emphasize, as we do, viewing missingness as interventions, and connect our perspective to censoring in survival models. Yang and Kim [2025] and Wang and van der Laan [2025] both examine challenges of applying m-DAGs in applications, particularly assumption validation, scalability, and sensitivity analysis. 2 Response to the discussions 2.1 On the discussion by Levis and Kennedy Levis and Kennedy [2025] draw attention to additional causal identification tools and highlight implications for estimation. In particular, they emphasize the relevance of instrumental variables, shadow variables, and SWIGs as complementary devices for reasoning about identification in missing data problems. They also consider how identification results derived from m-DAGs can be translated into practical estimation procedures with desirable asymptotic properties. Their discussion situates our contribution within a broader pipeline of causal inference methods and points toward promising directions for future methodological development. We appreciate Levis and Kennedy's thoughtful remarks. On their first point, we agree that instrumental variables [Sun et al., 2018], shadow variables [Miao et al., 2024], and related tools [Li et al., 2023] are useful complements. Our emphasis in the paper, however, was on a nonparametric identification framework: aside from the Markov restrictions encoded via independence assumptions in m-DAGs, we make no further distributional assumptions. In contrast, the use of IVs or shadow variables begins in settings where nonparametric identification fails, and proceeds by imposing additional restrictions (such as functional form constraints e.g., homogeneity assumptions when using of IVs), relevance assumptions which result in generic identification, or restrictions on the support of variables to restore identification. These extra ingredients move the analysis outside the purely nonparametric domain that was our focus. On estimation, we appreciate the worked example they provide. While our goal in this paper was primarily conceptual, highlighting the philosophical parallels and differences between missing data and causal identification, their discussion underscores the importance of connecting identification results to practical estimation and efficiency theory. We agree this is important future work. We also agree with Levis and Kennedy that causal inference problems in the presence of confounding often occur together with censoring, and that obtaining identification when both complications are present is challenging. We also appreciate their worked example illustrating these complications. 2 A Y A a Y (a) A 1 Y (1) + A 0 Y (0) R L R r = 1 L(1) A X Y A a X Y (a) A 1 X Y (1) + A 0 X Y (0) R X L R r = 1 X L(1) (a) (b) (c) (d) Figure 1: (a) Ignorable treatment model with SWIGs; (b) Its MCAR analogue, with a single relevant SWIG; (c) Conditionally ignorable treatment model with SWIGs; (d) Its MAR analogue, again with only one relevant SWIG. We would like to offer two notes of caution regarding generalizing the construction presented by Levis and Kennedy, first on the appropriate generalization of the SWIG splitting construction from causal diagrams to m-DAGs, and second on the general utility of using SWIGs for obtaining identification in missing data settings. On the translation of m-DAGs to SWIGs For simple MAR models where the m-DAG's structure resembles that of the standard conditionally ignorable model in causal inference, the translation of an m-DAG to an equivalent SWIG is fairly natural. There are, however, important distinctions even in these simple settings. In causal inference, SWIGs provide a template indexed by treatment interventions (two templates if the treatment is binary) [Richardson and Robins, 2013]; in the missing data analogue, only the template for R = 1 is meaningful, since the counterfactual L(0) is not defined. This difference is highlighted in Figures 1(a, b) for a causal model satisfying ignorability and an analogous missing data model that is MCAR. Similarly, Figures 1(c, d) highlight the same for a conditionally ignorable causal model and a MAR missing data model. Caution must be exercised when SWIGs are constructed from m-DAGs representing MNAR mechanisms. For instance, in a missing data model with self-censoring, a natural idea for what the SWIG should look like would be the graph shown in Figure 2(a). However, if we try to collapse the random and fixed nodes in this SWIG to reconstruct the observed data graph, we obtain the graph shown in Figure 2(b), which erroneously implies a cycle in the underlying data generating process. The problem arises from conflating the full data variable L(1) in m-DAGs, and the observed data variable L relabeled to be counterfactual in the SWIG construction. An appropriate view of m-DAGs that avoids these types of difficulties is to view full data variables such as L(1) as unobserved confounders U, with additional structure imposed by missing data consistency. In this view, the SWIG corresponding to the self-censoring model is better represented by Figure 2(c), where the variable L(1) is viewed as an unobserved confounder U with special structure, which influences both the observability indicator R, and the observed proxy variable L. The m-DAG corresponding to this view of the self-censoring model is shown in Figure 2(d). SWIGs do not make identification arguments for MNAR problems clearer The primary motivation for SWIGs in the causal inference context is to provide a graphical representation for independences that arise in identification arguments. The function of m-DAGs is to provide precisely the same graphical representation in missing data problems, rendering SWIGs unnecessary. 3 R r = 1 L(1) (a) R L (b) R r = 1 U ≡L(1) L(1) (c) R U ≡L(1) L (d) Figure 2: (a) A possible SWIG representation of a self-censoring missing data model; (b) Stitching SWIGs into an observed-data model produces a cycle; (c) A SWIG that draws a distinction between the full data variable L(1), viewed as an unobserved confounder U with extra restrictions, and the observed variable L under an intervention where R is set to 1; (d) The full data graph equating the full data variable L(1) with an unobserved confounder U with special structure. U1 ≡L(1) 1 U2 ≡L(1) 2 R1 R2 L1 L2 (a) U1 ≡L(1) 1 U2 ≡L(1) 2 R1 R2 L(1) 1 L(1) 2 r1 = 1 r2 = 1 (b) Figure 3: (a) The block-parallel MNAR model, labeling missing variables as unmeasured for the purposes of constructing a SWIG; (b) The corresponding SWIG obtained from (a). Take, for example, the independences that can be read from the m-DAG in Figure 3(a) and the corresponding SWIG in Figure 3(b). From either graph, we are able to extract independences of the form Rk ⊥⊥L(1) k , R-k | L(1) -k, for k ∈{1, 2} that define the binary block-parallel MNAR model. That is, the m-DAG itself is just as expressive as the SWIG for reading off independences important for identification in missing data models. In fact, we now argue that attempting identification using standard causal inference arguments from the SWIG may be problematic for missing data problems due to the single counterfactual nature of missing data, which as we discussed in Section 6 of our paper is more akin to the rank preservation assumption in causal inference. To obtain the identification of the target law p(l(1) 1 , l(1) 2 ) of the block-parallel MNAR model, we showed in Section 5.2 of our paper that a parallel use of the g-formula was required, which never arose in standard causal inference settings. If just the marginal p(l(1) 2 ) is desired, this is obtained by marginalization. Now suppose instead, we attempted to perform identification arguments for p(l(1) 2 ) using the SWIG, in a manner similar to the derivation of the adjustment formula for causal inference. This would proceed as follows. First notice from the SWIG in Figure 3(b) that L(1) 2 ⊥⊥R2 | U1, R1 - this is analogous to conditional ignorability except U1 is observed only when R1 = 1. Thus we have, p(l(1) 2 ) = X r1,u1 p(r1, u1, l(1) 2 ) = X r1,u1 p(r1, u1) p(l(1) 2 | r1, u1) = X r1,u1 p(r1, u1) p(l(1) 2 | r1, u1, r2 = 1) = X r1,u1 p(r1, u1) p(l2 | r1, u1, r2 = 1). Despite the final expression above appearing to be a function that is devoid of counterfactuals, the appearance of the variable U1 in the expression prevents identification, as U1 is observed 4 only when R1 = 1. Since R1 cannot be set to the value 1 in the formula above, we are unable to establish identification via the arguments presented. In fact, no sequential strategy that we are aware of based on the SWIG in Figure 3(b) would help establish identification. That is, the parallel fixing arguments presented in Section 5.2 of our paper are required here. In short, while applying the SWIG construction to m-DAGs offers useful insights in causal problems with simple types of missingness, such as MCAR or MAR, we caution that such constructions must be done with care, as Levis and Kennedy have done. To illustrate, Figure 4(a) extends the example of Levis and Kennedy to include missingness in both the outcome Y and treatment A. The goal here is to identify the average causal effect of A on Y in the absence of censoring. Just as in the previous example shown in Figures 3(a, b), an argument based solely on SWIGs does not yield identification. Instead, identification strategies that leverage restrictions encoded in both m-DAGs and SWIGs are necessary. In particular, we first obtain the target law p(X, A(1), Y (1)) by applying inverse probability weighting with the product of the propensity scores for R1 and R2, resulting in the graph in Figure 4(b). We then apply the g-formula on A(1) to adjust for confounding by X, taking advantage of the restrictions in the SWIG shown in Figure 4(c). This yields p(Y (1,a)), the distribution of the outcome Y , had it not been censored and had treatment been set to value a, from which the average causal effect of interest may be obtained. Finally, we note that there is currently no complete graphical identification theory for causal parameters associated with arbitrary m-DAGs that encode both MNAR and (possible) confounding by unmeasured common causes U. We expect this theory to employ both SWIG and m-DAG constructions, as in the worked example of Levis and Kennedy, and our examples above. 2.2 On the discussion by Luo and Geng Luo and Geng [2025] extend our framework by focusing on self-censoring MNAR mechanisms, where the missingness of a variable depends directly on its unobserved value. They study such models with binary outcomes and establish identifiability results under explicit, testable conditions. Their discussion highlights how auxiliary variables, either baseline or follow-up measurements, can be leveraged to restore identification. In doing so, they illustrate that self-censoring structures, which were not emphasized in our paper, can still yield identification in important cases. They also point to future directions for extending these ideas beyond the binary case to more general settings. We thank Luo and Geng for their examples of self-censoring mechanisms. In our paper, we noted self-censoring as a form of MNAR but did not explore it further, since m-DAG models with self centering are never nonparametrically identified [Mohan et al., 2013]. Since our emphasis was on nonparametric identification, we did not employ any restrictions not implied by the m-DAG factorization, including (i) relevant restrictions needed by instrumental variable or proxy methods, or (ii) constraints implied by state space restrictions. As Luo and Geng [2025] illustrated, such assumptions can sometimes yield identifiability, and their results highlight concrete testable conditions under which this occurs. We fully agree that self-censoring is of practical importance, with income surveys, sensitive health questions, and other social science settings providing common examples. Their discussion usefully demonstrates how auxiliary information, through baseline or follow-up variables, can be leveraged to address such scenarios. More broadly, we believe that in MNAR submodels that identify the target law, restrictions implied by the m-DAG always yield testable implications for the observed data law, with the 5 A(1) X Y (1) RA RY A Y (a) A(1) = A X Y (1) = Y rA = 1 rY = 1 A Y (b) A(1) = A a X Y (1,a) = Y (a) rA = 1 rY = 1 A Y (c) Figure 4: (a) Extension of Levis and Kennedy's example to have two missing variables. (b) Graph obtained after fixing RA and RY in parallel. (c) SWIG obtained by splitting the treatment variable A after already having fixed RA and RY . exception of the permutation model [Robins, 1997]. Examples of such tests have been developed in prior work on graphical models [Mohan and Pearl, 2014, Nabi and Bhattacharya, 2023, Guo et al., 2023, Chen et al., 2023]. Overall, we view their discussion as complementary to our focus: while our framework aimed to catalog nonparametric identification results, their work illustrates how additional structure can both sharpen identifiability and connect to practical applications. 2.3 On the discussion by Wang and van der Laan Wang and van der Laan [2025], like us, approach missingness through the lens of interventions. They emphasize the close relationship between missing data models and multivariate censoring models in survival analysis. They carefully prove that the permutation missingness model is MNAR rather than MAR and flesh out our example of why the permutation model is substantively plausible in some real world applications. They also discuss substantive settings where the block sequential model is plausible. They are less certain that the other identifiable MNAR models discussed in our paper are substantively plausible. In our view, it is difficult to establish plausibility of any missing data model in practice, whether it is formulated graphically or not. One advantage of the sort of theory we present is a single formulation for a large class of identifiable models (including existing models in the literature such as those described in [Zhou et al., 2010, Robins, 1997]). Since no single model may be plausible in applications, an alternative strategy is to perform the analysis of interest under a wide class of identifiable missingness models, as a form of sensitivity analysis. We make one final observation regarding the plausibility of (or the lack there of) the identifiable MNAR models considered in the paper. Identification in every case we discuss relies on an assumption that one or more counterfactuals L(1) j suffice to control confounding of the "effect" of Rk on Lk where j ̸= k. But why should we privilege L(1) j (or a set of such variables) as a set sufficient for adjustment? It seems much more plausible that there exist other unmeasured common causes of Rk and Lk (equivalently L(1) k ) that would also need to be adjusted for to eliminate confounding. Are we assuming the L(1) j suffice to control confounding because we believe it to be so or rather because we want to achieve identification for identification's sake? In fact, similar plausibility concerns arise in non-graphical identifiable models, such as non-monotone MAR. In our view, plausible models of missingness feature a description of data generation that follows a temporal order (often representable as a DAG). Unfortunately, to be realistic, even models of this type would feature unobserved confounding of the sort that would prevent identifiability. 6 2.4 On the discussion by Yang and Kim Yang and Kim [2025] emphasize the practical challenges of applying the m-DAG framework. They point out that while m-DAGs provide a principled way to encode assumptions and extend identification theory, their utility in practice depends critically on correct graph specification, which may be difficult to achieve in applied settings. They raise concerns about scalability in high-dimensional problems, where the number of variables and missingness patterns grows quickly, and about the feasibility of validating the conditional independence assumptions encoded in an m-DAG. They further stress the importance of developing systematic approaches to sensitivity analysis, noting that small misspecifications of the graph can have consequences for identification and downstream tasks. We thank Yang and Kim for raising important questions about the practical utility of m-DAGs, and graphical modeling more broadly. Our goal in this paper was primarily conceptual: to provide a general framework that connects causal and missing data perspectives and to catalog identification results that arise from this connection. Their emphasis on practical feasibility is appreciated, and complementary to our focus. That said, continuing work in the graphical modeling community over the last two decades has rendered much of the critiques of the graphical modeling approach to causal inference and missing data problems out of date. Identification of causal and missing data parameters must, by necessity, rely on equality restrictions in the full data distribution. Graphical models aim to encode these restrictions in a way that allows the construction of a plausible data generating mechanism consistent with a temporal order of events. Models that obtain identifiability without a corresponding graphical representation, for instance the non-monotone missing at random (MAR) model, often lack such a plausible mechanism. Indeed, existing efforts that argue for the plausibility of models such as non-monotone MAR embed them into a graphical model or a mixture of such models [Robins and Gill, 1997]. Over the last decade, an explosion of easy to use open-source packages that take advantage of graphical models have been developed for all tasks in causal inference and missing data, including establishing identification, constructing estimators and applying them to data, conducting sensitivity analyses, establishing bounds on non-identified parameters, and model selection. A non-exhaustive list of these packages includes: Ananke [Lee et al., 2023], autobounds [Duarte et al., 2024], dosearch [Tikka et al., 2021], DAGitty [Textor et al., 2016], software developed as part of the Tetrad project (equipped with Python and R interfaces) [Ramsey and Andrews, 2023], as well the pcalg package [Kalisch et al., 2012], and extensions to deal with missing values [Andrews et al., 2024]. Further, sensitivity analyses results can be fruitfully applied to graphical models. For example, nongraphical results in Robins et al. [1999] can easily be represented in graphical structures that encode MAR, conditional or sequential ignorability, and other types of Markov restrictions. For instance, the permutation model in Robins [1997] is shown to be a graphical model in our paper. In addition, graphical models allow a particularly powerful form of nonparametric sensitivity analysis, where consistent inferences are made in union models defined over a potentially large class of graphs that the analyst is uncertain about; see for example, [VanderWeele and Shpitser, 2011, Yang et al., 2024, Shpitser and VanderWeele, 2010, Shpitser et al., 2010, Chang et al., 2024, Wang et al., 2025]. For instance, the result in [VanderWeele and Shpitser, 2011] states that identification of the causal effect by covariate adjustment may be formulated without precise knowledge of the graph, but only via the set of common causes of the treatment and outcome, provided an adjustment set exists. The method developed by Yang et al. [2024] provide similar robustness guarantees with more complex types of identifying functionals, including the front-door functional, and the ratio functional arising in instrumental variable analysis. 7 We share the authors' view that algorithmic and computational advances will be needed to bring graphical identification methods to bear on high-dimensional problems with complex missingness patterns. We see this as an exciting frontier for future work, and one where continued integration of causal inference tools with missing data methodology will be especially fruitful. 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2510.14693
|
FibRace: a large-scale benchmark of client-side
proving on mobile devices
Simon Malatrait
simon@kakarot.org
KKRT Labs
Alex Sirac
alex@hyli.org
Hyli
October 9, 2025
Abstract
FibRace, jointly developed by KKRT Labs and Hyli, was the first large-scale
experiment to test client-side proof generation on smartphones using Cairo M.
Presented as a mobile game in which players proved Fibonacci numbers and
climbed a leaderboard, FibRace served a dual purpose: to engage the public
and to provide empirical benchmarking. Over a three-week campaign (September
11-30, 2025), 6,047 players across 99 countries generated 2,195,488 proofs on
1,420 unique device models. The results show that most modern smartphones can
complete a proof in under 5 seconds, confirming that mobile devices are now
capable of producing zero-knowledge proofs reliably, without the need
for remote provers or specialized hardware. Performance was correlated primarily
with RAM capacity and SoC (System on Chip) performance: devices with at least
3 GB of RAM proved stably, when Apple’s A19 Pro and M-series chips achieved
the fastest proving times. Hyli’s blockchain natively verified every proof onchain
without congestion. FibRace provides the most comprehensive dataset to date on
mobile proving performance, establishing a practical baseline for future research in
lightweight provers, proof-powered infrastructure, and privacy-preserving mobile
applications.
Keywords zero-knowledge proofs · client-side proving · mobile cryptography ·
blockchain · Hyli · KKRT Labs
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 The FibRace gamified campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Proof generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 Data collection and dataset structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.5 Benchmark parameters and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.6 Ethical and experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.7 Data availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Benchmark results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Global participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Proving performance distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Hardware correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 Reliability and crashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.5 Summary interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Discussion and implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.1 Viability of client-side proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Implications for developers and ecosystem builders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Broader impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.4 Implications for Hyli and KKRT Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Limitations of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.1 Technical limitations . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .. .. .. .. . . . . . .. .. .. .. . . . . . . . . . .. . . . 10
5.2 Game design limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Tables
Table 1 Collected datasets during FibRace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Table 2 FibRace benchmarking assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Figures
Figure 1 FibRace downloads by country. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Figure 2 FibRace Cairo M proof generation duration distribution (central 99%). . . . . . . . . . . . 5
Figure 3 Key performance indicators per RAM capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Figure 4 Proof volume per device manufacturer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Figure 5 Best 50 SoCs for Cairo M proof generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 Introduction
Zero-knowledge proofs have become a cornerstone of blockchain scalability and privacy. In recent
years, most progress in proving technology has focused on server-side proving, powering large-
scale systems such as Layer 2 roll-ups and blockchain settlement layers. These developments,
spurred by launches of high-throughput proving networks in 2024 and 2025, have dramatically
reduced latency and cost for infrastructure-level proofs.
Yet this focus leaves a gap in a critical area: client-side proving.
Server-based models assume proofs are generated in centralized environments or distributed
networks, but not on user devices. This architecture works well for throughput, not for privacy.
Some use cases, especially those involving personal identity and private data, can only be
realized if proofs are generated locally. Confidentiality is no longer guaranteed when sensitive
data leaves the user’s device for processing elsewhere.
Client-side proving minimizes friction for both developers and users. Developers no longer need
to maintain or pay for remote-proving infrastructure: computation is delegated to end-user
devices. Users, in turn, gain stronger privacy guarantees: proofs are generated and owned locally,
without ever exposing private data to external servers or custodial intermediaries.
Recent incidents highlight this need vividly: as we write this report in October 2025, millions of
Discord users’ government IDs were leaked following a third-party verification breach, reigniting
debates about how identity management should be handled online (Chia 2025). Similarly,
systems like Google’s zkWallet with Ligero and other emerging proof-of-age or proof-of-identity
applications show that for many privacy-preserving use cases, the proof must be generated by
the user, not by a remote prover (Kuhn 2025).
Despite the progress in ZK proving systems, client-side proving remains a technical challenge.
For instance, a 2025 benchmark across several proving schemes showed that in-browser proving
for p256 ECDSA signatures was still largely impractical, with only Noir achieving limited
usability (Vlad 2025).
To achieve privacy and scalability at a global level, proof generation must become viable across
the full spectrum of consumer devices, including low-end smartphones. KKRT Labs’s Cairo
M, a ZK-VM optimized for mobile hardware, was designed to address this challenge (KKRT
Labs 2025).
Developing for client-side proving inevitably raises a central question: Can we generate
proofs directly on consumer devices, at scale, and under real-world conditions?
In September 2025, FibRace was created as a joint initiative by KKRT Labs, developers
of the Cairo M prover, and Hyli, the proof-powered blockchain for the next generation of
applications. By disguising a client-side proving benchmark as a mobile game, it allowed us to
test thousands of device models worldwide and begin answering this question experimentally.
1
2 Methodology
2.1 Overview
The FibRace campaign was designed as a public mobile game and a controlled benchmark
of client-side proving using Cairo M.
Players interacted with a game interface, computing Fibonacci numbers and climbing
leaderboards (Sirac 2025). Each interaction generated a data point for the study. This dual
framing allowed the experiment to reach thousands of devices under natural, real-world usage
conditions.
2.2 The FibRace gamified campaign
FibRace was a mobile game designed to turn a serious experiment in zero-knowledge
benchmarking into an accessible and playful experience. Its core challenge was simple: prove
Fibonacci numbers as quickly as possible and climb the leaderboard.
Each proof was generated locally on the player’s smartphone using Cairo M, KKRT Labs’s
prover optimized for mobile hardware (Walter 2025). Once a proof was produced, it was
transmitted to Hyli’s proof-powered blockchain, which natively verifies Cairo M proofs and
settles them onchain. Every successful proof advanced the player in the game and added a real
data point to the benchmark.
The campaign ran from September 11 to 30, 2025, and reached a diverse global audience. Players
from 99 countries participated across Android and iOS devices, 5,156 of them generating over
2 million proofs on 1,420 unique device models.
2.2.1 Gameplay
Participants could:
1. Generate a proof of the computation of the 𝑛-th term of the Fibonacci sequence, for a
randomly chosen 𝑛.
2. Mint a collector card that displayed their number and proving time, designed for easy
sharing on social media.
3. Compete on leaderboards tracking three metrics (fastest proof, highest proving
frequency, and largest collection) to encourage repeated participation.
No financial rewards, airdrops, or tradeable assets were offered. This ensured that gameplay
reflected genuine engagement and natural device diversity rather than speculative incentives.
2.2.2 Benchmark connection
Every player session generated measurable data collected automatically through the app’s
leaderboard and device-logging modules. This structure enabled KKRT Labs and Hyli to gather
an unprecedented dataset on real-world proving performance.
By merging gaming dynamics with experimental rigor, FibRace shows that large-scale
benchmarking of zero-knowledge provers can occur in open, decentralized conditions, engaging
users not as test subjects, but as participants in a game.
2
2.3 Proof generation
Each game session consisted of the following sequence:
1. Sampling: The app pseudo-randomly sampled a 𝑛 in [1..100, 000] for which to compute the
𝑛-th term of the Fibonacci sequence 𝐹𝑛 on the M31 prime field.
2. Proof generation: The player’s smartphone executed the Cairo M program locally to prove
the correct computation of 𝐹𝑛, with 96 security bits. The higher the number 𝑛, the more
resource-intensive the proving.
3. Verification and settlement: The proof was sent to Hyli, where it was natively verified and
settled on the Hyli testnet blockchain.
4. Data logging: Metadata, including device model, OS version, RAM, SoC information, proof
duration, and frequency, was recorded client-side and uploaded once verification succeeded.
All computation and proof generation occurred entirely on-device. No remote provers, cloud
relays, or trusted execution environments were used.
2.4 Data collection and dataset structure
Four datasets were collected during the campaign, described in Table 1. The initial analysis
was performed in a Jupyter Notebook, as detailed in (Malatrait 2025):
Dataset
Description
mintedItems
One record per successfully generated and verified proof (2,195,488 rows)
players
Player profiles, including historical best proof and mint counters
devices
Hardware and OS metadata for each registered device (1,420 unique models)
crashLogs
Reports of failed or incomplete proof attempts, including device identifiers
Table 1: Collected datasets during FibRace.
Each entry is timestamped.
2.5 Benchmark parameters and assumptions
The campaign defined a set of baseline assumptions to guide analysis and interpretation,
summarized in Table 2.
A proof is deemed valid if successfully generated, sent to Hyli, verified by Hyli, and settled
onchain on Hyli. The sampling window was the duration of the FibRace campaign, September 11
to 30, 2025. The program used for the benchmark would be a fixed Cairo M program computing
the randomly chosen 𝑛-th term of the Fibonacci sequence.
3
Parameter
Description
Assumed / Target Value
Minimum usable device RAM
We
consider
that
client-side
proving is viable if successful
on enough devices with a RAM
higher than this.
≥ 3 GB
Acceptable crash rate
We considered Cairo M was
successful on device models under
a certain failure threshold.
< 1% failure rate
Target proof duration
We considered Cairo M offered
a smooth user experience if the
median proving time was under
this threshold.
≤ 5 s per proof
Table 2: FibRace benchmarking assumptions.
KKRT Labs used these thresholds to evaluate Cairo M’s performance. Hyli used them to
characterize onchain verification throughput.
2.6 Ethical and experimental considerations
• Voluntary participation: Players opted in by installing the application. No rewards,
airdrops, or financial incentives were offered, and this was made clear in the launch posts as
well as in the application’s interface.
• Data minimization: Only technical and performance metrics were collected. No personal
identifiers were stored. FibRace did not enable account creation or data portability.
• Cross-platform coverage: Android and iOS were supported, ensuring SoC diversity: Apple
Silicon, Qualcomm Snapdragon, MediaTek, Huawei Kirin, and Samsung Exynos.
2.7 Data availability
The performance report and raw datasets are publicly available at https://github.com/kkrt-
labs/cairo-m/tree/main/docs/fibrace.
3 Benchmark results
3.1 Global participation
Across the 19-day campaign, 6,047 player names were registered. While some players
registered multiple player names (due to multi-device use or reinstalling the application), we
will refer to them as « players » in the remainder of this report. Of these players, 5,156 generated
at least one valid proof.
The players used 1,420 unique device models. Device models are provided by the
manufacturers themselves and read from the OS API. According to the app stores’ data, the
game was installed by players in 99 countries. The Figure 1 is an heatmap of the number of
FibRace downloads per country.
In total, 2,195,488 proofs were generated on FibRace during the campaign.
4
Figure 1: FibRace downloads by country.
3.2 Proving performance distribution
The duration of proof generation followed a right-skewed distribution, shown in Figure 2, with
most devices completing a proof within 3 to 10 seconds. The average frequency (Cairo M
cycles per second) was 30 kHz, but the median was closer to 10 kHz, reflecting the heavy tail
of slower devices.
High-performance clusters (≤ 3 s proofs) were led by the most recent iPhone and iPad models
running Apple A19 Pro, M1–M4 chips, followed by older ones and high-end Android phones.
Mid-range Android devices populated the 5–10 s band. Older devices (with an average proving
time of more than 10s) showed increased crash incidence and throttling effects.
Figure 2: FibRace Cairo M proof generation duration distribution (central 99%).
5
3.3 Hardware correlations
3.3.1 RAM Capacity
Figure 3 displays key performance indicators, grouped by RAM buckets.
At least 3 GB of available memory is required for successful proving on Cairo M.
Devices with 3–4 GB RAM were the most prone to crashes due to Cairo M’s memory footprint,
while stability scaled steadily with RAM until plateauing beyond 6 GB.
Figure 3: Key performance indicators per RAM capacity.
3.3.2 Device and manufacturer trends
Most proofs were generated on Xiaomi, Apple, and Samsung devices, as per Figure 4, reflecting
global market share rather than platform bias (Kuzmin 2025). Within each brand, newer chip
generations demonstrated predictable improvements in both duration and frequency.
Figure 4: Proof volume per device manufacturer.
6
Figure 5 ranks the 50 best SoC, performance wise. It has been sorted based on proof duration
as well as frequency, with best score as well as best average.
Apple SoCs dominated the upper performance tier, with the latest iPhone chips and M-series
on iPad. iPads with the Apple M4 SoC and iPhones with the A19 Pro achieved the fastest
proving times, approaching the limits of what Cairo M can deliver on mobile hardware.
Following the most recent Apple SoCs, high-end SoCs from Qualcomm, MediaTek, Kirin, and
Exynos achieved great performance, with a proving time under 3 seconds.
Figure 5: Best 50 SoCs for Cairo M proof generation.
3.4 Reliability and crashes
3.4.1 Reliability per device
Stable devices maintained success rates above 95%, confirming that Cairo M can operate
reliably on modern smartphones without external proving support.
7
The overall crash rate during the FibRace campaign is 0.72%, representing the number of
recorded crash events (15,762 occurrences) over the total number of proof successfully generated.
The majority of these occurred on devices with limited RAM (3-4 GB) as shown in Figure 3,
which are often over five years old.
Crash rates correlated strongly with memory pressure rather than CPU limitations, suggesting
that future optimizations should prioritize reducing the RAM footprint.
3.4.2 Temporal trends
Hourly aggregates showed distinct activity peaks corresponding to regional evening hours,
demonstrating genuine player engagement rather than automated testing.
Proof throughput and crash incidence followed inversely proportional patterns: as participation
increased, minor rises in crash frequency indicated thermal throttling or sustained high-load
conditions on some devices.
We also encountered a crash error on higher-RAM devices when minting many times in a row
(usually more than a hundred). While unable to identify the root cause of these crashes with
certainty, we assume that, although the spawned Rust thread ought to be killed at the end
of each generated proof, it may not have been properly done, which could lead to OOM-type
crashes. The short delay incurred when verifying and settling the proof onchain between two
proof generation might be the cause. If this delay is too short, the thread may not properly
close. Due to the short campaign duration and to the benchmark’s focus on one-time proving,
we did not investigate the issue further.
3.5 Summary interpretation
• Performance: Modern smartphones can generate Cairo M proofs in under 5 seconds on
average.
• Accessibility: Proofs succeeded on 1,400+ device models, proving wide feasibility.
• Reliability: Devices with ≥ 3 GB RAM show consistent stability; below that threshold,
proving fails systematically.
• Hardware Insight: CPU frequency correlates strongly with proof frequency, confirming
Cairo M’s effective scaling on mobile hardware.
4 Discussion and implications
4.1 Viability of client-side proving
The FibRace experiment demonstrates that client-side proving using Cairo M is technically
viable and globally accessible.
Over 2 million proofs were generated, totaling 333 billions Cairo M cycles on more than 1,400
distinct device models, confirming that local proof generation can operate reliably across diverse
hardware and network environments.
The results validate that the majority of modern smartphones can generate zero-knowledge
proofs in under 5 seconds, which is well within the user-acceptable latency for casual or
transactional applications. Devices with less than 3 GB of RAM consistently failed to complete
a proof, delineating a clear lower bound for current implementations.
8
From a research perspective, FibRace provides the first large-scale, open dataset that quantifies
the distribution of proof generation times across consumer hardware. This enables empirical
calibration of future prover optimizations and helps refine benchmarks for memory footprint,
cycle counts, and acceptable crash rates.
4.2 Implications for developers and ecosystem builders
For developers, these results are a strong signal that end-user devices can handle proving
workloads previously reserved for servers or specialized hardware.
Client-side proving simplifies architecture, reduces cost, and improves privacy, aligning with the
long-term goal of self-sovereign computation.
In practical terms, this means applications can now:
• Deploy zero-knowledge solutions without relying on remote provers.
• Execute verifiable interactions offline, syncing proofs asynchronously to Hyli.
• Build new interaction models (e.g., games, attestations, on-device identity proofs) that no
longer depend on cloud resources.
For ecosystem builders, FibRace establishes a methodology for gamified benchmarking,
collecting real-world performance data in a playful and opt-in format. This approach can be
replicated for other proving schemes or for measuring device-specific bottlenecks, such as GPU
acceleration or energy efficiency.
4.3 Broader impact
FibRace bridges the gap between cryptographic research and user experience. Re-framing
benchmarking as a public game showcased how ZK technology can reach mainstream audiences
without compromising scientific rigor.
Future iterations of this model could benchmark additional proving schemes, introduce
controlled parameters (e.g., energy consumption, thermal throttling), and work upon different
circuits.
4.4 Implications for Hyli and KKRT Labs
For Hyli, the experiment validated the scalability of its onchain proof verification layer. Over
two million verifications were processed without congestion, even during peaks with 260,000
proofs per hour, demonstrating that Hyli’s implementation of the Autobahn consensus can
absorb high-frequency proof submissions from a large user base (Giridharan et al. 2024).
For KKRT Labs, FibRace served as a proof-of-concept for Cairo M, showing that its
mobile optimization can sustain multi-million-proof throughput across everyday devices. The
benchmark offers valuable feedback for future improvements.
Together, Hyli and KKRT Labs demonstrated that a proof-powered L1 and a mobile-optimized
prover can function seamlessly, paving the way for scalable, privacy-preserving applications that
can be deployed to consumer hardware.
9
5 Limitations of the experiment
While FibRace demonstrates the viability of client-side proving on mobile devices, several
limitations must be acknowledged to correctly interpret the dataset and results.
5.1 Technical limitations
5.1.1 Benchmark circuit scope
The experiment focused exclusively on proving computations of terms of the Fibonacci sequence,
an arithmetic circuit with low memory usage.
While representative for testing Cairo M baseline performance, and while we benched a wide
range of witnesses, resulting in traces of 8 to 800,000 Cairo M cycles, it does not capture all of the
Cairo M capabilities that can be found in real-world proving workloads, such as SHA2 hashes.
5.1.2 Memory footprint
Cairo M’s memory consumption constrained participation on low-end devices. Proofs could not
be generated on devices with less than 3 GB of RAM, and devices with 3–4 GB of RAM showed
the highest crash rate, respectively 4.2 and 4.4 crashes per generated proof, due to OOM errors
on the largest witnesses (𝑛 near the 100,000 upper-bound).
It showcases that devices with limited available memory can generate proofs for most workloads
of the benchmark, but not for the most expensive ones, due to memory constraints.
5.1.3 Energy and thermal effects
FibRace did not record power draw or thermal throttling. Anecdotal player feedback suggested
noticeable heat and battery drain during extended play sessions, but these factors were not
quantified.
Other performance factors were not measured, including battery state, energy saving mode
activation, and CPU temperature.
Therefore we do not have a quantitative performance degradation ratio on our assumption that
these factors lead to degraded performance.
Future benchmarks should include energy-use metrics to more effectively evaluate sustainability,
user comfort and performance impact.
5.2 Game design limitations
5.2.1 Gameplay constraints
Because FibRace was designed as a game, participants could interrupt or close the app at any
point, including by force-quitting the app while generating or sending a proof. Such interruptions
are indistinguishable from technical crashes in the dataset, introducing ambiguity in interpreting
some failure events.
5.2.2 No user accounts
The game operated without persistent user accounts to reduce friction and prevent airdrop
speculation. While this ensured privacy and simplicity, it made longitudinal tracking of player
behavior and device switching impossible. This did not affect the benchmark, but made it less
enjoyable for some players.
10
5.3 Summary
These limitations do not diminish the core finding that modern smartphones can locally perform
zero-knowledge proving. They do, however, highlight key areas for future experiments:
• measuring more complex circuits,
• measuring circuits with real-life applications,
• quantifying energy and heat effect on performance.
Together, these refinements would strengthen the empirical foundation laid by FibRace and
support the next generation of client-side proving research.
6 Conclusion
FibRace demonstrates that client-side proving is no longer theoretical: it is operational,
measurable, and already viable across a wide range of consumer devices.
Over the course of three weeks, more than 5,000 participants generated 2.2 million Cairo M
proofs on 1,420 unique device models, confirming that proof generation can occur reliably on
modern smartphones without the need for centralized infrastructure or trusted hardware.
The results validate a key assumption behind both KKRT Labs and Hyli: that distributed
proving and native onchain verification can coexist efficiently. Cairo M made on-device proof
generation accessible even to mid-range devices, while Hyli’s proof-powered Layer 1 absorbed
millions of verifications without network congestion.
Beyond its immediate findings, FibRace established a new model for open benchmarking
through gamified participation. By blending experimentation and gameplay, it produced one of
the most extensive public datasets on client-side proving to date, an approach that future ZK
ecosystems can replicate for other circuits, provers, or device classes.
The experiment’s limitations define clear next steps. Expanding to real-world use cases, and
integrating thermal and battery instrumentation.
Ultimately, FibRace offers a simple but powerful insight: zero-knowledge proofs can now
run on users’ own devices.
This shift opens a new frontier for privacy-preserving applications and proof-native blockchains,
where the path to scalable cryptography becomes user-driven.
11
Bibliography
[1] O. Chia, “ID photos of 70,000 users may have been leaked, Discord says,” BBC News,
Oct. 2025.
[2] D. Kuhn, “Google Wallet integrates ZK-proofs, a tech incubated by crypto industry,” The
Block, May 2025.
[3] Vlad, “Benchmarking in-browser p256 ECDSA proving systems,” Hyli, Mar. 2025.
[4] KKRT Labs, “Kakarot presents Cairo M: The Frontier zkVM Built for Your Phone.” Oct.
08, 2025.
[5] A. Sirac, “Launching FibRace: Prove to win!,” Hyli, Sep. 2025.
[6] C. Walter, “Cairo M Design Document.” Sep. 29, 2025.
[7] S. Malatrait, “FibRace performance report.” Oct. 01, 2025.
[8] A. Kuzmin, “Baseline Benchmarking Hardware.” Mar. 31, 2025.
[9] N. Giridharan, F. Suri-Payer, I. Abraham, L. Alvisi, and N. Crooks, “Autobahn: Seamless
high speed BFT,” in Proceedings of the ACM SIGOPS 30th Symposium on Operating
Systems Principles, New York, NY, USA, Nov. 2024, pp. 1–23.
12
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FibRace: a large-scale benchmark of client-side proving on mobile devices Simon Malatrait KKRT Labs Alex Sirac Hyli October 9, 2025 Abstract FibRace, jointly developed by KKRT Labs and Hyli, was the first large-scale experiment to test client-side proof generation on smartphones using Cairo M. Presented as a mobile game in which players proved Fibonacci numbers and climbed a leaderboard, FibRace served a dual purpose: to engage the public and to provide empirical benchmarking. Over a three-week campaign (September 11-30, 2025), 6,047 players across 99 countries generated 2,195,488 proofs on 1,420 unique device models. The results show that most modern smartphones can complete a proof in under 5 seconds, confirming that mobile devices are now capable of producing zero-knowledge proofs reliably, without the need for remote provers or specialized hardware. Performance was correlated primarily with RAM capacity and SoC (System on Chip) performance: devices with at least 3 GB of RAM proved stably, when Apple's A19 Pro and M-series chips achieved the fastest proving times. Hyli's blockchain natively verified every proof onchain without congestion. FibRace provides the most comprehensive dataset to date on mobile proving performance, establishing a practical baseline for future research in lightweight provers, proof-powered infrastructure, and privacy-preserving mobile applications. Keywords zero-knowledge proofs · client-side proving · mobile cryptography · blockchain · Hyli · KKRT Labs Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 The FibRace gamified campaign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 Proof generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Data collection and dataset structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5 Benchmark parameters and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.6 Ethical and experimental considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.7 Data availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Benchmark results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1 Global participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Proving performance distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Hardware correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.4 Reliability and crashes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.5 Summary interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Discussion and implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.1 Viability of client-side proving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Implications for developers and ecosystem builders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.3 Broader impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4 Implications for Hyli and KKRT Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 Limitations of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.1 Technical limitations . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . .. .. .. .. . . . . . .. .. .. .. . . . . . . . . . .. . . . 10 5.2 Game design limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Tables Table 1 Collected datasets during FibRace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Table 2 FibRace benchmarking assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Figures Figure 1 FibRace downloads by country. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 2 FibRace Cairo M proof generation duration distribution (central 99%). . . . . . . . . . . . 5 Figure 3 Key performance indicators per RAM capacity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 4 Proof volume per device manufacturer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 5 Best 50 SoCs for Cairo M proof generation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1 Introduction Zero-knowledge proofs have become a cornerstone of blockchain scalability and privacy. In recent years, most progress in proving technology has focused on server-side proving, powering largescale systems such as Layer 2 roll-ups and blockchain settlement layers. These developments, spurred by launches of high-throughput proving networks in 2024 and 2025, have dramatically reduced latency and cost for infrastructure-level proofs. Yet this focus leaves a gap in a critical area: client-side proving. Server-based models assume proofs are generated in centralized environments or distributed networks, but not on user devices. This architecture works well for throughput, not for privacy. Some use cases, especially those involving personal identity and private data, can only be realized if proofs are generated locally. Confidentiality is no longer guaranteed when sensitive data leaves the user's device for processing elsewhere. Client-side proving minimizes friction for both developers and users. Developers no longer need to maintain or pay for remote-proving infrastructure: computation is delegated to end-user devices. Users, in turn, gain stronger privacy guarantees: proofs are generated and owned locally, without ever exposing private data to external servers or custodial intermediaries. Recent incidents highlight this need vividly: as we write this report in October 2025, millions of Discord users' government IDs were leaked following a third-party verification breach, reigniting debates about how identity management should be handled online (Chia 2025). Similarly, systems like Google's zkWallet with Ligero and other emerging proof-of-age or proof-of-identity applications show that for many privacy-preserving use cases, the proof must be generated by the user, not by a remote prover (Kuhn 2025). Despite the progress in ZK proving systems, client-side proving remains a technical challenge. For instance, a 2025 benchmark across several proving schemes showed that in-browser proving for p256 ECDSA signatures was still largely impractical, with only Noir achieving limited usability (Vlad 2025). To achieve privacy and scalability at a global level, proof generation must become viable across the full spectrum of consumer devices, including low-end smartphones. KKRT Labs's Cairo M, a ZK-VM optimized for mobile hardware, was designed to address this challenge (KKRT Labs 2025). Developing for client-side proving inevitably raises a central question: Can we generate proofs directly on consumer devices, at scale, and under real-world conditions? In September 2025, FibRace was created as a joint initiative by KKRT Labs, developers of the Cairo M prover, and Hyli, the proof-powered blockchain for the next generation of applications. By disguising a client-side proving benchmark as a mobile game, it allowed us to test thousands of device models worldwide and begin answering this question experimentally. 1 2 Methodology 2.1 Overview The FibRace campaign was designed as a public mobile game and a controlled benchmark of client-side proving using Cairo M. Players interacted with a game interface, computing Fibonacci numbers and climbing leaderboards (Sirac 2025). Each interaction generated a data point for the study. This dual framing allowed the experiment to reach thousands of devices under natural, real-world usage conditions. 2.2 The FibRace gamified campaign FibRace was a mobile game designed to turn a serious experiment in zero-knowledge benchmarking into an accessible and playful experience. Its core challenge was simple: prove Fibonacci numbers as quickly as possible and climb the leaderboard. Each proof was generated locally on the player's smartphone using Cairo M, KKRT Labs's prover optimized for mobile hardware (Walter 2025). Once a proof was produced, it was transmitted to Hyli's proof-powered blockchain, which natively verifies Cairo M proofs and settles them onchain. Every successful proof advanced the player in the game and added a real data point to the benchmark. The campaign ran from September 11 to 30, 2025, and reached a diverse global audience. Players from 99 countries participated across Android and iOS devices, 5,156 of them generating over 2 million proofs on 1,420 unique device models. 2.2.1 Gameplay Participants could: 1. Generate a proof of the computation of the n-th term of the Fibonacci sequence, for a randomly chosen n. 2. Mint a collector card that displayed their number and proving time, designed for easy sharing on social media. 3. Compete on leaderboards tracking three metrics (fastest proof, highest proving frequency, and largest collection) to encourage repeated participation. No financial rewards, airdrops, or tradeable assets were offered. This ensured that gameplay reflected genuine engagement and natural device diversity rather than speculative incentives. 2.2.2 Benchmark connection Every player session generated measurable data collected automatically through the app's leaderboard and device-logging modules. This structure enabled KKRT Labs and Hyli to gather an unprecedented dataset on real-world proving performance. By merging gaming dynamics with experimental rigor, FibRace shows that large-scale benchmarking of zero-knowledge provers can occur in open, decentralized conditions, engaging users not as test subjects, but as participants in a game. 2 2.3 Proof generation Each game session consisted of the following sequence: 1. Sampling: The app pseudo-randomly sampled a n in [1..100, 000] for which to compute the n-th term of the Fibonacci sequence Fn on the M31 prime field. 2. Proof generation: The player's smartphone executed the Cairo M program locally to prove the correct computation of Fn, with 96 security bits. The higher the number n, the more resource-intensive the proving. 3. Verification and settlement: The proof was sent to Hyli, where it was natively verified and settled on the Hyli testnet blockchain. 4. Data logging: Metadata, including device model, OS version, RAM, SoC information, proof duration, and frequency, was recorded client-side and uploaded once verification succeeded. All computation and proof generation occurred entirely on-device. No remote provers, cloud relays, or trusted execution environments were used. 2.4 Data collection and dataset structure Four datasets were collected during the campaign, described in Table 1. The initial analysis was performed in a Jupyter Notebook, as detailed in (Malatrait 2025): Dataset Description mintedItems One record per successfully generated and verified proof (2,195,488 rows) players Player profiles, including historical best proof and mint counters devices Hardware and OS metadata for each registered device (1,420 unique models) crashLogs Reports of failed or incomplete proof attempts, including device identifiers Table 1: Collected datasets during FibRace. Each entry is timestamped. 2.5 Benchmark parameters and assumptions The campaign defined a set of baseline assumptions to guide analysis and interpretation, summarized in Table 2. A proof is deemed valid if successfully generated, sent to Hyli, verified by Hyli, and settled onchain on Hyli. The sampling window was the duration of the FibRace campaign, September 11 to 30, 2025. The program used for the benchmark would be a fixed Cairo M program computing the randomly chosen n-th term of the Fibonacci sequence. 3 Parameter Description Assumed / Target Value Minimum usable device RAM We consider that client-side proving is viable if successful on enough devices with a RAM higher than this. ≥ 3 GB Acceptable crash rate We considered Cairo M was successful on device models under a certain failure threshold. < 1% failure rate Target proof duration We considered Cairo M offered a smooth user experience if the median proving time was under this threshold. ≤ 5 s per proof Table 2: FibRace benchmarking assumptions. KKRT Labs used these thresholds to evaluate Cairo M's performance. Hyli used them to characterize onchain verification throughput. 2.6 Ethical and experimental considerations • Voluntary participation: Players opted in by installing the application. No rewards, airdrops, or financial incentives were offered, and this was made clear in the launch posts as well as in the application's interface. • Data minimization: Only technical and performance metrics were collected. No personal identifiers were stored. FibRace did not enable account creation or data portability. • Cross-platform coverage: Android and iOS were supported, ensuring SoC diversity: Apple Silicon, Qualcomm Snapdragon, MediaTek, Huawei Kirin, and Samsung Exynos. 2.7 Data availability The performance report and raw datasets are publicly available at https://github.com/kkrtlabs/cairo-m/tree/main/docs/fibrace. 3 Benchmark results 3.1 Global participation Across the 19-day campaign, 6,047 player names were registered. While some players registered multiple player names (due to multi-device use or reinstalling the application), we will refer to them as « players » in the remainder of this report. Of these players, 5,156 generated at least one valid proof. The players used 1,420 unique device models. Device models are provided by the manufacturers themselves and read from the OS API. According to the app stores' data, the game was installed by players in 99 countries. The Figure 1 is an heatmap of the number of FibRace downloads per country. In total, 2,195,488 proofs were generated on FibRace during the campaign. 4 Figure 1: FibRace downloads by country. 3.2 Proving performance distribution The duration of proof generation followed a right-skewed distribution, shown in Figure 2, with most devices completing a proof within 3 to 10 seconds. The average frequency (Cairo M cycles per second) was 30 kHz, but the median was closer to 10 kHz, reflecting the heavy tail of slower devices. High-performance clusters (≤ 3 s proofs) were led by the most recent iPhone and iPad models running Apple A19 Pro, M1-M4 chips, followed by older ones and high-end Android phones. Mid-range Android devices populated the 5-10 s band. Older devices (with an average proving time of more than 10s) showed increased crash incidence and throttling effects. Figure 2: FibRace Cairo M proof generation duration distribution (central 99%). 5 3.3 Hardware correlations 3.3.1 RAM Capacity Figure 3 displays key performance indicators, grouped by RAM buckets. At least 3 GB of available memory is required for successful proving on Cairo M. Devices with 3-4 GB RAM were the most prone to crashes due to Cairo M's memory footprint, while stability scaled steadily with RAM until plateauing beyond 6 GB. Figure 3: Key performance indicators per RAM capacity. 3.3.2 Device and manufacturer trends Most proofs were generated on Xiaomi, Apple, and Samsung devices, as per Figure 4, reflecting global market share rather than platform bias (Kuzmin 2025). Within each brand, newer chip generations demonstrated predictable improvements in both duration and frequency. Figure 4: Proof volume per device manufacturer. 6 Figure 5 ranks the 50 best SoC, performance wise. It has been sorted based on proof duration as well as frequency, with best score as well as best average. Apple SoCs dominated the upper performance tier, with the latest iPhone chips and M-series on iPad. iPads with the Apple M4 SoC and iPhones with the A19 Pro achieved the fastest proving times, approaching the limits of what Cairo M can deliver on mobile hardware. Following the most recent Apple SoCs, high-end SoCs from Qualcomm, MediaTek, Kirin, and Exynos achieved great performance, with a proving time under 3 seconds. Figure 5: Best 50 SoCs for Cairo M proof generation. 3.4 Reliability and crashes 3.4.1 Reliability per device Stable devices maintained success rates above 95%, confirming that Cairo M can operate reliably on modern smartphones without external proving support. 7 The overall crash rate during the FibRace campaign is 0.72%, representing the number of recorded crash events (15,762 occurrences) over the total number of proof successfully generated. The majority of these occurred on devices with limited RAM (3-4 GB) as shown in Figure 3, which are often over five years old. Crash rates correlated strongly with memory pressure rather than CPU limitations, suggesting that future optimizations should prioritize reducing the RAM footprint. 3.4.2 Temporal trends Hourly aggregates showed distinct activity peaks corresponding to regional evening hours, demonstrating genuine player engagement rather than automated testing. Proof throughput and crash incidence followed inversely proportional patterns: as participation increased, minor rises in crash frequency indicated thermal throttling or sustained high-load conditions on some devices. We also encountered a crash error on higher-RAM devices when minting many times in a row (usually more than a hundred). While unable to identify the root cause of these crashes with certainty, we assume that, although the spawned Rust thread ought to be killed at the end of each generated proof, it may not have been properly done, which could lead to OOM-type crashes. The short delay incurred when verifying and settling the proof onchain between two proof generation might be the cause. If this delay is too short, the thread may not properly close. Due to the short campaign duration and to the benchmark's focus on one-time proving, we did not investigate the issue further. 3.5 Summary interpretation • Performance: Modern smartphones can generate Cairo M proofs in under 5 seconds on average. • Accessibility: Proofs succeeded on 1,400+ device models, proving wide feasibility. • Reliability: Devices with ≥ 3 GB RAM show consistent stability; below that threshold, proving fails systematically. • Hardware Insight: CPU frequency correlates strongly with proof frequency, confirming Cairo M's effective scaling on mobile hardware. 4 Discussion and implications 4.1 Viability of client-side proving The FibRace experiment demonstrates that client-side proving using Cairo M is technically viable and globally accessible. Over 2 million proofs were generated, totaling 333 billions Cairo M cycles on more than 1,400 distinct device models, confirming that local proof generation can operate reliably across diverse hardware and network environments. The results validate that the majority of modern smartphones can generate zero-knowledge proofs in under 5 seconds, which is well within the user-acceptable latency for casual or transactional applications. Devices with less than 3 GB of RAM consistently failed to complete a proof, delineating a clear lower bound for current implementations. 8 From a research perspective, FibRace provides the first large-scale, open dataset that quantifies the distribution of proof generation times across consumer hardware. This enables empirical calibration of future prover optimizations and helps refine benchmarks for memory footprint, cycle counts, and acceptable crash rates. 4.2 Implications for developers and ecosystem builders For developers, these results are a strong signal that end-user devices can handle proving workloads previously reserved for servers or specialized hardware. Client-side proving simplifies architecture, reduces cost, and improves privacy, aligning with the long-term goal of self-sovereign computation. In practical terms, this means applications can now: • Deploy zero-knowledge solutions without relying on remote provers. • Execute verifiable interactions offline, syncing proofs asynchronously to Hyli. • Build new interaction models (e.g., games, attestations, on-device identity proofs) that no longer depend on cloud resources. For ecosystem builders, FibRace establishes a methodology for gamified benchmarking, collecting real-world performance data in a playful and opt-in format. This approach can be replicated for other proving schemes or for measuring device-specific bottlenecks, such as GPU acceleration or energy efficiency. 4.3 Broader impact FibRace bridges the gap between cryptographic research and user experience. Re-framing benchmarking as a public game showcased how ZK technology can reach mainstream audiences without compromising scientific rigor. Future iterations of this model could benchmark additional proving schemes, introduce controlled parameters (e.g., energy consumption, thermal throttling), and work upon different circuits. 4.4 Implications for Hyli and KKRT Labs For Hyli, the experiment validated the scalability of its onchain proof verification layer. Over two million verifications were processed without congestion, even during peaks with 260,000 proofs per hour, demonstrating that Hyli's implementation of the Autobahn consensus can absorb high-frequency proof submissions from a large user base (Giridharan et al. 2024). For KKRT Labs, FibRace served as a proof-of-concept for Cairo M, showing that its mobile optimization can sustain multi-million-proof throughput across everyday devices. The benchmark offers valuable feedback for future improvements. Together, Hyli and KKRT Labs demonstrated that a proof-powered L1 and a mobile-optimized prover can function seamlessly, paving the way for scalable, privacy-preserving applications that can be deployed to consumer hardware. 9 5 Limitations of the experiment While FibRace demonstrates the viability of client-side proving on mobile devices, several limitations must be acknowledged to correctly interpret the dataset and results. 5.1 Technical limitations 5.1.1 Benchmark circuit scope The experiment focused exclusively on proving computations of terms of the Fibonacci sequence, an arithmetic circuit with low memory usage. While representative for testing Cairo M baseline performance, and while we benched a wide range of witnesses, resulting in traces of 8 to 800,000 Cairo M cycles, it does not capture all of the Cairo M capabilities that can be found in real-world proving workloads, such as SHA2 hashes. 5.1.2 Memory footprint Cairo M's memory consumption constrained participation on low-end devices. Proofs could not be generated on devices with less than 3 GB of RAM, and devices with 3-4 GB of RAM showed the highest crash rate, respectively 4.2 and 4.4 crashes per generated proof, due to OOM errors on the largest witnesses (n near the 100,000 upper-bound). It showcases that devices with limited available memory can generate proofs for most workloads of the benchmark, but not for the most expensive ones, due to memory constraints. 5.1.3 Energy and thermal effects FibRace did not record power draw or thermal throttling. Anecdotal player feedback suggested noticeable heat and battery drain during extended play sessions, but these factors were not quantified. Other performance factors were not measured, including battery state, energy saving mode activation, and CPU temperature. Therefore we do not have a quantitative performance degradation ratio on our assumption that these factors lead to degraded performance. Future benchmarks should include energy-use metrics to more effectively evaluate sustainability, user comfort and performance impact. 5.2 Game design limitations 5.2.1 Gameplay constraints Because FibRace was designed as a game, participants could interrupt or close the app at any point, including by force-quitting the app while generating or sending a proof. Such interruptions are indistinguishable from technical crashes in the dataset, introducing ambiguity in interpreting some failure events. 5.2.2 No user accounts The game operated without persistent user accounts to reduce friction and prevent airdrop speculation. While this ensured privacy and simplicity, it made longitudinal tracking of player behavior and device switching impossible. This did not affect the benchmark, but made it less enjoyable for some players. 10 5.3 Summary These limitations do not diminish the core finding that modern smartphones can locally perform zero-knowledge proving. They do, however, highlight key areas for future experiments: • measuring more complex circuits, • measuring circuits with real-life applications, • quantifying energy and heat effect on performance. Together, these refinements would strengthen the empirical foundation laid by FibRace and support the next generation of client-side proving research. 6 Conclusion FibRace demonstrates that client-side proving is no longer theoretical: it is operational, measurable, and already viable across a wide range of consumer devices. Over the course of three weeks, more than 5,000 participants generated 2.2 million Cairo M proofs on 1,420 unique device models, confirming that proof generation can occur reliably on modern smartphones without the need for centralized infrastructure or trusted hardware. The results validate a key assumption behind both KKRT Labs and Hyli: that distributed proving and native onchain verification can coexist efficiently. Cairo M made on-device proof generation accessible even to mid-range devices, while Hyli's proof-powered Layer 1 absorbed millions of verifications without network congestion. Beyond its immediate findings, FibRace established a new model for open benchmarking through gamified participation. By blending experimentation and gameplay, it produced one of the most extensive public datasets on client-side proving to date, an approach that future ZK ecosystems can replicate for other circuits, provers, or device classes. The experiment's limitations define clear next steps. Expanding to real-world use cases, and integrating thermal and battery instrumentation. Ultimately, FibRace offers a simple but powerful insight: zero-knowledge proofs can now run on users' own devices. This shift opens a new frontier for privacy-preserving applications and proof-native blockchains, where the path to scalable cryptography becomes user-driven. 11 Bibliography [1] O. Chia, "ID photos of 70,000 users may have been leaked, Discord says," BBC News, Oct. 2025. [2] D. Kuhn, "Google Wallet integrates ZK-proofs, a tech incubated by crypto industry," The Block, May 2025. [3] Vlad, "Benchmarking in-browser p256 ECDSA proving systems," Hyli, Mar. 2025. [4] KKRT Labs, "Kakarot presents Cairo M: The Frontier zkVM Built for Your Phone." Oct. 08, 2025. [5] A. Sirac, "Launching FibRace: Prove to win!," Hyli, Sep. 2025. [6] C. Walter, "Cairo M Design Document." Sep. 29, 2025. [7] S. Malatrait, "FibRace performance report." Oct. 01, 2025. [8] A. Kuzmin, "Baseline Benchmarking Hardware." Mar. 31, 2025. [9] N. Giridharan, F. Suri-Payer, I. Abraham, L. Alvisi, and N. Crooks, "Autobahn: Seamless high speed BFT," in Proceedings of the ACM SIGOPS 30th Symposium on Operating Systems Principles, New York, NY, USA, Nov. 2024, pp. 1-23. 12
|
2510.14696
|
High-Resolution PTDF-Based Planning of Storage
and Transmission Under High Renewables
Kevin Wu
Pascal Van Hentenryck
H. Milton Stewart School of Industrial and Systems Engineering
Georgia Institute of Technology, Atlanta, GA, United States
{kwu381, pvh}@gatech.edu
Rabab Haider
Civil and Environmental Engineering
University of Michigan, Ann Arbor, MI, USA
rababh@umich.edu
Abstract—Transmission Expansion Planning (TEP) optimizes
power grid upgrades and investments to ensure reliable, effi-
cient, and cost-effective electricity delivery while addressing grid
constraints. To support growing demand and renewable energy
integration, energy storage is emerging as a pivotal asset that
provides temporal flexibility and alleviates congestion. This paper
develops a multiperiod, two-stage PTDF formulation that co-
optimizes transmission upgrades and storage siting/sizing. To
ensure scalability, a trust-region, multicut Benders scheme warm-
started from per-representative-day optima is proposed. Applied
to a 2,000-bus synthetic Texas system under high-renewable
projections, the method attains final optimality gaps below 1%
and yields a plan with storage at about 180 nodes (32% of peak
renewable capacity). These results demonstrate that the proposed
PTDF-based methodology efficiently handles large distributed
storage fleets, demonstrating scalability at high spatial resolution.
Index Terms—transmission expansion planning, storage plan-
ning, co-optimization, PTDF, renewables.
NOMENCLATURE
a) Sets:
N
Set of buses; N = {1, ..., N}; indexed by i
E
Set of branches; E = {1, ..., E}; indexed by ij
S
Set of scenarios; S = {1, ..., S}; indexed by s
T
Set of hours; T = {1, ..., T}; indexed by t
b) Parameters:
pd
i,s,t
Load at node i in scenario s at hour t
¯
pg
i,s,t
min output of generator i in scenario s at hour t
¯pg
i,s,t
max output of generator i in scenario s at hour t
ccap
ij
Line capacity upgrade cost of edge ij
cstor
i
Storage energy rating cost at node i
cG
i (·)
Production cost function of generator i
ΩS(·)
Probability mass function for set of scenarios S
λ
Load shedding penalty cost
∆c
ij
Capacity upgrade increment of branch ij
¯pf
ij
Thermal limit of branch ij
This work was partially funded by NSF award 2112533.
c) Variables:
γij
Capacity upgrade level of branch ij
σi
Storage energy rating at node i
soci,s,t
Storage SoC at node i in scenario s at hour t
chi,s,t
Storage charge at node i in scenario s at hour t
disi,s,t
Storage discharge at node i in scenario s at hour t
pg
i,s,t
Output of generator i in scenario s at hour t
pf
ij,s,t
Power flow on branch ij in scenario s at hour t
ξi,s,t
Load shed at node i in scenario s at hour t
I. INTRODUCTION
The U.S. electric grid is rapidly evolving with rising elec-
tricity demand and renewable energy penetration, requiring
substantial transmission infrastructure investments to maintain
reliable and affordable service [1]. This presents a critical
challenge for system operators who must solve Transmission
Expansion Planning (TEP) problems to identify cost-effective
strategies for grid upgrades to ensure capability of delivering
power from generators to load centers. TEP is modeled as
a mixed-integer program (MIP) that is difficult to solve for
large-scale systems due to discrete investment decisions. They
are made more computationally challenging by including
investment decisions for generation [2], FACTS devices [3],
and energy storage [4]. Given the critical role of storage in
providing flexibility under high renewable penetration, this
work focuses on TEP with storage.
The TEP+Storage literature typically adopts at least one
of three simplifications: (i) reduced spatial granularity; (ii)
relaxation of DC power flow to a transport model; or (iii)
settings whose economics yield sparse storage deployments.
These choices reduce problem dimensionality and computa-
tional burden. This work address these limits: a PTDF-based,
high-resolution model under high-renewable scenarios that
endogenously deploys a large distributed storage fleet, creat-
ing a more demanding computational setting. A trust-region-
stabilized Benders scheme addresses this regime, attaining
final optimality gaps below 1% on a 2,000-bus system with
storage at about 180 nodes (32% of peak renewable capacity).
A. Related Work
High-fidelity TEP+Storage formulations are often limited to
small systems due to computational complexity. For example,
arXiv:2510.14696v1 [eess.SY] 16 Oct 2025
[5] uses a linearized AC model on a 24-bus system, and
[6] solves a security-constrained DC model on a 53-bus
system; these preserve grid physics but do not readily scale
to thousands of buses. To reach larger systems, many studies
adopt transport relaxations; e.g., [7] scales to the 8,870-bus
CATS using progressive hedging. Storage deployment is also
frequently sparse: [8] studies merchant storage with a tri-level
DC model and, under their economic assumptions and 240-
bus case, report profitable deployment at ≤7 sites (3% of
peak renewables); while [9] solves a DC model with binary
storage siting/sizing and line undergrounding on CATS using
Benders under wildfire risk and report deployments at ≤9 sites
totaling 2.6 GWh (≈11% of peak renewables). Collectively,
these works show large-scale DCOPF feasibility, but it has
typically involved reduced flow-model fidelity and/or sparse
storage arising from the assumed economics or risk setting,
which can alter congestion patterns and investment signals
while easing computation burden.
B. Contributions
To address these limitations, this paper proposes a com-
bined a PTDF-based DC formulation with a stabilized (trust-
region) Benders approach. The methodology is demonstrated
in a high-renewables setting on a 2,000-bus network, thereby
addressing simultaneous gaps in spatial resolution, network-
fidelity, and storage scalability. The contributions are:
• A multiperiod, two-stage PTDF model that co-optimizes
line upgrades and storage siting/sizing with hourly dis-
patch and storage dynamics, preserving DC physics while
remaining tractable at scale
• A trust-region, multicut Benders scheme, warm-started
from per-representative day optima, that mitigates dual
degeneracy and oscillatory master-iterates, strengthens
cuts, and improves convergence predictability
• A multiyear evaluation on a 2,000-bus synthetic Texas
system under high-renewable scenarios that endogenously
yields a large distributed storage fleet (≥179 devices, 32%
of peak renewables) while attaining final optimality gaps
≤1%, demonstrating quality of solution.
The paper proceeds as follows. §II formulates the two-stage
TEP+Storage model; §III details the Benders methodology;
§IV describes the Texas case study; §V reports optimality
gaps, runtime, and investment decisions; and §VI concludes.
II. FORMULATION
PTDF-based DCOPF is well established for TEP [10],
yet large-scale storage integration within PTDF frameworks
remains comparatively unexplored. This section presents the
PTDF-based TEP+Storage formulation. The formulation as-
sumes that at most one generator is connected to each bus,
without loss of generality.
A. The Two-Stage TEP+Storage PTDF Formulation
The TEP+Storage model is a two-stage program. The
investment stage selects transmission upgrades and storage
siting/sizing to minimize capital expenditures (CapEx). The
recourse stage fixes those investments and minimizes opera-
tional costs (OpEx) over the investment period H, the plan-
ning horizon during which CapEx decisions apply, thereby
capturing the coupling between long-term planning and short-
term operations.
1) Investment decisions and variables: This study prior-
itizes capacity upgrades of existing transmission lines over
new line additions, as new construction often faces economic,
regulatory, and social barriers [11]. Upgrades are represented
by the discrete decision variable γij for each branch ij ∈E,
with allowable levels of enhancement ∆c
ij (MW per line).
Storage is modeled as short-duration lithium-ion batteries
due to their technological maturity and widespread deploy-
ment [12]. Sizing decisions are captured by σi, denoting the
installed energy capacity (MWh) at node i. Each unit has a
4-hour duration, giving a power rating of 1
4σi. Hourly storage
operations are modeled by chi,t, disi,t, and soci,t, representing
charge, discharge, and state of charge (SoC), respectively.
Other key variables include nodal generation dispatch pg
and line power flows pf (auxiliary via PTDF).
2) Objective: The TEP+Storage model minimizes the sum
of CapEx and OpEx. The CapEx is a linear function of first-
stage decisions (γ, σ) covering line and storage investments:
CapEx = f(γ, σ) := P
ij∈E ccap
ij γij + P
i∈N cstor
i
σi
(1)
Given investments (γ, σ), the operating cost OpExs(γ, σ) of
scenario-s includes daily generation expenses, storage charge
and discharge costs, and penalties for load shedding using a
per-unit penalty λ for load shed ξi,s,t. It is computed over
the investment period via scaling factor α (e.g., α = 365
for one year). The aggregate operating cost OpEx(γ, σ) is
a probability-weighted sum across scenarios s ∈S.
OpExs(γ, σ) := GenEx + StorEx + Penalties
(2)
= α P
i∈N ,t∈T [cG
i (pg
i,s,t) + cS
i (chi,s,t + disi,s,t) + λ ξi,s,t],
OpEx(γ, σ) := P
s∈S ΩS(s) OpExs(γ, σ)
(3)
3) Constraints: The DC power flow model uses the Power
Transfer Distribution Factor (PTDF) matrix Φ, which maps
net nodal injections to line flows. For scenario s and hour t,
let injs,t := [pg
i,s,t +disi,s,t −pd
i,s,t −chi,s,t +ξi,s,t]N
i=1 denote
the vector of net injections. The operational constraints are:
[pf
ij,s,t]E
ij=1 = Φ × injs,t
(4)
1⊤injs,t = 0
(5)
−¯pf
ij −γij∆c
ij ≤pf
ij,s,t ≤¯pf
ij + γij∆c
ij
(6)
¯
pg
i,s,t ≤pg
i,s,t ≤¯pg
i,s,t
(7)
soci,s,t = soci,s,t−1 + chi,s,tη −disi,s,t/η
(8)
soci,s,1 = chi,s,1η −disi,s,1/η
(9)
soci,s,T = 0
(10)
0 ≤soci,s,t ≤σi
(11)
0 ≤chi,s,t ≤σi/4
(12)
0 ≤disi,s,t ≤σi/4
(13)
Model 1 Two-Stage TEP+Storage Formulation
min
(1) + (3)
s.t.
(4) −(5),
∀s ∈S, t ∈T
(6),
∀ij ∈E, s ∈S, t ∈T
(7),
∀i ∈N, s ∈S, t ∈T
(8)
∀i ∈N, s ∈S, t ∈{2, ..., T}
(9) −(10),
∀i ∈N, s ∈S
(11) −(13),
∀i ∈N, s ∈S, t ∈T
where (4) computes the vector of line flows; (5) imposes
power balance; (6) enforces thermal flow limits; (7) bounds
generator outputs, with renewable generators modeled with
¯
pg = 0 and scenario-dependent ¯pg (curtailment allowed), and
conventional units dispatchable within their limits; (8)–(13)
govern storage, including SoC with efficiency losses, a zero-
SoC boundary at the start and end of each representative day,
and charge/discharge limits. Model 1 outlines the full two-
stage TEP+Storage formulation.
Remark. To maintain recourse feasibility, load shed slack
variables ξi,s,t are included and penalized in OpEx by a large
coefficient λ (see §II-A2). Any plan with ξ > 0 is rejected;
λ is increased (doubled) and the model is re-solved until zero
load shed is achieved. This approach preserves feasibility and
is guaranteed to terminate when a no-shed solution exists,
consistent with industry practice where TEP must serve future
load by system design.
B. Scalable PTDF Reformulation
The PTDF model is scaled to high resolution systems
using a sparse representation of the PTDF matrix and lazy
constraint generation. A PTDF cutoff is a common sparsifi-
cation approach, where entries of Φ with magnitudes below
a threshold kp (typically 0.01–0.05) are set to zero, thereby
sparsifying the matrix and improving computational efficiency
by removing negligible effects [10]. For very large systems,
however, even the sparse PTDF can remain expensive. To
address this, the PTDF flow (4) and thermal-limit constraints
(6) are enforced lazily: the model is first solved without these
constraints, then any violated line-flow inequalities are itera-
tively added to model until no violations remain. Specifically,
if branch ij in scenario s at time t violates its upper bound,
the following inequality constraint is added to the model:
pf
ij,s,t = P
k∈N PTDFij,kinjs,t
k
≤¯pf
ij + γij∆c
ij,
with the inequality reversed for lower bound violations. Lazy
constraints preserve model equivalence while omitting non-
binding limits, yielding smaller and more tractable formula-
tions. The set of lazy constraints is further reduced by adding
only the kv most severe violations per iteration, where hyper-
parameter kv can be tuned: smaller values produce a sparser
model at the cost of more iterations to convergence. Both
techniques (cutoff and lazy constraints) are used throughout
the remainder of the paper.
Fig. 1. Benders Schematic Flow Diagram.
Model 2 Benders Master Problem (MP)
zMP
k
= min
(1) + P
s∈S ΩS(s)θs + ρ
s.t.
ρ ≤0
(14)
S
s∈S Ck
s
III. METHODOLOGY
The two-stage TEP+Storage model grows linearly with
the number of second-stage scenarios; combined with the
network’s high spatial resolution, even a modest number
of scenarios yields very large instances. To address this, a
Benders decomposition (BD) is employed. The baseline BD
formulation (§III-A) decomposes the problem into a master
problem (MP) and subproblems (SPs) linked by investment
and operational decisions. Two methodological enhancements
are introduced to improve convergence and stability: a warm-
start module (§III-B) that initializes the MP using relaxed in-
vestment solutions, and a trust-region Benders module (§III-C)
that stabilizes the iterative process using a proximal constraint
and restricted candidate set. §III-D presents a valid global
lower bound used to assess optimality gaps.
A. Multicut Benders for TEP+Storage
The BD algorithm alternates between the MP, which selects
first-stage investments to minimize CapEx, and scenario SPs,
which–given (γ, σ)–optimize continuous operational variables
and minimize OpEx. Each SP is a linear program, and SPs
are solved in parallel. From each SP, feasibility and optimality
cuts are generated and added to the MP. Under the multicut
scheme, the MP retains all accumulated cuts per scenario. Let
C k
s denote the set of cuts for scenario s generated through
iteration k −1 (with C 1
s = ∅); at iteration k the MP imposes
C k
s for all s ∈S. Figure 1 summarizes the algorithm.
1) Benders MP: The MP co-optimizes line and storage
investments to minimize CapEx. To capture operational and
feasibility constraints, the MP includes proxy variables θs
an underestimator of OpExs for each scenario s, and ρ
an underestimator of scenario-wide load-shed. The MP at
iteration k is presented in Model 2, where (14) prevents load
shedding in the final solution. After solving the MP, the new
investment point (γk, σk) is broadcast to the SPs, which are
solved in parallel to generate the next set of cuts in C k+1
s
.
2) Benders SP: The scenario-s SP optimizes operations
under the MP decisions (γk, σk), fixed in the SP with dual
dual multipliers πs,k
γ
and πs,k
σ
in (15)-(16). Each SP runs
two phases: Phase I enforces load feasibility and Phase II
Model 3 Benders Subproblem (SP)
min
OpExs(γk, σk)
s.t.
(4) −(5),
∀t ∈T
(6),
∀ij ∈E, t ∈T
(7),
∀i ∈N, t ∈T
(8),
∀i ∈N, t ∈{2, ..., T}
(9) −(10),
∀i ∈N
(11) −(13),
∀i ∈N, t ∈T
γij = γk
ij
(πs,k
γ ),
∀ij ∈E
(15)
σi = σk
i
(πs,k
σ ),
∀i ∈N
(16)
optimizes dispatch. In Phase I, the SP minimizes load shed,
P
i∈N,t∈T ξi,s,t. If the optimum exhibits load shed, the SP
terminates and a feasibility cut is generated for shed under-
estimator ρ. If no shed occurs, the SP proceeds to Phase II
to minimize OpExs (Model 3) and generate an optimality cut
for θs. Both feasibility and optimality cuts take an affine form,
where ψ ∈{I, II} indicates the Phase, ζs = ρ in Phase I and
ζs = θs in Phase II:
ζs ≥βk
s + (πs,k
γ )⊤γ + (πs,k
σ )⊤σ
βk
s := vk,ψ
s
−(πs,k
γ )⊤γk −(πs,k
σ )⊤σk
vk,I
s
= P
i,t ξi,s,t,
vk,II
s
= OpExs(γk, σk)
This two-phase design prioritizes feasibility (no shed) while
maintaining complete recourse via ξ, avoiding classical
infeasibility-ray cuts which can be numerically fragile [13].
3) Convergence of Benders: At iteration k, the MP opti-
mum zMP
k
is a valid lower bound LBk for Model 1, since the
variables {θs} underestimate the true scenario costs {OpExs},
giving LBk := zMP
k
. If all SPs are feasible (Phase I yields
P
i∈N,t∈T ξi,s,t = 0 for every s ∈S), a feasible two-stage
objective at (γk, σk) is ˆzk = f(γk, σk) + OpEx(γk, σk), and
is used to maintain a non-increasing upper bound UBk :=
min{UBk−1, ˆzk}. The algorithm terminates when the relative
gap falls below a threshold ϵ: (UBk −LBk)/UBk ≤ϵ.
Remark. Lazy constraints integrate naturally with BD: per
scenario flow/limit constraints identified in one iteration are
retained for subsequent iterations, accelerating SP solves.
B. Warm-Start from Relaxed Two-Stage Model (WS Module)
In the baseline BD formulation, early iterations are
feasibility-cut dominated: with no initial cuts, the MP selects
a no-expansion plan, causing heavy load shedding in the SPs
until a feasible investment set is learned. The WS accelerates
this by solving, in parallel for each s ∈S, the continuous
relaxation of Model 1 to obtain (γs, σs). A conservative, non-
anticipative design is then formed by component-wise aggre-
gation of the per-scenario optimal investments, and rounded
up to the admissible investment granularity:
¯γ :=
maxs∈S γs
,
¯σ :=
maxs∈S σs
.
Fig. 2. Workflow of the proposed methodology.
The Benders MP is initialized at (¯γ, ¯σ), which reduces early
infeasibility cuts and accelerates convergence.
C. Trust-Region-Stabilized Benders (TR-Benders Module)
Once feasibility is achieved, the BD formulation exhibits
strong dual degeneracy–many line-storage configurations with
nearly identical operating costs–yielding flat recourse value
functions, oscillatory MP iterates, and weak cuts that slow
convergence. The TR-Benders addresses this issue by aug-
menting the BD framework with a proximal trust-region (TR)
constraint on first-stage decisions and a restricted candidate
set. This stabilization mechanism limits drastic investment
changes between iterations, improving numerical stability and
ensuring more consistent convergence behavior.
The MP is stabilized around an anchor point (˘γ, ˘σ) defined
as the incumbent investment that attains the best known upper
bound; it is initialized at (¯γ, ¯σ) and updated only upon a strict
improvement in the upper bound. Inspired by stabilized BD
(e.g., [14]), the method adapts to discrete first-stage variables
with an ℓ1 trust-region (linear, sparsity-promoting):
P
ij∈E
γij −˘γij
≤rγ
k
(17)
P
i∈N
σi −˘σi
≤rσ
(18)
Absolute values are linearized in the standard way for MIPs.
A fixed storage radius rσ = 2 (permitting limited re-siting) is
used, while the transmission radius rγ
k is dynamic: (i) resetting
to 1 whenever the anchor point updates, and (ii) increasing
by 1 whenever the MP revisits the same solution, to limit
local oscillations. To further reduce degeneracy, investment
decisions are restricted to a candidate set derived from the
per-scenario relaxations:
Ec := { ij ∈E | ∃s ∈S : γs
ij > 0}
Nc := { i ∈N | ∃s ∈S : σs
i > 0}
Finally, a level-set constraint prunes regions already known to
be suboptimal:
f(γ, σ) + P
s∈S ΩS(s) θs ≤UBk
(19)
Together, the TR, candidate sets, and level-set constraint damp
oscillations and weaken the effects of dual degeneracy. Figure
2 summarizes the full workflow of the proposed methodology.
D. Global Lower Bound and Optimality Assessment
A limitation of the TR stabilization is that the master’s value
no longer certifies a global lower bound: with the TR active,
LBk is a local lower bound over the current region around
the anchor point. However, a valid global lower bound can be
constructed as below.
1) First-Stage Lower Bound: With load shed disallowed by
(14), the minimum feasible investment decision is constrained
by the first-stage investments of the most challenging repre-
sentative day (typically characterized by high load and/or low
renewable generation). This is formalized in Theorem 1.
Theorem 1. For each s ∈S, define the day-s feasible set and
the all-scenarios feasible set:
Σs
feas := {(γ, σ) : recourse is feasible for scenario s}
Σfeas := T
s∈S Σs
feas
Given the feasible sets, define the corresponding minimum
first stage costs for feasible day-s and all-scenarios:
cfeas
s
:=
min
(γ,σ)∈Σs
feas
f(γ, σ),
cfeas :=
min
(γ,σ)∈Σfeas f(γ, σ),
and first-stage optimal investments for scenario s:
(γs
feas, σs
feas) ∈arg min(γ,σ)∈Σs
feas f(γ, σ)
Let ˆs ∈arg maxs∈S cfeas
s
. Then
maxs∈S cfeas
s
≤f(γ, σ)
∀(γ, σ) ∈Σfeas
In particular, f(γˆs
feas, σˆs
feas) = maxs cfeas
s
is a valid lower
bound on any year-feasible first-stage plan’s cost.
Proof. Since Σfeas ⊆Σs
feas for every s,
cfeas =
min
(γ,σ)∈Σfeas f(γ, σ) ≥
min
(γ,σ)∈Σs
feas
f(γ, σ) = cfeas
s
∀s,
yielding maxs cfeas
s
≤cfeas. Choosing ˆs ∈arg maxs cs gives
the stated bound with (γˆs
feas, σˆs
feas).
2) Second-Stage Lower Bound: The per-scenario opera-
tional cost is monotone nonincreasing in investment capac-
ity, as per Theorem 2. Let (γmax, σmax) denote the com-
ponentwise upper bounds on line and storage capacities
(not restricted to the candidate set) of Model 1. Then a
valid lower bound on the aggregated second-stage cost is
P
s∈S ΩS(s) qs(γmax, σmax).
Theorem 2. Let qs(γ, σ) be the optimal value of Model 3.
If γ′ ⪰γ and σ′ ⪰σ (componentwise), then qs(γ′, σ′) ≤
qs(γ, σ). In particular, qs(·) is monotone nonincreasing in γ
and σ.
Proof. For fixed s ∈S, increasing γ relaxes the thermal limit
constraints (6), and increasing σ relaxes the storage constraints
(11)-(13). Hence the feasible region under (γ′, σ′) contains
that under (γ, σ). Because the SP objective is minimized and
depends on γ, σ only through these constraints, enlarging the
feasible set cannot increase the optimal value, yielding the
claim.
Fig. 3. Generator sites of the system. Black circles indicate nonrenewable,
blue circles indicate wind, and red circles indicate solar generators.
3) Combined Bound: Define z∗as the optimal final so-
lution’s objective, with optimal line and storage investments
(γ∗, σ∗) ∈Σfeas.
z∗:= f(γ∗, σ∗) + P
s∈S ΩS(s) qs(γ∗, σ∗)
Since Theorem 1 and 2 show, respectively, that
f(γˆs
feas, σˆs
feas) ≤f(γ∗, σ∗),
qs(γmax, σmax) ≤qs(γ∗, σ∗)
∀s ∈S,
a valid combined lower bound can be constructed:
LB∗:= f(γˆs
feas, σˆs
feas) + P
s∈S ΩS(s) qs(γmax, σmax) ≤z∗
4) Optimality Gap: Each cfeas
s
is computed under the integer
relaxation of the first-stage investments (as in the WS). Being
a relaxation, this provides a valid lower bound for the year-
wide plan, which must satisfy discretized first-stage decisions.
The reported optimality gaps are therefore conservative upper
bounds: (UBk −LB∗)/UBk.
IV. CASE STUDY
The proposed methodology is tested on a large-scale power
system case study: the ACTIVSg2000 test case, a 2000-bus
model of the U.S. Texas grid. The TEP+Storage model is
applied in a rolling-horizon manner. The formulation itself has
year-agnostic investment variables, but is solved sequentially
at 5-year decision points Y = {2030, 2035, 2040, 2045} with
investment periods of length α = 1825 days. Investments
(and their capital costs) are cumulative: capacity installed
in an earlier year remains available in subsequent years,
ensuring monotone, non-decreasing expansion across planning
periods. This application reflects practical staging, but it is not
guaranteed optimal relative to an omniscient, all-years model;
quantifying any cross-period optimality gap is left for future
work.
a) Network Data: The synthetic Texas system is built
from public information and statistical analysis of real power
systems [15]. The system has 2000 buses, 3206 lines, 423
non-renewable generators, and 175 renewable generators.
Figure 3 shows the location of solar, wind, and nonrenew-
able generators, overlaid on the grid topology. The future
generation mix from 2030 to 2045 follows EIA outlook
TABLE I
GENERATION BASE CAPACITY (GW) AND GENERATION/LOAD SCALING
Type
Capacity
2022
2030
2035
2040
2045
Coal
14.5
1.00
0.82
0.82
0.82
0.82
Natural Gas
75.1
1.00
0.79
0.73
0.71
0.72
Nuclear
7.1
1.00
0.98
0.90
0.80
0.80
Solar
2.5
1.00
4.51
6.00
6.87
8.04
Wind
15.5
1.00
2.02
2.23
2.26
2.32
Load
-
1.00
1.13
1.21
1.31
1.41
Fig. 4. Representative days by average demand vs. wind in 2022. Red circles
indicate selected representative days. Transparent blue circles indicate non-
selected days.
projections [16]. All generation resources are uniformly scaled
to reflect projected growth or planned decommissioning. Load
growth is modeled with a year-to-year increase of 1.5% and
is scaled uniformly across all buses. Table I shows the total
generator capacities (GW) of the network and summarizes the
scaling for each generation type and load relative to the base
year, 2022.
b) Scenario selection: Scenarios are 24-h representative
days sampled from a year of load/wind/solar data. This re-
duces the computational time required for evaluating a full
year of operations, while still capturing intertemporal storage
dynamics. In this case study, kr = 18 days are selected by
a max-min diversity criterion on normalized daily profiles–
maximizing the minimum pairwise Euclidean distance–and
assigned equal weights. Figure 4 shows the chosen days
(red circles) in average wind-load space–the main drivers of
investment (solar capacity is comparatively smaller). Days that
require storage operation to avoid load shed are labeled; these
generally coincide with higher demand. August 11 (08–11)
is the most challenging day, exhibiting the highest average
demand.
c) Investments parameters: Table II lists capital and
operating costs for line upgrades and storage. Line upgrades
are discretized to capacity increases of 30, 60, and 90%,
reflecting the enhancements achievable through typical line
reconductoring [11]. Storage units are 250 MWh each, limited
to 12 per node (3 GWh total), with charge and discharge
efficiencies of η = 95%. Line upgrade costs are adapted from
[17]. Storage costs are adapted from [18].
d) Computational Setup and Hyperparameters:
The
TEP+Storage model is implemented in Julia with JuMP [19]
and solved with Gurobi [20] on Georgia Tech’s PACE Phoenix
TABLE II
COST CONFIGURATION
Costs
Capacity Upgrades
$1243/MW-km
Storage Cost
$1,000,000/MWh
Charge/Discharge Cost
$68.50/MWh
cluster [21] (24-core Intel Xeon, Linux); each run used 24
threads and 288 GB RAM. TR-Benders employed a 0.1% MP
optimality-gap tolerance, with wall-clock limits of 24 hours
for 2030/2035/2040 and 72 hours for 2045 to accommodate
longer per-iteration solves. The per-iteration added-violations
quota was set to kv = 32 and the PTDF cutoff to kp = 0.005,
reducing nonzeros in Φ by 85.5%.
V. RESULTS
With the proposed methodology, the year-wise TEP+Storage
planning problem is both tractable and achieves optimality
guarantees of ≤1%. Results are organized as follows: op-
timality gaps (§V-A), computational performance (§V-B), and
investment outcomes (§V-C).
A. Optimality Gaps
All optimality gaps are computed relative to the lower
bound defined in §III-D, which bounds the best achievable
objective for each 5-year decision given previously committed
investments. Final gap values are reported in the rightmost
column of Table III. Across all investment periods, the WS
initializes runs near optimality, yielding initial gaps below
10%. The subsequent TR-Benders reduces these gaps to
below 1% in every period, validating the approach and the
investment solutions presented below.
B. Computational Results
Under
the
scalable
PTDF
reformulation
(§II-B),
the
TEP+Storage model is tractable at scale. A tunable PTDF
cutoff kp provides planners with a direct accuracy-scalability
trade-off (less straightforward in phase-angle DC formula-
tions). The following sections describe the computational
results from the WS and TR-Benders modules, respectively.
1) WS Module: Figure 5 summarizes computation times
for the relaxed two-stage model in the WS, reported per
representative day and 5-year investment period. Runtime is
dominated by the iterative discovery of violated thermal-limit
constraints; consequently, more congested days take longer to
solve. Average time-to-solve falls from 2030 to 2040 because
investments made in earlier 5-year periods remain in place
in subsequent periods, relieving congestion and simplifying
decisions for most representative days. Outliers persist on
days that combine heavy storage utilization with elevated
congestion (e.g., 08–11 with its high average demand).
2) TR-Benders Module: The TR-Benders demonstrates im-
proved tractability by decoupling first-stage investment de-
cisions from per-scenario operations. Table IV reports time
limits, iteration counts, and average seconds per iteration
for each 5-year investment period. Iteration time increases
TABLE III
INVESTMENT DECISIONS AND COSTS FOR 2030-2045 FOR MOST CHALLENGING REP. DAY, WS, AND TR-BENDERS PLANS.
Mode
# Lines
# Storage units /
Total capacity (GWh)
CapEx
OpEx
Opt. Gap
Lines ($B)
Storage ($B)
Gen ($B)
StorOpEx ($B)
2030
08-11
183
0 / 0
0.14
0
46.49
0
-
Warm-start
291
0 / 0
0.37
0
28.80
0
1.3%
TR-Benders
238
0 / 0
0.24
0
28.86
0
1.0%
2035
08-11
334
33 / 28.91
0.36
28.92
50.91
7.27
-
Warm-start
381
33 / 32.75
0.53
32.77
30.39
0.40
6.3%
TR-Benders
352
32 / 29.25
0.40
29.27
30.42
0.41
0.8%
2040
08-11
425
118 / 98.07
0.47
98.12
56.60
24.71
-
Warm-start
462
118 / 111.75
0.66
111.81
33.35
1.64
9.4%
TR-Benders
441
108 / 98.50
0.53
98.55
33.38
1.65
0.4%
2045
08-11
476
184 / 167.30
0.58
167.40
62.6
42.01
-
Warmstart
508
184 / 183.75
0.72
183.84
36.63
3.66
7.4%
TR-Benders
499
179 / 167.75
0.66
167.84
36.64
3.66
0.3%
Fig. 5.
WS computation times by representative day for each 5-year
investment period.
TABLE IV
TR-BENDERS COMPUTATIONAL PERFORMANCE FOR EACH 5-YEAR
INVESTMENT PERIOD.
2030
2035
2040
2045
Time Limit (Hrs.)
24
24
24
72
Iterations
1964
1198
332
102
Avg. Seconds/Iter
44
72
260
2541
over the planning horizon because progress each round is
paced by the most challenging SP/representative day: the
algorithm advances only after all cuts are generated. The
slowest iterations coincide with intensive storage operation
and pronounced congestion, which prolong SP solves due
to storage time-coupling and ongoing discovery of violated
thermal-limit constraints. Further computational improvement
may be attained by pre-screening thermal violations.
C. Investment Results
The TR-Benders produces a feasible, near-optimal year-
wide plan for 2030–2045 with zero load shed across all
representative days. Figure 6 maps the geographic distribution
of investments. Transmission upgrades concentrate in West
Texas, strengthening transfer from wind/solar regions, while
storage appears beginning in 2035 and grows considerably
with increasing load. Figure 7 reports storage energy through-
puts for each planning year, defined as the total GWh charged
and discharged during the representative day. Only a small
fraction of days require storage operation to eliminate load
shedding; 08–11 exhibits the largest throughput and aligns
with its computational difficulty (Fig. 5). Consequently, mul-
tiyear investment choices are largely driven by this hardest
day.
Table III summarizes the selected upgrades, storage sit-
ing/sizing, and costs; its rightmost column displays the fi-
nal optimality gaps referenced earlier. For completeness, the
single-day (08-11) WS results are also listed by period.
The WS intentionally over-invests. After the TR-Benders, the
near-optimal plan reduces CapEx relative to the WS, while
incurring only negligible increases in OpEx, yielding a lower
total cost. The 08–11 single-day results show comparatively
higher OpEx than both the WS and TR-Benders plans because
they optimize solely for the highest-demand day rather than the
equally weighted set of representative days. Because storage
dominates investment costs, most of the residual gap is driven
by reconciling storage capacity rather than line upgrades.
The storage capacity of TR-Benders approaches–but remains
above–the requirement implied by the hardest day, consistent
with Theorem 1 ensuring zero load-shed feasibility.
By 2045, the optimized plan yields 167.75 GWh of storage
distributed across 179 nodes, which is equivalent to 32%
of the scaled peak renewable capacity. This underscores the
robustness of the proposed formulation and its tractability at
high spatial and storage resolution.
VI. CONCLUSION
This paper presents a PTDF-based TEP+Storage for joint
transmission and storage expansion and applies it to a 2,000-
bus synthetic Texas system. To ensure computational feasi-
bility, it introduces a trust-region, multicut Benders scheme
warm-started from per-representative-day optima. The method
attains final optimality gaps ≤1% and remains tractable for a
large distributed storage fleet deployed at ≥179 nodes (≈32%
of peak renewable capacity). These results demonstrate the
proposed PTDF-based methodology effectively handles high
spatial resolution and large distributed storage deployments in
joint TEP+Storage planning.
Table IV shows that computational challenges persist, par-
ticularly under high storage penetration and network conges-
(a) 2030 TEP+Storage
(b) 2035 TEP+Storage
(c) 2040 TEP+Storage
(d) 2045 TEP+Storage
Fig. 6.
Upgrades from 2030 to 2045 in 5-year intervals. Red lines denote capacity upgrade locations, with their thickness corresponding to one of three
possible upgrade levels. Circles indicate storage locations, with their size and color representing the number of installed units (250MWh each).
Fig. 7.
Storage energy throughput (GWh) for representative days with
nonzero utilization across the 2030–2045 planning horizon. Values reflect the
representative day under the investment plan in effect for that year.
tion, and will intensify as temporal, operational, and spatial
fidelity increase. Priorities include accelerating SP solves–
e.g., pre-screening and dynamic pruning of thermal/flow
constraints–and exploring unified Benders variants that com-
bine feasibility and optimality cuts. Another direction is to
integrate generation expansion, which may relieve conges-
tion and further improve lazy-constraint performance. Finally,
certifying cross-period optimality, testing multistage formula-
tions, and quantifying the value of deferred investments are
natural extensions.
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|
High-Resolution PTDF-Based Planning of Storage and Transmission Under High Renewables Kevin Wu Pascal Van Hentenryck H. Milton Stewart {kwu381, Rabab Haider Civil and Environmental Engineering -Transmission Expansion Planning (TEP) optimizes power grid upgrades and investments to ensure reliable, efficient, and cost-effective electricity delivery while addressing grid constraints. To support growing demand and renewable energy integration, energy storage is emerging as a pivotal asset that provides temporal flexibility and alleviates congestion. This paper develops a multiperiod, two-stage PTDF formulation that cooptimizes transmission upgrades and storage siting/sizing. To ensure scalability, a trust-region, multicut Benders scheme warmstarted from per-representative-day optima is proposed. Applied to a 2,000-bus synthetic Texas system under high-renewable projections, the method attains final optimality gaps below 1% and yields a plan with storage at about 180 nodes (32% of peak renewable capacity). These results demonstrate that the proposed PTDF-based methodology efficiently handles large distributed storage fleets, demonstrating scalability at high spatial resolution. Index Terms-transmission expansion planning, storage planning, co-optimization, PTDF, renewables. NOMENCLATURE a) Sets: N Set of buses; N = {1, ..., N}; indexed by i E Set of branches; E = {1, ..., E}; indexed by ij S Set of scenarios; S = {1, ..., S}; indexed by s T Set of hours; T = {1, ..., T}; indexed by t b) Parameters: pd i,s,t Load at node i in scenario s at hour t ̄ pg i,s,t min output of generator i in scenario s at hour t ̄pg i,s,t max output of generator i in scenario s at hour t ccap ij Line capacity upgrade cost of edge ij cstor i Storage energy rating cost at node i cG i (·) Production cost function of generator i ΩS(·) Probability mass function for set of scenarios S λ Load shedding penalty cost ∆c ij Capacity upgrade increment of branch ij ̄pf ij Thermal limit of branch ij This work was partially funded by NSF award 2112533. c) Variables: γij Capacity upgrade level of branch ij σi Storage energy rating at node i soci,s,t Storage SoC at node i in scenario s at hour t chi,s,t Storage charge at node i in scenario s at hour t disi,s,t Storage discharge at node i in scenario s at hour t pg i,s,t Output of generator i in scenario s at hour t pf ij,s,t Power flow on branch ij in scenario s at hour t ξi,s,t Load shed at node i in scenario s at hour t I. INTRODUCTION The U.S. electric grid is rapidly evolving with rising electricity demand and renewable energy penetration, requiring substantial transmission infrastructure investments to maintain reliable and affordable service [1]. This presents a critical challenge for system operators who must solve Transmission Expansion Planning (TEP) problems to identify cost-effective strategies for grid upgrades to ensure capability of delivering power from generators to load centers. TEP is modeled as a mixed-integer program (MIP) that is difficult to solve for large-scale systems due to discrete investment decisions. They are made more computationally challenging by including investment decisions for generation [2], FACTS devices [3], and energy storage [4]. Given the critical role of storage in providing flexibility under high renewable penetration, this work focuses on TEP with storage. The TEP+Storage literature typically adopts at least one of three simplifications: (i) reduced spatial granularity; (ii) relaxation of DC power flow to a transport model; or (iii) settings whose economics yield sparse storage deployments. These choices reduce problem dimensionality and computational burden. This work address these limits: a PTDF-based, high-resolution model under high-renewable scenarios that endogenously deploys a large distributed storage fleet, creating a more demanding computational setting. A trust-regionstabilized Benders scheme addresses this regime, attaining final optimality gaps below 1% on a 2,000-bus system with storage at about 180 nodes (32% of peak renewable capacity). A. Related Work High-fidelity TEP+Storage formulations are often limited to small systems due to computational complexity. For example, 16 Oct 2025 [5] uses a linearized AC model on a 24-bus system, and [6] solves a security-constrained DC model on a 53-bus system; these preserve grid physics but do not readily scale to thousands of buses. To reach larger systems, many studies adopt transport relaxations; e.g., [7] scales to the 8,870-bus CATS using progressive hedging. Storage deployment is also frequently sparse: [8] studies merchant storage with a tri-level DC model and, under their economic assumptions and 240bus case, report profitable deployment at ≤7 sites (3% of peak renewables); while [9] solves a DC model with binary storage siting/sizing and line undergrounding on CATS using Benders under wildfire risk and report deployments at ≤9 sites totaling 2.6 GWh (≈11% of peak renewables). Collectively, these works show large-scale DCOPF feasibility, but it has typically involved reduced flow-model fidelity and/or sparse storage arising from the assumed economics or risk setting, which can alter congestion patterns and investment signals while easing computation burden. B. Contributions To address these limitations, this paper proposes a combined a PTDF-based DC formulation with a stabilized (trustregion) Benders approach. The methodology is demonstrated in a high-renewables setting on a 2,000-bus network, thereby addressing simultaneous gaps in spatial resolution, networkfidelity, and storage scalability. The contributions are: • A multiperiod, two-stage PTDF model that co-optimizes line upgrades and storage siting/sizing with hourly dispatch and storage dynamics, preserving DC physics while remaining tractable at scale • A trust-region, multicut Benders scheme, warm-started from per-representative day optima, that mitigates dual degeneracy and oscillatory master-iterates, strengthens cuts, and improves convergence predictability • A multiyear evaluation on a 2,000-bus synthetic Texas system under high-renewable scenarios that endogenously yields a large distributed storage fleet (≥179 devices, 32% of peak renewables) while attaining final optimality gaps ≤1%, demonstrating quality of solution. The paper proceeds as follows. §II formulates the two-stage TEP+Storage model; §III details the Benders methodology; §IV describes the Texas case study; §V reports optimality gaps, runtime, and investment decisions; and §VI concludes. II. FORMULATION PTDF-based DCOPF is well established for TEP [10], yet large-scale storage integration within PTDF frameworks remains comparatively unexplored. This section presents the PTDF-based TEP+Storage formulation. The formulation assumes that at most one generator is connected to each bus, without loss of generality. A. The Two-Stage TEP+Storage PTDF Formulation The TEP+Storage model is a two-stage program. The investment stage selects transmission upgrades and storage siting/sizing to minimize capital expenditures (CapEx). The recourse stage fixes those investments and minimizes operational costs (OpEx) over the investment period H, the planning horizon during which CapEx decisions apply, thereby capturing the coupling between long-term planning and shortterm operations. 1) Investment decisions and variables: This study prioritizes capacity upgrades of existing transmission lines over new line additions, as new construction often faces economic, regulatory, and social barriers [11]. Upgrades are represented by the discrete decision variable γij for each branch ij ∈E, with allowable levels of enhancement ∆c ij (MW per line). Storage is modeled as short-duration lithium-ion batteries due to their technological maturity and widespread deployment [12]. Sizing decisions are captured by σi, denoting the installed energy capacity (MWh) at node i. Each unit has a 4-hour duration, giving a power rating of 1 4σi. Hourly storage operations are modeled by chi,t, disi,t, and soci,t, representing charge, discharge, and state of charge (SoC), respectively. Other key variables include nodal generation dispatch pg and line power flows pf (auxiliary via PTDF). 2) Objective: The TEP+Storage model minimizes the sum of CapEx and OpEx. The CapEx is a linear function of firststage decisions (γ, σ) covering line and storage investments: CapEx = f(γ, σ) := P ij∈E ccap ij γij + P i∈N cstor i σi (1) Given investments (γ, σ), the operating cost OpExs(γ, σ) of scenario-s includes daily generation expenses, storage charge and discharge costs, and penalties for load shedding using a per-unit penalty λ for load shed ξi,s,t. It is computed over the investment period via scaling factor α (e.g., α = 365 for one year). The aggregate operating cost OpEx(γ, σ) is a probability-weighted sum across scenarios s ∈S. OpExs(γ, σ) := GenEx + StorEx + Penalties (2) = α P i∈N ,t∈T [cG i (pg i,s,t) + cS i (chi,s,t + disi,s,t) + λ ξi,s,t], OpEx(γ, σ) := P s∈S ΩS(s) OpExs(γ, σ) (3) 3) Constraints: The DC power flow model uses the Power Transfer Distribution Factor (PTDF) matrix Φ, which maps net nodal injections to line flows. For scenario s and hour t, let injs,t := [pg i,s,t +disi,s,t -pd i,s,t -chi,s,t +ξi,s,t]N i=1 denote the vector of net injections. The operational constraints are: [pf ij,s,t]E ij=1 = Φ × injs,t (4) 1⊤injs,t = 0 (5) - ̄pf ij -γij∆c ij ≤pf ij,s,t ≤ ̄pf ij + γij∆c ij (6) ̄ pg i,s,t ≤pg i,s,t ≤ ̄pg i,s,t (7) soci,s,t = soci,s,t-1 + chi,s,tη -disi,s,t/η (8) soci,s,1 = chi,s,1η -disi,s,1/η (9) soci,s,T = 0 (10) 0 ≤soci,s,t ≤σi (11) 0 ≤chi,s,t ≤σi/4 (12) 0 ≤disi,s,t ≤σi/4 (13) Model 1 Two-Stage TEP+Storage Formulation min (1) + (3) s.t. (4) -(5), ∀s ∈S, t ∈T (6), ∀ij ∈E, s ∈S, t ∈T (7), ∀i ∈N, s ∈S, t ∈T (8) ∀i ∈N, s ∈S, t ∈{2, ..., T} (9) -(10), ∀i ∈N, s ∈S (11) -(13), ∀i ∈N, s ∈S, t ∈T where (4) computes the vector of line flows; (5) imposes power balance; (6) enforces thermal flow limits; (7) bounds generator outputs, with renewable generators modeled with ̄ pg = 0 and scenario-dependent ̄pg (curtailment allowed), and conventional units dispatchable within their limits; (8)-(13) govern storage, including SoC with efficiency losses, a zeroSoC boundary at the start and end of each representative day, and charge/discharge limits. Model 1 outlines the full twostage TEP+Storage formulation. Remark. To maintain recourse feasibility, load shed slack variables ξi,s,t are included and penalized in OpEx by a large coefficient λ (see §II-A2). Any plan with ξ > 0 is rejected; λ is increased (doubled) and the model is re-solved until zero load shed is achieved. This approach preserves feasibility and is guaranteed to terminate when a no-shed solution exists, consistent with industry practice where TEP must serve future load by system design. B. Scalable PTDF Reformulation The PTDF model is scaled to high resolution systems using a sparse representation of the PTDF matrix and lazy constraint generation. A PTDF cutoff is a common sparsification approach, where entries of Φ with magnitudes below a threshold kp (typically 0.01-0.05) are set to zero, thereby sparsifying the matrix and improving computational efficiency by removing negligible effects [10]. For very large systems, however, even the sparse PTDF can remain expensive. To address this, the PTDF flow (4) and thermal-limit constraints (6) are enforced lazily: the model is first solved without these constraints, then any violated line-flow inequalities are iteratively added to model until no violations remain. Specifically, if branch ij in scenario s at time t violates its upper bound, the following inequality constraint is added to the model: pf ij,s,t = P k∈N PTDFij,kinjs,t k ≤ ̄pf ij + γij∆c ij, with the inequality reversed for lower bound violations. Lazy constraints preserve model equivalence while omitting nonbinding limits, yielding smaller and more tractable formulations. The set of lazy constraints is further reduced by adding only the kv most severe violations per iteration, where hyperparameter kv can be tuned: smaller values produce a sparser model at the cost of more iterations to convergence. Both techniques (cutoff and lazy constraints) are used throughout the remainder of the paper. Fig. 1. Benders Schematic Flow Diagram. Model 2 Benders Master Problem (MP) zMP k = min (1) + P s∈S ΩS(s)θs + ρ s.t. ρ ≤0 (14) S s∈S Ck s III. METHODOLOGY The two-stage TEP+Storage model grows linearly with the number of second-stage scenarios; combined with the network's high spatial resolution, even a modest number of scenarios yields very large instances. To address this, a Benders decomposition (BD) is employed. The baseline BD formulation (§III-A) decomposes the problem into a master problem (MP) and subproblems (SPs) linked by investment and operational decisions. Two methodological enhancements are introduced to improve convergence and stability: a warmstart module (§III-B) that initializes the MP using relaxed investment solutions, and a trust-region Benders module (§III-C) that stabilizes the iterative process using a proximal constraint and restricted candidate set. §III-D presents a valid global lower bound used to assess optimality gaps. A. Multicut Benders for TEP+Storage The BD algorithm alternates between the MP, which selects first-stage investments to minimize CapEx, and scenario SPs, which-given (γ, σ)-optimize continuous operational variables and minimize OpEx. Each SP is a linear program, and SPs are solved in parallel. From each SP, feasibility and optimality cuts are generated and added to the MP. Under the multicut scheme, the MP retains all accumulated cuts per scenario. Let C k s denote the set of cuts for scenario s generated through iteration k -1 (with C 1 s = ∅); at iteration k the MP imposes C k s for all s ∈S. Figure 1 summarizes the algorithm. 1) Benders MP: The MP co-optimizes line and storage investments to minimize CapEx. To capture operational and feasibility constraints, the MP includes proxy variables θs an underestimator of OpExs for each scenario s, and ρ an underestimator of scenario-wide load-shed. The MP at iteration k is presented in Model 2, where (14) prevents load shedding in the final solution. After solving the MP, the new investment point (γk, σk) is broadcast to the SPs, which are solved in parallel to generate the next set of cuts in C k+1 s . 2) Benders SP: The scenario-s SP optimizes operations under the MP decisions (γk, σk), fixed in the SP with dual dual multipliers πs,k γ and πs,k σ in (15)-(16). Each SP runs two phases: Phase I enforces load feasibility and Phase II Model 3 Benders Subproblem (SP) min OpExs(γk, σk) s.t. (4) -(5), ∀t ∈T (6), ∀ij ∈E, t ∈T (7), ∀i ∈N, t ∈T (8), ∀i ∈N, t ∈{2, ..., T} (9) -(10), ∀i ∈N (11) -(13), ∀i ∈N, t ∈T γij = γk ij (πs,k γ ), ∀ij ∈E (15) σi = σk i (πs,k σ ), ∀i ∈N (16) optimizes dispatch. In Phase I, the SP minimizes load shed, P i∈N,t∈T ξi,s,t. If the optimum exhibits load shed, the SP terminates and a feasibility cut is generated for shed underestimator ρ. If no shed occurs, the SP proceeds to Phase II to minimize OpExs (Model 3) and generate an optimality cut for θs. Both feasibility and optimality cuts take an affine form, where ψ ∈{I, II} indicates the Phase, ζs = ρ in Phase I and ζs = θs in Phase II: ζs ≥βk s + (πs,k γ )⊤γ + (πs,k σ )⊤σ βk s := vk,ψ s -(πs,k γ )⊤γk -(πs,k σ )⊤σk vk,I s = P i,t ξi,s,t, vk,II s = OpExs(γk, σk) This two-phase design prioritizes feasibility (no shed) while maintaining complete recourse via ξ, avoiding classical infeasibility-ray cuts which can be numerically fragile [13]. 3) Convergence of Benders: At iteration k, the MP optimum zMP k is a valid lower bound LBk for Model 1, since the variables {θs} underestimate the true scenario costs {OpExs}, giving LBk := zMP k . If all SPs are feasible (Phase I yields P i∈N,t∈T ξi,s,t = 0 for every s ∈S), a feasible two-stage objective at (γk, σk) is ˆzk = f(γk, σk) + OpEx(γk, σk), and is used to maintain a non-increasing upper bound UBk := min{UBk-1, ˆzk}. The algorithm terminates when the relative gap falls below a threshold ε: (UBk -LBk)/UBk ≤ε. Remark. Lazy constraints integrate naturally with BD: per scenario flow/limit constraints identified in one iteration are retained for subsequent iterations, accelerating SP solves. B. Warm-Start from Relaxed Two-Stage Model (WS Module) In the baseline BD formulation, early iterations are feasibility-cut dominated: with no initial cuts, the MP selects a no-expansion plan, causing heavy load shedding in the SPs until a feasible investment set is learned. The WS accelerates this by solving, in parallel for each s ∈S, the continuous relaxation of Model 1 to obtain (γs, σs). A conservative, nonanticipative design is then formed by component-wise aggregation of the per-scenario optimal investments, and rounded up to the admissible investment granularity: ̄γ := maxs∈S γs , ̄σ := maxs∈S σs . Fig. 2. Workflow of the proposed methodology. The Benders MP is initialized at ( ̄γ, ̄σ), which reduces early infeasibility cuts and accelerates convergence. C. Trust-Region-Stabilized Benders (TR-Benders Module) Once feasibility is achieved, the BD formulation exhibits strong dual degeneracy-many line-storage configurations with nearly identical operating costs-yielding flat recourse value functions, oscillatory MP iterates, and weak cuts that slow convergence. The TR-Benders addresses this issue by augmenting the BD framework with a proximal trust-region (TR) constraint on first-stage decisions and a restricted candidate set. This stabilization mechanism limits drastic investment changes between iterations, improving numerical stability and ensuring more consistent convergence behavior. The MP is stabilized around an anchor point ( ̆γ, ̆σ) defined as the incumbent investment that attains the best known upper bound; it is initialized at ( ̄γ, ̄σ) and updated only upon a strict improvement in the upper bound. Inspired by stabilized BD (e.g., [14]), the method adapts to discrete first-stage variables with an l1 trust-region (linear, sparsity-promoting): P ij∈E γij - ̆γij ≤rγ k (17) P i∈N σi - ̆σi ≤rσ (18) Absolute values are linearized in the standard way for MIPs. A fixed storage radius rσ = 2 (permitting limited re-siting) is used, while the transmission radius rγ k is dynamic: (i) resetting to 1 whenever the anchor point updates, and (ii) increasing by 1 whenever the MP revisits the same solution, to limit local oscillations. To further reduce degeneracy, investment decisions are restricted to a candidate set derived from the per-scenario relaxations: Ec := { ij ∈E | ∃s ∈S : γs ij > 0} Nc := { i ∈N | ∃s ∈S : σs i > 0} Finally, a level-set constraint prunes regions already known to be suboptimal: f(γ, σ) + P s∈S ΩS(s) θs ≤UBk (19) Together, the TR, candidate sets, and level-set constraint damp oscillations and weaken the effects of dual degeneracy. Figure 2 summarizes the full workflow of the proposed methodology. D. Global Lower Bound and Optimality Assessment A limitation of the TR stabilization is that the master's value no longer certifies a global lower bound: with the TR active, LBk is a local lower bound over the current region around the anchor point. However, a valid global lower bound can be constructed as below. 1) First-Stage Lower Bound: With load shed disallowed by (14), the minimum feasible investment decision is constrained by the first-stage investments of the most challenging representative day (typically characterized by high load and/or low renewable generation). This is formalized in Theorem 1. Theorem 1. For each s ∈S, define the day-s feasible set and the all-scenarios feasible set: Σs feas := {(γ, σ) : recourse is feasible for scenario s} Σfeas := T s∈S Σs feas Given the feasible sets, define the corresponding minimum first stage costs for feasible day-s and all-scenarios: cfeas s := min (γ,σ)∈Σs feas f(γ, σ), cfeas := min (γ,σ)∈Σfeas f(γ, σ), and first-stage optimal investments for scenario s: (γs feas, σs feas) ∈arg min(γ,σ)∈Σs feas f(γ, σ) Let ˆs ∈arg maxs∈S cfeas s . Then maxs∈S cfeas s ≤f(γ, σ) ∀(γ, σ) ∈Σfeas In particular, f(γˆs feas, σˆs feas) = maxs cfeas s is a valid lower bound on any year-feasible first-stage plan's cost. Proof. Since Σfeas ⊆Σs feas for every s, cfeas = min (γ,σ)∈Σfeas f(γ, σ) ≥ min (γ,σ)∈Σs feas f(γ, σ) = cfeas s ∀s, yielding maxs cfeas s ≤cfeas. Choosing ˆs ∈arg maxs cs gives the stated bound with (γˆs feas, σˆs feas). 2) Second-Stage Lower Bound: The per-scenario operational cost is monotone nonincreasing in investment capacity, as per Theorem 2. Let (γmax, σmax) denote the componentwise upper bounds on line and storage capacities (not restricted to the candidate set) of Model 1. Then a valid lower bound on the aggregated second-stage cost is P s∈S ΩS(s) qs(γmax, σmax). Theorem 2. Let qs(γ, σ) be the optimal value of Model 3. If γ′ ⪰γ and σ′ ⪰σ (componentwise), then qs(γ′, σ′) ≤ qs(γ, σ). In particular, qs(·) is monotone nonincreasing in γ and σ. Proof. For fixed s ∈S, increasing γ relaxes the thermal limit constraints (6), and increasing σ relaxes the storage constraints (11)-(13). Hence the feasible region under (γ′, σ′) contains that under (γ, σ). Because the SP objective is minimized and depends on γ, σ only through these constraints, enlarging the feasible set cannot increase the optimal value, yielding the claim. Fig. 3. Generator sites of the system. Black circles indicate nonrenewable, blue circles indicate wind, and red circles indicate solar generators. 3) Combined Bound: Define z∗as the optimal final solution's objective, with optimal line and storage investments (γ∗, σ∗) ∈Σfeas. z∗:= f(γ∗, σ∗) + P s∈S ΩS(s) qs(γ∗, σ∗) Since Theorem 1 and 2 show, respectively, that f(γˆs feas, σˆs feas) ≤f(γ∗, σ∗), qs(γmax, σmax) ≤qs(γ∗, σ∗) ∀s ∈S, a valid combined lower bound can be constructed: LB∗:= f(γˆs feas, σˆs feas) + P s∈S ΩS(s) qs(γmax, σmax) ≤z∗ 4) Optimality Gap: Each cfeas s is computed under the integer relaxation of the first-stage investments (as in the WS). Being a relaxation, this provides a valid lower bound for the yearwide plan, which must satisfy discretized first-stage decisions. The reported optimality gaps are therefore conservative upper bounds: (UBk -LB∗)/UBk. IV. CASE STUDY The proposed methodology is tested on a large-scale power system case study: the ACTIVSg2000 test case, a 2000-bus model of the U.S. Texas grid. The TEP+Storage model is applied in a rolling-horizon manner. The formulation itself has year-agnostic investment variables, but is solved sequentially at 5-year decision points Y = {2030, 2035, 2040, 2045} with investment periods of length α = 1825 days. Investments (and their capital costs) are cumulative: capacity installed in an earlier year remains available in subsequent years, ensuring monotone, non-decreasing expansion across planning periods. This application reflects practical staging, but it is not guaranteed optimal relative to an omniscient, all-years model; quantifying any cross-period optimality gap is left for future work. a) Network Data: The synthetic Texas system is built from public information and statistical analysis of real power systems [15]. The system has 2000 buses, 3206 lines, 423 non-renewable generators, and 175 renewable generators. Figure 3 shows the location of solar, wind, and nonrenewable generators, overlaid on the grid topology. The future generation mix from 2030 to 2045 follows EIA outlook TABLE I GENERATION BASE CAPACITY (GW) AND GENERATION/LOAD SCALING Type Capacity 2022 2030 2035 2040 2045 Coal 14.5 1.00 0.82 0.82 0.82 0.82 Natural Gas 75.1 1.00 0.79 0.73 0.71 0.72 Nuclear 7.1 1.00 0.98 0.90 0.80 0.80 Solar 2.5 1.00 4.51 6.00 6.87 8.04 Wind 15.5 1.00 2.02 2.23 2.26 2.32 Load - 1.00 1.13 1.21 1.31 1.41 Fig. 4. Representative days by average demand vs. wind in 2022. Red circles indicate selected representative days. Transparent blue circles indicate nonselected days. projections [16]. All generation resources are uniformly scaled to reflect projected growth or planned decommissioning. Load growth is modeled with a year-to-year increase of 1.5% and is scaled uniformly across all buses. Table I shows the total generator capacities (GW) of the network and summarizes the scaling for each generation type and load relative to the base year, 2022. b) Scenario selection: Scenarios are 24-h representative days sampled from a year of load/wind/solar data. This reduces the computational time required for evaluating a full year of operations, while still capturing intertemporal storage dynamics. In this case study, kr = 18 days are selected by a max-min diversity criterion on normalized daily profilesmaximizing the minimum pairwise Euclidean distance-and assigned equal weights. Figure 4 shows the chosen days (red circles) in average wind-load space-the main drivers of investment (solar capacity is comparatively smaller). Days that require storage operation to avoid load shed are labeled; these generally coincide with higher demand. August 11 (08-11) is the most challenging day, exhibiting the highest average demand. c) Investments parameters: Table II lists capital and operating costs for line upgrades and storage. Line upgrades are discretized to capacity increases of 30, 60, and 90%, reflecting the enhancements achievable through typical line reconductoring [11]. Storage units are 250 MWh each, limited to 12 per node (3 GWh total), with charge and discharge efficiencies of η = 95%. Line upgrade costs are adapted from [17]. Storage costs are adapted from [18]. d) Computational Setup and Hyperparameters: The TEP+Storage model is implemented in Julia with JuMP [19] and solved with Gurobi [20] on Georgia Tech's PACE Phoenix TABLE II COST CONFIGURATION Costs Capacity Upgrades 1,000,000/MWh Charge/Discharge Cost B) Storage ( B) StorOpEx ($B) 2030 08-11 183 0 / 0 0.14 0 46.49 0 - Warm-start 291 0 / 0 0.37 0 28.80 0 1.3% TR-Benders 238 0 / 0 0.24 0 28.86 0 1.0% 2035 08-11 334 33 / 28.91 0.36 28.92 50.91 7.27 - Warm-start 381 33 / 32.75 0.53 32.77 30.39 0.40 6.3% TR-Benders 352 32 / 29.25 0.40 29.27 30.42 0.41 0.8% 2040 08-11 425 118 / 98.07 0.47 98.12 56.60 24.71 - Warm-start 462 118 / 111.75 0.66 111.81 33.35 1.64 9.4% TR-Benders 441 108 / 98.50 0.53 98.55 33.38 1.65 0.4% 2045 08-11 476 184 / 167.30 0.58 167.40 62.6 42.01 - Warmstart 508 184 / 183.75 0.72 183.84 36.63 3.66 7.4% TR-Benders 499 179 / 167.75 0.66 167.84 36.64 3.66 0.3% Fig. 5. WS computation times by representative day for each 5-year investment period. TABLE IV TR-BENDERS COMPUTATIONAL PERFORMANCE FOR EACH 5-YEAR INVESTMENT PERIOD. 2030 2035 2040 2045 Time Limit (Hrs.) 24 24 24 72 Iterations 1964 1198 332 102 Avg. Seconds/Iter 44 72 260 2541 over the planning horizon because progress each round is paced by the most challenging SP/representative day: the algorithm advances only after all cuts are generated. The slowest iterations coincide with intensive storage operation and pronounced congestion, which prolong SP solves due to storage time-coupling and ongoing discovery of violated thermal-limit constraints. Further computational improvement may be attained by pre-screening thermal violations. C. Investment Results The TR-Benders produces a feasible, near-optimal yearwide plan for 2030-2045 with zero load shed across all representative days. Figure 6 maps the geographic distribution of investments. Transmission upgrades concentrate in West Texas, strengthening transfer from wind/solar regions, while storage appears beginning in 2035 and grows considerably with increasing load. Figure 7 reports storage energy throughputs for each planning year, defined as the total GWh charged and discharged during the representative day. Only a small fraction of days require storage operation to eliminate load shedding; 08-11 exhibits the largest throughput and aligns with its computational difficulty (Fig. 5). Consequently, multiyear investment choices are largely driven by this hardest day. Table III summarizes the selected upgrades, storage siting/sizing, and costs; its rightmost column displays the final optimality gaps referenced earlier. For completeness, the single-day (08-11) WS results are also listed by period. The WS intentionally over-invests. After the TR-Benders, the near-optimal plan reduces CapEx relative to the WS, while incurring only negligible increases in OpEx, yielding a lower total cost. The 08-11 single-day results show comparatively higher OpEx than both the WS and TR-Benders plans because they optimize solely for the highest-demand day rather than the equally weighted set of representative days. Because storage dominates investment costs, most of the residual gap is driven by reconciling storage capacity rather than line upgrades. The storage capacity of TR-Benders approaches-but remains above-the requirement implied by the hardest day, consistent with Theorem 1 ensuring zero load-shed feasibility. By 2045, the optimized plan yields 167.75 GWh of storage distributed across 179 nodes, which is equivalent to 32% of the scaled peak renewable capacity. This underscores the robustness of the proposed formulation and its tractability at high spatial and storage resolution. VI. CONCLUSION This paper presents a PTDF-based TEP+Storage for joint transmission and storage expansion and applies it to a 2,000bus synthetic Texas system. To ensure computational feasibility, it introduces a trust-region, multicut Benders scheme warm-started from per-representative-day optima. The method attains final optimality gaps ≤1% and remains tractable for a large distributed storage fleet deployed at ≥179 nodes (≈32% of peak renewable capacity). These results demonstrate the proposed PTDF-based methodology effectively handles high spatial resolution and large distributed storage deployments in joint TEP+Storage planning. Table IV shows that computational challenges persist, particularly under high storage penetration and network conges- (a) 2030 TEP+Storage (b) 2035 TEP+Storage (c) 2040 TEP+Storage (d) 2045 TEP+Storage Fig. 6. Upgrades from 2030 to 2045 in 5-year intervals. Red lines denote capacity upgrade locations, with their thickness corresponding to one of three possible upgrade levels. Circles indicate storage locations, with their size and color representing the number of installed units (250MWh each). Fig. 7. Storage energy throughput (GWh) for representative days with nonzero utilization across the 2030-2045 planning horizon. Values reflect the representative day under the investment plan in effect for that year. tion, and will intensify as temporal, operational, and spatial fidelity increase. Priorities include accelerating SP solvese.g., pre-screening and dynamic pruning of thermal/flow constraints-and exploring unified Benders variants that combine feasibility and optimality cuts. Another direction is to integrate generation expansion, which may relieve congestion and further improve lazy-constraint performance. 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Quantum beats of exciton-polarons in CsPbI3 perovskite nanocrystals
A. V. Trifonov,1, 2, ∗M. O. Nestoklon,1 M.-A. Hollberg,1 S. Grisard,1 D. Kudlacik,1 E. V. Kolobkova,3, 4
M. S. Kuznetsova,2 S. V. Goupalov,5 J. M. Kaspari,6 D. E. Reiter,6 D. R. Yakovlev,1 M. Bayer,1, 7 and I. A. Akimov1, †
1Experimentelle Physik 2, Technische Universit¨at Dortmund, 44227 Dortmund, Germany
2Spin Optics Laboratory, St. Petersburg State University, Peterhof, 198504 St. Petersburg, Russia
3ITMO University, 199034 St.
Petersburg, Russia
4St. Petersburg State Institute of Technology, 190013 St. Petersburg , Russia
5Department of Physics, Jackson State University, Jackson, 39217 Mississippi, USA
6Condensed Matter Theory, Department of Physics,
Technische Universit¨at Dortmund, 44227 Dortmund, Germany
7Research Center FEMS, Technische Universit¨at Dortmund, 44227 Dortmund, Germany
(Dated: October 17, 2025)
Exciton–phonon interactions govern the energy level spectrum and thus the optical response in
semiconductors.
In this respect, lead-halide perovskite nanocrystals represent a unique system,
for which the interaction with optical phonons is particularly strong, giving rise to a ladder of
multiple exciton states which can be optically excited with femtosecond pulses. We establish a new
regime of coherent exciton-polaron dynamics with exceptionally long coherence times (T2 ≈300 ps)
in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix.
Using transient two-pulse
photon echo at 2 K temperature, we observe quantum beats between the exciton-polaron states.
Within a four-level model, we directly quantify the exciton–phonon coupling strength through the
Huang–Rhys factors of 0.05 ÷ 0.1 and 0.02 ÷ 0.04 for low-energy optical phonons with energies of
3.2 and 5.1 meV, respectively. The pronounced size dependence of both coupling strengths and
phonon lifetimes offers a path to tune the optical transitions between polaron states and to tailor
the coherent optical dynamics in perovskite semiconductors for solid-state quantum technologies.
I.
INTRODUCTION
Lead halide perovskite semiconductors attract close at-
tention due to their intriguing optical and electrical prop-
erties with huge potential for photovoltaic and light emit-
ting applications [1, 2]. These materials provide a unique
playground for polaron physics in the solid state gov-
erned by the interaction of electronic excitations with the
phonons of the crystal lattice [3]. In particular, several
studies indicate that low-energy optical phonons in halide
perovskites have an important influence on the conduc-
tivity, while acoustic phonons have a minor effect [4, 5].
This is in contrast to conventional semiconductors such
as GaAs, where the interaction of electrons with acous-
tic phonons is dominant at cryogenic temperatures [6, 7].
Furthermore, the carrier mobility is determined not by a
few specific optical phonons, but by a large multitude of
modes due to the complex phonon spectrum [8].
Perovskite nanocrystals (NC) belong to the class of
quantum emitters with discrete energy level spectrum,
large spectral tunability, and high quantum yield, which
makes them attractive as single photon sources [9, 10].
At cryogenic temperatures, the elementary optical exci-
tation in NCs is an exciton (electron-hole pair) which
possesses long optical and spin coherence times up to
hundreds of ps, comparable to its radiative lifetime [11–
13].
Long-lived coherent excitons are often considered
as qubits and are appealing for possible applications
∗correspondence address: artur.trifonov@tu-dortmund.de
† correspondence address: ilja.akimov@tu-dortmund.de
in quantum communication [14].
However, the strong
exciton-phonon interaction leads to formation of exciton-
polarons which modifies the energy level spectrum. In
particular, the photoluminescence spectra of single NCs
show pronounced optical phonon replicas which confirm
the substantial exciton-phonon coupling with Huang-
Rhys factors in the range from 0.05 to 0.4 in lead halide
perovskite NCs [15–19].
Recently, electron-phonon in-
teraction dominated by optical phonons with a similar
Huang-Rhys factor of about 0.4 was demonstrated in
CsPbBr3 NCs [20] and other inorganic lead halide per-
ovskite NCs [21] using Raman spectroscopy. Compared
to InGaAs quantum dots [22, 23], these values are sig-
nificantly larger and comparable to those reported in
II-VI nanostructures [24]. A distinctive feature of per-
ovskite nanocrystals is the existence of low-energy optical
phonons with a strong coupling to phonons. This combi-
nation substantially modifies the exciton level spectrum
in the vicinity of the ground state [25] and represents an
attractive testbed for investigation of the quantum dy-
namics under resonant femtosecond excitation markedly
different from previously studied systems.
Time-resolved studies can provide direct access to the
exciton dynamics. It is well established that scattering
on incoherent phonons with a thermal distribution causes
loss of exciton coherence [26, 27].
Yet, less is known
about the quantum dynamics of exciton-polarons, i.e.,
the hybrid excitations of an exciton and a phonon. Re-
cently, transient four-wave mixing and two-dimensional
Fourier spectroscopy revealed the importance of coherent
exciton-phonon coupling for the ground and excited (ex-
citon and biexciton) states in lead halide perovskites [28–
arXiv:2510.14695v1 [cond-mat.mes-hall] 16 Oct 2025
2
31]. However, most of these studies were limited to the
case of short optical coherence of excitons (≤100 fs) and
therefore reported on the coherent evolution of either the
exciton or the ground electronic state modulated by the
frequencies of phonons with much slower decoherence.
In other words, the lack of optical coherence restricted
the observations to the phonon dynamics, while the gen-
uine coherent evolution of the coupled exciton–phonon
system has remained inaccessible. This scenario was also
documented for bulk semiconductors, nanostructures and
organic molecules [32–36]. However, the opposite quan-
tum dynamics limit, where pronounced quantum beats of
exciton-polarons are expected, has not yet been uncov-
ered. Furthermore, another general question arises, con-
nected to the limitations of coherent control for resonant
excitation with femtosecond pulses, since time-resolved
studies reported quite short coherence times up to 20 ps
in lead halide perovskites [28, 37], in contrast with pho-
toluminescence data of single NCs [11–13].
In this work we establish a new regime of coherent dy-
namics governed by exciton-polarons in NCs with an ex-
ceptionally long coherence time of the zero-phonon exci-
ton (T2,X ≈300 ps) on the one hand and a discrete spec-
trum of spectrally narrow optical phonon modes on the
other hand. The regime manifests in long-lived exciton-
polaron quantum beats, which are detected by tran-
sient two-pulse photon echoes in an ensemble of CsPbI3
nanocrystals embedded in a glass matrix at low temper-
ature, T = 2 K. The oscillatory signal is dictated by
optical phonons with low frequencies corresponding to
3.2 and 5.1 meV energy, and can be described by the
solution of the Lindblad equations for a set of four-level
systems that include a phonon ladder in the ground and
excited exciton-polaron states. Different sizes, ranging
from 10 to 12 nm, were selectively addressed within the
same ensemble of nanocrystals by varying the laser pho-
ton energy. From the relative amplitude of oscillations
we directly evaluate the strength of the exciton-phonon
coupling with Huang–Rhys factors of 0.06 and 0.02 for
optical phonons with 3.2 meV and 5.1 meV energy, re-
spectively.
Correspondingly, the electron–phonon cou-
pling strength doubles reaching values of 0.12 and 0.04
as the NC size is reduced from 12 to 10 nm. The theo-
retical analysis of the size dependence shows that both
the deformation potential and the Fr¨ohlich mechanism
of electron-phonon interaction qualitatively agree with
the experimental data in the regime of weak exciton con-
finement.
Our study demonstrates that the dynamics
of a pure quantum state following resonant optical ex-
citation requires consideration of polaron effects, whose
strength and coherence can be controlled by composition
and size of the NCs. This opens up new possibilities for
on-demand coherent control and coherent phonon gener-
ation in lead-halide perovskite NCs.
II.
PHOTON ECHO AND LONG COHERENCE
OF ZERO-PHONON EXCITON
The ensemble of CsPbI3 NCs is synthesized in a flu-
orophosphate glass matrix by rapid cooling of a glass
melt enriched with the materials needed for the per-
ovskite crystallization. The details of synthesis are given
in Ref. [38]. The NC size is about 9 ÷ 13 nm in diameter
(see Fig. S6 in Ref. 39). The low-temperature photolumi-
nescence (PL) spectrum is dominated by a 60 meV broad
band centered around 1.75 eV, as shown in Fig. 1(a). The
broadening originates from fluctuations of the NC size,
which we estimate to be in the order of 20%. A single NC
is schematically illustrated in the inset of Fig. 1(b). Note
that the cubic crystal structure is shown for illustrative
purposes as the actual crystallographic phase depends on
the growth conditions and the size of the nanocrystals
[40–43].
We perform transient four-wave mixing (FWM) exper-
iments in transmission geometry as shown schematically
in Fig. 1(b). The NCs are resonantly excited with a se-
quence of two 100 fs laser-pulses with photon energies
hν <1.76 eV in the low energy flank of the PL band.
The sample temperature is kept at T = 2 K. The electric
field amplitude of the FWM signal EFWM(t) is resolved in
time using heterodyne detection where the signal field is
temporally overlapped with a strong reference pulse with
amplitude Eref(t−τref) (see Methods and Refs. 44–47 for
details). Here, time t = 0 corresponds to excitation with
the first pulse, while τref is the delay of the reference
pulse with respect to the first pulse in the excitation se-
quence. The resulting FWM signal is shown in Fig. 1(c),
which gives a two-dimensional plot of EFWM as function
of the delay time between the first and second pulse τ12
(vertical axis) and the reference delay time τref (horizon-
tal axis). The FWM signal demonstrates the expected
peak centered at time τref = 2τ12, corresponding to emis-
sion of a photon echo (PE) from the NC ensemble [48].
This behavior arises from the inhomogeneous broaden-
ing of the optical transitions resulting from fluctuations
of the NC size as demonstrated for similar halide per-
ovskite NCs [28, 37]. Interestingly, during the first tens
of ps the PE amplitude shows pronounced high frequency
oscillations with increasing τ12 which we will discuss be-
low.
On a longer time scale, the amplitude of the photon
echo decays exponentially with increasing delay time τ12
as shown in Fig. 1(d).
Using exponential decay fits,
shown with the dashed lines, we obtain T2 = 230÷330 ps
for photon energies in the range 1.724 ÷ 1.75 eV, show-
ing a weak decrease with increasing hν.
The popula-
tion relaxation time T1, measured in three-pulse experi-
ments, is found to be T1 = 800 ÷ 850 ps, and is compa-
rable to the exciton lifetime measured by time-resolved
PL on the same sample.
Therefore, we conclude that
the signal at τref > 50 ps is due to the zero-phonon ex-
citon transition with long-lived optical coherence time
T2 and exciton recombination lifetime T1.
If the co-
3
1.65
1.7
1.75
1.8
1.85
0
0.5
1
Energy (eV)
PL intensity (arb. units)
τ12
τref
Lead
Iodine
Caesium
1
2
3
4
5
6
7
−0.5
0
0.5
0.5
1
1.5
2
τ12 (ps)
τref - 2τ12 (ps)
E *
FWM E ref (arb. units)
(a)
(b)
(c)
0
50
100
1
234
0.1
τref (ps)
E *
FWM E ref (arb. units)
1.75 eV
1.74 eV
1.724 eV
T2 = 276 ps
(d)
10
12
8
9
15
NC diameter (nm)
T2 = 330 ps
T2 = 230 ps
1
2
ref
PE
k2
k1
2k2 - k1
T = 2K
Figure 1. Photon echoes in CsPbI3 nanocrystals. (a) Typical photoluminescence (PL) spectrum for excitation with photon
energy of 2.33 eV. Vertical dashed line indicates the highest photon energy used in the four-wave mixing FWM experiment.
The scale on top corresponds to the NC diameter. (b) Schematic representation of the experimental geometry, where the
laser pulses hit the sample with wave vectors k1, k2.
The photon echo is detected along the direction 2k2 −k1 using a
heterodyning technique by overlapping it with a reference pulse. Inset shows schematically a CsPbI3 perovskite nanocrystal.
(c) Two-dimensional plot of the FWM electric field amplitude E∗
FWM as function of delay between the two pulses τ12 and the
reference time τref. The photon echo (PE) signal forms at the time τref = 2τ12. The amplitude of the PE shows oscillations
during the initial evolution when τ12 is scanned. The signal is recorded in linearly co-polarized configuration. Photon energy
hν = 1.736eV. (d) Decay of two-pulse photon echo amplitudes for excitation with different photon energies. Black dashed lines
are fits with exponential functions, from which the labeld exciton coherence times T2 are extracted. Temperature T = 2 K.
herent dynamics would be governed only by the pop-
ulation decay we would expect to the T2 = 2T1 rela-
tion to hold, which typically holds for excitons in self-
assembled InGaAs quantum dots at T = 2 K [49]. Here,
we observe a different situation, where elastic scattering
processes (”pure dephasing”) with a decay constant of
Tp = [1/T2 −1/(2T1)]−1 = 330 ps mainly govern the ex-
citon coherence. Nevertheless, to the best of our knowl-
edge, the homogeneous linewidth of the zero-phonon ex-
citon of Γ2 = 2ℏ/T2 = 4.8 µeV demonstrated here has a
record low value for perovskite nanocrystals [11–13].
III.
COHERENT EXCITON-PHONON
DYNAMICS
As follows from Fig. 1(c) for short delay times τ12 ≲
10 ps, oscillations of the PE signal with multiple frequen-
cies are observed. For deeper insight into the origin of
the oscillations, we analyze polarization-resolved PE sig-
nals. These results are summarized in Fig. 2. We employ
two configurations, where the first and second pulses are
linearly co- or cross- polarized, while the detection polar-
ization in all cases coincides with that of the first pulse
(see Methods). The PE amplitudes as function of τ12 in
co-(∥) and cross-(×) polarized configurations are shown
in Fig. 2(a). The oscillations can be grouped into two
categories: slow oscillations with a period of 1–10 ps and
fast oscillations with a period of less than 1 ps. It is evi-
dent that the fast oscillations in the co-polarized config-
uration are in phase and nearly identical in both curves.
We will show below that these oscillations correspond to
quantum beats between exciton-polaron states.
By contrast, the slow oscillations possess opposite
phases.
The co-polarized A∥-signal starts maximum
value, while the A×-signal starts from zero.
Subse-
quently, the minimum of the slow oscillations in ∥-
configuration around τ12 = 5 ps corresponds to a maxi-
mum in the ×-configuration. This antiphase behavior is
related to quantum beats between the orthogonally po-
larized bright exciton states [37]. In perovskite nanocrys-
tals, the bright exciton fine structure comprises three lin-
early polarized states [15, 50], split by the energies δ1 and
δ2 as shown in Fig. 3(a). This energy level scheme is sim-
ilar to the fine structure splitting in self-assembled quan-
tum dots [51, 52] and in II-VI colloidal NCs [53, 54]. In
order to isolate the oscillations due to the fine structure
splitting in the data, we model the expected polariza-
tion dependence of the PE signal, accounting also for the
random orientation of the nanocrystals (for details see
Sec. S1 of the Supplementary Information). We obtain
the following equations for the amplitude of the phonon
4
0
10
20
30
40
0
0.2
0.4
0.6
0.8
1
τ12 (ps)
PE ampl. (arb. units)
co-linear (||)
cross-linear (x)
0
10
20
30
40
0
0.5
1
1.5
τ12 (ps)
Amplitude (arb. units)
ρ
Σ
Fit
0
2
4
6
8
0
2
4
6
8
10
Energy (meV)
Intensity (arb. units)
0
2
4
6
8
0
0.5
1
1.5
2
Energy (meV)
FFT (arb. units)
ρ
Σ
Fit
(a)
(b)
(c)
(d)
Raman
(x2)
Figure 2. Quantum beats of exciton-polarons. (a) Initial range of the PE dynamics in the co-linear (∥) and cross-linear
(×) polarization configurations shown with blue and red lines, respectively. Photon energy hν = 1.746 eV. The amplitude of
the × signal is multiplied by two for clarity. (b) Dynamics of ρ (black) and Σ (green) as defined by Eqs. 2 and 3, respectively.
The dashed red curve is a fit using the polaron model with Eq. 5 with the following parameters for two phonon modes:
ℏΩ1 = 3.2 meV, SHR1 = 0.097, τph1 = 5.1 ps; ℏΩ2 = 5.1 meV, SHR2 = 0.032, τph2 = 10 ps. (c) Fast Fourier transform (FFT)
amplitude spectra of ρ (black), Σ (green), and the fit curves in panel (b). Vertical dashed lines mark the peaks corresponding to
optical phonons. (d) Raman spectrum measured at photon energy hν = 1.734 eV. Vertical dashed lines indicate the positions
of peaks corresponding to the optically active optical phonon modes that couple most strongly to the exciton.
echo signal in two polarization configurations:
A∥=
"
18
15 +
X
i
4
15 cos (δiτ12) e
−2τ12
t∗
i
#
Ψ0(τ12) ,
(1a)
A× =
"
6
15 −
X
i
2
15 cos (δiτ12) e
−2τ12
t∗
i
#
Ψ0(τ12) ,
(1b)
where the index i = 1, 2, 3 corresponds to the beats be-
tween the three optical transitions (δ3 = δ1 + δ2), and t∗
i
is the dephasing time of the beats caused by the disper-
sion of the splitting energies ℏδi in the ensemble. The
common factor Ψ0(τ12) is the same for both polariza-
tion configurations and will be discussed below, see also
Supplementary Information S1. Following Eqs. (1) it is
possible to distinguish between quantum beats related to
the exciton fine structure and other polarization insen-
sitive contributions to the PE signal by introducing the
polarization contrast
ρ = A∥−3A×
A∥+ 2A×
,
(2)
and the polarization sum
Σ = A∥+ 2A×
2
= Ψ0(τ12) .
(3)
It follows that the temporal evolution of ρ exhibits oscil-
lations at frequencies corresponding to the energy split-
tings δi of the exciton fine structure and is independent
of Ψ0(τ12). In contrast, the polarization sum Σ is given
by the intrinsic coherent dynamics Ψ0(τ12) only.
Figure 2(b) shows the dynamics of these quantities,
calculated from the data in Fig. 2(a).
In full accord
with our expectations we obtain that the high-frequency
oscillations are absent in the dynamics of ρ, while the
low-frequency oscillations are still present. On the other
hand, the low-frequency oscillations vanish in the Σ tran-
sient in contrast to the high-frequency components. This
becomes even more clear from the fast Fourier transform
(FFT) spectra of ρ and Σ shown in Fig. 2(c). From the
fit of the ρ transient we evaluate δ1 = 0.25 meV and
δ2 = 0.55 meV, which are in agreement with the exci-
ton fine structure splitting evaluated by two-dimensional
Fourier spectroscopy on CsPbI3 NCs of similar size with
a diameter of about 9 nm [37]. A detailed discussion of
the fine exciton structure combining pump-probe and PE
studies will be published elsewhere. In what follows, we
focus on the dynamics of Ψ0(τ12), which is independent
of the spin level structure of the exciton.
The FFT spectrum of Σ, shown by the green line in
Figure 2(c), exhibits several spectrally narrow features.
These features are consistent with the peaks in the Ra-
5
man spectrum in Figure 2(d). The latter originates from
light scattering on optically active phonons.
The fea-
tures at energies below 0.5 meV are attributed to con-
fined acoustic phonons [55]. Below we focus on the two
most prominent features marked by the vertical dashed
lines with energies of ℏΩ1 = 3.2 meV and ℏΩ2 = 5.1meV,
corresponding to the energies of optically active optical
phonons in the vicinity of the Γ point. Indeed optical
phonon modes with energies close to 3 and 5 meV in
CsPbI3 were observed in Ref. [21] and calculated [8]. Op-
tical phonons with similar energies were also detected by
Raman spectroscopy in other lead halide perovskites such
as FAPbI3 [56, 57], CsPbBr3 [19], and CsPbCl3 [17, 58].
Thus, we conclude that the high frequency THz oscilla-
tions in the two-pulse coherent optical response are due to
the coherent evolution of the coupled, hybridized exciton-
phonon system.
IV.
EXCITON-POLARON QUANTUM BEATS
In order to describe the quantum dynamics of the cou-
pled exciton-phonon system we consider the structure of
the optically excited energy levels in a single NC, which
requires the involvement of polaron states. To that end,
we introduce the four-level scheme shown in Fig. 3(b)
which comprises the ground state |0⟩, the phonon mode
|0′⟩, the exciton polaron state |X⟩(“exciton”) and the
first excited exciton polaron state |X′⟩(see details in the
Supplementary Information S2 S2.A). We neglect the ex-
citon fine structure splitting to concentrate on the deriva-
tion of Ψ0(τ12) in Eq. (1).
Initially, we consider the
exciton coupling to a single phonon mode in a NC. At
the next stage, to compare with the experimental data,
we perform summation over different phonon modes Ωi
where we neglect the interaction between them
Ψ0(τ12) =
X
i
Ψ0(τ12; Ωi) .
(4)
The optical transitions between all energy levels are
allowed using the same polarization.
In the polaron
model [59], the transition probability amplitude is pro-
portional to the product of the dipole matrix element and
the overlap of the wavefunctions associated with vibronic
modes shifted due to polaron formation (see Fig. 3(c) and
Eq. S16), which depends on the strength of the exciton-
phonon interaction given by the Huang-Rhys factor SHR
(Condon principle). The pulse duration is assumed to be
short as compared with all other characteristic times in
the system, i.e. we can assume the δ-pulse limit. This is
justified because the laser pulse duration is τd ≈100 fs
(see Methods) and the fit to experimental data gives re-
laxation times in the order of 10 ps. The spectral width of
the laser pulse ∼h/τd is about 15 meV, which is larger
than the phonon energies ℏΩi.
Therefore, we assume
that all four transitions between states |0⟩, |0′⟩and |X⟩,
|X′⟩are covered by the spectral width of the laser. Note
that a not too large Huang-Rhys factor means that states
involving multiple phonons (i.e. 2ℏΩ1, etc.) may be ne-
glected. However, the spectral width of the laser pulse is
large enough to cover them as well. We neglect the possi-
ble difference of phonon energy in the ground and polaron
state as we do not find corresponding frequency changes
in our experimental data and in previous reports in other
perovskites [56, 57]. Similar assumptions were made in
the model developed for desscribing 2D spectroscopy in
Ref. 60.
In the polaron model, we trace the evolution of the
density matrix components which contribute to the PE.
The result arises from the coherent off-diagonal terms
and all levels contribute to the result in first order in the
Huang-Rhys factor SHR, which quantifies the exciton-
phonon interaction. Solution of the Lindblad equation
in Supplementary Information S2 gives the following ex-
pression for the amplitude of the phonon echo signal from
a NC with single phonon mode Ωi
Ψ0(τ12; Ωi) = e−(1+SHR)γ0τ12
"
1 + SHRe−(γph−SHRγ0)τ12
+SHR cos (Ωiτ12)e−
γph−2SHRγ0
2
τ12h
2+e−SHRγ0τ12+e−γphτ12i
+ SHR cos (2Ωiτ12)e−(γph−SHRγ0)τ12
#
.
(5)
Here T2,0 = 1/γ0 is the coherence time associated with
the zero-phonon optical transition, and τph = 1/γph is
the lifetime of the optical phonon.
In the simplified
model we do not include other sources of decoherence.
The analytical expression fully supports the experimen-
tal observations for Σ = Ψ0(τ12) shown in Fig. 2(b). The
first term in Eq. 5 corresponds to the long-lived zero-
phonon coherence which decays exponentially with the
time T2 = T2,0/(1 + SHR) ∼300 ps, as evaluated in
section
II. The last two terms correspond to high fre-
quency oscillations at the single and double frequency of
the phonon mode Ωi, respectively. They appear due to
quantum interference of excitations of different exciton-
polaron states, i.e.
due to quantum beats of exciton-
polarons.
These oscillations are superimposed on the
long-lived signal and decay with a shorter time τph. The
relative amplitude of the oscillatory signal and the long-
lived plateau allows one to measure the Huang-Rhys fac-
tor SHR.
Using Eqs. 4 and 5 for two independent phonon modes
with ℏΩ1 = 3.2 meV and ℏΩ2 = 5.1 meV we obtain
excellent agreement with the experimental data.
The
phonon frequencies are taken from the peak positions in
the Fourier spectrum of Fig. 2(c). The Huang-Rhys fac-
tors and phonon lifetimes are evaluated from the best fit
to the experimentally measured transients, see Fig. 2(b).
We emphasize that in contrast to previous reports [32–
36], our system demonstrates long-lived coherent dynam-
ics, which is attributed to an exceptionally long zero-
phonon exciton coherence (∼300 ps) and a relatively
long phonon lifetime (∼10 ps). Here, we stress that the
6
low-temperature regime with T = 2 K is essential, as the
coherence of the zero-phonon excitons rapidly vanishes
with increasing temperature. Furthermore, the optical
phonons each are represented by well-defined frequen-
cies due to their flat dispersion around the Γ-point, as
confirmed by the Raman spectrum in Fig. 2(d), mak-
ing CsPbI3 NCs embedded in a glass matrix particularly
interesting for investigating coupled exciton-phonon dy-
namics. The peak widths in the Raman spectrum can be
recalculated into phonon lifetimes using the following re-
lation τph = 2ℏ/∆ER [61], where ∆ER is the full width
at half maximum in energy units. This yields phonon
lifetimes of approximately 7 ps and 11 ps for the 3.2 and
5.1 meV modes, respectively. These values are in excel-
lent agreement with the decay times of the PE oscillatory
signals.
It should be noted that the Raman peaks in
Fig. 2(d) appear narrower than those in the FFT spectra
in Fig. 2(c), which originates from the difference between
the power spectral and the amplitude spectral represen-
tations, respectively.
The proposed polaron model (i) provides an intuitive
picture of the microscopic processes involved, (ii) has
a simple analytical solution, and (iii) accounts explic-
itly for the coherence decay through simple equations.
Equation (5) gives a result that closely matches that in
Refs. [32, 62] and is in effect equivalent within the accu-
racy limits of the models. Note that a direct comparison
between the results obtained by the two approaches is
not straightforward: Eq. (5) is obtained from the exact
solution of the Lindblad equations for a four-level sys-
tem, while Ref. 62 considers the quasi-classical evolution
of a complex system averaged over the phonon subsys-
tem.
For a detailed discussion and comparison of the
two approaches see e.g. Ref. 63.
V.
NC SIZE DEPENDENCE
Above, we discussed the PE dependence measured for
the fixed laser pulse photon energy of hν = 1.734 eV
and explained how the phonon parameters can be eval-
uated from the PE signal. However, the sample under
study contains NCs with different sizes which can be se-
lectively excited by tuning the laser photon energy. We
measured PE transients at photon energies in the range
of 1.72÷1.76 eV. The photon energy can be recalculated
into a NC diameter using the parameters from Ref. 39,
where the same sample was studied (sample #3). Fig-
ures 4(a) and
4(b) show the dependence of the evalu-
ated Huang-Rhys factors SHR and phonon lifetimes τph
as function of the NC diameter a, for phonon modes with
energies ℏΩ1 = 3.2 meV and ℏΩ2 = 5.1 meV. We note
that the frequencies of these modes do not depend on a
within the accuracy of the experiment. The Huang–Rhys
factors for both phonon modes exhibit a clear size de-
pendence, increasing with decreasing NC size.
Corre-
spondingly, the phonon lifetimes decrease with decreas-
ing a.
An increase in the electron–phonon coupling
|0>
|0'>
|X'>
|X>
(b)
γph
γph
γ0
SHRγ0
(1-SHR)2γ0
|Z>
|Y>
|X>
|0>
γ0
γ0
γ0
(a)
δ2
δ1
ħΩ
ħΩ
Configuration coordinate Q
Energy
EX
SHRħΩ
ħΩ
|0>
|0'>
|X'>
|X>
(c)
Figure 3. Level schemes. (a) Energy levels of the bright ex-
citon fine structure. (b) Energy level structure of the excitons
interacting with optical phonons. We assume this level struc-
ture for each phonon mode, independent of the exciton po-
larization. The probability of the diagonal transitions is pro-
portional to the Huang-Rhys factor SHR. γph is the phonon
decay rate and γ0 is the exciton decay rate, mostly determined
by its oscillator strength. (c) Sketch of adiabatic potential of
an exciton interacting with a particular phonon mode. The
solid parabolas represent the potential for phonons for the
ground state of the crystal (no exciton, bottom) and the ex-
cited state (with one exciton, top). The solid parabolas are
separated by the energy difference EX corresponding to the
energy of zero-phonon exciton transition. ℏΩdenotes the en-
ergy of the optical phonon.
strength with decreasing NC size was previously reported
for CdSe nanocrystals [64], PbS quantum dots [65], and
perovskite nanocrystals, both for optical and acoustic
phonons [16, 19]. However, to the best of our knowledge,
direct measurements of the associated phonon relaxation
times have not been reported.
It is also important to
note that, in contrast to previous studies, our PE exper-
iments selectively probe the coherent dynamics of each
individual phonon mode.
In Ref. 19, the dominant mechanism of electron-
phonon coupling is assigned to the optical deformation
potential [66]. In this case, the size dependence of the
interaction can be estimated from the phonon normal-
ization condition [67] which leads to SHR ∼a−3.
For
completeness, let us discuss the size dependence of the
interaction between charge carriers and optical phonons
7
for the Fr¨holich mechanism following Takagahara [68].
In the original work, the interaction of charge carriers
with phonons is rewritten as potential for an electron
in the field induced by the phonon mode polarization
P(r). This potential is found from the Poisson equation
∇2φ(r) = 4 π∇· P(r) which leads to
φ(re) =
Z
dr ∇· P(r)
|r −re| = −
Z
dr P(r) · ∇
1
|r −re| . (6)
From Eq. (6) the strength of electron-phonon interaction
∆e ∼a−1 which leads to Se
HR = ∆2
e/2 ∼a−2. This esti-
mate is valid also for excitons in the strong confinement
regime.
For weakly confined excitons, the change of exciton
energy ∆is proportional to the difference of the electro-
static potential for electron and hole φ(re) −φ(rh). It is
found to be the sum of Eq. (6) for electron and hole:
∆∼φ(re)−φ(rh) =
Z
dr P(r)·∇
1
|r −rh| −
1
|r −re|
.
(7)
In the weak confinement regime when the exciton Bohr
radius aB is small compared with the NC diameter aB ≪
a,
1
|r −rh| ≈
1
|r −re| + (re −rh) · ∇
1
|r −re| .
(8)
Substituting Eq. (8) into Eq. (7) we obtain
φ(re) −φ(rh) ≈
Z
dr P(r) · ∇
(re −rh) · ∇
1
|r −re|
.
As a result, in the weak confinement regime, the size
dependence of this expression is estimated as
∆∼φ(re) −φ(rh) ∝a3 · a−3/2 · aB
a3 ∝a−3/2.
Here, the first size factor comes from the integral, the
second one from the normalization of the phonon mode,
and the last one from the gradient of the expression in
the square brackets. As a result the Huang-Rhys factor
scales as SHR = ∆2/2 ∼a−3.
Thus, in the weak confinement regime both mecha-
nisms, the deformation potential and the Fr¨ohlich inter-
action, lead to similar dependences of the Huang-Rhys
factor on the NC size SHR ∼a−3. In the strong con-
finement regime, the Fr¨ohlich interaction is expected to
have a weaker size dependence, while the deformation
potential mechanism should show the same scaling. The
dashed lines in Fig. 4 represent fits of the experimental
data by the empirical relation SHR ∝a−3. The experi-
mental data in Fig. 4(a) even shows a steeper dependence
of the Huang–Rhys factor on NC size, with pronounced
deviation from the a−3 dependence, particularly for small
NCs. This observation suggests that further analysis is
required for smaller NCs to identify the mechanisms of
electron–phonon interaction in perovskite NCs.
0
0.02
0.04
0.06
0.08
0.1
0.12
SHR
ħΩ1 = 3.2 meV
ħΩ2 = 5.1 meV
10
10.5
11
11.5
12
0
5
10
15
20
NC diameter (nm)
τph (ps)
(a)
(b)
Figure 4.
NC size dependence.
Dependence of (a) the
Huang-Rhys factor, SHR, and (b) the phonon lifetime, τph,
for the two phonon modes with energies ℏΩ1 = 3.2 meV
(blue dots) and ℏΩ2 = 5.1 meV (red dots) on NC diameter a.
Dashed lines in (a) are fits proportional to a−3. Dashed lines
in (b) are guides to the eye.
VI.
DISCUSSION
To summarize, we have revealed a new regime of co-
herent exciton-phonon dynamics following resonant opti-
cal excitation with femtosecond pulses in an ensemble of
CsPbI3 nanocrystals embedded in a glass matrix. Quan-
tum beats between exciton-polaron states manifest them-
selves in two-pulse photon echo experiments as oscilla-
tions with high frequencies corresponding to two optical
phonon modes at 3.2 and 5.1 meV. A polarization re-
solved analysis of the photon echo signal has allowed us
to unambiguously isolate the quantum beat signatures of
exciton-polarons from slower beats arising from exciton
fine-structure splittings. This unique regime of exciton-
polaron quantum beats is established at the low temper-
ature of 2 K, where an exceptionally long zero-phonon
exciton coherence time is achieved (T2 ≈300 ps). The
presence of spectrally narrow optical phonon modes with
lifetimes of about 10 ps, combined with a relatively strong
exciton-phonon coupling (Huang-Rhys factors up to 0.1),
leads to a pronounced modulation of the photon echo sig-
nal at the phonon frequencies. Furthermore, the strong
size dependence of the Huang–Rhys factor makes it pos-
sible to tune the strength of the optical transitions be-
tween different polaron states, offering a route to control
the quantum dynamics for targeted applications.
We emphasize that the observed optical response
markedly differs from previous studies in, e.g., III–V self-
assembled quantum dots, where the dynamics are mainly
governed by exciton–acoustic phonon coupling [26, 69,
70].
There, the continuous phonon spectrum leads to
a partial loss of coherence within the first few picosec-
8
onds after pulsed excitation.
In our case no such be-
havior is observed in the temporal evolution of Ψ0(τ12)
[see Fig. 2(b)]. Instead, the coherent dynamics is dom-
inated by exciton–polaron states, i.e., interactions with
discrete optical phonon modes, which can be consistently
described within a four-level model for each phonon fre-
quency. The desired quantum evolution can be achieved
by tuning both the exciton–phonon coupling strength
and, most importantly, the phonon lifetime, which limits
the evolution of the pure quantum state following res-
onant excitation with a short optical pulse.
Notably,
the phonon lifetime exhibits a pronounced dependence on
nanocrystal size. For the optical phonon mode with an
energy of 5.1 meV, we observe an almost twofold increase
of τph as the size increases from 10 to 12 nm. In layered
perovskites, phonon lifetimes on the order of ∼10 ps were
reported [71], with values extended up to 75 ps [72], ap-
proaching the record-high ∼200 ps observed in bulk ZnO
[61].
Therefore, the strong size dependence of phonon
lifetimes, along with reports of extended phonon coher-
ence in layered perovskites, gives a promising outlook for
long-lived and tunable coherent control of the quantum
dynamics in perovskite nanocrystals.
VII.
METHODS
Samples. The studied CsPbI3 nanocrystals embed-
ded in fluorophosphate Ba(PO3)2-AlF3 glass were syn-
thesized by rapid cooling of a glass melt enriched with
the components needed for the perovskite crystallization.
Details of the method are given in Refs. [38, 73]. The
data on NC sizes evaluated from scanning transmission
electron microscopy can be found in the supporting in-
formation section S5 in Ref. [39]. Samples of fluorophos-
phate (FP) glass with the composition 35P2O5–35BaO–
5AlF3–10Ga2O3–10PbF2–5Cs2O (mol.
%) doped with
BaI2 were synthesized using the melt-quench technique.
The glass synthesis was performed in a closed glassy car-
bon crucible at the temperature of T = 1050◦C. Techno-
logical code of the studied sample is EK8. More infor-
mation on its optical and spin-dependent properties can
be found in Refs. 39 and 74.
The sample is placed in the variable temperature insert
of a bath cryostat and kept in superfluid helium at the
temperature of 2 K. The low energy exciton optical tran-
sitions in nanocrystals are located in the spectral range of
1.7–1.8 eV as follows from the photoluminescence spec-
trum, see Fig. 1(a).
Photon echo. Two-pulse photon echoes were mea-
sured using a transient four-wave mixing technique with
heterodyne detection [44–47].
We used a Ti:Sapphire
laser as source of optical pulses with duration of ∼100 fs
at the repetition rate of 75.75 MHz. All optical experi-
ments were carried out at photon energies below 1.76 eV.
The sample is excited resonantly by a sequence of two
pulses separated in time by the delay τ12, giving rise to
the formation of a four-wave mixing signal with electric
field amplitude EFWM(t). The pulses are focused into a
spot of about 200 µm diameter using a 0.5-m parabolic
aluminum mirror. The incidence angles of the pulses are
close to normal and equal to approximately 1/50 rad and
2/50 rad, corresponding to the in-plane wavevectors k1
and k2. An additional reference pulse is used for detec-
tion and its delay with respect to the first pulse is given
by the time τref. The delay times τ12 and τref are con-
trolled using mechanical delay lines. The four-wave mix-
ing signal is measured in transmission geometry in the
direction ≈3/50 rad, which corresponds to the phase-
matching direction 2k2 −k1. Optical heterodyne detec-
tion is used to resolve the enhanced signal in time. By
mixing with a relatively strong reference pulse and scan-
ning τref, the temporal profile of the photon echo pulse
is measured.
Polarization sensitive measurements are performed
by
selecting
proper
polarization
configurations
(H-
horizontal or V-vertical) for each of the beams using
Glan-Thompson prism in conjunction with half-wave
plates.
In the detection path we also select the linear
polarization by setting the polarization of the reference
beam. We adopt a three-letter notation for the PE polar-
ization configuration. The first two symbols correspond
to the polarizations of the pulses exciting the sample,
and the third symbol corresponds to the detection po-
larization. For example, in the HHH configuration, the
polarizations of both excitation pulses and the detection
are horizontal (H). In the HVH configuration, the sec-
ond excitation pulse is vertically (V) polarized, while the
first pulse and detection are horizontally polarized. HHH
is accordingly termed co-linearly polarized (∥) and HVH
cross-linearly polarized (×) in Fig. 2. We note that rota-
tion of the sample does not lead to changes in the signal,
i.e. the orientation of the sample is not important.
Raman spectroscopy.
The Raman scattering sig-
nal was excited by a single-frequency Ti:Sapphire laser.
The laser beam with the power of about 1 mW was
focused on the sample to a spot with a diameter of
about 300 µm. The scattered light was analyzed by a
Jobin-Yvon U1000 double monochromator with 1-meter
focal length, allowing the high resolution of 0.2 cm−1
(0.024 meV). The Raman signal was detected by a cooled
GaAs photomultiplier and conventional photon-counting
electronics. The Raman spectra were measured for co-
polarized linear polarizations of excitation and detection.
VIII.
ACKNOWLEDGEMENTS
The authors are thankful to I. A. Yugova and N.
E. Kopteva for fruitful discussions.
We acknowledge
the financial support by the Deutsche Forschungsge-
meinschaft:
A.V.T., M.A.H, S.G., M.O.N and I.A.A
(project AK40/13-1, no. 506623857) and D.R.Y. (project
YA65/28-1, no.
527080192).
The work of S.V.G. was
supported by the NSF through DMR-2100248. A.V.T,
E.V.K. and M.S.K. acknowledge the Saint-Petersburg
9
State University (Grant No. 125022803069-4).
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S1
Supplementary Information:
Quantum beats of exciton-polarons in CsPbI3 perovskite nanocrystals
S1.
FINE STRUCTURE OF EXCITON
To take into account the fine structure of the exciton, we assume that the three-fold bright exciton state is split
into mutually orthogonal linearly polarized components. For the photon echo calculations the following procedure
is used: the dynamics is calculated in the eigenbasis of NC {|0⟩, |x′⟩, |y′⟩, |z′⟩} then the equations for the density
matrix components are rotated to the laboratory frame {|0⟩, |x⟩, |y⟩, |z⟩}. The 1st pulse is assumed to be polarized
along z axis, the 2nd pulse at time τ is either polarized along z or x axis for HHH and HVH schemes respectively.
The polarization along z is detected at the photon echo time 2τ. The orientation of a nanocrystal in the laboratory
reference frame is described by three Euler angles α, β, and γ. The rotation matrix relating coordinates in xyz and
x′y′z′ bases is [75]:
R(α, β, γ) =
cαcβcγ −sαsγ −cαcβsγ −sαcγ cαsβ
sαcβcγ + cαsγ −sαcβsγ + cαcγ sαsβ
−sβcγ
sβsγ
cβ
,
(S1)
here and below in this section we use sξ ≡sin ξ, cξ ≡cos ξ (ξ = α, β, γ) to shorten the notation.
In the eigenbasis of NC, the dynamics is given by (we skip common factor e−γt):
ρ0x′ = e−t/T2eiω1tρ0
0x′ ,
ρ0y′ = e−t/T2eiω2tρ0
0y′ ,
ρ0z′ = e−t/T2eiω3tρ0
0z′ ;
(S2)
ρx′0 = e−t/T2e−iω1tρ0
x′0 ,
ρy′0 = e−t/T2e−iω2tρ0
y′0 ,
ρz′0 = e−t/T2e−iω3tρ0
z′0 .
(S3)
Here ℏωi (i = 1, 2, 3) are the three exciton energies for excitons linerly polarized along x′, y′, z′ respectively. We
assume that the diplole matrix element (and, consequently, radiative decay time) as well as the decoherence time T2
is the same for all three exciton states. After 1st pulse the components of density matrix are
ρ1+
0z = 1 ,
ρ1+
0y = ρ1+
0x = 0 .
(S4)
After 2nd pulse we have two options: for HHH scheme
ρ2+
z0 = ρ2−
0z ,
ρ2+
y0 = ρ2+
x0 = 0 .
(S5)
and for HVH scheme
ρ2+
x0 = ρ2−
0x ,
ρ2+
y0 = ρ2+
z0 = 0 .
(S6)
Rotating the dynamics equations (S2) into laboratory coordinate frame using (S1) gives rather lengthy equations for
the amplitude of the signal:
A∥(α, β, γ) ∝ρz0 =
c4
αs4
β + s4
αs4
β + c4
β
+ 2s2
αc2
αs4
β cos(ω1 −ω2)τ + 2c2
αs2
βc2
β cos(ω1 −ω3)τ + 2s2
αs2
βc2
β cos(ω2 −ω3)τ
e−2τ/T2
(S7)
A×(α, β, γ) ∝ρz0 =
c2
αs2
β(cαcβcγ −sαsγ)2 + s2
αs2
β(sαcβcγ + cαsγ)2 + s2
βc2
βc2
γ
+ 2sαcαs2
β(cαcβcγ −sαsγ)(sαcβcγ + cαsγ) cos(ω1 −ω2)τ −2cαs2
βcβcγ(cαcβcγ −sαsγ) cos(ω1 −ω3)τ
−2sαs2
βcβcγ(sαcβcγ + sαsγ) cos(ω2 −ω3)τ
e−2τ/T2
(S8)
The result should be averaged over directions using:
⟨A⟩ang ≡
1
8π2
Z 2π
0
dα
Z π
0
sin βdβ
Z 2π
0
dγ A(α, β, γ)
(S9)
S2
Assuming the constant energies of three exciton components, but the arbitrary direction of the NCs, the amplitude
of the echo is proporitional to
A∥
ang ∝
6
5 + 4
15
X
i>j
cos(ωi −ωj)τ
e−2τ
T2 ,
(S10)
⟨A×⟩ang ∝
2
5 −2
15
X
i>j
cos(ωi −ωj)τ
e−2τ
T2 .
(S11)
Now we also take into account the possible deviation of exciton energies. The absolute value of the exciton energy is
not important for the phonon echo, only the splittings. E.g., the amplitude averaged over distribution of splittings
between 1st and 2nd levels ω12 = ω1 −ω2 is given by
⟨A⟩ω12 =
Z
W(ω12)A(ω12, ω13, ω23 = ω12 −ω13)dω12
(S12)
Assuming the distribution function of the splittings to be Lorentzian,
W(ω) = t∗
i
π
1
(ω −δi)2 +
√
2/(t∗
i )2 ,
(S13)
and that all three splittings are not correlated, the PE signal averaged over both the angles and the splittings
distribution is given by
A∥
∝
18
15 + 4
15
X
i=1,2,3
cos(δiτ)e
−2τ
t∗
i
e−2τ
T2 ,
(S14)
⟨A×⟩∝
6
15 −2
15
X
i=1,2,3
cos(δiτ)e
−2τ
t∗
i
e−2τ
T2 ,
(S15)
where the parameters t∗
i work as an effective signal dephasing times and δi are the average splittings. An interesting
result is that even though the PE signal is stable with respect to the level disorder, the disorder of its fine structure
gives the contribution to the additional decay, not of the full signal, but of its oscillating components.
S2.
MODEL OF PHOTON ECHO OF EXCITON-POLARON STATES.
In this section we outline the derivation of PE signal in the polaron model. We assume the exciton-polaron with
relatively small Huang-Rhys factor which allows to perform calculations in four (six with account on light polarization)
level scheme.
S2.A.
Scheme of the levels and dynamics of the system
We consider the four-level system shown in Fig. 3b. There are: ground level |0⟩, statwe with one phonon |0′⟩,
polaron state |X⟩, and excited state of polaron |X′⟩. The energies of corresponding states are 0, ℏΩ, ωX, ωX + ℏΩ.
In the calculations below we assume that spectral width of the laser pulse covers at least |X⟩and |X′⟩levels, this is
in contrast to standard approximations for three-level atoms [76].
Due to polaron formation, allowed transitions are not only X →0 (with probability γ0) and X′ →0′ (with
probability γ1), but also X →0′ and X′ →0 (in this model, probability is the same γ′). In polaron model [59], the
transition probability is proportional to product of dipole matrix element |d|2 and the overlap of oscillator functions
shifted due to exciton formation which leads to extra factor
γ0 ∝|d|2e−2SHR
(S16a)
γ′ ∝|d|2SHRe−2SHR
(S16b)
γ1 ∝|d|2(1 −SHR)2e−2SHR
(S16c)
S3
where d is the dipole matrix elements of the optical transition, and we defined Huang-Rhys factor as SHR = ϵpol/(ℏΩ)
(ϵpol is the polaron formation energy). Note that in this approximation polaron formation energy should be small
which means that γ′ ≪γ0, γ1.
In addition, we assume that the phonon may be emitted which leads to transitions 0′ →0 and X′ →X with the
same probability γph.
To trace the polarization of the signal, we need to consider two polarization states of exciton, from now on the
enumeration 0 and 0′ are ground state and state with phonon, 1(2) and 1′(2′) are right(left) circularly polarized states
of exciton(polaron) and first oscillator level of exciton(polaron), making the model effectively six-level. Still, we call
the model four-level as here we neglect the splittings between levels (as the role of the fine structure is excluded, see
Sec. S1) and we need it only to demonstrate that the phonon structure does not change the polarization structure of
the signal. Below, we extensively omit states 2 and 2′ when equations are same.
The Lindblad equation, with all levels considered is:
dρ
dt = −i
ℏ[H0, ρ] + Ld[ˆρ] + Γ ⊙ˆρ .
(S17)
For convenience and compactness of the notation, with the use of ⊙for the Hadamard product (element-wise product
or Schur product).
The Hamiltonian is
H0 =
0
0
0
0
0
0
0 ℏΩ
0
0
0
0
0
0
ℏωX
0
0
0
0
0
0
ℏωX
0
0
0
0
0
0
ℏωX + ℏΩ
0
0
0
0
0
0
ℏωX + ℏΩ
(S18)
and the “inflow”diagonal part of superoperator is
Ld[ˆρ] =
γphρ0′0′ + γ0ρX + γ′ρX′
0
0
0
0 0
0
γ1ρX′ + γ′ρX
0
0
0 0
0
0
γphρ1′1′
0
0 0
0
0
0
γphρ2′2′ 0 0
0
0
0
0
0 0
0
0
0
0
0 0
(S19)
where for convenience we introduced ρX = ρ11 + ρ22 and ρX′ = ρ1′1′ + ρ2′2′, and the matrix in the “outflow” part is
Γ = −1
2
0
γph
γ′
0
γ′
0
γph + γ′
1
γph + γ′
1
γph
2γph
γph + γ′
0
γph + γ′
0
2γph + γ′
1
2γph + γ′
1
γ′
0
γph + γ′
0
2γ′
0
2γ′
0
γph + γ′
1 + γ′
0 γph + γ′
0 + γ′
1
γ′
0
γph + γ′
0
2γ′
0
2γ′
0
γph + γ′
1 + γ′
0 γph + γ′
0 + γ′
1
γph + γ′
1 2γph + γ′
1 γph + γ′
0 + γ′
1 γph + γ′
0 + γ′
1
2γph + 2γ′
1
2γph + 2γ′
1
γph + γ′
1 2γph + γ′
1 γph + γ′
0 + γ′
1 γph + γ′
0 + γ′
1
2γph + 2γ′
1
2γph + 2γ′
1
(S20)
where for convenience we defined
γ′
0 = γ0 + γ′ ,
γ′
1 = γ1 + γ′ .
(S21)
S2.B.
Effect of the pulse on density matrix
Now we consider light-mediated transitions in the system shown in Fig. 3b. This is again generalization of the
two-level system, check e.g. chapters 2, 6 and 10 of [76]. The pulse is approximated by the plane wave with the
smooth enveope E0(z, t) and electric field given by
E(r, t) = E0(k · r/k, t)ei(k·r−ωt) + c.c.
(S22)
Assuming the pulse duration is short compared with all coherence and relaxation times, the evolution of density
matrix (DM) during the pulse may be found from the Schr¨odinger equation
dρ
dt = −i
ℏ[H0 + V, ρ] ,
(S23)
S4
where, in rotating wave approximation,
V = ℏ
0
0
˜f ∗
+
˜f ∗
−
s · ˜f ∗
+ s · ˜f ∗
+
0
0
s · ˜f ∗
+ s · ˜f ∗
−
˜ff ∗
+
˜f ∗
+
˜f+
s · ˜f+
0
0
0
0
˜f−
s · ˜f−
0
0
0
0
s · ˜f+
˜f+
0
0
0
0
s · ˜f−
˜f−
0
0
0
0
,
(S24)
˜f± = dE±
0 e−iωt ≡f±e−iωt ,
(S25)
where d is the dipole matrix element for the transitions 1, 2 →0, s = √SHR, E±
0 are circular components of the
envelope (S22).
Solution of Eq. (S23) is surprisingly complex in general case, for two-level system it may be found in Ref. 76, but
we are interested in four-level system and the case of small pulse area when the full solution is surplus. When the
pulse area θ± (equal to f±tp for square pulse of duration tp) is small compared with Rabi frequency, the first order
correction to the density matrix may be obtained as
ρ+ = −i
ℏ
H0 + V, ρ−
tp ≡ˆ
Aρ−,
(S26)
where ρ−is the density matrix before pulse and ρ+ is the density matrix after pulse. Full solution of Eq. (S26) is too
lengthy for our needs, below we will only give the results relevant for the calculations.
For the PE we will need the action of the light pulse in the second order over pulse area. It may be shown that up
to second order, it may be calculated as
ρ+ = ˆ
A ˆ
Aρ−.
(S27)
Below we also give only part of this result relevant for photon echo calculations.
S2.C.
Photon echo
The signal in two circular polarizations is proportional [76] to the corresponding components of polarization vector
which is given by the following components of the density matrix:
P+ ∼ρ10 + ρ1′0′ + sρ1′0 + sρ10′ ,
(S28a)
P−∼ρ20 + ρ2′0′ + sρ2′0 + sρ20′ .
(S28b)
Note the extra factor s in Eq. (S28) which originates from the ratio of diplole matrix elements for phonon-assisted
transitions to the phonon-less transitions. Thus, we are interested in the following components of the density matrix
at times after second pulse: ρ10, ρ20, ρ1′0, ρ2′0, ρ10′, ρ20′, ρ1′0′, ρ2′0′.
a.
Density matrix after 1st pulse
The components of DM after 1st pulse are given by (S26) which leads to:
ρ1+
01 = iθ∗
1+ ,
ρ1+
01′ = isθ∗
1+ ,
ρ1+
02 = iθ∗
1−,
ρ1+
02′ = isθ∗
1−.
(S29)
b.
Relaxation of DM components after 1st impulse
One needs to trace the evolution of the following DM compo-
nents: ρ01, ρ01′, ρ02, ρ02′. Below we give results only for right polarization, dynamics of other components is the same.
Writing the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components:
ρ01 (t ∈(0, τ12)) = ρ1+
01 eiωXte−
γ′
0
2 t ,
(S30a)
ρ01′ (t ∈(0, τ12)) = ρ1+
01′ei(ωX+Ω)te−
γ′
1+γph
2
t ,
(S30b)
This leads to the following components of the DM at the time of 2nd pulse t = τ12
ρ2−
01 = ρ1+
01 εXΓ0 ,
ρ2−
01′ = ρ1+
01′εXεΓ1Γp ,
(S31)
where for convenience we defined
εX = eiωXτ12 , ε = eiΩτ12 , Γ0 = e−
γ′
0
2 τ12 ,
Γ1 = e−
γ′
1
2 τ12 ,
Γp = e−
γph
2 τ12 .
(S32)
Note that Γ0 and Γ1 are almost equal and defined by the exciton coherence time (under our assumptions half of its
lifetime). The difference between them is Γ1/Γ0 = e−γ1−γ0
2
τ12 = e
SHRγ0
2
τ12 which is much smaller than all typical
times in the system since SHR is small.
S5
c.
DM after 2nd pulse
From (S27) we may compute the components relevant for the PE signal after the 2nd
pulse:
ρ2+
10 =
θ2
2+(ρ2−
01 + sρ2−
01′) + θ2+θ2−(ρ2−
02 + sρ2−
02′)
(S33a)
ρ2+
10′ =
θ2
2+(sρ2−
01 + ρ2−
01′) + θ2+θ2−(sρ2−
02 + ρ2−
02′)
(S33b)
ρ2+
1′0 = s
θ2
2+(ρ2−
01 + sρ2−
01′) + θ2+θ2−(ρ2−
02 + sρ2−
02′)
(S33c)
ρ2+
1′0′ = s
θ2
2+(ρ2−
01′ + sρ2−
01 ) + θ2+θ2−(ρ2−
02′ + sρ2−
02 )
(S33d)
and
ρ2+
20 =
θ2+θ2−(ρ2−
01 + sρ2−
01′) + θ2
2−(ρ2−
02 + sρ2−
02′)
(S34a)
ρ2+
2′0 =
θ2+θ2−(sρ2−
01 + ρ2−
01′) + θ2
2−(sρ2−
02 + ρ2−
02′)
(S34b)
ρ2+
20′ = s
θ2+θ2−(ρ2−
01 + sρ2−
01′) + θ2
2−(ρ2−
02 + sρ2−
02′)
(S34c)
ρ2+
2′0′ = s
θ2+θ2−(ρ2−
01′ + sρ2−
01 ) + θ2
2−(ρ2−
02′ + sρ2−
02 )
(S34d)
d.
Relaxation of DM components after 2nd impulse
We we need to know the evolution of the DM components
which enter (S28). Below we give only components entering P+. Dynamics of components entering P−is the same.
Writing the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components:
ρ10 (t > t2) = ρ2+
10 e−iωX(t−t2)e−
γ′
0
2 (t−t2) ,
(S35a)
ρ1′0 (t > t2) = ρ2+
1′0e−i(ωX+Ω)(t−t2)e−
γ′
1+γph
2
(t−t2) ,
(S35b)
ρ10′ (t > t2) = ρ2+
10′e−i(ωX−Ω)(t−t2)e−
γ′
0+γph
2
(t−t2) ,
(S35c)
ρ1′0′ (t > t2) = ρ2+
1′0′e−iωX(t−t2)e−
γ′
1+2γph
2
(t−t2) ,
(S35d)
Which gives the DM components at the time of echo signal t −t2 = τ12:
ρP E
10 = ρ2+
10 ε∗
XΓ0 ,
ρP E
1′0 = ρ2+
1′0ε∗
Xε∗Γ1Γp ,
(S36a)
ρP E
10′ = ρ2+
10′ε∗
XεΓ0Γp ,
ρP E
1′0′ = ρ2+
1′0′ε∗
XΓ1Γ2
p .
(S36b)
The evolution of components corresponding to second circular polarization is the same.
e.
Contribution of different paths to PE
To get the final result, we substitude (S29) into (S31) and then the result
into (S33) and then from (S36) we get the DM at the time of PE. The amplitude of the signal in two polarizations is
then obtained from Eqs. (S28):
P+ ∼ρPE
10 + ρPE
1′0′ + sρPE
1′0 + sρPE
10′ ,
(S37a)
P−∼ρPE
20 + ρPE
2′0′ + sρPE
2′0 + sρPE
20′ .
(S37b)
Collecting all terms in the result we have for contributions to P+ from ρ10, ρ1′0′, ρ10′, ρ1′0 originating from the DM
components after 1st pulse given below in the beginning of each line:
ρ01,02 →ρPE
10
Γ2
0A+ ,
(S38a)
ρ01′,02′ →ρPE
10
s2εΓ0Γ1ΓpA+ ,
(S38b)
ρ01,02 →ρPE
10′
s2εΓ2
0ΓpA+ ,
(S38c)
ρ01′,02′ →ρPE
10′
s2ε2Γ0Γ1Γ2
pA+ ,
(S38d)
ρ01,02 →ρPE
1′0
s2ε∗Γ0Γ1ΓpA+ ,
(S38e)
ρ01′,02′ →ρPE
1′0
s4Γ2
1Γ2
pA+ ,
(S38f)
ρ01,02 →ρPE
1′0′
s2Γ0Γ1Γ2
pA+ ,
(S38g)
ρ01′,02′ →ρPE
1′0′
s2εΓ2
1Γ3
pA+ ,
(S38h)
where
A+ = θ2+
θ2+θ∗
1+ + θ2−θ∗
1−
.
(S39a)
S6
Note that the term (S38f) may be dropped as it exceeds precision of equations.
The contribution to P−from ρ20, ρ2′0′, ρ20′, ρ2′0 is the same as (S38) but instead of A+ the polarization dependence
is given by
A−= θ2−
θ2−θ∗
1−+ θ2+θ∗
1+
.
(S39b)
It is convenient to write both as
A± = θ2±A0 ,
A0 = θ2−θ∗
1−+ θ2+θ∗
1+ ,
(S40)
to see that the polarization dependence of the PE signal coincides with second impulse polarization while its amplitude
depends on respective polarization of first and second pulses.
As a result, the amplitude of the non-oscillating signal is
PP E,0
±
∼
Γ2
0 + SHRΓ0Γ1Γ2
p
A± .
(S41)
The amplitude of the signal oscillating at Ωis
PP E,Ω
±
∼SHRΓp
Γ0Γ1 (ε + ε∗) + ε
Γ2
0 + Γ2
1Γ2
p
A± .
(S42)
and of the signal oscillating at 2Ωis
PP E,2Ω
±
∼SHRΓ0Γ1Γ2
pε2A± .
(S43)
f.
Result
Let us first consider the general scheme of the polarization dependence of PE signal. Following [47],
we consider linearly polarized excitation pulses with second pulse rotated to angle ϕ and linearly polarized detection
with polarized rotated to ϕd. Then
θ1+ = θ1−= 1/
√
2 , θ2+ = eiϕ/
√
2 , θ2−= e−iϕ/
√
2 .
(S44)
Which gives A0 = cos ϕ and
A± = e±iϕ
√
2 cos ϕ .
(S45)
Amplitude of linearly-polarized heterodyne detection of main PE signal is proportional to
P 0
d =
ei(ϕ−ϕd) + e−i(ϕd−ϕ)
2
cos ϕ
= |cos (ϕ −ϕd) cos ϕ|
(S46)
We have two configurations: HHH and HVH. First is ϕ = ϕd = 0 and second is ϕd = 0, ϕ = π/2. This gives
P 0,HHH
d
= 1 ,
P 0,HV H
d
= 0.
(S47)
As long as polarization dependence on the excitation/detection angles (S46) is real (which is the case for HHH
and HVH configurations), the absolute value of product is the product of absolute values and signal is zero for HVH
scheme and for HHH scheme (we omit index HHH below) is:
PP E
d
= Γ2
0
h
1 + SHReΓΓ2
p
i
+ SHRΓp
h
eΓ (ε + ε∗) + ε(1 + eΓ2Γ2
p)
i
+ SHReΓΓ2
pε2
(S48)
where we introduced
eΓ = Γ1/Γ0 = eSHRγ0τ12 .
(S49)
Note that eΓ is 1 as compared with both, Γ0 and Γp.
To write explicitly the result for the PE amplitude, we use the fact that for t ≪1 and arbitrary complex ϵ:
|1 + tϵ| = 1 + tRe(ϵ) + t2
2 (Im(ϵ))2 + O
t3
.
(S50)
We keep second order terms to highlight the fact that even for single frequency of the complex amplitude, the absolute
value in (S48) will result in second order in SHR signal at double frequency.
S7
Using (S50) we may write the final result (keeping only terms to the first order in SHR)
PP E
d
= Γ2
0
n
1 + SHRΓp
h
eΓΓp +
1 + 2eΓ + eΓ2Γ2
p
cos(Ωτ12) + eΓΓp cos(2Ωτ12)
io
(S51)
Or, explicitly,
PP E
d
= e−γ′
0τ12
"
1 + SHRe−
γph
2 τ12
e−
γph−2SHRγ0
2
τ12
+
1 + 2eSHRγ0τ12 + e−(γph−2SHRγ0)τ12
cos(Ωτ12) + e−
γph−2SHRγ0
2
τ12 cos(2Ωτ12)
!#
,
(S52)
and, finally, with the help of Eqs.(S16,S21):
PP E
d
= e−(1+SHR)γ0τ12
"
1 + SHRe−(γph−SHRγ0)τ12
+ SHR cos(Ωτ12)e−
γph−2SHRγ0
2
τ12 e−SHRγ0τ12 + 2 + e−γphτ12
+ SHR cos(2Ωτ12)e−(γph−SHRγ0)τ12
#
≡Ψ0(τ12; Ω) .
(S53)
|
Quantum beats of exciton-polarons in CsPbI3 perovskite nanocrystals A. V. Trifonov,1, 2, ∗M. O. Nestoklon,1 M.-A. Hollberg,1 S. Grisard,1 D. Kudlacik,1 E. V. Kolobkova,3, 4 M. S. Kuznetsova,2 S. V. Goupalov,5 J. M. Kaspari,6 D. E. Reiter,6 D. R. Yakovlev,1 M. Bayer,1, 7 and I. A. Akimov1, † 1Experimentelle Physik 2, Technische Universit ̈at Dortmund, 44227 Dortmund, Germany 2Spin Optics Laboratory, St. Petersburg State University, Peterhof, 198504 St. Petersburg, Russia 3ITMO University, 199034 St. Petersburg, Russia 4St. Petersburg State 190013 St. Petersburg , Russia 5 39217 Mississippi, USA 6Condensed Matter Theory, ̈at Dortmund, 44227 Dortmund, Germany 7Research Center FEMS, Technische Universit ̈at Dortmund, 44227 Dortmund, Germany (Dated: October 17, 2025) Exciton-phonon interactions govern the energy level spectrum and thus the optical response in semiconductors. In this respect, lead-halide perovskite nanocrystals represent a unique system, for which the interaction with optical phonons is particularly strong, giving rise to a ladder of multiple exciton states which can be optically excited with femtosecond pulses. We establish a new regime of coherent exciton-polaron dynamics with exceptionally long coherence times (T2 ≈300 ps) in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix. Using transient two-pulse photon echo at 2 K temperature, we observe quantum beats between the exciton-polaron states. Within a four-level model, we directly quantify the exciton-phonon coupling strength through the Huang-Rhys factors of 0.05 ÷ 0.1 and 0.02 ÷ 0.04 for low-energy optical phonons with energies of 3.2 and 5.1 meV, respectively. The pronounced size dependence of both coupling strengths and phonon lifetimes offers a path to tune the optical transitions between polaron states and to tailor the coherent optical dynamics in perovskite semiconductors for solid-state quantum technologies. I. INTRODUCTION Lead halide perovskite semiconductors attract close attention due to their intriguing optical and electrical properties with huge potential for photovoltaic and light emitting applications [1, 2]. These materials provide a unique playground for polaron physics in the solid state governed by the interaction of electronic excitations with the phonons of the crystal lattice [3]. In particular, several studies indicate that low-energy optical phonons in halide perovskites have an important influence on the conductivity, while acoustic phonons have a minor effect [4, 5]. This is in contrast to conventional semiconductors such as GaAs, where the interaction of electrons with acoustic phonons is dominant at cryogenic temperatures [6, 7]. Furthermore, the carrier mobility is determined not by a few specific optical phonons, but by a large multitude of modes due to the complex phonon spectrum [8]. Perovskite nanocrystals (NC) belong to the class of quantum emitters with discrete energy level spectrum, large spectral tunability, and high quantum yield, which makes them attractive as single photon sources [9, 10]. At cryogenic temperatures, the elementary optical excitation in NCs is an exciton (electron-hole pair) which possesses long optical and spin coherence times up to hundreds of ps, comparable to its radiative lifetime [1113]. Long-lived coherent excitons are often considered as qubits and are appealing for possible applications ∗correspondence address: † correspondence address: in quantum communication [14]. However, the strong exciton-phonon interaction leads to formation of excitonpolarons which modifies the energy level spectrum. In particular, the photoluminescence spectra of single NCs show pronounced optical phonon replicas which confirm the substantial exciton-phonon coupling with HuangRhys factors in the range from 0.05 to 0.4 in lead halide perovskite NCs [15-19]. Recently, electron-phonon interaction dominated by optical phonons with a similar Huang-Rhys factor of about 0.4 was demonstrated in CsPbBr3 NCs [20] and other inorganic lead halide perovskite NCs [21] using Raman spectroscopy. Compared to InGaAs quantum dots [22, 23], these values are significantly larger and comparable to those reported in II-VI nanostructures [24]. A distinctive feature of perovskite nanocrystals is the existence of low-energy optical phonons with a strong coupling to phonons. This combination substantially modifies the exciton level spectrum in the vicinity of the ground state [25] and represents an attractive testbed for investigation of the quantum dynamics under resonant femtosecond excitation markedly different from previously studied systems. Time-resolved studies can provide direct access to the exciton dynamics. It is well established that scattering on incoherent phonons with a thermal distribution causes loss of exciton coherence [26, 27]. Yet, less is known about the quantum dynamics of exciton-polarons, i.e., the hybrid excitations of an exciton and a phonon. Recently, transient four-wave mixing and two-dimensional Fourier spectroscopy revealed the importance of coherent exciton-phonon coupling for the ground and excited (exciton and biexciton) states in lead halide perovskites [2816 Oct 2025 2 31]. However, most of these studies were limited to the case of short optical coherence of excitons (≤100 fs) and therefore reported on the coherent evolution of either the exciton or the ground electronic state modulated by the frequencies of phonons with much slower decoherence. In other words, the lack of optical coherence restricted the observations to the phonon dynamics, while the genuine coherent evolution of the coupled exciton-phonon system has remained inaccessible. This scenario was also documented for bulk semiconductors, nanostructures and organic molecules [32-36]. However, the opposite quantum dynamics limit, where pronounced quantum beats of exciton-polarons are expected, has not yet been uncovered. Furthermore, another general question arises, connected to the limitations of coherent control for resonant excitation with femtosecond pulses, since time-resolved studies reported quite short coherence times up to 20 ps in lead halide perovskites [28, 37], in contrast with photoluminescence data of single NCs [11-13]. In this work we establish a new regime of coherent dynamics governed by exciton-polarons in NCs with an exceptionally long coherence time of the zero-phonon exciton (T2,X ≈300 ps) on the one hand and a discrete spectrum of spectrally narrow optical phonon modes on the other hand. The regime manifests in long-lived excitonpolaron quantum beats, which are detected by transient two-pulse photon echoes in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix at low temperature, T = 2 K. The oscillatory signal is dictated by optical phonons with low frequencies corresponding to 3.2 and 5.1 meV energy, and can be described by the solution of the Lindblad equations for a set of four-level systems that include a phonon ladder in the ground and excited exciton-polaron states. Different sizes, ranging from 10 to 12 nm, were selectively addressed within the same ensemble of nanocrystals by varying the laser photon energy. From the relative amplitude of oscillations we directly evaluate the strength of the exciton-phonon coupling with Huang-Rhys factors of 0.06 and 0.02 for optical phonons with 3.2 meV and 5.1 meV energy, respectively. Correspondingly, the electron-phonon coupling strength doubles reaching values of 0.12 and 0.04 as the NC size is reduced from 12 to 10 nm. The theoretical analysis of the size dependence shows that both the deformation potential and the Fr ̈ohlich mechanism of electron-phonon interaction qualitatively agree with the experimental data in the regime of weak exciton confinement. Our study demonstrates that the dynamics of a pure quantum state following resonant optical excitation requires consideration of polaron effects, whose strength and coherence can be controlled by composition and size of the NCs. This opens up new possibilities for on-demand coherent control and coherent phonon generation in lead-halide perovskite NCs. II. PHOTON ECHO AND LONG COHERENCE OF ZERO-PHONON EXCITON The ensemble of CsPbI3 NCs is synthesized in a fluorophosphate glass matrix by rapid cooling of a glass melt enriched with the materials needed for the perovskite crystallization. The details of synthesis are given in Ref. [38]. The NC size is about 9 ÷ 13 nm in diameter (see Fig. S6 in Ref. 39). The low-temperature photoluminescence (PL) spectrum is dominated by a 60 meV broad band centered around 1.75 eV, as shown in Fig. 1(a). The broadening originates from fluctuations of the NC size, which we estimate to be in the order of 20%. A single NC is schematically illustrated in the inset of Fig. 1(b). Note that the cubic crystal structure is shown for illustrative purposes as the actual crystallographic phase depends on the growth conditions and the size of the nanocrystals [40-43]. We perform transient four-wave mixing (FWM) experiments in transmission geometry as shown schematically in Fig. 1(b). The NCs are resonantly excited with a sequence of two 100 fs laser-pulses with photon energies hν 50 ps is due to the zero-phonon exciton transition with long-lived optical coherence time T2 and exciton recombination lifetime T1. If the co3 1.65 1.7 1.75 1.8 1.85 0 0.5 1 Energy (eV) PL intensity (arb. units) τ12 τref Lead Iodine Caesium 1 2 3 4 5 6 7 -0.5 0 0.5 0.5 1 1.5 2 τ12 (ps) τref - 2τ12 (ps) E * FWM E ref (arb. units) (a) (b) (c) 0 50 100 1 234 0.1 τref (ps) E * FWM E ref (arb. units) 1.75 eV 1.74 eV 1.724 eV T2 = 276 ps (d) 10 12 8 9 15 NC diameter (nm) T2 = 330 ps T2 = 230 ps 1 2 ref PE k2 k1 2k2 - k1 T = 2K Figure 1. Photon echoes in CsPbI3 nanocrystals. (a) Typical photoluminescence (PL) spectrum for excitation with photon energy of 2.33 eV. Vertical dashed line indicates the highest photon energy used in the four-wave mixing FWM experiment. The scale on top corresponds to the NC diameter. (b) Schematic representation of the experimental geometry, where the laser pulses hit the sample with wave vectors k1, k2. The photon echo is detected along the direction 2k2 -k1 using a heterodyning technique by overlapping it with a reference pulse. Inset shows schematically a CsPbI3 perovskite nanocrystal. (c) Two-dimensional plot of the FWM electric field amplitude E∗ FWM as function of delay between the two pulses τ12 and the reference time τref. The photon echo (PE) signal forms at the time τref = 2τ12. The amplitude of the PE shows oscillations during the initial evolution when τ12 is scanned. The signal is recorded in linearly co-polarized configuration. Photon energy hν = 1.736eV. (d) Decay of two-pulse photon echo amplitudes for excitation with different photon energies. Black dashed lines are fits with exponential functions, from which the labeld exciton coherence times T2 are extracted. Temperature T = 2 K. herent dynamics would be governed only by the population decay we would expect to the T2 = 2T1 relation to hold, which typically holds for excitons in selfassembled InGaAs quantum dots at T = 2 K [49]. Here, we observe a different situation, where elastic scattering processes ("pure dephasing") with a decay constant of Tp = [1/T2 -1/(2T1)]-1 = 330 ps mainly govern the exciton coherence. Nevertheless, to the best of our knowledge, the homogeneous linewidth of the zero-phonon exciton of Γ2 = 2ħ/T2 = 4.8 μeV demonstrated here has a record low value for perovskite nanocrystals [11-13]. III. COHERENT EXCITON-PHONON DYNAMICS As follows from Fig. 1(c) for short delay times τ12 ≲ 10 ps, oscillations of the PE signal with multiple frequencies are observed. For deeper insight into the origin of the oscillations, we analyze polarization-resolved PE signals. These results are summarized in Fig. 2. We employ two configurations, where the first and second pulses are linearly co- or cross- polarized, while the detection polarization in all cases coincides with that of the first pulse (see Methods). The PE amplitudes as function of τ12 in co-(∥) and cross-(×) polarized configurations are shown in Fig. 2(a). The oscillations can be grouped into two categories: slow oscillations with a period of 1-10 ps and fast oscillations with a period of less than 1 ps. It is evident that the fast oscillations in the co-polarized configuration are in phase and nearly identical in both curves. We will show below that these oscillations correspond to quantum beats between exciton-polaron states. By contrast, the slow oscillations possess opposite phases. The co-polarized A∥-signal starts maximum value, while the A×-signal starts from zero. Subsequently, the minimum of the slow oscillations in ∥- configuration around τ12 = 5 ps corresponds to a maximum in the ×-configuration. This antiphase behavior is related to quantum beats between the orthogonally polarized bright exciton states [37]. In perovskite nanocrystals, the bright exciton fine structure comprises three linearly polarized states [15, 50], split by the energies δ1 and δ2 as shown in Fig. 3(a). This energy level scheme is similar to the fine structure splitting in self-assembled quantum dots [51, 52] and in II-VI colloidal NCs [53, 54]. In order to isolate the oscillations due to the fine structure splitting in the data, we model the expected polarization dependence of the PE signal, accounting also for the random orientation of the nanocrystals (for details see Sec. S1 of the Supplementary Information). We obtain the following equations for the amplitude of the phonon 4 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 τ12 (ps) PE ampl. (arb. units) co-linear (||) cross-linear (x) 0 10 20 30 40 0 0.5 1 1.5 τ12 (ps) Amplitude (arb. units) ρ Σ Fit 0 2 4 6 8 0 2 4 6 8 10 Energy (meV) Intensity (arb. units) 0 2 4 6 8 0 0.5 1 1.5 2 Energy (meV) FFT (arb. units) ρ Σ Fit (a) (b) (c) (d) Raman (x2) Figure 2. Quantum beats of exciton-polarons. (a) Initial range of the PE dynamics in the co-linear (∥) and cross-linear (×) polarization configurations shown with blue and red lines, respectively. Photon energy hν = 1.746 eV. The amplitude of the × signal is multiplied by two for clarity. (b) Dynamics of ρ (black) and Σ (green) as defined by Eqs. 2 and 3, respectively. The dashed red curve is a fit using the polaron model with Eq. 5 with the following parameters for two phonon modes: ħΩ1 = 3.2 meV, SHR1 = 0.097, τph1 = 5.1 ps; ħΩ2 = 5.1 meV, SHR2 = 0.032, τph2 = 10 ps. (c) Fast Fourier transform (FFT) amplitude spectra of ρ (black), Σ (green), and the fit curves in panel (b). Vertical dashed lines mark the peaks corresponding to optical phonons. (d) Raman spectrum measured at photon energy hν = 1.734 eV. Vertical dashed lines indicate the positions of peaks corresponding to the optically active optical phonon modes that couple most strongly to the exciton. echo signal in two polarization configurations: A∥= " 18 15 + X i 4 15 cos (δiτ12) e -2τ12 t∗ i # Ψ0(τ12) , (1a) A× = " 6 15 - X i 2 15 cos (δiτ12) e -2τ12 t∗ i # Ψ0(τ12) , (1b) where the index i = 1, 2, 3 corresponds to the beats between the three optical transitions (δ3 = δ1 + δ2), and t∗ i is the dephasing time of the beats caused by the dispersion of the splitting energies ħδi in the ensemble. The common factor Ψ0(τ12) is the same for both polarization configurations and will be discussed below, see also Supplementary Information S1. Following Eqs. (1) it is possible to distinguish between quantum beats related to the exciton fine structure and other polarization insensitive contributions to the PE signal by introducing the polarization contrast ρ = A∥-3A× A∥+ 2A× , (2) and the polarization sum Σ = A∥+ 2A× 2 = Ψ0(τ12) . (3) It follows that the temporal evolution of ρ exhibits oscillations at frequencies corresponding to the energy splittings δi of the exciton fine structure and is independent of Ψ0(τ12). In contrast, the polarization sum Σ is given by the intrinsic coherent dynamics Ψ0(τ12) only. Figure 2(b) shows the dynamics of these quantities, calculated from the data in Fig. 2(a). In full accord with our expectations we obtain that the high-frequency oscillations are absent in the dynamics of ρ, while the low-frequency oscillations are still present. On the other hand, the low-frequency oscillations vanish in the Σ transient in contrast to the high-frequency components. This becomes even more clear from the fast Fourier transform (FFT) spectra of ρ and Σ shown in Fig. 2(c). From the fit of the ρ transient we evaluate δ1 = 0.25 meV and δ2 = 0.55 meV, which are in agreement with the exciton fine structure splitting evaluated by two-dimensional Fourier spectroscopy on CsPbI3 NCs of similar size with a diameter of about 9 nm [37]. A detailed discussion of the fine exciton structure combining pump-probe and PE studies will be published elsewhere. In what follows, we focus on the dynamics of Ψ0(τ12), which is independent of the spin level structure of the exciton. The FFT spectrum of Σ, shown by the green line in Figure 2(c), exhibits several spectrally narrow features. These features are consistent with the peaks in the Ra5 man spectrum in Figure 2(d). The latter originates from light scattering on optically active phonons. The features at energies below 0.5 meV are attributed to confined acoustic phonons [55]. Below we focus on the two most prominent features marked by the vertical dashed lines with energies of ħΩ1 = 3.2 meV and ħΩ2 = 5.1meV, corresponding to the energies of optically active optical phonons in the vicinity of the Γ point. Indeed optical phonon modes with energies close to 3 and 5 meV in CsPbI3 were observed in Ref. [21] and calculated [8]. Optical phonons with similar energies were also detected by Raman spectroscopy in other lead halide perovskites such as FAPbI3 [56, 57], CsPbBr3 [19], and CsPbCl3 [17, 58]. Thus, we conclude that the high frequency THz oscillations in the two-pulse coherent optical response are due to the coherent evolution of the coupled, hybridized excitonphonon system. IV. EXCITON-POLARON QUANTUM BEATS In order to describe the quantum dynamics of the coupled exciton-phonon system we consider the structure of the optically excited energy levels in a single NC, which requires the involvement of polaron states. To that end, we introduce the four-level scheme shown in Fig. 3(b) which comprises the ground state |0⟩, the phonon mode |0′⟩, the exciton polaron state |X⟩("exciton") and the first excited exciton polaron state |X′⟩(see details in the Supplementary Information S2 S2.A). We neglect the exciton fine structure splitting to concentrate on the derivation of Ψ0(τ12) in Eq. (1). Initially, we consider the exciton coupling to a single phonon mode in a NC. At the next stage, to compare with the experimental data, we perform summation over different phonon modes Ωi where we neglect the interaction between them Ψ0(τ12) = X i Ψ0(τ12; Ωi) . (4) The optical transitions between all energy levels are allowed using the same polarization. In the polaron model [59], the transition probability amplitude is proportional to the product of the dipole matrix element and the overlap of the wavefunctions associated with vibronic modes shifted due to polaron formation (see Fig. 3(c) and Eq. S16), which depends on the strength of the excitonphonon interaction given by the Huang-Rhys factor SHR (Condon principle). The pulse duration is assumed to be short as compared with all other characteristic times in the system, i.e. we can assume the δ-pulse limit. This is justified because the laser pulse duration is τd ≈100 fs (see Methods) and the fit to experimental data gives relaxation times in the order of 10 ps. The spectral width of the laser pulse ∼h/τd is about 15 meV, which is larger than the phonon energies ħΩi. Therefore, we assume that all four transitions between states |0⟩, |0′⟩and |X⟩, |X′⟩are covered by the spectral width of the laser. Note that a not too large Huang-Rhys factor means that states involving multiple phonons (i.e. 2ħΩ1, etc.) may be neglected. However, the spectral width of the laser pulse is large enough to cover them as well. We neglect the possible difference of phonon energy in the ground and polaron state as we do not find corresponding frequency changes in our experimental data and in previous reports in other perovskites [56, 57]. Similar assumptions were made in the model developed for desscribing 2D spectroscopy in Ref. 60. In the polaron model, we trace the evolution of the density matrix components which contribute to the PE. The result arises from the coherent off-diagonal terms and all levels contribute to the result in first order in the Huang-Rhys factor SHR, which quantifies the excitonphonon interaction. Solution of the Lindblad equation in Supplementary Information S2 gives the following expression for the amplitude of the phonon echo signal from a NC with single phonon mode Ωi Ψ0(τ12; Ωi) = e-(1+SHR)γ0τ12 " 1 + SHRe-(γph-SHRγ0)τ12 +SHR cos (Ωiτ12)eγph-2SHRγ0 2 τ12h 2+e-SHRγ0τ12+e-γphτ12i + SHR cos (2Ωiτ12)e-(γph-SHRγ0)τ12 # . (5) Here T2,0 = 1/γ0 is the coherence time associated with the zero-phonon optical transition, and τph = 1/γph is the lifetime of the optical phonon. In the simplified model we do not include other sources of decoherence. The analytical expression fully supports the experimental observations for Σ = Ψ0(τ12) shown in Fig. 2(b). The first term in Eq. 5 corresponds to the long-lived zerophonon coherence which decays exponentially with the time T2 = T2,0/(1 + SHR) ∼300 ps, as evaluated in section II. The last two terms correspond to high frequency oscillations at the single and double frequency of the phonon mode Ωi, respectively. They appear due to quantum interference of excitations of different excitonpolaron states, i.e. due to quantum beats of excitonpolarons. These oscillations are superimposed on the long-lived signal and decay with a shorter time τph. The relative amplitude of the oscillatory signal and the longlived plateau allows one to measure the Huang-Rhys factor SHR. Using Eqs. 4 and 5 for two independent phonon modes with ħΩ1 = 3.2 meV and ħΩ2 = 5.1 meV we obtain excellent agreement with the experimental data. The phonon frequencies are taken from the peak positions in the Fourier spectrum of Fig. 2(c). The Huang-Rhys factors and phonon lifetimes are evaluated from the best fit to the experimentally measured transients, see Fig. 2(b). We emphasize that in contrast to previous reports [3236], our system demonstrates long-lived coherent dynamics, which is attributed to an exceptionally long zerophonon exciton coherence (∼300 ps) and a relatively long phonon lifetime (∼10 ps). Here, we stress that the 6 low-temperature regime with T = 2 K is essential, as the coherence of the zero-phonon excitons rapidly vanishes with increasing temperature. Furthermore, the optical phonons each are represented by well-defined frequencies due to their flat dispersion around the Γ-point, as confirmed by the Raman spectrum in Fig. 2(d), making CsPbI3 NCs embedded in a glass matrix particularly interesting for investigating coupled exciton-phonon dynamics. The peak widths in the Raman spectrum can be recalculated into phonon lifetimes using the following relation τph = 2ħ/∆ER [61], where ∆ER is the full width at half maximum in energy units. This yields phonon lifetimes of approximately 7 ps and 11 ps for the 3.2 and 5.1 meV modes, respectively. These values are in excellent agreement with the decay times of the PE oscillatory signals. It should be noted that the Raman peaks in Fig. 2(d) appear narrower than those in the FFT spectra in Fig. 2(c), which originates from the difference between the power spectral and the amplitude spectral representations, respectively. The proposed polaron model (i) provides an intuitive picture of the microscopic processes involved, (ii) has a simple analytical solution, and (iii) accounts explicitly for the coherence decay through simple equations. Equation (5) gives a result that closely matches that in Refs. [32, 62] and is in effect equivalent within the accuracy limits of the models. Note that a direct comparison between the results obtained by the two approaches is not straightforward: Eq. (5) is obtained from the exact solution of the Lindblad equations for a four-level system, while Ref. 62 considers the quasi-classical evolution of a complex system averaged over the phonon subsystem. For a detailed discussion and comparison of the two approaches see e.g. Ref. 63. V. NC SIZE DEPENDENCE Above, we discussed the PE dependence measured for the fixed laser pulse photon energy of hν = 1.734 eV and explained how the phonon parameters can be evaluated from the PE signal. However, the sample under study contains NCs with different sizes which can be selectively excited by tuning the laser photon energy. We measured PE transients at photon energies in the range of 1.72÷1.76 eV. The photon energy can be recalculated into a NC diameter using the parameters from Ref. 39, where the same sample was studied (sample #3). Figures 4(a) and 4(b) show the dependence of the evaluated Huang-Rhys factors SHR and phonon lifetimes τph as function of the NC diameter a, for phonon modes with energies ħΩ1 = 3.2 meV and ħΩ2 = 5.1 meV. We note that the frequencies of these modes do not depend on a within the accuracy of the experiment. The Huang-Rhys factors for both phonon modes exhibit a clear size dependence, increasing with decreasing NC size. Correspondingly, the phonon lifetimes decrease with decreasing a. An increase in the electron-phonon coupling |0> |0'> |X'> |X> (b) γph γph γ0 SHRγ0 (1-SHR)2γ0 |Z> |Y> |X> |0> γ0 γ0 γ0 (a) δ2 δ1 ħΩ ħΩ Configuration coordinate Q Energy EX SHRħΩ ħΩ |0> |0'> |X'> |X> (c) Figure 3. Level schemes. (a) Energy levels of the bright exciton fine structure. (b) Energy level structure of the excitons interacting with optical phonons. We assume this level structure for each phonon mode, independent of the exciton polarization. The probability of the diagonal transitions is proportional to the Huang-Rhys factor SHR. γph is the phonon decay rate and γ0 is the exciton decay rate, mostly determined by its oscillator strength. (c) Sketch of adiabatic potential of an exciton interacting with a particular phonon mode. The solid parabolas represent the potential for phonons for the ground state of the crystal (no exciton, bottom) and the excited state (with one exciton, top). The solid parabolas are separated by the energy difference EX corresponding to the energy of zero-phonon exciton transition. ħΩdenotes the energy of the optical phonon. strength with decreasing NC size was previously reported for CdSe nanocrystals [64], PbS quantum dots [65], and perovskite nanocrystals, both for optical and acoustic phonons [16, 19]. However, to the best of our knowledge, direct measurements of the associated phonon relaxation times have not been reported. It is also important to note that, in contrast to previous studies, our PE experiments selectively probe the coherent dynamics of each individual phonon mode. In Ref. 19, the dominant mechanism of electronphonon coupling is assigned to the optical deformation potential [66]. In this case, the size dependence of the interaction can be estimated from the phonon normalization condition [67] which leads to SHR ∼a-3. For completeness, let us discuss the size dependence of the interaction between charge carriers and optical phonons 7 for the Fr ̈holich mechanism following Takagahara [68]. In the original work, the interaction of charge carriers with phonons is rewritten as potential for an electron in the field induced by the phonon mode polarization P(r). This potential is found from the Poisson equation ∇2φ(r) = 4 π∇· P(r) which leads to φ(re) = Z dr ∇· P(r) |r -re| = - Z dr P(r) · ∇ 1 |r -re| . (6) From Eq. (6) the strength of electron-phonon interaction ∆e ∼a-1 which leads to Se HR = ∆2 e/2 ∼a-2. This estimate is valid also for excitons in the strong confinement regime. For weakly confined excitons, the change of exciton energy ∆is proportional to the difference of the electrostatic potential for electron and hole φ(re) -φ(rh). It is found to be the sum of Eq. (6) for electron and hole: ∆∼φ(re)-φ(rh) = Z dr P(r)·∇ 1 |r -rh| - 1 |r -re| . (7) In the weak confinement regime when the exciton Bohr radius aB is small compared with the NC diameter aB ≪ a, 1 |r -rh| ≈ 1 |r -re| + (re -rh) · ∇ 1 |r -re| . (8) Substituting Eq. (8) into Eq. (7) we obtain φ(re) -φ(rh) ≈ Z dr P(r) · ∇ (re -rh) · ∇ 1 |r -re| . As a result, in the weak confinement regime, the size dependence of this expression is estimated as ∆∼φ(re) -φ(rh) ∝a3 · a-3/2 · aB a3 ∝a-3/2. Here, the first size factor comes from the integral, the second one from the normalization of the phonon mode, and the last one from the gradient of the expression in the square brackets. As a result the Huang-Rhys factor scales as SHR = ∆2/2 ∼a-3. Thus, in the weak confinement regime both mechanisms, the deformation potential and the Fr ̈ohlich interaction, lead to similar dependences of the Huang-Rhys factor on the NC size SHR ∼a-3. In the strong confinement regime, the Fr ̈ohlich interaction is expected to have a weaker size dependence, while the deformation potential mechanism should show the same scaling. The dashed lines in Fig. 4 represent fits of the experimental data by the empirical relation SHR ∝a-3. The experimental data in Fig. 4(a) even shows a steeper dependence of the Huang-Rhys factor on NC size, with pronounced deviation from the a-3 dependence, particularly for small NCs. This observation suggests that further analysis is required for smaller NCs to identify the mechanisms of electron-phonon interaction in perovskite NCs. 0 0.02 0.04 0.06 0.08 0.1 0.12 SHR ħΩ1 = 3.2 meV ħΩ2 = 5.1 meV 10 10.5 11 11.5 12 0 5 10 15 20 NC diameter (nm) τph (ps) (a) (b) Figure 4. NC size dependence. Dependence of (a) the Huang-Rhys factor, SHR, and (b) the phonon lifetime, τph, for the two phonon modes with energies ħΩ1 = 3.2 meV (blue dots) and ħΩ2 = 5.1 meV (red dots) on NC diameter a. Dashed lines in (a) are fits proportional to a-3. Dashed lines in (b) are guides to the eye. VI. DISCUSSION To summarize, we have revealed a new regime of coherent exciton-phonon dynamics following resonant optical excitation with femtosecond pulses in an ensemble of CsPbI3 nanocrystals embedded in a glass matrix. Quantum beats between exciton-polaron states manifest themselves in two-pulse photon echo experiments as oscillations with high frequencies corresponding to two optical phonon modes at 3.2 and 5.1 meV. A polarization resolved analysis of the photon echo signal has allowed us to unambiguously isolate the quantum beat signatures of exciton-polarons from slower beats arising from exciton fine-structure splittings. This unique regime of excitonpolaron quantum beats is established at the low temperature of 2 K, where an exceptionally long zero-phonon exciton coherence time is achieved (T2 ≈300 ps). The presence of spectrally narrow optical phonon modes with lifetimes of about 10 ps, combined with a relatively strong exciton-phonon coupling (Huang-Rhys factors up to 0.1), leads to a pronounced modulation of the photon echo signal at the phonon frequencies. Furthermore, the strong size dependence of the Huang-Rhys factor makes it possible to tune the strength of the optical transitions between different polaron states, offering a route to control the quantum dynamics for targeted applications. We emphasize that the observed optical response markedly differs from previous studies in, e.g., III-V selfassembled quantum dots, where the dynamics are mainly governed by exciton-acoustic phonon coupling [26, 69, 70]. There, the continuous phonon spectrum leads to a partial loss of coherence within the first few picosec8 onds after pulsed excitation. In our case no such behavior is observed in the temporal evolution of Ψ0(τ12) [see Fig. 2(b)]. Instead, the coherent dynamics is dominated by exciton-polaron states, i.e., interactions with discrete optical phonon modes, which can be consistently described within a four-level model for each phonon frequency. The desired quantum evolution can be achieved by tuning both the exciton-phonon coupling strength and, most importantly, the phonon lifetime, which limits the evolution of the pure quantum state following resonant excitation with a short optical pulse. Notably, the phonon lifetime exhibits a pronounced dependence on nanocrystal size. For the optical phonon mode with an energy of 5.1 meV, we observe an almost twofold increase of τph as the size increases from 10 to 12 nm. In layered perovskites, phonon lifetimes on the order of ∼10 ps were reported [71], with values extended up to 75 ps [72], approaching the record-high ∼200 ps observed in bulk ZnO [61]. Therefore, the strong size dependence of phonon lifetimes, along with reports of extended phonon coherence in layered perovskites, gives a promising outlook for long-lived and tunable coherent control of the quantum dynamics in perovskite nanocrystals. VII. METHODS Samples. The studied CsPbI3 nanocrystals embedded in fluorophosphate Ba(PO3)2-AlF3 glass were synthesized by rapid cooling of a glass melt enriched with the components needed for the perovskite crystallization. Details of the method are given in Refs. [38, 73]. The data on NC sizes evaluated from scanning transmission electron microscopy can be found in the supporting information section S5 in Ref. [39]. Samples of fluorophosphate (FP) glass with the composition 35P2O5-35BaO5AlF3-10Ga2O3-10PbF2-5Cs2O (mol. %) doped with BaI2 were synthesized using the melt-quench technique. The glass synthesis was performed in a closed glassy carbon crucible at the temperature of T = 1050◦C. Technological code of the studied sample is EK8. More information on its optical and spin-dependent properties can be found in Refs. 39 and 74. The sample is placed in the variable temperature insert of a bath cryostat and kept in superfluid helium at the temperature of 2 K. The low energy exciton optical transitions in nanocrystals are located in the spectral range of 1.7-1.8 eV as follows from the photoluminescence spectrum, see Fig. 1(a). Photon echo. Two-pulse photon echoes were measured using a transient four-wave mixing technique with heterodyne detection [44-47]. We used a Ti:Sapphire laser as source of optical pulses with duration of ∼100 fs at the repetition rate of 75.75 MHz. All optical experiments were carried out at photon energies below 1.76 eV. The sample is excited resonantly by a sequence of two pulses separated in time by the delay τ12, giving rise to the formation of a four-wave mixing signal with electric field amplitude EFWM(t). The pulses are focused into a spot of about 200 μm diameter using a 0.5-m parabolic aluminum mirror. The incidence angles of the pulses are close to normal and equal to approximately 1/50 rad and 2/50 rad, corresponding to the in-plane wavevectors k1 and k2. An additional reference pulse is used for detection and its delay with respect to the first pulse is given by the time τref. The delay times τ12 and τref are controlled using mechanical delay lines. The four-wave mixing signal is measured in transmission geometry in the direction ≈3/50 rad, which corresponds to the phasematching direction 2k2 -k1. Optical heterodyne detection is used to resolve the enhanced signal in time. By mixing with a relatively strong reference pulse and scanning τref, the temporal profile of the photon echo pulse is measured. Polarization sensitive measurements are performed by selecting proper polarization configurations (Hhorizontal or V-vertical) for each of the beams using Glan-Thompson prism in conjunction with half-wave plates. In the detection path we also select the linear polarization by setting the polarization of the reference beam. We adopt a three-letter notation for the PE polarization configuration. The first two symbols correspond to the polarizations of the pulses exciting the sample, and the third symbol corresponds to the detection polarization. For example, in the HHH configuration, the polarizations of both excitation pulses and the detection are horizontal (H). In the HVH configuration, the second excitation pulse is vertically (V) polarized, while the first pulse and detection are horizontally polarized. HHH is accordingly termed co-linearly polarized (∥) and HVH cross-linearly polarized (×) in Fig. 2. We note that rotation of the sample does not lead to changes in the signal, i.e. the orientation of the sample is not important. Raman spectroscopy. The Raman scattering signal was excited by a single-frequency Ti:Sapphire laser. The laser beam with the power of about 1 mW was focused on the sample to a spot with a diameter of about 300 μm. The scattered light was analyzed by a Jobin-Yvon U1000 double monochromator with 1-meter focal length, allowing the high resolution of 0.2 cm-1 (0.024 meV). The Raman signal was detected by a cooled GaAs photomultiplier and conventional photon-counting electronics. The Raman spectra were measured for copolarized linear polarizations of excitation and detection. VIII. ACKNOWLEDGEMENTS The authors are thankful to I. A. Yugova and N. E. Kopteva for fruitful discussions. We acknowledge the financial support by the Deutsche Forschungsgemeinschaft: A.V.T., M.A.H, S.G., M.O.N and I.A.A (project AK40/13-1, no. 506623857) and D.R.Y. 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Khersonskii, Quantum theory of angular momentum (World Scientific, 1988). [76] P. R. Berman and V. S. Malinovsky, Principles of Laser Spectroscopy and Quantum Optics (Princeton University Press, 2011). S1 Supplementary Information: Quantum beats of exciton-polarons in CsPbI3 perovskite nanocrystals S1. FINE STRUCTURE OF EXCITON To take into account the fine structure of the exciton, we assume that the three-fold bright exciton state is split into mutually orthogonal linearly polarized components. For the photon echo calculations the following procedure is used: the dynamics is calculated in the eigenbasis of NC {|0⟩, |x′⟩, |y′⟩, |z′⟩} then the equations for the density matrix components are rotated to the laboratory frame {|0⟩, |x⟩, |y⟩, |z⟩}. The 1st pulse is assumed to be polarized along z axis, the 2nd pulse at time τ is either polarized along z or x axis for HHH and HVH schemes respectively. The polarization along z is detected at the photon echo time 2τ. The orientation of a nanocrystal in the laboratory reference frame is described by three Euler angles α, β, and γ. The rotation matrix relating coordinates in xyz and x′y′z′ bases is [75]: R(α, β, γ) = cαcβcγ -sαsγ -cαcβsγ -sαcγ cαsβ sαcβcγ + cαsγ -sαcβsγ + cαcγ sαsβ -sβcγ sβsγ cβ , (S1) here and below in this section we use sξ ≡sin ξ, cξ ≡cos ξ (ξ = α, β, γ) to shorten the notation. In the eigenbasis of NC, the dynamics is given by (we skip common factor e-γt): ρ0x′ = e-t/T2eiω1tρ0 0x′ , ρ0y′ = e-t/T2eiω2tρ0 0y′ , ρ0z′ = e-t/T2eiω3tρ0 0z′ ; (S2) ρx′0 = e-t/T2e-iω1tρ0 x′0 , ρy′0 = e-t/T2e-iω2tρ0 y′0 , ρz′0 = e-t/T2e-iω3tρ0 z′0 . (S3) Here ħωi (i = 1, 2, 3) are the three exciton energies for excitons linerly polarized along x′, y′, z′ respectively. We assume that the diplole matrix element (and, consequently, radiative decay time) as well as the decoherence time T2 is the same for all three exciton states. After 1st pulse the components of density matrix are ρ1+ 0z = 1 , ρ1+ 0y = ρ1+ 0x = 0 . (S4) After 2nd pulse we have two options: for HHH scheme ρ2+ z0 = ρ20z , ρ2+ y0 = ρ2+ x0 = 0 . (S5) and for HVH scheme ρ2+ x0 = ρ20x , ρ2+ y0 = ρ2+ z0 = 0 . (S6) Rotating the dynamics equations (S2) into laboratory coordinate frame using (S1) gives rather lengthy equations for the amplitude of the signal: A∥(α, β, γ) ∝ρz0 = c4 αs4 β + s4 αs4 β + c4 β + 2s2 αc2 αs4 β cos(ω1 -ω2)τ + 2c2 αs2 βc2 β cos(ω1 -ω3)τ + 2s2 αs2 βc2 β cos(ω2 -ω3)τ e-2τ/T2 (S7) A×(α, β, γ) ∝ρz0 = c2 αs2 β(cαcβcγ -sαsγ)2 + s2 αs2 β(sαcβcγ + cαsγ)2 + s2 βc2 βc2 γ + 2sαcαs2 β(cαcβcγ -sαsγ)(sαcβcγ + cαsγ) cos(ω1 -ω2)τ -2cαs2 βcβcγ(cαcβcγ -sαsγ) cos(ω1 -ω3)τ -2sαs2 βcβcγ(sαcβcγ + sαsγ) cos(ω2 -ω3)τ e-2τ/T2 (S8) The result should be averaged over directions using: ⟨A⟩ang ≡ 1 8π2 Z 2π 0 dα Z π 0 sin βdβ Z 2π 0 dγ A(α, β, γ) (S9) S2 Assuming the constant energies of three exciton components, but the arbitrary direction of the NCs, the amplitude of the echo is proporitional to A∥ ang ∝ 6 5 + 4 15 X i>j cos(ωi -ωj)τ e-2τ T2 , (S10) ⟨A×⟩ang ∝ 2 5 -2 15 X i>j cos(ωi -ωj)τ e-2τ T2 . (S11) Now we also take into account the possible deviation of exciton energies. The absolute value of the exciton energy is not important for the phonon echo, only the splittings. E.g., the amplitude averaged over distribution of splittings between 1st and 2nd levels ω12 = ω1 -ω2 is given by ⟨A⟩ω12 = Z W(ω12)A(ω12, ω13, ω23 = ω12 -ω13)dω12 (S12) Assuming the distribution function of the splittings to be Lorentzian, W(ω) = t∗ i π 1 (ω -δi)2 + √ 2/(t∗ i )2 , (S13) and that all three splittings are not correlated, the PE signal averaged over both the angles and the splittings distribution is given by A∥ ∝ 18 15 + 4 15 X i=1,2,3 cos(δiτ)e -2τ t∗ i e-2τ T2 , (S14) ⟨A×⟩∝ 6 15 -2 15 X i=1,2,3 cos(δiτ)e -2τ t∗ i e-2τ T2 , (S15) where the parameters t∗ i work as an effective signal dephasing times and δi are the average splittings. An interesting result is that even though the PE signal is stable with respect to the level disorder, the disorder of its fine structure gives the contribution to the additional decay, not of the full signal, but of its oscillating components. S2. MODEL OF PHOTON ECHO OF EXCITON-POLARON STATES. In this section we outline the derivation of PE signal in the polaron model. We assume the exciton-polaron with relatively small Huang-Rhys factor which allows to perform calculations in four (six with account on light polarization) level scheme. S2.A. Scheme of the levels and dynamics of the system We consider the four-level system shown in Fig. 3b. There are: ground level |0⟩, statwe with one phonon |0′⟩, polaron state |X⟩, and excited state of polaron |X′⟩. The energies of corresponding states are 0, ħΩ, ωX, ωX + ħΩ. In the calculations below we assume that spectral width of the laser pulse covers at least |X⟩and |X′⟩levels, this is in contrast to standard approximations for three-level atoms [76]. Due to polaron formation, allowed transitions are not only X →0 (with probability γ0) and X′ →0′ (with probability γ1), but also X →0′ and X′ →0 (in this model, probability is the same γ′). In polaron model [59], the transition probability is proportional to product of dipole matrix element |d|2 and the overlap of oscillator functions shifted due to exciton formation which leads to extra factor γ0 ∝|d|2e-2SHR (S16a) γ′ ∝|d|2SHRe-2SHR (S16b) γ1 ∝|d|2(1 -SHR)2e-2SHR (S16c) S3 where d is the dipole matrix elements of the optical transition, and we defined Huang-Rhys factor as SHR = εpol/(ħΩ) (εpol is the polaron formation energy). Note that in this approximation polaron formation energy should be small which means that γ′ ≪γ0, γ1. In addition, we assume that the phonon may be emitted which leads to transitions 0′ →0 and X′ →X with the same probability γph. To trace the polarization of the signal, we need to consider two polarization states of exciton, from now on the enumeration 0 and 0′ are ground state and state with phonon, 1(2) and 1′(2′) are right(left) circularly polarized states of exciton(polaron) and first oscillator level of exciton(polaron), making the model effectively six-level. Still, we call the model four-level as here we neglect the splittings between levels (as the role of the fine structure is excluded, see Sec. S1) and we need it only to demonstrate that the phonon structure does not change the polarization structure of the signal. Below, we extensively omit states 2 and 2′ when equations are same. The Lindblad equation, with all levels considered is: dρ dt = -i ħ[H0, ρ] + Ld[ˆρ] + Γ ⊙ˆρ . (S17) For convenience and compactness of the notation, with the use of ⊙for the Hadamard product (element-wise product or Schur product). The Hamiltonian is H0 = 0 0 0 0 0 0 0 ħΩ 0 0 0 0 0 0 ħωX 0 0 0 0 0 0 ħωX 0 0 0 0 0 0 ħωX + ħΩ 0 0 0 0 0 0 ħωX + ħΩ (S18) and the "inflow"diagonal part of superoperator is Ld[ˆρ] = γphρ0′0′ + γ0ρX + γ′ρX′ 0 0 0 0 0 0 γ1ρX′ + γ′ρX 0 0 0 0 0 0 γphρ1′1′ 0 0 0 0 0 0 γphρ2′2′ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (S19) where for convenience we introduced ρX = ρ11 + ρ22 and ρX′ = ρ1′1′ + ρ2′2′, and the matrix in the "outflow" part is Γ = -1 2 0 γph γ′ 0 γ′ 0 γph + γ′ 1 γph + γ′ 1 γph 2γph γph + γ′ 0 γph + γ′ 0 2γph + γ′ 1 2γph + γ′ 1 γ′ 0 γph + γ′ 0 2γ′ 0 2γ′ 0 γph + γ′ 1 + γ′ 0 γph + γ′ 0 + γ′ 1 γ′ 0 γph + γ′ 0 2γ′ 0 2γ′ 0 γph + γ′ 1 + γ′ 0 γph + γ′ 0 + γ′ 1 γph + γ′ 1 2γph + γ′ 1 γph + γ′ 0 + γ′ 1 γph + γ′ 0 + γ′ 1 2γph + 2γ′ 1 2γph + 2γ′ 1 γph + γ′ 1 2γph + γ′ 1 γph + γ′ 0 + γ′ 1 γph + γ′ 0 + γ′ 1 2γph + 2γ′ 1 2γph + 2γ′ 1 (S20) where for convenience we defined γ′ 0 = γ0 + γ′ , γ′ 1 = γ1 + γ′ . (S21) S2.B. Effect of the pulse on density matrix Now we consider light-mediated transitions in the system shown in Fig. 3b. This is again generalization of the two-level system, check e.g. chapters 2, 6 and 10 of [76]. The pulse is approximated by the plane wave with the smooth enveope E0(z, t) and electric field given by E(r, t) = E0(k · r/k, t)ei(k·r-ωt) + c.c. (S22) Assuming the pulse duration is short compared with all coherence and relaxation times, the evolution of density matrix (DM) during the pulse may be found from the Schr ̈odinger equation dρ dt = -i ħ[H0 + V, ρ] , (S23) S4 where, in rotating wave approximation, V = ħ 0 0 ̃f ∗ + ̃f ∗ - s · ̃f ∗ + s · ̃f ∗ + 0 0 s · ̃f ∗ + s · ̃f ∗ - ̃ff ∗ + ̃f ∗ + ̃f+ s · ̃f+ 0 0 0 0 ̃fs · ̃f0 0 0 0 s · ̃f+ ̃f+ 0 0 0 0 s · ̃f- ̃f0 0 0 0 , (S24) ̃f± = dE± 0 e-iωt ≡f±e-iωt , (S25) where d is the dipole matrix element for the transitions 1, 2 →0, s = √SHR, E± 0 are circular components of the envelope (S22). Solution of Eq. (S23) is surprisingly complex in general case, for two-level system it may be found in Ref. 76, but we are interested in four-level system and the case of small pulse area when the full solution is surplus. When the pulse area θ± (equal to f±tp for square pulse of duration tp) is small compared with Rabi frequency, the first order correction to the density matrix may be obtained as ρ+ = -i ħ H0 + V, ρ- tp ≡ˆ Aρ-, (S26) where ρ-is the density matrix before pulse and ρ+ is the density matrix after pulse. Full solution of Eq. (S26) is too lengthy for our needs, below we will only give the results relevant for the calculations. For the PE we will need the action of the light pulse in the second order over pulse area. It may be shown that up to second order, it may be calculated as ρ+ = ˆ A ˆ Aρ-. (S27) Below we also give only part of this result relevant for photon echo calculations. S2.C. Photon echo The signal in two circular polarizations is proportional [76] to the corresponding components of polarization vector which is given by the following components of the density matrix: P+ ∼ρ10 + ρ1′0′ + sρ1′0 + sρ10′ , (S28a) P-∼ρ20 + ρ2′0′ + sρ2′0 + sρ20′ . (S28b) Note the extra factor s in Eq. (S28) which originates from the ratio of diplole matrix elements for phonon-assisted transitions to the phonon-less transitions. Thus, we are interested in the following components of the density matrix at times after second pulse: ρ10, ρ20, ρ1′0, ρ2′0, ρ10′, ρ20′, ρ1′0′, ρ2′0′. a. Density matrix after 1st pulse The components of DM after 1st pulse are given by (S26) which leads to: ρ1+ 01 = iθ∗ 1+ , ρ1+ 01′ = isθ∗ 1+ , ρ1+ 02 = iθ∗ 1-, ρ1+ 02′ = isθ∗ 1-. (S29) b. Relaxation of DM components after 1st impulse One needs to trace the evolution of the following DM components: ρ01, ρ01′, ρ02, ρ02′. Below we give results only for right polarization, dynamics of other components is the same. Writing the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components: ρ01 (t ∈(0, τ12)) = ρ1+ 01 eiωXteγ′ 0 2 t , (S30a) ρ01′ (t ∈(0, τ12)) = ρ1+ 01′ei(ωX+Ω)teγ′ 1+γph 2 t , (S30b) This leads to the following components of the DM at the time of 2nd pulse t = τ12 ρ201 = ρ1+ 01 εXΓ0 , ρ201′ = ρ1+ 01′εXεΓ1Γp , (S31) where for convenience we defined εX = eiωXτ12 , ε = eiΩτ12 , Γ0 = eγ′ 0 2 τ12 , Γ1 = eγ′ 1 2 τ12 , Γp = eγph 2 τ12 . (S32) Note that Γ0 and Γ1 are almost equal and defined by the exciton coherence time (under our assumptions half of its lifetime). The difference between them is Γ1/Γ0 = e-γ1-γ0 2 τ12 = e SHRγ0 2 τ12 which is much smaller than all typical times in the system since SHR is small. S5 c. DM after 2nd pulse From (S27) we may compute the components relevant for the PE signal after the 2nd pulse: ρ2+ 10 = θ2 2+(ρ201 + sρ201′) + θ2+θ2-(ρ202 + sρ202′) (S33a) ρ2+ 10′ = θ2 2+(sρ201 + ρ201′) + θ2+θ2-(sρ202 + ρ202′) (S33b) ρ2+ 1′0 = s θ2 2+(ρ201 + sρ201′) + θ2+θ2-(ρ202 + sρ202′) (S33c) ρ2+ 1′0′ = s θ2 2+(ρ201′ + sρ201 ) + θ2+θ2-(ρ202′ + sρ202 ) (S33d) and ρ2+ 20 = θ2+θ2-(ρ201 + sρ201′) + θ2 2-(ρ202 + sρ202′) (S34a) ρ2+ 2′0 = θ2+θ2-(sρ201 + ρ201′) + θ2 2-(sρ202 + ρ202′) (S34b) ρ2+ 20′ = s θ2+θ2-(ρ201 + sρ201′) + θ2 2-(ρ202 + sρ202′) (S34c) ρ2+ 2′0′ = s θ2+θ2-(ρ201′ + sρ201 ) + θ2 2-(ρ202′ + sρ202 ) (S34d) d. Relaxation of DM components after 2nd impulse We we need to know the evolution of the DM components which enter (S28). Below we give only components entering P+. Dynamics of components entering P-is the same. Writing the components of the Lindblad equation (S17), we obtain the following dynamics of the DM components: ρ10 (t > t2) = ρ2+ 10 e-iωX(t-t2)eγ′ 0 2 (t-t2) , (S35a) ρ1′0 (t > t2) = ρ2+ 1′0e-i(ωX+Ω)(t-t2)eγ′ 1+γph 2 (t-t2) , (S35b) ρ10′ (t > t2) = ρ2+ 10′e-i(ωX-Ω)(t-t2)eγ′ 0+γph 2 (t-t2) , (S35c) ρ1′0′ (t > t2) = ρ2+ 1′0′e-iωX(t-t2)eγ′ 1+2γph 2 (t-t2) , (S35d) Which gives the DM components at the time of echo signal t -t2 = τ12: ρP E 10 = ρ2+ 10 ε∗ XΓ0 , ρP E 1′0 = ρ2+ 1′0ε∗ Xε∗Γ1Γp , (S36a) ρP E 10′ = ρ2+ 10′ε∗ XεΓ0Γp , ρP E 1′0′ = ρ2+ 1′0′ε∗ XΓ1Γ2 p . (S36b) The evolution of components corresponding to second circular polarization is the same. e. Contribution of different paths to PE To get the final result, we substitude (S29) into (S31) and then the result into (S33) and then from (S36) we get the DM at the time of PE. The amplitude of the signal in two polarizations is then obtained from Eqs. (S28): P+ ∼ρPE 10 + ρPE 1′0′ + sρPE 1′0 + sρPE 10′ , (S37a) P-∼ρPE 20 + ρPE 2′0′ + sρPE 2′0 + sρPE 20′ . (S37b) Collecting all terms in the result we have for contributions to P+ from ρ10, ρ1′0′, ρ10′, ρ1′0 originating from the DM components after 1st pulse given below in the beginning of each line: ρ01,02 →ρPE 10 Γ2 0A+ , (S38a) ρ01′,02′ →ρPE 10 s2εΓ0Γ1ΓpA+ , (S38b) ρ01,02 →ρPE 10′ s2εΓ2 0ΓpA+ , (S38c) ρ01′,02′ →ρPE 10′ s2ε2Γ0Γ1Γ2 pA+ , (S38d) ρ01,02 →ρPE 1′0 s2ε∗Γ0Γ1ΓpA+ , (S38e) ρ01′,02′ →ρPE 1′0 s4Γ2 1Γ2 pA+ , (S38f) ρ01,02 →ρPE 1′0′ s2Γ0Γ1Γ2 pA+ , (S38g) ρ01′,02′ →ρPE 1′0′ s2εΓ2 1Γ3 pA+ , (S38h) where A+ = θ2+ θ2+θ∗ 1+ + θ2-θ∗ 1- . (S39a) S6 Note that the term (S38f) may be dropped as it exceeds precision of equations. The contribution to P-from ρ20, ρ2′0′, ρ20′, ρ2′0 is the same as (S38) but instead of A+ the polarization dependence is given by A-= θ2- θ2-θ∗ 1-+ θ2+θ∗ 1+ . (S39b) It is convenient to write both as A± = θ2±A0 , A0 = θ2-θ∗ 1-+ θ2+θ∗ 1+ , (S40) to see that the polarization dependence of the PE signal coincides with second impulse polarization while its amplitude depends on respective polarization of first and second pulses. As a result, the amplitude of the non-oscillating signal is PP E,0 ± ∼ Γ2 0 + SHRΓ0Γ1Γ2 p A± . (S41) The amplitude of the signal oscillating at Ωis PP E,Ω ± ∼SHRΓp Γ0Γ1 (ε + ε∗) + ε Γ2 0 + Γ2 1Γ2 p A± . (S42) and of the signal oscillating at 2Ωis PP E,2Ω ± ∼SHRΓ0Γ1Γ2 pε2A± . (S43) f. Result Let us first consider the general scheme of the polarization dependence of PE signal. Following [47], we consider linearly polarized excitation pulses with second pulse rotated to angle φ and linearly polarized detection with polarized rotated to φd. Then θ1+ = θ1-= 1/ √ 2 , θ2+ = eiφ/ √ 2 , θ2-= e-iφ/ √ 2 . (S44) Which gives A0 = cos φ and A± = e±iφ √ 2 cos φ . (S45) Amplitude of linearly-polarized heterodyne detection of main PE signal is proportional to P 0 d = ei(φ-φd) + e-i(φd-φ) 2 cos φ = |cos (φ -φd) cos φ| (S46) We have two configurations: HHH and HVH. First is φ = φd = 0 and second is φd = 0, φ = π/2. This gives P 0,HHH d = 1 , P 0,HV H d = 0. (S47) As long as polarization dependence on the excitation/detection angles (S46) is real (which is the case for HHH and HVH configurations), the absolute value of product is the product of absolute values and signal is zero for HVH scheme and for HHH scheme (we omit index HHH below) is: PP E d = Γ2 0 h 1 + SHReΓΓ2 p i + SHRΓp h eΓ (ε + ε∗) + ε(1 + eΓ2Γ2 p) i + SHReΓΓ2 pε2 (S48) where we introduced eΓ = Γ1/Γ0 = eSHRγ0τ12 . (S49) Note that eΓ is 1 as compared with both, Γ0 and Γp. To write explicitly the result for the PE amplitude, we use the fact that for t ≪1 and arbitrary complex ε: |1 + tε| = 1 + tRe(ε) + t2 2 (Im(ε))2 + O t3 . (S50) We keep second order terms to highlight the fact that even for single frequency of the complex amplitude, the absolute value in (S48) will result in second order in SHR signal at double frequency. S7 Using (S50) we may write the final result (keeping only terms to the first order in SHR) PP E d = Γ2 0 n 1 + SHRΓp h eΓΓp + 1 + 2eΓ + eΓ2Γ2 p cos(Ωτ12) + eΓΓp cos(2Ωτ12) io (S51) Or, explicitly, PP E d = e-γ′ 0τ12 " 1 + SHReγph 2 τ12 eγph-2SHRγ0 2 τ12 + 1 + 2eSHRγ0τ12 + e-(γph-2SHRγ0)τ12 cos(Ωτ12) + eγph-2SHRγ0 2 τ12 cos(2Ωτ12) !# , (S52) and, finally, with the help of Eqs.(S16,S21): PP E d = e-(1+SHR)γ0τ12 " 1 + SHRe-(γph-SHRγ0)τ12 + SHR cos(Ωτ12)eγph-2SHRγ0 2 τ12 e-SHRγ0τ12 + 2 + e-γphτ12 + SHR cos(2Ωτ12)e-(γph-SHRγ0)τ12 # ≡Ψ0(τ12; Ω) . (S53)
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2510.14691
|
If You Hold Me Without Hurting Me: Pathways to Designing
Game Audio for Healthy Escapism and Player Well-being∗
CAIO NUNES, Federal University of Ceará, Brazil
BOSCO BORGES, Federal University of Ceará, Brazil
GEORGIA CRUZ, Federal University of Ceará, Brazil
TICIANNE DARIN, Federal University of Ceará, Brazil
Escapism in games can support recovery or lead to harmful avoidance. Self-regulation, understood as combining
autonomy with positive outcomes, is key to this distinction. We argue that audio, often overlooked, plays a
central role in regulation. It can modulate arousal, mark transitions, and provide closure, yet its contribution
to well-being remains underexplored. This paper identifies methodological and accessibility gaps that limit
recognition of audio’s potential and outlines ways to address them. We aim to encourage researchers and
developers to integrate audio more deliberately into the design and study of healthier escapist play.
CCS Concepts: • Human-centered computing →HCI design and evaluation methods; • Applied
computing →Sound and music computing.
Additional Key Words and Phrases: Escapism, Emotions, Self-Regulation, Game Audio
1
INTRODUCTION
Escapism has long been recognized as one of the main motivations for gameplay, appearing across
models of player motivation and media use [4, 7, 19]. It is often described as the tendency to
detach from everyday concerns or stressors, yet its consequences are ambivalent. On the one
hand, escapism can offer distraction, stress relief, and opportunities for affective regulation [2]. On
the other hand, it can reinforce avoidance and interfere with everyday functioning. Recent work
emphasizes this tension by distinguishing between healthy escapism, when play supports recovery
and well-being, and subversive escapism, when it undermines them [9]. This perspective shifts the
debate from whether escapism is good or bad to a more pressing question: "Which factors determine
when escapist play sustains well-being and when it becomes harmful?"
One of the most decisive factors in this distinction is self-regulation. In Self-Determination
Theory (SDT), self-regulation refers to the extent to which behavior is experienced as autonomous,
reflecting choice and volition rather than external pressure or compulsion [17]. Applied to games,
this perspective captures whether players can autonomously decide when to begin, how long
to continue, and when to disengage. Research consistently shows that difficulties in regulation
are linked to excessive or compulsive gaming [6]. At the same time, successful self-regulation
allows games to function as bounded recesses that facilitate recovery, coping, and positive mood
management [9, 13]. Framing escapism through this lens highlights autonomy as a key condition
for distinguishing between healthy and subversive play. However, a player could autonomously
choose to play for a duration that negatively impacts their life responsibilities. In this case, the
behavior is self-determined, but the outcome is detrimental. Hence, a healthy escapism also requires
positive outcomes, where psychological relief (e.g., reported mood improvement) is supported by
physiological recovery (e.g., stress marker reduction), independent of the feeling of volition during
play. Consequently, we draw attention to how the design of game elements can either scaffold a
player’s ability to self-regulate or undermine it.
We argue that audio is a key game element that can be intentionally designed to struc-
ture the escapist experience itself, facilitating autonomous self-regulation and positive
∗Presented and discussed at the CHI PLAY 2025 Workshop Exploring Future Directions for Healthy Escapism and Self-Regulation
in Games, Pittsburgh, USA, October 13, 2025.
1
arXiv:2510.14691v1 [cs.HC] 16 Oct 2025
Nunes et al.
outcomes. While often perceived as secondary to visual or mechanical elements, audio possesses
unique qualities that make it a potent regulatory channel. Unlike visual cues, which require focused
attention, sound is pervasive and processed with a lower cognitive load, capable of influencing
arousal and emotional states subconsciously, creating a direct tension between systemic regulation
and player autonomy that design must address. The well-documented link between music and
physiological responses, from emotion regulation and stress recovery to everyday relaxation [5, 8].
This makes audio a prime channel for influencing the very recovery and emotion regulation that
are central to healthy escapism [9]. Besides, audio’s ability to structure time and rhythm transitions
offers a subtle yet material means of scaffolding player awareness without disrupting immersion.
Yet, despite its unique potential, the role of audio in player well-being remains poorly understood.
Much of the game-audio literature concentrates on immersion, presence, arousal, and affect, seldom
linking these outcomes to well-being or self-regulatory processes [10]. This gap is especially evident
in discussions of escapism and self-regulation, which often emphasize constructs such as absorption,
affective modulation, stress relief, and distraction [3, 18]. While these emphases are not identical,
they share a concern with how players become deeply involved and how their emotional states are
shaped during play.
This lack of synthesis leaves a critical gap, preventing designers and researchers from systemat-
ically using audio as a tool for player well-being. This position paper moves to fill that gap. We
contend that audio is not a peripheral feature but a central, designable mechanism for scaffolding
healthy escapism in games. To substantiate this claim, the remainder of this paper will first identify
methodological and accessibility gaps previously identified in game audio research that perpetuate
this problem. We then propose three concrete directions for future work focused on using audio to
support self-regulation and positive emotional outcomes.
2
HOW RESEARCH GAPS OBSCURE AUDIO’S ROLE IN HEALTHY ESCAPISM
In a previous study, we reviewed how audio and player experience are evaluated [10]. The results
show a fragmented evidence base where game audio is routinely linked to immersion and affect, but
its deeper impact is unclear. A key problem is that self-report measures often fail to converge with
physiological data; for example, players may report feeling calm while their arousal markers remain
elevated [10]. These inconsistencies make it challenging to draw conclusions about outcomes
such as coping, stress recovery, or emotion regulation. This methodological fragmentation is
compounded by other persistent issues, including a lack of personalization and the tendency for
audio to be overshadowed in high-sensory contexts [11, 14–16]. Together, these findings reveal that
audio is seen as an immersive aesthetic but not perceived as a potential lever for self-regulatory
outcomes.
The real-world impact of this research gap is starkly visible in games designed for sensitive
audiences. Our literature review (n=49) on games for autistic players [1], for instance, shows
that audio is often absent from game design descriptions and from user evaluation, despite its
potential to support engagement or, if mishandled, trigger overload in audiences with auditory
hypersensitivity. Across the surveyed studies, only 14% (n=7) explicitly evaluated audio; when
they did, its uses ranged from background music to movement sonification, yet the impacts on
the target audience were seldom reported in depth [12]. Such an absence of systematic evaluation
and reporting reveals a double gap: a lack of design transparency and potentially a neglect of
accessibility. For that audience, without customization options or sensitivity to diverse auditory
needs, audio risks becoming a barrier instead of a resource. These findings highlight how sound
design, if overlooked, can miss therapeutic opportunities and expose players to overstimulation or
discomfort. This case starkly illustrates how neglecting audio can undermine a player’s ability to
self-regulate, turning a potentially healthy escape into a subversive, dysregulating experience.
2
If You Hold Me Without Hurting Me
While these examples are not exhaustive, the gaps identified in the literature prevent us from
measuring whether audio supports well-being or causes distress. Methodological blind spots mean
that we lack ways to evaluate if audio is contributing to the positive outcome of healthy escapism,
whether audio supports coping and recovery or simply intensifies immersion. Accessibility gaps
can imply that sound becomes overwhelming or exclusionary, rather than restorative. In both cases,
escapism risks sliding toward avoidance and overuse – not because audio cannot support healthier
outcomes, but because research and design fall short in providing the tools to differentiate them.
The absence of variation, closure, and inclusive controls leaves players to self-manage through
workarounds such as muting or replacing game soundtracks [16], undermining the potential of
audio to contribute directly to constructive escapist experiences.
Ultimately, players are left without the necessary scaffolds to exercise their autonomy due to a
lack of key tools, such as methods for measuring player responses to audio cues or designs that
utilize audio to create natural stopping points. For instance, if safe exit points are signaled by a
shift to calmer music in a hub area, it creates a clear closing motif after a mission, or even a brief
"breathing window"" of silence after a moment of high tension, all of which provide a sense of
closure that could help players disengage with a sense of accomplishment. Without such intentional
design, poorly tested or repetitive audio can instead reinforce compulsive immersion or force
avoidance, both of which undermine self-regulation. These blind spots leave players without the
necessary scaffolds to disengage autonomously, showing how overlooked audio can shift escapism
from recovery to overuse.
3
TOWARDS SELF-REGULATION AND HEALTHY ESCAPISM WITH AUDIO
The gaps highlighted reveal a significant methodological deficiency: despite audio’s clear influence,
we remain unable to reliably determine its impact on recovery versus its potential to undermine
regulation. Strategies are absent that repurpose sound as a design utility for player empowerment
and healthy boundaries. To resolve this issue, we propose three core directions focusing on ad-
dressing methodological pitfalls, enhancing reporting transparency, and solidifying collaborative
ties between research and industry.
First, adopt multi-modal evaluation: self-reports, behavioral traces, and psychophysiological
markers each provide partial insights, but their combination offers a fuller picture of audio’s
regulatory role. Logs of muting, playlist swaps, or sudden volume drops can reveal when players
regulate their own auditory load. Coupled with physiological markers and short probes, such as “Do
you feel calmer?” or “Are you ready to stop?” one can test whether sound genuinely supports coping
or merely shifts physiology without providing felt relief. These behaviors can act as practical,
scalable proxies for laboratory measures. Convergences and divergences are equally informative,
allowing researchers to identify when design succeeds and giving developers concrete indicators
of overload or recovery.
Second, report design intentions transparently. Findings remain anecdotal when studies only state
that sound was present. Reporting must clarify how audio was crafted and with what regulatory
purpose to subconsciously guide players toward recovery, or to provide explicit cues for conscious
self-regulation: a low-tempo loop added after combat to support recovery, a brief silence window
following repeated failures to encourage pause, or a closing motif marking the transition to a
safe hub area. By grounding technical decisions in regulatory intent, researchers can accumulate
evidence across contexts and developers can align design choices with player well-being, especially
in sensitive cases where unexamined audio risks overstimulation [1].
Third, foster a shared agenda between research and industry. Rather than prescribing isolated
techniques, advancing this field requires collaboration and teamwork. Initiatives such as joint
3
Nunes et al.
workshops, repositories of customizable regulatory audio patterns and adaptive sound systems, or
co-produced guidelines, similar to existing accessibility standards, can foster cumulative knowledge.
Together, these commitments elevate audio from a mere backdrop to an active mediator of
escapist play. They reposition sound as a critical lever for discerning the balance between recovery
and over-engagement, underscoring that without systematic evidence, designers still lack the means
to use it intentionally for player well-being. To drive this agenda, we invite the community to
consider:
(1) Which combinations of self-report, behavioral, and physiological measures best capture
audio’s regulatory role?
(2) How might soundscapes, finality cues in UI sounds, and music be designed to make transi-
tions between immersion and disengagement more intentional?
(3) What forms of player control over audio foster genuine autonomy in shaping escapist
experiences?
(4) Which standardized or recurring sound patterns are being used unreflectively, and how
might their effects on well-being be better understood?
AUTHOR BIOGRAPHIES
Caio Nunes is a researcher with a Master’s degree in Computer Science from Federal University of
Ceará, Brazil. His work focuses on Human–Computer Interaction (HCI) and Game User Research,
exploring audio, musical visualizations, motivation, and well-being in emotional communication.
Bosco Borges is a Master’s student at Federal University of Ceará, Brazil. His work focuses on
HCI and Game User Research, examining how sound and interaction design can support player
experience, promote healthier engagement, and cater to diverse audiences.
Georgia Cruz is an Adjunct Professor at Federal University of Ceará, Brazil. Her work focuses on
UX, Audiovisual, and Communicability, exploring user experience, gender in games, and critical
play, with an emphasis on how interactive media can foster reflection and influence well-being.
Ticianne Darin is an Associate Professor of HCI at the Federal University of Ceará. Her game
research examines the foundational components of player experience, from motivation and engage-
ment to the pursuit of eudaimonia through reflexive games. She is currently focused on developing
ethical alternatives to manipulative design to foster digital well-being. Her leadership in the Brazil-
ian research community includes chairing the nation’s primary HCI symposia and founding the
WIPlay workshop series, a dedicated venue to unite the country’s HCI and games researchers. As a
UNESCO consultant, she helps shape national policy on digital rights, focusing on implementing
"Child and Adolescent Rights by Design" frameworks for games and online services.
REFERENCES
[1] Bosco Borges and Ticianne Darin. 2025. Sounds in Game Design and Evaluation for Autistic People: A Literature
Review. In International Workshop on Interaction and Player Research in Game Development. Springer, 29–48.
[2] Lorelle Bowditch, Janine Chapman, and Anjum Naweed. 2018. Do coping strategies moderate the relationship between
escapism and negative gaming outcomes in World of Warcraft (MMORPG) players? Computers in Human Behavior 86
(2018), 69–76.
[3] Gordon Calleja. 2010. Digital games and escapism. Games and Culture 5, 4 (2010), 335–353.
[4] Scott Caplan, Dmitri Williams, and Nick Yee. 2009. Problematic Internet use and psychosocial well-being among MMO
players. Computers in human behavior 25, 6 (2009), 1312–1319.
[5] Hiroharu Kamioka, Kiichiro Tsutani, Minoru Yamada, Hyuntae Park, Hiroyasu Okuizumi, Koki Tsuruoka, Takuya
Honda, Shinpei Okada, Sang-Jun Park, Jun Kitayuguchi, et al. 2014. Effectiveness of music therapy: a summary of
systematic reviews based on randomized controlled trials of music interventions. Patient preference and adherence
(2014), 727–754.
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If You Hold Me Without Hurting Me
[6] Daniel Kardefelt-Winther. 2014. A conceptual and methodological critique of internet addiction research: Towards a
model of compensatory internet use. Computers in human behavior 31 (2014), 351–354.
[7] Christoph Klimmt, Dorothée Hefner, and Peter Vorderer. 2009. The video game experience as “true” identification: A
theory of enjoyable alterations of players’ self-perception. Communication theory 19, 4 (2009), 351–373.
[8] Stefan Koelsch. 2015. Music-evoked emotions: principles, brain correlates, and implications for therapy. Annals of the
New York Academy of Sciences 1337, 1 (2015), 193–201.
[9] Mehmet Kosa and Ahmet Uysal. 2020. Four pillars of healthy escapism in games: Emotion regulation, mood manage-
ment, coping, and recovery. In Game user experience and player-centered design. Springer, 63–76.
[10] Caio Nunes and Ticianne Darin. 2024. Echoes of player experience: A literature review on audio assessment and
player experience in games. Proceedings of the ACM on Human-Computer Interaction 8, CHI PLAY (2024), 1–27.
[11] Anthony Prechtl, Robin Laney, Alistair Willis, and Robert Samuels. 2014. Methodological approaches to the evaluation
of game music systems. In Proceedings of the 9th audio mostly: a conference on interaction with sound. 1–8.
[12] Grazia Ragone, Judith Good, and Katherine Howland. 2020. OSMoSIS: Interactive sound generation system for children
with autism. In Proceedings of the 2020 ACM Interaction Design and Children Conference: Extended Abstracts. 151–156.
[13] Leonard Reinecke. 2009. Games and recovery: The use of video and computer games to recuperate from stress and
strain. Journal of media psychology 21, 3 (2009), 126–142.
[14] Giovanni Ribeiro, Katja Rogers, Maximilian Altmeyer, Thomas Terkildsen, and Lennart E Nacke. 2020. Game atmo-
sphere: effects of audiovisual thematic cohesion on player experience and psychophysiology. In Proceedings of the
annual symposium on computer-human interaction in play. 107–119.
[15] Katja Rogers, Giovanni Ribeiro, Rina R Wehbe, Michael Weber, and Lennart E Nacke. 2018. Vanishing importance:
studying immersive effects of game audio perception on player experiences in virtual reality. In Proceedings of the 2018
CHI conference on human factors in computing systems. 1–13.
[16] Katja Rogers and Michael Weber. 2019. Audio habits and motivations in video game players. In Proceedings of the 14th
International Audio Mostly Conference: A Journey in Sound. 45–52.
[17] Richard M Ryan and Edward L Deci. 2017. Self-determination theory: Basic psychological needs in motivation, development,
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[19] Nick Yee. 2006. Motivations for play in online games. CyberPsychology & behavior 9, 6 (2006), 772–775.
Received 20 February 2025; revised 12 March 2025; accepted 5 June 2025
5
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If You Hold Me Without Hurting Me: Pathways to Designing Game Audio for Healthy Escapism and Player Well-being∗ CAIO NUNES, Federal á, Brazil BOSCO BORGES, Federal á, Brazil GEORGIA CRUZ, Federal á, Brazil TICIANNE DARIN, Federal á, Brazil Escapism in games can support recovery or lead to harmful avoidance. Self-regulation, understood as combining autonomy with positive outcomes, is key to this distinction. We argue that audio, often overlooked, plays a central role in regulation. It can modulate arousal, mark transitions, and provide closure, yet its contribution to well-being remains underexplored. This paper identifies methodological and accessibility gaps that limit recognition of audio's potential and outlines ways to address them. We aim to encourage researchers and developers to integrate audio more deliberately into the design and study of healthier escapist play. CCS Concepts: • Human-centered computing →HCI design and evaluation methods; • Applied computing →Sound and music computing. Additional Key Words and Phrases: Escapism, Emotions, Self-Regulation, Game Audio 1 INTRODUCTION Escapism has long been recognized as one of the main motivations for gameplay, appearing across models of player motivation and media use [4, 7, 19]. It is often described as the tendency to detach from everyday concerns or stressors, yet its consequences are ambivalent. On the one hand, escapism can offer distraction, stress relief, and opportunities for affective regulation [2]. On the other hand, it can reinforce avoidance and interfere with everyday functioning. Recent work emphasizes this tension by distinguishing between healthy escapism, when play supports recovery and well-being, and subversive escapism, when it undermines them [9]. This perspective shifts the debate from whether escapism is good or bad to a more pressing question: "Which factors determine when escapist play sustains well-being and when it becomes harmful?" One of the most decisive factors in this distinction is self-regulation. In Self-Determination Theory (SDT), self-regulation refers to the extent to which behavior is experienced as autonomous, reflecting choice and volition rather than external pressure or compulsion [17]. Applied to games, this perspective captures whether players can autonomously decide when to begin, how long to continue, and when to disengage. Research consistently shows that difficulties in regulation are linked to excessive or compulsive gaming [6]. At the same time, successful self-regulation allows games to function as bounded recesses that facilitate recovery, coping, and positive mood management [9, 13]. Framing escapism through this lens highlights autonomy as a key condition for distinguishing between healthy and subversive play. However, a player could autonomously choose to play for a duration that negatively impacts their life responsibilities. In this case, the behavior is self-determined, but the outcome is detrimental. Hence, a healthy escapism also requires positive outcomes, where psychological relief (e.g., reported mood improvement) is supported by physiological recovery (e.g., stress marker reduction), independent of the feeling of volition during play. Consequently, we draw attention to how the design of game elements can either scaffold a player's ability to self-regulate or undermine it. We argue that audio is a key game element that can be intentionally designed to structure the escapist experience itself, facilitating autonomous self-regulation and positive ∗Presented and discussed at the CHI PLAY 2025 Workshop Exploring Future Directions for Healthy Escapism and Self-Regulation in Games, Pittsburgh, USA, October 13, 2025. 1 16 Oct 2025 Nunes et al. outcomes. While often perceived as secondary to visual or mechanical elements, audio possesses unique qualities that make it a potent regulatory channel. Unlike visual cues, which require focused attention, sound is pervasive and processed with a lower cognitive load, capable of influencing arousal and emotional states subconsciously, creating a direct tension between systemic regulation and player autonomy that design must address. The well-documented link between music and physiological responses, from emotion regulation and stress recovery to everyday relaxation [5, 8]. This makes audio a prime channel for influencing the very recovery and emotion regulation that are central to healthy escapism [9]. Besides, audio's ability to structure time and rhythm transitions offers a subtle yet material means of scaffolding player awareness without disrupting immersion. Yet, despite its unique potential, the role of audio in player well-being remains poorly understood. Much of the game-audio literature concentrates on immersion, presence, arousal, and affect, seldom linking these outcomes to well-being or self-regulatory processes [10]. This gap is especially evident in discussions of escapism and self-regulation, which often emphasize constructs such as absorption, affective modulation, stress relief, and distraction [3, 18]. While these emphases are not identical, they share a concern with how players become deeply involved and how their emotional states are shaped during play. This lack of synthesis leaves a critical gap, preventing designers and researchers from systematically using audio as a tool for player well-being. This position paper moves to fill that gap. We contend that audio is not a peripheral feature but a central, designable mechanism for scaffolding healthy escapism in games. To substantiate this claim, the remainder of this paper will first identify methodological and accessibility gaps previously identified in game audio research that perpetuate this problem. We then propose three concrete directions for future work focused on using audio to support self-regulation and positive emotional outcomes. 2 HOW RESEARCH GAPS OBSCURE AUDIO'S ROLE IN HEALTHY ESCAPISM In a previous study, we reviewed how audio and player experience are evaluated [10]. The results show a fragmented evidence base where game audio is routinely linked to immersion and affect, but its deeper impact is unclear. A key problem is that self-report measures often fail to converge with physiological data; for example, players may report feeling calm while their arousal markers remain elevated [10]. These inconsistencies make it challenging to draw conclusions about outcomes such as coping, stress recovery, or emotion regulation. This methodological fragmentation is compounded by other persistent issues, including a lack of personalization and the tendency for audio to be overshadowed in high-sensory contexts [11, 14-16]. Together, these findings reveal that audio is seen as an immersive aesthetic but not perceived as a potential lever for self-regulatory outcomes. The real-world impact of this research gap is starkly visible in games designed for sensitive audiences. Our literature review (n=49) on games for autistic players [1], for instance, shows that audio is often absent from game design descriptions and from user evaluation, despite its potential to support engagement or, if mishandled, trigger overload in audiences with auditory hypersensitivity. Across the surveyed studies, only 14% (n=7) explicitly evaluated audio; when they did, its uses ranged from background music to movement sonification, yet the impacts on the target audience were seldom reported in depth [12]. Such an absence of systematic evaluation and reporting reveals a double gap: a lack of design transparency and potentially a neglect of accessibility. For that audience, without customization options or sensitivity to diverse auditory needs, audio risks becoming a barrier instead of a resource. These findings highlight how sound design, if overlooked, can miss therapeutic opportunities and expose players to overstimulation or discomfort. This case starkly illustrates how neglecting audio can undermine a player's ability to self-regulate, turning a potentially healthy escape into a subversive, dysregulating experience. 2 If You Hold Me Without Hurting Me While these examples are not exhaustive, the gaps identified in the literature prevent us from measuring whether audio supports well-being or causes distress. Methodological blind spots mean that we lack ways to evaluate if audio is contributing to the positive outcome of healthy escapism, whether audio supports coping and recovery or simply intensifies immersion. Accessibility gaps can imply that sound becomes overwhelming or exclusionary, rather than restorative. In both cases, escapism risks sliding toward avoidance and overuse - not because audio cannot support healthier outcomes, but because research and design fall short in providing the tools to differentiate them. The absence of variation, closure, and inclusive controls leaves players to self-manage through workarounds such as muting or replacing game soundtracks [16], undermining the potential of audio to contribute directly to constructive escapist experiences. Ultimately, players are left without the necessary scaffolds to exercise their autonomy due to a lack of key tools, such as methods for measuring player responses to audio cues or designs that utilize audio to create natural stopping points. For instance, if safe exit points are signaled by a shift to calmer music in a hub area, it creates a clear closing motif after a mission, or even a brief "breathing window"" of silence after a moment of high tension, all of which provide a sense of closure that could help players disengage with a sense of accomplishment. Without such intentional design, poorly tested or repetitive audio can instead reinforce compulsive immersion or force avoidance, both of which undermine self-regulation. These blind spots leave players without the necessary scaffolds to disengage autonomously, showing how overlooked audio can shift escapism from recovery to overuse. 3 TOWARDS SELF-REGULATION AND HEALTHY ESCAPISM WITH AUDIO The gaps highlighted reveal a significant methodological deficiency: despite audio's clear influence, we remain unable to reliably determine its impact on recovery versus its potential to undermine regulation. Strategies are absent that repurpose sound as a design utility for player empowerment and healthy boundaries. To resolve this issue, we propose three core directions focusing on addressing methodological pitfalls, enhancing reporting transparency, and solidifying collaborative ties between research and industry. First, adopt multi-modal evaluation: self-reports, behavioral traces, and psychophysiological markers each provide partial insights, but their combination offers a fuller picture of audio's regulatory role. Logs of muting, playlist swaps, or sudden volume drops can reveal when players regulate their own auditory load. Coupled with physiological markers and short probes, such as "Do you feel calmer?" or "Are you ready to stop?" one can test whether sound genuinely supports coping or merely shifts physiology without providing felt relief. These behaviors can act as practical, scalable proxies for laboratory measures. Convergences and divergences are equally informative, allowing researchers to identify when design succeeds and giving developers concrete indicators of overload or recovery. Second, report design intentions transparently. Findings remain anecdotal when studies only state that sound was present. Reporting must clarify how audio was crafted and with what regulatory purpose to subconsciously guide players toward recovery, or to provide explicit cues for conscious self-regulation: a low-tempo loop added after combat to support recovery, a brief silence window following repeated failures to encourage pause, or a closing motif marking the transition to a safe hub area. By grounding technical decisions in regulatory intent, researchers can accumulate evidence across contexts and developers can align design choices with player well-being, especially in sensitive cases where unexamined audio risks overstimulation [1]. Third, foster a shared agenda between research and industry. Rather than prescribing isolated techniques, advancing this field requires collaboration and teamwork. Initiatives such as joint 3 Nunes et al. workshops, repositories of customizable regulatory audio patterns and adaptive sound systems, or co-produced guidelines, similar to existing accessibility standards, can foster cumulative knowledge. Together, these commitments elevate audio from a mere backdrop to an active mediator of escapist play. They reposition sound as a critical lever for discerning the balance between recovery and over-engagement, underscoring that without systematic evidence, designers still lack the means to use it intentionally for player well-being. To drive this agenda, we invite the community to consider: (1) Which combinations of self-report, behavioral, and physiological measures best capture audio's regulatory role? (2) How might soundscapes, finality cues in UI sounds, and music be designed to make transitions between immersion and disengagement more intentional? (3) What forms of player control over audio foster genuine autonomy in shaping escapist experiences? (4) Which standardized or recurring sound patterns are being used unreflectively, and how might their effects on well-being be better understood? AUTHOR BIOGRAPHIES Caio Nunes is a researcher with a Master's degree in Computer Science from Federal á, Brazil. His work focuses on Human-Computer Interaction (HCI) and Game User Research, exploring audio, musical visualizations, motivation, and well-being in emotional communication. Bosco Borges is a Master's student at Federal á, Brazil. His work focuses on HCI and Game User Research, examining how sound and interaction design can support player experience, promote healthier engagement, and cater to diverse audiences. Georgia Cruz is an Adjunct Professor at Federal á, Brazil. Her work focuses on UX, Audiovisual, and Communicability, exploring user experience, gender in games, and critical play, with an emphasis on how interactive media can foster reflection and influence well-being. Ticianne Darin is an Associate Professor of HCI at the Federal á. Her game research examines the foundational components of player experience, from motivation and engagement to the pursuit of eudaimonia through reflexive games. She is currently focused on developing ethical alternatives to manipulative design to foster digital well-being. Her leadership in the Brazilian research community includes chairing the nation's primary HCI symposia and founding the WIPlay workshop series, a dedicated venue to unite the country's HCI and games researchers. As a UNESCO consultant, she helps shape national policy on digital rights, focusing on implementing "Child and Adolescent Rights by Design" frameworks for games and online services. REFERENCES [1] Bosco Borges and Ticianne Darin. 2025. Sounds in Game Design and Evaluation for Autistic People: A Literature Review. In International Workshop on Interaction and Player Research in Game Development. Springer, 29-48. [2] Lorelle Bowditch, Janine Chapman, and Anjum Naweed. 2018. Do coping strategies moderate the relationship between escapism and negative gaming outcomes in World of Warcraft (MMORPG) players? Computers in Human Behavior 86 (2018), 69-76. [3] Gordon Calleja. 2010. Digital games and escapism. Games and Culture 5, 4 (2010), 335-353. [4] Scott Caplan, Dmitri Williams, and Nick Yee. 2009. Problematic Internet use and psychosocial well-being among MMO players. Computers in human behavior 25, 6 (2009), 1312-1319. [5] Hiroharu Kamioka, Kiichiro Tsutani, Minoru Yamada, Hyuntae Park, Hiroyasu Okuizumi, Koki Tsuruoka, Takuya Honda, Shinpei Okada, Sang-Jun Park, Jun Kitayuguchi, et al. 2014. Effectiveness of music therapy: a summary of systematic reviews based on randomized controlled trials of music interventions. Patient preference and adherence (2014), 727-754. 4 If You Hold Me Without Hurting Me [6] Daniel Kardefelt-Winther. 2014. A conceptual and methodological critique of internet addiction research: Towards a model of compensatory internet use. Computers in human behavior 31 (2014), 351-354. [7] Christoph Klimmt, Dorothée Hefner, and Peter Vorderer. 2009. The video game experience as "true" identification: A theory of enjoyable alterations of players' self-perception. Communication theory 19, 4 (2009), 351-373. [8] Stefan Koelsch. 2015. Music-evoked emotions: principles, brain correlates, and implications for therapy. Annals of the New York Academy of Sciences 1337, 1 (2015), 193-201. [9] Mehmet Kosa and Ahmet Uysal. 2020. Four pillars of healthy escapism in games: Emotion regulation, mood management, coping, and recovery. In Game user experience and player-centered design. Springer, 63-76. [10] Caio Nunes and Ticianne Darin. 2024. Echoes of player experience: A literature review on audio assessment and player experience in games. Proceedings of the ACM on Human-Computer Interaction 8, CHI PLAY (2024), 1-27. [11] Anthony Prechtl, Robin Laney, Alistair Willis, and Robert Samuels. 2014. Methodological approaches to the evaluation of game music systems. In Proceedings of the 9th audio mostly: a conference on interaction with sound. 1-8. [12] Grazia Ragone, Judith Good, and Katherine Howland. 2020. OSMoSIS: Interactive sound generation system for children with autism. In Proceedings of the 2020 ACM Interaction Design and Children Conference: Extended Abstracts. 151-156. [13] Leonard Reinecke. 2009. Games and recovery: The use of video and computer games to recuperate from stress and strain. Journal of media psychology 21, 3 (2009), 126-142. [14] Giovanni Ribeiro, Katja Rogers, Maximilian Altmeyer, Thomas Terkildsen, and Lennart E Nacke. 2020. Game atmosphere: effects of audiovisual thematic cohesion on player experience and psychophysiology. In Proceedings of the annual symposium on computer-human interaction in play. 107-119. [15] Katja Rogers, Giovanni Ribeiro, Rina R Wehbe, Michael Weber, and Lennart E Nacke. 2018. Vanishing importance: studying immersive effects of game audio perception on player experiences in virtual reality. In Proceedings of the 2018 CHI conference on human factors in computing systems. 1-13. [16] Katja Rogers and Michael Weber. 2019. Audio habits and motivations in video game players. In Proceedings of the 14th International Audio Mostly Conference: A Journey in Sound. 45-52. [17] Richard M Ryan and Edward L Deci. 2017. Self-determination theory: Basic psychological needs in motivation, development, and wellness. Guilford Publications. [18] Frode Stenseng, Jonas Falch-Madsen, and Beate Wold Hygen. 2021. Are there two types of escapism? Exploring a dualistic model of escapism in digital gaming and online streaming. Psychology of Popular Media 10, 3 (2021), 319. [19] Nick Yee. 2006. Motivations for play in online games. CyberPsychology & behavior 9, 6 (2006), 772-775. Received 20 February 2025; revised 12 March 2025; accepted 5 June 2025 5
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2510.14690
|
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
ON COMPACT HERMITIAN MANIFOLDS
YUETONG FANG
Abstract. Let (X, ω) be a compact Hermitian manifold of dimension n. We derive an
L∞-estimate for bounded solutions to the complex m-th Hessian equations on X, assuming
a positive right-hand side in the Orlicz space L
n
m (log L)n(h◦log ◦log L)n, where the associ-
ated weight satisfies Ko lodziej’s Condition. Building upon this estimate, we then establish
the existence of continuous solutions to the complex Hessian equation under the prescribed
assumptions.
Introduction
Let (X, ω) be a compact complex Hermitian manifold of dimension n equipped with a
Hermitian metric ω. Fix 0 < m ≤n and let dVX be a smooth volume form. The problem
of interest in this paper is to find a constant c > 0 and continuous (ω, m)-subharmonic
solutions to the following complex m-Hessian equation
(ω + ddcu)m ∧ωn−m = ceλufdVX, λ ≥0,
(Hessian)
under the assumption that the densities fn/m belong to an Orlicz space Lχ with the weight
χ satisfying Ko lodziej’s condition, which will be explained later.
When m = 1, this simplifies to the Poisson equation, while when m = n, it reduces to the
Monge–Amp`ere equation. Yau’s solution [Yau78] of the Calabi Conjecture has motivated
many subsequent studies on Monge–Amp`ere equations over the past fifty years. As presented
by B locki [B lo12], using the simplifications due to Kazdan and to Aubin–Bourguignon, Yau’s
a priori estimate can be extended to cases where the densities f belong to Lp for p > n.
One of the breakthrough was achieved by Ko lodziej [Ko l98], who applied pluripotential
theory to obtain an L∞-estimate for Monge–Amp`ere equations when f belongs to some
Orlicz space, with the associated weight specifically satisfying Condition (K). As remarked
by Ko lodziej [Ko l98], such an a priori estimate is ”almost sharp”.
This integrability condition on densities f can be extended in the context of complex
Hessian equations. We say that a convex weight χ satisfies Ko lodziej’s condition (Condition
(K)) if it is increasing and
χ(t)
∼
t→+∞t(log(t + 1))n (h ◦log ◦log(t + 3))n ,
where the function h : R+ →R+ satisfying
´ +∞h−1(t)dt < +∞. In this paper, we focus on
the case where fn/m ∈Lχ. We refer the reader to Section 1 for more detailed discussions
on Orlicz spaces and Condition (K).
2020 Mathematics Subject Classification. 32W20, 32U05, 32Q15, 35A23.
Key words and phrases. Hermitian Geometry, Monge–Amp`ere equations and complex Hessian equations,
weak solution, a priori estimate, stability estimate.
1
arXiv:2510.14690v1 [math.CV] 16 Oct 2025
2
YUETONG FANG
The study of weak solutions to complex Monge–Amp`ere equations plays an important role
in the analysis of Ricci flows (see [TW15]). On compact K¨ahler manifolds, one can study the
modulus of continuity for solutions to complex Monge–Amp`ere equations. H¨older regularity
for such equations with densities in Lp was established in [DDG+14], and has been further
generalized to the case where the densities belong to certain Orlicz spaces, in [GPTW21]
via a PDE approach, and in [GGZ25] using pluripotential theory.
To prove the existence of continuous solution to the complex Monge–Amp`ere equation,
a key ingredient is the L∞-estimate.
Various techniques have been developed to obtain
L∞-estimate for Monge–Amp`ere equations on compact manifolds, including Yau’s [Yau78]
original proof, which employs Moser’s iteration scheme. Later, Ko lodziej [Ko l98] developed
L∞-estimates using properties of capacity and established an optimal integrability condition
for the density f (see also [EGZ08, DP10] for related generalizations).
Inspired by the
work of Chen–Cheng [CC21] on constant scalar curvature K¨ahler metrics, a very different
approach was introduced by Guo–Phong–Tong [GPT23], by comparing the given equation
to an auxiliary complex Monge–Amp`ere equation.
This PDE technique has since been
generalized in [GP24, Qia24].
An efficient approach was developed by Guedj–Lu [GL23,
GL24], using properties of envelopes, which can be adapted to the case where f ∈L logp L
with p > n, as noted in [GL24, Section 2]. More recently, they generalized this result to
Orlicz spaces satisfying Condition (K) on compact K¨ahler manifolds in [GL25b], using a
new approach that reduces Ko lodziej’s criterion to Yau’s theorem.
On the other hand, complex m-Hessian equations are natural generalizations of Monge–
Amp`ere equations. B locki [B lo05] established potential theory for m-Hessian equations in
domains of Cn. Later, Dinew–Ko lodziej [DK14] obtained a priori estimates and proved a
stability result for weak solutions of the complex Hessian equation in domains of Cn and
compact K¨ahler manifolds.
The modulus of continuity of solutions to complex Hessian
equations has been studied in [Ngu14,Cha16] with densities f ∈Lp, p > n/m, and has been
extended recently in [˚AC25] to the cases where f ∈Ln/m logp(L), p > 2n. On compact
Hermitian manifolds, Ko lodziej and Nguyen proved the existence of weak solutions with a
nonnegative right-hand side belonging to Lp, p > n/m, in [KN16], while the corresponding
result for Monge–Amp`ere equations was established in [KN15].
The purpose of this paper is to extend the above results to prove the continuity of bounded
solutions to the complex Hessian equation. We begin by establishing a subsolution lemma for
the Monge–Amp`ere equation, generalizing Lemma 2.1 of [GL23] to the case where f ∈Lχ,
with χ satisfying Condition (K). The lemma is stated as follows:
Lemma A (Subsolution Lemma). Assume χ satisfies Condition (K). Then there exist a
uniform constant s = s(χ) > 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω),
−1 ≤ψ ≤0, satisfying
(ω + ddcψ)n ≥s(χ)
g
∥g∥χ
dVX.
We obtain the Subsolution Lemma by gluing the solutions to the Dirichlet problem in
local context:
(ddcu)n =
f
∥f∥χ
dVX, in Ω;
u = 0, on ∂Ω.
(0.1)
Compared with the case where the densities f ∈Lp, working with Orlicz norms is more
involved, as it requires controlling the Orlicz norm of the solution u associated to the convex
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
3
conjugate of χ. To simplify the argument, we adapt an approach from the recent article by
Guedj–Lu [GL25b] to derive a local L∞-estimate for problem 0.1.
As a consequence of Lemma A, we obtain a uniform estimate for the complex Hessian
equation on a Hermitian manifold, where the densities f satisfy Ko lodziej’s optimal condi-
tion.
Theorem B (L∞-estimate). Let φ be a bounded (ω, m)-subharmonic function satisfying
(ω + ddcφ)m ∧ωn−m ≤fdVX,
(Hess)
where 0 ≤fn/m ∈Lχ with the weight χ satisfying Condition (K). Then
OscX(φ) ≤C,
where C depends on ω, n, m and
f
n
m
χ.
By generalizing an idea in [GL25a], we demonstrate that the L∞-estimate for complex
Hessian equations can be reduced to the corresponding estimate for an associated Monge–
Amp`ere equation:
(ω + ddcφ)n ≤fdVX,
(0.2)
where f ∈Lχ, and χ satisfies Condition (K). To obtain the L∞-estimate for the Monge–
Amp`ere equation, our main tool is the domination principle. The key novelty of our approach
lies in constructing an appropriate subsolution. Our approach avoids potential computa-
tional difficulties associated with Orlicz norms and presents a different method compared to
Ko lodziej’s original proof.
Once the L∞-estimate is obtained, we can derive the existence and regularity of bounded
solutions. This leads us to the second main result of the study:
Theorem C. Let 0 ≤fn/m ∈Lχ, with the weight function χ satisfying Condition (K).
Then there exist a constant c > 0 and a solution u ∈SHm(X, ω) ∩C0(X) satisfying the
complex Hessian equation
(ω + ddcu)m ∧ωn−m = ceλufdVX, λ ≥0.
Moreover, all bounded solutions to the above complex Hessian equation are continuous.
To prove the first statement of Theorem C, an essential component is to establish a
stability theorem (Theorem 3.1). For f ∈Lp, with p > n/m, such a stability result was
proven in [KN16]. In the case where f ∈Lχ and the weight function χ satisfies Condition
(K), the continuity of solutions to Monge–Amp`ere equations was studied in [KN21] by
adapting the method in [KN19, Theorem 3.1].
We point out that our approach to the
stability estimate is fundamentally different and remains of interest in the case where the
densities f ∈Lp, as discussed in Remark 3.2. Our main tool will be the Domination Principle
and the subsolution (Lemma A). Having established the stability estimate (Theorem 3.1),
we approximate the density f by the sequence of bounded functions min(f, j). We show
that the corresponding solutions uj converge uniformly to an (ω, m)-subharmonic function
u solving the equation (Hessian), thereby obtaining a continuous solution. The continuity
of all bounded solutions follows from the Domination Principle (Proposition 1.8).
4
YUETONG FANG
Organization of the paper. The paper is organized as follows. Section 1 gives preliminary
material on (ω, m)-subharmonic functions, as well as key concepts of Orlicz spaces. A local
L∞-estimate is established, and Lemma A and Theorem B are proved in Section 2.
In
section 3, we establish a stability estimate and subsequently investigate the continuity of
bounded solutions, leading to the proof of Theorem C.
Acknowledgment. The author thanks her supervisor, Hoang-Chinh Lu, for suggesting the
problem, valuable discussions during the preparation of the work, and useful comments that
improved the presentation. This research is part of a PhD program funded by the PhD
scholarship CSC-202308070110, and is partially supported by the projects Centre Henri
Lebesgue ANR-11-LABX-0020-01 and the KRIS project of Fondation Charles Defforey.
1. Preliminaries
In the whole paper, we let (X, ω) denote a compact Hermitian manifold of complex
dimension n ∈N∗, equipped with a Hermitian form ω. We denote by dVX a smooth volume
form on X. We use differential operators d = ∂+ ¯∂, and dc = i(¯∂−∂), so that ddc = 2i∂¯∂.
1.1. (ω, m)-subharmonic functions.
1.1.1. Definition. We recall the definition of (ω, m)-subharmonic functions following [GN18]
and briefly review some results related to both ω-plurisubharmonic and (ω, m)-subharmonic
functions that will be used later.
Fix an integer 1 ≤m ≤n. Fix Ωan open set in Cn. Given a hermitian metric ω on Ω, a
real (1, 1)-form α on X is called m-positive with respect to ω if at all points in X,
αk ∧ωn−k ≥0, ∀k = 1, · · · , m.
A C2(Ω) function u : Ω→R is called harmonic with respect to ω if ddcu ∧ωn−1 = 0 at all
points in Ω.
Definition 1.1. A function u : Ω→{−∞} ∪R is subharmonic with respect to ω if:
(a) u is upper semicontinuous and u ∈L1
loc(Ω);
(b) for every relatively compact open set D ⋐Ωand every function h ∈C0(D) that is
harmonic with respect to ω on D the following implication holds:
u ≤h on ∂D =⇒u ≤h in D.
Definition 1.2. A function φ : Ω→{−∞} ∪R is quasi-subharmonic with respect to ω if
locally φ = u + ρ, where u is subharmonic with respect to ω and ρ is smooth.
A function φ is α-subharmonic with respect to ω if φ is quasi-subharmonic with respect
to ω and (α + ddcφ) ∧ωn−1 ≥0 in the sense of distributions.
The positive cone Γm(ω) associated with the metric ω is defined by
n
γ real (1, 1)-form : γk ∧ωn−k > 0, k = 1, · · · , m
o
.
It follows from G˚arding’s inequality [G˚ar59] that if γ0, γ1, · · · , γm−1 ∈Γm(ω), then
γ0 ∧γ1 ∧· · · ∧γm−1 ∧ωn−m > 0.
One can write eω = γ1 ∧· · · ∧γm−1 ∧ω, and it is a strictly positive (n −1, n −1)-form on Ω.
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
5
Definition 1.3. A function φ : Ω→[−∞, +∞) is called (α, m)-subharmonic with respect to
ω if φ is α-subharmonic with respect to eω in Ωfor all eω of the form eωn−1 = γ1∧· · ·∧γm−1∧ω,
where γ1, · · · , γm−1 ∈Γm(ω).
A function u : X →R ∪{−∞} is called (α, m)-subharmonic on X if u is (α, m)-
subharmonic on each local chart U of X.
The set of all locally integrable functions on U which are (α, m)-subharmonic with respect
to ω in U is denoted by SHω,m(U, α). In many contexts, such as [GL25a], the forms ω and
α may differ.
However, in this paper, we focus on the set SHω,m(U, ω). To simplify the notations, we
denote by SHm(U, ω) the set of all (ω, m)-subharmonic functions with respect to ω in U.
The set of all ω-plurisubharmonic functions on U is denoted by PSH(U, ω) = SHn(U, ω).
For notational convenience, we denote ωu := ω + ddcu.
Remark 1.4. By G˚arding’s inequality [G˚ar59], if u ∈C2(X), then u is (ω, m)-subharmonic
with respect to ω on X if and only if the associated form ωu belongs to the closure of Γm(ω).
It is well known that when n = m, the set PSH(X, ω) is a closed subset of Lr(X), for all
r ≥1. For (ω, m)-subharmonic functions, the following L1-compactness result was proved
in [KN16, lemma 3.3].
Lemma 1.5. Let u ∈SHm(X, ω) be normalized by supX u = 0. Then there exists a uniform
constant A > 0 depending only on X, ω such that
ˆ
X
|u|ωn ≤A.
1.1.2. Domination principle. Analogous to the Monge–Amp`ere operators, the complex Hes-
sian operators also satisfy a domination principle. The domination principle for Hessian op-
erator associated with continuous (ω, m)-subharmonic functions is established by Guedj–Lu
in [GL25a, Section 2]. Following the demonstration that the Hessian operators for bounded
functions are well-defined, we show the domination principle associated with bounded (ω, m)-
sh functions here.
Lemma 1.6 (Maximum principle). Assume that u and v are bounded (ω, m)-subharmonic
functions. Then
(ω + ddc(max(u, v)))m ∧ωn−m ≥1{u>v}(ω + ddcu)m ∧ωn−m + 1{v≥u}(ω + ddcv)m ∧ωn−m.
In particular, if u ≤v, then,
1{v=u}(ω + ddcu)m ∧ωn−m ≤1{v=u}(ω + ddcv)m ∧ωn−m
In the case where u and v are continuous, the above lemma is proven in [GL25a, Lemma
1.10]. When u is bounded, thanks to the continuity of complex Hessian operators ( [KN23]),
we can approximate u by a sequence of decreasing smooth (ω, m)-subharmonic functions.
The comparison principle for u, v ∈SHm(Ω) ∩L∞(Ω) is introduced in [KN23, Section 6].
Proposition 1.7 (Comparison principle). Let u, v be bounded m-subharmonic functions
with respect to ω and such that lim infz→∂Ωu(z) −v(z) ≥0. Assume
(ddcv)m ∧ωn−m ≥(ddcu)m ∧ωn−m in Ω.
Then it follows that u ≥v in Ω.
Proof. See [KN23, Corollary 6.2].
□
6
YUETONG FANG
We will need a global version of domination principle:
Proposition 1.8 (Domination principle). Assume that u, v ∈PSH(X, ω) ∩L∞(X) satisfy
1{u<v} (ω + ddcu)m ∧ωn−m ≤c1{u<v} (ω + ddcv)m ∧ωn−m,
for some constant 0 ≤c < 1. Then u ≥v.
The case in which u, v are continuous has been treated in [GL25a, Section 2]. In the case
where u, v ∈L∞, our proof is essentially analogous to the one developed for the Monge–
Amp`ere operators in [BGL25].
Proof. we may assume v ≥1. We fix a constant a such that c < am < 1, and we prove that
u ≥av. The result then follows by letting a →1. Assume it is not the case and let {xj} be
a sequence converging to x0 ∈X with
lim
j→+∞(u (xj) −av (xj)) = ma := inf
X (u −av) < 0.
Let B be a small neighborhood of x0 and fix a smooth strictly plurisubharmonic function
ρ such that ρ = 0 on ∂B, |ρ| ≤v, and ddcρ ≤ω. One can take a holomorphic coordinate
chart around x0 which is biholomorphic to the unit ball and ρ(x) = b
|x|2 −1
, for a small
constant b. Observe that ρ < 0 in B. Consider now φ = av −(1−a)ρ. A direct computation
then shows that ω + ddcφ ≥0 in B. Also, by construction φ ≤v and so
{u < φ} ∩B ⊂{u < av −(1 −a)ρ} ⊂{u < v}.
Hence by assumption and using the fact that ω + ddc(−ρ) ≥0, we get
1{u<φ} (ω + ddcu)m ∧ωn−m ≤c1{u<φ} (ω + ddcv)m ∧ωn−m
≤am1{u<φ} (ω + ddcv)m ∧ωn−m
≤1{u<φ} (ω + ddcφ)m ∧ωn−m.
From this and the maximum principle (Lemma 1.6) we infer
(ω + ddcu)m ∧ωn−m ≤(ω + ddc max(u, φ))m ∧ωn−m in B.
It thus follows from the comparison principle (Proposition 1.7) that
inf
B (u −max(u, φ)) ≥lim inf
x→∂B (u −max(u, φ))(x)
Evaluating this inequality at xj and letting j →+∞, we arrive at
lim
j→+∞min (u (xj) −av (xj) + (1 −a)ρ (xj) , 0) ≥lim inf
x→∂B min (u (x) −av (x) , 0) .
Since by construction,
lim
j→+∞u (xj) −av (xj) + (1 −a)ρ (xj) = ma −(1 −a)b,
we infer that
ma −(1 −a)b ≥lim
x→∂B (u (x) −av (x)) ≥ma,
which leads to a contradiction.
□
As a direct consequence of the domination principle, we have
Corollary 1.9. Fix λ > 0. If e−λvωm
v ∧ωn−m ≥e−λuωm
u ∧ωn−m, then u ≥v.
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
7
Proof. Fix C > 0. On the set {u < v −C} we have
ωm
u ∧ωn−m ≤eλ(u−v)ωm
v ∧ωn−m ≤e−λCωm
v ∧ωn−m.
It follows from Proposition 1.8 that u ≥v −C. Since this holds for all C > 0, we conclude
that u ≥v.
□
Corollary 1.10. Assume u, v are bounded (ω, m)-subharmonic functions on X such that
(ω + ddcu)m ∧ωn−m ≤c (ω + ddcv)m ∧ωn−m
for some positive constant c. Then c ≥1.
1.1.3. Cegrell classes. In Section 2, we study the Monge–Amp`ere equation in a local context.
Here, we recall some essential definitions of the Cegrell classes [Ceg98,Ceg04].
Fix a bounded hyperconvex domain Ω⊂Cn. Let T (Ω) denote the set of bounded plurisub-
harmonic functions u in Ωsuch that limz→ζ u(z) = 0 for every ζ ∈∂Ω, and
´
Ω(ddcu)n < +∞.
A function u belongs to F(Ω) iff there exists a sequence uj ∈T (Ω) decreasing to u in all
of Ω, which satisfies supj
´
Ω(ddcuj)n < +∞.
As demonstrated by Cegrell [Ceg98,Ceg04], the complex Monge–Amp`ere operator can be
defined for functions belonging to the class F(Ω) even when these functions are unbounded.
The Dirichlet problem in F(Ω) was solved in [Ceg04]:
Theorem 1.11. Assume that µ is a positive measure on Ω. If µ(Ω) < +∞and µ vanishes
on all pluripolar sets, then there exists a unique function u ∈F(Ω) such that (ddcu)n = µ.
Proof. See [Ceg04, Lemma 5.14].
□
1.2. Orlicz space. We recall some essential facts about Orlicz spaces that will be used
throughout, following [Dar19]. Let µ be a probability measure on X. A weight χ : R∗→
R∗∪{+∞} is called normalized Young weight if it is convex, lower semi-continuous, non-
trivial satisfying the normalizing condition
χ(0) = 0 and 1 ∈∂χ(1).
Recall that ∂χ(l) ⊂R denotes the set of subgradients of χ at l, which means that a ∈∂χ(t)
if and only if χ(t) + ab ≤χ(t + b) for all b ∈R. Its conjugate convex weight χ∗is the
Legendre transform of χ:
χ∗(h) = sup
t∈R∗{ht −χ(t)}.
One can easily verify that the following propositions hold:
Proposition 1.12. Let χ be a normalized Young weight. Then
(i) χ∗is also a normalized Young weight;
(ii) (Fenchel–Moreau theorem). The biconjugate weight χ∗∗is also a normalized Young
weight, and χ∗∗= χ.
One can define a function space with respect to the Young function. Let Lχ(µ) denote
the space of measurable functions defined as follows:
Lχ(µ) :=
f : X →R∗∪{+∞} : ∃s > 0,
ˆ
X
χ(sf) < +∞
.
One may also define the norm (also known as Luxembourg norm) on Lχ(µ).
∥f∥χ,µ := inf
r > 0 :
ˆ
X
χ
f
r
≤χ(1)
.
8
YUETONG FANG
We remark that (Lχ(µ), ∥·∥χ,µ) constitutes a normed space and is complete, hence a Banach
space. In the remainder of the paper, we fix µ = dVX, a smooth probability measure, and
write (Lχ, ∥· ∥χ) for simplicity.
In [Ko l98], S.Ko lodziej developped a technique using pluripotential theory that enables
one to bound the solution to the Monge–Amp`ere equation by the Orlicz norm, ∥· ∥χ, of
the right-hand side, associated with a weaker integrable condition. We thus introduce the
Condition (K).
Definition 1.13 (Condition (K)). A weight χ : R+ →R+ satisfies condition (K) if it is
convex increasing and
χ(t)
∼
t→+∞tn/m (log(1 + t))n (h ◦log ◦log(1 + t))n , where
ˆ +∞dt
h(t) < +∞.
Example 1.14. We discuss and present a few typical examples:
(i) When χ(t) = tp
p for p > 1, its convex conjugate is given by χ∗(t) = tq
q , where q =
p
p−1
is the H¨older conjugate of p. The Orlicz norm ∥· ∥χ here is the usual Lp norm.
(ii) The weight χ defined in the Condition (K) above is a Young function hence induce
an Orlicz space.
In particular, the weight χ(t) = t logp(1 + t) with p > n is a
Young function. One can check that as t →+∞, the conjugate weight grows like
χ∗(t) = t1−1
p exp(t
1
p ).
We will focus on the Orlicz space (Lχ, ∥· ∥χ) in section 3. It is important to note that in
such an Orlicz space, the H¨older inequality (or H¨older–Young inequality) holds.
Proposition 1.15 (H¨older–Young inequality). For f ∈Lχ and g ∈Lχ∗, we have
ˆ
X
|fg|dVX ≤∥f∥χ∥g∥χ∗.
Proof. See [Dar19, Proposition 1.3].
□
We will need the following additive Young inequality (also known as Fenchel–Young in-
equality).
Proposition 1.16. For every a, b ∈R+, it holds that
ab ≤χ(a) + χ∗(b),
with equality if and only if b ∈∂χ(a).
In particular, let f, g : X →R+ be two measurable functions. Then
f
(χ∗)−1 ◦g
≤χ ◦f + g.
1.3. Envelopes. Given a Lebesgue measurable function l : X →R, the ω-plurisubharmonic
envelope of l is defined by
Pω(l) := sup{u ∈PSH(X, ω) : u ≤l}.
The (ω, m)-subharmonic envelope of l is defined by
Pω,m(l) := sup{u ∈SHm(X, ω) : u ≤l}.
We now summarize the basic properties of these envelopes, following [GL25a].
Proposition 1.17. If l : X →R is continuous, then
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
9
(1) The complex Monge–Amp`ere measure (ω +ddcPω(l))n is concentrated on the contact
set {Pω(l) = l}.
(2) The complex Hessian measure (ω + ddcPω,m(l))m ∧ωn−m is concentrated on the
contact set {Pω,m(l) = l}.
(3) If l is C1,1-smooth, then Pω(l) is C1,1-smooth and
(ω + ddcPω(l))n = 1{Pω(l)=l}(ω + ddcl)n.
(4) If l is C1,1-smooth, then Pω,m(l) is C1,1-smooth and
(ω + ddcPω,m(l))m ∧ωn−m = 1{Pω,m(l)=l}(ω + ddcl)m ∧ωn−m.
For detailed proofs and additional discussion, we refer the reader to [Ber19,CM21,CZ19,
GL22, GLZ19, Tos18]. As a consequence of the preceding results, we will make use of the
following statements in the sequel.
Lemma 1.18. Fix λ > 0. Let u, v ∈SHm(X, ω) ∩L∞(X).
(1) If (ω + ddcu)m ∧ωn−m ≥eλufdVX and (ω + ddcv)m ∧ωn−m ≥eλugdVX, then
(ω + ddc(max(u, v)))m ∧ωn−m ≥eλ max(u,v) min(f, g)dVX.
(2) If (ω + ddcu)m ∧ωn−m ≤eλufdVX, and (ω + ddcv)m ∧ωn−m ≤eλvgdVX, then
(ω + ddc(Pω,m(min(u, v)))m ∧ωn−m ≤eλPω,m(min(u,v)) max(f, g)dVX.
The proof follows the same strategy as that for the Monge–Amp`ere measures presented
in [GL22, Lemma 1.9].
Proof. Let ψ := max(u, v). As a consequence of the maximum principle (Lemma 1.6), we
have
(ω + ddcψ)m ∧ωn−m ≥1{u≥v}(ω + ddcu)m ∧ωn−m + 1{v>u}(ω + ddcv)m ∧ωn−m
≥1{u≥v}eλufdVX + 1{v>u}eλvgdVX
=1{u≥v}eλψfdVX + 1{v>u}eλψgdVX ≥eλψ min(f, g)dVX.
Set φ := Pω,m(min(u, v)). We prove the second statement. It follows from the proposi-
tion 1.17 that the complex Hessian measure (ω + ddcφ)m ∧ωn−m has support in
{Pω,m(min(u, v)) = min(u, v)} = {Pω,m(min(u, v)) = u < v} ∪{Pω,m(min(u, v)) = v}.
We therefore obtain that
(ω + ddcφ)m ∧ωn−m ≤1{φ=u<v}(ω + ddcφ)m ∧ωn−m + 1{φ=v}(ω + ddcφ)m ∧ωn−m. (1.1)
Moreover, since φ ≤u and φ ≤v, the maximum principle (Lemma 1.6) yields
1{φ=u}(ω + ddcφ)m ∧ωn−m ≤1{φ=u}(ω + ddcu)m ∧ωn−m,
and
1{φ=v}(ω + ddcφ)m ∧ωn−m ≤1{φ=v}(ω + ddcv)m ∧ωn−m.
Together with (1.1), we deduce that
(ω + ddcφ)m ∧ωn−m ≤1{φ=u<v}(ω + ddcu)m ∧ωn−m + 1{φ=v}(ω + ddcv)m ∧ωn−m
≤1{φ=u<v}eλufdVX + 1{φ=v}eλvgdVX
≤1{φ=u<v}eλφfdVX + 1{φ=v}eλφgdVX ≤eλφ max(f, g)dVX.
□
10
YUETONG FANG
2. Uniform L∞-estimates
In this section, we aim to establish a uniform L∞-estimate for solutions φ ∈PSH(X, ω)∩
L∞(X) to the Monge–Amp`ere equation:
(ω + ddcφ)n ≤fdVX,
(MA)
assuming f ∈Lχ, and the associated weight function χ satisfies Condition (K). We then
generalize it to the complex Hessian equation.
For the sake of convenience, we fix
χ(t) = t logn(t + 1)(h ◦log ◦log(t + 3))n.
We fix a constant C0 > 0 such that ∥f∥χ ≤C0
´
X χ(f)dVX for all f ∈Lχ.
2.1. Local L∞-bound of Monge–Amp`ere operators. Let Ωbe the unit ball in Cn. We
define g :=
f
∥f∥χ to normalize f. We now consider the following Dirichlet problem to find a
plurisubharmonic solution u:
(ddcu)n = gdVX, in Ω;
u = 0, on ∂Ω.
(LocMA)
We now establish a local uniform estimate for u.
We start by establishing a useful re-
sult, which, for K¨ahler manifold, was shown in [DDL21, Theorem 3.3] and [GL25b, Lemma
2.2]. We present here a local version. Its proof is simpler than that of the global case, as
plurisubharmonicity is preserved under addition.
Lemma 2.1. Assume that f ∈Lp(Ω, dVX) for some p > 1, and that u is normalized so that
supΩu = 0 and u = 0 on ∂Ω. Suppose that there exists a bounded function v ∈PSH(Ω)
normalized so that supΩv = 0 and v = 0 on ∂Ω, such that
(ddcu)n ≤(ddcv)n + fdVX,
then u ≥−C, where C = C(n, p, ∥f∥p, ∥v∥∞).
Proof. Let ρ ∈PSH(Ω) be the solution to the Monge–Amp`ere equation
(ddcρ)n = fdVX in Ω; ρ = 0 on ∂Ω.
The uniform bound for ρ can be deduced from the results in [Ko l98,GL23]. Consider now
ψ = v + ρ. It is sufficient to show that ψ ≤u. Indeed, ψ satisfies
(ddcψ)n ≥(ddcv)n + (ddcρ)n = (ddcv)n + fdVX ≥(ddcu)n.
It thus follows from the Comparison Principle (Proposition 1.7) that u ≥ψ on Ω.
□
We are now prepared to prove a local uniform estimate.
Lemma 2.2 (Local uniform estimate for Monge–Amp`ere equations). Assume that u is a
bounded plurisubharmonic solution to the Dirichlet problem (LocMA). Then
OscΩu ≤C,
where C = C(n, Ω).
Proof. Let v ∈F(Ω) be such that
(ddcv)n = χ (g) dVX, in Ω; v = 0, on ∂Ω.
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
11
The existence of v ∈F(Ω) is ensured by Theorem 1.11. Let α > 0 be fixed (to be specified
later). On the set {log (g + 3) < −αv} we have
(ddcu)n ≤(e−αv −3)dVX.
We are going to show that e−αv ∈L2. To see this, observe that
ˆ
Ω
(ddcv)n =
ˆ
Ω
χ (g) dVX ≤χ(1).
(2.1)
We now choose α =
1
χ(1). It follows from [˚ACK+09, Theorem B] that
ˆ
Ω
e−2αv ≤C0(n, Ω).
We proceed to bound 1{log(g+3)≥−αv}(ddcu)n. Let Θ : R−→R−be a convex increasing
function. Then Θ ◦v satisfies
Θ ◦v ∈PSH(Ω), and (ddc(Θ ◦v))n =
Θ′′(v)dv ∧dcv + Θ′(v)ddcv
n ≥(Θ′ ◦v)n(ddcv)n.
On the set {log (g + 3) ≥−αv}, we deduce that
(ddc(Θ ◦v))n ≥(Θ′ ◦v)n(ddcv)n ≥
Θ′
−log (g + 3)
α
n
χ (g) dVX.
(2.2)
We aim to construct a uniformly bounded function Θ such that, for every t > 1, the following
inequality is satisfied:
Θ′
−t
α
n
tn(h ◦log t)n ≥2.
(2.3)
To see this, let x = log t and let
b(x) = −Θ
−ex
α
.
Then the condition (2.3) is equivalent to b′(x)h(x) ≥21/n
α , for all x > 0. We define b by
b(x) :=
ˆ x
0
21/n
αh(s)ds.
Since h−1 is integrable, the function b(x) remains bounded, which in turn implies that Θ is
uniformly bounded, as claimed. Taking t = log (g + 3) in (2.3) yields that
Θ′
−log (g + 3)
α
n
(log (g + 3))n (h ◦log log (g + 3))n ≥2.
(2.4)
Without loss of generality, we assume that there exist k0 > 0 such that
χ(k) ≥1
2k logn(k + 3)(h ◦log ◦log(k + 3))n, ∀k ≥k0.
From (2.2) and (2.4), it follows that on the set {log (g + 3) ≥−αv} ∩{g ≥k0},
(ddc(Θ ◦v))n ≥gdVX = (ddcu)n.
On the other hand, on the set {log (g + 3) ≥−αv} ∩{g < k0}, we deduce that
(ddcu)n < k0dVX.
From the preceding arguments, we arrive at the result that
(ddcu)n ≤(e−αv + k0)dVX + (ddc(Θ ◦v))n .
(2.5)
12
YUETONG FANG
It follows from Lemma 2.1 that OscΩu ≤C(n, Ω).
□
2.2. Global L∞- bounds. We now provide a proof of the global L∞bound for the solution
to the Monge–Amp`ere equations on Hermitian manifolds, that only relies on local solutions.
The strategy is to construct a bounded subsolution by solving equations on each small ball,
and then to show that the uniform boundedness of these subsolutions implies a uniform
bound for the solution, using domination principle. This method is adapted from the work
of Guedj–Lu, in [GL23]. We start by establishing a subsolution lemma.
Lemma 2.3 (Subsolution Lemma). Assume χ satisfies Condition (K). Then there exist a
uniform constant s = s(χ) > 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω),
−1 ≤ψ ≤0, with
(ω + ddcψ)n ≥s(χ)
g
∥g∥χ
dVX.
Proof. Cover X by balls, X ⊂∪N
j=1Bj. Choose B′
j = {ρj < 0}, where ρj : X →R are
smooth functions such that ω + ddcρj > 0 in a neighborhood of B′
j, and such that Bj ⋐B
′
j.
In B′
j, let vj ∈PSH(B′
j) be solution to Monge–Amp`ere equation
(ddcvj)n =
g
∥g∥χ
dVX in B′
j; vj = −1 on ∂Ω.
The local L∞-estimate (Lemma 2.2) implies OscB′
jvj ≤C′
j. We define ψj := max(vj, ξjρj),
where ξj > 1 for all j. Thus, ψj coincides with vj in Bj, and coincides with ξjρj in X \ B′
j
and in the neighborhood of ∂B′
j. Moreover, ψj is δ−1ω-plurisubharmonic for some uniform
constant δ > 0 in Bj. Taking ψ := 1
N
PN
j=1 δψj, and taking 0 < s(χ) ≤δn
Nn , we infer that in
Bj,
ωn
ψ ≥δn
Nn (ω + ddcψj)n ≥s(χ)(ddcvj)ndVX = s(χ)
g
∥g∥χ
dVX.
Therefore, ψ is the subsolution that we are looking for.
□
Theorem 2.4 (A priori estimates for Monge–Amp`ere equations). Let χ be as above, and
let f ∈Lχ with f ≥0. Suppose that φ ∈PSH(X, ω) ∩L∞(X) satisfies
(ω + ddcφ)n ≤fdVX.
Let s := s(χ) be the constant in Lemma 2.3 and assume that A > 0 satisfies
∥1{f>A}f∥χ ≤s(χ)
4 .
Then OscX(φ) ≤C, where C depends on A, and χ.
Proof. We consider the function
g =
1{f>A}f
∥1{f>A}f∥χ
.
Applying the Subsolution Lemma (Lemma 2.3) to g yields that there exists subsolution
ψ ∈PSH(X, ω), such that
(ω + ddcψ)n ≥sgdVX.
The dominated convergence theorem implies that
lim
t→+∞
ˆ
X
χ(1{f>t}f)dVX =
ˆ
X
lim
t→+∞χ
1{f>t}f
dVX = 0.
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
13
We can thus choose A sufficiently large, depending only on f and s, such that
∥1{f>A}f∥χ ≤s
4.
It follows that
(ω + ddcψ)n ≥41{f>A}(ω + ddcφ)n.
Therefore,
(ω + ddcφ)n ≤1
4(ω + ddcψ)n + AdVX.
Fix B > 0 large enough. It follows from [GL23, Lemma 2.5] that there exist ρ ∈PSH(X, ω), −1 ≤
ρ ≤0 such that
(ω + ddcρ)n ≥4A1{φ<−B}dVX.
Normalize φ such that supX φ = 0. Then on the set {φ < ψ+ρ
2
−B} ⊂{φ < −B}, we have
(ω + ddcφ)n ≤1
4(ω + ddcψ)n + 1
4(ω + ddcρ)n ≤1
2
ω + ddc ψ + ρ
2
n
.
We conclude from the domination principle (Proposition 1.8) that φ ≥ψ+ρ
2 −B ≥−1−B.
□
2.3. Global L∞- estimates for Hessian equations. The uniform bound for the Hessian
equation follows the method of [GL25a, Theorem 3.1], which assumes f ∈Lp with p > n/m.
A generalization to Orlicz spaces is outlined in Remark 3.2 therein. It has also been shown in
the recent article [GL25b, Theorem B] that this approach can be adapted in certain general
complex geometric PDEs. We provide a more detailed presentation in the proof of Theorem
B here.
The following lemma will be used to obtain a uniform estimate for bounded solutions.
Lemma 2.5. Assume f ∈L1(dVX). Let φ ∈SHm(X, ω) ∩L∞(X) be such that
(ω + ddcφ)m ∧ωn−m = fωn.
Then, we have (ω + ddcu)n ≤fn/mωn, where u = Pω(φ).
Proof. The proof proceeds in two steps.
We first treat the case where the density f is
bounded. In the general case, we approximate a given f ∈L1 by a sequence of bounded
densities. Normalize the solution φ such that supX φ = 0.
Step 1. Assume that f is bounded. Let {fj} be a sequence of smooth, uniformly bounded
functions such that fj →f in L2n. It follows from [KN16, Lemma 3.20] that there exists a
decreasing sequence of (ω, m)-subharmonic functions {ψj} such that ψj →φ. We consider
the following complex Hessian equation:
(ω + ddcφj)m ∧ωn−m = eφj−ψjfjωn.
The existence of smooth, (ω, m)-subharmonic solution φj follows from [Sz´e18]. Then, from
[KN16, Theorem 3.9] we deduce that φj converges uniformly to eφ ∈SHm(X, ω) ∩C0(X),
which is the solution to
(ω + ddc eφ)m ∧ωn−m = eeφ−φfωn.
By the domination principle (Proposition 1.8), we obtain that eφ = φ. Taking uj = Pω(φj),
it follows from Proposition 1.17 that uj ∈C1,1(X). The mixed Monge–Amp`ere inequality
[Ngu16, Lemma 1.19] implies that
(ω + ddcuj)n ≤e
n(φj−ψj)
m
fn/m
j
ωn.
14
YUETONG FANG
Since uj converges uniformly to u, passing to the limit, we conclude that
(ω + ddcu)n ≤fn/mωn.
Step 2. Assume now that f ∈L1(dVX). We consider the equation given by
(ω + ddcφj)m ∧ωn−m = eφj−φ min(f, j)ωn.
Theorem 3.19 of [KN15] ensures the existence of a bounded solution φj, satisfying φj ≥φ.
The domination principle yields that φj is decreasing to a (ω, m)-subharmonic function eφ.
It follows from the domination principle again that eφ = φ. Taking uj := Pω(φj), then uj
decreases to u. Moreover, Step 1 yields
(ω + ddcuj)n ≤en/m(φj−φ)fn/m
j
ωn.
Applying the continuity of Monge–Amp`ere operator along monotone sequences [BT76,BT82],
we obtain
(ω + ddcu)n ≤fn/mωn,
as required.
□
Proof of Theorem B. Normalize φ such that supX φ = 0. Let u = Pω (φ) and let C denote
the contact set C := {u = φ}. We remark that u ∈PSH(X, ω) ⊂SHm(X, ω). Therefore, it
follows from the mixed Monge–Amp`ere inequality [Ngu16, Lemma 1.19] and proposition 1.17
that
1C(ω + ddcu)m ∧ωn−m ≤1C(ω + ddcφ)m ∧ωn−m ≤fdVX
We remark that hX := ωn/dVX ≥δ0 is positive on X. Lemma 2.5 ensures us to take
g := (ω + ddcu)n /dVX. Then, the mixed Monge–Amp`ere inequality yields that
δ1−m/n
0
gm/n ≤h1−m/n
X
gm/n ≤f,
hence g ≤C0fn/m, so that g ∈Lχ0, where χ0 = t logn(1 + t)[h ◦log ◦log(3 + t)]n satisfies
Condition (K). It follows from Theorem 2.4 that
OscX(u) ≤A.
It suffices to bound supX u from below. Set
v−
A(ω) := inf
ˆ
X
(ω + ddcφ)n : φ ∈L∞(X) ∩PSH(X, ω), −A ≤φ ≤0
.
Recall that v−
A(ω) is strictly positive by [GL22, Proposition 3.4]. We denote by (χ∗
0)−1 the
inverse of χ∗
0. Then it follows from additive H¨older–Young inequality (Proposition 1.16) that
v−
A(ω)(χ∗
0)−1
−sup
X
u
≤
ˆ
X
(χ∗
0)−1(−u) (ω + ddcu)n
=
ˆ
X
(χ∗
0)−1(−u)gdVX
≤
ˆ
X
χ0(g)dVX +
ˆ
X
(−u)dVX.
We remark that
´
X(−u)dVX is uniformly bounded from above by Lemma 1.5. Thus we
obtain a uniform lower bound of supX u, which yields together with OscXu ≤A that |u| has
a uniform bound, and since u ≤φ, the solution φ is also uniformly bounded.
□
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
15
3. Weak solution
3.1. Stability estimates.
Theorem 3.1. Let u, v ∈SHm(X, ω)∩L∞(X) be such that supX u = 0 and v ≤0. Assume
that u ∈SHm(X, ω) is a solution to the complex Hessian equation
ωm
u ∧ωn−m = eλufωn,
(3.1)
where λ ≥0 and fn/m ∈Lχ with χ satisfying Condition (K). Then,
sup
X
(v −u)+ ≤C∥(v −u)+fn/m∥
1
n+1
χ
,
where the positive constant C depends only on n, m, ω, ∥f
n
m ∥χ, and supX |v|.
Proof. We consider only the case λ = 0; when λ > 0, the L∞-estimate implies that eλuf ≤f,
allowing the problem to be reduced to the case λ = 0. We now fix ϵ > 0, to be chosen later,
and define gϵ := 1{u<v−ϵ}fn/m.
By the subsolution lemma (Lemma 2.3), there exist an uniform constant s := s(χ), and
−1 ≤ψ ≤0 such that
(ω + ddcψ)n ≥sgϵdVX
∥gϵ∥χ
.
It follows from mixed Monge–Amp`ere inequality [Ngu16, Lemma 1.9] that
(ω + ddcψ)m ∧ωn−m ≥sm/n 1{u<v−ϵ}f
∥gϵ∥m/n
χ
.
We set φ := (1 −δ)v + δψ, where 0 < δ < 1 is to be specified later. Then there exists a
positive constant C2, depending only on ∥ψ∥∞and supX |v| such that
{φ > u + C2δ + ϵ} ⊂{u < v −ϵ}.
From the (ω, m)-subharmonicity of v and ψ, and applying [KN15, Lemma 2.3], we derive
that
1{φ>u+C2δ+ϵ}ωm
φ ∧ωn−m =1{φ>u+C2δ+ϵ} [(1 −δ)ωv + δωψ]m ∧ωn−m
=1{φ>u+C2δ+ϵ}
m
X
k=0
m
k
[(1 −δ)ωv]k ∧(δωψ)m−k ∧ωn−m
≥1{φ>u+C2δ+ϵ}δm(ωψ)m ∧ωn−m.
If ∥gϵ∥χ is not small, then the stability property follows trivially by adjusting C. Hence, we
can assume that ∥gϵ∥χ <
s
2n/m . Choosing δ = 21/m∥gϵ∥1/n
χ
s1/n
, we obtain that
1{φ>u+C2δ+ϵ}δm(ωψ)m ∧ωn−m ≥21{φ>u+C2δ+ϵ}fωn
=21{φ>u+C2δ+ϵ}ωm
u ∧ωn−m.
It follows from domination principle (Proposition 1.8) that
φ ≤u + C2δ + ϵ.
16
YUETONG FANG
Hence,
(v −u)+ ≤C3
21/m
s1/n
!
∥gϵ∥1/n
χ
+ ϵ
≤C4∥gϵ∥1/n
χ
+ ϵ
≤C4(ϵ)−1/n∥(v −u)+fn/m∥1/n
χ
+ ϵ.
Optimizing the right hand side by choosing
ϵ =
∥C4(v −u)+fn/m∥1/n
χ
n
!
n
n+1
,
we obtain
(v −u)+ ≤C
∥(v −u)+fn/m∥χ
1
n+1 .
The conclusion follows by taking the supremum of the left-hand side.
□
Remark 3.2. When χ(t) = tp/p, with p > 1, the Orlicz norm ∥· ∥χ is simply the Lebesgue
Lp-norm. Assume fn/m ∈Lp, for p > 1. Let χq(t) = tq/q for some 1 < q < p. Then
ψ ∈PSH(X, ω) is a subsolution such that
(ω + ddcψ)n ≥s′gωn, where g =
1{u<v−Bϵ}fn/m
∥1{u<v−Bϵ}fn/m∥q
.
Following the same procedure and applying the Chebyshev inequality to
(v −u)+ ≤C4∥(v −u)+gϵ∥1/n
q
+ ϵ,
we obtain that
(v −u)+ ≤
C′
(ϵ)
p−q
pqn
∥(v −u)+∥
p−q
pqn
1
∥fn/m∥1/n
p
+ ϵ.
Choosing an optimal ϵ = C′
1∥(v −u)+∥
p−q
(npq+p−q)
1
∥fn/m∥
pq
(npq+p−q)
p
, we deduce a stability esti-
mate
(v −u)+ ≤C
′′∥(v −u)+∥
p−q
(npq+p−q)
1
f
n
m
pq
(npq+p−q)
p
,
which slightly generalize the stability estimate in [KN16].
The following result is derived by applying the above theorem twice.
Corollary 3.3. Suppose that u, v ∈SHm(X, ω), normalized such that supX u = 0, supX v =
0, satisfy the complex Hessian equations:
ωm
u ∧ωn−m = eλufωn, ωm
v ∧ωn−m = eλvgωn,
where 0 < f
n
m , g
n
m ∈Lχ, and λ ≥0. Then,
∥u −v∥∞≤C
∥(v −u)+f
n
m ∥
1
n+1
χ
+ ∥(u −v)+g
n
m ∥
1
n+1
χ
,
where the constant C depends on n, m, ω, ∥f
n
m ∥χ and ∥g
n
m ∥χ.
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
17
3.2. Continuity of solutions. This subsection is devoted to the proof of Theorem C. We
approximate f by positive densities
fj = min(f, j).
It follows from the main theorem of [KN16] that for each j, there exist a continuous solution
uj and a constant cj > 0 such that
(ω + ddcuj)m ∧ωn−m = cjeλujfjdVX, λ ≥0.
We normalized uj such that supX uj = 0. Our first step is to bound the coefficients cj from
above and below. This requires an inequality involving mixed Hessian measures. It’s well
known that if φj ∈SHm(X, ω)∩C∞(X) and (ω +ddcφj)m ∧ωn−m = fjdVX, then G˚arding’s
inequality [G˚ar59] holds:
(ω + ddcφ1) ∧· · · ∧(ω + ddcφm) ∧ωn−m ≥(f1 · · · fm)1/mdVX.
In the case where φj’s are not smooth, we can apply an approximation argument to obtain
the following inequality.
Proposition 3.4. Let u ∈SHm(X, ω) be bounded functions such that
(ω + ddcφ)m ∧ωn−m ≥fωn.
Then we have that (ω + ddcφ) ∧ωn−1 ≥f1/mωn.
Proof. Let {gj} be a sequence of bounded, positive and smooth functions converging to f.
Let vj be a sequence of smooth, (ω, m)-subharmonic functions converging to φ. We consider
the following Hessian equation:
(ω + ddcφj)m ∧ωn−m = eφj−vjgjωn.
The existence of smooth solution φj follows from [Sz´e18]. Applying G˚arding’s inequality
[G˚ar59] yields
(ω + ddcφj) ∧ωn−1 ≥e
φj−vj
m
g1/m
j
ωn.
Moreover, we deduce from [KN16, Theorem 3.19] that φj converges uniformly to eφ ∈
SHm(X, ω). Passing to the limit as j →+∞, we obtain
(ω + ddceu)m ∧ωn−m = eeφ−φfωn.
The domination principle (Proposition 1.8) then implies eφ = φ. Consequently, we conclude
that
(ω + ddcφ) ∧ωn−1 ≥f1/mωn.
□
We now turn to obtaining uniform bounds for cj.
Lemma 3.5. The sequence of constant {cj} is uniformly bounded. In particular, the Orlicz
norm ∥cjfj∥χ is uniformly bounded from above.
Proof. We first bound cj from above. Fix δ0 > 0 such that δ0 ≤ωn/dVX. Proposition 3.4
implies that for each j,
(ω + ddcuj) ∧ωn−1 ≥c1/m
j
f1/m
j
ωn ≥δ0c1/m
j
f1/m
j
dVX.
18
YUETONG FANG
Thus, by taking the integral of both sides:
δ0c1/m
j
ˆ
X
f1/m
j
dVX ≤
ˆ
X
(ω + ddcuj) ∧ωn−1.
Note that uj ∈SHm(X, ω) ⊂SH1(X, ω), and the sequence (uj) is relatively compact in L1.
Moreover, we have
´
X f1/m
j
ωn →
´
X f1/mωn > 0 as j →+∞. It follows that there exist j0
large enough such that for each j > j0, the following inequality holds:
c1/m
j
≤
2
δ0
´
X f1/mdVX
ˆ
X
(ω + ddcuj) ∧ωn−1.
It suffices to bound the right hand side. Indeed, it follows from Stokes’ theorem that
ˆ
X
(ω + ddcuj) ∧ωn−1 =
ˆ
X
ωn +
ˆ
X
ujddc(ωn−1) ≤
ˆ
X
ωn + B
ˆ
X
|uj|ωn.
By the L1-compactness result for (ω, 1)-subharmonic functions (Lemma 1.5), we conclude
that the right hand side above is bounded. Thus cj is bounded from above.
We next show that {cj} is bounded away from 0. Let gj = fn/m
j
∈Lχ. By the subsolution
Lemma (Lemma 2.3), there exists a uniform constant δ > 0, such that
(ω + ddcvj)n ≥δjgjdVX.
Then, it follows from mixed Monge–Amp`ere inequality [Ngu16, Lemma 1.9] that
ωm
vj ∧ωn−m ≥δm/n
j
gm/n
j
ωn ≥δm/n
j
fjωn.
The domination principle (Proposition 1.8) yields that
δm/n
j
≤cj.
We thus obtain a uniform lower bound for cj’s, since δj’s are strictly positive, uniform
constants.
The uniqueness of cj for each fj follows directly from domination principle
(Proposition 1.8).
□
Uniform bound of uj. It follows from Theorem B that {uj} have uniform bound:
−C ≤uj ≤0.
By extracting, we can assume that {uj} converges in L1(X) to a bounded (ω, m)-subharmonic
function u, as j →+∞.
Existence of solutions. We now show that, by construction, the limit u is a solution to
the Hessian equation
(ω + ddcu)m ∧ωn−m = ceλufdVX, λ ≥0.
We first consider the simpler case in which λ > 0.
Theorem 3.6. Let λ > 0. Then u ∈SHm(X, ω) ∩L∞(X) is the unique solution such that
(ω + ddcu)m ∧ωn−m = eλufdVX.
Moreover u is uniformly bounded.
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
19
Proof. To simplify the notation, we assume that λ = 1. Then each uj satisfies
(ω + ddcuj)m ∧ωn−m = eujfjdVX.
From the domination principle (Proposition 1.8), we deduce that uj decreases to some
functions u ∈SHm(X, ω). Passing to the limit, it follows from the continuity of Hessian
operator along decreasing sequence ( [KN23, Section 5]) that
(ω + ddcu)m ∧ωn−m = eufdVX.
Applying Theorem B, we obtain a uniform bound for u.
□
Monotonicity of the sequence uj is not guaranteed when λ = 0.
Our approach is to
construct a supersolution and a subsolution and demonstrate that they coincide, as in [GL24,
Section 3] and [BGL25, Theorem 4.5].
Theorem 3.7 (Existence of bounded solution). Let f, u and the constant c be as above.
Then u ∈SHm(X, ω) ∩L∞(X) is a solution to
(ω + ddcu)m ∧ωn−m = cfdVX.
Moreover, u is uniformly bounded.
Proof. We define
ρj :=
sup
k≥j
uk
!∗
, vj := Pω,m
inf
k≥j uk
,
where Pω,m (infk≥j uk) denotes the largest (ω, m)-subharmonic function lying below infk≥j uk.
Observe that ρj is a decreasing sequence that converges to u, and vj is an increasing se-
quence that converges to an (ω, m)-subharmonic function v. In particular, by construction,
we have u ≥v. We are going to show that u = v.
Set c−
j f−
j := infk≥j ckfk. Then it follows from Lemma 1.18 that
(ω + ddcρj)m ∧ωn−m ≥c−
j f−
j dVX.
Then we deduce from the continuity of Hessian operator along monotone sequences (see
[KN23, Lemma 5.1]) that
(ω + ddcu)m ∧ωn−m ≥cfdVX.
(3.2)
On the other hand, we write (ω + ddcvj)m ∧ωn−m = evj−vjcjfjdVX.
Define c+
j f+
j
:=
supk≥j ckfk. It also follows from Lemma 1.18 that
(ω + ddcvj)m ∧ωn−m ≤evj−infk≥j ukc+
j f+
j dVX.
Let j →+∞. Together with (3.2) we deduce that
(ω + ddcv)m ∧ωn−m ≤ev−ucfdVX ≤ev−u(ω + ddcu)m ∧ωn−m.
By the domination principle (Proposition 1.8), we conclude that v ≥u. Hence, v = u is a
bounded solution to the complex Hessian equation:
(ω + ddcu)m ∧ωn−m = cfdVX.
The uniform bound for u follows from Theorem B.
□
20
YUETONG FANG
Proof of Theorem C. Existence of continuous solutions. We now prove that the sequence
{uj} converges uniformly to u as j →+∞. We recall that when λ > 0, the L∞-estimate
implies that eλuf ≤f, allowing us to reduce to the case where λ = 0. Fix j0 large enough.
For each j > j0, we infer by Theorem 3.1 that
(uj −u)+ ≤C1
(uj −u)+f
n
m
1
n+1
χ
, (u −uj)+ ≤C1
(u −uj)+f
n
m
j
1
n+1
χ
.
We are going to show that
(uj −u)+f
n
m
χ →0 as j →+∞. Fixing s > 0, we proceed to
compute the corresponding integrability.
ˆ
X
χ
(uj −u)+f
n
m
s
dVX
=
ˆ
X
(uj −u)+f
n
m
s
logn
1 + (uj −u)+f
n
m
s
h ◦log ◦log
3 + (uj −u)+f
n
m
s
n
dVX.
As u and uj’s are uniformly bounded, it follows that (uj −u)+ ≤C0 for some uniform constant C0,
and since h is increasing, we deduce that
ˆ
X
χ
(uj −u)+f
n
m
s
dVX ≤
ˆ
X
(uj −u)+f
n
m
s
logn
1 + C0f
n
m
s
·
h ◦log ◦log
3 + C0f
n
m
s
n
dVX.
By the dominated convergence theorem, we conclude that
(uj −u)+f
n
m
s
χ
→0, for all 0 < s < +∞,
and consequently
∥(uj −u)+f
n
m ∥
1
n+1
χ
→0.
Applying the same argument, we obtain that
(u −uj)+f
n
m
j
1
n+1
χ
≤
(u −uj)+f
n
m
1
n+1
χ
→0. There-
fore, {uj}∞
j=1 is a Cauchy sequence in SHm(X, ω) ∩C0(X). It follows that the bounded solution u
to the complex Hessian equation
(ω + ddcu)m ∧ωn−m = ceλufωn, where f ∈Lχ,
is continuous.
Continuity of bounded solutions. Assume now that v ∈SHm(X, ω) is another bounded solution
to the complex Hessian equation
(ω + ddcv)m ∧ωn−m = cfdVX.
Then v is continuous. Indeed, the uniform bound for v ensures that ef = e−vf ∈Lχ. We deduce
from the previous argument that there exist eu ∈SHm(X, ω) ∩C0(X) solving
(ω + ddceu)m ∧ωn−m = eeu efdVX.
It follows that
(ω + ddceu)m ∧ωn−m = eeu−vfdVX = eeu−v(ω + ddcv)m ∧ωn−m.
Applying the Domination Principle (Proposition 1.8) yields that v = eu. Therefore, v ∈C0(X), as
claimed.
□
CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS
21
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Email address: yuetong.fang@univ-angers.fr
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CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS ON COMPACT HERMITIAN MANIFOLDS YUETONG FANG Abstract. Let (X, ω) be a compact Hermitian manifold of dimension n. We derive an L∞-estimate for bounded solutions to the complex m-th Hessian equations on X, assuming a positive right-hand side in the Orlicz space L n m (log L)n(h◦log ◦log L)n, where the associated weight satisfies Ko lodziej's Condition. Building upon this estimate, we then establish the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions. Introduction Let (X, ω) be a compact complex Hermitian manifold of dimension n equipped with a Hermitian metric ω. Fix 0 0 and continuous (ω, m)-subharmonic solutions to the following complex m-Hessian equation (ω + ddcu)m ∧ωn-m = ceλufdVX, λ ≥0, (Hessian) under the assumption that the densities fn/m belong to an Orlicz space Lχ with the weight χ satisfying Ko lodziej's condition, which will be explained later. When m = 1, this simplifies to the Poisson equation, while when m = n, it reduces to the Monge-Amp`ere equation. Yau's solution [Yau78] of the Calabi Conjecture has motivated many subsequent studies on Monge-Amp`ere equations over the past fifty years. As presented by B locki [B lo12], using the simplifications due to Kazdan and to Aubin-Bourguignon, Yau's a priori estimate can be extended to cases where the densities f belong to Lp for p > n. One of the breakthrough was achieved by Ko lodziej [Ko l98], who applied pluripotential theory to obtain an L∞-estimate for Monge-Amp`ere equations when f belongs to some Orlicz space, with the associated weight specifically satisfying Condition (K). As remarked by Ko lodziej [Ko l98], such an a priori estimate is "almost sharp". This integrability condition on densities f can be extended in the context of complex Hessian equations. We say that a convex weight χ satisfies Ko lodziej's condition (Condition (K)) if it is increasing and χ(t) ∼ t→+∞t(log(t + 1))n (h ◦log ◦log(t + 3))n , where the function h : R+ →R+ satisfying ́ +∞h-1(t)dt n, as noted in [GL24, Section 2]. More recently, they generalized this result to Orlicz spaces satisfying Condition (K) on compact K ̈ahler manifolds in [GL25b], using a new approach that reduces Ko lodziej's criterion to Yau's theorem. On the other hand, complex m-Hessian equations are natural generalizations of MongeAmp`ere equations. B locki [B lo05] established potential theory for m-Hessian equations in domains of Cn. Later, Dinew-Ko lodziej [DK14] obtained a priori estimates and proved a stability result for weak solutions of the complex Hessian equation in domains of Cn and compact K ̈ahler manifolds. The modulus of continuity of solutions to complex Hessian equations has been studied in [Ngu14,Cha16] with densities f ∈Lp, p > n/m, and has been extended recently in [ ̊AC25] to the cases where f ∈Ln/m logp(L), p > 2n. On compact Hermitian manifolds, Ko lodziej and Nguyen proved the existence of weak solutions with a nonnegative right-hand side belonging to Lp, p > n/m, in [KN16], while the corresponding result for Monge-Amp`ere equations was established in [KN15]. The purpose of this paper is to extend the above results to prove the continuity of bounded solutions to the complex Hessian equation. We begin by establishing a subsolution lemma for the Monge-Amp`ere equation, generalizing Lemma 2.1 of [GL23] to the case where f ∈Lχ, with χ satisfying Condition (K). The lemma is stated as follows: Lemma A (Subsolution Lemma). Assume χ satisfies Condition (K). Then there exist a uniform constant s = s(χ) > 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω), -1 ≤ψ ≤0, satisfying (ω + ddcψ)n ≥s(χ) g ∥g∥χ dVX. We obtain the Subsolution Lemma by gluing the solutions to the Dirichlet problem in local context: (ddcu)n = f ∥f∥χ dVX, in Ω; u = 0, on ∂Ω. (0.1) Compared with the case where the densities f ∈Lp, working with Orlicz norms is more involved, as it requires controlling the Orlicz norm of the solution u associated to the convex CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 3 conjugate of χ. To simplify the argument, we adapt an approach from the recent article by Guedj-Lu [GL25b] to derive a local L∞-estimate for problem 0.1. As a consequence of Lemma A, we obtain a uniform estimate for the complex Hessian equation on a Hermitian manifold, where the densities f satisfy Ko lodziej's optimal condition. Theorem B (L∞-estimate). Let φ be a bounded (ω, m)-subharmonic function satisfying (ω + ddcφ)m ∧ωn-m ≤fdVX, (Hess) where 0 ≤fn/m ∈Lχ with the weight χ satisfying Condition (K). Then OscX(φ) ≤C, where C depends on ω, n, m and f n m χ. By generalizing an idea in [GL25a], we demonstrate that the L∞-estimate for complex Hessian equations can be reduced to the corresponding estimate for an associated MongeAmp`ere equation: (ω + ddcφ)n ≤fdVX, (0.2) where f ∈Lχ, and χ satisfies Condition (K). To obtain the L∞-estimate for the MongeAmp`ere equation, our main tool is the domination principle. The key novelty of our approach lies in constructing an appropriate subsolution. Our approach avoids potential computational difficulties associated with Orlicz norms and presents a different method compared to Ko lodziej's original proof. Once the L∞-estimate is obtained, we can derive the existence and regularity of bounded solutions. This leads us to the second main result of the study: Theorem C. Let 0 ≤fn/m ∈Lχ, with the weight function χ satisfying Condition (K). Then there exist a constant c > 0 and a solution u ∈SHm(X, ω) ∩C0(X) satisfying the complex Hessian equation (ω + ddcu)m ∧ωn-m = ceλufdVX, λ ≥0. Moreover, all bounded solutions to the above complex Hessian equation are continuous. To prove the first statement of Theorem C, an essential component is to establish a stability theorem (Theorem 3.1). For f ∈Lp, with p > n/m, such a stability result was proven in [KN16]. In the case where f ∈Lχ and the weight function χ satisfies Condition (K), the continuity of solutions to Monge-Amp`ere equations was studied in [KN21] by adapting the method in [KN19, Theorem 3.1]. We point out that our approach to the stability estimate is fundamentally different and remains of interest in the case where the densities f ∈Lp, as discussed in Remark 3.2. Our main tool will be the Domination Principle and the subsolution (Lemma A). Having established the stability estimate (Theorem 3.1), we approximate the density f by the sequence of bounded functions min(f, j). We show that the corresponding solutions uj converge uniformly to an (ω, m)-subharmonic function u solving the equation (Hessian), thereby obtaining a continuous solution. The continuity of all bounded solutions follows from the Domination Principle (Proposition 1.8). 4 YUETONG FANG Organization of the paper. The paper is organized as follows. Section 1 gives preliminary material on (ω, m)-subharmonic functions, as well as key concepts of Orlicz spaces. A local L∞-estimate is established, and Lemma A and Theorem B are proved in Section 2. In section 3, we establish a stability estimate and subsequently investigate the continuity of bounded solutions, leading to the proof of Theorem C. Acknowledgment. The author thanks her supervisor, Hoang-Chinh Lu, for suggesting the problem, valuable discussions during the preparation of the work, and useful comments that improved the presentation. This research is part of a PhD program funded by the PhD scholarship CSC-202308070110, and is partially supported by the projects Centre Henri Lebesgue ANR-11-LABX-0020-01 and the KRIS project of Fondation Charles Defforey. 1. Preliminaries In the whole paper, we let (X, ω) denote a compact Hermitian manifold of complex dimension n ∈N∗, equipped with a Hermitian form ω. We denote by dVX a smooth volume form on X. We use differential operators d = ∂+ ̄∂, and dc = i( ̄∂-∂), so that ddc = 2i∂ ̄∂. 1.1. (ω, m)-subharmonic functions. 1.1.1. Definition. We recall the definition of (ω, m)-subharmonic functions following [GN18] and briefly review some results related to both ω-plurisubharmonic and (ω, m)-subharmonic functions that will be used later. Fix an integer 1 ≤m ≤n. Fix Ωan open set in Cn. Given a hermitian metric ω on Ω, a real (1, 1)-form α on X is called m-positive with respect to ω if at all points in X, αk ∧ωn-k ≥0, ∀k = 1, · · · , m. A C2(Ω) function u : Ω→R is called harmonic with respect to ω if ddcu ∧ωn-1 = 0 at all points in Ω. Definition 1.1. A function u : Ω→{-∞} ∪R is subharmonic with respect to ω if: (a) u is upper semicontinuous and u ∈L1 loc(Ω); (b) for every relatively compact open set D ⋐Ωand every function h ∈C0(D) that is harmonic with respect to ω on D the following implication holds: u ≤h on ∂D =⇒u ≤h in D. Definition 1.2. A function φ : Ω→{-∞} ∪R is quasi-subharmonic with respect to ω if locally φ = u + ρ, where u is subharmonic with respect to ω and ρ is smooth. A function φ is α-subharmonic with respect to ω if φ is quasi-subharmonic with respect to ω and (α + ddcφ) ∧ωn-1 ≥0 in the sense of distributions. The positive cone Γm(ω) associated with the metric ω is defined by n γ real (1, 1)-form : γk ∧ωn-k > 0, k = 1, · · · , m o . It follows from G ̊arding's inequality [G ̊ar59] that if γ0, γ1, · · · , γm-1 ∈Γm(ω), then γ0 ∧γ1 ∧· · · ∧γm-1 ∧ωn-m > 0. One can write eω = γ1 ∧· · · ∧γm-1 ∧ω, and it is a strictly positive (n -1, n -1)-form on Ω. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 5 Definition 1.3. A function φ : Ω→[-∞, +∞) is called (α, m)-subharmonic with respect to ω if φ is α-subharmonic with respect to eω in Ωfor all eω of the form eωn-1 = γ1∧· · ·∧γm-1∧ω, where γ1, · · · , γm-1 ∈Γm(ω). A function u : X →R ∪{-∞} is called (α, m)-subharmonic on X if u is (α, m)- subharmonic on each local chart U of X. The set of all locally integrable functions on U which are (α, m)-subharmonic with respect to ω in U is denoted by SHω,m(U, α). In many contexts, such as [GL25a], the forms ω and α may differ. However, in this paper, we focus on the set SHω,m(U, ω). To simplify the notations, we denote by SHm(U, ω) the set of all (ω, m)-subharmonic functions with respect to ω in U. The set of all ω-plurisubharmonic functions on U is denoted by PSH(U, ω) = SHn(U, ω). For notational convenience, we denote ωu := ω + ddcu. Remark 1.4. By G ̊arding's inequality [G ̊ar59], if u ∈C2(X), then u is (ω, m)-subharmonic with respect to ω on X if and only if the associated form ωu belongs to the closure of Γm(ω). It is well known that when n = m, the set PSH(X, ω) is a closed subset of Lr(X), for all r ≥1. For (ω, m)-subharmonic functions, the following L1-compactness result was proved in [KN16, lemma 3.3]. Lemma 1.5. Let u ∈SHm(X, ω) be normalized by supX u = 0. Then there exists a uniform constant A > 0 depending only on X, ω such that ˆ X |u|ωn ≤A. 1.1.2. Domination principle. Analogous to the Monge-Amp`ere operators, the complex Hessian operators also satisfy a domination principle. The domination principle for Hessian operator associated with continuous (ω, m)-subharmonic functions is established by Guedj-Lu in [GL25a, Section 2]. Following the demonstration that the Hessian operators for bounded functions are well-defined, we show the domination principle associated with bounded (ω, m)- sh functions here. Lemma 1.6 (Maximum principle). Assume that u and v are bounded (ω, m)-subharmonic functions. Then (ω + ddc(max(u, v)))m ∧ωn-m ≥1{u>v}(ω + ddcu)m ∧ωn-m + 1{v≥u}(ω + ddcv)m ∧ωn-m. In particular, if u ≤v, then, 1{v=u}(ω + ddcu)m ∧ωn-m ≤1{v=u}(ω + ddcv)m ∧ωn-m In the case where u and v are continuous, the above lemma is proven in [GL25a, Lemma 1.10]. When u is bounded, thanks to the continuity of complex Hessian operators ( [KN23]), we can approximate u by a sequence of decreasing smooth (ω, m)-subharmonic functions. The comparison principle for u, v ∈SHm(Ω) ∩L∞(Ω) is introduced in [KN23, Section 6]. Proposition 1.7 (Comparison principle). Let u, v be bounded m-subharmonic functions with respect to ω and such that lim infz→∂Ωu(z) -v(z) ≥0. Assume (ddcv)m ∧ωn-m ≥(ddcu)m ∧ωn-m in Ω. Then it follows that u ≥v in Ω. Proof. See [KN23, Corollary 6.2]. □ 6 YUETONG FANG We will need a global version of domination principle: Proposition 1.8 (Domination principle). Assume that u, v ∈PSH(X, ω) ∩L∞(X) satisfy 1{u 0. If e-λvωm v ∧ωn-m ≥e-λuωm u ∧ωn-m, then u ≥v. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 7 Proof. Fix C > 0. On the set {u 0, we conclude that u ≥v. □ Corollary 1.10. Assume u, v are bounded (ω, m)-subharmonic functions on X such that (ω + ddcu)m ∧ωn-m ≤c (ω + ddcv)m ∧ωn-m for some positive constant c. Then c ≥1. 1.1.3. Cegrell classes. In Section 2, we study the Monge-Amp`ere equation in a local context. Here, we recall some essential definitions of the Cegrell classes [Ceg98,Ceg04]. Fix a bounded hyperconvex domain Ω⊂Cn. Let T (Ω) denote the set of bounded plurisubharmonic functions u in Ωsuch that limz→ζ u(z) = 0 for every ζ ∈∂Ω, and ́ Ω(ddcu)n 0, ˆ X χ(sf) 0 : ˆ X χ f r ≤χ(1) . 8 YUETONG FANG We remark that (Lχ(μ), ∥·∥χ,μ) constitutes a normed space and is complete, hence a Banach space. In the remainder of the paper, we fix μ = dVX, a smooth probability measure, and write (Lχ, ∥· ∥χ) for simplicity. In [Ko l98], S.Ko lodziej developped a technique using pluripotential theory that enables one to bound the solution to the Monge-Amp`ere equation by the Orlicz norm, ∥· ∥χ, of the right-hand side, associated with a weaker integrable condition. We thus introduce the Condition (K). Definition 1.13 (Condition (K)). A weight χ : R+ →R+ satisfies condition (K) if it is convex increasing and χ(t) ∼ t→+∞tn/m (log(1 + t))n (h ◦log ◦log(1 + t))n , where ˆ +∞dt h(t) 1, its convex conjugate is given by χ∗(t) = tq q , where q = p p-1 is the H ̈older conjugate of p. The Orlicz norm ∥· ∥χ here is the usual Lp norm. (ii) The weight χ defined in the Condition (K) above is a Young function hence induce an Orlicz space. In particular, the weight χ(t) = t logp(1 + t) with p > n is a Young function. One can check that as t →+∞, the conjugate weight grows like χ∗(t) = t1-1 p exp(t 1 p ). We will focus on the Orlicz space (Lχ, ∥· ∥χ) in section 3. It is important to note that in such an Orlicz space, the H ̈older inequality (or H ̈older-Young inequality) holds. Proposition 1.15 (H ̈older-Young inequality). For f ∈Lχ and g ∈Lχ∗, we have ˆ X |fg|dVX ≤∥f∥χ∥g∥χ∗. Proof. See [Dar19, Proposition 1.3]. □ We will need the following additive Young inequality (also known as Fenchel-Young inequality). Proposition 1.16. For every a, b ∈R+, it holds that ab ≤χ(a) + χ∗(b), with equality if and only if b ∈∂χ(a). In particular, let f, g : X →R+ be two measurable functions. Then f (χ∗)-1 ◦g ≤χ ◦f + g. 1.3. Envelopes. Given a Lebesgue measurable function l : X →R, the ω-plurisubharmonic envelope of l is defined by Pω(l) := sup{u ∈PSH(X, ω) : u ≤l}. The (ω, m)-subharmonic envelope of l is defined by Pω,m(l) := sup{u ∈SHm(X, ω) : u ≤l}. We now summarize the basic properties of these envelopes, following [GL25a]. Proposition 1.17. If l : X →R is continuous, then CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 9 (1) The complex Monge-Amp`ere measure (ω +ddcPω(l))n is concentrated on the contact set {Pω(l) = l}. (2) The complex Hessian measure (ω + ddcPω,m(l))m ∧ωn-m is concentrated on the contact set {Pω,m(l) = l}. (3) If l is C1,1-smooth, then Pω(l) is C1,1-smooth and (ω + ddcPω(l))n = 1{Pω(l)=l}(ω + ddcl)n. (4) If l is C1,1-smooth, then Pω,m(l) is C1,1-smooth and (ω + ddcPω,m(l))m ∧ωn-m = 1{Pω,m(l)=l}(ω + ddcl)m ∧ωn-m. For detailed proofs and additional discussion, we refer the reader to [Ber19,CM21,CZ19, GL22, GLZ19, Tos18]. As a consequence of the preceding results, we will make use of the following statements in the sequel. Lemma 1.18. Fix λ > 0. Let u, v ∈SHm(X, ω) ∩L∞(X). (1) If (ω + ddcu)m ∧ωn-m ≥eλufdVX and (ω + ddcv)m ∧ωn-m ≥eλugdVX, then (ω + ddc(max(u, v)))m ∧ωn-m ≥eλ max(u,v) min(f, g)dVX. (2) If (ω + ddcu)m ∧ωn-m ≤eλufdVX, and (ω + ddcv)m ∧ωn-m ≤eλvgdVX, then (ω + ddc(Pω,m(min(u, v)))m ∧ωn-m ≤eλPω,m(min(u,v)) max(f, g)dVX. The proof follows the same strategy as that for the Monge-Amp`ere measures presented in [GL22, Lemma 1.9]. Proof. Let ψ := max(u, v). As a consequence of the maximum principle (Lemma 1.6), we have (ω + ddcψ)m ∧ωn-m ≥1{u≥v}(ω + ddcu)m ∧ωn-m + 1{v>u}(ω + ddcv)m ∧ωn-m ≥1{u≥v}eλufdVX + 1{v>u}eλvgdVX =1{u≥v}eλψfdVX + 1{v>u}eλψgdVX ≥eλψ min(f, g)dVX. Set φ := Pω,m(min(u, v)). We prove the second statement. It follows from the proposition 1.17 that the complex Hessian measure (ω + ddcφ)m ∧ωn-m has support in {Pω,m(min(u, v)) = min(u, v)} = {Pω,m(min(u, v)) = u 0 such that ∥f∥χ ≤C0 ́ X χ(f)dVX for all f ∈Lχ. 2.1. Local L∞-bound of Monge-Amp`ere operators. Let Ωbe the unit ball in Cn. We define g := f ∥f∥χ to normalize f. We now consider the following Dirichlet problem to find a plurisubharmonic solution u: (ddcu)n = gdVX, in Ω; u = 0, on ∂Ω. (LocMA) We now establish a local uniform estimate for u. We start by establishing a useful result, which, for K ̈ahler manifold, was shown in [DDL21, Theorem 3.3] and [GL25b, Lemma 2.2]. We present here a local version. Its proof is simpler than that of the global case, as plurisubharmonicity is preserved under addition. Lemma 2.1. Assume that f ∈Lp(Ω, dVX) for some p > 1, and that u is normalized so that supΩu = 0 and u = 0 on ∂Ω. Suppose that there exists a bounded function v ∈PSH(Ω) normalized so that supΩv = 0 and v = 0 on ∂Ω, such that (ddcu)n ≤(ddcv)n + fdVX, then u ≥-C, where C = C(n, p, ∥f∥p, ∥v∥∞). Proof. Let ρ ∈PSH(Ω) be the solution to the Monge-Amp`ere equation (ddcρ)n = fdVX in Ω; ρ = 0 on ∂Ω. The uniform bound for ρ can be deduced from the results in [Ko l98,GL23]. Consider now ψ = v + ρ. It is sufficient to show that ψ ≤u. Indeed, ψ satisfies (ddcψ)n ≥(ddcv)n + (ddcρ)n = (ddcv)n + fdVX ≥(ddcu)n. It thus follows from the Comparison Principle (Proposition 1.7) that u ≥ψ on Ω. □ We are now prepared to prove a local uniform estimate. Lemma 2.2 (Local uniform estimate for Monge-Amp`ere equations). Assume that u is a bounded plurisubharmonic solution to the Dirichlet problem (LocMA). Then OscΩu ≤C, where C = C(n, Ω). Proof. Let v ∈F(Ω) be such that (ddcv)n = χ (g) dVX, in Ω; v = 0, on ∂Ω. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 11 The existence of v ∈F(Ω) is ensured by Theorem 1.11. Let α > 0 be fixed (to be specified later). On the set {log (g + 3) 1, the following inequality is satisfied: Θ′ -t α n tn(h ◦log t)n ≥2. (2.3) To see this, let x = log t and let b(x) = -Θ -ex α . Then the condition (2.3) is equivalent to b′(x)h(x) ≥21/n α , for all x > 0. We define b by b(x) := ˆ x 0 21/n αh(s)ds. Since h-1 is integrable, the function b(x) remains bounded, which in turn implies that Θ is uniformly bounded, as claimed. Taking t = log (g + 3) in (2.3) yields that Θ′ -log (g + 3) α n (log (g + 3))n (h ◦log log (g + 3))n ≥2. (2.4) Without loss of generality, we assume that there exist k0 > 0 such that χ(k) ≥1 2k logn(k + 3)(h ◦log ◦log(k + 3))n, ∀k ≥k0. From (2.2) and (2.4), it follows that on the set {log (g + 3) ≥-αv} ∩{g ≥k0}, (ddc(Θ ◦v))n ≥gdVX = (ddcu)n. On the other hand, on the set {log (g + 3) ≥-αv} ∩{g 0 such that for all 0 ≤g ∈Lχ, we can find ψ ∈PSH(X, ω), -1 ≤ψ ≤0, with (ω + ddcψ)n ≥s(χ) g ∥g∥χ dVX. Proof. Cover X by balls, X ⊂∪N j=1Bj. Choose B′ j = {ρj 0 in a neighborhood of B′ j, and such that Bj ⋐B ′ j. In B′ j, let vj ∈PSH(B′ j) be solution to Monge-Amp`ere equation (ddcvj)n = g ∥g∥χ dVX in B′ j; vj = -1 on ∂Ω. The local L∞-estimate (Lemma 2.2) implies OscB′ jvj ≤C′ j. We define ψj := max(vj, ξjρj), where ξj > 1 for all j. Thus, ψj coincides with vj in Bj, and coincides with ξjρj in X \ B′ j and in the neighborhood of ∂B′ j. Moreover, ψj is δ-1ω-plurisubharmonic for some uniform constant δ > 0 in Bj. Taking ψ := 1 N PN j=1 δψj, and taking 0 0 satisfies ∥1{f>A}f∥χ ≤s(χ) 4 . Then OscX(φ) ≤C, where C depends on A, and χ. Proof. We consider the function g = 1{f>A}f ∥1{f>A}f∥χ . Applying the Subsolution Lemma (Lemma 2.3) to g yields that there exists subsolution ψ ∈PSH(X, ω), such that (ω + ddcψ)n ≥sgdVX. The dominated convergence theorem implies that lim t→+∞ ˆ X χ(1{f>t}f)dVX = ˆ X lim t→+∞χ 1{f>t}f dVX = 0. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 13 We can thus choose A sufficiently large, depending only on f and s, such that ∥1{f>A}f∥χ ≤s 4. It follows that (ω + ddcψ)n ≥41{f>A}(ω + ddcφ)n. Therefore, (ω + ddcφ)n ≤1 4(ω + ddcψ)n + AdVX. Fix B > 0 large enough. It follows from [GL23, Lemma 2.5] that there exist ρ ∈PSH(X, ω), -1 ≤ ρ ≤0 such that (ω + ddcρ)n ≥4A1{φ n/m. A generalization to Orlicz spaces is outlined in Remark 3.2 therein. It has also been shown in the recent article [GL25b, Theorem B] that this approach can be adapted in certain general complex geometric PDEs. We provide a more detailed presentation in the proof of Theorem B here. The following lemma will be used to obtain a uniform estimate for bounded solutions. Lemma 2.5. Assume f ∈L1(dVX). Let φ ∈SHm(X, ω) ∩L∞(X) be such that (ω + ddcφ)m ∧ωn-m = fωn. Then, we have (ω + ddcu)n ≤fn/mωn, where u = Pω(φ). Proof. The proof proceeds in two steps. We first treat the case where the density f is bounded. In the general case, we approximate a given f ∈L1 by a sequence of bounded densities. Normalize the solution φ such that supX φ = 0. Step 1. Assume that f is bounded. Let {fj} be a sequence of smooth, uniformly bounded functions such that fj →f in L2n. It follows from [KN16, Lemma 3.20] that there exists a decreasing sequence of (ω, m)-subharmonic functions {ψj} such that ψj →φ. We consider the following complex Hessian equation: (ω + ddcφj)m ∧ωn-m = eφj-ψjfjωn. The existence of smooth, (ω, m)-subharmonic solution φj follows from [Sz ́e18]. Then, from [KN16, Theorem 3.9] we deduce that φj converges uniformly to eφ ∈SHm(X, ω) ∩C0(X), which is the solution to (ω + ddc eφ)m ∧ωn-m = eeφ-φfωn. By the domination principle (Proposition 1.8), we obtain that eφ = φ. Taking uj = Pω(φj), it follows from Proposition 1.17 that uj ∈C1,1(X). The mixed Monge-Amp`ere inequality [Ngu16, Lemma 1.19] implies that (ω + ddcuj)n ≤e n(φj-ψj) m fn/m j ωn. 14 YUETONG FANG Since uj converges uniformly to u, passing to the limit, we conclude that (ω + ddcu)n ≤fn/mωn. Step 2. Assume now that f ∈L1(dVX). We consider the equation given by (ω + ddcφj)m ∧ωn-m = eφj-φ min(f, j)ωn. Theorem 3.19 of [KN15] ensures the existence of a bounded solution φj, satisfying φj ≥φ. The domination principle yields that φj is decreasing to a (ω, m)-subharmonic function eφ. It follows from the domination principle again that eφ = φ. Taking uj := Pω(φj), then uj decreases to u. Moreover, Step 1 yields (ω + ddcuj)n ≤en/m(φj-φ)fn/m j ωn. Applying the continuity of Monge-Amp`ere operator along monotone sequences [BT76,BT82], we obtain (ω + ddcu)n ≤fn/mωn, as required. □ Proof of Theorem B. Normalize φ such that supX φ = 0. Let u = Pω (φ) and let C denote the contact set C := {u = φ}. We remark that u ∈PSH(X, ω) ⊂SHm(X, ω). Therefore, it follows from the mixed Monge-Amp`ere inequality [Ngu16, Lemma 1.19] and proposition 1.17 that 1C(ω + ddcu)m ∧ωn-m ≤1C(ω + ddcφ)m ∧ωn-m ≤fdVX We remark that hX := ωn/dVX ≥δ0 is positive on X. Lemma 2.5 ensures us to take g := (ω + ddcu)n /dVX. Then, the mixed Monge-Amp`ere inequality yields that δ1-m/n 0 gm/n ≤h1-m/n X gm/n ≤f, hence g ≤C0fn/m, so that g ∈Lχ0, where χ0 = t logn(1 + t)[h ◦log ◦log(3 + t)]n satisfies Condition (K). It follows from Theorem 2.4 that OscX(u) ≤A. It suffices to bound supX u from below. Set vA(ω) := inf ˆ X (ω + ddcφ)n : φ ∈L∞(X) ∩PSH(X, ω), -A ≤φ ≤0 . Recall that vA(ω) is strictly positive by [GL22, Proposition 3.4]. We denote by (χ∗ 0)-1 the inverse of χ∗ 0. Then it follows from additive H ̈older-Young inequality (Proposition 1.16) that vA(ω)(χ∗ 0)-1 -sup X u ≤ ˆ X (χ∗ 0)-1(-u) (ω + ddcu)n = ˆ X (χ∗ 0)-1(-u)gdVX ≤ ˆ X χ0(g)dVX + ˆ X (-u)dVX. We remark that ́ X(-u)dVX is uniformly bounded from above by Lemma 1.5. Thus we obtain a uniform lower bound of supX u, which yields together with OscXu ≤A that |u| has a uniform bound, and since u ≤φ, the solution φ is also uniformly bounded. □ CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 15 3. Weak solution 3.1. Stability estimates. Theorem 3.1. Let u, v ∈SHm(X, ω)∩L∞(X) be such that supX u = 0 and v ≤0. Assume that u ∈SHm(X, ω) is a solution to the complex Hessian equation ωm u ∧ωn-m = eλufωn, (3.1) where λ ≥0 and fn/m ∈Lχ with χ satisfying Condition (K). Then, sup X (v -u)+ ≤C∥(v -u)+fn/m∥ 1 n+1 χ , where the positive constant C depends only on n, m, ω, ∥f n m ∥χ, and supX |v|. Proof. We consider only the case λ = 0; when λ > 0, the L∞-estimate implies that eλuf ≤f, allowing the problem to be reduced to the case λ = 0. We now fix ε > 0, to be chosen later, and define gε := 1{u u + C2δ + ε} ⊂{u u+C2δ+ε}ωm φ ∧ωn-m =1{φ>u+C2δ+ε} [(1 -δ)ωv + δωψ]m ∧ωn-m =1{φ>u+C2δ+ε} m X k=0 m k [(1 -δ)ωv]k ∧(δωψ)m-k ∧ωn-m ≥1{φ>u+C2δ+ε}δm(ωψ)m ∧ωn-m. If ∥gε∥χ is not small, then the stability property follows trivially by adjusting C. Hence, we can assume that ∥gε∥χ u+C2δ+ε}δm(ωψ)m ∧ωn-m ≥21{φ>u+C2δ+ε}fωn =21{φ>u+C2δ+ε}ωm u ∧ωn-m. It follows from domination principle (Proposition 1.8) that φ ≤u + C2δ + ε. 16 YUETONG FANG Hence, (v -u)+ ≤C3 21/m s1/n ! ∥gε∥1/n χ + ε ≤C4∥gε∥1/n χ + ε ≤C4(ε)-1/n∥(v -u)+fn/m∥1/n χ + ε. Optimizing the right hand side by choosing ε = ∥C4(v -u)+fn/m∥1/n χ n ! n n+1 , we obtain (v -u)+ ≤C ∥(v -u)+fn/m∥χ 1 n+1 . The conclusion follows by taking the supremum of the left-hand side. □ Remark 3.2. When χ(t) = tp/p, with p > 1, the Orlicz norm ∥· ∥χ is simply the Lebesgue Lp-norm. Assume fn/m ∈Lp, for p > 1. Let χq(t) = tq/q for some 1 0 such that (ω + ddcuj)m ∧ωn-m = cjeλujfjdVX, λ ≥0. We normalized uj such that supX uj = 0. Our first step is to bound the coefficients cj from above and below. This requires an inequality involving mixed Hessian measures. It's well known that if φj ∈SHm(X, ω)∩C∞(X) and (ω +ddcφj)m ∧ωn-m = fjdVX, then G ̊arding's inequality [G ̊ar59] holds: (ω + ddcφ1) ∧· · · ∧(ω + ddcφm) ∧ωn-m ≥(f1 · · · fm)1/mdVX. In the case where φj's are not smooth, we can apply an approximation argument to obtain the following inequality. Proposition 3.4. Let u ∈SHm(X, ω) be bounded functions such that (ω + ddcφ)m ∧ωn-m ≥fωn. Then we have that (ω + ddcφ) ∧ωn-1 ≥f1/mωn. Proof. Let {gj} be a sequence of bounded, positive and smooth functions converging to f. Let vj be a sequence of smooth, (ω, m)-subharmonic functions converging to φ. We consider the following Hessian equation: (ω + ddcφj)m ∧ωn-m = eφj-vjgjωn. The existence of smooth solution φj follows from [Sz ́e18]. Applying G ̊arding's inequality [G ̊ar59] yields (ω + ddcφj) ∧ωn-1 ≥e φj-vj m g1/m j ωn. Moreover, we deduce from [KN16, Theorem 3.19] that φj converges uniformly to eφ ∈ SHm(X, ω). Passing to the limit as j →+∞, we obtain (ω + ddceu)m ∧ωn-m = eeφ-φfωn. The domination principle (Proposition 1.8) then implies eφ = φ. Consequently, we conclude that (ω + ddcφ) ∧ωn-1 ≥f1/mωn. □ We now turn to obtaining uniform bounds for cj. Lemma 3.5. The sequence of constant {cj} is uniformly bounded. In particular, the Orlicz norm ∥cjfj∥χ is uniformly bounded from above. Proof. We first bound cj from above. Fix δ0 > 0 such that δ0 ≤ωn/dVX. Proposition 3.4 implies that for each j, (ω + ddcuj) ∧ωn-1 ≥c1/m j f1/m j ωn ≥δ0c1/m j f1/m j dVX. 18 YUETONG FANG Thus, by taking the integral of both sides: δ0c1/m j ˆ X f1/m j dVX ≤ ˆ X (ω + ddcuj) ∧ωn-1. Note that uj ∈SHm(X, ω) ⊂SH1(X, ω), and the sequence (uj) is relatively compact in L1. Moreover, we have ́ X f1/m j ωn → ́ X f1/mωn > 0 as j →+∞. It follows that there exist j0 large enough such that for each j > j0, the following inequality holds: c1/m j ≤ 2 δ0 ́ X f1/mdVX ˆ X (ω + ddcuj) ∧ωn-1. It suffices to bound the right hand side. Indeed, it follows from Stokes' theorem that ˆ X (ω + ddcuj) ∧ωn-1 = ˆ X ωn + ˆ X ujddc(ωn-1) ≤ ˆ X ωn + B ˆ X |uj|ωn. By the L1-compactness result for (ω, 1)-subharmonic functions (Lemma 1.5), we conclude that the right hand side above is bounded. Thus cj is bounded from above. We next show that {cj} is bounded away from 0. Let gj = fn/m j ∈Lχ. By the subsolution Lemma (Lemma 2.3), there exists a uniform constant δ > 0, such that (ω + ddcvj)n ≥δjgjdVX. Then, it follows from mixed Monge-Amp`ere inequality [Ngu16, Lemma 1.9] that ωm vj ∧ωn-m ≥δm/n j gm/n j ωn ≥δm/n j fjωn. The domination principle (Proposition 1.8) yields that δm/n j ≤cj. We thus obtain a uniform lower bound for cj's, since δj's are strictly positive, uniform constants. The uniqueness of cj for each fj follows directly from domination principle (Proposition 1.8). □ Uniform bound of uj. It follows from Theorem B that {uj} have uniform bound: -C ≤uj ≤0. By extracting, we can assume that {uj} converges in L1(X) to a bounded (ω, m)-subharmonic function u, as j →+∞. Existence of solutions. We now show that, by construction, the limit u is a solution to the Hessian equation (ω + ddcu)m ∧ωn-m = ceλufdVX, λ ≥0. We first consider the simpler case in which λ > 0. Theorem 3.6. Let λ > 0. Then u ∈SHm(X, ω) ∩L∞(X) is the unique solution such that (ω + ddcu)m ∧ωn-m = eλufdVX. Moreover u is uniformly bounded. CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 19 Proof. To simplify the notation, we assume that λ = 1. Then each uj satisfies (ω + ddcuj)m ∧ωn-m = eujfjdVX. From the domination principle (Proposition 1.8), we deduce that uj decreases to some functions u ∈SHm(X, ω). Passing to the limit, it follows from the continuity of Hessian operator along decreasing sequence ( [KN23, Section 5]) that (ω + ddcu)m ∧ωn-m = eufdVX. Applying Theorem B, we obtain a uniform bound for u. □ Monotonicity of the sequence uj is not guaranteed when λ = 0. Our approach is to construct a supersolution and a subsolution and demonstrate that they coincide, as in [GL24, Section 3] and [BGL25, Theorem 4.5]. Theorem 3.7 (Existence of bounded solution). Let f, u and the constant c be as above. Then u ∈SHm(X, ω) ∩L∞(X) is a solution to (ω + ddcu)m ∧ωn-m = cfdVX. Moreover, u is uniformly bounded. Proof. We define ρj := sup k≥j uk !∗ , vj := Pω,m inf k≥j uk , where Pω,m (infk≥j uk) denotes the largest (ω, m)-subharmonic function lying below infk≥j uk. Observe that ρj is a decreasing sequence that converges to u, and vj is an increasing sequence that converges to an (ω, m)-subharmonic function v. In particular, by construction, we have u ≥v. We are going to show that u = v. Set cj fj := infk≥j ckfk. Then it follows from Lemma 1.18 that (ω + ddcρj)m ∧ωn-m ≥cj fj dVX. Then we deduce from the continuity of Hessian operator along monotone sequences (see [KN23, Lemma 5.1]) that (ω + ddcu)m ∧ωn-m ≥cfdVX. (3.2) On the other hand, we write (ω + ddcvj)m ∧ωn-m = evj-vjcjfjdVX. Define c+ j f+ j := supk≥j ckfk. It also follows from Lemma 1.18 that (ω + ddcvj)m ∧ωn-m ≤evj-infk≥j ukc+ j f+ j dVX. Let j →+∞. Together with (3.2) we deduce that (ω + ddcv)m ∧ωn-m ≤ev-ucfdVX ≤ev-u(ω + ddcu)m ∧ωn-m. By the domination principle (Proposition 1.8), we conclude that v ≥u. Hence, v = u is a bounded solution to the complex Hessian equation: (ω + ddcu)m ∧ωn-m = cfdVX. The uniform bound for u follows from Theorem B. □ 20 YUETONG FANG Proof of Theorem C. Existence of continuous solutions. We now prove that the sequence {uj} converges uniformly to u as j →+∞. We recall that when λ > 0, the L∞-estimate implies that eλuf ≤f, allowing us to reduce to the case where λ = 0. Fix j0 large enough. For each j > j0, we infer by Theorem 3.1 that (uj -u)+ ≤C1 (uj -u)+f n m 1 n+1 χ , (u -uj)+ ≤C1 (u -uj)+f n m j 1 n+1 χ . We are going to show that (uj -u)+f n m χ →0 as j →+∞. Fixing s > 0, we proceed to compute the corresponding integrability. ˆ X χ (uj -u)+f n m s dVX = ˆ X (uj -u)+f n m s logn 1 + (uj -u)+f n m s h ◦log ◦log 3 + (uj -u)+f n m s n dVX. As u and uj's are uniformly bounded, it follows that (uj -u)+ ≤C0 for some uniform constant C0, and since h is increasing, we deduce that ˆ X χ (uj -u)+f n m s dVX ≤ ˆ X (uj -u)+f n m s logn 1 + C0f n m s · h ◦log ◦log 3 + C0f n m s n dVX. By the dominated convergence theorem, we conclude that (uj -u)+f n m s χ →0, for all 0 < s < +∞, and consequently ∥(uj -u)+f n m ∥ 1 n+1 χ →0. Applying the same argument, we obtain that (u -uj)+f n m j 1 n+1 χ ≤ (u -uj)+f n m 1 n+1 χ →0. Therefore, {uj}∞ j=1 is a Cauchy sequence in SHm(X, ω) ∩C0(X). It follows that the bounded solution u to the complex Hessian equation (ω + ddcu)m ∧ωn-m = ceλufωn, where f ∈Lχ, is continuous. Continuity of bounded solutions. Assume now that v ∈SHm(X, ω) is another bounded solution to the complex Hessian equation (ω + ddcv)m ∧ωn-m = cfdVX. Then v is continuous. Indeed, the uniform bound for v ensures that ef = e-vf ∈Lχ. We deduce from the previous argument that there exist eu ∈SHm(X, ω) ∩C0(X) solving (ω + ddceu)m ∧ωn-m = eeu efdVX. It follows that (ω + ddceu)m ∧ωn-m = eeu-vfdVX = eeu-v(ω + ddcv)m ∧ωn-m. Applying the Domination Principle (Proposition 1.8) yields that v = eu. Therefore, v ∈C0(X), as claimed. □ CONTINUITY OF SOLUTIONS TO COMPLEX HESSIAN EQUATIONS 21 References [ ̊AC25] Per ̊Ahag and Rafa l Czy ̇z. Continuity of solutions to complex Hessian equations via the dinew-Ko lodziej estimate. Ann. Fenn. Math., 50:201-214, 2025. [ ̊ACK+09] Per ̊Ahag, Urban Cegrell, S lawomir Ko lodziej, Hoang-Hiep Pha.m, and Ahmed Zeriahi. Partial pluricomplex energy and integrability exponents of plurisubharmonic functions. Adv. Math., 222:2036-2058, 01 2009. [Ber19] Robert J. Berman. From Monge-Amp`ere equations to envelopes and geodesic rays in the zero temperature limit. Math. Z., 291:365-394, 2019. 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Magnetic D-brane solitons: skyrmion strings ending on a N´eel wall in chiral magnets
Sven Bjarke Gudnason1, 2 and Muneto Nitta3, 4
1Institute of Contemporary Mathematics, School of Mathematics and Statistics,
Henan University, Kaifeng, Henan 475004, P. R. China
2Department of Physics, Chemistry and Pharmacy,
University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark
3Department of Physics & Research and Education Center for Natural Sciences,
Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
4International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2),
Hiroshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8511, Japan
(Dated: October, 2025)
Magnetic skyrmions extended to three dimensions form string-like objects whose fundamental role
remains largely unexplored. We show that skyrmion strings can terminate on a N´eel-type domain
wall (DW), realizing a magnetic analogue of a Dirichlet(D)-brane soliton. While an isolated N´eel DW
tends to rotate into a Bloch DW, the N´eel DW is stabilized when a skyrmion string ends on it. Unlike
field-theory D-branes, the Bloch-type DMI produces linear rather than logarithmic DW bending,
and the strings retain finite width far from the DW, circumventing singular behavior. Furthermore,
the repulsive interaction between strings allows periodic multi-junction solutions, yielding a square
lattice of alternating strings and local DW deformations. These results establish magnetic skyrmion
strings as fundamental strings that can end on a D-brane.
Superstring theory predicts the fundamental con-
stituents not to be point particles, but strings, which
are generally found in two varieties: Closed strings and
open strings [1, 2]. Excitations of the strings correspond
to point particles at low energies. The former need no
extra ingredients in the theory and can e.g. describe the
propagation of gravitons in spacetime. The latter, how-
ever, predict and require the existence of hypersurfaces
called Dirichlet(D)-branes on which strings terminate, pi-
oneered and much developed by Polchinski [3, 4]. String
excitations induce a gauge theory on D-branes, described
by the Dirac-Born-Infeld (DBI) action. The discovery of
D-branes was a key ingredient for the 2nd revolution of
string theory in the ’90ies. A field theoretical analog of
D-branes was proposed in Ref. [5] as a string ending on
a domain wall (DW), referred to as a D-brane soliton,
in the O(3) nonlinear sigma model (NLσM) or the CP 1
model with a quadratic potential, known as the easy-axis
anisotropy. In fact, the D-brane soliton coincides with
the so-called BIon [6] when it is viewed from the DBI ac-
tion on the DW, thereby justifying to be identified with
a D-brane. D-brane solitons were further generalized to
a U(1) gauge theory [7] and the CP N or Grassmannian
model [8, 9]; see also Refs. [10–12].
This brings us to consider analogous systems of D-
branes in more down-to-earth systems. The purpose of
this Letter is to propose a D-brane-string junction in
chiral magnets with a Dzyaloshinskii-Moriya-Interaction
(DMI) [13, 14] in three dimensions. Chiral magnets are
known to admit various topological defects and solitons
[15]. With an easy-axis potential, they admit magnetic
DWs. These have, in particular, been a subject of intense
study due to their application to magnetic memory (stor-
age) [16, 17]. The ground state is in the ferromagnetic
phase for small DMI while for large enough DMI it is
in a spiral phase or a chiral soliton lattice (CSL) [18–23].
The other more well-studied topological solitons are mag-
netic skyrmions [24–30], which are a 2-dimensional vari-
ant of the original 3-dimensional skyrmions proposed by
Skyrme himself as a simple model for nucleons [31, 32].1
With a certain Zeeman magnetic field, a skyrmion lattice
is the ground state [26–30, 34]. Isolated skyrmions have
been also studied [35]. They have been extensively stud-
ied in particular for applications to nanotechnology such
as information carriers in magnetic storage devices [36].
See Ref. [36] for a review of magnetic skyrmions. The co-
existence of a DW and skyrmions in two dimensions gives
rise to their composite states called DW-skyrmions [37–
52] (see also Refs. [53, 54]) which are experimentally ob-
served [55–57].
Similar DW-skyrmions (bimerons) are
proposed for an easy-plane anisotropy [58]. Such DW-
skyrmions are expected to be useful for manupulating
skyrmions circumventing the skyrmion Hall effect [59–
62].
Recently, much attention has been paid to topological
defects in three spatial dimensions; Hopfions [63, 64] and
monopoles [65–68] have been studied in chiral magnets.
In addition, DWs and skyrmions are extended objects
in three dimensions, which are two dimensional hyper-
surfaces and strings [69], respectively. Several aspects of
skyrmion strings have been studied thus far, such as a
1 Baby-skyrmions were latter considered in the 1990’ies and on-
ward [33], which differ from the magnetic skyrmions in the way
they are stabilized: The baby-skyrmions have a Skyrme term like
3D skyrmions and the magnetic skyrmions are stabilized by the
DMI. An important consequence is that the DMI breaks parity,
unlike the Skyrme term, and this implies that only skyrmions
and not anti-skyrmions exist in the chiral magnets.
arXiv:2510.14689v1 [cond-mat.mes-hall] 16 Oct 2025
2
skyrmion string ending on a monopole (torons) [65, 66],
Kelvin modes propagating along a string [70, 71], braid-
ing skyrmion strings [72], etc. One of fundamental ques-
tions may be how skyrmion strings differ from or are sim-
ilar to fundamental strings in string theory. We should
ask whether skyrmion strings can end on a D-brane or
some alternative extended structure. One hint to attack
this problem is a D-brane soliton in field theory [5, 7, 8].
In this Letter, we show that indeed skyrmion strings
can end on a N´eel DW, forming a stable D-brane soliton,
and thus can be regarded as fundamental strings. While
a single N´eel DW is unstable to rotate to a Bloch DW, it
can be stabilized once a string terminates on it. There are
outstanding differences between the field theory D-brane
soliton and the magnetic D-brane soliton. In the former,
the string gives rise to a logarithmic bending of the DW,
while in the latter we find a linear bending of the DW
due to the Bloch-type DMI. We also have stable finite-
width strings far from the DW, in contrast to a singular
behaviour in the former. We also consider the possibil-
ity of having more than one string-DW junction. Due
to the repuslive nature of the force between strings, we
construct a solution with periodic boundary conditions
in the (x, y)-plane and find a square lattice of strings
pointing up- and downwards from the DW, that locally
has a linear bending around each of the strings. We thus
establish the notion that magnetic skyrmion strings are
fundamental strings that can end on a D-brane.
The ferromagnetic chiral magnetic system is described
by a gradient term (Heisenberg interaction), the DMI
and a potential (on-site interaction) that we take here to
be the easy-axis anisotropy term
E = 1
2∂in·∂in+κn·∇×n+ 1
2(1−n2
3),
i = 1, 2, 3, (1)
where the total energy is given by E =
R
E d3x and
the magnetization vector or sigma model field, n =
(n1, n2, n3), is subject to the usual sigma model con-
straint: n · n = 1.
The energy and length units are
rescaled so as to have a canonical kinetic term and a po-
tential coupling constant of unity: The only remaining
coupling constant of the theory (κ) is a combination of
the physical constants. In this Letter, we consider the
ferromagnetic phase κ ∈[0.3, κcrit] with κcrit ≈0.6 being
the critical coupling beyond which a CSL-type phase is
the ground state (for κ > κcrit) [18–22]. We also concen-
trate on static configurations and do not discuss dynam-
ics.
The topology of the theory can be classified into codi-
mensions. First, drawing a line in the z-direction, the
importance of the potential comes into light: The DW
(of codimension one) is supported by the existence of
two generate and disconnected vacua
π0(Mvac) = π0
{n3 = 1 ∪n3 = −1}
≃Z2.
(2)
Taking instead a slice, say in the (x, y)-plane, the topol-
ogy of 2-dimensional skyrmions is given by the 2nd ho-
motopy group that supports also NLσM lumps:
π2(S2) ≃Z ∋Qskyrmion.
(3)
This π2 corresponds to the point-compactified (x, y)-
plane, also known as the extended complex plane, and
is topologically identical to the Riemann sphere (S2).
Finally, the 3-dimensional space mapped to the sigma
model target space being a 2-sphere, gives rise to the
Hopf invariant π3(S2) = Z ∋QHopfion, which supports
Hopfions that are knotted string-like solutions [63, 64] –
we will not discuss those further here. In this Letter, we
construct the string-DW junction, which is combination
of the two former types of soliton.
Let us briefly review the field theory D-brane soliton
solution, which is the string-DW junction for κ := 0,
that endows the DW with a logarithmic bending. The
solution is
η = ρeiθ±z,
(4)
where we have used the Riemann sphere coordinate
η = n1 + in2
1 + n3
,
(5)
on the sigma model target space and cylindrical coordi-
nates on space: x+iy = ρeiθ. To observe the logarithmic
being of the DW, we can simply analyze the iso-curve
corresponding to the field at a fixed level set, say |η| = 1:
log ρ = ±z.
(6)
This equation also reveals that far from the junction, on
the string side, the string becomes infinitesimally small
in size.
We now turn to the case of κ ̸= 0 and will for concrete-
ness work in the ferromagnetic phase with κ ∈[0.3, κcrit].
Firstly, we have stable skyrmion strings in this case with
κ > 0 turned on.
This can be understood simply by
comparing each slice at a fixed z to the usual 2D mag-
netic skyrmions and see that up to a choice of potential,
these are identical. The DWs perpendicular to a vector
in the (x, y)-plane are of the Bloch-type, for which DMI
contributes negatively to the energy, and are stable. In
the presence of a Bloch DW, the skyrmion string is re-
pelled [43] and does not automatically flow to the DW to
become a DW-skyrmion [37–52, 55–57].2 On the other
hand, the DW is of N´eel-type when perpendicular to the
z-direction, which is actually unstable and tends to ro-
tate to a Bloch DW, as the N´eel DW’s DMI energy is
2 In the CSL phase, the Bloch DWs have negative energy and
constitute a CSL [18–22], and the DW-skyrmion string, if formed,
is unstable to split into a bimeron (pair) [42].
3
positive. Nevertheless, we show the N´eel DW is stabi-
lized in the ferromagnetic phase when skyrmion strings
terminate on it. One essential point is that the N´eel DW
has an exact U(1) modulus on it while the Bloch DW
does not – this facilitates that the winding of the string
can be connected to the DW.
First, to establish that the bending of the wall will be
linear, we cannot simply fix the field value of the solution,
since we are not aware of analytic exact solutions to the
equations of motion in this case. However, using the idea
of fixing a curve in the equation of motion, we arrive at
leading order at the solution of the curve for |η| = 1:
κr + cr log r + czz = 0,
(7)
where dimensional analysis3 fixes cr/cz = 1 when κ = 0.
This reproduces the logarithmic bending of the DW when
κ = 0 and predicts a linear bending of the DW when
κ ̸= 0.
In order to confirm our prediction, we turn to numeri-
cal computations of the equation of motion coming from
varying the energy functional (1):
eom := ∂2
i η −2(∂iη)2¯η
1 + η¯η
−(1 −η¯η)η
1 + η¯η
= 0.
(8)
We discretize a box typically of the size 511 × 511 × 767
where the derivatives of the equation of motion are ap-
proximated by a 4th order 5-point stencil finite-difference
scheme, with lattice spacing h = 0.15. The method for
finding solutions utilized here is the so-called arrested-
Newton flow, where we solve
∂2
τη = eom,
(9)
monitoring the static energy given by Eq. (1) at every
step of the flow in τ (τ is a fictitious time parameter).
If the static energy rises, we stop the flow by setting
∂τη = 0 and then restart the algorithm.
In Fig. 1, we display the numerical results for the
string-DW junction (trumpet) for various values of κ =
0.35, 0.4, 0.45, 0.5, illustrating the dependence of the
slope of the linear bending of the wall on the DMI cou-
pling κ, with the steepest slopes corresponding to the
largest values of κ. The figure shows the isosurface of
the n3 field at n3 = 0; this is the midway between the
two vacua of the potential in Eq. (1). Clearly the DW
obeys a linear relation between the z-coordinate and the
radius from the string junction’s center when the radius
is sufficiently large (say of the order of 10 or larger). The
size (thickness) of the string also depends approximately
linearly on κ, which can be seen from Derrick’s theorem
3 More specifically, by reinstating units into the equations, the
ratio cz/cr = m with m2 being the coupling of the potential
term. Our dimensionless equations correspond to m = 1.
FIG. 1. String-DW junctions (trumpets) for various strength
of the DMI κ = 0.35 (blue), 0.4 (red), 0.45 (magenta), 0.5
(yellow). The slope of the linear bending is nearly linearly
proportional to κ.
(also known as a virial law in other fields of physics) [73].
In a linear theory or an integrable theory, we would ex-
pect the constants cr and cz to truly be constants (like
in the field theory D-brane soliton case), but due to non-
linearities of system we find they have some dependence
on κ. Fitting to the slope data of the solutions in Fig. 1,
we find
cz ≈3.4 −5.2κ,
cr/cz ≈1 + 12.6κ.
(10)
which is consistent with the logarithmic bending with
coefficient cr/cz = 1 when κ →0.
Interestingly, the
approximate relation between cz and κ predicts a criti-
cal value of the coupling, which indeed exists due to the
transition to the CSL: The linear fit predicts κcrit ≈0.65,
whereas in the full nonlinear theory we find it to be closer
to κcrit ∼0.6.
We find also a lower critical value for
κ, where the string-DW junction ceases to exist near
κcrit,junc ≳0.3, although the skyrmion string itself still
exists at κ = 0.3.
Fig. 2 shows the detailed information of the entire 3-
vector field, n, at the string-slice in the inlet at the top-
right corner, at the wall-slice at the bottom of the figure
and finally in a vertical cross section in the top of the fig-
ure. In this figure, the vector n is mapped onto real space
4
FIG. 2.
String-DW junction with κ = 0.5.
Arrows are
mapped from n to x and the colors are in one-to-one cor-
respondence with the arrows: n1 = 1 7→red, n2 = 1 7→green,
n1 = −1 7→cyan, n2 = −1 7→magenta, n3 = 1 7→white and
n3 = −1 7→black. The inlet and the bottom are horizontal
cross sections, whereas the top is a vertical cross section of
the string-DW junction.
(R3) using arrows, with the up vacuum (n3 = 1) marked
by ⊙, the down vacuum (n3 = −1) marked by ⊗and the
vectors in the plane corresponding to (n1, n2) when n3 is
small. The color scheme is in one-to-one correspondence
with the arrows and is described in the caption of the
figure. There are about 15 times more lattice points in
the solution, than arrows depicted on the figure.
This string junction turns out to repel other string
junctions, both of strings pointing upwards (skyrmion
strings) and strings pointing downwards (anti-skyrmion
strings). Packing the string junctions into a given vol-
ume, however, has solutions which can most easily be
found by considering periodic boundary conditions in the
(x, y)-plane.
Such a configuration of multi-string-DW
junctions is shown in Fig. 3, with the upper part of the
space being mostly in the up vacuum (n3 = 1) and the
lower part mostly in the down vacuum (n3 = −1). The
right-hand side of the figure shows the spin interpreta-
tion of the map from n to real space, see the caption of
Fig. 2 for details of the color scheme.
Although we mentioned in the introduction, that the
chiral DMI supports only skyrmion strings and no anti-
skyrmion string, this is indeed the case in the up-vacuum
(upper part of Fig. 3). However, in the down-vacuum
(lower part of Fig. 3), the situation is reversed and the
DMI supports only anti-skyrmion strings.
Interestingly, if we impose periodic boundary condi-
tions also in the z-direction, we obtain a fully periodic
configuration just like that of Fig. 3, but with the anti-
DW placed equidistantly from the DW (the one visible
in Fig. 3). This means that one could have n layers of
cyclic DW-anti-DW configurations with skyrmion strings
and anti-skyrmion strings stretched between them.
Real-world experimentally realizable physical systems
for studying D-brane analogs etc. are hard to come by.
It is yet unclear if and how this could be utilized for
applications in real-world spintronics. However, the pos-
sibility of realizing a D-brane-string junction in a chi-
ral magnet is very interesting for fundamental physics.
One of the important directions to explore is how to
create a D-brane soliton. If a N´eel wall moving under
an applied current passes through a fixed impurity or
monopole, it will be bent and eventually would create
a D-brane soliton. Another direction entails excitations
on a D-brane soliton. Kelvin modes propagating along a
skyrmion string [70, 71] are the so-called type-B Nambu-
Goldstone modes [74, 75] with a fractional dispersion re-
lation [76, 77] as well as ripple modes on a domain wall
[77–79].
It will be worthwhile to study how they are
transformed to each other when propagating along a D-
brane soliton. D-brane analogs have been also proposed
in Bose-Einstein condensates [80–83]. Condensed matter
D-branes open up a new avenue of composite topologi-
cal objects to simulate fundamental physics, which could
be realized in other systems such as liquid crystals and
optics.
Acknowledgments
S. B. G. thanks the Outstanding Talent Program of
Henan University for partial support.
The work of
M. N. is supported in part by JSPS KAKENHI [Grants
5
FIG. 3. Multi-string-DW junctions with κ = 0.4. Periodic boundary conditions are imposed in the x- and y-directions. Left
panel: the two isosurfaces corresponding to n3 = 1/2 (red) and n3 = −1/2 (green). Right panel: the corresponding details of
the entire field using arrows, which are mapped from n to x ∈R3 and the colors corresponding to the arrows are described in
Fig. 2. In the top-part of the configuration, the spins circle counter-clockwise corresponding to skyrmion strings, whereas in
the bottom-part of they circle clockwise corresponding to anti-skyrmion strings.
No. JP22H01221 and No. JP23K22492] and the WPI pro-
gram “Sustainability with Knotted Chiral Meta Matter
(WPI-SKCM2)” at Hiroshima University.
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Magnetic D-brane solitons: skyrmion strings ending on a N ́eel wall in chiral magnets Sven Bjarke Gudnason1, 2 and Muneto Nitta3, 4 1 475004, P. R. China 2 55, 5230 Odense M, Denmark 3 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan 4International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2), Hiroshima University, 1-3-2 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8511, Japan (Dated: October, 2025) Magnetic skyrmions extended to three dimensions form string-like objects whose fundamental role remains largely unexplored. We show that skyrmion strings can terminate on a N ́eel-type domain wall (DW), realizing a magnetic analogue of a Dirichlet(D)-brane soliton. While an isolated N ́eel DW tends to rotate into a Bloch DW, the N ́eel DW is stabilized when a skyrmion string ends on it. Unlike field-theory D-branes, the Bloch-type DMI produces linear rather than logarithmic DW bending, and the strings retain finite width far from the DW, circumventing singular behavior. Furthermore, the repulsive interaction between strings allows periodic multi-junction solutions, yielding a square lattice of alternating strings and local DW deformations. These results establish magnetic skyrmion strings as fundamental strings that can end on a D-brane. Superstring theory predicts the fundamental constituents not to be point particles, but strings, which are generally found in two varieties: Closed strings and open strings [1, 2]. Excitations of the strings correspond to point particles at low energies. The former need no extra ingredients in the theory and can e.g. describe the propagation of gravitons in spacetime. The latter, however, predict and require the existence of hypersurfaces called Dirichlet(D)-branes on which strings terminate, pioneered and much developed by Polchinski [3, 4]. String excitations induce a gauge theory on D-branes, described by the Dirac-Born-Infeld (DBI) action. The discovery of D-branes was a key ingredient for the 2nd revolution of string theory in the '90ies. A field theoretical analog of D-branes was proposed in Ref. [5] as a string ending on a domain wall (DW), referred to as a D-brane soliton, in the O(3) nonlinear sigma model (NLσM) or the CP 1 model with a quadratic potential, known as the easy-axis anisotropy. In fact, the D-brane soliton coincides with the so-called BIon [6] when it is viewed from the DBI action on the DW, thereby justifying to be identified with a D-brane. D-brane solitons were further generalized to a U(1) gauge theory [7] and the CP N or Grassmannian model [8, 9]; see also Refs. [10-12]. This brings us to consider analogous systems of Dbranes in more down-to-earth systems. The purpose of this Letter is to propose a D-brane-string junction in chiral magnets with a Dzyaloshinskii-Moriya-Interaction (DMI) [13, 14] in three dimensions. Chiral magnets are known to admit various topological defects and solitons [15]. With an easy-axis potential, they admit magnetic DWs. These have, in particular, been a subject of intense study due to their application to magnetic memory (storage) [16, 17]. The ground state is in the ferromagnetic phase for small DMI while for large enough DMI it is in a spiral phase or a chiral soliton lattice (CSL) [18-23]. The other more well-studied topological solitons are magnetic skyrmions [24-30], which are a 2-dimensional variant of the original 3-dimensional skyrmions proposed by Skyrme himself as a simple model for nucleons [31, 32].1 With a certain Zeeman magnetic field, a skyrmion lattice is the ground state [26-30, 34]. Isolated skyrmions have been also studied [35]. They have been extensively studied in particular for applications to nanotechnology such as information carriers in magnetic storage devices [36]. See Ref. [36] for a review of magnetic skyrmions. The coexistence of a DW and skyrmions in two dimensions gives rise to their composite states called DW-skyrmions [3752] (see also Refs. [53, 54]) which are experimentally observed [55-57]. Similar DW-skyrmions (bimerons) are proposed for an easy-plane anisotropy [58]. Such DWskyrmions are expected to be useful for manupulating skyrmions circumventing the skyrmion Hall effect [5962]. Recently, much attention has been paid to topological defects in three spatial dimensions; Hopfions [63, 64] and monopoles [65-68] have been studied in chiral magnets. In addition, DWs and skyrmions are extended objects in three dimensions, which are two dimensional hypersurfaces and strings [69], respectively. Several aspects of skyrmion strings have been studied thus far, such as a 1 Baby-skyrmions were latter considered in the 1990'ies and onward [33], which differ from the magnetic skyrmions in the way they are stabilized: The baby-skyrmions have a Skyrme term like 3D skyrmions and the magnetic skyrmions are stabilized by the DMI. An important consequence is that the DMI breaks parity, unlike the Skyrme term, and this implies that only skyrmions and not anti-skyrmions exist in the chiral magnets. 16 Oct 2025 2 skyrmion string ending on a monopole (torons) [65, 66], Kelvin modes propagating along a string [70, 71], braiding skyrmion strings [72], etc. One of fundamental questions may be how skyrmion strings differ from or are similar to fundamental strings in string theory. We should ask whether skyrmion strings can end on a D-brane or some alternative extended structure. One hint to attack this problem is a D-brane soliton in field theory [5, 7, 8]. In this Letter, we show that indeed skyrmion strings can end on a N ́eel DW, forming a stable D-brane soliton, and thus can be regarded as fundamental strings. While a single N ́eel DW is unstable to rotate to a Bloch DW, it can be stabilized once a string terminates on it. There are outstanding differences between the field theory D-brane soliton and the magnetic D-brane soliton. In the former, the string gives rise to a logarithmic bending of the DW, while in the latter we find a linear bending of the DW due to the Bloch-type DMI. We also have stable finitewidth strings far from the DW, in contrast to a singular behaviour in the former. We also consider the possibility of having more than one string-DW junction. Due to the repuslive nature of the force between strings, we construct a solution with periodic boundary conditions in the (x, y)-plane and find a square lattice of strings pointing up- and downwards from the DW, that locally has a linear bending around each of the strings. We thus establish the notion that magnetic skyrmion strings are fundamental strings that can end on a D-brane. The ferromagnetic chiral magnetic system is described by a gradient term (Heisenberg interaction), the DMI and a potential (on-site interaction) that we take here to be the easy-axis anisotropy term E = 1 2∂in·∂in+κn·∇×n+ 1 2(1-n2 3), i = 1, 2, 3, (1) where the total energy is given by E = R E d3x and the magnetization vector or sigma model field, n = (n1, n2, n3), is subject to the usual sigma model constraint: n · n = 1. The energy and length units are rescaled so as to have a canonical kinetic term and a potential coupling constant of unity: The only remaining coupling constant of the theory (κ) is a combination of the physical constants. In this Letter, we consider the ferromagnetic phase κ ∈[0.3, κcrit] with κcrit ≈0.6 being the critical coupling beyond which a CSL-type phase is the ground state (for κ > κcrit) [18-22]. We also concentrate on static configurations and do not discuss dynamics. The topology of the theory can be classified into codimensions. First, drawing a line in the z-direction, the importance of the potential comes into light: The DW (of codimension one) is supported by the existence of two generate and disconnected vacua π0(Mvac) = π0 {n3 = 1 ∪n3 = -1} ≃Z2. (2) Taking instead a slice, say in the (x, y)-plane, the topology of 2-dimensional skyrmions is given by the 2nd homotopy group that supports also NLσM lumps: π2(S2) ≃Z ∋Qskyrmion. (3) This π2 corresponds to the point-compactified (x, y)- plane, also known as the extended complex plane, and is topologically identical to the Riemann sphere (S2). Finally, the 3-dimensional space mapped to the sigma model target space being a 2-sphere, gives rise to the Hopf invariant π3(S2) = Z ∋QHopfion, which supports Hopfions that are knotted string-like solutions [63, 64] - we will not discuss those further here. In this Letter, we construct the string-DW junction, which is combination of the two former types of soliton. Let us briefly review the field theory D-brane soliton solution, which is the string-DW junction for κ := 0, that endows the DW with a logarithmic bending. The solution is η = ρeiθ±z, (4) where we have used the Riemann sphere coordinate η = n1 + in2 1 + n3 , (5) on the sigma model target space and cylindrical coordinates on space: x+iy = ρeiθ. To observe the logarithmic being of the DW, we can simply analyze the iso-curve corresponding to the field at a fixed level set, say |η| = 1: log ρ = ±z. (6) This equation also reveals that far from the junction, on the string side, the string becomes infinitesimally small in size. We now turn to the case of κ ̸= 0 and will for concreteness work in the ferromagnetic phase with κ ∈[0.3, κcrit]. Firstly, we have stable skyrmion strings in this case with κ > 0 turned on. This can be understood simply by comparing each slice at a fixed z to the usual 2D magnetic skyrmions and see that up to a choice of potential, these are identical. The DWs perpendicular to a vector in the (x, y)-plane are of the Bloch-type, for which DMI contributes negatively to the energy, and are stable. In the presence of a Bloch DW, the skyrmion string is repelled [43] and does not automatically flow to the DW to become a DW-skyrmion [37-52, 55-57].2 On the other hand, the DW is of N ́eel-type when perpendicular to the z-direction, which is actually unstable and tends to rotate to a Bloch DW, as the N ́eel DW's DMI energy is 2 In the CSL phase, the Bloch DWs have negative energy and constitute a CSL [18-22], and the DW-skyrmion string, if formed, is unstable to split into a bimeron (pair) [42]. 3 positive. Nevertheless, we show the N ́eel DW is stabilized in the ferromagnetic phase when skyrmion strings terminate on it. One essential point is that the N ́eel DW has an exact U(1) modulus on it while the Bloch DW does not - this facilitates that the winding of the string can be connected to the DW. First, to establish that the bending of the wall will be linear, we cannot simply fix the field value of the solution, since we are not aware of analytic exact solutions to the equations of motion in this case. However, using the idea of fixing a curve in the equation of motion, we arrive at leading order at the solution of the curve for |η| = 1: κr + cr log r + czz = 0, (7) where dimensional analysis3 fixes cr/cz = 1 when κ = 0. This reproduces the logarithmic bending of the DW when κ = 0 and predicts a linear bending of the DW when κ ̸= 0. In order to confirm our prediction, we turn to numerical computations of the equation of motion coming from varying the energy functional (1): eom := ∂2 i η -2(∂iη)2 ̄η 1 + η ̄η -(1 -η ̄η)η 1 + η ̄η = 0. (8) We discretize a box typically of the size 511 × 511 × 767 where the derivatives of the equation of motion are approximated by a 4th order 5-point stencil finite-difference scheme, with lattice spacing h = 0.15. The method for finding solutions utilized here is the so-called arrestedNewton flow, where we solve ∂2 τη = eom, (9) monitoring the static energy given by Eq. (1) at every step of the flow in τ (τ is a fictitious time parameter). If the static energy rises, we stop the flow by setting ∂τη = 0 and then restart the algorithm. In Fig. 1, we display the numerical results for the string-DW junction (trumpet) for various values of κ = 0.35, 0.4, 0.45, 0.5, illustrating the dependence of the slope of the linear bending of the wall on the DMI coupling κ, with the steepest slopes corresponding to the largest values of κ. The figure shows the isosurface of the n3 field at n3 = 0; this is the midway between the two vacua of the potential in Eq. (1). Clearly the DW obeys a linear relation between the z-coordinate and the radius from the string junction's center when the radius is sufficiently large (say of the order of 10 or larger). The size (thickness) of the string also depends approximately linearly on κ, which can be seen from Derrick's theorem 3 More specifically, by reinstating units into the equations, the ratio cz/cr = m with m2 being the coupling of the potential term. Our dimensionless equations correspond to m = 1. FIG. 1. String-DW junctions (trumpets) for various strength of the DMI κ = 0.35 (blue), 0.4 (red), 0.45 (magenta), 0.5 (yellow). The slope of the linear bending is nearly linearly proportional to κ. (also known as a virial law in other fields of physics) [73]. In a linear theory or an integrable theory, we would expect the constants cr and cz to truly be constants (like in the field theory D-brane soliton case), but due to nonlinearities of system we find they have some dependence on κ. Fitting to the slope data of the solutions in Fig. 1, we find cz ≈3.4 -5.2κ, cr/cz ≈1 + 12.6κ. (10) which is consistent with the logarithmic bending with coefficient cr/cz = 1 when κ →0. Interestingly, the approximate relation between cz and κ predicts a critical value of the coupling, which indeed exists due to the transition to the CSL: The linear fit predicts κcrit ≈0.65, whereas in the full nonlinear theory we find it to be closer to κcrit ∼0.6. We find also a lower critical value for κ, where the string-DW junction ceases to exist near κcrit,junc ≳0.3, although the skyrmion string itself still exists at κ = 0.3. Fig. 2 shows the detailed information of the entire 3vector field, n, at the string-slice in the inlet at the topright corner, at the wall-slice at the bottom of the figure and finally in a vertical cross section in the top of the figure. In this figure, the vector n is mapped onto real space 4 FIG. 2. String-DW junction with κ = 0.5. Arrows are mapped from n to x and the colors are in one-to-one correspondence with the arrows: n1 = 1 7→red, n2 = 1 7→green, n1 = -1 7→cyan, n2 = -1 7→magenta, n3 = 1 7→white and n3 = -1 7→black. The inlet and the bottom are horizontal cross sections, whereas the top is a vertical cross section of the string-DW junction. (R3) using arrows, with the up vacuum (n3 = 1) marked by ⊙, the down vacuum (n3 = -1) marked by ⊗and the vectors in the plane corresponding to (n1, n2) when n3 is small. The color scheme is in one-to-one correspondence with the arrows and is described in the caption of the figure. There are about 15 times more lattice points in the solution, than arrows depicted on the figure. This string junction turns out to repel other string junctions, both of strings pointing upwards (skyrmion strings) and strings pointing downwards (anti-skyrmion strings). Packing the string junctions into a given volume, however, has solutions which can most easily be found by considering periodic boundary conditions in the (x, y)-plane. Such a configuration of multi-string-DW junctions is shown in Fig. 3, with the upper part of the space being mostly in the up vacuum (n3 = 1) and the lower part mostly in the down vacuum (n3 = -1). The right-hand side of the figure shows the spin interpretation of the map from n to real space, see the caption of Fig. 2 for details of the color scheme. Although we mentioned in the introduction, that the chiral DMI supports only skyrmion strings and no antiskyrmion string, this is indeed the case in the up-vacuum (upper part of Fig. 3). However, in the down-vacuum (lower part of Fig. 3), the situation is reversed and the DMI supports only anti-skyrmion strings. Interestingly, if we impose periodic boundary conditions also in the z-direction, we obtain a fully periodic configuration just like that of Fig. 3, but with the antiDW placed equidistantly from the DW (the one visible in Fig. 3). This means that one could have n layers of cyclic DW-anti-DW configurations with skyrmion strings and anti-skyrmion strings stretched between them. Real-world experimentally realizable physical systems for studying D-brane analogs etc. are hard to come by. It is yet unclear if and how this could be utilized for applications in real-world spintronics. However, the possibility of realizing a D-brane-string junction in a chiral magnet is very interesting for fundamental physics. One of the important directions to explore is how to create a D-brane soliton. If a N ́eel wall moving under an applied current passes through a fixed impurity or monopole, it will be bent and eventually would create a D-brane soliton. Another direction entails excitations on a D-brane soliton. Kelvin modes propagating along a skyrmion string [70, 71] are the so-called type-B NambuGoldstone modes [74, 75] with a fractional dispersion relation [76, 77] as well as ripple modes on a domain wall [77-79]. It will be worthwhile to study how they are transformed to each other when propagating along a Dbrane soliton. D-brane analogs have been also proposed in Bose-Einstein condensates [80-83]. Condensed matter D-branes open up a new avenue of composite topological objects to simulate fundamental physics, which could be realized in other systems such as liquid crystals and optics. Acknowledgments S. B. G. thanks the Outstanding Talent Program of Henan University for partial support. The work of M. N. is supported in part by JSPS KAKENHI [Grants 5 FIG. 3. Multi-string-DW junctions with κ = 0.4. Periodic boundary conditions are imposed in the x- and y-directions. 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Online Reliable Anomaly Detection via
Neuromorphic Sensing and Communications
Junya Shiraishi, Member, IEEE, Jiechen Chen, Member, IEEE, Osvaldo Simeone, Fellow, IEEE,
and Petar Popovski, Fellow, IEEE
Abstract—This paper proposes a low-power online anomaly
detection framework based on neuromorphic wireless sensor net-
works, encompassing possible use cases such as brain–machine
interfaces and remote environmental monitoring. In the con-
sidered system, a central reader node actively queries a subset
of neuromorphic sensor nodes (neuro-SNs) at each time frame.
The neuromorphic sensors are event-driven, producing spikes in
correspondence to relevant changes in the monitored system. The
queried neuro-SNs respond to the reader with impulse radio (IR)
transmissions that directly encode the sensed local events. The
reader processes these event-driven signals to determine whether
the monitored environment is in a normal or anomalous state,
while rigorously controlling the false discovery rate (FDR) of
detections below a predefined threshold. The proposed approach
employs an online hypothesis testing method with e-values
to maintain FDR control without requiring knowledge of the
anomaly rate, and it dynamically optimizes the sensor querying
strategy by casting it as a best-arm identification problem in a
multi-armed bandit framework. Extensive performance evalua-
tion demonstrates that the proposed method can reliably detect
anomalies under stringent FDR requirements, while efficiently
scheduling sensor communications and achieving low detection
latency.
Index Terms—anomaly detection, false discovery rate control,
hypothesis test, multi-armed bandit, neuromorphic sensing and
communication.
I. INTRODUCTION
Neuromorphic technologies – encompassing sensing [1],
[2], computing [3], [4], and communication [5], [6], [7],
[8], [9], [10] – are emerging as a promising paradigm for
ultra-low-power, low-latency sensor networks. Inspired by the
brain’s event-driven information processing, a neuromorphic
sensor records information by producing spikes at times
marking significant changes in the monitored environment
[11]. Thanks to the inherent capacity of neuromorphic sensors
to respond only to relevant events, incorporating neuromorphic
sensors in a remote monitoring system can drastically reduce
J. Shiraishi and J. Chen contributed equally to this work. The work of
J. Shiraishi was supported by European Union’s Horizon Europe research
and innovation funding programme under Marie Skłodowska-Curie Action
(MSCA) Postdoctoral Fellowship, “NEUTRINAI” with grant agreement
No. 101151067. The work of J. Chen and O. Simeone was supported by the
EPSRC project EP/X011852/1, and O. Simeone was also supported by an
Open Fellowships of the EPSRC (EP/W024101/1). The work of P. Popovski
was supported by the Velux Foundation, Denmark, through the Villum
Investigator Grant WATER, nr. 37793. (Corresponding author: Jiechen Chen)
J. Shiraishi and P. Popovski are with the Department of Electronic
Systems, Aalborg University, 9220 Aalborg, Denmark (e-mail: {jush,
petarp}@es.aau.dk). J. Chen and O. Simeone are with the King’s Com-
munications, Learning and Information Processing Laboratory, King’s
College London, WC2R 2LS London, U.K. (e-mail: {jiechen.chen, os-
valdo.simeone}@kcl.ac.uk)
Neuro-SN 1
1
𝐾
𝑆
𝑞ௌ
ଵ
1. Query (𝒩)
2. IR transmission
Reader
Environment state
t
t
Neuro-SN 𝐾
⋮
𝑞ௌ
⋮
𝑆መ
anomaly
detection
Fig. 1: In the setup under study, a reader interrogates a subset Nf
of the K neuromorphic sensor nodes (neuro-SN) at each frame f in
order to determine whether the behavior of the monitored system is
normal or anomalous. The binary variable Sf indicates the presence
or absence of an anomaly at frame f, while ˆSf represents the
corresponding estimate of the reader.
redundant transmissions, leading to substantial energy and
bandwidth savings and supporting the deployment of large
populations of wireless sensors [5], [10].
For example, reference [10] demonstrated an asynchronous
wireless network of salt-grain-sized sensors that mimic neu-
ral spikes, showing that hundreds of microscale nodes can
communicate sporadic bursts of data to a central receiver
using backscattering. Such neuromorphic networks have been
highlighted for their potential in brain–machine interfaces,
where implantable sensors would relay neural spikes in real
time, as well as in other domains like wearable devices and
environmental sensing [12].
In applications such as remote surveillance and autonomous
systems, timely detection of abnormal events is critical. No-
tably, neuromorphic vision sensors can detect sudden motion
or the appearance of an intruder – e.g., an unauthorized drone
– by producing rapid spikes, while remaining largely inactive
under normal conditions [1]. In environmental monitoring, dis-
tributed sensors might pick up abrupt changes in temperature
or smoke indicative of a forest fire [13]. In these scenarios,
the system must provide reliable anomaly detection despite
limited energy supply and communication bandwidth.
To this end, the detection algorithm needs to be statistically
robust, minimizing false alarms to avoid wasting attention
and energy on benign events. Furthermore, the network must
intelligently decide which sensors to query and when, so
as to maximize the chance of detecting an anomaly while
conserving resources. In this regard, a common reliability
criterion is maintaining the false discovery rate (FDR) below
arXiv:2510.14688v1 [cs.LG] 16 Oct 2025
a target level, ensuring that only a small fraction of reported
anomalies are false positives [14], [15], [16]. Achieving a low
FDR is especially important in safety-critical applications, like
health monitoring or security, to maintain trust in the system’s
alarms.
This paper addresses these challenges by proposing a novel
solution that integrates tools from statistical FDR control
[17] and adaptive learning [18] into a neuromorphic wireless
sensing framework.
A. Related Work
Neuromorphic sensing: Neuromorphic sensing aims to em-
ulate biological sensory systems by encoding information
through sparse, event-driven signals rather than continuous
streams. A prominent category is vision sensing, where dy-
namic vision sensors (DVSs) detect asynchronous changes in
brightness at each pixel, achieving low latency and high en-
ergy efficiency. Prophesee and IniVation are among the leading
companies in this area, offering state-of-the-art event-based
cameras for machine vision and robotic applications [19], [20].
Beyond vision, companies like SynSense are developing ultra-
low-power neuromorphic processors such as Speck, which en-
able local event-driven processing for embedded intelligence
[21].
Neuromorphic communication: Recent work [6] has intro-
duced end-to-end neuromorphic communication systems that
replaces conventional sensing and frame-based radio with
a fully spike-driven pipeline. In it, neuromorphic sensors
produce asynchronous event spikes which are encoded by a
spiking neural network (SNN) and transmitted as impulse ra-
dio (IR) pulses, then decoded by another SNN at the receiver.
This all-spike design was shown to significantly lower latency
and improve efficiency compared to conventional synchronous
digital communication.
Building on this, reference [5] extended the approach to
a multi-user, fading-channel scenario. In it, each internet of
things (IoT) device uses a neuromorphic event sensor, on-
device SNN, and IR transmitter, communicating over a shared
wireless channel to a receiver that also employs an SNN. A
key principle in these systems is event-driven data acquisition,
which allows the wireless network to inherently focus on
the “semantics” of the data – i.e. changes that carry new
information – rather than sending redundant raw readings.
Further energy savings can be obtained by combining IR-
based communication with wake-up radio triggers [7], adopt-
ing different design criteria [22], and leveraging multi-level
spikes [9]. In particular, reference [9] shows that associating
each spike with a small payload can improving inference
accuracy without increasing spike count. This study also
includes an experimental evaluation using software-defined
radios.
Online anomaly detection: Online anomaly detection fo-
cuses on identifying abnormal patterns in streaming data
in real time, without requiring access to the entire dataset
beforehand. For example, reference [14] proposed online false
discovery rate control methods for anomaly detection in time
series to address the challenges of rare anomalies and temporal
dependencies. Reference [15] proposed an unsupervised real-
time anomaly detection algorithm for streaming data, which
is capable of continuous learning and early detection of both
spatial and temporal anomalies without requiring labeled data
or batch processing.
Multi-armed
bandit
optimization:
Multi-armed
bandit
(MAB) optimization has been extensively studied in previous
works. For example, reference [18] proposed the track-and-
stop strategy for best-arm identification under fixed-confidence
settings. Reference [23] studied the complexity of best-arm
identification in stochastic MAB under both fixed-budget and
fixed-confidence settings. To speed up the convergence of
MAB in unseen tasks, [24] proposed for the first time the
use of contextual meta-learning that incorporates auxiliary
optimization tasks knowledge.
B. Main Contributions
Building on the above context and prior art, this pa-
per develops a protocol for online anomaly detection via
neuromorphic sensor networks. The main contributions are
summarized as follows:
• Online anomaly detection framework based on neuromor-
phic sensors: We introduce a novel online anomaly detec-
tion system based on neuromorphic sensor networks. As
shown in Fig. 1, this model implements a pull-based com-
munication strategy, where a central reader intermittently
queries neuromorphic sensor nodes and receives event-
driven IR spike signals. The model explicitly incorporates
the sparse, asynchronous nature of neuromorphic data
and the requirement to control the false discovery rate in
detection decisions.
• Adaptive FDR control with e-values: We design an adap-
tive statistical testing framework that guarantees FDR
control at any desired level in an online setting. In
contrast to traditional methods that rely on fixed p-
value thresholds, our framework utilizes e-values [25] to
accumulate evidence over each frame. This allows the
detector to flexibly adjust its confidence as new spike
data arrive, ensuring that the expected proportion of false
alarms remains below the target level even when the
anomaly occurrence rate is unknown and nonstationary.
• Best-arm identification for sensor scheduling: We in-
troduce a sensor scheduling policy based on best-arm
identification principles from multi-armed bandit theory
[18]. At each time frame, the reader must decide which
sensors to query next. Our policy dynamically learns
the sensors that are most likely to yield an anomaly
indication by treating each sensor as an “arm’ with an
unknown reward. By continuously updating estimates
and concentrating queries on the most promising sensor,
the policy achieves efficient allocation of limited query
opportunities, accelerating the detection of anomalies
under uncertainty.
• Extensive performance evaluation: We provide a thor-
ough performance evaluation of the proposed system
through analysis and simulations. The results demon-
strate that our approach can detect anomalies quickly and
reliably under a variety of conditions. In particular, we
show that the FDR is maintained below the target level
in all tested scenarios, validating the effectiveness of the
e-value based control. Moreover, the adaptive scheduling
policy is shown to significantly reduce detection delay
and resource usage compared to non-adaptive or random
scheduling baselines.
The rest of the paper is organized as follows. Section
II describes the pull-based neuromorphic wireless commu-
nications system, while the online anomaly detection based
on hypothesis test framework is described in Section III.
Section IV explains the scheduling policy optimization based
on best-arm identification. Experimental setting and results
are described in Section V. Finally, Section VI concludes the
paper.
II. SYSTEM MODEL
A. Setting
Fig. 1 shows the basic setup considered in this paper.
We consider a pull-based neuromorphic wireless communica-
tions scenario where the reader conducts a continuous online
anomaly detection at each time frame based on the deployment
of neuromorphic sensor nodes (neuro-SN) [10].
The K neuro-SNs observe a common environment via
local neuromorphic sensors [5], and communicate with the
reader via IR. We define the set of node indices by K =
{1, 2, . . . , K}. Through communication with the neuro-SNs,
the reader continuously monitors the environment to detect
anomalies, while ensuring that the FDR is maintained below
a pre-determined level, denoted as α. The FDR measures
the fraction of detected anomalies that do not correspond to
actual anomalies. Specific applications include real-time forest
fire detection [13] or drone activity monitoring [5] based on
wireless sensor networks.
As in [14], anomalies are defined as rare events that
significantly deviate from expected patterns. In our setting,
an anomaly causes neuromorphic sensors to produce spikes,
also known as events, at a significantly higher rate than
under normal conditions. For example, neuromorphic cameras
produce spikes at times in which activity is recorded in the
monitored scene [5]. Thus, in the presence of significant
activity, e.g., from unauthorized drones, the spike rate is
increased.
As illustrated in Fig. 2, we adopt a discrete time model, in
which time is divided into frames indexed by f = 1, 2, . . . , F.
Each frame consists of three phases:
1) Downlink query signal transmission: At the beginning
of each frame, the reader first sends a query signal to a
selected subset of devices.
2) Sensing and uplink IR transmission: Upon detecting
the query signal, e.g., via a wake-up radio [7], [26],
a neuro-SN activates its neuromorphic sensor and its IR
transmitter. The IR transmission phase is divided into
t
⋯
t
1-st frame
2nd frame
𝐹-th frame
Sensing and IR transmission
Anomaly detection
𝐿uplink slots
Query transmission
downlink
anomaly
detection
Fig. 2: Frame structure illustrating the query transmission from the
reader (downlink), sensing and impulse radio (IR) transmission by
the selected neuro-SN during L uplink slots, and anomaly detection
at the reader.
L time slots, during which the neuro-SNs sense and
transmit in an online and event-driven manner.
3) Anomaly detection at the reader: Finally, the reader
determines whether the environment is in an anomalous
state based on the received signals.
B. Observation Models
Following [14], we model the presence or absence of an
anomaly in a frame indexed by f as an arbitrary, unknown
sequence of binary state Sf, which are defined as
Sf =
(
0,
no anomaly,
1,
anomaly.
(1)
We model the observations of each neuromorphic sensor
k at the f-th frame via a Bernoulli process with spiking
rate determined by the current state Sf. Accordingly, for
each sensor k, we set two spiking rates corresponding to the
absence or presence of an anomaly, namely, qk
0 and qk
1. As
illustrated in Fig. 3, we assume the inequality qk
0
≤qk
1 for
all k
∈
{1, 2, . . . , K}. This indicates that anomalies are
characterized by larger spiking rates.
Accordingly, each node k obtains a spike sequence sk
f =
[sk
f,1, . . . , sk
f,L] ∈{0, 1}L, where sk
f,l = 1 indicates that node
k records a spike at time instant l of the f-th frame, and
sk
f,l = 0 indicates that no spike was recorded. This means
that we have
sk
f,l
∼
i.i.d. Bern(qk
f) with qk
f =
(
qk
0
if Sf = 0
qk
1
if Sf = 1.
(2)
Through an initial calibration phase, the reader can ob-
tain the statistics on the spike rates in normal conditions.
Therefore, the reader is assumed to know the value of the
probability qk
0 for all k
∈
{1, 2, . . . , K}. In contrast, we
assume the reader does not know the spiking probability qk
1
in the presence of anomalies.
C. Scheduling and Impulse Radio Transmission
In this work, we assume time division multiple access
(TDMA)-based communication between the reader and neuro-
SNs, which consists of downlink phase and uplink phases.
Spike rate
Neuro-SN
1
2
𝐾
⋯
⋯
Spike rate for the normal state
1
0
Spike rate for the anomalous state
𝑞ଵ
ଵ
𝑞ଵ
ଶ
𝑞
ଵ
𝑞
ଶ
𝑞ଵ
𝑞
Fig. 3: Each k-th sensor is characterized by a known spiking
probability qk
0 in normal conditions, and a higher, unknown spiking
probability qk
1 ≥qk
0 in the presence of an anomaly.
0
1
Transmitted bit
Received bit
𝜀ଵ
1 −𝜀ଵ
0
1
1 −𝜀ଵ
𝜀ଵ
Fig. 4: Illustration of a binary asymmetric channel.
At each time frame f, the reader requests data from a subset
Nf ⊆{1, . . . , K} of neuro-SN by multicasting a query signal
that encodes the target node IDs and its schedule information.
The reader schedules |Nf| = Cmax nodes, and transmits this
control information during the downlink phase. In practice, the
cardinality Cmax of the scheduled set depends on the available
uplink capacity.
The nodes that receive the query signal begin sensing over
the L time instants within the frame. The spike sequence sk
f is
transmitted by all scheduled sensors in Nf in an online and
event-driven manner using IR by directly mapping a spike
from the sensor to an electromagnetic pulse [5].
As illustrated in Fig. 4, we model the channel as a binary
asymmetric channel, where transmitted bits may be flipped at
the receiver, with bit 0 flipping to 1 with probability ϵ01, and
bit 1 flipping to 0 with probability ϵ10. The IR receiver at
the reader estimates the spike sequence transmitted by each
neuro-SN. After receiving the estimated spike sequence ˆsk
f =
[ˆsk
f,1, . . . , ˆsk
f,L] ∈{0, 1}L for each node k ∈Nf at time frame
f, the reader uses this information to determine whether the
current state Sf is normal or anomalous, as described in the
next section.
III. ONLINE ANOMALY DETECTION
In this section, we first present the hypothesis test frame-
work, assuming that the reader has collected data from the
subset of nodes Nf at the f-th frame. Then, we describe the
proposed anomaly detection mechanism for online anomaly
detection. The problem of optimizing node scheduling will be
addressed in the next section.
A. Hypothesis Testing Framework
At the end of each frame f, the reader tests whether the
current system state is normal or anomalous. Specifically, it
considers the following hypothesis
Hf : Sf = 0 (normal state)
(3)
against the alternative hypothesis that there is an anomaly, i.e.,
Sf = 1. Equivalently, the reader evaluates the null hypothesis
for the requested sensors in Nf as follows
Hf : sk
f ∼
i.i.d. Bern(qk
0), ∀k ∈Nf
(4)
against the composite alternative hypothesis
¯Hf : sk
f ∼
i.i.d. Bern(qk
1), for some qk
1 > qk
0, ∀k ∈Nf.
(5)
In order to conduct a hypothesis test during the f-th frame,
the reader uses the spike sequences ˆsf = {ˆsk
f}k∈Nf observed
during this frame and creates the test statistics Xf. In addition,
the reader selects the rejection threshold αf by considering the
target metrics, such as FDR. The null hypothesis Hf is that
there is no anomaly, such that if this hypothesis is rejected,
the reader declares an anomalous state. The decision at frame
f is denoted as ˆSf ∈{0, 1}, which can be described as
ˆSf = h(Xf, αf). Here, h(Xf, αf) is an indicator function
representing the rejection decision based on Xf and αf. In
the following subsection, we first describe our target metrics
and use test statistics based on e-values. Finally, we describe
how to select the rejection thresholds αf considering the target
metrics.
B. Decaying Memory Time-Averaged False Discovery Rate
Let Ff|0
represent the subset of the frame indices
{1, 2, . . . , f} for which the environment is in a normal state,
i.e., Ff|0 = {f ′ ∈{1, 2, . . . , f} : Sf ′ = 0}. At frame f,
the cumulative number of anomalies declared by the reader,
averaged with a forgetting factor 0 < δ < 1 over the past
frames, is
ˆAf =
f
X
f ′=1
δf−f ′1( ˆSf ′ = 1),
(6)
where 1(·) is the indicator function, which equals to 1 if the
argument is true, and equals to 0 otherwise. In a similar way,
the weighted time-average of incorrectly detected anomalies
is
Ff =
f
X
f ′=1
δf−f ′1( ˆSf ′ = 1)1(Sf ′ = 0).
(7)
We are interested in controlling the decaying-memory time-
averaged FDR, which is defined as the ratio [14]
FDRf = E
Ff
max( ˆAf, 1)
,
(8)
This quantity measures the fraction of estimated anomalies
that were actually normal states. Thus, controlling the FDR
in (8) ensures that the system does not cause an excessive
fraction of false alarms. Accordingly, the design target is
to determine a sequence of rejection thresholds {αf}∞
f=1
ensuring that the metric FDRf is controlled below a pre-
determined level α, as
FDRf ≤α, for all frames f = 1, . . . , F.
(9)
C. Decaying Memory Time-Averaged True Discovery Rate
The bound in (9) can be trivially guaranteed by setting αf =
0 for all frames f. In this case, no anomalies are detected,
and thus there is no false detection of anomalies. However,
this approach clearly fails to detect any potential anomalies.
Therefore, a testing strategy must be also evaluated in terms
of its ability to detect true anomalies. To account for this, we
consider the true discovery rate (TDR) as the average fraction
of truly anomalous states that are correctly detected.
In a similar way to the decaying-memory time-averaged
FDR, let Af denote the cumulative number of anomalies
that occur in the environment at frame f, averaged with a
forgetting factor 0 < δ < 1, which is defined as
Af =
f
X
f ′=1
δf−f ′1(Sf ′ = 1).
(10)
Similarly, the weighted time-averaged of correctly detected
anomaly, denoted as Tf can be expressed as
Tf =
f
X
f ′=1
δf−f ′1( ˆSf ′ = 1)1(Sf ′ = 1).
(11)
Finally, we define decayed-memory time-averaged TDR,
which is defined as the ratio, as described below:
TDRf = E
Tf
max(Af, 1)
.
(12)
This quantity measures how many actual anomaly events the
reader successfully detects. This depends on the scheduling
policy, which will be described in Sec. IV.
The metrics FDR in (8) and TDR in (12) will be used as a
basis for designing rejection threshold αf. Specifically, in each
frame, the reader sets a rejection threshold αf based on the
past rejection history so that FDR condition in (9) is always
guaranteed while providing high TDR. Specific algorithms
should be designed by taking into account the well-known
piggybacking and α death problem [14], whose details are
specified in Sec. III-E.
D. Construction of E-Values
An e-value ef
∈[0, ∞] is a function of the current
observations Sf that satisfies the inequality E[ef] ≤1 when
the null hypothesis Hf is true, i.e., when the state is normal
(Sf = 0) [25]. To construct an e-value ef for each time frame
f, the reader estimates the spike rate from node k at the f-th
frame as
ˆqk
f = 1
L
L
X
l=1
ˆsk
f,l.
(13)
Intuitively, a larger estimated rate ˆqk
f suggests a higher likeli-
hood that the environment is experiencing an anomaly.
Since the anomalous spike rates are unknown to the reader,
an e-value can be constructed by relying on the plug-in
method [25] along with the merging of e-values for the
scheduled sensors k ∈Nf. Specifically, for each sensor
k ∈Nf, an e-value is obtained via the likelihood ratio
ek
f =
ˆψk
f|1
ψk
f|0
!Lˆqk
f
1 −ˆψk
f|1
1 −ψk
f|0
!L(1−ˆqk
f )
,
(14)
where
ψk
f|0 = qk
0(1 −ϵ10) + (1 −qk
0)ϵ01
(15)
is the spiking probability that the reader receives from the
neuro-SN k at the f-th frame when the environment is normal
state and ˆψk
f|1 is the maximum likelihood of received spike
rate at the reader from neuro-SN k at the f-th frame when
the environment is anomalous state. This is obtained as
ˆψk
f|1 = ˆqk
1(1 −ϵ10) + (1 −ˆqk
1)ϵ01,
(16)
where ˆqk
1 is the maximum likelihood estimate of the spike rate
qk
1, which is given by
ˆqk
1 = max(qk
0, ˆqk
f).
(17)
Finally, the reader merges the e-values {ek
f}k∈Nf to obtain
a single e-value via the arithmetic mean [25]
¯ef(Nf) =
1
|Nf|
X
k∈Nf
ek
f.
(18)
E. Design of the Rejection Thresholds
In order to ensure FDR condition (9), we adopt the al-
gorithm detailed in [14]. Accordingly, at the end of each
time frame f, the reader dynamically controls the decision
threshold αf based on the past rejection history. An anomaly
is declared if the statistic ef is larger than a threshold 1/αf,
i.e.,
ˆSf = 1
ef > 1
αf
.
(19)
To describe the update of the threshold αf, let ρj denote the
time of j-th detected anomaly, i.e.,
ρj = min
f ≥0
f
X
f ′=1
1
ef ′ >
1
αf ′
≥j
,
(20)
and let {γf}∞
f=1 denote a non-increasing sequence summing
to 1, where γf = 0 for all f ≤0. We specifically use the
default sequence [14]
γf ∝
log (max(f, 2))
f exp
p
log(f)
.
(21)
Finally, let ˜γf = max(γf, 1−δ) as in [14]. Then, the rejection
threshold αf is updated as
αf = α η ˜γf + α
X
j≥1
δf−ρjγf−ρj,
(22)
where η is smoothing parameter η > 0.
Proposition 1 [14]. When the decision threshold αf is up-
dated according to (22) at each frame f, the anomaly de-
tection rule in (19) ensures that the decaying-memory time-
averaged FDR defined in (8) is controlled at level α for all
frames f, thus satisfying condition (9).
IV. SCHEDULING POLICY OPTIMIZATION
In order to increase the TDR while maintaining FDR
control as in (9), an ideal scheduling policy would select at
each frame f the subset Nf of sensors k with the highest
spiking rates qk
1 under an anomalous state. In fact, this would
maximize the chance of correctly detecting an anomaly via
the test (19). However, the true spiking rates qk
1, for k ∈K,
are not known by the reader. Furthermore, anomalies occur
arbitrarily and unpredictably, making it challenging to estimate
the probabilities qk
1.
In this section, we address the problem of optimizing the
set Nf by tackling these challenges. To start, we study the
problem of identifying the best sensor
k∗= arg max
k∈K
qk
1,
(23)
thus concentrating on the case Cmax = |Nf| = 1, in which the
uplink channel supports the transmission of a single neuro-
SN. In this context, formally, the goal in this section is to
minimize the number of frames used to identify the best neuro-
SN k∗with a probability of error not exceeding a threshold
δs ∈(0, 1). An extension to any uplink capacity Cmax ≥1
will be also provided at the end of this section.
A. Multi-Armed Bandit Formulation
To facilitate the design of a scheduling policy, we make the
working assumption that the anomaly process is i.i.d. across
the frame index f, with an unknown anomalous probability
π1, i.e., Sf ∼Bern(π1). Furthermore, we also assume that
the normal spiking probabilities of all neuro-SNs are equal,
i.e.,
qk
0 = q0,
(24)
for all k = 1, . . . , K. Under these simplified assumptions, the
spike rates
1
L
PL
l=1 sk
f,l of each neuro-SN k is i.i.d. across
frames f, with an average
µk = (1 −π1)q0 + π1qk
1.
(25)
This implies that the problem (23) of identifying the best
sensor becomes equivalent to the problem of finding a node
whose average spike rate µk is the highest. i.e.,
k∗= arg max
k∈K
µk.
(26)
This optimization can be formulated as the fastest best
arm identification problem, studied in [18]. As demonstrated
therein, a strategy known as track-and-stop reviewed next, is
optimal.
B. Track-and-Stop Strategy
At each f-th frame, the reader estimates the spike rate µk
of neuro-SN k as
ˆµk
f =
1
N k
f
f
X
f ′=1
ˆqk
f ′1 (k ∈Nf ′) ,
(27)
with
N k
f =
f
X
f ′=1
1 (k ∈Nf ′)
(28)
being the number of frames in which neuro-SN k was sched-
uled and with ˆqk
f ′ defined in (13).
Define as Uf = {k : N k
f < √f −K
2 } the subset of nodes
that have been scheduled fewer than √f−K
2 times up to frame
f. If this set is not empty, the track-and-stop strategy selects
one of the neuro-SNs that was scheduled least frequently,
i.e., in the smallest number of frames, emphasizing explo-
ration [18]. Otherwise, the scheduling selection accounts also
for the current spiking rate estimates ˆµf = {ˆµ1
f, ˆµ2
f, . . . , ˆµK
f },
considering both exploitation and exploration. Overall, the
scheduling decision is
nf =
arg min
k∈K
N k
f ,
if |Uf| > 0,
arg min
k∈K
N k
f −f · wσ(k)(ˆµf)
,
otherwise,
(29)
where σ : {1, 2, . . . , K} →{1, 2, . . . , K} is a function that
returns the rank of each node k in terms of the estimated spike
rate ˆµk in descending order, satisfying the inequalities ˆµσ(1)
f
≥
ˆµσ(2)
f
. . . ≥ˆµσ(K)
f
. The function wσ(k)(ˆµf), which returns
positive value, is detailed in [18] and satisfies the following
property is true: w1(µ) ≥. . . ≥wK(µ). This way, neuro-
SNs with a larger estimated spiking rate are prioritized by the
selection rule (29).
C. Extension of the Scheduling for Any Scheduling Capacity
In this subsection, we extend the track-and-stop strategy to
the case where it is possible to schedule Cmax = |Nf| ≥1
neuro-SNs. To this end, we propose that, at the beginning
of each frame, the reader selects Cmax = |Nf| nodes for
scheduling by recursively applying the track-and-stop method
in (29). Specifically, the reader selects the scheduled nodes
one-by-one, obtaining the set Nf = {ni
f}Cmax
i=1 . In more detail,
define U i
f = {k ∈K \ {nj
f}i−1
j=1 : N k
f < √f −K
2 } as the
set of neuro-SNs not yet included in the set Nf that have
been scheduled in the past frames fewer than √f −K
2 times.
Then, the i-th scheduling node at the f-th frame is selected
by applying the rule (29) as
ni
f =
arg min
k∈K\{nj
f }j=i−1
j=1
N k
f ,
if |U i
f| > 0,
arg min
k∈K\{nj
f }j=i−1
j=1
N k
f −fwσ(k)(ˆµf),
otherwise,
(30)
where function wσ(k)(·) is as defined in (29).
V. NUMERICAL RESULTS
In this section, we present numerical results to validate the
performance of the proposed online anomaly detection system.
A. Simulation Setting
In our experiments, the anomaly process Sf is i.i.d. across
all frames f, so that at each frame f the environment is in
the anomalous state with probability π1 and in the normal
state with probability 1 −π1, independently from frame to
frame. We simulate a setting in which, as in (24), the spike
rate qk
0 of a normal state is the same across all nodes, i.e.,
qk
0 = q0 for all k ∈K. In contrast, the spiking rate qk
1 under
the anomalous state is uniformly generated within the interval
[q0, q0+∆max] for each neuro-SN k. Table I shows our default
parameters setting, which are used throughout the experiments
unless stated otherwise.
TABLE I: Default parameter setting.
Parameters
Values
Number of neuro-SN K
5
Forgetting factor δ
0.99 [14]
η
0.99
Number of uplink slots L
50
Target FDR α
0.1 [14]
Simulation runs
103
Spiking rate under the normal state q0
0.10
Anomaly ratio π1
0.05
Error rates (ϵ01, ϵ10)
(0, 0)
Upper bound on the spiking probability ∆max
0.5
Channel capacity Cmax
1
B. On the Validity of Online Anomaly Detection
We first investigate the validity of the online anomaly
detection algorithm introduced in Sec. III. To this end, we
compare the performance of the proposed scheme with adap-
tive threshold αf described in (22) with a more conventional
fixed-threshold strategy, in which we apply the threshold
αf = α for all f ∈[1, F]. Note that this combined approach
does not satisfy the theoretical statistical guarantee in (9), and
its performance can only be evaluated via numerical results.
Fig. 5 shows FDR and TDR versus the time frame f
for fixed anomaly threshold method and for the proposed
dynamic anomaly detection method in (22), where we set
the number of neuro-SN K = 1. As anticipated, the fixed-
threshold is not able to always satisfies the FDR constraint
in (9). This is because this approach does not take into account
the past rejection history for the current anomaly decision
tasks. A fixed threshold can achieve a higher TDR than the
proposed method, but only by violating the FDR constraint
in (9). In contrast, the introduced methodology, dynamic
threshold always satisfies the FDR constraint α, validating
Proposition 1.
Fig. 6 shows the FDR and TDR as a function of the anomaly
probability π1, where we set K = 1 and F = 1000. We
observe that the fixed-threshold approach suffers from the
higher FDR when the anomaly probability π1 is small, while
the proposed dynamic anomaly detection threshold method
can always satisfy the FDR constraint. When the anomaly
probability π1 is small, with a dynamic threshold, the reader
takes a conservative decision by setting a relatively smaller
threshold αf, will a fixed-threshold method cannot adjust to
the anomaly probability.
C. Random v.s. Track-and-Stop Scheduling
We now study the impact of the scheduling strategy. To
this end, Fig. 7 shows the FDR and the TDR versus the time
frame f for the proposed anomaly detection scheme under
a random scheduling strategy that selects Cmax neuro-SNs
uniformly at random at every frame and for the track-and-stop
method presented in Sec. IV. We set the parameters described
in Table I with the uplink channel capacity Cmax = {1, 2}
and anomaly probability π1 = 0.1.
All schedulers, when coupled with the proposed anomaly
detection, are seen to satisfy the FDR requirements, confirm-
ing the theoretical results in Proposition 1. Moreover, the
figure highlights the benefits of track-and-stop, increasing the
TDR for both uplink capacities Cmax = 1 and Cmax = 2.
This advantage stems from the capacity of track-and-stop to
select the neuro-SNs that are most likely to contribute to the
identification of the neuro-SNs with highest spiking rates.
From Fig. 7 we also see that TDR becomes larger as the
number of scheduled nodes Cmax increases, demonstrating
the importance of collecting information from multiple neuro-
SNs. This is because by calculating e-value (18) based on the
spike rate from multiple nodes, the reader can mitigate the
negative impact of small e-value caused by the selection of
nodes having small spike rate in anomalous state, qk
1.
D. Impact of System Parameters
In this subsection, we focus on the performance of track-
and-stop scheduling, evaluating the impacts of different sys-
tem parameters.
1) On the FDR constraint α: Fig. 8 shows the FDR and
TDR as a function of FDR constraint α. The figure highlights
the trade-off between TDR and FDR: as the allocated FDR
constraint α becomes larger, the reader can more easily detect
anomalies, increasing the TDR. We can also see that TDR
becomes larger as the forgetting factor δ becomes smaller. In
fact, with a smaller δ, the FDR constraint puts higher weight
on the results of the latest decisions, decreasing the importance
of past false alarms.
2) On the sensitivity of the sensors: We now study the
dependence of the system performance on the parameter
defining the quality of the observations of sensors. First, we
100
200
300
400
500
0
0.1
0.2
Fixed αf = α
Dynamic αf
FDR constraint
f
FDR
100
200
300
400
500
0.0
0.25
0.50
0.75
1.0
Fixed αf = α
Dynamic αf
f
TDR
Fig. 5: FDR and TDR versus the time frame f for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed dynamic
anomaly decision threshold based on (22), where we set K = 1 and π1 = 0.05.
0.025
0.05
0.075
0.10
0
0.1
0.2
0.3
Fixed αf = α
Dynamic αf
FDR constraint
π1
FDR
0.025
0.05
0.075
0.10
0.0
0.25
0.50
0.75
1.0
Fixed αf = α
Dynamic αf
π1
TDR
Fig. 6: FDR and TDR versus the anomaly probability π1 for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed
dynamic anomaly decision threshold based on (22), where we set K = 1 and F = 1, 000.
vary the upper bound ∆max on the spiking rate in the presence
of anomalies. From Fig. 9, we can see that TDR is improved as
the upper bound ∆max becomes larger, that is, the sensitivity
improves. This is because a larger upper bound ∆max tends
to increase the difference between the spike rate of neuro-SN
k in the normal state qk
0 and that in the anomalous state qk
1,
enabling the reader to detect anomalies with higher accuracy
(see Fig. 3).
Fig. 10 shows the TDR as a function of the frame index f
for different numbers of uplink slots L. From Fig. 10, we can
see that TDR becomes larger as L increases because the reader
can execute anomaly detection task based on more number
of received spikes, realizing accurate decision based on the
e-value in (18) and decision threshold defined in (22). For
example, to achieve TDR larger than 0.9 under ∆max = 0.5,
there is a need for L = 100 uplink slots.
3) Impact of anomaly probability π1: Fig. 11 shows TDR
against f for track-and-stop, where we set the anomaly
probability to π1 = {0.05, 0.10, 0.15}. As π1 increases, the
probability that the reader obtains the higher e-value in the
single frame also increases, making it more likely that the
reader rejects the hypothesis at the current frame. As a result,
we can see that TDR increases as π1 becomes larger.
4) On the channel error rate: To evaluate the impact of
channel error in the binary asymmetric channel described in
Sec. II-C, we vary the error probability ϵ, by setting: (i)
ϵ01 = ϵ and ϵ10 = 0; (ii) ϵ10 = ϵ and ϵ01 = 0; (iii)
ϵ01 = ϵ10 = ϵ. From Fig. 12, we see that the error probability
ϵ10 has the largest impact on TDR. In fact, as increase in
ϵ10 causes spike losses, which may be critical for anomaly
detection. Conversely, as the error probability ϵ01 increases, it
becomes more likely that the reader mistakenly detects false
spikes, causing the reader to mistakenly detect an anomaly
event when the environment is in a normal state. As the reader
must satisfy the FDR constraint in (9), in this case the reader
must take a conservative decision when the actual anomaly
event happens, reducing the TDR. When both errors have a
non-zero probability, it is affected by both spike losses and
falsely detected spikes, causing further decreases in the TDR.
VI. CONCLUSION
This paper proposed a low-power online anomaly detection
framework for neuromorphic wireless sensor networks. We
have introduced an algorithm that can control the false dis-
covery rate by incorporating e-values test statistics calculated
based on the received spike sequences at each frame. We have
also introduced a track-and-stop methods for scheduling nodes
200
400
600
800
1,000
0
0.1
FDR constraint
f
FDR
200
400
600
800
1,000
0.50
0.75
1.0
Random (Cmax = 1)
track-and-stop (Cmax = 1)
Random (Cmax = 2)
track-and-stop (Cmax = 2)
f
TDR
Fig. 7: FDR and TDR versus the time frame f for random sampling scheme and track-and-stop.
0.05
0.10
0.15
0.20
0
0.1
0.2
FDR constraint
α
FDR
0.05
0.10
0.15
0.20
0.50
0.75
1.0
δ : {0.8, 0, 9, 0.99}
α
TDR
Fig. 8: FDR and TDR versus the FDR constraint α for track-and-stop
(the forgetting factor δ = {0.8, 0.9, 0.99}).
100
200
300
400
500
0
0.2
0.4
0.6
0.8
1
∆max = 0.2
∆max = 0.3
∆max = 0.4
∆max = 0.5
f
TDR
Fig. 9: TDR versus the time frame f for track-and-stop with different
values of upper bound of sensor reading ∆max.
based on the channel constraint. Our numerical results show
that the proposed approach can track the nodes having the
largest spike efficiently with high TDR, while maintaining the
FDR below the predetermined threshold.
Future work includes designing pull-based communication
protocol in neuromorphic communication while taking into
account aspects such as information freshness and value of
information. Designing neuromorphic communication proto-
100
200
300
400
500
0
0.2
0.4
0.6
0.8
1
L = 25
L = 50
L = 75
L = 100
f
TDR
Fig. 10: TDR versus the time frame f for track-and-stop with
different values of the number of uplink slots L.
100
200
300
400
500
0.6
0.7
0.8
0.9
1
π1 = 0.05
π1 = 0.10
π1 = 0.15
f
TDR
Fig. 11: TDR versus the time frame f for track-and-stop scheme
with different values of anomaly potion π1 = {0.05, 0.10, 0.15}
cols for heterogeneous neuro-SNs with different capabilities
in terms of sensing and communication is also an interesting
research avenue.
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Online Reliable Anomaly Detection via Neuromorphic Sensing and Communications Junya Shiraishi, Member, IEEE, Jiechen Chen, Member, IEEE, Osvaldo Simeone, Fellow, IEEE, and Petar Popovski, Fellow, IEEE Abstract-This paper proposes a low-power online anomaly detection framework based on neuromorphic wireless sensor networks, encompassing possible use cases such as brain-machine interfaces and remote environmental monitoring. In the considered system, a central reader node actively queries a subset of neuromorphic sensor nodes (neuro-SNs) at each time frame. The neuromorphic sensors are event-driven, producing spikes in correspondence to relevant changes in the monitored system. The queried neuro-SNs respond to the reader with impulse radio (IR) transmissions that directly encode the sensed local events. The reader processes these event-driven signals to determine whether the monitored environment is in a normal or anomalous state, while rigorously controlling the false discovery rate (FDR) of detections below a predefined threshold. The proposed approach employs an online hypothesis testing method with e-values to maintain FDR control without requiring knowledge of the anomaly rate, and it dynamically optimizes the sensor querying strategy by casting it as a best-arm identification problem in a multi-armed bandit framework. Extensive performance evaluation demonstrates that the proposed method can reliably detect anomalies under stringent FDR requirements, while efficiently scheduling sensor communications and achieving low detection latency. Index Terms-anomaly detection, false discovery rate control, hypothesis test, multi-armed bandit, neuromorphic sensing and communication. I. INTRODUCTION Neuromorphic technologies - encompassing sensing [1], [2], computing [3], [4], and communication [5], [6], [7], [8], [9], [10] - are emerging as a promising paradigm for ultra-low-power, low-latency sensor networks. Inspired by the brain's event-driven information processing, a neuromorphic sensor records information by producing spikes at times marking significant changes in the monitored environment [11]. Thanks to the inherent capacity of neuromorphic sensors to respond only to relevant events, incorporating neuromorphic sensors in a remote monitoring system can drastically reduce J. Shiraishi and J. Chen contributed equally to this work. The work of J. Shiraishi was supported by European Union's Horizon Europe research and innovation funding programme under Marie Skłodowska-Curie Action (MSCA) Postdoctoral Fellowship, "NEUTRINAI" with grant agreement No. 101151067. The work of J. Chen and O. Simeone was supported by the EPSRC project EP/X011852/1, and O. Simeone was also supported by an Open Fellowships of the EPSRC (EP/W024101/1). The work of P. Popovski was supported by the Velux Foundation, Denmark, through the Villum Investigator Grant WATER, nr. 37793. (Corresponding author: Jiechen Chen) J. Shiraishi and P. Popovski are with the 9220 Aalborg, Denmark (e-mail: {jush, ). J. Chen and O. Simeone are with the King's Communications, Learning and Information Processing Laboratory, King's College London, WC2R 2LS London, U.K. (e-mail: {jiechen.chen, os- ) Neuro-SN 1 1 K S qௌ ଵ 1. Query (N) 2. IR transmission Reader Environment state t t Neuro-SN K ⋮ qௌ ⋮ Sመ anomaly detection Fig. 1: In the setup under study, a reader interrogates a subset Nf of the K neuromorphic sensor nodes (neuro-SN) at each frame f in order to determine whether the behavior of the monitored system is normal or anomalous. The binary variable Sf indicates the presence or absence of an anomaly at frame f, while ˆSf represents the corresponding estimate of the reader. redundant transmissions, leading to substantial energy and bandwidth savings and supporting the deployment of large populations of wireless sensors [5], [10]. For example, reference [10] demonstrated an asynchronous wireless network of salt-grain-sized sensors that mimic neural spikes, showing that hundreds of microscale nodes can communicate sporadic bursts of data to a central receiver using backscattering. Such neuromorphic networks have been highlighted for their potential in brain-machine interfaces, where implantable sensors would relay neural spikes in real time, as well as in other domains like wearable devices and environmental sensing [12]. In applications such as remote surveillance and autonomous systems, timely detection of abnormal events is critical. Notably, neuromorphic vision sensors can detect sudden motion or the appearance of an intruder - e.g., an unauthorized drone - by producing rapid spikes, while remaining largely inactive under normal conditions [1]. In environmental monitoring, distributed sensors might pick up abrupt changes in temperature or smoke indicative of a forest fire [13]. In these scenarios, the system must provide reliable anomaly detection despite limited energy supply and communication bandwidth. To this end, the detection algorithm needs to be statistically robust, minimizing false alarms to avoid wasting attention and energy on benign events. Furthermore, the network must intelligently decide which sensors to query and when, so as to maximize the chance of detecting an anomaly while conserving resources. In this regard, a common reliability criterion is maintaining the false discovery rate (FDR) below 16 Oct 2025 a target level, ensuring that only a small fraction of reported anomalies are false positives [14], [15], [16]. Achieving a low FDR is especially important in safety-critical applications, like health monitoring or security, to maintain trust in the system's alarms. This paper addresses these challenges by proposing a novel solution that integrates tools from statistical FDR control [17] and adaptive learning [18] into a neuromorphic wireless sensing framework. A. Related Work Neuromorphic sensing: Neuromorphic sensing aims to emulate biological sensory systems by encoding information through sparse, event-driven signals rather than continuous streams. A prominent category is vision sensing, where dynamic vision sensors (DVSs) detect asynchronous changes in brightness at each pixel, achieving low latency and high energy efficiency. Prophesee and IniVation are among the leading companies in this area, offering state-of-the-art event-based cameras for machine vision and robotic applications [19], [20]. Beyond vision, companies like SynSense are developing ultralow-power neuromorphic processors such as Speck, which enable local event-driven processing for embedded intelligence [21]. Neuromorphic communication: Recent work [6] has introduced end-to-end neuromorphic communication systems that replaces conventional sensing and frame-based radio with a fully spike-driven pipeline. In it, neuromorphic sensors produce asynchronous event spikes which are encoded by a spiking neural network (SNN) and transmitted as impulse radio (IR) pulses, then decoded by another SNN at the receiver. This all-spike design was shown to significantly lower latency and improve efficiency compared to conventional synchronous digital communication. Building on this, reference [5] extended the approach to a multi-user, fading-channel scenario. In it, each internet of things (IoT) device uses a neuromorphic event sensor, ondevice SNN, and IR transmitter, communicating over a shared wireless channel to a receiver that also employs an SNN. A key principle in these systems is event-driven data acquisition, which allows the wireless network to inherently focus on the "semantics" of the data - i.e. changes that carry new information - rather than sending redundant raw readings. Further energy savings can be obtained by combining IRbased communication with wake-up radio triggers [7], adopting different design criteria [22], and leveraging multi-level spikes [9]. In particular, reference [9] shows that associating each spike with a small payload can improving inference accuracy without increasing spike count. This study also includes an experimental evaluation using software-defined radios. Online anomaly detection: Online anomaly detection focuses on identifying abnormal patterns in streaming data in real time, without requiring access to the entire dataset beforehand. For example, reference [14] proposed online false discovery rate control methods for anomaly detection in time series to address the challenges of rare anomalies and temporal dependencies. Reference [15] proposed an unsupervised realtime anomaly detection algorithm for streaming data, which is capable of continuous learning and early detection of both spatial and temporal anomalies without requiring labeled data or batch processing. Multi-armed bandit optimization: Multi-armed bandit (MAB) optimization has been extensively studied in previous works. For example, reference [18] proposed the track-andstop strategy for best-arm identification under fixed-confidence settings. Reference [23] studied the complexity of best-arm identification in stochastic MAB under both fixed-budget and fixed-confidence settings. To speed up the convergence of MAB in unseen tasks, [24] proposed for the first time the use of contextual meta-learning that incorporates auxiliary optimization tasks knowledge. B. Main Contributions Building on the above context and prior art, this paper develops a protocol for online anomaly detection via neuromorphic sensor networks. The main contributions are summarized as follows: • Online anomaly detection framework based on neuromorphic sensors: We introduce a novel online anomaly detection system based on neuromorphic sensor networks. As shown in Fig. 1, this model implements a pull-based communication strategy, where a central reader intermittently queries neuromorphic sensor nodes and receives eventdriven IR spike signals. The model explicitly incorporates the sparse, asynchronous nature of neuromorphic data and the requirement to control the false discovery rate in detection decisions. • Adaptive FDR control with e-values: We design an adaptive statistical testing framework that guarantees FDR control at any desired level in an online setting. In contrast to traditional methods that rely on fixed pvalue thresholds, our framework utilizes e-values [25] to accumulate evidence over each frame. This allows the detector to flexibly adjust its confidence as new spike data arrive, ensuring that the expected proportion of false alarms remains below the target level even when the anomaly occurrence rate is unknown and nonstationary. • Best-arm identification for sensor scheduling: We introduce a sensor scheduling policy based on best-arm identification principles from multi-armed bandit theory [18]. At each time frame, the reader must decide which sensors to query next. Our policy dynamically learns the sensors that are most likely to yield an anomaly indication by treating each sensor as an "arm' with an unknown reward. By continuously updating estimates and concentrating queries on the most promising sensor, the policy achieves efficient allocation of limited query opportunities, accelerating the detection of anomalies under uncertainty. • Extensive performance evaluation: We provide a thorough performance evaluation of the proposed system through analysis and simulations. The results demonstrate that our approach can detect anomalies quickly and reliably under a variety of conditions. In particular, we show that the FDR is maintained below the target level in all tested scenarios, validating the effectiveness of the e-value based control. Moreover, the adaptive scheduling policy is shown to significantly reduce detection delay and resource usage compared to non-adaptive or random scheduling baselines. The rest of the paper is organized as follows. Section II describes the pull-based neuromorphic wireless communications system, while the online anomaly detection based on hypothesis test framework is described in Section III. Section IV explains the scheduling policy optimization based on best-arm identification. Experimental setting and results are described in Section V. Finally, Section VI concludes the paper. II. SYSTEM MODEL A. Setting Fig. 1 shows the basic setup considered in this paper. We consider a pull-based neuromorphic wireless communications scenario where the reader conducts a continuous online anomaly detection at each time frame based on the deployment of neuromorphic sensor nodes (neuro-SN) [10]. The K neuro-SNs observe a common environment via local neuromorphic sensors [5], and communicate with the reader via IR. We define the set of node indices by K = {1, 2, . . . , K}. Through communication with the neuro-SNs, the reader continuously monitors the environment to detect anomalies, while ensuring that the FDR is maintained below a pre-determined level, denoted as α. The FDR measures the fraction of detected anomalies that do not correspond to actual anomalies. Specific applications include real-time forest fire detection [13] or drone activity monitoring [5] based on wireless sensor networks. As in [14], anomalies are defined as rare events that significantly deviate from expected patterns. In our setting, an anomaly causes neuromorphic sensors to produce spikes, also known as events, at a significantly higher rate than under normal conditions. For example, neuromorphic cameras produce spikes at times in which activity is recorded in the monitored scene [5]. Thus, in the presence of significant activity, e.g., from unauthorized drones, the spike rate is increased. As illustrated in Fig. 2, we adopt a discrete time model, in which time is divided into frames indexed by f = 1, 2, . . . , F. Each frame consists of three phases: 1) Downlink query signal transmission: At the beginning of each frame, the reader first sends a query signal to a selected subset of devices. 2) Sensing and uplink IR transmission: Upon detecting the query signal, e.g., via a wake-up radio [7], [26], a neuro-SN activates its neuromorphic sensor and its IR transmitter. The IR transmission phase is divided into t ⋯ t 1-st frame 2nd frame F-th frame Sensing and IR transmission Anomaly detection Luplink slots Query transmission downlink anomaly detection Fig. 2: Frame structure illustrating the query transmission from the reader (downlink), sensing and impulse radio (IR) transmission by the selected neuro-SN during L uplink slots, and anomaly detection at the reader. L time slots, during which the neuro-SNs sense and transmit in an online and event-driven manner. 3) Anomaly detection at the reader: Finally, the reader determines whether the environment is in an anomalous state based on the received signals. B. Observation Models Following [14], we model the presence or absence of an anomaly in a frame indexed by f as an arbitrary, unknown sequence of binary state Sf, which are defined as Sf = ( 0, no anomaly, 1, anomaly. (1) We model the observations of each neuromorphic sensor k at the f-th frame via a Bernoulli process with spiking rate determined by the current state Sf. Accordingly, for each sensor k, we set two spiking rates corresponding to the absence or presence of an anomaly, namely, qk 0 and qk 1. As illustrated in Fig. 3, we assume the inequality qk 0 ≤qk 1 for all k ∈ {1, 2, . . . , K}. This indicates that anomalies are characterized by larger spiking rates. Accordingly, each node k obtains a spike sequence sk f = [sk f,1, . . . , sk f,L] ∈{0, 1}L, where sk f,l = 1 indicates that node k records a spike at time instant l of the f-th frame, and sk f,l = 0 indicates that no spike was recorded. This means that we have sk f,l ∼ i.i.d. Bern(qk f) with qk f = ( qk 0 if Sf = 0 qk 1 if Sf = 1. (2) Through an initial calibration phase, the reader can obtain the statistics on the spike rates in normal conditions. Therefore, the reader is assumed to know the value of the probability qk 0 for all k ∈ {1, 2, . . . , K}. In contrast, we assume the reader does not know the spiking probability qk 1 in the presence of anomalies. C. Scheduling and Impulse Radio Transmission In this work, we assume time division multiple access (TDMA)-based communication between the reader and neuroSNs, which consists of downlink phase and uplink phases. Spike rate Neuro-SN 1 2 K ⋯ ⋯ Spike rate for the normal state 1 0 Spike rate for the anomalous state qଵ ଵ qଵ ଶ q ଵ q ଶ qଵ q Fig. 3: Each k-th sensor is characterized by a known spiking probability qk 0 in normal conditions, and a higher, unknown spiking probability qk 1 ≥qk 0 in the presence of an anomaly. 0 1 Transmitted bit Received bit εଵ 1 -εଵ 0 1 1 -εଵ εଵ Fig. 4: Illustration of a binary asymmetric channel. At each time frame f, the reader requests data from a subset Nf ⊆{1, . . . , K} of neuro-SN by multicasting a query signal that encodes the target node IDs and its schedule information. The reader schedules |Nf| = Cmax nodes, and transmits this control information during the downlink phase. In practice, the cardinality Cmax of the scheduled set depends on the available uplink capacity. The nodes that receive the query signal begin sensing over the L time instants within the frame. The spike sequence sk f is transmitted by all scheduled sensors in Nf in an online and event-driven manner using IR by directly mapping a spike from the sensor to an electromagnetic pulse [5]. As illustrated in Fig. 4, we model the channel as a binary asymmetric channel, where transmitted bits may be flipped at the receiver, with bit 0 flipping to 1 with probability ε01, and bit 1 flipping to 0 with probability ε10. The IR receiver at the reader estimates the spike sequence transmitted by each neuro-SN. After receiving the estimated spike sequence ˆsk f = [ˆsk f,1, . . . , ˆsk f,L] ∈{0, 1}L for each node k ∈Nf at time frame f, the reader uses this information to determine whether the current state Sf is normal or anomalous, as described in the next section. III. ONLINE ANOMALY DETECTION In this section, we first present the hypothesis test framework, assuming that the reader has collected data from the subset of nodes Nf at the f-th frame. Then, we describe the proposed anomaly detection mechanism for online anomaly detection. The problem of optimizing node scheduling will be addressed in the next section. A. Hypothesis Testing Framework At the end of each frame f, the reader tests whether the current system state is normal or anomalous. Specifically, it considers the following hypothesis Hf : Sf = 0 (normal state) (3) against the alternative hypothesis that there is an anomaly, i.e., Sf = 1. Equivalently, the reader evaluates the null hypothesis for the requested sensors in Nf as follows Hf : sk f ∼ i.i.d. Bern(qk 0), ∀k ∈Nf (4) against the composite alternative hypothesis ̄Hf : sk f ∼ i.i.d. Bern(qk 1), for some qk 1 > qk 0, ∀k ∈Nf. (5) In order to conduct a hypothesis test during the f-th frame, the reader uses the spike sequences ˆsf = {ˆsk f}k∈Nf observed during this frame and creates the test statistics Xf. In addition, the reader selects the rejection threshold αf by considering the target metrics, such as FDR. The null hypothesis Hf is that there is no anomaly, such that if this hypothesis is rejected, the reader declares an anomalous state. The decision at frame f is denoted as ˆSf ∈{0, 1}, which can be described as ˆSf = h(Xf, αf). Here, h(Xf, αf) is an indicator function representing the rejection decision based on Xf and αf. In the following subsection, we first describe our target metrics and use test statistics based on e-values. Finally, we describe how to select the rejection thresholds αf considering the target metrics. B. Decaying Memory Time-Averaged False Discovery Rate Let Ff|0 represent the subset of the frame indices {1, 2, . . . , f} for which the environment is in a normal state, i.e., Ff|0 = {f ′ ∈{1, 2, . . . , f} : Sf ′ = 0}. At frame f, the cumulative number of anomalies declared by the reader, averaged with a forgetting factor 0 1 αf . (19) To describe the update of the threshold αf, let ρj denote the time of j-th detected anomaly, i.e., ρj = min f ≥0 f X f ′=1 1 ef ′ > 1 αf ′ ≥j , (20) and let {γf}∞ f=1 denote a non-increasing sequence summing to 1, where γf = 0 for all f ≤0. We specifically use the default sequence [14] γf ∝ log (max(f, 2)) f exp p log(f) . (21) Finally, let ̃γf = max(γf, 1-δ) as in [14]. Then, the rejection threshold αf is updated as αf = α η ̃γf + α X j≥1 δf-ρjγf-ρj, (22) where η is smoothing parameter η > 0. Proposition 1 [14]. When the decision threshold αf is updated according to (22) at each frame f, the anomaly detection rule in (19) ensures that the decaying-memory timeaveraged FDR defined in (8) is controlled at level α for all frames f, thus satisfying condition (9). IV. SCHEDULING POLICY OPTIMIZATION In order to increase the TDR while maintaining FDR control as in (9), an ideal scheduling policy would select at each frame f the subset Nf of sensors k with the highest spiking rates qk 1 under an anomalous state. In fact, this would maximize the chance of correctly detecting an anomaly via the test (19). However, the true spiking rates qk 1, for k ∈K, are not known by the reader. Furthermore, anomalies occur arbitrarily and unpredictably, making it challenging to estimate the probabilities qk 1. In this section, we address the problem of optimizing the set Nf by tackling these challenges. To start, we study the problem of identifying the best sensor k∗= arg max k∈K qk 1, (23) thus concentrating on the case Cmax = |Nf| = 1, in which the uplink channel supports the transmission of a single neuroSN. In this context, formally, the goal in this section is to minimize the number of frames used to identify the best neuroSN k∗with a probability of error not exceeding a threshold δs ∈(0, 1). An extension to any uplink capacity Cmax ≥1 will be also provided at the end of this section. A. Multi-Armed Bandit Formulation To facilitate the design of a scheduling policy, we make the working assumption that the anomaly process is i.i.d. across the frame index f, with an unknown anomalous probability π1, i.e., Sf ∼Bern(π1). Furthermore, we also assume that the normal spiking probabilities of all neuro-SNs are equal, i.e., qk 0 = q0, (24) for all k = 1, . . . , K. Under these simplified assumptions, the spike rates 1 L PL l=1 sk f,l of each neuro-SN k is i.i.d. across frames f, with an average μk = (1 -π1)q0 + π1qk 1. (25) This implies that the problem (23) of identifying the best sensor becomes equivalent to the problem of finding a node whose average spike rate μk is the highest. i.e., k∗= arg max k∈K μk. (26) This optimization can be formulated as the fastest best arm identification problem, studied in [18]. As demonstrated therein, a strategy known as track-and-stop reviewed next, is optimal. B. Track-and-Stop Strategy At each f-th frame, the reader estimates the spike rate μk of neuro-SN k as ˆμk f = 1 N k f f X f ′=1 ˆqk f ′1 (k ∈Nf ′) , (27) with N k f = f X f ′=1 1 (k ∈Nf ′) (28) being the number of frames in which neuro-SN k was scheduled and with ˆqk f ′ defined in (13). Define as Uf = {k : N k f 0, arg min k∈K N k f -f · wσ(k)(ˆμf) , otherwise, (29) where σ : {1, 2, . . . , K} →{1, 2, . . . , K} is a function that returns the rank of each node k in terms of the estimated spike rate ˆμk in descending order, satisfying the inequalities ˆμσ(1) f ≥ ˆμσ(2) f . . . ≥ˆμσ(K) f . The function wσ(k)(ˆμf), which returns positive value, is detailed in [18] and satisfies the following property is true: w1(μ) ≥. . . ≥wK(μ). This way, neuroSNs with a larger estimated spiking rate are prioritized by the selection rule (29). C. Extension of the Scheduling for Any Scheduling Capacity In this subsection, we extend the track-and-stop strategy to the case where it is possible to schedule Cmax = |Nf| ≥1 neuro-SNs. To this end, we propose that, at the beginning of each frame, the reader selects Cmax = |Nf| nodes for scheduling by recursively applying the track-and-stop method in (29). Specifically, the reader selects the scheduled nodes one-by-one, obtaining the set Nf = {ni f}Cmax i=1 . In more detail, define U i f = {k ∈K \ {nj f}i-1 j=1 : N k f 0, arg min k∈K\{nj f }j=i-1 j=1 N k f -fwσ(k)(ˆμf), otherwise, (30) where function wσ(k)(·) is as defined in (29). V. NUMERICAL RESULTS In this section, we present numerical results to validate the performance of the proposed online anomaly detection system. A. Simulation Setting In our experiments, the anomaly process Sf is i.i.d. across all frames f, so that at each frame f the environment is in the anomalous state with probability π1 and in the normal state with probability 1 -π1, independently from frame to frame. We simulate a setting in which, as in (24), the spike rate qk 0 of a normal state is the same across all nodes, i.e., qk 0 = q0 for all k ∈K. In contrast, the spiking rate qk 1 under the anomalous state is uniformly generated within the interval [q0, q0+∆max] for each neuro-SN k. Table I shows our default parameters setting, which are used throughout the experiments unless stated otherwise. TABLE I: Default parameter setting. Parameters Values Number of neuro-SN K 5 Forgetting factor δ 0.99 [14] η 0.99 Number of uplink slots L 50 Target FDR α 0.1 [14] Simulation runs 103 Spiking rate under the normal state q0 0.10 Anomaly ratio π1 0.05 Error rates (ε01, ε10) (0, 0) Upper bound on the spiking probability ∆max 0.5 Channel capacity Cmax 1 B. On the Validity of Online Anomaly Detection We first investigate the validity of the online anomaly detection algorithm introduced in Sec. III. To this end, we compare the performance of the proposed scheme with adaptive threshold αf described in (22) with a more conventional fixed-threshold strategy, in which we apply the threshold αf = α for all f ∈[1, F]. Note that this combined approach does not satisfy the theoretical statistical guarantee in (9), and its performance can only be evaluated via numerical results. Fig. 5 shows FDR and TDR versus the time frame f for fixed anomaly threshold method and for the proposed dynamic anomaly detection method in (22), where we set the number of neuro-SN K = 1. As anticipated, the fixedthreshold is not able to always satisfies the FDR constraint in (9). This is because this approach does not take into account the past rejection history for the current anomaly decision tasks. A fixed threshold can achieve a higher TDR than the proposed method, but only by violating the FDR constraint in (9). In contrast, the introduced methodology, dynamic threshold always satisfies the FDR constraint α, validating Proposition 1. Fig. 6 shows the FDR and TDR as a function of the anomaly probability π1, where we set K = 1 and F = 1000. We observe that the fixed-threshold approach suffers from the higher FDR when the anomaly probability π1 is small, while the proposed dynamic anomaly detection threshold method can always satisfy the FDR constraint. When the anomaly probability π1 is small, with a dynamic threshold, the reader takes a conservative decision by setting a relatively smaller threshold αf, will a fixed-threshold method cannot adjust to the anomaly probability. C. Random v.s. Track-and-Stop Scheduling We now study the impact of the scheduling strategy. To this end, Fig. 7 shows the FDR and the TDR versus the time frame f for the proposed anomaly detection scheme under a random scheduling strategy that selects Cmax neuro-SNs uniformly at random at every frame and for the track-and-stop method presented in Sec. IV. We set the parameters described in Table I with the uplink channel capacity Cmax = {1, 2} and anomaly probability π1 = 0.1. All schedulers, when coupled with the proposed anomaly detection, are seen to satisfy the FDR requirements, confirming the theoretical results in Proposition 1. Moreover, the figure highlights the benefits of track-and-stop, increasing the TDR for both uplink capacities Cmax = 1 and Cmax = 2. This advantage stems from the capacity of track-and-stop to select the neuro-SNs that are most likely to contribute to the identification of the neuro-SNs with highest spiking rates. From Fig. 7 we also see that TDR becomes larger as the number of scheduled nodes Cmax increases, demonstrating the importance of collecting information from multiple neuroSNs. This is because by calculating e-value (18) based on the spike rate from multiple nodes, the reader can mitigate the negative impact of small e-value caused by the selection of nodes having small spike rate in anomalous state, qk 1. D. Impact of System Parameters In this subsection, we focus on the performance of trackand-stop scheduling, evaluating the impacts of different system parameters. 1) On the FDR constraint α: Fig. 8 shows the FDR and TDR as a function of FDR constraint α. The figure highlights the trade-off between TDR and FDR: as the allocated FDR constraint α becomes larger, the reader can more easily detect anomalies, increasing the TDR. We can also see that TDR becomes larger as the forgetting factor δ becomes smaller. In fact, with a smaller δ, the FDR constraint puts higher weight on the results of the latest decisions, decreasing the importance of past false alarms. 2) On the sensitivity of the sensors: We now study the dependence of the system performance on the parameter defining the quality of the observations of sensors. First, we 100 200 300 400 500 0 0.1 0.2 Fixed αf = α Dynamic αf FDR constraint f FDR 100 200 300 400 500 0.0 0.25 0.50 0.75 1.0 Fixed αf = α Dynamic αf f TDR Fig. 5: FDR and TDR versus the time frame f for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed dynamic anomaly decision threshold based on (22), where we set K = 1 and π1 = 0.05. 0.025 0.05 0.075 0.10 0 0.1 0.2 0.3 Fixed αf = α Dynamic αf FDR constraint π1 FDR 0.025 0.05 0.075 0.10 0.0 0.25 0.50 0.75 1.0 Fixed αf = α Dynamic αf π1 TDR Fig. 6: FDR and TDR versus the anomaly probability π1 for fixed anomaly decision threshold αf = α for all f ∈[1, F] and for the proposed dynamic anomaly decision threshold based on (22), where we set K = 1 and F = 1, 000. vary the upper bound ∆max on the spiking rate in the presence of anomalies. From Fig. 9, we can see that TDR is improved as the upper bound ∆max becomes larger, that is, the sensitivity improves. This is because a larger upper bound ∆max tends to increase the difference between the spike rate of neuro-SN k in the normal state qk 0 and that in the anomalous state qk 1, enabling the reader to detect anomalies with higher accuracy (see Fig. 3). Fig. 10 shows the TDR as a function of the frame index f for different numbers of uplink slots L. From Fig. 10, we can see that TDR becomes larger as L increases because the reader can execute anomaly detection task based on more number of received spikes, realizing accurate decision based on the e-value in (18) and decision threshold defined in (22). For example, to achieve TDR larger than 0.9 under ∆max = 0.5, there is a need for L = 100 uplink slots. 3) Impact of anomaly probability π1: Fig. 11 shows TDR against f for track-and-stop, where we set the anomaly probability to π1 = {0.05, 0.10, 0.15}. As π1 increases, the probability that the reader obtains the higher e-value in the single frame also increases, making it more likely that the reader rejects the hypothesis at the current frame. As a result, we can see that TDR increases as π1 becomes larger. 4) On the channel error rate: To evaluate the impact of channel error in the binary asymmetric channel described in Sec. II-C, we vary the error probability ε, by setting: (i) ε01 = ε and ε10 = 0; (ii) ε10 = ε and ε01 = 0; (iii) ε01 = ε10 = ε. From Fig. 12, we see that the error probability ε10 has the largest impact on TDR. In fact, as increase in ε10 causes spike losses, which may be critical for anomaly detection. Conversely, as the error probability ε01 increases, it becomes more likely that the reader mistakenly detects false spikes, causing the reader to mistakenly detect an anomaly event when the environment is in a normal state. As the reader must satisfy the FDR constraint in (9), in this case the reader must take a conservative decision when the actual anomaly event happens, reducing the TDR. When both errors have a non-zero probability, it is affected by both spike losses and falsely detected spikes, causing further decreases in the TDR. VI. CONCLUSION This paper proposed a low-power online anomaly detection framework for neuromorphic wireless sensor networks. We have introduced an algorithm that can control the false discovery rate by incorporating e-values test statistics calculated based on the received spike sequences at each frame. We have also introduced a track-and-stop methods for scheduling nodes 200 400 600 800 1,000 0 0.1 FDR constraint f FDR 200 400 600 800 1,000 0.50 0.75 1.0 Random (Cmax = 1) track-and-stop (Cmax = 1) Random (Cmax = 2) track-and-stop (Cmax = 2) f TDR Fig. 7: FDR and TDR versus the time frame f for random sampling scheme and track-and-stop. 0.05 0.10 0.15 0.20 0 0.1 0.2 FDR constraint α FDR 0.05 0.10 0.15 0.20 0.50 0.75 1.0 δ : {0.8, 0, 9, 0.99} α TDR Fig. 8: FDR and TDR versus the FDR constraint α for track-and-stop (the forgetting factor δ = {0.8, 0.9, 0.99}). 100 200 300 400 500 0 0.2 0.4 0.6 0.8 1 ∆max = 0.2 ∆max = 0.3 ∆max = 0.4 ∆max = 0.5 f TDR Fig. 9: TDR versus the time frame f for track-and-stop with different values of upper bound of sensor reading ∆max. based on the channel constraint. Our numerical results show that the proposed approach can track the nodes having the largest spike efficiently with high TDR, while maintaining the FDR below the predetermined threshold. 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Astronomy & Astrophysics manuscript no. Article_v1
©ESO 2025
October 17, 2025
Search for exocomets transits in Kepler light curves
Ten new transits identified
P. Dumond1, 2, A. Lecavelier des Etangs1, F. Kiefer1, 3, G. Hébrard1, and V. Caillé1, 4
1 Institut d’astrophysique de Paris, CNRS, Sorbonne Université, 98bis boulevard Arago, 75014 Paris, FRANCE
2 CRAL, Ecole normale supérieure de Lyon, Université de Lyon, UMR, CNRS 5574, F-69364 Lyon Cedex 07, France
3 LIRA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195
Meudon, France
4 Institut Pierre-Simon Laplace, Sorbonne-Université, CNRS, Paris, France,
October 17, 2025
ABSTRACT
The Kepler mission, despite its conclusion over a decade ago, continues to offer a rich dataset for uncovering new astrophysical
objects and phenomena. In this study, we conducted a comprehensive search for exocometary transit signatures within the Kepler
light curves, using a machine learning approach based on a neural network trained on a library of theoretical exocomet transit light
curves. By analyzing the light curves of 201,820 stars, we identified candidate events through the neural network and subjected the
output to filtering and visual inspection to mitigate false positives.
Our results are presented into three catalogs of increasing ambiguity. The first-tier catalog includes 17 high-confidence exocometary
transit events, comprising 7 previously reported events and 10 newly identified ones, each associated with a different host star. The
second-tier catalog lists 30 lower-confidence events that remain consistent with possible exocometary transits. The third-tier catalog
consists of 49 more symmetric photometric events that could be either exocometary transits, exoplanet mono-transits, or false positives
due to eclipsing binaries mimicking transits.
Contrary to previous studies, which suggested that the cometary activity was favored by stellar youth, we find a broad age distribution
among candidate host stars, including several red giants. This challenges the general idea of a decline in cometary activity with stellar
age and underlines the need for further investigation into the temporal evolution of exocometary activity in planetary systems.
Key words. Methods: data analysis - Techniques: photometric - Surveys - Comets: general - Planetary systems - Exocomets - Transit
photometry
1. Introduction
Exocomets are rocky or icy minor bodies that become active
when they approach their parent star on a highly eccentric or-
bit (Strøm et al. 2020). Producing a huge tail of gas and dust
close to the periastron, when they transit in front of the star,
they become detectable using spectroscopy (for the gaseous tail)
and photometry (for the dust tail). The first exocomets were thus
identified in the 1980s in the β Pic system, through the detec-
tion of variable absorption features in the Ca ii lines of the stellar
spectrum (Ferlet et al. 1987; Kiefer et al. 2014a). Later on, a few
other exocometary systems have been found using spectroscopy,
like HD 172555 (Kiefer et al. 2014b; Grady et al. 2018), and
49 Cet (Montgomery & Welsh 2012; Miles et al. 2016). A list of
spectroscopically identified exocometary systems can be found
in Strøm et al. (2020).
For the detection of exocomets using photometry, one needs
to obtain photometric measurements at high accuracy, typically
at 10−3–10−4 level over the full transit duration, that is, for one to
three days (Lecavelier des Etangs et al. 1999, hereafter LdE99a).
This capability has been reached only during the last decade with
the space missions like Corot, Kepler, TESS, and Cheops. Detec-
tions have thus been obtained with Kepler toward KIC 3542116
and KIC 11084727 (Rappaport et al. 2018), and KIC 8027456
(Kennedy et al. 2019), with TESS toward β Pic (Zieba et al.
2019; Pavlenko et al. 2022; Lecavelier des Etangs et al. 2022),
and with Cheops toward HD 172555 (Kiefer et al. 2023). Here
it must also be mentioned the detection with Kepler of a photo-
metric event, possibly periodic, that is interpreted by the transit
of a string of 5 to 7 exocomets in front of KIC 8462852 (Kiefer
et al. 2017).
All these events have been identified thanks to the particular
shape of the light curve of an exocomet transit, as theoretically
predicted at the end of the 90’s in LdE99a : the characteristic
of an exocomet transit light curve is the asymmetry caused by
the cometary tail passing in front of the star after the nucleus.
This causes a sharp decrease of the star light followed by a slow
return to the normal brightness (see also Lecavelier des Etangs
1999, hereafter LdE99b). However, exocomet transit light curves
can also be more symmetric when the longitude of the periastron
of the orbit is about 90◦from the line of sight, because here the
cometary tail is aligned with the line of sight. In this case, the
shape of the transit light curve is similar to that of an exoplanet
transit (LdE99a).
The detection of exocometary transits based on photometry
is complementary to the detection through spectroscopy. It al-
lows exploring the physical characteristics of dust particles, such
as the dust particle size distribution, the distance to the host star
at the time of transit, and the dust production rate (Luk’yanyk
et al. 2024). Indeed, because transits give direct access to the ge-
ometrical extent and the optical thickness of the transiting dust
cloud, coupled with models of dust production, this allows the
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estimate of the dust production rate and the size of the comets
nucleus. In the case of β Pic, statistical analysis of photometric
events allowed the estimate of the nucleus size distribution that
is found to be similar to that of asteroids and comets in the Solar
system as set by collisional equilibrium (Lecavelier des Etangs
et al. 2022).
Although the characterization of exocomets paves the way
to understanding the dynamical and chemical processes occur-
ring in young planetary systems, only a very limited number
of exocometary systems have been identified to date. As a re-
sult, our current understanding of the diversity, frequency, and
evolutionary significance of exocometary activity remains in-
complete. Systematic searches have been performed using spec-
troscopy in the Ca ii line (Montgomery & Welsh 2012; Welsh &
Montgomery 2013, 2018; Rebollido et al. 2020; Bendahan-West
et al. 2025), resulting in a few detections (see review in Strøm
et al. 2020). Spectroscopy is efficient in detecting tiny amount
of gas and thus allows sensitive search ; the result is that de-
tectable spectroscopic transits seem to be more frequent than de-
tectable photometric events: for instance in the case of β Pic, sev-
eral exocomet transits are routinely detected every day in spec-
troscopy (Kiefer et al. 2014a), while only 30 photometric transits
have been detected in 156 days of TESS observations (Lecave-
lier des Etangs et al. 2022). However, spectroscopic searches for
exocomets are inherently limited by their need to observe each
target star individually and the need for long observation time
with high spectral resolution. In contrast, photometric surveys
can monitor several thousands of stars simultaneously, making
them significantly more efficient for large-scale searches for ex-
ocometary transits.
In that spirit, Rappaport et al. (2018) and Kennedy et al.
(2019) have undertaken a deep search for exocometary transits
in the Kepler light curves. To identify exocometary transits, they
used a modified model of planetary transit or a combination of
exponential and Gaussian tails, as also done later on by Zieba
et al. (2019), Pavlenko et al. (2022), and Lecavelier des Etangs
et al. (2022). Although this empirical approach does not carry
physical information about the cometary bodies and tails, these
are simple and easy to implement to identify candidates in the
large number (∼200 000) of light curves provided by Kepler.
Here we propose to use an automated search without the use
of analytical functions to describe the exocometary transit light
curves. For that we combined a search algorithm based on a neu-
ral network together with a library of photometric transits to train
the network. The library is an updated version of the library of
LdE99b, which has been recalculated using faster CPUs than the
ones used 20 years ago, covering a wide range of parameters and
therefore a wide range of transit shapes that need to be identified.
We apply this automated search to the Kepler data (Borucki et al.
2010) with the aim of producing a new catalog of exocometary
systems candidates. The methods are described in Sect. 2, 3 and
4, the search in Kepler data is presented in Sect. 5 and the final
catalog of candidates, divided in three tiers, is given in Sect.6.
The results are discussed in Sect.7.
2. Training of the neural network
2.1. Training principle
The method used for the detection of exocomets in Kepler data
is based on a neural network using supervised learning. For the
network learning, we used a library of theoretical light curves di-
vided into three disjoint sets of labeled light curves: the training
set, the validation set, and the test set. Each set contains the same
number of light curves with and without an exocomet transit (see
Sect. 2.2). Each light curve is labeled to indicate if it includes a
transit or not.
Our objective is to make the algorithm identify the curves
containing an exocomet transit among all the light curves col-
lected by the Kepler satellite and to recover the transit time when
a transit is present. Using an iterative procedure, the algorithm
optimizes its parameters thanks to the training set. At the end of
each iteration step called an "epoch", the network performance
is tested using the validation set. This allows us to control the
overfitting of the algorithm (e.g. it becomes specific to the train-
ing set and loses in generality) and to adjust the hyperparameters.
The learning phase is stopped when the network shows the best
performance in the validation set. The final performance of the
algorithm is given by its evaluation on the test set.
2.2. Light Curve production
The light curves used to build the training, validation, and test
set are the Presearch Data Conditioning (PDC) data from Kepler
Data Release 25 (Thompson et al. 2016). We used the Quarters 1
to 17 and divided them into sub-quarters in case of interruption
of the acquisition within a given quarter: two sub-quarters are
separated by at least four consecutive photometric measurements
missing in the light curve.
For each sub-quarter, a training, validation, and test set were
constructed to overcome their possible specificity. For each sub-
quarter, the training set consists of 36 000 curves, while the val-
idation and test sets contain 12 000 each. A curve consists on 10
days (480 photometric measurements at a cadence of one mea-
surement every ∼30 minutes) of acquisition of the luminosity
flux from a star randomly chosen without repetition.
To ensure efficient learning and prevent the algorithm from
falling into obvious traps, several modifications have been made
to the Kepler PDC data. First, all points outside the mean value
of the light curve by more than 5-sigmas while the neighboring
points are not are considered to be outliers and removed. In ad-
dition, each curve was normalized to have a zero mean and a
standard deviation of 1, as needed for optimal learning by the al-
gorithm. Finally, the possible missing measurements, i.e. at most
three consecutive missing measurements within the same sub-
quarter, were filled using a linear interpolation.
For the learning process, half of the 10-day light curves have
been used with a label "0", indicating the absence of an exocomet
transit: we assume that the probability that any randomly chosen
light curve in the Kepler data includes an exocomet transit is ex-
tremely low. This set of 0-labeled light curves without exocomet
transit is used to learn the noise patterns in the Kepler data. For
the other half of the 10-day light curves, we add one exocomet
transit by multiplying by a light curve from the theoretical tran-
sit library (Sect. 2.3) and we label them with "1", indicating the
presence of an exocomet transit.
2.3. Simulation of exocomet transits
In order for the algorithm to learn to identify the shape of an ex-
ocomet transit in the light curves, we add theoretical exocomet
transits to half of the curves that make up the training data. These
theoretical exocomet transit light curves have been calculated us-
ing the method describes in LdE99a. We produced a library of
2200 exocomet transit light curves, which is an updated version
of the library published twenty years ago by LdE99b. Among all
available simulations, we selected only those with a gas produc-
Article number, page 2 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
tion rate log10(P/(1kg · s−1)) between 6 and 7.5 and an impact
parameter b < R∗, where R∗is the radius of the star. This choice
guarantees that the used transit light curves are sufficiently deep.
Indeed, an impact parameter greater than one stellar radius or a
too low production rate leads to a noisy and very weak extinc-
tion and would therefore have no interest for the training. Before
adding the simulated transits to the Kepler light curves, we mul-
tiplied the theoretical transit light curve by a random constant
such that the depth of the transits is in the interval [2σ, K + 2σ]
where K is the depth of the simulated transits and σ the standard
deviation of the Kepler light curves. We thus train our model for
a signal-to-noise ratio higher than 2 to exclude the cases where
the transit would be non detectable. The transit times of the the-
oretical transits added to the 10-day light curves are chosen ran-
domly, with a uniform distribution between the beginning and
the duration of the simulated transit before the end of the light
curve.
Finally, in order to ensure that the training data are repre-
sentative of the diversity of possible transits, we use a uniform
random distribution of the distance of the periastron q in the set
{0.2; 0.3,; 0.5; 0.7; 1.0} astronomical unit and of the longitude
of the periastron ω from -157.5◦to +157.5◦with a step of 22.5◦,
as in LdE99b.
The theoretical transits that have been used are totally dif-
ferent for the training, validation and the test sets. To guarantee
that the evaluation of the algorithm’s performance is unbiased,
the choice of the transits for each set has been made randomly
from the whole set of theoretical transits.
3. The neural network model
3.1. Architecture
The deep learning algorithm developed in this study is mainly
based on the convolution layer sequence adapted for pattern
recognition. We relied on the architecture of existing algorithms
for the detection of exoplanet transits (Shallue & Vanderburg
2018; Dattilo et al. 2019; Zucker & Giryes 2018; Chintarun-
gruangchai & Jiang 2019). These algorithms, which all have a
similar architecture, are optimized for periodic transit detection.
They have been modified to be efficient for the detection of sin-
gle transits, as the orbital period of an exocomet is expected to be
much longer than the Kepler mission duration. Two major mod-
ifications have been made. "Squeeze-excited blocks" (Hu et al.
2018) have been added to the convolution blocks. The authors
indeed show that such blocks can improve the performance of
a pattern matching algorithm by improving the interdependence
relations between the convolution layers without adding com-
putational cost. The addition of the LSTM (Long Short-Term
Memory) (Hochreiter & Schmidhuber 1997) and GRU (Gated
Recurrent Unit) (Cho et al. 2014) layers also resulted in sig-
nificant performance improvements. The algorithm was imple-
mented in TensorFlow, an open-source machine learning frame-
work (Abadi et al. 2016). We optimized the neural network pa-
rameters thanks to KerasTuner (O’Malley et al. 2019).
Figure 1 describes our final model, which is mainly a one-
dimensional convolutional network. The activation function of
all the convolutional layers is the ReLu (Rectified Linear unit)
function (Jarrett et al. 2009). The output is a list of two reals.
The first one uses a sigmoid function whose range is (0,1). It is
the probability returned by the algorithm that the 10 days input
light curve contains an exocomet transit. An output value close
to 1 indicates high confidence that the considered light curve has
one transit, while an output value close to 0 means that there is
Fig. 1: Architecture of our best performing neural network
model. The input data is a light curve of 480 timesteps, the
output is a list of two reals between 0 and 1: one for the
probability of detection of an exocometary transit and one for
the position of the transit in the light curve.
Convolutional
layers
are
noted
Conv <kernel size>
<number of feature maps>, max pooling layers are noted
MaxPool <window length> <stride length>, fully con-
nected layers are noted Dense <number of units>, LSTM
layers are noted LSTM <units> <dropout>, GRU layers are
noted GRU <unit> <droupout> <recurent dropout> and
the squeezed-excitation blocks are noted SEB.
none. The second output layer is activated by a linear function
in range (0,1). It gives the position of the identified transit in the
light curve. If no transit is detected, the layer should return a
value close to zero.
3.2. Training
For each sub-quarter, we trained the model using 36 000 light
curves for 150 training epochs, that means that we performed
150 complete pass through the entire training dataset to update
the model weight parameters. We used the Adam optimization
algorithm (Kingma & Ba 2015) to minimize the cross-entropy
error function for the classification of the light curves and the
mean absolute error function for the position of the transit. We
used a learning rate of 10−4 and a batch size of 5 000. The saved
model in the one for which the validation loss reaches its mini-
mum ; usually it is obtained after between 50 and 80 iterations
("epochs"). Therefore, training the model over 150 epochs is
considered sufficient.
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4. Evaluation of the neural network performance
We evaluated the performance of our neural network with re-
spect to several metrics in the same way as Dattilo et al. (2019)
did for their model. We computed all metrics over the test set
(rather than the training or validation sets) to avoid using any
data that were used to optimize the parameters or hyperparame-
ters of the model.
4.1. The metrics
The metrics used to evaluate the performance of a classifier are
the following:
1. Accuracy: the fraction of curves correctly classified by the
model.
2. Precision (reliability): the fraction of correctly classified ex-
ocomets, i.e., the fraction of curves classified as exocomet
candidates (False positives + True positives NFp + NTp) that
are indeed true positives (labeled 1):
P =
NTp
NFp + NTp
(1)
3. Recall (completeness or True-positive rate): the fraction of
the curves labeled 1, which therefore includes an exocomet
transit (True positives + False negatives NTp + NFn), that the
model classifies as exocomet candidates (NTp):
R =
NTp
NTp + NFn
(2)
4. False-positive rate: the fraction of total labeled 0 curves,
which therefore do not include any exocomet transit (False
Positives +True Negatives NFp + NTn) that the model classi-
fies as exocomet candidates.
FPR =
NFp
NFp + NTn
(3)
5. AUC (area under the receiver-operator characteristic curve;
see Figure 2): the probability that the model, if it receives
two light curves, one 1-labeled curve including an exocomet
transit and one 0-labeled curve without transit, would rank
the 1-labeled curve higher than the 0-labeled curve.
6. Mean Absolute Error: the absolute error for all labeled 1
curves between the true position of the transit and the pre-
dicted one by the model.
The first five metrics are related to the detection of the pres-
ence of an exocomet in the light curves, while the last metric is
related to the position in time of the transit in the light curve.
Except for the AUC, the values of these metrics depend on the
classification threshold chosen for the model. This threshold is
the value of the probability of detection yield by the algorithm
above which we consider that there is a detection. In Fig. 2, we
show the evolution of the recall as a function of the FPR for vari-
ous thresholds, usually called the receiver-operator characteristic
(ROC) curve. It can be seen that the algorithm differentiates well
between true transits and noise, as it achieves very low FPR re-
call while keeping a reasonably large recall: our model reaches
an AUC of 98.88% for the quarter Q1. We also plot the preci-
sion as a function of the recall. This curve shows the trade-off
between having no false positives (high precision) and identify-
ing all exocomet transits (high recall).
Fig. 2: The receiver-operator characteristic (ROC) curve (left
panel) and the precision versus recall curve (right panel). The
receiver-operator characteristic (ROC) curve shows the recall
(true-positive rate) of the model against the ability to recognize
false positives (the false-positive rate) for different classification
thresholds. Our model is highly successful at identifying false
positives as shown by the high AUC value (see Table A.1). The
plot of the fraction of exocomets that the model classified as exo-
comets (recall) versus the fraction of correctly classified planets
(precision) shows the trade-off between having no false positives
(high precision) and identifying all exocomet transits (high re-
call).
Considering the huge number of light curves in the Kepler
data, the goal is to drastically reduce the number of false posi-
tive. We thus chose a classification threshold of 0.99. With this
threshold, our model reaches an accuracy of 89.5% for the quar-
ter Q1. The precision of the model is 99.8%, which means that
we reach a high true positive rate, and the recall is 79,1%, which
means that about 20% of the true exocomet transits are miss-
ing in the final classification. This trade-off aims to accept some
loss in the finding of exocomet transits of about 20% to avoid a
large number of false positives. Indeed, given a large number of
light curves in the Kepler data and a small number of exocomet
transits in these data, even a tiny fraction of false positives can
significantly pollute the results. The metrics values are found to
be similar for all quarters (see Table A.1 in Appendix A).
4.2. Quality of the output
The quality of the model classification can also be visualized by
the histogram of the results on the test set (top panel of Fig. 3).
The closer the output from the neural network is to 1 (resp. 0),
the higher the probability that the 10-day light curve (resp. does
not) contains an exocomet transit. The histogram shows that the
neural network works as expected: on the left-hand side, most
of the light curves characterized by a low probability are mostly
noise (orange histogram), while on the right-hand side, the large
majority of the curves characterized by a probability close to 1
included a simulated exocomet transit (label 1; blue histogram)
and with very few false positives. The bottom graph of Fig. 3
shows the mean error of the position of the transit as a function
of the probability assigned by the algorithm when it is applied
to the curves containing a simulated transit (label 1, with known
position). One can see that when the transit is not found (prob-
ability close to 0), the error on the predicted position is large,
about 3 days, whereas it is of the order of a few hours only when
the probability is above 0.99. Thus, one can be confident in the
time of the transit given by the algorithm for curves with a prob-
ability higher than this threshold.
Article number, page 4 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
Fig. 3: Histogram of the results on the test set and error on the
transit position found by the network.
Top panel: Histogram of the result of the neural network applied
to the test data. Most of the 1-labeled light curves with an exo-
comet transit yield a probability close to 1, while there are only a
few false positives (0-labeled light curves yielding a high proba-
bility). Above the chosen threshold, there is more than two order
of magnitude between the number of true positives and false pos-
itive.
Bottom panel: the mean error of the position of the transit as a
function of the probability of the presence of a transit assigned
by the network to the light curves. This error is of a few hours at
most for the transits identified by the algorithm.
5. Search for exocomets in Kepler data
5.1. Application of the model to Kepler data
To apply the algorithm presented in the previous sections, all the
light curves of the Kepler data were cut into 10-day windows
obtained in the same way as the samples used to construct the
training, validation, and test sets described in Sec. 2.2. In order
not to lose any transits located at the edge of the windows, an
overlap of 2 days was applied between two consecutive 10-day
time intervals.
However, despite the level of precision achieved, the algo-
rithm provides a large number of false positives due to the large
number of 10-day windows: approximately 106 windows per
quarter lead to about 103 positive candidate detections, among
which less than a dozen real transits should be identified (see
LdE99a and discussion in Sect. 7). To reduce the number of false
positives, we decided to filter the candidates using constraints on
the shape of the light curves of the detected photometric events.
5.2. Additional criteria to filter false positives
After all the Kepler light curves have passed through the neural
network, we end up with a set of 56525 exocomet candidates.
To filter these candidates, we decided to add constraints on the
shape and on the context on the transit (noise, quality flags).
5.2.1. Filtering through the shape of the transits candidates
The shape of the candidate transits were determined by fitting
them by the model proposed by Lecavelier des Etangs et al.
(2022) together with a second order polynomial baseline:
fexo(t) =
Ct2 + Dt + E
t < t0,
(Ct2 + Dt + E)(1 + K(e−β(t−t0) −1))
t1 > t > t0
(Ct2 + Dt + E)(1 + K(e−β(t−t0) −e−β(t−t1)))
t > t1
(4)
Fig. 4: Distribution of the value of the parameters obtained by
fitting the 2163 light curves in the library of the simulated ex-
ocomet transits. The parameters K, ∆T = t1 −t0, and 1/β cor-
respond to the depth, ingress duration, and crossing time of the
transit, respectively.
The parameters t0 and t1 represent respectively the time at which
the transit starts and the time at which it reaches its minimum.
K represents the characteristic depth of the transit, while 1/β
corresponds to the characteristic crossing time of the transit. C, D
and E are the coefficients of the polynomial baseline. The fit
of the light curve by the transit model can be easily performed
using the position of the detected candidate in the light curve as
provided by the network output.
To accept or reject a proposed detection given by the neural
network, we fit the light curve of the corresponding photometric
event to obtain the values of the parameters K, β, and ∆T = t1−t0
that characterize the shape of the event. We then check that these
values are in the domain of validity for an exocometary transit.
The domain of validity is obtained by considering the parame-
ters space occupied by the fits of the theoretical light curves in
the library of numerical simulations of exocometary transits, as
shown in Fig. 4 and summarized in the first three lines of Table 1.
It is found that in the theoretical light curves K is mostly
between 10−4 and 5×10−3, ∆T is between 1.7 and 17 hours (cor-
responding to a periastron between ∼0.02 to 2 au, Lecavelier des
Etangs et al. 2022), and β−1 is in the domain above 0.5 hours, and
between 0.3∆T −1.7 and 0.5∆T + 12. Moreover, the parameters
C and D have to verify the condition |C|, |D| < 10−3. This ensures
that the background is stable enough in the neighborhoods of the
transit. When this is not the case, the detected transit appears to
be only noise.
5.2.2. Quality flags
We noticed that the presence of certain quality flags indicates an
alteration of the light curve leading to a systematic detection by
the network. Thus, candidates whose transit position is less than
1 day away from a quality flag 3 ("Spacecraft is in coarse point")
or 12 ("Impulsive outlier removed before cotrending") have been
discarded from the candidate list.
5.2.3. Presence in the list of KOIs
To avoid periodic transits, which constitute roughly half of the
candidate transits output by the algorithm, stars belonging to the
list of KOIs (Kepler Object of Interest) have also been removed.
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Table 1: Parameters values for filtering the candidate transits
Parameter
Lower bound
Upper bound
Definition
K
10−4
5 × 10−3
Transit depth
β−1
(0.5h
0.3∆T −1.7
0.5∆T + 12
Characteristic crossing time
∆T
1.7h
17h
Ingress duration
|C|, |D|
10−3
Baseline coefficients
∆RMS
36
∆χ2
pl
5
∆χ2
rp
10
5.2.4. Other cases of false positives
We have found three other circumstances that can mislead the
algorithm and then result in obvious false positives: (1) a signifi-
cant background noise on a time scale close to that of a cometary
transit; (2) a ramp-like shape in the light curve, i.e. a very sud-
den decrease in flux followed by a more or less rapid return
to the normal; (3) the presence of a mono-transit of a possible
exoplanet that results in a very symmetrical brightness dip. In
the following, we suggest criteria to address each of these three
cases.
To characterize the presence of noise anomaly, the ∆RMS was
calculated. It is defined as:
∆RMS =
X
i
( fexo(ti) −¯F3)/σ
(5)
where fexo(ti) is the exocomet model fit to the flux where the
candidate transit is found, ¯F3 is the mean of the flux without
the three lowest points and σ is the standard deviation of the
flux. Only candidate transits with ∆RMS > 36 were conserved.
This criterion was very useful to remove candidates that were
detected due to a few points significantly lower than the average.
To detect the presence of red-noise that mimics an exocomet
transit, we calculated the correlation product between the exo-
comet model fit to the candidate transit and the light curve over
all the considered sub-quarters. If the maximum of the correla-
tion max1 is reached more than one day away from the transit
time or if its second maximum max2 is significantly close to the
first maximum ((max1−max2)/max1 < 0.15), the candidate is re-
moved, considering that in this case the noise in the light curve
resembles the identified transit, which is therefore probably not
a real exocomet transit, but a background red-noise.
The cases of the ramp-like light curves mimicking an exo-
cometary transit have been discarded by calculating the differ-
ence between the fit with a comet model (Eq. 4) and a fit with a
simple ramp model defined by:
framp(t) =
(Ct2 + Dt + E
t < t0,
(Ct2 + Dt + E)(1 −K e−β(t−t0))
t > t0
(6)
We calculate ∆χ2
rp defined by :
∆χ2
rp = N
P
i( framp(ti) −F(ti))2
P
i( fexo(ti) −F(ti))2 −1
!
,
(7)
where N = 120 is the number of points in the fitted window.
The photometric events that are almost as well fitted with the
ramp model as with the exocomet model (∆χ2
rp < 10), have been
discarded from the candidate list.
Finally, to identify the cases of a symmetrical transit that
could be due to the passage of an exoplanet with a long period
(a monotransit), we calculate the difference ∆χ2
pl between the χ2
pl
of a model of a planetary transit calculated using the equations
of Mandel & Agol (2002) and the χ2
exo of an exocomet transit
model (Eq. 4). It can be written as:
∆χ2
pl = N
P
i(fpl(ti) −F(ti))2
P
i(fexo(ti) −F(ti))2 −1
!
.
(8)
The photometric events having ∆χ2
pl < 5 have a symmetrical
shape and have therefore not been taken into account in our final
selection of exocomets. Such transits have been taken into ac-
count to establish the list of symmetrical mono-transits (see Sec.
6.3). This criterion is, however, not used as a sharp cut to distin-
guish the exocomets transits from the symmetrical mono-transits
but only as a help. The final distinction has been made through
visual inspection.
After applying these criteria, the number of candidate tran-
sits per quarter varies between 50 and 100 per quarter, leading
to a total of 1349 transits. It then becomes possible to proceed
for each of them to a visual inspection to know if the candi-
date is indeed or not an exocomet transit. This final inspection is
necessary because, although the criteria put in place make it pos-
sible to eliminate the vast majority of false positives, there still
remain obvious false positives. Note here that the criteria have
been deliberately chosen to be conservative so as not to acciden-
tally eliminate a true exocomet transit. The number of criteria
was chosen so that the final number of transits to be visually
inspected would be no more than a few thousand. Otherwise,
visual inspection would be impractical. It would be more ineffi-
cient and inaccurate to add extra criteria than to visually check
the current list of 1,349 transits.
To verify the robustness of the detected transits, we perform
an additional test. We apply our detection procedure to the data
after reversing the time. If the proportion of detected exocomets
is the same in the reversed time data as in the original data, this
may question the relevance of the detected transits. However,
as this analysis is time-consuming, we performed this analysis
on half of the quarters. Of the 17 quarters, we analysed eight
randomly chosen quarters: Q1, Q2, Q6, Q9, Q10, Q12, Q15 and
Q16. We found only one transit that clearly mimics an exocomet
transit: KIC_3129239 in Q9. Given that approximately 17 robust
transits have been detected in the Kepler data (see Sec. 6), we
can infer from this study that about 10% to 15% of these transits
can be false positives. Although the same number of detections
in the time-reversed data would have disproven the exocomet
hypothesis, the result of this test supports the idea that there is an
astrophysical signal in the data that is consistent with exocomet
transits.
Article number, page 6 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
6. A new catalog of transiting exocomets
In this section, we present the final list of exocomet transits re-
sulting from the selection procedure. We divided the list in three
different tiers based on the likelihood of a genuine exocometary
detection. In the first tier, the photometric events are the ones that
we consider as the most likely due to the transit of an exocomet ;
in the second tier are the events possibly due to comets, while
the third tier gathers the most symmetric dip in the light curve,
for which the transit of an exoplanet remains a possibility.
6.1. The first-tier catalog of exocometary transits
The first-tier catalog of detected exocometary transits is given in
Table 2 and the plots of the corresponding light curves are given
in Appendix B. It is particularly noteworthy that our neural net-
work is able to find all the transits already identified by previous
works: KIC 3542116 in Quarters 1, 10 and 12 and KIC 11084727
as found by Rappaport et al. (2018), and KIC 8027456 as found
by Kennedy et al. (2019). This independent retrieval of the same
exocometary transits provides confidence in our procedure.
Nonetheless, after application of all the search procedure de-
scribed above, it appeared that two transits of KIC 3542116 iden-
tified by Rappaport et al. (2018) in Quarters 8 were not in our
list of detections. These transits were eliminated by the criterion
based on the maximum of the correlation product that aims to
remove the detections whose pattern occurs several times in the
same subquarter (Sect. 5.2.4). In fact, this criterion also has the
disadvantage of eliminating active cometary systems where at
least two comets transits occurred in the same quarter, as is the
case for KIC 3542166 in Quarter 8. Therefore, we made a vi-
sual inspection of all photometric events eliminated because of
this criterion in order to recover possible obvious exocometary
transits that were wrongly rejected. We thus found the two tran-
sits of KIC 3542116 in the Quarter 8, and in addition identify a
new transit in front of KIC 6263848. This last case does not cor-
respond to a star with frequent cometary transits, but had been
deleted due to an edge effect on the calculation of the correlation
function. These two cases are identified by an asterisk in Table 2.
For all of these detection, we calculate the ∆χ2
polynomial de-
fined as
∆χ2
polynomial = N
P
i( fpolynomial(ti) −F(ti))2
P
i(fexo(ti) −F(ti))2
−1
!
.
(9)
where fpolynomial is the second order polynomial that best fits
the detection. For all detections, we obtained large ∆χ2
polynomial
(larger than about 100). This highlights that the detection cannot
be interpreted just as noise patterns.
The properties of the parent stars of the identified exocomets
as tabulated by Kepler are given in Table 3 (Brown et al. 2011).
For KIC 3542116 and KIC 3662483, Teff, log10 g, metallicity
and radius are from Zhang et al. (2025). From the Teff and
log10 g we can obtain an estimate of the stellar type (Gray 2008),
which is given in the last column of Table 3. It appears that
most of the stars are main-sequence stars, with the exception
of KIC 4078638, which is likely a red giant. For these stars, the
upper limits on the age estimate given by Zhang et al. (2025)
are all above 109 years, with the exception of KIC 4481029 with
an upper limit of 9 × 108 years, and KIC 8027456 with an age
estimate of 5 × 108 years.
6.2. The second-tier catalog of possible exocometary
After visual inspection of the list of candidates produced by
our algorithm, we came up with a list of interesting photomet-
ric events that can be due to transiting exocomets. This includes
transits that pass through all the criteria described in Sect. 5, but
whose shape or signal-to-noise ratio makes it difficult to decide
whether they are actually transits. The resulting list of possible
exocometary transits in this second-tier catalog is given in Ta-
ble 4 and the plots of the corresponding light curves are given in
Appendix C.
The properties of the corresponding parent stars in the sec-
ond tier catalog are given in Table 5 (Brown et al. 2011).
Here again, estimates of the stellar types given in the last col-
umn of the table show that most of the stars are on or close
to the main sequence, with the exception of KIC 2984102 and
KIC 11153134, which have a log10 g below 3 and are therefore
likely red giants. For all stars in this second-tier catalog, the up-
per limit of the age estimate given by Zhang et al. (2025) is above
109 years, with the exception of KIC 2984102, KIC 5294231 and
KIC 7183123 with age estimates of 5×108, 1×108 (with an upper
limit of 3 × 108) and 9 × 108 years, respectively.
6.3. The third-tier catalog of symmetric transits
After visual inspection of the list of candidates produced by our
algorithm, we also came up with a list of interesting photomet-
ric events that appears to be symmetrical. This is not surpris-
ing as some of the exocomets transits in the library used for
training the network are indeed symmetrical (see Fig. 2 and 3
in LdE99a). However, we cannot exclude that these are due to
mono-transits of exoplanets or to false positives mimicking plan-
etary or cometary transits. Several scenarios might cause such
false positives, including diluted or undiluted eclipsing binaries.
To identify these cases one needs to carry out additional anal-
yses that are beyond the scope of the present paper (see, e.g.,
Crossfield et al. 2016). Finally, although an exocometary tran-
sit classification cannot be made, these detections deserve to be
mentioned. They are given in the third-tier catalog of symmetric
transits listed in Table 6.
We compared these detected photometric events with the
lists of single transits published by Huang et al. (2013), Wang
et al. (2015), Foreman-Mackey et al. (2016), and Herman et al.
(2019). It appears that four events in our list were already identi-
fied by Foreman-Mackey et al. (2016) and Herman et al. (2019).
The transit in front of KIC 8410697 was found in both studies.
It is interpreted as due to an exoplanet with a radius of 0.7 times
that of Jupiter. The photometric event of KIC 10321319 was
found by Foreman-Mackey et al. (2016) and interpreted as due
to the transit of an exoplanet of 0.16 Jupiter radius. The photo-
metric event of KIC 6196417 was found by Herman et al. (2019)
and interpreted as due to the transit of an exoplanet of about 0.7
Jupiter radius. The transit in front of KIC 10668646 at the time of
BKJD=1449.3 was already identified by Foreman-Mackey et al.
(2016) but the exoplanet transit scenario was rejected because of
a centroid shift in the data. However, we also identified another
photometric event in the same target at BKJD=196.3 with a dif-
ferent shape and a lower absorption depth. These few examples
show that our list of newly identified photometric events which
could be due to transits of exocomet with symmetrical shape of
the light curve deserves further investigations which are beyond
the scope of the present paper.
Article number, page 7 of 18
A&A proofs: manuscript no. Article_v1
Table 2: First-tier catalog of exocometary transits.
Stars (KIC)
Time (BKJD)
Quarter
K(×10−4)
β−1(h)
∆T(h)
AD(×10−4)
∆χ2
pl
∆χ2
polynomial
†3542116
161.5
1
4.74
6.5
9.92
3.71
18.32
279.74
†*3542116
742.6
8
4.17
1.36
6.05
4.12
0.6
348.83
†*3542116
792.9
8
8.62
4.16
4.77
5.88
15.57
285.27
†3542116
992.1
10
10.12
3.62
14.68
9.94
28.67
2993.05
†3542116
1175.8
12
12.98
3.40
9.35
12.15
8.58
2670.38
3662483
1132.1
12
6.81
6.38
6.87
4.49
12.98
204.52
4078638
355.5
4
83.21
6.30
3.22
33.30
47.89
358.90
4481029
172.5
2
9.08
8.57
4.64
3.80
14.31
235.74
5206257
925.0
10
10.98
4.75
4.47
6.69
5.04
322.26
5514200
670.1
7
34.48
12.13
7.27
15.53
19.15
155.20
*6263848
775.9
8
10.22
3.29
4.90
7.92
7.96
151.93
6927963
416.8
4
33.60
9.13
2.83
8.96
6.54
115.89
7660548
966.2
10
15.74
5.95
5.09
9.05
26.10
219.47
†8027456
1449.0
15
8.0
9.90
14.12
6.08
16.78
401.18
8738545
238.1
2
17.0
4.92
3.67
8.94
7.64
92.53
10484683
309.5
3
29.85
14.05
8.01
12.97
87.66
853.82
†11084727
1076.2
11
15.29
3.92
8.20
13.40
88.14
2368.80
Notes. The first column gives the name of the star by its KIC identifier. The second and third columns indicate the position of the transit. Columns
4 to 6 give the fits parameters associated with the exocomet model (Eq. 4). The seventh column gives the absorption depth. The last two columns
present the difference of χ2 between the cometary model and, first, the planetary model (Mandel & Agol 2002) and, second, a second order
polynomial fit.
The transits that were first eliminated by the criterion based on the maximum of the correlation product (see Sect. 5.2.4) are identified by an
asterisk (*). The five transits already known are identified by the † symbol.
Table 3: Information on the stars of the first-tier catalog.
Kepler ID
RA
Dec
Kepler Mag
Teff
Log g
Metallicity
Radius
Stellar
(J2000)
(J2000)
(K)
(cm/s2)
(R⊙)
type
3542116
19 22 52.939
+38 41 41.51
9.979
6766
4.208
-0.190
1.454
F2 V
3662483
19 42 09.199
+38 43 01.38
10.541
5848
3.945
0.107
1.786
G1 IV
4078638
19 46 16.611
+39 06 15.23
13.156
4783
2.758
0.150
8.331
K3 III
4481029
19 43 13.210
+39 32 32.03
11.303
8865
3.968
0.010
2.310
A0 IV
5206257
19 46 33.826
+40 22 57.97
12.051
6017
4.435
-0.105
1.052
F6 V
5514200
18 58 08.306
+40 46 29.32
14.377
5651
4.500
-0.243
0.955
G4 V
6263848
18 52 30.566
+41 36 33.34
13.172
5850
4.365
-0.224
1.137
G2 V
6927963
18 56 15.946
+42 27 52.20
13.807
5711
4.441
-0.012
1.030
G4 V
7660548
18 47 22.807
+43 23 05.39
13.094
6158
4.369
-0.050
1.148
F5 V
8027456
19 25 15.838
+43 51 33.55
9.697
8732
3.766
0.065
2.974
A0 IV
8738545
18 58 37.262
+44 55 15.24
13.227
5091
3.581
-0.162
3.101
K1 IV
10484683
19 47 54.377
+47 40 14.16
12.588
6326
4.543
-0.009
0.934
F5 V
11084727
19 28 41.191
+48 41 15.14
9.987
6762
4.067
-0.152
1.726
F3 V
7. Conclusion
Although the Kepler mission ended more than 10 years ago, the
available data is still worth deep data mining in search of new
discoveries. Here we present a new search for exocometary tran-
sits in the Kepler light curves. We used machine learning tech-
nique with a neural network that has been trained using a li-
brary of theoretical exocomets transit light curves inherited from
the work of LdE99b. After parsing the light curves of close to
200 000 stars through the neural network, despite the application
of several filters to eliminate most of the false positive, a visual
inspection of the outcome of the network was still needed. We
ended up with three catalogs of interesting objects. The first-tier
catalog is composed of a total of 17 exocometary transits, includ-
ing 7 previously identified transits and 10 new transits in front of
10 different stars. The second-tier catalog is a list of 30 photo-
metric events that appear of second quality and that we qualify
as possible exocometary transits. Finally, the third-tier catalog
presents the list of interesting photometric events that are sym-
metrical and may be due to transits of either an exoplanet or an
exocomet with a periastron at 90◦from the line of sight.
The complete Kepler data represent a 4-year photometry sur-
vey of more than 170 000 distinct stars. However, as noted in
Kennedy et al. (2019), not all stars were observed for the full
duration of the mission. As a result, we can consider 150 000 as
the approximate number of equivalent stars that were observed
for the full duration of the mission. Or, in other words, our anal-
ysis covers the equivalent of about 600 000 stars-years. LdE99a
showed that with 30 000 stars-years survey of stars with a solar
system cometary activity and a few 10−4 photometric accuracy,
the number of exocomets transit detections should be around 10.
Therefore, assuming that each star harbors a cometary system
similar to our own, we could expect roughly 200 detections with
Article number, page 8 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
Table 4: Second-tier catalog of possible exocometary transits. Same legend as in Table 2.
Stars (KIC)
Time (BKJD)
Quarter
K(×10−4)
β−1(h)
∆T(h)
AD(×10−4)
∆χ2
pl
∆χ2
polynomial
2984102
345.7
3
36.42
15.9
10.12
17.14
61.90
430.26
4826941
385.8
4
21.58
15.22
7.85
8.69
3.76
126.17
5195476
1508.8
16
16.80
5.09
3.67
8.31
3.42
45.92
5294231
411.4
4
4.92
3.26
2.13
2.36
31.78
285.46
5511746
1105.8
12
13.31
12.33
5.06
4.48
2.41
44.54
5514277
670.4
7
63.11
13.57
9.88
32.64
13.96
160.95
5534083
1167.7
12
9.34
10.45
3.33
2.55
7.22
43.16
6119914
747.2
8
31.41
9.36
6.63
15.95
3.54
50.65
7183123
336.4
3
8.43
8.36
5.98
4.31
28.32
321.42
7201827
409.9
4
43.64
4.83
2.83
19.34
5.77
48.98
7220968
1325.7
14
18.60
5.50
6.23
12.61
2.61
46.42
7611858
1275.8
14
6.46
9.77
5.19
2.67
7.63
109.47
7669613
315.9
3
13.24
13.66
6.31
4.89
12.77
121.72
8111649
934.0
10
83.04
23.69
10.28
29.24
16.36
123.61
8144412
639.1
7
86.39
12.00
2.95
18.84
7.06
140.49
8374877
769.1
8
7.78
7.04
8.46
5.44
13.67
300.35
8489495
1351.8
14
16.43
8.44
15.04
13.67
10.70
47.92
8625941
1505.0
16
5.16
8.44
9.80
3.54
8.10
66.25
8683719
1340.2
14
19.21
6.43
14.25
17.12
7.59
195.79
8738569
1462.3
15
13.18
8.30
4.30
5.33
8.84
56.11
8843356
702.2
7
8.84
10.54
2.99
2.18
4.66
46.51
9025688
1078.2
11
13.29
7.51
3.57
5.03
58.03
58.03
9388975
680.3
7
18.41
3.36
2.47
9.59
6.76
60.28
9531080
748.0
8
10.23
9.31
6.56
5.18
16.13
53.73
10329957
1385.1
15
18.47
12.90
4.10
5.03
9.62
64.04
11124157
704.2
7
15.01
10.93
2.00
2.51
11.48
45.20
11153134
739.7
8
18.93
3.15
7.99
17.44
11.42
345.83
11246607
696.4
7
18.06
15.10
8.84
8.00
7.04
165.29
11515196
644.3
7
16.40
7.98
3.84
6.26
34.15
138.30
11701976
931.7
10
23.79
11.98
2.70
4.80
13.24
40.42
a perfect search in the Kepler data. Finding a few dozens of ex-
ocometary transits can therefore be considered as a satisfactory
result in agreement with what could be expected using reason-
able estimates of the algorithm efficiency and cometary activity
of the surveyed stars.
Unfortunately, the Kepler stars are rather faint, and our cat-
alogs contain stars with magnitude between 9.7 and 15.9. It is
therefore not possible to undertake a spectroscopic follow up
with the hope to confirm the exocometary nature of the photo-
metric events. All the more that, with only one transit per star
in about four years, exocometary transits are rare and not pre-
dictable in time. However, we can anticipate that the same tech-
niques could be applied to search for exocometary transits in
current TESS data and in upcoming PLATO data. For these two
instruments, the surveyed star in the input catalog are signifi-
cantly brighter that the Kepler stars. We can thus expect that the
exocometary systems to be discovered in near future can be stud-
ied in detail, particularly through spectroscopy.
Previous searches (Rappaport et al. 2018; Kennedy et al.
2019) provided a list of 3 stars that seem to be rather young.
This supported the anticipated conclusion that cometary activity
is likely correlated with stellar age, with an expected decline over
time. However, our catalogs do not support this idea, as we found
a wide spread of age for both the first- and second-tier catalogs,
with possible exocometary transits in front of stars classified as
red giants. The issue of cometary activity with age thus remains
open, and further studies will be needed to clarify this important
question of the evolution of planetary systems.
Note added in manuscript: After this manuscript was sub-
mitted we became aware of a paper by Norazman et al. (2025)
that used Machine Learning techniques to search exocometary
transits in the sectors 1 to 22 of the TESS light curves. They
found three additional transits compared to those already known,
which underlines the utility of this technique for the search of
such transits.
Acknowledgements. We thank Dr. Neda Heidari for her help in identifying al-
ready known mono-transits. We acknowledge support from the CNES (Centre
national d’études spatiales, France). This work has made use of the Infinity Clus-
ter hosted by Institut d’Astrophysique de Paris. We thank Dr. Guilhem Lavaux
for his invaluable help in making efficient use of the GPUs in this cluster. This pa-
per includes data collected by the Kepler mission and obtained from the MAST
data archive at the Space Telescope Science Institute (STScI). Funding for the
Kepler mission is provided by the NASA Science Mission Directorate. STScI
is operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555. This research has made use of the NASA
Exoplanet Archive, which is operated by the California Institute of Technology,
under contract with the National Aeronautics and Space Administration under
the Exoplanet Exploration Program.
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A&A proofs: manuscript no. Article_v1
Table 5: Information on the stars of the second-tier catalog.
Kepler ID
RA
Dec
Kepler Mag
Teff
Log g
Metallicity
Radius
Stellar
(J2000)
(J2000)
(K)
(cm/s2)
(R⊙)
type
2984102
19 20 24.302
+38 09 41.26
11.271
5064
2.543
-0.167
10.715
K1III
4826941
19 16 42.927
+39 58 17.69
13.917
5748
4.425
-0.257
1.051
G3V
5195476
19 36 47.573
+40 19 41.48
13.925
4848
3.504
-0.214
3.406
K2IV
5294231
19 46 54.941
+40 26 00.17
9.842
9552
4.082
0.032
2.107
A0V
5511746
18 53 09.113
+40 46 26.87
13.216
5928
4.266
-0.277
1.290
G1V
5514277
18 58 16.380
+40 47 18.02
15.366
5401
4.601
-0.242
0.824
G7V
5534083
19 27 57.571
+40 45 16.38
12.602
6416
4.186
-0.261
1.458
F4V
6119914
19 22 29.443
+41 24 30.31
15.882
6167
4.819
-0.036
0.660
F5V
7183123
19 02 20.712
+42 47 30.55
11.207
8005
3.703
-0.145
3.020
A0IV
7201827
19 29 03.487
+42 44 03.41
15.409
5562
4.815
-0.098
0.638
G5V
7220968
19 48 59.026
+42 44 39.52
15.463
5015
4.483
0.35
0.921
K1V
7611858
19 32 36.857
+43 17 11.54
11.187
5684
4.477
-0.315
0.984
G4V
7669613
19 06 56.856
+43 18 44.17
13.044
5754
4.246
-0.382
1.313
G3V
8111649
19 45 47.110
+43 58 36.44
15.776
6080
4.576
-0.044
0.888
F6V
8144412
18 47 41.122
+44 04 04.40
15.213
5721
4.428
0.043
1.047
G3V
8374877
19 40 27.934
+44 19 09.62
12.083
6510
4.021
-0.505
1.809
F4V
8489495
19 18 40.234
+44 33 44.75
15.214
4912
4.328
0.068
1.112
K2V
8625941
19 28 55.855
+44 44 53.77
12.783
5689
4.602
-0.318
0.844
G4V
8683719
19 18 58.464
+44 49 28.02
14.717
5989
4.398
-0.035
1.100
G0V
8738569
18 58 41.198
+44 57 51.12
12.771
4907
3.548
0.173
3.227
K2IV
8843356
19 58 40.882
+45 01 02.64
12.039
5419
3.694
-0.387
2.687
G7IV
9025688
19 32 42.634
+45 23 41.57
13.382
6445
4.165
-1.598
1.500
F4V
9388975
18 56 00.802
+45 57 05.90
14.292
5163
4.493
-0.938
0.923
K0V
9531080
19 35 07.706
+46 09 35.64
13.861
6280
4.268
-0.346
1.306
F5V
10329957
19 10 18.223
+47 27 04.43
13.58
5002
4.567
-0.024
0.824
K1V
11124157
18 56 39.605
+48 45 44.21
12.244
4775
3.954
0.071
1.869
K3IV
11153134
19 53 11.854
+48 46 20.82
12.007
4941
2.719
-0.037
8.746
K2III
11246607
19 32 21.149
+48 58 47.78
13.169
6251
4.387
-0.253
1.127
F5V
11515196
19 42 06.725
+49 26 57.88
13.568
4754
4.15
0.345
1.403
K3V
11701976
19 02 02.592
+49 49 17.65
13.403
6210
4.398
-0.014
1.111
F5V
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Article number, page 10 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
Table 6: Third-tier catalog of symmetric transits that could be either exocometary transits or exoplanet mono-transits.
Stars (KIC)
Time (BKJD)
Quarter
K(×10−4)
β−1(h)
∆T(h)
AD(×10−4)
∆χ2
pl
∆χ2
polynomial
2983000
759.1
8
18.38
0.84
16.33
18.38
-1.94
483.82
2993038
550.9
6
15.22
2.46
4.72
12.99
-1.57
1165.8
3222471
673.1
7
97.7
5.49
14.59
90.83
-70.29
2436.25
3346436
270.4
3
8.48
3.03
3.05
5.38
3.42
182.43
3755854
1492.3
16
3.53
1.07
6.11
3.51
1.03
45.73
5184479
534.6
5
8.87
1.04
19.61
8.87
-11.23
316.53
5184479
1203.0
13
12.17
1.97
17.91
12.17
-3.69
409.71
5305217
673.1
7
33.0
5.46
20.87
32.28
-7.41
170.59
5456365
270.8
3
35.42
3.85
10.37
33.03
-2.28
323.73
5905878
1343.3
14
2.25
1.46
7.55
2.24
-7.35
353.12
5905878
1441.3
15
0.89
1.49
8.14
0.89
-2.01
60.16
5967153
1394.5
15
5.06
4.05
7.9
4.34
-22.09
141.91
5975275
213.3
2
9.78
2.57
10.57
9.62
2.7
179.4
b6186417
959.1
10
33.19
1.09
18.42
33.19
-2.36
645.89
6387193
651.7
7
78.79
2.73
12.43
77.96
-48.64
3817.28
6515488
1420.4
15
28.85
2.98
9.54
27.67
-7.02
359.44
6804821
1009.1
11
12.97
1.6
22.41
12.97
-6.09
317.96
7047396
298.8
3
11.35
2.0
5.57
10.65
-4.56
305.5
7105703
1509.0
16
20.54
2.86
6.53
18.45
-4.32
186.11
7213651
1239.2
13
21.53
1.66
6.78
21.17
-9.23
214.24
7465971
342.7
3
11.14
1.14
9.75
11.14
-1.07
200.89
7983622
471.9
5
65.06
4.63
13.76
61.73
-15.71
732.08
8005892
608.5
6
96.11
1.2
9.78
96.09
-75.32
9905.87
8007462
1458.6
15
2.44
2.56
5.4
2.14
-7.15
93.52
8159297
440.7
4
18.1
1.87
4.62
16.57
-1.87
132.24
8313257
1149.1
12
25.05
2.17
17.4
25.05
-8.72
320.72
a,b8410697
542.5
6
54.16
1.0
17.87
54.16
-12.07
5694.83
8496108
277.7
3
19.41
9.57
3.83
6.4
-6.55
62.5
8617888
171.8
2
55.2
1.83
5.44
52.4
-4.99
149.3
9016734
666.0
7
61.61
5.57
13.7
56.35
-48.25
1624.94
9016734
1230.1
13
27.52
5.5
14.31
25.48
-7.51
522.79
9413755
1564.2
17
19.5
1.63
4.4
18.18
-16.75
217.06
9775416
1398.1
15
5.04
4.99
10.63
4.44
-2.36
141.64
10024862
1494.0
16
20.95
1.68
13.29
20.94
-2.57
121.87
10157075
369.5
4
26.75
1.93
5.57
25.27
-2.17
114.74
10192453
757.5
8
18.64
5.63
1.73
4.94
-29.2
76.45
a10321319
554.5
6
4.76
4.77
13.56
4.48
-29.18
174.87
10334763
549.7
6
7.23
0.74
10.88
7.23
-3.39
412.26
10397849
506.5
5
19.93
2.35
7.95
19.25
0.15
289.02
10450889
239.6
2
60.03
1.58
2.35
46.44
1.45
192.69
10556420
756.3
8
3.59
0.64
6.03
3.59
-1.17
364.07
10643786
1421.2
15
12.62
3.23
5.12
10.04
3.16
81.51
10668646
196.3
2
42.81
0.93
8.74
42.8
-20.57
1376.58
a10668646
1449.3
15
57.08
1.36
8.64
56.98
-38.4
3632.55
10909733
898.7
9
3.41
1.37
6.17
3.37
4.64
67.65
11561379
137.7
1
13.51
2.11
7.45
13.12
-1.64
103.03
11921843
1234.8
13
3.85
0.62
14.83
3.85
-2.49
287.33
12066509
632.3
7
38.71
0.78
12.41
38.71
-0.31
1883.85
12644038
584.7
6
36.26
1.49
6.37
35.77
3.56
199.23
Notes. Four transits were already identified by Foreman-Mackey et al. (2016) (labeled with "a" in the first column) and Herman et al. (2019)
(labeled with "b").
Article number, page 11 of 18
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Appendix A: Performance of the algorithm over all the quarters
Table A.1: Performance of the neural network.
Quarter
Subquarter
Accuracy
Recall
Precision
FPR
AUC
MAE
1
0
0.8945
0.7910
0.9975
0.0020
0.9888
2.3404
2
0
0.8665
0.7347
0.9977
0.0017
0.9907
2.2237
2
1
0.8616
0.7245
0.9982
0.0013
0.9900
2.0792
2
2
0.8743
0.7498
0.9984
0.0012
0.9893
2.0455
3
0
0.9108
0.8240
0.9972
0.0023
0.9896
2.0329
3
1
0.9068
0.8162
0.9969
0.0025
0.9905
2.1186
3
2
0.8903
0.7835
0.9964
0.0028
0.9895
2.1638
4
0
0.8862
0.7745
0.9972
0.0022
0.9909
2.3073
4
1
0.9017
0.8065
0.9961
0.0032
0.9924
2.4661
4
2
0.8854
0.7728
0.9974
0.0020
0.9920
1.9370
5
0
0.8994
0.8013
0.9969
0.0025
0.9919
1.9726
5
1
0.8898
0.7818
0.9972
0.0022
0.9916
2.1857
5
2
0.8951
0.7923
0.9973
0.0022
0.9919
2.5873
6
0
0.8979
0.7972
0.9983
0.0013
0.9906
2.2020
6
1
0.8908
0.7833
0.9979
0.0017
0.9911
1.9720
6
2
0.8860
0.7738
0.9976
0.0018
0.9910
2.4570
7
0
0.8792
0.7605
0.9972
0.0022
0.9899
2.1862
7
1
0.8752
0.7522
0.9978
0.0017
0.9909
1.7915
7
2
0.9128
0.8267
0.9986
0.0012
0.9935
2.1101
8
0
0.9004
0.8018
0.9988
0.0010
0.9911
2.0729
8
1
0.9027
0.8065
0.9986
0.0012
0.9926
2.1139
9
0
0.9068
0.8157
0.9976
0.0020
0.9930
2.1293
9
1
0.9017
0.8062
0.9965
0.0028
0.9916
2.3157
9
2
0.8983
0.7992
0.9969
0.0025
0.9918
2.3348
9
3
0.8798
0.7610
0.9980
0.0015
0.9916
2.2389
10
0
0.8755
0.7532
0.9971
0.0022
0.9896
2.7444
10
1
0.8758
0.7538
0.9971
0.0022
0.9889
2.0156
10
2
0.8829
0.7672
0.9983
0.0013
0.9890
2.7258
11
0
0.8830
0.7685
0.9968
0.0025
0.9908
2.0722
11
1
0.8941
0.7900
0.9977
0.0018
0.9913
1.9576
11
2
0.8907
0.7828
0.9981
0.0015
0.9921
2.4971
12
0
0.8876
0.7767
0.9981
0.0015
0.9921
2.3743
12
1
0.9028
0.8075
0.9977
0.0018
0.9930
2.3531
12
2
0.9038
0.8103
0.9967
0.0027
0.9913
2.4029
13
0
0.8981
0.8000
0.9952
0.0038
0.9913
2.8353
13
1
0.8431
0.6875
0.9981
0.0013
0.9872
2.4796
13
2
0.9037
0.8117
0.9947
0.0043
0.9915
2.5127
14
0
0.8654
0.7342
0.9955
0.0033
0.9877
2.4772
14
1
0.8801
0.7632
0.9961
0.0030
0.9897
2.4096
14
2
0.8788
0.7592
0.9978
0.0017
0.9915
3.1279
15
0
0.8363
0.6743
0.9973
0.0018
0.9869
2.5724
15
1
0.8378
0.6780
0.9966
0.0023
0.9826
3.3750
15
2
0.8490
0.7000
0.9972
0.0020
0.9862
3.1479
16
0
0.8238
0.6483
0.9987
0.0008
0.9870
2.4968
16
1
0.8665
0.7355
0.9966
0.0025
0.9868
2.1849
16
2
0.8445
0.6907
0.9976
0.0017
0.9858
2.4707
17
0
0.8744
0.7510
0.9971
0.0022
0.9899
2.3075
Article number, page 12 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
Appendix B: Cometary transits light curves
Plots of the light curves of the detected cometary transits in the first tier catalog. For each transit, the light curve is shown in the
top panels over a 5 days duration centered on the transit time and with an exocomet transit fit superimposed (red line). The bottom
panels show the light curves over a larger time frame.
Article number, page 13 of 18
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Article number, page 14 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
Article number, page 15 of 18
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Appendix C: Possible exocometary transits light curves
Plots of the light curves of the possibly detected cometary transits in the second tier catalog. The plots cover a 5 days time frame
centered on the transit time and with an exocomet transit fit superimposed (red line).
Article number, page 16 of 18
P. Dumond et al.: Search for exocomets transits in Kepler light curves
Article number, page 17 of 18
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Article number, page 18 of 18
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Astronomy & Astrophysics manuscript no. Article_v1 October 17, 2025 Search for exocomets transits in Kepler light curves Ten new transits identified P. Dumond1, 2, A. Lecavelier des Etangs1, F. Kiefer1, 3, G. Hébrard1, and V. Caillé1, 4 1 Institut d'astrophysique de Paris, CNRS, Sorbonne Université, 98bis boulevard Arago, 75014 Paris, FRANCE 2 CRAL, Ecole normale supérieure de Lyon, Université de Lyon, UMR, CNRS 5574, F-69364 Lyon Cedex 07, France 3 LIRA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France 4 Institut Pierre-Simon Laplace, Sorbonne-Université, CNRS, Paris, France, October 17, 2025 ABSTRACT The Kepler mission, despite its conclusion over a decade ago, continues to offer a rich dataset for uncovering new astrophysical objects and phenomena. In this study, we conducted a comprehensive search for exocometary transit signatures within the Kepler light curves, using a machine learning approach based on a neural network trained on a library of theoretical exocomet transit light curves. By analyzing the light curves of 201,820 stars, we identified candidate events through the neural network and subjected the output to filtering and visual inspection to mitigate false positives. Our results are presented into three catalogs of increasing ambiguity. The first-tier catalog includes 17 high-confidence exocometary transit events, comprising 7 previously reported events and 10 newly identified ones, each associated with a different host star. The second-tier catalog lists 30 lower-confidence events that remain consistent with possible exocometary transits. The third-tier catalog consists of 49 more symmetric photometric events that could be either exocometary transits, exoplanet mono-transits, or false positives due to eclipsing binaries mimicking transits. Contrary to previous studies, which suggested that the cometary activity was favored by stellar youth, we find a broad age distribution among candidate host stars, including several red giants. This challenges the general idea of a decline in cometary activity with stellar age and underlines the need for further investigation into the temporal evolution of exocometary activity in planetary systems. Key words. Methods: data analysis - Techniques: photometric - Surveys - Comets: general - Planetary systems - Exocomets - Transit photometry 1. Introduction Exocomets are rocky or icy minor bodies that become active when they approach their parent star on a highly eccentric orbit (Strøm et al. 2020). Producing a huge tail of gas and dust close to the periastron, when they transit in front of the star, they become detectable using spectroscopy (for the gaseous tail) and photometry (for the dust tail). The first exocomets were thus identified in the 1980s in the β Pic system, through the detection of variable absorption features in the Ca ii lines of the stellar spectrum (Ferlet et al. 1987; Kiefer et al. 2014a). Later on, a few other exocometary systems have been found using spectroscopy, like HD 172555 (Kiefer et al. 2014b; Grady et al. 2018), and 49 Cet (Montgomery & Welsh 2012; Miles et al. 2016). A list of spectroscopically identified exocometary systems can be found in Strøm et al. (2020). For the detection of exocomets using photometry, one needs to obtain photometric measurements at high accuracy, typically at 10-3-10-4 level over the full transit duration, that is, for one to three days (Lecavelier des Etangs et al. 1999, hereafter LdE99a). This capability has been reached only during the last decade with the space missions like Corot, Kepler, TESS, and Cheops. Detections have thus been obtained with Kepler toward KIC 3542116 and KIC 11084727 (Rappaport et al. 2018), and KIC 8027456 (Kennedy et al. 2019), with TESS toward β Pic (Zieba et al. 2019; Pavlenko et al. 2022; Lecavelier des Etangs et al. 2022), and with Cheops toward HD 172555 (Kiefer et al. 2023). Here it must also be mentioned the detection with Kepler of a photometric event, possibly periodic, that is interpreted by the transit of a string of 5 to 7 exocomets in front of KIC 8462852 (Kiefer et al. 2017). All these events have been identified thanks to the particular shape of the light curve of an exocomet transit, as theoretically predicted at the end of the 90's in LdE99a : the characteristic of an exocomet transit light curve is the asymmetry caused by the cometary tail passing in front of the star after the nucleus. This causes a sharp decrease of the star light followed by a slow return to the normal brightness (see also Lecavelier des Etangs 1999, hereafter LdE99b). However, exocomet transit light curves can also be more symmetric when the longitude of the periastron of the orbit is about 90◦from the line of sight, because here the cometary tail is aligned with the line of sight. In this case, the shape of the transit light curve is similar to that of an exoplanet transit (LdE99a). The detection of exocometary transits based on photometry is complementary to the detection through spectroscopy. It allows exploring the physical characteristics of dust particles, such as the dust particle size distribution, the distance to the host star at the time of transit, and the dust production rate (Luk'yanyk et al. 2024). Indeed, because transits give direct access to the geometrical extent and the optical thickness of the transiting dust cloud, coupled with models of dust production, this allows the Article number, page 1 of 18 16 Oct 2025 A&A proofs: manuscript no. Article_v1 estimate of the dust production rate and the size of the comets nucleus. In the case of β Pic, statistical analysis of photometric events allowed the estimate of the nucleus size distribution that is found to be similar to that of asteroids and comets in the Solar system as set by collisional equilibrium (Lecavelier des Etangs et al. 2022). Although the characterization of exocomets paves the way to understanding the dynamical and chemical processes occurring in young planetary systems, only a very limited number of exocometary systems have been identified to date. As a result, our current understanding of the diversity, frequency, and evolutionary significance of exocometary activity remains incomplete. Systematic searches have been performed using spectroscopy in the Ca ii line (Montgomery & Welsh 2012; Welsh & Montgomery 2013, 2018; Rebollido et al. 2020; Bendahan-West et al. 2025), resulting in a few detections (see review in Strøm et al. 2020). Spectroscopy is efficient in detecting tiny amount of gas and thus allows sensitive search ; the result is that detectable spectroscopic transits seem to be more frequent than detectable photometric events: for instance in the case of β Pic, several exocomet transits are routinely detected every day in spectroscopy (Kiefer et al. 2014a), while only 30 photometric transits have been detected in 156 days of TESS observations (Lecavelier des Etangs et al. 2022). However, spectroscopic searches for exocomets are inherently limited by their need to observe each target star individually and the need for long observation time with high spectral resolution. In contrast, photometric surveys can monitor several thousands of stars simultaneously, making them significantly more efficient for large-scale searches for exocometary transits. In that spirit, Rappaport et al. (2018) and Kennedy et al. (2019) have undertaken a deep search for exocometary transits in the Kepler light curves. To identify exocometary transits, they used a modified model of planetary transit or a combination of exponential and Gaussian tails, as also done later on by Zieba et al. (2019), Pavlenko et al. (2022), and Lecavelier des Etangs et al. (2022). Although this empirical approach does not carry physical information about the cometary bodies and tails, these are simple and easy to implement to identify candidates in the large number (∼200 000) of light curves provided by Kepler. Here we propose to use an automated search without the use of analytical functions to describe the exocometary transit light curves. For that we combined a search algorithm based on a neural network together with a library of photometric transits to train the network. The library is an updated version of the library of LdE99b, which has been recalculated using faster CPUs than the ones used 20 years ago, covering a wide range of parameters and therefore a wide range of transit shapes that need to be identified. We apply this automated search to the Kepler data (Borucki et al. 2010) with the aim of producing a new catalog of exocometary systems candidates. The methods are described in Sect. 2, 3 and 4, the search in Kepler data is presented in Sect. 5 and the final catalog of candidates, divided in three tiers, is given in Sect.6. The results are discussed in Sect.7. 2. Training of the neural network 2.1. Training principle The method used for the detection of exocomets in Kepler data is based on a neural network using supervised learning. For the network learning, we used a library of theoretical light curves divided into three disjoint sets of labeled light curves: the training set, the validation set, and the test set. Each set contains the same number of light curves with and without an exocomet transit (see Sect. 2.2). Each light curve is labeled to indicate if it includes a transit or not. Our objective is to make the algorithm identify the curves containing an exocomet transit among all the light curves collected by the Kepler satellite and to recover the transit time when a transit is present. Using an iterative procedure, the algorithm optimizes its parameters thanks to the training set. At the end of each iteration step called an "epoch", the network performance is tested using the validation set. This allows us to control the overfitting of the algorithm (e.g. it becomes specific to the training set and loses in generality) and to adjust the hyperparameters. The learning phase is stopped when the network shows the best performance in the validation set. The final performance of the algorithm is given by its evaluation on the test set. 2.2. Light Curve production The light curves used to build the training, validation, and test set are the Presearch Data Conditioning (PDC) data from Kepler Data Release 25 (Thompson et al. 2016). We used the Quarters 1 to 17 and divided them into sub-quarters in case of interruption of the acquisition within a given quarter: two sub-quarters are separated by at least four consecutive photometric measurements missing in the light curve. For each sub-quarter, a training, validation, and test set were constructed to overcome their possible specificity. For each subquarter, the training set consists of 36 000 curves, while the validation and test sets contain 12 000 each. A curve consists on 10 days (480 photometric measurements at a cadence of one measurement every ∼30 minutes) of acquisition of the luminosity flux from a star randomly chosen without repetition. To ensure efficient learning and prevent the algorithm from falling into obvious traps, several modifications have been made to the Kepler PDC data. First, all points outside the mean value of the light curve by more than 5-sigmas while the neighboring points are not are considered to be outliers and removed. In addition, each curve was normalized to have a zero mean and a standard deviation of 1, as needed for optimal learning by the algorithm. Finally, the possible missing measurements, i.e. at most three consecutive missing measurements within the same subquarter, were filled using a linear interpolation. For the learning process, half of the 10-day light curves have been used with a label "0", indicating the absence of an exocomet transit: we assume that the probability that any randomly chosen light curve in the Kepler data includes an exocomet transit is extremely low. This set of 0-labeled light curves without exocomet transit is used to learn the noise patterns in the Kepler data. For the other half of the 10-day light curves, we add one exocomet transit by multiplying by a light curve from the theoretical transit library (Sect. 2.3) and we label them with "1", indicating the presence of an exocomet transit. 2.3. Simulation of exocomet transits In order for the algorithm to learn to identify the shape of an exocomet transit in the light curves, we add theoretical exocomet transits to half of the curves that make up the training data. These theoretical exocomet transit light curves have been calculated using the method describes in LdE99a. We produced a library of 2200 exocomet transit light curves, which is an updated version of the library published twenty years ago by LdE99b. Among all available simulations, we selected only those with a gas producArticle number, page 2 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves tion rate log10(P/(1kg · s-1)) between 6 and 7.5 and an impact parameter b , max pooling layers are noted MaxPool , fully connected layers are noted Dense , LSTM layers are noted LSTM , GRU layers are noted GRU and the squeezed-excitation blocks are noted SEB. none. The second output layer is activated by a linear function in range (0,1). It gives the position of the identified transit in the light curve. If no transit is detected, the layer should return a value close to zero. 3.2. Training For each sub-quarter, we trained the model using 36 000 light curves for 150 training epochs, that means that we performed 150 complete pass through the entire training dataset to update the model weight parameters. We used the Adam optimization algorithm (Kingma & Ba 2015) to minimize the cross-entropy error function for the classification of the light curves and the mean absolute error function for the position of the transit. We used a learning rate of 10-4 and a batch size of 5 000. The saved model in the one for which the validation loss reaches its minimum ; usually it is obtained after between 50 and 80 iterations ("epochs"). Therefore, training the model over 150 epochs is considered sufficient. Article number, page 3 of 18 A&A proofs: manuscript no. Article_v1 4. Evaluation of the neural network performance We evaluated the performance of our neural network with respect to several metrics in the same way as Dattilo et al. (2019) did for their model. We computed all metrics over the test set (rather than the training or validation sets) to avoid using any data that were used to optimize the parameters or hyperparameters of the model. 4.1. The metrics The metrics used to evaluate the performance of a classifier are the following: 1. Accuracy: the fraction of curves correctly classified by the model. 2. Precision (reliability): the fraction of correctly classified exocomets, i.e., the fraction of curves classified as exocomet candidates (False positives + True positives NFp + NTp) that are indeed true positives (labeled 1): P = NTp NFp + NTp (1) 3. Recall (completeness or True-positive rate): the fraction of the curves labeled 1, which therefore includes an exocomet transit (True positives + False negatives NTp + NFn), that the model classifies as exocomet candidates (NTp): R = NTp NTp + NFn (2) 4. False-positive rate: the fraction of total labeled 0 curves, which therefore do not include any exocomet transit (False Positives +True Negatives NFp + NTn) that the model classifies as exocomet candidates. FPR = NFp NFp + NTn (3) 5. AUC (area under the receiver-operator characteristic curve; see Figure 2): the probability that the model, if it receives two light curves, one 1-labeled curve including an exocomet transit and one 0-labeled curve without transit, would rank the 1-labeled curve higher than the 0-labeled curve. 6. Mean Absolute Error: the absolute error for all labeled 1 curves between the true position of the transit and the predicted one by the model. The first five metrics are related to the detection of the presence of an exocomet in the light curves, while the last metric is related to the position in time of the transit in the light curve. Except for the AUC, the values of these metrics depend on the classification threshold chosen for the model. This threshold is the value of the probability of detection yield by the algorithm above which we consider that there is a detection. In Fig. 2, we show the evolution of the recall as a function of the FPR for various thresholds, usually called the receiver-operator characteristic (ROC) curve. It can be seen that the algorithm differentiates well between true transits and noise, as it achieves very low FPR recall while keeping a reasonably large recall: our model reaches an AUC of 98.88% for the quarter Q1. We also plot the precision as a function of the recall. This curve shows the trade-off between having no false positives (high precision) and identifying all exocomet transits (high recall). Fig. 2: The receiver-operator characteristic (ROC) curve (left panel) and the precision versus recall curve (right panel). The receiver-operator characteristic (ROC) curve shows the recall (true-positive rate) of the model against the ability to recognize false positives (the false-positive rate) for different classification thresholds. Our model is highly successful at identifying false positives as shown by the high AUC value (see Table A.1). The plot of the fraction of exocomets that the model classified as exocomets (recall) versus the fraction of correctly classified planets (precision) shows the trade-off between having no false positives (high precision) and identifying all exocomet transits (high recall). Considering the huge number of light curves in the Kepler data, the goal is to drastically reduce the number of false positive. We thus chose a classification threshold of 0.99. With this threshold, our model reaches an accuracy of 89.5% for the quarter Q1. The precision of the model is 99.8%, which means that we reach a high true positive rate, and the recall is 79,1%, which means that about 20% of the true exocomet transits are missing in the final classification. This trade-off aims to accept some loss in the finding of exocomet transits of about 20% to avoid a large number of false positives. Indeed, given a large number of light curves in the Kepler data and a small number of exocomet transits in these data, even a tiny fraction of false positives can significantly pollute the results. The metrics values are found to be similar for all quarters (see Table A.1 in Appendix A). 4.2. Quality of the output The quality of the model classification can also be visualized by the histogram of the results on the test set (top panel of Fig. 3). The closer the output from the neural network is to 1 (resp. 0), the higher the probability that the 10-day light curve (resp. does not) contains an exocomet transit. The histogram shows that the neural network works as expected: on the left-hand side, most of the light curves characterized by a low probability are mostly noise (orange histogram), while on the right-hand side, the large majority of the curves characterized by a probability close to 1 included a simulated exocomet transit (label 1; blue histogram) and with very few false positives. The bottom graph of Fig. 3 shows the mean error of the position of the transit as a function of the probability assigned by the algorithm when it is applied to the curves containing a simulated transit (label 1, with known position). One can see that when the transit is not found (probability close to 0), the error on the predicted position is large, about 3 days, whereas it is of the order of a few hours only when the probability is above 0.99. Thus, one can be confident in the time of the transit given by the algorithm for curves with a probability higher than this threshold. Article number, page 4 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Fig. 3: Histogram of the results on the test set and error on the transit position found by the network. Top panel: Histogram of the result of the neural network applied to the test data. Most of the 1-labeled light curves with an exocomet transit yield a probability close to 1, while there are only a few false positives (0-labeled light curves yielding a high probability). Above the chosen threshold, there is more than two order of magnitude between the number of true positives and false positive. Bottom panel: the mean error of the position of the transit as a function of the probability of the presence of a transit assigned by the network to the light curves. This error is of a few hours at most for the transits identified by the algorithm. 5. Search for exocomets in Kepler data 5.1. Application of the model to Kepler data To apply the algorithm presented in the previous sections, all the light curves of the Kepler data were cut into 10-day windows obtained in the same way as the samples used to construct the training, validation, and test sets described in Sec. 2.2. In order not to lose any transits located at the edge of the windows, an overlap of 2 days was applied between two consecutive 10-day time intervals. However, despite the level of precision achieved, the algorithm provides a large number of false positives due to the large number of 10-day windows: approximately 106 windows per quarter lead to about 103 positive candidate detections, among which less than a dozen real transits should be identified (see LdE99a and discussion in Sect. 7). To reduce the number of false positives, we decided to filter the candidates using constraints on the shape of the light curves of the detected photometric events. 5.2. Additional criteria to filter false positives After all the Kepler light curves have passed through the neural network, we end up with a set of 56525 exocomet candidates. To filter these candidates, we decided to add constraints on the shape and on the context on the transit (noise, quality flags). 5.2.1. Filtering through the shape of the transits candidates The shape of the candidate transits were determined by fitting them by the model proposed by Lecavelier des Etangs et al. (2022) together with a second order polynomial baseline: fexo(t) = Ct2 + Dt + E t t > t0 (Ct2 + Dt + E)(1 + K(e-β(t-t0) -e-β(t-t1))) t > t1 (4) Fig. 4: Distribution of the value of the parameters obtained by fitting the 2163 light curves in the library of the simulated exocomet transits. The parameters K, ∆T = t1 -t0, and 1/β correspond to the depth, ingress duration, and crossing time of the transit, respectively. The parameters t0 and t1 represent respectively the time at which the transit starts and the time at which it reaches its minimum. K represents the characteristic depth of the transit, while 1/β corresponds to the characteristic crossing time of the transit. C, D and E are the coefficients of the polynomial baseline. The fit of the light curve by the transit model can be easily performed using the position of the detected candidate in the light curve as provided by the network output. To accept or reject a proposed detection given by the neural network, we fit the light curve of the corresponding photometric event to obtain the values of the parameters K, β, and ∆T = t1-t0 that characterize the shape of the event. We then check that these values are in the domain of validity for an exocometary transit. The domain of validity is obtained by considering the parameters space occupied by the fits of the theoretical light curves in the library of numerical simulations of exocometary transits, as shown in Fig. 4 and summarized in the first three lines of Table 1. It is found that in the theoretical light curves K is mostly between 10-4 and 5×10-3, ∆T is between 1.7 and 17 hours (corresponding to a periastron between ∼0.02 to 2 au, Lecavelier des Etangs et al. 2022), and β-1 is in the domain above 0.5 hours, and between 0.3∆T -1.7 and 0.5∆T + 12. Moreover, the parameters C and D have to verify the condition |C|, |D| 36 were conserved. This criterion was very useful to remove candidates that were detected due to a few points significantly lower than the average. To detect the presence of red-noise that mimics an exocomet transit, we calculated the correlation product between the exocomet model fit to the candidate transit and the light curve over all the considered sub-quarters. If the maximum of the correlation max1 is reached more than one day away from the transit time or if its second maximum max2 is significantly close to the first maximum ((max1-max2)/max1 t0 (6) We calculate ∆χ2 rp defined by : ∆χ2 rp = N P i( framp(ti) -F(ti))2 P i( fexo(ti) -F(ti))2 -1 ! , (7) where N = 120 is the number of points in the fitted window. The photometric events that are almost as well fitted with the ramp model as with the exocomet model (∆χ2 rp < 10), have been discarded from the candidate list. Finally, to identify the cases of a symmetrical transit that could be due to the passage of an exoplanet with a long period (a monotransit), we calculate the difference ∆χ2 pl between the χ2 pl of a model of a planetary transit calculated using the equations of Mandel & Agol (2002) and the χ2 exo of an exocomet transit model (Eq. 4). It can be written as: ∆χ2 pl = N P i(fpl(ti) -F(ti))2 P i(fexo(ti) -F(ti))2 -1 ! . (8) The photometric events having ∆χ2 pl < 5 have a symmetrical shape and have therefore not been taken into account in our final selection of exocomets. Such transits have been taken into account to establish the list of symmetrical mono-transits (see Sec. 6.3). This criterion is, however, not used as a sharp cut to distinguish the exocomets transits from the symmetrical mono-transits but only as a help. The final distinction has been made through visual inspection. After applying these criteria, the number of candidate transits per quarter varies between 50 and 100 per quarter, leading to a total of 1349 transits. It then becomes possible to proceed for each of them to a visual inspection to know if the candidate is indeed or not an exocomet transit. This final inspection is necessary because, although the criteria put in place make it possible to eliminate the vast majority of false positives, there still remain obvious false positives. Note here that the criteria have been deliberately chosen to be conservative so as not to accidentally eliminate a true exocomet transit. The number of criteria was chosen so that the final number of transits to be visually inspected would be no more than a few thousand. Otherwise, visual inspection would be impractical. It would be more inefficient and inaccurate to add extra criteria than to visually check the current list of 1,349 transits. To verify the robustness of the detected transits, we perform an additional test. We apply our detection procedure to the data after reversing the time. If the proportion of detected exocomets is the same in the reversed time data as in the original data, this may question the relevance of the detected transits. However, as this analysis is time-consuming, we performed this analysis on half of the quarters. Of the 17 quarters, we analysed eight randomly chosen quarters: Q1, Q2, Q6, Q9, Q10, Q12, Q15 and Q16. We found only one transit that clearly mimics an exocomet transit: KIC_3129239 in Q9. Given that approximately 17 robust transits have been detected in the Kepler data (see Sec. 6), we can infer from this study that about 10% to 15% of these transits can be false positives. Although the same number of detections in the time-reversed data would have disproven the exocomet hypothesis, the result of this test supports the idea that there is an astrophysical signal in the data that is consistent with exocomet transits. Article number, page 6 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves 6. A new catalog of transiting exocomets In this section, we present the final list of exocomet transits resulting from the selection procedure. We divided the list in three different tiers based on the likelihood of a genuine exocometary detection. In the first tier, the photometric events are the ones that we consider as the most likely due to the transit of an exocomet ; in the second tier are the events possibly due to comets, while the third tier gathers the most symmetric dip in the light curve, for which the transit of an exoplanet remains a possibility. 6.1. The first-tier catalog of exocometary transits The first-tier catalog of detected exocometary transits is given in Table 2 and the plots of the corresponding light curves are given in Appendix B. It is particularly noteworthy that our neural network is able to find all the transits already identified by previous works: KIC 3542116 in Quarters 1, 10 and 12 and KIC 11084727 as found by Rappaport et al. (2018), and KIC 8027456 as found by Kennedy et al. (2019). This independent retrieval of the same exocometary transits provides confidence in our procedure. Nonetheless, after application of all the search procedure described above, it appeared that two transits of KIC 3542116 identified by Rappaport et al. (2018) in Quarters 8 were not in our list of detections. These transits were eliminated by the criterion based on the maximum of the correlation product that aims to remove the detections whose pattern occurs several times in the same subquarter (Sect. 5.2.4). In fact, this criterion also has the disadvantage of eliminating active cometary systems where at least two comets transits occurred in the same quarter, as is the case for KIC 3542166 in Quarter 8. Therefore, we made a visual inspection of all photometric events eliminated because of this criterion in order to recover possible obvious exocometary transits that were wrongly rejected. We thus found the two transits of KIC 3542116 in the Quarter 8, and in addition identify a new transit in front of KIC 6263848. This last case does not correspond to a star with frequent cometary transits, but had been deleted due to an edge effect on the calculation of the correlation function. These two cases are identified by an asterisk in Table 2. For all of these detection, we calculate the ∆χ2 polynomial defined as ∆χ2 polynomial = N P i( fpolynomial(ti) -F(ti))2 P i(fexo(ti) -F(ti))2 -1 ! . (9) where fpolynomial is the second order polynomial that best fits the detection. For all detections, we obtained large ∆χ2 polynomial (larger than about 100). This highlights that the detection cannot be interpreted just as noise patterns. The properties of the parent stars of the identified exocomets as tabulated by Kepler are given in Table 3 (Brown et al. 2011). For KIC 3542116 and KIC 3662483, Teff, log10 g, metallicity and radius are from Zhang et al. (2025). From the Teff and log10 g we can obtain an estimate of the stellar type (Gray 2008), which is given in the last column of Table 3. It appears that most of the stars are main-sequence stars, with the exception of KIC 4078638, which is likely a red giant. For these stars, the upper limits on the age estimate given by Zhang et al. (2025) are all above 109 years, with the exception of KIC 4481029 with an upper limit of 9 × 108 years, and KIC 8027456 with an age estimate of 5 × 108 years. 6.2. The second-tier catalog of possible exocometary After visual inspection of the list of candidates produced by our algorithm, we came up with a list of interesting photometric events that can be due to transiting exocomets. This includes transits that pass through all the criteria described in Sect. 5, but whose shape or signal-to-noise ratio makes it difficult to decide whether they are actually transits. The resulting list of possible exocometary transits in this second-tier catalog is given in Table 4 and the plots of the corresponding light curves are given in Appendix C. The properties of the corresponding parent stars in the second tier catalog are given in Table 5 (Brown et al. 2011). Here again, estimates of the stellar types given in the last column of the table show that most of the stars are on or close to the main sequence, with the exception of KIC 2984102 and KIC 11153134, which have a log10 g below 3 and are therefore likely red giants. For all stars in this second-tier catalog, the upper limit of the age estimate given by Zhang et al. (2025) is above 109 years, with the exception of KIC 2984102, KIC 5294231 and KIC 7183123 with age estimates of 5×108, 1×108 (with an upper limit of 3 × 108) and 9 × 108 years, respectively. 6.3. The third-tier catalog of symmetric transits After visual inspection of the list of candidates produced by our algorithm, we also came up with a list of interesting photometric events that appears to be symmetrical. This is not surprising as some of the exocomets transits in the library used for training the network are indeed symmetrical (see Fig. 2 and 3 in LdE99a). However, we cannot exclude that these are due to mono-transits of exoplanets or to false positives mimicking planetary or cometary transits. Several scenarios might cause such false positives, including diluted or undiluted eclipsing binaries. To identify these cases one needs to carry out additional analyses that are beyond the scope of the present paper (see, e.g., Crossfield et al. 2016). Finally, although an exocometary transit classification cannot be made, these detections deserve to be mentioned. They are given in the third-tier catalog of symmetric transits listed in Table 6. We compared these detected photometric events with the lists of single transits published by Huang et al. (2013), Wang et al. (2015), Foreman-Mackey et al. (2016), and Herman et al. (2019). It appears that four events in our list were already identified by Foreman-Mackey et al. (2016) and Herman et al. (2019). The transit in front of KIC 8410697 was found in both studies. It is interpreted as due to an exoplanet with a radius of 0.7 times that of Jupiter. The photometric event of KIC 10321319 was found by Foreman-Mackey et al. (2016) and interpreted as due to the transit of an exoplanet of 0.16 Jupiter radius. The photometric event of KIC 6196417 was found by Herman et al. (2019) and interpreted as due to the transit of an exoplanet of about 0.7 Jupiter radius. The transit in front of KIC 10668646 at the time of BKJD=1449.3 was already identified by Foreman-Mackey et al. (2016) but the exoplanet transit scenario was rejected because of a centroid shift in the data. However, we also identified another photometric event in the same target at BKJD=196.3 with a different shape and a lower absorption depth. These few examples show that our list of newly identified photometric events which could be due to transits of exocomet with symmetrical shape of the light curve deserves further investigations which are beyond the scope of the present paper. Article number, page 7 of 18 A&A proofs: manuscript no. Article_v1 Table 2: First-tier catalog of exocometary transits. Stars (KIC) Time (BKJD) Quarter K(×10-4) β-1(h) ∆T(h) AD(×10-4) ∆χ2 pl ∆χ2 polynomial †3542116 161.5 1 4.74 6.5 9.92 3.71 18.32 279.74 †*3542116 742.6 8 4.17 1.36 6.05 4.12 0.6 348.83 †*3542116 792.9 8 8.62 4.16 4.77 5.88 15.57 285.27 †3542116 992.1 10 10.12 3.62 14.68 9.94 28.67 2993.05 †3542116 1175.8 12 12.98 3.40 9.35 12.15 8.58 2670.38 3662483 1132.1 12 6.81 6.38 6.87 4.49 12.98 204.52 4078638 355.5 4 83.21 6.30 3.22 33.30 47.89 358.90 4481029 172.5 2 9.08 8.57 4.64 3.80 14.31 235.74 5206257 925.0 10 10.98 4.75 4.47 6.69 5.04 322.26 5514200 670.1 7 34.48 12.13 7.27 15.53 19.15 155.20 *6263848 775.9 8 10.22 3.29 4.90 7.92 7.96 151.93 6927963 416.8 4 33.60 9.13 2.83 8.96 6.54 115.89 7660548 966.2 10 15.74 5.95 5.09 9.05 26.10 219.47 †8027456 1449.0 15 8.0 9.90 14.12 6.08 16.78 401.18 8738545 238.1 2 17.0 4.92 3.67 8.94 7.64 92.53 10484683 309.5 3 29.85 14.05 8.01 12.97 87.66 853.82 †11084727 1076.2 11 15.29 3.92 8.20 13.40 88.14 2368.80 Notes. The first column gives the name of the star by its KIC identifier. The second and third columns indicate the position of the transit. Columns 4 to 6 give the fits parameters associated with the exocomet model (Eq. 4). The seventh column gives the absorption depth. The last two columns present the difference of χ2 between the cometary model and, first, the planetary model (Mandel & Agol 2002) and, second, a second order polynomial fit. The transits that were first eliminated by the criterion based on the maximum of the correlation product (see Sect. 5.2.4) are identified by an asterisk (*). The five transits already known are identified by the † symbol. Table 3: Information on the stars of the first-tier catalog. Kepler ID RA Dec Kepler Mag Teff Log g Metallicity Radius Stellar (J2000) (J2000) (K) (cm/s2) (R⊙) type 3542116 19 22 52.939 +38 41 41.51 9.979 6766 4.208 -0.190 1.454 F2 V 3662483 19 42 09.199 +38 43 01.38 10.541 5848 3.945 0.107 1.786 G1 IV 4078638 19 46 16.611 +39 06 15.23 13.156 4783 2.758 0.150 8.331 K3 III 4481029 19 43 13.210 +39 32 32.03 11.303 8865 3.968 0.010 2.310 A0 IV 5206257 19 46 33.826 +40 22 57.97 12.051 6017 4.435 -0.105 1.052 F6 V 5514200 18 58 08.306 +40 46 29.32 14.377 5651 4.500 -0.243 0.955 G4 V 6263848 18 52 30.566 +41 36 33.34 13.172 5850 4.365 -0.224 1.137 G2 V 6927963 18 56 15.946 +42 27 52.20 13.807 5711 4.441 -0.012 1.030 G4 V 7660548 18 47 22.807 +43 23 05.39 13.094 6158 4.369 -0.050 1.148 F5 V 8027456 19 25 15.838 +43 51 33.55 9.697 8732 3.766 0.065 2.974 A0 IV 8738545 18 58 37.262 +44 55 15.24 13.227 5091 3.581 -0.162 3.101 K1 IV 10484683 19 47 54.377 +47 40 14.16 12.588 6326 4.543 -0.009 0.934 F5 V 11084727 19 28 41.191 +48 41 15.14 9.987 6762 4.067 -0.152 1.726 F3 V 7. Conclusion Although the Kepler mission ended more than 10 years ago, the available data is still worth deep data mining in search of new discoveries. Here we present a new search for exocometary transits in the Kepler light curves. We used machine learning technique with a neural network that has been trained using a library of theoretical exocomets transit light curves inherited from the work of LdE99b. After parsing the light curves of close to 200 000 stars through the neural network, despite the application of several filters to eliminate most of the false positive, a visual inspection of the outcome of the network was still needed. We ended up with three catalogs of interesting objects. The first-tier catalog is composed of a total of 17 exocometary transits, including 7 previously identified transits and 10 new transits in front of 10 different stars. The second-tier catalog is a list of 30 photometric events that appear of second quality and that we qualify as possible exocometary transits. Finally, the third-tier catalog presents the list of interesting photometric events that are symmetrical and may be due to transits of either an exoplanet or an exocomet with a periastron at 90◦from the line of sight. The complete Kepler data represent a 4-year photometry survey of more than 170 000 distinct stars. However, as noted in Kennedy et al. (2019), not all stars were observed for the full duration of the mission. As a result, we can consider 150 000 as the approximate number of equivalent stars that were observed for the full duration of the mission. Or, in other words, our analysis covers the equivalent of about 600 000 stars-years. LdE99a showed that with 30 000 stars-years survey of stars with a solar system cometary activity and a few 10-4 photometric accuracy, the number of exocomets transit detections should be around 10. Therefore, assuming that each star harbors a cometary system similar to our own, we could expect roughly 200 detections with Article number, page 8 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Table 4: Second-tier catalog of possible exocometary transits. Same legend as in Table 2. Stars (KIC) Time (BKJD) Quarter K(×10-4) β-1(h) ∆T(h) AD(×10-4) ∆χ2 pl ∆χ2 polynomial 2984102 345.7 3 36.42 15.9 10.12 17.14 61.90 430.26 4826941 385.8 4 21.58 15.22 7.85 8.69 3.76 126.17 5195476 1508.8 16 16.80 5.09 3.67 8.31 3.42 45.92 5294231 411.4 4 4.92 3.26 2.13 2.36 31.78 285.46 5511746 1105.8 12 13.31 12.33 5.06 4.48 2.41 44.54 5514277 670.4 7 63.11 13.57 9.88 32.64 13.96 160.95 5534083 1167.7 12 9.34 10.45 3.33 2.55 7.22 43.16 6119914 747.2 8 31.41 9.36 6.63 15.95 3.54 50.65 7183123 336.4 3 8.43 8.36 5.98 4.31 28.32 321.42 7201827 409.9 4 43.64 4.83 2.83 19.34 5.77 48.98 7220968 1325.7 14 18.60 5.50 6.23 12.61 2.61 46.42 7611858 1275.8 14 6.46 9.77 5.19 2.67 7.63 109.47 7669613 315.9 3 13.24 13.66 6.31 4.89 12.77 121.72 8111649 934.0 10 83.04 23.69 10.28 29.24 16.36 123.61 8144412 639.1 7 86.39 12.00 2.95 18.84 7.06 140.49 8374877 769.1 8 7.78 7.04 8.46 5.44 13.67 300.35 8489495 1351.8 14 16.43 8.44 15.04 13.67 10.70 47.92 8625941 1505.0 16 5.16 8.44 9.80 3.54 8.10 66.25 8683719 1340.2 14 19.21 6.43 14.25 17.12 7.59 195.79 8738569 1462.3 15 13.18 8.30 4.30 5.33 8.84 56.11 8843356 702.2 7 8.84 10.54 2.99 2.18 4.66 46.51 9025688 1078.2 11 13.29 7.51 3.57 5.03 58.03 58.03 9388975 680.3 7 18.41 3.36 2.47 9.59 6.76 60.28 9531080 748.0 8 10.23 9.31 6.56 5.18 16.13 53.73 10329957 1385.1 15 18.47 12.90 4.10 5.03 9.62 64.04 11124157 704.2 7 15.01 10.93 2.00 2.51 11.48 45.20 11153134 739.7 8 18.93 3.15 7.99 17.44 11.42 345.83 11246607 696.4 7 18.06 15.10 8.84 8.00 7.04 165.29 11515196 644.3 7 16.40 7.98 3.84 6.26 34.15 138.30 11701976 931.7 10 23.79 11.98 2.70 4.80 13.24 40.42 a perfect search in the Kepler data. Finding a few dozens of exocometary transits can therefore be considered as a satisfactory result in agreement with what could be expected using reasonable estimates of the algorithm efficiency and cometary activity of the surveyed stars. Unfortunately, the Kepler stars are rather faint, and our catalogs contain stars with magnitude between 9.7 and 15.9. It is therefore not possible to undertake a spectroscopic follow up with the hope to confirm the exocometary nature of the photometric events. All the more that, with only one transit per star in about four years, exocometary transits are rare and not predictable in time. However, we can anticipate that the same techniques could be applied to search for exocometary transits in current TESS data and in upcoming PLATO data. For these two instruments, the surveyed star in the input catalog are significantly brighter that the Kepler stars. We can thus expect that the exocometary systems to be discovered in near future can be studied in detail, particularly through spectroscopy. Previous searches (Rappaport et al. 2018; Kennedy et al. 2019) provided a list of 3 stars that seem to be rather young. This supported the anticipated conclusion that cometary activity is likely correlated with stellar age, with an expected decline over time. However, our catalogs do not support this idea, as we found a wide spread of age for both the first- and second-tier catalogs, with possible exocometary transits in front of stars classified as red giants. The issue of cometary activity with age thus remains open, and further studies will be needed to clarify this important question of the evolution of planetary systems. Note added in manuscript: After this manuscript was submitted we became aware of a paper by Norazman et al. (2025) that used Machine Learning techniques to search exocometary transits in the sectors 1 to 22 of the TESS light curves. They found three additional transits compared to those already known, which underlines the utility of this technique for the search of such transits. Acknowledgements. We thank Dr. Neda Heidari for her help in identifying already known mono-transits. We acknowledge support from the CNES (Centre national d'études spatiales, France). This work has made use of the Infinity Cluster hosted by Institut d'Astrophysique de Paris. We thank Dr. Guilhem Lavaux for his invaluable help in making efficient use of the GPUs in this cluster. This paper includes data collected by the Kepler mission and obtained from the MAST data archive at the Space Telescope Science Institute (STScI). Funding for the Kepler mission is provided by the NASA Science Mission Directorate. 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Stars (KIC) Time (BKJD) Quarter K(×10-4) β-1(h) ∆T(h) AD(×10-4) ∆χ2 pl ∆χ2 polynomial 2983000 759.1 8 18.38 0.84 16.33 18.38 -1.94 483.82 2993038 550.9 6 15.22 2.46 4.72 12.99 -1.57 1165.8 3222471 673.1 7 97.7 5.49 14.59 90.83 -70.29 2436.25 3346436 270.4 3 8.48 3.03 3.05 5.38 3.42 182.43 3755854 1492.3 16 3.53 1.07 6.11 3.51 1.03 45.73 5184479 534.6 5 8.87 1.04 19.61 8.87 -11.23 316.53 5184479 1203.0 13 12.17 1.97 17.91 12.17 -3.69 409.71 5305217 673.1 7 33.0 5.46 20.87 32.28 -7.41 170.59 5456365 270.8 3 35.42 3.85 10.37 33.03 -2.28 323.73 5905878 1343.3 14 2.25 1.46 7.55 2.24 -7.35 353.12 5905878 1441.3 15 0.89 1.49 8.14 0.89 -2.01 60.16 5967153 1394.5 15 5.06 4.05 7.9 4.34 -22.09 141.91 5975275 213.3 2 9.78 2.57 10.57 9.62 2.7 179.4 b6186417 959.1 10 33.19 1.09 18.42 33.19 -2.36 645.89 6387193 651.7 7 78.79 2.73 12.43 77.96 -48.64 3817.28 6515488 1420.4 15 28.85 2.98 9.54 27.67 -7.02 359.44 6804821 1009.1 11 12.97 1.6 22.41 12.97 -6.09 317.96 7047396 298.8 3 11.35 2.0 5.57 10.65 -4.56 305.5 7105703 1509.0 16 20.54 2.86 6.53 18.45 -4.32 186.11 7213651 1239.2 13 21.53 1.66 6.78 21.17 -9.23 214.24 7465971 342.7 3 11.14 1.14 9.75 11.14 -1.07 200.89 7983622 471.9 5 65.06 4.63 13.76 61.73 -15.71 732.08 8005892 608.5 6 96.11 1.2 9.78 96.09 -75.32 9905.87 8007462 1458.6 15 2.44 2.56 5.4 2.14 -7.15 93.52 8159297 440.7 4 18.1 1.87 4.62 16.57 -1.87 132.24 8313257 1149.1 12 25.05 2.17 17.4 25.05 -8.72 320.72 a,b8410697 542.5 6 54.16 1.0 17.87 54.16 -12.07 5694.83 8496108 277.7 3 19.41 9.57 3.83 6.4 -6.55 62.5 8617888 171.8 2 55.2 1.83 5.44 52.4 -4.99 149.3 9016734 666.0 7 61.61 5.57 13.7 56.35 -48.25 1624.94 9016734 1230.1 13 27.52 5.5 14.31 25.48 -7.51 522.79 9413755 1564.2 17 19.5 1.63 4.4 18.18 -16.75 217.06 9775416 1398.1 15 5.04 4.99 10.63 4.44 -2.36 141.64 10024862 1494.0 16 20.95 1.68 13.29 20.94 -2.57 121.87 10157075 369.5 4 26.75 1.93 5.57 25.27 -2.17 114.74 10192453 757.5 8 18.64 5.63 1.73 4.94 -29.2 76.45 a10321319 554.5 6 4.76 4.77 13.56 4.48 -29.18 174.87 10334763 549.7 6 7.23 0.74 10.88 7.23 -3.39 412.26 10397849 506.5 5 19.93 2.35 7.95 19.25 0.15 289.02 10450889 239.6 2 60.03 1.58 2.35 46.44 1.45 192.69 10556420 756.3 8 3.59 0.64 6.03 3.59 -1.17 364.07 10643786 1421.2 15 12.62 3.23 5.12 10.04 3.16 81.51 10668646 196.3 2 42.81 0.93 8.74 42.8 -20.57 1376.58 a10668646 1449.3 15 57.08 1.36 8.64 56.98 -38.4 3632.55 10909733 898.7 9 3.41 1.37 6.17 3.37 4.64 67.65 11561379 137.7 1 13.51 2.11 7.45 13.12 -1.64 103.03 11921843 1234.8 13 3.85 0.62 14.83 3.85 -2.49 287.33 12066509 632.3 7 38.71 0.78 12.41 38.71 -0.31 1883.85 12644038 584.7 6 36.26 1.49 6.37 35.77 3.56 199.23 Notes. Four transits were already identified by Foreman-Mackey et al. (2016) (labeled with "a" in the first column) and Herman et al. (2019) (labeled with "b"). Article number, page 11 of 18 A&A proofs: manuscript no. Article_v1 Appendix A: Performance of the algorithm over all the quarters Table A.1: Performance of the neural network. Quarter Subquarter Accuracy Recall Precision FPR AUC MAE 1 0 0.8945 0.7910 0.9975 0.0020 0.9888 2.3404 2 0 0.8665 0.7347 0.9977 0.0017 0.9907 2.2237 2 1 0.8616 0.7245 0.9982 0.0013 0.9900 2.0792 2 2 0.8743 0.7498 0.9984 0.0012 0.9893 2.0455 3 0 0.9108 0.8240 0.9972 0.0023 0.9896 2.0329 3 1 0.9068 0.8162 0.9969 0.0025 0.9905 2.1186 3 2 0.8903 0.7835 0.9964 0.0028 0.9895 2.1638 4 0 0.8862 0.7745 0.9972 0.0022 0.9909 2.3073 4 1 0.9017 0.8065 0.9961 0.0032 0.9924 2.4661 4 2 0.8854 0.7728 0.9974 0.0020 0.9920 1.9370 5 0 0.8994 0.8013 0.9969 0.0025 0.9919 1.9726 5 1 0.8898 0.7818 0.9972 0.0022 0.9916 2.1857 5 2 0.8951 0.7923 0.9973 0.0022 0.9919 2.5873 6 0 0.8979 0.7972 0.9983 0.0013 0.9906 2.2020 6 1 0.8908 0.7833 0.9979 0.0017 0.9911 1.9720 6 2 0.8860 0.7738 0.9976 0.0018 0.9910 2.4570 7 0 0.8792 0.7605 0.9972 0.0022 0.9899 2.1862 7 1 0.8752 0.7522 0.9978 0.0017 0.9909 1.7915 7 2 0.9128 0.8267 0.9986 0.0012 0.9935 2.1101 8 0 0.9004 0.8018 0.9988 0.0010 0.9911 2.0729 8 1 0.9027 0.8065 0.9986 0.0012 0.9926 2.1139 9 0 0.9068 0.8157 0.9976 0.0020 0.9930 2.1293 9 1 0.9017 0.8062 0.9965 0.0028 0.9916 2.3157 9 2 0.8983 0.7992 0.9969 0.0025 0.9918 2.3348 9 3 0.8798 0.7610 0.9980 0.0015 0.9916 2.2389 10 0 0.8755 0.7532 0.9971 0.0022 0.9896 2.7444 10 1 0.8758 0.7538 0.9971 0.0022 0.9889 2.0156 10 2 0.8829 0.7672 0.9983 0.0013 0.9890 2.7258 11 0 0.8830 0.7685 0.9968 0.0025 0.9908 2.0722 11 1 0.8941 0.7900 0.9977 0.0018 0.9913 1.9576 11 2 0.8907 0.7828 0.9981 0.0015 0.9921 2.4971 12 0 0.8876 0.7767 0.9981 0.0015 0.9921 2.3743 12 1 0.9028 0.8075 0.9977 0.0018 0.9930 2.3531 12 2 0.9038 0.8103 0.9967 0.0027 0.9913 2.4029 13 0 0.8981 0.8000 0.9952 0.0038 0.9913 2.8353 13 1 0.8431 0.6875 0.9981 0.0013 0.9872 2.4796 13 2 0.9037 0.8117 0.9947 0.0043 0.9915 2.5127 14 0 0.8654 0.7342 0.9955 0.0033 0.9877 2.4772 14 1 0.8801 0.7632 0.9961 0.0030 0.9897 2.4096 14 2 0.8788 0.7592 0.9978 0.0017 0.9915 3.1279 15 0 0.8363 0.6743 0.9973 0.0018 0.9869 2.5724 15 1 0.8378 0.6780 0.9966 0.0023 0.9826 3.3750 15 2 0.8490 0.7000 0.9972 0.0020 0.9862 3.1479 16 0 0.8238 0.6483 0.9987 0.0008 0.9870 2.4968 16 1 0.8665 0.7355 0.9966 0.0025 0.9868 2.1849 16 2 0.8445 0.6907 0.9976 0.0017 0.9858 2.4707 17 0 0.8744 0.7510 0.9971 0.0022 0.9899 2.3075 Article number, page 12 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Appendix B: Cometary transits light curves Plots of the light curves of the detected cometary transits in the first tier catalog. For each transit, the light curve is shown in the top panels over a 5 days duration centered on the transit time and with an exocomet transit fit superimposed (red line). The bottom panels show the light curves over a larger time frame. Article number, page 13 of 18 A&A proofs: manuscript no. Article_v1 Article number, page 14 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Article number, page 15 of 18 A&A proofs: manuscript no. Article_v1 Appendix C: Possible exocometary transits light curves Plots of the light curves of the possibly detected cometary transits in the second tier catalog. The plots cover a 5 days time frame centered on the transit time and with an exocomet transit fit superimposed (red line). Article number, page 16 of 18 P. Dumond et al.: Search for exocomets transits in Kepler light curves Article number, page 17 of 18 A&A proofs: manuscript no. Article_v1 Article number, page 18 of 18
|
2510.14684
|
The geometry of magnitude for finite metric spaces
Karel Devriendt
Abstract
The main result of this article is a geometric interpretation of magnitude, a real-valued
invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric
space X such that the magnitude of X can be expressed in terms of the ‘circumradius’ of
its embedding S.
The circumradius is the smallest r for which the r-thickening of S is
contractible. We give three applications: First, we describe the asymptotic behaviour of the
magnitude of tX as t →∞, in terms of the circumradius. Second, we develop a matrix
theory for magnitude that leads to explicit relations between the magnitude of X and the
magnitude of its subspaces. Third, we identify a new regime in the limiting behaviour of tX,
and use this to show submodularity-type results for magnitude as a function on subspaces.
1
Introduction
1.1
Background
Magnitude is a real-valued invariant for enriched categories (in general) and metric spaces (in
particular) that was introduced by Leinster in 2006, in [11].
Before giving a definition, we
mention two slogans that are helpful in guiding one’s intuition on magnitude. The first slogan
says that
“Magnitude is like an Euler characteristic.”
This statement reflects the history of magnitude [11], motivates its definition, and guides the
development of its theory, for instance in the generalization to magnitude homology [9, 16].
In this paper, the connection to Euler characteristics shows itself in our study of valuative
properties, i.e., how magnitude behaves as a function on subspaces Y ⊆X and what kind of
inclusion–exclusion-type relations it satisfies. A second slogan says that
“Magnitude counts the effective number of points.”
Perhaps more appropriate for finite spaces, this intuition explains the success of magnitude in
applications such as quantifying (bio-)diversity [14] and in data analysis [1, 10]. The second
slogan comes forward in this paper in our study of the asymptotic convergence of magnitude
to the point count #X; see also Examples 1.1 and 1.4.
At the time of writing, an online
bibliography [15] maintained by Leinster & Meckes counts more than 120 papers on magnitude.
1.2
Summary of results
In this article, we focus on magnitude for finite metric spaces. This setting was treated early
on in the development of magnitude, for instance by Leinster in [12, §2] and by Meckes in [18].
In particular, both works found that so-called positive definite metric spaces are particularly
well-behaved (see Definition 2.2). Our work builds forth on this analysis by drawing a strong
connection between magnitude of finite positive definite metric spaces and Euclidean geometry,
via matrix theory. We briefly describe our main result and its applications below.
1
arXiv:2510.14684v1 [math.MG] 16 Oct 2025
The magnitude of a finite metric space (X, d) is defined based on the similarity matrix Z with
entries zij = e−d(i,j). When Z is positive definite, we call X positive definite and define:
The magnitude of X = (X, d) is |X| :=
X
i,j∈X
Z−1
ij.
We will come to a more generally applicable definition later. The main new idea in this article is
that there is a Euclidean embedding ϕ : X →RX−1, which we call the similarity embedding, such
that the magnitude of X can be expressed in terms of the circumradius R(S) of its embedding
S = ϕ(X); the circumradius is the smallest r such that the r-thickening S
s∈S Br(s) of S is
contractible. More precisely, we find:
Theorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then
|X| =
1
1 −2R(S)2 .
(1)
In fact, requiring expression (1) for all subspaces Y ⊆X characterizes the embedding S (see
Theorem 2.15). So, in a sense, the embedding ϕ naturally appears in the context of magnitude.
Example 1.1. Let us illustrate Theorem 2.12 by a classical example in magnitude: the metric
space X(2) consisting of two points at distance d > 0. By inverting the positive definite similarity
matrix Z =
1
e−d
e−d
1
and summing the entries of Z−1, one directly computes the magnitude
|X(2)| = 1 + tanh(d/2).
The embedding of X(2) that we introduce gives two points S ⊆R at distance
√
1 −e−d. Since
the circumradius of S is half this distance, Theorem 2.12 gives the relation
1
1 −2R(S)2 =
1
1 −1−exp(−d)
2
=
2
1 + exp(−d) = 1 + tanh(d/2) = |X(2)|.
What lies behind Theorem 2.12? To lift some of the mystery, let us reveal the underlying
linear algebra and construct the embedding ϕ of X explicitly, starting from its matrix Z:
Step 1: Construct the centered matrix K = 1
2(I −11T
n )Z(I −11T
n ), where n = #X.
Step 2: Because Z is positive definite, K admits a square root
√
K.
Step 3: Define the embedding ϕ(i) of i ∈X as the ith column of
√
K.
In other words, K is the Gram matrix of S when the origin is placed at the centroid of S. Based
on this embedding, Theorem 2.12 then follows from the observation that the circumradius of S
is governed by very similar “equilibrium” equations to those of magnitude; see Remark 2.13.
We continue with a brief description of the three main applications of Theorem 2.12.
As a first application, we consider the asymptotic behaviour of magnitude. Much of the
theory of magnitude is concerned with studying the magnitude |tX| across different scales t > 0,
based on the metric transformation d 7→t · d. Leinster & Willerton showed in [17, Thm. 3] that
magnitude asymptotically counts the number of points n in the metric space X, as follows
|tX| = n −q(tX) with q(tX) →0 as t →∞.
We give a description of the asymptotics of the error term q in this formula; let St := ϕ(tX).
Theorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence
n −|tX| = q(tX) ∼n2
n −1
n
−2R(St)2
.
2
Geometrically, Theorem 3.1 states that the magnitude of tX tends to the magnitude of the
discrete metric space in the same way as the circumradius of St tends to the circumradius of the
regular simplex with unit-length edges. Let us return to the example of the two-point metric
space to illustrate the theorem.
Example 1.2. For the two-point metric space X(2), Theorem 3.1 correctly gives
q(tX(2)) = 2 −|tX(2)| =
2
1 + exp(t · d) ∼4(1
2 −1
2 exp(−t · d)).
In other words, we find that |tX(2)| ≈2e−td asymptotically. This is illustrated in Figure 1.
Figure 1: The magnitude |tX(2)| of the metric space with two points at distance d = 1 is well-
approximated by 2e−t at large scales t ≫0. This is explained by Theorem 3.1.
As a second application, we take inspiration from Fiedler’s matrix theory for Euclidean
simplices in [5], and develop a matrix theory for magnitude. Generalizing the relation between
the pseudoinverse Gram matrix and distance matrix of a simplex (see, e.g., [2, §3.2]), we find:
Theorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then
0
1T
1
Z
!−1
=
−|X|−1
wT /|X|
w/|X|
1
2K†
!
.
(3)
In equation (3), the vector 1 = (1, . . . , 1)T is the all-ones vector, w := (Z−1)1 is the weighting
on the metric space X and K† is the Moore–Penrose pseudoinverse of the Gram matrix K of
the centered point set S; more results on the matrices Z, K and K† are given in Section 4.
Example 1.3. For the two-point metric space with distance d = −ln(δ), Theorem 4.11 reads
0
1
1
1
1
δ
1
δ
1
−1
=
1
2
−(1 + δ)
1
1
1
(1 −δ)−1
−(1 −δ)−1
1
−(1 −δ)−1
(1 −δ)−1
.
What is the use of this identity? In this article, it mainly serves as a tool to study the
magnitude of subspaces of X. Since any subspace Y ⊆X of a positive definite metric space is
also positive definite, Theorem 4.11 applies to Y as well. The subspace relation between Y and
X translates nicely to an algebraic relation between their respective matrices, and we arrive at:
Corollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have
|X\{x}| = |X|
1 + 2w2
x
¯cx|X|
!−1
and
w′
i
|X\{x}| = wi
|X| + cix
¯cx
wx
|X|,
for any i ̸= x, and where w′ is the weighting on X\{x}.
3
Here, we use the notation cij = −(K†)ij and ¯ci = P
j cij. This is a corollary of Theorem
4.16, which is an analogous statement for general subspaces. We illustrate the relation between
|X| and |X\{x}| in Corollary 4.18 by looking at the quantity
2w2
x
¯cx|X| that controls this relation.
Example 1.4. We consider the metric space on X = {1, 2, 3} with distances
d(1, 2) = 2
and
d(1, 3) = 100
and
d(2, 3) = 100.
The points 1 and 2 are close together and equidistant from point 3. Since every three-point
metric space is positive definite [12, Prop. 2.4.15], its magnitude exists for all t > 0. The black
lines in Figure 2 plot the magnitude of X: for small t we effectively see one point, for medium
t we see two points, i.e., {1, 2} and 3, and for large t we see three points.
Figure 2: Both panels: The magnitude |tX| (black line) of the three-point metric space in Exam-
ple 1.4; note the logarithmic scale for the horizontal axis. Left panel: The relative contribution
wx/|tX| for points 1 and 2 (blue line) and 3 (red line). Right panel: The change contribution
2w2
x/(¯cx|tX|) for points 1 and 2 (blue line) and 3 (red line).
The left and right panel in Figure 2 compare two possible ways to measure the contribution
of a point x to the magnitude: wx/|X| on the left, and
2w2
x
¯cx|X| on the right, inspired by Corollary
4.18. Overall, the latter appears to be more sensitive to the changing geometry across scales.
As a third application, we initiate the study of a new class of metric spaces; we call these
spaces strongly positive definite and they are defined by the additional positivity conditions
w > 0 and cij = −(K†)ij > 0, for all i ̸= j. We show that this class is closed under taking
subspaces, and that every metric space tX becomes strongly positive definite when t ≫0.
As the main application, we prove two valuative results on the magnitude of a strongly posi-
tive definite metric space X. These results are an attempt to reproduce results for magnitude in
the spirit of the inclusion–exclusion property of the Euler characteristic: for (suitable) subspaces
A, B ⊆X of a topological space X, the Euler characteristic χ satisfies
χ(A) + χ(B) = χ(A ∪B) + χ(A ∩B).
This fact has inspired research into valuative properties of magnitude: Leinster showed an
inclusion–exclusion result for magnitude of graphs in [13], Leinster and Willerton showed in [17]
that the magnitude of subsets of Rn satisfies inclusion–exclusion asymptotically, and Gimperlein,
Goffeng and Louca further improved on these geometric results in [6]. Here, we consider a variant
of inclusion–exclusion; a function f : 2X →R is called strictly submodular if1
f(A) + f(B) > f(A ∪B) + f(A ∩B),
for all A, B ⊆X, and f is called increasing (resp. decreasing) if it is increasing (resp. decreasing)
with respect to the subset partial order. We show the following two results:
1Later in this article we will work with a different, but equivalent definition (4).
4
Theorem 5.9. Let X be strongly positive definite. Then the function
f : Y 7−→
(
−|Y |−1, if Y ̸= ∅
α
, if Y = ∅
is increasing if α < −1 and strictly submodular if α < −3
2. In particular, f is strictly submodular
on the subspaces of tX′, for any X′ and t ≫0 and α < −3
2.
Theorem 5.10. Let X be any metric space and t ≫0. Then the function
f : Y 7−→
( m−|tY |
m2
+ m−1
m
if m := #Y ̸= 0
α
if Y = ∅
is increasing if α < 1
2 and strictly submodular if α < −1
2.
Organization: Section 2 starts with the definition of magnitude, positive definite metric spaces,
and the similarity embedding ϕ; this leads to the proof of Theorem 2.12, which relates the
magnitude |X| and circumradius R(S). Section 3 is rather short and deals with the asymptotics
of q(tX), in Theorem 3.1. In Section 4, we develop the matrix theory of the matrices Z, K and
K†: we study their relations, their connections to magnitude and weightings and prove Theorem
4.11. This is then used to derive Theorem 4.16 and its corollaries, which describe the magnitude
and weighting of subspaces Y ⊆X in terms of the data of X. Finally, Section 5 deals with
the new class of strongly positive definite metric spaces. We define this class, study some of
its properties and show that it describes the asymptotics of tX for general X. These result are
then used to prove the submodularity results, Theorem 5.9 and 5.10.
2
Magnitude and circumradius
2.1
Definitions: magnitude, weighting & positive definite
We start with the definition of magnitude and positive definite metric spaces; recall that the
similarity matrix Z = Z(X) of a finite metric space (X, d) has entries defined as zij = e−d(i,j).
Definition 2.1 (Weighting, magnitude). Let X be a finite metric space.
• A vector w ∈RX that satisfies Zw = 1 is called a weighting on X.
• The magnitude of X is the sum |X| = 1T w, for any weighting w on X.
If the system has no solutions, we say that the magnitude and weighting do not exist.
By
definition, we set the magnitude of the empty metric space X = ∅to zero. We use n or #X to
denote the cardinality of X, and we assume that 0 < #X < ∞unless explicitly stated otherwise.
Definition 2.2. A metric space X is positive definite if Z(X) is positive definite.
The magnitude of a positive definite metric space always exists. In fact, since the similarity
matrix Z is invertible in this case, there is a unique weighting w = Z−11 on X, and the
magnitude of a positive definite metric space X has the explicit form:
|X| =
X
i,j∈X
(Z−1)ij = 1T Z−11.
Positive definite metric spaces play an important role in the theory of magnitude. Not only
is much of the theory of magnitude easier to develop for positive definite metric spaces, as is
the case in this article, but the following result due to Leinster states that any metric space
eventually becomes positive definite when scaling its metric d 7→t · d; we write tX = (X, t · d).
5
Proposition 2.3 ([12, Prop. 2.4.6]). For any metric space X and t ≫0, tX is positive definite.
The following closure result will be useful later on.
Proposition 2.4. If X is positive definite then any subspace Y ⊆X is positive definite.
Proof. Any principal submatrix Z(Y ) of a positive definite matrix Z(X) is positive definite.
To conclude, we give an important class of examples of positive definite metric spaces.
Example 2.5 (Negative type metric). A metric space (X, d) has negative type if (X,
√
d) admits
an isometric Euclidean embedding. As shown by Meckes in [18], a metric space X has negative
type if and only if tX is positive definite for all t > 0; this is also called stably positive definite.
There are many natural examples of negative type metric spaces, for instance point sets in
Euclidean, spherical or hyperbolic space. See [18, Thm. 3.6] for a list of examples.
2.2
Similarity embedding of positive definite metric spaces
In this section, we introduce an embedding of a positive definite metric space and show its
relation to magnitude. This embedding makes critical use of positive definiteness.
Definition 2.6 (Similarity embedding). A similarity embedding of a positive definite metric
space X is an embedding ϕ : X →RX−1 that satisfies
∥ϕ(i) −ϕ(j)∥2 = 1 −e−d(i,j) = 1 −zij
for all i, j ∈X.
Recall the construction from the introduction, which proves existence of these embeddings
Proposition 2.7. The similarity embedding exists and is unique up to rigid transformations.
Proof. Let X be a positive definite metric space and construct the centered similarity matrix
K := 1
2
I −11T
n
Z
I −11T
n
,
where n = #X is the cardinality of the metric space and I is the identity matrix.
Since
Z is positive definite, the matrix K is positive semidefinite with ker(K) = span(1) (see also
Proposition 4.3) and thus there exists an (n−1)×n square root
√
K, i.e., such that
√
K
T √
K =
K. The embedding ϕ : X ∋i 7→(ith column of
√
K) ∈Rn−1 is a similarity embedding, since
∥ϕ(i) −ϕ(j)∥2 = 1
2(ei −ej)T √
K
T √
K(ei −ej) = 1
2(zii + zjj −2zij) = 1 −e−d(i,j),
where ei is the ith basis vector. Since the embedding ϕ is defined via all pairwise distances
between the points ϕ(X), it is unique up to rigid transformations, i.e., isometries of Rn−1.
Remark 2.8 (Computation). The similarity embedding ϕ can be computed efficiently using
standard linear algebra methods. For instance, one can compute
√
K via the Cholesky decom-
position or an eigenvalue decomposition, for which highly optimized algorithms exist.
Proposition 2.7 associates to each positive definite metric space X a set of points S := ϕ(X)
in RX−1. As introduced, we will show that a certain geometric invariant of this point set is
related to magnitude. We first establish another important fact about the geometry of S.
Proposition 2.9. Let X be positive definite. Then S = ϕ(X) are the vertices of a simplex.
Proof. This is a consequence of positive definiteness: since Z has full rank, the matrix K has
rank (n −1) and its square root, the (n −1) × n matrix
√
K, has full rank n −1. It follows that
the columns of
√
K and thus the points S are affinely independent.
6
2.3
Magnitude and circumradius
We now come to the definition of circumradius. This is the geometric invariant of the point set
S that is closely related to magnitude; we note that, by construction, the matrix (11T −Z) is
the matrix of squared distances between pairs of points in S, and we may thus write Z(S).
Theorem 2.10. The following numbers are equal for the vertices S of a simplex:
1. The radius r of the unique sphere in RS−1 that goes through S;
2. The smallest r such that the r-thickening of S is contractible;
3. The unique value r for which (11T −Z(S))x = 2r2 · 1 with xT 1 = 1 has a solution.
This common value is called the circumradius of S and is denoted by R(S).
Proof.
From Euclidean geometry (or algebraic geometry), we know that there is a unique
sphere in RS−1 that goes through the vertices S of a simplex. This is called the circumsphere
of the simplex, and it is the smallest sphere that encloses the points S.
(1. ⇔2.) The r-thickening of S is contractible if and only if S is contained in some radius-r
sphere. Since the circumsphere of a simplex is the smallest sphere that contains S, its radius
R(S) is the smallest r for which the r-thickening of S is contractible, as required.
(1. ⇔3.) The circumsphere has center c (= the circumcenter), which satisfies the equations:
∥c −s∥2 = r2
for each s ∈S,
where r is the radius of the circumsphere. If we write p for the barycentric coordinate of the
circumcenter, i.e., the unique vector such that 1T p = 1 and
√
Kp = c, and use the fact that for
each i ∈X, the vector ei is the barycentric coordinate of the ith point of S, then we can rewrite
(p −ei)T K(p −ei) = r2
for each i ∈X
⇐⇒1
2(p −ei)T Z(p −ei) = r2
for each i ∈X
(since 1T (p −ei) = 0)
⇐⇒eT
i Zp = −r2 + 1
2pT Zp + 1
2
for each i ∈X
⇐⇒Zp = (1
2 −r2 + 1
2pT Zp) · 1.
⇐⇒Zp = (1 −2r2) · 1
(since 1T p = 1)
⇐⇒(11T −Z)p = 2r2 · 1.
Thus the radius r of the circumsphere satisfies the required equation (3.). Uniqueness of r in
(3.) follows from uniqueness of the sphere that contains all vertices S of a simplex, or it can be
derived from invertibility of Z as implied by positive definiteness.
Remark 2.11 (Scaled embeddings). Any rescaling α · ϕ for α > 0 of the similarity embedding
still embeds the metric space X as the vertices of a simplex, with circumradius αR(S). Different
choices than α = 1 may be useful in different applications, but some of the nice theory may be
lost, for instance, when choosing a scaling that depends on X. For instance, for α2 = |X| or
α2 = #X, the subspace characterization in Theorem 2.15 below no longer holds. In a sense,
X-dependent scalings are not functorial in the same way as independent scalings are.
We now arrive at the first main result from the introduction, and its proof.
Theorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then
|X| =
1
1 −2R(S)2 .
(1)
7
Proof. This follows immediately from the third expression for the circumradius in Theorem
2.10. From Zx = (1−2R(S)2)1 with xT 1 = 1, we find that w = x/(1−2R(S)2) is the weighting
on X. Since x is normalized, the claimed expression for the magnitude |X| follows.
Remark 2.13 (Equilibrium equations). With the proofs of Theorem 2.10 and Theorem 2.12
in place, we can now appreciate where the relation between magnitude and circumradius comes
from: both quantities are defined via an ‘equilibrium equation’. In the case of magnitude, the
equilibrium equation Zw = 1 is part of the definition which Leinster arrived at by generalizing
the Euler characteristic. In the geometric setting, the equilibrium equation (11T −Z)p = 2r21
encodes the fact that the circumcenter is equidistant to all vertices of a simplex.
Corollary 2.14. Let X be a positive definite metric space. Then |X| > 1.
Proof. This follows from |X| = 1 + 2|X|R2(S) with |X| > 0 and R(S) > 0.
We can in fact turn Theorem 2.12 around and use it to characterize the similarity embedding.
Theorem 2.15. Let X be a positive definite metric space. The similarity embedding ϕ is the
unique embedding of X into RX−1, up to rigid transformation, such that
|Y | =
1
1 −2R(ϕ(Y ))2 for all nonempty Y ⊆X.
Proof. We recall from Example 1.1 that the magnitude of a two-point metric space is 2/(1+e−d).
Furthermore, the circumradius of two points at Euclidean distance d′ is d′/2.
For every two-point subspace Yij = {i, j} ⊆X, the requirement on the embedding ϕ gives
2
1 + e−d(i,j) = |Yij| =
1
1 −2R(ϕ(Yi,j))2 =
2
2 −∥ϕ(i) −ϕ(j)∥2 ⇐⇒∥ϕ(i) −ϕ(j)∥2 = 1−e−d(i,j).
The latter equation shows that any embedding with the prescribed conditions must be the
similarity embedding, which is unique. It remains to prove existence, i.e., that the similarity
embedding ϕ satisfies the stated relation between |Y | and R(ϕ(Y )) for subspaces Y of any
cardinality. This follows because the restriction ϕ|Y of the similarity embedding of X onto any
subspace Y is the similarity embedding of Y , and R(ϕ(Y )) = R(ϕ|Y (Y )). The expression for
|Y | in terms of R(ϕ(Y )) is then simply Theorem 2.12.
Another way to think about Theorem 2.12 is that it expresses functoriality of the similarity
embedding with respect to the subspace relation. This is also reflected in the matrix identity
(3) and its implication, e.g., Theorem 4.16.
To end this section, we compute the similarity embedding, magnitude and circumradius of
a three-point metric space in a numerical example.
Example 2.16. Consider the three-point metric space X = {1, 2, 3} with metric d(1, 2) =
ln(2) ≈0.69 and d(1, 3) = d(2, 3) = ln(10) ≈4.61. Its similarity matrix, centered similarity
matrix and a similarity embedding are:
Z =
1
0.5
0.1
0.5
1
0.1
0.1
0.1
1
K = 1
9
1.9
−0.35
−1.55
−0.35
1.9
−1.55
−1.55
−1.55
3.1
√
K =
ϕ(1) ϕ(2) ϕ(3)
=
√
2/4
−
√
2/4
0
−0.2934
−0.2934
0.5869
!
3
Magnitude asymptotics
As explained in the introduction, much of the theory of magnitude is concerned with studying
how the magnitude |tX| behaves as a function of the scale t > 0. Our goal here is to refine the
following asymptotic formula, due to Leinster and Willerton in [17]:
|tX| = n −q(tX) with q(tX) →0 as t →∞,
8
where n is the cardinality of the metric space. As a direct application of Theorem 2.12, we give
an asymptotic expression for the remainder q(tX) in this expression; we write St := ϕ(tX).
Theorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence
n −|tX| = q(tX) ∼n2
n −1
n
−2R(St)2
.
Proof. As t →∞, tX is eventually positive definite (Proposition 2.3) and its similarity matrix
Z(tX) converges to the identity matrix I. As a result, the simplex with vertices St converges to
the regular simplex with all edge lengths equal to p1 −zij = 1. From the equilibrium equation,
one computes that R(St)2 converges to n−1
2n , the squared circumradius of the regular simplex.
Next, based on Theorem 2.12, we can write the remainder q(tX) = n −|tX| as follows:
|tX| =
1
1 −2R(St)2 =
n
1 + n
n−1
n
−2R(St)2 ⇒q(tX) = n
1 −
1
1 + n
n−1
n
−2R(St)2
!
Using the asymptotic equivalence 1 −
1
1+x ∼x for x →0 and the earlier result that
n−1
n
−
2R(St)2 →0 as t →0, we then find the claimed asymptotic equivalence.
Theorem 3.1 states that the magnitude of a metric space tX converges to the magnitude of
the discrete metric space (X, ∞) in the same way as the circumradius of the simplex St converges
to the circumradius of the regular simplex. We continue with magnitude asymptotics in Section
5, where we will further refine our understanding of the asymptotics by looking more closely at
some implications of the convergence of St to the regular simplex.
4
Matrix theory & magnitude of subspaces
In this section, we look more closely at the matrices Z and K that appeared in Section 2, and
introduce and study the matrix K†. This section is partly an intermezzo, where we develop the
matrix theory of magnitude for its own sake, and partly a setup for Section 5.
4.1
Matrix theory of Z, K and K†
The Moore–Penrose pseudoinverse of a matrix M is the unique matrix M† that satisfies
MM†M = M
and
M†MM† = M†
and
MM†, M†M are symmetric.
We point out two consequences of this definition: (i) if M is real and symmetric, then so is M†
and (ii) the matrices MM† and M†M are orthogonal projection matrices onto ker(M) = ker(M†)
(see, e.g., [2, §A.3]). We recall the definition of matrices K and K†:
Definition 4.1. Let X be a finite metric space with similarity matrix Z. Then
• The centered similarity matrix is K := 1
2
I −11T
n
Z
I −11T
n
.
• The pseudoinverse centered similarity matrix is K†.
By their construction, we immediately find the following properties of K and K†
Proposition 4.2. K and K† are real, symmetric matrices with ker(K) = ker(K†) ⊇span(1).
Proposition 4.3 (Interlacing). Let λ1 ≥λ2 ≥· · · ≥λn be the real eigenvalues of Z, and
µ1 ≥µ2 ≥· · · ≥µn−1 the real eigenvalues of K, with one zero eigenvalue omitted2. Then
λ1 ≥2µ1 ≥λ2 ≥2µ2 ≥· · · ≥λn−1 ≥2µn−1 ≥λn,
with equality throughout if Z1 = |X|−11. We have rank(K) = rank(K†) = rank(Z) −1 and if Z
is positive definite, then K and K† are positive semidefinite with ker(K) = ker(K†) = span(1).
2By construction we know that K has at least one eigenvalue. The eigenvalues µ1, . . . , µn−1 are all other
eigenvalues except for this one zero eigenvalue; there may be other zero eigenvalues.
9
Proof. The eigenvalue inequalities are precisely the interlacing theorem for matrix projections
[7, Thm. 2.1] applied to the matrices 2K and Z. If Z1 = |X|−1 · 1 then |X|−1 is an eigenvalue
of Z with (normalized) eigenvector 1/√n; let’s say that it is the jth eigenvalue.
Then Z
has an eigendecomposition Z = P
i̸=j λixxT + |X|−1 11T
n
and thus by its construction K has
eigendecomposition K = P
i̸=j
1
2λixxT , which proves that 2µi = λi for all i ̸= j.
The implication (Z positive definite) =⇒(K and K† positive semidefinite) follows from
the interlacing inequalities.
The rank statement holds because 2K arises from Z through a
projection onto span(1)⊥, where furthermore ker(Z) ̸⊇span(1) since Z is a nonnegative matrix
with unit-diagonal. By the rank statement, K and K† have rank n −1 when Z is positive
definite, and thus ker(K) = ker(K†) = span(1). This completes the proof.
One example where Z1 = |X|−11 holds is when the isometry group of X acts transitively on
its points; these are called homogeneous spaces. We continue by showing some relations between
Z, K and K†. We start with a useful lemma:
Lemma 4.4. Let Z be invertible with nonzero magnitude. Then
ZK† = 2I −21wT
|X| .
(2)
Proof. We start with the definition of K in terms of Z and use KK† =
I −11T
n
:
K = 1
2
I −11T
n
Z
I −11T
n
⇒KK† = 1
2
I −11T
n
ZK† ⇔
I −11T
n
= 1
2
I −11T
n
ZK†.
The third equation implies that ZK† must be of the form ZK† = 2I −21yT for some y ∈Rn.
Using the additional fact that wT ZK† = 0, we find that y = w/|X| as required.
We now show some relations between Z, K and Z−1, K†.
Proposition 4.5. The similarity and centered similarity matrices satisfy:
2K −Z = γ1T + 1γT ,
where γ := −1
nZ1 +
1
2n21T Z1 · 1 = diag(K) −1
21.
Proof. We note that Z may be singular. Starting from the definition of K, we find
2K −Z = −1
nZ11T −1
n11T Z + 1
n2 11T Z11T ,
which proves the expression for 2K−Z. The second expression of γ follows from diag(Z) = 1.
Proposition 4.6. Let Z be invertible with nonzero magnitude. Then
2Z−1 −K† = 2wwT
|X| ,
where
w
|X| = 1
2K†γ + 1
n1;
|X|−1 = −1
2γT K†γ −2
n1T γ.
Proof. Right-multiplying equation (2) by Z−1 proves the expression for 2Z−1 −K†. We now
prove the expression for w/|X| in terms of K† and γ.
K† = 2Z−1 −2wwT
|X| ⇒K†γ =
2Z−1 −2wwT
|X|
−1
nZ1 +
1
2n21T Z1 · 1
= −2
n1 +
2
|X|w ⇒1
2K†γ = −1
n1 + w
|X|.
Finally, we prove the expression for |X|−1 in terms of K† and γ.
1
2K†γ + 1
n1 =
w
|X| ⇒1
2γT K†γ + 1
nγT 1 = wT γ
|X|
= wT
|X|
−1
nZ1 +
1
2n2 1T Z1 · 1
= −1
|X| +
1
2n2 1T Z1
= −1
|X| −1
nγT 1,
10
which leads to the claimed expression for |X|−1 and completes the proof.
We focus on one further ‘object’ that turns out to play an important role in the matrix theory
and the asymptotics of magnitude: the off-diagonal entries of matrix K†. Later, in Section 5,
we will show that large scales t ≫0 are marked by negativity of these entries, which in turn
leads to additional algebraic and geometric properties. We introduce the notation
cij := −(K†)ij = 2wiwj
|X| −2(Z−1)ij, for i ̸= j ∈X
and
¯ci := (K†)ii =
X
j∈X
cij.
The relation between ¯ci and P
j cij follows because the rows of K† sum to zero. As a notational
convention, when summing over j ∈X or i, j ∈X with cij in the sum, we consider each pair
only once (i.e., with (i, j) = (j, i)); think of c as a function on unordered pairs of distinct points.
We give expressions for magnitude and weightings in terms of c.
Proposition 4.7. Let Z be invertible with nonzero magnitude. Then
wi
|X| = 1 −1
2
X
j∈X
cij(1 −zij)
and
|X|−1 = 1 −
X
i,j∈X
cij(zix −zjx)2 for any x ∈X.
Proof. The expression for wi follows by taking the ith diagonal entry of equation (2). For the
expression of |X|−1 we start by the observation that, due to ker(K†) = span(1), we can write
K† =
X
i,j∈X
cij(ei −ej)(ei −ej)T ⇒1
2yT K†y =
X
i,j∈X
cij(yi −yj)2 for any y ∈RX.
Right-multiplying equation (2) by Z and looking at the xth diagonal entry, we then find
(ZK†Z)xx = 2 −2|X|−1 ⇔|X|−1 = 1 −
X
i,j∈X
cij(zix −zjx)2.
We note that a factor of 1
2 is incorporated in the sum because of our notational convention.
Corollary 4.8. Let Z be invertible with nonzero magnitude. Then P
i,j∈X cij(1 −zij) = n −1.
Proof. This follows by summing wi/|X| in Proposition 4.7, and 1T w = |X|.
Remark 4.9 (Foster’s theorem). The direct analogue of this corollary is called Foster’s Theorem
in other settings. One hint at the depth of this theorem is that the integer on the righthandside
is known to count, depending on the situation, a dimension, a degree or an Euler characteristic.
Corollary 4.10. Let Z be invertible with nonzero magnitude. Then
|X|−1 =
1
n tr
I −1
2ZK†Z
.
Proof. This follows by summing the expression for |X|−1 in Proposition 4.7 over all x ∈X.
To end the section, we prove a matrix identity that relates the similarity and pseudoinverse
centered similarity matrices of a metric space. This identity summarizes several of the expres-
sions above, and it will be the basis to derive explicit relations between different subspaces.
Theorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then
0
1T
1
Z
!−1
=
−|X|−1
wT /|X|
w/|X|
1
2K†
!
.
(3)
11
Proof. We verify that multiplying the two sides of equation (3) gives the identity matrix:
1Tw/|X|
1
21T K†
|X|−1 · (Zw −1)
1wT/|X| + 1
2ZK†
!
=
1
0
0
I
!
.
The top-left equation holds by definition of magnitude. The top-right equation holds because
ker(K†) = span(1).
The bottom-left equation holds because Zw = 1.
The bottom-right
equation holds by equation (2).
Remark 4.12 (Cayley–Menger matrix). Theorem 4.11 mimics a matrix identity in simplex
geometry and algebraic graph theory; we called this the Fiedler–Bapat identity in [2, §3]. In
that context, the matrix
0 1T
1 D
is called the Cayley–Menger matrix, where D contains the
squared pairwise distances between a set of points. This matrix is an important algebraic object
in geometry; for instance, its determinant encodes affine (in-)dependence of the set of points.
We note the following rational expression for |X| in variables (zij) that follows from (3):
Proposition 4.13. Let X be a positive definite metric space. Then
|X| = −det
0
1T
1
Z
!
/ det(Z).
4.2
Magnitude and weightings for subspaces
We now apply Theorem 4.11 to study the magnitude of subspaces of a metric space; we start
by fixing some notation. We write Y ⊆X to denote the metric subspace (Y, d|Y ) ⊆(X, d), and
write d for the metric d|Y ; unless stated otherwise, we assume Y nonempty. We write Y c = X\Y
and use the following subscript notation for submatrices:
Z =
ZY Y
ZY Y c
ZY cY
ZY cY c
!
and
w =
wY
wY c
!
.
Since the metric on Y is simply the restriction of d, it follows that the similarity matrix Z(Y )
is the principal submatrix ZY Y of Z(X). We recall the definition of Schur complements:
Definition 4.14 (Schur complement). Let M =
MY Y
MY Y c
MY cY MY cY c
be a matrix such that MY cY c
is invertible. The Schur complement of M with respect to Y c is the matrix3 M/Y c, with rows
and columns indexed by Y , defined by
M/Y c := MY Y −MY Y c(MY cY c)−1MY cY .
Proposition 4.15 ([8]). Let M be a matrix such that M/Y c is well-defined. Then
• If M is invertible, then (M−1)Y Y = (M/Y c)−1.
• M/Y c = ((M/x1)/ . . . )/xℓwith entries Y c = {x1, . . . , xℓ} in any order.
We now arrive at the main result of this section.
Theorem 4.16. Let X be a positive definite metric space. Then for any Y ⊆X we have
|Y | = |X|
1 + 2wT
Y c(K†
Y cY c)−1wYc
|X|
!−1
and
w′
|Y | = wY
|X| −K†
Y Y c(K†
Y cY c)−1 wY c
|X| ,
where w′ is the weighting on Y . Furthermore, K†(Y ) = K†(X)/Y c.
3This notation is not standard; we use it to keep the focus on the underlying metric space X and its subspaces.
12
Proof. Starting from Theorem 4.11 for X, combined with the expression of inverse matrix
submatrices in terms of Schur complements (Proposition 4.15), we find
0
1T
1
ZY Y
!−1
=
−|X|−1
wT
Y /|X|
wT
Y c/|X|
wY /|X|
1
2K†
Y Y
1
2K†
Y Y c
wY c/|X|
1
2K†
Y cY
1
2K†
Y cY c
.
Y c.
=
−|X|−1
wT
Y /|X|
wY /|X|
1
2K†
Y Y
!
−2
wT
Y c/|X|
1
2K†
Y Y c
!
(K†
Y cY c)−1
wY c/|X|
1
2K†
Y cY
.
Next, Theorem 4.11 for Y states that
0
1T
1
Z(Y )
!−1
=
−|Y |−1
w′T /|Y |
w′/|Y |
1
2K(Y )†
!
.
The theorem then follows by equating these two expressions, using the equality Z(Y ) = ZY Y .
Corollary 4.17. Let X be positive definite. Then |Y | ≤|X| for all Y ⊂X, with equality if and
only the weighting w on X satisfies wy = 0 for all y ∈Y .
Proof. We first establish that the matrix K†
Y cY c, and thus its inverse, is positive definite:
xT K†
Y cY cx =
x
0Y
!T
I −11T
n
!
K†
I −11T
n
!
x
0Y
!
> 0 for all nonzero x ∈RY c,
which follows since K† is positive definite on span(1)⊥. Thus K†
Y cY c is positive definite. The
corollary then follows from the expression for |Y | in Theorem 4.16, and |X| > 0.
The special case of the subspace Y = X\{x} for some x ∈X is particularly useful:
Corollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have
|X\{x}| = |X|
1 + 2w2
x
¯cx|X|
!−1
and
w′
i
|X\{x}| = wi
|X| + cix
¯cx
wx
|X|,
for any i ̸= x, and where w′ is the weighting on X\{x}.
For the entries c of the pseudoinverse centered similarity matrix K†, we find:
Corollary 4.19. Let X be a positive definite metric space. Then for any x ∈X we have
c′
ij = cij + cixcjx
¯cx
and
¯c′
i = ¯ci −c2
ix
¯cx
,
for all i, j ̸= x and where c′ is defined on X\{x}, i.e., via the entries of K(X\{x})†.
5
Strongly positive definite metric spaces
In this section, we return to the asymptotics of |tX|. Our main contribution is to identify a new
property of metric spaces, which we call “strongly positive definite”, that is satisfied by every
metric space tX for t ≫0. This new regime refines our understanding of the asymptotics of tX.
13
5.1
Definition and first properties
We start by defining the new class of metric spaces; recall that cij = −(K†)ij = 2wiwj
|X| −2(Z−1)ij.
Definition 5.1 (Strongly positive definite). A finite metric space X is called strongly positive
definite if it is positive definite with c > 0 and w > 0.
Question 5.2. What is the right generalization of Definition 5.1 when X is not finite?
Strongly positive definite metric spaces are related to other known objects and properties.
We summarize some of these equivalences in the two propositions below.
Proposition 5.3. The following are equivalent for a positive definite metric space X:
1. The weighting is nonnegative w ≥0 (resp. positive w > 0);
2. S(X) are the vertices of a simplex that contains (resp. strictly contains) its circumcenter.
Proof. In the proof of Theorem 2.12, we found that the normalized weighting w/|X| is the
barycentric coordinate of the circumcenter of S(X). The circumcenter lies in the simplex (resp.
in its interior) if and only if these coordinates are nonnegative, w ≥0 (resp. positive, w > 0).
In the following, the terms “M-matrix, Laplacian and nonobtuse” are defined in the proof.
Proposition 5.4. The following are equivalent for a positive definite metric space X.
1. K† has nonpositive (resp. negative) off-diagonal entries, equivalently, c ≥0 (resp. c > 0).
2. K† is an M-matrix (resp. strict M-matrix);
3. K† is the Laplacian matrix of a connected, positively weighted graph (resp. complete graph).
4. S(X) are the vertices of a nonobtuse (resp. acute) simplex.
Proof. (1.⇔2.) An M-matrix is a matrix whose eigenvalues have nonnegative real part and
whose off-diagonal entries are non-positive. Since K† is positive semidefinite when X is positive
definite, K† is indeed an M-matrix if and only if its off-diagonal entries are non-positive.
(1.⇔3.) The Laplacian matrix of a graph G = (V, E) with edge weights ω : E →R>0 is the
matrix L = P
ij∈E ω(ij)(ei −ej)(ei −ej)T . We have established before that K† can be written
in this form, with ω(ij) = cij, as a consequence of span(1) ⊆ker(K†). Thus, if c > 0 then the
matrix K† is a Laplacian matrix. A graph is connected if and only if ker(L) = span(1) (see,
e.g., [5, §6]), which completes the equivalence.
(3.⇔4.) A simplex is nonobtuse if every angle between facets is at most π/2 (i.e., nonobtuse).
The equivalence between Laplacian matrices of connected graphs L and nonobtuse simplices is
due to Fiedler [5, §6.2]; the equivalence is established via (Moore–Penrose pseudoinverse of
Laplacian matrix)←→(centered Gram matrix of simplex).
If c > 0 in (1.) this corresponds to G being the complete graph in (3.) and the simplex
having angles smaller than π/2 (i.e., acute) in (4.). The terminology “strict M-matrix” in (2.)
is not standard, but we use it here for an M-matrix with negative off-diagonal entries.
Remark 5.5 (Graphs & discrete curvature). We highlight in particular the connection with
graph Laplacian matrices. Many of the techniques in this article are inspired by the author’s
previous work in that context; see for instance [2, §2–4,§6]. Relatedly, we point out that similar
objects to w have recently been defined on graphs in the context of discrete curvature [3, 4, 19];
there, w > 0 has the natural interpretation of a positive curvature condition.
The following result shows that strongly positive definite metric spaces can be parametrized
(via Z−1) by an open semi-algebraic subset of GLn(R), the space of real, invertible n×n matrices.
14
Proposition 5.6. A metric space X is strongly positive definite if and only if its similarity
matrix Z is invertible with positive magnitude, and satisfies
(Z−11)i(Z−11)j > (1T Z−11) · (Z−1)ij
and
(Z−11)i(Z−11)j > 0, for all distinct i, j ∈X.
Proof. This follows from the identities K† = 2Z−1 −2wwT /|X| and w = Z−11. Introducing
K† and w in the inequalities above implies that c > 0 is equivalent to the first inequalities,
whereas the second inequalities are equivalent to w > 0. For wi · wj > 0 to imply that w > 0,
we use that w cannot be all negative because |X| = 1T w is positive.
As a setup for the next subsection, we establish two further results:
Proposition 5.7. For any metric space X and t ≫0, tX is strongly positive definite.
Proof. The result for w > 0 was shown by Leinster in [12, Prop. 2.2.6]. We use the same proof
strategy to show c > 0. First, let t be sufficiently large such that tX is positive definite, and
represent a metric space by its invertible similarity matrix Z ∈GLn(R) ⊂Rn×n; here n = #X.
On the subset of invertible matrices GLn(R), the following function is continuous
GLn(R) ∋Z 7→cij(Z) = −2 · adjij
0
1T
1
Z
!
/ det
0
1T
1
Z
!
for any i ̸= j ∈X, where adjij denotes the (i, j)th entry of the adjugate matrix. By continuity
and cij(I) > 0 for the identity matrix I, there is a neighbourhood U ⊂GLn(R) of the identity
matrix such that cij(Z) > 0 for each Z ∈U.
Since Z(tX) →I as t →∞, we know that
Z(tX) ∈U and thus that cij(Z(tX)) > 0 for t ≫0. This completes the proof.
Proposition 5.8. Strong positive definiteness and c > 0 are preserved under taking subspaces.
Proof. This follows from Corollaries 4.18 and 4.19, which express c′ and w′ for a subspace
Y = X \ {x} in terms of c and w of the metric space X, or by repeated application of this
argument following the composition property of Schur complements (Proposition 4.15).
5.2
Magnitude submodularity
This section deals with valuative properties of magnitude. We show that two magnitude-based
set functions on the subspaces of a strongly positive definite metric space X are submodular;
recall that a function f on the subsets of X is called strictly submodular if it satisfies4
f(Y ) −f(Y \{y}) < f(Y \{x}) −f(Y \{x, y}), for all Y ⊆X and distinct x, y ∈Y ,
(4)
and f is called increasing (resp. decreasing) if it is increasing (resp. decreasing) with respect to
the subset partial order on 2X. We start with the inverse of magnitude as a set function:
Theorem 5.9. Let X be strongly positive definite. Then the function
f : Y 7−→
(
−|Y |−1, if Y ̸= ∅
α
, if Y = ∅
is increasing if α < −1 and strictly submodular if α < −3
2. In particular, f is strictly submodular
on the subspaces of tX′, for any X′ and t ≫0 and α < −3
2.
4As noted, this is a different but equivalent definition from the one given in the introduction.
15
Proof. Because strong positive semidefiniteness is conserved for subspaces (Proposition 5.8), we
may assume without loss of generality that Y = X when checking the submodularity inequalities
(4). For #X = 1 the statement is true since |X| = 1 and (4) holds vacuously; we distinguish
the two remaining cases, #X = 2 and #X > 2, and start with the latter.
(#X > 2): We can rewrite the submodularity inequalities for f as
−|X|−1 + |X\{x}|−1 < −|X\{y}|−1 + |X\{x, y}|−1
⇐⇒
2w2
x
¯cx · |X|2 <
2w′2
x
¯c′x · |X\{y}|2
(Corollary 4.18),
where c, w are defined on X and c′, w′ on X\{y}. For strongly positive definite X, Corollaries
4.18 and 4.19 imply ¯c′
x < ¯cx and w′
x/|X\{y}| > wx/|X|, which confirms the inequality above.
(#X = 2): We can rewrite the submodularity inequality for f as
−|X|−1 + |X\{x}|−1 < −|X\{y}|−1 −α
⇐⇒1 −1 + exp(−d(x, y))
2
< −1 −α
(Example 1.1)
⇐⇒α < −3
2
This confirms strict submodularity for #X = 2 if α < −2/3, and thus completes the proof for
strongly positive definite X. By Corollary 4.18, we know that f is increasing for nonempty
subsets and that f(Y ) ≥−1 with equality for #Y = 1; it follows that f is increasing on 2X if
α < −1. The asymptotic result follows from Proposition 5.7.
Equivalently, one can say that −f in Theorem 5.9 is strictly supermodular and decreasing,
for appropriate α. To end this section, we prove a second submodularity result. The function
considered in the theorem below is inspired by the asymptotic expression in Theorem 3.1 for
q(tX) = #X −|X| in terms of R(St)2.
Theorem 5.10. Let X be any metric space and t ≫0. Then the function
f : Y 7−→
( m−|tY |
m2
+ m−1
m
if m := #Y ̸= 0
α
if Y = ∅
is increasing if α < 1
2 and strictly submodular if α < −1
2.
Proof. We follow the strategy from the proof of Proposition 5.7 and represent a metric space
X by its invertible similarity matrix Z ∈GLn(R). We consider three families of functions: the
family FY,x,y, labeled by x ̸= y ∈Y ⊆X, the family Gx,y labeled by x ̸= y ∈X, and the family
HY ′,Y labeled by Y ′ ⊂Y ⊆X; these are defined as
FY,x,y = f(Y ) −f(Y \{x}) −f(Y \{y}) + f(Y \{x, y})
Gx,y = f({x, y}) −f({x}) −f({y}) + f(∅)
HY ′,Y = f(Y ′) −f(Y )
Since |Y | = −det
0
1T
1 ZY Y
/ det(Z) by Proposition 4.13, these are continuous functions on
GLn(R). Next, we confirm that each of these functions evaluates negatively for the discrete
metric space; in that case, Z = I and thus |Y | = m and f(Y ) = (m −1)/m. We find
FY,x,y(I) = m−1
m
−2m−2
m−1 + m−3
m−2 =
−2
m(m−1)(m−2) < 0
Gx,y(I) = 1
2 + α
HY ′,Y (I) =
( m′−1
m′
−m−1
m
= m′−m
m·m′ < 0
if Y ′ ̸= ∅
α −m−1
m
≤α −1
2
if Y ′ = ∅.
16
Thus F(I) < 0, G(I) < 0 if α < −1
2 and H(I) < 0 if α < 1
2; fix a choice of α. By continuity
of F, G, H and negativity at the identity matrix I, there is a neighbourhood U ⊂GLn(R) of I,
such that F, G, H are negative on each Z ∈U. Since Z(tX) →I as t →∞, we thus know that
Z(tX) ∈U for t ≫0 and thus that F, G, H are negative. These are the strict submodularity
inequalities (F, G) and the increasing inequalities (H), so this completes the proof.
Acknowledgments
I thank Mark Meckes for helpful discussions and encouragements. Part of this article was written
during a research stay at the Max Planck Institute of Molecular Cell Biology and Genetics in
Dresden. For the purpose of open access, the author has applied a CC BY public copyright li-
cence to any author accepted manuscript arising from this submission.
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Authors’ address:
Karel Devriendt, University of Oxford
devriendt@maths.ox.ac.uk
18
|
The geometry of magnitude for finite metric spaces Karel Devriendt Abstract The main result of this article is a geometric interpretation of magnitude, a real-valued invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric space X such that the magnitude of X can be expressed in terms of the 'circumradius' of its embedding S. The circumradius is the smallest r for which the r-thickening of S is contractible. We give three applications: First, we describe the asymptotic behaviour of the magnitude of tX as t →∞, in terms of the circumradius. Second, we develop a matrix theory for magnitude that leads to explicit relations between the magnitude of X and the magnitude of its subspaces. Third, we identify a new regime in the limiting behaviour of tX, and use this to show submodularity-type results for magnitude as a function on subspaces. 1 Introduction 1.1 Background Magnitude is a real-valued invariant for enriched categories (in general) and metric spaces (in particular) that was introduced by Leinster in 2006, in [11]. Before giving a definition, we mention two slogans that are helpful in guiding one's intuition on magnitude. The first slogan says that "Magnitude is like an Euler characteristic." This statement reflects the history of magnitude [11], motivates its definition, and guides the development of its theory, for instance in the generalization to magnitude homology [9, 16]. In this paper, the connection to Euler characteristics shows itself in our study of valuative properties, i.e., how magnitude behaves as a function on subspaces Y ⊆X and what kind of inclusion-exclusion-type relations it satisfies. A second slogan says that "Magnitude counts the effective number of points." Perhaps more appropriate for finite spaces, this intuition explains the success of magnitude in applications such as quantifying (bio-)diversity [14] and in data analysis [1, 10]. The second slogan comes forward in this paper in our study of the asymptotic convergence of magnitude to the point count #X; see also Examples 1.1 and 1.4. At the time of writing, an online bibliography [15] maintained by Leinster & Meckes counts more than 120 papers on magnitude. 1.2 Summary of results In this article, we focus on magnitude for finite metric spaces. This setting was treated early on in the development of magnitude, for instance by Leinster in [12, §2] and by Meckes in [18]. In particular, both works found that so-called positive definite metric spaces are particularly well-behaved (see Definition 2.2). Our work builds forth on this analysis by drawing a strong connection between magnitude of finite positive definite metric spaces and Euclidean geometry, via matrix theory. We briefly describe our main result and its applications below. 1 16 Oct 2025 The magnitude of a finite metric space (X, d) is defined based on the similarity matrix Z with entries zij = e-d(i,j). When Z is positive definite, we call X positive definite and define: The magnitude of X = (X, d) is |X| := X i,j∈X Z-1 ij. We will come to a more generally applicable definition later. The main new idea in this article is that there is a Euclidean embedding φ : X →RX-1, which we call the similarity embedding, such that the magnitude of X can be expressed in terms of the circumradius R(S) of its embedding S = φ(X); the circumradius is the smallest r such that the r-thickening S s∈S Br(s) of S is contractible. More precisely, we find: Theorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then |X| = 1 1 -2R(S)2 . (1) In fact, requiring expression (1) for all subspaces Y ⊆X characterizes the embedding S (see Theorem 2.15). So, in a sense, the embedding φ naturally appears in the context of magnitude. Example 1.1. Let us illustrate Theorem 2.12 by a classical example in magnitude: the metric space X(2) consisting of two points at distance d > 0. By inverting the positive definite similarity matrix Z = 1 e-d e-d 1 and summing the entries of Z-1, one directly computes the magnitude |X(2)| = 1 + tanh(d/2). The embedding of X(2) that we introduce gives two points S ⊆R at distance √ 1 -e-d. Since the circumradius of S is half this distance, Theorem 2.12 gives the relation 1 1 -2R(S)2 = 1 1 -1-exp(-d) 2 = 2 1 + exp(-d) = 1 + tanh(d/2) = |X(2)|. What lies behind Theorem 2.12? To lift some of the mystery, let us reveal the underlying linear algebra and construct the embedding φ of X explicitly, starting from its matrix Z: Step 1: Construct the centered matrix K = 1 2(I -11T n )Z(I -11T n ), where n = #X. Step 2: Because Z is positive definite, K admits a square root √ K. Step 3: Define the embedding φ(i) of i ∈X as the ith column of √ K. In other words, K is the Gram matrix of S when the origin is placed at the centroid of S. Based on this embedding, Theorem 2.12 then follows from the observation that the circumradius of S is governed by very similar "equilibrium" equations to those of magnitude; see Remark 2.13. We continue with a brief description of the three main applications of Theorem 2.12. As a first application, we consider the asymptotic behaviour of magnitude. Much of the theory of magnitude is concerned with studying the magnitude |tX| across different scales t > 0, based on the metric transformation d 7→t · d. Leinster & Willerton showed in [17, Thm. 3] that magnitude asymptotically counts the number of points n in the metric space X, as follows |tX| = n -q(tX) with q(tX) →0 as t →∞. We give a description of the asymptotics of the error term q in this formula; let St := φ(tX). Theorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence n -|tX| = q(tX) ∼n2 n -1 n -2R(St)2 . 2 Geometrically, Theorem 3.1 states that the magnitude of tX tends to the magnitude of the discrete metric space in the same way as the circumradius of St tends to the circumradius of the regular simplex with unit-length edges. Let us return to the example of the two-point metric space to illustrate the theorem. Example 1.2. For the two-point metric space X(2), Theorem 3.1 correctly gives q(tX(2)) = 2 -|tX(2)| = 2 1 + exp(t · d) ∼4(1 2 -1 2 exp(-t · d)). In other words, we find that |tX(2)| ≈2e-td asymptotically. This is illustrated in Figure 1. Figure 1: The magnitude |tX(2)| of the metric space with two points at distance d = 1 is wellapproximated by 2e-t at large scales t ≫0. This is explained by Theorem 3.1. As a second application, we take inspiration from Fiedler's matrix theory for Euclidean simplices in [5], and develop a matrix theory for magnitude. Generalizing the relation between the pseudoinverse Gram matrix and distance matrix of a simplex (see, e.g., [2, §3.2]), we find: Theorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then 0 1T 1 Z !-1 = -|X|-1 wT /|X| w/|X| 1 2K† ! . (3) In equation (3), the vector 1 = (1, . . . , 1)T is the all-ones vector, w := (Z-1)1 is the weighting on the metric space X and K† is the Moore-Penrose pseudoinverse of the Gram matrix K of the centered point set S; more results on the matrices Z, K and K† are given in Section 4. Example 1.3. For the two-point metric space with distance d = -ln(δ), Theorem 4.11 reads 0 1 1 1 1 δ 1 δ 1 -1 = 1 2 -(1 + δ) 1 1 1 (1 -δ)-1 -(1 -δ)-1 1 -(1 -δ)-1 (1 -δ)-1 . What is the use of this identity? In this article, it mainly serves as a tool to study the magnitude of subspaces of X. Since any subspace Y ⊆X of a positive definite metric space is also positive definite, Theorem 4.11 applies to Y as well. The subspace relation between Y and X translates nicely to an algebraic relation between their respective matrices, and we arrive at: Corollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have |X\{x}| = |X| 1 + 2w2 x ̄cx|X| !-1 and w′ i |X\{x}| = wi |X| + cix ̄cx wx |X|, for any i ̸= x, and where w′ is the weighting on X\{x}. 3 Here, we use the notation cij = -(K†)ij and ̄ci = P j cij. This is a corollary of Theorem 4.16, which is an analogous statement for general subspaces. We illustrate the relation between |X| and |X\{x}| in Corollary 4.18 by looking at the quantity 2w2 x ̄cx|X| that controls this relation. Example 1.4. We consider the metric space on X = {1, 2, 3} with distances d(1, 2) = 2 and d(1, 3) = 100 and d(2, 3) = 100. The points 1 and 2 are close together and equidistant from point 3. Since every three-point metric space is positive definite [12, Prop. 2.4.15], its magnitude exists for all t > 0. The black lines in Figure 2 plot the magnitude of X: for small t we effectively see one point, for medium t we see two points, i.e., {1, 2} and 3, and for large t we see three points. Figure 2: Both panels: The magnitude |tX| (black line) of the three-point metric space in Example 1.4; note the logarithmic scale for the horizontal axis. Left panel: The relative contribution wx/|tX| for points 1 and 2 (blue line) and 3 (red line). Right panel: The change contribution 2w2 x/( ̄cx|tX|) for points 1 and 2 (blue line) and 3 (red line). The left and right panel in Figure 2 compare two possible ways to measure the contribution of a point x to the magnitude: wx/|X| on the left, and 2w2 x ̄cx|X| on the right, inspired by Corollary 4.18. Overall, the latter appears to be more sensitive to the changing geometry across scales. As a third application, we initiate the study of a new class of metric spaces; we call these spaces strongly positive definite and they are defined by the additional positivity conditions w > 0 and cij = -(K†)ij > 0, for all i ̸= j. We show that this class is closed under taking subspaces, and that every metric space tX becomes strongly positive definite when t ≫0. As the main application, we prove two valuative results on the magnitude of a strongly positive definite metric space X. These results are an attempt to reproduce results for magnitude in the spirit of the inclusion-exclusion property of the Euler characteristic: for (suitable) subspaces A, B ⊆X of a topological space X, the Euler characteristic χ satisfies χ(A) + χ(B) = χ(A ∪B) + χ(A ∩B). This fact has inspired research into valuative properties of magnitude: Leinster showed an inclusion-exclusion result for magnitude of graphs in [13], Leinster and Willerton showed in [17] that the magnitude of subsets of Rn satisfies inclusion-exclusion asymptotically, and Gimperlein, Goffeng and Louca further improved on these geometric results in [6]. Here, we consider a variant of inclusion-exclusion; a function f : 2X →R is called strictly submodular if1 f(A) + f(B) > f(A ∪B) + f(A ∩B), for all A, B ⊆X, and f is called increasing (resp. decreasing) if it is increasing (resp. decreasing) with respect to the subset partial order. We show the following two results: 1Later in this article we will work with a different, but equivalent definition (4). 4 Theorem 5.9. Let X be strongly positive definite. Then the function f : Y 7-→ ( -|Y |-1, if Y ̸= ∅ α , if Y = ∅ is increasing if α 0; this is also called stably positive definite. There are many natural examples of negative type metric spaces, for instance point sets in Euclidean, spherical or hyperbolic space. See [18, Thm. 3.6] for a list of examples. 2.2 Similarity embedding of positive definite metric spaces In this section, we introduce an embedding of a positive definite metric space and show its relation to magnitude. This embedding makes critical use of positive definiteness. Definition 2.6 (Similarity embedding). A similarity embedding of a positive definite metric space X is an embedding φ : X →RX-1 that satisfies ∥φ(i) -φ(j)∥2 = 1 -e-d(i,j) = 1 -zij for all i, j ∈X. Recall the construction from the introduction, which proves existence of these embeddings Proposition 2.7. The similarity embedding exists and is unique up to rigid transformations. Proof. Let X be a positive definite metric space and construct the centered similarity matrix K := 1 2 I -11T n Z I -11T n , where n = #X is the cardinality of the metric space and I is the identity matrix. Since Z is positive definite, the matrix K is positive semidefinite with ker(K) = span(1) (see also Proposition 4.3) and thus there exists an (n-1)×n square root √ K, i.e., such that √ K T √ K = K. The embedding φ : X ∋i 7→(ith column of √ K) ∈Rn-1 is a similarity embedding, since ∥φ(i) -φ(j)∥2 = 1 2(ei -ej)T √ K T √ K(ei -ej) = 1 2(zii + zjj -2zij) = 1 -e-d(i,j), where ei is the ith basis vector. Since the embedding φ is defined via all pairwise distances between the points φ(X), it is unique up to rigid transformations, i.e., isometries of Rn-1. Remark 2.8 (Computation). The similarity embedding φ can be computed efficiently using standard linear algebra methods. For instance, one can compute √ K via the Cholesky decomposition or an eigenvalue decomposition, for which highly optimized algorithms exist. Proposition 2.7 associates to each positive definite metric space X a set of points S := φ(X) in RX-1. As introduced, we will show that a certain geometric invariant of this point set is related to magnitude. We first establish another important fact about the geometry of S. Proposition 2.9. Let X be positive definite. Then S = φ(X) are the vertices of a simplex. Proof. This is a consequence of positive definiteness: since Z has full rank, the matrix K has rank (n -1) and its square root, the (n -1) × n matrix √ K, has full rank n -1. It follows that the columns of √ K and thus the points S are affinely independent. 6 2.3 Magnitude and circumradius We now come to the definition of circumradius. This is the geometric invariant of the point set S that is closely related to magnitude; we note that, by construction, the matrix (11T -Z) is the matrix of squared distances between pairs of points in S, and we may thus write Z(S). Theorem 2.10. The following numbers are equal for the vertices S of a simplex: 1. The radius r of the unique sphere in RS-1 that goes through S; 2. The smallest r such that the r-thickening of S is contractible; 3. The unique value r for which (11T -Z(S))x = 2r2 · 1 with xT 1 = 1 has a solution. This common value is called the circumradius of S and is denoted by R(S). Proof. From Euclidean geometry (or algebraic geometry), we know that there is a unique sphere in RS-1 that goes through the vertices S of a simplex. This is called the circumsphere of the simplex, and it is the smallest sphere that encloses the points S. (1. ⇔2.) The r-thickening of S is contractible if and only if S is contained in some radius-r sphere. Since the circumsphere of a simplex is the smallest sphere that contains S, its radius R(S) is the smallest r for which the r-thickening of S is contractible, as required. (1. ⇔3.) The circumsphere has center c (= the circumcenter), which satisfies the equations: ∥c -s∥2 = r2 for each s ∈S, where r is the radius of the circumsphere. If we write p for the barycentric coordinate of the circumcenter, i.e., the unique vector such that 1T p = 1 and √ Kp = c, and use the fact that for each i ∈X, the vector ei is the barycentric coordinate of the ith point of S, then we can rewrite (p -ei)T K(p -ei) = r2 for each i ∈X ⇐⇒1 2(p -ei)T Z(p -ei) = r2 for each i ∈X (since 1T (p -ei) = 0) ⇐⇒eT i Zp = -r2 + 1 2pT Zp + 1 2 for each i ∈X ⇐⇒Zp = (1 2 -r2 + 1 2pT Zp) · 1. ⇐⇒Zp = (1 -2r2) · 1 (since 1T p = 1) ⇐⇒(11T -Z)p = 2r2 · 1. Thus the radius r of the circumsphere satisfies the required equation (3.). Uniqueness of r in (3.) follows from uniqueness of the sphere that contains all vertices S of a simplex, or it can be derived from invertibility of Z as implied by positive definiteness. Remark 2.11 (Scaled embeddings). Any rescaling α · φ for α > 0 of the similarity embedding still embeds the metric space X as the vertices of a simplex, with circumradius αR(S). Different choices than α = 1 may be useful in different applications, but some of the nice theory may be lost, for instance, when choosing a scaling that depends on X. For instance, for α2 = |X| or α2 = #X, the subspace characterization in Theorem 2.15 below no longer holds. In a sense, X-dependent scalings are not functorial in the same way as independent scalings are. We now arrive at the first main result from the introduction, and its proof. Theorem 2.12. Let X be a positive definite metric space with similarity embedding S. Then |X| = 1 1 -2R(S)2 . (1) 7 Proof. This follows immediately from the third expression for the circumradius in Theorem 2.10. From Zx = (1-2R(S)2)1 with xT 1 = 1, we find that w = x/(1-2R(S)2) is the weighting on X. Since x is normalized, the claimed expression for the magnitude |X| follows. Remark 2.13 (Equilibrium equations). With the proofs of Theorem 2.10 and Theorem 2.12 in place, we can now appreciate where the relation between magnitude and circumradius comes from: both quantities are defined via an 'equilibrium equation'. In the case of magnitude, the equilibrium equation Zw = 1 is part of the definition which Leinster arrived at by generalizing the Euler characteristic. In the geometric setting, the equilibrium equation (11T -Z)p = 2r21 encodes the fact that the circumcenter is equidistant to all vertices of a simplex. Corollary 2.14. Let X be a positive definite metric space. Then |X| > 1. Proof. This follows from |X| = 1 + 2|X|R2(S) with |X| > 0 and R(S) > 0. We can in fact turn Theorem 2.12 around and use it to characterize the similarity embedding. Theorem 2.15. Let X be a positive definite metric space. The similarity embedding φ is the unique embedding of X into RX-1, up to rigid transformation, such that |Y | = 1 1 -2R(φ(Y ))2 for all nonempty Y ⊆X. Proof. We recall from Example 1.1 that the magnitude of a two-point metric space is 2/(1+e-d). Furthermore, the circumradius of two points at Euclidean distance d′ is d′/2. For every two-point subspace Yij = {i, j} ⊆X, the requirement on the embedding φ gives 2 1 + e-d(i,j) = |Yij| = 1 1 -2R(φ(Yi,j))2 = 2 2 -∥φ(i) -φ(j)∥2 ⇐⇒∥φ(i) -φ(j)∥2 = 1-e-d(i,j). The latter equation shows that any embedding with the prescribed conditions must be the similarity embedding, which is unique. It remains to prove existence, i.e., that the similarity embedding φ satisfies the stated relation between |Y | and R(φ(Y )) for subspaces Y of any cardinality. This follows because the restriction φ|Y of the similarity embedding of X onto any subspace Y is the similarity embedding of Y , and R(φ(Y )) = R(φ|Y (Y )). The expression for |Y | in terms of R(φ(Y )) is then simply Theorem 2.12. Another way to think about Theorem 2.12 is that it expresses functoriality of the similarity embedding with respect to the subspace relation. This is also reflected in the matrix identity (3) and its implication, e.g., Theorem 4.16. To end this section, we compute the similarity embedding, magnitude and circumradius of a three-point metric space in a numerical example. Example 2.16. Consider the three-point metric space X = {1, 2, 3} with metric d(1, 2) = ln(2) ≈0.69 and d(1, 3) = d(2, 3) = ln(10) ≈4.61. Its similarity matrix, centered similarity matrix and a similarity embedding are: Z = 1 0.5 0.1 0.5 1 0.1 0.1 0.1 1 K = 1 9 1.9 -0.35 -1.55 -0.35 1.9 -1.55 -1.55 -1.55 3.1 √ K = φ(1) φ(2) φ(3) = √ 2/4 - √ 2/4 0 -0.2934 -0.2934 0.5869 ! 3 Magnitude asymptotics As explained in the introduction, much of the theory of magnitude is concerned with studying how the magnitude |tX| behaves as a function of the scale t > 0. Our goal here is to refine the following asymptotic formula, due to Leinster and Willerton in [17]: |tX| = n -q(tX) with q(tX) →0 as t →∞, 8 where n is the cardinality of the metric space. As a direct application of Theorem 2.12, we give an asymptotic expression for the remainder q(tX) in this expression; we write St := φ(tX). Theorem 3.1. Let X be a metric space on n points. Then we have the asymptotic equivalence n -|tX| = q(tX) ∼n2 n -1 n -2R(St)2 . Proof. As t →∞, tX is eventually positive definite (Proposition 2.3) and its similarity matrix Z(tX) converges to the identity matrix I. As a result, the simplex with vertices St converges to the regular simplex with all edge lengths equal to p1 -zij = 1. From the equilibrium equation, one computes that R(St)2 converges to n-1 2n , the squared circumradius of the regular simplex. Next, based on Theorem 2.12, we can write the remainder q(tX) = n -|tX| as follows: |tX| = 1 1 -2R(St)2 = n 1 + n n-1 n -2R(St)2 ⇒q(tX) = n 1 - 1 1 + n n-1 n -2R(St)2 ! Using the asymptotic equivalence 1 - 1 1+x ∼x for x →0 and the earlier result that n-1 n - 2R(St)2 →0 as t →0, we then find the claimed asymptotic equivalence. Theorem 3.1 states that the magnitude of a metric space tX converges to the magnitude of the discrete metric space (X, ∞) in the same way as the circumradius of the simplex St converges to the circumradius of the regular simplex. We continue with magnitude asymptotics in Section 5, where we will further refine our understanding of the asymptotics by looking more closely at some implications of the convergence of St to the regular simplex. 4 Matrix theory & magnitude of subspaces In this section, we look more closely at the matrices Z and K that appeared in Section 2, and introduce and study the matrix K†. This section is partly an intermezzo, where we develop the matrix theory of magnitude for its own sake, and partly a setup for Section 5. 4.1 Matrix theory of Z, K and K† The Moore-Penrose pseudoinverse of a matrix M is the unique matrix M† that satisfies MM†M = M and M†MM† = M† and MM†, M†M are symmetric. We point out two consequences of this definition: (i) if M is real and symmetric, then so is M† and (ii) the matrices MM† and M†M are orthogonal projection matrices onto ker(M) = ker(M†) (see, e.g., [2, §A.3]). We recall the definition of matrices K and K†: Definition 4.1. Let X be a finite metric space with similarity matrix Z. Then • The centered similarity matrix is K := 1 2 I -11T n Z I -11T n . • The pseudoinverse centered similarity matrix is K†. By their construction, we immediately find the following properties of K and K† Proposition 4.2. K and K† are real, symmetric matrices with ker(K) = ker(K†) ⊇span(1). Proposition 4.3 (Interlacing). Let λ1 ≥λ2 ≥· · · ≥λn be the real eigenvalues of Z, and μ1 ≥μ2 ≥· · · ≥μn-1 the real eigenvalues of K, with one zero eigenvalue omitted2. Then λ1 ≥2μ1 ≥λ2 ≥2μ2 ≥· · · ≥λn-1 ≥2μn-1 ≥λn, with equality throughout if Z1 = |X|-11. We have rank(K) = rank(K†) = rank(Z) -1 and if Z is positive definite, then K and K† are positive semidefinite with ker(K) = ker(K†) = span(1). 2By construction we know that K has at least one eigenvalue. The eigenvalues μ1, . . . , μn-1 are all other eigenvalues except for this one zero eigenvalue; there may be other zero eigenvalues. 9 Proof. The eigenvalue inequalities are precisely the interlacing theorem for matrix projections [7, Thm. 2.1] applied to the matrices 2K and Z. If Z1 = |X|-1 · 1 then |X|-1 is an eigenvalue of Z with (normalized) eigenvector 1/√n; let's say that it is the jth eigenvalue. Then Z has an eigendecomposition Z = P i̸=j λixxT + |X|-1 11T n and thus by its construction K has eigendecomposition K = P i̸=j 1 2λixxT , which proves that 2μi = λi for all i ̸= j. The implication (Z positive definite) =⇒(K and K† positive semidefinite) follows from the interlacing inequalities. The rank statement holds because 2K arises from Z through a projection onto span(1)⊥, where furthermore ker(Z) ̸⊇span(1) since Z is a nonnegative matrix with unit-diagonal. By the rank statement, K and K† have rank n -1 when Z is positive definite, and thus ker(K) = ker(K†) = span(1). This completes the proof. One example where Z1 = |X|-11 holds is when the isometry group of X acts transitively on its points; these are called homogeneous spaces. We continue by showing some relations between Z, K and K†. We start with a useful lemma: Lemma 4.4. Let Z be invertible with nonzero magnitude. Then ZK† = 2I -21wT |X| . (2) Proof. We start with the definition of K in terms of Z and use KK† = I -11T n : K = 1 2 I -11T n Z I -11T n ⇒KK† = 1 2 I -11T n ZK† ⇔ I -11T n = 1 2 I -11T n ZK†. The third equation implies that ZK† must be of the form ZK† = 2I -21yT for some y ∈Rn. Using the additional fact that wT ZK† = 0, we find that y = w/|X| as required. We now show some relations between Z, K and Z-1, K†. Proposition 4.5. The similarity and centered similarity matrices satisfy: 2K -Z = γ1T + 1γT , where γ := -1 nZ1 + 1 2n21T Z1 · 1 = diag(K) -1 21. Proof. We note that Z may be singular. Starting from the definition of K, we find 2K -Z = -1 nZ11T -1 n11T Z + 1 n2 11T Z11T , which proves the expression for 2K-Z. The second expression of γ follows from diag(Z) = 1. Proposition 4.6. Let Z be invertible with nonzero magnitude. Then 2Z-1 -K† = 2wwT |X| , where w |X| = 1 2K†γ + 1 n1; |X|-1 = -1 2γT K†γ -2 n1T γ. Proof. Right-multiplying equation (2) by Z-1 proves the expression for 2Z-1 -K†. We now prove the expression for w/|X| in terms of K† and γ. K† = 2Z-1 -2wwT |X| ⇒K†γ = 2Z-1 -2wwT |X| -1 nZ1 + 1 2n21T Z1 · 1 = -2 n1 + 2 |X|w ⇒1 2K†γ = -1 n1 + w |X|. Finally, we prove the expression for |X|-1 in terms of K† and γ. 1 2K†γ + 1 n1 = w |X| ⇒1 2γT K†γ + 1 nγT 1 = wT γ |X| = wT |X| -1 nZ1 + 1 2n2 1T Z1 · 1 = -1 |X| + 1 2n2 1T Z1 = -1 |X| -1 nγT 1, 10 which leads to the claimed expression for |X|-1 and completes the proof. We focus on one further 'object' that turns out to play an important role in the matrix theory and the asymptotics of magnitude: the off-diagonal entries of matrix K†. Later, in Section 5, we will show that large scales t ≫0 are marked by negativity of these entries, which in turn leads to additional algebraic and geometric properties. We introduce the notation cij := -(K†)ij = 2wiwj |X| -2(Z-1)ij, for i ̸= j ∈X and ̄ci := (K†)ii = X j∈X cij. The relation between ̄ci and P j cij follows because the rows of K† sum to zero. As a notational convention, when summing over j ∈X or i, j ∈X with cij in the sum, we consider each pair only once (i.e., with (i, j) = (j, i)); think of c as a function on unordered pairs of distinct points. We give expressions for magnitude and weightings in terms of c. Proposition 4.7. Let Z be invertible with nonzero magnitude. Then wi |X| = 1 -1 2 X j∈X cij(1 -zij) and |X|-1 = 1 - X i,j∈X cij(zix -zjx)2 for any x ∈X. Proof. The expression for wi follows by taking the ith diagonal entry of equation (2). For the expression of |X|-1 we start by the observation that, due to ker(K†) = span(1), we can write K† = X i,j∈X cij(ei -ej)(ei -ej)T ⇒1 2yT K†y = X i,j∈X cij(yi -yj)2 for any y ∈RX. Right-multiplying equation (2) by Z and looking at the xth diagonal entry, we then find (ZK†Z)xx = 2 -2|X|-1 ⇔|X|-1 = 1 - X i,j∈X cij(zix -zjx)2. We note that a factor of 1 2 is incorporated in the sum because of our notational convention. Corollary 4.8. Let Z be invertible with nonzero magnitude. Then P i,j∈X cij(1 -zij) = n -1. Proof. This follows by summing wi/|X| in Proposition 4.7, and 1T w = |X|. Remark 4.9 (Foster's theorem). The direct analogue of this corollary is called Foster's Theorem in other settings. One hint at the depth of this theorem is that the integer on the righthandside is known to count, depending on the situation, a dimension, a degree or an Euler characteristic. Corollary 4.10. Let Z be invertible with nonzero magnitude. Then |X|-1 = 1 n tr I -1 2ZK†Z . Proof. This follows by summing the expression for |X|-1 in Proposition 4.7 over all x ∈X. To end the section, we prove a matrix identity that relates the similarity and pseudoinverse centered similarity matrices of a metric space. This identity summarizes several of the expressions above, and it will be the basis to derive explicit relations between different subspaces. Theorem 4.11. Let X be a metric space with invertible Z and nonzero magnitude. Then 0 1T 1 Z !-1 = -|X|-1 wT /|X| w/|X| 1 2K† ! . (3) 11 Proof. We verify that multiplying the two sides of equation (3) gives the identity matrix: 1Tw/|X| 1 21T K† |X|-1 · (Zw -1) 1wT/|X| + 1 2ZK† ! = 1 0 0 I ! . The top-left equation holds by definition of magnitude. The top-right equation holds because ker(K†) = span(1). The bottom-left equation holds because Zw = 1. The bottom-right equation holds by equation (2). Remark 4.12 (Cayley-Menger matrix). Theorem 4.11 mimics a matrix identity in simplex geometry and algebraic graph theory; we called this the Fiedler-Bapat identity in [2, §3]. In that context, the matrix 0 1T 1 D is called the Cayley-Menger matrix, where D contains the squared pairwise distances between a set of points. This matrix is an important algebraic object in geometry; for instance, its determinant encodes affine (in-)dependence of the set of points. We note the following rational expression for |X| in variables (zij) that follows from (3): Proposition 4.13. Let X be a positive definite metric space. Then |X| = -det 0 1T 1 Z ! / det(Z). 4.2 Magnitude and weightings for subspaces We now apply Theorem 4.11 to study the magnitude of subspaces of a metric space; we start by fixing some notation. We write Y ⊆X to denote the metric subspace (Y, d|Y ) ⊆(X, d), and write d for the metric d|Y ; unless stated otherwise, we assume Y nonempty. We write Y c = X and use the following subscript notation for submatrices: Z = ZY Y ZY Y c ZY cY ZY cY c ! and w = wY wY c ! . Since the metric on Y is simply the restriction of d, it follows that the similarity matrix Z(Y ) is the principal submatrix ZY Y of Z(X). We recall the definition of Schur complements: Definition 4.14 (Schur complement). Let M = MY Y MY Y c MY cY MY cY c be a matrix such that MY cY c is invertible. The Schur complement of M with respect to Y c is the matrix3 M/Y c, with rows and columns indexed by Y , defined by M/Y c := MY Y -MY Y c(MY cY c)-1MY cY . Proposition 4.15 ([8]). Let M be a matrix such that M/Y c is well-defined. Then • If M is invertible, then (M-1)Y Y = (M/Y c)-1. • M/Y c = ((M/x1)/ . . . )/xlwith entries Y c = {x1, . . . , xl} in any order. We now arrive at the main result of this section. Theorem 4.16. Let X be a positive definite metric space. Then for any Y ⊆X we have |Y | = |X| 1 + 2wT Y c(K† Y cY c)-1wYc |X| !-1 and w′ |Y | = wY |X| -K† Y Y c(K† Y cY c)-1 wY c |X| , where w′ is the weighting on Y . Furthermore, K†(Y ) = K†(X)/Y c. 3This notation is not standard; we use it to keep the focus on the underlying metric space X and its subspaces. 12 Proof. Starting from Theorem 4.11 for X, combined with the expression of inverse matrix submatrices in terms of Schur complements (Proposition 4.15), we find 0 1T 1 ZY Y !-1 = -|X|-1 wT Y /|X| wT Y c/|X| wY /|X| 1 2K† Y Y 1 2K† Y Y c wY c/|X| 1 2K† Y cY 1 2K† Y cY c . Y c. = -|X|-1 wT Y /|X| wY /|X| 1 2K† Y Y ! -2 wT Y c/|X| 1 2K† Y Y c ! (K† Y cY c)-1 wY c/|X| 1 2K† Y cY . Next, Theorem 4.11 for Y states that 0 1T 1 Z(Y ) !-1 = -|Y |-1 w′T /|Y | w′/|Y | 1 2K(Y )† ! . The theorem then follows by equating these two expressions, using the equality Z(Y ) = ZY Y . Corollary 4.17. Let X be positive definite. Then |Y | ≤|X| for all Y ⊂X, with equality if and only the weighting w on X satisfies wy = 0 for all y ∈Y . Proof. We first establish that the matrix K† Y cY c, and thus its inverse, is positive definite: xT K† Y cY cx = x 0Y !T I -11T n ! K† I -11T n ! x 0Y ! > 0 for all nonzero x ∈RY c, which follows since K† is positive definite on span(1)⊥. Thus K† Y cY c is positive definite. The corollary then follows from the expression for |Y | in Theorem 4.16, and |X| > 0. The special case of the subspace Y = X\{x} for some x ∈X is particularly useful: Corollary 4.18. Let X be a positive definite metric space. Then for any x ∈X we have |X\{x}| = |X| 1 + 2w2 x ̄cx|X| !-1 and w′ i |X\{x}| = wi |X| + cix ̄cx wx |X|, for any i ̸= x, and where w′ is the weighting on X\{x}. For the entries c of the pseudoinverse centered similarity matrix K†, we find: Corollary 4.19. Let X be a positive definite metric space. Then for any x ∈X we have c′ ij = cij + cixcjx ̄cx and ̄c′ i = ̄ci -c2 ix ̄cx , for all i, j ̸= x and where c′ is defined on X\{x}, i.e., via the entries of K(X\{x})†. 5 Strongly positive definite metric spaces In this section, we return to the asymptotics of |tX|. Our main contribution is to identify a new property of metric spaces, which we call "strongly positive definite", that is satisfied by every metric space tX for t ≫0. This new regime refines our understanding of the asymptotics of tX. 13 5.1 Definition and first properties We start by defining the new class of metric spaces; recall that cij = -(K†)ij = 2wiwj |X| -2(Z-1)ij. Definition 5.1 (Strongly positive definite). A finite metric space X is called strongly positive definite if it is positive definite with c > 0 and w > 0. Question 5.2. What is the right generalization of Definition 5.1 when X is not finite? Strongly positive definite metric spaces are related to other known objects and properties. We summarize some of these equivalences in the two propositions below. Proposition 5.3. The following are equivalent for a positive definite metric space X: 1. The weighting is nonnegative w ≥0 (resp. positive w > 0); 2. S(X) are the vertices of a simplex that contains (resp. strictly contains) its circumcenter. Proof. In the proof of Theorem 2.12, we found that the normalized weighting w/|X| is the barycentric coordinate of the circumcenter of S(X). The circumcenter lies in the simplex (resp. in its interior) if and only if these coordinates are nonnegative, w ≥0 (resp. positive, w > 0). In the following, the terms "M-matrix, Laplacian and nonobtuse" are defined in the proof. Proposition 5.4. The following are equivalent for a positive definite metric space X. 1. K† has nonpositive (resp. negative) off-diagonal entries, equivalently, c ≥0 (resp. c > 0). 2. K† is an M-matrix (resp. strict M-matrix); 3. K† is the Laplacian matrix of a connected, positively weighted graph (resp. complete graph). 4. S(X) are the vertices of a nonobtuse (resp. acute) simplex. Proof. (1.⇔2.) An M-matrix is a matrix whose eigenvalues have nonnegative real part and whose off-diagonal entries are non-positive. Since K† is positive semidefinite when X is positive definite, K† is indeed an M-matrix if and only if its off-diagonal entries are non-positive. (1.⇔3.) The Laplacian matrix of a graph G = (V, E) with edge weights ω : E →R>0 is the matrix L = P ij∈E ω(ij)(ei -ej)(ei -ej)T . We have established before that K† can be written in this form, with ω(ij) = cij, as a consequence of span(1) ⊆ker(K†). Thus, if c > 0 then the matrix K† is a Laplacian matrix. A graph is connected if and only if ker(L) = span(1) (see, e.g., [5, §6]), which completes the equivalence. (3.⇔4.) A simplex is nonobtuse if every angle between facets is at most π/2 (i.e., nonobtuse). The equivalence between Laplacian matrices of connected graphs L and nonobtuse simplices is due to Fiedler [5, §6.2]; the equivalence is established via (Moore-Penrose pseudoinverse of Laplacian matrix)←→(centered Gram matrix of simplex). If c > 0 in (1.) this corresponds to G being the complete graph in (3.) and the simplex having angles smaller than π/2 (i.e., acute) in (4.). The terminology "strict M-matrix" in (2.) is not standard, but we use it here for an M-matrix with negative off-diagonal entries. Remark 5.5 (Graphs & discrete curvature). We highlight in particular the connection with graph Laplacian matrices. Many of the techniques in this article are inspired by the author's previous work in that context; see for instance [2, §2-4,§6]. Relatedly, we point out that similar objects to w have recently been defined on graphs in the context of discrete curvature [3, 4, 19]; there, w > 0 has the natural interpretation of a positive curvature condition. The following result shows that strongly positive definite metric spaces can be parametrized (via Z-1) by an open semi-algebraic subset of GLn(R), the space of real, invertible n×n matrices. 14 Proposition 5.6. A metric space X is strongly positive definite if and only if its similarity matrix Z is invertible with positive magnitude, and satisfies (Z-11)i(Z-11)j > (1T Z-11) · (Z-1)ij and (Z-11)i(Z-11)j > 0, for all distinct i, j ∈X. Proof. This follows from the identities K† = 2Z-1 -2wwT /|X| and w = Z-11. Introducing K† and w in the inequalities above implies that c > 0 is equivalent to the first inequalities, whereas the second inequalities are equivalent to w > 0. For wi · wj > 0 to imply that w > 0, we use that w cannot be all negative because |X| = 1T w is positive. As a setup for the next subsection, we establish two further results: Proposition 5.7. For any metric space X and t ≫0, tX is strongly positive definite. Proof. The result for w > 0 was shown by Leinster in [12, Prop. 2.2.6]. We use the same proof strategy to show c > 0. First, let t be sufficiently large such that tX is positive definite, and represent a metric space by its invertible similarity matrix Z ∈GLn(R) ⊂Rn×n; here n = #X. On the subset of invertible matrices GLn(R), the following function is continuous GLn(R) ∋Z 7→cij(Z) = -2 · adjij 0 1T 1 Z ! / det 0 1T 1 Z ! for any i ̸= j ∈X, where adjij denotes the (i, j)th entry of the adjugate matrix. By continuity and cij(I) > 0 for the identity matrix I, there is a neighbourhood U ⊂GLn(R) of the identity matrix such that cij(Z) > 0 for each Z ∈U. Since Z(tX) →I as t →∞, we know that Z(tX) ∈U and thus that cij(Z(tX)) > 0 for t ≫0. This completes the proof. Proposition 5.8. Strong positive definiteness and c > 0 are preserved under taking subspaces. Proof. This follows from Corollaries 4.18 and 4.19, which express c′ and w′ for a subspace Y = X \ {x} in terms of c and w of the metric space X, or by repeated application of this argument following the composition property of Schur complements (Proposition 4.15). 5.2 Magnitude submodularity This section deals with valuative properties of magnitude. We show that two magnitude-based set functions on the subspaces of a strongly positive definite metric space X are submodular; recall that a function f on the subsets of X is called strictly submodular if it satisfies4 f(Y ) -f(Y \{y}) 2, and start with the latter. (#X > 2): We can rewrite the submodularity inequalities for f as -|X|-1 + |X\{x}|-1 wx/|X|, which confirms the inequality above. (#X = 2): We can rewrite the submodularity inequality for f as -|X|-1 + |X\{x}|-1 < -|X\{y}|-1 -α ⇐⇒1 -1 + exp(-d(x, y)) 2 < -1 -α (Example 1.1) ⇐⇒α < -3 2 This confirms strict submodularity for #X = 2 if α < -2/3, and thus completes the proof for strongly positive definite X. By Corollary 4.18, we know that f is increasing for nonempty subsets and that f(Y ) ≥-1 with equality for #Y = 1; it follows that f is increasing on 2X if α < -1. The asymptotic result follows from Proposition 5.7. Equivalently, one can say that -f in Theorem 5.9 is strictly supermodular and decreasing, for appropriate α. To end this section, we prove a second submodularity result. The function considered in the theorem below is inspired by the asymptotic expression in Theorem 3.1 for q(tX) = #X -|X| in terms of R(St)2. Theorem 5.10. Let X be any metric space and t ≫0. Then the function f : Y 7-→ ( m-|tY | m2 + m-1 m if m := #Y ̸= 0 α if Y = ∅ is increasing if α < 1 2 and strictly submodular if α < -1 2. Proof. We follow the strategy from the proof of Proposition 5.7 and represent a metric space X by its invertible similarity matrix Z ∈GLn(R). We consider three families of functions: the family FY,x,y, labeled by x ̸= y ∈Y ⊆X, the family Gx,y labeled by x ̸= y ∈X, and the family HY ′,Y labeled by Y ′ ⊂Y ⊆X; these are defined as FY,x,y = f(Y ) -f(Y \{x}) -f(Y \{y}) + f(Y \{x, y}) Gx,y = f({x, y}) -f({x}) -f({y}) + f(∅) HY ′,Y = f(Y ′) -f(Y ) Since |Y | = -det 0 1T 1 ZY Y / det(Z) by Proposition 4.13, these are continuous functions on GLn(R). Next, we confirm that each of these functions evaluates negatively for the discrete metric space; in that case, Z = I and thus |Y | = m and f(Y ) = (m -1)/m. We find FY,x,y(I) = m-1 m -2m-2 m-1 + m-3 m-2 = -2 m(m-1)(m-2) < 0 Gx,y(I) = 1 2 + α HY ′,Y (I) = ( m′-1 m′ -m-1 m = m′-m m·m′ < 0 if Y ′ ̸= ∅ α -m-1 m ≤α -1 2 if Y ′ = ∅. 16 Thus F(I) < 0, G(I) < 0 if α < -1 2 and H(I) < 0 if α < 1 2; fix a choice of α. By continuity of F, G, H and negativity at the identity matrix I, there is a neighbourhood U ⊂GLn(R) of I, such that F, G, H are negative on each Z ∈U. Since Z(tX) →I as t →∞, we thus know that Z(tX) ∈U for t ≫0 and thus that F, G, H are negative. These are the strict submodularity inequalities (F, G) and the increasing inequalities (H), so this completes the proof. Acknowledgments I thank Mark Meckes for helpful discussions and encouragements. Part of this article was written during a research stay at the Max Planck . For the purpose of open access, the author has applied a CC BY public copyright licence to any author accepted manuscript arising from this submission. References [1] M. F. Adamer, E. D. Brouwer, L. O'Bray, and B. Rieck. The magnitude vector of images. Journal of Applied and Computational Topology, 8(2):251-274, 2024. 1 [2] K. Devriendt. Graph geometry from effective resistances. PhD thesis, 2022. 3, 9, 12, 14 [3] K. Devriendt and R. Lambiotte. Discrete curvature on graphs from the effective resistance. Journal of Physics: Complexity, 3(2):025008, 2022. 14 [4] K. Devriendt, A. Ottolini, and S. Steinerberger. Graph curvature via resistance distance. Discrete Applied Mathematics, 348:68-78, 2024. 14 [5] M. Fiedler. Matrices and graphs in geometry. Encyclopedia of mathematics and its applications ; 139. Cambridge University Press, Cambridge, 2011. 3, 14 [6] H. Gimperlein, M. Goffeng, and N. Louca. The magnitude and spectral geometry, 2024. . 4 [7] W. H. Haemers. Interlacing eigenvalues and graphs. Linear Algebra and its Applications, 226-228:593-616, 1995. 10 [8] E. V. Haynsworth. On the Schur complement. Basel Mathematical Notes, (20), 1968. 12 [9] R. Hepworth and S. Willerton. Categorifying the magnitude of a graph. Homology, Homotopy and Applications, 19(2):31-60, 2017. 1 [10] S. Huntsman. Diversity enhancement via magnitude. In Evolutionary Multi-Criterion Optimization (EMO 2023), volume 13994 of Lecture Notes in Computer Science, pages 420-432. Springer, 2023. 1 [11] T. Leinster. The Euler characteristic of a category. Documenta Mathematica, 13:21-49, 2008. 1 [12] T. Leinster. The magnitude of metric spaces. Documenta Mathematica, 18:857-905, 2013. 1, 4, 6, 15 [13] T. Leinster. The magnitude of a graph. Mathematical Proceedings of the Cambridge Philosophical Society, 166(2):247-264, 2019. 4 [14] T. Leinster and M. W. Meckes. Maximizing diversity in biology and beyond. Entropy, 18(3):88, 2016. 1 17 [15] T. Leinster and M. W. Meckes. Magnitude: a bibliography. https://webhomes.maths.ed.ac.uk/ ̃tl/magbib/, 2025. Accessed: 2025-06-23. 1 [16] T. Leinster and M. Shulman. Magnitude homology of enriched categories and metric spaces. Algebraic & Geometric Topology, 21(5):2175-2221, 2021. 1 [17] T. Leinster and S. Willerton. On the asymptotic magnitude of subsets of Euclidean space. Geometriae Dedicata, 164:287-310, 2013. 2, 4, 8 [18] M. W. Meckes. Positive definite metric spaces. Positivity : an international journal devoted to the theory and applications of positivity in analysis, 17(3):733-757, 2012. 1, 6 [19] S. Steinerberger. Curvature on graphs via equilibrium measures. Journal of Graph Theory, 103:415-436, 2022. 14 Authors' address: Karel Devriendt, 18
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2510.14692
|
Unique Hierarchical Rotational Dynamics Induces Ultralow Lattice Thermal
Conductivity in Cyanide-bridged Framework Materials
Zhunyun Tang,1 Xiaoxia Wang,1 Jin Li,1 Chaoyu He,1 Mingxing Chen,2 Chao Tang,1, ∗and Tao Ouyang1, †
1School of Physics and Optoelectronics & Hunan Provincial Key Laboratory of
Computational Condensed Matter Physics and Quantum Materials Engineering,
Xiangtan University, Xiangtan 411105, Hunan, China
2School of Physics and Electronics, Hunan Normal University, Changsha 410081, Hunan, China
(Dated: October 17, 2025)
The pursuit of materials combining light constituent elements with ultralow lattice thermal con-
ductivity (κL) is crucial to advancing technologies like thermoelectrics and thermal barrier coatings,
yet it remains a formidable challenge to date. Herein, we achieve ultralow κL in lightweight cyanide-
bridged framework materials (CFMs) through the rational integration of properties such as the
hierarchical vibrations exhibited in superatomic structures and rotational dynamics exhibited in
perovskites. Unique hierarchical rotation behavior leads to multiple negative peaks in Gr¨uneisen
parameters across a wide frequency range, thereby inducing pronounced negative thermal expan-
sion and strong cubic anharmonicity in CFMs.
Meanwhile, the synergistic effect between large
four-phonon scattering phase space (induced by phonon quasi-flat bands and wide bandgaps) and
strong quartic anharmonicity (associated with rotation modes) leads to giant quartic anharmonic
scattering rates in these materials. Consequently, the κL of these CFMs decreases by one to two
orders of magnitude compared to the known perovskites or perovskite-like materials with equiva-
lent average atomic masses. For instance, the Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 (X
= B, Al, Ga, In) exhibit ultralow room-temperature κL values ranging from 0.35 to 0.81 W/mK.
This work not only establishes CFMs as a novel and rich platform for studying extreme phonon
anharmonicity, but also provides a new paradigm for achieving ultralow thermal conductivity in
lightweight materials via the conscious integration of hierarchical and rotational dynamics.
I.
INTRODUCTION
The pursuit of materials with ultralow lattice thermal
conductivity (κL) is a central endeavor in condensed mat-
ter physics and materials science, driven by applications
in thermoelectrics, thermal insulation, and barrier coat-
ings [1–4].
Conventional strategies for suppressing κL
often rely on the incorporation of heavy constituent el-
ements or structural complexity [5–8], which typically
conflicts with the demand for lightweight functional ma-
terials in applications ranging from aerospace compo-
nents to flexible electronics.
Consequently, the explo-
ration of lightweight materials that simultaneously ex-
hibit ultralow κL remains a formidable goal, one that
requires moving beyond conventional phonon scattering
paradigms toward designs that leverage emergent lattice
dynamics and anharmonicity.
Recent advances highlight the role of specific bonding
characteristics and dynamic structural features in driv-
ing phonon anharmonicity and collapse of thermal trans-
port [9–16].
Specifically, the presence of anti-bonding
valence states leads to bond weakening and formation of
soft optical phonons, resulting in intrinsic phonon scat-
tering rates far exceeding those in conventional semicon-
ductors [9, 10]. Alternatively, the concept of hierarchical
vibrational architectures (inspired by superatomic crys-
tals) has emerged as a potent design strategy [11]. Hier-
∗tang chao@xtu.edu.cn.
† ouyangtao@xtu.edu.cn.
archical vibrations endow additional rotational or trans-
lational degrees of freedom, leading to strong coupling
between internal optical phonons and acoustic modes,
further generating extreme phonon anharmonic scatter-
ing [12–16]. In a parallel development, it has been found
that low-energy optical modes associated with octahedral
rotations in perovskites (perovskite-like materials) often
exhibit strong anharmonicity, leading to low κL and pro-
nounced wave-like tunneling behavior [17–21].
Although these mechanisms have been individually es-
tablished in disparate material classes, a fundamental
question remains: can these distinct phonon-suppressing
features be intentionally combined within a lightweight
material system to achieve ultralow thermal conductivity
through synergistic effects? Based on this design princi-
ple, we perform a systematical search in the Inorganic
Crystal Structure Database (ICSD) and discover that
cyanide-bridged framework materials (CFMs) present an
exceptionally promising platform for such an integrated
approach [22].
Their framework structures, compris-
ing M—C≡N—M′ linkages, the flexibility of the cyanide
bridge allows for dynamic disorder and local distortions,
reminiscent of perovskites, while the subunits within the
framework suggests a pathway toward hierarchical vi-
bration [23–25]. Despite these intriguing features, their
thermal transport properties remain largely unexplored,
particularly in relation to the interplay among local ro-
tational degrees of freedom, hierarchical vibrations, and
collective low-energy dynamics.
In this study, we fill this gap by employing first-
principles calculations combined with machine learning
arXiv:2510.14692v1 [cond-mat.mtrl-sci] 16 Oct 2025
2 of 10
II
RESULTS AND DISCUSSION
potentials.
To ensure reliability, the thermal conduc-
tivity values are cross-validated through unified phonon
theory and large-scale molecular dynamics simulations.
The results demonstrate that this family of materials
uniquely combines the key phonon-suppressing mecha-
nisms previously identified in separate systems: the hier-
archical vibrational architecture characteristic of super-
atomic crystals and the rotational dynamics typical of
perovskites. This cooperative mechanism leads to emer-
gent phenomena, including multiple negative Gr¨uneisen
parameter peaks across a broad frequency range, signal-
ing pronounced negative thermal expansion and strong
cubic anharmonicity. Most notably, we identify a syn-
ergistic interplay between the large four-phonon scatter-
ing phase space (arising from phonon quasi-flat bands
and wide bandgaps) and the strong quartic anharmonic-
ity (associated with rotation modes), leading to giant
four-phonon scattering rates that dominate thermal re-
sistance. As a result, the κL of these CFMs is suppressed
by one to two orders of magnitude compared to the
previously known perovskite and perovskite-like mate-
rials with equivalent average atomic mass. Specific com-
pounds, such as Cd(CN)2, NaB(CN)4, LiIn(CN)4, and
AgX(CN)4 (X = B, Al, Ga, In), exhibit ultralow room-
temperature κL values ranging from 0.35 to 0.81 W/mK,
positioning them among the lowest known thermal con-
ductivity materials for their weight class. These findings
provide a promising strategy for designing lightweight
materials with ultralow thermal conductivity.
II.
RESULTS AND DISCUSSION
Figure 1(a) illustrates a simplified model of cubic-
phase atomic structure, where four distinct Wyckoff po-
sitions are represented by differently colored spheres.
When positions 1 and 2 are occupied by specific tetra-
hedral superatomic clusters ZnC4 or CdC4, and posi-
tions 3 and 4 by ZnN4 or CdN4 clusters, these units
connect through —C≡N— bridges to form the classic
Zn(CN)2 or Cd(CN)2 framework materials with P−43m
space group, as depicted in Fig. 1(b). If positions 1 and
2 are occupied by BC4 clusters and positions 3 and 4
by AgN4, CuN4, or LiN4 clusters, the resulting structure
corresponds to AgB(CN)4, CuB(CN)4, or LiB(CN)4, re-
spectively, all crystallizing in the P−43m space group,
as shown in Fig. 1(c).
In contrast, when positions 1
and 4 are occupied by BC4 clusters and positions 2 and
3 by NaN4 clusters, the framework adopts the Fd−3m
space group, characteristic of NaB(CN)4, see Fig. 1(d).
Previous experimental and theoretical studies have ex-
tensively explored the thermal expansion properties of
these cyanide-bridged framework materials (CFMs) [25–
30]. The results reveal that these compounds generally
exhibit negative thermal expansion (NTE) behavior. Fig-
ure S2 of the Supporting Information (SI) demonstrates
that the calculated NTE behavior exhibits markedly im-
proved agreement with experimental data when the van
der Waals (vdW) correction term is included in molecular
dynamics (MD) simulations. This indicates that inter-
cluster vdW interactions are crucial for accurately cap-
turing the lattice dynamics and anharmonicity that drive
the pronounced NTE in these cyanide frameworks. No-
tably, despite their low average atomic mass, Zn(CN)2,
Cd(CN)2, AgB(CN)4, and NaB(CN)4 hold exceptionally
pronounced NTE effects, surpassing most other known
NTE materials [31].
Such remarkable behavior fore-
shows the presence of intriguing lattice dynamics in these
CFMs. Therefore, we select Zn(CN)2, AgB(CN)4, and
NaB(CN)4 as representative systems and further com-
puted their phonon spectra. As illustrated in Figs. 1(e-
g), the presence of light elements leads to remarkably
high optical phonon cutoff frequencies.
The hierarchi-
cal vibrational behavior inherent to superatomic struc-
tures gives rise to a plethora of localized phonon modes,
inducing numerous phonon quasi-flat bands and wide
bandgaps. Moreover, certain phonon modes exhibit pro-
nounced temperature-dependent behavior, particularly
the optical branches around 2–6 THz and 10 THz, sug-
gesting strong anharmonic effects in these specific modes.
In conventional perovskite or perovskite-like NTE ma-
terials, great NTE coefficients mainly originate from
large negative Gr¨uneisen parameters associated with ex-
tremely low-frequency soft modes linked to octahedral
rotations [21, 32, 33]. However, these traditional NTE
materials typically exhibit only a limited set of rotation
phonon modes due to constrained rotational degrees of
freedom. In contrast, the unique hierarchical vibrations
confer additional rotational freedom in CFMs, resulting
in richer rotation phonon modes, as depicted in the insets
of Fig. 2 (and Fig. S7 of the SI). The positions of these
rotation modes are marked on the phonon dispersion in
Fig. S8 of the SI, with AgB(CN)4 as an example. The
distinctive hierarchical rotation behavior leads to multi-
ple negative peaks in the Gr¨uneisen parameter (γ) across
a broad frequency range, as shown in the middle pan-
els of Figs. 1(e-g). This widespread negative γ distribu-
tion significantly enhances the overall cubic anharmonic-
ity and is a key driver of the pronounced NTE observed in
these CFMs. Moreover, by mapping the potential energy
curves along the coordinates of these rotation modes, one
can find that the actual potential deviates significantly
from the harmonic approximation (quadratic fitting), as
demonstrated in Figs. 2(a–h).
It is worth noting that
the fitted curves agree well with the DFT data points
when the quartic term is introduced.
This deviation
provides direct evidence of strong quartic anharmonic-
ity associated with these rotation modes. Additionally,
the phonon quasi-flat bands and wide bandgaps prevalent
in these materials generally expand the available phase
space for four-phonon scattering processes [34, 35], as
shown in Fig. S9 of the SI. As a result, the synergistic
effect between large four-phonon scattering phase space
and intrinsically strong quartic anharmonicity leads to
giant four-phonon anharmonic scattering rates (SRs) in
these materials, as quantitatively shown in the right pan-
2
3 of 10
II
RESULTS AND DISCUSSION
FIG. 1. (a) A simplified model of cubic crystal structure, with each sphere consisting of a tetrahedral superatom cluster.
Scheme of unit cell structure of (b) Zn(CN)2 and Cd(CN)2, (c) LiB(CN)4, CuB(CN)4, and AgB(CN)4, (d) and NaB(CN)4.
(e-g) Calculated finite-temperature phonon dispersion, accounting for thermal expansion effects, for Zn(CN)2, AgB(CN)4, and
NaB(CN)4, respectively. The middle and right panels of (e-g) present the mode γ and SRs (including three- and four-phonon
processes) at room temperature, respectively.
els of Figs. 1(e-g).
We now turn our attention to the thermal transport
properties of these CFMs. First, the lattice thermal con-
ductivity (κL) of Zn(CN)2, AgB(CN)4, and NaB(CN)4
in the temperature range of 100 to 400 K is obtained
based on the self-consistent phonon (SCP) approach com-
bined with the unified phonon transport theory [36, 37],
accounting of the thermal expansion effect.
The re-
sults are decomposed into particle-like (κP
L) and wave-
like coherence (κC
L) contributions, with and without four-
phonon (4ph) scattering. All three materials exhibit low
κL values, which is consistent with the strong anhar-
monicity and enhanced scattering discussed above.
In
Zn(CN)2, the inclusion of four-phonon processes dras-
tically reduces κP
L from 4.84 (3ph only; κP
3ph) to 1.74
W/mK (3+4ph; κP
3+4ph) at 300 K, with a 64% sup-
pression [Fig. 3(a)]. Such suppression is even more pro-
nounced in AgB(CN)4 and NaB(CN)4, where κP
L is re-
duced by 75% (from 1.35 to 0.35 W/mK) and 90% (from
2.15 to 0.22 W/mK), respectively [Figs. 3(b-c)]. The co-
herence contribution κC
3+4ph remains minor in Zn(CN)2,
accounting for only
10% of the total κP+C
3+4ph at 300
K. However, in AgB(CN)4 and NaB(CN)4, κC
3+4ph con-
tributes significantly to the total thermal conductivity,
even exceeding κP
3+4ph at elevated temperatures. Con-
sequently, κL in the two systems is increasingly domi-
nated by wave-like phonon tunneling, leading to a weak
temperature dependence.
Remarkably, NaB(CN)4 ex-
hibits a positive temperature dependence of κL at high
temperatures, a behavior previously observed in other
ultralow κL materials [38–41].
This is mainly because
the stronger anharmonic scattering and smaller average
phonon band spacing in NaB(CN)4 induce more phonon
modes with lifetimes below the Wigner limit [42] (see Fig.
3
4 of 10
II
RESULTS AND DISCUSSION
FIG. 2. The potential well for different rotation modes. The quadratic and quartic fitting curves are represented by gray dashed
lines and yellow solid lines, respectively. The insets label the vibration eigenvectors for different rotation modes
S10 of the SI), resulting in glass-like thermal transport.
However, we also observe that certain extreme phonon
modes of NaB(CN)4 enter the overdamped regime at
high temperatures (Fig.
S10 of the SI), which might
render the standard phonon gas model invalid [43, 44].
Therefore, we further employ homogeneous nonequilib-
rium molecular dynamics (HNEMD) simulations to pro-
vide a more robust description for thermal transport
[45]. The HNEMD results are in good agreement with
the unified transport theory calculations at low tempera-
tures, but reveal a continuing decrease in κL at high tem-
peratures, underscoring the limitations of the standard
phonon gas model in giant anharmonic systems. Finally,
we extend HNEMD simulations to 15 CFMs, including
Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4, CuX(CN)4,
and AgX(CN)4 (X=B, Al, Ga, In). As summarized in
Fig. 4, thermal transport is further suppressed as the
X atom transitions from B to In.
Several compounds
[Cd(CN)2, NaB(CN)4, LiIn(CN)4, and all AgX(CN)4
members] exhibit ultralow room-temperature values be-
tween 0.35 and 0.81 W/mK. Notably, NaX(CN)4 (X=Al,
Ga, In) are excluded due to their complex thermal ex-
pansion behavior (possibly involving multiple structural
phase transitions), as shown in Fig. S3 of the SI, which is
beyond the scope of this study. In contrast, AgIn(CN)4
shows monotonic positive thermal expansion at low tem-
peratures, but transitions to negative thermal expan-
sion above 200 K. This appears to arise from the low-
temperature instability of the cubic phase, as evidenced
by imaginary phonon modes below 200 K that vanish
at higher temperatures [Fig. S11 of the SI], mirroring
behavior observed in other cubic frameworks [18, 46].
To elucidate the microscopic mechanism underlying
the gradual suppression of thermal conductivity as the
X atom transitions from B to In in these CFMs, we an-
alyze the evolution of chemical bonding and its impact
on lattice anharmonicity. Figure 5(a) presents the elec-
tron localization function (ELF) for AgX(CN)4 (X=B,
Al, Ga, In). The intense localization between B and C
atoms in AgB(CN)4 signifies strong bonding character of
the B–C bond. However, as B transitions to In, this local-
ization gradually diminishes, suggesting that the bond-
ing strength of X–C is progressively weakened. To fur-
ther quantify the bonding strength, the integrated crys-
tal orbital Hamilton population (iCOHP) is calculated
[47], a measure of bond strength, for key atomic pairs,
as shown in inset of Fig. 5(b). The –iCOHP values for
the Ag–N and C–N bonds remain essentially constant
throughout the series. However, the –iCOHP for the X–C
bond decreases substantially from B to In, confirming a
systematic chemical bond weakening. As previously es-
tablished, the strong anharmonicity in these materials is
intimately linked to rotation modes, which primarily in-
volve large-amplitude transverse vibrations of the N and
4
5 of 10
III
CONCLUSION
FIG. 3. The temperature-dependent κL of (a) Zn(CN)2, (b)
AgB(CN)4, and (c) NaB(CN)4 is calculated based on different
models are compared with molecular dynamics simulations.
C atoms. The root-mean-square displacements (RMSD)
of C and N atoms, calculated at 300 K, are presented in
Fig. 5(b). The RMSD values are significantly enhanced
in AgIn(CN)4 compared to AgB(CN)4, indicating more
pronounced vibrational amplitudes and thus larger dy-
namic tetrahedral distortion. This enhanced flexibility
directly leads to stronger lattice anharmonicity, which is
quantitatively confirmed by the markedly increased neg-
ative Gr¨uneisen parameters in AgIn(CN)4 [see Fig. 5(c)].
Consequently, AgIn(CN)4 exhibits more pronounced neg-
ative thermal expansion (Fig. S3 of the SI) and stronger
phonon scattering rates (SRs) [Fig. 5(d)], thereby further
suppression of thermal transport properties.
Finally, we plot the room-temperature thermal con-
ductivity (κL) of these CFMs as a function of average
atomic mass ( ¯
M) in Fig. 6, alongside data for classic
perovskites and perovskite-like materials for comparison
[17, 19–21, 38, 48–64].
Obviously, the
¯
M–κL trend of
CFMs deviates significantly from that of perovskite ma-
terials. Despite possessing relatively low ¯
M, the CFMs
still exhibit ultralow κL, roughly 1-2 orders of magnitude
lower than those of perovskites with comparable ¯
M.
FIG. 4.
The temperature-dependent κL calculated using
the HNEMD method for (a) Zn(CN)2 and Cd(CN)2, (b)
AgX(CN)4, (c) CuX(CN)4, and (d) LiX(CN)4 (X=B, Al, Ga,
In).
FIG. 5. (a) The ELF, (b) RMSD of N and C atoms, (c) γ,
and (d) phonon SRs for AgX(CN)4 (X=B, Al, Ga, In). The
inset of (b) shows the –iCOHP for different atomic bonds.
III.
CONCLUSION
In summary, we have unveiled the microscopic ori-
gin of ultralow thermal conductivity (κL) in lightweight
cyanide-bridged framework materials (CFMs), challeng-
5
6 of 10
IV
METHODS
FIG. 6. The dependence of room temperature κL on average
mass.
The red spheres mark the 15 CFMs studied in this
paper. The blue squares mark some typical perovskite and
perovskite-like materials.
ing the conventional trade-off between low average
atomic mass and efficient thermal transport.
Our re-
sults reveal that the suppressed thermal transport be-
havior in these materials stems from exceptionally strong
three- and four-phonon anharmonic scattering, driven
by abundant hierarchical rotation modes.
Remark-
ably, four-phonon scattering rates even surpass those
of three-phonon processes for numerous modes, which
significantly suppresses thermal transport.
Such be-
havior originates from the large four-phonon scatter-
ing phase space induced by phonon quasi-flat bands
and wide bandgaps as well as the strong quartic an-
harmonicity accompanied by rotation modes. Moreover,
coherent thermal conductivity contributes significantly
in AgB(CN)4 and NaB(CN)4, with its contribution ex-
ceeding that of the particle-like term at elevated tem-
peratures.
This leads to a positive temperature de-
pendence of thermal conductivity in NaB(CN)4 at high
temperature regions. However, certain extreme phonon
modes of NaB(CN)4 enter the overdamped regime at
high temperatures, which render the standard phonon
gas model invalid. Therefore, we further employ homo-
geneous nonequilibrium molecular dynamics (HNEMD)
simulations to provide a more robust description for
thermal transport.
The HNEMD simulations confirm
the unified transport theory results at low tempera-
tures but reveal a continuing decrease in κL at high
temperatures, underscoring the limitations of the stan-
dard phonon gas model in giant anharmonic systems.
Finally, we extend HNEMD simulations to 15 CFMs,
including Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4,
CuX(CN)4, and AgX(CN)4 (X=B, Al, Ga, In).
As
the X atom transitions from B to In, the corresponding
system exhibits weaker bonding and greater root-mean-
square displacement, thereby inducing stronger anhar-
monic scattering and lower κL. Among them, Cd(CN)2,
NaB(CN)4, LiIn(CN)4, and AgX(CN)4 exhibit ultralow
room-temperature κL values ranging from 0.35 to 0.81
W/mK. These results shed light on the physical mecha-
nisms of thermal transport in CFMs, and suggest a prac-
ticable strategy for developing novel lightweight materi-
als with ultralow thermal conductivity.
IV.
METHODS
All density-functional theory (DFT) calculations are
implemented using the Vienna Ab initio Simulation
Package (VASP) [65] with the projector augmented wave
(PAW) [66] method. The PBE [67] exchange-correlation
functional is used to relax the geometric structure. The
kinetic energy cutoff for wave function is set to 500 eV,
and the k-mesh of 8 × 8 × 8 is used to sample the Bril-
louin Zone. The DFT-D3 correction [68] is adopted to ac-
count for van der Waals (vdW) interactions. The ALAM-
ODE package is used to compute both the harmonic
second-order interatomic force constants (IFCs) via fi-
nite displacement method and the anharmonic higher-
order IFCs through compressive sensing lattice dynamics
[36, 50, 69]. High precision multi-atomic cluster expan-
sion (MACE) machine learning potential is constructed
to capture IFCs more efficiently [70].
Self-consistent
phonon (SCP) [36, 50] calculation is performed to obtain
the temperature-dependent phonon dispersion and corre-
sponding second-order IFCs. The lattice thermal conduc-
tivity is calculated by a unified thermal transport theory
and implemented in the modified FOURPHONON soft-
ware [37, 71–73]. The equation for κL is as follows
κP/C
L
=
ℏ2
kBT 2V N
X
q
X
j,j′
ωqj + ωqj′
2
νqjj′ ⊗νqjj′
×ωqjnqj(nqj + 1) + ωqj′nqj′ (nqj′ + 1)
4 (ωqj −ωqj′)2 + (Γqj + Γqj′)2
(Γqj + Γqj′) ,
(1)
where the superscripts P and C represent the contribu-
tions from populations (particle-like propagation) and
coherences (wave-like phonon tunneling), respectively.
When j=j′, Eq. 1 is equivalent to the Peierls-Boltzmann
transport equation, i.e., the particle-like transport term.
Otherwise, it corresponds to the wave-like transport
term.
Moreover, the κL is also obtained using homo-
geneous nonequilibrium molecular dynamics (HNEMD)
simulations based on the trained neuroevolution poten-
tial (NEP) and implemented in the GPUMD software
[45, 74, 75]. To eliminate the size effects, the large su-
percell exceeding 85 × 85 × 85 ˚A3 (containing 40960
atoms) is adopted. Corresponding supercell is also ap-
plied to simulate the thermal expansion behavior. More
computational details are provided in the Supporting In-
formation (SI).
6
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IV
METHODS
ASSOCIATED CONTENT
Supporting Information The Supporting Information
is available free of charge at ***
Detailed method description, lattice constant, root-
mean-square error of machine learning potential models,
training parameters for fitting machine learning potential
models, thermal expansion, phonon spectrum, rotation
mode eigenvectors, position markers of rotation modes
within the phonon spectrum, weighted scattering phase
space, and phonon lifetime.
DATA AVAILABILITY
The data supporting the findings of this study are
available within this article and its Supporting Infor-
mation.
The trained NEP and MACE models are
openly available in the repository https://github.com/
ZhunyunTang/CFMs-ML.
Additional data are available
from the corresponding author on reasonable request.
CODE AVAILABILITY
The source code for GPUMD is available at https:
//github.com/brucefan1983/GPUMD,
for ALAMODE
is available at https://github.com/ttadano/alamode,
for SHENGBTE is available at https://www.shengbte.
org,
and
FOURPHONON
is
available
at
https:
//github.com/FourPhonon/FourPhonon,
for
PYNEP
is
available
at
https://github.com/bigd4/PyNEP,
for
the
modified
version
of
FOURPHONON
is
available
at
https://github.com/ZhunyunTang/
FourPhonon--wigner-sampling.
ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-
ence Foundation of China (Grant No.
52372260), the
Science Fund for Distinguished Young Scholars of Hu-
nan Province of China (No.
2024JJ2048), the Hunan
Provincial Innovation Foundation for Postgraduate (No.
CX20240058), and the Youth Science and Technology
Talent Project of Hunan Province (No.
2022RC1197).
The calculation in this work is partly supported by
the high-performance computing platform of school of
physics and optoelectronics.
AUTHOR CONTRIBUTIONS
Z.T. contributed to the methodology, software, all cal-
culations, formal analysis, visualization, and writing of
original draft. X.W., J.L., C.H., and M.C. contributed
to discussions and writing of manuscript. C.T. and T.O.
were responsible for supervision, writing and discussion.
COMPETING INTERESTS
The authors declare no competing interests.
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10
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Unique Hierarchical Rotational Dynamics Induces Ultralow Lattice Thermal Conductivity in Cyanide-bridged Framework Materials Zhunyun Tang,1 Xiaoxia Wang,1 Jin Li,1 Chaoyu He,1 Mingxing Chen,2 Chao Tang,1, ∗and Tao Ouyang1, † 1 411105, Hunan, China 2 410081, Hunan, China (Dated: October 17, 2025) The pursuit of materials combining light constituent elements with ultralow lattice thermal conductivity (κL) is crucial to advancing technologies like thermoelectrics and thermal barrier coatings, yet it remains a formidable challenge to date. Herein, we achieve ultralow κL in lightweight cyanidebridged framework materials (CFMs) through the rational integration of properties such as the hierarchical vibrations exhibited in superatomic structures and rotational dynamics exhibited in perovskites. Unique hierarchical rotation behavior leads to multiple negative peaks in Gr ̈uneisen parameters across a wide frequency range, thereby inducing pronounced negative thermal expansion and strong cubic anharmonicity in CFMs. Meanwhile, the synergistic effect between large four-phonon scattering phase space (induced by phonon quasi-flat bands and wide bandgaps) and strong quartic anharmonicity (associated with rotation modes) leads to giant quartic anharmonic scattering rates in these materials. Consequently, the κL of these CFMs decreases by one to two orders of magnitude compared to the known perovskites or perovskite-like materials with equivalent average atomic masses. For instance, the Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 (X = B, Al, Ga, In) exhibit ultralow room-temperature κL values ranging from 0.35 to 0.81 W/mK. This work not only establishes CFMs as a novel and rich platform for studying extreme phonon anharmonicity, but also provides a new paradigm for achieving ultralow thermal conductivity in lightweight materials via the conscious integration of hierarchical and rotational dynamics. I. INTRODUCTION The pursuit of materials with ultralow lattice thermal conductivity (κL) is a central endeavor in condensed matter physics and materials science, driven by applications in thermoelectrics, thermal insulation, and barrier coatings [1-4]. Conventional strategies for suppressing κL often rely on the incorporation of heavy constituent elements or structural complexity [5-8], which typically conflicts with the demand for lightweight functional materials in applications ranging from aerospace components to flexible electronics. Consequently, the exploration of lightweight materials that simultaneously exhibit ultralow κL remains a formidable goal, one that requires moving beyond conventional phonon scattering paradigms toward designs that leverage emergent lattice dynamics and anharmonicity. Recent advances highlight the role of specific bonding characteristics and dynamic structural features in driving phonon anharmonicity and collapse of thermal transport [9-16]. Specifically, the presence of anti-bonding valence states leads to bond weakening and formation of soft optical phonons, resulting in intrinsic phonon scattering rates far exceeding those in conventional semiconductors [9, 10]. Alternatively, the concept of hierarchical vibrational architectures (inspired by superatomic crystals) has emerged as a potent design strategy [11]. Hier- ∗tang . † . archical vibrations endow additional rotational or translational degrees of freedom, leading to strong coupling between internal optical phonons and acoustic modes, further generating extreme phonon anharmonic scattering [12-16]. In a parallel development, it has been found that low-energy optical modes associated with octahedral rotations in perovskites (perovskite-like materials) often exhibit strong anharmonicity, leading to low κL and pronounced wave-like tunneling behavior [17-21]. Although these mechanisms have been individually established in disparate material classes, a fundamental question remains: can these distinct phonon-suppressing features be intentionally combined within a lightweight material system to achieve ultralow thermal conductivity through synergistic effects? Based on this design principle, we perform a systematical search in the Inorganic Crystal Structure Database (ICSD) and discover that cyanide-bridged framework materials (CFMs) present an exceptionally promising platform for such an integrated approach [22]. Their framework structures, comprising M-C≡N-M′ linkages, the flexibility of the cyanide bridge allows for dynamic disorder and local distortions, reminiscent of perovskites, while the subunits within the framework suggests a pathway toward hierarchical vibration [23-25]. Despite these intriguing features, their thermal transport properties remain largely unexplored, particularly in relation to the interplay among local rotational degrees of freedom, hierarchical vibrations, and collective low-energy dynamics. In this study, we fill this gap by employing firstprinciples calculations combined with machine learning 16 Oct 2025 2 of 10 II RESULTS AND DISCUSSION potentials. To ensure reliability, the thermal conductivity values are cross-validated through unified phonon theory and large-scale molecular dynamics simulations. The results demonstrate that this family of materials uniquely combines the key phonon-suppressing mechanisms previously identified in separate systems: the hierarchical vibrational architecture characteristic of superatomic crystals and the rotational dynamics typical of perovskites. This cooperative mechanism leads to emergent phenomena, including multiple negative Gr ̈uneisen parameter peaks across a broad frequency range, signaling pronounced negative thermal expansion and strong cubic anharmonicity. Most notably, we identify a synergistic interplay between the large four-phonon scattering phase space (arising from phonon quasi-flat bands and wide bandgaps) and the strong quartic anharmonicity (associated with rotation modes), leading to giant four-phonon scattering rates that dominate thermal resistance. As a result, the κL of these CFMs is suppressed by one to two orders of magnitude compared to the previously known perovskite and perovskite-like materials with equivalent average atomic mass. Specific compounds, such as Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 (X = B, Al, Ga, In), exhibit ultralow roomtemperature κL values ranging from 0.35 to 0.81 W/mK, positioning them among the lowest known thermal conductivity materials for their weight class. These findings provide a promising strategy for designing lightweight materials with ultralow thermal conductivity. II. RESULTS AND DISCUSSION Figure 1(a) illustrates a simplified model of cubicphase atomic structure, where four distinct Wyckoff positions are represented by differently colored spheres. When positions 1 and 2 are occupied by specific tetrahedral superatomic clusters ZnC4 or CdC4, and positions 3 and 4 by ZnN4 or CdN4 clusters, these units connect through -C≡N- bridges to form the classic Zn(CN)2 or Cd(CN)2 framework materials with P-43m space group, as depicted in Fig. 1(b). If positions 1 and 2 are occupied by BC4 clusters and positions 3 and 4 by AgN4, CuN4, or LiN4 clusters, the resulting structure corresponds to AgB(CN)4, CuB(CN)4, or LiB(CN)4, respectively, all crystallizing in the P-43m space group, as shown in Fig. 1(c). In contrast, when positions 1 and 4 are occupied by BC4 clusters and positions 2 and 3 by NaN4 clusters, the framework adopts the Fd-3m space group, characteristic of NaB(CN)4, see Fig. 1(d). Previous experimental and theoretical studies have extensively explored the thermal expansion properties of these cyanide-bridged framework materials (CFMs) [2530]. The results reveal that these compounds generally exhibit negative thermal expansion (NTE) behavior. Figure S2 of the Supporting Information (SI) demonstrates that the calculated NTE behavior exhibits markedly improved agreement with experimental data when the van der Waals (vdW) correction term is included in molecular dynamics (MD) simulations. This indicates that intercluster vdW interactions are crucial for accurately capturing the lattice dynamics and anharmonicity that drive the pronounced NTE in these cyanide frameworks. Notably, despite their low average atomic mass, Zn(CN)2, Cd(CN)2, AgB(CN)4, and NaB(CN)4 hold exceptionally pronounced NTE effects, surpassing most other known NTE materials [31]. Such remarkable behavior foreshows the presence of intriguing lattice dynamics in these CFMs. Therefore, we select Zn(CN)2, AgB(CN)4, and NaB(CN)4 as representative systems and further computed their phonon spectra. As illustrated in Figs. 1(eg), the presence of light elements leads to remarkably high optical phonon cutoff frequencies. The hierarchical vibrational behavior inherent to superatomic structures gives rise to a plethora of localized phonon modes, inducing numerous phonon quasi-flat bands and wide bandgaps. Moreover, certain phonon modes exhibit pronounced temperature-dependent behavior, particularly the optical branches around 2-6 THz and 10 THz, suggesting strong anharmonic effects in these specific modes. In conventional perovskite or perovskite-like NTE materials, great NTE coefficients mainly originate from large negative Gr ̈uneisen parameters associated with extremely low-frequency soft modes linked to octahedral rotations [21, 32, 33]. However, these traditional NTE materials typically exhibit only a limited set of rotation phonon modes due to constrained rotational degrees of freedom. In contrast, the unique hierarchical vibrations confer additional rotational freedom in CFMs, resulting in richer rotation phonon modes, as depicted in the insets of Fig. 2 (and Fig. S7 of the SI). The positions of these rotation modes are marked on the phonon dispersion in Fig. S8 of the SI, with AgB(CN)4 as an example. The distinctive hierarchical rotation behavior leads to multiple negative peaks in the Gr ̈uneisen parameter (γ) across a broad frequency range, as shown in the middle panels of Figs. 1(e-g). This widespread negative γ distribution significantly enhances the overall cubic anharmonicity and is a key driver of the pronounced NTE observed in these CFMs. Moreover, by mapping the potential energy curves along the coordinates of these rotation modes, one can find that the actual potential deviates significantly from the harmonic approximation (quadratic fitting), as demonstrated in Figs. 2(a-h). It is worth noting that the fitted curves agree well with the DFT data points when the quartic term is introduced. This deviation provides direct evidence of strong quartic anharmonicity associated with these rotation modes. Additionally, the phonon quasi-flat bands and wide bandgaps prevalent in these materials generally expand the available phase space for four-phonon scattering processes [34, 35], as shown in Fig. S9 of the SI. As a result, the synergistic effect between large four-phonon scattering phase space and intrinsically strong quartic anharmonicity leads to giant four-phonon anharmonic scattering rates (SRs) in these materials, as quantitatively shown in the right pan2 3 of 10 II RESULTS AND DISCUSSION FIG. 1. (a) A simplified model of cubic crystal structure, with each sphere consisting of a tetrahedral superatom cluster. Scheme of unit cell structure of (b) Zn(CN)2 and Cd(CN)2, (c) LiB(CN)4, CuB(CN)4, and AgB(CN)4, (d) and NaB(CN)4. (e-g) Calculated finite-temperature phonon dispersion, accounting for thermal expansion effects, for Zn(CN)2, AgB(CN)4, and NaB(CN)4, respectively. The middle and right panels of (e-g) present the mode γ and SRs (including three- and four-phonon processes) at room temperature, respectively. els of Figs. 1(e-g). We now turn our attention to the thermal transport properties of these CFMs. First, the lattice thermal conductivity (κL) of Zn(CN)2, AgB(CN)4, and NaB(CN)4 in the temperature range of 100 to 400 K is obtained based on the self-consistent phonon (SCP) approach combined with the unified phonon transport theory [36, 37], accounting of the thermal expansion effect. The results are decomposed into particle-like (κP L) and wavelike coherence (κC L) contributions, with and without fourphonon (4ph) scattering. All three materials exhibit low κL values, which is consistent with the strong anharmonicity and enhanced scattering discussed above. In Zn(CN)2, the inclusion of four-phonon processes drastically reduces κP L from 4.84 (3ph only; κP 3ph) to 1.74 W/mK (3+4ph; κP 3+4ph) at 300 K, with a 64% suppression [Fig. 3(a)]. Such suppression is even more pronounced in AgB(CN)4 and NaB(CN)4, where κP L is reduced by 75% (from 1.35 to 0.35 W/mK) and 90% (from 2.15 to 0.22 W/mK), respectively [Figs. 3(b-c)]. The coherence contribution κC 3+4ph remains minor in Zn(CN)2, accounting for only 10% of the total κP+C 3+4ph at 300 K. However, in AgB(CN)4 and NaB(CN)4, κC 3+4ph contributes significantly to the total thermal conductivity, even exceeding κP 3+4ph at elevated temperatures. Consequently, κL in the two systems is increasingly dominated by wave-like phonon tunneling, leading to a weak temperature dependence. Remarkably, NaB(CN)4 exhibits a positive temperature dependence of κL at high temperatures, a behavior previously observed in other ultralow κL materials [38-41]. This is mainly because the stronger anharmonic scattering and smaller average phonon band spacing in NaB(CN)4 induce more phonon modes with lifetimes below the Wigner limit [42] (see Fig. 3 4 of 10 II RESULTS AND DISCUSSION FIG. 2. The potential well for different rotation modes. The quadratic and quartic fitting curves are represented by gray dashed lines and yellow solid lines, respectively. The insets label the vibration eigenvectors for different rotation modes S10 of the SI), resulting in glass-like thermal transport. However, we also observe that certain extreme phonon modes of NaB(CN)4 enter the overdamped regime at high temperatures (Fig. S10 of the SI), which might render the standard phonon gas model invalid [43, 44]. Therefore, we further employ homogeneous nonequilibrium molecular dynamics (HNEMD) simulations to provide a more robust description for thermal transport [45]. The HNEMD results are in good agreement with the unified transport theory calculations at low temperatures, but reveal a continuing decrease in κL at high temperatures, underscoring the limitations of the standard phonon gas model in giant anharmonic systems. Finally, we extend HNEMD simulations to 15 CFMs, including Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4, CuX(CN)4, and AgX(CN)4 (X=B, Al, Ga, In). As summarized in Fig. 4, thermal transport is further suppressed as the X atom transitions from B to In. Several compounds [Cd(CN)2, NaB(CN)4, LiIn(CN)4, and all AgX(CN)4 members] exhibit ultralow room-temperature values between 0.35 and 0.81 W/mK. Notably, NaX(CN)4 (X=Al, Ga, In) are excluded due to their complex thermal expansion behavior (possibly involving multiple structural phase transitions), as shown in Fig. S3 of the SI, which is beyond the scope of this study. In contrast, AgIn(CN)4 shows monotonic positive thermal expansion at low temperatures, but transitions to negative thermal expansion above 200 K. This appears to arise from the lowtemperature instability of the cubic phase, as evidenced by imaginary phonon modes below 200 K that vanish at higher temperatures [Fig. S11 of the SI], mirroring behavior observed in other cubic frameworks [18, 46]. To elucidate the microscopic mechanism underlying the gradual suppression of thermal conductivity as the X atom transitions from B to In in these CFMs, we analyze the evolution of chemical bonding and its impact on lattice anharmonicity. Figure 5(a) presents the electron localization function (ELF) for AgX(CN)4 (X=B, Al, Ga, In). The intense localization between B and C atoms in AgB(CN)4 signifies strong bonding character of the B-C bond. However, as B transitions to In, this localization gradually diminishes, suggesting that the bonding strength of X-C is progressively weakened. To further quantify the bonding strength, the integrated crystal orbital Hamilton population (iCOHP) is calculated [47], a measure of bond strength, for key atomic pairs, as shown in inset of Fig. 5(b). The -iCOHP values for the Ag-N and C-N bonds remain essentially constant throughout the series. However, the -iCOHP for the X-C bond decreases substantially from B to In, confirming a systematic chemical bond weakening. As previously established, the strong anharmonicity in these materials is intimately linked to rotation modes, which primarily involve large-amplitude transverse vibrations of the N and 4 5 of 10 III CONCLUSION FIG. 3. The temperature-dependent κL of (a) Zn(CN)2, (b) AgB(CN)4, and (c) NaB(CN)4 is calculated based on different models are compared with molecular dynamics simulations. C atoms. The root-mean-square displacements (RMSD) of C and N atoms, calculated at 300 K, are presented in Fig. 5(b). The RMSD values are significantly enhanced in AgIn(CN)4 compared to AgB(CN)4, indicating more pronounced vibrational amplitudes and thus larger dynamic tetrahedral distortion. This enhanced flexibility directly leads to stronger lattice anharmonicity, which is quantitatively confirmed by the markedly increased negative Gr ̈uneisen parameters in AgIn(CN)4 [see Fig. 5(c)]. Consequently, AgIn(CN)4 exhibits more pronounced negative thermal expansion (Fig. S3 of the SI) and stronger phonon scattering rates (SRs) [Fig. 5(d)], thereby further suppression of thermal transport properties. Finally, we plot the room-temperature thermal conductivity (κL) of these CFMs as a function of average atomic mass ( ̄ M) in Fig. 6, alongside data for classic perovskites and perovskite-like materials for comparison [17, 19-21, 38, 48-64]. Obviously, the ̄ M-κL trend of CFMs deviates significantly from that of perovskite materials. Despite possessing relatively low ̄ M, the CFMs still exhibit ultralow κL, roughly 1-2 orders of magnitude lower than those of perovskites with comparable ̄ M. FIG. 4. The temperature-dependent κL calculated using the HNEMD method for (a) Zn(CN)2 and Cd(CN)2, (b) AgX(CN)4, (c) CuX(CN)4, and (d) LiX(CN)4 (X=B, Al, Ga, In). FIG. 5. (a) The ELF, (b) RMSD of N and C atoms, (c) γ, and (d) phonon SRs for AgX(CN)4 (X=B, Al, Ga, In). The inset of (b) shows the -iCOHP for different atomic bonds. III. CONCLUSION In summary, we have unveiled the microscopic origin of ultralow thermal conductivity (κL) in lightweight cyanide-bridged framework materials (CFMs), challeng5 6 of 10 IV METHODS FIG. 6. The dependence of room temperature κL on average mass. The red spheres mark the 15 CFMs studied in this paper. The blue squares mark some typical perovskite and perovskite-like materials. ing the conventional trade-off between low average atomic mass and efficient thermal transport. Our results reveal that the suppressed thermal transport behavior in these materials stems from exceptionally strong three- and four-phonon anharmonic scattering, driven by abundant hierarchical rotation modes. Remarkably, four-phonon scattering rates even surpass those of three-phonon processes for numerous modes, which significantly suppresses thermal transport. Such behavior originates from the large four-phonon scattering phase space induced by phonon quasi-flat bands and wide bandgaps as well as the strong quartic anharmonicity accompanied by rotation modes. Moreover, coherent thermal conductivity contributes significantly in AgB(CN)4 and NaB(CN)4, with its contribution exceeding that of the particle-like term at elevated temperatures. This leads to a positive temperature dependence of thermal conductivity in NaB(CN)4 at high temperature regions. However, certain extreme phonon modes of NaB(CN)4 enter the overdamped regime at high temperatures, which render the standard phonon gas model invalid. Therefore, we further employ homogeneous nonequilibrium molecular dynamics (HNEMD) simulations to provide a more robust description for thermal transport. The HNEMD simulations confirm the unified transport theory results at low temperatures but reveal a continuing decrease in κL at high temperatures, underscoring the limitations of the standard phonon gas model in giant anharmonic systems. Finally, we extend HNEMD simulations to 15 CFMs, including Zn(CN)2, Cd(CN)2, NaB(CN)4, LiX(CN)4, CuX(CN)4, and AgX(CN)4 (X=B, Al, Ga, In). As the X atom transitions from B to In, the corresponding system exhibits weaker bonding and greater root-meansquare displacement, thereby inducing stronger anharmonic scattering and lower κL. Among them, Cd(CN)2, NaB(CN)4, LiIn(CN)4, and AgX(CN)4 exhibit ultralow room-temperature κL values ranging from 0.35 to 0.81 W/mK. These results shed light on the physical mechanisms of thermal transport in CFMs, and suggest a practicable strategy for developing novel lightweight materials with ultralow thermal conductivity. IV. METHODS All density-functional theory (DFT) calculations are implemented using the Vienna Ab initio Simulation Package (VASP) [65] with the projector augmented wave (PAW) [66] method. The PBE [67] exchange-correlation functional is used to relax the geometric structure. The kinetic energy cutoff for wave function is set to 500 eV, and the k-mesh of 8 × 8 × 8 is used to sample the Brillouin Zone. The DFT-D3 correction [68] is adopted to account for van der Waals (vdW) interactions. The ALAMODE package is used to compute both the harmonic second-order interatomic force constants (IFCs) via finite displacement method and the anharmonic higherorder IFCs through compressive sensing lattice dynamics [36, 50, 69]. High precision multi-atomic cluster expansion (MACE) machine learning potential is constructed to capture IFCs more efficiently [70]. Self-consistent phonon (SCP) [36, 50] calculation is performed to obtain the temperature-dependent phonon dispersion and corresponding second-order IFCs. The lattice thermal conductivity is calculated by a unified thermal transport theory and implemented in the modified FOURPHONON software [37, 71-73]. The equation for κL is as follows κP/C L = ħ2 kBT 2V N X q X j,j′ ωqj + ωqj′ 2 νqjj′ ⊗νqjj′ ×ωqjnqj(nqj + 1) + ωqj′nqj′ (nqj′ + 1) 4 (ωqj -ωqj′)2 + (Γqj + Γqj′)2 (Γqj + Γqj′) , (1) where the superscripts P and C represent the contributions from populations (particle-like propagation) and coherences (wave-like phonon tunneling), respectively. When j=j′, Eq. 1 is equivalent to the Peierls-Boltzmann transport equation, i.e., the particle-like transport term. Otherwise, it corresponds to the wave-like transport term. Moreover, the κL is also obtained using homogeneous nonequilibrium molecular dynamics (HNEMD) simulations based on the trained neuroevolution potential (NEP) and implemented in the GPUMD software [45, 74, 75]. To eliminate the size effects, the large supercell exceeding 85 × 85 × 85 ̊A3 (containing 40960 atoms) is adopted. Corresponding supercell is also applied to simulate the thermal expansion behavior. More computational details are provided in the Supporting Information (SI). 6 7 of 10 IV METHODS ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge at *** Detailed method description, lattice constant, rootmean-square error of machine learning potential models, training parameters for fitting machine learning potential models, thermal expansion, phonon spectrum, rotation mode eigenvectors, position markers of rotation modes within the phonon spectrum, weighted scattering phase space, and phonon lifetime. DATA AVAILABILITY The data supporting the findings of this study are available within this article and its Supporting Information. The trained NEP and MACE models are openly available in the repository https://github.com/ ZhunyunTang/CFMs-ML. Additional data are available from the corresponding author on reasonable request. CODE AVAILABILITY The source code for GPUMD is available at https: //github.com/brucefan1983/GPUMD, for ALAMODE is available at https://github.com/ttadano/alamode, for SHENGBTE is available at https://www.shengbte. org, and FOURPHONON is available at https: //github.com/FourPhonon/FourPhonon, for PYNEP is available at https://github.com/bigd4/PyNEP, for the modified version of FOURPHONON is available at https://github.com/ZhunyunTang/ FourPhonon--wigner-sampling. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant No. 52372260), the Science Fund for Distinguished Young Scholars of Hunan Province of China (No. 2024JJ2048), the Hunan Provincial Innovation Foundation for Postgraduate (No. CX20240058), and the Youth Science and Technology Talent Project of Hunan Province (No. 2022RC1197). The calculation in this work is partly supported by the high-performance computing platform of school of physics and optoelectronics. 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|
2510.14685
|
Bosonic Laughlin and Moore-Read states from non-Chern flat bands
Hongyu Lu1, 2, 3 and Wang Yao1, 2, 3, ∗
1New Cornerstone Science Lab, Department of Physics, The University of Hong Kong, Hong Kong, China
2HK Institute of Quantum Science & Technology, The University of Hong Kong, Hong Kong, China
3State Key Laboratory of Optical Quantum Materials, The University of Hong Kong, Hong Kong, China
(Dated: October 17, 2025)
The rapid advances in the study of fractional Chern insulators (FCIs) raise a fundamental question: while
initially discovered in flat Chern bands motivated by their topological equivalence to Landau levels, is single-
particle band topology actually a prerequisite for these many-body topological orders emergent at fractional
fillings? Here, we numerically demonstrate bosonic FCIs in two types of non-Chern flat bands in honeycomb
lattices, using exact diagonalization and density matrix renormalization group calculations. In a gapless flat
band with a singular band touching, we observe a Laughlin state at half filling, stabilized by onsite interactions
from the hard-core limit down to arbitrarily small strength. Furthermore, we report the first example of a non-
Abelian FCI in a non-Chern band system: a Moore-Read state at ν = 1 filling of the same singular flat band
with hard-core bosons. Under lattice parameters that realize a gapped trivial band (C = 0) of exact flatness,
we also find the Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much
smaller than the band gap. In this case, the FCI forms as interacting bosons spontaneously avoid the peaks
in quantum metric and Berry curvature, preferentially occupying Brillouin zone region with relatively uniform
quantum geometry. Our work significantly expands the landscape for (non-)Abelian FCIs and broadens the
understanding of their formation beyond the Chern band paradigm.
Introduction.— The recent experimental discovery of frac-
tional Chern insulators (FCI) in correlated moir´e materials
[1–7], together with quantum simulations of their bosonic
analogues in artificial lattices [8, 9], have sparked widespread
interest in this exotic topological matter.
The FCI phase
was first conceived theoretically by translating the integer-to-
fractional hierarchy of the quantum Hall effect in Landau lev-
els (LLs) to lattices systems [10–15]. Initial explorations nat-
urally focused on the scenario of isolated flat Chern band that
resembles the LLs in topology, quantum geometry, and en-
ergy landscape [10–15]. This appears to be sufficient but not
necessary condition for FCI states to emerge in a Bloch band
context. While having ideal quantum geometry (like that of
LLs) is favorable for stabilizing FCI states [16–22], it is not a
strict requirement for their existence in a flat Chern band [23–
25]. FCIs have also been found in models where interaction
strength well exceeds the band gap separating the flat Chern
band [12, 26]. Notably, in some cases, the Chern bands them-
selves are not present in the non-interacting limit but instead
arise from interaction-driven band renormalization [27–32].
The study of FCIs has also moved beyond the flat Chern
band paradigm along different lines. Opposite to the flat band
limit, interaction-driven high-angular-momentum band inver-
sion has been proposed for realizing Laughlin states at inte-
ger filling [33]. Within the flat band limit, exact diagonaliza-
tion (ED) and density matrix renormalization group (DMRG)
calculations have unambiguously demonstrated FCI phases in
fractionally filled non-Chern bands [34, 35], referring to the
situations where band Chern number is either undefinable or
zero. For the former, a notable example is the singular flat
band, with band touching to a dispersive band dictated by real-
space topology [36, 37]. Despite the gapless band structure
and discontinuity of Bloch function at the singular touching,
Laughlin-like FCI has been found in such band hosted by a
fluxed dice lattice [34]. Even more surprisingly, FCI phases
are also discovered in a trivial flat band of zero Chern num-
ber, which is well isolated by a band gap much larger than
the interaction [35]. The FCI state is shown to develop an
inhomogeneous carrier distribution over the Brillouin zone
(BZ), avoiding the peaks of Berry curvature and quantum met-
ric, and preferentially occupying regions of relatively uniform
quantum geometry. Because of the exact flatness allowed in
non-Chern bands [38], the FCI phases are shown to persist
down to an arbitrarily small nearest-neighbor (NN) interaction
in both cases [34, 35]. Non-Chern bands may greatly expand
the landscape for exploring FCI physics, but the findings are
so far limited to Abelian Laughlin states, and to fermionic sys-
tems. Non-Abelian FCIs [39–44] is of paramount importance,
both for fundamental interest and for fault-tolerant quantum
computation [45]. And whether bosonic FCIs [46–48] can
form in such non-Chern bands and the effect of characteris-
tic onsite repulsion are still open questions. Addressing them
is critical for elucidating the mechanism of FCI beyond the
Chern band paradigm and for broadening the potential exper-
imental platforms.
In this work, we demonstrate bosonic FCIs in a honeycomb
lattice that can host either type of non-Chern flat bands, us-
ing ED and iDMRG calculations. At ν = 1/2 filling of the
singular flat band, we identify the bosonic Laughlin state sta-
bilized by onsite interactions from hard-core limit down to
arbitrarily small strength, with the weak-interaction stability
enabled by the exact flatness of the band. In addition, we
show the existence of bosonic Moore-Read state at ν = 1 fill-
ing of the singular flat band with hard-core bosons, the first
example of non-Abelian FCI in non-Chern band systems. Un-
der lattice parameters that realize a gapped trivial flat band,
we further demonstrate the bosonic Laughlin FCI of soft-core
bosons in the isolated band limit where onsite interaction is
much smaller than the band gap. In such a C = 0 band of
exact flatness, we show that the interacting soft-core bosons
arXiv:2510.14685v1 [cond-mat.str-el] 16 Oct 2025
2
⃗a 1
⃗a 2
t1
ηt1
t2ei 2π
3
ηt2ei 2π
3
t2ei 2π
3
ηt2ei 2π
3
t3
η2t3
(a)
η = 0.8
(c)
Isolated trivial flat band
C = 0
-30
0
0.167
Ω
Ω
0.167
0
Singular flat band
η = 1
(b)
Gapless
Gap:
Δ = 0.08
FIG. 1. Non-interacting model. (a) Two orbital model on honeycomb lattice including up to third NN hoppings with −t1 = t2 = t3 = 1.
(b) When η = 1, the lowest band has exact flatness and a singular quadratic touching point to a dispersive band (grey) at Γ. (c) When η < 1,
the flat band is isolated from the dispersive band by a band gap ∆, while retaining the exact flatness, which necessarily means the band Chern
number C = 0. Both (b) the singular flat band and (c) the isolated trivial flat band are color-coded by their Berry curvature distribution Ω(k).
tend to avoid the peak of quantum metric and Berry curva-
ture, and the FCI state is formed primarily in the rest BZ re-
gion of relatively uniform quantum geometry. This finding
implies the generality of the mechanism for forming bosonic
and fermionic FCIs in such isolated C = 0 flat bands.
Models and methods.— The two-orbital model on the honey-
comb lattice is:
H =
X
ij
tij(b†
ibj + h.c.) + µ
X
i
ni +
X
n
Un
n!
X
i
(b†
i)n(bi)n.
(1)
The non-interacting part including NN, second NN, and third
NN hoppings is illustrated in Fig.1 (a). The hopping ampli-
tude |tij| = 1 serves as the energy unit. When η = 1, this non-
interacting part describes a singular flat band model with the
lower band being exactly flat [Fig.1(b)] [35]. The Bloch wave-
function and quantum geometry is ill-defined at the touching
point at Γ, while the quantum geometry elsewhere is relatively
uniform, and we show the Ωdistribution of the singular flat
band in Fig.1 (b). When η < 1, the flat band is isolated
from the upper band by a band gap ∆≈2(η −1)2 [Fig.1
(c)], while retraining the exact flatness [35]. This necessarily
means the band Chern number C = 0. The Ωdistribution
of the C = 0 flat band has a negative peak at Γ, and a pos-
itive and relatively uniform background in the rest of the BZ
[Fig.1 (c)]. µ = (2 + η2)t fixes the non-interacting flat band
at zero energy. In the main text, we will focus on the effect
of onsite interactions by considering both the hard-core (with
some Un being infinite) and soft-core (finite Un) cases, and
will discuss the NN interactions in the supplementary infor-
mation (SI) [49].
We label the number of two-orbital unit cells along the two
primitive vectors as N1 (⃗a1) and N2 (⃗a2) with a total of N1 ×
N2 × 2 sites. The two-band ED simulations only take the
soft-core case such that the Hilbert space with Nb bosons is
C(N1 × N2 × 2 + Nb −1, Nb). The iDMRG simulations
are based on the real-space basis with the maximum onsite
occupation as nmax and the local Hilbert-space dimension is
d = nmax + 1. The hard-core condition here is realized by
limiting nmax, which means Un>nmax = ∞. To address the
0
2
4
6
8
10
0.05
0.1
0.15
0
5
10
15
0.08
0.12
0.16
-1
0
1
2
3
4
0
5
10
NL = 0
1
1
2
3
5
ϕy
0
2π
4π
0
-0.5
-1
2
4
6
ES
ES
⟨ΔQL(ϕy)⟩
(a)
(b)
(d)
(e)
NL
Soft-core
Soft-core
(c) Hard-core
Hard-core
FIG. 2. Laughlin FCI in the singular flat band with arbitrarily
small and large onsite interactions. We consider ν = 1/2 filling
of the singular flat band (η = 1). ED spectra of soft-core bosons
with a very small U2 = 0.001 interaction from N = 4 × 3 × 2 and
N = 4 × 4 × 2 tori are shown in panels (a,b), respectively. For each
system size, the spectral flow of the well gapped two-fold ground
states is shown in the inset. The iDMRG results of hard-core bosons
(Un≥2 = ∞) are shown in panel (c-e). In panel (c), the momentum-
resolved entanglement spectrum (ES) of the charge sector (NL =
0) with the largest eigenvalue of the reduced density matrix shows
the characteristic counting {1, 1, 2, 3, 5, ...} of the edge conformal
field theory. We further show (d) the evolution of ES of different
charge sectors (NL = −2, −1, 0, 1) and (e) charge pumping results
by adiabatically inserting 4π fluxes.
soft-core case in iDMRG, we do not need infinite local Hilbert
space since the maximal number of bosons in the iDMRG unit
cell is N1 × N2 × ν, bounded by filling factor ν of the flat
band. We keep up to D = 3000 bond dimensions in iDMRG
simulations with maximum truncation errors around 10−5 and
10−6 for the Abelian and non-Abelian FCIs respectively.
Bosonic Laughlin FCI in the singular flat band.— We first
3
consider ν = 1/2 filling of the singular flat band (η = 1) and
start with soft-core bosons and only very small U2 = 0.001
(Un>2 = 0) interaction. The spectra from two-orbital ED
simulations of N = 4 × 3 × 2 and N = 4 × 4 × 2 tori
are shown in Fig.2 (a,b), respectively. The two-fold ground
states (with their spectral flow in the insets) are well gapped
from the excited states and their momenta are in agreement
with the generalized Pauli principle for the bosonic 1/2 FCI
on torus [14, 50, 51]. The many-body Chern number of the
ground states gives the fractional quantized Hall conductivity
σH = 1
2
e2
h , further confirming the bosonic Laughlin-like FCI.
In fact, due to the exact flatness of the singular flat band, ar-
bitrarily small U2 could stabilize the FCI, which has not been
numerically reported in Chern-band systems. The considered
U2 = 0.001 is already very small and the low-energy spectra
have well converged with decreasing U2 in the unit of En/U2
(c.f. SI [49]).
We then go to the opposite limit, i.e.
hard-core bosons
with Un≥2 = ∞. The iDMRG results with circumference
N1 = 6 are shown in Fig.2(c-e). The momentum-resolved
entanglement spectrum (ES) shows the characteristic edge-
mode counting {1, 1, 2, 3, 5, ...}, in agreement with the edge
conformal field theory, supporting the topological nature of
the ground state [Fig.2(c)] [52, 53]. After adiabatically in-
serting 4π fluxes, the ES goes back to itself while shifted by
one charge sector and there is one boson pumped from the
left to the right side of the system, suggesting the FCI ground
state with σH =
1
2
e2
h [54]. Therefore, we have shown that
the Laughlin FCI could be stabilized by arbitrarily small and
large onsite two-body interactions. In the SI, we further show
that this FCI could survive with very large NN interactions as
well [49].
Bosonic Moore-Read state in the singular flat band.— Fur-
thermore, we demonstrate the existence of the MR state in the
singular flat band, where we consider the ν = 1 filling with
three-body hard-core bosons (nmax = 2), which is numeri-
cally easier than the soft-core case for this high filling. The
ES of the identity sector ΨI ground state is shown in Fig.3(a),
with the characteristic edge-mode of counting {1, 2, 4, 7, ...}
for odd charge sectors and {1, 1, 3, 5, 10...} for even charge
sectors. The even-odd difference can be understood from the
root configuration “..020202..” of ΨI depending on how it is
cut [51, 52, 55–58]. To obtain the Ising anyon sector Ψσ, we
consider an iDMRG unit cell with N1 = 5 and N2 = 1 such
that only the root configuration “..11111..” is allowed. The
ES of the Ψσ ground state is shown in Fig.3(b), and the edge
mode degeneracy pattern is {1, 2, 4, 8, ...} for all charge sec-
tors, consistent with the analytical prediction of Ising anyon
primary field [55, 57, 58].
Then we perform the adiabatic flux insertion. The charge
pumping results starting from the ΨI or the Ψσ ground states
are shown in Fig.3(c). After inserting 2π (4π), there is 1 (2)
net charges pumped from the left to the right edge of the sys-
tems for both cases. However, the underlying fusion rules
of the quasiparticles are different [58]. Starting from the ΨI
0
2
4
-2
0
2
0
5
10
1
1
3
5
10
1
2
4
7
NL = 0
NL = −1
(a) Identity sector |ΨI⟩
(b) Ising anyon sector |Ψσ⟩
-1
0
1
2
3
NL = 0
1
2
4
8
0
5
10
1
2
4
8
NL = −1
NL
ϕy
0
2π
4π
0
2
4
6
-1
0
1
2
3
ϕy
0
2π
4π
0
-1
-2
⟨ΔQL(ϕy)⟩
Starting from |ΨI⟩
(c)
(d)
FIG. 3. Moore-Read state in the singular flat band. We show the
iDMRG results of ν = 1 filling of the singular flat band (η = 1)
with hard-core bosons (nmax = 2).
(a) The ES of ΨI ground
state from the N1 = 6 cylinder, showing the edge mode counting
{1, 2, 4, 7, ...} and {1, 1, 3, 5, 10...} for the NL = −1 and NL = 0
charge sectors, respectively.
(b) The ES of the Ψσ ground state
from the N1 = 5 cylinder, showing the same degeneracy pattern
{1, 2, 4, 8, ...} in all charge sectors (NL = −1, 0 plotted for exam-
ple). (c) Charge pumping results starting from the ΨI or the Ψσ
ground state. (d) The evolution of ES after flux insertion, starting
from the ΨI ground state.
ground state, the evolution of ES with flux insertion is shown
in Fig.3(d). Although there is a net charge pumped after inser-
tion of ϕy = 2π flux, the ES does not go back to itself and is
different from that at ϕy = 0. This is because the ground state
adiabatically evolves into the fermion anyon sector Ψf, where
the ES would exhibit edge-mode counting {1, 2, 4, 7, ...} for
even charge sectors and {1, 1, 3, 5, 10...} for odd charge sec-
tors. The even-odd counting is shifted by one charge for the
ΨI and Ψf ground states, which is also manifested from the
4
comparison of different ES counting in the NL = −1, 0 sec-
tors in Fig.3(a). After inserting 4π flux, the ES goes back
to itself with the charge sectors shifted by 2, and the ground
state goes back to the ΨI again. These relate to the fusion
rules: I × f = f and f × f = I. If starting from the Ψσ,
just as the same edge mode counting in different charge sec-
tors (Fig.3 (b)), the ES would be the same with each integer
number of 2π flux. This relates to σ × f = σ and the Ψσ
ground state always evolves into itself with flux insertion of
2π, unlike the ΨI and Ψf ground states that evolve into each
other. This also demonstrates that the Ising anyon does not
respond to the U(1) charge flux.
Bosonic Laughlin FCI in the isolated C = 0 flat band.— At
η = 0.8, the (exactly) flat band is gapped with ∆= 0.08. Like
the Berry curvature, the trace of quantum metric also shows a
peak at Γ [inset of Fig. 4(a)]. When spinless fermions form
FCI in such C = 0 band [35], the nearest-neighbor repulsion
tends to drive carriers away from such peak, thereby occu-
pying BZ region with relatively uniform quantum geometry.
This has restricted the filling factor to the low side (ν = 1/3),
otherwise the peak of quantum geometry would be less avoid-
able. It is unclear whether such mechanism allows bosonic
FCI to form with the characteristic on-site interaction. We ad-
dress this open issue by considering ν = 1/2 filling of the
isolated trivial flat band with the onsite two-body repulsion
much weaker than the band gap.
The soft-core-boson condition is realized with onsite
nmax = N1 × N2 × ν since each iDMRG unit cell can not
have more bosons. Without explicit interactions, the system
is basically non-interacting and the ground state is topolog-
ically trivial.
We will consider U2 = 0.02 (Un>2 = 0),
which is much smaller than the band gap and thus ensuring
the isolated-band limit. The results of N1 = 6 (circumfer-
ence of the infinite cylinder) and N2 = 1 (we have checked
that there will be no translation symmetry breaking along ⃗a2
with only small U2) are shown in Fig. 4. With random ini-
tial conditions and optimization, we always obtain two nearly
degenerate ground states [53, 58], which actually refer to the
two topological sectors of the 1/2 Laughlin state.
To reveal the mechanism of forming this FCI in an isolated
trivial band, we compute the momentum-space boson distri-
bution n(k) = P
α
P
j⟨b†
i,αbj,α⟩e−ikrij by considering a fi-
nite region of the infinite cylinder, where α refers to the A/B
sublattice. The results from both topological sectors show that
n(k) drops to 0 when approaching Γ, suggesting that this FCI
is realized by occupying the BZ away from Γ with relatively
uniform Ω. The n(k) from the identity sector along one k-path
is shown in Fig.4 (a). The ES of the charge sector with largest
eigenvalue of the reduced density matrix from the identity-
sector ground state is further shown in Fig.4 (b), with the char-
acteristic edge mode counting {1, 1, 2, 3, 5, 7, ...}. Moreover,
we show the ES from the semion-sector ground state in Fig.4
(c) with all the charge sectors having the same degeneracy pat-
tern {1, 1, 2, 3, 5, ...}. The ES is symmetric about NL = 0.5,
confirming the semion anyon at each edge of the cylinder, car-
rying the fractional charge e/2 [52, 53, 59].
0
2
4
2
4
6
8
10
(a)
(b)
1
1
2
3
5
7
Identity sector
NL = 0
0
3
6
9
0
2
4
6
Semion sector
1
1
2
3
1
1
2
3
5
1
1
2
3
5
1
1
2
3
(c)
Tr(g)
1.1
34
FIG. 4. Laughlin FCI in the isolated C = 0 flat band with soft-
core bosons. The iDMRG results at ν = 1/2 filling of the isolated
flat band (η = 0.8 with band gap ∆= 0.08) with only U2 = 0.02.
The inset of panel (a) shows the trace of quantum metric of the trivial
flat band with a peak at Γ, and the blue dashed line shows the path
along which we compute the momentum-space boson distribution
n(k) in (a). n(k) is measured from the ground state in the identity
sector for example and the bosons tend to avoid occupying around
Γ. (b) The ES of the NL = 0 charge sector of the ground state in
the identity sector, showing the characteristic edge-mode counting
{1, 1, 2, 3, 5, 7, ...}. (c) The ES of the ground state in the semion
sector, where all charge sectors show the same degeneracy pattern
and the ES is symmetric about NL = 0.5.
Discussions.— Our demonstrations of FCIs in non-Chern flat
band systems broaden the understanding of the origins and
mechanisms of FCI. Whereas many-body interactions are of-
ten crucial, we provide a distinct perspective centered on flat-
band quantum geometry, pointing to FCI physics beyond band
topology. Due to the exact flatness of these non-Chern bands,
FCIs could be induced by arbitrarily small onsite interactions.
This feature surpasses the limitations of Chern bands, which
must have a finite bandwidth in a finite range hopping lat-
tice [38], thereby requiring a minimum interaction strength
to overcome their dispersion. This enables investigation in a
broad range of platforms - including photonic, phononic, ex-
citonic and polaritonic systems [44, 60–67] - thereby signifi-
cantly expanding the landscape for exploring bosonic FCIs.
We also note that in an extended Bose-Hubbard model on
the Kagome lattice with infinite on-site interaction and com-
peting different-range interactions, the bosonic Laughlin state
has been reported [68], which is a chiral spin liquid in spin
language as the model can be mapped onto the spin-1/2 XXZ
5
model [69, 70]. Upon spontaneous time reversal symmetry
breaking, the 1/2 Laughlin state emerges at an integer filling
of bosons per unit cell [68], which is intrinsically different
from the Laughlin states in our scenarios at half flat-band fill-
ings.
Acknowledgments — We thank Jie Wang and Nicolas Reg-
nault for helpful discussions. We thank Zuzhang Lin, Wenqi
Yang, and Dawei Zhai for earlier collaborations. The work
is supported by the National Natural Science Foundation of
China (No.
12425406), Research Grant Council of Hong
Kong (AoE/P-701/20, HKU SRFS21227S05), and New Cor-
nerstone Science Foundation. We thank Beijing PARATERA
Tech Co., Ltd. (https://cloud.paratera.com) for providing HPC
resources that supported the research results reported in this
paper. TeNPy library is used in the iDMRG simulations [71].
∗wangyao@hku.hk
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7
SUPPLEMENTARY INFORMATION
Supplementary spectra
(a)
(b)
FIG. S1. Supplementary spectral flow of the Laughlin FCI in the singular flat band. In Figs. 2(a-b) of the main text, the spectral flow of
the degenerate ground states are shown in the insets. Here, we zoom out and add the low-energy excited states to further support the robust gap.
The parameters are the same as Fig. 2(a-b) of the main text with ν = 1/2 of the singular flat band (η = 1), and only very small U2 = 0.001
interaction. Panels (a) and (b) are from N = 3 × 4 × 2 and N = 4 × 4 × 2 tori, respectively.
0
5
10
0.05
0.1
0.15
0
5
10
0.05
0.1
0.15
0
5
10
0.05
0.1
0.15
0
5
10
0.05
0.1
0.15
(a) U2 = 0.1
(b) U2 = 0.01
(c) U2 = 0.001
(d) U2 = 0.0001
FIG. S2. Laughlin FCI in the singular flat band with arbitrarily small onsite interactions. We show the spectra of the FCI at ν = 1/2 of
the singular flat band with different U2 from N = 3 × 4 × 2 torus. Panel (c) has been showed in Fig.2(a) of the main text.
In Fig. 2(a-b) of the main text, we have shown the spectral flow of the two degenerate ground states in the insets. To further
support the robust many-body gap upon flux insertion, we zoom out and and add the low-energy excited states as well in Fig.
S1.
In the main text, we have mentioned that due to the exact flatness, arbitrarily small onsite interactions among the soft-core
bosons can stabilize the FCI. In the main text, the example is taken as U2 = 0.001. To further support this, we show more
ED spectra with different U2 in Fig. S2. It is clear that with the decreasing U2, the spectra almost converge, supporting that
arbitrarily small U2 could stabilize this FCI at ν = 1/2 of the singular flat band.
Robust FCI with NN repulsion
In the main text, we only consider the onsite interactions. Here, in the singular flat band model, we take the hard-core boson
case for example to explore the NN repulsion: HV = V1
P
⟨ij⟩ninj. At ν = 1/2, we find that the FCI is stable even when V1
is larger than the total width of the two bands (W = 9). We show the iDMRG results with V1 = 9 in Fig.S3. The characteristic
edge-mode counting from the ES and the charge pumping with one boson pumped after adiabatically inserting 4π fluxes confirm
the robustness of the FCI with very large V1 interactions and suggest that there might be no upper limit of V1 for this FCI.
8
0
1
2
3
4
0
5
10
1
1
2
3
5
(a)
(b)
FIG. S3. Robust FCI with strong NN interaction. Here, we consider the singular flat band at ν = 1/2 filling with hard-core bosons, and
V1 = 9 is taken for these results. (a) The momentum-resolved ES. (b) The charge pumping result.
|
Bosonic Laughlin and Moore-Read states from non-Chern flat bands Hongyu Lu1, 2, 3 and Wang Yao1, 2, 3, ∗ 1New Cornerstone Science Lab, 2HK 3State Key Laboratory of Optical Quantum Materials, The (Dated: October 17, 2025) The rapid advances in the study of fractional Chern insulators (FCIs) raise a fundamental question: while initially discovered in flat Chern bands motivated by their topological equivalence to Landau levels, is singleparticle band topology actually a prerequisite for these many-body topological orders emergent at fractional fillings? Here, we numerically demonstrate bosonic FCIs in two types of non-Chern flat bands in honeycomb lattices, using exact diagonalization and density matrix renormalization group calculations. In a gapless flat band with a singular band touching, we observe a Laughlin state at half filling, stabilized by onsite interactions from the hard-core limit down to arbitrarily small strength. Furthermore, we report the first example of a nonAbelian FCI in a non-Chern band system: a Moore-Read state at ν = 1 filling of the same singular flat band with hard-core bosons. Under lattice parameters that realize a gapped trivial band (C = 0) of exact flatness, we also find the Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much smaller than the band gap. In this case, the FCI forms as interacting bosons spontaneously avoid the peaks in quantum metric and Berry curvature, preferentially occupying Brillouin zone region with relatively uniform quantum geometry. Our work significantly expands the landscape for (non-)Abelian FCIs and broadens the understanding of their formation beyond the Chern band paradigm. Introduction.- The recent experimental discovery of fractional Chern insulators (FCI) in correlated moir ́e materials [1-7], together with quantum simulations of their bosonic analogues in artificial lattices [8, 9], have sparked widespread interest in this exotic topological matter. The FCI phase was first conceived theoretically by translating the integer-tofractional hierarchy of the quantum Hall effect in Landau levels (LLs) to lattices systems [10-15]. Initial explorations naturally focused on the scenario of isolated flat Chern band that resembles the LLs in topology, quantum geometry, and energy landscape [10-15]. This appears to be sufficient but not necessary condition for FCI states to emerge in a Bloch band context. While having ideal quantum geometry (like that of LLs) is favorable for stabilizing FCI states [16-22], it is not a strict requirement for their existence in a flat Chern band [2325]. FCIs have also been found in models where interaction strength well exceeds the band gap separating the flat Chern band [12, 26]. Notably, in some cases, the Chern bands themselves are not present in the non-interacting limit but instead arise from interaction-driven band renormalization [27-32]. The study of FCIs has also moved beyond the flat Chern band paradigm along different lines. Opposite to the flat band limit, interaction-driven high-angular-momentum band inversion has been proposed for realizing Laughlin states at integer filling [33]. Within the flat band limit, exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations have unambiguously demonstrated FCI phases in fractionally filled non-Chern bands [34, 35], referring to the situations where band Chern number is either undefinable or zero. For the former, a notable example is the singular flat band, with band touching to a dispersive band dictated by realspace topology [36, 37]. Despite the gapless band structure and discontinuity of Bloch function at the singular touching, Laughlin-like FCI has been found in such band hosted by a fluxed dice lattice [34]. Even more surprisingly, FCI phases are also discovered in a trivial flat band of zero Chern number, which is well isolated by a band gap much larger than the interaction [35]. The FCI state is shown to develop an inhomogeneous carrier distribution over the Brillouin zone (BZ), avoiding the peaks of Berry curvature and quantum metric, and preferentially occupying regions of relatively uniform quantum geometry. Because of the exact flatness allowed in non-Chern bands [38], the FCI phases are shown to persist down to an arbitrarily small nearest-neighbor (NN) interaction in both cases [34, 35]. Non-Chern bands may greatly expand the landscape for exploring FCI physics, but the findings are so far limited to Abelian Laughlin states, and to fermionic systems. Non-Abelian FCIs [39-44] is of paramount importance, both for fundamental interest and for fault-tolerant quantum computation [45]. And whether bosonic FCIs [46-48] can form in such non-Chern bands and the effect of characteristic onsite repulsion are still open questions. Addressing them is critical for elucidating the mechanism of FCI beyond the Chern band paradigm and for broadening the potential experimental platforms. In this work, we demonstrate bosonic FCIs in a honeycomb lattice that can host either type of non-Chern flat bands, using ED and iDMRG calculations. At ν = 1/2 filling of the singular flat band, we identify the bosonic Laughlin state stabilized by onsite interactions from hard-core limit down to arbitrarily small strength, with the weak-interaction stability enabled by the exact flatness of the band. In addition, we show the existence of bosonic Moore-Read state at ν = 1 filling of the singular flat band with hard-core bosons, the first example of non-Abelian FCI in non-Chern band systems. Under lattice parameters that realize a gapped trivial flat band, we further demonstrate the bosonic Laughlin FCI of soft-core bosons in the isolated band limit where onsite interaction is much smaller than the band gap. In such a C = 0 band of exact flatness, we show that the interacting soft-core bosons 16 Oct 2025 2 ⃗a 1 ⃗a 2 t1 ηt1 t2ei 2π 3 ηt2ei 2π 3 t2ei 2π 3 ηt2ei 2π 3 t3 η2t3 (a) η = 0.8 (c) Isolated trivial flat band C = 0 -30 0 0.167 Ω Ω 0.167 0 Singular flat band η = 1 (b) Gapless Gap: Δ = 0.08 FIG. 1. Non-interacting model. (a) Two orbital model on honeycomb lattice including up to third NN hoppings with -t1 = t2 = t3 = 1. (b) When η = 1, the lowest band has exact flatness and a singular quadratic touching point to a dispersive band (grey) at Γ. (c) When η nmax = ∞. To address the 0 2 4 6 8 10 0.05 0.1 0.15 0 5 10 15 0.08 0.12 0.16 -1 0 1 2 3 4 0 5 10 NL = 0 1 1 2 3 5 φy 0 2π 4π 0 -0.5 -1 2 4 6 ES ES ⟨ΔQL(φy)⟩ (a) (b) (d) (e) NL Soft-core Soft-core (c) Hard-core Hard-core FIG. 2. Laughlin FCI in the singular flat band with arbitrarily small and large onsite interactions. We consider ν = 1/2 filling of the singular flat band (η = 1). ED spectra of soft-core bosons with a very small U2 = 0.001 interaction from N = 4 × 3 × 2 and N = 4 × 4 × 2 tori are shown in panels (a,b), respectively. For each system size, the spectral flow of the well gapped two-fold ground states is shown in the inset. The iDMRG results of hard-core bosons (Un≥2 = ∞) are shown in panel (c-e). In panel (c), the momentumresolved entanglement spectrum (ES) of the charge sector (NL = 0) with the largest eigenvalue of the reduced density matrix shows the characteristic counting {1, 1, 2, 3, 5, ...} of the edge conformal field theory. We further show (d) the evolution of ES of different charge sectors (NL = -2, -1, 0, 1) and (e) charge pumping results by adiabatically inserting 4π fluxes. soft-core case in iDMRG, we do not need infinite local Hilbert space since the maximal number of bosons in the iDMRG unit cell is N1 × N2 × ν, bounded by filling factor ν of the flat band. We keep up to D = 3000 bond dimensions in iDMRG simulations with maximum truncation errors around 10-5 and 10-6 for the Abelian and non-Abelian FCIs respectively. Bosonic Laughlin FCI in the singular flat band.- We first 3 consider ν = 1/2 filling of the singular flat band (η = 1) and start with soft-core bosons and only very small U2 = 0.001 (Un>2 = 0) interaction. The spectra from two-orbital ED simulations of N = 4 × 3 × 2 and N = 4 × 4 × 2 tori are shown in Fig.2 (a,b), respectively. The two-fold ground states (with their spectral flow in the insets) are well gapped from the excited states and their momenta are in agreement with the generalized Pauli principle for the bosonic 1/2 FCI on torus [14, 50, 51]. The many-body Chern number of the ground states gives the fractional quantized Hall conductivity σH = 1 2 e2 h , further confirming the bosonic Laughlin-like FCI. In fact, due to the exact flatness of the singular flat band, arbitrarily small U2 could stabilize the FCI, which has not been numerically reported in Chern-band systems. The considered U2 = 0.001 is already very small and the low-energy spectra have well converged with decreasing U2 in the unit of En/U2 (c.f. SI [49]). We then go to the opposite limit, i.e. hard-core bosons with Un≥2 = ∞. The iDMRG results with circumference N1 = 6 are shown in Fig.2(c-e). The momentum-resolved entanglement spectrum (ES) shows the characteristic edgemode counting {1, 1, 2, 3, 5, ...}, in agreement with the edge conformal field theory, supporting the topological nature of the ground state [Fig.2(c)] [52, 53]. After adiabatically inserting 4π fluxes, the ES goes back to itself while shifted by one charge sector and there is one boson pumped from the left to the right side of the system, suggesting the FCI ground state with σH = 1 2 e2 h [54]. Therefore, we have shown that the Laughlin FCI could be stabilized by arbitrarily small and large onsite two-body interactions. In the SI, we further show that this FCI could survive with very large NN interactions as well [49]. Bosonic Moore-Read state in the singular flat band.- Furthermore, we demonstrate the existence of the MR state in the singular flat band, where we consider the ν = 1 filling with three-body hard-core bosons (nmax = 2), which is numerically easier than the soft-core case for this high filling. The ES of the identity sector ΨI ground state is shown in Fig.3(a), with the characteristic edge-mode of counting {1, 2, 4, 7, ...} for odd charge sectors and {1, 1, 3, 5, 10...} for even charge sectors. The even-odd difference can be understood from the root configuration "..020202.." of ΨI depending on how it is cut [51, 52, 55-58]. To obtain the Ising anyon sector Ψσ, we consider an iDMRG unit cell with N1 = 5 and N2 = 1 such that only the root configuration "..11111.." is allowed. The ES of the Ψσ ground state is shown in Fig.3(b), and the edge mode degeneracy pattern is {1, 2, 4, 8, ...} for all charge sectors, consistent with the analytical prediction of Ising anyon primary field [55, 57, 58]. Then we perform the adiabatic flux insertion. The charge pumping results starting from the ΨI or the Ψσ ground states are shown in Fig.3(c). After inserting 2π (4π), there is 1 (2) net charges pumped from the left to the right edge of the systems for both cases. However, the underlying fusion rules of the quasiparticles are different [58]. Starting from the ΨI 0 2 4 -2 0 2 0 5 10 1 1 3 5 10 1 2 4 7 NL = 0 NL = -1 (a) Identity sector |ΨI⟩ (b) Ising anyon sector |Ψσ⟩ -1 0 1 2 3 NL = 0 1 2 4 8 0 5 10 1 2 4 8 NL = -1 NL φy 0 2π 4π 0 2 4 6 -1 0 1 2 3 φy 0 2π 4π 0 -1 -2 ⟨ΔQL(φy)⟩ Starting from |ΨI⟩ (c) (d) FIG. 3. Moore-Read state in the singular flat band. We show the iDMRG results of ν = 1 filling of the singular flat band (η = 1) with hard-core bosons (nmax = 2). (a) The ES of ΨI ground state from the N1 = 6 cylinder, showing the edge mode counting {1, 2, 4, 7, ...} and {1, 1, 3, 5, 10...} for the NL = -1 and NL = 0 charge sectors, respectively. (b) The ES of the Ψσ ground state from the N1 = 5 cylinder, showing the same degeneracy pattern {1, 2, 4, 8, ...} in all charge sectors (NL = -1, 0 plotted for example). (c) Charge pumping results starting from the ΨI or the Ψσ ground state. (d) The evolution of ES after flux insertion, starting from the ΨI ground state. ground state, the evolution of ES with flux insertion is shown in Fig.3(d). Although there is a net charge pumped after insertion of φy = 2π flux, the ES does not go back to itself and is different from that at φy = 0. This is because the ground state adiabatically evolves into the fermion anyon sector Ψf, where the ES would exhibit edge-mode counting {1, 2, 4, 7, ...} for even charge sectors and {1, 1, 3, 5, 10...} for odd charge sectors. The even-odd counting is shifted by one charge for the ΨI and Ψf ground states, which is also manifested from the 4 comparison of different ES counting in the NL = -1, 0 sectors in Fig.3(a). After inserting 4π flux, the ES goes back to itself with the charge sectors shifted by 2, and the ground state goes back to the ΨI again. These relate to the fusion rules: I × f = f and f × f = I. If starting from the Ψσ, just as the same edge mode counting in different charge sectors (Fig.3 (b)), the ES would be the same with each integer number of 2π flux. This relates to σ × f = σ and the Ψσ ground state always evolves into itself with flux insertion of 2π, unlike the ΨI and Ψf ground states that evolve into each other. This also demonstrates that the Ising anyon does not respond to the U(1) charge flux. Bosonic Laughlin FCI in the isolated C = 0 flat band.- At η = 0.8, the (exactly) flat band is gapped with ∆= 0.08. Like the Berry curvature, the trace of quantum metric also shows a peak at Γ [inset of Fig. 4(a)]. When spinless fermions form FCI in such C = 0 band [35], the nearest-neighbor repulsion tends to drive carriers away from such peak, thereby occupying BZ region with relatively uniform quantum geometry. This has restricted the filling factor to the low side (ν = 1/3), otherwise the peak of quantum geometry would be less avoidable. It is unclear whether such mechanism allows bosonic FCI to form with the characteristic on-site interaction. We address this open issue by considering ν = 1/2 filling of the isolated trivial flat band with the onsite two-body repulsion much weaker than the band gap. The soft-core-boson condition is realized with onsite nmax = N1 × N2 × ν since each iDMRG unit cell can not have more bosons. Without explicit interactions, the system is basically non-interacting and the ground state is topologically trivial. We will consider U2 = 0.02 (Un>2 = 0), which is much smaller than the band gap and thus ensuring the isolated-band limit. The results of N1 = 6 (circumference of the infinite cylinder) and N2 = 1 (we have checked that there will be no translation symmetry breaking along ⃗a2 with only small U2) are shown in Fig. 4. With random initial conditions and optimization, we always obtain two nearly degenerate ground states [53, 58], which actually refer to the two topological sectors of the 1/2 Laughlin state. To reveal the mechanism of forming this FCI in an isolated trivial band, we compute the momentum-space boson distribution n(k) = P α P j⟨b† i,αbj,α⟩e-ikrij by considering a finite region of the infinite cylinder, where α refers to the A/B sublattice. The results from both topological sectors show that n(k) drops to 0 when approaching Γ, suggesting that this FCI is realized by occupying the BZ away from Γ with relatively uniform Ω. The n(k) from the identity sector along one k-path is shown in Fig.4 (a). The ES of the charge sector with largest eigenvalue of the reduced density matrix from the identitysector ground state is further shown in Fig.4 (b), with the characteristic edge mode counting {1, 1, 2, 3, 5, 7, ...}. Moreover, we show the ES from the semion-sector ground state in Fig.4 (c) with all the charge sectors having the same degeneracy pattern {1, 1, 2, 3, 5, ...}. The ES is symmetric about NL = 0.5, confirming the semion anyon at each edge of the cylinder, carrying the fractional charge e/2 [52, 53, 59]. 0 2 4 2 4 6 8 10 (a) (b) 1 1 2 3 5 7 Identity sector NL = 0 0 3 6 9 0 2 4 6 Semion sector 1 1 2 3 1 1 2 3 5 1 1 2 3 5 1 1 2 3 (c) Tr(g) 1.1 34 FIG. 4. Laughlin FCI in the isolated C = 0 flat band with softcore bosons. The iDMRG results at ν = 1/2 filling of the isolated flat band (η = 0.8 with band gap ∆= 0.08) with only U2 = 0.02. The inset of panel (a) shows the trace of quantum metric of the trivial flat band with a peak at Γ, and the blue dashed line shows the path along which we compute the momentum-space boson distribution n(k) in (a). n(k) is measured from the ground state in the identity sector for example and the bosons tend to avoid occupying around Γ. (b) The ES of the NL = 0 charge sector of the ground state in the identity sector, showing the characteristic edge-mode counting {1, 1, 2, 3, 5, 7, ...}. (c) The ES of the ground state in the semion sector, where all charge sectors show the same degeneracy pattern and the ES is symmetric about NL = 0.5. Discussions.- Our demonstrations of FCIs in non-Chern flat band systems broaden the understanding of the origins and mechanisms of FCI. Whereas many-body interactions are often crucial, we provide a distinct perspective centered on flatband quantum geometry, pointing to FCI physics beyond band topology. Due to the exact flatness of these non-Chern bands, FCIs could be induced by arbitrarily small onsite interactions. This feature surpasses the limitations of Chern bands, which must have a finite bandwidth in a finite range hopping lattice [38], thereby requiring a minimum interaction strength to overcome their dispersion. This enables investigation in a broad range of platforms - including photonic, phononic, excitonic and polaritonic systems [44, 60-67] - thereby significantly expanding the landscape for exploring bosonic FCIs. We also note that in an extended Bose-Hubbard model on the Kagome lattice with infinite on-site interaction and competing different-range interactions, the bosonic Laughlin state has been reported [68], which is a chiral spin liquid in spin language as the model can be mapped onto the spin-1/2 XXZ 5 model [69, 70]. Upon spontaneous time reversal symmetry breaking, the 1/2 Laughlin state emerges at an integer filling of bosons per unit cell [68], which is intrinsically different from the Laughlin states in our scenarios at half flat-band fillings. Acknowledgments - We thank Jie Wang and Nicolas Regnault for helpful discussions. We thank Zuzhang Lin, Wenqi Yang, and Dawei Zhai for earlier collaborations. 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In Figs. 2(a-b) of the main text, the spectral flow of the degenerate ground states are shown in the insets. Here, we zoom out and add the low-energy excited states to further support the robust gap. The parameters are the same as Fig. 2(a-b) of the main text with ν = 1/2 of the singular flat band (η = 1), and only very small U2 = 0.001 interaction. Panels (a) and (b) are from N = 3 × 4 × 2 and N = 4 × 4 × 2 tori, respectively. 0 5 10 0.05 0.1 0.15 0 5 10 0.05 0.1 0.15 0 5 10 0.05 0.1 0.15 0 5 10 0.05 0.1 0.15 (a) U2 = 0.1 (b) U2 = 0.01 (c) U2 = 0.001 (d) U2 = 0.0001 FIG. S2. Laughlin FCI in the singular flat band with arbitrarily small onsite interactions. We show the spectra of the FCI at ν = 1/2 of the singular flat band with different U2 from N = 3 × 4 × 2 torus. Panel (c) has been showed in Fig.2(a) of the main text. In Fig. 2(a-b) of the main text, we have shown the spectral flow of the two degenerate ground states in the insets. To further support the robust many-body gap upon flux insertion, we zoom out and and add the low-energy excited states as well in Fig. S1. In the main text, we have mentioned that due to the exact flatness, arbitrarily small onsite interactions among the soft-core bosons can stabilize the FCI. In the main text, the example is taken as U2 = 0.001. To further support this, we show more ED spectra with different U2 in Fig. S2. It is clear that with the decreasing U2, the spectra almost converge, supporting that arbitrarily small U2 could stabilize this FCI at ν = 1/2 of the singular flat band. Robust FCI with NN repulsion In the main text, we only consider the onsite interactions. Here, in the singular flat band model, we take the hard-core boson case for example to explore the NN repulsion: HV = V1 P ⟨ij⟩ninj. At ν = 1/2, we find that the FCI is stable even when V1 is larger than the total width of the two bands (W = 9). We show the iDMRG results with V1 = 9 in Fig.S3. The characteristic edge-mode counting from the ES and the charge pumping with one boson pumped after adiabatically inserting 4π fluxes confirm the robustness of the FCI with very large V1 interactions and suggest that there might be no upper limit of V1 for this FCI. 8 0 1 2 3 4 0 5 10 1 1 2 3 5 (a) (b) FIG. S3. Robust FCI with strong NN interaction. Here, we consider the singular flat band at ν = 1/2 filling with hard-core bosons, and V1 = 9 is taken for these results. (a) The momentum-resolved ES. (b) The charge pumping result.
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2510.14686
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xLLM Technical Report
Tongxuan LiuB, Tao Peng, Peijun Yang, Xiaoyang Zhao, Xiusheng Lu∗, Weizhe Huang,
Zirui Liu††, Xiaoyu Chen, Zhiwei Liang, Jun Xiong, Donghe Jin, Minchao Zhang,
Jinrong Guo, Yingxu Deng, Xu Zhang, Xianzhe Dong†, Siqi Wang‡, Siyu Wu‡, Yu Wu†,
Zihan Tang∗, Yuting Zeng†, Yanshu Wang††, Jinguang Liu, Meng Kang, Menxin Li,
Yunlong Wang, Yiming Liu∗, Xiaolong Ma, Yifan Wang, Yichen Zhang∗, Jinrun Yin††,
Keyang Zheng††, Jiawei Yin†, Jun Zhang†, Ziyue Wang†, Xiaobo Lin, Liangyu Liu†,
Liwei Lan∗, Yang Liu†, Chunhua Peng, Han Liu, Songcheng Ren††, Xuezhu Wang‡,
Yunheng Shen∗, Yi Wang, Guyue Liu††B, Hui Chen∗B, Tong Yang††B, Hailong Yang‡B,
Jing Li†B, Guiguang Ding∗B, Ke ZhangB
JD.com
THU ∗
USTC †
BUAA ‡
PKU ††
Abstract
We introduce xLLM, an intelligent and efficient Large Language Model (LLM) inference frame-
work designed for high-performance, large-scale enterprise-grade serving, with deep optimizations
for diverse AI accelerators. Current mainstream inference frameworks face practical challenges. On
the one hand, enterprise-grade serving struggles with hybrid and dynamic workloads, strict demand
for high availability of services, and distributed storage management. On the other hand, infer-
ence execution is bottlenecked by underutilized AI accelerators due to new paradigms of hardwares,
model architectures and inference algorithms.
To address these challenges, xLLM builds a novel decoupled service-engine architecture. At
the service layer, xLLM-Service features an intelligent scheduling module that efficiently processes
multimodal requests and co-locates online and offline tasks through unified elastic scheduling to
maximize cluster utilization.
This module also relies on a workload-adaptive dynamic Prefill-
Decode (PD) disaggregation policy for instance scheduling and a novel Encode-Prefill-Decode
(EPD) disaggregation policy designed for multimodal inputs. Furthermore, it incorporates a dis-
tributed architecture to provide global KV Cache management for efficient AI accelerator memory
handling and robust fault-tolerant capabilities for high availability. At the engine layer, xLLM-
Engine co-optimizes system and algorithm designs to fully saturate computing resources. This is
achieved through comprehensive multi-layer execution pipeline optimizations, including overlap-
ping CPU scheduling with AI accelerator operations to minimize computational bubbles, employing
dual-stream parallelism to overlap computation with communication, and fine-grained overlapping
of various computational units to maximize hardware utilization. These are complemented by an
adaptive graph mode that drastically reduces kernel launch overhead, alongside the innovative "log-
ically contiguous, physically discrete" xTensor memory management which resolves memory allo-
cation conflicts. xLLM-Engine also further integrates algorithmic enhancements such as optimized
speculative decoding and dynamic Expert Parallel Load Balance (EPLB), collectively serving to
substantially boost throughput and inference efficiency.
Extensive evaluations demonstrate that xLLM delivers significantly superior performance and
resource efficiency. Under identical TPOT constraints, xLLM achieves throughput up to 1.7× that
of MindIE and 2.2× that of vLLM-Ascend with Qwen-series models, while maintaining an average
throughput of 1.7× that of MindIE with Deepseek-series models. We have deployed xLLM in
production to support a range of core business scenarios at JD.com, covering areas including LLM,
Multimodal Large Language Model (MLLM), and generative recommendation. These applications
encompass the JingYan AI chatbot, marketing recommendations, product understanding, customer
service assistants, and more. xLLM framework is publicly available at https://github.com/
jd-opensource/xllm and https://github.com/jd-opensource/xllm-service.
BCorresponding authors: {liutongxuan1, zhangke323}@jd.com, {guyue, yangtong}@pku.edu.cn, hai-
long.yang@buaa.edu.cn, lj@ustc.edu.cn, {dinggg, huichen}@tsinghua.edu.cn.
1
arXiv:2510.14686v1 [cs.DC] 16 Oct 2025
xLLM Technical Report
Contents
1
Introduction
3
2
System Overview
5
2.1
xLLM-Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
xLLM-Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3
xLLM-Service Designs
7
3.1
Online-Offline Co-location Scheduler Policy . . . . . . . . . . . . . . . . . . . . .
7
3.2
Dynamic PD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . .
9
3.3
Hybrid EPD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . .
10
3.4
Global KV Cache Management . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.5
Fast Fault Recovery Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4
xLLM-Engine Designs
13
4.1
Multi-layer Pipeline Execution Engine . . . . . . . . . . . . . . . . . . . . . . . .
13
4.2
Adaptive Graph Mode
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.3
Efficient Memory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.4
Algorithm Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.4.1
Optimized Speculative Decoding . . . . . . . . . . . . . . . . . . . . . . .
19
4.4.2
Dynamic EP Load Balance . . . . . . . . . . . . . . . . . . . . . . . . . .
20
4.4.3
Hierarchical DP Load Balance . . . . . . . . . . . . . . . . . . . . . . . .
21
4.5
Generative Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
4.5.1
Host-side Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
4.5.2
Device-side Optimization
. . . . . . . . . . . . . . . . . . . . . . . . . .
24
5
Evaluations
24
5.1
Main Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.1.1
Benchmarking Performance . . . . . . . . . . . . . . . . . . . . . . . . .
25
5.1.2
Business Serving Scenarios
. . . . . . . . . . . . . . . . . . . . . . . . .
26
5.2
Ablation Study
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6
Future Work
30
6.1
Fostering an Open and Diverse Hardware Ecosystem . . . . . . . . . . . . . . . .
31
6.2
Cultivating a Vibrant and Responsive Model Ecosystem . . . . . . . . . . . . . . .
31
6.3
Evolving into an AI-Native Application Framework . . . . . . . . . . . . . . . . .
32
7
Conclusion
32
2
xLLM Technical Report
1
Introduction
In recent years, large language models (LLMs) with parameters ranging from billions to trillions
(GPT [1], Claude [2], DeepSeek [3], LLaMA [4], etc.) have achieved breakthrough progress in the
fields of natural language processing and multimodal interaction, which drives an urgent demand
in the industry for efficient inference engines and service systems. These models are being rapidly
deployed in core business scenarios such as intelligent customer service [5], real-time recommen-
dation [6], and content generation [7]. However, how to reduce the cost of model inference and
improve computing efficiency remains a key challenge for large-scale commercial serving.
Current mainstream LLM inference frameworks (vLLM [8], SGLang [9], TensorRT-LLM [10], etc.)
face four key challenges in enterprise-level serving scenarios: First, in an inference cluster with hy-
brid deployment, online inference requests exhibit significant tidal characteristics [11, 12]. Current
scheduling systems fail to meet the service level objective (SLO) for online services while fully
leveraging idle periods of online services to increase the throughput of offline tasks. Second, ex-
isting Prefill-Decode (PD) disaggregation architecture [13–15] assumes static resource allocation
for the two phases, which cannot adapt to the dynamically changing request loads (i.e., input/output
lengths fluctuate) in real-world applications, resulting in low AI accelerator utilization and increased
SLO violation risks. Third, there is a lack of strategy to efficiently service multimodal requests
(i.e., image, voice and text input [16, 17]), including parallel processing for the encode phase and
fine-grained resource allocation accordingly. Fourth, as the scale of the inference cluster increases,
ensuring fast fault detection and service recovery for nodes or instances is critical to maintaining the
stability of inference services.
The evolving computing paradigms also present significant performance challenges on existing LLM
inference engines: First, they struggle to fully utilize the computing units of modern AI accelera-
tors [18, 19]. Second, the All-to-All communication overhead [20] and expert parallel (EP) load
imbalance in the Mixture of Experts (MoE) architecture [21, 22] restrict the scalability of the sys-
tem. Third, as the model context window continues to expand, efficient KV Cache management
becomes critical to inference performance [23]. Forth, Due to the unpredictable nature of infer-
ence requests, conventional static scheduling and operator strategies face difficulties in effectively
balancing workloads across computing units in Data Parallelism (DP).
To address above challenges , we propose xLLM, an efficient and intelligent LLM inference frame-
work featuring a service-engine decoupled design. xLLM achieves efficient support for enterprise-
level inference through the following innovations: at the service layer, xLLM has made ground-
breaking achievements in 1) unified elastic scheduling for online/offline requests, 2) workload-
adaptive dynamic PD disaggregated architecture, 3) novel Encode-Prefill-Decode (EPD) disaggrega-
tion for multimodal requests, and 4) a distributed cache management and fault-tolerance framework;
at the engine layer, xLLM enhances the resource efficiency across the full-stack of “communication-
computation-storage”, including 1) a multi-layer pipeline execution mechanism, 2) efficient comput-
ing and memory optimization, and 3) intelligent algorithm designs.
Specifically, xLLM-Service schedules online requests with preemptive execution and offline re-
quests in a best effort manner, to maximize resource utilization while strictly ensuring SLOs for
online services. To address the inherent limitations of the static PD configuration, xLLM incorpo-
rates an adaptive scheduler to dynamically adjust the proportion of PD instances for each request
and supports fast role reversal for instances, by monitoring key metrics such as Time to First Token
(TTFT) and Time Per Output Token (TPOT) [24]. For multimodal visual requests, xLLM auto-
matically selects the optimal EPD phase-disaggregation strategy based on pre-profiling, for the best
performance trade-off between throughput and latency. To handle failures such as hardware prob-
lems, network faults, or software errors [25], xLLM collects node error information, evaluates KV
recomputation or migration costs of interrupted requests, and makes optimal global rescheduling
decisions. Across multiple instances, xLLM supports KV offloading and routing migration within a
hybrid storage architecture, improving KV storage capacity and cache hit rates.
xLLM-Engine employs a multi-layer execution pipeline that incorporates hardware-specific op-
timizations: (i) at the framework scheduling layer, it implements asynchronous CPU-accelerator
scheduling to minimize computational idle time; (ii) at the model graph layer, it utilizes dual-stream
micro-batch parallelism to overlap computation with communication; (iii) at the operator level, it
achieves kernel computation and memory access overlapping. For computational efficiency, xLLM-
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xLLM Technical Report
Multi-modal
Requests
xLLM Service
Fault Recovery
Global Scheduler
Hybrid EPD Disagg.
Dynamic PD Disagg.
On/Offline Co-location
Global Cache Mgr
xLLM Engine
Adaptive Graph
Operator Optim.
xTensor Memory
drive
drive
manage
schedule
Speculative Inference
Dynamic EPLB
Prefill
Local KV
Cache
KV Cache
Transfer
…
Prefill
Local KV
Cache
KV Cache
Transfer
…
Prefill Instance
Local KV
Cache
KV Cache
Transfer
…
Decode
Local KV
Cache
KV Cache
Transfer
…
Decode
Local KV
Cache
KV Cache
Transfer
…
Decode Instance
Local KV
Cache
KV Cache
Transfer
…
System-level Designs
Batching Scheduler
Algorithm-driven Tech.
co-optimize
Hierarchical DPLB
Multi-layer Pipeline
Figure 1: Overview of xLLM’s capabilities, with decoupled service-engine architecture.
Engine automatically fuses small kernels into a unified computation graph and dispatches them to the
accelerator in a single operation, significantly reducing kernel launch overhead. Regarding memory
management efficiency, xLLM-Engine introduces the xTensor memory management scheme, which
employs a “logically contiguous, physically discrete"" KV cache storage structure that resolves the
conflict between memory contiguity requirements and dynamic allocation needs. To further enhance
performance, xLLM-Engine incorporates an adaptive speculative decoding mechanism, a redundant
expert-based load balancing algorithm for EP, and a hierarchical load balancing algorithm for DP.
Additionally, xLLM-Engine provides scenario-specific optimizations, such as for generative recom-
mendation – one of JD.com’s core businesses – where it achieves 23% performance improvement
through host-kernel operation overlapping.
In summary, we make the following contributions when developing xLLM:
xLLM Intelligent Service Capabilities
• We design a unified scheduling algorithm for online/offline workloads (§3.1).
• We implement a workload-adaptive PD disaggregation policy to address scenarios with rapidly
changing traffic load and request input/output lengths (§3.2).
• We propose a hybrid EPD disaggregation policy for multimodal requests, achieving intelligent
resource allocation across different phases (§3.3).
• We leverage multi-level KV Cache management and global KV Cache routing strategy to expand
KV Cache capacity and improve cache hit rates (§3.4).
• We design a multi-node fault tolerance architecture to ensure high service availability (§3.5).
xLLM Intelligent Engine Capabilities
• We achieve intelligent and efficient inference, through hardware-software co-design to improve
hardware computing efficiency, including pipeline execution (§4.1), graph optimization (§4.2),
and memory optimization (§4.3).
• We enhance inference performance through algorithmic optimizations (§4.4), including optimized
speculative decoding (§4.4.1), dynamic expert parallel load balance (§4.4.2), and hierarchical data
parallel load balance (§4.4.3).
• We optimize online inference for generative recommendation scenarios (§4.5).
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xLLM Technical Report
Predictor
Profiler
Dynamic PD
Disaggregation
Hybrid EPD
Disaggregation
On/Offline
Co-locatiom
Prefill Pool
Decode pool
Encode Pool
Global Cache Management
Fault Recovery Framework
Pre-Processing
Layer
Scheduling
Layer
online
preempt
support
disaggregate
dynamic PD
allocation
KV & Images
Resource
Layer
Offline Requests
Online Requests
Figure 2: System workflow of xLLM-Service.
2
System Overview
The overall architecture of xLLM is depicted in Figure 1. Upon request arrival, xLLM-Service
performs intelligent scheduling to distribute each request to one of three elastic instance pools and
manages instance migration across these pools during runtime. xLLM-Engine then drives efficient
request inference by orchestrating system- and algorithm-level optimizations.
2.1
xLLM-Service
We describe the workflow of xLLM-Service in Figure 2. The system comprises three primary lay-
ers: 1) a preprocessing layer consisting predictor and profiler, 2) a scheduling layer integrating
three policies (Dynamic PD Disaggregation Policy, Hybrid EPD Disaggregation Policy, and Online-
Offline Co-location Policy), and 3) a resource layer consisting of three heterogeneous instance pools.
Specifically, the Dynamic PD Disaggregation Policy leverages information from the predictor to
dynamically convert PD instances based on the workload status of the two instance types. The Hy-
brid EPD Disaggregation Policy utilizes the profiler to determine the optimal EPD disaggregation
strategy for multimodal requests. Concurrently, the Online-Offline Co-location Policy dispatches re-
quests according to their online or offline attributes. KV and image caches are offloaded and routed
among distributed instances, while the fault recovery framework ensures the high availability of the
entire service.
Elastic Instance Pools.
Instances in a cluster are partitioned into three elastic pools:
▷Prefill Instance Pool: This pool handles the prefill phase of text requests.
▷Decode Instance Pool: This pool handles the decode phase of text requests.
▷Encode Instance Pool: This pool handles the encode phase of multimodal requests.
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xLLM Technical Report
We design the instances in Prefill and Decode pools as stateless, i.e., instances do not require phys-
ical migration between instance pools. Instead, they achieve flexible switching between Prefill-
Decode roles based on the type of request being processed.
Request Preprocessing.
xLLM-Service implements unified resource management for both online
and offline requests. Online requests are submitted in a preemptive and deadline-prioritized manner,
and are co-deployed with best-effort offline tasks within the shared resource pool. During the run-
time of an offline request, the Unified-Scheduler dynamically scales it up/down based on the tidal
traffic characteristics of online requests. When an online request is submitted, it first passes through
the pre-processing layer, which consists of two modules:
▷TTFT Predictor: A TTFT prediction model built for text requests. It evaluates SLO fulfillment
by analyzing queueing delays from each prefill instance queue and request input lengths, thereby
guiding instance allocation for the Dynamic PD Disaggregation Policy in the scheduling layer.
▷
EPD Profiler: A profiler for multimodal requests that uses binary search to identify optimal
deployment configurations: (1) EPD separation strategy, choosing from three approaches: EP-D
(i.e., aggregated execution of Encode and Prefill phases, with Decode phase executed separately),
ED-P, or E-P-D; (2) The maximum batch size for the Encoder phase; (3) The maximum number of
tokens for Prefill/Decode’s inputs. The Hybrid EPD Disaggregation Policy in the scheduling layer
will use the optimal configuration determined for phase disaggregation and task dispatching.
Intelligent Scheduling.
The intelligent scheduling layer adjusts resource allocation for requests
across their full lifecycles. It contains three major scheduling policies designed for various scenarios:
▷Online-Offline Co-location Policy: This policy implements a preemptive scheduler for managing
online and offline requests. When the load of online requests reaches the peak, they preempt some
offline requests on PD instances. To surrender resources to offline requests, when the load on P
instances decreases (usually earlier than on D instances), P instances continue to process offline
prefill requests and also migrate decode offline requests from D instances to P instances.
▷Dynamic PD Disaggregation Policy: This adaptive scheduling policy is responsible for dynam-
ically managing P and D instance allocation. It intelligently assigns requests to suitable instances
using a heuristic algorithm guided by the TTFT Predictor. Furthermore, it implements a feedback
mechanism by continuously collecting performance data from computing instances, enabling run-
time monitoring and adjustment of allocation decisions to maintain system efficiency.
▷Hybrid EPD Disaggregation Policy: This multimodal policy executes the three-phase disaggre-
gation based on strategies searched by the EPD Profiler. For the EP-D disaggregation, the fused EP
phase executes in the P instance pool; for the ED-P disaggregation, the fused ED phase executes in
the D instance pool; for the E-P-D disaggregation, the three phases execute separately in the three
instance pools. This deployment also enables multimodal requests to benefit from the adjustment of
Dynamic PD Disaggregation Policy.
KV-centric Storage Architecture.
The instance storage employs a hybrid architecture (HBM-
DRAM-SSD) to cache KV values and image tokens. At the global level, xLLM takes the idea from
Mooncake Store [23] and extends it to domestic accelerators with specific optimizations. Routing
and reuse of caches across instances are determined by embedded intelligent routing strategies.
Efficient Fault-tolerant.
The fault recovery framework of xLLM-Service supports fault detection
and fast recovery of instances from the three elastic pools (E, P, D). For requests on failed instances,
the architecture manages the migration of image caches between instances, and automatically de-
cides the optimal KV recomputation or migration strategy for handling affected KV caches.
2.2
xLLM-Engine
During request execution, the xLLM-Engine layer provides intelligent computing capabilities. We
achieve various joint inference accelerations between the computing system layer and the algorithm-
driven layer in the following ways:
Computing System Layer
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xLLM Technical Report
Latency-relaxed instance
Online Prefill
Latency-strict instance
Online / Offline
Mixed Decode
Offline Prefill
Offline Decode
Resource Plane
Control Plane
On-demand
Preemption
Offline Request
Migration
Offline Request
Gating
Mix decoding
Selection
Figure 3: Overview design of Online-Offline Co-location Scheduler Policy.
▷Multi-layer Pipeline Execution Engine: In the framework layer, CPU tasks are scheduled asyn-
chronously to create a pipeline with inference computation, reducing computational bubbles; in the
model graph layer, a single batch is split to form a pipeline between two micro-batches, effectively
overlapping computation and communication; in the operator kernel layer, operations are pipelined
across different computational units, enabling computation and memory access to overlap.
▷
Graph Optimization for Dynamic Inputs: Small kernels in the decoding stage are fused into a
single computational graph. To handle variable sequence lengths and batch sizes, we parameterize
the input dimensions, and employ a multi-graph caching scheme to reduce compilation overhead.
▷xTensor Memory Management: It uses a “logically continuous, physically discrete” KV storage
structure. It allocates physical memory space on-demand during token generation for each request,
while asynchronously predicting and intelligently mapping the physical pages required for the next
token. When a request completes, the existing physical memory will be reused to execute the next
request.
Algorithm-Driven Layer
▷Speculative Decoding: xLLM-Engine incorporates an optimized speculative inference algorithm
that generates multiple tokens at once to boost throughput [26]. It further optimizes the computing
architecture through ways like asynchronous CPU processing and reduced data transfer.
▷EP Load Balance: For MoE models, xLLM implements expert weight updates based on historical
expert load statistics, enabling effective dynamic load balancing during inference.
▷DP Load Balance: For data parallel (DP) deployment, xLLM achieves fine-grained load balance
by kvcache-aware instance allocation, inter-DP requests migration, and intra-DP computing units
allocation.
We will elaborate on the detailed design of xLLM-Service in §3 and outline the optimizations within
the xLLM-Engine in §4. A comprehensive evaluation of xLLM will be presented in §5.
3
xLLM-Service Designs
3.1
Online-Offline Co-location Scheduler Policy
Online/Offline Request Characteristic.
LLM services can be categorized into two types based
on their service modes: online and offline requests. Online requests, including those from chat-
bots [1, 2, 27], code completion [28–30], and recommendation systems [31, 32], constitute latency-
sensitive workloads. These services must respond immediately upon request arrival, often returning
each generated token in real time via streaming output. Consequently, they impose strict SLO re-
quirements on TTFT or TPOT to ensure a satisfactory user experience. In contrast, offline services,
such as document analysis [33] and intelligent data annotation [34], are non-real-time workloads
with minimal latency constraints and thus no stringent SLO requirements.
Furthermore, we observe that the request traffic for online services typically exhibits significant
volatility, including tidal variations at hourly or daily scales and sudden bursts at minute-level inter-
vals [11, 12]. While cluster auto-scaling [35] could theoretically mitigate resource underutilization
caused by tidal patterns, the slow cold-start latency of instances—involving model loading and com-
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xLLM Technical Report
plex initialization—renders it ineffective for responding to rapid traffic spikes [36], increasing the
risk of service-level agreement violations.
Online/Offline Request Co-location Deployment.
To address these practical challenges, we
adopt a hybrid deployment strategy that co-locates online and offline requests. In such a system,
offline requests can utilize idle resources during off-peak periods of online traffic. Conversely, dur-
ing online traffic peaks, offline tasks can be preempted, as they are not bound by strict SLOs. This
approach significantly enhances aggregate resource utilization and mitigates idleness during traffic
troughs. Although some recent work [37–39] also attempts to co-locate offline requests, they have
not explored the multi-instance scenario, especially for PD disaggregation.
Actually, the PD disaggregation architecture has demonstrated superior latency performance and is
increasingly becoming a mainstream design paradigm in industry [13, 23]. However, directly apply-
ing online-offline co-location to PD-disaggregated systems introduces critical PD load imbalance
issues: (1) such systems require the load ratio between prefill and decoding stages to align with their
respective resource allocation ratios; otherwise, one stage may become a bottleneck, causing block-
ing or resource underutilization in the other. (2) Furthermore, as these PD load variations exhibit
high volatility and burstiness similar to online traffic patterns, existing PD disaggregation techniques
struggle to effectively address this challenge.
Latency-Constrained Decoupled Architecture.
We rethink the design of PD disaggregation ar-
chitecture under online-offline co-location deployment. The latency advantage of PD disaggregation
essentially stems from the separation of latency constraints: the decoding phase is highly sensitive
to per-step latency and cannot be blocked by long-duration operations, thus necessitating decou-
pling from the prefill phase. Inspired by this insight, we propose a latency-constrained decoupled
architecture, as illstrated in Figure 3. This design regards cluster resources as two pools: a latency-
relaxed pool (corresponding to original Prefill instances) and a latency-strict pool (corresponding to
original Decode instances). All tasks are then reassigned to one of the two resource pools based on
their inherent latency characteristics and requirements. Within this architecture, the decoding phase
of offline requests can be executed in either resource pool. This flexibility allows us to dynamically
adjust the load ratio between the two pools, thereby maximizing the overall resource utilization of
the cluster.
However, it also introduces another two challenges: (1) Complex Scheduling Space: since the de-
coding of offline requests can be performed on either type of instance, how to leverage this flexibility
to design new intelligent scheduling remains unknown. (2) Strict SLO Guarantee: the execution of
offline requests consumes resources, their prefill phase may block newly arrived online requests,
and their decode phase on latency-strict nodes may slow down the overall response speed. Both
scenarios can compromise the SLO satisfaction of online requests.
Solution 1 - Performance Bottleneck Analysis.
We construct an LLM inference performance
model based on the Roofline Model [40] and online factor learning. This model is designed to
predict the latency, computation utilization, and memory utilization of both the prefill and decode
phases. Since decoding operations executed on latency-strict instances typically account for a large
proportion of workloads and are highly performance-sensitive, we set balancing computational and
memory resources as the optimization objective. By analyzing performance bottlenecks through
this model, we can select more appropriate offline requests to merge into decoding batches, thereby
improving resource utilization efficiency.
Solution 2 - Efficient Preemption Mechanism.
To strictly ensure that the SLO of online requests
remains unaffected, we introduce a preemption mechanism that allows online requests to preempt
offline requests. For offline prefill tasks running on latency-relax nodes, we propose a model execu-
tion interruption technique, which enables preemption within an acceptable latency range without
incurring additional model maintenance overhead. For decoding tasks running on latency-strict
nodes, we leverage the performance model to dynamically select requests for decoding batching,
ensuring that decoding latency always meets SLO constraints.
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xLLM Technical Report
Stateless Instance
Local Scheduler
request queue
migration queue
Execution
Engine
Schedule Request
& Cache Migration
P
D
prefill
unfinished
P
D
decode
unfinished
decode
finished
prefill
finished
D
P
Global Scheduler
Instance Scheduler
Request Scheduler
Instance Monitor
TTFT / TPOT, Req Len, Memory Util., etc.
schedule
schedule
report status
Figure 4: Overview design of Dynamic PD Disaggregation Scheduler Policy.
3.2
Dynamic PD Disaggregation Scheduler Policy
Inefficiency of Existing PD Disaggregation Policy under Workload Fluctuations.
The Prefill-
Decode (PD) disaggregated inference architecture [13, 24, 23, 41] partitions computing instances
into dedicated prefill and decode instances, each handling their respective processing phases. This
design mitigates interference between prefill and decode requests, achieving superior performance
compared to PD co-located architectures [13].
However, we observe that most existing PD disaggregated systems employing static instance parti-
tioning schemes suffer from low hardware utilization and inadequate responsiveness to traffic bursts.
On one hand, existing analysis-based methods [13, 24, 23, 41] typically determine the PD ratio using
profiling or simulator data. These approaches remain effective only when request arrival patterns and
length distributions remain relatively stable. Under significant workload fluctuations, pre-collected
analysis data often fails to accurately capture real-time request characteristics, leading to mismatches
between preset load-balancing ratios and actual load requirements. On the other hand, when con-
fronting substantial variations in input/output lengths within production workloads, existing PD dis-
aggregation architectures [13, 42] generally adopt dynamic instance type adjustment strategies [43].
Nevertheless, online switching between prefill and decode instances typically involves multiple
steps—including monitoring, waiting for flip conditions, and instance restarting—which introduces
substantial latency overhead [15]. To address these limitations, we propose a Dynamic PD Dis-
aggregation Policy that adaptively adjusts resource allocation in response to real-time workload
characteristics.
Runtime Instance Monitor.
Performance indicators such as TTFT and TPOT directly reflect the
capability of instances to process requests in the prefill and decode phases, respectively. They serve
as the core basis for evaluating whether an inference system can meet SLO requirements. Ideally,
prediction-based methods could be adopted to dynamically assess these indicators. However, while
TTFT exhibits relatively predictable characteristics (as its computation time is proportional to the
square of the input sequence length [44, 41]), due to the uncertainty in the length of output tokens
and transfer overhead, TPOT is difficult to accurately predict using traditional input/output features
and cluster load metrics. To address this issue, we deploy additional instance monitors to collect
real-time performance data from each computing instance. This data includes metrics such as the
number and length of prefill and decode requests, memory usage, TTFT, TPOT, and token generation
intervals. The system can further use these real-time performance indicators to dynamically evaluate
instance loads and adjust scheduling strategies.
Stateless Instance and Elastic Pool.
As illustrated in Figure 3, Dynamic PD Disaggregation Pol-
icy adopts a design of stateless instances and elastic instance pools to enable fast and dynamic
switching of instance roles. First, it treats prefill or decode phase as the request attribute rather than
instance attribute. Instances are designed to be stateless, allowing each instance to process both
prefill and decode requests simultaneously. Additionally, to facilitate the management of multiple
instances, we further extend PD pools to four elastic instance pools (i.e., P, D, P→D, D→P) as de-
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xLLM Technical Report
scribed in §2.1. When flipping an instance (switching its role), we only needs to remove the instance
from its original pool and move it to the new pool. This achieves zero-wait-time instance scheduling,
avoiding the overhead of instance restart or model reloading incurred in traditional systems.
SLO-aware Instance Role Switching.
The instance scheduling strategy is dynamically adjusted
strictly based on SLO objectives: during the prefill phase, if it is predicted that existing instances
cannot meet the TTFT requirements, the conversion of decode instances is triggered; while during
the decode phase, when resource shortage occurs, the average token generation interval exceeds the
TPOT threshold, or prefill instances are idle, the conversion of prefill instances to decode instances
will be initiated to cope with sudden traffic surges. Specifically, when decode instances are reallo-
cated to prefill instances, the scheduler prioritizes selecting the instance with the lightest load (i.e.,
the fewest tokens being processed) from the P→D pool for role conversion, and always ensures
that at least two decode instances are available; conversely, when prefill instances are reallocated
to decode instances, the scheduler prioritizes scheduling from the D→P pool, which avoids local
overload and maximizes resource utilization.
SLO-aware Request Scheduling.
The request scheduling scheme adopts a two-level architecture:
▷Global Request Scheduler: The scheduler implements a greedy strategy of prioritizing the light-
est load, while being strictly restricted by SLO constraints. For prefill requests, the scheduler first
evaluates the estimated queuing latency of each instance in the Prefill pool, selects the candidate in-
stance with the smallest latency, and then invokes the TTFT prediction model for verification: if the
estimated TTFT can still meet the SLO requirements after assigning the request to this instance, the
request is allocated immediately; otherwise, it continues to find a suitable instance in the D→P pool.
If no instance can meet the TTFT requirements, the instance scheduling mechanism is triggered to
allocate resources from the decode side. For decode requests, the scheduler gives priority to having
the original prefill instance continue processing (to avoid KV Cache transfer overhead); secondly,
it selects the instance with the fewest running tokens in the Decode pool, and checks whether the
total number of tokens in its current batch is below the memory capacity upper limit and comput-
ing throughput limit determined by pre-analysis, so as to ensure that the new request will not cause
TPOT to exceed the standard.
▷Local Request Scheduler: Each instance adopts a refined queue management strategy internally.
KV Cache transfer events are placed in an independent migration queue and processed sequen-
tially in accordance with the FCFS principle; for forward requests, an innovative scheme combining
Chunked Prefill [45] and Continuous Batching [8] is adopted: on the premise of ensuring that de-
code requests are prioritized to enter the running batch, the remaining computing resources are used
to process chunked prefill requests in parallel [46].
3.3
Hybrid EPD Disaggregation Scheduler Policy
Challenges of Multi-modal Inference.
The inference process of multimodal large language mod-
els (MLLMs) [47–52] typically consists of three phases: the image encoding phase (for extracting
image features), the prefill phase (for encoding images and text prompts, feeding them into the
language model to generate the first output token, and caching intermediate states), and the de-
code phase (for iteratively generating subsequent tokens based on the cached data). Existing main-
stream inference engines, such as vLLM [8], Text-Generation-Inference [53], SGLang [9], and Dist-
Serve [13] are all tailored for LLMs, and thus face several challenges in handling inference tasks for
MLLMs:
▷Insufficient Parallelism: For example, the inference of visual models and language models can
be executed in parallel, thus improving the utilization of computing resources. However, most ex-
isting inference engines [54] adopt a serial strategy, failing to exploit the inter-request parallelism
effectively.
▷
Coarse-grained Scheduling: The decode phase is a memory-intensive task, which is suitable
for batch processing to improve throughput [55]; the prefill phase is computationally intensive,
and it is appropriate to adopt the chunked prefill [45] together with the decode phase to balance
latency and throughput [56]. The computation and memory access overhead of the encode phase
lies between the two, and it can also benefit from independent scheduling and batch processing.
Nevertheless, existing engines process encode and prefill in a combined manner, failing to perform
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xLLM Technical Report
TPOT & TTFT SLO Target
ED
D
P
+
EP +
E
P
D
+
+
EPD Profiler
Encode Instance
Vision Model
Image Cache
Batch
Scheduler
Migrate
Scheduler
Prefill Instance
Language Model
Image Cache
Batch
Scheduler
Migrate
Scheduler
KV Cache
Decode Instance
Language Model
Batch
Scheduler
Migrate
Scheduler
KV Cache
Hybrid EPD Disaggregation
async transfer
async transfer
Figure 5: Overview design of Hybrid EPD Disaggregation Policy.
phase-specific batch processing and lacking support for chunked prefill. Coarse-grained scheduling
makes it difficult to finely control the execution time.
▷Disaggregation Strategy: Existing architectures such as DistServe [13] reduce resource interfer-
ence by decoupling the prefill and decode phases. However, in multimodal inference, encoding and
prefill jointly affect TTFT, while prefill and decode jointly determine TPOT. Under different loads,
how to select the optimal decoupling strategy remains a challenge [57]. For example, the perfor-
mance of strategies such as E+P+D, EP+D, and ED+P in different tasks has not yet been evaluated,
and in-depth analysis is required to guide system design.
Dual-stream Parallel.
As illustrated in Figure 5, we adopt a dual-stream parallel mode, where the
visual model and language model are assigned to separate execution streams: the visual stream is
dedicated to performing computationally intensive image encoding tasks, while the language stream
handles the prefill and decode operations for language generation. By isolating workloads into
distinct streams, our system achieves concurrent execution of heterogeneous phases from different
requests.
Three-Phase Disaggregation.
To address the differences in computation and memory access
characteristics among tasks in each phase of multimodal inference, we propose a phase-aware
scheduling strategy. Requests are subdivided into three phases within an instance: encode, prefill,
and decode. Batch processing and scheduling optimizations are performed for each phase respec-
tively:
▷Optimized Batch Processing: The maximum batch size for image encoding and the token budget
for the language model are set according to the user’s SLO. Specifically, during system startup,
we use a binary search method to profile the maximum encoding batch size and the model’s token
budget, ensuring that the execution time of subsequent batch processing tasks in each iteration is
less than the TPOT SLO. In each iteration, we (i) first add all running decode requests to the current
batch; (ii) then check for any chunked prefill tasks that have been partially computed. If such tasks
exist, we add them to the batch; (iii) if there are no chunked prefill tasks, we check for pending
encoding tasks and add them if any exist. This approach aims to complete requests in the prefill
phase as quickly as possible, reducing their TTFT. New requests’ encoding phases are processed
only when no requests are in the prefill phase.
▷
Phase-aware Scheduling: To address the difficulty in selecting decoupling strategies for mul-
timodal inference tasks, xLLM proposes the Hybrid Encode-Prefill-Decode Disaggregation archi-
tecture, an innovative multi-instance collaborative inference framework. In this architecture, each
instance only executes part of the subtasks among the three phases, while the remaining phases are
processed by migrating requests to other instances, thereby avoiding resource waste and interfer-
ence. During runtime, our system automatically selects the optimal disaggregation strategy based
on historical request profiling (via EPD Profiler) to dynamically balance throughput and latency ob-
jectives. Compared with the traditional approach of binding instances to single-phase or full-phase
tasks, this architecture significantly improves the overall system processing capability and resource
utilization efficiency while meeting SLO requirements.
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xLLM Technical Report
Master
Election
Cache & Load
Metric Sync
Service
Discovery
ETCD
Global Multi-Level KV Cache Management
Master
Global
Scheduler
Global
KVCache Mgr
Select Master
Write Cache &
Load Metrics
Watch & Notify
HeartBeat (Lease)
Report Cache &
Load Metrics
Req
Req
Resp
Prefill
Local KV
Cache
KV Cache
Transfer
…
Prefill
Local KV
Cache
KV Cache
Transfer
…
Prefill Instance
Local KV
Cache
KV Cache
Transfer
…
Decode
Local KV
Cache
KV Cache
Transfer
…
Decode
Local KV
Cache
KV Cache
Transfer
…
Decode Instance
Local KV
Cache
KV Cache
Transfer
…
Slave
Slave
Slave
Figure 6: The framework of Global Multi-Level KV Cache Management.
3.4
Global KV Cache Management
During the decode phase of LLMs, subsequent tokens are generated autoregressively one by one. Al-
though the computational cost per step is relatively low, frequent access to the historical KV Cache
is required, making memory bandwidth the primary bottleneck. As model size expands and con-
text window grows, the memory consumption of the KV Cache exhibits an exponentially increasing
trend. For instance, a context of 128K tokens may consume over 40GB of memory, severely strain-
ing the memory resources of single-GPU devices [3].
Although current mainstream optimization solutions like vLLM [8] and FasterTransformer [58] have
made significant progress in single-instance environments, many critical issues remain unresolved.
In long-context scenarios, the time required for the prefill phase increases dramatically, while the
decode phase suffers from intense competition for memory bandwidth. To meet stringent SLO
requirements (e.g., TTFT < 2s, TBT < 100ms), one instance often has to reserve excessive re-
sources, instead of leveraging resources in other instances. This significantly restricts the overall
cluster resource utilization. To address these challenges, we propose a global multi-level KV Cache
management system with a memory-compute integrated architecture, as illustrated in Figure 6.
Distributed Storage and KV Transfer.
We adopt the Mooncake Store [23], a KV cache-centric
storage engine, as the underlying storage infrastructure for xLLM’s KV Cache, along with the Moon-
cake Transfer Engine as the transmission component. Mooncake Store leverages striping and paral-
lel I/O techniques to fully utilize the aggregated bandwidth of multiple network cards. It provides
three persistence strategies—Eager, Lazy, and None—to meet data durability requirements across
various scenarios. Additionally, Mooncake Store offers flexible object storage capabilities, support-
ing multiple replicas and eventual consistency, which effectively mitigates hot-spot access pressure.
As the core transmission engine of the system, Mooncake Transfer Engine [23] automatically selects
the optimal transmission path based on data location and abstracts low-level complexities through a
unified Segment and BatchTransfer interface.
Multi-Level KV Cache Management.
At the global level, the system employs ETCD [59] as a
metadata service middleware to achieve cluster service registration, load information synchroniza-
tion, and global cache state management. Each computing instance maintains a local multi-level
KV Cache pool (HBM > DRAM > SSD) adhering to the strict consistency rule: “if data resides in
HBM, it must also be present in DRAM”. Specifically, when local operations involving KV cache
(including prefix cache [8]) loading and offloading occur, these operational events are aggregated at
regular intervals and transmitted to the xLLM-Service via ETCD heartbeat mechanisms, enabling
unified global monitoring and management.
12
xLLM Technical Report
KV Cache-aware Scheduling.
In terms of scheduling strategy, the system implements a decision-
making mechanism based on KV cache utilization, which operates through three key steps: (1)
Prefix Matching Detection: calculating the KV cache reuse rate for each candidate node via prefix
matching analysis; (2) Performance Estimation: estimating the expected latency for different nodes
according to current load conditions and cache hit rates; (3) Optimal Node Selection: identifying
the node with the optimal overall performance for request processing, thereby enabling dynamic
offloading and migration of KV cache.
3.5
Fast Fault Recovery Architecture
Current fault handling mechanisms are mainly designed for model training [60–62] and cannot be
directly applied to large model inference due to its low-latency requirements. Specifically, existing
approaches primarily adopt a checkpoint-then-recovery method [63–65] for fault handling, which
periodically stores model data as checkpoints in distributed storage and reloads the most recent
checkpoint after a failure occurs [66]. As model parameters increase, the overhead of storage and
loading gradually grows. This is still acceptable in training since there are no strict latency require-
ments [67], but in inference, it may cause all high-priority requests on the failed instance to time
out, resulting in severe losses.
To address this issue, we propose an efficient failover architecture specifically for large model infer-
ence, with targeted optimizations in two aspects: fast request migration and fast instance recovery.
Fast request migration ensures the SLO of high-priority tasks mainly through an efficient kv cache
quick recovery strategy, while fast instance recovery achieves low-overhead recovery through effi-
cient masking of computation and communication. Through the above optimization methods, we
achieve fast fault recovery, greatly reducing the performance and economic losses caused by faults.
4
xLLM-Engine Designs
4.1
Multi-layer Pipeline Execution Engine
Traditional autoregressive LLM inference relies on single-stream sequential execution. This conven-
tional approach fails to fully exploit the parallelism capabilities of modern hardware, often leading
to computational resource underutilization, communication stalls, and cascading blocking delays in
heterogeneous computing environments [68, 69]. As illustrated in Figure 7, the system’s perfor-
mance is hindered by three primary sources of inefficiency: (1) CPU-Accelerator Dependency: At
the framework level, a rigid dependency between the CPU and the accelerator forces the accelerator
to remain idle during task scheduling and data processing phases, creating significant “computation
bubbles” [68, 3]. (2) Communication Latency: In both distributed workloads and complex model
layers, communication delays interrupt continuous computation, preventing the full utilization of
available hardware resources [69, 70]. (3) Architectural Bottlenecks: On certain AI accelerators,
the compute-focused (tensor cores) and general-purpose (vector cores) computation units lack a
shared high-level cache (e.g., L1 or SRAM). This architectural separation necessitates additional
data transfers between them, which introduces latency and leads to the underutilization of the spe-
cialized computation units.
To address these challenges, we propose a three-tier asynchronous pipeline parallelism design span-
ning the framework scheduling layer, the model graph layer, and the operator layer, as shown in
Figure 7, with the goal of maximizing hardware efficiency.
Framework-Layer Scheduling-Execution Overlap.
In a typical sequential execution procedure,
the accelerator remains idle while the CPU performs scheduling to prepare the input data batch for
the next computation cycle. This serial dependency creates significant latency and underutilizes the
accelerator.
To eliminate this bottleneck, we introduce an asynchronous pipelined scheduling mechanism that
decouples CPU scheduling from accelerator execution. The core idea is to overlap these two stages:
while the accelerator is executing the forward pass for the current batch, the CPU concurrently
assembles the batch and prepares the metadata for the next batch. This parallel workflow effectively
hides the latency of CPU-side scheduling and data preparation. The process is as follows: (1)
Accelerator Compute: The accelerator is busy executing the forward pass for the current iteration,
13
xLLM Technical Report
w/o Overlap
w/ Overlap
CPU
AI Accelerator
AI accelerator bubble
AI accelerator bubble
… …
…
…
Schedule 0
Schedule 1
Schedule 2
Schedule N
Forward 0
Forward 1
Forward 2
Forward N
Schedule 3
Schedule 0
Forward 0
Schedule 1
Forward 1
Schedule N
Forward N
Framework Layer
Communication
Stream
… …
Model Layer
No bubble
Attn 0
Dispatch 0
MoE 0
Combine 0
Attn 1
Dispatch 1
MoE 1
Combine 1
Computation resource waste
Computation resource waste
Attn 0
Attn 0
MoE 0
MoE 0
Attn 1
Attn 1
MoE 1
MoE 1
Full utilization
Computation
Stream
Dispatch 0
Dispatch 0
Combine 0
Combine 0
Dispatch 1
Dispatch 1
Combine 1
Combine 1
Computation
Stream - 1
Operator Layer
Vector
Matrix
Matrix Comp. Unit idle
Matrix Comp. Unit idle
Attn 1-0
Attn 1-1
Attn 1-2
Attn 1-N
MoE 1-0
MoE 1-1
MoE 1-2
MoE 1-N
Full utilization
Computation
Stream - 2
Stream 0
Attn 1-0
Attn 1-1
Attn 1-2
Attn 1-N
MoE 1-0
MoE 1-1
MoE 1-2
MoE 1-N
Vector
Matrix
Vector
Matrix
Vector
Matrix
AI Comp. Unit
CPU
AI Accelerator
Batch 0
Batch 1
Matrix Comp. Unit
Vector Comp. Unit
Matrix Comp. Unit idle
w/o Overlap
w/ Overlap
w/o Overlap
w/ Overlap
Figure 7: Multi-layer pipeline execution engine.
which will eventually produce output. (2) CPU Schedule: Instead of waiting, the CPU immediately
begins scheduling the batch for the next iteration. To do so, it uses a set of placeholder tokens that
stand in for the yet-to-be-computed output. This allows all CPU-side batch preparations to occur
concurrently with the accelerator’s computation. (3) Seamless Transition: Once the accelerator
completes the forward pass and the real output is generated, a fast replacement operation swaps the
placeholder tokens with the real generated tokens. Since all scheduling is already complete, the
accelerator can begin the next forward pass with minimal delay.
By transforming the conventional serial “prepare-then-compute"" workflow into a parallelized
pipeline, our method effectively masks scheduling latency behind accelerator computation time.
This design eliminates idle execution “bubbles"" in the accelerator timeline, thereby maximizing
hardware utilization and enhancing overall system throughput.
Model-Layer Computation-Communication Overlap.
Maximizing the overlap between com-
putation and communication is a critical design principle in recent LLM frameworks [3, 71]. This
strategy is essential for hiding the significant latency of data transfers and achieving high hard-
ware utilization. However, architectural constraints can make this challenging. For instance, on a
typical GPU, compute-focused units (Tensor Cores) are tightly coupled with general-purpose units
(CUDA Cores) within a Streaming Multiprocessor (SM). Consequently, when an SM is allocated
to a communication-related task, its powerful Tensor Cores often sit idle, leading to wasted com-
pute potential. In contrast, our target accelerator architecture allows for more flexible, independent
allocation of its Cube (matrix) and Vector (general-purpose) units. We leverage this flexibility to
address the aforementioned challenges by introducing a dual-stream pipeline architecture that uses
micro-batch splitting to effectively overlap computation and communication.
Specifically, our architecture consists of a Computation Stream for compute-bound tasks (Attention,
ExpertForward, etc.) and a Communication Stream for data distribution and collection tasks (MoE
Dispatch and Combine). To enable pipelining and maximize overall pipeline throughput, we parti-
tion a macro-batch B into n micro-batches [68] {b1, b2, · · · , bn}. The two streams asynchronously
execute tasks for different micro-batches. On the one hand, our scheduling policy dynamically deter-
mines the optimal execution order of the micro-batches. On the other hand, our resource scheduling
policy adaptively allocates an appropriate number of AI Cores (Vector Cores and Cube Cores) to
the Communication Stream and Computation Stream respectively. Figure 7 illustrates the case of
n = 2, where the Communication Stream performs the Dispatch operation for micro-batch bk while
the Computation Stream executes the ExpertForward pass for the preceding micro-batch bk−1.
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xLLM Technical Report
Operator-Layer Matrix-Vector Units Overlap.
On heterogeneous AI accelerators, the serial
scheduling of matrix and vector computation units often results in significant underutilization, as
one class of units remains idle while the other is active. This inefficiency motivates a strategy of
operator-level matrix-vector overlap to fully exploit the hardware’s parallel capabilities. However,
a naive implementation via coarse-grained parallel scheduling, without a systemic resource coordi-
nation mechanism, is also problematic. This approach frequently leads to disordered contention for
limited compute units, causing resource fragmentation and access conflicts that ultimately degrade
overall performance.
To address these challenges, we propose a dynamic resource allocation mechanism for computa-
tion units (Cube and Vector Units) based on real-time load monitoring. This mechanism enables
a deep pipeline across heterogeneous compute units by dynamically and adaptively assigning the
precise type and quantity of resources required for each concurrent operator. By doing so, it miti-
gates resource contention and ensures that parallel operators execute within highly overlapping time
windows, achieving precise computational overlap and maximizing hardware utilization.
We formulate the dynamic resource allocation as an optimization problem. Let C and V be the sets
of matrix and vector operators, respectively. Let xi be the number of matrix units (Cube) allocated
to the ith matrix operator (i ∈C), and yj be the number of vector units (Vector) allocated to the jth
vector operator (j ∈V). For simplicity, we assume all operators begin execution simultaneously,
without considering data dependencies or communication latency. Based on the known computa-
tional workload of each operator, our mechanism seeks to find an optimal resource allocation that
minimizes the maximum difference in execution times between any two operators. This objective,
which we term the alignment loss (Lalign), effectively synchronizes the completion time of all par-
allel kernels. The optimization problem is defined as:
argminxi,yjLalign =
max
i∈C, j∈V |Ti −Tj| ,
Ti =
Wi
γCube · xi
,
Tj =
Wj
γVector · yj
,
X
i∈C
xi ≤NCube,
X
j∈V
yj ≤NVector,
(1)
where T is the operator execution time, W is its computational workload, γcube and γvector are the
peak performance per unit, and Ncube and Nvector are the total available matrix and vector units.
The meaning of this optimization objective is to minimize the maximum difference in execution
time between any pair of concurrent matrix and vector operators.
4.2
Adaptive Graph Mode
The performance of LLM inference deployment is often impeded by Host-side CPU overhead, par-
ticularly when the computation graph is composed of many fine-grained operators [72]. This bottle-
neck manifests as both significant CPU-Accelerator synchronization latency, measured at 5 ∼50µs
per invocation from frequent kernel launches, and suboptimal hardware utilization due to idle accel-
erator cycles between these intermittent operator executions.
Mainstream AI accelerators such as NVIDIA GPUs (with CUDAGraph) [73] and Ascend NPUs
(with ACLGraph) [74] employ computation graphs to try to handle above issues and enhance host-
side scheduling performance. As illustrated in Figure 8, traditional Eager Mode relies on the CPU
to submit a multitude of small, intensive tasks, leading to frequent launches of small kernels on the
accelerator. In contrast, the Graph Mode enables the CPU to submit one large task, after which the
accelerator internally executes the small kernels in a streamlined fashion. This method significantly
reduces both launch overhead and “accelerator bubbles” (idle cycles). Specifically, the Graph Mode,
such as ACLGraph [74], comprises two distinct phases: graph capture and graph execution. During
the graph capture phase, the entire computation flow is recorded to capture the sequence of kernel
calls and their dependencies, including kernel launch parameters. This recorded workflow is then
pre-compiled into a replayable directed acyclic graph (DAG) object. It is important to note that
during this phase, tasks are merely staged within the model’s runtime instance and are not actually
executed. In the subsequent graph execution phase, the entire graph is launched via a single CPU
call. The accelerator then follows the pre-defined process autonomously, without real-time interven-
tion from the CPU. This effectively consolidates many individual kernel launches into one single
graph launch.
15
xLLM Technical Report
A
Launch A
Launch B
Launch C
Launch D
Launch E
B
C
D
E
Kernel Execution
Launch Overhead
Launch graph
D
Kernel Execution
Launch Overhead
Build Graph
cudaGraphExecKernelNodeSetParams
Speed Up
E
C
B
A
w/o Graph
w/ Graph
Figure 8: Pipeline comparison between Eager Mode and Graph Mode.
Building upon this foundation, we further advance the practical implementation of the ACL-
Graph [74] and propose our Adaptive Graph Mode through addressing key challenges including
dynamic shape adaptation, memory reuse conflicts across multiple graphs, and compatibility with
custom operators. This enables the pre-compilation of kernel sequences, memory copy operations,
and synchronization points into a single computational graph, which is then dispatched to the AI ac-
celerator in one execution. Consequently, our approach significantly reduces kernel launch overhead
and minimizes accelerator idle periods, maximizing overall hardware efficiency.
Dynamic Shape Adaptation via Partial Graph and Dimension Parameterization.
ACL-
Graph [74] fixes kernel parameters and memory sizes during the graph capturing phase. However,
in real-world scenarios, LLMs typically process inputs with variable sequence lengths and batch
sizes, making this static capturing characteristic difficult to adapt to dynamic input requirements.
To address this issue, dimension parameterization can be employed, treating key dynamic dimen-
sions—such as batch size and sequence length—as input parameters to the entire graph, thereby
enhancing flexibility. During memory allocation and kernel configuration, these dynamic param-
eters are used to compute actual required values; for example, the memory size can be calculated
using the formula: alloc_size = batch_size × seq_len × hidden_size. At the graph launch stage,
the actual batch size and sequence length are passed as parameters to ensure the graph adapts to
varying input dimensions.
Although parameterization can handle variations in dynamic dimensions, certain operator imple-
mentations on different hardware may not support dynamic dimension parameterization. To over-
come this, we propose the Partial Graph Mode: the model is partitioned into modules with simple
dynamic shapes (such as FFN and LayerNorm, which only have the num_tokens dimension as dy-
namic) and modules with complex dynamic shapes (such as Multi-Head Attention, which involves
multiple dynamic dimensions like batch_size, kv_seq_len, and q_seq_len). Modules with simple
dynamic shapes are extracted and compiled for execution using Partial Graph Mode, while those
with complex dynamic shapes are executed in Eager mode. Our Adaptive Graph Mode dynamically
selects the most appropriate mode above based on the current input shapes. This intelligent selection
ensures optimal performance across varying workload conditions.
Table 1 compares the key characteristics of three dynamic shape processing mode. The traditional
Eager Mode requires no pre-compilation but incurs high scheduling overhead at runtime due to N
kernel launches (where N is the number of operators). The Full Graph Mode, by fixing shapes,
allows single compilation and very low launch overhead, yet its lack of dynamic adaptability limits
its applicability. In contrast, the proposed parameterized and multi-caching Partial Graph Mode
strikes an optimal balance. It achieves execution performance comparable to that of a full graph
while maintaining high flexibility, by trading a manageable number of pre-compilations (M ≪
Nreq, where M is the number of cached graphs and Nreq is the number of actual requests) for the
high efficiency of a single graph launch.
16
xLLM Technical Report
Method
Eager Mode
Full Graph Mode
Partial Graph Mode
Compilation Times
0
1
M
Low Launch Overhead
×
✓
✓
Low Memory Usage
✓
×
✓/×
High Flexibility
✓
×
✓
Table 1: Comparison of different shape handling solutions. Our Adaptive Graph Mode dynamically
selects the most appropriate mode above based on dynamic shapes.
Efficient and Reusable HBM Memory Pool.
During the graph capture process, the specific
virtual addresses of input, output, and intermediate tensors are recorded. To prevent illegal memory
access, which can occur when the address of an input tensor changes during an actual inference
request, we develop a shared HBM memory pool. First, during graph initialization, a sufficiently
large, contiguous block of HBM memory is pre-allocated to serve as the graph’s memory pool.
Subsequently, for internal address management, computation graph intercepts all tensor memory
operations and re-describes them using offsets relative to the pool’s base address, rather than absolute
virtual addresses.
Before the graph is launched, the user-provided input tensor is copied into the HBM memory pool at
the offset pre-defined for the graph’s input. After graph execution is complete, the data from the out-
put tensor is copied to the user-specified output buffer address. Furthermore, internal tensors within
the graph are also managed using fixed offsets within the pool, ensuring safe reuse and efficient
memory management. This managed memory pool guarantees that all addresses used internally by
the graph are known and safe, which adapts to external address changes with only the overhead of
memory copies before and after launch.
Integrating Custom Operators.
Integrating performance-critical custom operators, such as
PageAttention and AllReduce, into the Adaptive Graph Mode presents a unique challenge. The
internal implementation of these operators often relies on CPU to perform just-in-time shape calcu-
lations and kernel configurations based on runtime inputs. This dynamic behavior conflicts with the
pre-compiled nature of computation graph.
To resolve this, we modified the implementation of such operators. The first step is to identify
custom operators within the graph that have dynamic shape dependencies. We then refactor these
operators to accept shape-related parameters (e.g., dimension sizes or loop counts, which can only
be determined at runtime) directly as kernel arguments. This approach avoids hard-coding these
values on the host side during the graph capture phase. As a result of this modification, these custom
operators can be successfully captured by our computation graph. By leveraging the graph’s param-
eterization mechanism, the required dynamic arguments for these kernels can be derived from the
graph’s main input parameters and passed along during execution. This method achieves seamless
integration of essential custom operators within the Adaptive Graph Mode framework.
4.3
Efficient Memory Management
LLM inference deployment faces a critical balance between memory utilization and computational
efficiency, particularly in autoregressive generation tasks where efficient KV Cache management
has become a key challenge. Traditional solutions fall into two categories: First, contiguous mem-
ory allocation approaches that statically pre-allocate memory space based on maximum sequence
length before inference, ensuring physical contiguity of KV Cache. While this improves compu-
tational efficiency, it results in low memory utilization. Second, PagedAttention [75] approaches
that support larger batch sizes through paging mechanisms, but frequent access to block tables sac-
rifices computational efficiency, and the increased parameters complicate operator development and
debugging. To address this challenge, inspired by virtual memory management in operating sys-
tem field [76, 77], we propose the xTensor memory management scheme, which adopts a "logically
contiguous, physically discrete" KV Cache storage structure that resolves the contradiction between
17
xLLM Technical Report
Req
Virt_start ... Virt_end
Virtual Address Space
Request R1：
MaxSeqLen
Virt_start ... Virt_end
Request R2：
Page1:
-PageID: P1
-Status: Reusable
-Owner: R1
Physical Page Pool
Page2:
-PageID: P2
-Status: Mapped
-Owner: R2
Scheduler
On-demand Mapping
Request R2: Write a short story about a young explorer discovering a hidden ancient civilization.
Request R1: I am a 25-year-old woman. Please give some recommendations for moderately priced clothes.
Smart Scheduling:
Request ID Reuse
Async Pre-mapping
…
…
Inference
Requests
Addr
Mapping
Figure 9: The framework of xTensor memory management.
Features
Contiguous Allocation
Paged Attention
xTensor (Ours)
Efficient Memory Usage
×
✓
✓
Efficient Computation
✓
×
✓
Large Batch Support
×
✓
✓
Operator Development Complexity
✓
×
✓
Table 2: Comparison of memory management strategies.
memory contiguity and dynamic allocation, thereby achieving both high computational efficiency
and high memory utilization (as shown in Table 2).
Physical Memory Page Management and Virtual Memory Tensor Management.
Figure 9 il-
lustrates the xTensor memory management framework. During service initialization, based on the
available memory size for KV Cache, a large number of fixed-size discrete physical memory pages
are pre-allocated. Each physical page maintains a triple state ⟨PageID, Status, OwnerSession⟩,
where PageID represents the page identifier, OwnerSession indicates the service owning the
page, and Status ∈{Free, Allocated, Mapped, Reusable} represents states of free, allocated,
mapped, and reusable to dynamically track physical page usage. Subsequently, each inference re-
quest is pre-allocated with a logically contiguous virtual address space with an address range equal
to MaxSeqLen, where this virtual address space is not actually associated with physical pages dur-
ing allocation. This decoupled design provides operators with a virtual view of KV Cache stored in
contiguous memory, thereby masking the discrete nature of underlying physical pages [78, 79].
On-demand Memory Mapping.
During actual allocation, the scheduler dynamically maps phys-
ical memory pages on-demand based on the actual sequence length of requests. When a sequence
generates new tokens and KV Cache needs expansion, the scheduler retrieves one or more free
pages from the pre-allocated physical page pool and maps these physical pages to the next available
contiguous address location in the request’s virtual address space. Since physical memory is gradu-
ally mapped as sequences grow, short sequences only require a small amount of physical memory,
thus avoiding the memory waste of contiguous allocation strategies that still require reserving space
according to maximum sequence length for short sequences.
After memory mapping is completed, kernel access to virtual addresses can automatically locate the
corresponding physical page phypageidx and offset:
phypageidx =
virt_addr −virt_start
page_size
,
offset = (virt_addr −virt_start) mod page_size,
(2)
where virt_addr and virt_start represent the current virtual address and starting virtual address
respectively, and page_size represents the physical page size.
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xLLM Technical Report
Requests Schedule
Draft model
Forward
Input
Target model
Forward
Input
Validate
Output
Requests Schedule
Draft model
Forward
Input
Target model
Forward
Input
Validate
CPU thread1
NPU
CPU thread2
Requests Schedule
Draft model
Forward
Input
Target model
Forward
Input
Validate
Draft model
Forward
Output
Target model
Forward
Input
CPU
NPU
Schedule
…
…
w/o Overlap
w/ Overlap
Figure 10: Comparison between original and our optimized speculative decoding.
Low Overhead Allocation.
If virtual address to physical page mapping is immediately released
upon request completion, new requests require remapping physical pages, but the Unmap oper-
ations on accelerators incur significant overhead. Therefore, under high load conditions, frequent
Map/Unmap operations become performance bottlenecks, especially for numerous short-term con-
current requests. We adopt request ID reuse and asynchronous pre-mapping designs to reduce la-
tency from Map/Unmap operations.
• Physical Page Reuse. Upon request completion, physical pages are not immediately released
but marked as Reusable. When new requests arrive, if their required KV Cache size matches
some Reusable physical page set, that page set is remapped to the new request’s virtual address
space. This saves expensive Map and Unmap operations through fast virtual address remapping,
particularly suitable for scenarios with concentrated request length distributions.
• Asynchronous Pre-mapping. During current token decoding, the next token’s required physical
pages are asynchronously predicted and mapped. Since new token writes to KV Cache occur
after current token decoding computation, this design can partially hide physical page mapping
operations, significantly reducing page mapping latency.
Operator Adaptation.
The decoupled design of xTensor’s physical memory pages and virtual
contiguous memory requires the adaptation of existing operators. On one hand, to accommodate
KV Cache contiguity in virtual address space, operators no longer require block_table parameter
input. Whether for attention operators or write_kvcache operators responsible for writing KV
Cache during Prefill and Decode phases, only the starting virtual address and related offsets need to
be passed in, with the system automatically associating virtual addresses to corresponding physical
pages during operation. On the other hand, accelerators lack native contiguous KV FlashMLA [80]
operators and only support PagedFlashMLA [80] operators optimized for paging. Therefore, by
reconstructing the PagedMLAttention operator in the accelerator operator library, removing Block
Table-related logic such as block table queries and cross-page boundary judgments, and adding
functionality for direct computation based on input virtual addresses, we implement FlashMLA
operators supporting contiguous virtual address input on accelerators.
4.4
Algorithm Optimizations
4.4.1
Optimized Speculative Decoding
Traditional autoregressive inference requires LLMs to generate tokens sequentially one by one, re-
sulting in high computational latency and limited throughput. Speculative decoding technology the-
oretically breaks through this performance bottleneck through a paradigm of parallelly predicting
candidate token sequences and quickly verifying them [81, 82, 26]. However, in distributed de-
ployment environments, this technology faces some core challenges: First, synchronous scheduling
between CPUs and accelerators in traditional multi-step inference frameworks results in insufficient
utilization of computing resources, with accelerators often remaining idle while waiting for data
preparation and processing. Second, existing attention mechanisms are not optimized for the char-
acteristics of speculative inference, causing a large amount of redundant data movement. To address
these challenges, xLLM proposes a systematic optimization scheme based on existing frameworks.
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xLLM Technical Report
Compute
Load
decode
EPLB Controller
Global Expert Mapping
Planner
Checker
Compute
Load
decode
decode
decode
decode
decode
Layer-0 Loader
Layer-0
Weight
Old
Buffer
New
Buffer
Layer-0 Rebalance
Layer-0
Weight
Old
Buffer
New
Buffer
Double Buffer
Double Buffer
Layer-1
Rebalance
…
DB Policy
Expert
Routing Map
Expert
Load
Worker-0
Ready
All
Ready
Expert
Load
…
Thread-0
Worker-0
…
Thread-1
Layer-1
Loader
…
…
Worker-0
Ready
All
Ready
…
…
Figure 11: Overview design of EPLB.
Asynchronous Decoding.
Inspired by [83], xLLM implements a lightweight asynchronous frame-
work. For timing overlap optimization, while the accelerator executes the main model inference for
the current batch, the CPU processes the output decoding of the previous batch and the input prepa-
ration for the next batch in parallel—thereby minimizing the idle time of the CPU and accelerator
as much as possible.
MLA Optimization.
xLLM focuses on improving the MLA [84, 3] computation process. Based
on the characteristic that speculative inference needs to process multiple tokens simultaneously on
a single sequence, we conducted in-depth optimizations on the self-attention computation in MLA.
Specifically, when speculatively inferring m tokens, the self-attention computation of MLA involves
product operations between m+1 Q matrices and the same K matrix. By reconstructing the compu-
tation process and optimizing the tiling strategy, we effectively reduced the data movement overhead
of Q/K matrices. The main optimization measures include:
▷Optimization of K matrix loading times: By adjusting the L1 cache allocation scheme to enable
parallel loading of multiple Q matrices, and adopting a sliding window-based K matrix loading
method that allows consecutive rows of the K matrix to multiply with multiple Q matrices, the
number of K matrix loading operations is significantly reduced.
▷
Q matrix cache residency mechanism: Since there are m+1 Q matrices in speculative infer-
ence scenarios, the time to move Q matrices to L1 accounts for a larger proportion of the matrix
multiplication process. xLLM redesign the computing scheme to prevent softmax-V product opera-
tions from overwriting Q matrices in the L1 cache, enabling Q matrices to remain in the cache and
significantly reducing the overhead of repeated data movement, thereby effectively improving the
utilization of the tensor core’s arithmetic logic units.
4.4.2
Dynamic EP Load Balance
With the large-scale application of MoE models [85–88] in production environments, their efficient
inference capability relies on a expert routing mechanism, which allocates input tokens to differ-
ent experts for computation [3, 18, 89]. However, in practical deployment, due to the distribution
characteristics of input data and the influence of the model’s own structure, there may be significant
differences in the number of tokens received by different experts [90, 91]. This imbalance leads
to low computing resource utilization: some devices are overloaded due to processing too many
tokens, while others remain idle, affecting the overall inference efficiency.
To optimize resource utilization, the industry has proposed the Expert Redundancy strategy, which
means replicating hot experts and deploying replicas on multiple devices to distribute computing
loads. Currently, DeepSeek [3] adopts two load balancing strategies: group-limited load balance
and global load balance, which are optimized respectively for the characteristics of the prefill and
decode phases. Adjustments to expert redundancy (such as adding/deleting replicas) require ad-
20
xLLM Technical Report
Preprocess
MLA
Preprocess
MLA
MLA
MLA
MLA
MLA
MLA
MLA
Preprocess
MLA
Preprocess
MLA
MLA
MLA
MLA
MLA
MLA
MLA
Aggregate
transfer
transfer
1K tokens
3K tokens
3K tokens
DP-1
(seq-1)
DP-2
(seq-2)
DP-1
(seq-1)
DP-2
(seq-2)
w/o DP Load
Balance
w/o DP Load
Balance
Figure 12: Inter-DP group load migration at MLA block granularity.
ditional memory and may affect inference latency due to weight migration. How to achieve such
adjustments efficiently and smoothly is a major challenge. We have made some adaptive improve-
ments based on DeepSeek, realizing dynamic EP load balance.
Expert Load Statistics.
When the Router distributes tokens, it records the load status of each
expert in real time and returns the statistical results through the model output. Each Worker asyn-
chronously starts a thread to periodically aggregate expert loads and report them to the Controller of
the Master. The Controller calculates a new routing table based on the load data and then distributes
the updated routing table to each Worker.
Double Buffer Mechanism for Weight Update.
In general, each Worker starts an asynchronous
thread to update the weights of single-layer experts during each decoding process. Specifically, after
a Device completes the preparation of new weights, it proactively sends a readiness notification to
the Controller. The Controller then verifies the weight readiness status of all Worker nodes to ensure
global consistency before broadcasting an update instruction, upon which each Worker executes the
weight update immediately. Here, we adopt a Double Buffer mechanism, meaning that the old and
new expert weights are stored in two separate buffers respectively. The new expert weights are
preloaded in the spare memory space, and after the preloading is completed, the address switching
is performed to realize the unperceived update of experts.
4.4.3
Hierarchical DP Load Balance
In typical MoE model architectures like DeepSeek [3], the MoE layers use expert parallelism (EP),
while the attention layers use data parallel (DP). This model requires all DP groups to synchronously
complete their attention calculations in a single inference step before initiating the MoE all-to-all
dispatch operation. This synchronization barrier means the total time for the attention phase is
determined by the slowest DP group (the “straggler”). Faster DP groups cannot immediately proceed
to the next stage after finishing their tasks; instead, they are forced into an idle state, waiting for the
straggler to complete [92]. This waiting directly translates into wasted computational resources and
an increase in overall inference latency.
In practical scenarios, due to the dynamic and unpredictable nature of inference requests, the load
within different DP groups is often difficult to balance. Particularly in the decoding phase with
large-scale DP (e.g., 80 groups), the load difference between DP groups can reach tens of thousands
of tokens, causing fast DP groups to wait for slow ones and wasting approximately 5 milliseconds of
delay [93]. Experimental data shows that although the difference in the number of requests processed
between DP groups may be as small as four, the corresponding difference in KV Cache token count
during the decoding phase can be as high as 20,000. The actual computational workload is directly
related to the total number of tokens the system needs to process, as this determines the memory
footprint of the KV Cache and the scale of matrix operations in the attention mechanism. This
further establishes DP load balancing as a critical factor in determining overall system efficiency.
Current mainstream frameworks like vLLM [8] and SGLang [9] use a simple static round-robin
scheduling strategy. Once a request is assigned, its subsequent computations are fixed to a specific
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xLLM Technical Report
DP group, making it unable to adapt to dynamic load changes. Furthermore, at the hardware level,
the implementation of operators like the MLA on certain accelerators employs a “one request per
tensor compute core” partitioning strategy. This can lead to idle compute cores within the same DP
group due to differences in request lengths.
To address these challenges, xLLM proposes a hierarchical “defense-in-depth” strategy to tackle DP
load imbalance issues across different time scales and granularities. The first layer is preventative,
KV Cache-aware request scheduling; the second layer performs macroscopic correction through
inter-DP group load balancing; and the third layer carries out microscopic correction via intra-DP
group load balancing. This layered design enables the system to flexibly address the root causes of
various performance degradations, thereby building a robust and efficient inference service.
Layer 1: KV Cache-Aware Scheduling.
xLLM’s first-layer strategy implements request load dis-
tribution through KV Cache-aware scheduling. This mechanism moves beyond simple round-robin
methods. When a new request arrives, the scheduler comprehensively checks the status of all DP
groups, paying special attention to the remaining KV Cache capacity in each. It then preferentially
assigns the new request to the group with the most available space. By balancing the total token load
and its memory footprint at the system level over the long term, this strategy prevents the formation
of severe load imbalances, thereby achieving intelligent resource allocation.
Layer 2:
Reactive Inter-DP Group Workload Migration.
xLLM’s second-layer strategy
achieves reactive balancing through workload migration between DP groups. During the decoding
process, the varying prompt and KV Cache lengths of selected requests cause different computa-
tion latencies among DP groups. To counter this, xLLM’s central scheduler evaluates the current
computational load of each DP group during every inference round. If a significant imbalance is
detected, it initiates a workload migration process, moving tasks from overloaded to under-loaded
groups. The scheduler also determines the migration granularity – whether to move an entire batch,
a single sequence, or a partial MLA block of a sequence.
Figure 12 illustrates this process at the MLA block granularity. First, xLLM dispatches the input
tokens for the migration to both the source and destination DP groups. Then, as all groups execute
the MLA Preprocess operation, the request’s KV Cache is transferred asynchronously, overlapping
communication with computation. Next, all groups begin the attention calculation. The underloaded
group prioritizes the migrated attention task, allowing it to immediately send the resulting token
back to the source group upon completion while concurrently processing its own native attention
operations, further overlapping overhead. Finally, all DP groups use an aggregation operator to
gather the external computation results.
Layer 3: Fine-Grained Kernel-Level Optimization.
xLLM’s third-layer strategy achieves fine-
grained optimization through kernel-level migration within a DP group. This optimization is per-
formed inside the matrix computation kernels of a single DP group. On one hand, during scheduling,
it reorders requests based on their load, replacing the original compute core’s round-robin alloca-
tion strategy. This aims to keep the total number of computation tokens assigned to each matrix
computation unit as consistent as possible. On the other hand, for requests with extremely long
sequences, it explicitly splits their computation sequence, migrating parts of the long-sequence re-
quest’s computation to be calculated with other short-sequence computation units. Through this
fine-grained request splitting, it directly resolves the issue of idle compute cores caused by load
imbalance within the DP group.
4.5
Generative Recommendation
Recent advancements in generative models have spurred significant interest in generative recom-
mendation [31, 94–98]. Beyond enhancing the retrieval and ranking stages of traditional multi-stage
pipelines [98, 94], single-stage generative recommendation frameworks directly generate a candi-
date item list in an auto-regressive manner [31]. These frameworks usually utilize beam search
decoding to directly produce diverse recommendation results, and xLLM implements extensive op-
timizations for these single-stage frameworks. As shown in Figure 13, in the generative recommen-
dation scenario, xLLM is dedicated to overlapping host and device operations as much as possible to
improve overall execution efficiency. When a request arrives, xLLM preemptively prepares the in-
formation required during the forward process on the CPU side. After the preparation is completed,
22
xLLM Technical Report
Pre-allocate
Scheduler
Schedule
Beam
Schedule
Pre-allocate
Beam
Pre-allocate
Schedule
Beam
Pre-allocate
Schedule
Beam
w/o Overlap
w/ Overlap
Prefill
Beam
Decode
Beam
Filter Mask
Worker
Forward
Add
Sampler
Logits
w/o Overlap
w/ Overlap
Engine
Decode
Filter Mask
Forward
Logits
Add
Sampler
Figure 13: Overview design of generative recommendation process.
relevant tasks are handed over to the engine for processing. The Engine executes three forward
passes in one go. During this process, the intermediate beam search is a host operation, which by
its nature cannot be overlapped with device operations. To optimize the processing pipeline, xLLM
internally employs a multi-micro-batch, multi-stream approach. During the Worker’s execution, the
CPU operation for generating the filter mask is overlapped with the device operation for logits com-
putation. Through an addition operation, a masked logits is generated before the sampler, thereby
achieving data filtering and improving the accuracy and effectiveness of the recommendation results.
4.5.1
Host-side Optimization
In scenarios with large beam_width and top_k parameters, the significant increase in the number
of candidate sequences results in substantial sorting and filtering operations on the host side. This
shifts the computational bottleneck of the beam search algorithm from the AI accelerator to the CPU,
ultimately leading to a severe CPU-bound issue. The following outlines xLLM’s efforts to optimize
the beam search process in generative recommendation scenario.
Beam Search Optimization.
In many fields such as natural language processing, beam search
optimization is an efficient algorithm widely used in sequence generation tasks [99]. When process-
ing a large number of candidate sequences, one of the core operations of beam search is to select
beam_width sequences from numerous candidates. This process is essentially a partial sorting op-
eration, with the unique characteristic that it only needs to identify the top beam_width elements
without fully sorting all candidate sequences, thereby significantly reducing computational com-
plexity. In another key step of beam search, generating beam_width × top_k candidate sequences
from the existing beam_width sequences, there exists an important property: log_probs of each se-
quence are arranged in descending order. Leveraging this property, an optimized early termination
mechanism can be adopted, where computations are halted prematurely if log_probs does not meet
specific criteria, further enhancing computational efficiency. The specific operational procedure is
as follows:
First, the existing beam_width sequences are accessed in order. For each accessed sequence, we
create a small min-heap of size beam_width, which is used to dynamically maintain the set of
locally optimal elements currently selected. Next, the subsequent sequences are traversed. During
this process, if the subsequent log_probs of a sequence is smaller than the top element of the min-
heap, it implies that, based on the current filtering criteria, the subsequent elements of this sequence
cannot enter the current optimal set. At this point, the filtering operation for the current sequence
can be terminated directly, and the next existing sequence processed. Conversely, if the subsequent
log_probs is larger than the top element of the min-heap, the element is inserted into the min-heap,
23
xLLM Technical Report
and the heap structure is adjusted according to the properties of the min-heap to maintain its order.
After the traversal of all relevant sequences is completed, the top elements are sequentially extracted
from the min-heap. These elements, in the order of extraction, form the beam_width candidate
sequences sorted in descending order. Through this procedure, the approach ensures the selection of
high-quality candidate sequences while significantly reducing unnecessary computational overhead,
thereby improving the operational efficiency of the beam search algorithm in practical applications.
Resource Reuse and Pre-allocation.
During the beam search process, the beam_width is pre-
determined and fixed for each forward pass of the model, which implies that the computational
resources required per forward propagation remains relatively constant. However, an excessively
large beam_width can lead to significant wastage of both CPU computational and memory re-
sources. To effectively mitigate this issue, xLLM incorporates a carefully designed resource reuse
strategy. Specifically, during the generation of new candidate sequences, the system reuses resources
previously occupied by older sequences without allocating new space for each candidate. This ap-
proach avoids the overhead associated with frequently creating new data structures. Based on the
final search results, the system then updates the storage areas for the old sequences with the content
of the newly generated ones upon completion of the beam search algorithm. By doing so, xLLM
not only reduces memory usage but also minimizes the additional overhead on the CPU for resource
management and allocation, significantly improving the efficiency of computational resource usage.
4.5.2
Device-side Optimization
Valid Item Filtering.
In typical single-stage generative recommendation frameworks, the valid
item filtering mechanism plays a crucial role. For example, OneRec [31] framework uses an ordered
combination of three token IDs to uniquely represent a valid product item ID. However, due to
the vast number of possible token ID combinations, not all of them correspond to actual and valid
items [100, 101]. To ensure that the results generated by the model are all valid product items, the
system generates a valid mask asynchronously during the forward pass of the model. This mask is
based on a pre-constructed vocabulary of valid items, which is then element-wise added to the logits
output by the model. Through this clever design, the logits corresponding to invalid token IDs are
adjusted, ensuring that these invalid token IDs are almost never selected in subsequent calculations.
This effectively filters out invalid token IDs, thereby guaranteeing the validity and accuracy of the
final recommendation results.
5
Evaluations
In §5.1, we evaluate the full implementation of our xLLM framework against several base-
line inference systems, specifically vLLM-Ascend1(v0.10.rc1) and MindIE2(v2.1.rc1), on Ascend
910B/910C [18] instances across a range of scenarios. xLLM, MindIE, and vLLM-Ascend refer
to the default deployments on Ascend 910B. Additionally, we denote the deployments of xLLM,
MindIE and vLLM-Ascend on Ascend 910C as xLLM‡, MindIE‡ and vLLM-Ascend‡, respectively.
The testing scenarios are organized into multiple categories reflecting diverse online serving appli-
cations, such as the JingYan AI chatbot, customer service assistants, merchant assistants, product
understanding and marketing recommendation systems. Additionally, in §5.2, we conduct a detailed
ablation study to assess the individual contributions of each optimized module.
5.1
Main Results
This section benchmarks xLLM against mainstream inference frameworks to demonstrate its
superior inference efficiency across diverse models, including Qwen2/3-series [102, 87] and
Deepseek [3] models, and datasets. §5.1.1 details a fair comparison using the ShareGPT3 dataset.
To ensure an equitable comparison across all LLM inference frameworks, the key feature of our
experimental setup is that the input and output sequence lengths are fixed, while the request
1https://github.com/vllm-project/vllm-ascend
2https://www.hiascend.com/en/software/mindie
3https://huggingface.co/datasets/anon8231489123/ShareGPT_Vicuna_unfiltered/blob/
main/ShareGPT_V3_unfiltered_cleaned_split.json
24
xLLM Technical Report
1
2
4
Number of Accelerators
2000
4000
6000
8000
10000
Throughput (tokens/s)
Qwen3-0.6B
1
2
4
Number of Accelerators
2000
4000
6000
8000
Qwen3-1.7B
1
2
4
Number of Accelerators
2000
3000
4000
5000
6000
Qwen3-4B
1
2
4
Number of Accelerators
1000
2000
3000
4000
5000
Throughput (tokens/s)
Qwen3-8B
1
2
4
Number of Accelerators
1000
2000
3000
4000
Qwen3-14B
2
4
8
Number of Accelerators
1000
2000
3000
Qwen3-32B
xLLM
xLLM
MindIE
MindIE
vLLM-Ascend
vLLM-Ascend
Figure 14: Throughput comparison of LLM inference frameworks evaluated on Qwen3-series mod-
els using the ShareGPT dataset, under the constraint of TPOT=50ms and input/output length=2048.
16×910B
8×910C
0
100
200
300
400
Throughout (tokens/s)
[2048,2048] TPOT:50ms
16×910B
8×910C
0
500
1000
1500
[2048,2048] TPOT:100ms
16×910B
8×910C
0
100
200
300
400
[2500,1500] TPOT:50ms
16×910B
8×910C
0
250
500
750
1000
1250
1500
[2500,1500] TPOT:100ms
xLLM
xLLM
MindIE
MindIE
vLLM-Ascend
Figure 15: Throughput comparison of LLM inference frameworks using DeepSeek-R1 on the
ShareGPT dataset constrained by given TPOT and input/output length, where [2500, 1500] means
input length=2500 and output length=1500. vLLM-Ascend on the 910C is excluded from the com-
parison, as its performance with Deepseek-R1 fails to meet the required TPOT threshold.
rate is dynamically adjusted to match the target SLO (e.g., TPOT) threshold for each frame-
work. We also evaluate this setup under different node configurations, including both single-node
and multi-node setups with PD disaggregation. In §5.1.2, we evaluate xLLM on various real-world
business scenarios from JD.com, where it is currently deployed, showcasing its performance under
practical deployment conditions.
5.1.1
Benchmarking Performance
Qwen3-series.
As shown in Figure 14, we comprehensively compares the throughput performance
of four inference frameworks across the Qwen3-series (from 0.6B to 32B parameters) on the bench-
mark. All tests are conducted under uniform configuration with input/output lengths set to 2048
tokens and a TOPT constraint of 50 ms. The experimental results demonstrate that while all eval-
uated frameworks show improved throughput with additional accelerators, xLLM and its Ascend
910C implementation (xLLM‡) consistently deliver superior performance, confirming near-linear
strong scalability across various model sizes. Specifically, xLLM achieves throughput improve-
ments of up to 1.9× and 1.7× compared to vLLM-Ascend and MindIE, respectively. Similarly,
xLLM‡ outperforms vLLM-Ascend‡ and MindIE‡ by up to 2.2× and 1.5×, respectively. Further-
25
xLLM Technical Report
Method
Prompt Length
Output Length
Throughput (tokens/s)
Request Rate (req/s)
MindIE
2048
2048
8476.44
4.14
xLLM
2048
2048
11351.58
5.54
Table 3: Comparison of DeepSeek-R1 with PD disaggregation on the ShareGPT dataset constrained
by TPOT=100ms.
1
2
4
Number of Accelerators
400
600
800
1000
1200
1400
Throughput (tokens/s)
Qwen2-7B TPOT:40ms
1
2
4
Number of Accelerators
0
500
1000
1500
2000
Qwen2-14B TPOT:90ms
1
2
4
Number of Accelerators
2000
3000
4000
5000
Qwen3-8B TPOT:80ms
2
4
8
Number of Accelerators
1000
2000
3000
Qwen3-32B TPOT:80ms
xLLM
xLLM
MindIE
vLLM-Ascend
Figure 16: Throughtput comparison of Qwen2-series and Qwen3-series models with various infer-
ence frameworks in the JingYan scenario.
more, xLLM‡ demonstrates steady performance improvements over the Ascend 910B-based xLLM
in most scenarios, validating the software stack’s effective utilization of the new hardware.
DeepSeek-R1.
Figure 15 presents the throughput performance of various inference frameworks
for the DeepSeek-R1 model on the benchmark, using 16 accelerators for Ascend 910B and 8 accel-
erators for Ascend 910C, respectively. The results demonstrate that the proposed xLLM framework
achieves exceptional throughput on the Ascend 910 series. Quantitatively, xLLM on Ascend 910B
delivers an average throughput improvement of approximately 1.7× over MindIE and a significant
12× enhancement over vLLM-Ascend. Furthermore, xLLM‡ achieves an average throughput in-
crease of approximately 1.4× compared to MindIE‡.
PD Disaggregation Settings.
Table 3 benchmarks the inference performance of the MindIE and
xLLM frameworks on the DeepSeek-R1 model using a PD disaggregation architecture. Under iden-
tical conditions with TPOT controlled at 100ms for 2048-length inputs/outputs, xLLM achieves
approximately 34% higher throughput (11,351.58 vs. 8,476.44 tokens/s) and request rate (5.54 vs.
4.14 req/s), marking a significant efficiency improvement.
5.1.2
Business Serving Scenarios
JingYan.
JingYan is an AI shopping assistant designed to help users discover new products, find
inspiration, and get answers to their questions. The dataset for JingYan, consequently, consists of
conversational logs between the model and its users, capturing these rich interactions. As illustrated
in Figure 16, we systematically evaluate the inference throughput of four frameworks for serving
the Qwen2-series and Qwen3-series model in the JingYan scenario. The results demonstrate that
both xLLM and xLLM‡ maintain superior throughput and exhibit better scaling efficiency across
all model sizes compared to MindIE and vLLM-Ascend. For instance, when serving the Qwen3-8B
model with 4 accelerators, xLLM delivers a throughput approximately 1.6 times that of vLLM-
Ascend and significantly surpasses MindIE. The robust performance of xLLM on the 910B, and its
enhanced results on the 910C hardware, highlight its effective adaptation to successive hardware
generations. As shown in Table 4, a similar trend is observed for the DeepSeek-V3 model, where
xLLM achieves a throughput that is over 9 times greater than vLLM-Ascend and surpasses MindIE
by 36%.
Customer Service.
The Customer Service dataset comprises the interactive dialogues between
customers and support agents. Figure 17 details the performance differences among inference frame-
26
xLLM Technical Report
Method
Prompt Length
Output Length
Throughput (tokens/s)
Request Rate (req/s)
vLLM-Ascend
6800
400
21.17
0.11
MindIE
6800
400
144.40
0.67
xLLM
6800
400
196.45
0.89
Table 4: Comparison of Deepseek-V3 model with various frameworks in the JingYan scenario con-
strained by TPOT=80ms.
1
2
4
Number of Accelerators
50
100
150
Throughput (tokens/s)
Qwen3-8B
2
4
8
Number of Accelerators
20
40
60
80
Qwen3-32B
xLLM
xLLM
MindIE
vLLM-Ascend
Figure 17: Throughtput comparison of Qwen3-series models with various inference frameworks in
the customer service scenario constrained by End-to-End latency(E2E)=10s.
works for the Qwen-8B and Qwen-32B models in the customer service scenario. The results high-
light that our xLLM, particularly the xLLM‡ running on Ascend 910C, provide significantly better
throughput across all tested configurations. For instance, with Qwen3-32B on 8 accelerators, the
throughput of xLLM is 3.1 and 1.2 times greater than that of vLLM-Ascend and MindIE, respec-
tively. It is important to note that the vLLM-Ascend framework shows a clear scaling bottleneck
as the number of accelerators increases, whereas xLLM maintains near-linear efficiency scaling.
This validates the high efficiency and superiority of the xLLM framework in managing distributed
inference for large-scale models.
Merchant Assistant.
Table 18 benchmarks the throughput of various inference frameworks on
three tasks (i.e., search terms, arrangement, intent recognition) within the merchant assistant sce-
nario. The proposed xLLM framework achieves better or comparable performance to MindIE and
demonstrates a significant lead over vLLM-Ascend. Specifically, for the search terms task with four
accelerator cards, xLLM delivers 34% higher throughput than MindIE and roughly 3.4× that of
vLLM-Ascend.
Product Understanding.
For the product understanding scenario, the throughput comparison of
the Qwen2-7B model with several frameworks is shown in Table 5. The experimental results indicate
that xLLM outperforms MindIE and vLLM-Ascend by an average of 25% and 56%, respectively,
across different accelerator card counts. Moreover, the superiority of xLLM scales with the num-
ber of cards, demonstrating its effective utilization of large-scale parallel computing resources and
thereby offering a robust solution for high-performance LLM inference.
Generative Recommendation.
As illustrated in Figure 19, the evaluation results on the Qwen-8B
model demonstrate that xLLM consistently achieves lower mean end-to-end latency than MindIE
across various request rates and beam widths, except under very low load condition. Notably, the
performance advantage of xLLM becomes increasingly pronounced as the beam width (from 4 to
128) and the request rate escalate. For instance, under the most challenging scenario with a beam
width of 128 and a request rate of 8, xLLM reduces latency by approximately 23% relative to
MindIE. This significant improvement validates that our xLLM’s host-side and device-side opti-
mizations effectively alleviate the computational bottlenecks in generative recommendation tasks,
markedly enhancing inference efficiency and scalability under heavy loads.
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xLLM Technical Report
1
2
4
Number of Accelerators
100
200
300
Throughput (tokens/s)
Search Term Qwen2.5-14B
xLLM
MindIE
vLLM-Ascend
1
2
4
Number of Accelerators
50
75
100
125
Arrangement Qwen2.5-14B
xLLM
MindIE
vLLM-Ascend
1
2
4
Number of Accelerators
100
150
200
250
300
Intent Recognition Qwen3-14B
xLLM
MindIE
vLLM-Ascend
Figure 18: Throughtput comparison of Qwen-series models with various inference frameworks in
the merchant assistant scenario constrained by End-to-End latency(E2E)=1s.
Method
Prompt Length
Output Length
Throughput (tokens/s)
#accelerator=1
#accelerator=2
#accelerator=4
vLLM-Ascend
1200
40
795.77
874.97
1272.52
MindIE
1200
40
944.81
1051.44
1693.45
xLLM
1200
40
1001.91
1323.90
2425.13
Table 5: Throughtput comparison of Qwen2-7B model with various inference frameworks in the
product understanding scenario.
5.2
Ablation Study
Impact of MTP.
As shown in Figure 20, under the configuration of 1500 input length and 2500
output length for the DeepSeek-R1 model, enabling Multi-Token Prediction (MTP) technology sig-
nificantly optimizes inference performance. As max concurrency increases, the MTP-enabled ver-
sion consistently exhibits lower TPOT compared to the baseline, indicating reduced generation la-
tency. Meanwhile, its throughput is markedly higher than the non-MTP version, with particularly
notable advantages beyond 32 concurrent requests. This demonstrates that MTP effectively en-
hances computational efficiency and system throughput under high concurrency conditions.
Impact of Dynamic PD Disaggregation Scheduler Policy.
We evaluate our proposed SLO-aware
Dynamic PD Disaggregation Policy against the Minimal Load and Round Robin strategies, as illus-
trated in Figure 21. On the Azure Code dataset characterized significant bursty traffic, our SLO-
aware policy achieves a request serving rate 1.67 times that of the Minimal Load strategy. Mean-
while, relative to the Round Robin strategy, the Minimal Load strategy improves the SLO attainment
by up to 4.3%. For the Azure Conversation dataset with stable input/output length variations, our
SLO-aware policy results in a 1.1 times higher request serving rate compared to the Minimal Load
strategy. Although the Minimal Load strategy performs similarly to Round Robin, it still enhances
the SLO attainment by up to 2.4%. These results indicate that minimal-load scheduling is closer to
the optimal scheduling strategy than round-robin approach, while our adaptive instance scheduling
delivers the best overall performance.
Impact of Hybrid EPD Disaggregation Scheduler Policy.
Figure 22 demonstrates the effective-
ness of the proposed Hybrid EPD Disaggregation Policy on the TextCaps dataset, specifically in
controlling TPOT and improving the SLO attainment rate. Under the configuration of 8 general-
purpose inference instances, the removal of the hybrid EPD disaggregation method results in a drop
in goodput from 9.5 req/s to 7.2 req/s, and a subsequent additional removal of stage-level scheduling
policy further decreases the goodput to 5.1 req/s. This indicates that a well-designed disaggregation
strategy can effectively mitigate inter-stage interference, while our stage-level scheduling policy
contributes to finer control over the execution time of each batch. The integration of both compo-
nents significantly enhances the overall performance and stability of the system, particularly under
high-concurrency scenarios.
Impact of Online-Offline Co-location Scheduler Policy.
As shown in Figure 23, we evaluated
three scheduling strategies, namely the baseline P/D, online priority and our proposed Online-Offline
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xLLM Technical Report
1
4
8
Request Rate (req/s)
90
100
110
Mean E2E (ms)
Beam Width:4 Top-k:4
xLLM
MindIE
1
4
8
Request Rate (req/s)
100
120
140
Beam Width:32 Top-k:32
xLLM
MindIE
1
4
8
Request Rate (req/s)
125
150
175
200
Beam Width:64 Top-k:64
xLLM
MindIE
1
4
8
Request Rate (req/s)
150
200
250
300
350
Beam Width:128 Top-k:128
xLLM
MindIE
Figure 19: Comparison of mean End-to-End latency(E2E) with various inference frameworks in the
generative recommendation scenario. Since vLLM-Ascend does not support beam width>10, the
corresponding results are not plotted in the figure. Even with beam width=4 and request rate=1, its
mean E2E far exceeds that of our framework.
2
8
32
Max Concurrency
50
100
150
TPOT (ms)
DeepSeek-R1 [2500, 1500]
w/ MTP
w/o MTP
2
8
32
Max Concurrency
100
200
Throughput (tokens/s)
DeepSeek-R1 [2500, 1500]
w/ MTP
w/o MTP
Figure 20: Impact of MTP on the concurrent performance of DeepSeek-R1 model.
Co-location Scheduler Policy (denoted as xLLM-OOC), to assess the maximum achievable through-
put for offline requests without violating the SLO of online requests. The green shaded region in the
figure indicates the sustainable range of offline request throughput that remains within the acceptable
online SLO violation threshold. When the offline query-per-second (QPS) exceeds a certain level,
both the baseline P/D and online priority strategies lead to a sharp increase in the online SLO viola-
tion rate, indicating interference caused by the offline workload. In contrast, xLLM-OOC maintains
stable SLO compliance even as offline QPS continues to rise. On our proprietary dataset, xLLM-
OOC achieves a throughput that is three times higher than the other two methods. Furthermore, it
demonstrates improvements of 75% and 17% over online priority and baseline P/D, respectively, on
the Azure Code dataset.
Impact of Multi-layer Pipeline Execution.
As evidenced by the results in Table 6, our proposed
asynchronous scheduling mechanism delivers consistent throughput improvements across all evalu-
ated model scales. The most substantial relative gain of 17.4% is observed for the 1.5B parameter
model, highlighting the method’s particular efficacy for smaller architectures where scheduling over-
head constitutes a larger portion of total computation time. While relative improvements moderate
for larger models (reaching 0.6% for 7B, 3.7% for 14B, and 6.6% for 32B), all configurations exhibit
statistically significant absolute gains. These results robustly validate that our method successfully
masks scheduling latency and eliminates computational bubbles, through leveraging placeholder
tokens to decouple and overlap CPU scheduling with NPU execution.
We further assess the effectiveness of the proposed dual-stream architecture for the DeepSeek-R1
model in Table 7. Experimental results from a single decoder layer show that the total communica-
tion time increases to 12.4ms in dual-stream mode, up from 9.3ms in single-stream mode. However,
the computation-communication overlap mechanism successfully hides 80% of the communication
time, which brings the exposed communication time down to just 2.5ms, saving 6.8ms per layer. De-
spite introducing a computational overhead of 4ms for each layer, the dual-stream strategy yields a
net performance gain of 172.0ms across the entire 61-layer model, clearly illustrating the capability
of our scheduling strategy in real-world workloads.
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xLLM Technical Report
10
20
30
40
50
Request Rate (req/s)
40
60
80
100
SLO Attainment (%)
Azure Code
25
50
75
100
Request Rate (req/s)
40
60
80
100
Azure Conversation
Our SLO Aware
Minimal Load
Round Robin
Figure 21: Performance of Dynamic PD Disaggregation Policy with different scheduling strategies.
1
2
3
4
5
6
7
8
9
10
11
12
Request Rate (req/s)
0
20
40
60
80
100
SLO Attainment (%)
Ours
w/o hybrid EPD
w/o hybrid EPD and sched
Figure 22: Impact of Hybrid EPD Disaggregation Scheduler Policy.
Impact of Adaptive Graph Mode.
To validate the effectiveness of the Adaptive Graph Mode opti-
mization technique, we conduct controlled experiments on the Qwen3-1.7B and Qwen3-4B models.
As shown in Table 8, with both the prompt and output lengths set to 2048 tokens, enabling Adaptive
Graph Mode result in significant performance gains for both models. The throughput of Qwen3-
1.7B increases from 2385.58 to 3038.88 tokens/s (a 27.4% improvement), while its mean TPOT
decreases by 22.0%. Correspondingly, Qwen3-4B achieved an 8.5% increase in throughput and an
8.8% reduction in latency. These results robustly demonstrate the effectiveness of Adaptive Graph
Mode as a general-purpose inference acceleration technique, with more pronounced optimization
effects observed on the model with a smaller parameter size.
Impact of Hierarchical DP Load Balancing.
Our proposed hierarchical DP load balancing
scheme is projected to increase total throughput by 5%. The kernel-level optimization yields the
most significant latency savings; for example, for an ultra-long 32k-token request, reordering and
splitting can reduce a single core’s computational load from 32k to 1.3k tokens, saving approxi-
mately 800 microseconds. In contrast, the latency savings from inter-DP group migration are more
modest. Even after balancing a 20k-token difference, the total time saved over 61 Transformer lay-
ers is only about 600 microseconds. This indicates that the greatest performance bottleneck and
optimization potential lie within the computation units themselves.
6
Future Work
Although proposed xLLM framework has demonstrated potential in enhancing LLM inference ef-
ficiency and reducing operational cost, achieving truly efficient and inclusive intelligent computing
infrastructure requires deeper collaborative innovation across Hardware, Model, and Application
Layers. Our future research will advance xLLM from a high-performance Inference Engine toward
a comprehensive Operating System for AI that supports next-generation intelligent applications. We
outline our planned efforts in three key directions:
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xLLM Technical Report
0
3%
5%
10%
OOC on Qwen2.5-7B
0
1
2
3
0
3%
5%
10%
Online SLOs Violation Rate
0
1
2
3
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Offline Request QPS (req/s)
0
3%
5%
10%
0
1
2
3
0
3%
5%
10%
Azure Code on Qwen2.5-7B
0
1
2
3
0
3%
5%
10%
0
1
2
3
Offline Throughput (req/s)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Offline Request QPS (req/s)
0
3%
5%
10%
0
1
2
3
base P/D - Online SLOs Violation
base P/D - Offline Throughput
online priority - Online SLOs Violation
online priority - Offline Throughput
xLLM-OOC (ours) - Online SLOs Violation
xLLM-OOC (ours) - Offline Throughput
Figure 23: Impact of our Online-Offline Co-location Scheduler Policy.
Model
Prompt Length
Output Length
Async Scheduling
Throughput (tokens/s)
DS-Distill-Qwen-1.5B
1000
1000
×
8709.90
1000
1000
✓
10223.15
DS-Distill-Qwen-7B
1000
1000
×
3183.68
1000
1000
✓
3201.83
DS-Distill-Qwen-14B
1000
1000
×
1472.22
1000
1000
✓
1527.23
DS-Distill-Qwen-32B
1000
1000
×
1415.20
1000
1000
✓
1508.76
Table 6: Impact of asynchronous scheduling mechanism in Multi-layer Pipeline Execution.
6.1
Fostering an Open and Diverse Hardware Ecosystem
Current AI infrastructure often suffers from tight coupling with specific hardware architectures,
introducing computational stability risks and limiting efficient deployment in heterogeneous envi-
ronments such as edge computing. To address this, the first pillar aims to cultivate an open and
diverse hardware ecosystem through the following initiatives:
Unified Hardware Abstraction Layer.
We will develop a high-performance, hardware-agnostic
runtime abstraction layer that encapsulates instruction sets and memory architectures across compu-
tational units—from domestic cloud chips to edge-side accelerators. This layer will provide unified
operator interfaces and memory management APIs, enabling seamless migration and efficient ex-
ecution across heterogeneous hardware without code modifications, thereby significantly reducing
integration barriers for emerging hardware.
Software-Hardware Co-Design and Ecosystem Synergy.
We will collaborate with hardware
partners to define more efficient and open interface standards. This co-design approach will not only
drive hardware innovation from the software perspective but also help establish a self-sustaining and
competitive computational supply ecosystem, ultimately providing users with optimized choices
across cost, performance, and security considerations.
6.2
Cultivating a Vibrant and Responsive Model Ecosystem
The system’s core value derives from its supported model diversity and integration efficiency. Our
future efforts will focus on building an inclusive and agile model ecosystem through:
Multi-Scenario Support.
We will extend the platform’s optimization and deployment capabilities
beyond large language models to systematically support various generative AI scenarios, including
generative recommendation, text-to-image, and text-to-video generation. We will optimize execu-
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xLLM Technical Report
Metric
Single-Stream (ms)
Dual-Stream (ms)
Total Communication Time
9.3
12.4
Overlapped Communication Ratio
-
80%
Exposed Communication Time
9.3
2.5
Total Computation Time
13.0
17.0
Reduced Time per Layer
-
2.8
Total Reduced Time (61 layers)
-
172.0
Table 7: Communication and computation overheads in a single decoder layer of DeepSeek-R1
using Multi-layer Pipeline Execution.
Model
Prompt Length
Output Length
Adaptive Graph Mode
Throughput (tokens/s)
Mean TPOT (ms)
Qwen3-1.7B
2048
2048
×
2385.58
39.27
2048
2048
✓
3038.88
30.63
Qwen3-4B
2048
2048
×
1539.93
55.44
2048
2048
✓
1671.39
50.58
Table 8: Impact of Adaptive Graph Mode.
tion engines, request scheduling, and memory management specifically for different workloads to
ensure optimal performance across diverse applications.
“Zero-Day” Model Integration.
To accommodate rapidly evolving model architectures, we will
implement a unified framework combining model graph representation, an extensible operator li-
brary, and automated compilation optimization. This will enable rapid “zero-day” integration of
newly released models, reducing deployment cycles from weeks to hours.
6.3
Evolving into an AI-Native Application Framework
To democratize AI adoption and accelerate value delivery, we will evolve the system from an infer-
ence engine into a full-stack AI-native application framework by:
Framework-Native AI Middleware.
We will design high-level, framework-native APIs and ab-
stractions that package complex distributed inference, multi-model orchestration, and stateful ses-
sion management capabilities into out-of-the-box middleware services. This will enable application
developers to build sophisticated AI applications (e.g., multimodal AI agents) without managing
underlying infrastructure complexities.
Rapid Application Integration and Deployment.
Building upon this AI-native framework, we
will deliver a comprehensive toolchain — including encapsulated SDKs, application templates, and
integrated CI/CD pipelines — to streamline development and deployment processes. Our goal is
to enable application teams to integrate and deploy AI services within hours rather than weeks,
significantly enhancing business innovation agility and fully bridging the “last mile” from model
development to value creation.
7
Conclusion
We introduce xLLM, an intelligent and efficient LLM inference framework, featuring a service-
engine decoupled architecture.
The framework consists of two main components: (1) xLLM-
Service, a versatile service layer designed for efficient instance and cluster management as well
as request scheduling. It incorporates unified elastic scheduling for co-located online and offline re-
quests to maximize cluster utilization, a workload-adaptive PD disaggregation architecture for SLO-
aware dynamic instance scheduling, a novel Encode-Prefill-Decode (EPD) disaggregation mecha-
nism for multimodal requests, a global KV Cache manager for efficient memory management in-
cluding KV Cache upload and offload and a distributed fault-tolerance design to ensure high avail-
ability. (2) xLLM-Engine, a high-performance inference engine optimized for accelerating LLM
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xLLM Technical Report
inference across various AI accelerators. xLLM-Engine employs full-stack multi-layer execution
pipeline optimizations through asynchronizing CPU-side scheduling and AI accelerator-side model
forwarding to minimize computational bubbles, utilizing dual-stream parallelism of micro-batches
to overlap computation with all-to-all communication, and further overlapping various AI compu-
tation units at the operator level. xLLM-Engine also implements an adaptive graph mode which
pre-compiles kernel sequences, memory operations, and synchronization into a single computation
graph to drastically reduce launch overhead and accelerator idle time. Additionally, it adopts a
"logically contiguous, physically discrete" KV Cache storage strategy via proposed xTensor Mem-
ory Management, which resolves the tension between memory contiguity and dynamic allocation.
To further boost hardware utilization, xLLM-Engine integrates algorithmic enhancements such as
asynchronous pipelined adaptive speculative decoding, KV cache-aware scheduling, reactive inter-
DP group workload migration for dynamic load balancing, and dynamic MoE load balancing based
on real-time expert workload statistics and transparent weight updates. We have further extended
xLLM to emerging generative recommendation scenarios, improving both recommendation accu-
racy and efficiency.
Extensive experiments demonstrate that xLLM achieves a consistent improvement compared to
leading inference systems such as MindIE and vLLM-Ascend, evaluated across mainstream Qwen-
series and Deepseek-series models as well as public and real-world industrial datasets. In partic-
ular, xLLM outperforms MindIE by up to 1.7× and vLLM-Ascend by up to 2.2× in throughput.
Comprehensive ablation studies further validate the effectiveness of key components, including the
proposed scheduling modules, multi-layer execution pipeline, optimized model tensor parallelism,
adaptive graph mode, and others.
By releasing the xLLM framework as an open-source project, we intend to stimulate further in-
novation in developing robust, enterprise-scale inference solutions, optimizing performance on a
diverse range of AI accelerators, and creating tightly integrated service and engine architectures for
next-generation AI applications.
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xLLM Technical Report
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xLLM Technical Report Tongxuan LiuB, Tao Peng, Peijun Yang, Xiaoyang Zhao, Xiusheng Lu∗, Weizhe Huang, Zirui Liu††, Xiaoyu Chen, Zhiwei Liang, Jun Xiong, Donghe Jin, Minchao Zhang, Jinrong Guo, Yingxu Deng, Xu Zhang, Xianzhe Dong†, Siqi Wang‡, Siyu Wu‡, Yu Wu†, Zihan Tang∗, Yuting Zeng†, Yanshu Wang††, Jinguang Liu, Meng Kang, Menxin Li, Yunlong Wang, Yiming Liu∗, Xiaolong Ma, Yifan Wang, Yichen Zhang∗, Jinrun Yin††, Keyang Zheng††, Jiawei Yin†, Jun Zhang†, Ziyue Wang†, Xiaobo Lin, Liangyu Liu†, Liwei Lan∗, Yang Liu†, Chunhua Peng, Han Liu, Songcheng Ren††, Xuezhu Wang‡, Yunheng Shen∗, Yi Wang, Guyue Liu††B, Hui Chen∗B, Tong Yang††B, Hailong Yang‡B, Jing Li†B, Guiguang Ding∗B, Ke ZhangB JD.com THU ∗ USTC † BUAA ‡ PKU †† Abstract We introduce xLLM, an intelligent and efficient Large Language Model (LLM) inference framework designed for high-performance, large-scale enterprise-grade serving, with deep optimizations for diverse AI accelerators. Current mainstream inference frameworks face practical challenges. On the one hand, enterprise-grade serving struggles with hybrid and dynamic workloads, strict demand for high availability of services, and distributed storage management. On the other hand, inference execution is bottlenecked by underutilized AI accelerators due to new paradigms of hardwares, model architectures and inference algorithms. To address these challenges, xLLM builds a novel decoupled service-engine architecture. At the service layer, xLLM-Service features an intelligent scheduling module that efficiently processes multimodal requests and co-locates online and offline tasks through unified elastic scheduling to maximize cluster utilization. This module also relies on a workload-adaptive dynamic PrefillDecode (PD) disaggregation policy for instance scheduling and a novel Encode-Prefill-Decode (EPD) disaggregation policy designed for multimodal inputs. Furthermore, it incorporates a distributed architecture to provide global KV Cache management for efficient AI accelerator memory handling and robust fault-tolerant capabilities for high availability. At the engine layer, xLLMEngine co-optimizes system and algorithm designs to fully saturate computing resources. This is achieved through comprehensive multi-layer execution pipeline optimizations, including overlapping CPU scheduling with AI accelerator operations to minimize computational bubbles, employing dual-stream parallelism to overlap computation with communication, and fine-grained overlapping of various computational units to maximize hardware utilization. These are complemented by an adaptive graph mode that drastically reduces kernel launch overhead, alongside the innovative "logically contiguous, physically discrete" xTensor memory management which resolves memory allocation conflicts. xLLM-Engine also further integrates algorithmic enhancements such as optimized speculative decoding and dynamic Expert Parallel Load Balance (EPLB), collectively serving to substantially boost throughput and inference efficiency. Extensive evaluations demonstrate that xLLM delivers significantly superior performance and resource efficiency. Under identical TPOT constraints, xLLM achieves throughput up to 1.7× that of MindIE and 2.2× that of vLLM-Ascend with Qwen-series models, while maintaining an average throughput of 1.7× that of MindIE with Deepseek-series models. We have deployed xLLM in production to support a range of core business scenarios at JD.com, covering areas including LLM, Multimodal Large Language Model (MLLM), and generative recommendation. These applications encompass the JingYan AI chatbot, marketing recommendations, product understanding, customer service assistants, and more. xLLM framework is publicly available at https://github.com/ jd-opensource/xllm and https://github.com/jd-opensource/xllm-service. BCorresponding authors: {liutongxuan1, , {guyue, , hai- , , {dinggg, . 1 16 Oct 2025 xLLM Technical Report Contents 1 Introduction 3 2 System Overview 5 2.1 xLLM-Service . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 xLLM-Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 xLLM-Service Designs 7 3.1 Online-Offline Co-location Scheduler Policy . . . . . . . . . . . . . . . . . . . . . 7 3.2 Dynamic PD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . . 9 3.3 Hybrid EPD Disaggregation Scheduler Policy . . . . . . . . . . . . . . . . . . . . 10 3.4 Global KV Cache Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.5 Fast Fault Recovery Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 xLLM-Engine Designs 13 4.1 Multi-layer Pipeline Execution Engine . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Adaptive Graph Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Efficient Memory Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4 Algorithm Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.1 Optimized Speculative Decoding . . . . . . . . . . . . . . . . . . . . . . . 19 4.4.2 Dynamic EP Load Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4.3 Hierarchical DP Load Balance . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5 Generative Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5.1 Host-side Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5.2 Device-side Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5 Evaluations 24 5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.1.1 Benchmarking Performance . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.1.2 Business Serving Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.2 Ablation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Future Work 30 6.1 Fostering an Open and Diverse Hardware Ecosystem . . . . . . . . . . . . . . . . 31 6.2 Cultivating a Vibrant and Responsive Model Ecosystem . . . . . . . . . . . . . . . 31 6.3 Evolving into an AI-Native Application Framework . . . . . . . . . . . . . . . . . 32 7 Conclusion 32 2 xLLM Technical Report 1 Introduction In recent years, large language models (LLMs) with parameters ranging from billions to trillions (GPT [1], Claude [2], DeepSeek [3], LLaMA [4], etc.) have achieved breakthrough progress in the fields of natural language processing and multimodal interaction, which drives an urgent demand in the industry for efficient inference engines and service systems. These models are being rapidly deployed in core business scenarios such as intelligent customer service [5], real-time recommendation [6], and content generation [7]. However, how to reduce the cost of model inference and improve computing efficiency remains a key challenge for large-scale commercial serving. Current mainstream LLM inference frameworks (vLLM [8], SGLang [9], TensorRT-LLM [10], etc.) face four key challenges in enterprise-level serving scenarios: First, in an inference cluster with hybrid deployment, online inference requests exhibit significant tidal characteristics [11, 12]. Current scheduling systems fail to meet the service level objective (SLO) for online services while fully leveraging idle periods of online services to increase the throughput of offline tasks. Second, existing Prefill-Decode (PD) disaggregation architecture [13-15] assumes static resource allocation for the two phases, which cannot adapt to the dynamically changing request loads (i.e., input/output lengths fluctuate) in real-world applications, resulting in low AI accelerator utilization and increased SLO violation risks. Third, there is a lack of strategy to efficiently service multimodal requests (i.e., image, voice and text input [16, 17]), including parallel processing for the encode phase and fine-grained resource allocation accordingly. Fourth, as the scale of the inference cluster increases, ensuring fast fault detection and service recovery for nodes or instances is critical to maintaining the stability of inference services. The evolving computing paradigms also present significant performance challenges on existing LLM inference engines: First, they struggle to fully utilize the computing units of modern AI accelerators [18, 19]. Second, the All-to-All communication overhead [20] and expert parallel (EP) load imbalance in the Mixture of Experts (MoE) architecture [21, 22] restrict the scalability of the system. Third, as the model context window continues to expand, efficient KV Cache management becomes critical to inference performance [23]. Forth, Due to the unpredictable nature of inference requests, conventional static scheduling and operator strategies face difficulties in effectively balancing workloads across computing units in Data Parallelism (DP). To address above challenges , we propose xLLM, an efficient and intelligent LLM inference framework featuring a service-engine decoupled design. xLLM achieves efficient support for enterpriselevel inference through the following innovations: at the service layer, xLLM has made groundbreaking achievements in 1) unified elastic scheduling for online/offline requests, 2) workloadadaptive dynamic PD disaggregated architecture, 3) novel Encode-Prefill-Decode (EPD) disaggregation for multimodal requests, and 4) a distributed cache management and fault-tolerance framework; at the engine layer, xLLM enhances the resource efficiency across the full-stack of "communicationcomputation-storage", including 1) a multi-layer pipeline execution mechanism, 2) efficient computing and memory optimization, and 3) intelligent algorithm designs. Specifically, xLLM-Service schedules online requests with preemptive execution and offline requests in a best effort manner, to maximize resource utilization while strictly ensuring SLOs for online services. To address the inherent limitations of the static PD configuration, xLLM incorporates an adaptive scheduler to dynamically adjust the proportion of PD instances for each request and supports fast role reversal for instances, by monitoring key metrics such as Time to First Token (TTFT) and Time Per Output Token (TPOT) [24]. For multimodal visual requests, xLLM automatically selects the optimal EPD phase-disaggregation strategy based on pre-profiling, for the best performance trade-off between throughput and latency. To handle failures such as hardware problems, network faults, or software errors [25], xLLM collects node error information, evaluates KV recomputation or migration costs of interrupted requests, and makes optimal global rescheduling decisions. Across multiple instances, xLLM supports KV offloading and routing migration within a hybrid storage architecture, improving KV storage capacity and cache hit rates. xLLM-Engine employs a multi-layer execution pipeline that incorporates hardware-specific optimizations: (i) at the framework scheduling layer, it implements asynchronous CPU-accelerator scheduling to minimize computational idle time; (ii) at the model graph layer, it utilizes dual-stream micro-batch parallelism to overlap computation with communication; (iii) at the operator level, it achieves kernel computation and memory access overlapping. For computational efficiency, xLLM3 xLLM Technical Report Multi-modal Requests xLLM Service Fault Recovery Global Scheduler Hybrid EPD Disagg. Dynamic PD Disagg. On/Offline Co-location Global Cache Mgr xLLM Engine Adaptive Graph Operator Optim. xTensor Memory drive drive manage schedule Speculative Inference Dynamic EPLB Prefill Local KV Cache KV Cache Transfer ... Prefill Local KV Cache KV Cache Transfer ... Prefill Instance Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Instance Local KV Cache KV Cache Transfer ... System-level Designs Batching Scheduler Algorithm-driven Tech. co-optimize Hierarchical DPLB Multi-layer Pipeline Figure 1: Overview of xLLM's capabilities, with decoupled service-engine architecture. Engine automatically fuses small kernels into a unified computation graph and dispatches them to the accelerator in a single operation, significantly reducing kernel launch overhead. Regarding memory management efficiency, xLLM-Engine introduces the xTensor memory management scheme, which employs a "logically contiguous, physically discrete"" KV cache storage structure that resolves the conflict between memory contiguity requirements and dynamic allocation needs. To further enhance performance, xLLM-Engine incorporates an adaptive speculative decoding mechanism, a redundant expert-based load balancing algorithm for EP, and a hierarchical load balancing algorithm for DP. Additionally, xLLM-Engine provides scenario-specific optimizations, such as for generative recommendation - one of JD.com's core businesses - where it achieves 23% performance improvement through host-kernel operation overlapping. In summary, we make the following contributions when developing xLLM: xLLM Intelligent Service Capabilities • We design a unified scheduling algorithm for online/offline workloads (§3.1). • We implement a workload-adaptive PD disaggregation policy to address scenarios with rapidly changing traffic load and request input/output lengths (§3.2). • We propose a hybrid EPD disaggregation policy for multimodal requests, achieving intelligent resource allocation across different phases (§3.3). • We leverage multi-level KV Cache management and global KV Cache routing strategy to expand KV Cache capacity and improve cache hit rates (§3.4). • We design a multi-node fault tolerance architecture to ensure high service availability (§3.5). xLLM Intelligent Engine Capabilities • We achieve intelligent and efficient inference, through hardware-software co-design to improve hardware computing efficiency, including pipeline execution (§4.1), graph optimization (§4.2), and memory optimization (§4.3). • We enhance inference performance through algorithmic optimizations (§4.4), including optimized speculative decoding (§4.4.1), dynamic expert parallel load balance (§4.4.2), and hierarchical data parallel load balance (§4.4.3). • We optimize online inference for generative recommendation scenarios (§4.5). 4 xLLM Technical Report Predictor Profiler Dynamic PD Disaggregation Hybrid EPD Disaggregation On/Offline Co-locatiom Prefill Pool Decode pool Encode Pool Global Cache Management Fault Recovery Framework Pre-Processing Layer Scheduling Layer online preempt support disaggregate dynamic PD allocation KV & Images Resource Layer Offline Requests Online Requests Figure 2: System workflow of xLLM-Service. 2 System Overview The overall architecture of xLLM is depicted in Figure 1. Upon request arrival, xLLM-Service performs intelligent scheduling to distribute each request to one of three elastic instance pools and manages instance migration across these pools during runtime. xLLM-Engine then drives efficient request inference by orchestrating system- and algorithm-level optimizations. 2.1 xLLM-Service We describe the workflow of xLLM-Service in Figure 2. The system comprises three primary layers: 1) a preprocessing layer consisting predictor and profiler, 2) a scheduling layer integrating three policies (Dynamic PD Disaggregation Policy, Hybrid EPD Disaggregation Policy, and OnlineOffline Co-location Policy), and 3) a resource layer consisting of three heterogeneous instance pools. Specifically, the Dynamic PD Disaggregation Policy leverages information from the predictor to dynamically convert PD instances based on the workload status of the two instance types. The Hybrid EPD Disaggregation Policy utilizes the profiler to determine the optimal EPD disaggregation strategy for multimodal requests. Concurrently, the Online-Offline Co-location Policy dispatches requests according to their online or offline attributes. KV and image caches are offloaded and routed among distributed instances, while the fault recovery framework ensures the high availability of the entire service. Elastic Instance Pools. Instances in a cluster are partitioned into three elastic pools: ▷Prefill Instance Pool: This pool handles the prefill phase of text requests. ▷Decode Instance Pool: This pool handles the decode phase of text requests. ▷Encode Instance Pool: This pool handles the encode phase of multimodal requests. 5 xLLM Technical Report We design the instances in Prefill and Decode pools as stateless, i.e., instances do not require physical migration between instance pools. Instead, they achieve flexible switching between PrefillDecode roles based on the type of request being processed. Request Preprocessing. xLLM-Service implements unified resource management for both online and offline requests. Online requests are submitted in a preemptive and deadline-prioritized manner, and are co-deployed with best-effort offline tasks within the shared resource pool. During the runtime of an offline request, the Unified-Scheduler dynamically scales it up/down based on the tidal traffic characteristics of online requests. When an online request is submitted, it first passes through the pre-processing layer, which consists of two modules: ▷TTFT Predictor: A TTFT prediction model built for text requests. It evaluates SLO fulfillment by analyzing queueing delays from each prefill instance queue and request input lengths, thereby guiding instance allocation for the Dynamic PD Disaggregation Policy in the scheduling layer. ▷ EPD Profiler: A profiler for multimodal requests that uses binary search to identify optimal deployment configurations: (1) EPD separation strategy, choosing from three approaches: EP-D (i.e., aggregated execution of Encode and Prefill phases, with Decode phase executed separately), ED-P, or E-P-D; (2) The maximum batch size for the Encoder phase; (3) The maximum number of tokens for Prefill/Decode's inputs. The Hybrid EPD Disaggregation Policy in the scheduling layer will use the optimal configuration determined for phase disaggregation and task dispatching. Intelligent Scheduling. The intelligent scheduling layer adjusts resource allocation for requests across their full lifecycles. It contains three major scheduling policies designed for various scenarios: ▷Online-Offline Co-location Policy: This policy implements a preemptive scheduler for managing online and offline requests. When the load of online requests reaches the peak, they preempt some offline requests on PD instances. To surrender resources to offline requests, when the load on P instances decreases (usually earlier than on D instances), P instances continue to process offline prefill requests and also migrate decode offline requests from D instances to P instances. ▷Dynamic PD Disaggregation Policy: This adaptive scheduling policy is responsible for dynamically managing P and D instance allocation. It intelligently assigns requests to suitable instances using a heuristic algorithm guided by the TTFT Predictor. Furthermore, it implements a feedback mechanism by continuously collecting performance data from computing instances, enabling runtime monitoring and adjustment of allocation decisions to maintain system efficiency. ▷Hybrid EPD Disaggregation Policy: This multimodal policy executes the three-phase disaggregation based on strategies searched by the EPD Profiler. For the EP-D disaggregation, the fused EP phase executes in the P instance pool; for the ED-P disaggregation, the fused ED phase executes in the D instance pool; for the E-P-D disaggregation, the three phases execute separately in the three instance pools. This deployment also enables multimodal requests to benefit from the adjustment of Dynamic PD Disaggregation Policy. KV-centric Storage Architecture. The instance storage employs a hybrid architecture (HBMDRAM-SSD) to cache KV values and image tokens. At the global level, xLLM takes the idea from Mooncake Store [23] and extends it to domestic accelerators with specific optimizations. Routing and reuse of caches across instances are determined by embedded intelligent routing strategies. Efficient Fault-tolerant. The fault recovery framework of xLLM-Service supports fault detection and fast recovery of instances from the three elastic pools (E, P, D). For requests on failed instances, the architecture manages the migration of image caches between instances, and automatically decides the optimal KV recomputation or migration strategy for handling affected KV caches. 2.2 xLLM-Engine During request execution, the xLLM-Engine layer provides intelligent computing capabilities. We achieve various joint inference accelerations between the computing system layer and the algorithmdriven layer in the following ways: Computing System Layer 6 xLLM Technical Report Latency-relaxed instance Online Prefill Latency-strict instance Online / Offline Mixed Decode Offline Prefill Offline Decode Resource Plane Control Plane On-demand Preemption Offline Request Migration Offline Request Gating Mix decoding Selection Figure 3: Overview design of Online-Offline Co-location Scheduler Policy. ▷Multi-layer Pipeline Execution Engine: In the framework layer, CPU tasks are scheduled asynchronously to create a pipeline with inference computation, reducing computational bubbles; in the model graph layer, a single batch is split to form a pipeline between two micro-batches, effectively overlapping computation and communication; in the operator kernel layer, operations are pipelined across different computational units, enabling computation and memory access to overlap. ▷ Graph Optimization for Dynamic Inputs: Small kernels in the decoding stage are fused into a single computational graph. To handle variable sequence lengths and batch sizes, we parameterize the input dimensions, and employ a multi-graph caching scheme to reduce compilation overhead. ▷xTensor Memory Management: It uses a "logically continuous, physically discrete" KV storage structure. It allocates physical memory space on-demand during token generation for each request, while asynchronously predicting and intelligently mapping the physical pages required for the next token. When a request completes, the existing physical memory will be reused to execute the next request. Algorithm-Driven Layer ▷Speculative Decoding: xLLM-Engine incorporates an optimized speculative inference algorithm that generates multiple tokens at once to boost throughput [26]. It further optimizes the computing architecture through ways like asynchronous CPU processing and reduced data transfer. ▷EP Load Balance: For MoE models, xLLM implements expert weight updates based on historical expert load statistics, enabling effective dynamic load balancing during inference. ▷DP Load Balance: For data parallel (DP) deployment, xLLM achieves fine-grained load balance by kvcache-aware instance allocation, inter-DP requests migration, and intra-DP computing units allocation. We will elaborate on the detailed design of xLLM-Service in §3 and outline the optimizations within the xLLM-Engine in §4. A comprehensive evaluation of xLLM will be presented in §5. 3 xLLM-Service Designs 3.1 Online-Offline Co-location Scheduler Policy Online/Offline Request Characteristic. LLM services can be categorized into two types based on their service modes: online and offline requests. Online requests, including those from chatbots [1, 2, 27], code completion [28-30], and recommendation systems [31, 32], constitute latencysensitive workloads. These services must respond immediately upon request arrival, often returning each generated token in real time via streaming output. Consequently, they impose strict SLO requirements on TTFT or TPOT to ensure a satisfactory user experience. In contrast, offline services, such as document analysis [33] and intelligent data annotation [34], are non-real-time workloads with minimal latency constraints and thus no stringent SLO requirements. Furthermore, we observe that the request traffic for online services typically exhibits significant volatility, including tidal variations at hourly or daily scales and sudden bursts at minute-level intervals [11, 12]. While cluster auto-scaling [35] could theoretically mitigate resource underutilization caused by tidal patterns, the slow cold-start latency of instances-involving model loading and com7 xLLM Technical Report plex initialization-renders it ineffective for responding to rapid traffic spikes [36], increasing the risk of service-level agreement violations. Online/Offline Request Co-location Deployment. To address these practical challenges, we adopt a hybrid deployment strategy that co-locates online and offline requests. In such a system, offline requests can utilize idle resources during off-peak periods of online traffic. Conversely, during online traffic peaks, offline tasks can be preempted, as they are not bound by strict SLOs. This approach significantly enhances aggregate resource utilization and mitigates idleness during traffic troughs. Although some recent work [37-39] also attempts to co-locate offline requests, they have not explored the multi-instance scenario, especially for PD disaggregation. Actually, the PD disaggregation architecture has demonstrated superior latency performance and is increasingly becoming a mainstream design paradigm in industry [13, 23]. However, directly applying online-offline co-location to PD-disaggregated systems introduces critical PD load imbalance issues: (1) such systems require the load ratio between prefill and decoding stages to align with their respective resource allocation ratios; otherwise, one stage may become a bottleneck, causing blocking or resource underutilization in the other. (2) Furthermore, as these PD load variations exhibit high volatility and burstiness similar to online traffic patterns, existing PD disaggregation techniques struggle to effectively address this challenge. Latency-Constrained Decoupled Architecture. We rethink the design of PD disaggregation architecture under online-offline co-location deployment. The latency advantage of PD disaggregation essentially stems from the separation of latency constraints: the decoding phase is highly sensitive to per-step latency and cannot be blocked by long-duration operations, thus necessitating decoupling from the prefill phase. Inspired by this insight, we propose a latency-constrained decoupled architecture, as illstrated in Figure 3. This design regards cluster resources as two pools: a latencyrelaxed pool (corresponding to original Prefill instances) and a latency-strict pool (corresponding to original Decode instances). All tasks are then reassigned to one of the two resource pools based on their inherent latency characteristics and requirements. Within this architecture, the decoding phase of offline requests can be executed in either resource pool. This flexibility allows us to dynamically adjust the load ratio between the two pools, thereby maximizing the overall resource utilization of the cluster. However, it also introduces another two challenges: (1) Complex Scheduling Space: since the decoding of offline requests can be performed on either type of instance, how to leverage this flexibility to design new intelligent scheduling remains unknown. (2) Strict SLO Guarantee: the execution of offline requests consumes resources, their prefill phase may block newly arrived online requests, and their decode phase on latency-strict nodes may slow down the overall response speed. Both scenarios can compromise the SLO satisfaction of online requests. Solution 1 - Performance Bottleneck Analysis. We construct an LLM inference performance model based on the Roofline Model [40] and online factor learning. This model is designed to predict the latency, computation utilization, and memory utilization of both the prefill and decode phases. Since decoding operations executed on latency-strict instances typically account for a large proportion of workloads and are highly performance-sensitive, we set balancing computational and memory resources as the optimization objective. By analyzing performance bottlenecks through this model, we can select more appropriate offline requests to merge into decoding batches, thereby improving resource utilization efficiency. Solution 2 - Efficient Preemption Mechanism. To strictly ensure that the SLO of online requests remains unaffected, we introduce a preemption mechanism that allows online requests to preempt offline requests. For offline prefill tasks running on latency-relax nodes, we propose a model execution interruption technique, which enables preemption within an acceptable latency range without incurring additional model maintenance overhead. For decoding tasks running on latency-strict nodes, we leverage the performance model to dynamically select requests for decoding batching, ensuring that decoding latency always meets SLO constraints. 8 xLLM Technical Report Stateless Instance Local Scheduler request queue migration queue Execution Engine Schedule Request & Cache Migration P D prefill unfinished P D decode unfinished decode finished prefill finished D P Global Scheduler Instance Scheduler Request Scheduler Instance Monitor TTFT / TPOT, Req Len, Memory Util., etc. schedule schedule report status Figure 4: Overview design of Dynamic PD Disaggregation Scheduler Policy. 3.2 Dynamic PD Disaggregation Scheduler Policy Inefficiency of Existing PD Disaggregation Policy under Workload Fluctuations. The PrefillDecode (PD) disaggregated inference architecture [13, 24, 23, 41] partitions computing instances into dedicated prefill and decode instances, each handling their respective processing phases. This design mitigates interference between prefill and decode requests, achieving superior performance compared to PD co-located architectures [13]. However, we observe that most existing PD disaggregated systems employing static instance partitioning schemes suffer from low hardware utilization and inadequate responsiveness to traffic bursts. On one hand, existing analysis-based methods [13, 24, 23, 41] typically determine the PD ratio using profiling or simulator data. These approaches remain effective only when request arrival patterns and length distributions remain relatively stable. Under significant workload fluctuations, pre-collected analysis data often fails to accurately capture real-time request characteristics, leading to mismatches between preset load-balancing ratios and actual load requirements. On the other hand, when confronting substantial variations in input/output lengths within production workloads, existing PD disaggregation architectures [13, 42] generally adopt dynamic instance type adjustment strategies [43]. Nevertheless, online switching between prefill and decode instances typically involves multiple steps-including monitoring, waiting for flip conditions, and instance restarting-which introduces substantial latency overhead [15]. To address these limitations, we propose a Dynamic PD Disaggregation Policy that adaptively adjusts resource allocation in response to real-time workload characteristics. Runtime Instance Monitor. Performance indicators such as TTFT and TPOT directly reflect the capability of instances to process requests in the prefill and decode phases, respectively. They serve as the core basis for evaluating whether an inference system can meet SLO requirements. Ideally, prediction-based methods could be adopted to dynamically assess these indicators. However, while TTFT exhibits relatively predictable characteristics (as its computation time is proportional to the square of the input sequence length [44, 41]), due to the uncertainty in the length of output tokens and transfer overhead, TPOT is difficult to accurately predict using traditional input/output features and cluster load metrics. To address this issue, we deploy additional instance monitors to collect real-time performance data from each computing instance. This data includes metrics such as the number and length of prefill and decode requests, memory usage, TTFT, TPOT, and token generation intervals. The system can further use these real-time performance indicators to dynamically evaluate instance loads and adjust scheduling strategies. Stateless Instance and Elastic Pool. As illustrated in Figure 3, Dynamic PD Disaggregation Policy adopts a design of stateless instances and elastic instance pools to enable fast and dynamic switching of instance roles. First, it treats prefill or decode phase as the request attribute rather than instance attribute. Instances are designed to be stateless, allowing each instance to process both prefill and decode requests simultaneously. Additionally, to facilitate the management of multiple instances, we further extend PD pools to four elastic instance pools (i.e., P, D, P→D, D→P) as de9 xLLM Technical Report scribed in §2.1. When flipping an instance (switching its role), we only needs to remove the instance from its original pool and move it to the new pool. This achieves zero-wait-time instance scheduling, avoiding the overhead of instance restart or model reloading incurred in traditional systems. SLO-aware Instance Role Switching. The instance scheduling strategy is dynamically adjusted strictly based on SLO objectives: during the prefill phase, if it is predicted that existing instances cannot meet the TTFT requirements, the conversion of decode instances is triggered; while during the decode phase, when resource shortage occurs, the average token generation interval exceeds the TPOT threshold, or prefill instances are idle, the conversion of prefill instances to decode instances will be initiated to cope with sudden traffic surges. Specifically, when decode instances are reallocated to prefill instances, the scheduler prioritizes selecting the instance with the lightest load (i.e., the fewest tokens being processed) from the P→D pool for role conversion, and always ensures that at least two decode instances are available; conversely, when prefill instances are reallocated to decode instances, the scheduler prioritizes scheduling from the D→P pool, which avoids local overload and maximizes resource utilization. SLO-aware Request Scheduling. The request scheduling scheme adopts a two-level architecture: ▷Global Request Scheduler: The scheduler implements a greedy strategy of prioritizing the lightest load, while being strictly restricted by SLO constraints. For prefill requests, the scheduler first evaluates the estimated queuing latency of each instance in the Prefill pool, selects the candidate instance with the smallest latency, and then invokes the TTFT prediction model for verification: if the estimated TTFT can still meet the SLO requirements after assigning the request to this instance, the request is allocated immediately; otherwise, it continues to find a suitable instance in the D→P pool. If no instance can meet the TTFT requirements, the instance scheduling mechanism is triggered to allocate resources from the decode side. For decode requests, the scheduler gives priority to having the original prefill instance continue processing (to avoid KV Cache transfer overhead); secondly, it selects the instance with the fewest running tokens in the Decode pool, and checks whether the total number of tokens in its current batch is below the memory capacity upper limit and computing throughput limit determined by pre-analysis, so as to ensure that the new request will not cause TPOT to exceed the standard. ▷Local Request Scheduler: Each instance adopts a refined queue management strategy internally. KV Cache transfer events are placed in an independent migration queue and processed sequentially in accordance with the FCFS principle; for forward requests, an innovative scheme combining Chunked Prefill [45] and Continuous Batching [8] is adopted: on the premise of ensuring that decode requests are prioritized to enter the running batch, the remaining computing resources are used to process chunked prefill requests in parallel [46]. 3.3 Hybrid EPD Disaggregation Scheduler Policy Challenges of Multi-modal Inference. The inference process of multimodal large language models (MLLMs) [47-52] typically consists of three phases: the image encoding phase (for extracting image features), the prefill phase (for encoding images and text prompts, feeding them into the language model to generate the first output token, and caching intermediate states), and the decode phase (for iteratively generating subsequent tokens based on the cached data). Existing mainstream inference engines, such as vLLM [8], Text-Generation-Inference [53], SGLang [9], and DistServe [13] are all tailored for LLMs, and thus face several challenges in handling inference tasks for MLLMs: ▷Insufficient Parallelism: For example, the inference of visual models and language models can be executed in parallel, thus improving the utilization of computing resources. However, most existing inference engines [54] adopt a serial strategy, failing to exploit the inter-request parallelism effectively. ▷ Coarse-grained Scheduling: The decode phase is a memory-intensive task, which is suitable for batch processing to improve throughput [55]; the prefill phase is computationally intensive, and it is appropriate to adopt the chunked prefill [45] together with the decode phase to balance latency and throughput [56]. The computation and memory access overhead of the encode phase lies between the two, and it can also benefit from independent scheduling and batch processing. Nevertheless, existing engines process encode and prefill in a combined manner, failing to perform 10 xLLM Technical Report TPOT & TTFT SLO Target ED D P + EP + E P D + + EPD Profiler Encode Instance Vision Model Image Cache Batch Scheduler Migrate Scheduler Prefill Instance Language Model Image Cache Batch Scheduler Migrate Scheduler KV Cache Decode Instance Language Model Batch Scheduler Migrate Scheduler KV Cache Hybrid EPD Disaggregation async transfer async transfer Figure 5: Overview design of Hybrid EPD Disaggregation Policy. phase-specific batch processing and lacking support for chunked prefill. Coarse-grained scheduling makes it difficult to finely control the execution time. ▷Disaggregation Strategy: Existing architectures such as DistServe [13] reduce resource interference by decoupling the prefill and decode phases. However, in multimodal inference, encoding and prefill jointly affect TTFT, while prefill and decode jointly determine TPOT. Under different loads, how to select the optimal decoupling strategy remains a challenge [57]. For example, the performance of strategies such as E+P+D, EP+D, and ED+P in different tasks has not yet been evaluated, and in-depth analysis is required to guide system design. Dual-stream Parallel. As illustrated in Figure 5, we adopt a dual-stream parallel mode, where the visual model and language model are assigned to separate execution streams: the visual stream is dedicated to performing computationally intensive image encoding tasks, while the language stream handles the prefill and decode operations for language generation. By isolating workloads into distinct streams, our system achieves concurrent execution of heterogeneous phases from different requests. Three-Phase Disaggregation. To address the differences in computation and memory access characteristics among tasks in each phase of multimodal inference, we propose a phase-aware scheduling strategy. Requests are subdivided into three phases within an instance: encode, prefill, and decode. Batch processing and scheduling optimizations are performed for each phase respectively: ▷Optimized Batch Processing: The maximum batch size for image encoding and the token budget for the language model are set according to the user's SLO. Specifically, during system startup, we use a binary search method to profile the maximum encoding batch size and the model's token budget, ensuring that the execution time of subsequent batch processing tasks in each iteration is less than the TPOT SLO. In each iteration, we (i) first add all running decode requests to the current batch; (ii) then check for any chunked prefill tasks that have been partially computed. If such tasks exist, we add them to the batch; (iii) if there are no chunked prefill tasks, we check for pending encoding tasks and add them if any exist. This approach aims to complete requests in the prefill phase as quickly as possible, reducing their TTFT. New requests' encoding phases are processed only when no requests are in the prefill phase. ▷ Phase-aware Scheduling: To address the difficulty in selecting decoupling strategies for multimodal inference tasks, xLLM proposes the Hybrid Encode-Prefill-Decode Disaggregation architecture, an innovative multi-instance collaborative inference framework. In this architecture, each instance only executes part of the subtasks among the three phases, while the remaining phases are processed by migrating requests to other instances, thereby avoiding resource waste and interference. During runtime, our system automatically selects the optimal disaggregation strategy based on historical request profiling (via EPD Profiler) to dynamically balance throughput and latency objectives. Compared with the traditional approach of binding instances to single-phase or full-phase tasks, this architecture significantly improves the overall system processing capability and resource utilization efficiency while meeting SLO requirements. 11 xLLM Technical Report Master Election Cache & Load Metric Sync Service Discovery ETCD Global Multi-Level KV Cache Management Master Global Scheduler Global KVCache Mgr Select Master Write Cache & Load Metrics Watch & Notify HeartBeat (Lease) Report Cache & Load Metrics Req Req Resp Prefill Local KV Cache KV Cache Transfer ... Prefill Local KV Cache KV Cache Transfer ... Prefill Instance Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Local KV Cache KV Cache Transfer ... Decode Instance Local KV Cache KV Cache Transfer ... Slave Slave Slave Figure 6: The framework of Global Multi-Level KV Cache Management. 3.4 Global KV Cache Management During the decode phase of LLMs, subsequent tokens are generated autoregressively one by one. Although the computational cost per step is relatively low, frequent access to the historical KV Cache is required, making memory bandwidth the primary bottleneck. As model size expands and context window grows, the memory consumption of the KV Cache exhibits an exponentially increasing trend. For instance, a context of 128K tokens may consume over 40GB of memory, severely straining the memory resources of single-GPU devices [3]. Although current mainstream optimization solutions like vLLM [8] and FasterTransformer [58] have made significant progress in single-instance environments, many critical issues remain unresolved. In long-context scenarios, the time required for the prefill phase increases dramatically, while the decode phase suffers from intense competition for memory bandwidth. To meet stringent SLO requirements (e.g., TTFT DRAM > SSD) adhering to the strict consistency rule: "if data resides in HBM, it must also be present in DRAM". Specifically, when local operations involving KV cache (including prefix cache [8]) loading and offloading occur, these operational events are aggregated at regular intervals and transmitted to the xLLM-Service via ETCD heartbeat mechanisms, enabling unified global monitoring and management. 12 xLLM Technical Report KV Cache-aware Scheduling. In terms of scheduling strategy, the system implements a decisionmaking mechanism based on KV cache utilization, which operates through three key steps: (1) Prefix Matching Detection: calculating the KV cache reuse rate for each candidate node via prefix matching analysis; (2) Performance Estimation: estimating the expected latency for different nodes according to current load conditions and cache hit rates; (3) Optimal Node Selection: identifying the node with the optimal overall performance for request processing, thereby enabling dynamic offloading and migration of KV cache. 3.5 Fast Fault Recovery Architecture Current fault handling mechanisms are mainly designed for model training [60-62] and cannot be directly applied to large model inference due to its low-latency requirements. Specifically, existing approaches primarily adopt a checkpoint-then-recovery method [63-65] for fault handling, which periodically stores model data as checkpoints in distributed storage and reloads the most recent checkpoint after a failure occurs [66]. As model parameters increase, the overhead of storage and loading gradually grows. This is still acceptable in training since there are no strict latency requirements [67], but in inference, it may cause all high-priority requests on the failed instance to time out, resulting in severe losses. To address this issue, we propose an efficient failover architecture specifically for large model inference, with targeted optimizations in two aspects: fast request migration and fast instance recovery. Fast request migration ensures the SLO of high-priority tasks mainly through an efficient kv cache quick recovery strategy, while fast instance recovery achieves low-overhead recovery through efficient masking of computation and communication. Through the above optimization methods, we achieve fast fault recovery, greatly reducing the performance and economic losses caused by faults. 4 xLLM-Engine Designs 4.1 Multi-layer Pipeline Execution Engine Traditional autoregressive LLM inference relies on single-stream sequential execution. This conventional approach fails to fully exploit the parallelism capabilities of modern hardware, often leading to computational resource underutilization, communication stalls, and cascading blocking delays in heterogeneous computing environments [68, 69]. As illustrated in Figure 7, the system's performance is hindered by three primary sources of inefficiency: (1) CPU-Accelerator Dependency: At the framework level, a rigid dependency between the CPU and the accelerator forces the accelerator to remain idle during task scheduling and data processing phases, creating significant "computation bubbles" [68, 3]. (2) Communication Latency: In both distributed workloads and complex model layers, communication delays interrupt continuous computation, preventing the full utilization of available hardware resources [69, 70]. (3) Architectural Bottlenecks: On certain AI accelerators, the compute-focused (tensor cores) and general-purpose (vector cores) computation units lack a shared high-level cache (e.g., L1 or SRAM). This architectural separation necessitates additional data transfers between them, which introduces latency and leads to the underutilization of the specialized computation units. To address these challenges, we propose a three-tier asynchronous pipeline parallelism design spanning the framework scheduling layer, the model graph layer, and the operator layer, as shown in Figure 7, with the goal of maximizing hardware efficiency. Framework-Layer Scheduling-Execution Overlap. In a typical sequential execution procedure, the accelerator remains idle while the CPU performs scheduling to prepare the input data batch for the next computation cycle. This serial dependency creates significant latency and underutilizes the accelerator. To eliminate this bottleneck, we introduce an asynchronous pipelined scheduling mechanism that decouples CPU scheduling from accelerator execution. The core idea is to overlap these two stages: while the accelerator is executing the forward pass for the current batch, the CPU concurrently assembles the batch and prepares the metadata for the next batch. This parallel workflow effectively hides the latency of CPU-side scheduling and data preparation. The process is as follows: (1) Accelerator Compute: The accelerator is busy executing the forward pass for the current iteration, 13 xLLM Technical Report w/o Overlap w/ Overlap CPU AI Accelerator AI accelerator bubble AI accelerator bubble ... ... ... ... Schedule 0 Schedule 1 Schedule 2 Schedule N Forward 0 Forward 1 Forward 2 Forward N Schedule 3 Schedule 0 Forward 0 Schedule 1 Forward 1 Schedule N Forward N Framework Layer Communication Stream ... ... Model Layer No bubble Attn 0 Dispatch 0 MoE 0 Combine 0 Attn 1 Dispatch 1 MoE 1 Combine 1 Computation resource waste Computation resource waste Attn 0 Attn 0 MoE 0 MoE 0 Attn 1 Attn 1 MoE 1 MoE 1 Full utilization Computation Stream Dispatch 0 Dispatch 0 Combine 0 Combine 0 Dispatch 1 Dispatch 1 Combine 1 Combine 1 Computation Stream - 1 Operator Layer Vector Matrix Matrix Comp. Unit idle Matrix Comp. Unit idle Attn 1-0 Attn 1-1 Attn 1-2 Attn 1-N MoE 1-0 MoE 1-1 MoE 1-2 MoE 1-N Full utilization Computation Stream - 2 Stream 0 Attn 1-0 Attn 1-1 Attn 1-2 Attn 1-N MoE 1-0 MoE 1-1 MoE 1-2 MoE 1-N Vector Matrix Vector Matrix Vector Matrix AI Comp. Unit CPU AI Accelerator Batch 0 Batch 1 Matrix Comp. Unit Vector Comp. Unit Matrix Comp. Unit idle w/o Overlap w/ Overlap w/o Overlap w/ Overlap Figure 7: Multi-layer pipeline execution engine. which will eventually produce output. (2) CPU Schedule: Instead of waiting, the CPU immediately begins scheduling the batch for the next iteration. To do so, it uses a set of placeholder tokens that stand in for the yet-to-be-computed output. This allows all CPU-side batch preparations to occur concurrently with the accelerator's computation. (3) Seamless Transition: Once the accelerator completes the forward pass and the real output is generated, a fast replacement operation swaps the placeholder tokens with the real generated tokens. Since all scheduling is already complete, the accelerator can begin the next forward pass with minimal delay. By transforming the conventional serial "prepare-then-compute"" workflow into a parallelized pipeline, our method effectively masks scheduling latency behind accelerator computation time. This design eliminates idle execution "bubbles"" in the accelerator timeline, thereby maximizing hardware utilization and enhancing overall system throughput. Model-Layer Computation-Communication Overlap. Maximizing the overlap between computation and communication is a critical design principle in recent LLM frameworks [3, 71]. This strategy is essential for hiding the significant latency of data transfers and achieving high hardware utilization. However, architectural constraints can make this challenging. For instance, on a typical GPU, compute-focused units (Tensor Cores) are tightly coupled with general-purpose units (CUDA Cores) within a Streaming Multiprocessor (SM). Consequently, when an SM is allocated to a communication-related task, its powerful Tensor Cores often sit idle, leading to wasted compute potential. In contrast, our target accelerator architecture allows for more flexible, independent allocation of its Cube (matrix) and Vector (general-purpose) units. We leverage this flexibility to address the aforementioned challenges by introducing a dual-stream pipeline architecture that uses micro-batch splitting to effectively overlap computation and communication. Specifically, our architecture consists of a Computation Stream for compute-bound tasks (Attention, ExpertForward, etc.) and a Communication Stream for data distribution and collection tasks (MoE Dispatch and Combine). To enable pipelining and maximize overall pipeline throughput, we partition a macro-batch B into n micro-batches [68] {b1, b2, · · · , bn}. The two streams asynchronously execute tasks for different micro-batches. On the one hand, our scheduling policy dynamically determines the optimal execution order of the micro-batches. On the other hand, our resource scheduling policy adaptively allocates an appropriate number of AI Cores (Vector Cores and Cube Cores) to the Communication Stream and Computation Stream respectively. Figure 7 illustrates the case of n = 2, where the Communication Stream performs the Dispatch operation for micro-batch bk while the Computation Stream executes the ExpertForward pass for the preceding micro-batch bk-1. 14 xLLM Technical Report Operator-Layer Matrix-Vector Units Overlap. On heterogeneous AI accelerators, the serial scheduling of matrix and vector computation units often results in significant underutilization, as one class of units remains idle while the other is active. This inefficiency motivates a strategy of operator-level matrix-vector overlap to fully exploit the hardware's parallel capabilities. However, a naive implementation via coarse-grained parallel scheduling, without a systemic resource coordination mechanism, is also problematic. This approach frequently leads to disordered contention for limited compute units, causing resource fragmentation and access conflicts that ultimately degrade overall performance. To address these challenges, we propose a dynamic resource allocation mechanism for computation units (Cube and Vector Units) based on real-time load monitoring. This mechanism enables a deep pipeline across heterogeneous compute units by dynamically and adaptively assigning the precise type and quantity of resources required for each concurrent operator. By doing so, it mitigates resource contention and ensures that parallel operators execute within highly overlapping time windows, achieving precise computational overlap and maximizing hardware utilization. We formulate the dynamic resource allocation as an optimization problem. Let C and V be the sets of matrix and vector operators, respectively. Let xi be the number of matrix units (Cube) allocated to the ith matrix operator (i ∈C), and yj be the number of vector units (Vector) allocated to the jth vector operator (j ∈V). For simplicity, we assume all operators begin execution simultaneously, without considering data dependencies or communication latency. Based on the known computational workload of each operator, our mechanism seeks to find an optimal resource allocation that minimizes the maximum difference in execution times between any two operators. This objective, which we term the alignment loss (Lalign), effectively synchronizes the completion time of all parallel kernels. The optimization problem is defined as: argminxi,yjLalign = max i∈C, j∈V |Ti -Tj| , Ti = Wi γCube · xi , Tj = Wj γVector · yj , X i∈C xi ≤NCube, X j∈V yj ≤NVector, (1) where T is the operator execution time, W is its computational workload, γcube and γvector are the peak performance per unit, and Ncube and Nvector are the total available matrix and vector units. The meaning of this optimization objective is to minimize the maximum difference in execution time between any pair of concurrent matrix and vector operators. 4.2 Adaptive Graph Mode The performance of LLM inference deployment is often impeded by Host-side CPU overhead, particularly when the computation graph is composed of many fine-grained operators [72]. This bottleneck manifests as both significant CPU-Accelerator synchronization latency, measured at 5 ∼50μs per invocation from frequent kernel launches, and suboptimal hardware utilization due to idle accelerator cycles between these intermittent operator executions. Mainstream AI accelerators such as NVIDIA GPUs (with CUDAGraph) [73] and Ascend NPUs (with ACLGraph) [74] employ computation graphs to try to handle above issues and enhance hostside scheduling performance. As illustrated in Figure 8, traditional Eager Mode relies on the CPU to submit a multitude of small, intensive tasks, leading to frequent launches of small kernels on the accelerator. In contrast, the Graph Mode enables the CPU to submit one large task, after which the accelerator internally executes the small kernels in a streamlined fashion. This method significantly reduces both launch overhead and "accelerator bubbles" (idle cycles). Specifically, the Graph Mode, such as ACLGraph [74], comprises two distinct phases: graph capture and graph execution. During the graph capture phase, the entire computation flow is recorded to capture the sequence of kernel calls and their dependencies, including kernel launch parameters. This recorded workflow is then pre-compiled into a replayable directed acyclic graph (DAG) object. It is important to note that during this phase, tasks are merely staged within the model's runtime instance and are not actually executed. In the subsequent graph execution phase, the entire graph is launched via a single CPU call. The accelerator then follows the pre-defined process autonomously, without real-time intervention from the CPU. This effectively consolidates many individual kernel launches into one single graph launch. 15 xLLM Technical Report A Launch A Launch B Launch C Launch D Launch E B C D E Kernel Execution Launch Overhead Launch graph D Kernel Execution Launch Overhead Build Graph cudaGraphExecKernelNodeSetParams Speed Up E C B A w/o Graph w/ Graph Figure 8: Pipeline comparison between Eager Mode and Graph Mode. Building upon this foundation, we further advance the practical implementation of the ACLGraph [74] and propose our Adaptive Graph Mode through addressing key challenges including dynamic shape adaptation, memory reuse conflicts across multiple graphs, and compatibility with custom operators. This enables the pre-compilation of kernel sequences, memory copy operations, and synchronization points into a single computational graph, which is then dispatched to the AI accelerator in one execution. Consequently, our approach significantly reduces kernel launch overhead and minimizes accelerator idle periods, maximizing overall hardware efficiency. Dynamic Shape Adaptation via Partial Graph and Dimension Parameterization. ACLGraph [74] fixes kernel parameters and memory sizes during the graph capturing phase. However, in real-world scenarios, LLMs typically process inputs with variable sequence lengths and batch sizes, making this static capturing characteristic difficult to adapt to dynamic input requirements. To address this issue, dimension parameterization can be employed, treating key dynamic dimensions-such as batch size and sequence length-as input parameters to the entire graph, thereby enhancing flexibility. During memory allocation and kernel configuration, these dynamic parameters are used to compute actual required values; for example, the memory size can be calculated using the formula: alloc_size = batch_size × seq_len × hidden_size. At the graph launch stage, the actual batch size and sequence length are passed as parameters to ensure the graph adapts to varying input dimensions. Although parameterization can handle variations in dynamic dimensions, certain operator implementations on different hardware may not support dynamic dimension parameterization. To overcome this, we propose the Partial Graph Mode: the model is partitioned into modules with simple dynamic shapes (such as FFN and LayerNorm, which only have the num_tokens dimension as dynamic) and modules with complex dynamic shapes (such as Multi-Head Attention, which involves multiple dynamic dimensions like batch_size, kv_seq_len, and q_seq_len). Modules with simple dynamic shapes are extracted and compiled for execution using Partial Graph Mode, while those with complex dynamic shapes are executed in Eager mode. Our Adaptive Graph Mode dynamically selects the most appropriate mode above based on the current input shapes. This intelligent selection ensures optimal performance across varying workload conditions. Table 1 compares the key characteristics of three dynamic shape processing mode. The traditional Eager Mode requires no pre-compilation but incurs high scheduling overhead at runtime due to N kernel launches (where N is the number of operators). The Full Graph Mode, by fixing shapes, allows single compilation and very low launch overhead, yet its lack of dynamic adaptability limits its applicability. In contrast, the proposed parameterized and multi-caching Partial Graph Mode strikes an optimal balance. It achieves execution performance comparable to that of a full graph while maintaining high flexibility, by trading a manageable number of pre-compilations (M ≪ Nreq, where M is the number of cached graphs and Nreq is the number of actual requests) for the high efficiency of a single graph launch. 16 xLLM Technical Report Method Eager Mode Full Graph Mode Partial Graph Mode Compilation Times 0 1 M Low Launch Overhead × ✓ ✓ Low Memory Usage ✓ × ✓/× High Flexibility ✓ × ✓ Table 1: Comparison of different shape handling solutions. Our Adaptive Graph Mode dynamically selects the most appropriate mode above based on dynamic shapes. Efficient and Reusable HBM Memory Pool. During the graph capture process, the specific virtual addresses of input, output, and intermediate tensors are recorded. To prevent illegal memory access, which can occur when the address of an input tensor changes during an actual inference request, we develop a shared HBM memory pool. First, during graph initialization, a sufficiently large, contiguous block of HBM memory is pre-allocated to serve as the graph's memory pool. Subsequently, for internal address management, computation graph intercepts all tensor memory operations and re-describes them using offsets relative to the pool's base address, rather than absolute virtual addresses. Before the graph is launched, the user-provided input tensor is copied into the HBM memory pool at the offset pre-defined for the graph's input. After graph execution is complete, the data from the output tensor is copied to the user-specified output buffer address. Furthermore, internal tensors within the graph are also managed using fixed offsets within the pool, ensuring safe reuse and efficient memory management. This managed memory pool guarantees that all addresses used internally by the graph are known and safe, which adapts to external address changes with only the overhead of memory copies before and after launch. Integrating Custom Operators. Integrating performance-critical custom operators, such as PageAttention and AllReduce, into the Adaptive Graph Mode presents a unique challenge. The internal implementation of these operators often relies on CPU to perform just-in-time shape calculations and kernel configurations based on runtime inputs. This dynamic behavior conflicts with the pre-compiled nature of computation graph. To resolve this, we modified the implementation of such operators. The first step is to identify custom operators within the graph that have dynamic shape dependencies. We then refactor these operators to accept shape-related parameters (e.g., dimension sizes or loop counts, which can only be determined at runtime) directly as kernel arguments. This approach avoids hard-coding these values on the host side during the graph capture phase. As a result of this modification, these custom operators can be successfully captured by our computation graph. By leveraging the graph's parameterization mechanism, the required dynamic arguments for these kernels can be derived from the graph's main input parameters and passed along during execution. This method achieves seamless integration of essential custom operators within the Adaptive Graph Mode framework. 4.3 Efficient Memory Management LLM inference deployment faces a critical balance between memory utilization and computational efficiency, particularly in autoregressive generation tasks where efficient KV Cache management has become a key challenge. Traditional solutions fall into two categories: First, contiguous memory allocation approaches that statically pre-allocate memory space based on maximum sequence length before inference, ensuring physical contiguity of KV Cache. While this improves computational efficiency, it results in low memory utilization. Second, PagedAttention [75] approaches that support larger batch sizes through paging mechanisms, but frequent access to block tables sacrifices computational efficiency, and the increased parameters complicate operator development and debugging. To address this challenge, inspired by virtual memory management in operating system field [76, 77], we propose the xTensor memory management scheme, which adopts a "logically contiguous, physically discrete" KV Cache storage structure that resolves the contradiction between 17 xLLM Technical Report Req Virt_start ... Virt_end Virtual Address Space Request R1: MaxSeqLen Virt_start ... Virt_end Request R2: Page1: -PageID: P1 -Status: Reusable -Owner: R1 Physical Page Pool Page2: -PageID: P2 -Status: Mapped -Owner: R2 Scheduler On-demand Mapping Request R2: Write a short story about a young explorer discovering a hidden ancient civilization. Request R1: I am a 25-year-old woman. Please give some recommendations for moderately priced clothes. Smart Scheduling: Request ID Reuse Async Pre-mapping ... ... Inference Requests Addr Mapping Figure 9: The framework of xTensor memory management. Features Contiguous Allocation Paged Attention xTensor (Ours) Efficient Memory Usage × ✓ ✓ Efficient Computation ✓ × ✓ Large Batch Support × ✓ ✓ Operator Development Complexity ✓ × ✓ Table 2: Comparison of memory management strategies. memory contiguity and dynamic allocation, thereby achieving both high computational efficiency and high memory utilization (as shown in Table 2). Physical Memory Page Management and Virtual Memory Tensor Management. Figure 9 illustrates the xTensor memory management framework. During service initialization, based on the available memory size for KV Cache, a large number of fixed-size discrete physical memory pages are pre-allocated. Each physical page maintains a triple state ⟨PageID, Status, OwnerSession⟩, where PageID represents the page identifier, OwnerSession indicates the service owning the page, and Status ∈{Free, Allocated, Mapped, Reusable} represents states of free, allocated, mapped, and reusable to dynamically track physical page usage. Subsequently, each inference request is pre-allocated with a logically contiguous virtual address space with an address range equal to MaxSeqLen, where this virtual address space is not actually associated with physical pages during allocation. This decoupled design provides operators with a virtual view of KV Cache stored in contiguous memory, thereby masking the discrete nature of underlying physical pages [78, 79]. On-demand Memory Mapping. During actual allocation, the scheduler dynamically maps physical memory pages on-demand based on the actual sequence length of requests. When a sequence generates new tokens and KV Cache needs expansion, the scheduler retrieves one or more free pages from the pre-allocated physical page pool and maps these physical pages to the next available contiguous address location in the request's virtual address space. Since physical memory is gradually mapped as sequences grow, short sequences only require a small amount of physical memory, thus avoiding the memory waste of contiguous allocation strategies that still require reserving space according to maximum sequence length for short sequences. After memory mapping is completed, kernel access to virtual addresses can automatically locate the corresponding physical page phypageidx and offset: phypageidx = virt_addr -virt_start page_size , offset = (virt_addr -virt_start) mod page_size, (2) where virt_addr and virt_start represent the current virtual address and starting virtual address respectively, and page_size represents the physical page size. 18 xLLM Technical Report Requests Schedule Draft model Forward Input Target model Forward Input Validate Output Requests Schedule Draft model Forward Input Target model Forward Input Validate CPU thread1 NPU CPU thread2 Requests Schedule Draft model Forward Input Target model Forward Input Validate Draft model Forward Output Target model Forward Input CPU NPU Schedule ... ... w/o Overlap w/ Overlap Figure 10: Comparison between original and our optimized speculative decoding. Low Overhead Allocation. If virtual address to physical page mapping is immediately released upon request completion, new requests require remapping physical pages, but the Unmap operations on accelerators incur significant overhead. Therefore, under high load conditions, frequent Map/Unmap operations become performance bottlenecks, especially for numerous short-term concurrent requests. We adopt request ID reuse and asynchronous pre-mapping designs to reduce latency from Map/Unmap operations. • Physical Page Reuse. Upon request completion, physical pages are not immediately released but marked as Reusable. When new requests arrive, if their required KV Cache size matches some Reusable physical page set, that page set is remapped to the new request's virtual address space. This saves expensive Map and Unmap operations through fast virtual address remapping, particularly suitable for scenarios with concentrated request length distributions. • Asynchronous Pre-mapping. During current token decoding, the next token's required physical pages are asynchronously predicted and mapped. Since new token writes to KV Cache occur after current token decoding computation, this design can partially hide physical page mapping operations, significantly reducing page mapping latency. Operator Adaptation. The decoupled design of xTensor's physical memory pages and virtual contiguous memory requires the adaptation of existing operators. On one hand, to accommodate KV Cache contiguity in virtual address space, operators no longer require block_table parameter input. Whether for attention operators or write_kvcache operators responsible for writing KV Cache during Prefill and Decode phases, only the starting virtual address and related offsets need to be passed in, with the system automatically associating virtual addresses to corresponding physical pages during operation. On the other hand, accelerators lack native contiguous KV FlashMLA [80] operators and only support PagedFlashMLA [80] operators optimized for paging. Therefore, by reconstructing the PagedMLAttention operator in the accelerator operator library, removing Block Table-related logic such as block table queries and cross-page boundary judgments, and adding functionality for direct computation based on input virtual addresses, we implement FlashMLA operators supporting contiguous virtual address input on accelerators. 4.4 Algorithm Optimizations 4.4.1 Optimized Speculative Decoding Traditional autoregressive inference requires LLMs to generate tokens sequentially one by one, resulting in high computational latency and limited throughput. Speculative decoding technology theoretically breaks through this performance bottleneck through a paradigm of parallelly predicting candidate token sequences and quickly verifying them [81, 82, 26]. However, in distributed deployment environments, this technology faces some core challenges: First, synchronous scheduling between CPUs and accelerators in traditional multi-step inference frameworks results in insufficient utilization of computing resources, with accelerators often remaining idle while waiting for data preparation and processing. Second, existing attention mechanisms are not optimized for the characteristics of speculative inference, causing a large amount of redundant data movement. To address these challenges, xLLM proposes a systematic optimization scheme based on existing frameworks. 19 xLLM Technical Report Compute Load decode EPLB Controller Global Expert Mapping Planner Checker Compute Load decode decode decode decode decode Layer-0 Loader Layer-0 Weight Old Buffer New Buffer Layer-0 Rebalance Layer-0 Weight Old Buffer New Buffer Double Buffer Double Buffer Layer-1 Rebalance ... DB Policy Expert Routing Map Expert Load Worker-0 Ready All Ready Expert Load ... Thread-0 Worker-0 ... Thread-1 Layer-1 Loader ... ... Worker-0 Ready All Ready ... ... Figure 11: Overview design of EPLB. Asynchronous Decoding. Inspired by [83], xLLM implements a lightweight asynchronous framework. For timing overlap optimization, while the accelerator executes the main model inference for the current batch, the CPU processes the output decoding of the previous batch and the input preparation for the next batch in parallel-thereby minimizing the idle time of the CPU and accelerator as much as possible. MLA Optimization. xLLM focuses on improving the MLA [84, 3] computation process. Based on the characteristic that speculative inference needs to process multiple tokens simultaneously on a single sequence, we conducted in-depth optimizations on the self-attention computation in MLA. Specifically, when speculatively inferring m tokens, the self-attention computation of MLA involves product operations between m+1 Q matrices and the same K matrix. By reconstructing the computation process and optimizing the tiling strategy, we effectively reduced the data movement overhead of Q/K matrices. The main optimization measures include: ▷Optimization of K matrix loading times: By adjusting the L1 cache allocation scheme to enable parallel loading of multiple Q matrices, and adopting a sliding window-based K matrix loading method that allows consecutive rows of the K matrix to multiply with multiple Q matrices, the number of K matrix loading operations is significantly reduced. ▷ Q matrix cache residency mechanism: Since there are m+1 Q matrices in speculative inference scenarios, the time to move Q matrices to L1 accounts for a larger proportion of the matrix multiplication process. xLLM redesign the computing scheme to prevent softmax-V product operations from overwriting Q matrices in the L1 cache, enabling Q matrices to remain in the cache and significantly reducing the overhead of repeated data movement, thereby effectively improving the utilization of the tensor core's arithmetic logic units. 4.4.2 Dynamic EP Load Balance With the large-scale application of MoE models [85-88] in production environments, their efficient inference capability relies on a expert routing mechanism, which allocates input tokens to different experts for computation [3, 18, 89]. However, in practical deployment, due to the distribution characteristics of input data and the influence of the model's own structure, there may be significant differences in the number of tokens received by different experts [90, 91]. This imbalance leads to low computing resource utilization: some devices are overloaded due to processing too many tokens, while others remain idle, affecting the overall inference efficiency. To optimize resource utilization, the industry has proposed the Expert Redundancy strategy, which means replicating hot experts and deploying replicas on multiple devices to distribute computing loads. Currently, DeepSeek [3] adopts two load balancing strategies: group-limited load balance and global load balance, which are optimized respectively for the characteristics of the prefill and decode phases. Adjustments to expert redundancy (such as adding/deleting replicas) require ad20 xLLM Technical Report Preprocess MLA Preprocess MLA MLA MLA MLA MLA MLA MLA Preprocess MLA Preprocess MLA MLA MLA MLA MLA MLA MLA Aggregate transfer transfer 1K tokens 3K tokens 3K tokens DP-1 (seq-1) DP-2 (seq-2) DP-1 (seq-1) DP-2 (seq-2) w/o DP Load Balance w/o DP Load Balance Figure 12: Inter-DP group load migration at MLA block granularity. ditional memory and may affect inference latency due to weight migration. How to achieve such adjustments efficiently and smoothly is a major challenge. We have made some adaptive improvements based on DeepSeek, realizing dynamic EP load balance. Expert Load Statistics. When the Router distributes tokens, it records the load status of each expert in real time and returns the statistical results through the model output. Each Worker asynchronously starts a thread to periodically aggregate expert loads and report them to the Controller of the Master. The Controller calculates a new routing table based on the load data and then distributes the updated routing table to each Worker. Double Buffer Mechanism for Weight Update. In general, each Worker starts an asynchronous thread to update the weights of single-layer experts during each decoding process. Specifically, after a Device completes the preparation of new weights, it proactively sends a readiness notification to the Controller. The Controller then verifies the weight readiness status of all Worker nodes to ensure global consistency before broadcasting an update instruction, upon which each Worker executes the weight update immediately. Here, we adopt a Double Buffer mechanism, meaning that the old and new expert weights are stored in two separate buffers respectively. The new expert weights are preloaded in the spare memory space, and after the preloading is completed, the address switching is performed to realize the unperceived update of experts. 4.4.3 Hierarchical DP Load Balance In typical MoE model architectures like DeepSeek [3], the MoE layers use expert parallelism (EP), while the attention layers use data parallel (DP). This model requires all DP groups to synchronously complete their attention calculations in a single inference step before initiating the MoE all-to-all dispatch operation. This synchronization barrier means the total time for the attention phase is determined by the slowest DP group (the "straggler"). Faster DP groups cannot immediately proceed to the next stage after finishing their tasks; instead, they are forced into an idle state, waiting for the straggler to complete [92]. This waiting directly translates into wasted computational resources and an increase in overall inference latency. In practical scenarios, due to the dynamic and unpredictable nature of inference requests, the load within different DP groups is often difficult to balance. Particularly in the decoding phase with large-scale DP (e.g., 80 groups), the load difference between DP groups can reach tens of thousands of tokens, causing fast DP groups to wait for slow ones and wasting approximately 5 milliseconds of delay [93]. Experimental data shows that although the difference in the number of requests processed between DP groups may be as small as four, the corresponding difference in KV Cache token count during the decoding phase can be as high as 20,000. The actual computational workload is directly related to the total number of tokens the system needs to process, as this determines the memory footprint of the KV Cache and the scale of matrix operations in the attention mechanism. This further establishes DP load balancing as a critical factor in determining overall system efficiency. Current mainstream frameworks like vLLM [8] and SGLang [9] use a simple static round-robin scheduling strategy. Once a request is assigned, its subsequent computations are fixed to a specific 21 xLLM Technical Report DP group, making it unable to adapt to dynamic load changes. Furthermore, at the hardware level, the implementation of operators like the MLA on certain accelerators employs a "one request per tensor compute core" partitioning strategy. This can lead to idle compute cores within the same DP group due to differences in request lengths. To address these challenges, xLLM proposes a hierarchical "defense-in-depth" strategy to tackle DP load imbalance issues across different time scales and granularities. The first layer is preventative, KV Cache-aware request scheduling; the second layer performs macroscopic correction through inter-DP group load balancing; and the third layer carries out microscopic correction via intra-DP group load balancing. This layered design enables the system to flexibly address the root causes of various performance degradations, thereby building a robust and efficient inference service. Layer 1: KV Cache-Aware Scheduling. xLLM's first-layer strategy implements request load distribution through KV Cache-aware scheduling. This mechanism moves beyond simple round-robin methods. When a new request arrives, the scheduler comprehensively checks the status of all DP groups, paying special attention to the remaining KV Cache capacity in each. It then preferentially assigns the new request to the group with the most available space. By balancing the total token load and its memory footprint at the system level over the long term, this strategy prevents the formation of severe load imbalances, thereby achieving intelligent resource allocation. Layer 2: Reactive Inter-DP Group Workload Migration. xLLM's second-layer strategy achieves reactive balancing through workload migration between DP groups. During the decoding process, the varying prompt and KV Cache lengths of selected requests cause different computation latencies among DP groups. To counter this, xLLM's central scheduler evaluates the current computational load of each DP group during every inference round. If a significant imbalance is detected, it initiates a workload migration process, moving tasks from overloaded to under-loaded groups. The scheduler also determines the migration granularity - whether to move an entire batch, a single sequence, or a partial MLA block of a sequence. Figure 12 illustrates this process at the MLA block granularity. First, xLLM dispatches the input tokens for the migration to both the source and destination DP groups. Then, as all groups execute the MLA Preprocess operation, the request's KV Cache is transferred asynchronously, overlapping communication with computation. Next, all groups begin the attention calculation. The underloaded group prioritizes the migrated attention task, allowing it to immediately send the resulting token back to the source group upon completion while concurrently processing its own native attention operations, further overlapping overhead. Finally, all DP groups use an aggregation operator to gather the external computation results. Layer 3: Fine-Grained Kernel-Level Optimization. xLLM's third-layer strategy achieves finegrained optimization through kernel-level migration within a DP group. This optimization is performed inside the matrix computation kernels of a single DP group. On one hand, during scheduling, it reorders requests based on their load, replacing the original compute core's round-robin allocation strategy. This aims to keep the total number of computation tokens assigned to each matrix computation unit as consistent as possible. On the other hand, for requests with extremely long sequences, it explicitly splits their computation sequence, migrating parts of the long-sequence request's computation to be calculated with other short-sequence computation units. Through this fine-grained request splitting, it directly resolves the issue of idle compute cores caused by load imbalance within the DP group. 4.5 Generative Recommendation Recent advancements in generative models have spurred significant interest in generative recommendation [31, 94-98]. Beyond enhancing the retrieval and ranking stages of traditional multi-stage pipelines [98, 94], single-stage generative recommendation frameworks directly generate a candidate item list in an auto-regressive manner [31]. These frameworks usually utilize beam search decoding to directly produce diverse recommendation results, and xLLM implements extensive optimizations for these single-stage frameworks. As shown in Figure 13, in the generative recommendation scenario, xLLM is dedicated to overlapping host and device operations as much as possible to improve overall execution efficiency. When a request arrives, xLLM preemptively prepares the information required during the forward process on the CPU side. After the preparation is completed, 22 xLLM Technical Report Pre-allocate Scheduler Schedule Beam Schedule Pre-allocate Beam Pre-allocate Schedule Beam Pre-allocate Schedule Beam w/o Overlap w/ Overlap Prefill Beam Decode Beam Filter Mask Worker Forward Add Sampler Logits w/o Overlap w/ Overlap Engine Decode Filter Mask Forward Logits Add Sampler Figure 13: Overview design of generative recommendation process. relevant tasks are handed over to the engine for processing. The Engine executes three forward passes in one go. During this process, the intermediate beam search is a host operation, which by its nature cannot be overlapped with device operations. To optimize the processing pipeline, xLLM internally employs a multi-micro-batch, multi-stream approach. During the Worker's execution, the CPU operation for generating the filter mask is overlapped with the device operation for logits computation. Through an addition operation, a masked logits is generated before the sampler, thereby achieving data filtering and improving the accuracy and effectiveness of the recommendation results. 4.5.1 Host-side Optimization In scenarios with large beam_width and top_k parameters, the significant increase in the number of candidate sequences results in substantial sorting and filtering operations on the host side. This shifts the computational bottleneck of the beam search algorithm from the AI accelerator to the CPU, ultimately leading to a severe CPU-bound issue. The following outlines xLLM's efforts to optimize the beam search process in generative recommendation scenario. Beam Search Optimization. In many fields such as natural language processing, beam search optimization is an efficient algorithm widely used in sequence generation tasks [99]. When processing a large number of candidate sequences, one of the core operations of beam search is to select beam_width sequences from numerous candidates. This process is essentially a partial sorting operation, with the unique characteristic that it only needs to identify the top beam_width elements without fully sorting all candidate sequences, thereby significantly reducing computational complexity. In another key step of beam search, generating beam_width × top_k candidate sequences from the existing beam_width sequences, there exists an important property: log_probs of each sequence are arranged in descending order. Leveraging this property, an optimized early termination mechanism can be adopted, where computations are halted prematurely if log_probs does not meet specific criteria, further enhancing computational efficiency. The specific operational procedure is as follows: First, the existing beam_width sequences are accessed in order. For each accessed sequence, we create a small min-heap of size beam_width, which is used to dynamically maintain the set of locally optimal elements currently selected. Next, the subsequent sequences are traversed. During this process, if the subsequent log_probs of a sequence is smaller than the top element of the minheap, it implies that, based on the current filtering criteria, the subsequent elements of this sequence cannot enter the current optimal set. At this point, the filtering operation for the current sequence can be terminated directly, and the next existing sequence processed. Conversely, if the subsequent log_probs is larger than the top element of the min-heap, the element is inserted into the min-heap, 23 xLLM Technical Report and the heap structure is adjusted according to the properties of the min-heap to maintain its order. After the traversal of all relevant sequences is completed, the top elements are sequentially extracted from the min-heap. These elements, in the order of extraction, form the beam_width candidate sequences sorted in descending order. Through this procedure, the approach ensures the selection of high-quality candidate sequences while significantly reducing unnecessary computational overhead, thereby improving the operational efficiency of the beam search algorithm in practical applications. Resource Reuse and Pre-allocation. During the beam search process, the beam_width is predetermined and fixed for each forward pass of the model, which implies that the computational resources required per forward propagation remains relatively constant. However, an excessively large beam_width can lead to significant wastage of both CPU computational and memory resources. To effectively mitigate this issue, xLLM incorporates a carefully designed resource reuse strategy. Specifically, during the generation of new candidate sequences, the system reuses resources previously occupied by older sequences without allocating new space for each candidate. This approach avoids the overhead associated with frequently creating new data structures. Based on the final search results, the system then updates the storage areas for the old sequences with the content of the newly generated ones upon completion of the beam search algorithm. By doing so, xLLM not only reduces memory usage but also minimizes the additional overhead on the CPU for resource management and allocation, significantly improving the efficiency of computational resource usage. 4.5.2 Device-side Optimization Valid Item Filtering. In typical single-stage generative recommendation frameworks, the valid item filtering mechanism plays a crucial role. For example, OneRec [31] framework uses an ordered combination of three token IDs to uniquely represent a valid product item ID. However, due to the vast number of possible token ID combinations, not all of them correspond to actual and valid items [100, 101]. To ensure that the results generated by the model are all valid product items, the system generates a valid mask asynchronously during the forward pass of the model. This mask is based on a pre-constructed vocabulary of valid items, which is then element-wise added to the logits output by the model. Through this clever design, the logits corresponding to invalid token IDs are adjusted, ensuring that these invalid token IDs are almost never selected in subsequent calculations. This effectively filters out invalid token IDs, thereby guaranteeing the validity and accuracy of the final recommendation results. 5 Evaluations In §5.1, we evaluate the full implementation of our xLLM framework against several baseline inference systems, specifically vLLM-Ascend1(v0.10.rc1) and MindIE2(v2.1.rc1), on Ascend 910B/910C [18] instances across a range of scenarios. xLLM, MindIE, and vLLM-Ascend refer to the default deployments on Ascend 910B. Additionally, we denote the deployments of xLLM, MindIE and vLLM-Ascend on Ascend 910C as xLLM‡, MindIE‡ and vLLM-Ascend‡, respectively. The testing scenarios are organized into multiple categories reflecting diverse online serving applications, such as the JingYan AI chatbot, customer service assistants, merchant assistants, product understanding and marketing recommendation systems. Additionally, in §5.2, we conduct a detailed ablation study to assess the individual contributions of each optimized module. 5.1 Main Results This section benchmarks xLLM against mainstream inference frameworks to demonstrate its superior inference efficiency across diverse models, including Qwen2/3-series [102, 87] and Deepseek [3] models, and datasets. §5.1.1 details a fair comparison using the ShareGPT3 dataset. To ensure an equitable comparison across all LLM inference frameworks, the key feature of our experimental setup is that the input and output sequence lengths are fixed, while the request 1https://github.com/vllm-project/vllm-ascend 2https://www.hiascend.com/en/software/mindie 3https://huggingface.co/datasets/anon8231489123/ShareGPT_Vicuna_unfiltered/blob/ main/ShareGPT_V3_unfiltered_cleaned_split.json 24 xLLM Technical Report 1 2 4 Number of Accelerators 2000 4000 6000 8000 10000 Throughput (tokens/s) Qwen3-0.6B 1 2 4 Number of Accelerators 2000 4000 6000 8000 Qwen3-1.7B 1 2 4 Number of Accelerators 2000 3000 4000 5000 6000 Qwen3-4B 1 2 4 Number of Accelerators 1000 2000 3000 4000 5000 Throughput (tokens/s) Qwen3-8B 1 2 4 Number of Accelerators 1000 2000 3000 4000 Qwen3-14B 2 4 8 Number of Accelerators 1000 2000 3000 Qwen3-32B xLLM xLLM MindIE MindIE vLLM-Ascend vLLM-Ascend Figure 14: Throughput comparison of LLM inference frameworks evaluated on Qwen3-series models using the ShareGPT dataset, under the constraint of TPOT=50ms and input/output length=2048. 16×910B 8×910C 0 100 200 300 400 Throughout (tokens/s) [2048,2048] TPOT:50ms 16×910B 8×910C 0 500 1000 1500 [2048,2048] TPOT:100ms 16×910B 8×910C 0 100 200 300 400 [2500,1500] TPOT:50ms 16×910B 8×910C 0 250 500 750 1000 1250 1500 [2500,1500] TPOT:100ms xLLM xLLM MindIE MindIE vLLM-Ascend Figure 15: Throughput comparison of LLM inference frameworks using DeepSeek-R1 on the ShareGPT dataset constrained by given TPOT and input/output length, where [2500, 1500] means input length=2500 and output length=1500. vLLM-Ascend on the 910C is excluded from the comparison, as its performance with Deepseek-R1 fails to meet the required TPOT threshold. rate is dynamically adjusted to match the target SLO (e.g., TPOT) threshold for each framework. We also evaluate this setup under different node configurations, including both single-node and multi-node setups with PD disaggregation. In §5.1.2, we evaluate xLLM on various real-world business scenarios from JD.com, where it is currently deployed, showcasing its performance under practical deployment conditions. 5.1.1 Benchmarking Performance Qwen3-series. As shown in Figure 14, we comprehensively compares the throughput performance of four inference frameworks across the Qwen3-series (from 0.6B to 32B parameters) on the benchmark. All tests are conducted under uniform configuration with input/output lengths set to 2048 tokens and a TOPT constraint of 50 ms. The experimental results demonstrate that while all evaluated frameworks show improved throughput with additional accelerators, xLLM and its Ascend 910C implementation (xLLM‡) consistently deliver superior performance, confirming near-linear strong scalability across various model sizes. Specifically, xLLM achieves throughput improvements of up to 1.9× and 1.7× compared to vLLM-Ascend and MindIE, respectively. Similarly, xLLM‡ outperforms vLLM-Ascend‡ and MindIE‡ by up to 2.2× and 1.5×, respectively. Further25 xLLM Technical Report Method Prompt Length Output Length Throughput (tokens/s) Request Rate (req/s) MindIE 2048 2048 8476.44 4.14 xLLM 2048 2048 11351.58 5.54 Table 3: Comparison of DeepSeek-R1 with PD disaggregation on the ShareGPT dataset constrained by TPOT=100ms. 1 2 4 Number of Accelerators 400 600 800 1000 1200 1400 Throughput (tokens/s) Qwen2-7B TPOT:40ms 1 2 4 Number of Accelerators 0 500 1000 1500 2000 Qwen2-14B TPOT:90ms 1 2 4 Number of Accelerators 2000 3000 4000 5000 Qwen3-8B TPOT:80ms 2 4 8 Number of Accelerators 1000 2000 3000 Qwen3-32B TPOT:80ms xLLM xLLM MindIE vLLM-Ascend Figure 16: Throughtput comparison of Qwen2-series and Qwen3-series models with various inference frameworks in the JingYan scenario. more, xLLM‡ demonstrates steady performance improvements over the Ascend 910B-based xLLM in most scenarios, validating the software stack's effective utilization of the new hardware. DeepSeek-R1. Figure 15 presents the throughput performance of various inference frameworks for the DeepSeek-R1 model on the benchmark, using 16 accelerators for Ascend 910B and 8 accelerators for Ascend 910C, respectively. The results demonstrate that the proposed xLLM framework achieves exceptional throughput on the Ascend 910 series. Quantitatively, xLLM on Ascend 910B delivers an average throughput improvement of approximately 1.7× over MindIE and a significant 12× enhancement over vLLM-Ascend. Furthermore, xLLM‡ achieves an average throughput increase of approximately 1.4× compared to MindIE‡. PD Disaggregation Settings. Table 3 benchmarks the inference performance of the MindIE and xLLM frameworks on the DeepSeek-R1 model using a PD disaggregation architecture. Under identical conditions with TPOT controlled at 100ms for 2048-length inputs/outputs, xLLM achieves approximately 34% higher throughput (11,351.58 vs. 8,476.44 tokens/s) and request rate (5.54 vs. 4.14 req/s), marking a significant efficiency improvement. 5.1.2 Business Serving Scenarios JingYan. JingYan is an AI shopping assistant designed to help users discover new products, find inspiration, and get answers to their questions. The dataset for JingYan, consequently, consists of conversational logs between the model and its users, capturing these rich interactions. As illustrated in Figure 16, we systematically evaluate the inference throughput of four frameworks for serving the Qwen2-series and Qwen3-series model in the JingYan scenario. The results demonstrate that both xLLM and xLLM‡ maintain superior throughput and exhibit better scaling efficiency across all model sizes compared to MindIE and vLLM-Ascend. For instance, when serving the Qwen3-8B model with 4 accelerators, xLLM delivers a throughput approximately 1.6 times that of vLLMAscend and significantly surpasses MindIE. The robust performance of xLLM on the 910B, and its enhanced results on the 910C hardware, highlight its effective adaptation to successive hardware generations. As shown in Table 4, a similar trend is observed for the DeepSeek-V3 model, where xLLM achieves a throughput that is over 9 times greater than vLLM-Ascend and surpasses MindIE by 36%. Customer Service. The Customer Service dataset comprises the interactive dialogues between customers and support agents. Figure 17 details the performance differences among inference frame26 xLLM Technical Report Method Prompt Length Output Length Throughput (tokens/s) Request Rate (req/s) vLLM-Ascend 6800 400 21.17 0.11 MindIE 6800 400 144.40 0.67 xLLM 6800 400 196.45 0.89 Table 4: Comparison of Deepseek-V3 model with various frameworks in the JingYan scenario constrained by TPOT=80ms. 1 2 4 Number of Accelerators 50 100 150 Throughput (tokens/s) Qwen3-8B 2 4 8 Number of Accelerators 20 40 60 80 Qwen3-32B xLLM xLLM MindIE vLLM-Ascend Figure 17: Throughtput comparison of Qwen3-series models with various inference frameworks in the customer service scenario constrained by End-to-End latency(E2E)=10s. works for the Qwen-8B and Qwen-32B models in the customer service scenario. The results highlight that our xLLM, particularly the xLLM‡ running on Ascend 910C, provide significantly better throughput across all tested configurations. For instance, with Qwen3-32B on 8 accelerators, the throughput of xLLM is 3.1 and 1.2 times greater than that of vLLM-Ascend and MindIE, respectively. It is important to note that the vLLM-Ascend framework shows a clear scaling bottleneck as the number of accelerators increases, whereas xLLM maintains near-linear efficiency scaling. This validates the high efficiency and superiority of the xLLM framework in managing distributed inference for large-scale models. Merchant Assistant. Table 18 benchmarks the throughput of various inference frameworks on three tasks (i.e., search terms, arrangement, intent recognition) within the merchant assistant scenario. The proposed xLLM framework achieves better or comparable performance to MindIE and demonstrates a significant lead over vLLM-Ascend. Specifically, for the search terms task with four accelerator cards, xLLM delivers 34% higher throughput than MindIE and roughly 3.4× that of vLLM-Ascend. Product Understanding. For the product understanding scenario, the throughput comparison of the Qwen2-7B model with several frameworks is shown in Table 5. The experimental results indicate that xLLM outperforms MindIE and vLLM-Ascend by an average of 25% and 56%, respectively, across different accelerator card counts. Moreover, the superiority of xLLM scales with the number of cards, demonstrating its effective utilization of large-scale parallel computing resources and thereby offering a robust solution for high-performance LLM inference. Generative Recommendation. As illustrated in Figure 19, the evaluation results on the Qwen-8B model demonstrate that xLLM consistently achieves lower mean end-to-end latency than MindIE across various request rates and beam widths, except under very low load condition. Notably, the performance advantage of xLLM becomes increasingly pronounced as the beam width (from 4 to 128) and the request rate escalate. For instance, under the most challenging scenario with a beam width of 128 and a request rate of 8, xLLM reduces latency by approximately 23% relative to MindIE. This significant improvement validates that our xLLM's host-side and device-side optimizations effectively alleviate the computational bottlenecks in generative recommendation tasks, markedly enhancing inference efficiency and scalability under heavy loads. 27 xLLM Technical Report 1 2 4 Number of Accelerators 100 200 300 Throughput (tokens/s) Search Term Qwen2.5-14B xLLM MindIE vLLM-Ascend 1 2 4 Number of Accelerators 50 75 100 125 Arrangement Qwen2.5-14B xLLM MindIE vLLM-Ascend 1 2 4 Number of Accelerators 100 150 200 250 300 Intent Recognition Qwen3-14B xLLM MindIE vLLM-Ascend Figure 18: Throughtput comparison of Qwen-series models with various inference frameworks in the merchant assistant scenario constrained by End-to-End latency(E2E)=1s. Method Prompt Length Output Length Throughput (tokens/s) #accelerator=1 #accelerator=2 #accelerator=4 vLLM-Ascend 1200 40 795.77 874.97 1272.52 MindIE 1200 40 944.81 1051.44 1693.45 xLLM 1200 40 1001.91 1323.90 2425.13 Table 5: Throughtput comparison of Qwen2-7B model with various inference frameworks in the product understanding scenario. 5.2 Ablation Study Impact of MTP. As shown in Figure 20, under the configuration of 1500 input length and 2500 output length for the DeepSeek-R1 model, enabling Multi-Token Prediction (MTP) technology significantly optimizes inference performance. As max concurrency increases, the MTP-enabled version consistently exhibits lower TPOT compared to the baseline, indicating reduced generation latency. Meanwhile, its throughput is markedly higher than the non-MTP version, with particularly notable advantages beyond 32 concurrent requests. This demonstrates that MTP effectively enhances computational efficiency and system throughput under high concurrency conditions. Impact of Dynamic PD Disaggregation Scheduler Policy. We evaluate our proposed SLO-aware Dynamic PD Disaggregation Policy against the Minimal Load and Round Robin strategies, as illustrated in Figure 21. On the Azure Code dataset characterized significant bursty traffic, our SLOaware policy achieves a request serving rate 1.67 times that of the Minimal Load strategy. Meanwhile, relative to the Round Robin strategy, the Minimal Load strategy improves the SLO attainment by up to 4.3%. For the Azure Conversation dataset with stable input/output length variations, our SLO-aware policy results in a 1.1 times higher request serving rate compared to the Minimal Load strategy. Although the Minimal Load strategy performs similarly to Round Robin, it still enhances the SLO attainment by up to 2.4%. These results indicate that minimal-load scheduling is closer to the optimal scheduling strategy than round-robin approach, while our adaptive instance scheduling delivers the best overall performance. Impact of Hybrid EPD Disaggregation Scheduler Policy. Figure 22 demonstrates the effectiveness of the proposed Hybrid EPD Disaggregation Policy on the TextCaps dataset, specifically in controlling TPOT and improving the SLO attainment rate. Under the configuration of 8 generalpurpose inference instances, the removal of the hybrid EPD disaggregation method results in a drop in goodput from 9.5 req/s to 7.2 req/s, and a subsequent additional removal of stage-level scheduling policy further decreases the goodput to 5.1 req/s. This indicates that a well-designed disaggregation strategy can effectively mitigate inter-stage interference, while our stage-level scheduling policy contributes to finer control over the execution time of each batch. The integration of both components significantly enhances the overall performance and stability of the system, particularly under high-concurrency scenarios. Impact of Online-Offline Co-location Scheduler Policy. As shown in Figure 23, we evaluated three scheduling strategies, namely the baseline P/D, online priority and our proposed Online-Offline 28 xLLM Technical Report 1 4 8 Request Rate (req/s) 90 100 110 Mean E2E (ms) Beam Width:4 Top-k:4 xLLM MindIE 1 4 8 Request Rate (req/s) 100 120 140 Beam Width:32 Top-k:32 xLLM MindIE 1 4 8 Request Rate (req/s) 125 150 175 200 Beam Width:64 Top-k:64 xLLM MindIE 1 4 8 Request Rate (req/s) 150 200 250 300 350 Beam Width:128 Top-k:128 xLLM MindIE Figure 19: Comparison of mean End-to-End latency(E2E) with various inference frameworks in the generative recommendation scenario. Since vLLM-Ascend does not support beam width>10, the corresponding results are not plotted in the figure. Even with beam width=4 and request rate=1, its mean E2E far exceeds that of our framework. 2 8 32 Max Concurrency 50 100 150 TPOT (ms) DeepSeek-R1 [2500, 1500] w/ MTP w/o MTP 2 8 32 Max Concurrency 100 200 Throughput (tokens/s) DeepSeek-R1 [2500, 1500] w/ MTP w/o MTP Figure 20: Impact of MTP on the concurrent performance of DeepSeek-R1 model. Co-location Scheduler Policy (denoted as xLLM-OOC), to assess the maximum achievable throughput for offline requests without violating the SLO of online requests. The green shaded region in the figure indicates the sustainable range of offline request throughput that remains within the acceptable online SLO violation threshold. When the offline query-per-second (QPS) exceeds a certain level, both the baseline P/D and online priority strategies lead to a sharp increase in the online SLO violation rate, indicating interference caused by the offline workload. In contrast, xLLM-OOC maintains stable SLO compliance even as offline QPS continues to rise. On our proprietary dataset, xLLMOOC achieves a throughput that is three times higher than the other two methods. Furthermore, it demonstrates improvements of 75% and 17% over online priority and baseline P/D, respectively, on the Azure Code dataset. Impact of Multi-layer Pipeline Execution. As evidenced by the results in Table 6, our proposed asynchronous scheduling mechanism delivers consistent throughput improvements across all evaluated model scales. The most substantial relative gain of 17.4% is observed for the 1.5B parameter model, highlighting the method's particular efficacy for smaller architectures where scheduling overhead constitutes a larger portion of total computation time. While relative improvements moderate for larger models (reaching 0.6% for 7B, 3.7% for 14B, and 6.6% for 32B), all configurations exhibit statistically significant absolute gains. These results robustly validate that our method successfully masks scheduling latency and eliminates computational bubbles, through leveraging placeholder tokens to decouple and overlap CPU scheduling with NPU execution. We further assess the effectiveness of the proposed dual-stream architecture for the DeepSeek-R1 model in Table 7. Experimental results from a single decoder layer show that the total communication time increases to 12.4ms in dual-stream mode, up from 9.3ms in single-stream mode. However, the computation-communication overlap mechanism successfully hides 80% of the communication time, which brings the exposed communication time down to just 2.5ms, saving 6.8ms per layer. Despite introducing a computational overhead of 4ms for each layer, the dual-stream strategy yields a net performance gain of 172.0ms across the entire 61-layer model, clearly illustrating the capability of our scheduling strategy in real-world workloads. 29 xLLM Technical Report 10 20 30 40 50 Request Rate (req/s) 40 60 80 100 SLO Attainment (%) Azure Code 25 50 75 100 Request Rate (req/s) 40 60 80 100 Azure Conversation Our SLO Aware Minimal Load Round Robin Figure 21: Performance of Dynamic PD Disaggregation Policy with different scheduling strategies. 1 2 3 4 5 6 7 8 9 10 11 12 Request Rate (req/s) 0 20 40 60 80 100 SLO Attainment (%) Ours w/o hybrid EPD w/o hybrid EPD and sched Figure 22: Impact of Hybrid EPD Disaggregation Scheduler Policy. Impact of Adaptive Graph Mode. To validate the effectiveness of the Adaptive Graph Mode optimization technique, we conduct controlled experiments on the Qwen3-1.7B and Qwen3-4B models. As shown in Table 8, with both the prompt and output lengths set to 2048 tokens, enabling Adaptive Graph Mode result in significant performance gains for both models. The throughput of Qwen31.7B increases from 2385.58 to 3038.88 tokens/s (a 27.4% improvement), while its mean TPOT decreases by 22.0%. Correspondingly, Qwen3-4B achieved an 8.5% increase in throughput and an 8.8% reduction in latency. These results robustly demonstrate the effectiveness of Adaptive Graph Mode as a general-purpose inference acceleration technique, with more pronounced optimization effects observed on the model with a smaller parameter size. Impact of Hierarchical DP Load Balancing. Our proposed hierarchical DP load balancing scheme is projected to increase total throughput by 5%. The kernel-level optimization yields the most significant latency savings; for example, for an ultra-long 32k-token request, reordering and splitting can reduce a single core's computational load from 32k to 1.3k tokens, saving approximately 800 microseconds. In contrast, the latency savings from inter-DP group migration are more modest. Even after balancing a 20k-token difference, the total time saved over 61 Transformer layers is only about 600 microseconds. This indicates that the greatest performance bottleneck and optimization potential lie within the computation units themselves. 6 Future Work Although proposed xLLM framework has demonstrated potential in enhancing LLM inference efficiency and reducing operational cost, achieving truly efficient and inclusive intelligent computing infrastructure requires deeper collaborative innovation across Hardware, Model, and Application Layers. Our future research will advance xLLM from a high-performance Inference Engine toward a comprehensive Operating System for AI that supports next-generation intelligent applications. We outline our planned efforts in three key directions: 30 xLLM Technical Report 0 3% 5% 10% OOC on Qwen2.5-7B 0 1 2 3 0 3% 5% 10% Online SLOs Violation Rate 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Offline Request QPS (req/s) 0 3% 5% 10% 0 1 2 3 0 3% 5% 10% Azure Code on Qwen2.5-7B 0 1 2 3 0 3% 5% 10% 0 1 2 3 Offline Throughput (req/s) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Offline Request QPS (req/s) 0 3% 5% 10% 0 1 2 3 base P/D - Online SLOs Violation base P/D - Offline Throughput online priority - Online SLOs Violation online priority - Offline Throughput xLLM-OOC (ours) - Online SLOs Violation xLLM-OOC (ours) - Offline Throughput Figure 23: Impact of our Online-Offline Co-location Scheduler Policy. Model Prompt Length Output Length Async Scheduling Throughput (tokens/s) DS-Distill-Qwen-1.5B 1000 1000 × 8709.90 1000 1000 ✓ 10223.15 DS-Distill-Qwen-7B 1000 1000 × 3183.68 1000 1000 ✓ 3201.83 DS-Distill-Qwen-14B 1000 1000 × 1472.22 1000 1000 ✓ 1527.23 DS-Distill-Qwen-32B 1000 1000 × 1415.20 1000 1000 ✓ 1508.76 Table 6: Impact of asynchronous scheduling mechanism in Multi-layer Pipeline Execution. 6.1 Fostering an Open and Diverse Hardware Ecosystem Current AI infrastructure often suffers from tight coupling with specific hardware architectures, introducing computational stability risks and limiting efficient deployment in heterogeneous environments such as edge computing. To address this, the first pillar aims to cultivate an open and diverse hardware ecosystem through the following initiatives: Unified Hardware Abstraction Layer. We will develop a high-performance, hardware-agnostic runtime abstraction layer that encapsulates instruction sets and memory architectures across computational units-from domestic cloud chips to edge-side accelerators. This layer will provide unified operator interfaces and memory management APIs, enabling seamless migration and efficient execution across heterogeneous hardware without code modifications, thereby significantly reducing integration barriers for emerging hardware. Software-Hardware Co-Design and Ecosystem Synergy. We will collaborate with hardware partners to define more efficient and open interface standards. This co-design approach will not only drive hardware innovation from the software perspective but also help establish a self-sustaining and competitive computational supply ecosystem, ultimately providing users with optimized choices across cost, performance, and security considerations. 6.2 Cultivating a Vibrant and Responsive Model Ecosystem The system's core value derives from its supported model diversity and integration efficiency. Our future efforts will focus on building an inclusive and agile model ecosystem through: Multi-Scenario Support. We will extend the platform's optimization and deployment capabilities beyond large language models to systematically support various generative AI scenarios, including generative recommendation, text-to-image, and text-to-video generation. We will optimize execu31 xLLM Technical Report Metric Single-Stream (ms) Dual-Stream (ms) Total Communication Time 9.3 12.4 Overlapped Communication Ratio - 80% Exposed Communication Time 9.3 2.5 Total Computation Time 13.0 17.0 Reduced Time per Layer - 2.8 Total Reduced Time (61 layers) - 172.0 Table 7: Communication and computation overheads in a single decoder layer of DeepSeek-R1 using Multi-layer Pipeline Execution. Model Prompt Length Output Length Adaptive Graph Mode Throughput (tokens/s) Mean TPOT (ms) Qwen3-1.7B 2048 2048 × 2385.58 39.27 2048 2048 ✓ 3038.88 30.63 Qwen3-4B 2048 2048 × 1539.93 55.44 2048 2048 ✓ 1671.39 50.58 Table 8: Impact of Adaptive Graph Mode. tion engines, request scheduling, and memory management specifically for different workloads to ensure optimal performance across diverse applications. "Zero-Day" Model Integration. To accommodate rapidly evolving model architectures, we will implement a unified framework combining model graph representation, an extensible operator library, and automated compilation optimization. This will enable rapid "zero-day" integration of newly released models, reducing deployment cycles from weeks to hours. 6.3 Evolving into an AI-Native Application Framework To democratize AI adoption and accelerate value delivery, we will evolve the system from an inference engine into a full-stack AI-native application framework by: Framework-Native AI Middleware. We will design high-level, framework-native APIs and abstractions that package complex distributed inference, multi-model orchestration, and stateful session management capabilities into out-of-the-box middleware services. This will enable application developers to build sophisticated AI applications (e.g., multimodal AI agents) without managing underlying infrastructure complexities. Rapid Application Integration and Deployment. Building upon this AI-native framework, we will deliver a comprehensive toolchain - including encapsulated SDKs, application templates, and integrated CI/CD pipelines - to streamline development and deployment processes. Our goal is to enable application teams to integrate and deploy AI services within hours rather than weeks, significantly enhancing business innovation agility and fully bridging the "last mile" from model development to value creation. 7 Conclusion We introduce xLLM, an intelligent and efficient LLM inference framework, featuring a serviceengine decoupled architecture. The framework consists of two main components: (1) xLLMService, a versatile service layer designed for efficient instance and cluster management as well as request scheduling. It incorporates unified elastic scheduling for co-located online and offline requests to maximize cluster utilization, a workload-adaptive PD disaggregation architecture for SLOaware dynamic instance scheduling, a novel Encode-Prefill-Decode (EPD) disaggregation mechanism for multimodal requests, a global KV Cache manager for efficient memory management including KV Cache upload and offload and a distributed fault-tolerance design to ensure high availability. (2) xLLM-Engine, a high-performance inference engine optimized for accelerating LLM 32 xLLM Technical Report inference across various AI accelerators. xLLM-Engine employs full-stack multi-layer execution pipeline optimizations through asynchronizing CPU-side scheduling and AI accelerator-side model forwarding to minimize computational bubbles, utilizing dual-stream parallelism of micro-batches to overlap computation with all-to-all communication, and further overlapping various AI computation units at the operator level. xLLM-Engine also implements an adaptive graph mode which pre-compiles kernel sequences, memory operations, and synchronization into a single computation graph to drastically reduce launch overhead and accelerator idle time. Additionally, it adopts a "logically contiguous, physically discrete" KV Cache storage strategy via proposed xTensor Memory Management, which resolves the tension between memory contiguity and dynamic allocation. To further boost hardware utilization, xLLM-Engine integrates algorithmic enhancements such as asynchronous pipelined adaptive speculative decoding, KV cache-aware scheduling, reactive interDP group workload migration for dynamic load balancing, and dynamic MoE load balancing based on real-time expert workload statistics and transparent weight updates. We have further extended xLLM to emerging generative recommendation scenarios, improving both recommendation accuracy and efficiency. Extensive experiments demonstrate that xLLM achieves a consistent improvement compared to leading inference systems such as MindIE and vLLM-Ascend, evaluated across mainstream Qwenseries and Deepseek-series models as well as public and real-world industrial datasets. In particular, xLLM outperforms MindIE by up to 1.7× and vLLM-Ascend by up to 2.2× in throughput. 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|
2510.14680
|
KA-TP-31-2025, COMETA-2025-47
Complete two-loop Yukawa-induced running of the
Higgs-gluon coupling in SMEFT
Stefano Di Noi a , Barbara Anna Erdelyi b,c and Ramona Gr¨ober b,c
aInstitute for Theoretical Physics, Karlsruhe Institute of Technology (KIT),
Wolfgang-Gaede Straße 1, Karlsruhe D-76131, Germany
bDipartimento di Fisica e Astronomia ”G. Galilei”, Universit`a degli Studi di Padova,
Via F. Marzolo 8, 35131 Padova, Italy
cIstituto Nazionale di Fisica Nucleare, Sezione di Padova,
Via F. Marzolo 8, 35131 Padova, Italy
E-mail: stefano.dinoi@kit.edu, barbaraanna.erdelyi@phd.unipd.it,
ramona.groeber@pd.infn.it
Abstract:
We compute the two-loop renormalisation group equation for the effective
Higgs to gluon coupling in Standard Model Effective Field Theory. Concretely, we present
the contributions generated by the operators belonging to class 3 and 7 in the Warsaw basis,
completing the two-loop renormalization program of the Higgs-gluon coupling proportional
to the top Yukawa coupling for potentially tree-level generated operators. We investigate
the phenomenological impact of the contributions in fits to Higgs data both in a bottom-up
approach and a top-down approach in terms of UV models with vector-like quarks.
arXiv:2510.14680v1 [hep-ph] 16 Oct 2025
Contents
1
Introduction
1
2
Notation
3
3
Computation and result
4
4
Phenomenology
8
4.1
Fitting framework
10
4.2
Results
10
4.2.1
Bottom-up results
10
4.2.2
Top-down results
12
5
Conclusions
15
A Feynman Rules
17
1
Introduction
There are several reasons to believe that the Standard Model (SM) of particle physics
requires an extension. Despite this, no direct evidence of new physics has been observed
so far, motivating the use of the Effective Field Theory (EFT) framework to describe
potential beyond-the-SM (BSM) effects in the most model-independent way possible. This
approach is valid under the assumption that the scale of new physics lies well above the
electroweak scale. Assuming that the Higgs boson transforms as in the SM, namely as part
of an SU(2)L doublet, writing down all possible higher-dimensional operators consistent
with the symmetries of the SM leads to the so-called Standard Model EFT (SMEFT).
Concerning collider physics, the most relevant contributions are expected to arise from
dimension-six operators, for which a non-redundant basis was first established in Ref. [1],
the so-called Warsaw basis.
Despite not being the only possible EFT extension of the SM,1 the SMEFT has become
a popular framework for global analyses [9–13], providing a self-consistent way to correlate
various experimental measurements across different energy regimes.
The connection between the Wilson Coefficients (WCs) at different energy scales is
encoded in the renormalisation group equations (RGEs). The complete one-loop RGEs of
dimension-six operators have been computed in [14–16] and implemented for systematic
1A different philosophy is embodied by the Higgs EFT [2–8] (HEFT or electroweak chiral Lagrangian),
where the Goldstone bosons and the Higgs boson do not transform in the same multiplet, but the physical
Higgs boson transforms as a singlet.
– 1 –
inclusion into phenomenological analyses in several computer codes [17–21]. Even though
the full set of two-loop RGEs for the dimension-six SMEFT has not been computed yet,
many partial results have been presented in the last years and progress towards the two-loop
SMEFT RGEs has been reported [22–35].
The RGEs are thus crucial for global analyses that combine observables measured at
different energy scales [36–38], exploiting fully the variety of available data from different
experiments. Moreover, with the increasing precision of collider measurements, RGE effects
can also become sizeable in processes where the renormalisation scale varies dynamically
over a wide range of energies. In this context, including the RGE running of WCs is essen-
tial, as explicitly demonstrated in [39–43]. These works focus on the running proportional
to the strong coupling constant αs, which is expected to give the dominant contributions.
Moreover, restricting to these effects allows to solve the RGEs analytically, greatly improv-
ing the time efficiency of the computer programs employed for phenomenological studies.
However, also the running effects generated by the top-quark Yukawa coupling Yt can
produce significant effects [27, 44]. Despite the strength of the interaction being na¨ıvely
smaller than in the strong case, WCs entering the RGEs via Yukawa-induced contributions
are generally less constrained [11], potentially enhancing their impact.
In this work we complete the effort started in Refs. [24, 27] to determine the two-
loop O
g2
sY 2
t
RGE of the Higgs-gluon coupling, which enters the dominant gluon-fusion
production channel for Higgs production. Within the SMEFT framework, this coupling is
parametrized by the operator OHG = (H†H)GA
µνGA,µν where H denotes the Higgs doublet
and GA
µν the gluon field strength. We stress that this interaction generates a tree-level
coupling between the Higgs bosons and the gluons, which instead is loop induced in the
SM. In weakly interacting and renormalizable UV models, the corresponding WC CHG is
loop-generated [45, 46]. Moreover, at one-loop level, the RGE of CHG involves only WCs
of operators that are likewise loop-generated under the same assumptions. Consequently,
OHG does not undergo one-loop renormalisation, even when considering dimension-eight
effects [47]. It follows that, to ensure consistency in the perturbative expansion in the loop
order, the two-loop running effects proportional to tree-level generated operators must be
considered when including RGE effects for Higgs production. Contributions belonging to
this category include the ones from four-quark operators [24] and Yukawa-like operators
[27].
In this paper, we present the contributions from the operators with the schematic
structure D2H4 and D H2 ψ2. Such operators belong to, respectively, class 3 and 7 in
the classification of the Warsaw basis notation [1] and are the last missing contributions
for the complete two-loop RGE proportional to the third family Yukawa couplings Yt and
Yb involving operators that are potentially tree-level generated.
Conversely, we do not
consider chromomagnetic operators as they can only be generated at loop-level in weakly-
interacting theories [45, 46] and are expected to give a contribution at O(g2
sYt) that would
be a three-loop effect.
We investigate the phenomenological impact of the new contributions to the RGE run-
ning by performing a fit to Higgs data both in the bottom-up and top-down approach, using
simplified models with heavy Vector-Like Quarks (VLQs). Such states naturally arise in a
– 2 –
variety of frameworks proposed to address open issues of the Standard Model, such as the
hierarchy problem, with Composite Higgs Models providing a prominent example [48–50].
In these models, operators belonging to class 7 are generated at tree level, which lead to a
modification of the couplings of quarks to massive vector, eventually in association with a
Higgs boson. In particular, the couplings of top quarks to Z bosons are weakly constrained
so far [37, 51], as a direct constraint comes from t¯tZ production, single top production [52]
or Zh production [53], whereas other probes need to rely on RGE effects. Other operators
that couple to the third generation SU(2)L doublet can also be constrained by electroweak
precision data.
Conversely, class 3 operators can be constrained by electroweak preci-
sion data and by measurements of the Higgs coupling to massive vector bosons, currently
available at few percent level [54, 55].
The paper is structured as follows. In Sec. 2 we present our notation. In Sec. 3 we
describe our computation and present the result for the RGE of the Higgs-gluon coupling
in the SMEFT. In Sec. 4 we study the phenomenological implications of the inclusion of
two-loop running effects in single Higgs physics. Finally, in Sec. 5 we present our summary.
2
Notation
The SMEFT extends the SM by a tower of gauge and Lorentz invariant higher-dimensional
operators as a series in inverse powers of a high-energy new physics scale Λ. In this work, we
neglect lepton and baryon number violation and focus exclusively on the leading dimension-
six terms in the expansion:
L = LSM +
X
Di=6
CiOi.
(2.1)
The operators are denoted by Oi and the associated WCs by Ci. We note that the WCs in
the previous formula have dimension -2 in units of energy. The operators in the previous
expressions are invariant under the unbroken SM group GSM = SU(3)C × SU(2)L × U(1)Y.
We follow the Warsaw basis of Ref. [1] for the dimension-six operators and Ref. [14]
for the definitions of the SM Lagrangian.
We consider only the third family of quarks, leading to a minimal modification of the
notation with respect to the original publication. In particular, we use tR, bR to denote the
right-handed component and QL to denote the left-handed component of the third family
of quarks. The relevant class 7 operators are defined as:
O(1)
HQ =
iH†←
→
DµH
¯QLγµQL
,
O(3)
HQ =
iH†←
→
DI
µH
¯QLγµτ IQL
,
(2.2)
OHt =
i H†←
→
DµH
(¯tRγµtR) ,
OHb =
i H†←
→
DµH
¯bRγµbR
,
(2.3)
OHtb =
i ˜H†DµH
(¯tRγµbR) ,
(2.4)
where ˜Hi = ϵijH†,j (with ϵ12 = +1) and
i H†←
→
DµH
= iH†(DµH) −i(DµH)†H,
(2.5)
i H†←
→
DI
µH
= iH†τ I(DµH) −i(DµH)†τ IH.
(2.6)
– 3 –
The operators in class 3 are defined as
OH□=
H†H
□
H†H
,
OHD =
H†DµH
H†DµH
∗
.
(2.7)
These two operators in the broken phase modify the kinetic term of the Higgs boson, thus
requiring a field redefinition to reobtain the canonical normalisation of the Higgs kinetic
term. In the unitary gauge, the Higgs doublet H is related to the physical Higgs field h via
H =
1
√
2
0
v + h
1 + v2 CH, kin
!
,
where CH, kin = CH□−1
4CHD .
(2.8)
Finally, we recall that the Higgs-gluon coupling in the SMEFT is parametrized by the
operator
OHG =
H†H
GA
µνGA µν
.
(2.9)
This operator belongs to class 4 in the Warsaw basis. Other class 4 operators that will
appear in the phenomenological study performed in this work are defined below
OHW =
H†H
W I
µνW I µν
,
OHB =
H†H
(BµνBµν) ,
(2.10)
OHWB =
H†τ IH
W I
µνBµν
.
(2.11)
Furthermore, the following class 5 operators will be considered, often referred to as Yukawa-
like operators:
OtH =
H†H
¯QL ˜HtR
,
ObH =
H†H
¯QLHbR
.
(2.12)
We remind the reader that the SM Yukawa coupling is not defined in direct analogy to the
Yukawa-like operator.
All the previous operators are Hermitian with the exclusion of OHtb, OtH and ObH,
which enforce the addition of their complex conjugate to the Lagrangian. We use T A for
the SU(3)C generators and τ I for the Pauli matrices. We define the gluon field-strength
tensor as GA
µν = ∂µGA
ν −∂νGA
µ −gsfABCGB
µ GC
ν .
3
Computation and result
We describe in this section the computation of the two-loop contributions of the operators in
Eqs. (2.2)-(2.4) and Eq. (2.7) to the RGE of CHG. Practically, this requires the computation
of the divergent parts of the two-loop corrections to a process where OHG, defined in
Eq. (2.9), enters at tree-level. We perform our computation in the unbroken phase and
we consider the process gg →H∗H. In dimensional regularization, the divergences are
parametrized by poles in ϵ = 0, being D = 4 −2ϵ the number of space-time dimensions.
We will discuss here in detail the calculation of the terms proportional to OHt. All
the other operators of class 7 are obtained in analogy. At one-loop order, the contributions
to gg →H∗H can only arise via triangle diagrams as in Fig. 1a. However, once all the
relevant diagrams are considered their contribution vanishes [16]. The reason is that the
– 4 –
H
H∗
g
g
(a)
H
H∗
g
g
(b)
H
H∗
g
g
(c)
Figure 1: Sample of the diagrams with a single insertion of the operator OHt which vanish because
of kinematics, see main text for details. We use a thick (thin) line to denote a left-(right-)handed
field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt and
a black dot to denote a SM vertex insertion.
g
H
H∗
g
(a)
g
H
H∗
g
(b)
g
H
H∗
g
(c)
Figure 2: Sample of the two-loop order diagrams with a single insertion of the operator OHt.
We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs
doublet, a black square to denote an insertion of OHt, a black dot to denote a SM vertex insertion
and a white star to denote the Yukawa coupling counterterm.
g
g
H
H∗
(a)
g
g
H
H∗
(b)
Figure 3: Sample of the two-loop order diagrams with a single insertion of the operators OH□and
OHD. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the
Higgs doublet, a white square to denote an insertion of OH□and OHD and a black dot to denote
a SM vertex insertion.
diagrams are purely s-channel, depending on the linear combination q1+q2 = p1+p2 where
we have used p1,2 (q1,2) to denote the incoming (outgoing) momenta of the gluons (Higgs
doublets). The scalar current gives rise to a contraction with (p1 −p2), yielding a vanishing
result.
Following this argument, we can safely ignore all the two-loop diagrams where the
two external Higgs bosons are connected to the effective vertex. This applies both to pure
QCD (see Fig. 1b) and to mixed Yukawa-QCD (see Fig. 1c) contributions.2 We have cross-
checked the correctness of our reasoning by explicitly computing the O
g4
s
diagrams with
2As a consequence of this statement, the class 7 pure QCD contribution to the RGE of CHG vanishes at
all loop orders at dimension-six level.
– 5 –
a QCD correction to the propagator connecting the two QCD vertices and verifying that
their sum vanishes after a trivial change of variable in the momentum integration.
Our argument allows to restrict our attention to a total of 24 diagrams containing an
insertion of OHt, where one of the two Higgs doublets in the EFT vertex is external and
the other is internal. A sample of the two families of diagrams is displayed in Figs. 2a and
2b. Moreover, class 7 operators correct the Yukawa coupling at one-loop [14], calling for
the inclusion of the one-loop insertions of the one-loop Yukawa counterterms as in Fig. 2c.
Class 3 operators modify the quadrilinear coupling, yielding 12 diagrams, of which we
present a sample in Fig. 3.
We have performed two independent computations of the class 7 operator contributions
to check the set-up of our computation. The first one relies on FeynRules [56, 57] for the
generation of the Feynman rules, where the model file is obtained by extending the SM
model file in the unbroken phase provided in the additional material to MatchMakerEFT [58].
The resulting Feynman rules are presented in App. A. FeynArts [59] and FeynCalc [60–
63], interfaced via FeynHelpers [64], are used to generate the diagrams and to simplify
the Dirac algebra. The numerical integration of the two-loop integrals is performed with
AMFlow [65] supplemented with Blade [66].
The second computation relies on the same pipeline of Ref. [27]. The diagrams are
generated with qgraf-3.6.10 [67] and simplified with FeynCalc. The numeric integration
is performed with AMFlow supplemented with LiteRed2 [68, 69] and FiniteFlow [70].
Both computations use the Na¨ıve Dimensional Regularization (NDR) [71, 72] scheme
for the treatment of γ5 in a non-integer number of dimensions. This prescription is known
to be algebraically inconsistent in presence of traces involving an odd number of γ5 in
combination with at least six γ matrices as in the diagrams we compute. However, we
are interested in the contribution to the RGE of CHG, meaning that we can ignore the
contribution of the traces involving γ5, which would renormalize its CP-odd counterpart,
namely ˜CHG.
The Breitenlohner-Maison-t’Hooft-Veltman (BMHV) scheme [73, 74] has recently gained
popularity as an alternative to the NDR scheme, being the only known continuation scheme
proved to be algebraically consistent [75–79]. However, it highly increases the algebraic
difficulty of loop computation. We have checked that, for the present computation, no
difference arises between the two schemes by employing the map presented in Ref. [80].
This procedure is legit in this context since the leading divergence is of O (1/ϵ), as recently
showcased in Ref. [33].
We note that, in principle, all the operators considered in this work can renormalize
redundant operators, for which we follow the notation of Ref. [58]. This renormalization
arises from subamplitudes with schematic structure ¯ψψ →H∗H, ψ = QL, tR, bR, which
affect the operators R′
Hψ, R′′
Hψ,3 and from H →¯QLχR, χ = b, t (possibly in association
with one or two gluons), which impact the operators RuHDi, i = 1, 2, 3, 4. Since some of
the external states in these subprocesses are not on-shell, the relevant one-loop insertions
3When ψ = QL, two possible isospin contractions are possible, generating a total of four redundant
operators involving the left-handed quarks.
– 6 –
of redundant operators must be included in the total contribution to gg →H∗H. We have
explicitly checked that these contributions vanish in the case where only OHt is present.
Moreover, we do not need to consider contributions coming from one-particle reducible
diagrams to OHG.
This conclusion stems from the fact that no redundant operator is
reduced to OHG when the SM equations of motions are used [1].
We obtain
µdCHG
dµ
⊃
g2
s
(16π2)2
h
+ YbY ∗
b (CHb −C(1)
HQ −3C(3)
HQ) + YtY ∗
t (C(1)
HQ −3C(3)
HQ −CHt)
+ (CHtbYbY ∗
t + C∗
HtbY ∗
b Yt)
−
CH□−1
2CHD
(YbY ∗
b + YtY ∗
t )
+ 3
2 [CtHYt + C∗
tHY ∗
t + CbHYb + C∗
bHY ∗
b ]
−4 δNDR
YtY ∗
t
C(1)
Qt −1
6C(8)
Qt
+ YbY ∗
b
C(1)
Qb −1
6C(8)
Qb
i
,
(3.1)
δNDR =
(
1
(NDR)
0
(BMHV).
(3.2)
We use the same conventions as the one-loop results of [14–16]. The first two lines represent
our novel result, while the contributions generated by four-quark operators and Yukawa-like
operators (see Eq. (2.12)) have been first derived in Ref. [24] and in Ref. [27],4 respectively,
which we point to for the precise definition of the operators.
We stress that only the
four-top contribution depends on the γ5 scheme, while the others do not.
The final amplitude cannot exhibit infrared (IR) divergences, since no corresponding
real-emission diagrams exist at the considered order in the coupling constants. However,
IR divergences can still appear at intermediate stages of the calculation. As they must
ultimately cancel, one has to verify that this cancellation does not occur against ultraviolet
(UV) divergences, as can happen, for instance, in the case of scaleless integrals.
We note that our result can be used to determine the analogous two-loop contributions
from the operators in class 3 and 7 to OHW , OHB, OHWB (defined in Eqs. (2.10)–(2.11)),
where the gluon field strength is replaced by the corresponding quantities involving elec-
troweak bosons. However, this would represent just a partial result, since new diagrams
arise as a consequence of the non-vanishing SU(2)L × U(1)Y charge of the Higgs doublet.
Obtaining the RGEs for such operators is hence beyond the scope of this paper. The phe-
nomenological relevance of these operators lies in their impact in the interactions involving
photons or a photon and Z bosons and (at least) one Higgs boson and they lead to a shift
in the electroweak input [81]. In the SM, such interactions are numerically dominated by
W loops, so we expect the two-loop running contributions to be less numerically important
than in the gluon case such that the phenomenological study we perform in the next section
4In the previous expression, the contribution from the operators OtH and ObH presents an extra 1/2
with respect to the result of Ref. [27], which will be updated.
– 7 –
remains valid omitting these sub-leading contributions. Furthermore, their contributions
to the Higgs couplings to massive vector bosons is suppressed in the loop counting with
respect to the tree-level contributions from class 3 operators.
4
Phenomenology
In this section, we study the effect of the inclusion of two-loop running effects with fits to
Higgs data at the LHC. In particular, we focus on the modifications to the signal strengths
of a set of Higgs decay final states due to the presence of SMEFT operators.
We perform two different fits. In first instance, we adopt a bottom-up perspective by
performing the analysis in terms of the WCs to observe how their bounds are modified
by the inclusion of the running. Secondly, we consider a top-down approach by selecting
UV models with pairs of Vector-like Quarks (VLQs) coupled to the third-generation SM
quarks, which generate, at tree-level, at least one of the class 7 operators considered in this
paper.
The WCs that enter the observables in this section are computed at the low scale
µR = mh = 125 GeV. Their expression in terms of the WCs at the high energy scale
Λ = 2 TeV can be obtained with the evolution matrix formalism [20]
Ci(µR) = Uij(µR, Λ)Cj(Λ).
(4.1)
In the present case, the evolution matrix Uij(µR, Λ) has been obtained with a modified
version of RGESolver [21] relying on a numeric solution of the RGEs. All the operators
included in this section have been defined in Sec. 2.
Cross-section
The SM value for the Higgs boson production cross section in the dom-
inant gluon-gluon fusion contribution is σSM = 48.68 pb at a center of mass energy of
√s = 13 TeV. In the SMEFT, the cross-section reads
σ =
48.68 + 2.83 · 104v2CHG + 97.36v2CH,kin −112.25v2
Yt
CtH
pb ,
(4.2)
where CH,kin was defined in Eq. (2.8). We have obtained this parameteric dependence on
the SMEFT WCs using the EFT implementation [41] in higlu [82].
Higgs total and partial decay widths
The Higgs total decay width in the SMEFT
at LO in the U(3)5 flavour-symmetric limit was computed in Ref. [83]. We adapt those
results, noting that quark operators of class 7 contribute only via corrections to the gauge
boson total width except for the cases with quarks in final states from Z boson decays.
In this case there are some direct contributions where the Higgs boson and one Z boson
couples via a class 7 operator to bottom quarks, and the Z boson decays subsequently.
This results in
Γh = ΓSM
h
1 + 50.6 v2CHG −1.50 v2CHB −1.21 v2CHW + 1.21 v2CHWB + 1.83 v2CH□
−0.43 v2CHD −48.5 v2Yb CbH + 0.00016v2 C(1)
Hq + 0.0015v2 C(3)
Hq −0.00023v2 CHb
+2 v2
BRh→γγ
CH,kin + 0.53
Yt
CtH
+ BRh→gg
CH,kin −2.12
Yt
CtH
,
(4.3)
– 8 –
using the {MW , MZ, GF } input scheme for the electroweak parameters.
The last line
encodes the higher-order rescaling of the Higgs decays into photon and gluon pairs through
CH,kin defined in Eq. (2.8), together with the additional modifications induced by shifts in
the top Yukawa coupling. In the UV models we consider, also the bottom Yukawa coupling
can be modified but we do not consider its effect in the h →gg and h →γγ decays as
it is negligible with respect to the top Yukawa contribution. We note already from this
formula that generically our two-loop contributions can be of the same order than the
direct contributions. Indeed, since the CHG has a large pre-factor, even the two-loop factor
1/(16π2)2 log(Λ/mh) can be compensated.
We obtain the partial decay widths in the SMEFT rescaling the results in Ref. [83]
to take into account the modifications of the decay into photon pairs and the top Yukawa
coupling, analogously to the case of the total width discussed previously. We have validated
these expressions to those reported in Ref. [84].5
For the decays into fermion pairs, introducing l ∈{µ−, τ −} , the partial decay widths
are
Γh→l¯l
ΓSM
h→l¯l
= 1 + 2v2 CH,kin,
(4.4)
Γh→b¯b
ΓSM
h→b¯b
= 1 + v2 2CH,kin −2 CbH
.
(4.5)
For the decays into photon pairs we get
Γh→γγ
ΓSM
h→γγ
= 1 + v2
2 CH,kin + 0.53
Yt
CtH −231 CHW −805 CHB + 431 CHWB
.
(4.6)
As for decays into pairs of W and Z pairs, we further consider their decay products in
order to compare with experimental results. In particular, Ref. [55], whose experimental
results we will be using later, considers the processes h →W +W −→2l 2ν and h →ZZ →
4l. In order to get the partial decay width for these processes, we start again from Ref. [83]
and focus on their results for partial decay widths of the Higgs boson into four-fermion final
states mediated by ”only neutral”, ”only charged” and ”neutral plus charged” currents.
In our scenario, the first case reduces to the 4l final state, while the second and third
correspond to the 2l 2ν final state. The SM partial decay width for h →ZZ →4l is then
ΓSM
4l
= 1.1 keV, the partial decay width for h →W +W −→2l 2ν reads ΓSM
2l2ν = 90 keV ,
and the corresponding SMEFT expressions read
Γ4l
ΓSM
4l
=1 + v2 (2CH□+ 0.27 CHD + 0.17 CHW −1.66 CHB −0.51 CHWB
−0.27 C(1)
HQ −0.27 C(3)
HQ + 0.048 CHb
,
(4.7)
Γ2l2ν
ΓSM
2l2ν
=1 + v2 (2CH□−0.52 CHD −1.5 CHW + 0.003 CHB −0.023 CHWB
−0.0046 C(1)
HQ −0.0046 C(3)
HQ + 0.00081 CHb
.
(4.8)
5Ref. [84] uses the SILH basis [85], whereas Ref. [83] uses the Warsaw basis. To compare the results, we
translate both sets of results into the Greens’ basis by means of Ref. [86] and compare the terms of interest.
– 9 –
4.1
Fitting framework
Adopting the symbol Oα to identify an observable α, the χ2-function can be defined as
χ2 =
X
α, β
Oexp
α
−Oth
α
C−2
αβ
Oexp
β
−Oth
β
,
(4.9)
where C−2 is the inverse of the covariance matrix and the superscript ’th’ (’exp’) means
that the theoretical (experimental) value of the observable Oα is being considered. The
theoretical value accounts for both the SM prediction and the deviations due to the insertion
of SMEFT operators.
Following Ref. [87], we construct a fit to Higgs data using the measurements of signal
strengths with their covariance matrix as provided in Ref. [55]. Considering the Higgs
decay channels h →α with α ∈{W +W −, ZZ, b¯b, τ +τ −, µ+µ−, γγ},6 for each channel it
is given by
µα =
σ × BRh→α
σSM × BRSM
h→α
=
σ
σSM
ΓSM
h
ΓSM
h→α
Γh→α
Γh
=
σ
σSM
ΓSM
h
Γh
Γh→α
ΓSM
h→α
,
(4.10)
in which σ(SM) and BR(SM) denote, respectively, the value in the SMEFT (SM) of the Higgs
production cross-section and the branching ratio into the specific channel. We consider
up to order O(1/Λ2) in the signal strength. Of the three multiplied ratios in the final
expression, the first is known from Eq. (4.2), the second from Eq. (4.3) and the third is
selected from Eqs. (4.4)-(4.8) depending on the decay channel of interest.
The allowed region satisifies the condition
χ2 < χ2
min + ∆χ2 (n, CL) .
(4.11)
In the previous expression, n denotes the degrees of freedom (number of couplings in the
top-down approach, number of involved WCs in the bottom-up approach) while CL stands
for Confidence Level. The value χ2
min is the minimum of the χ2.
4.2
Results
We now present our results. We notice that the operators OHtb and OHt do not enter di-
rectly any of the Eqs. (4.2)-(4.8), so they enter only via mixing effects with other operators.
4.2.1
Bottom-up results
In this section we report our one- and two-parameter results for the bottom-up fits. We
stress that these parameters are set to be non-zero at the high-scale Λ = 2 TeV, meaning
that several other operators are generated via operator mixing, governed by Eq. (4.1).
Hence, the bounds are reported for the operators at such scale.
One parameter fits
We present the individual 2σ CL interval for each WC considered
at the energy scale Λ = 2 TeV in Fig. 4.
We note that the intervals for the class 3
6For the decays into pairs of W and Z bosons, Ref. [55] considers the bosonic decay channels ZZ →4l
and WW →lνlν. Thus, for the first two decay channels, we can use the decay widths of Eqs. (4.7) and
(4.8) respectively.
– 10 –
−12
−10
−8
−6
−4
−2
0
2
4
6
8
10
12
Intervals
CHD(Λ)
CH□(Λ)
CtH(Λ)
CbH(Λ)
CHt(Λ) · 10−1
CHb(Λ) · 10−1
C(1)
HQ(Λ)
C(3)
HQ(Λ)
CHtb(Λ) · 10−2
2σ 1L RGEs
2σ 2L RGEs
Figure 4: Allowed ranges at 2σ for the WCs obtained by performing one parameter fits. The
yellow (orange) line is obtained by considering one-loop (two-loop) running effects.
coefficients are not sizeably affected by the inclusion of the two-loop effects in the RGEs.
These operators are already constrained by direct contributions to the couplings with the
Higgs bosons, reducing the impact of higher order corrections. Similar conclusions apply
to the WCs of the Yukawa-like operators CtH and CbH, which enter directly in the Higgs
production cross section and in several decay channels.
As for the class 7 operators, CHb is not sizeably modified by the inclusion of the two-
loop running effects, due to the bottom Yukawa suppression. Conversely, C(1)
HQ and CHt
show a moderate reduction when two-loop running effects are accounted, while for C(3)
HQ
a quite sizable reduction of the allowed range can be observed once two-loop running is
included. Moreover, we note that the bounds for CHtb are looser at two-loop level, hinting
for some cancellations happening with respect to one-loop running effects.
Finally, we note that the bounds we present here are much weaker than the ones
presented in Ref. [37]. Since our focus is on the impact of the two-loop running effects in
the fit, we only accounted for inclusive Higgs data and not, for instance, for electroweak
precision data or top data as Ref. [37]. Those measurements are though sensitive to the
operators considered here.
Two parameter fits
In Fig. 5 we allow two WCs to be non-vanishing at the energy scale
Λ = 2 TeV. We present the results of three two-parameter fits, comparing the impact of
including only one-loop or two-loop running effects. In Fig. 5a, the two class 3 operators
are considered. The correlation between the two arises from the fact that their contribution
appears mostly in the linear combination CH, kin = CH□−1
4CHD. Moving on to Fig. 5b,
we consider the two WCs C(1)
HQ and C(3)
HQ which, in motivated UV extensions of the SM,
– 11 –
(a) Class 3 operators
(b) Class 7 operators with quark doublets
(c) Class 7 operators with quark singlets
Figure 5: Allowed regions at 2σ obtained via a two parameter fit including one-loop (in yellow)
and two-loop (in orange) running effects. We consider in pairs the operators belonging to class
3 and to class 7. For class 7, we further separate between the cases in which the quarks in the
operators are the doublets or the singlets under SU(2)L.
appear together, as shown in Tab. 1. Finally, in Fig. 5c we compare CHt and CHb. In this
case, there is a significant improvement in the bounds considering that CHt does not enter
any of the observables defined in Eqs. (4.2)–(4.8), while CHb enters but is very suppressed
compared to the other WCs.
4.2.2
Top-down results
In this second approach we consider UV models featuring Vector-like Quarks (VLQs) as
they generate at tree level both the operators in Eqs. (2.2)-(2.4) and the Yukawa-like
– 12 –
GSM
O(1)
HQ
O(3)
HQ
OHt
OHb
OHtb
U
(3, 1) 2
3
✓
✓
Q1
(3, 2) 1
6
✓
✓
✓
Q7
(3, 2) 7
6
✓
T1
(3, 3)−1
3
✓
✓
T2
(3, 3) 2
3
✓
✓
Model
Particle Content
1
U + Q1
2
U + Q7
3
Q1 + T1
4
Q1 + T2
5
Q7 + T2
Table 1: Class 7 operators generated at the tree-level by the VLQs considered in this work, along
with the charges under the SM gauge group of the VLQs, with GSM = (SU(3)C, SU(2)L)U(1)Y (left).
The presented VLQs will be used to construct models with pairs of these particles (right).
operators OtH and ObH [88]. The operators OHG, OHD and OH□are instead generated at
one-loop level.
In a similar spirit to Ref. [87], we consider models with pairs of VLQs which give rise
to unsuppressed contributions to the operator OtH. In Tab. 1 we list the VLQs considered,
their charges under the SM gauge group GSM, the class 7 operators they generate at tree-
level and how the five models considered in this work are defined.
Given that we are interested in the RGE effects due to terms proportional to third
generation Yukawa couplings, we make the simplifying assumption that the VLQs only
couple to the third generation SM quarks. Furthermore, we set their masses to be Λ =
2 TeV. This value is in agreement with lower bounds on singly-produced VLQs coupled to
third generation quarks [89, 90] and with the bounds from VLQ pair production [90–92].
The interaction Lagrangians for the considered models are given for a general flavour
structure in Ref. [88]
−L(int)
M.1 = [λU]rp ¯Ur
R ˜H†qp
L +
λu
Q1
rp ¯Qr
1L ˜Hup
R +
h
λd
Q1
i
rp
¯Qr
1LHdp
R+
+ [λUQ1]rp ¯Ur ˜H†Qp
1 + H.c. ,
(4.12)
−L(int)
M.2 = [λU]rp ¯Ur
R ˜H†qp
L + [λQ7]rp ¯Qr
7LHup
R + [λUQ7]rp ¯UrH†Qp
7 + H.c. ,
(4.13)
−L(int)
M.3 =
λu
Q1
rp ¯Qr
1L ˜H up
R +
h
λd
Q1
i
rp
¯Qr
1LH dp
R +
[λT1]rp
2
¯T I r
1RH†τ I qp
L+
+
[λT1Q1]rp
2
¯T I r
1 H†τ I Qp
1 + H.c. ,
(4.14)
−L(int)
M.4 =
λu
Q1
rp ¯Qr
1L ˜H up
R +
h
λd
Q1
i
rp
¯Qr
1LH dp
R +
[λT2]rp
2
¯T I r
2R ˜H†τ I qp
L+
+
[λT2Q1]rp
2
¯T I r
2
˜H†τ I Qp
1 + H.c. ,
(4.15)
– 13 –
(a) Model 2.
(b) Model 3.
(c) Model 4.
(d) Model 5.
Figure 6: Allowed regions at 2σ for pairs of couplings for the considered models. The orange
(yellow) coloured areas correspond to allowed values derived accounting for two-loop (one-loop)
running effects.
−L(int)
M.5 = [λQ7]rp ¯Qr
7LHup
R +
[λT2]rp
2
¯T I r
2R ˜H†τ I qp
L +
[λT2Q7]rp
2
¯T I r
2 H†τ IQp
7 + H.c. .
(4.16)
For these equations we follow the notation of Ref. [88] and use up
R and dp
R for right-handed
quark singlets and qp
L for the left-handed quark doublets, being p, r a flavour index. For our
phenomenological study, we only consider third generation SM quarks, namely p = r = 3.
For the tree-level expressions we refer to Ref. [88], whereas for the one-loop matching
we use SOLD [93] and Matchete [94].
In all models and for all operators entering Eqs. (4.2)-(4.8) we match up to the one-loop
level.
– 14 –
(a) Model 1 with complete one-loop matching
(b) Model 1 using tree-level matching
Figure 7: Two fits performed for Model 1, considering either the complete matching up to one
loop for all the considered operators (left) or only the tree-level matching results (right). In both
cases, the allowed parameter space at 2σ is plotted in yellow (orange) when the one-loop (two-loop)
running effects are included.
For each model the χ2 is a function of either three or four couplings in the UV model.
In Fig. 6 and Fig. 7a we present our results as two-dimensional CL regions, setting the
remaining parameters to their best fit value.7 As can be inferred in Figs. 6–7a, the inclusion
of two-loop running in the RGEs makes only a minor difference. While the model at tree-
level matches only into operators of class 7 and the Yukawa-type operators, at one-loop level
it matches into the class 3 operators and OHG, OHW , OHB and OHWB, which enter with
large coefficients into the signal strength. This reduces the impact of the two-loop running
effects considered here. A greater impact is observed when only the tree-level matching
results are considered. In order to showcase this aspect, in Fig. 7, we compare the complete
fit performed for model 1 with a second fit in which only the tree-level generated operators
are considered.
5
Conclusions
We have analyzed the two-loop RGE of the effective Higgs-gluon coupling in the SMEFT,
parametrized by the operator OHG. Several single and double Higgs production channels
are directly sensitive to this coupling, that enters already at tree level whereas the SM
signal arises at one loop.
At one loop, CHG renormalizes itself and also mixes with the chromomagnetic operator.
Adopting a loop counting in weakly coupled, renormalizable UV completions of the SM
[45, 46], these operators are expected to arise only at the one-loop level, pushing their
7For some models, we have explicitly checked that this approximation reproduces the behavior of the
profiled results.
– 15 –
RGE effects at two-loop order. Consequently, operators that can be generated at tree level
and mix with CHG at two loop must also be consistently included under this counting.
Our computation therefore extends the previous efforts for the computation of the two-
loop contributions of the potentially tree-level generated four-top [24] and Yukawa-like
operators [27], presenting for the first time the two-loop contributions of the operators
with schematic structure D2H4 and D H2 ψ2.
We addressed the phenomenological impact of the two-loop running in a fit to inclusive
Higgs data. Indeed, we could find that bounds on the class 7 operators can shrink sizably
due to the two-loop running effects, if their contribution is proportional to the top Yukawa
coupling. In concrete models, such as the ones with VLQs presented in our phenomenolog-
ical study, this effect is though reduced, as many more operators arise when matching up
to one-loop order. For instance, OHG is generated at one-loop level in this type of models.
Nevertheless, our results indicate that two-loop running effects can have a sizable
impact on global fits, once completely available. This applies in particular for operators so
far loosely constrained. For instance, in the case of four-top operators, it was shown that
two-loop corrections to electroweak precision observables [32] can, in some cases, limit the
currently allowed range [95].
In summary, our results complete an important piece of the two-loop RGE framework
for Higgs–gluon interactions, setting the stage for consistent and precise SMEFT analyses
of Higgs data with potential phenomenological impact.
Note added
During the completion this work, we became aware of [96]. Our results are in agreement.
Acknowledgements
We thank Javier Fuentes-Mart´ın for cross-checking our results with [96] and the discussions
that made us realise that we need to clearly state the definition of the SM Lagrangian
that we adopt. BE would like to thank Jonathan Ronca for several useful discussions on
numerical integration and Alejo N. Rossia for comments on marginalisation and profiling.
Also, we would like to acknowledge the Mainz Institute for Theoretical Physics (MITP)
of the Cluster of Excellence PRISMA+ (Project ID 390831469) for its hospitality and its
partial support during part of the completion of this work. This work received funding
by the INFN Iniziativa Specifica APINE and by the University of Padua under the 2023
STARS Grants@Unipd programme (Acronym and title of the project: HiggsPairs – Precise
Theoretical Predictions for Higgs pair production at the LHC). This work was also partially
supported by the Italian MUR Departments of Excellence grant 2023-2027 “Quantum
Frontiers”. We acknowledge support by the COST Action COMETA CA22130.
CloudVeneto is acknowledged for the use of computing and storage facilities. The Feynman
diagrams in this work were made using TikZ-Feynman [97].
– 16 –
A
Feynman Rules
In this Appendix we present the Feynman rules associated with the effective operators
defined in Eqs. (2.2)-(2.4). mj (Aj) identifies an index in the fundamental (adjoint) rep-
resentation of SU(3)C whereas ij denotes an index in the fundamental representation of
SU(2)L. The numbers are associated to the particles in the figure on the left.
H3,∗
H4
g1
g2
= +4i CHGδi3i4δA1A2 (pµ2
1 pµ1
2 −gµ1µ2p1 · p2) .
(A.1a)
H1
H2, ∗
Q4
L
Q3
L
= +i
C(1)
Hqδi1i2δi3i4 + C(3)
Hqτ I
i2i1τ I
i4i3
δm3m4
/p1 −/p2
PL,
(A.1b)
H1
H2, ∗
ψ3
R
ψ4
R
= +i CHψδi1i2δm3m4
/p1 −/p2
PR,
ψ = t, b ,
(A.1c)
H1
H2
d4
R
u3
R
= +i CHtbδm3m4ϵi1i2
/p1 −/p2
PR,
(A.1d)
H1
H2
H4,∗
H3,∗
= See Eq. (A.2).
(A.1e)
Given their length, we report here the Feynman rules for the last diagrams above:
−i
"
δi1i4δi2i3
CHD (p1 · p3 + p2 · p4) + CH□
X
i
p2
i + 2p1 · p4 + 2p2 · p3
!!
+ (4 ↔3)
#
.
(A.2)
For the sake of completeness we present the Feynman rules of the relevant SM interac-
tions. We remind the reader that the for the Yukawa sector of the SM we adopt the same
– 17 –
notation of Refs. [14–16].
g1
Q2
L
Q3
L
= igs γµ1PLT A1
m3m2δi3i2,
g1
t2
R/b2
R
t3
R/b3
R
= igs γµ1PRT A1
m3m2,
(A.3a)
H1
Q2
L
t3
R
= +iYt PRδm3m2εi1i2,
H1,∗
t2
R
Q3
L
= −iY ∗
t PLδm3m2εi3i1, (A.3b)
H1,∗
Q2
L
b3
R
= −iYb PRδm3m2δi1i2,
H1
b2
R
Q3
L
= −iY ∗
b PLδm3m2δi3i1.
(A.3c)
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KA-TP-31-2025, COMETA-2025-47 Complete two-loop Yukawa-induced running of the Higgs-gluon coupling in SMEFT Stefano Di Noi a , Barbara Anna Erdelyi b,c and Ramona Gr ̈ober b,c aInstitute for Theoretical Physics, Karlsruhe (KIT), Wolfgang-Gaede Straße 1, Karlsruhe D-76131, Germany bDipartimento di Fisica e Astronomia "G. Galilei", Universit`a degli Studi di Padova, Via F. Marzolo 8, 35131 Padova, Italy cIstituto Nazionale di Fisica Nucleare, Sezione di Padova, Via F. Marzolo 8, 35131 Padova, Italy E-mail: , , Abstract: We compute the two-loop renormalisation group equation for the effective Higgs to gluon coupling in Standard Model Effective Field Theory. Concretely, we present the contributions generated by the operators belonging to class 3 and 7 in the Warsaw basis, completing the two-loop renormalization program of the Higgs-gluon coupling proportional to the top Yukawa coupling for potentially tree-level generated operators. We investigate the phenomenological impact of the contributions in fits to Higgs data both in a bottom-up approach and a top-down approach in terms of UV models with vector-like quarks. 16 Oct 2025 Contents 1 Introduction 1 2 Notation 3 3 Computation and result 4 4 Phenomenology 8 4.1 Fitting framework 10 4.2 Results 10 4.2.1 Bottom-up results 10 4.2.2 Top-down results 12 5 Conclusions 15 A Feynman Rules 17 1 Introduction There are several reasons to believe that the Standard Model (SM) of particle physics requires an extension. Despite this, no direct evidence of new physics has been observed so far, motivating the use of the Effective Field Theory (EFT) framework to describe potential beyond-the-SM (BSM) effects in the most model-independent way possible. This approach is valid under the assumption that the scale of new physics lies well above the electroweak scale. Assuming that the Higgs boson transforms as in the SM, namely as part of an SU(2)L doublet, writing down all possible higher-dimensional operators consistent with the symmetries of the SM leads to the so-called Standard Model EFT (SMEFT). Concerning collider physics, the most relevant contributions are expected to arise from dimension-six operators, for which a non-redundant basis was first established in Ref. [1], the so-called Warsaw basis. Despite not being the only possible EFT extension of the SM,1 the SMEFT has become a popular framework for global analyses [9-13], providing a self-consistent way to correlate various experimental measurements across different energy regimes. The connection between the Wilson Coefficients (WCs) at different energy scales is encoded in the renormalisation group equations (RGEs). The complete one-loop RGEs of dimension-six operators have been computed in [14-16] and implemented for systematic 1A different philosophy is embodied by the Higgs EFT [2-8] (HEFT or electroweak chiral Lagrangian), where the Goldstone bosons and the Higgs boson do not transform in the same multiplet, but the physical Higgs boson transforms as a singlet. - 1 - inclusion into phenomenological analyses in several computer codes [17-21]. Even though the full set of two-loop RGEs for the dimension-six SMEFT has not been computed yet, many partial results have been presented in the last years and progress towards the two-loop SMEFT RGEs has been reported [22-35]. The RGEs are thus crucial for global analyses that combine observables measured at different energy scales [36-38], exploiting fully the variety of available data from different experiments. Moreover, with the increasing precision of collider measurements, RGE effects can also become sizeable in processes where the renormalisation scale varies dynamically over a wide range of energies. In this context, including the RGE running of WCs is essential, as explicitly demonstrated in [39-43]. These works focus on the running proportional to the strong coupling constant αs, which is expected to give the dominant contributions. Moreover, restricting to these effects allows to solve the RGEs analytically, greatly improving the time efficiency of the computer programs employed for phenomenological studies. However, also the running effects generated by the top-quark Yukawa coupling Yt can produce significant effects [27, 44]. Despite the strength of the interaction being na ̈ıvely smaller than in the strong case, WCs entering the RGEs via Yukawa-induced contributions are generally less constrained [11], potentially enhancing their impact. In this work we complete the effort started in Refs. [24, 27] to determine the twoloop O g2 sY 2 t RGE of the Higgs-gluon coupling, which enters the dominant gluon-fusion production channel for Higgs production. Within the SMEFT framework, this coupling is parametrized by the operator OHG = (H†H)GA μνGA,μν where H denotes the Higgs doublet and GA μν the gluon field strength. We stress that this interaction generates a tree-level coupling between the Higgs bosons and the gluons, which instead is loop induced in the SM. In weakly interacting and renormalizable UV models, the corresponding WC CHG is loop-generated [45, 46]. Moreover, at one-loop level, the RGE of CHG involves only WCs of operators that are likewise loop-generated under the same assumptions. Consequently, OHG does not undergo one-loop renormalisation, even when considering dimension-eight effects [47]. It follows that, to ensure consistency in the perturbative expansion in the loop order, the two-loop running effects proportional to tree-level generated operators must be considered when including RGE effects for Higgs production. Contributions belonging to this category include the ones from four-quark operators [24] and Yukawa-like operators [27]. In this paper, we present the contributions from the operators with the schematic structure D2H4 and D H2 ψ2. Such operators belong to, respectively, class 3 and 7 in the classification of the Warsaw basis notation [1] and are the last missing contributions for the complete two-loop RGE proportional to the third family Yukawa couplings Yt and Yb involving operators that are potentially tree-level generated. Conversely, we do not consider chromomagnetic operators as they can only be generated at loop-level in weaklyinteracting theories [45, 46] and are expected to give a contribution at O(g2 sYt) that would be a three-loop effect. We investigate the phenomenological impact of the new contributions to the RGE running by performing a fit to Higgs data both in the bottom-up and top-down approach, using simplified models with heavy Vector-Like Quarks (VLQs). Such states naturally arise in a - 2 - variety of frameworks proposed to address open issues of the Standard Model, such as the hierarchy problem, with Composite Higgs Models providing a prominent example [48-50]. In these models, operators belonging to class 7 are generated at tree level, which lead to a modification of the couplings of quarks to massive vector, eventually in association with a Higgs boson. In particular, the couplings of top quarks to Z bosons are weakly constrained so far [37, 51], as a direct constraint comes from t ̄tZ production, single top production [52] or Zh production [53], whereas other probes need to rely on RGE effects. Other operators that couple to the third generation SU(2)L doublet can also be constrained by electroweak precision data. Conversely, class 3 operators can be constrained by electroweak precision data and by measurements of the Higgs coupling to massive vector bosons, currently available at few percent level [54, 55]. The paper is structured as follows. In Sec. 2 we present our notation. In Sec. 3 we describe our computation and present the result for the RGE of the Higgs-gluon coupling in the SMEFT. In Sec. 4 we study the phenomenological implications of the inclusion of two-loop running effects in single Higgs physics. Finally, in Sec. 5 we present our summary. 2 Notation The SMEFT extends the SM by a tower of gauge and Lorentz invariant higher-dimensional operators as a series in inverse powers of a high-energy new physics scale Λ. In this work, we neglect lepton and baryon number violation and focus exclusively on the leading dimensionsix terms in the expansion: L = LSM + X Di=6 CiOi. (2.1) The operators are denoted by Oi and the associated WCs by Ci. We note that the WCs in the previous formula have dimension -2 in units of energy. The operators in the previous expressions are invariant under the unbroken SM group GSM = SU(3)C × SU(2)L × U(1)Y. We follow the Warsaw basis of Ref. [1] for the dimension-six operators and Ref. [14] for the definitions of the SM Lagrangian. We consider only the third family of quarks, leading to a minimal modification of the notation with respect to the original publication. In particular, we use tR, bR to denote the right-handed component and QL to denote the left-handed component of the third family of quarks. The relevant class 7 operators are defined as: O(1) HQ = iH†← → DμH ̄QLγμQL , O(3) HQ = iH†← → DI μH ̄QLγμτ IQL , (2.2) OHt = i H†← → DμH ( ̄tRγμtR) , OHb = i H†← → DμH ̄bRγμbR , (2.3) OHtb = i ̃H†DμH ( ̄tRγμbR) , (2.4) where ̃Hi = εijH†,j (with ε12 = +1) and i H†← → DμH = iH†(DμH) -i(DμH)†H, (2.5) i H†← → DI μH = iH†τ I(DμH) -i(DμH)†τ IH. (2.6) - 3 - The operators in class 3 are defined as OH□= H†H □ H†H , OHD = H†DμH H†DμH ∗ . (2.7) These two operators in the broken phase modify the kinetic term of the Higgs boson, thus requiring a field redefinition to reobtain the canonical normalisation of the Higgs kinetic term. In the unitary gauge, the Higgs doublet H is related to the physical Higgs field h via H = 1 √ 2 0 v + h 1 + v2 CH, kin ! , where CH, kin = CH□-1 4CHD . (2.8) Finally, we recall that the Higgs-gluon coupling in the SMEFT is parametrized by the operator OHG = H†H GA μνGA μν . (2.9) This operator belongs to class 4 in the Warsaw basis. Other class 4 operators that will appear in the phenomenological study performed in this work are defined below OHW = H†H W I μνW I μν , OHB = H†H (BμνBμν) , (2.10) OHWB = H†τ IH W I μνBμν . (2.11) Furthermore, the following class 5 operators will be considered, often referred to as Yukawalike operators: OtH = H†H ̄QL ̃HtR , ObH = H†H ̄QLHbR . (2.12) We remind the reader that the SM Yukawa coupling is not defined in direct analogy to the Yukawa-like operator. All the previous operators are Hermitian with the exclusion of OHtb, OtH and ObH, which enforce the addition of their complex conjugate to the Lagrangian. We use T A for the SU(3)C generators and τ I for the Pauli matrices. We define the gluon field-strength tensor as GA μν = ∂μGA ν -∂νGA μ -gsfABCGB μ GC ν . 3 Computation and result We describe in this section the computation of the two-loop contributions of the operators in Eqs. (2.2)-(2.4) and Eq. (2.7) to the RGE of CHG. Practically, this requires the computation of the divergent parts of the two-loop corrections to a process where OHG, defined in Eq. (2.9), enters at tree-level. We perform our computation in the unbroken phase and we consider the process gg →H∗H. In dimensional regularization, the divergences are parametrized by poles in ε = 0, being D = 4 -2ε the number of space-time dimensions. We will discuss here in detail the calculation of the terms proportional to OHt. All the other operators of class 7 are obtained in analogy. At one-loop order, the contributions to gg →H∗H can only arise via triangle diagrams as in Fig. 1a. However, once all the relevant diagrams are considered their contribution vanishes [16]. The reason is that the - 4 - H H∗ g g (a) H H∗ g g (b) H H∗ g g (c) Figure 1: Sample of the diagrams with a single insertion of the operator OHt which vanish because of kinematics, see main text for details. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt and a black dot to denote a SM vertex insertion. g H H∗ g (a) g H H∗ g (b) g H H∗ g (c) Figure 2: Sample of the two-loop order diagrams with a single insertion of the operator OHt. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a black square to denote an insertion of OHt, a black dot to denote a SM vertex insertion and a white star to denote the Yukawa coupling counterterm. g g H H∗ (a) g g H H∗ (b) Figure 3: Sample of the two-loop order diagrams with a single insertion of the operators OH□and OHD. We use a thick (thin) line to denote a left-(right-)handed field, a dashed line to denote the Higgs doublet, a white square to denote an insertion of OH□and OHD and a black dot to denote a SM vertex insertion. diagrams are purely s-channel, depending on the linear combination q1+q2 = p1+p2 where we have used p1,2 (q1,2) to denote the incoming (outgoing) momenta of the gluons (Higgs doublets). The scalar current gives rise to a contraction with (p1 -p2), yielding a vanishing result. Following this argument, we can safely ignore all the two-loop diagrams where the two external Higgs bosons are connected to the effective vertex. This applies both to pure QCD (see Fig. 1b) and to mixed Yukawa-QCD (see Fig. 1c) contributions.2 We have crosschecked the correctness of our reasoning by explicitly computing the O g4 s diagrams with 2As a consequence of this statement, the class 7 pure QCD contribution to the RGE of CHG vanishes at all loop orders at dimension-six level. - 5 - a QCD correction to the propagator connecting the two QCD vertices and verifying that their sum vanishes after a trivial change of variable in the momentum integration. Our argument allows to restrict our attention to a total of 24 diagrams containing an insertion of OHt, where one of the two Higgs doublets in the EFT vertex is external and the other is internal. A sample of the two families of diagrams is displayed in Figs. 2a and 2b. Moreover, class 7 operators correct the Yukawa coupling at one-loop [14], calling for the inclusion of the one-loop insertions of the one-loop Yukawa counterterms as in Fig. 2c. Class 3 operators modify the quadrilinear coupling, yielding 12 diagrams, of which we present a sample in Fig. 3. We have performed two independent computations of the class 7 operator contributions to check the set-up of our computation. The first one relies on FeynRules [56, 57] for the generation of the Feynman rules, where the model file is obtained by extending the SM model file in the unbroken phase provided in the additional material to MatchMakerEFT [58]. The resulting Feynman rules are presented in App. A. FeynArts [59] and FeynCalc [6063], interfaced via FeynHelpers [64], are used to generate the diagrams and to simplify the Dirac algebra. The numerical integration of the two-loop integrals is performed with AMFlow [65] supplemented with Blade [66]. The second computation relies on the same pipeline of Ref. [27]. The diagrams are generated with qgraf-3.6.10 [67] and simplified with FeynCalc. The numeric integration is performed with AMFlow supplemented with LiteRed2 [68, 69] and FiniteFlow [70]. Both computations use the Na ̈ıve Dimensional Regularization (NDR) [71, 72] scheme for the treatment of γ5 in a non-integer number of dimensions. This prescription is known to be algebraically inconsistent in presence of traces involving an odd number of γ5 in combination with at least six γ matrices as in the diagrams we compute. However, we are interested in the contribution to the RGE of CHG, meaning that we can ignore the contribution of the traces involving γ5, which would renormalize its CP-odd counterpart, namely ̃CHG. The Breitenlohner-Maison-t'Hooft-Veltman (BMHV) scheme [73, 74] has recently gained popularity as an alternative to the NDR scheme, being the only known continuation scheme proved to be algebraically consistent [75-79]. However, it highly increases the algebraic difficulty of loop computation. We have checked that, for the present computation, no difference arises between the two schemes by employing the map presented in Ref. [80]. This procedure is legit in this context since the leading divergence is of O (1/ε), as recently showcased in Ref. [33]. We note that, in principle, all the operators considered in this work can renormalize redundant operators, for which we follow the notation of Ref. [58]. This renormalization arises from subamplitudes with schematic structure ̄ψψ →H∗H, ψ = QL, tR, bR, which affect the operators R′ Hψ, R′′ Hψ,3 and from H → ̄QLχR, χ = b, t (possibly in association with one or two gluons), which impact the operators RuHDi, i = 1, 2, 3, 4. Since some of the external states in these subprocesses are not on-shell, the relevant one-loop insertions 3When ψ = QL, two possible isospin contractions are possible, generating a total of four redundant operators involving the left-handed quarks. - 6 - of redundant operators must be included in the total contribution to gg →H∗H. We have explicitly checked that these contributions vanish in the case where only OHt is present. Moreover, we do not need to consider contributions coming from one-particle reducible diagrams to OHG. This conclusion stems from the fact that no redundant operator is reduced to OHG when the SM equations of motions are used [1]. We obtain μdCHG dμ ⊃ g2 s (16π2)2 h + YbY ∗ b (CHb -C(1) HQ -3C(3) HQ) + YtY ∗ t (C(1) HQ -3C(3) HQ -CHt) + (CHtbYbY ∗ t + C∗ HtbY ∗ b Yt) - CH□-1 2CHD (YbY ∗ b + YtY ∗ t ) + 3 2 [CtHYt + C∗ tHY ∗ t + CbHYb + C∗ bHY ∗ b ] -4 δNDR YtY ∗ t C(1) Qt -1 6C(8) Qt + YbY ∗ b C(1) Qb -1 6C(8) Qb i , (3.1) δNDR = ( 1 (NDR) 0 (BMHV). (3.2) We use the same conventions as the one-loop results of [14-16]. The first two lines represent our novel result, while the contributions generated by four-quark operators and Yukawa-like operators (see Eq. (2.12)) have been first derived in Ref. [24] and in Ref. [27],4 respectively, which we point to for the precise definition of the operators. We stress that only the four-top contribution depends on the γ5 scheme, while the others do not. The final amplitude cannot exhibit infrared (IR) divergences, since no corresponding real-emission diagrams exist at the considered order in the coupling constants. However, IR divergences can still appear at intermediate stages of the calculation. As they must ultimately cancel, one has to verify that this cancellation does not occur against ultraviolet (UV) divergences, as can happen, for instance, in the case of scaleless integrals. We note that our result can be used to determine the analogous two-loop contributions from the operators in class 3 and 7 to OHW , OHB, OHWB (defined in Eqs. (2.10)-(2.11)), where the gluon field strength is replaced by the corresponding quantities involving electroweak bosons. However, this would represent just a partial result, since new diagrams arise as a consequence of the non-vanishing SU(2)L × U(1)Y charge of the Higgs doublet. Obtaining the RGEs for such operators is hence beyond the scope of this paper. The phenomenological relevance of these operators lies in their impact in the interactions involving photons or a photon and Z bosons and (at least) one Higgs boson and they lead to a shift in the electroweak input [81]. In the SM, such interactions are numerically dominated by W loops, so we expect the two-loop running contributions to be less numerically important than in the gluon case such that the phenomenological study we perform in the next section 4In the previous expression, the contribution from the operators OtH and ObH presents an extra 1/2 with respect to the result of Ref. [27], which will be updated. - 7 - remains valid omitting these sub-leading contributions. Furthermore, their contributions to the Higgs couplings to massive vector bosons is suppressed in the loop counting with respect to the tree-level contributions from class 3 operators. 4 Phenomenology In this section, we study the effect of the inclusion of two-loop running effects with fits to Higgs data at the LHC. In particular, we focus on the modifications to the signal strengths of a set of Higgs decay final states due to the presence of SMEFT operators. We perform two different fits. In first instance, we adopt a bottom-up perspective by performing the analysis in terms of the WCs to observe how their bounds are modified by the inclusion of the running. Secondly, we consider a top-down approach by selecting UV models with pairs of Vector-like Quarks (VLQs) coupled to the third-generation SM quarks, which generate, at tree-level, at least one of the class 7 operators considered in this paper. The WCs that enter the observables in this section are computed at the low scale μR = mh = 125 GeV. Their expression in terms of the WCs at the high energy scale Λ = 2 TeV can be obtained with the evolution matrix formalism [20] Ci(μR) = Uij(μR, Λ)Cj(Λ). (4.1) In the present case, the evolution matrix Uij(μR, Λ) has been obtained with a modified version of RGESolver [21] relying on a numeric solution of the RGEs. All the operators included in this section have been defined in Sec. 2. Cross-section The SM value for the Higgs boson production cross section in the dominant gluon-gluon fusion contribution is σSM = 48.68 pb at a center of mass energy of √s = 13 TeV. In the SMEFT, the cross-section reads σ = 48.68 + 2.83 · 104v2CHG + 97.36v2CH,kin -112.25v2 Yt CtH pb , (4.2) where CH,kin was defined in Eq. (2.8). We have obtained this parameteric dependence on the SMEFT WCs using the EFT implementation [41] in higlu [82]. Higgs total and partial decay widths The Higgs total decay width in the SMEFT at LO in the U(3)5 flavour-symmetric limit was computed in Ref. [83]. We adapt those results, noting that quark operators of class 7 contribute only via corrections to the gauge boson total width except for the cases with quarks in final states from Z boson decays. In this case there are some direct contributions where the Higgs boson and one Z boson couples via a class 7 operator to bottom quarks, and the Z boson decays subsequently. This results in Γh = ΓSM h 1 + 50.6 v2CHG -1.50 v2CHB -1.21 v2CHW + 1.21 v2CHWB + 1.83 v2CH□ -0.43 v2CHD -48.5 v2Yb CbH + 0.00016v2 C(1) Hq + 0.0015v2 C(3) Hq -0.00023v2 CHb +2 v2 BRh→γγ CH,kin + 0.53 Yt CtH + BRh→gg CH,kin -2.12 Yt CtH , (4.3) - 8 - using the {MW , MZ, GF } input scheme for the electroweak parameters. The last line encodes the higher-order rescaling of the Higgs decays into photon and gluon pairs through CH,kin defined in Eq. (2.8), together with the additional modifications induced by shifts in the top Yukawa coupling. In the UV models we consider, also the bottom Yukawa coupling can be modified but we do not consider its effect in the h →gg and h →γγ decays as it is negligible with respect to the top Yukawa contribution. We note already from this formula that generically our two-loop contributions can be of the same order than the direct contributions. Indeed, since the CHG has a large pre-factor, even the two-loop factor 1/(16π2)2 log(Λ/mh) can be compensated. We obtain the partial decay widths in the SMEFT rescaling the results in Ref. [83] to take into account the modifications of the decay into photon pairs and the top Yukawa coupling, analogously to the case of the total width discussed previously. We have validated these expressions to those reported in Ref. [84].5 For the decays into fermion pairs, introducing l ∈{μ-, τ -} , the partial decay widths are Γh→l ̄l ΓSM h→l ̄l = 1 + 2v2 CH,kin, (4.4) Γh→b ̄b ΓSM h→b ̄b = 1 + v2 2CH,kin -2 CbH . (4.5) For the decays into photon pairs we get Γh→γγ ΓSM h→γγ = 1 + v2 2 CH,kin + 0.53 Yt CtH -231 CHW -805 CHB + 431 CHWB . (4.6) As for decays into pairs of W and Z pairs, we further consider their decay products in order to compare with experimental results. In particular, Ref. [55], whose experimental results we will be using later, considers the processes h →W +W -→2l 2ν and h →ZZ → 4l. In order to get the partial decay width for these processes, we start again from Ref. [83] and focus on their results for partial decay widths of the Higgs boson into four-fermion final states mediated by "only neutral", "only charged" and "neutral plus charged" currents. In our scenario, the first case reduces to the 4l final state, while the second and third correspond to the 2l 2ν final state. The SM partial decay width for h →ZZ →4l is then ΓSM 4l = 1.1 keV, the partial decay width for h →W +W -→2l 2ν reads ΓSM 2l2ν = 90 keV , and the corresponding SMEFT expressions read Γ4l ΓSM 4l =1 + v2 (2CH□+ 0.27 CHD + 0.17 CHW -1.66 CHB -0.51 CHWB -0.27 C(1) HQ -0.27 C(3) HQ + 0.048 CHb , (4.7) Γ2l2ν ΓSM 2l2ν =1 + v2 (2CH□-0.52 CHD -1.5 CHW + 0.003 CHB -0.023 CHWB -0.0046 C(1) HQ -0.0046 C(3) HQ + 0.00081 CHb . (4.8) 5Ref. [84] uses the SILH basis [85], whereas Ref. [83] uses the Warsaw basis. To compare the results, we translate both sets of results into the Greens' basis by means of Ref. [86] and compare the terms of interest. - 9 - 4.1 Fitting framework Adopting the symbol Oα to identify an observable α, the χ2-function can be defined as χ2 = X α, β Oexp α -Oth α C-2 αβ Oexp β -Oth β , (4.9) where C-2 is the inverse of the covariance matrix and the superscript 'th' ('exp') means that the theoretical (experimental) value of the observable Oα is being considered. The theoretical value accounts for both the SM prediction and the deviations due to the insertion of SMEFT operators. Following Ref. [87], we construct a fit to Higgs data using the measurements of signal strengths with their covariance matrix as provided in Ref. [55]. Considering the Higgs decay channels h →α with α ∈{W +W -, ZZ, b ̄b, τ +τ -, μ+μ-, γγ},6 for each channel it is given by μα = σ × BRh→α σSM × BRSM h→α = σ σSM ΓSM h ΓSM h→α Γh→α Γh = σ σSM ΓSM h Γh Γh→α ΓSM h→α , (4.10) in which σ(SM) and BR(SM) denote, respectively, the value in the SMEFT (SM) of the Higgs production cross-section and the branching ratio into the specific channel. We consider up to order O(1/Λ2) in the signal strength. Of the three multiplied ratios in the final expression, the first is known from Eq. (4.2), the second from Eq. (4.3) and the third is selected from Eqs. (4.4)-(4.8) depending on the decay channel of interest. The allowed region satisifies the condition χ2 < χ2 min + ∆χ2 (n, CL) . (4.11) In the previous expression, n denotes the degrees of freedom (number of couplings in the top-down approach, number of involved WCs in the bottom-up approach) while CL stands for Confidence Level. The value χ2 min is the minimum of the χ2. 4.2 Results We now present our results. We notice that the operators OHtb and OHt do not enter directly any of the Eqs. (4.2)-(4.8), so they enter only via mixing effects with other operators. 4.2.1 Bottom-up results In this section we report our one- and two-parameter results for the bottom-up fits. We stress that these parameters are set to be non-zero at the high-scale Λ = 2 TeV, meaning that several other operators are generated via operator mixing, governed by Eq. (4.1). Hence, the bounds are reported for the operators at such scale. One parameter fits We present the individual 2σ CL interval for each WC considered at the energy scale Λ = 2 TeV in Fig. 4. We note that the intervals for the class 3 6For the decays into pairs of W and Z bosons, Ref. [55] considers the bosonic decay channels ZZ →4l and WW →lνlν. Thus, for the first two decay channels, we can use the decay widths of Eqs. (4.7) and (4.8) respectively. - 10 - -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Intervals CHD(Λ) CH□(Λ) CtH(Λ) CbH(Λ) CHt(Λ) · 10-1 CHb(Λ) · 10-1 C(1) HQ(Λ) C(3) HQ(Λ) CHtb(Λ) · 10-2 2σ 1L RGEs 2σ 2L RGEs Figure 4: Allowed ranges at 2σ for the WCs obtained by performing one parameter fits. The yellow (orange) line is obtained by considering one-loop (two-loop) running effects. coefficients are not sizeably affected by the inclusion of the two-loop effects in the RGEs. These operators are already constrained by direct contributions to the couplings with the Higgs bosons, reducing the impact of higher order corrections. Similar conclusions apply to the WCs of the Yukawa-like operators CtH and CbH, which enter directly in the Higgs production cross section and in several decay channels. As for the class 7 operators, CHb is not sizeably modified by the inclusion of the twoloop running effects, due to the bottom Yukawa suppression. Conversely, C(1) HQ and CHt show a moderate reduction when two-loop running effects are accounted, while for C(3) HQ a quite sizable reduction of the allowed range can be observed once two-loop running is included. Moreover, we note that the bounds for CHtb are looser at two-loop level, hinting for some cancellations happening with respect to one-loop running effects. Finally, we note that the bounds we present here are much weaker than the ones presented in Ref. [37]. Since our focus is on the impact of the two-loop running effects in the fit, we only accounted for inclusive Higgs data and not, for instance, for electroweak precision data or top data as Ref. [37]. Those measurements are though sensitive to the operators considered here. Two parameter fits In Fig. 5 we allow two WCs to be non-vanishing at the energy scale Λ = 2 TeV. We present the results of three two-parameter fits, comparing the impact of including only one-loop or two-loop running effects. In Fig. 5a, the two class 3 operators are considered. The correlation between the two arises from the fact that their contribution appears mostly in the linear combination CH, kin = CH□-1 4CHD. Moving on to Fig. 5b, we consider the two WCs C(1) HQ and C(3) HQ which, in motivated UV extensions of the SM, - 11 - (a) Class 3 operators (b) Class 7 operators with quark doublets (c) Class 7 operators with quark singlets Figure 5: Allowed regions at 2σ obtained via a two parameter fit including one-loop (in yellow) and two-loop (in orange) running effects. We consider in pairs the operators belonging to class 3 and to class 7. For class 7, we further separate between the cases in which the quarks in the operators are the doublets or the singlets under SU(2)L. appear together, as shown in Tab. 1. Finally, in Fig. 5c we compare CHt and CHb. In this case, there is a significant improvement in the bounds considering that CHt does not enter any of the observables defined in Eqs. (4.2)-(4.8), while CHb enters but is very suppressed compared to the other WCs. 4.2.2 Top-down results In this second approach we consider UV models featuring Vector-like Quarks (VLQs) as they generate at tree level both the operators in Eqs. (2.2)-(2.4) and the Yukawa-like - 12 - GSM O(1) HQ O(3) HQ OHt OHb OHtb U (3, 1) 2 3 ✓ ✓ Q1 (3, 2) 1 6 ✓ ✓ ✓ Q7 (3, 2) 7 6 ✓ T1 (3, 3)-1 3 ✓ ✓ T2 (3, 3) 2 3 ✓ ✓ Model Particle Content 1 U + Q1 2 U + Q7 3 Q1 + T1 4 Q1 + T2 5 Q7 + T2 Table 1: Class 7 operators generated at the tree-level by the VLQs considered in this work, along with the charges under the SM gauge group of the VLQs, with GSM = (SU(3)C, SU(2)L)U(1)Y (left). The presented VLQs will be used to construct models with pairs of these particles (right). operators OtH and ObH [88]. The operators OHG, OHD and OH□are instead generated at one-loop level. In a similar spirit to Ref. [87], we consider models with pairs of VLQs which give rise to unsuppressed contributions to the operator OtH. In Tab. 1 we list the VLQs considered, their charges under the SM gauge group GSM, the class 7 operators they generate at treelevel and how the five models considered in this work are defined. Given that we are interested in the RGE effects due to terms proportional to third generation Yukawa couplings, we make the simplifying assumption that the VLQs only couple to the third generation SM quarks. Furthermore, we set their masses to be Λ = 2 TeV. This value is in agreement with lower bounds on singly-produced VLQs coupled to third generation quarks [89, 90] and with the bounds from VLQ pair production [90-92]. The interaction Lagrangians for the considered models are given for a general flavour structure in Ref. [88] -L(int) M.1 = [λU]rp ̄Ur R ̃H†qp L + λu Q1 rp ̄Qr 1L ̃Hup R + h λd Q1 i rp ̄Qr 1LHdp R+ + [λUQ1]rp ̄Ur ̃H†Qp 1 + H.c. , (4.12) -L(int) M.2 = [λU]rp ̄Ur R ̃H†qp L + [λQ7]rp ̄Qr 7LHup R + [λUQ7]rp ̄UrH†Qp 7 + H.c. , (4.13) -L(int) M.3 = λu Q1 rp ̄Qr 1L ̃H up R + h λd Q1 i rp ̄Qr 1LH dp R + [λT1]rp 2 ̄T I r 1RH†τ I qp L+ + [λT1Q1]rp 2 ̄T I r 1 H†τ I Qp 1 + H.c. , (4.14) -L(int) M.4 = λu Q1 rp ̄Qr 1L ̃H up R + h λd Q1 i rp ̄Qr 1LH dp R + [λT2]rp 2 ̄T I r 2R ̃H†τ I qp L+ + [λT2Q1]rp 2 ̄T I r 2 ̃H†τ I Qp 1 + H.c. , (4.15) - 13 - (a) Model 2. (b) Model 3. (c) Model 4. (d) Model 5. Figure 6: Allowed regions at 2σ for pairs of couplings for the considered models. The orange (yellow) coloured areas correspond to allowed values derived accounting for two-loop (one-loop) running effects. -L(int) M.5 = [λQ7]rp ̄Qr 7LHup R + [λT2]rp 2 ̄T I r 2R ̃H†τ I qp L + [λT2Q7]rp 2 ̄T I r 2 H†τ IQp 7 + H.c. . (4.16) For these equations we follow the notation of Ref. [88] and use up R and dp R for right-handed quark singlets and qp L for the left-handed quark doublets, being p, r a flavour index. For our phenomenological study, we only consider third generation SM quarks, namely p = r = 3. For the tree-level expressions we refer to Ref. [88], whereas for the one-loop matching we use SOLD [93] and Matchete [94]. In all models and for all operators entering Eqs. (4.2)-(4.8) we match up to the one-loop level. - 14 - (a) Model 1 with complete one-loop matching (b) Model 1 using tree-level matching Figure 7: Two fits performed for Model 1, considering either the complete matching up to one loop for all the considered operators (left) or only the tree-level matching results (right). In both cases, the allowed parameter space at 2σ is plotted in yellow (orange) when the one-loop (two-loop) running effects are included. For each model the χ2 is a function of either three or four couplings in the UV model. In Fig. 6 and Fig. 7a we present our results as two-dimensional CL regions, setting the remaining parameters to their best fit value.7 As can be inferred in Figs. 6-7a, the inclusion of two-loop running in the RGEs makes only a minor difference. While the model at treelevel matches only into operators of class 7 and the Yukawa-type operators, at one-loop level it matches into the class 3 operators and OHG, OHW , OHB and OHWB, which enter with large coefficients into the signal strength. This reduces the impact of the two-loop running effects considered here. A greater impact is observed when only the tree-level matching results are considered. In order to showcase this aspect, in Fig. 7, we compare the complete fit performed for model 1 with a second fit in which only the tree-level generated operators are considered. 5 Conclusions We have analyzed the two-loop RGE of the effective Higgs-gluon coupling in the SMEFT, parametrized by the operator OHG. Several single and double Higgs production channels are directly sensitive to this coupling, that enters already at tree level whereas the SM signal arises at one loop. At one loop, CHG renormalizes itself and also mixes with the chromomagnetic operator. Adopting a loop counting in weakly coupled, renormalizable UV completions of the SM [45, 46], these operators are expected to arise only at the one-loop level, pushing their 7For some models, we have explicitly checked that this approximation reproduces the behavior of the profiled results. - 15 - RGE effects at two-loop order. Consequently, operators that can be generated at tree level and mix with CHG at two loop must also be consistently included under this counting. Our computation therefore extends the previous efforts for the computation of the twoloop contributions of the potentially tree-level generated four-top [24] and Yukawa-like operators [27], presenting for the first time the two-loop contributions of the operators with schematic structure D2H4 and D H2 ψ2. We addressed the phenomenological impact of the two-loop running in a fit to inclusive Higgs data. Indeed, we could find that bounds on the class 7 operators can shrink sizably due to the two-loop running effects, if their contribution is proportional to the top Yukawa coupling. In concrete models, such as the ones with VLQs presented in our phenomenological study, this effect is though reduced, as many more operators arise when matching up to one-loop order. For instance, OHG is generated at one-loop level in this type of models. Nevertheless, our results indicate that two-loop running effects can have a sizable impact on global fits, once completely available. This applies in particular for operators so far loosely constrained. For instance, in the case of four-top operators, it was shown that two-loop corrections to electroweak precision observables [32] can, in some cases, limit the currently allowed range [95]. In summary, our results complete an important piece of the two-loop RGE framework for Higgs-gluon interactions, setting the stage for consistent and precise SMEFT analyses of Higgs data with potential phenomenological impact. Note added During the completion this work, we became aware of [96]. Our results are in agreement. Acknowledgements We thank Javier Fuentes-Mart ́ın for cross-checking our results with [96] and the discussions that made us realise that we need to clearly state the definition of the SM Lagrangian that we adopt. BE would like to thank Jonathan Ronca for several useful discussions on numerical integration and Alejo N. Rossia for comments on marginalisation and profiling. Also, we would like to acknowledge the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 390831469) for its hospitality and its partial support during part of the completion of this work. This work received funding by the INFN Iniziativa Specifica APINE and by the 2023 STARS Grants@Unipd programme (Acronym and title of the project: HiggsPairs - Precise Theoretical Predictions for Higgs pair production at the LHC). This work was also partially supported by the Italian MUR Departments of Excellence grant 2023-2027 "Quantum Frontiers". We acknowledge support by the COST Action COMETA CA22130. CloudVeneto is acknowledged for the use of computing and storage facilities. The Feynman diagrams in this work were made using TikZ-Feynman [97]. - 16 - A Feynman Rules In this Appendix we present the Feynman rules associated with the effective operators defined in Eqs. (2.2)-(2.4). mj (Aj) identifies an index in the fundamental (adjoint) representation of SU(3)C whereas ij denotes an index in the fundamental representation of SU(2)L. The numbers are associated to the particles in the figure on the left. H3,∗ H4 g1 g2 = +4i CHGδi3i4δA1A2 (pμ2 1 pμ1 2 -gμ1μ2p1 · p2) . (A.1a) H1 H2, ∗ Q4 L Q3 L = +i C(1) Hqδi1i2δi3i4 + C(3) Hqτ I i2i1τ I i4i3 δm3m4 /p1 -/p2 PL, (A.1b) H1 H2, ∗ ψ3 R ψ4 R = +i CHψδi1i2δm3m4 /p1 -/p2 PR, ψ = t, b , (A.1c) H1 H2 d4 R u3 R = +i CHtbδm3m4εi1i2 /p1 -/p2 PR, (A.1d) H1 H2 H4,∗ H3,∗ = See Eq. (A.2). (A.1e) Given their length, we report here the Feynman rules for the last diagrams above: -i " δi1i4δi2i3 CHD (p1 · p3 + p2 · p4) + CH□ X i p2 i + 2p1 · p4 + 2p2 · p3 !! + (4 ↔3) # . (A.2) For the sake of completeness we present the Feynman rules of the relevant SM interactions. We remind the reader that the for the Yukawa sector of the SM we adopt the same - 17 - notation of Refs. 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